Past exam of the quantum condensed matter course of the University of Cambridge / 2007

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

For a narrowly aligned nematic liquid crystal, the scalar nematic order parameter is $S=⟨(3{\cos }^2\theta -1)/2⟩\simeq 1-\frac{3}{2}⟨{\theta }^2⟩$. Hence the small-angle width of nematic orientation isThis is a root-mean-square angular width about the unoriented nematic director. The single value of $S$ does not determine a complete distribution or its full width at half maximum. For an isotropic two-component Gaussian tilt, each transverse component would have standard deviation ${\theta }_{\mathrm{rms}}/\sqrt{2}$.

For a spherical micelle of sodium dodecyl sulfate with $n$ molecules and a sharp tail-filled core of radius $R$, volume and head-group area give $nv=4\pi R^3/3$ and $n{a}_0=4\pi R^2$. Dividing gives the spherical micelle geometryThere is a consistency limit: $R$ exceeds the supplied maximum tail length $1.67\mathrm{nm}$. Equivalently the amphiphile packing parameter is $v/(a_0{l}_c)=0.368>1/3$, outside the strict spherical-core bound. Thus the requested spherical assumption gives these estimates, but the three molecular dimensions cannot all be exact within that ideal sharp-core geometry. Effective head area, core structure or aggregate shape must adjust; silently replacing $R$ by $l_c$ would no longer satisfy both packing equations.

For the polybutadiene chain in 1,4-dioxane, define $R_x$ from the negative end charge toward the positive one. The electric dipole energy is $-eE{R}_x$, so the force conjugate to end separation is $eE$, even though the two endpoint forces are opposite. Balancing it against the entropic spring force givesNumerically,The zero-field rms of just the $x$ component is $60/\sqrt{3}=34.6\mathrm{nm}$, a different comparison. The mean extension is still much smaller than the contour length $Na=6\mu \mathrm{m}$, consistent with the Gaussian ideal chain force law. This is an end-charged ideal polymer in an electric field; the direct endpoint Coulomb interaction is omitted as stipulated.

Long anisometric molecules cannot pack independently. In the Onsager excluded-volume mechanism, two slender rods of length $L$ and diameter $D$ exclude a relative-position volume approximately $2L^{2}D|\sin \gamma |$, where $\gamma$ is their mutual angle. Aligning rods reduces this excluded volume, increasing the translational entropy available at a fixed number density. It simultaneously reduces orientational entropy. At sufficiently high concentration the positional gain wins, producing a nematic liquid crystal even for purely repulsive rods.

The nematic director is apolar: $\mathbf{n}$ and $-\mathbf{n}$ describe the same orientation. The molecules retain liquid-like positional disorder, in contrast to a smectic phase with layered order. A typical dimensionless concentration controlling the slender-rod competition is $c{L}^{2}D$, although finite aspect ratio, flexibility and attractions change the transition. The mechanism illustrates that ordering can increase total entropy: the lost orientational freedom is compensated by less mutual obstruction.

At a micrometre scale, viscous forces normally dominate inertia. As an illustrative Newtonian estimate, $L=1\mu \mathrm{m}$, $U=10\mu \mathrm{m}{\mathrm{s}}^{-1}$, $\rho =10^3\mathrm{kg}{\mathrm{m}}^{-3}$ and $\eta =10^{-3}\mathrm{Pa}\mathrm{s}$ give a Reynolds number $\rho UL/\eta =10^{-5}$. The surrounding flow is approximately a Stokes flow, satisfying $-\nabla p+\eta {\nabla }^2\mathbf{u}=0$ and $\nabla \cdot \mathbf{u}=0$.

A body of radius $0.5\mu \mathrm{m}$ translated at that speed in an unbounded Newtonian liquid would need Stokes drag $6\pi \eta aU\simeq 9.4\times 10^{-14}\mathrm{N}$. It stops almost immediately when propulsion stops. The scallop theorem rules out propulsion by a single reciprocal stroke in this ideal regime; a rotating helix or a nonreciprocal cycle of shapes can propel it. Brownian motion also randomizes position and orientation: Stokes–Einstein relation scales as $a^{-1}$ and rotational diffusion as $a^{-3}$, so retaining heading becomes harder as size decreases.

The Newtonian calculation is a scale estimate, not a complete model of blood. Vessel walls alter drag, the imposed flow advects the body, and nearby red blood cells make the medium heterogeneous at this scale. Non-Newtonian fluid effects and deformable obstacles can matter; steering must be assessed against those effects rather than inertial coasting.

A lipid bilayer is fluid within its plane, so a bare equilibrated bilayer has no static in-plane shear modulus. Stretching its area is nevertheless expensive, whereas bending is much softer. The Helfrich membrane energy expresses this distinction throughwith membrane bending rigidity $\kappa$, mean curvature $H$, Gaussian curvature $K$, spontaneous curvature $C_0$ and membrane tension $\sigma$. Area dilation additionally costs approximately $K_A(\Delta A{)}^2/(2A_0)$. For a closed surface of fixed topology, the Gauss-Bonnet theorem makes the integrated Gaussian-curvature term constant.

These energies control vesicle shapes, budding and the ease of forming membrane tubes. Asymmetric leaflet composition or bound proteins can change the preferred curvature. Real cell membranes can also be coupled to a cytoskeleton, which supplies shear elasticity absent from a free lipid bilayer. Thus a red-cell shape cannot generally be inferred from the bare bilayer model alone.

Two useful measurements probe different terms. Optical tracking of thermal membrane flickering measures fluctuation amplitudes; for small height modes, a normalization with $h_q=A^{-1}\int h{e}^{-i\mathbf{q}\cdot \mathbf{x}}{d}^{2}x$ gives $⟨|h_q{|}^2⟩=k_{B}T/[A(\kappa q^4+\sigma q^2)]$. Micropipette aspiration controls pressure while measuring the membrane drawn into a narrow pipette, linking apparent area change to tension and, at larger strain, area dilation. The observables test the elastic coefficients rather than merely showing a membrane image.

In diffusion-controlled aggregation, particles meet through Brownian motion and stick with probability close to one on encounter. Diffusion-limited cluster aggregation allows already formed clusters to diffuse and join; diffusion-limited aggregation in its usual single-seed model instead adds diffusing individual particles to one growing cluster. First-contact sticking shields inner regions from later arrivals, favouring open branched structures. A reaction-limited cluster aggregation model has a small sticking probability and permits many failed encounters before attachment.

For two dilute spherical species, the Smoluchowski diffusion-controlled encounter rate is $k_{12}=4\pi (D_1+D_2)(a_1+a_2)$. In relative coordinates, solve the steady diffusion equation with an absorbing surface at $R=a_1+a_2$ and concentration $c_{\infty }$ at infinity: $c(r)=c_{\infty }(1-R/r)$. The inward diffusive flux is $4\pi (D_1+D_2)R{c}_{\infty }$, proving the rate coefficient. Interaction barriers and non-spherical aggregates require changes to this ideal encounter calculation.

Cluster growth can be tested by microscopy and light scattering. A mass fractal dimension $d_f$ relates particle number to cluster size through $N\propto R^{d_f}$; a scattering regime between the monomer and cluster length scales can reveal that scaling. The measured morphology depends on whether whole clusters move, on the sticking rule, and on possible bond rearrangement.

Use polymer volume fraction $\varphi$ and interaction parameter $\chi$, increasing toward poor-solvent conditions. The Flory–Huggins solution theory free energy per lattice site isThe common-tangent construction gives the binodal; $f^{\prime \prime }=0$ gives the spinodal, $2{\chi }_s=1/(N\varphi )+1/(1-\varphi )$. Its critical point is ${\varphi }_c=1/(1+\sqrt{N})$, ${\chi }_c=\frac{1}{2}(1+N^{-1/2}{)}^2$. Above the binodal in this $\chi$ plot, the solution separates into polymer-poor and polymer-rich phases; between binodal and spinodal it is metastable, while inside the spinodal a homogeneous state is unstable.

The line $\chi =1/2$ is the theta solvent condition for a long dilute chain. Below it is the good solvent side; above it is the poor solvent side. Poor solvent does not imply that every composition is already phase separated. Independently, a homogeneous solution is dilute for $\varphi \ll {\varphi }^{\ast }$, semidilute for $\varphi \gg {\varphi }^{\ast }$ below close packing, and concentrated as $\varphi$ approaches one.

Here $R$ denotes a typical chain size, such as an rms end-to-end distance of a polymer chain, rather than the mean end-to-end vector of an isotropic solution, which is zero. The Flory approximation gives $R\sim a{N}^{3/5}$ in a good solvent, an ideal chain has $R\sim a{N}^{1/2}$ in a theta solvent, and a compact polymer globule has size $R\sim a{N}^{1/3}$ in a strongly poor solvent. These are scaling estimates with order-one prefactors omitted, treating the stated $N$ as the segment count associated with the supplied statistical length.

In ethylbenzene, the good solvent makes effective segment repulsions swell the coil. With the Flory approximation,

At the specified theta condition in cyclohexane, the leading effective excluded volume vanishes and the ideal chain estimate is

In water, the strongly poor solvent favours a compact polymer globule. A site-volume estimate givesThe exact numerical conversion between globule radius and rms end-to-end distance depends on its density and shape; this calculation supplies the characteristic size scale.

The polymer overlap concentration corresponds to about one chain per coil volume. Approximating the occupied volume of one statistical segment by $a^3$ givesThus the site-volume model predicts overlap at a few percent volume fraction. A separately specified molecular segment volume or a coil-volume prefactor would change this numerical prefactor.

For a concentration blob with $g$ ideal segments, $\xi \sim a\sqrt{g}$ and concentration matching gives $\varphi \sim g{a}^3/{\xi }^3\sim g^{-1/2}$. Thus the concentration-blob size in a theta solvent isThe whole chain contains about ten blobs, so the separation of scales is moderate. Below $\xi$, theta-solvent segments have ideal chain statistics; above it, screening also gives Gaussian random-walk statistics for the sequence of blobs. There is no good-solvent self-avoiding regime to insert in this particular theta-solvent calculation. Density correlations and segment dynamics can still cross over at the blob scale.

In a dilute solution, solvent-mediated hydrodynamic interactions couple distant chain segments. The Zimm model describes the resulting cooperative dynamics: an intermediate-time segment mean-square displacement scales as $t^{2/3}$, while whole-chain translation has $D_Z\sim k_{B}T/({\eta }_{s}R)$. Once internal modes relax, each tagged segment follows the chain's ordinary long-time diffusion, with mean-square displacement asymptotic to $6D_{Z}t$.

In a concentrated solution or polymer melt, hydrodynamic coupling is screened. The unentangled Rouse model has an intermediate $t^{1/2}$ segment regime and whole-chain diffusivity proportional to $N^{-1}$. Long chains with polymer entanglement are additionally confined by neighbouring chains: reptation gives a slower intermediate $t^{1/4}$ regime and an asymptotic chain diffusivity scaling approximately as $N^{-2}$. The time windows depend on segment, entanglement and terminal relaxation times, so these powers do not all describe the same time interval.

After sufficiently long times both dilute and concentrated chains undergo ordinary translational diffusion. The main long-time difference is the coefficient and its chain-length dependence, with entanglement strongly reducing mobility when the chains are long enough; concentration alone does not guarantee an entangled regime.

Creating a liquid surface changes the molecular environment and costs interfacial free energy. In the dispersion-only cleavage model, the attractive Van der Waals force per area has magnitude $A/(6\pi h^3)$. Separating two contacting halves from a molecular cutoff $h_0$ to infinity takes work per areaHere $A_H$ denotes the Hamaker constant, to distinguish it from an area. Cleavage creates two free surfaces, giving the Hamaker estimate of liquid surface tensionThe continuum attraction diverges at zero separation and neglects molecular impenetrability and short-range repulsion. The cutoff is therefore an effective contact separation of molecular size, commonly of order $0.1$–$0.3\mathrm{nm}$, not a limit that can be taken to zero.

An experimental limitation is the strong temperature dependence of surface tension. For water in equilibrium with its vapour, the recommended experimental values decrease from $71.98\mathrm{mN}{\mathrm{m}}^{-1}$ at $25{}^{\circ }\mathrm{C}$ to $58.92\mathrm{mN}{\mathrm{m}}^{-1}$ at $100{}^{\circ }\mathrm{C}$, and the fitted tension vanishes at the critical point. A fixed pairwise cleavage energy misses interfacial entropy, changing density and many-body correlations.^[ref] The dispersion-only model is unsuitable when cohesion is dominated by hydrogen bonds, ionic interactions or metallic bonding; liquid metals are a clear example of the latter failure.

For a smooth homogeneous solid and a partially wetting liquid at equilibrium, tangential balance at the contact line gives Young's equationThe angle is measured through the liquid. Replacing solid-vapour area by solid-liquid plus liquid-vapour area changes free energy by $-S$ per wetted area, where the spreading coefficient isFor partial wetting, $S={\gamma }_{LV}(\cos \theta -1)<0$. Positive $S$ favours a spreading film and zero macroscopic equilibrium contact angle; the finite-angle Young construction cannot be extrapolated to $\cos \theta >1$.

The intended chemically matched surface approximation is ${\gamma }_{SL}\simeq 0$. It gives $S\simeq 23-21.3=1.7\mathrm{mN}{\mathrm{m}}^{-1}>0$, so increasing the wetted area lowers free energy. The initially steep drop spreads and flattens toward complete wetting:The initial $90^{\circ }$ angle describes a nonequilibrium shape and must not be inserted into equilibrium Young's equation to fix ${\gamma }_{SL}$. More precisely, complete wetting requires ${\gamma }_{SL}\le 1.7\mathrm{mN}{\mathrm{m}}^{-1}$ in this surface-energy model; chemical similarity alone is not a measurement of that interfacial energy.

Water on such a low-energy, oil-like surface is expected to bead into a comparatively rounded, high-angle drop because the substrate does not compensate its cohesive interactions. Its numerical contact angle needs the water-solid interfacial energy, which is not supplied.

The liquid does not wet the glass because $\cos 140^{\circ }<0$. The liquid meniscus is convex, with the capillary mercury level depressed below the reservoir. Balancing vertical surface-tension force against the hydrostatic column gives Jurin's law, using the inner radius $r=0.5\mathrm{mm}$:Thus the depression is approximately $10.1\mathrm{mm}$. The simple height law neglects small meniscus-volume corrections. Mercury's high surface tension reflects strong metallic bonding, not merely dispersion between weakly interacting molecular units.

A thin soap bubble has two interfaces, so the Young–Laplace equation gives $\Delta p=4\gamma /R=8\gamma /d$. HenceAt equal surface tension, the smaller bubble has the higher internal pressure. Air diffuses through their separating film from the smaller to the larger bubble, so the smaller shrinks and the larger grows. The pressure difference increases as the smaller shrinks. This is gas transfer driven by curvature, not necessarily immediate merging of the interfaces; coalescence additionally requires film rupture.

The mobile electrons in a metal screen an incident optical field. In the weak-damping Drude model, $\epsilon (\omega )\simeq {\epsilon }_{\infty }-{\omega }_p^2/{\omega }^2$. Below the screened plasma frequency, the real dielectric response is negative, the wave is evanescent inside the metal, and a clean surface is highly reflective. Above the plasma edge, the response can become positive and propagating waves are allowed, explaining the qualitative transmission window at sufficiently high, often ultraviolet, frequencies.

This is a free-carrier argument, not a statement that every metal is transparent throughout the ultraviolet. Interband absorption, damping and sample thickness can still produce strong absorption above the plasma edge.

The hydrogenic donor state radius is the Bohr radius rescaled by screening and effective mass:The donor-state overlap estimate compares the mean spacing $n_D^{-1/3}$ to this radius. Taking the spacing as $a_B^{\ast }$ gives $n_D\sim (a_B^{\ast }{)}^{-3}=1.32\times 10^{25}{\mathrm{m}}^{-3}=1.32\times 10^{19}\mathrm{c}{\mathrm{m}}^{-3}$. Taking it as the orbital diameter gives a factor eight less, $1.65\times 10^{18}\mathrm{c}{\mathrm{m}}^{-3}$. Thus a geometric estimate is roughly $10^{18}$–$10^{19}\mathrm{c}{\mathrm{m}}^{-3}$, with its contact convention stated. The exponential tails do not have a sharp overlap boundary, and this estimate should not be identified with a precise metal-insulator transition threshold.

Let the missing electron occupy the Bloch state of crystal momentum $\hslash {\mathbf{k}}_e$ and energy $E_e({\mathbf{k}}_e)$. Relative to the filled band, define ${\mathbf{k}}_h=-{\mathbf{k}}_e$ modulo a reciprocal-lattice vector and $E_h({\mathbf{k}}_h)=C-E_e(-{\mathbf{k}}_h)$, with a fixed reference $C$. ThenThe velocity equality follows by differentiating the hole energy with respect to ${\mathbf{k}}_h$; two minus signs cancel. The signed band effective mass is therefore opposite, becoming a positive hole mass near an electron-band maximum. The hole electric charge is $+e$ when the electron charge is $-e$. These momenta are crystal momenta, and the electron mass being compared is the signed band mass, not the bare electron mass. The positive hole current equals the current lost by removing that electron.

Angle-resolved photoemission spectroscopy maps occupied energy bands: ultraviolet or x-ray photons eject electrons, and an energy- and angle-resolving analyzer measures $E_{\mathrm{kin}}$ and emission direction. The photoelectric effect gives the binding energy $E_B=h\nu -\Phi -E_{\mathrm{kin}}$; the parallel momentum follows from $k_{\parallel }=\sqrt{2m{E}_{\mathrm{kin}}}\sin \theta /\hslash$, up to surface reciprocal vectors. Clean surfaces and controlled photon energy are needed, and an occupied-state measurement alone does not map an empty conduction band.

Scanning tunneling spectroscopy instead brings a conducting tip close to a surface and modulates the bias while recording the differential tunnel current. At low temperature with a smooth tip density of states and tunnelling matrix element, $dI/dV$ probes the local density of states near energy $E_F+eV$. The signal is thermally broadened and depends on tunnelling selection, so it is not an unqualified map of the whole bulk band structure.

Bulk probes complement these surface measurements. In the de Haas-van Alphen effect, a low-temperature magnetization or torque measurement versus magnetic field gives quantum-oscillation frequencies $F=\hslash A_{\mathrm{ext}}/(2\pi e)$, measuring extremal Fermi surface sections; temperature damping gives a cyclotron effective mass. The linear electronic heat capacity gives the density of low-energy excitations at the Fermi level. Optical absorption measures transition thresholds and matrix-element-weighted joint densities, rather than a unique full dispersion by itself.

In a heavy fermion material, interactions of narrow, often $f$-derived states with conduction electrons produce low-energy quasiparticles with very large thermodynamic effective mass. A coherent Kondo lattice can form a narrow low-energy band, so the large mass is an interaction-renormalized response rather than a larger bare electron mass.

Two concrete experimental signatures are an enormous linear electronic heat capacity coefficient and a strongly enhanced quadratic low-temperature electrical resistivity. Calorimetry separates $C=\gamma T+\beta T^3$ by plotting $C/T$ against $T^2$; the intercept measures the electronic coefficient rather than a lattice contribution. Four-terminal transport tests the Fermi liquid law $\rho ={\rho }_0+A{T}^2$. The low-temperature calorimetry and resistivity of CeAl3 found $\gamma =1.62\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-2}$ and $A=35\mu \mathrm{\Omega }\mathrm{cm}{\mathrm{K}}^{-2}$ below $0.2\mathrm{K}$, making both signatures quantitative.^[ref]

An enhanced nearly temperature-independent spin susceptibility and a large quantum-oscillation cyclotron effective mass provide additional tests. Not every heavy-fermion compound has the same ordered ground state; magnetism or superconductivity can intervene before a simple normal Fermi-liquid limit is reached.

A partly filled one-dimensional energy band is especially sensitive to a lattice modulation at $2k_F$. At half filling, $2k_F=\pi /a$, so the Peierls distortion alternates the bond lengths and doubles the primitive unit cell. Mixing states at the Fermi points opens a band gap, lowering the energy of occupied states. In the ideal weak-coupling, zero-temperature chain, the electronic gain behaves as $-u^2\ln (1/|u|)$ while the elastic cost is proportional to $u^2$, favouring a small distortion. The dimerized tight-binding chain gives an explicit band calculation.

The structural signature is extra diffraction intensity at the modulation wavevector; optical absorption can detect the gap and transport can detect loss of metallic conduction. A dynamic precursor is a Kohn anomaly, with phonon softening near $2k_F$. Inelastic neutron scattering of TTF-TCNQ, using a deuterated sample over $60$–$300\mathrm{K}$, found an anomaly near $2k_F$ developing below about $150\mathrm{K}$ as the $54\mathrm{K}$ transition was approached. This probes a lattice mode coupled to the electronic instability rather than inferring distortion from resistance alone.^[ref]

Away from half filling, the modulation need not simply double the lattice. Finite-temperature ordering in real quasi-one-dimensional solids also involves interchain coupling and fluctuations; the ideal isolated-chain argument alone does not establish a finite-temperature transition.

The nearly-free electron model treats a weak periodic potential as a perturbation of plane waves. Near a Bragg plane, states with wavevectors $\mathbf{k}$ and $\mathbf{k}-\mathbf{G}$ are nearly degenerate and must be mixed together. Their two-state Hamiltonian isAt the unperturbed crossing, the band gap is $2|V_{\mathbf{G}}|$. The corresponding standing waves place different charge densities near the ions and between them, giving different potential energies. Far from a crossing, ordinary perturbation theory is adequate; near it, the degenerate two-wave calculation is essential. In three dimensions, gaps at individual zone boundaries do not guarantee a global insulating gap because bands from different directions can overlap.

A direct band gap has its conduction minimum and valence maximum at the same crystal momentum. A photon can then connect the band-edge states with negligible momentum change. An indirect band gap requires an additional phonon to supply or remove momentum, making radiative recombination much less probable. Direct-gap materials such as gallium arsenide or gallium nitride are therefore usually better for a light-emitting diode than silicon.

For the graded junction, let $q=|e|>0$, ${\epsilon }_s={\epsilon }_0{\epsilon }_r$, and use electrostatic potential $\phi$ with $E_x=-{\phi }^{\prime }$. Fully ionized dopants have charge density $\rho =q(N_D-N_A)=q\alpha x$, so Poisson's equation is ${\phi }^{\prime \prime }=-q\alpha x/{\epsilon }_s$. The paper's displayed plus sign corresponds to using the negative potential $V=-\phi$, or a signed electron charge in that formula. The positive barrier magnitude and charge magnitude must be used consistently in the width below.

The intended compensation point is $x=0$, where $N_D=N_A$ and the net doping changes sign. A locally neutral model with comparable electron and hole mobilities has its lowest carrier conductivity near there. The depletion approximation neglects mobile-carrier conductivity throughout the layer, so it does not by itself locate an exact conductivity minimum; unequal mobilities or asymmetric carrier profiles can shift it.

Let the depletion edges be $x=\pm d$. With zero exterior field, integrating $E_x^{\prime }=\rho /{\epsilon }_s$ and imposing $E_x(\pm d)=0$ gives the linearly graded p-n junction fieldIts integral gives $\phi (x)=\phi (0)+q\alpha (d^{2}x-x^3/3)/(2{\epsilon }_s)$, so the electric potential increases toward the n-type side while the electron band edges $E_c,E_v$ bend downward together. Their separation is the local band gap, and the equilibrium electrochemical potential is flat. The built-in potential arises when carrier diffusion exposes ionized dopants until the opposing field balances further diffusion.

Finally the positive barrier isWith total width $w=2d$, this yields the linearly graded junction depletion widthHere $\alpha >0$ labels p-to-n progression with increasing $x$; otherwise use its magnitude in the width formula.

Bloch's theorem states that an eigenfunction in a periodic potential can be written ${\psi }_{\mathbf{k}}(\mathbf{r})=e^{i\mathbf{k}\cdot \mathbf{r}}{u}_{\mathbf{k}}(\mathbf{r})$, with $u_{\mathbf{k}}$ lattice-periodic. In the orthogonal tight-binding approximation, choose localized orthonormal orbitals $|j⟩$ with on-site matrix element $E_0$ and real nearest-neighbour hopping integral $t$. The Bloch sum $|k⟩=N^{-1/2}{\sum }_j{e}^{ikja}|j⟩$ is an eigenstate because each neighbour shift contributes a phase $e^{\pm ika}$. ThusThe simple-cubic nearest-neighbour tight-binding band has six neighbours, giving $E(\mathbf{k})=E_0+2t[\cos (k_{x}a)+\cos (k_{y}a)+\cos (k_{z}a)]$. The cosine sum ranges from $-3$ to $3$, so the positive bandwidth is $W=12|t|$. The stated hopping formula is negative when $E_1>0$, so the printed $12t$ must be read as a bandwidth magnitude.

Near the zone centre, $E=E_0+6t-t{a}^2{k}^2+O(k^4)$. To leading order the constant-energy surfaces are spheres, and comparison with $E_{\Gamma }+{\hslash }^2{k}^2/(2m^{\ast })$ givesThis is the effective mass for overlap-weighted exponential hopping. With $s=a/a_1$, its logarithmic derivative is $d\ln m^{\ast }/ds=1-2/s-1/(1+s)$. The positive stationary point is $s=1+\sqrt{3}$, and it is a minimum. The formula grows at both very small and very large $s$; the small-separation divergence is a mathematical extrapolation into a regime where localized tight binding itself becomes unreliable.

For the bandwidth, $W=24E_1(1+s)e^{-s}$ and $d\ln W/d\ln a=-s^2/(1+s)$. At $s=1$, a fractional spacing change $-0.01$ therefore givesThe exact ratio $W(0.99a_1)/W(a_1)=(1.99/2)e^{0.01}=1.004999916\dots$ checks the first-order result. Compression strengthens the hopping and widens the band.