List of equations in nuclear and particle physics

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

Definitions[edit]

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 = \mathrm{d} N /\mathrm{d} t \,\! 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 \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_jW_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[edit]

Nuclear structure[edit]

Physical situation Nomenclature Equations
Mass number
  • A = (Relative) atomic mass = Mass number = Number 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_\Sigma = Zm_p + Nm_n \,\!
  • M_\Sigma > M_N \,\!
  •  \Delta M = M_\Sigma - M_\mathrm{nuc} \,\!
  •  \Delta E = \Delta M c^2\,\!
Nuclear radius

r0 ≈ 1.2 fm

r=r_0A^{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{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} 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[edit]

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_0e^{-\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_0e^{-\mu x}\,\!

Nuclear scattering theory[edit]

The following apply for the nuclear reaction:

a + bRc

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{\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^2dr\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 coloumb 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[edit]

Name Equations
Strong force 
\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}
\,\!
Electroweak interaction :\mathcal{L}_{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^c_i+ \overline{d}_i^c iD\!\!\!\!/\; d^c_i+ \overline{L}_i iD\!\!\!\!/\; L_i+ \overline{e}^c_i iD\!\!\!\!/\; e^c_i \,\!
\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^c_j - y_{e\,ij} \,h\, \overline{L}_i e^c_j + h.c.\,\!
Quantum electrodynamics \mathcal{L}=\bar\psi(i\gamma^\mu D_\mu-m)\psi -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\;,\,\!

See also[edit]

Footnotes[edit]

Sources[edit]

  • 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 9-781429-202657. 
  • J.R. Forshaw, A.G. Smith (2009). Dynamics and Relativity. Wiley,. ISBN 978-0-470-01460-8. 

Further reading[edit]

  • 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.