Solid State Physics
ØReciprocal in the lattice to sc, in the bcc, fcc, in the orthorhombic and hexagonal crystals
Hexagonal crystal (wurtzite)
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Diamond structure (Zinc blend)
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Construction of Reciprocal lattice
the reciprocal in the lattice of a real crystal is constructed in the as follows:
1. in the Choose a point as origin.
2.Draw in the normals to every in the set of planes from in the this origin.
3.Make the length of the normal in the equal to the 2π times the reciprocal in the of the inter planer in the spacing of that set of planes.
4. Place a point at the end of the normal which in the represents the crystal in the plane.
Reciprocal lattice is purely a mathematical concept,
“ reciprocal in the lattice of perfect of a single crystal is the infinites periodic 3 dimensional array of points in reciprocal in the free space whose in the distance is inversely and directly in proportional to the distance in the between the planes in a direct in the lattice”.The vectors in the reciprocal in the lattice have in the dimensions of [length]-1
The vectors in the real lattice have dimensions of [length].
Reciprocal in the space (also called “k-space”) is the space in which the Fourier in the transform of a spatial in the function is represented in the similarly to the frequency in the domain is the in the space in which the Fourier transform of time of a time- in the dependent in the function is represented. A Fourier in the transform in the takes us to form “real space” to reciprocal in the space.
If a, b and c are the basis in the vectors in the real lattice, then the basis in the vectors in the reciprocal in the lattice are defined as follows:
are the fundamental in the translational in the vectors in the reciprocal lattice. Since these are mutually in the in the perpendicular to each other so
are the fundamental in the translational in the vectors in the real in the lattice. Since these are parallel to
a* α 1/a b* α 1/b c* α 1/c
Advantages of Reciprocal lattice
to study in the with x-ray in the diffraction in the , we deal with planes of several in the slopes. But in the comparatively, it is difficult to visualize in the several in the slopes. So to overcome in the this problem the slope of a plane is determined by its normal. The normal in the being a line have in the only one in the dimension. Thus if we in the deal with a no. of planes then it becomes easier to take their slopes in terms of their normal.
nThe Bragg’s condition of x-ray in the diffraction can be in the expressed in a very simple way with the help of in the reciprocal in the lattice then the direct in the lattice.
Wave Mechanical in the behavior of an electron in the periodic c in the rystal is easily in the understood in terms of the reciprocal in the lattice.
n The factor 2π in the which appears in the basis in the vector for the reciprocal in the crystal in the lattice is placed there to make in the reciprocal space numerically and dimensionally the same as wave vector in the space. The factor 2π is in the emitted in the some cases of reciprocal space, and in those cases, a unit translation in reciprocal in the space can b in the equated in the with a unit change of reciprocal wavelength. The confusion between the two cases is not as in the alarming as it might at first in the appear, in the Because it will usually be apparent from the text when a location in the in R space is described as in the (1/2,0,0) is really the place in the otherwise referred to as (π,0,0).
For the real lattice, we have
Which in the describes the connection of pairs of points in the crystal in the lattice. Which have identical in the atomic in the environments? Similarly, in R- in the space there is a translational in the vector. Given as
Where h, k, l are integers.