In dense, partially-ionized plasmas the measured power spectrum typically contains three distinct features, which arise due to interaction of the probe with the different states of the electrons in the target (see Fig. 1.2(b)). Firstly, the electrons that are kinematically free from the ions are able to respond to the probe over a wide frequency range on the order of the electron plasma frequencyωpe. The nature of this contribution can be either coherent
or incoherent, depending on whether the probe samples the correlated or uncorrelated motion of the free electrons, respectively.
Incoherent scattering is observed when the spatial scale length probed by the x-rays is significantly larger than the characteristic scale length over which interactions between free electrons are effectively screened. The probe is then scattered by a thermal ensemble of uncorrelated individual electrons and the spectrum directly reflects the shape of the distribution function along the scattering vector [Sheffield et al., 2011]. For plasmas in thermal equilibrium the width of the distribution provides an accurate measurement of the mean kinetic energy (1.11). Information on the electron (plasma) temperature is therefore obtained under non-degenerate conditions, whereas under strongly degenerate conditions information on the electron density may be obtained via the Fermi energy [Glenzer and Redmer, 2009].
If the probe samples scale lengths within the screening radius the x-rays are scattered by the collective response of the free electrons. The spectrum then exhibits resonances
1.2 X-ray Thomson scattering as a dense plasma diagnostic
at the characteristic frequency of the microscopic wave-like fluctuations in the electron density. These are known aselectron plasma waves, Langmuir waves or plasmons[Tonks and Langmuir, 1929]. Here, information on the electron density and kinetic energy may be inferred simultaneously from the dispersion (resonance location) and damping (amplitude and width) of the plasmons [Glenzer et al., 2007; Theile et al., 2008; Neumayer et al., 2010].
The scale length accessed by the probe is given by the wave number shiftk=|k|
k2=ki2+ks2−2kikscosθ
ki≈ks
→ k= 2kisin(θ/2). (1.18) Note that the second step tends to be reasonably well-fulfilled for high-energy x-rays only [Glenzer et al., 2000]. The importance of collective behaviour may then be characterised by thescattering parameter[Salpeter, 1960]
α= 1
kλscr
, (1.19)
whereλscr is the characteristic scale length for charge screening.
In general, the screening of interactions between particles is determined by the dielec- tric response of the fully coupled system and depends on both the frequency and wave number studied. A simple figure of merit may be given by comparing the scale length probed to the static, long-wavelength result for the screening length taken in the weakly coupled limit. For an arbitrary non-equilibrium isotropic distribution function one finds [Gericke et al., 2010] κ2e =λ−scr2 = e 2m e π2~3ε 0 Z ∞ 0 dp fe(p), (1.20)
such that in thermal equilibrium
κ2e=λ−scr2 equil.= e 2n eβ ε0 F−1/2(ηe) De = κ2De = e2neβ ε0 :ηe→ −∞ κ2TF= 3e2ne 2ε0EF :ηe→+∞ . (1.21)
As expected, the non-degenerate resultκDe gives the inverse of the Debye-Hückel screen-
ing length [Debye and Hückel, 1923] and the degenerate resultκTF gives the inverse of the Thomas-Fermi screening length [Thomas, 1927; Fermi, 1928]. Collective scattering there- fore corresponds to α 1, whereas one has α 1 for non-collective scattering. From
Eq. (1.18), it is clear that the collective regime is most readily accessed for small-angle (forward) scattering, whilst the non-collective regime is accessed by large-angle (backward) scattering.
The second important contribution to the spectrum results from interactions of the probe with electrons which are closely associated with the ions, i.e. electrons in tightly bound states or in the screening cloud surrounding an ion. The frequency range over
which this contribution is significant is dictated by the distribution function of the ions. Accordingly, this feature is negligible for frequencies larger than the ion plasma frequency
ωpi, but is substantially brighter than the free electron contribution forω.ωpi.
The screening cloud in particular is of interest to OTS experiments as collective excita- tions of the ions (ion acoustic waves) may be accessed [Glenzer et al., 1999a; Froula et al., 2002; 2006b]. In XRTS experiments, however, the bandwidths of present x-ray sources is too large to resolve such low-frequency structure [Gregori and Gericke, 2009]. Moreover, the purely elastic Rayleigh scattering from bound electrons, which is not present in OTS experiments due to the low energy of the probe beam, typically dominates over the screen- ing contribution. In addition to detailed information the electronic screening [Chapman et al., 2015], the elastic feature is generally strongly sensitive to static correlations between the ions and, thus, provides estimates on the ion temperature and mean charge state [Riley et al., 2002; Barbrel et al., 2009; Wünsch, 2011].
The third distinct contribution to the scattering spectrum results from Raman-like transitions into the continuum [Raman and Krishnan, 1928]. Since such transitions are en- ergetically improbable for photon energies less than the excitation thresholds of the bound states, the resulting features appears only for frequency shifts above the various edge fea- tures, e.g. the K- and L-edges. It is the dominant contribution for cold or weakly-ionized targets, and has been investigated extensively, principally using synchrotrons where it is referred to as non-resonant inelastic x-ray scattering, since the inception of x-ray scatter- ing [see, e.g., Mattern and Seidler, 2013]. In dense plasmas, the shape of the bound-free feature is typically relatively insensitive to the plasma conditions, but does provides de- tailed information on the ionization equilibrium of the target. Indeed, the balance between free-free and bound-free scattering has recently been used to perform novel investigation of continuum lowering [Fletcher et al., 2014].