30
R. Rüffer and A. I. Chumakov
Fig. 1.10 Expansion of the energy spectrum of nuclear inelastic scattering of synchrotron radiation
in α-iron in multi-phonon terms. The data were taken at room temperature. Different symbols show
the regions of the spectra, where the corresponding contributions are dominant. The lines are
the calculations according to Eqs. 1.32–1.34 and convoluted with the instrumental function of the
monochromator
The normalized probability of nuclear inelastic scattering W (E) can be decomposed in terms of a multiphonon expansion [90]
W (E) = f LM
δ(E) +
∞
n=1
S n (E)
.
(1.32)
The Dirac δ-function δ(E) describes the elastic part of scattering (zero-phonon term),
and the n-th term of the series S n (E) represents the inelastic scattering accompanied
by creation (annihilation) of n phonons. The one-phonon term is given by
S 1 (E) =
E R · g(|E|)
E(1 − e −β E )
,
(1.33)
and the subsequent terms under harmonic approximation may be found through the
recursive relation:
S n (E) =
1
n
∞
−∞
S 1 (E
) · S n−1 (E − E
) dE
.
(1.34)
Here β = (k B T )
−1 with k B the Boltzmann constant, T the temperature; E R =
2 k
2
/2M the recoil energy of a free nucleus; k the wave vector of the x-ray quantum; M the mass of the atom. The function g(E) is the normalized phonon density
of states