List of equations in nuclear and particle physics

(common name/s)

! scope="col" width="100" | (Common) symbol/s

! scope="col" width="300" | Defining equation

! scope="col" width="125" | SI units

! scope="col" width="100" | Dimension

! scope="col" width="250" | Nomenclature

! scope="col" width="10" | Equations

|{{plainlist}}

• A = (Relative) atomic mass = Mass number = Sum of protons and neutrons
• N = Number of neutrons
• Z = Atomic number = Number of protons = Number of electrons

{{endplainlist}}

|$A\; =\; Z+N\backslash ,\backslash !$

|{{plainlist}}

• 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

{{endplainlist}}

|{{plainlist}}

• $M\_\backslash Sigma\; =\; Zm\_p\; +\; Nm\_n\; \backslash ,\backslash !$
• $M\_\backslash Sigma\; >\; M\_N\; \backslash ,\backslash !$
• $\backslash Delta\; M\; =\; M\_\backslash Sigma\; -\; M\_\backslash mathrm\{nuc\}\; \backslash ,\backslash !$
• $\backslash Delta\; E\; =\; \backslash Delta\; M\; c^2\backslash ,\backslash !$

{{endplainlist}}

|r₀ ≈ 1.2 fm

|$r=r\_0A^\{1/3\}\; \backslash ,\backslash !$

hence (approximately)

{{plainlist}}

• nuclear volume ∝ A
• nuclear surface ∝ A^2/3

{{endplainlist}}

|Dimensionless parameters to fit experiment:

{{plainlist}}

• 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,{{endplainlist}}

|

& -a_a (N-Z)^2 A^{-1} + 12\delta(N,Z)A^{-1/2} \\

\end{align}

where (due to pairing of nuclei) {{plainlist}}

• δ(N, Z) = +1 even N, even Z,
• δ(N, Z) = −1 odd N, odd Z,
• δ(N, Z) = 0 odd A

{{endplainlist}}

! scope="col" width="250" | Nomenclature

! scope="col" width="10" | Equations

|{{plainlist}}

• N₀ = Initial number of atoms
• N = Number of atoms at time t
• λ = Decay constant
• t = Time

{{endplainlist}}

|Statistical decay of a radionuclide:

$\backslash frac\{\backslash mathrm\{d\}\; N\}\{\backslash mathrm\{d\}\; t\}\; =\; -\; \backslash lambda\; N$

$N\; =\; N\_0e^\{-\backslash lambda\; t\}\backslash ,\backslash !$

|$c\_i\; =\; \backslash prod\_\{j=1,\; i\backslash neq\; j\}^D\; \backslash frac\{\backslash lambda\_j\}\{\backslash lambda\_j\; -\; \backslash lambda\_i\}$

|$N\_D\; =\; \backslash frac\{N\_1(0)\}\{\backslash lambda\_D\}\; \backslash sum\_\{i=1\}^D\; \backslash lambda\_i\; c\_i\; e^\{-\backslash lambda\_i\; t\}$

|{{plainlist}}

• I₀ = Initial intensity/Flux of radiation
• I = Number of atoms at time t
• μ = Linear absorption coefficient
• x = Thickness of substance

{{endplainlist}}

|$I\; =\; I\_0e^\{-\backslash mu\; x\}\backslash ,\backslash !$

! scope="col" width="250" | Nomenclature

! scope="col" width="10" | Equations

|{{plainlist}}

• E₀ = 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

{{endplainlist}}

|Cross-section:

$\backslash sigma(E)\; =\; \backslash frac\{\backslash pi\; g\}\{k^2\}\backslash frac\{\backslash Gamma\_\{ab\}\backslash Gamma\_c\}\{(E-E\_0)^2+\backslash Gamma^2/4\}$

Spin factor:

$g\; =\; \backslash frac\{2J+1\}\{(2s\_a+1)(2s\_b+1)\}$

Total width:

$\backslash Gamma\; =\; \backslash Gamma\_\{ab\}\; +\; \backslash Gamma\_c$

Resonance lifetime:

$\backslash tau\; =\; \backslash hbar/\backslash Gamma$

|{{plainlist}}

• 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

{{endplainlist}}

|Differential cross-section:

$\backslash frac\{d\backslash sigma\}\{d\backslash Omega\}\; =\; \backslash left|\backslash frac\{2\backslash mu\}\{\backslash hbar^2\}\backslash int\_0^\backslash infty\backslash frac\{\backslash sin(\backslash Delta\; kr)\}\{\backslash Delta\; kr\}V(r)r^2dr\backslash right|^2$

|{{plainlist}}

• χ = reduced mass of a and b
• v = incoming velocity

{{endplainlist}}

|Differential cross-section (for identical particles in a coulomb potential, in centre of mass frame):

$\backslash frac\{d\backslash sigma\}\{d\backslash Omega\}=\backslash left(\backslash frac\{\backslash alpha\}\{4E\}\backslash right)\backslash left[\backslash csc^\{4\}\backslash frac\{\backslash chi\}\{2\}+\backslash sec^\{4\}\backslash frac\{\backslash chi\}\{2\}+\backslash frac\{A\backslash cos\backslash left(\backslash frac\{\backslash alpha\}\{\backslash hbar\backslash nu\}\backslash ln\backslash tan^\{2\}\backslash frac\{\backslash chi\}\{2\}\backslash right)\}\{\backslash sin^\{2\}\backslash frac\{\backslash chi\}\{2\}\backslash cos\backslash frac\{\backslash chi\}\{2\}\}\backslash right]^\{2\}$

Scattering potential energy (α = constant):

$V\; =\; -\backslash alpha/r$

|

|Differential cross-section (non-identical particles in a coulomb potential):

$\backslash frac\{d\backslash sigma\}\{d\backslash Omega\}=\backslash left(\backslash frac\{1\}\{n\}\backslash right)\backslash frac\{dN\}\{d\backslash Omega\}\; =\; \backslash left(\backslash frac\{\backslash alpha\}\{4E\}\backslash right)^2\; \backslash csc^4\backslash frac\{\backslash chi\}\{2\}$

! Equations

||$$

\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}

\,\!

||$\backslash mathcal\{L\}\_\backslash mathrm\{EW\}\; =\; \backslash mathcal\{L\}\_g\; +\; \backslash mathcal\{L\}\_f\; +\; \backslash mathcal\{L\}\_h\; +\; \backslash mathcal\{L\}\_y.\backslash ,\backslash !$

:$\backslash mathcal\{L\}\_g\; =\; -\backslash frac\{1\}\{4\}W\_a^\{\backslash mu\backslash nu\}W\_\{\backslash mu\backslash nu\}^a\; -\; \backslash frac\{1\}\{4\}B^\{\backslash mu\backslash nu\}B\_\{\backslash mu\backslash nu\}\backslash ,\backslash !$

:$\backslash mathcal\{L\}\_f\; =\; \backslash overline\{Q\}\_i\; iD\backslash !\backslash !\backslash !\backslash !/\backslash ;\; Q\_i+\; \backslash overline\{u\}\_i^c\; iD\backslash !\backslash !\backslash !\backslash !/\backslash ;\; u^c\_i+\; \backslash overline\{d\}\_i^c\; iD\backslash !\backslash !\backslash !\backslash !/\backslash ;\; d^c\_i+\; \backslash overline\{L\}\_i\; iD\backslash !\backslash !\backslash !\backslash !/\backslash ;\; L\_i+\; \backslash overline\{e\}^c\_i\; iD\backslash !\backslash !\backslash !\backslash !/\backslash ;\; e^c\_i\; \backslash ,\backslash !$

:$\backslash mathcal\{L\}\_h\; =\; |D\_\backslash mu\; h|^2\; -\; \backslash lambda\; \backslash left(|h|^2\; -\; \backslash frac\{v^2\}\{2\}\backslash right)^2\backslash ,\backslash !$

:$\backslash mathcal\{L\}\_y\; =\; -\; y\_\{u\backslash ,\; ij\}\; \backslash epsilon^\{ab\}\; \backslash ,h\_b^\backslash dagger\backslash ,\; \backslash overline\{Q\}\_\{ia\}\; u\_j^c\; -\; y\_\{d\backslash ,\; ij\}\backslash ,\; h\backslash ,\; \backslash overline\{Q\}\_i\; d^c\_j\; -\; y\_\{e\backslash ,ij\}\; \backslash ,h\backslash ,\; \backslash overline\{L\}\_i\; e^c\_j\; +\; h.c.\backslash ,\backslash !$

||$\backslash mathcal\{L\}\_\backslash mathrm\{QED\}=\backslash bar\backslash psi(i\backslash gamma^\backslash mu\; D\_\backslash mu-m)\backslash psi\; -\backslash frac\{1\}\{4\}F\_\{\backslash mu\backslash nu\}F^\{\backslash mu\backslash nu\}\backslash ;,\backslash ,\backslash !$