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

| N0=N+ND | dimensionless  | dimensionless
Decay rate, activity of a radioisotope  | A | A=λN | Bq = Hz = s−1 | [T]−1
Decay constant  | λ | λ=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->t+T1/2
N->N/2

| s  | [T]
Number of half-lives  | n (no standard symbol)  | n=t/T1/2 | dimensionless  | dimensionless
Radioisotope time constant, mean lifetime of an atom before decay  | τ (no standard symbol)  | τ=1/λ | 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=∑jHjWj

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

∑jWj=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

* MΣ=Zmp+Nmn
* MΣ>MN
* ΔM=MΣ−Mnuc
* ΔE=ΔMc2

Nuclear radius  |

r0 ≈ 1.2 fm

| r=r0A1/3 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,

| EB=avA−asA2/3−acZ(Z−1)A−1/3−aa(N−Z)2A−1+12δ(N,Z)A−1/2 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:

dNdt=−λN

N=N0e−λt

Bateman's equations  | ci=∏j=1,i≠jDλjλj−λi | ND=N1(0)λD∑i=1Dλicie−λit
Radiation flux  |

* I0 = Initial intensity/Flux of radiation
* I = Number of atoms at time t
* μ = Linear absorption coefficient
* x = Thickness of substance

| I=I0e−μx

### 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:

σ(E)=πgk2ΓabΓc(E−E0)2+Γ2/4

Spin factor:

g=2J+1(2sa+1)(2sb+1)

Total width:

Γ=Γab+Γc

Resonance lifetime:

τ=ℏ/Γ

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:

dσdΩ=|2μℏ2∫0∞sin⁡(Δkr)ΔkrV(r)r2dr|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):

dσdΩ=(α4E)[csc4χ2+sec4χ2+Acos⁡(αℏνln⁡tan2χ2)sin2χ2cos⁡χ2]2

Scattering potential energy (α = constant):

V=−α/r

Rutherford scattering |  | Differential cross-section (non-identical particles in a coulomb potential):

dσdΩ=(1n)dNdΩ=(α4E)2csc4χ2

### 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  | ℒQCD=ψ¯i(iγμ(Dμ)ij−mδij)ψj−14GμνaGaμν=ψ¯i(iγμ∂μ−m)ψi−gGμaψ¯iγμTijaψj−14GμνaGaμν,
Electroweak interaction | :ℒEW=ℒg+ℒf+ℒh+ℒy.

ℒg=−14WaμνWμνa−14BμνBμν
ℒf=Q‾iiD/Qi+u‾iciD/uic+d‾iciD/dic+L‾iiD/Li+e‾iciD/eic
ℒh=|Dμh|2−λ(|h|2−v22)2
ℒy=−yuijϵabhb†Q‾iaujc−ydijhQ‾idjc−yeijhL‾iejc+h.c.
Quantum electrodynamics | ℒ=ψ¯(iγμDμ−m)ψ−14FμνFμν,

## 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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