# X-ray methods in correlated electron science

The relevance for correlated-electron materials of ps dynamics at the nanoscale, together with strong interaction of X-ray photons with all four of the C-S-O-L degrees of freedom, promises important applications of the SwissFEL. To begin the discussion of relevant X-ray techniques, we consider the Hamiltonian describing an X-ray photon field interacting with the electrons in the sample [6, 7]. For the moment, we treat a single free electron, without spin:where describes photons in the radiation field, with wavevector and polarization ε. Expanding the expression for H, we obtain:

The interaction Hamiltonian, Hinteraction, is responsible for producing transitions from an initial state to a final state of the combined system of X-rays and sample. According to Fermi’s Golden Rule, the transition rate is given by:

Here we are interested in “photon-in” (k) to “photon-out” (k’) transitions – thus we include only those terms which are quadratic in the vector potential A. We disregard the linear terms, which are responsible for photoemission. The transition rate is proportional to the square of the “scattering factor”, , which, in turn, is a function of the momentum and energy transfers and .

Consider now the following cases:

### Hard X-ray diffraction

If the photon energy is much larger than the excitation energies En-E0 of the system, we need only consider the first matrix element in the expression for w:where is the ground-state of the system, and where we now sum over all the electrons, with coordinates. For a crystalline sample, the scattering factor , is written

i.e., as a lattice sum over the unit cells and a “structure-factor” sum over the I atoms per unit cell. The quantity is known as the atomic scattering factor. For a perfect crystal, the lattice sum dictates that F is non-zero only if , a reciprocal lattice vector. This is the condition for Bragg scattering, which, using the relations and , can be expressed as Bragg’s law: . We thus see that diffraction at the Bragg angle θ is possible when the X-ray wavelength λ is shorter than twice the lattice-plane spacing (2d). The scattered intensity at the Bragg condition is proportional to . The sensitivity of the Bragg condition to the lattice parameters implies that the diffraction of short X-ray pulses can be used to directly observe lattice phonons.

### Soft X-ray resonant diffraction

_{2}which dominates the scattering rate. The case of elastic scattering, , is treated via an energy-dependent correction to the atomic scattering factor:

where the correction term obeys:

We see that the scattering is sensitive to the electronic structure of the ground- and intermediate states and to the polarization of the incoming and outgoing photons (see Fig. V.6 a). The existence of a “core-hole” in the intermediate state is responsible for introducing the linewidth parameter Γ, representing the lifetime of the state |n>.

For soft X-rays, where the photon wave-vector k is significantly larger than the atomic dimensions, one can expand the exponentials into dipole, quadrupole and octupole terms:

### Resonant inelastic X-ray scattering (RIXS)

_{14}Cu

_{24}O

_{41}, taken at the oxygen K-edge and at the Cu L3-edge [11], showing structures due to the transfer of charge between atoms, the Cu crystal-field splitting and collective spin-flip excitations.

^{2}. For RIXS, because the vir tual excitation of different atoms leads to different final states, the contributions add incoherently, resulting in an intensity propor tional to N [7]. Performing RIXS is an extremely challenging undertaking, due to the low scattered intensity and because of the necessity of per forming an energy (and scattering-angle) analysis of the scattered radiation. For pump-probe RIXS experiments at the SwissFEL, it would therefore be par ticularly interesting to realize a single-shot mode of measuring, either in the frequency (see Infobox) or in the time (ditto) domains.

### The intermediate scattering function

At the beginning of this Section, we saw how the transition rate for X-ray photon scattering, with momentum transfer and energy transfer , is related to the scattering function . This function shows peaks as a function of ω corresponding, for example, to longlived oscillations (quasipar ticles), such as phonons, spin-waves, etc. But finite lifetime effects will broaden these quasipar ticle peaks, and in the limit of strong damping, it may be advantageous to observe the timedependent fluctuations of the system directly – i.e., to measure instead the so-called time-domain or intermediate scattering function [12]:The intermediate scattering function basically provides the correlation time (over which S(Q,t) decays to the value 1/e) for the equilibrium fluctuations of a system, as a function of the fluctuation length scale 1/Q. By monitoring the scattered intensity I(θ,t) at a par ticular scattering angle 2θ (related to the momentum-transfer by Q = 4π sinθ/λ), one has access to S(Q,t) via the intensity correlation function g

_{2}(θ,t) and the “Sieger t relation” [13]:

A measurement of I(θ,t) on the ultrafast time scale, pertinent to correlated electron materials, would require reading out a detector at an impossible rate of GHz -THz. Two realistic alternatives, however, which are well-suited to the characteristics of the SwissFEL, are provided by the “split-pulse” and “Mössbauer filter foil” techniques and are described in Infoboxes.

### A single-shot RIXS spectrometer

_{in}, E

_{out}. A method of per forming single-shot RIXS has been proposed by V. Strocov [25] (see Fig. V.i2). The Swiss- FEL pulse is dispersed vertically by a monochromator (upper right in the figure) and brought to a line focus on the (homogeneous) sample. Scattered light corresponding to the various incoming photon energies is then focused to a vertical line and dispersed in Eout horizontally onto a CCD detector. The result is the desired two-dimensional plot.

### RIXS in the time-domain

### Split-pulse XPCS

Since this requires a detector bandwidth which exceeds that of the fluctuations, one is generally limited to times τ longer than 10 nsec. The use the SwissFEL to probe ps-dynamics will require a different approach – “split-pulse XPCS” (see Fig. V.i4) [11]. Here a single SwissFEL pulse is split and delayed, producing a pair of pulses with tunable separation τ. The 2d-detector then registers a doubleexposure speckle pattern, and the speckle contrast will decrease when τ exceeds the fluctuation correlation time τ

_{c}. That one indeed can measure g(τ) with split-pulse XPCS has been demonstrated by Gutt et al. [29]: The double exposure delivers the intensity , and the speckle contrast is given by the variance:

We note that:

and that, for a fully coherent beam,

We thus obtain:

.

## The Mössbauer filter-foil technique

The Mössbauer filter-foil technique provides an alternate method for measuring the intermediate scattering function [30]. Foils containing a stable Mössbauer isotope are placed in front of and behind the scattering sample and, at time t=0, a resonant Swiss- FEL pulse excites the isotope. The subsequent decay is then monitored by the total transmitted counting rate, measured in the forward direction (see Fig. V.i5). Denoting by ρ(Q,t) the spatial Fourier transform of the electron density in the sample at the time t, the intermediate scattering function is given by S(Q,t) = 〈ρ(Q,0)ρ*(Q,t)〉. Further, we let |g(t)|^{2}denote the decay probability of the Mössbauer isotope a time t after excitation. The signal I(t) at the detector is the square of the coherent sum of the probability amplitudes for a) radiation scattered by the sample at time zero and emitted by the nuclei of the second foil at time t and b) emitted by the first foil at time t and directly scattered into the detector:

From this expression, the intermediate scattering function can be extracted. By far the most popular Mössbauer isotope is^{57}Fe, with a resonant energy of 14.4 keV and an excited state lifetime of 141 ns. An upgrade option for the SwissFEL will make this photon energy accessible with the fundamental undulator radiation, as well as still higher energies using XFEL harmonics [31]. Advantages of the filter-foil method in combination with the SwissFEL are: a) extremely intense X-ray pulses with zero background during the nuclear decay, and b) measurements of S(Q,t) for times far longer than those accessible with the split-pulse XPCS method. (The maximum feasible split-pulse delay will be ns for hard X-rays and tens of ps for soft X-rays.)