SOLID STATE PHYSICS

By definition, solid state is that particular aggregation form of matter characterized by strong
interaction forces between constituent particles (atoms, ions, or molecules). As a result, a
solid state material has an independent geometric form (in contrast to liquids, which take the
form of the container) and an invariant volume (in contrast to gases/vapors) in given
temperature and pressure conditions. As temperature increases, a solid state material can
evolve into another aggregation form (liquid or gas). Solid state physics studies the structural,
mechanical, thermodynamic, electrical, magnetic, and optical properties of (poly-)crystalline
and non-crystalline solids (for example, amorphous materials, such as glass).

Crystal structure

The properties of crystalline solids are determined by the symmetry of the crystalline lattice,
because both electronic and phononic systems, which determine, respectively, the electric/
magnetic and thermal response of solids, are very sensitive to the regular atomic order of
materials and to any (local or non-local) perturbation of it. The crystalline structure can be
revealed by the macroscopic form of natural or artificially-grown crystals (see the pictures
below), or can be inferred from the resulting debris after cleaving a crystalline material.

(a) (b) (c)

(d) (e) (f)

Crystals of (a) baryt, (b) salt, (c) hexagonal beryl, (d) trigonal quartz, (e) monoclinic gypsum,
and apatite (f)
 Crystal Structure 2

Non-crystalline materials have no long-range order, but at least their optical properties
are similar to that of crystalline materials because the wavelength of the incident photons (of
the order of 1 m) is much larger than the lattice constant of crystals and so, photons “see” an
effective homogeneous medium. Other properties of non-crystalline materials are derived
based on concepts proper to crystalline solids and, therefore, the crystal structure is extremely
important in understanding the properties of solid state materials.
The macroscopic, perfect crystal is formed by adding identical building blocks (unit
cells) consisting of atoms or groups of atoms. A unit cell is the smallest component of the
crystal that, when stacked together with pure translational repetition, reproduces the whole
crystal. The periodicity of the crystalline structure that results in this way is confirmed by X-
ray diffraction experiments. The figures below illustrate crystals in which the basis consists of
(a) one atom and (b) two atoms.

(a) (b)

The group of atoms or molecules that forms, by infinite repetition, the macroscopic
crystal is called basis. The basis is positioned in a set of mathematical/abstract points that
form the lattice (also called Bravais lattice). So, a crystal is a combination of a basis and a
lattice. Although usually the basis consists of only few atoms, it can also contain complex
organic or inorganic molecules (for example, proteins) of hundreds and even thousands of
atoms.
In three dimensions, all Bravais lattice points

Rmnp  ma1  na 2  pa3 (1)

can be obtained as superpositions of integral multiples of three non-coplanar primitive

translation vectors a1 , a 2 and a 3 (m, n, and p are arbitrary integers). A basis consisting of s

atoms is then defined by the set of vectors r j  m j a1  n j a 2  p j a3 , j = 1,2,…,s, that

describe the position of the centers of the basis atoms with respect to one point of the Bravais
lattice. In general, 0  m j , n j , p j  1 .
 Crystal Structure 3

Every point of a Bravais lattice is equivalent to every other point, i.e. the arrangement
of atoms in the crystal is the same when viewed from different lattice points. The Bravais
lattice defined by (1) is invariant under the operation of discrete translation
Tqrs  qa1  ra 2  sa3 along integer multiples q, r and s of vectors a1 , a 2 and a 3 ,

respectively, because

Tqrs ( Rmnp )  Tqrs  Rmnp  Rqm,r n,s p (2)

is again a Bravais lattice point. In fact, since the translation operation is additive, i.e.
TqrsTuvw  Tqu,r v,sw , associative, i.e. Tqrs (TuvwTmnp )  (TqrsTuvw )Tmnp , commutative, i.e.
1
TqrsTuvw  TuvwTqrs , and has an inverse, such that Tqrs  Tq,r , s and TqrsTq,r , s  I with

I the identity transformation, it follows that the translations form an abelian (commutative)
group. Because condition (2) is satisfied for all Bravais lattice points, a1 , a 2 and a 3 are
called primitive translation vectors, and the unit cell determined by them is called primitive
unit cell. The modulus of these vectors, a1 | a1 | , a2 | a 2 | and a3 | a3 | , are the lattice
constants along the respective axes, and the volume of the primitive unit cell, which in this
case is a parallelepiped, is  | (a1  a 2 )  a3 | .

It is important to notice that the set of vectors a1 , a 2 and a 3 is not unique (see the
figures below for a two-dimensional lattice), but all primitive unit cells have the same
volume.

The primitive unit cell covers the whole lattice once, without overlap and without
leaving voids, if translated by all lattice vectors. An equivalent definition of the primitive unit
cell is a cell with one lattice point per cell (each lattice point in the figures above belong to
 Crystal Structure 4

four adjacent primitive unit cells, so that each primitive unit cell contains 4(1/4) = 1 lattice
point). Non-primitive (or conventional) unit cells are larger than the primitive unit cells, but
are sometimes useful since they can exhibit more clearly the symmetry of the Bravais lattice.
Besides discrete translations, the Bravais lattice is invariant also to the point group
operations, which are applied around a point of the lattice that remains unchanged. These
operations are:

 Rotations by an angle 2 / n about a specific axis, denoted by C n , and its multiples,

Cnj  (Cn ) j . Geometric considerations impose that n = 1, 2, 3, 4 and 6, and that

repeating the rotation n times one obtains C nn  E , where E is the identity operation,

which acts as r  r . Moreover, C1  2  E does not represent a symmetry element.

C D

 
A B

The allowed values of n can be determined assuming that we apply a rotation with an
angle  around an axis that passes first through a point A and then through an adjacent
lattice point B. The points A and B are separated by the lattice constant a. If C and D
are the resulting points, they should also be separated by an integer multiple of a. From
the requirement that CD = a  2a sin(   / 2)  a  2a cos  = ma, or  1  cos  
(1  m) / 2  1 , with m integer, it follows that m can only take the values 1, 0, 1, 2, and
3, the corresponding n  2 /  taking the values specified above. As for translations,
the rotations also form an abelian group.
Examples of two-dimensional figures with different rotation symmetries:

C2 C3 C4 C6

 Inversion I, which is defined by the operation r  r if applied around the origin.

 Crystal Structure 5

 Reflection  j , which can be applied around the horizontal plane (j = h), the vertical
plane (j = v), or the diagonal plane (j = d).

 Improper rotation S n , which consists of the rotation C n followed by reflection in

the plane normal to the rotation axis. Note that S 2  I .

When we combine the point group symmetry with the translational symmetry, we
obtain the space-group symmetry. It is important to notice that the basis can introduce
additional symmetry elements, such as helicoidal symmetry axes and gliding reflection
planes. The figure bellow represents several symmetry operations: (a) translations, (b)
rotation, (c) inversion, and reflection with respect to a (d) vertical, and (e) horizontal plane.

(a) (b) (c) (d) (e)

Crystal lattices are classified according to their symmetry properties at point group
operations. There are 14 three-dimensional Bravais lattices, which belong to 7 crystal
systems, as can be seen from the figure below, where the primitive translation vectors are
denoted by a, b, c (with respective lengths a, b, and c), and , ,  are the angles between b
and c, c and a, and a and b, respectively. These crystal systems, which are different point
groups endowed with a spherical symmetric basis, are:

 cubic, for which a = b = c,  =  =  = 90°. It consists of three non-equivalent space-

group lattices: simple cubic, body-centered cubic, and face-centered cubic. This
crystal system has the highest symmetry and is characterized by the presence of four
C 3 axes (the diagonals of the cube)

 tetragonal, for which a = b  c,  =  =  = 90°. It encompasses the simple and body-

centered Bravais lattices and contains one C 4 symmetry axis.

 orthorhombic, for which a  b  c,  =  =  = 90°. It incorporates the simple, body-

centered, face-centered, and side-centered lattices and has more than one C 2
symmetry axis or more than one reflection plane (actually, three such axes/planes,
perpendicular to each other).
 Crystal Structure 6

 hexagonal, for which a = b  c,  =  = 90°,  = 120°. It

is characterized by the existence of a single C 6
symmetry axis. The conventional hexagonal unit cell
(see the figure at right) is composed of three primitive
cells.

 trigonal, for which a = b = c,  =  =   90°. It contains a single C 3 axis.

 monoclinic, for which a  b  c,  =  = 90° . It includes the simple and side-

centered lattices, and has one C 2 symmetry axis and/or one reflection plane
perpendicular to this axis.

 triclinic, for which a  b  c,       90°. This is the crystal system with the
lowest symmetry. It has no rotation axis or reflection plane.
 Crystal Structure 7

The different crystal systems have different numbers of unit cell types because other
possible unit cell types cannot represent new Bravais lattices. For example, both the body-
centered and the face-centered monoclinic lattices can be reduced to the side-centered lattice
by appropriately choosing the primitive translation vectors.
Examples of two sets of primitive translation vectors for a body-centered cubic (bcc)
lattice are represented in the figure below at left and center, while the figure at right displays a
set of primitive translation vectors for a face-centered cubic (fcc) lattice.

The primitive translation vectors for the left figure above can be expressed as

a1  (a / 2)( x  y  z) , a 2  (a / 2)( x  y  z) , a3  (a / 2)( x  y  z) , (3)

while those for the right figure are

a1  (a / 2)( x  y) , a 2  (a / 2)( y  z) , a3  (a / 2)( z  x) (4)

and the angles between these vectors are 60°.

A simple lattice has lattice points only at the corners, a body-centered lattice has one
additional point at the center of the cell, a face-centered lattice has six additional points, one
on each side, and a side-centered lattice has two additional points, on two opposite sides. The
simple lattices are also primitive lattices and have one lattice point per cell, since the eight
sites at the corners are shared by eight adjacent unit cells, so that 8(1/8) = 1. The non-simple
lattices are non-primitive. The volume of the primitive unit cell in these lattices is obtained by
dividing the volume of the conventional unit cell by the number of lattice points. In particular,
the body-centered lattices have two points per unit cell: the eight at the corners which
contribute with 8(1/8) = 1, and the one in the center, which belongs entirely to the unit cell.
The face-centered lattices have 4 lattice points per cell: those in the corners contribute with
 Crystal Structure 8

8(1/8) = 1, and those on the faces contribute with 6(1/2) = 3, since they are shared by two
adjacent cells. Finally, the side-centered lattices have two lattice points per cell: the points at
the corner contribute with 8(1/8) = 1, and those on the faces with 2(1/2) = 1. If each lattice
point is expanded into a sphere with a radius equal to half of the distance between nearest
neighbors, such that adjacent spheres touch each other, then a packing fraction can be defined
as the fraction between the volume of the spheres contained in the conventional unit cell and
the volume of the unit cell. Note that in the volume between the spheres one can always insert
smaller spheres, which can stand for other atom types.
The 14 Bravais lattices incorporate all possible crystalline structures; they result by
taking into consideration the space-group symmetry, i.e. the symmetry at translations and the
point group symmetry of the lattice (the symmetry with respect to rotation, reflexion or
inversion). All other crystalline structures can be identified with already mentioned Bravais
lattices. For instance, a base-centered monoclinic structure
can be identified with a monoclinic side-centered crystal with
a different choice of crystal axes, whereas a side-centered
cubic lattice is identical to a primitive tetragonal crystal with
a smaller unit cell, as can be seen from the figure at right.
When the basis consists of only one atom, the Bravais lattice is identical to the
crystalline structure. But when the basis is complex and consists of several atoms, say s, the
crystalline structure can be seen as formed by the interpenetration of s Bravais lattices. The
Bravais lattices have always an inversion center in one of the lattice points, whereas such an
inversion center can lack in crystals with complex bases.
By counting the point groups of the possible different crystals (which have bases with
different symmetries), one ends with 32 crystalline classes that can be accommodated by the 7
crystal systems. Also, there are 230 space groups that result from the combination of the 32
crystalline structures with the translational symmetry.

Index system for lattice points, directions and planes

When the origin of the primitive translation vectors is a lattice point, another lattice point
with a position Rmnp  ma1  na 2  pa3 is simply specified by the set of numbers [[m,n,p]]. A

negative integer m, n or p is denoted by a  sign placed on top of it. For example, [[ mn p ]]

stays for the lattice point specified by the integers m, n and p, with m, n and p positive
numbers. In particular, for the three-dimensional primitive Bravais lattices the coordinates of
 Crystal Structure 9

the lattice point at the origin are [[0,0,0]], the other lattice points differing only through
discrete translations along the three coordinate axis. The number of non-equivalent lattice
points in a Bravais lattice is given by the number of lattice points per unit cell. In particular,
for the body-centered lattice, the position of the lattice point at the center of the cube is
denoted by [[1/2,1/2,1/2]], the three additional lattice points in face-centered lattices having
coordinates [[0,1/2,1/2]], [[1/2,0,1/2]], [[1/2,1/2, 0]]. In a similar manner, depending on the
set of opposite sites they can occupy, the additional site in a face-centered lattice has the
coordinates [[0,1/2,1/2]], [[1/2,0,1/2]] or [[1/2,1/2,0]].
A direction, by definition, passes through two lattice points. To specify a direction in
a crystalline lattice, one uses the symbol [mnp], where m, n and p are three integers
determined by the following rule: since one can specify a direction by the coordinates
[[ m1 , n1 , p1 ]] and [[ m2 , n2 , p2 ]] of two points through which it passes, the indices m, n and p
are defined as the smallest integer numbers that satisfy the proportionality relations

m m2  m1 n n2  n1 p p 2  p1
 ,  ,  , (5)
n n2  n1 p p 2  p1 m m2  m1

or

m : n : p  (m2  m1 ) : (n2  n1 ) : ( p2  p1 ) . (6)

If one of the integers is negative, the  sign is placed on top of the integer. For example,
[ mn p ] stays for the direction specified by the integers m, n and p. If the direction is not
considered as an oriented axis but as a simple line, the direction specified by the integers m, n,
and p is the same as that specified by m, n, and p (otherwise, the change of all signs
means a change of direction of the same line). If there are several equivalent directions
(equivalent, from the point of view of crystal symmetry), they are denoted as mnp . A
particular situation is encountered in the hexagonal lattice, in which lattice directions are
labeled by four numbers (this situation is not
further discussed in this course). a3

Examples: The a1 axis is the [100] direction. The

a2
[110]
 a 2 axis is the [ 0 1 0 ] direction. Other examples
are illustrated in the figure at right. a1 [011]
[101]
[011]
 Crystal Structure 10

In three-dimensional lattices, the orientation of a crystal plane is determined by three

non-collinear points in the plane. If each point is situated on a different crystal axis, the plane
is specified by the coordinates of the points in terms of the lattice constants a1 , a 2 , and a 3 .
Another way to specify the orientation of a plane, which is more useful for structure analysis,
involves the determination of three indices, called Miller indices, according to the rule:

 Find first the intercepts of the plane on the axes in terms of lattice constants a1 , a 2 ,

and a 3 , irrespective of the nature (primitive or non-primitive) of the unit cell.

 Take the reciprocal of these numbers.

 If fractional, reduce these numbers to the smallest three integers, say m, n, p, with the
same ratio. The result, symbolized by (mnp) (or (mn p) if the second index, for
example, is negative), is the Miller index system of the plane.

The Miller index for an intercept at infinity is zero. The faces of a cubic crystal, for
example, are denoted by (100), (010), (001), ( 1 00) , (0 1 0) , and (00 1 ) . Moreover, the plane

(200) is parallel to (100), but cuts the a1 axis at a / 2 . If, from the point of view of crystal
symmetry, there is a set of nonparallel equivalent planes, they are symbolized as {mnp}. For
example, the set of faces of a cubic crystal is {100}. Again, for the hexagonal lattice there are
four Miller indices instead of three. Examples of Miller indices are given in the figures below.

(001) (101) (111)

 Crystal Structure 11

Note that the Miller indices determine not only one plane but a family of parallel
planes, since there is an infinite number of planes with the same indices, all of which cut the
coordinate axes at s / m , s / n , and s / p , with s integer. The plane that cuts the axes at 1 / m ,
1 / n , and 1 / p is the closest to the origin from the family of parallel planes.
Note also that the planes with Miller indices (sm,sn,sp) are parallel with the plane
(mnp), but the distance between them is s times smaller. For example, the set of planes (222)
is parallel to but twice as close as the (111) set of planes.
In cubic crystals, the plane (mnp)
is perpendicular to the direction [mnp]
with the same indices (see the example
in the figure at right), but this result
cannot be extended to other crystal
systems.

Simple crystal structures

One of the most simple crystal structures and, at the same time, of general interest, is that of
NaCl (sodium chloride). It is illustrated below. The lattice is face-centered cubic, with a basis
consisting of one Cl ion (blue) at [[000]] and a Na+ ion (green) at [[1/2,1/2,1/2]]. As can be
seen from the figure below, a unit cube consists of four NaCl units, with Na + ions at positions
[[1/2,1/2,1/2]], [[0,0,1/2]], [[0,1/2,0]], and [[1/2,0,0]] and Cl ions at [[000]], [[1/2,1/2,0]],
[[1/2,0,1/2]], and [[0,1/2,1/2]]. Each atom has as nearest neighbors six atoms of opposite kind.
Example of crystals with this structure and their lattice constants are given below.

Crystal a(Å) Crystal a (Å) Crystal a (Å)

LiF 4.02 KBr 6.60 MgO 4.21
LiBr 5.50 AgBr 5.77 MnO 4.43
NaCl 5.64 AgF 4.92 MgS 5.20
NaI 6.47 CaSe 5.91 PbS 5.92
KCl 6.29 BaO 5.52 SrTe 6.47
 Crystal Structure 12

Another common structure is that of CsCl (other crystals with the same structure are
given in the table below). The lattice is in this case simple cubic, with a basis consisting of
one Cs+ ion (red) at [[000]], and one Cl ion (green) at [[1/2,1/2,1/2]]. The number of nearest
neighbors (of opposite kind) is eight.

Crystal a (Å) Crystal a (Å) Crystal a (Å)

AlNi 2.88 CsCl 4.12 TlCl 3.83
CuZn (-brass) 2.94 CsBr 4.29 TlBr 3.97
AgMg 3.28 CsI 4.57 TlI 4.20

The crystal structure of diamond (and also of Si and Ge semiconductors) is

represented below.

Crystal a (Å)
C (diamond) 3.57
Si 5.43
Ge 5.66
-Sn (grey) 6.49

It is a face-centered cubic (fcc) lattice with a basis consisting of two identical atoms, with
coordinates [[000]] and [[1/4,1/4,1/4]]. Alternatively, diamond can be viewed as being formed
from two interpenetrating fcc lattices, displaced by 1/4 of the volume diagonal. Since the
conventional unit cell of the fcc lattice contains 4 lattice points, the conventional unit cell of
diamond has 24 = 8 atoms. No primitive cell exists that contains only one atom. In diamond,
each atom has 4 nearest neighbors and 12 next nearest neighbors. It is usually encountered in
materials where the covalent bonding prevails. Note that, although a fcc lattice, the packing
fraction of the diamond structure is only 0.34.
A closely related crystal structure to that of the diamond is the cubic zinc sulfide (zinc
blende structure). It differs from diamond in that the two atoms of the basis are different (in
 Crystal Structure 13

this case, Zn and S). The conventional unit cell contains four molecules, the Zn atoms (dark
blue in the figure below) being placed at the positions [[000]], [[0,1/2,1/2]], [[1/2,0,1/2]] and
[[1/2,1/2,0]], whereas the S atoms (green) occupy the positions [[1/4,1/4,1/4]], [[1/4,3/4,3/4]],
[[3/4,1/4,3/4]], and [[3/4,3/4,1/4]]. Each atom is surrounded by four equally distant atoms of
the opposite kind, placed in the corners of a regular tetrahedron.

Crystal a (Å) Crystal a (Å) Crystal a (Å)

SiC 4.35 AlP 5.45 InAs 6.04
ZnS 5.41 AlAs 5.66 InSb 6.48
ZnSe 5.67 GaAs 5.65 SiC 4.35
MnS (red) 5.60 GaSb 6.12 CuCl 5.41
CdS 5.82 GaP 5.45 CuBr 5.69
CdTe 6.48 AgI 6.47 HgSe 6.08

Unlike in the diamond structure, where there is a center of inversion at the midpoint of every
line between nearest-neighbor atoms, such inversion centers are absent in the zinc blende
structure. This is an example of additional symmetry operations related to the basis of the
crystal structure.
The hexagonal close-packed (hcp) crystal structure can be obtained from the
hexagonal Bravais lattice if the basis consists of two atoms (blue and red in the figure below,
left) and if the atoms in one plane, which touch each other, also touch the atoms in adjacent
planes. The packing fraction in this case is 0.74 (as in fcc lattices), and is maximum. This
crystal structure is found in the solid state of many elements, as can be seen from the table
below. The hcp structure can be viewed as vertical arrangement of two-dimensional
hexagonal structures, such as the spherical atoms in the second layer are placed in the
depressions left in the center of every other triangle formed by the centers of the spherical
atoms in the first layer. The third layer of atoms is then placed exactly above the first, the
fourth above the second, and so on. This kind of arrangement is called ABAB… In an ideal
hcp structure, the height between the first and the third layers (the height along the c axis in
the figure below) is c  8 / 3a = 1.63a. Because the symmetry of the hcp lattice is
independent of the ratio c/a, in real hcp structures this ratio can take values close to, but not
exactly identical to the ideal 1.63 value (see the table below).
 Crystal Structure 14

Crystal a (Å) c/a Crystal a (Å) c/a

He 3.57 1.63 Mg 3.21 1.62
Be 2.29 1.58 Ti 2.95 1.58
Nd 3.66 1.61 Zr 3.23 1.59
Zn 2.66 1.86 Y 3.65 1.57
Cd 2.98 1.88 Gd 3.64 1.59
-Co 2.61 1.62 Lu 3.50 1.58

The hcp and fcc structures have the same packing fraction because they are in fact
related to one another. As can be seen from the figures below, they differ only in respect to
the way in which the atoms on the third layer is arranged: above those in the first layer for hcp
(ABAB… arrangement), above the C positions for fcc (ABCABC… arrangement, designated
by ccp – cubic closed packed in the figure below).

If the c/a ratio in a hexagonal lattice differs

considerably from the ideal 1.63 value, the crystalline
structure is no longer close-packed. This is the case of
graphite, for example (see the figure at right), with a =
1.42Å and c = 3.40 Å, which implies that c/a = 2.39.
 Crystal Structure 15

Another structure closely related to hcp is wurtzite,

generally encountered in binary compound
semiconductors such as ZnS (wurtzite), ZnO, BN, CdS,
CdSe, GaN, AlN, but sometimes also in ternary
compounds such as Al0.25Ga0.5N. In binary compounds
(see the figure at right), each element has a hcp structure,
and the crystal is formed by interpenetrating two such
structures, so that an atom in one hcp lattice is equally
distanced from the atoms in the other hcp lattice.
The crystal structure of the elements in the periodic table is indicated in the figure
below. Note that several elements can suffer transitions from one crystalline structure to
another depending on the external conditions: temperature, pressure, etc. In the table below
dhcp stands for double hexagonal closed-packed (the height of the cell along the direction
normal to the hexagonal planes is twice that in the hcp structure)

The fact that the crystalline structure is the same along some columns in the periodic table of
elements suggests that there is a link between the specific periodic arrangement of atoms and
their properties/the type of binding.
 Crystal Structure 16

Lattice constants of some elements that crystallize in the fcc crystal structure:
Crystal a (Å) Crystal a (Å) Crystal a (Å) Crystal a (Å) Crystal a (Å)
Ar 5.26 Au 4.08 Cu 3.61 Ni 3.52 Pt 3.92
Ag 4.09 Ca 5.58 Kr 5.72 Pb 4.95 Sr 6.08
Al 4.05 -Co 3.55 Ne 4.43 Pd 3.89 Xe 6.2

Lattice constants of some elements that crystallize in the bcc crystal structure:
Crystal a (Å) Crystal a (Å) Crystal a (Å) Crystal a (Å)
Ba 5.26 Fe 4.08 Mo 3.61 Rb 3.52
Cr 4.09 K 5.58 Na 5.72 Ta 4.95
Cs 4.05 Li 3.55 Nb 4.43 V 3.92
W 6.08
 Reciprocal lattice

The concept of reciprocal lattice is directly connected with the periodicity of crystalline
materials and of their physical properties (such as charge density, electric field distribution,
etc.). Since the crystal is invariant under any translation with a Bravais lattice vector

Rmnp  ma1  na 2  pa3 (1)

for any integers m, n or p, any function  with the same periodicity as the crystalline lattice
must satisfy the relation

 (r )   (r  Rmnp ) , (2)

where r  ( x1 , x2 , x3 ) is an arbitrary position vector with coordinates x1 , x 2 , and x3 in the

(generally non-orthogonal) system of coordinates determined by a1 , a 2 , and a 3 . This implies

 ( x1 , x2 , x3 )   ( x1  ma1 , x2  na2 , x3  pa3 ) (3)

or, for a function that can be expanded in a Fourier series

 ( x1 , x2 , x3 )   G exp[i(G1 x1  G2 x2  G3 x3 )] (4)
G1 ,G2 ,G3

it follows that, for any m, n, and p,

exp(imG1a1 )  1, exp(inG2 a2 )  1 , exp(ipG3 a3 )  1 . (5)

Thus, Gi , with i = 1, 2, 3, can only take discrete values Gi  2si / ai , and (4) becomes

 (r )   G exp(iG  r ) (6)
s1 ,s2 ,s3

where

G  s1b1  s2 b2  s3b3 (7)

is a vector in a coordinate system defined by the vectors bi , i = 1,2,3, such that

 Reciprocal lattice 2

bi  a j  2 ij . (8)

Due to (6), the vectors G can be understood as wavevectors of plane waves with the periodicity
of the reciprocal lattice and wavelengths 2 / | G | , similar to wavevectors in optics that are
perpendicular to wavefronts and have wavenumbers 2 /  , where  is the optical wavelength.
Similar to the Bravais lattices that are constructed starting with the primitive vectors a i ,
one can define a reciprocal lattice in terms of the primitive vectors bi , such that G in (7) are
points in the reciprocal lattice. A reciprocal lattice can only be defined with respect to a given
direct lattice. As demonstrated in the following, the G vectors have dimensions (and meaning
of) wavevectors related to plane waves with the periodicity of the direct lattice.
If the vectors a i are chosen and the volume of the primitive cell in the direct space is
 | (a1  a 2 )  a3 | , the vectors bi can be chosen as

b1  (2 / )(a 2  a3 ), b2  (2 / )(a3  a1 ), b3  (2 / )(a1  a 2 ) . (9)

It follows then that the volume of the primitive cell of the reciprocal lattice is given by

 rec | b1  (b2  b3 ) | (2 ) 3 /  . (10)

An example of direct and corresponding reciprocal lattices in two dimensions is given below.

For a1  d ( x  y) , a 2  d ( x  y) , from (8) we get b1  ( / d )( x  y) , b2  ( / d )( x  y) .

In three dimensions, the reciprocal lattices for the Bravais lattices in the cubic system
are summarized in the table below. The reciprocal lattice of a cubic lattice is also cubic since,
in this case, if x, y, z are orthogonal vectors of unit length, a1 = ax, a 2 = ay, a 3 = az and

  a 3 , from (10) it follows that b1  (2 / a) x, b2  (2 / a) y, b3  (2 / a) z , i.e. the

reciprocal lattice is simple cubic with a lattice constant 2 / a .
 Reciprocal lattice 3

Real space Reciprocal space

Lattice Lattice constant Lattice Lattice constant
SC a SC 2 / a
BCC a FCC 4 / a
FCC a BCC 4 / a

Analogously, the reciprocal lattice to the bcc lattice with (see the first part of the
course) a1  (a / 2)( x  y  z) , a 2  (a / 2)( x  y  z) , a3  (a / 2)( x  y  z) , and   a 3 / 2
has primitive vectors b1  (2 / a)( x  y) , b2  (2 / a)( y  z ) , b3  (2 / a)( z  x) , i.e. is a

fcc lattice with a volume (of the primitive unit cell) in reciprocal state of  rec  2(2 / a) 3 ,

whereas the reciprocal lattice of the fcc lattice, with a1  (a / 2)( x  y) , a 2  (a / 2)( y  z ) ,

a3  (a / 2)( z  x) , and   a 3 / 4 is a bcc lattice with  rec  4(2 / a) 3 and primitive vectors
b1  (2 / a)( x  y  z ) , b2  (2 / a)( x  y  z) , b3  (2 / a)( x  y  z ) . In both cases the
cubic structure of the reciprocal lattice has a lattice constant of 4 / a .

Observation: The reciprocal lattice of a reciprocal lattice is the direct lattice.

Because the product of a primitive Bravais lattice vector and of a primitive vector of the
reciprocal cell is an integer multiple of 2 , i.e. that

Gmnp  Rhkl  2 (mh  nk  pl ) , (11)

for all integers m, n, p and h, k, l, it follows that exp(iG  R)  1 for any vector R in the Bravais
lattice and any vector G in the reciprocal lattice. This implies that the function exp(iG  r ) has
the same periodicity as the crystal because exp[iG  (r  R)]  exp(iG  r ) exp(iG  R) 
exp(iG  r ) . As a consequence,

cell exp(iG  r )dV   G ,0 (12)

and the set of functions exp(iG  r ) form a complete, orthonormal basis for any periodic
function which has the same periodicity as the crystal, i.e. which can be written as

 (r )   G exp(iG  r ) . (13)
G
 Reciprocal lattice 4

If the formula above is regarded as a Fourier transformation of the periodic function , the
coefficients  G can be retrieved by performing an inverse Fourier transformation:

G   1 cell  (r ) exp( iG  r )dV . (14)

Relations between the direct and reciprocal lattices

One geometrical property that can be easily shown is that the reciprocal lattice vector

Gmnp  mb1  nb2  pb3 (15)

is perpendicular to the plane (actually, to the set of parallel planes) with Miller indices (mnp) in
the Bravais lattice. The closest plane to the origin from the set of planes (mnp) cuts the a i
coordinate axes at a1 / m , a 2 / n , and a3 / p , respectively.
To show that (mnp) is perpendicular to G mnp it is sufficient to demonstrate that G mnp is

perpendicular to two non-collinear vectors in the (mnp) plane, which can be chosen as

u  a 2 / n  a1 / m , v  a3 / p  a1 / m , (16)

and satisfy, indeed, the relations

a3
u  Gmnp  v  Gmnp  0 (17) a3/p

because of (8). Then, it follows that the normal to

n a2/n
the (mnp) plane that passes through the origin can
a2
be expressed as (see the figure at right)
a1/m
nmnp  Gmnp / | Gmnp | . (18)
a1

A consequence of this result is that the distance between two consecutive planes with
the same Miller indices (mnp) is inversely proportional to the modulus of G mnp . Since we can
always draw a plane from the (mnp) family through the origin, the distance between two
successive planes is equal to the distance between the origin and the closest plane to origin
from the (mnp) family. This distance is obtained by calculating the projection on the normal to
 Reciprocal lattice 5

the (mnp), i.e. on nmnp  Gmnp / | Gmnp | , of any of the vectors a1 / m , a 2 / n , or a 3 / p . Using
(18) it is found that

a1 a a 2
d mnp  n   n 2  n 3  . (19)
m n p | G mnp |

So,

2
d mnp  . (20)
m 2 b12  n 2 b22  p 2 b32  2mn(b1  b2 )  2np(b2  b3 )  2 pm(b3  b1 )

As already pointed out in the discussion about Miller indices, the distance between any
two planes in the family (sm,sn,sp), is s times smaller than between any two planes in the
family (mnp). The two families/sets of planes are parallel.
In particular, for the simple, body-centered and face-centered cubic Bravais lattices
with coresponding primitive translation vectors given in the first part of the course, the distance
between two consecutive planes with the same Miller indices is, respectively,

a
sc
d mnp  , (21a)
m2  n2  p 2

a
bcc
d mnp  , (21b)
(n  p) 2  ( p  m) 2  (m  n) 2

a
fcc
d mnp  (21c)
(n  p  m) 2  ( p  m  n) 2  (m  n  p) 2

Similarly, for the tetragonal, hexagonal and orthorhombic crystals, the distance between two
consecutive planes with the same Miller indices can be found to be, respectively:
 Reciprocal lattice 6

1
tetra
d mnp 
m2  n2 p2
 2
a2 c

1
hexa
d mnp 
4  m 2  mn  n 2  p 2
 
3  a2  c2


1
ortho
d mnp 
m2 n2 p 2
 
a2 b2 c2

The first Brillouin zone

We have seen that the point group symmetry is important in the direct lattice; it is also
important in the reciprocal lattice. In order to incorporate the information about the point group
symmetry in the primitive cell of the reciprocal lattice, we construct a primitive unit cell called
the first Brillouin zone following the steps:

1) draw lines to connect a given lattice in the reciprocal lattice point to all nearby lattice points,

2) draw new lines (or planes, in three-dimensional lattices) at the mid point and normal to the
first lines. These lines (planes) are called Bragg planes since (as we will see later) all k vectors
that finish on these surfaces satisfy the Bragg condition.

3) the first Brillouin zone is the area (volume) in reciprocal space that can be reached from the
origin, without crossing any Bragg planes.

An example of the construction of the first Brillouin zone for a two-dimensional

oblique lattice is illustrated in the figure below. The first Brillouin zone is always centered on a
lattice point and incorporates the volume of space which is closest to that lattice point rather
than to any other point. The faces of the first Brillouin zone satisfy the relation k cos   G / 2 ,
where G | G | is the distance to the nearest neighbor in the reciprocal space and  is the angle

between k and G. This relation can be rewritten as 2k  G | G | 2 or, since the equation is

equivalent to the replacement of G with  G , we obtain 2k  G  G 2  0 and finally,

(k  G ) 2  k 2 . In other words, the the first Brillouin zone is the intersection of spheres with
the same radius centered at nearest neighbor points in the reciprocal lattice.
 Reciprocal lattice 7

k


Higher-order Brillouin zones, say the nth Brillouin zone, are then defined as the area
(volume) in reciprocal space that can be reached from the origin by crossing exactly n  1
Bragg planes. The construction of the first (light blue), second (light brown) and third (dark
blue) Brillouin zones for a two-dimensional lattice is illustrated in the figure below. The Bragg
planes enclosing the nth Brillouin zone correspond to the nth order X-ray diffraction.

Although higher order Brillouin zones are fragmented, the fragments, if translated, look
like the first Brillouin zone. This process is called reduced zone scheme. All Brillouin zones,
irrespective of the order, have the same volume.

For a two-dimensional square lattice the

first Brillouin zone is also a square, as
illustrated in the figure at right. The higher-
order Brillouin zones for such a square lattice
are also shown (numbered) in this figure.
 Reciprocal lattice 8

Observation: the analogous primitive cell constructed in the direct instead of the reciprocal
lattice (in the same manner as the first Brillouin zone) is called Wigner-Seitz cell.

In three dimensions, since the reciprocal lattice of the bcc lattice is a fcc lattice, the first
Brillouin zone of the bcc lattice (see the polyhedron in the figure (a) above) is the Wigner-Seitz
cell of the fcc. The reverse is also true: the first Brillouin zone of a fcc lattice (the truncated
octahedron/rhombododecahedron in figure (b) above) is the Wigner-Seitz cell of the bcc
lattice.
For certain Bravais lattice, in particular bcc, fcc and hexagonal, the points of highest
symmetry in the reciprocal lattice are labeled with certain letters. The center of the Brillouin
zone is in all cases denoted by , while other symmetry points are denoted as follows

sc lattice: M – center of an edge

R – corner point
X – center of a face

bcc lattice: H – corner point joining four edges

N – center of a face
P – corner point joining three edges
 Reciprocal lattice 9

fcc lattice: K – middle of an edge joining two

hexagonal faces
L – center of a hexagonal face
U – middle of an edge joining a hexagonal
and a square face
W – corner point
X – center of a square face

hexagonal : A – center of a hexagonal face

H – corner point
K – middle of an edge joining two
rectangular faces
L – middle of an edge joining a hexagonal
and a rectangular face
M – center of a rectangular face

Dispersion relations of electrons and phonons for different crystal directions use this labeling
(see the figures below), which indicates the direction and the symmetry of the crystal, since
different labels are used for different symmetries.
 X-ray diffraction 1

X-ray diffraction on crystalline structures

The direct observation of the periodicity of atoms in a crystalline material relies on the X-ray
or particle (electron or neutron) diffraction/scattering on these spatially periodic structures,
since the wavelength of the incident beam is in these cases comparable to the typical
interatomic distance of a few Å. Optical diffraction is not suitable for this purpose since the
wavelength of photons is much too long (about 1 m) in comparison to the lattice constant (a
few Angstroms). On the contrary, for X-rays the wavelength is determined from the relation
E  h  hc /  or   hc / E , which equals a few Å if the energy E is of the order of few keV.
In fact, (Å) = 12.4/E(keV). X-rays are scattered mostly by the electronic shells of atoms in a
solid, since the nuclei are too heavy to respond. (Similarly, for electron diffraction
E  (h /  ) 2 / 2m , with  the de Broglie wavelength. An electron with E = 6 eV corresponds to

  h / 2mE = 5 Å, while for neutron diffraction (Å) = 0.28/[E(eV)]1/2, the electron mass m
being replaced by the neutron mass M.)
In an X-ray diffraction experiment, both the source and the detector are placed in
vacuum and sufficiently far away from the sample such that, for monochromatic radiation, the
incident and outgoing X-ray beams can be approximated by plane waves. The X-rays can be
used in either transmission or reflection configurations. The diffraction picture offers
information on the symmetry of the crystal along a certain axis; the positions and intensities of
the spots (see the figures below) provide information on the lattice and the basis, respectively.
 X-ray diffraction 2

The X-rays penetrate deep in the material, so that many layers contribute to the
reflected intensity and the diffracted peak intensities are very sharp (in angular distribution). As
in optics, to obtain sharp intensity peaks the X-rays scattered by all atoms in the crystalline
lattice must interfere, and the problem is to determine the Bravais lattice (including the lattice
constants) and the basis from the interference patterns. Because a crystal structure consists of a
lattice and a basis, the X-ray diffraction is a convolution of diffraction by the lattice points and
diffraction by the basis. Generally, the latter term modulates the diffraction patterned of the
lattice points.
The wave diffracted in a certain direction is a sum of the waves scattered by all atoms.
Higher diffraction intensities are observed along directions of constructive interference, which
are determined by the crystal structure itself.
The diffraction of X-rays by crystals is elastic, the X-rays having the same frequency
(and wavelength) before and after the reflection. The path difference between two consecutive
planes separated by d is 2·AB = 2d sin  . First-order constructive interference occurs if

k k’
G

2d sin    , (1)

condition known as Bragg’s law.

The Bragg law is a consequence of the periodicity of the crystal structure and holds
only if   2d . This is the reason why the optical radiation is not suitable to detect the
crystalline structure, but only X-rays and electron or neutron beams can perform this task.
Higher order diffraction processes are also possible. The Bragg relation determines,
through the angle , the directions of maximum intensity. These directions are identified as
high-intensity points on the detection screen, the position of which reveal the crystal structure.
For example, if the sample has a cubic crystal structure oriented such that the direction [111]
(the diagonal of the cube) is parallel to the incident beam, the symmetry of the points on the
 X-ray diffraction 3

detector screen will reveal a C 3 symmetry axis. On the contrary, if the diffraction pattern has a
C 6 symmetry axis, the crystal is hexagonal, if it has a C 4 symmetry axis it is a tetragonal
crystal, whereas it is cubic if it shows both a C 4 and a C 3 symmetry axis.
In precise X-ray diffraction measurements, the result is not a diffraction pattern, but a
so-called diffractogram (see the figure below), obtained by detecting the scattered radiation at
an angle 2 with respect to the incident X-ray direction.

For a fixed wavelength of X-rays, maxima of scattered radiation/diffraction peaks on a family

of hkl planes are obtained for  angles at which the normal to the family of planes is parallel to
the diffraction vector/the difference between the incident and scattered wavevector, denoted by
s in the figure above. This condition is not satisfied, for instance, for [110] planes, so that no
diffraction peak is observed.
The Bragg formula says nothing about the intensity and width of the X-ray diffraction
peaks, assumes a single atom in every lattice point, and neglects both differences in scattering
from different atoms and the distribution of charge around atoms.
A closer look at the interaction between the X-rays and the crystal of volume V reveals
that the amplitude of the scattered radiation F (which is proportional to the amplitude of the
oscillation of the electric and magnetic fields of the total diffracted ray) is determined by the
local electron concentration n(r )   nG exp(iG  r ) , which is a measure of the strength of the
G

interaction, and has the same periodicity as the crystalline lattice. The diffraction intensity
 X-ray diffraction 4

I | F | 2 . For elastic X-ray scattering, the phase of the outgoing beam, with wavevector k ' ,
differs from that of the incoming beam that propagates with a wavevector k through
exp[i(k  k ' )  r ] , so that

F   n(r ) exp[i(k  k ' )  r ]dV   n(r ) exp( ik  r )dV   nG  exp[i(G  k )  r ]dV (2)
G

where k  k 'k is the scattering vector, which expresses the change in wavevector. The
result in the above integral depends on the volume of the crystal. If the crystal has length Li

and N i primitive cells in the i direction (i = 1,2,3) of an orthogonal coordinate system with

x  x1 , y  x2 , z  x3 , the integral along the i direction is given by

Li / 2
 2  sin[ ( si   i ) N i ]
 exp i ( si   i ) xi dxi  ai  Li sinc[ ( si   i ) N i ] (3)
 Li / 2  a i   ( s i   i )

where s i ,  i are the components of G and k on the i axis and ai  Li / N i is the lattice
constant on the same direction. The function sinc( x)  sin x / x has a maximum value for x = 0,
and tends to the Dirac delta function for large x.
Therefore, in large-volume crystals scattering occurs only if

k  G , (4)

case in which F  VnG . (In finite-volume crystals there is a sort of “uncertainty” in the angular
range of k around G for which the scattering amplitude takes significant values: as the
volume decreases, the angular range increases.) The above condition suggests that X-ray
diffraction experiments reveal the reciprocal lattice of a crystal, in opposition to microscopy,
which exposes the direct lattice (if performed with high-enough resolution).
The diffraction condition k  k 'k  G can be rewritten as k'  k  G or
k ' 2  k 2  G 2  2k  G . In particular, the form k '  k  G of the diffraction condition
represents the momentum conservation law of the X-ray photon in the scattering process; the
crystal receives the momentum  G . For elastic scattering | k ' || k | and thus G 2  2k  G  0 ,

or k  G | G | 2 / 2 , equation that defines the faces of the first Brillouin zone (the Bragg planes).
The geometric interpretation of this relation (see the figure below) is that constructive
 X-ray diffraction 5

interference/diffraction is the strongest on the faces of the first Brillouin zone. In other words,
the first Brillouin zone exhibits all the k wavevectors that can be Bragg-reflected by the crystal.

Bragg
plane

The diffraction condition is equivalent to Bragg’s law, which can be written for a
certain set of planes separated by the distance d  d mnp as 2(2 /  ) sin   2 / d mnp , or

2k G  G 2 , with G  mb1  nb2  pb3 (for the direction of G with respect to the set of planes,
see the figure illustrating the Bragg law).

The Ewald sphere

The direction of interference peaks can be B
easily determined also via a simple
geometrical construction suggested by
Ewald. Namely, one constructs a sphere
(a circle in two dimensions – see the red
circle in the figure at right) around a point O

O in the reciprocal lattice chosen such

A
that the incident wavevector with O as
origin, ends on an arbitrary lattice point
A. The origin of the Ewald sphere (or
circle) is not necessary a lattice point.
The radius of the sphere (circle) is the wavenumber of the incident (and outgoing)
radiation k | k || k '| . A maximum intensity is found around a direction k ' if and only if the
Ewald sphere (circle) passes through another point B of the reciprocal lattice. The direction k '
is determined by the origin O of the Ewald sphere and this lattice point on the surface
(circumference), which is separated from the tip of k (from A) by a reciprocal lattice vector. It
 X-ray diffraction 6

is possible that for certain incidence angles and wavelengths of the X-rays no such preferential
direction k ' exists.
Therefore, to obtain peaks in the scattered intensity it is necessary to vary either the
wavelength or the incidence angle of the incoming X-rays such that a sufficient number of
reciprocal lattice points find themselves on the Ewald sphere (circle), in order to determine
unambiguously the crystal structure.
In the first method, called Laue
method, the radius of the Ewald sphere
(circle) is varied continuously (see, for
example, the green circle in the figure
below), while in the second method,
called the rotating crystal method or
Debye-Scherrer-Hull method, the Ewald
sphere (circle) is rotated around the
original lattice point with respect to which
the Ewald sphere (circle) was constructed.
The result is represented with the dark
blue circle in the figure below.
In polycrystalline samples, the incident beam is scattered by only those crystallites
(randomly oriented) with planes that satisfy the Bragg condition. Because the sample contains
crystallites with all orientations, the diffraction pattern on the screen is no longer formed from
discrete points, but from concentric circles (see the figure below).
 X-ray diffraction 7

The influence of the basis on the scattered amplitude

If the Laue/diffraction condition k  G is satisfied, an explicit account of the basis influence
implies that the assumption of point/spherical sources at the lattice points have to be modified.
In this case, we have found that

F  VnG  N cell n(r ) exp( iG  r )dV  NS G , (6)

where nG   1 cell n(r ) exp( iG  r )dV , N is the total number of lattice points, and

SG  cell n(r ) exp( iG  r )dV (7)

is the structure factor. It is defined as an integral over a single cell, with r = 0 at one corner. If
there is only one lattice point in the basis and the electron distribution n(r )   (r ) , S G  1 .
If there are s atoms in the basis at positions r j , j = 1,2,..,s, the total electron density can

be expressed as a superposition of electron concentration functions n j at each atom j in the

basis, so that the structure factor is expressed as integrals over the s atoms of a cell:

s  s 
S G    n j (r  r j ) exp( iG  r )dV    n j ( ρ) exp( iG  ρ) exp( iG  r j )dV
 j 1   j 1  (8)
s s
  exp( iG  r j )  n j ( ρ) exp( iG  ρ)dV   f j exp( iG  r j )
j 1 j 1

where ρ  r  r j and f j   n j ( ρ) exp(iG  ρ)dV is the atomic form factor, which depends
only on the type of element that the atom belongs to. The integral has to be taken over the
electron concentration associated with a single atom.
The atomic form factor is a measure of the scattering power of the jth atom in the unit
cell. If the charge distribution has a spherical symmetry, one can use spherical coordinates with
the polar direction along G. In this case,


f j  4  n j ( ρ)  2 (sin G / G )d . (9)
0

where  is the radial coordinate. The atomic form factor decreases rapidly with the distance
and, in the limit   0 , when sin G / G  0 ,
 X-ray diffraction 8


f j  4  n j ( ρ)  2 d  Z , (10)
0

where Z is the number of electrons in an atom. Also, f j  Z when G  k  0 (for a

diffracted ray collinear with the incident ray). f can be viewed as the ratio of the radiation
amplitude scattered by the electron distribution in an atom to that scattered by one electron
localized at the same point as the atom.

Example: consider a bcc lattice as a sc lattice with a basis consisting of two atoms at [[000]]
and [[1/2,1/2,1/2]]. The primitive lattice vectors for the Bravais and the reciprocal lattices are
in this case a1 = ax, a 2 = ay, a 3 = az, and b1  (2 / a) x, b2  (2 / a) y, b3  (2 / a) z ,
respectively. The diffraction peak of the sc lattice that is labeled by (mnp) corresponds to
G  mb1  nb2  pb3  (2 / a)(mx  ny  pz) and for this diffraction peak

2
S mnp   f j exp(iG  r j )  f1 exp[i(2 / a)(mx  ny  pz )  0 ]
j 1 (11)
 f 2 exp[i(2 / a)(mx  ny  pz )  (a / 2)( x  y  z )]  f1  f 2 exp[i (m  n  p)]

The bcc diffraction intensity is I mnp | S mnp | 2  f12  f 22  2 Re[ f1 f 2 exp[i (m  n  p)]] . If

f1  f 2  f ,

4 f 2 , if m  n  p  even
I mnp  2 f [1  exp[i (m  n  p)]]  
2
(12)
 0, if m  n  p  odd

So, for the bcc structure with the same type of

atoms, the (mnp) diffraction peaks of the sc
lattice disappear whenever m  n  p is an odd
integer. In particular, it disappears for the (100)
reflection (see the figure at right) since the
phase difference between successive planes is
, and the reflected amplitudes from adjacent
planes are out-of-phase/destructive interference
occurs.
 X-ray diffraction 9

Observation: for a sc lattice with one atom in the basis, the diffraction intensity is the same,
irrespective of the parity (even or odd) of m  n  p . You could calculate similarly the
diffraction intensity for side-centered cubic and fcc structures, obtaining each time different
results. This example, showing the effect of the basis on the diffraction intensity, is illustrated
in the figure below, which emphasized the difference between diffraction patterns on bcc, fcc,
and diamond-like crystals formed from the same type of atoms.

The figures below show that X-ray diffraction can provides information on
- composition and crystalline structure (each crystal has its own established set of diffraction
peaks; different crystallization forms/allotropic forms of the same material have different
diffractograms)
 X-ray diffraction 10

- dimensionality (bulk crystals/with large dimensions have sharp diffraction peaks in

comparison with thin films or nanoparticles from the same material)

- defects/impurities in the material, including amorphous phases (non-periodic arrangements of

atoms), leading to widening of the diffraction peak
 X-ray diffraction 11

- presence of strain in the sample. A uniform strain leads to a shift in the diffraction peaks since
the lattice constant of the crystal changes, whereas a non-uniform strain induces a symmetric
broadening of the diffraction peaks

More information on X-ray diffraction – at the Solid State Lab

 Crystal binding

The stability of solid state materials is assured by the existing interactions (attractive and
repulsive) between the atoms in the crystal. The crystal itself is definitely more stable than the
collection of its constituent atoms. This means that there exist attractive interatomic forces
and that the energy of the crystal is lower than the energy of the free atoms. On the other
hand, repulsive forces must exist at small distances in order to prevent the collapse of the
material. One measure of the strength of the interatomic forces is the so-called cohesive
energy of the crystal, defined as the difference between the energy of free atoms and the
crystal energy. Similarly, the cohesive energy per atom U 0 is defined as the ratio between the
cohesive energy of the crystal and the number of atoms. Typical values of the cohesive energy
per atom range from 1 to 10 eV/atom, with the exception of inert gases, where the cohesive
energy is about 0.1 eV/atom. In particular, the cohesive energy determines the melting
temperature of solid state materials. Crystals with | U 0 | < 0.5 eV have weak bindings between
constituents, while the others are characterized by strong crystal bindings.

As shown in the figure above, the potential/binding energy U, which describes the interaction
between two atoms, approaches 0 for an interatomic distance R   and tends to infinity at
R  0, having a minimum at a certain distance R  R0 . It is composed of an attractive energy

part, dominant at R  R0 , and a repulsive energy part that prevails at R  R0 . Then, the most
stable state of the system, which occurs at the lowest possible energy, is characterized by the
cohesive energy U 0 , the corresponding interatomic distance, R0 , being known as the
equilibrium interatomic distance. The last parameter has typical values of 23 Å, which
implies that the stability of the crystal is determined by short-range forces.
 Crystal binding 2

The interatomic force, defined as

F ( R)  U / R , (1)

is negative (attractive) for R  R0 , and positive (repulsive) for R  R0 . The attractive and
repulsive forces, which have different origins, cancel each other at the equilibrium interatomic
distance.
The general form of the potential energy is

A B
U (r )   , with n > m. (2)
rn rm

The repulsive force between atoms in the solid has the same origin in all crystals: the
Pauli exclusion principle, which forbids two electrons to occupy the same orbital (the same
quantum state). The repulsive force is characterized (see the formula above) by the power-law
expression U  A / r n , with n > 6 or, sometimes, by the exponential expression
U   exp( r /  ) , where  and  are empirical constants that can be determined from the
lattice parameters and the compressibility of the material. Which expression is better suited to
describe the repulsive force depends on which one better fits with experimental values. The
repulsive potential is short-ranged and thus it is effective only for nearest neighbors.
The attractive forces create bonds between atoms/molecules in the solid, which
guarantee the crystal stability and are of different types depending on the crystal. Only the
outer (valence) electrons, i.e. the electrons on the outer shells, participate in the bonding;
these electrons are responsible also for the physical properties of the material. In contrast,
core electrons are those on inner shells. There are several types of bonding, depending on the
mechanism responsible for crystal cohesion: ionic, covalent and metallic, which give rise to
strong crystal bindings, and hydrogen bonding and van der Waals interactions, which
determine weak crystal bindings.

Crystal binding in inert/noble gases. Van der Waals-London interaction

The crystals of inert gases have low cohesion energies and melting temperatures, and high
ionization energies. They are the simplest crystals, with an electron distribution close to that
of free atoms. From an electrical point of view they are isolators, and from an optical point of
view, are transparent in the visible domain. The weak binding between the constituent atoms
 Crystal binding 3

favors compact crystalline structures, in particular fcc Bravais lattices with one atom in the
basis (the only exceptions are He3 and He4, which crystallize in the hcp crystal structure).
Individual atoms of Ne, Ar, Kr, or Xe have completely occupied external shells, with a
spherically symmetric electronic charge distribution. In crystals, the presence of other atoms
induces a redistribution of the electric charge and a perturbation of the spherical charge
symmetry that can be described within the model of fluctuating dipoles. Coulomb attraction
can occur between two neutral spheres, as long as their internal charges polarize the spheres.
In a classical formalism (valid since electrostatic forces have a long range), this model
assumes that the movement of the electron in atom 1 induces an instantaneous dipole moment
p1 which generates an electric field E (r12 ) at the position of atom 2 separated from atom 1
through a distance r12 | r12 | .

r12

r2

r1

This electric field induces a fluctuating dipole in atom 2 (the distance between the atoms as
well as the magnitude and direction of p1 fluctuate in time), with a moment p2  E (r12 ) ,
where  is the atomic polarizability. The energy of the dipole-dipole interaction between the
two dipoles is minimum when p1 || p2 || r12 , case in which

2
 1  4p12 C
U attr,min (r12 )     6 . (3)
 4 0  r12
6
r12

This van der Waals (or London) interaction is the dominant attractive interaction in noble
gases. The higher-order contributions of the dipole-quadrupole and quadrupole-quadrupole
interactions are characterized by the respective potentials  C1 / r128 and C 2 / r1210 , and do not

contribute significantly to the cohesion energy of the noble gases crystals. The same  C / r126
 Crystal binding 4

dependence of the energy is recovered in a quantum treatment, in the second-order

perturbation theory.
Assuming a power-law expression for the repulsive forces with n = 12, the interaction
potential is given by the Lenard-Jones formula

  12    6 
U (r12 )  4       , (4)
 r12   r12  

where the parameters  and  are determined from X-ray and cohesion energy experiments.
The interaction energy of atom 1 (atom i, in general) with all other atoms in the crystal
is then

  
12
  
6

U i   U (rij )   4     
 r   (5)
j i j i  rij   ij  


and the energy of the crystal composed of N atoms is U cryst  ( N / 2)U i . For a periodic

arrangement of atoms in the lattice, with nearest-neighbors at a distance R, rij  pij R and

   12   
6

U cryst  2 N  S12    S 6    (6)

  R   R  

where

6 12
 1   1 
S 6     ,
 S12    
 (7)
j  i p ij j  i p ij
   

are rapid convergent series, that can be calculated after the crystalline structure is determined
by X-ray measurements. Their values are, respectively, 12.132 and 14.454 for the fcc
structure, with almost the same values for hcp structures.
The crystal energy is minimum for the R value which is the solution of
U cryst / R  2 N [12S12 ( / R)11  6S 6 ( / R) 5 ]  0 , i.e. for

R0   (2S12 / S 6 )1 / 6 . (8)
 Crystal binding 5

The ratio R0 /   1.09 for a fcc Bravais lattice, the corresponding cohesion energy per atom
(at zero temperature and pressure) being

  S 2  S  S2
U 0  U cryst ( R0 ) / N  2 S12  6   S 6  6    6  8.6 . (9)
  2S12   2S12  2S12

Quantum corrections reduce the binding energy above by 28%, 10%, 6%, and 4% for Ne, Ar,
Kr, and Xe, respectively. The quantum corrections are more important for inert gas crystals
with smaller equilibrium interatomic distance (smaller lattice constants).

Ne Ar Kr Xe
R0 (Å) 3.05 3.74 4 4.34
U0 (eV) -0.024 -0.085 -0.118 -0.171
Tmelt (K) 24 84 117 161
 (eV) 0.031 0.01 0.014 0.02
 (Å) 2.74 3.4 3.05 3.98

Ionic binding
The ionic binding is found in ionic crystals formed
from positive and negative ions, for example Na+
and Cl in NaCl. In this bonding type, electrons are
transferred from the low electronegative atom,
which becomes a positive ion, to the high
electronegative atom, which is transformed into a
negative ion (see the figure on the right).
The electronegativity measures the ability of an atom or molecule to attract electrons
in the context of a chemical bond. Because the electronegativities of Na+ and Cl- differ, the
ionic bonding is realized at a cost denoted by E . In NaCl, the net energy cost of the ionic
bonding (i.e. the difference between the energy of the ions and that of the two atoms) is 1.58
eV per pair of ions, without taking into account the Coulomb energy between the ions. In
other words,
 Crystal binding 6

 
Na  Cl  
Na
Cl
  U 0  E
crystal

where U 0 is the cohesion energy.

In general, the electronegativity increases with the group number in the periodic
element table, from the first to the seventh group (elements in the eight group have complete
shells). Depending on the difference in electronegativity between two atoms, the bonding
between them is
 Ionic (for large difference). Example: Na-Cl.
 Polar covalent bonding (for moderate difference). Example: H-O.
 Covalent bonding (for small difference). Examples: C-O, O-O

In ionic crystals the bonding is achieved by the long-range electrostatic force and so, a
classical treatment is meaningful. The electronic configuration of the ions is similar to that of
inert/noble gases, i.e. the electronic charge has a spherical symmetry, which is only slightly
perturbed in crystal. The perturbations are localized in the regions in which the ions are
closer. In particular, in NaCl the electronic configurations of the Na + and Cl ions are similar
to that of noble gases Ne10 (1s22s22p6) and Ar18 (1s22s22p63s23p6), respectively (see below).

= Ne = Ar

The Coulomb force between one positive Na ion and one negative Cl ion, separated by
a distance R is given by

e2
FCoulomb   (10)
4 0 R 2

with R = 2.81 Å the nearest-neighbor distance in NaCl, so that the respective attractive
potential energy,
 Crystal binding 7

e2
U Coul   , (11)
4 0 R

equals 5.12 eV per pair. It follows then that the net energy gain in the ionic bonding, is
5.12 eV  1.58 eV = 3.54 eV per pair of ions.
The electrostatic energy gain per NaCl molecule in a fcc crystal is obtained by adding
different contributions:
e2
 that of the (opposite type) 6 nearest-neighbors of a certain ion, U 1  6 ,
4 0 R
e2
 that of the 12 second nearest-neighbors (of the same ion type), U 2  12 ,
4 0 R 2
e2
 that of the 8 third nearest neighbors of opposite type, U 3  8 , and so on.
4 0 R 3
The result is


e2 12 8  e2 ()
U ion    6    .....    (12)
4 0 R  2 3  4 0 R j i pij

The series above converge eventually to U ion  1.748e 2 /( 4 0 R)  Me 2 /( 4 0 R) ,
where M is the Madelung constant, which takes specific values for each crystal structure. For
other crystal structures: CsCl, zinc blende, and wurtzite, we have, respectively, M = 1.763,
1.638 and 1.641. (If the series is slowly convergent or even divergent, the terms in the sum
are rearranged such that the terms corresponding to each cell cancel each other – the cell
remains neutral in charge.) The total attractive energy in a NaCl crystal with N ion pairs is
given by U attr  2U ion  N / 2 , where the factor 2 in the numerator accounts for the fact that
there are two types of ions: Na and Cl, and the factor 2 in the denominator is introduced in
order to count every ion pair only once. For NaCl, Uattr = 861 kJ/mol (experiments give 776
kJ/mol). The discrepancy (of about 10%) between the experimental and theoretical values is
explained by the existence of the (non-classical) repulsive forces.
Similarly, if we add up the repulsive potential felt by an atom from all others (the
exponential form is used now), we obtain

U rep    exp( rij /  )  z exp(  R /  ) (13)

j i
 Crystal binding 8

where we consider that   R and z is the number of nearest neighbors. The interaction
energy of the whole crystal, which consists of N ion pairs/molecules is

 Me 2 
U cryst ( R)  N  z exp(  R /  )  , (14)
 4 0 R 

and its minimum value per molecule,

U cryst,0 e2M   
U min   1   , (15)
N 4 0 R0  R0 

occurs for the equilibrium interatomic distance R0 found from the condition dU cryst / dR 

 ( z /  ) exp(  R /  )  Me 2 /( 4 0 R 2 )  0 . The first term (the Madelung term) in (15),

which expresses the electrostatic contribution of the interactions, is dominant since   R0 .

LiF LiCl NaF NaCl KF KCl RbF RbCl

R0 (Å) 2.01 2.56 2.31 2.82 2.67 3.15 2.82 3.29
U0 (eV) -10.5 -8.45 -9.31 -7.86 -8.23 -7.1 -7.85 -6.84
 (102 eV) 3.08 5.1 6.24 10.93 13.63 21.35 18.54 33.24
 (Å) 0.291 0.33 0.29 0.321 0.298 0.326 0.301 0.323

Covalent bonding
The covalent bonding forms in molecules composed of identical particles, for example
hydrogen. In this case two atoms form a (homopolar) bond by sharing a pair of electrons (one
from each atom, with opposite spins). Most atoms can form more than one covalent bond. For
instance, C has four outer electrons (of 2sp3 type) and thus can form 4 covalent bonds. The
covalent bond is highly directional and different bonds repel each other. Therefore, the
corresponding crystal has generally a low packing ratio. For example, C and Si can have a
diamond structure, with atoms joined to 4 nearest neighbors at tetrahedral angles; this
structure has a packing ratio of only 0.34 compared to 0.74 for close-packed structures. The
electrons in covalent bonds are strongly localized along the bond, so that the crystals are
semiconductors or isolators, with not very good electrical conductivity.
 Crystal binding 9

To describe the covalent bonding in hydrogen, we introduce the normalized atomic

orbitals 1s for the j (j = 1,2) electron that can belong to either atom A or B as  1As, B ( j ) , so that
the normalized wavefunction of the total system can be either symmetric (labeled with +) or
antisymmetric (labeled with )

  [1 / 2(1  S 2AB ) ][ 1As (1) 1Bs (2)   1Bs (1) 1As (2)] (16a)

  [1 / 2(1  S 2AB ) ][ 1As (1) 1Bs (2)  1Bs (1) 1As (2)] (16b)

where S AB   ( 1As (1))* 1Bs (2)dr is the overlap integral. Note that the symmetric

wavefunction for ionic crystals can be expressed as   [ A (1) A (2)   B (1) B (2)] .

E- +
-

|+|2 |-|2
E+

The symmetric wavefunction (also called singlet) corresponds to two antiparallel spin,
with quantum number S = 0 of the operator S 2 (with S the total spin), while the antisymmetric
wavefunction (also called triplet) corresponds to parallel spins, i.e. S = 1 (with the spin
projection quantum number ms = 1, 0, and 1; there are three antisymmetric wavefunctions!).
The form of the wavefunctions above is determined from the condition that the total wave
function for fermions (including spin) must be antisymmetric upon particle exchange.
The energy eigenvalues are represented above as a function of the distance between
the atoms. A bound state can exist in the singlet state, with E+ = 3.14 eV if the covalent
bonding forms between H atoms, i.e. the strongest binding occurs if the spins of the two
electrons are antiparallel.
 Crystal binding 10

E-

atom A E+ atom B

To characterize the crystalline structure of diamond one must generalize the previous
formula in order to incorporate the p atomic orbitals. Indeed, the last occupied orbitals of
these materials are: C(2s22p2), Si(3s23p2), and Ge(4s24p2). When both s and p-type orbitals are
involved, they hybridize (see figures below). [The s atomic orbitals have quantum numbers n
= 1,2,3… (principal quantum number), l = 0 (orbital quantum number), and m = 0 (magnetic
quantum number; the projection of l). The p orbitals have n = 1,2,3…, l = 1, and m = 1,0,1.]
The s and p atomic orbitals hybridize when the energy difference between them is much
smaller than the binding energy.

In particular, when one s orbital with wavefunction s and one p orbital, say the p x
orbital, with wavefunction  px , hybridize, the result is two linear sp orbitals (see figure

above) with wavefunctions

1  2 1/ 2 (s   px ) , 2  2 1/ 2 (s   px ) . (17)

On the contrary, sp2 hybrid orbitals form between one s orbital and two p orbitals, the
resulting planar structure (see figure above) having orbitals at an angle of 120° between them.
The electrons in the hybrid orbitals are strongly localized and form  bonds; they do not
participate in electrical conduction. One p orbital remains perpendicular to the plane, where it
forms a  bond with other out-of-plane p orbitals from neighboring atoms; this is the case of
graphite or graphene (bidimensional crystal). The electrons in the  orbitals are delocalized
and participate in electrical conduction. The three hybrid orbitals are given by
 Crystal binding 11

1  31/ 2 (s  2 px ) , (18)

2  31/ 2 [s  (1 / 2 ) px  3 / 2 p y ] , 3  31/ 2 [s  (1 / 2 ) px  3 / 2 p y ] .

Similarly, the electronic configurations that forms from one s orbital and three p
orbitals is called sp3. This electronic configuration is characteristic for diamond. The four
hybrid atomic orbitals that are linear combinations of atomic orbitals form a tetrahedron (see
figure below) and are given by

1  (1 / 2)(s   px   p y   pz ) , (19a)

2  (1 / 2)(s   px   p y   pz ) , (19b)

3  (1 / 2)(s   px   p y   pz ) , (19c)

4  (1 / 2)(s   px   p y   pz ) . (19d)

C, Si and Ge form crystals in which the covalent binding is dominant, the van der
Waals contribution to the cohesion energy, also encountered in crystals from a single element,
being negligible. However, in crystals with a basis composed of two atoms A and B, with n
and, respectively, 8  n valence electrons, the covalent binding is accompanied by an ionic
contribution. The resulting bond is called polar covalent bond. The fraction of the ionic
contribution is of 0.18 in SiC, 0.26 in GaSb, 0.32 in GaAs, and 0.44 in InP. Similarly, in ionic
crystals the covalent binding can also contribute to the cohesion energy, the fraction of the
ionic contribution being only 0.86 in AgCl, 0.94 in NaCl, and 0.96 in RbF. When covalent
 Crystal binding 12

bonding forms between different atoms, the hybrid orbitals considered above are modified, as
can be seen from the figures below.

Electronic configuration in the CH4 molecule.

Bonding between the 1s orbitals of the H atom and the 2px and 2py orbitals of O atom
in H2O (a) without, and (b) with hybridization

Covalent crystals are characterized by:

 high melting temperatures (the cohesion energy per atom is about 10 eV)
 hardness (but also friable)
 their conductivity depends strongly on temperature and impurity atoms
 high value of the dielectric constant
 generally transparent in IR, but strongly absorbent in visible and near-IR.
 Crystal binding 13

Hydrogen binding of crystals

Because neutral hydrogen has only one electron, it should form a covalent bond with only one
other atom. However, just as oppositely charged ions are attracted to one another and can
form ionic bonds, the partial charges that exist at different atoms in polar covalent bonds can
interact with other partially charged atoms/molecules. Particularly strong polar covalent
bonds are found, for example, when a hydrogen atom bonds to extremely electronegative ions
such as O in water/ice (see the figure below, left), F (see the figure below, right), N or Cl. The
partial charges in the figures below are denoted by . The hydrogen bond forms between the
hydrogen atom with a strong partial positive charge and electronegative ions with strong
partial negative charges in neighboring molecules. The binding energy is of the order of 0.1
eV. For example, the cohesion energy per molecule in ice is 0.3 eV.

hydrogen bond

hydrogen bond

The hydrogen bond is weaker than, although similar to, ionic bond since it forms
between partial charges rather than full (complete) charges. In hydrogen bonds the hydrogen
atom is the hydrogen bond donor and the electronegative ion is the hydrogen bond acceptor.
As the polar covalent binding, the hydrogen bond can be viewed as a mixture of ionic and
covalent bonding, the ionic bonding being dominant.
In hydrogen, the proton radius is with five orders of magnitude smaller than the radius
of any other ion, and so it allows the existence of only two nearest neighbors of the proton
(more than two atoms would get in each other’s way), i.e. the hydrogen bond is directional.
Despite being weak, the hydrogen bond is extremely important in living organisms,
which are mainly composed of water, since water as well as proteins and nucleic acids posses
a great capacity to form hydrogen bonds. In particular, the hydrogen binding occurs as intra-
molecular binding between the organic complementary bases thymine and adenine, and
cytosine and guanine in DNA. It can also be encountered between constituents of crystals
such as KH2PO4, KD2PO4 (KDP), Ca(OH)2, or Mg(OH)2.
 Crystal binding 14

Metallic bonding
The metallic bonding can be understood as the bonding
between positively charged metallic nuclei/ions and
delocalized conduction electrons, seen as a “sea of free
electrons”. It prevails in elements in which the valence
electrons are not tightly bound with the nucleus (in metals,
for example). However, in the metallic bond we cannot
speak about ions, since there is no particular electron that
is “lost” to another ion.
Unlike other bonding types, the metallic bonding is collective in nature, so that no
single “metallic bond” exists. The metallic bonding can be understood as an extremely
delocalized form of covalent bonding. The delocalization is most pronounced for s and p
electrons, with l = 0 and l = 1, respectively, being much weaker for d and f electrons, which
have quantum numbers l = 2 and l = 3, respectively.
In metals, an atom achieves a more stable configuration by sharing all its valence
electrons with all other atoms in the crystal. However, besides delocalization, metallic
bonding also requires the availability of a far larger number of delocalized energy states than
of delocalized electrons. These states are referred to as electron deficiency; they assure the
kinetic energy for delocalization.
The metallic bonding is encountered, for example, in alkaline metals such as Li, K, Na,
with electronic configurations that resemble those of noble gases with an additional s electron
on the outer shell. Having few electrons on their outer shells, alkali metals have only partially
filled energy levels, and so are electron deficient. In forming the crystal, the wavefunctions of
the outer s electrons overlap with those of their nearest neighbors, and the electrons become
delocalized. Their dynamics resembles that of free electrons. In alkaline metals the lattice is
occupied by the positively charged ions with the noble gas structure (they occupy in fact only
about 20% of the crystal volume), while the valence electrons occupy the remaining volume.
Unlike in covalent crystals, where the electronic charge is distributed in a strongly
nonuniform manner (the bonds are spatially oriented), the electronic density in metallic
crystals is highly uniform. This explains the high elasticity and malleability of these materials.
The total Coulomb potential, which includes electron-electron, electron-ion and ion-
ion interactions, is U Coul  U ee  U ei  U i i < 0 (the first and third terms on the rhs are
 Crystal binding 15

positive, the middle one is negative). Therefore, the attractive potential is of electrostatic
nature, being balanced by the repulsive interaction due to the Pauli exclusion principle.

Observation: Not all metals have metallic bonding. For example, many transition metals
show covalent properties (not all electrons participate in covalent bonds, and are good
electrical conductors). In transition metals (Fe, Co, Ni, Cu, Zn, Ag, Au, Mn, etc.) the d
orbitals are only partially occupied and the outermost s orbitals are fully occupied.
Example: 4s2–full (lower in energy), 3d–incomplete (higher in energy) – see figure below.
 Crystal binding 16

Crystals with metallic bonding are usually characterized by

 high electrical and thermical conductivity, with weak temperature dependence

 high elasticity
 high optical reflectivity in a large frequency bandwidth
 broad range of melting temperatures: low melting temperatures for alkaline metals (Li,
Na, K, Rb, Cs), intermediate for noble metals (Cu, Ag, Au), and high values for metals
such as Ti, Zr, Mo, W. The corresponding cohesion energies vary between 1 eV and
5 eV.
 Lattice oscillations. Phonons

Let us consider a crystalline material consisting of a large number N ion of heavy positively-
charged ions (composed of the nucleus and the valence electrons on the inner atomic orbitals)
with masses M  and situated at positions R ,  = 1,.., N ion , surrounded by and in interaction

with N el electrons on the outer atomic orbitals with masses m and at positions denoted by ri ,

i = 1,.., N el . The total Hamiltonian of the system is then

H  Tel  Tion  Vel  Vion  Vel ion

2 2 2 1 1 (1)
   i    2   U el (ri  r j )   U ion ( R  R )   V (ri  R )
i 2m  2M  2 i, j 2  , i ,
i j  

The terms of the right-hand-side denote, in order, the kinetic energy of the electrons, the
kinetic energy of the ions, the (Coulomb) interaction energy of electron pairs, the interaction
energy of ion pairs, and the interaction energy between electrons and ions.
Since m  M  , the electron velocities are much higher than the ion velocities, so that
the electrons “see” a “frozen” distribution of ions, while the ions can only sense the average
(not instantaneous!) spatial distribution of electrons. In other words, for a given ion
configuration the electrons are in a quasi-equilibrium state that is slowly varying in time due
to ion’s motion, whereas the ions evolve slowly in a potential distribution generated by the
average configuration of the electrons. This adiabatic approximation, known also as the Born-
Oppenheimer approximation, allows a factorization of the total wavefunction of the system,
(r , R) with r  {r1 , r2 ,..., rNel } , R  {R1 , R2 ,..., RNion } into an electronic part,  (r ; R) , in

which the ion’s positions are considered as parameters, and into an ionic part,  (R) :

(r , R)   (r; R) ( R) . (2)

These electronic and ionic parts satisfy the following equations:

 
 2 2 1 
   i   U el (ri  r j )   V (ri  R )  (r ; R)  Eel ( R) (r ; R) (3)
 i 2m 2 i, j i , 
 i j 
 Lattice oscillations. Phonons 2

 
 2 1 
     ion     ( R)  E ( R) ,
2
 U ( R R  ) E el ( R ) (4)
  2M  2  , 
   

where E is the energy of the whole system and E el is the energy of the (sub)system of
electrons.
Let us assume further that in a crystalline lattice with s atoms in the basis, the ions
move around their equilibrium positions R0 , so that | R  R0  u | R0 . Then, the
interaction energy between pairs of ions can be expanded in a Taylor series around the
equilibrium positions. Taking into account that

 U ion ( R  R ) U ion ( R  R ) 
U ion ( R  R )   u  u 0
 (5)
 R R 

since the force that acts upon an ion at equilibrium (which is proportional to this derivative)
vanishes, we find that

  
U ion ( R  R )  U ion ( R0  R 0 )   A u u  , (6)
 ,  ,  ,  ,  ,
   

where

  2U ion ( Ra  R ) 
A 
    

 (7)
  R  R   R ,   R0 , 

and the indices  = x,y,z (and ) denote the projections of the position vectors on a Cartesian
coordinate system, the first spatial derivative of U ion vanishes due to the requirement that the
force (which is proportional to this derivative) that acts upon an ion at equilibrium vanishes,
and higher-order terms in the Taylor expansion are neglected. The last approximation is called
harmonic. Because the first term in the Taylor expansion of U ion is constant, it can (together
with E el ) be included in the reference energy of the system, so that in the harmonic
approximation the lattice dynamics is described by the Hamiltonian

( p ) 2 1   
H ion    A u u  . (8)
 ,  2M  2  ,  ,  ,
 Lattice oscillations. Phonons 3

This Hamiltonian describes in fact a set of coupled harmonic oscillators. The coupling

strength with neighboring ions is characterized by the coefficients A . A harmonic potential

energy corresponds to forces that are linear in the displacements. In particular, the force that
acts on an ion  from other ions , given by F  M  u    A
 
u , must be
 ,

proportional to the relative displacement, u  u , i.e. it should be expressed as

F    A u     A

   
(u   u ) , (9)
 ,  ,

which implies that

 A 
 A   A 0 (10)
  

Thus, in the harmonic approximation one can view the lattice vibration as an
interaction between connected elastic springs (classical harmonic oscillators), shown in the
figure below. The lattice oscillations are similar to elastic waves that propagate through such
a chain of springs. If an atom is displaced from its equilibrium site by a small amount, it will
tend to return to its equilibrium position due to the force acting on it. This results in lattice
vibrations. Due to interactions between atoms, various atoms move simultaneously, so we
have to consider the motion of the entire lattice.

The physical relevant solutions for the system of harmonic oscillators are of plane-
wave type, i.e. are oscillatory in time, with the same frequency for all ions. These are the
normal oscillations.
 Lattice oscillations. Phonons 4

Oscillations of an infinite atomic chain with one atom in the basis

To gain a deeper insight into the lattice oscillations, let us consider first a simple one-
dimensional infinite lattice (an “atomic chain”) consisting of identical atoms (more precisely,
ions) with mass M, separated by the lattice constant a, as in the figure below.

n–1 n n+1

For (thermal) vibrations of the crystalline lattice, in which the ions move slightly around their
equilibrium positions Rn0  na , their actual positions Rn satisfy the relations

| Rn  Rn0  u n | Rn0 , where the displacements can occur either along the chain (for
longitudinal vibrations) or transverse to the chain of atoms.

It should be noted that one-dimensional lattice vibrations are not encountered only in
atomic chains. For example, in a simple cubic crystal with one atom in the primitive cell,
when a wave propagates along the directions of the cube edge, face diagonal, or body
diagonal, entire planes of atoms move in phase with displacements either parallel or
perpendicular to the direction of the wavevector (see the figures above). We can describe the
displacement of the plane n from its equilibrium position with a single coordinate, u n . The
problem becomes in this way one-dimensional. For each wavevector u n there is one solution
with longitudinal polarization and two with transverse polarization. The parameter A is
different for longitudinal and transverse waves.
 Lattice oscillations. Phonons 5

Because in the harmonic approximation the dynamics of the system is equivalent to

that of coupled harmonic oscillators, the force acting on an ion is linear in the (relative)
displacement. For simplicity, we assume further that only the interaction between nearest
neighbors is significant, case in which the force exerted on th n-th atom in the lattice is

Fn  A(u n1  u n )  A(u n1  u n ) (11)

where A denotes the interatomic force (or, equivalently, the elastic constant between nearest-
neighbor ions). Note that equation (11) is compatible with (9) and (10) since, if An,m  0 for

m  n, n  1, and An,n1  An,n1   A , An,n  2 A , we obtain (10).

Applying Newton’s second law to the motion of the n-th atom with mass M,
M (d 2 u n / dt 2 )  Fn , we obtain

d 2un
M  A(u n1  u n )  A(u n1  u n )   A(2u n  u n1  u n1 ) . (12)
dt 2

A similar equation should be written for each atom in the lattice.

The normal solutions of the equation above have the form

u n (t )  u0 exp[i(kRn0  t )]  u0 exp[i(kan  t )] , (13)

and represent traveling waves, in which all atoms have the same wavevector k and oscillate
with the same frequency and the same amplitude u 0 . Solutions of this form are only
possible because of the translational symmetry of the lattice.
Inserting (13) into the equation of motion (12) we obtain

 M 2   A[2  exp( ika)  exp(ika)] (14)

or

 exp(ika)  exp( ika) 

M 2  2 A1    2 A[1  cos(ka)]  4 A sin (ka / 2) .
2
(15)
 2 

and the dispersion relation, i.e. the k dependence of the oscillation frequency, represented in
the figure below, is
 Lattice oscillations. Phonons 6

 (k )  2 A / M | sin(ka / 2) | . (16)


2(A/M)1/2

k
st
1 Brillouin zone

Because the dispersion relation is periodic: (k )  (k  2 / a) , with the periodicity
given by the reciprocal lattice vector, all distinct frequency values can be found in the k
interval

 / a  k   / a , (17)

which corresponds to the first Brillouin zone. The maximum (cut-off) frequency  max

 2 A/ M is obtained for the minimum wavelength of min  2 /( / a)  2a . The existence

of a minimum wavelength can be understood as resulting from the condition that waves with
wavelengths smaller than 2a cannot propagate in the lattice, being reflected at the boundaries
of the first Brillouin zone.
The significance of the periodicity of the dispersion relation is evident from the figure
below: changing k by one reciprocal lattice vector gives exactly the same movement of atoms.
 Lattice oscillations. Phonons 7

In the long-wavelength limit ka / 2  1, we have sin(ka / 2)  ka / 2 , and

 (k )  A / M ak  vac k . (18)

The oscillations with a linear dispersion relation in the long-wavelength limit are called
acoustic and are characterized by the acoustic velocity, which in this case is vac  a A / M .
A similar dispersion relation as that for acoustic oscillations holds for acoustic waves
propagating in a continuum, elastic and isotropic medium.
Moreover, since the oscillation frequency does not depend linearly with k, we can
define separately the phase velocity, i.e. the velocity of the phase of the wavefront, and the
group velocity, i.e. the propagation velocity of the wavepacket and of the wave energy. Their
modulus are given, respectively, by

 A sin(ka / 2) sin(ka / 2)
v
v ph   a  vac , (19)
k M ka / 2 ka / 2
vac vph
d A
v gr   a cos(ka / 2)  vac cos(ka / 2) . (20)
dk M 2vac/
vgr
In the long-wavelength range, k  0, v ph  v gr  vac ,
k
while at the edges of the first Brillouin zone, for 0 /a
k   / a , v ph  2vac /  and v gr  0 .

Finite lattices
For finite one-dimensional lattices consisting of N
identical atoms, the requirement of symmetry (of
equivalence of physical properties) when the equation of
motion refers to different atoms imposes the cyclic
boundary condition u n  u n N . This so-called Born-
Karman condition expresses the independence of the
properties on the surface, i.e. we have a finite solid, with
no surfaces; a finite chain with no end. From the cyclic
boundary condition it follows that exp(ikNa)  1, or
 Lattice oscillations. Phonons 8

2
k m, (21)
Na

with m an integer. There are N allowed m values for k in the first Brillouin zone:

 N /2  m  N /2, (22)

which correspond to the N degrees of freedom of the system. Because N is usually a large
number, the discrete nature of the wavenumber is disregarded and it is considered as a
continuous variable. Below: example for N = 10.



Density of oscillations in a simple finite one-dimensional lattice

How many oscillations (with different k values) exist in the frequency interval (,   d ) ?
This number is referred to as the density of oscillations per unit frequency and is denoted by
dN osc / d  D( ) . Because for the finite one-dimensional lattice the wavenumber varies
only in discrete steps of k  2 / Na , there is only one oscillation possible in this
wavenumber range, so that

dN osc 1 Na
  (23)
dk k 2

and, taking into account the double degeneracy due to the symmetry of  (k ) (two k values
correspond to the same , we finally obtain
 Lattice oscillations. Phonons 9

D
dN osc dN dk
D( )   2 osc
d dk d
N 1 N 1
  (24)
 A / M | cos(ka / 2) |  A / M 1  sin (ka / 2)
2
2N/max
N 1 2N 1
 
 A/ M 1 2
/  max
2   max
2
2 
max

Oscillations of an infinite atomic chain with two atoms in the basis

Let us consider now an infinite one-dimensional lattice (“chain”) with lattice constant a,
consisting of equally spaced ions with different masses, M 1 and M 2  M 1 , as shown below.

The basis has thus two atoms, at equilibrium positions Rn0,1  na and Rn0,2  (n  1 / 2)a .

n-1 n n+1 n+2

un-1,1 un-1,2 un,1 un,2 un+1,1 un+1,2 un+2,2

Similarly to the atomic chain with one atom in the basis, we have now two equations
of motion of the general form F  M (d 2 u / dt 2 ) , one for each type of atom. To distinguish

between the displacements of the two atoms, we denote with u n,1 (t )  u1 exp[i(kan  t )] the

displacement of atoms with mass M 1 (the yellow ones in the figure above) and with
u n,2 (t )  u 2 exp[i(ka(n  1 / 2)  t )] that of atoms with mass M 2 (the green ones). So, we

have

d 2 u n,1
M1  A(u n1, 2  u n,1 )  A(u n, 2  u n,1 )   A(2u n,1  u n, 2  u n1, 2 ) , (25a)
dt 2
 Lattice oscillations. Phonons 10

d 2 u n, 2
M2  A(u n,1  u n, 2 )  A(u n1,1  u n, 2 )   A(2u n, 2  u n,1  u n1,1 ) , (25b)
dt 2

or

 M 1 2 u1  2 Au1  Au 2 [exp(ika / 2)  exp( ika / 2)]  2 Au1  2 Au 2 cos(ka / 2) , (26a)

 M 2 2 u 2  2 Au 2  Au1[exp(ika / 2)  exp(ika / 2)]  2 Au 2  2 Au1 cos(ka / 2) . (26b)

Again, the sum rule (10) is satisfied if An,1;n1,2  An,1;n1,2  An,2;n1,1  An,2;n1,1   A and

An,1;n,1  An,2;n,2  2 A . The system of coupled equations (26) has solution only when its
determinant vanishes, i.e. when

2 A  M 1 2  2 A cos(ka / 2)
 0, (27)
 2 A cos(ka / 2) 2 A  M 2 2

or

04 2  ka 
 4   2 02  sin 2    0 (28)
4  2

M1  M 2 M 1M 2
with 02  2 A , 2 4 .
M 1M 2 (M 1  M 2 ) 2

From (28) it follows that there are two solutions (two types of lattice oscillations) for every
value of k; these are called the optical and the acoustic branches. The two solutions are

 02   ka    02  2  ka 

 (k ) 
2
1 1  1   sin    ,
2 2
 (k ) 
2
2 1  1   sin    .
2
(29)
2   2   2   2  

As for the one-dimensional lattice with one atom in the basis, 1, 2 (k  2 / a)  1, 2 (k ) , so

that all relevant values are found in the first Brillouin zone.
From the solutions (29) one can identify the oscillation branches: the acoustic one,
characterized by a linear dispersion relation in the long-wavelength limit (and a corresponding
vanishing oscillation frequency at the center of the first Brillouin zone), corresponds to the
first solution, for which
 Lattice oscillations. Phonons 11

   0
ac (0)  1 (0)  0 ,  ac   1  1   2   ac (0) , (30)
 a 2

while the second solution is associated to the optical branch, for which

   0
opt (0)  2 (0)  0 , opt    1  1   2  opt (0) . (31)
 a  2

-2/a 2/a

Because for the acoustic branch ac (0)  0 for k = 0, from the system of coupled
equations (26) it follows that (u1 / u 2 ) ac  1 , which implies that the displacement of the two
types of ions is the same/occurs in the same direction and the unit cell moves as a whole; it
oscillates in phase (see the figure above, bottom, right). On the other hand, in the long-
wavelength limit of the optical branch, (u1 / u 2 ) opt  M 2 / M1 , i.e. the ions are displaced in

opposite direction and we have out-of-phase oscillations. The oscillations occur such that the
center of mass of each ion pair is fixed, i.e. M1u1  M 2u 2  0 (see the figure above, top,
right). This oscillation branch is called optical because, when the unit cell consists of ions
with different type of charges (positive and negative), the oscillations form an instantaneous
dipolar moment that interacts strongly with the electromagnetic radiation. Unlike the acoustic
branch, the oscillation branch has no analog in the dynamic of continuum media.
In the long-wavelength limit, when sin(ka / 2)  ka / 2 , the dispersion relation of the
acoustic branch can be approximated as
 Lattice oscillations. Phonons 12

0   2  ka  2   0
 ac  1  1      ka , (32)
2  2  2   4

i.e.

1 A
ac (k )  vac k , vac   0a  a, (33)
4 2( M 1  M 2 )

while the dispersion relation for the optical branch becomes

0  2  ka 
2
  2  ka 
2

 opt  1  1       0 1     (34)
2  2  2   8  2 

or

opt (k )  0  k 2 ,   0 2 a 2 / 32 . (35)

In the long-wavelength limit, the dispersion relation of optical oscillations is parabolic.

Note that, to calculate the density of states for the finite one-dimensional lattice with
two atoms in the basis, we can follow the same treatment as for the atomic chain with one
atom in the basis, taking into account that we must calculate separately the density of states
for the two oscillation types, which have different dispersion relations.
In general, in a three-dimensional crystal with s atoms in the basis there are 3s
solutions/branches for a given k and the dispersion relation of the normal oscillation of the
-th branch satisfies the relation  (k )   (k ) since the oscillation frequencies are real
and positive. From these 3s branches, three are acoustic. The acoustic oscillation branches are
characterized by a linear dispersion relation in the limit k  0 and correspond to the situation
in which all atoms in the lattice have the same displacements (oscillate in phase), so that the
complex structure of the lattice is not manifest. The remaining 3s  3 oscillation branches
with a parabolic dispersion relation in the long-wavelength limit of the form

 (k )  0     k 2 (36)

are optical branches, characterized by the cut-off oscillation frequencies  0 ; the parameters

  are generally positive. In this case, ions with different signs oscillate in anti-phase, i.e. are
 Lattice oscillations. Phonons 13

displaced in opposite directions. As for the acoustic oscillations, in three dimensions we have
one longitudinal and two transverse optical oscillation branches for each s value.
In d dimensions, for a crystal with s atoms in the basis there are ds oscillation
branches, from which d are acoustic oscillations and d (s  1) are optical oscillations. These
general considerations are in agreement with the two examples considered above since, for
the case of the atomic chain with one atom in the basis we should have ds = 1 (for d = 1 and s
= 1), which corresponds to a single acoustical oscillation branch, whereas for the atomic chain
with two atoms in the basis we have d = 1, s = 2, and therefore expect two oscillation
branches, one acoustical and one optical.
The figure below shows the dispersion relation of a crystal, in which we can identify
only acoustical branches, meaning that there is only one atom in the basis of this crystal.

On the contrary, if optical branches exist in the phonon spectrum, the basis is usually formed
from more than one atom. In particular, as can be seen from the figure below, Si has optical
phonon branches; it crystallizes in a diamond-like structure, with two atoms (although
identical) in the basis.
 Lattice oscillations. Phonons 14

Density of states in a finite three-dimensional crystal

The density of states/oscillations in a three-dimensional crystal is obtained, as in the one-
dimensional crystal, by imposing the appropriate boundary conditions for u  un , the

index n being related to the unit cell and the index  labeling the atom in the basis. Because
there is a large number of atoms in a crystal, which interact strongly with their neighbors, the
contribution of the atoms at the surface of the crystal to any physical phenomena is negligible.
Then, we can employ again the Born-Karman cyclic condition for each atom in the basis:

u n  u (n N  )  (37)

with u n  u exp(ik  Rn0  it ) , where N  is the number of atoms in the x  direction,  =

1,2,3, with x1  x , x2  y , x3  z . As in the one-dimensional case, the wavevector

component along the  direction in the first Brillouin zone has a discrete spectrum,

2
k  m , (38)
N  a

with

N N
  m  (39)
2 2
 Lattice oscillations. Phonons 15

where a  is the lattice constant along x  , and m  are integers, i.e. it varies in steps of

k   2 / L , with L  N  a  the length of the crystal along the x  direction. The discrete
k values in a two-dimensional lattice are represented by points in the figure below.

ky

kx

allowed k values

It follows then that a state/oscillation occupies a volume in the k space given by

(2 ) 3
k  k1 k 2 k 3  , (40)
V

where V  L1 L2 L3  N is the volume of the crystal with sN  sN1 N 2 N 3 atoms that form a
lattice with a primitive cell of volume   a1a2 a3 and s atoms in the basis.
The density of states/oscillations in the k space is then defined as

dN osc 1 V
  . (41)
dk k (2 ) 3

The density of oscillations in the frequency space, defined as

dN osc dN osc dk
D( )   , (42)
d dk d

represents the number of oscillations (with different k values) that exist in the frequency
interval (,   d ) . In this formula dk is the volume in k space between the surfaces  (k )
and  (k )  d (k ) (see the figure below). The density of oscillations is discrete (as for one-
dimensional crystals) but, for sufficiently large crystals, the sum over the discrete states can
be replaced by an integral. We can calculate it observing that
 Lattice oscillations. Phonons 16

dk

dS

d
dk   dS dk    dS
| k |
, (43)
 ( k ) const  ( k ) const

from which it follows that

V dS
D( )   . (44)
(2 )  ( k )const |  k  |
3

We can express also the density of states as D( )d  [V /( 2 ) 3 ]dk , or

V
 D( )d  (2 ) 3  dk , (45)
1st BZ

which represents a particular case of approximating a sum over k in the first Brillouin zone by
an integral, approximation that for an arbitrary function f (k ) is

V V
 f (k )   f (k )k   f (k )dk . (46)
k (2 ) 3 k (2 ) 3 1st BZ

For f (k )  F ( (k )) , we have

 F ( (k ))   F ( ) D( )d (47)

k

if the density of states is normalized in each branch ,  = 1,…,s such that

max

 D( )d  3N (48)

0
 Lattice oscillations. Phonons 17

with  max the maximum value of the oscillation frequency in the branch.
Note that for the three-dimensional crystals, we have not defined the density of states
normalized at unit volume (as for the atomic chain with one atom in the basis), but have kept
the crystal volume throughout the calculations!

Quantized oscillations/phonons in a one-dimensional finite lattice with one

atom in the basis
We have seen that in the one-dimensional lattice with one atom in the basis, the ions act as
coupled harmonic oscillators. Here we show that this system of coupled oscillators can be
reduced to an equivalent system of independent harmonic oscillators by the introduction
of normal coordinates. Then, we associate a normal oscillation to each normal coordinate. The
Hamiltonian of the finite one-dimensional lattice with one atom in the basis can be written as

Np n2 1 N
H ion     Ann'u n u n' . (49)
n 12M 2 n,n'1

In the quantum treatment of the system of coupled harmonic oscillators, the position and
momentum coordinates become (conjugate) operators, such that

[uˆ n , pˆ n' ]  uˆ n pˆ n'  pˆ n'uˆ n  i nn' , [uˆ n , uˆ n' ]  0 , [ pˆ n , pˆ n' ]  0 . (50)

To proceed further, we must first note that the traveling-wave solutions considered in
the case of an infinite lattice studied above, i.e. u n (t )  u0 exp[i(kan  t )] , do not lead to real
displacements when finite lattices are accounted for. In this case, real displacements can be
obtained only by superimposing plane waves that travel in opposite directions, reflected at the
boundary of the finite one-dimensional crystal. For example, a solution could be

u n (t )  exp( it )[u0 exp(ikan)  u0* exp( ikan)] . Keeping this in mind, and replacing the
“amplitude” of the displacement and its complex conjugate with annihilation and creation
operators, we obtain the following form for the position and momentum operators:

1 
uˆ n   [aˆ k exp(ikna)  aˆ k exp( ikna)] , (51)
N k 2M (k )
 Lattice oscillations. Phonons 18

 i 2M (k )
pˆ n   [aˆ k exp(ikna)  aˆ k exp( ikna)] , (52)
2 N k 

where â k and â k are annihilation and creation operators of lattice oscillations corresponding
to the wavenumber k, which satisfy the commutation relations

[aˆ k , aˆ k' ]  aˆ k aˆ k'  aˆ k' aˆ k   kk ' , [aˆ k , aˆ k ' ]  0 , [aˆ k , aˆ k' ]  0 . (53)

The corresponding inverse relations are:

1 i 
aˆ k    M (k )uˆ n  pˆ n  exp( ikna)

(54)
2N n  M (k ) 
1  i 
aˆ k    M (k )uˆ n  pˆ n  exp(ikna)

(55)
2N n  M (k ) 

It can be easily shown that (50) are indeed satisfied by these expressions, since

[uˆ n , pˆ n' ] 
 i

 (k ' )
2 N k ,k '  ( k )

 [aˆ k , aˆ k' ]e i ( knk 'n') a  [aˆ k , aˆ k ' ]e i ( knk 'n') a 
(56)

i
e
2N k

ik ( n n ') a
 e ik ( nn') a  i nn'

The last equality follows from the fact that, for k  2m / Na in the first Brillouin zone, with
m an integer the sums in the equation above are given by

N exp(i 2l )  1
 exp( ikla)   exp(i 2ml / N )  exp(i 2l / N )  N l  N nn' (57)
k m1 exp(i 2l / N )  1

where l  n  n' is an integer. One can show, similarly, that the other two commutation
relations in (50) are satisfied, as well as the commutations relations in (53), using the
expressions in (54) and (55).
Then, using (57) and the symmetry relation  (k )   (k ) , we obtain for the kinetic
energy of the ensemble of oscillations the relation
 Lattice oscillations. Phonons 19

N
pˆ n2   N N
 2M   4 N   (k ) (k ' )  aˆ k aˆ k '  exp[i (k  k ' )na]  aˆ k aˆ k'  exp[i (k  k ' )na]
n 1 k ,k '  n 1 n 1

N N 
 aˆ k aˆ k '  exp[ i (k  k ' )na]  aˆ k aˆ k'  exp[ i (k  k ' )na]  (58)
n 1 n 1 

 
4 k
 (k )(aˆ k aˆ k  aˆ k aˆ k  aˆ k aˆ k  aˆ k aˆ k )

whereas the potential energy becomes, after similar calculations,

1
 Ann ' uˆ n uˆ n ' 


Ann '
aˆ aˆ
k k 'e
i ( kn k 'n ') a
 aˆ k aˆ k' e i ( knk 'n ') a
2 n,n ' 4 NM n ,n ', k ,k '  (k ) (k ' )
 aˆ k aˆ k ' e i ( knk 'n ') a  aˆ k aˆ k' e i ( kn k 'n ') a 
  2 (k ' )  N N N (59)

4N
  aˆ k aˆ k '  e i ( k  k ') na  aˆ k aˆ k'  e i ( k k ') na  aˆ k aˆ k '  e i ( k k ') na
 (k ) (k ' ) 
k ,k ' n 1 n 1 n 1

N  
 aˆ k aˆ k'  e i ( k  k ') na     (k )(aˆ k aˆ k  aˆ k aˆ k  aˆ k aˆ k  aˆ k aˆ k )
n 1  4 k

In (59) we performed first the sum over n' and used the equality  Ann' exp[ik ' (n'n)a] 
n'

M 2 (k ' ) , and then performed the sum over n using (57).

Observation: The equality  Ann' exp[ik ' (n'n)a]  M 2 (k ' ) is obtained introducing the
n'

solution u n (t )  u0 exp[i(kan  t )] in the (semi-)classical equation of motion:

Fn  Mun  M 2 u0 exp(ik ' na)   Ann'u n'   Ann'u0 exp(ik ' n' a)
n' n'

A fully quantum derivation of this equality, more exactly of the relation

M 2 uˆ n   Ann'uˆ n'
n'

is also possible starting from

uˆ n i ˆ
 [ H ion , uˆ n ]
t 
 Lattice oscillations. Phonons 20

N pˆ2 1 N
with Hˆ ion   n   Ann'uˆ n uˆ n' and employing the commutation relations. Then,
n12M 2 n,n'1

uˆ n i 1 pˆ
  [ pˆ n2' , uˆ n ]  n
t  n' 2M M

since

[ pˆ n2' , uˆ n ]  pˆ n2'uˆ n  uˆ n pˆ n2'  pˆ n' ( pˆ n'uˆ n )  (uˆ n pˆ n' ) pˆ n'  pˆ n' (uˆ n pˆ n'  i nn' )  (i nn'  pˆ n'uˆ n ) pˆ n'
 2ipˆ n' nn'

Analogously,

 2 uˆ n   pˆ n  i  ˆ pˆ n  i  An' j pˆ  A
  2 uˆ n      H ion ,     uˆ n'uˆ j , n     nn' uˆ n'
t 2 t  M    M   n', j 2 M n,n ' M

since Ann'  An'n due to its definition and

[  An' j uˆ n'uˆ j , pˆ n ]   An' j [uˆ n' ( pˆ n uˆ j  i nj )  (uˆ n' pˆ n  i n'n )uˆ j ]  2i Ann'uˆ n'
n', j n', j n'

Finally, the Hamiltonian of the system of ions becomes

 (k )  1  1
Hˆ ion   Hˆ k   (aˆ k aˆ k  aˆ k aˆ k )    (k ) aˆ k aˆ k      (k ) Nˆ k   , (60)
k k 2 k  2 k  2

with [ Hˆ k , Hˆ k ' ]  0 . Nˆ k  aˆ k aˆ k is the hermitic number operator, for which [ Nˆ k , Nˆ k ' ]  0 and

[ Nˆ k , Hˆ k ]  0 .
For each k mode, the eigenstates (Fock states) and eigenvalues of the Hamiltonian and
the number operators are

(aˆ k ) n
| nk   | 0k  , (61a)
nk !

 1
E k   (k ) nk   , n k = 0, 1, 2,…… (61b)
 2

where | 0 k  is the fundamental state of the oscillator, and

 Lattice oscillations. Phonons 21

aˆ k | nk   nk | nk  1 , aˆ k | nk   nk  1 | nk  1 , Nˆ k | nk   nk | nk  (62)

with aˆ k | 0 k   0 . The eigenstate of the lattice Hamiltonian for the N discrete k values is then

| nk1 , nk2 ,..., nk N  | nk1  | nk2   ... | nk N  (63)

and the energy of the collection of harmonic oscillators is

 1
E   E k    (k ) nk    E0  E ph . (64)
k k  2

where E0  k  (k ) / 2 is the zero (fundamental) energy, and E ph  k  (k )nk is the

energy of the quantized oscillations of the lattice in an excited state.

The state of each quantum oscillator k is the same as that of n k excitation quanta, each
with an associated energy  (k ) . This excitation quantum is associated to a quasi-particle
named phonon, in analogy with the photon, which is the quantum of the electromagnetic field.
In the one-dimensional lattice with one atom in the basis considered here, the phonon is called
acoustic phonon since there are only acoustic oscillation branches in this case, and the lattice
has n k1 phonons with wavevector k1 , nk2 phonons with wavevector k 2 , and so on.
Similarly, in a one-dimensional finite lattice with two atoms in the basis and N values
for k,

 1
Hˆ ion    (k ) aˆ k, aˆ k ,   , (65a)
k , 1, 2  2

(aˆ k1 ,1 ) (aˆ k1 , 2 ) ...(aˆ kN ,1 ) (aˆ kN , 2 )

nk 1,1 nk 1, 2 nkN ,1 nkN , 2

| nk1 , , nk2 , ,..., nk N ,   | 0 (65b)

(nk1 , )!(nk2 , )!...(nk N , )!

with | 0 | 0,0,...,0 , and

 1
E    (k ) nk ,   , (66)
k , 1, 2  2

where the phonons associated with the   1 branch are the acoustic phonons, and those
associated with   2 are the optical phonons.
 Lattice oscillations. Phonons 22

optical branches

acoustic branches

In a three-dimensional lattice with s atoms in the basis, there are 3 acoustic phonons
and 3s  3 optical phonons. A typical phonon dispersion spectrum for s = 2 is illustrated in
the figure above.
Phonons are quanta of the collective/thermal lattice oscillations. The crystalline lattice
can be viewed either as a collection of coupled harmonic oscillators or as a gas of free/non-
interacting phonons, which obey the laws of quantum statistics. In particular, phonons are
bosons and obey the Bose-Einstein (BE in the figure below) statistics. However, since they
are not real particles, their number is not independent of temperature and volume, so that the
electro-chemical potential of the phonon gas must be zero. Then, the thermal equilibrium
number of phonons with frequency  (k ) is given by the Planck distribution

1
n ph  nk ,  . (67)
exp[ (k ) / k BT ]  1

So, the number of phonons is small at low temperatures   k BT , for which
n ph  exp[  / k BT ] , but becomes high at large temperatures   k BT , for which

n ph  1 /[1  ( / k BT )  1]  k BT /  .

 Phononic heat capacity

The thermal properties of solids, and in particular the heat capacity, are determined by both
phonons and electrons. We refer now to the phonon, or lattice, contribution to the heat
capacity. The heat capacity is defined as the heat Q required to raise the temperature by
T , i.e. C  Q / T . If the process is carried out at constant volume, Q must be replaced
by U , which represents the increase in the internal energy U of the system. Then, the heat
capacity at constant volume is

 U 
CV    . (1)
 T V

The phonon contribution at the heat capacity is obtained from the lattice energy term

 1  1 1
E    k ,  nk ,      k ,    (2)
k ,  2  k ,  exp( k , / k BT )  1 2 

of the internal energy U  Eeq  E , with E eq the energy in the equilibrium configuration of
the system. Actually, expressing the internal energy of the system in terms of the free energy
F and the entropy S as

U  F  TS , (3)

apart from the heat capacity at constant volume CV  (U / T )V  T (S / T )V we can define
also a heat capacity at constant pressure, C P  T (S / T ) P . These two parameters are related

through C P  CV  T (V / T ) 2P /(V / P)T , and one can be determined from the other.
These parameters are the same only in the harmonic approximation of lattice oscillations.
In a classical statistical theory, based on the classical partition function, the mean
energy of a one-dimensional oscillator (resulting equally from its kinetic and potential energy
parts) is k B T , value that becomes 3k B T for a three-dimensional oscillator. Then, for sN

three-dimensional oscillators  E  3sNk BT , and the phononic heat capacity at constant

volume is
 Phononic heat capacity 2

 E 
C ph     3sNk B , (4)
 T V

i.e. C ph  3k B per atom, or C ph  3N A k B  3R  6 cal/molK per mole, with N A the Avogadro

number. This is the Dulong-Petit law, and it predicts a temperature-independent heat

capacity. This prediction agrees with experimental data at high temperatures, but not at low
temperatures, where experiments indicate that C ph  T 3  0 as T  0.

To explain the low-temperature behavior of the heat capacity, one should disregard the
classical statistical theory, which is no longer valid when the separation between the energy
levels of the oscillator is comparable to or higher than k B T , and use instead the quantum
statistical mechanics. The specific heat of the lattice is then defined as

  k , exp( k , / k BT )
2
dE 
C ph   k B    . (5)
 [exp( k , / k B T )  1]
2
dT k ,  k B T

This expression does not involve the zero energy of the lattice and is called for this reason the
phononic heat capacity. To specifically calculate the phononic heat capacity we need to know
the phonon dispersion relation. This relation is quite complicated for three-dimensional
crystals and therefore approximations are generally made.

The Einstein model

In this model each atom or molecule is considered as a particle that oscillates in the average
field of its neighbors. Therefore in the system with 3sN degrees of freedom all particles have
the same oscillation frequency  E . The phonon heat capacity is in this case given by

C ph (T )  3sNk B B2 ( E / T ) , (6)

where  E is the Einstein temperature, defined by  E  k B  E and

 2 exp( )
B2 ( )  (7)
[exp( )  1]2

is the Einstein function.

 Phononic heat capacity 3

 2 exp( )
At high temperatures, for    E / T  1 , B2 ( )   exp( )  1 , so that
(1    1) 2

C ph (T )  3sNk B , (8)

as in the Dulong-Petit law, but at low temperatures, for   1 , where B2 ( )   2 exp(  ) ,

the heat capacity has an exponential temperature dependence of the form

 
2

C ph (T )  3sNk B  E  exp(  E / T ) . (9)

 T 

Although lim C ph (T )  0 , the low-temperature dependence of the heat capacity is not

T 0

proportional to T 3 (see the figure above, right). The discrepancy is due to the inappropriate
treatment of the acoustic phonon contribution to the heat capacity. Unlike for optical phonons,
for which the frequency is almost constant as a function of k (see the figure below), the
frequency of acoustic phonons has a much wider interval of variation and the oscillations in
different lattice cells must be considered as correlated (the atoms oscillate in phase!).
Therefore, since the Einstein model describes in a satisfactory manner the optical phonon
contribution to C ph , the heat capacity is expressed as

C ph (T )  C ph
opt
(T )  C ph
ac
(T ) (10)

where
 Phononic heat capacity 4

opt
C ph (T )  3(s  1) Nk B B2 ( E / T ) (11)

and the contribution of the acoustic phonons is estimated from the Debye model.

The Debye model

In the Debye model the frequency of acoustic phonons in a general, anisotropic crystal is
written as

k ,   (k )  vac, ( ,  )k , = 1,2,3 (12)

with ,  the polar angles, and their contribution to the heat capacity is given by

  k , exp( k , / k B T )
2
3 
C ac
 k B    
 [exp( k , / k B T )  1]
ph 2
 1 k  k B T
(13)
  k , exp( k , / k B T )
2
3 
 k B     D( k , λ )d k , λ
 [exp( k , / k B T )  1]
2
 1  k B T

where

V dS  V k 2 d 
D( k , )    
(2 ) 3  ( k )  const |  k  | (2 ) 3  ( k )const | d k , / dk |
(14)
V  k , 2 d 
 
(2 ) 3  ( k )  const v ac, ( ,  )
3

ac
is the density of states. The frequency integral in the expression of C ph is performed between

0 and  max, ( ,  ) . If

1 3 d 3
  3   3 (15)
4  1 const vac, ( ,  ) vac

is an angular average of the acoustic velocity (the equality holds as identity in the isotropic
crystal), then we can introduce also an angle-independent maximum oscillation frequency (the
Debye frequency) max   D which, in the Debye model, is also independent on the
 Phononic heat capacity 5

polarization . This maximum oscillation frequency follows from the normalization condition
of the 3N acoustic oscillation branches:

3 D  1 3 d   V D 2 V D3
      
D ( k , )d k ,   4   v 3 ( ,  )  2 2  d 
 2 3
 3N , (16)
 1 0  1   const ac,   0 2 v ac

and so

 D  vac 3 6 2 N / V . (17)

In this case

2 D 4
3 V k B  k BT     exp( / k B T )
ac
C ph  3      d
2  2 v ac    0  k B T  [exp( / k B T )  1]
2
(18)
3
 T 
 9k B N   J 4 ( D / T )
 D 

where  D defined through  D  k B  D is the Debye temperature, and


x n exp( x)
J n ( )   dx (19)
0 [exp( x)  1]
2

is the Debye-Grüneisen integral, which has no analytical solution. The Debye temperature is
proportional to the acoustic velocity, and so is higher for high Young modulus values and for
lower crystal densities. It is usually determined by measuring the temperature dependence of
the resistance around the Debye temperature.
At high temperatures, for T   D , the argument in the J 4 integral is very small,
since x  1 , and after expanding it in series one obtains

3 3
 T  D / T
x 4 exp( x)  T  D / T
C ac
 9k B N    (1  x  1) 2 dx  9k B N     x exp( x)dx
2

 D
ph
 0  D  0
3
(20)
 T  D / T
 9k B N    x dx  3k B N
2

 D  0

while at low temperatures T   D the upper limit of the integral can be extended to , and
 Phononic heat capacity 6

3 3
 T   x 4 exp( x) 12 4  T 
C ac
 9k B N    dx  k B N    T 3 (21)
 D  0 [exp( x)  1]  D
ph 2
5 

This temperature dependence can be understood from a qualitative argument: at low

temperatures only the phonon modes with energy   k BT are excited. These modes are, in
the k space, inside a sphere (the thermal sphere), so that the number of modes is proportional
to k 3   3  T 3 . If each mode has an average excitation energy of k B T , the total energy of

excitation is proportional to T 4 and hence the heat capacity is proportional to T 3 .

The total phononic heat capacity, C ph  C ph
ac
 C ph
opt
, is now in agreement with

experiments for both high and low temperatures (see the figure below).

The Debye temperatures for some elemental crystals are given in the table below

Element D (K) Element D (K) Element D (K)

Li 344 Be 1440 Cu 343
Na 158 Mg 400 Ag 225
K 91 Ca 230 C 2230
Rb 56 Sr 177 Si 645
Cs 38 Ba 110 Ge 374
 Phononic heat capacity 7

As can be seen from the figure below, the Debye temperature in semiconductors increases as
the bandgap increases.

Although the Debye model is in good agreement with experimental data in semiconductors, it
cannot fit the measurements in metals, where the electron concentration is high, since in this
case the electon specific heat has a non-negligible contribution (see figure below). We will
refer to electron specific heat later.