Tabulación de los resultados de las encuestas

So far, the microscopic structure of simple materials, that is, plasmas with a single ion species, has been considered. However, most materials in nature consist of ions in different charge states or with multiple chemical elements. Accordingly, the partial ionic structure factors are required (see Eq. (3.73)) to account for all mutual corre- lations and to allow for an interpretation of the x-ray scattering signal in mixtures or composite materials.

The HNC approach can be used to study multicomponent effects as all the various ion species can be included as further components in the generalised mul- ticomponent version. Fig. 4.12 shows the microscopic ionic structure of a strongly coupled CH plasma obtained by a two-component HNC calculation. Here, hydro- gen is fully ionised and the carbon ions are fourfold charged. Both were taken as separate components. The densities of hydrogen and carbon ions are nH = nC = 2.5×1023cm−3 and the temperature is set to beT = 2×104K. As a comparison, the results from a further HNC run for an isolated hydrogen plasma under the same conditions are plotted in the figure as a green dashed line.

All three partial pair distribution functions show the typical behaviour of a strongly coupled system: a correlation hole, a sharp rise and well-pronounced

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 gab (r) 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 r [di] HH CH CC H, Z=1

(a) partial pair distribution functions

-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Sab (k) 0 1 2 3 4 5 6 7 8 9 k [di -1 ] HH CH CC H, Z=1

(b) partial static structure factors

Figure 4.12: Partial pair distribution functions and partial static structure factors for ions in a CH plasma obtained by HNC calculations. The plasma parameters are given by T = 2_×104_{K and n}

H =nC = 2.5×1023cm−3 whereas the hydrogen and carbon ions are fully and fourfold ionised, respectively. The dashed line, labelled “H, Z = 1”, shows results for an isolated hydrogen plasma under the same conditions.

oscillations indicating the occurrence of a short-range structure in the system. As the carbon ions are fourfold ionised, the carbon-carbon coupling is the strongest, yielding a pair distribution function and a structure factor with the most pronounced maxima. In contrast, the pair distribution function obtained by the one-component HNC calculation of isolated hydrogen plasma presents only a moderately coupled system with a monotonically rising function without further oscillations. Thus, the comparison in Fig. 4.12 shows that the highly charged carbon ions imprint their structure onto the proton subsystem in the CH mixture. These results illustrate the requirement for a full multicomponent description as such an effect cannot be described by a single-ion approach, using an average state of the system.

The partial structure factors, shown in Fig. 4.12 (b), illustrate the highly non-linear coupling of the carbon ions in the mixture as well. Whereas the large

k-behaviour of the hydrogen structure factor obtained by a single-ion approach is in agreement with the data from the multicomponent version, the smallk characteris-

tics presents qualitative differences. The screening of the hydrogen ions due to the carbon ions modifies the long-range part of the proton-proton potential yielding an increase of the hydrogen-hydrogen structure factor,SHH(k), for small wavenumbers. The partial structure factor,SCH(k), between protons and carbon ions presents the typical characteristic with the large k limit of zero. Due to the definition of the partial structure factors (see Eq. (2.48)), negative values for small k can occur for

As the HNC approach self-consistently includes the screening contributions of all species considered, all mutual correlations can be described within the system. This is significant as highly charged ions imprint their structure onto the subsystem of the ions with lower charge states. The generalisation of the HNC approach to multiple ion species is theoretically unlimited. The only practical restriction might come from run time issues which will be significant for a large number of species (K &100).

Although a full multicomponent description is required, present theoretical descriptions of the microscopic structure in complex systems often rely on an ap- proximate treatment based on a one component system via relation [Gregori et al., 2006] Sαβ(k) =δαβ + √_n αnβ n ZαZβ Z2 f h S_ii1comp(k)−1i . (4.45)

S_ii1comp(k)characterises the single-ion structure factor calculated for an average state of Zf =PαnαZα/

P

αnα. The expression (4.45) is exact in the limit of weakly coupled systems where the random phase approximation is valid.

Most x-ray scattering experiments are nowadays performed in moderately to strongly coupled systems. Therefore, the approximation (4.45) should be vali- dated by comparisons with results from full multicomponent calculations. Such a comparison can be found in Fig.4.13 where a CH plasma is considered again. The different temperatures generate moderately coupled (Fig.4.13(a)) and strongly cou- pled (Fig. 4.13(b)) systems. The effective inter-ionic potential used here is a linearly screened Coulomb potential. The dashed lines are the partial structure factors ob- tained by relation (4.45) after a single-ion HNC calculation was performed for the system with an average ion charge state of Zf = 2.5. In contrast, the solid lines illustrate the outcome from the multicomponent HNC calculations.

In the moderately coupled system, that is Fig. 4.13 (a), differences can be observed in the carbon-carbon structure factor, SCC(k), only. Here, the function is shifted down and to the right. For small k values, the partial structure factor

obtained by the approximation is negative, which is an unphysical behaviour. This effect gets more pronounced for the strongly coupled case presented in Fig.4.13(b). Furthermore, the position of the peak is significantly shifted to the right. For strong coupling, the hydrogen-hydrogen structure factor presents qualitative differences as well. Whereas the structure factor obtained by the multicomponent approach has a minimum, the approximate treatment predicts a maximum. These differences occur as the intrinsic mutual correlations in the complex CH plasma cannot be described by the relation (4.45). Again, this approach being based on the one-component

0.0 0.5 1.0 1.5 SCC (k) (a) T=10eV 0.5 1.0 SHH (k) 0.0 0.5 1.0 SCH (k) 0 1 2 3 4 5 6 7 8 9 10 k [A-1] Sabfrom 2-comp HNC

Sabfrom average 1-comp HNC

(a) moderately coupled system

0.0 0.5 1.0 1.5 SCC (k) (b) T=1eV 0.5 1.0 SHH (k) 0.0 0.5 1.0 SCH (k) 0 1 2 3 4 5 6 7 8 9 10 k [A-1] Sabfrom 2-comp HNC

Sabfrom average 1-comp HNC

(b) strongly coupled system

Figure 4.13: Comparison of partial static structure factors for a CH plasma using the full multicomponent description and calculations applying an average charge state. The plasma has a density ofni = 5×1023cm−3and an average ion charge ofZf = 2.5. In the two-component calculation, fully ionised hydrogen and fourfold ionised carbon ions with nH = nC = 2.5×1023cm−3 are considered. The temperatures used are

T = 10 eV(left) and T = 1 eV (right).

treatment is unable to account for the non-linear effects in strongly coupled plasmas. As a result, we find that the partial structure factors can be deduced from single-ion calculations only in weakly to moderately coupled plasmas. In strongly coupled systems, however, a multicomponent description is essential.