Hull-Davey chart

Using the Hull-Davey chart (Fig. 21.7) determine the Miller indices of the reflections, the ratio b/a, and the corresponding ratio a. 

Fig. 21.7 Hull-Davey chart for determining the unit cell dimensions of an orthorhombic lattice.

To use Fig. 21.7, one plots the values of log dm on the same scale, although not necessarily on the same range as the Hull-Davey chart using a sheet of tracing paper. Place this paper parallel to each axis until all the observed log d values fall on the chart [3]. The indices of the lines are the h, k, and / indices of the corresponding spots [3], The log d value corresponding to the 1,0 line is log a, whereas log b is 0,1[3]. 

Some Hull-Davey charts, like the one shown in Fig. 10-4, are designed for use with sin 6 values rather than d values. No change in the chart itself is involved, only a change in the accompanying scale. This is possible because an equation similar to Eq. (10-4) can be set up in terms of sin 9 rather than d, by combining Eq. (10-3) with the Bragg law. This equation is... 

Another graphical method of indexing tetragonal patterns has been devised by Bunn [10.2]. Like the Hull-Davey chart, a Bunn chart consists of a network of curves, one for each value of hkl, but the curves are based on somewhat different functions of hkl and cja than those used by Hull and Davey, with the result that... 

Fig. 10-3 Partial Hull-Davey chart for simple tetragonal lattices.

Fig. 10-4 Complete Hull-Davey chart for body-centered tetragonal lattices.

The powder pattern of zinc made with Cu A a radiation (Fig. 3-13) will serve to illustrate how the pattern of a hexagonal substance is indexed. Thirteen lines were observed on this pattern their sin 0 values and relative intensities are listed in Table 10-2. A fit was obtained on a Hull-Davey chart for hexagonal close-packed lattices at an approximate cja ratio of 1.87. The chart lines disclosed the indices listed in the fourth column of the table. In the case of line 5, two chart lines (10- 3 and 11-0) almost intersect at cja = 1.87, so the observed line is evidently the sum of two lines, almost overlapping, one from the (10 3) planes and the other from (11 -0) planes. The same is true of line 11. Four lines on the chart, namely, 20 0, 10 4, 21 -0, and 20 4, do not appear on the pattern, and it must be inferred that these are too weak to be observed. On the other hand, all the observed lines are accounted for, so we may conclude that the lattice of zinc is actually hexagonal close-packed. The next step is to calculate the lattice parameters. Combination of the Bragg law and the plane-spacing equation gives... 

Rhombohedral crystals are also characterized by unit cells having two parameters, in this case a and a. No new chart is needed, however, to index the patterns of rhombohedral substances, because, as mentioned in Sec. 2-4, any rhombohedral crystal may be referred to hexagonal axes. A hexagonal Hull-Davey or Bunn chart may therefore be used to index the pattern of a rhombohedral crystal. The indices so found will, of course, refer to a hexagonal cell, and the method of converting them to rhombohedral indices is described in Appendix 4. 

We can conclude that the pattern of any two-parameter crystal (tetragonal, hexagonal, or rhombohedral) can be indexed on the appropriate Hull-Davey or Bunn chart. If the structure is known, the procedure is quite straightforward. The best method is to calculate the cja ratio from the known parameters, lay a straightedge on the chart to discover the proper line sequence for this value of c/a, calculate the value of sin 6 for each line from the indices found on the chart, and then determine the indices of the observed lines by a comparison of calculated and observed sin 6 values. 

Since a rhombohedral lattice may be referred to hexagonal axes, it follows that the powder pattern of a rhombohedral substance can be indexed on a hexagonal Hull-Davey or Bunn chart. How then can we recognize the true nature of the lattice From the equations given above, it follows that... 

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