List of equations in nuclear and particle physics

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

## Contents

* 1 Definitions
* 2 Equations
* 2.1 Nuclear structure
* 2.2 Nuclear decay
* 2.3 Nuclear scattering theory
* 2.4 Fundamental forces
* 3 See also
* 4 Footnotes
* 5 Sources
* 6 Further reading

## 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]\displaystyle{ N_0 = N + N_D \,\\! }[/math] | dimensionless  | dimensionless
Decay rate, activity of a radioisotope  | A | [math]\displaystyle{ A = \lambda N\,\\! }[/math] | Bq = Hz = s−1 | [T]−1
Decay constant  | λ | [math]\displaystyle{ \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]\displaystyle{ t \rightarrow t + T_{1/2} \,\\! }[/math]
[math]\displaystyle{ N \rightarrow N / 2 \,\\! }[/math]

| s  | [T]
Number of half-lives  | n (no standard symbol)  | [math]\displaystyle{ n = t / T_{1/2} \,\\! }[/math] | dimensionless  | dimensionless
Radioisotope time constant, mean lifetime of an atom before decay  | τ (no standard symbol)  | [math]\displaystyle{ \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]\displaystyle{ H = DQ \,\\! }[/math]

Q = radiation quality factor (dimensionless)

| Sv = J kg−1 (Sievert)  | [L]2[T]−2
Effective dose  | E | [math]\displaystyle{ E = \sum_j H_jW_j \,\\! }[/math]

Wj = weighting factors corresponding to radiosensitivities of matter (dimensionless)

[math]\displaystyle{ \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]\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.

* v
* t
* e

SI units

Base units|

* ampere
* candela
* kelvin
* kilogram
* {{{sp}}}}}|us|er|re}}
* 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
* {{{sp}}}}}|us|er|re}}
* minute
* minute and second of arc
* neper
* tonne

See also|

* Conversion of units
* Metric prefixes
* 2005–2019 definition
* 2019 redefinition
* Systems of measurement

* Book
* Category

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