Auger cascade processes in xenon and krypton studied by electron ...

AUGER CASCADE PROCESSES IN XENON AND KRYPTON STUDIED BY ELECTRON AND ION SPECTROSCOPY LEENA PARTANEN

REPORT SERIES IN PHYSICAL SCIENCES Report No. 46 (2007)

AUGER CASCADE PROCESSES IN XENON AND KRYPTON STUDIED BY ELECTRON AND ION SPECTROSCOPY LEENA PARTANEN Department of Physical Sciences University of Oulu Finland

REPORT SERIES IN PHYSICAL SCIENCES OULU 2007 • UNIVERSITY OF OULU

Report No. 46

Opponent Dr. Antti Kivim¨ aki, TASC-INFM National Laboratory, Trieste, Italy Reviewers Dr. Antti Kivim¨ aki, TASC-INFM National Laboratory, Trieste, Italy Dr. Pascal Lablanquie, LCP-MR CNRS and University Pierre et Marie Curie, Paris, France Custos Prof. H. Aksela, Department of Physical Sciences University of Oulu, Finland

ISBN 978-951-42-8662-9 (Paperback) ISBN 978-951-42-8663-6 (PDF) http://herkules.oulu.fi/isbn9789514286636/ ISSN 1239-4327 OULU UNIVERSITY PRESS Oulu 2007

i Partanen, Leena: Auger cascade processes in xenon and krypton studied by electron and ion spectroscopy Department of Physical Sciences, University of Oulu, Finland Report No. 46 (2007)

Abstract In this thesis electronic structure and the dynamics of selected rare gases were investigated. Auger electron and ion spectroscopies were employed in the studies for the de-excitation processes of an inner-shell vacancy excited by synchrotron radiation. The collected spectra were analyzed by combining information from the Auger spectra created with different excitation methods and from theoretical calculations. The effects of electron correlation on the Auger spectra were emphasized in the studies. Keywords: Auger decay, satellite Auger process, electron spectroscopy, ion spectroscopy, photon excitation, rare gases

ii

Acknowledgments This work was carried out at the Department of Physical Sciences, University of Oulu. I would like to thank the Head of the Department, Professor Jukka Jokisaari, for placing the facilities at my disposal. I am most grateful to my supervisor Professor Helena Aksela. She has guided me from the very first step to the world of atomic physics. It has been inspiring to follow her aspiration to find new challenges and knowledge. In addition, she has supported me in many ways during these years. I would also like to thank Professor Seppo Aksela for his guidance and instruction in the experimental electron spectroscopy. I am grateful to the reviewers of the thesis, Dr. Antti Kivimäki and Dr. Pascal Lablanquie. Their constructive comments made me to understand more about the research addressed in this study. This thesis bases on the great power of teamwork. I wish to express my sincerest gratitude to the experts on atomic physics, Dr. Marko Huttula, Dr. Rami Sankari, Dr. Valdas Jonauskas, Professor Edwin Kukk, and Dr. Sami Heinäsmäki, who were the co-authors of the original publications and who gave me invaluable guidance and comments during the research work. It is my pleasure to thank also other co-authors and collaborators who participated the measurements and calculations of the original publications. I would like to thank former and present members of the Electron Spectroscopy group for pleasant teamwork. In particular, I would like to thank Anna Sankari and Minna Patanen with whom I have shared the office. They have had time for discussions, for all the sense and nonsense. Dr. Tommi Matila is acknowledged for teaching me everything I ever needed to know about Cowan’s code. I want to thank the staff of the Department of Physical Sciences as well for enjoyable moments with teaching and coffee breaks. I want to acknowledged the financial sources. This work was supported by the Vilho, Yrjö and Kalle Väisälä Foundation, the Tauno Tönning Foundation, and the National Graduate School in Material Physics. Finally, I would like to express my deepest gratitude to my relatives and friends. Especially I want thank my loving husband Antti, who stood by me, and my childen, Taru and Juho, who have kept me busy outside of work. I am grateful to my parents Eila and Kalevi, brother Esa and Jenny, brother Antti, my parents-in-law Eila and Esko, and Maija and her family. Thank iii

iv you for being there always when I needed you.

Oulu, December 2007

Leena Partanen

List of original papers This thesis contains an introductory part and the following five papers which will be referred in the text by their Roman numbers. I. L. Partanen, R. Sankari, S. Osmekhin, Z. F. Hu, E. Kukk, and H. Aksela, Multiple ionization of Xe — comparison of de-excitation pathways following 3d5/2 ionization and 3d5/2 → 6p resonance excitation, J. Phys. B: At. Mol. Opt. Phys. 38, 1881–1893 (2005). II. V. Jonauskas, L. Partanen, S. Kuˇcas, R. Karazija, M. Huttula, S. Aksela, and H. Aksela, Auger cascade satellites following 3d ionization in xenon, J. Phys. B: At. Mol. Opt. Phys. 36, 4403–4416 (2003). III. L. Partanen, M. Huttula, S.-M. Huttula, H. Aksela, and S. Aksela, Effects of electron correlation on the M4,5 NN4,5 Auger transitions in xenon, J. Phys. B: At. Mol. Opt. Phys. 39, 4515–4524 (2006). IV. L. Partanen, M. Huttula, H. Aksela, and S. Aksela, The M4,5 N-NNN Auger transitions in Kr, J. Phys. B: At. Mol. Opt. Phys. 40, 3795– 3805 (2007). V. L. Partanen, M. Huttula, S. Heinäsmäki, H. Aksela, and S. Aksela, Xe N4,5 O-OOO satellite Auger spectrum, J. Phys. B: At. Mol. Opt. Phys. 40, 4605 - 4615 (2007). All Papers I–V were prepared as teamwork. The present author is responsible for the majority of text and data handling in all papers. All the theoretical calculations using the Cowan’s code package were performed by the author, excluding the calculations done with the method of global characteristics in Paper II and the MCDF calculation in Papers III, IV, and V. The present author participated in the experimental work for Papers II and III.

v

vi

Contents Abstract

i

Acknowledgments

iii

List of original papers

v

Contents

vii

1 Introduction

1

2 Overview of theoretical methods 2.1 Hartree-Fock method . . . . . . . . . . . . . . . . . . . . . . . 2.2 Atomic energy states . . . . . . . . . . . . . . . . . . . . . . . 2.3 Configuration interaction . . . . . . . . . . . . . . . . . . . . .

3 3 5 7

3 Electronic transitions 3.1 Selection rules for dipole transition . 3.2 Transitions following photoabsorption 3.3 Normal Auger transitions . . . . . . . 3.4 Cascade of Auger transitions . . . . . 3.5 Satellite Auger transitions . . . . . . 3.6 Method of global characteristics . . .

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9 9 10 13 15 17 18

4 Experiment 19 4.1 Synchrotron radiation . . . . . . . . . . . . . . . . . . . . . . 19 4.2 Electron spectroscopy . . . . . . . . . . . . . . . . . . . . . . . 21 4.3 Ion spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . 23 5 Summary of included papers 5.1 Papers I–III: Multiple ionization of Xe . . . . . . . . . . . 5.1.1 Partial ion yields following 3d excitation . . . . . . 5.1.2 Paper III: M4,5 -NN4,5 Auger transitions . . . . . . . 5.1.3 Paper II: Interpretation of Auger cascade satellites 5.2 Paper IV–V: Satellite Auger spectra . . . . . . . . . . . . vii

. . . . .

. . . . .

25 25 26 30 31 34

CONTENTS

viii 5.3

Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Bibliography

39

Chapter 1 Introduction Humankind has been concerned with the structure of matter for thousands of years, and the term atomos, ”uncuttable”, dates back to around 450 BCE (coined by Democritus). However, the scientific studies of atoms did not begin until the 19th century. Atoms were thought of as indivisible units, until in 1897, J. J. Thomson discovered the electron and its subatomic nature [1]. In 1909, Ernest Rutherford conducted experiments which proved that the positive charge and the majority of atomic mass was concentrated in a nucleus at the center of an atom [2]. Electrons were thought to orbit around a nucleus like planets around the Sun. The most important starting points for the development of the quantum mechanical model of the atom were the Max Planck’s idea of quantized energy in 1900 [3], the photoelectric effect and the quantized energy of light discovered by Albert Einstein in 1905 [4], and the atom model with stationary energy states proposed in 1913 by Niels Bohr [5]. Consequently, the revolution in physics occurred during the 3-year period from 1925 to 1928, when the foundations of quantum mechanics were developed by a new generation of young physicists. In those years, Wolfgang Pauli proposed the exclusion principle [6]; Werner Heisenberg, with Max Born and Pascual Jordan, discovered matrix mechanics [7], [8]; Erwin Schrödinger invented wave mechanics [9]; Werner Heisenberg enunciated the Uncertainty Principle [10]; and Paul A. M. Dirac developed a relativistic wave equation for the electron [11]. The current model of the atom is based on the quantum mechanics, and it is usually referred to as an atomic orbital model or sometimes as a wave mechanics model. According to this model, electrons are described by wave functions surrounding a nucleus. Electrons are located in atomic orbitals, which consist of a set of quantum states. The ”tools” for the ab initio calculations of atomic states were available after Hartree formulated a computational method by the year 1935 [12], but applications for multi-electron systems were too complicated. Not until the end of 1960’s, the calculation of complex systems was possible when the first effective codes were developed 1

2

CHAPTER 1. INTRODUCTION

for computers. It is evident, that the calculations, which in those days lasted for months, can be done instantly with the computers of today. Now, computational efficiency is challenged by the inclusion of other interactions, like relativistic effects and electron correlation, and the simultaneous optimization of several variables. The most striking evidence supporting the atomic orbital model is that the computational codes which are based on this theory predict very well measured atomic energy levels and the transitions between them. Calculation of atomic energy levels is briefly discussed in chapter 2, and chapter 3 specifies the electronic transitions studied in this thesis. The studies related to electron spectroscopy started at the University of Oulu in the late 1960’s, when an electron energy analyzer was constructed [13], [14]. From the start, the electronic structure of vapors and gases have been studied mainly by means of Auger electron spectroscopy, but also by photoelectron spectroscopy and ion spectroscopy. The five articles included in this thesis, referred to as Papers I–V, are good representations of studies done by the electron spectroscopy group at the University of Oulu. The papers deal with electronic transitions in free xenon and krypton atoms studied by means of Auger electron spectroscopy and ion spectroscopy. Synchrotron radiation was used to induce inner-shell excitations, and an overview of the experimental methods used in this thesis is given in chapter 4. In chapter 5, the summary of Papers I–V is given. The original articles follow chapter 5. The first study, including Papers I, II and III, deals with the de-excitation processes following the 3d excitation in xenon. In this study, it was demonstrated how to combine the Auger and ion spectroscopic results in order to determine the cascade of Auger transitions. Two computational methods, namely the method of global characteristics and the level-to-level calculations, were used in predicting the branching ratios of ions and identifying the experimental electron spectra. Another study reported in Papers IV and V was about the satellite Auger transitions in krypton and xenon, in which the initial states were populated via two different channels, i.e., an Auger cascade and a direct photo-double-ionization channel. Krypton and xenon were compared to each other, and similarities were found between these two gases. In all the studies, the Auger electron spectra were selected on the grounds of their sensitiveness to electron correlation, because in this thesis the spectra were analyzed by taking into account the many-electron effects within the configuration-interaction treatment.

Chapter 2 Overview of theoretical methods In this thesis, the energy states of many-electron atoms, as well as the Auger transition energies and rates, were computed using Cowan’s code [15, 16]. Within Cowan’s code, the energy levels of a given configuration are calculated with the Hartree-Fock (HF) method. The Hartree-Fock method is the result of the work done during the years 1927 to 1935 by many physicists of which D. R. Hartree, J. C. Slater, J. A. Gaunt, and V. A. Fock were the most prominent [17]. In the following, the HF method, the calculation of atomic energy levels, and the configuration-interaction (CI) method are discussed briefly. The handling is based on the books authored by Cowan [15], but it is covered by many other books of quantum chemistry or computational physics/chemistry (e.g. [17, 18, 19]). Electronic transitions are discussed in chapter 3.

2.1

Hartree-Fock method

Generally, the determination of the total wave function Ψk and the energy E k of stationary quantum state k for an atom containing N electrons (N ≥ 2) is based on the solving of the time-independent Schrödinger equation [9] HΨk = E k Ψk .

(2.1)

The appropriate Hamiltonian operator for the many-electron atom is H=−

i

∇2i −

2Z i

ri

+

2 + ξi (ri )(li · si ), r j i>j ij i

(2.2)

where ri is the distance of the ith electron from the nucleus and rij the distance between the ith and jth electrons. The terms of equation (2.2) consists of the 3

4

CHAPTER 2. OVERVIEW OF THEORETICAL METHODS

kinetic energy of each electron, the Coulomb interaction energy between each electron and the nucleus, the electron-electron Coulomb interaction, and the spin-orbit interaction, respectively. The spin-orbit interaction describes the magnetic interaction between the electron’s spin magnetic moment and the magnetic field that the electron sees as a result of its orbital motion through the electric field of the nucleus. The difficulty in solving the eigenvalue equation (2.1) arises from the Coulomb interaction between electrons and a nucleus, as movement of a single particle is coupled with a position and movement of all other particles. The first approximation made was to neglect the kinetic energy of the nucleus in the Hamiltonian (2.2). The HF method can be used in solving the Schrödinger equation (2.1) with the Hamiltonian (2.2). The method is non-relativistic, but the relativistic corrections can be taken into account by the perturbation calculation, when the mass-velocity and Darwin terms are included in the Hamiltonian equation. This method is referred to as an HFR method [15]. To simplify a solution to the Schrödinger equation, a central-field model of a multi-electron atom is applied. According to the central-field approximation, any given electron i moves independently from the others in the electrostatic field of the stationary nucleus and the other N − 1 electrons. Within the central-field model, electrons are arranged around the nucleus so that every electron has a different set of the quantum numbers nlms [6] (see table 2.1). A set of quantum numbers nlms characterizes the stationary energy state of an electron, which is described by the wave function referred to as a one-electron spin-orbital 1 ϕi (ri) = Pni li (r)Yli mli (θi , φi )σmsi (siz ), (2.3) ri where r i is electron’s position (r, θ, φ) with respect to the nucleus and the orientation of a spin s, Pnl is the radial wave function, Yml is the spherical harmonic, and σms is the spin orientation symbol. The HF method assumes that the wave function of any multi-electron atom can be approximated by a single Slater determinant constructed by a set of spin-orbitals (2.3) ϕ1 (r1 ) ϕ1 (r2 ) ϕ1 (r3 ) · · · 1 ϕ2 (r1 ) ϕ2 (r2 ) ϕ2 (r3 ) · · · Ψb = √ ϕ (r ) ϕ (r ) ϕ (r ) · · · 3 2 3 3 N ! 3 1 .. .. .. ... . . .

.

(2.4)

The Slater determinants satisfy the condition that the wave function of an atom must be antisymmetric upon the interchange of coordinates of two electrons. Within the HF method, the average energy of an arbitrary configuration (n1 l1 )w1 (n2 l2 )w2 · · · (nq lq )wq , qi=1 wi = N , is calculated with the aid of the

2.2. ATOMIC ENERGY STATES

5

Table 2.1: Quantum numbers valid within central-field approximation

Principal quantum number Orbital quantum number Magnetic quantum number Spin quantum number

symbol n l ml ms

values 1, 2, 3, . . . 0, 1, 2, . . . , n − 1 0, ±1, ±2, . . . , ±l ±1/2

diagonal matrix elements of the Hamiltonian operator (2.2) b Ψb |H|Ψb . Eav = number of basis functions

(2.5)

The matrix elements for the determinantal functions in equation (2.5) can be written as a summation of the matrix elements for the spin-orbitals. For the spherically averaged atom, the diagonal matrix elements of the spin-orbit operator in (2.2) cancel each other out from the summation, because for any specific value of ni li mli there will be two matrix elements with equal magnitudes but opposite in signs corresponding to msi = ±1/2. Thus, the Hamiltonian operates only to parameter r appearing only in the radial wave function Pnl (r) of the spin-orbital (2.3). So, only the radial factor Pni li for each q subshell of an electron configuration has to be solved in order to determine the HF energies. A set of q HF equations [12, 17] is derived with the aid of the Variational principle [20]. The HF equations are of a similar form as the single-electron Schrödinger equation, so they can be solved quite easily by the self-consistent field method (SCF) [18]: As a starting point, approximate radial functions Pj (r), which are typically the radial functions for the orbitals of a hydrogenic atom, are set. With these radial functions, the potential energy and some coefficients are computed for each i. Then, a new set of radial functions Pi (r) is yield by solving the HF equations. This new set of radial functions is then used as a set of trial functions Pj (r) to construct the new set of HF equations beginning the cycle again. The procedure is stopped when the solutions Pi (r) are self-consistent, in other words, sufficiently close to the trial functions Pj (r) between two iterations.

2.2

Atomic energy states

In the Slater-Condon theory [20, 21], the unknown wave function Ψk of an atom is expanded in terms of a set of known basis functions Ψb , which are derived with the method described above. In the matrix method, a set of

6


expansion coefficients are written in the form of a column vector Yk , and the Schrödinger equation can be expressed as a single matrix equation HYk = E k Yk ,

(2.6)

where the energy matrix H consists of the matrix elements of the Hamiltonian operator between two basis functions Hbb ≡ Ψb |H|Ψb .

(2.7)

The standard techniques are used in finding the matrix T which diagonalizes the Hamiltonian matrix H ≡ (Hbb ): (2.8) T−1 HT = E k δkb . The k th diagonal element of the diagonalized Hamiltonian matrix is the eigenvalue E k and the k th column of T represents the corresponding eigenvector Yk of the matrix H. The level structure relative to the average energy (2.5) of a configuration arises from the interaction between the orbital angular momentum and spin of each electron and from the residual Coulomb interaction between electrons within an atom. So, the energies of atomic states can be found by calculating the matrix elements of the spin-orbit and Coulomb interaction. The relative magnitudes of different interactions define the coupling conditions for product wave functions which gives the form of the level structure. The relative magnitudes of the different interactions define the coupling scheme for how the resultant total angular momentum J = N i=1 (li + si ) of the atom containing N electrons (N > 1) is calculated. Generally, the stronger interaction is coupled first followed by the other interactions in an order of decreasing strength. The determinantal function (2.4) is already an eigenfunction of Jz with eigenvalue N (mli + msi ). (2.9) M= i=1

An eigenfunction of J 2 is formed by taking an appropriate linear combination of determinantal functions: Each determinant has to involve the same set of quantum numbers ni li , (1 ≤ i ≤ N ) but different sets of values mli and msi so that the sum (2.9) of all magnetic quantum numbers have the same value M for each determinant. The LS-coupling scheme is valid especially in light atoms, in which the spin-orbit interaction tends to be small and the electrostatic interactions between electrons are strong. In a close approximation to pure LS-coupling, there is no interaction between the coupled total orbital angular momentum L and total spin angular momentum S, that results in L and S being good quantum numbers or constants of the motion. Another important coupling

2.3. CONFIGURATION INTERACTION

7

scheme is the jj-coupling scheme, which is valid in heavy atoms. In the jjcoupling scheme, basis functions are formed by first coupling the spin of each electron to its own orbital angular momentum, and then coupling the various resultants ji to obtain the total angular momentum J . Since Cowan’s code uses only LS-coupled wave functions, they are the only one discussed below. The antisymmetrization and coupling of wave functions involve summations over different parameters so the order in which they are performed is interchangeable. However, since the Pauli exclusion principle restricts the possible values of mli and msi of equivalent electrons, it is more convenient to couple first and antisymmetrize second. Coupled functions can be formed by taking appropriate linear combinations of the uncoupled products of the spin-orbitals ϕnlms . The coupled functions for two electrons are C(LSML MS ; JM ) C(s1 s2 ms1 ms2 ; SMS ) × ψnik1 l1 n2 l2 LSJM = ML M S

×

ms1 ms2

C(l1 l2 ml1 ml2 ; LML ) ϕ1 (ri )ϕ2 (rk ), (2.10)

ml1 ml2

where the coefficients C are known as Clebsch-Gordan (CG), vector-coupling, or Wigner coefficients. The antisymmetric coupled basis functions are in the LS-coupling scheme 1 ik Ψik ψn1 l1 n2 l2 LSJM − ψnki1 l1 n2 l2 LSJM . n1 l1 n2 l2 LSJM = √ 2

(2.11)

For the equivalent electrons, the orthonormality relations lead to that antisymmetric LS-coupled wave functions (2.11) are null if L + S is odd.

2.3

Configuration interaction

The term electron correlation refers to the interaction between the electrons of an atom. Within the central-field approximation, the spherically symmetric potential field ignores that the positions of other electrons are correlated by the position of the ith electron via their mutual Coulomb repulsion. However, this electron correlation plays an important role especially in orbitals which have a prominent probability distribution spread over a wide space, e.g., the 4s/5s orbital of Kr/Xe which interacts with the close proximity 4p/5p orbitals [22, 23]. Then, the HF energies deviate from the experimental values significantly if the antisymmetric wave functions are approximated by the single-configuration Slater determinants. This was demonstrated in paper III, where the single- and multi-configuration calculations were compared with each other for the 4p−1 4d−1 and 4d−3 4f electron configurations. In order to improve the accuracy of the wave functions of atomic energy states, the CI method is applied. The interaction of configurations means

8


that the states belonging to different configurations are mixed while constructing the total wave function of an atom by taking a linear combination of basis functions (Slater determinants) for two or more configuration. For instance within a two-configuration approximation, the Hamiltonian matrix for any given value of J has the form c1 c1 − c2 H= . (2.12) c2 − c1 c2 The square diagonal blocks c1 and c2 are calculated in a separate SCF calculation for each configuration, which leads to the breakdown of the orthogonality conditions between the basis functions of different configurations. The offdiagonal blocks c1 − c2 and c2 − c1 include the configuration-interaction (CI) matrix elements. The CI matrix elements are calculated in the same way as the single-configuration matrix elements, except that the bra and ket functions in the expression (2.7) are the basis functions of different configurations. If there are more than two configurations in an approximation, the energy matrix (2.12) includes additional diagonal blocks c3 , c4 , . . . and and CI matrix element blocks c1 − c3 , c1 − c4 , . . . corresponding to single-configuration matrix elements and interaction energies between various configurations, respectively. Within Auger transitions, CI is referred to as IISCI (initial ionic state configuration interaction) or FISCI (final ionic state configuration interaction) when CI between initial state or final state configurations is accounted for. In the single-configuration case, an abbreviation SC is used. There are a few fundamental selection rules, which limit the set of configurations included in multi-configuration calculation. First, since the Hamiltonian operator (2.2) has an even parity, the CI matrix elements are zero unless the bra and ket functions have a common parity. Thus, only the configurations having the same parity need to be considered. Second, the interactions can occur only between the confiurations that differ in, at most, two orbitals because the Hamiltonian operator involves only one- and twoelectron operators. Finally, the matrix of the Coulomb operator in the LS representation is diagonal in the quantum numbers LSJM . Therefore, in order to obtain nonzero Coulomb CI matrix elements, each configuration must contain a basis state with some common value of LS. In addition to fundamental selection rules, a few qualitative remarks can be made: The CI effects tend to be largest between configurations whose center-of-gravity energies Eav are not greatly different and whose Coulomb matrix elements are large in magnitude. [15]

Chapter 3 Electronic transitions When electromagnetic radiation interacts with matter, photons can be either scattered or absorbed. In this chapter, the photoabsorption processes which lead to a rearrangement in the electronic core of a multi-electron atom are considered within the frame of the production of an inner-shell vacancy and its decay. The photoexcitation processes are approximated as dipole transitions, and the selection rules for the dipole transitions are given in section 3.1. The most essential electronic transitions within this thesis are introduced by using a step-by-step model consisting of at least two steps. The first step defines a primary excitation process, which is discussed in section 3.2. After an inner-shell vacancy is produced, an atom remains in an excited state which decays by a filling process in the course of a second-step decay. The possible decay processes are a fluorescent emission of characteristic radiation or a non-radiative ejection of an electron. Since fluorescence falls outside of the scope of this study, only an electron emission by the Auger transition is discussed. Normal Auger decay is discussed in section 3.3, the cascades of Auger transitions in section 3.4, and satellite Auger transitions in section 3.5. In Paper II, the Auger transitions were calculated with the method of global characteristics, which is briefly described in section 3.6.

3.1

Selection rules for dipole transition

Electromagnetic radiation consists of alternating, mutually-supporting electric and magnetic fields. The electric field experienced by target atoms is usually approximated by the electric field of a electric dipole. The electric dipole approximation can be assumed to be valid if the wavelength of incoming radiation is much larger than the diameter of a target atom. Electronic transitions following photoabsorption obey then the dipole transition selection rules (3.1). During a dipole transitions, the total energy and momentum of a photon will be transferred to a target atom, and consequently the total angular momentum quantum number J, the total magnetic quantum number 9

CHAPTER 3. ELECTRONIC TRANSITIONS

10

MJ and the parity π of an atomic state are allowed to change as ΔL = 0, ±1 (not from 0 to 0); ΔJ = 0, ±1 (not from 0 to 0);

ΔS = 0; ΔMJ = 0, ±1;

πi = −πf . (3.1)

A selection rule for one electron jump, e.g., in a resonant excitation or a photoionization, is Δl = ±1, (3.2) where l is the orbital quantum number of the transferring electron.

3.2

Transitions following photoabsorption

Resonant excitation The resonant excitation from an electronic bound state i to another bound state j can occur if the energy of an absorbed photon equals with the energy difference of two states as hν = Ei − Ef . In a closed-shell atom, it is possible for an electron to transit only to an initially non-occupied bound state, which is usually referred to as a Rydberg state. The orbital quantum number of an excited electron follows the selection rule (3.2). An atom (A) is left in an excited state (A∗ ) by a resonant excitation, and a reaction equation can be written as A + hν → A∗ . (3.3) The de-excitation following the resonant excitation from the 3d5/2 inner-shell to the 6p3/2 Rydberg orbital of Xe was studied in Paper I. The schematic picture of the resonant excitation is displayed in figure 3.1 (a) and the possible resonant Auger de-excitation processes are displayed in figures 3.1 (b)-(d) (discussed in section 3.5).

Photoionization In a photoionization, an incoming photon is absorbed by an atom (A) with the emission of an electron p called as a photoelectron. If the photoelectron is emitted from an inner-shell, an ion remains at an excited state (A+∗ ). The reaction function is then A + hν → A+∗ + p .

(3.4)

The photon energy hν must be (at least) as high as the energy difference between the initial and final states of an atom. If the photon energy exceeds this energy difference, the rest is left as the kinetic energy of the photoelectron as hν = Ef − Ei + εp . (3.5) The 3d photoionization is illustrated in figure 3.2 (a).

3.2. TRANSITIONS FOLLOWING PHOTOABSORPTION

Orbital energy

0

6p 5p 5s

11

7p

4d 4p 4s

3d (a)

(b)

(c)

(d)

Figure 3.1: Schematic picture of 3d → 6p resonant electronic transitions for Xe. (a) Inner-shell resonant excitation. Possible resonant Auger deexcitation processes are displayed in (b)–(d). (b) Participator resonant Auger transition (autoionization), (c) spectator resonant Auger transition, and (d) spectator resonant Auger transition with 6p → 7p monopole shake-up transition. A closed shell includes 2(2l + 1) electrons of which only two have been depicted.

Monopole shake-up and shake-off, sudden approximation When there is a sudden change in an atomic potential, e.g., during the ionization of an inner-shell electron, another atomic electron has a small probability to be excited from an outer-shell to an unoccupied bound state (shake-up) or ejected into the continuum (shake-off) via a monopole transition1 . The shake-up states are located in the binding energies above the resonance states and the single-ionization threshold. The monopole shake transition probabilities can be calculated quite easily with the sudden approximation [24]. If it is assumed that an atom has a set of one-electron atomic orbitals {ψnl } in its ground state, after the creation of an inner-shell n0 l0 -vacancy, the atomic orbitals of the core relax whereupon a new set of one-electron states {ψñl } describes an ion. Within the sudden 1 In the monopole transition, all the other quantum numbers except the principal quantum number are not allowed to change.

12


approximation, the square of the overlap integral ψ˜{n ,ε}l |ψnl 2 is the probability of a monopole excitation of the nl-electron into the states of discrete (n l) or continuous (εl) spectra. Thus, the probability for another (outershell) electron staying in its n1 l1 state after the relaxation of the atomic core is ψñ1 l1 |ψn1 l1 2 . The probability of the n1 l1 electron shaking up or off from the ion can be expressed as [25] Pn1 l1 = Nn1 l1 (1 − ψñ1 l1 |ψn1 l1 2 ),

(3.6)

where Nn1 l1 is the occupation of the n1 l1 subshell. Dipole approximation calculations for shake transitions in the rare gases have been reported for over the last four decades (see e.g. [26, 27] and references therein.)

Photo-double ionization Different nominations for single-photon double-ionization transitions can be found in the literature. While the meaning of a shake-off transition is well defined in the framework of the sudden approximation in the asymptotic high-energy limit [24], its meaning at finite energies is less clear and has been the subject of debate recently (see e.g. [28] and references therein). In Papers IV and V, the one-step double-ionization processes were referred to as direct photo-double ionization (PDI) transitions according to the recommendation of Briggs et al [29]. PDI describes the one-photon two-electron nature more aptly than a double photoionization (DPI) transition which is also used in the literature. The reaction equation for PDI can be written as A + hν → A++ + 1p + 2p .

(3.7)

An ion is in an excited state (A++∗ ) if there is a hole in the inner-shell after PDI. The PDI process competes with the photoionization and the shake-up transitions with the photon energies above a certain photo-double ionization threshold. When a photoabsorption takes place in the outer-shell of an atom, the change in the potential experienced by other electrons of that shell is small, and the ejection of two valence electrons simultaneously would be expected to be very rare. However, much larger double-ionization probabilities have been observed than what was predicted by calculations based on the singleparticle model (see e.g. [30]-[33] and references therein), which indicates that the electron correlation effects must be included [34]. Theoretically the problem of the three-body Coulomb continuum is very challenging, and so far one, universally applicable, theory does not exist [28]. The review article written by Briggs and Schmidt [29] about helium atom summarizes rather completely the theoretical aspects related to the PDI process. Spectroscopic techniques which allow a simultaneous detection of two particles have been

3.3. NORMAL AUGER TRANSITIONS

13

developed, which makes it possible to study PDI experimentally (see e.g. the electron-electron coincidence measurements in [35] and the recoil ion momentum spectroscopic technique in [36]).

3.3

Normal Auger transitions

The one-step model of the photoabsorption and the decay is based on the unified theory of inelastic x-ray scattering which has been developed by ˚ Aberg et al [37, 38] in terms of the time-independent resonant scattering theory. This gives a very general and exact description for the Auger effect, among others. Within the two-step model, the Auger electron emission is usually considered to be independent of the x-ray absorption process and the decay process is not affected by the leaving photoelectron. In Papers I–V, the photoionization is, however, considered to have an effect on the intensities of detected Auger lines because the population of the initial states depends on the photon energy used. Also, the other indications of the limitations of the two-step model were seen. A part of the measured Auger lines were affected by the post-collicion interaction (PCI) line profile, which was not taken into account by the theory. Also, the partial Auger rates are known to depend on the photon energy [41], which complicated the analysis in Papers IV and V. Within the two-step model of the normal Auger transition2 , the initial state is generated by an ionization of an inner-shell electron according to the reaction equation (3.4). During the Auger decay, an electron from an outer shell fills the initial vacancy while another outer-shell electron is emitted from the ion due to the Coulombic forces, which results in a doubly ionized final state. The reaction equation is A+∗ → A++ + A .

(3.8)

The initial vacancy and the two final vacancies in the core or valence region are usually labeled by the X-ray spectroscopic labels (K1 , L1,2,3 , M1,2,3 , . . .) corresponding to the atomic orbitals (1s1/2 , 2s1/2 , 2p1/2,3/2 , 3s1/2 , 3p1/2,3/2 , . . .) involved in the process. If the initial vacancy is filled by an electron from a higher subshell of the same shell, i.e. the vacancy and the electron have the same principal quantum number, the transition is referred to as a CosterKronig (CK) transition. If also the Auger electron is ejected from the same shell, the Auger transition is referred to as a super-Coster-Kronig (SCK) transition. In figure 3.2 (b), the initial core hole is created in the 3d subshell whereas the final vacancies are in the 4p and 4d subshells, in which case the Auger transition is denoted as M4,5 -N2,3 N4,5 . 2 The radiationless transitions was first discovered by L. Meitner [39] but it was named after P. Auger who developed the Auger spectroscopy [40].


14

0

Orbital energy

5p 5s

4.

3.

4d 2.

4p 4s 1.

3d (a)

(b)

(c)

(d)

Figure 3.2: Schematic picture of photoionization and subsequent decay processes. (a) 3d inner-shell photoionization. Possible de-excitation processes are displayed in (b)–(d). (b) M4,5 -N2,3 N4,5 normal Auger transition, (c) 5p shake-off during Auger decay and (d) cascade of Auger transitions: three consecutive steps following the 3d photoionization (a) and the first step transition (b), which leads to the 5p−5 final configuration. Every 2(2l + 1) electrons of closed shells have not been depicted. By the energy conservation law, the kinetic energy of the Auger electron EA is given by EA = Ei+ − Ef++ , where Ei+ is the total energy of the single ionized initial state and Ef++ is the total energy of the double ionized final state. Since the energy levels of an atom are discrete, the Auger energies are characteristic of the emitting atom. The Auger transition rate (probability per unit time) can be determined with the aid of the matrix elements of the electron-electron Coulombic interaction as [42] Wi→f

2π =

2 1 Ψ Ψi , rij

(3.9)

where the wave functions for the discrete initial state Ψi and the continuum final state Ψ is used. The Auger transitions obey the following selection rules for quantum numbers: ΔL = ΔS = ΔJ = 0;

πi = πf .

(3.10)

3.4. CASCADE OF AUGER TRANSITIONS

15

To obtain Jf , the quantum numbers of the continuum electron and the remaining vacancies of the ion are coupled together. Detected Auger lines are broadened by the natural lifetime widths of initial and final states, which originate from the uncertainty principle [7]. A lifetime of an atomic state means that if a system was prepared in the state Ψi at a certain instant, it would decay or become excited during the time interval (in atomic units) τi =

1 = , Γi Wi

(3.11)

where τi is a natural lifetime of that energy state, Γi is the FWHM (full width at half maximum) of a Lorentzian line profile describing the probability distribution as a function of energy, and Wi is the total Auger transition probability rate of the state i.

PCI The post-collision interaction (PCI) effect is a special form of electron correlation which involves interaction between the photoelectron and the Auger electron. The shape of the detected photoelectron line is distorted by PCI in the near threshold as the ionic field influences the photoelectron so that the kinetic energy of the photoelectron is decreased. Also the detected Auger line is distorted by PCI, and it cannot be described by a Lorentzian line shape. The PCI energy distribution of an Auger line is asymmetric and broadened, and the maximum is shifted towards high kinetic energies. A slow photoelectron can even be captured into a discrete state of the residual ion. [43]

3.4

Cascade of Auger transitions

During the Auger transition, an inner-shell hole is most probably filled by an electron from the next highest subshells. Thus, the probabilities of the CK and SCK transitions are usually high (see e.g. [57]). For example in the case of the 3d excitation, the de-excitation of the primary inner-shell vacancy leaves two vacancies most probably in the 4s, 4p, and 4d subshells, which are filled during the subsequent Auger transitions (Papers I and II). Consequently, the de-excitation of an inner-shell hole leads to a complicated cascade of Auger transitions increasing the ionization stage at each step and transferring vacancies to outer shells (illustrated in figure 3.2 (d)). Electrons are emitted from a vast variety of configurations with spectator vacancies left by the preceding transitions, and a cascade can continue until all the vacancies are in the valence-shell. The charge of an ion can be very high after the Auger cascade, e.g., the de-excitation of the 3d vacancy in Xe


16

b excited state first step

f shake-off + Auger second step

e

h third step

d

i

g

c a ground state

A

A

+

A

2+

A

3+

Figure 3.3: Cascade of Auger transitions following excitation of atom A. A+ ions are produced via b→c,d transitions, A2+ ions via b→e,f→g transitions, and A3+ ions are produced via b→f→h→i and b→f → i transitions. A shake-off transition occurs during the step f→i, and two electrons are emitted simultaneously.

ends up with the final states having the ionization stages of Xe3+ –Xe8+ . Thus, a cascade of Auger transitions has been traditionally studied by ion spectroscopic techniques (see e.g. [53]-[56].) The schematic illustration of the production of multiple ionized atoms via the cascade of Auger transitions following the excitation of an arbitrary atom is shown in figure 3.3. Auger transitions between electron configurations with multiple holes may be weak because the population of initial states is redistributed between numerous final states in the course of the subsequent Auger transitions. A spectrum of low energy Auger electrons consists of many overlapping lines from different steps of an Auger cascade (see Paper II). As it is also considered that, besides normal Auger transitions, also shake-up and multiple shake-off transitions are possible during the de-excitation of an inner-shell vacancy, it is evidenced that the Auger electron spectrum produced by a cascade of Auger transitions is very complicated.

3.5. SATELLITE AUGER TRANSITIONS

3.5

17

Satellite Auger transitions

Resonant Auger transitions The resonance excitation of an inner-shell electron leaves an atom in an excited state, which may decay by a resonant Auger transition. In the participator resonant Auger transition, the excited electron is ejected from the atom as an Auger electron. This transition is also called an autoionization since it leads to the same final states as the direct photoionization process. For example the participator resonant Auger decay shown in figure 3.1 (b) and the direct 4p photoionization produce the same final states. Two continuum channels may interfere either destructively or constructively with each other (see e.g. [44] and references therein). In the spectator resonant Auger transition, the initially excited electron does not participate in the subsequent Auger decay (figure 3.1 (c)). In that case, a resonant Auger spectrum is usually very complicated as compared to a normal Auger spectrum since the coupling of an electron in a Rydberg orbital splits both initial and final states leading to numerous overlapping resonant Auger transitions (see e.g. [45, 46]). Also shake transitions are more probable during resonant Auger transitions. In a shake transition, the excited electron is shaken down or up to other orbital during the Auger decay (illustrated in figure 3.1 (d)).

Simultaneous emission of two Auger electrons In the literature, the ejection of two electrons during an Auger decay is referred to as a double-Auger transition or a shake-off transition during the Auger transition (illustrated in figure 3.2 (c)). The latter mentioned designation is usually used in Papers I–V. Only the total kinetic energy of two outgoing electrons is fixed. The available energy is shared in an arbitrary ratio by two outgoing electrons, and they show in measured Auger electron spectrum as continuous background instead of characteristic lines. However, the shake-off electrons can be discriminated from other sources, particularly from background electrons, with the coincidence methods, which have been applied effectively e.g. in [47]-[52]. Besides the measurement, also the theoretical investigation, of two-electron emission is challenging because the process has to be described by a correlated two-electron wave function in the continuum. In the independent electron model (with the frozen orbitals), the emission of two electrons is forbidden. Therefore, a shake-off during the Auger transition is usually assumed to be a weak process. Even so, the investigations of the Auger cascades following the inner-shell excitation have proved the shake-off processes to be essential. For instance, only the shake-off transitions can produce the measured yield

18


of highly charged ions following the 3d excitation in studies [53]-[56].

3.6

Method of global characteristics

The detailed level-to-level calculations of the Auger energies and transition rates between hundreds of overlapping electron configurations are very timedemanding because they demand a lot of manual work. The method of global characteristics allows the determination of the configurations populated during a de-excitation process without the need to perform detailed calculations of lines. The method estimates the average energy and variance of transitions between two configurations using their explicit expressions [58]. The wave functions of every configuration are taken, e.g., from the HF calculations. The array of Auger transitions between two configurations is considered as an integral transition with energy equal to the average one and with the total transition rate [58, 59]. Within Auger cascade calculations, every configuration in each step is characterized by its population depending on the de-excitation history. The intensity of an Auger transition is obtained by multiplying the population of an initial configuration by the total transition rate to a final configuration. The probabilities of forbidden transitions by the energy conservation law are subtracted from the total transition rate [60]. Also, the multi-electron Auger transitions can be taken into account in a sudden perturbation approximation. The disadvantages of the method are briefly discussed in Paper II. The method was used succesfully in predicting the de-excitation pathways following the 3d ionization of Xe as well as the configurations participating in the Auger spectrum located at the low kinetic energy region (reported in Paper II). The method has also been applied in the theoretical investigation of the Auger and photoion-yield spectra resulting from the 3d photoionization and the 3d → 4f excitation of atomic Eu [60].

Chapter 4 Experiment 4.1

Synchrotron radiation

The electronic structure of a target atom can be studied only by exciting an atom by photon or particle bombardment. Particles can be e.g. high-kinetic energy electrons or ions, and a primary particle can give any amount of its kinetic energy to a target atom. This produces a large number of different excitations, which complicates resulting spectra. Instead, a target atom absorbs the total energy of an incident photon. There are many different devices applied as photon sources, for example X-ray tubes, which provide the characteristic Al or Mg Kα radiation, and Helium discharge lamps, which produce vacuum ultraviolet radiation. However, they can produce only certain wavelengths, which limits their usability. The synchrotron radiation provides a modern way to produce photons having energies from vacuum ultraviolet to hard X-rays. Synchrotron radiation has very important spectral characteristics such as a high brilliance and a natural collimation on the orbit plane of the electrons [61]. Synchrotron radiation is produced when the trajectory of free electrons moving with a relativistic speed is curved by a special magnetic device such as a bending magnet (second-generation storage ring) or a wiggler and an undulator (third-generation storage ring) according to the Lorentz force [61]. Wigglers and undulators consist of a periodic structure of dipole magnets. In figure 4.1 (a), a static magnetic field is alternating along the length of an undulator with the wavelength λ0 . The electrons traversing through a periodic magnet structure are forced to undergo oscillations and they start to emit X-ray radiation. The illustrative figure 4.1 (b) shows a characteristic peak structure of an undulator spectrum. Generally, undulator sources provide increased brilliance and tunability of radiation as compared to a continuous spectrum emitted by second-generation sources, because the most intense radiation peak can be moved to a chosen wavelength by changing a gap between the magnetic poles of an undulator. Usually a desired, narrow energy band is 19

CHAPTER 4. EXPERIMENT

20

separated from an emitted undulator spectrum with a monochromator. The undulator sources in two international, third-generation synchrotron radiation facility, MAX-II in Sweden and BESSY II in Germany, were used in the experiments of Papers I-V. These beamlines are briefly described below.

(b) Photons

(a)

Photon energy

Figure 4.1: (a) Schematic picture of the undulator’s magnetic structure and (b) photon spectrum.

MAX-II The MAX-II storage ring in Lund, Sweden, operates at 1.5 GeV electron energy providing photon energies from 10 eV to 20 keV. The beamline I411 was used in Papers I, IV, and V. The beamline is equipped with an 87 pole undulator, which gives radiation between 60 to 1500 eV. A Zeiss SX-700 plane grating monochromator, consisting of a plane mirror and a plane grating, is used. A more detailed description of the beamline is presented in [62]. The I411 beamline is equipped with a hemispherical SES-200 electron energy analyzer, which is briefly described in chapter 4.2 and more precisely in [63]. A one-meter-long section in front of the experimental station of the beamline can host other types of setups for atomic and molecular spectroscopy, e.g., a time-of-flight (TOF) mass spectrometer for ion spectroscopy measurements. A special differential pumping section is applied between the last re-focusing mirror and the sample chamber because the high vacuum of the beamline must be separated from the working pressure of about 10−6 mbar in the experimental chamber.

Bessy II The Bessy II storage ring in Berlin, Germany, operates at 1.7 GeV electron energy, and the photon energy provided is between 6 eV to 10 keV. The undulator beamline U49-1 (undulator period λ0 = 49.4 mm with 84 periods) was used in the measurements in Papers II and III. The beamline was equipped with a spherical grating monochromator and a refocusing mirror. For a more detailed description of the beamline, see [64]. During the gas

4.2. ELECTRON SPECTROSCOPY

21

phase measurements, a differential pumping section was applied between the beamline and the experimental chamber. A hemispherical SES-100 electron energy analyzer (see chapter 4.2) was used in data collection.

4.2

Electron spectroscopy

Atomic spectra of different elements and compounds have been measured using photon absorption and emission since the late 19th century. The electron spectroscopy technique was initiated by Kai M. Siegbahn, who was awarded the Nobel Prize in physics in 1981, and his coworkers at the University of Uppsala, Sweden. They published the first electron spectra in 1957 [65, 66]. The ESCA (Electron Spectroscopy for Chemical Analysis) technique proved to be feasible in studying the properties of gases, liquids as well as solids, and it provided a new, interesting method to study atomic and molecular physics, cluster physics, surface physics, liquid interfaces, magnetic materials, molecular materials etc. [67, 68]. Consequently, many other research groups all over the world also started to use and develop ESCA. The kinetic energies of electrons emitted by a target are analyzed by electron spectrometers. The measured electron spectrum depicts the counts (intensity) as a function of the kinetic energy of detected electrons. Electron energy analyzers separate electrons having different kinetic energies by magnetic or electric fields with respect to the Lorentz force. The kinetic energies between 0–2 keV are usually measured with analyzers which operate using electric field. In the following, the effects of the electrostatic electron spectrometers on measured electron spectra are discussed.

Dispersion, resolution and transmission of an electron analyzer Electron analyzers are characterized by dispersion, resolution, and transmission, which all have an influence also on the measured electron spectra. The dispersion of an analyzer is defined by an ability to separate electrons having different energies from each other as ∂L , (4.1) ∂E where L is the projection of the flight inside an analyzer and E is the kinetic energy of electrons. Energy resolution is usually defined by D=E

R=

ΔE , EP

(4.2)

where ΔE is the FWHM of the spectrum line produced by same energetic electrons and EP is the pass-energy (fixed through measurements to have a

22


constant resolution). Also the inverse of equation (4.2) is sometimes called resolution. In this thesis, resolution has a meaning of ΔE, which gives the separating capacity for the smallest energy difference. During measurements, an analyzer is mounted in a certain direction, usually in 54.7◦ versus the plane of the storage ring1 . The transmission of an analyzer is defined by a ratio between the number of detected electrons and all emitted electrons. The transmission of the constant pass energy spectrometers varies along the measured kinetic energies due to the changing retardation ratio of the electron optics. The transmission function T (Ek ) at a certain pass energy can be measured with the method in which the intensity ratio of a photoelectron and the corresponding normal Auger lines is determined with the photon energies between Eb + Ekmin and Eb + Ekmax , where Eb is the binding energy of the photoelectron and Ekmin – Ekmax is the desired kinetic energy region [69]. The ratio varies along the kinetic energy region due to the transmission of a spectrometer, and, consequently, the correction curve for measured data is determined by assuming that the ratio is independent of the photon energies. In fact, this is not valid because Auger transition rates have been found to depend on the photon energy [41]. Evidently, the method gives good results if the intensity of the whole Auger group is taken into account instead of only a few Auger lines. Electron’s flight inside an analyzer depends on its kinetic energy, the strength of the electric field as well as the angle between the entrance slit and the electron’s trajectory. In order to detect data with sufficient statistics, the variation of the flight and the angle must be allowed, which worsens the resolution of an analyzer. Analyzer’s contribution to detected line profiles is approximated with a Gaussian distribution. An ideal analyzer has a large dispersion and small resolution values, in which case narrow analyzer line widths are detected. In addition to the natural lifetime width of an ionic state and the spectrometer function, the Doppler broadening and, for the photolines only, the distribution function of the bandwidth of a light source are also disturbing a detected line profile. A final line profile is a convolution of all the functions, and it is usually described with a Voigt function.

SES-100 and SES-200 analyzers The commercial hemispherical Scienta SES-100 and SES-200 electron energy analyzers consist of an entrance lens, hemispherically-shaped metallic electrodes with the mean radius of 100 or 200 mm, and an electron detector. The electron lens focuses incoming electrons onto the entrance aperture, and at the same time it retards/accelerates electrons to a fixed pass energy. An ana1 The total intensity of a transition is distributed to different angles as I(θ) = Itotal [1 + βP2 (cos θ)]. Measuring in ”magic angle” allows the detection of intensities independently from β because P2 (cos θ) = 12 (3 cos2 θ − 1) is zero with θ = 54.7◦ ).

4.3. ION SPECTROSCOPY

23

lyzer having the spherical symmetry of electrostatic field has a so-called focus plane, which means that electrons having a small variance of kinetic energies are converged to different spots after deflection through the angle of 180◦ . This enables efficient data collection with a multichannel plate (MCP), since many slightly different energies (channels) can be separated and detected simultaneously. During the measurements reported in this thesis, the detector systems of the SES-200 and SES-100 analyzers consisted of an MCP combined with a fluorescent screen. The intensity was recorded by monitoring the fluorescence produced by the hitting electrons using a CCD camera.

4.3

Ion spectroscopy

Even if the electron spectroscopy method has proven to be very effective in studying electronic properties, in several cases electron spectra are composed of many overlapping structures which may not be resolved. These processes can be studied by measuring ion spectra, i.e., the yield of differently charged residual ions. As demonstrated in Papers I and II, ion spectroscopy offers complementary methods for determinining the probabilities of different decay channels.

TOF mass spectrometer A TOF analyzer measures the time required for a particle to travel a fixed distance. A TOF mass spectrometer is used to measure the distribution of particles with respect to their mass m and charge q. A Wiley-McLaren [71] type of linear TOF mass spectrometer used in Paper I consists of three sections, which are a source region, an acceleration region, and a field-free drift tube [70]. When working in a pulse-mode, the pulsed extraction field pushes ions towards the drift tube and a time measurement starts. When ions reach the detector, a spectrum is calculated from their time of flight. In a TOF spectrum of a single-atomic sample, the ions having different charges are resolved as the flight-time is dependent on a charge within the relation m T ∝ . (4.3) q In a partial ion yield (PIY) spectrum, the yield of ions having a given charge stage is determined as a function of incoming photon energy. The yield spectrum of all ions as a function of photon energy is referred to as a total ion yield (TIY) spectrum. It can be determined either by measuring a mass spectrum or simply detecting an ion current as a function of photon energy.

24


Chapter 5 Summary of included papers 5.1

Papers I–III: Multiple ionization of Xe following 3d excitation

The de-excitation pathways and the ion production following the decay of the 3d hole in xenon were studied in Papers I and II. Ion and electron spectroscopies were combined successfully with theoretical calculations computed using the method of global characteristics and detailed level-to-level calculations in order to determine the branching scheme in detail. The de-excitation pathways of the 3d−1 initial states in Xe were selected as a topic of our research for many reasons. Firstly, the dominance of the spin-orbit interaction in the 3d shell of Xe allows the separation of the various excitation and decay channels originated from the 3d3/2,5/2 states, which are not affected by IISCI. Instead, the 3d5/2 photoionization cross section shows a second maximum some 30 eV above threshold [72], which was predicted to originate from the spin-orbit activated interchannel coupling [73]. In our studies, the 3d−1 single-configuration approximation was applied in the calculations and the results were compared with the experiments and the previous studies. Secondly, Xe has a closed shell electron configuration as a ground state, so the spectral fine structures are created by the vacancies involved in the de-excitation processes. On the other hand, the complex de-excitation pathways of the inner-shell 3d excitation consist of several consecutive steps including thousands of possible participating states, which are, however, quite easily produced by the HF wave functions with Cowan’s code [15]. Thus, the topic was suitable for testing the capability of the calculation methods. Thirdly, even if there are other theoretical [74, 75] and experimental [55, 56, 76, 77] studies of the ion yields following the 3d ionization to be compared with our results, instead, there are no previous studies for the ion yields or the de-excitation pathways following the 3d−1 5/2 6p resonance excitation in the literature. Only the first-step resonant Auger transitions were 25

CHAPTER 5. SUMMARY OF INCLUDED PAPERS

26

studied previously [45, 78, 79]. Fourthly, the photon energy region of the 3d ionization threshold and above (from 650 eV to 740 eV) is within the reach of the beamlines U49-1 at Bessy II and I411 at Max II, where opportunities were available to carry out measurements with a good resolution and a flux of the photon beam.

5.1.1

Partial ion yields following 3d excitation

In Paper I, the PIYs of Xen+ (n = 1 − 8) ions were measured using the TOF mass spectrometer [70] in the photon energy range of 664–707 eV and the branching ratios of ions were determined. The TIY spectrum depicted in figure 5.1 was obtained by adding together the measured PIYs, which corresponded to the absorption spectrum of Xe at the 3d threshold region [72]. The branching ratios including the ions from all the processes following the 3d5/2 → 6p (1 P1 ) excitation and the 3d5/2 ionization were determined. The experimental branching ratios are depicted in figure 5.2 (a) and (b) labeled as Exp. 1 and Exp. 2. The experimental branching ratios for the Xen+ (n = 3−8) were found to be almost the same in the resonance excitation and the ionization. Ions having the charge of up to 8+ were detected but the yields of Xe4+ and Xe5+ ions were dominant in both cases.

log(Total ion yield)[*104]

10

III 3d3/2

3d5/2

9 8 7

III

3d5/2 –> 6p

6 5 4 3

2

670

680

690

700

Photon energy [eV]

Figure 5.1: TIY spectrum of Xe at the 3d threshold region. The 3d3/2,5/2 ionization thresholds, the 3d−1 5/2 6p resonance energy, and photon energies used in Paper III are depicted with bars. The decay of the 3d excitation leads to the cascade of Auger transitions by transferring the vacancies to the valence shell and increasing the charge of ions during each step. In order to solve the origin of the measured ion yields, the kinetic energies and the transition probabilities of possible subsequent

5.1. PAPERS I–III: MULTIPLE IONIZATION OF XE 0.5

(a)

Exp.1 Calc.1 Semiemp.1

Branching ratios

0.4

27

(b)

Exp.2 Calc.2 Semiemp.2

0.3 0.2 0.1 0.0

Xe

2+

Xe

3+

Xe

4+

Xe

5+

Xe

6+

Xe

7+

Xe

8+

Xe

2+

Xe

3+

Xe

4+

Xe

5+

Xe

6+

Xe

7+

Xe

8+

Figure 5.2: Branching ratios of Xen+ (n = 2 − 8) ions following (a) the 3d5/2 ionization and (b) the 3d5/2 → 6p resonance excitation. The experimental values Exp. 1 and Exp. 2, the results of the single-configuration calculations Calc. 1 and Calc. 2 as well as the semi-empirical results Semiemp. 1 and Semiemp. 2 have been given. The details of the calculated and semiempirical values are given in Papers I and II. configurations in the cascade of Auger transitions following the 3d ionization and the 3d5/2 → 6p resonant excitation were calculated. In Paper II, the branching scheme of the Auger transitions following the 3d ionization involving 130 pairs of configurations was determined by the method of global characteristics. Such an approximative method was found to give sufficient results for ionization. For resonance excitation, electron configurations are split into a large number of energy states by a single Rydberg electron and configurations are overlapping in energies. Thus, the levelto-level transition energies and probabilities were computed for the Auger cascades following the 3d5/2 → 6p resonant excitation. The monopole shakeup transition probabilities between the 6p → 7p, 8p and 7p → 8p orbitals were calculated with the sudden approximation and the probabilities were taken into account in a single-configuration model for the de-excitation of the resonantly excited state. Many Auger transitions which are energetically forbidden between the non-resonant states were found to be allowed when the Rydberg electron generates more energy states, which are shifting towards higher binding energies and spreading out to a wider energy region. The predicted branching ratios are presented in figure 5.2 with labels Calc. 1 and Calc. 2. The single-configuration calculations were found to underestimate the ion yield of Xe6+ , and the Xe7+ and Xe8+ states were found to be energetically forbidden. Obviously, multi-electron processes must be involved in calculations in order to improve the prediction for the branching ratios of highly charged

28


ions. In Paper I, semiempirical branching ratios were produced by taking into account the two most important electron correlations as follows: 1) The 5s−1 ↔ 5p−2 εl electron correlation was included by describing 10% of the 5s−1 core hole states with the 5p−2 states. Instead, electron correlation between the 5s−1 and 5p−2 nl, (n = 5d, 6s) configurations [80], which splits and shifts the energy levels towards higher binding energies with respect to the single-configuration levels [81]–[83], was neglected. −2 2) Electron correlation of the 4p−1 1/2 and 4d εf configurations [84, 85] was taken into account by estimating that 35% of the 4p−1 4d−1 population transfers to the 4d−3 states via CI.

Additionally, the shake-off transition probability from the 5p orbital at the first step of the Auger cascade following the 3d−1 ionization was estimated to be about 10%. In Paper II, the shake-off probabilities were calculated by using the sudden approximation, and the total shake-off probability was found to be 6% for the first-step 3d−1 → (4s, 4p, 5s, 5p)−2 Auger transitions, while the probabilities for the subsequent steps were even lower. In Paper I, the 3d−1 6p → (4s, 4p, 4d, 5s, 5p)−2 shake-off transitions with the estimated probability of 10% were included to compensate for all the possible transitions in which the Rydberg electron leaves the ion during the cascade of Auger transitions. Figures 5.2 (a) and (b) depict with labels Semiemp. 1 and Semiemp. 2 the semi-empirical branching ratios which take into account the correlations 1 and 2 as well as the shake-off transitions together with other Auger transitions, which were found to be energetically possible in more detailed calculations. The semi-empirical model was found to predict the branching ratios for the de-excitation pathways of the 3d−1 state better than the single-configuration model. The Xe6+ ion yield was still overestimated, so part of these ions obviously further decay to produce Xe7+ and Xe8+ states. The prediction for the ion yields at the 3d−1 5/2 6p resonance was, however, only slightly improved by the semi-empirical model. The Xe3+ ion production was overestimated while the yields of Xe4+ to Xe8+ ions were underestimated. In spite of the deviation between the semi-empirical and experimental results, the electron correlation effects were concluded to partly explain the discrepancies between calculations and experiments. The significance of shake processes was previously (e.g. in papers [76], [77]) estimated to be very high. Electron-electron coincidence measurements published recently by Hikosaka et al [86] revealed a prominent intensity of the core-core PDI into Xe2+ 4d−2 states. The results confirmed that the two continuum electrons are indeed produced by the electron correlation with the 4p−1 configuration rather than the 4d shake-off process.

5.1. PAPERS I–III: MULTIPLE IONIZATION OF XE

29

700 680

3d

-1

-1

Paper II Paper III Paper V

-1

4s 4d

300

Energy relative to ground state (eV)

-3

4d

-1

4p 4d-1

250

-2

200

4d 5p-1 -1

4p

4d 5p-3

-1

5p-5

4d-2

150 -1

4d 5p-2 -1

5p-4

4d 5s-1 -1

100

4d 5p-1 4d

-1

5s-2 -1

5s 5p-1

50

-1

5s 5p-2 5p-3

5p-2

+

Xe

2+

Xe

3+

Xe

4+

Xe

5+

Xe

Figure 5.3: Average energies of the Xe configurations calculated with the HF wavefunctions. The Auger transitions studied in detail in Papers II, III, and V are marked with the arrows.

30

5.1.2


Paper III: M4,5 -NN4,5 Auger transitions

In Paper III, the M4,5 -NN4,5 Auger transitions were studied in detail. The studied Auger transitions, displayed in figure 5.3 with dotted arrows, are located at 320 to 550 eV. The spectrum was measured with two different photon energies in order to separate the Auger transitions originated from the 3d−1 3/2,5/2 initial states. The pure M5 -NN4,5 Auger spectrum was recorded at the photon energy of 687.4 eV, and the M4,5 -NN4,5 Auger transitions were measured at 699.3 eV. The photon energies used in Paper III and the 3d ionization thresholds are shown in the TIY spectrum in figure 5.1. Three line groups were resolved in the experimental spectra, and the intensity ratio of the M4,5 -N1 N4,5 , M4,5 -N2,3 N4,5 and M4,5 -N4,5 N4,5 Auger groups was determined to be 15% : 30% : 55% with the error limit of ±2%. The partial Auger rates were calculated by taking the wave functions from three different calculations: the single-configuration HF, the single-configuration Dirac-Fock and the multi-configurational Dirac-Fock (MCDF)1 . All the methods gave almost similar results, which were in good agreement with the experimental ratios. It could be concluded that, within the error limits, the Auger transition probabilities predicted in Paper II agree well with the experiments. Each Auger group was found to display a completely distinct behavior. The M4,5 -N4,5 N4,5 Auger group, which was already well interpreted in other studies [87, 88], shows a well-resolved sharp line structure. In the M4,5 N2,3 N4,5 Auger group, the sharp lines h–k and n–s shown in figure 5.4 are followed by the broad structureless features e–g and l–m, which were found to gain almost 50% of the total group intensity. These lines are broadened by the subsequent SCK transitions to the 4d−3 states, which are energetically possible only from the most uppermost energy states having binding energies over 254.8 eV (calculated value). The Lorentzian lifetime width of 8 eV was estimated for the broad lines while the sharp lines have the lifetime width of about 1 eV. The SCK probability of 35% was found to be underestimated in Paper I since almost 50% of the 4p−1 4d−1 states were now found to decay to the 4d−3 states. The M4,5 -N1 N4,5 Auger spectrum also consists of the intense sharp lines a–d and the following broad structure (no labels in figure 5.4). In this case, the broad structure was found, however, to originate from the numerous weak correlation satellites. The M4,5 -N2,3 N4,5 and M4,5 -N1 N4,5 Auger transitions were simulated with the aid of the calculated Auger transitions. In the level-to-level calculations made with Cowan’s code, FISCI was taken into account with the CI treatment by allowing the following configurations to mix with each other:

1 The MCDF [89, 90] calculations for this paper and all the other papers discussed from here forward were generated using the GRASP92 [91] code.


31

(a) M5N1,2,3N4,5

k b

i j e

Intensity (arb. units)

a

f g

h

(b) M4,5N1,2,3N4,5

p

rs o

cd l

320

340

360

m n

380 400 420 Kinetic energy [eV]

440

q

460

Figure 5.4: Experimental (upper panel, markers) and simulated (lower panel, solid line) (a) M5 N1,2,3 N4,5 and (b) M4,5 N1,2,3 N4,5 Auger spectra. The calculated Auger transitions are illustrated with vertical bars. Assignments for the Auger transitions a–s are given in Paper III. 1◦ 4s−1 4d−1 , 4p−1 4d−2 4f , and 4d−4 4f 2 with positive parities, and 2◦ 4p−1 4d−1 , 4d−3 4f , and 4s−1 4p−1 with negative parities. For the configuration set 1◦ , the 4s−1 4d−1 states were found to contribute to the main lines (labeled as a–d) by about 80% while the correlation satellites are identified by other states. The 4s−1 4p−1 configuration in configuration set 2◦ was found to give only a marginal contribution to the predicted M4,5 N2,3 N4,5 spectrum. Instead, a prominent mixing between the 4p−1 4d−1 and 4d−3 4f configurations was observed. It could be concluded that the ratio of group rates is not sensitive to electron correlation whereas the mixing of configurations plays an important role in the spectral features.

5.1.3

Paper II: Interpretation of Auger cascade satellites

In Paper II, the Auger electron spectrum in the kinetic energy region of 8–40 eV was measured with the photon energies of 650 and 740 eV , which are below and above the 3d ionization thresholds of 676.7 and 689.4 eV [92],

32


respectively. Measured spectra are depicted in figure 5.5 (a). Many different Auger transitions, e.g. the N4,5 -OO Auger transitions among others, are located at the same kinetic energy region. In Paper II, only the Auger cascades produced during the de-excitation of the 3d ionization (referred to in Paper II as Auger cascade satellites) were studied, and they were separated from the other structure by subtracting the Auger spectra measured at the lower photon energy from the high energy data. The data after the subtraction is depicted in figure 5.5 (b). The Auger cascade satellites which have kinetic energies between 8–40 eV were determined with calculations using the method of global characteristics. Figure 5.5 (c) shows with the bars the calculated average energies and the relative intensities of the Auger cascade satellites with the envelope spectrum simulated with the aid of the calculated results. The Auger cascade satellites located in the studied kinetic energy region were second-, third- and fourthstep transitions. The most intense satellite transitions, labeled as 2, 8, 10, and 16 in figure 5.5 (c), respectively, are 2 4d−1 5p−3 → 5p−5 , 8 4d−1 5p−2 → 5p−4 , 10 4d−2 5p−1 → 4d−1 5p−3 , and 16 4d−2 → 4d−1 5p−2 . In order to interpret the fine structure of the subtracted spectrum, the levelto-level calculations were carried out for two different paths of Auger transitions, namely 3d−1 → 4p−1 4d−1 → 4d−2 5p−1 → 4d−1 5p−3 → 5p−5 and 3d−1 → 4d−2 → 4d−1 5p−2 → 5p−4 . The calculated Auger transitions are displayed in figure 5.3 with solid arrows. The electron spectrum shown in figure 5.5 (d) is a sum of the calculated fine structures of the Auger transitions 2, 8, 10 and 16 together with the envelope curve shown in figure 5.5 (c) excluding the structures 2, 8, 10 and 16. Three line groups can be distinguished in both the experimental spectrum in figure 5.5 (b) and the simulated spectrum in figure 5.5 (c). The calculations underestimates slightly the intensity ratio between the second and third line group as the ratios are 1.31 and 1.40 according to the calculations and experiments, respectively. This indicates that the calculations underestimate the transition probabilities of the last step transitions locating at the lower kinetic energy region. This was already comprehensively established by the PIY measurements discussed in section 5.1.1.


Intensity (arb. units)

3000

33

(a)

2000 1000 0

(b)

2000 1500 1000 500

Intensity relative to creation of one 3d hole

0 0.8

(c)

0.6 0.4 0.2

8

16

10

2

5 4 6 7

9

11

0.0

15 12 13

17

0.16

(d) 0.12 0.08 0.04 0.00 10

15

20

25

30

35

40

Kinetic energy [eV]

Figure 5.5: (a) Electron spectra measured with photon energy of 650 eV (dashed line) and 740 eV (solid line). (b) Difference spectrum of the detected spectra in figure (a). (c) Spectrum given by calculations with the method of global characteristics. The vertical bars show the calculated average energies and intensities of the Auger transitions following the 3d ionization. (d) Same as in figure (c) except that the spectra given by detailed level-to-level calculations have been substituted for the structures 2, 8, 10, and 16.


34

The correspondence between the subtracted spectrum in figure 5.5 (b) and the calculated spectrum in figure 5.5 (d) was found to be excellent especially in the high kinetic energy region. This allowed the identification of the main lines in the experiment in Paper II. The assignments for the 4d−2 → 4d−1 5p−2 and 4d−1 5p−2 → 5p−4 Auger transitions were later confirmed by the measurements of Hikosaka et al [86]. They measured two Auger electrons originated from the 4d−2 (1 D2 /1 G4 ) → 4d−1 5p−2 and 4d−1 5p−2 → 5p−4 Auger transitions in coincidence, which allowed them to connect the initial states to the final states.

5.2

Paper IV–V: Satellite Auger spectra

In Paper IV, the M4,5 N-NNN Auger transitions of Kr were studied in detail. The transitions appear as a satellite structure in the M4,5 -NN Auger spectra. The 3d−1 4l−1 (l = s, p) doubly ionized states of Kr are produced via CK decay after the 3p ionization [93], which leads to the M2,3 -M4,5 N-NNN cascade of Auger transitions. The M4,5 N-NNN satellite spectrum appears also when the incident photon energy is below the 3p ionization threshold. In that case, the doubly ionized states are produced via direct PDI from the ground state [95]. In Paper IV, it was possible to separately investigate the PDI and Auger cascade channels as the satellite spectra were measured both above (at 217.6 and 280 eV) and below (at 208 eV) the 3p ionization threshold of Kr. In figure 5.6, the calculated energy levels of Kr are shown, and arrows labeled with channel I and II depict the PDI and Auger cascade channels, respectively. The contribution of the Auger cascade channel in the satellite production was determined by subtracting the satellite spectrum originated purely from the PDI channel. The kinetic energies and intensities of the M2,3 -M4,5 N CK and M4,5 NNNN Auger transitions of Kr were calculated by using the HF wave functions obtained for the following configurations with the CI treatment: Double ionized configurations were 1◦ 3d−1 4s−1 and 3d−1 4p−2 4d with positive parities and 2◦ 3d−1 4p−1 with negative parities and the triply ionized configurations were 3◦ 4s−1 4p−2 and 4p−4 4d with positive parities and 4◦ 4s−2 4p−1 , 4s−1 4p−3 4d and 4p−3 with negative parities. The 3p−1 configuration of Kr was found to mix with the 3d−2 5p configuration according to the MCDF calculations, but the mixing was not as prominent as the CI between the 4p−1 and 4d−2 4f configurations of Xe [57]. The electron

5.2. PAPER IV–V: SATELLITE AUGER SPECTRA

35

correlation related to the 3p orbital was seen to affect the M2 -M4,5 N1,2,3 CK spectrum. The M4,5 N-NNN satellite transitions were found to originate mainly from the 3d−1 4p−1 states. Thus, the population of the 3d−1 4p−1 initial states produced by PDI was estimated semiempirically. The most intense satellite transitions were assigned with the aid of the calculations. Many new satellite lines were identified. Especially, the M4,5 N2,3 -N1 N2,3 N2,3 satellite lines locating at the low kinetic energy region, which were neglected in the previous studies [95, 96], were now assigned. The studies for the satellite transitions were continued in Paper V, as the N4,5 O-OOO satellite transitions of xenon were measured using two photon energies of 130 eV and 150 eV which are above the 4d−1 5l−1 (l = s, p) and 4p−1 ionization thresholds, respectively. The N2,3 -N4,5 O2,3 -O2,3 O2,3 O2,3 cascade of Auger transitions is illustrated in figure 5.3 with dashed arrows. The cascade and PDI channels were found to be important in satellite production at the photon energies just above the 3p/4p ionization threshold both in the Kr and Xe case. In the case of xenon, the 4p−1 initial state is highly correlated, which influences the N2,3 -N4,5 O CK spectrum [57, 94]. However, as according to [94] most of the total intensity of the N2,3 -N4,5 O Auger transitions originates from the most populated initial state. So, only the SC 4p−1 3/2 initial state was included in our calculations. The N2,3 -N4,5 O CK transitions were calculated by using the SC 4d−1 5p−1 configuration and by including FISCI between the 4d−1 5s−1 and 4d−1 5p−2 5d configurations. These intermediate states were also used as initial states in the calculations for the subsequent N4,5 O-OOO Auger transitions. The final states were constructed by the SC 5p−3 configuration and the mixed 5s−1 5p−2 ↔ 5p−4 5d configurations. The most intense N4,5 O2,3 O2,3 O2,3 O2,3 satellite lines were assigned with the aid of the calculations, while the N4,5 O2,3 -O1 O2,3 O2,3 transitions show only as a background overlapping with the intense main lines. Indications that electron correlation effects are more prominent in the 5s orbital of Xe than in the 4s orbital of Kr were found. Our results were compared to the previous studies [95] and [98]. The assignments for the N4,5 O2,3 -O2,3 O2,3 O2,3 satellite lines were revised from the previous study [95], and our assignments were found to agree well with the measurements [98].


36

230

3p

-1

-2

3d 5p

220 210

Ma,b 2 M3

200

channel II

Binding energy (eV)

150

-1

-1

3d 4s -1 -2 3d 4p 4d

140 -2

130

3d 4p

-1

-1

-1

-2

-1

4s 4p -1 -3 4s 4p 4d

120

channel I 110 100

4s 4p

-4

4p 4d

90

4p

80 70

-3

from the ground state

Kr+

Kr2+

Kr3+

Figure 5.6: Energy level diagram of Kr. Kr+ states were calculated with the MCDF method, and Kr2+ and Kr3+ states were computed using the HF method. The Auger transitions studied in Paper IV have been marked with arrows.

5.3. EPILOGUE

5.3

37

Epilogue

This thesis consists of a summary and five articles which deal with the electron spectra of Xe and Kr analyzed by combining experiments with theoretical predictions. Using electron and ion spectroscopical methods and computational tools, it has been possible to obtain new information from structures and decay dynamics of Xe and Kr core excited states. Most of the electronic structures studied in this thesis were well-predicted by the pseudorelativistic HF wave functions. Still, the limitations of the approximation were seen especially in predicting highly correlated energy states, in which cases the use of the relativistic MCDF approximation was needed in order to properly account for the spin-orbit interaction. However, as the MCDF calculations are usually time-consuming and may encounter convergence problems, multi-configurational HF calculations are more suitable when energy states of dozens or hundreds of electron configurations must be computed. The efficiency of the multicoincidence technique in separating the different decay processes, in which many electrons are ejected simultaneously, was demonstrated recently for various atoms and molecules [35, 86, 97]. By analyzing conventional Auger electron spectra, information about transition energies and probabilities is obtained. However, a direct observation of multiple ionizations or transitions in which many electrons are emitted during the Auger decay is impossible. Instead, they can be studied indirectly by measuring their decay spectra, like it was done in this thesis.

38


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