some Carbonates of the Calcite Group. 343 



where the summation is taken from each atom m in the 

 unit of structure. p m is the scattering power of the atom 

 m; its coordinates are x m , y m , s m . If the inverse-sine-square 



law held, then / (-) would of course be {-)% where d is 

 n n 



the spacing of planes hklm the unit and n is the order of 

 the reflection. 



In arrangement (/) if the calcium atoms are placed at 

 (1/4 1/4 1/4) and (3/4 3/4 3/4) and the carbon atoms 

 at (0 0) and (1/2 1/2 1/2), the first order reflection by a 

 plane (hkl) in the crystal can then be written: 

 A 2 + B 2 oc Intensity 



A=20a|cos|(h4-k4-l) j +0 j 14-costt (h+k + 1) j +0 

 | cos I (8li+k+.31) 4- cos |(h+3k+l) 4- cos| (31i + k+l) + cos 



7T 



2 



W (h 4- k 4- 31) 4- cos 1 (h + 3k + 31) + cos - (3h 4- 3k 4- 1) 



B = 2Ca (0)4- 20 (0)4-0 j sin |(h 4- 3k 4- 31) +sin| 

 (3h + 3k + l) + sin | (3h + k +31) 4- sin | (h 4- 3k +1) 

 4- sin | (3h 4- k +1) 4- sin | (h 4- k 4- 31) | • 



Ca, C and are the scattering powers of calcium, carbon 

 and oxygen respectively. In case all the indices are odd, 

 or two are even and one is odd (the sum h + k + I is odd), 

 both the A and the B terms disappear. This is also true 

 if the positions of calcium and carbon are interchanged. 

 But since several first order reflections are found from 

 planes having one odd and two even indices, this arrange- 

 ment must not be the correct one. 



A term which will be proportional to the intensity of 

 reflection if the atoms are arranged according to (g) may 

 similarly be written : 



A 2 4- B 2 x Intensity 



A=-2Ca | cos | (h + k+Z) I + ] 1 4- cos tt (A+ifc+Z) j 4- O 



{cos 2 7r u (h—k) (14- cos 7r s^ 4- cos 2 tt u (l — h) (14- cos tt s 2 ) 4- 



COS 2 TT U (k — I) (1 4" COS 7T S 3 ) } . 



B=0 {sin 2 7r u (k—k) (1 — cos it s^ 4- sin 2 tt u {l—h) 



(1 — cos it s 9 ) 4- sin 2 it u (Jc—V) (1 — cos it *,)}, where 



Si=h — Jc-\-l, s 2 =Jc—h+l, s 3 =h+k—l. 



