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Condensed Matter operators

26 operators in the condensed_matter category of the live registry. Compose them in a contract or call POST /api/zeq/compute — KO42 is always on, plus up to three more per call.

How it computes: dedicated closed-form solver solveCDM — real equations over CODATA-2018 constants; where an operator needs a series/matrix/field a scalar cannot express, it refuses honestly. See the solvers.

OperatorDescriptionEquation
BLOCH1Blochs theorem: eigenstates of a periodic potential are a plane wave times a lattice-periodic function, ψ(r) = e^{ik·r} u(r) — the foundation of electronic band structure.ψ(r) = e^{ik·r} u(r)
CDO1NFW dark matter halo density profile and enclosed mass for galaxy rotation curves.\rho(r) = \frac{\rho_0}{\frac{r}{r_s}\left(1 + \frac{r}{r_s}\right)^2}, \quad M(<r) = 4\pi \int_0^r \rho(r') r'^2 dr'
CDO3CMB temperature anisotropy expanded in spherical harmonics with angular power spectrum.\frac{\Delta T}{T}(\theta,\phi) = \sum_{l=2}^\infty \sum_{m=-l}^l a_{lm} Y_{lm}(\theta,\phi), \quad C_l = \langle |a_{lm}|^2 \rangle
CDO5Inflationary cosmology: Friedmann equation with scalar field driving exponential expansion.H^2 = \frac{8\pi G}{3} \left[\frac{1}{2} \dot{\phi}^2 + V(\phi)\right], \quad \ddot{\phi} + 3H\dot{\phi} + V'(\phi) = 0
CDO7Cosmological distance operators: luminosity distance and angular diameter distance as integrals over redshift.d_L(z) = (1+z) \int_0^z \frac{c dz'}{H(z')}, \quad d_A(z) = \frac{d_L(z)}{(1+z)^2}
CDO8Linear structure formation equation: growth of density perturbations in an expanding universe.\frac{d^2\delta}{dt^2} + 2H\frac{d\delta}{dt} = 4\pi G \bar{\rho} \delta, \quad \delta(\vec{x},t) = \frac{\rho(\vec{x},t) - \bar{\rho}(t)}{\bar{\rho}(t)}
CMP1Energy band structure operator computing electron energy as a function of crystal momentum in periodic potentials.E = \frac{\hbar^2 k^2}{2m^*}
CMP10Band gap energy operator computing the forbidden energy range between valence and conduction bands.E_g = E_c - E_v
CMP11Carrier concentration operator computing the density of electrons or holes in a semiconductor.n = N_c e^{-(E_c - E_F)/k_B T}
CMP12Mass action law n*p = n_i^2 relating electron and hole concentrations to the intrinsic carrier density.p = N_v e^{-(E_F - E_v)/k_B T}
CMP13Drift-diffusion equation combining carrier drift in electric fields with diffusion down concentration gradients.np = n_i^2
CMP14Heisenberg exchange interaction J*S_i dot S_j modeling spin-spin coupling in magnetic materials.J = qn\mu_n E + qD_n\frac{dn}{dx}
CMP15Magnetization operator M computing the net magnetic moment per unit volume in ferromagnetic materials.H = -J\sum_{\langle i,j \rangle} \vec{S}_i \cdot \vec{S}_j
CMP16Curie-Weiss law chi = C/(T-Tc) describing magnetic susceptibility near the ferromagnetic phase transition.M = M_s\tanh\left(\frac{\mu_0 m H}{k_B T}\right)
CMP17BCS superconducting gap equation determining the energy gap from electron-phonon pairing interaction.\chi = \frac{C}{T - T_C}
CMP18BCS critical temperature Tc expression relating superconducting transition temperature to Debye frequency and coupling strength.\Delta = 2\hbar\omega_D e^{-1/N(0)V}
CMP19Josephson junction current-phase relation I = I_c * sin(delta_phi) for supercurrent across a weak link.T_c = 1.13\hbar\omega_D e^{-1/N(0)V}/k_B
CMP2Electronic density of states g(E) counting the number of available quantum states per unit energy interval.g(E) = \frac{1}{2\pi^2}\left(\frac{2m^*}{\hbar^2}\right)^{3/2}\sqrt{E}
CMP20Condensed matter operator 20, an extended solid-state transport operator for multi-band systems.j_s = j_c\sin\phi
CMP3Fermi-Dirac distribution f(E) = 1/(e^((E-mu)/kT)+1) giving the occupation probability of fermion energy levels.f(E) = \frac{1}{e^{(E-\mu)/k_B T} + 1}
CMP4Electrical conductivity sigma relating current density to applied electric field in conducting materials.\sigma = ne\mu
CMP5Hall effect operator computing the transverse voltage arising from charge carriers moving in a magnetic field.R_H = \frac{1}{ne}
CMP6Resistivity operator relating electric field to current density, the inverse of conductivity.\rho(T) = \rho_0 + AT^2 + BT^5
CMP7Thermal conductivity operator relating heat flux to temperature gradient via Fourier's law.\kappa = \frac{1}{3}C_v v_s l
CMP8Specific heat capacity operator modeling the temperature dependence of heat capacity in solids.C_V = \gamma T + \beta T^3
CMP9Phonon dispersion relation omega(k) describing lattice vibration frequencies as a function of wavevector.\omega = \sqrt{\frac{4K}{M}}|\sin(ka/2)|

Compute one — a real call​

curl -sS -X POST https://zeqsdk.com/api/zeq/compute \
  -H "Authorization: Bearer $ZEQ_KEY" \
  -H "Content-Type: application/json" \
  -d '{"operators":["CMP4"],"inputs":{"n":1e+23,"mu":0.1}}'

Returns 1602.18 S/m — with unit, uncertainty, the generated master equation, and a signed envelope you can verify.

See also​