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.


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

dimensionless dimensionless
Decay rate, activity of a radioisotope A Bq = Hz = s−1 [T]−1
Decay constant λ 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

s [T]
Number of half-lives n (no standard symbol) dimensionless dimensionless
Radioisotope time constant, mean lifetime of an atom before decay τ (no standard symbol) 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

Q = radiation quality factor (dimensionless)

Sv = J kg−1 (Sievert) [L]2[T]−2
Effective dose E

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

Sv = J kg−1 (Sievert) [L]2[T]−2


Nuclear structure[edit]

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
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
Nuclear radius

r0 ≈ 1.2 fm

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,
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:

Bateman's equations
Radiation flux
  • I0 = Initial intensity/Flux of radiation
  • I = Number of atoms at time t
  • μ = Linear absorption coefficient
  • x = Thickness of substance

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

Spin factor:

Total width:

Resonance lifetime:

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:

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):

Scattering potential energy (α = constant):

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

Fundamental forces[edit]

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
Electroweak interaction :
Quantum electrodynamics

See also[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 978-1-4292-0265-7. 
  • 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.