Resonant Bound-Free Contributions to Thomson Scattering of X-rays by Warm Dense Matter

Resonant Bound-Free Contributions to Thomson Scattering of X-rays by Warm Dense Matter

W. R. Johnson University of Notre Dame, Notre Dame, IN 46556 J. Nilsen K. T. Cheng Lawrence Livermore National Laboratory, Livermore, CA 94551

Recent calculations [Nilsen et al. arXiv:1212.5972] predict that contributions to the scattered photon spectrum from and bound states in chromium () at metallic density and  eV resonate below the respective bound-state thresholds. These resonances are shown to be closely related to continuum lowering, where bound states in the free atom dissolve into a resonant partial wave in the continuum. The resulting -state resonance dominates contributions to the bound-free dynamic structure function, leading to the predicted resonances in the scattered X-ray spectrum. Similar resonant features are shown to occur in all elements in the periodic table between Ca and Mn ().

52.25.Os: Emission, absorption and scattering of radiation, 52.38.-r: Laser-plasma interaction, 52.70.-m: Plasma diagnostic techniques, 56.65.Rr: Particle in cell method.
journal: High Energy Density Physics\biboptions


As discussed in the accompanying article Nilsen et al. ar:12 (), resonances in bound-free contributions to Thomson scattering of X-rays for a Cr plasma can complicate the determination of plasma density and temperature from measurements of the scattered X-ray spectrum. Inasmuch as bound-state features in dense plasmas of lighter elements show up as smooth broad features, it is of interest to explore the origin of the resonances that show up in Cr. To this end, we first note that the free Cr atom has the configuration [Ar] . In a dense plasma, the bound electrons outside the Ar-like ionic core dissolve into the continuum. In particular, the electrons in Cr form a resonant state near 12 eV above threshold for the conditions comsidered here. It is the contribution of this -state resonance that leads to resonances in the Thomson scattering cross section. In the following paragraphs, we study these resonances in greater detail for Cr and other elements between Ca and Mn ().

Ca Sc Ti V Cr Mn
20 21 22 23 24 25
40.080 44.960 47.870 50.940 52.000 51.940
1.56 2.99 4.54 6.11 7.19 7.47
- 35.16 38.94 43.68 49.50 56.22 63.66
- 16.99 18.69 21.26 24.84 29.26 34.37
17.67 17.91 17.96 17.97 17.98 17.99
2.33 3.09 4.04 5.03 6.02 7.01
1.45 2.00 2.24 2.56 2.65 2.32
0.298 0.389 0.419 0.474 0.511 0.507
0.459 0.572 0.606 0.691 0.744 0.709
1.343 1.933 2.852 3.698 4.611 5.676
Table 1: Properties of elements at  eV. is the atomic weight, (g/cc) is plasma density, and are average-atom eigenvalues (eV) of and bound states, and are the number of bound and continuum electrons per ion. is the average ion charge. , and are the number of continuum electrons in , and states, respectively.

In Table 1, we show results from average-atom calculations of plasmas of elements in row four of the periodic table. In each case, we assume that the plasma density is the metallic density and that the plasma temperature is  eV. In the first four rows of the table, we list the atomic symbol, atomic number , atomic weight and density  (g/cc). In the next two lines, we show average-atom eigenvalues and of and states (eV). The theoretical inner shell ionization thresholds - are 20-30% smaller than measured values PRD (). Rows seven and eight list and , the average number of bound and continuum electrons inside the Wigner-Seitz (WS) sphere. The average-atom model predicts that there are approximately 18 bound electrons and continuum electrons for each element. The ninth row gives , the average ionic charge or, equivalently, the number of free electrons per ion. For elements listed in the table, , which is near 2, is significantly smaller than owing to the fact that the continuum charge is concentrated inside the WS sphere. In the final three rows of the table, we list , and , contributions to from , and partial waves, respectively. It should be noted that these three partial-wave contributions account almost completely for . Moreover, for the elements in question, -states provide the dominant contribution to the sum. In summary, the continuum is concentrated inside the WS sphere and is dominated by -state partial waves.

Figure 1: The -state phase shift and twice the continuum -state distribution function are shown as functions of electron energy  (eV) for 5 eV plasmas at metallic density of elements in row four of the periodic table.

In Fig. 1, we plot the -state phase shift and the continuum -wave distribution inside the WS sphere for elements in Table 1. The distribution function for a continuum state with angular momentum is defined by


where is the radial wave function for a continuum state with momentum and angular momentum . When integrated over energy,


gives the total number of continuum electrons for the partial waves listed in the final three rows of Table 1. In all cases, the continuum -waves exhibit resonant behavior near 12 eV, with strong peaks in the distribution functions and corresponding changes in the phase shifts by almost a factor of .

The contribution to the Thomson scattering dynamic structure function from bound states is the sum over contributions from individual subshells with quantum numbers :


where is the momentum of the continuum electron, is the fractional occupation number of subshell , and and are the momentum and energy transfer from the incident photon to the scattered photon , respectively. Atomic units in which are used here. In Eq. (4), is an average-atom wave function that approaches a plane wave asymptotically and is the wave function for a bound state with quantum numbers . The bound-state wave function is expressed in terms of the radial wave function as


Moreover, the scattering wave function, which consists of a plane wave plus an incoming spherical wave is expanded as


where the radial function is normalized to a phase-shifted sine wave asymptotically:


With the above definitions in mind, Eq. (4) can be expressed as






As shown earlier, the dominant contribution to the continuum wave function is from waves () and occurs near continuum energy  eV above threshold. It follows that the dominant contribution to is from the two terms with and , while the dominant contribution to is from the single term . Energy conservation implies that the resonant contribution to the scattered photon spectrum from bound state occurs at scattered X-ray energy , with  eV.

Figure 2: Left panel: for Cr at metallic density and  eV is plotted against , where is the scattered photon energy. The incident photon energy is  eV and the scattering angle is 40. Resonance peaks and thresholds associated with transitions from and bound states are shown by the vertical construction lines. Right panel: Contributions from bound-free , free-free , and the total structure function , which includes elastic scattering, are plotted against .

As a specific example, we consider scattering of a 4750 eV photon at 40 from Cr at metallic density and  eV. In the left panel of Fig. 2, we show the bound-free contribution to the dynamic structure function. The vertical construction lines show the and resonance peaks and thresholds. The resonance peaks are shifted 12 eV below the thresholds of 29.3 and 56.2 eV for the and electrons, respectively, as expected from the position of the peak in shown in the Cr panel of Fig. 1.

Figure 3: The dynamic structure function (short black dashes), bound-free contributions including and resonances (solid blue lines) and free contributions (long red dashes) for scattering from elements listed in Table 1 are plotted as functions of , where is the scattered X-ray energy and  eV is the incident X-ray energy. The scattering angle for the illustrated cases is 130.

In Fig. 3, we present results of calculations of the inelastic bound-free contribution to the dynamic structure function , the inelastic free-free contribution and the total dynamic structure function including elastic scattering for elements listed in Table 1. The incident X-ray energy in these examples is  eV and the scattering angle is 130. The relative amplitude of the resonant peaks is seen to increase systematically throughout the sequence as expected from the -electron distributions shown in Fig. 1. By contrast, the relative amplitude of the free-free contribution remains relatively constant from element to element as a consequence of the fact that , the number of free electrons per ion, is relatively constant throughout the sequence.

In conclusion, the resonances predicted in the bound-free contribution to the Thomson scattering X-ray spectrum arise from a -state resonance in the continuum associated with continuum lowering in a dense plasma. For the examples considered herein, elements between Ca and Mn at metallic density and  eV, the -state resonances occur at continuum-electron energy  eV. The amplitude of the -state resonances increase systematically from element to element. The -state resonance dominate the continuum and the bound-free contribution to the dynamic-structure function . The resonance associated with transitions from subshell occurs at scattered X-ray energy , where the resonance energy for the cases considered here is  eV.


The authors owe a debt of gratitude to T. Döppner for bringing up the question of the origin of the resonances predicted in average-atom calculations. The work of J.N. and K.T.C. was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.


  • (1) J. Nilsen, W. Johnson, K. T. Cheng, The effect of bound states on X-ray Thomson scattering for partially ionized plasmas, HEDP submitted for publication, arXiv:1212.5972.
  • (2)
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