Crystal lattices and reciprocal lattice

Università Cattolica del Sacro Cuore A.A. 2005-06

Crystal lattices
and

reciprocal lattice

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Crystal Lattices

Solids: Amorphous
Crystalline: ions are arranged in a periodic array on a microscopic scale.

BravRra=is lattices consist of all lattice points generated by:

n1ar1 + n2ar2 + n3ar3

anr11 , nar22 , nar33 are positive or negative integers the same plane
, , are 3D vectors (primitive) not in

2D example

Pr = ar1 + 2ar2
Qr = −ar1 + ar2

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Examples 3D: Simple Cubic

The most familiar of 3D Bravais lattice is the simple cubic. It can be spanned by three
mutually perpendicular primitive vectors of equal length.

SC

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Examples in 3D: BBC

Body Centered Cubic Lattices (BCC)
are formed by adding to simple cubic
lattice (A) an additional point at the center
of each cube (B).
Points A and B are equivalent: both can
be taken from the origin of the simple
cubic lattice.

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If the original simple cubic lattice is generated by primitive vectors ax,ay,az (a is
called lattice parameter) a set of primitive vectors for a BBC coud be

aarr12 = axˆ
= ayˆ

ar3 = a ( xˆ + yˆ + zˆ)
2

A more symmetric set is

ar1 = a ( yˆ + zˆ − xˆ)
2

ar2 = a ( zˆ + xˆ − yˆ )
2

ar3 = a ( xˆ + yˆ − zˆ)
2

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Examples in 3D: FCC

A simple cubic lattice with a point in the center of
each square face. All points are equivalent and can
be taken as the origin of the simple cubic lattice.
A symmetric set of primitive vectors for FCC lattice is:

ar1 = a ( yˆ + zˆ)
2

ar2 = a ( zˆ + xˆ)
2

ar3 = a ( xˆ + yˆ )
2

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Primitive cell

Primitive Cell: a volume which, when traslated, fills exactly the space (no overlaps, no
voids) containing one lattice point. Not unique!

Fig. 4.10

Several possible choices of primitive cell for 2D Bravais lattice

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Unit Cell

The unit cell (non primitive unit cell) is a conventional cell which contains more then
one point in the Bravais lattice.

Fig. 4.12 e 4.13

Primitive (parallelogram) and conventional Primitive and conventional unit cell for
unit (large cube) cell for FCC Bravais lattice. BCC Bravais lattice.

The conventional unit cell is generally chosen to be bigger than the primitive cell.

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Wigner-Seitz Cell

The most common choice for a primitive cell is the Wigner-Seitz cell. The Wigner-
Seitz cell about a lattice point is the region of space that is closer to that point than to
any other lattice. It is obtained by bisecting with a plane each line connecting a point
with its neighbors and taking the smallest polyhedron containing the point.

BCC: truncated octahedron FCC: rhombic dodecahedron
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Each lattice point may consists of a single atom (monoatomic lattice) or 2 set of
atoms (lattice with a basis).

Also monoatomic non-primitive Bravais lattices are often described as a lattice
with a basis (to emphasize a given symmetry):

BCC: simple cubic + (0,0,0); a/2(1,1,1)
FCC: simple cubic + (0,0,0); a/2(0,1,1); a/2(1,1,0); a/2(1,0,1)

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Diamond Crystal Structure Fig. 2.12 luth
For the most usual semiconductors the cell
consists of two atoms, one at the position (0,0,0) Reticoli
and the other¼(1,1,1). The primitive cell contains a
basis of two atoms. Two FCC compenetrating
lattices
Elements a
C 3,57 Å
Si 5,43 Å
Ge 5,66 Å

In III-V and II-VI the two atoms are different
species: Zincblende Crystal Structure

GaAs 5.653 Å
ZnSe 5.666 Å

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The reciprocal lattice is defined by all points generated by the primitive vectors b1, b2,
b3 such that

brj ⋅ari = 2πδi,j Reciprocal Lattice

Three of such vectors are ( V = ar1 ⋅ (ar2 × ar3 ) )

br1 = 2π (ar2 × ar3) Reciprocal Lattice GeGRrrriG⋅==rRr⋅nhR11r=abrr112=+π+1(hnn221barrh221+++hnn33b2rar3h3 w2wi+itthhnn3h11h,,3nh)22,,nh33 integers
Direct Lattice integers
V

br2 = 2π (ar3 × ar1)

V

br3 = 2π (ar1 × ar2 )

V

This properties is usually taken as defining reciprocal lattice points (G).

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Reciprocal lattice vectors

e = eiGr (rr+Rr) iGrrr

=>The G represent all wave vectors that yield plane waves with the periodicity of the real
space lattice.

⇒the reciprocal lattice of a reciprocal lattice is the direct lattice.
⇒Direct and reciprocal lattices give complete, alternative and equivalent geometrical
description of the crystalline solid, either in a real or Fourier space.

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Examples

Relevant examples

Direct lattice: SC (with a cubic primitive cell of side a);
Reciprocal lattice: SC (with a cubic primitive cell of side 2π/a).

Direct lattice: FCC (conventional cubic cell a); Reciprocal lattice: BCC (conventional
cubic cell 4π/ a).

Direct lattice: BCC (conventional cubic cell a); Reciprocal lattice: FCC (conventional
cubic cell 4π/ a).

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1st Brillouin Zone

The first Brillouin zone is the volume obtained in the same way as the Wigner-Seitz
cell, but in the reciprocal space.

=> the first Brillouin zone of the FCC lattice is just the BCC Wigner-Seitz cell

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Miller indices of lattice planes

The correspondence between reciprocal lattice vectors and families of lattice planes
provides a convenient way to specify the orientation of a lattice plane.

Usually the orientation of the plane is described by giving a vector normal to the plane.
Since we know there are reciprocal lattice vectors normal to any family of lattice planes
it is natural to use a reciprocal lattice vector to represent the normal.
To make the choice unique one uses the shortest.

The Miller indices of a lattice plane are the coordinates of the shortest reciprocal lattice
vector normal to the plane.

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Theorem Lattice Planes

Any reciprocal lattice vector is normal to a plane of the direct lattice; its components are

the Miller indices of the plane

Proof: Gr = hbr1 + kbr2 + lbr3 is normal to a plane with intercepts ma1, na2, pa3 if
A vector

{Gr ⋅ (mar1 − nar2 ) = 0
{Gr ⋅ (mar1 − par3 ) = 0
{Gr ⋅ (nar2 − par3 ) = 0

Therefore we need to satisfy

{mh = nk
{mh = pl
{nk = pl

For example h=1/m,k=1/n,l=1/p => m,n,l are the Miller indices except for an integer
common factor.

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Miller indices

Miller indices are three integers representing a family of real lattice planes by their normal
axis:

1) find the intercepts of the plane with axis along a1,a2, a3 in terms of lattice constants,
2) find the smallest integers which are in the same ratio as their reciprocals.

Example: Intercepts: 3a1,2a2,2a3
Reciprocals: 1/3 , ½ , ½
.
Miller indices: 2, 3, 3

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Lattice planes

Since the reciprocal lattice of a simple cubic lattice is again a simple cubic lattice and
the Miller indices are the coordinates of a vector normal to the planes, their use is very
simple in lattices with cubic symmetry.

Lattice planes are usually specified by giving their Miller indices in parentheses: (h,k,l)

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