List of equations in nuclear and particle physics

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

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	${\displaystyle A=Z+N}$
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	${\displaystyle M_{\mathrm{\Sigma }}=Z{m}_p+N{m}_n}$ ${\displaystyle M_{\mathrm{\Sigma }}>M_N}$ ${\displaystyle \mathrm{\Delta }M=M_{\mathrm{\Sigma }}-M_{\mathrm{n}\mathrm{u}\mathrm{c}}}$ ${\displaystyle \mathrm{\Delta }E=\mathrm{\Delta }M{c}^2}$
Nuclear radius	r0 ≈ 1.2 fm	${\displaystyle r=r_0{A}^{1/3}}$ 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,	${\displaystyle \begin{array}{rl}{E}_B=& a_{v}A-a_s{A}^{2/3}-a_{c}Z(Z-1)A^{-1/3}\\ & -a_a(N-Z{)}^2{A}^{-1}+12\delta (N,Z)A^{-1/2}\\ \end{array}}$ 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

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: ${\displaystyle \frac{\mathrm{d}N}{\mathrm{d}t}=-\lambda N}$ ${\displaystyle N=N_0{e}^{-\lambda t}}$
Bateman's equations	${\displaystyle c_i={\displaystyle \prod _{j=1,i\ne j}^D}\frac{{\lambda }_j}{{\lambda }_j-{\lambda }_i}}$	${\displaystyle N_D=\frac{N_1(0)}{{\lambda }_D}{\displaystyle \sum _{i=1}^D}{\lambda }_i{c}_i{e}^{-{\lambda }_{i}t}}$
Radiation flux	I0 = Initial intensity/Flux of radiation I = Number of atoms at time t μ = Linear absorption coefficient x = Thickness of substance	${\displaystyle I=I_0{e}^{-\mu x}}$

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: ${\displaystyle \sigma (E)=\frac{\pi g}{k^2}\frac{{\mathrm{\Gamma }}_{ab}{\mathrm{\Gamma }}_c}{(E-E_0{)}^2+{\mathrm{\Gamma }}^2/4}}$ Spin factor: ${\displaystyle g=\frac{2J+1}{(2s_a+1)(2s_b+1)}}$ Total width: ${\displaystyle \mathrm{\Gamma }={\mathrm{\Gamma }}_{ab}+{\mathrm{\Gamma }}_c}$ Resonance lifetime: ${\displaystyle \tau =\hslash /\mathrm{\Gamma }}$
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: ${\displaystyle \frac{d\sigma }{d\mathrm{\Omega }}={|\frac{2\mu }{{\hslash }^2}{\displaystyle \underset{0}{\overset{\mathrm{\infty }}{\int }}}\frac{\sin (\mathrm{\Delta }kr)}{\mathrm{\Delta }kr}V(r)r^{2}dr|}^2}$
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): ${\displaystyle \frac{d\sigma }{d\mathrm{\Omega }}=\left(\frac{\alpha }{4E}\right){[{\csc }^4\frac{\chi }{2}+{\sec }^4\frac{\chi }{2}+\frac{A\cos (\frac{\alpha }{\hslash \nu }\ln {\tan }^2\frac{\chi }{2})}{{\sin }^2\frac{\chi }{2}\cos \frac{\chi }{2}}]}^2}$ Scattering potential energy (α = constant): ${\displaystyle V=-\alpha /r}$
Rutherford scattering		Differential cross-section (non-identical particles in a coulomb potential): ${\displaystyle \frac{d\sigma }{d\mathrm{\Omega }}=\left(\frac{1}{n}\right)\frac{dN}{d\mathrm{\Omega }}={\left(\frac{\alpha }{4E}\right)}^2{\csc }^4\frac{\chi }{2}}$

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	${\displaystyle \begin{array}{rl}{{ℒ}}_{\mathrm{Q}\mathrm{C}\mathrm{D}}& ={\overline{\psi }}_i(i{\gamma }^{\mu }(D_{\mu }{)}_{ij}-m{\delta }_{ij}){\psi }_j-\frac{1}{4}{G}_{\mu \nu }^a{G}_a^{\mu \nu }\\ & ={\overline{\psi }}_i(i{\gamma }^{\mu }{\partial }_{\mu }-m){\psi }_i-g{G}_{\mu }^a{\overline{\psi }}_i{\gamma }^{\mu }{T}_{ij}^a{\psi }_j-\frac{1}{4}{G}_{\mu \nu }^a{G}_a^{\mu \nu },\\ \end{array}}$
Electroweak interaction	:${\displaystyle {{ℒ}}_{EW}={{ℒ}}_g+{{ℒ}}_f+{{ℒ}}_h+{{ℒ}}_y.}$ ${\displaystyle {{ℒ}}_g=-\frac{1}{4}{W}_a^{\mu \nu }{W}_{\mu \nu }^a-\frac{1}{4}{B}^{\mu \nu }{B}_{\mu \nu }}$ ${\displaystyle {{ℒ}}_f={\bar{Q}}_{i}iD/Q_i+{\bar{u}}_i^{c}iD/u_i^c+{\bar{d}}_i^{c}iD/d_i^c+{\bar{L}}_{i}iD/L_i+{\bar{e}}_i^{c}iD/e_i^c}$ ${\displaystyle {{ℒ}}_h=|D_{\mu }h{|}^2-\lambda {(|h{|}^2-\frac{v^2}{2})}^2}$ ${\displaystyle {{ℒ}}_y=-y_{uij}{ϵ}^{ab}{h}_b^{†}{\bar{Q}}_{ia}{u}_j^c-y_{dij}h{\bar{Q}}_i{d}_j^c-y_{eij}h{\bar{L}}_i{e}_j^c+h.c.}$
Quantum electrodynamics	${\displaystyle {ℒ}=\overline{\psi }(i{\gamma }^{\mu }{D}_{\mu }-m)\psi -\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu },}$

• 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

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