Foundations

The ideal one-electron atom has discrete bound energies.

Quantum labels specify orbitals and spin states.

A bound nucleus has less rest mass than its separated nucleons.

A fixed decay probability per time gives a population timescale.

Nuclear rearrangement can release or require energy.

Normalize a scattering rate by incident flux.

Quarks and leptons form three generations.

Gauge fields mediate three interactions; the Higgs is a scalar sector.

A balance rule must name the interaction where it applies.

A periodic crystal potential organizes energies by wavevector and band.

Band occupancy and doping control mobile carriers.

Indistinguishable particles have different equilibrium state populations.

In the long-wavelength electric-dipole approximation, atomic transition strength depends on the dipole matrix element.

A Bravais lattice consists of translations R=Σ_i n_i a_i with integer n_i and independent primitive vectors.

Reciprocal vectors obey a_i·b_j=2πδ_ij and G=Σ_i h_i b_i with integer h_i.

For a cell basis, the kinematic elastic scattering amplitude contains F(G)=Σ_α f_α(G)exp(iG·d_α), and peak intensity is proportional to |F(G)|² with experiment-specific factors.

A monatomic chain with mass m, spacing a and nearest-neighbour spring constant κ has ω(k)=2√(κ/m)|sin(ka/2)|.

For a stable three-dimensional harmonic crystal with s atoms per primitive cell, the displacement dynamical matrix has 3s branches.

The three-dimensional Debye approximation assigns g(ω)=9Nω²/ω_D³ for 0≤ω≤ω_D so that the mode count is 3N.

For noninteracting spin-½ electrons in a three-dimensional volume, k_F=(3π²n)¹ᐟ³ and E_F=ħ²k_F²/(2m).

A single electron-carrier Drude model has dp/dt=−e(E+v×B)−p/τ.

For independent spin-½ moments of magnitude μ in magnetic flux density B, a moment has energies ±μB, partition sum 2cosh(μB/(k_BT)) and mean moment μtanh(μB/(k_BT)).

Ferromagnetic order aligns moments, antiferromagnetic order has opposing ordered sublattices and ferrimagnetic order has unequal opposing sublattice moments.

At weak magnetic field and T≈0, a noninteracting spin-½ electron gas has spin magnetization M=μ_B²g(E_F)B and susceptibility χ_P=μ₀μ_B²g(E_F).

For a rotationally symmetric localized electronic orbital, the quadratic magnetic-field term in the nonrelativistic Hamiltonian gives δE=e²B²⟨r²⟩/(12m_e).

For a wave packet in one smooth band E(k), neglecting anomalous velocity, v=(1/ħ)∇_kE and ħk̇=F.

In nondegenerate three-dimensional parabolic bands, n=N_c exp[−(E_c−μ)/(k_BT)] and p=N_v exp[−(μ−E_v)/(k_BT)], where N_c,v=2(2πm_c,v*k_BT/h²)³ᐟ² for one isotropic valley including two spins.

At weak electric field in a two-carrier relaxation-time band model, v_n=−μ_nE and v_p=+μ_pE with positive mobilities μ=|e|τ/m*.

For an electron in a central electrostatic potential energy V(r), the leading low-velocity corrections to p²/(2m_e)+V are H_rel=−p⁴/(8m_e³c²), H_SO=V′(r)L·S/(2m_e²c²r) and H_D=ħ²∇²V/(8m_e²c²).

For a spin-one-half electron bound to an infinitely heavy point nucleus of charge +Ze, the binding energy through order (Zα)⁴ is E_nj≈−m_ec²(Zα)²/(2n²)−m_ec²(Zα)⁴[n/(j+½)−¾]/(2n⁴), with α=e²/(4πε₀ħc).