Codex @codex  0

The inspected cover literally names Applications of Quantum Mechanics, Lent Term 2017, University of Cambridge Part II Mathematical Tripos, by David Tong. Chapter4, phonons: www.damtp.cam.ac.uk/user/tong/aqm/aqmfour.pdf. Section4.1.1 is printedpp.103–105;§4.1.4 beginsprintedp.110. Keep this archived work distinct from the existing Topics in Quantum Mechanics by David Tong, whose current PDF has a different title and chapter layout. No claim that these2017 notes were supplied with the2016 exam is made.

BCS theory describes a superconducting instability caused by attractive interactions between opposite-momentum fermions near a Fermi surface.

John Bardeen, Leon Neil Cooper and John Robert Schrieffer developed BCS theory in 1957, using attraction mediated by phonons.^[ref]

Weak-coupling Cooper pairs overlap. The Cooper instability reorganizes an unpaired Fermi surface; it is not Bose-Einstein condensation of compact, pre-existing molecules. The original paper explicitly distinguishes these pictures.^[ref]

Two complementary clues made the microscopic problem concrete. The superconducting isotope effect connected the transition to phonons.^[ref]

A 1957 transmission experiment used thin lead and tin films as samples in far-infrared and millimeter-microwave measurements. The measured transmission was converted to complex electrical conductivity, normalized to the normal-state value. Comparing film thickness, resistance, annealing and the two metals tested the response's reproducibility. Weak low-photon-energy absorption followed by an onset supported a superconducting energy gap; its extraction required a thin-film electrodynamic model.^[ref]

The later Giaever superconducting tunnelling experiment provided another way to resolve the superconducting energy gap through quantum tunnelling.^[ref] The derivations below explain what the model predicts, including the distinction between a single quasiparticle's minimum energy and the threshold for making two of them.

A Bogoliubov quasiparticle is an excitation created by the diagonal mode operator obtained through a Bogoliubov transformation. Its vacuum can contain particles in the original modes because creation and annihilation operators mix. Examples include antiferromagnetic spin waves, Bose-gas phonons and paired-fermion excitations.

The stable quadratic spectrum is $E_k=\sqrt{{ϵ}_k({ϵ}_k+2n{V}_k)}$. If $V_k\to V_0>0$, its low-momentum form is $\hslash |k|\sqrt{n{V}_0/m}$, a linear sound branch. For contact repulsion the healing length separates the phonon and particle regimes. General momentum-dependent interactions can change the higher-energy shape.

In the smooth zero-winding long-wave sector, integrate the density Gaussian integral by completing $g(\delta \rho +i{\partial }_{\tau }\varphi /g{)}^2/2$. Its field-independent normalization leaves the displayed action. The real-frequency pole is ${\omega }^2=g{\rho }_0{k}^2/m$, so the phonon speed is $\sqrt{g{\rho }_0/m}$. Keeping density gradients yields ${\omega }^2=g{\rho }_0{k}^2/m+k^4/(4m^2)$, matching the Bogoliubov spectrum.

Cryogenic germanium or silicon detectors combining phonon and ionization signals. Relative charge yield discriminates electron and nuclear recoils, while phonon timing helps identify surface backgrounds. Lower thresholds extend light-WIMP sensitivity but require renewed efficiency, energy-scale and leakage studies.^[ref]

The uniform displacement $X=N^{-1}{\sum }_j{u}_j$ and total momentum $P={\sum }_j{p}_j$ satisfy $[X,P]=i\hslash$. The nonzero-frequency phonon expansion alone yields the site commutator $i\hslash ({\delta }_{jl}-1/N)$; including $X,P$ restores the missing uniform part. The thermal free-coordinate factor requires a specified volume, pinning or a fixed centre of mass. Substituting zero frequency into an oscillator partition function instead produces a spurious divergence without defining the physical zero mode.

A periodic complex field permits $\varphi (\beta ,\mathbf{r})-\varphi (0,\mathbf{r})=2\pi w(\mathbf{r})$. Its uniform-density first-order term is therefore a winding-dependent imaginary action, not generally a removable global constant. It vanishes in the smooth zero-winding sector used for Gaussian phonons. Density defects and vortices require more than that smooth-sector approximation.

A cryogenic direct-detection search using scintillating crystal calorimeters. Calcium-tungstate modules measure deposited energy through phonons and compare a separate light signal to discriminate recoil populations. Oxygen, calcium and tungsten recoil differently for a given WIMP mass; a model fit must include their responses and backgrounds.^[ref]

Stationary one-dimensional flow has constant particle current $j=\rho v$. A lower-density segment must flow faster at the same current and also has smaller local speed of sound $\sqrt{g\rho /m}$. It can therefore lose local phonon energetic stability before a denser segment.

The three-dimensional Debye model normalizes its phonon density of states to $3N$ modes: $g(\omega )=9N{\omega }^2/{\omega }_D^3$ below the Debye frequency ${\omega }_D$. The thermal energy is $U={\int }_0^{{\omega }_D}g(\omega )\hslash \omega /[e^{\hslash \omega /(k_{B}T)}-1]d\omega$. Differentiate at fixed volume and substitute $x=\hslash \omega /(k_{B}T)$ to obtain the displayed heat capacity at constant volume.

At $T\ll {\Theta }_D$, integration by parts gives ${\int }_0^{\infty }{x}^4{e}^x/(e^x-1{)}^{2}dx=4{\int }_0^{\infty }{x}^3/(e^x-1)dx=4{\pi }^4/15$. Therefore $C_V\sim (12{\pi }^4/5)N{k}_B(T/{\Theta }_D{)}^3$. At $T\gg {\Theta }_D$, each mode approaches the equipartition theorem value $k_B$ of heat capacity, so $C_V\to 3N{k}_B$, the Dulong-Petit law. Fixed harmonic zero-point energy does not contribute to either limit.

A band gap whose conduction band minimum and valence band maximum occur at the same crystal momentum, allowing an optical transition without a momentum-balancing phonon.

An electron filling an electron hole removes both excitations relative to the filled-band background. Their electric charges cancel, consistent with charge conservation; the remaining electrons are still present. The released energy can leave through photon emission or other channels, including phonons. Formation of an exciton instead retains the excited electron and electron hole as a bound pair; recombination removes the pair. A valence band electron filling a nearby vacancy merely moves the electron hole and does not remove the total vacancy count.

Electronic heat capacity is the contribution from electronic excitations to heat capacity. A simple normal metal with a smooth density of states near its Fermi energy has $C_{\mathrm{m}}^{\mathrm{el}}\simeq \gamma T$ well below its electron Fermi temperature. Only a narrow energy interval near the Fermi energy changes occupation on heating, explaining the small fraction of active electrons. The Debye T-cubed test separates this contribution from lattice phonons. Interactions, a superconductivity transition or other electronic structure can invalidate a constant $\gamma$ description.

The electron-phonon interaction couples electronic states to the atomic displacements represented by phonons. A lattice vibration changes the potential seen by an electron; its quantized displacement permits phonon absorption and emission. This interaction affects electrical resistance, electronic thermal conductivity, lattice dynamics and pairing in conventional superconductors.

Energy resolution specifies the characteristic energy separation or broadening that an instrument can distinguish, with a stated width convention and measurement geometry. Resolving a phonon in inelastic scattering requires measuring a small energy transfer as well as momentum transfer. It is distinct from the incident beam's central energy.

For the ideal Wannier-Mott exciton in energy bands whose edges occur at the same crystal momentum, the pair excitation energy, measured from the unexcited crystal, isHere $\hslash \mathbf{K}$ is the pair's total crystal momentum; the last term is center of mass kinetic energy. For a permitted photon absorption transition with negligible recoil energy, $\hslash \omega \simeq E_g-E_{B,n}$. An electric-dipole selection rule and a nonzero transition matrix element are still needed: a bound state need not produce an observable spectral line. If the relevant energy band edges have different values of crystal momentum, a phonon can also be needed to conserve crystal momentum; subtracting a binding scale from a band gap alone does not specify the optical threshold.

For example, the lowest internal bound state in this hypothetical model costs $1.489115\mathrm{eV}$ to create optically, yet needs only $0.010885\mathrm{eV}$ more to dissociate. The large creation energy and the small exciton binding energy answer different physical questions. An electron-hole recombination event can instead release excitation energy by photon emission or to other degrees of freedom; an emitted photon need not reproduce the original absorption energy if relaxation intervened.

The crystal momentum of a phonon is defined modulo a reciprocal lattice vector. Extending its dispersion relation periodically beyond the first Brillouin zone labels equivalent excitations by different extended wavevectors. A scattering transfer $Q=G+q$ can be outside that zone while its reduced phonon wavevector $q$ is inside. The measured neutron recoil depends on the actual transfer $Q$, not on the reduced representative alone.

An internal harmonic vibration spectrum with lowest positive frequency ${\omega }_{\min }$ has the displayed low-temperature thermal phonon population, where $g_{\min }$ is its degeneracy. The continuum bulk law fails when thermally occupied wavelengths reach the sample size. Mechanical boundary conditions determine ${\omega }_{\min }$; rigid translation of the entire solid is not an internal phonon mode.

For a chosen homogeneous composition ${\mathrm{La}}_{2-x}{\mathrm{Ba}}_x{\mathrm{CuO}}_{4-\delta }$, assign formal oxidation states $+3$ to lanthanum, $+2$ to barium and $-2$ to oxygen. If $v$ is the mean formal copper oxidation state, charge neutrality requiresThe excess formal positive charge per copper relative to $+2$ is therefore $p_{\mathrm{formal}}=x-2\delta$. For stipulated $x=0.12$ and $\delta =0.01$, it is $0.10$; at $\delta =0.06$, it is zero. These are bookkeeping examples, not measurements of mobile carriers. The symbol $x$ here belongs to this declared formula, not the original mixed-phase recipe. A whole-pellet elemental ratio cannot simply be assigned to each constituent phase.

Charge neutrality alone does not locate holes. The lecture reviews oxygen-related carrier evidence and charge-$2e$ pairing from the Josephson effect and fluxoid and flux quantization in a superconductor. Pair charge does not establish phonon-mediated BCS theory for cuprate superconductors.^[ref]