A Bravais lattice

Chap. 1 CRYSTAL STRUCTURE
1.1 Basic Concepts
• Materials: Large quantity of particular atoms or molecules.
Atoms : electrons + nucleus
Schrodinger equation describes the behavior of the electrons

• Hydrogen like atom : nucleus + single electron
The absolute square of the wave function is interpreted as the probability density for
finding the associated particle.

Small Corrections :
Spin –orbit effect
Relativistic effect
Hyperfine structure : (nuclear spin)
Lamb shift : (electrodynamics)

• Differences in energy of some particular pairs of states in the hydrogen atom. The state of lower energy is listed first.
The hydrogen atom is one of the most important dynamical systems in all of physics, for several reasons:
1.Hydrogen is the most abundant stuff in the known universe. About 92% by number of the nuclei
in the universe are hydrogen, 75% by mass.
2.Even though it is a relatively simple system, the physics of the hydrogen atom contains many
important quantum mechanical concepts that extend to more complex atoms and other systems.
3.Because of its relative simplicity, the hydrogen atom can be solved theoretically to very high
precision. Experimental measurements involving hydrogen thus offer very sensitive tests of
modern physical theories, like quantum electrodynamics.
Ref : http://www.pha.jhu.edu/~rt19/hydro/node5.html
To see the wave functions, refer to
http://www.webelements.com

• Atoms with many electrons
Electron – electron interaction
V
int
2
e
= ∑
r i − r
i,
j
j
Hatree approximation
Pauli exclusion principle
No two electrons in an atom can be in the same quantum state
Total electronic wave function must be antisymmetric
Exchange interaction
Hatree –Fock approximation
1s
2s 2p
3s 3p 3d
4s 4p 4d 4f
5s 5p 5d 5f 5g
Not exact
Ex : Co
Atomic number : 27
1s(2) 2s(2) 2p(6) 3s(2) 3p(6) 4s(2) 3d(7)

* Atomic orbitals
http://www.shef.ac.uk/chemistry/orbitron/AOs/1s/index.html
1s
2s
2p
3s
3p
3d

• What is Solid ?
Crystal: Periodic array of atoms, Translational symmetry
• Solid state Physics :
- Study of what may be called the material properties of solid bodies.
- Physics relevant to crystals and electrons in crystals.
Ex: · dielectric constant of a crystal versus electrostatic field distribution
of a specimen of particular shape.
· Electronic band structure of a semiconductor crystal
Electron’s spin
Crystal binding
Electronic energy
Crystal structure (real, reciprocal)
Lattice vibration
• Why Crystals ?
Minimum energy configuration of the interaction between atoms
=> Equilibrium structure at zero temperature.
Non crystalline solid : glass, amorphous

1.2 Concepts of Lattice
Crystal: Periodic array of atoms.
Lattice: Mathematical (Geometrical) structure of crystal
(Constructed by the equilibrium position of atoms)
Physical Crystal Structure = lattice + basis
basis : atoms or molecules attached to every lattice point identically
• Fundamental Translational vector, Basic vectors a 1 , a 2 , a 3
atomic structure remains invariant under translation through any vector
which is the sum of integral multiples of these vectors.
• Primitive Translational Vectors : R lmn
= r o
+ la 1
+ ma 2
+ na 3
• Primitive Cell: constitute building block (cell) with the smallest volume

• Unit Vectors : Obvious, convenient choice of translational vectors. – show crystal symmetry
• Unit Cell : parallelepiped constructed with unit vectors
Basis vectors:
Volume of the unit cell
• Wigner-Seitz Cell :
A primitive cell that shows the crystal symmetry related to a crystal clearly.
Parallelepiped constructed by perpendicular bisector planes of the translation
vectors from the chosen center to the nearest equivalent lattice sites.
1. Draw lines to connect all nearby lattice points.
2. At the mid point and normal to these lines, draw new lines
perpendicular to the lines.
3. The smallest volume enclosed is Wigner-Seitz Cell.
Wigner-Seitz Cell of BCC

1.3 Classification of Bravais Lattice
• A Bravais lattice
An infinite array of discrete points with an arrangement and orientation
that appears exactly same from whichever of the points the array is viewed.
Primitive cell
Honey comb lattice is not a Bravais Lattice
FCC is a Bravais Lattice

• Symmetry Operations : (Space Group)
A Bravais lattice is characterized by the specifications of all rigid operations
that takes the lattice into itself.
=> needs group theory (we would not go into the details)
•Crystal structures are classified by their symmetry operations :
- translation by fundamental lattice vectors
- rotation about an axis
- reflection in a plane
- inversion
v v
r r
→ −
Understanding Crystals would be much easier if we group similar crystals together
according to their symmetry property.
Ex : cubic
: rotation through 90 about a line of lattice points in a direction
: rotation through 120 about a line of lattice point in a direction
: reflection of all points in a {100} lattice plane
2π/4
2π/3

• Point operations : (Point Group)
Symmetry operations that leave at least one lattice point fixed.
Theorem : Any symmetry operation of a Bravais lattice can be compounded out of a
translation T and a point operation.
Proof : If a symmetry operation S takes O ⌫ R, Consider operation T -R
T -R
S is also symmetry operation that leaves origin fixed => point operation
p
90º rotation
S = (T -R
S)T R
p
p
Rotation around
translation by a
(Point operation)

• 5 Bravais Lattice in Two-Dimensions
a) square lattice a = b α = 90
b α
a
b) rectangular lattice a b α = 90
c) hexagonal lattice a = b α = 60
Bravais lattice can contain only 2-,
3-, 4-, 6-fold axes.
d) Centered rectangular lattice a ≠ b α = 90
e) Oblique lattice a b α = arbitrary

• Bravais Lattice in Three Dimensions
Number of Point Group : the 7 crystal systems
Number of Space Group : the 14 Bravais lattices
- Basis of spherical symmetry
* Crystal Structure including non Bravais lattices
Number of Point Group : the 32 crystallographic point groups
Number of Space Group : the 230 space groups
- Basis of arbitrary symmetry
a
β
c
γ
α
b

• Cubic lattice : a=b=c α=β=γ=90
simple cubic (SC) face centered cubic (FCC) body centered cubic (BCC)
- Simple Cubic
* Primitive lattice vectors :
ax$ ) )
, ay,
az
* Millar Indices
(100) (110) (111)
* Crystal Planes : (100) (010) (001) (-100) (0-10) (00-1)
Planes of equivalent symmetry {100}
Crystalline axis directions : [100] [010] [001] [-100] [0-10] [00-1]
Directions of equivalent symmetry

Body Centered Cubic (BCC)
Wigner-Zeits Cell
- BCC : Li, Na, K, Rb, CS, Fe , Mo, W
* basis : (000) (1/2 1/2 1/2), primitive lattice vector : a/2 (x+y-z), a/2(x-y+z), a/2(-x+y+z)
* Unit cell : Simple Cubic # of atoms in a unit cell : 2
- Face Centered Cubic (FCC)
Rhombohedral Primitive cell
Wigner-Zeit Primitive Cell
FCC : Ne, Ar, Kr, Xe, Ag, Cu, Au, Pt
basis : (000) (0 1/2 1/2) (1/2 0 1/2) (1/2 1/2 0) primitive lattice vector : a/2 (x+y), a/2(x+z), a/2(y+z)
# of atoms in a unit cell : 4

ABCABC stacking
First layer
Second layer
Third layer
b) Tetragonal lattice a = b ≠ c α=β=γ=90
simple tetragonal
centered tetragonal
In, Sn
* face centered tetragonal is identical to body centered tetragonal
* FCC and BCC is a special case of the centered tetragonal

c) Orthorhombic a ≠ b ≠ c α=β=γ=90
simple orthorhombic
base centered orthorhombic
body centered orthorhombic
face centered orthorhombic
d) monoclinic a ≠ b ≠ c α=β=γ≠90
simple monoclinic
centered monoclinic
e) triclinic a ≠ b ≠ c α≠β≠γ
f) trigonal a=b=c α=β=γ

1.4 Important Crystal Structure
• Diamond Structure
- Bravais Lattice : FCC
- 2 Basis at (000), (1/4 1/4 1/4)
- Group IV elements, semiconductors : C, Si, Ge, Sn
- Relatively empty, Directional covalent bonding , Tetrahedral
• ZincBlend Structure
Diamond Structure with two different atoms in each FCC lattice
Compount Semiconductors
GaAs, ZnS, SiC, InAs

• Hexagonal Close-packed Structure
Bravais Lattice : Hexagonal Lattice
He, Be, Mg, Hf, Re (Group II elements)
ABABAB Type of Stacking
a=b a=120, c=1.633a,
basis : (0,0,0) (2/3a ,1/3a,1/2c)

•Sodium Chloride Structure
Bravais Lattice : Face Centered Cubic
Basis : (000) (1/2 1/2 1/2)
NaCl, MgO, KCL, MnO (Ionic bonding materials: Rock Salts)
• Cesium Chloride Structure
Bravais Lattice : Simple Cubic
Basis : (000) (1/2 1/2 1/2)
BeCu, CsCl, CuZn