Past exam of the quantum condensed matter course of the University of Cambridge / 2005

Official past-paper archive: www.sp.phy.cam.ac.uk/drp2/home. The direct PDF link has not yet been verified.

A nematic liquid crystal has long-range orientational order without a three-dimensional positional lattice. Its nematic director $\mathbf{n}$ identifies the preferred molecular axis, with $\mathbf{n}$ and $-\mathbf{n}$ equivalent for apolar molecules. The scalar nematic order parameter $S=⟨(3{\cos }^2\theta -1)/2⟩$ is zero for isotropic orientations and one for perfectly parallel axes. The tensor $Q_{ij}=⟨u_i{u}_j-{\delta }_{ij}/3⟩$ describes alignment without choosing a sign for the director.

Ordering balances orientational entropy against interactions and packing. For long rods, alignment can increase available translational configurations by reducing mutual excluded volume; thermotropic materials additionally have temperature-dependent energetic preferences. Spatially varying alignment costs Frank elastic energy through splay, twist and bend. Surface anchoring and electric or magnetic fields select the director in a finite specimen.

Crossed polarizers reveal birefringence and defect textures, while scattering tests positional order independently. A particularly direct response experiment is the Fréedericksz transition: confine a planar-aligned layer between anchored plates, apply a transverse field and monitor the transmitted polarization as the director tilts above an elastic threshold. This tests orientational elasticity and dielectric anisotropy rather than presuming that an optically anisotropic fluid has crystalline positional order.

A lipid bilayer self-assembles from amphiphiles: hydrophilic head groups face the surrounding water, while hydrocarbon tails form a hydrophobic interior. In a fluid membrane, molecules diffuse laterally within each membrane leaflet. Closing the bilayer into a vesicle avoids an exposed edge. The interior is a substantial barrier to isolated ions and many polar solutes; membrane proteins provide selective transport pathways in a cell membrane.

The Helfrich energy describes long-wavelength bending:Here $H=(c_1+c_2)/2$ and $K=c_1{c}_2$ are mean curvature and Gaussian curvature, $C_0$ is spontaneous curvature, ${\kappa }_b$ the bending modulus, $\bar{\kappa }$ the Gaussian-curvature coefficient and $\sigma$ the membrane tension. Bending is comparatively inexpensive, whereas stretching the molecular area strongly is costly. Leaflet asymmetry can favor a nonzero spontaneous curvature.

In micropipette aspiration, controlled suction draws part of a large vesicle into a narrow pipette. Pressure and geometry determine membrane tension, and the aspirated length measures apparent area change. At low tension, smoothing thermal undulations supplies much of that area; stronger tension probes actual stretching. Tracking vesicle shape fluctuations provides a complementary estimate of the bending modulus.

Here the complex viscoelastic modulus is a stress-to-strain response coefficient, rather than a complex number's absolute value. Use strain $\gamma (t)=\mathrm{Re}({\gamma }_0{e}^{i\omega t})$ and stress $\tau (t)=\mathrm{Re}[G^{\ast }(\omega ){\gamma }_0{e}^{i\omega t}]$, withThe storage modulus $G^{\prime }$ measures the elastic component; the loss modulus $G^{\prime \prime }$ measures dissipation. The mean power per volume is $\omega {\gamma }_0^2{G}^{\prime \prime }/2$, and the energy dissipated per cycle is $\pi {\gamma }_0^2{G}^{\prime \prime }$, fixing the sign convention.

A Hookean solid has $G^{\ast }=G_0$, while a Newtonian fluid has $G^{\ast }=i\omega \eta$. The Maxwell model interpolates with $G^{\ast }=G_{0}i\omega {\tau }_R/(1+i\omega {\tau }_R)$, giving $G^{\prime }=G_0(\omega {\tau }_R{)}^2/[1+(\omega {\tau }_R{)}^2]$ and $G^{\prime \prime }=G_0\omega {\tau }_R/[1+(\omega {\tau }_R{)}^2]$. It is fluid-like at low frequency and elastic on times short compared with relaxation time ${\tau }_R$.

A rheometer imposes small-amplitude oscillatory shear and resolves the torque amplitude and phase relative to the imposed motion. The torsional measurement of a complex shear modulus makes the conversion from torque and twist to $G^{\ast }$ explicit. A frequency sweep locates relaxation scales; keeping the strain small tests the linear-response assumption.

Wetting concerns how a liquid covers a solid in contact with another phase, such as its vapor. For an ideal flat homogeneous surface, balance of the three interfacial tensions gives Young's equationwhere the contact angle $\theta$ is measured through the liquid. Small angle means favorable contact; partial wetting gives a finite angle. The spreading coefficient $S={\gamma }_{SV}-{\gamma }_{SL}-{\gamma }_{LV}$ compares a covered surface with a dry one. Positive $S$ favors complete wetting, while negative $S$ permits a finite-angle droplet in the partial-wetting regime.

A sessile-drop experiment images the tangent at the contact line on a clean level substrate. Varying drop volume slowly distinguishes advancing and receding angles; their difference is contact-angle hysteresis from pinning and heterogeneity, so a single observed angle need not equal the ideal equilibrium value. A capillary-rise experiment gives a second consequence: the Young–Laplace equation and hydrostatic balance predict $h=2{\gamma }_{LV}\cos \theta /(\rho gr)$ in a narrow circular tube when meniscus-volume corrections are negligible. A wetting liquid rises, whereas a sufficiently nonwetting one is depressed.

For a dilute suspended sphere in Stokes flow, the translational friction coefficient is $f_h=6\pi \eta R$, with dynamic viscosity $\eta$ and hydrodynamic radius $R$. To derive the diffusion-friction relation, put particles in a weak conservative force $F=-U^{\prime }$. Their drift velocity is $F/f_h$, while the diffusive flux is $-Ddc/dx$. At equilibrium the Boltzmann distribution gives $c^{\prime }=-c{U}^{\prime }/(k_{B}T)=cF/(k_{B}T)$. Zero total flux therefore gives $cF/f_h-DcF/(k_{B}T)=0$, hence the Stokes–Einstein relationHere $k_B$ is the Boltzmann constant, $T$ absolute temperature and $D$ the translational diffusion coefficient. The sphere, no-slip continuum-fluid and dilute thermal-equilibrium assumptions enter through the drag law and flux balance.

At $T=293.15\mathrm{K}$ and the supplied viscosity, the predicted product is $DR=2.1472\times 10^{-19}{\mathrm{m}}^3{\mathrm{s}}^{-1}$. The derived predictions and radii inferred from the measured diffusion coefficients are:

The last three agree closely at the precision of the supplied data. The ribonuclease diffusion coefficient is about $16\%$ below the spherical prediction, corresponding to a larger frictional radius than the stated radius. Hydration, shape, a different operational radius and coarse measurement precision can matter; molar mass is not an independent parameter in the spherical diffusion law. A compact constant-density globule would have $R\propto M^{1/3}$, whereas collagen's large frictional radius relative to its mass is consistent with an extended nonspherical molecule. The inferred Stokes radius is a friction-equivalent radius, not automatically a geometric half-length.

In analytical ultracentrifugation, scan the radial protein concentration while a rotor spins at known angular speed $\omega$. Optical absorbance or refractive-index interference records a moving sedimentation boundary and its diffusive broadening.^[ref] With solvent density ${\rho }_s$, partial specific volume $\bar{v}$ and molar mass $M$ in $\mathrm{kg}\mathrm{mo}{\mathrm{l}}^{-1}$, the buoyant mass is $m_b=M(1-\bar{v}{\rho }_s)/N_A$. Force balance gives $f_{h}v=m_b{\omega }^{2}r$, so the sedimentation coefficient is $s=v/({\omega }^{2}r)=m_b/f_h$. A boundary trajectory gives $s={\omega }^{-2}d\ln r/dt$; diffusion gives $f_h=k_{B}T/D$.

Combining them gives the molar mass from sedimentation and diffusion:Known mass and buoyancy can instead determine friction from $s$ alone, and comparison with a compact sphere tests shape or hydration. In sedimentation equilibrium, the stationary radial profile obeys $d\ln c/d(r^2)=m_b{\omega }^2/(2k_{B}T)$, providing buoyant mass without a friction model. This is a sedimentation-diffusion balance in a uniform solvent, rather than density-gradient banding at an isopycnic point.

An ideal chain can be treated as a random walk when effective segment orientations are independent and isotropic, segment length is fixed, and interactions such as long-range excluded volume are absent or effectively screened. A large number of segments is needed for the Gaussian continuum approximation. Write $\mathbf{R}={\sum }_{j=1}^N{\mathbf{a}}_j$. Each Cartesian component has mean zero and variance $N{a}^2/3$, so the central limit theorem gives the Gaussian limit of the end-to-end distribution of a freely jointed chainEquivalently, one step has characteristic function $\sin (ka)/(ka)$; taking its $N$th power and expanding its logarithm gives $e^{-N{a}^2{k}^2/6}$, whose inverse Fourier transform is this density. It is a density per vector volume $d^{3}R$. The scalar end-to-end-distance density is $4\pi R^{2}p(\mathbf{R},N)$. Thus the printed expression must be interpreted as a vector density; it is also a large-$N$ approximation, not the exact finite-chain distribution near $R=Na$.

For a freely rotating chain, neighboring forward bond vectors have fixed angle $\theta$, and independently uniform torsion angles give $⟨{\mathbf{a}}_i\cdot {\mathbf{a}}_j⟩=a^2{\alpha }^{|j-i|}$ with $\alpha =\cos \theta$. Squaring the sum of bonds therefore givesThere are $N-k$ pairs separated by $k$ bonds, so the characteristic ratio of a freely rotating chain isFor fixed $|\alpha |<1$, the geometric sums give $C_N=(1+\alpha )/(1-\alpha )-2\alpha (1-{\alpha }^N)/[N(1-\alpha {)}^2]$. The correction vanishes as $N\to \infty$, yieldingFor the specified polyethylene parameters, $\alpha =\cos (-70.5^{\circ })\simeq 0.333807$, so $C_{\infty }\simeq 2.00213$ and $C_{9160}\simeq 2.00197$. HenceThe negative sign of the stated angle does not affect its cosine. The forward-bond angle is supplementary to the chemical internal bond angle. This freely rotating estimate includes fixed-angle correlations but omits torsional energy preferences; the Kuhn length of a freely rotating chain expresses the same long-chain persistence as $b_K=a{C}_{\infty }$.

Exchange interaction arises from Coulomb interaction together with the antisymmetry required for identical electrons. Different spin states require different spatial symmetry and therefore different pair correlations and Coulomb energy; exchange is not the classical magnetic dipole force between spins. For electrons in distinct atomic orbitals, positive direct exchange favors parallel spins, underlying Hund's rules and the formation of local moments.

In a solid with localized moments, a common convention is $H=-J{\sum }_{⟨ij⟩}{\mathbf{S}}_i\cdot {\mathbf{S}}_j$. Then $J>0$ favors ferromagnetism and $J<0$ favors antiferromagnetism. Superexchange through a ligand and other virtual hopping processes can produce either effective sign depending on orbital occupancy and geometry; half-filled orbitals often give antiferromagnetic coupling. For itinerant electrons, the Stoner criterion describes a competition between exchange energy gain and the kinetic cost of splitting the spin populations.

Experiments distinguish these descriptions through both static and dynamic response. Magnetic neutron diffraction locates magnetic Bragg peaks and therefore the ordering wavevector; an antiferromagnet can have such peaks despite zero bulk moment. Inelastic neutron scattering measures spin-wave dispersion, from which an exchange scale can be fitted. A Curie-Weiss susceptibility intercept gives complementary information about dominant coupling, but its sign alone does not establish the actual ordered structure.

A Peierls transition is a lattice distortion coupled to a gap in an initially partly filled quasi-one-dimensional electronic band. A modulation at $2k_F$ couples the opposite Fermi points. At half filling with one orbital per atom, it doubles the cell and alternates strong and weak bonds; the dimerized tight-binding chain shows explicitly how the gap opens at the Fermi energy. In the ideal weak-coupling zero-temperature model, the Peierls energy gain has a negative term proportional to $-{\delta }^2\ln (t/|\delta |)/t$, which can outweigh a positive quadratic elastic cost.

Observation should connect electronic and structural evidence: cooling through the transition can reduce conductivity as the gap develops, while diffraction satellites identify the ordering wavevector. A softened phonon near $2k_F$ tests the precursor electron-lattice coupling. In inelastic neutron scattering of TTF-TCNQ, measurements from $60$ to $300\mathrm{K}$ found an increasing Kohn anomaly near $2k_F$ as the $54\mathrm{K}$ transition was approached. Those particular measurements were above the transition, so they probe the precursor dynamics rather than a static low-temperature diffraction pattern.^[ref]

A real finite-temperature transition also depends on interchain coupling and fluctuations. The ideal one-dimensional energy argument by itself does not prove finite-temperature long-range order.

A p-n junction joins p-type and n-type semiconductors. Initial carrier diffusion exposes ionized acceptors and donors, creating a depletion region and an internal electric field. At equilibrium the Fermi level is uniform and drift balances diffusion, so the built-in potential produces no net terminal current. For nondegenerate abrupt doping, its magnitude is $V_{\mathrm{bi}}=(k_{B}T/e)\ln (N_A{N}_D/n_i^2)$.

Forward bias lowers the barrier and injects minority carriers across the junction. In the ideal low-injection diffusion model, the Shockley diode equation is $I=I_s[e^{eV/(k_{B}T)}-1]$. Reverse bias broadens the depletion region and leaves a small leakage current until breakdown mechanisms become important.

The asymmetric current-voltage curve enables rectification. Illumination produces electron-hole pairs that the junction field can separate, giving a photodiode or solar cell. Forward injection and radiative recombination in a suitable direct-gap material give a light-emitting diode. Measuring dark and illuminated current-voltage curves distinguishes electrical injection from photogenerated current; an ideal junction need not have zero reverse leakage or identical behavior in every material.

In a heavy fermion material, correlated low-energy quasiparticles have a large electronic density of states and effective mass. A Kondo lattice with hybridization between localized f-electron states and conduction states is an important setting. The large mass is a low-energy dynamical property, not a change in the electron's bare mass or the ionic mass.

Low-temperature calorimetry measures $C/T\simeq \gamma +\beta T^2$ in a regime where an electronic linear term and a Debye lattice term are appropriate. The large intercept $\gamma =({\pi }^2{k}_B^2/3)D(E_F)$ measures the enhanced quasiparticle density of states, with volume or molar conventions kept consistent. Apply a known heat pulse to a thermally characterized sample platform and measure the temperature rise, then vary the base temperature to separate contributions.

In low-temperature calorimetry and resistivity of CeAl3, measurements below $0.2\mathrm{K}$ gave $\gamma =1.62\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-2}$ and $\rho ={\rho }_0+A{T}^2$ with $A=35\mu \Omega \mathrm{cm}{\mathrm{K}}^{-2}$. The large thermal coefficient and enhanced quadratic transport coefficient are complementary evidence for a strongly renormalized Fermi liquid.^[ref] Enhanced spin susceptibility provides another probe. Where coherent quantum oscillations can be resolved, their thermal damping determines a cyclotron effective mass, giving a band-sensitive comparison with the thermodynamic mass inferred from heat capacity.

In a direct optical transition, a photon promotes an electron between the valence band and conduction band at essentially the same crystal wavevector: optical photon momentum is small on the scale of the Brillouin zone. An indirect optical transition also needs a phonon or another momentum source. Near an indirect gap, phonon emission requires photon energy approximately $E_g+\hslash \Omega$, while phonon absorption permits $E_g-\hslash \Omega$ with a thermally occupied phonon.

Strong direct interband absorption needs both many allowed transitions and a nonzero optical matrix element. The joint density of states counts pairs with $E_c(\mathbf{k})-E_v(\mathbf{k})=\hslash \omega$; nearly parallel bands and critical points of this energy difference can enhance it. Dipole selection rules and polarization weight that count, so a large joint density of states alone does not guarantee a strong absorption peak.

The supplied infrared spectrum lies below the interband threshold and instead probes free-carrier response. In the Drude model, $ϵ(\omega )={ϵ}_{\infty }-{\omega }_p^2/[\omega (\omega +i/\tau )]$, with ${\omega }_p^2=n{e}^2/({ϵ}_0{m}^{\ast })$. At long wavelengths the mobile electrons screen the field and the reflectivity is high. As wavelength decreases through the screened plasma edge, the real dielectric function becomes positive and propagating light can enter. The reflectivity reaches its minimum near index matching, $ϵ\simeq 1$, then rises toward the bound-electron background reflectance.

For background ${ϵ}_{\infty }=16$, its refractive index is four and the normal-incidence Fresnel equations give $R_{\infty }=[(4-1)/(4+1){]}^2=9/25=0.36$. The short-wavelength side of the figure is approaching a reflectance of this order, consistent with that background; its shortest plotted wavelength still has a carrier correction. Here “infinity” means high frequency relative to the free-carrier scale while remaining below the remote interband resonances.

The Drude-Lorentz reflectivity minimum is near $ϵ=1$, not the plasma-edge zero $ϵ=0$. Reading the deep minimum as ${\lambda }_{\min }\simeq 22\mu \mathrm{m}$ and neglecting damping there gives ${\omega }_p^2=({ϵ}_{\infty }-1)(2\pi c/{\lambda }_{\min }{)}^2$. The carrier effective mass from a reflectivity minimum is thereforeThe graph supports only a rough read-off: shifting the minimum by $0.5\mu \mathrm{m}$ changes this mass by about five percent. The deep narrow minimum supports the weak-loss assumption $\omega \tau \gg 1$ in this region, although finite scattering explains why the long-wavelength reflectivity is below unity. The estimate also assumes one dominant carrier species, a nonmagnetic bulk sample and a nearly constant background dielectric function. If the plasma-edge zero is used instead, its denominator is ${ϵ}_{\infty }$ and its wavelength is slightly longer; the two wavelengths must not be interchanged.

In the Bloch wave sum, $N$ is the number of sites, $R_j=ja$ the equilibrium position of site $j$, $\varphi (r-R_j)$ a localized atomic orbital there, and $k$ the crystal wavevector. The phase factor gives the translation character of the Bloch state. The factor $1/\sqrt{N}$ normalizes it when the site orbitals are treated as orthonormal; appreciable intersite overlap requires an overlap-dependent normalization.

For a tight-binding model, choose localized orbitals and calculate the on-site and intersite Hamiltonian matrix elements. Form Bloch sums for each orbital type and solve $H(k)c=E(k)S(k)c$ at each $k$, where $S(k)$ is the orbital-overlap matrix. In the orthogonal tight-binding approximation, $S=I$. The eigenvalues give the energy bands and the eigenvectors their orbital composition; the nonorthogonal nearest-neighbour tight-binding band illustrates how retained overlap modifies this procedure.

For one orthonormal s orbital per site, keep on-site energy ${ϵ}_s$ and real nearest-neighbor hopping integral $t=⟨{\varphi }_j|H|{\varphi }_{j+1}⟩$. The two neighbors contribute phases $e^{\pm ika}$, giving $E_s(k)={ϵ}_s+2t\cos ka$. The band-curvature definition $1/m^{\ast }=E^{\prime \prime }(k)/{\hslash }^2$ givesWith the usual bonding convention $t<0$ for these s orbitals, the center is a minimum with positive electron effective mass and the edge a maximum with negative electron effective mass. Hole masses near a filled-band maximum have the opposite sign.

For p orbitals aligned along the chain, the orbital phase and the sign of one-dimensional hopping reverse the usual center-versus-edge ordering. Choose every $p_x$ orbital with its positive lobe on the right. Equal neighboring Bloch coefficients then bring opposite-sign lobes together between sites, giving an antibonding state at $k=0$. Alternating coefficients at $k=\pi /a$ make those facing lobes have the same sign, producing the lower bonding state. Thus a nearest-neighbor model has $E_p(k)={ϵ}_p+2t_{\sigma }\cos ka$ with the usual $t_{\sigma }>0$, maximum at zero and minima at the zone edges. This uses a fixed common orbital-phase convention at all sites.