4. HYDROGEN ATOM
Hydrogen atom consists of a proton and a single electron. The electron is bound to the
proton by Coloumbic potential, V(r). The wavefunction (r,) of the electron can be
best described in the spherical coordinate system. This wave function
8. BOSE-EINSTEIN CONDENSATES
BOSE EINSTEIN CONDENSATES
Fermi-Dirac distribution defines the average occupation number of free fermions at
state a in equilibrium temperature T and chemical potential . The Bose-Einstein
distribution defines the average occu
6. METALS: Fermi-Dirac Distribution
Metals: Fermi electron gas model
So far we have looked at systems that obey the Boltzmann distribution, however since electrons
must also obey the Pauli Exclusion Principle then we should really use the Fermi-Dirac
dist
6. METALS: Thermal Properties
Drude model: Electronic Heat capacity
Classical heat capacity for solids was shown to be 3R (Dulong-Petit) from the equiparition
theorem. Similarly every free electron in metals have average kinetic energy of 3/2 kBT.
E
n
3
N
7. SEMICONDUCTOR: Hall Effect
When a current carrying material is placed under a magnetic field, its carriers will
initially deflected and accumulate on one side until there is enough induced electric field
to counter further deflection. This is Hall effe
6. METALS
Metals: Drude model
1900 DRUDE proposed that the high thermal and electrical conductivities of metals are due to
the large concentrations of carriers within the material.
1909 Lorentz improved on the Drude model by treating the free electrons as
6. METALS: Hall Effect
Hall Effect
Hall measurement or four-point probe can be used to measure the above mentioned electrical
properties. If electric field and magnetic field is applied in a material, the carriers would
experience Lorentz force. Consider
5. ENERGY BANDS: Nearly Free Electron
Nearly Free Electron: Bragg Condition and Energy gap
The origin of the band gap can be based on electron wave diffraction at the BZ edges.
Recall that the wave vectors from the BZ center and to its edge must always sa
6. DENSITY OF STATES
From classical mechanics the total energy of the system is described by the Hamiltonian
equation (H) which is the sum of the kinetic (T) and potential energies (V).
H T V
T
p2
2m
H
p2
V
2m
classical particle
Under quantum mechanics t
5. ENERGY BANDS: Tight Binding Model
Energy Band: Tight binding (linear combination of atomic orbitals LCAO)
Recall from the quantum treatment of the hydrogen molecule that the electronic
wavefunctions were combined, (via linear combination of atomic orbi
7. INTRINSIC SEMICONDUCTOR
SEMICONDUCTORS IN EQUILIBRIUM
A semiconductor is a crystalline material having an energy structure wherein the
electronic states are completely filled (valence band) at zero temperature and those states
that are completely empty
6. SCATTERING MECHANISMS
Scattering times
From the electrical and thermal conductivities derived in terms of the scattering times, we can
show that the Wiedemann-Franz ratio and the Lorentz number would now become;
1 2 2 T 2
n k B
vF
6
F
and
1
T 2 m *
n
7. EXTRINSIC SEMICONDUCTOR
Impurity semiconductors have excess carriers coming from the dopants (impurities)
incorporated into the material. These dopants affect the band structure and thus cause
many interesting properties of semiconductors. A dopant is
8. SUPERCONDUCTIVITY: Ginsburg-Landau
Thermodynamics studies of the phase transition at T=TC would yield the heat capacity
change. We start with the discussion of the Helmholtz Free Energy (F) and then the
Gibbs Free Energy (G) function.
The Helmholtz Fre
8. SUPERCONDUCTOR PROPERTIES
SUPERCONDUCTIVITY
BASIC EXPERIMENTAL RESULTS
A. Zero resistance
Kamerlingh Onnes discovered in 1911 that when mercury (Hg) was cooled to 4.2K, its
resistance becomes extremely small (zero). Most elemental solids exhibit such z
3. HEAT CAPACITY
HEAT CAPACITY of a crystal lattice:
A crystal can be considered as a solid wherein the movements of the atoms influence the
properties of the solid.
The total thermal energy E1 of a single vibrating atom (oscillator) is the collection of
4. ATOMIC BONDING
Atomic Bonding in Solids
Crystals are held together by the electrostatic forces between the electrons and the nuclei.
This cohesive energy often dictates the properties of the solid; such as melting point and
bulk moduli. Here the contri
3. THERMAL EXPANSION
ANHARMONICITY: Thermal Expansion
Due to the anharmonicity of the potential energy function some properties are observed.
Ideally the potential energy function of a harmonic oscillator is parabolic with displacement.
U ( x) cx 2
U ( x
4. Review: Quantum Mechanics I
Free electron:
From classical mechanics, the total energy of the system is described by the Hamiltonian
equation H which is the sum of the kinetic T and potential energies V.
H T V
T
p2
2m
p2
H
V
2m
classical particle
Under
5. ENERGY BANDS: Free Electron Model
Energy Bands: Free Electron, empty lattice model
Recall that for isolated atoms, the probability density * decreases with distance and
the discrete energy states are created. And when two such atoms are brought togethe
2. RECIPROCAL SPACE
RECIPROCAL SPACE
We can take advantage of the periodic nature of a crystal to define its physical properties.
Physical properties like local electron density, electrostatic potential can be represented
by some periodic function in real
4. HYDROGEN MOLECULE
We begin our study of molecules with the Hydrogen molecule (H2). For simplicity, let us
assume that the nuclei and electrons are point-masses and neglect spin. The molecular
Hamiltonian would be;
Z Z q 2
2
1 2
2
Z q 2
q2
2
H
i
r
1. CRYSTALS
CRYSTAL STRUCTURE
Solid state physics is largely concerned with crystals and electrons in the crystals and
their effect on the material characteristics.
An ideal crystal is an infinite repetition of identical structure or UNIT CELL in a certai
3. LATTICE VIBRATIONS
PHONONS: Lattice vibrations
Elastic vibrations propagate in a continuous media (for instance, a crystal). Consider an
isotropic slab of length x of uniform cross sectional area yz. Assuming an elastic
strain is applied along the x-ax
8. SUPERCONDUCTIVITY: London Theory
LONDON THEORY proposed in 1935 used a two fluid model at T=0, wherein the
average density of particles consists of super fluid (nS) and normal fluid (nN) and
with associated average speeds vS and vN, which resulted into
6. ELECTRONIC HEAT CAPACITY
Fermi Model: Electronic Heat Capacity
In the Drude model, electrons are treated as free electrons that have 3 degrees of freedom. From
3
the equipartition theorem, the classical electronic heat capacity is CV ,el Nk B .
2
Assum
Smart Materials
AN D ITS M E C H AN IC AL B E H AVIO R
MSE 251 Report
Alexander C. Guevara (201491234)Alberto Amorsolo PhD
Smart Materials and its mechanical
behavior
Background
Types of Smart Materials
Smart Materials deep dive studies
Applications o