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Centric and acentric reflections: Difference between revisions
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== Real space == | == Real space == | ||
Let's first look at real space. A special position results if there exists one or more symmetry operators, other than the trivial operator {x,y,z}, which map this position upon itself. As a example: take spacegroup P2 with its symmetry operators {x,y,z} and {-x,y,-z}. Now consider any point with x=0 and z=0, and some value of y. Obviously this point, when transformed with -x,y,-z , yields 0,y,0 - just the same point! Thus this is a special position. | Let's first look at real space. A special position results if there exists one or more symmetry operators, other than the trivial operator {x,y,z}, which map this position upon itself. As a example: take spacegroup P2 with its symmetry operators {x,y,z} and {-x,y,-z}. Now consider any point with x=0 and z=0, and some value of y. Obviously this point, when transformed with -x,y,-z , yields 0,y,0 - just the same point! Thus this is a special position. Generally, positions on n-fold symmetry axes are special. | ||
Mathematically, to find special positions we have to solve the Eigenproblem A v = v where A is the symmetry operator (expressed as rotation matrix and translation vector), and v, the Eigenvector, represents the special position(s). For a given space group, we need to check all symmetry operators. | Mathematically, to find special positions we have to solve the Eigenproblem A v = v where A is the symmetry operator (expressed as rotation matrix and translation vector), and v, the Eigenvector, represents the special position(s). For a given space group, we need to check all symmetry operators. | ||
An atom at a special position usually has an occupancy of 0.5. However, it may happen that more than one symmetry operator maps the special position upon itself; in that case the occupancy is 1/(number of symmetry operators mapping point onto itself). | An atom at a special position usually has (at most) an occupancy of 0.5. However, it may happen that more than one symmetry operator maps the special position upon itself; in that case the occupancy is 1/(number of symmetry operators mapping point onto itself). Ths, a point on a n-fold rotation axis has (maximum) occupancy of 1/n. | ||
In space group P2_1, there are no special positions - the Eigenproblem has no solution. | In space group P2_1, there are no special positions - the Eigenproblem has no solution. | ||