PHYSICS 231

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

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

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

where

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

where

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).

Note:

• even at T=0 because of the zero point motion of the atoms.
• In three dimensions, W is finite so the intensity of Bragg peaks is finite, as observed.
• In two dimensions, W is finite at T=0 but infinite at finite-T. (You should show that the integral diverges logarithmically as .) Since the amplitude of the Bragg peak is proportional to then there are no (delta-function) Bragg peaks in two dimensions. In other words there is no long range crystalline order at finite-T in two dimensions. Because of thermal fluctuations, positional order is eventually lost at long distances, though locally there will be a good crystalline arrangement. This was known in the 1930's through the work of Landau (see Landau and Lifshitz, Statistical Mechanics), and Peierls, (Quantum Theory of Solids) and was made rigorous in the 1960's through the work of Mermin, Wagner, and Hohenberg, see especially N.D. Mermin. J. Math. Phys. 8 1061 (1967). Since the divergence is weak, and since it turns out that there is long range orientational order, see Halperin and Nelson, Phys. Rev. 19, 2467 (1979), the delta function Bragg peaks are replaced by cusp-like divergences of the form , where is an exponent which varies with T and G.
• In one dimension, not only is there no Bragg peak (long range positional order) at finite-T but now, additionally, zero point fluctuations destroy long range order even at T=0. You should show that the divergence is power law at finite-T but only logarithmic at T=0.
• These last results just depend on there being at least one acoustic phonon branch with a linear dispersion relation. It is quite easy to see that they do not depend on the assumption made above of one atom per unit cell. However, all our calculations have used the harmonic approximation. One might worry that the conclusions would be changed if we relax this. Although, it might seem impossibly difficult to to get exact results when anharmonic interactions are included, this turns out not to be the case, and the work of Mermin, Wagner and Hohenberg shows rigorously that the above conclusions about lack of translational order in low dimensions, found within the harmonic approximation, are correct.

Appendix

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).

Peter Young
Wed Oct 23 14:45:34 PDT 1996