Implementation of a fully numerical model

Substituting Eqs. (A.9 - A.11) into Eq. (A.8), one has

fe(p) = ˜xc h 1 +ep2/2pc2−ηci−1_+x h neΛ3h 2 e −p2_/2p2 h+x_b neΛ 3 b 2 Ψ(p0/√2pb) e−(p−p0)22p2b, (A.12)

where x˜c = xcneΛ3c/2F1/2(ηc). In this form, the model distribution function for a given electron density is fully characterised by seven free parameters;{Tc, ηc, Th, p0, pb, xh, xb}, withxc= 1−(xh+xb). For the more simple case of a cold bulk component and a single hot tail, i.e. xb = 0, only four free parameters are needed;{Tc, ηc, Th, xh}, andxc= 1−xh. For simplicity, it is also convenient to remove ηc as a free parameter by noting that the shape of the low-momentum portion of the cold component will not be greatly disturbed from it’s equilibrium form, provided thatxh 1. One can then use the well-known Padé fit [Ichimaru, 1994b] to determineηcin terms ofneandTc. Only under this approximation may one takeF1/2(ηc) = 1₂neΛ3c, such that x˜c=xc.

A.2 Implementation of a fully numerical model

Often, a simple analytic form such as Eq. (A.12) may not be appropriate, and instead the DSF must be calculated in a fully numerical framework. In this case, the fundamental equations are given by Eqs. (4.17 - 4.18), in addition to the electron density Eq. (A.2). The evaluation of these equations gives rise to several important criteria, which warrant brief discussion.

A.2.1 Density convergence check

Firstly, due to factor of p2 in the integrand, the electron density only converges for an

arbitrary distribution function providing sufficiently large momenta are considered. For equilibrium-like distributions, convergence to within a suitably small tolerance may be achieved for a few times the normalisation momentum. The accuracy of the estimate then rapidly improves as exponentially less weight is contributed by higher-momentum states. For strongly non-equilibrium distributions with significant structure deep in the tail, the maximum momentum considered to achieve a converged electron density may need to be far larger. Moreover, evaluating Eqs. (4.17) and (4.18) sets limits on the momentum at |1_±z|, which scale as ω/k2_{. Thus, as the wave number of the fluctuations probed by the}

incident radiation gets smaller, e.g. using a soft x-ray source such as FLASH or going to extremely small scattering angles, then the minimum momentum required to evaluate Eq. (4.18) can potentially exceed the upper limit used to determine the density.

A simple self-consistency check can be used to safeguard against this possibility (see Fig. A.1). The density is first evaluated within a loop which runs until convergence to within a pre-defined accuracy is reached, yielding an estimate for xmax. The latter is

range of interest (both must be considered sinceω can be both positive and negative), and

repeated until the condition

xmax>max (|1−z|,|1 +z|) (A.13)

is satisfied. Note that because k changes non-linearly over the spectral range, in general

giving k(_|ω_|) < k(_{− |}ω_|), symmetry relations for frequency shifts of equal magnitude but

opposite sign cannot be relied upon.

A.2.2 Evaluation of the Kramers-Kronig integral

The fact that the limits of the momentum integration are determined not only by con- vergence considerations, but also by the range of frequencies of interest, raises another important consideration. Namely, in constructing the real part of the dielectric function using the Kramers-Kronig relation, the imaginary part must be calculated for a sufficiently large range of frequency shifts. In the context of the present problem, this means that the weight added to the total integral by considering larger frequency shifts must decrease below some required tolerance. For example, the constraint can be satisfied by requiring that the magnitude of the imaginary part at the limits of the spectral range is less than some small fraction of the value at the global maximum

{|Imεee(k, ωmin)| and |Imεee(k, ωmax)|} ≤10−6max (|Imεee(k, ω)|). (A.14)

The absolute values are taken as the imaginary part changes sign with ω. If this test is

not passed, ωmin and ωmax are increased (with the spectral resolution held constant) and

the calculation is re-run from the beginning.

Provided that the final check on the frequency range is passed, the real part can be easily calculated by transforming the usual representation of the Kramers-Kronig relation

Reεee(k, ω)−1 = Z _∞ −∞ dω0 π Imεee(k, ω) ω0_−ω =1 π Z _∞ 0 dω0 ω0 Imεee(k, ω+ω0) + Z 0 −∞ dω0 ω0 Imεee(k, ω+ω0) =1 π Z _∞ 0 dω0 ω0 Imεee(k, ω+ω0)−Imεee(k, ω−ω0) . (A.15)

It is then clear thatImεee(k, ω) must be indexed into at frequencies outside the spectral

range initially supplied; the largest value being2ωmax. However, since the frequency range

convergence test guarantees that frequencies beyondωmax contribute (normalised) weight

on the order of, e.g.,.10−6, then one can safely extend the frequency range as required

and simply pad with zeros, such that

A.2 Implementation of a fully numerical model

Initialise calculation:

Select/read distribution:

Calculate density:

Density converged?

Dispersion and renormalisation:

Momentum range large enough?

Noninteracting DSF and imaginary dielectric function:

Frequency range large enough?

Real dielectric function from Kramers Kronig integral:

Dynamic structure factor in RPA:

YES YES YES NO NO NO ωi,θ,p,ωmin,ωmax: ωs=ωi−ω k0≈(k2i+k2s−2kikscosθ)1/2 fe(p) x= 2p/~k0 x0

max= max(x),x1max=δ1x0max

n0,1 e = k03/8π2 Rx0,1 max 0 dx x 2_fe(x) 1−n0_e/n1 e≤δ2 x 0 max=x1max ˜ ki,s= (ωi,s2 −ω2pe)1/2/c k= (˜k2 i + ˜k2s−2˜ki˜kscosθ)1/2 x= 2p/~k,z=ω/ωC x1 max>max |1 +z(ω)|,|1−z(ω)| S0 e(k, ω), Imεee(k, ω)

|Imεee(k, ωmin)|and|Imεee(k, ωmax)| ≤δ3max

|Imεee(k, ω)|

ωmin=δ4ωmin

ωmax=δ4ωmax

ωs=ωi−ω

ω={ωmin, ωmax} → {ωmin,2ωmax} Imεee(k, ω) = 0 forωmax< ω≤2ωmax

εee(k, ω) = Reεee(k, ω) +iImεee(k, ω)

See(k, ω) = S

0

e(k, ω)

|εee(k, ω)|2

Figure A.1: Flow diagram for evaluating the non-equilibrium dynamic structure model as described by Eqs. (4.17 - 4.18). Typical values of the control constants are δ1 = 1.5,

The integration can then be performed without further hindrance by employing a suitable variable transformation and using a general adaptive integration algorithm. Here, an effec- tive transformation is given byexp-sinh quadrature, i.e. one write ω0 = Ω exp(πsinh(t)).

The integral then has the form I(ω) =

Z _∞

−∞

dtcosh(t)hf(ω+ Ωeπsinh(t))₋f(ω₋Ωeπsinh(t))i, (A.17)

whereΩ is some suitable normalisation frequency, e.g. Ω =ωpe. The double-exponential

decay of this transformation ensures that the integrand is effectively bounded on a vastly reduced range, such as −3 _≤ t _≤ 3. Subsequently, the complex dielectric function is

easily constructed and used to compute the interacting DSF. A flow diagram describing the evaluation of the non-equilibrium model is shown in Fig. A.1.

A.2.3 Additional considerations for dispersion curves

Finally, using the general algorithm described here, one may also compute dynamic re- sponse quantities independently resolved ink and ω. An important example is the locus

describing the dispersion curves for collective modes, which is given by Reε(k, ω) = 0.

This is especially useful for non-equilibrium plasmas since the plasmon resonance location deviates from the modified Bohm-Gross relation at relatively small k. In this case, one

performs nested loops over independent frequency and wave number arrays, ignoring the step which refines the momentum normalisation. Moreover, from the fourth moment of the distribution function, i.e. the mean kinetic energy, is also important to the plasmon dispersion. It is therefore also necessary to ensure convergence ofp4_f

e(p) for distributions

with significant weight in the high-energy tail. A second convergence test is therefore performed with respect tohKei before the momentum range is checked.

Appendix B

Reduced approach

to the coupled modes model

for temperature relaxation

In Chapter 5 it was shown that temperature relaxation in dense plasmas is strongly sensitive to the electron-ion energy transfer rate. Unlike the Landau-Spitzer (LS) model Eq. (5.4), the coupled mode (CM) energy transfer rate Eq. (5.14) must be evaluated numerically. A reduced approach to the coupled modes rate [Chapman et al., 2013a] is therefore required to facilitate highly-resolved relaxation calculations with hundreds of time steps.

B.1

f-sum rule approach

In the case of uncoupled collective modes, i.e. the Fermi golden rule (FGR), a reduced approach which considerably simplifies numerical evaluation has been derived by Hazak et al. [Hazak et al., 2001]. The reduced FGR (RFGR) is based on three approxima- tions which are well-fulfilled under a wide range of conditions. Firstly, the structure of the frequency integral is modulated by the spectral function of the ions, Imε−_ii1(k, ω) =

−Imεii(k, ω)/|εii(k, ω)|2. This function decays exponentially over significantly smaller fre-

quency scales than the equivalent electronic term due to the large mass differencemi me.

Taylor expanding the electronic term aboutω= 0 is therefore justified Imε−_ee1(k, ω)_≈ω ∂ ∂ωImε −1 ee(k, ω) ω=0 . (B.1)

Secondly, provided thatβa~ωpa1, one may also Taylor expand the Bose functions

nBi(ω)−nBe(ω)≈

kB

~ω(Ti−Te). (B.2)

Eq. (B.2) is valid for non-degenerate states and for small frequencies. Furthermore, this approximation is always expected to hold for the ions since they are ubiquitously non-

degenerate and their mode spectrum is always downshifted due to screening [Vorberger and Gericke, 2014]. Thirdly, assuming only static screening, then the derivative in Eq. (B.1) can be simplified further by approximatingImε−1

ee(k, ω)≈ −Imεee(k, ω)/|εee(k,0)|2. For

arbitrary degeneracies one finds

Imε−_ee1(k, ω)_≈ √ πω √ 2kve k2_κ2 e (k2_+κ2 e)2 fe(~k/2) De , (B.3)

wherefe(~k/2)is the distribution function of the electrons evaluated at p=~k/2. Here,

the static screening has been taken in the long-wavelength limitεee(k,0) = 1 + (κe/k)2.

Applying the approximations Eqs. (B.1 - B.3) to Eq. (5.13), and making use of the f-

sum rule (2.81), the frequency integration can be performed analytically. The result is ZeiRFGR= V ω2 peω2pikB(Ti−Te) (2πv2 e)3/2 Z _∞ 0 dk k k2 k2_+κ2 e 2 fe(~k/2) De . (B.4)

The integration overkis inexpensive to perform numerically, e.g. using theexp-sinhtrans- form discussed in Appendix A.

It is tempting to follow a similar approach to the CM energy transfer rate. One then readily derives [Daligault and Mozyrsky, 2008]

ZeiDM= V ω2_peω_p2_ikB(Ti−Te) (2πv2 e)3/2 Z _∞ 0 dk k k2 k2_+κ2 e 2 fe(~k/2) De × 1 πω2 pi Z _∞ −∞ dω ωIm Vii(k)ΠRii(k, ω) 1₋VD ii(k)ΠRii(k, ω) . (B.5)

Since thef-sum rule holds for all systems in which the potential energy commutes with the

density operator, i.e.[ˆn,Vˆ] = 0, the frequency integral term in Eq. (B.5) yields unity, and

the RFGR result is recovered exactly. This suggests that a coupled mode effect does not occur when the approximations made here are well-fulfilled, in contradiction to previous numerical results [see, e.g., Dharma-wardana and Perrot, 1998; Vorberger et al., 2010].