Evaluación de Factores Externos: Oportunidades y

The form of the scattered power spectrum can be derived within a relativistic, fully quan- tum mechanical framework by considering the behaviour of an interacting ensemble of elec- trons and ions subjected to the time-dependent scalar and vectors potentials associated with the incident x-ray photons [Crowley and Gregori, 2013; 2014]. However, a qualitative understanding of the expected properties of the power spectrum can be achieved using classical electrodynamics [Jackson, 1962; Sheffield et al., 2011].

For simplicity, consider a homogeneous volume of plasma V containing a number of charged particlesNa=Vna. The x-ray probe is modelled as an incident plane wave

Ei=Eiei(ki·r−ωit)pˆi, (1.22) where pˆi denotes the unit vector in the direction of the electric field polarisation (see

1.3 Scattered power spectrum and dynamic structure factor

Fig. 1.2). The relevant quantity is the time-averaged scattered power per unit solid angle observed by a detector instrument located at Rdet, which depends on the distance from the source to the detector and the energy flux scattered in its direction

∂Ps(Rdet, t)

∂Ω =R

2

detSs·eˆs=R2detε0c|Es(Rdet, t)|2. (1.23) Here,Ss=ε0c2Es×Bs is the Poynting vector of the scattered wave. The time averaging is performed over the period of the radiationT, which is assumed to be large compared to

the characteristic life time of inter-particle correlations

∂Ps(Rdet) ∂Ω = ∂Ps(Rdet, t) ∂Ω =R 2 detε0c lim T→∞ 1 T Z _∞ −∞ dt_|Es(Rdet, t)|2. (1.24) The electric field emitted from a single moving (non-relativistic) charge in the system (labelledj) can be found by solving the Ampere-Maxwell equation [Jackson, 1962]

Ej_s(Rdet, t) =

Qa

4πε0c2Rdet

[ˆes×(ˆes×v˙j)]_ret, (1.25)

where v˙j denotes the acceleration of the particle induced by the E-field of the incident

wave. The quantity in square brackets must be evaluated at the retarded time t0 _≈ t₋ (Rdet−rj(t0)·ˆes)/cto account for the time delay related to observations of the state of the particle made at the distant detector. The acceleration is given by the equation of motion under the action of the incident EM wave

˙

vj = ˆpi

QaEi

ma

ei(ki·rj(t0)−ωit0)_, (1.26)

whererj(t0) describes the time-dependent trajectory of the charge as seen from the per-

spective of the detector instrument.

Since the time-dependent trajectories of all the particles are correlated due to their mu- tual interactions, the total scattered field cannot be obtained by simply scaling Eq. (1.25) by the number of scatterers. Instead, the scattered field samples a distribution function containing the statistical information on the many-particle system [Klimontovich, 1975]

Fa(r,v, t) = Na

X j=1

δ(r₋rj(t))δ(v−vj(t)). (1.27)

The evolution of the distribution function obeys the Boltzmann equation [Hansen and McDonald, 1990] and, thus, encodes the information on the inter-particle correlations. The total scattered electric field is then given by averaging over all possible configurations

Es(Rdet, t) = reEi Rdet Z drna(r, t0) [ˆes×(ˆes×pˆi)]ei(ki·r−ωit 0₎ . (1.28)

In Eq. (1.28), the velocity integral has been performed to give the density distribution na(r, t0) = Z dvFa(r,v, t0) = Z _dk (2π)3 Z _∞ −∞ dω 2π e i(k·r−ωt0)_n a(k, ω). (1.29)

For frequency-discriminating detectors such as spectrometers, the spectral informa- tion of the intensity of the scattered field is naturally of principal interest. Appealing to Parseval’s theorem [Riley et al., 1998], the average of the intensity in the time domain is equivalent to averaging the intensity of the Fourier transform over scattered frequencies. Thus, by introducing the Fourier transform of the scattered field one may include the frequency dependence of the time-averaged signal Eq. (1.24). Subsequently, the scattered power spectrum at the detector is defined as

∂2Ps(Rdet, ωs) ∂Ω∂ωs =R2_detε0c lim T→∞ 1 πT Z _∞ 0 dt eiωst_E s(Rdet, t) 2 . (1.30)

The factor of1/πstems from the fact that the integration range over scattered frequencies

is halved since only positive values are sampled.

Substituting Eqs. (1.28) and (1.29) into Eq. (1.30) and writing the time at the source in terms of the retarded time, i.e. t_≈t0+R/c₋r_·ˆes, the various phase factors may be collected to yieldδ-functions which act to select the possible wave vectors and frequencies

ofna(k, ω) that can be interrogated. The result is

∂2Ps(Rdet, ωs) ∂Ω∂ωs =Z 4 am2e m2 a 3σT 2π IiP(θ, φ) limT→∞ 1 2π_VT |na(ks−ki, ωs−ωi)| 2_{. (1.31)}

Here,Ii= 1₂cε0Ei2 is the intensity of the incident x-rays and P(θ, φ) =|ˆes×(ˆes×pˆi)|2 =

1₋(ˆes·pˆi)2. The result Eq. (1.31) highlights the dominance of the scattering due to electrons over ions since the ratio isZ_a4(me/ma)2 1for all elements. Furthermore, it demonstrates

that the magnitude of the scattered power detected depends on the polarisation of the probing radiation [Sheffield et al., 2011]

P(θ, φ) =

  

1−sin2θ cos2φ :linearly polarised

1₋1₂sin2θ= 1₂(1 + cos2_{θ) :unpolarised} . (1.32)

The optimum configuration is given for linearly-polarised x-rays with φ = 90◦, which

may be achieved using FELs [Höll et al., 2007]. In comparison, using unpolarised x-rays, e.g. produced by thermal line emission from laser-heated foil backlighters, the amplitude of the signal is sensitive to the scattering geometry only.

It is important to note that a statistically significant number of random fluctuations will be sampled by the probe in the scattering volume. The measured power spectrum

1.3 Scattered power spectrum and dynamic structure factor

therefore reflects the ensemble average of the spectral electron density

∂2_P s(Rdet, ωs) ∂Ω∂ωs =3σT 2π IiP(θ, φ) ωs ωi 2 NeSee(k, ω), (1.33) See(k, ω)≡ lim V,T→∞ 1 VT hδne(k, ω)δne(−k,−ω)i 2πne , (1.34)

in which the angular bracketsh. . .i denotes the ensemble average. The quantitySee(k, ω)

is the autocorrelation function of the density fluctuations and is known as the dynamic structure factor(DSF); the ensemble average over the density ne(k, ω) and density fluctu-

ationsδne(k, ω) =ne(k, ω)− hne(k, ω)i are interchangeable sincehne(k, ω)ne(−k,−ω)i=

hδne(k, ω)δne(−k,−ω)i. The DSF contains all the information on microscopic spatio-

temporal correlations between density fluctuations, and therefore gives rise to spectral fea- tures associated with collective excitations such as plasmons. It is the principal quantity of interest to the study of dense matter using scattering diagnostics and will be exam- ined extensively in this thesis. Note that the factor (ωs/ωi)2 appears due to momentum conservation in a fully quantum mechanical treatment [Crowley and Gregori, 2013].

The form of the power spectrum (1.33) assumes that the incident radiation is monochro- matic. In reality, however, sources of probe x-rays have a finite bandwidth. If a distribution of incident frequencies interacts with the electrons in the target the scattered power spec- trum becomes convolved with the source shape

∂2_Pobs s (R, ωs) ∂Ω∂ωs ≡ Σ(ω)∗ ∂ 2_P s(R, ωs) ∂Ω∂ωs = Z _∞ −∞ dω0Σ(ω−ω0)∂ 2_P s(R, ω0) ∂Ω∂ω0 (1.35) whereω0 =ω0_s−ωi. Best practice for modelling XRTS experiments requires that the source be well-characterised or actively monitored during the experiment, although it often suf- fices to use a simple model of the source function. Commonly used examples are Gaussian- or Lorentzian-shaped profiles [Glenzer and Redmer, 2009]. One may also consider combi- nations of functional forms, results of detailed atomic physics modelling from codes such as FLY/FLYCHK [Lee and Larsen, 1996; Chung et al., 2005] and SPECT3D [MacFarlane et al., 2007], and even experimentally characterised x-ray source data. In general, the source function must be normalised according to

Z _∞

−∞

dωΣ(ω) = 1. (1.36)

Furthermore, the response of the detector as a function of frequency also contributes to the broadening of the spectrum. This is typically characterised prior to an experimental campaign and modelled with a Gaussian function. For XRTS experiments the instrument response is typically a small contribution compared to the bandwidth of the source itself. Further modelling considerations include source broadening, the angular divergence of the source (k-blurring), the effect of plasma gradients and temporal blurring of the

spectrum over the measurement time and the influence of non-equilibrium distribution functions. These will each be addressed throughout the course of this thesis.