Under the one-carrier Drude model, the magnitude of the Hall coefficient gives an effective carrier densityConvert the material mass density to $1850\mathrm{kg}{\mathrm{m}}^{-3}$ and its molar mass to $0.009\mathrm{kg}\mathrm{mo}{\mathrm{l}}^{-1}$. The atomic number density isHere $N_A=6.02214076\times 10^{23}\mathrm{mo}{\mathrm{l}}^{-1}$ is the Avogadro constant. The positive Hall coefficient gives a hole-like response in a one-carrier interpretation. The Hall carrier count versus valence distinction matters: beryllium has two valence electrons, but the result is neither two free electrons nor a literal fractional atom donating charge. Its energy bands and Fermi surface invalidate the simple isotropic one-carrier inference. Electron-like and hole-like contributions with different charge carrier mobilities can yield this sign and effective count.

Take nearest-neighbour separation $d$, a common spring constant $K$, and one scalar displacement per atom. For a monatomic chain, the harmonic approximation near a potential minimum, Newton's second law and a normal mode giveThe monatomic-chain dispersion relation has one acoustic phonon branch, with $\omega \simeq d\sqrt{K/m}|k|$ near the origin. The first Brillouin zone ends at $\boxed{k=\pm \pi /d}$, where $\omega =2\sqrt{K/m}$ and the group velocity vanishes.

For alternating masses $m,M$, choose a two-atom cell of length $2d$. The coupled normal modes give the diatomic-chain dispersion relation with equal springs,The first Brillouin zone now ends at $\boxed{k=\pm \pi /(2d)}$. The minus branch is the acoustic phonon branch; its two atoms move together at long wavelength. The plus branch is the optical phonon branch; at $k=0$ the atoms move oppositely with a stationary centre of mass and ${\omega }_{+}^2=2K(1/m+1/M)$. For $M>m$ the edge frequencies are $\sqrt{2K/M}$ and $\sqrt{2K/m}$, with a gap between them.

If $M=m$, the two branches reduce to $2\sqrt{K/m}|\sin (kd/2)|$ and $2\sqrt{K/m}|\cos (kd/2)|$, touching at $\sqrt{2K/m}$. This is zone folding in an equal-mass diatomic chain: identical sites actually have a primitive unit cell of length $d$. A doubled cell folds the monatomic branch but opens no gap. Equal masses alone do not specify all possible spring models; the sketches assume equal nearest-neighbour spring constants.

At normal incidence between lossless media with real positive acoustic impedances $Z_1,Z_2$, continuity of pressure and normal particle velocity gives the pressure transmission amplitude $t_p=2Z_2/(Z_1+Z_2)$. Since the mean acoustic intensity is $|p{|}^2/(2Z)$, the fraction of incident energy transmitted isThis is acoustic reflection and transmission at an interface. With the supplied acoustic impedances,A thin water-based gel layer excludes air gaps and approximately matches the body's acoustic impedance. About $99.90\%$ of incident energy would otherwise reflect at the air-body interface. The coupling is also needed on the return journey, so it strongly improves the detectable echo. These figures concern interface transmission, before absorption within tissue; the gel is approximated by the stated water value.

Both the incoming and outgoing spherical wavefronts contribute to the path excess. At aperture radius $u$, with axial distances $s$ before and $r$ after the aperture,The outer edge of the $n$th Fresnel half-period zone has $\Delta \ell =n\lambda /2$. HenceTherefore the outer radius is $\boxed{u_2=4.47\times 10^{-4}\mathrm{m}=0.447\mathrm{mm}}$. The paraxial approximation is excellent because $u_2\ll s,r$. Treating the illumination as a plane wave would incorrectly omit the source-side path curvature.

For a photon, $E=hc/\lambda$. Using Planck constant $h=6.62607015\times 10^{-34}\mathrm{J}\mathrm{s}$, speed of light $c=2.99792458\times 10^8\mathrm{m}{\mathrm{s}}^{-1}$ and $1\mathrm{eV}=1.602176634\times 10^{-19}\mathrm{J}$ givesThe Rayleigh criterion for a circular aperture of diameter $D$ is ${\theta }_R=1.22\lambda /D$, set by the first zero of its Airy disk. Thus the ideal diffraction-limited resolution isThe conversion to transverse separation uses the small-angle approximation. An order-of-magnitude estimate $\lambda /D$ instead gives $2.07\times 10^{-7}\mathrm{rad}$ and $4.13\times 10^{10}\mathrm{m}$; the distinction is the chosen resolution criterion. These are aperture limits, before instrumental aberrations and detector sampling.