Past exam of the physics A course of the University of Cambridge / 2014 / Paper 2 / Section A

For a diffraction-limited circular aperture, the Rayleigh criterion gives $\theta =1.22\lambda /D$. The photon wavelength is $\lambda =hc/E=1.24\mu \mathrm{m}$, hence $\theta \simeq 1.5\times 10^{-6}\mathrm{rad}$. This assumes adequate optical quality and sampling; being in space removes atmospheric seeing, not diffraction.

For a lossless single interface at normal incidence, $R=1-T=0.04$ and the Fresnel equations give $R=[(n-1)/(n+1){]}^2$, taking air's refractive index as one. Selecting the glass branch $n>1$ givesThe reciprocal root $2/3$ is the mathematical lower-index branch, not ordinary glass.

The power linewidth is $\Delta f=f_0/Q\simeq 0.044\mathrm{Hz}$. Fourier frequency resolution is of order $1/T_{\mathrm{obs}}$, so $T_{\mathrm{obs}}\gtrsim Q/f_0\simeq 23\mathrm{s}$, and preferably longer to distinguish the intrinsic linewidth from the observation-window width. zero padding does not improve this physical resolution. A freely decaying signal also needs sufficient microphone dynamic range.

The mean energy of a two-dimensional Fermi gas follows from its constant density of states $g$ for nonrelativistic quadratic dispersion. At zero temperature, states below ${ϵ}_F$ are filled, so

Translating a scattering centre by any lattice translation $\mathbf{R}$ changes the scattered amplitude by $e^{i\mathbf{G}\cdot \mathbf{R}}$, up to the immaterial overall sign convention. constructive interference requires $\mathbf{G}\cdot \mathbf{R}=2\pi$ times an integer. Applying this to $a\hat{\mathbf{x}},a\hat{\mathbf{y}},a\hat{\mathbf{z}}$ yieldsThus $\mathbf{G}$ lies in the reciprocal lattice. elastic scattering additionally requires $|{\mathbf{k}}_i+\mathbf{G}|=|{\mathbf{k}}_i|=2\pi /\lambda$; not every reciprocal vector is accessible for a fixed incident beam.