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	<math> N_0 = N + N_D \,\!</math>	dimensionless	dimensionless
Decay rate, activity of a radioisotope	A	<math> A = \lambda N\,\!</math>	Bq = Hz = s−1	[T]−1
Decay constant	λ	<math> \lambda = A/N \,\!</math>	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 <math> t \rightarrow t + T_{1/2} \,\!</math> <math> N \rightarrow N / 2 \,\!</math>	s	[T]
Number of half-lives	n (no standard symbol)	<math> n = t / T_{1/2} \,\!</math>	dimensionless	dimensionless
Radioisotope time constant, mean lifetime of an atom before decay	τ (no standard symbol)	<math> \tau = 1 / \lambda \,\!</math>	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	<math> H = DQ \,\!</math> Q = radiation quality factor (dimensionless)	Sv = J kg−1 (Sievert)	[L]2[T]−2
Effective dose	E	<math> E = \sum_j H_jW_j \,\!</math> Wj = weighting factors corresponding to radiosensitivities of matter (dimensionless) <math> \sum_j W_j = 1 \,\!</math>	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	<math>A = Z+N\,\!</math>
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	<math>M_\Sigma = Zm_p + Nm_n \,\!</math> <math>M_\Sigma > M_N \,\!</math> <math> \Delta M = M_\Sigma - M_\mathrm{nuc} \,\!</math> <math> \Delta E = \Delta M c^2\,\!</math>
Nuclear radius	r0 ≈ 1.2 fm	<math>r=r_0A^{1/3} \,\!</math> 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,	<math>\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}</math> 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: <math>\frac{\mathrm{d} N}{\mathrm{d} t} = - \lambda N </math> <math>N = N_0e^{-\lambda t}\,\!</math>
Bateman's equations	<math> c_i = \prod_{j=1, i\neq j}^D \frac{\lambda_j}{\lambda_j - \lambda_i} </math>	<math> N_D = \frac{N_1(0)}{\lambda_D} \sum_{i=1}^D \lambda_i c_i e^{-\lambda_i t} </math>
Radiation flux	I0 = Initial intensity/Flux of radiation I = Number of atoms at time t μ = Linear absorption coefficient x = Thickness of substance	<math>I = I_0e^{-\mu x}\,\!</math>

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: <math>\sigma(E) = \frac{\pi g}{k^2}\frac{\Gamma_{ab}\Gamma_c}{(E-E_0)^2+\Gamma^2/4}</math> Spin factor: <math>g = \frac{2J+1}{(2s_a+1)(2s_b+1)}</math> Total width: <math>\Gamma = \Gamma_{ab} + \Gamma_c</math> Resonance lifetime: <math>\tau = \hbar/\Gamma </math>
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: <math>\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 </math>
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): <math>\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} </math> Scattering potential energy (α = constant): <math>V = -\alpha/r</math>
Rutherford scattering		Differential cross-section (non-identical particles in a coulomb potential): <math>\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}</math>

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	<math> \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} \,\!</math>
Electroweak interaction	<math>\mathcal{L}_\mathrm{EW} = \mathcal{L}_g + \mathcal{L}_f + \mathcal{L}_h + \mathcal{L}_y.\,\!</math> <math>\mathcal{L}_g = -\frac{1}{4}W_a^{\mu\nu}W_{\mu\nu}^a - \frac{1}{4}B^{\mu\nu}B_{\mu\nu}\,\!</math> <math>\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 \,\!</math> <math>\mathcal{L}_h = |D_\mu h|^2 - \lambda \left(|h|^2 - \frac{v^2}{2}\right)^2\,\!</math> <math>\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.\,\!</math>
Quantum electrodynamics	<math>\mathcal{L}_\mathrm{QED}=\bar\psi(i\gamma^\mu D_\mu-m)\psi -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\;,\,\!</math>

See also

• Defining equation (physical chemistry)
• 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

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.

Further reading

• L.H. Greenberg (1978). Physics with Modern Applications. Holt-Saunders International W.B. Saunders and Co. ISBN 0-7216-4247-0.
• J.B. Marion, W.F. Hornyak (1984). Principles of Physics. Holt-Saunders International Saunders College. ISBN 4-8337-0195-2.
• A. Beiser (1987). Concepts of Modern Physics (4th ed.). McGraw-Hill (International). ISBN 0-07-100144-1.
• H.D. Young, R.A. Freedman (2008). University Physics – With Modern Physics (12th ed.). Addison-Wesley (Pearson International). ISBN 978-0-321-50130-1.