7.4 Main Theoretical Expressions
173
I inelasic (s) =
i
4
S k (s)
s 4 .
(7.5)
Thus, I B (s) = I A (s) + I inelasic (s) is a function of 1/s
4 , i.e., steeply decreasing with
increasing scattering variable.
However, the interference effects are observed only for scattering by pairs of
nuclei ij, i.e., by molecules, and depend on the magnitudes of internuclear distances
r ij . Thus, only this component of the total scattering intensity contains information
about molecular structure.
For a rigid N-atomic molecule, I M (s) is expressed as follows:
I M (s) =
K
2
L 2 I 0
N
i=1
N
j=1
| f i (s)|
f j (s)
cos[η i (s) − η j (s)]
sin sr i j
sr i j
,
(7.6)
where N is the number of atoms, i = j, and η i (s) and η j (s) are the atomic electron
scattering phases for the ith and jth atoms, respectively.
In a non-rigid molecule, the vibrational effects can be taken into account by
employing the probability distribution function P(r) because P(r)dr is probability
that the internuclear distance r ij is located in the interval between r and r + dr. In
this case, I M (s) can be written as:
I M (s) =
K
2
L 2 I 0
N
i=1
N
j=1
| f i (s)|
f j (s)
cos[η i (s) − η j (s)]
∞
0
P i j (r )
sin sr i j
sr i j
dr . (7.7)
The sine Fourier transformation of the I M (s) function leads to the so-called radial
distribution function f (r):
f (r ) =
∞
0
s I M (s) sin(sr)ds,
(7.8)
which is related to the probability distribution of the r ij distances.
For temperature T, the P(r) function can be obtained by averaging the P υ (r)
function among all vibrational states employing the Boltzmann distribution:
P(r ) =
υ
P υ (r ) exp
−
E υ
kT
υ
exp
−
E υ
kT
,
(7.9)
where υ is the vibrational quantum number, P υ (r) = |ψ υ (r)|
2 , ψ υ (r) is the
wavefunction, E υ is the vibrational energy, and k is the Boltzmann constant.
For the harmonic oscillator with E υ = hv(υ+1/2), the P(r) function can be
presented as a Gaussian function: