Systemmatic absences: response to Structure Factor question
Diane Hope Peapus
peapud at aix02.ecs.rpi.edu
Tue Oct 26 16:42:52 EST 1993
>Newsgroups: sci.techniques.xtallography
>Subject: Structure Factor question
In sci.techniques.xtallography, v128fm4v at ubvms.cc.buffalo.edu (Scott W
Parks) asked:
>Could someone please explain why for example: In a B.C.C structure the
>structure factor vanishes for m1+m2+m3=odd integers and =2 SA for
>m1+m2+m3=even integers???
eoh at mace.cc.purdue.edu (Thom Hendrixson) and gerard at rigel.bmc.uu.se (Gerard
Kleijwegt) answered mathmatically by showing that the structure factor
formula goes to zero in both the exponetial and the sin/cos forms when you
substitute 2n+1 into the equation for m1+m2+m3, or whatever translational
symmetry operation you happen to have. While this was completely
instructive for me when I first learned crystallography 8yrs ago, I'm
finding that teaching diffraction to freshman materials engineers requires
a little more imagination then just a mathematical proof, so I draw
pictures.
(Speaking of imagination, the most common error I get in lab
reports, where they have to show understanding rather than just math
proficiency which will get them through an exam, is that the extra atom in
the middle of a centered cell absorbs radiation and causes an absense. ie:
they think that the lattice points block impinging radiation and the
diffraction pattern is the transmission.)
I can't draw very well using ASCII, so please try to follow my
discussion.
I start with Bragg's law, n(lamda) = 2d sin(theta)
I draw out the standard pix of ray tracing two rays through a 2-d
array of dots. For illustration, I'm careful to have the rays bounce offa
the dots and not offa the planes drawn through the dots, which is what is
standardly shown.
I show them that the path difference of one ray with respect to the
other is derived trigonemetrically using sin(theta) and the distance, d,
between the dots.
I then draw a sine wave onto the ray-tracing arrows, careful to
make the impinging wavelets in phase and the wave length = 2dsin(theta),
ie: first order diffraction. That clarifies to them that the reflective
wavelets are traced in phase only when the path difference is an integral
multiple of the wavelength. I then talk about constructive and destructive
interference.
Then I introduce centering. I make it fcc. I put a dot at 200,
draw a set of 200 planes and draw a wavelet bouncing offa the 200 dot. Low
and behold!! The reflective wavelet from the centered dot is 180degrees
outta phase from the other two offa the 100 dot!!! How did that happen???
I tell them that for every time that n(lamda) = 2dsin(theta) offa
the 100 atoms, there is always reflection at n(lamda) = (1/2) 2dsin(theta)
from the centering operation which puts an atom at 200. ie: when the new
centered atom is at an angle that puts it inbetween two latice points,
there will always be destructive interference.
Most of them seem to get it when it is drawn out.
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