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Reference: IUCr Computing Commission Newsletter No. 2, July 2003

Each cut plane of an asymmetric unit is defined by a condition of the form

  h*x+k*y+l*z+c>=0

h,k,l are Miller indices and define the normal vector of the cut plane, c is a constant which determines the distance from the origin. x,y,z are fractional coordinates in direct space. The expression h*x+k*y+l*z+c is

exactly zero for points in the cut plane.
greater than zero for points inside the asymmetric unit.
less than zero for points outside the asymmetric unit.

If all points that are exactly in a cut plane are not inside the asymmetric unit, the condition changes from h*x+k*y+l*z+c>=0 to h*x+k*y+l*z+c>0.

To enhance readability the asymmetric unit conditions (shown under the pictures in the gallery) are simplified by omitting terms with zeros (e.g. 0*x) and unit factors (e.g. x instead of 1*x). The constant term c is moved to the right-hand side. For example:

  x>=0
  y<=1/4
  z<1
  x-y<=1/2

A point x,y,z is inside the asymmetric unit only if all conditions are simultaneously true.

Often a face or edge on the surface of the asymmetric unit is only partially inside. The dividing lines are defined by face- or edge-specific sub-conditions. For example:

  y<=1/4 [z<=1/2]

The first condition defines the face as before. The second condition in square brackets only applies if y=1/4. This notation is recursive. For example:

  y<=1/4 [z<=1/2 [x<=1/4]]

The third condition only applies if y=1/4 and z=1/2.

Some asymmetric units require the combination of conditions with the boolean operators and or or. For example:

  y>=0 [x<=0 | x>=1/4]
  y<=1/4 [z>=1/8 & z<=5/8]

In words:

If y is exactly zero, a point is inside the asymmetric unit only if x is less than or equal to zero or greater than or equal to 1/4.
If y is exactly 1/4, a point is inside the asymmetric unit only if z is greater than or equal to 1/8 and at the same time less than or equal to 5/8.

The boolean operators may occur at any level in the recursive hierarchy defined by nested square brackets. An example is the asymmetric unit of space group I 41/a (No. 88).

Gallery of direct-space asymmetric units

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