Resonance Ionization Spectroscopy of Protactinium Studies on Intrinsic Quantum Chaos and the Ionization Potential Dissertation zur Erlangung des Grades ”Doktor der Naturwissenschaften“ am Fachbereich Physik, Mathematik und Informatik (FB 08) der Johannes Gutenberg-Universität Mainz Pascal Naubereit geb. in Aschaffenburg LAR SSA Mainz, den 30. Mai 2020 Tag der Prüfung: 2. Oktober 2020 Pascal Naubereit WA QUANTUM/LARISSA Institut für Physik Staudingerweg 7 Johannes Gutenberg-Universität D-55128 Mainz naubereit@uni-mainz.de Abstract The rare and all-radioactive actinide element protactinium (Pa) is one of the very few examples where even the most powerful experimental technique of resonance ionization spectroscopy (RIS) ultimately reaches its limits. On the one hand, the chemical properties of this element hinders the efficient production of a stable ion beam using RIS in a hot cavity ion source. Additionally, only small sample sizes of Pa can be used for the measurements due to its radioactivity and due its availability at all. On the other hand, even in the case one does really succeed to record atomic spectra, the analysis is strongly hampered due to the extraordinary complexity of the observed level structure. The present thesis is addressing exactly these challenges. This is why the spec- troscopy and especially the subsequent analysis of the spectra make up the main part of this work. Beforehand, the method RIS as well as the apparatus used need to be introduced. Following a generalized introduction, an explanation of RIS, the laser system and the spectrometer setup used, a first featured article, Resonance ionization spectroscopy of sodium Rydberg levels using difference frequency generation of high-repetition-rate pulsed Ti:sapphire lasers, validates RIS in combination with Ryd- berg analysis as method of choice for extraction of the ionization potential (IP) of almost any element. Additionally, the tool of difference frequency generation was applied to the Mainz Ti:sapphire lasers in order to significantly enlarge their avail- able wavelength range. Most often, highly elaborate experiments are only possible if a functional and diversified research network of expert groups is working together. Luckily, the LARISSA working group has built up a set of several valuable collaborations. Exem- plarily, the outcome of one of those collaborations is presented in the second featured article, Developments towards in-gas-jet laser spectroscopy studies of actinium isotopes at LISOL, describing preparatory activities, which will give rise to many upcoming col- laborative publications. Here, the novel technique of RIS in a supersonic gas jet was tested and approved to be highly efficient as well as to offer extremely high spectral resolution in the obtained data, which both is necessary for on-line production and investigations of rare isotopes far-off stability. The main part of the present thesis comprises different studies on the actinide element protactinium. At first, a sketch of the history of its discovery and the chal- lenges one has to face for its study are presented. In the following third publication, iii Excited atomic energy levels in protactinium by resonance ionization spectroscopy, the RIS measurements on protactinium are presented. A rather large contribution to this article is also the level analysis which was necessary to arrange the vast collection of energy levels in proper order. More than 1500 energy levels with excitation energies of up to 50000 cm−1 were extracted from the extremely dense spectra. After spectroscopy, the subsequent chapter addresses the search for the IP of Pa. Since all conventional methods could not be successfully applied to the withstanding element, a procedure implying a combination of two different methods of analysis has been developed and is presented in detail. With this procedure the extraction of the IP from the complex level structure of the RIS spectra was achieved and led to an IP of EPaIP = 49034(10) cm −1. Due to the extraordinary complexity of the Pa spectra, the presence of intrinsic quantum chaos (IQC) was speculated. Despite the fact, that this introduces a com- pletely new research area to the LARISSA group, the last chapter of the thesis’ main part covers this assumption including a qualitative introduction to quantum chaos and a complete analysis of the spectra regarding IQC aspects. The results of this analysis are comprised in a forth featured publication, Intrinsic quantum chaos and spectral fluctuations within the protactinium atom. Herein, the spectral statistics were investigated thoroughly and the influence of missing levels or mixed level sets was analyzed in detail. iv List of Included Articles Four peer-reviewed publications together with Chapter 4 form the basis of this cu- mulative thesis. They show the outcome of three years of research, mainly done in the LARISSA1 laboratories in Mainz, but also within several international collabo- rations. Even though the data for Article I was already gathered before the PhD studies were started, the main analysis was carried out during this period. It gives an overview of the method of resonance ionization spectroscopy in general, includ- ing the laser sources and the analysis methods applied within the LARISSA group, whereas Article II is an example of the excellent collaborative work marking the LARISSA network. Articles III and IV illustrate the extensive work invested into the spectroscopy and analysis of the protactinium atom. Chapter 4 also belongs to the latter research topic, but, unfortunately, a successful preparation and submission of a publication was not yet feasible withing the timeframe of this work, however it is still planned for the near future, based on the considerations given here. Article I: Resonance ionization spectroscopy of sodium Rydberg levels using difference fre- quency generation of high-repetition-rate pulsed Ti:sapphire lasers. P. Naubereit, J. Marı́n-Sáez, F. Schneider, A. Hakimi, M. Franzmann, T. Kron, S. Richter, and K. Wendt. Phys. Rev. A 93, 052518 (2016). Article II: Developments towards in-gas-jet laser spectroscopy studies of actinium isotopes at LISOL. S. Raeder, B. Bastin, M. Block, P. Creemers, P. Delahaye, R. Ferrer, X. Fléchard, S. Franchoo, L. Ghys, L.P. Gaffney, C. Granados, R. Heinke, L. Hijazi, M. Huyse, T. Kron, Yu. Kudryavtsev, M. Laatiaoui, N. Lecesne, F. Luton, I.D. Moore, Y. Martinez, E. Mogilevskiy, P. Naubereit, J. Piot, S. Rothe, H. Savajols, S. Sels, V. Sonnenschein, E. Traykov, C. Van Beveren, P. Van den Bergh, P. Van Duppen, K. Wendt, and A. Zadvornaya. Nucl. Instrum. Meth. B 376, 382 (2016). 1LAser Resonance Ionization Spectroscopy and Selective Applications v Article III: Excited atomic energy levels in protactinium by resonance ionization spec- troscopy. P. Naubereit, T. Gottwald, D. Studer, and K. Wendt. Phys. Rev. A 98, 022505 (2018). Article IV: Intrinsic quantum chaos and spectral fluctuations within the protactinium atom. P. Naubereit, D. Studer, A.V. Viatkina, A. Buchleitner, B. Dietz, V.V. Flambaum, and K. Wendt. Phys. Rev. A 98, 022506 (2018). vi Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Included Articles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Resonance ionization spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1 Mainz Ti:sapphire laser system . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.1 Specialized versions of the Mainz lasers . . . . . . . . . . . . . . . . 5 2.1.2 Frequency conversion . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Experiment and technique . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.1 Mainz atomic beam unit . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.2 RIS technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Publication: RIS of sodium Rydberg levels using DFG of high-repetition- rate Ti:sapph lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4 Collaborative activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.5 Publication: Developments towards in-gas-jet laser spectroscopy studies of actinium isotopes at LISOL . . . . . . . . . . . . . . . . . . . . . . . . . 20 3 Spectroscopy of the protactinium atom . . . . . . . . . . . . . . . . . . . . . . 27 3.1 The element 91 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 Publication: Excited atomic energy levels in protactinium by resonance ionization spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4 Ionization potential of protactinium . . . . . . . . . . . . . . . . . . . . . . . . 37 4.1 Direct measurement techniques for the IP . . . . . . . . . . . . . . . . . . 38 4.1.1 Rydberg convergences and separation of Rydberg levels . . . . . . 38 4.2 Analytical extraction of the IP from complex atomic spectra . . . . . . . 39 4.2.1 Level density collapse . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.2.2 Rydberg correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.2.3 Combination of LDC & RC method . . . . . . . . . . . . . . . . . . 48 4.3 Extracting the IP of protactinium . . . . . . . . . . . . . . . . . . . . . . . 49 5 Intrinsic quantum chaos within atomic protactinium . . . . . . . . . . . . . . 53 5.1 Quantum chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5.1.1 Access through classical mechanics . . . . . . . . . . . . . . . . . . . 53 5.1.2 From classical to quantum chaos . . . . . . . . . . . . . . . . . . . . 58 vii Contents 5.1.3 Random matrix theory . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.2 Publication: Intrinsic quantum chaos and spectral fluctuations within the protactinium atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 viii List of Figures 2.1 Properties of Ti:sapphire as laser crystal . . . . . . . . . . . . . . . . . . 4 2.2 Schematics of the most common types of the Mainz Ti:sapphire laser system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Ti:sapphire wavelength ranges accessible involving frequency conver- sion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4 Photographs of the Mainz atomic beam unit . . . . . . . . . . . . . . . 7 2.5 Generalized RIS ionization scheme . . . . . . . . . . . . . . . . . . . . . 8 4.1 High-energy excitation spectrum of protactinium . . . . . . . . . . . . 40 4.2 High-energy excitation spectrum of sodium . . . . . . . . . . . . . . . . 41 4.3 High-energy excitation spectrum of holmium . . . . . . . . . . . . . . . 42 4.4 Cumulative sum of levels for Na and Ho. . . . . . . . . . . . . . . . . . 43 4.5 Unified Rydberg spectrum of sodium . . . . . . . . . . . . . . . . . . . 44 4.6 Rydberg spectrum of sodium in dependency from n . . . . . . . . . . . 45 4.7 Autocorrelation function for sodium . . . . . . . . . . . . . . . . . . . . 45 4.8 3D correlation density for sodium . . . . . . . . . . . . . . . . . . . . . 46 4.9 Projection of the 3D correlation density for sodium . . . . . . . . . . . 47 4.10 RC spectrum of holmium . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.1 Instabilities of trajectories under small perturbations . . . . . . . . . . 54 5.2 Trajectory of a double pendulum . . . . . . . . . . . . . . . . . . . . . . 55 5.3 Phase space and trajectory of a Benoulli system . . . . . . . . . . . . . 56 The figures appearing in the featured publications are not given in this listing. ix List of Abbreviations AC Autocorrelation AI Auto-Ionizing state CERN Former french Conseil européen pour la recherche nucléaire, now Euro- pean Organization for Nuclear Research CM Curved Mirror CRC French: Centre de Recherche du Cyclotron CRIS Collinear Resonance Ionization Spectroscopy DFG Difference Frequency Generation DFI Delayed Field Ionization EA Electron Affinity EM End Mirror FHG Fourth Harmonic Generation FPE Fabry-Pérot Etalon G (Refraction) Grating GANIL French: Grand Accélérateur National d’Ions Lourds GOE Gaussian Orthogonal Ensemble GISELE Ganil Ion Source using Electron Laser Excitation HWHM Half Width at Half Maximum IAESTE International Association for the Exchange of Students for Technical Experience ICE Isolated Core Excitation xi IGISOL Ion Guide Isotope Separation On-Line IGLIS In-Gas-Laser Ionization and Spectroscopy IP Ionization Potential IQC Intrinsic Quantum Chaos ISAC Isotope Separator and Accelerator ISOLDE Isotope Separator On Line DEvice JYFL Jyväskylä accelerator laboratory LARISSA LAser Resonance Ionization Spectroscopy and Selective Applications LDC Level Density Collapse LF Lyot Filter, synonymous birefringent filter LISOL Leuven Isotope Separator On-Line MABU Mainz Atomic Beam Unit OC Output Coupler PALIS PArasitic Laser Ion-Source PBE Prism Beam Expander PL Pump Lens PM Pump Mirror RC Rydberg Correlation RIB Radioactive Ion Beam RIBF Radioactive Ion Beam Factory RIKEN From Japanese: Institute of physics and chemistry RILIS Resonance Ionization Laser Ion Source RIS Resonance Ionization Spectroscopy RMT Random Matrix Theory S3 Super Separator Spectrometer SFG Sum Frequency Generation xii SHG Second Harmonic Generation SPIRAL2 French: Système de Production d’Ions RAdioactifs en Ligne generation 2 SPM Saddle Point Model Ti:sapphire Titanium doped Al2O3 (sapphire) THG Third Harmonic Generation TRILIS TRIUMF Resonance Ionization Laser Ion Source TRIUMF TRI University Meson Facility UKAEA United Kingdom Atomic Energy Authority xiii Chapter1 Introduction The LARISSA group is looking back onto a long history of research studies in the field of resonance ionization spectroscopy (RIS). During almost three decades, the group around Prof. Dr. Klaus Wendt gained widespread experiences not only in the general field of RIS, but also in a variety of adjacent research areas. RIS can be used as a powerful tool for fundamental investigations in atomic physics, but also serves as methodical link to the large field of studies on ground state properties of atomic nuclei in nuclear physics. Based on an active worldwide research network, the group takes part in many collaborative activities at several Radioactive ion beam (RIB) facilities located usually at specific large scale research institutions. RIS is the tool, par excellence, to explore the formation of matter in the early universe. In this way, it is also the basis for this thesis. Its specific properties as well as the dedicated laser system developments by the group for different applications and collaborations within the LARISSA network are introduced in the first chapter of this thesis. Besides this, the treatment and removal of still open spaces of knowledge regard- ing fundamental atomic properties, i.e. the ionization potential (IP) or electron affin- ity (EA), evolved quickly to the pet passion of the LARISSA group. During the years, many of such left-over gaps in the list of data on atomic elements could be closed, including, e.g. the direct measurement of the IP and EA of astatine, often referred to as the rarest naturally occurring element in the earth’s crust [1,2], or the IP of various members of the lanthanide and actinide groups as well as further elements of the periodic table, e.g. in the region of the transition metals, see for example [3–5,22,37]. Many of the elements investigated in these studies are rare, radioactive and exhibit specifically complex atomic spectra. Nonetheless, the ionization potential could be determined, either by a direct measurement studying the convergence of Rydberg series or by an indirect approach applying e.g. field ionization within the saddle point model. The element protactinium has shown to be a very peculiar, unmanage- able case: The IP of this actinide could not be determined by using the standard RIS techniques, neither via a direct nor an indirect approach. Even after several elabo- 1 1 Introduction rate measurement campaigns, this fundamental atomic property of Pa could not be unveiled due to unpleasant chemical properties in combination with an exceptional complex atomic spectrum do far. On the other hand, this atomic structure, especially the high level density, is used within this thesis to elicit the IP of protactinium. Ana- lytical methods dedicatedly developed for this purpose were applied to the different atomic RIS excitation schemes of protactinium, which lead to a final value for the IP in the end. The final determination of the IP is not the only scientifically relevant information obtained from the complex RIS spectra of protactinium. A rather different research field, completely new for the doctoral candidate as well as for the whole LARISSA group, was initiated by the statement, how chaotic those spectra look. Since studies on quantum chaos experienced a renaissance in the 1990’s and early 2000’s by inves- tigating microwave systems, also studies on many other complex quantum chaot- ical systems became possible by more sophisticated research techniques. In this context, the protactinium spectrum was investigated regarding intrinsic quantum chaos (IQC) during the PhD studies documented here. It is the first time, that an atomic system of such complexity was accessible for quantum chaos research, mak- ing it even more interesting. Nonetheless, with stepping into this rather advanced, very theoretical research field, the whole LARISSA group break fresh ground. Even though it was somewhat difficult and time consuming to get used to the new topic, it delivered a valuable outcome with many stimulating ideas and new scientific contacts. The topic gained interest by several experts in the field requesting pre- sentations and discussion meetings about the quantum chaos in protactinium. All results found during this lively period were put together in a publication featured in the last chapter of this thesis. Lastly it should be mentioned, that this thesis is not foreseen to replace elaborate textbooks. Therefore, in each of the sections not the full-depth theory, which can anyhow easily be found in these books for further reading, is given. Rather, the introductory parts of different chapters are addressed and compiled for the non- expert readers, so that the featured publications can be understood. Nonetheless, for interested readers, who want to develop deeper insight into the individual topics, further reading is given as reference. 2 Chapter2 Resonance ionization spectroscopy With the first description and utilization of RIS by Ambartzumian, Letokhov & Mishin [6,7], a new versatile technique for investigation not only on atomic but also on nuclear fundamental properties was born. It is based on a successive excitation of one (or more) valence electron(s) via laser radiation precisely in tune with atomic transitions. A last excitation step, resonant or non-resonant, finally ionizes the atom and, thus, the resulting ion can be manipulated by electric or magnetic fields and then be detected with highest probability. Exceptional elemental selectivity, exceed- ing other excitation or ionization methods, is ensured by the resonant excitation since the atomic transitions used are unique for every element. Even isotopic selec- tivity can be achieved if the laser linewidth is below the isotope shift between the transitions of specific isotopes of one element. Resonance ionization is most often used in combination with a hot-cavity source for efficient atomization followed by a conventional mass separator. Thus, isobaric contaminants, stemming e.g. from surface ionization or non-resonant laser ioniza- tion, can be successfully eliminated. For more information on RIS, laser spectroscopy in general and correlated experiments in the atomic physics as well as the nuclear physics sector, please see the reviews [8, 9]. The resonance ionization spectroscopy performed in the low-energy mass spec- trometer MABU in the laboratory of the LARISSA group in Mainz serves as starting point for all publications this thesis is based on. Therefore, in the following sec- tions a short summary of the utilized laser systems, the mass separator and the spectroscopic technique is comprised. Thereafter, the first publication Resonance ionization spectroscopy of sodium Rydberg levels using difference frequency generation of high-repetition-rate pulsed Ti:sapphire lasers is presented. This work explains the en- largement of the accessible Ti:sapphire wavelength range by difference frequency generation (DFG) as well as validates our method of Rydberg spectroscopy by com- paring the measured ionization potential to the well-known IP of sodium. 3 2 Resonance ionization spectroscopy 2.1 Mainz Ti:sapphire laser system For RIS, a broadly tunable, narrow bandwidth, highly stable and maintenance-free, pulsed laser system with high output power is essential. Since a laser system com- prising all mentioned properties is not or at least not affordably available commer- cially, the laser systems were originally developed and stepwise updated and opti- mized within the Mainz LARISSA working group. Today, they are used at almost exclusively all leading RIB facilities worldwide. Ti:sapphire as active medium is most suitable for matching the above listed requirements. Sapphire, an Al2O3 crystal, is therefore doped with Ti3+-ions, so that the vibra- tionally broadened 2T2 triplet and 2E doublet of the Ti3+-ion embedded in the lattice structure form the emission and absorption bands, respectively, of this quasi-four- level laser system. Fig. 2.1(a) shows a schematic view of the quasi-four-level scheme of a Ti:sapphire laser, where the absorption band is given as red shaded area allow- ing for pump wavelengths between 420 nm and 600 nm and the emission band as green shaded area allowing for stimulated emission in a range between 650 nm and 1100 nm, while Fig. 2.1(b) gives the absorption and emission spectra of Ti:Sapphire. 1.0 0.8 │2 2E 0.6 │3 0.4 Pumping Stimulated emission Absorption Fluorescence 420 – 600 nm 650 – 1100 nm 0.2 2T2 │4 0 400 500 600 700 800 900 │1 Wavelength (nm) (a) Quasi-four-level laser system of (b) Absorption and emission spectrum of Ti:sapphire. The red and green shaded Ti:sapphire. Figure adapted from [13]. areas depict the absorption and emission band of the Ti3+-ion embedded in the sapphire crystal lattice. Graphic taken from [10]. Figure 2.1: Properties of Ti:sapphire as laser crystal. The basic laser theory and principle of the Ti:sapphire laser shall not be given here since it can be found in great detail in numerous textbooks, theses or scientific articles [11–17]. 4 Intensity (arb. Units) 2.1 Mainz Ti:sapphire laser system 2.1.1 Specialized versions of the Mainz lasers All types of the Mainz Ti:sapphire lasers have some general properties in common, which are the repetition rate of typically 10 kHz, the pulse length between 30 ns and 80 ns, the high beam quality with M2 < 1.3 and the average output powers between 1 W and 4 W. Nevertheless, different experimental requirements and needs led to specialized versions of the Mainz Ti:sapphire laser system. The main properties of a laser are primarily defined by the shape of the resonator cavity and the installed frequency selective elements. In this way, also the unique properties of the three main types of the Mainz Ti:sapphire lasers are defined: Standard Ti:sapphire laser: The standard version of the Mainz Ti:sapphire laser is a high-stable low-maintenance system that is mostly used. A Z-shaped resonator with birefringent filter and Fabry-Pérot etalon enables high average output powers with suitable low laser linewidths. More details can be found in [18]. Grating-assisted Ti:sapphire laser: This version is based on a refraction grating as frequency selective element, which allows continuous and mode-hop-free wave- length tuning over almost the complete Ti:sapphire emission range. It is the most versatile system ensuring low linewidth and high output power aside of the wide-range scan-ability. The current stage of this system was developed in [15]. Injection-locked Ti:sapphire laser: Injection locking of a bow-tie traveling-wave res- onator with a diode laser enables very narrow output linewidths with reason- able average output powers. This most elaborate system is used for experimen- tal approaches relying on extremely narrow linewidths. [16] gives an overview on this Ti:sapphire laser. Since the first two versions are frequently used throughout this work, their schemat- ics are displayed in Fig. 2.2. Characteristics of all types of the Mainz Ti:sapphire laser systems can be found e.g. in [15, 16]. 2.1.2 Frequency conversion Most optical atomic transitions between low-lying levels lie in the visible up to the blue or ultraviolet spectral range. Unfortunately, many of those are not directly reachable with the near-infrared radiation of the fundamental Ti:sapphire wave- length. Making use of non-linear optical processes the frequency of (pulsed) laser radiation can easily be converted, what can be applied for a very useful extension of the achievable wavelength range for Ti:sapphire lasers as well. Non-linear optical processes like second/third/fourth harmonic generation (SHG/THG/FHG), sum frequency generation (SFG) or difference frequency generation (DFG) require crys- tals with a non-linear optical susceptibility and additional birefringent properties 5 2 Resonance ionization spectroscopy EM FPE LF CM G PBE CM Ti:Sapphire OC Ti:Sapphire OC PM PM CM CM PL PL PM PM (a) Standard laser. (b) Grating-assisted laser. Figure 2.2: Schematics of the most common types of the Mainz Ti:sapphire laser system. Graphics adapted from [15], the explanation of the abbreviations can be found in the List of Abbreviations. for enabling the necessary phase matching. Note, that for the sake of efficiency, the third harmonic is usually generated via a SFG process and the fourth harmonic via applying the SHG process twice. Optical media fulfilling these requirements are for example β-barium borate (BBO), bismuth borate (BiBO) or lithium triborate (LBO). The theory of non-linear optics is primarily given in [19] while the basics for SHG and DFG especially with pulsed Ti:sapphire lasers can also be found in [15, 20] as well as in the publication Resonance ionization spectroscopy of sodium Rydberg levels using difference frequency generation of high-repetition-rate pulsed Ti:sapphire lasers. Fig. 2.3 shows a compilation of all wavelength ranges that are reachable with Ti:sapphire lasers involving frequency conversion processes. Figure 2.3: Compilation of wavelength ranges accessible with the Mainz Ti:sapphire lasers involving frequency conversion. Graphic adapted from [16]. 6 2.2 Experiment and technique 2.2 Experiment and technique 2.2.1 Mainz atomic beam unit The Mainz atomic beam unit (MABU) is a low-energy mass spectrometer equipped with a radio-frequency quadrupole for mass filtering. This table-top experiment, as given in Fig. 2.4(a), needs only very low voltages in the order of a few 100 V for ion extraction and ion beam shaping which makes it comfortable to use. Free atoms for resonant laser ionization are generated in a resistively heated atomizer furnace as shown in Fig. 2.4(b). After ionization and extraction, beam-shaping, mass-filtering and final detection of the ions on a channel electron multiplier in single ion counting mode is carried out. The apparatus is described in great detail in [21]. More Informa- tion can also be found in the upcoming publications Resonance ionization spectroscopy of sodium Rydberg levels using difference frequency generation of high-repetition-rate pulsed Ti:sapphire lasers and Excited atomic energy levels in protactinium by resonance ionization spectroscopy. (a) Photograph of the MABU. (b) Photograph of an atomizer furnace for the MABU. Figure 2.4: Photographs of the Mainz atomic beam unit. Pictures taken from [21]. 2.2.2 RIS technique Within the atomizer furnace, which serves as atom-laser-interaction region, the evap- orated atoms of an sample are excited by a set of two or three Ti:sapphire lasers. Obeying the transition rules for optical dipole transitions, the valence electron(s) are stepwise excited into high-lying energy states before the final ionization takes place. Fig. 2.5 shows a generalized ionization scheme with typical excitation steps and three different possibilities for the last ionization step. Note, that an excitation via ν02 and via ν01 and ν12 into the same energy level |2〉 is not possible, because a change in parity is mandatory for every transition. Ionization takes place as soon as the energy of the valence electron is raised above the IP of the investigated element. In RIS, a distinction is made between resonant and non-resonant ionization. For 7 2 Resonance ionization spectroscopy E Continuum |AIۧ EIP IP Rydberg ν ν Levels2nr 2AI ν2Ryd |2ۧ ν12 |1ۧ ν02 ν01 0 |0ۧ Figure 2.5: Generalized RIS ionization scheme. For details, please see text. Graphic adapted from [15]. the latter, an ionization laser excites the electron into the continuum above the IP. Much higher ionization efficiency can be achieved in the resonant case, namely if an auto-ionizing state (AI) is excited. Such an AI is a multi-electron excitation that transfers its energy with rather short decay times to a single electron whose energy is than high enough to exit the atom. A third also highly efficient ionization process implies the intermediate excitation of Rydberg states. Rydberg states are highly excited, only weakly bound energy levels with high principle quantum numbers n located just below the IP. These states are easily ionized by “third-party” processes as i.e. black-body radiation, electric or magnetic fields and collisions with other atoms, ions or electrons. A series of Rydberg levels converges towards the IP proportional to 1/n2, correspondingly the spectroscopy of Rydberg levels is often used for the direct determination of the IP, as also done in the publication presented in Sec. 2.3. More information on Rydberg spectroscopy and other methods for determining a value for the IP can be found e.g. in [22] or briefly in Sec. 4. In spectra determined via RIS, the measured resonances exhibit typical line shapes. Several broadening effects determine the observed linewidth of the resonances. The most important underlying effects are discussed in the publication Resonance ioniza- tion spectroscopy of sodium Rydberg levels using difference frequency generation of high- repetition-rate pulsed Ti:sapphire lasers. Very detailed information on the RIS technique and selected experiments can be found for example in [8, 9, 23, 24]. 8 2.3 Publication: RIS of sodium Rydberg levels using DFG of high-repetition-rate Ti:sapph lasers 2.3 Publication: Resonance ionization spectroscopy of sodium Ryd- berg levels using difference frequency generation of high-repetition- rate pulsed Ti:sapphire lasers The article was published in 2016 in Physical Review A although the data was already collected between autumn 2013 and spring 2014. In the meantime a proceeding [20] was published presenting details on the DFG with pulsed Ti:sapphire lasers and demonstrating the ionization of sodium via its famous D-doublet. The objective of the article presented herein was to demonstrate the RIS on transitions, where formerly dye-lasers were necessary. With the obvious spectroscopy of sodium via the famous D-lines in the orange wavelength regime, a showcase for the application and evaluation of the spectroscopic technique and the verification of the IP determination method via Rydberg convergences was given at the same time. In the context of this thesis, the article gives a very valuable overview on the laser system involving frequency conversion, on the mass spectrometer MABU, as well as on several issues concerning the RIS method in general, i.e. the resonance broadening effects, on Rydberg spectroscopy and the therewith given possibility of a direct determination of the ionization potential. As main author, the doctoral candidate not only prepared the manuscript of this publication, but also supervised the Spanish student Julia Marı́n-Sáez during her IAESTE internship and was responsible for the entire measurement campaign on NA and prepared the analysis given in this article. 9 PHYSICAL REVIEW A 93, 052518 (2016) Resonance ionization spectroscopy of sodium Rydberg levels using difference frequency generation of high-repetition-rate pulsed Ti:sapphire lasers P. Naubereit,* J. Marı́n-Sáez,† F. Schneider, A. Hakimi, M. Franzmann,‡ T. Kron, S. Richter, and K. Wendt Institute of Physics, University of Mainz, D-55128 Mainz, Germany (Received 8 March 2016; published 31 May 2016) The generation of tunable laser light in the green to orange spectral range has generally been a deficiency of solid-state lasers. Hence, the formalisms of difference frequency generation (DFG) and optical parametric processes are well known, but the DFG of pulsed solid-state lasers was rarely efficient enough for its use in resonance ionization spectroscopy. Difference frequency generation of high-repetition-rate Ti:sapphire lasers was demonstrated for resonance ionization of sodium by efficiently exciting the well-known D1 and D2 lines in the orange spectral range (both ≈ 589 nm). In order to prove the applicability of the laser system for its use at resonance ionization laser ion sources of radioactive ion beam facilities, the first ionization potential of Na was remeasured by three-step resonance ionization into Rydberg levels and investigating Rydberg convergences. A result of EIP = 41449.455(6)stat(7) −1sys cm was obtained, which is in perfect agreement with the literature value of ElitIP = 41449.451(2) cm−1. A total of 41 level positions for the odd-parity Rydberg series nf 2 oF 5/2,7/2 for principal quantum numbers of 10  n  60 were determined experimentally. DOI: 10.1103/PhysRevA.93.052518 I. INTRODUCTION corresponding duty cycle losses of such low-repetition-rate operation are a huge disadvantage concerning the efficiencies Powerful tunable high-repetition-rate solid-state lasers are of ion production at RIB facilities. Dye lasers instead can a most useful tool for spectroscopy and ion production at provide both high repetition rate and high pulse energies radioactive ion beam (RIB) facilities, which use resonance with a broad range of output wavelengths [5]. Unlike the ionization laser ion sources (RILIS). Those lasers may be used high-repetition-rate solid-state lasers, however, dye lasers need in long-term stable and almost maintenance-free continuous a lot of maintenance and care during long-term operation in operation and, moreover, they permit a rather simple spectral order to guarantee suitable performance and stability as needed range extension by generation of the second, third, or fourth at RIB facilities. The principle of the DFG approach has been harmonic. Using a titanium:sapphire crystal as active laser demonstrated using high-repetition-rate ns Ti:sapphire lasers medium a wavelength coverage of 680 nm to 1000 nm in in [6]. In this work, the reliable application in high-resolution fundamental mode and 210 nm to 450 nm by using frequency multistep resonance ionization spectroscopy is shown. doubling, tripling and quadrupling, has been demonstrated. As an evaluation case, Rydberg spectroscopy on the element Nevertheless, a remaining gap in the visible spectral range with the probably best known transitions in the orange between the fundamental (> 680 nm) and the second harmonic wavelength regime around 590 nm was chosen by addressing (< 450 nm) output is remaining. The second-order nonlinear the famousD andD doublet lines of sodium. By using three- process of difference frequency generation (DFG) allows to 1 2step resonance ionization, involving DFG in the first excitation close that gap. The theory of this process can be found step, spectroscopy around the sodium ionization potential for example in [1]. The desired wavelength range is also was performed and its value was redetermined. Although this achievable with a variety of different laser systems with or value has been known with high precision since 1992 (Elit = without DFG. Most of them are even commercially available IP41449.451(2) cm−1 [7,8]), energies of higher lying levels of for many years. However, those systems do not really fulfill odd-parity Rydberg series like nf 2 o were just published the requirements at on-line facilities as pointed out explicitly F 5/2,7/2for the first time during the analysis of our data in 2015 by above. Continuous-wave laser systems, for example, operating Nadeem and co-workers [9]. These authors give an elaborate as Raman lasers [2] or using sum frequency generation [3], overview on the atomic spectroscopy of high-lying levels of are not able to provide either the high output energies or the sodium, to which we refer here for a general overview. They spectral coverage needed for saturation of a broadened excita- have carried out two-color, three-step resonance excitation tion line for a thermal atom ensemble, in particular regarding based upon the 3s-3d two-photon transition from the ground weaker atomic transitions. In contrast, high pulse energies are state and finally populate nf levels in the range 15  n  51. needed for optical parametric oscillators or amplifiers [4] and Results include level energies and oscillator strengths. They are in most cases only achieved at low repetition rates. The report an uncertainty of their level energies of 0.2 cm−1 due to strong broadening effects in their thermionic diode setup, which is almost one order of magnitude less precise than the *naubereit@uni-mainz.de results of our measurements with in-source laser geometry on †Present address: Applied Physics Section of the Environmental a well-collimated atomic beam. In addition, our data, resulting Department, University of Lleida, Escuela Politécnica Superior, from three-color, three-step resonance excitation along two Jaume II 69, 25001 Lleida, Spain. different excitation ladders, which were based upon the D1 ‡Institute for Radioecology and Radiation Protection, University of and D2 ground-state transitions, respectively, extend to higher Hannover, D-30419 Hannover, Germany. principal quantum numbers of up to n = 60. Correspondingly, 2469-9926/2016/93(5)/052518(9) 052518-1 ©2016 American Physical Society P. NAUBEREIT et al. PHYSICAL REVIEW A 93, 052518 (2016) Laser system with DFG Mainz Atomic Beam Unit of less than 1 GHz a day. Nevertheless, the wavelengths of all Ti:sapph 1 lasers were actively monitored during the experiment.DFG Ion optics Atomizer furnace Ti:sapph 2 SHG Quadrupole deflector Ion 2. Considerations for difference frequency generation Quadrupole extraction Ti:sapph 3 mass filter In [6] a high-repetition-rate, pulsed, narrow-bandwidth Channeltron Ti:sapphire laser involving injection locking was used for Grating Nd:YAG Nd:YAG ion detectorTi:sapph DFG. This laser has both a narrow bandwidth of 20 MHz and a suitable output power of 2.6 W. Frequency doubling FIG. 1. Sketch of the experimental setup including the laser was carried out in a very efficient but somewhat delicate system and the atomic beam mass spectrometer unit MABU. intracavity approach. Despite some advantages of this setup, we followed a different approach here and used the standard Mainz Ti:sapphire lasers as described above to guarantee we have attempted a full Rydberg-Ritz analysis of our data the universality of the DFG system without involving and including earlier known literature data as bandheads and controlling the sophisticated injection locking procedure. In redetermined the ionization potential with high accuracy. This addition we abstained from intracavity doubling for the DFG approach resulted in a confirmation of the high-precision pump photon generation to provide a more simple frequency- literature value of lit = 41449 451(2) cm−1 from [7,8] in an tuning procedure for both the second harmonic light and theEIP . independent measurement. DFG light at the expense of a reduced output power. Based on calculations for optimization of the mixing process with the software SNLO [12], we expected considerable losses in the II. APPARATUS DFG process, as the different available BBO crystals had to be tilted quite far off from orthogonal orientation due to imperfect Figure 1 gives an overview of the experimental apparatus cutting angles for proper phase matching. After the mixing used. It consists of two distinct components: the dedicated stage the laser beam in addition had to pass several dispersive high-repetition-rate solid-state laser system including the DFG optics, e.g., dichroitic mirrors and prisms, in order to separate unit and the atomic beam apparatus with single ion detection the DFG light from the remaining blue and infrared parts. MABU (Mainz atomic beam unit). Regarding those limitations we expected rather low output power of the DFG light of just a few mW. Keeping in mind the pulsed structure of the laser light with a duty cycle of 10−3 this A. Laser system value should nevertheless ensure saturation of all reasonably 1. General layout strong optical transitions under our experimental conditions. For a resonant three-step ionization process involving DFG, four Ti:sapphire lasers are necessary: Two of them provide the input for the DFG, which is used for the first excitation B. Atomic beam unit MABU step. The higher energetic pump wave of the DFG process is The atomic spectroscopy, mass separation, and detection generated by frequency doubling of the fundamental laser light of the sodium ions were performed in the atomic beam unit in a BBO crystal. The lower energetic, subtracted idler wave is MABU. This machine is a low-energy mass spectrometer with provided in the fundamental infrared range of the Ti:sapphire an extraction field of well below 10 V/mm. For chemical laser. The radiation of the third laser excites the atoms in a decomposition of the sample molecule NaHCO3, reduction second resonant step into the second excited state. Those three and final atomization, a microscopic sample of a few mg lasers are based on the standard Z-shaped cavity design of the was placed in a resistively heated graphite furnace with a Mainz Ti:sapphire lasers in the latest design according to [10]. diameter of 2.2 mm and a length of 50 mm. The laser The ionization step is then provided from a fourth laser, which beams were directed into the atomic vapor, which formed is a grating-assisted Ti:sapphire laser [11] to guarantee a wide, inside the hot cavity, to induce resonant multistep excitation continuously tunable scanning range. of the sodium atoms into Rydberg levels, from which they To achieve sufficient pump power and temporal syn- ionize through collisions, black-body radiation from the heated chronous output pulses of the Ti:sapphire lasers, two furnace, as well as influences from the electric extraction commercial frequency-doubled Nd:YAG pump lasers with field. Field ionization of Rydberg atoms is rather strong, but 10-kHz repetition rates (Quantronix Hawk-Pro 532-60-M and in our case gives only small contributions due to the low Photonics Industries DM 80-532) were used. The standard tension applied in the extraction region of our apparatus. Mainz Ti:sapphire lasers provide an average power between 2 The individual strengths of the remaining processes cannot and 4 W at pump powers of 15 W. The pulse lengths span from easily be disentangled. After acceleration and beam shaping, 25 ns at the gain maximum of ≈ 810 nm to 100 ns at the edges the ions were guided through an electrostatic 90◦ quadrupole of the tuning range around 700 and 1000 nm, respectively, deflector to remove background from neutral species and a whereas the spectral bandwidth amounts to typically νfund = radio-frequency quadrupole mass filter to isolate 23Na from 4(1) GHz. In the experiment the grating-assisted laser had an other disturbing masses. Subsequently the highly isotopic output power of ≈ 280 mW with a bandwidth of 2 GHz. The pure ion beam was detected by an off-axis-mounted channel lasers show a good long-term stability with frequency drifts electron multiplier in single-ion counting operation. 052518-2 RESONANCE IONIZATION SPECTROSCOPY OF SODIUM . . . PHYSICAL REVIEW A 93, 052518 (2016) III. RESULTS A. Difference frequency generation -1 2 6 One of the two lasers necessary for DFG operated in its 41449.451 cm IP 2s 2p fundamental wavelength regime at a wavelength of 920 nm Rydberg with an average output power of 2500 mW (idler). The second levels laser operated at 718 nm with 2900 mW and was externally frequency doubled using a BBO crystal cut at 32.8◦ for type I phase matching (pump). We achieved 700 mW in the blue 810 - 895 nm spectral range around at 359 nm. The light was collimated, 29172.887 cm-1 2p63d 2D separated from the remaining fundamental light, and then 3/229172.837 cm-1 2p63d 2D5/2 focused together with the infrared light into another BBO crystal for DFG. Here, we also implemented type I phase 818.55 nm 819.70 nm 819.71 nmA = 4.3·107 s-1 A = 8.6·106 s-1 A = 5.1·107 s-1 matching at a BBO angle of 32◦, which means that both -1 6 2 o the infrared idler wave and the DFG wave were in ordinary 16973.366 cm 2p 3p P316956.170 cm-1 2p63p 2Po /2 polarization (o) while the frequency-doubled pump light was 1/2 introduced as an extraordinary (e) wave (oo → e). After its generation, the DFG light was collimated and separated from 589.76 nm 589.16 nm the remaining idler and pump parts by a combination of A=6.14·107 s-1 A=6.16·107 s-1 different dichroitic mirrors and dispersive prisms, primarily to allow for power determination. With this simplified setup we obtained up to 11 mW of average power for the DFG 0 cm-1 2p63s 2S1/2 light at 589 nm in good agreement with the expectations from the calculations. The overall conversion efficiency is FIG. 2. Sodium excitation schemes investigated in this work. The IDFG/(Iidler + Ipump) = 0.3 %, which may be increased by energy level positions are taken from [13]. The lowest transition on using, e.g., the type II phase matching with a BBO suitably cut the left side is classically denoted as the D1 line, the right one as the at 41.5◦ [12]. Due to the high transition strength of the sodium D2 line. D lines, less than 2 mW of the DFG light were used for most of the spectroscopic measurements. For the DFG laser rad√iation a linewidth (FWHM) of they were not resolved when scanning the laser frequencyacross the second transition. Thus, it was assumed that in the ν 2DFG = √νfund +ν 2 SHG D2 scheme the majority of the atoms were excited by the √ stronger transitions to the J = 5/2, according to the A factors = ν2 2fund + ( 2νfund) in the excitation scheme of Fig. 2. For the data analysis, the ≈ energetic position of the second excited level during the D26.9(17) GHz (1) excitation was set to a properly weighted intermediate value is expected, as only Gaussian contributions determine the between the two fine structure sublevels. width. The range of spectral coverage was not investigated To verify the energy position of the two first and second because this is strongly dependent on the available crystals for excited states and to get information about the bandwidth of DFG, which were not optimum during this study. With a set the individual excitation steps, frequency scans on the different of four properly cut BBO crystals the range of 520− 680 nm transitions were performed while the third step laser was tuned should be easily accessible by DFG of our Ti:sapphire lasers. to above the IP for nonresonant ionization. Due to its single Access to the remaining gap of 450− 520 nm will require valence electron, the spectrum of sodium just above the IP is frequency tripling or quadrupling for the DFG pump wave, completely unstructured. Correspondingly, in the third step, which will severely complicate the process and reduce the no autoionizing state is accessible with our laser system and achievable output power. for excitation into the continuum no resonance structure was observed. Figure 3 shows on the left the binned frequency scans for B. Two-step excitation into the 3d 2DJ configuration the first and second excitation steps of the right excitation The three-step excitation scheme used in this work is ladder of Fig. 2 along the D2 line and the saturation curves on shown in Fig. 2. It is based on the energy levels as given the right side for all three transitions. in [13]. The DFG process was used for the first transition, As expected, the saturation power of the first excitation step from the ground state 3s 2S1/2 to either the level 3p 2P1/2 at of 1.1(3) mW, as measured under our experimental conditions, 16956.170 cm−1 (via the D1 line of the doublet) or to level is very low and the transition is easily saturated. The just 3p 2P −13/2 at 16973.366 cm (via D2). The second transition slightly higher saturation power of 2.6(5) mW for the second is induced by a Ti:sapphire laser in fundamental mode to the step is also well expected due to a comparable oscillator levels 3d 2D −1 −13/2,5/2 at 29172.887 cm and 29172.837 cm , strength as given in Fig. 2; the factor of 2 increase may even respectively, regarding the selection rule of J = ±1,0. be ascribed to an imperfect spatial overlapping of the laser However, these two second excited levels are very close lying beams. Due to the different contributing broadening effects, (ν ≈ 1.5 GHz) and due to the laser linewidth of 3− 5 GHz, a Voigt profile was fitted to the data points of the frequency 052518-3 P. NAUBEREIT et al. PHYSICAL REVIEW A 93, 052518 (2016) Transition 3 C. Linewidths and saturation 1 For the experimental linewidth of the transitions, we have to consider several contributions. Aside from the negligible natural linewidth (FWHM) of 9.8 MHz for the first step excitation [15] most of the broadening effects contribute in the GHz range. The minimal measured linewidth is limited by the laser linewidth given above. Besides this the Doppler 29.2(9) GHz Transition 2 broadening (FWHM) w√as taken into account with1 @ 255 mW = ν0 8kT ln 2νdop = 2.5(1) GHz , (5) P c matomsat = 2.6(5) mW for a frequency of the transition of ν0 = 508.8487 THz, 0 temperature in the hot furnace of T ≈ 1100 K, as extrapolated -2 -1 0 1 2 0 50 100 150 200 from a pyrometric measurement, and the sodium massmatom = 24.6(8) GHz Transition 1 23 u. The overall Gaussian contribution considers the linewidth 1 @ 10.7 mW of the DFG laser and the Doppler broadening, which is a small contribution due to√the low oven temperatures: P = 1.1(3) mW ν = ν2G DFG +ν2dop = 7(2) GHz . (6)sat 0 The saturation broadening can be approximated by a -2 -1 0 1 2 0.0 0.5 1.0 -1 Lorentzian distribution and gives a significant contribution toν-νlit [cm ] Laser Power [mW] the overall linewidth at a maximum laser power of 10.7(10) mW. It is calculated as (FWHM) FIG. 3. Frequency scans and saturation curves of the three √ transitions of the excitation scheme via the D2 line. Since the third νL = νG 1 + S0 = 24(6) GHz , (7) step is nonresonantly ionizing, the resonance curve was omitted and the saturation curve has a constant slope. The extracted linewidths where νG is the Gaussian contribution FWHM of the are noted in the graphs. transition and S0 = I = 10(3), with laser intensity I andIsat saturation intensity of the transition Isat. For combining Gaussian and Lorentzian contributions we used Eq. (3) with a scans: resulting overall Voigt profile linewidth of the transition of 2 ln(2) ν νcalc = 26(3) GHz. (8) I (ν) = A+ LB ∫ π3/2 ν2G For the second transition the same calculation with adapted∞ ( ) ex(p(− 2) ) parameters leads to a linewidth expectation of× √ √ t dt, (2) −∞ 2 2ln 2νL + 4 ln 2 ν−νcenter − t 32(5) GHz, (9) νG νG where the saturation parameter S = I0 × 10 = 48(14) mustI where νG and νL give the linewidths of the Gaussian and sat additionally be scaled by the saturation of the first excitation the Lorentzian contribution, respectively. The flattened top of step. Notice that there is no additional contribution from the the peak which results from the saturation is not included in excitation of the pair of energy levels in that step, because this the fit model. Nevertheless, the curves perfectly reproduce the is fully covered by the five times larger laser linewidth. outer wings of the peaks, which contain the most important The calculated linewidths of the two lower excitation information for the precise extraction of energy positions νcenter steps reproduce the experimental values rather well. When and FWHMν, as obtained using the numerical method given addressing higher spectral resolutions, there are several ways in [14] by to reduce the linewidths. The biggest contribution results from √ the large oversaturation, which easily is compensated for by = 0 5346 + 0 2166 2 + 2 (3) adjusting the intensities of the lasers accordingly. The partν . νL . νL νG. resulting from the laser linewidth itself is more complicated to address and would require, e.g., the use of an injection locked For the saturation powers Psat, as quoted, a fit function with system or alternatively a further passive wavelength selection the form within the laser resonator as discussed, e.g., in [6]. P I (P ) = A+ B + (4) D. Rydberg spectroscopyP Psat For the spectroscopy on sodium Rydberg levels, the laser was applied to all three saturation curves. For the nonresonant for the third excitation step was continuously scanned using ionization step an absolutely linear behavior was observed, as the wide-range tunable, grating-assisted Ti:sapphire laser to expected, for laser powers of up to 175 mW. cover the energy range just below the literature value for 052518-4 Normalized Countrate [arb. units] RESONANCE IONIZATION SPECTROSCOPY OF SODIUM . . . PHYSICAL REVIEW A 93, 052518 (2016) (D1) F 10000 S P D 5/21/2 1/2 3/2 (P5/2) 1000 21 - 60 1000 10 11 12 13 14 15 16 17 18 19 20 (D2) F S P D 5/2, 7/2 100 1/2 3/2 3/2, 5/2 (P5/2, 7/2) 10 1 40400 40600 40800 41000 41200 41400 Wave Number [cm-1] FIG. 4. Two examples of frequency scans of the third step for sodium resonance ionization via the D1 transition (upper graph) and the D2 transition (lower graph) as the first step. The excitation scheme and the principal quantum numbers of the determined states are indicated. It should be mentioned that the significant difference in the count rates of the two scans is due to variations in experimental conditions, i.e., settings of the ion optics, and not related to individual transition strengths. the ionization potential of Elit = 41449.451(2) cm−1 [7,8]. A The statistical error of each level energy consists of theIP range ofEstart ≈ 40300 cm−1 toE ≈ 41500 cm−1 of overall statistical readout error of the wave meter, the uncertainty ofend excitation energy was used in the case of the excitation via the the fitted position and a very small error, which accounts for D and a slightly narrower range for theD transition. Figure 4 the correction of systematical shifts in wave numbers by the1 2 shows representative scans for both schemes in logarithmic data acquisition system when scanning upward or downward. scale, where the investigated Rydberg states are numbered A systematic error of 0.007 cm −1, which is the 1 σ absolute according to their principal quantum numbers and Elit is accuracy of the wavelength meter (High Finesse WS6-600),IP marked with a dashed line. was added afterward to each value. This systematic shift is We identified Rydberg states of the f series with principal only applied once, because for the first and second excitation quantum numbers from n = 10 to n = 60 for the excitation via step the literature values, not the herein measured ones, were D1 and from n = 12 to n = 59 for the excitation via D . Only used for calculation of the total energy. The data in Table I2 peaks of one Rydberg series, i.e., belonging to the unresolved are in perfect agreement with the earlier literature values nf 2 oF , is visible. This is due to the very low transition for nine lower lying levels but slightly disagree for the two5/2,7/2 strengths into the p series, which is three to four orders of cases of n = 15 and n = 19. The minor discrepancies of 0.05−1 magnitude lower than that into the f states according to [16]. and 0.03 cm , respectively, are well below the uncertainty The peaks in Fig. 4 show typical linewidths of 15− 24 GHz, of the recent measurement of [9] and only slightly exceed which are slightly lower than the linewidth obtained in the the statistical uncertainty of our data, while being covered frequency scans in Fig. 3 due to the avoidance of strong by the total error. It is thus ascribed to the fact that the oversaturation in any of the three consecutive excitation steps literature values from n = 15 to n = 20 are not measured during these measurements. but calculated using interpolation. Further details of this Several scans, upward and downward in energy, were approach are given in [17], which is the reference quoted averaged for both excitation schemes. The energy positions in [13]. The data used for the interpolation was taken from found with excitation via both the D line and D line are in [18,19]. Possibly the uncertainties given therein were also1 2 a good agreement and perfectly overlap within the statistical somewhat underestimated. In addition, we considered shifts error bars. It should be mentioned that the measured deviation caused by the Autler-Townes effect (ac Stark effect) [20] for the first ten levels is with 0.09 cm−1 in average larger than to be responsible for deviating level energies. In multistep for the rest of the levels with a deviation of only 0.03 cm−1 resonance ionization, highly intense light fields can cause a in average. This is due to the count rate, which was in the broadening and a splitting of energy levels in the range of D scan very low for the first ten peaks. This results in very several GHz [21]. If the frequencies of the exiting lasers are2 few data points in each of those peaks, which occasionally furthermore detuned regarding the resonance frequencies, an led to a shifted energy position in the fit. The fitted and error asymmetry of the split structure can occur. In our case, none weighted average energy positions for each Rydberg peak are of the measured peaks are apparently broadened or split which listed together with the available high-precision literature data is ascribed to relatively low laser intensities in the interaction from [13] and the most recent data from [9] in Table I. This region in comparison to those in [21]. Moreover, we set the table also gives the quantum defects calculated with Eqs. (10) lasers always at the resonance frequencies, so that only a and (11) for each Rydberg level. symmetrical splitting could occur, which would not shift the 052518-5 Count Rate [counts/s] P. NAUBEREIT et al. PHYSICAL REVIEW A 93, 052518 (2016) TABLE I. Energetic positions (in cm−1) and quantum defects of the measured Rydberg levels of the f series together with literature values a b from [13,9]. The values marked with ∗ were used as bandheads for the Rydberg-Ritz fitting routine, those marked with ∗ were only interpolated according to [17]. Configuration Ref. [13] Ref. [9] Weighted average Quantum defect δ(n) a 2p64f 34586.92(2)∗ 6 ∗a2p 5f 37057.65(2) a 2p66f 38399.79(2)∗ a 2p67f 39208.98(2)∗ 6 a2p 8f 39734.16(2)∗ 6 a2p 9f 40094.19(2)∗ 2p610f 40351.761(2) 40351.751(21)stat(7)sys 0.0016(1) 2p611f 40542.293(2) 40542.293(21)stat(7)sys 0.0016(1) 2p612f 40687.203(2) 40687.19(20) 40687.193(26)stat(7)sys 0.0017(2) 2p613f 40799.974(2) 40799.92(20) 40799.964(26)stat(7)sys 0.0017(2) 2p614f 40889.452(2) 40889.39(20) 40889.438(26)stat(7)sys 0.0018(3) 2p6 b 15f 40961.637(2)∗ 40961.52(20) 40961.588(26)stat(7)sys 0.0024(3) b 2p616f 41020.714(2)∗ 41020.69(20) 41020.707(26)stat(7)sys 0.0018(4) b 2p617f 41069.674(2)∗ 41069.62(20) 41069.662(26)stat(7)sys 0.0019(5) 2p6 b 18f 41110.703(2)∗ 41110.65(20) 41110.692(15)stat(7)sys 0.0020(4) 2p6 b 19f 41145.425(2)∗ 41145.44(20) 41145.454(15)stat(7)sys 0.0008(5) 2p6 b 20f 41175.070(2)∗ 41174.98(20) 41175.067(15)stat(7)sys 0.0019(6) 2p621f 41200.56(20) 41200.576(15)stat(7)sys 0.0020(6) 2p622f 41222.59(20) 41222.669(15)stat(7)sys 0.0030(7) 2p623f 41241.95(20) 41241.980(15)stat(7)sys 0.0021(8) 2p624f 41258.89(20) 41258.912(15)stat(7)sys 0.0020(10) 2p625f 41273.80(20) 41273.825(15)stat(7)sys 0.0039(11) 2p626f 41286.91(20) 41287.083(15)stat(7)sys 0.0034(12) 2p627f 41298.76(20) 41298.880(15)stat(7)sys 0.0043(14) 2p628f 41309.40(20) 41309.473(15)stat(7)sys 0.0015(15) 2p629f 41318.89(20) 41318.919(15)stat(7)sys 0.0061(17) 2p630f 41327.41(20) 41327.512(15)stat(7)sys 0.0020(19) 2p631f 41335.14(20) 41335.246(15)stat(7)sys 0.0029(21) 2p632f 41342.20(20) 41342.283(15)stat(7)sys 0.0015(23) 2p633f 41348.67(20) 41348.682(15)stat(7)sys 0.0011(25) 2p634f 41354.48(20) 41354.507(15)stat(7)sys 0.0040(27) 2p635f 41359.77(20) 41359.867(15)stat(7)sys 0.0018(30) 2p636f 41364.71(20) 41364.764(15)stat(7)sys 0.0041(32) 2p637f 41369.22(20) 41369.264(15)stat(7)sys 0.0080(35) 2p638f 41373.43(20) 41373.490(15)stat(7)sys −0.0071(38) 2p639f 41377.20(20) 41377.332(15)stat(7)sys −0.0061(41) 2p640f 41380.81(20) 41380.861(15)stat(7)sys 0.0029(44) 2p641f 41384.12(20) 41384.204(15)stat(7)sys −0.0088(48) 2p642f 41387.16(20) 41387.196(15)stat(7)sys 0.0174(51) 2p643f 41390.06(20) 41390.158(18)stat(7)sys −0.0182(65) 2p644f 41392.71(20) 41392.813(18)stat(7)sys −0.0151(70) 2p645f 41395.22(20) 41395.280(18)stat(7)sys −0.0060(75) 2p646f 41397.53(20) 41397.615(18)stat(7)sys −0.0082(80) 2p647f 41399.71(20) 41399.788(18)stat(7)sys −0.0042(85) 2p648f 41401.75(20) 41401.831(18)stat(7)sys −0.0019(91) 2p649f 41403.66(20) 41403.763(18)stat(7)sys −0.0061(97) 2p650f 41405.47(20) 41405.577(18)stat(7)sys −0.0090(103) 2p651f 41407.18(20) 41407.253(18)stat(7)sys 0.0082(109) 2p652f 41408.912(18)stat(7)sys −0.0248(115) 2p653f 41410.417(18)stat(7)sys −0.0182(122) 2p654f 41411.821(18)stat(7)sys 0.0019(129) 2p655f 41413.189(18)stat(7)sys −0.0075(137) 2p656f 41414.495(18)stat(7)sys −0.0254(144) 052518-6 RESONANCE IONIZATION SPECTROSCOPY OF SODIUM . . . PHYSICAL REVIEW A 93, 052518 (2016) TABLE I. (Continued.) Configuration Ref. [13] Ref. [9] Weighted average Quantum defect δ(n) 2p657f 41415.675(18)stat(7)sys 0.0049(152) 2p658f 41416.859(18)stat(7)sys −0.0209(160) 2p659f 41417.944(18)stat(7)sys −0.0112(169) 2p660f 41418.982(25)stat(7)sys −0.0118(248) center of the symmetrical fit function applied to the observed two approaches for analysis are truly minor and do not data. significantly exceed the statistical error bars, while the much To extract the ionization potential from the energy positions, lower precision in the A and particularly the B parameters the Rydberg-Ritz formula is obvious when omitting the bandheads. The corresponding distortion in the Rydberg-Ritz function even accidentally 1 E(n) = EIP − RM ∗ , (10) compensates for the slight discrepancy of 0.023 cm −1 obtained n 2 between the two excitation schemes by inclusion of the data with the reduced-mass Rydberg constant RM and the effective from [13]. Nevertheless, for both series, via D1 and D2, quantum number n∗ as given by the principal quantum number the energy of the average IP value matches the literature n and the Ritz expansion of the quantum defect δ(n) in second value of ElitIP = 41449.451(2) cm−1 perfectly even within the order ( ) statistical uncertainties. The trend of the values indicatesthe possibility that they are systematically shifted by about ∗ = 1 −1n n− δ(n) = n− A+ B , (11) ±0.007 cm , which fully confirms the quoted uncertainty (n− A)2 of our measurement. Correspondingly, we give the average was fitted to the data points as shown in Fig. 5. This graph, of both measurements obtained with the inclusion of band- with residuals well below 0.05 cm−1, is a representative plot heads as our final value for the first ionization potential of for the different fits, which were also performed individually sodium: for the series with excitation via D1 and D2, respectively. −1 Since the peak positions for lower principal quantum EIP = 41449.455(6)stat(7)sys cm . numbers could not be measured experimentally in this work, we included the literature values for = 4 to = 9 as This value is in good agreement with the literature value. Then n bandheads for the fit. This enables a better convergence of statistical uncertainty given accounts for the fit error which the fit, especially regarding the uncertainties of the and covers all errors in the energy positions of the experimentalA B parameters of the Rydberg-Ritz expansion. peaks addressed by the fit. On top of that a systematic error The fit parameters and their uncertainties are given in- was added as discussed before. dividually for both excitation schemes in Table II. For Figure 6 shows the quantum defects for the f series in comparison also the parameters for a fit ignoring the bandheads dependence of the effective quantum numbern ∗. Similar to that are included. The variation in the IP values between the visible in the residuals of Fig. 5, in Fig. 6 minor irregularities are observed in the range of 38  n  44. In that range an underlying peak structure which is also well visible in the base 42000 line of Fig. 4 but could not be identified, seemingly affects the peak positions and the fitting procedure. The situation is shown 40000 in detail in the magnification given in Fig. 7. Nevertheless, the additional peaks observed around n = 38 and n = 43 have no significant influence on the fit results for the IP. 38000 36000 TABLE II. Fit parameters for the Rydberg-Ritz fits of the different series. The literature value is from [7,8]. 34000 0.05 With bandheads Without bandheads 0.00 D1 E −1 IP [cm ] 41449.443(5)stat(7)sys 41449.456(6)stat(7)sys A 0.00169(3) 0.0030(4) -0.05 B −0.0079(6) −0.153(45) 0 5 10 15 20 25 30 35 40 45 50 55 60 D2 EIP [cm−1] 41449.466(3)stat(7)sys 41449.463(6)stat(7)sys Principal Quantum Number n A 0.00172(2) 0.0011(9) B −0.0083(3) −0.068(240) FIG. 5. Energy positions of the Rydberg peaks in dependence of −1 the principal quantum number. The Rydberg-Ritz formula was fitted Avg. EIP [cm ] 41449.455(6)stat(7)sys 41449.460(8)stat(7)sys to the data (red curve). The resulting residuals are given in the lower Lit. E [cm−1IP ] 41449.451(2) graph. 052518-7 Residuals [cm-1] Energy Position [cm-1] P. NAUBEREIT et al. PHYSICAL REVIEW A 93, 052518 (2016) 0.03 0.02 10000 0.01 1000 0.00 -0.01 1000 36 37 38 39 40 41 42 43 44 45 46 47 48 4950 0.005 -0.02 100 -0.03 0.000 10 -0.04 10 15 20 25 30 35 1 5 10 15 20 25 30 35 40 45 50 55 60 65 41370 41380 41390 41400 Effective Quantum Number n* Wave Number [cm-1] FIG. 6. Quantum defects of the Rydberg levels of the f series FIG. 7. Enlarged view of the wide range scans in Fig. 4 from measured. The defects were calculated from the measured IP and the n = 36 to n = 50. In this region an underlying peak structure slightly energy positions of the resonances. The red curve shows the error affects the energy positions of the Rydberg peaks. Excitation in the weighted average for the quantum defect of δ(n) = 0.0017(1). upper graph via D1, in the lower via D2. IV. CONCLUSION AND OUTLOOK using excitation schemes via the well-known D1 and D2 lines of sodium, we determined the energetic position of A. Difference frequency generation 41 high-lying Rydberg states with a high precision of about We showed that a laser system based upon the Mainz 0.03 cm−1. We confirmed the already precisely measured first Ti:sapphire lasers including a DFG stage is ready for ionization potential of sodium with a precision of 0.02 cm−1 midresolution atomic spectroscopy on excitation schemes and in order to demonstrate the accuracy of our system and high-lying atomic levels and is particularly well suited for the spectroscopic method of in-source multistep resonance operation in the frame of laser ion sources of on-line RIB ionization spectroscopy, which is becoming more and more facilities. Although the present output power of the DFG popular at on-line laser ion sources. stage of only about 10 mW is already suitable for reasonably A minor, so far unexplained finding is the appearance of strong first-step transitions, it is still possible to increase unexpected humps in the count rate, seen particularly well this value with simple methods as discussed above, e.g., by pronounced in the third step scans in Fig. 4 at approximately involving intracavity frequency doubling for generation of 41440 cm−1, just very shortly below the IP. This hump appears the DFG pump wave. For optimum performance of such a in all recorded scans, but has not been observed in other system suitable mixing crystals must be used to deliver a truly spectroscopic measurement of any of the various elements significant gain in output power and beam profile. In addition studied by Ti:sapphire laser resonance ionization so far. It the bandwidth can be lowered to about 50 MHz by using more appears sometimes more or less pronounced, which might narrow-band linewidth laser radiation for the fundamental and be ascribed to the conditions of the extraction field being the frequency-doubled input lasers for DFG. Additionally, this responsible for additional ionization slightly below the IP. This will result in a higher spectral power density and therefore aspect will be investigated systematically in the near future. increase the output power. ACKNOWLEDGMENTS B. Rydberg spectroscopy of sodium J.M.-S. thanks the German Academic Exchange Service Using the DFG system discussed above, we performed (DAAD) for support in the framework of the IAESTE program. spectroscopic measurements on the odd-parity Rydberg series P.N. gratefully thanks the Carl Zeiss Stiftung for financial nf 2 oF 5/2,7/2 of sodium. By three-step resonance ionization support. [1] Y. B. Band, C. Radzewicz, and J. S. Krasinski, Phys. Rev. A 49, [6] V. 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Vogel, and J. L. Hall, Phys. Rev. Lett. 76, [21] S. E. Moody and M. Lambropoulos, Phys. Rev. A 15, 1497 2866 (1996). (1977). 052518-9 2.4 Collaborative activity 2.4 Collaborative activity Atomic and nuclear properties of exotic radioisotopes at the borderline of stabil- ity are of major interest for several important research fields, e.g. research on nu- cleosynthesis, nuclear structre and nuclear stability of matter and is therewith as- sociated with the initial formation of matter in the early universe. To gain infor- mation on ground state configurations and the composition of rare short-lived nu- clei far off stability, it is crucial to produce such species on-line at radioactive ion beam (RIB) facilities. The majority of RIB facilities worldwide utilizes the RIS tech- nique either as ion source or for laser mass spectrometric measurements itself, see cf. [8, 9]. The Mainz LARISSA group is involved in several collaborations on cor- responding experiments, for example RILIS [25] and CRIS [26] at ISOLDE (CERN, Switzerland), GISELE [27] at SPIRAL2/S3 (GANIL, France), TRILIS [28] at ISAC (TRIUMF, Canada), IGISOL [29] (JYFL, Finland), PALIS [30] at RIBF (RIKEN, Japan) or LISOL [31] (CRC, Belgium), which has been shut down recently and will be re- placed by the GISELE/S3 activities. The latter one will be presented in detail within the following publication Developments towards in-gas-jet laser spectroscopy studies of actinium isotopes at LISOL. 19 2 Resonance ionization spectroscopy 2.5 Publication: Developments towards in-gas-jet laser spectroscopy studies of actinium isotopes at LISOL This article was published in early 2016 and presents preparatory experiments for several important studies at the LISOL separator in Louvain-La-Neuve (Belgium). It comprises information on the experimental setup and first results on the novel tech- nique of resonant ionization spectroscopy in a supersonic gas jet. The combination of RIS with a supersonic gas jet combines highest spectral resolution with highest efficiencies, two mandatory requirements for studies on rare isotopes, that are com- monly regarded as contradictory. This work paved the way for revolutionary studies presented in [32] and [33]. The doctoral candidate co-authors this article for his contributions to the mea- surement campaign. Specifically, he was one together with T. Kron and S. Reader responsible for setting up and operating the laser system for resonance ionization spectroscopy during the measurement campaign at the RIB facility in Louvain-La- Neuve. 20 Nuclear Instruments and Methods in Physics Research B 376 (2016) 382–387 Contents lists available at ScienceDirect Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier .com/locate /n imb Developments towards in-gas-jet laser spectroscopy studies of actinium isotopes at LISOL S. Raeder a,b,c,⇑, B. Bastin d, M. Block b,c,f, P. Creemers a, P. Delahaye d, R. Ferrer a, X. Fléchard e, S. Franchoo g, L. Ghys a,l, L.P. Gaffney a,1, C. Granados a, R. Heinke h, L. Hijazi d, M. Huyse a, T. Kron h, Yu. Kudryavtsev a, M. Laatiaoui b,c, N. Lecesne d, F. Luton d, I.D. Moore i, Y. Martinez a, E. Mogilevskiy a,j, P. Naubereit h, J. Piot d, S. Rothe k, H. Savajols d, S. Sels a, V. Sonnenschein i, E. Traykov d, C. Van Beveren a, P. Van den Bergh a, P. Van Duppen a, K. Wendt h, A. Zadvornaya a aKU Leuven, Instituut voor Kern- en Stralingsfysica, Celestijnenlaan 200D, B-3001 Leuven, Belgium bHelmholtz-Institut Mainz, 55128 Mainz, Germany cGSI Helmholtzzentrum für Schwerionenforschung GmbH, Planckstraße 1, 64291 Darmstadt, Germany dGANIL, CEA/DSM-CNRS/IN2P3, B.P. 55027, 14076 Caen, France e LPC Caen, ENSICAEN, Université de Caen, CNRS/IN2P3, Caen, France f Institut für Kernchemie, Johannes Gutenberg Universität, 55128 Mainz, Germany g Institute de Physique Nucléaire (IPN) d’Orsay, 91406 Orsay, Cedex, France h Institut für Physik, Johannes Gutenberg Universität, 55128 Mainz, Germany iDepartment of Physics, University of Jyväskylä, P.O. Box 35 (YFL), Jyväskylä, FI40014, Finland j Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Leninskie gory, 1, 119992 Moscow, Russia kCERN, CH-1211 Genève, Switzerland l SCK-CEN, Belgian Nuclear Research Center, Boeretang 200, 2400 Mol, Belgium a r t i c l e i n f o a b s t r a c t Article history: To study exotic nuclides at the borders of stability with laser ionization and spectroscopy techniques, Received 6 September 2015 highest efficiencies in combination with a high spectral resolution are required. These usually opposing Received in revised form 20 November 2015 requirements are reconciled by applying the in-gas-laser ionization and spectroscopy (IGLIS) technique in Accepted 10 December 2015 the supersonic gas jet produced by a de Laval nozzle installed at the exit of the stopping gas cell. Carrying Available online 4 February 2016 out laser ionization in the low-temperature and low density supersonic gas jet eliminates pressure broad- ening, which will significantly improve the spectral resolution. This article presents the required modi- Keywords: fications at the Leuven Isotope Separator On-Line (LISOL) facility that are needed for the first on-line Resonance ionization spectroscopy Gas cell studies of in-gas-jet laser spectroscopy. Different geometries for the gas outlet and extraction ion guides Gas jet have been tested for their performance regarding the acceptance of laser ionized species as well as for Actinium their differential pumping capacities. The specifications and performance of the temporarily installed high repetition rate laser system, including a narrow bandwidth injection-locked Ti:sapphire laser, are discussed and first preliminary results on neutron-deficient actinium isotopes are presented indicating the high capability of this novel technique.  2016 Elsevier B.V. All rights reserved. 1. Introduction where highly energetic projectiles impinge on thin solid targets. This technique was previously employed within the high- At the Leuven Isotope Separator On-Line (LISOL) facility the in- pressure environment of the stopping gas cell to study ground gas laser ionization and spectroscopy (IGLIS) technique is used to and isomeric state properties of neutron-deficient 57—59Cu [1] and study short-lived nuclides produced in heavy-ion fusion reactions 97—101Ag [2] isotopes by resonant laser excitation and ionization. To improve the spectral resolution (typically 4–10 GHz due to ⇑ Corresponding author at: GSI Helmholtzzentrum für Schwerionenforschung pressure broadening) and the selectivity of this technique, off- GmbH, Planckstraße 1, 64291 Darmstadt, Germany. line studies on resonant ionization in the low-temperature and E-mail address: s.raeder@gsi.de (S. Raeder). low-density regime of the supersonic free jet formed at the gas cell 1 Present address: School of Engineering, University of the West of Scotland, Paisley exit hole were successfully performed using stable Cu isotopes [3]. PA1 2BE, United Kingdom. http://dx.doi.org/10.1016/j.nimb.2015.12.014 0168-583X/ 2016 Elsevier B.V. All rights reserved. S. Raeder et al. / Nuclear Instruments and Methods in Physics Research B 376 (2016) 382–387 383 Here a resolution down to 450 MHz for the 327 nm optical ground FC: Faraday cup state transition in Cu could be demonstrated using pulsed amplifi- IC SEM: Secondary electron multiplier SPIG: Sextupole ion guide cation of a cw diode laser an improvement of one order of magni- dual chamber gas cell IC: Ion collector tude compared to in-gas-cell spectroscopy. To explore the full SEM capability of the in-gas-jet ionization technique a new off-line lab- orifice oratory is being commissioned at KU Leuven [4]. Finally, this tech- filament nique will be used at the low energy front-end of the S3 in-flight Ionization volume FC separator at GANIL [5] to study rare isotopes at the extremes of target bell shape existence. This technique was now applied for the first time on- SPIG Lasers longitudinal line at the LISOL facility to study neutron-deficient actinium iso- CYCLONE 110 215 cyclotron beam straight accel. dipole topes including the semi-magic Ac nucleus. nozzle d optics magnet The first attempt to perform resonant laser ionization in the gas Lasers transverse Extraction cell on actinium isotopes was reported in [5]. This research was Gas cell chamber chamber Beamline motivated by studying the evolution of the N ¼ 126 shell closure towards heavier Z systems by investigating electrical quadrupole Fig. 1. Schematic diagram of the experimental set-up at LISOL illustrating different and magnetic dipole moments, as well as by the investigation on options investigated in preparation for the on-line studies. In-gas-cell (longitudinal) shape effects occurring in this region of the nuclear chart through and in-gas-jet (transverse) ionization for the off-line (filament) and on-line measurements are indicated. The two options under investigation for the gas cell the observation of changes in charge radii. Complementary, exit, a free jet from an orifice and a shaped gas jet from a de Laval nozzle are neutron-rich actinium isotopes, located in a region with expected depicted in the blue circled areas. The green circled area indicates the tested SPIG octupole deformation [6], are accessible at ISOL facilities [7]. configurations for an enhanced ion capture efficiency. (For interpretation of the Atomic levels of actinium for resonant excitation are known from references to color in this figure legend, the reader is referred to the web version of the measurements of arc and hollow-cathode spectra using long- this article.) lived 227Ac [8,9], while investigations on the first ionization poten- tial (IP) as well as on auto-ionizing (AI) resonance above the first IP collector. The neutralized atoms are ionized using resonant laser have been carried out in a previous work within the collaboration ionization in a two-step ionization scheme [14] allowing for an [10]. Additional spectroscopic off-line studies focused on the iden- improved selectivity in radioactive ion beam (RIB) production tification of levels featuring a wide hyperfine structure (HFS) split- [15] as well as for laser spectroscopic investigations. ting with a suitable transition at 438 nm [11]. Using this transition, For the laser ionization in the gas jet a high repetition rate laser the HFS splitting of the neutron deficient actinium isotopes system as well as an extended laser-atom ionization region are 212—215Ac was partly resolved in in-gas-cell laser spectroscopic required for highest efficiencies [3,4]. As the argon jet velocity is studies at LISOL [12]. Nevertheless, the spectral resolution of about about 550 m/s an ionization region of 55 mm length is needed 6 GHz did not allow the extraction of all required parameters with assuming a laser repetition rate of 10 kHz. The schematic setup an adequate precision. Especially the information on the spectro- for the experiment in combination with investigated options is scopic quadrupole moment and therefore the nuclear deformation depicted in Fig. 1. Using the dual-chamber gas cell [16] the gas cell could not be determined. chamber and the position of the primary cyclotron beam restrict The in-gas-jet ionization technique overcomes these restric- the distance between gas cell exit and the beginning of the sex- tions and the production rates of actinium isotopes made an on- tupole ion guide (SPIG) to about 22 mm. The SPIG is required for line application at LISOL feasible. For the investigations on rare iso- ion extraction and differential pumping between the gas cell topes the total efficiency of the method is crucial as the production chamber and the extraction chamber. To test the capturing effi- rates are small. During the previous off-line tests on copper ciency of the photo ions in the gas jet by the SPIG two different reported in [3] the total efficiency could not be estimated due to options for the gas outlet at the gas cell exit were tested. The stan- the unknown evaporation rate of stable Cu from the filament. dard configuration is a simple orifice with a hole diameter of However, the duty cycle when using the existing 200 Hz 0.8 mm, leading to a free expanding gas jet. For in-gas-cell ioniza- Excimer-pumped dye laser system for ionization in the fast propa- tion the SPIG is located within a few mm after the orifice to ensure gating gas jet as well as the large divergence angle of the freely an efficient capturing while a DC gradient is applied to dissociate expanding jet limit the potential efficiency. The existing front- molecules formed during the ion transport in the gas cell. The sec- end at LISOL furthermore constrained the implementation of this ond option is a de Laval nozzle with a throat diameter of 1 mm as technique limiting the final performance. indicated in the blue encircled areas in Fig. 1. The gas dynamic cal- This article summarizes the preparation work for the on-line culations indicate that the argon boundary layer inside the nozzle application of the in-gas-jet ionization involving the gas-jet forma- limits the effective Mach number of M  5:5 and a jet diameter of tion, the ion extraction as well as the performance and character- about 3.5 mm. istics of the high-repetition laser system that was installed at the Fig. 2 shows the measured extraction efficiency for ions created LISOL facility to improve the total performance. The first prelimi- inside the gas cell and transported to the first Faraday cup (FC) nary results will also be summarized here. after acceleration as a function of the distance for both options. For these measurements stable Cu isotopes were evaporated from a filament and were ionized inside the gas cell using an Excimer- 2. Production, ion extraction and interaction region pumped (LPX 240i, Lambda Physik) dye laser system (Scanmate, Lambda Physik) at a repetition rate of 50 Hz [17]. The extraction At LISOL the radioactive isotopes are produced using the fusion efficiency for the free jet clearly drops when the distance d (see reaction from 20;22Ne projectiles (0.16 plA, 145 MeV) onto a 1 lm Fig. 1) between orifice and SPIG is increased, limiting the effective thick 197Au target, with production cross sections as low as distance to values below 10 mm. The extraction efficiency for the 1.6 mbarn for 212Ac [5,13]. The target is located inside a gas cell de Laval nozzle shows a better performance and is constant at filled with about 500 mbar of argon for thermalization and neutral- the level of about 80% within the range of the possible nozzle – ization of the reaction products. The neutralized products are SPIG distance, indicating that the gas jet is more forward directed guided by the gas flow towards the exit nozzle of the gas cell. compared to the free-expanding gas jet from the orifice. In contrast Remaining charged particles are removed by an electrostatic ion to the low divergence shape of the jet from the de Laval nozzle an 384 S. Raeder et al. / Nuclear Instruments and Methods in Physics Research B 376 (2016) 382–387 3 mm, a rod diameter of 1.5 mm and a length of 126 mm [18] by a bell-shaped SPIG. Here the inner diameter towards the gas cell is increased to about 6 mm which then smoothly reduces to the normal inner diameter of 3 mm. This geometry is sketched in the green encircled area in Fig. 1 and was intended to improve the ion extraction. Due to the larger acceptance area of the SPIG this geometry was found to be less sensitive to the alignment of the gas cell with the de Laval nozzle, while the total performance in extraction efficiency and in-gas-cell to in-gas-jet ionization was comparable to a well aligned normal SPIG geometry. A clear disad- vantage precluding the bell-shaped SPIG from further considera- tions was the reduced differential pumping capability. The pressure exceeded 1  104 mbar in the extraction chamber at the Fig. 2. Dependence of the transmission of in-gas-cell ionized stable Cu isotopes on nominal gas cell pressure of 350 mbar for all distances, indicating the distance d for the two options of gas jets. The FC signal is referenced to the ion current captured directly onto the SPIG rods. The points are connected to guide the that the gas jet is well captured. eyes. Another study during these tests was to investigate the forma- tion of molecular ions after the laser ionization of copper atoms in the gas jet environment. It is known that the atomic ions created in the gas cell can form molecular ions with impurity molecules pre- sent in the buffer gas. Most of these ions can be cracked by collision-induced-dissociation (CID) when applying a sufficient voltage between the gas cell and the SPIG [19]. Fig. 3 shows the mass spectra obtained for in-gas-cell ionization and in-gas-jet ion- ization and the additional effect of a dissociation voltage for the latter. It is clearly visible that argon clusters and nitrogen com- pounds are still formed in the gas jet while in the gas cell signifi- cant amounts of water compounds can be additionally found. This observation can be explained by the difference in gas pressure and thus in the change in total ion-neutral collision rate which is orders of magnitude reduced in the gas jet environment and will therefore decrease the number of molecular loss channels in com- parison to the gas cell. This observation is in agreement with ear- lier measurements on the time structure of different molecular Fig. 3. Mass spectra obtained from in-gas-cell (black curve) and in-gas-jet ionization (red curve) show the molecular formation of Cu ions. The blue curve compounds indicating that the formation of argon clusters has represents the case of in-gas-jet ionization in combination with a SPIG offset the fastest time scale [20]. Nevertheless the formation of clusters voltage of 300 V for CID. The red and blue curves are shifted for visibility. (For between argon (and possibly nitrogen) and the photo-ions in the interpretation of the references to color in this figure legend, the reader is referred low density region of the gas jet is somewhat unexpected and to the web version of this article.) has to be considered in further investigations on the in-gas-jet ion- ization technique. Applying a voltage of 300 V for the in-jet ioniza- opening angle of about 30was estimated for the free jet in off-line tion, most but not all clusters can be dissociated and the experiments [3]. This conclusion is supported by the observations abundance of copper ions bound in molecular sidebands is reduced from the differential pumping capability of the SPIG. In contrast to from 46% to 19%. In addition, the time structures of the ions from the orifice, a larger gas cell-SPIG distance is required for the de the different ionization regions have been measured and conform Laval nozzle in order to maintain a working pressure below with the results from previous investigations considering in-SPIG 1  104 ionization [21].mbar in the extraction chamber when operating the gas cell at the nominal pressure of 350 mbar, which is required to effi- ciently stop the actinium fusion products. Additionally, the ratio of 3. Laser system in-gas-cell to in-gas-jet ionized Cu was determined to estimate the efficiency performance. For a laser beam diameter of 5 mm irradi- Efficient and selective laser ionization of short-lived species in ating the gas jet in a transversal arrangement for both laser beams the gas jet requires a high-repetition rate laser system, which in and laser repetition rate of 100 Hz a ratio of about 1000 is roughly addition must include one laser with a narrow bandwidth match- expected from duty cycle and geometrical considerations. For the ing the reduced Doppler broadening in the gas jet to gain in spec- orifice a ratio of 500 was observed while for the de Laval nozzle tral resolution. A high-repetition rate Titanium:sapphire (Ti:Sa) a ratio below 200 was repeatedly measured. This exceeds the laser system with a broad bandwidth has been installed temporar- expectations by a factor of 4–5 which could be explained by a ily at LISOL in 2011 to test the performance for in-gas-cell laser reduction of collisional de-excitation compared to the high pres- ionization [22]. It was shown that at repetition rates of 10 kHz a sure environment inside the gas cell. In summary the available similar in-gas-cell ionization efficiency was obtained compared de Laval nozzle, although not fully characterized and optimized, to the Excimer pumped dye laser system operated at 200 Hz but showed a superior performance in comparison to the orifice. The at higher pulse energies. length of the diverging part of the de Laval nozzle is 10.5 mm. This For the in-gas-jet laser ionization and spectroscopy of the limits the effective length of the gas jet determined by the nozzle – neutron-deficient actinium isotopes a similar solid-state laser sys- SPIG distance to 12 mm in on-line operation, which is still larger tem was commissioned at LISOL in a joint collaboration between than the laser beam diameter of 5 mm realized at the experiment. GANIL [23], Mainz University [24] and University of Jyväskylä A second test was performed in which we investigated the [25]. The layout of the laser system is schematically shown in replacement of the straight SPIG with an inner diameter of Fig. 4. It consists of a Nd:YAG pump laser (DM-YAG 60-532, Pho- S. Raeder et al. / Nuclear Instruments and Methods in Physics Research B 376 (2016) 382–387 385 2x broad bandwidth Ti:Sa laser transport to linear beta barium borate (BBO) crystal used for SHG is installed experiment inside the cavity. This allows an increase of the conversion effi- SHG SHG ciency as the cavity enhancement of the fundamental radiation telescopes can be efficiently used. The second harmonic is coupled out using mode matching a dichroic mirror inside the cavity. This configuration generated SHG up to 1.8 W of laser light in the 430 nm region using a pump power of about 14.5 W. The third laser, pumped by the same Nd:YAG laser, was a pulsed narrow bandwidth Ti:Sa laser with a bow-tie geometry. This laser Nd:YAG pump beam injection-locked was developed and operated off-line [27,11] but so far never used SHG: second harmonic generation pulsed Ti:Sa laser for on-line operation. The cavity was injection-locked by narrow ECDL: external cavity diode laser FPI: Fabry Perot interferometer Dither lock bandwidth light from a cw external cavity diode laser (ECDL) seed- ing the pulsed laser radiation at 877 nm. A home-built ECDL in Lit- Fringe Offset Stabilization stabilized trow geometry was used with an anti-reflection (AR) coated laser He:Ne 2 diode (EYP-RWE-0860, Eagleyard Photonics) for an extended scan- ning range. Frequency stabilization and control was achieved using 60 dB scanning FPI a fringe-off-set lock technique. A homemade scanning Fabry–Perot cw master laser Interferometer (sFPI) with a free spectral range (FSR) of 40 dB 40 dB 299.979 MHz – referenced to a stabilized He:Ne laser (SL03, SIOS ECDL iScan Messtechnik) for relative and stable frequency control – was used in combination with a commercial quadrature interferometer sta- Fig. 4. Schematic setup of the laser system consisting of two broad bandwidth bilization (iScan, TEM Messtechnik) for short-term stabilization. pulsed Ti:Sa lasers and a narrow bandwidth injection-locked Ti:Sa laser. The narrow This provided laser light with a bandwidth of a few MHz which bandwidth laser is seeded by a cw ECDL as master laser. Additionally, the components for the active stabilization with a He:Ne are shown, see text for details. could be scanned over several GHz [28]. To protect the diode laser from back reflections, as well as from the pulsed laser radiation, a total of 140 dB of optical isolation was installed using three optical tonics Industries Inc.) with an average output power up to 60 W at Faraday isolators. Due to the stabilization, monitoring and protec- 532 mn operated at a repetition rate of 10 kHz. This light was used tion measures a laser power of only 6 mW, from the original to simultaneously pump three tunable Ti:Sa lasers of which two 26 mW measured directly after the diode laser, was available for have a Z-shaped resonator featuring a broad bandwidth of about seeding the pulsed Ti:Sa laser. Nevertheless, this power was suffi- 4 GHz and a pulse length of 35 ns as used for resonant ionization cient to stabilize the cavity and seed the wavelength of the pulsed laser ion sources [24]. They can be tuned in the range of 690– Ti:Sa laser. After mode matching, the cw laser light was injected 990 nm (fundamental output) with extension to shorter wave- into the bow-tie Ti:Sa ring resonator which itself was stabilized length by generation of higher harmonics. One of the lasers is used onto the diode laser radiation using a dither lock (LaseLock, TEM to excite the 424 nm transition into an AI resonance as shown in Messtechnik). As the fundamental wavelength of 877 nm is rela- the ionization scheme in Fig. 5. The other laser is operated at tively far off the Ti:Sa gain maximum around 800 nm, an appropri- 438 nm for a broadband first step excitation but also can be oper- ate mirror set with high reflectivity (HR) coating from 850 to ated at 434 nm to increase the ionization efficiency by exciting a 950 nm was used to suppress self stimulated lasing at wavelengths second AI transition. As the required laser light is in the blue wave- possessing a higher gain. In total about 1.5 W of fundamental laser length region around 430 nm, the broad bandwidth lasers have radiation with a spectral band width of about 10–20 MHz, which is been operated with intra-cavity second harmonic generation a typical value for this laser [11], was available resulting in about (SHG) [26]. In this method all resonator mirrors are highly reflec- 80–100 mW at 438 nm after external single pass SHG. tive for the fundamental laser radiation and the optical non- Ac 45 815.4 cm-1 46 347.0 cm -1 IP 43 394.5 cm-1 424.69 nm ionization & 434.51 nm 22 801.1 cm-1 J1= 5/2° 438.58 nm 0 cm-1 6d 7s2 2D3/2 I ≥ 5/2 Fig. 5. Left: ionization scheme with a schematic HFS splitting for a nuclear spin I > 5=2 as it is the case for 214;215Ac. The transition wavelengths are given in vacuum and the AI states have been identified in a previous work [7]. Right: Observed signal for two partially resolved HFS transitions as indicated in the ionization scheme in 214Ac. 386 S. Raeder et al. / Nuclear Instruments and Methods in Physics Research B 376 (2016) 382–387 Fig. 6. Left: transmission of the laser radiation through the optical elements up to the ionization volume in dependence of the distance to the front of the focusing telescopes. Right: spot size, shape and orientation of the laser beams at the location of the ionization volume. The ellipticity of the first step is caused by an imperfectly compensated astigmatism from the external SHG, while the orientation is due to the beam transport. The laser radiation of all three lasers was then shaped using the ionization region. The total efficiency of this first on-line use of telescopes and guided towards the gas cell located 17 m away from the in-gas-jet technique seems to follow the trend observed in the the laser table. For transport 2” high-reflecting dielectric mirrors off-line tests. An evaluation of the efficiency as well as of the spec- and anti-reflection coated quartz prisms were used. After the troscopic results is ongoing. experiment and all corresponding optimization procedures, the transmission of the laser radiation was measured as shown in the left part of Fig. 6. The ionization lasers show a rather steep drop Acknowledgments in transmitted power at a distance of around 15 m where a pair of prisms was mounted on a translation stage allowing a remote hor- This work was supported by FWO-Vlaanderen (Belgium), by izontal tuning of all three lasers simultaneously. Unfortunately, it GOA/2010/010 (BOF KU Leuven), by the IAP Belgian Science Policy was set up close to the limit of total internal reflection and opti- (BriX network P7/12) and by a Grant from the European Research mization during the experiment resulted in additional losses, Council (ERC-2011-AdG-291561-HELIOS). S.S acknowledges a Ph. which have to be taken into account when estimating the effi- D. Grant of the belgian Agency for Innovation by Science and Tech- ciency and saturation powers. The first step shows significant nology (IWT). L.P.G. acknowledges FWO-Vlaanderen (Belgium) as losses during laser beam transport which is attributed to the an FWO Pegasus Marie Curie Fellow. imperfectly compensated astigmatism introduced by the external SHG. As the laser power was more than sufficient for spectroscopy and even additional attenuation of a factor of ten was required, this References loss in laser power was not further optimized but the quantifica- [1] T. Cocolios, A. Andreyev, B. Bastin, N. Bree, J. Büscher, J. Elseviers, J. Gentens, M. tion is important for reporting the applied laser pulse energies. Huyse, Yu. Kudryavtsev, D. Pauwels, et al., Phys. Rev. Lett. 103 (2009) 102501, The right part of Fig. 6 shows the final laser beam size of the first http://dx.doi.org/10.1103/PhysRevLett. 103.102501. and second step at the gas jet position, thus defining the ionization [2] R. Ferrer, N. Bree, T. Cocolios, I. Darby, H. De Witte, W. Dexters, J. Diriken, J. volume. 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To become familiar with the in many regards very specific element 91Pa, the next section introduces the element including the history of its discovery, some of its properties, the efforts needed for its extraction and chemical separation and the first attempts for laser spectroscopic investigation in general as well as to measure the first ionization potential. The publication Excited atomic energy levels in protactinium by resonance ionization spectroscopy at the end of this chapter provides a detailed view on the spectroscopic work on protactinium in the frame of the present work, including the extraction of many hundreds of high-lying energy levels and tedious approaches for their identification and separation. 3.1 The element 91 The knowledge today on protactinium is comprised in a chapter within a book on the actinide and transactinide elements, from which the following quotation sums it up in a nutshell: “Protactinium, element 91, is one of the most rare of the naturally occurring elements and may well be the most difficult of all to extract from natural sources. Protactinium is, formally, the third element of the actinide series and the first having a 5 f electron. The superconducting properties of protactinium metal provide clear evidence that Pa is a true actinide element [...]. Its chemical behavior in aqueous solution, however, would seem to place it in group VB of the Mendeleev’s table, below Ta and Nb. The predominant oxidation state is 5+. Pa(V) forms no simple cations in 27 3 Spectroscopy of the protactinium atom aqueous solution and, like Ta, it exhibits an extraordinarily high tendency to undergo hydrolysis, to form polymers, and to be adsorbed on almost any available surface. These tendencies undoubtedly account for the many reports of erratic and irreproducible behavior of protactinium as well as for its frustrating habit of disappearing in the hands of inexperienced or unwary investigators.” From: The Chemistry of the Actinides and Transactinide Elements [34]. These paragraphs illustrate in a perfect manner, which difficulties a researcher will face when trying to spectroscopically investigate the protactinium atom: • small sample size due to the difficult production of protactinium and, of course, due to its radioactivity • occurrence of extremely complex atomic spectra due to several open shells and up to five valence electrons • refractory element behavior leading to a high melting point and therewith pre- venting easy vaporization as necessary in RIS • formation of complex polymers hindering the reduction to atomic protactinium • absorption on almost every surface preventing formation of a dense atomic vapor as prerequisite for a stable atomic/ion beam Due to these complications and its general chemical behavior [34, 35], there is not much knowledge on the atomic or nuclear properties of protactinium. Thus, the ab- sence of a value for the IP of protactinium is not surprising anymore. Even though a theoretically extrapolated value was predicted in the 1970’s [36], a direct measure- ment of the IP is still pending. Just a very rough value could be extracted from systematics along the series of actinides [22, 37]. Already the discovery process of Pa was comparatively tedious. In 1872 Mendeleev proposed a chemical element 91 named “eka-tantalum” with an atomic mass around 235 and chemical properties similar to Nb and Ta [34]. From the radioactive displace- ment rules proposed 1913 [38, 39] as well as from the missing mother nuclide of Ac in its decay chain [40] it was clear, that an actinide element between thorium and uranium had yet to be discovered. In 1913 Göhring & Fajans discovered a short- lived radio-isotope in the U–Ra decay chain (it was 234mPa) which they associated correctly with the element 91, and named it “brevium” (Latin brevis: short, brief) because of its short lifetime [34]. In 1918 Meitner & Hahn discovered the natural and thus more relevant isotope 231Pa and determined a lifetime between 1200 and 180000 years, which is not assigned as very “brevis” anymore. Thus, they suggested 28 3.1 The element 91 to name it “protactinium” from greek protos (first) and “actinium” meaning “prede- cessor of actinium” in the U–Ac decay chain [41, 42]. The name “protoactinium” is wrongly claimed in tertiary literature [43]: It was givens later by linguists, because it was “better greek” [34, 42], but was officially restored in 1949. As already mentioned, the production of atomic protactinium is rather sophisti- cated, expensive and work consuming. Extracting it from natural pitchblende for example is not economically viable at all. Luckily, the United Kingdom Atomic En- ergy Authority (UKAEA) decided in the late 50’s to extract the remaining uranium from a huge amount of around 60 t of radioactive waste. This “sludge” contained aside from around 12 t of high grade uranium also some ppm pf Pa, what is more than ten times the percentage contained in natural pitchblende. Since the costs of US$ 500 000 were covered by the U recovery it was economically worthwhile to ex- tract both, U and Pa. This effort leads to 127 g of 231Pa – showing an remarkable activity of 229 GBq despite the long lifetime – with a high purity of 99.9 % [34]. The material was generously distributed between research facilities around the world. The Pa samples for the spectroscopy described in the following, were kindly pro- vided by the Mainz institute of nuclear chemistry. It is plausible and highly likely, that the Mainz institute of nuclear chemistry, with its founder and first head Fritz Straßmann, also received a fraction of the Pa from UKAEA. In spite of the expected difficulties, extensive spectroscopic studies were con- ducted. Objectives were a precise determination of the IP in first place and secondly a gain of information on the overall atomic structure of this extraordinary complex atomic system. The following publication Excited atomic energy levels in protactinium by resonance ionization spectroscopy comprises the experimental data of several mea- surement campaigns, while the analysis on the determination of the IP and the in- vestigation on the atomic structure regarding quantum chaos are treated separately, namely in Sec. 4 and Sec. 5 of this thesis. 29 3 Spectroscopy of the protactinium atom 3.2 Publication: Excited atomic energy levels in protactinium by res- onance ionization spectroscopy The article was published in August 2018 in Physical Review A as an ex-ante pre- sentation of the experimental method, the apparatus used and not at least to make available the vast amount of atomic physics data as basis for a final conclusive anal- ysis of the atomic structure presented separately. The data was gathered within three independent measurement campaigns, which all were held in the LARISSA laboratory in Mainz during the years 2016 and 2017 and also earlier data from T. Gottwald was inspected. In the article a special focus lies on the correct assignment of energy levels, since individual resonances in a spectrum could very well result from different initial states. The final energy values extracted for individual lev- els are listed in the supplemental material of the article and can be found under https://journals.aps.org/pra/supplemental/10.1103/ PhysRevA.98.022505. The extensive work load in the experimental spectroscopy was carried out by the thesis author and first author of the publication with strong assistance from Dominik Studer. The entire analysis of the spectra as well as the preparation of the manuscript was done by the doctoral candidate. 30 PHYSICAL REVIEW A 98, 022505 (2018) Excited atomic energy levels in protactinium by resonance ionization spectroscopy Pascal Naubereit,* Tina Gottwald, Dominik Studer, and Klaus Wendt Institute of Physics, University of Mainz, 55128 Mainz, Germany (Received 18 May 2018; published 10 August 2018) We present high-resolution data of the single-excitation spectrum of protactinium, reaching slightly beyond the first-ionization threshold. Within this work, more than 1500 energy levels are recorded in different excitation energy ranges below 50 000 cm−1. Our experimental results show that the tabulated data in the literature severely underestimate the density of states particularly regarding the highly excited spectral range. DOI: 10.1103/PhysRevA.98.022505 I. INTRODUCTION electrons. The electronic structure of Pa with an even-parity ground-state configuration 7s25f 26d involves in relativistic Laser resonance ionization spectroscopy is an utterly ver- notation 7s , 5f , 6d , and additionally 7p satile technique for investigations on the atomic shell [1] 1/2 5/2,7/2 3/2,5/2 1/2,3/2orbitals. These N = 32 possible single-electron states for each as well as for gaining knowledge of the characteristics of individual of the n = 5 active valence electrons in Pa lead to the nuclear structure and properties of rare species [2,3]. It ∼(N )n/n! ≈ 2.8 × 105 possible electron configurations [7,8]. has been used throughout the Periodic Table of elements for Given this rough estimate, we note that the 156 even and 494 high-precision studies up to fermium; therefore, it is denotable = odd energy levels available in the literature [9] are by far notthat protactinium (Z 91) is the only actinide below element complete but strongly suggest unobserved levels particularly number 100, the actinide element fermium, for which no spec- situated at increasing excitation energies. Furthermore, no level troscopic measurements were performed until today [4]. To be at all has been tabulated so far for excitation energies above more precise, it is, besides a few chalcogenic, halogenic, and 34 500 cm−1 (38 500 cm−1) for even (odd) parity [9]. refractory elements, which are known to be rather difficult to In the measurements presented here, we studied far more investigate with laser spectroscopic methods, the only element than 2000 resonances in the bound spectrum of the Pa atom, at all [4]. The only ever laser-ionized protactinium beam, covering selected energy ranges and states of different total without advanced spectroscopy, was demonstrated in [5]. In angular momentum and parity. Making use of multistep laser order to prepare comprehensive studies on the atomic system resonance ionization spectroscopy [10,11] with wide-range as well as for nuclear structure research via resonance ion- tunable Ti:sapphire lasers, extensive scans on resonances and ization spectroscopy it is necessary to identify and investigate level positions in different energy ranges of the spectrum were efficient optical excitation schemes to provide highly resolved recorded. spectra. Here we present high-resolution spectroscopic data of the excitation spectrum of protactinium. Protactinium is also one II. EXPERIMENTAL SETUP of the very few remaining elements in the Periodic Table for which the fundamental atomic quantity of the first-ionization For a fully resonant two-step (three-step) excitation, two potential has not yet been precisely measured [6]; a value of (three) lasers of the Mainz titanium:sapphire laser sys- E = 49 000(110) cm−1 [6.075(14) eV] has been inferred tem, involving second-harmonic generation (SHG) and third-IP from systematic comparison to the other actinides and iso- harmonic generation (THG), were used. The standard Z- electronic lanthanides [5]. This shortcoming is not only due to shaped cavity lasers according to [12] have a typical linewidth the unpleasant radiological and chemical properties of protac- of 4(1) GHz (fundamental) to 6(1) GHz (SHG) and 7(2) GHz tinium, which make sample preparation and the production of a (THG), respectively, and provide an average output power of stable atomic beam, as required in such experiments, very com- tunable laser light between 2 and 4 W. The power output for plicated. Also the complexity of its atomic spectrum has so far frequency-doubled and -tripled lasers is somewhat lower, de- prevented a conclusive analysis towards a determination of the pending on wavelength and adjustment, lying somewhat below ionization potential via Rydberg convergences: Rydberg series 500 mW. A Ti:sapphire laser with a grating-assisted resonator could not be identified due to the exceptionally high level den- served as the scanning laser for spectroscopy. This laser type, sity of other excited states below the ionization potential. The a modified development of [13], allows continuous scanning level density in the protactinium spectrum even exceeds that of without mode-hop covering almost the complete Ti:sapphire most other actinide and the isoelectronic lanthanide elements. gain range from 650 to 1000 nm. Due to wavelength selection Both groups exhibit several open shells and numerous “active” via a refraction grating, a linewidth below 2(1) GHz, with a slightly reduced output power of approximately 2 W, can be achieved. All lasers have a typical pulse length between 30 and 90 ns. A commercially available frequency-doubled *naubereit@uni-mainz.de diode-pumped solid-state Nd:YAG laser (Photonics Industries 2469-9926/2018/98(2)/022505(6) 022505-1 ©2018 American Physical Society NAUBEREIT, GOTTWALD, STUDER, AND WENDT PHYSICAL REVIEW A 98, 022505 (2018) FES (i) (ii) (iii) (iv) (v) (vi) SES (vii) (viii) 49000 cm-1 IP 5f 7s H 36035 cm-1 J = ??? 35943 cm-1 J = 13/2o 35926 cm-1 J = ??? 25724 cm-1 J = 11/2o 25448 cm-1 J = 9/2o 24778 cm-1 J = 13/2o 23807 cm-1 J = 11/2o 11445 cm-1 J = 9/2o 0 cm-1 5f 6d7s K FIG. 2. Compilation of excitation schemes for resonance ioniza- tion spectroscopy used for protactinium within this work. For further FIG. 1. Sketch of the experimental setup including the Ti:sapphire details see the text. laser system on the left-hand side and the mass spectrometer system on the right-hand side. For details, see the text. several excitation schemes in order to investigate different total angular momenta of both parities. Figure 2 gives an DM 100-532) at 10-kHz repetition rate delivers the necessary overview on all investigated excitation schemes. Herein, the pump power of 10–15 W for each Ti:sapphire laser. For more arrows in out-fading colors depict the scanned step and the detailed information on the laser system used, see, for example, color itself gives a hint to the laser wavelength range in each [14] and references therein. step. Levels where the configuration or a J value is indicated Figure 1 gives an overview of the apparatus; the laser system can also be found in the literature [9]. All schemes where the is depicted schematically on the left-hand side. The right-hand ionization step was scanned are labeled in roman numerals, side shows a sketch of the low-energy mass spectrometer while FES (SES) describes the search for a first (second) ex- system, the Mainz Atomic Beam Unit. The lasers are guided citation step using a nonresonant ionization step. All schemes inside, anticollinearly overlapped to the atomic beam into the start from the even-parity atomic ground-state configuration 2 2 4 −1 atomizer furnace. Samples of typically 1014 atoms of 231Pa 7s 5f 6d K11/2 located at zero energy (0 cm ). To ensure dissolved in nitric acid are crystallized on a zirconium foil that the excitation does not start from a thermally populated acting as reduction agent, which is afterward placed in the fine-structure component slightly above the lowest ground−1 atomizer furnace. The resistively heated graphite furnace with state, found at 825.42, 1618.3, or 1978.2 cm (odd parity), an inner diameter of 2.2 mm and a length of 50 mm is internally we choose transitions into energy levels, listed in the literature fully lined by tantalum to prevent formation of PaC on the walls with an unambiguously assigned value for the total angular [15]. After vaporization at temperatures above the melting momentum [9]. Based on the selection rule J = ±1, 0, point of protactinium at 1568 ◦C and reduction of PaO, PaO2, fulfilled for every optically allowed dipole transition, and the and especially the high-stable Pa2O5 molecules, the Pa atoms known J value of the initial state, a range of just three [five are ionized via stepwise excitation by the laser radiation, drift for the three-step schemes (vii) and (viii)] neighboring J towards the exit hole of the furnace, and are extracted and values can be inferred for our measured resonances. Beyond accelerated with low electric fields on the order of 10 V/mm. that, some total angular momenta of resonances could be After passing ion optics for beam shaping, the ions are assigned by comparing several scans of excitation schemes separated from evaporated neutral species by bending the ion with different J values (see the Supplemental Material [16]). beam in a 90◦ electrostatic quadrupole deflector. Subsequent The uncertainties of all energy levels measured in this work mass separation with a radio-frequency quadrupole mass filter are calculated in a similar way. The statistical error accounts separates the 231Pa ions from other ionic species before the for fitting errors in the first place. A smaller contribution is ions are detected by a channel electron multiplier in single-ion ascribed to the data acquisition, which slightly shifts the spectra counting mode. depending on scan direction and speed. The latter can be easily Several effects, e.g., Doppler broadening and broadening corrected, but result in a small contribution to the overall due to the laser linewidth, increase the width of measured statistical uncertainty. The systematic uncertainty is produced resonances compared to their natural linewidth, but an achieved by the wavelength measurement using a High Finesse WS6- resolution in the range below 20 GHz (FWHM) for most 600 wavelength meter. Fast statistical scattering is averaged out transitions is still well suited for resolving individual states due to a rather low scanning speed and the fitting procedure. with high precision. However, we have to mention that some Long-term drifts and absolute measurement uncertainties are transitions into autoionizing resonances may exceed this value covered within the specified 1σ absolute accuracy. Thus it must by far. be applied for every measured wavelength contributing to the total excitation energy of each level. In the following sections level energies are compared to the III. WIDE-RANGE HIGH-RESOLUTION LASER levels available in the literature [9]. According to [17–19], i.e., RESONANCE IONIZATION SPECTROSCOPY the primary resources of [9], the resonance energies might be The overall scanning range of the Ti:sapphire laser system slightly shifted due to variations in excitation probability of is spanning only 1500 cm−1. Thus, we have probed the the underlying hyperfine-structure components. As a conse- spectrum of Pa I in several ranges, from 23 600 cm−1 up quence, we added half of the hyperfine-structure width given to the first-ionization potential and slightly above. We used in [9] as additional uncertainty of the literature values for 022505-2 EXCITED ATOMIC ENERGY LEVELS IN PROTACTINIUM … PHYSICAL REVIEW A 98, 022505 (2018) TABLE I. Compilation of energy ranges, parity, total angular considered as first excited states, as long as the total angular momenta, and number of atomic transitions in protactinium as momentum is suitable. identified in the different excitation schemes: FES, SES, and (i)–(viii). (b) All transitions that appear in more than one scan are Energy levels available in literature [9] are given for comparison. considered as leading into a first excited state; again this is valid − for both excitations considered as starting from the ground stateScheme Energy range (cm 1) Parity Range of J Transitions or from the thermally populated state. FES 23600, . . . , 26000 odd 9 , . . . , 13 88 (c) If for one resonance neither the transition starting from2 2 SES 35800, . . . , 36400 even 9 , . . . , 13 32 the ground state nor from the thermally populated state fulfills2 2 7 11 rule (a) or (b), this resonance line is considered as leading into(i) 48600, . . . , 49100 even 2 , . . . , 2 215 11 15 a second excited state. In contrast, the remaining transitions in(ii) 48900, . . . , 49500 even 2 , . . . , 2 67 the FES scheme are considered to lead into first excited states. (iii) 48100, . . . , 49500 even 9 , . . . , 132 2 424 (d) For any transition matching one of the rules above, the (iv) 49200, . . . , 50000 even 7 , . . . , 112 2 159 corresponding “other” transition, i.e., starting either from the (v) 48600, . . . , 49500 even 112 , . . . , 15 2 119 ground state or from the thermally populated state, is discarded. (vi) 48100, . . . , 49400 even 9 , . . . , 13 432 This ensures that for each resonance in a spectrum only one2 2 (vii) 47900, . . . , 49100 odd 7 , . . . , 15 472 energy level remains in the end.2 2 (viii) 48500 49700 odd 7 15 316 (e) The few remaining resonances that fulfill more than one, . . . , 2 , . . . , 2 3 17 of the rules above and resonances where transitions starting[9] 0, . . . , 34500 even 2 , . . . , 2 156 both from the ground state and from the thermally populated [9] 2000, . . . , 38500 odd 3 , . . . , 172 2 494 state match a rule are treated as special cases, which are analyzed separately. After this separation procedure the total excitation energy for every transition was calculated. In total 239 first excited comparison. In cases where no width was indicated, we took states were found and are displayed in Table III in the the half mean of all given widths as the uncertainty. Supplemental Material [16]. Obeying rule (a), 72 of them can Table I gives a compilation of the number of individual directly be found in [9] and thus a total angular momentum can resonances which were determined in the ten different two- be assigned. For two of them, numbers 145 and 159 in Table III and three-step excitation schemes given in Fig. 2. We have in the Supplemental Material [16], the range of possible total to mention that in this compilation several schemes contain a angular momenta given in [9] could be limited to the value 11 . number of identical energy levels. For a complete list of levels 2Rule (b) can be applied to 167 levels, while for 67 of them the observed in this work we refer to the Supplemental Material energies appear only in scans starting from the ground state or [16]. For comparison, the numbers of hitherto tabulated levels from the thermally populated state. These 67 level energies are in the literature [9] are included in Table I, covering a range somewhat unreliable as we cannot state explicitly from which about half of the excitation energies up to the first-ionization state the excitation starts; they are thus labeled with a question potential. For most of these levels, parity and total angular mark in the Supplemental Material [16]. For these presently momentum have been assigned. detected energy levels, there is a chance between 47% (at 1500 ◦C) and 35% (at 2000 ◦C) for excitation from the ground state, depending on the temperature in the source region, and A. Search for first and second excitation steps between about 20% (at 1500 ◦C) and 18% (at 2000 ◦C) from the There is no simple procedure to experimentally distinguish state at 825.42 cm−1. For the other 100 levels it was possible between excitations starting from the ground state or, alter- to restrict the range for the total angular momentum. One natively, from a thermally populated state located slightly transition with an energy of 27 812.68(19)stat(1) −1sys cm is above. Already at a moderate atomizer furnace temperature supposed to start from the thermally populated odd state at of 1500 ◦C, the state at 825.42 cm−1 has a population ratio of 1978.22 cm−1 leading to an even-parity level, while all other 20%. Another problem appears since for the two-step schemes transitions lead to odd-parity levels. Level number 119 in the (i)–(vi) the radiation of the first and the scanning laser could table for odd levels (Table III of the Supplemental Material swap regarding the consecutive steps of the ladder of excitation. [16]), would match a level at 25 891.71(1) −1stat(1)sys cm from Correspondingly, the scanning laser might excite the atoms [9] if we consider an excitation from the state at 825.42 cm−1, into a first excited state and the actual first-step laser serves but should not be accessible due to the assigned total angular for nonresonant ionization. The following procedure is used momentum of 152 [9]. Despite our confidence in the correctness to circumvent these difficulties during data analysis: At first, of the energy level assignments, there is still a non-negligible for every detected resonance, two energies are calculated, one possibility that a second-step transition coincidentally matches expecting the transition to start from the ground state and the energy of a first excited state. the other expecting the transition to start from the thermally Figure 3 shows the scan of the search for a first excitation populated state at 825.42 cm−1. To the energies obtained (note step in the lower trace. The poor statistics especially at lower that every transition is now doubly existent), a set of five rules excitation energies in this scan are caused by low laser powers is applied. and an almost depleted sample. (a) The available literature data found in [9] are correct. That In a further step of the excitation scheme development means matching energies in any scan for either value, starting we scanned with an infrared laser to reach an energy range from the ground state or from the thermally excited state, are around 36 000 cm−1 by a two-step excitation. Therefore, we 022505-3 NAUBEREIT, GOTTWALD, STUDER, AND WENDT PHYSICAL REVIEW A 98, 022505 (2018) 1.0 0.8 0.6 (vi) from 23807 cm-1 0.4 0.2 0.0 48100 48200 48300 48400 48500 48600 48700 48800 48900 49000 49100 49200 49300 1.0 0.8 0.6 SES from 23807 cm-1 0.4 0.2 0.0 35800 35850 35900 35950 36000 36050 36100 36150 36200 36250 36300 36350 1.0 0.8 0.6 FES 0.4 0.2 0.0 23200 23400 23600 23800 24000 24200 24400 24600 24800 25000 25200 25400 25600 25800 26000 Energy (cm-1) FIG. 3. Different wide-range scans with normalized count rates. The energy scale shows the calculated total excitation energy. Every resonance is indicated by a black bar above the spectra, while arrows indicate resonances used in the various excitation schemes of Fig. 2 or for the frequency scans of Fig. 4. The bottom graph shows the search for a first excitation step, the middle the search for a second excitation step, and the top the scan of scheme (vi). In this scan, the orange line represents the value for the expected ionization potential of EIP = 49 000(110) cm−1 with its uncertainty range visualized as the light orange area. used the odd-parity energy level at 23 807 cm−1 with a total was used. All of the first excited states can additionally be angular momentum of J = 112 as the first excited state. Of all found in [9] and have a clearly assigned value for the total the levels with sufficient excitation probability and assigned angular momentum, which once again results in a range of J value detected in the FES scheme, this state has one of three consecutive J values for the measured resonances within the lowest energies [9]. A nonresonant one-color two-photon the second step. The upper trace of Fig. 3 shows the very ionization above an estimated ionization potential of EIP = dense scan of scheme (vi) as an example for the highly excited 49 000(110) cm−1 is not possible. A range of J = 9 , . . . , 13 spectra.2 2 for the second excited states is accessible. Additional to the Schemes (i) and (ii) use laser light in the ultraviolet range laser for excitation into the first excited state and the scanning for either the scanning or the first-step excitation, respectively. laser, a third laser, also operating in the infrared range, was In both cases the sum frequency generation technique was utilized to ionize from the second excited states nonresonantly. applied [20]. For the scanning uv laser two individual lasers To ensure that the measured resonances represent second are needed. One intracavity frequency-doubled laser [21] at a excited states, and do not accidentally coincide with alternative fixed wavelength is sum frequency mixed with the scanning first excited states, the scan was repeated with three different laser running in fundamental wavelength operation. The uv nonresonant ionization steps. Consequently, only levels ap- laser with fixed wavelength, as needed for scheme (ii), is pearing in at least two scans were accepted as second excited a frequency-tripled laser, where only one laser is frequency states. Those 28, out of 32 in total, energy levels are listed in doubled and then again mixed with its own fundamental the compilation for even-parity energy levels in Table I of the wavelength output. Unfortunately, for scheme (i) we cannot Supplemental Material [16]. The middle trace of Fig. 3 shows check if the resonances found are first-step transitions, because a typical scan of the SES scheme. we did not perform first-step searches in that energy range due to the experimental challenge for the further excitation steps nor did the literature cover this energy range. In addition, B. Two-step excitation schemes the number of detected energy levels is very low compared In addition to the previously described schemes, we inves- to other schemes of the same parity and energy range, as tigated highly excited energy levels of protactinium via the six depicted in Table I. Due to this fact, a presumably low transition different two-step excitation schemes (i)–(vi) in energy ranges strength for the first excitation step may cause many resonances around 49 000 cm−1 situated below to slightly beyond the of scheme (ii) to remain undetected. Nevertheless, none of expectation for the first-ionization potential. The first excited the detected transitions of scheme (ii) seems to belong to states for schemes (i) and (ii) are taken from the literature first excited states comparing the fundamental energies of the [9], while for the others a state found via the FES scheme scanning laser to the literature data [9]. 022505-4 Ion Signal (arb. units) EXCITED ATOMIC ENERGY LEVELS IN PROTACTINIUM … PHYSICAL REVIEW A 98, 022505 (2018) −1 ν = 49219.29(6) cm-1 Transition 2 at 23 807 cm was used as the first excitation step. The secondcenter 1 excited state for both schemes was initially detected via scans 24.8(9) GHz using the SES scheme, as shown in the center trace of Fig. 3. @ 30 mW For these states with energies of 35 926 and 36 035 cm−1, respectively, a range for the total angular momentum of J = 9 2 , . . . , 13 2 is suitable. Consequently, this results in a range of J = 7 , . . . , 150 2 2 , which is permitted for energy levels detected νcenter = 23806.86(4) cm -1 Transition 1 by the scanned third excitation step. Note that the third excited 1 states may also lead into one of three consecutive J values. 15.6(4) GHz Since we do not know the J value of the intermediate second @ 3 mW excited state, the range of these three values is comprised Psat = 3.0(4) mW within the interval of five consecutive J values given. First and second excited states are chosen in such a way that 0 direct nonresonant ionization from the first excited state with -2 -1 0 1 2 0 20 40 60 80 photons of the third laser is very unlikely to occur without a ν-ν (cm-1) Laser Power (mW) necessary intermediate resonant laser step. Nonetheless, othercenter combinations could be possible, but are not expected to be FIG. 4. Frequency scans and saturation curves for the two-step strong. To verify this statement, we performed an exclusion excitation scheme (vi). Here the second transition yields for an scan, where the intermediate second-step laser was blocked. autoionizing state at 49 219 cm−1, hence the saturation shows linear The remaining resonances in this scan were compared with behavior. Both linewidths and the saturation power of the first schemes (vii) and (viii), here with a 3σ systematic uncertainty, transition are noted in the graph. See the text for further details. and matching levels were directly sorted out from the scans for schemes (vii) and (viii). Similar to the two-step scans, also a comparison of possible transitions from the ground state with Applying the rules described in Sec. III A, the entirety of the literature levels [9] was performed during which further second excited even-parity energy levels with their own range matching transitions were again sorted out. After these proce- of J values found via the two-step schemes (i)–(vi) was finally dures, we are confident that the energies of the levels given for compared after calculating their total excitation energy. They schemes (vii) and (viii) in the table for odd-parity energy levels are given in Table II in the Supplemental Material [16] for even- in the Supplemental Material [16] are correct. The odd-parity parity energy levels. For levels matching each other within levels in this list were also compared with their total energy their uncertainties, the weighted averages are given as the final and if two levels matching each other within their uncertainties level energy. Wherever possible, the values of the total angular were found, the weighted averages were calculated. momenta were limited due to a procedure of exclusion. Figure 4 shows frequency scans and saturation curves of one exemplary pair of transitions measured along scheme (vi), IV. CONCLUSION AND OUTLOOK where the second step populates an autoionizing state at an excitation energy of 49 219 cm−1. On the left-hand side of this An extensive search for resonance ionization schemes was resonance, the tailing from a neighboring resonance is visible. performed in protactinium. We identified far more than 2000 In order to optimize the ionization efficiency, the saturation be- resonances and found about 1500 so far undocumented energy havior was investigated for every transition [14]. The resulting levels. Most of these levels are located around the expected saturation curve of the first transition shows a clear saturation value for the first-ionization potential at 49 000(110) cm−1 and behavior followed by a somewhat unexpected linear slope. cover both parities and several different total angular momenta. Due to the high level density in protactinium, a second-step The achieved resolution is limited by the experimental transition might lead nearly resonantly to an energy level linewidth of typically below 20 GHz for levels below the first- located around 2 × 23 807 cm−1 followed by nonresonant ionization potential with significantly broader autoionizing ionization. The second transition exhibits a linearly increasing resonances above. Some of the detected levels indicate signs count rate with laser power due to a nonsaturated autoionizing of broad hyperfine splittings which need to be investigated resonance. The resonance scans on the left show nonsaturated more thoroughly. In this work every resolved peak was fitted peak shapes, namely, a Gaussian profile for transition 1 and separately, because it was not possible to define whether it is an a rather symmetric Fano profile for transition 2, respectively. individual level or a hyperfine component. Since we have not Note that the linewidth for the first transition with a width of found clear Rydberg series in any of the spectra, we have inves- 15.6 GHz is much broader than most of the other transitions tigated the level structure more deeply in order to nevertheless measured below the expected first-ionization potential. This extract a reasonably precise value for the first-ionization fact is ascribed to the relatively broad hyperfine structure potential applying different analytical approaches. In parallel, involved: A width of 11.5 GHz is given in [9]. we evaluated this extraordinary complex atomic structure concerning indications of intrinsic quantum chaos through analyzing the measures of spectral fluctuations (cf. [22,23]) C. Three-step excitation schemes and compared our findings to recently published simulated data Schemes (vii) and (viii) are three-step excitation schemes, [24] and theoretical predictions as given by the random matrix where the transition from the ground state into the J = 112 state theory. 022505-5 Ion Signal (arb. units) NAUBEREIT, GOTTWALD, STUDER, AND WENDT PHYSICAL REVIEW A 98, 022505 (2018) ACKNOWLEDGMENTS fully acknowledges the Carl-Zeiss-Stiftung and D.S. the EU through ENSAR2 RESIST (Grant No. 654002) for financial The authors want to thank N. Trautmann and P. Thörle- support. Pospiech from the Mainz Institute of Nuclear Chemistry for preparation of the protactinium samples. P.N. grate- [1] M. Laatiaoui et al., Nature (London) 538, 495 (2016). [14] P. Naubereit, J. Marín-Sáez, F. Schneider, A. Hakimi, M. [2] K. M. Lynch et al., Phys. Rev. C 97, 024309 (2018). Franzmann, T. Kron, S. Richter, and K. Wendt, Phys. Rev. A [3] C. Granados et al., Phys. Rev. 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Levels with an uncertain 2015). energy position are marked with “?”. For more information on [7] V. V. Flambaum, A. A. Gribakina, G. F. Gribakin, and M. G. the determination of energy positions, see full text of the article. Kozlov, Phys. Rev. A 50, 267 (1994). [17] A. Giacchetti, J. Opt. Soc. Am. 56, 653 (1966). [8] V. A. Dzuba and V. V. Flambaum, Phys. Rev. Lett. 104, 213002 [18] E. W. T. Richards, I. Stephen, and H. S. Wise, Spectrochim. Acta (2010). B 23, 635 (1968). [9] J. Blaise and J.-F. Wyart, Energy Levels and Atomic Spectra of [19] J. Blaise, A. Ginibre, and J. F. Wyart, Z. Phys. A 321, 61 (1985). Actinides, available at http://web2.lac.u-psud.fr/lac/Database/ [20] Y. B. Band, C. Radzewicz, and J. S. Krasinski, Phys. Rev. A 49, Contents.html (accessed 29 September 2016) (TIC, Paris, 1992). 517 (1994). [10] R. V. Ambartzumian and V. S. Letokhov, Appl. Opt. 11, 354 [21] V. Sonnenschein, I. D. Moore, I. Pohjalainen, M. Reponen, S. (1972). Rothe, and K. Wendt, JPS Conf. Proc. 6, 030126 (2015). [11] V. S. Letokhov and V. I. Mishin, Opt. Commun. 29, 168 (1979). [22] F. Haake, Quantum Signatures of Chaos, Springer Series in [12] S. Rothe, B. A. Marsh, C. Mattolat, V. N. Fedosseev, and K. Synergetics Vol. 54 (Springer, Berlin, 2013). Wendt, J. Phys.: Conf. Ser. 312, 052020 (2011). [23] T. Guhr, A. Müller-Groeling, and H. A. Weidenmüller, Phys. [13] A. Teigelhöfer, P. Bricault, O. Chachkova, M. Gillner, J. Lassen, Rep. 299, 189 (1998). J. P. Lavoie, R. Li, J. Meißner, W. Neu, and K. D. A. Wendt, [24] A. V. Viatkina, M. G. Kozlov, and V. V. Flambaum, Phys. Rev. Hyperfine Interact. 196, 161 (2010). A 95, 022503 (2017). 022505-6 Chapter4 Ionization potential of protactinium Within the measurement campaigns of resonance ionization spectroscopy on pro- tactinium, a direct extraction of the IP has unfortunately not been achieved due to the complexity and mutual intractability of any regularity in the observed spectrum. Hence, dedicated analytical methods providing the ability to extract the IP from dense atomic spectra were developed and are presented in this chapter. After de- termination of the functionality of the methods on less complex atomic systems, the extensive spectroscopic data pointed out in the previous chapter was analyzed to extract a value for the IP of protactinium. As qualitative measure for the complexity of an atomic system and as estimate for the expectable number of energy levels, the calculation of possible multi-electron states is used [44]. Discussed already in publi- cation III, protactinium involves approximately 2.8× 105 possible electron configura- tions. As system with intermediate complexity the lanthanide element holmium was chosen. It exhibits an electronic ground state configuration of 4 f 11 6s2 and involves in relativistic notation 4 f5/2, 7/2, 6s1/2 as well as 5d3/2, 5/2 orbitals. These N = 26 single electron states for each of the n = 5 lesser bound “active” electrons – the three, which are not paired in the 4 f -shell, together with the two easily accessible 6s electrons – lead to approximately (N)n/n! ≈ 9.9× 104 possible electron configu- rations. Sodium serves as an atomic system with lower complexity. The calculation mentioned here is not necessary for Na with its only one single valence electron in its ground state configuration of 3s1, but it is a model case for IP extraction from Ry- dberg convergences as pointed out in publication I. To motivate and justify the new analytical methods that were developed and are tested here, the conventional spec- troscopic procedures that could not be applied for the IP determination for Pa are introduced briefly. Afterwards the procedures for the work in complex atomic spec- tra are worked out and applied to protactinium. With that, the first ever precisely assigned value for its IP is extracted. 37 4 Ionization potential of protactinium 4.1 Direct measurement techniques for the IP RIS is a most powerful tool also for the direct measurement of ionization potentials. Two very precise techniques have been used throughout the last decades, namely the determination via Rydberg convergences and application of the saddle point model, respectively. Both methods are discussed in full detail for example in [22] and are introduced briefly in this section. 4.1.1 Rydberg convergences and separation of Rydberg levels The ideal case of Rydberg spectroscopy was presentend in the publication Resonance ionization spectroscopy of sodium Rydberg levels using difference frequency generation of high-repetition-rate pulsed Ti:sapphire lasers. There, sodium, as an element with only one valence electron, was excited into Rydberg levels. These weakly-bound states decay rapidly into an ion and an electron under the influence of black-body radia- tion, collisions or external fields. The ion is detected and appears as signal in the laser scan, if the laser wavelength hits a transition into a Rydberg level. With a con- tinuous scan of the laser wavelength, a Rydberg series converging towards the IP can be recorded. According to the well-known Rydberg-Ritz formula, see publication I and [45], a value for IP can be extracted from such Rydberg convergences. Regarding Rydberg spectra of most of the investigated elements, the unambiguous identification of a series of unperturbed Rydberg levels is not as clear as in the case of sodium or even not possible. If there are obvious series, they are often concealed by disturbers or other strong or stronger resonances. In some cases, the spectra are even that dense, that no clear Rydberg series at all can be identified, cf. the spectra of protactinium as presented in the publication Excited atomic energy levels in protactinium by resonance ionization spectroscopy. Hence, methods for separating Rydberg levels from other valence states need to be applied. Two powerful attempts are discussed here, while a detailed discussion on those techniques can be found e.g. in [22]. Delayed field ionization The Delayed field ionization (DFI) takes advantage of the relatively long lifetime of Rydberg states reaching into the µs range. To prevent that an “undefined” ionization takes place to early, the atoms are excited within a well-shielded area without any external fields or heat radiation – in RIS, this is only achievable if the excitation takes place outside the atomizer furnace, cf. the experimental setup described in publications I or III. Once excited into states close to the ionization potential, the population in all other valence states starts to decay rather quickly while the one in Rydberg levels still remains in its high excitation. After a waiting time of up to the lifetime of Rydberg levels, an intense electric field pulse is applied to the excitation region ionizing the remaining highly excited atoms, which are now only 38 4.2 Analytical extraction of the IP from complex atomic spectra atoms in Rydberg states. With this procedure, the resulting spectra ideally contain only Rydberg levels and can thus be analyzed and further evaluated easily. Isolated core excitation The Isolated core excitation (ICE) is a second method for separating Rydberg levels from other states but must also be carried out in regions without fields or heat radiation to prevent ionization. Here, the basic idea is that the Rydberg electron is far away from the remaining atom, so that no shielding occurs and the remaining atom resembles a singly ionized ion. In addition to the lasers exciting the Rydberg – and unwanted other – levels, another laser induces a transition from the ionic ground stated into a known excited ionic state. This ionic transition can only be resonant if an “ion” is present, i.e. in case of an excited Rydberg atom. Thus, a Rydberg atom produces a signal on the ionic transition while this is not possible for other excited valence states. Scanning of the Rydberg excitation laser again delivers spectra ideally only containing Rydberg levels that can be easily analyzed and used for extracting a value for the ionization potential. Saddle point model In some specific cases, the IP determination via Rydberg convergences may not be possible at all due to the complexity of a spectrum – like in the case of protactinium – and even with the separation methods discussed so far, accidentally remaining levels could be falsely assigned to the Rydberg series. In these cases, a determination of the ionization potential by RIS can be attempted by applying the saddle point model (SPM). This method was used in Mainz for the first-time determination of the IP of ten actinides [46, 47] and most recently in a refined version for the IP determination of the also exceptional complex promethium atom [5], discussed in detail in the thesis of D. Studer [48]. Within the SPM an electric field with strength√E applied to the excitation region lowers the ionization threshold proportional to E. Thus, an excited energy level below the IP will not ionize and contribute to the detected signal until E is high enough. With a determination of the ionization threshold as function of the external e√lectrical field, the IP can be extracted via extrapolation of the resulting curve along E → 0. Accordingly, a dense spectrum as e.g. in protactinium is an advantage regarding the precision of determination of the IP for the SPM. 4.2 Analytical extraction of the IP from complex atomic spectra During the different measurement campaigns, for which the results are presented in the publication Excited atomic energy levels in protactinium by resonance ionization spectroscopy, all of the methods discussed above in Sec. 4.1 were applied in order 39 4 Ionization potential of protactinium to directly measure the ionization potential of protactinium. Unfortunately, none of them worked out. Primarily due to the chemical properties of protactinium as discussed in Sec. 3.1, the formation of a stable atomic beam effusion from a surface failed. Thus, RIS was only possible with sufficient efficiency inside an atomizer furnace where the atom vapor density was high enough but not outside in a field- free region, as would be needed for all those measurement and spectroscopic line separation techniques. Nevertheless, as a promising alternative, the numerous spectra of protactinium, obtained via RIS, cf. publication III, were approached by specific statistical and an- alytical methods for an extraction of the ionization potential as presented in the following sections. 4.2.1 Level density collapse A first approach exploits the level density collapse (LDC) above the ionization poten- tial. Regarding high-energy excitation spectra of protactinium – examples are given in publication III, Fig. 3, top panel or in publication IV, Fig. 1, top panel – a signifi- cant reduction, drop or collapse of the level density is expected and finely apparent above the ionization potential. One exemplary spectrum of protactinium is shown in Fig. 4.1. It visualizes the spectroscopic data obtained via excitation scheme (viii) as named in publication III. Note, that the peak heights are arbitrarily scaled to gain visibility especially of the smallest resonances. The gray line represents the value for the expected ionization potential of EPaIP = 49000(110) cm −1 [37] with its uncertainty range featured as light gray shaded area. Additionally, the peak positions, displayed as black bars above the spectrum, emphasize a LDC being located just slightly above the expected ionization potential well within its error range. 1.0 0.8 0.6 0.4 0.2 0.0 48500 48600 48700 48800 48900 49000 49100 49200 49300 49400 49500 49600 49700 49800 Energy (cm-1) Figure 4.1: Exemplary high-energy excitation spectrum of protactinium via excita- tion scheme (viii) from publication III. The gray line represents the value for the expected ionization potential of EPaIP = 49000(110) cm −1 [37] with its un- certainty range visualized as light gray shaded area. The peak positions are indicated by a black bar above the spectrum. This trend of a decreasing level density just above the IP is not unique for protac- 40 Ion Signal (arb. units) 4.2 Analytical extraction of the IP from complex atomic spectra tinium but can be traced back and verified also in other complex atomic spectra. In the following analysis, three cases of different atomic complexity will be compared in order to prove the universality of the method. Protactinium constitutes the most complex system whereas sodium, as presented in publication I, provides the most simple system and the spectra of holmium, as discussed in detail in [15], represent an atomic system of intermediate complexity. In the case of sodium – the high- energy excitation spectrum is shown in Fig. 4.2 representing data of publication I – the LDC is not surprising since no resonance at all is expected above the IP until very high excitation energies are reached. 1 0.1 0.01 40300 40400 40500 40600 40700 40800 40900 41000 41100 41200 41300 41400 Energy (cm-1) Figure 4.2: High-energy excitation spectrum of sodium. The graph shows a modified version of the upper panel in Fig 4 of publication I. The gray line represents the value for the ionization potential of ENaIP = 41449.451(2) cm −1 [50,51], whereas the uncertainty range as small as 0.002 cm−1 is not visible in this depiction. The peak positions are indicated by a black bar above the spectrum. For a detailed discussion on the graph, please see publication I. Fig. 4.3 shows the RIS spectrum of holmium around the IP as obtained in [15]. Again, the peak heights are scaled to gain visibility. For holmium a LDC just above the IP appears as clear as for protactinium at first sight. To quantitatively analyze the LDC regarding the value for the IP, the level density has to be plotted as function of the excitation energy. The typical approach yields to the so-called cumulative number of levels, a staircase-like function of the excita- tion energy, as often used for describing spectral statistical properties – this topic is specifically addressed in Sec. 5 and in publication IV. The cumulative number of levels Nc (E) is given as ∫ E ( ) N (E) = ∑ δ E′ − E dE′c i (4.1) 0 i with Ei the energy of the i-th resonance. The level density is given by the derivative of the cumulative number of levels: ρ (E) = ddE Nc (E). Fig. 4.4(b) shows the cumulative number of levels for the holmium spectrum. The cumulative number of levels for the least complex atomic system of sodium is 41 Ion Signal (arb. units) 4 Ionization potential of protactinium 1.0 0.8 0.6 0.4 0.2 0.0 48100 48200 48300 48400 48500 48600 48700 48800 48900 49000 Energy (cm-1) Figure 4.3: High-energy excitation spectrum of holmium as obtained in [15]. The gray line represents the value for the ionization potential of EHoIP = 48565.910(3) cm−1 [49], whereas the uncertainty range as small as 0.003 cm−1 is not visible in this depiction. The peak positions are indicated by a black bar above the spectrum. included in Fig. 4.4(a). Since the level density collapses and thus the staircase plot for Nc (E) changes its shape by exhibiting a sharp kink at a certain energy, namely the IP, a stepwise defined function NLDCc (E) serves as fitting function for the cumulative sum of levels. It is defined as { y + aE + bE2, E ≤ E NLDCc (E 0 = LDC) , (4.2) y0 + aELDC + bE2LDC − c ln (ELDC) + c ln (E) , E > ELDC where y0, a, b and c are free parameters. The additional fitting parameter ELDC represents the energy, where the LDC sets in. Since the shape of the staircase plot above the IP is sometimes more logarithmic-like, this type of function was chosen in Eq. 4.2, even though the fits shown in this work look rather linear above ELDC. For the lower part, a parabola fit was sufficient, except for the holmium case, where a hyperbola function was used. In Fig. 4.4(b) a value of EHoLDC = 48553.5(12) cm −1 was extracted by fitting Eq. 4.2 to the data. This value is about 10 cm−1 below the IP of EHoIP = 48565.910(3) cm −1 [49], which gives already a rough estimation of the error produced by the LDC method. Note, that it is expected to deduce a LDC value lying below the IP, which is caused by the detection method: With increasing excitation energy approaching the IP, the level density gets larger and larger, which causes the energy levels to be indistinguishable for RIS. The result is an onset of the LDC already at lower energies. Enhancing the effect, the electrical fields present in the ionization region are regarded as second reason: Electrical fields cause ionization of excited atoms already at excitation energies somewhat lower than the actual IP so that the expected peaks just below the IP “disappear” in a continuously raised background signal. For a deviation of LDC to the IP in the order of ≈ 10 cm−1, voltages of around ≈ 100V would be necessary. As fields of this strength may be 42 Ion Signal (arb. units) 4.2 Analytical extraction of the IP from complex atomic spectra Na Ho -1 40 ELDC = 41421.4(14) cm-1 200 ELDC = 48553.5(12) cm 150 20 100 50 0 0 40400 40800 41200 48200 48400 48600 48800 49000 Energy (cm-1) Energy (cm-1) (a) Sodium. The data is fitted by a (b) Holmium. The dashed black curve piecewise defined function (black curve), represents the piecewise defined funcion which leads to a value for the LDC onset Eq. 4.2 fitted to the cumulative sum of lev- in sodium of ENaLDC = 41421.4(14) cm −1. els depicted as red staircase-like plot. The fit leads to an energy of the LDC onset of EHoLDC = 48553.5(12) cm −1. Figure 4.4: Cumulative sum of levels for sodium and holmium as systems of least and intermediate complexity, respectively. found in the ionization region of the RIS apparatus in use, this seems to be plausible. The described effects are even more important in the case of sodium where no levels above the IP are present to improve the fit. The extracted value of ENaLDC = 41421.4(14) cm−1 is about 28 cm−1 below the actual IP of ENaIP = 41449.451(2) cm −1 [50, 51]. The LDC method is applied to the system of protactinium in Sec. 4.3, but before that, a second analytical method for the extraction of the IP from dense atomic spectra is presented. With the combination of both methods, the extraction of a more precise value for the IP will be enabled. 4.2.2 Rydberg correlation The second analytical method for an extraction of the IP from a dense atomic spec- trum implies deeper knowledge on Rydberg levels, more precisely the correlation of Rydberg levels belonging to specific series. The existence of unperturbed Rydberg series forms the fundamental hypothesis of the Rydberg correlation (RC) method. Albeit Rydberg levels can not be distinguished from other excited valence states in the case of very dense spectra via the typical methods presented in Sec. 4.1.1, there is no reason to question their existence. Even if intrinsic quantum chaos is involved in the system – see Sec. 5 for details on this topic – Rydberg levels with higher principal quantum numbers n lying in the typical detection range for RIS with pulsed lasers in a hot cavity approach of n ≈ 20 – 60 should not be affected by chaotic behavior [52]. The RC method is probably best explained by means of a very simple spectrum with a clearly identifiable Rydberg series – as in the case of sodium. Therefore the 43 Nc(E) Nc(E) 4 Ionization potential of protactinium spectrum given in the upper trace of Fig. 4 in publication I is used. In more complex systems it is necessary to unify the resonances in a spectrum concerning peak heights and widths to induce visibility. Fig. 4.5 shows the sodium Rydberg spectrum with this procedure applied. 1.0 0.8 0.6 0.4 0.2 0.0 40400 40500 40600 40700 40800 40900 41000 41100 41200 41300 41400 Energy (cm-1) Figure 4.5: High-energy Rydberg spectrum of sodium as obtained in publication I, but with unified peak heights and widths. The gray line represents the value for the ionization potential of ENaIP = 41449.451(2) cm −1 [50, 51]. If the energy of the IP EIP is known, the energetic positions of the Rydberg levels En with certain principal quantum number n are given by the Rydberg formula R En (n) = E M IP − 2 , (4.3)n where RM is the mass-reduced Rydberg constant for the requested element1. Using Eq. 4.3, the simple transformation √ → RE M (4.4) EIP − E enables the presentation of the Rydberg spectrum in dependency of n. In this vi- sualization the Rydberg levels become equidistant, in the case that a correct and precise value of the IP is known. For sodium this requirement is fulfilled. Insert- ing the well-known value ENaIP = 41449.451(2) cm −1 [50, 51], the Rydberg spectrum of Fig. 4.5 can be plotted in dependency of the principal quantum number giving perfectly equidistant peaks as shown in Fig. 4.6. As next step the autocorrelation (AC) function of the spectrum in Fig. 4.6 is calcu- lated. For the AC, a point-by-point multiplication of the spectrum I(n) with a shifted 1Usually, n in this formula has to be replaced by n∗, the effective principal quantum number, which also includes the n-dependent quantum defect δ (n). Since the error produced by omitting the quantum defect is much smaller than the uncertainty produced by the RC method itself, the usage of just n is acceptable. 44 Ion Signal (arb. units) 4.2 Analytical extraction of the IP from complex atomic spectra C 1.0 0.8 0.6 0.4 0.2 0.0 10 20 30 40 50 60 Principal Quantum Number n Figure 4.6: Presentation of the unified Rydberg spectrum of Fig. 4.5 in dependency from the principal quantum number n. In this visualization the Rydberg peaks appear equidistant. version of itself I(n + δn) is integrat∫ed over all n, or as mathematical expression∞ A(δn) = I(n)I(n + δn)dn , (4.5) 0 which gives the AC of the spectrum depending on the shift δn. This function has a maximum whenever the peaks of the shifted spectrum overlap with peaks from the original spectrum. Thus, for equidistant peaks with distance n a maximum appears for all shifts that fulfill δn = m · n , m ∈N . (4.6) The AC function is visualized in Fig. 4.7. The AC function is normalized to the global maximum at δn = 0 and because the overlap of the spectra becomes smaller for larger shifts, also the local maximums of the AC function become smaller quite quickly. Therefore, in Fig. 4.7, a magnification of the relevant data range is shown. correlation of B, B, Y Mean(X Average) Stats Reduction of "Corr Y1" 0.1 0.0 0 10 20 30 40 50 Shift δn (n) Figure 4.7: Autocorrelation function for sodium for different shifts δn. A magnifica- tion of the relevant data range is shown. See text for more information. Note, that such a clear structure can only appear even for one single clear Rydberg series, if the peaks are really equidistant for which the IP has to be known rather 45 A(δn) (arb. units) Ion Signal (arb. units) 4 Ionization potential of protactinium precisely. However, this method helps to localize the IP. For that, the AC function has to be calculated for a large number of different IP values. This results in a three dimensional graph, where a multitude of individual AC functions, as given in Fig. 4.7, are stacked next to each other depending in the second dimension on the value of the IP inserted in Eq. 4.4. As visualization, the resulting three dimensional correlation density plot for sodium is depicted in Fig. 4.8 while a two dimensional contour plot as projection is given above. As expected from the proper choice of the well known IP value, the AC functions show strong global maxima in the center corresponding to the IP of ENaIP = 41449.451(2) cm −1 [50, 51]. 1.0 0.5 0.0 41500 5 Sh 10 4i 1ft 4 5n 0δ (arb. un 1i 5ts) 20 41400 EIP (cm -1) Figure 4.8: Three dimensional correlation density for sodium. Many AC functions for different inserted IP values are stacked next to each other forming the peak structure. Above this structure the two dimensional projection is shown. The correct value for the IP lies at ENaIP = 41449.451(2) cm −1, where all the AC functions reach their maximum values. For the final IP evaluation a second projection is carried out by integrating the three dimensional peak structure of Fig∫. 4.8 over all shifts δn. The projection∞ Aint(EIP) = A(δn,EIP)dδn (4.7) 0 results. Of course, in the actual calculation, the infinite integral is replaced by a finite, point-wise sum over all shifts δn. Note, if the bins are all of same size like here, taking the total or the mean of each energy bin results in the same curve after normalization. Similarly, instead of the total of each energy bin, the root-mean- square in some cases may help to lower the background without shifting the peak 46 A(δn) (arb. units) 4.2 Analytical extraction of the IP from complex atomic spectra position. It slightly favors larger values of A(δn). Fig. 4.9 shows the projection Aint(EIP) for sodium for 100 different set values of E −1IP in steps of 1 cm . The error bars represent the standard deviation in each of the energy bins. Although contain- 1.2 ENaRC = 41449.5(3) cm-1 1.0 0.8 0.6 0.4 0.2 41400 41450 41500 E (cm-1IP ) Figure 4.9: Projection of the three dimensional correlation density of Fig. 4.8. The er- ror bars represent the standard deviation in each energy bin. A Lorentzian was fitted to the peak structure leading to a center value of ENaRC = 41449.5(3) cm −1, which agrees to the IP value for sodium as extracted with the RC method. See text for more details. ing only few data points, a clear peak around an set energy of EIP ≈ 41450 cm−1 is visible. A Lorentzian was fitted to this structure, yielding a center energy of ENa −1RC = 41449.5(3) cm . Albeit in perfect agreement with the literature value for the IP of sodium ENaIP = 41449.451(2) cm −1 [50, 51], the IP value obtained with the RC method is obviously less accurate compared to the literature value. The accu- racy would be strongly enhanced, if more Rydberg levels would be available for the AC function and also if more and smaller steps for the IP variation would be cho- sen. Therewith, the number of data points, and the corresponding accuracy, may be increased, but in this case, it was actively limited due to the time-consuming calculation of the three dimensional correlation density as it was shown in Fig. 4.8. Moving on to a more complex atomic spectrum, i.e. the one of holmium as exem- plary candidate with intermediate complexity, the situation is somewhat different. Instead of having one structure of equidistant peaks for a correct value of the ioniza- tion potential, in a more dense spectrum, some peaks may be equidistant for almost every choice of the ionization potential. This results in the appearance of not only one single peak in the projection of the 3D Rydberg correlation, but in numerous or even broad structures with different intensities and widths. To further analyze these structures, a combination of both, the LDC and the RC method, has been employed as demonstrated in the upcoming section. 47 Aint (arb. units) 4 Ionization potential of protactinium 4.2.3 Combination of LDC & RC method Fig. 4.10 visualizes the RC spectrum for holmium. The underlying spectroscopic data again stem from [15]. In Fig. 4.10(a), the RC spectrum is given for the entire 1.0 1.0 0.8 0.6 0.8 Ho 0.4 -1EHoLDC = 48553.5(12) cm-1 ERC = 48561.1(3) cm 0.6 48450 48550 48650 48530 48550 48570 E (cm-1) E (cm-1IP IP ) (a) Full view. The arrow indicates the (b) Magnification of the area around value for the IP as extracted via the LDC EHoLDC. The red line shows a double- method. Gaussian fit, from which the center posi- tion of the relevant second peak is indi- cated as EHoRC = 48561.1(3) cm −1. Figure 4.10: Rydberg correlation spectrum for holmium as obtained with the proce- dure described in Sec. 4.2.2. See text for more details. IP range of relevance. In contrast to the sodium case many peaks, even approxi- mately equal in height and width, are visible. Only with preceding application of the LDC method, one peak can be identified to yield the correct value for the ion- ization potential. In doing so, the energy range around the LDC value of Holmium, EHoLDC = 48553.5(12) cm −1, is plotted in magnification in Fig. 4.10(b). The remain- ing double-peak structure was fitted by a sum of two Gaussians, where the second one is the most pronounced peak and the closest to EHoLDC and is thus assumed to be the peak resulting from the correct ionization potential. The fit yields a value of EHoRC = 48561.1(3) cm −1 for the ionization potential of holmium. The indicated uncertainty only accounts for the fit error and not for total uncertainty as produced by the RC method. However, this value is very close to the literature value for the ionization potential for holmium of EHoIP = 48565.910(3) cm −1 [49]. Because of that, for similar complex atomic structures as in the holmium case, the uncertainty produced by the combination of the LDC and RC method is assumed to be in the order of approximately 10 cm−1. A prove is pending since many more spectra for different atomic systems would be needed to fully characterize the quality of this method. Nevertheless, protactinium with its exceptional atomic complexity maybe seen as perfect test candidate to prove, whether the analytic method is able to extract a value for the ionization potential or not. 48 Aint (arb. units) Aint (arb. units) 4.3 Extracting the IP of protactinium 4.3 Extracting the IP of protactinium With the tools obtained in the preceding sections, the attempt was made to extract the ionization potential of protactinium from several dense RIS spectra as obtained in the measurement campaigns for publication III. As shown before, first the LDC method and thereafter the RC method is applied to each spectrum. With the com- bination of both methods, each spectrum leads to a specific, slightly different value for the expected ionization potential of protactinium. The graphs belonging to the LDC and the RC analysis are comprised for the tested schemes (i) - (iv) in Tab. 4.1 and for schemes (v) - (viii) in Tab. 4.2. The schemes (ii), (iv) and (v) cannot be used for an evaluation with the RC method. Even though for schemes (ii) and (v) a value could be assigned applying the LDC method, in both schemes the energy lev- els below the assigned values are too few to achieve a reliable analysis with the RC approach. The range of excitation energy covered by scheme (iv) is already above the expected value for the ionization potential of 49000(110) cm−1 [37] and thus, no result can be inferred via the methods of investigation. All results inferred via the LDC and the RC method for the ionization potential of protactinium are comprised in Tab. 4.3. For those excitation schemes, where more peaks could be fitted dur- ing the RC analysis, all relevant energies are given in this column. The values that match the inferred LDC values withing their uncertainties are marked in bold font. The uncertainties are given by the fit errors for the LDC values and the half width at half maximum (HWHM) of the fitted peaks for the RC values, respectively. For scheme (vii) two values lie within the uncertainty range of the relevant LDC energy, so that their average of 49027(2) cm−1 is considered for the further analysis. Before calculating a final value for the IP, it must be mentioned that the results obtained in scheme (i) are not considered as appropriate. In this scheme, with the RC method a strongly pronounced peak was obtained. Unfortunately, the position of the peak is not coinciding with the energy inferred via the LDC method. The curve and the fit of the LDC, especially the collapse itself is not as clear as for the remaining schemes. Thus, it is believed, that the RC result is more trustable than the LDC result. However, for the sake of consistence, only the four remaining schemes (iii), (vi), (vii) and (viii) with a coincidence between LDC and RC are used for the IP calculation. For the extraction of a final value for the ionization potential of protactinium the mean of the energies from the four different schemes is calculated. This leads to EPaIP = 49034(10) cm −1 (4.8) as value for IP of protactinium obtained with the analytical method as described in this work. The uncertainty of the IP is given as the standard deviation of the averaged energies. This extracted IP lies well within the uncertainty range of the expected IP value of 49000(110) cm−1 by Wendt et al. [37]. 49 4 Ionization potential of protactinium Table 4.1: Schemes (i) - (iv) Level Density Collaps Rydberg Correlation Rydberg Correlation (zoom) B 200 EILDC = 48983(5) cm-1 1.0 1.0 150 0.8 100 0.8 0.6 50 EI = 48983(5) cm-1 EI -1 0.4 LDC 0.6 LDC = 48983(5) cm 0 48600 48800 49000 48900 49000 49100 48950 49000 49050 Energy (cm-1) E (cm-1) E (cm-1IP IP ) B B 60 EII = 49025(24) cm-1 1.0 1.0LDC 0.8 40 0.6 0.8 0.4 20 0.2 0.6 0.0 EIILDC = 49025(24) cm-1 E II LDC = 49025(24) cm-1 0 0.4 49000 49200 49400 48900 49000 49100 49020 49040 49060 Energy (cm-1) E -1 -1IP (cm ) E B IP (cm ) 350 III 300 ELDC = 49044(3) cm-1 1.0 1.0 250 0.8 0.8 200 150 0.6 0.6 100 50 0.4 EIIILDC = 49044(3) cm-1 0.4 E III -1 LDC = 49044(3) cm 0 48000 48400 48800 49200 48900 49000 49100 49000 49050 49100 Energy (cm-1) EIP (cm-1) EIP (cm-1) 60 40 N.A. N.A. 20 EIVLDC = N.A. 0 49200 49400 49600 49800 Energy (cm-1) 50 Scheme (iv) Scheme (iii) Scheme (ii) Scheme (i) Nc(E) Nc(E) Nc(E) Nc(E) Aint (arb. units) Aint (arb. units) Aint (arb. units) Aint (arb. units) Aint (arb. units) Aint (arb. units) 4.3 Extracting the IP of protactinium Table 4.2: Schemes (v) - (viii) Level Density Collaps Rydberg Correlation Rydberg Correlation (zoom) B B 50 EVLDC = 49183(230) cm-1 1.0 EVLDC = 49183(230) cm-1 1.0 40 0.8 0.8 30 0.6 0.6 20 0.4 0.4 10 V0.2 0.2 E = 49183(230) cm-1LDC 0 48600 48800 49000 49200 49400 48900 49000 49100 49040 49060 49080 Energy (cm-1) E (cm-1IP ) EIP (cm-1) B EVI = 49034(2) cm-1 1.0 1.0 200 LDC 0.8 100 0.8 0.6 EVILDC = 49034(2) cm-1 E VI -1 0 0.6 LDC = 49034(2) cm 48400 48800 49200 48900 49000 49100 49000 49050 Energy (cm-1) EIP (cm-1) E (cm-1) B IP 200 EVIILDC = 49028(5) cm-1 1.0 1.0 150 0.8 100 0.6 0.8 50 0.4 EVII = 49028(5) cm-1 EVII -1 0.2 LDC 0.6 LDC = 49028(5) cm 0 48700 48800 48900 49000 48900 49000 49100 49020 49040 49060 Energy (cm-1) EIP (cm-1) EIP (cm-1) B 300 250 E VIII LDC = 49062(2) cm-1 1.0 1.0 200 0.8 150 0.6 0.8 100 50 0.4 EVIII -1 VIIILDC = 49062(2) cm 0.6 ELDC = 49062(2) cm-10 48400 48800 49200 49600 48900 49000 49100 49020 49040 49060 Energy (cm-1) E (cm-1 -1IP ) EIP (cm ) 51 Scheme (viii) Scheme (vii) Scheme (vi) Scheme (v) Nc(E) Nc(E) Nc(E) Nc(E) Aint (arb. units) Aint (arb. units) Aint (arb. units) Aint (arb. units) Aint (arb. units) Aint (arb. units) Aint (arb. units) Aint (arb. units) 4 Ionization potential of protactinium Table 4.3: Compilation of the results from the LDC and RC method for the eight protactinium excitation schemes which were studied. The uncertainty given for the LDC values is given by the fit error. The RC uncertainty is the half width at half maximum of the fitted peak structure. For the schemes where different RC peaks could be fitted, all relevant energy values are given. There, the bold values mark the energies matching the LDC values within their uncertainties. Scheme LDC (cm−1) RC (cm−1) 48985(5) 48993(2) (i) 48983(5) 49004(7) 49021(7) (ii) 49025(24) N.A. (iii) 49044(3) 49031(16) (iv) N.A. N.A. (v) 49183(230) N.A. 49006(5) 49026(11) (vi) 49034(2) 49040(3) 49050(4) 49023(1) (vii) 49028(5) 49031(3) 49038(3) 49017(1) 49022(1) (viii) 49062(2) 49024(4) 49031(3) 49052(10) 52 Chapter5 Intrinsic quantum chaos within atomic protactinium The exceptional complexity of the protactinium spectra as investigated in the pre- vious sections leads to a well visible “chaoticity” within the spectra. Not only this subjective impression, but more an expected strong disturbance of the individual energy levels leads to the assumption of a truly chaotic electronic level structure for the protactinium atom. This prediction is thoroughly examined on the basis and theory of quantum chaos involving measures known from the theory of random matrices [53, 54]. The first subsection here introduces the basics of quantum chaos and few examples of its manifestation. It was aimed to present the physical concepts without the need of advanced mathematics, which would complicate this overview and can easily be found in detail elsewhere. The ideas and explanations given in the following are partly inspired by the lecture notes of T. Guhr [55] and the book by H.-J. Stöckmann [56] as well as by personal discussions with A. Buchleitner, B. Dietz, T. Guhr and F. Haake. Further relevant literature will be given later in this chapter. The derivation of the contiguities, which required a precise analysis of the protactinium spectra, are described in publication IV featuring the main part of this chapter. 5.1 Quantum chaos 5.1.1 Access through classical mechanics In classical physics, unlike the impressions given in most classical mechanics lec- tures, most systems existing in nature are chaotic. The definition of a chaotic system lies in its non-integrability. Obviously, a system called non-integrable if it is not analytically or numerically integrable; i.e. a conservative Hamiltonian system is in- tegrable as long as it has as many independent constants of motion as it has degrees 53 5 Intrinsic quantum chaos within atomic protactinium of freedom. This means the constants of motion have to Poisson commute pairwise and every conservative system – we focus on conservative systems within this work – with only one degree of freedom is per definitionem integrable since the conserva- tion of energy is the one necessary constant of motion. With increasing complexity, more constants of motion are getting relevant. The three dimensional Kepler prob- lem with its constants of motion (total energy E = H(~q,~p), angular momentum ~L2, its projection Lz, and the Runge-Lenz-Vector M~ = 1m~p×~L− κr~r) is an example that is even over-integrable. While for integrable systems the equation of motion leads to a stable solution under small perturbations the same small perturbations lead to instabilities or di- vergent integrals for the solutions of the (Hamiltonian) equations of motion for non-integrable or chaotic systems. Such instabilities can be easily visualized with a system of three defocusing elements, e.g. disks in the unit plane, like it is depicted in Fig. 5.1. The lines may be for example the trajectories of ideally reflected billiard balls or light rays. In this setup, strong deviating trajectories result due to only slight perturbation of the input parameters. 32123231 3 1 2 3212 Figure 5.1: Instabilities of neighboring trajectories within a system of three disks in the unit plane under small perturbations of the input parameters. The number chains denote the specific path of the trajectories. Adapted from [55]. Hamiltonian systems like the one in the example above are deterministic despite their chaotic properties. That means, as long as the input parameters in phase space and the Hamiltonians are known, every trajectory can be calculated exactly at any time. Additionally, these trajectories are explicit, what is given by the theorem of Picard and Lindelöf [57], and, moreover gives the explanation for the non-crossing of trajectories in phase space. If trajectories could cross, the crossing point could be taken as starting point and the trajectory could never know which path to take from 54 5.1 Quantum chaos there. This would be in complete contradiction to the uniqueness of the trajectories in phase space. Having said this, it becomes obvious, that the herein discussed chaos is a “deterministic chaos”. Chaotic systems are classified according to their emphasis of chaos. Systems that are generally accepted as chaotic systems are called K-systems after the mathemati- cian Andrey Kolmogorov [58]. One specific illustrative property pf K-systems is that the trajectories in phase space are exponentially departing from their neighboring one once they came close to each other. This property holds for the mean so that trajectories can proceed some time next to each other before diverging, but the very most depart quickly after approaching. An example for visualizing this “repulsion”, even though not visualized in phase space, is the chaotic double pendulum. Imag- ine the trajectory the outer end of the second arm describes: most of the time, it is departing from its prior path very quickly and in arbitrary shape, direction and amplitude. Fig. 5.2 shows an exemplary trajectory of the second arm of a double pendulum. Figure 5.2: Trajectory of the outer and of the second arm of a double pendulum [59]. The chaotic path arbitrarily changes direction and amplitude, even though it is deterministic. Specific K-systems showing the strongest chaotic properties are the so-called Bernoulli systems after the famous Swiss mathematician Jacob Bernoulli. Their supplemen- tary property is that the exponentially departing phase space trajectories are pro- ducing perfect randomness, although still deterministic. This can be imagined as follows [55]: At all times, phase space is divided into small cells numbered with integers from 1 to n as shown in Fig. 5.3. As time evolves, a trajectory passes differ- ent cells described by a series of numbers, i.e. {6, 2, 7}. For Bernoulli systems, these numbers are distributed as produced by a perfect random number generator. This equals the production of perfect randomness out of a deterministic propagation of time. 55 5 Intrinsic quantum chaos within atomic protactinium n = 1 2 3 4 … 5 6 7 8 t = 0 t = t1 t = t2 n = 6 n = 2 n = 7 Figure 5.3: Phase space and trajectory of a Benoulli system. The phase space is di- vided into small cells and the trajectory acts as a perfect random number gen- erator by passing the numbered cells. For more information see text. Adapted from [55]. Billards are often regarded as ideal examples of chaotic (Bernoulli) systems for research and teaching purposes and shall also act herein as examples. First of all, a Billiard is a two-dimensional domain with a closed curve acting as border. Inside the area, a particle, i.e. the “Billiard ball”, can move without friction and is ideally reflected on the border. No further requirement on the shape of the closed border is necessary. Rectangle, circle or ellipse are examples for integrable, non-chaotic billiards. They have symmetries and therewith resulting constants of motion for at least every degree of freedom, which are (figures adapted from [55]): • Rectangle Besides the total energy, the absolute values for the two components of the momentum |px| and |py| for both directions ~x and ~y. • Circle Total energy and the total angular momentum relative to the circle center. 56 … … … … 5.1 Quantum chaos • Ellipse Total energy and the product of the two angular momenta relative to the two focal points L~1L~2. There are endless possibilities to create a chaotic system from a former regular bil- liard, by simply reducing the number of symmetries to a number smaller than the degrees of freedom. Three examples shall be given, adapted figures from [55, 60]: • Sinai Billiard The momentum constants of motion of the rectangle billiard can easily be de- stroyed for example by placing a defocussing element like a circle inside the rectangle. In the resulting Sinai billiard, named after the Russian mathemati- cian Yakov Sinai, the trajectories of the particle are defocused resulting in the exponential repulsion within this Bernoulli system. • Bunimovich Stadium Stretching the circular billiard by inserting straight parts between the two half circles destroys the radial symmetry and thus the angular momentum constant of motion. The Bernoullian chaos of this so-called Bunimovich stadium, after 57 5 Intrinsic quantum chaos within atomic protactinium the Russian mathematician Leonid Bunimovich, results not from a defocussing of the trajectories but from over-focussing similar to an unstable laser resonator. • Africa Billiard The Africa billiard – named Africa, because some representations look similar to the continent – is a more generalized chaotic billiard initially investigated by Berry and Robnik in 1986 [61]. It shall underline, that almost every thinkable billiard is chaotic while regular ones are very exceptional. Please note, even though these examples are completely chaotic Bernoulli systems, stable regular or periodic trajectories are possible and allowed. One very simple example is a particle bouncing straight back and forth from the lower to the upper boundary in the Sianai billiard just next to the defocussing element. Actually there are countable infinite ways to realize such trajectories, but there are much more chaotic ones, namely of uncountable infinite number [55]. Additionally, in most natural chaotic systems chaotic and regular structures can coexist. This feature will be investigated later on in this work. 5.1.2 From classical to quantum chaos Billiards are very useful in many aspects. With them even the transition from clas- sical chaos to quantum chaos can be easily illustrated. One has just to replace the particles for example by electromagnetic radiation like microwaves or laser light to observe chaos within quantum mechanical systems. There, the billiard is nothing else than a arbitrarily shaped resonator. To start one step earlier, Bohr formulated in his correspondence principle, that classical mechanics is correct for macroscopic systems in the similar way as quantum mechanics for microscopic systems [62]. 58 5.1 Quantum chaos Consequently, quantum theory has to merge asymptotically into classical theory for large quantum numbers [55]. From this we can infer, that every quantum chaotic system has its classical analog – as for the billiard example. Unfortunately, the simple concept of exponentially departing phase space trajec- tories is not applicable in quantum mechanics because of Heisenberg’s uncertainty principle, which prevents the simultaneous measurement of position and momen- tum for a quantum particle. Nevertheless, the requirement for an integrable system remains: Like in the classical analog, a quantum mechanical system is integrable and therewith non-chaotic and fully solvable, if it has at least one constant of motion for every degree of freedom. In quantum mechanics, the classical constant of motion is replaced by its operator and the Poisson bracket by the quantum mechanical com- mutator. Once the number of degrees of freedom exceeds the number of constants of motion, the system is no longer integrable, thus resulting in chaotic behavior. In- stead of phase space investigations – they are not applicable anymore as mentioned above – a different measure for quantifying chaotic behavior is needed. Therefore the focus lies now on the so-called spectral statistics, the properties of quantum spectra. Here, not a single spectral component, e.g. a resonance or energy level, is investigated, but the statistical interplay between all of them. For atomic systems, interestingly, the expected correlations appear only for spectra or level sets showing identical “good” quantum numbers of total angular momentum and parity [63]. 5.1.3 Random matrix theory In the 1950’s Wigner was concerned with investigations on spectra of neutron scat- tering resonances. Due to the complexity of these spectra he started to describe the spectra according to their statistical properties. He studied the properties of the nowadays well established random matrix theory (RMT)1 and proved that the spectral statistics of nuclear resonances behave identical to the Gaussian orthogonal ensemble (GOE) of the RMT [64, 65]. Here, the eigenvalues of many large GOE ma- trices form the complement to nuclear resonances. For the detailed mathematical background and the generation of the eigenvalues of the different ensembles, the reader is referred to more specialized books on this topic, in particular the book by M.L. Mehta [53]. In this place it shall just be mentioned, that the GOE matri- ces are invariant under orthogonal transformations and the entries of the symmetric matrices are real and random, but follow a Gaussian distribution. An analog to the repulsion of trajectories in classical chaotic systems can also be found in the case of GOE statistics: In RMT the eigenvalues are “repelling” each other. This correlation can be quantified with different measures. But before these measures can be applied, the eigenvalues have to be “unfolded”. This unfolding 1The RMT appeared already in the 1930’s in mathematical statistics due to the work of Hsu, Wishart and others, but took not much attention until Wigner connected it to the field of nuclear physics [53]. 59 5 Intrinsic quantum chaos within atomic protactinium is necessary in order to make the sequence of values independent from the specific eigenvalue density at any position in the sequence. After a proper unfolding process, measures with different correlation lengths can be investigated. The three most common measures shall be briefly mentioned here: • Nearest neighbor spacing distribution This gives the “repulsion” of the nearest neighboring eigenvalues and thus is the measure with the shortest possible correlation length. • Number variance Σ2(L) Long-range measure, which is comparable with an error square depending on the correlation length L. • Spectral rigidity ∆3(L) Most common long-range measure, which is comparable with an χ2 measure- ment depending on the correlation length L. The measures and the unfolding process are addressed in full detail in the publica- tion Intrinsic quantum chaos and spectral fluctuations within the protactinium atom. Until today, no proof is found, that the statistics as inferred from random matrix theory should be expected for every chaotic system. Nonetheless, every investiga- tion, from billiards and quantum billiards over complex nuclei and atoms to ultra- cold gases or quartz oscillations, revealed significant coincidence with GOE statistics. This inspired Bohigas, Giannoni and Schmit in 1984 to their famous conjecture, that, due to its importance, shall be given here in full [66]: “Spectra of timereversal-invariant systems whose classical analogs are K systems show the same fluctuation properties as predicted by GOE (alternative stronger conjectures that cannot be excluded would apply to less chaotic systems, provided that they are ergodic). If the conjecture happens to be true, it will then have been established the universality of the laws of level fluctuations in quantal spectra already found in nuclei and to a lesser extent in atoms. Then, they should also be found in other quantal systems, such as molecules, hadrons, etc.” This statement initiated a huge hype in quantum chaos research [56] and thus, the conjecture is probably the most often verified statement in chaos research. The fol- lowing investigation in the protactinium atom regarding quantum chaos also relies on the correctness of the universality of the GOE statistics. Beside several examples of different manifestations of quantum chaos given in the publication, almost every atomic or nuclear system – except for hydrogen – reveals chaotic properties. For all these systems there is a very limited number of quantum numbers available for a much larger number of degrees of freedoms of nucleons or electrons. Similar as for billiards, even a hydrogen atom can “be made” chaotic if the number of constants 60 5.1 Quantum chaos of motion is reduced via symmetry breaking – e.g. a strong magnetic field destroys the spherical symmetry of the atomic system and the angular momentum no longer is a constant of motion anymore [67]. It must be mentioned, that the transition from regular to chaotic behavior need not to be abrupt like in the classical billiard, which is either integrable or fully chaotic [67]. In a complex many-body system regularity and chaos can coexist and the amount of both can vary arbitrarily. However, the con- jecture, that the chaotic behavior increases with excitation energy, e.g., in an atomic system should be pointed out here, but still lacks any proof. As a last point before the quantum chaos research on the atomic protactinium is presented, an attempt of a qualitative explanation of the level repulsion in atomic systems, which is responsible for all the spectral correlation measures, shall be given: The regular level structuring is disturbed due to a residual interaction. This distur- bance causes the repulsion of levels that can easily be seen in the nearest neighbor distribution of energy levels. The residual interaction may be reducible to the chaotic mixing of several thousands of possible electronic eigenstate configurations (≈ 3 · 105 in Pa). In quantum mechanics, this can be interpreted as overlap of wave functions for all the different possible configurations. A non-vanishing probability results, that the arising “compound” level contains part of the same wave function as the adja- cent one. For close lying energy levels, this leads to a strong repulsion of these states caused by Pauli’s exclusion principle. 61 5 Intrinsic quantum chaos within atomic protactinium 5.2 Publication: Intrinsic quantum chaos and spectral fluctuations within the protactinium atom The last article of this work was published in August 2018 in Physical Review A together with the previous publication III. Besides the IP extraction in Sec. 4, it forms the second analytical research topic derived from the spectroscopic data of the protactinium measurements. The main objective of this article is the proof of full expression of quantum chaos within the protactinium atom. Additionally, it is investigated how variations in the ratio of energy levels with different total angular momentum or missing levels influence the spectral statistics. These findings are compared with ideal level sets from RMT. Because the field of quantum chaos was completely new for the candidate as well as for his research group LARISSA, lots of efforts were invested for gaining enough knowledge to seriously contribute to this research area of sophisticated theoretical depth. The thesis author and first author of the publication carried out the entire analysis and also prepared the manuscript, while the co-authors supported the ex- perimental work and further consultatively contributed to this endeavor. 62 PHYSICAL REVIEW A 98, 022506 (2018) Intrinsic quantum chaos and spectral fluctuations within the protactinium atom Pascal Naubereit,1,* Dominik Studer,1 Anna V. Viatkina,2 Andreas Buchleitner,3 Barbara Dietz,4 Victor V. Flambaum,2,5 and Klaus Wendt1 1Institute of Physics, University of Mainz, 55128 Mainz, Germany 2Helmholtz Institute, University of Mainz, 55128 Mainz, Germany 3Physikalisches Institut, Albert-Ludwigs-Universität Freiburg, 79104 Freiburg, Germany 4School of Physical Science and Technology and Key Laboratory for Magnetism and Magnetic Materials of MOE, Lanzhou University, Lanzhou, Gansu 730000, China 5School of Physics, University of New South Wales, Sydney 2052, New South Wales, Australia (Received 18 May 2018; published 10 August 2018) Recently, spectroscopic investigations of the protactinium atom applying resonant laser ionization spectroscopy revealed high-resolution data of the single-excitation spectrum of protactinium, reaching slightly beyond the first ionization potential [P. Naubereit et al., preceding paper, Phys. Rev. A 98, 022505 (2018)]. The more than 1500 recently detected energy levels contain several complete sequences of levels. In this work we study the spectral fluctuations of these data exhibiting clear signatures of intrinsic quantum chaos within the protactinium atom. In order to obtain an estimate on possibly missing levels, simulations were performed based on large ensembles of random matrices from the Gaussian orthogonal ensemble. Our experimental results show that tabulated data in the literature are far from completeness and atomic structure calculations severely underestimate the density of states in the spectral range of highly excited states. However, the statistical analysis of our data as well as of the data from literature and calculations predict a level statistics close to that of fully developed chaos at energies well below the single-ionization threshold. DOI: 10.1103/PhysRevA.98.022506 I. INTRODUCTION because complete high-resolution spectroscopic data from the ground state up to the first-ionization potential were and still As much as classical systems, like, e.g., the double pendu- are unavailable. Even for lighter elements, very few cases lum, can exhibit a transition from regular to chaotic dynamics, of investigation of intrinsic quantum chaos (IQC), meaning the spectral properties of quantum systems can change drasti- without outer influence of artificially applied electromagnetic cally [1]. This is well established by now both theoretically and fields or scattering processes by specific projectiles, in complex experimentally for few-degrees-of-freedom systems [2–11]. atomic systems are available [28–30]. In contrast, evidence for quantum chaos in systems existing Here we analyze spectral fluctuation properties in the recent on higher-dimensional (classical) phase spaces which also spectroscopic data collected for protactinium and presented in have a much more intricate topology, giving rise, e.g., to Ref. [31] using as statistical measures the nearest-neighbor Arnold diffusion [12–15], and likely are at the origin of spacing distribution (NNSD), the number variance 2, and many-body localization [16,17], is rather scarce, with only a the spectral rigidity  . For further explanations see Sec. II. few systematic results [18–24] so far. This fact is due to the 3According to the Bohigas-Giannoni-Schmit conjecture [32], unfavorable scaling properties of the density of states with the spectral fluctuation properties are universal and coincide increasing excitation energy, accompanied by strong coupling with those of uncorrelated random numbers exhibiting Poisson between the various degrees of freedom. On the theoretical statistics for classically integrable systems and with those as well as on the experimental side, this defines substantial of random matrices from the Gaussian orthogonal ensemble challenges for the resolution of the relevant spectral structures (GOE) if the classical dynamics is fully chaotic. For a proper [7]. Atomic many-body systems constitute ideal, naturally analysis we have to deal with problems stemming primarily occurring test cases for such proliferation of complexity, where from missing or spurious levels. Therefore, large ensembles the generic presumption holds that the chaotic proportion of of random matrices from the GOE were generated and a phase space increases with the excitation energy, while an certain fraction of eigenvalues was randomly deleted in order to unambiguous designation of the demarcation line between simulate missing levels. The resulting statistics are compared regularity and chaos has so far remained elusive [15,20,25–27]. to the experimental data. In truly complex many-body systems, e.g., an actinide atom, Recently [31], about 1500 hitherto unknown resonances the transition point from regularity to significant chaoticity is were detected in the bound spectrum of the Pa atom, covering expected already at low excitation energies. A proof is pending, selected energy ranges, different total angular momentum states, and both parities have been tabulated. An exemplary spectrum, corresponding to scheme (iii) of [31], is shown in *naubereit@uni-mainz.de the top panel of Fig. 1, covering excitation energies from below 2469-9926/2018/98(2)/022506(11) 022506-1 ©2018 American Physical Society PASCAL NAUBEREIT et al. PHYSICAL REVIEW A 98, 022506 (2018) 1.0 Scheme (iii) wide-range scan Level distribution 0.5 0.0 5.96 5.98 6.00 6.02 6.04 6.06 6.08 6.10 6.12 2500 640 2600 J = 9/2 - 13/2 2000 100 Literature Simulation Scheme SES Fit function 1500 Scheme (iii) 620 1000 22000 500 0 E 1 E 2 3 4.45 E 4.50 6.0 E 6.1 0 0 EA 1 EB 2 3 4 EC 5 6ED Energy (eV) FIG. 1. The top graph shows the highly resolved excitation spectrum of protactinium for excitation scheme (iii) of [31], complemented by the corresponding level distribution (black lines above spectrum). The bottom graph shows the staircase functions for the cumulative number of resonances, as inferred from the literature (purple) [33], simulation (light gray) [34], and the present experimental data sets (green and blue) around EC and ED , respectively. The dashed red line represents the self-consistent fit to the spectral density over the entire energy range; see the main text for details. The orange vertical line in all plots indicates the estimated ionization potential, with its uncertainty range identified by the orange-shaded region [35]. For further explanations also on the three insets, see the text. to slightly above the first ionization potential. The data of [31] For the simulation data, a loss of accuracy and completeness serve as a basis for the analysis of intrinsic quantum chaos is anticipated already well below threshold, because only and its spectral characteristics in this highly complex atomic low-energy configurations had so far been included in these system. calculations [34]. In addition, the model neglects correlations between core and valence electrons, which are known to II. LEVEL STATISTICS AND LEVEL DENSITY contribute substantially at higher excitation energies. Conse- FLUCTUATIONS quently, the experimental staircase function is expected to grow faster than the one predicted by the simulation, which exhibits A. Level density in general an approximately quadratic energy dependence (see the gray Let us first compare the density of states of the experimental line in the leftmost inset of the bottom panel of Fig. 1). data, the available literature, and simulation data. The bottom In order to compare the level densities of the experimental panel of Fig. 1, together with the insets which zoom into three data [31] with those of the literature and simulation data and to selected energy ranges, shows the cumulated number of energy localize strong deviations between them we performed a fit of tot levels ∫ Nc (E) to the overall experimental density using the prediction∑ for its averageE N (E) = δ(E′ ′ √c − Ei )dE (1) d ′ a E 0 N (E) = ρ0e (2)i dE c as a staircase function of the excitation energy, with Ei the for the density of states of an interacting many-body spectrum energy of the ith resonance. As explained in Ref. [31], our [29], which neglects Rydberg excitations of either one of the experimental data are limited to a range of three subsequent electrons. Thus, in the ansatz for N totc (E), a phenomenological total angular momenta. Therefore, we took into account in the Rydberg term is included: analysis of the simulated and the literature data only energy √ levels with J = 92 , . . . , 132 . The apparent significant decrease tot ′ hcR of the level density just above the first ionization potential, Nc (E) = Nc(E, ρ0, a) + r − . (3)EIP E as evident in the rightmost inset, can be explained by our experimental method: While resonances in the continuum still HerehcR is the mass-reduced Rydberg energy andρ0, a, r , and may bind all five electrons, many of these resonances are EIP are free fitting parameters, where r could be interpreted hidden by a continuum background due to broad autoionizing as the number of Rydberg series involved and EIP as the or ionic resonances, which becomes manifest in the top panel ionization potential. This procedure is legitimate since no of Fig. 1 as a clearly visible broadening of the resonances chaotic perturbation is expected for high principal quantum above approximately 6.07 eV. This behavior was generally numbers of single-electron Rydberg levels according to [36]. observed for all scans reaching beyond the expectation value However, the data sets of scans in the energy range between of the ionization potential. A second reason is the termination 4.48 and 6.05 eV, denoted by SES (for second excitation step) of various Rydberg series converging towards the ionization and (iii), which are exemplarily considered from [31] and potential. shown in the middle and right insets of the bottom panel 022506-2 Nc(E) Ion Signal (arb. units) INTRINSIC QUANTUM CHAOS AND SPECTRAL … PHYSICAL REVIEW A 98, 022506 (2018) of Fig. 1, provide no information on how many levels lie of normalized, dimensionless energy spacings below and between these associated energy ranges. To extract a consistent offset for the disjoint data sets, we fit s = ξ∂N tot(E)/∂E i i+1 − ξi, (4)c to the slopes of the cumulated density of states, as visualized with the unfolded energy in Fig. 1, within four predefined energy intervals EA, EB , EC , fit and ED , centered around the excitation energies 0.46, 1.52, ξi = Nc (Ei ), (5) 4.48, and 6.05 eV, respectively. These regions most optimally obtained by utilizing the smooth part of the staircase function, incorporate the different available data sets, stemming either i.e., its fit function Nfitc (E) [38]. Second-order polynomials from the literature at low excitation energies (EA and EB , serve as fit functions for the unfolding of all experimental level where only the three J values studied in the experiment sets as well as of the numerically simulated spectral data from are accounted for) or from the experimental data of [31] at [34]. medium or high excitation energies (EC and ED). By this Random matrix theory predicts a Poisson distribution procedure, initially the offsets of the SES and (iii) scans were −s determined and the specific subsets of the three different data PP(s) = e (6) sets depicted in the three insets, now properly leveled, were for the individual si in the limit of regular spectra, i.e., in fitted by Nfitc (E). The resulting fitting parameters are reason- systems of well-preserved quantum numbers, and a Wigner- able: A value of r = 16(6) for the different involved Rydberg Dyson distribution [39] series is realistic and EIP = 6.11(1) eV matches the estimate π 2 of 6.075(14) eV for the ionization potential derived from P (s) = se−(π/4)sWD (7) systematics [35]. 2 The resulting curve for the overall level density is shown in the limit of fully broken integrability, synonymous with as a red dashed line in the bottom panel of Fig. 1 as the complete destruction of good quantum numbers, within well as in the close-ups of the three insets. The com- subspaces defined by one specific value J of the total angular parison to the experimental data reveals three remarkable momentum [37]. Both these idealized limits are interpolated facts. by the Brody distribution (a) The theoretical simulations, which only incorporate ( + )2 η+1η+1 η low-energy configurations, give correct and complete level PB(s, η) = asηe−bs , a = (η + 1)b, b =  , densities up to excitation energies around 1.5 eV and slowly η + 1 start to deviate at higher energies. (8) (b) Literature data appear to be complete up to energies around 2 eV, with a sharp cutoff at this value. with Euler’s Gamma function . The Brody parameter η (c) The experimental data taken around E and E are well controls this interpolation, with the limiting cases η = 0 forC D matched by the fit. Both data sets thus confirm the completeness the Poisson and η = 1 for the Wigner-Dyson distribution [40]. of the levels detected in these energy ranges. Since our experimental spectra are superpositions of three The issue of missing levels will be addressed Sec. III. Here independent J manifolds, we furthermore need to account for “completeness” refers to unexpected deviations of the level the thereby induced convolution of distinct distributions. This density from its energy-dependent average, for example visible can be achieved with the help of the superposition formula in the bottom panel of Fig. 1 for the literature data above initially suggested by Rosenzweig and Porter [28]. Here we 2 eV or for scheme (iii) above 6.07 eV. The slight deviations utilize Eq. (3.69) from [38], giving the spacing distribution observed at the edges of both data sets (middle and right P3B(s, η) for superpositions of three independent subspectra. inset) are ascribed to a signal depletion at the edges of our In doing so, we make two assumptions. laser scan ranges, due to decreasing laser power, as well as (i) The level density in all three J manifolds is equal. to the emerging continuum background discussed above for (ii) The manifolds exhibit the same energy dependence of the energy range beyond the ionization threshold. Therefore, η(E). only the following parts of the individual data sets were A comparison with simulation and literature data validates employed in our subsequent analysis: For the even literature both assumptions as reasonable first-order approximations data a range of 0.00, . . . , 2.00 eV, for the SES scheme a [33,34]. A second interpolating function applicable to super- range of 4.44, . . . , 4.51 eV, and for scheme (iii) a range of imposed spectra is the Abul-Magd distribution [24,41] 5.96, . . . , 6.08 eV were evaluated. ( )π P −(1−f )s−(π/4)Q(f )s 2 AM(s, f ) = 1 − f + Q(f )s e , (9) 2 B. Nearest-neighbor spacing distribution with Q(f ) = 0.7f + 0.3f 2. The parameter f of this descrip- Based on the above confirmation of the essential com- tion similarly approaches the Poisson distribution for f → 0 pleteness of our experimental spectra in the inspected en- and the Wigner-Dyson distribution forf → 1, but the meaning ergy intervals, we can now proceed towards an analysis of is different as for the superposition of three Brody distributions: the spectral structure of protactinium in terms of statistical While the η parameter of the Brody convolution gives a measures commonly used in random matrix theory (RMT). hint at a close-to-GOE behavior of the three independent We extract the nearest-neighbor spacing distribution, as one of subspectra, the f parameter expresses a prediction for the the fundamental quantifiers of regularity-to-chaos transition in number of superimposed fully chaotic GOE spectra. Thus, the complex quantum systems [37], as area-normalized histograms number of independent GOE subspectra is given by n = 1 .f 022506-3 PASCAL NAUBEREIT et al. PHYSICAL REVIEW A 98, 022506 (2018) 1.0 (a) NNSD 1.0 (b) 0.8 PP(s) 0.8 I (s, 1.02) PWD(s) Cumulative NNSD 5 3B0.6 0.6 P3B(s, 1.02) IP(s) IAM(s, 0.35)0.4 PAM(s, 0.35) 0.4 IWD(s) 0.2 0.2 I3B(s, 1.02) IAM(s, 0.35) 0.0 0.0 0 1 2 3 4 5 0 1 2 3 4 5 s s 0 FIG. 2. (a) Nearest-neighbor spacing distribution P (s ) and (b) corresponding cumulative NNSD I (s ) for simulated data with J = 92 , . . . , 132 . The Poisson and Wigner-Dyson distributions are shown as gray dash-dotted and dashed lines, respectively. The fitted Abul-Magd distribution is displayed as a blue line and the convoluted Brody distribution as a dotted red line. 0 1 2 3 4 5 s The special case of three superimposed subspectra yields FIG. 3. Residuals of the fitted curves for the CNNSD. Here I3B − P3B(s, 1) ≡ PAM(s, 13 ). I (s ) is given as a dotted red line and IAM − I (s ) as a blue solid line. To determine the best-fit Brody parameter η or Abul-Magd parameter f , the maximum-likelihood (MLH) method [42] was used. This method is, in contrast to a least-squares-fitting For the experimental data, the NNSDs are conclusive only method, completely independent from the binning procedure, after accounting for missing levels. Because of that, we proceed since it is directly applied to the raw data. The uncertainty of similarly in the following two sections about the statistical and is conservatively approximated by the half-width at measure’s number variance and spectral rigidity and explainη f half maximum (HWHM) of the likelihood distribution. them by means of the simulated data first. The experimental Figure 2(a) shows an exemplary NNSD. Especially for measurements will be discussed afterward in detail in Sec. IV small spacings and/or a small number of spacings, the cumu- involving all statistical measures. lative nearest-neighbor spaci∫ng distribution (CNNSD) C. Number variance 2(L)s I (s) = P (s ′)ds ′ (10) The NNSD gives information on the repulsion of energy 0 levels and shows correlations on the shortest possible scale, is more reliable for comparing the fits with the data. In Fig. 2(b) namely, on the scale of one or two mean spacings. Accordingly, the CNNSD is shown. The NNSD is obtained from the simu- the NNSD is sensitive with respect to the unfolding procedure, lated data with superimposed subspectra with J = 92 , . . . , 132 . missing levels and the fitting procedure used for its description, Besides the two fitted distributions P3B and PAM for the NNSD however, by far not as sensitive as long-range correlations. and I3B and IAM for the CNNSD, the curves for Poisson (PP Thus the NNSD itself does not serve as a significant measure and IP) and Wigner-Dyson (PWD and IWD) distributions are for GOE or Poisson behavior. One additional measure for indicated as gray dash-dotted and dashed lines, respectively. spectral statistics, which gives information on long-range The histograms are depicted for illustration; however, they correlations, is given by the variance 2(L) of the number were not used for the fitting procedure. The NNSD and ν(L) of unfolded levels in an interval with length L [38], CNNSD for the simulated data show a level repulsion close 2 2 2 to that of chaotic GOE spectra. Both repulsion parameters  (L) = 〈ν (L)〉 − 〈ν(L)〉 . (11) η = 1.02(30) and f = 0.35(14) match the expected values Here the angular brackets stand for a spectral average. Like a of η = 1 and f = 1 , respectively, for a convolution of three variance in stochastics, also the quantity23 (L) gives the mean- GOE spectra very well. The rather large uncertainties of these square deviation of the number of energy levels in an interval values are caused by a broad likelihood distribution because the L from their mean L. For uncorrelated Poissonian spectra, convoluted Brody distribution and the Abul-Magd distribution 2P(L) grows linearly with the correlation length L. For highly barely change in their shapes when varying the parameters η correlated GOE spectra, the variance 2GOE(L) grows slower and f around a given value. according to a logarithmic slope for large L. For our case of For an estimation of the quality of the fits for the CNNSD, three superimposed subspectra, the spectral correlation is lower the residuals I3B − I (s) and IAM − I (s) are shown in Fig. 3 as in the GOE case. The resulting curve for23GOE(L) can easily as the difference between the resulting fit curves and the data. be calculated [38]: This form of presentation gives an idea of the shape of the ( ) NNSD and CNNSD by regarding the distribution parameters ∑32 2 L η and f and therefore a hint of the chaotic repulsion of the 3GOE(L) = GOE . (12)3 neighboring energy levels. In addition, it provides information m=1 on the quality of the individual fits of the CNNSD. Thus, in the Figure 4(a) shows the level number variance for the simulated following analysis only these residuals will be discussed when data withJ = 9 , . . . , 132 2 and the theoretical curves for the cases analyzing the level repulsion in the experimental data sets as of Poissonian statistics, GOE statistics, and the statistics of short-range correlation. three convoluted GOE spectra. The data matches the expected 022506-4 P(s) I(s) CNNSD Residual (%) INTRINSIC QUANTUM CHAOS AND SPECTRAL … PHYSICAL REVIEW A 98, 022506 (2018) 3.0 0.6 standard deviations of each bin. Like the number variance, (a) (b) on ss also the spectral rigidity confirms chaotic GOE behavior of Po i 2.0 0.4 3GO E the simulated data for every involved J subset. The data fit the 3GOE expected theory curve up to correlation lengths of L  5 and 1.0 0.2 GOE start to slightly deviate above. GOE As described earlier in this section, the spectral rigidity 0.0 0.0 is calculated from the number variance, which makes both 0 1 2 3 4 5 6 0 2 4 6 8 10 quantities very similar by definition. Therefore, and because L L of its more smooth character due to the “smoothing” integral FIG. 4. (a) Number variance 2(L) and (b) spectral rigidity transform in Eq. (14), only 3(L) will be utilized as a measure 3(L) for the simulated data with J = 9 , . . . , 132 2 . The error bars for long-range correlations of the energy levels in the following display the standard deviation of the binning process. The theory analysis of the experimental data. curves for Poissonian, GOE, and 3GOE statistics are given as gray lines. III. MISSING LEVELS curve for three superimposed GOE spectra (3GOE) perfectly For an accurate analysis of the experimental data sets it is up to a correlation length of  3 5. Only for higher , mandatory to take into account the possibility of missing levelsL . L the data points gradually start to deviate from the theoretical leading to incomplete spectra. The incompleteness of a spec- expectation of 3GOE, but still are significantly off from the trum will influence the spectral statistics leading to either more Poissonian curve. This finding can be interpreted as GOE-like GOE-like or more Poissonian-like behavior. Imagine levels are statistics for each of the involved subspectra as already “overlooked” due to the obviously finite spectral resolutionJ predicted in [34]. Nevertheless, another observable for long- of the experiment, where two or more levels may overlap, range correlation statistics, the spectral rigidity ( ), will be especially with increasing level density. Also resonances with3 L investigated as a further preparation step for the data analysis intensities below the detection threshold may be missed. If presented in Sec. IV. levels are randomly extracted from the spectra, all statistical measures for the spectral properties of such incomplete spectra will show a displacement towards Poissonian statistics, simply D. Spectral rigidity 3(L) because the correlation between levels is decreased. In our The spectral rigidity or Dyson-Metha statistic 3(L) is experiment, which utilizes resonance ionization spectroscopy, defined as the least-squares deviation of the unfolded staircase such situations cannot be avoided. In addition, especially function ν(ξ ) from the best linear fit. Note that after proper as three J submanifolds are superimposed, the strength for unfolding the smooth part of the staircase function is by transitions in subspectra of one specific J value might be definition a straight line. The spectral rigidity 3(L) is given significantly suppressed compared to the others. This causes as 〈 ∫ 〉 the missing of levels in the respective J submanifold. In this 1 ξs+L case, the fluctuation properties would exhibit a displacement 3(L) = min [ν ′(ξ ) − Aξ − B]2dξ , (13) towards GOE-like behavior for every subspectrum, if the L A,B ξs complete spectra exhibit GOE, because the correlations in each where ξs defines the first unfolded level energy of an interval independent subspectrum seem to be higher if levels from of length L [38,39]. Due to its definition, the spectral rigidity only one J submanifold are missing. One possible method is comparable to a χ2 depending on the correlation length L. for characterizing the consequences for a certain number of Since 3 is rather similar to 2, the spectral rigidity can also missing levels in the experimental data is to randomly delete be expressed as an integral transform of the number variance a specific percentage of levels from a set of levels from which [38] ∫ we know that it shows GOE statistics. 2 L  33(L) = (L − 2L2r + r3)2(r )dr. (14) L4 A. Random matrices0 Similar to the number variance, for the Poissonian case of The members of the Gaussian orthogonal ensemble are noncorrelated spectra (14) also leads to a linear expression of real symmetric matrices with Gaussian-distributed entries that P3 (L) = L15 , while GOE3 (L) for correlated GOE spectra again are invariant under real orthogonal transformations [43]. We follows a logarithmic trend for large L. The resulting curve generated such matrices with dimension N = 300. In order to for three superimposed GOE spectra is simply calculated in simulate the experimental situation, where the level sequences analogy to Eq. (12) by [38] are composed of three independent subspectra corresponding ∑ ( ) to different values of J , we merged the eigenvalues of three3 3GOE = GOE L random matrices into one sequence before applying the unfold-3 (L) 3 . (15)3 ing procedure. In order to improve the statistical significance m=1 we considered ensembles composed of five such sequences, Figure 4(b) shows the spectral rigidity for the three super- thus yielding a set of 4500 levels. imposed simulated subspectra with J = 9 132 , . . . , 2 as well as Figure 5 shows the statistical measure for long-range the theory curves for Poisson, GOE, and convoluted GOE correlation 3(L) as well as the residuals from fitting the statistics. The rather small error bars again stem from the CNNSD with the convolution of three Brody functions and 022506-5 Σ2(L) Poisson Δ3(L) PASCAL NAUBEREIT et al. PHYSICAL REVIEW A 98, 022506 (2018) 0.6 1.2 0.4 I3B(s, 1.00) 0.3 0.5 5 I (s, 0.34) o n 0.8 AM ois s P 0.2 0.4 3GOE 0.4 0 (a) 0.1 (b) 0.3 0 1 2 3 4 5 0.0s 0 20 40 60 80 0 20 40 60 80 OE 0.2 G Missing Levels (%) Missing Levels (%) FIG. 6. Dependence of (a) the Brody parameter and (b) the Abul- 0.1 Magd parameter on the percentage of missing levels regardless of the level subset. For each data point the distribution of the parameters for 0.0 100 different level sets has been evaluated. The red lines show linear 0 2 4 6 8 10 fits to the data with fit parameters of (a) 0.9902(54) and −0.0095(1) L and (b) 0.3474(21) and −0.0027(1) for the intercepts and slopes, respectively. FIG. 5. Spectral rigidity for the RMT eigenvalues. The error bars display the standard deviation of the binning process. The theory curves for Poissonian, GOE, and 3GOE statistics are given as gray from fits to the NNSD of 100 level sets for various values of lines. The inset comprises the residuals of the CNNSD for the the percentage of missing levels. The resulting distributions fitted Brody convolution (red dashed line) and the fitted Abul-Magd are of Gaussian shape and the center positions together with distribution (blue solid line). the HWHM values as uncertainties are displayed in Fig. 6 for various percentages of missing levels reaching from 2% to 90%. Both parameters are linearly decreasing as a function the Abul-Magd distribution for this set of eigenvalues. In the = of omission percentage. Thus, as expected, the correlationsinset of Fig. 5, a Brody parameter of η 1.00(23) and an = between the remaining neighboring levels are reduced withAbul-Magd parameter of f 0.34(4) illustrate the expected increasing fraction of missed eigenvalues. repulsion of the nearest neighbors in a convolution of three For each fraction of missing levels a representative set of GOE spectra. Even though the fits are lying at slightly higher eigenvalues was identified for which a fit of the convoluted values than the data, the small residuals also confirm the Brody and Abul-Magd distributions to the NNSD yields values two extracted repulsion parameters and their uncertainties. In of η and f as evaluated via Fig. 6. For these data sets also the addition, the RMT data coincide perfectly with the theoretical dependence of the spectral rigidity  (L) on the fraction of results for the spectral rigidity. Thus, the dimension of the 3missing levels has been obtained. Figure 7 shows the spectral random matrices is large enough to ensure good agreement rigidity for a choice of percentages of missing levels regardless of the numerical simulations with the theoretical results for of their J value, i.e., subset. The plots concerning that type of correlation lengths in ranges relevant for the experimental data. missing level are labeled with “All J missing.” The data for the B. Missing level statistics 0.6 As mentioned above, we have to take care of two types of All J missing One J missing n missing levels. First, there may be randomly missing levels so0.5 80 % 99 % ois regardless of the subset, or in our case of the J value. Second, P60 % 80 % levels of a specific subset, or synonymous J manifold, might 0.4 40 % 60 % GOE be suppressed at random and thus not detected. There actually 3 exist exact analytical results for incomplete eigenvalue spectra GOE0.3 2 of random matrices from the Gaussian ensembles [44] which can be generalized to the case of a superposition of three GOE0.2 independent GOE matrices. However, because of the fact that we here have to deal with the above-mentioned cases of missing 0.1 levels, the chosen way of using random matrix ensembles is more straightforward. In the following we will focus on 0.0 the first case. The second case will be analyzed in detail in 0 2 4 6 8 10 Sec. III C. L We simulated the first case by removing a specific percent- age of randomly chosen levels from the total set of eigenvalues. FIG. 7. Spectral rigidity for different percentages of missing To match the experimental situation, this has to be done before levels from all subsets as colored dashed lines with increasing the unfolding process takes place. We created an ensemble of percentages from top to bottom. The colored solid lines represent the random matrices which will have slightly differing spectral spectral rigidity for different percentages of missing levels from one properties. To extract the universal fluctuation behavior for certain subset with increasing percentages from bottom to top. The a specific percentage of missing levels, we analyzed the curves for Poissonian, GOE, 2GOE, and 3GOE behavior are given as distributions of the Brody and Abul-Magd parameters deduced gray dotted lines. For more information see the text. 022506-6 Δ3(L) CNNSD Res. (%) Δ (L) η-Parameter3 f-Parameter INTRINSIC QUANTUM CHAOS AND SPECTRAL … PHYSICAL REVIEW A 98, 022506 (2018) delivers the expected value for a convolution of now only two (a) 0.5 (b) GOE subspectra with n = 1 ≈ 2. 1.6 f Again, representative level sets are found with NNSDs 0.4 predicted by the distribution of η and f as given in Fig. 8.1.2 From these representative sets the spectral rigidity 3(L) was calculated for several percentages of missing levels. Figure 7 0.8 0.3 illustrates the results for this type of missing levels as graphs 0 20 40 60 80 100 0 20 40 60 80 100 Missing Levels (%) Missing Levels (%) labeled with “One J missing.” The theory curves for one, two convoluted, and three convoluted GOE spectra are given FIG. 8. Dependence of (a) the Brody parameter and (b) the Abul- together with the curve for the Poissonian case as gray dotted Magd parameter on the percentage of missing levels in one specific lines. The analyzed data are given as solid lines, differently level subset. For each data point the distribution of the parameters for colored according to the number of missing levels. At first 100 different level sets has been evaluated. The red lines show a cubic sight it is recognizable that the colored solid lines of Fig. 7 lie fit to the data. between the theory curves of 2GOE and 3GOE. Accordingly, the spectral rigidity 3(L) shows the expected trend: Starting different percentages of missing levels are given as differently at the 3GOE curve, it approaches the 2GOE curve where it colored dashed lines, while the curves for Poissonian and finally ends up. Note that the curve for 99% missing levels convoluted GOE behavior are given as gray dotted lines. The also fully agrees with the 2GOE curve within its uncertainty error bars are omitted in these graphs for the sake of clarity. The range, which was omitted here for the sake of clarity. That remaining graphs will be discussed in Sec. III C. As expected behavior corresponds to the increase of spectral correlation as from the features of the NNSDs, the correlations between the seen before in the NNSDs of Fig. 8. Remarkably, 3(L) does remaining levels dwindle the more levels have been taken out, not change very much for percentages of missing levels below thus the corresponding spectral rigidity approaches the curve 40%–50%, like it was already suggested by the behavior of the for Poissonian statistics accordingly. In contrast to the NNSDs, NNSD parameters in Fig. 8. this approach does not take place in a linear manner: A clear Note that in a spectroscopic experiment, naturally a mixture deviation of 3(L) from the 3GOE curve is observed not of both types of missing levels will occur. As described in before approximately 30% of missing levels, at least within Secs. III B and III C, either type has a more or less opposite these rather short correlation lengths. influence on the measures for spectral correlations. Thus, a specific ratio of the occurrence of both types might lead to a compensation of the effects on one of the measures for C. J suppression spectral correlations. Since the effects caused by missing levels Comparable to the approach in Sec. III B, here also specific are strongly nonlinear for the various statistical measures, we amounts of levels are deleted, but now from only one subset expect that a simultaneous cancellation for all of them is not in order to simulate the suppression of levels with a specific possible. Hence, in the following analysis of experimental data, J value. We proceeded as in Sec. III B and thus obtained for always the residuals of the fits for the CNNSD and the spectral the NNSD parameters a dependence on η and f as illustrated rigidity are evaluated. in Fig. 8. Unlike in Fig. 6, here the parameters η and f are well described by a cubic increase with an increasing number of missing levels.1 Both starting at the values for IV. SPECTROSCOPIC DATA NNSDs of three convoluted GOE spectra around η = 1 and Based upon the statistical measures for spectral fluctuation f = 13 , the parameters remain fairly unchanged until about properties introduced in Sec. II and the results on missing 30% of missing levels. From this point on the parameters level statistics presented in Sec. III, we can now begin with are quickly approaching their final values of η = 1.71(2) and a detailed analysis of the vast spectroscopic data of the pro- f = 0.51(0.2) at 99% missing levels. Of course, a Brody tactinium atom. We will analyze several spectra composed of parameter larger than η = 1 is not very meaningful if a single mainly three subspectra with individual total angular momenta spectrum is evaluated. Contrarily, it accentuates in our case covering different ranges of excitation energy and both parities. the increase in level correlation due the reduced influence of one of the independent subsets. Also the Abul-Magd parameter A. Literature data Data available in the literature [33] cover excitation energies 1Cubic and quadratic fits have been tested, both with an fixed from the ground state at 0 eV up to about 4.5 eV and both apex at 0% missing levels to guarantee a monotonically increasing parities. We furthermore merged the levels with total angular9 13 function. Because of the worse adjusted R2 for the quadratic fits momenta ofJ = 2 , . . . , 2 into one data set. Even-parity levels of R2η = 0.9972 and R2f = 0.9955, respectively, in addition to larger with an excitation energy above 2 eV have been omitted be- residuals in comparison to the cubic fits—here the adjusted R2 were cause of incompleteness indicated by the sharp cutoff observed R2 = 0.9975 and R2η f = 0.9999, respectively—we decided to rely on in Fig. 1 in the left inset. For odd-parity levels we proceeded the cubic fits. We presume that the dependence would be described similarly by also omitting levels for energy ranges where best by a quadratic function if levels from two subsets, not one or all obviously levels are missing according to the level density; three, would be taken out. A proof for that conjecture as well as a in this case levels with excitation energies above 3.7 eV are reason for the found dependences is pending. neglected. Due to the low total count of only 46 energy levels, 022506-7 η-Parameter f-Parameter PASCAL NAUBEREIT et al. PHYSICAL REVIEW A 98, 022506 (2018) 0.6 0.6 5 I3B(s, 1.45) n 10 I3B(s, 1.46)0.5 o onIAM(s, 0.33) iss 0.5 sPo 5 is Po 0 0.4 0 GOE 0.43 -5 EIAM(s, 0.58) 3G O 0.3 0 1 2 3 4 5 0.3 0 1 2 3 2GOE s s OE OE 0.2 G 0.2 G 0.1 0.1 0.0 0.0 0 2 4 6 8 10 0 2 4 6 8 10 L L FIG. 9. Spectral rigidity for low-energy odd-parity levels of [33]. FIG. 10. Spectral rigidity for the midenergy even-parity levels The error bars display the standard deviation of the binning process. of the SES scheme from [31]. The error bars display the standard The theory curves for Poissonian, GOE, and 3GOE statistics are given deviation of the binning process. The theory curves for Poissonian, as gray lines. The inset comprises the residuals of the CNNSD for the GOE, 2GOE, and 3GOE statistics are given as gray lines. The fitted Brody convolution (red dashed line) and the fitted Abul-Magd inset comprises the residuals of the CNNSD for the fitted Brody distribution (blue solid line). convolution (red dashed line) and the fitted Abul-Magd distribution (blue solid line). the even spectrum is not significant and is thus only discussed in Appendix A. statistically significant information on the spectral properties Within the superimposed odd-parity spectrum, we analyzed of this region. a total of 217 levels. Figure 9 shows the results of this analysis. The parameters of the Abul-Magd function and the composition of three Brody functions of f = 0.33(9) and η = 1.45(49), respectively, predict 3GOE level statistics B. The SES scheme of these energetically low-lying energy levels. However, the In the following we discuss the spectral statistics of MLH fit somewhat overestimated the Brody parameter. This is three representative spectra of even-parity levels identified in probably caused by the midsize level spacings, where both Ref. [31], covering different energy ranges and both types of functions have problems to fit the data as visible in the missing levels; an example of spectral statistics for odd-parity residual plot of the CNNSD in the inset of Fig. 9. The spectral levels is additionally given in Appendix B. The spectrum rigidity 3(L) perfectly coincides with the 3GOE curve up of the SES scheme of [31], also shown in Fig. 1, covers a to correlation lengths of about L = 4.5, which is comparable medium-energy range around 4.5 eV. Since excitation starts to the correlation length of the simulated data as analyzed in from an excited level with J = 112 , the detected levels may Fig. 4. have values for total angular momentum in the range J = From these level statistics we learn two important things. 9 , . . . , 132 2 . With 28 energy levels this spectrum constitutes the (i) The analyzed level sequence tabulated in [33] shows shortest level sequence evaluated here and thus suffers from no apparently missing levels. This seems to be the fact low statistical significance, which also results in very large also for the case of the even-parity levels analyzed in uncertainties. Nonetheless, the analysis of the observables for Appendix A. level fluctuations, as comprised in Fig. 10, reveals spectral (ii) Already at these very low energies (the spectrum statistical properties that can be well understood if missing reaches from 0.8 to 3.7 eV) the level fluctuations are not levels are taken into account. Already theη andf parameters of distinguishable anymore from 3GOE statistics, at least within the CNNSD, η = 1.46(103) and f = 0.58(29), respectively, the correlation lengths mentioned above. predict an overestimation of chaotic behavior which would be Especially the second point is remarkable, as it affects a caused by missing levels of one specific submanifold, even detailed investigation of the transition region into the chaotic though the large residuals as shown on the inset of Fig. 10 regime. As pointed out in Sec. I, regularity and intrinsic suffer by the low statistical significance. Comparing the data quantum chaos are coexisting in a specific energy region until in Fig. 10 with the curves in Figs. 7 and 8, one can estimate the chaotic behavior finally becomes prevalent above. In order that a very high percentage of one J manifold of more than to analyze the spectral statistics in this region, the spectrum 80% must be missing due to the J suppression discussed in has to be divided into several parts which need to be analyzed Sec. III C. Most important for the SES scheme is that the data separately. Unfortunately, the transition is apparently located clearly deviate from nonchaotic Poissonian behavior, which at such low excitation energies that the number of levels in testifies again to the presence of IQC in the protactinium atom the individual parts of the spectrum is too low to extract already at low excitation energies. 022506-8 Δ3(L) CNNSD Res. (%) Δ3(L) CNNSD Res. (%) INTRINSIC QUANTUM CHAOS AND SPECTRAL … PHYSICAL REVIEW A 98, 022506 (2018) 0.6 0.6 % I (s, 1.19) I (s, 0.64) 80 0.5 3B5 I so n 0.5 3B AM(s, 0.49) ois 5 IAM(s, 0.27)P n % ois so 70 0 0.4 P3GOE 0.4 0 GOE 0.3 0 1 2 3 4 5 2GOE 0.3 0 1 2 3 4 5 3 s s GOE 0.2 0.2 GOE 0.1 0.1 0.0 0.0 0 2 4 6 8 10 0 2 4 6 8 10 L L FIG. 11. Spectral rigidity for the high-energy even-parity levels of FIG. 12. Spectral rigidity for the high-energy even-parity levels scheme (iii) from [31]. The error bars display the standard deviation of of scheme (vi) from [31]. The error bars display the standard deviation the binning process. The theory curves for Poissonian, GOE, 2GOE, of the binning process. The theory curves for Poissonian, GOE, and and 3GOE statistics are given as gray lines. The inset comprises the 3GOE statistics are given as gray solid lines. Additionally, the curves residuals of the CNNSD for the fitted Brody convolution (red dashed for 70% and 80% of missing levels regardless of their total angular line) and the fitted Abul-Magd distribution (blue solid line). momentum are included as gray dashed lines. The inset comprises the residuals of the CNNSD for the fitted Brody convolution (red dashed line) and the fitted Abul-Magd distribution (blue solid line). For more C. Scheme (iii) information see the text. The spectrum of excitation scheme (iii) in [31] with a total of 226 spectrally investigated energy levels involving total angular momenta of J = 92 , . . . , 132 is much more significant result for 3(L) in Fig. 12 lies exactly between the two gray in terms of statistics. The spectral properties of this scheme dashed curves for 70%−80% missing levels as evaluated in covering high-energy regions just below the ionization poten- Sec. III B. The very long correlation lengths of this coincidence tial are summarized in Fig. 11. The spectral rigidity follows the together with the consistency of the distribution parameters η curve for 2GOE similarly to the SES midenergy scheme up to and f , the corresponding residuals, and the evaluated spectral correlation lengths around L = 5. For higher L the spectral rigidity validates the applied methods of analysis. Despite the rigidity starts to slowly fluctuate between the curves for 2GOE rather large amount of missing levels obtained in this excitation and 3GOE. Regarding the inset of Fig. 11, the CNNSD is scheme (vi), a full expression of intrinsic quantum chaos seems well described by the fitted distributions with η = 1.19(44) nonetheless confirmed. and f = 0.49(10), also predicting a slight overestimation of the GOE behavior of each involved J subset. Only for the V. CONCLUSION AND OUTLOOK smallest spacings, the residuals are somewhat larger, which can be explained by missing levels. Once again, the observables We have analyzed several sets of energy levels of the for the CNNSD and 3(L) clearly deviate from Poissonian protactinium atom concerning their spectral fluctuation prop- statistics consistently exhibiting an overestimation of GOE erties. The sets are of different origins, stemming from the statistics due to suppression of transitions leading into levels literature [33], calculations [34], and recent experimental data with a certain total angular momentum . [31]. Since the experimental data were not separable into setsJ with only one total angular momentum J , all analyzed data, separable for J or not, were composed for incorporating the D. Scheme (vi) same range of J levels as the experimental data. Moreover, For scheme (vi) from [31] with a total of 173 analyzed for the spectroscopic data, it was essential to investigate the energy levels also lying in an energy range just below the influence of missing levels on the spectral fluctuations, which ionization potential and having total angular momenta of was performed accurately. Therefore, randomly levels were J = 92 , . . . , 132 , the situation regarding the spectral properties taken out either from the whole spectrum or only from one is consistent and convincing, although it is not satisfactory subspectrum of three superimposed submanifolds simulated for the applied spectroscopic method: Inspecting the spectral by three GOE matrices in order to correctly simulate the fluctuation properties in Fig. 12, the results imply a high experimental situations. In addition, the mathematical analysis number of missing levels in this scan, here regardless of of the statistical measures had to be customized for the special J . A Brody parameter of η = 0.64(28) and an Abul-Magd difficulty of nonseparability for the “good” numbers, or at least parameter of f = 0.27(11) already suggest a percentage of for the total angular momentum J . more than 30%−40% missing levels if compared to Fig. 6. In As already suggested in [34], besides the short-range addition, the residuals depicted on the inset of Fig. 12 show correlation of the therein investigated NNSD, 3(L) clearly strong deviation of both fitted distribution only for very small indicate agreement with GOE behavior. This was confirmed spacings, which is also clearly provoked by missing levels. The using the available data from the literature [33] for odd-parity 022506-9 Δ3(L) CNNSD Res. (%) Δ3(L) CNNSD Res. (%) PASCAL NAUBEREIT et al. PHYSICAL REVIEW A 98, 022506 (2018) levels as well as even-parity levels with lower significance. 0.6 For the experimental data recently obtained in [31], it was 10 I (s, 0.91) possible with extensive analysis of the missing level problem 3B0.5 sso n 5 i also to extract level statistics that coincide well with those of Po a composition of three independent GOE spectra. The spectral 0.4 0 I (s, 0.31) statistics for all cases studied strongly deviate from Poissonian AM 3GOE statistics, or nonchaotic behavior, and therefore emphasize 0 1 2 30.3 s the prognosticated occurrence of IQC in the protactinium E atom. O0.2 G At excitation energies below 2 eV the energy levels avail- able in the literature [33] already show chaotic level statistics. 0.1 This means that the onset of chaos, or synonymously the transition point from regular to chaotic behavior, must be 0.0 located at even lower energies. In such low-energy regimes, 0 2 4 6 8 10 the level density is too small to extract the spectral properties L of this transition region with high statistical significance. As a future prospect, one approach to account for the low number FIG. 13. Spectral rigidity for low-energy even-parity levels of of energy levels is to combine several unfolded level sets with [33]. The error bars display the standard deviation of the binning excitation energies centered in this region of different elements process. The theory curves for Poissonian, GOE, and 3GOE statistics with comparable atomic properties, i.e., a similar number of are given as gray lines. The inset comprises the residuals of the open shells and active electrons. CNNSD for the fitted Brody convolution (red dashed line) and the For a quantification of the missing levels of either type, fitted Abul-Magd distribution (blue solid line). an empirical function that describes the influence of missing levels on the fluctuation laws would be the method of choice. results seem to be consistent with those for the odd-parity Fitting such a function to the NNSD and the spectral rigidity energy levels of [33] in Sec. IV A. of experimentally determined data sets could unveil even more details of the spectral properties of the underlying level APPENDIX B: SPECTRAL STATISTICS OF ODD-PARITY subsets. LEVELS FROM SCHEME (viii) With scheme (viii) from [31], a spectrum with 168 odd- ACKNOWLEDGMENTS parity energy levels lying in the energy region just below the The authors want to thank T. Guhr, F. Haake, and R. Heinke ionization potential is investigated regarding level correlations for many fruitful discussions. P.N. gratefully acknowledges and spectral statistics. The corresponding observables for the the Carl-Zeiss-Stiftung, D.S. gratefully acknowledges the EU spectral statistics, namely, the η and f parameters for fitting through ENSAR2 RESIST (Grant No. 654002), and A.V.V. the CNNSD with the corresponding residuals and the spectral and V.V.F. gratefully acknowledge the Australian Research rigidity 3(L), are comprised in Fig. 14. For the explanation Council and Gutenberg Fellowship for financial support. B.D. of the results, the consideration of missing levels is again thanks the NSF of China for financial support under Grant No. 11775100. 0.6 10 I3B(s, 1.18) n APPENDIX A: SPECTRAL STATISTICS OF EVEN-PARITY 0.5 5 IAM(s, 0.64) ois so LITERATURE DATA P 0 0.4 E Figure 13 shows the spectral statistics for the even-parity -5 3GO energy levels that can be found in the literature [33]. Even 0.3 0 1 2 3 4 5 2GOE though the results are not significant due to too few levels (only s 46) involved, the observables for the short-range correlation OE0.2 G between the energy levels promise a full expression of chaos: While the parameters of the composition of three Brody 0.1 functions and the Abul-Magd function of η = 0.91(78) and f = 0.31(21), respectively, are close to the ideal values of 0.0 f = 13 and η = 1 and thus already suggest GOE behavior, 0 2 4 6 8 10 the spectral rigidity is more sensitive to non-GOE features. L Here, similar to the short-range correlation due to their large uncertainties, also the spectral rigidity is not very conclusive, FIG. 14. Spectral rigidity for the high-energy odd-parity levels of which is also confirmed by the rather large residuals in the scheme (viii) from [31]. The error bars display the standard deviation inset of Fig. 13: Regarding small correlation lengthsL, it might of the binning process. The theory curves for Poissonian, GOE, 2GOE, coincide with regular just as well as with 3GOE statistics. Only and 3GOE statistics are given as gray lines. The inset comprises the for largerL, the spectral rigidity starts to deviate more strongly residuals of the CNNSD for the fitted Brody convolution (red dashed from the Poissonian statistics. Despite the poor statistics, the line) and the fitted Abul-Magd distribution (blue solid line). 022506-10 Δ3(L) Δ3(L) CNNSD Res. (%) CNNSD Res. (%) INTRINSIC QUANTUM CHAOS AND SPECTRAL … PHYSICAL REVIEW A 98, 022506 (2018) mandatory. The spectral rigidity 3(L) exhibits a behavior the missing levels. Even though the residuals shown in the very close to the curves for 2GOE, which would be explained inset of Fig. 14 are at least for small spacings quite large, by the case of missing levels from only one J manifold. 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Pato, Phys. Lett. B 595, 171 (2004). 022506-11 Chapter6 Conclusion The four publications comprised in this cumulative thesis address a series of sci- entific topics, starting from experimental to more theoretical nature, all applying resonance ionization spectroscopy as general base. With this regard, the choice of Resonance ionization spectroscopy of sodium Rydberg levels using difference frequency gener- ation of high-repetition-rate pulsed Ti:sapphire lasers as initial article had two scopes: The first was to verify and validate the method of RIS regarding Rydberg spectroscopy and the general approach and precision of the Rydberg analysis for extracting the ionization potential of an element. The second aspect of the article, somewhat less highlighted in the frame of this thesis, was the extension of the accessible wave- length range for RIS via establishing a difference frequency generation process for the Mainz Ti:sapphire lasers. For a proper validation of both aspects of the tech- nique, the element sodium was the ideal candidate. It is an element with a very precisely known ionization potential and, more over, its very simple atomic level structure is perfectly suited for RIS. For the laser excitation, the famous 589 nm fine structure doublet was chosen, clearly lying in the DFG range of a Ti:sapphire laser. The task could successfully be mastered by performing RIS on both, the sodium D1 and D2 line applying an DFG-extended wavelength range and re-measuring the ionization potential with ENa −1IP = 41449.455(6)stat(7)sys cm in very good agreement with the precise literature value of ENaIP = 41449.451(2) cm −1 [50, 51]. The second article, co-authored by the author of this thesis, Developments towards in-gas-jet laser spectroscopy studies of actinium isotopes at LISOL explores an important part of the LARISSA collaboration network with regard to on-line RIS applications. Most importantly, it demonstrated the high specifications and broad capabilities of the novel in-gas-laser ionization and spectroscopy (IGLIS) technique and therewith paved the way for a number of follow-up studies in this field, see e.g. [32, 33]. As main part of this thesis, the resonance ionization spectroscopy of protactinium is presented. This work included extensive and time consuming studies in the spec- trum of this rare element as well as sophisticated analyses of the recorded atomic 75 6 Conclusion spectra. Despite many complications and challenges which appeared during the handling of protactinium, Excited atomic energy levels in protactinium by resonance ion- ization spectroscopy comprises all information on the atomic data and the complex separation procedure of energy levels. The compilation of more than 1500 newly obtained energy levels in Pa I is given in the supplementary material to this article. Since a determination of the ionization potential was impossible by just applying conventional evaluation techniques, the focus was shifted towards the development and understanding of refined theoretical methods for extracting the IP from ultra dense atomic spectra. In chapter 4 of this thesis two of such methods are derived and elucidated in detail. Named Level Density Collapse and Rydberg Correlation, both analytical techniques can be used to extract estimates for the ionization potential from atomic spectra. As shown, a combination of both methods gives a reliable IP value even for the very complex and strongly disturbed spectra as found in the case of protactinium. With the aid of these newly developed analytical methods it was finally possible to extract a consistent value for the ionization potential of protactinium from the different spectra, whereas only a very rough estimate from comparison with other actinides was known before. The newly found value of EPa −1IP = 49034(10) cm (6.1) lies perfectly within the uncertainty range of the earlier estimation of 49000(110) cm−1 by Wendt et al. [37]. Despite of its scientific relevance, a publication of this finding and the involved analytical methods is still pending due to time constraints during the finalization of the present work. With the last topic of this thesis, a completely new field of research was worked out and established within the LARISSA group in Mainz. The more theoretical ap- proach of investigating intrinsic quantum chaos is rather far from the central area of expertise within the group. Nonetheless, the studies aroused deep interest among specialists in chaos research. This was accompanied by the high visibility and long- standing expertise of the group around Prof. Dr. Klaus Wendt: Based upon several decades of experience in lasers and specifically resonance ionization spectroscopy, the IQC studies in such exceptionally complex atomic systems like protactinium became possible at first. The article Intrinsic quantum chaos and spectral fluctuations within the protactinium atom comprises the findings and results of the investigations on quantum chaos in the Pa I spectrum. Amongst other insights, the proof was de- livered that protactinium is a fully chaotic many-body system as presumed, e.g., by A.V. Viatkina et al. [68]. Due to the extremely high level density, full expression of chaotic behavior is visible already at excitation energies as low as 2 eV, implying that the onset of chaos must be located at even lower energies close to the ground state. Moreover, the influence of missing levels, which cannot be avoided in spectroscopy with finite resolution, on spectral fluctuations was investigated in great detail within this article. 76 With the results presented in the featured articles, this thesis contributes to the broad oeuvre of the Mainz LARISSA group and extends it into a new direction. On the one hand, the method of RIS and the validation of the Rydberg analysis was proven at highest precision as well as a most useful technique for enhancement of the accessible wavelength range for the Mainz Ti:sapphire lasers was accomplished. On the other hand, two new research fields with promising application areas were established: The development and application of specific analytic methods for the determination of ionization potentials applicable wherever classical procedures fail. On top of that, intrinsic quantum chaos was investigated as self-contained research field. 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