1. Miller Indices
The aim of this section is to understand how the Miller indices (the
[h,k,l], or [h,k] in two dimensions) relate to the
electron density in the unit cell. Additionally we will see the effect
of the structure factor magnitude and phase.
First, select the (1,0) reflection. (Either enter 1 and 0 in
the h and k boxes, then press `Enter', or click on the
first reflection on the a* axis in the structure factor
Look at the map canvas. It shows a green line, along the direction of
the a* axis (not the a axis), and grey lines parallel to
the b axis. Note that the grey lines are separated by one unit
Now click on the positive-real axis in the reflection window to give
this reflection a value (or type a number in the magnitude window and
press `Enter'). A cosine-wave will be superimposed on the green
line. This wave represents the electron density which would give rise
to a diffraction pattern containing that reflection alone. The grey
lines correspond to the positive peaks of the density - these are the
Bragg planes. If all the atoms lie on these planes, then this
reflection will be very strong.
Try altering the phase of the reflection, by clicking elsewhere in the
reflection window, or editing the phase box. Note that the peaks of
the cosine wave move as the phase changes, the phase is giving
positional information concerning areas of high density. As the phase
changes through 360 degrees, the cosine wave advances exactly one
Now select the (2,0) reflection. Note that this corresponds to
a wave with twice the frequency or half the wavelength - there are two
complete waves across the unit cell. Try varying the phase of this
wave. Again, as the phase advances by 360 degrees, the cosine wave
advances exactly one cycle (not one unit cell).
What would a reflection with an index of (1.5,0) look like?
Imagine the cell to the right of the one shown. Both the reflections
we have seen so far repeat exactly from one unit cell to the
next. If there were 1.5 repeats of the cosine wave across the cell,
then the wave would have different values in different copies of the
unit cell. The repeating nature of the unit cell leads to reflections
with integer indices only, and the shape and spacing of the grid of
reflections is determined by the shape and size of the cell repeat.
Now find the (2,1) reflection, and give it a magnitude and
phase. Now press the `Set SF' button. This will set this structure
factor in the diffraction pattern, and show the corresponding electron
density in the map. The map should now show red and blue stripes, with
dark red representing the most positive density and dark cyan
representing the most negative, with white as zero. Notice that the
(-2,-1) reflection has also been set, in accordance with the
Friedel relationship for coherent scattering. Try some different
phases for the (2,1) reflection and confirm that the
(-2,-1) reflection always has the same magnitude, but the
Trace the variation in density along the cell edges. Note that there
are exactly two complete waves along the a axis and one
complete wave along the b axis. There must be a whole number of
repeats along each axis to preserve the unit cell repeat. The number
of repeats along each axis give us a label to describe the reflection,
thus this is called the (2,1) reflection.
The positions of the reflections in the diffraction pattern (i.e. the
grid on which the diffraction pattern spots lie) are also closely
related to map features. Note that the direction from the origin to
the (2,1) reflection in the diffraction pattern is identical to
the direction of the green line in the map.
If you try some other reflections and put a ruler on the screen, you
can also discover that the distance from the origin of the diffraction
pattern to a reflection is the reciprocal of the distance between wave
peaks (grey lines) in the map. The distance between wave peaks in the
map is called the `resolution' of a reflection, because it is the
distance between features which can be represented by that reflection.