Past exam of the condensed matter physics major option course of the University of Oxford / 2015 / 2

A neutron can exchange both energy and crystal momentum with a lattice vibration. Determine its incoming and outgoing wave vectors ${\mathbf{k}}_i,{\mathbf{k}}_f$ from neutron wavelengths and directions. Its nonrelativistic kinetic energy is $E_n={\hslash }^2{k}_n^2/(2m_n)$, so the measured transfers areFor phonon creation, momentum conservation reads $\mathbf{Q}=\mathbf{G}+\mathbf{q}$ and the neutron loses energy $\hslash {\omega }_{\nu }(\mathbf{q})$. For phonon annihilation the corresponding phonon energy is gained by the neutron. Reduce $\mathbf{Q}$ by a reciprocal lattice vector $\mathbf{G}$ to obtain the phonon wave vector, and trace a branch by repeating this measurement at several reduced wave vectors.

A triple-axis spectrometer provides a suitable arrangement: a monochromator selects the incident wavelength, the oriented sample defines the momentum transfer through the scattering angles, and an analyzer crystal selects the final wavelength before detection. A time-of-flight spectrometer instead uses pulse timing and flight paths to obtain the energy change. In either case instrumental energy and angular resolution must be separated from intrinsic line widths. Nuclear scattering from atomic displacements couples neutrons to phonons; the one-phonon intensity contains a polarization factor involving $\mathbf{Q}\cdot {\mathbf{e}}_{\nu }$, atomic scattering lengths, and thermal occupation factors. Intensities can therefore differ even at equivalent phonon frequencies.

For the body-centered tetragonal direct lattice, conventional reciprocal labels $(h,k,l)$ are integer triples satisfying $h+k+l$ even. This follows because the extra translation $(1/2,1/2,1/2)$ must have phase one for a reciprocal lattice vector. In the $k=0$ section the condition is $h+l$ even. The reciprocal steps areWithin the strict index window $-2<h,l<2$, the reciprocal points are $(h,l)=(0,0)$ and $(\pm 1,\pm 1)$. The neighboring $(\pm 2,0)$ and $(0,\pm 2)$ points lie on the excluded boundary but are useful for constructing the zone bisectors.

At the requested drawing scale, one conventional step in $h$ is $a^{\ast }/5=3.560$ cm and one step in $l$ is $c^{\ast }/5=1.197$ cm. Thus the four off-origin included points have drawing coordinates $(\pm 3.560,\pm 1.197)$ cm. Keeping these unequal physical scales avoids drawing a fictitious square reciprocal section.

Construct the first Brillouin zone by the perpendicular bisectors between the origin and reciprocal lattice points. For a test point $(h{a}^{\ast },0,l{c}^{\ast })$, the nearest relevant reciprocal points are $(0,0,\pm 2c^{\ast })$ and $(\pm a^{\ast },0,\pm c^{\ast })$. Their inequalities areSet $\rho =(c^{\ast }/a^{\ast }{)}^2=(a/c{)}^2=0.113024$. The six vertices in conventional $(h,l)$ coordinates areNumerically these are $(\pm 0.55651,0)$ and $(\pm 0.44349,\pm 1)$. The $k\ne 0$ reciprocal lattice vectors must in principle be checked because the requested object is a section of a three-dimensional zone. For example $(\pm 1,\pm 1,0)$ gives the weaker bound $|h|\le 1$, and $(0,\pm 1,\pm 1)$ gives a weaker vertical bound; neither clips this hexagon. Direct comparison with surrounding three-dimensional reciprocal lattice vectors confirms the displayed section. It is not the rectangular reciprocal conventional cell.

The difference of the two wave-vector labels is $(1,0,-1)$, an allowed reciprocal lattice vector because $1+0-1$ is even. Bloch's theorem of the lattice dynamical matrix therefore givesfor corresponding phonon branches. Neither individual odd-sum point needs to be a reciprocal lattice point. It is their difference that establishes equivalence. Structure factors and neutron polarization factors can still change the observed intensities at those two scattering vectors, so equal frequencies do not require identical measured spectra peak by peak in intensity.

There are three acoustic branches, not two. For propagation parallel to the tetragonal $c$ axis, one branch is a longitudinal wave, polarized parallel to $c$. The other branches are transverse waves whose polarizations lie along the equivalent basal directions. The fourfold symmetry about $c$ makes those transverse modes degenerate along that axis.

Consequently at a generic nonzero wave vector along $c$ there are only two distinct acoustic frequencies: one longitudinal frequency and one doubly degenerate transverse frequency. At the zone center all acoustic frequencies go to zero as required by uniform translation invariance. The degeneracy would generally lift if propagation moved away from the symmetry axis or the basal tetragonal symmetry were broken.

Subtract the reciprocal lattice vector $(0,0,8)$, leaving a reduced wave vector $(0.2,0,0)$ along $a$. A linear-acoustic estimate givesInterpret the quoted intrinsic width as the full width at half maximum in energy, $\Gamma =0.1$ meV. The corresponding energy-relaxation mean lifetime is $\tau =\hslash /\Gamma =6.58\times 10^{-12}$ s, giving mean free path $v_g\tau \simeq 19.7$ nm. If the given width instead denotes a half width, the inferred mean lifetime differs by a factor of two; the convention must be specified.

Approximating the thermal average by this velocity and mean lifetime, and using the isotropic angular factor $⟨v_a^2⟩=v_g^2/3$, kinetic theory givesUsing a single purely $a$-directed effective channel without angular averaging gives $2.17$ W m${}^{-1}$ K${}^{-1}$, still far below the observed total $39$ W m${}^{-1}$ K${}^{-1}$. The isotropic estimate is roughly $54$ times smaller than the measured total.

The calculation is not a complete thermal-transport average. Electrons in this metallic compound can carry substantial heat, the supplied total heat capacity need not be entirely phononic, and one $7$ meV phonon need not represent all thermally active branches and mean lifetimes. At $20$ K, $k_{B}T\simeq 1.72$ meV, so that measured mode is relatively high in the thermal distribution; lower-energy modes can have much longer mean lifetimes. A measured phase velocity is also only an estimate of the group velocity if the branch is nonlinear.

electron-phonon interaction is a plausible damping mechanism in a phonon-mediated superconductor, but the measurement is above its $15$ K transition and probes the normal state. It cannot be attributed directly to decay into an already established superconducting condensate. Below the transition an energy gap can modify electron-hole damping. The weak-coupling scale $2\Delta \simeq 3.52k_B{T}_c\simeq 4.55$ meV is below the $7$ meV phonon energy, so even superconductivity need not prohibit all pair-breaking damping of this mode. Temperature-dependent spectra would distinguish that mechanism from defects or anharmonic phonon decay.