196 The Crystalline Structure of Bismuth. 



order spectra from the face (111), which will be called the 

 spectra (111), (222), and (333), are in the ratio 29 : 100 : 24. 

 Parallel to the (111) face, the planes containing the atoms 

 occur in pairs corresponding to the two face-centred lattices. 

 The contribution to the first-order spectrum from the planes 

 belonging to one lattice differs in phase from that of the 

 planes belonging to the other lattice bj an amount depending 

 on the displacement of the atoms along the diagonals. 

 Assuming that the intensities of a normal series of spectra 

 are inversely proportional to the squares of the orders of the 

 spectra, we find that in order to get the ratio (111) •' (222) : 

 (333) = 29 : 100 : 24 we must take the phase difference 

 between the contributions from the two sets of planes to be 

 155°. If, on the other hand, we assume that the normal 

 intensities fall off more slowly, for example 100 : 34 : 14, we 

 find the phase difference which fits the observed intensities 

 most closely is about 150°. Quite a large variation in the 

 normal intensity series makes very little difference in ihe 

 calculated value of the phase angle 8. The value of 8 taken is 



152° + 3°, 



which corresponds to a displacement along the diagonal of 

 0*052 of its length away from the unoccupied corner. 

 Since the length of the diagonal is 5'92 x 10~ 8 cm., the 

 atoms are displaced from the corners by a distance 

 0*307 x 10~ 8 cm., and the range of distance within which 

 the atoms probably lie is about 0'0'65 x 10" 8 cm. 



6. Distance between the Atoms. 



Any given atom in one of the (111) planes is equidistant 

 from three atoms in each of the (HI) planes on either side 

 of it, but the three atoms in one of these planes are much 

 nearer the given atom than those in the other. In fact, the 

 closest approach between two atomic centres in the structure 

 is the distance between two atoms in these close pairs of 

 planes. This distance may be considered as the " atomic- 

 diameter " of bismuth, using the term in the sense in which 

 it is employed by W. L. Bragg *. This closest approach of 

 the atoms, is found to o be 3*11 A.U. The corresponding value 

 for antimony is 2'87 A.U. For bismuth the nearest distance 

 of approach for the wider pairs is 3'47 A.U. The (111) 

 planes thus occur in pairs in which the atoms approach one 

 another closely, separated by a wider spacing, and, just as in 

 the case of antimony, the good cleavage of the crystal is 

 parallel to these pairs of planes. 

 Manchester University. 

 .April 22, 1921. 



* W. L. Bragg, Phil. Mag. vol. xi. p. 169 (Aug. 1920). 



