Example 2. Determining the space group

The reflection conditions (systematic absences) and the crystal system generally do not uniquely determine the space group but, instead, reveal a small number of possible space groups. The Patterson map peaks can be used to determine the true space group from the list of possible space groups.


Example:

  1. molecule (formula unit) contains one "heavy" atom
  2. crystal system is monoclinic (b-axis unique)
  3. no reflection conditions
  4. unit cell contains four molecules (Z = 4)

From b. and c.: the possible space groups are P2 (#3), Pm (#6), and P2/m (#10).
From a. and d.: there will be four "heavy" atoms in the unit cell, either as one unique "heavy" atom in space group P2/m (#10) or as two unique "heavy" atoms in space group P2 (#3) or in space group Pm (#6).


  1. The Patterson map peaks (coordinates, multiplicities) are:
  2.                       Observed:      Expected for P2/m (#10):
      1.    0.00   0.00   0.00   7.          0,     0,     0   4.
      2.    0.20   0.00   0.30   2.         2X,     0,    2Z   2.
      3.    0.00   0.40   0.00   2.          0,    2Y,     0   2.
      4.    0.20   0.40   0.30   1.         2X,    2Y,    2Z   1.
             etc.
    

    This peak list shows (with the same multiplicities) the Harker line peak 0,V,0 for a mirror plane perpendicular to the b axis and the Harker plane peak U,0,W for a 2-fold axis parallel to the b axis and (with half the above multiplicities) a general peak U,V,W. This pattern is consistent with space group P2/m (#10) and the "heavy" atom at 0.10, 0.20, 0.15.

     
  3. The Patterson map peaks (coordinates and multiplicities) are:
  4.                       Observed:         Expected for Pm (#6):
      1.    0.00   0.00   0.00   7.          0,     0,     0   2.
      2.    0.00   0.40   0.00   1.          0,    2Y,     0   1.
      3.    0.00   0.30   0.00   1.
      4.    0.25   0.35   0.35   1.
      5.    0.25   0.05   0.35   1.
             etc.
    

    This peak list shows (all with the same multiplicities) two Harker line peaks 0,V,0 for a mirror plane perpendicular to the b axis and two general peaks U,V,W. Note the absence of Harker plane peaks. This pattern is consistent with space group Pm (#6) and two independent "heavy" atoms at 0.10, 0.20, 0.15 and 0.85, 0.85, 0.80.

     
  5. The Patterson map peaks (coordinates and multiplicities) are:
  6.                       Observed:         Expected for P2 (#3):
      1.    0.00   0.00   0.00   7.          0,     0,     0   2.
      2.    0.20   0.00   0.30   1.         2X,     0,    2Z   1.
      3.    0.30   0.00   0.40   1.
      4.    0.25   0.35   0.35   1.
      5.    0.05   0.35   0.05   1.
             etc.
    

    This peak list shows (all with the same multiplicities) two Harker plane peaks U,0,W for a 2-fold axis parallel to the b axis and two general peaks U,V,W. Note the absence of Harker line peaks. This pattern is consistent with space group P2 (#3) and two independent "heavy" atoms at 0.10, 0.20, 0.15 and 0.85, 0.85, 0.80.


The above demonstrate that the presence or absence of special (Harker line and Harker plane) peaks in the Patterson map can help to indicate the space group.