X-ray spectroscopy

X-ray spectroscopy is a gathering name for several spectroscopic techniques for determining the electronic structure of materials by using x-ray excitation.

X-ray emission spectroscopy
Karl Manne Georg Siegbahn from Uppsala, Sweden (Nobel Prize 1924), who painstakingly produced numerous diamond-ruled glass diffraction gratings for his spectrometers, was one of the pioneers in developing X-ray emission spectroscopy (also called X-ray fluorescence spectroscopy). He measured the X-ray wavelengths of many elements to high precision, using high-energy electrons as excitation source.

Intense and wavelength-tunable X-rays are now typically generated with synchrotrons. In a material, the X-rays may suffer an energy loss compared to the incoming beam. This energy loss of the re-emerging beam reflects an internal excitation of the atomic system, an X-ray analogue to the well-known Raman spectroscopy that is widely used in the optical region.

In the x-ray region there is sufficient energy to probe changes in the electronic state (transitions between orbitals; this is in contrast with the optical region, where the energy loss is often due to changes in the state of the rotational or vibrational degrees of freedom). For instance, in the ultra soft X-ray region (below about 1 keV), crystal field excitations give rise to the energy loss.

We may think of the photon-in-photon-out process as a scattering event. When the x-ray energy corresponds to the binding energy of a core level electron this scattering process is resonantly enhanced by many orders of magnitude. This type of X-ray emission spectroscopy is often referred to as resonant inelastic x-ray scattering (RIXS).

Due to the wide separation of orbital energies of the core levels, it is possible to select a certain atom of interest. The small spatial extent of core level orbitals forces the RIXS process to reflect the electronic structure in close vicinity of the chosen atom. Thus RIXS experiments give valuable information about the local electronic structure of complex systems, and theoretical calculations are relatively simple to perform.

Instrumentation
There exist several efficient designs for analyzing an x-ray emission spectrum in the ultra soft X-ray region. The figure of merit for such instruments is the spectral throughput, i.e. the product of detected intensity and spectral resolving power. Usually, it is possible to change the parameters within a certain range while keeping their product constant.

Grating spectrometers
Typically, the X-rays emerging from a sample must pass a source-defining slit, then optical elements (mirrors and/or gratings)  disperse them by diffraction according to their wavelength and, finally, a detector is placed at their focal points.

Spherical grating mounts
Henry Augustus Rowland (1848-1901) devised an instrument that allowed the use of a single optical element that combines diffraction and focusing: a spherical grating. Reflectivity of X-rays is low regardless of the used material and therefore grazing incidence upon the grating is necessary. X-ray beams impinging on a smooth surface at a few degrees glancing angle of incidence undergo external total reflection which is taken advantage of to enhance the instrumental efficiency substantially.

Denote by R the radius of a spherical grating. Imagine a circle with half the radius R tangent to the center of the grating surface. This small circle is called the Rowland circle. If the entrance slit is anywhere on this circle, then a beam passing the slit and striking the grating will be split into a specularly reflected beam, and beams of all diffraction orders, that come into focus at certain points on the same circle.

Plane grating mounts
Similar as in optical spectrometers, a plane grating spectrometer first needs optics that turns the divergent rays emitted by the x-ray source into a parallel beam. This may be achieved by using a parobolic mirror. The parallel rays emerging from this mirror strike a plane grating (with constant groove distance) at the same angle and are diffracted according to their wavelength. A second parabolic mirror then collects the diffracted rays at a certain angle and creates an image on a detector. A spectrum within a certain wavelength range can be recorded simultaneously by using a 2-dimensional position sensitive detector such as a microchannel photomultiplier plate or an x-ray sensitive CCD chip (film plates are also possible to use).

Interferometers
Instead of using the concept of multiple beam interference that gratings produce, one may simply let two rays interfere. By recording the intensity of two such co-linearly at some fixed point and changing their relative phase one obtains an intensity spectrum as a function of path length difference. One can show that this is equivalent to a Fourier transformed spectrum as a function of frequency. The highest recordable frequency of such a spectrum is dependent on the minimum step size chosen in the scan and the frequency resolution (i.e. how well a certain wave can be defined in terms of its frequency) depends on the maximum path length difference achieved. The latter feature allows a much more compact design for achieving high resolution than for a grating spectrometer because x-ray wavelengths are small compared to attainable pathlength differences.

Other types of X-ray spectroscopy

 * X-ray absorption spectroscopy
 * X-ray magnetic circular dichroism