[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?
Eric
>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.
>
>stefano
>
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