Geos 306, Fall 2011, Lecture 11
Crystallography III, Xray Diffraction
 One of the most important consequences of the translational periodicity displayed by crystals is that they
produce Xray diffraction patterns.
In fact, the International Union of Crystallography defines crystals as substances that produce Xray diffraction patterns.
The study and understanding of the unit cell of a crystal is largely done through the Xray diffraction experiment.
In this lecture we introduce the basic concepts involved with Xray diffraction.
As part of your term project, you will be given the opportunity to conduct an Xray diffraction experiment.
Coordinate systems

There are two coordinate systems that are commonly used in crystallography,
(1) direct space, and (2) reciprocal space.

Direct space uses the unit cell edges as its basis vectors. The positions
of atoms are given in this coordinate system. They are referred to by the
symbol [xyz].

Reciprocal space is the system that is used to describe planes of atoms. Reciprocal
vectors are oriented perpendicular to the planes that they describe. The
planes are denoted with the symbol (hkl), where h, k, and l are integers.
A given point in space, [xyz],
is on a plane defined by indices (hkl)
that passes through the origin, if
xh + yk + zl = 0.
Planes are known as lattice planes if a lattice point is on the plane.
So the plane that passes through the origin is a lattice plane because the lattice point [000] is on the plane.
The next parallel lattice plane has the same values for (hkl), but satisfies the equation:
xh + yk + zl = 1.

Both systems can be used to describe a direction.
Link to a lattice.

Associated with each plane is its dspacing. This is the distance
between successive, parallel planes of atoms.
In particular, it is the distance between the planes described by
xh + yk + zl = 0 and
xh + yk + zl = 1.
 This mathematical relationship implies that the first plane from the origin (hkl) intercepts the crystallographic axes at a/h, b/k and c/l.
So, for example, (100) intercepts the aaxis at [100], but never intercepts b or c because 1/0 = ∞. Another example is the plane (111).
It intercepts the crystallographic axes at [100], [010], and [001].
The plane (210) intercepts the crystallographic axes at [1/2 0 0], [010] and does not intercept the caxis.

Lattice planes are very important in that they can act as diffraction
gratings to radiation that has a wavelength comparable in size to the
spacing between planes.

Origin of Xrays, Xray tube, create a high voltage and send
electrons across the potential. These electrons will travel at a very high
speed if in a vacuum. They collide with a target material, typically Cu
or Mo. The incoming electrons slow down as they hit the target. Accelerating
electrons create electromagnetic waves proportional in wavelength to the
change in speed. This process produces white radiation (radiation of all
wavelengths). However, another process is also at work. The incoming electrons
collide with electrons that are already in the target and knock them out
of their orbits. If a 1s electron is knocked out of orbit, then an electron
from further out (We are most interested in the ones from 2p) will drop into its place. The change
in energy is very sharp, and this energy change causes the emission of
electromagnetic waves with a very narrow wavelength range. These are known
as Ka radiation. The wavelength is around 1.5
Å for a Cu target and 0.7 Å for a Mo target.

If we shine a beam of these Xrays, called incident Xrays, at a crystal then the Xrays will excite
the electrons in the crystal and they will oscillate with a frequency that
is equal to that of the Xrays. Since all these oscillating electrons are accelerating, they then create electromagnetic
waves equal in wavelength to that of the incident Xray, and these emitted Xrays
radiate from the entire crystal volume in all directions. Most of these emmitted
waves interfere in a destructive way with the other emmitted waves that are being produced throughout
the crystal. However, in certain directions the Xrays interfere with each
other in a constructive way. This results in a beam of Xrays being emitted
by the crystal in a direction that is different from that of the incident
beam.

Bragg determined this arrangement in the early 1900's and described it
with an equation now known as Bragg's equation.
nl = 2d sin(q),
where l is the wavelength of the Xray,
d is the spacing between planes, and q is the
angle of incidence that the incident Xray beam makes with the plane of atoms (hkl).
You should be able to derive Bragg's equation from this drawing.
Indexing a diffraction pattern

When a powdered sample of crystalline material is placed in a diffractometer and bathed in Xrays then a
characteristic pattern can be recorded, known as the diffraction
pattern.
Illustrated below is a powder diffraction pattern of quartz recorded with Cu radiation.

The diffraction pattern is unique for each crystalline material. The positions of
the peaks gives us information that can be used to determine the cell parameters,
and the intensities of the peaks gives us information about the chemical elements
that are present in the crystal, as well as their locations. Each peak present in the pattern has its own
set of indices. More than one diffraction peak can be located at the exact same position.

The American Society for Testing Materials (ASTM) maintains a large database
of diffraction patterns for most of the known crystalline substances. By
comparison with their database we should be able to identify most minerals.

Furthermore, the ASTM also keep a record of the indices of the each diffraction
peaks that can be matched up with their associated dspacing. The process of determining the (hkl) associated with a individual peak
is known as indexing the pattern. Aside from looking up the indices of a peak in the ASTM tables,
we can also theoretically figure out the indices if we know the crystal structure.
In general, it can be extremely difficult to index a pattern without any other information.
Refining the cell parameters

A knowledge of the indices and their experimentally determined dspacings
(or 2q positions) can be used to refine the
cell parameters. For instance, here is diffraction data recorded for hexagonal
quartz with CuKa radiation (l
= 1.541838 Å).
2q

INTENSITY

dspacing

h

k

l

20.88

20

4.2554

1

0

0

26.66

70

3.3434

0

1

1

26.66

30

3.3434

1

0

1

36.57

7

2.4569

1

1

0

39.50

1

2.2812

0

1

2

39.50

6

2.2812

1

0

2

40.32

3

2.2366

1

1

1

42.49

5

2.1277

2

0

0

45.83

2

1.9798

2

0

1

50.18

13

1.8179

1

1

2

50.67

1

1.8016

0

0

3


The hexagonal cell parameters, a and c, can be directly obtained from the dspacings of (100)
and (003) repectively.

From the dspacing of (100), d = 4.2554 Å, we obtain a = 4.2554/cos(30)
= 4.9137 Å.

From (003) (d = 1.8016 Å) we obtain c = 3 * 1.8016 = 5.4048 Å.
Here we used the fact that (hkl) intersect the axes at a/h,
b/k, and c/l, so (003) intersects c
1/3 of the way along the axis.

In such a way we are able to detemine the lengths of the cell edges.

Each measurement has some error associated with it, so, in order to obtain
the best estimate of the cell parameters, we can use as many of the measured diffraction peaks as possible.

The math to do this is tedious so computer algorithms have been devised
to compute the cell parameters from measured 2q values.
One such program, called
CrystalSleuth,
is available on the course website.
This software will help you determine peak positions, identify the powder pattern, index the peaks and
refine your cell parameters.

The diffraction profile is read into the program as a set of x,y data that
corresponds to a 2theta position and intensity. The program is used to find peaks, and fit them with certain mathematical functions (Gaussian)
in order to determine their positions. The program can search the American Mineralogist Crystal Structure Database and identify the pattern.
You can download the crystal structure data from the database and index your pattern and then refine the cell parameters.
You will be taught how to do this in detail by your lab TA.

After all the data has been refined, you can create a summary page (click on Output File button). Typical output is displayed below.
******************************************************************************
**Program REFINE Version 3.0** Kurt Bartelmehs and Bob Downs, 1998
******************************************************************************
quartz: Kihara: E. J. Mineral. 2 (1990) 6377
Symmetry constraint is: HEXAGONAL
Wavelength #1 = 1.541838
Wavelength #2 = 1.540562
Wavelength #3 = 1.544390
Refinement after correcting for machine error
MACHINE ERROR: 0.010
OBSERVED CALCULATED DIFFERENCE
2theta d h k l Wave# 2theta d 2theta d
20.850 4.26043 1 0 0 1 20.886 4.25314 0.036 0.00729
20.900 4.24683 1 0 0 2 20.869 4.25314 0.031 0.00631
26.630 3.34738 0 1 1 1 26.677 3.34156 0.047 0.00583
26.700 3.33600 0 1 1 2 26.655 3.34156 0.045 0.00555
26.630 3.34738 1 0 1 1 26.677 3.34156 0.047 0.00583
26.700 3.33600 1 0 1 2 26.655 3.34156 0.045 0.00555
36.530 2.45976 1 1 0 1 36.595 2.45555 0.065 0.00421
36.630 2.45124 1 1 0 2 36.563 2.45555 0.067 0.00431
39.460 2.28360 1 0 2 1 39.527 2.27989 0.067 0.00371
39.560 2.27618 1 0 2 2 39.493 2.27989 0.067 0.00372
39.460 2.28360 0 1 2 1 39.527 2.27989 0.067 0.00371
39.560 2.27618 0 1 2 2 39.493 2.27989 0.067 0.00372
40.280 2.23899 1 1 1 1 40.348 2.23539 0.068 0.00360
40.380 2.23183 1 1 1 2 40.313 2.23539 0.067 0.00357
rmse = 0.00053932607
A = 4.9110(39) B = 4.9110(39) C = 5.4013(57)
Alpha = 90.0 Beta = 90.0 Gamma = 120.0
Volume = 112.82(16)
Reading:
Wenk and Bulakh, chapter 7