[Pw_forum] Neutrons, Xrays, and Phonons, Oh, My!

degironc degironc at sissa.it
Thu Jun 29 11:30:08 CEST 2006

dear Eric,

My  understanfing of the matter is the following:

Let's have a system that is described by an harmonic hamiltonian

H = \sum_i 1/(2M_i) P_i^2 + 1/2 \sum_ij U_i Phi_ij U_j

and let's consider the eigenvalue problem
\sum_j (Phi_ij/sqrt(M_i M_j)  v_j^nu =  (omega^nu) ^2 v_i^nu

The canonical transformation from ordinary cartesian coordinates
(U_i,P_i)  to normal mode coordinates (csi_nu, pi_nu)

U_i = \sum_nu  v_i^nu / sqrt(M_i)  csi_nu
P_i = \sum_nu  v_i^nu * sqrt(M_i) pi_nu

with its inverse

csi_nu = sum_i v_i^nu * sqrt(M_i) U_i
pi_nu  = sum_i v_i^nu / sqrt(M_i) P_i

the unitary matrix v_i^nu has no "units" .... it is a rotation in
coordinate space.

For instance, using this change of variables the hamiltonian is

H = sum_nu 1/2 pi_nu ^2  + 1/2 (omega^nu)^2 csi_nu^2

In order to calculate whatever cross-section you are interested in, you
must similarly use  the same tranformation to write things in terms of
normal-mode amplitudes and then plug in the value corresponding to the
temperature you are interested (for instance at zero temperature:
1/2 pi_nu^2 = 1/2 omega_nu^2 csi_nu^2 = hbar omega^nu / 4) .

At least that would be what I would try to do.

stefano