t 

 SEPT. 19, 1921 WYCKOFF: WAVE LENGTHS OF X-RAYS 371 



Placing sodium and chlorine (or chlorine and sodium) atoms at the 

 positions of (1) and (2) yields the "sodium chloride arrangement." 

 The next simplest structure that can be developed is obtained from 

 arrangement (3) by placing atoms of sodium at the 24 points corre- 

 sponding to a value of ii = Ui and chlorine atoms at the points where 



For such a structure the intensity of a reflection from a plane whose 

 indices are h, k and / is proportional to (compare equation (1)) : 



A = pj,a [2cos2irnhui + 2c.os2irnkui +2cos2xw/wi + 



cos7rw(2/'2M] -^h-\- k) -\- coswJtih-hui + ^) + costtw {h + 2kni + h) + 

 cosrw(/i + k-2kui) + cosTTiiQi + ^ + 2hti) + costtw {h + k-2lui) + 

 cos7rn(2/zMi + fe + /) + cosTruih - 2hui + /) + cost;z {h + 2kui + /) + 

 cosirn(h - 2kui + /) + cos7r?i(/i + 2/wi + /) + costtw {h + l-2lui) + 



QOS-Kn{2hUi -\-k -\- I) -f C0S7rw(/j + I ~2hUi) + COS7rw(2feM.i + ^ + /) + 

 COSx«(fe - 2kUi + + COS7rn(fe + 2/Mi + /) + COSttW (k + /-.2/?ii)] + 



Pci[a similar set of terms involving Wo] . 

 The B term, a similar sine expression, is not in this instance qualita- 

 tively important. When h, k and / are two odd and one even {h = 2m 

 -\- 1, k = 2p -\- 1, I = 2q, w^here m, and p and q are any integers) 

 or when they are two even and one odd (/i = 2m -f l,k = 2p,l = 2g) 

 this expression is invariably equal to zero forn = 1. When, however, 

 the indices are all odd, the intensity becomes 



A = 8 cos 2Tru (2m + 1) + 8 cos 2xzi {2p -fl) + 8 cos 27rM {2q + 1), 

 for 11 = 1. Except perhaps for very special values of i/i and ii2, 

 this expression will invariably have a value other than zero. In 

 case M =2 all three types of planes will give factors of appreciable 

 value. This arrangement is consequently in entire agreement with 

 the data that we now possess^ upon which to base the first determination 

 of the structure of a crystal.^ 



* This need be no longer true if we possessed accurate measurements of the relative 

 intensities of the X-rays reflected from different planes of this type, combined with a knowl- 

 edge of the positions of the electrons as the scattering centers within the atom. 



^ The agreement with the spectrometer experiments is readily shown. By substituting 



the appropriate values for h, k and / in expression (1), the ratio of the spacings for the 



1 1 

 (100), (110) and (111) planes in the cubic lattice is seen to be 1 '■~T'' ;-■ Since the calcu- 

 lations given above have shown that the first order reflections from the first two of these 

 planes is blotted out, the first observable reflections would occur at twice the angle of 

 the first reflection and would consequently give apparent spacings of 



1 _J_ J_ J_ . _? 



2 2V2 "VS^"^^ ■V2 ■ V3' 



