Resonant elastic soft x-ray scattering

In the following we will consider a crystal that is formed by a perfect periodic arrangement of lattice sites, which act as scattering centres for the incident x-ray field. The scattering from site n in the crystal is described in terms of a scattering length f_n, which is also called the form factor, and can be represented as [15]

$$\begin{equation} f_n=f_n^{\rm T}+f_n^{\rm M}+\Delta f_n'+i \Delta f_n''. \end{equation} \tag{ 1 }$$

The first two terms $f_n^{\rm T}$ and $f_n^{\rm M}$ represent the non-resonant charge and magnetic scattering, respectively, where the scattering described by $f_n^{\rm T}$, which is proportional to the total number of electrons of the scatterer, is called Thomson scattering. Δf_n = Δf'_n + iΔf''_n denotes the so-called dispersion correction, which is not only a function of the photon energy ℏω, but also of the polarization of the incoming (e) and scattered beam (e'). This correction becomes very important close to absorption edges and describes the resonant scattering processes illustrated in figure 1.

Physically the scattering length f_n describes the change in amplitude and phase suffered by the incident wave during the scattering process: a scatterer with scattering length f_n exposed to an incident plane wave ∝e^ik·r causes a scattered radial wave ∝f_ne^ikr/r, which, far away from the scattering center, can be approximated by a plane wave.

In order to calculate the intensity of the total scattered wave field with wave vector k', all the radial waves have to be summed with the correct relative phases. As illustrated in figure 2, the geometric relative phase of two scatterers is given by Q · r, where Q = k' − k is the scattering vector and r is the difference in position. The total intensity detected in a distant detector is therefore

$$\begin{eqnarray} \fl I({\bit Q}) &\propto \Big| \sum_{n,m} f_n(\hbar \omega, {\bit e}, {\bit e}') {\rm e}^{{\rm i}{\bit Q}\cdot({\bit R}_m+{\bit r}_n)}\Big|^2 \nonumber\\ \fl & = \Big|\underbrace{\sum_n f_n(\hbar \omega, {\bit e}, {\bit e}') {\rm e}^{{\rm i} {\bit Q}\cdot{\bit r}_n}}_{\doteq F(\hbar \omega, {\bit e}, {\bit e}')}\Big|^2 \times \Big|\underbrace{\sum_m {\rm e}^{{\rm i} {\bit Q}\cdot{\bit R}_m}}_{\doteq L({\bit Q})}\Big|^2 \, , \end{eqnarray} \tag{ 2 }$$

where R_m is the vector pointing to the origin of unit cell m and r_n is the position of the scatterer f_n measured from that origin. F(ℏω, e, e') is called the unit cell structure factor and describes the interference of the waves scattered from the different sites within a unit cell. The lattice sum L (Q) is due to the interference of the scattering from the different unit cells at R_m. Its Q-dependence therefore provides information about the number of sites scattering coherently, i.e. it is related to the correlation length of the studied order. For an infinite number of coherently scattering unit cells, $L({\bit Q}) \rightarrow \sum_{{\bit G}\in \mathcal{B}^*}\delta({\bit G}-{\bit Q})$, where $\mathcal{B}^*$ represents the reciprocal lattice. This corresponds to the well-known Laue condition, which states that a reflection can only be observed if Q is a reciprocal lattice vector.

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The treatment described above is known as the kinematic approximation. This approximation is valid as long as the intensity of the scattered wave is much weaker than that of the incidence wave, which means that the interaction between the incoming and outgoing waves or multiple scattering events can be neglected. Since in resonant x-ray diffraction experiments one usually deals with weak superlattice reflections, this approximation is valid in most cases. If, however, the scattered wave becomes too strong, then one has to resort to the so-called dynamical theory of diffraction. This description will not be presented here and the interested reader is referred to the literature [12, 16]. It should be mentioned, however, that dynamical effects and multiple scattering play a very prominent role in reflectivity studies using soft x-rays, where the intensities of the scattered waves can be very strong.

But even if the scattered intensities are small, i.e. the kinematical approach is in principle valid, the refraction of the soft x-rays can be significant around an absorption edge. The aforementioned kinematical treatment neglects this, i.e. the index of refraction is assumed to be n = 1. While this assumption is very well fulfilled for non-resonant hard x-ray scattering, n can deviate significantly from 1 close to absorption edges (see section 2.2.4). Physically this means that the direction of the photon wave vectors changes upon entering or exiting the material that contains the resonant scattering centres. Since the corresponding scattering vector Q = k' − k inside the material enters the phases between the various scattered waves, the refraction will change the scattering geometry for a given peak. A more detailed description of the refraction in scattering experiments can be found in [17].

Another effect that needs to be considered when analysing the resonant scattering from a crystal is the energy dependence of the photon penetration depth [12]. When the photon energy is scanned across an absorption edge, the penetration depth can vary greatly and with it also the volume and the number of coherent scatterers. This dominates in many cases the peak width in RSXS, which then changes a lot as a function of photon energy. A more detailed discussion of this behaviour, its consequences and the corresponding correction methods can be found in [18].

A typical geometry of a resonant scattering experiment is shown in figure 3. An incoming photon beam with a defined wave vector k impinges on the sample and is scattered elastically into the direction defined by k', corresponding to a scattering vector Q = k' − k. In the so-called specular geometry the incoming and the outgoing beam are at an angle Θ with the sample surface. In this case, the scattering angle between k and k is ∠(k, k') = 2Θ.

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The wave vectors of the incident and scattered beam define the scattering plane, which is vertical in the present example. Polarization directions parallel or perpendicular to this scattering plane are referred to as π- and σ-polarization, respectively. Correspondingly, the incoming and outgoing polarizations in figure 3 are denoted as e_σ,π and e'_σ,π.

An important aspect of resonant x-ray scattering is given by the polarization dependence of Δf_n at resonance. This can cause the polarization of the outgoing beam to be different from that of the incoming beam. A typical example, which is frequently encountered in resonant scattering experiments, is a change of the incoming polarization from e_σ to an outgoing polarization e'_π; so-called σπ-scattering. Such changes in the polarization contain important information about the symmetry of the studied order, as will be discussed in more detail later on.

Controlling both the incoming and the outgoing polarization can therefore be a big advantage. The incoming polarization can be routinely controlled by modern insertion devices at synchrotron radiation sources like elliptical undulators. The determination of the outgoing polarization in RSXS is currently less common, but can be achieved using a polarization analyser as sketched in figure 3. By switching between the two configurations shown in the figure, for example, it is possible to observe either the e'_π- or the e'_σ-component of the scattered beam.

In the following discussion of the various case studies, a number of different scan-types will be described, which we will introduce briefly here:

• Radial scan or Θ2Θ scan. This is a scan along a fixed direction defined by the scattering vector Q perpendicular to the sample surface. In practice this scan is done by rotating the sample and the detector in steps of δΘ and 2δΘ about u₂, respectively, which is equivalent to changing Θ of the incoming and outgoing beam by the same amount.
• Transverse scan. This is a scan perpendicular to the direction defined by Q. It contains information about correlation in the plane perpendicular to Q, but is also often affected by the sample mosaic. A transverse scan can be done by rotating the sample about u₁ or u₂, while keeping the detector position fixed.
• Azimuth scan. This is another way of investigating the polarization dependence of the resonant scattering. In this type of scan, the resonant scattered intensity is recorded while rotating the sample by an angle ψ about the scattering vector Q (see figure 3). Azimuthal scans will be described in more detail in section 2.3.
• Energy scan. Here the photon energy dependence of a given reflection at a fixed Q is measured. In many cases this measurement is performed by scanning the photon energy, while recording the scattered intensity at Q = k' − k. Since both k and k' change with the photon energy, this means that for each photon energy the scattering geometry has to be adjusted such as to keep Q constant.

In general, the scattering of photons caused by matter is due to the interactions between the electromagnetic wave and the particles in a solid. In the present case only the interactions between the electromagnetic wave and the electrons in a solid are important. The corresponding coupling term $\mathcal H_{\rm int}$ is obtained by replacing the momentum operator p_l in the free-electron Hamiltonian $\mathcal H_0=\sum_l p_l^2/2m$ by $({\bit p}_l-e \hat{\bit A}/c)$, which gives

$$\begin{eqnarray} &&\fl \mathcal H_{\rm int}=\sum_l \left( -\frac{e}{m c} {\bit p}_l\cdot\hat{\bit A}({\bit r}_l,t)+\frac{{\rm e}^2}{2 m c^2} \hat{\bit A}({\bit r}_l,t)\cdot\hat{\bit A}({\bit r}_l,t) \right).\nonumber\\ \end{eqnarray} \tag{ 3 }$$

Here, p_l and r_l correspond to the l th electron of the atom. Furthermore, ${\bit p}_l\cdot\hat{\bit A}=\hat{\bit A}\cdot{\bit p}_l$ for a transversal radiation field $\hat{\bit A}$ has been used. Note also that the spin dependent part of $\mathcal H_{\rm int}$, which is proportional to ${\bit s}_l\cdot(\nabla \times \hat{\bit A})$ and ${\bit s}_l\cdot(\hat{\bit E} \times \hat{\bit A})$, has been neglected, because these relativistic terms scale as ℏω/mc² and hence are very small compared to the terms in equation (3). The quantized electromagnetic field that couples to the electrons can be expressed as

$$\begin{eqnarray} &&\fl \hat{\bit A}({\bit r},t)=\sqrt{\frac{2\pi \hbar c}{V}} \sum_{i,{\bit k}} \frac{1}{\sqrt{k}} {\bit e}_i\, \left( a_{{\bit k},i}(t)\, {\rm e}^{{\rm i}{\bit k}\cdot{\bit r}} + a_{{\bit k},i}^+(t)\, {\rm e}^{-{\rm i}{\bit k}\cdot{\bit r}} \right),\nonumber\\ \end{eqnarray} \tag{ 4 }$$

where a_k,i(t) = a_k,i(0)exp[−iω_kt] and $a^+_{{\bit k},i}(t)= a_{{\bit k},i}^+(0)\exp{[{\rm i}\omega_{{\bit k}}t]}$. The operators $a_{{\bit k},i}^+$ and a_k,i, respectively, create and annihilate a photon with polarization e_i and wave vector k.

Since the first term in equation (3) is proportional to ${\bit p}\cdot\hat{\bit A}$ it is linear in $a_{{\bit k},i}^+$ and a_k,i and therefore describes processes involving the emission and absorption of one photon. In other words, the ${\bit p}\cdot\hat{\bit A}$ term changes the number of photons by ±1. The second term in equation (3) is proportional to $\hat{\bit A}\cdot \hat{\bit A}$ and is therefore quadratic in the $a_{{\bit k},i}^+$ and a_k,i. As a result, this term changes the number of photons by 0 or ±2.

As introduced above, the scattering of a photon from a specific site n in the crystal is described by the scattering length f_n, which is determined by the interactions given by equation (3). The most important terms entering f_n can be calculated by means of time-dependent perturbation theory up to second order in $\mathcal H_{\rm int}$. In the following we will briefly sketch the main steps of this calculation and will present the final result. A detailed derivation of the expression for f_n can be found, for example, in [19]. Since in the following we will consider the scattering from a single site, we will drop the site index n.

The following scattering event will be considered: a single photon with wave vector k and polarization e impinges on a lattice site in the initial state |G〉 and is scattered into a state with wave vector k' and polarization e'. Since we are dealing with elastic scattering, the lattice site will remain in state |G〉; i.e.

$$\begin{equation} \fl |G,{\bit k},{\bit e}\rangle\doteq|G\rangle\cdot |{\bit k}, {\bit e}\rangle \rightarrow |G\rangle \cdot |{\bit k}',{\bit e}'\rangle \doteq |G,{\bit k}',{\bit e}'\rangle. \end{equation} \tag{ 5 }$$

Before and after the scattering event only one photon exists and, therefore, the only non-vanishing contributions of $\mathcal H_{\rm int}$ (equation (3)) to the scattering process must contain one creation and one annihilation operator corresponding to |k', e'〉 and |k, e〉, respectively.

The terms first order in $\mathcal H_{\rm int}$ are given by the matrix element $M_1= \langle G,{\bit k}',{\bit e}'|\mathcal H_{\rm int} |G,{\bit k},{\bit e}\rangle$. Since the interaction term ${\bit p}\cdot \hat{\bit A}$ is linear in the a_k,i and $a^+_{{\bit k},i}$, it changes the net number of photons and therefore does not contribute to M₁. Only the term proportional to $\hat{\bit A}\cdot \hat{\bit A}$ contains products of $a^+_{{\bit k}',i'}$ and a_k,i, which do not change the number of photons and, hence, only this term gives non-vanishing contributions to M₁.

While the ${\bit p} \cdot \hat{\bit A}$ term does not contribute to the first-order matrix element M₁, there are contributions of this term in second-order perturbation theory, which are of the same order of magnitude as M₁. These second-order contributions have to be considered as well. The second-order matrix element M₂ involves two successive ${\bit p}\cdot \hat{\bit A}$ interactions. The dominant contributions to M₂ are due to terms, where the incoming photon is annihilated first and then the scattered photon created: $M_2\sim \langle G,{\bit k}',{\bit e}'|({\bit p}\cdot \hat{\bit A})^+ |I\rangle \langle I| {\bit p}\cdot \hat{\bit A}|G,{\bit k},{\bit e}\rangle$. Here, |I〉 denotes an intermediate state of the system without a photon.

Taken together, the terms given by M₁ and M₂ yield the following expression for the differential cross-section dσ/dΩ, i.e. the probability that a photon is scattered into the solid angle dΩ:

$$\begin{eqnarray} &&\fl \frac{\rmd\sigma}{\rmd\Omega}= r_0^2 \Bigg| {\bit e}\cdot{\bit e}'\, \langle G|\rho({\bit Q})|G\rangle \nonumber\\ &&\quad-\,m \sum_{I} \frac{\langle G|{\bit e}'\cdot {\bit J}({\bit k}')|\,I\,\rangle \langle \,I\,|{\bit e}\cdot{\bit J}(-{\bit k})|G\rangle}{E_I-E_{G}-\hbar \omega- {\rm i} \Gamma_I/2} \Bigg|^2. \end{eqnarray} \tag{ 6 }$$

In this equation, r₀ = e²/(m c²) ≃ 2.82 × 10⁻¹³ cm is the classical electron radius, ρ (Q) corresponds to the Fourier amplitude at Q = k' − k of the charge density, |G〉 (|I〉) is the ground state (intermediate state) of the scatterer (see figure 1), E_G(E_I) is the energy of |G〉 (|I〉), Γ_I is the life time of |I〉 and ${\bit J}({\bit k})=1/m\sum_l {\bit p}_l\,{\rm e}^{{\rm i} {\bit k}\cdot{\bit r}_l}$ is the current operator that describes the virtual transitions between |G〉 and |I〉.

Equation (6) is the famous Kramers–Heisenberg formula applied to elastic scattering of photons. By definition the differential cross-section is related to the scattering length by dσ/dΩ = |f|², i.e. the above equation also provides the expression for f_n. The first term proportional to 〈G|ρ(Q)|G〉 describes the non-resonant Thomson scattering of photons from the total charge density ρ. The scattering due to the Thomson term is given by the first-order matrix element M₁, which describes the direct scattering of a photon caused by the $\hat{\bit A}\cdot \hat{\bit A}$ interaction. This non-resonant scattering scales with the total number of electrons and, hence, usually does not provide high sensitivity to electronic ordering phenomena, which typically affect only a very small fraction of ρ.

The second-order matrix element M₂ results in the second term in equation (6). This term describes the resonant scattering processes of the kind illustrated in figure 1. The matrix elements in the nominators describe the virtual |G〉 → |I〉 → |G〉 processes, which correspond to the two successive ${\bit p}\cdot \hat{\bit A}$ interactions. Close to an absorption edge, the photon energy ℏω is close to some of the E_I − E_G, in which case the corresponding contribution to the sum becomes large and can completely dominate the scattering.

According to the above discussion, the scattering length f can be written as

$$\begin{equation} f(\omega,{\bit e}, {\bit e'})=f^{\rm T}({\bit Q},{\bit e}, {\bit e}')+\Delta f(\hbar \omega,{\bit e}, {\bit e}',{\bit k},{\bit k}'), \end{equation} \tag{ 7 }$$

where f_T represents the non-resonant Thomson scattering and Δf is the anomalous dispersion correction. Equation (7) corresponds exactly to equation (1) given above with the magnetic scattering neglected.

In many cases, the current operator J is treated in the dipole approximation, which means that for 〈k·r_l〉 ≃ 0 one can use ${\rm e}^{{\rm i} {\bit k}\cdot{\bit r}_l}\simeq1$ and therefore J(k) → 1/m∑_lp_l. In the dipole approximation the dependence of Δf on k and k' is therefore neglected. Furthermore, since $2m [\mathcal{H}_0,{\bit r}]=-2{\rm i}\hbar {\bit p}$, one can replace the momentum operators p_l appearing in J by the commutator $[\mathcal{H}_0,{\bit r}_l]$. This together with the resonant term in equation (6) then yields the following expression for the resonant scattering length:

$$\begin{eqnarray} \Delta f & =& k^2 \sum_{I}\frac{\langle G|({\bit e}'\cdot {\bit D})^+|\,I\,\rangle \langle \,I\,|{\bit e}\cdot{\bit D}|G\rangle}{E_I-E_{G}-\hbar \omega- {\rm i} \Gamma_I/2}. \end{eqnarray} \tag{ 8 }$$

Here, D = e∑_l r_l is the dipole operator. The polarization dependence in the above equation can also be expressed using the tensor $\Delta \hat f$, which is defined by ${\bit e}'^+\cdot\Delta \hat f(\hbar\omega)\cdot {\bit e}=\Delta f(\hbar\omega,{\bit e},{\bit e}')$. This tensor formalism is often found in the literature and will also be used in the following.

As can be already realized in figure 1, the excitation from the ground state into the intermediate state corresponds to an x-ray absorption process. Therefore, there should be a relation between the resonant scattering length Δf and the x-ray absorption cross-section. That such a direct relation indeed exists can be seen in the following way: firstly, Δf can also be written more compactly as

$$\begin{equation} \Delta f =r_0\,m\,\langle G|\,[{\bit e}'\cdot{\bit J}({\bit k}')]^+ \, \mathcal{G}(\hbar \omega)\, [{\bit e}\cdot{\bit J}(-{\bit k})]\,|G\rangle \end{equation} \tag{ 9 }$$

using the Greens function

$$\begin{equation} \mathcal{G}(\hbar \omega)=\left(\mathcal{H}_I-E_G-\hbar \omega-{\rm i}\Gamma/2\right)^{-1}. \end{equation} \tag{ 10 }$$

Note that $\mathcal{G}$ is an operator, which is related to the hamiltonian $\mathcal{H}_I$ describing the intermediate state, i.e. the state with the core hole.

Secondly, the transition probability per unit time w_abs for the x-ray absorption is given by Fermi's golden rule:

$$\begin{equation} w_{\rm abs}=\frac{2 \pi}{\hbar} \, \sum_F \left|\, \langle F | \mathcal{H}_{\rm int} |G \rangle\,\right|^2 \, \delta (E_F-E_G-\hbar \omega). \end{equation} \tag{ 11 }$$

As explained in section 2.2.1, only the ${\bit p}\cdot \hat{\bit A}$ is linear in the photon annihilators and, hence, it is the only term active in the x-ray absorption process. The effective vector field A^abs, which couples |G〉 to |F〉 can be expressed as [19]

$$\begin{equation} {\bit A}^{\rm abs}=c \sqrt{\frac{2 \pi n_{{\bit k},i}\, \hbar}{\omega_{{\bit k}}\, V}} {\bit e} \ {\rm e}^{{\rm i}({\bit k.r}-\omega_{{\bit k}}\,t)}. \end{equation} \tag{ 12 }$$

The action of the operator a_k,i on the multiphoton state yields the factor $\sqrt{n_{{\bit k},i}}$, where n_k,i is the number of photons with wave vector k and polarization e. Furthermore, w_abs together with the photon flux Φ = c n_k,i/V gives the absorption cross-section dσ_abs/dΩ = w_abs/Φ. This relation together with equations (11) and (12) gives the following result:

$$\begin{eqnarray} \fl \frac{\rmd\sigma_{\rm abs}}{\rmd\Omega}&=\frac{4 \pi^2}{k}\, r_0\,m \sum_F|\langle F|\, {\bit e}\cdot{\bit J}({\bit k})\,|G\rangle|^2\delta (E_F-E_G-\hbar \omega)\nonumber\\\ms \fl &= \frac{4 \pi}{k}\, r_0\,m \,\mathrm{Im}\left[\!\langle G|[{\bit e}\cdot{\bit J}({\bit k})]^+ \mathcal{G}(\hbar \omega) [{\bit e}\cdot{\bit J}(-{\bit k})]|G\rangle\!\right].\nonumber\\ \end{eqnarray} \tag{ 13 }$$

Since the Greens function entering dσ_abs/dΩ is also given by equation (10), the comparison of equations (9) and (13) shows that

$$\begin{equation} \frac{\rmd\sigma_{\rm abs}}{\rmd\Omega}= \frac{4 \pi}{k}\,{\rm Im}\left[\Delta f\right] \end{equation} \tag{ 14 }$$

for k = k' and e = e'. This important relation is known as the optical theorem. It enables one to determine the imaginary part of the resonant scattering length from x-ray absorption. Once Im[Δf] is known, the corresponding real part can be calculated by means of the Kramers–Kronig relations, which read

$$\begin{eqnarray} {\rm Re}[\Delta f(\epsilon)]&=&\frac{2}{\pi} \mathcal P \int_0^{\infty} \frac{\epsilon'{\rm Im}[\Delta f(\epsilon')]}{\epsilon'^2-\epsilon^2}\,\rmd\epsilon'\quad {\rm and}\nonumber\\\ms {\rm Im}[\Delta f(\epsilon)]&=& -\frac{2\epsilon}{\pi} \mathcal P \int_0^{\infty} \frac{{\rm Re}[\Delta f(\epsilon')]}{\epsilon'^2-\epsilon^2}\,\rmd\epsilon', \end{eqnarray} \tag{ 15 }$$

with $\mathcal P \int_0^{\infty}$ the principal part $\lim_{\delta\rightarrow 0} \int_0^{\omega-\delta}+ \int_{\omega+\delta}^{\infty}$ of the integral and  = ℏω the photon energy.

Using these relations the energy dependence of $\Delta f(\hbar\omega,{\bit e},{\bit e})={\bit e}^+\cdot\Delta \hat f(\hbar\omega)\cdot{\bit e}$ can be calculated, where e is the polarization of the absorbed photon. This is demonstrated in figure 4, where such an analysis is shown for the high-temperature cuprate superconductor YBa₂Cu₃O_6+δ. It is important to realize however, that the above analysis allows one to obtain only matrix elements of the form ${\bit e}^+\cdot\Delta \hat f\cdot {\bit e}$. The matrix elements ${\bit e}'^+\cdot\Delta \hat f\cdot {\bit e}$ with e ≠ e' can therefore not directly be determined by x-ray absorption. In principle it is possible, however, to measure the x-ray absorption with three different polarizations with respect to the crystal axes and six other linear combinations of those. This enables one to retrieve all nine matrix elements of $\Delta \hat f$, by solving a set of linear equations.

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The scattering length can also be related to the index of refraction n. We will only briefly discuss the origin of this relation. Readers interested in the detailed derivation are referred to the literature [12, 17]. The relation between n and f is essentially given by the fact that the transmission of a beam through a plate of some material can be described in two equivalent ways: (i) behind the plate there will be an additional phase shift that was suffered by the wave during the passage through the material of the plate. At normal incidence, this phase shift is simply given by (1 − n) k W, where k and W are the wave vector of the light and the thickness of the plate, respectively. (ii) The plate can also be considered as a set of scatterers, which cause an additional scattered wave behind the plate. The total wave field behind the plate is therefore a superposition of the incident wave and the scattered wave, where the latter is related to the scattering length f of the scatterers within the plate. Only identical scattering centres are assumed. Since both descriptions must give the same result for the wave field behind the plate one can deduce that

$$\begin{equation} n=1-\frac{2\pi\,\rho_a\,r_0}{k^2}\times(f^{\rm T}+\Delta f), \end{equation} \tag{ 16 }$$

where ρ_a is the number density of the material. Since Δf is a complex number, the above equation (16) relates the real and imaginary part of f to the dispersion terms δ and β of n = 1 − δ + iβ.

The general expressions derived in the previous sections are useful to understand the general process of resonant diffraction. However, for the analysis of experimental data it is necessary to evaluate Δf for a specific situation. The simplest case for which explicit expressions can be derived is one of a free atom with a magnetic moment. In this situation only the magnetic moment breaks the spherical symmetry of the free atom. For this special case the resonant scattering length in the dipole approximation can be expressed as [6, 15]

$$\begin{equation} \Delta f = R^{(0)}\,{\bit e}'^*\cdot{\bit e}-{\rm i} \,R^{(1)}\,({\bit e}'^*\times{\bit e})\cdot \hat{\bit s}+R^{(2)}\,({\bit e}'^*\cdot \hat{\bit s})\,({\bit e}\cdot \hat{\bit s}), \end{equation} \tag{ 17 }$$

where $\hat{\bit s}$ is the unit vector defining the local spin moment and the R^(j) = R^j)(ℏω) (j = 0, 1, 2) are photon-energy-dependent resonance factors. These factors depend on the products 〈G|(e' · D)⁺| I 〉〈 I |e · D|G〉 appearing in equation (8) and define the strength of the resonant scattering.

There are three different terms in equation (17): the first term does not depend on the spin direction and corresponds to non-magnetic resonant scattering. This term simply adds to the Thomson scattering in equations (1) and (6), which has the same polarization dependence. The second term is proportional to $({\bit e}'^*\times{\bit e})\cdot \hat{\bit s}$ and is linear in $\hat{\bit s}$ and therefore describes magnetic resonant scattering. By virtue of the optical theorem this part of Δf can be related to the x-ray magnetic circular dichroism in absorption [11]. The third term is proportional to $({\bit e}'^*\cdot \hat{\bit s})\,({\bit e}\cdot \hat{\bit s})$, i.e. this term is quadratic in $\hat{\bit s}$ and hence also corresponds to resonant magnetic x-ray scattering. This scattering is related to the magnetic linear dichroism in x-ray absorption [11].

Equation (17) is very often used for the analysis of resonant scattering from magnetically ordered materials [11, 21]. However, it is important to remember that this expression has been derived for a free atom, where only the magnetic moment breaks the otherwise spherical symmetry.

Whenever an atom is embedded in a crystal, the chemical environment will break the spherical symmetry that was used to derive equation (17). This equation has therefore to be considered as an approximation for describing magnetic scattering in a crystal lattice. For instance, the x-ray absorption of a non-magnetic and non-cubic material will change with the polarization e pointing along the inequivalent lattice directions. The optical theorem therefore immediately implies that Δf(ℏω, e, e) also varies with e. This polarization dependence is not captured by equation (17), because in the non-magnetic case, this formula implies Δf(ℏω, e, e) = R⁽⁰⁾, which is isotropic and the same for all e.

In the following we will provide a simplified description of the effects described by Carra and Thole [22], and by Haverkort et al [23]. In particular, we will restrict ourselves to the dipole approximation and to magnetic scattering terms that are linear in the spin direction $\hat{\bit s}=(x,y,z)$. This is sufficient to illustrate the effects of the local environment, but in general the higher order terms can be relevant as well. For the complete expressions including higher order terms in $\hat{\bit s}$ the reader is referred to [23].

In the most general case of a magnetic site in a low symmetry environment, the tensor $\Delta \hat f(\hbar \omega)=\{\Delta f(\hbar \omega,{\bit e},{\bit e}')\}_{i,j}$ (see section 2.2.3), describing the polarization and energy dependence of the resonant scattering has the form

$$\begin{equation*} \Delta \hat f = \left( \begin{array}{@{}ccc@{}} R_{xx} & R_{xy} & R_{xz} \\ R_{yx} & R_{yy} & R_{yz} \\ R_{zx} & R_{zy} & R_{zz} \end{array}\right) \end{equation*}$$

with complex energy-dependent matrix elements R_ij = R_ij(ℏω). However, depending on the local point group symmetry of the scatterer, this tensor can be simplified and in high symmetries especially the simplifications are significant: in the extreme case of a free non-magnetic atom with spherical symmetry one has $\Delta \hat f=R^{(0)}\times I$ with the identity matrix I. If the spherical symmetry is broken only by the spin moment of the scatterer, the first two terms of equation (17) translate to

$$\begin{equation} \Delta \hat f = R^{(0)}\,I+ R^{(1)}\,\left( \begin{array}{@{}ccc@{}} 0 & - z & y\\ z & 0 & -x \\ -y &x & 0 \end{array}\right). \end{equation} \tag{ 18 }$$

It is this expression that is very often used for the analysis of resonant x-ray scattering data. It is important to know in which situations this approximate expression can be used and in which situation it yields incorrect results. We therefore discuss two other cases to clarify this point. First, we consider a local environment of O_h-symmetry. In this situation $\Delta \hat f$ can be expressed as

$$\begin{equation} \Delta \hat f = R^{(0)}_{a1g}\,I+R^{(1)}_{t1u}\, \left( \begin{array}{@{}ccc@{}} 0 & - z & y\\ z & 0 & -x \\ -y &x & 0 \end{array}\right). \end{equation} \tag{ 19 }$$

Similar to what has been described in [22] the subscript of the resonance factors in equation (19) corresponds to their symmetry representation in the O_h-point group. Comparing equations (18) and (19) it can be seen that the spherical approximation may still be used, as long as the magnetic term linear in $\hat{\bit s}$ dominates the resonant scattering signal. We mention that already for O_h-symmetry the terms quadratic in $\hat{\bit s}$ differ from equation (17). As a second example we consider a local D_4h symmetry. Depending on the deviation from O_h, the resonant scattering length can now deviate significantly from the spherical approximation:

$$\begin{eqnarray} \Delta \hat f &=& \left( \begin{array}{@{}ccc@{}} R^{(0)}_{a1g, B} & 0 & 0\\ 0 & R^{(0)}_{a1g,B} & 0 \\ 0 &0 & R^{(0)}_{a1g,A} \end{array} \right) \nonumber\\ &&+ \left( \begin{array}{@{}ccc@{}} 0 & - z R^{(1)}_{a2u} & y R^{(1)}_{eu}\\ z R^{(1)}_{a2u}& 0 & -x R^{(1)}_{eu}\\ -y R^{(1)}_{eu}&x R^{(1)}_{eu}& 0 \end{array}\right). \end{eqnarray} \tag{ 20 }$$

As before, the subscripts of the resonance factors in the above equation give their symmetry representation, this time in the D_4h-point group. The superscripts A and B indicate the different fundamental spectra due to the natural linear dichroism, which is also present in a paramagnetic sample. It can be seen in the above equation that the energy dependence of the magnetic resonant scattering now depends on the spin direction, i.e. depending on the direction of $\hat{\bit s}$, $R^{(1)}_{a2u}$ and $R^{(1)}_{eu}$, contribute with different weights. This effect is therefore given by the difference between $R^{(1)}_{a2u}$ and $R^{(1)}_{eu}$. It is also important to note that even the non-magnetic scattering in equation (20) is no longer isotropic. This means that for local symmetries lower than cubic the non-magnetic resonant scattering also becomes polarization dependent, in agreement with the qualitative arguments given in the beginning of this section.

Before we discuss our example, it is useful to introduce an efficient way to describe polarization-dependent resonant scattering. The resonant scattering length as a function of the incoming and outgoing polarization can be expressed using a 2 × 2 scattering matrix $\mathcal{S}$ [24]:

$$\begin{equation} \mathcal{S}_n=\left( \begin{array}{@{}cc@{}} \epsilon_{\sigma}^t\cdot(\Delta \hat f_n)\cdot\epsilon_{\sigma}' & \epsilon_{\sigma}^t\cdot(\Delta \hat f_n)\cdot\epsilon_{\pi}'\\ \epsilon_{\pi}^t\cdot(\Delta \hat f_n)\cdot\epsilon_{\sigma}' & \epsilon_{\pi}^t\cdot(\Delta \hat f_n)\cdot\epsilon_{\pi}' \end{array}\right) \end{equation} \tag{ 21 }$$

The matrix elements of $\mathcal{S}_n$ therefore correspond to σσ-, σπ-, πσ- and ππ-scattering of lattice site n. Using these $\mathcal{S}_n$, the corresponding structure factor matrix $\mathcal{F}$ can be calculated according to equation (2): $\mathcal{F}({\bit Q})=\sum_n \mathcal{S}_n {\rm e}^{{\rm i} {\bit Q}\cdot{\bit r}_n}$.

For the special case described by equation (17), the scattering matrix as a function of the local spin direction reads [15]

$$\begin{eqnarray} &&\mathcal{S}= R^{(0)} \left(\begin{array}{@{}cc@{}}1&0 \\ 0&\cos(2\Theta)\end{array}\right)\nonumber \\ &&- i R^{(1)} \left(\begin{array}{@{}cc@{}}0 &x \cos(\Theta) + z \sin(\Theta) \\ z \sin(\Theta)-x \cos(\Theta) &-y \sin(2\Theta)\end{array}\right) \nonumber \\ &&+\,R^{(2)}\! \left(\begin{array}{@{}cc@{}}y^2 & -y(x \sin(\Theta) - z \cos(\Theta)) \\ y(x \sin(\Theta)+z \cos(\Theta)) & -\cos^2(\Theta)(x^2 \tan^2(\Theta)+z^2)\end{array}\!\right)\nonumber \\ \end{eqnarray} \tag{ 22 }$$

where the spin direction $\hat{\bit s}=(x,y,z)$ is expressed with respect to the u_1,2,3 and Θ denotes the scattering angle (see figure 3).

In order to illustrate the polarization dependence of resonant magnetic x-ray scattering, we consider the example of a bcc antiferromagnet. The unit cell contains two inequivalent lattice sites: one at r₁ = (0, 0, 0) with $\hat{\bit s}_1=(1,0,0)$ and one at r₂ = (0.5, 0.5, 0.5) with $\hat{\bit s}_2=-(1,0,0)$. For the sake of simplicity we assume that the two lattice sites differ only in the local spin moment and are identical otherwise. Note that the above coordinates refer to the so-called crystal frame, which is defined by the crystal axes.

We will consider the (0 0 1)-reflection for which $\mathcal{F}(0\,0\,1)=\mathcal{S}_1-\mathcal{S}_2$. Since the system is cubic and only the terms linear in $\hat{\bit s}$ contribute to resonant $\mathcal{F}(0\,0\,1)$, we can use the expression given in equation (19) which is proportional to $({\bit e}'\times{\bit e})\cdot \hat{\bit s}$, i.e. the $\mathcal{S}_{1,2}$ are simply given by the term proportional to R⁽¹⁾ in equation (22). This then yields

$$\begin{equation} \mathcal{F}(0\,0\,1)=-2 R^{(1)} i \left( \begin{array}{@{}cc@{}} 0 & \cos{\psi}\cos{\Theta}\\ -\cos{\psi}\cos{\Theta} & \sin{\psi}\sin{2\Theta} \end{array}\right). \end{equation} \tag{ 23 }$$

In this equation ψ is the azimuthal angle and describes a rotation of the sample about u₃ as indicated in figure 3. When the sample is rotated about u₃, the crystal frame and with it the spin directions rotate within the u_1,2-plane. In the current setting the spin directions are along u₂ for ψ = 0. The square modulus of the matrix elements of $\mathcal{F}$ is proportional to the intensities in the different scattering channels. For instance, the intensities in the σπ- and the ππ-channels are I_σπ ∝ (cos ψ cos Θ)² and I_ππ ∝ (sin ψ sin 2Θ)², respectively. The calculated azimuth-dependences for I_σπ and I_ππ are shown in the top panel figure 5. The azimuthal dependence of I_σπ is due to the fact that $({\bit e}_{\sigma}\times{\bit e}_\pi')\cdot \hat{\bit s}\propto {\bit k}'\cdot \hat{\bit s}$, i.e. this intensity is maximal if the spin directions are parallel to the scattering plane. Similarly, it is easy to see that $({\bit e}_{\pi}\times{\bit e}_\pi')\cdot \hat{\bit s}\propto ({\bit k}\times{\bit k}')\cdot \hat{\bit s}$, which means that I_ππ maximal for spins aligned perpendicular to the scattering plane. Figure 5 also shows the azimuthal dependences of I_σπ and I_ππ for spins along ±(sin Φ, 0, cos Φ). It can be seen that the azimuthal dependence changes drastically with the inclination angle Φ between u₃ and $\hat{\bit s}$.

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In this simple example of a cubic antiferromagnetic (AFM) system, the polarization dependence of the resonant scattering provides a powerful means to identify magnetic scattering and to determine spin directions: (i) in a cubic system, the observation of σπ-scattering directly implies magnetic scattering. (ii) Analyzing the azimuthal dependences enables one to determine the direction of the spins.

However, great care must be taken when dealing with non-cubic systems. In this case, σπ-scattering no longer unambiguously identifies magnetic scattering. A very prominent example is the orbitally ordered and magnetically disordered phase of LaMnO₃, where the σπ-scattering is caused by orbital ordering and not by spin order [25]. The σπ-scattering in this case is related to the polarization dependence of the first term in equation (20). In materials with low local symmetries, the analysis of the azimuthal dependences is also more involved and the cubic approximation might lead to incorrect results [23].

In the TM oxides the O 2p states play an important role in the chemical bonding. It is therefore of great interest to investigate their role for electronic ordering phenomena. By performing a scattering experiment at the O K edge, the scattering process involves 1s → 2 p transitions, i.e. virtual excitations directly into the 2p valence shell. In this way, the resonant scattering at the O K edge becomes very sensitive to spatial modulations of the O 2p states. There is no spin–orbit interaction of the 1s core hole and the spin–orbit interaction is usually weak in the 2p final state also. Resonant scattering at the O K edge is therefore typically rather insensitive towards the ordering of spin moments. However, the hybridization of the O 2p levels with the TM 3d states can induce magnetic sensitivity, as discussed in section 4.5. In addition to this, and in contrast to the TM L_2,3 and RE M_4,5 edges described below, the 1s core hole does not produce a strong multiplet splitting in the intermediate state.

To date, a number of experiments on different materials, including the cuprates and the manganites, have demonstrated a strong resonant enhancement of superlattice reflections at the O K edge, revealing the active role of the oxygen 2p states in the electronic order (see section 4.3). Nearly all the resonances observed so far are peaked at the low-energy side of the oxygen K edge, showing that the probed electronic modulations mainly involve the electronic states close to the Fermi level.

At the transition metal L_2,3 edges the resonant scattering process involves 2p → 3d transitions. This makes it possible to study ordering phenomena related to the 3d valence states of the TM. The strong spin–orbit interaction of the 2p core hole splits the x-ray absorption spectra into the so-called L₃ edge at lower energies (2p_3/2 core hole) and the L₂ edge at higher energies (2p_1/2 core hole). This is shown for the Cu L_2,3-edge in figure 4. Furthermore, the coupling of the 2p core hole with a partially filled 3d-shell in the intermediate state results in a very pronounced multiplet structure of the L_2,3 edges, not only in absorption, but also in resonant scattering, as can be inferred from the optical theorem.

The spin–orbit interaction causes a greatly enhanced sensitivity towards spin-order [26], which, by means of the optical theorem, is directly related to the magnetic circular and linear dichroism in absorption [11]. RSXS at the TM L_2,3 edges therefore allows the study of magnetic order of the TM-sites. In addition other electronic degrees of freedom of the 3d electrons, like orbital and charge, can be probed at these edges in a very direct way, since the virtual transitions depend directly on the configuration of the 3d shell.

As will be described in detail in section 4.1, RSXS at the RE M_4,5 edges plays an important role in determining the magnetic structure of thin films containing REs. At the RE M_4,5 edge, the resonant scattering process involves 3d → 4f transitions. One therefore probes the 4f states of the RE elements, which play the most important role for their magnetism.

In the 3d⁹4fⁿ⁺¹ final state of XAS there is a strong spin–orbit coupling that splits the absorption edge into the M₅ edge at lower energies (3d_5/2 core hole) and the M₄ edge at higher energies (3d_3/2 core hole). Furthermore, the coupling between the core hole and the valence shell, and the spin–orbit interaction acting on the 4f-electrons, causes a multiplet structure. Both the M_4,5 splitting and the multiplet structure are also reflected in RSXS as dictated by the optical theorem. Similar to the case of the TM L_2,3 edge, the spin–orbit splitting in the intermediate state of the RSXS process again results in greatly enhanced magnetic sensitivity, which enables the study of the magnetic order in extremely small sample volumes. A particular feature of the RE spectra is the small crystal field splitting. Hence, the spectral shape at the M_4,5 resonances is largely determined by the atomic multiplet structure independent of the crystal symmetry. Effects of the local environment and its symmetry, as discussed in section 2.2.6 are therefore not so pronounced as in the case of 3d TM, which permits an interpretation of RSXS data without recourse to detailed multiplet calculations [27–29].

The RSXS intensity as a function of photon energy and polarization contains very detailed information about the type of electronic order and the involved electronic states. The microscopic nature of the studied electronic order is hence encoded in the spectroscopic part of RSXS and, consequently, there have been considerable efforts to decode this spectroscopic information. However, in most cases this cannot be achieved easily by only examining the experimental data. Instead, it is almost always necessary to compare the experimental data to theoretical models.

For the above-mentioned TM L_2,3 edges and RE M_4,5, where the multiplet splittings in the intermediate states are important, local cluster calculations have proven very successful in analysing the XAS spectra [26]. Based on the optical theorem it is therefore natural to extend this approach to the RSXS lineshape analysis. In the following section 4 examples are provided, where experimental RSXS data is compared to such models. The main assumption of these multiplet calculations is that the resonant scattering process is local. In practice this means that typically a local cluster containing only the resonant scatterer and its ligands is considered. For this cluster it is then possible to treat the many-body interactions and the resulting multiplets exactly, which is the strength of this approach.

Multiplet calculations are therefore very well suited for materials where the various intermediate states of RSXS are strongly localized. Charge transfer processes, which involve sites that are outside the local cluster, are neglected. But this is not always a good approximation and for itinerant systems in particular this assumption is not valid. For less localized systems many-body calculations for larger clusters are therefore necessary. In a few cases this has already been done (e.g. in [30]), but treating the full many-body physics of large clusters is computationally and numerically still challenging.

Even for small multiplet splittings, as is the case for the O K edge, non-local effects in the intermediate state are still important. In particular the non-local screening of the core hole can play an important role for the RSXS lineshape. Also in such cases many-body calculations of an extended cluster, which take into account the effects of the core hole, would be extremely valuable for the RSXS lineshape analysis.

The power of magnetic RSXS to study AFM Bragg reflections from thin single films of only a few magnetic layers has been demonstrated for thin Holmium metal films [75, 76]. Metallic Holmium possesses the largest magnetic moment of all elements coupled by oscillatory long-range RKKY interactions. Below the bulk ordering temperature of 132 K, Ho develops FM order within the closed-packed planes of the hexagonal closed packed (hcp) structure but with a certain angle between the moments of neighbouring planes resulting in a long-period AFM structure along the [0 0 1] direction with an incommensurate temperature-dependent period of about 10 monolayers at 40 K. In a magnetic diffraction experiment, this magnetic structure causes magnetic superstructure reflections well separated from the crystallographic reflections (0 0 2L ± τ) with τ ≈ 0.2 at 40 K as observed from Ho single crystals by neutron scattering [77–79]. The potential of magnetic off-resonant and magnetic hard x-ray resonant scattering has been demonstrated in a series of pioneering magnetic structure studies on Ho by Gibbs et al [5, 80]. In RSXS experiments at the Ho M₅ soft x-ray resonance, due to the long period of the magnetic structure, the (0 0 τ) magnetic reflection of Ho is well within the Ewald sphere. Exploiting the huge magnetic scattering strength of RSXS, this magnetic structure could be readily studied in thin films down to the thickness of a single magnetic period [27, 75, 76] as can been seen from the intense and well-developed magnetic superstructure reflections obtained from a 11 monolayer thin Ho metal film shown in figure 16.

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While for RSXS the relation between magnetic ordering parameters and integrated diffraction peak intensity can be more involved, the integrated superstucture peak intensity of Ho has been shown to be, in good approximation, a measure of the helical AFM order parameter [27] and thus allows one to determine the ordering temperatures for various film thicknesses d. For small d, a finite-size scaling of T_N would have been anticipated and was indeed observed. However, this study found a much faster decrease of T_N with d (see figure 16) than known from simple antiferromagnets and ferromagnets [81–84]. The observed modified finite-size effect in the ordering temperature is characterized by an offset thickness d₀ representing a minimum sample thickness below which no helical order can be established. This finding has been related to the long period of the AFM structure [76], in line with results obtained for the long-period spin-density wave of Cr in Cr/Fe superlattices [85]. As shown in figure 16, even above T_N, finite magnetic intensity could be observed characterized by a diverging peak width with increasing temperature. These remaining broad intensity distributions are caused by short-range magnetic correlations persisting above T_N, with the peak width being an inverse measure of the magnetic correlation length. This observation shows the potential of RSXS for studying magnetic short-range correlations in films of only a few monolayer thickness, even with the perspective to employ coherent scattering as discussed later.

The helical magnetic structure of bulk Ho was already well established by neutron scattering. For some materials, however, no large single crystals can be synthesized, which limits the capability of neutron scattering for magnetic structure determination. In such cases, RSXS can provide detailed information on the magnetic-moment directions exploiting the polarization dependence of the scattered intensity as discussed in section 2.3. Here, the example of thin epitaxial NdNiO₃ (NNO) films [86, 87] is presented. In this work the element specificity of RSXS to study the magnetic structure of Ni and Nd moments in NNO separately is exploited. NNO shows a metal-to-insulator transition at about 210 K. At the same temperature AFM ordering occurs. From the observed (0.5 0 0.5) magnetic reflection, initially, an unusual collinear up–up–down–down magnetic structure connected with orbital ordering of the Ni³⁺ e_g1 electrons has been proposed on the basis of neutron powder diffraction data [88, 89]. These results have been challenged by the later observation of charge disproportionation without any indication for orbital order using hard x-ray diffraction [90, 91]. However, off-resonant and resonant scattering at the Ni K edge can suffer from weak signals and only indirect sensitivity to orbital order. Therefore the (0.5 0 0.5) reflection has been studied by RSXS in the vicinity of the Ni L_2,3 and Nd M₄ resonance. Figure 17 shows the observed azimuthal dependence of the scattered intensity compared to model calculations based on the formalism developed by Hill and McMorrow [15]. Analogous to the case of the simple bcc antiferromagnet (see figure 5), either a collinear up–up–down–down or a non-collinear magnetic structure was assumed.

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The results clearly reveal a non-collinear magnetic structure of the Ni and Nd moments at low temperatures. In addition, the observed polarization dependence of scattered intensity, including polarization analysis of the scattered light at the Ni L_2,3 resonance, has been found to be inconsistent with orbital order and the energy-dependent line shape of the reflection could only be reproduced assuming charge order. Thus these studies fully support a picture of combined charge and non-collinear magnetic order of the Ni sites but are inconsistent with orbital order suggesting that the metal–insulator transition is driven by charge disproportionation in NNO.

At this point it is appropriate to stress again a caveat: the model calculations for NNO used equation (17), which strictly applies to the case in which only the spin breaks the otherwise spherical local symmetry. This approach is commonly used and yields a reasonable data interpretation in many cases. Nonetheless, we need to bear in mind that this is an approximation which might not always be valid in a real material (see section 2.2.6). In particular, for systems with symmetry lower than cubic, the standard treatment can yield completely incorrect results. This is demonstrated in a study on NaCu₂O₂ by Leininger et al [92]. NaCu₂O₂ belongs to the class of edge-sharing copper-oxide chain compounds which have attracted considerable interest in the past few years due to their diversity of ground states. The related compound LiCu₂O₂, e.g. has been found to exhibit ferroelectricity. At present it is not clear whether this ferroelectricity is caused by the complex magnetic structure or the strong tendency of Li and Cu intersubstitution. A non-collinear spiral AFM structure was derived by neutron diffraction on polycrystalline samples and NMR measurements on NaCu₂O₂ [93, 94] in line with results from susceptibility measurements [92]. Serving as an isostructural and isoelectronic reference compound without complicated intersubstitution of alkaline and 3d TM ions [93, 94], NaCu₂O₂ is only available in form of tiny single crystals. Therefore, RSXS is the method of choice to study details of the magnetic structure. However, according to the simple treatment of magnetic RSXS data, the azimuthal angle dependence of the magnetic reflection suggested a magnetic structure made of only one magnetic-moment direction—in stark contrast to the results of the susceptibility measurements and neutron powder diffraction data. This inconsistency could be explained on the basis of the strong anisotropy observed in the polarization-dependent absorption probability. The anisotropy is caused by a selection rule precluding excitations into the partially occupied planar $3{\rm d}_{x^{2}-y^{2}}$ orbitals of the Cu²⁺ ions which modifies the polarization-dependent magnetic scattering strength according to the Kramers–Heisenberg formula and hence masks the azimuthal angle dependence of the magnetic signal. These findings are in excellent agreement with the theoretical predictions by Haverkort et al [23], which shows that in general the local orbital symmetry has to be taken into account for magnetic structure determination by magnetic RSXS.

The first application of RSXS to study long-period AFM structures in bulk materials has been reported by Wilkins et al [95] showing the potential of RSXS to study magnetic structures and, even more importantly, other electronic ordering phenomena in bulk oxides. For magnetic structure determination, however, neutron scattering will remain an unrivaled method. On the other hand, despite the limited Ewald sphere, the potential of magnetic RSXS for exploring magnetic structure has been clearly shown and its importance will increase with growing interest in thin transition metal oxide films and superstructures that can be obtained with very high quality. Films of REMnO₃ perovskites represent a prominent example, where long-period complex magnetic order is strongly coupled to ferroelectricity. Such a thin improper multiferroic has been studied for the first time by RSXS very recently by Wadati et al [96].

While RSXS is particularly useful in the case of long-period magnetic structures, RSXS can also provide detailed information on the microscopic magnetic properties of ferromagnets and simple antiferromagnets, in particular when applied to artificial structures that produce an additional periodicity of the order of the wavelength of soft x-rays. Such structures are also naturally obtained by the formation of domains in chemically homogeneous materials, as has been observed for thin films of FePd. FM FePd films are characterized by perpendicular magnetic anisotropy leading to the formation of well-ordered stripe domains as shown in figure 18. In order to reduce the stray field outside the sample, closure domains can be formed at the surface, which link the magnetizations of the neighbouring stripes, resulting in an overall domain structure of a chiral nature. X-ray scattering experiments are not usually sensitive to the phase of the scattered wave and, hence, cannot easily distinguish left and right-handed magnetic structures. However, for circular polarized incident light, the resonant magnetic structure factor differs for structures with a different sense of rotation [97] giving rise to circular dichroism in the scattered intensity not connected with absorption effects but depending on the sense of rotation of a magnetic structure. This property has been exploited by Dürr et al [40] to study closure domains in thin FePd films by RSXS. Here, the regular domain pattern at the surface does not cause superstructure reflections in terms of a single peak but rod-like off-specular magnetic scattering intensity with a maximum intensity at a transverse momentum transfer determined by the inverse of the mean distance of the stripe domains as shown in figure 18. The intensity of these off-specular magnetic rods showed pronounced circular dichroism, in this way proving the existence of the aforementioned closure domains. By modelling the Q-dependent magnetic rod intensity, shown in figure 19, the depth of the closure domains could be estimated to be about 8.5 nm in a 42 nm thick film [98].

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Pure in-plane momentum transfer can be achieved by performing experiments in transmission geometry. As demonstrated for soft-x-ray small-angle scattering from Co/Pt multilayers and nanoparticle assemblies in symmetric scattering geometry, this technique is very well suited to explore lateral magnetic and charge heterogeneities like mean grain- and magnetic-domain sizes [99, 100]. Therefore, it has been intensively applied to study domain configurations of Co/Pt multilayers and FePd alloy films depending on external magnetic fields in this way generating a deeper understanding of the microscopic details of the magnetic reversal processes in these materials [101–107].

In addition to such 'natural' in-plane structures, RSXS can be exploited to study artificially designed lateral structures on a nanometre-length scale, such as regular or irregular line and dot arrays. Regular patterns of magnetic materials are of interest in connection with future data storage technologies. The typical length scales (nm) of nanostructures perfectly match the wavelengths of soft x-rays. The first application was presented by Chesnel et al monitoring the magnetization reversal of a regular arrangement of magnetic lines [108]. These samples have been made of Si lines of about 200 nm width and 300 nm height with a line spacing of 75 nm covered by Co/Pt multilayers. In contrast to the FePd stripe domains discussed above, this structure gives rise to off-specular intensity rods even without magnetism (see figure 20). Analyzing the in-plane momentum transfer q_x as well as the width and the relative intensities of the superstructure rods of different order yields a detailed characterization of the chemical structure. At the Co L_2,3 resonance, additional magnetic information can be extracted from the scattered intensities. Here, ferromagnetically aligned neighbouring stripes contribute magnetic intensity to the chemical superstructure rods, while AFM aligned lines give rise to pure magnetic satellites at half-order positions. Measuring hysteresis loops by recording the scattered intensity at various q_x as a function of an external field as shown in figure 20, yields information about the magnetic coupling of such nanolines and the magnetization reversal process can be modelled in detail. While the hysteresis for q_x = 0 displays the average magnetic behaviour, also accessible by macroscopic techniques, the off-specular magnetic rod intensity reflects the magnetic behaviour of the well-ordered part of the sample. The occurrence of significant magnetic intensity at half-order positions for the external field close to the coercive field shows that the demagnetized state of this sample is in fact characterized by an AFM alignment of neighbouring lines with a correlation length of about four lines. Similar studies have been performed on a regular dot array of FM permalloy dots [109] as well as permalloy rings, where the RSXS intensity distribution has been shown to be sensitive to the possible magnetization states of the single rings, i.e. vortex or so-called onion states [110].

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As already shown in the previous section, RSXS is capable of studying ordering phenomena with spatial resolution. In addition to domain configurations discussed above, spatially varying order can be a consequence of the broken symmetry at thin-film surfaces and multilayer interfaces. One way of achieving spatial resolution makes use of local probes deposited into the system of interest which can be easily measured at the corresponding resonance of the probe material as a consequence of the element specificity of the method and the strong optical effects at resonance [111]. However, even without a probe material the information on spatially varying ordering phenomena is contained in the dependence of scattered intensity on the chosen incident energy and momentum transfer which can be very sensitive even to tiny modifications of order. Hence, RSXS has been applied very successfully to obtain magnetization depth profiles of thin films and multilayers with very high depth resolution. In the past few years, three different ways have been introduced to extract depth-dependent information on magnetic ordering, exploiting

• (i)  
  the change of the sample volume probed through the resonance due to strong incident energy dependence of the photon penetration depth [35, 112, 113],
• (ii)  
  the sensitivity of the shape of a magnetic reflection from very thin samples on the spatial varying magnetic ordering parameter [114], and
• (iii)  
  the incident energy- and momentum-transfer dependent reflectivity [46, 52, 65, 115–117].

While the first two approaches allow one to draw qualitative conclusions from the raw data without involved data analysis they, however, require a magnetic reflection inside the Ewald sphere at resonance. Approach (i) has been applied to study depth-dependent AFM ordering in thin Dy metal films grown on W(1 1 0). Metallic Dy develops a long-period helical AFM superstructure below T_N = 179 K, which undergoes a first-order transition into a FM structure at about 80 K. This transition is characterized by a pronounced thickness dependence of its hysteretic behaviour, i.e. the hysteresis smears out with decreasing film thickness. The long-period AFM structure gives rise to magnetic superstructure reflections well separated from the structural Bragg reflections similar to those observed from Ho metal described above, which can be easily measured in the vicinity of the Dy M_4,5 resonance. Through this resonance not only does the magnetic scattering strength vary but also the probing depth of the photons due to the energy dependence of the imaginary part of the index of refraction. Figure 21 shows the temperature-dependent intensity of this AFM reflection through the AFM-FM transition measured for two different photon energies, i.e. with different probing depths of the photons. Distinctly different hysteretic behaviour was observed, which readily indicates a pronounced depth dependence of the growth of the helical AFM order with temperature. From modelling these energy-dependent hysteresis loops, taking the energy-dependent absorption into account, a temperature-dependent depth profile through this first-order phase transition could be extracted, shown in figure 21 [35, 118]. Obviously, upon heating the helical order starts to grow in the near surface region of the film and develops deeper towards the W interface with temperature. This behaviour has been attributed to pinning of the FM phase at the interface by strain effects. Interestingly, also at the top surface, a phase of modified AFM order characterized by a tendency towards ferromagnetism survives over a large range of temperature.

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This method of achieving depth-dependent information requires a film thickness much larger than the minimum photon penetration depth at resonance and hence cannot be applied to ultra-thin samples. In this latter case, however, according to approach (ii), the shape of a superstructure reflection measured at a fixed energy already contains information on the spatial modulation of magnetic order, as has been shown for magnetic scattering from the magnetic semiconducter EuTe [114]. Below T_N = 9.8 K, EuTe develops a simple AFM structure with ferromagnetically ordered (1 1 1) planes but alternating magnetization along the [1 1 1] direction. By virtue of the strong magnetic Eu M_4,5 resonance and the very high quality of the epitaxially grown [1 1 1] EuTe films, a very well-resolved magnetic half-order reflection could be observed from a film of only 20 monolayers, with a peak intensity in the order of almost 1% of the incident intensity. This magnetic reflection is already comparable to the off-resonant charge scattering from the superlattice peak of figure 11. At resonance, this magnetic reflection appears very close to the Brewster angle where the charge scattering background for π polarized incident light is almost completely suppressed. This ideal combination of magnetic period length and resonance energy in EuTe therefore leads to a magnetic signal-to-background ratio of several orders of magnitude, allowing measurements of the magnetic superstructure reflection with unprecedented quality. As can be seen in figure 22, the intensity of this reflection is not contained in a single peak but distributed over several side maxima caused by the finite size of the system (so-called Laue oscillations). The period of these oscillations displays a measure of the inverse number of contributing magnetic layers and the overall envelope functions depends sensitively on the magnetization depth profile. By virtue of the high magnetic contrast in this study, the side maxima could be resolved over a large range of momentum transfer revealing distinct changes with temperature: with increasing temperature the period of the Laue oscillations increases and the intensity of the side maxima compared to the central peak decreases. This behaviour signals a non-homogeneous decrease of the magnetic order through the film, such that the effective magnetic thickness decreases with increasing temperatures, i.e. the order decays faster at the film boundaries.

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From a kinematical modelling of the observed intensity distributions, temperature-dependent atomic-layer-resolved magnetization profiles across the entire film could be extracted. Even at low temperatures, these profiles displayed reduced order at the film interfaces, which is due to some chemical intermixing. In addition, the profiles reveal a faster decrease of the order close to the film boundaries with temperature, i.e. a layer-dependent temperature dependence of the magnetic order is observed. Due to the half filled 4f shell of Eu²⁺ ions, EuTe exhibits almost pure spin magnetism caused by strongly localized moments and can therefore be regarded as a Heisenberg model system, for which this type of surface-modified order was theoretically predicted [119].

Modeling of Q-dependent intensity distributions is not restricted to magnetic superstructure reflections but can also be applied in a more general way to the specular reflectivity from systems without any magnetic or chemical reflection inside the Ewald sphere at a soft x-ray resonance, e.g. FM films, in this way widening the method to a broader field of applications. This has been successfully demonstrated for hard x-ray resonant scattering [73, 121, 122]. In the soft x-ray range, in particular at resonance, specular reflectivity typically does not fall off so rapidly and hence can be measured over a large range of momentum transfer due to the strong optical contrast at the interfaces and surfaces. Information on FM depth profiles usually requires one to compare reflectivities measured with different polarizations of the incident x-ray beam or for opposite magnetization directions. Interpretation of such reflected intensities is often less intuitive and requires detailed theoretical modelling as well as precise knowledge of the energy-dependent optical parameters. Based on an magneto-optical matrix algorithm developed by Zak et al [123, 124], several computer codes have been developed within the last decade for this purpose. In pioneering early experiments, modelling of energy-dependent reflectivity data has been exploited to extract averaged magnetic properties, like the Fe magnetic moment in Fe_xMn_1−x thin films, which undergo a magneto-structural transition at x = 0.75 [65]. Later studies improved data analysis to incorporate the effects of magnetic and chemical roughness as well as magnetization depth profiles [52, 116, 120, 125, 126]. Examples of interest are FM half-metals, like La_1−xSr_xMnO₃ (LSMO) close to x = 0.3 and a number of Heusler compounds. Half-metals are characterized by a band gap for only one spin direction and therefore are candidates to show 100% spin polarization at the Fermi level, thus being of high interest for spintronic devices. Functionality of such materials crucially depends on the magnetic properties at interfaces, which can be strongly altered by extrinsic effects like electronic reconstruction, disorder, roughness, or strain, but also by intrinsic changes caused by the broken translational symmetry. Therefore, obtaining depth-dependent information on the FM properties of such materials at various interfaces is greatly needed but also challenging.

The interface magnetization depth profiles for Co and Mn moments in Co₂MnGe/Au multilayers were obtained from the analysis of element-specific RSXS [120]. By fitting the XMCD spectra measured at the angular positions of the first three superlattice reflections, the individual magnetization profiles shown in figure 23 for the Co and Mn moments through the magnetic layers can be reconstructed. They are characterized by non-magnetic interface regions of different extension for the upper and lower boundaries and also differ for the two magnetic species leading to a rather complicated behaviour of the magnetization through a magnetic layer. This observation has been attributed to structural disorder caused by strain at the interfaces which affects Co and Mn spin in a different way since Co on a regular Mn position keeps its FM spin orientation and full moment, but Mn on a Co position has an antiparallel spin orientation and a reduced moment [120, 127].

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With a similar scope, RSXS was applied to thin manganite films. Manganites show one of the richest set of phase diagrams among the transition metal oxides and therefore offer great potential for application in future functional heterostructures. Freeland et al [128] showed the existence of a thin non-FM and insulating surface layer with a thickness of only one Mn–O bilayer on a single crystal of layered LSMO by analysing its energy-dependent reflectivity at the Mn L_2,3 resonance combined with XMCD and tunnelling probe techniques [128]. This finding renders the material a natural magnetic tunnel junction. The sensitivity of RSXS to weak magnetic modifications like a thin non-magnetic layer in manganites is nicely demonstrated in a study of La_1−xCa_xMnO₃ (LCMO) films for different thicknesses and substrates by Valencia et al [116]. By analysing the XMCD spectra, measured on the specular reflectivity for fixed scattering angles, detailed knowledge of the chemical and magnetic surface properties could be obtained. In particular, all samples studied developed a thin surface region ranging from 0.5 to 2 nm with depressed magnetic properties. Figure 24 shows simulations of the XMCD spectra depending on the presence of a magnetic dead layer, revealing distinctly different energy profiles irrespective of the film thickness. In the same way, Verna et al very recently reconstructed the magnetization depth profile with high spatial resolution from thin LSMO samples, again, observing a non-FM surface region of about 1.5 nm thickness which expands with increasing temperature [129]. The observation of such non-FM dead layers at manganite surfaces for LCMO as well as LSMO single crystals and films of different thickness, independent of the particular epitaxial strain, strongly suggests that the observed modified surface magnetization is an intrinsic property of manganite surfaces [116]. This conclusion is further supported by an XAS study exploiting linear dichroism at Mn L_2,3 resonance from thin LSMO films which observed an orbital reconstruction, i.e. a rearrangement of the orbital occupation at the surface consistent with a suppression of the double-exchange mechanism responsible for the FM properties [130].

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The latter examples of modified, mostly suppressed magnetic properties at surfaces and interfaces, are important results in view of possible application of materials in heterostructures. However, the proximity of two materials in heterostructures can cause much stronger and more complex changes of magnetic properties at buried interfaces, which has been studied by RSXS intensively in the last decade.

Famous examples of interface-generated magnetic phenomena with high technological relevance are giant magnetoresistance and exchange bias. The experimental challenge of characterizing such phenomena is due to the extremely small amount of contributing material, the presence of more than one type of magnetic moment, and the buried character of the region of interest. Also here, RSXS has been shown to be a very sensitive tool to study such interfaces since its huge charge and magnetic scattering strength is accompanied by a moderate penetration depth, element specificity and access to structural information. One crucial parameter that can influence interface-generated functionality is chemical and/or magnetic roughness, which for a buried interface can be characterized best by scattering techniques measuring the related off-specular scattered intensity distributions. Such diffuse scattering contributions, however, are typically several orders of intensity weaker than the specular reflectivity and require an extraordinary magnetic scattering strength to be detectable.

The first demonstration of how diffuse RSXS intensities can be exploited to track magnetic roughness was reported by McKay et al for Co films and Co/Cu/Co trilayers [131]. In this study the diffuse intensity around the specular reflectivity was measured at the Co L₃ resonance containing contributions from magnetic and chemical roughness. In order to separate the two quantities, the intensity was measured with circular polarized x-rays for two opposite magnetizations of the sample. In this case the sum of the two measurements represents the chemical roughness while the difference was interpreted as a measure of the magnetic roughness. From the intensity and width of such broad intensity distributions, properties like the root-mean-square roughness (intensity) and correlation lengths (width) can be extracted independently for the magnetic and chemical interface. Interestingly, the magnetic signal drops much faster with increasing momentum transfer S_y than the chemical intensity as can be seen in figure 25. This observation yields a surprising result, namely a much smoother magnetic interface if compared with the chemical roughness. It was explained by a weaker coupling of the Co moments in the intermixing region compared to that of the homogeneous part of the film. This means that the moments connected with chemical roughness do not contribute to the signal from the film magnetization and the magnetic interface therefore appears smoother than the chemical one. Similar results have been obtained by Freeland et al who studied systematically the relation between magnetic and chemical interfacial roughness by monitoring the diffuse scattering from a series of CoFe alloy films with varying chemical roughness [132, 133]. Here, it was also shown that the diffuse intensity can be used to measure magnetic hysteresis loops of bulk and interfacial magnetization independently [132]. It has been shown later that the difference signal contains charge-magnetic interference contributions, such that it does not represent the pure magnetic roughness [134]. The general result, however, that the magnetic and chemical roughness can be very different, is not affected. This was supported later in a study by Hase et al, who separated chemical and magnetic roughness in a different way by comparing the diffuse scattering in the vicinities of either a chemical or half-order pure magnetic reflection from an antiferromagnetically coupled [Co/Cu]_n multilayer [135]. The observation of different roughnesses of the chemical and magnetic interfaces shows that the understanding of macroscopic magnetic phenomena and their relation to roughness cannot be simply obtained from the knowledge of the chemical roughness alone. Nevertheless, Grabis et al observed almost identical chemical and magnetic roughness parameters in a Co₂MnGe/Au multilayer, showing that different roughnesses are not a general feature of magnetic interfaces [120]. A theory describing diffuse resonant scattering from chemically and magnetically rough interfaces beyond the kinematical approximation has been published by Lee et al [136].

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The proximity of two different magnetic systems can cause more complex changes of the magnetic properties than just introducing disorder, as discussed in the previous paragraphs. A prominent example, connected with functionality already applied in technological applications is the exchange bias effect. Exchange bias is the observation of a shift of the hysteresis along the field axis of a FM material in contact with an antiferromagnet (AF) after field-cooling through the Néel temperature. The general mechanism was identified to be the exchange interaction between the spins of the FM and the AF at the interface, which generates an additional unidirectional anisotropy for the FM spins [137, 138]. However, the original models assuming perfect interfaces failed to describe exchange bias and related phenomena in a quantitative way. Rather, the exchange bias strongly depends on details of the magnetic structure at the interface. Exploiting XMCD techniques, rich knowledge on the magnetic behaviour of the interface was obtained. In particular the presence of a FM component within the AF including a pinned fraction that is related to the exchange bias was observed [139]. Due to the fact that interfaces are buried, techniques not limited to the top sample surface region are advantageous. Therefore, a large number of experiments exploited the XMCD on the specular reflectivity to characterize the element-specific behaviour of spins in exchange bias systems [68, 140–142]. Such studies revealed the complex magnetic situation at the interface connected with exchange bias. In particular, several different types of spins are involved, namely the FM spins, the compensated AFM moments, as well as uncompensated AFM moments that can be divided into a pinned and a rotatable fraction. By exploiting the structural information contained in reflectivity data, detailed characterization of the depth dependence of the different magnetic contributions at the interface could be achieved. By varying the absorption threshold, magnetization direction, photon polarization and scattering geometry, RSXS can be tuned to be sensitive to almost all the magnetic ingredients of the magnetic interface. In this way Roy et al determined the depth profile of FM and unpinned uncompensated AFM spins of the exchange bias system Co/FeF₂ by analysing the specular reflectivities measured with constant helicity of the incident photon beam for opposite magnetization directions [115]. They found the majority of the uncompensated unpinned AFM spins in a region of a few nanometres below the interface, characterized by AFM coupling to the spins of the ferromagnet. In addition to the magnetization direction, elliptical undulators allow one to switch the light helicity, providing a detailed picture of the magnetic structure at the interface. In this way, Brück et al studied the interface of the MnPd/Fe exchange bias system [52]. Figure 26 shows a scenario resulting from an analysis of respective reflectivity difference curves for either opposite external field directions or light polarizations, separating the pinned and unpinned uncompensated spins within the antiferromagnet. According to this study, the major fraction of the uncompensated AFM spins located very close to the ferromagnet is rotatable and antiferromagnetically coupled to the ferromagnet, while the uncompensated AFM spins deeper in the AFM are predominantly pinned to the AF which finally generates the exchange bias effect. Similar results were obtained by Tonnerre et al who extended the method to study the interface of a system with perpendicular exchange bias [117].

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Interesting, interface-generated functionality in magnetic heterostructures can emerge from a magnet in contact with a non-magnetic material. In a very recent study, Valencia et al explored the influence of a ferromagnet in contact with a ferroelectric [70]. In this experiment, the structural information that can be extracted from scattering experiments was disregarded, still, it can be considered a text-book example showing the sensitivity of XMCD-like experiments exploiting specular reflectivity rather than TEY or FY methods to study tiny magnetic effects at buried interfaces. In addition, it represents an important example of interface-generated material properties [70].

In present solid state research, magnetic control of ferroelectric properties and, in particular, electric control of magnetic order are phenomena with very promising possibilities for applications. Unfortunately, most of the known ferroelectrics are either not magnetically ordered or magnetism and ferroelectricity are weakly coupled. Exceptions are the so-called improper ferroelectrics, where ferroelectricity is induced by complex mostly AFM structures, which is discussed for bulk materials in more detail in section 4.5. These materials, however, are often connected with very low ordering temperatures. In the quest for room-temperature multiferroicity, one way to overcome this limitation is to combine a room-temperature ferromagnet and a room-temperature ferroelectric in a heterostructure. Here, the chosen heterostructures consist of Fe or Co on BaTiO₃ (BTO) [70]. BTO is a robust diamagnetic room-temperature ferroelectric. The study could show that the tunnel-magnetic resistance of Fe or Co/BTO/LSMO trilayers strongly depends on the direction of the ferroelectric polarization in the insulating diamagnetic BTO layer. Thus, these heterostructures possess coupled magnetic and ferroelectric, i.e. multiferroic, properties. Element-selective XMCD in reflection geometry could identify the key mechanism: the ferromagnet generates ferromagnetism in the topmost layer of BTO by inducing a magnetic moment in the formally non-magnetic O and Ti⁴⁺ ions as revealed by a seizable asymmetry A_XRMS displayed in figure 27. The corresponding magnetic interface properties are very weak and hardly detectable by conventional XMCD techniques. Exploiting the dichroism in specular reflectivity through the Ti L_2,3 and O K resonance, Valencia et al have been able to unambiguously prove the existence of FM moments at the Ti and O ions coupled to the magnetization of the ferromagnetic layer. This observation shows the huge advantage of performing XMCD-like experiments exploiting scattered photons to study tiny effects at buried interfaces and, in addition to these methodological aspects, opens the door for future application of interface-generated room temperature multiferroicity. Modified macroscopic and/or microscopic properties at heterostructure interfaces are by far not limited to magnetic behaviour but can be found for other electronic phenomena as well, which offers even more fascinating new possibilities for future nanoscale devices.

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In ultra-thin samples or at heterostructure interfaces completely new phenomena may arise that cannot be found in the corresponding bulk materials. An important and fast-developing field of research was triggered by the discovery that interfaces of transition metal oxides can be grown with very high structural quality down to a flatness on the atomic level. With these structurally perfect interfaces, fundamental studies of new interface properties have become feasible. Examples are the discovery of a highly mobile electron gas [144] and even superconductivity [145] at the interface of the two insulating perovskite oxides LAO and SrTiO₃ (STO). A recent example is the observation of insulating and AFM properties of LaNiO₃ (LNO) in LNO/LaAlO₃ (LAO) superlattices, as soon as the LNO thickness falls below 3 unit cells, while thicker LNO samples are metallic and paramagnetic [146]. Finally, a major reconstruction of the orbital occupation and orbital symmetry in the interfacial CuO₂ layers at the (Y,Ca)Ba₂Cu₃O₇/LCMO interface has been observed [147]. It is a matter of debate to what extent such properties are caused by either electronic reconstruction, strain, oxygen vacancies, or disorder. Hence, microscopic understanding of macroscopic interface properties may require detailed knowledge on microscopic electronic properties with high spatial resolution. With its strong sensitivity not only to magnetic but also to charge and orbital order, RSXS is well suited to extract quantitative information on even subtle changes of these electronic degrees of freedom at interfaces.

The general strategy here is very similar to the case of magnetic depth profiling: the information on spatially modulated orbital occupation or charge distribution is contained in the reflectivities, measured as a function of momentum transfer and incident photon energy. This has been demonstrated by Abbamonte et al while studying the carrier distribution in thin films of La₂CuO_4+δ grown on SrTiO₃ [45]. The thin-film reflectivities of these samples are characterized by well-developed thickness oscillations when measured at the Cu L_2,3 resonance, proving high sample homogeneity and surface/interface quality (see figure 28). In contrast, with a photon energy corresponding to the mobile carrier peak (MCP), the oscillations vanish at higher angles of incidence. This MCP in the O K pre-edge region represents a characteristic energy for scattering from the doped carriers with an enhancement of the scattering strength of a single hole by about two orders of magnitude. Hence, the reflectivity measured at this photon energy is almost entirely determined by the distribution of the doped carriers. The strong damping of the thickness oscillations at this specific energy is readily explained by assuming a smoothing of the carrier density at the interface towards the substrate.

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In a very similar way Smadici et al studied the distribution of doped holes in superlattices consisting of double layers of insulating La₂CuO₄ (LCO) and overdoped La_1.64Sr_0.36CuO₄ [148]. While none of these materials is superconducting, the heterostructure shows superconductivity below T_C = 35 K suggesting charge redistribution at the interface. In RSXS with the photon energy tuned to the La M₅ resonance, the superstructure gives rise to a series of reflections detectable up to the 5th order. These reflections mainly contain information about the distribution of the Sr ions that induce the hole doping. In contrast, at the MCP energy only the first superstructure reflection could be observed. This result directly shows that although being modulated with the superlattice period, the doped hole distribution does not follow that of the Sr ions. From a quantitative analysis, an average hole density in the insulating LCO layers of 0.18 holes per Cu site has been deduced, suggesting that superconductivity occurs in the formally insulating LCO layers.

In order to clearly separate interface behaviour, often it is of an advantage to design a multilayer such that a specific superstructure reflection is interface sensitive, i.e. its structure factor essentially is given by the difference of the optical properties of an interfacial layer and a bulk-like layer, in that way directly representing the changes at the interface. This approach has been demonstrated by Smadici et al [143] who studied the interfacial electronic properties of SrMnO₃ (SMO)/LaMnO₃ (LMO). While the two single materials are a Mott insulator (LMO) and a band insulator (SMO), repectively, FM and metallic interface properties had been predicted [149] and macroscopically observed [150]. In this experiment, (0 0 L) superstructure reflections were studied in a heterostructure consisting of m × LMO/n × SMO double layers. m and n were chosen such that the L = 3 reflection perpendicular to the surface only occurs if scattering from the interfacial and inner MnO₂ planes of LMO or SMO differ. A pronounced L = 3 reflection from this structure could be observed only in the vicinity of the Mn L_2,3 resonance and in a very narrow energy region at the onset of the O K resonance (see figure 29), i.e. involving electronic states in the vicinity of the Fermi level. The observation of no broken symmetry in the atomic lattice (i.e. no L = 3 reflection off-resonance), but an interfacial reflection induced by the unoccupied density of states near E_F gives strong evidence that the interface is characterized by electronic reconstruction. The intensity of the L = 3 reflection at the Mn L_2,3 resonance was identified to be of magnetic origin from a rough azimuthal dependence of the scattered intensity. Comparison of the temperature dependence of the macroscopic properties (conductivity and magnetization) with the observed L = 3 intensities at the O K and Mn L resonances suggests that metallic and FM behaviour is indeed interface-generated driven by electronic reconstruction. A very similar approach has also been used to characterize the STO/LAO interface in detail very recently by Wadati et al [151].

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While the latter studies revealed interface-driven modifications in a more qualitative way, it has been shown very recently that quantitative information from such superstructure reflections can be derived for each individual atomic plane. This, however, requires detailed modelling of energy- and polarization-dependent reflected intensities as in the case of magnetic depth profiling discussed above. The studied system is a multilayer made of repeated bilayers of four unit cells of LAO and four unit cells of LNO [152]. Ni in cubic symmetry is characterized by a doubly degenerate e_g level filled with one electron. According to model calculations, epitaxial strain can be utilized to favour the occupation of the in-plane x²–y² orbital in a superlattice geometry, in this way matching the electronic structure of cuprate high-T_c superconductors, which may be a route to generate high-T_c superconductivity in artificial nanostructures [153–155]. While magnetic depth profiling used the contrast obtained by circularly polarized light, light with linear polarizations π and σ (see figure 3) was used to measure the linear dichroism of the orbital scattering according to the last term in equation (17). By changing the light polarization from in-plane to normal to the superlatice plane, the sensitivity to the 3z²–r² orbital could be varied. The observed linear dichroism in absorption yields the average difference in occupation of the two e_g orbitals of 5.5%. In addition, the linear dichroism at the second superlattice reflection, sensitive to the difference of the scattering strength from the inner LNO unit cell and the interfacial LNO unit cell, was monitored. As shown in figure 30, the observed linear dichroism here is much stronger than expected for a homogeneous system with the same orbital occupancy for the inner and interfacial LNO layers. Hence, an inhomogeneity of the orbital occupation through this four unit cells thin LNO layer can be readily deduced. Modeling the observed energy dependence for the two different linear polarizations, taking the average dichroism as a fixed input, the data could be explained by a distinct higher occupation of the x²–y² orbitals at the interface (about 7%) compared to the inner layers (about 4%), which is explained by the reduced gain of kinetic energy for the electrons in the interface layer by hopping across the interface due to the closed shell of the neighbouring Al³⁺ ions.

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HTSC in cuprates occurs after doping the AFM Mott–Hubbard insulator [156, 157] with holes or electrons. Since the on-site Coulomb interaction U on the Cu sites is considerably larger than the charge transfer energy Δ between Cu 3d states and O 2p states, in the undoped system the gap is essentially determined by Δ and the holes are not formed on the Cu sites but on the O sites [158]. The doping-induced mobile holes are shared by four oxygen sites surrounding a divalent Cu site with one hole in the 3d shell, in this way forming the antiferromagnetically coupled Zhang–Rice singlet [159] state. This state may be considered as an effective lower Hubbard band (LHB) state. The effective upper Hubbard band (UHB) is formed predominantly by Cu $3{\rm d}_{x^2-y^2}$ states hybridized with some O 2p_x,y states. The doping, which leads to a controlled metal–insulator transition, occurs by block layers between which the CuO₂ layer are embedded. In La_2−xSr_xCuO₂ hole doping of the CuO₂ layers occurs via replacing the trivalent La ions in the LaO block layers by divalent Sr ions. Long-range AFM order disappears at x = 0.05 and the highest superconducting transition temperature T_c is reached at x = 0.18. In the overdoped case (x > 0.18) the normal state can be well described by a Fermi liquid. For x < 0.15 the normal state properties are far from being understood. There is no agreement on what should be the minimal model which contains the basic physics of underdoped CuO₂ layers. Several phases are found in this doping regime: an AFM insulating phase, the high-T_c superconducting phase, and a charge and spin-ordered phase [160]. Regarding the latter, two different phases are discussed in the literature: the checkerboard phase and the stripe-like phase. In the stripe-like phase, AFM antiphase domains are separated by periodically spaced domain walls along the Cu–O directions in which the holes are situated (see figure 31). Partially motivated by the detection of incommensurate low-energy spin excitations, the existence of a stripe-like state was proposed by theoretical work on the basis of a Hartree–Fock analysis of the one-band or three-band Hubbard model [161–164]. The proposed ground state resembles a soliton state in doped conjugated polymers [165]. The theoretical predictions of a stripe phase were based on a mean-field approximation and the importance of the long-range Coulomb interaction, not taken into account in the Hubbard model, was pointed out [166]. The phase separation, in which it seems that the AFM background expels holes, was supposed to have a strong influence on the physical properties of doped cuprates: (i) a central ingredient to the pairing mechanism in HTSC, (ii) the non-Fermi-liquid behaviour around optimal doping and (iii) the existence of the pseudogap in the underdoped region.

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The stripe-like structure consists of three concomitant modulations of the spin density, of the charge density, and of the lattice which is coupled to the charge modulation. The spin-density modulation has twice the wavelength of the charge and lattice modulation, and in a scattering experiment, these two modulations give rise to superstructure reflections around the AFM Bragg peaks at (0.5 $\pm\frac{1}{2}\epsilon$,0.5,L) and around the structural reflection at (±,0,L), respectively, and symmetry related positions (see figure 31). Assuming half-doped stripes, independent of the average doping concentration, the propagation vector is determined by 1/n where n is the average stripe distance given in d-spacings of the lattice. Using the same assumption about the constant doping concentration in the stripes, should be equal to 0.5n_h. Actually this is observed for $n_h\leq\frac{1}{8}$ [167], indicating that in real-space incommensurate stripes exist, where the stripe distance varies on a local scale and where only the mean distance between the stripes is determined by n_h.

In several doped cuprates such as La_2−xBa_xCuO₂ (LBCO), (La,Nd)_2−xSr_xCuO₂, and (La,Eu)_2−xSr_xCuO₂, the stripe-like order is stabilized near $x=n_h=\frac{1}{8}$, concomitant with a suppression of superconductivity. It occurs in the so-called low-temperature tetragonal (LTT) phase, which is characterized by a corrugated pattern caused by rotation of the CuO₆ octahedra. Evidence for static stripes was first detected in an elastic neutron scattering study on the system (La,Nd)_2−xSr_xCuO₂, where the superstructure reflection for the spin and the lattice modulation have been detected [169].

Neutron scattering [169] and non-resonant hard x-ray scattering [170] monitor the ordering of the charges indirectly by the associated lattice distortion. The reason for this is that these techniques are mainly sensitive to the nuclear scattering and the core electron scattering, respectively. Detection of charge order by these methods is hence not fully conclusive, since lattice distortions may also occur with very small or even no charge order of the valence electrons: they may be caused as a result of a bond-length mismatch between different units of a solid (e.g. planes or chains) or by a spin-density wave alone, like e.g. in chromium metal. On the other hand, RSXS is a method which can directly probe the existence of the charge modulation of the conduction electrons. Furthermore, this method enables one to study the wave length of the modulation, the coherence length, the temperature dependence of the order parameter, and in principle also the momentum dependence of the form factor which would give the detailed spatial dependence of the charge modulation. This has been demonstrated for the doped cuprates by RSXS at the O K and the Cu L_2,3 edges by Abbamonte et al [45, 171]. In figure 32 we show data from this work on LBCO [171], comparing x-ray absorption data, measured with the fluorescence method, with the resonance profile, i.e. the photon energy dependence of the intensity of the charge-order superstructure reflection at energies near a core excitation.

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Absorption edges at the O K and Cu L₃ edges in cuprates have been extensively studied by electron energy–loss spectroscopy (EELS) [173–175] and XAS [175–178]. The O K edge spectra of hole-doped cuprates show two pre-peaks: the lower one at 528.6 eV, the MCP (see section 4.2.2), is assigned to transitions into empty O 2p states related to the Zhang–Rice singlet states. The intensity of this pre-peak is directly related to the number of the mobile charge carriers. The second pre-peak (UHB) at 530.4 eV is related to the UHB. With increasing doping concentration there is a spectral weight transfer from the upper UHB peak to the MCP peak which is a classical signature of a doped correlated system [179], not observed in doped semiconductors. The resonance profile for the ( ${\approx}\frac{1}{4}$,0,L) superstructure reflection shown in figure 32(a) for the O K edge exhibits a prominent resonance at the MCP, in this way providing a direct link to the doped holes and hence identifying the charge order. Interestingly, the reflection also resonates at the UHB energy, which means that also the 'Mottness' is spatially modulated. 'Mottness' in this context means that the system acts like a Mott insulator. The resonance profile for the ( ${\approx}\frac{1}{4}$,0,L) superstructure reflection shown in figure 32(b) shows also a strong enhancement near the Cu L₃ absorption peak. This resonance probably arises from the modulation of the Cu lattice [180]. The charge-order reflections are sharp along H but rod-like along L, indicating quasi-long-range order in the CuO₂ plane but weak coupling between planes.

In the work by Abbamonte et al [171] not only has the existence of a charge modulation in LBCO been directly demonstrated but also a quantitative analysis has been presented. Analogous to the procedure outlined in section 2.2.4, the imaginary part of the form factor can be derived from x-ray absorption data and finally the real part can be derived via a Kramers–Kronig transformation. The form factor for that part which is related to doped holes is shown in figure 32(a) by the blue curve. One realizes a strong enhancement of the calculated form factor near the two pre-peaks although the line shape does not perfectly agree with the measured resonance profile. According to this analysis, the form factor for the scattering of a doped hole at resonance has a value of 82, which means that a single hole in this compound scatters like a Pb atom off-resonance. Since the scattered intensity is determined by the form factor squared, this also means that at resonance the scattered intensity of a hole is amplified by a factor of about 7000. This fact demonstrates the enormous sensitivity of RSXS to the charge modulation of valence electrons which renders this method an ideal tool to study the charge modulation of doped holes in transition metal oxides. The amplitude of the charge modulation was estimated to be 0.063 holes. In a one-band model, which only contains Cu sites, the charge modulation for $x=\frac{1}{8}$ should be close to 0.5 holes. As outlined above, this value is expected for the doping concentration in a simple model in which three Cu–O lines are not doped and the fourth line is 50% doped. The relatively small experimental value of 0.063 hole modulation can be explained by the fact that the holes are distributed among many different (oxygen) sites.

In figure 32(b) XAS data at the Cu L₃ edge together with a resonance profile is presented. Here the enhancement of the scattering length was estimated to be about 300.

Finally, the temperature dependence of the charge ordering in LBCO has been studied yielding a transition temperature T_CO = 60 K which is the same as the transition temperature T_LTT for a lattice transition from the low-temperature orthorhombic (LTO) to the LTT phase. In the latter structure, the CuO₆ octahedra are tilted along the Cu–O bond direction stabilizing the stripe formation.

A similar RSXS study has been performed on the system La_1.8−xEu_0.2Sr_xCuO₄ (LESCO) [181, 182]. Generally, with decreasing ionic radius of the substitutes of La (here Eu), the tilt angle Φ in the LTT phase increases leading to a higher lattice transition temperature T_LTT. This stabilizes the stripe order and leads to a stronger suppression of high-T_c superconductivity [183]. Charge-order diffraction peaks for LESCO could be observed over a range of doping levels x. In addition to information about the phase diagram discussed below, data for different x allow one to extract more detailed information about the amplitude of the hole doping modulation from the photon energy dependences according to the following considerations.

In figures 33(a) and (b) LESCO data for x = 0.15 are reproduced, showing XAS results, calculated intensities, and a RSXS scattering profile. The calculated intensities were derived from the calculated form factors presented in figures 33(c) and (d) which in turn were evaluated from doping dependent XAS data as described above. Assuming a linear variation of the atomic form factor f_j as a function of the doping concentration x one can expand f_j(E, x + δx_j) = f(E, x) + (Δf(E, x)/Δx)δx_j. Then the intensity ratio between the LHB and the UHB is I_LHB/I_UHB = |[Δf(E, x)/Δx]_LHB/[Δf(E, x)/Δx]_UHB|². A remarkable result, typical of a correlated electron system, is that due to a spectral weight transfer, in the XAS data the intensity of the UHB decreases proportional to 1 − x while the intensity of the MCP peak increases proportional to 2x [179]. Since the absorption is related to the form factor, in first approximation the ratio for the form factor of the UHB to the LHB should be near 2, i.e. the intensity ratio in the resonant diffraction profile should be four, in agreement with the calculated ratio (see figure 33(a)), but in clear disagreement with the measured resonance profile presented in the same figure. This discrepancy was explained in terms of a non-linear change of absorption at the UHB and the MCP observed for doping concentrations x > 0.2 [174, 177, 178, 184, 185]. From this analysis it was concluded that the hole doping modulation per Cu site was larger than 20% in agreement with the modulation of about 50% derived for LBCO mentioned above [171].

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In figure 34 we illustrate that other resonances near the O K edge can be used to obtain information on the structure of cuprates. In addition to the first two pre-peaks in the XAS spectrum discussed already, near 533.2 eV a further peak appears due to a hybridization of O 2p states with rare earth (RE = La and Eu) 5d and/or 4f states [173]. In figure 34 we also show the photon energy dependence of the (0 0 1) reflection measured in the LTT phase at T = 6 K. In this phase, neighbouring CuO₂ planes are rotated by 90°, yielding O sites with different (rotated) local environments. As a result, the (0 0 1) reflection is allowed at resonance [186]. The strong resonance at 533.2 eV is due to octahedral tilts, which cause different local environments and affect the hybridization between the apical O and the RE orbitals. In the LTO phase, neighbouring CuO₂ planes are just shifted, not rotated, with respect to each other. In this case the (0 0 1) reflection remains forbidden even at resonance. Using the resonance at 533.2 eV it is possible the detect the LTO–LTT phase transition with soft x-rays. This transition is not detectable by the orthorhombic strain (a-b splitting), because the (1 0 0)/(0 1 0) reflections cannot be reached due to the limited range in momentum space in RSXS.

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With the help of this resonance feature, it is possible to study the temperature-dependence of the structural LTT order and the charge order in one experiment, as shown in figure 35. Here the temperature-dependent RSXS data of LESCO x = 0.15 of the stripe superstructure reflection and the (0 0 1) reflection are depicted. These data indicate a first-order LTO–LTT transition at T_LTT = 135 K and a charge-order transition at T_CO = 65 K, clearly showing that the two phase transition are well separated. A systematic RSXS study of the doping dependence of charge order yielded the phase diagram for LESCO shown in figure 36.

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In contrast to LBCO, the LESCO results represent the first example in which the lattice transition temperature T_LTT is so high that the charge order can no longer be stabilized at this temperature and therefore a gap of 55 K exists between T_LTT and T_CO. This is a remarkable result since it demonstrates that the charge order is at least partially electronic in origin.

The width of the superstructure reflections, being about five times larger than the experimental resolution, indicate disorder effects and/or glassy behaviour. From the widths the coherence lengths of the charge order have been determined. These lengths are of the order of about 100 lattice constants and increase, at least for smaller x, linearly with increasing Sr concentration. As a function of doping no maximum of the coherence lengths has been detected for integer numbers of stripe separation in units of the in-plane lattice spacing corresponding to n_h = 1/2n with n being the integer. This is different from doped nickelates where a clear maxinum was detected for $n_h=\frac{1}{3}$ [189]. The results on the cuprates indicate that in these systems the coherence length is not only related to the commensurability of the stripe lattice with the underlying CuO lattice. Furthermore the results point to the fact that the coherence lengths in these systems are not determined by the impurity potential of the doping atoms but is probably related to an increasing tilt angle with increasing Sr content.

Finally the wave vector of the superstructure reflection increases with increasing Sr content, at least up to x = 0.125. Since the nesting vector between parallel segments in the Fermi surface decreases with increasing doping [190], this indicates that stripes are not conventional charge density waves caused by nesting. Thus, these results point to more strong coupling scenarios for the stripe formation.

In a more recent work the stripe modulations in LBCO and La_1.48Nd_0.4Sr_0.12CuO₄ (LNSCO) were compared using RSXS and hard x-ray scattering [56]. Making use of a two-dimensional detector for RSXS an isotropic coherence length of the charge modulation of the hole density in the CuO₂ planes has been detected. Also the strain modulation of the lattice shows an isotropic coherence length. These results are surprising given that the stripes in a single CuO₂ layer are highly anisotropic. In the direction perpendicular to the CuO₂ planes, the stripes are weakly correlated giving rise to the uniform streak of intensity along L. Both the charge and the strain modulation is better correlated in LBCO than in LNSCO. The in-plane hole correlation lengths for LBCO and LNSCO are 255 ± 5 Å and 111 ± 7 Å, respectively. Similar to LESCO, in LNSCO the hole density modulation sets in well below the transition into the LTT phase while for LBCO T_CO is equal to T_LTT and below T_CO the amplitude of the modulation is independent of the temperature. This suggests that the LTT transition occurred at a higher temperature in LBCO, then it is likely that both the electronic and structural modulations would also have persisted in higher temperatures. A further result from this investigation is that the electronic charge stripe modulation in LBCO is 10 times larger in amplitude than in LNSCO. This result is surprising since from various other experiments, for LBCO, LNSCO and LESCO a similar amplitude for the charge modulation is expected. Further work is required to solve this puzzle.

Until now we have reviewed RSXS studies on static stripe-like CDW order in LBCO, LNSCO and LESCO. In all these compounds the CDW is stabilized by an LTT lattice distortion. An important issue in this context is whether static stripes also exist in other two-dimensional cuprates. Checkerboard-like static charge order has been reported in Ca_2−xNa_xCuO₂Cl₂ (NCCOC) using surface sensitive scanning tunnelling experiments [191]. CDW order could be related to the electron pockets detected in underdoped YBCO in quantum oscillation measurements [192]. The observation of a complete wipe-out of the Cu nuclear quadrupole resonance signal at low temperatures in Ni-substituted NdBa₂Cu₃O_6+δ [193], similar to that of stripe-ordered underdoped cuprates, suggests the existence of static stripes in this compound. Thus, one may think that static stripes are generic to the cuprates and it is very important to look at stripe-like CDW order in other systems which are not stabilized by an LTT lattice distortion.

A RSXS study on a tetragonal compound, already discussed in section 4.2.2, has been performed on an excess O doped La₂CuO_4+δ layer (n_h = 2δ) epitaxially grown on a SrTiO₃ crystal [45]. In this work the resonance effects at the pre-edge of the O K shell excitation and near the Cu L edge have been used to exploit charge modulation in the CuO₂ layers. It was there where it was demonstrated that at the pre-edge of the O K shell excitations, the scattering amplitude for mobile holes is enhanced by a factor of 82 which leads to an amplification of diffraction peaks related to these holes by more than 10³. Extensive non-successful searching for superstructure reflections due to a stripe-like CDW lead to the suggestion of an absence of static stripes in this high-T_c superconductor.

A further RSXS study on a possible charge modulation in a tetragonal cuprate has been performed on NCCOC [194]. From experiments at the MCP peak at the O K edge or at the Cu L₃ edge no evidence for a checkerboard-like charge modulation has been detected. From this null experiment the authors have concluded that the checkerboard order observed in the STS experiments is either glassy or nucleated by the surface. A further RSXS experiment on Ni-substituted NdBa₂Cu₃O_6+δ was stimulated by the above-mentioned Cu nuclear quadrupole resonance experiment [193]. No superstructure reflection due to stripe-like charge order could be detected at low temperatures in this compound [195].

More RSXS experiments on charge order have been performed on the most studied high-T_c superconductor YBCO which contains a CuO₂ double layer and in addition CuO₃ chains along the b-axis. In a first paper on the ortho-II YBa₂Cu₃O_6.5 phase, in which the CuO₃ chain layers are ordered into alternating full and empty chains, charge order in the planes and in the chains were reported [196]. This observation was later questioned, partially by the same authors, in a more refined RSXS study on ortho-II and ortho-VIII oxygen ordered YBCO [20]. However, charge order in underdoped YBCO was postulated on the basis of NMR studies in high magnetic fields [197] and was eventually detected in very recent experiments by RSXS [198] and also with hard x-ray diffraction [199]. The former study showed by the resonance profile near the Cu L₃ resonance, that the charge order in fact takes place in the CuO₂ planes rather than the chains. The temperature dependence of the charge order found in both studies clearly revealed competition with superconductivity. This result has important implications for the understanding of the material and the cuprates in general. It readily explains the anomalously low superconducting temperature in underdoped YBCO and shows that charge order seems to be a generic feature of CuO₂ planes in layered cuprates.

Among the various mechanisms discussed for high-T_c superconductivity in cuprates, the electronic model based on ladder-like structures in which Cu and O atoms are ordered in two chains, coupled by rungs, has been heavily discussed [200, 201]. In this model, depending on the exchange coupling along the chains and that along the rungs, singlets can exist on the rungs of a doped ladder compound and exchange-driven superconductivity can be formed. Depending on the size of parameters or doping concentration, an insulating hole crystal in which the carriers crystallize into a static Wigner crystal may also form the ground state. Note, this would be an electronic charge density wave (CDW), driven by the Coulomb interaction and not by a coupling to the lattice (Peierls transition). The competition of the two phases is similar to that believed to occur between ordered stripes and high-T_c superconductivity in two-dimensional cuprates.

Indeed the only known doped ladder compound Sr_14−xCa_xCu₂₄O₄₁ (SCCO) exhibits both phases: superconductivity has been detected for x = 13.6 below T_c = 12 K at hydrostatic pressures larger than 3.5 GPa [202], while for x = 0 SCCO exhibits a CDW. Therefore, because the interplay between charge and spin degrees of freedom can be easier studied theoretically in quasi one-dimensional systems, experimental studies of charge density modulations in SCCO by RSXS are extremely important [172, 203–205].

SCCO consists of two different alternating types of copper-oxide sheets perpendicular to the b-axis. One with chains, formed out of edge-sharing CuO₄ plaquettes and one with weakly coupled two-leg ladders, i.e. two parallel adjacent chains formed out of corner-sharing CuO₄ plaquettes. Both the chains and the ladders are aligned along the c axis. The sheets are separated by Sr/Ca ions. Since the CuO₄ plaquettes in the chains and those in the ladders are rotated by 45° with respect to each other, the ratio of the lattice constant for the ladder c_L to that of the chain c_C is about $10/7 \approx \sqrt 2$. This means that one unit cell is composed out of ten CuO₄ plaquettes forming the chains and two times seven equal to 14 CuO₄ plaquettes forming the two-leg ladders. This leads to a "misfit compound" which is structurally incommensurate and has a large unit cell along the c axis with the length $c=27.3\,{\AA} \approx 7c_{\rm L} \approx 10 c_{\rm C}$. According to the chemical formula and assuming that Cu is divalent, there are six holes per formula unit. The distribution of the holes among chains and ladders is still under debate. Earlier polarization-dependent XAS measurements for x = 0 [206] suggested that 0.8 holes are on the ladder and the rest is on the chains. Replacing the Sr ions by the smaller isovalent Ca ions, i.e. by chemical pressure, there is a hole transfer from the chains to the ladders which according to the earlier XAS results [206], for x = 12 increases the number of holes on the ladders to 1.1, i.e. 0.08 holes per Cu site. Optical spectroscopy [207], ARPES experiments [208] and recent XAS experiments [204] showed for high Ca replacements values of 0.2, 0.15–0.2 and 0.31 holes per Cu site, respectively, with no sign of convergence with time.

In figure 37 we reproduce SCCO data showing XAS spectra for x = 0 and a photon polarization parallel to c together with resonance profiles for various Ca concentrations [203]. In agreement with previous XAS data near the O K edge [206] the first pre-peak at 528.4 eV and the shoulder at 528.7 eV were assigned to hole states at the chains and the ladders, respectively. The next peak at 530 eV was interpreted in terms of the UHB. As in other doped cuprates, the first peak at 931.5 eV in the Cu L₃ XAS spectrum is assigned to a transition into empty Cu 3d states while the shoulder at 933 eV is related to ligand (O) hole states. For x = 0, a resonance profile measured at a photon energy of 528.7 eV and a wave vector $Q=(0, 0, \frac{2\pi}{c_{\rm L}}L_{\rm L})$ shows a clear resonance for L_L = 1/5 (see figures 37 and 38). This observation, together with the fact that away from the ladder hole shoulder no scattering intensity has been detected, signals the existence of a commensurate hole crystal in the ladders with a wave length λ = 2π/|Q| = c_L/L_L = 5c_L = 19.5 Å. At higher Ca concentrations (10 ⩽ x ⩽ 12) a resonance is observed for L_L = 1/3 (see figures 37 and 38). On the other hand, no resonance has been detected for L_L = 1/4. This could indicate that the hole crystal is stable for odd, though not even, multiples of the ladder period. The resonance profiles at the Cu L₃ edge shows, in particular for higher x values two peaks, at 930 eV and at 933 eV. The latter peak coincidences with the ligand-hole shoulder and thus indicates the presence of a hole modulation. The former indicates a lattice modulation [180] which is more pronounced for larger hole (Ca) concentrations. The hole crystal intensities show both for L_L = 1/5 and 1/3 a strong temperature dependence which points to melting of the hole crystal at higher temperatures and also signals that the modulation is not caused by the misfit between the chains and the ladders. The non-existence of hole crystals with non-integer L values signals that incommensurate hole crystals melt even at very low temperatures.

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Hole crystallization in ladder compounds was predicted by a t-J model [200] and density matrix renormalization group calculations [209]. In particular it was found that holes like to pair up on the rungs of the ladder. Thus, doped rungs are separated by the wave length λ and the doping concentration p per Cu site should then be p = c_L/λ = L_L or the number of holes on the ladder per unit cell should be n_L = 14L_L. From these considerations we expect for zero Ca concentration x = 0 p_L = 0.2 and n_L = 2.8 and for x = 11 p_L = 0.33 and n_L = 4.7. These values are in good agreement with the values derived in the recent XAS evaluation [204] and thus offer strong support for a pairing of holes along the rungs and thus also support models on the mechanism of high-T_c superconductivity in cuprates based on ladder systems. On the other hand, these results are in striking disagreement with other results on the number of holes an the ladders mentioned above.

The absence of an L_L = 1/4 periodicity was explained in terms of resonant valence bond calculations indicating that the Wigner hole crystal is stable for odd, but not for even multiples of the ladder period. Further theoretical work showed that the widely used ladder t–J model is not sufficient and has to be supplemented by Coulomb repulsion between the holes on neighbouring ladders to explain the existence of a hole crystallization on the ladders. A mean-field calculation of the extended model could explain a charge density wave with an odd period [210].

At the end of the paragraph reviewing charge order in the system SCCO, we report on RSXS results on the chains of these compounds [204]. For the undoped compound in which divalent Sr was replaced by trivalent La, a superstructure reflection for L_C = 0.30 has been detected for photons with an energy near the Cu L₃ and L₂ edges. Since in this compound, no holes can form a charge density wave and since the superstructure reflection was independent of the temperature, it has been interpreted in terms of a pure strain wave formed by the misfit between the chains and the ladders. For doped SCCO an incommensurate superstructure reflection at L_C = 0.318 has been detected. It shows a remarkable temperature dependence and appears mainly near the ligand-hole shoulder. From this it was concluded that in SCCO a strain-stabilized charge density wave is formed.

The first transition metal oxide system in which a static stripe-like order was detected was La_2−xSr_xNiO_4+y (LSNO). In contrast to LSCO, upon doping, i.e. replacing the trivalent La ions by divalent Sr ions or by adding oxygen, the nickelates remain insulating except for very high doping concentrations, and superconductivity has not been detected so far. Both in the electron diffraction [211] and neutron diffraction [212] data, superstructure reflections were observed, which were interpreted in terms of coupled charge and spin-density modulations in the NiO₂ planes, similar to the case depicted for the cuprates in figure 31. In contrast to the cuprates, however, the stripes are rotated by 45°. In an orthorhombic unit cell with axes rotated by 45° with respect to the Ni-O bonds, the magnetic peaks are split about the AFM position (1,0,0) along the [1 0 0] and [0 1 0] directions by , while the charge order are split about the fundamental Bragg peaks by 2, as in the case of cuprates. Also, similar to the cuprates, for hole concentrations $n_h<\frac{1}{8}$, in the nickelates  ≈ 0.5n_h = 0.5(x + 2y). Stripe-like charge and spin order in LSNO is observed for 0.15 ≦ n_h ≦ 0.5. The stripe order is most stable at a doping level $n_h=\frac{1}{3}$ where it shows the highest charge and spin ordering temperatures and the longest correlation length. Differing from the cuprates in La₂NiO₄ there are two intrinsic holes on each Ni site in the $3{\rm d}_{x^2-y^2}$ and the $3{\rm d}_{3z^2-r^2}$ state [213].

RSXS experiments were performed on a La_2−xSr_xNiO₄ single crystal [214]. The energy profiles at the Ni L_2,3 edges, shown in figure 39, exhibit resonances at the charge-order wave vector (2,0,1) and at the spin-order wave vector (1-,0,0) with  = 0.196 in the above notation. A resonance enhancement is also observed at the La M₄ edge at 849.2 eV, however, this enhancement is much weaker than at the Ni L_2,3 edges indicating that the modulation occurs mainly in the NiO₂ planes. The resonances show a strong polarization dependence, e.g. the intensity of the magnetic superstructure peak for σ polarization is only 10% of that for π polarization. Since only a magnetic moment perpendicular to the polarization of the incoming light is probed and because in the chosen experimental geometry the π polarization is perpendicular to the stripes, one can conclude that the Ni spins are essentially collinear with the stripes. Further conclusions can be drawn from a comparison with calculations based on a configuration-interaction model (see figure 39). The difference of about 1 eV for the charge modulation resonances for π and σ polarization indicates a large energy splitting between the $3{\rm d}_{x^2-y^2}$ and the $3{\rm d}_{3z^2-r^2}$ levels. The comparison also signals that the holes are going mainly to the O sites on a ligand molecular orbital with x² − y² symmetry. Due to the strong on-site Coulomb repulsion of two holes on the Ni sites, the Ni 3d count is not strongly modulated in the stripe structure, ruling out a Ni²⁺/Ni³⁺ charge-order scenario. In contrast, the situation is very close to that of stripe structures in cuprates, except that in the nickelates there is an additional intrinsic hole in the Ni $3{\rm d}_{3z^2-r^2}$ states.

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In summary, the comparison with the RSXS data with the cluster calculations yields interesting results on the stripe structure and the electronic structure of nickelates: (i) both the charge order and the spin order resides in the NiO₂ layers, (ii) the doped holes are mainly located on O sites and the spin of these holes are coupled antiparallel to the spins of the intrinsic hole on the Ni $3{\rm d}_{x^2-y^2}$ states in close analogy to the Zhang-Rice singlets in the cuprates.

Magnetite, Fe₃O₄, is another prototype correlated transition metal oxide, in which a subtle interplay of lattice, charge, spin, and orbital degrees of freedom determines the physical properties [215]. It was the first magnetic material known to mankind. Verwey discovered in the late 1930s that upon lowering the temperature below the Verwey temperature T_V = 123 K, Fe₃O₄ undergoes a first-order transition connected with a conductivity decrease by two orders of magnitude. Magnetite has been considered as a mixed-valence system. In this compound at high temperatures, the tetrahedral A sites are occupied by Fe³⁺ ions while the octahedral B sites are occupied by an equal number of randomly distributed Fe²⁺ and Fe³⁺ cations. According to Verwey the transition into the low conducting state is caused by an ordering of the Fe valency on the B sites accompanied by a structural transition. Although numerous experimental [216] and theoretical studies [217, 218] have been performed on magnetite, no full consensus over all aspects of the order transition exists. With its particular sensitivity to the Fe 3d electronic structure, recent RSXS studies at the Fe L_2,3 edges contributed significantly to the understanding of the Verwey transition [219]. XAS data of Fe₃O₄ at the Fe L_2,3 edge together with resonance profiles of the (0,0, $\frac{1}{2}$) and the (0,0,1) superstructure reflections are displayed in figure 40.

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Based on previous interpretations of the Fe L_2,3 absorption spectra and on simulations it was possible to decompose the absorption structure and to assign the different peaks to specific Fe sites (see figure 40(a)). The low-energy features (red solid line) were assigned to Fe²⁺. Remarkably, the (0,0, $\frac{1}{2}$) shows a resonance exactly at these energies which means that the (0,0, $\frac{1}{2}$) peak is due to an order which involves only B-site Fe²⁺ ions. This implies further that there is an orbital order of the t_2g electrons that distinguishes different B-site Fe²⁺ ions. This experimental result on the orbital ordering of the t_2g electrons of the Fe²⁺ ions agrees with the theoretical predictions on the basis of LDA+U calculations [217, 218]. The (0,0,1) superstructure reflection shows maxima at two energies corresponding to the B-site Fe²⁺ and the B-site Fe³⁺ ions. This is precisely expected for a charge order involving two B-site Fe ions. In figure 40(c) the measured resonance profile is compared with simulations for a charge-ordered state (solid line) and a homogeneous Fe^2.5+ configuration (dashed). The very good agreement with the charge-order state and the disagreement with the homogeneous state clearly supports charge order on the B sites in magnetite. In this context, one should mention that the charge order of the Fe 3d electrons on the B-site is small and that one should rather speak of a modulation of the t_2g count which is partially screened by a charge transfer from the oxygen neighbours to the empty e_g states [217].

Classical examples of orbitally ordered systems are KCuF₃ [221] and LaMnO₃. In the case of manganites, in particular, orbital order has attracted considerable attention and controversy since they exhibit a wide diversity of ground states including phases with colossal magnetic resistance [9, 10] or charge and orbitally ordered ground states. In many cases small changes in some parameters such as the charge-carrier doping or temperature can lead to transitions between disparate ground states. The origin of this rich physics is widely believed to be due to a complex interplay of charge, magnetic, orbital, and lattice degrees of freedom, which results in different strongly competing electronic phases.

Two prototypical examples are the manganites RE_1−xA_xMnO₃ and RE_1−xA_x+1MnO₄ (RE = trivalent rare-earth ions, A = divalent alkaline-earth ions), which belong to the famous Ruddlesden-Popper series. In all these compounds each Mn ion is surrounded by an O octahedron. The interaction with the O ions leads to a partial lifting of the degeneracy of the 3d states into the lower t_2g states and the twofold degenerate e_g states at higher energies. In the undoped case (x = 0), the Mn ions have a formal valence of 3+ with four electrons in the 3d shell. Since the Hund's rule energy is much larger than the crystal field splitting, the Mn 3d⁴ is in a very stable high-spin state that corresponds to three spin-up electrons in the t_2g↑ states and a single spin-up electron in the e_g↑ state, which results in a S = 2 state of Mn. The t_2g states, being less hybridized with the O ions than the e_g states, are assumed to be more localized, also by strong correlation effects. The e_g electrons are supposed to be more delocalized due to the stronger hybridization with the O ions. On the other hand, for a cubic MnO₆ octahedron, a single electron in the degenerate e_g levels can occupy any linear combination of the x² − y² and 3z² − r² orbitals. This is referred to as the orbital degree of freedom, which plays a very prominent role for the physics of manganites. The ground state of a cubic octahedron is therefore degenerate and, according to the theorem of Jahn and Teller, this degeneracy will always be lifted by a symmetry reduction, i.e. a distortion of the octahedron. The Mn 3d⁴ is therefore strongly Jahn–Teller active causing local lattice distortions.

When the filling of the e_g states is close to 1 or close to a commensurate value, the distortions of the interconnected octahedra can occur in a cooperative way and one is then talking about a collective Jahn–Teller distortion. This collective ordering of Jahn–Teller distorted octahedra also corresponds to an orbital ordering. In other words the collective Jahn–Teller effect is one way to stabilize an orbitally ordered state. There is, however, another mechanism that can stabilize orbital order, even in the absence of electron-phonon coupling, which is given by the so-called Kugel–Khomskii exchange [220, 222]. It was discussed controversially which of the two mechanisms is the main driving force for orbital order in the manganites.

A particular example for this controversy was given by the half-doped manganite La_1−xSr_xMnO₄, which is composed of MnO₂ planes separated by a cubic (LaSr)O layer. Upon hole doping, more and more Mn ions with a formal valence of 4+ (3d³) are created, which suppresses the collective Jahn–Teller effect and causes an increased itineracy of the e_g electrons. The e_g electrons thus play the role of conduction electrons coupled to the localized S = 3/2 t_2g↑ electrons. As discussed by Zener [223], the mobility of e_g conduction electrons is strongly affected by the spin degrees of freedom: due to the so-called double-exchange mechanism, the e_g electrons can delocalize only for FM alignment of neighbouring spins. Upon increasing the temperature above the FM transition temperature the configuration of the spins gets disordered, which causes a strong spin-charge scattering and thus an enhancement of the resistivity close to the Curie temperature. Applying an external magnetic field at this temperature can easily align the local spins leading to a strong magnetoresistance. This is a simple qualitative explanation of the colossal magnetoresistance observed e.g. in La_1−xSr_xMnO₃(x = 0.3) [10] which is not correct on a quantitative level, since it neglects the orbital and lattice degrees of freedom.

The relevance of the latter becomes particular evident near x = 0.5 where an electron ordering takes place which has attracted much attention and controversy. Several studies have been performed on the compound La_0.5Sr_1.5MnO₄ which is composed of MnO₂ planes separated by a cubic (LaSr)O layer. At room temperature the Mn sites in this system are all crystallographically equivalent and have an average valency of +3.5. At T_CO ≈ 240 K a charge disproportionation occurs creating two inequivalent Mn sites. Originally the two sites were assigned to Mn³⁺ and Mn⁴⁺ ions, but later on it turned out that the charge difference of the two sites is much less than 1. Below T_N = 120 K a long-range AFM ordering of the Mn ions into a CE-Type structure is realized, originally detected in the related compound La_1−xCa_xMnO₃ by neutron scattering [224]. It is believed that this complex electronic order is stabilized by several interactions, but it was discussed controversially whether orbital order really exists in this system and, if it does, whether it is driven by collective Jahn–Teller distortions or by superexchange interactions.

Synchrotron-based investigations of orbital order in the manganites were initiated by a seminal study of La_0.5Sr_1.5MnO₄ using hard x-rays at the Mn K edge [25]. Superstructure reflections were detected, which could not be explained on the basis of the high-temperature crystal structure and were therefore explained in terms of an orbital ordering of the Mn 3d states. While there was consensus about the orbital origin of these reflections, the direct observation by resonant hard x-ray scattering was challenged, as the M K edge is dominated by dipole-allowed transitions to the 4p states, which are not sensitive to orbital order but rather lattice distortions, as pointed out in a theoretical study [225]. Another theoretical paper [43] thus suggested the use of RSXS experiments at the Mn L edges, which are directly related to virtual excitations into Mn 3d states and thus to orbital ordering. The calculations predicted different photon energy dependencies of the scattered intensity for orbital ordering being stabilized by superexchange interaction compared to the Jahn–Teller effect. Such RSXS studies on single-layered manganite La_0.5Sr_1.5MnO₄ at the Mn L edges were subsequently reported in [226–231].

A report was given by Wilkins et al [226], who studied the orbital and magnetic reflections at wave vectors of ${\bit q}_{\rm OO}=(\frac{1}{4},\frac{1}{4},0)$ and $({\bit q}_{\rm AF}=\frac{1}{4},\frac{1}{4},\frac{1}{2})$, respectively. We note that the occurrence of orbital order causes the unit cell to quadruple in the a − b plane. Figure 41 shows the energy dependence of the scattered intensity at the orbital order wave vector q_OO through the Mn L₃ (near 640 eV) and L₂ edges (near 650 eV) which are related to virtual electric-dipole transitions between the Mn 2p_3/2 and 2p_1/2 core levels to the unoccupied 3d states. Figure 42 shows analogous data for the AFM order reflection q_AF. It can be clearly observed in these figures that the orbital and magnetic reflection exhibit strong resonances at the Mn L_2,3 edges. We note that these measurements provide the most direct proof of orbital ordering in these materials. In figures 41 and 42 the insets show the temperature dependences of the orbital and magnetic order parameters, respectively. The orbital order parameter decreases continuously with increasing temperature and disappears at T_OO = 230 K. The intensity of the AFM reflection shows a similar behaviour but disappears at a lower temperature of T_N = 120 K.

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In both figures the experimental data are compared with theoretical calculations of the RSXS spectra based on atomic multiplet calculations in a crystal field. In the calculations the crystal fields were modified in such a way as to fit the experimental data. In the panels (a) and (b) the cubic and the tetragonal crystal field parameters were adjusted for a ${\rm d}_{x^2-z^2}$ and ${\rm d}_{3x^2-r^2}$ type of orbital ordering, respectively. In the panel (c) for the ${\rm d}_{x^2-z^2}$ type ordering a small orthorhombic crystal field component was added. From the comparison of the experimental data with the theoretical calculations the authors concluded that the orbital ordering below T_OO is predominantly of ${\rm d}_{x^2-z^2}$ type. The inclusion of a small orthorhombic crystal field component moderately improved the fits, although many details of the complicated experimental lineshape could not be reproduced by the calculation.

Notwithstanding these difficulties, calculations of the resonance profiles predicted a drastic reduction of the L₃/L₂ intensity ratio with decreasing tetragonal crystal field. This theoretical prediction was used for the interpretation of experimental results. The experimental temperature-dependent RSXS data shown in figures 43(a) and (b) indicate that below the orbital ordering temperature T_OO the L₃/L₂ ratio increases, suggesting an increase of the tetragonal crystal field, i.e. an increase of the Jahn–Teller distortion with cooling. Approaching T_N the ratio L₃/L₂ increases further and saturates below T_N.

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From these findings the authors concluded that there are two separate contributions causing the orbital ordering: a dominant mechanism related to the cooperative Jahn–Teller distortion of the O ions around the Mn³⁺ ions and the direct magnetic Kugel–Khomskii superexchange mechanism which further strengthens orbital ordering by short-range AFM order and another increase by long-range AFM order below T_N. Furthermore the authors concluded a strong interaction between orbital and magnetic correlations.

Nearly at the same time another group performed similar RSXS experiments on the same compound La_0.5Sr_1.5MnO₄ [229]. The diffraction profile at the Mn L_2,3 edges for the orbital order $(\frac{1}{4},\frac{1}{4},0)$ superstructure reflection were very close to those shown in figure 41. In a following paper [30], the experimental data were compared to calculations of a small planar cluster consisting of a central active Mn site with a first neighbour shell comprising O and Mn sites. Thus by allowing a hopping between the Mn and the O ions an integer charge ordering of the Mn ions could be abandoned. In addition to the Jahn–Teller distortion of the O ions, the spin magnetization of the inactive Mn⁴⁺ sites were explored as adjustable parameters. The calculations based on a 3z² − r²/3x² − r² orbital ordering of the e_g electrons reproduced all spectral features of the resonance profile while calculations on a x² − y²/z² − y² order did not, although the differences were not dramatic enough to provide a definite answer. A further result of the RSXS studies of Dhesi et al [229] was the detection of a complicated temperature dependence of the individual features of the L₃ edge indicating that the complete L₃ edge intensities cannot be assigned to one order parameter (e.g. orbital ordering by a collective Jahn–Teller effect).

These findings were supported by a following RSXS study of the same compound by Staub et al [230, 231]. They realized that the features observed in the resonance profiles of the magnetic reflection show the same temperature dependence while those of the orbital order reflection show a strong energy and temperature dependence. The high-energy features of the L_2,3 edges (in figure 41 at 647 and 656 eV) which were assigned to a Jahn–Teller ordering parameter saturate below T_N. The other features assigned to a direct superexchange ordering parameter still increases below T_N. From this experimental result together with a polarization dependence of the resonance profiles they came to the conclusion that there are two interactions leading to the orbital order. It was concluded that the orbital superexchange interaction dominates over the Jahn–Teller distortion strain field interaction and drives the transitions.

The RSXS results on the manganites described above demonstrate that the lineshapes at the Mn L_2,3 edges display rich spectral features, which contain a lot of information about the underlying order phenomena. To extract this information and to interpret the spectra is, however, challenging as it requires extensive theoretical modelling. Here it should be mentioned that in addition various RSXS studies have been performed on double and multilayer manganites [95, 232–238].

Although largely discussed in connection with 3d transition metal compounds, orbital order is a phenomenon that also occurs in other materials, such as RE compounds. Here, the shielding of the 4f shell by the outer 5d electrons causes a weaker coupling of the electronic degrees of freedom of the 4f electrons to the lattice and hence allows the study of orbital order mechanisms in a less complex situation. In a series of publications on orbital ordering of 4f electrons in REB₂C₂ compounds Mulders et al show that RSXS performed at the RE−M_4,5 resonances is not just able to observe orbital order of the 4f electrons but can in addition quantify the contributions of the different higher order multipole moments to the electronic ordering [239–242].