XUV laser studies of Rydberg-valence states in N 2 and H + H heavy Rydberg states

Transcription

1 VRIJE UNIVERSITEIT XUV laser studies of Rydberg-valence states in N 2 and H + H heavy Rydberg states ACADEMISCH PROEFSCHRIFT ter verkrijging van de graad Doctor aan de Vrije Universiteit Amsterdam, op gezag van de rector magnificus door Maria Ofelia Vieitez Hornos geboren te Buenos Aires, Argentina

2 promotor: prof. dr. W. M. G. Ubachs promotor: prof. dr. C. A. de Lange promotor: prof. dr. L. E. Berg copromotor: dr. O. Launila

3 Reading committee: Dr. ir. G.C. Groenenboom (Radboud University Nijmegen) Prof. dr. R.A. Hoekstra (University of Groningen) Prof. dr. Th. Lindblad (Royal Institute of Technology, Sweden) Prof. dr. H.B. van Linden van den Heuvell (University of Amsterdam) Vrije Universiteit Amsterdam The investigations described in this thesis were partly carried out in the Laser Centre Vrije Universiteit (De Boelelaan 1083, 1081 HV Amsterdam, The Netherlands) and partly carried out at the Department of Physics of the Royal Institute of Technology (AlbaNova University Centrum, SE Stockholm, Sweden).

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5 Contents Contents v Introduction vii Experimental: the XUV laser setup vii Rydberg states ix Rydberg-valence state interactions xi Laser induced breakdown spectroscopy xii Outline of this thesis xiv 1 On the complexity of the absorption spectrum of molecular nitrogen Introduction The complexity of the spectrum Experimental Illustrative examples Conclusions and outlook Quantum-interference effects in the o 1 Π u (v = 1) b 1 Π u (v = 9) Rydberg-valence complex of molecular nitrogen Introduction Experiments Results and discussion Summary and conclusions Interactions of the 3pπ u c 1 Π u (v = 2) Rydberg-complex in N Introduction Experimental Methods

6 Contents 3.3 CSE calculations Results and discussion Summary and conclusions Observation of a Rydberg series in a heavy Bohr atom 83 5 Spectroscopic observation and characterization of H + H heavy Rydberg states Introduction Experiment and observations Analysis Conclusion Elemental analysis of steel scrap metals and minerals by laser-induced breakdown spectroscopy Introduction Experimental Optimization of experimental parameters Results and discussion Conclusions Samenvatting 131 Bibliography 137 List of publications 153 vi

7 Introduction This thesis is based on a number of experimental investigations in the field of laser spectroscopy that were carried out at two different institutes. The work began at the Royal Institute of Technology in Sweden in the period , focusing on the technique of laserinduced breakdown spectroscopy (LIBS). Thereafter a number of studies were performed with the Amsterdam extreme ultraviolet (XUV) laser setup, starting with Rydberg-valence state interactions in the nitrogen molecule. Afterwards it proceeded to the characterization of the socalled heavy Rydberg states in the H + H composite. In the following sections of this chapter a few key concepts of the experiments will be explained, as an introduction to the chapters contained in this thesis. Experimental: the XUV laser setup The majority of the experimental work presented in this thesis (Chapters 1 to 5) has been performed in the extreme ultraviolet (XUV) region of the spectrum. A review on the nature of XUV light and a historical perspective on its applications in spectroscopy can be found in [1]. The presently used instrument, the Amsterdam narrowband and tunable extreme ultraviolet laser facility, and its applications to gas-phase atomic and molecular spectroscopy has been described in a number of PhD thesis at the Vrije Universiteit [2]. Some of its main characteristics will be detailed briefly below. The tunable, narrowband XUV laser source is based on harmonic upconversion. To achieve this, narrowband visible radiation is generated in a pulse-dye laser (PDL) pumped by the second harmonic of a Nd:YAG laser. The visible radiation is frequency doubled using a non-

8 Introduction linear crystal (BBO, KDP). The resulting ultraviolet (UV) output is guided by dichroic mirrors (that filter the remnant of visible light) into a vacuum chamber. The XUV radiation is generated by a third harmonic generation (THG) process; this is achieved by focusing the UV light in an inert gas (Xe, Kr) [3, 4]. The need of a windowless vacuum system arises as XUV radiation is absorbed in all materials below 110 nm, and at these wavelengths the use of cells sealed off by some optical material is prohibited. The solution is to use a pulsed gas jet as non-linear medium [5]. The pulsed gas jet, in combination with differential pumping provides an solution because the density of inert gas in the focus of the UV laser can be made large without having too much absorption for the generated XUV radiation, as the molecular density is restricted to the small path length across the molecular jet close to the nozzle. The differential pumping provides the conditions of vacuum outside the THG region, and the XUV radiation is not absorbed and can propagate into another (differentially pumped) vacuum chamber to be used for spectroscopy experiments. The efficiency of the tripling process is rather low ( ) and therefore there is a strong remnant beam of UV light that travels collinearly to the XUV beam generated. The physics of the harmonic generation is well understood [3, 6, 7, 8, 9], and following the notation of [3], the equation that governs the frequency generation process, in terms of the component i of the polarization and the susceptibility tensor is: P i = ε 0 χ (1) ij E j + P NL = ε 0 ( χ (1) ij E j + χ (2) ijk E je k + χ (3) ijkl E je k E l ) (1) where χ (1) ij is the linear susceptibility, χ (2) ijk is responsible for the frequency doubling generation and χ (3) ijkl is responsible for the third harmonic generation. In isotropic homogeneous media (such as a gas jet) a reversal in the sign of E j and E k must cause a reversal in the sign of P i. This condition results in χ (2) ijk = 0. This means that in a gas jet it is not possible to generate the second harmonic of the initial frequency. Note that the above equation describes harmonic conversion in the perturbative regime (i.e. field densities of < W/cm 2 ); these conditions are typically met in a setup with nanosecond pulsed laser systems. The amount of XUV light produced by these means is low because it relies viii

9 Introduction on a third-order perturbative process while the material density in the focus (gas-phase) is low: hence the conversion efficiency is 10 6 or less. The bandwidth achieved in the XUV range is determined by that of the incident visible pulsed laser beam. For a Gaussian beam (the ideal case) the XUV-bandwidth is ν XUV = 6 ν vis, but in practical cases the bandwidth is slightly larger. In a scanning experiment the wavelength can be calibrated in the visible domain by on-line monitoring the Doppler-limited absorption spectrum of I 2 in an absorption cell; XUV frequencies relate to the visible by the exact relation ν XUV = 6ν vis. The used excitation scheme is resonantly enhanced multiphoton ionization, combined with a time-of-flight detection system (REMPI- TOF). The XUV+UV radiation is perpendicularly intersected by a pulsed molecular beam. The XUV photon is resonantly absorbed by the molecules in the beam (H 2, N 2 ) and the UV photon excites the molecule (non-resonantly) further above the ionization energy. The generated ion is detected by a time-of-flight spectrometer that allows for mass identification. This scheme is particularly useful to study isotope effects. In the experimental scheme described above, the spatial and temporal overlap (within 1 ns) of the XUV and UV laser pulses is assured by the THG process. However, in Chapters 4 and 5 a slightly different experimental approach is used: only the XUV photons are selected and the UV photon necessary for the non-resonant ionization is provided by the frequency doubled output of a second PDL. In this case, the XUV frequency was kept constant, and the UV frequency was tuned. The XUV radiation was kept fixed to an intermediate state of the H 2 molecule that has a short lifetime (> 0.5 ns), and therefore careful alignment and precise triggering was necessary to achieve spatial and temporal overlap of the two laser pulses. Rydberg states Most of this thesis work deals with Rydberg states, and in the following paragraphs some basic ideas about them will be explained. Rydberg states are excited states of the molecule (or atom) for which the energy of the states can be expressed as: E n = IE ion R m (n µ l ) 2 (2) ix

10 Introduction where n is called the principal quantum number having positive integer values, µ l is the quantum defect, which is a characteristic of a particular Rydberg series and it depends on the orbital angular momentum quantum number l of the Rydberg electron. R m is the mass-scaled Rydberg constant ( ) taking into account the finite mass of the nucleus R m = R M where R = cm 1, M is the mass of m e +M the molecule (or atom) and m e is the mass of the electron. IE ion is the ionization energy of the neutral molecule (or atom) and is called the Rydberg series limit. In a pure Coulombic potential, classical mechanics gives the orbit of the electron as ellipses, and the quantum defect is related on how much the electron penetrates the core [10]. Due to the l(l+1) centrifugal r 2 potential, in the higher l states the electron does not penetrate the core as much, and therefore the quantum defect becomes smaller when the l values increase. Also, the mean radius of the orbit is proportional to the principal quantum number squared. In atomic units [11]: r n,l = n 2 Z { [ 1 2 ]} l (l + 1) n 2 where Z is the charge of the ion core (Z = 1 for neutral molecules) and n = n µ l is the effective quantum number. This means that even for rather low quantum number (n 10), the size of the radius of the orbit is rather large. When the electron is at such a large distance, the core electrons shield the excited electron of the charge of the nucleus, so that the electric potential provided by the core is similar to that of a hydrogen atomic ion. Therefore the energy dependence is hydrogenatom like ( 1/n 2 ), but the principal quantum number n = n µ l takes into account (via the quantum defect) the deviation from a purely Coulombic potential. Electronic Rydberg series can converge to excited rotational or vibrational states of the ion and even to electronically excited states of the ion. In the high energy regions (close to the series limit), it becomes crowded with Rydberg series converging to the various limits. In most cases, Rydberg-Rydberg state interactions occur. Such high- n value electronic Rydberg series are observed in Chapters 4 and 5 for the H 2 molecule. In Chapters 1, 2 and 3, low- n Rydberg states of N 2 were studied. In the latter cases, the Rydberg states were heavily perturbed x

11 Introduction by neighboring valence states. In Chapter 4, a different Rydberg series is studied: a Rydberg series of the H + H ionic pair. In this series, the Rydberg electron is substituted by an H composite particle, and in spite of being a different type of state altogether, the energy spacing of the levels and its quantum mechanical treatment still follows the simple eq. (2), but with a new R m value defined [12]. The description of Rydberg states, as well as the interactions between them and with other bound states and the continua above them, are the main subject of quantum defect theory (QDT) [10, 13]. Rydberg-valence state interactions Rydberg-valence interactions and perturbations arise whenever the approximations (in most cases the Born-Oppenheimer (BO) approximation) used to derive wave functions associated with these states are not sufficiently accurate. In the BO approximation an approximate Hamiltonian of the system (H BO ) is built, and its orthonormal solutions (φ BO j ) obey: φ BO i H BO φ BO i = Ei BO φ BO i H BO φ BO j = 0 The total Hamiltonian of the system can be written as H T = H BO + H negl [11] and the H negl is a neglected term in the BO approximation. In most cases, the interactions are local and they involve only two states (sometimes three). This means that the matrix element: φ BO i H T φ BO j = φ BO i H negl φ BO j = 0 (3) where the term neglected (H negl ) in the H BO Hamiltonian couple the different φ BO j states to each other. In principle it is possible to express any exact solution (Ψ i ) of the total Hamiltonian (H T ) as an (infinite) expansion of the approximated BO functions: Ψ i = j c ij φ BO j If one term of the expansion is sufficient, this means that the BO approximation is reasonable. In the work presented in Chapters 1, 2 and xi

12 Introduction 3, the states have to be expressed as expansions of two and sometimes three terms of the BO expansion. Usually these interactions (perturbations) are detected in terms of irregularities in the quantum level structure, as the spacing between recorded spectral lines becomes irregular. We also investigate the interactions in terms of constructive and destructive interference effects, affecting the intensities of the lines in the spectrum. The quantum interference effect manifest itself by the intensity borrowing phenomenon and unusual predissociation J- dependence of the linewidths. The first three chapters deal with Rydberg-valence interaction between states, and specially homogeneous and heterogeneous interactions. Homogeneous interactions are those where Ω = 0 with Ω the value of the projection of the total angular momentum (exclusive of nuclear spin) J z of the molecule onto the internuclear axis. It fulfills Ω = Λ+Σ, where Λ is the value of the projection of the total electronic orbital angular momentum L z onto the internuclear axis, and Σ is the value of the projection of the total electron spin S z onto the internuclear axis. Both Σ and Λ values are (in most cases) labels of the eigenfunction. A heterogeneous perturbation is characterized by Ω = ±1. The homogeneous interactions dealt with here are of electrostatic nature, and therefore φ BO i H negl φ BO j = H ij = constant [11]. The important feature in heterogeneous interactions is that the interaction matrix element depends on the quantum number J. In all perturbations, the total angular momentum quantum number J remains defined (because J 2 commutes with H), and the selection rule common to all perturbations is J = 0. Laser induced breakdown spectroscopy In laser-induced breakdown spectroscopy (LIBS), the idea is to focus a powerful pulsed laser onto a target, ablating the surface of this target and creating a plume of plasma. This plasma emits light as the excited atoms (and molecules) decay to their ground states. The main goal is to record this emission spectrum and determine the composition of the original target. Apart from being able to know which elements are present in the target, the objective is to quantify the elemental abundance. Normally one uses samples of known composition to calibrate the setup. xii

13 Introduction The calibration is done by determining the relationship between the light intensity measured and the amount of various elements present in the sample. In the calibration process, standard samples with known chemical composition are measured. After selecting the standards, it is necessary to select the spectral lines to use for the analysis of the element of interest (the analyte). The desirable situation for spectral lines used to detect the concentration of the analyte is that the total (integrated) intensity of the line varies linearly with the concentration of the element. In reality it is found that this intensity versus concentration curve is not linear for all concentrations. At high concentrations of the analyte, re-absorption in the cooler, outer parts of the plasma takes place, diminishing the intensity of some (more sensitive) lines. Therefore, a high sensitivity spectral line must be used for low concentrations, while a low sensitivity line should be used for high concentrations. For the purposes of detection, it means that different lines should be used for calibration for different concentration ranges. Therefore a pre-existent knowledge of the range of concentrations to be measured in the sample must be established before LIBS studies take place. Another major drawback of this detection system is what is called the matrix effect. The matrix is the composition of the substance in which the analyte is found. Usually, the matrix refers to the major component or base element (for example, iron in steel). The introduction of additional elements into the sample in large amounts (> 5%) can affect the slope of the calibration curve. This is called interelement effect or matrix effect. In many cases the matrix effect is independent of the spectral line, i.e. the same matrix effect on many different spectral lines of the same element in a given matrix can be found. The phenomenon is not well understood from a physical point of view [14]. For the purposes of detection, this means that for each specific study, the target s main composition should be known, in order to perform a calibration based on reference samples of the same matrix composition. What makes the LIBS detection system attractive is the fact that it can be used in harsh environments, such as inside melting furnaces of metals. Or in places where a fast selection of the target is important, such as a scrap metal yard, where the value of the different pieces of metal depends greatly on their specific composition. The paper presented here was part of a project together with Stena Metall AB, a xiii

14 Introduction metal recycling company present in Stockholm, in which the aim was to assess the possibility to use LIBS to discriminate between metallic pieces to be recycled according to their composition. This work was used for the initiation of a study on a larger scale: Laser-Induced Breakdown Spectroscopy for Advanced Characterization and Sorting of Steel Scrap (LCS) (EU Research Programme of the Research Fund for Coal and Steel, Grant Agreement Number: RFSR-CT ). The goal of the latter project was to industrially evaluate the use of LIBS for scrap analysis and sorting. Outline of this thesis Chapter 1 is an introduction to the problems and challenges that must be dealt with when studying the spectrum of molecular nitrogen. From a general perspective, it describes the contribution of the Amsterdam XUV laser facility to the understanding of the complex N 2 energy levels and their interactions. This chapter also serves as an introduction to the next two chapters. Chapter 2 is about Rydberg-valence states interacting and perturbing each other. In this Chapter one exemplary case, prototypical for the entire N 2 spectrum, is investigated in detail. The Rydberg state o 1 Π u (v = 1) is perturbed (homogeneous perturbation) at low- J values by the valence state b 1 Π u (v = 9) and at high- J values by the b 1 Σ + u (v = 6) valence state (heterogeneous perturbation). The o 1 Π u (v = 1) b 1 Π u (v = 9) mixing is so pronounced that it has led in the past to incorrect assignment of the spectral lines. The mixing of states gives rise to effects of quantum interference on the oscillator strength of the observed lines. Chapter 3 deals with the perturbations of the Rydberg complex 3pπ u c 1 Π u (v = 2) with other valence singlet states and with triplet states. The perturbations with the singlet valence state b 1 Σ + u (v = 7) results in a strong Λ-doubling between the e and f states of the c(v = 2) and a P/R branch intensity anomaly for the b X(7, 0) band. Strong local perturbations in energy and line width of the c(v = 2) are attributed to a heterogeneous perturbation of the C 3 Π u (v = 17) state. Chapter 4 is about the first spectral observation of a series that was baptized heavy Rydberg series, a short for Rydberg series in a xiv

15 Introduction heavy Bohr atom. A heavy Bohr atom is a hydrogen atom where the electron is replaced by a H composite-like particle, forming the ionpair H + +H. The remarkable result is that the energy dependence of the heavy Rydberg series is similar to that of a regular (electronic) Rydberg series, but with rescaling of the Rydberg constant due to the difference in mass between the electron and the H. Chapter 5 deals with the characterization of the heavy Rydberg series excitation and observation mechanism, the complex resonances. The linewidths and line positions of the heavy Rydberg series are analyzed. The resulting quantum defects are related to the line positions using an (over-simplified) quantum defect theory model. Chapter 6 deals with using laser-induced breakdown spectroscopy detection system to measure the trace amounts of nickel, copper and other metals in steel targets. As an example of an industrial application, the concentration of copper in scrap metals is studied, which is an important factor to determine the quality of the samples to recycle. Another application of the LIBS method is the study of the nickel and copper concentrations in a sample of iron-rich magma. xv

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17 Chapter 1 On the complexity of the absorption spectrum of molecular nitrogen The spectral properties of molecular nitrogen are crucial to a better understanding of radiative-transfer phenomena and activated N/N 2 chemistry in the Earth s upper atmosphere. Excited states of N 2 are difficult to access experimentally, and analysis of its electric dipole-allowed spectrum is notoriously complex. In this paper, we give an overview of these complexities and of the power of extreme ultraviolet ionization spectroscopy in unraveling many of the observed features. Some illustrative examples from our own research will be discussed.

18 1. On the complexity of the absorption spectrum of N Introduction The importance of molecular nitrogen as the most abundant species in the Earth s atmosphere is evident. The strong absorption bands in the range nm shield the Earth s surface from the extreme ultraviolet (XUV) part of the solar radiation [15]. In fact, even the entire troposphere and stratosphere are free from this hazardous radiation that penetrates only some 150 km above the Earth s surface. Absorption of the short-wavelength light leads to molecular dissociation, and for N 2 this process is via predissociation, with ground- and excited-state atoms as products. Clearly, an understanding of the spectroscopy of N 2 in this wavelength range is essential for a better understanding of radiativetransfer phenomena and activated N/N 2 chemistry in the Earth s upper atmosphere. Similar processes are expected to take place in our solar system in the upper atmospheres of Jupiter, Saturn and its moon Titan, and Triton, the largest moon of Neptune [16]. Molecular nitrogen, N 2, together with the isoelectronic carbon monoxide CO, is one of the most stable molecules in nature. The electronic configuration of homonuclear N 2 in its X 1 Σ + g ground state is: (1sσ g ) 2 (1sσ u ) 2 (2sσ g ) 2 (2sσ u ) 2 (2pπ u ) 4 (2pσ g ) 2, corresponding to a triple chemical bond. For 14 N 15 N, the g (gerade) and u (ungerade) symmetry assignments for the orbitals hold only in approximation. The triple chemical bond explains the large dissociation limit of N 2 ( cm 1 [17]). Removal of an electron from the highest occupied molecular orbital leads to the lowest X 2 Σ + g ionic state, with configuration (1sσ g ) 2 (1sσ u ) 2 (2sσ g ) 2 (2sσ u ) 2 (2pπ u ) 4 (2pσ g ) 1, and an ionization energy of cm 1 [18]. As a result, excited electronic states of molecular nitrogen are high lying and not easily accessible by normal experimental means. Focusing on optical transitions involving the ground state, the weak spin forbidden A 3 Σ + u X 1 Σ + g Vegard Kaplan bands, the weak symmetry-forbidden a 1 Σ u X 1 Σ + g Ogawa-Tanaka-Wilkinson-Mulliken and a 1 Π g X 1 Σ + g Lyman-Birge-Hopfield bands have been observed, 2

19 1. On the complexity of the absorption spectrum of N 2 both in emission and absorption in the (far) ultraviolet (UV) [17]. The weakness of these bands implies that N 2 is optically transparent in the visible and UV regions of the spectrum. The much stronger one-photon electric-dipole-allowed absorption features in the N 2 spectrum involve transitions to valence and Rydberg states of 1 Σ + u and 1 Π u symmetry from the ground state and are found in the extreme ultraviolet. In this paper, we shall focus on the complexities of the electricdipole-allowed spectrum of molecular nitrogen and on the role that XUV ionization spectroscopy can play in unraveling them. The N 2 spectrum, situated in the energy range just above cm 1, displays many irregularities due to strong global vibronic Rydberg-valence interactions between the singlet ungerade states. Other local and accidental perturbations in the rotational structure are also evident in many places, generally arising from heterogeneous interactions that are usually significantly dependent on the isotopomer involved. All these interactions strongly affect vibronic and rotational intensities, and can result in vibronic and rotational quantum interferences. Another key process is predissociation, which is mediated through the spin-orbit interaction with triplet states. This coupling between the singlet and triplet manifolds is another source of spectral complexity. The rate of predissociation in molecular nitrogen is often sufficiently slow not to wash out the rotational structure in highly-excited states completely, but, at the same time sufficiently fast to allow the observation of line broadening of individual rotational transitions. Because of its excellent spectral resolution, XUV laser spectroscopy is eminently suitable for resolving this rotational structure and for determining the degree of line broadening and the corresponding rate of predissociation. Various illustrative examples derived from our research in Amsterdam will be discussed. 1.2 The complexity of the spectrum The dipole-allowed absorption spectrum of molecular nitrogen in the XUV shows a very complex behavior. Initially, it was thought that the many bands in the spectrum were due to transitions involving a large number of excited electronic states [17], but later it was found that they arose as a result of Rydberg-valence and Rydberg-Rydberg interactions between a limited number of singlet ungerade states lying at excitation 3

20 1. On the complexity of the absorption spectrum of N 2 o 1 Π u Potential Energy (10 4 cm 1 ) c 1 Σ u + G 3 Π u C 3 Π u c 1 Π u F 3 Π u b 1 + Σ u b 1 Π u C 3 Π u Internuclear Distance (Å) Figure 1.1: Potential-energy curves for the ungerade singlet states that govern the dipole absorption spectrum, and the triplet (C, C, F and G) electronic states that govern the predissociation behavior of N 2. Full lines: 1 Π u states. Dashed lines: 1 Σ + u states. Dotted lines: 3 Π u states. energies just above cm 1 [19, 20, 21]. There are two valence states involved, one of 1 Σ + u and one of 1 Π u symmetry (designated as b 1 Σ + u and b 1 Π u, respectively). The relevant Rydberg states belong either to the series converging on the lowest X 2 Σ + g ionization limit (npσ u c n+1 1 Σ + u and npπ u c 1 n Π u, with principal quantum number n 3), or the nsσ g o 1 n Π u series, converging on the A 2 Π u ionic limit of N + 2. The relevant electronic configurations of the lowest-lying singlet states are: b 1 Σ + u...(2sσ u ) 2 (2pπ u ) 3 (2pσ g ) 2 (2pπ g ) 1 valence b 1 Π u...(2sσ u ) 1 (2pπ u ) 4 (2pσ g ) 2 (2pπ g ) 1 valence c 4 or c 1 Σ + u...(2sσ u ) 2 (2pπ u ) 4 (2pσ g ) 1 (3pσ u ) 1 Rydberg c 3 or c 1 Π u...(2sσ u ) 2 (2pπ u ) 4 (2pσ g ) 1 (3pπ u ) 1 Rydberg o 3 or o 1 Π u...(2sσ u ) 2 (2pπ u ) 3 (2pσ g ) 2 (3sσ g ) 1 Rydberg. Potential-energy curves of these states, together with the C, C, F and G states of triplet character (to be discussed later), are presented in Fig They have the following configurations: 4

21 1. On the complexity of the absorption spectrum of N 2 C 3 Π u...(2sσ u ) 1 (2pπ u ) 4 (2pσ g ) 2 (2pπ g ) 1 valence C 3 Π u...(2sσ u ) 2 (2pπ u ) 3 (2pσ g ) 1 (2pπ g ) 2 valence F 3 Π u...(2sσ u ) 2 (2pπ u ) 3 (2pσ g ) 2 (3sσ g ) 1 Rydberg G 3 Π u...(2sσ u ) 2 (2pπ u ) 4 (2pσ g ) 1 (3pπ u ) 1 Rydberg. We note, in particular, that the configurations listed above for the b, b, and C valence states are those predominating at smaller internuclear distances R. As R increases, other configurations become important [22], as evidenced by the unusual shapes of the potential-energy curves for these states in Fig A detailed understanding of the spectroscopy in this energy region has long been hampered by the complex nature of the observed spectra. In a benchmark paper by Stahel et al. [23], a model of Rydbergvalence interactions was developed that provides a quantitative explanation for the energy-level perturbations, the seemingly erratic behavior of the rotational constants, and the observed band intensities that deviate strongly from Franck-Condon predictions, due to vibronic quantuminterference effects. In particular, the homogeneous vibronic interactions between states of 1 Σ + u symmetry (the b valence and the c 4 and c 5 Rydberg states) and between states of 1 Π u symmetry (the b valence and the c 3 and o 3 Rydberg states) were treated [23]. These global perturbations have been crucial in understanding the key features of the allowed optical absorption spectrum of N 2. Later, Spelsberg and Meyer put forward a quantitatively improved model, based on ab initio calculations [24]. Edwards et al. [25] extended the model by incorporating heterogeneous interactions to treat the mixing of states with different symmetries. These rotationally-dependent perturbations are local in character, and experimental methods to study such interactions usually require rotational resolution. These perturbations may cause shifts in rotational energy levels and can affect rotational transition intensities and predissociation line widths. Similar to what has been observed in the case of vibronic levels, such interactions may also give rise to rotational quantum interferences. Photodissociation can occur directly, by photoexcitation from a bound state to a repulsive state or to a bound state above its dissociation limit. Dissociation can also be indirect, when photoexcitation takes place from a bound state to another bound state, which in turn predissociates through a perturbative interaction with the continuum 5

22 1. On the complexity of the absorption spectrum of N 2 of another electronic state. The importance of predissociation phenomena in the lowest-lying electric-dipole accessible states of the abundant 14 N 2 and its rarer stable isotopomers, 14 N 15 N and 15 N 2, has been apparent for decades [26]. A diversity of experimental techniques were exploited to chart the most prominent of the predissociation effects. Studies of fluorescence excitation, induced either by electron collisions [27] or by synchrotron-radiation absorption [28], revealed the subclass of states that are subject to radiative decay, while the early XUV-laser studies revealed the strongly varying predissociative behavior for vibrational levels within a single electronic manifold [29]. Complementary techniques employing neutralization in fast ion beams allowed for monitoring of the photo-fragments from predissociating N 2 [30, 31]. However, a detailed quantitative understanding of the mechanisms underlying N 2 predissociation was not achieved until the work of Lewis et al. [32, 33]. This work is based on a coupled-channel Schrödinger equation (CSE) model, and it should be emphasized that it achieves spectroscopic accuracy, therefore allowing for a close comparison with experiment. In essence, the homogeneous ( Ω = 0) spin-orbit coupling provides an interaction between accessible singlet states and the triplet manifold. The spin-orbit coupling takes place between 1 Π u and C 3 Π u, which in turn is coupled to C 3 Π u above its dissociation limit. This is a form of accidental (or indirect) predissociation and can be interpreted as a perturbation of a nominally bound rotational level by a predissociated level that lies nearby in energy. The above pathway for predissociation in molecular nitrogen in the region below cm 1 has been confirmed by a large body of experimental spectroscopic evidence, based on an analysis of global and local perturbations. We note that the present understanding of the predissociation mechanisms for the 1 Σ + u states (with smaller predissociation rates) is not as well developed. Since predissociation of molecular nitrogen is such a key process, experimental methods suitable for its detection are important. In this paper, we shall show how ionization of N 2 and its isotopomers in a 1 XUV + 1 UV ionization process is a very convenient way of monitoring predissociation with rotational resolution. The method allows for the reliable detection of predissociation rates from experimentally observed line broadening. The excited-state lifetimes in molecular nitrogen happen to be in the range where the rotational structure in the 6

23 1. On the complexity of the absorption spectrum of N 2 spectra can still be discerned, while the associated line broadening often exceeds the instrumental line widths if narrow-band laser systems are used. Moreover, the interplay between our XUV laser experiments and the theory, as treated in the CSE model, forms a very powerful and successful combination that has led to significant new insights. As examples, the CSE model was able to explain the strongly varying lifetimes for the b 1 Π u (v = 1) state in the three N 2 isotopomers [34], and it predicted the location of the very weak spectral lines probing the F 3 Π u (v = 0) state, that were indeed observed [35]. Since our 1 XUV + 1 UV laser experiments start with molecular nitrogen in its X 1 Σ + g ground state, we are principally limited in what we can learn to detailed studies of the singlet ungerade manifold. Our studies are concerned with rotationally-resolved interactions between states that lead to perturbed term energies and transition intensities. These perturbations can take the form of rotational quantum interferences, similar to the vibronic quantum interferences discussed in [23]. However, important information about the triplet manifold can also be collected, both directly and indirectly, since spin-orbit coupling with triplets can cause observable perturbations in the singlet manifold. When the focus is on rotationally-resolved phenomena, these perturbations are usually isotopomer-dependent. Hence, the experimental study of the different stable isotopomers tends to be very informative. As illustrative examples from our own research, we shall discuss the analysis of homogeneous and heterogeneous perturbations in the singlet manifold, rotationally resolved quantum interferences in oscillator strengths and predissociation line widths, and the direct and indirect observation of triplet states through their interactions with the singlet manifold. 1.3 Experimental Details of the experimental method, including a description of the lasers, vacuum setup, molecular-beam configuration, time-of-flight (TOF) detection scheme and calibration procedures, have been given previously [36]. A skimmed and well defined pulsed molecular beam of nitrogen is perpendicularly intersected by temporally and spatially overlapping the XUV and UV laser beams. Nitrogen molecules are resonantly excited 7

24 1. On the complexity of the absorption spectrum of N 2 Figure 1.2: Competing decay mechanisms in a 1 XUV + 1 UV ionization experiment. The XUV photon excites the molecule with rate k abs. From the excited state, the molecule can fluoresce to lower levels with rate k fluor, it can undergo predissociation (with rate k pred ) or, via a UV photon, it can become ionized. This ionization process is, in principle, non resonant and it occurs with rate k ion. by the XUV photons and subsequently ionized by the intense UV light. N + 2 ions are detected using a TOF mass selector. For the detailed investigation of the N 2 spectral features, a tunable light source in the extreme ultraviolet region with sufficiently narrow bandwidth is required. Tunability and narrow bandwidth are achieved using two types of laser sources which deliver energetic pulses in the visible wavelength domain. Harmonic generation in two steps, by frequency doubling in nonlinear crystals and subsequent frequency tripling in gas jets, provides coverage of the XUV wavelength range, while the bandwidth of the fundamental light sources is nearly retained in the conversion process. When using a commercially available Pulsed Dye Laser (PDL), the bandwidth in the XUV is 0.3 cm 1 full-width at half-maximum (FWHM) and the absolute wavenumber uncertainty in the XUV for this system is ± 0.1 cm 1 for fully-resolved lines. When using a home-built Pulsed Dye Amplification system, a bandwidth of 0.01 cm 1 FWHM is attained and the absolute calibration uncertainty is ± cm 1. Wavelength calibration can be performed in the visible range, since exact harmonics are produced in the nonlinear optical conversion process. The two-photon-ionization TOF experiment has some useful characteristics that are favorably employed. Mass separation can be combined with laser excitation to separate the contributions to the spectrum of 8

25 1. On the complexity of the absorption spectrum of N 2 the main 14 N 2 isotopomer from those of the mixed 14 N 15 N and 15 N 2 species. Furthermore, by changing the relative delay between the N 2 pulsed-valve trigger and the laser pulse, as well as varying the nozzleskimmer distance, the rotational temperature in the molecular beam can be selected to measure independent spectra of cold (10 20 K) and warm (up to 300 K) samples. This form of temperature tuning of the gaseous sample aids in the assignment of the spectral lines. In a 1 XUV + 1 UV two-photon-ionization experiment, the excited state is populated by the XUV absorption process and depopulated by decay mechanisms that all, in principle, lead to a shortened lifetime (see Fig. 1.2). This occurs through (i) fluorescence, (ii) predissociation, and (iii) through ionization by UV radiation. As we intend to measure linewidths with our setup, excessive UV radiation that would lead to depletion of the population of the intermediate state is avoided in our experiment. Information on predissociation can be obtained using several complementary methods. The lifetime τ (s) of the excited level, shortened due to predissociation, can be expressed as τ = (2πcΓ) 1, with Γ the natural (Lorentzian) line width (in cm 1 FWHM). Hence, the excited-state lifetime τ can be derived straightforwardly from line-width measurements. The shortening of the lifetime due to predissociation will not only cause line broadening, but also a decrease in signal intensity, since we detect ions that result from ionization of a decaying excited level. Using a rate equation model [37], it can be proved that the intensity of the signal is proportional to the lifetime of the excited state, when the laser line width exceeds the natural width Γ. Hence, predissociation can be monitored by (i) directly detecting the broadening of the XUV transition [29]; (ii) carrying out a pump-probe experiment on the excited state with a variable time delay between the pulses [38]; (iii) measuring the decrease in the ionization signal which is proportional to the decrease of the excited state lifetime; or (iv) measuring the decrease in the fluorescence signal. In our experiments, both the decrease in the ionization signal and the broadening of the excited-state line width are signatures of the occurrence of predissociation. 9

26 1. On the complexity of the absorption spectrum of N 2 Figure 1.3: PDL-based XUV-source spectra of the b 1 Π u X 1 Σ + g (9,0) and o 1 Π u X 1 Σ + g (1,0) bands of 14 N 2, with corresponding line assigments. Two separate scans are joined in the region marked with an asterisk (*), and their relative intensities should not be compared. Note that several lines are blended and many transitions are too weak to be observed. 1.4 Illustrative examples In this section we shall treat a number of examples of (i) homogeneous interactions between states of the same symmetry and heterogeneous interactions between states of different symmetry; (ii) quantum-interference effects occurring both in the oscillator strengths between electric-dipole allowed transitions, and in the line widths between predissociating levels; and (iii) evidence for triplet states through their coupling with the singlet manifold. Without trying to be exhaustive, we have selected these examples from our research as representative of the current experimental and theoretical state-of-the-art in studying perturbation and predissociation phenomena in molecular nitrogen. The o 1 Σ + u (v = 1) b 1 Π u (v = 9) b 1 Σ + u (v = 6) interaction complex in 14 N 2 In the energy region between and cm 1, the following energy levels in 14 N 2 are situated closely together, thus allowing for 10

27 1. On the complexity of the absorption spectrum of N 2 Figure 1.4: Term values of the b 1 Π u (v = 9) o 1 Π u (v = 1) b 1 Σ + u (v = 6) e- parity states, reduced such that the deperturbed b(v = 9) levels are on the zero line. Clear anti-crossings for the b(v = 9) o(v = 1) and o(v = 1) b (v = 6) levels are shown. Measured energy levels are displayed using black symbols, predicted levels using grey symbols. possible interactions: o 1 Σ + u (v = 1) (Rydberg), and the valence states b 1 Π u (v = 9) and b 1 Σ + u (v = 6). For 14 N 2 at low J-values, the o(v = 1) and b(v = 9) states cross, while at higher J-values this complex can interact with the b (v = 6) state. In Fig. 1.3, the b 1 Π u X 1 Σ + g (9, 0) and o 1 Π u X 1 Σ + g (1, 0) high-temperature spectrum is presented, showing the P, Q, and R branches. For the isotopomers 14 N 15 N and 15 N 2, the relative positions of the o(v = 1) and b(v = 9) states are such that significant interactions are not expected between these states. Rotational levels associated with excited states can be of e or f parity in the case of 1 Π u states, or only of e parity for 1 Σ + u. Moreover, in 1 Π u states, Λ-doubling occurs. These issues are discussed in detail in many places [11, 39]. In order to obtain the term energies and transition intensities, a detailed analysis must be carried out. First, the rotational transitions P, Q and R are assigned, guided by the nuclear spin statistics, the combination differences (for the P and R branches) and the differences in intensities from the cold and warm spectra. These experimental transition energies are then compared with theoretical values calculated using ground- and upper-state term values which are parametrized employing the usual spectroscopic parameters (rotational constant B, centrifugal 11

28 1. On the complexity of the absorption spectrum of N 2 distortion parameters D and H). The rotational level structure of the ground state is well understood and represented by the constants published by Trickl et al. [40] in the case of 14 N 2, and by the constants of Bendtsen et al. [41] for the other isotopomers. By adjusting the upperstate rotational constants to minimize the differences with respect to the experimental transitions, a least-squares fit of the rotational constants and therefore of the term values is obtained. The b 1 Π u (v = 9) and o 1 Π u (v = 1) levels in 14 N 2 undergo an avoided crossing in the rotational structure between J = 4 and J = 5. Since both states have the same symmetry, this interaction is homogeneous ( Ω = 0), electrostatic in character, J-independent and involves both the e- and f-parity levels. Moreover, at high J an interaction that only involves e-parity levels between b 1 Σ + u (v = 6) and o 1 Π u (v = 1) is apparent. This interaction is heterogeneous ( Ω 0), arises from L-uncoupling and is therefore J-dependent. For the e-parity levels, a complete three-state deperturbation was performed for each value of J by diagonalizing the matrix T b9 (J) H b9o1 0 H b9o1 T o1 (J) H o1b 6 J(J + 1) 0 H o1b 6 J(J + 1) Tb 6(J). (1.1) The diagonal elements are the term energies of the b (v = 9), o (v = 1) and b (v = 6) states. The off-diagonal element H b9o1 is the homogeneous interaction between the 1 Π u states, and H o1b 6 J(J + 1) represents the effective heterogeneous interaction matrix element between o (v = 1) and b (v = 6). For f-parity levels, Eq. (1.1) reduces to a 2 2 matrix. In Fig. 1.4, the e-parity term values, reduced such that the deperturbed b (v = 9) values lie on the zero line, are shown. The figure clearly shows an avoided crossing between J = 4 and J = 5 for the b (v = 9) and o (v = 1) states, with a maximum energy shift of 8.7 cm 1 at J = 4. A second avoided crossing, this time between o (v = 1) and b (v = 6), occurs between J = 24 and J = 25, with a maximum shift of 13.5 cm 1 at J = 25. A similar f-parity plot can be constructed for the coupling between the Rydberg and valence 1 Π u states. Moreover, around the rotational levels where the interaction takes place, the wave functions are strongly mixed. For each state, the wave 12

29 1. On the complexity of the absorption spectrum of N 2 function will be a linear combination of the unperturbed wave functions: Ψ = c 1 Φ o(1) + c 2 Φ b(9) + c 3 Φ b (6), (1.2) where the values of the c i coefficients are the components of the eigenvectors of Eq. (1.1). Near J = 4, o (v = 1) and b (v = 9) exchange electronic character, while the level of mixing of b (v = 6) is almost negligible. The wave functions of o (v = 1) and b (v = 9) are then expressed as: Ψ o(1) = cφ o(1) + 1 c 2 Φ b(9), Ψ b(9) = 1 c 2 Φ o(1) + cφ b(9). (1.3) Also, near J = 25, the same happens for the e-levels of o (v = 1) and b (v = 6), while the mixing of b (v = 9) is close to zero. For the intermediate J levels, a three-state problem can be solved. Rotational quantum-interference effects Oscillator strengths Because both the b X (9,0) and o X (1,0) transitions carry oscillator strength, and since both b (v = 9) and o (v = 1) levels of like symmetry interact, this leads to a classic situation where two-level quantuminterference effects are expected. As discussed by Lefebvre-Brion and Field [11], the perturbed vibronic oscillator strengths for transitions from a common level 0 (the X 1 Σ + g ground state in our case) to upper (+) and lower ( ) levels of the interacting pair are given by: f +0 = c 2 f 10 + (1 c 2 )f 20 ± 2c (1 c 2 )f 10 f 20, f 0 = (1 c 2 )f 10 + c 2 f 20 2c (1 c 2 )f 10 f 20, (1.4) where f 10 and f 20 are the vibronic oscillator strengths for transitions to the unperturbed levels 1 and 2, and c > 0 signifies the mixing coefficient that corresponds to the amount of character of the unperturbed level 1 in the perturbed upper-level wave function, as explained previously. Using the mixing coefficients that have been obtained from the deperturbation procedure in the previous section, we can now deperturb the experimental oscillator strengths through the application of Eq. (1.4). 13

30 1. On the complexity of the absorption spectrum of N Oscillator Strength J Figure 1.5: Rotational dependence of band oscillator strengths (obtained from synchrotron-based experiments [39]) in the mixed b X(9, 0) and o X(1, 0) transitions of 14 N 2, demonstrating a strong quantum-interference effect near J = 6, together with the results of a deperturbation analysis (see text). Open squares: experimental (perturbed) oscillator strengths for transitions to the higher-energy levels. Solid squares: experimental oscillator strengths for transitions to the lower-energy levels. Dot-dashed line: deperturbed oscillator strength for the o X(1, 0) transition. Long-dashed line: deperturbed oscillator strength for the b X(9, 0) transition. Dashed curve: calculated perturbed oscillator strength for the higher levels. Solid curve: calculated perturbed oscillator strength for the lower levels. Our 1 XUV + 1 UV experiments reveal perturbations in the intensity pattern of the observed transitions, but as the photon flux is not measured, and the ionization cross sections are not known a priori, absolute oscillator strengths are not obtainable with our setup. Therefore, recent synchrotron-based measurements are used [39] (see also [42]) and shown in Fig As is apparent from the figure, transitions to the higher-energy level, i.e., b (v = 9) for J 4 and o (v = 1) for J 5, show a strong rotational dependence and peak around J = 6. At the same time, oscillator strengths to the lower-energy level, i.e., o (v = 1) for J 4 and b (v = 9) for J 5, show a minimum at J = 6, to the extent that this transition was too weak to be observed. Deperturbation of 14

31 1. On the complexity of the absorption spectrum of N 2 Figure 1.6: 1 XUV + 1 UV ionization spectrum for the c 3 1 Π u X 1 Σ + g (2,0) band of 15 N 2, with corresponding line assigments. The asterisks (*) indicate how large transition intensities would have been (on a relative scale) if not affected by the singlet-triplet interaction. the experimental oscillator strengths according to Eq. (1.4) leads to the dot-dashed line in Fig. 1.5 for the o X (1,0) transition, and to the longdashed line for the b X (9,0) transition. In summary, the experimental oscillator strengths of both bands show clear evidence of rotationallydependent constructive and destructive quantum-interference effects in 14 N 2. A full account of this work is presented in Ref. [39]. Predissociation line widths If two energy levels predissociate via the same perturbative state, an interaction between these levels may result in a quantum-interference effect in the strength of the predissociation, in the same fashion as for the oscillator strengths. The predissociation strength can be detected via line broadening of the measured transitions. In this case, the width interference is described by equations similar to Eq. (1.4), but with the oscillator strength f replaced by the width Γ, and with 0 the perturbative state. The constructive and destructive interferences observed near J = 6 for the oscillator strengths of the b X (9,0) and o X (1,0) transitions support the notion of line-width interferences and associated predissociation rate modulations. This is actually observed for the 15

32 1. On the complexity of the absorption spectrum of N 2 Figure 1.7: Energies of the c 3 1 Π u (v = 2) f-parity state of 15 N 2, reduced to be positioned on the zero-energy line. The straight lines indicate the crossings with C 3 Π u (v = 17) as predicted by an extended model based on [32]. o (v = 1) and b (v = 9) levels. A complete description of this phenomenon is presented in Ref. [39]. Triplet-singlet interactions As mentioned above, in principle our setup only allows the direct measurement of singlet ungerade states of N 2. Nevertheless, direct measurement of transitions of the triplet manifold are possible if these transitions become visible via intensity borrowing [11]. This is the case for the F 3 Π u (v = 0) Rydberg state. Sprengers et al. presented the results of an ultrahigh-resolution laser-spectroscopic study of the F X(0,0) transition in 14 N 2 [35]. This dipole-forbidden transition became observable through the spin-orbit-induced intensity borrowing from the dipole-allowed b X(5,0) transition. This phenomenon was facilitated by the near-degeneracy of the F (v = 0) and b(v = 5) levels in the 14 N 2 isotopomer. Direct observation of the F state led to assignments of the R, P and Q branches for the Ω = 0, 1, 2 triplet sublevels, together with the predissociation widths of the transitions. The obtained rotational parameters, together with the term values of the predissociating levels, suggest strong interactions among the F and G 3 Π u Rydberg and 16

33 1. On the complexity of the absorption spectrum of N 2 Figure 1.8: 1 XUV + 1 UV ionization spectra of the Q(15) and Q(6) lines of the c 3 1 Π u X 1 Σ + g (2,0) band for 15 N 2, recorded using the PDL-based XUV source and showing that Q(6) is a factor of two broader than Q(15), due to the increase of predissociation. Note that the relative X-scales are the same in each subfigure. C 3 Π u valence states. Hence, the complexity of the triplet manifold is somewhat similar to that of the singlet states. Even when the fortuitous coincidence of energy levels does not allow their direct observation through intensity-borrowing effects, the triplet states may manifest themselves in an indirect way, via perturbations of the singlet states. In the energy range cm 1, several singlet states, i.e., the Rydberg states c 1 3 Π u (v = 2) and c 4 1 Σ + u (v = 2) and the valence states b 1 Σ + u (v = 7) and b 1 Π u (v = 11) are situated. The predictions of the CSE model indicate a crossing between the singlet c 1 3 Π u (v = 2) state and the three components (Ω = 2, 1, 0) of the triplet C 3 Π u (v = 17) valence state. For 15 N 2, the crossing is predicted to occur at relatively low J. This is favorable, because it means that, with the 1 XUV + 1 UV setup, where the lower-j lines show the most intensity, this phenomenon can be detected. For 14 N 15 N and 14 N 2, the crossing takes place at higher values of J, that are difficult to access with our XUV setup. In the following, we shall focus on 15 N 2. 17

34 1. On the complexity of the absorption spectrum of N 2 The c 1 3 Π u X 1 Σ + g (2,0) band between J = 5 and J = 13 shows large intensity deviations from those expected for a Boltzmann distribution. These deviations are apparent in the P, Q, and R branches, shown in Fig It was found that the c 3 (v = 2) state interacts with the b (v = 7) state via heterogeneous coupling and that the states exchange electronic character around J = 20. At higher J values, this complex possibly interacts with the b(v = 11) state, but the results at higher J for all states did not allow for definitive conclusions. As mentioned before, the heterogeneous interaction involves only e-parity levels, and therefore does not show up in Q-branch transitions. Hence, this intensity depletion is attributed to the crossing with the triplet state. After the perturbations in the singlet manifold have been accounted for, remaining spectral deviations can be ascribed to the interaction with C 3 Π u (v = 17). In Fig. 1.7, the reduced f-parity levels of c 3 (v = 2) are shown. Similar results were obtained for the e-parity levels. Between J = 7 and J = 9, and also between J = 11 and J = 12, two perturbations are clearly present. These J-value positions coincide approximately with the predicted crossings with C 3 Π u (v = 17). In principle, only C Ω=1 (v = 17) can interact via the homogeneous ( Ω = 0) spin-orbit coupling with c 3 (v = 2). However, since the S-uncoupling mechanism induces mixing between the three C Ω (v = 17) components, all three states can interact to some extent with c 3 (v = 2). The perturbation between J = 7 and J = 8 is assigned to C Ω=1 (v = 17), and the one between J = 11 and J = 12 to C Ω=0 (v = 17). No clear indications for a crossing with C Ω=2 (v = 17) are observed, but they are expected to occur at lower J values. In Fig. 1.8, the line widths for two transitions from the ground state to levels with f-parity of the c 1 3 Π u (v = 2) state in 15 N 2 are shown. Near J = 8 and J = 12, an increase in broadening, and hence in predissociation rates, is observed that is again ascribed to a coupling with the crossing, strongly-predissociated C Ω (v = 17) states. As an example, the line width of Q(6) is increased by predissociation and is approximately double that of Q(15) which shows no broadening beyond the normal Doppler width. Notably, the intensity depletions shown in Fig. 1.6 take place at the same J-values at which the crossings with the triplet states are predicted. Altogether, these results are taken as convincing cumulative evidence that the local interactions with the C 3 Π u (v = 17) triplet 18

35 1. On the complexity of the absorption spectrum of N 2 state can be observed indirectly, through their perturbative effects on the singlet manifold. Moreover, the predictions of the CSE model appear to be very adequate in this respect. 1.5 Conclusions and outlook In this paper, some of the complexities observed in the electric-dipoleallowed spectrum of molecular nitrogen are discussed. These complexities arise from global vibronic and local rotationally-dependent perturbations, causing energy level shifts, redistribution of intensities and intensity borrowing, and quantum-interference phenomena. Experimental methods, such as XUV ionization spectroscopy, that can achieve rotational resolution are a suitable tool to study this important molecule. The rate of predissociation in molecular nitrogen is often fast enough to cause line broadening that exceeds the narrow-band XUV laser instrumental line width, but predissociation rates are often slow enough not to remove the rotational structure completely. The triplet manifold plays a key role in the predissociation processes in molecular nitrogen. The continuous interplay with the theoretical CSE modeling that is essentially capable of spectroscopic accuracy has played a crucial role in the detailed analysis and understanding of our experimental results. The connection between high-resolution XUV spectroscopy of the lowest ungerade states of molecular nitrogen and atmospheric chemistry is a strong one. For example, the effective emission from c 4 1 Σ + u X 1 Σ + g (0,0) and (0,1) bands in the Earth s airglow are unusually weak. The radiation of the (0,0) band, which is in fact the strongest emission feature in the N 2 spectrum [27], is radiatively trapped and undergoes resonant fluorescent scattering under atmospheric conditions. In this process the (0,1) band acquires some intensity also. This (0,1) band is in accidental resonance with the transition b 1 Π u X 1 Σ + g (2,0). Since the b(v = 2) level is strongly predissociated, the observed overall emission is unexpectedly weak [43]. This remarkable coincidence in the complex N 2 spectrum is strongly dependent on isotopomer, as are many other observed predissociation phenomena in molecular nitrogen [34]. So far, isotopic fractionation in nitrogen-containing atmospheres has been studied principally from the perspective of gravitational phenomena. Dissociative recombination of N + 2 and electron-impact dissociation of neutral N 2 19

36 1. On the complexity of the absorption spectrum of N 2 produce kinetically-hot atomic nitrogen, that, depending on the planetary escape velocities, may lead to isotopic fractionation and strongly varying 15 N/ 14 N ratios [16, 44]. The isotope-dependent predissociation effects resulting from the complexities in the N 2 spectrum will add to this behavior, although the particularities of each case are to be explored in more detail. Although fundamental in character, studies of perturbation and predissociation processes in molecular nitrogen and their isotopomers are crucial to a better understanding of radiative-transfer phenomena and activated N/N 2 chemistry in the Earth s upper atmosphere. In a similar spirit, this type of research has implications for other planetary systems with nitrogen-containing atmospheres. For example, Liang et al. [45] have already used the results of a CSE model of N 2 photodissociation to explain nitrogen isotope anomalies in HCN in the atmosphere of Titan. Future work will attempt to expand both the experimental database and the accurate theoretical modeling to higher energies. This requires the incorporation into the model of a plethora of higher-lying Rydberg and valence states of singlet as well as triplet character, and the interactions among them. Acknowledgments Part of this research was supported by Australian Research Council Discovery Program Grant DP

37 Chapter 2 Quantum-interference effects in the o 1 Π u (v = 1) b 1 Π u (v = 9) Rydberg-valence complex of molecular nitrogen Two distinct high-resolution experimental techniques, 1 XUV + 1 UV laser-based ionization spectroscopy and synchrotronbased XUV photoabsorption spectroscopy, have been used to study the o 1 Π u (v = 1) b 1 Π u (v = 9) Rydberg-valence complex of 14 N 2, providing new and detailed information on the perturbed rotational structures, oscillator strengths, and predissociation linewidths. Ionization spectra probing the b 1 Σ + u (v = 6) state of 14 N 2, which crosses o 1 Π u (v = 1) between J = 24 and J = 25, and the o 1 Π u (v = 1), b 1 Π u (v = 9), and b 1 Σ + u (v = 6) states of 14 N 15 N, have also been recorded. In the case of 14 N 2, rotational and deperturbation analyses correct previous misassignments for the low-j levels of o(v = 1) and b(v = 9). In addition, a two-level quantum-mechanical interference effect has been found between the o X(1, 0) and b X(9, 0) transition amplitudes which is totally destructive for the lower-energy levels just above the level crossing, making it impossible to observe transitions to b(v = 9, J = 6). A similar interference effect is found to affect the o(v = 1) and b(v = 9) predissociation linewidths, but, in this case, a small non-interfering component of the b(v = 9) linewidth is indicated, attributed to an additional spin-orbit predissociation by the repulsive 3 3 Σ + u state.

38 2. Quantum-interference effects on a Rydber-valence complex of N Introduction The dipole-allowed absorption spectrum of molecular nitrogen in the extreme ultraviolet (XUV) wavelength region was initially thought to consist of a multitude of electronic band structures, until the true nature of the excited states was unravelled [19, 20, 21]. The apparent complexity of the XUV spectrum is a result of Rydberg-valence mixing between a limited number of singlet ungerade states lying at excitation energies just above cm 1. There are two valence states involved, one of 1 Σ + u and one of 1 Π u symmetry (referred to as the b 1 Σ + u and b 1 Π u states), and there exist singlet Rydberg series converging on the first X 2 Σ + g ionization limit in the N + 2 ion (the npσ u c n+1 1 Σ + u and npπ u c 1 n Π u series; principal quantum number n 3) and the nsσ g o 1 n Π u series converging on the A 2 Π u ionization limit. The vibrational numbering in the o 1 3 Π u state, of relevance to the present study, was determined by Ogawa et al. [46]. In the seminal paper by Stahel et al. [23], a model of Rydberg-valence interactions was presented that provides a quantitative explanation for the energy-level perturbations, the seemingly erratic behaviour of the rotational constants, and the observed pattern of band intensities which deviate strongly from Franck-Condon-factor predictions. A comprehensive ab initio study by Spelsberg and Meyer [24] later confirmed the main conclusions of the Stahel et al. [23] model. In addition to these homogeneous perturbations in which states of like symmetry are coupled, the effects of heterogeneous perturbations, i.e., coupling between states of 1 Π u and 1 Σ + u symmetry, were also included in subsequent analyses [25], thereby improving the quantitative agreement between theory and experiment. Recently, the inclusion of spin-orbit interactions between the 1 Π u and 3 Π u states in a coupled-channel model of the Rydberg-valence interactions has allowed the complex isotopic pattern of predissociation in the lower vibrational levels of the 1 Π u states to be explained [32], including rotational effects [47]. In addition to the comprehensive theoretical studies describing the overall excited-state structure for dipole-allowed transitions in N 2 [23, 24], several semiempirical local-perturbation analyses have been performed which focus on particular level crossings. Yoshino et al. [48] examined a number of such local perturbations, most of which had one 22

39 2. Quantum-interference effects on a Rydber-valence complex of N 2 of the o 1 3 Π u (v) or o 1 4 Π u (v) Rydberg states as a perturbation partner. Yoshino and Freeman [49] treated a multi-level local perturbation involving the Rydberg states c 5 1 Σ + u (v = 0) and c 1 5 Π u (v = 0) interacting with a number of valence states of 1 Π u and 1 Σ + u symmetry. A well-known perturbation, observed as a pronounced feature in N 2 spectra involving the c 4 1 Σ + u (v = 0) and b 1 Σ + u (v = 1) levels, was analysed by Yoshino and Tanaka, based on classical spectroscopic data [50], and later by Levelt and Ubachs, based on XUV-laser data [51]. In the 15 N 2 isotopomer, several local perturbation studies have also been performed, e.g., for the o 1 Π u (v = 0) b 1 Σ + u (v = 3) crossing [36]. (In the o 1 n Π u (v) series, the subscripts are commonly dropped from the state designation for n = 3.) Due to differing isotopic shifts in the Rydberg and the valence states, the accidental perturbations occur at different locations in the rovibronic structure of the three natural isotopomers of N 2. In this study, the o 1 Π u (v = 1) b 1 Π u (v = 9) Rydberg-valence complex of 14 N 2 is examined using two different experimental techniques, providing new and detailed information on the perturbed rotational structures, oscillator strengths, and predissociation linewidths. Rotational and deperturbation analyses are performed which correct previous misassignments [48] for transitions to the low-j levels of o(v = 1) and b(v = 9), and elucidate the quantum-interference effects occurring in oscillator strength, between these two electric-dipole-allowed transitions, and in linewidth, between these two predissociated levels. In addition to recording the b X(9, 0) and o X(1, 0) bands for the main 14 N 2 isotopomer, these bands were also investigated for 14 N 15 N and a rotational analysis performed. In the case of the mixed 14 N 15 N isotopomer, the rotational structure of each transition is unperturbed due to differing isotopic shifts. Since, for high J, the homogeneous perturbation complex of the two 1 Π u states undergoes a heterogeneous interaction with the b 1 Σ + u (v = 6) state, as already noticed in 14 N 2 by Yoshino et al. [48], the b (v = 6) level is also included in the present study for both 14 N 2 and 14 N 15 N. Christian Jungen has made significant contributions to the understanding of the excited states of N 2. In 1990, Huber and Jungen reported a high-resolution jet absorption study of N 2 in the region near 80 nm [52], unravelling the Rydberg structure and following the vibrational sequence of the b 1 Σ + u state even beyond the ionization potential. In 23

40 2. Quantum-interference effects on a Rydber-valence complex of N 2 a subsequent study [53], he was part of a team investigating the nf- Rydberg series in the last 6000 cm 1 below the ionization energy, based on high-resolution spectra of 14 N 2 and 15 N 2 recorded with the 10.6 m spectrograph at Ottawa and at the Photon Factory synchrotron facility in Tsukuba. Finally, this led to the development of a comprehensive multichannel quantum-defect analysis of the near-threshold spectrum of N 2 [54]. It is with great pleasure that we dedicate our present work to Dr Jungen. 2.2 Experiments Two distinct experimental techniques were employed in this work to study the interaction between the b(v = 9) and o(v = 1) levels of N 2. First, very-high-resolution laser-based ionization spectroscopy was used to determine the energy perturbations. Second, high-resolution synchrotron-based quantitative photoabsorption spectroscopy was used, primarily to study quantum-interference effects in the corresponding oscillator strengths. Laser-based 1 XUV + 1 UV two-photon ionization spectroscopy was employed to study the excitation spectrum of N 2, initially in the range λ = nm. Details of the experimental method, including a description of the lasers, vacuum setup, molecular beam configuration, time-of-flight (TOF) detection scheme and calibration procedures, have been given previously [36]. Two different laser systems were used, a pulsed dye laser (PDL)-based source, delivering an XUV bandwidth of 0.3 cm 1 full-width at half-maximum (FWHM), and a pulsed dye amplifier (PDA)-based source, delivering an XUV-bandwidth of 0.01 cm 1 FWHM. The wavelength range was later extended to λ = nm, to also cover excitation of the b 1 Σ + u (v = 6) state, which was investigated under similar molecular-beam conditions using the PDL-based source. The PDL-based system and its application to the spectroscopy of N 2 has been described in Refs. [36, 55]. Briefly, the sixth harmonic of a pulsed dye laser was employed, calibrated against the reference standard provided by the Doppler-broadened linear absorption spectrum of molecular iodine [56]. The absolute wavenumber uncertainty in the XUV for this system is ± 0.1 cm 1 for fully resolved lines. The PDA-based system was used in a frequency-mixing scheme: 24

41 2. Quantum-interference effects on a Rydber-valence complex of N 2 ω XUV = 3(ω PDA + ω 532 ), where ω 532 is the frequency-doubled output of an injection-seeded Nd:YAG laser. It has been documented previously how this frequency-mixing scheme produces a bandwidth of 0.01 cm 1 FWHM in the XUV [57, 58]. The efficiency for XUV production with this mixing scheme is much lower than for the PDL-based system, and signal levels decrease accordingly. Even though the molecular-beam densities were increased to compensate, this scheme could only be used for calibration (and linewidth measurements) of a few low-j lines. In view of the lifetime broadening encountered in the b(v = 9) and o(v = 1) levels of 14 N 2 (see also [38]), the absolute accuracy of the corresponding energy calibration was limited to 0.05 cm 1. The two-photon-ionization TOF experiment has some characteristics that were favourably employed in the present study. Mass separation can be combined with laser excitation to separate the contributions to the spectrum of the main 14 N 2 isotopomer from those of the mixed 14 N 15 N species. Even in the case of natural nitrogen, which contains only 0.74% 14 N 15 N, resolved isotopic lines at the bandheads of the o(v = 1), b(v = 9) X(v = 0) systems could be observed. However, in a later stage of the experiments, isotopically-enriched 14 N 15 N gas became available, yielding better quality spectra for the b X(9, 0), o X(1, 0), and b X(6, 0) transitions that were used in the final analysis. Furthermore, by changing the relative delay between the N 2 pulsed-valve trigger and the laser pulse, as well as varying the nozzle-skimmer distance, the rotational temperature in the molecular beam could be selected to measure independent spectra of cold (10 20 K) and warm (up to 180 K) samples. These options helped greatly in the assignment of the spectral lines. Photoabsorption spectroscopy on the o(v = 1), b(v = 9) X(v = 0) systems of 14 N 2 was performed at the 2.5 GeV storage ring of the Photon Factory, a synchrotron radiation facility at the High Energy Accelerator Research Organization in Tsukuba, Japan. Details of the experimental procedure can be found in Stark et al. [42]. Briefly, a 6.65 m spectrometer with a 1200 grooves per mm grating (blazed at 550 nm and used in the sixth order) provided an instrumental resolving power of 150, 000, equivalent to a resolution of 0.7 cm 1 FWHM. The spectrometer tank, at a temperature of 295 K, served as an absorption cell with a path length of m. A flowing gas configuration was used: N 2 of normal isotopic composition entered the spectrometer tank through a 25

42 2. Quantum-interference effects on a Rydber-valence complex of N 2 needle valve and was continuously pumped by a 1500 ls 1 turbomolecular pump. Absorption spectra were recorded at tank pressures ranging from torr to torr, corresponding to N 2 column densities ranging from cm 2 to cm 2. The nm spectral region was scanned, in three overlapping portions, at a speed of nm min 1. A signal integration time of about 1 s resulted in one data point for each nm interval of the spectrum. Signal rates from the detector, a windowless solar-blind photomultiplier tube with a CsI-coated photocathode, were about s 1 for the background continuum; the detector dark count rate was less than 2 s 1. A signal-to-noise ratio of about 250 was typically achieved for the continuum level. The experimental absorption spectra were converted into photoabsorption cross sections through application of the Beer-Lambert law. Non-statistical uncertainties in the experimental cross sections are estimated to be 10%, with contributions from uncertainties in the N 2 column density, variations in the signal background, and scattered light. 2.3 Results and discussion Energies Using the PDL-based XUV source, rotationally-resolved spectra of the b 1 Π u X 1 Σ + g (9,0) and o 1 Π u X 1 Σ + g (1,0) bands in 14 N 2 were recorded for mass 28 in the TOF spectrum at low and high rotational temperatures, achieved by varying the timing of the N 2 supersonic beam. In Fig. 2.1, the two spectra are compared and line assignments shown. High-rotational-temperature laser-excitation spectra were also recorded for 14 N 15 N using a molecular beam of isotopically enriched nitrogen. Figure 2.2 shows such a recording, displaying lines from the b 1 Π u X 1 Σ + g (9,0) and o 1 Π u X 1 Σ + g (1,0) bands in 14 N 15 N. Using the narrower-bandwidth frequency-mixing PDA-based XUV source, spectra of the b X(9,0) and o X(1,0) bands in 14 N 2 were recorded for some low-rotational lines. Recordings of the Q(2) line from the b X(9,0) band and the Q(1) line from the o X(1,0) band in 14 N 2, obtained with this source, are shown in Fig Signal levels for the transition to the broader b(v = 9, J = 2) level are low because of the competition between predissociation and ionization in the two- 26

43 2. Quantum-interference effects on a Rydber-valence complex of N 2 R o(1) Q o(1) 2 5 P o(1) 6 R b(9) P b (6) Q b(9) 1 P b(9) * (a) (b) 14 N Transition Energy (cm 1 ) Figure 2.1: 1 XUV + 1 UV ionization spectra and line assignments for the b 1 Π u X 1 Σ + g (9,0) and o 1 Π u X 1 Σ + g (1,0) bands of 14 N 2 in high [spectrum (a)] and in low [spectrum (b)] rotational-temperature molecular beams, recorded using the PDL-based XUV source. Note that severe blending occurs in the spectrum and several weak lines are not identified. In the region in spectrum (a) marked with an asterisk, two separate scans are joined and intensities of the two scans should not be compared. photon ionization detection scheme and the rather low XUV and UV intensity levels available with the frequency-mixing scheme employed. Analysis of the linewidths and discussion of the predissociation phenomena in the excited states are deferred to Sec Transition energies and line assignments for the o(v = 1) and b(v = 9) levels (see below) are presented in Tables 2.1 and 2.2, for 14 N 2, and in Tables 2.3 and 2.4, for 14 N 15 N. An excitation spectrum for the b 1 Σ + u X 1 Σ + g (6,0) band of 14 N 2, which lies just above the overlapped b X(9,0) and o X(1,0) bands, is shown in Fig Assignment of the lines in this 1 Σ + u 1 Σ + g band, containing only P and R branches, is straightforward, but the rotational constants for the ground and excited states are such that, for J 9, the P (J ) and R(J + 3) lines overlap within the resolution of the experiment. Observed line positions for the b 1 Σ + u X 1 Σ + g (6,0) band are listed in Table 2.5 for 14 N 2, and Table 2.6 for 14 N 15 N. 27

44 2. Quantum-interference effects on a Rydber-valence complex of N 2 14 N 15 N o X(1,0) Q 17 P 16 R b X(9,0) R 1 Q P Transition Energy (cm 1 ) Figure 2.2: 1 XUV + 1 UV ionization spectrum and line assignments for the b 1 Π u X 1 Σ + g (9,0) and o 1 Π u X 1 Σ + g (1,0) bands of 14 N 15 N, recorded using the PDL-based XUV source. Q(1) o X(1,0) 14 N 2 Q(2) b X(9,0) 14 N Transition Energy (cm 1 ) Figure 2.3: 1 XUV + 1 UV ionization spectra of the Q(2) b 1 Π u X 1 Σ + g (9,0) and Q(1) o 1 Π u X 1 Σ + g (1,0) lines of 14 N 2, recorded using the PDA-based XUV source and showing the different predissociation broadening observed for b(v = 9) and o(v = 1). 28

45 2. Quantum-interference effects on a Rydber-valence complex of N R P 1 14 N2 b X(6,0) Transition Energy (cm 1 ) Figure 2.4: 1 XUV + 1 UV ionization spectrum and line assignments for the b 1 Σ + u X 1 Σ + g (6,0) band of 14 N 2, recorded using the PDL-based XUV source. Term values and spectroscopic parameters were determined from the experimental transition energies using least-squares fitting procedures. The terms of the ground state X 1 Σ + g were represented as F (J ) = B[J (J + 1)] D[J (J + 1)] 2 + H[J (J + 1)] 3, (2.1) where B is the rotational constant, and D and H are the centrifugal distortion parameters. The spectroscopic parameters of Trickl et al. [40] and Bendtsen [41] were used for the X 1 Σ + g states of 14 N 2 and 14 N 15 N, respectively. The terms of the excited 1 Π u states, where, for simplicity, we denote the rotational quantum number by J, rather than J (and the same holds for v and v ), were taken to have the form T f (J) = ν 0 + B[J(J + 1) 1] D[J(J + 1) 1] 2, (2.2) for the f-parity [11] levels, where ν 0 is the band origin, and T e (J) = T f (J) + T ef (J), (2.3) for the e-parity levels, where the Λ-doubling is represented by T ef (J) = q[j(j + 1) 1], (2.4) 29

46 2. Quantum-interference effects on a Rydber-valence complex of N 2 Table 2.1: Observed transition energies (in cm 1 ) for the b 1 Π u X 1 Σ + g (9,0) band in 14 N 2. Deviations from transition energies calculated using the corresponding fitted term values are also shown ( o c = obs calc). Wave numbers given to three decimal places are from narrow-bandwidth PDA spectra, those to two decimal places are from PDL spectra. Wave numbers derived from blended lines are flagged with an asterisk (*) and those from shoulders in the spectra by an s. J R(J ) o c Q(J ) o c P (J ) o c * s * * * * * * * * * * * * and q is the Λ-doubling parameter. Rotational energy levels in the b (v = 6) state, of 1 Σ + u (e) symmetry, were represented by T (J) = ν 0 + B[J(J + 1)] D[J(J + 1)] 2. (2.5) The b(v = 9) and o(v = 1) levels in 14 N 2 give rise to an avoided crossing in the rotational structure between J = 4 and J = 5, resulting from a J-independent homogeneous perturbation ( Ω = 0, electro- 30

47 2. Quantum-interference effects on a Rydber-valence complex of N Reduced term value (cm 1 ) o(1) deperturbed b(9) deperturbed b (6) deperturbed b(9) o(1) b (6) e levels 14 N J(J+1) Figure 2.5: Reduced term values of the b 1 Π u (v = 9) o 1 Π u (v = 1) b 1 Σ + u (v = 6) crossings in 14 N 2 (e-levels only), including the results of an effective three-state deperturbation. The terms are reduced such that the deperturbed b(v = 9) levels lie on the zero line. The solid symbols represent measured energy levels, the open symbols predicted levels N2 o(1) mixing factors o(1) b(9) b (6) J Figure 2.6: Calculated o(v = 1) mixing factors for the o 1 Π u (v = 1), b 1 Π u (v = 9), and b 1 Σ + u (v = 6) levels of 14 N 2. Plotted is the absolute square c i 2 of the coefficients c i representing the projection of the wave function Ψ o(1) on a basis of unperturbed wave functions (φ o(1), φ b(9), φ b (6)). The data pertain to the e-parity component only. 31

48 2. Quantum-interference effects on a Rydber-valence complex of N 2 static) between these valence and Rydberg states. Furthermore, at high J, interaction between the b (v = 6) levels, of 1 Σ + u symmetry, and neardegenerate o(v = 1) levels, of 1 Π u symmetry, becomes significant. This interaction is heterogeneous ( Ω 0, L-uncoupling) and only involves the e-parity levels of the o(v = 1) state. For the e-parity manifold, a full three-state deperturbation analysis was performed for each J value by diagonalizing the matrix T b9 H b9o1 0 H b9o1 T o1 H o1b 6 J(J + 1) 0 H o1b 6 J(J + 1) Tb 6, (2.6) where the diagonal elements are the term energies of b(v = 9), o(v = 1) and b (v = 6), given by Eqs. ( ). The off-diagonal element H b9o1 is the two-level homogeneous interaction parameter between the 1 Π u states, and H o1b 6 J(J + 1) represents the effective heterogeneous interaction matrix element between the o(v = 1) and b (v = 6) states. Since the b (v = 6) state has 1 Σ + u symmetry, and therefore only e-parity levels, in the case of the f-parity manifold Eq. (2.6) reduces to a 2 2 matrix. The resulting deperturbed spectroscopic parameters for o(v = 1), b(v = 9), and b (v = 6), obtained from a comprehensive least-squares fit for all available spectral lines pertaining to the three band systems, are listed in Table 2.7. In the fitting procedure, the nominal uncertainty in the absolute transition energy for fully-resolved lines of reasonable strength was set at 0.1 cm 1. In the case of weak or blended lines, the uncertainty was set at an estimated value in the range cm 1. The lines obtained with the PDA-based laser system have an uncertainty of 0.05 cm 1. In Fig. 2.5, reduced experimental term values are plotted for the b(v = 9), o(v = 1) and b (v = 6) levels in 14 N 2, based on the assignments to be discussed below. An avoided crossing between b(v = 9) and o(v = 1) is clearly visible between J = 4 and 5, with a maximum energy shift of 8.7 cm 1 at J = 4, as a result of the homogeneous interaction between the two levels. A second avoided crossing occurs between J = 24 and 25, associated with the heterogeneous coupling between o(v = 1) and b (v = 6), with a maximum shift of 13.5 cm 1 at J = 25. In Fig. 2.5, only the e-parity levels of both 1 Π u states are displayed. For the f- 32

49 2. Quantum-interference effects on a Rydber-valence complex of N 2 parity components, a similar graph can be constructed, but in this case there is no interaction with the b (v = 6) state. Note that the observed levels in Fig. 2.5 are represented by solid symbols, the predicted levels by open symbols. The mixing factors following from the diagonalization procedure using Eq. (2.6) are presented in Fig The present line assignments for the b(v = 9) o(v = 1) complex in 14 N 2 differ from those in the previous study of Yoshino et al. [48]. (The assignments for rotational lines accessing b(v = 9) given by Carroll and Collins [21] are erroneous, as already discussed by Yoshino et al. [48].) First, we have adopted a different convention for the labelling of the levels before the crossing (J = 1 4), assigning the b(v = 9) and o(v = 1) labels to the levels with the highest mixing factor for the corresponding nominal level. Thus, as shown in Fig. 2.5, for J-levels before the crossing, the higher term values belong to b(v = 9) and, after the crossing (J 5), to o(v = 1), while the reverse is true for the lower terms. Yoshino et al. [48], in contrast, assigned all the higher and lower term values as o(v = 1) and b(v = 9), respectively. Apart from this unimportant difference in nomenclature, we have made some significant reassignments. For J 5 in the b(v = 9) o(v = 1) complex, the assignments of the current and previous (Yoshino et al. [48]) studies agree, but, for lines accessing the J = 1 4 levels of the excited states, the assignments differ. These reassignments were facilitated in the present work by the recording of a number of spectra under differing experimental conditions. Separate PDL-based spectra for relatively low and high rotational temperatures aided considerably in the assignment of lines, which may be blended in one or other spectrum, to low or high J. Furthermore, the PDA-based measurements were performed with a rather low rotational temperature, so that only lines originating from J = 0 2 appeared strongly, with possible weak lines from J = 3 and 4. This conclusion was confirmed independently in measurements of the b X(10, 0) band in 14 N 2 [58], recorded under the same experimental conditions. Of course, the most convincing argument for the present line assignments and analysis is the reproducibility of all lines in the fitting procedure using Eqs. ( ). With 163 lines included in the fit, a χ 2 of 93 is found. The deperturbed rotational constant for 3sσ g o 1 Π u (v = 1) in Table 2.7, B = cm 1, is in agreement with expectation for a Ryd- 33

50 2. Quantum-interference effects on a Rydber-valence complex of N 2 Table 2.2: Observed transition energies (in cm 1 ) for the o 1 Π u X 1 Σ + g (1,0) band in 14 N 2. Deviations from transition energies calculated using the corresponding fitted term values are also shown ( o c = obs calc). Wave numbers given to three decimal places are from narrow-bandwidth PDA spectra, those to two decimal places are from PDL spectra. Wave numbers derived from blended lines are flagged with an asterisk (*) and those from shoulders in the spectra by an s. J R(J ) o c Q(J ) o c P (J ) o c * * * * * * * * * s * * * * * * * * * * berg state converging on the A 2 Π u ionic state [B(A, v + = 1) = cm 1 [59]], with the residual difference occurring because the effects of 34

51 2. Quantum-interference effects on a Rydber-valence complex of N 2 Table 2.3: Observed transition energies (in cm 1 ) for the b 1 Π u X 1 Σ + g (9,0) band in 14 N 15 N, from spectra obtained with the PDL-based XUV source. Deviations from transition energies calculated using the corresponding fitted term values are also shown ( o c = obs calc). Wave numbers derived from blended lines are flagged with an asterisk (*) and those from shoulders in the spectra by an s. J R(J ) o c Q(J ) o c P (J ) o c s s s * * * * * * * perturbations by more remote levels are still included in the two-leveldeperturbed value. The deperturbed rotational constant for b(v = 9) is significantly lower, B = cm 1, appropriate for a valence b 1 Π u state. The deperturbed D value for b(v = 9) in Table 2.7 is small and negative, comparable to the value D = cm 1 for 15 N 2 reported in Ref. [36]. This negative D value is supported by the observations of Yoshino et al. [48], who find some even higher-j P - and Q-branch lines accessing b(v = 9), up to J = 28, which exhibit a further gradual shift upward in energy, by up to several cm 1, for both e- and f-parity components. It is likely that the negative D value for b(v = 9) is a result simply of the multilevel perturbation by more distant levels. The new assignments for the lowest-j levels of b(v = 9) and o(v = 1) in 14 N 2 indicate a greater separation in energy than in Ref. [48] (espe- 35

52 2. Quantum-interference effects on a Rydber-valence complex of N 2 Table 2.4: Observed transition energies (in cm 1 ) for the o 1 Π u X 1 Σ + g (1,0) band in 14 N 15 N, from spectra obtained with the PDL-based XUV source. Deviations from transition energies calculated using the corresponding fitted term values are also shown ( o c = obs calc). Wave numbers derived from blended lines are flagged with an asterisk (*) and those from shoulders in the spectra by an s. J R(J ) o c Q(J ) o c P (J ) o c * * * * * * * * * * * * * * * * * * * s * * s * * * * * * * * * s * * * s * 0.01 cially for J = 1f where the separation is larger by 3.4 cm 1 ) resulting in significant differences between the two-level-deperturbed parameters in Table 2.7 and previous values [48]. For example, the fitted homogeneous interaction matrix element, H b9o1 = 9.47 cm 1, should be compared with the previously accepted value of 8.10 cm 1 [48]. It has been pointed out elsewhere [32] that it was the incorrect use of the deperturbed experimental b(v = 9) and o(v = 1) spectroscopic parameters [21, 48] in comparisons with the theoretical results of Stahel et al. [23] and Spels- 36

53 2. Quantum-interference effects on a Rydber-valence complex of N 2 Table 2.5: Observed transition energies (in cm 1 ) for the b 1 Σ + u X 1 Σ + g (6,0) band of 14 N 2, from spectra obtained using the PDL-based XUV source. Deviations from transition energies calculated using the corresponding fitted term values are also shown ( o c = obs calc). Wave numbers derived from blended lines are flagged with an asterisk (*). J R(J ) o c P (J ) o c a a a a a * a * a * * * * * * * * * * * * * * 0.16 a Data taken from PDA-based measurements [58]. berg and Meyer [24] which led to the large discrepancies, especially in B, for these particular levels. Evidently, the present reassignments will also have some effect in resolving those discrepancies. 37

54 2. Quantum-interference effects on a Rydber-valence complex of N 2 Table 2.6: Observed transition energies (in cm 1 ) for the b 1 Σ + u X 1 Σ + g (6,0) band of 14 N 15 N, from spectra obtained using the PDL-based XUV source. Deviations from transition energies calculated using the corresponding fitted term values are also shown ( o c = obs calc). Wave numbers derived from blended lines are flagged with an asterisk (*). J R(J ) o c P (J ) o c * * * * * * * * * * * * * * * * * * * * * * The major contribution to the splitting between the e- and f-parity levels of o(v = 1) is due to the heterogeneous interaction with the b 1 Σ + u (v = 6) state, as already discussed by Yoshino et al. [48]. Rotational levels in the higher-lying b (v = 6) valence state approach the Rydberg o(v = 1) levels from above with increasing J, since b (v = 6) has a smaller B value than o(v = 1). This is illustrated in Fig For levels with J 20, the e-parity components of o(v = 1) are pushed down in energy, while the f-parity levels are unaffected. The R(24) line from the o X(1, 0) band, predicted from the perturbation analysis to 38

55 2. Quantum-interference effects on a Rydber-valence complex of N 2 lie near cm 1, was found in the experimental spectrum as a weak satellite, thereby pinpointing the culmination of the heterogeneous interaction. The R(22) line of o X(1, 0) is predicted to coincide with the P (7) line in b X(9, 0), while R(21) would coincide with Q(13) of o X(1, 0). R(23) is probably too weak to be observed under our experimental conditions. Unfortunately, lines accessing the b (v = 6) perturbation partner could only be followed up to R(21) and P (22), below the crossing point. We note that the P (26) line in o X(1, 0), reported by Yoshino et al. [48], does not match the present analysis and is reassigned as the Q(19) line of b X(9, 0). Despite the somewhat incomplete nature of the experimental data defining the b (v = 6) o(v = 1, e) crossing, it has been possible to determine a heterogeneous interaction matrix element H o1b 6 = cm 1, which is of a similar order to those found by Yoshino et al. [48] for crossings involving other vibrational levels of the same states: H o0b 3 = 0.29 cm 1, H o3b 11 = 0.44 cm 1, and H o4b 14 = 0.60 cm 1. The three-level-deperturbed Λ-doubling parameters q given in Table 2.7 for the o(v = 1) and b(v = 9) states, which are negative and positive, respectively, agree in sign with those determined for 15 N 2 by Sprengers et al. [36]. These residual Λ-doublings are likely caused, directly or indirectly, by heterogeneous interactions between o(v = 1) and other more remote levels of the b 1 Σ + u perturber. In 14 N 15 N, the b(v = 9) level shifts further down in energy than the o(v = 1) level following isotopic substitution. Therefore, the order of the two states is reversed and the lower-lying b(v = 9) level, with the smaller B value, does not cross the higher-lying o(v = 1). The same is true for 15 N 2 [36]. As a consequence, in the fitting procedure for 14 N 15 N it was not possible to determine a meaningful homogeneous interaction parameter H b9o1, which was therefore set to zero. Furthermore, due to the inferior signal-to-noise ratio for the 14 N 15 N spectra, the highest-j levels could not be observed, making it difficult to define the inherent Λ-doubling parameters for both the b(v = 9) and o(v = 1) states. Accordingly, these parameters were also set to zero, as including them in the fit of 132 lines did not improve the χ 2 value of 93. Although it was not possible experimentally to follow the b (v = 6) and o(v = 1) levels of 14 N 15 N to high enough J values to define the expected avoidedcrossing region, a significant reduction in χ 2 was obtained by including the corresponding heterogeneous interaction parameter in the fitting 39

56 2. Quantum-interference effects on a Rydber-valence complex of N 2 Table 2.7: Molecular parameters for the b 1 Π u (v = 9), o 1 Π u (v = 1) and the b 1 Σ + u (v = 6) states of 14 N 2 and 14 N 15 N. All values are in cm 1. 1σ statistical uncertainties, resulting from the fit, are shown in parentheses, in units of the last significant figure. Additional systematic uncertainties of order 0.05 cm 1 apply to the band origins ν 0. b 1 Π u (v = 9) a o 1 Π u (v = 1) a b 1 Σ + u (v = 6) a Species: 14 N 2 B (4) (3) (1) D (8) 6.9(5) 51.8(3) q (13) 0.81(10) ν (4) (3) (1) H b9o1 9.47(1) H o1b (4) Species: 14 N 15 N B 1.199(1) (6) (4) D (6) 2(2) 56(2) q b 0 b ν (4) (4) (3) H b9o1 0 b H o1b (3) a Deperturbed parameters. b Fixed parameter. No statistically significant value found. procedure, principally due to its ability to reproduce the o(v = 1)-state Λ-doubling observed experimentally. The fitted spectroscopic parameters for 14 N 15 N are given in Table 2.7. In the case of the heterogeneous interaction between b (v = 6) and o(v = 1, e), the interaction parameter, H o1b 6 = 0.67 cm 1, is of the same order as the value obtained for 14 N 2. A crossing of the b (v = 6) and o(v = 1, e) levels between J = 24 and 25 is predicted by the fit, with a maximum shift of 13 cm 1 at J = 24. The B values given in Table 2.7 are slightly lower in the mixed isotopomer due to the greater reduced mass. Oscillator strengths The Photon Factory experimental photoabsorption cross sections were analyzed using a least-squares fitting procedure in which each line was 40

57 2. Quantum-interference effects on a Rydber-valence complex of N 2 represented by a Voigt profile and account was taken of the effects of the finite experimental resolution. The line oscillator strength and the Lorentzian width component, corresponding to the predissociation linewidth, for each Voigt line were parameters of the fit, while the Gaussian width component was fixed at the room-temperature Doppler width of 0.24 cm 1 FWHM. The instrumental function was also defined by a Voigt profile, with Gaussian and Lorentzian width components of 0.60 and 0.20 cm 1 FWHM, respectively, determined by analyzing scans over the almost pure Doppler lines from the c 4 1 Σ + u X 1 Σ + g (0, 0) band [60]. The Voigt-model cross section was convolved, in the transmission domain, with the instrumental function, and compared iteratively with the experimental cross section. In the case of weak lines, it was not possible to independently determine the predissociation linewidths, which were fixed at realistic values interpolated from other known widths (see Sec. 2.3), but, in any case, the fitted oscillator strengths were not very sensitive to these adopted linewidths. In the case of overlapping lines, generally the line-strength ratios were fixed at the values expected from Hönl-London- and Boltzmann-factor considerations and an average oscillator strength determined. The fitting procedure is illustrated in Fig In this example, it was possible to determine independent oscillator strengths for the stronger, partially-overlapped P (11) and Q(16) lines from the o X(1, 0) band, but only a common predissociation linewidth. For the very weak P (8) line from the b X(9, 0) band, however, only the oscillator strength could be determined. The fitted line oscillator strengths were converted into band oscillator strengths by dividing by appropriately normalized 1 Π 1 Σ Hönl- London factors and fractional initial-state populations, the latter determined from T = 295 K Boltzmann factors based on the N 2 ground-state term values, and taking into account the 2:1 rotational intensity alternation caused by nuclear-spin effects. No significant systematic differences were found between the P -, Q-, or R-branch band oscillator strengths, for either the b X(9, 0) or o X(1, 0) transitions, over the range of rotation studied, J 20. The overall results, summarized in Table 2.8, and illustrated in Fig. 2.8 (circles) represent weighted means over each available branch. Inspection of Fig. 2.8 reveals an interesting effect: transitions to the higher-energy levels [b(v = 9) for J 4, o(v = 1) for J 5; solid 41

58 2. Quantum-interference effects on a Rydber-valence complex of N Cross Section (10 16 cm 2 ) o -X (1,0) P (11) Q (16) b -X (9,0) P (8) Transition Energy (cm 1 ) Figure 2.7: Experimental room-temperature photoabsorption cross section for a region of the overlapping b X(9, 0) and o X(1, 0) bands of 14 N 2 (open circles), together with the fitted cross section (solid curve). The P (8) line from the b X(9, 0) band, probing J = 7, is anomalously weak (see text). circles] peak in strength near J = 6, while those to the lower-energy levels [o(v = 1) for J 4, b(v = 9) for J 5; open circles] decrease rapidly in intensity in the same region, with transitions to J = 6 too weak to be observed. This is a classic example of a two-level quantummechanical interference effect, in the case where transitions to both levels, of like symmetry, carry an oscillator strength, as discussed in detail by Lefebvre-Brion and Field [11]. The perturbed vibronic oscillator strengths for transitions from a common level 0 to the upper (+) and lower ( ) levels of the interacting pair, respectively, are given by [11]: f +0 = c 2 f 10 + (1 c 2 )f 20 ± 2c (1 c 2 )f 10 f 20, f 0 = (1 c 2 )f 10 + c 2 f 20 2c (1 c 2 )f 10 f 20, (2.7) where f 10 and f 20 are the vibronic oscillator strengths for transitions to the unperturbed levels, 1 and 2, and c > 0 is the mixing coefficient corresponding to the presence of the unperturbed level 1 in the perturbed upper-level wave function. The sense of the interference effect, corresponding to the signs of the right-hand terms in Eq. (2.7), depends on 42

59 2. Quantum-interference effects on a Rydber-valence complex of N 2 Table 2.8: Experimental (perturbed) band oscillator strengths f for the b X(9, 0) and o X(1, 0) transitions of 14 N 2. Values flagged with an asterisk ( ) are derived only from blended lines. 1σ statistical uncertainties are shown in parentheses, in units of the last significant figure. Additional systematic uncertainties of 10% are applicable. J f b X(9,0) f o X(1,0) (8)* (6)* (3)* (2)* (3) (3)* (2)* (2)* (2)* (3) (2) (1) (3) (1) (2) (2) (3) (1) (2) (1)* (2) (2) (3) (1) (2) (1) (2) (2) (2) (1) (2)* (3) (4) (2) (3) (3) (5) (5) (5)* the sign of µ 10 µ 20 H 12, where the µ i0 are vibronic transition moments (f i0 µ 2 i0 ) and H 12 is the coupling matrix element [11]. The present case, where the higher-energy transitions are enhanced in strength by the interference effect, corresponds to µ bx(9,0) µ ox(1,0) H b9o1 > 0. Using the f-level mixing coefficients determined as part of the energy-level deperturbation procedure described in Sec. 2.3, together with the experimental (perturbed) oscillator strengths from Table 2.8, we have successfully deperturbed the oscillator strengths through the application of Eq. (2.7). In contrast to the perturbed case, the deperturbed oscillator strengths display only slight J dependences, yielding the following fits: f bx(9,0) = (2) 4.2(9) 10 6 J(J + 1) (dashed 43

60 2. Quantum-interference effects on a Rydber-valence complex of N Oscillator Strength J(J+1) Figure 2.8: Rotational [J(= J )] dependence of band oscillator strengths in the mixed b X(9, 0) and o X(1, 0) transitions of 14 N 2, demonstrating a strong quantum-interference effect near J = 6, together with the results of a deperturbation analysis (see text). Solid circles: experimental (perturbed) oscillator strengths for transitions to the higher-energy levels for a given J. Open circles: experimental oscillator strengths for transitions to the lowerenergy levels. Solid line: deperturbed oscillator strength for the o X(1, 0) transition. Dashed line: deperturbed oscillator strength for the b X(9, 0) transition. Solid curve: calculated perturbed oscillator strength for the higher levels. Dashed curve: calculated perturbed oscillator strength for the lower levels. Open squares: sum of experimental oscillator strengths for a given J. Long-dashed line: sum of calculated perturbed (or deperturbed) oscillator strengths. line in Fig. 2.8), f ox(1,0) = (3) + 1.1(11) 10 6 J(J + 1) (solid line in Fig. 2.8). Back-generating the perturbed oscillator strengths from these fits yields the results shown in Fig. 2.8 for the upper and lower levels (solid curve, dashed curve, respectively), which are seen to be in excellent agreement with the measurements. Furthermore, the summed deperturbed (or perturbed) fitted oscillator strengths (long-dashed line in Fig. 2.8) are in good agreement with the summed experimental values (open squares), neither exhibiting any strong J dependence, as would be expected in the case of the simple two-level interaction described by Eq. (2.7). While the b(v = 9) o(v = 1) crossing occurs between J = 4 5, the intensity minimum for transitions to the lower of the 44

61 2. Quantum-interference effects on a Rydber-valence complex of N 2 perturbed levels appears to occur just below J = 6. This difference is a consequence of the deperturbed f ox(1,0) significantly exceeding f bx(9,0). From Eq. (2.7), only in the specific case f 10 = f 20 will an intensity zero occur at the level crossing (c = 1/ 2). These results confirm that the b(v = 9) o(v = 1) level crossing in 14 N 2 produces a simple two-level quantum-mechanical interference effect in the corresponding perturbed oscillator strengths for transitions from the ground state, yielding a virtually complete destructive interference for transitions to the lower level with J = 6 [principally of b(v = 9) character]. As will be seen in Sec. 2.3, the situation for the corresponding predissociation linewidths is somewhat more complicated. Finally, we note that an ostensibly similar deep minimum in oscillator strength has been reported in 14 N 2 for transitions to b(v = 8) near J = 12 [47]. This latter case is distinguished, however, by no level crossing being involved, the destructive interference effect being a multilevel phenomenon. Predissociation linewidths Lorentzian linewidth components for individual rotational levels of b(v = 9) and o(v = 1) in 14 N 2, dominated by the contribution of predissociation over radiation, were determined from both the PDA laser-based ionization spectra and the Photon Factory photoabsorption spectra. Of course, the PDA-based system was to be preferred for the measurement of linewidths because of its superior resolution and Doppler-reduced character, but, in practice, these advantages applied only at low J in the cold spectra, since the room-temperature background gas contribution dominated the jet contribution for higher J. On the other hand, the high signal-to-noise ratio of the photoabsorption spectra made them suitable for the determination of Lorentzian linewidth components with Γ 0.15 cm 1 FWHM, albeit with significant uncertainty, even with an experimental resolution of only 0.7 cm 1 FWHM. Predissociation linewidths, determined from the PDA-based spectra by deconvolving the instrument width from the observed widths as explained in Ref. [34], and from the photoabsorption spectra, simultaneously with the oscillator strengths as described in Sec. 2.3, are summarised in Table 2.9 and Fig Since no significant systematic e/f parity dependence was found for either the b(v = 9) or o(v = 1) level widths, the val- 45

62 2. Quantum-interference effects on a Rydber-valence complex of N 2 ues presented represent weighted means over determinations from each available branch. The PDA-based system was used only to determine level widths for b(v = 9, J = 1 2) and o(v = 1, J = 1 3), i.e., restricted to J-levels below the crossing. The average predissociation linewidths in this region, Γ b(v=9) 0.24 cm 1 FWHM and Γ o(v=1) 0.05 cm 1 FWHM, differ substantially (see also Fig. 2.3). They are equivalent to lifetimes τ = 1/2πΓ of 23(5) ps and 105(30) ps, for b(v = 9) and o(v = 1), respectively, which agree with previous time-domain pump-probe lifetime measurements [38]. In Ref. [38], the rotational structure of the b(v = 9) o(v = 1) complex was not resolved and the lifetimes observed, determined mainly by predissociation, varied from < 50 ps to 110 ps. From the present study, it can be deduced unambiguously that the < 50-ps component arose from b(v = 9) and the 110-ps component from o(v = 1). The predissociation lifetimes of these two levels in 15 N 2 have been determined with the same frequency-mixing PDA-based XUV source in Ref. [58]. For b(v = 9) in 15 N 2, an f-parity lifetime of 46(7) ps was found, a factor of two higher than in 14 N 2. The lifetime of o(v = 1) is also isotope dependent: a lifetime of 27(6) ps was obtained for this level in 15 N 2 [58], a factor of four lower than in 14 N 2. All data pertaining to 14 N 15 N were obtained here with the PDL-based XUV source and do not, therefore, yield reliable information on the excited-state lifetimes. While it is not possible unambiguously to detect any systematic J dependence in the PDA linewidths, when taken together with the Photon Factory results, it is evident from Fig. 2.9, despite considerable uncertainty in the data, that the width of the lower-energy levels (open circles) increases substantially as J increases, while the width of the higher-energy levels (solid circles) decreases overall, exhibiting a maximum near J = 6. Because of line overlap and the weakness of the corresponding transitions (see Fig. 2.8), we have been unable to determine widths for the lower-energy levels with J = 4 8. In the case of a simple, two-level interaction where both (unperturbed) levels predissociate via the same route, i.e., coherently, a quantum-interference effect in the widths would be expected, analogous to that observed for the corresponding oscillator strengths. In this case, the width interference is described by expressions similar to Eq. (2.7), with the widths Γ replacing the oscillator strengths f [11]. The width 46

63 2. Quantum-interference effects on a Rydber-valence complex of N 2 Lorentz Linewidth (cm 1 FWHM) J(J+1) Figure 2.9: Rotational dependence of predissociation linewidths for the mixed b(v = 9) and o(v = 1) states of 14 N 2, together with the results of a deperturbation analysis (see text). Solid circles: experimental (perturbed) widths for the higher-energy levels. Open circles: experimental widths for the lower-energy levels. Solid line: deperturbed o(v = 1) widths. Dashed line: deperturbed b(v = 9) widths. Solid curve: calculated perturbed widths for the higher levels. Dashed curve: calculated perturbed widths for the lower levels. Open squares: sum of experimental widths for a given J. Long-dashed line: sum of calculated perturbed (or deperturbed) widths. maximum observed near J = 6 for the upper-energy levels in Fig. 2.9 supports the notion of a width quantum-interference effect associated with the b(v = 9) o(v = 1) level crossing. However, despite an inability to fully monitor the lower-energy widths in this region, there is no rapid decrease in the J = 1 3 level widths indicated by the PDA results, contrary to the case of the oscillator strengths in Fig Thus, while there may be a minimum in the lower-energy widths near the crossing region, that minimum is unlikely to be as deep as the effectively zero value exhibited by the corresponding oscillator strengths, and required for a coherent two-level interaction. An initial attempt to deperturb the level widths in Fig. 2.9, using the width analogue of Eq. (2.7), together with the same mixing coefficients used for the oscillator-strength deperturbation, failed because such a 47

64 2. Quantum-interference effects on a Rydber-valence complex of N 2 Table 2.9: Experimental (perturbed) predissociation linewidths Γ (in cm 1 FWHM) for the b 1 Π u (v = 9, J) and o 1 Π u (v = 1, J) levels of 14 N 2. Values flagged with an asterisk ( ) are derived only from blended lines. 1σ statistical uncertainties are shown in parentheses, in units of the last significant figure. Additional systematic uncertainties of 25% are applicable to the non-pda measurements. J Γ b(v=9) Γ o(v=1) (4) a 0.05(1) a (2) a 0.06(1) a (4) 0.04(1) a (2)* (2) (2) (2) (2) (5) 0.21(2) (2) 0.21(2) (4)* 0.22(2) (3) 0.15(2) (2) 0.18(2) (3) 0.18(2) (3) 0.14(2) (2) 0.22(2)* (5) 0.14(3) (4) 0.15(2) (5) 0.12(4) a PDA measurements totally destructive quantum interference for the lower-energy levels was incompatible with the experimental widths. Thus, the implied nonzero width minimum for the lower-energy levels suggests that an additional, incoherent, and thus additive, width component is applicable to these levels. If a J-independent value of 0.03 cm 1 FWHM is assumed for this component, together with coherent two-level interference for the remainder, then the realistic deperturbation shown in Fig. 2.9 results, yielding the following fits: Γ b(v=9) = 0.17(2)+2.3(8) 10 4 J(J +1) cm 1 FWHM (dashed line in Fig. 2.9), including the incoherent component, and Γ o(v=1) = 0.15(1) 1.0(4) 10 4 J(J + 1) cm 1 FWHM (solid 48

65 2. Quantum-interference effects on a Rydber-valence complex of N 2 line in Fig. 2.9). Back-generating the perturbed widths from these fits yields the results shown in Fig. 2.9 for the upper and lower levels (solid curve, dashed curve, respectively), which, as for the oscillator strengths, are in excellent agreement with the experimental measurements. The summed deperturbed (or perturbed) fitted widths (long-dashed line in Fig. 2.9) are in good agreement with the summed experimental values (open squares), neither exhibiting any strong J dependence. It is of interest that the two-level-deperturbed low-j predissociation lifetimes implied by the above linewidth analysis, 31(4) ps and 35(3) ps, for b(v = 9) and o(v = 1), respectively, are in much better agreement with the 15 N 2 experimental lifetimes, 46(7) ps and 27(6) ps [58], than are the perturbed lifetimes. This indicates that the bulk of the large experimental isotope effect is caused simply by the interference effect associated with the two-level crossing in 14 N 2. According to Lewis et al. [32], the lowest 1 Π u states of N 2 are predissociated, ultimately, by the C 3 Π u state, which correlates with the 4 S + 2 D dissociation limit at cm 1. On this basis, the fact that the higher-energy levels are enhanced in width by the interference effect indicates that the matrix-element product H b9c H o1c H b9o1 > 0, where the bound-free vibronic elements relate to the unperturbed widths (Γ i HiC 2 ). The question remains as to the source of the incoherent contribution to the b(v = 9) widths. We propose here that it is the interaction with a repulsive state, namely the second valence state of 3 Σ + u symmetry, labelled 3 3 Σ + u by Minaev et al. [61], which provides the additional incoherent predissociation of the b 1 Π u state. The potentialenergy curve for the 3 3 Σ + u state, which also correlates with the 4 S + 2 D limit, crosses that of the b 1 Π u state on its outer limb, near R = 1.75 Å, at an energy of cm 1 [62]. Thus, it is energetically possible for the 3 3 Σ + u state to predissociate b 1 Π u (v = 9), which lies near cm 1 (see Table 2.7). Furthermore, the spin-orbit coupling ( Ω = 0) between the b 1 Π 1u and 3 3 Σ + 1u states has been estimated ab initio to be on the order of 7 cm 1 [63], making this the likely incoherent predissociation mechanism supplying the additional 0.03 cm 1 FWHM width to b(v = 9). 49

66 2. Quantum-interference effects on a Rydber-valence complex of N Summary and conclusions Two distinct high-resolution experimental techniques, 1 XUV + 1 UV laser-based ionization spectroscopy and synchrotron-based XUV photoabsorption spectroscopy, have been used to study the o 1 Π u (v = 1) b 1 Π u (v = 9) Rydberg-valence complex of 14 N 2, providing new and detailed information on the perturbed rotational structures, oscillator strengths, and predissociation linewidths. Ionization spectra probing the b 1 Σ + u (v = 6) state of 14 N 2, which crosses o 1 Π u (v = 1) between J = 24 and J = 25, and the o 1 Π u (v = 1), b 1 Π u (v = 9), and b 1 Σ + u (v = 6) states of 14 N 15 N, have also been recorded. In the case of 14 N 2, rotational and deperturbation analyses correct previous misassignments for the low-j levels of o(v = 1) and b(v = 9). In addition, a two-level quantummechanical interference effect has been found between the o X(1, 0) and b X(9, 0) transition amplitudes which is totally destructive for the lower-energy levels just above the level crossing, making it impossible to observe transitions to b(v = 9, J = 6). A similar interference effect is found to affect the o(v = 1) and b(v = 9) predissociation linewidths, but, in this case, a small non-interfering component of the b(v = 9) linewidth is indicated, attributed to an additional spin-orbit predissociation by the repulsive 3 3 Σ + u state. The experimental linewidths, together with the corresponding interference effect, will provide a challenging test for coupled-channel models of the predissociation dynamics for the 1 Π u states of N 2. Acknowledgment The authors gratefully acknowledge the support provided for this research by the following bodies: the Molecular Atmospheric Physics (MAP) Program of the Netherlands Foundation for Fundamental Research on Matter (FOM); the Australian Research Council s Discovery Program (project number DP ); and NASA (grant NNG05GA03G). The assistance of the staff of the Photon Factory is also acknowledged. 50

67 Chapter 3 Interactions of the 3pπ u c 1 Π u (v = 2) Rydberg-complex in N 2 The 3pπ u c 1 Π u X 1 Σ + g (2, 0) Rydberg and b 1 Σ + u X 1 Σ + g (7, 0) valence transitions of 14 N 2, 14 N 15 N, and 15 N 2 are studied using laser-based 1 XUV + 1 UV two-photon-ionization spectroscopy, supplemented by synchrotron-based photoabsorption measurements in the case of 14 N 2. For each isotopomer, effective rotational interactions between the c(v = 2) and b (v = 7) levels are found to cause strong Λ-doubling in c(v = 2), and dramatic P/R-branch intensity anomalies in the b X(7, 0) band due to the effects of quantum interference. Local perturbations in energy and predissociation line width for the c(v = 2) Rydberg level are observed and attributed to a spin-orbit interaction with the crossing, short-lived C 3 Π u (v = 17) valence level.

68 3. Interactions of the 3pπ u c 1 Π u (v = 2) Rydberg-complex in N Introduction The level structure of the nitrogen molecule in the region of its dipoleallowed absorption spectrum (i.e., above cm 1 ) is one of severe complexity. First, the singlet states that can be accessed directly in the extreme ultraviolet (XUV) region via allowed transitions behave seemingly erratically. Numerous investigations have been performed to unravel these singlet structures. Through the semiempirical work of Stahel et al.,[23] the lowest-energy dipole-allowed spectrum of N 2 has been explained by considering the homogeneous electrostatic interactions between the Rydberg and valence states of 1 Σ + u and 1 Π u symmetry. In particular, the corresponding calculated vibronic band strengths[23] showed strong quantum-interference effects. Later, a quantitatively-improved model of the N 2 spectrum, based on ab initio calculations, was put forward by Spelsberg and Meyer.[24] Edwards et al.[25] extended the description by incorporating heterogeneous, rotationally-dependent interactions in the coupling model. Nevertheless, there remain many details in the structure of the singlet states to be discovered and explained. The theoretical and experimental study of rotationally-dependent perturbations in the Rydberg-valence singlet manifold of N 2 and its isotopomers ( 15 N 2 and 14 N 15 N) has been a topic of recent interest. Strong local perturbations can give rise, not only to level shifts, but also changes in intensities. By analogy with the vibronic case, such perturbations can cause rotational quantum-interference effects, leading, in some cases, to complete damping of intensity.[39] Second, the singlet states show predissociation behavior that is also erratic. Four decades ago, Carroll and Collins[21] noted that, of the vibrational levels of the b 1 Π u valence state, only v = 1, 5 and 6 could be observed in emission. The predissociation rates pertaining to the b 1 Π u (v) levels have been determined experimentally from line-width studies employing either synchrotron-radiation[42] or narrow-bandwidth XUV laser-radiation[29, 64] sources. The XUV laser technique is especially suitable for the determination of excited-state lifetimes with rotational resolution. In addition, complementary pump-probe experiments employing picosecond lasers have been carried out to determine lifetimes experimentally.[65] Numerous additional investigations have been performed in order to produce a quantitative experimental database for the 52

69 3. Interactions of the 3pπ u c 1 Π u (v = 2) Rydberg-complex in N 2 predissociation rates of the singlet states of N 2, including translationalspectroscopic studies,[66, 31] and investigations focusing specifically on the 15 N 2 and 14 N 15 N isotopomers.[36] Knowledge of the N 2 predissociation behaviour and mechanism is a prerequisite for a detailed understanding of radiative-transfer processes, as well as stratospheric chemistry, in nitrogen-rich planetary atmospheres such as those of the Earth and Titan. Over the years, a large effort has been invested towards an understanding of the origin of the predissociation in the lowest-lying electricdipole-accessible singlet states of N 2.[26] Since the singlet states couple via homogeneous spin-orbit interaction to the C 3 Π u state, which in turn is coupled to the C 3 Π u state above its dissociation limit, a detailed understanding of the predissociation of the singlet states requires consideration of the interactions within the triplet manifold. A recent study by Lewis et al.,[32] based on a coupled-channel Schrödinger equation (CSE) model for the interacting 1 Π u (b, c, o) and 3 Π u (C, C ) states, has provided a quantitative explanation for the predissociation mechanism of the 1 Π u states in the energy region below cm 1. This model achieves spectroscopic accuracy and builds upon the information gathered over the years concerning low-lying vibrational levels of the C 3 Π u and C 3 Π u states, starting with the benchmark study by Carroll and Mulliken.[67] This triplet data is reviewed in Ref. [32]. Additional information beyond C 3 Π u (v = 5) and C 3 Π u (v = 2) has been obtained indirectly by examining the predissociative effect of these states on the singlet manifold, guided by ab initio potential curves and an extrapolation from the known lower levels. The CSE model is capable of explaining the predissociation rates for b 1 Π u (v 6), with rotational resolution, for the three isotopomers 15 N 2, 14 N 15 N and 14 N 2.[47] In particular, the b 1 Π u (v = 1) level of 14 N 2 provides a good test for the model. For low J, the predissociation rate is very small and decay is mainly radiative, with an exceptionally long lifetime.[68] The predicted predissociation rate increases steeply with J, in good agreement with experiment.[33] In order to extend our understanding of the predissociation phenomenon in N 2 into the excitation region above cm 1, significant CSE-model development is necessary, facilitated by new experimental measurements. In particular, this requires knowledge of the locations of 53

70 3. Interactions of the 3pπ u c 1 Π u (v = 2) Rydberg-complex in N 2 higher vibrational levels of the C 3 Π u valence state, together with a consideration of the other triplet Rydberg states, F 3 Π u,[35] and G 3 Π u,[69] that play the role of inducing further predissociation through their interactions with the C 3 Π u and C 3 Π u states and direct spin-orbit coupling to the 1 Π u states.[70] Electric-dipole transitions from the X 1 Σ + g ground state of N 2 to the triplet manifold are spin-forbidden, thus limiting the information that can be obtained directly on these states using normal spectroscopic techniques. However, spin-orbit interactions cause coupling between the singlet and triplet manifolds, leading to local perturbations in the singlet states that can be observed using narrow-bandwidth spectroscopic techniques, providing useful information on the triplet states indirectly. For example, the C 3 Π u (v) valence-state levels, which have low rotational constants, are most likely to cross, and interact with, levels of the Rydberg states of N 2, which have significantly higher rotational constants. Indeed, spectroscopic parameters for the C 3 Π u (v = 14) level of 15 N 2 have been reported recently, determined by analyzing its perturbation of c 1 Π u (v = 1) near cm 1, including the observation of a few extra lines due to transitions to the perturbing level.[71, 72] The 3pπ u c 1 Π u state is a member of the 3p Rydberg complex of N 2,[11] which also includes the 3pπ u G 3 Π u, 3pσ u c 1 Σ + u, and 3pσ u D 3 Σ + u states. The c and G states exhibit a (small) spin-orbit interaction, while the G and D states, and the c and c states, exhibit strong rotational interactions which perturb the rotational constants and contribute to Λ-doubling in the Π u 3p-complex components, leading to a good deal of spectral complexity. In the present study, we investigate the spectroscopy and predissociation characteristics of the c 1 Π u (v = 2) 3p-complex member in the energetic region cm 1, for all three N 2 isotopomers, using principally 1 XUV + 1 UV twophoton-ionization spectroscopy. This level is perturbed by singlet valence states occurring in this energy region, most notably b 1 Π u (v = 11) and b 1 Σ u (v = 7), the latter of which contributes to observable Λ- doubling effects and quantum-interference phenomena. Local perturbations in the c 1 Π u (v = 2) level, caused by these singlet levels, as well as those due to C 3 Π u (v = 17), are studied, together with associated predissociative effects. 54

71 3. Interactions of the 3pπ u c 1 Π u (v = 2) Rydberg-complex in N Experimental Methods XUV laser spectroscopy Laser-based 1 XUV + 1 UV two-photon-ionization spectroscopy is the primary technique employed here to study the excitation spectrum of N 2, in the range λ = nm. Details of the experimental method, including a description of the lasers, vacuum setup, molecular-beam configuration, time-of-flight (TOF) detection scheme, and calibration procedures, have been presented previously.[55] Briefly, narrow bandwidth XUV radiation is produced by frequency tripling the UV light from a frequency-doubled pulsed dye laser (PDL) pumped by an injectionseeded Nd:YAG laser. Frequency tripling takes place in a free Xe-jet expansion, where the UV power (25 mj in 5 ns) is focused. The frequency in the XUV range is calibrated against a Doppler-broadened I 2 linear-absorption spectrum,[56] recorded simultaneously using part of the visible output of the dye laser. An absolute accuracy of 0.1 cm 1 for fully-resolved line positions is estimated. A skimmed, pulsed N 2 beam is perpendicularly intersected by temporally- and spatially-overlapping XUV and UV laser beams. Nitrogen molecules are excited resonantly by the XUV photons and subsequently ionized by the intense UV light. N + 2 ions are detected using a TOF mass selector. In addition to normal N 2, isotopically-enriched samples of 14 N 15 N and 15 N 2 are also studied. The TOF technique employed allows for mass identification of the N + 2 ions, which can thus be distinguished from ions arising from the background oil in the vacuum system. The experimental parameters were established to provide the highest N 2 rotational temperature in the interaction region. This was achieved by increasing the N 2 density ( mbar) and therefore increasing the population of the higher rotational levels through collisions between the molecular beam and the higher-pressure background N 2. To achieve the background pressure required, the nozzle-skimmer distance was kept to a minimum. We thereby recorded spectra associated with two distinct rotational temperatures, one belonging to the supersonic expansion of the molecular beam ( 10 K) and one belonging to the background gas at ambient temperature ( 300 K). Moreover, by changing the relative delay between the N 2 pulsed-valve trigger and the laser pulse, the rota- 55

72 3. Interactions of the 3pπ u c 1 Π u (v = 2) Rydberg-complex in N 2 tional temperature in the molecular beam could be selected to measure independent spectra of cold (10 50 K) and warm ( K) samples. These options helped greatly in the assignment of the spectral lines. The instrumental contribution to the measured line widths depends on the ambient gas pressure, the settings for measuring cold or warm spectra, the wavelength range, the dye used, and the alignment of the PDL.[55] In the case of the high-temperature spectra, an additional Doppler broadening amounting to 0.25 cm 1 full-width at half-maximum (FWHM) must be accounted for in the line-width analyses. The instrumental width is determined from observed transitions in the c 1 Σ + u X 1 Σ + g (2,0) band, because of its close proximity to the c 1 Π u X 1 Σ + g (2,0) transitions that are measured under virtually the same experimental conditions. The c (v = 2) level is known to be slightly predissociated,[65] but not to the extent that it will induce significant line broadening in the present experiments. Hence, the instrumental width can be taken to equal the observed width for these transitions. The values obtained for the instrumental contribution to the line width are 0.34±0.04 cm 1 FWHM for 15 N 2, 0.42±0.04 cm 1 FWHM for 14 N 15 N and 0.40±0.02 cm 1 FWHM for 14 N 2. Note that the instrumental width also includes the Doppler contribution. The differences between these values are not so much related to the isotopomers, but rather to the specific experimental conditions employed. Using this narrow-bandwidth XUV laser system, line positions, line widths, and relative intensity variations can be measured with rotational resolution. Information on the predissociation process can be obtained through several complementary means. The lifetime τ(s) of the excited level, shortened due to predissociation, can be expressed as: τ = 1 2πcΓ, (3.1) where Γ is the natural (Lorentzian) line width (in cm 1 FWHM). Under our experimental conditions, Doppler broadening is also present, contributing to the effective instrumental profile which can be taken to be Gaussian.[37] Thus, the measured line shape has a Voigt profile, with a width of ν obs FWHM. The Lorentzian line-width component, principally due to predissociation, can be obtained by deconvolution of the experimental Voigt profile using the appropriate Gaussian instrumen- 56

73 3. Interactions of the 3pπ u c 1 Π u (v = 2) Rydberg-complex in N 2 tal profile, of width ν inst FWHM. With some approximations,[73] the following relation is obtained: Γ = ν obs ( ν inst) 2 ν obs. (3.2) Hence, using Eqs. (3.1) and (3.2), the excited-state lifetime τ can be derived straightforwardly from the line-width measurements. In addition, the shortening of the lifetime due to predissociation will not only cause line broadening, but also a decrease in signal intensity, since the detected ions result from the ionization of a decaying excited level. The resulting ionization signal is proportional to the lifetime of the excited intermediate state that is ionized.[37] Thus, the intensity depletion also provides information on the predissociation behavior of the excited state, in fact more sensitively, in many cases, than from direct line-width measurements. XUV synchrotron spectroscopy Relative ionization signals in the spectra obtained using the two-photonionization spectroscopic technique described in Sec. 3.2 are influenced, not only by the aforementioned competition between predissociation and ionization, but also by the inherent oscillator strengths of the transitions. Thus, successful interpretation of the ionization spectra will, in some cases, require a detailed knowledge of the oscillator strengths, the measurement of which requires a different experimental technique. Therefore, in the case of the b X(7, 0) transition of 14 N 2, the present ionization spectra are interpreted using oscillator strengths derived from absolute XUV photoabsorption spectra obtained at the 2.5 GeV storage ring of the Photon Factory, a synchrotron radiation facility at the High Energy Accelerator Research Organization in Tsukuba, Japan. These measurements form part of the extensive experimental campaign of Stark et al.,[42, 74] who have described the apparatus in detail. Briefly, a 6.65 m spectrometer with a 1200 grooves per mm grating (blazed at 550 nm and used in the sixth order) provides an instrumental resolving power of 150, 000, equivalent to a resolution of 0.7 cm 1 FWHM. The spectrometer tank, filled with N 2 of normal isotopic composition in a flowing configuration, serves as an absorption cell with a 57

74 3. Interactions of the 3pπ u c 1 Π u (v = 2) Rydberg-complex in N 2 path length of 12.5 m and a temperature of 295 K. Rotational-line oscillator strengths and Lorentzian predissociation line-width components are obtained from the experimental photoabsorption cross sections using a nonlinear least-squares fitting procedure which takes account of the finite instrumental resolving power. The line oscillator strengths are converted into equivalent absolute band oscillator strengths using appropriate Hönl-London and Boltzmann factors. Predissociation linewidths determined from these synchrotron-based spectra for the c(v = 2) level of 14 N 2 are also used in Sec. 3.4 to supplement the present laser-based results which, for this isotopomer, do not extend to high-enough J values to access the principal region of interest, i.e., the crossing region with C(v = 17). 3.3 CSE calculations Due to the many interactions associated with levels of the 3p Rydberg complex, interpretation of the corresponding spectra is difficult without some recourse to the theoretical aspects. Therefore, although the present work is, in principle, an experimental study, some CSE calculations have been performed in order to supplement the analysis. The CSE model employed in this work, an extension of the ( 1 Π u + 3 Π u ) Rydberg-valence model of Lewis et al.[32] and Haverd et al.,[47] which includes 1 Σ + u Rydberg and valence states and their mutual electrostatic interactions, together with 1 Σ + u 1 Π u rotational interactions, is described and applied in Liu et al.[75] and Liang et al.[45] Briefly, the coupled-channel Schrödinger equation is solved for the radial wavefunctions of a series of interacting diabatic electronic molecular states defined by potential-energy curves and off-diagonal coupling parameters, and the corresponding photodissociation cross section from the ground state computed, yielding line positions and oscillator strengths. The formalism of the technique is described, e.g., by van Dishoeck et al.[76] and Lewis et al.[77] A major advantage of this method is that, after optimization to the well-known singlet energy levels of the 14 N 2 isotopomer, reliable interpolations and extrapolations may be made to previously unknown 15 N 2 and 14 N 15 N levels. Additionally, because unbound states are included among the coupled channels, predissociation line widths can be modeled, and so observed line-width perturbations of singlet states 58

75 3. Interactions of the 3pπ u c 1 Π u (v = 2) Rydberg-complex in N 2 Figure 3.1: 1 XUV + 1 UV ionization spectrum and line assignments for the c 1 Π u X 1 Σ + g (2,0) band of 15 N 2. The diamond-shaped symbols act as a guide to indicate the perturbed intensity pattern in the Q branch, where the 3:1 ratio between odd- and even-j transition lines, arising from nuclear-spin statistics, has been taken into account. may enable the indirect determination of potential-energy curves for the triplet manifold as well as singlet-triplet coupling parameters. The role of the present CSE modeling is to act as an intelligent extrapolator to support the positive identification of C(v = 17) as the perturber of c(v = 2). A comparison of experimental versus CSE-model band origins and rotational constants for the C 3 Π u (v 9) levels is given in Lewis et al.,[32] showing excellent agreement. Further confidence in the rather large extrapolation to v = 17 is found in the physically-based ab initio C 3 Π u potential-energy curve used within the model, as well as indirect verification of intermediate C(v) levels through the perturbative effects they produce on allowed singlet transitions.[72] 3.4 Results and discussion The c 1 Π u (v = 2) and b 1 Σ + u (v = 7) levels Rotationally-resolved spectra of the c 1 Π u X 1 Σ + g (2,0) and b 1 Σ + u X 1 Σ + g (7,0) bands in 15 N 2, 14 N 15 N and 14 N 2 were recorded, for a range 59

76 3. Interactions of the 3pπ u c 1 Π u (v = 2) Rydberg-complex in N 2 Table 3.1: Observed transition energies (in cm 1 ) for the c 1 Π u X 1 Σ + g (2,0) band in 15 N 2. Wave numbers derived from blended lines are flagged with an asterisk (*). Transitions that have not been included in the singlet-singlet deperturbation analysis are marked with a cross-hatch (#). J R(J ) Q(J ) P (J ) # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # 60

77 3. Interactions of the 3pπ u c 1 Π u (v = 2) Rydberg-complex in N 2 Table 3.2: Observed transition energies (in cm 1 ) for the b 1 Σ + u X 1 Σ + g (7,0) band in 15 N 2. For explanation of the annotations, see caption to Table 3.1. J R(J ) P (J ) of rotational temperatures. The corresponding transition energies and line assignments deduced from the spectra are presented in Tables 3.1 to 3.6. A high-temperature spectrum of the c 1 Π u X 1 Σ + g (2,0) band in 15 N 2 is shown in Fig. 3.1, where, with the aid of the diamond-shaped symbols, it can be seen that there is a spectacular deviation from a Boltzmann distribution in the Q-branch rotational intensities. This effect, present in all branches, made the line assignments somewhat difficult. Finally, these were established by using combination differences for the P - and R-branches, the 3:1 ratio between odd- and even-j lines associated with 15 N 2 nuclear-spin statistics, and the cold and warm spectral recordings. From Fig. 3.1, it can be seen that there is a significant intensity decrease between Q(6) and Q(13), with the Q(8) line too weak to be observed. 61

78 3. Interactions of the 3pπ u c 1 Π u (v = 2) Rydberg-complex in N 2 Table 3.3: Observed transition energies (in cm 1 ) for the c 1 Π u X 1 Σ + g (2,0) band in 14 N 15 N. For explanation of the annotations, see caption to Table 3.1. J R(J ) Q(J ) P (J ) # # # # # # # # # # # # # # # # # # 62

79 3. Interactions of the 3pπ u c 1 Π u (v = 2) Rydberg-complex in N 2 Transitions terminating on levels between J =5 and J =13 also show a marked intensity decrease in the P and R branches. As will be seen below, this intensity depletion is attributed to the effects of the crossing of c(v = 2) by the predissociated v = 17 level of the triplet state C 3 Π u. In the case of 14 N 15 N, small intensity deviations are observed for transitions to both e- and f-parity levels of c(v = 2) with J = 17 and 18. No unusual intensity behavior is observed in the case of 14 N 2, where J = 19 is the highest rotational level of the excited state probed experimentally. Table 3.4: Observed transition energies (in cm 1 ) for the b 1 Σ + u X 1 Σ + g (7,0) band in 14 N 15 N. For explanation of the annotations, see caption to Table 3.1. J R(J ) P (J ) The overall singlet structure in the region of the v = 2 level of the 3p Rydberg complex of N 2 is summarized in Fig. 3.2 for the e-parity levels of each isotopomer, the plotted points representing a combination of measurements taken using the present apparatus and data from the Harvard-Smithsonian molecular database.[78] This structure is com- 63

80 3. Interactions of the 3pπ u c 1 Π u (v = 2) Rydberg-complex in N 2 Figure 3.2: Rotational term values for b 1 Π u (v = 11), b 1 Σ + u (v = 7), c 1 Π u (v = 2) and c 1 Σ + u (v = 2), for the three isotopomers of N 2. For clarity, only the e-parity levels are displayed. The symbols in gray represent data from the Harvard-Smithsonian molecular database.[78] Indicative C 3 Π u (v = 17) energy levels (lines) were obtained using the CSE model. The missing rotational levels for C(v = 17) were difficult to determine due to excessive predissociation and additional perturbation by the F 3 Π u (v = 2) level. pletely analogous to that reported and discussed in detail by Sprengers et al.[36] for the v = 1 level of the same Rydberg complex in 15 N 2, where the interacting valence states are b(v = 8) and b (v = 4). In the present case, the diabatic c 1 Π u (v = 2) and the lower-lying c 1 Σ + u (v = 2) levels exhibit the first-order heterogeneous L-uncoupling interaction characteristic of p-complex members,[11] providing the root cause of the J- dependent Λ-doubling in c(v = 2). In addition, the higher-lying valence levels, b 1 Π u (v = 11) and b 1 Σ + u (v = 7), with lower rotational constants, interact homogeneously with the c 1 Π u (v = 2) and c 1 Σ + u (v = 2) Rydberg levels, respectively, more strongly as J increases. In the case of the f-parity levels, c 1 Π u (v = 2) b 1 Π u (v = 11) is the only operative interaction. In reality, in the case of the e levels, the four singlet states 64

81 3. Interactions of the 3pπ u c 1 Π u (v = 2) Rydberg-complex in N 2 form a Rydberg-valence complex, with mutual effective interactions arising from the basis couplings just detailed. For example, b (v = 7) and c(v = 2), the principal levels studied here, exhibit an effective rotational interaction which culminates in an avoided crossing at intermediate J values, providing a further dimension to the J-dependence of the c(v = 2) Λ-doubling. The observed Λ-doublings for the c(v = 2) levels of the three isotopomers, shown in Fig. 3.3, are seen to exhibit qualitatively similar behavior. In the case of 15 N 2, the Λ-doubling is very similar to that reported for the c(v = 1) p-complex member of the same isotopomer by Sprengers et al.[36] At low J, the effective c (v = 2) contribution to the Λ-doubling is a little more important, with the e levels higher in energy. As J increases, b (v = 7) approaches from above more closely and its contribution dominates, with the e levels now much lower in energy. Also shown in Fig. 3.2 is the indicative rotational structure of the C 3 Π u (v = 17) valence level that is responsible for the local-intensity perturbation of c(v = 2), computed using the CSE model. As can be seen from the figure, C 3 Π u (v = 17) crosses c 1 Π u (v = 2) at significantly different J values for each isotopomer. The model crossing points agree with the observed positions of maximum intensity perturbation for both 15 N 2 and 14 N 15 N. However, for 14 N 2, the C 3 Π u (v = 17) state is predicted to cross c 1 Π u (v = 2) at J 23, too high to be accessible with the current experimental setup. One of the main aims of the present work is to examine the character of the local perturbation of c(v = 2) by C(v = 17). In order to facilitate this in the case of the e-parity levels, we attempt to first remove the effects of the stronger perturbation by b (v = 7) by performing a twolevel deperturbation analysis. In the case of the f-parity levels, this procedure is not necessary, the avoided crossing between b(v = 11) and c(v = 2) occurring only at high J levels, beyond the range of interest for the C(v = 17) crossing. Effective two-level-deperturbed spectroscopic parameters have been determined from the experimental transition energies for the c X(2, 0) and b X(7, 0) bands, using a nonlinear least-squares fitting procedure, and omitting the J = 5 13 and J = c(v = 2)-state levels, for 15 N 2 and 14 N 15 N, respectively, that correspond to the regions of observed intensity perturbation. For 14 N 2, no unusual intensity behavior 65

82 3. Interactions of the 3pπ u c 1 Π u (v = 2) Rydberg-complex in N 2 Table 3.5: Observed transition energies (in cm 1 ) for the c 1 Π u X 1 Σ + g (2,0) band in 14 N 2. For explanation of the annotations, see caption to Table 3.1. J R(J ) Q(J ) P (J ) # # is observed, so all levels have been included in the analysis. The terms of the ground state, X 1 Σ + g, are represented by F (J ) = B[J (J + 1)] D[J (J + 1)] 2 + H[J (J + 1)] 3, (3.3) where J labels the rotational levels of the ground state, B is the rotational constant, and D and H are centrifugal distortion parameters. The spectroscopic parameters of Bendtsen[41] and Trickl et al.[40] are used for the X 1 Σ + g states of 15 N 2, 14 N 15 N and 14 N 2. The terms for the excited c 1 Π u state, where, for simplicity, we denote the rotational quantum number by J, rather than J (and the same holds for v and v ), are taken to have the form T f (J) = ν 0 + B[J(J + 1) 1] D[J(J + 1) 1] 2, (3.4) 66

83 3. Interactions of the 3pπ u c 1 Π u (v = 2) Rydberg-complex in N 2 Table 3.6: Observed transition energies (in cm 1 ) for the b 1 Σ + u X 1 Σ + g (7,0) band in 14 N 2. For explanation of the annotations, see caption to Table 3.1. Information derived from shoulders in the spectra is marked with an s. J R(J ) P (J ) s s for the f-parity[11] levels, where ν 0 is the band origin, and T e (J) = T f (J) + q[j(j + 1) 1], (3.5) for the e-parity levels, where q is the Λ-doubling parameter. The terms for b (v = 7), which has only e-parity levels, are represented by T (J) = ν 0 + B[J(J + 1)] D[J(J + 1)] 2. (3.6) As already noted, the effective interaction between the b (v = 7) levels, of 1 Σ + u symmetry, and c(v = 2) levels, of 1 Π u symmetry, is of a heterogeneous nature ( Ω = 1, L-uncoupling) and only involves the e-parity levels of the c(v = 2) state. For the e-parity manifold, a two-state deperturbation analysis was performed for each J value by diagonalizing the matrix ( ) T b (7) H b (7)c(2) J(J + 1), (3.7) J(J + 1) Tc(2) H b (7)c(2) 67

84 3. Interactions of the 3pπ u c 1 Π u (v = 2) Rydberg-complex in N 2 Figure 3.3: Observed Λ-doubling [ T e f = T e (J) T f (J)] for the c 1 Π u (v = 2) Rydberg levels of isotopic N 2. The symbols in gray represent data from the Harvard-Smithsonian molecular database.[78] where the diagonal elements are the term energies of b (v = 7) and c(v = 2), given by Eqs. (3.6) and (3.5), respectively. The off-diagonal elements represent the effective heterogeneous interaction between the c(v = 2) e-parity and b (v = 7) levels. In the weighted fitting procedure, the nominal uncertainty in the absolute transition energy for fully-resolved lines of reasonable strength was taken to be 0.1 cm 1. In the case of weak or blended lines, the uncertainty was set to an estimated value in the range cm 1. For all three isotopomers, the quality of the obtained fit was reasonable. The results of the two-level deperturbation analysis for the three isotopomers are given in Table 3.7. It should be noted that the two-level deperturbation performed here is only an approximation, and that, consequently, the spectroscopic parameters in Table 3.7 have limited physical significance. For example, from molecular-orbital configurational arguments, and experimentation with the CSE model, the direct heterogeneous electronic coupling between the b 1 Σ + u and c 1 Π u states is likely to be very small, so it is principally the indirect mechanism involving both the homogeneous b c and the heterogeneous c c electronic couplings that is responsible for the complexity of the observed c(v = 2)-level Λ-doubling in Fig. 3.3 and the anomalous D values for b (v = 7) in Table 3.7. Consequently, for 68

85 3. Interactions of the 3pπ u c 1 Π u (v = 2) Rydberg-complex in N 2 Table 3.7: Two-level-deperturbed molecular parameters for the c 1 Π u (v = 2) and b 1 Σ + u (v = 7) levels of 14 N 2, 14 N 15 N and 15 N 2. All values are in cm 1. Statistical uncertainties (1σ), resulting from the fit, are shown in parentheses, in units of the last significant figure. Additional systematic uncertainties of order 0.05 cm 1 apply to the band origins ν 0. Species 14 N 2 14 N 15 N 15 N 2 c 1 Π u (v = 2) B 1.821(1) 1.764(1) 1.720(3) D (4) 74(5) 151(9) q (2) 38(2) 17(3) ν (5) (4) (5) b 1 Σ + u (v = 7) B 1.327(2) 1.278(1) 1.243(2) D (7) 64(6) 11(9) ν (5) (5) (7) H b (7)c(2) 2.84(4) 2.76(5) 2.07(8) large J, the effective deperturbation parameter H b (7)c(2) will not be well represented as J-independent if c (v = 2) is not also explicitly deperturbed concurrently. However, additional possible perturbation partners could not be followed up to sufficiently high J, and an attempted three state analysis [b(v = 11), b (v = 7) and c(v = 2)] gave inconclusive results, as did the further inclusion of c (v = 2). Thus, only the two-state analysis is presented and the higher-j levels for which the model does not apply are left out of the fit. This limitation does not affect our main aim of studying the additional local perturbation of c(v = 2) by C(v = 17). P/R intensity anomalies in the b X(7, 0) band The effective heterogeneous interaction between c(v = 2) and b (v = 7) not only produces energy shifts, but also affects the intensities of transitions accessing the corresponding e-parity levels. In particular, the b X(7, 0) ionization spectra for each isotopomer display significant P/R-branch intensity anomalies, behavior characteristic of transitions 69

86 3. Interactions of the 3pπ u c 1 Π u (v = 2) Rydberg-complex in N 2 into states of mixed Σ and Π character, where each component of the transition is dipole-allowed. P/R anomalies and their causative quantum-interference mechanism have been described in detail by Lefebvre-Brion and Field.[11] Briefly, in the case of a transition from a 1 Σ initial state to a nominal 1 Σ level which is perturbed by a lower-lying 1 Π level, thus corresponding to the present situation, the J dependence of the perturbed R-branch intensity is given by: I R (J = J 1) Jc 2 Jµ 2 + (J + 1)(1 c2 J)µ 2 while the P -branch dependence is given by: ±2c J J(J + 1)(1 c 2 J )µ µ, (3.8) I P (J = J + 1) (J + 1)c 2 Jµ 2 + J(1 c2 J)µ 2 2c J J(J + 1)(1 c 2 J )µ µ, (3.9) where µ and µ are the unperturbed 1 Σ 1 Σ and 1 Π 1 Σ vibronic transition moments, respectively, and c J 0 is the J-dependent mixing coefficient corresponding to the amount of 1 Σ character in the perturbed 1 Σ wave function. In Eqs. ( ), the upper signs apply when the Σ Π vibronic interaction matrix element H ΣΠ > 0, the lower when H ΣΠ < 0. The sense of the quantum-interference effect, corresponding to the overall signs of the cross terms in Eqs. ( ), depends on the sign of the product µ µ H ΣΠ. In the present case, where the P branch of the b X(7, 0) transition is observed to decrease in intensity as a result of the interference effect, evidently, µ b X(7,0)µ cx(2,0) H b (7)c(2) > 0. Note that as this is an excited-state interaction, we have chosen to express the intensity of the R and P branches in terms of the excited-state rotational quantum number J. To study this phenomenon experimentally, it is necessary to extract information on the intensity of the transitions measured in the ionization spectra. Generally, it is risky to rely on intensities in a 1 XUV + 1 UV experiment. However, if one assumes that the ionization cross section and XUV intensity are constant during the measurement, some semiquantitative information can be extracted from the experimental results. 70

87 3. Interactions of the 3pπ u c 1 Π u (v = 2) Rydberg-complex in N 2 Figure 3.4: P/R-branch intensity anomalies in the b 1 Σ + u X 1 Σ + g (7, 0) bands of 15 N 2 and 14 N 15 N. Solid circles: Reduced R-branch intensities (from experimental ionization spectra, see text). Open circles: Reduced P -branch intensities. Solid curve: P -branch band oscillator strengths from CSE-model calculations. Dashed curve: R-branch band oscillator strengths. The experimental reduced intensities have been normalized to the CSE results. Relative rotational intensities in well-behaved, unperturbed vibronic transitions of N 2 are given by: I g i J S JJ (2J + 1) e B [J (J +1)] kt, (3.10) where gj i is the nuclear spin-statistics factor responsible for the wellknown rotational-intensity alternation in the homonuclear isotopomers, S JJ is the Hönl-London factor, B is the rotational constant of the ground state, k is the Boltzmann constant, and T is the absolute rotational temperature. In deriving Eq. (3.10), it is assumed that the ground state has a Boltzmann population distribution. From Eq. (3.10), by plotting ln[i/(gj i S JJ (2J + 1)] against J (J + 1), the molecularbeam temperature may be determined from the slope of B /(kt ). In many cases, Eq. (3.10) will not apply and the measured ionization signal will be influenced additionally, not only by genuine intensity interference effects, e.g., the P/R anomaly discussed above, but 71

88 3. Interactions of the 3pπ u c 1 Π u (v = 2) Rydberg-complex in N 2 Figure 3.5: P/R-branch intensity anomalies in the b 1 Σ + u X 1 Σ + g (7, 0) band of 14 N 2. Solid circles: Reduced R-branch intensities (from experimental ionization spectra, see text). Open circles: Reduced P -branch intensities. Solid curve: P -branch band oscillator strengths from CSE-model calculations. Dashed curve: R-branch band oscillator strengths from CSE-model calculations. Solid squares: R-branch band oscillator strengths from synchrotronbased experiments. Open squares: P -branch band oscillator strengths from synchrotron-based experiments. The experimental reduced intensities have been normalized to the CSE results. also by J-dependent competition between predissociation and ionization. In order to study these effects, it is instructive to consider the reduced intensity, i.e., the actual intensity divided by that implied by Eq. (3.10), where it is assumed that there is an independent estimate of the molecular-beam temperature(s) available, e.g., from a separate scan over a well-behaved band, taken under similar experimental conditions. The reduced intensity thus serves to highlight any strongly J-dependent interference or predissociation effects, with subsequent analysis required to decide between these two possibilities. In Fig. 3.4, the reduced intensities (circles) obtained with our 1 XUV + 1 UV setup for the P and R branches of the b X(7, 0) bands in 15 N 2 and 14 N 15 N are shown as a function of J, normalized against the corresponding absorption oscillator strengths predicted by our CSE model (curves). The molecular-beam temperature used to determine the reduced intensities was obtained from c X(2, 0) Q-branch lines, 72

89 3. Interactions of the 3pπ u c 1 Π u (v = 2) Rydberg-complex in N 2 measured under virtually the same conditions. For both isotopomers, Fig. 3.4 shows a remarkable difference in the behavior of the P - and R-branch reduced intensities, with the P -branch intensities exhibiting deep minima at intermediate J values, in good agreement with the CSEmodel oscillator strengths. This is a classic example of a P/R intensity anomaly due to the quantum-interference effect described by Eqs. (3.8) and (3.9). However, we note that attempts to fit the experimental results in detail using these equations were of limited success, due to the inherently multilevel nature of the N 2 spectrum in this region, which is embodied in the CSE results. In Fig. 3.5, the reduced intensities (circles) obtained with our 1 XUV + 1 UV setup in the case of 14 N 2 are shown, normalized against CSEmodel calculations (curves), and compared with absolute band oscillator strengths obtained from our synchrotron-based experiments.(these synchrotron-based results have also been reported in Ref. [74]) As for the other isotopomers, once again there is a strong P/R intensity anomaly. It is also of note that the excellent relative agreement between the reduced ionization intensities and the absolute optical oscillator strengths implies that J-dependent predissociation effects are not a factor for b (v = 7), at least for J 15. The quantum-interference effect also influences the e-levels of c(v = 2). A pair of equations complementary to Eqs.( ) can be written to describe the R- and P -branch intensities in the transition to the lower-lying 1 Π level. In the present case, the R branch of the c X(2, 0) band will suffer the intensity decrease and the P -branch intensity will be enhanced by the interference effect. However, this phenomenon is barely visible in our reduced intensities, since it is both masked by the strong intensity perturbations due to the crossing of c(v = 2) by C(v = 17), discussed in Sec. 3.4, and less prominent due to the much greater oscillator strength of the c X(2, 0) transition. The c 1 Π u (v = 2) C 3 Π u (v = 17) interaction Transitions from the X 1 Σ + g ground state to the 3 Π u manifold of N 2 are forbidden by the S = 0 selection rule. However, as mentioned in Sec. 3.1, the 3 Π u states may spin-orbit couple to the 1 Π u states. As a result, transitions to the 3 Π u states may become directly ob- 73

90 3. Interactions of the 3pπ u c 1 Π u (v = 2) Rydberg-complex in N 2 Figure 3.6: (a) Reduced terms T for the c 1 Π u (v = 2) level of 15 N 2. Solid circles: f-parity levels. Open circles: e-parity levels. Solid line: CSE prediction for crossing by Ω = 1 sublevel of C 3 Π u (v = 17). Dashed line: CSE prediction for crossing by Ω = 0 sublevel. (b) Predissociation widths for c 1 Π u (v = 2) in 15 N 2. Solid circles: f-parity levels, from fitting Q-branch line profiles in c X(2, 0) ionization spectra. Open circles: e-parity levels, weighted average of fits to P - and R-branch line profiles. Solid line: CSE-model widths for f levels. Dashed line: CSE-model widths for e levels. (c) Reduced intensities (see text), taking into account the high- (202 K) and low- (25 K) temperature components of the molecular beam. Solid circles: f-parity levels, from integrated Q-branch intensities in c X(2, 0) ionization spectra. Open circles: e-parity levels, weighted average of P - and R-branch reduced intensities. 74

91 3. Interactions of the 3pπ u c 1 Π u (v = 2) Rydberg-complex in N 2 servable through their intensity borrowing from dipole-allowed transitions, and/or the 3 Π u states may make their presence felt through energy-, intensity-, or line-width-perturbation of the 1 Π u states. In the present case, based on CSE-model predictions, we propose that it is the C 3 Π u (v = 17) level which is responsible for the energy perturbations observed here in the c(v = 2) levels of 15 N 2 and 14 N 15 N. Furthermore, since the C 3 Π u state couples electrostatically to the dissociative C 3 Π u state, leading to strong predissociation,[32] it is likely that the local perturbation of c(v = 2) by C(v = 17) is also responsible for the intensity anomalies observed in the corresponding ionization spectra, through accidental predissociation effects. Due to the complexity of the N 2 structure, however, these conclusions are by no means obvious from an examination of the experimental spectra alone. We attempt to justify them below. After the perturbations in the singlet manifold have been roughly accounted for and removed, as in Sec. 3.4, the residual local energy shifts can be examined. In Fig. 3.6(a), the reduced term values of the c(v = 2) rotational levels are shown for 15 N 2. Both e- and f-parity reduced terms exhibit a similar pattern, with local perturbations evident between J = 7 and 9, and, more clearly, between J = 11 and 12. In Fig. 3.6(b), the fitted Lorentzian line-width components (symbols) for isolated rotational transitions in the c X(2, 0) band of the ionization spectrum of 15 N 2 are presented, with only minor differences observed between the line-width patterns for transitions into the e- and f-parity levels of the c(v = 2) state. Considerable predissociation of the c(v = 2)- state levels is observed in the region of the local perturbations, with level-width maxima near J = 8 and 12. In the case of blended lines, term values could be extracted from the experimental spectra, but neither line-width nor intensity information could be obtained reliably. Also shown in Fig. 3.6(b) are the results of our preliminary CSE-model calculations (lines), which are in good overall agreement with the observed predissociation pattern, but with some discrepancy in the region of the higher-j crossing. Finally, in Fig. 3.6(c), the reduced intensities for transitions to the e- and f-parity levels of c(v = 2) in 15 N 2 are shown. For J > 14, these reduced intensities are essentially J-independent. However, at lower J, significant intensity depletion is observed, with deep intensity minima near J = 8 75

92 3. Interactions of the 3pπ u c 1 Π u (v = 2) Rydberg-complex in N 2 and 12. In the case of the missing J = 8f level, for example, the associated Q(8) line was too weak to be observed. As stated in Sec. 3.4, there is no clear evidence of P/R intensity anomalies in the c X(2, 0) spectrum, so the dramatic behavior of the reduced intensities in Fig. 3.6(c) is likely another signature of predissociation. It is notable that the positions of the intensity minima occur for the same values of J where the line widths and term values also show unusual behavior. The patterns in Fig. 3.6, containing only two distinct regions of perturbation, are not directly suggestive of perturbation by a triplet level, despite the CSE model indicating that C 3 Π u (v = 17) is primarily responsible. There are two reasons for this. First, only C Ω=1 (v = 17) can interact by spin-orbit coupling ( Ω = 0) directly with c Ω=1 (v = 2). However, as the J-dependent S-uncoupling mechanism ( Ω = ±1) induces mixing between C Ω=1 (v = 17) and both C Ω=0 (v = 17) and C Ω=2 (v = 17), the latter substates may also interact indirectly with c(v = 2). For the limiting case of a slowly rotating molecule, the Ω = 1 component of the triplet perturber will strongly spin-orbit couple to the c(v = 2) state, while the Ω = 0, 2 components will couple only weakly, corresponding to the Hund s case (a) limit. On the other hand, at high J values, the Hund s case (b) limit is approached, where the Ω = 0, 2 components interact strongly with c(v = 2), but the Ω = 1 component is only weakly coupled.[79] The local perturbation culminating near J = 8 coincides with the CSE-predicted crossing by C Ω=1 (v = 17), indicated by the solid line in Fig. 3.6(a), while that at J = is due to the Ω = 0 crossing,(the C 3 Π u state is normal for the lower vibrational levels, but inverted in this region (Ref. [72])) indicated by the dashed line. The C Ω=2 (v = 17) sublevel would be expected to cross c(v = 2) in the region of J = 5, but no evidence for a perturbation associated with this crossing is visible in Fig. 3.6(a). Part of the reason for this relates to the low J value for the crossing, suggesting that the Hund s case (a) limit will be approached, therefore implying only a weak indirect coupling between C Ω=2 (v = 17) and c Ω=1 (v = 2). This phenomenon was also observed in the k 3 Π state of CO, which is spin-orbit coupled to the E 1 Π state, causing predissociation in the same manner as in the present study (see Fig. 12 of Ref. [80]). Second, the characteristics of local perturbations in energy depend also on the predissociation width of the perturbing level. For example, 76

93 3. Interactions of the 3pπ u c 1 Π u (v = 2) Rydberg-complex in N 2 Figure 3.7: (a) Reduced terms T for the c 1 Π u (v = 2) level of 14 N 15 N. Solid circles: f-parity levels. Open circles: e-parity levels. Dashed line: CSE prediction for crossing by Ω = 0 sublevel. (b) Predissociation widths for c 1 Π u (v = 2) in 14 N 15 N. Solid circles: f-parity levels, from fitting Q-branch line profiles in c X(2, 0) ionization spectra. Open circles: e-parity levels, weighted average of fits to P - and R-branch line profiles. Solid line: CSE-model widths for f levels. Dashed line: CSE-model widths for e levels. (c) Reduced intensities (see text), taking into account the high- (275 K) and low- (72 K) temperature components of the molecular beam. Solid circles: f-parity levels, from integrated Q-branch intensities in c X(2, 0) ionization spectra. Open circles: e-parity levels, weighted average of P - and R-branch reduced intensities. perturbation by a long-lived level results in a perturbation with a rapid J-dependence and a maximum shift at the crossing, while perturbation by a short-lived level may result in a barely-noticeable perturbation with a slow J-dependence and zero shift at the crossing. In the case of crossing by a triplet level, the visibility and character of the associated perturbations will also depend on the relative predissociation widths of the triplet sublevels. In the present case, the CSE model predicts that the lowest C(v = 17) sublevel (Ω = 2) is very strongly predissociated, with the level widths, also strongly J-dependent, decreasing significantly for Ω = 1, and Ω = 0, respectively. Thus, primarily due to the large width of C Ω=2 (v = 17) relative to the spin-splitting of the triplet state, 77

94 3. Interactions of the 3pπ u c 1 Π u (v = 2) Rydberg-complex in N 2 only the highest two of the expected three energy perturbations in the c(v = 2) level are visible in Fig. 3.6(a). For the same reasons, only two clear peaks are visible in the c(v = 2) predissociation pattern in Fig. 3.6(b), and two minima in the associated intensity-depletion pattern in Fig. 3.6(c). In particular, the higher- J crossing, corresponding to the better defined local perturbation in Fig. 3.6(a), is seen in Fig. 3.6(b) to be associated with the most rapid variation in the computed predissociation line width which is borrowed by c(v = 2) from the C(v = 17) perturber. It is notable that, while the energy perturbations in the c(v = 2) level of 15 N 2 caused by the crossing with the short-lived C(v = 17) level are small, the corresponding indirect predissociations and intensity depletions are dramatic, despite the weak spin-orbit coupling, because of the otherwise long inherent lifetime of c(v = 2). Finally, we emphasize that caution is necessary when trying to infer detailed information on the perturbing C(v = 17) level from the experimental spectra, with simplistic level deperturbations of little use. Detailed examination of the CSE model indicates that the 3 Π u manifold in this energy region is of great complexity, with the F 3 Π u and G 3 Π u Rydberg states also having an active influence on the character of the c(v = 2) C(v = 17) perturbation. In fact, the very strongly predissociated F (v = 2) level, which also contains a significant G-state admixture,[70] crosses C(v = 17) in the region of the latter level s crossing with c(v = 2), for all isotopomers, leading, e.g., to the irregularities in the computed nominal C(v = 17)-level energies for 14 N 2 displayed in Fig In addition, electrostatic interactions within the 3 Π u manifold lead to strong J- and Ω-dependences in the C(v = 17) predissociation line widths, as mentioned above, which further complicate interpretation of the data. In particular, the determination of C(v = 17) sublevel rotational terms becomes impossible when the corresponding predissociation line widths are comparable to, or larger than, the spin splitting. The results for 14 N 15 N, summarized in Fig. 3.7, are more equivocal. For this isotopomer, the crossing of c(v = 2) by C(v = 17) is predicted by the CSE model to occur in the region of J = 17. In Fig. 3.7(a), no significant term-value shifts can be observed for the f-parity levels, but there is evidence of a weak local perturbation between J = 17 and 18 in the e-parity levels. Overall, the fitted Lorentzian line widths in 78

95 3. Interactions of the 3pπ u c 1 Π u (v = 2) Rydberg-complex in N 2 Fig. 3.7(b) (symbols) are not conclusive, because of large error bars in the crossing region, but are not inconsistent with the computed CSEmodel widths (lines), which predict a predissociation maximum in the J = region. The reduced intensities in Fig. 3.7(c), however, show fairly clear depletion, displaying a single minimum near J = 18, which is shaded towards lower J values, behavior consistent with that expected from the computed line widths. Taken together, the experimental results are consistent with the perturbation of c(v = 2) by C(v = 17) in the J = region, but imply such a strong predissociation of the C(v = 17) sublevels that (1) the associated energy shifts are only small, and (2) the predissociation signatures collapse into a single broad feature. This is supported by the CSE results, which suggest that it is the Ω = 0 sublevel of C(v = 17), indicated by the dashed line in Fig. 3.7(a), that has the lowest predissociation width and is responsible for the only local perturbation evident in the c(v = 2) terms, with the other sublevels so broad that no corresponding shifts occur. As stated in Sec. 3.4, for 14 N 2, the CSE model predicts the crossing between c(v = 2) and C(v = 17) to take place in the region of J = 23 (see Fig. 3.2). Since, under our experimental conditions, such highrotational levels cannot be accessed with the laser-based system, the fact that no energy-level shifts or intensity anomalies have been observed in the present data for this isotopomer is hardly surprising. Nevertheless, in Fig. 3.8, we are able to compare the synchrotron-based predissociation widths for the c(v = 2) level of 14 N 2 (circles) (These synchrotron-based results have also been reported in Ref. [74]) with the CSE-model values (lines). The CSE results indicate a marginally double-peaked structure, with predissociation maxima near J = and 23 24, in excellent agreement with the measurements. Once again, the short lifetime ol the C(v = 17) perturber, together with the complex interactions at play, combine to severely modify the expected triplet pattern of predissociation. For this isotopomer, it is the Ω = 2 sublevel of C(v = 17) which has the lowest predissociation width and is responsible for the first c(v = 2) predissociation maximum in Fig. 3.8, while the broader Ω = 1, 0 sublevels contribute to the second maximum, with the Ω = 1 contribution likely to be smaller because of the case-(b) nature of this high-j crossing. 79

96 3. Interactions of the 3pπ u c 1 Π u (v = 2) Rydberg-complex in N Γ ( cm 1 FWHM ) J (J +1) Figure 3.8: Comparison between experimental predissociation line widths for the c 1 Π u (v = 2) level of 14 N 2, derived from synchrotron-based photoabsorption spectroscopy of the c X(2, 0) band (Ref. [74]), and CSE-model calculations. Solid circles: Experimental values from fits to Q-branch transitions. Open circles: Weighted average of experimental values from P - and R-branch transitions. Solid line: CSE widths for f-parity levels. Dashed line: CSE widths for e-parity levels. 3.5 Summary and conclusions The 3pπ u c 1 Π u X 1 Σ + g (2, 0) Rydberg and b 1 Σ + u X 1 Σ + g (7, 0) valence transitions of 14 N 2, 14 N 15 N, and 15 N 2 have been studied using laser-based 1 XUV + 1 UV two-photon-ionization spectroscopy, supplemented by synchrotron-based XUV spectroscopy in the case of 14 N 2. Due to the inherent complexity and multilevel nature of the N 2 structure and spectrum in this energy region, few-level deperturbation techniques were found to be of limited use and interpretation of the results by means of coupled-channel Schrödinger-equation techniques was found necessary. For each isotopomer, effective rotational interactions between the c(v = 2) and b (v = 7) levels were found to cause strong Λ-doubling in c(v = 2), together with dramatic P/R-branch intensity anomalies in the 80

97 3. Interactions of the 3pπ u c 1 Π u (v = 2) Rydberg-complex in N 2 b X(7, 0) band due to the effects of quantum interference. The P/R intensity anomalies for all isotopomers, together with the synchrotronbased oscillator strengths for 14 N 2, were found to be well described by the present CSE calculations. Local perturbations in energy and predissociation line width for the c(v = 2) Rydberg level were observed and attributed primarily to a spinorbit interaction with the crossing, short-lived C 3 Π u (v = 17) valence level. However, the CSE calculations suggest that this is a complex interaction, also involving the even shorter-lived F (v = 2) Rydberg level in the same energy region. The characterization of these interactions should further help to elucidate the predissociation mechanism for the 1 Π u states of N 2,[32] extending the applicability of the corresponding CSE code towards higher energies. Acknowledgment This research was partially supported by Australian Research Council Discovery Program Grants DP and DP

98

99 Chapter 4 Observation of a Rydberg series in a heavy Bohr atom We report on the realization of a heavy Bohr atom, through the spectroscopic observation of a Rydberg series of bound quantum states at principal quantum numbers n = 140 to 230. The system is made heavy by replacing an electron inside a hydrogen atom by a composite H particle, thus forming a H + H Coulombically bound system obeying the physical laws of a generalized atom with appropriate mass scaling.

100 4. Observation of a Rydberg series in a heavy Bohr atom Bohr developed the atomic model by imposing an ad hoc quantization condition on the angular momentum of an electron orbiting in the Coulombic potential of a positively charged nucleus [81]. The same solutions for the level energies are found by solving the Schrödinger equation for a quantum system bound by a 1/r potential; Rydberg states are represented by [82] E n = E lim R A (n δ) 2 (4.1) where R A is the atomic Rydberg constant and n is the principal quantum number. For atoms with an extended core, i.e all atoms other than Hydrogen, a quantum defect δ should be included. The wide variety of atomic and molecular realizations of Rydberg systems all have in common that the fundamental scale factor, the Rydberg constant, is essentially the same, because it is governed by the mass of the electron bound to a much heavier core: R A = (µ/m e )R ; for all these systems the reduced mass of the electron µ is close to m e. Only few exotic systems like muonium (µ + e ) [83] or positronium (e + e ) [84] provide a different scale factor, the latter having exactly R /2. Ion-pair systems bound by their Coulombic 1/r potential provide a realistic extension of the Rydberg concept to heavier systems. Initially the process of ion-pair formation above threshold was investigated by Chupka et al.., [85] measuring H 2 + hν H + + H. Hepburn and coworkers used the existence of long-lived bound states immediately below the ion-pair limit in their threshold ion-pair production spectroscopy (TIPPS) to determine accurate values of ion-pair dissociation thresholds, applied amongst others to the H + H system [86]. Later Reinhold and Ubachs demonstrated the existence of bound heavy Rydberg states of the H + H system through the probing of wave packet dynamics in the densely structured quantum region several cm 1 below threshold [87]. These time-domain observations of heavy Rydberg states, conceptually analogous to observations of electronic wave packets [88, 89], reveal the characteristic signatures of such states without resolving individual quantum states. Fig. 4.2 displays the fundamental discovery of the present study. A long series of regularly spaced resonances is observed in an energy 84

101 4. Observation of a Rydberg series in a heavy Bohr atom Figure 4.1: Experimental setup. An H 2 molecular beam is perpendicularly intersected with a beam of XUV-radiation, obtained via third harmonic generation of a pulsed UV-laser beam underneath the orifice of a pulsed valve in krypton gas. The XUV-radiation is geometrically filtered from the incident fundamental UV-beam by selecting the phase-matched k XUV -vector in a UVdark region imposed by a post placed in the incident laser beam (see inset). H + and H + 2 are extracted from the interaction region by a pulsed electric field for signal detection. window of some 2000 cm 1. These resonances visibly follow the pattern of a Rydberg-like series indicated and numbered in the Figure. However, the Rydberg series is not as clean as usually observed in atoms; the intensities, widths and line shapes vary across the series superimposed on a structured continuum, while some unassigned resonances are present as well. The series of quantum states with a Rydberg electron bound by an H + 2 molecular core in quantum states v+ = 7 and N + = 0, 2 or 4 (only even rotational quantum numbers for para-h 2 ), expected in this energy range, do not match the observed features. Alternatively, the series can be fitted to an equation for heavy Rydberg states: E n = E IP R h (n δ) 2 (4.2) 85

102 4. Observation of a Rydberg series in a heavy Bohr atom H 2 + signal (arb. units) H 2 + (7,4) + + H H HR(n) 2 (7,0) 2 (7,2) Excitation Energy, cm -1 Figure 4.2: Characteristic spectrum after two-step laser excitation via C-X (3,0) R(0) in the para-h 2 molecule. The markers and dotted lines show the predicted positions in the heavy Bohr atom between n = 161 and n = 230 in the H + H system. In the range between n = the series is interrupted and overlaid by resonant structures, but appears again for n = 225. The Rydberg series converging to the H + 2 (v+ = 7, N + = 0, 2 and 4) states of the ion are denoted with open triangles, full triangles and open circles, respectively. where R h denotes the Rydberg constant for the H + H heavy Bohr atom. While in the experiment the X 1 Σ + g, v = 0, N = 0 ground level of the neutral H 2 molecule is taken as the zero energy level, the H + H (ionpair) dissociation limit E IP can be determined from a thermodynamic cycle [12], thus including the values for the ionization energy of the neutral H 2 molecule, the dissociation energy of the H + 2 ion and the electron affinity of the H-atom, yielding E IP = IE(H 2 )+D 0 (H + 2 ) EA(H) and resulting in E IP = cm 1. The Rydberg constant for the heavy Bohr atom H + H can be obtained by replacing the electron by an H particle, yielding the scaling factor (µ/m e ) = [12]. Note that the Bohr radius for the heavy system is a h 0 = m. Based on the numbers for E IP and R h the observed Rydberg series in parahydrogen can be assigned to principal quantum numbers n between 161 and 230 (with a gap between n = ). A fit to Eq. (4.2) yields a value for the quantum defect of δ = The observations displayed in Fig. 4.3 add to the findings. They show that the heavy Rydberg series can be observed at even lower energies, 86

103 4. Observation of a Rydberg series in a heavy Bohr atom where states of n = are located, that the B 1 Σ + u intermediate state of valence character can be used, as opposed to the C 1 Π u state of molecular Rydberg character, and that the series can also be observed in ortho-h 2, as opposed to para-h 2 in Fig A remarkable observation features at excitation energies above cm 1, where the assignment in terms of Rydberg states in the heavy Bohr atom ceases to match. From the more detailed upper part of Fig. 4.3 it becomes clear that in this energy range the ordinary or electronic Rydberg states converging to the v + = 6, N + = 1, 3 limits in the ortho-h + 2 core are excited. Also, some effects of mutual interaction between these Rydberg series, a well-known feature in molecular Rydberg series [90], seem to be present. From the perspective of electronic Rydberg spectroscopy, an intermediate state with a 2p-electron, 2pσ in case of the B 1 Σ + u state and 2pπ in case of the C 1 Π u state, should give rise to ns and nd-series, as are indeed observed [91] in such excitation schemes in H 2. Considering the angular momenta in a Hund s case (d) framework, mutually interacting ns and nd Rydberg series converging to several allowed N + series limits are expected. This is indeed observed in Fig. 4.3 with series converging to N + = 1 and 3, for v + = 6 in the ortho-ion core. We observed the same phenomenon in para-hydrogen, when exciting via the B 1 Σ + u, v = 12, J = 1 intermediate state in the energetic region near cm 1 (not shown here), where Rydberg series converging to N + = 0, 2 and 4, for v + = 6 in the para-ion core are observed. However, the observed spectral lines, indicated by HR(n) in Figs. 4.2 and 4.3, are not associated with electronic states, but with excitations in the heavy H + H system. An elucidating connection can be made between the quantum numbers that characterize the heavy Bohr atom, principal quantum number n and angular momentum quantum numbers l or J, and the usual molecular quantum numbers v and J associated with an intra-molecular potential-energy curve (in this case 1/R). This connection was established by Pan and Mies [92] to be n v + J + 1. It can be understood by considering the number of radial nodes in the wave function, v in the molecular system and n l 1, or n J 1 in the Rydberg system. In the present experiment for ortho-hydrogen only a J = 1 state is excited (J = 0 in the intermediate state), hence a single n series in the heavy Bohr atom results through jumps in the v quantum number. 87

104 4. Observation of a Rydberg series in a heavy Bohr atom H 2 + (6,1) H 2 + (6,3) H 2 + signal (arb. units) HR(n) Excitation Energy (cm -1 ) Figure 4.3: Lower panel: Characteristic spectrum after two-step laser excitation via B-X (12,0) P(1) in the ortho-h 2 molecule, showing n = Rydberg states in the heavy Bohr atom. In the upper panel the part of the spectrum between and cm 1 is enlarged, showing the electronic Rydberg series converging to the H + 2 (v+ = 6, N + = 1, 3) states of the ion. In para-hydrogen (a J = 1 intermediate state) J = 0 and 2 states can be excited, when applying the l = J = ±1 (Laporte) selection rule in the heavy Bohr atom. Hence, two series might be expected, possibly with different quantum defects δ J. Since this H particle is in a unique 1 S 0 state, there exists only a single series limit (neglecting hyperfine structure). In the potential-energy diagram (Fig. 4.4) a distinction is made between a short-range regime (R < 12a 0 ), where the covalent states in the H 2 molecule and the electronic Rydberg states are located, and a long range regime (R > 12a 0 ). At 10a 0 the ion-pair potential undergoes a strong interaction ( 3000 cm 1 ) with a dissociative state, thus giving rise to the characteristic double-well states in H 2, the H H 1 Σ + g state in the g-manifold [93] and the B B 1 Σ + u state in the u-manifold [94]. Heavy Rydberg states with small principal quantum numbers, thus having bond lengths of the order of the scaled Bohr radius, are unphysical because the H -ion itself is larger. Therefore, the principal quantum 88

105 4. Observation of a Rydberg series in a heavy Bohr atom number n of the lowest essentially pure ion-pair state in H + H is given by R = n 2 a h 0 12a m, i.e., n 100. All presently observed heavy Bohr atomic states fall in the energetic range between the H(1s)+H(3l) and H(1s)+H(4l) dissociation energies (see Fig. 4.4). Below n = 130 a perturbation in the heavy Rydberg series is expected, by about 60 cm 1 at 36a 0, due to the avoided crossing with H(3l) [95, 96, 97]. In this range and above n = 250, where another crossing with H(4l) occurs (presumably with a strength of only a few cm 1 ), the series in the heavy Bohr atom indeed dies out. Unresolved heavy Rydberg states were shown to exist above the n = 4 threshold by wave-packet experiments [12]. Photo-excitation of molecular states is governed by the Condon principle: Electronic orbitals are excited by the optical transition during which the nuclei remain essentially fixed in space. As a result, the strengths of transitions are governed by the square of the overlap integral of the initial and final ro-vibrational wave functions of nuclear motion (the Franck-Condon factor): I n = Ψ n(r)ψ B,C (R) dr 2 (4.3) By far the main part of the nuclear wave function of the heavy Rydberg states is at large internuclear distance (e.g., the outer turning point for n = 200 is R 90a 0 ), while the lower state is confined to R < 6a 0. The Franck-Condon factor of the transition should therefore be extremely small; further, excitation from the C state (which is confined to R < 3a 0 ) should be much weaker than from the B state, but this is not observed. From this we conclude that the heavy Rydberg states are not directly excited from the prepared intermediate B and C states via their Franck- Condon overlap. We invoke the mechanism of a complex resonance [98] to resolve this issue. A superposition quantum state: Ψ c = Ψ hr (n) + Ψ(H + 2, v+, J + ; e ) + Ψ(H, H ) (4.4) v +,J + diss with mixed character of electronic Rydberg channels (some 20 of those) associated with a Ψ(H + 2, v+, J + ) core representing the ionizationcontinua above each limit and a bound series below, dissociativecontinua (both n = 2 and n = 3 channels), and a series of Bohr atomic 89

106 4. Observation of a Rydberg series in a heavy Bohr atom Figure 4.4: Potential-energy diagram of the H 2 molecule with distinction between two separated regimes: for R < 12a 0 the range of internuclear distances where covalently bound states and the electronic Rydberg states converging to H + 2 (v+, J + ) dominate (dashed lines representing the potential curves of gerade symmetry in this region [93]), and the range R > 12a 0 where H + H heavy Rydberg states exist. The H + H Coulomb potential is shown in dark (red) and extended to smaller R by dots. Intermediate energy levels in the B and C states (at cm 1 excitation) are indicated and their accessible Franck-Condon ranges mapped (boxes). 90

107 4. Observation of a Rydberg series in a heavy Bohr atom states Ψ hr (n); in excitation the latter have a zero transition dipole moment. A crucial ingredient in the complex resonance model is a broad interloper state for which the transition dipole moment is large; a low n Rydberg state converging to a higher lying limit H + 2 (v+ ), hence a state in which v + is significantly larger than 6, takes on this role. As shown for another transition scheme in the H 2 molecule [98], under such conditions quantum interferences via different but indistinguishable pathways can lead to excitation of narrow series (in the example of [98] electronic Rydberg states), while excitation of the associated bound states may result in auto-ionizing decay. This closely matches the situation in the present experiment, although the number of channels is much larger than in an idealized example. The interpretation in terms of a complex resonance explains the occurrence of underlying (possibly structured) continua in the spectrum, the variation of widths over the bound series, and the variation of asymmetry in the line-shapes exhibiting differing Fano-parameters. At the same time it provides an explanation for the fact that H + H heavy Bohr atomic states are excited, while signal is detected in another channel, such as H + 2. A detailed quantitative analysis is beyond the scope of the present Letter It is interesting to note that the pp matter-antimatter protonium system [99] exhibits nearly the same Rydberg constant as the H + H system. Hence both systems have a similar energy level structure. In H + H the electrons have the positive effect to mediate a complex resonance, thus providing optical access to the series of heavy quantum states, a phenomenon which does not occur in pp. In conclusion, we have observed a frequency-resolved series of quantum resonances in a heavy Rydberg system by two-step excitation. In principle such series should be observable in any A + B diatomic molecule, or in A + (BC) poly-atomic molecules [92], as long as the negative particle is bound (which would exclude e.g. N ). In such systems a heavy Rydberg series is associated with each quantum state (rotational, vibrational, electronic and fine structure) in either of the two constituting particles. The presently studied H + H system is the simplest, not only because H 2 is the simplest neutral molecule, but in particular since H in the 1 S 0 state has no internal structure and therefore only a single Rydberg series limit exists, making the system truly a heavy Bohr atom. 91

108

109 Chapter 5 Spectroscopic observation and characterization of H + H heavy Rydberg states A series of discrete resonances was observed in the spectrum of H 2, which can be unambiguously assigned to bound quantum states in the 1/R Coulombic potential of the H + H ion-pair system. Two-step laser excitation was performed, using tunable extreme ultraviolet radiation at λ = nm in the first step, and tunable ultraviolet radiation in the range λ = nm in the second step. The resonances, detected via H + and H + 2 ions produced in the decay process, follow a sequence of principal quantum numbers (n = ) associated with a Rydberg formula in which the Rydberg constant is mass scaled. The series converges upon the ionic H + H dissociation threshold. This limit can be calculated without further assumptions from known ionization and dissociation energies in the hydrogen system, and the electronegativity of the hydrogen atom. A possible excitation mechanism is discussed in terms of a complex resonance. Detailed measurements are performed to unravel and quantify the decay of the heavy Rydberg states into molecular H + 2 ions, as well as into atomic fragments, both H(n = 2) and H(n = 3). Lifetimes are found to scale as n 3.

110 5. Characterization of H + H heavy Rydberg states 5.1 Introduction The presence of ion-pair states and ionic binding in molecules has long been recognized. In diatomic molecules potential energy curves are known whose outer limb of the potential curve closely overlaps with the Coulombic 1/R attraction of the ionic constituents. Such electronic states were investigated in ICl [100] and in I 2 [101]. Also, in H 2 molecular states were identified for which the outer limb follows the H + H potential. In the low-energy region this holds for the well-known EF 1 Σ + g and the B 1 Σ + u states. At higher excitation energies the H H 1 Σ + g and the B B 1 Σ + u state were identified with dominant ionic character at large internuclear separation [102, 94]. At even higher energies in the H 2 molecule additional electronic states of ion-pair character were predicted [95, 97], but experimental searches for bound levels are still inconclusive [96]. The photo-excitation, and the photo-physics including ionization and dissociation phenomena above the ion-pair threshold have been widely investigated over the years. Suits and Hepburn published a review on the spectroscopy and dynamics of ion-pair dissociation processes [103]. While many of the ion-pair phenomena have been observed in diatomic molecules (see e.g. the work on Cl 2 [104, 105, 106]), some of the studies have focused on poly-atomics as well: the excitation mechanism of ionpair states in bromomethane was recently investigated [107]. Previously, it had been demonstrated that ion-pair dissociation (in N 2 O) can also lead to an unstable fragment (N ) which then autodetaches [108]. The H 2 molecule is often chosen as the benchmark target for investigation of ion-pair phenomena in molecules. Already in the early studies involving classical light sources by Chupka et al. [85] and by McCulloh and Walker [109] a strong coupling was found in the H + H continuum above 17.3 ev between the ion-pair channel and electronic Rydberg series converging upon highly excited vibrational levels of H + 2. In the ion-pair photoionization production curve (obtained via H detection) the structure of predissociating and autoionizing Rydberg series was clearly observed. Pratt and coworkers applied a laser multi-step excitation scheme to investigate the same energy region in H 2 [110]; they found strong coupling of the H + H continuum to electronic Rydberg series converging upon v + = 9 in the H + 2 ion. In a subsequent study [111] 94

111 5. Characterization of H + H heavy Rydberg states it was found that the observed resonances above the H + H threshold decay via autoionization as well as dissociation involving H(n=3) and H(n=4) dissociation products. Kung and coworkers [112] approached the same energy region above the H + H limit via intermediate states (C 1 Π u, v = 2) and (B 1 Σ + u, v = 12) of ungerade symmetry. They observed the same Rydberg series converging to v + = 9 in the H + 2 ion, again employing H detection. This once more demonstrated the strong coupling between the H + H ion-pair and the electronic Rydberg channels. This coupling is of importance for the effective oscillator strength for quantum states in the H + H potential. The studies mentioned here were all focused on the energetic region above the ion-pair threshold. The energy region very near and just below the onset of the H + H threshold was investigated by various laser spectroscopic methods. Shiell et al. applied the threshold ion-pair production spectroscopy (TIPPS) technique to the H + H system [86], therewith demonstrating the existence of long-lived high-n ion-pair, or heavy Rydberg states. Reinhold and Ubachs provided further evidence for such high-n heavy Rydberg states by producing wave packets of angular momentum states in an electric field and following their time evolution before pulsed field dissociation (i.e ion-pair formation) [87]. The temporal behavior in the wave-packet dynamics can be quantitatively interpreted in terms of a heavy Rydberg system, or a heavy Bohr atom : a hydrogen atom, in which the negatively charged electron is exchanged for a composite H particle, which is treated as a point-like entity [12]. It was postulated that heavy Rydberg systems follow the physical laws of electronic Rydberg systems and that each system can be described by a single mass-scaling parameter that defines all properties of heavy Rydberg states. This was tested and quantitatively demonstrated in detecting wave packet dynamics in the H + F system [113], which has a reduced mass differing by a factor of two from the H + H system. As a follow-up on a recent Letter on the observation of a spectrum of energy-resolved heavy Rydberg states in the H + H system [114] we here report on a fuller description of these remarkable features. Regularly structured resonances are detected that obey a generalized Rydberg formula [12]: E n = E IP R h (n δ) 2 (5.1) 95

112 5. Characterization of H + H heavy Rydberg states with n the principal quantum number observed over the range , E IP the ion-pair threshold, R h the Rydberg constant of the heavy system, and δ a quantum defect, which is usual for a non-point-like system [92, 82]. Although the assignment of principal quantum numbers to the observed heavy Rydberg states is unambiguous, there still exist some open issues about the spectroscopic features. They concern the excitation mechanism of heavy Rydberg states, their quantum defects, and their interactions with electronic Rydberg states; these form the subject of the present paper. 5.2 Experiment and observations The range of excitation energies cm 1 in the H 2 molecule (above the v = 0, J = 0 ground state of para-h 2 ) is investigated via two-step laser excitation in an experimental configuration as schematically displayed in Fig The first step is induced by tunable extreme ultraviolet (XUV) radiation, which is produced by third harmonic generation in a krypton gas jet. The XUV-laser is tuned and fixed at some intermediate resonances in H 2. Here any strong line in the spectrum of the Lyman and Werner absorption bands can be chosen; we performed the studies at XUV-wavelengths at which resonanceenhanced third harmonic (THG) in krypton can be utilized to deliver abundant amounts of XUV radiation. Such resonances are found to exist at λ = 94.6, 95.1, 95.25, and 96.3 nm [9]. At the wavelength positions of these THG resonances the low rotational states in the B 1 Σ + u - X 1 Σ + g (12,0) and C 1 Π u - X 1 Σ + g (3,0) bands can be probed. A special experimental feature in the present study is the filtering of the powerful fundamental radiation (used in the third harmonic generation process) from the XUV-harmonic by a method of spatially-selective phase-matching as discussed in Ref. [115]. This is done to avoid non-resonant ionization of the excited H 2 molecule by the powerful UV pulses from the first laser. In a second excitation step a tunable ultraviolet (UV) laser pulse in the range nm, obtained from a frequency-doubled pulsed dye laser, is employed to transfer the population of the intermediate states to the energetic region, where the heavy Rydberg states are expected. Pulses from both lasers are spatially overlapped in a region where they perpendicularly intersect a pulsed and skimmed molecular beam of pure 96

113 5. Characterization of H + H heavy Rydberg states Figure 5.1: Experimental setup. An XUV laser beam, generated via third harmonic generation in a jet of krypton, intersects a molecular beam of H 2. Counter-propagating is a second UV laser beam, further exciting the H 2 to heavy Rydberg resonances. Ions (H + 2 and H+ ) produced after decay are accelerated by extraction field plates and detected after time-of-flight selection. H 2. Temporal overlap is also required in view of the short lifetimes of the intermediate states ( 0.5 ns). Signal is detected by either monitoring H + 2 ions generated from immediate decay of the H+ H resonances or H + ions, produced via dissociation involving H(n = 2) and H(n = 3) products; the latter are subsequently ionized by the UV laser. Time-of-flight mass selection permits distinguishing signals from either H + 2 or H+ ions. Pulse extraction takes place after both laser pulses so that excitation of the heavy Rydberg states is warranted under field-free conditions. In Fig. 5.2 a spectral recording of a heavy Rydberg series is shown when using B 1 Σ + u, v = 12, J = 1 in para-hydrogen as an intermediate state. The spectrum is calibrated to a total excitation energy (above X 1 Σ + g, v = 0, J = 0 in the neutral H 2 molecule) by adding the calibration of the second tunable UV-laser to the level energies of the intermediate states. The R(0) line in the B-X(12,0) band is at cm 1 [116], hence this is also the level energy of the B 1 Σ + u, v = 12, J = 1 intermediate state. This recording clearly shows 97

114 5. Characterization of H + H heavy Rydberg states H 2 + (6,4) H 2 + (6,0) H2 + (6,2) H 2 + signal (arb. units) 5 15 HR(n) Excitation Energy, cm -1 Figure 5.2: Lower panel: Characteristic spectrum after two-step laser excitation via B-X (12,0) R(0) in the para-h 2 molecule. The markers and dotted lines show the predicted positions of the heavy Rydberg series between n = 134 and n = 161. In the range between n = the series is interrupted. In the upper panel, the part of the spectrum between and cm 1 is enlarged, showing in detail the corresponding resonances for the electronic Rydberg series converging to the H + 2 (v+ = 6, N + = 0, 2 and 4) states; these states are denoted with open triangles, full triangles and open circles, respectively. the series of resonances that can be assigned to heavy Rydberg states of principal quantum numbers n = , indicated in Figure 5.2. In the region above cm 1 this series is interrupted and instead electronic Rydberg series appear that converge to v + = 6, N + = 0, 2 and 4 in H + 2. Fig. 5.3 shows a second spectral recording of the heavy Rydberg series in para-h 2, now using the R(0) line in the C-X(3,0) band in the first excitation step. Hence the C 1 Π u, v = 3, J = 1 level of (-) or (e) parity at cm 1 above the H 2 ground state acts as the intermediate state. This spectrum clearly shows resonances that can be assigned as heavy Rydberg states with principal quantum numbers n = In this spectrum no signatures of electronic Rydberg 98

115 5. Characterization of H + H heavy Rydberg states H 2 + signal (arb. units) Excitation Energy, cm -1 Figure 5.3: Spectrum obtained in two-step laser excitation via C-X (3,0) R(0) in the para-h 2 molecule. The markers and connected vertical lines show the predicted positions of the heavy Rydberg series between n = 162 and n = 192. series are discernable, although electronic Rydberg series converging to v + = 7 are expected in this energy range. In addition to the excitation spectra displayed in Figs. 5.2 and 5.3, both obtained in para-hydrogen, also spectra in ortho-hydrogen were recorded. In such spectra, located in the energy range cm 1, using the P(1) line in the B-X (12,0) band of H 2, heavy Rydberg series are observed as well; a recording is shown in Fig. 3 of Ref. [114]. In those observations excitation of the heavy Rydberg series competes with excitation of electronic Rydberg series converging to the v + = 6, N + = 1 and 3 limits in the H + 2 ion. An important experimental result when studying Rydberg series is a determination of quantum defects. Rearranging Eq. (5.1) yields an expression for an effective quantum defect: Rh δ n = n (5.2) E IP E n where E n is an observed line position and n is the (integer) quantum number assigned to that particular resonance. Determination of quantum defects can be accomplished by fitting of the spectral line shapes. While in principle the quantum defect can be larger than 1 in a system 99

116 5. Characterization of H + H heavy Rydberg states n (a) effective difference d(n) (b) -0.4 via J=1, C(v=3) via J=1, B(v=12) n Figure 5.4: (a) Calculated quantum defects δ n, from the complex resonance model, assuming the interlopers are the Rydberg states (n = 5, 6, 7) converging to H + 2 (v+ = 9) N + = 0 (marked with full lines), N + = 2 (marked with dashed lines) and N + = 4 (marked with dotted lines). The electronic Rydberg states are assumed to have zero quantum defect. (b) Observed quantum defects δ n obtained from spectra of para-hydrogen, using two different intermediate states: B 1 Σ + u, v = 12, J = 1 and C 1 Π u, v = 3, J = 1. with an extended core (as for the s-series in alkali atoms), here only the fractional part of the quantum defect δ n is deduced. Problems in determining the values for δ n lie in the treatment of the structured underlying continuum that gives rise to background slopes, in the moderate signalto-noise ratio, and in the occurrence of asymmetric Fano-type line profiles. These phenomena lead to uncertainties in the determination of the quantum defects. Analysis of the spectra obtained for para-hydrogen, for which long series of heavy Rydberg states are observed, results in values of quantum defects as plotted in the lower panel of Fig Another feature associated with the heavy Rydberg resonances is their width as observed in the spectral recordings. In many cases asymmetric line shapes were observed, and in other cases the underlying 100

117 5. Characterization of H + H heavy Rydberg states 40 Line width (cm -1 ) via J =1, B(v =12) via J =1, C(v =3) n Figure 5.5: Linewidths Γ for some of the heavy Rydberg resonances observed with good signal-to-noise ratio and symmetric line shapes as a function of principal quantum number n. The central (full) line represents a fit to a functional form Γ n 3 with the outer (dashed) curves representing the 1σ uncertainty. continuum was very structured, making it difficult to derive a reliable value for the resonance width. For those transitions observed at reasonable signal-to-noise ratio and with a symmetric line profile the widths Γ were determined and plotted in Fig The spectra displayed in Figs. 5.2 and 5.3 are recorded by monitoring H + 2 signal. Indeed, autoionization is the most significant decay channel, forming H + 2 ions in all quantum states below the total excitation energy, i.e, v + = 0 5. Detection of H + 2 is a means of recording the heavy Rydberg states, although in the present study we cannot determine the final ionic states H + 2 (v+, N + ). A competing channel (difficult to quantify but we estimate several 10 %) is photodissociation in which the heavy Rydberg states decay into H(n=1) + H(n=2) and into H(n=1) + H(n=3). The second UV-laser exciting the heavy Rydberg states further ionizes both H(n=2) and H(n=3), producing H + that can be detected separately from H + 2 in the time-of-flight setup. To further distinguish and characterize the dissociation channels the ion particle detector was replaced by a velocity map ion-imaging setup [117] to investigate if both dissociation pathways indeed occur. A result recorded upon excitation 101

118 5. Characterization of H + H heavy Rydberg states of the n = 191 heavy Rydberg resonance at cm 1 excited via the C 1 Π u, v = 3 intermediate state is shown in Fig This experiment indeed proves that both H(n=2) and H(n=3) are produced. The ring structures can be unambiguously assigned to the kinetic energies required for these channels. In the outer ring H(n = 2) atoms are detected that have higher kinetic energy from the dissociation process. In the inner ring the H(n = 3) atoms, that have higher internal energy and lower kinetic energy, are probed. No distinction is made between 2s and 2p atoms, or between 3s, 3p and 3d hydrogen atoms. The angular distributions can be interpreted to gain further understanding of the decay mechanism; this is outside the scope of the present work, but will be pursued in the future [118]. 5.3 Analysis Interpretation of observed features In the spectra two distinct phenomena are apparent. First, series of electronic Rydberg series are observed. From an intermediate state with a 2p-electron, 2pσ in case of the B 1 Σ + u state and 2pπ in case of C 1 Π u, ns and nd-series can be expected. Due to selection rules Rydberg series in ortho-h 2 can converge upon N + = 1 and 3, and series in para-h 2 upon N + = 0, 2 and 4 for each v + vibrational quantum number [91]. The (v +, N + ) ionization limits can be calculated from the value of the ionization energy of H 2 ( cm 1 [119]) and the level energies in the H + 2 ion [120]. In the spectrum of Fig. 5.2, excited via the B 1 Σ + u, v = 12, J = 1 intermediate state in para-h 2, three series are observed, converging to v + = 6, N + = 0, 2 and 4 states in the H + 2 ion. These states show the typical behavior of series interaction as often observed in electronic Rydberg series in the H 2 [91, 121] and HD [122] molecules. One of the visible effects of interaction of a series with continua is that the Rydberg states converging upon the v + = 6, N + = 4 limit clearly appear as window resonances in Fig It is noted that it requires an nd Rydberg series converging upon N + = 4 to be observed in excitation from an intermediate state of total angular momentum J = 1; selection rules do not permit excitation of an ns series converging to N + = 4. Similar 102

119 5. Characterization of H + H heavy Rydberg states Figure 5.6: Ion image representation of the kinetic energy distribution upon excitation of a heavy Rydberg resonance. The XUV beam passes from left to right, the second UV laser beam from right to left; polarizations are linear and parallel and in the plane of the paper. The outer ring corresponds to H + ions of high kinetic energy in decay to a H(n = 2) fragment, whereas the inner ring represents decay to H(n = 3). The off-center dot represents detection of H + 2 ions; the offset from the center is related to the fact that H + 2 are accelerated to lower velocities in a typical velocity-map-imaging configuration than H +. Therefore they travel a longer time along the direction of the molecular beam, which is from up to down in the figure. Note that we present raw data here, which are not 2D 3D inverted. electronic Rydberg series were observed in excitation via the B 1 Σ + u, v = 12, J = 0 state in ortho-h 2. Selection rules impose that only Rydberg states converging to v + = 6, N + = 1 and 3 should be observable, which is indeed the case. Again, qualitatively some features of series interaction are visible in the spectrum of Fig. 3 in Ref. [114]. In the Rydberg series converging to v + = 6, N + = 3 the Fano q-parameter is found to vary as a function of the principal quantum number, indicative of an interaction between multiple channels. However, the observations on electronic Rydberg states are not the predominant issue in the present study, and are left for future analysis. 103

120 5. Characterization of H + H heavy Rydberg states The second and key feature in the spectra is the observed series of broad resonances that follows Eq. (5.1), the heavy Rydberg series, as indicated in the spectra of Fig. 5.2 and 5.3 for para-h 2, and in Fig. 3 of Ref. [114] for ortho-h 2. They represent the first spectroscopic observation (i.e. frequency-domain resolved resonances) of heavy Rydberg states in a molecular system. The assignment in terms of heavy Rydberg states is unambiguous, since the spectral features follow the representation by Eq. (5.1) without invoking adjustable parameters. As discussed by Reinhold and Ubachs [12] the crucial parameters can be calculated in elementary fashion. The Rydberg constant R h for the H + H heavy Bohr atom can be obtained by replacing the electron by an H particle, yielding the scaling factor (µ/m e ) = Hence, via R h = (µ/m e )R one obtains: R h = cm 1. The other parameter in Eq. (5.1), the value for the ion-pair dissociation limit E IP, can be determined from E IP = IE(H 2 ) + D 0 (H + 2 ) EA(H), yielding E IP = cm 1. Based on these parameters, and choosing zero quantum defects (δ n = 0), the positions HR(n) of the heavy Rydberg states in Figs. 5.2 and 5.3 can be calculated in a straightforward manner. The positions of zero quantum defects are indicated in the figures with dotted lines to guide the eye. The observed resonances clearly follow the calculated regular structure of the heavy Rydberg states, making the assignment unambiguous. On quantum defects There are some striking observations in the spectra, which require further discussion and explanation. Here we address three: the fact that the heavy Rydberg states are only observed for principal quantum numbers n > 100, the issue of angular momentum quantum numbers and selection rules, and the question as to where the oscillator strength for exciting heavy Rydberg states originates. Effective quantum defects for the heavy Rydberg series are derived and plotted in Fig The Bohr radius in a two-particle quantum system is defined by [12]: a 0 = 4πɛ 0 2 µe 2 (5.3) 104

121 5. Characterization of H + H heavy Rydberg states H 2 + (8,0) H 2 + (5,0) H H + H H(n=4) H(n=3) Energy (cm 1 ) C H(n=2) ~ B ~ X Internuclear distance (a 0 ) Figure 5.7: Potential-energy diagram of the H 2 molecule with distinction between two separated regimes: for R < 12a 0 the range of internuclear distances where covalently bound states and the electronic Rydberg states converging upon H + 2 (v+, N + ) dominate (dashed lines representing the potential curves of gerade symmetry in this region [102, 94, 95, 97, 96]), and the range R > 12a 0 where H + H heavy Rydberg states exist. The H + H Coulomb potential is extended to smaller R by dots. Intermediate energy levels in the B and C states (at cm 1 excitation) are indicated and their accessible Franck-Condon ranges mapped (boxes). with µ the reduced mass of the heavy Bohr system ( m e ). Hence, the Bohr radius a h 0 of the heavy Bohr system (H+ H ) becomes as small as m. This is much smaller than the extension of the composite H particle [123]; consequently, the model of heavy Rydberg states breaks down at small internuclear separation, or at small principal quantum numbers n. The potential energy diagram (Fig. 5.7) shows how the H + H Coulombic 1/R potential is affected by perturbations of covalent states even at large internuclear separation. Near the H(n = 2) dissociation threshold there is a large interaction, with an avoided crossing of some 3000 cm 1 at 10a 0. If the H + H potential can be considered pure for R > 12a 0, this means R = n 2 a h 0 12a m, 105

122 5. Characterization of H + H heavy Rydberg states and the description in terms of heavy Rydberg states may be considered valid only above n 100. It is noted that perturbations with covalent states exist at internuclear separations of 36a 0 and at some 250a 0, where the H(n = 3) and (n = 4) dissociation curves cross [95, 97, 96]. These perturbations are much weaker, but should be included in more refined models of H + H heavy Rydberg states. The above considerations imply that quantum defects in the heavy Bohr system are large, much larger than in the s-series of alkali atomic systems. Pan and Mies [92] have calculated such quantum defects for the Li + I ionically bound system from first principles, and deduced that the lowest 177 angular momentum states are removed from the purely Coulombic spectrum. Previously Asaro and Dalgarno had found a number of 128 excluded levels in the Li + F system [124]. In the experimental analysis we will deal, as usual, with the fractional part of the quantum defects δ n. In the work of Pan and Mies [92] a relation was also established between sets of quantum numbers as commonly used for Rydberg states (n, l, m) and those usually invoked to describe molecular motion (v, J, M). The orbital angular momentum is either described as l equivalent to J, with projections of a magnetic quantum number m or M. A connection for the principal quantum number can be established through the relation n v + J + 1. Hence, the equation for the Rydberg energy levels can be written as [92]: E v,j = E IP R h (v + J + 1 δ J ) 2 (5.4) where the J-dependence of the quantum defect is explicitly written. In most of the spectra reported here, H 2 is excited in its para-configuration via a R(0) transition in the first step. Hence, from the intermediate J = 1 state, following the l = J = ±1 selection rule, in principle two heavy Rydberg series can be excited, with J = 0 and J = 2 angular momenta. These series may possess different quantum defects δ 0 and δ 2, since low angular momentum states are known to exhibit strongly varying quantum defects; an example is the strong difference between the s and p electronic Rydberg series in the Na atom. In our experiments we only observe a single heavy Rydberg series for para-h 2. Clearly, this requires an explanation. Notably, Pan and Mies [92] have shown in a theoretical analysis of the Li + I heavy Rydberg system, that for the low angular momentum 106

123 5. Characterization of H + H heavy Rydberg states states the quantum defects δ J can vary as by much as 1 for each step in J. If such a situation would also prevail in the para-h + H system, the two heavy Rydberg series that are expected would approximately coincide. The same would be true if both quantum defects δ 0 and δ 2 would differ by zero or any integer number. Without any knowledge about quantum defects, an explanation for the observation of only a single series must remain tentative. In ortho-h 2, when using the P(1) transition in the first step to excite the molecule from X 1 Σ + g, v = 0, J = 1 to a J = 0 intermediate state, only a single heavy Rydberg state with J = 1 is expected and observed. It is important to note that the heavy Rydberg series in both paraand ortho-h 2 possess the same ion-pair dissociation limit. This is not necessarily true for every heavy Rydberg series, but is a consequence of the fact that both the H + and H fragments possess no internal electronic structure. In addition, the distinction between para- and ortho-h 2 remains unaffected even if the nuclear spins have an internuclear separation of 12a 0 or more (several hundred a 0 in case of n = 200). This once more underlines the fact that it is not the distance-dependent interaction between the nuclear spins, but only the different symmetries of the nuclear spin wave functions that cause the ortho-para distinction. Another point of interest concerns the fact that situations in which both electrons are positioned on either nucleus A or B should be indistinguishable. If we define the corresponding localized wave functions as A and B, the true degenerate solutions of the Schrödinger equation are 1/ 2(Ψ A ± Ψ B ). However, even very small stray electric fields will lift this degeneracy due to the large polarizability of heavy Rydberg states [12]. Intensities via channel interaction: complex resonance The mechanism for exciting molecular ion-pair states has been discussed in the literature [101, 107] and Franck-Condon arguments play a role. High-n states in H + H cannot be directly excited from the intermediate states B 1 Σ + u, v = 12 and C 1 Π u, v = 3, because their wave function density is confined to R < 6a 0 and R < 3a 0, respectively. The internuclear distances at which the intermediate states exhibit wave function density is mapped onto the energy region of the heavy Rydberg states with 107

124 5. Characterization of H + H heavy Rydberg states color-(grey) shaded rectangles in Fig Intensities I n for excitation of heavy Rydberg states, based on Franck-Condon factors, should follow: I n = Ψ n(r)ψ B,C (R) dr 2 (5.5) Similar to usual Rydberg wave functions of high principal quantum number, heavy Rydberg states exhibit a wave function with their density entirely located at large separation from the core, in this case the H + H separation. In the language of molecular structure, the wave function is concentrated at the outer turning point of the H + H potential. Inspection of Fig. 5.7 indicates that this is at R a 0. This implies that direct excitation of the heavy Rydberg states should be Franck-Condon forbidden. In order to provide an explanation for their excitation, the model of a complex resonance is invoked. The observation of Rydberg series to which a direct transition is highly improbable has been reported for the atomic case [125]. For the molecular case this phenomenon was observed and analyzed for the case of H 2. Jungen and Raoult [98] developed a model of channel interaction that explains how a Rydberg series with zero transition dipole moment can borrow effective oscillator strength, which they baptized as a complex resonance. Later this phenomenon was modeled in basic form by Giusti-Suzor and Lefebvre-Brion [126]. Before givivg details we consider if simpler models might explain the observed features; interaction between a heavy Rydberg series (without oscillator strength) with a broad continuum (which exhibits oscillator strength) would in a Fano-type analysis lead to window resonances, contrary to what is observed. The complex resonance model involves (at least) three channels: one open channel represented by an ionization continuum, and two closed channels, viz. an interloper (i.e. a strong resonance, excited with a large transition dipole moment) and a (Rydberg) series of close-lying states. Couplings are assumed between each of the closed channels with the continuum, and between both closed channels. From this simple model it is shown that an interference takes place between on the one hand the direct interaction between both closed channels, and on the other hand the indirect interaction between the closed channels via the continuum. This interference effect can enhance or diminish the intensities of the observed transitions to both interloper and members of the Rydberg series, and also affects line positions, line shapes and widths. In general, 108

125 5. Characterization of H + H heavy Rydberg states v + =9, N + =4 v + =9, N + =2 v + =9, N + =0 H + H - continuum channel 5 channel 4 channel 3 channel 2 channel 1 R 25 R 12 H 2 + R 24 R 23 R 13 R 15 R 14 intermediate state Figure 5.8: Interaction diagram for the model of a complex resonance in the case of the H + H series. The interloper states are assumed to be members of the electronic Rydberg series converging upon the H + 2 (v+ = 9, N + = 0, 2 and 4) states of the ion (for the case of para-hydrogen; in case of ortho-hydrogen N + = 1 and 3). The upward pointing arrows represent transition dipole moments giving rise to direct excitation, while the sideways pointing arrows refer to channel interactions. Note that in this 5-channel model the electronic Rydberg series converging to v + = 6, as observed in the spectrum of Fig. 5.2 are excluded; this is one of the many assumptions made. an essential feature of the model is that any channel i, for which the transition dipole moment D i is zero, can obtain an effective oscillator strength σ i due to interference, such that it can be excited and observed. In our case, the simple model of three interacting channels as presented in Ref. [126] is not exactly valid, but can easily be extended to a situation that involves more than one interloper. In a five-channel description, graphically displayed in Fig. 5.8, we assume that the closed bound H + H heavy Rydberg series (channel 2) possesses a zero transition dipole moment (D 2 = 0). A further simplification of this model is that the electronic Rydberg series as observed in Figs. 5.2 are left out. As far as interlopers are concerned, they are assumed to be low-n members of electronic Rydberg series that converge upon high v + levels in the H + 2 ion. From the work of Chupka et al. [85], McCulloh and Walker [109], Pratt and coworkers [110, 111], and Kung et al. [112] a 109

126 5. Characterization of H + H heavy Rydberg states strong coupling between the H + H continuum and the H + 2 (v+ = 9) electronic Rydberg series is apparent. Therefore we hypothesize that our interloper states are members of this particular v + = 9 Rydberg series. Taking the oscillator strength to be continuous across the ionpair dissociation limit, we assume that this coupling persists below the threshold. Following angular momentum selection rules, from a p intermediate state ns and nd electronic Rydberg states can be excited. In view of the fact that p nd transition dipole moments are larger than those for p ns, we limit ourselves to nd states. In a Hund s case (d) coupling scheme for para-hydrogen nd electronic Rydberg series can converge upon three different limits (N + = 0, 2 and 4) of the ionic ground state. Low-n members of these series will act as interloper states and will be denoted by the closed channels 3, 4 and 5 in Fig. 5.8, respectively. As for the open continuum channel, at an excitation energy of cm 1 the system is above the v + = 5 level of the ion. Hence, all the electronic Rydberg states converging upon the v + = 0 5 levels can contribute via autoionization to a background continuum signal of H + 2. We consider only a single continuum as channel 1 in our model. This is another crude approximation of the true situation at hand. Following a multi-channel quantum defect theory (MQDT) approach similar to Ref. [126], we introduce a set of five adjusted channel wave functions, including an open channel (the interacting continuum, ϕ 1 ) and four closed channels: the H + H series (ϕ 2 ) and three electronic Rydberg (interloper) series converging to v + = 9, N + = 0, 2 and 4 (ϕ 3, ϕ 4 and ϕ 5 respectively). The total wave function of the system can be written as: ψ = 5 Z i cos [π (ν i + µ i )] ϕ i (5.6) i=1 with Z i the amplitudes, ν 1 = τ the open channel phase, and for i = 2 ν 2 = ( ) 1/2 Rh (5.7) E IP E the phase of the H + H series, where R h is the Rydberg constant for the heavy Bohr atom ( cm 1 ) and E IP is the ion-pair dissociation 110

127 5. Characterization of H + H heavy Rydberg states limit ( cm 1 ) [12]. For the three interloper channels i = 3, 4, 5: ν i = ( Re E i lim E ) 1/2 (5.8) and R e is the Rydberg constant for the electronic Rydberg system R e = µ(h e )/m e ( cm 1 ), and Elim i is the ionization limit of channel i, calculated using the values of [120] (Elim 3 = cm 1, Elim 4 = cm 1, Elim 5 = cm 1 ). The effective quantum defects µ i have been all set to zero. Note that in this 5-channel model we assume that the three interloper states do not mutually interact, and the observed electronic Rydberg series converging to v + = 6 are not included. Similar as in Ref. [126], where a calculation was performed for the quantum defects of an electronic Rydberg series interacting with an interloper, we derive for the quantum defects of the heavy Rydberg series (channel 2): tan(x 2 ) = R 2 23 cot(x 3 ) + R 2 24 cot(x 4 ) + R 2 25 cot(x 5 ) (5.9) where x i = π(ν i + µ i ) is the notation for the quantum defects and R ij are the R-matrix elements representing the channel interactions. This is an important result: the resonance peaks will be found at the energies where this equation is fulfilled. The strength of the interaction between the interlopers and the H + H series (R 23, R 24 and R 25 in our notation), as well as the exact location of the interloper states greatly affect the position of the observed H + H lines. This will be reflected in the values for the quantum defects δ n of the H + H series. These values are calculated, assuming zero quantum defects (µ i ) for the interloper states converging to H + 2 (v+ = 9, N + = 0, 2 and 4). We have attempted to reconstruct the findings on the experimental quantum defects from this model; the small amount of experimental information and the large error bars on the one hand, and the large number of parameters on the other hand do not allow for a fit or a one-to-one correspondence of the model. However, by scanning the parameter space the solution of R23 2 = 0.01π, R2 24 = 0.01π and R2 25 = 0.03π somewhat mimics the observed trend in the observed quantum defects, as is displayed in Fig This merely serves as an illustration on how the complex resonance model might explain trends in observed quantum 111

128 5. Characterization of H + H heavy Rydberg states defects of the heavy Rydberg series. By comparing Figs. 5.4 and 5.5 it may be verified that at the locations of the interactions also resonances in the observed lifetimes seem to occur. In view of all the assumptions made and the limited amount of spectroscopic information, we cannot provide a conclusive model to quantitatively reproduce the entire spectrum of heavy Rydberg resonances, and the electronic Rydberg series observed in the same energy region. We have presented a general framework that gives some insight in the variation of the quantum defects as in Fig Furthermore from the generalized model of Ref. [126] several conclusions relevant for our purpose can be drawn. Whereas in the previous investigations of complex resonances [98, 126] the model was applied to electronic Rydberg series, here it describes properties of heavy Rydberg series: (i) Even if the members of a Rydberg series cannot be optically excited directly, coupling to an interloper and a continuum can still lead to their observation; (ii) The spectral range covered by a complex resonance can be much larger than the interloper width; (iii) The interaction strength between the Rydberg series and the interloper is of great importance, since it affects the widths and positions of the peak maxima of the series; (iv) Decay of the Rydberg resonances can occur into various channels including the continua. The latter is the subject of the next section. Resonance widths and decay The heavy Rydberg series are observed as short-lived resonances having typical widths of some 5 25 cm 1, corresponding to a lifetime of a picosecond or even a fraction thereof. The observed linewidths are displayed in Fig. 5.5 as a function of principal quantum number. The uncertainties in the data points are large, due to limited signal-to-noise ratio and asymmetries in the line shapes, but nevertheless some interesting conclusions can be drawn from these observations. Lifetimes of Rydberg states are known to scale as τ n 3 [82]. This is reflected in the observed scaling of the linewidths of Fig. 5.5 as n 3. Notwithstanding the large error margins the data in Fig. 5.5 somewhat follow the expected scaling law. In addition there is at least indication of an increased linewidth near n = 163. This resonance may hint at the location of the effective interloper. If this is true the interferences 112

129 5. Characterization of H + H heavy Rydberg states in the multichannel problem give rise to enhanced widths, where in principle the complex resonance model allows for both the possibility of enhancement or reduction of the linewidth. At n 200 a linewidth of Γ(200) = 10 cm 1 is observed. Following the usual n 3 dependence for Rydberg states, this would scale to Γ(2000) = 0.01 cm 1, which would correspond to a lifetime of 0.5 ns. However, in the studies on heavy Rydberg wave packets [12] for n = 2000 a lifetime of 90 ns was actually observed, deviating by over two orders of magnitude from the n 3 -scaling based on lower n-values. In addition, for these higher-lying heavy Rydberg states decay into the H(n = 4) dissociation channel was observed [12]; this was accomplished by selectively ionizing the H(n = 4) fragments by laser pulses of 1064 nm. The presently observed heavy Rydberg states of n < 230 are below the H(n = 4) dissociation threshold and therefore this decay channel is closed. Nevertheless, the lifetimes in the range n = are much shorter than those in the higher energy region probed in the wave packet experiments. This discrepancy may be resolved by considering the fact that the wave packet experiments were conducted in electric fields [87, 12]. As is extensively discussed in the literature on zero-kinetic energy (ZEKE) spectroscopy [127, 128] high-n Rydberg states are known to exhibit longer lifetimes in the presence of electric fields. Heavy Rydberg states have a large polarizability; even in small stray electric fields of < 0.1 V/cm at high n values full l-mixing (or rather J mixing) occurs [12]. In the oscillatory motion of the angular wave packets population is transferred to high angular momentum states of the heavy Rydberg system. This gives rise to longer lifetimes than expected from n 3 scaling. Indeed, as shown in Fig. 11 of Ref. [12], lifetimes were found to scale as n 4, a clear indication of angular momentum mixing in the electric field. Heavy Rydberg states in other systems Heavy Rydberg states are expected to exist in other molecular systems and a viable question is whether the presently observed rather short lifetimes are typical for heavy Rydberg states, or whether they depend on the specifics of the level structure and associated decay channels of the H 2 molecule. In H 2 heavy Rydberg states can be observed in 113

130 5. Characterization of H + H heavy Rydberg states the range where vibrational autoionization occurs (into v + = 0 5 vibrational levels) and dissociation into channels with H(n = 2) and H(n = 3); this may seem an unfortunate situation giving rise to strong decay and broad resonances. Experimentally H 2 is the only system where frequency-resolved heavy Rydberg series are observed. During the course of the investigations it was attempted to register similar heavy Rydberg series in D 2, however unsuccessfully: only electronic Rydberg series were observed in the D + 2 and D+ signal channels. This absence might be attributed to the lack of a suitably located interloper state in D 2. Stark wave packets were observed in both H + H and in the H + F system. In the latter system lifetimes of typically an order of magnitude larger were found [113] in the high n region where l-mixing is important. Hence, the H + F system might be a candidate for observing frequency resolved heavy Rydberg series. We note that the Rydberg constant for heavy Rydberg series scales with µ/m e, where µ is the reduced mass of the molecule. Therefore, for all systems heavier than H 2 the density of states for heavy Rydberg states increases, and it will be more difficult to observe them, in particular if the lifetimes are as short as in H + H. Further it is noted that in poly-atomic systems there is a multitude of series limits for heavy Rydberg states. Each bound quantum state (rovibrational, spin-orbit) in the positively or negatively charged ionpair fragments provides such a limit. This will further enhance the density of heavy Rydberg states. Finally we note that the p p protonium system, consisting of a proton electromagnetically bound to an antiproton should exhibit a series of quantum states of the same density and spacing as in the H + H system, since the Rydberg constants of those systems are similar. Experiments for detecting and investigating protonium are under way [99, 129]. 5.4 Conclusion The first spectroscopic observation of frequency-resolved heavy Rydberg states in a molecular system is reported. Bound quantum states in the 1/R potential in H + H are detected for principal quantum numbers in the range , as predicted from a generalized Rydberg formula, which is mass-scaled for the orbital motion of the H particle replacing 114

131 5. Characterization of H + H heavy Rydberg states the electron. The resonances decay via autoionization forming H + 2 ions as well as via dissociation forming H(n = 3) and H(n = 2) products. Lifetimes scale with n 3 and are typically shorter than 1 ps for the quantum states investigated (n = ). A complex resonance model is invoked, reproducing the observed quantum defects, and explaining the origin of the effective oscillator strengths of the heavy Rydberg series. In H + H the series could be observed in excitation from states of 1 Σ + u and of 1 Π u symmetry, and both in para- and ortho-hydrogen. Such heavy Rydberg resonances are hypothesized to exist in all A + B diatomic molecules, as well as in polyatomic molecules. 115

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133 Chapter 6 Elemental analysis of steel scrap metals and minerals by laser-induced breakdown spectroscopy The atomic emission of laser-induced plasma on steel samples has been studied for quantitative elemental analysis. The plasma has been created with 8 ns wide pulses using the second-harmonic from a Q-switched Nd:YAG laser, in air at atmospheric pressure. The plasma emission is detected with temporal resolution, using an Echelle spectrometer of wide spectral range ( nm) combined with an intensified charge coupled device camera. A plasma temperature of 7800 ± 400 K is determined using the Boltzmann plot method, from spectra obtained under optimized experimental conditions. As an example of an industrial application the concentration of copper in scrap metals is studied, which is an important factor to determine the quality of the samples to recycle. Cu concentrations down to 200 ppm can be detected. Another application of the laser-induced plasma spectroscopy method is the measurement of the nickel and copper concentrations in an iron-containing sample of reduced magma from the 1870s expedition to western Greenland by Adolf Erik Nordenskiöld. Different spectral lines of nickel are used for calibration, and their results are compared.

134 6. LIBS on scrap metals and minerals 6.1 Introduction Laser-induced breakdown spectroscopy (LIBS) is a powerful analysis method which has undergone a rapid development during the past years. The power of LIBS lies in its non-intrusive character, permitting on-line analyses to be made in harsh or otherwise inaccessible environments [130],[131], [132], and [133]. The range of applications that utilizes LIBS is growing rapidly, driven by the materials processing industries [134] and [135] but also by the environmental monitoring fields [136], [137], and [138] The technique can be used in a variety of more complex analyses such as determination of alloy composition, origin of manufacture (by monitoring trace components), and molecular analysis (unknown identification). There is also a wide range of applications for in vivo medical investigations [139], analysis of biological and pharmaceutical samples [140], archaeological investigations [141], art restoration and paint pigment analysis etc. There is also a possibility of using LIBS as an analytical technique for corrosive or hazardous environments (such as space and nuclear reactors) preventing risk to the operator. Although the sensitivity of the LIBS method cannot yet compete with some other analysis methods, e.g. inductively coupled plasmamass spectrometry, the rapid development of dedicated LIBS systems is constantly narrowing this gap, with sub-ppm levels as a clearly realistic goal. There are several kinds of LIBS-systems on the market. The ones that can combine rapidity with high resolution make use of large spectrometers. These systems are already now capable of few-ppm sensitivity for some elements [142], particularly in combination with double-pulse excitation [143]. The portable ICCD-based systems, as the one used in the present work [144], are in general somewhat less sensitive with lower resolution. A portable system has been compared with a laboratory system by Wainner et al. [138]. They found that their laboratory system was only about factor of 2 more sensitive than the portable one. In the present work, we are focusing on the use of LIBS as an on-line analysis tool in metal recycling industry, where there is an urgent need to get rid of traces of certain elements in order to obtain better raw materials for steel plants. As another application, we investigate the 118

135 6. LIBS on scrap metals and minerals possibilities of LIBS in detailed spatial analysis of mineral surfaces, in this case a lava sample found on Greenland for more than 100 years ago. The potential of LIBS in the mining industry as a field tool for on-site characterization of ore quality constitutes yet another important use of LIBS in the future. We are carrying out these studies as a preparation for the construction of a pilot plant where the techniques can be applied in realistic environments, as has been described by Stepputat et al.[145]. The first application is being carried out in collaboration with Stena Metall AB, Sweden [146]. The aim is to identify scrap metal samples with small concentrations of chromium and copper, for subsequent removal. The concentrations studied in this work range from about 0.04% to 1.6% (Cr) and 0.09% to about 0.7% (Cu). The other application, which also shows the power of the LIBS method, is the study of a lava sample. The lava sample originates from the island of Disko in Greenland and is now in the possession of the Swedish Museum of Natural History. During one of his expeditions to west Greenland in the 1870s, Adolf Erik Nordenskiöld collected the frozen magma sample with the belief that the metallic piece was a meteorite. Unfortunately for A.E. Nordenskiöld, the nickel contents reveal the sample s origin as terrestrial. Metallic iron on earth possesses a nickel concentration in the range 1% to 4%, while meteorites contain a much higher nickel concentration (5-16%) [147]. Using the LIBS method, the nickel concentration of the piece is studied using several calibration spectral lines of nickel. The main focus of this study is to observe if the different calibration lines will lead to similar results. For this purpose, the nickel concentration as well as the copper content of the lava sample is studied spatially. 6.2 Experimental Fig. 6.1 shows the experimental setup schematically. A pulsed Nd:YAG laser (Lumonics HY500) placed in air at atmospheric pressure creates the laser-induced plasma. The laser produces 8 ns wide laser pulses at its second harmonic wavelength (532 nm) of about 10 mj, with an adjustable repetition rate up to 10 Hz. The laser beam is focused perpendicular to the sample surface, producing plasma via a plano-convex 119

136 6. LIBS on scrap metals and minerals Figure 6.1: Experimental setup used in LIBS experiment. lens of 50 mm nominal focal length. Placing the focal plane 1 mm below the sample surface stabilizes plasma intensity and avoids plasma formation in air as the crater deepens [148]. In order to maintain constant conditions for the sample during different measurements, the sample is displaced each time to guarantee a surface free of impurities coming from sputtering of previous laser shots. The samples surfaces had been etched using a mixture of nitric acid and alcohol at the Swedish Museum of Natural History. A low laser repetition rate improves furthermore the data reproducibility as it matches the spectrometer readout speed. A fiber optic cable collects the emission from the formed plasma and transmits the collected light to the entrance slit of an echelle spectrometer (Mechelle 7500, Multichannel instruments, Sweden [149]). Echelle grating spectrometers are designed to work at high orders and high angles of incidence and diffraction. The Mechelle 7500 provides high constant spectral resolution (7500) over a wide spectral interval ( nm) displayable in one single spectrum. This allows the acquisition of a spectrum containing all used spectral lines generated in one single laser shot. An intensified CCD camera (Andor) coupled to the spectrometer finally detects the dispersed light. For the construction of the calibration curves, reference steel samples having representative contents of copper, nickel and chromium are used and are summarized in Table

137 6. LIBS on scrap metals and minerals Table 6.1: Reference samples used to create the calibration curves. Ref. samples Cu, % Ni, % Cr, % SS-CRM 408/ SS-CRM 410/ SS-CRM 402/ SS-CRM 409/ SS-CRM 401/ SS-CRM 215/ SS-CRM 416/ The mass contents of the elements are given in wt.% 6.3 Optimization of experimental parameters Experimental parameters such as delay time, gate time or number of shots to average vary from experimental group or publication [150]. Therefore, special care is taken in this work to find the optimum experimental parameters. Internal normalization reduces the influence of the sample properties and laser shot-to-shot variation [151]. Fig. 6.2 illustrates the improvement using internal normalization (Fe) on the standard deviation of the integrated line intensity of the nickel line at nm. The relative standard deviation (RSD) diminishes as much as 50%. Moreover, special care is taken to determine the optimum number of single-shot spectra to be averaged. As a representative example, Fig. 6.2 shows the behavior of the relative standard deviation of the nm Ni integrated line intensity versus the number of averaged laser shots. Here, the RSD value does not diminish significantly by increasing the number of averaged laser shots to more than 60. Therefore, the reproducible average spectrum is acquired using the spectra generated in 50 or 60 single shot laser pulses. All spectra are improved by subtraction of the dark current of the detector. At early stages of the plasma evolution, the continuum emission is overwhelming and covers many spectral lines. As the delay time increases, the continuum reduces strongly with a pronounced improvement of the signal-to-noise ratio. However, the self-absorption (optically thick conditions) of the lines emitted by the laser induced plasma increases with the delay time with respect to the laser pulse [152]. The visual 121

138 6. LIBS on scrap metals and minerals Figure 6.2: Relative standard deviation of integrated spectral line of nickel at nm. The value of the RSD reduces strongly using the normalization procedure. inspection of the spectra revealed that the self-absorption reversal of the copper line became more pronounced after 3 µs. Thus, the time interval chosen to perform the measurements is from 2 µs to 3 µs. This gate time of 1 µs is chosen because the plasma temperature does not decrease substantially within this time interval leading to changes in the equilibrium conditions of the plasma. 6.4 Results and discussion Calibration curves The spectroscopic lines studied in this work are: in the case of copper nm, chromium nm (resonance lines intended to study low concentrations) and nm, nm nm, nm, nm in the case of nickel [153] and [154]. A plasma temperature of 7800 K is calculated using six low self-absorption iron peaks free of overlap in the wavelength interval 379 to 443 nm by means of equation [155, 156]: ln( Iλ 0 q j Aij ) = E j kt + x ln[hcdn10 2 ] Z(T ) 122

139 6. LIBS on scrap metals and minerals The calculated temperatures in all measurements vary from 7400 to 8200 K, typical values for plasma generated in these experimental conditions [151], [152] and [157]. As the expected copper and chromium concentrations are very low (Cu 0.3%, Cr 0.1%), their resonance spectral lines are used to create the calibration curves [132]. In the case of nickel, the expected concentrations are higher, about 1% to 5%. For these concentrations, the resonance lines of nickel experience a high level of self-absorption and they are therefore not the most suitable lines to use. Instead, spectral lines free of overlap are used to create the calibration curves for nickel. The experimental spectral lines shapes are mainly determined by the instrumental line profile. Calibration curves are the plots of the integrated line intensity versus the (reference) sample elemental concentration. The integration of the spectral line intensity is performed using the measured data points. It results in a similar outcome compared to the integral using for instance a Voigt function fit of this same spectral line profile. The difference between these two results is ten times smaller than the standard deviation of the integrated line profile. Thus, the integral of the measured data points is used for all the calibration curves. To calculate the standard deviation of the integrated line intensities, 10 measurements are applied to each reference sample. Fig. 6.3 shows the spectra obtained for the copper line at nm for different elemental contents. In this case an interfering iron line at nm makes the used integration method inaccurate. Under the assumption that the line profile is symmetric, only the left half of the measured intensity line is integrated. Fig. 6.3 also shows that the background (baseline) is the same for different concentrations. As a result, the integration over the measured spectral line without correction for the baseline value only leads to a constant value added to the integral. This constant value is the same for all samples with different concentrations. The variation of the integrated line intensity with the certified sample elemental concentration is presented in Fig. 6.4, Fig. 6.5, Fig. 6.6 and Fig In these calibration graphs, the element concentrations are expressed as a percentage of the weight. For most of the analyzed lines, the integrated line intensity grows non-linearly with increasing certified concentration values. Fig. 6.4 and 123

140 6. LIBS on scrap metals and minerals Figure 6.3: Spectra of the copper line at nm, plotted for different concentrations. Figure 6.4: Calibration curve of copper spectral line at nm. The measurements are performed using 1 µs gate time and with 2 µs delay between measurement and laser firing. 124

141 6. LIBS on scrap metals and minerals Figure 6.5: Chromium calibration curve using the spectral line at nm. Figure 6.6: Calibration curves using nickel spectral lines at nm and nm. The dashed line shows the optically thin linear limit. 125

142 6. LIBS on scrap metals and minerals Figure 6.7: Calibration curves for Ni spectral lines at nm and nm. Fig. 6.5 show that the resonance lines of copper and chromium present strong self-absorption, as the optically thin linear limit is only valid for concentrations less than 0.101% in the case of copper, and 0.138% for chromium, unfortunately leaving only three of our reference samples on the linear range interval. For most used nickel lines, the optically thin linear limit curve can be used as calibration curve for concentrations up to 2% Ni. All calibration curves are fitted using the relation y = a + bc(1 exp( x/c)) where y is the integrated line intensity and x is the certified concentration value (wt.%) [158]. The limit of detection (LOD) is calculated using the 3σ criterium, using the standard deviation of the available samples with the lowest concentrations (0.0091% Cu, 0.019% Ni, and 0.040%Cr). The obtained values for the LOD of copper and chromium are 0.02% and 0.025% respectively. In the case of nickel, the different spectral lines give the following limits of detection: 0.5% (341.5 nm), 1.3% (346.2 nm), 1.6% (351.5 and nm) and 4% (351.0 nm). The LOD obtained are not as good as for example in Ref. [143]. The cause for these poor values lies probably in the unexpectedly large shot-to-shot variations. 126

143 6. LIBS on scrap metals and minerals Scrap metal samples The possibility to use LIBS for the on-line detection of low concentration of copper and chromium in scrap metal samples depends on the minimum amount of laser shots. In the optimum case this would only be one or two laser shots. Unfortunately, Fig. 6.2 shows that the relative standard deviation for an integrated line in one or two laser shots is at least 35%. This fluctuation is unacceptable and the possibility of measuring copper concentrations with only one laser shot is therefore discarded. The fluctuation in the calculated concentration is greatly reduced by averaging spectra. The results are presented for 10 and 50 single laser shots respectively. In the case of scrap metal #1, the copper contents are found to be 0.12 ± 0.03% for 10 laser shots, and 0.10 ± 0.02% for 50 laser shots. The results for copper of scrap metal #2 show a more clear discrepancy; as for 10 laser shots the copper concentration is 0.13 ± 0.03 % while for 50 laser shots this value is 0.09 ± 0.02%. The amount of chromium for both scrap metal pieces is not detected. Disko-lava sample A two dimensional study of the frozen magma piece is performed, and the locations of the measurements are presented in Fig For each of the measured points, the nickel and copper concentrations are obtained. The matrix effect is not accounted for in these measurements and can then affect their reliability. The composition of the lava sample obtained by analysis methods other than LIBS is to our knowledge unavailable and therefore the correctness of the LIBS technique cannot be tested. Fig. 6.9 shows the results for nickel with their uncertainties using different spectral lines for calibration for 2 of the measured points. The uncertainties in the obtained concentrations are calculated using the propagation of the uncertainties of the fitting parameters a, b, c of the calibration curves. All these calibration curves are checked by re-measuring the concentrations of the certified reference samples. The curves give correct values within the calculated uncertainties. However, the accuracy of the calculated nickel contents differs with the sample concentration. For example, using the reference sample with 3.02 % nickel leads to similar and correct results (within the uncertainty interval) for all the chosen refer- 127

144 6. LIBS on scrap metals and minerals Figure 6.8: Locations of the measurements performed on the lava sample. Figure 6.9: Nickel concentrations obtained with measurements performed onto the Disko sample. 128