Università degli Studi di Milano

FACOLTÀ DI SCIENZE E TECNOLOGIECorso di Laurea Magistrale in Fisica

Ultrafast X-ray scattering

beyond linear-response theory

Relatore: Prof. Nicola MANINICorrelatore: Prof. Giovanni ONIDA

Tesi di Laurea di:Mattia MANTOVANI

matr. 859695

PACS: 42.55.Vc78.70.Ck

Anno Accademico 2015-2016

Ultrafast X-ray scatteringbeyond linear-response theory

Mattia MantovaniDipartimento di Fisica, Università degli Studi di Milano

Via Celoria 16, I-20133 Milano (Italy)

October 6th, 2016

Abstract

In the past few decades, a special effort has been made toward the developmentof new X-ray sources, aiming at higher efficiency, monochromaticity, coherence, bril-liance and lower beam divergence. The fourth-generation of X-ray sources employsfree-electron lasers (XFELs) whose peak intensity reaches values of order 1020 W m−2

and pulse duration of a few femtoseconds. The single-molecule imaging offered byXFELs is currently addressed within linear-response theory, but there is no guaranteethat the perturbative limit is fulfilled at the XFEL intensities. Nonlinear or collec-tive electronic effects may arise. On the other hand, non-perturbative methods forradiation-matter interaction often rely on the dipole approximation, which fails forhard X-rays.

We address the scattering of X-ray pulses by a one-electron atom within a Maxwell-Schrödinger numerical scheme. We solve the time-dependent Schrödinger equation(TDSE) for one electron subject to both the Coulomb potential and the electromag-netic field of a ultraintense, ultrashort laser pulse, treated classically. The coupling ofthe electron dynamics with the vector potential does not rely on the dipole approx-imation, making this method suitable for hard X-rays. We use the current densitydistribution arising from the solution of the TDSE to compute the scattered radia-tion, by solving Maxwell’s equations on the Yee grid (finite-difference time-domainmethod). We obtain the scattered power and hence the scattering cross section of theatom, and compare the values with the weak-coupling results in linear-response theory,for different pulse intensities, probing nonlinear effects. We find a disagreement withthe perturbative behavior for intensities beyond 1025 W m−2. The method providesthe basis for a consistent treatment of ultrafast scattering from a nano-object beyondperturbation theory, and can in principle be extended to handle many-electron atomsand molecules within time-dependent density-functional theory.

Advisor: Prof. Nicola ManiniCo-advisor: Prof. Giovanni Onida

Mr. Praline: «He’s not pinin’! He’s passed on! This parrot is no more!He has ceased to be! He’s expired and gone to meet his maker! He’s a stiff!Bereft of life, he rests in peace! If you hadn’t nailed him to the perch he’dbe pushing up the daisies! His metabolic processes are now history! He’s offthe twig! He’s kicked the bucket, he’s shuffled off his mortal coil, run downthe curtain and joined the bleedin’ choir invisible! This is an ex-parrot!»

— Monty Python’s Flying Circus, season 1, episode 8

Contents

1 Introduction 1

2 The theory of X-ray scattering 52.1 Equations for the electromagnetic field . . . . . . . . . . . . . . . . . . 5

2.1.1 Gauge choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.2 Transverse and longitudinal fields . . . . . . . . . . . . . . . . . 62.1.3 Energy flow and the Poynting vector . . . . . . . . . . . . . . . 72.1.4 Free classical field . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Quantization of the electromagnetic field . . . . . . . . . . . . . . . . . 92.3 Hamiltonian for a molecule-field system . . . . . . . . . . . . . . . . . . 10

2.3.1 Perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 X-ray scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4.1 Scattering cross section . . . . . . . . . . . . . . . . . . . . . . . 132.4.2 Atomic form factor . . . . . . . . . . . . . . . . . . . . . . . . . 162.4.3 Compton scattering . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Radiation from an assigned current density 193.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 FDTD method and Perfectly Matched Layer . . . . . . . . . . . . . . . 19

3.2.1 The Yee algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 193.2.2 Discretized equations . . . . . . . . . . . . . . . . . . . . . . . . 203.2.3 Perfectly matched layer . . . . . . . . . . . . . . . . . . . . . . . 22

3.3 Trial current density distributions . . . . . . . . . . . . . . . . . . . . . 233.4 Spatial discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.5.1 Simple-harmonic current . . . . . . . . . . . . . . . . . . . . . . 243.5.2 Gaussian-modulated current . . . . . . . . . . . . . . . . . . . . 253.5.3 Oscillation along the x axis . . . . . . . . . . . . . . . . . . . . 25

4 Schrödinger equation in a laser pulse 314.1 Hamiltonian and Schrödinger equation . . . . . . . . . . . . . . . . . . 314.2 Charge density and current density operators . . . . . . . . . . . . . . . 32

4.2.1 Observation on the gauge-invariance of current density . . . . . 334.3 Numerical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.3.1 Integration method and spatial discretization . . . . . . . . . . 334.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.4.1 Field-free hydrogen atom . . . . . . . . . . . . . . . . . . . . . . 33

vii

viii CONTENTS

4.4.2 Interaction with the field . . . . . . . . . . . . . . . . . . . . . . 374.4.3 Probability density and current density distribution . . . . . . . 374.4.4 Radiation intensity and total scattering cross section . . . . . . 37

5 Conclusions and outlook 39

A Minimal coupling Hamiltonian 41

B The Keldysh parameter 43B.1 Radiation beam parameters . . . . . . . . . . . . . . . . . . . . . . . . 43B.2 The Keldysh parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

B.2.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44B.2.2 At European-xfel . . . . . . . . . . . . . . . . . . . . . . . . . 46B.2.3 At fermi and flash . . . . . . . . . . . . . . . . . . . . . . . . 46B.2.4 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

B.3 Relativistic effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

C Atomic units 49

Acknowledgements 51

Bibliography 53

Chapter 1

Introduction

X-ray physics is an ever-growing field with relevant connections to vast areas of natu-ral sciences, from crystallography to medical applications, through accelerator physics,astrophysics and airport security systems. The interaction of X-rays with matter pro-vides a useful probe for the investigation of structural and electronic properties ofmatter through scattering and spectroscopy, respectively. Since the typical wavelengthof X-rays ranges from 0.01 to 10 nm, this type of radiation yields the adequate spatialresolution for nanoscale objects (single atoms, molecules, polymers) and crystals com-posed of such building blocks, with lattice parameter falling in this range. Moreover,the photon energy of the X-rays (100 eV – 100 keV) matches the binding energy ofinner-shell electrons in atoms enabling us to do core absorption and photoemissionspectroscopy.

In the past decades, a special effort has been made toward the development ofnew X-ray sources, aiming at higher efficiency, monochromaticity, coherence, brillianceand lower beam divergence [1]. X-rays were originally produced as a collateral effectof accelerators, due to the power radiated by electrons in early synchrotrons – firstgeneration of X-ray sources. Next, dedicated storage rings were built to maximize thisparasitic effect; magnetic insertion devices such as undulators and wigglers were laterintroduced to increase spectral brightness and brilliance by several orders of magnitude(second and third generation sources). Lastly, the fourth generation of X-ray sourcesemploys free-electron lasers (FELs), usually based on a very long wiggler in a high-energy electron linear accelerator. Such a device has a peak brightness many ordersof magnitude beyond that of the previous-generation sources (up to 1020 W m−2), aswell as pulse duration of 100 fs or shorter, and is spatially fully coherent.

X-ray free-electron lasers (XFELs) have begun to influence many areas of sciencein a revolutionary way: the unprecedentedly high brilliance permits us in principleto image small, non-periodic objects such as single atoms and molecules, where thecoherent superposition of scattered waves cannot be exploited [2, 3]; the ultrafast timeresolution (pulse duration < 1 fs) promises to peek into the dynamics through directobservation of nuclear motion on the pico- to femtosecond time scale or even electronicmotion on the femto- to attosecond time scale through pump-probe experiments [4–6].

Experiments at XFELs can exploit the ultrafast time resolution as well as the in-herent properties of X-rays: short wavelength, high penetration-depth, access to coreand valence electrons alike, element specificity and even orbital specificity, sensitivityto chemical environment and molecular geometry, sensitivity to magnetic moment.

1

2 CHAPTER 1. INTRODUCTION

The disciplines with applications range from atomic and molecular physics, plasmaphysics, femtochemistry and chemical analysis to structural biology and biochemistry.The experimental methods to utilize XFEL radiation are manifold: resonant, non-resonant and magnetic X-ray scattering, absorption and fluorescence spectroscopies,photoelectron and ion spectroscopies.

Several XFELs are already in operation, with more currently under construction[7–12]. Currently (2016), in the hard X-ray regime, lcls (USA, 2009), sacla (Japan,2012) and pal-xfel (South Korea, 2016) are in operation, while the European-xfel(Germany, 2017), the Swissfel (Switzerland, expected 2017) are under construction.lcls will undergo a major upgrade to lcls-ii on a long-term basis (expected in 2022).In the soft X-ray and XUV regime flash (Germany, since 2005) and fermi (Italy,since 2010) are operational with extensions in commissioning.

The possibilities of single-molecule imaging offered by XFELs are however hinderedby the radiation damage effects due to the extremely high field intensity of the pulse[13–16]. Radiation absorption leads to photoionization, Auger cascades, fluorescenceand ultimately Coulomb explosion for molecules at the femtosecond scale, resulting inthe sample destruction and possibly also in a deteriorated or even useless scatteringsignal. In order to understand these effects, and interpret correctly the scatteringdata, a number of theoretical approaches has been developed in recent years [17–22],mainly based on the standard perturbative approach used traditionally to describe thecoupling of radiation and matter. Although these methods may be justified with thesmall scattering cross section of X-rays from electrons, or with the value of Keldyshparameter [23], there is no guarantee that the perturbative limit is really fulfilled atthe XFEL intensity level. Nonlinear or collective electronic effects may arise [24, 25].On the other hand, non-perturbative methods for radiation-matter interaction usuallyrely on the dipole approximation. The latter fails for hard X-rays. Moreover, dipolemethods do not address scattering explicitly [26–28]. The interaction of ultraintensehard X-rays with atoms is thus a largely unexplored territory.

The aim of the present thesis is to provide the basis for a Maxwell-Schrödingerapproach to scattering of a ultraintense, ultrafast X-ray pulse from a nano-object.For simplicity, we investigate the case of the one-electron atom, leaving more complexmany-body problems to future work. The main goal is to identify strong-field effectsleading to differences in the scattering cross section compared with the one calculatedperturbatively in linear-response theory. To this end, we set up a finite-difference time-domain (FDTD) numerical approach to compute the radiated power from an arbitrarycurrent density distribution, which derives from the solution of the Time-DependentSchrödinger Equation (TDSE) for an electron, fully coupled to the vector potential ofthe electromagnetic field, in a Coulomb potential.

This work is structured as follows:

• In Chapter 2, after a brief review of classical and quantum electrodynamics, wepresent the current theoretical framework for X-ray scattering, and derive theweak-coupling scattering cross section.

• Chapter 3 is dedicated to the presentation of the FDTD method and of somebenchmark calculations with trial current density distributions.

• In Chapter 4 we report the solution the TDSE for one electron in an electro-

3

magnetic field with the full minimal coupling to a vector potential, given as afunction of time and space. We then obtain the time-dependent current densityand compute the scattered radiation.

• Chapter 5 reports a discussion, conclusions and a general outlook for the method.

• In the Appendices we gather a few useful formal treatments.

Chapter 2

The theory of X-ray scattering

In this Chapter we summarize the state-of-the-art weak-coupling theory of X-raysinteraction with matter. We first review basic tools such as electrodynamics andquantization of the electromagnetic field. We discuss X-ray scattering and derive thescattering cross section by standard first-order time-dependent perturbation theory.Compton scattering is also considered. We focus on the interaction with finite materialsystems throughout the Chapter, which are the main targets of scattering experimentsat XFELs.

2.1 Equations for the electromagnetic fieldWe start with the microscopic Maxwell’s equations [29]:

∇ · E =ρ

ε0

∇× E = −∂B∂t

∇ ·B = 0

∇×B =1

c2

∂E

∂t+ µ0J.

(2.1a)

(2.1b)

(2.1c)

(2.1d)

The current density J and charge density ρ are those of the electrons and the nuclei.These densities, together with electric field E and magnetic induction field B, arefunctions of both time and space. By introducing auxiliary potentials Φ and A, relatedto E and B by

E = −∇Φ− ∂A

∂t, (2.2a)

B = ∇×A, (2.2b)

one rewrites Eqns. (2.1) as second-order equations for A and Φ:∇2Φ +

∂

∂t(∇ ·A) = − ρ

ε0

∇2A−∇(∇ ·A +

1

c2

∂Φ

∂t

)− 1

c2

∂2A

∂t2= −µ0J.

(2.3a)

(2.3b)

5

6 CHAPTER 2. THE THEORY OF X-RAY SCATTERING

2.1.1 Gauge choice

Equations (2.3) can be decoupled in the Lorenz gauge (∇ · A + 1c2∂Φ∂t

= 0) or inthe Coulomb gauge (∇ · A = 0). The former is covariant, while the latter is thenatural framework for quantum electrodynamics and radiation-matter interaction ina given frame of reference. We will adopt Coulomb gauge throughout this thesis.Equations (2.3) in the Coulomb gauge read:

∇2Φ = − ρ

ε0

∇2A− 1

c2

∂∇Φ

∂t− 1

c2

∂2A

∂t2= −µ0J.

(2.4a)

(2.4b)

Equation (2.4a) is the Poisson’s equation and handles the instantaneous electrostaticpotential. The formal solution is [29]

Φ(r) =1

4πε0

∫d3r′

ρ(r′)

|r− r′|. (2.5)

The charge density to be used in the integral is evaluated at the time for which Φ isrequired: the scalar potential in Coulomb gauge behaves as if it had an instantaneousknowledge of the charge density everywhere in space, with no retardation effects. Forradiation-matter interaction purposes, the Coulomb gauge is especially useful, sincethe scalar potential Φ of the field can be taken to be zero very far from the source,where the experiment is performed.

2.1.2 Transverse and longitudinal fields

According to Helmholtz’ theorem [30], any vector field can be written as the sum oftwo auxiliary fields, one with null divergence and the other with vanishing curl. Forthe current density J one can write

J = JT + JL, (2.6)

where

∇ · JT = 0,

∇× JL = 0.(2.7)

JT is the transverse field and JL is the longitudinal field. In the Coulomb gauge,Eq. (2.4b) can be shown to be separable into [31]

∇2A− 1

c2

∂2A

∂t2= −µ0JT,

1

c2

∂∇Φ

∂t= µ0JL.

(2.8a)

(2.8b)

Equations (2.4a)-(2.8b) and the identity c2µ0 = ε−10 imply the continuity equation:

∇ · JL = −∂ρ∂t. (2.9)

2.1. EQUATIONS FOR THE ELECTROMAGNETIC FIELD 7

In the same fashion one can decompose the electric field E in transverse and longitu-dinal components:

E = ET + EL, (2.10)

and the two parts can be expressed as

ET = −∂A∂t

(2.11)

andEL = −∇Φ. (2.12)

The magnetic field B is entirely transverse since its divergence vanishes. The Coulombgauge allows the division of Maxwell’s equations into two distinct sets. The transverseequations are those associated with the vector potential A of Eq. (2.8a) and describeelectromagnetic waves determined by JT:

∇× ET = −∂B∂t, (2.13a)

∇×B =1

c2

∂ET

∂t+ µ0JT. (2.13b)

The longitudinal equations are associated with the scalar potential obeying Eq. (2.4a),and describe the unretarded electric fields arising from the charge density:

∇ · EL =ρ

ε0

, (2.14a)

∂EL

∂t= −JL

ε0

, (2.14b)

consistently with Eq. (2.9).

2.1.3 Energy flow and the Poynting vector

The Poynting vector is defined through microscopic fields E and B as

S =E×B

µ0

. (2.15)

Poynting’s theorem states the conservation of energy for the electromagnetic field, andis usually written in its local, differential form

− ∂u

∂t= ∇ · S + J · E, (2.16)

where J is the current density and

u =1

2

(ε0E

2 +1

µ0

B2

)(2.17)

is the energy density of the electromagnetic field. Equation (2.16) indicates that thelocal decrease of electromagnetic energy density is due to the combination of twoeffects: the flux of electromagnetic radiation energy expressed by the divergence of

8 CHAPTER 2. THE THEORY OF X-RAY SCATTERING

the Poynting vector S, plus the work done by the electric field on the moving charges.The Poynting vector represents the instantaneous power flow due to the instantaneouselectric and magnetic fields. For a monochromatic plane wave in the form

E(r, t) = E0 cos(k · r− ωt), (2.18)

B(r, t) =κ

c× E(r, t), (2.19)

with ω = c|k| and κ = k/|k|, the time average over a cycle 2π/ω of the Poyntingvector at an arbitrary point can be written as

〈κ · S〉 =1

2cε0E

20 = cε0E

2rms, (2.20)

where Erms = E0/√

2 is the root-mean square value of |E|. S is a power per unitsurface, and accordingly is measured in W m−2 in SI units.

2.1.4 Free classical field

In a region where JT = 0, Eq. (2.8a) becomes

∇2A− 1

c2

∂2A

∂t2= 0, (2.21)

which corresponds to the homogeneous wave equation (free electromagnetic field). Wecan expandA in terms of plane waves in a cubic region of side L with periodic boundaryconditions:

A(r, t) =∑k

∑λ=1,2

εk,λAkλ(r, t), (2.22)

whereAk,λ(r, t) = αk,λ(t)e

ik·r + α∗k,λ(t)e−ik·r. (2.23)

The wave vector components have the usual values

kx =2π

Lnx, ky =

2π

Lny, kz =

2π

Lnz; ni ∈ Z. (2.24)

In the Coulomb gauge, the unit polarization vectors εk,λ satisfy

εk,λ · k = 0, ε∗k,1 · εk,2 = 0. (2.25)

Substitution of Eq. (2.23) in Eq. (2.21) yields the mode equation (together with itscomplex conjugate one)

∂2αk,λ(t)

∂t2+ ω2

kαk,λ(t) = 0, ωk = c|k|. (2.26)

Equation (2.26) is a simple-harmonic equation of motion with solution

αk,λ(t) = αk,λe−iωkt. (2.27)

2.2. QUANTIZATION OF THE ELECTROMAGNETIC FIELD 9

The complete form of the vector potential is obtained by substitution of Eq. (2.27)into Eq. (2.23):

A(r, t) =∑k,λ

εk,λ[αk,λe

i(k·r−ωkt) + α∗k,λe−i(k·r−ωkt)

]. (2.28)

The transverse electric field and the magnetic field are obtained through Eq. (2.11)and Eq. (2.2b), respectively:

ET(r, t) =∑k,λ

εk,λ iωk

[αk,λe

i(k·r−ωkt) − α∗k,λe−i(k·r−ωkt)], (2.29)

B(r, t) =∑k,λ

εk,λ i|k|[αk,λe

i(k·r−ωkt) − α∗k,λe−i(k·r−ωkt)]. (2.30)

We can now express the classical energy of the radiating electromagnetic field in termsof the amplitudes [31]:

EEM =1

2

∫d3r[ε0|ET(r, t)|2 + µ−1

0 |B(r, t)|2]

=∑k,λ

ε0V ω2k

(αk,λα

∗k,λ + α∗k,λαk,λ

)=∑k,λ

2ε0V ω2k|αk,λ|2.

(2.31)

2.2 Quantization of the electromagnetic fieldThe electromagnetic field is quantized by promoting each classical harmonic oscillatorfor the individual mode (k, λ) of the radiation field, Eq. (2.26), to a quantum harmonicoscillator. Each mode has then creation and destruction operators in the Fock space,

ak,λ|nk,λ〉 =√nk,λ|nk,λ − 1〉, (2.32)

a†k,λ|nk,λ〉 =√nk,λ + 1|nk,λ + 1〉, (2.33)

with the usual bosonic commutation relation[ak,λ, a

†k,λ

]= δk,k′δλ,λ′ . (2.34)

The total Hamiltonian for the quantum field in vacuum is obtained by summating overthe single-mode Hamiltonians

HEM =∑k,λ

Hk,λ

=∑k,λ

~ωk

2

(ak,λa

†k,λ + a†k,λak,λ

)=∑k,λ

~ωk

(a†k,λak,λ +

1

2

).

(2.35)

10 CHAPTER 2. THE THEORY OF X-RAY SCATTERING

Let us compare the second line of Eq. (2.35) with the second line of Eq. (2.31). Asthe ladder operators a, a† are dimensionless and do not commute, the conversion fromclassical vector potential amplitudes to quantum-mechanical operators is implementedby promoting

αk,λ −→√

~2ε0V ωk

ak,λ, α∗k,λ −→√

~2ε0V ωk

a†k,λ. (2.36)

The quantum mechanical expression for the vector potential is therefore:

A(r) =∑k,λ

√~

2ε0V ωk

[ak,λεk,λe

ik·r + a†k,λε∗k,λe

−ik·r]. (2.37)

2.3 Hamiltonian for a molecule-field systemWe now proceed to analyze the interaction of a piece of matter, e.g. a molecule, withradiation. We assume that the photon energy is much smaller than the electron restenergy (511 keV) so that relativistic QED effects (pair production) are neglected. Thestandard Hamiltonian approach is suitable since we neglect relativistic effects and donot seek a covariant theory.

The molecular Hamiltonian is

Hmol = TN + VNN + Hel, (2.38)

whereTN = −1

2

∑n

1

Mn

∇2n (2.39)

is the nuclear kinetic energy and

VNN =1

2

e2

4πε0

∑n 6=n′

ZnZn′

|Rn −Rn′|(2.40)

describes the nucleus-nucleus repulsion. We denote position, mass and charge of then-th nucleus with Rn,Mn and Zn, respectively. The electronic Hamiltonian

Hel =

∫d3r ψ†(r)

(− ~2

2m∇2 − e2

4πε0

∑n

Zn|r−Rn|

)ψ(r)

+1

2

e2

4πε0

∫d3r

∫d3r′ ψ†(r)ψ†(r′)

1

|r− r′|ψ(r′)ψ(r)

(2.41)

includes the electron kinetic energy, the electron-nucleus attraction, and the electron-electron repulsion. It is here written using the formalism of second quantization [32] interms of the electron field operators ψ(r), ψ†(r). The interaction Hamiltonian betweenthe photon and the electron fields follows from the principle of minimal coupling (seeAppendix A) and the Coulomb gauge (∇ ·A = 0):

Hint = −ie~m

∫d3r ψ†(r)

[A(r) · ∇

]ψ(r) +

e2

2m

∫d3r ψ†(r) A2(r) ψ(r), (2.42)

2.3. HAMILTONIAN FOR A MOLECULE-FIELD SYSTEM 11

where A(r) was obtained in Eq. (2.37).Adding the Hamiltonian for the electromagnetic field, Eq. (2.35), we obtain the

total Hamiltonian of the interacting molecule-field system:

H = Hmol + HEM + Hint. (2.43)

2.3.1 Perturbation theory

In the present Section we employ standard time-dependent perturbation theory andnonrelativistic quantum electrodynamics to understand X-ray scattering. For our pur-poses, we can neglect the nuclear motion, since it is much slower and contributes veryweakly to the scattering signal because of the large mass of the nuclei compared toelectrons. Hence, the total Hamiltonian is rewritten as

H = H0 + Hint, (2.44)

whereH0 = Hel + HEM (2.45)

is the unperturbed system and Hint is a small perturbation. Let us assume that theatom or molecule is in the electronic many-body ground state

∣∣ΨN0

⟩, where N is the

number of electrons. The associated ground state energy is EN0 . The electromagnetic

field is in the Fock state |Nph〉, which contains Nph photons in the mode (ki, λi) andzero photons in all other modes, i.e.

a†k,λak,λ |Nph〉 = δk,kiδλ,λiNph |Nph〉 . (2.46)

The initial state of the total system is hence

|i〉 =∣∣ΨN

0

⟩|Nph〉 , (2.47)

with energyEi = EN

0 + ~ωiNph, (2.48)

where ωi = c|ki|. The zero-point energy∑

k,λ ωk/2 is dropped as a standard practice(renormalization of the vacuum energy). We now turn to the interaction picture [33, 34]and write the generic state vector of the system as

|Ψ, t〉int = ei~ H0t |Ψ, t〉 , (2.49)

which satisfies the equation of motion

i~∂

∂t|Ψ, t〉int = e

i~ H0tHinte

− ε~ |t|e−i~ H0t |Ψ, t〉int . (2.50)

The infinitesimal energy ε > 0 ensures that the perturbation disappears long before(t→ −∞) and long after (t→ +∞) the collision between the photons and the molecule(adiabatic switching of the interaction). Together with the initial condition

limt→−∞

|Ψ, t〉int = |i〉 (2.51)

12 CHAPTER 2. THE THEORY OF X-RAY SCATTERING

the integration of Eq. (2.50) leads to the well-known Dyson expansion [33]

|Ψ, t〉int =|i〉 − i

~

∫ t

−∞dt′e

i~ H0t′Hinte

− ε~ |t′|e−

i~ H0t′ |i〉

− 1

~2

∫ t

−∞dt′e

i~ H0t′Hinte

− ε~ |t′|e−

i~ H0t′

∫ t′

−∞dt′′e

i~ H0t′′Hinte

− ε~ |t′′|e−

i~ H0t′′ |i〉

+ · · ·(2.52)

Let us derive the probability amplitude of a transition from an initial state |i〉 to afinal state |f〉 long after the interaction with the field; we can assume that |f〉 is aneigenstate of H0 with energy Ef . In the following we take the ε→ 0 limit:

limε→0

1

~

∫ ∞−∞

dt ei~ (Ef−Ei)t− ε~ |t| = 2πδ(Ef − Ei). (2.53)

We obtain a transition amplitude:

si→f = limε→0

limt→∞〈f |Ψ, t〉int

=− limε→0

[i

~

∫ ∞−∞

dt′ ei~ (Ef−Ei)t′− ε~ |t

′|〈f |Hint|i〉

− 1

~2

∫ ∞−∞

dt′∑m

ei~ (Ef−Em)t′− ε

2~ |t′|〈f |Hint|m〉

∫ t′

−∞dt′′

∑m

ei~ (Em−Ei)t′′− ε

2~ |t′′|〈m|Hint|i〉

+ · · · ]=− 2πi〈f |Hint|i〉δ(Ef − Ei)

− limε→0

1

~

∫ ∞−∞

dt∑m

ei~ (Ef−Em)t− ε~ |t|

〈f |Hint|m〉〈m|Hint|i〉i(Ei − Em − iε)

+ · · ·

= −2πiδ(Ef − Ei)

(〈f |Hint|i〉+

1

~∑m

〈f |Hint|m〉〈m|Hint|i〉Ei − Em + iε

+ · · ·

).

(2.54)

By squaring and expressing one of the deltas in terms of the duration τ of the collision[35]

[δ(Ef − Ei)]2 =1

~2δ(ωf − ωi)

∫ τ/2

−τ/2

dt2πei(ωf−ωi)t =

1

~δ(Ef − Ei)

τ

2π, (2.55)

we obtain the transition rate (probability of transition per unit time):

Γi→f =|si→f |2

τ

=2π

~δ(Ef − Ei)

∣∣∣∣∣〈f |Hint|i〉+1

~∑m

〈f |Hint|m〉〈m|Hint|i〉Ei − Em + iε

+ · · ·

∣∣∣∣∣2

.

(2.56)

2.4. X-RAY SCATTERING 13

By truncating Eq. (2.56) to the leading term in the square modulus we recover Fermi’sGolden Rule. In general, at a given Ei, many processes can take place, provided thatthey satisfy energy conservation and have nonzero transition matrix element. Themeasured signal for the desired process is proportional to

∑f Γi→f , where the sum

runs over all possible final states. In the case of a continuum of final states the sum isoften replaced by an integral over a density of states.

2.4 X-ray scattering

2.4.1 Scattering cross section

Consider now the expression of Eq. (2.56) for elastic (coherent) X-ray scattering froma system of electrons, in order to derive the differential scattering cross section. Anelastic scattering event leads to a final state of the type

|f〉 =∣∣ΨN

0

⟩a†kf ,λf |Nph − 1〉, (2.57)

with |kf | = |ki|. In the final state the total electromagnetic field energy is the sameas in the initial state. Due to energy conservation required by Eq. (2.56), also theenergy of the material system must remain the same: the summation over the finalstates involves only those which are exactly degenerate with the initial state

∣∣ΨN0

⟩.

For simplicity, we assume a non-degenerate ground state. To first order in Hint thetransition from |i〉 [Eq. (2.47)] to |f〉 reads:

Γi→f =2π

~δ(Ef − Ei)|〈f |Hint|i〉|2

=2π

~δ(Ef − Ei)

(e2

2m

)2 ∣∣⟨ΨN0

∣∣ 〈Nph − 1|akf ,λf

×∫

d3r ψ†(r) A2(r) ψ(r)∣∣ΨN

0

⟩|Nph〉

∣∣∣∣2=

2π

~δ(Ef − Ei)

(~2

4V 2ωiωfε20

)(e2

2m

)2

|ε∗kf ,λf · εki,λi |2

×∣∣∣⟨Nph − 1

∣∣∣akf ,λf (aki,λi a†kf ,λf + a†kf ,λf aki,λi

)∣∣∣Nph

⟩∣∣∣2︸ ︷︷ ︸=|2√Nph|2

×∣∣∣∣∫ d3r

⟨ΨN

0

∣∣∣ψ†(r)ei(ki−kf )·rψ(r)∣∣∣ΨN

0

⟩∣∣∣∣2=

(2π)3

V

(e2

4πε0m

)2Nph

V

1

ωfωi|ε∗kf ,λf · εki,λi |

2∣∣f 0(ki − kf)

∣∣2 δ(ωf − ωi).

(2.58)

In Eq. (2.58), we considered the A2 operator but not the p · A operator in theinteraction Hamiltonian, since the latter does not conserve the total number of photons(to first order in Hint). The second order term in Eq. (2.56) becomes relevant in thecase of resonant scattering, i.e. when Em − Ei matches the photon energy for some

14 CHAPTER 2. THE THEORY OF X-RAY SCATTERING

state |m〉. In this case we have contributions to the transition amplitude from boththe A2 and the p · A term. For our purposes, the second-order matrix elements tendto be negligible, and we can stop at the leading order.

In Eq. (2.58) we introduced the atomic form factor for elastic scattering:

f 0(q) =

∫d3rρ(r)e−iq·r, (2.59)

whereρ(r) = 〈ΨN

0 |ψ†(r)ψ(r)|ΨN0 〉 (2.60)

is the ground state electron number density and q = kf−ki is the momentum transfer.Defining the X-ray photon flux as the number of incident photons per unit time andunit area

Φ =Nph

τ · A=cNph

V(2.61)

we define the (total) scattering cross section:

σ =1

Φ

∑kf ,λf

Γi→f . (2.62)

To obtain the differential cross section, it is sufficient to restrict the summation in Eq.(2.62) to an infinitesimal solid angle dΩ for kf , assuming that the polarization of thescattered photon remains undetected:

dσ =1

Φ

∑kf∈dΩ,λf

Γi→f

=1

Φ

∑λf

V

(2π)3dΩ

∫ ∞0

dkfk2fΓi→f

=1

Φ

∑λf

V

(2π)3

1

c3dΩ

∫ ∞0

dωfω2fΓi→f .

(2.63)

Substituting Eq. (2.58) into Eq. (2.63) yields

dσ(q)

dΩ= r2

e

∣∣f 0(q)∣∣2∑

λf

|ε∗kf ,λf · εki,λi |2. (2.64)

We denoted with re = e2

4πε0mc2the classical radius of the electron. Recall that for a

single electron |f 0(0)|2 = 1. It is now convenient to introduce the differential crosssection for Thomson scattering :(

dσdΩ

)T

= r2e

∑λf

|ε∗kf ,λf · εki,λi |2. (2.65)

This quantity describes the scattering from one electron with negligible momentumtransfer (q ' 0). In terms of (dσ/dΩ)T , we can rewrite Eq. (2.64) as

dσ(q)

dΩ=

(dσdΩ

)T

∣∣f 0(q)∣∣2 . (2.66)

2.4. X-RAY SCATTERING 15

Figure 2.1: Coordinate system for the evaluation of Eqns. (2.70) - (2.72).

The polarization-dependent part of Eq. (2.65) can be evaluated upon choosing asuitable coordinate system, see Fig. 2.1. The incident and scattered wave vectors canbe taken as

ki =ωic

001

, kf =ωic

cosφf sin θfsinφf sin θf

cos θf

. (2.67)

The photon momentum transfer is therefore

q =2ωic

sinθf2

− cosφf cosθf2

− sinφf cosθf2

sinθf2

. (2.68)

Its module |q| = 2ki sinθf2

ranges from 0 at θf = 0 (forward scattering) to 2|ki| forθf = π (backward scattering). We define the unit polarization vectors

εkf ,1 =

cosφf cos θfsinφf cos θf− sin θf

, εkf ,2 =

− sinφfcosφf

0

, (2.69)

mutually orthogonal and both orthogonal to kf . It follows immediately that for anincident X-ray photon linearly polarized along x we have(

dσdΩ

)pol−x

T

= r2e

(1− cos2 φf sin2 θf

), (2.70)

while for y-polarization the differential cross section is(dσdΩ

)pol−y

T

= r2e

(1− sin2 φf sin2 θf

). (2.71)

For unpolarized incoming X-rays (statistical mixture of polarizations along x and y)we thus obtain (

dσdΩ

)unpol

T

= r2e

(1 + cos2 θf

2

). (2.72)

16 CHAPTER 2. THE THEORY OF X-RAY SCATTERING

Integration of Eq. (2.72) over the whole solid angle yields the total Thomsonscattering cross section for the single electron [29, 36]:

σT =8π

3r2e ' 0.665 b = 6.65× 10−29 m2. (2.73)

The following assumptions have been made in deriving Eqns. (2.59)-(2.65)-(2.66): (a)only one electron is present; (b) there is no multiple scattering of the X-ray photoninside the molecule; (c) the photon-molecule interaction is treated as a weak pertur-bation; (d) scattering is nonresonant.

2.4.2 Atomic form factor

Elastic X-ray scattering experiments provide access only to the modulus of the complex-valued form factor f 0(q), Eq. (2.59). Its phase φ(q) can be retrieved with specificmethods [3, 37]. The form factor determines in this way the atomic structure of asample. Figure 2.2 shows the atomic form factor for the ground state of hydrogen,which can be evaluated analytically and equals

f 0(q) =16

(4 + q2a20)2. (2.74)

Since the electron density distribution is spherically symmetric, the form factor onlydepends on the magnitude of the momentum transfer. When the momentum transferis small with respect to the inverse of the characteristic size of the atom, all parts of theelectron wave distribution scatter in phase, and f 0 equals the number N of electrons(N = 1 for hydrogen). When momentum transfer is increased, spatial resolution isincreased; the limit to spatial resolution depends on the maximum momentum transfer2|ki|. In practice this quantity depends not only on the X-ray wavelength, but alsoon the range of scattering angles that the setup can observe, and on how many X-rayphotons can be detected before the sample undergoes radiation damage. Equations(2.66)-(2.74) allow us to calculate explicitly the total scattering cross section for hy-drogen, without the need for the Thomson small-|q| limit. For a photon energy of10 keV this cross section is significantly smaller than σT [38]:

σH(10 keV) ' 0.042 b ' 0.06 σT , (2.75)

while for optical photons of 5 eV we find σH ' 0.665 b ' σT .

2.4.3 Compton scattering

We now consider briefly nonrelativistic inelastic (incoherent) Compton scattering by afree electron initially at rest [39]. The p · A second-order contribution vanishes, sincethe target electron has zero initial momentum. Only the A2 term produces a nonzerocontribution in the nonrelativistic theory. We denote the initial and final states by

|i〉 = |qi = 0〉|Nph〉, |f〉 = |qf〉a†kf ,λf |Nph − 1〉. (2.76)

2.4. X-RAY SCATTERING 17

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

q[bohr−1

]

f0

Figure 2.2: The atomic form factor for the ground state hydrogen as a functionof the momentum transfer q = |q|. A photon energy of 10 keV is required tosample all q-points in the graph.

Following the same treatment of the previous Section, one obtains the inelastic tran-sition rate:

Γi→f =2π

~δ(Ef − Ei)|〈f |Hint|i〉|2

=(2π)6

V 2

(e2

4πε0m

)2Nph

V

1

ωfωi|ε∗kf ,λf · εki,λi |

2

× δ(~q2

f

2m− ωf − ωi

)δ(3)(qf + kf − ki).

(2.77)

The resulting differential cross section for Compton scattering (in the nonrelativisticapproximation) is [36](

dσCdΩ

)free'(

e2

4πε0mc2

)2(ωfωi

)2∑λf

|ε∗kf ,λf · εki,λi|2. (2.78)

In the elastic limit of vanishing momentum transfer to the electron, Eq. (2.78) matchesEq. (2.65)1. For molecular samples of electron number N (i.e. scattering from boundelectrons), Eq. (2.78) is further corrected with the incoherent scattering functionS(q, N) [38]: (

dσCdΩ

)bound

=

(dσCdΩ

)free

S(q, N). (2.79)

The incoherent scattering function may be expressed in terms of a generalized formfactor which include excited states:

fε(q, N) = 〈ε|N∑j=1

eiq·rj |ΨN0 〉, (2.80)

1The fully relativistic QED description of Compton scattering results in the Klein-Nishina formula[40].

18 CHAPTER 2. THE THEORY OF X-RAY SCATTERING

where ε indicates the energy of the excited state measured from the ground state |ΨN0 〉.

The incoherent scattering function is then

S(q, N) =∑ε>0

|fε(q, N)|2, (2.81)

where the sum means either a sum over the discrete states, excluding the groundstate, or an integral over the continuum states. Equation (2.81) can be rewritten as afunction of the ground state only:

S(q, N) =∑ε

〈ΨN0 |

N∑n=1

e−iq·rn|ε〉〈ε|N∑m=1

eiq·rm|ΨN0 〉 − |〈ΨN

0 |N∑j=1

eiq·rj |ΨN0 〉|2

=N∑n=1

N∑m=1

〈ΨN0 |eiq·(rm−rn)|ΨN

0 〉 − |f 0(q, N)|2.

(2.82)

In the special case of hydrogen the double sum in Eq. (2.82) reduces to unity, hence:

S(q, 1) = 1− |f 0(q)|2. (2.83)

For |q| → 0 the incoherent scattering function vanishes, while it approaches N for|q| → ∞. For lighter atoms, Compton scattering plays a role comparable to elasticscattering: in the extreme case of hydrogen and 10 keV photons, σC ' 0.64 b andCompton scattering is therefore the dominant contribution to the scattering signal[41]. The incoherent scattering function for hydrogen S(q, 1) is plotted in Fig. 2.3.

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

q[bohr−1

]

S(q,1

)

Figure 2.3: The incoherent scattering function for ground state hydrogen as afunction of momentum transfer q [38].

Chapter 3

Radiation from an assigned currentdensity

3.1 Purpose

To investigate the scattered radiation from an atom, illuminated by an ultrafast, ul-traintense X-ray pulse, we employ a Maxwell-Schrödinger calculation, as opposed tostandard perturbation theory. The electromagnetic field is treated classically : this isexpected to be accurate when treating scattering from a stationary sample, but suffersfrom the incorrect analysis of the inelastic Compton component, which constitutes arelevant component for light atoms such as hydrogen (see Section 2.4.3).

We set up a numerical simulation to compute the radiated fields E and H — andhence the radiation intensity — generated by an arbitrary current density distributionJ(r, t) depending on both space and time, by solving Maxwell’s equations (2.1). Thisserves as a needed tool for the simulation of the pulse-atom interaction, since the(time-dependent) current density arising from the charges determines the scatteredradiation.

3.2 FDTD method and Perfectly Matched Layer

To solve Eqns. (2.1) we use a standard and efficient computational technique calledfinite-difference time-domain method (FDTD) [42]. The method is based on the KanyeYee’s idea to apply centered finite-difference operators on staggered grids in space andtime for each electric and magnetic vector field component in Maxwell’s curl equations[43].

3.2.1 The Yee algorithm

Rather than solving for electric field E or magnetic field H alone through the waveequation, the Yee method solves for both fields in time and space through the coupledMaxwell’s curl equations (2.1b)-(2.1d). This approach represents robustly both thedifferential and integral forms of Maxwell’s equations.

We now recall Maxwell’s curl equations in the time domain. Since we deal with asingle atom or a molecule in vacuum, we have no dielectric function nor permeability

19

20 CHAPTER 3. RADIATION FROM AN ASSIGNED CURRENT DENSITY

function. The medium is lossless, so that the conductivity σ vanishes; the only currentdensity is that of the source and it is assumed to be known everywhere at any time:

µ0∂H

∂t= −∇× E, (3.1a)

ε0∂E

∂t= ∇×H− J. (3.1b)

These two vector equations are equivalent to the six equations for the components:

µ0∂Hx

∂t=∂Ey∂z− ∂Ez

∂y, (3.2a)

µ0∂Hy

∂t=∂Ez∂x− ∂Ex

∂z, (3.2b)

µ0∂Hz

∂t=∂Ex∂y− ∂Ey

∂x, (3.2c)

ε0∂Ex∂t

=∂Hz

∂y− ∂Hy

∂z− Jx, (3.2d)

ε0∂Ey∂t

=∂Hx

∂z− ∂Hz

∂x− Jy, (3.2e)

ε0∂Ez∂t

=∂Hy

∂x− ∂Hx

∂y− Jz. (3.2f)

To solve these equations in a volume V we enclose the volume in a box of arbitraryshape and then divide the box into many small parallelepipedic cells. The Yee’s gridis then built by assigning the electric field components at the center of each face ofthe cell, and the magnetic field components at the center of each edge of the cell (Fig.3.1). Then, if one offsets the entire grid by half cell in each direction, we would haveelectric field components on the edges and magnetic field components on the faces.This provides a beautiful representation of an interlinked array of Faraday’s Law andAmpere’s law contours: E components are associated with displacement current fluxthrough H loops, and H components are identified with magnetic flux through Eloops. Yee’s algorithm simultaneously simulates the differential and the integral formof Maxwell’s equations.

3.2.2 Discretized equations

Following Fig. 3.1 we denote a space point in a uniform, parallelepipedic lattice as

(x, y, z) = (i∆x, j∆y, k∆z), (3.3)

where i, j, k are integers and ∆x,∆y,∆z are the lattice spacings in the three directions.Any function u of space and time evaluated at a discrete point in the grid and at adiscrete point in time is written as

u(i∆x, j∆y, k∆z, n∆t) = un(i, j, k), (3.4)

where ∆t is the time step and n is an integer. Yee wrote the discretized form of Eqns.(3.2) by applying centered finite-difference for space and time derivatives. In our casethe first three equations read:

3.2. FDTD METHOD AND PERFECTLY MATCHED LAYER 21

Figure 3.1: The positions of the electric and magnetic field components relativeto a cubic unit cell of the Yee lattice. The two coloured squares highlight theintegral representation.

Hn+ 1

2x

(i, j +

1

2, k +

1

2

)= H

n− 12

x

(i, j +

1

2, k +

1

2

)− ∆t

µ0∆y

[Enz

(i, j + 1, k +

1

2

)− Enz

(i, j, k +

1

2

)]+

∆t

µ0∆z

[Eny

(i, j +

1

2, k + 1

)− Eny

(i, j +

1

2, k

)];

(3.5)

Hn+ 1

2y

(i+

1

2, j, k +

1

2

)= H

n− 12

y

(i+

1

2, j, k +

1

2

)− ∆t

µ0∆z

[Enx

(i+

1

2, j, k + 1

)− Enx

(i+

1

2, j, k

)]+

∆t

µ0∆x

[Enz

(i+ 1, j, k +

1

2

)− Enz

(i, j, k +

1

2

)];

(3.6)

Hn+ 1

2z

(i+

1

2, j +

1

2, k

)= H

n− 12

z

(i+

1

2, j +

1

2, k

)− ∆t

µ0∆x

[Eny

(i+ 1, j +

1

2, k

)− Eny

(i, j +

1

2, k

)]+

∆t

µ0∆y

[Enx

(i+

1

2, j + 1, k

)− Enx

(i+

1

2, j, k

)].

(3.7)

Similarly, we discretize Eqns. (3.2d)-(3.2f). The electric fields are evaluated at half-time step offset with respect to the magnetic fields, according to a leapfrog time march-ing scheme:

22 CHAPTER 3. RADIATION FROM AN ASSIGNED CURRENT DENSITY

En+1x

(i+

1

2, j, k

)= Enx

(i+

1

2, j, k

)+

∆t

ε0∆y

[Hn+ 1

2z

(i+

1

2, j +

1

2, k

)−Hn+ 1

2z

(i+

1

2, j − 1

2, k

)]− ∆t

ε0∆z

[Hn+ 1

2y

(i+

1

2, j, k +

1

2

)−Hn+ 1

2y

(i+

1

2, j, k − 1

2

)]− Jn+ 1

2x

(i+

1

2, j, k

);

(3.8)

En+1y

(i, j +

1

2, k

)= Eny

(i, j +

1

2, k

)+

∆t

ε0∆z

[Hn+ 1

2x

(i, j +

1

2, k +

1

2

)−Hn+ 1

2x

(i, j +

1

2, k − 1

2

)]− ∆t

ε0∆x

[Hn+ 1

2z

(i+

1

2, j +

1

2, k

)−Hn

z

(i− 1

2, j +

1

2, k

)]− Jn+ 1

2y

(i, j +

1

2, k

);

(3.9)

En+1z

(i, j, k +

1

2

)= Enz

(i, j, k +

1

2

)+

∆t

ε0∆x

[Hn+ 1

2y

(i+

1

2, j, k +

1

2

)−Hn+ 1

2y

(i− 1

2, j, k +

1

2

)]− ∆t

ε0∆y

[Hn+ 1

2x

(i, j +

1

2, k +

1

2

)−Hn+ 1

2x

(i, j − 1

2, k +

1

2

)]− Jn+ 1

2z

(i, j, k +

1

2

).

(3.10)

Given the source current distribution J, the initial values for E and H and appropriateboundary conditions one can use the equations above to calculate the fields at thesuccessive time step. It can be shown that the Yee’s discretization is of second-orderin accuracy, and the method is stable if the time step ∆t satisfies

∆t ≤ 1

c

√1

(∆x)2+

1

(∆y)2+

1

(∆z)2

. (3.11)

Equation (3.11) is a form of the Courant-Friedrichs-Lewy stability condition for finite-difference methods1 [44].

3.2.3 Perfectly matched layer

A major challenge when solving unbounded electromagnetic problems is the trunca-tion of the infinite space into a finite computational domain. The truncation can beaccomplished by enclosing the computational domain with an artifical absorbing mate-rial which ideally prevents any radiation to be reflected at the boundaries. A popular

1The CFL condition is actually only a necessary condition, and may not be sufficient for theconvergence of finite-difference methods.

3.3. TRIAL CURRENT DENSITY DISTRIBUTIONS 23

absorber model proposed for FDTD simulations is the perfectly matched layer (PML)[45]. The PML is made of an artificial material having a matched impedance to themedium in the simulation domain, and additionally has non-zero conductivity, causingattenuation of the waves inside it. One way of implementing this material is using theidea of coordinate stretching [46]. The Maxwell’s equations are rewritten in the new(source-free) medium in frequency domain as

∇s × E = iωµ0H, (3.12a)∇s ×H = −iωε0E, (3.12b)∇s · E = 0, (3.12c)∇s ·B = 0. (3.12d)

where the modified nabla operator reads

∇s = x1

sx

∂

∂x+ y

1

sy

∂

∂y+ z

1

sz

∂

∂z. (3.13)

with stretching parameters sx, sy, sz. The effective wave vector in this medium thenbecomes

ks =kxsx

x +kysy

y +kzszz

so that, if one of sx, sy, or sz is complex with a negative imaginary part, the wave isdamped in the x−, y− or z−direction. The stretching parameters are chosen as

sx = 1 + iσxωε0

, sy = 1 + iσyωε0

, sx = 1 + iσzωε0

. (3.14)

where σx, σy and σz represent the conductivity of the artificial medium along the threeaxes. In an isotropic case it is sufficient to put σx = σy = σz. The discretized equationsfor the PML region can be found in their most general form in Ref. [42], Chapter 7.

3.3 Trial current density distributionsAs a benchmark we use a trial current field only in the x-direction, with a sphericallysymmetric intensity and simple-harmonic oscillation, in the form

J1(r, t) = J0 exp

(− r

a0

)sinωt x. (3.15)

J0 is an arbitrary amplitude to ensure correct dimensionality, a0 is the Bohr radius,r =

√x2 + y2 + z2 and ω is the angular frequency of oscillations. Figure 3.2 shows

the current distribution at its maximum intensity.We also simulate a current distribution obtained by multiplying Eq. (3.15) by a

gaussian time-envelope of width τ :

J2(r, t) = J0 exp

(− r

a0

)sinωt exp

[−(t− t0)2

2τ 2

]x. (3.16)

Additionally, we run a simulation for the distribution

J3(r, t) = J0 exp

[−√

(x− x0 sinωt)2 + y2 + z2

a0

]x, (3.17)

which reproduces a current cloud performing harmonic oscillations along the x-direction.

24 CHAPTER 3. RADIATION FROM AN ASSIGNED CURRENT DENSITY

−2 −1 0 1 2−2

−1

0

1

2

y [a0]

z [a0]

J1(r) (A m−2)

2

4

6

8

·1017

Figure 3.2: x = 0 cut of the current density distributions J1, Eq. (3.15) att = π/ω.

3.4 Spatial discretization

After several tests, we decided to adopt for the computational cell a sphere of radiusR = 10a0 (physical domain) surrounded by a perfectly matched layer of thicknessRPML = 5a0. The physical domain is then discretized with tetrahedral elements,while the PML domain mesh consists of a free triangular mesh swept along the radialdirection to meet the surface of the physical domain. Figure 3.3 reports a sketch ofthe mesh.

3.5 Results

3.5.1 Simple-harmonic current

For the current distribution (3.15) we assume J0 = 1018 A m−2, which is comparableto the current density values inside the hydrogen atom, when subject to ultraintenseX-ray pulses (see Chapter 4 and Table C.3). We choose ω = 1.8837× 1018 s−1: thisangular frequency corresponds to a wavelength 2πc/ω = 1 nm. Time stepping iscarried from 0 to 3× 2π/ω ' 10 as with time-step 0.05 as.

Figure 3.4 reports the radiation intensity (projection of the Poynting vector) inte-grated over the surface of the physical domain as a function of time. Figure 3.5 reportsthe instantaneous intensity on the surface at different times. The artifacts are due tothe spatial discretization of the sphere.

3.5. RESULTS 25

3.5.2 Gaussian-modulated current

For the current distribution (3.16) we choose t0 = 10× 2π/ω, τ = 10× 2π/ω and runthe simulation from t = 0 to t = 2τ , with time-step 0.05 as. We report the intensityintegrated over the cell surface, as a function of time (Fig. 3.6).

3.5.3 Oscillation along the x axis

For current distribution J3(r, t), Eq. 3.17, we use x0 = a0 and run the simulation fromt = 0 to t = 3 × 2π/ω ' 10 as, with time-step 0.05 as. We plot again the integratedintensity (Fig. 3.7) as a function of time along with a few snapshots of it at the cellsurface (Fig. 3.8).

26 CHAPTER 3. RADIATION FROM AN ASSIGNED CURRENT DENSITY

Figure 3.3: An example of spatial discretization for the solution of Maxwell’sequations (2.1). The complete mesh consists of 8279 elements, of which 4109belong to the physical domain. The radius of the inner sphere is R = 10a0 '5.29 Å, and the thickness of the PML domain is RPML = 5a0 ' 2.65 Å. Themaximum mesh side length is of 1.06 Å.

0 0.5 1 1.5 2 2.5

0

1

2

3

Number of periods

Pow

er(m

W)

Figure 3.4: The total power radiating outward across the interface between thecomputational cell and the PML domain for current density J1, Eq. (3.15) as afunction of time. The period of oscillations for the current density is T ' 3.336 as,and the intensity oscillates with half this period.

3.5. RESULTS 27

(a) t/T = 0 (b) t/T = 0.15

(c) t/T = 0.30 (d) t/T = 0.45

(e) t/T = 0.60 (f) t/T = 0.75

(g) t/T = 0.90 (h) t/T = 1.05

Figure 3.5: Successive snapshots of the intensity radiated outward at the surfaceof the computational cell of radius R = 10a0, at times expressed as fractions ofthe period of oscillation, for the J1 current density distribution of (3.15). Theperiod is T ' 3.336 as. Values are given in W m−2.

28 CHAPTER 3. RADIATION FROM AN ASSIGNED CURRENT DENSITY

0 2 4 6 8 10 12 14 16 18

0

1

2

3

Number of periods

Pow

er(m

W)

Figure 3.6: The total power radiated on the surface of the computational cell forcurrent density J2 of Eq. (3.16) as a function of time. The period is T ' 3.336 as.The pulse width is τ = 10T and integration is carried out from t = 0 to t = 20T .

0 0.5 1 1.5 2 2.5

3.58

3.59

3.59

Number of periods

Pow

er(m

W)

Figure 3.7: The total power radiated on the surface of the computational cellfor the current density J3 of Eq. (3.17) as a function of time. The period isT ' 3.336 as.

3.5. RESULTS 29

(a) t/T = 0 (b) t/T = 0.12

(c) t/T = 0.24 (d) t/T = 0.36

(e) t/T = 0.48 (f) t/T = 0.60

(g) t/T = 0.72 (h) t/T = 0.84

Figure 3.8: Snapshots of the intensity radiated on the surface of the compu-tational cell, in W m−2, for current density distribution J3 of Eq. (3.17). Theperiod of oscillation is T ' 3.336 as.

Chapter 4

The Schrödinger equation inelectromagnetic fields

In this Chapter we implement the numeric solution of the Time-Dependent SchrödingerEquation (TDSE) for one electron in the field of an electromagnetic pulse. This ap-proach does not employ the dipole approximation, but rather couples the full vector-potential of the electromagnetic field to the electron dynamics. It is therefore adequateto treat X-ray radiation. We recall that the electromagnetic field is treated classically.

4.1 Hamiltonian and Schrödinger equation

We study the semiclassical Hamiltonian of an electron subject to an electromagneticfield in the Coulomb gauge, and to the attraction of the nucleus:

H =1

2m(p + eA)2 − eΦ− e2

4πε0

1

|r|

=p2

2m− e

2m(p ·A + A · p) +

e2

2mA2 − eΦ− e2

4πε0

1

|r|.

(4.1)

We denote with A and Φ the vector and scalar potentials. From the point of view ofthe quantum-mechanical motion of the electron, A(r) and Φ(r) are functions of theposition operator. The TDSE for Hamiltonian (4.1) reads

i~∂ψ

∂t=

[p

2m− e

2m(p ·A + A · p) +

e2

2mA2 − eΦ− e2

4πε0

1

|r|

]ψ. (4.2)

We now recall that in the Coulomb gauge ∇ ·A = 0 and in the absence of charges wecan put Φ = 0. We have

(p + eA)2 ψ = (p + eA) · (p + eA)ψ

= (p + eA) · (−i~∇ψ + eAψ)

= −~2∇2ψ − ie~∇ · (Aψ)− ie~A · (∇ψ) + e2A2ψ

= −~2∇2ψ − ie~(∇ ·A)ψ − ie~A · (∇ψ)− ie~A · (∇ψ) + e2A2ψ

= −~2∇2ψ − 2ie~A · (∇ψ) + e2A2.

(4.3)

31

32 CHAPTER 4. SCHRÖDINGER EQUATION IN A LASER PULSE

The TDSE simplifies to:

i~∂ψ

∂t=

[− ~2

2m∇2 − i~ e

mA · ∇+

e2

2mA2 − e2

4πε0|r|

]ψ, (4.4)

which is to be solved numerically. Crucially A = A(r, t), as opposed to the dipoleapproximation where the vector potential is assumed to be spatially uniform.

4.2 Charge density and current density operatorsTo compute the scattered radiation from the electron through the FDTD methoddescribed in Chapter 3 we need to evaluate the electric current density of the electron,which in the semiclassical picture is the expectation value of the electric current densityoperator on the electron wavefunction. For a chargeless particle, probability densityand number current density operators in the Schrödinger picture are defined as

ρ(r) = δ (r− r) , (4.5)

j(r) =1

2mp, ρ(r) , (4.6)

where p is the momentum operator, r is the position operator and · denotes theanticommutator. It is easy to show that for a given normalized state |ψ〉 the followingrelations hold:

〈ψ|ρ(r)|ψ〉 = |ψ(r)|2 , (4.7)

〈ψ|j(r)|ψ〉 = − i~2m

[ψ∗(r)∇ψ(r)− ψ(r)∇ψ∗(r)] . (4.8)

Consider now a charged particle (electron) in electromagnetic field. The Hamiltonianis given by Eq. (4.1). The number current density operator modifies as follows1:

jem(r) =1

2m[p + eA(r)] , ρ(r)

= j(r) +e

mA(r)ρ(r),

(4.9)

and it automatically satisfies continuity equation in the Heisenberg picture, namely

∇ · jem, H(r, t) +∂

∂tρH(r, t) = 0. (4.10)

The electric current density operator J is simply obtained multiplying jem by the charge−e. The expectation value on the state |ψ〉 is evaluated through Eqns. (4.7)-(4.8):

J(r, t) = −e〈ψ|jem(r)|ψ〉

=ie~2m

[ψ∗(r, t)∇ψ(r, t)− ψ(r, t)∇ψ∗(r, t)]− e2

mA(r, t) |ψ(r, t)|2 .

(4.11)

1A consistent expression of the current density within a quantum description of the electromagneticfield involves the A operator, which acts in the Fock space. The semiclassical expression is formallyobtained by taking the expectation value of A on the Fock state.

4.3. NUMERICAL METHOD 33

4.2.1 Observation on the gauge-invariance of current density

Since J is an observable, the quantity (4.11) is gauge-invariant even if ψ and A arenot, as it is readily verified. Note that the vector potential is in principle the sum ofthe external field Aext and the “feedback” vector potential generated by the movingelectron, Ael. In the calculation of the current J we choose to neglect this lattercontribution, so that the resulting current density is not really gauge-invariant. Ael ishowever orders of magnitude smaller than Aext for sufficiently intense laser fields.

4.3 Numerical methodFor convenience, we adopt atomic units (e = m = ~ = 1) for the numerical solution ofEq. (4.4), which becomes

i∂ψ

∂t=

[−1

2∇2 − iA · ∇+

A2

2− 1

|r|

]ψ. (4.12)

Current density (4.11) transforms similarly to

J =i

2[ψ∗∇ψ − ψ∇ψ∗]−A|ψ|2. (4.13)

The atomic units and their values in SI system are listed in Tables C.1-C.3. Asdiscussed above, we take A = Aext in Eqns. (4.12) and (4.13).

4.3.1 Integration method and spatial discretization

Equation (4.12) is solved through finite-element analysis (FEA), employing backwarddifferentiation formulas (BDFs) for time-stepping [47] and software mumps for solutionof large sparse systems of linear algebraic equations [48].

The computational cell is a sphere of radiusR = 30a0, discretized with a tetrahedralmesh at two different fineness levels: an inner sphere of radius R0 = 3a0 with a finermesh and an outer shell of thickness R − R0 with a coarser mesh (Fig. 4.4). Thereason for this approach is that most of the wavefunction remains localized within aradius R0 from the origin, even in the presence of the field, and is here subject to itsmost significant modifications, while it is much smaller outside this range. The finerinner mesh accounts for these important effects. The total radius of the cell R allowsus to support even higher-excited eigenstates of the hydrogen atom, which are moreextended spatially.

4.4 Results

4.4.1 Field-free hydrogen atom

We first run a simulation for the electron subject only to the Coulomb potential, withnull vector potential, as a preliminary test. The equation solved is

i∂ψ

∂t= −1

2∇2ψ − 1

|r|ψ, (4.14)

34 CHAPTER 4. SCHRÖDINGER EQUATION IN A LASER PULSE

Figure 4.1: Tetrahedral mesh used for solving Eq. (4.12). The mesh totals20301 elements, with maximum side length of 0.42a0 (inner mesh) and 1.2a0(outer mesh).

with the boundary condition ψ = 0 on the sphere surface. The initial condition

ψ(r, 0) =1√πe−|r| (4.15)

is the ground-state wavefunction of hydrogen, normalized to the entire space. Thisintroduces an error of ' 10−4 on the actual normalization in our spherical box. How-ever, the integration of Eq. (4.14) is not dramatically affected by this, since the initialvalue of the wavefunction at the boundary of the simulation sphere is quite small.

We run the simulation for a total time T = 4π a.u. (period corresponding to theground-state energy of hydrogen) with the BDF time-stepping method. The time stepsize is adaptive. To monitor convergence of the numerical algorithm we evaluate thewavefunction norm over as a function of time. We can improve its conservation bytuning the initial time-step size ∆t0, as shown in Fig. 4.2, top panel. We find theoptimal trend for ∆t0 = 0.1 a.u. Additionally, we select a set of five points within 2a0

from the origin and evaluate the standard deviation σφ of the wavefunction phase, asa function of time. As we are evolving a stationary state, this deviation should vanish,and indeed it remains quite small, see Fig. 4.2, bottom panel.

Figure 4.3 compares the real and imaginary parts of ψ in the x = 0 plane at theinitial time t = 0 and after one full period, at t = 4π.

The code generates a drop in the wavefunction norm during the first few time steps,which is independent on the initial time step size. After t = 4π, the imaginary part,which should be zero everywhere, is of order 10−2 at a radius of ' a0 from the origin.This indicates a slight error in the integration of Eq. (4.14) which is also responsiblefor a non-zero residual current even in the ground-state, of order |J| ' 10−4 a.u.

4.4. RESULTS 35

0.9

0.92

0.94

0.96

0.98

∫ |ψ|2

∆t0 = 0.001∆t0 = 0.01∆t0 = 0.05∆t0 = 0.1∆t0 = 0.2

0 2 4 6 8 10 120

2

4

6·10−3

Time (a.u.)

σφ

[10−

3ra

d]

Figure 4.2: Top panel: numerical time evolution of the wavefunction norm, forthe field-free ground-state, for a few values of the initial time step ∆t0, keeping themaximum time step allowed to the adaptive algorithm to 0.2 a.u. Bottom panel:standard deviation of field-free ground-state wavefunction phase calculated overfive points near the origin, as a function of time.

36 CHAPTER 4. SCHRÖDINGER EQUATION IN A LASER PULSE

−2 0 2

−2

0

2

y

z

t = 0

(a)

−2 0 2y

t = 4π

0.1

0.2

0.3

0.4

0.5

(b)

−2 0 2

−2

0

2

y

z

t = 0

(c)

−2 0 2y

t = 4π

1

2

3

4

·10−2

(d)

Figure 4.3: yz-view of ground-state wavefunction at t = 0 and t = 4π. (a)-(b):real part; (c)-(d): imaginary part. After one period of oscillation we observe anon-negligible imaginary part, which is responsible for a noise current of order10−4 a.u. This behaviour is irrespective of the time step size.

4.4. RESULTS 37

4.4.2 Interaction with the field

We now turn on the electromagnetic field. We consider the vector potential for asine-square laser pulse linearly polarized along the x-axis and propagating along thez-axis:

A(r, t) =E0

ωsin2

[ πωτ

(kz − ωt)]

sin(kz − ωt) x, (4.16)

We choose a sine-squared pulse rather than a gaussian pulse, since the former vanishesexactly at fixed time instants and is hence more stable numerically, especially at thehigher intensities. For a pulse intensity of 2 a.u. (maximum value achieved with aXFEL) at a photon energy of 365 a.u. (' 10 keV) the order of magnitude of A isabout 10−3 a.u. Hence, the A2 potential term in Eq. (4.12) is of order ' 10−6 a.u.It is likely that the vector-potential term becomes comparable to the Coulomb termonly for extremely high intensities (' 105 a.u.).

We run simulations for pulses containing 10 periods of field oscillation (correspond-ing to τ = 10 × 2π/ω ' 4.164 as). The intensity of the incident pulse I is graduallyincreased from 50 to 109 a.u.

4.4.3 Probability density and current density distribution

We observe that for intensities up to I ' 107 a.u. the probability density of the electronis indistinguishable from the ground state throughout the simulation, in agreementwith the perturbative behavior. By further cranking up the intensity (up to I ' 109

a.u.), the electron cloud starts to undergo substantial modifications. For intensitieshigher than 109 the numerical integration of the equation fails to converge.

We compute the time-dependent current density distribution using Eq. (4.13).This current density is affected by the intrinsic noise deriving from the solution of theSchrödinger equation in the stationary case (see previous Section).

4.4.4 Radiation intensity and total scattering cross section

Combining the Schrödinger simulation with the solution of Maxwell’s equations (seeChapter 3) we obtain the electromagnetic field generated by the electron cloud underthe influence of the incoming pulse. The Poynting vector is calculated, and we integrateits flux across spheres centered in the origin, of different radii. We divide this valueby the intensity I of the incoming pulse, to obtain the total cross section of scattering.We observe that the total cross section agress with the perturbative value to withina few percent, and this is mainly due to the non-monochromaticity of the pulse. Forintensities higher than 106 a.u. we report a sensitive increase in the scattering crosssection, witnessing the onset of nonlinearity.

103 104 105 106 107 108

42

43

44

45

46

Onset of nonlinearity

Intensity (a.u.)

Total

crosssection(m

b) Our model

Linear response

Figure 4.4: Total scattering cross section of the hydrogen atom, as a function ofthe pulse intensity. Dashed (green) line: perturbative result, equal to 41.705 mbat 10 keV. Solid (blue) line: calculations with the Maxwell-Schrödinger approachfor several values of pulse intensity. The breakdown of linear response is observedfor intensities higher than 106 a.u.

Chapter 5

Conclusions and outlook

In this thesis we address the scattering of ultrafast and ultraintense X-ray pulses froma one-electron atom with a Maxwell-Schrödinger model. We solve the time-dependentSchrödinger equation for the electron in the electromagnetic field of a laser pulse byusing finite-element analysis, in order to obtain the time-dependent current densitydistribution. This current density is responsible for the scattered radiation, which wecompute through the FDTD method on the Yee grid [42]. The scattered radiation al-lowes us to obtain the scattering cross section (both total and angle-resolved) from theelectron, as a function of the pulse intensity. Our model addresses light scattering froma single electron semiclassically, thus neglecting the Compton scattering contribution.The latter is actually dominant for a few-electron atom at hard X-ray energies.

The present model provides the basis for a consistent treatment of ultrafast scat-tering from a nano-object beyond perturbation theory. As future development steps,it will need to be extended to a many-body approach (e.g. time-dependent density-functional theory) for many-electron atoms and molecules, which are the objects ofexperimental interest, and where Compton scattering is comparably smaller.

A further extension involves the investigation of the effect of the radiated poweron the radiating electron itself, amounting to release the approximation A ' Aext inEqns. (4.12) and (4.13).

The numerical scheme should be further improved, especially for the integration ofthe Schrödinger equation. The quantity of interest to be addressed is the differentialscattering cross section, which can be easily calculated from our simulations. Themodel can be extended to handle longer pulses, and to evaluate the ionization rateof the atom as well. This can be accomplished by implementing absorbing boundarycondition in the solution of the Schrödinger equation, without the need to enlargearbitrarily the computational cell, which would eventually be numerically unfeasible.

39

Appendix A

Minimal coupling Hamiltonian

Recall the classical Lagrange equations of motion for a particle [49]:

ddt

(∂L∂ri

)− ∂L∂ri

= 0, (A.1)

where i = 1, 2, 3 denotes the cartesian component of the position r. We want to expressa suitable Lagrangian function L that generates the equations of motion for a particleof mass m, charge q and velocity v = r in an electromagnetic field. The particleexperiences the Lorentz force

F = q (E + r×B) = q

[−∇Φ− ∂A

∂t+ r× (∇×A)

], (A.2)

where Φ and A are the electromagnetic potentials. Such a Lagrangian is

L =1

2mr2 − qΦ + qr ·A (A.3)

as is shown directly by applying Lagrange equations (A.1). The canonical conjugated

momenta are defined as pi =∂L∂ri

[49]. As a result, the momentum vector is

p = mr + qA. (A.4)

We introduce the standard definition for the Hamiltonian:

H = p · r− L, (A.5)

that we express in terms of p and functions of the position r:

H =1

2m(p− qA)2 + qΦ. (A.6)

This expression is the Hamiltonian function for a classical charged particle, with theminimal coupling to the electromagnetic field.

Equation (A.6) is readily generalized for a system of N charges:

H =N∑j=1

1

2mj

(pj − qjA)2 +N∑j=1

qjΦ + V (r1, . . . , rN), (A.7)

where V describes all non-electromagnetic conservative interactions among the parti-cles and with the surrounding environment. Hamiltonian (A.7) is adopted in its exactform to describe the quantum mechanics of microscopical particles.

41

Appendix B

The Keldysh parameter at currentradiation sources

This Appendix introduces a dimensionless parameter which allows us to compare thecharacteristic times of radiation and of matter subject to the same radiation. We thencompare these characteristic quantities in present and future radiation sources in theUV to X-ray region.

B.1 Radiation beam parameters

Compared to the ideal plane wave field of Eq. (2.18), real life radiation sources sufferfrom intrinsic finite bandwidth, beam divergence and imperfect spatial coherence. Atarget of free-electron lasers is to improve these quantities, together with brilliance,peak power and number of photons per pulse. In recent years, a number of free-electron laser facilities started to come online: flash at desy in Hamburg (Germany,2005) and fermi at the elettra synchrotron in Trieste (Italy, 2010), in the UV-XUVphoton wavelength regime, and lcls in Stanford, California, in the hard X-ray regime(2012). The European-xfel is a free-electron laser research facility currently (2016)under construction in the Hamburg area, in Germany. It should be up and runningin 2017 and will allow the study of ultrafast phenomena (in the time frame of fewfemtoseconds) through the generation of ultrashort, relatively coherent, ultraintenseX-ray pulses [9]. These phenomena include previously hard-to-investigate processessuch as photosynthesis, ultrafast magnetization in solids, the formation of molecules,electron-hole excitations. We briefly report here the design parameters of European-xfel, together with the current data for fermi and flash (Table B.1).

The design peak spectral brilliance for e-xfel is

B = 5× 1033 photonss mm2 mrad2 0.1% bandwidth

, (B.1)

where “0.1% bandwidth” specifies that the number of photons emitted falls in thefrequency range ∆ω

ω= 0.1% around the central frequency. The brilliance is a standard

quantity, characterizing radiation sources. Bandwidth from synchrotron radiation israrely narrower than 0.1%, therefore this unit definition is appropriate. Consider ae-xfel pulse with photon energy ~ω = 12 keV; using design parameters (brilliance

43

44 APPENDIX B. THE KELDYSH PARAMETER

and divergence, see Table B.1) the expected peak radiance is

I ' 5× 1033 photons s−1 × 12 keV × 10−12 sr1

10−6 m2 × 10−6 sr' 1019 W m−2. (B.2)

fermi flash e-xfel

wavelength (nm) 20-65 4-45 0.05-5photon energy (eV) 19-62 28-295 0.25− 25 (×103)pulse length (fs) 30-100 30-300 50-100peak power (GW) 1-5 1-5 72photons per pulse 5× 1014 1013 1.2× 1012

bandwidth (%) 0.15 0.1-1 0.1beam size (µm) 290 100 110beam divergence (µrad) 50 90 0.8peak brilliance (∗) 6× 1029 1031 5× 1033

pulse energy (mJ) 0.7 0.5 1-3

Table B.1: XFEL vs. FERMI. Data from elettra.trieste.it/FERMI, and xfel.desy.de.

(∗) Expressed as number of photons s−1 mm−2 mrad−2 (0.1% bandwidth)−1.

Repeating the calculation for fermi (~ω = 40 eV) we obtain:

I ' 6× 1029 photons s−1 × 40 eV × 2.5× 10−9 sr1

10−6 m2 × 10−6 sr' 1016 W m−2. (B.3)

These estimates are obtained using formulas related to a continuous beam, but theyexpress the peak power of a single XFEL pulse. A pulse of 1012 photons at 12 keVspread over an area of 104 µm2 with duration 100 fs yields roughly the same intensityas Eq. (B.2); a similar result holds for fermi.

B.2 The Keldysh parameterThe Keldysh adiabaticity parameter [23] characterizes the dominant ionization processfor an atom or molecule in an oscillating electromagnetic field. It is defined as the ratiobetween the mean tunneling time τ , obtained below, and one half of the the field periodT/2. When τ decreases below T/2 the tunneling rate becomes dependent on the fieldfrequency. When τ T/2 non-linear processes such as multiphoton ionization (MPI),above threshold ionization (ATI) or above barrier ionization (ABI) can dominate theradiation-matter interaction, but the tunneling ionization rate can be safely neglected.In the following section we express explicitly the Keldysh parameter and evaluate itfor fermi/flash and xfel pulses.

B.2.1 Derivation

Consider a single bound electron in an atom, subject to an external oscillating electricfield with peak amplitude E0. Let the electron ionization potential be Ip, due to

B.2. THE KELDYSH PARAMETER 45

`

Ip

r

V (r)

Figure B.1: Sketch of the instantaneous potential energy (solid line) for anelectron in electromagnetic field, due to the sum of field potential (dashed line)and Coulomb potential (dash-dotted line). The dotted horizontal line correspondsto the binding energy of the electron.

nuclear Coulomb attraction, possibly screened by the other electrons in the case of amany-electron atom. The instantaneous resulting potential energy is sketched in Fig.B.1. We can define the tunneling length as

` =IpeE0

, (B.4)

that is, the distance from the origin at which the energy gained under the action ofthe electric force leads to a potential energy lowering exceeding Ip. In the classicallyforbidden region (between 0 and `) we can estimate the mean velocity with the aid ofWKB approximation [50] as

〈v〉 =

√2Ipm. (B.5)

The mean tunneling time amounts approximately to

τ =`

〈v〉=

1

eE0

√mIp

2. (B.6)

The tunneling is possible only if τ < T/2, where T = 2π/ω is the duration of afield cycle. We can express the Keldysh parameter in terms of the radiation angularfrequency ω and of a tunneling “angular frequency” ωτ = 2π/τ :

γ =τT2

=2ω

ωτ=ω√

2Ipm

eE0

. (B.7)

The physical meaning of this ratio is straightforward: a small γ tells us how fast is theelectric potential oscillating compared to the mean tunneling time. The latter growswith the radiation amplitude. As a result, tunneling ionization dominates for small γ,resulting from low frequency and high field intensity.

46 APPENDIX B. THE KELDYSH PARAMETER

B.2.2 At European-xfel

Consider a FEL pulse with photon energy ~ω = 12 keV (mid-range value at European-xfel, see Table B.1), corresponding to ω = 2× 1019 s−1 (hard X-rays). FollowingEquation (B.7), the condition γ 1 (perturbative limit) is fulfilled for

E0 ω

e

√2mIp =⇒ E0 2× 1014 V m−1, (B.8)

where we used Ip ' 10 eV (outer-shell electrons). For core electrons one can haveIp up to 105 eV, yielding E0 1016 V m−1. Recovering basic electrodynamics (seeSection 2.1) the corresponding condition for the radiation intensity is

I 1025 W m−2 (valence electrons), (B.9)I 1029 W m−2 (core electrons). (B.10)

Comparing these values with those of the previous Section, we can see that the XFELpulse satisfies γ 1.

B.2.3 At fermi and flash

We can perform the same estimate on the fermi free-electron laser in Trieste, where~ω = 40 eV (UV region): the conditions for γ 1 are:

I 1020 W m−2 (valence electrons), (B.11)I 1024 W m−2 (core electrons), (B.12)

again within the working parameters of fermi. However, for UV-XUV free-electronlasers (fermi and flash) a strong focusing of the pulse can be achieved by means ofoptics, increasing intensity by several orders of magnitude: an intensity of 1020 W m−2

with 93-eV photons has been recently reached at flash, resulting in γ ' 1 [51].

B.2.4 Limitations

The Keldysh parameter indicates that tunneling ionization should be of negligibleimportance at XFEL intensities. However, this indication has some limitations: (a) itis relative to a single electron, treated semiclassically. Therefore, it does not accountfor collective many-body effects that can favour tunneling ionization in many-electronsystems; (b) it is not guaranteed to be scalable to the X-ray energies and intensities,since it is used mainly for infrared, optical and near-UV applications; (c) at microwave-infrared frequencies, the electron cannot usually be excited to higher states, while withoptical and UV photons the electron can pass through several different states andmodify the wavefunction in a sensitive manner [52]. For X-ray photons this last effectis again negligible, at least if we do not consider core electrons.

B.3 Relativistic effectsIn this Section we address relativistic effects within our treatment. The relativisticLorentz force for an electron in a field is

dpdt

= −e (E + v ×B) (B.13)

B.3. RELATIVISTIC EFFECTS 47

where p = mγv (relativistic momentum). Since the magnetic field amplitude is oforder |E|/c, the magnetic component of the force is comparable to the electric oneonly if v/c for the electron is not negligible. The mean square velocity for an electronin a uniform electric field can be calculated as follows. Consider an electron interactingwith an electric field linearly polarized along x, of the form E(t) = E0e

−iωtx. The forceexperienced by the electron is

F = −eE0e−iωtx = mr. (B.14)

The solution for r(t) is readily found to be

r(t) =eE0

mω2e−iωtx =⇒ v(t) = −ieE0

mωe−iωtx (B.15)

Squaring the resulting velocity and averaging over a period 2π/ω we obtain:

〈|v(t)|2〉 =

(eE0

mω

)2

(B.16)

The ratio v/c is of order ' 10−6 for ~ω = 12 keV and E0 = 1014 V m−1, the valuesappropriate for XFEL pulses, and is at most of order ' 10−2 for fermi and flashsituations, indicating that relativistic effects are certainly negligible for X-rays.

Appendix C

Atomic units

We collect here the fundamental atomic units (Table C.1) and express a number ofphysical constant in this system of units (Table C.2). We adopt this system for thecomputer evaluation of the quantum mechanics of the electron. The main derivedunits, and those useful for this thesis are collected in Table C.3. All quantities arerelated to the international system of units (SI), which we adopt through this work.We recall the value of the dimensionless fine structure constant :

α =e2

4πε0~c= 0.007 297 352 57. (C.1)

Dimension Name Symbol Value (SI)

mass electron mass m 9.109 382× 10−31 kgcharge elementary charge e 1.602 176× 10−19 Caction reduced Planck’s constant ~ 1.054 571× 10−34 J s

Coulomb force constant 14πε0

8.987 551× 109 kg m3 s−2 C−2

Table C.1: The fundamental physical quantities that are taken as unity in the atomic system ofunits. The SI value is given by comparison.

Constant Symbol Value (a.u.) Value (SI)

speed of light c 137.036 299 792 458 m s−1

vacuum permittivity ε0 0.079 577 8.854 188× 10−12 F m−1

vacuum permeability µ0 6.691 762× 10−4 4π × 10−7 V s A−1 m−1

classical electron radius re 5.325 135× 10−5 2.817 940× 10−15 mBohr magneton µB 0.5 9.274 009× 10−24 J T−1

Table C.2: A few physical constants expressed in atomic units, and in the SI.

49

50 APPENDIX C. ATOMIC UNITS

Dimension Name Symbol Expression Value (SI)

length Bohrradius

a04πε0~2me2

5.291 771× 10−11 m

energy Hartreeenergy

EHae2

4πε0a04.459 744× 10−18 J

time t0~

EHa2.418 884× 10−17 s

frequency t0−1 4.134 138× 1016 s−1

angular frequency 2πt0

2.597 555× 1017 s−1

power EHat0

0.180 237 8 W

electric current density et0a20

2.365 337× 1018 A m−2

electric field EHaea0

5.142 207× 1011 V m−1

electric potential EHae 27.211 385 V

magnetic induction ~ea20

2.350 517× 105 T

vector potential ~ea0

1.243 840× 10−5 V s m−1

energy flux EHat0a20

6.436 409× 1019 W m−2

magnetic permeability ma0e2

1.8779× 10−3 V s A−1 m−1

Table C.3: Derived atomic units for the physical quantities appearing in the present work, withtheir SI value.

Acknowledgements

First and foremost, I would like to express my deepest gratitude to my advisors, Prof.Dr. Nicola Manini and Prof. Dr. Giovanni Onida, for giving me the opportunityof working on this thesis and opening me to the field of ultrafast radiation-matterinteraction. Their guidance and their patience have been fundamental for this work.They infused me with their enthusiasm and their scientific curiosity. They opened meseveral new perspectives, taught me how to do research, and encouraged me to takeup my academic career by starting a Doctoral training. I also want to thank themexplicitly for inviting me to attend the noxss Workshop in Erice, Sicily (5-10 June,2016) where I met several scientists working in the field of ultrafast phenomena andhad the chance to deepen my knowledge in a stimulating environment.

I am very grateful to my colleagues at the University of Milan Nicolas Trojani,Tommaso Seresini, Stefano Verrastro, Riccardo Porotti and Stefano Minuti for themoments of discouragement and optimism that we shared, and for the feedback theyprovided on the manuscript.

I am honored to acknowledge Prof. Dr. Robin Santra, head of the Theory Divisionat the Center for Free Electron Laser in Hamburg, for the helpful discussions duringthe noxss Workshop and his valuable suggestions on our research.

I would like to thank all the Professors, researchers and Ph.D. students worldwidethat I contacted for clarifications and doubts, and who replied always with kindnessand generosity. I care to mention among them Prof. G. Dixit, Prof. A. Fratalocchi,Prof. J.M. McMahon.

I thank the people of the Finite Systems Division of Prof. Rost at the Max-Planck-Institut für Phisik Complexer Systeme in Dresden, and the Quantum Transport Groupof Prof. Belzig at Universität Konstanz for hosting me, enabling me to present myresearch activity, offering me their suggestions and for being the starting point of myfuture career.

I thank my sister Monia, and my friends Nata, Alex, Alice, Vekki, Emilio, Federica,Gaia, Gigio, Jio, Giuditta, Giulia. The fun moments with them during all these yearsgave me energy and will to move on and fulfill my goals.

I thank my former Classical Guitar teacher, Mo Guido Fichtner, for his pricelesslessons about life and about Music, for his patience and generosity, and for havingshowed me the right path to follow.

The major debt of gratitude goes to my dearest Mai. Her unconditional, endlesslove and support, her constant stimulus and the meticulous proofreading have beenessential to the success of this thesis. I dedicate this goal to her, hoping she will alwaysstand by my side with her wisdom and her beautiful soul.

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