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

• 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 d-spacing. 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 a-axis 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 c-axis.
• 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 X-rays, X-ray 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 X-rays, called incident X-rays, at a crystal then the X-rays will excite the electrons in the crystal and they will oscillate with a frequency that is equal to that of the X-rays. Since all these oscillating electrons are accelerating, they then create electromagnetic waves equal in wavelength to that of the incident X-ray, and these emitted X-rays 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 X-rays interfere with each other in a constructive way. This results in a beam of X-rays 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.

• When a powdered sample of crystalline material is placed in a diffractometer and bathed in X-rays 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 d-spacing. 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.

• A knowledge of the indices and their experimentally determined d-spacings (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	d-spacing	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 d-spacings of (100) and (003) repectively.

• From the d-spacing 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 2-theta 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) 63-77

 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)