282 H. K. SCHACHMAN AND R. C. WILLIAMS 



atoms in the unit cell may be considered to be the equivalent of N equal 

 and parallel, interpenetrating space lattices, one for each kind of atom of 

 the unit. Such a lattice is the tjTpe usually encountered and is known as a 

 compound lattice. The ultimate purpose of X-ray crystal structure analysis 

 is to determine the relative position of each of the N atoms, or more pre- 

 cisely, of their electron clouds. But the X-ray data furnish direct information 

 only about the positions and the intensities of the X-ray spots. In a com- 

 pound lattice (suppose iV = 2) all planes with the same {h k I) value will be 

 involved in producing the observed intensity of one spot. Depending upon 

 the spacing between the h k I planes of atom 1 and of atom 2, in terms of 

 the spacing of the planes of atoms 1 or 2 alone, the coherent scattermg from 

 the two sets of planes will have differing phase relations. For example, if the 

 planes through atoms 1 are midway between those through atoms 2, the 

 scattering amplitudes from the two sets of planes will be completely out of 

 phase for m = 1, 3, . . . In a very complex lattice, where N might be 1000, 

 the intensity of every X-ray spot wiU be the square of the vector sum of some 

 1000 scattering amplitudes all having different phase relations. But these 

 phass relations camiot be calculated precisely until the structure is known — ■ 

 the very problem being investigated. 



c. Calculation of X-ray Intensities. In order to attempt to miderstand the 

 methods employed to resolve the dilemma of unknown phases it is useful to 

 approach the inverse problem: that is, to assume that the positions and 

 X-ray scattering powers for each of the N atoms are known, and to calculate 

 the predicted intensities of the X-ray spots. If in the unit cell there are 

 atoms of kinds A, B, . . . , there is associated with each kind a scattering 

 power, /^,/g, .... The value of/ depends upon the radial density distribu- 

 tion of the scattering electrons within the atom, upon the wavelength of the 

 X-rays, and upon the angle of scattering, 6. It can be shown that 



f" -^r, . sin ^ , , , 47Trsin^ ,„^, 



/= U{r) —j^dr, where </. = (36) 



Jo ^ 



U{r) is a fmiction of the radial density distribution of the electron cloud and 

 may be written: U{r) = irrr^ I */* l"^ where | p is a solution of Schrodinger's 

 equation, and where | i/j \^dv is proportional to the chance of finding an elec- 

 tron within a small volume, dv. Values of/ for many kinds of atoms have 

 been calculated from their wave functions. 



The total amplitude scattered from a unit ceU and forming a single X-ray 

 spot will depend upon the scattering factors for the atoms within the unit 

 cell and upon their spatial distribution. The spatial distribution wiU govern 

 the phase relations among the amplitudes of scattering from the parallel 

 planes (of a given h k I) that go through all the kinds of the N atoms. The 



