Physics 489 September 8, 2015
Summary of Reciprocal lattices:


3
(1) Definition and properties: The reciprocal lattice is the lattice G = ∑ i=1 mi bi formed by the
set of primitive vectors:
 2π a × a
 2π a × a
 2π a × a
2
3
3
1
b1 =    , b2 =    , b3 =  1 2 .
a1 ⋅ a2 × a3
a1 ⋅ a2 × a3
a1 ⋅ a2 × a3
[1]
The reciprocal lattice forms a "dual lattice", that is related to the Bravais lattice (not including the
basis). The reciprocal lattice by construction is also a Bravais lattice, however notice that the
reciprocal space has dimensions measured in wave-numbers (1/m units). In contrast I will
sometimes refer to the normal Bravais lattice for the actual crystal in real space (m units) as the
direct lattice.
From the above we can readily establish that the reciprocal lattice:
 
a) Is orthogonal to the direct lattice in a particular way: ai ⋅ b j = 2πδ ij . Or, considering the
 
complete set of lattice vectors we have, G ⋅ R = 2πM , with M an integer.
 
b) As a consequence of the orthogonality results just stated, we also must have exp[iG ⋅ R] = 1

for all points of the direct lattice (positions R ).
 
c) More generally we can see that the wave-fronts of each plane wave exp[iG ⋅ r ] constructed
!
from a reciprocal lattice wave-vector G coincide
with a set of lattice planes, if we define the
 
wave-fronts to be the planes where exp[iG ⋅ r ] = 1 . (Like the lattice planes, these wave-fronts
will each coincide with an equivalent density of direct-lattice points as long as m1, m2, and m3
have no common divisors, otherwise the wave-fronts also include additional parallel planes
between the lattice planes.)
(2) Fourier
  transformation: As might be guessed based on the !properties of the plane waves
exp[iG ⋅ r ] described above, the set of reciprocal lattice vectors G is the set of wavevectors used

to Fourier transform the crystal. For example if n(r ) is periodic with a particular Bravais lattice

symmetry [ n(r ) could be the electron density, or electric potential of a crystal, etc.], then one

can always express n(r ) as
 

 exp[iG ⋅ r ] ,
n(r ) = ∑
n
G

[2]
G
for a particular set of values nG which are the Fourier components. [2] is an alternative way to

define the reciprocal lattice. Also note that the sum over G is equivalent to a triple sum over the
3 integer indices,
∑

G
=
∞
∞
∞
∑ ∑ ∑
m1 =−∞ m2 =−∞ m3 =−∞
.
(3) Some other properties that can be shown to follow:
a) The reciprocal of the reciprocal lattice is the same as the direct lattice.
b) The reiprocal lattice for a given direct lattice is also uniquely associated with a given Bravais
lattice.
 
c) Because of the association with lattice planes [1.c above], the plane waves exp[iG ⋅ r ] have

the same translational symmetry as the direct lattice. This is also clearly the reason that the G vectors are the set used for Fourier transformation of the crystal.
(4) Association with lattice planes:
 
• By construction for the plane waves, the wavevector exp[iG ⋅ r ] is perpendicular to the
wavefronts, hence we find that each set of lattice planes has associated with it a perpendicular
reciprocal lattice vector G .
• Miller indices: we can also show that, for the (h, k, l) set of planes the associated wavevector




is G = hb1 + kb2 + lb3 .
• Another way to express this correspondence, we can show that if d is the spacing of a set of
Bragg planes, and nˆ is the perpendicular unit vector, then (2πnˆ d ) is the G -vector. This means,

d = 2π / Ghkl .
(5) Notation, and other comments:
(i) Round and square brackets etc.:
• By convention the round-bracket notation (hkl) denotes a set of parallel planes, or sometimes a
single plane.
• Also as is sometimes used, {hkl} denotes a larger set of planes connected by symmetry: for
example if cubic, {100} includes (100), (010), and (001).




• Finally note that [hkl] refers to a direction in direct space, parallel to R = ha1 + ka2 + la3 . This

is not the same as the direction of the G -vector with reciprocal-space indices h, k, l.
(ii) The cubic lattices FCC and BCC are always indexed according to the conventional cubic cell,
not their primitive cells. This means that reciprocal space is also mapped as simple cubic (which
makes life simpler), although as a consequence not all of the reciprocal lattice points are "real"
lattice points.
(iii) The hexagonal lattice is also common in materials systems, and is usually indexed in a
slightly different way, with a redundant set of four indices instead of three, based on a
conventional cell having a hexagon base. I won't cover this notation, since it isn't a necessity for
our course.