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	$N_0=N+N_D$	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	t1/2, T1/2	Time taken for half the number of atoms present to decay $t\to t+T_{1/2}$ $N\to N/2$	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	$E=\sum _j{H}_j{W}_j$ Wj = 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 MΣ = Sum of masses for isolated nucleons mp = proton rest mass mn = neutron rest mass	$M_{\mathrm{\Sigma }}=Z{m}_p+N{m}_n$ $M_{\mathrm{\Sigma }}>M_N$ $\mathrm{\Delta }M=M_{\mathrm{\Sigma }}-M_{\mathrm{n}\mathrm{u}\mathrm{c}}$ $\mathrm{\Delta }E=\mathrm{\Delta }M{c}^2$
Nuclear radius	r0 ≈ 1.2 fm	$r=r_0{A}^{1/3}$ 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,	$\begin{array}{rl}{E}_B=& a_{v}A-a_s{A}^{2/3}-a_{c}Z(Z-1)A^{-1/3}\\ & -a_a(N-Z{)}^2{A}^{-1}+12\delta (N,Z)A^{-1/2}\\ \end{array}$ 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

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	$c_i=\prod _{j=1,i\ne j}^D\frac{{\lambda }_j}{{\lambda }_j-{\lambda }_i}$	$N_D=\frac{N_1(0)}{{\lambda }_D}\sum _{i=1}^D{\lambda }_i{c}_i{e}^{-{\lambda }_{i}t}$
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, 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: $\sigma (E)=\frac{\pi g}{k^2}\frac{{\mathrm{\Gamma }}_{ab}{\mathrm{\Gamma }}_c}{(E-E_0{)}^2+{\mathrm{\Gamma }}^2/4}$ Spin factor: $g=\frac{2J+1}{(2s_a+1)(2s_b+1)}$ Total width: $\mathrm{\Gamma }={\mathrm{\Gamma }}_{ab}+{\mathrm{\Gamma }}_c$ Resonance lifetime: $\tau =\hslash /\mathrm{\Gamma }$
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: $\frac{d\sigma }{d\mathrm{\Omega }}={|\frac{2\mu }{{\hslash }^2}\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\sin (\mathrm{\Delta }kr)}{\mathrm{\Delta }kr}V(r)r^{2}dr|}^2$
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): $\frac{d\sigma }{d\mathrm{\Omega }}=\left(\frac{\alpha }{4E}\right){[{\csc }^4\frac{\chi }{2}+{\sec }^4\frac{\chi }{2}+\frac{A\cos (\frac{\alpha }{\hslash \nu }\ln {\tan }^{}\frac{\chi }{2})}{{\sin }^2\frac{\chi }{2}\cos \frac{\chi }{2}}]}^2$ Scattering potential energy (α = constant): $V=-\alpha /r$
Rutherford scattering		Differential cross-section (non-identical particles in a coulomb potential): $\frac{d\sigma }{d\mathrm{\Omega }}=\left(\frac{1}{n}\right)\frac{dN}{d\mathrm{\Omega }}={\left(\frac{\alpha }{4E}\right)}^2{\csc }^4\frac{\chi }{2}$

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

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.