Zone-Indexing Methods

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Zone-Indexing Methods

The methods that have been described so far work well for the higher-symmetry crystal systems, but are less suitable for the lower-symmetry ones. Zone indexing is a method that works extremely well for low-symmetry systems since it makes no prior assumptions about degrees of symmetry. Further tests are then required to check whether the low symmetry cell corresponds to a higher symmetry system, e.g. rhombohedral, trigonal, and hexagonal cells are nearly always indexed as monoclinic.

A zone is a plane in reciprocal space passing through the origin. In general 3 points in space are required to uniquely define a plane, but for a zone plane only 2 act as variable parameters, since the third point is fixed at the origin. Since the unit-cell axes can be arbitrarily chosen, one can choose 2 of the axes, say a and b, to be in the zone plane. Points in this plane therefore have indices hk0 and the equation for the Q values reduces to

with only three constants of the metric tensor constants A, B, and F. If one takes two of the low Q values, then these can arbitrarily be assigned as Q_h00 and Q_0k0, giving the equations:

Combining these with the initial equation, the following equation is obtained:

In order to find zone planes, the program permutes, say, 3 low Q values with, say, the next 6 lowest Q values with h,k = ±1 or ±2, and stores calculated values of F. Values that occur frequently are stored as likely values for this variable. Hence this method of indexing should not be thought of as exhaustive: it simply works on the most frequently-occurring solutions.

Once a set of possible zone planes has been found, the next step is to combine the possible zone planes together. Arbitrarily, one of the zone planes can be considered as hk0 with parameters A, B and F, while the second one can be treated as hl0 with parameters A, C, and E. Hence the only unknown parameter is D, where

D is again found by permutations using different values for Q_hkl, and the most likely solutions are again kept.

Although the method is clearly not exhaustive, it is very quick since only 3 defining parameters are required for each zone, and only low values are needed in practice for h and k. Also, it has the advantage that it is less sensitive to impurity reflections in the powder pattern than many other methods.

In order to demonstrate the zone-indexing method, some laboratory data of a chloro-organic sodium salt is shown below, plotted is Q space:

The next diagram shows the position of 20 strong and well-defined reflections that will be chosen for indexing:

The best way to check the solutions is to generate the complete list of reflections for the likely unit cells and compare them to the observed data. The result, plotted again in Q space, is shown in the red and cyan line diagram, which should be compared to the data, now shown in expanded format below:

The red lines are those used as input data to the zone-indexing program, while cyan lines are additional ones generated by the hkl-generating program. You should see that the cyan lines account for all the weak peaks that were excluded at the indexing stage.

A complete list of reflections corresponding to the above can be obtained by clicking on the icon for generation. You can compare this list to the list of 2θ values used as input to the indexing program.

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