### Notes about the lectures, Spring Semester, 2001

Updated 21.05.01
```          Questions for exercises from 1998
quest98.ps (Postscript file)

22.01.01  Introduction.
Walking through the topics, about molecular physics
Interesting phenomena
Quantum Mechanics, Statistical Physics
Structure

24.01.01  Quantum Mechanics Highlights
Density - squre modulus of wave function
Expectation values in classical probability
Quant.Mech. expectation values
To every quantity - an operator
Hamilton operator
H Psi = E Psi
Eigenvalues - discrete solutions (sometimes continuous)

Commutator
Heisenberg (Heisenberg in acoustics)
(Born in optics ..... sideremarks)

Harmonic Oscillator - Newton
Harmonic Oscillator - Quant. Ladder spectrum (equidistant)
hbar omega (circular frequency vs. frequency)

Sideremarks, illustrations:
Rydberg atoms - see later impurity states in semiconducters

Crystals as huge molecules
Born-Oppenheimer approximation
Fix the nuclei, do quant. mech. for electrons
Electron energies depend on position of nuclei
(for 2 atomic nuclei just the distance)

Splitting of levels, origins of bands
This might be seen as origin of the bonding
The lower level goes down (the united nuclei)

29.01.01  The linear chain of masses on a spring
Dispersion relation
Ashcroft-Mermin chapter 22 pages 430-433
Mentioned phase velocity, group velocity (will be done later)
relation between omega and k - the 'wave vector'

Relation for the density of modes (a note will be prepared) -NOTE
Densities (for lattice vibrations; explanation to be added)
modedens.gif
modedens.m

String with two types of atoms (two masses, two branches)
Ashcroft-Mermin chapter 22 pages 433-435,
see fig. 22.10; two spring constants, not two masses

31.01.01  Vibrations of 2-dim and 3-dim lattices
Density of modes goes as omega squared
Einsteins model for heat capacity
(note, picture of the page, jpg 148kB)
calculation details, jpg 81kB

05.02.01  Debye's model calculation  note, jpg 127kB
Table of Debye temperatures table30.jpg, 80kB
High resolution, same table table50.jpg 150 kB
Origin of the Boltzmann factor NOTE
Einstein vs. Debye model: Ashcroft-Mermin page 462
.

07.02.01  Elementary transport theory
Fourier's law - heat conductivity
(deriving expression for kappa in Fourier's law)
Group velocity

Algebraic Method - harmonic oscillator (for phonons)   *NOTE*
Eigenmodes (diagonalization of frequencies square...)  *NOTE*

12.02.01  Computer (Matlab) Simulations
Chains - showing the addition of wave numbers
types of vibrations - waves - demonstration
Diagonalization of the coupled Harm.Osc. matrix

back to phonons
mean free path
```
```	  Umklapp process - see chains above
Unharmonic effects: terms of u^3 type
Heat conductivity: T-dependence ( T^3 low T, T^(-1) high)

Thermal expansion
.

End of first part

14.02.01  Heat and electrical conductivity of electrons
CLASSICAL MODELS
a) HEAT CONDUCTIVITY - see phonons
b) electrical conductivity
Drift velocity from acceleration
Conductivity (relation to resistance)
Conductivity (sigma) for Cu as an example
Wiedemann-Franz law derived
Lorenz number
CLASSICAL ASSUMPTIONS:
a) the gas heat capacity (related to Boltzmann)
b) velocity squared replaced by kT

19.02.01  FERMI GAS MODEL
Pauli principle
Independent particles-independent probabilities
Indistinguishable particles
-- new form of probability P(r1,r2)=P(r2,r1)
Slater Determinant
Available states (density of states in k-space)
NOTE (scanned) fermi-density.jpg
Fermi momentum, Fermi energy, Fermi temperature

Lu: No two leaves are the same (Chinese saying)
(just opposite of the Indistinguishable particles)

21.02.01  Fermi gas - at nonzero T
Heat capacity etc  (1 hour only)
Sommerfeld integrals: Fermi Gas
In Ashcroft-Mermin: Sommerfeld integrals are pages 45-47
FermiSom.ps  (postscript file, overview)
Scanned:
sommer-meth-1.jpg (Sommerfeld's method)
sommer-meth-2.jpg
sommer-eval.jpg (Evaluation details)

21.02.01  Fermi gas - at nonzero T
Heat capacity etc  (1 hour only)
Wiedemann-Franz law in Fermi-gas version

26.02.01  Bloch theorem

28.02.01  Bloch theorem via Fourier
NOTE: Bloch.ps (postscript file)

05.03.01  Fourier, bands weak potential limit
periodicity in the k-space

07.03.01  started semiclassical motion (forces) (1 hour only)
equation for u-function (see note)

12.03.01  semiclassical motion
Selfconsistent method (background information - see textbook)

14.03.01  3-dim crystals electronic bands
the shape of bands
Notes for the model construction of 3-dim bands
3dim-1.jpg
3dim-2.jpg
3dim-3.jpg

19.03.01  -- no lecture
21.03.01  -- no lecture

26.03.01  Semiconductors
Energy Gap: diagram of resistivity, photon excitation
Evaluation of number of electrons in bands (textbook)
Semiconductor equation (textbook)

28.03.01  Semiconductor equation (textbook)
Impurity states (textbook)
a) Donors (P in Si)
(the 'quasiatoms'; see discussion of Rydberg atoms,
sideremark 24.01.01)
b) Acceptors (Al in Si)
HOLES and ELECTRONS

02.04.01  p-n junction 7 page NOTE
The placement of Fermi level in acceptor and donor doped
semiconductors. The mechanisms leading to the common
position of Ferni level.
Diffusion current and drift current.
Fick's law for diffusion. Diffusion constant.
Mobility.
Einstein-Nernst relation

04.04.01  p-n junction 7 page NOTE, continued
Model for charge density distribution
Model for the depletion zone
The rectifying function of p-n junction

Questions for exercises from 1998
quest98.ps (Postscript file)

18.04.01  Metals, Fermi surface, thermoelectricity ....
Explaining the fermi surface in 2 dimensions
Electron motion in magnetic field
Cyclotron Frequency ... add formulae here (new *NOTE* ... missing)

Metals as poly-micro-crystalline
Matthiesens rule for conductivity (temperature dependence)
.
To be added: typical values of v_F, l, tau (relax. time)
to be added: Matth. rule relations
( rho sigma l) (sigma cross section)
(sigma conductivity; sigma dep. on tau, tau dep. 1/T)
(sigma dep. sigma_0 / ( CONST + T )
.
23.04.01  Hall effect   NOTE Hall effect, 2 pages
Hall effect in semiconductors, mobility: Not in Ashcroft-Mermin
.
Magnetic field/Magnetism basics    NOTE (background information 4 pages)
(Table of systems of units;  JACKSON's book)
System of units (physics the same, but even concepts different)
.
25.04.01  Magnetic properties of materials
Diamagnetism and Paramagnetism      NOTE  3 pages
Diamagnetism-atomic origin
Understanding of negative susceptibility
Ashcroft-Mermin chapter 31 pages 648-649
.
30.04.01  Paramagnetic response
The g-factor
Thermic effects, evaluation of average spin projection
Brillouin function for general J and for J=1/2
Ashcroft-Mermin chapter 31 pages 653-656
Paramagnetism of metals (picture explains easily)
Ashcroft-Mermin chapter 31 pages 661-662
(Ash.Mer. too detailed on this point)
.
02.05.01  Ferromagnetism          NOTE 3 pages
The effective spin-spin interaction
Mean Field theory
(the explanation of permanent magnetization)
.
07.05.01  Superconductivity
.
Ashcroft-Mermin chapter 34.
09.05.01  Crystal systems, Crystal Structure
Crystal lattices
.Ashcroft-Mermin chapter 4, chapter 7; parts only
Determination of crystal structures
Ashcroft-Mermin chapter 6, parts only

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