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 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	r₀ ≈ 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	N₀ = Initial number of atoms N = Number of atoms at time t λ = Decay constant t = Time	Statistical decay of a radionuclide: ${\frac {\mathrm {d} N}{\mathrm {d} t}}=-\lambda N$ $N=N_{0}e^{-\lambda t}\,\!$
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	I₀ = 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	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	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	${\mathcal {L}}_{\mathrm {QED} }={\bar {\psi }}(i\gamma ^{\mu }D_{\mu }-m)\psi -{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }\;,\,\!$

Footnotes

Sources

B. R. Martin, G.Shaw (3 December 2008). Particle Physics (3rd ed.). Manchester Physics Series, John Wiley & Sons. ISBN 978-0-470-03294-7.
D. McMahon (2008). Quantum Field Theory. Mc Graw Hill (USA). ISBN 978-0-07-154382-8.
P.M. Whelan, M.J. Hodgeson (1978). Essential Principles of Physics (2nd ed.). John Murray. ISBN 0-7195-3382-1.
G. Woan (2010). The Cambridge Handbook of Physics Formulas. Cambridge University Press. ISBN 978-0-521-57507-2.
A. Halpern (1988). 3000 Solved Problems in Physics, Schaum Series. Mc Graw Hill. ISBN 978-0-07-025734-4.
R.G. Lerner, G.L. Trigg (2005). Encyclopaedia of Physics (2nd ed.). VHC Publishers, Hans Warlimont, Springer. pp. 12–13. ISBN 978-0-07-025734-4.
C.B. Parker (1994). McGraw Hill Encyclopaedia of Physics (2nd ed.). McGraw Hill. ISBN 0-07-051400-3.
P.A. Tipler, G. Mosca (2008). Physics for Scientists and Engineers: With Modern Physics (6th ed.). W.H. Freeman and Co. ISBN 978-1-4292-0265-7.
J.R. Forshaw, A.G. Smith (2009). Dynamics and Relativity. Wiley. ISBN 978-0-470-01460-8.

SI units
Base units	ampere candela kelvin kilogram metre mole second
Derived units with special names	becquerel coulomb degree Celsius farad gray henry hertz joule katal lumen lux newton ohm pascal radian siemens sievert steradian tesla volt watt weber
Other accepted units	astronomical unit dalton day decibel degree of arc electronvolt hectare hour litre minute minute and second of arc neper tonne
See also	Conversion of units Metric prefixes Historical definitions of the SI base units 2019 revision of the SI System of units of measurement
Category