to study of the Structure of Crystals. 133 



The unit corresponding to the first cubic lattice is a 

 simple cube with points of the lattice at its corners. The 

 center of the single crystal molecule may be taken at 

 0, the origin (figure 4b). The unit of structure for 

 the body-centered cubic lattice ( T/ ) is a cube having 

 points of the lattice at its corners and center. It thus 

 contains two crystal molecules, one having its center at 

 the origin 0, (000) and the other at the point P,,, (tx, Ty, 

 Tz) ^g. 4:0. The face-centered unit (T/) with points 

 of the lattice at the corners and the centers of the faces 

 of the unit cube, has four crystal molecules associated 

 Avith it. Their centers are at the origin 0, (000) and 

 at the points P (0, r,, r,), P (r,, r^., 0), P ,/t„ 0, T,), fig. U. 



This kind of a unit wherewith the crystal can be built 

 up by simple translation of the unit involving only one 

 coordinate axis at a time, is desired for the present 

 purposes because calculations of relative spacings of 

 like planes and of interference effects from different 

 planes can be made upon it as typical of the crystal as a 

 whole. 



It is, then, essential to be able to write down the posi- 

 tions of all the atoms which lie within the unit prism. 

 This can be done readily if the nature of the space group 

 is known. If the unit is a simple prism having a single 

 crystal molecule associated with it, then the atoms in the 

 unit, of which the center of this molecule is a corner, 

 can be represented by the coordinates of the atoms in the 

 crystal molecule. ^^ 



The coordinates of the equivalent points in a unit prism 

 for the space group C21/ may consequently be written as 



xyz, xy2, ~xyz, xyz. 



If the unit has n crystal molecules associated with it, then 

 of course it will contain n groups of equivalent points, 

 that is, t^ X p equivalent points, if |; is the number of 

 equivalent points of the underlying point group. In the 

 case of the space group Csn^ which is formed by placing a 

 group of points having the symmetry of the point group 

 C (holohedry of the monoclinic system) at each point 

 of the second monoclinic lattice (symbol P'.n), the 



^^ This will be clear if it is remembered that since neighboring points of 

 the lattice are all alike, corresponding points of neighboring crystal mole- 

 cules are identical. This leads to the fact that a translation along the X-axis 

 of -X is equal to the translation 2Tx—:r; likewise —yz=2Ty — y, ^\\(\. — z=^2t ■^— z, 

 where y and z are translations along the Y- and Z-axes. 



