320 Mr. C. G. Darwin on the 



so that the whole effect is 



f(20, k) ik( ct-p) P tc^- ik 9 - (£ 2 sin 2 + ?, 2 ) 



Let M be the number of atoms per unit area. Then the 

 number in an area d^dr] is Md^drj, Since the phase 

 variation between neighbouring atoms is small we can 

 replace the sum by an integral, and get as the reflected 

 wave 



GO 



f(20, Je) wot- P ) M P (Y* »* P r& a :„2a, iwtj 



W),V- ri ^ 



?7T 



7T<? 2 



p & sin ^ 



I£ N be the number of atoms per unit volume and a the 

 distance between successive planes of the crystal, M = Na, 

 and we have as reflexion coefficient, 



_ ?7r 



fQW-nzw-—* (1) 



This expression is not perfectly general since q might be 

 made greater than unity by increasing N. This would 

 violate the conservation of energy. In actual matter this 

 would be prevented, because the vibration of each atom 

 would diminish those of its neighbours, so that we should 

 have to regard /as dependent on N. Numerical calculation 

 shows that q is of the order 10 ~ 4 , so that the simple form 



?v 

 probably stands. The factor e ~ 2 is the converse of the 

 quarter wave which has to be introduced into diffraction 

 problems. 



6. The Refractive Index. 



If the factor /(20, k) is replaced by /(0, Jc) the same 

 radiation, represented by — iq , is scattered on the other 

 side of the plane, so that the wave there is of the form 



e ik(Ct - E) e ik(Qt - ■ E) - iq Q 



(l-*0o) B or H 



since q is small. This neglects the absorption in the plane 



