284 



X-ray Analyses of Proteins and Nucleic Acids /1 5 : 4 



r 



(100) 



(110) 



proteins and nucleic acids, it is necessary to use one of these schemes. 



The reader is referred to Reference 4 for details of these techniques. 



Instead of rocking the crystal, a 

 powder of small crystallites can be 

 used. Then the two-dimensional pic- 

 ture consists of a series of concentric 

 circles. The powder pattern is harder 

 to interpret but easier to obtain. It 

 has been widely used in metallurgy 

 and mineral sciences but has had few 

 biological applications. 



A sensitive test of an assumed atomic 

 structure is to compare the relative 

 intensities of the X-ray diffraction 



. maxima observed with those computed 



from the model. The relative inten- 

 sities are found by adding together the 

 contributions of each atom, taking 

 into account the phase differences 

 due to the difference in pathlength 

 to each atom. This is usually ex- 

 pressed by a crystal structure factor, 

 F hkl , for the beam perpendicular to 

 the (hkl) planes. It can be found 

 from 



+ Y 



Wm.m 



(322) 



Figure 8. Miller indices (hkl) for 

 some crystal planes illustrated for 

 cubic crystals. The Miller indices 

 are inversely proportional to the 

 distance from the origin to the inter- 

 sections with the crystal axes when 

 these distances are expressed in terms 

 of the lengths of the unit cell. The 

 proportionality constant is so chosen 

 that the Miller indices are the 

 smallest possible whole numbers. 



N 



F, 



hkl 



2 In e 27imu " 



+ kv„+lw„) 



(2) 



where 

 N = number of atoms in a unit cell of the crystal 

 n = a particular atom 

 f n = atomic structure factor defined below 

 (u n , v n , w n ) = coordinates of the nth. atom expressed as fractions of the 

 unit crystal lattice lengths 



The atomic structure factor/ is defined by 



r _ amplitude of the wave scattered by an atom 

 amplitude of the wave scattered by an electron 



In general, / depends both on the particular element (for example, Zn) 

 and on the angle between the incident and the scattered beam; tables of 

 /for various elements are available. 



