Statistical Mechanics and Thermal Properties Flashcards by Owen Lawton

What is an eigenvalue equation?

An equation where an observable operator acts on a function (eigenstate) to produce a scalar eigenvalue and the eigenstate.

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Solve the Schödinger Equation for the infinite square well.

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How can we tell if eigenstates are orthogonal?

Integrate the an eigenstate multiplied by the conjugate of another over all space and if equal to zero then orthogonal.

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What does it mean for two eigenstates to be orthogonal?

They are independent.

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How many quantum numbers are associated with each degree of freedom?

1 per degree of freedom.

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Derive the constraint for a 1D particle that the separation of independent states in k-space is 2π/L.

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Define a microstate.

A microstate of a system is a unique quantum state of the system - it defiens the eigenstate of each particle in the system.

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Define a macrostate.

A macrostate of a system gives the macroscopic properties that come from the aggregate of all the particles in the system.

eg, internal energy U tells us the total energy of particles in the system but not the energies of the individual particles.

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What is the main apriori assumption of statistical mechanics?

Every microstate that is consistent with the constraints placed in the system is equally probable.

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State the Boltzmann Hypothesis.

Entropy is related to the probability of a system being in a certain microscopic quantum state, given its macroscopic equation of state.

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Is entropy extensive and why?

Yes it is, Stot = Sa + Sb = f(Wa) + f(Wb)

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Is W extensive and why?

No, Wtot = WaWb

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Boltzmann’s equation for entropy.

S = klnW

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Using the volume of a molecule and the volume of a box, find a value for the Boltzmann constant.

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State the second law of thermodynamics.

The equilibrium state of a system is the most probable macrostate and is therefore the one with the highest entropy. A non-equilibrium state will always have a lower entropy and will evolve in time to maximise its entropy and thereby attain equilibrium.

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Use Stirling’s approximation to find an expression for entropy in terms of the fraction of atoms.

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For large systems, what macrostate is most probable?

The one with the highest entropy.

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Derive an expression for temperature in the microcanonical ensemble.

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What is the microcanonical ensemble?

N isolated two level system.

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What is the canonical ensemble?

N two level systems in thermal equilibrium with a heat bath at temperature T.

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Give the expression for internal energy in the canonical ensemble.

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Give the low T limit for internal energy in the canonical ensemble.

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Derive the Schottky heat capacity for a two level system of spin in a magnetic field B.

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Find an expression for the internal energy and entropy of a system with n vacancies.

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Show using U and S that two systems thermally equilibrate.

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State the equation for Boltzmann distribution.

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Define the partition function.

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Give the probability of finding the system in an energy level (inc degeneracy).

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Give the energy eigenvalue solution to the Schrödinger eqn for the harmonic oscillator potential.

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Give the partition function for the harmonic oscillator approximation.

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Derive the mean value of n for the harmonic oscillator approximation,

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Derive the Einstein model for specific heat.

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Give the high and low T limits for the Einstein model for specific heat.

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Define the Einstein Temperature.

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What are the limitations of the Einstein Model?

The assumption that atoms act independently of each other is false. Near T = 0, heat capacity tends to be proportional to t^3 not exponential.

How can we generally estimate the specific heat capacity of a material?

1) Identify all degrees of freedom of the atoms/molecules/solid. 2) Work out the energy eigenvalues for those degrees of freedom (usually the degrees of freedom can be treated as independent so eigenvalues can be separated). 3) Establish how many degrees of freedom have quantisation energies less than kT and allocate k/2 of heat capacity to each.

Derive an expression for entropy in terms of probability.

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By substituting probability into the entropy expression, find an expression for Helmholtz free energy F, in terms of the partition function.

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Give expressions for entropy, mean internal energy and pressure in terms of F and then use these to give them in terms of Z.

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Give expressions for Cv, Bulk modulus and thermal expansion coefficient in terms of F.

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Give the expression for KE in a free electron gas.

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Using the partition function for free particles, derive an expression for Helmholtz free energy for N particles in a box.

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Using the einstein model, derive an expression for Helmholtz free energy.

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Give an equation that defines the Grüneisen parameter.

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How many bosons can be put into any specific quantum state?

Unlimited as not subject to the Pauli exclusion principle.

Give the equation of the Bose Einstein distribution.

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State the equation for the Fermi Dirac distribution.

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Which particles is the Bose Einstein distribution valid for?

Bosons

Which particles is the Fermi Dirac distribution valid for?

Fermions

What can be noted about the Fermi Dirac distribution?

At zero T, all the quantum states with EiEf At any temperature the a quantum state with Ei = Ef being occupied is 1/2 At a temperature T, the energies at which a large number of both occupied and unoccupied states coexist are in the range Ef±kT

Derive the density of states per unit volume in k-space in 3D.

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What are the assumptions of the freee electron model?

The electrons experience no interaction with the lattice points; only a uniform negative potential that extends over the volume of the crystal. That the electrons experience no interaction with each other. The electrons are in thermal equilibrium.

What is the total number of electrons per unit energy and per unit volume?

The product of DoS with the Fermi Dirac distribution function.

Give equations for the Fermi energy and the Fermi wave vector.

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Give the Dos in terms of the Fermi energy.

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In k-space, the thing that connects the ends of all the Fermi wave vectors is called what?

The Fermi surface.

Give the equation for the speed of electrons at the Fermi energy.

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Do free electrons near the Fermi energy in metals provide a contribution to the specific heat capacity?

Yes as the higher T changes the Fermi Dirac distribution in such a way to increase the total energy.

Define the Fermi temperature

Tf = Ef/kB

Give the expression for Cv found by arguing that on average all electrons with kBT of the Fermi energy are excited above their ground state.

Check (the one that used a Taylor expansion).

Find the dispersion relation for phonons in a chain and plot the result of w against k.

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Define the 1st BZ in terms of k.

-π/a < k < π/a

How many longitudinal vibrational modes are contained in the first BZ?

How many vibrational modes are there along a 1D chain in the 1st BZ?

What are the assumptions of the Debye model?

w = vk | The 1st BZ is approximately a sphere.

Define the Debye frequency.

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Use the Debye model to calculate the total energy of a lattice at thermal equilibrium and its Cv.

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Find approximate expressions for Cv from the Debye model in the high and low T limits.

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Give an expression for thermal conductivity from first year.

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Using the assumption that only the electrons within kT of the Fermi energy can be thermally excited, find an expression for thermal conductivity.

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Of what order is the mean free path for electron-electron scattering?

1µm | scattering lifetime ≈10^-12 (metals)

What is the relation between thermal and electrical conductivity?

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What is the order of the lifetime of scattering for electron-phonon scattering?

≈10^-14s (metals)

What are Umklapp processes?

Phonon-phonon scattering events where one phonon is kicked outside the 1st BZ since the conservation of momentum then includes a reciprocal lattice vector.

What do Umklapp processes depend on?

At least one phonon must have a wavevector that reaches at least halfway to the BZ boundary and the population of these phonons

Why do polymers make for good insulators?

They have a high degree of disorder which reduces the phonon scattering time. The strong and disordered intramolecular bonds impede phonon motion.

Why are metals good conductors?

They conduct by the thermal conduction of electrons.

Crystals with what also make good conductors?

High Debye temperature as this minimises Umklapp processes.

We expect Einstein temperatures to be higher for what types of atoms and bonds?

Higher for stiffer bonds and lighter atoms.