Subject: orientational averaging

From: "Dale A. Braden" <[email-masked]>
Date: Thu, 8 Apr 1999 10:14:45 -0700 (PDT)

Dear CCL,

I want to write a program to calculate neutron scattering intensities from
a sample of vibrating molecules.  The molecules are randomly oriented, so
an orientational average of the molecule with respect to the scattering
vector, Q, must be performed.  Due to certain experimental details, the
direction of Q is actually fixed, and only its magnitude changes.  Because
the specification of the molecule's orientation and normal mode
eigenvectors require many coordinates, it is easier to perform the
orientational average by rotating Q with respect to a fixed molecular
frame, rather than vice versa.

Normally, an orientational average requires the use of 3 Euler angles, but
in this case, I think only 2 angles are needed, because Q is a vector and
not another 3-dimensional entity.  Is this correct?  So if my scattering
function is S(theta,phi,nu), theta is the angle between Q and the z-axis
of my molecular reference frame, phi is the precession angle about z, and
nu is the vibrational frequency, then the orientational average should be
just

 Integral[ S(theta,phi,nu) sin(theta) d_theta d_phi ]

with theta integrated from 0 to Pi and phi integrated from 0 to 2*Pi.
Right?

I've looked through many books and articles on this subject, and all agree
that the orientational average must be performed, but none give any
details. 

Best wishes,

Dale

Dale Braden
Department of Chemistry
University of Oregon
Eugene, OR 97403-1253
[email-masked]