The coordinates u , v , w in direct space of the zone axis intersection of two families of lattice planes of Miller indices h 1 , k 1 , l 1 and h 2 , k 2 , l 2 , respectively, are proportional to the coordinates of the vector product of the reciprocal lattice vectors associated with these two families:.
An elementary proof that the reciprocal lattice of a face-centred lattice F is a body-centred lattice I and, reciprocally, is given in The Reciprocal Lattice Teaching Pamphlet No. It is called the hkl reflection. It is equivalent to Bragg's law , as can be seen in Fig. It can be seen from the figure that. This sphere is called the Ewald sphere. The notion of reciprocal vectors was introduced in vector analysis by J. Those reach only the lattice points at the vertices of the cubic structure but not the ones at the faces.
If the origin of the coordinate system is chosen to be at one of the vertices, these vectors point to the lattice points at the neighboured faces. Now we apply eqs. Your browser does not support all features of this website! Solid State Physics Crystal Geometry. The Reciprocal Lattice 1. Motivation 2. Introduction of the Reciprocal Lattice 2. Starting Point 2. Definition 2. Basis Representation of the Reciprocal Lattice Vectors 2.
Position vector for a non-reticular point black circle. Following with the argument given above, each motif in a repetitive distribution generates its own lattice, although all these lattices are identical red and blue.
Of the two families of equivalent lattices shown red and blue we can choose only one of them, on the understanding that it also represents the remaining equivalent ones. Note that the distance between the planes drawn on each lattice interplanar spacing is the same for the blue or red families. However, the family of red planes is separated from the family of blue planes by a distance that depends on the separation between the objects which produced the lattice.
This distance between the planes of different families can be called the geometric out-of-phase distance. Left: Family of reticular planes cutting the vertical axis of the cell in 2 parts and the horizontal axis in 1 part. These planes are parallel to the third reticular axis not shown in the figure.
Right: Family of reticular planes cutting the vertical axis of the cell in 3 parts and the horizontal axis in 1 part. The number of parts in which a family of planes cut the cell axes can be associated with a triplet of numbers that identify that family of planes. In the three previous figures, the number of cuts, and therefore the numerical triplets would be , and , respectively, according to the vertical, horizontal and perpendicular-to-the-figure axes.
In this figure, the numerical triplets for the planes drawn are , that is, the family of planes does not cut the a axis, but cuts the b and c axes in 2 identical parts, respectively.
0コメント