A 



A' 



272 PEOF. W. H. BEAGG ON X-EAYS AND CEYSTAL STEUCTUEE. 



Let the curve in fig. 23 represent one of these terms, in fact, the second. The 

 existence of this term implies that the medium, so far as regards the reflection 

 of a wave in the second order, may be looked on as a harmonic 

 medium of amplitude AA'. The rise and fall of the curve above 

 the line AB implies that the medium has a certain distribution 

 of density which will give a second order spectrum if the rays 

 fall on the medium at the proper inclination to the planes. The 

 area ABCD represents the mass of one atom (there being one 

 jj,. (r L atom to eacli spacing) ; and of the whole, the portion A'B'CD is 



ineffective so far as this reflection is concerned. The ratio of 



AA' to AD represents the fraction of the atom which is effective. Now the X-rays, 

 however they strike the medium, always traverse the same number of atoms. The 

 ratio of AA' to AD therefore represents that portion of the medium which is effective 

 in the production of the spectrum, and must be proportional to its intensity. 



If the distance apart of the planes is not varied the amplitude of the harmonic, 

 that is to say, AA' must be proportional to the intensity. Experiment shows that 

 after allowing for temperature effects the intensity is nearly proportional to the 

 inverse square of the order. The periodic function which represents the density of 

 the medium must therefore be of the form 



cos lirxld cos 4TTX/d cos nirxld 



constant + - ^ -^-i - + ...- ^ + ... 



(Q_ \3 



= constant + ' , where 6 is put for 2-n-x/d, 



that is to say, a series of parabolas arranged as in the upper curve of fig. 22. 



We have, therefore, to find such a form of density curve for the individual atom, 

 that when it is combined with others the resulting curve is a series of parabolas, or 

 something very near to it. 



Suppose the density of the atom to be represented by y = ke~" for x positive, and 

 y = Jce CI for x negative. Of course, this implies that the atom is of infinite extent, but 

 it will appear that in practice no real disadvantage arises. 



The ordinate of the compounded curve is given by 



y = k{e- cz + e- ct -' Jae + e- cz -' af + ... e <*-*" + e ei -' M + ... } 



e - cz + ac + ^x-ac 



= k - , where 2a = a. 



e _ e -ac 



This is very nearly parabolic in form, as can be found on trial, except when the 

 individual exponential curves of fig. 22 do not overlap sufficiently. 



