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
fi exp( 2πi (hx+ky) )
Since the symmetry operator maps every atom onto an identical atom, if
the coordinates x
in this equation are replaced by
the coordinates of the equivalent position, -x
, then the result should be unchanged. Substituting these
into the structure factor equation we get:
fi exp( 2πi (h(-x)+k(y+1/2)) )
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
Thus for k even, F(h,k)=F(-h,k). For k odd,
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?