La aplicación del sistema de ejercicios de Educación Vial para los escolares de cuarto grado de la Enseñanza Primaria mostró su efectividad reflejada en los

In chapter 4, the effect of different inter-particle potentials on the ionic structure in warm dense matter was discussed. Now, the influence of the various models on the elastic scattering feature will be studied. Here, three models will be considered: (i) the one component plasma (OCP) model (ii) the Yukawa model (Y) and (iii) the Yukawa model with an additional short range repulsion (Y+SRR). The first two model systems are widely used. The OCP model, which considers the ions interacting by a pure Coulomb potential (see section 4.2), often yields similar results as an electron-ion system when the electron-ion interaction is described by a weak pseudo- potential (see section4.3). In the second model, the electrons are taken into account and a linearly screened ion-ion Coulomb potential is used. The third model includes an additional short-range repulsion term to the Debye potential to mimic the effect of bound electrons in partially ionised systems (see section4.4).

Fig.6.12 shows the weight of the Rayleigh peak,WR(k), for a silicon plasma calculated in the three different model systems described above. In the upper right panel of the figure, the related ion-ion structure factors are presented. Here, the

0 20 40 60 80 100 120 140 W R (k) [ryd -1 ] 0 1 2 3 4 5 6 7 8 9 k [1/A] DFT-MD HNC-Y+SRR HNC-Y HNC-OCP = 1.9 0.6 0.5 1.0 1.5 _S ii(k) 0 5 10 0 2 4 6 8 10 k [1/A] q(k) f(k)

Figure 6.12: The weight of the Rayleigh peak, WR(k), for a silicon plasma with

T = 4.7 eV,̺= 2.33 g/cm3 and Z = 4 calculated for three different model systems applied in the HNC approach, namely, the OCP, the Yukawa (Y) and the Yukawa model with a short-range repulsion (Y+SRR). The related ion-ion structure factors are displayed in the top right panel, showing furthermore the structure obtained by DFT-MD simulations. The electronic structure, that is, the form factor, f(k), and

the screening function,q(k), are presented in the bottom right picture for the system considered. The first quantity is obtained from DFT simulations of an isolated ion and the second is calculated after Eq. (5.23). The gray lines in WR(k) indicate the wave vectors which will be used to discuss the behaviour of the x-ray scattering signal in the non-collective and collective regime.

results from the HNC approach are compared with data obtained by DFT-MD sim- ulations. It can be seen, that the HNC calculations applying the OCP model can reproduce the first peak in the ionic structure as shown in the data gained by the simulations. However, the smallk-behaviour is not correctly described as screening

effects are neglected in this model. The use of the Debye potential in the HNC method (line labelled HNC-Y), however, cannot described the correct spatial cor- relations in this strongly coupled system. Following the discussions of section 4.4, the bound electrons of the fourfold charged silicon ions cause a further repulsion for small distances which can be mimicked by an additional short-range repulsion term added to the Debye potential. Applying this model potential in the HNC code can reproduce the ionic structure obtained by DFT-MD very well.

The electron distribution around the silicon ions is displayed in the lower right panel of the Fig. 6.12. Here, the form factor, f(k), which is obtained by

0 10 20 30 40 50 W R (k, ) [ryd -1 ] -40 -20 0 20

energy shift [eV] = 30

k = 1.24 A-1

-40 -20 0 20 40

energy shift [eV] = 95

k = 3.55 A-1

-40 -20 0 20 40 60

energy shift [eV] = 120

k = 4.17 A-1

HNC-Y+SRR HNC-Y HNC-OCP

Figure 6.13: Calculated Rayleigh peak for the silicon system as described in Fig.6.12 broadened by a Gaussian response function for various wavenumbers as indicated in

WR(k). The incident photon energy is Ei = 4.75 keV. The bandwidths of the instrument function are set to∆E/E = 0.002 and ∆E/E = 0.01 for the collective and the non-collective scattering regime, respectively.

shell, respectively. The screening function,q(k), describes the four valence electrons

contributing to the screening cloud. The latter one is calculated in linear response to a Coulomb field (5.23).

The weight of the Rayleigh peak varies significantly for the different models applied. For experimental verifications, the wave vectors probed should be chosen at the positions where the most discrepancies arise. These positions are given for small distances aroundk= 1.24Å−1 _{and around the peak position atk= 3.55Å}−1

(corresponding to the first gray lines in WR(k) presented in Fig. 6.12). The first wave vector probes the system in the collective mode with a scattering parameter of

α = 1.9 whereas the second wavenumber yields α = 0.6, that is, probing occurs at the border of the non-collective scattering regime.

To compare with experimental results, the Rayleigh peak has to be broad- ened with an instrument response function to account for the energy blurring (see section 6.1.1). Fig. 6.13 presents the resulting elastic Rayleigh peak for the three different wave vectors highlighted as gray lines in Fig. 6.12. The discrepancies be- tween the different models for the ion structure are small for the largest wave vector at k= 4.17Å−1 and will lie below the experimental noise level. This characteristic can be traced back to the form of the related ion-ion structure factors which show a

0 10 20 30 40 50 W R (k, ) [ryd -1 ] -40 -30 -20 -10 0 10

energy shift [eV] = 10

= 15 = 20

HNC-Y+SRR HNC-Y HNC-OCP

% % % (a) collective 0 5 10 15 20 25 30 35 40 W R (k, ) [ryd -1 ] -120 -100 -80 -60 -40 -20 0 20 40 60

energy shift [eV] = 10

= 15 = 20 = 25

HNC-Y+SRR HNC-Y HNC-OCP

% % % %

(b) non-collective

Figure 6.14: The theoretically generated dynamic Rayleigh peak for the silicon system as described in Fig. 6.12 and Fig. 6.13 now convolved furthermore with a Gaussian instrument response function with various FWHM for∆θ to include the k-vector blurring. The scattering angles are θ = 30° and θ = 95° to access the collective and non-collective regime, respectively.

similar behaviour for all models considered at this wavenumber.

Directly on the position with the peak in the ion-ion structure factor at

k= 3.55Å−1, the differences between the various models can be clearly seen. It still

might be challenging for current experimental accuracy to distinguish the various models as the height of the Rayleigh peak varies around 15%. However, between the Yukawa model, which predicts the smallest peak, and the OCP model, which leads to the largest value, differences of 30% arise which might allow the experimental validation of the influence of screening effects in WDM.

In the collective scattering regime, under a scattering angle of θ = 30°, the

influence of the various ionic models strongly affects the predictions of the scattering signal. Here, discrepancies up to 75% arise between the OCP model and the Y+SRR model and differences of 50% between the Y+SRR model and the pure Yukawa model. These effects are caused by the different small k-behaviour of the related

ion-ion structure factors whose absolute values vary by a factor of two between the models in the selected scattering regime. These large differences in the Rayleigh peak makes the different theories experimentally testable.

Besides the broadening in the energy space, an uncertainty in the scattering angle needs to be considered for experimental comparisons. Therefore, the theo- retical scattering spectra from Fig. 6.13 will be further convolved with a Gaussian function to account for thek-vector blurring. The results are presented in Fig. 6.14

for the two scattering angles with the largest discrepancies between the various ionic models, that is, θ = 30° and θ = 95°. Here, errors up to _±12.5% (or ∆θ = 25%)

are investigated, as these uncertainties might occur in current x-ray experiments [Glenzer and Redmer, 2009]. Interestingly, the k-vector blurring does not change the scattering signal in the collective regime. The reason for such a behaviour lies in the characteristics of the weight of the Rayleigh peak, which is approximately linear around the wave vector considered. Therefore, thek-vector blurring causes no significant contribution. In contrast, WR(k) is strongly non-linear around the peak position. Here, thek-vector blurring yields a reduction of the elastic Rayleigh peak,

most significantly for the data applying the OCP model. An error up to ±12.5% in the determination of the scattering angle, that means, θ = 95±12°, can smear out the results in such a way, that the previous differences of 30% between the OCP and the Yukawa model will be reduced to 15%. Obviously this fact, will make an experimental verification even more challenging.

In summary, the different ion structure models significantly change the theo- retically predicted scattering spectra and, thus, the interpretation of the experimen- tal data. This is particularly important in the collective scattering regime, as the height of the elastic Rayleigh peak is currently used to extract the temperature and the ionisation degree of the system. Therefore, an experimental verification of the various theories is highly desirable for the advanced development of x-ray Thomson scattering as a reliable diagnostics tool in WDM. For an optimal testing ground, a strongly coupled, partially ionised system should be considered as the different theories will predict the largest differences. Furthermore, the wave vector probed should be chosen carefully to access the regions with the most discrepancies under consideration of experimental uncertainties due to sample size, scattering geometry, bandwidth of x-ray source and finite detector resolution.