Download Spontaneous and stimulated X-ray Raman scattering

Spontaneous and stimulated X-ray Raman
scattering
YU-PING SUN
Department of Theoretical Chemistry and Biology
School of Biotechnology
Royal Institute of Technology
Stockholm, Sweden 2011
c Yu-Ping Sun, 2011
°
ISBN 978-91-7415-925-7
ISSN 1654-2312
TRITA-BIO Report 2011:8
Printed by Universitetsservice US AB,
Stockholm, Sweden, 2011
Abstract
The present thesis is devoted to theoretical studies of resonant X-ray scattering and propagation of strong X-ray pulses.
In the first part of the thesis the nuclear dynamics of different molecules is studied using resonant X-ray Raman and resonant Auger scattering techniques. We show that the
shortening of the scattering duration by the detuning results in a purification of the Raman spectra from overtones and soft vibrational modes. The simulations are in a good
agreement with measurements, performed at the MAX-II and the Swiss Light Source with
vibrational resolution. We explain why the scattering to the ground state nicely displays the
vibrational structure of liquid acetone in contrast to excited final state. Theory of resonant
X-ray scattering by liquids is developed. We show that, contrary to aqueous acetone, the
environmental broadening in pure liquid acetone is twice smaller than the broadening by
soft vibrational modes significantly populated at room temperature. Similar to acetone,
the “elastic” band of X-ray Raman spectra of molecular oxygen is strongly affected by the
Thomson scattering. The Raman spectrum demonstrates spatial quantum beats caused by
two interfering wave packets with different momenta as the oxygen atoms separate. It is
found that the vibrational scattering anisotropy caused by the interference of the “inelastic”
Thomson and resonant scattering channels in O2 . A new spin selection rule is established
in inelastic X-ray Raman spectra of O2 . It is shown that the breakdown of the symmetry selection rule based on the parity of the core hole, as the core hole and excited electron
swap parity. Multimode calculations explain the two thresholds of formation of the resonant
Auger spectra of the ethene molecule by the double-edge structure of absorption spectrum
caused by the out-of- and in-plane modes. We predict the rotational Doppler effect and
related broadening of X-ray photoelectron and resonant Auger spectra, which has the same
magnitude as its counterpart–the translational Doppler effect.
The second part of the thesis explores the interaction of the medium with strong X-ray
free-electron laser (XFEL) fields. We perform simulations of nonlinear propagation of femtosecond XFEL pulses in atomic vapors by solving coupled Maxwell’s and density matrix
equations. We show that self-seeded stimulated X-ray Raman scattering strongly influences
the temporal and spectral structure of the XFEL pulse. The generation of Stokes and
four-wave mixing fields starts from the seed field created during pulse propagation due to
the formation of extensive ringing pattern with long spectral tail. We demonstrate a compression into the attosecond region and a slowdown of the XFEL pulse up to two orders of
magnitude. In the course of pulse propagation, the Auger yield is strongly suppressed due to
the competitive channel of stimulated emission. We predict a strong X-ray fluorescence from
the two-core-hole states of Ne created in the course of the two-photon X-ray absorption.
Preface
The work presented in this thesis has been carried out (May 2008-May 2011) at the Department of Theoretical Chemistry and Biology, Royal Institute of Technology, Stockholm,
Sweden.
List of papers included in the thesis
Paper I Y.-P. Sun, Q. Miao, A. Mohammed, H. Ågren, and F. Gel’mukhanov, Shortening scattering duration by detuning purifies Raman spectra of complex systems, manuscript
(2011).
Paper II Y.-P. Sun, F. Hennies, A. Pietzsch, B. Kennedy, T. Schmitt, V.N. Strocov,
M. Berglund, J.-E. Rubensson, K. Aidas, F. Gel’mukhanov, M. Odelius and A. Föhlisch,
Intramolecular Soft Modes and Intermolecular Interactions in Liquid Acetone, manuscript
(2011).
Paper III A. Pietzsch, Y.-P. Sun, F. Hennies, Z. Rinkevicius, H.O. Karlsson, T. Schmitt,
V.N. Strocov, J. Andersson, B. Kennedy, J. Schlappa, A. Föhlisch, J.-E. Rubensson, and
F. Gel’mukhanov, Spatial Quantum Beats in Vibrational Resonant Inelastic Soft X-Ray
Scattering at Dissociating States in Oxygen, Phys. Rev. Lett., accepted (2011).
Paper IV Y.-P. Sun, A. Pietzsch, F. Hennies, Z. Rinkevicius, H.O. Karlsson, T. Schmitt,
V.N. Strocov, J. Andersson, B. Kennedy, J. Schlappa, A. Föhlisch, F. Gel’mukhanov, and
J.-E. Rubensson, Hidden Symmetry in Molecular Oxygen, manuscript (2011).
0
Paper V J.-C Liu, C. Nicolas, Y.-P Sun, R. Flammini, P. O Keeffe, L. Avaldi, P. Morin,
V. Kimberg, N. Kosugi, F. Gelmukhanov and C. Miron, Multimode Resonant Auger Scattering from the Ethene Molecule, J. Phys. Chem. B, accepted (2011).
Paper VI Y.-P. Sun , C.-K. Wang and F. Gel’mukhanov, Rotational Doppler effect in
x-ray photoionization, Phys. Rev. A, 82, 052506 (2010).
Paper VII Y.-P. Sun , J.-C. Liu, C.-K. Wang and F. Gel’mukhanov, Propagation of a
strong x-ray pulse: Pulse compression, stimulated Raman scattering, amplified spontaneous
emission, lasing without inversion, and four-wave mixing, Phys. Rev. A, 81, 013812 (2010).
Paper VIII Y.-P. Sun, J.-C. Liu and F. Gel’mukhanov, Slowdown and compression of
a strong X-ray free-electron pulse propagating through the Mg vapors, EuroPhys. Lett., 87,
64002 (2009).
III
Paper IX J.-C. Liu, Y.-P. Sun, C.-K. Wang, H. Ågren and F. Gel’mukhanov, Auger
effect in the presence of strong x-ray pulses, Phys. Rev. A, 81, 043412 (2010).
Paper X Y.-P. Sun , Z. Rinkevicius, C.-K. Wang, S. Carniato, M. Simon, R. Taı̈eb and
F. Gel’mukhanov, Two-photon-induced x-ray emission in neon atoms, Phys. Rev. A, 82,
043430 (2010).
List of papers that are not included in the thesis
Paper I S. Gavrilyuk, Y.-P. Sun, S. Levin, H. Ågren and F. Gel’mukhanov, Recoil splitting of x-ray-induced optical fluorescence, Phys. Rev. A, 81, 035401 (2010).
Paper II Y.-P. Sun, J.-C. Liu and F. Gel’mukhanov, The propagation of a strong xray pulse followed by pulse slowdown and compression, amplified spontaneous emission and
lasing without inversion, J. Phys. B: At. Mol. Opt. Phys., 42, 201001 (2009).
Paper III Y.-P. Sun, S. Gavrilyuk, J.-C. Liu, C.-K. Wang, H. Ågren, and F. Gel’mukhanov,
Optical limiting and pulse reshaping of picosecond pulse trains by fullerene C60, J. Electron.
Spectrosc. Relat. Phenom, 174, 125 (2009).
IV
Comments on my contributions to the papers included
• I was responsible for part of the calculations in Papers I, and contributed to the
discussion of the results and writing of the manuscript.
• I was responsible for the theoretical work in Papers II, III, IV, V, and participated
in the discussion, writing and editing of the manuscript.
• I was responsible for the theory and calculations, and participated in the discussion,
writing of the manuscript in Paper VI, X.
• I was responsible for most of the calculations, and participated in discussion, writing
and editing of the manuscript in Papers VII, VIII, IX.
V
Acknowledgments
This present thesis can not be written without the generous help of many people:
First of all, I like to express my deep gratitude to my supervisor Prof. Dr. Faris Gel’mukhanov.
Thank him for everything I learned from him, his guidance, the constant help, and the essential supervising work dedicated during the elaboration of this thesis. His exceptional
energy and enthusiasm for science are indeed inspiring. It was a pleasure and honor to work
under his careful supervision.
I want to thank Prof. Dr. Hans Ågren for giving me the opportunity to develop my PHD
studies in Theoretical Chemistry group. I thank him for creating such a good atmosphere
for doing scientific work and fruitful collaboration.
Special thanks to my supervisor in China, Chuan-Kui Wang who introduced me to my
research field and gave me great guidance. I thank him for giving me the opportunity to
study here in Stockholm and his help both in scientific work and my personal life.
I am very grateful to Prof. Yi Luo for many suggestions about the work and life. I also
want to appreciate Luo’s family for the happy memories we shared together. I thank Prof.
Zivilnas Rinkevicius and Prof. Ying Fu for their constructive discussion and help in program
problems in my research work.
I would like to acknowledge the theoreticians with whom I had the pleasure to work - Dr.
Ji-Cai Liu, Dr. Sergey Gavrilyuk, Quan Miao, Dr. Abdelsalam Mohammed, Dr. Victor Kimberg, Dr. Michael Odelius, Dr. Kestutis Aidas, Prof. Nobuhiro Kosugi, Prof.
Boris Minaev, Prof. Stephane Carniato, and Prof. Richard Taı̈eb. A good deal of my
projects concerning resonant X-ray scattering were done in close collaboration with excellent experimentalists- Prof. Jan-Erik Rubensson, Prof. Alexander Föhlisch, Dr. Annette
Pietzsch, Dr. Franz Hennies, Dr. Hans O. Karlsson, Dr. Thorsten Schmitt, Dr. Vladimir N.
Strocov, Dr. Joakim Andersson, Dr. Brian Kennedy, Dr. Justine Schlappa, Prof. Catalin
Miron, Prof. Paul Morin, Prof. Roberto Flammini, Prof. Lorenzo Avaldi, Prof. Marc
Simon, and Dr. Christophe Nicolas. I would like to acknowledge them for everything I
learned in our common works.
Special thanks to Prof. Dr. Hans Ågren, Prof. Marten Ahlquist, Dr. Michael Odelius,
Prof. Zivilnas Rinkevicius, Prof. Olav Vahtras and Dr. Kestutis Aidas for careful reading
the preprint and providing valuable suggestions.
I want to thank my colleagues Yue-Jie Ai, Xing Chen, Guang-Jun Tian, Wei-Jie Hua, Xin
Li, Li-Li Lin, Xiu-Neng Song, Yong Ma, Qiong Zhang, Qiang Fu, Sai Duan, Ying Zhang,
Fu-Ming Ying and all the present and former members of the Department of Theoretical
VI
Chemistry and Biology at KTH for creating such a friendly, warm and stimulating working
atmosphere and for all the help.
I want to acknowledge the financial support (2008-2010) from the China Scholarship Council.
Finally, last but not least, Great thanks to my parents and my brother for their endless love
and support wherever I go. I wish to thank Xin-Ming Du for his sweetest love and support.
Yu-Ping Sun
Stockholm, 2011-05
Contents
1 Introduction
1
2 Overview of some basic principles
3
2.1
2.2
Weak X-ray field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.1.1
X-ray absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.1.2
Resonant X-ray scattering . . . . . . . . . . . . . . . . . . . . . . . .
4
2.1.3
Resonant X-ray scattering as a dynamical process. Scattering duration
7
Strong X-ray field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.2.1
Maxwell’s equations
. . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.2.2
Amplitude equations . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.2.3
Density matrix equation . . . . . . . . . . . . . . . . . . . . . . . . .
10
2.2.4
Rabi oscillations and McCall-Hahn Area Theorem . . . . . . . . . . .
11
3 Short-time approximation for resonant Raman scattering
13
4 Resonant X-ray Raman scattering from liquid acetone
17
4.1
Scheme of transitions and vibrational modes . . . . . . . . . . . . . . . . . .
18
4.2
Resonant X-ray Raman scattering: f = n−1 π ∗ . . . . . . . . . . . . . . . . .
20
4.2.1
Environmental broadening in liquid . . . . . . . . . . . . . . . . . . .
20
4.2.2
Thermal population of torsion modes . . . . . . . . . . . . . . . . . .
22
4.2.3
Broadening of RIXS band by soft vibrational modes . . . . . . . . . .
22
Resonant elastic X-ray scattering: f = 0 . . . . . . . . . . . . . . . . . . . .
24
4.3
VIII
CONTENTS
5 RXS spectra of the O2 molecule near the σ ∗ = 3σu resonance
27
5.1
Potential surfaces and scheme of transitions . . . . . . . . . . . . . . . . . .
27
5.2
REXS channel: Thomson scattering, spatial quantum beats . . . . . . . . . .
28
5.3
RIXS channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
5.3.1
Hidden selection rules (f = 13 Πg (1πu−1 3σu1 )) . . . . . . . . . . . . . .
32
5.3.2
Opening of the symmetry forbidden RIXS channel due to core-hole
−1 1
jump (f =3 Σ−
g (3σg 3sg )) . . . . . . . . . . . . . . . . . . . . . . . .
33
6 Resonant Auger scattering from the ethene molecule
37
6.1
Scheme of transitions and vibrational modes . . . . . . . . . . . . . . . . . .
37
6.2
Multimode resonant Auger scattering . . . . . . . . . . . . . . . . . . . . . .
40
7 Rotational Doppler effect in X-ray photoionization and resonant Auger
scattering
43
7.1
Theoretical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
7.2
Rotational Doppler effect in the X 2 Σ+ band of CO molecule . . . . . . . . .
46
7.3
Rotational Doppler effect in photoionization of the 5σ electrons of HCl molecule 47
8 Nonlinear propagation of strong XFEL pulses
49
8.1
Theoretical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
8.2
The Burnham-Chiao modulation and self-seeding . . . . . . . . . . . . . . .
51
8.3
Quenching of the population inversion and lasing without inversion . . . . .
52
8.4
Pulse compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
8.5
Slowdown of the XFEL pulse . . . . . . . . . . . . . . . . . . . . . . . . . .
55
8.6
Suppression of the Auger effect in the propagation of XFEL pulses . . . . . .
56
9 Two-photon induced X-ray fluorescence
59
9.1
Bound-bound-bound-bound (BBBB) TPA . . . . . . . . . . . . . . . . . . .
60
9.2
Bound-continuum-bound-bound (BCBB) TPA . . . . . . . . . . . . . . . . .
61
10 Summary of results
63
Chapter 1
Introduction
There is no royal road to science, and only those who do not dread the
fatiguing climb of its steep paths have a chance of gaining its luminous summits.
— Karl Marx
The discovery of X-rays by W.K. Röngen in 1895 is one of the notable landmarks in the
history of light. One of the greatest applications of X-rays is X-ray crystallography triggered by K. von Laue and W.H. and W.L. Bragg. Soon later, various X-ray spectroscopy
techniques like X-ray absorption, fluorescence, photoelectron and Auger spectroscopy were
developed to study the structure of matter. During the last two decades we have evidenced
a fast development of Resonant inelastic X-ray scattering (RIXS) and resonant Auger scattering (RAS) methods with numerous applications in investigations of electronic structure
and nuclear dynamics of gases and condensed matter. These processes are site specific on
the atomic length scale and time specific on the timescale for nuclear and electronic rearrangements (femto- to attoseconds), which makes resonant scattering techniques very useful
in atomic, chemical and condensed matter physics. However, RIXS techniques have suffered
from the lack of adequate radiation sources for a long time. In practice this has limited
the spectral quality and only a fraction of the inherent advantages have been exploited.
Recently, the availability of high-flux X-ray sources–3rd generation of synchrotron radiation
light sources, combined with the drastic improvement of the spectral resolution of spectrometers (E/∆E ∼ 104 ) allows to resolve fine electronic structure and even single vibrational
modes. In this thesis, we investigate the Resonant X-ray scattering from different typical
molecules (like O2 , C2 H4 and CO(CH3 )2 ) in gas or liquid phases measured recently by our
experimental colleagues with superhigh resolution.
The advent of coherent and brilliant X-ray free-electron lasers (XFELs)1–6 has brought a
qualitative change in X-ray science, and has promoted X-ray science from the linear into the
2
Chapter 1 Introduction
high nonlinear regime. The peak intensity of short X-ray free-electron laser pulses is enough
to saturate bound-bound resonant transitions. When the pulse duration of the XFEL pulse
exceeds 100 fs, the atoms and molecules can be fully ionized and the target becomes a
plasma. Due strong ionization the molecules dissociate due to the Coulomb explosion. The
high intensity of the X-rays make stimulated emission important, which starts to compete
with fast Auger decay (or even quench it). Other nonlinear processes, like four wave mixing,
strongly affect the propagation of the XFEL pulse. Strong nonlinearity of the light-matter
interaction changes the refraction index in X-ray spectral region. One of the manifestations
of this is a strong slowdown of X-ray pulses. All of this indicate that strong field X-ray
physics is a new fast developing branch of physics with many applications and new effects.
To describe quantitatively the interaction of XFEL pulses with the matter, we solve in
present thesis coupled equations for field and matter, namely Maxwell’s and density matrix
equations.
Chapter 2
Overview of some basic principles
2.1
Weak X-ray field
X-rays have wavelengths from tenths of Angstroms to hundreds of Angstroms, which corresponds to energies in the range of 100 eV to 100 keV. When X-rays with sufficient energies
are exposed on the molecules, the X-ray photons can knock the inner orbital electrons and
induce spectral transitions between different quantum levels of the molecule. Absorption
transitions are followed by emission of either an X-ray photon or an Auger electron. Xray spectroscopies have been used widely in chemistry and material analysis to determine
structure and element composition of matter. Among X-ray techniques, the resonant X-ray
scattering (RXS) spectroscopy has become a major tool for probing and studying electronic structure of the matter owing to the rapid development of synchrotron radiation light
sources.
To explain the basic processes studied in the thesis we use in this Chapter the BornOppenheimer (BO) approximation which says that the total molecular wave function is
the product of electronic and nuclear wave functions
|Ψi i ≈ |ii|νi i.
(2.1)
One should notice that this approximation is broken near the crossing of the potentials of
different states.7, 8
2.1.1
X-ray absorption
The basic process for different X-ray spectroscopies is the absorption of an X-ray photon
followed by excitation of the molecule to an excited electronic-vibrational state (2.1). The
4
Chapter 2 Overview of some basic principles
cross section of absorption of the X-ray photon with the frequency ω and the polarization
vector e is given by the Fermi Golden rule9
σXAS = 4παωΓ
X
hν i |(e · di0 )|ν 0 i2
.
(ω − ωi0 − ∆i0 )2 + Γ2
i,ν i
(2.2)
Here α ≈ 1/137 is the fine structure constant, dij and ωij are the transition dipole moment
and the adiabatic transition energy, ∆ij = ²νi − ²νj is the change of the vibrational energy
under the electronic transition i − j, Γ is the lifetime broadening of the core excited state,
which determines the wave packet evolution in the core excited state.
2.1.2
Resonant X-ray scattering
Resonant X-ray scattering occurs when the photon energy of incident light coincides with an
electronic transition of the molecule. The molecule on the ground state |0i can be excited to
the core excited state |ii by absorbing an incoming X-ray photon (ω). The core excited state
is metastable with a lifetime about a few femtoseconds. Therefore, the molecule will decay
to one of many possible final states |f i. There exists two distinct decay channels: radiative
and nonradiative decay channels. The radiative decay channel is accompanied by emitting
i
i
Figure 2.1: Scheme of RXS: radiative RXS and nonradiative RXS.
a final X-ray photon (ω1 ), while in the nonradiative (Auger) scattering channel the energy
of the core excited state will be released by ionization of one electron (e− ) from the molecule
(Fig. 2.1). The physical mechanisms of the emission steps are qualitatively different for these
two RXS processes. In the radiative RXS process, the interaction leading to the emission
of X-ray photons is the electromagnetic interaction (df i ), which normally in the soft X-ray
region is governed by the dipole selection rules. For Auger scattering, the interaction of the
2.1 Weak X-ray field
5
Auger decay is Coulomb interelectron interaction (Qf i ), in which the symmetry selection
rules are invalid. There is a competition between radiative (fluorescence) and non-radiative
(Auger) decay processes (see Fig. 2.2). One can see that a clear transition from electron to
photon emission occurs when increasing the atomic number.
1.0
Auger
Fluorescence
0.5
K-shell
0.0
10
20
30
40
50
60
Atomic Number
Figure 2.2: Comparison of Auger yield and fluorescence yield as a function of atomic number.10
X-ray Raman, Thomson and Compton Scattering
The radiative RXS process consists of two distinct scattering channels (Fig. 2.1 left panel):
the resonant inelastic X-ray scattering (RIXS) channel in which the final electronic state |f i
differs from the ground electronic state |0i and the resonant elastic X-ray scattering (REXS)
channel which ends up in the ground electronic state. According to standard perturbation
theory,11–13 the radiative RXS cross section is given by the Kramers-Heisenberg formula14
ω1 X
σ(ω, ∆ω) = r02
|Fνf |2 Φ(ω − ω1 − ωf 0 − ∆f 0 , γ), ∆ω = ω1 − ω,
(2.3)
ω ν
f
where r0 = α2 = 2.28 × 10−13 cm is the classical radius of the electron. The width γ of
the spectral function Φ includes the instrumental broadening as well as the spectral width
of incident light. The scattering amplitude Fνf consists of two parts: Thomson scattering
amplitude FT and resonant scattering amplitude FR :
Fνf = FT + FR = (e1 · e)hf |
X
eıq·r |0ihνf |ν0 i +
X
i,νi
ωf i ωi0
(e1 · df i )(di0 · e)hνf |νi ihνi |ν0 i
.
ω − ωi0 − ∆i0 + iΓ
(2.4)
6
Chapter 2 Overview of some basic principles
Here q = k1 − k is the change of the X-ray photon momentum, ω, ω1 , k, k1 and e, e1
are the frequencies, momenta and the polarizations of the incoming and emitted photons,
|ν0 i, |νi i and |νf i are the vibrational wave functions of the ground, core excited and final
states. The small second off-resonant term11–13 is neglected in Eq.(2.4). The first term at the
right-hand side of this equation is well known in REXS as the Thomson scattering. When
the photon frequency is much larger than the ionization potential of the bound electron,
this term describes inelastic scattering followed by ionization of the bound electron. This is
well known Compton scattering of X-rays from bound electrons.13, 15 The discussed term in
Eq.(2.4) results also in inelastic X-ray scattering accompanied by excitation of a core electron
into a bound state when the photon frequency is smaller than the ionization potential of this
electron. We will show in Sec. 5.3 that this term can compete with the resonant contribution.
We name below the first contribution at the right-hand side of Eq.(2.4) as the Thomson
term.
Nonlinear polyatomic molecules have the 3n − 6 normal vibrational modes, νi = {νim } =
{νi1 , νi2 , · · · , νi3n−6 }, where νim is the vibrational quantum number of the m−th mode of the
electronic state i. This means that the multimode Franck-Condon (FC) amplitude hνf |νi i
is the product of the FC amplitudes of each particular mode, and the vibrational energy
²νi is the sum of the vibrational energies ²νim of each mode:
X
νi
→
3n−6
XX
m=1 νim
,
hνf |νi i =
3n−6
Y
hνf m |νim i,
m=1
²νi →
3n−6
X
²νim .
(2.5)
m=1
Resonant Auger scattering and direct ionization
Resonant Auger scattering (RAS),16 being very similar to the radiative RXS, has two qualitatively different resonant channels (Fig. 2.1 right panel). One is the so-called “spectator”
channel in which the core excited electron remains in the same unoccupied molecular orbital
where it was excited. The other possible alternative is the “participator” channel. In this
process the core excited electron participates in the Auger transition. It returns to its initial
core orbital, filling the hole. The final state of the Auger scattering can be also populated by
direct photoionization, whose counterpart in the radiative RXS is the Thomson scattering.
The resonant Auger scattering (RAS) cross section and amplitude are similar to the equations (2.3, 2.4) except
ω1 → E, df i → Qf i , FT → FD .
(2.6)
Here E is the energy of Auger electron, FD = (df 0 · e)hνf |ν0 i is the amplitude of direct
photoionization. When the photon energy ω of incident X-ray is tuned far from the resonant
frequency ωi0 , the direct radiative transition from the ground state to the final ionic state
2.1 Weak X-ray field
7
(with amplitude FD ) becomes dominant ionization channel. The picture is inverted when
the photon energy ω approaches the resonance at ωi0 . The amplitudes of resonant (FR ) and
direct (FD ) scattering channels can be comparable in the vicinity of the resonance. Ugo
Fano has developed a theory which treats both ionization channels as a single scattering
process.17
2.1.3
Resonant X-ray scattering as a dynamical process. Scattering duration
To understand the dynamics of RXS process, it is instructive to rewrite the resonant part
of Kramers-Heisenberg scattering amplitude (2.4) in time-dependent representation
X hν f |ν i ihν i |ν 0 i Z ∞
FR ∝
=
F (t)dt.
(2.7)
Ω − ∆i0 + ıΓ
0
i,νi
Here Ω = ω−ωi0 is the detuning of the photon frequency relative to the absorption resonance.
The scattering amplitude in time domain
F (t) = −ıe−t/τ hν f |ψ(t)i,
|ψ(t)i = e−ıHi t |ν 0 i
(2.8)
is the projection of the wave packet of the intermediate state |ψ(t)i to the final vibrational
state |ν f i. Here Hi is the nuclear Hamiltonian of the core excited state. According to the
uncertainty principle, the photon with well defined frequency is unrestricted in time domain,
which means the time of the photon absorption is unknown. Due to this the integration
over time in Eq.(2.7) shows the contribution of all coherent absorption-emission events (see
Fig. 2.3).
The effective scattering duration
τ = τ (Γ, Ω) =
1
Γ − ıΩ
(2.9)
depends on both the lifetime broadening of the core excited state Γ and the detuning Ω.
The qualitatively different roles of Γ and Ω can be seen clearly in Fig. 2.4. The lifetime
broadening Γ corresponds to the irreversible decay of the core excited state, which quenches
the long-term contribution (t > 1/Γ) to the scattering amplitude. The detuning |Ω| plays
qualitatively a different role. It gives the destructive interference of different scattering
channels with different absorption times tabs
n (Fig. 2.3). When the detuning |Ω| is large, the
long-time contribution to the scattering amplitude will be quenched due to the fast signchanging oscillations exp(ıΩt). One can vary the effective scattering duration continuously
8
Chapter 2 Overview of some basic principles
temission
field
t1abs t2abs t3abs t4abs t5abs
t
Figure 2.3: The scattering is going through the coherent superposition of the core excited
states created at different absorption times tabs
n .
by changing the frequency detuning, and hence control the nuclear dynamics, which acts
as a camera shutter for the measurement (Fig. 2.4). This makes it possible to select the
subprocesses with different time scales.
Figure 2.4: Shortening of the scattering duration by detuning.
2.2 Strong X-ray field
2.2
9
Strong X-ray field
When the X-ray field is strong the interaction between the field and the matter becomes
nonlinear and we should solve coupled equations for field and matter. We will treat the
medium explicitly using the quantum density matrix equations while the evolution of the
electromagnetic field will be described semiclassically using Maxwell’s equations.
2.2.1
Maxwell’s equations
The propagation of the field through the medium is governed by the Maxwell’s equations
(in SI units),
∇ · D(r, t) = ρ0 (r, t),
∂H(r, t)
∇ × E(r, t) = −µ
,
∂t
∇ · B(r, t) = 0,
∂D(r, t)
.
∇ × H(r, t) = J 0 (r, t) +
∂t
(2.10)
where E, H, D and B are electric, magnetic, electric displacement and magnetic induction
fields, respectively. ρ0 (r, t) and J 0 (r, t) are the densities of free charges and free currents,
respectively, which are absent in vacuum, µ is the permeability of the medium.
The electric displacement field is related to the polarization P.
D(r, t) = ²0 E(r, t) + P(r, t),
(2.11)
where ²0 is the free space permittivity. The polarization P(r, t) being the dipole moment
of the medium acts as a source term in the equation for the radiation field. It depends
nonlinearly on the electric field E(r, t).
2.2.2
Amplitude equations
The dynamics of the quantum system in the electromagnetic field is described by the timedependent Schrödinger equation
ı~
∂ψ(r, t)
= Ĥψ(r, t),
∂t
Ĥ = Ĥ0 + V̂ (r, t).
(2.12)
Here Ĥ0 is the unperturbed Hamiltonian of the system, and
V̂ (r, t) = −d · E(r, t)
(2.13)
10
Chapter 2 Overview of some basic principles
is the interaction Hamiltonian and d =
P
ei ri is the total electric dipole moment of the
i
system.
The starting point to solve Eq.(2.12) is the solution
ψn (r, t) = e−ıEn t/~ ψn (r),
(2.14)
of the Schrödinger equation in the absence of the external field, V̂ (r, t) = 0,
ı~
∂ψn (r, t)
= Ĥ0 ψn (r, t) = En ψn (r, t),
∂t
(2.15)
where En are the energy levels of the system with the eigenfunctions ψn (r) obeying the
orthonormality relation
Z
ψn∗ (r)ψk (r)dr = δnk .
(2.16)
According to well-known superposition principle, the solution of Eq.(2.12) with the external
field, ψ(r, t), can be expressed in the form of linear combination of the eigenfunctions ψn (r)
(Eq.(2.15)).
X
X
an (t)ψn (r)e−ıEn t/~ .
(2.17)
an (t)ψn (r, t) =
ψ(r, t) =
n
n
Here an (t) is the amplitude of probability to find the system in the n th quantum state.
Substituting (2.17) into (2.12) and using the orthonormality of the eigenfunctions (2.16),
one can get the amplitude equation18
ı~
∂an (t) X
ak (t)Vnk (r, t).
=
∂t
k
(2.18)
R
where Vnk (r, t) = ψn∗ (r)V̂ (r, t)ψk (r)eıωnk t dr, and ωnk = (En − Ek )/~ is the resonant transition frequency between levels k and n.
2.2.3
Density matrix equation
We intend to study the dynamics short-living core excited states where the role of relaxation
is very important. The proper theoretical tool to describe the evolution of quantum system
with dissipation is the density matrix formalism which is more general than the Schrödinger
P
equation. In general, the density matrix operator is defined as ρ = |ψihψ| =
ρkn ψk ψn∗ ,
kn
which describes a coherent mixture of pure unperturbed states ψn via the density matrix
ρkn = ak a∗n .
2.2 Strong X-ray field
11
The interaction between the electromagnetic field and a quantum system is accompanied
by different decay processes caused by spontaneous decay, collision and other dephasing
mechanisms. The density matrix equations after introducing the relaxation can be written
as
∂ρij
ı
= − [Ĥ, ρ]ij − γij ρij (i 6= j),
∂t
~
X
X
ı
∂ρii
= − [Ĥ, ρ]ii +
Γij ρjj −
Γji ρii .
(2.19)
∂t
~
E >E
E <E
j
i
j
i
where the diagonal matrix element ρii is the population of the i-th level, and the off-diagonal
matrix element ρij gives the coherence between the i-th and j-th quantum levels. Γij is the
decay rate of the population from the energy level j to i. The relaxation rate of the coherence
γij ,
X
Γi + Γj
γij =
+ γijdeph , Γi =
Γji .
(2.20)
2
E <E
j
i
γijdeph .
Usually the dephasing rate is caused by collisions or
includes the dephasing rate
interruptions, which is proportional to the concentration of the molecules. In gas phase, the
pressure or collisional broadening is smaller than the total decay rate Γi of core excited state.
It should be noted that in the case of XFEL pulse, the dephasing rate can be produced by
the random fluctuations of the amplitude and the phase of the field itself, which result in
a spectral broadening in a similar way as the collisional broadening (see Paper X). When
a quantum system interacts with a strong XFEL field, the direct ionization channel should
be taken into account. In this case, we need introduce the time-dependent photoionization
rates into the density matrix equations (see Chapter 8).
The Maxwell’s equations (2.10) and density matrix equations (2.19) are coupled with each
other through the macroscopic polarization P of the medium, P = N Tr(ρd). N is the
concentration of the molecules.
2.2.4
Rabi oscillations and McCall-Hahn Area Theorem
Rabi oscillations is an important physical process in quantum optics and nuclear magnetic
resonance.19 When a two-level quantum system is exposed to an oscillatory driving field,
the wave function can be written as
ψ(t) = a0 (t)ψ0 (t) + a1 (t)ψ1 (t)
(2.21)
where ψ0 (t) and ψ1 (t) represent the lower and upper eigenstates of the unperturbed atom,
a0 (t) and a1 (t) are the probability amplitudes of finding the atom in states ψ0 (t) and ψ1 (t)
mixed by the field.
12
Chapter 2 Overview of some basic principles
Substituting the wave function (2.21) into the Schrödinger equation, we can get the amplitude equations for a0 (t) and a1 (t) as follows
¢
¡
∂a0 (t)
G10
=ı
a1 (t) e−ı(ω+ω10 )t + eı(ω−ω10 )t ,
∂t
2
¡
¢
∂a1 (t)
G01
=ı
a0 (t) eı(ω+ω10 )t + e−ı(ω−ω10 )t .
(2.22)
∂t
2
Here, ω is the frequency of the incident field, and ω10 is the resonant transition frequency.
In resonant case, we can apply the rotating-wave approximation (RWA)20 and neglect the
fast oscillation terms e±ı(ω+ω10 )t on the right side of Eq.(2.22). It is easy to get the equations
for the population of the ground and excited states
G10 t
G10 t
, ρ1 (t) = |a1 (t)|2 = sin2
.
(2.23)
2
2
One can see that the populations of the ground and excited states experience Rabi oscillations caused by the cyclical absorption and stimulated emission processes. Here G10 (t) =
d10 · E(t)/~ is the Rabi frequency and E(t) is the envelop of the laser field E.
ρ0 (t) = |a0 (t)|2 = cos2
In 1967 McCall and Hahn21 invented the notion of the pulse area
Z
d10 · E(t)
,
ϑ(z) = G10 (t)dt, G10 (t) =
~
(2.24)
and obtained the Area Theorem which explains both the dynamics of the populations as well
as the pulse propagation through the resonant medium. The McCall-Hahn Area Theorem
says the ”pulse area” obeys the following equation
1
d
ϑ(z) = − ξ sin ϑ(z),
dz
2
(2.25)
where ξ = 4π 2 N ωd210 /~c is the absorption coefficient. The most striking consequences of
the Area Theorem are: (i) Pulses with initial area equal to nπ are predicted to maintain the
same area during propagation. The 2π pulse will remain unchanged in shape and energy
through the absorbing media, which is so-called ”Self-induced transparency”.22 (ii) Pulses
with other values of area must change during propagation until their area reaches one of the
special values. For example, the pulses with the area slightly different from the odd multiples of π are unstable. The pulse areas will evolve into the neighbor even multiples of π.
The Area Theorem describes successfully some interesting nonlinear optical phenomena for
resonant one-photon interaction in a two-level atomic medium such as Self-induced transparency, pulse splitting and pulse compression. It should be pointed out that the derivation
of McCall-Hahn Area Theorem is based on the traditional slowing varying amplitude approximation (SVEA)23 and rotating wave approximation (RWA), thus the application of
Area Theorem is invalid for few-cycle laser pulses.24–26
Chapter 3
Short-time approximation for
resonant Raman scattering
As it was shown in Chapter 2 the resonant Raman scattering (RRS) is a dynamical process
with the characteristic scattering duration12, 27–29 τ = τ (Γ, Ω) (2.9), which is determined
by the frequency detuning from the absorption resonance Ω and the lifetime broadening
of intermediate electronic state Γ. The concept of the scattering duration and the notion
of fast scattering 12, 27–29 was invented in resonant X-ray Raman scattering in 1996. In
2005 Jensen and coworkers30, 31 revived the notion of the scattering duration in the narrow
sense τ → 1/Γ and introduced short-time approximation ST(Γ) in the theory of RRS in
optical and UV regions. They assumed the dephasing time 1/Γ is much shorter than the
characteristic time of nuclear dynamics. However, analysis of available experimental data
indicates that this is not the case of optical and UV spectral regions, and even not in the
soft X-ray region (ω ≤ 3 keV). Moreover, Kane and Jensen32 analyzed the accuracy of the
ST(Γ) approximation and observed that it significantly overestimated the absolute value of
the cross section. Nevertheless, the spectral shape of the resonant Raman scattering can be
obtained with reasonable accuracy when the displacements of excited state potentials from
the ground state are small (small Huang-Rhys parameter). Quite often the deviation of the
nuclear potential in the excited state from the ground state potential surface is significant
and the role of the nuclear dynamics becomes important. This leads to the appearance
of overtones in the Raman spectrum and to a strong variation of the spectral profile with
respect to photon energy, which evidences the breakdown of the ST(Γ) approximation.
In this chapter we discuss the evolution of the RRS profile in the course of shortening of
scattering duration by detuning and show the practical application of this technique to
purify the RRS spectra by depleting higher overtones and soft vibrational modes (Paper I).
14
Chapter 3 Short-time approximation for resonant Raman scattering
It is useful to write down the scattering amplitude (2.7) in semi-classical approximation (see
Paper I),
Fνf ≈ hνf |Fνf (Q)|0i,
F νf =
D0 (Q)D(Q)
−ıτ D0 (Q)D(Q)
=
.
Ω − (²i (Q) − ²0 (Q)) + ıΓ
1 + ıτ (²i (Q) − ²0 (Q))
(3.1)
Nevertheless, this approximation is too rough, it allows to understand the formation of the
RRS spectrum in the course of shortening of the scattering duration. One should note that
the main Q-dependence arises from the resonant denominator. In the case of quite weak
anharmonic potentials, the difference ²i (Q) − ²0 (Q) is the linear function of the normal
coordinates Q, and hence the scattering amplitude Fνf (3.1) can be expanded in series over
power of Q,
p
∞
X
νf !
F (n)
Fνf ≈
h0|Qn |νf i ≈ −ıτ D0 (Q)D(Q)(−ıζτ ω0 )νf νf
(3.2)
n!
2
n=0
Here ζ = −∆Q/a, ∆Q is the displacement of the potential of the excited state relative
√
to the ground state, and a = 1/ µω0 is the size of the vibrational wave function. When
the scattering duration τ is much shorter than the characteristic time of nuclear motion,
|τ |ω0 ¿ 1, the intensity of the higher overtones are suppressed by the factor (τ ω0 )νf ¿ 1.
(a)
1.0
<0|υi>
2
(b)
Γ=0.1eV
Γ=0.01eV
0.8
0.2
D
P2
FC factor
0.3
0.6
0.1
0.4
0.0
-1
0
1
2
0.2
3
-0.20
-0.15
Ω (eV)
1.0
f
-0.05
0.00
νf = 2
(c)
Pν
-0.10
Ω (eV)
νf = 3
νf = 4
0.5
Γ=0.1eV
ω0
0.0
-1
0
D
1
2
3
Ω (eV)
Figure 3.1: The ratio of the peak intensities Pνf = |Fνf |2 /|F1 |2 versus the detuning Ω for
Γ = 0.1 eV and 0.01 eV. One-mode model.
15
To gain insight in the dynamics of resonant scattering, first let us analyze the fast scattering
for model systems starting with a one-mode molecule (ω = 0.2 eV). The relative peak
intensity Pνf (Ω, Γ) of the overtones versus Ω is shown in Fig. 3.1. One can see that the
shortening of the scattering happens only beyond the region of strong photoabsorption.
The scattering becomes fast when Ω ≤ −ω0 or Ω ≥ ∆, where the higher overtones will
be quenched faster. This means the short-time limit ST (Ω, Γ) can be easily approached
by changing the frequency detuning Ω. The main role of Γ is to increase the interference
between the vibrational levels of the intermediate electronic state. The decrease of Γ does
not affect significantly blue and red wings of Pνf (Fig. 3.1(b)). Now let us consider the
RRS spectrum of two-mode molecule with high-frequency (ω10 = 0.2 eV) and low-frequency
(ω20 = 0.06 eV). Fig. 3.2 shows the RRS cross section for different detunings and lifetime
broadenings. It is clear that in the course of shortening of scattering duration by increasing
the detuning |Ω|, the contribution of the soft mode is washed out faster from the RRS
spectra than the high-frequency mode. Therefore, the detuning gives a convenient tool to
purify the Raman spectrum from the ”contamination” by higher overtones and soft modes,
which is very important for the analysis of the resonant Raman spectra of complex systems.
In contrast, the “purification” of the spectrum by Γ occurs for unrealistically large (for UV
region) Γ ≥ 0.1 eV.
W
Ω=-0.1eV
Ω=-0.03eV
Intensity
Ω=-0.01eV
Ω=0.0eV
Γ=0.01eV
Γ=0.02eV
Γ=0.05eV
Γ=0.1eV
0.0
0.2
0.4
0.6
0.8
1.0
1.2
G
ω−ω’ (eV)
Figure 3.2: The RRS spectrum of two-mode model for narrow band excitation γ = 0. The
bars show the peak intensities σνf = |Fνf |2 . ω10 = 0.2 eV, ∆Q1 /a10 = 2.07, ω20 = 0.06 eV,
∆Q2 /a20 = 2.27.
16
Chapter 3 Short-time approximation for resonant Raman scattering
As it was shown already for model two-mode molecule (Fig. 3.2), the ST(Γ) approximation
based on large Γ is very dangerous. The same conclusion we get analyzing both experimental
and theoretical UV RRS spectra of a relatively large organic molecule, trans-1,3,5-hexatriene
(THT) (Fig. 3.3). In contrast to the detuning, the shortening of the scattering duration
solely by Γ does not allow to approach the region of the fast scattering, which is free from
both overtones and soft modes. The experimental homogeneous broadening of the THT
molecule Γ ≈ 0.0079 eV ¿ 0.1 eV33 is too small to make the ST(Γ) approximation valid.
Indeed, in the resonant region (λexp = 251 nm), the intensities of the overtones (2ν5 ) and
combination modes (ν5 + ν10 ) are comparable with the intensity of the fundamental mode
ν10 . However, the high overtones (2ν5 ) are quenched quickly and the contamination of soft
modes (ν13 , ν9 ) are gradually eliminated from the RRS spectra in the case of fast scattering
reached by the detuning (λexp = 280 nm). Thus, the shortening of the RRS duration by the
detuning provides a unique opportunity to interpret the experimental RRS spectra of large
molecules with numerous overlapping resonances.
exp
theo
(a)
n13
n10
n9
n5
n5+n10
2n5
Intensity
(b)
(c)
0
500
1000
1500
2000
2500
3000
3500
ω−ω’ (cm )
-1
Figure 3.3: The experimental33 and theoretical Raman spectra of the trans-1,3,5-hexatriene
(C6 H8 ) molecule for different excitation wavelengths. (a) λexp = 251 nm. (b) λexp = 257
nm. (c) λexp = 280 nm. The theoretical excitation wavelengths are red shifted by 6 nm to
match the experimental spectra. The theoretical spectra are based on linear coupling model
(LCM), and have been broadened by a Gaussian function with a width of γ = 15 cm−1 .
Chapter 4
Resonant X-ray Raman scattering
from liquid acetone
In this chapter, we developed the RXS theory for liquid phase and apply it to the resonant X-ray Raman scattering from the liquid acetone excited at oxygen K edge. Acetone
((CH3 )2 CO) is an important trace gas in the background troposphere. One should mention
that the studied system with rather weak intermolecular dipole-dipole interaction differs
strongly from “strong” liquids like water investigated earlier.34–39 The structure of the acetone molecule together with the scheme of transitions are shown in Fig. 4.1. In the ground
state 1 A1 , acetone has C2v symmetry with a planar CCCO fragment.
π∗
LUMO
n
HOMO
ω1
ω
REXS
RIXS
π∗
n
O1s
O1s
Figure 4.1: Scheme of REXS and RIXS transitions in the acetone molecule.
The experiments are performed at the ADRESS beamline40 at the SLS with a resolution
of E/∆E ∼ 10000. The incoming photon energy is tuned near the first core excited state
∗
ψi = |1s−1
O π i. Fig. 4.2 shows the X-ray Raman spectra for the different excitation energies.
One can see that the total RXS profile displays three spectral bands: the REXS band with
18
Chapter 4 Resonant X-ray Raman scattering from liquid acetone
nicely resolved vibrational structures and two unresolved inelastic bands. One RIXS band
is located at the energy loss ∆ω ≈ −5 ∼ −4 eV. The origin of this band is the excitation of
the electron from the lone pair HOMO (5b2 = n) to LUMO (b1 = π ∗ ), which is related to
|n−1 π ∗ i final state. Another RIXS band (∆ω ≈ −9 ∼ −8 eV) arises from the excitation of
the electrons from the HOMO-1, HOMO-3, HOMO-5 to LUMO, which are related to the
−1 ∗
−1 ∗
∗
final states |7a−1
1 π i, |8a1 π i and |2b1 π i, respectively.
RXS (arb.units)
ω=530.9eV
ω=530.5eV
ω=530.2eV
ω=529.9eV
-10
-8
-6
-4
-2
0
∆ω (eV)
Figure 4.2: The RXS spectra of acetone.
4.1
Scheme of transitions and vibrational modes
Here we only focus on the REXS band and the first RIXS band (∆ω ≈ −5 ∼ −4 eV) related
to |n−1 π ∗ i final state (see Fig. 4.1 and Fig. 4.2).
½
X
ω1 + |n−1 π ∗ ; ν f i, RIXS
−1 ∗
ω + |ψ0 i →
|1s π ; ν i i →
(4.1)
ω
+
|0;
ν
i,
REXS
1
0
νi
Acetone molecule has 24 vibrational modes. Ab-initio calculations of the potential energy
surfaces for each vibrational mode were performed using STOBE code.41 The analysis of
the Frank-Condon (FC) factors shows that mainly 8 modes (see Fig. 4.3) are active in XAS
4.1 Scheme of transitions and vibrational modes
19
Q 1 : CH3 d-torsion
Q 2 : CH3 s-torsion
Q 3 : CCC deformation
Q 6 : CC stretching
Q 9 : CH3 rocking
Q 12 : CH3 s-deformation
Q 4 : CO out-of-plane wagging
Q 18 : CO stretching
Figure 4.3: 8 active modes in XAS and RXS: ν1 , ν2 , ν3 , ν4 , ν6 , ν9 , ν12 and ν18 .
and RXS. Among these modes, the most active modes are CO stretching mode, CO out-ofplane wagging mode and CH3 torsion modes. The corresponding potentials for these three
-4710
-4728.22
(b)
(a)
ground
-4715
core-excited
-4728.23
-4720
final
-4728.24
Potential Energy (eV)
-4725
0.10
-5240
-5254.50
0.08
-5245
(c)
0.06
-5254.55
0.04
-5250
0.02
-5255
-5254.60
0.00
0.03
-5245
-5258.2
0.02
-5250
-5260
0.01
-5258.4
-5255
0.00
-0.5
0.0
0.5
1.0
Q (au)
1.5
2.0
-1.0
-0.5
0.0
Q (au)
0.5
1.0
-15
-10
-5
0
5
10
15
Q (au)
Figure 4.4: The potential energy surfaces of the most active three modes. (a) CO stretching
mode. (b) CO out-of-plane wagging mode. (c) CH3 torsion mode. The ground and final
state potentials in (b) and (c) are extracted from the UV experimental data.42
20
Chapter 4 Resonant X-ray Raman scattering from liquid acetone
modes are shown in Fig. 4.4. It should be noted that the shape of the spectral profile is
very sensitive to the quality of the potentials. A comparison with the UV experimental
data shows that the quality of the potentials of the CO out-of-plane wagging mode and of
the CH3 torsion modes are not enough. Thus, in our simulations we used the potentials of
these modes extracted from the UV experimental data42 (Fig. 4.4), in which the coupling
between two methyl groups is neglected. To reduce computational costs, the simulations
are performed by taking into account of 8 active modes and the number of the vibrational
states are chosen different for these modes according to the analysis of the FC factors.
4.2
Resonant X-ray Raman scattering: f = n−1π ∗
Let us start from the analysis of the scattering to the excited final state f = n−1 π ∗ which,
in contrast to the REXS band, does not display vibrational structure.
4.2.1
Environmental broadening in liquid
At first impression the broadening of the RIXS band is due to the liquid phase. Inhomogeneous broadening caused by solute-solvent interaction is the major broadening mechanism in
condensed phase spectroscopy. For a dipolar liquid the dominant dipole-dipole interaction
P µS ·∆µi0 −3(µS ·R̂)(∆µi0 ·R̂)
shifts the transition energy ωi0 by ∆ωi0 =
, i = i, f . It is natural
R3
to expect that the shifts of transition energies ωf 0 → ωf 0 + ∆ωf 0 and ωi0 → ωi0 + ∆ωi0
result in a broadening. Let us assume that the dipole moments are parallel to each other
µ0 k µi k µf . For example, this is the case of studied acetone molecule. This was as a
consequence that
∆µi0
∆ωi0 (R) = η∆ωf 0 (R), η =
= const.
(4.2)
∆µf 0
In order to predict how the intermolecular dipole-dipole interaction affects the scattering,
the RIXS cross section should be averaged over all possible instantaneous structures of the
liquid around the solute acetone
σ̄(ω, ∆ω) =
YZ
α
Z∞
dRα g(Rα )σ(ω, ∆ω) = Ŝ
Z∞
dt
0
Z∞
dt1
0
dτ
−∞
£
¤
×Φ̃(τ + η(t1 − t)) exp ı (Ω(ν i ) + ıΓ) t − ı (Ω(ν 0i ) − ıΓ) t1 + ıΩν f τ ,
(4.3)
assuming that the probability of each configurations is the product of two-particle distribuQ
PPP
1
hν f |ν i ihν i |ν 0 ihν 0 |ν 0i ihν 0i |ν f i.
tion functions, P (R1 · · · RN ) = g(Rα ). Here Ŝ = 2π
0
νf νi νi
4.2 Resonant X-ray Raman scattering: f = n−1 π ∗
21
The interaction between the solute and solvent molecules is included in the autocorrelation
function
Z
¤
£
−ρJ(t)
Φ̃(t) = e
, J(t) = dRg(R) 1 − eıt∆ωf 0 (R) ,
(4.4)
which modifies the scattering amplitude43 via the parameter η (4.2) and leads to a additional
broadening γ → γ + γS :
X
(4.5)
σ̄(ω, ∆ω) ∝
|Fν f |2 Φ(Ων f , γ),
νf
X hν f |ν i ihν i |ν 0 i
Fν f =
(4.6)
,
ν i Ω(ν i ) + ηΩν f + iΓ
r
Z∞
1 ln 2 −²2 ln 2/γ 2
1
−ı²t −ρJ(t)
dte e
≈
e
Φ(², γ) =
2π
γ
π
−∞
Here Ω(ν i ) = ω − ωi0 − (²νi − ²ν0 ), Ων f = ∆ω + ωf 0 + (²νf − ²ν0 ). The environmental
broadening
p
4√
γS ≈
π ln 2|∆µf 0 ||µ0 | ρ/a3
(4.7)
3
is sensitive to the solvent. The theoretical dipole moments µ0 = 3.038D and µf = 1.958D
give the environmental broadening γS (HW HM ) ≈ 0.04 eV. Instantaneous structures can
be explicitly sampled in MD calculations, which results in almost the same broadening in
pure liquid acetone γsM D (HW HM ) ≈ 0.04 eV (Fig. 4.5 (b)), which is not enough to make
the vibrational structures in RIXS band vanish. However, it is expected that the liquid
Figure 4.5: The distribution of transition energy ωf 0 for pure liquid acetone (a) and aqueous
acetone (b) from MD simulations. The blue arrow corresponds to the vertical transition
energy for gas acetone.
22
Chapter 4 Resonant X-ray Raman scattering from liquid acetone
broadening for aqueous acetone (γS ≈ 0.11 eV, Fig. 4.5 (a)) will play a more important
role on the spectral broadening due to the effect of the hydrogen bond between the ”solute”
acetone molecule and ”solvent” water molecules.
4.2.2
Thermal population of torsion modes
Let us remind the experiments are performed at the room temperature. Acetone has two
torsion modes (the internal rotation of methyl groups) with frequencies ω1 ≈ ω2 ≈ 0.01 eV
which are comparable with the thermal energy kB T ≈ 0.026 eV. Due to this a few torsion
vibrational levels of the ground state are thermal populated according to the Boltzmann distribution (∝ exp(−²µ0 /kB T )), which can contribute to the broadening of the RXS spectra.
We should take into account this effect in the sum over initial torsion levels
X −² /k T
e µ0 B σ(ω, ∆ω).
(4.8)
σ(ω, ∆ω) →
µ0
The introduction of thermally populated torsion levels makes the calculation of the RXS
cross section very time consuming. However, the CH3 group has no time to rotate in the
core excited state because ω1 ≈ ω2 ¿ Γ, and the ground state torsion wave packet is
almost immediately transferred to the final n−1 π ∗ state. Therefore, we can apply the shorttime approximation for these soft modes by removing them from intermediate states. The
cross section has the same expression as Eq.(2.3) except for the spectral function. Now the
effective spectral function includes the broadening by the soft torsion modes
µmax
eff
Φ (Ων f , γ) =
X
µ0 =0
−²
/k T
e µ0 B hµf |µ0 i2 Φ(Ων f + ωµf ,µ0 , γ).
(4.9)
It should be noted that this thermal broadening mechanism by soft torsion modes is absent
for REXS band because the final electronic state coincides with the ground state in the REXS
channel. The transitions only occur between vibrational states with the same vibrational
quantum number, which prevents a broadening of the REXS profile by soft modes.
4.2.3
Broadening of RIXS band by soft vibrational modes
Broadening caused by the thermal population of the torsion vibrational levels for RIXS is
very sensitive to temperature (Fig. 4.6 (b),(c)). One can see that this broadening is reduced
by more than two times when lowering to T = 0K. In this case the vibrational structure
is nicely resolved contrary to room temperature T = 300K. The CO out-of-plane wagging
4.2 Resonant X-ray Raman scattering: f = n−1 π ∗
23
mode plays also an important role in the broadening of RIXS. This mode has a doublewell potential in the final state (Fig. 4.4(b)), which gives two strong peaks in the direct
transition from the ground state to the final state (Fig. 4.6 (a)). Our simulations show that
the main reason for the broadening of the RIXS profile is the interplay of the CO out-ofplane wagging mode and the torsion modes. The overall broadening due to these modes
γ(HW HM ) ≈ 0.075 eV (see Fig. 4.6) is enough to form a quasi-continuum distribution of
vibrational states.
(a) CO out-of-plane wagging
γ=0.001eV
γ=0.025eV
(b) CH3 torsion
HWHM:0.075eV
Intensity
T=300K
(c) CH3 torsion
T=0K
3.8
3.9
4.0
4.1
4.2
ωf0 (eV)
Figure 4.6: The spectra of the direct transition 0 → f (which is in fact forbidden) for the
CO out-of-plane wagging mode (a), torsion mode at T = 300K (b) and at T = 0K (c). The
blue line in panel (b) displays a spectrum taking into account both modes for γ = 0.025 eV.
Fig. 4.7 shows the experimental and theoretical RIXS spectral profiles for different excitation
energies. Here we take into account all active eight modes. One can see that our theoretical
simulations reproduce the experimental RIXS profile very well. The shape of the spectral
profile is very sensitive to the quality of the potential of CO stretching mode. As mentioned
already, the total broadening of the profile is caused by following three different broadening
mechanisms: (1) the width of incident light and instrumental broadening (∼0.025 eV), (2)
environmental broadening (∼0.04 eV) and (3) multimode excitation including CO out-of-
24
Chapter 4 Resonant X-ray Raman scattering from liquid acetone
plane wagging mode and torsion modes (∼0.075 eV). It is clearly seen that narrowing of the
RIXS band in the course of shortening of scattering duration.
exp
theo
T=300K
Intensity
ω=530.9eV
ω=530.5eV
ω=530.2eV
-5.5
-5.0
-4.5
-4.0
-3.5
-3.0
∆ω (eV)
Figure 4.7: The experimental and theoretical RIXS profile. γ = 0.025 eV. θ = ∠(k1 , e) =
90o . The simulations are performed taking into account eight active modes.
4.3
Resonant elastic X-ray scattering: f = 0
The REXS cross section (Eqs.(2.3) and (2.4)) is the sum of two contributions related to the
Thomson and resonant scattering channels
σ(ω, ∆ω) = σT (ω, ∆ω) + σR (ω, ∆ω),
¶
h
iµ
2
2 ω1 1
2
2
2
σT (ω, ∆ω) = r0
1 − P2 (k̂1 · e) Z + |di0 | Zωi0 <e(Fν 0 ) Φ(Ων 0 , γ),
ω 3
3
·
¸
X¯
¯
2
2 ω1 |di0 |
4
¯Fν ¯2 Φ(Ων , γ).
σR (ω, ∆ω) = r0
1 − P2 (k̂1 · e) ωi0
f
f
ω 9
5
ν
νf = ν0
(4.10)
4
f
Here σT (ω, ∆ω) term includes the interference between the Thomson and resonant scattering
channels. The experiment detects the photons summed over final polarization vectors. Due
4.3 Resonant elastic X-ray scattering: f = 0
25
to this in Eq.(4.10) the polarization vectors of the final photons are averaged using
e1i e1j =
i
1h
δij − k̂1i k̂1j ,
2
(4.11)
which gives
i 1
1h
1 − (k̂1 · e)2 = sin2 θ.
(4.12)
2
2
Here θ = ∠(k1 , e) is the angle between the momentum of the final photon and the polarization of the incident light.
(e1 · e)2 =
The REXS band is lying in the region of strong absorption. Hence, the self-absorption
should be taken into account using equation σ(ω, ∆ω)/(1 + µ(ω1 )/µ(ω)). µ(ω) = (3 × 12 ×
µC (ω) + 16 × µO (ω))/58 is the mass absorption coefficient of the acetone molecule. Here we
took into account also the off-resonant absorption by carbon atoms. It should be mentioned
that the environmental broadening in liquid phase is absent for REXS channel because the
REXS channel ends up in the ground electronic state. The broadening of REXS spectra
REXS (arb.units)
(a)
exp
theo
(b)
ω=530.9eV
ω=530.9eV
ω=530.5eV
ω=530.5eV
ω=530.2eV
ω=530.2eV
ω=529.9eV
ω=529.9eV
-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5
Dw (eV)
Dw (eV)
Figure 4.8: Comparison of the experimental and theoretical spectra of resonant elastic Xray scattering. (a) Without Thomson scattering and self-absorption. (b) The effects of
Thomson scattering and self-absorption are included. θ = 90o .
26
Chapter 4 Resonant X-ray Raman scattering from liquid acetone
is caused by the narrow spectral function of the incident light as well as the instrumental
broadening (γ ≈ 0.025 eV). The REXS spectra for different excitation energies are shown in
Fig. 4.8. Our multimode calculations are in close agreement with the experiment only when
the Thomson scattering and self-absorption effect are taken into account (Fig. 4.8(b)). The
Thomson scattering enhances the intensity of unshifted ”0-0” line, while the self-absorption
effect is rather weak. One can see that the nuclear dynamics affects differently the Thomson
and resonant scattering channels. The resonant scattering channel results in population of
many vibrational levels of the ground state near the resonant region (ω = 530.9 eV or 530.5
eV ). When the incoming photon energy is tuned far away from the absorption resonance
(ω = 529.9 eV), the REXS band almost collapses to the single ”0-0” line. It should be noted
that the CO stretching mode gives the dominant contribution to the REXS profile except
the quasi-continuum. The quasi-continuum ”bump” located at −2.5 eV ≤ ∆ω ≤ −2.0 eV
(ω = 530.9 eV) is formed due to the multimode excitation of other modes.
One should notice that contrary to any other spectroscopies, vibrationally resolved REXS
gives a unique opportunity to obtain the element and chemical specific information on the
local vibrational structure.
Chapter 5
RXS spectra of the O2 molecule near
the σ ∗ = 3σu resonance
In this chapter, the Resonant soft X-ray scattering spectra of oxygen molecule near σ ∗
resonance in gas phase are studied. The experiments are carried out using the SAXES
spectrometer44 at the ADRESS beam line40 of the Swiss Light Source, Paul Scherrer Institut
Villigen with a energy resolution around 50 meV. The simulations are performed using the
time-dependent wave packet technique.45
5.1
Potential surfaces and scheme of transitions
The ground state of O2 molecule has the following electronic configuration, |0i = |X 3 Σ−
gi =
2
2
2
2
2
4
2
1σg 1σu 2σg 2σu 3σg 1πu 1πg . We study here the RXS of O2 molecules near the region of the
1
2
4
1
1s−1 3σu resonance. The core excited state |ii = |3 Σ−
u i = 1σg · · · 1πu 1πg 3σu with a core-hole
in the gerade 1σg molecular orbital leads to two decay channels 3σu , 1πu → 1σg . However,
the experimental data (see Paper III and Paper IV) show much more rich spectra as a
function of photon energy. Some unexpected RIXS bands related to the decays from the
gerade molecular orbitals are also observed in the experiment. Our study (see below) shows
the symmetry forbidden RIXS channels are also opened due to the core-hole jump in the
mixing of the Rydberg and valence core excited states. The potential surfaces are collected
in Fig. 5.1. Both the valence and Rydberg core excited states have two triplet states with
distinct spins on the parent ion, S = 3/2 (quartet (Q)) and S = 1/2 (doublet (D)):46–49
28
Chapter 5 RXS spectra of the O2 molecule near the σ ∗ = 3σu resonance
E (eV)
3
544
6 u 1V g1 3V u1 3s(D)
540
3
3s(Q)
6 u 1V u1 3sg1 536
3V u (Q)
3
P 1s,
1 22p +3 P g 3V u (D)
Z
40
30
atomic
c peaks
molecular band
ds
532
3
20
6 g 3V g1 3sg1 3
10
3 g 1S u1 3V u1 X 3 6 g
0
2
3
4
5
3
P+3 P g 6
7
R (au)
Figure 5.1: Scheme of transitions.
1
Q : |11i = √ [−3αx αy α1σ βu + αx αy β1σ αu + αx βy α1σ αu + βx αy α1σ αu ] ,
2 3
1
D : |11i = √ [2αx αy β1σ αu − αx βy α1σ αu − βx αy α1σ αu , ]
6
(5.1)
here the notations x, y, 1σ, u in the spin wave functions correspond to 1πgx , 1πgy , 1σg , 3σu
MOs for the valence core excited state and 1πgx , 1πgy , 1σu , 3sg in the case of the Rydberg
state. In general the triplet wave function is the mixture of the Q- and D-states. However,
simulations show47, 50 that this mixing is very weak.
5.2
REXS channel: Thomson scattering, spatial quantum beats
The REXS cross section (Eq.(4.10)) can be rewritten as
¸
¶
X · sin2 θ µ
2Z
1 + sin2 θ
2
2
2 ω1
Z +
ReF0 δν,0 +
|FR,ν | Φ(Ων , γ).
σ(ω, ∆ω) = r0
ω ν
2
3
15
(5.2)
5.2 REXS channel: Thomson scattering, spatial quantum beats
E (eV)
3
29
6 u
544
540
\Q
536
\D
RD RQ
'-
Q
532
\ Q \ D
528
2
HD
D
HQ
12
Z
3
6
-'Z (eV)
10
6 g
Q+D
8
Q
6
4
4
2
2
vibrational levels
0
0
2
3
4
R ((au))
5
6
intensity
Figure 5.2: Spatial quantum beats caused by interfering wave packets at the dissociated
core excited states.
where ν is used for the vibrational levels of the final electronic state instead of ν f in
Eq.(4.10), and the adiabatic transition energies as well as the transition dipole moments
are included in the scattering amplitudes. The Thomson scattering which affects only unshifted line ν = 0 vanishes when θ = ∠(e, k1 ) = 0o .
The scattering of the oxygen molecule is going through two indistinguishable intermediate
dissociative states (doublet and quartet). Therefore, the resonant scattering amplitude FR,ν
is written as
Q
D
+ FR,ν
.
(5.3)
FR,ν = FR,ν
The scattering amplitude of the j th channel is the integral over the energy of the dissociative
state Ej relative to the energy of the vertical transition ωj0 :
Z
hφj² |0iφj² (R)
j
2
FR,ν = ωω1 dj0 hν|ψj i, ψj (R) = d²
.
(5.4)
Ωj − ² + ıΓ
The main contribution of the integral comes from the pole ² = Ωj + ıΓ.
As one can see from Fig. 5.2 both wave packets ψQ (R) and ψD (R) propagate along the same
resonant energy surface with the total energy ω, but with different kinetic energies εj (R)
Chapter 5 RXS spectra of the O2 molecule near the σ ∗ = 3σu resonance
30
and hence different phases ϑj (R). The phase difference between the Q and D wave packets
∆ϑ = ϑQ − ϑD ≈ (kQ − kD )R + ϕ = −τ ∆ + ϕ
(5.5)
results in a quantum beating of two wave packets which leads to modulation of the spectra
in the energy domain (see Fig. 5.3), where τ is the propagation time of the wave packet to
point R and ϕ is the constant phase shift (see Paper III). It should be noted that the number
of oscillations for continuum-continuum transitions is very sensitive to the QD splitting ∆.
The role of Thomson scattering channel (Eq.(4.10)) should deserve a special comment. In
this case Thomson scattering is the dominant scattering channel for the elastic (ν = 0)
peak. Both experiment and theory (Fig. 5.3) show the cross section of Thomson scattering
is about 7 times larger than the resonant scattering cross section for θ = 90o . The main
reason for this is the amplitude of the resonant scattering through the continuum state is
strongly suppressed (more than 10 times) in comparison with bound intermediate state.
(a)
O 2 a b s o rp tio n
(b)
(c)
Z=539eV
T=90
T=0
528
532
536
540
544
RXS (arb
b.units)
0
exp
Z 539 eV
Z ((e V )
0
theo
Z 538.75 eV
Z 538.5eV
Z 538eV
-12
-10
-8
-6
'Z (eV)
-4
-2
0
-12
-10
-8
-6
-4
'Z (eV)
-2
0
-0.3
03
00
0.0
-0.3
03
00
0.0
'Z (eV)
Figure 5.3: Experimental (a) and theoretical (b) RIXS spectra for θ = 90o excited in the
537-539 eV energy range. The theoretical “atomic” peak includes also contribution from the
dissociative 13 Πg final state which converges to the same dissociative limit as the ground
state. The gray bar shows the intensity of the elastic peak intensity for θ = 90o without
Thomson scattering.
5.3 RIXS channel
5.3
31
RIXS channel
Now let us turn to the inelastic scattering channels which end up in the final electronic
states different from the initial ground electronic state. The experimental and theoretical
RIXS spectra are shown in Fig. 5.4. The origin of the bands (Πmol , Πat ) and Σ are the decay
(a)
Σ
Πat
Πmol
(b)
θ=90
θ=0
o
o
Q
D
Intensity
Intensity
539ev
538.75ev
538.5ev
-16
-15
-14
-13
∆ω (eV)
-12
-11
-15
-14
-13
-12
-11
-10
∆ω (eV)
Figure 5.4: The experimental (a) and theoretical (b) RIXS spectra of molecular oxygen
excited in the 537-539 eV energy range measured in two geometries, θ = ∠(e, k1 ) = 0o , 90o .
The gray shaded area shows the theoretical spectrum when we assume both decay transitions
ψi (Q) → 13 Πg and ψi (D) → 13 Πg are allowed (θ = 0o ).
transitions 1πu → 1σg and 3σg → 1σu , corresponding to the final states 13 Πg (1πu−1 3σu1 ) and
3 −
Σg (3σg−1 3s1g ), respectively. The angular anisotropy is different for different final states.
The orientational averaging of the resonant term for studied diatomic molecule is straightforward12, 51
½
r02 ω1 X
1 + sin2 θ, f = Σ,
2
σν (ω, ∆ω) =
|FR,ν | Φ
(5.6)
4 − sin2 θ, f = Π,
15ω ν
where the resonant scattering amplitude reads
Z
df i di0 hν|²ih²|0g i
FR,ν = ω1 ω d²
(5.7)
Ω² + ıΓ
Here Φ is the spectral function, di0 and df i are the absolute values of transition dipole
moment of core excitation and of the decay, Ω² = ω − (Ei,² − Eg,0 ), Ef,ν is the total energy
of the f th electronic state in the ν th vibrational level.
32
5.3.1
Chapter 5 RXS spectra of the O2 molecule near the σ ∗ = 3σu resonance
Hidden selection rules (f = 13 Πg (1πu−1 3σu1 ))
The formation of the peaks Πmol and Πat (−15 eV ≤ ∆ω ≤ −11 eV) corresponds to the
3
−
−1
1
3
−1
1
scattering channel X 3 Σ−
g → Σu (1σg 3σu ) → Πg (1πu 3σu ) involving the dissociative core
excited and final states. Such kind of scattering channel involving the dissociative states
(Fig. 5.1) results in the broad molecular and narrow atomic peaks.12, 52 The first impression
says the splitting (about 1.4 eV) between the Q and D core excited states will result in
two molecular peaks. Indeed, this doubling is seen in our wave packet simulations if we
assume that both decay transitions Q → f and D → f are allowed (see Fig.5.4(b)). The
reason for this doubling is the potential of the final 13 Πg state which is very similar to the
potentials of Q and D core excited states (Fig. 5.1), and due to this the emission energy of the
molecular peak is close to the spacing between the potentials of core excited and final states
at equilibrium.53 However, the measurements indicate that nevertheless of core excitation
in both Q and D states, the decay transition from one of these state should be forbidden
(Fig.5.4). To give insight into the physics, we analyzed the final state wave function and
recognized that the main contribution to this function (13 Πg = 0.96 × 13 Πg (Q)) comes from
the Q-state of the parent ion. This results in additional (hidden) selection rules: the parent
ion does not change the spin in the course of the decay transition when the hole is promoted
from the core level to the valence shell (Fig.5.5). The decay transition ψi (D) → 13 Πg is
forbidden, which is nicely confirmed by the measurements (Fig.5.4).
T( D)
T(Q)
y
x
Q
x
y
D
Figure 5.5: Q (quartet, S = 3/2) and D (doublet, S = 1/2) correspond to the spin of the
parent ion.
5.3 RIXS channel
5.3.2
33
Opening of the symmetry forbidden RIXS channel due to
−1 1
core-hole jump (f =3 Σ−
g (3σg 3sg ))
Let us pay attention to the sharp double-peak feature (∆ω ≈ −15 eV) when the photon
frequency is tuned near 539 eV (Fig.5.4). The energy position and the polarization dependence of these narrow peaks say that their origin is the decay transition 3σg → 1σu to the
−1 1
bound final state 3 Σ−
g = |3σg 3sg i. However, the first impression is that this scattering
channel is symmetry forbidden. Indeed, the direct population of the Rydberg core excited
state 1σg,u → 3sg is rather weak (see Fig.5(a) in ref 48 ), while the core excitation 1σg → 3σu
creates only the gerade core hole. One should notice also another problem: Nevertheless
of dissociative core excited state the discussed sharp feature arises only in the vicinity of
ω = 539 eV.
Selection rules
The main physical mechanism of discussed effect is as follows. According to the previous
study,49 the excitation energy ω = 539 eV is close to the crossing point between the valence
|1σg−1 3σu1 i and Rydberg |1σu−1 3s1g i core excited Q-states. Due to the Coulomb interactions,
these two core excited states are coupled with each other near this crossing point when the
light is tuned in the vicinity of the vibrational level of the bound Rydberg state. Thus,
the light populates the Rydberg state |1σu−1 3s1g i directly or via strong core excitation to
the dissociative state |1σg−1 3σu1 i followed by the core hole hopping (Fig.5.6). The symmetry
1
Figure 5.6: The change of the core hole parity caused by core hole hopping in core excited
states.
Chapter 5 RXS spectra of the O2 molecule near the σ ∗ = 3σu resonance
34
forbidden dissociative RIXS channel |0i → |1σg−1 3σu1 i → |3σg−1 3s1g i is opened because the
hole and the electron swap parity.
Vibrational scattering anisotropy due to the Thomson scattering in RIXS channel
Q=0
(a)
a
(b)
b
exp
0
T=90
3
6 u (1V g1 3V u1 )
0
n=1
n
1
Q=1
-15.0
15 0
-14.9
14 9
-14.8
14 8
-14.7
14 7
theo
En
nergy p
potentia
als
RX
XS (arrb.units
s)
T=0
3
n=0
6u (1V u1 3sg1 )
coupling
Resonant
3
6g (3V g1 3sg1 )
Q=11
Q=0
Thomson
0g
-14.3
-14.2
'Z (eV)
-14.1
X3 6g
2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
R (au)
Figure 5.7: Inelastic Thomson scattering. (a) Experimental and theoretical profiles. (b) The
final vibrational state is populated through two interfering paths, resonant and Thomson
channels which have different scattering anisotropy.
Let us now pay attention to another unusual effect seen in the experimental spectra, Fig. 5.7
(a). One can see clearly that two vibrational peaks have different vibrational scattering
anisotropy. The only reason for this is the Thomson scattering which has qualitatively
different angular anisotropy and the FC amplitude than the resonant one. It was recognized
recently the importance of the Thomson scattering for the REXS54, 55 (see also Sec.4.3).
It is a common belief that the Thomson contribution FT is negligibly small when the final
electronic state differs from the ground one. However, this is not the case for studied system
because the resonant scattering channel is suppressed due to the dissociative intermediate
state. Thus the final state can be reached through two interfering scattering channels
5.3 RIXS channel
35
(Fig. 5.7 (b)) with the total amplitude
F = FT + (e1 · n)(e · n)FR,ν ,
(5.8)
where only the resonant contribution FR,ν is affected by the nuclear dynamics in the core
excited state.
The Thomson scattering amplitude for RIXS band (with the final state |3σg−1 3s1g i) can be
written as
£
¤
(5.9)
FT = (e1 · e)h3σg |eıq·r |3sg ihν|0i ≈ −(e1 · e) FTis + (q̂ · n)2 FTa hν|0i,
where the isotropic and anisotropic parts of the Thomson amplitude read
(3σg )
FTa = C2p
(3sg ) 2
C3s
q Rd2pz ,3s hν|0i,
(3σg )
FTis = C2s
(3sg ) 2
C3s
q h3s|r2 |2sihν|0i.
(5.10)
In our studied case, the admixture of the Rydberg state to the dissociative state modifies
the resonant amplitude (Eq.(5.7)),
Z
df i di0 hν|niKn,² h²|0g i
βhn|²i
FR,ν → ω1 ω d²
, Kn,² =
,
(5.11)
Ω² + ıΓ
ω − (Er,n − Eg,0 ) + ıΓ
where Kn,² is the propagator related to the coupling jump from the dissociative state
|1σg−1 3σu1 i to the bound Rydberg state |1σu−1 3s1g i. One can see that (Eq.(5.11)) this interaction takes sharp maximum at the resonance ω = Er,n − Eg,0 with the Rydberg state.
The predicted sharp dependence of the Σ band on the excitation energy is nicely confirmed
by experiment. Simulations show that ω = 539 eV corresponds to the lowest vibrational
level of the Rydberg well n = 0.
The RIXS cross section averaged over molecular orientations
σν = (r02 ω1 /15ω)|FR,ν |2 (1 + %ν sin2 θ)Φ,
(5.12)
!
Ã
½
¾2
1 a
1
5 a
0
00
is 2
is
−
)2 , (5.13)
)
+
F
) + (FR,ν
−
F
%ν =
(F
(F + 5FT,ν
R,ν
T,ν
T,ν
|FR,ν |2 4
2 T,ν
strongly resembles Eq.(5.6) for the Σ final state, except the scattering anisotropy coefficient
%ν . In our studied case, the resonant scattering contribution is strongly suppressed due
to the scattering through the dissociative states. Thus, the competitive inelastic Thomson
scattering channel can not be neglected. When the resonant term approaches the Thomson
a
is
0
amplitude: FR,ν
→ 12 (FT,ν
+ 5FT,ν
), the anisotropy coefficient can be suppressed. Here
0
00
FR,ν = FR,ν + ıFR,ν . Our simulations show %0 = 0.8 and %1 = 0.3 for the final vibrational
states 0 and 1, respectively, which gives a good agreement with the measurements (Fig. 5.7
36
Chapter 5 RXS spectra of the O2 molecule near the σ ∗ = 3σu resonance
(a)). In comparison with the main peak ν = 0, the θ−dependence of the scattering to the
final vibrational state ν = 1 is strongly suppressed. The main reason for this is the similarity
of the potentials of core excited and final states, which suppresses the resonant term FR,1
because of small FC amplitude hν|ni = h1|0i.
Chapter 6
Resonant Auger scattering from the
ethene molecule
The availability of the electron spectrometers with high-resolution and the soft X-ray sources
with high brilliance makes it possible to measure Resonant Auger scattering (RAS) spectra
of molecules with vibrational resolution, which provides a unique opportunity to study fast
multimode nuclear dynamics. The RAS process is very similar to the resonant Raman
scattering studied in the previous chapters except the final particle which is now the Auger
electron (see Fig.2.1). In this chapter, we study the RAS spectra of ethene molecule.
6.1
Scheme of transitions and vibrational modes
The ground state of ethene (C2 H4 ) molecule has the following electronic configuration,
|0i = |X 1 Ag i = (1ag )2 (1b1u )2 (2ag )2 (2b1u )2 (1b2u )2 (3ag )2 (1b3g )2 (1b3u )2 . C2 H4 molecule has
been intensively studied through different X-ray spectroscopies.56–60 The RAS spectra of
ethene molecule following C1s excitation to an unoccupied valence orbital as well as to
different Rydberg states were measured by Sorensen et al.58 The nuclear motion of ethene
molecule was also investigated using resonant X-ray Raman scattering.61 We investigate here
−1
the RAS of C2 H4 molecules (Fig. 6.1) in the region of the C1s
1b2g (π ∗ ). The participator
Auger decay of the core excited state leads to the final state X 2 B3u = |1b−1
3u i with a single
hole in the 1b3u molecular orbital,
−
−1
ω + |0i → |1b−1
1u 1b2g i → |1b3u i + e .
(6.1)
38
Chapter 6 Resonant Auger scattering from the ethene molecule
RAS
E
i
z
1b2g
y
1b3u X
υi
e-
w
1b3g A
f
3ag B
υf
1b2u C
0
1b1u
1ag
υ0
core orbitals
Figure 6.1: Geometry and scheme of the participator resonant Auger scattering process for
the ethene molecule.
Table 6.1: Vibrational modes of ethene and related FC amplitudes. Numbers in brackets
show the frequencies and FC amplitudes of the 0-2 transitions to the core excited state. The
most active seven modes in the X-ray absorption are marked by stars. The mode notations
used in Refs.59, 60 are shown in square brackets. (Taken from Paper V. Reprinted with
permission of American Chemical Society.)
mode sym ground(eV) core-exc(eV)
FC amplitude 0-1(0-2)
∗
5 [ν2 ] Ag
0.2084
0.2112
0.490
∗
7 [ν3 ] Ag
0.1713
0.1743
0.478
∗
3 [ν1 ] Ag
0.3887
0.4254
0.467
∗
4 [ν11 ] B1u 0.3926
0.4172
0.372
6
B1u 0.1865
0.1996
0.117
12
B2u 0.1046
0.1302
0
8
B1g 0.1543
0.1663
0
1
B2u 0.4064
0.4127
0
2
B1g 0.4029
0.4028
0
∗
9 [ν4 ] Au
0.1338
0.0217 (0.0883) 0 (0.567)
∗
10 [ν8 ] B2g 0.1183
0.0254 (0.1009) 0 (0.503)
∗
11 [ν7 ] B3u 0.1218
0.0602 (0.1596) 0 (-0.26)
6.1 Scheme of transitions and vibrational modes
39
Potential Energy (eV)
Potential Energy (eV)
8
4
0
i
f
0
-2
1
Potential Energy (eV)
-1
0
Q (a.u.)
5:Ag
4
2
0
-2
3
-1
0
Q (a.u.)
9:Au
1
-1
8
0
Q (a.u.)
1
2
0.0
0.5
1.0
4:B1u
6
4
2
0
-1.0
4
3
-0.5
0.0
0.5
1.0
Q (a.u.)
7:Ag
2
1
0
1
2
0
-2
Potential Energy (eV)
3:Ag
-1
Potential Energy (eV)
12
Potential Energy (eV)
Potential Energy (eV)
Ethene molecule has 12 vibrational modes, which are collected in Table 6.1. The analysis of
FC amplitudes of core excitation (Table 6.1) and Auger decay shows that only seven modes
are active in the XAS and RAS processes. The potential energy surfaces of these modes are
shown in Fig. 6.2, including four in-plane modes (3,4,5,7) and three torsion modes (9,10,11).
0
1
Q (a.u.)
2
0
Q (a.u.)
2
3 10:B2g
2
1
0
-2
-1
1
0.6 11:B
3u
0.4
0.2
0.0
-1.0
-0.5
Q (a.u.)
Figure 6.2: The ground (0), core excited (i) and final (f) state potentials for the most active
modes (3,4,5,7,9,10,11) in XAS and RAS. We use mass weighted normal coordinates in a.u..
40
6.2
Chapter 6 Resonant Auger scattering from the ethene molecule
Multimode resonant Auger scattering
To get a deeper insight into the formation of the RAS spectral shape, it is necessary to
analyze the excitation probability of different vibrational states in the core excited state.
The XAS spectra profile of the ethene molecule is shown in Fig. 6.3, and the corresponding
assignment is given in Paper V. The natural width of the core excited state (HWHM) is
about 0.05 eV.62, 63 We use Γ = 0.05 eV and Γ = 0.07 eV for RAS and XAS simulations,
respectively. The larger Γ value for XAS includes the additional instrumental broadening.
One can see clearly that the XAS profile (Fig. 6.3) has a double-edged structure. The first
edge at peak A corresponds to the 0-0 vibrational transition followed by mainly singlemode excitations (7,9,10,11,5) and weak double-mode excitations (Region ”B” and ”Top”).
The single-mode excitations of modes 3 and 4 form the second edge (peak C), which is
followed by mainly double-mode excitations. The spectral tail for the high energy region
corresponds to the multimode excitations. Such a double-edged structure of XAS profile
plays an important role in the evolution of the RAS spectra (Fig. 6.4).
B Top
XAS (arb.units)
A
Left
C
B1
B2
B3
T1
T2
C
D
T3
D 3
D1 2
T4
exp
Right
R
0
-0.5
Γ = 0.07eV
Γ = 0.005eV
D
0.0
1i3,1i4
0.5
1.0
1.5
Ω (eV)
Figure 6.3: X-ray absorption spectrum represented in a relative energy scale (Ω = ω − ωA )
with respect to peak A (all-mode calculation). The experimental spectrum (crosses) is
shifted in order to match the theoretical profile. The labels Left, Top, Right, A, B, C, and
D correspond to the photon energies of the RAS spectra measured at Max-Lab. (Taken
from Paper V. Reprinted with permission of American Chemical Society.)
One can see that the theoretical RAS spectra reproduce the global behavior of the experi-
6.2 Multimode resonant Auger scattering
41
mental spectra (Fig. 6.4). The formation of the RAS spectra is very sensitive to the dynamics
of the nuclear wave packet on the core excited potential energy surfaces. Compared with
the photoelectron spectrum (PES) ,56, 57 the RAS spectra show more extended vibrational
progressions due to multimode core excitation. Both theoretical and experimental spectra
show qualitatively different spectral features that follow the Raman and non-Raman dispersions, respectively. Two non-Raman bands are related to the slow resonant scattering
channel, and the Raman (vertical) band corresponds to the fast scattering channel. The
rich behavior of the complex features in RAS spectra can be explained through the selective
resonant excitation and the richness of the potential energy surfaces of different vibrational
modes. Two non-Raman spectral bands are attributed to different vibrational modes, which
are related to the ”double-edge” structure of the XAS spectrum. For large detunings the
RAS spectra coincide with the PES because of the shortening of the scattering duration.
1.0
0.9
R I R II V
0.8
R I R II
V
0.7
0.6
0.6
D3
D2
0.5
D1
C
0.4
(eV)
D
C
0.4
2f9,2f10
1i3,1i4
W
W (eV)
R
0.8
Right
0.3
T4
T3
Top
0.2
T1
B3
B2
0.2
B
A
0.0
Left
-0.2
2.0
1.5
1.0
0.5
0.0
-0.5
Relative Binding Energy (eV)
0.1
B1
A
0
Left
0.0
-0.1
2.0
1.5
1.0
0.5
0.0
-0.5
Relative Binding Energy (eV)
Figure 6.4: The experimental (left panel) and theoretical (right panel) resonant Auger
spectra of the X 2 B3u final state of C2 H4 for different excitation energies. The experimental
spectra are measured at Max-lab by M. Catalin et al. The calculations are performed taking
into account seven active modes (3,4,5,7,9,10,11) with five numbers of vibrational levels for
each mode. The labels correspond to those in Fig. 6.3. (Taken from Paper V. Reprinted
with permission of American Chemical Society.)
42
Chapter 6 Resonant Auger scattering from the ethene molecule
Chapter 7
Rotational Doppler effect in X-ray
photoionization and resonant Auger
scattering
7.1
Theoretical model
When X-rays ionize the valence electrons with the binding energy about EB ∼ 10 eV,
the kinetic
energy of the photoelectron E = ω − EB and hence the electron momentum
√
k = 2E are large. This results in a rather big translational Doppler broadening of the
X-ray photoelectron line,64
Dtr = kv̄,
(7.1)
√
where v̄ = 2kB T is thermal velocity. For example, Dtr ≈ 53 meV for CO molecule when
the kinetic energy of the photoelectron is 500 eV. However, the molecules have also rotational
degrees of freedom. Let us consider the ionization of electron from the molecular orbital
(MO)
ψ = ψ1 + ψ2
of the diatomic molecule, where ψn is the contribution of the n atom to MO. The ejected
photoelectron will transfer the angular momentum to the molecule due to the rotational
recoil effect (see Fig. 7.1).
jn = [Rn × k],
jn = αn kR sin θ,
θ = ∠(R, k)
(7.2)
where Rn is the coordinate of the ionized atom n relative to the center of gravity: R1 = α1 R,
R2 = −α2 R, α1 = m2 /M , α2 = m1 /M . One should point out that rather high angular
44
Chapter 7 Rotational Doppler effect in X-ray photoionization and resonant Auger
scattering
Figure 7.1: The initial angular momentum J of the molecule changes by recoil angular
momentum jn due to the ejection of the photoelectron from the n atom.
momenta J ∼ 10 are populated at room temperature. This one can see from
Erot = BJ(J + 1) ≈ BJ 2 = kB T ≈ 0.027eV,
(7.3)
where the rotational constant B = 1/2I is inversely proportional to the momenta of inertia
of the molecule I = µR2 . This means rather high thermal angular velocity of the molecules,
w=
J
.
I
Taking into account the close analogy between translational and rotational motions
m ↔ I,
v ↔ w,
k ↔ j,
one should expect the rotational counterpart of the translational Doppler shift and related
broadening,
Drot = jn w̄.
(7.4)
Due to high angular momentum caused by the thermal motion, the rotational degrees of
freedom can be described classically. Similar to the translational Doppler effect we should
use the energy and momentum conservation laws as well the conservation of the total angular
momentum
M v = M v0 + k, J = J0 + jn .
(7.5)
Taking into account the expression for the initial and final energies
E0 = ω + Eg +
M v2
+ BJ 2 ,
2
Ef = E + Ef +
M v 02
+ BJ 02 ,
2
(7.6)
7.1 Theoretical model
45
the conservation laws for the momentum and angular momentum (7.5) gives
k2
j2
− jn · w + n .
(7.7)
2M
2I
One can see clearly that the total shift of the resonance includes both the translational and
rotational recoil effects.
Ef − E0 = E − (ω − EB ) − k · v +
k2
j2
, ∆nrot = n .
2M
2I
The matrix element of the photoionization reads in the high energy approximation
∆n = ∆tr + ∆nrot ,
∆tr =
(7.8)
hψ|e · r|ψk− i ∝ d1 eıα1 k·R + d2 e−ıα2 k·R .
(7.9)
The matrix element dn of the photoionization of the n th atom is related to the partial
ionization cross section of this atom as
σn(ph) ∝ |dn |2 .
This expression says that the amplitude of the ionization is the sum of the atomic ionization
amplitudes dn with different phase factors,
F = F1 + F2 ,
Fn ≈
√
Γ
dn e±ıαn k·R
,
∆E − k · v + k 2 /2M − jn · ω + jn2 /2I + ıΓ
(7.10)
where ∆E = E −(ω −EB ). The origin of the resonant denominator can be easily understood
from Eq.(7.7). The ionization cross section in the classical limit is given by squared scattering amplitude averaged over molecular orientations and over initial Boltzmann distribution.
Z
Z
Z
1
σ=
dΩR dvW (v) dωW (ω)|F |2
(7.11)
2
2
2
Here W (ω) = W (ω )W (ω ) = e−ω /ω̄ /π ω̄ 2 is the rotational Boltzmann distribution. The
1
2
interference term
sin kR
¿1
kR
is small and can be neglected. Due to this the total cross section is the sum of the independent contribution from different atoms
F1∗ F2 ∼ cos(k · R) =
σ ≈ σ1 + σ2 .
(7.12)
For the valence shell ionization, the lifetime broadening of the final ionic state is much
smaller than the total Doppler broadening. In this case the partial ionization cross section
from atom n can be described by Gaussian function.
µ
¶
√ Zπ
sin θ
[∆E + ∆n (θ)]2
(ph) π
dθ
exp −
, Γ ¿ Dn .
(7.13)
σn = σn
2
Dn (θ)
Dn2 (θ)
0
46
Chapter 7 Rotational Doppler effect in X-ray photoionization and resonant Auger
scattering
Here the total Doppler broadening Dn (θ) includes both the translational
q Doppler broad(n)2
2
ening (TDB) and the rotational Doppler broadening (RDB). Dn (θ) = Dtr
+ Drot (θ) =
p
k 2 v̄ 2 + jn2 ω̄ 2 . The rotational Doppler broadening is different for different atoms because
the recoil angular momentum jn depends on the mass ratio.
r
m2
(1)
Drot (θ) = Dtr
sin θ.
(7.14)
m1
Heavier atom has smaller jn and hence smaller rotational Doppler broadening.
7.2
Rotational Doppler effect in the X 2Σ+ band of CO
molecule
To exemplify our theory, let us consider the single vibrational line of the X 2 Σ+ (5σ −1 ) band
of the gas phase CO molecule. Here we only analyze the partial ionization cross section
related to C atom. Eq.(7.14) says that the similar results should be obtained for O atom
due to the close mass. Fig. 7.2 shows the energy dependence of the total broadening D, the
partial broadenings Dtr , Drot , and the rotational recoil energy ∆rot on the photoelectron
0.20
D
Dtr
D, ∆rot (eV)
0.15
Drot
∆rot
0.10
0.05
0.00
500
1000
1500
2000
2500
3000
E (eV)
Figure 7.2: The total (D), translation (Dtr ) and rotational (Drot ) broadenings and the shift
∆rot of the XPE line caused by the rotational recoil effect. Here the broadening D is the
full width at half maximum (FWHM) contrary to Eq.(7.13). The arrow shows the point
(E ≈ 2.5 keV) where Dtr = Drot . Calculations are based on Eq.(7.13). (Taken from Paper
VI. Reprinted with permission of American Physical Society.)
energy. One can see that the broadenings (D, Dtr and Drot ) as well as the rotational recoil
7.3 Rotational Doppler effect in photoionization of the 5σ electrons of HCl molecule
47
shift grow when the energy of the photoelectron increases. It is clear that the RDB has the
same order of magnitude as the TDB. When the energy of the photoelectron is high enough
(E > 2.5keV ), the RDB even exceeds the TDB due to an additional broadening caused by
the rotational recoil effect. Therefore, the effect of the RDB on the spectral line should be
taken into account in analysis of the experimental spectra.
7.3
Rotational Doppler effect in photoionization of the
5σ electrons of HCl molecule
Cross section (arb.units)
The rotational Doppler effect is very sensitive to the ionization site. One can enhance or
suppress this rotational Doppler effect by choosing different molecules. It is expected that
the rotational Doppler broadening can be measured for molecules with light atoms. In this
case one can enhance the rotational Doppler effect in comparison with the translational
one when the electron is ejected from the light atom. From example, we studied here the
photoionization of the 5σ electrons of HCl molecule. The molecular orbital 5σ of HCl has
a large contribution from the 1s function of H atom, 5σ = C1s 1sH + C3s 3sCl + C3p 3pCl ,
and the photoionization probabilities of the H and Cl atoms are comparable when 100 eV
≤ E ≤ 200 eV (see Fig.4 in Paper VI). Fig. 7.3 shows the XPE line of the A(2 Σ+ ) band of
(a) σH
-0.3
-0.2
(b) σCl
tr+rot
tr
-0.1
0.0
0.1
0.2
0.3
-0.3 -0.2 -0.1 0.0
∆E (eV)
Cross section (arb.units)
tr+rot
tr
(c) σ=σH+σCl
-0.3
-0.2
0.1
0.2
0.3
∆E (eV)
tr+rot
-0.1
0.0
0.1
0.2
0.3
∆E (eV)
Figure 7.3: The spectral shape of the single vibrational line (A2 Σ+ (5σ −1 ) band) of the HCl
molecule. The photoelectron energy E = 200 eV. The effective ionization cross sections of
hydrogen and chlorine atoms in molecule are assumed the same. The narrow peak (approximately Dtr ) is caused by the ionization of Cl atom, while large broadening is caused mainly
by Drot related to the ionization of H atom.
48
Chapter 7 Rotational Doppler effect in X-ray photoionization and resonant Auger
scattering
HCl molecule for E = 200 eV. One can see that when the photoelectron is ejected from H
atom the XPE line experiences an obvious additional broadening caused by the rotational
recoil effect (Fig. 7.3 (a)). The total Doppler broadening for the ionization channel related
to hydrogen atom is almost the same as the RDB due to the large factor mCl /mH = 35.3.
Contrarily, the RDB is suppressed when the photoelectron is ejected from relative heavier
Cl atom (Fig. 7.3 (c)). In this case the contribution of RDB to the spectral broadening is
almost zero. Our calculations show that the total broadening DH ≈ 147 meV and DCl ≈ 30
meV for E = 200 eV. Here, we assume the photoionization probabilities of H and Cl atoms
are the same, hence σ = σH +σCl . We predict that the A(2 Σ+ ) ionized band of HCl molecule
consists of a narrow peak and a broad pedestal related to the photoionization of the Cl and
H atoms, respectively (Fig. 7.3 (c)). The similar measurement can be performed for HF
molecule. It is worth to mention that the rotational Doppler effect can be suppressed for
the nonlinear molecules like N O2− in the case of ionization of the 4a1 electron localized on
the nitrogen atom (lone pair).
The X-ray free-electron laser (XFEL) with high brilliance makes it possible to observe the
rotational Doppler effect. Here, we suggest an X-ray pump-probe experiment where strong
IR laser rotationally accelerates jet cooled molecules. The RDB can be extracted from the
experimental spectral by calculating the slope of the broadening versus the photoelectron
energies.
Apparently, the rotational Doppler broadening should be taken into account in the analysis
of the RAS spectra.
Chapter 8
Nonlinear propagation of strong
XFEL pulses
The invention of the lasers has opened unique opportunities to study the nonlinear processes
in IR and optical regions. However, the research of the nonlinear phenomena in X-ray region
is hampered by the limitations of current synchrotron radiation sources like relatively low
intensity and long pulse duration. The advent of the short and intense X-ray free-electron
laser (XFEL) opens a new field - strong field X-ray physics, with nonlinear X-ray optics
and spectroscopy as important applications of this new light source. The recently opened
Linac Coherent Light Source (LCLS) at the SLAC National Accelerator Laboratory (SNAL)
provides a qualitatively new tool to study fundamental properties of atoms and materials at
previously unreachable dimensions. One should mention very recent results related to the
X-ray pump-probe spectroscopy,65 femtosecond X-ray protein nanocrystallography66 and
X-ray imaging of single mimivirus particles.67
In this chapter, we study the propagation of strong XFEL pulses through the resonant atomic
vapors. The propagation of XFEL pulses is accompanied by interesting nonlinear processes
including amplified spontaneous emission (ASE), lasing without inversion (LWI), stimulated
resonant raman scattering (SRRS) and four-wave mixing. During pulse propagation, the
compression and slowdown of the XFEL pulse occur due to the reshaping of the pulse.
8.1
Theoretical model
Our theoretical model is based on the strict numerical solution of coupled Maxwell-Bloch
equations. To exemplify our theory, we select the gas phase Ar and Mg atoms, which can
50
Chapter 8 Nonlinear propagation of strong XFEL pulses
Auger
photoionization
4s
3p
3s
Z
2p 3
2
−1
Figure 8.1: Scheme of transitions. |1i = |2p−1
3/2 4si, |2i = |3s 4si.
be described by a three-level scheme (Fig. 8.1): the ground state |0i, core excited state
−1
|1i = |2p−1
3/2 4si and two final states, |0i and |2i = |3s 4si. The frequency of the incident
light is tuned in resonance with the 2p3/2 − 4s transition. The propagation of the strong
XFEL pulses is followed by two ASE channels: 4s − 2p3/2 and 3s − 2p3/2 .
To describe the propagation of the pulse, we solve numerically the coupled Bloch
¶
µ
X
∂
(1)
+ Γ11 + γph (t) ρ11 = 2=
(V1n ρn1 ) ≡ W,
∂t
n=0,2
µ
¶
∂
(2)
+ Γ22 + γph (t) ρ22 = 2= (V21 ρ12 ) ,
∂t
µ
¶
∂
(0)
+ Γ00 + γph (t) ρ00 = 2= (V01 ρ10 ) ,
∂t
¶
µ
∂
(10)
+ Γ10 + γph (t) ρ10 = ıV10 (ρ11 − ρ00 ) − ıV12 ρ20 ,
∂t
µ
¶
∂
(12)
+ Γ12 + γph (t) ρ12 = ıV12 (ρ11 − ρ22 ) − ıV10 ρ02 ,
∂t
µ
¶
∂
(20)
+ Γ20 + γph (t) ρ20 = −ıV21 ρ10 + ıρ21 V10 ,
∂t
(8.1)
and 1D Maxwell’s equations for the XFEL propagating along the z axis
∂E
∂H
+ µ0
= 0,
∂z
∂t
∂H
∂E
∂P
+ ²0
=−
,
∂z
∂t
∂t
(8.2)
8.2 The Burnham-Chiao modulation and self-seeding
51
which ignore the transverse inhomogeneity of the pulse. Here SI units are used. The
(i)
quantities Γii and γph (t) are the Auger lifetime broadening and the depopulation rate caused
by the direct photoionization, respectively. The decay rate of the coherences ρij : Γij =
(ij)
(i)
(j)
(Γii + Γjj )/2, γph (t) = (γph (t) + γph (t))/2. The small spontaneous decay rates are neglected
(i)
in Eqs. (8.1). Γ00 = 0. The photoionization rate, γph (t) = σi I(t)/~ω, depends on the
photoionization cross section σi and the intensity of the pulse I(t). In our cases the role
of the direct photoionization is very weak because the photoionization time is much longer
than the pulse duration. The Bloch equations (8.1) and Maxwell’s equations (8.2) are
coupled through the interaction Vnm = −E(t, z)dnm eıωnm t /~ and the macroscopic lightinduced polarization P (t, z) = N tr(dρ̂). In simulations the envelope of the input pulse
E0 (t, z) = E(t, z) cos(ωt − kz) is modeled by the Gaussian or sech shape.
8.2
The Burnham-Chiao modulation and self-seeding
According to the Area Theorem, the pulse with the area θ ≥ π can invert the medium, which
leads to two time-delayed ASE channels: 1-0 and 1-2 (see Fig. 8.1). To give insight into the
physics of pulse reshaping, let us analyze the propagation of the 2 fs 3π XFEL Gaussian
pulse. One can see that during propagation the front part of the pulse is compressed, which
is followed by the formation of the Burnham-Chiao (BC) ringing68, 69 (Fig. 8.2 (a)). In the
2.0
(a)
(b)
3.81µm
1.5
0.36fs
E(t)/

0
0.5
0.0
-0.5
0µm
1.27µm
2.54µm
3.81µm
5.08µm
6.35µm
-1.0
-1.5
-2.0
10 20 30 40 50 60 70 80
t (fs)
|E(ω)| (arb.units)
1.0
5.08µm
6.35µm
ωs
ω
ωa
ωs*
180
200
220
240
260
280
ω (eV)
Figure 8.2: Gaussian input pulse. θ = 3π. τ = 2 fs. The temporal shape (a) and the
corresponding spectrum (b) of the pulse for different propagation lengths. I0 = 5.03 × 1017
W/cm2 . (Taken from Paper VII. Reprinted with permission of American Physical Society.)
52
Chapter 8 Nonlinear propagation of strong XFEL pulses
z 1.27 P m
z
0
stokes
Figure 8.3: The self-seeding. Two-level system. θ = 3π. τ = 2 fs. The initial narrow band
pulse gets the long spectral tail during propagation.
course of pulse propagation, the intensity of the Stokes field grows and two weak sidebands
(ωa = 2ω − ωs , ωs∗ = 2ωs − ω) are formed (Fig. 8.2 (b)) due to four-wave mixing70 (see
Paper VII). The interference between the coherent pump and Stokes fields results in the
fast beating of the total field with the period δt = 2π/(ω10 − ω12 ).
One should mention that the spontaneous radiation is not accounted for in the classical
Maxwell’s equations. The question arises: What is the origin of the seed radiation needed
for generation of the Stokes and four-wave mixing fields? Initial Gaussian pulse is unable to
create this seed field near the Stokes line due to the lack of sufficient bandwidth (see Fig. 8.3).
However, this seed field can arise during propagation due to the temporal reshaping of the
pulse (already in two-level media), which results in the broadening of the central part
and the formation of the long weak tail (see Fig. 8.3). The origin of this spectral tail
(E(ω) ∝ (ω − ω10 )−3/4 ) is the BC pattern (E ∝ t−1/4 ). These BC oscillations create the
spectral tail which triggers the ASE Stokes channel. This self-seeding mechanism is crucial
for the SRRS in the case of longer input pulses.
8.3
Quenching of the population inversion and lasing
without inversion
The population inversion plays an important role in the dynamics of the propagation of the
XFEL pulse, which is crucial for the competition between ASE and LWI processes. The
8.3 Quenching of the population inversion and lasing without inversion
6
zc
6
(a)
z (Pm)
-1.0
-1.0
4
-0.60
-0.20
3
3
0.20
2
(b)
5
5
4
53
-0.60
-0.20
0.20
2
0.60
1.0
0.60
1.0
1
1
20
30
40
50
t (fs)
60
70
20
30
40
50
60
70
t (fs)
Figure 8.4: Gaussian input pulse. θ = 3π. τ = 2 fs. The 2D maps of the population
inversions, which are quenched in the region z > zc . (a) ρ11 (t, z) − ρ00 (t, z), (b) ρ11 (t, z) −
ρ22 (t, z). (Taken from Paper VII. Reprinted with permission of American Physical Society.)
2D map of the population inversion (Fig. 8.4) shows that the XFEL creates population
inversions (ρ11 − ρ00 ) and (ρ11 − ρ22 ), which opens two ASE channels and induces the Stokes
field. The inversions are quenched abruptly in the region z > zc ≈ 5.5 µm. The reason for
this is as follows. The dynamics of the population inversion is governed by the field work
W (t, z) (Eq.(8.1) and Fig. 8.5):
Z t
G(t, z)
ρ11 =
W (t, z)e−Γ1 (t−t1 ) dt1 , W (t, z) =
sin θ(t, z).
(8.3)
2
−∞
Rt
Here the Rabi frequency G(t, z) = d10 E(t, z)/~ and the pulse area θ(t, z) = −∞ G(t1 , z)dt1 .
In the region z < zc (see insert of Fig. 8.5) the absorption (W > 0) of the front part of
the pulse populates the core excited state and inverts the medium, which is followed by
the stimulated emission (W < 0). This behavior of the field work is very similar as that
for two-level atoms because the Stokes ASE channel |1i → |2i is weaker in this region.
However, the field work of the three-level atoms changes drastically for z > zc where the
strengths of two ASE channels become comparable. The interference between coherent
pump and Stokes fields results in severe sub-femtosecond modulation (δt = 0.17 fs) of the
field work W (t, z), which prevents the population of the core excited state and quenches the
population inversion in the medium. Despite the quenching of the population inversion, the
54
Chapter 8 Nonlinear propagation of strong XFEL pulses
z
z
3.81P m
6.35P m ! zc
Figure 8.5: Field work W (t, z) for the two-level and three-level atoms. The insert shows the
field work before the critical distance zc ≈ 5.5 µm. Gaussian input pulse. θ = 3π. τ = 2 fs.
(Reprinted with permission of Institute of Physics and IOP Publishing Limited.)
gain of the Stokes components does not stop. The SRRS proceeds in the regime of lasing
without inversion (LWI) instead of amplified spontaneous emission (ASE).
8.4
Pulse compression
As seen already in Fig. 8.2, the 2 fs input XFEL pulse is compressed to 0.36 fs pulse during
propagation. The main reason for this is the intrinsic chirp produced in the course of pulse
propagation. Our simulations show that the compression of X-ray pulse occurs only when
its duration is comparable to the lifetime of the core excited state. The compression rate
of the X-ray pulse is also sensitive to its area and shape. The pulse compression is stronger
for Gaussian pulse than sech pulse, and the 3π pulse is strongly compressed compared to
the pulses with other areas. It should be noted that the width of the X-ray pulse is not a
monotonic function of the propagation lengths (Fig. 8.6).
One can see the pulse is strongly compressed in region I followed by the pulse broadening
in the second region (II). The reason for the compression of the 3π pulse in region I can be
qualitatively explained through the McCall-Hahn Area Theorem, which says the pulse with
an initial area slightly smaller than 3π will evolve to 2π pulse. Indeed, we get τ2π /τ3π = 4/9
8.5 Slowdown of the XFEL pulse
55
2.0
3-level
2-level
τ (fs)
1.5
I
1.0
III
II
0.5
0.0
0
1
2
3
4
5
6
z (µm)
Figure 8.6: Gaussian input pulse. θ = 3π. τ = 2 fs. The width of the pulse as a function of
the propagation lengths.
using estimations for the areas 3π = E3π τ3π , 2π = E2π τ2π and taking into account the
2
2
energy conservation law in the course of pulse propagation, W = E3π
τ3π = E2π
τ2π . This
estimation reveals that the width of the initial 3π pulse is compressed to 4/9 of the initial
width. When the pulse goes further into region II, the intensity of the pulse becomes weak
while its duration becomes slightly wider due to the dispersion of the medium. This trend
is completely broken in region III for the three-level system, where the pulse is sharply
compressed again. In contrast, no pulse compression occurs for the two-level model in
this region. We conclude that the pulse compression in region III must be related to the
lasing without inversion. Our studies show that a 50 fs XFEL pulse can experience a 6-fold
compression when it passes through the resonant atomic Mg vapor (see Paper VIII).
8.5
Slowdown of the XFEL pulse
It is usually accepted that X-rays travel in different media with the speed very close to the
speed of light in vacuum c. This is the reason why it is difficult to focus X-rays with mirrors
and lenses of various media. However, our simulations show the strong slowdown of the
XFEL pulses before the critical distance zc . Fig. 8.7 displays the X-ray pulse is decelerated
by two orders of magnitude in Mg vapor. The reason for this is the large nonlinear refractive
index caused by the high intensity of XFEL pulses. The slowdown of the XFEL pulse by
resonant medium can be used as the X-ray delay time. A slowdown of the XFEL pulse in
56
Chapter 8 Nonlinear propagation of strong XFEL pulses
Figure 8.7: (a) The trajectory of the XFEL pulse. The insert shows the slowdown of the
pulse in the region 0 ≤ z ≤ zc ≈ 0.2 µm. (b) The velocity of the leading edge of the pulse
vg versus the propagation length. vg /c ≈ 0.007 for z = 3.77 × 10−4 µm. (Taken from Paper
VIII. Reprinted with permission of European Physical Society.)
the Ar atomic medium is shown up to 78% of the speed of light in vacuum (see Paper VII).
8.6
Suppression of the Auger effect in the propagation
of XFEL pulses
Auger effect is a common analytical tool in surface and material science. However, this effect
is undesirable in the propagation of the XFEL through the resonant medium. The Auger
ionization followed by Coulomb explosion71 is the main reason for the radiative damage. In
this section we will study the effect of the propagation of strong XFEL pulse on the relative
Auger yield.
Let us first rewrite the equations for the population of the core excited state and for the
total population of the intact atoms (Eqs.(8.1)):
∂
ρ11 = −Γ11 ρ11 + W = −(Γ11 ρ11 + WE ) + WA ,
∂t
∂
(ρ00 + ρ11 + ρ22 ) = −Γ11 ρ11 − Γ22 ρ22 .
∂t
(8.4)
(8.5)
8.6 Suppression of the Auger effect in the propagation of XFEL pulses
57
For Ar atoms, the lifetime broadenings of the core excited state and final state are Γ11 = 0.12
eV, Γ22 = 0.076 eV, respectively. As already shown in Sec. 8.3, the field work (Eq.(8.3))
experiences a sign-changing modulation during pulse propagation. The positive and negative
field work are responsible for the absorption (WA ) and stimulated emission (WE ) induced
by the XFEL pulse, respectively. Eq.(8.4) says the population of the core excited state,
populated by the X-ray absorption (WA ), can be depopulated through the Auger decay
(Γ11 ρ11 ) and stimulated emission (WE ) processes. There will be a competition between the
stimulated emission and Auger decay channels, which will influence the Auger yield during
pulse propagation. For example, if the X-ray field is strong enough, the Auger process will
have no time to occur because the stimulated emission will be faster than the Auger decay.
Let us introduce the relative Auger yield as
ρAug (z)
ω(z) =
,
ρAug (z) + ρE (z)
Z∞
ρE (z) =
Z∞
WE (t, z)dt,
ρAug (z) ≈
∞
Γ11 ρ11 (t, z)dt.
(8.6)
∞
Here we will focus only on the Auger yield caused by the Auger decay of the core excited
state because the Auger lifetime broadening of the final state is smaller. It is convenient to
define the integral Auger yield from the whole sample with the length L as
NAug (L)
q(L) =
,
NAug (L) + NE (L)
ZL
Ni (L) =
ρi (z)dz,
i = Aug, E.
(8.7)
0
Fig. 8.8 shows that the propagation of the XFEL pulse affects intensively the Auger yields
Auger yield
0.8
ω (3-level)
ω (2-level)
0.6
q (3-level)
q (2-level)
0.4
0.2
0.0
0
1
2
3
4
5 zC 6
7
8
z (µm)
Figure 8.8: Comparison of the Auger yield for 3-level and 2-level models. Gaussian input
pulse. θ = 3π. τ = 2 fs.
58
Chapter 8 Nonlinear propagation of strong XFEL pulses
ω(z) and q(L). Compared to the two-level model, the Auger branching ratio ω(z) drops
down fast near the critical distance zc for the three-level system. As we have discussed in
Sec. 8.3, the interference between comparable pump and Stokes fields quenches abruptly the
population inversions in this region and hence suppresses the Auger yields. For the longer
distances, the relative Auger yield ω(z) changes smoothly because the population of the
core excited state is very small. Contrary to ω(z), the change of the integral Auger yield
q(z) is not so strong since q(z) gives the relative Auger yield collected from the entry up to
z.
Chapter 9
Two-photon induced X-ray
fluorescence
The intensity of XFEL is high enough to create double-core-hole excited states. This state
was already earlier observed in electron impact experiments by H. Ågren et al72 in 1978, and
its first theoretical study was carried out by Bhalla.73 The cross sections of the double-corehole formation via the two-photon absorption (TPA) were recently calculated for atoms74–76
and molecules.77 The first experimental observation of TPA in extreme-ultraviolet region
was performed on helium.78 The double-core-hole formation via TPA in X-ray region was
observed this year at LCLS.65, 79, 80
The usual experimental method to observe X-ray TPA is photoemission.81 In this chapter
we investigate an alternative scheme based on the TPA induced X-ray fluorescence (TPAXF)
or lasing. The general theory developed in Paper X is applied to TPAXF of neon vapor
with the frequencies of incident X-ray photon of 927.7 eV and 984.7 eV, which correspond to
two different schemes (Fig. 9.1). In the first scheme, we consider the TPA double-core-hole
excitation 1s2 → 1s1 3p1 → 1s0 3p2 with off-resonant intermediate state. The second one is
based on the following process: 1s2 → 1s1 ψE1 → 1s0 ψE1 3p1 .
Let us analyze only the population of two-core-hole state because the cross section of X-ray
emission is proportional to the population of the corresponding decaying state. The main
theoretical tool to solve this problem is based on the density matrix equations (see Paper X,
and Eq.(2.19)). Taking into account the initial condition ρ00 (−∞) = 1, the density matrix
equations are solved numerically at strict two-photon resonance:
2ω = ω20 ,
ω − ω10 = −(ω − ω21 ) = Ω > 0.
(9.1)
It should be noted that important dynamical aspect of studied schemes, namely, the lifetime
60
Chapter 9 Two-photon induced X-ray fluorescence
(a)
(b)
1s 0 3p1 ψ E1
1s 0 3p2
h ω =927.66 eV
59.27 eV
hω
984.7 eV
986.93 eV
1s 1 ψ E1
1s 1 3p1
868.39 eV
h ω =984.7 eV
hω
1s 2 3p0
1s 2 3p0
Figure 9.1: Scheme of the TPA induced fluorescence in the Ne atom. (a) bound-boundbound-bound (BBBB) transitions. (b) bound-continuum-bound-bound (BCBB) transitions.
The experimental value of the 1s-ionization potential of Ne is 870.2 eV. (Taken from Paper
X. Reprinted with permission of American Physical Society.)
broadenings (FWHM) of the single 1s hole and of the double 1s hole are different:82
Γ11 ≈ 0.27eV,
Γ22 ≈ 0.802eV.
(9.2)
The major reason for this is that the number of the Auger channels of the double 1s-core
hole state is twice than that of the single 1s-core hole state. Another reason for increased
value of Γ22 is the contraction of the atomic orbitals due to the smaller electronic screening
of the nucleus in the presence of extra core hole.
9.1
Bound-bound-bound-bound (BBBB) TPA
We first consider the dynamics of population of double-core-hole state due to the twophoton absorption along the scheme shown in Fig. 9.1 (a). The first photon enforces the
transition from the ground (bound) state 1s2 3p0 to the intermediate bound state 1s1 3p1 .
The second photon leads to the transition between the 1s1 3p1 and 1s0 3p2 bound states. We
label this TPA scheme as BBBB standing for the bound-bound-bound-bound transition:
9.2 Bound-continuum-bound-bound (BCBB) TPA
61
|0i → |1i → |2i, where |0i = 1s2 2s2 2p6 , |1i = 1s1 2s2 2p6 3p1 and |2i = 1s0 2s2 2p6 3p2 denote
the ground, intermediate and final states of neon, respectively.
9.2
Bound-continuum-bound-bound (BCBB) TPA
Let us consider now the second scheme of the TPA population of the two-core-hole final state
(Fig. 9.1 (b)). The first absorbed photon induces the transition from the ground bound state
to the core ionized continuum state, while the second one results in a bound-bound transition
1s → 3p between the core ionized states. This TPA scheme corresponds to the boundcontinuum-bound-bound (BCBB) transition: |0i → |1Ei → |2Ei, where |0i = 1s2 2s2 2p6 ,
|1Ei = 1s1 2s2 2p6 ψE1 and |2Ei = 1s0 2s2 2p6 3p1 ψE1 are the ground, intermediate and final
states of neon, respectively. Here, ψE is the continuum state of electron. It should be noted
that, contrary to the BBBB scheme, the intermediate continuum state in BCBB scheme is
in resonance with the X-ray photon.
Fig. 9.2 shows the maximum population of double-core-hole states for different pulse durations and different peak intensities. One can see that the population of the double-core-hole
0.03
0.3
(b) BCBB
(a) BBBB
1fs
5fs
10fs
30fs
100fs
0.2
ρ22
ρ22
0.02
0.01
0.1
0.00
0.0
1E15
1E16
1E17
2
I0 (W/cm )
1E18
1E15
1fs
5fs
10fs
30fs
100fs
1E16
1E17
1E18
2
I0 (W/cm )
Figure 9.2: Maximum population of two-core-hole state ρ22 as a function of XFEL pulse
duration and peak intensity. (a) BBBB scheme (Fig. 9.1 (a)). ρ22 ≈ 0.001 (τ = 30 fs), and
ρ22 ≈ 0.0001 (τ = 100 fs) for I = 1 × 1018 W/cm2 . (b) BCBB scheme (Fig. 9.1 (b)). (Taken
from Paper X. Reprinted with permission of American Physical Society.)
62
Chapter 9 Two-photon induced X-ray fluorescence
state grows faster for higher intensities in both schemes. In comparison with BBBB scheme
(Fig.9.2 (a)), the BCBB TPA scheme (Fig.9.2 (b)) gives significant population of the TPA
state, which will result in a rather strong two-photon absorption induced X-ray emission.
According to the report of LCLS website,83 the intensity of XFEL can be achieved as high
as 5×1017 W/cm2 for photon energies of 900 eV. It is predicted that the two-photon-induced
fluorescence would give an experimentally observable signal.
One should pay attention to the decrease of the concentration of intact molecules due
to the direct photoionization. Numerical analysis (see Paper X) shows that the direct
photoionization is more important in the BBBB scheme, where it strongly suppresses the
population of the TPA state for long pulses τ ≥ 30 fs.
Chapter 10
Summary of results
The main results of this thesis can be summarized as follows:
1. Resonant UV and X-ray Raman scattering
• It is shown that the shortening of the scattering duration by frequency detuning from
absorption resonance results in a purification of the resonant Raman spectra from
contamination by low frequency vibrational modes and higher overtones. This makes
simpler interpretation of spectra of complex systems. The numerical analysis shows
that the widely used short-time approximation based on the assumption of a short
lifetime of the intermediate state is not applicable for real molecular systems.
• We develop a theory of resonant X-ray Raman scattering in the liquid phase, and
show that the environmental broadening in liquid phase is sensitive to the solvent.
For example, the environmental broadening for pure liquid acetone is much weaker
than that for aqueous acetone.
• The liquid phase theory, in agreement with the experiment, shows that the X-ray
Raman spectra of liquid and gas phase acetone are very similar. The scattering to the
ground state, in contrast to the inelastic scattering channel, displays nicely resolved
vibrational structure, which is strongly affected by the Thomson scattering. Strong
broadening of the n−1 π ∗ band of acetone is caused by the C = O out-of-plane wagging
and CH3 torsion modes as well as the thermal population of the torsion modes.
• The time-dependent wave packet technique is applied to explain X-ray Raman spectra
of gas phase oxygen in the region of the dissociative O1s → σ∗ resonance. The spectra
of scattering to the ground electronic state demonstrate spatial quantum beats caused
64
Chapter 10 Summary of results
by two interfering wave packets in the quartet and doublet states with different momenta as the atoms separate. Simulations show that the Thomson scattering strongly
affects both the spectral profile and the scattering anisotropy of this ”elastic” channel.
• Despite the quartet-doublet splitting of core excited states the related splitting of the
molecular band is absent in the experimental spectra. The reason for this effect is a
hidden selection rule: The spin of a sub-set of the electrons of the parent ion does not
change in the course of the decay transition.
• We show that the core hole and the excited electron in the intermediate state exchange parity, thereby violating a widely accepted symmetry selection rule based on
the concept of the core-hole parity. This explains the symmetry forbidden band in the
resonant X-ray Raman scattering spectrum of the oxygen molecule.
• We have found a new effect of vibrational scattering anisotropy in resonant X-ray
Raman scattering from molecular oxygen caused by the interference of the inelastic
Thomson and resonant scattering channels. This happens because the intermediate
dissociative state weakens the resonant channel.
2. Resonant Auger scattering
• Our ab-initio multimode calculations quantitatively explain the origin of extended
vibrational structure and its evolution as a function of excitation energy in spectra of
resonant Auger scattering of ethene molecule.
• It is shown that Raman and non-Raman dispersions seen in the experimental Auger
spectra of ethene are due to resonant and vertical scattering channels, respectively.
• Calculations show that two observed thresholds of formation of non-Raman spectral
bands are related to the double-edge structure of the X-ray absorption spectrum caused
by the out-of- and in-plane modes.
3. Rotational Doppler effect
• The rotational Doppler effect and related broadening in the X-ray photoelectron and
resonant Auger spectra are predicted. The magnitude of the rotational Doppler broadening is of the same order as its counterpart–translational Doppler broadening.
• The rotational Doppler broadening is very sensitive to the ionization site. It is shown
that the rotational Doppler broadening can be enhanced for molecules with light
atoms, like HCl and HF molecules.
65
• We suggest an X-ray pump-probe experiment to measure the rotational Doppler broadening making use of the synchrotron radiation light sources or X-ray free-electron
lasers.
4. Nonlinear propagation of XFEL pulses
• The interaction between strong XFEL pulses and atomic media is investigated by
solving coupled Maxwell and density matrix equations. Our simulations show that
the nonlinear propagation of XFEL pulses is accompanied by self-seeded stimulated
resonant Raman scattering (SRRS). The SRRS starts with the amplified spontaneous
emission (ASE) followed by lasing without inversion (LWI).
• The XFEL pulses experience a strong compression (up to attosecond region) during
propagation when the duration of the pulse is comparable or shorter than the lifetime
of the core excited state. Due to strong nonlinear interaction with the media, the
XFEL pulses is decelerated up to two orders of magnitude in comparison with the
speed of light in vacuum.
• The rate of Auger decay is strongly suppressed in the course of XFEL pulse propagation due to enhancement of the stimulated emission.
5. X-ray fluorescence induced by two-photon absorption
• It is predicted a rather strong X-ray fluorescence from the two-core-hole states created
in the course of two-photon X-ray absorption in neon vapor.
66
Chapter 10 Summary of results
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