418 BELL SYSTEM TECHNICAL JOURNAL 



which yield other small integer values for the sum {hi, h^, hs) — but 

 not in every case. There is no triad of integers of which the squares 

 add up to seven, and there is none for which they add to fifteen. 

 If the radii of the rings are plotted as a function of their ordinal 

 number, there are breaks in the smooth curve beyond the sixth and 

 the thirteenth, as if two were absent from the regular sequence. 

 To express it more graphically than accurately, the seventh and the 

 fifteenth rings are missing. Other rings also may be wanting, for the 

 atoms in the atom-groups may be so disposed that in certain directions 

 the individual groups scatter no waves whatever; and even if the 

 lattice were so proportioned that waves in such a direction would be 

 tremendously amplified, there would be nothing to amplify and no 

 diffraction-spot. But this also is a subject for later study. 



If we measure the radius of any ring and calculate sin ^$, and then 

 change the wave-length and repeat the process, the values so obtained 

 for the sine should stand to one another in the ratio of the wave- 

 lengths. If the waves in question are electron-waves, then since 

 their wave-length is inversely as the speed of the electrons, the radius 

 of each ring should vary inversely as the speed of the electrons falling 

 upon the crystals.^ This was verified by G. P. Thomson. Knowing 

 the values of the spacing a of the metal crystals which he had used — 

 these values having been deduced in earlier days from the diffraction- 

 circles produced by X-rays of known wave-length, scattered by the 

 crystals of such metals — Thomson also determined by equation (8) 

 the actual wave-lengths of the electron-waves. With X-rays the 

 powder-method is seldom used to evaluate wave-lengths, Bragg's 

 being the better when available; it serves chiefly for the study of 

 lattices. But with X-rays and electrons both, the splendid array of 

 the diffraction circles which spring forth when a beam of either kind 

 is sent against a mass of tiny crystals is the most easily adducible, 

 the simplest and perhaps the most vivid and striking evidence that 

 there is something wave-like in the nature of either. 



So much for the fundamental equation of the powder method! 

 We will now derive, from equations (6a • • • 6e), the fundamental 

 equation of the method invented by Bragg. 



We have seen that the diffracted beam forming the spot with 



^ More precisely, the radii should vary inversely as the momentum of the electrons. 

 The true formula for the wave-length as function of the electron-speed is probably 



X = {h/mov)yll -i^Vc- 



instead of X = h/mov (here v stands for the speed and nto for the mass-at-zero-speed 

 of the electrons) ; but as yet the speeds employed have not been great enough nor 

 the measurements exact enough to distinguish between the formulae. 



