A.—_MATHEMATICS AND PHYSICS. 35 
substance) which has all the properties of the crystal. There are atoms of 
silicon and of oxygen: there is a molecule of silicon dioxide, and a crystal 
unit of quartz containing three molecules of silicon dioxide. The separate 
atoms of silicon and oxygen are not silicon dioxide, of course : in the same 
way the molecule of silicon dioxide is not quartz ; the crystal unit consist- 
ing of three molecules arranged in a particular way 7s quartz. 
The final aim of the X-ray analysis of crystals is to determine the 
arrangement of the atoms and the molecules in the crystal unit, and to 
account for the properties of the crystal in terms of that arrangement. 
The first step is the determination of the dimensions of the crystal unit 
cell: any one of the possible ways in which the cell can be drawn will do. 
When this has been completed it is a simple calculation in geometry to 
find the distance between any atom and any other atom in the crystal of 
like kind and condition, or, in other words, the distance an observer would 
have to travel from any point within the crystal to any other point from 
which the outlook would be exactly the same and would be similarly 
oriented. This isthe only measurement which the X-rays make directly : 
any other measurement of distance is made indirectly, by aid of some 
additional physical or chemical reasoning. It is not possible by direct 
X-ray measurement to determine the distance between any two points— 
atom centres, for example—within the same cell. 
Let us take an example. The crystal unit of naphthalene has the 
dimensions defined in the usual way by the statement :— 
a=8.34h $=6.05A c=8.69A B=122°49' a=y~=90°. 
It contains two molecules: an integral number, as always. These facts 
are given directly by the X-ray measurements. But there is no direct 
determination of the distance between any carbon atom and any other 
carbon atom contained within the same cell: the measurements given 
are those of the distances between any atom and the nearest neighbours, 
in three principal directions, which are exactly like itself, these distances 
being the lengths of the edge of the cell. There is not even a measurement 
of the distance between the two molecules in the same cell, because they are 
not similarly oriented. In fact, there is no clear meaning in the term 
‘distance ’ in this case, just as we cannot state the distance between an 
object and its image in a mirror, unless the object isa pomtof no dimensions. 
If the molecule of naphthalene has a centre of symmetry, as is indeed 
indicated during the development of the results of the X-ray analysis, it 
is possible to state the distance between the centres of symmetry of the 
two molecules in the same cell, but this does not define the distance between 
any atom in one of the two molecules and any atom in the other. All 
such distances, if they are to be defined and measured, can only be found 
by the aid of fresh considerations, 
Or again, let us take the case of rock-salt. The crystal unit cell of rock- 
salt contains one molecule: one form of the cell has for its eight corners 
the six middle points of the faces of a certain cube (edge=5.62 A.U.) and 
_ two of the opposite ends of any diagonal of the cube. The so-called face- 
centred cube is four times as large as the cell, and contains four molecules. 
The dimensions of the cell are determined directly by the X-rays, which 
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