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ENE 311 Lecture 3
Bohr’s model
• Niels Bohr came out with a model for hydrogen atom from emission spectra experiments.
• The simplest Bohr’s model is that the atom has a positively charged nucleus and negatively charged electrons.
• The total charge of all electrons is equal to that of the nucleus. Therefore, the whole atom is neutral.
Bohr’s model
Bohr made some statements about electronsas
1 . Electrons exist in certain stable and orbit circularly around the nucleus without radi
ating.
Bohr’s model
2. The electron could shift to a higher or low er level of energy by gaining or losing en
ergy equal to the difference in the energylevels.
Bohr’s model
3. The angular momentum p of electron in an orbit is
p n ; n = 1,2,3,...
Bohr’s model
• Hydrogen atom is the best model to study S chrödinger’s equation in3 dimensions.
• It consists of an electron and a proton. The p otential energy of electron at distance r fro
m the proton is
where ε0 = permittivity of free space = 8.85 x 10-12 F/m
2
0
( )4
eV r
r
Hydrogen atom
- We consider only time independent Schrödi nger’s equation since we are now interested
in the probability of electron to be found at a distance r from the nucleus and its allowe d energy levels.
22
2V E
m
Hydrogen atom
Use spherical coordinate (r,, ), ψ only depend s
on r. 2 2 2 22
2 2 2 2
2
x y z r r r
2 2 2
20
20
2 4
eE
m r r r r
Hydrogen atom
A solution of this equation is
Ψ -= exp( c0
r)
The energy of hydrogen can be calculated as
2
0 204
e mc
4
2 208
meE
h
Hydrogen atom
• Bohr further said that a stationary orbit is determined by the quantization of energy represented by 4 quantum numbers (n, l, m, s).
• Any combination of quantum number specif ies one allow state or energy level which is “
Pauli exclusion principle”.
Hydrogen atom
For hydrogen atom, quantized energy is
*For n = 1, this is called a “ground state”.
4
2 2 2 20
1 13.6 eV.
8n
meE
h n n
Basic Crystal Structure
Solids may be classified into3 groups:
- Amorphous
- Polycrystalline
- Crystalline
Basic Crystal Structure
1 .Amorphous: Atoms are randomly arra nged (formlessness).
Basic Crystal Structure
2. Polycrystalline: Atoms have many sm all regions. Each region has a well organ
ized structure but differs from its neighb oring regions.
Basic Crystal Structure
3.Crystalline: Atoms are arranged in a
n orderly array.
Basic Crystal Structure
• A collection of point periodically arranged is called a lattice .
• Lattice contains a volume called a unit cell which is representative of the entire lattice.
• By repeating the unit cell throughout the cry stal, one can generate the entire lattice.
Unit Cell
• Unit cell has an atom at each corner shared with adjacent cells.
• This unit cell is characterized by integral multiples of vectors such as a, b, and c called basis vectors.
• Basis vectors are not normal to each other and not necessarily equal in length.
Unit Cell
Every equivalent lattice point in the crystal
can be expressed by• - Two dimensional case:
R = ma + nb• - Three dimensional case:
R = ma + nb +pc
where m , n , and p are integers
Unit Cell
The smallest unit cell that can be repeated to form the lattice is called a primitive cell
shown below.
Unit Cell
Ex. 2-D lattice
r = 3a+2b
Basic Crystal Structure
• - There are three basic cubic crystal structures:
(a) simple cubic
- (b) body centered cubic, and
- (c) face centered cubic.
Simple Cubic (SC)
18 1 atom/unit cell
8
• Each corner of cubic lattice is occupied by an atom which is shared equally by eight adjacent unit cells.
• The total number of atoms is equal to
Simple Cubic (SC)volume of atom in unit cell
Atomic packing factor(PF) = volume of unit cell
3 3
3 3
4 41
3 30.52
(2 ) 6
r rPF
a r
For the case of maximum packing (at oms in the corners touching each oth
er), about 52% of the SC unit cell volu me is filled with hard spheres, and ab out 48% of the volume is empty.
Body-centered cubic (BCC)
• An additional atom is located at the center of the cube.
• The total number of atoms is equal to
18 1 2 atoms/unit cell
8
Body-centered cubic (BCC)
Ex. Ba, Ce, Cr, Mo, Nb, K
3 3
33
4 82
33 30.68
84
3
r rPF
a r
Face-centered cubic (FCC)
• One atom is added at each of six cubic faces in addition to the eight corner atoms.
• The total number of atoms is equal to
1 18 6 4 atoms/unit cell
8 2
3 3
33
4 164
23 30.742
62 2
r rPF
a r
Ex. Al, Ag, Au, Cu, Ne
The Diamond Structure
• This structure is like the FCC crystal family.
• It results from the inter penetration of two FCC
lattices with one displa ced from the other by o
- ne quarter of the distan ce along a body diagon
al of the cube or a displ acement of . 3 / 4a
The Diamond Structure
• If a corner atom has one nearest neighbor in the body diagonal, then it will have no nearest neighbor in the reverse direction.
• Therefore, there are eig ht atoms in one unit cell
.
• Examples for this structure are Si and Ge.
Zinc blende structure
• For the Zinc blende structure, this results from the diamond structure with mixed atoms such that one FCC sublattice has column III (or V) atoms and the other has Column V (or III) atoms.
• This is a typical structure of III-V compounds.
Ex. GaAs, GaP, ZnS, CdS
Fourteen Bravais lattices
Lattice systems
Lattice System Unit cell properties
Triclinic a b c β
Monoclinic a b c = β = 90°
Orthorhombic a b c = β = = 90°
Tetragonal a = b c = β = = 90°
Cubic a = b = c = β = = 90°
Hexagonal a = b c = β = 90°, = 120°
Trigonal a = b = c = β = 90°
Crystal Planes and Miller Indices
• The crystal properties along different planes are different.
• The device characteristics (not only mechanical properties but also electrical properties) are dependent on the crystal orientation.
• The way to define crystal planes is to use Miller indices.
• The Miller indices are useful to specify the orientation of crystal planes and directions.
Crystal Planes and Miller Indices
Miller indices can be obtained using this followi ng procedure:
1 . Determine the intercepts of the plane with crys tal axes (three Cartesian coordinates).
2. Take the reciprocals of the numbers from1.
3. Multiply the fractions by the least common mul tiple of the intercepts.
4. Label the plane in parentheses (hkl ) as the Mill er indices for a single plane.
Crystal Planes and Miller Indices
Ex. Find the Miller indices of the below plane.
Crystal Planes and Miller Indices
• Intercepts are a,3 a,2a:
(132)
• Take the reciprocals of these intercepts:
1,1/3,1/2
• Multiply the fractions b y the least common mu
ltiple, which in this cas e is6 , we get 6,2,3 .
• Label the plane as(623)
Crystal Planes and Miller Indices
The Miller indices of important planes in a cubic crystal.
Crystal Planes and Miller Indices
There are some other conventions as:
1. For a plane that intercepts the negative x-axis, for example,
2. {hkl}: For planes of equivalent symmetry depending only on the orientation of the axes, e.g. {100} for (100), (010), (001), , ,
hkl 100
100 010 001
Crystal Planes and Miller Indices
3. [hkl ]: For a crystal direction, such as[010] - for the y axis. By definition, the[010] direction is normal to (010) plane, a
nd the [111] direction is perpendicular to the (111) plane.
Crystal Planes and Miller Indices
4. <hkl>: For a full set of equivalent directions, e.g. <100> for [100], [010], [001], , ,100 010 001
Density of crystal
• Crystal with a lattice constant “a” and “n” atoms within unit cell, then the weight of unit cell is equal to the weight of atoms per unit cell.
• The density of the crystal will be expressedas
3
1.
A
nM
N a
Density of crystal
where = density of crystal [g/cm3]
M = atomic weight [g/mole]
NA = ========== ====== = 6.022 x 1023 atoms/mole
= number of atoms per unit volume
3
1.
A
nM
N a
3
n
a
Density of crystal
Ex. At 300 K the lattice constant for Si is 5.43 Å. Calculate the number of Si atoms per cub ic centimeter and the density of Si at room t
emperature. Note: Atomic weight of Si =28.09 g/mole
Density of crystal
Ex. At 300 K the lattice constant for Si is 5.43 Å. Calculate the number of Si atoms per cub ic centimeter and the density of Si at room t
emperature. Note: Atomic weight of Si =28.09 g/mole
Soln There are eight atoms per unit cell. Therefore, there are 8/a3 = 8/(5.43 x 10-8)3 = 5 x 1022 atoms/cm3
22 33
3 23
1 5 10 atoms/cm 28.09 g/mole. 2.33g/cm
6.02 10 atoms/moleA
nM
N a
Density of crystal
Ex. Find the density of Cu in [g/cm3 ]. Cu has a n atomic radius of 1.278 Å and atomic weig ht of 63.5 g/mole.
Density of crystal
Ex. Find the density of Cu in [g/cm3 ]. Cu has a n atomic radius of 1.278 Å and atomic weig ht of 63.5 g/mole.
• Soln Cu has an FCC structure.
Å
Number of atoms/unit cell is 4.
4 4 1.2783.615
2 2
ra
3
3323 -8
1.
4 63.58.928g/cm
6.02 10 3.615 10
A
nM
N a