A.—_MATHEMATICS AND PHYSICS. 43 
Similar methods are applicable to the three-dimensional crystal. If, 
for example, we consider the case of C},, or D*EY we can show that, whereas 
in general the spacings of planes are such as are proper to a cell of the dimen- 
sions and form drawn in the figure, all planes of the form lz/a-+-mz/e=an 
integer, show halved spacings, unless / is odd and m is even: which is 
sufficient identification of the mode of arrangement. The symbols a and 
c denote edges of the cell. 
If we follow this line of reasoning through all the thirty-two classes, 
we end, of course, with the discovery of the 230 modes which are known to 
exist: and with the identification marks of each, with certain qualifica- 
tions. These last are of two kinds. One of them is general in nature and 
is a consequence of the fact that the X-rays can measure only the distance 
between two like points in neighbouring cells, say A and B. But they do 
A 
‘ 
A 
U] 
B 
B 
Fia. 8. 
not indicate any difference that may exist between AB and BA. IH such 
a difference exists it may be expected to show in the external characteristics 
of the cell, giving it polarity. A good example is tu be found in zinc blende. 
Layers of zinc and of sulphur atoms alternate with one another as in fig. 9, 
all of them being perpendicular to a trigonal axis of the crystal. The 
distance between a zinc atom in the layer A to a zinc atom in the layer B 
is found without question by the X-ray method. Now we know from 
observation of the crystal that there is a difference between AB and BA : the 
crystal is polar. A crystal plate cut so that its faces are perpendicular to 
the axis shows different properties on its two sides: if heated, one face 
becomes positively and one negatively electrified. Whichever face we use 
in the X-ray spectrometer we obtain the same value for the spacing, and 
we find ourselves unable to detect any difference between the two aspects 
_ by means of the spectrometer observations. 
We may see this point in another way. Suppose that fig. 9 
represents a section of a crystal consisting of two kinds of atoms, indicated 
respectively by full and empty circles. The arrangement clearly has no 
_ symmetry about a vertical line in the plane of the paper. But if X-rays 
were incident from above, as shown there would be equal reflections from 
the planes 11’ and 22’. If the incident rays were heterogeneous and a 
photographic plate were placed to receive the Laue reflections in the usual 
way, there would be a symmetry distribution of spots on either side of A, 
although there is no symmetry in the crystal to correspond. 
It is only when we have taken other considerations into account and 
have determined the structure that we can establish the polarity of the 
crystal. We may take, for example, the fact that zinc blende is cubic, 
and therefore has four trigonal axes, a fact which we may discover from 
X-rays as well as from the external form. Also, the unit cell contains only 
one molecule of zinc sulphide, and may be drawn of the same form as in 
diamond: that is to say, its eight corners can consist of the six centres 
