Reciprocal Lattice Vector-Solid State Physics-Lecture Slides, Slides of Solid State Physics

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In The Name of ALLAH

Scattering Wave Amplitude

• The Bragg derivation of diffraction condition gives neat

statement of constructive interference

• Constructive Interfernce waves scatter from the lattice points
• We need a deeper analysis on scattering intensity from the

basis of atoms

• Spatial distribution of electrons within each cell

Fourier Analysis

• We consider first a function n(x) with period a in the direction

x, we expand n(x) in a Fourier series of sines and cosines.

• Where the p’s are +ve integers and Cp and Sp are real constants
• We say that the 2πp/a is a point in the reciprocal lattice of the

crystal

Reciprocal Lattice

• A diffraction of x-rays occurs from various sets of parallel

plane having different orientations and different inter-planer

spacing

• It is difficult to visualize all such planes b/c of their two

dimensional nature

• P. P. Ewald simplified the problem by developing a new type of

lattice known as “reciprocal lattice”

Reciprocal Lattice Vector

• A reciprocal lattice vector ‘бhkl’ is defined as a

vector having magnitude equal to the reciprocal

of the inter-planar spacing ‘dhkl’ and direction

coinciding with normal to the (hkl) planes. Thus

we have

Continue

• Like a direct lattice, a reciprocal lattice also has a unit cell
• The structure is like parallelepiped
• The unit cell is formed by the shortest normals along the three dimensions
• e.g. along the normals to the planes (100), ( and (001)
• These normals produce reciprocal lattice vectors
• These are denoted by б 100 , б 010 , б 001 and known as fundamental reciprocal lattice vectors

Continue

• a, b and c be the primitive translation vectors of

the direct lattice

• The base of the unit cell is b and c and its height

is equal to d 100. Then the volume will be

• In vector form

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 The fundamental reciprocal lattice vectors б 100 ,

б 010 and б 001 are denoted by a, b and c*

• a* = б 100 =
• b* = б 010
• c* = б 001

Continue

• This appears that a, b and c* are parallel to a, b and c.

however, this is not always true. In non-cubic or monoclinic

crystal system, it is different

• It means that a* is the reciprocal of acosθ
• In S.S.P, the there is a relation b/w direct lattice and reciprocal

lattice

• a.a = b.b = c*.c = 2π

Conclusion

• It is now concluded that every crystal structure is associated with two important lattices
• Direct lattice and reciprocal lattice
• The fundamental translation vector of crystal lattice and reciprocal lattice have dimensions of
• [length] and [length]-1^ respectively
• The volume of the reciprocal lattice vector is inversely proportional to the volume of the direct lattice