The Brillouin function is odd, increases monotonically through zero, and approaches $\pm 1$ for $x\to \pm \infty$. Its small-$x$ slope is $(J+1)/(3J)$. For $x\gg 1$, the magnetic ions overwhelmingly occupy the largest field-aligned moment projection, so the magnetization reaches magnetic saturation:The sketch shows several $J$ values; all have the same limiting values, while their slopes differ.

There are two electrons per primitive unit cell; a spin-degenerate energy band accommodates exactly that number. A semiconductor results when the first band is full and does not overlap the second. In the leading two-wave/four-wave approximation, compare the first-band maximum at $M$ with the second-band minimum at $X$:This is an indirect band gap; a nonzero local splitting at $X$ alone is insufficient. At the onset,With $a=0.3\mathrm{nm}$, ${\hslash }^2/(2m_e)=0.03810\mathrm{eV}\mathrm{n}{\mathrm{m}}^2$ gives $E_X\simeq 4.178\mathrm{eV}$, henceFor $V_{10}$ more negative than this value the estimated indirect band gap is positive; at equality the bands just touch. This is the onset estimate within the source's nearly-free electron model truncation, not an exact critical potential after coupling all plane waves.

Normalize the plane waves in a periodic crystal of area or volume $\mathcal{V}$. The Fourier coefficients of the periodic potential obeyTranslation of the integral by any lattice vector $\mathbf{R}$ leaves it invariant but multiplies it by $e^{-i(\mathbf{k}-{\mathbf{k}}^{\prime })\cdot \mathbf{R}}$. A nonzero result therefore requires $\mathbf{k}-{\mathbf{k}}^{\prime }$ to be a reciprocal lattice vector.

Near an isolated Bragg scattering degeneracy, ordinary nondegenerate perturbation theory fails because the free energies coincide. Retain the resonant plane waves $|\mathbf{k}⟩$, $|\mathbf{k}+\mathbf{G}⟩$ and apply degenerate perturbation theory. Absorb the constant $V_0$ into the energy origin. Since a real potential has $V_{-\mathbf{G}}=V_{\mathbf{G}}^{\ast }$, the projected Schrödinger equation has coefficient matrixA nonzero eigenstate requiresOff-resonant plane waves give higher-order energy shifts omitted in this leading nearly-free electron model.

Put $w=V_{11}<0$. At $M=(\pi /a,\pi /a)$, the free energy is $2E_X$. The four cyclic combinations in the supplied secular equation give the four-plane-wave splitting at a square-zone cornerFor example, the normalized coefficient vectors in the source's cyclic basis are $(1,1,1,1)/2$, $(1,0,-1,0)/\sqrt{2}$, $(0,1,0,-1)/\sqrt{2}$, and $(1,-1,1,-1)/2$. Because $u<w<0$, these are ordered as written: a lower singlet, a central doublet and an upper singlet. The sketch puts the $X$ and $M$ energies on one common scale; its representative negative couplings are stated in the caption.

Let $u=V_{10}<0$ andAt $X=(\pi /a,0)$ the two degenerate plane waves have momenta $(\pi /a,0)$ and $(-\pi /a,0)$ and coupling $u$. ThusTheir symmetric and antisymmetric combinations are the lower and upper states respectively. The local band gap is $2|u|$.

For the Lennard-Jones potential,Its minimum and harmonic spring constant areUsing the monatomic-chain dispersion relation,The maximum is at $|k|=\pi /a$. Taking the argon atomic mass as $m\simeq 39.95\mathrm{u}=6.63\times 10^{-26}\mathrm{kg}$ and $ϵ=1.602\times 10^{-21}\mathrm{J}$ givesThese are the stipulated one-dimensional nearest-neighbour model estimates; a three-dimensional argon crystal has additional neighbours and phonon polarizations.

With atomic displacement $u_j$, the harmonic potential energy is $\frac{1}{2}K{\sum }_j(u_{j+1}-u_j{)}^2$. Newton's second law givesFor a periodic chain of $N$ atoms, its normal modes are $u_j=U{e}^{i(kja-\omega t)}$, with $k=2\pi s/(Na)$ and one representative per Brillouin zone. Substitution yields the monatomic-chain dispersion relationA real general displacement is a superposition of these normal modes and their complex conjugates. The $k=0$ mode is a rigid translation. At small $|k|$, the acoustic phonon branch is linear, with speed of soundFor $0<k<\pi /a$, the group velocity is $a\sqrt{K/m}\cos (ka/2)$ and vanishes at the zone boundary. The dashed initial tangent in the sketch has slope $v$.

Let $a$ be a stable equilibrium separation: $V^{\prime }(a)=0$, $K=V^{\prime \prime }(a)>0$. With bond extension $\delta =x-a$, the Taylor expansion isThe harmonic approximation applies when the typical thermal and zero-point bond fluctuations satisfy $|\delta |\ll a$ and the higher terms are negligible, for example $|V^{\prime \prime \prime }(a)\delta |\ll 3K$ and $|V^{\prime \prime \prime \prime }(a){\delta }^2|\ll 12K$. The relevant small quantity is the relative displacement of neighbouring atoms. Small bond excursions about a stable minimum give an amplitude-independent spring constant $K=V^{\prime \prime }(a)$. Large thermal excursions, bond breaking or appreciable anharmonicity invalidate the approximation.

In the electron-count estimate $f_{\mathrm{Ga}}\simeq 31$, $f_{\mathrm{As}}\simeq 33$, the structure factors are $F_{111}=4(31-33i)$ and $F_{200}=4(31-33)$. Thus a single-directed-reflection estimate isA powder X-ray diffraction peak collects all equivalent reflections. The powder diffraction multiplicities are eight for $\{111\}$ and six for $\{200\}$, so the multiplicity-and-electron-count estimate isAt $(200)$ the two sublattice amplitudes are opposite in phase and nearly cancel; at $(111)$ their relative phase is a quarter cycle, so this cancellation does not occur. In silicon, the two atomic scattering factors are identical and the diamond cubic structure has an exactly forbidden $(200)$ reflection, while $(111)$ remains allowed in the kinematic approximation.

The numerical ratio neglects the angular dependence of the atomic scattering factors, the Lorentz-polarization factor and thermal-motion corrections. No wavelength or measurement geometry is supplied to refine those factors. This is therefore a stated structural estimate, rather than a unique precision ratio of measured peak areas.

In the conventional cubic unit cell, the four face-centered cubic lattice positions contributeThe two-atom atomic basis of a crystal gives the zinc blende structure factorMixed-parity Miller indices have $S_{\mathrm{FCC}}=0$. With all indices even or all odd, $S_{\mathrm{FCC}}=4$. For real atomic scattering factors,Hence all same-parity reflections are generally permitted in GaAs, although the last class is weak when the two atomic scattering factors nearly coincide. These are the structural selection rules; a particular experiment must also satisfy the elastic Bragg scattering condition.

Use the scattering vector $\mathbf{q}={\mathbf{k}}_{\mathrm{in}}-{\mathbf{k}}_{\mathrm{out}}$; reversing its sign conjugates the following amplitudes without changing the intensities. In the kinematic approximation, the atomic scattering factor of atom $s$ isAtoms at $\mathbf{R}+{\mathbf{d}}_s$ therefore give the scattering amplitudeHere $A_e$ is the single-electron amplitude; $F$ is the unit cell structure factor, and $S_L$ is the geometrical lattice sum. If the crystal has $N_i$ repeats along ${\mathbf{a}}_i$, the latter factor is a product of geometric series, whose magnitudes containThe sharp constructive peaks occur when every $\mathbf{q}\cdot {\mathbf{a}}_i=2\pi h_i$. Thus a permitted reflection has wavevector change equal to a reciprocal lattice vector, with $F(\mathbf{G})\ne 0$ and the additional elastic condition $|{\mathbf{k}}_{\mathrm{out}}|=|{\mathbf{k}}_{\mathrm{in}}|$. A finite crystal broadens these peaks; it does not impose an exact delta-function selection on every scattered photon.

Write $\mathbf{r}={\sum }_i{s}_i{\mathbf{a}}_i$. The electron density is periodic with period one in each $s_i$, so its three-dimensional Fourier series has integer harmonics:If $\Omega$ is a primitive unit cell of volume ${\Omega }_c$, Fourier orthogonality givesConversely, translation by ${\mathbf{a}}_i$ multiplies a general plane wave by $e^{i\mathbf{q}\cdot {\mathbf{a}}_i}$. Periodicity permits only $\mathbf{q}\cdot {\mathbf{a}}_i\in 2\pi \mathbb{Z}$, precisely the reciprocal lattice vectors. For a physical bounded periodic electron density, the expansion exists in the usual square-integrable sense.

For the mutually orthogonal direct primitive lattice translation vectors ${\mathbf{a}}_i$, define the reciprocal lattice basis byThus $\mathbf{G}={\sum }_i{h}_i{\mathbf{b}}_i$, with integer $h_i$, satisfies $\exp (i\mathbf{G}\cdot \mathbf{R})=1$ for every lattice vector $\mathbf{R}$.

The official notes, PDF p. 27, handwritten p. 20 derive the early-time photon count, its vanishing limit, and the failure of thermal equilibrium when the count is of order one. Two details need supplementing.

First, their constant $N_{\gamma 0}$ is the normalization obtained by extrapolating the early-time particle horizon formula to $z=0$. It is not the count within today's actual particle horizon. Let $N_0=N_{\gamma }(t_0)$ and let $r_{h0}$ be that horizon's physical radius. Using the paper's early-time approximation givesFor a present matter-dominated approximation, $r_{h0}\simeq 2c/H_0$. Alternatively, retaining both components in the flat model exactly, ${\Omega }_R=1/(2+z_{\mathrm{eq}})$, and integrating the particle horizon givesThe second boxed formula uses the exact two-component normalization and the radiation-dominated early-time limit. Its large-$z_{\mathrm{eq}}$ leading term agrees with the first expression and $r_{h0}\simeq 2c/H_0$.

Second, a particle horizon containing only order-one photons cannot support a smooth, locally representative radiation density: discrete-particle fluctuations are of the same order as the mean, and a single photon's direction is strongly anisotropic. Thus the fluid description assuming spatial homogeneity and cosmological isotropy is not self-consistent on that causal scale. This does not by itself disprove an ensemble-averaged cosmological symmetry; it limits the extrapolation of the smooth thermal equilibrium description.

The official notes, PDF pp. 24–27, handwritten pp. 17–20 obtain the photon number density by integrating the Planck distribution, explain the temperature and dilution scalings, and compare the interaction time with the expansion time for thermal equilibrium. They also correct the missing cube and factor-of-two error in the paper's printed photon number density.

The official notes, PDF p. 24, handwritten p. 17 integrate the physical particle horizon and derive its early-time redshift dependence, explicitly stating the present matter-dominated approximation used in the printed expression.

The official notes, PDF pp. 22–23, handwritten pp. 15–16 give both density scalings, integrate the radiation- and matter-dominated Friedmann equations, normalize the late solution today, and match the two approximations at equality.

The official notes, PDF p. 21, handwritten p. 14 give the acceleration signs and the instability conclusion for the Einstein static universe. One wording correction is necessary: their word “oscillatory” for the overdense perturbation is a slip. Starting near rest, an overdense model contracts and its matter density rises, whereas an underdense model expands and its matter density falls; both move away from the static balance.

The official notes, PDF pp. 20–21, handwritten pp. 13–14 combine the Friedmann equations and Raychaudhuri equation to derive the density and scale factor of the Einstein static universe.

The official notes, PDF p. 19, handwritten p. 12 integrate the radial null geodesic, express the radial coordinate in terms of comoving distance, and give its maximum value in that coordinate chart. The maximum radial coordinate is not the total path length around a closed spatial slice.