X-Ray Crystallography of Biological Macromolecules 



whose distance from the origin corresponds to the distance between 

 two or more peaks in the electron density projection. 



The matter becomes more complex if a crystal structure in three 

 dimensions is considered. Just as the results of a three-dimensional 

 Fourier summation with the amplitudes as coefficients of the terms 

 can be expressed as a series of contour maps showing the distribution 

 of electron density in sections through the unit cell, the Patterson 

 function when calculated in three dimensions leads to a series of 

 contour maps showing the density distribution within a cloud of 

 ' something ' in the unit cell. Patterson called this * something ' the 

 weighted density distribution about any point in the crystal structure. 

 What does this mean ? By analogy with the one-dimensional example 

 just given the density of this ' something ' at any point of the unit cell 

 having the coordinates u, v, w, should be equal to the integral of the 

 product of p at any point x, y, z in the crystal structure and p at any 

 other point having the coordinates x + u, y + v, z + w. 



1 a b c 



P(w, v, w) = y \ \ \ [ P (x, y, z)] [p (x+u), (v+v), (z+w)] du. dv. dw 



o o o 



Supposing there were two atoms, one at x, y, z and the other at 

 x -f u, y + v, z + w. The electron density at both these points will 

 be large and hence their product 



[p (x, y, z)] [p (x + u), ( y -f v), (z + w)] 



will also be large ; the Patterson function will therefore show a peak 

 at the point u, v, w. The line joining this peak to the origin (the centre 

 of the unit cell, say) will correspond in length and direction to the line 

 joining the two atoms in the crystal structure. Since a quantity having 

 both length and direction is known as a vector, the density of ' some- 

 thing ' in the Patterson synthesis is now generally called vector density. 

 In a sense this name is misleading, because the peak in a Patterson 

 synthesis is characterized not only by its position with respect to the 

 origin, defining the length and direction of the vector, but also by the 

 magnitude of the density at the centre of the peak, usually called its 

 height. The height of a vector peak depends on two factors : one is 

 the number of pairs of atoms in the unit cell which are joined by corre- 

 sponding lines of the same length and direction, and the other the 

 atomic number of the atoms concerned. 



Consider now the relationship between the distribution of ' ideal ' 

 atoms in a crystal structure and the distribution of ' ideal ' vectors in 

 the corresponding Patterson synthesis, which follows from the abstract 

 reasoning just given. For this purpose it will be convenient to think 

 of point atoms which scatter x-rays only at their centres and give rise 



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