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

Now trivially,

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).