Reciprocal lattice ( $Λ^{*}$ )

The reciprocal lattice is

Λ^{*} = {q : q \cdot l \in 2 π Z for every l \in Λ} .

(1)

Ω = a_{1} \cdot (a_{2} \times a_{3})

, its primitive vectors are

b_{1} = 2 π \frac{a _{2} \times a _{3}}{Ω}, b_{2} = 2 π \frac{a _{3} \times a _{1}}{Ω}, b_{3} = 2 π \frac{a _{1} \times a _{2}}{Ω},

(2)

and satisfy

a_{i} \cdot b_{j} = 2 π δ_{ij}

Table of contents

Reciprocal lattice rod

 0  0

An elongated reciprocal-space reflection caused by finite extent in the corresponding real-space direction. A foil of thickness

t

has reflection breadth of order

1/ t

along its normal, increasing the range over which a reflection can intersect an Ewald sphere. The exact profile depends on the specimen shape and scattering model.

Lattice-periodic plane wave ( $e^{i k \cdot R} = 1$ )

 0  0

A plane wave whose phase is unchanged by every direct lattice translation:

e^{i k \cdot R} = 1

for all

R \in Λ

. Its wavevector is a reciprocal lattice vector. In one dimension

ka = 2 πm

Reciprocal lattice vector ( $G \cdot R \in 2 π Z$ )

 0  0

A reciprocal lattice vector

G

has the displayed property for every direct lattice vector

R

. Thus

e^{i G \cdot R} = 1

. Fourier components of a lattice-periodic potential carry these wavevectors and couple plane-wave electronic states whose momenta differ by

ℏ G

. This is the origin of the Bragg-plane mixing in a nearly-free electron model.

Bragg plane ( $k \cdot G = G^{2} /2$ )

 1  0

For free-electron dispersion proportional to

k^{2}

, states at

k

and

k - G

are degenerate on this plane, obtained by equating their squared norms. A nonzero periodic-potential Fourier component can mix them and open an avoided crossing in the nearly-free electron model. First-zone boundaries are selected planes of this kind.

Reciprocal lattice change in a structural phase transition

 0  0

The reciprocal lattice of a body-centered cubic lattice is face-centred cubic, and that of a face-centered cubic lattice is body-centred cubic. In both cases its conventional cubic side is

4 π / a

if the direct conventional side is

a

. Thus a direct BCC-to-FCC phase transition reverses the reciprocal lattice types. The reciprocal primitive cell volume is

(2 π)^{3} /Ω

, so thermal expansion contracts reciprocal spacings, while a discontinuous change in direct primitive unit cell volume gives an inverse change in reciprocal cell volume.

A first-order structural transition need not continuously deform one perfect reciprocal array into another. Phase coexistence, new orientation variants, domains and loss of single-crystal alignment can produce two sets of diffraction peaks or many differently oriented sets. The two endpoint lattice types alone do not specify the transformation path or orientation relationship.