Indexing patterns of cubic crystals

A cubic crystal gives diffraction lines whose sin 6 values satisfy the following equation, obtained by combining the Bragg law with the plane-spacing equation for the cubic system, as in Eq. (3-10)  

The proper set of integers s is not hard to find because there are only a few possible sets. Each of the four common cubic lattice types has a characteristic sequence of diffraction lines, described by their sequential s values  

Each set can be tried in turn. Longer lists can be prepared from Appendix 10. If a set of integers satisfying Eq. (10-2) cannot be found, then the substance involved does not belong to the cubic system, and other possibilities (tetragonal, hexagonal, etc.) must be explored. 

We can also determine the Bravais lattice of the specimen by observing which lines are present and which absent. Examination of the sixth column of Table 10-1 

The characteristic line sequences for cubic lattices are shown graphically in Fig. 10-2, in the form of calculated diffraction patterns. The calculations are made for Cu Kol radiation and a lattice parameter a of 3.50 A. The positions of all the diffraction lines which would be formed under these conditions are indicated as they would appear on a film or chart of the length shown. (For comparative purposes, the pattern of a hexagonal close-packed structure is also illustrated, since this structure is frequently encountered among metals and alloys. The line positions are calculated for Cu Kol radiation, a = 2.50 A, and cja = 1.633, which corresponds to the close packing of spheres.) 

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For reasons to be discussed in Chap. 11, the observed values of sin 6 always contain small systematic errors. These errors are not large enough to cause any difficulty in indexing patterns of cubic crystals, but they can seriously interfere with the determination of some noncubic structures. The best method of removing such errors from the data is to calibrate the camera or diffractometer with a substance of known lattice parameter, mixed with the unknown. The difference between the observed and calculated values of sin 6 for the standard substance gives the error in sin 9, and this error can be plotted as a function of the observed values of sin 6. Figure 10-1 shows a correction curve of this kind, obtained with a particular specimen and a particular Debye-Scherrer camera. The errors represented by the ordinates of such a curve can then be applied to each of the observed values of sin 0 for the diffraction lines of the unknown substance. For the particular determination represented by Fig. 10-1, the errors shown are to be subtracted from the observed values. 

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