Farrell–Markushevich theorem

Template:Nuclear 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	$A=\mathrm{d}N/\mathrm{d}t$	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 $t\to t+T_{1/2}$ $N\to N/2$	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	Sv = J kg−1 (Sievert)	[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 = 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	$M_{\mathrm{\Sigma }}=Z{m}_p+N{m}_n$ $M_{\mathrm{\Sigma }}>M_N$ $\mathrm{\Delta }M=M_{\mathrm{\Sigma }}-M_{\mathrm{n}\mathrm{u}\mathrm{c}}$ $\mathrm{\Delta }E=\mathrm{\Delta }M{c}^2$
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 coloumb 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

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

See also

• Defining equation (physics)
• Defining equation (physical chemistry)
• List of equations in classical mechanics
• List of equations in wave theory
• List of relativistic equations
• List of electromagnetism equations
• List of photonics equations
• List of equations in quantum mechanics
• Relativistic wave equations

Footnotes

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Further reading

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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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