Structural resolution. The
Patterson function and the Patterson method
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The impossibility
of measuring the relative phases
among the diffracted beams, Φ(hkl), makes a direct calculation of the
electron density function
(Formula 1, below, which would provide us the atomic positions within
the unit cell) unfeasible. This was remedied only after 1934
when Arthur
Lindo Patterson
(19021966) introduced his brilliant idea, thereby obtaining the first solution
to the phase problem..., as he demonstrated in his article entitled A
Fourier Series Method for the Determination of the Components of
Interatomic Distances in Crystals, A.L. Patterson (1934) Phys. Rev.,
46, 372376. Patterson derived hus function (P(uvw), Formula 2 below) by intruoducing some modifications into the electron density function, so that the structure factors, represented by their amplitudes, [F(hkl)] and phases Φ(hkl), are replaced by the squared amplitudes whose values are proportional to the diffracted intensities (Formula 3 below). With these modifications, the Patterson function can be directly calculated from the experimental data obtained in the diffraction experiment. 
Formula 1. Electron density function  
Formula 2. The Patterson Function  
Formula 3. The relationship between structure factor amplitude and intensity. K is a scale factor, A is the absorption factor, L is the Lorentz factor, and p represents the polarization factor 
P2_{1}  ( x, y, z ) ( x, 1/2+y, z ) 
Patterson 
(
x, y, z )
( x, y, z
)
+
( x, y, z ) ( x, y, z)

To
solve a Patterson Function means to derive the atomic
positions
(coordinates) in the crystal (usually those of the atoms with
more
electrons) from the coordinates of the maxima of the Patterson map, and
generally it is not easy. At least it was not so until 1935,
when David
Harker (19061991), a "trainee" at that time, discovered an
"easy" way to solve the Patterson functions, a special circumstance
that Arthur
L. Patterson had not been aware of. What Harker "saw" is that certain locations (lines or planes) in the Patterson Function contain information about the interatomic vectors between equivalent atoms (atoms related by symmetry operations), and therefore to locate an interatomic vector (between equivalent atoms) one has not to look around in the whole Patterson space, but just in these special locations. 
Space group  Equivalent atomic positions in the crystal 
Patterson vectors (crystal coordinate differences) 
Pm 
( x, y, z ) ( x, y, z )  <0, 2y, 0> this a Harker line 
P2_{1}  ( x, y, z ) ( x, 1/2+y, z )  <2x, 1/2, 2z> this is a Harker plane 
#  u  v  w  Relative value of Patterson Function 
1  0  0  0  999 
2  0.50  0.50  0.45  342 
3  0  0.05  0.50  337 
4  0.51  0.45  0.95  137 
5  0.26  0.92  0.14  129 
Symmetry operations of the space group P2_{1}/c  
( x, y, z )  ( x, 1/2+y, 1/2z )  ( x, y, z )  ( x, 1/2y, 1/2+z ) 
Harker positions  ( x, 1/2+y, 1/2z )  ( x, y, z )  ( x, 1/2y, 1/2+z ) 
( x, y, z )  <2x, 1/2, 1/2+2z>  <2x, 2y, 2z>  <0, 1/2+2y, 1/2> 
Harker positions  <2x, 1/2, 1/2+2z>  <2x, 2y, 2z>  <0, 1/2+2y, 1/2> 
Symmetry operations of the space group P2_{1}/c  
( x, y, z )  ( x, 1/2+y, 1/2z )  ( x, y, z )  ( x, 1/2y, 1/2+z ) 
Symmetry of the corresponding Patterson map  
( x,
y, z ) 
( x, y, z )  ( x, y, z )  ( x, y, z ) 
