[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
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