Análisis de los resultados de las encuestas

An experimentally obtained scattering spectrum in backscattering geometry is shown in Fig.7.1[Glenzer et al.,2003a]. The scattering profile shows two peaks according to the radiation lines from the titanium foil at 4.75 keV and 4.92 keV. The latter

corresponds to the low intensity Ly-α radiation line. The photons scatter on both lines elastically causing a Rayleigh peak at the initial photon energies. As the Ly-α

emission is lower in intensity, the related scattering spectra is also reduced in com- parison with the scattering contribution at the strongHe-αradiation line. The latter line also produces a clearly visible inelastic Compton shifted feature. In the original publication, a scattering spectrum of cold beryllium was also presented highlighting the broadening of the Compton feature with the increase of the temperature and thus the increase of the thermal motion of the electrons in the case of heated beryllium.

To extract the plasma parameters, one generates synthetic scattering profiles taking the probe spectrum and the finite resolution of the detector into account. These profiles will then be matched to experimentally measured data by adjusting the plasma parameters. The best fit is then considered to yield the conditions of the material investigated. The applied theory is presented in Ref. [Gregori et al., 2003], which is based on the RPA approximation. Following this routine, the sample is characterised byTe= 53 eV,Z = 2.7,ne= 3.3×1023cm−3 and the known mass density of ̺= 1.85 g/cm3. An error range for the temperature of10−20%and for the density of20% is given for the heated beryllium.

Based on these plasma parameters, a theoretical scattering spectrum based on Eq. (6.3) was generated here as well. This spectrum applies the theories presented in this thesis. That is, the free electron feature is described in RPA and the inelastic excitations are neglected as their contribution is small in comparison with the free electron dynamic structure factor [Gregori et al., 2003; Glenzer et al., 2003b]. To calculate the elastic Rayleigh peak, an ion structure factor obtained by the HNC approach is used. Here, the applied effective inter-particle potential is less relevant, as the structure factor is already close to unity (wavenumber of k = 4.27Å−1) for all ionic structure models. The form factor is obtained from the simulations, averaged to the non-integer ionisation degree of Z = 2.7 from results for double and threefold charged beryllium ions. The contribution of the screening cloud can be neglected for the large wave vector considered. The result of this procedure is shown in Fig.7.1comparing it with the experimental data. Obviously, the synthetic scattering spectrum agrees very well with the experimental results. As only theHe-α

line at4.75 keV is considered, the second peak is not fitted by this approach. The used theories are similar to those applied in the original publication [Glenzer et al.,2003a]. Although the method for the calculation of the ion structure factor is different, the results are insensitive to these changes in the backscattering geometry (see section6.2.2).

In the following, the fitting process to extract the plasma parameters will be considered in more detail. As discussed in section6.1.2, the Compton down-shifted line is sensitive to the electron temperature as long as the system is in a non- or par- tially degenerate state. This condition is fulfilled here as the degeneracy parameter isneΛ3e≈0.28 for the plasma conditions considered. In Fig.7.2, the experimentally measured Compton feature is compared with several synthetic scattering spectra generated for various temperatures. It can be seen that the shape of the Compton feature is sensitive to the temperature of the system. For instance, the slope for

0.0 0.5 1.0 1.5 2.0 2.5 3.0 Intensity [arb . units] 4400 4500 4600 4700 4800 energy [eV] T=30eV T=40eV T=53eV T=60eV T=70eV

Figure 7.2: Experimentally measured inelastic Compton feature from Fig. 7.1and theoretical scattering profiles calculated for various temperatures. The electron den- sity ofne= 3.3×1023cm−3 and the ionisation degree ofZ = 2.7are fixed. The same methods of generating the synthetic scattering spectra are applied as in Fig.7.1.

is too shallow. In the case of cold beryllium, the extraction of the temperature is less accurate as the system is in a highly degenerate state withneΛ3e ≈23if a tem- perature ofT = 2.5 eV is assumed. In this state, the Compton feature presents the parabolic Fermi distribution which is roughly temperature independent.

Furthermore, the ionisation degree can be inferred by comparing the ratio of the intensities of the inelastic to the elastic scattering feature as discussed in sec- tion6.1.2. Fig.7.3displays several synthetic scattering profiles calculated for various ionisation degrees and compares them with the experimental data. The Rayleigh peak significantly changes by varying the ionisation degree. This can be understood as the Rayleigh peak reflects the scattering on bound electrons characterised by the form factor, which are shown in Fig.7.3bfor different charge states of the beryllium ions. The best fit yields Z = 2.7 and, as the mass density is known, the electron density can be determined to bene= 3.3×1023cm−3.

To accurately describe a system with a non-integer ionisation degree, a two- component model is required. Here, the double and threefold charged beryllium ions are taken into account as individual components. However, as the charge states are similar and the system is in a weakly coupled state with a classical electron cou- pling parameter ofΓee= 0.3, the outcome of the two-component approach is barely different from results applying an average state of the system (see section 6.3.1).

0 1 2 3 4 5 6 7 Intensity [arb . units] 4400 4500 4600 4700 4800 4900 energy [eV] Z=2 Z=2.3 Z=2.5 Z=2.7 Z=3

(a) scattering spectrum

0.0 0.5 1.0 1.5 2.0 f(k) 0 5 10 15 k [A-1] Be2+ Be2.7+ Be3+ (b) form factor

Figure 7.3: Experimentally measured x-ray scattering profile from Fig.7.1and the- oretical scattering profiles calculated for various ionisation degrees. An ion density ofni = 1.23×1023cm−3and temperature ofT = 53 eVare used. The same methods to generate the synthetic scattering spectra are applied as in Fig. 7.1. The small panel presents the form factors for the different charge states of the beryllium ions. As shown above, the plasma parameters can be extracted with an accuracy of 10 −20% from the non-collective scattering spectrum for isochorically heated beryllium in the warm dense matter region. However, in a system with highly de- generate electrons, the x-ray scattering spectra is roughly temperature independent and, thus, further methods are required to fully determine the state of the system. The analysis is also more complicated in the case of unknown mass density. Here, it might be likely that several sets of basic plasma parameters, that is, electron density, temperature and ionisation degree, can be fitted to the experimental x-ray scattering spectrum and further investigations are required to accurately extract the plasma conditions.