Crystal structure

UEEP2024 Solid State Physics
Topic 1 Crystal Structure

Introduction
Solid state physics :
‒ The physics in condensed matter
"condensed matter" :
‒ a collection of atoms (or molecules) arranged
in a well defined lattice with long range order.

Definitions
• Crystal structure = lattice + basis
– A lattice is a set of regular and periodic
geometrical points in space
– A basis is a collection of atoms or molecules at a
lattice point
A crystal is a collection of atoms or
molecules arranged at all the lattice points.

Example
Given that the grains in a polycrystalline metal
are typically 50 m across and that metal ions
have a radius of 0.15 nm, estimate the
average number of ions in a grain and the
proportion of these ions which are adjacent to
a grain boundary. (Assume the grain is
roughly cubic in shape)

Solution
Volume of grain is = (50.0×10-6 m)3 = 1.25×10-13m3.
Volume of ion is = (0.30×10-9 m)3 = 2.70×10-29m3.
Number of ions per gain is
Surface area of grain≈(6)(50× 10-6 )2 = 1.5×10-8m2.
Area corresponding to one ion ≈(0.3× 10-9 )2= 9×10-20m2.
Number of ions adjacent to surface of grain is
Proportion of ions adjacent to grain boundary
.1063.4
1070.2
1025.1 15
329
313



 

m
m
.1067.1
109
105.1 11
220
28



 

m
m
.106.3
1063.4
1067.1 5
15
11






translation operation
• long range order
– One symmetry that all lattices must have is the
translation symmetry. This means that if one
moves along some axis by a certain distance, one
reaches another lattice point which looks the
same as the first point in all respects. This
movement is known as the translation operation
and is also the definition of long range order in a
crystal

Translation vectors
Mathematically, the crystal translation operation may be defined as:
r’ = r + l a1 + m a2 + n a3
(l, m, n are integers)
The quantities a1, a2 and a3 are the smallest vectors called the primitive
translation vectors.
T = l a1 + m a2 + n a3
T is the translation vector and any two points are connected by a vector of this
form

primitive and conventional cells
• A lattice can be formed by repetition of a cell
and the cell can be either primitive or
conventional
Note : The ways to define a primitive cell or conventional cell are not unique

Primitive Lattice cell
• A primitive cell is a minimum-volume cell.
• There is always one lattice point per primitive
cell.
• The volume of primitive cell with axes a1, a2
and a3 is
• The basis associated with a primitive cell is
called a primitive basis.
321 aaa cV

Wigner-Seitz cell
• A Wigner-Seitz cell is a primitive cell constructed by the following method:
– (i) draw lines to connect a given lattice point to all nearby lattice
points;
– (ii) at the mid point and normal to these lines, draw new lines or
planes;
– (iii) the smallest volume enclosed by these new lines or planes is the
Wigner-Seitz cell.

3-D Lattice types
• seven major groups of lattice types

• if non-primitive
cells (or
conventional cell)
are allowed, the
number will expand
to 14
Bravais
lattice
diamond

Diamond lattice Zincblende lattice
Si, Ge GaAs, GaP, AlAs, InSb

Lattice planes
• Lattice planes are flat parallel planes
separated by equal distance. All the lattice
points lye on these lattice planes

Miller indices
• Orientation of the lattice planes is specified by the
Miller indices (hkl).
• To determine the Miller indices:
1. Find the intercepts on the axes in terms of the lattice
constants a1, a2, a3. (The axes may be those of a primitive
or nonprimitive cell.)
2. Take the reciprocals of these numbers.
3. Reduce the numbers to three smallest integers by
multiplying the number with the same integral multipliers.
4. The results, enclosed in parenthesis (hkl), are called the
Miller indices.

Miller indices
This plane intercepts the a, b, c axes at 3a, 2b, 2c. The reciprocals
of these numbers are 1/3,1/2, 1/2. The smallest three integers
having the same ratio are 2, 3, 3, and thus the Miller indices of the
plane are (233).

Cubic crystal system
Simple cubic Body-centered
cubic
Face-centered
cubic

Simple Cubic (SC)
• Primitive translation
vectors
zac
yab
xaa
ˆ'
'
ˆ'







Body-centered cubic (BCC)
vectors
)ˆˆˆ(
2
'
)ˆˆˆ(
2
'
)ˆˆˆ(
2
'
zyx
a
c
zyx
a
b
zyx
a
a






orthogonal vectors of unit length

Face-centered cubic (FCC)
vectors
).ˆˆ(
2
'
)(
2
'
);ˆˆ(
2
'
xz
a
c
zy
a
b
yx
a
a







Example
Determine the actual volume occupied by the
spheres in a simple cubic structure as a
percentage of the total volume.

Solution
• Volume of cube is Vc = (2r)3 = 8r3.
• There are eight spheres of radius r each of
volume 1/8 of the sphere.
• Volume of sphere Vs =
• Percentage of volume occupied
 
3
4
3
4
8
1
8
33
rr 












%4.52%100
8
3
4
%100 3
3

r
r
V
V
c
s


Real Crystal Structures
(NaCl)
• FCC
• The basis consists
of one Na atom
and one Cl atom
Na Cl

(NaCl)
A 3x3x3 lattice of NaCl

(NaCl)
The (111) plane of NaCl The (100) plane of NaCl

(CsCl)
• Simple
cubic
A single unit cell of CsCl

(CsCl)
A 3x3x3 lattice of CsCl

(CsCl)
The (111) plane of CsCl The (100) plane of CsCl

• Zincblende
(GaAs)
A single unit cell of GaAs

(GaAs)
A 3x3x3 lattice of GaAs

(GaAs)
The (111) plane of GaAs The (110) plane of GaAs

X-ray Diffraction
• X-ray diffraction is
the most commonly
used technique for
studying the
structure of crystal
lattice.

X-ray Diffraction
Typical x-ray diffraction data

X-ray Diffraction
(Bragg Law)
• For constructive
interference  md sin2

X-ray Diffraction
(Bragg Law)
• the longest possible wavelength is for sinθ=1
and m = 1,
λmax = 2d
• This means that no diffraction is possible if the
wavelength is greater than this maximum. This
explains why we cannot study crystal structure
with visible or I.R. radiation.

X-ray Diffraction
For crystal
f(r) = f(r+T)
T = l a1 + m a2 + n a3
• Any local physical property of the crystal is
invariant under T, such as the electron number
density, magnetic moment density and etc.

X-ray Diffraction
1-D system
n(x) = n(x+pa)
p : arbitraty integers
Expand n(x) in a Fourier series

p
p apxinxn )/2exp()( 

X-ray Diffraction
3-D system
n(r) = n(r+T)
Expand n(r) in a Fourier series




cellc
G
G
G
rGirdVn
V
n
rGinrn
)exp()(
1
)exp()(



Reciprocal lattice vectors
If a1, a2 and a3 are primitive vectors of crystal lattice,
then b1, b2 and b3 are primitive vectors of
reciprocal lattice.
Reciprocal lattice vector G= v1 b1 + v2 b2 + v3 b3
(v1, v2, v3 are any integers)
b1 = 2π(a2xa3)/(a1•a2xa3)
b2 = 2π(a3xa1)/(a1•a2xa3)
b3 = 2π(a1xa2)/(a1•a2xa3)
primitive vectors of the reciprocal lattice

Reciprocal lattice vectors
• Properties of Reciprocal lattice vector G .
From these equations we observe the
following properties:
1. The vector b1 is normal to both a2 and a3 .
This is particularly simple for a cubic system
in which case we can see that the reciprocal
lattice is also a cubic system.
• For any component, say b1 , we have the
relations:
211  ab
01312  abab

X-ray Diffraction
• Every crystal structure has two lattices
associated with it, the crystal lattice and the
reciprocal lattice. A diffraction pattern of a
crystal is a map of the reciprocal lattice of the
crystal. A microscope image is a map of the
crystal structure in real space.

Diffraction condition
• Theorem- The set of reciprocal lattice vectors G
determines the possible x-ray reflections.
• The Diffraction condition is written as
where k is the wavevector of the beam.
2
2
2
02
G
or
G


Gk
Gk

X-ray Diffraction
• diffraction pattern of salt crystal

X-ray Diffraction
• Diffraction pattern of crystallized enzyme.

Brillouin Zones
• Brillouin zones is defined as a Wigner-Seitz primitive
cell in reciprocal lattice.
• The central cell in the reciprocal lattice is of special
importance in theory of solids, and is called the first
Brillouin zone.
• The first Brillouin zone is the smallest volume entirely
enclosed by planes that are the perpendicular bisectors
of the reciprocal lattice vectors drawn from the origin.
• Only waves whose wavevector k drawn from the origin
terminates on a surface of the Brillouin zone can be
difffracted by the crystal.

Reciprocal lattice to SC lattice
• The primitive translation vectors of a simple
cubic lattice may be taken as
• are orthogonal vectors of unit length.
• The volume of the cell is
• The primitive translation vectors of reciprocal
lattice are
.,, 321 zayaxa

aaa 
zyx

,,
.3
321 a aaa
.
2
,
2
,
2
321 zbybxb




















aaa


Reciprocal lattice to SC lattice
• The reciprocal lattice is itself a simple cubic lattice
of lattice constant 2/a.
• The boundaries of the first Brillouin zones are
planes normal to the six reciprocal lattice vectors
b1, b2, b3 at their midpoints:
• The six planes bound a cube of edge 2/a and of
volume (2/a)3. This cube is the first Brillouin
zone of the sc crystal lattice.
.
2
1
,
2
1
,
2
1
321 zbybxb




















aaa


Reciprocal lattice to bcc lattice
• Primitive translation vectors of bcc lattice are
where a is the side of the conventional cube.
• The volume of primitive cell is
• The primitive translations of the reciprocal
lattice are
);(
2
1
);(
2
1
);(
2
1
321 zyxazyxazyxa

 aaa
.
2
1 3
321 aV  aaa
     .
2
;
2
;
2
321 yxbzxbzyb




















aaa


Reciprocal lattice to fcc lattice
• Primitive translation vectors of fcc lattice are
where a is the side of the conventional cube.
• The volume of primitive cell is
• The primitive translations of the reciprocal
lattice are
);(
2
1
);(
2
1
);(
2
1
321 yxazxazya

 aaa
.
4
1 3
321 aV  aaa
     .
2
;
2
;
2
321 zyxbzyxbzyxb




















aaa


Crystal binding
• What holds a crystal together?
– Electrons and electrostatic forces play an important role in
binding atoms together to form a solid (crystal).
• Common types of crystal bindings:
– (i) Ionic bonding
– (ii) Covalent bonding
– (iii) Metallic bonding
– (iv) Hydrogen bonding
– (v) Van der Waals interaction

Crystal binding
• Cohesive energy (u)
– the energy required to disassemble the solid into its
constituent part (e.g. atoms of the chemical elements out
of which the solid is composed)
• For a stable, the cohesive energy has an attractive
term when the inter atomic distance is large (so that
the crystal can be formed), and a repulsive term
when the inter atom distance is short (so the crystal
will not collapse).

Crystal binding
The equilibrium distance between two atoms is given by

Ionic bonding
• When the difference in electronegativity between two
different types of atom is large, electrons will be transferred
from the low electronegative atom to the high electronegative
atom. The low electronegative atom will become a positive
ion and the high electronegative atom will become a negative
ion (e.g. Na + Cl → Na+ + Cl-). These ions will attract each
other by electrostatic force to form a solid.
• The repulsive force is due to the Pauli exclusion principle –
this prevents the crystal from collapsing.
• The attractive force is due to the Coulomb attraction between
the ions.

electronegativity
• Electronegativity is the average of the first ionization energy
and the electron afinity. It is the measure of the ability of an
atom or molecule to attract electrons in the context of a
chemical bond.

Covalent bonds
• When the electronegativiy between two atoms is
small, the two atoms can form covalent bond by
sharing a pair of electrons (one from each atom).
• Most atoms can form more than one covalent bond.
For example, C has four outer electrons and hence it
can form 4 covalent bonds.
• A crystal can be formed with one atom forming
covalent bonds with several other atoms.

Metallic bonding
• Atoms bounded by “free electrons”. Good
example is alkali metals (Li, K, Na, etc.)

Van der Waals interaction
• Coulomb attraction can occur between two neutral spheres,
as long as they have some “internal charges” so that the
neutral spheres can be polarized.
• The repulsive force is due to the Pauli exclusion principle.
• The attractive force is due to the Coulomb attraction

Van der Waals interaction
• Larger molecule ⇒ stronger Van der
Waals force
⇒ higher melting point.
For example:
He Ne Ar Kr Xe Rn
Increasing melting point
This is also true for many organic molecules.

Hydrogen bonding
• First ionization energy of atomic hydrogen is very high (13.6
eV). It is highly unlikely for hydrogen to form ionic bonding.
• The complete shell of hydrogen atom is 2 electrons and a
hydrogen atom has only one electron. It can form only one
covalent bond and it does not have sufficient bond to bind the
whole crystal together with covalent bond.
• However, the covalent bond between hydrogen and the other
atom (e.g. oxygen) can often be polarized,

Hydrogen bonding
• These polarized molecules will “stick” to each other
by Coulomb attraction. This is possible because the
hydrogen size is very small. For example, for water:

Crystal structure

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Crystal structure