Crystalline Structure of Bismuth. 195 



The sines of these angles are in the ratio 



1 : 1-206 : 1-060, 



while for a face-centred rhombohedral lattice the corre- 

 sponding ratios should be 



1 : 1-204 : 1*055 



The underlying structure is evidently that of the face-centred 

 lattice. 



Taking the most likely value for the glancing angle of the 

 (100) spectrum as 5° 24 ; , the edge of the face-centred rhombo- 

 hedral lattice is found to be 6*56 X 10 -8 cm., and using this 

 value, taking the density of bismuth as 9 '80, the atomic 

 weight as 208, and the mass of the hydrogen atom as 

 1'650 x 10~ 24 gm., the number of bismuth atoms in the unit 

 face-centred rhombohedron comes out as 8 "03. This shows 

 that the unit of the structure contains two face-centred 

 lattices, since a single lattice will contain four atoms only. 



The very small first-order spectra for the faces (111) and 

 (111) show that the atoms lie very nearly on a simple rhombo- 

 hedral lattice, the length of the edge of the unit structure 

 being 3*28 x 10" 8 cm. 



4. The structure which will account for the observed spectra 

 is exactly similar to that already given for antimony *. The 

 bismuth atoms lie on two interpenetrating face-centred 

 rhombohedral lattices. For one of the lattices, suppose the 

 long diagonals, parallel to the trigonal axis, are drawn for 

 each of the eight rhombohedral cells into which the lattice 

 may be divided. Considerations of symmetry show that the 

 atoms of the second lattice must lie on these diagonals. 

 Moreover, since the structure is nearly a simple rhombohedral 

 one, they must lie close to the unoccupied corners of the 

 first lattice. But to account for the presence of the first- 

 order spectrum for the (111) and (111) faces we must suppose 

 the atoms of the second lattice all to be displaced in the same 

 direction along the diagonals by a distance equal to 0'052 of 

 the diagonal of one of the small cells. 



5. The amount of displacement along the diagonal may be 

 calculated from the relative intensities of the different orders 

 of spectra from the (111) face. There is some uncertainty 

 about the exact displacement, since it is very "difficult to 

 obtain intensity measurements from a crystal such as bismuth, 

 and since the law according to which the spectra from a face, 

 in which all the planes contain an equal number of atoms and 

 are evenly spaced, is not known with any approach io accuracy . 



The observed intensities of the first, second, and third 



* Loc oit. p. 235. 



