# List of equations in nuclear and particle physics

This article summarizes equations in the theory of nuclear physics and particle physics.

## Definitions

| Quantity (common name/s) | (Common) symbol/s | Defining equation | SI units | Dimension |
| Number of atoms | N = Number of atoms remaining at time t N$_0$ = Initial number of atoms at time t = 0 N$_D$ = Number of atoms decayed at time t | $N_{0}=N+N_{D}\,\!$ | dimensionless | dimensionless |
| Decay rate, activity of a radioisotope | A | $A=\lambda N\,\!$ | Bq = Hz = s$^{−1}$ | [T]$^{−1}$ |
| Decay constant | λ | $\lambda =A/N\,\!$ | Bq = Hz = s$^{−1}$ | [T]$^{−1}$ |
| Half-life of a radioisotope | t$_{1/2}$, T$_{1/2}$ | Time taken for half the number of atoms present to decay $t\rightarrow t+T_{1/2}\,\!$ $N\rightarrow N/2\,\!$ | s | [T] |
| Number of half-lives | n (no standard symbol) | $n=t/T_{1/2}\,\!$ | dimensionless | dimensionless |
| Radioisotope time constant, mean lifetime of an atom before decay | τ (no standard symbol) | $\tau =1/\lambda \,\!$ | s | [T] |
| Absorbed dose, total ionizing dose (total energy of radiation transferred to unit mass) | D can only be found experimentally | N/A | Gy = 1 J/kg (Gray) | [L]$^2$[T]$^{−2}$ |
| Equivalent dose | H | $H=DQ\,\!$ Q = radiation quality factor (dimensionless) | Sv = J kg$^{−1}$ (Sievert) | [L]$^2$[T]$^{−2}$ |
| Effective dose | E | $E=\sum _{j}H_{j}W_{j}\,\!$ W$_j$ = weighting factors corresponding to radiosensitivities of matter (dimensionless) $\sum _{j}W_{j}=1\,\!$ | Sv = J kg$^{−1}$ (Sievert) | [L]$^2$[T]$^{−2}$ |

## Equations
### Nuclear structure

| Physical situation | Nomenclature | Equations |
| Mass number | A = (Relative) atomic mass = Mass number = Sum of protons and neutrons N = Number of neutrons Z = Atomic number = Number of protons = Number of electrons | $A=Z+N\,\!$ |
| Mass in nuclei | M'$_{nuc}$ = Mass of nucleus, bound nucleons M$_Σ$ = Sum of masses for isolated nucleons m$_p$ = proton rest mass m$_n$ = neutron rest mass | $M_{\Sigma }=Zm_{p}+Nm_{n}\,\!$ $M_{\Sigma }>M_{N}\,\!$ $\Delta M=M_{\Sigma }-M_{\mathrm {nuc} }\,\!$ $\Delta E=\Delta Mc^{2}\,\!$ |
| Nuclear radius | r$_0$ ≈ 1.2 fm | $r=r_{0}A^{1/3}\,\!$ hence (approximately) nuclear volume ∝ A nuclear surface ∝ A$^{2/3}$ |
| Nuclear binding energy, empirical curve | Dimensionless parameters to fit experiment: E$_B$ = binding energy, a$_v$ = nuclear volume coefficient, a$_s$ = nuclear surface coefficient, a$_c$ = electrostatic interaction coefficient, a$_a$ = symmetry/asymmetry extent coefficient for the numbers of neutrons/protons, | ${\begin{aligned}E_{B}=&a_{v}A-a_{s}A^{2/3}-a_{c}Z(Z-1)A^{-1/3}\\&-a_{a}(N-Z)^{2}A^{-1}+12\delta (N,Z)A^{-1/2}\\\end{aligned}}$ where (due to pairing of nuclei) δ(N, Z) = +1 even N, even Z, δ(N, Z) = −1 odd N, odd Z, δ(N, Z) = 0 odd A |

### Nuclear decay

| Physical situation | Nomenclature | Equations |
| Radioactive decay | N$_0$ = Initial number of atoms N = Number of atoms at time t λ = Decay constant t = Time | Statistical decay of a radionuclide: ${\frac {\mathrm {d} N}{\mathrm {d} t}}=-\lambda N$ $N=N_{0}e^{-\lambda t}\,\!$ |
| Bateman's equations | $c_{i}=\prod _{j=1,i\neq j}^{D}{\frac {\lambda _{j}}{\lambda _{j}-\lambda _{i}}}$ | $N_{D}={\frac {N_{1}(0)}{\lambda _{D}}}\sum _{i=1}^{D}\lambda _{i}c_{i}e^{-\lambda _{i}t}$ |
| Radiation flux | I$_0$ = Initial intensity/Flux of radiation I = Number of atoms at time t μ = Linear absorption coefficient x = Thickness of substance | $I=I_{0}e^{-\mu x}\,\!$ |

### Nuclear scattering theory

The following apply for the nuclear reaction:
    a + b ↔ R → c
in the centre of mass frame, where a and b are the initial species about to collide, c is the final species, and R is the resonant state.
| Physical situation | Nomenclature | Equations |
| Breit-Wigner formula | E$_0$ = Resonant energy Γ, Γ$_{ab}$, Γ$_c$ are widths of R, a + b, c respectively k = incoming wavenumber s = spin angular momenta of a and b J = total angular momentum of R | Cross-section: $\sigma (E)={\frac {\pi g}{k^{2}}}{\frac {\Gamma _{ab}\Gamma _{c}}{(E-E_{0})^{2}+\Gamma ^{2}/4}}$ Spin factor: $g={\frac {2J+1}{(2s_{a}+1)(2s_{b}+1)}}$ Total width: $\Gamma =\Gamma _{ab}+\Gamma _{c}$ Resonance lifetime: $\tau =\hbar /\Gamma$ |
| Born scattering | r = radial distance μ = Scattering angle A = 2 (spin-0), −1 (spin-half particles) Δk = change in wavevector due to scattering V = total interaction potential V = total interaction potential | Differential cross-section: ${\frac {d\sigma }{d\Omega }}=\left|{\frac {2\mu }{\hbar ^{2}}}\int _{0}^{\infty }{\frac {\sin(\Delta kr)}{\Delta kr}}V(r)r^{2}dr\right|^{2}$ |
| Mott scattering | χ = reduced mass of a and b v = incoming velocity | Differential cross-section (for identical particles in a coulomb potential, in centre of mass frame): ${\frac {d\sigma }{d\Omega }}=\left({\frac {\alpha }{4E}}\right)\left[\csc ^{4}{\frac {\chi }{2}}+\sec ^{4}{\frac {\chi }{2}}+{\frac {A\cos \left({\frac {\alpha }{\hbar \nu }}\ln \tan ^{2}{\frac {\chi }{2}}\right)}{\sin ^{2}{\frac {\chi }{2}}\cos {\frac {\chi }{2}}}}\right]^{2}$ Scattering potential energy (α = constant): $V=-\alpha /r$ |
| Rutherford scattering | | Differential cross-section (non-identical particles in a coulomb potential): ${\frac {d\sigma }{d\Omega }}=\left({\frac {1}{n}}\right){\frac {dN}{d\Omega }}=\left({\frac {\alpha }{4E}}\right)^{2}\csc ^{4}{\frac {\chi }{2}}$ |

### Fundamental forces

These equations need to be refined such that the notation is defined as has been done for the previous sets of equations.
| Name | Equations |
| Strong force | ${\begin{aligned}{\mathcal {L}}_{\mathrm {QCD} }&={\bar {\psi }}_{i}\left(i\gamma ^{\mu }(D_{\mu })_{ij}-m\,\delta _{ij}\right)\psi _{j}-{\frac {1}{4}}G_{\mu \nu }^{a}G_{a}^{\mu \nu }\\&={\bar {\psi }}_{i}(i\gamma ^{\mu }\partial _{\mu }-m)\psi _{i}-gG_{\mu }^{a}{\bar {\psi }}_{i}\gamma ^{\mu }T_{ij}^{a}\psi _{j}-{\frac {1}{4}}G_{\mu \nu }^{a}G_{a}^{\mu \nu }\,,\\\end{aligned}}\,\!$ |
| Electroweak interaction | ${\mathcal {L}}_{\mathrm {EW} }={\mathcal {L}}_{g}+{\mathcal {L}}_{f}+{\mathcal {L}}_{h}+{\mathcal {L}}_{y}.\,\!$ ${\mathcal {L}}_{g}=-{\frac {1}{4}}W_{a}^{\mu \nu }W_{\mu \nu }^{a}-{\frac {1}{4}}B^{\mu \nu }B_{\mu \nu }\,\!$ ${\mathcal {L}}_{f}={\overline {Q}}_{i}iD\!\!\!\!/\;Q_{i}+{\overline {u}}_{i}^{c}iD\!\!\!\!/\;u_{i}^{c}+{\overline {d}}_{i}^{c}iD\!\!\!\!/\;d_{i}^{c}+{\overline {L}}_{i}iD\!\!\!\!/\;L_{i}+{\overline {e}}_{i}^{c}iD\!\!\!\!/\;e_{i}^{c}\,\!$ ${\mathcal {L}}_{h}=|D_{\mu }h|^{2}-\lambda \left(|h|^{2}-{\frac {v^{2}}{2}}\right)^{2}\,\!$ ${\mathcal {L}}_{y}=-y_{u\,ij}\epsilon ^{ab}\,h_{b}^{\dagger }\,{\overline {Q}}_{ia}u_{j}^{c}-y_{d\,ij}\,h\,{\overline {Q}}_{i}d_{j}^{c}-y_{e\,ij}\,h\,{\overline {L}}_{i}e_{j}^{c}+h.c.\,\!$ |
| Quantum electrodynamics | ${\mathcal {L}}_{\mathrm {QED} }={\bar {\psi }}(i\gamma ^{\mu }D_{\mu }-m)\psi -{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }\;,\,\!$ |