X-ray line shapes of metals: Exact solutions of a final-state interaction model

By means of model calculations for an independent-electron metal, we obtain exact line shapes for the photon absorption, emission, and photoemission spectra of core states, including electronic relaxation. In all cases we find an x-ray edge anomaly. For the absorption and emission spectra this anomaly is superposed on a continuum resembling Elliott exciton theory. We display how the spectra evolve from the exciton limit to the free-electron limit as the final-state interaction strength is decreased or the Fermi energy increased. We compare the spectra obtained for different final-state interactions and find that different types of interactions produce different spectral shapes. Away from threshold the absorption and emission profiles show an enhancement of the free-electron result, as predicted by the screened-exciton theory. Our results offer potential explanations for (i) incompatibilities between threshold exponents and exponents extracted from other data, (ii) the occurrence of nearly symmetric x-ray photoemission lines, and (iii) the lack of mirror symmetry of absorption and emission edges.

Figure 1

Single-particle energy levels and their occupancies in the initial state $\mid i⟩$ and various final states $\mid f\nu ⟩$ for $N=3$. The lower portion of the figure gives the numerical recoil profile for this case, as well as the ND-DS (Nozières–de Dominicis or Doniach-Sunjic) profile (dashed).

Figure 2

Typical integrated recoil spectrum $J\left(E\right)$ versus energy in units of the Fermi energy, for $N=3$, 6, 12, 24, 46, and 80. These spectra for a square well with phase shift ${\delta }_0=0.35\pi$ illustrate how rapidly the calculations for finite $N$ approach the infinite-$N$ limit.

Figure 3

Recoil profiles $E_{F}I\left(E\right)$ for barriers of infinite height, with phase shifts $\mid {\delta }_0\mid =0.125\pi$, $0.25\pi$, $0.50\pi$, $0.75\pi$, $0.90\pi$, and $1.05\pi$. Profiles for positive phase shifts are displayed to the right of the threshold (turning off the barrier), those for negative phase shifts are displayed to the left of the threshold (turning on the barrier). Solid lines, exact results; chained lines, ND theory.

Figure 4

Recoil profiles $E_{F}I\left(E\right)$ for turning on a square well of radius $a=1.5a_B$ $(a∕r_s=0.38a_B)$; as in Fig. 3. The phase shifts are $\mid {\delta }_0\mid =0.125\pi$, $0.25\pi$, $0.35\pi$, $0.45\pi$, and $0.75\pi$. The profiles for turning off the wells are given on the left.

Figure 5

Recoil profiles $E_{F}I\left(E\right)$ for turning on $({\delta }_0>0)$ and off $({\delta }_0<0)$ an attractive $\delta$ shell. The calculations have employed $a∕r_s=0.14a_B$, $0.18a_B$, and $0.047a_B$ for ${\delta }_0∕\pi =0.25$, 0.50, and 0.75 respectively, as in Fig. 3.

Figure 6

Recoil profiles $E_{F}I\left(E\right)$ for turning on $({\delta }_0>0)$ and off $({\delta }_0<0)$ a repulsive $\delta$ shell. The calculations have employed $a∕r_s=0.32a_B$, $0.55a_B$, $1.00a_B$, and $1.27a_B$ for $\mid {\delta }_0\mid ∕\pi =0.125$, 0.25, 0.50, and 0.75 respectively. As in Fig. 3.

Figure 7

Plot of $A=({\pi }^2∕192)(M^2∕a_c^5)E^{1∕2}$ versus energy for various values of the core radius $a_c$, showing the falloff of the free-electron square root caused by the finite core radius. $A$ is given by [compare with Eq. (4.4)] $A={(1-a_c^{2}E)}^2{(1+a_c^{2}E)}^{-8}(E^{1∕2}∕4{\pi }^2)$.

Figure 8

Integrated absorption spectrum $Y\left(E\right)$ for a square well with ${\delta }_0=0.25\pi$ with ${\delta }_0=0.25\pi$ versus energy in units of the Fermi energy, for $N=5$, 10, 20, 40, and 80.

Figure 9

Emission and absorption profiles $E_F\kappa \left(E\right)∕M^2$ for a barrier of infinite height. Emission profiles are displayed to the left of the common threshold, absorption profiles to the right. Solid lines, exact results; chained lines, ND theory; dotted lines, free-electron theory. Not in this figure, but in some of the following figures, broken lines, screened-exciton approximation.

Figure 10

Emission and absorption profiles $E_F\kappa \left(E\right)∕M^2$ for a square well of width $a=1.5a_B$ $(a∕r_s=0.38a_B)$; as in Fig. 9.

Figure 11

Emission and absorption profiles $E_F\kappa \left(E\right)∕M^2$ for an attractive $\delta$ shell. The calculations have employed $a∕r_s=0.085a_B$, $0.11a_B$, $0.18a_B$, and $0.047a_B$ for ${\delta }_0∕\pi =0.05$, 0.10, 0.50, and 0.75, respectively. As in Fig. 9.

Figure 12

Emission and absorption profiles $E_F\kappa \left(E\right)∕M^2$ for a repulsive $\delta$ shell. The calculations have employed $a∕r_s=0.27a_B$, $0.73a_B$, and $1.27a_B$ for ${\delta }_0∕\pi =0.10$, 0.35, and 0.75 respectively. As in Fig. 9.

Figure 13

Comparison of the absorption and emission spectra $E_F\kappa \left(E\right)∕M^2$ for various potentials with ${\delta }_0=0.25\pi$, namely, barrier, square well, and attractive $\delta$ shell. As in Fig. 9.