362 SCIENCE PROGRESS 



Equivalence Parameters 



The conclusion reached as to the internal structure of a 

 crystal is then that it consists of a homogeneous closest-packed 

 arrangement of spheres of atomic influence, the volumes of 

 these spheres of influence being approximately in the ratio 

 of the valencies of the respective atoms : all this is deduced 

 from the fundamental hypothesis. We cannot learn anything 

 about the internal structure of a crystal by direct observation, 

 but from direct measurements of the interfacial angles of the 

 crystal we can obtain, as it were, a numerical statement of its 

 external form. And the next step is to discover a connecting 

 link between the external form and internal structure of the 

 crystal, in order that by observation of the former we may 

 gain information regarding the latter. 



On the assumption that the spheres of atomic influence 

 fill space without interstices and that the volume of each 

 sphere is proportional to the valency of the atom, the total 

 volume appropriated by the molecule can be represented by 

 the sum of the valencies of all the atoms comprising it. 

 Further, since we know the relative dimensions of the crystal 

 structure in the three axial directions of the crystal, these 

 being given by the axial ratios, it is possible, by choosing 

 suitable fractions of these axial ratios, to determine the 

 dimensions of the molecule in these three directions in terms 

 of the valency-volume unit. The sum of the valencies of the 

 constituent atoms is termed the " valency volume " of the 

 substance under consideration, and is denoted by the symbol W. 

 The molecular dimensions referred to are called " equivalence 

 parameters," and are denoted by the symbols a:, j/ and 0. They 

 are calculated from formulae similar to those already given for 

 calculating topic parameters, substituting W for V. 



.r=^l 



anv X 



y=T. 2 = cy. 



sin A sin /3 sin y a 



It will be readily understood that the equivalence para- 

 meters represent three translations in the three selected axial 

 directions of the crystal structure. They may be defined as the 

 dimensions of the sides of a parallelopipedal cell equal in 

 volume to the valency volume, the sides of the cell being 

 proportional to and parallel with the three axes. It may 

 be inquired, of what greater service are these equivalence 



