Debye Waller factor
The intensity of Bragg scattering is reduced, due to motion of atoms about their equilibrium positions, by the Debye-Waller factor, generally written as , where
The average is evaluated in the harmonic approximation, is a reciprocal lattice vector, and is the displacement of the l-th atom. The average in Eq. (1) is trivially found from the result
where denotes a quantum statistical mechanical average at a temperature for a simple harmonic oscillator (or set of oscillators), and A is a linear combination of the displacements and momenta of the oscillators (or, equivalently, a linear combination of creation and destruction operators), i.e.
We now proceed to prove Eq. (2).
The most commonly given proofs of Eq. (2) are quite cumbersome, but Mermin, J. Math. Phys. 7, 1038 (1966) has a compact derivation which we follow here. To begin with, consider just a single oscillator, and let us evaluate
where a and b are constants and c and are creation and annihilation operators for an oscillator with Hamiltonian
We start with the well known formula
assuming that the commutator is a c-number. A derivation of this is given in the Appendix. Applying Eq. (6) to Eq. (4) gives
We can also apply Eq. (6) to Eq. (4) but with the opposite identification of A and B to give
Eqs. (7) and (8) are consistent only if
where the last line follows from the cyclic invariance of the trace. Expanding out the in the last line one can sandwich factors of and around each factor of c and so
where the last line is derived in the Appendix. Hence we have
Iterating this procedure another n times we have
and so taking the limit we have
since all terms in the expansion of the exponential give zero except the first. Hence
is the Planck distribution, and so from Eqs. (7), (9) and (19), we obtain
The derivation goes over straightforwardly to the case where there are many operators and so we obtain Eq. (2).
As a result of Eq. (2), the Debye Waller factor is given by
This expression is evaluated by transforming to k-space, so
where s denotes a branch. For simplicity we assume a Bravais lattice, ( i.e. just one atom per unit cell) and so writing
where is the polarization vector and M is the mass of the ion, we have
Our final result, then, is that the intensity of a Bragg peak is reduced, due to the motion of the atoms, by the Debye Waller factor, , where W is given by Eq. (26).
First of all we will derive Eq. (14), i.e. \
where is the free particle Hamiltonian in Eq. (5). Defining
and differentiating with respect to gives
Solving this equation with the boundary condition gives Eq. (14).
Finally, we derive Eq. (6). Let us write
where is a c-number. Then
and, differentiating with respect to gives
We evaluate the commutator using the result
valid if is a c-number, which is obtained by successively moving the factor of B through the n factors of A one place at a time. Hence
where F is any function, and the prime denotes a derivative. For the case of interest here, this gives
Substituting into Eq. (34) then gives
Integrating, and using the boundary condition that , which is obvious, gives the desired result, Eq. (6).