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

Eric Abel etabel at hotmail.com
Fri Jun 30 04:08:08 CEST 2006

Dear all,

Thank you all for your input.  I found the solution to my problem.  First of 
all, it seems that "cross-section" may have been a poor choice of words.  
What I really want to calculate is the dynamical structure factor, 
S(Q,\omega).  This is simply (for mode s):
S(Q,\omega)=\sum_j b_j/sqrt(m_j) * (Q \dot v_{js})exp(i Q \dot R_j)exp(W_j)
where b_j is the neutron scattering cross-section of atom j, m_j is the mass 
of atom j, Q is the reciprocal space vector, v_{js} is the eigenvector and 
R_j is the position in the unit cell of atom j.  The exp(W_j) is the debye 
waller factor. For x-rays I simply need to replace the b_j with the x-ray 
form factor f_j(Q).

In the end, the unitless v_{js} of pwscf is what I need.  Thank you for your 
valuable input.  I learned a lot from this discussion.  In the end, this is 
what being a grad student is all about, right?


>From: degironc <degironc at sissa.it>
>Reply-To: pw_forum at pwscf.org
>To: pw_forum at pwscf.org
>Subject: Re: [Pw_forum] Neutrons, Xrays, and Phonons, Oh, My!
>Date: Thu, 29 Jun 2006 11:30:08 +0200
>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.
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