CONTEMPORARY ADVANCES IN PHYSICS 419 



indices (/?i, h-i, Jh) is inclined to the primary beam at a certain angle $ 

 for which we have found the formulae (8) and (9). We may conceive 

 that this deflection is due to a reflection of part of the incident wave- 

 motion from a mirror or mirrors traversing the crystal, so tilted that 

 their plane bisects the angle $. The picture would be legitimate 

 even if there were nothing physical corresponding to these "mirrors"; 

 but we shall presently see that they are not imaginary. The normal 

 to their plane makes supplementary angles ^o and = tt — ^o with 



Fig. 20 — "Powder method" diffraction-rings obtained with electron-waves and a 

 thin film of gold. (G. P. Thomson, Proc. Roy. Soc.) 



the primary and the diffracted beams, respectively; and (^o — 6) is 

 the angle of deflection ^J Combining these statements, and choosing 

 the positive sense of the normal so that it shall make an acute angle 

 with the diffracted beam, we find: 



d ^ \tV - H; ^0 = f TT + i$. (10) 



We wish to deduce the direction-cosines of the normal — denote 

 them for the moment by the symbols 71, 72, 73 — from the conditions 

 that it makes the angles Bq and 6 with the rays having the direction- 

 cosines cci, 0:2, 0:3 and /3i, 182, /Ss respectively. These conditions are 

 thus expressed by the aid of equation (9) : 



ai7i + «272+a373 = cos 0o= — sin-$= — — V//l- + /^2^+/^3^ (H) 



/3i7i + /3272 + /3373 = cos0-+sin^$= + ^V^7+I?+^, (12) 



and the reader can easily show by means of equations (6a, 6&, 6c) 

 that they are satisfied when : 



hi h-i hz 



'V 1 ^ — — - • "Y t> ^ — — • -y o = — - • ( 1 ) 



V/Zl- + /^2- + /^3' ' " V/2l2 + //2' + //3- ' " V//l'-^ + //2- + /i3- 



^ It is the custom to say that the normal to a mirror makes equal angles with the 

 incident and the reflected beams, but this manner of statement implies that the 

 positive senses along the directions of the beams are defined oppositely — one with, 

 the other against the sense in which the waves are advancing. The convention 

 adopted here is the more logical, and leads to more symmetrical equations. 



