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h k
|F| phi

4. Symmetry (1)

In this section we will look at the effect of crystallographic symmetry on the structure factors, with a single 2-fold screw axis. (or equivalently in 2-dimensions, a glide plane).

The map and structure factors now show a 3-atom structure with a 2-fold screw axis parallel to the b axis. The symmetry equivalent positions are:

The symmetry in real space gives rise to relationships between reflections in reciprocal space. For example, it is immediately apparent that the diffraction pattern now has mirror symmetry about the b* axis. Check some reflections and confirm that F(h,k)=F(-h,k) both in magnitude. (The phases are also related).

Examine all the reflections with k=0. What do you notice about their phases? What values can the phases take and still be consistent with the space group?

Look at the (0,1) and (0,3) reflections. What is significant about their values. Can you alter these in any way which is consistent with the space group?

The phase relationships can be understood by examining the structure factor equation:

F(h,k)=Σi fi exp( 2πi (hx+ky) )

Since the symmetry operator maps every atom onto an identical atom, if the coordinates x and y in this equation are replaced by the coordinates of the equivalent position, -x and y+1/2, then the result should be unchanged. Substituting these into the structure factor equation we get:

F(h,k) =Σi fi exp( 2πi (h(-x)+k(y+1/2)) )
=Σi fi exp( 2πi (-hx+ky+k/2) )
=F(-h,k) exp( πi k )

Since k is an integers, and exp( πi )=-1, the exponential is +/-1 for k even/odd respectively.

Thus for k even, F(h,k)=F(-h,k). For k odd, F(h,k)=-F(-h,k).

What happens when h is 0 and k is odd? Is what what you see in the diffraction pattern consistent with this equation?

Remember the Friedel relationship dictates that F(h,k)=F*(-h,-k). Does this explain the values of the k=0 reflections?