Past exam of the quantum condensed matter course of the University of Cambridge / 2009

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

The Stokes–Einstein relation gives the translational diffusion coefficientat $T=298\mathrm{K}$. For isotropic three-dimensional Brownian motion, the mean squared displacement is $⟨|\Delta \mathbf{r}{|}^2⟩=6Dt$. Thus after one hour the root mean square displacement isThe phrase “on average” needs a convention: the mean displacement vector is zero; the mean displacement magnitude of three-dimensional Brownian motion is $⟨|\Delta \mathbf{r}|⟩=4\sqrt{Dt/\pi }=1.22\mathrm{mm}$, obtained by integrating the three-dimensional Gaussian distribution. These are displacements, not the total microscopic path length. The estimate assumes dilute unconfined water, a spherical hydrodynamic radius and constant temperature.

For a spherical micelle with $N$ tails, core radius $r$, molecular tail volume $v$ and head-group area $a$, volume and surface counting give $Nv=4\pi r^3/3$ and $Na=4\pi r^2$. Consequently $a=3v/r$. A tail cannot span a core radius larger than its maximum length $\ell$, so the amphiphile packing parameter condition isThis is a minimum area, not a maximum. The printed request reverses the geometric inequality. The limiting value corresponds to fully extended tails with $r=\ell$; larger head groups permit smaller spherical aggregates in this simple packing argument. The supplied data alone impose no finite maximum head-group area, and geometry alone does not guarantee thermodynamic micelle formation.

In the planar Debye–Hückel approximation, a surface potential decays as $\psi (x)={\psi }_s{e}^{-\kappa x}$, with $\kappa \propto \sqrt{I}$ at fixed temperature and solvent permittivity; $I$ is the ionic strength. For a monovalent salt, reducing its concentration by ten reduces $\kappa$ by $\sqrt{10}$. Calibrating the first decay from the supplied potential ratio, and assuming unchanged ${\psi }_s$, givesHere the printed meV unit is interpreted as the electrostatic potential energy of a singly charged probe; the same ratio calculation applies to potentials in mV.

This numerical answer uses an effective exponential decay calibrated from the data. The inferred initial Debye length is $5/\log (35/6.6)=3.00\mathrm{nm}$, whereas the standard dilute aqueous-electrolyte expression at room temperature gives about $0.96\mathrm{nm}$ for $0.1\mathrm{M}$ monovalent salt. The numbers therefore are not consistent with that absolute ideal-electrolyte model. Moreover a spherical colloid has an additional radius-dependent geometric factor, and its surface potential need not remain constant when salt is changed. Without a surface boundary condition and geometry, the new potential is not uniquely specified beyond the stated planar, fixed-potential approximation.

The Flory interaction parameter measures the energetic preference for unlike rather than like contacts in Flory–Huggins solution theory. For polymer volume fraction $\varphi$ and $N$ lattice sites per chain, the dimensionless mixing Helmholtz free energy per site isThe logarithmic terms represent combinatorial entropy; polymer connectivity reduces the chain's translational mixing entropy by the factor $N$. In a simple contact model $\chi =z[{ϵ}_{ps}-({ϵ}_{pp}+{ϵ}_{ss})/2]/(k_{B}T)$, where $z$ is the coordination number. Positive $\chi$ penalizes polymer-solvent contacts. Increasing it can overcome mixing entropy and cause phase separation; a negative second derivative of the mixing free energy marks the spinodal region. For a long dilute chain, $\chi <1/2$ corresponds to a good solvent, $\chi =1/2$ to the idealized theta condition, and $\chi >1/2$ to a poor solvent. Real effective parameters can depend on temperature and concentration, rather than being universal chemical constants.

A lipid bilayer has two opposed leaflets: hydrophilic heads contact the water on either side, while hydrophobic tails meet in its interior. This arrangement reduces exposure of the tails to water through the hydrophobic effect. Approximately cylindrical molecular packing favors a lipid bilayer; a strongly wedge-shaped amphiphile can instead form a micelle. Closing a sheet into a lipid vesicle removes its costly exposed edge.

Within a fluid membrane, lipids diffuse laterally, while transfer of a polar head through the hydrophobic core is much slower without assistance. The core limits passive transport of ions and many polar solutes; membrane proteins provide selective routes. Bending costs free energy but does not require the large area compression associated with stretching the molecular packing, so a thin membrane can bend readily while strongly resisting area change. Cholesterol and tail saturation alter packing and fluidity. In a cell, the bilayer also supports embedded proteins and a difference of electric potential across the membrane.

A polymer melt has an observation-time-dependent response because chains need time to rearrange. Short compared with their terminal relaxation time, entangled chains constrain one another and can store elastic free energy. Long compared with that time, chains escape their original constraints by reptation and the uncrosslinked melt flows. A permanently crosslinked elastomer instead retains a long-time elastic modulus.

A single-time approximation is the Maxwell fluid: a spring of shear modulus $G$ and a viscous dashpot of viscosity $\eta$ in series. Their common stress and additive shear strains give $\dot{\gamma }=\dot{\sigma }/G+\sigma /\eta$, with relaxation time $\tau =\eta /G$. For sinusoidal deformation proportional to $e^{i\omega t}$, the storage modulus and loss modulus areAt low frequency $G^{\prime \prime }\simeq \eta \omega$ dominates; at high frequency within this coarse-grained model $G^{\prime }\simeq G$ dominates. Real melts have a spectrum of relaxation times and additional segmental or glassy regimes, so the single-time model does not describe every frequency. A step-strain experiment measures the decay of stress, whereas oscillatory rheometry separates stored energy from dissipated energy through the phase of the response.

In a nematic liquid crystal, elongated molecules have orientational order around a nematic director $\mathbf{n}$, but lack the long-range positional order of a crystal. Head-tail symmetry identifies $\mathbf{n}$ and $-\mathbf{n}$. Molecular anisotropy and sufficient density favor collective alignment: for long hard rods this can increase translational entropy by reducing excluded-volume constraints, while attractive interactions can also favor alignment. Thermal orientational disorder competes with these effects.

The scalar nematic order parameter convention here isThus $S=0.44$ means $⟨{\cos }^2\theta ⟩=0.6267$. The effective orientation angle from nematic order isThis is not enough information to determine the literal mean angle $⟨\theta ⟩$: different orientation distributions can have the same second moment. Isotropic orientations have $S=0$, and perfect axial alignment has $S=1$.

In the supplied threshold expression, $d$ is the cell thickness; $K_1,K_2,K_3$ are the splay, twist and bend elastic constants of the Frank elastic energy. The dimensionless relative dielectric anisotropy of a nematic is $\Delta ϵ={ϵ}_{\parallel }-{ϵ}_{\perp }$, and ${ϵ}_0$ is the vacuum permittivity. Positive anisotropy favors a director along the applied electric field. Strong surface anchoring fixes orthogonal in-plane directions on the two plates. The low-field director therefore twists by a quarter-turn through the cell. Between crossed polarizers, with the entrance polarization matched to the first surface director, adiabatic optical following in a twisted nematic rotates the polarization and allows transmission. A sufficiently large normal field tilts the interior director towards the field, suppressing that rotation and making the crossed-polarizer cell dark; the anchored boundary layers remain distorted. Reversing the polarizer arrangement can reverse the optical contrast.

The threshold is a Fréedericksz transition: the electric energy gain overcomes the restoring elastic energy of an allowed distortion. Using the expression explicitly supplied by the paper givesThe thickness cancels for this strong-anchoring model. This is the onset of director reorientation, not an assertion that every molecule instantaneously becomes perfectly aligned. The quoted elastic combination is taken as given for this numerical task; a threshold derived from a specified full director-energy model must use that model's own elastic combination consistently. The companion illustration in A2(b) shows the plate anchoring, polarizers and bulk field-aligned state.

For a freely jointed Gaussian chain, independent isotropic segment vectors have zero mean and coordinate variance ${\ell }^2/3$. The central limit theorem gives the end-to-end vector densityThe number of conformations in a fixed small vector-volume element is proportional to this density. Taking its logarithm gives the configurational entropyThis is the vector-constrained entropy appropriate to an anchored strand. A density for the scalar length alone also has a $4\pi R^2$ shell factor and is a different constraint. The approximation requires many segments and extensions well below $N\ell$.

If the internal energy has negligible extension dependence, the Helmholtz free energy is $F=F_0+3k_{B}T{R}^2/(2N{\ell }^2)$. Differentiation gives an entropic force with effective spring constant $3k_{B}T/(N{\ell }^2)$. In an affine isotropic network, averaging over initially oriented strands gives rubber elasticityfor an incompressible material; ${\nu }_s$ is the number of elastically active strands per volume. For example a volume-preserving stretch with principal factors $\lambda ,{\lambda }^{-1/2},{\lambda }^{-1/2}$ has strain-energy density $W=G({\lambda }^2+2/\lambda -3)/2$. Its nominal tensile stress is $G(\lambda -{\lambda }^{-2})=3G(\lambda -1)+O((\lambda -1{)}^2)$, proving the stated small-strain Young's modulus.

Chemically, increasing the effective cross-link density, for example by changing vulcanization, generally increases the modulus by shortening strands between cross-links. Dangling ends, loops and changes of network structure mean the chemical junction count need not equal ${\nu }_s$.

The reference area is $A_0=10^{-5}{\mathrm{m}}^2$. Neglect self-weight, thermal expansion and energetic elasticity, keep the same permanent network, and heat quasistatically in the rubbery regime. Then $G$ increases from $20$ to $22\mathrm{MPa}$. With a constant suspended load, the exact affine-network balance isIt gives ${\lambda }_{300}=1.01662$ and ${\lambda }_{330}=1.01509$. ThusLinear elasticity instead gives extensions $16.35$ and $14.86\mathrm{mm}$, a $1.49\mathrm{mm}$ rise, consistent at this small strain. Incompressibility makes the area $A_0/\lambda$, already included by using nominal stress. Heating increases entropic stiffness, so the loaded rubber contracts and performs work on the mass. Strong thermal expansion, changes in cross-linking or leaving the rubbery regime would require additional material data.

The attractive part of the Lennard-Jones potential represents the London dispersion force between fluctuating atomic electric dipoles. The dipole-dipole interaction is proportional to $r^{-3}$; for nonpolar ground-state atoms its first-order expectation vanishes. In second-order perturbation theory, virtual excitations give a negative energy shift proportional to the square of that interaction, hence $-A/r^6$ with $A>0$. This applies at separations large compared with atomic size but small enough that electromagnetic retardation is negligible.

The repulsive term models the steep energy increase when closed electron shells overlap, involving the Pauli exclusion principle and increased electronic energy. The $r^{-12}$ power is an empirical convenient approximation, not a universal power derived from the exclusion principle; its being the square of $r^{-6}$ makes computation convenient. At still larger separations retardation changes the attractive asymptotics, outside this model.

A wave packet in an energy band has group velocity $v={\hslash }^{-1}dE/dk$. The semiclassical force law $\hslash \dot{k}=F$ then gives $\dot{v}={\hslash }^{-2}(d^{2}E/d{k}^2)F$. The curvature therefore defines the one-dimensional effective mass by $1/m^{\ast }=E^{\prime \prime }/{\hslash }^2$. Here $E^{\prime \prime }(k)=A{a}^2\cos (ka)$, so at the band minimum $k=0$,It measures acceleration within the band, rather than a change in the electron's rest mass. At a band maximum the electron curvature mass can be negative; a missing electron is then described by a positively charged hole with a positive curvature mass.

For elastic scattering, $|{\mathbf{k}}_{\mathrm{in}}|=|{\mathbf{k}}_{\mathrm{out}}|=2\pi /\lambda$. Contributions from sites differing by any Bravais lattice translation $\mathbf{R}$ add in phase exactly whenfor every $\mathbf{R}$, namely when the momentum-transfer wavevector is a reciprocal lattice vector $\mathbf{G}$. For a family of planes of spacing $d$, the $n$th reciprocal vector normal to those planes has magnitude $2\pi n/d$. The scattering geometry gives $|\mathbf{G}|=2k\sin \theta$. Combining the two expressions provesConversely, reflection from that plane family satisfying Bragg's law supplies the corresponding normal reciprocal vector. The scalar angle condition alone needs this plane-normal geometry; it does not specify an arbitrary outgoing direction. A nontrivial atomic basis can extinguish an otherwise allowed reciprocal-lattice reflection through its structure factor.

A charge density wave is a spatial modulation of the electron charge density, often accompanied by a periodic lattice distortion. Write a weak modulation as $\rho (x)={\rho }_0+{\rho }_1\cos (Qx+\phi )$. In a one-dimensional metal, the Peierls distortion couples states near opposite Fermi points most strongly for $Q=2k_F$. Opening a gap near the Fermi energy lowers occupied electronic energies, while the lattice deformation has a positive elastic energy cost. Their competition can make a periodic state favorable below a transition temperature.

The modulation can be commensurate or incommensurate with the underlying Bravais lattice. Diffraction detects additional satellite peaks, while transport can show reduced low-energy carrier density. Impurities can pin its phase; above a depinning field a collectively sliding modulation can contribute nonlinear current. These effects distinguish collective density modulation from simply assigning independent static charges to isolated lattice sites.

Electrons contribute magnetic response through both spin and orbital motion. In an ordinary weakly interacting metal, a field splits the two spin energies. Only states near the Fermi energy can change occupancy because of the Pauli exclusion principle, producing Pauli paramagnetism. For a total two-spin density of states $g(E_F)$, the simple result is ${\chi }_P={\mu }_0{\mu }_B^{2}g(E_F)$. Orbital cyclotron motion gives Landau diamagnetism; for an isotropic parabolic free-electron band it contributes $-{\chi }_P/3$. Filled shells can also supply a diamagnetic background.

Interactions can favor parallel spin occupancy. In an itinerant description, the exchange-energy gain competes with the kinetic-energy cost of redistributing electrons between spin bands. The Stoner criterion describes when an unpolarized state becomes unstable, with the precise factor in $Ig(E_F)$ depending on whether the density of states counts one spin or both. This mechanism can yield spontaneous ferromagnetism and magnetic domains. Alternatively, sufficiently localized moments in a metal can interact through conduction electrons. A high density of states alone is therefore not a complete microscopic explanation of a material's magnetism; band structure, electron interactions and temperature matter.

In a Hall effect measurement, a current flows along $x$ and a magnetic field is applied along $z$. The transverse Lorentz force initially deflects the mobile charges, building a transverse electric field until the net transverse current vanishes. Defining $R_H=E_y/(j_x{B}_z)$, the one-carrier Drude model gives $R_H=1/(nq)$. Thus electron-like carriers give $R_H=-1/(ne)$ and hole-like carriers give $R_H=+1/(pe)$.

A nearly filled band can conduct through holes: its missing electrons behave as positive mobile charges. Metals can have both electron-like and hole-like Fermi surface sheets. In the weak-field two-carrier approximation the Hall coefficient isThe sign depends on the carrier densities and squared mobilities; it need not identify the most numerous carrier. Experimentally the Hall voltage changes sign when the field or current reverses, allowing subtraction of unwanted longitudinal-voltage pickup. Anisotropic bands and different scattering rates complicate a literal conversion of $R_H$ into a carrier count.

The de Haas-van Alphen effect is an oscillation of equilibrium magnetization, and hence magnetic susceptibility, as Landau levels pass through the Fermi energy when the field changes. For a fixed slice perpendicular to the field, Onsager quantization givesAt $E_F$, successive crossings change $n$ by one, soHere $F=1/\Delta (1/B)$ is the oscillation frequency in inverse field. In three dimensions, integrating the oscillatory response over the momentum parallel to the field causes most sections to cancel by phase averaging. The leading surviving contributions come from stationary, hence extremal, cross-sectional areas. Low temperature and sufficiently weak scattering are needed to resolve the levels. Measuring several field directions therefore probes the shape of the Fermi surface rather than only a single scalar electron density.

For the cylindrical $\gamma$ sheet, $A_k=\pi k_F^2$. The measured frequency givesThere are $4/3$ electrons per site in this band, with two spin states per wavevector. Thus the occupied disk takes fraction $2/3$ of the square Brillouin zone:This uses the paper's circular, electron-occupied disk approximation and equal band populations.

In the tight-binding approximation, hopping to the two neighbors at $\pm a\hat{x}$ adds phase factors $e^{i{k}_{x}a}$ and $e^{-i{k}_{x}a}$. Their sum is $2\cos (k_{x}a)$; absorb twice the hopping matrix element into $t$. With no $y$ or $z$ hopping this yields $E=E_0+t\cos (k_{x}a)$. For $t<0$, occupancy is centered on $k_x=0$. The filling fraction of an independent cosine band is $2/3$, requiring the occupied strip $|k_x|<2\pi /(3a)$, so sheet $A$ consists of the two planes $k_x=\pm 2\pi /(3a)$; sheet $B$ similarly has $k_y=\pm 2\pi /(3a)$. Each extends along $k_z$, with no dispersion in that direction. The $\gamma$ sheet is a cylinder along $k_z$, of radius $k_F$.

Numerically the planar offsets are $5.35\times 10^9{\mathrm{m}}^{-1}$, the zone boundary is $\pi /a=8.03\times 10^9{\mathrm{m}}^{-1}$, and the cylinder radius is $k_{F}a=2.894$ in dimensionless coordinates. If $t>0$ instead, the band minimum is at the zone edge; occupying its lowest two-thirds puts the planes at $k_x=\pm \pi /(3a)$ or $k_y=\pm \pi /(3a)$, with the central third unoccupied. The sign of the hopping must therefore accompany an absolute placement of these open sheets. The following sketch uses $t<0$ and the stipulated independent-band model, before any avoided crossings caused by additional interorbital hybridization.

In a conductor, mobile electrons redistribute in response to an electric potential, producing an induced charge density that opposes its source. Finite electronic compressibility means a finite density change requires a change of local kinetic Fermi energy. Screening is therefore not perfect at every microscopic distance. On scales comparable to atomic structure or the Fermi wavelength, a continuum local-density model itself fails.

Thomas-Fermi screening treats electrons as a locally uniform degenerate free electron gas, replaces the ionic background by a smooth neutralizing charge, and neglects nonlocal response, exchange and correlation corrections. The electrochemical chemical potential is uniform in equilibrium. Because an electron has charge $-e$, its electrostatic energy is $-eV$, givingAt zero potential $\mu =E_{F0}$. Linearizing at fixed $\mu$, with $|eV|\ll E_{F0}$, givesInsert this in Poisson's equation to obtain $({\nabla }^2-{\xi }^{-2})V=-{\rho }_{\mathrm{free}}/{ϵ}_0$, where the Thomas-Fermi screening length isAway from a point source, direct radial differentiation shows ${\nabla }^2(e^{-r/\xi }/r)={\xi }^{-2}{e}^{-r/\xi }/r$. The decaying solution is therefore $V=C{e}^{-r/\xi }/r$. Integrating Poisson's equation over a small sphere containing charge $Q$ fixes $C=Q/(4\pi {ϵ}_0)$. For a sufficiently large source, the linearized formula is only a far-field approximation, not a justified description of its singular core.

For monovalent sodium, the electron number density is its atomic number density,This gives $E_{F0}=3.15\mathrm{eV}$ andRoom temperature is far below the corresponding Fermi temperature, so degeneracy is well justified, and sodium's simple conduction band makes a free-electron estimate useful. However $k_F\xi =0.615$ and the mean atomic spacing is $n_0^{-1/3}=0.340\mathrm{nm}$. The potential varies on a scale shorter than $k_F^{-1}$ and the atomic spacing, so the slowly varying local-density and continuum assumptions are not quantitatively controlled at the screening length itself. A nonlocal treatment includes Friedel oscillations, absent from the pure exponential.

Nearly unchanged interlayer force constants in two-layer and twenty-layer films are consistent with very local screening and bonding forces: reducing the distant bulk need not strongly change the force between adjacent layers. They also indicate that metallic electronic response can survive in the thin film. The observation does not by itself validate the numerical Thomas-Fermi approximation, establish an identical full phonon spectrum, or rule out surface-induced electronic changes; it directly constrains the measured local force constant.