Magnetism and Accelerator-Based Light Sources.
Proceedings of the 7th International School
“Synchrotron Radiation and Magnetism”, Mittelwihr
(France), 2018

Hervé Bulou, Loïc Joly, Jean-Michel Mariot, Fabrice Scheurer

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Hervé Bulou, Loïc Joly, Jean-Michel Mariot, Fabrice Scheurer. Magnetism and Accelerator-Based Light Sources. Proceedings of the 7th International School “Synchrotron Radiation and Magnetism”, Mittelwihr (France), 2018. Springer International Publishing, 262, 2021, Springer Proceedings in Physics, 978-3-030-64622-6. 10.1007/978-3-030-64623-3. hal-03943933

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Hervé Bulou

Loïc Joly

Jean-Michel Mariot

Fabrice Scheurer *Editors*

# Magnetism and
Accelerator-Based
Light Sources

Proceedings of the 7th International
School "Synchrotron Radiation and
Magnetism", Mittelwihr (France), 2018

OPEN ACCESS

<figure><img src="image_2.png" /></figure>

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Hervé Bulou · Loïc Joly ·
Jean-Michel Mariot · Fabrice Scheurer
Editors

Magnetism
and Accelerator-Based Light
Sources

Proceedings of the 7th International School
“Synchrotron Radiation and Magnetism”,
Mittelwihr (France), 2018

<figure><img src="image_3.png" /></figure>

Springer
Editors

Hervé Bulou
IPCMS
Université de Strasbourg, CNRS
Strasbourg, France

Jean-Michel Mariot
LCPMR
Sorbonne Université
Paris, France

Loïc Joly
IPCMS
Université de Strasbourg, CNRS
Strasbourg, France

Fabrice Scheurer
IPCMS
Université de Strasbourg, CNRS
Strasbourg, France

<figure><img src="image_4.png" /></figure>

ISSN 0930-8989
Springer Proceedings in Physics
ISBN 978-3-030-64622-6
https://doi.org/10.1007/978-3-030-64623-3

ISSN 1867-4941 (electronic)
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To Eric Beaurepaire
Foreword

In October 2018, as the last epiphany of a long-standing tradition, the 7th International School “Magnetism and Synchrotron Radiation”, most commonly known as “Mittelwihr School of Magnetism”, gathered more than 100 young (and older) scientists from the whole Europe as well as from the USA and Japan. For the first time of this long series, one of the organizers of the very first Mittelwihr schools, the recently departed Eric Beaurepaire was not attending the school and, in many occasions, lectures were given with a very special attention to Eric’s pioneering work. This textbook is dedicated to Eric and can be regarded as a tribute to his scientific achievements.

This version of the “Mittelwihr School” was not much different from the previous ones so that the Lecture Notes contain most of the expected, basic ingredients. It starts with an introduction to the physics of modern X-ray sources (Chap. 1) coupled to a profound deep presentation of light/matter interaction in the X-ray range (Chap. 3) with a complete overview of all different types of angular dependence for X-ray absorption spectroscopy (Chap. 4). A broad and general introduction to magnetism (Chap. 2) is followed by an involved description of spintronics (Chap. 5) and physics of superconductivity (Chap. 6) with emphasis brought to how spintronics or superconductivity can be understood thanks to X-ray spectroscopies and X-ray scattering.

The 7th International School “Magnetism and Synchrotron Radiation” took place in a context where many storage rings were either upgraded or looking forward to be upgraded. For most X-ray beamlines on third-generation storage rings, the vertical emittance of the electron beam was close to the diffraction limit imposed by the X-ray wavelength, i.e. $\lambda/4\pi$. Not much could be gained in this direction. On the contrary, for most 3rd generation storage rings, the horizontal emittance was much larger than the diffraction limit. At SOLEIL, the horizontal emittance of the electrons was 4000 pm rad so that it was more than one order of magnitude larger than the diffraction limit for X-rays with energy larger than 250 eV. At ESRF, a similar situation existed. Reducing the horizontal emittance was then regarded as a target to follow in order to increase the brilliance and this was first obtained at MAX-IV by replacing the “long” dipole bending magnets by a
series of smaller dipole magnets yielding the multibend achromat lattice. Brilliance is defined as the number of photons per 0.1% bandwidth, per second, per surface of the source and per horizontal and vertical angular divergences of the source. Reducing the size of the source was an efficient way to increase the brilliance. Brilliance is a driving parameter for experiments where the photon source is focused on the sample and in such cases, the increase of brilliance is fully related to an increase of useful photons. It should nevertheless be kept in mind that for experiments where the beam is not refocused on the sample, the gain in brilliance is not accompanied by a high gain in useful photons: this is indeed what is observed for experiments where the figure of merit is the flux at the sample position (the flux, expressed in J m⁻² s⁻¹, is the energy of all the photons crossing a surface unit per second).

As a consequence of increasing the brilliance, the new X-ray beams of the diffraction-limited storage rings have a much higher level of coherence since one estimates that a source with size σ and divergence σ' has a 50% coherence for λ such that $\sigma\sigma' = \lambda/4\pi$. Not regarding the controversy between brilliance and flux, the reduction of the emittance in the horizontal plane opens new perspectives for experiments taking advantage of coherence such as X-ray phase-contrast imaging, ptychography, Fourier transform holography, X-ray photon-correlation spectroscopy, coherent diffractive imaging, or fluctuation microscopy.

Chapter 1, “X-ray sources at large scale facilities”, gives a brief introduction to the physics of storage rings with special attention to the physics of electron accelerators and to the devices developed to produce X-rays: bending magnets, wigglers, undulators. One can notice an introduction to the diffraction-limited storage rings that are presently blooming in Lund (MAX-IV) or Grenoble (upgraded ESRF) and that will replace the old, common structure of conventional storage rings. The X-ray free-electron lasers in conjunction with the self-amplified spontaneous emission of X-rays are also presented and the characteristics of these new X-ray beams are described so that it is accessible to most, future potential users. Based on the knowledge of the X-ray beams, Chap. 3, “Electronic structure theory for X-ray absorption and photoemission spectroscopy” presents the connection between the electronic structure of matter and core-hole spectroscopies such as X-ray absorption spectroscopy and X-ray photoemission spectroscopy. I would like to attract the attention of the reader concerning the recently developed, state of the art pieces of theory that go beyond the common density functional theory (DFT). The time-dependent density functional theory, the Green's function approach (with special care to treatment of the self-energy within the GW approximation), or the Bethe–Salpeter equation are presented in a concise way so that the reader interested in learning beyond the DFT can get a flavour of these actual topics, still under development. Chapter 4, “X-ray dichroisms in spherical tensor and Green’s function formalism”, calculates the various types of angular dependence present in XAS. It first starts with a presentation of the multi-electronic, many-body calculation of the X-ray absorption cross sections. Then connections are made with the atomic multi-electronic approach as it is developed in the ligand-field
multiplet approach or with the monoelectronic DFT approach up to a brief survey of the many-body extended theories. The second part is dedicated to the angular dependence of electric dipole, *E1*.E1 and electric quadrupole *E2*.E2 terms, present in the X-ray absorption cross sections where the emphasis is brought to their tensor expansions. Various symmetries are considered, either octahedral or tetrahedral, as well as the presence of an external or molecular magnetic field along with the directions of high symmetry. The presentation is exhaustive but for the absence of the crossed-terms necessary for the calculation of X-ray natural circular dichroism and X-ray magneto-chiral dichroism.

Chapter 2, “Concepts in magnetism”, is a nice and concise introduction to magnetism, necessary to make the present book a self-contained piece of work. The basic, theoretical framework is presented so that spectroscopists with no knowledge of magnetism can grasp the general picture. A special attention is devoted to the role played by the orbital part in the building of the magnetic properties. This emphasis needs to be acknowledged because the X-ray/matter interaction can be fully described in the electric dipole and electric quadrupole approximations, where no spin variables are present, so that X-ray absorption can be highly sensitive to the orbital part of the electron wave function and more specifically to the orbital magnetic moment.

Chapters 5 and 6 are descriptions of special states of matter where chemists and physicists are tailoring matter so that it presents original, unprecedented characteristics. In the first part of Chap. 5, “Spintronics and synchrotron radiation”, the authors give an overall view of the concept of magnetoresistance and they explore how magnetization can be triggered in the absence of an external magnetic field, with a special note on spin-orbitronics and the fascinating skyrmions. Then the authors present the information that can be gained by advanced X-ray spectroscopies such as angle-resolved photoemission spectroscopy, X-ray magnetic circular dichroism and scanning transmission X-ray microscopy. In Chap. 6, “p-wave superconductivity and d-vector representation”, the author presents advanced concepts in the field of superconductivity. Through a special attention paid to the role played by the orbital part of the electrons responsible for superconductivity, the reader can make a connection with the information that can be extracted from advanced, polarized X-ray spectroscopies.
# Preface

This volume contains the lecture notes of the seventh school on Magnetism and Synchrotron Radiation held in *Mittelwihr*, France, from 7 to 12 October 2018.

This school has been contributing since 1989 to federate the community which uses synchrotron radiation for magnetic investigations, starting with first-generation synchrotrons, until today where new opportunities open up with the access to free-electron lasers. We would like to dedicate this school to the late Eric Beaurepaire (1959–2018) from IPCMS-CNRS for his tribute to the existence of this school and to the science of magnetism. He was one of the founders and passionate supporter of this school. He set several milestones in the field of femtomagnetism, first using pump-probe laser experiments to unravel the magnetization dynamics with a founding paper in 1996, and several years later, making use of time-resolved experiments with synchrotron radiation. We will also remember him as a simple and very human person.

Strasbourg, France

Strasbourg, France

Paris, France

Strasbourg, France

Hervé Bulou

Loïc Joly

Jean-Michel Mariot

Fabrice Scheurer
# Teachers

A. Bonnani, University of Linz, Austria

S. Blundell, University of Oxford, United Kingdom

J.-P. Brison, CEA Grenoble, France

Ch. Brouder, IMPMC, Paris, France

H. Dürr, University of Uppsala, Sweden

A. Juhin, IMPMC, Paris, France

P. Krüger, Chiba University, Japan

C. Mathonière, ICMCB, Bordeaux, France

R. Mattana, CNRS, Thales, Université Paris-Saclay, Palaiseau, France

P. Ohresser, Synchrotron SOLEIL, Gif-sur-Yvette, France

E. Otero, Synchrotron SOLEIL, Gif-sur-Yvette, France

M. Retegan, ESRF, Grenoble, France

A. Rogalev, ESRF, Grenoble, France

Ph. Willmott, SLS, Villigen, Switzerland

## Scientific Committee

S. Andrieu, IJL, Nancy, France

H. Ebert, University of Munich, Germany

F. Kronast, HZB, Berlin, Germany

J. Lüning, Synchrotron SOLEIL, France

G. Panaccione, TASC-INFM, Trieste, Italy

C. Piamonteze, SLS, Villigen, Switzerland

Ph. Sainctavit, IMPMC, Paris, France

D. Sébilleau, IPR, Rennes, France

G. van der Laan, Diamond Light Source, Didcot, United Kingdom

F. Wilhelm, ESRF, Grenoble, France
Acknowledgements

We thank our scientific committee for their choice of programme and their selection of lecturers. We are also grateful to the lecturers for their commitment to preparing interesting and pedagogical talks as well as the manuscripts presented in this book. They are also warmly thanked for their lively contribution throughout the duration of the school.

This school benefited from the personal investment of J.-L. Bubendorff (IS2M), C. Guillet and V. Wernher (IPCMS) for the organization.

We are very grateful to Ph. Sainctavit (IMPMC, CNRS-Sorbonne Université) for writing the foreword of this book. It is also a great pleasure to acknowledge here the kind hospitality of the accommodation centre "Le Mittel" and the Communauté des Communes de Ribeauvillé.

We gratefully acknowledge financial support from the following institutions

* Formation Permanente du CNRS,

* Université Franco-Allemande/Deutsch-Französische Hochschule,

* Université de Strasbourg, Investissements d'Avenir,

* Université de Haute Alsace & Conseil Général du Haut-Rhin,

* Synchrotron SOLEIL,

* Institut de Physique et Chimie des Matériaux de Strasbourg (IPCMS),

* Institut des Sciences des Matériaux de Mulhouse (IS2M),

* Labex *Nanostructures in Interaction with their Environment* (ANR-11-LABX-0058 NIE),

* CNRS, Groupement de Recherche GDR REST,

* CNRS, Groupement de Recherche International GDRi XFEL Science, and companies

* CRYOCONCEPT,

* NEYCO,

* AXESS-TECH,

* AGILENT,

* MATECK,

* WORLD SCIENCEPRESS.
# Contents

1 **X-Ray Sources at Large-Scale Facilities** 1
Philip R. Willmott

2 **Concepts in Magnetism** 39
Stephen J. Blundell

3 **Electronic Structure Theory for X-Ray Absorption and Photoemission Spectroscopy** 63
Peter Krüger

4 **X-ray Dichroisms in Spherical Tensor and Green's Function Formalism** 83
Hebatalla Elnaggar, Pieter Glatzel, Marius Retegan, Christian Brouder, and Amélie Juhin

5 **Spintronics and Synchrotron Radiation** 131
Richard Mattana, Nicolas Locatelli, and Vincent Cros

6 **p-Wave Superconductivity and d-Vector Representation** 165
Jean-Pascal Brison

**Index** 205
Contributors

**Stephen J. Blundell** Clarendon Laboratory, Oxford University, Department of Physics, Oxford, UK

**Jean-Pascal Brison** Université Grenoble Alpes, CEA, IRIG, Pheliqs, Grenoble, France

**Christian Brouder** Institut de Minéralogie, Physique des Matériaux et Cosmochimie, CNRS-Sorbonne Université, Paris, France

**Vincent Cros** Unité Mixte de Physique, CNRS, Thales, Université Paris-Saclay, Palaiseau, France

**Hebatalla Elnaggar** Debye Institute for Nanomaterials Science, Utrecht University, Utrecht, The Netherlands

**Pieter Glatzel** European Synchrotron Radiation Facility, Grenoble, France

**Amélie Juhin** Institut de Minéralogie, Physique des Matériaux et Cosmochimie, CNRS-Sorbonne Université, Paris, France

**Peter Krüger** Graduate School of Engineering and Molecular Chirality Research Center, Chiba University, Chiba, Japan

**Nicolas Locatelli** Unité Mixte de Physique, CNRS, Thales, Université Paris-Saclay, Palaiseau, France

**Richard Mattana** Unité Mixte de Physique, CNRS, Thales, Université Paris-Sud, Université Paris-Saclay, Palaiseau, France

**Marius Retegan** European Synchrotron Radiation Facility, Grenoble, France

**Philip R. Willmott** Swiss Light Source, Paul Scherrer Institute, Villigen, Switzerland
# Chapter 1
## X-Ray Sources at Large-Scale Facilities

<figure><img src="image_5.png" /></figure>

Check for
updates

Philip R. Willmott

**Abstract** High-brilliance X-ray sources are powerful probes to investigate the prop-
erties of matter down to the sub-angstrom scale and on time scales that can extend
below a femtosecond. In this chapter, an introductory overview of the physics behind
storage ring-based synchrotrons and linear accelerator-based X-ray free-electron
lasers is presented, while the properties of the radiation they produce are explained.

**1.1 Introduction**

Since their discovery by Wilhelm Röntgen in the last decade of the nineteenth century, X-rays have played a central role in all branches of the natural sciences and medicine. The primary reasons for this are threefold. Firstly, ‘hard’ X-rays (that is, those with photon energies in excess of a few keV and up to approximately 50 keV, see Fig. 1.1), have interaction strengths that, on the one hand, are small enough to allow them to penetrate deeply into solid matter, while, on the other, are sufficiently large that these interactions are easily observable. Secondly, X-rays have wavelengths λ on the nanometre to angstrom scale, meaning that, according to the Abbe limit, they can be used to image objects composed of features down to sizes comparable to λ. Finally, the binding energies of electrons, from weakly bound valence electrons to very strongly bound core electrons in heavy elements, lie in the ultraviolet to hard X-ray regime, allowing detailed studies of these bonds through spectroscopy, thereby providing insights into chemistry, electronic structure and magnetic properties.

Modern scientific disciplines are increasingly concerned with correlating physi-
cal structure with physical properties. This has been long recognized in the lock-
and-key functionality of biological structures such as enzymes and proteins. In
condensed matter physics, the properties of many emergent materials, in particu-
lar (though not exclusively) transition metal oxides, depend exceedingly sensitively

<footnote>P. R. Willmott (✉)
Swiss Light Source, Paul Scherrer Institute, 5232 Villigen, Switzerland
e-mail: philip.willmott@psi.ch
URL: http://www.psi.ch/sls</footnote>
<figure><img src="image_6.png" /><figcaption>**Fig. 1.1** Photon energies and the electromagnetic spectrum. Above the visible regime, spanning only an energy range of 1.77–3.1 eV, the electromagnetic spectrum is divided into UV (up to approximately 140 eV), soft X-rays (up to 2 keV), tender X-rays (to 4 keV), hard X-rays (to 50 keV) and gamma rays (above 50 keV). Important photon energies are highlighted, green (K-edge) and red (L-edge) arrows pointing down imply absorption energies, while those pointing up imply emission lines</figcaption></figure>

on the structure—even changes of a few picometres in bond length, or one degree in bond angle, can have fundamental consequences on the material’s electronic character [1].

Protein crystals can often only be grown with linear dimensions of a few tens of micrometres, while interfacial regions between different oxide materials which exhibit unexpected properties [2] may only be a few nanometres thick. Any signal from irradiation of such samples using X-rays, be it the degree of absorption, the amount of elastically scattered radiation, fluorescence or the photoelectron yield, is likely to be very weak. This sets a premium on finding a very intense X-ray source—synchrotrons and X-ray free-electron lasers (XFELs) have been developed to fulfil this need.

The broad range of applications of X-rays has manifested itself in the last two decades in the heterogeneity of scientific fields served and the broad spectrum of techniques now available at synchrotron facilities, representing perhaps the clearest illustration of multidisciplinary research, covering the natural and medical sciences and many aspects of engineering and technology. Nowadays, there are worldwide more than seventy facilities in operation, or under construction, providing sophisticated investigative tools for well over 110 000 users from virtually every field of the natural and engineering sciences.

These users need to understand the operating principles of synchrotrons and their generation of X-radiation in order to best prepare, firstly, proposals to be submitted to the highly competitive review procedure at synchrotrons, and secondly, the beamtimes themselves. This brief overview of synchrotrons, synchrotron sources, and XFELs draws substantially from chapters on synchrotron and XFEL physics in [3] and as such is intended as an accessible primer to the interested reader from any branch of the natural and engineering sciences.

In the next section, the architecture and operating principles of synchrotrons are described and the standard figure of merit for synchrotron light, called the brilliance, is introduced. The three different types of source (bending magnets, wigglers and undulators) are presented in Sect. 1.3. We finish this section by describing ways to control the polarization of X-rays at synchrotrons.
Sections 1.4 and 1.5 outline the most pertinent features of the latest generation of storage rings, so-called diffraction-limited storage rings (DLSRs), and X-ray free-electron lasers (XFELs), respectively.

## 1.2 A Brief Description of Synchrotrons

### 1.2.1 Introduction

A synchrotron consists of a ring-shaped evacuated vessel (the storage ring, having a circumference measured typically in a few hundreds of metres, Fig. 1.2) in which high-energy electrons circulate at highly relativistic velocities, and so-called ‘beamlines’, that extract and use the radiation emitted by the electrons tangentially to their orbital path, at positions defined by components known as bending magnets (BMs) and insertion devices (IDs).

The electron energy $\mathcal{E}$ at synchrotrons is typically of the order of a few GeV. The emitted photons, on the other hand, have energies measured anywhere between a few eV (just above the visible) and several hundred keV, in the ultrahard X-ray regime.<fnref n="1" /> Note, however, that even the photon energies of the latter are still some four orders of magnitude smaller than the electrons’ kinetic energy $\mathcal{E}$ in the storage ring.

The electrons are forced into a closed path by bending magnets, which exert a centripetal Lorentz force on them. It is here, and in straight sections in which the insertion devices are installed (see Sect. 1.3), that they emit electromagnetic (EM) radiation.

The electrons lose energy due to their emitting EM radiation. This must be replenished and is achieved via axial acceleration through one or more radio-frequency (RF) cavities installed in the storage ring.

### 1.2.2 The Lorentz Factor

Before we proceed further, the dimensionless parameter $\gamma$ is introduced. This so-called ‘Lorentz factor’ expresses the ratio of the electron energy $\mathcal{E}$ to the rest mass energy of the electrons $m_e c^2 = 511$ keV ($m_e = 9.109 \times 10^{-31}$ kg is the electron rest mass and $c = 2.9979 \times 10^8$ m s$^{-1}$ the speed of light), that is

$$ \gamma = \frac{\mathcal{E}}{m_e c^2}. \qquad (1.1) $$

<footnote n="1">¹Some facilities host beamlines that extend down into the far-infrared regime; these relatively uncommon sources are not discussed here.</footnote>
<figure><img src="image_7.png" /><figcaption>**Fig. 1.2** A schematic of the basic components of a modern synchrotron facility. Electrons from a source such as a heated filament in an electron gun are accelerated by a linear accelerator (LINAC) and then injected into a booster ring, where they are further accelerated. They are then further injected into the so-called storage ring. There, they are maintained in a closed path using bending-magnet achromats at arc sections. The beamlines use the radiation emitted from insertion devices (IDs, either wigglers or, more commonly, undulators) placed at the straight sections between the arcs, and from the bending magnets (BM), on the axes of emission. The energy lost by the electrons through the emission of synchrotron light is replenished by particular parts of the cycle of one or more radio frequency (RF) supplies. This forces the electrons within the storage ring to separate into discrete bunches. Each bunch contains of the order of $10^9$ electrons and has a full width at half maximum duration of the order of 100 ps. Reproduced from [3] with permission (Copyright 2019, John Wiley and Sons)</figcaption></figure>

Consequently, for typical storage ring energies of the order of a few GeV, $\gamma$ is of the order of a few thousand to a little over 15 000 (for the Advanced Light Source in Berkeley, $\mathcal{E} = 1.9$ GeV and hence $\gamma = 3718$, while the highest energy storage ring, SPring8, has a storage ring energy of $\mathcal{E} = 8$ GeV, leading to $\gamma = 15656$). The Lorentz factor crops up in many equations related to synchrotron radiation (SR), including the beam divergence, relativistic electron mass, electron emittance and the radiative power output.

From the special theory of relativity, it emerges that

$$ \gamma = [1 - (v/c)^2]^{-1/2}, \qquad (1.2) $$

where $v$ is the electron's velocity. We re-arrange this to obtain

$$ v = c (1 - 1/\gamma^2)^{1/2}. \qquad (1.3) $$
We know $\gamma$ is of the order of several thousand, hence $1/\gamma^2$ is a very small number, of the order of $10^{-8}$. We use the approximation $(1-x)^n = 1-nx$ for $x \ll 1$ to obtain

$$ \frac{c-v}{c} \approx \frac{1}{2\gamma^2}. \qquad (1.4) $$

In other words, the difference between c and v is very small, of the order of a few m s$^{-1}$.

Lastly, the mass of the electron from the perspective of a stationary observer is equal to $\gamma m_e$. We will return to these findings later.

### 1.2.3 Dipole Radiation and Synchrotron Radiation

Why do electrons emit EM radiation at all? First, it should be stressed that electrons, or indeed any charged particles, only emit EM radiation when accelerated. ‘Accelerated’ can include the conventional meaning of an increase in speed but no change in direction; its opposite, that is, a deceleration; or a change in the electrons’ direction, such as in centripetal acceleration. The second case corresponds to the so-called ‘Bremsstrahlung’ (German expression for braking radiation), while the third is associated with SR.

Given this, why then does the action of accelerating electrons cause them to emit light? Consider Fig. 1.3. EM radiation is a form of transverse wave in which the oscillations (of the electric and magnetic fields) are at right angles to the direction of motion. For the sake of simplicity, we consider here only the electric field component of the EM radiation. The electric field lines of the electrostatic field of a stationary and isolated electron emanate out radially from the electron. Any observer looking at the electron, therefore, sees no transverse component of the field, which thus implies she sees no radiation [Fig. 1.3a].

If, however, the electron is made to execute oscillatory motion, the electric field lines, which are anchored to the electron and emanate out from it at the speed of light, will reflect this motion and thus also oscillate accordingly [Fig. 1.3b]. In all directions except that exactly along the axis of acceleration, our observer will ‘see’ a transverse component to the electric field and therefore perceive that light is being emitted. The amplitude of the radiation is proportional to $\cos \chi$, where $\chi$ is the polar angle between the axis of acceleration and the observer’s direction. The intensity distribution, shown in Fig. 1.3b, is proportional to $\cos^2 \chi$. This so-called ‘dipole radiation’ is the reason why mirrors reflect visible light, radio antennae capture or emit radio waves and undulators produce X-radiation.

SR is highly collimated, with natural divergences of the order of 0.1 mrad (mrad; 1 mrad is approximately equal to 0.06°). A detailed derivation of the spatial distribution of SR lies beyond the brief of this introductory overview (see, for example, [3]). Suffice it to say, it differs substantially from the dipole distribution shown in Fig. 1.3b; this is due to relativistic Doppler shifting. The angular power distribution
<figure><img src="image_8.png" /><figcaption>**Fig. 1.3** Generation of EM radiation through the acceleration of charged particles. **a** A charged particle at rest or moving at a constant velocity will not emit light, as an observer of the particle will detect no lateral component of the electric field lines. **b** If, however, the particle is accelerated, an observer positioned anywhere except along the axis of that acceleration (position A) will experience a shift in the position and direction of the electric field lines as the event horizon washes over them at the speed of light (for example, at position B). A simple harmonic driving force will generate radiation from the electrons at the same frequency. Adapted from [3] with permission (Copyright 2019, John Wiley and Sons)</figcaption></figure>

per solid angle $dP/d\Omega$ for an electron moving with a velocity $v$ is given without derivation as

$$ \frac{dP}{d\Omega} = \kappa \frac{a^2}{(1 - \beta \cos \theta)^3} \left[ 1 - \frac{\sin^2 \theta \cos^2 \phi}{\gamma^2 (1 - \beta \cos \theta)^2} \right] \quad (1.5) $$

where $\phi$ and $\theta$ are the polar (out of the orbital plane) and azimuthal (in the orbital plane) angles, respectively, $\kappa = e^2/(16\pi^2\epsilon_0 c) = 6.124 \times 10^{-38} \text{ kg m}^2 \text{ s}^{-1}$ and $a = \text{Bec}/(\gamma m_e)$ is the acceleration perpendicular to the direction of motion due to the Lorentz force exerted on the relativistic electron by the magnetic field $B$. Note that in the nonrelativistic limit of $v \ll c$ ($\beta = v/c \ll 1$), (1.5) reduces to the simple $\cos^2 \theta$ dependence of dipole radiation. The progression from dipole to synchrotron radiation for different values of $\beta$ is shown in Fig. 1.4).

For a given centripetal acceleration $a$, the maximum power (in the forward direction) scales with the fourth power of the electron-beam energy. At highly relativistic velocities, it emerges that

$$ \frac{dP}{d\Omega} = \kappa' B^2 \gamma^4 \frac{[1 - (\gamma\theta)^2]^2}{[1 + (\gamma\theta)^2]^5} \quad (1.6) $$

where $\kappa' = e^4/(2\pi^2 m_e^2 \epsilon_0 c) = 1.5156 \times 10^{-14} \text{ m}^2 \text{C}^2 \text{kg}^{-1} \text{s}^{-1}$ (or W T$^{-2}$). In the frame of reference of the electron, the distribution remains pure dipole radiation
<figure><img src="image_9.png" /><figcaption>**Fig. 1.4** From dipole radiation to synchrotron beams. The progression of the angular power distribution of radiation from an electron travelling at a fraction of the speed of light $\beta = v/c$ while experiencing a centripetal acceleration $a$ perpendicular to its motion. The case $v = 0$ corresponds to dipole radiation. As $\beta$ increases, the radiation distribution is swept in the forward direction</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Fig. 1.5** Plot of the exact (blue solid curve) and approximate (red dashed curve) expressions given in (1.8), as a function of $\beta$ up to $\beta = 0.9999$. More typical values of $\beta$ at modern synchrotrons are $1 - 10^{-8}$ (0.999 999 999)</figcaption></figure>

($\beta = 0$ in Fig. 1.4) and the opening angle is $\theta' = \pm\pi/2$. From the perspective of a stationary observer, however, the angular distribution is modified such that

$$ \theta = \sin^{-1} \left[ \frac{\sin \theta'}{\gamma(1 + \beta \cos \theta')} \right]. \qquad (1.7) $$

Thus, the opening angle changes from $\pm\pi/2$ in the electrons' frame of reference to $\pm 1/\gamma$ in the laboratory frame. The entire beam therefore lies within $\pm 1/\gamma$, and has a full width at half maximum of approximately $1/\gamma$. This 'natural opening angle' (or divergence) of the narrow radiation cone, for typical storage ring energies of 1–8 GeV, is equal to 0.5–0.06 mrad (0.028–0.0034°), respectively; SR is highly collimated.

Lastly, it can be simply demonstrated from (1.2) and (1.5) that, for a given acceleration $a$, the ratio of the power in the forward direction ($\theta = 0$) for an electron beam travelling at a non-zero speed perpendicular to $a$, to that of the maximum of dipole radiation for an electron with zero velocity perpendicular to the acceleration is

$$ \frac{P(\theta = 0, \beta \neq 0)}{P(\theta = 0, \beta = 0)} = \frac{(1 - \beta^2)}{(1 - \beta)^3} \approx 8\gamma^4 \qquad (1.8) $$
where the approximation on the right is valid for relativistic velocities. Both the exact and approximate expressions are shown in Fig. 1.5 up to $\beta = 0.9999$. Increases in power of the order of $10^{16}$ compared to electrons undergoing stationary dipole oscillations can thus be expected—synchrotrons truly do deliver powerful beams!

### 1.2.4 Spectral Flux, Emittance, and Brilliance

Flux and brilliance are figures of merit of the quality of a synchrotron facility. The spectral flux is defined as the number of photons per second per unit bandwidth (BW), normally given as 0.1%, and is the appropriate measure for experiments that use the entire, unfocussed X-ray beam. Brilliance essentially states how tightly the spectral flux is collimated and how small the source size is. It is defined as

$$ \text{Brilliance} = \frac{\text{photons/second}}{(\text{mrad})^2 (\text{mm}^2 \text{ source area}) (0.1\% \text{ BW})}, \quad (1.9) $$

and is therefore equal to the flux per unit source cross-sectional area and unit solid angle. Note that the flux (as against the spectral flux) is simply measured in photons per second. Doubling the transmitted BW from a broadband source thus doubles the flux, but leaves the spectral flux unchanged. From (1.9), it is seen that the brilliance is inversely proportional to both the source size and the beam divergence. The product of the linear source size $\sigma$ and the beam divergence $\sigma'$ in the same plane is known as $\epsilon$, the emittance in that plane, that is

$$ \epsilon_x = \sigma_x \sigma'_x, \quad (1.10) $$

$$ \epsilon_y = \sigma_y \sigma'_y. \quad (1.11) $$

The goal of the machine physicist designing a synchrotron magnet lattice is to provide as low an emittance as possible, in other words, a source with an exceedingly small cross section emitting X-rays that are highly collimated. For a given synchrotron storage ring, the emittance in each transverse direction (x, in the orbital plane, or y, perpendicular to the orbital plane) is, according to Liouville's theorem, a constant. The emittance will be different for different facilities, in each case being determined primarily by the degree of sophistication and perfection of the magnet lattice.

Importantly, the total emittance in a given plane is a convolution of the contribution from the electron beam and that from the emitted photons. It follows that the total source size $\sigma_{x,y}$ and divergence $\sigma'_{x,y}$ in the x- and y-planes perpendicular to the direction of propagation of a given storage ring are also convolutions of contributions from the electron and photon beams, that is

$$ \sigma_{x,y} = [(\sigma_{x,y}^e)^2 + (\sigma^p)^2]^{1/2}, \quad (1.12) $$

$$ \sigma'_{x,y} = [(\sigma'_{x,y}^e)^2 + (\sigma'^p)^2]^{1/2}. \quad (1.13) $$
While the electron contribution can be minimized by sophisticated electron optics (see Sect. 1.4), the photon emittance is an intrinsic property defined by Heisenberg's uncertainty principle, and is equal to $\lambda/4\pi$ in both the x- and y-plane. In third-generation storage rings, the electron emittance in the orbital plane $\epsilon_x^e$ dominates the total emittance and is thus the limiting factor for the brilliance. In fourth-generation DLSRs, the benchmark for modern storage ring designs, $\epsilon_x^e$ has been reduced to values close to or even below (in the case of soft X-rays) the intrinsic photon emittance, which we consider in detail in Sect. 1.4. In other words, the emittance is no longer limited by the electron optics, but by the fundamental optical diffraction limit. This is the meaning of the moniker 'diffraction-limited storage ring' defining the fourth-generation synchrotron facilities now coming online.

Synchrotrons have brilliances using modern undulators of approximately $10^{22}$ photons s$^{-1}$ mrad$^2$ mm$^{-2}$ 0.1% bandwidth$^{-1}$. This is some 12 orders of magnitude higher than that of a standard laboratory-based Cu $K\alpha$ source and less than a factor of 100 lower than high-quality visible laser sources. The main reasons for this are the size of the radiation source, of the order of ten micrometres at fourth-generation DLSRs, the high collimation of the beams, being of the order of 10 $\mu$rad in the orbital plane and the fact that synchrotrons emit an enormous amount of light. The power emitted by an electron is proportional to the square of the electron's centripetal acceleration $a$, and the fourth power of the storage ring energy.

### 1.2.5 The Radio-Frequency Power Supply

Conservation of energy dictates that the kinetic energy of the electrons is dissipated due to emission of radiation at the bending magnets and insertion devices. This energy must be replenished, or otherwise, the electrons would spiral into the inner wall of the storage ring. This is achieved by boosting the electrons' energy at one or more positions along the storage ring as they pass through RF cavities [Fig. 1.6a]. This requires that the electrons enter the cavity at a certain point of the RF cycle.

Because the electrons can only receive the correct amount of energy at very specific and narrowly defined values in the RF cycle, they are separated into a series of packets, or 'bunches'. The energy loss of the electrons for each cycle around the ring is given by the total power loss of the storage ring divided by the storage ring current and is equal to approximately 0.2–1 MeV, or of the order of 0.05% of the nominal electron energy. Depending on the size of the facility, most storage rings host between two and eight RF cavities. Between them, they must be able to replenish this loss.

Consider Fig. 1.6b. On average, the electrons require a certain energy boost in order to keep them on a stable path, given by an amount $eV_{\text{ref}}$. If an electron loses more than this amount of energy, it will enter the RF cavity somewhat earlier at point A. This might sound counterintuitive—surely if the electron has less energy, it will be slower and enter the cavity later. But because it takes a shorter path in the bends according to the linear dependence of the bending radius and the electron energy it
<figure><img src="image_11.png" /><figcaption>**Fig. 1.6** Replenishing the electron energy in a storage ring. **a** Electrons entering the resonant RF cavity at the correct moment in its voltage cycle are accelerated by a suitable amount by the electric field within the cavity. Note that the field lines point in the opposite direction to the acceleration, as the electric force is **F**<sub>E</sub> = *q***E** but an electron has a negative charge —*e*. **b** 'Slow' electrons entering the RF cavity at **A** will be given more of a boost than 'fast' electrons at **B**. Adapted from [3] with permission (Copyright 2019, John Wiley and Sons)</figcaption></figure>

indeed will arrive earlier than a higher energy electron, and will thus experience a
larger acceleration than if it were at the reference voltage. Likewise, if the electron is
too fast, it will receive less of a boost. Any electrons entering the RF cavity outside
this narrow range above and below the reference voltage will not gain the correct
energy and will be lost to the system. The electrons therefore quickly bunch into
packets associated with each cycle of the RF cavity.

The short bunch lengths allow users to exploit the time structure of SR down to
well below the nanosecond time scale for time-resolved experiments. These types
of experiment became increasingly important in third-generation synchrotron facil-
ities, in areas as diverse as molecular biology, catalysis, condensed matter physics
and domain flipping in nanomagnetism and, for DLSRs, they promise to be comple-
mentary to XFEL investigations.

**1.2.6 Radiation Equilibrium**

What determines the electron emittance? The emittance of a storage ring is deter-
mined by the opposing influences of two phenomena: radiation damping (some-
thing you want) and quantum excitation (something you don't). At the NSLS II in
Brookhaven, the machine performance is optimized by maximizing radiation damp-
ing, while in the next generation of DLSRs such as MAX-IV, quantum excitation
has been minimized. As we have already stated, radiation damping improves the
emittance by reducing the transverse momentum component [see Fig. 1.7a]. When
an electron emits a photon, it loses the energy of the photon. This causes it to oscil-
<figure><img src="image_12.png" /><figcaption>**Fig. 1.7** Radiation equilibrium between quantum excitation and radiation damping. **a** Radiation damping. An electron traversing a magnet in an insertion device is made to deviate from the central axis by an angle $\theta$, due to the Lorentz force. Emission of a photon with momentum $h/\lambda = \hbar k$ will be in the direction of the electron at that instant in time. Conservation of momentum dictates that the electron's momentum will be reduced to $p' = p - h/\lambda$. The same electron will regain this momentum loss $dp$ after travelling through the RF cavity; importantly, this will now be parallel to the central axis, thus reducing the angle of the electron's momentum $p''$ to the central axis and hence also the electron beam's emittance. **b** Quantum excitation. An electron loses energy due to the emission of a photon and begins to oscillate around a new reference orbital path with a smaller radius. These oscillations induce a stochastic distribution of transverse momenta, thereby increasing the emittance. Reproduced from [3] with permission (Copyright 2019, John Wiley and Sons)</figcaption></figure>

late around a new reference orbit, thus broadening the beam and thereby increasing the emittance [Fig. 1.7b]. Moreover, the dispersion of the electron beam increases. Quantum excitation can be reduced by designing the magnet lattice so that the electron energy dispersion is minimized at the main locations of radiation, namely the bending magnets. This is achieved by horizontal focusing at the bends and the use of many small deflection angle bends in multibend achromat lattices (see Sect. 1.4) to limit dispersion growth.

### 1.2.7 Coherence

We now consider coherence, including both longitudinal and transverse coherence. The latter depends on the source size and divergence of the photon beam, in other words, it depends on the emittance; longitudinal coherence depends on the bandwidth. The coherent fraction of a beam is critically important in lensless imaging techniques and photon correlation spectroscopy, plus also in phase-contrast tomography. Note also that bound up in the figure of merit of brilliance are the above parameters that quantitatively define coherence: the emittance and the relative spectral BW. Figure 1.8 provides a schematic summary of coherence. Brilliance really does encompass the most important qualities of synchrotron light; because however, it combines flux, spatial coherence and longitudinal coherence, it is important to
<figure><img src="image_13.png" /><figcaption>**Fig. 1.8** Coherent radiation. The coherent fraction of a broadband, spatially extended, source can be selected as follows. Firstly, a pinhole selects a small, spatially constrained fraction of the radiation thereby acting as a secondary, quasi-pointlike, source. This secondary source has an improved transverse (or spatial) coherence, as the emittance, which is the product of the radiation's divergence and source size, is now much smaller. Next, a filter (which is normally a monochromator) selects a narrow BW which is much narrower than the original source. Now, the radiation is spatially and longitudinally (or temporally) coherent. Note that both the emittance and relative spectral BW are included in the definition of brilliance. Reproduced from [3] with permission (Copyright 2019, John Wiley and Sons)</figcaption></figure>

know which factors determine a given value. For example, is the brilliance high because there are more photons on the sample, or because the emittance is small?

No X-ray source has an infinitely narrow bandwidth. Consequently, the different frequency components within the beam will sooner or later drift out of phase with one another. The time for the phase between two waves differing in frequency by an amount $\Delta\nu$ but which are initially in phase to differ by $\pi$ radians (i.e. from fully constructive to fully destructive) is simply $1/2\Delta\nu$. This is known as the so-called longitudinal coherence time, $\Delta\tau_c^{(l)}$. During this time, the waves have travelled in vacuum a distance $l_c^{(l)} = c\Delta\tau_c^{(l)}/2$, known as the longitudinal (or temporal) coherence length [see Fig. 1.9a], given by

$$l_c^{(l)} = \frac{\lambda^2}{2\Delta\lambda}. \qquad (1.14)$$

The longitudinal coherence after a monochromator is usually determined by the rocking curve of the crystal or grating used in the monochromator, which defines $\lambda/\Delta\lambda$. For a perfect crystal with insignificant mosaicity, $\Delta\lambda$ is limited by the so-called Dar-
win width, and $\lambda/\Delta\lambda$ can easily exceed $10^4$—the relative bandwidth of a Si(111) single crystal is determined by the extinction depth and is approximately $1.4 \times 10^{-4}$. The longitudinal coherence length can thus be several micrometres, even in the hard X-ray regime.

The transverse coherence length $l_c^{(t)}$ (also called the spatial coherence length) results from the interference of waves having the exact same wavelength but with slightly different directions of propagation. This arises because all sources have a finite size $D$ and a non-zero divergence $\Delta\theta$ (that is, a non-zero emittance), as shown in Fig. 1.9b. In this case,

$$l_c^{(t)} = \lambda/2\Delta\theta = \lambda R/2D, \qquad (1.15)$$

where $D$ is the linear size of the finite source and $R$ is the distance from the source to the observation point. If we assume the source has a Gaussian profile, determination of $D$ requires integration of interference contributions across the entire source's intensity distribution. It emerges that $l_c^{(t)}$ is related to the standard deviation of the beam size $\sigma_{x,y}$ by

$$l_c^{(t)} = \lambda R/(2\pi^{1/2}\sigma_{x,y}), \qquad (1.16)$$

or, in practical units

$$l_c^{(t)}[\mu\text{m}] = 28.21 \frac{\lambda \stackrel{\circ}{\text{A}} R [\text{m}]}{\sigma_{x,y} [\mu\text{m}]} . \qquad (1.17)$$

Note that the transverse coherence length can be made larger by judicious use of slits limiting the apparent source size and divergence, but obviously at the cost of flux. Beamlines such as coherent lensless imaging beamlines tend to be very long in order to maximize $R$.

In the vertical direction, the source size at an undulator of, say, 2-m length, is of the order of $\sigma_y = 2 \mu\text{m}$. For 1-Å radiation, this yields a vertical spatial coherence for an observer at 40 m of $l_c^{(t,y)} = 564 \mu\text{m}$. The horizontal spatial coherence has traditionally been two orders of magnitude smaller than this, due to the very much larger extent of the electron beam in the orbital plane. The electron beam source size in DLSR storage rings in the horizontal direction may, however, be as small as $\sigma_x = 10 \mu\text{m}$; the corresponding coherence length $l_c^{(t,x)}$ is of the order of 0.1 mm.

## 1.3 Sources of Synchrotron Radiation

In this section, the three sources of SR are semi-quantitatively described, while the differences between the radiation produced by bending magnets and wigglers on the one hand, and undulators on the other, are presented in a heuristic manner.
<figure><img src="image_14.png" /><figcaption>**Fig. 1.9** Beam coherence. **a** The temporal, or longitudinal, coherence length is determined by the monochromaticity of the source, while **b** the transverse, or spatial, coherence length depends on the photon beam emittance. Reproduced from [3] with permission (Copyright 2019, John Wiley and Sons)</figcaption></figure>

### 1.3.1 Bending Magnets and Wigglers

As mentioned above, the electrons are maintained on a closed path via magnetic, or Lorentz, forces. The Lorentz force, **F**_L, is proportional to the cross-product of the magnetic field strength, **B**, and the charged particle's velocity, **v**, that is

$$ \mathbf{F}_L = e\mathbf{B} \times \mathbf{v}, \qquad (1.18) $$

where $e = 1.6022 \times 10^{-19}$ C is the elementary charge. It acts perpendicular to the plane defined by **B** and **v**. For the sake of simplicity, we only consider those cases where **F**_L, **B**, and **v** are mutually orthogonal, and drop the bold face implying their vectorial nature. Now,

$$ F_L = eBv. \qquad (1.19) $$

We equate this with a centripetal force $mv^2/\rho$, where $m = \gamma m_e$ is the relativistic mass of the electron travelling at a speed $v \approx c$. The bending radius of the centripetal force, $\rho$, is equal to the bending magnet radius. Therefore, to a high degree of accuracy,

$$ eBc \approx \frac{\gamma m_e c^2}{\rho} \qquad (1.20) $$

$$ \Rightarrow \rho = \frac{\varepsilon}{ceB}. \qquad (1.21) $$

In practical units, we obtain

$$ \rho = 3.3 \frac{\varepsilon[\text{GeV}]}{B[\text{T}]} . \qquad (1.22) $$
<figure><img src="image_15.png" /><figcaption>**Fig. 1.10** Double-bend achromats. The angular dispersion of electrons of different energies as they pass through a bending magnet and the consequent increase in the electron beam emittance can be corrected in a DBA by placing a focusing quadrupole magnet (FQM) symmetrically in between two identical bends (BM). Lower energy electrons are bent through larger angles than are those of higher energy. Adapted from [3] with permission (Copyright 2019, John Wiley and Sons)</figcaption></figure>

For typical storage ring energies of a few GeV and magnetic field strengths of the order of 1 T, we obtain bending magnet radii of the order of 10 m.

The relative spread in electron energy $\Delta\epsilon/\epsilon$ in a storage ring is of the order of $10^{-3}$. The bending radius $\rho$ is directly proportional to the electron energy [see (1.21)]. Therefore, the path of those electrons with more (less) than the central energy will have a larger (smaller) radius, resulting in an unwanted increase in emittance. This problem is resolved by using an arrangement of bending and quadrupole magnets is known as a double-bend achromat (DBA, also called a Chasman–Green lattice, after its inventors [4]), shown in Fig. 1.10.

The natural (i.e. minimum) horizontal electron emittance of a DBA with bending angle $2\theta$ (that is, $\theta$ for each dipole pair) is given by

$$ \epsilon_{x,\text{DBA}} = C_{\text{DBA}}\gamma^2\theta^3, \qquad (1.23) $$

where $C_{\text{DBA}} = 11\sqrt{5\hbar}/384 m_e c = 2.474 \times 10^{-5} \text{ nm}$ [5]. So, for example, the lower limit emittance of a 3 GeV storage ring containing 20 DBAs would be 3.3 nm rad, larger by nearly three orders of magnitude than the photons’ diffraction-limited value of $\lambda/4\pi = 8$ pm rad calculated for 1-Å radiation. Efforts to approach the diffraction limit by using multibend achromats (MBAs) are discussed in Sect. 1.4.

The primary purpose of bending magnets is to maintain the electrons in the storage ring on a closed path. Bending magnets have typical magnetic field strengths of the order of 1 T. They produce bending magnet radiation in a flattened cone with a fan angle equal to that swept out by the path of the electrons due to the Lorentz forces they are subjected to. The relatively large subtended angle of bending magnet
**Fig. 1.11** $K_{2/3}(x)$, the modified Bessel function of the second kind for order 2/3

<figure><img src="image_16.png" /></figure>

**Fig. 1.12** Bending-magnet spectra for on-axis horizontally polarized radiation for three different combinations of the storage ring energy and magnetic field strength. Reproduced from [3] with permission (Copyright 2019, John Wiley and Sons)

<figure><img src="image_17.png" /></figure>

radiation, measured in degrees, allows one to accommodate more than one so-called ‘bending-magnet beamline’ at a single bending magnet.

The spectral flux distribution is given by

$$ \frac{\text{ph/s}}{(\text{mrad})^2 0.1\% \text{BW}} = 1.325 \times 10^{10} I_e [\text{mA}] \varepsilon^2 [\text{GeV}] \left(\frac{E}{E_c}\right)^2 K_{2/3}\left(\frac{E}{2E_c}\right), \quad (1.24) $$

where $E_c = 3\hbar e B \gamma^2 / 2m_e$ is the so-called critical, or characteristic, energy (which in practical units can be expressed as $E_c$[eV] = 665.023 $B$[T] $\varepsilon^2$[GeV$^2$]), and $K_{2/3}(x = E/2E_c)$ is the modified Bessel function of the second kind for nonintegral order (in this instance, 2/3), shown in Fig. 1.11.

The spectral flux is thus determined by the storage ring energy and the magnetic field strength. Increasing $B$ (but keeping $\varepsilon$ constant) shifts the maximum of the spectrum to higher photon energies but does not increase the spectral flux's maximum. In contrast, only increasing $\varepsilon$ both shifts the spectral maximum to higher photon energies and higher values.
The broadband bending magnet spectra for three different combinations of storage ring energy and magnetic field strength are shown in Fig. 1.12. Particularly at low- and medium-energy facilities, the maximum values of both these quantities are too small to extend the spectrum far into the hard X-ray regime. However, if superconducting magnets with larger magnetic field strengths are employed, the photon energy range can be extended to harder X-radiation, as the critical energy is proportional to the magnetic field strength. Moreover, the radiative power increases with the square of the magnetic field strength B. These so-called ‘superbends’ can have magnetic field strengths as high as 8 T.

There are two types of insertion devices, distinguished from each other by the amount that the electrons are forced to deviate in a slalom-like path from a purely straight path. This at first seemingly subtle distinction has a fundamental effect on the nature of the radiation, however. For angular excursions substantially larger than the synchrotron radiation’s natural opening angle $\gamma^{-1}$, the radiation cones from each magnet in the insertion device do not overlap. Under these conditions, the intensities produced from each dipole are added and the ID is referred to as a *wiggler*, which is briefly described below.

For gentler excursions of the order of $\gamma^{-1}$, the ID is called an *undulator*, described in Sect. 1.3.2.

The maximum angular deviation $\phi_{max}$ of the electron oscillations in an ID is defined by the dimensionless ‘magnetic deflection parameter’ K, given by

$$ \phi_{max} = K/\gamma . \qquad (1.25) $$

K can be expressed in terms of the maximum magnetic field $B_0$ as

$$ K = \frac{eB_0}{m_eck_{u,w}} = 0.934 \lambda_{u,w} [\text{cm}] B_0 [\text{T}], \qquad (1.26) $$

where $\lambda_u$ or $\lambda_w$ are the periods of the oscillations in the undulator or wiggler, respectively, and $k_{u,w} = 2\pi/\lambda_{u,w}$. For a wiggler, K is typically between 10 and 50, while for undulators, K is close to unity and changes according to the size of the gap between the upper and lower magnet arrays. The horizontal spread in the electron beam divergence is

$$ \theta_x = 2K/\gamma . \qquad (1.27) $$

So, for example, a wiggler having K = 20, operating in a 4 GeV storage ring would have a horizontal divergence of 5.2 mrad (0.30°).

A wiggler can be thought of as being a series of bending magnets within a straight section of the storage ring that turns the electrons alternately to the left and to the right. The maximum angular excursion from the central axis is larger than the natural opening angle of the radiation, $\gamma^{-1}$. For each oscillation, the electrons are twice moving parallel to (and in reality also very close to) this axis. The radiation is therefore enhanced by a factor of 2N, where N is the total number of wiggler periods and is of
<figure><img src="image_18.png" /><figcaption>**Fig. 1.13** Comparison of brilliances at a 3 GeV DLSR running at 400 mA between a U14 undulator with *K* = 1.6 (blue), a bending magnet with *B* = 1.41 T (red), a superbend with *B* = 4 T (yellow), and a wiggler with the same field strength as the bending magnet and 100 periods (green). Note that the peak brilliance of synchrotron sources can be calculated from the average brilliance shown in this figure by multiplying by the ratio of the pulse separation *Δt* ~ 5 ns to the pulse width *Δτ* ~ 50 ps, that is, approximately a factor 100. The peak brilliance of XFELs is of the order of 10³⁴ ph s⁻¹ mm⁻² mrad⁻² 0.1% BW⁻¹, some ten orders of magnitude greater than that produced by DLSR undulators. Reproduced from [3] with permission (Copyright 2019, John Wiley and Sons)</figcaption></figure>

the order of 20. The spectrum from a wiggler has the same form as that from a bending magnet—it is broadband and thus produces a large amount of integrated radiative power, of the order of several kW. Thermal management of optical components is thus critical. Wigglers are therefore becoming fairly uncommon, particularly in fourth-generation DLSRs.

### 1.3.2 Undulators

In undulators, the angular deviation of the electrons away from the central axis is of the order of $1/\gamma$; the radiation cones emitted by the electrons thus overlap as they execute the slalom motion. Consequently, radiation from the dipoles interferes with one another. As such, the field *amplitudes* are added vectorially (i.e. including the phase difference from each contribution) and the sum of this is squared to produce the intensity, which peaks at those wavelengths where interference is constructive.

Undulators therefore differ fundamentally from bending magnets and wigglers in that their spectral flux reflects this interference phenomenon and is hence concentrated in evenly separated, narrow bands of radiation (Fig. 1.13). The first practical undulator device to operate in the X-ray regime was constructed by Klaus Halbach and co-workers at the Lawrence-Berkeley National Laboratory and tested at the SSRL synchrotron at Stanford in 1981. This breakthrough was thanks on the one hand to the development of novel magnetic alloys such as SmCo₅ [6], allowing the
construction of magnet arrays with the required small periodicities and high magnetic field strengths [7]; and on the other, to a clever arrangement of pole orientations (referred to as the ‘Halbach array’) which effectively suppresses the field strength on one side of the array and doubles it on the other, thus maximizing the magnetic flux where it is needed.

The four basic parameters for undulator radiation from a device of length L are the relativistic Lorentz factor $\gamma$, the undulator spatial period $\lambda_u$, the number of periods in the magnet array $N = L/\lambda_u$, and K. As already stated, for an undulator, K is about unity. K can be varied by changing the gap size between the upper and lower arrays of magnets; this tunes the spectrum so that a suitable near-lying spectral maximum sits at the desired photon energy. The transformation from wiggler to undulator radiation is achieved in practice not by reducing the lateral excursions through reduction of the magnetic field strength between the magnetic pole pairs—this would result in an unacceptable drop in flux—but instead by reducing the magnetic pole spatial periodicity $\lambda_u$ [see (1.26)].

It emerges that the condition for constructive interference is given for the mth harmonic by

$$m\lambda_m = \frac{\lambda_u}{2\gamma^2} \left(1 + \frac{K^2}{2}\right), \qquad (1.28)$$

or in practical units

$$m\lambda_m \AA = \frac{13.056 \lambda_u [\text{cm}]}{\varepsilon^2 [\text{GeV}]} \left(1 + \frac{K^2}{2}\right). \qquad (1.29)$$

The intrinsic source size and divergence of undulators associated exclusively with photon emission (i.e. ignoring the electron emittance) are given by

$$\sigma^p = \frac{1}{4\pi} \sqrt{\lambda L} \qquad (1.30)$$

and

$$\sigma'^p = \sqrt{\frac{\lambda}{L}}, \qquad (1.31)$$

resulting in an intrinsic emittance of

$$\epsilon^p [\text{pm rad}] = \frac{\lambda}{4\pi} = \frac{98.66}{E[\text{keV}]}. \qquad (1.32)$$

Note that the divergence $\sigma'^p$ can also be expressed in terms of the harmonic number m and number of periods N, and is approximately equal to $\sigma'^p = 1/(\gamma\sqrt{mN})$. The
<figure><img src="image_19.png" /><figcaption>**Fig. 1.14** Measurements of optimized gap positions for different photon energies and utilized harmonics of the U14 undulator of the Materials Science beamline, SLS, from the second to nineteenth harmonic. In normal operation, scanning to higher energies within any given harmonic is achieved by opening the undulator gap (the progression of this between the third and fifth harmonics is shown as the dot-dashed blue lines). One moves from a given harmonic to the next higher (red dot-dashed lines) when the desired photon energy can be accessed at the higher harmonic with a gap size no smaller than the minimum allowed value, here 4 mm, shown as the blue horizontal dashed line. Adapted from [8], with permission (Copyright 2013, IUCr)</figcaption></figure>

divergence thus becomes smaller with harmonic number and number of periods in the undulator.

The interference spectrum at an angle $\theta$ away from the central axis of the undulator is shifted towards lower energies, and is given by

$$m\lambda_m(\theta) = \frac{\lambda_u}{2\gamma^2} \left( 1 + \frac{K^2}{2} + \gamma^2\theta^2 \right). \qquad (1.33)$$

For example, an observer positioned off axis by half the natural opening angle $\theta = 1/2\gamma$ (approximately 3.5 mm at a distance of 40 m for a facility with a 3 GeV storage ring energy) would, for $K = 1$, see a spectrum shifted by a factor 7/6 to the red.

The spectral width of the undulator harmonics is inversely proportional to the number of periods, $N$. As in any interference or diffraction set-up, the condition for constructive interference becomes increasingly strict the larger the number of involved 'scatterers' (in this instance, the $2N$ magnet pairs). Hence, the inverse of the relative bandwidth, called variously the monochromaticity or the quality factor $\lambda_m/\Delta\lambda_m = \nu_m/\Delta\nu_m$, is equal to the number of periods $N$ multiplied by the harmonic number $m$. As an example, the tenth harmonic of an undulator consisting of $N = 70$ periods has a relative bandwidth $\Delta\lambda_m/\lambda_m$ of $1.4 \times 10^{-3}$.

The undulator spectrum is tuned by varying $K$. This is achieved by changing the gap between the two sets of magnetic poles and thereby the magnetic field strength $B_0$ [see (1.26) and Fig. 1.14].

Note that, for reasons of symmetry, even harmonics are in general weaker than odd harmonics. Higher $K$ undulators provide both more intense higher-energy harmonics than lower $K$ devices, while the difference in intensities between even and odd harmonics is less pronounced.
### 1.3.3 Polarization of Synchrotron Radiation

Polarization of light in spectroscopy is a highly valuable tool, as certain selection rules imply that certain electronic features can be distinguished merely by using differently polarized X-radiation, for example, in the study of surface absorbates on crystalline surfaces, or in distinguishing between differently oriented ferromagnetic or antiferromagnetic domains. Magnetic dichroism using tunable and polarized soft or hard X-rays at synchrotron sources offers the unique feature of chemical specificity. Magnetic features down to below the 100-nm scale can be imaged by exploiting the dependence of the absorption on the polarization of X-rays in ferromagnetic and antiferromagnetic materials. The methods are called X-ray magnetic circular dichroism (XMCD) and X-ray magnetic linear dichroism (XMLD), respectively, for ferromagnetic and antiferromagnetic materials (see Chap. 4). Imaging down to the nanometre range is very important for magnetic structures, as at this scale, the influence of domain boundaries between one magnetic direction and another becomes significant and new phenomena can occur which would be negligible in larger structures. An understanding of the energetics of nanomagnetism is thus essential in the drive to further miniaturize magnetic memory storage devices.

The polarization of undulator radiation can be controlled and varied in so-called APPLE (Advanced Planar Polarized Light Emitter) undulators. APPLE undulators have undergone successive refinements (reflected by their generational names of APPLE-I, II, III and X), based on the flexibility of movements of their magnet arrays; here, we consider only the basic principles.

Instead of the two Halbach magnet arrays found in 'normal' undulators discussed thus far, APPLE undulators consist of four arrays (two above, two below) that can be longitudinally shifted relative to each other [A1–A4, Fig. 1.15a] [9]. Each array consists of a periodic repetition of four different orientations [down (blue ⊗); reverse (green arrow); up (yellow ◦); and forward (red arrow), see Fig. 1.15b]. With all array components aligned longitudinally, normal linear horizontal (LH) radiation as found in 'normal' undulators is produced. Antisymmetric shifts (that is, one array moving in the positive z-direction, the other in the negative) of A1 and A4 between $-\lambda_u/2$ and $+\lambda_u/2$ will maintain linearly polarized radiation but will vary the tilt angle from linear vertical (LV at $-\lambda_u/2$), through LH (zero shift) back to LV ($+\lambda_u/2$). Conversely, symmetric movements of A1 and A4 by approximately $\pm\lambda_u/4$ will produce right circularly polarized light (at approximately $-\lambda_u/4$), or left circularly polarized light (at $+\lambda_u/4$). In this last instance, the exact shift in the arrays depends on the gap size and details of the magnet strengths. Symmetric shifts smaller than this result in elliptical polarization, that is, radiation with both a linear and circularly polarized component.

Synchrotron sources provide fluxes and brilliances many orders of magnitude greater than those from laboratory-based sources. Third-generation facilities were defined by their use of insertion devices, most importantly undulators. The fourth generation of synchrotrons, just beginning to come online at the time of writing, increase brilliances by two orders of magnitude by reducing the horizontal (orbital
<figure><img src="image_20.png" /><figcaption>**Fig. 1.15** APPLE undulators. **a** APPLE undulators consist of four magnet arrays A1, A2, A3 and A4. **b** The magnet periodicity $\lambda_u$ is composed of four magnet orientations. Viewed from above, these are down (blue $\otimes$); reverse horizontal (green arrow); up (yellow $\odot$); and forward horizontal (red arrow). When all four arrays are aligned longitudinally, linear horizontal polarization is produced. **c** By shifting arrays A1 and A4 either symmetrically or antisymmetrically by $\pm\lambda_u/2$, linear vertically polarized radiation is produced. **d** Linear polarization of any desired tilt angle can be selected by moving A1 and A4 antisymmetrically relative to A2 and A3. **e** and **f** Circularly polarized radiation is generated by symmetric shifts of A1 and A4 by approximately $\lambda_u/4$, the exact shift depending on the gap size and details of the magnetic field strengths. Reproduced from [3] with permission (Copyright 2019, John Wiley and Sons)</figcaption></figure>

plane) emittance through the implementation of MBAs. We discuss this in more detail now.

## 1.4 Diffraction-Limited Storage Rings

As we have already intimated, the main limit to the emittance in storage rings is due to the electron spread induced at bending magnet achromats as a consequence of quantum excitation (Sect. 1.2.6). In our discussion of DBAs in Sect. 1.3.1, the lower limit to the electron emittance, its so-called ‘natural emittance’, was given by (1.23). However, if we extend an arc section simply by increasing the number of dipoles from 2 to M in a so-called MBA, and *keeping the swept angle per dipole constant*, the ratio between the natural emittances of the DBA and MBA is

$$ \frac{\epsilon_{x, \text{DBA}}}{\epsilon_{x, \text{MBA}}} = 3 \frac{M - 1}{M + 1}. \qquad (1.34) $$
Such an MBA, subtending an arc of angle $M/2$ times larger than that of the DBA, will have a superior emittance up to a factor of three for large $M$. So, for example, a 10° DBA will have a natural emittance 2.25 times that of a 35° 7BA ($M = 7$) for a fixed swept angle per dipole.

This is clearly not the way to proceed, as an increase from two bends to $M$ will reduce the number of straights around the ring by a factor of $M/2$. A far more effective way to improve the natural emittance of an MBA can be achieved by reducing the swept angle $\theta$ per dipole. This is due to the cubic dependence of the former on the latter [see (1.23)]. So, if in the example above, the 7BA is designed to sweep the same total angle as the DBA, the reduction in emittance will now be $(4/9) \times (2/7)^3 = 32/3087 \approx 1/100$, thus producing emittances which assume values of the order of 100 pm rad. This is only an order of magnitude larger than the diffraction-limited photon emittance for hard X-rays of approximately 10 pm rad. Indeed, soft X-ray beamlines, for which $\lambda/4\pi \sim \epsilon_{x,MBA}$, are already diffraction-limited when using MBAs.

Note also that the natural emittance scales with the square of the storage ring energy $\mathcal{E}$. The two highest-energy third-generation storage rings, namely the APS (7 GeV) and SPring8 (8 GeV) are both planning to decrease $\mathcal{E}$ to 6 GeV in their upgrades, despite the associated decrease in the highest accessible photon energies.

In most instances, because an achromat (be it a DBA or MBA) requires about the same length of 'real estate' within the storage ring, the ring's circumference $C$ is approximately inversely proportional to $\theta$—large rings have bend achromats that subtend smaller angles than those at small rings. This is why the large APS facility has 36 sectors, while the smaller ALS in Berkeley only has 12. Thus, for a fixed cell design, a convenient figure of merit

$$M = \frac{\epsilon_x C^3}{\mathcal{E}^2} \qquad (1.35)$$

describes how well optimized the magnet lattice is [10], as this takes into account the available real estate and electron beam energy and weighs the emittance by this, accordingly.

The newest (fourth) generation synchrotron facilities are thus dubbed 'diffraction-limited storage-rings' (DLSRs) [10–13]. A light source is said to be diffraction-limited if the horizontal electron beam emittance is smaller than that of the radiated photon beam ($\lambda/4\pi$). In practical terms, the diffraction-limited photon energy $E_{DL}$ with a wavelength equal to $4\pi\epsilon_x$, is given by

$$E_{DL}[\text{keV}] = \frac{98.66}{\epsilon_x[\text{pm rad}]} \qquad (1.36)$$

By this metric, even a second-generation facility would be a DLSR for a beam-line generating infrared radiation; in contrast, most third-generation facilities are diffraction-limited in the very soft X-ray regime at around 25 eV. Note that beam-lines that use photon energies that lie near to or below the diffraction limit for the
facility where they are installed gain little or nothing from the emittance of the stor-
age ring being further improved (i.e. reduced by more sophisticated electron optics).
This is the true meaning of the diffraction limit: the fundamental lower value set by
the photon beam rather than the electron beam.

DLSRs are thus defined by their use of MBAs. They gain most at large-
circumference storage rings, at medium storage ring energies and indeed the two
greenfield DLSR facilities to date (MAX-IV and Sirius) plus the first major upgrade
from third to fourth generation (ESRF-EBS) reflect these characteristics. This seem-
ingly obvious and long-understood approach to improve the horizontal emittance via
the use of MBAs and thereby the brilliance has only recently been pursued because,
until now, the costs (both in hardware and real estate) and the introduction of poten-
tial mechanical misalignments associated with increasing the number of elements in
the magnet lattice, were considered unacceptable. Miniaturization of these magnet
lattice components and the development of multifunctional magnets machined from
a single yoke block have decreased both costs and the necessary circumference of
the ring to accommodate the MBAs and thus achieve these goals. The PETRA-IV
project is especially interesting, as the circumference is 2.3 km, allowing exception-
ally shallow angles for the arcs, promising diffraction-limited photon energies well
into the hard X-ray regime, at approximately 6 keV.

Another serious obstacle to reducing the size and separation of the magnets is
that the lateral dimensions (that is, the cross-section) of the storage ring vacuum
vessels containing the electron beam need to become so small that pumping them with
traditional pumping equipment, in particular, ion-getter pumps, becomes increasingly
difficult. In recent years, however, a novel approach to achieve ultrahigh vacuum
(UHV) conditions has been developed, namely the use of non-evaporable getter
to the order of a micrometre, or even as little as 100–200 nm.
After installation, the vacuum vessel is pumped to a moderately high vacuum using
traditional pumps and then heated out to temperatures below 200 °C. This activates the
NEG material, allowing pressures to drop to approximately 10<sup>-10</sup> mbar in the UHV
regime. Importantly, even very narrow spaces can be readily coated. Moreover, recent
developments in computer numerical control of machining storage ring components
allow micrometre accuracy such that geometrically near-perfect miniature vacuum
vessels with cross-sections of the order of a square centimetre can be constructed.

With the advent of DLSRs and improvements in magnetic materials, undulator
spectra have undergone important transformations. Figure 1.16 compares the brill-
liance of the same undulator at a third-generation source and at a DLSR having a
40 times smaller total horizontal emittance. In addition to the expected 40 times
increase in peak brilliance, the DLSR-spectrum is substantially cleaner: the lobes
seen on the low-energy flanks of the spectral peaks for the third-generation source
are completely absent in the DLSR spectrum. This is because the horizontal width
of the electron beam is much smaller (typically by an order of magnitude). In third-
generation facilities, the horizontal electron beam width is, at approximately 100 µm,
two orders of magnitude larger than the oscillation amplitude A, which is of the order
<figure><img src="image_21.png" /><figcaption>**Fig. 1.16** Comparison of the brilliance of undulator spectra at third- and fourth-generation facilities. Top: the lateral extent of the electron beam passing through an undulator at third-generation facilities is approximately two orders of magnitude larger than the oscillation amplitude A, while at DLSRs, it might only be approximately 10Å. Bottom: as a consequence, an observer on-axis will ‘see’ less off-axis radiation, given by (1.33), thereby suppressing the lobes on the low-energy flanks of the main spectral maxima. Note also the enhanced brilliance at the spectral peaks for the DLSR. Both simulated spectra were generated for a U12 undulator (that is, $\lambda_u = 12$ mm) containing 120 magnet periods, for $K = 1.6$, 400 mA and a storage ring energy of 2.4 GeV. Courtesy Marco Calvi, Paul Scherrer Institute. Reproduced from [3] with permission (Copyright 2019, John Wiley and Sons)</figcaption></figure>

of a micrometre. This means that an observer on axis at the DLSR-undulator will see a much smaller contribution from emission from off-axis electrons, and it is these that produce the low-energy lobes in the spectra, according to (1.33). Many experiments do not require the relative BWs of the order of $10^{-4}$ provided by crystal monochromators and would profit from using the entire flux from any given harmonic. The relative BW of the $m$th undulator harmonic is $1/mN$, where $N$ is the number of undulator periods. For the lower harmonics, this is of the order of 0.005–0.01; for higher harmonics, the relative BW does not drop as steeply as $1/m$, as gradually, the off-axis contributions do begin to leak into the peaks, causing them to broaden marginally. Nonetheless, the spectral quality remains sufficiently high to use the entire flux of any given harmonic for small-period undulators at DLSRs. Note, however, spectral filtering is still required to remove the other harmonics. This
can be effectively achieved using multilayer monochromators or, in some instances,
refractive optics such as compound refractive lenses or prisms [16].

Improvements don't stop here, however. A critical aspect of undulator design is
that the magnetic field must be exceedingly homogeneous in the *x*-direction (that
is, in the horizontal plane, perpendicular to the undulator axis) in the region where
the electron beam propagates. This has meant that, for third-generation facilities, the
magnet yokes need to be a few centimetres wide or more in the *x*-direction. The
approximately five to ten times smaller extent of electron beams passing through
undulators installed in DLSRs means that the yokes can be of the order of one
centimetre. This allows the design of compact magnetic 'funnels', concentrating
magnetic field lines and thus increasing the magnetic field strength, which, in turn,
allows for a more compact design and shorter undulator periods. Moreover, this
means that more periods (*N*) can be fit into a given undulator length *L*.

The reduction of the horizontal electron emittance in DLSRs thus lends many
potential scientific opportunities and will drive exciting innovations along the full
technology chain, from the sources (in particular, undulators), through improved X-
ray optics (regarding minimization of optical imperfections and aberrations), detector
technology, to handling of large data volumes. The first DLSR, MAX-IV in Lund,
Sweden, came online in Summer 2016, while a second greenfield facility. Sirius in
Campinas, Brazil and the ESRF, the first facility to undergo an upgrade from third
generation to DLSR status, both began pilot experiments in early 2020.

Several orders of magnitude higher peak brilliance are provided by high-gain
X-ray free-electron lasers (XFELs). The machine science and technologies for this
paradigm shift in X-ray sources are based on fundamentally different principles, and
consequently, XFELs are discussed separately in the next section.

1.5 X-Ray Free-Electron Lasers

Radiation from fourth-generation DLSRs has some important common features with
laser radiation: it is very intense, collimated and, in the case of radiation from undula-
tors, partially monochromatic. An important distinction, however, is that the degree
of transverse (spatial) coherence of synchrotron radiation, although much improved
at DLSRs compared to that produced by third-generation facilities, is still only of the
order of a few percent in the hard X-ray regime. This is because, although radiation
from any single electron travelling along an undulator is coherent, there is no spatial
(i.e. phase) correlation between different electrons, and hence their combined output
remains largely incoherent. In contrast, visible lasers are normally close to being
100% coherent.

Moreover, the shortest pulse duration of X-rays from synchrotrons is a few tens
of picoseconds and may be as large as a few hundred picoseconds in the case of
DLSRs. Lasers in the visible and near-visible regimes can have pulse lengths as small
as 70 attoseconds in some exceptional cases; more representatively, pulse lengths of
a few femtoseconds are routinely achieved. The production of femtosecond light
pulses has been motivated by studies of the dynamics of chemical reactions—this
is one of the most important goals in science, as a biomolecular understanding of
the processes of life, plus most industrial chemical processes, depend intimately
on understanding transient intermediate states. Indeed, the holy grail of physical
chemistry is the direct observation of the making and breaking of chemical bonds
and following reactions through the reconfiguration of the atomic structure. In order
to achieve this, one needs a probe that can spatially resolve the atomic positions,
measured in angstroms, thus necessitating the use of hard X-rays. Moreover, in order
to record a movie of chemical processes, each ‘frame’ must be shorter than the speed
of the process divided by the desired resolution, which we have just stated should
be of the order of an angstrom. Typical velocities associated with atomic motion
are of the order of 1000 m s⁻¹, hence each movie frame should be separated by
no more than approximately 100 fs. For example, phonons and molecular rotations
have time scales of approximately 100–1000 fs, while molecular vibrations, crucial
to the process of chemical reactions, are shown to have periodicities of the order of
10–100 fs.

X-ray free-electron lasers (XFELs) were developed to fulfil this goal. They provide extremely intense and short pulses of X-rays. XFELs and synchrotron facilities differ substantially in this respect (see Table 1.1). The total average number of photons per second delivered to a third- or fourth-generation synchrotron beamline after monochromatization (typically between 10¹³ and 10¹⁴ s⁻¹) is comparable to that of an XFEL beamline, although the latter can vary by orders of magnitude, depending on the facility repetition rate and whether the beam is monochromatized or not. In contrast, their time structures are very different: synchrotrons deliver pulses with the same frequency as the RF supply, typically several hundred million pulses per second (1/Δτ); each pulse is maybe a few tens of picosecond long (Δτ) and contains the order of 10⁴–10⁵ photons. XFELs deliver anything between approximately 100 and several hundreds of thousands of pulses per second, depending on the electron gun source and the properties of the accelerating LINAC. Most importantly, each XFEL pulse is only a few tens of femtoseconds long (and can be even shorter, down to a femtosecond) and contains the order of 10¹² photons. The peak arrival rate of photons at XFELs is thus approximately 10 billion times higher than that at synchrotron beamlines.

XFELs thus provide the tools needed to study the dynamics, physical properties
and structure of materials with unsurpassed spatial detail and on a time scale over a
thousand times shorter than that which is possible at synchrotron facilities.

The beam quality in storage rings is limited by the stochastic competing processes described above in radiation equilibrium. High-gain XFELs are made possible by a runaway process called self-amplified spontaneous emission (SASE). Because, in contrast to synchrotrons, the electrons in XFELs are not stored and require only a few microseconds to traverse the length of the entire XFEL facility, they are far less perturbed than are the equilibrated electrons in a storage ring, meaning they can maintain a very low emittance defined by the electron gun, acceleration and bunch compression mechanism.
<table><caption>**Table 1.1** Comparison of orders of magnitude synchrotron and XFEL radiation† properties. †XFEL values derive from LCLS unless otherwise stated. 8-keV photons assumed. *EuroXFEL time structure: 2700 pulses at 4.5 MHz, 10 such bursts per second. **photons s⁻¹ 0.1% BW⁻¹. ‡After Si(111) monochromator, Δv/v = 1.4 × 10⁻⁴. ¶Unmonochromatized, full SASE spectrum. §23 kW during pulse burst</caption><thead><tr><th>Property</th><th>Synchrotron</th><th>XFEL</th></tr></thead><tbody><tr><td>Δτ</td><td>50 – 400 ps</td><td>1 – 100 fs</td></tr><tr><td>Δt</td><td>5 ns</td><td>10<sup>-2</sup> – 2 × 10<sup>-7</sup> s*</td></tr><tr><td>Average flux**</td><td>2 × 10<sup>14</sup></td><td>10<sup>14</sup></td></tr><tr><td>Peak flux**</td><td>6 × 10<sup>15</sup></td><td>2 × 10<sup>25</sup></td></tr><tr><td>#hv/pulse</td><td>4 × 10<sup>4</sup> ‡</td><td>4 × 10<sup>12</sup> ¶</td></tr><tr><td>Peak power</td><td>1 W ‡</td><td>10<sup>11</sup> W¶</td></tr><tr><td>Average power</td><td>25 mW‡</td><td>600 mW¶ – 140 W¶*§</td></tr></tbody></table>

The linear architecture of XFELs is very different than that of synchrotron facilities; it is summarized in the following.

## 1.5.1 XFEL Architecture

The fate of electrons in a high-gain XFEL proceeds as follows. In order to generate pulses of electrons, a ps laser is focussed on to the surface of a photoemitter. The resulting ps-duration electron pulse contains a charge of the order of 300 pC (or 2 × 10⁹ electrons) and is accelerated in a first LINAC to energies of the order of several hundred MeV (see Fig. 1.17). Shorter laser pulse durations containing the same total number of electrons are excluded due to Coulomb repulsion, which would blow up the beam size and thus spoil the emittance. However, once the electrons become relativistic after acceleration in the first, short, LINAC, the bunch can be compressed without spoiling the emittance. This is achieved in one or more bunch compressors; thereafter, the electrons are further accelerated in a longer LINAC, before entering a very long undulator array. It is while the electrons are travelling along the undulator that the SASE process takes place, resulting in the emission of femtosecond-duration pulses of X-rays.

The initial few-ps-duration electron bunch produced by the low-emittance gun has a transient peak current of about 50 A (300 pC/6 ps). Bunch compression is required in order to reduce the pulse duration to approximately 300 fs and produce the several thousand amperes peak current required to induce SASE (Fig. 1.18).

The necessary factor of approximately 20–50 increase is realized by adjusting the RF phase in the first accelerator module (LINAC 1 in Fig. 1.17) to allow the electrons to ‘surf’ down the slope of the sinusoidal RF field—the leading electrons experience a slightly smaller acceleration by the sinusoidal electric field of the RF cavity and are thus accelerated less than those electrons towards the back of the
<figure><img src="image_22.png" /></figure>

**Fig. 1.17** Schematic of XFEL architecture. Electron bunches are emitted from a low-emittance gun (LEG) irradiated by picosecond laser pulses. They are then accelerated in a short LINAC (LINAC 1), then compressed using one or more bunch compressor magnet chicanes (BC); they are then further accelerated using a much longer LINAC (LINAC 2) before entering a long undulator, typically a few hundred metres in length. The SASE process along the undulator produces extremely intense X-ray pulses with durations of the order of 50 fs. Reproduced from [3] with permission (Copyright 2019, John Wiley and Sons)

<figure><img src="image_23.png" /></figure>

**Fig. 1.18** Bunch compression in high-gain XFELs. Before entering the first LINAC, the electron density $n_e$ and spread in kinetic energy $\Delta E$ are both relatively low. The phase of the LINAC RF-field relative to the passage of the bunch is so adjusted to induce a larger spread of the electrons' kinetic energies, whereby the bunch's trailing edge is made to be more energetic than the leading edge. Compression is achieved by allowing the bunch to pass through a four-dipole chicane, where the faster electrons at the back can catch up with the slower electrons at the front, thanks to the shorter path that they execute. Reproduced from [3] with permission (Copyright 2019, John Wiley and Sons)

bunch (a phenomenon referred to as 'chirping'). However, the electrons' velocities are already so close to the speed of light that any differences are far too small to allow the faster electrons at the back of the bunch to catch up with the slower electrons positioned further forward and thereby squeeze the bunch length. Instead, the bunch passes through a magnetic four-dipole chicane. The trailing (high-energy) electrons execute a shorter path through the chicane because they are less deviated by the chicane's magnetic field. This shorter path means that they catch up with the less energetic leading electrons, which are more deviated by the magnet chicane. This compression thus shortens the bunch duration to approximately 300 fs (equating to a
bunch length of 100 µm) and increases $n_e$ by a factor of approximately 20–50. The increased energy spread of the chirped electron beam after bunch compression must be removed through ‘dechirping’, which can be achieved in one of several ways [18, 19].

### 1.5.2 The SASE Process

As the electrons begin to propagate down the undulator, they initially emit X-rays independently and stochastically and are bathed in this radiation. They will interact with this EM field. We begin our discussion of SASE by determining the magnitude of these forces in conventional undulators at synchrotrons.

Classical electromagnetism tells us that the power transmitted per unit area by an EM plane wave is given by

$$ \frac{P}{A} = \frac{E_0 H_0}{2}, \qquad (1.37) $$

where A is the cross-section of the beam, $E_0$ is the amplitude of the electric field component and $H_0$ is the amplitude of the magnetic intensity, which, in vacuum, is related to the magnetic field strength amplitude $B_0$ by

$$ H_0 = \frac{B_0}{\mu_0}, \qquad (1.38) $$

where $\mu_0 = 4\pi \times 10^{-7} \text{ mkgC}^{-2}$ is the permeability of free space (given here in SI units). Also from classical electromagnetic theory, it emerges that

$$ B_0 = \frac{E_0}{c}. \qquad (1.39) $$

From (1.37) to (1.39), we thus obtain an areal power density

$$ \frac{P}{A} = \frac{E_0^2}{2\mu_0 c}. \qquad (1.40) $$

A typical synchrotron undulator may have a source size of $A = 100 \times 10 \text{ µm}^2$, produce light bunches of 50-ps duration, each containing $5 \times 10^6 \text{ 1-Å photons}$ (within the full width of a harmonic). This equates to an areal power density of the order of $1.6 \times 10^{11} \text{ Wm}^{-2}$. From (1.39) and (1.40), one calculates an electric field amplitude $E_0 \approx 10^7 \text{ Vm}^{-1}$ and $B_0 \approx 40 \text{ mT}$. This latter value is nearly two orders of magnitude smaller than that imposed by the undulator’s magnet array. We can thus conclude that, in a conventional third- or fourth-generation synchrotron facility, the
forces of the electromagnetic field generated by undulator radiation on the electrons' trajectory are negligible.

In the case of XFELs, however, the beam is tailored to have as low an emittance and as high an electron density as possible, through the low-emittance gun and bunch compression, respectively. The motivation for this is exactly to induce forces through the generated radiation that are sufficiently large that they do indeed have an impact on the energy and spatial distribution of the electron bunch. Although at the upstream end of the XFEL undulator, this interaction is still very weak, it is sufficient to seed the runaway process of SASE, which is now briefly described.

The ratio of the Lorentz force and electric field force experienced by an electron bathed in EM radiation is

$$F_L = eE\frac{v}{c} = -F_E\beta. \qquad (1.41)$$

The electrons within XFELs are highly relativistic and hence to a high degree of accuracy, $\beta \approx 1$ and $F_L \approx -F_E$ (the fractional imbalance in the forces is equal to $\Delta F/F_E = 1/2\gamma^2 \sim 2 \times 10^{-9}$). Thus, for an electron moving exactly along the axis of the EM radiation (that is, ignoring for the time being the oscillations induced by the magnet array of an undulator), the electric and Lorentz forces are equal and opposite and point perpendicularly to the beam propagation.

If the electrons move in a straight line parallel to the EM plane wave, both the electric and magnetic forces (which anyway cancel each other out) always act at right angles to the electrons' direction of propagation and thus could transfer energy to or from the electrons. What happens if we now allow the electrons to move in directions that are not precisely parallel to the EM radiation, such as in the slalom path induced by an undulator's magnet array? Let us now consider those positions exactly in between the magnet poles, where the electrons have their maximal transverse velocity.

The first observation to make is that, because the magnetic field of the emitted EM radiation always lies perpendicular to the plane of the electrons' trajectory, it cannot transfer energy, but only vary the direction of the electrons' motion. In contrast, the electric field part of the emitted radiation has a component parallel to the electrons' trajectory and can either decelerate them or accelerate them. Consequently, some electrons will be accelerated while others are decelerated. These two opposing interactions cause the electrons to form microbunches within the 'normal' bunch, separated by a distance equal to the wavelength of the light they both generate and are bathed in. Although, for hard X-rays, the 'normal' bunch length of approximately 100 µm equates to approximately $10^6$ microbunches, SASE only begins in the coincidentally most intense portion of the bunch. As such, SASE is a stochastic process. The duration of the SASE radiation therefore depends on the degree of bunch compression and the integrated charge of the initial bunch as it enters the undulator array. Low-charge bunches will thus produce XFEL pulses with lower peak brilliance but with durations that can be shorter than 1 fs. More commonly, 'standard' bunch durations are a few tens of femtoseconds.
<figure><img src="image_24.png" /><figcaption>**Fig. 1.19** Incoherent and coherent radiation from an ensemble (bunch) of electrons. If the relative longitudinal positions of the electrons bear no correlation to the radiation they emit, the relative phases of that radiation are likewise random. From normal statistics, it can be shown that their summed amplitude is equal to the amplitude of the radiation emitted from one electron multiplied by the square root of the number of electrons ($n_e$) in the bunch. The intensity (proportional to the square of the amplitude) is therefore proportional to $n_e$. In contrast, the radiation induced by SASE is fully coherent, as the emission from all electrons is in phase. The amplitudes of the radiation from each of the $n_e$ electrons thus add linearly, resulting in an increase in intensity by $n_e^2$. Reproduced from [3] with permission (Copyright 2019, John Wiley and Sons)</figcaption></figure>

SASE is a runaway effect. The more the electrons bunch together into microbunches, the stronger is the radiation they produce, leading to stronger associated forces on those electrons and still tighter microbunching.

An important property of the microbunches produced by SASE is the following. In the case of normal SR, the emission of radiation from $n_e$ electrons within a bunch is stochastic and uncorrelated, and hence the vector addition of the individual amplitudes will follow normal statistics and, on average, be equal to $A_{tot} = \sqrt{n_e}A$, where A is the amplitude of the contribution to the radiation of any one electron. But the intensity is proportional to the square of the amplitude and is therefore proportional to $n_e$.

In the case of XFEL SASE radiation, the initial bunch is modified so that it becomes a set of microbunches separated from their neighbours by the wavelength of light produced by the undulator interference. It follows that the individual contributions to the radiation amplitude will all be more or less in phase and the electrons will all emit coherently. Thus, the amplitudes add linearly as they all have the same
phase and hence $A_{tot} = n_e A$. The intensity will be proportional to $n_e^2$ (see Fig. 1.19). This is the key to the ‘runaway’ amplification in SASE: the more the electrons bunch and radiate in phase (that is, coherently), the stronger is the EM-field interaction with them. This in turn further enhances the microbunching phenomenon associated with coherent emission, thus increasing the degree of coherent emission in a positive feedback loop.

This runaway instability causes the light intensity to grow exponentially along the undulator until the process saturates [20]. The microbunches produced by the SASE process each contain approximately $n_e = 10^9$ electrons, resulting in a peak brilliance for XFELs approximately one billion times larger than that from fourth-generation synchrotron sources!

The quality of a high-gain XFEL is encapsulated in the dimensionless so-called Pierce, or FEL, parameter, $\rho_{FEL}$, given approximately by

$$ \rho_{FEL} = \left[ \frac{\lambda_u^2 r_0 n_e K^2}{32 \pi \gamma^3} \right]^{1/3}, \qquad (1.42) $$

where $\lambda_u$ is the undulator period length, $r_0 = 2.82 \times 10^{-15}$ m is the Thomson scattering length (also known as the classical electron radius), $N_e$ is the electron density of the pulse bunch as it enters the undulator, $K$ is the deviation parameter, and $\gamma = \mathcal{E}/m_e c^2$ is the Lorentz factor. Equation (1.42) can be re-expressed in practical units as

$$ \rho_{FEL} = \frac{1.55 \times 10^{-5}}{\mathcal{E}[\text{GeV}]} \left[ (\lambda_u[\text{mm}])^2 N_e[\text{mm}^{-3}] K^2 \right]^{1/3} . \qquad (1.43) $$

Electron densities in the compressed electron bunch entering the undulator are typically $10^4 \mu\text{m}^{-3}$ (that is, $10^9$ electrons in a volume of the order of $10^5 \mu\text{m}^3$); the undulator period $\lambda_u = 15-30$ mm, and K-values of approximately 2–3 are normal. The LINACs produce electron energies of the order of 10 GeV. Because of the weak cube-root dependency of $\rho_{FEL}$ on all the design variables except $\mathcal{E}$, its spread is fairly narrow across all facilities designed or commissioned to date, and normally assumes values of around $5 \times 10^{-4}$.

$\rho_{FEL}$ determines three important properties of high-gain XFEL radiation. Firstly, it describes the fraction of the electrons' power converted at SASE saturation to photon power, that is

$$ \rho_{FEL} = \frac{P_{ph}}{P_e}. \qquad (1.44) $$

The larger is $\rho_{FEL}$, the more efficiently electron energy is converted into photon energy. $\rho_{FEL}$ also describes the relative spectral BW at saturation,

$$ \rho_{FEL} = \frac{\Delta\omega}{\omega}, \qquad (1.45) $$
which means that at photon energies of the order of 10 keV, the instantaneous BW (that is, the BW of any one FEL pulse) is of the order of 5 eV. This is over an order of magnitude larger than the transmitted BW after monochromatization with a diamond (004) single crystal. In addition, the stochastic nature of the initial spontaneous generation of SASE radiation is responsible for the central position of the SASE spectrum jittering from pulse to pulse by a few tens of eV. This is acceptable for many types of experiment, such as in serial femtosecond crystallography, but can be problematic in others, especially in spectroscopy. The bandwidth can be reduced in several ways including self-seeding, or the use of chicanes in so-called high-brightness SASE. It lies outside the remit of this overview to detail these here [3].

In contrast to the transverse coherence length, the longitudinal coherence length of SASE pulses [which depends inversely on the BW, see (1.14)] is small, of the order of 200 nm in the hard X-ray regime.

Thirdly, the distance $L_G$ within an XFEL undulator required to obtain a gain in SASE radiation by a factor $e \approx 2.72$ is inversely proportional to $\rho_{FEL}$ and is given by

$$L_G = \frac{\lambda_u}{4\pi\sqrt{3}\rho_{FEL}}. \qquad (1.46)$$

Inserting our known values for $\lambda_u$ and $\rho_{FEL}$, we obtain values for $L_G$ of the order of a few metres. Saturation of SASE occurs after a gain of approximately five orders of magnitude. But $10^5 \approx e^{11}$, and hence the total undulator length should exceed $L_G$ by a factor of 11 or more. The LCLS hard X-ray undulator is over 110 m long.

Hard XFEL radiation can therefore directly track atomic motions in condensed matter and vibrations with periods of the order of picoseconds down to tens of femtoseconds, a hitherto inconceivable scientific endeavour. XFELs deliver peak brilliances many orders of magnitude greater than radiation from third- and fourth-generation synchrotron sources, 100% transverse coherent radiation, and pulse durations typically of a few tens of femtoseconds, but which can also be tailored to be less than one femtosecond [21, 22].

### 1.5.3 Concluding Remarks

The main objective of this section was to summarize the machine physics of XFELs in a manner that is accessible to a wide spectrum of potential users. XFEL technology and science are bound to further develop and morph in the next decades, as, in contrast to synchrotron science and technology, XFELs are still very much in their infancy. Nonetheless, they are maturing rapidly, thanks in no small part to the knowledge base already established for synchrotron facilities. Because the radiation produced by XFELs differs so greatly from SR in the peak brilliance and associated very short pulse durations, experimental methods commonly used at synchrotrons need more often than not to be entirely rethought in order to be carried out at XFELs; indeed, some experimental methods at synchrotrons are wholly excluded at XFELs.
On the other hand, some experimental techniques first developed at XFELs, in particular serial femtosecond crystallography, are now being adopted and adapted for synchrotron light. XFELs should thus not be considered as superior alternatives to synchrotrons, but more as complementary facilities.

Many experimental disciplines, particularly in the fields of time resolved studies and nonlinear light-matter interactions, have emerged using XFELs and are still rapidly evolving; this is the reason, in fact, why XFELs were developed in the first place. It will be intriguing to see how these develop and expand in the forthcoming years and decades. The future of photon science indeed looks bright!

## 1.6 Summary

This chapter has introduced the reader to the basic physics of radiation produced by accelerated electrons and the recent advances in machine technologies which have resulted in the development of DLSRs and XFELs. The number of dedicated X-ray sources has burgeoned over the last three decades and has opened the field of synchrotron science, transforming it into a multidisciplinary enterprise, nowadays attracting some hundred thousand scientists across the broadest spectrum of the natural sciences.

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<figure><img src="image_25.png" /></figure>
# Chapter 2
Concepts in Magnetism

<figure><img src="image_26.png" /></figure>

Check for
updates

Stephen J. Blundell

**Abstract** I review some general concepts in magnetism including the nature of magnetic exchange (direct, indirect and superexchange), and how exchange interactions play out in multiple spin systems. The nature of atomic orbitals and the way in which they interact with the spin system is also considered. Several examples are also treated, including the Jahn-Teller interaction and its role in the properties in layered manganites.

## 2.1 Introduction

Magnetic properties are found in a wide variety of materials. In order to explain magnetism we need to consider a range of different behaviours in many different types of magnetic system. Consider the following: Fe and Ni are both metallic elements and exhibit ferromagnetism; MnO is an insulating oxide with a three-dimensional antiferromagnetic structure; La₂CuO₄ is a layered material which exhibits antiferromagnetism but, when doped, becomes superconducting; some compounds do not order magnetically but show frustrated effects with an abundance of slow dynamics; some molecules become single-molecule magnets in which the individual molecules show quantum tunnelling of magnetization and a range of other interesting properties. Theories of magnetism have to explain all these materials and more.

For a start, we must realize that a classical approach will not work. The Bohr-van Leeuwen theorem [1] states that in a classical system there is no thermal equilibrium magnetization. We can prove this in outline as follows: in classical statistical mechanics the partition function Z for N particles, each with charge q, is proportional to

$$ \iiint \dots \int \exp[-\beta E(\{r_i, p_i\})] dr_1 \dots dr_N dp_1 \dots dp_N, \quad (2.1) $$

<footnote>S. J. Blundell (✉)
Clarendon Laboratory, Oxford University, Department of Physics, Parks Road,
Oxford OX1 3PU, UK</footnote>

<footnote>e-mail: stephen.blundell@physics.ox.ac.uk</footnote>
where $\beta = 1/k_B T$, $k_B$ is the Boltzmann constant, $T$ is the temperature and $i = 1, \dots, N$. Here $E(\{\mathbf{r}_i, \mathbf{p}_i\})$ is the energy associated with the $N$ charged particles having positions $\mathbf{r}_1, \mathbf{r}_2, \dots, \mathbf{r}_N$, and momenta $\mathbf{p}_1, \mathbf{p}_2, \dots, \mathbf{p}_N$. The integral is, therefore, over a $6N$-dimensional phase space ($3N$ position coordinates, $3N$ momentum coordinates). The effect of a magnetic field is to shift the momentum of each particle by an amount $q\mathbf{A}$. We must, therefore, replace $\mathbf{p}_i$ by $\mathbf{p}_i - q\mathbf{A}$. The limits of the momentum integrals go from $-\infty$ to $+\infty$ so this shift can be absorbed by shifting the origin of the momentum integrations. Hence the partition function is not a function of magnetic field, and so neither is the free energy $F = -k_B T \log Z$. Thus the magnetic moment $m = -(\partial F / \partial B)_T$ must be zero in a classical system.

Thus, we need quantum mechanics to make further progress. In this chapter, I will not provide an exhaustive review of magnetism (fuller treatments can be found elsewhere, e.g. [2–5]) but focus on a few key issues and some selected examples. To begin our discussion, it is helpful to note the energy scales inherent in magnetic problems. First, there is the kinetic energy which is on the eV scale. This typically takes a value like $\hbar^2\pi^2/(2mL^2)$, where $L$ is a length scale and this expression is the familiar one for particle in a box. This is an energy cost and arises because it takes energy to put an electron in a small box. Second comes the potential energy which is also on a similar scale and takes a form such as $e^2/(4\pi\epsilon_0 L)$. This will be a negative energy if considering the attraction between an electron and a nucleus (and becomes larger and more negative as $L$ decreases) and positive if considering electron-electron repulsion. Atoms are the size they are because of a compromise between kinetic energy wanting the atom to be infinite size and the potential energy wanting the atom to be zero size. Because one energy goes as $L^{-2}$ and the other as $-L^{-1}$ a compromise can be reached (and this is essentially the derivation of the Bohr radius). Both the kinetic and potential energies are large and are typically $\gg k_B T$. Next, we have to add the spin-orbit interaction which is typically much smaller, usually in the meV, and the magnetocrystalline anisotropy, which in cubic materials is in the $\mu$eV. These effects will turn out to be very important in magnetic materials, but they are small perturbations to the main interactions and will mainly come into play only once the magnetic order is established by the dominant interactions.

## 2.2 Exchange

The exchange interaction arises from the kinetic and potential energy in bonds between atoms. To see how this comes about, we begin by recalling simple results for the molecular orbitals in $H_2$ [see Fig. 2.1a]. We label the two hydrogen atoms A and B and write the wave function $|\psi\rangle$ as a linear combination of atomic orbitals $|\psi_A\rangle$ and $|\psi_B\rangle$ so that

$$|\psi\rangle = c_A|\psi_A\rangle + c_B|\psi_B\rangle. \qquad (2.2)$$
**Fig. 2.1 a** Two hydrogen atoms A and B can lower their energy by forming a hydrogen molecule H₂.

**b** The bonding and antibonding molecular orbitals $\sigma$ and $\sigma^*$

<figure><img src="image_27.png" /></figure>

<figure><img src="image_28.png" /></figure>

The Hamiltonian can be written as a sum of the kinetic energy and two terms for the potential energy due to the attraction to each hydrogen nucleus so that

$$ \hat{\mathcal{H}} = -\frac{\hbar^2}{2m} \nabla^2 + V_A + V_B. \qquad (2.3) $$

We then need to solve the equation $\hat{\mathcal{H}}|\psi\rangle = E|\psi\rangle$. The diagonal integral $E_0$, which can be approximated by the binding energy of the electron at one of the centres for a hydrogen *atom*, is given by

$$ E_0 = \langle \psi_A | \hat{\mathcal{H}} | \psi_A \rangle . \qquad (2.4) $$

The transfer integral (also known as the hopping integral or resonance integral) $t$ is given by

$$ t = \langle \psi_A | \hat{\mathcal{H}} | \psi_B \rangle . \qquad (2.5) $$

In the simplest approximation (the Hückel approximation), the overlap integrals are given by $S_{ij} = \langle \psi_i | \psi_j \rangle = \delta_{ij}$ and hence the secular equation $|H_{ij} - ES_{ij}| = 0$ can be written as

$$ \left| \begin{array}{cc} E_0 - E & t \\ t & E_0 - E \end{array} \right| = 0, \qquad (2.6) $$

and hence

$$ E = E_0 \pm t . \qquad (2.7) $$

The eigenfunctions for these solutions are the symmetric solution

$$ |\sigma\rangle = \frac{|\psi_A\rangle + |\psi_B\rangle}{\sqrt{2}} \qquad (2.8) $$

which costs energy $E_0 - t$ and the antisymmetric solution

$$ |\sigma^*\rangle = \frac{|\psi_A\rangle - |\psi_B\rangle}{\sqrt{2}} \qquad (2.9) $$

which costs energy $E_0 + t$. These are known as the bonding and antibonding states, respectively [see Fig. 2.1b]. The hydrogen molecule has two electrons so the $\sigma$ level
is full and the $\sigma^*$ level is empty, thus saving the energy overall and leading to the H$_2$ molecule being a stable entity. The molecule He$_2$ does not form because it has four electrons and would, therefore, involve filling both $\sigma$ and $\sigma^*$ and thus saves no energy (and in fact, outside the Hückel approximation, it turns out that $\sigma^* - E_0 > E_0 - \sigma$ and so helium bonding costs more energy than it saves).

### 2.2.1 Direct Exchange

Exchange interactions are nothing more than a consequence of electrostatic interactions and the familiar interplay between potential energy and kinetic energy that we see in chemical bonds. Consider a simple model with just two electrons which have spatial coordinates $\mathbf{r}_1$ and $\mathbf{r}_2$, respectively. The wave function for the joint state can be written as a product of single electron states, so that if the first electron is in state $\psi_a(\mathbf{r}_1)$ and the second electron is in state $\psi_b(\mathbf{r}_2)$, then the joint wave function is $\psi_a(\mathbf{r}_1)\psi_b(\mathbf{r}_2)$. However, this product state does not obey exchange symmetry, since if we exchange the two electrons we get $\psi_a(\mathbf{r}_2)\psi_b(\mathbf{r}_1)$, which is not a multiple of what we started with. Therefore, the only states which we are allowed to make are symmetrized or antisymmetrized product states which behave properly under the operation of particle exchange.

For electrons, the overall wave function must be antisymmetric so the spin part of the wave function must either be an antisymmetric singlet state $\chi_S$ ($S = 0$) in the case of a symmetric spatial state or a symmetric triplet state $\chi_T$ ($S = 1$) in the case of an antisymmetric spatial state. Therefore, we can write the wave function for the singlet case $\Psi_S$ and the triplet case $\Psi_T$ as

$$ \begin{aligned} \Psi_S &= \frac{1}{\sqrt{2}} [\psi_a(\mathbf{r}_1)\psi_b(\mathbf{r}_2) + \psi_a(\mathbf{r}_2)\psi_b(\mathbf{r}_1)] \chi_S \\ \Psi_T &= \frac{1}{\sqrt{2}} [\psi_a(\mathbf{r}_1)\psi_b(\mathbf{r}_2) - \psi_a(\mathbf{r}_2)\psi_b(\mathbf{r}_1)] \chi_T, \end{aligned} \quad (2.10) $$

where both the spatial and spin parts of the wave function are included. The energies of the two possible states are

$$ \begin{aligned} E_S &= \int \Psi_S^* \hat{H} \Psi_S dr_1 dr_2 \\ E_T &= \int \Psi_T^* \hat{H} \Psi_T dr_1 dr_2, \end{aligned} $$

with the assumption that the spin parts of the wave function $\chi_S$ and $\chi_T$ are normalized. The difference between the two energies is

$$ E_S - E_T = 2 \int \psi_a^*(\mathbf{r}_1) \psi_b^*(\mathbf{r}_2) \hat{H} \psi_a(\mathbf{r}_2) \psi_b(\mathbf{r}_1) d\mathbf{r}_1 d\mathbf{r}_2. \quad (2.11) $$
For a singlet state $S_1 \cdot S_2 = -\frac{3}{4}$ while for a triplet state $S_1 \cdot S_2 = \frac{1}{4}$. Hence the Hamiltonian can be written in the form of an 'effective Hamiltonian'

$$ \hat{\mathcal{H}} = \frac{1}{4}(E_S + 3E_T) - (E_S - E_T)S_1 \cdot S_2. \qquad (2.12) $$

This is the sum of a constant term and a term which depends on spin. The constant can be absorbed into other constant energy terms, but the second term is more interesting. The exchange constant $J$ is defined by

$$ J = \frac{E_S - E_T}{2} = \int \psi_a^*(\mathbf{r}_1) \psi_b^*(\mathbf{r}_2) \hat{\mathcal{H}} \psi_a(\mathbf{r}_2) \psi_b(\mathbf{r}_1) \, d\mathbf{r}_1 d\mathbf{r}_2, \qquad (2.13) $$

and hence the spin-dependent term in the effective Hamiltonian can be written as

$$ \hat{\mathcal{H}}^{\text{spin}} = -2J S_1 \cdot S_2. \qquad (2.14) $$

If $J > 0$, $E_S > E_T$ and the triplet state $S = 1$ is favoured. If $J < 0$, $E_S < E_T$ and the singlet state $S = 0$ is favoured. Thus, the exchange interaction compares two different configurations that are tied to the singlet and triplet spin states, but the energy difference associated with exchange comes from the difference in those two configurations worked out from an integral [see (2.13)] over the spatial coordinates. Thus the spins are really there just to label the two different spatial states and are inextricably tied to the spatial wave functions by the Pauli principle; the exchange interaction is *really* between spatial wave functions, even though we tend to think about it as between the spin parts that really just come along for the ride!

Equation (2.14) is relatively simple to derive for two electrons, but generalizing to a many-body system is far from trivial. It motivates the Hamiltonian of the Heisenberg model:

$$ \hat{\mathcal{H}} = -\sum_{ij} J_{ij} S_i \cdot S_j, \qquad (2.15) $$

where $J_{ij}$ is the exchange constant between the $i$th and $j$th spins. The factor of 2 is omitted because the summation includes each pair of spins twice. Another way of writing (2.15) is

$$ \hat{\mathcal{H}} = -2 \sum_{i>j} J_{ij} S_i \cdot S_j, \qquad (2.16) $$

where the $i > j$ avoids the 'double-counting' and hence the factor of two returns. It is worth noting that there are different conventions for the definition of $J$ that are in use in the literature. I call these the $J$-convention and the $2J$-convention and they are summarized in Fig. 2.2. Note that it is also possible to choose the sign of $J$ so that $J > 0$ means ferromagnetic (as here) or antiferromagnetic. Both choices are found in the literature.
**Fig. 2.2** The two different conventions used for the definition of *J*. In this chapter (and in [2]), we are using the 2*J* convention (so that 2*J* is the energy associated with a single pairwise interaction between two spins). The various alternative expressions that one can use for the Heisenberg interaction are shown under the heading ‘many spins’, as well as an expression for a one-dimensional (1D) chain of spins

<figure><img src="image_29.png" /></figure>

<figure><img src="image_30.png" /></figure>

### 2.2.2 Indirect Exchange

If the electrons on neighbouring magnetic atoms interact via an exchange interaction, this is known as direct exchange. This is because the exchange interaction proceeds directly without the need for an intermediary, and this was considered in the previous section.

Very often direct exchange cannot be an important mechanism in controlling the magnetic properties because there is insufficient direct overlap between neighbouring magnetic orbitals. For example, in rare earths the 4f electrons are strongly localized and lie very close to the nucleus, with little probability density extending significantly further than about a tenth of the interatomic spacing. This means that the direct exchange interaction is unlikely to be very effective in rare earths. Even in transition metals, such as Fe, Co and Ni, where the 3d orbitals extend further from the nucleus, it is extremely difficult to justify why direct exchange should lead to the observed magnetic properties. These materials are metals which means that the role of the conduction electrons should not be neglected, and a correct description needs to be taken into account of both the localized and band character of the electrons.

In metals the exchange interaction between magnetic ions can be mediated by the conduction electrons. A localized magnetic moment spin-polarizes the conduction electrons and this polarization in turn couples to a neighbouring localized magnetic moment a distance *r* away. The exchange interaction is thus indirect because it does not involve direct coupling between magnetic moments. It is known as the RKKY interaction (or also as itinerant exchange). The name RKKY is used because of the initial letters of the surnames of the discoverers of the effect, Ruderman, Kittel, Kasuya and Yosida [6–8]. The coupling takes the form of an *r*-dependent exchange interaction $J_{RKKY}(r)$ given by
<figure><img src="image_31.png" /><figcaption>**Fig. 2.3** The real space susceptibility of a free electron gas is given by $\chi(r) = 2k_F^3\mu_0\mu_B^2g(E_F)F(2k_{F\text{R}})/\pi$ where $F(x) = (-x \cos x + \sin x)/x^4$ is the function illustrated. A localized spin in a free electron gas, therefore, gives rise to an effective exchange $J_{\text{RKKY}} \propto F(2k_{F\text{R}})$ and this is proportional to $\cos(2k_{F\text{R}}r)/r^3$ when $r \gg k_F^{-1}$</figcaption></figure>

$$J_{\text{RKKY}}(r) \propto \frac{\cos(2k_{\text{F}}r)}{r^3}, \qquad (2.17)$$

at large $r$ (assuming a spherical Fermi surface of radius $k_F$). The interaction is long range and has an oscillatory dependence on the distance between the magnetic moments (see Fig. 2.3). Hence depending on the separation it may be either ferromagnetic or antiferromagnetic. The coupling is oscillatory with wavelength $\pi/k_F$ because of the sharpness of the Fermi surface.

### 2.2.3 Superexchange

A number of ionic solids, including some oxides and fluorides, have magnetic ground states. For example, MnO [see Fig. 2.4a] and MnF₂ are both antiferromagnets, though this observation appears at first sight rather because there is no direct overlap between the electrons on Mn²⁺ ions in each system. The exchange interaction is normally very short ranged so that the longer ranged interaction that is operating in this case must be in some sense ‘super’ (think of Superman leaping over buildings, a skill not afforded to ordinary mortals).

The origin of superexchange is the possibility of mixing in excited states to lower the energy. The favouring of antiferromagnetic superexchange in a linear Mn–O–Mn bond arises from the fact that the excited states are allowed, while for the ferromagnetic arrangement these excited states are forbidden [see Fig. 2.4b]. One can consider this problem with a toy model based on a Hubbard-style Hamiltonian (see, e.g. [9]) which may be written as

$$\hat{\mathcal{H}} = -t \sum_{\langle ij \rangle \sigma} \hat{c}_{i\sigma}^{\dagger} \hat{c}_{j\sigma} + U \sum_{i} \hat{n}_{i\uparrow} \hat{n}_{i\downarrow}, \qquad (2.18)$$
<figure><img src="image_32.png" /></figure>

<table><caption>**Fig. 2.4** a The crystal structure of MnO. Nearest neighbour pairs of Mn<sup>2+</sup> (manganese) ions are connected via O<sup>2-</sup> (oxygen) ions. b A simple model of superexchange for a Mn–O–Mn bond. I: the antiferromagnetic ground state with opposite spins on the two Mn ions and a pair of electrons on the oxygen anion. II and III: two excited states of the antiferromagnetic ground state in which the electrons from (I) hop back and forth. IV: the competing ferromagnetic ground state. This is energetically more costly because the excited states analogous to (II) and (III) are not available because of the Pauli exclusion principle</caption><thead><tr><th></th><th>Mn</th><th>O</th><th>Mn</th></tr></thead><tbody><tr><td>I</td><td>↑</td><td>↑↓</td><td>↓</td></tr><tr><td>II</td><td>↑↓</td><td>↑↓</td><td></td></tr><tr><td>III</td><td>↑↓</td><td></td><td>↑↓</td></tr><tr><td>IV</td><td>↑</td><td>↑↓</td><td>↑</td></tr></tbody></table>

where the first sum is over nearest neighbours; thus energy is lowered by hopping (the first term on the right) but there is an energy penalty for double occupancy (the second term on the right) due to the Coulomb repulsion energy U. Let us now restrict this model to a system with two possible sites for electrons (here we are ignoring the intermediate oxygen to for simplicity). We can start by putting a single electron with spin ↑ into the system. Using a basis |↑, 0⟩ and |0, ↑⟩, the Hamiltonian is given by

$$ \hat{\mathcal{H}} = \begin{pmatrix} 0 & -t \\ -t & 0 \end{pmatrix}, \qquad (2.19) $$

because with one electron there is no possibility of a Coulomb penalty, and so the only energy to worry about is the energy saving you get from hopping. This is the same as the H<sub>2</sub> problem we considered earlier and the eigenvalues are ±t and so the lowest energy state is the bonding state, just as before.

Now let us put a second electron into the system with opposite spin to the first. Now using a basis such that a general state can be written as

$$ |\psi\rangle = a|\uparrow\downarrow, 0\rangle + b|\uparrow, \downarrow\rangle + c|\downarrow, \uparrow\rangle + d|0, \uparrow\downarrow\rangle, \qquad (2.20) $$

and we can easily show that in this basis the Hubbard Hamiltonian is

$$ \hat{\mathcal{H}} = \begin{pmatrix} U & t & -t & 0 \\ t & 0 & 0 & t \\ -t & 0 & 0 & -t \\ 0 & t & -t & U \end{pmatrix}, \qquad (2.21) $$
where the minus signs appear because of the exchange symmetry. The eigenvalues are 0, U and $(U/2) \pm \sqrt{(U/2)^2 + 2t^2}$, so in the limit that $t/U \ll 1$ the last pair of eigenvalues are $U + 2t^2/U + O(t^4/U^3)$ and $-2t^2/U + O(t^4/U^3)$. Thus, the ground state has energy $-2t^2/U$. If we try the same problem again with two electrons with the same spin then they cannot sit on the same site because of the Pauli exclusion principle. Thus the only state possible is $|↑, ↑⟩$ and this has energy $E = 0$. Thus, there is an energy saving in having the two electrons with opposite spin because you can go lower than $E = 0$ and have $E = -2t^2/U + O(t^4/U^3)$. This means that the exchange interaction has a magnitude $J \approx 2t^2/U$. The moral of the story is that by having the possibility to mix in the higher energy states in which two spins sit on the same site (costing $U$), it is possible to lower the overall energy. The antiferromagnetic arrangement allows this process to happen; the ferromagnetic arrangement forbids it. Superexchange can be considered in more detail [10] and can in certain circumstances be ferromagnetic. The size and sign of the superexchange interaction is codified in the Goodenough-Kanamori-Anderson rules [11–14].

## 2.3 Consequences of the Heisenberg Exchange Interaction

We have seen that at the heart of the exchange interaction is a term $\hat{S}_a \cdot \hat{S}_b$, a simple scalar product between two spin operators. If that scalar product is expanded, we have

$$ \hat{S}_a \cdot \hat{S}_b = \hat{S}_a^x \hat{S}_b^x + \hat{S}_a^y \hat{S}_b^y + \hat{S}_a^z \hat{S}_b^z = \hat{S}_a^z \hat{S}_b^z + \frac{1}{2}(\hat{S}_a^+ \hat{S}_b^- + \hat{S}_a^- \hat{S}_b^+) , \quad (2.22) $$

where the raising and lowering operators $\hat{S}^+$ and $\hat{S}^-$ are defined by

$$ \begin{aligned} \hat{S}^+ &= \hat{S}^x + i\hat{S}^y \\ \hat{S}^- &= \hat{S}^x - i\hat{S}^y . \end{aligned} \quad (2.23) $$

Although the term $\hat{S}_a^z \hat{S}_b^z$ in (2.22) seems to be simple enough to handle, the term $\frac{1}{2}(\hat{S}_a^+ \hat{S}_b^- + \hat{S}_a^- \hat{S}_b^+)$ will give rise to flip-flop processes in which simultaneously an up-spin labelled $a$ is lowered and a down-spin labelled $b$ is raised, or vice versa. This part of the interaction has profound effects.

### 2.3.1 Two Interacting Spin-$\frac{1}{2}$ Particles

In this section, we will consider two spin-$\frac{1}{2}$ particles coupled by a scalar interaction described by a Hamiltonian $\hat{H} = A\hat{S}_a \cdot \hat{S}_b$, where $\hat{S}_a$ and $\hat{S}_b$ are the operators for the spins for the two particles. We can also write the total spin operator $\hat{S}^{\text{tot}} = \hat{S}_a + \hat{S}_b$ so that
<table><caption>**Table 2.1** The eigenstates of $\hat{S}_a \cdot \hat{S}_b$ for a two-spin system and the corresponding values of $m_s$, $s$ and the eigenvalue of $\hat{S}_a \cdot \hat{S}_b$</caption><thead><tr><th>Eigenstate</th><th>$m_s$</th><th>$s$</th><th>$\hat{S}_a \cdot \hat{S}_b$</th></tr></thead><tbody><tr><td>$|\uparrow\uparrow\rangle$</td><td>1</td><td>1</td><td>$\frac{1}{4}$</td></tr><tr><td>$\frac{|\uparrow\downarrow\rangle + |\downarrow\uparrow\rangle}{\sqrt{2}}$</td><td>0</td><td>1</td><td>$\frac{1}{4}$</td></tr><tr><td>$|\downarrow\downarrow\rangle$</td><td>-1</td><td>1</td><td>$\frac{1}{4}$</td></tr><tr><td>$\frac{|\uparrow\downarrow\rangle - |\downarrow\uparrow\rangle}{\sqrt{2}}$</td><td>0</td><td>0</td><td>$-\frac{3}{4}$</td></tr></tbody></table>

$$(\hat{S}^{\text{tot}})^2 = (\hat{S}_a)^2 + (\hat{S}_b)^2 + 2\hat{S}_a \cdot \hat{S}_b. \qquad (2.24)$$

In quantum mechanics, when you combine the angular momentum of two spin-$\frac{1}{2}$ particles you have the 'addition law' that $\frac{1}{2} + \frac{1}{2} = 0, 1$. You can think of this simply as arising from the fact that you can combine the two moments together constructively or destructively. Alternatively, imagine adding two classical vectors $J_1$ and $J_2$ together but varying the angle between them. In that case, the resulting vector $J_1 + J_2$ would have length ranging from $|J_1 - J_2|$ to $J_1 + J_2$ (where $J_1 = |J_1|$ and $J_2 = |J_2|$). More formally, combining the representations of two spin-$\frac{1}{2}$ together yields a representation

$$D^{(\frac{1}{2})} \otimes D^{(\frac{1}{2})} = D^{(0)} \oplus D^{(1)}. \qquad (2.25)$$

In other words, the result of combining two spin-$\frac{1}{2}$ particles is a combined object with spin quantum number $s = 0$ or 1. The eigenvalue of $(\hat{S}^{\text{tot}})^2$ is $s(s+1)$ which is therefore either 0 or 2 for the cases of $s = 0$ or 1. The eigenvalues of both $(\hat{S}_a)^2$ and $(\hat{S}_b)^2$ are $\frac{3}{4}$. Hence from (2.24)

$$\hat{S}_a \cdot \hat{S}_b = \begin{cases} +\frac{1}{4} & \text{if } s = 1 \\ -\frac{3}{4} & \text{if } s = 0. \end{cases} \qquad (2.26)$$

The system, therefore, has two energy levels for $s = 0$ and 1 with energies given by

$$E = \begin{cases} +\frac{A}{4} & \text{if } s = 1 \\ -\frac{3A}{4} & \text{if } s = 0. \end{cases} \qquad (2.27)$$

The degeneracy of each state is $2s + 1$, so that the $s = 0$ state is a singlet (a single energy level) and the $s = 1$ state is a triplet (three energy levels). The $z$ component of the spin of this state, $m_s$, can only equal 0 for the singlet, but can be $-1, 0$, or 1 for the triplet. Thus the product of two spin-$\frac{1}{2}$ representations, which have a dimensionality of $2 \times 2 = 4$, gives rise to states $s = 0$ (singlet) and $s = 1$ (triplet), which have a total dimensionality of $1 + 3 = 4$.
We have considered the eigenvalues of $\hat{S}_a \cdot \hat{S}_b$, but what about the eigenstates? The most straightforward basis to consider is

$$|\uparrow\uparrow\rangle, \ |\uparrow\downarrow\rangle, \ |\downarrow\uparrow\rangle, \ |\downarrow\downarrow\rangle. \qquad (2.28)$$

The first arrow refers to the z component of the spin labelled a and the second arrow refers to the z component of the spin labelled b. The eigenstates of $\hat{S}_a \cdot \hat{S}_b$ are linear combinations of these basis states and are listed in Table 2.1. The value of $m_s$ is equal to the sum of the z components of the individual spins. Since the eigenstates are a mixture of states in the original basis, we cannot know both the z components of the original spins and the total spin of the resultant entity. This is a general feature which will become more important in more complicated situations.

The basis in (2.28) also fails to satisfy the condition that the overall wave function must be antisymmetric with respect to exchange of the two electrons. Since the wave function is a product of a spatial function $\psi_{\text{space}}(\mathbf{r}_1, \mathbf{r}_2)$ and the spin function $\chi$, the spatial wave function can be either symmetric or antisymmetric with respect to exchange of electrons. For example, the spatial wave function

$$\psi_{\text{space}}(\mathbf{r}_1, \mathbf{r}_2) = \frac{\phi(\mathbf{r}_1)\xi(\mathbf{r}_2) \pm \phi(\mathbf{r}_2)\xi(\mathbf{r}_1)}{\sqrt{2}} \qquad (2.29)$$

is symmetric (+) or antisymmetric (−) with respect to exchange of electrons depending on the $\pm$. This type of symmetry is known as exchange symmetry. In (2.29), $\phi(\mathbf{r}_i)$ and $\xi(\mathbf{r}_i)$ are single-particle wave functions for the ith electron. Whatever is the exchange symmetry of the spatial wave function, the spin-wave function $\chi$ must have the opposite exchange symmetry. Hence $\chi$ must be antisymmetric when the spatial wave function is symmetric and vice versa. This is in order that the product $\psi_{\text{space}}(\mathbf{r}_1, \mathbf{r}_2) \times \chi$ is antisymmetric overall.

States such as $|\uparrow\uparrow\rangle$ and $|\downarrow\downarrow\rangle$ are clearly symmetric under exchange of electrons, but exchanging the two electrons in $|\uparrow\downarrow\rangle$ yields $|\downarrow\uparrow\rangle$ which is not a multiple of $|\uparrow\downarrow\rangle$. Thus $|\uparrow\downarrow\rangle$, and also by an identical argument $|\downarrow\uparrow\rangle$, are both neither symmetric nor antisymmetric under exchange of the two electrons. The true eigenstates must, therefore, be linear combinations of these two states (see Table 2.1). The state $(|\uparrow\downarrow\rangle + |\downarrow\uparrow\rangle)/\sqrt{2}$ is symmetric under exchange of electrons (in common with the other two $s=1$ states) while the state $(|\uparrow\downarrow\rangle - |\downarrow\uparrow\rangle)/\sqrt{2}$ (the $s=0$ state) is antisymmetric under exchange of electrons.

The energy levels are shown in Fig. 2.5. Without the flip-flop term $\frac{1}{2}(\hat{S}_a^+ \hat{S}_b^- + \hat{S}_a^- \hat{S}_b^+)$ in (2.22), the Hamiltonian is simply $\hat{\mathcal{H}} = A\hat{S}_a^z \hat{S}_b^z$ and this leads to two degenerate doublets as shown. The upper doublet (consisting of the states $|\uparrow\uparrow\rangle$ and $|\downarrow\downarrow\rangle$) is unchanged when the flip-flop terms are switched on. The lower doublet (consisting of the states $|\uparrow\downarrow\rangle$ and $|\downarrow\uparrow\rangle$, although strictly speaking it should of course be the symmetric and antisymmetric combinations of these two states) splits with the addition of the flip-flop terms to make $\hat{\mathcal{H}} = A\hat{S}_a \cdot \hat{S}_b$ and the symmetric combination
<figure><img src="image_33.png" /><figcaption>Fig. 2.5 The energy levels for the two Hamiltonians: $\hat{\mathcal{H}} = A\hat{S}_a^z\hat{S}_b^z$ and $\hat{\mathcal{H}} = A\hat{S}_a \cdot \hat{S}_b$</figcaption></figure>

rises up in energy to $A/4$ and the antisymmetric combination is lowered to $-3A/4$, becoming the ground state when $A > 0$.

### 2.3.2 A Chain of Spins

Let us now consider not just two spins but a one-dimensional ferromagnetic chain of spin-$\frac{1}{2}$ moments described by the Heisenberg model. In one dimension, the Hamiltonian for the Heisenberg model can be written as

$$ \hat{\mathcal{H}} = -2J \sum_i \left[ \hat{S}_i^z \hat{S}_{i+1}^z + \frac{1}{2} (\hat{S}_i^+ \hat{S}_{i+1}^- + \hat{S}_i^- \hat{S}_{i+1}^+) \right], \quad (2.30) $$

where $J > 0$. The ground state $\Phi$ consists of all spins aligned [see Fig. 2.6a] and this is an eigenstate of $\hat{\mathcal{H}}$ so that $\hat{\mathcal{H}}|\Phi\rangle = -NS^2J|\Phi\rangle$. Now to create an excitation, let us flip a spin at site $j$, so let us now consider a state

$$ |j\rangle = \hat{S}_j^- |\Phi\rangle \quad (2.31) $$
$$
(a) \quad |\Phi\rangle = |\cdots \uparrow\uparrow\uparrow\uparrow\uparrow\cdots\rangle
$$

$$
|j\rangle = |\cdots \uparrow\uparrow\uparrow\downarrow\uparrow\uparrow\cdots\rangle
$$

$$
|q\rangle = \frac{1}{\sqrt{N}} \sum_j e^{iqR_j} |j\rangle
$$

<figure><img src="image_34.png" /><figcaption>**Fig. 2.6** a The ground state of a one-dimensional ferromagnet in the Heisenberg model is $|\Phi\rangle$. An excited state $|j\rangle$ has a single flipped spin at site $j$. The spin-wave state $|q\rangle$ is a delocalized spin flip. b The dispersion relation for the spin waves</figcaption></figure>

which is simply the ground state with the spin at site *j* flipped [see Fig. 2.6a]. By
flipping a spin, we have changed the total spin of the system by 1/2 - (1/2) = 1. This
excitation, therefore, has integer spin and is a boson. If we apply the Hamiltonian to
this new state, we get

$$
\hat{H}|j\rangle = 2 [(-NS^2J + 2SJ)|j\rangle - SJ|j+1\rangle - SJ|j-1\rangle], \quad (2.32)
$$

which is not a constant multiplied by |j⟩, so this state is not an eigenstate of the
Hamiltonian. Nevertheless, we can diagonalize the Hamiltonian by looking for plane
wave solutions of the form

$$
|q\rangle = \frac{1}{\sqrt{N}} \sum_j e^{iqR_j} |j\rangle . \qquad (2.33)
$$

The state $|q\rangle$ is essentially a flipped spin delocalized (smeared out) across all the sites [see Fig. 2.6a] and is known as a spin wave or a magnon. The state $|q\rangle$ is also an eigenstate of an operator exchanging any two spins, which is not the case for $|j\rangle$. Since $|q\rangle$ is a linear combination of states like $|j\rangle$ which represent a single flipped spin, the total spin in the z-direction of $|q\rangle$ itself has the value $NS - 1$. It is then straightforward to show that

$$
\hat{H}|q\rangle = E(q)|q\rangle, \tag{2.34}
$$

where

$$
E(q) = -2NS^2J + 4JS(1 - \cos qa). \quad (2.35)
$$

The energy of the excitation is then $\hbar\omega = 4JS(1 - \cos qa)$ and is plotted in Fig. 2.6b.
At small $q$, $\hbar\omega \approx 2JSq^2a^2$, so that $\omega \propto q^2$. In three dimensions, the density of states
is given by $g(q)dq \propto q^2 dq$, which leads to
$$g(\omega)d\omega \propto \omega^{1/2} d\omega \qquad (2.36)$$

at low temperature where only small $q$ and small $\omega$ are important. The spin waves are quantized in the same way as lattice waves. The latter are termed phonons, and so in the same way the former are termed magnons. They are bosons and have a spin of one.

The number of magnon modes excited at temperature $T$, $n_{\text{magnon}}$, is calculated by integrating the magnon density of states over all frequencies after multiplying by the Bose factor, $[\exp(\hbar\omega/k_{\text{B}}T) - 1]^{-1}$, which must be included because magnons are bosons. Thus the result is given by

$$n_{\text{magnon}} = \int_{0}^{\infty} \frac{g(\omega) \, d\omega}{\exp(\hbar\omega/k_{\text{B}}T) - 1}, \qquad (2.37)$$

which can be evaluated using the substitution $x = \hbar\omega/k_{\text{B}}T$. At low temperature, where $g(\omega) \propto \omega^{1/2}$ in three dimensions, this yields the result

$$n_{\text{magnon}} = \left(\frac{k_{\text{B}}T}{\hbar}\right)^{3/2} \int_{0}^{\infty} \frac{x^{1/2} \, dx}{e^x - 1} \propto T^{3/2}. \qquad (2.38)$$

Since each magnon mode which is thermally excited reduces the total magnetization by one (because each magnon mode is a delocalized single reversed spin), then at low temperature the reduction in the spontaneous magnetization from the $T = 0$ value is given by

$$\frac{M(0) - M(T)}{M(0)} \propto T^{3/2}. \qquad (2.39)$$

This result is known as the Bloch $T^{3/2}$ law. If one repeats this calculation in two dimensions (rather than three), the integral diverges, showing that magnons spontaneously form at all non-zero temperatures, thereby destroying any magnetization. The impossibility of spontaneous magnetization in two dimensions for the Heisenberg model is known as the Mermin–Wagner theorem [15–17] (see also [9]).

### 2.3.3 Three Spins

Let us return now to an apparently simpler system and consider three spins on the corners of an equilateral triangle. We will put the exchange interaction to be negative and thus the system is frustrated. If we put the first spin up, the next one down, then we have a dilemma of how to arrange the third one because we cannot satisfy the antiferromagnetic interactions on every bond. The solution has to be one of compromise and in fact the ground state of the classical Heisenberg model on a triangle is the so-called 120° state shown in Fig. 2.7a.
<figure><img src="image_35.png" /><figcaption>**Fig. 2.7** The ground state of the classical Heisenberg model is the 120° state shown in (a), though a version with opposite chirality (b) is also possible. If each of the three spins are rotated by a constant angle [(c) and (d)] then additional ground state configurations can be obtained</figcaption></figure>

In fact, there are some other possible solutions since we can choose to wind the spins round the triangle in two different ways. The configuration in Fig. 2.7b also has a 120° angle between adjacent spins but has the opposite chirality to that of Fig. 2.7a. Moreover, the Heisenberg model only cares about the relative angle between spins, not their absolute orientation, and therefore the configurations in Fig. 2.7c and d are also part of the ground state manifold.

Let us now solve the problem quantum mechanically. The law of addition of angular momentum now gives $\frac{1}{2} + \frac{1}{2} + \frac{1}{2} = \frac{1}{2}, \frac{1}{2}, \frac{3}{2}$. Now three two-dimensional representations (for three spin-$\frac{1}{2}$) have a dimensionality of $2^3 = 8$ which is equal to two two-dimensional representation and a four-dimensional representation (for two spin-$\frac{1}{2}$ and a single spin-$\frac{3}{2}$, so $2^3 = 2 + 2 + 4$). Another way of writing this combination is

$$D^{(\frac{1}{2})} \otimes D^{(\frac{1}{2})} \otimes D^{(\frac{1}{2})} = 2D^{(\frac{1}{2})} \oplus D^{(\frac{3}{2})}. \qquad (2.40)$$

For three spins we have that

$$\hat{S}^{\text{tot}} = \hat{S}_1 + \hat{S}_2 + \hat{S}_3 \qquad (2.41)$$

and hence

$$(\hat{S}^{\text{tot}})^2 = \hat{S}_1^2 + \hat{S}_2^2 + \hat{S}_3^2 + 2 \sum_{(i,j)} \hat{S}_i \cdot \hat{S}_j, \qquad (2.42)$$

and so using the facts that the eigenvalue of $(\hat{S}^{\text{tot}})^2$ is $S^{\text{tot}}(S^{\text{tot}} + 1)$ and the eigenvalue of $\hat{S}_i^2$ is $\frac{1}{2}(\frac{1}{2} + 1) = \frac{3}{4}$, we have

$$\sum_{(i,j)} \hat{S}_i \cdot \hat{S}_j = \frac{1}{2} \left( S^{\text{tot}}(S^{\text{tot}} + 1) - 3 \times \frac{3}{4} \right). \qquad (2.43)$$

We have two cases: (i) $S^{\text{tot}} = \frac{3}{2}$ implies that $\sum_{(i,j)} \hat{S}_i \cdot \hat{S}_j = \frac{3}{4}$; (ii) $S^{\text{tot}} = \frac{1}{2}$ implies that $\sum_{(i,j)} \hat{S}_i \cdot \hat{S}_j = -\frac{3}{4}$. The energy levels are drawn in Fig. 2.8 and consist of two degenerate doublets at $E = -3A/4$ ($S = \frac{1}{2}$) and a quartet at $E = 3A/4$.
<figure><img src="image_36.png" /><figcaption>**Fig. 2.8** The energy levels for a triangle of spins consist of two degenerate doublets and a quartet</figcaption></figure>

It is interesting to consider the 8 states that make up these energy levels [18]. We
can write these as follows

$$
\begin{align*}
|\Psi_{M=\frac{3}{2}}\rangle &= |\uparrow\uparrow\uparrow\rangle \\
|\Psi_{M=\frac{1}{2}}^{(k)}\rangle &= \frac{1}{\sqrt{3}} \sum_{j=0}^{2} e^{2\pi i j k / 3} C_3^j |\downarrow\uparrow\uparrow\rangle \\
|\Psi_{M=-\frac{1}{2}}^{(k)}\rangle &= \frac{1}{\sqrt{3}} \sum_{j=0}^{2} e^{2\pi i j k / 3} C_3^j |\uparrow\downarrow\downarrow\rangle \\
|\Psi_{M=-\frac{3}{2}}\rangle &= |\downarrow\downarrow\downarrow\rangle \tag{2.44}
\end{align*}
$$

Two states are obvious. These are the ‘ferromagnetic’ configurations $|\uparrow\uparrow\uparrow\rangle$ and $|\downarrow\downarrow\downarrow\rangle$. The other six contributions have a single spin-flip with respect to these ‘ferromagnetic configurations’ and so are made up of states like $|\downarrow\uparrow\uparrow\rangle$. However, a state like $|\downarrow\uparrow\uparrow\rangle$ is not exchange symmetric or antisymmetric, so you have to make linear combinations such as $|\downarrow\uparrow\uparrow\rangle + |\uparrow\downarrow\uparrow\rangle + |\uparrow\uparrow\downarrow\rangle$. This is exactly what is achieved by the sums in (2.44) (and note that the $1/\sqrt{3}$ factor is simply a normalization). The operators $C_3^j$ are threefold rotations of order $j$, and $j, k = 0, 1, 2$.

What is more, we can recover the chiral nature noted in the classical solutions. If
we define a chirality operator $\hat{C}_z$ given by

$$
\hat{C}_z = \frac{1}{4\sqrt{3}} S_1 \cdot (S_2 \times S_3), \qquad (2.45)
$$
then our states are eigenstates of the chirality operator

$$ 
\begin{align*}
\hat{C}_z |\Psi_{M=\frac{3}{2}}\rangle &= 0 \\
\hat{C}_z |\Psi_{M=\frac{1}{2}}^{(0)}\rangle &= 0 \\
\hat{C}_z |\Psi_{M=\frac{1}{2}}^{(1)}\rangle &= |\Psi_{M=\frac{1}{2}}^{(1)}\rangle \\
\hat{C}_z |\Psi_{M=\frac{1}{2}}^{(2)}\rangle &= -|\Psi_{M=\frac{1}{2}}^{(2)}\rangle. \tag{2.46}
\end{align*}
$$

Note that the quantum numbers $k = 0, 1, 2$, and chirality $C_z = 0, \pm 1$ describe the same states [18]. The states with non-zero chirality are in the pair of doublet ground states (for $A > 0$) while the excited state quartet has zero chirality.

## 2.4 Orbitals

### 2.4.1 Transition Metal Ions

The Heisenberg model only depends on the relative orientation of spins, not on their absolute orientation. It, therefore, seems to take no account of the lattice in which spins are embedded. However, spins are ‘aware’ of the lattice via the spin-orbit interaction. We now turn to consider the electronic orbitals that may be occupied in real systems. We will here particularly focus on transition metal compounds in which localized moments occur. (For metallic systems a band-like description would be more appropriate.)

Commonly occurring first-row transition metal ions are shown in Fig. 2.9 (taken from [19]), illustrating the range in occupancy of the 3d shell that can be obtained using transition metal ions in different oxidation states. I also show their electronic configuration for the case in which the ion experiences an octahedral crystal field (which splits the ten d-levels into a sixfold $t_{2g}$ level and a fourfold $e_g$ level [2]). Octahedral environments are very common in a wide variety of compound. This figure also shows that some ions are very particular about their oxidation state; for energetic reasons, Sc<sup>III</sup> and Zn<sup>II</sup> are the only stable states. On the other hand, an ion such as Mn can have a very large range of oxidation states from Mn<sup>II</sup> (in MnO) to Mn<sup>VII</sup> (in potassium permanganate KMnO<sub>4</sub>). In fact, with clever chemistry even Mn<sup>I</sup> is possible [20]! This range of occupancy leads to different spin and orbital moments of the ion and hence can be used to control the magnetic properties of the resulting system. For ions with $d^4$, $d^5$, $d^6$ and $d^7$ configuration, there is the possibility of both low-spin and high-spin states.
<figure><img src="image_37.png" /><figcaption>**Fig. 2.9** Electronic states of the *d*-electrons in octahedrally coordinated first-row transition metal ions in commonly found oxidation states. The *d*-shell contains 10 electrons and so configurations *d*⁰, *d*¹, ..., *d*¹⁰ are possible, indicated by the vertical lines. Depending on the oxidation state chosen (and not all possibilities are chemically available) these configurations can be achieved using different first-row transition metal ions (Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn). The oxidation state is indicated by the roman superscript, so that Cu¹¹ is copper in its doubly ionized oxidation state (i.e. Cu²⁺). An octahedral crystal field splits the ten *d*-levels into a sixfold *t*₂g level and a fourfold *e*₉ level and the electron configurations are shown schematically at the bottom of the figure together with their description using the conventional spectroscopic notation ('A' signifies an orbitally non-degenerate state, 'E' doubly degenerate and 'T' triply degenerate). For ions with *d*⁴, *d*⁵, *d*⁶ and *d*⁷ configuration, there is the possibility of low-spin and high-spin configurations (Figure from [19])</figcaption></figure>
## 2.4.2 Spin-Orbit Interaction and Crystal Fields

Now that we have the possibility of some partially filled d-levels, there can be magnetism. Next let us turn to the spin-orbit interaction which has the form

$$ \lambda \hat{S} \cdot \hat{L}, \qquad (2.47) $$

so that the $\hat{S}$ operator acts on the spin part of the wave function, while the $\hat{L}$ operator acts on the spatial part of the wave function. If the states are approximately atomic states, and the spin-orbit interaction acts as a perturbation, one can focus on the $\lambda S^z L^z$ part and note that

$$ \hat{L}^z = -i\hbar \frac{\partial}{\partial \phi}, \qquad (2.48) $$

which has eigenfunctions $e^{im\phi}$, i.e.

$$ \hat{L}^z e^{im\phi} = -i\hbar \frac{\partial}{\partial \phi} e^{im\phi} = m\hbar e^{im\phi}. \qquad (2.49) $$

Now the crystal field is a real potential which is due to electrostatic fields from neighbouring ions. The eigenfunctions of the crystal field cannot be proportional to $e^{im\phi}$ because we require real solutions. (Recall the problem of particle in a box where the solutions are real and take the form $\cos kx$ or $\sin kx$, but not $e^{ikx}$, but of course we can make real functions by making linear combinations such as $e^{ikx} \pm e^{-ikx}$ and making a wave function out of something proportional to that.) To make a crystal field state we must, therefore, look for linear combinations of eigenfunctions such as

$$ |\psi\rangle = e^{im\phi} \pm e^{-im\phi}. \qquad (2.50) $$

This kind of state though will automatically have zero angular momentum along the z-direction because it is made up of an equal contribution of a state with $\langle L^z \rangle = m\hbar$ and $\langle L^z \rangle = -m\hbar$. In fact, this idea works for all directions and

$$ \langle \hat{L} \rangle = 0. \qquad (2.51) $$

For example, if $l = 1$, there are three states with $m = 1, 0, -1$ with wave functions given by the spherical harmonics $Y_{1m}(\theta, \phi)$, and are therefore proportional to $\sin\theta e^{i\phi}$, $\cos\theta$ and $\sin\theta e^{-i\phi}$, respectively. We could write these states as $|1\rangle$, $|0\rangle$ and $|-1\rangle$. However, these are not the famous p-orbitals familiar from chemistry books. These arise in the formation of chemical bonds due to (real) electrostatic effects and must, therefore, be linear combinations of exactly the kind we are talking about. Thus the p-orbitals that line up along the x-, y- and z-directions [see Fig. 2.10a] are the zero-angular-momentum linear combinations given below:
<figure><img src="image_38.png" /><figcaption>Fig. 2.10 Angular representation of a *p*-orbitals and b *d*-orbitals</figcaption></figure>

$$ 
\begin{aligned}
|p_x\rangle &= \frac{|1\rangle + |-1\rangle}{\sqrt{2}} \\
|p_y\rangle &= \frac{|1\rangle - |-1\rangle}{\sqrt{2i}} \\
|p_z\rangle &= |0\rangle.
\end{aligned} 
\qquad (2.52) $$

What we see here is the phenomenon of quenching of angular momentum and it is interesting that it shows up even in the unfamiliar setting of the ‘balloon animals’ of *p*-orbitals. Of course in magnetism we are usually more interested in considering *d*-orbitals or *f*-orbitals but the same principles hold. For example, the equivalent equations for the *d*-orbitals [see Fig. 2.10b] are

$$ 
\begin{aligned}
|d_{xy}\rangle &= \frac{|2\rangle - |-2\rangle}{\sqrt{2i}} \\
|d_{x^2-y^2}\rangle &= \frac{|2\rangle + |-2\rangle}{\sqrt{2}} \\
|d_{yz}\rangle &= \frac{-|1\rangle - |-1\rangle}{\sqrt{2i}} \\
|d_{zx}\rangle &= \frac{-|1\rangle + |-1\rangle}{\sqrt{2}} \\
|d_{z^2}\rangle &= |0\rangle.
\end{aligned} 
\qquad (2.53) $$

### 2.4.3 Jahn-Teller Effect

It can sometimes be energetically favourable for, say, an octahedron to spontaneously distort as shown in Fig. 2.11 because the energy cost of increased elastic energy is balanced by a resultant electronic energy saving due to the distortion. This phe-
<figure><img src="image_39.png" /><figcaption>**Fig. 2.11** The Jahn–Teller effect for Mn³⁺ (3d⁴). An octahedral complex (left) can distort (right), thus splitting the t₂g and e_g levels. The distortion lowers the energy because the singly occupied e_g level is lowered in energy. The saving in energy from the lowering of the dₓz and dᵧz levels is exactly balanced by the raising of the dₓᵧ level</figcaption></figure>

nomenon is known as the Jahn–Teller effect [21]. The distortion lowers the overall energy by breaking an orbital degeneracy. For example, Mn³⁺ ions (which have a configuration 3d⁴) in an octahedral environment show this kind of behaviour (see Fig. 2.11) because the distortion can break the orbital degeneracy in the e_g levels. In contrast, Mn⁴⁺ ions (3d³) would not show this effect because there is no net lowering of the electronic energy by a distortion.

To describe the effect, at least at the phenomenological level, we will assume that the distortion of the system can be quantified by a parameter *Q*, which denotes the distance of distortion along an appropriate normal mode coordinate. This gives rise to an energy cost which is quadratic in *Q* and can be written as

$$E(Q) = \frac{1}{2} M \omega^2 Q^2, \qquad (2.54)$$

where *M* and *ω* are, respectively, the mass of the anion and the angular frequency corresponding to the particular normal mode. Clearly the minimum distortion energy is zero and is obtained when *Q* = 0 (no distortion).

The distortion also raises the energy of certain orbitals while lowering the energy of others. If all orbitals are either completely full or completely empty, this does not matter since the overall energy is simply given by (2.54). However, in the cases of partially filled orbitals this effect can be highly significant since the system can have a net reduction in total energy. The electronic energy dependence on *Q* could be rather complicated, but one can write it as a Taylor series in *Q* and provided the distortion is small it is legitimate to keep only the term linear in *Q*. Let us, therefore, suppose that the energy of a given orbital has a term either +*AQ* or −*AQ* corresponding to a raising or a lowering of the electronic energy, where *A* is a suitable constant, assumed
to be positive. Then the total energy $E(Q)$ is given by the sum of the electronic energy and the elastic energy

$$E(Q) = \pm A Q + \frac{1}{2} M \omega^2 Q^2, \qquad (2.55)$$

where the two possible choices of the sign of the $AQ$ term give rise to two separate curves. If we consider only one of them we can find the minimum energy for that orbital using $\partial E/\partial Q = 0$ which yields a value of $Q$ given by

$$Q_0 = \frac{A}{M\omega^2} \qquad (2.56)$$

and a minimum energy which is given by $E_{min} = -A^2/2M\omega^2$ which is less than zero. If only that orbital is full, then the system can make a net energy saving by spontaneously distorting.

LaMnO$_3$ contains Mn in the Mn$^{3+}$ state which is a Jahn-Teller ion. LaMnO$_3$ shows A-type antiferromagnetic ordering. If a fraction $x$ of the trivalent La$^{3+}$ ions are replaced by divalent Sr$^{2+}$, Ca$^{2+}$ or Ba$^{2+}$ ions, holes are introduced on the Mn sites. This results in a fraction $1-x$ of the Mn ions remaining as Mn$^{3+}$ ($3d^4$, $t_{2g}^3 e_g^1$) and a fraction $x$ becoming Mn$^{4+}$ ($3d^4$, $t_{2g}^3 e_g^0$). When $x = 0.2$ the Jahn-Teller distortion vanishes and the system becomes ferromagnetic with a Curie temperature ($T_C$) around room temperature. Above $T_C$, the material is insulating and non magnetic, but below $T_C$, it is metallic and ferromagnetic. Particularly near $T_C$, the material shows an extremely large magnetoresistive effect which has been called colossal magnetoresistance.

The situation is actually more complicated because the carriers interact with phonons because of the Jahn-Teller effect. The strong electron-phonon coupling in these systems implies that the carriers are actually polarons above $T_C$, i.e. electrons accompanied by a large lattice distortion. These polarons are magnetic and are self-trapped in the lattice. The transition to the magnetic state can be regarded as an unbinding of the trapped polarons. There are other signatures of the electron-phonon couplings, including magnetic-field dependent structural transitions and charge ordering.

## 2.5 Conclusion

This chapter has discussed a number of important concepts in magnetism. Clearly this has just scratched the surface and for more details the reader should look elsewhere [2-5]. Nevertheless, these principles are helpful in understanding the wide variety of magnetic materials that are being studied, from frustrated magnets [22] to molecular magnets [19, 23] and from permanent magnetic materials [24] to spintronics [25].
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13. J.B. Goodenough, An interpretation of the magnetic properties of the perovskite-type mixed crystals $La_{1-x}Sr_xCoO_{3-\lambda}$. J. Phys. Chem. Solids **6**, 287 (1958). https://doi.org/10.1016/0022-3697(58)90107-0

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15. N.D. Mermin, H. Wagner, Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models. Phys. Rev. Lett. **17**, 1133 (1966). [Erratum: Phys. Rev. Lett. **17**, 1307 (1966).] https://doi.org/10.1103/PhysRevLett.17.1133

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20. E. Dixon, J. Hadermann, S. Ramos, A.L. Goodwin, M.A. Hayward, Mn(I) in an extended oxide: the synthesis and characterization of $La_{1-x}Ca_xMnO_{2+\delta}$ ($0.6 \le x \le 1$). J. Amer. Chem. Soc. **133**, 18397 (2011). https://doi.org/10.1021/ja207616c

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<figure><img src="image_40.png" /></figure>
<figure><img src="image_41.png" /></figure>

Chapter 3

Electronic Structure Theory for X-Ray
Absorption and Photoemission
Spectroscopy

Peter Krüger

**Abstract** The principles of X-ray absorption and photoemission spectroscopy calculations are introduced and the basics of electronic structure theory, including the Hartree–Fock approximation, density functional theory, its time-dependent version and quasiparticle theory are reviewed on an elementary level. Emphasis is put on polarization effects and the role played by electron correlation.

**3.1 Introduction**

In this chapter, the basic theory of X-ray absorption spectroscopy (XAS) and photoemission spectroscopy (PES) is introduced and popular computational methods are reviewed. Since XAS and PES mainly probe electronic excitations, a thorough understanding of electronic structure theory is mandatory. We shall review the standard theoretical methods for ground state electronic structure calculations, namely, Hartree–Fock (HF) and density functional theory (DFT). Among the various excited state theories, we focus on time-dependent DFT and briefly touch upon Green’s function quasiparticle methods and the Bethe–Salpeter equation approach. We do not discuss ligand-field atomic multiplet theory, because this important method for transition metal L-edge calculations is covered in Chap. 4.

**3.2 Light–Matter Interaction**

As light is an electromagnetic wave, it interacts with all charged particles. In the
visible to X-ray regime, the interaction with the electrons hugely dominates the
interaction with the atomic nuclei. We shall therefore disregard the nuclear degrees

<footnote>P. Krüger (✉)
Graduate School of Engineering and Molecular Chirality Research Center, Chiba University,
Chiba 263-8522, Japan
e-mail: pkruger@chiba-u.jp</footnote>
of freedom in the following, focusing on the electronic state of the system. When an X-ray photon impinges on an atom it can be either absorbed or scattered, energy, momentum and spin being, of course, conserved in the whole process. In absorption, the photon vanishes: all its energy is transferred to an electron which is excited to an empty state above the vacuum level. In scattering, the energy of the photon can remain the same [elastic (Thomson) scattering]; it can also be partly transferred to the atom (inelastic scattering) as in Compton scattering, which leads to the ejection of an electron, or in a Raman-like scattering, in which the energy lost by the photon brings the atoms in an excited state, without any ionization. Neglecting relativistic effects and treating the X-ray field classically, the light-electron interaction is obtained by replacing, in the electronic Hamiltonian, the electron momentum operator **p** by $\mathbf{p} - e\mathbf{A}/c$, where $\mathbf{A}(\mathbf{r}, t)$ is the vector potential of the light [1]. In Coulomb gauge $(\nabla \cdot \mathbf{A} = 0)$, the interaction Hamiltonian then becomes

$$H_{\text{int}} = -\frac{e}{mc}\mathbf{A} \cdot \mathbf{p} + \frac{e^2}{2mc^2}A^2. \qquad (3.1)$$

The first-order term in A describes light absorption and stimulated emission while the second-order term is responsible for (non-resonant) light scattering. Here we focus on the absorption process and neglect the generally much weaker $A^2$ term. First-order perturbation theory (Fermi golden rule) leads to the following expression for the absorption intensity from an initial state $|i\rangle$ of energy $E_i$:

$$I(\omega) \propto \sum_f |\langle f | \mathbf{A} \cdot \mathbf{p} | i \rangle|^2 \delta(\hbar\omega - E_f + E_i), \qquad (3.2)$$

where the sum runs over all possible final states $|f\rangle$ with energy $E_f$. It is common to make the dipole approximation, i.e. to neglect the spatial variation of the X-ray field $\mathbf{A}(\mathbf{r})$. We may also replace the transition operator $\mathbf{A} \cdot \mathbf{p}$, by $\mathbf{e} \cdot \mathbf{r}$, where $\mathbf{e} = \mathbf{A}/|\mathbf{A}|$ is the light polarization vector and the change from **p** to **r** is possible by exploiting commutation relations between **r**, **p** and **H**, and the fact that $|i\rangle$ and $|f\rangle$ are energy eigenstates [2]. Equation (3.2) is often interpreted in a single-particle picture, in which case $|i\rangle$ is an atomic core state and $|f\rangle$ are unoccupied states above the Fermi level. However, electrons interact with each other through the Coulomb interaction, such that the excitation of one electron affects the motion of the others. Therefore, the correct use of (3.2) is in a many-particle sense, where $|i\rangle = |\Phi_g\rangle$ is the many-electron ground state, and $|f\rangle = |\Phi_f\rangle$ are many-electron excited states with a core hole. Putting the constants we have

$$I(\omega) = 4\pi^2\alpha\hbar\omega \sum_f |\langle\Phi_f|\mathbf{e} \cdot \sum_j \mathbf{r}_j|\Phi_g\rangle|^2 \delta(\hbar\omega - E_f + E_g), \qquad (3.3)$$

where $\alpha = e^2/\hbar c$ is the fine structure constant and $j$ counts the electrons. Having established the expressions of the absorption intensity, the remaining task is to calculate the eigenstates of the (unperturbed) electronic system, $|\Phi_g\rangle$ and $|\Phi_f\rangle$, and their
energies. Thus, the main theoretical problem of XAS is the accurate description of the electronic structure of the system, both for the ground and core-excited states. We, therefore, start by reviewing the basics of (ground state) electronic structure theory before turning to the specific methods for handling core-excited states.

## 3.3 Ground State Electronic Structure Theory

Consider $N$ electrons interacting with each other and the atomic nuclei. Following the Born–Oppenheimer approximation, we neglect the coupling between the nuclear and electronic dynamics. For the electronic problem, this means that the nuclei are at fixed positions and can be described by a static external potential $V_{\text{ext}}(\mathbf{r})$. The electronic Hamiltonian is then given by

$$H = T + V_{\text{ext}} + V_{ee} = \sum_{i} \frac{1}{2} \nabla_{i}^{2} + \sum_{i} V_{\text{ext}}(\mathbf{r}_{i}) + \sum_{i<j} \frac{1}{|\mathbf{r}_{i} - \mathbf{r}_{j}|}, \quad (3.4)$$

where $i, j$ count the electrons and atomic units are used ($\hbar = m = e = 1$). The kinetic energy $T$ and the external potential $V_{\text{ext}}$ are one-particle operators whereas the electron–electron interaction $V_{ee}$ is a two-particle operator. Because of $V_{ee}$, the electronic motion is correlated and the many-electron problem cannot be solved exactly (except for a few electrons). Drastic approximations need to be made. The most important ground state electronic structure methods are HF and DFT.

### 3.3.1 Hartree–Fock Approximation

Historically, the first accurate electronic structure method is the Hartree–Fock approximation (HFA) [3]. It is still widely used for single molecule calculations and as a starting point for more advanced schemes. The basic assumption of the HFA is that the many-electron ground state wave function is a Slater determinant, i.e. an antisymmetrized product of single-electron states (spin-orbitals). By applying the Rayleigh–Ritz variational principle, the HF equations are obtained, whose solutions are the HF orbitals $\phi_n(\mathbf{r})$ and energies $\epsilon_n$. For convenience, we suppress the spin part of the single-particle wave functions. The HF equations are

$$\left(-\frac{1}{2}\nabla^2 + V_{\text{ext}}(\mathbf{r}) + V_{\text{H}}(\mathbf{r}) + V_{X}\right) \phi_n(\mathbf{r}) = \epsilon_n \phi_n(\mathbf{r}). \quad (3.5)$$

This is a one-electron Schrödinger equation where the pair-wise electron–electron interaction is replaced by an effective potential $V_H + V_X$.
$$V_H(\mathbf{r}) = \sum_m^{\text{occ}} \int d\mathbf{r}' \frac{|\phi_m(\mathbf{r}')|^2}{|\mathbf{r} - \mathbf{r}'|} = \int d\mathbf{r}' \frac{n(\mathbf{r}')}{|\mathbf{r} - \mathbf{r}'|} \quad (3.6)$$

is called the Hartree potential and corresponds to the classical electrostatic potential due to the electronic charge density $n(\mathbf{r}) = \sum_m^{\text{occ}} |\phi_m(\mathbf{r})|^2$ of the occupied orbitals.

$$V_X \phi_n(\mathbf{r}) = - \sum_m^{\text{same spin}} \int d\mathbf{r}' \frac{\phi_m^*(\mathbf{r}') \phi_m(\mathbf{r}) \phi_n(\mathbf{r}')}{|\mathbf{r} - \mathbf{r}'|} \quad (3.7)$$

is the exchange potential which is due to the electron-electron interaction together with the antisymmetry of Slater determinants under permutation of two electrons. $V_X$ is a non-local potential and has no classical analogue. Both $V_H$ and $V_X$ are static ‘mean-field’ potentials, obtained from the time-averaged orbital motion of the electrons. Dynamical effects are neglected. The exchange interaction induces some correlation between electrons of same spin, which avoid each other due to the Pauli principle. Correlation between electrons of opposite spin is completely absent in the HFA. By definition, the difference between the exact ground state and the HF ground state is called the electron correlation effect (even though mathematically speaking, there is correlation between same spin electrons in the HFA).

There are various methods to take account of electron correlation, often termed collectively as ‘post-HF’ methods in the chemical literature. The conceptually most simple way to include electron correlation is the configuration interaction (CI) method. In CI, a set of Slater determinants is generated from the HF ground state by (multiple) particle-hole excitations. The CI wave function is a linear combination of these many-electron basis states and the coefficients and total energy levels are determined variationally by diagonalizing the Hamiltonian in this sub-space. CI can be very accurate for atoms and small molecules, but cannot directly be applied to large molecules and materials because the number of Slater determinants grows exponentially with system size. For X-ray absorption spectra, CI effects, i.e. mixing between Slater determinants, are especially strong at transition metal L-edges and lanthanide M-edges, which correspond to excitations into the localized 3d- and 4f-orbitals. For these spectra, CI must be taken into account. This can be done with the ligand-field multiplet method which is based on CI of a single atom or a very small cluster (see Chap. 4 for details).

### 3.3.2 Density Functional Theory

Nowadays, most electronic structure methods are based on DFT [4, 5]. In DFT, one does not try to find approximations to the many-electron wave function. Instead, the idea is to directly find the exact electronic density $n(\mathbf{r})$ and total energy, which is expressed as a functional of the density. DFT is based on two theorems due to Hohenberg and Kohn [4] about the (non-degenerate) ground state of the interacting,
inhomogeneous electron gas. The first theorem states that the external potential $V_{\text{ext}}$ is uniquely determined by the ground state electronic density $n(\mathbf{r})$ and that the total energy $E$ (minus the external potential energy) is a unique and universal functional of $n$.<fnref n="1" /> The consequence of the theorem is that knowledge of the ground state density alone is, in principle, sufficient to determine all properties of the system. The second theorem states that the exact ground state density $n_0(\mathbf{r})$ minimizes the total energy functional $E[n]$ in the space of all possible functions $n(\mathbf{r})$. Thus, approximations to $E$ and $n$ can be found variationally.

The Hohenberg–Kohn theorems are exact mathematical theorems. If the universal functional $E[n]$ were known, DFT would yield the exact total energy and electron density of the interacting electron system. But the exact functional $E[n]$ is unknown. Various approximate functionals have been proposed such as the local density approximation (LDA), generalized gradient approximations (GGA) and hybrid functionals, i.e. mixtures of GGA and HF exchange. In practical DFT calculations, Kohn–Sham theory is employed, which introduces an auxiliary, non-interacting system which, by definition, has the same electronic density as the real, interacting system. In the auxiliary system, the external potential is called Kohn–Sham potential $V_{\text{KS}}$ and it is the sum of the true external (nuclei) potential and an effective one-electron potential which replaces the electron–electron interaction. The Kohn–Sham potential $V_{\text{KS}}$ is given by the functional derivative of the total energy functional $E[n]$ with respect to the electron density. The effective electron–electron potential is written as the sum of the Hartree potential $V_{\text{H}}$ and a rest, which is called exchange–correlation potential $V_{\text{XC}}$. As the exact energy functional $E[n]$ is unknown, so is $V_{\text{XC}}$, and the actual expression depends on the approximation used (LDA, GGA, hybrid).

As the Kohn–Sham auxiliary system is non-interacting, its eigenstates are Slater determinants made of orbitals $\phi_n$ that are solutions of the Kohn–Sham equations

$$ \left(-\frac{1}{2}\nabla^2 + V_{\text{ext}}(\mathbf{r}) + V_{\text{H}}(\mathbf{r}) + V_{\text{XC}}\right) \phi_n(\mathbf{r}) = \epsilon_n \phi_n(\mathbf{r}). \qquad (3.8) $$

The Kohn–Sham equations (3.8) are similar to the HF equations (3.5) except that the exchange potential $V_X$ is replaced by the exchange–correlation potential $V_{\text{XC}}$ and the expression of the total energy as a function of the orbitals is different.

DFT takes account of electron correlation through $V_{\text{XC}}$ and generally performs better than the HFA for ground state properties. This is, however, not necessarily true for excited states for which DFT should, in principle, not be used, because the Kohn–Sham orbitals and levels $\epsilon_n$ describe the auxiliary system and have, strictly speaking, no direct physical meaning for the real system. In practice, however, the orbitals and energy levels are used in the same way as the HF orbitals, namely, as a first-order approximation for the one-electron or one-hole excitations of the system.

<footnote n="1">¹The opposite is obvious because when $V_{\text{ext}}$ is fixed, the Hamiltonian is known and so all properties, including the electronic density, are determined.</footnote>
## 3.4 Absorption Spectra in the Independent Particle Approximation

Recalling (3.3), the absorption intensity is determined by the transition amplitude $M_{fg} = \langle \Phi_f | \mathbf{e} \cdot \sum_n \mathbf{r}_n | \Phi_g \rangle$. If both $\Phi_f$ and $\Phi_g$ are Slater determinants made of orbitals which are eigenstates of the same one-electron Hamiltonian, then it is easy to see that $M_{fg}$ reduces to a one-particle transition matrix element between the core orbital $|\phi^c\rangle$ with energy $\epsilon_c$ and an unoccupied orbital $|\phi_k\rangle$ with energy $\epsilon_k$ and (3.3) simplifies to

$$I(\omega) = 4\pi^2\alpha\hbar\omega \sum_{k}^{\epsilon_k > \epsilon_F} |\langle\phi_k|\mathbf{e} \cdot \mathbf{r}|\phi^c\rangle|^2 \delta(\hbar\omega - \epsilon_k + \epsilon_c). \quad (3.9)$$

This is the basic equation of XAS in the independent particle approximation.

So far we have implicitly assumed that $\phi_k$ are the unoccupied orbitals of a ground state calculation. However, $\Phi_f$ is an excited state with a core hole. The core hole acts as a local positive charge which modifies the effective potential ($V_H + V_{X/XC}$) and so the best Slater determinant for $\Phi_f$ is made of a different set of orbitals $\tilde{\phi}_k$ than the ground state orbitals $\phi_k$. Accordingly, better results are usually obtained with ‘relaxed’ orbitals $\tilde{\phi}_k$, corresponding to a constraint HF or DFT calculation with a core hole. As core holes are localized on one atomic site, the symmetry of the system is generally lowered in a core hole calculation and the computational cost increases. For crystals, in particular, a supercell calculation is needed in order to effectively separate the artificially repeated core hole sites. In the following, we shall write $\phi_k$ regardless for relaxed and unrelaxed orbitals.

### 3.4.1 Dipole Selection Rules and Density of States

For the calculation of the dipole transition matrix elements $\langle \phi_k | \mathbf{e} \cdot \mathbf{r} | \phi^c \rangle$, it is useful to expand the states $|\phi_k\rangle$ in a spherical harmonics basis centred at the atomic sites $\mathbf{R}_i$. By doing so, the dipole transition selection rules known from atomic physics can be exploited. This simplifies the calculation and yields an interpretation of the spectra in terms of projected density of states as we shall see. We write

$$\phi_k = \sum_{ilm} B_{ilm}^k \chi_{ilm}, \quad \chi_{ilm}(\mathbf{r}) = R_{il}(\mathbf{r}_i) Y_{lm}(\mathbf{r}_i), \quad (3.10)$$

where $\mathbf{r}_i \equiv \mathbf{r} - \mathbf{R}_i$, $Y_{lm}$ are spherical harmonics, $R_{il}$ radial functions and $B_{ilm}$ complex coefficients. The core orbital is localized at some site ($i_c$). Therefore, only orbitals $\chi_{ilm}$ with $i = i_c$ give a non-zero contribution to the matrix element. Next we write the dipole operator as a spherical tensor product $\mathbf{e} \cdot \mathbf{r} = \sum_q (-1)^q e_{-q} r_q$, where $q = 0, \pm 1$ are the spherical components of a vector $\mathbf{a}$, given by $a_0 = a_z$,
$a_± = (\mp a_x - ia_y)/\sqrt{2}$. The angular integrals of the matrix elements can then be simplified with the help of the Wigner-Eckart theorem [2]

$$ \langle n'l'm'|r_q|nlm \rangle = (-1)^{l'-m'} \begin{pmatrix} l' & 1 & l \\ -m' & q & m \end{pmatrix} \langle n'l' ||r||nl \rangle, \quad (3.11) $$

where (·, ·) are Wigner-3j symbols and $\langle n'l' ||r||nl \rangle$ are reduced matrix elements, which are independent of $m, m', q$. The Wigner-3j symbol is non-zero only for $l' = l \pm 1$ and $m' = m + q$. These are the dipole selection rules. For example, for K-edge spectra $l = m = 0$ and thus only $l' = 1$, i.e. p-type final states can be reached. If polarized light is used we further have $m' = q$, e.g. in z-polarization only $p_z$ states are probed. We thus see that XAS is a local probe of the unoccupied electronic states, where different orbital symmetries can be projected out by appropriately choosing the absorption edge $l$ value and the light polarization $q$.

Using the expansion (3.10) and the dipole selection rules (3.11), we find for the transition matrix elements from a core orbital $\phi_{ilm}^c$

$$ \langle \phi_k | r_q | \phi_{ilm}^c \rangle = \sum_{\pm} B_{i,l \pm 1, m+q}^{k*} \langle \chi_{i,l \pm 1, m+q} | r_q | \phi_{ilm}^c \rangle . \quad (3.12) $$

The absorption intensity (3.9), dropping constants, from a core shell with angular momentum $l$, located at site $i$, for light polarization $q$ becomes

$$ I_q(\omega) = \sum_{km} \left| \sum_{\pm} B_{i,l \pm 1, m+q}^{k*} \langle \chi_{i,l \pm 1, m+q} | r_q | \phi_{ilm}^c \rangle \right|^2 \delta(\hbar\omega - \epsilon_k + \epsilon_c) . \quad (3.13) $$

As defined in (3.10), the orbitals $R_{il}(r)$ and the expansion coefficients $B$ are, in principle, energy dependent. This is the choice in multiple scattering theory which allows a minimal basis set (one orbital for each site and $l$). In the following, we neglect the slow energy dependence of the radial waves. We then obtain

$$ I_q(\omega) \approx \sum_{m,a,b=\pm} M_{ilm,q}^{a*} M_{ilm,q}^{b*} \sum_k B_{il_a m+q}^k B_{il_b m+q}^{k*} \delta(\hbar\omega - \epsilon_k + \epsilon_c) , \quad (3.14) $$

where $l_a, l_b = l \pm 1$ and $M_{ilm,q}^b = \langle \chi_{i,l_b, m+q} | r_q | \phi_{ilm}^c \rangle$. We introduce the local, orbital projected density of states matrix

$$ \rho_{ilm,l'm'}(\epsilon) = \sum_k \langle \chi_{ilm} | \phi_k \rangle \delta(\epsilon - \epsilon_k) \langle \phi_k | \chi_{il'm'} \rangle = \sum_k B_{ilm}^k \delta(\epsilon - \epsilon_k) B_{il'm'}^{k*} . \quad (3.15) $$

Note that the usual partial density of states (DOS) is given by the diagonal elements ($lm = l'm'$). So (3.14) can be written as
$$I_q(\omega) \approx \sum_{m,a,b=\pm} M_{ilm,q}^{a*} M_{ilm,q}^{b} \rho_{il_a m+q, l_b m+q} (\hbar\omega + \epsilon_c). \quad (3.16)$$

We see that the absorption intensity is a weighted sum of a few partial DOS components with angular momentum $l_a = l \pm 1$. For high enough symmetry, the interference terms $a \neq b$ vanish, leaving only the diagonal, usual partial DOS. In some cases, e.g. for the linear dichroism at the sulphur $L_{2,3}$-edges in MoS$_2$, it was found that interference between $p \to s$ and $p \to d$ transitions is non-negligible [6]. For the special case of s-wave core states ($K, L_1, M_1$ edges) where $l = m = 0$, the selection rules (3.11) give $l' = 1, m = q$, such that the absorption spectrum for q-polarized light is directly proportional to the $p_q$-DOS (where $q = 0, \pm 1$ or $q = x, y, z$).

In this section, we have seen that in the independent particle and dipole approximation, the X-ray absorption spectra are approximately given by a weighted sum of partial DOS with momenta $l \pm 1, m + q$. The weighting factors are local transition matrix elements and reflect the light polarization and orbital symmetry. As a consequence, XAS can be used to probe the unoccupied DOS of the material in a site and orbital-resolved way, which gives detailed insight into the local bonding properties [7].

## 3.5 Absorption Spectra in Linear Response TDDFT

### 3.5.1 Time-Dependent Density Functional Theory

DFT is a ground state theory whose application to excited states is ill-founded. However, a large class of excitations can be computed using the time-dependent version of DFT. Time-dependent DFT (TDDFT) is the generalization of standard DFT to time-dependent external potentials $V_{\text{ext}}(\mathbf{r}, t)$. It was pioneered by Zangwill and Soven in 1980 [8], who developed a linear response theory for optical absorption spectroscopy of atoms using a time-dependent version of the LDA. In 1984, Runge and Gross [9] generalized the Hohenberg-Kohn theorems of DFT to the case of time-dependent systems, thus putting TDDFT on a rigorous theoretical ground. TDDFT has been applied to XAS of solids for the first time in 1998 by Schwitalla and Ebert [10] and to molecules in 2003 by Stener et al. [11].

The problem at hand is to find the time-dependent electron density $n(\mathbf{r}, t)$ of an interacting electron system subject to a time-dependent external field. In TDDFT, the exact time-dependent electron density $n(\mathbf{r}, t)$ can, in principle, be found from the knowledge of the external field, the universal energy functional $E[n(\mathbf{r}, t)]$ and the initial density $n(\mathbf{r}, 0)$. Linear response functions, including absorption coefficients, can be expressed as integrals over the electron density change induced by a time-dependent external field. Thus, if the exact functional $E[n(\mathbf{r}, t)]$ were known, TDDFT would allow to obtain exact absorption spectra. As in the case of time-independent
DFT, however, the exact functional is unknown. Moreover, finding good approximate functionals is even more difficult in TDDFT than in standard DFT.

### 3.5.2 Linear Response Theory

Here we shall outline the theory of absorption spectroscopy in linear response following Zangwill and Soven [8]. We consider an interacting electron system as described by the unperturbed Hamiltonian $H$ in (3.4), and try to find its response to a time-dependent applied field $\varphi_{\text{ext}}(\mathbf{r}, t)$ such as the electromagnetic field of an X-ray beam. The perturbation Hamiltonian is written as

$$H'(t) = \int \varphi_{\text{ext}}(\mathbf{r}, t)n(\mathbf{r}, t)d\mathbf{r}, \qquad (3.17)$$

where $n(\mathbf{r}, t)$ is the electron density. It differs from the density of the unperturbed system $n^0(\mathbf{r})$ by the induced density

$$\delta n(\mathbf{r}, t) = n(\mathbf{r}, t) - n^0(\mathbf{r}). \qquad (3.18)$$

The fundamental assumption of linear response theory is that the response of the system, $\delta n$, is proportional to the applied field $\varphi_{\text{ext}}$, i.e.

$$\delta n(\mathbf{r}, t) = \int d\mathbf{r}' dt' \chi(\mathbf{r}, \mathbf{r}', t-t') \varphi_{\text{ext}}(\mathbf{r}', t'), \qquad (3.19)$$

where $\chi$, the response function, is an intrinsic property of the unperturbed system. In the frequency domain, this relation reads

$$\delta n(\mathbf{r}, \omega) = \int d\mathbf{r}' \chi(\mathbf{r}, \mathbf{r}', \omega) \varphi_{\text{ext}}(\mathbf{r}', \omega). \qquad (3.20)$$

It can be shown that $\chi$ is given by the retarded density–density Green’s function

$$\chi(\mathbf{r}, \mathbf{r}', t-t') = -i\theta(t-t')\langle 0 | [\hat{n}(\mathbf{r}, t), \hat{n}(\mathbf{r}', t')] | 0 \rangle, \qquad (3.21)$$

where $\hat{n}(t) = e^{iHt}\hat{n}e^{-iHt}$ is the density operator in Heisenberg representation, $|0\rangle$ is the exact ground state of $H$ with energy $E_0$, $[$, $]$ denotes the commutator and $\theta(x)$ is the Heaviside step function $[\theta(x) = 1 \text{ for } x > 0 \text{ and } \theta(x) = 0 \text{ for } x < 0]$. By inserting a complete set of excited states $\sum_m |m\rangle\langle m|$ and performing a time-frequency Fourier transformation, we obtain the following exact (‘Lehmann’) representation:

$$\chi(\mathbf{r}, \mathbf{r}', \omega) = \sum_m \frac{\langle 0 | \hat{n}(\mathbf{r}) | m \rangle \langle m | \hat{n}(\mathbf{r}') | 0 \rangle}{\hbar\omega - E_m + E_0 + i\eta} - \sum_m \frac{\langle 0 | \hat{n}(\mathbf{r}') | m \rangle \langle m | \hat{n}(\mathbf{r}) | 0 \rangle}{\hbar\omega + E_m - E_0 + i\eta}, \qquad (3.22)$$
where $|m\rangle$ are excited states with energy $E_m$ and $\eta$ is an infinitesimal positive number. The exact eigenstates and energies of the interacting electron systems are unknown, so (3.22) cannot be evaluated directly. For a non-interacting electron gas, however, all eigenstates are Slater determinants, and (3.22) can be calculated. The only excitations which give non-zero matrix elements are single particle-hole excitations $|m\rangle = c_p^+ c_h|0\rangle$ with energy $\epsilon_p - \epsilon_h$, where $p$ and $h$ label states above and below the Fermi level, respectively. This gives the response function in the independent particle approximation

$$ \chi_0(\mathbf{r}, \mathbf{r}', \omega) = \sum_{hp} \frac{\phi_h^*(\mathbf{r})\phi_p(\mathbf{r})\phi_p^*(\mathbf{r}')\phi_h(\mathbf{r}')}{\hbar\omega - \epsilon_p + \epsilon_h + i\eta} - [p \leftrightarrow h]. \quad (3.23) $$

If the electrons did not interact we would have $\chi = \chi_0$. But they do interact. In TDDFT, the interaction is handled as in DFT, by introducing an auxiliary, non-interacting system with the same electron density $n(\mathbf{r}, t)$ which corresponds to a time-dependent effective potential. In the real system, the density change $\delta n(\mathbf{r}, t)$ is induced by the external perturbation $\varphi_{\text{ext}}(\mathbf{r}, t)$. In the auxiliary system, however, the density $n(\mathbf{r}, t)$ corresponds to the sum of the Kohn-Sham potential $V_{\text{KS}}$ and the perturbation $\varphi_{\text{ind}}$. As the Kohn-Sham potential depends on the density, a density change $\delta n$ gives rise to an induced field $\varphi_{\text{ind}}(\mathbf{r}, t) = \delta V_{\text{KS}}[n(\mathbf{r}, t)]$. Thus, the density change $\delta n(\mathbf{r}, t)$ is due not only to the true external potential $\varphi_{\text{ext}}$ but also to the induced field $\varphi_{\text{ind}}$. Note that there is a feedback effect: $\varphi_{\text{ext}} \rightarrow \delta n \rightarrow \varphi_{\text{ind}} \rightarrow \delta^2 n \rightarrow \delta \varphi_{\text{ind}} \dots$, so we need to solve for $\delta n$ and $\varphi_{\text{ind}}$ self-consistently. Further, in linear response theory, a linear relation between the induced charge density and the induced field is assumed

$$ \varphi_{\text{ind}}(\mathbf{r}t) = \int d\mathbf{r}' dt' K(\mathbf{r}t, \mathbf{r}'t') \delta n(\mathbf{r}'t') \quad (3.24) $$

which defines the interaction kernel $K$. We have $\varphi_{\text{ind}} \equiv \delta V_{\text{KS}} = \delta V_H + \delta V_{\text{XC}}$, where

$$ \delta V_H(\mathbf{r}t) = \int d\mathbf{r}' \frac{\delta n(\mathbf{r}'t)}{|\mathbf{r} - \mathbf{r}'|}, \quad \delta V_{\text{XC}}(\mathbf{r}t) = \int d\mathbf{r}' dt' \frac{\delta V_{\text{XC}}(\mathbf{r}t)}{\delta n(\mathbf{r}'t')} \delta n(\mathbf{r}'t'). \quad (3.25) $$

The total time-dependent perturbation in the auxiliary system is often called the 'local field', $\varphi_{\text{loc}} = \varphi_{\text{ext}} + \varphi_{\text{ind}}$. As the electrons of the auxiliary system are independent, they respond to the perturbation $\varphi_{\text{loc}}$ with the free response function $\chi_0$, i.e. $\delta n(\mathbf{r}t) = \int d\mathbf{r}' dt' \chi_0(\mathbf{r}, \mathbf{r}', t-t')\phi_{\text{loc}}(\mathbf{r}'t')$. By construction, the charge densities of the real and auxiliary systems are the same, so we have

$$ \chi \varphi_{\text{ext}} = \delta n = \chi_0 \varphi_{\text{loc}} = \chi_0(\varphi_{\text{ext}} + K \delta n) = \chi_0(1 + K \chi) \varphi_{\text{ext}}, \quad (3.26) $$

where arguments and integration symbols have been suppressed to simplify the notation. Since $\varphi_{\text{ext}}$ is arbitrary, we have

$$ \chi = \chi_0 + \chi_0 K \chi \quad \Leftrightarrow \quad \chi = (\chi_0^{-1} - K)^{-1}. \quad (3.27) $$