PROF. W. H. BRAGG ON X-RAYS AND CRYSTAL STRUCTURE. 273 



Proceeding in the usual way, we find that this function from to 2a can be expressed 

 by the Fourier series 



2k 



ac I ac +TT a ac + n TT a 



If a 2 c 2 is small compared to V 2 , the harmonic coefficient is nearly proportional to 

 l/n 2 , so long as a is unchanged. That is to say, the intensities of the various orders 

 for a given spacing are inversely as the square of the number of the order, which 

 means no more than that the exponential curve we chose to express the variation of 

 atomic density was sufficiently correct. It gives a compound curve whicli is nearly 

 a series of parabolas. 



The quantity ac is really a measure of the overlapping of two atoms, because the 

 stratum density of an atom in the plane containing the centre of the next is ke'' 2ac . 

 If, for example, e" 2 = O'Ol, ac = 2" 3 ; if e- 2nc = O'l, ac = 1'15. Overlappings of this 

 magnitude are quite to be expected, because unless the planes are very widely 

 separated, the atoms of one plane penetrate some distance into the interstices 

 of the next. 



If ac = 1'15 the values of the coefficient a 2 c 2 /( 2 c 2 + wV 2 ) for increasing values of n 

 are 1 .* , , ^-^, 9 \ } 4 , -, .-,%., or very nearly as the inverse squares of 1, 2, 3, 4. 



If ac = 2 '3 the departure from the inverse square law is still not very great. The 

 numbers are then Y^~, 4^T> T^l.> 16 1 3 .,. 



Now let us consider the effect of altering the spacing. As we bring the planes 

 together, let us say, the area ABCD of fig. 23 must always represent the weight of one 

 atom because there is one atom to each spacing. The amplitude of any order, say 

 the n th , is 



OZ. ri'*f 



.A Lv L/ 



ac ' a 2 " 



The effective portion of the atom is therefore 



2k aV 



AD x ac ' 



of the whole. But AD . x a = the mass of the atom. Also the mass of the atom 



2k 



r 



= 2 lce~" 

 Jo 



-"dx 



c 



Hence, finally, the fraction of the atom which is effective is aV/(a 2 c 3 + nV 3 ). 



Now we have to explain two things : in the first place, the fact that for a given 

 spacing the intensities of the various orders fall off as the inverse square, in the second 

 the fact that the intensities of two spectra of the same order belonging to different 

 spacings are proportional to the squares of the spacings. . For in practice we find 



