International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C, ch. 7.4, pp. 660-661

Section 7.4.3.4. Plasmon, Raman, and resonant Raman scattering

N. G. Alexandropoulosa and M. J. Cooperb

7.4.3.4. Plasmon, Raman, and resonant Raman scattering

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In typical X-ray experiments, as is evident from Table 7.4.3.1[link], the energy transfer may be so low that Compton scattering will be inhibited from all but the most loosely bound electrons. Indeed, in the situation in metals where K, the momentum transfer, is less than kF (the Fermi momentum), Compton scattering from the conduction electrons may be restricted by exclusion because of the lack of unoccupied final states [see Bushuev & Kuz'min (1977[link])]. Fortunately, in these uncertain circumstances, the incoherent intensities are low. In this regime, the electron gas may be excited into collective motion. For almost all solids, the plasmon excitation energy is 20–30 eV and, in the random phase approximation, the incoherent scattering factor becomes SE, K) ∝ (K2/wp)δ(ΔEhωp), where ωp is the plasma frequency.

At slightly higher energies [(\Delta E \ge E_B)], Compton scattering and Raman scattering can coexist, though the Raman component is only evident at low momentum transfer (Bushuev & Kuz'min, 1977[link]). The resultant spectrum is often referred to as the Compton–Raman band. In semi-classical radiation theory, Raman scattering is usually differentiated from Compton scattering by dropping the requirement for momentum conservation between the photon and the individual target electron, the recoil being absorbed by the atom. The Raman band corresponds to transitions into the lowest unoccupied levels and these can be calculated within the dipole approximation as long as |K|a [\lt1], where K is the momentum transfer and a the orbital radius of the core electron undergoing the transition. The transition probability in equation (7.4.3.4)[link] becomes [\sum_f|\langle \psi_f|{\bf r}|\psi_i\rangle|^2\delta (E_f-E_i-\Delta E), \eqno (7.4.3.7)]which implies that the near-edge structure is similar to the photoelectric absorption spectrum.

Whereas plasmon and Raman scattering are unlikely to make dramatic contributions to the total incoherent intensity, resonant Raman scattering (RRS) may, when [E_1\le E_B]. The excitation involves a virtual K-shell vacancy in the intermediate state and a vacancy in the L (or M or N) shell and an electron in the continuum in the final state. It has now been observed in a variety of materials [see, for example, Sparks (1974[link]), Eisenberger, Platzman & Winick (1976[link]), Schaupp et al. (1984[link])]. It was predicted by Gavrila & Tugulea (1975[link]) and the theory has been treated comprehensively by Åberg & Tulkki (1985[link]). The effect is the exact counterpart, in the inelastic spectrum, of anomalous scattering in the elastic spectrum. It is important because, as the resonance condition is approached, the intensity will exceed that due to Compton scattering and therefore play havoc with any corrections to total intensities based solely on the latter.

Although systematic tabulations of resonance Raman scattering do not exist, Fig. 7.4.3.3[link], which is based on the calculations of Bannett & Freund (1975[link]), shows how the intensity of RRS clearly exceeds that of the Compton scattering for incident energies just below the absorption edge. However, since the problems posed by anomalous scattering and X-ray fluorescence are generally appreciated, the energy range [0.9\lt E_1 /E_B \lt 1.1] is wisely avoided by crystallographers intent upon absolute intensity measurements.

[Figure 7.4.3.3]

Figure 7.4.3.3 | top | pdf |

The cross section for resonant Raman scattering (RRS) and fluorescence (F) as a function of the ratio of the incident energy, E, and the K-binding energy, EB. The units of dσ/dΩ are (e2/mc2)2 and the data are taken from Bannett & Freund (1975[link]). For comparison, the intensity of Compton scattering (C) from copper through an angle of 30° is also shown [data taken from Hubbell et al. (1975[link])].

References

Åberg, T. & Tulkki, J. (1985). Inelastic X-ray scattering including resonance phenomena. Atomic inner shell physics, edited by B. Crasemann, Chap. 10. New York: Plenum.Google Scholar
Bannett, Y. B. & Freund, I. (1975). Resonant X-ray Raman scattering. Phys. Rev. Lett. 34, 372–376.Google Scholar
Bushuev, V. A. & Kuz'min, R. N. (1977). Inelastic scattering of X-ray and synchrotron radiation in crystals, coherent effects in inelastic scattering. Sov. Phys. Usp. 20, 406–431.Google Scholar
Eisenberger, P., Platzman, P. M. & Winick, M. (1976). Resonant X-ray Raman scattering studies using synchrotron radiation. Phys. Rev. B, 13, 2377–2380.Google Scholar
Gavrila, M. & Tugulea, M. N. (1975). Compton scattering by L shell electrons. Rev. Roum. Phys. 20, 209–230.Google Scholar
Hubbell, J. H., Veigele, W. J., Briggs, E. A., Brown, R. T., Cromer, D. T. & Howerton, R. J. (1975). Atomic form factors, incoherent scattering functions and photon scattering cross-sections. J. Phys. Chem. Ref. Data, 4, 471–538.Google Scholar
Schaupp, D., Czerwinski, H., Smend, F., Wenskus, R., Schumacher, M., Millhouse, A. H. & Strauss, S. (1984). Resonant Raman scattering of synchrotron X-rays by neodymium: observation of fine structure in K-L-RRS and of K-N-RRS. Z. Phys. A, 319, 1–7.Google Scholar
Sparks, C. J. (1974). Inelastic resonance emission of X-rays: anomalous scattering associated with anomalous dispersion. Phys. Rev. Lett. 33, 262–265.Google Scholar








































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