Re: [ccp4bb]: anisotropic ellipsoids

[Ian J Tickle <i.tickle@mail.cryst.bbk.ac.uk>]

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On Thu, 8 Mar 2001, Norbert Straeter wrote:

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> 
> According to many textbooks the first three of the thermal parameters
> U11 U22 U33 U12 U13 and U23 describe the displacements along the
> perpendicular principal axis of the ellipsoid and the latter three give
> the orientation of the principal axes with respect to the unit cell axes.
> However, I can't find anywhere how U12 U13 and U23 (apparently as
> direction cosini) exactly describe the orientation of the ellipsoid, say
> in a cartesian system.
> 
> Any hint is appreciated (but don't suggest to try to follow the ortep
> code)...
> 
> Thanks, Norbert

Hi, it's not correct to say that the diagonal elements Uii describe the
displacements (actually mean square displacements) along the principal axes
of the thermal ellipsoid.  They are actually the mean square displacements
along the base vectors of the co-ordinate system in use.  This is easy to see
from the equation for the mean square displacement:

	<u^2> = U11.l^2 + U22.m^2 + U33.n^2 + 2U23.m.n ... etc

so for the base vector (l,m,n) = (1,0,0) we get just:

	<u^2> = U11

Unfortunately there's no simple relationship in the general case between the
orientations of the principle axes (i.e. the eigenvectors of the U matrix)
and the components of U.  If there were we wouldn't need numerical methods to
compute the eigenvalues & eignevectors!

-- Ian