Simple Methods

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The equations on this page are to be treated as reference material with the exception of the differential form of the Bragg equation [1].

Simple Methods

Pattern Matching

The assignment of hkl triplets to the peaks of a silicon powder diffraction has already been demonstrated in the previous section: this can be considered as a form of indexing, for which the best descriptive term is pattern matching. Pattern matching uses the hkl indices generated from approximate known values of the wavelength and unit-cell parameters to index the pattern.

Generally the problem is approached by indexing the low-angle peaks (or large d spacing peaks) first. One might ask why, since these reflections ultimately provide the least accurate wavelength or unit-cell parameters upon refinement. To answer this question, we will consider the case of angle-dispersive diffraction and a cubic unit-cell material; the arguments can be extended to any crystal symmetry.

The Bragg equation relating d spacing to scattering angle may be differentiated at constant wavelength to give its differential form:

The minus sign just indicates that an increase in d gives a decrease in the angle θ.

For a cubic system, the d spacing is given by the equation:

so by differentiation, one can obtain the following equation:

The combination of equations [1] and [2] yields the final equation:

How does the equation function in practice? The tangent term implies that any error in a results in a very large position error for peaks at high scattering-angles. Suppose we are working with Cu Kα radiation and a cubic material with a unit-cell parameter of, say, about 8 Å. If the true value of a is smaller (i.e. negative δa) by 1%, what is the difference between the observed and calculated 2θ positions for, say, the 100 and 844 reflections? The answer is given in the table below, where you can check the results (if you wish) by clicking on the icons for hkl generation:

hkl	2θ	2θ′	2 δθ		8.00 Å cell	7.92 Å cell
100	11.059°	11.171°	0.112°
844	141.519°	144.986°	3.467°

The strip diagram below shows how the observed and calculated 2θ positions vary for all of the reflections:

This illustrates the importance of indexing the low-angle reflections first even though they provide the least accurate lattice parameters. If one assigns hkl triplets to high-angle data based on poor values of the lattice parameters, then the probability of assigning incorrect values increases rapidly the higher the 2θ value: in the above example, the observed peak at, say, 158.86° would be matched to the 862 reflection of the 8 Å cell, which is calculated to be at 158.65°, whereas the correct indices are 772.

There is another reason why it is best to index the low-angle reflections first: the density of reflections as a function of scattering angle is greatest when 2θ ≈ 90°. Again consider the cubic crystal system: since h² + k² + l² is always an integer, n, then the Bragg equation may be written as:

√n = 2a sin θ / λ       [3]

Differentiation of n with respect to angle leads to the equation:

dn/dθ = (2a / λ)² sin 2θ

dn/dθ is a measure of the density of reflections, i.e. the number of reflections in an interval δθ. This equation shows that for a cubic material, dn/dθ is greatest when sin 2θ = 1, i.e. at 2θ = 90°. This effect is illustrated in the reflection diagram shown above. Conversely, the density of reflections is lowest at 2θ = 0° and 180°. Hence low-angle reflections are easier to index!

For the cubic crystal system, the density of reflections is symmetric about 2θ = 90°, but for a lower symmetry crystal systems this is not the case and the density of reflections is skewed to slightly higher angle: the maximum density is for 2θ > 90°, but the minimum is still at 0° (and 180°). (As an aside, this result has some bearing on the design of high-resolution angle-dispersive powder neutron diffractometers since it is clearly desirable to match the resolution function to dn/dθ: hence they are designed so that the best resolution is around 90° to 120° in 2θ.)

Indexing Cubic Materials

From the 2θ positions of the lines, we can generate a table of sin²θ values (as in the table below). From equation [3] above, for a cubic system we can write

from which we can obtain the equation:

This equation shows that for a material with cubic symmetry, the ratio of pairs of sin²θ values must be equal to a rational fraction. Thus powder patterns of cubic materials may be indexed by simply examining the ratios of pairs of sin²θ values to see if the ratio is equivalent to the ratio of two integer numbers. This can be done by trial and error, and is readily done by hand. The ratios are easier to spot if the lowest-angle reflections are used since these correspond to the smallest values of n. Once values have been ascertained for n₁ and n₂, values of n for the higher-angle reflections may be obtained since:

This approach avoids errors in the low-angle reflections propagating through the calculation. An accurate value of the lattice parameter a can then be calculated by finding the correct value n for one of the highest-angle / best-determined peaks.

The method is demonstrated in the table below for the yttria data shown in the previous section: the table lists values for the ratios R_m:

where N is the peak number and m is the initial value chosen for n₁.

Since R must be integer, then it is clear that the value of n for the first reflection cannot correspond to 1, 2, or 4. The lowest value that it can have is 3, but note that higher multiples such as n₁ = 6 result in equally valid solutions. Once n₁ is known, then a can be determined and the pattern can be indexed in full. Referring to the complete table, the peak at 164.3450° can be indexed with n = 93 from which a value for a equal to 7.498 Å may be calculated. This solution, however, is wrong! The reason is that the reflection at 46.9° has n = 15, and this value is impossible to achieve from the sum of squares of three integer hkl values.
The generation program can be used to demonstrate the above point.

This demonstrates one pitfall with indexing: it is sometimes possible to index a pattern with the wrong unit cell. The correct solution is with n₁ = 6, which gives a = 10.604 Å.

In this very simple example, it is possible to take the analysis a step further since the values of n that are obtained depend upon the symmetry of the material. For primitive cubic symmetry (e.g. Pm3m), there are only a a few missing values of n. For body-centred cubic symmetry (e.g. Im3m), all the odd values of n are missing. Finally, for face-centred cubic symmetry (e.g. Fm3m), the pattern is more complex: typical values of n observed are 3, 4, 8, 11, 12, etc.

Although these methods may be useful for materials of cubic symmetry, they are clearly less well adapted to the indexing of patterns from lower-symmetry crystals. A more general approach will be discussed on the next page.

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