List of equations in nuclear and particle physics

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

Definitions

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]\displaystyle{ 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]\displaystyle{ M_\Sigma = Zm_p + Nm_n \,\! }[/math] [math]\displaystyle{ M_\Sigma \gt M_N \,\! }[/math] [math]\displaystyle{ \Delta M = M_\Sigma - M_\mathrm{nuc} \,\! }[/math] [math]\displaystyle{ \Delta E = \Delta M c^2\,\! }[/math]
Nuclear radius	r0 ≈ 1.2 fm	[math]\displaystyle{ 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]\displaystyle{ \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]\displaystyle{ \frac{\mathrm{d} N}{\mathrm{d} t} = - \lambda N }[/math] [math]\displaystyle{ N = N_0e^{-\lambda t}\,\! }[/math]
Bateman's equations	[math]\displaystyle{ c_i = \prod_{j=1, i\neq j}^D \frac{\lambda_j}{\lambda_j - \lambda_i} }[/math]	[math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ \sigma(E) = \frac{\pi g}{k^2}\frac{\Gamma_{ab}\Gamma_c}{(E-E_0)^2+\Gamma^2/4} }[/math] Spin factor: [math]\displaystyle{ g = \frac{2J+1}{(2s_a+1)(2s_b+1)} }[/math] Total width: [math]\displaystyle{ \Gamma = \Gamma_{ab} + \Gamma_c }[/math] Resonance lifetime: [math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ V = -\alpha/r }[/math]
Rutherford scattering		Differential cross-section (non-identical particles in a coulomb potential): [math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \mathcal{L}_{EW} = \mathcal{L}_g + \mathcal{L}_f + \mathcal{L}_h + \mathcal{L}_y.\,\! }[/math] [math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \mathcal{L}_h = |D_\mu h|^2 - \lambda \left(|h|^2 - \frac{v^2}{2}\right)^2\,\! }[/math] [math]\displaystyle{ \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]\displaystyle{ \mathcal{L}=\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)
• 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

Footnotes

Sources

• B. R. Martin, G.Shaw. 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 0-321-50130-6. 

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