CONTEMPORARY ADVANCES IN PHYSICS 425 



Now the principle emphasized by Bragg is this, in full: any 

 diffraction-spot results from reflection by the strata having the same 

 indices as the spot, at the angle of selective reflection deflned by equation 



{IS). 



The proof of this statement depends on the following formula ^ 

 for the distance between adjacent strata of the family having the 

 Millerian indices HiH^Hz : 



d = aHH.^ + W + W. (16) 



Substituting into equation (12) the value of a given by this formula, we 

 find that: 



^ X -^h^' + h2' + hs' ,.^-. 



cosB ^ wi , (17) 



and since the quantities H are integers without a common divisor, 

 while the quantities hi are integers for which hi/IIi = h2/H2 = h/Hs, 

 the ratio of the two radicals must be an integer. 



The mirrors which I introduced at a previous page as a way of 

 accounting for the spots do actually exist. They are the strata into 

 which the groups of atoms fall. Each one by itself, however, re- 

 flects so little that in effect there is no reflection to speak of, unless 

 and until an entire procession of parallel strata is brought into play. 

 Earlier we located the diffraction-beams as the directions in which 

 the scattered waves from four adjacent atom-groups enhance one 

 another most by constructive interference. Multiplication of atom- 

 groups beyond the first four merely made the beams sharper and 

 more intense. Alternatively we may now locate these beams as the 

 directions in which the reflected waves from two adjacent parallel 

 strata enhance each other most. Multiplication of strata beyond 

 the first two intensifies and sharpens them. 



This theorem of Bragg's thus gives a remarkably helpful picture of 



" the way of a crystal with a beam of waves" ; a picture most valuable, 



when one has a single large crystal of which the surfaces are natural 



facets. Suppose for instance one has a cubic crystal of rocksalt, one of 



L^ Let and P stand for two planes of the family {H1H2II3), one being drawn 

 through the origin, the other through any other lattice-point. The components ot 

 the vector r from the origin to this other lattice point must be integer multiples 

 lia, lia, Isa of the spacing a. The projection of this vector on the direction of the 

 normal drawn from the origin to the plane P is equal to (71/1 -\- 72/2 -\- y3lz)a; the 

 values of 717273 are to be taken from (14). This projection is equal to the distance D 

 between the planes and P, for which we therefore have: 



D 



Hih + H2I2 + H,h = — <m -f Hi + m. 



The least value (except for zero) which the quantity on the left can and does assume 

 is unity; whence equation (16), 



