Basic concepts

In order for EM radiation to propagate through a plasma its frequency must exceed the plasma frequency ω_p2 =X a ω_p2_a=X a Q2_ana ε0ma . (1.12)

This quantity is dominated by the electrons due to their small mass, i.e.ωp = ωpe. The

dispersion relation in the material relating the wave number ki and frequency ωi is then changed from the usualin vacuo relation ωi=ckiaccording to

k2_i = ω 2 i c2 1₋ ω 2 pe ω2 i = ω 2 i c2 1₋ ne ncrit = ω 2 iµ2 c2 . (1.13)

For radiation with frequencies below the plasma frequency, one hasµ2 <0 and the wave

suffers strong damping and reflection [Kruer, 1992]. Settingµ= 0, on finds that radiation

of a given energy can only freely propagate up to a critical electron density

ncrit=

ε0meωi2

e2 ≈7642×10 23_E2

keVcm−3, (1.14)

where EkeV is the x-ray energy in units of keV. The use of OTS is therefore clearly

restricted to electron densities several orders of magnitude less than those present in the dense plasmas of interest (see the vertical colour-coded lines in Fig. 1.1). On the other hand, high-energy x-rays with0.1 .~ωi .10 keV are easily capable of penetrating such dense states of matter.

The probability that the radiation will be scattered in any direction is given by the low-energy limit of the Klein-Nishina cross section [Klein and Nishina, 1929]

σKN=σT " 1−2 ~ωi mec2 +26 5 ~ωi mec2 2 +. . . # . (1.15)

1.2 X-ray Thomson scattering as a dense plasma diagnostic x y z ˆ pi φ b r b v ki ks k θ b Rdet

(a) Geometry of scattering experiments. (b) Sources of scattering in partially-ionized plasmas.

Figure 1.2: (a): Diagrammatic representation of the scattering geometry for which the incident probe with wave vectorki is scattered by an electron at rwith initial velocity v into a detector instrument located atRdetthrough an angleθ. Wave vector of the scattered radiation isks, giving a shift ofkfrom the incident probe. The angleφdenotes the rotation of the scattering plane (cyan) relative to the plane defined by polarisation vector of the incident EM wavepˆi(grey). (b): Cartoon illustration of the different sources of scattering from a partially-ionized plasma showing elastic, inelastic free-free, and inelastic bound-free contributions.

Here, σT = 6.65×10−29m2 is the famous Thomson cross section. Corrections beyond the leading order are negligible for ~ωi . 10 keV. In addition to having sufficiently high energies to penetrate the target, the small value ofσTrequires that the probe pulse contains a sufficiently large number of photons if a useful scattering signal is to be observed.

The development of high-intensity, high-energy x-ray sources suitable for XRTS have progressed significantly since early platforms were proposed. For long (∼ns) duration

sources, the thermal line emission from laser-heated foils can be used. These have steadily advanced in both x-ray energy and peak intensity from Ly-α radiation at ∼3 keV from

thin Cl-doped plastic targets [Glenzer et al., 2003; Urry et al., 2006; García Saiz et al., 2008] to He-α radiation between_∼5–18 keV from metal foils, such as Ti, Mn, Zn and Mo

[Gregori et al., 2004; Lee et al., 2009; Kritcher et al., 2011b; Ma et al., 2013]. Furthermore, short (∼ps) duration measurements have been made using short-pulse laser-driven K-α

fluorescence [see, e.g., Kritcher et al., 2009; Barbrel et al., 2009; Le Pape et al., 2010; Neumayer et al., 2010].

More recently, use of FELs has realised the possibility of probing matter on ultra- short ∼10 fs time scales, which is relevant to atomic processes [Gaffney and Chapman,

2007; Ziaja and Medvedev, 2012] and the dynamics associated with energy absorption and thermalisation [Medvedev et al., 2011] and phase transitions [Medvedev et al., 2013]. The theoretical descriptions of matter probed with FELs may be significantly more complex

than those required to describe laser-produced sources as the probe is often sufficiently intense to strongly drive the system under study. Accordingly, an evolving, strongly time- dependent non-equilibrium state may be produced, which must be rigorously accounted for when modelling the observed scattering spectrum [see, e.g., Chapman and Gericke, 2011]. A schematic representation of the geometry for scattering from a single moving electron is shown in Fig. 1.2(a). The incident radiation is incoming from the left with wave vector

ki and frequencyωi. The radiation is then scattered through an angle θtowards a distant detector instrument located atRdet, with a wave vectorksand frequencyωs. The observed shifts in the wave vector and frequency are given by

k=ki−ks, (1.16)

ω=ωi−ωs≈k·v+ωCe, (1.17)

respectively. The shift in frequency originates from the Doppler shift induced by the parti- cle motion,k_·v, and the Compton shiftωCe=~k2/2meresulting from momentum transfer

from the probe to the electron. From Eq. (1.17), the spectrum of frequency shifts observed by a spectrally-resolving detector yields a direct measurement of the momentum distri- bution of the electrons in the target. Moreover, information on the dynamic correlations between electrons can also be accessed.