On 03/02/2019 09:59 PM, Ronald E. Stenkamp wrote:
I was taught about centric reflections using different words from those in the
wiki.
If you look at your crystal structure in projection and the planar view looks
centrosymmetric, the zone of reflections corresponding to that projection will
have centric phases, i.e., their phases will be restricted to one of two
values. Sometimes those phases are restricted to 0 and 180 degrees, other
times, depending on the location of the pseudo-inversion center, the phases
might be 90 or 270 degrees.
So if you look down the two-fold axis in P2, the coordinates of equivalent
positions become x,0,z and (-x,0,-z). The zone perpendicular to the two-fold
contains the h0l reflections, and they end up with restricted phases. For
orthorhombic structures, all three zones (h0l, hk0, 0kl) are centric. And if
you look at trigonal structures, as in P3(1), there are no centric reflections.
(In P3(1)21, there are centric zones, but they aren't the hk0 reflections.
Ron Stenkamp
Thanks, I think I see that. Any time you have a two-fold axis, proper or screw, the
projection of density onto a plane perpendicular to that axis and passing through the
origin will be 2-fold symmetric. Any reflection whose scattering vector lies within that
plane will be taken to its Friedel mate by the reciprocal space version of the operator,
and that reflection's scattering vector will lie along the same line but in the opposite
direction. The further projection of the density onto one of those scattering vectors
will obey the 2-fold and be centrosymetric. Centrosymmetry in one-dimension is also
called "even function" (vs odd function). The fourier components of even
functions are non-zero only for the cos terms (sin is an odd function), taking zero at
the point of centrosymmetry. If the center of symmetry is offset from the origin then
there is a phase shift equal to 2pi times the fractional distance from origin to center
along the scattering vector. If we don't like negative am
p
litudes, we make them positive and add pi to the phase. So we get 2 possible
phases, separated by 180*. (Or something like that.)
-----Original Message-----
From: CCP4 bulletin board <[email protected]> On Behalf Of Edward A. Berry
Sent: Saturday, March 2, 2019 2:00 PM
To: [email protected]
Subject: [ccp4bb] Confused about centric reflections
The wiki:
https://strucbio.biologie.uni-konstanz.de/ccp4wiki/index.php/Centric_and_acentric_reflections
says:
"A reflection is centric if there is a reciprocal space symmetry operator which
maps it onto itself (or rather its Friedel mate).
. . .
Centric reflections in space group P2 and P21 are thus those with 0,k,0."
The operator -h,k,-l does NOT take 0,k,0 to its Friedel mate.
it takes h,0,k to their Friedel mates. In other words the plane perpendicular
to the 2-fold axis, at 0 along the axis
Or am I missing something?
eab
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