CONTEMPORARY ADVANCES IN PHYSICS 415 



substituting from (6^, 6e), we obtain: 



2 - 2(ati3, + a,i32 + a3/33) = ^, (//r + /^a" + /^2-). (7) 



Now by the second of the theorems concerning" direction-cosines, the 

 quantity in parentheses on the left is none other than the cosine of 

 the angle between the direction of advance of the primary beam, and 

 the direction of the scattered waves which go to form the spot or 



Fig. 16 — Diffraction-peaks obtained with a fixed crystal and a fixed collector, 

 i.e. with a constant value of the angle of deflection $, by varying the wave-length. 

 (Davisson and Germer.) 



diffraction-maximum of the indices hi, ho, hs. If we conceive the 

 diffraction-beam as the path of a portion of the energy which came 

 with the primary stream and was deflected out of it, then this is the 

 angle of deflection. Call it $. We have: 



2a~ 



sin^<J> =T^V^7TI7+1?. 

 2 2a 



(8) 

 (9) 



As the reader will observe, there is no allusion here, explicit or 

 implicit, to the orientation of the crystal. This is therefore the 

 appropriate equation for the "powder method," in which crystals 

 turned every way are presented all together to the primary stream, 

 and no one knows the orientation of any particular one — indeed the 

 individuals are often too small to be seen. 



Equation (9) describes a cone, having for its origin the mass of 

 assembled crystals, for its axis the direction of the primary beam, 

 for its apical semi-angle the angle $. Such a cone intersects any 

 sphere centred at the crystals (our imaginary bulb), or any plane at 

 right angles to the primary wave-stream, in a ring. There are in 

 principle as many of these rings as there are triads of integers {hi, h^, hz) 

 except that when two different triads have the same value of the 



