Past exam of the physics A course of the University of Cambridge / 2013 / Paper 2 / Section D

The Debye model replaces the crystal by a harmonic elastic continuum with acoustic dispersion $\omega =v_{s}k$. It counts three polarizations and introduces a cutoff so that the total number of vibrational modes is $3N$, where $N$ is the atom count. Periodic-box counting givesFor an isotropic average speed, $3/v_D^3=1/v_L^3+2/v_T^3$. With cutoff frequency ${\omega }_D$ fixed by the mode count,This density of states is quadratic because a three-dimensional shell has volume proportional to $k^{2}dk$ and the acoustic dispersion is linear. The cutoff is a mode-counting device; it is not a claim that every actual branch stays linear to the zone boundary.

Each phonon mode has mean thermal energy $\hslash \omega /(e^{\beta \hslash \omega }-1)$, using Bose-Einstein statistics with zero phonon chemical potential. The fixed harmonic zero-point contribution has zero temperature derivative. With $x=\hslash \omega /(k_{B}T)$,and differentiating at fixed volume givesFor $T\ll {\Theta }_D$, extend the upper limit to infinity. Expanding $1/(e^x-1)={\sum }_{j\ge 1}{e}^{-jx}$ gives ${\int }_0^{\infty }{x}^3/(e^x-1)dx=6\sum j^{-4}={\pi }^4/15$. integration by parts makes the heat-capacity integral $4{\pi }^4/15$, soOnly modes with $\hslash \omega \lesssim k_{B}T$ are appreciably excited; their number scales as $T^3$, explaining the power law physically. At high temperature every mode contributes $k_B$, giving the Dulong-Petit limit $C_V\to 3N{k}_B$.

The model assumes the harmonic approximation, a macroscopic three-dimensional solid and acoustic continuum behaviour for the thermally populated modes. Anisotropic sound speeds change the low-temperature coefficient rather than its cubic power. Real optical branches and nonlinear acoustic dispersion limit the full-temperature approximation; electronic excitations can add a linear term in a metal, and anharmonicity produces thermal expansion and distinctions between $C_P$ and $C_V$. At temperatures below the lowest finite-sample mode, the bulk continuum law also fails. Thus the $T^3$ result is the low-temperature lattice contribution, not a universal total heat capacity for every solid.

For a circular structure, cylindrical symmetry reduces the scalar Fresnel-Kirchhoff diffraction integral to concentric annuli. With plane-wave illumination, radius $a$ and distance $z$, the relevant Fresnel number is $N_F=a^2/(\lambda z)$. Successive Fresnel half-period zones have radii $r_j^2\simeq j\lambda z$: their areas are approximately equal while their phases alternate by $\pi$, causing neighbouring-zone cancellation.

For Fresnel diffraction by a circular aperture, the on-axis field follows directly from the radial integral,Odd zone counts reinforce on axis; even counts cancel in this ideal scalar model. An annular opening gives the difference of the corresponding two edge exponentials. A zone plate alternately blocks or phase-shifts zones so that otherwise cancelling contributions reinforce at a focus, with $r_j^2\simeq j\lambda f$ for its principal focal distance.

Babinet's principle gives $U_{\mathrm{disk}}=U_{\mathrm{free}}-U_{\mathrm{ap}}=U_{\mathrm{free}}{e}^{i\pi N_F}$ for Fresnel diffraction by an opaque disk. Thus its ideal on-axis intensity equals the unobstructed intensity, the Arago spot, even within the geometric shadow. This concerns a narrow axial feature, not the whole shadow. Finite source size, imperfect circularity, bandwidth and alignment reduce its visibility. When $N_F\ll 1$, the circular-aperture far-field pattern becomes the Airy disk intensity pattern $[2J_1(u)/u{]}^2$, with $u=ka\sin \theta$ and first minimum near $\theta =1.22\lambda /(2a)$.

Two parallel partially reflecting surfaces in a Fabry–Pérot interferometer produce a sequence of coherent transmitted beams. For equal lossless mirror intensity reflectance $\mathcal{R}$ and round-trip phase $\delta =4\pi nd\cos {\theta }_t/\lambda$, summing the geometric series of amplitudes givesresonances have $2nd\cos {\theta }_t=m\lambda$ and unit peak transmission in this symmetric lossless model. Varying angle gives concentric equal-inclination rings; changing spacing scans the resonance wavelengths.

At normal incidence with negligible dispersion, the free spectral range is $\Delta {\nu }_{\mathrm{FSR}}=c/(2nd)$. For high reflectance, the peak full width gives finesse and resolving powerThe coefficient $\mathcal{F}$ in the transmission formula and the finesse $\mathcal{F}$ are different quantities. Multiple round trips sharpen a peak, allowing high-resolution spectroscopy, frequency selection and cavity measurements. Absorption, mirror mismatch, finite angular spread, imperfect parallelism and surface errors reduce usable transmission or finesse. Adjacent orders can overlap in a broadband measurement, requiring order selection or prior spectral information.

For $N$ uniformly illuminated identical slits of a diffraction grating, with width $b$ and period $d$, the far-field amplitude is a single-slit transform times a finite phasor sum. At normal incidence, write $\beta =\pi d\sin \theta /\lambda$. ThenPrincipal maxima obey $d\sin \theta =m\lambda$ and have $N^2$ times the single-slit intensity at that angle. Nearby first minima are separated from a principal maximum by $\delta \theta \simeq \lambda /(Nd\cos {\theta }_m)$, so more illuminated slits sharpen each order. The broad single-slit envelope suppresses high angles; an order is missing if its principal maximum coincides with an envelope zero.

Differentiating the grating equation gives angular dispersion $d\theta /d\lambda =m/(d\cos {\theta }_m)$. Comparing the separation of nearby wavelengths with the first-minimum width yields the Rayleigh resolving powerOnly propagating orders with $|m\lambda |\le d$ exist. Different spectral orders can overlap, and finite source-slit width, limited coherence, aberrations and grating errors worsen ideal resolution. A spectrometer therefore collimates light from an entrance slit and focuses each diffracted direction onto a detector; the slit image width and illuminated grating width must both support the desired spectral separation.