Solid State Physics
5th Semester B.Sc Physics
UNIT 1 –CRYSTAL PHYSICS
LECTURE 13
Reciprocal Lattice
Dr. Ambika Pradhan
Dept. Of Physics
SRM University Sikkim
1
Reciprocal Lattice
• Every crystal has two lattices associated with it-crystal lattice
(or direct space lattice) and reciprocal lattice.
• The concept of reciprocal lattice was devised for the purpose
of tabulating two important properties of crystal planes; their
slope and interplanar spacing.
• Each set of parallel planes in a direct lattice can be
represented by a normal to these planes having length equal
to the reciprocal of the interplanar spacing.
• The normal are drawn with reference to any arbitrary origin
and points are marked at their ends.
Reciprocal Lattice
• These points form a regular arrangement which is called
reciprocal lattice.
• Thus each point in a reciprocal lattice is a representative point
of a particular parallel set of planes.
• Reciprocal lattice vector is a vector whose magnitude is equal
1
to the reciprocal of the inter planar spacing i.e., and
𝑑ℎ𝑙𝑘
direction is parallel to the normal to (h k l).
• Thus reciprocal lattice is named because the length assigned
to each normal representing the reciprocal lattice is
proportional to the reciprocal of the interplanar spacing of
that plane in ordinary space.
Primitive translation vectors of reciprocal lattice
• Let 𝑎, 𝑏, 𝑐Ԧ be the primitive translation
vectors of a direct space lattice for a crystal forming a
primitive unit cell.
• The volume of this unit cell is given by
are known as reciprocal lattice vectors or primitive translation vectors of
reciprocal lattice.
Reciprocal lattice vectors are orthogonal to two axis
vectors
• Each of the reciprocal lattice vectors is orthogonal to two of
the axis vectors of a direct space lattice of the crystal.
Reciprocal lattice vectors are orthogonal to two axis vectors
Ԧ 𝐵 and 𝐶 be the primitive translation vectors of the
• Let 𝐴,
reciprocal lattice (also known as reciprocal axes).
• The reciprocal lattice points or reciprocal lattice vectors can
be constructed given by
where hkl are integers and define the co-ordinates of the point
in the reciprocal space.
Reciprocal lattice vectors are orthogonal to two axis vectors
• The crystal lattice is a lattice in real or ordinary space whereas
reciprocal lattice is a lattice in the reciprocal space or
associated k space or Fourier space. The wave vector k is
always drawn in Fourier space.
• Vectors in the crystal lattice have dimensions of length [L1]
and vectors in the reciprocal lattice have dimensions of
1/Length [L-1]
Construction of reciprocal lattice
A reciprocal lattice to direct lattice is constructed by the
following method:
• Take origin at some arbitrary point.
• From the origin draw normals to every set of parallel
planes of direct lattice. The direction of the normal
specifies the orientation of the plane and thus the two
dimensional plane is represented by the normal which has
only one dimension.
1
• Take length of each normal equal to times the reciprocal
𝑑
of the interplanar spacing for the corresponding set of
planes.
Construction of reciprocal lattice
• The terminal points of these normals form the reciprocal
lattice because the distances in these lattices are the
reciprocal to those in the direct crystal lattice.
Properties of reciprocal lattice.
• The direct lattice is the reciprocal lattice to its own reciprocal
lattice.
• Simple cubic lattice is self reciprocal whereas bcc and fcc
lattices are reciprocal of each other.
• Each point in a reciprocal lattice corresponds to a particular
set of parallel planes of the direct lattice.
Properties of reciprocal lattice.
• The distance of a reciprocal lattice point from an arbitrarily
fixed origin is inversely proportional to the interplanar spacing
of the corresponding parallel planes of the direct lattice.
• The unit cell of the reciprocal lattice is not necessarily a
parallelopiped.
• The volume of a unit cell of the reciprocal lattice is inversely
proportional to the volume of the corresponding unit cell of the
direct lattice.
Reciprocal lattice to sc lattice.
• Let the primitive translation vectors of a simple cubic
cell be 𝑎, 𝑏, 𝑐.
Ԧ We can write the vectors as
taking the vectors 𝑎, 𝑏, 𝑐Ԧ each of magnitude 𝑎 along the
co-ordinate axes X, Y and Z, 𝑥,
ො 𝑦ෝ and 𝑧Ƹ being unit vectors in
these directions respectively
• The volume of the cubic cell
Reciprocal lattice to sc lattice
• The primitive translation vectors of the reciprocal sc lattice
are given by
• Thus we find that the reciprocal lattice to the sc direct
space lattice is itself a simple cubic Lattice with lattice .
2𝜋
constant .
𝑎
Reciprocal lattice to bcc lattice
• The primitive translation vectors of
body centred cubic (bcc) lattice in
terms of the cube edge a the primitive
translation vectors are
• The volume of the primitive cell is
given by
Reciprocal lattice to bcc lattice
• Now
• The primitive translation vectors of the reciprocal bcc lattice are
given by
Thus we find that the reciprocal bcc lattice vectors are just the
2𝜋
primitive vectors of fcc lattice with lattice constant .
𝑎
Reciprocal lattice to fcc lattice
• The primitive translation
vectors of the face centred
cubic (fcc) lattice 𝑎, 𝑏, 𝑐Ԧ .
• In terms of the cube edge a
the primitive translation
vectors are:
RECIPROCAL LATTICE TO FCC LATTICE
• The volume of the primitive cell is given by
RECIPROCAL LATTICE TO FCC LATTICE
• The primitive translation vectors of the reciprocal fcc
lattice are given by
• Thus we find that reciprocal fcc lattice are just the
2𝜋
primitive vectors of bcc lattice with lattice constant .
𝑎
RECIPROCAL LATTICE
Show that every reciprocal lattice vector is perpendicular to a
direct lattice plane
• Let the reciprocal space lattice be given by
• Let (hkl) be the plane of the direct lattice, then
•
RECIPROCAL LATTICE
•
RECIPROCAL LATTICE
Thus the vector 𝐺Ԧ is perpendicular to two linearly independent
vectors 𝐴𝐶 and 𝐴𝐵which lie in the (hkl) plane. This means that
every reciprocal lattice vector 𝐺Ԧ is perpendicular to a direct lattice
plane (hkl)
