Theory of X-Ray Reflexion. 687 



depth such that the crystal is twisted through an amount 

 sufficient to allow of a new reflexion. Roughly speaking, 

 then, at every successive d we shall get a reflexion, and the 

 intensities of these reflexions will be 1, ^-V*eosec<^ ^-4^cosee^ 

 ■<fcc. The whole reflexion formula should then be multiplied 



or '9 _ Tt — X ^ ^ ne crvs t a l is so badly 



bv — 



J 1— e-^eosectfP 2/JLdC0SeG(j) 



twisted that there are a number o£ reflexions. It appears 

 that as it describes a property of the crystal, d ought to 

 be taken constant. For a not very great distortion this 

 might be justifiable, but we have strong experimental reason 

 to believe that the crystals are even more imperfect than 

 this. For when the reflexion is evaluated with this factor it 

 will be found that the second order of reflexion is as strong 

 as the first, a result known to be untrue. This must be 

 taken to indicate that crystals are so badly twisted that their 

 planes do not remain parallel even long enough to produce 

 a single perfect reflexion. 



Suppose, therefore, that the crystal is composed of pieces 

 each of depth d small compared with the amount necessary 

 to produce a perfect reflexion. At the depth d the trans- 

 mitted wave has on the average an intensity e~ 4qd ' ,7Ta (see (8)), 

 :and the wave reflected by the thickness d has intensity 



A. 7 



l-. e -tydna or _J/_ . Suppose one of the reflecting pieces is 

 -at depth z. Then the amount reflected from it is propor- 

 tional to — — <?-2^ cosec^ The number of such pieces in a 



rra 



length dz is dz/d. so that the reflexion formula is to be 

 .multiplied by a factor 



V* f ,-*.««♦ dzjd or to li^*. 



If we multiply (11) and (12) by this we see that apart from 

 a numerical factor they lead to the old expressions for the 

 reflexion*. That this should be so is not remarkable, since 

 ■ each reflecting piece of the crystal consists only of a few 

 planes, so tbat the mutual influence of the atoms becomes 

 unimportant. The chief difference is that the whole reflexion 

 now no longer takes place in a band 5" broad, so that the 

 argument f which pointed to the insufficiency of the earlier 

 formulae loses its validity. The displacement of the reflexion 

 * Loc. cit. pp. 332, 334. f Loc. cit. p. 331. 



2Z2 



