I. Introduction
The determination of the chemical structure of molecules is indispensable to
chemists in their effort to gain insight into chemical problems. Only a few physical
methods are capable of determining chemical structure, and amongst these methods,
diffraction methods have been the most successful. Diffraction methods are capable of
defining bond lengths, bond angles and the spatial proximity of non-bonded atoms for
materials capable of forming crystalline solids. For example, the models of myoglobin
which were displayed in the fluoride-myoglobin lab are representations of the myoglobin
structure found using x-ray diffraction. The importance of structural determination to
science has been recognized with several Nobel Prizes for x-ray diffraction related
research.
Year Prize Awardee Topic
1914 Physics M. Von Laue Discovery of x-ray diffraction
1915 Physics W.H. Bragg
W.L. Bragg
Analysis of crystal structure using x-ray
diffraction
1962 Chemistry J.C. Kendrew
M.F. Perutz
Structural determination of globular proteins
(myoglobin and hemoglobin)
1964 Chemistry D.C. Hodgkin Structure determinations of important
biochemical molecules
1962 Medicine F.H. Crick
J.D. Watson
M.H. Wilkins
Structure of DNA
199? Chemistry J. Deisenhofer
R. Huber
H. Michel
Structure of photosynthetic reaction center
In this final laboratory exercise, you will determine the structure and unit cell dimension of
an alkali halide crystal using powder x-ray diffraction, and use the data to construct a chart
of ionic radii.
Experiment 4: X-ray Diffraction
2 Larsen 4:20 Spring 2000
To explain x-ray diffraction it is convenient to think of light as a wave with a
wavelength that is related to the energy of the light by the equation, E=hc/l, where h is
Plank’s constant, c is the speed of light and l is the wavelength of light. For light within
the x-ray region of the spectrum, the wavelength of the light is within the range 0.1? to
100?. One of the fundamental properties of waves is that two waves can interfere with
each other if they are in the same spatial region. If the phases of the two waves are
coincident they will constructively interfere (increased amplitude) and if their phases are
opposed they will destructively interfere (decreased amplitude).
The interference patterns formed in x-ray diffraction originates from the x-rays that are
scattered by the electrons of atoms within a crystalline solid. A simple picture of a solid
(composed of only one type of atom) is a collection of
balls packed together to form a crystal lattice. In a
crystalline material the packing of the atoms is well
defined, and in fact, the entire crystal can be constructed
by repeating a pattern of atoms that define the unit cell
(see Chang, chapter 11, Kotz, chapter 13). This is
analogous to the fact that a wall can be constructed by
stacking bricks; the brick is the unit cell of the wall. The
simplest case (the only one that we will consider) is
when the unit cell is cubic, that is to say that all of the
edges of the cube are equal and the corners form right
angles. For cubic unit cells the length of the edge is
usually given the symbol ‘a’.
Due to the regular spacing of the atoms, imaginary planes can be pictured which contain
the nuclei. For example, each face of the cube in the illustration above is one such plane.
The electrons of each of these imaginary planes of atoms scatter the x-rays giving a result
that can be interpreted as set of parallel partial mirrors; the light (x-rays) is partially
“reflected” by each plane.
Although the x-rays originate
in-phase from the x-ray
source, the scattered x-rays are
not necessarily in phase when
they emerge from the crystal
and reach the detector. The
relative phases of scattered xrays
depend critically on the
separation distance between
these imaginary planes. This
is due to the fact that x-rays
that penetrate deeper into the
Constructive Interference
Wave 1
Wave 2
Sum 1+2
Destructive Interference
Wave 1
Wave 2
Sum 1+2
Experiment 4: X-ray Diffraction
3 Larsen 4:20 Spring 2000
crystal lattice before they are reflected, travel further. Simple geometry demonstrates that
for two waves reflected by adjacent planes, the wave that is reflected by the second plane
has traveled a distance of (2d sinq) further than the wave that is reflected by the first plane.
In general, the reflected waves from successive planes destructively interfere, however, if
the pathlength difference between reflected waves is an integer multiple of the wavelength
(l), constructive interference is observed (a bright spot). This condition (the Bragg’s
condition for constructive interference) is summarized by Bragg’s Law:
nl = 2d sinq,
where l is the wavelength, d is the interplanar separation, and q is the Bragg’s angle.
The n defines the order of the reflection, with n=1 giving first-order, n=2 giving secondorder,
etc.
From the above equation, it is evident that the structure of the solid (i.e. the distance
between the atoms) can be obtained if a monochromatic x-ray source is used (only one l)
and the intensity of x-rays is measured as a function of q. The constructive interference
that results at specific values of theta (q) map out the interplanar spacing of the solid.
Further, the total intensity at a spot, not only depends on the fulfillment of the Bragg’s
condition, but also on the number of electrons, which depends on the chemical identity of
the atoms.
There is one further complication however, the planes of atoms can be constructed in
several different ways as illustrated below. Each imaginary plane can be indexed using
Miller indices. The Miller index is written as three numbers, (h k l), one for each
coordinate in three dimensional space. The protocol for writing these indices are given in
the reference1, but for now just consider them as labeling scheme for all possible imaginary
planes. Each plane can satisfy the Bragg’s condition and give a bright spot due to
constructive interference. To account for all possible planes, the Bragg’s Law is usually
rewritten (for a cubic system) as,
sin2q =
l2
4a2
?
è
ç ?
?
÷ h2 + k2 + l 2 ( )
1 Start at the origin of the unit cell, then determine where the point where the plane intersects each of the
three axes (a,b,c) of the cell. (abc is used for crystal coordinate rather than xyz) If a plane is parallel to an
axis it is assigned an intercept of infinity (¥). The Miller index is form by dividing the cell dimension
along each axis (a,b,c), by the intercept to general the Miller indices (h,k,l) respectively. For example, if
the intersection occur half way alone a then h=a/(a/2)=2.