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The reciprocal lattice of a reciprocal lattice is equivalent to the original direct lattice, because the defining equations are symmetrical with respect to the vectors in real and reciprocal space. Mathematically, direct and reciprocal lattice vectors represent covariant and contravariant vectors, respectively.
That is, (hkℓ) simply indicates a normal to the planes in the basis of the primitive reciprocal lattice vectors. Because the coordinates are integers, this normal is itself always a reciprocal lattice vector. The requirement of lowest terms means that it is the shortest reciprocal lattice vector in the given direction.
The third parameter specifying the reciprocal lattice vector is the angle formed by the X-ray beam and the plane containing and . The vertical coordinate ζ {\displaystyle \zeta } has a special significance, since all the reciprocal lattice points which have a constant ζ {\displaystyle \zeta } value lie in the plane normal to the rotation axis.
The first Brillouin zone is the locus of points in reciprocal space that are closer to the origin of the reciprocal lattice than they are to any other reciprocal lattice points (see the derivation of the Wigner–Seitz cell). Another definition is as the set of points in k-space that can be reached from the origin without crossing any Bragg plane.
Reciprocal space, containing the reciprocal lattice of a spatial lattice; Momentum space, or wavevector space, the vector space of possible values of momentum for a particle; k-space (magnetic resonance imaging) Another name for a compactly generated space in topology
By definition, the syntax (hkℓ) denotes a plane that intercepts the three points a 1 /h, a 2 /k, and a 3 /ℓ, or some multiple thereof. That is, the Miller indices are proportional to the inverses of the intercepts of the plane with the unit cell (in the basis of the lattice vectors).
Hence we identify = =, means that allowed scattering vectors = are those equal to reciprocal lattice vectors for a crystal in diffraction, and this is the meaning of the Laue equations. This fact is sometimes called the Laue condition .
The vectors are the reciprocal lattice vectors, and the discrete values of are determined by the boundary conditions of the lattice under consideration. Before doing the perturbation analysis, let us first consider the base case to which the perturbation is applied.