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 N0 = Initial number of atoms at time t = 0 ND = Number of atoms decayed at time t	$N_0 = N + N_D \,\!$	dimensionless	dimensionless
Decay rate, activity of a radioisotope	A	$A = \mathrm{d} N /\mathrm{d} t \,\!$	Bq = Hz = s−1	[T]−1
Decay constant	λ	$\lambda = A/N \,\!$	Bq = Hz = s−1	[T]−1
Half-life of a radioisotope	t1/2, T1/2	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	$E = \sum_j H_jW_j \,\!$ Wj = 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 MΣ = Sum of masses for isolated nucleons mp = proton rest mass mn = neutron rest mass	$M_\Sigma = Zm_p + Nm_n \,\!$ $M_\Sigma > M_N \,\!$ $\Delta M = M_\Sigma - M_\mathrm{nuc} \,\!$ $\Delta E = \Delta M c^2\,\!$
Nuclear radius	r0 ≈ 1.2 fm	$r=r_0A^{1/3} \,\!$ hence (approximately) nuclear volume ∝ A nuclear surface ∝ A2/3
Nuclear binding energy, empirical curve	Dimensionless parameters to fit experiment: EB = binding energy, av = nuclear volume coefficient, as = nuclear surface coefficient, ac = electrostatic interaction coefficient, aa = symmetry/asymmetry extent coefficient for the numbers of neutrons/protons,	$\begin{align} E_B = & a_v A - a_s A^{2/3} - a_c Z(Z-1)A^{-1/3} \\ & -a_a (N-Z)^2 A^{-1} + 12\delta(N,Z)A^{-1/2} \\ \end{align}$ 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	N0 = Initial number of atoms N = Number of atoms at time t λ = Decay constant t = Time	Statistical decay of a radionuclide: $\frac{\mathrm{d} N}{\mathrm{d} t} = - \lambda N$ $N = N_0e^{-\lambda t}\,\!$
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	I0 = Initial intensity/Flux of radiation I = Number of atoms at time t μ = Linear absorption coefficient x = Thickness of substance	$I = I_0e^{-\mu x}\,\!$

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	E0 = Resonant energy Γ, Γab, Γc 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: $\frac{d\sigma}{d\Omega} = \left|\frac{2\mu}{\hbar^2}\int_0^\infty\frac{\sin(\Delta kr)}{\Delta kr}V(r)r^2dr\right|^2$
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	$\begin{align} \mathcal{L}_\mathrm{QCD} & = \bar{\psi}_i\left(i \gamma^\mu (D_\mu)_{ij} - m\, \delta_{ij}\right) \psi_j - \frac{1}{4}G^a_{\mu \nu} G^{\mu \nu}_a \\ & = \bar{\psi}_i (i \gamma^\mu \partial_\mu - m )\psi_i - g G^a_\mu \bar{\psi}_i \gamma^\mu T^a_{ij} \psi_j - \frac{1}{4}G^a_{\mu \nu} G^{\mu \nu}_a \,,\\ \end{align} \,\!$
Electroweak interaction	:$\mathcal{L}_{EW} = \mathcal{L}_g + \mathcal{L}_f + \mathcal{L}_h + \mathcal{L}_y.\,\!$ $\mathcal{L}_g = -\frac{1}{4}W_a^{\mu\nu}W_{\mu\nu}^a - \frac{1}{4}B^{\mu\nu}B_{\mu\nu}\,\!$ $\mathcal{L}_f = \overline{Q}_i iD\!\!\!\!/\; Q_i+ \overline{u}_i^c iD\!\!\!\!/\; u^c_i+ \overline{d}_i^c iD\!\!\!\!/\; d^c_i+ \overline{L}_i iD\!\!\!\!/\; L_i+ \overline{e}^c_i iD\!\!\!\!/\; e^c_i \,\!$ $\mathcal{L}_h = |D_\mu h|^2 - \lambda \left(|h|^2 - \frac{v^2}{2}\right)^2\,\!$ $\mathcal{L}_y = - y_{u\, ij} \epsilon^{ab} \,h_b^\dagger\, \overline{Q}_{ia} u_j^c - y_{d\, ij}\, h\, \overline{Q}_i d^c_j - y_{e\,ij} \,h\, \overline{L}_i e^c_j + h.c.\,\!$
Quantum electrodynamics	$\mathcal{L}=\bar\psi(i\gamma^\mu D_\mu-m)\psi -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\;,\,\!$

See also

• Defining equation (physical chemistry)
• Defining equation (physics)
• List of electromagnetism equations
• List of equations in classical mechanics
• List of equations in quantum mechanics
• List of equations in wave theory
• List of photonics equations
• List of relativistic equations
• Relativistic wave equations