Lattice plane

As a grid or network level is referred to in the crystallography a plane which is defined by points of the lattice. Your position in space is described by the Miller indices (hkl ).

Description

A crystal lattice can be described as integer linear combination of the basis vectors, and describe (in the direction of the crystal axes ). A lattice plane is defined by its intersections with the crystal axes. Miller indices (hkl ) represent the plane passing through the three points, and. So the crystal axes of each crystal system cut the levels just to the reciprocals of the individual indices. An index of zero denotes an intersection point at infinity, that is, the corresponding base vector is parallel to the plane.

The reciprocal lattice vector is perpendicular to the through the Miller indices (hkl ) lattice plane defined. The vectors, and are the basis vectors of the reciprocal lattice.

A lattice plane group consists of all parallel lattice planes with the interplanar spacing. This can be calculated from the Miller indices and the reciprocal lattice vectors:

For crystal systems with orthogonal axes, ie orthorhombic and higher symmetry lattice ( tetragonal and cubic systems ), the following formula is valid (, , are the lattice constants ):

This simplifies, for example, for cubic systems by equating further:

Derivations

A layer is clearly not defined by three points lying on a straight line. Here these are the intersections with the crystal axes: , and.

Let the points on the plane described by the parametric form ( by start point and two direction vectors lying in the plane and are not collinear ). If two points in the plane, its connection vector also lies in the plane. Here over the direction vectors can be constructed (and). As a start point choose any point lying in the plane (here):

By forming the scalar product between the reciprocal lattice vector and taking advantage of the relation, we obtain:

For a normal vector of the plane of the scalar products are the direction vectors equal to zero (and). Is exactly the case on, so this is perpendicular to the plane ( hkl).

Due to the lattice point at the origin is parallel to the plane under consideration by a level with the indices (hkl ). Their distance is the projection of a vector connecting both levels () on the normalized normal vector (). This results together with the above account of the interplanar spacing:

In the denominator of both the contact length of reciprocal lattice vectors ( ), and the projections of the grating vectors of each other (co ) in the amount of formation. The latter are in non- orthogonal crystal systems equal to zero:

An orthorhombic crystal system is a rectangular crystal system with three 90 ° angles, but without the same long axis. The lattice vectors for this option are expressed respect to the canonical unit basis:

And the corresponding reciprocal lattice vectors are also orthogonal ( to ):

Put this one in the above general formula for the interplanar spacing:

The cubic crystal system is also rectangular, but in addition, the lattice constants of each crystal axis with respect to the same and the formula simplifies to continue: