THE PHYSICAL PROPERTIES OF INFECTIVE PARTICLES 283 



problem is solely a geometrical one, and it turns out that the entire structure 

 amplitude, F, of a unit cell may be written as: 



I ^AH I cos a =/^ cos (^^ +/g cos 0J5 + . . . = P 

 I Fni^i I sin a =/^ siji 0^ +/b sin 0^ + . . . = Q 

 Fl,= P-'+QMs.n^=QjP 



where the sine and cosine terms represent the phase relations that result 

 from the relative positions of the N kind of atoms within the unit cell. The 

 ^'s are simple functions of the atomic positions, of the value of (fi k I) for the 

 planes giving rise to the X-ray spot, and of the order of the X-ray reflection. 

 It might be noted that the/'s are dimensionless quantities, and consequently 

 so is F. F^ represents the ratio of the expected intensity of the X-ray spot to 

 that resulting from the scattering of X-rays by a single classic electron. 

 Since the latter value is calculable, the expected intensities may be calcu- 

 lated and may be compared with the observed ones. This method of com- 

 parison is universally employed in checking the reliability of any X-ray 

 analysis. 



d. Fourier Summations. The practical problem in X-ray crystallography is 

 formally the inverse of the above-described operation: the calculation of the 

 atomic positions from the observed intensities and {h k I) designations of 

 the X-ray spots. Since a crystal is a periodic arrangement, in three dimen- 

 sions, of electron densities (atomic positions) it is possible to represent the 

 electron density at any point by a triple Fourier summation: 



Pxvz = 7; III I Ff,T,i I cos [2TT{hx + % + h) — a^ifcj] 



Pxvz is the electron density at the point x, y, z in the miit cell, and V is the 

 volume of the cell. The coefficients of the summation are the absolute 

 values of the structure amplitude factors discussed previously. The summa- 

 tion is formally an infinite one over all values of h, k, and I; in j^ractice it 

 would be limited to those values for which corresponding X-ray spots are 

 measured. The evaluation of the triple sum is a formidable task, but, what 

 is worse, for the case of complex molecules it cannot in principle even be 

 attempted. The difficulty is that only quantities proportional to F^ can be 

 measured, but F itself is a quantity having both a magnitude and a phase 

 angle. Hence, in the summation the value of | F,i^i \ is simply the square 

 root of F-, but the value of a^^j is usually miknown. Another way to express 

 this fact is to say that the Fourier sum represents the integrated information 

 contained in the entire set of X-ray spots from all measured j^lanes, and that 

 this information cannot be evaluated until the phase relations among the 

 X-ray beams forming all spots are known. The a^^tj term represents these 

 phase relations. 



