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 N_D = 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	t_1/2, T_1/2	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	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	$E=\sum _{j}H_{j}W_{j}\,\!$ W_j = 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 M_Σ = Sum of masses for isolated nucleons m_p = proton rest mass m_n = neutron rest mass	$M_{\Sigma }=Zm_{p}+Nm_{n}\,\!$ $M_{\Sigma }>M_{N}\,\!$ $\Delta M=M_{\Sigma }-M_{\mathrm {nuc} }\,\!$ $\Delta E=\Delta Mc^{2}\,\!$
Nuclear radius	r₀ ≈ 1.2 fm	$r=r_{0}A^{1/3}\,\!$ hence (approximately) nuclear volume ∝ A nuclear surface ∝ A^2/3
Nuclear binding energy, empirical curve	Dimensionless parameters to fit experiment: E_B = binding energy, a_v = nuclear volume coefficient, a_s = nuclear surface coefficient, a_c = electrostatic interaction coefficient, a_a = symmetry/asymmetry extent coefficient for the numbers of neutrons/protons,	${\begin{aligned}E_{B}=&a_{v}A-a_{s}A^{2/3}-a_{c}Z(Z-1)A^{-1/3}\\&-a_{a}(N-Z)^{2}A^{-1}+12\delta (N,Z)A^{-1/2}\\\end{aligned}}$ 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	$c_{i}=\prod _{j=1,i\neq j}^{D}{\frac {\lambda _{j}}{\lambda _{j}-\lambda _{i}}}$	$N_{D}={\frac {N_{1}(0)}{\lambda _{D}}}\sum _{i=1}^{D}\lambda _{i}c_{i}e^{-\lambda _{i}t}$
Radiation flux	I₀ = Initial intensity/Flux of radiation I = Number of atoms at time t μ = Linear absorption coefficient x = Thickness of substance	$I=I_{0}e^{-\mu x}\,\!$

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, Γ_ab, Γ_c 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: ${\frac {d\sigma }{d\Omega }}=\left\|{\frac {2\mu }{\hbar ^{2}}}\int _{0}^{\infty }{\frac {\sin(\Delta kr)}{\Delta kr}}V(r)r^{2}dr\right\|^{2}$
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

Name	Equations
Strong force	${\begin{aligned}{\mathcal {L}}_{\mathrm {QCD} }&={\bar {\psi }}_{i}\left(i\gamma ^{\mu }(D_{\mu })_{ij}-m\,\delta _{ij}\right)\psi _{j}-{\frac {1}{4}}G_{\mu \nu }^{a}G_{a}^{\mu \nu }\\&={\bar {\psi }}_{i}(i\gamma ^{\mu }\partial _{\mu }-m)\psi _{i}-gG_{\mu }^{a}{\bar {\psi }}_{i}\gamma ^{\mu }T_{ij}^{a}\psi _{j}-{\frac {1}{4}}G_{\mu \nu }^{a}G_{a}^{\mu \nu }\,,\\\end{aligned}}\,\!$
Electroweak interaction	: ${\mathcal {L}}_{EW}={\mathcal {L}}_{g}+{\mathcal {L}}_{f}+{\mathcal {L}}_{h}+{\mathcal {L}}_{y}.\,\!$ ${\mathcal {L}}_{g}=-{\frac {1}{4}}W_{a}^{\mu \nu }W_{\mu \nu }^{a}-{\frac {1}{4}}B^{\mu \nu }B_{\mu \nu }\,\!$ ${\mathcal {L}}_{f}={\overline {Q}}_{i}iD\!\!\!\!/\;Q_{i}+{\overline {u}}_{i}^{c}iD\!\!\!\!/\;u_{i}^{c}+{\overline {d}}_{i}^{c}iD\!\!\!\!/\;d_{i}^{c}+{\overline {L}}_{i}iD\!\!\!\!/\;L_{i}+{\overline {e}}_{i}^{c}iD\!\!\!\!/\;e_{i}^{c}\,\!$ ${\mathcal {L}}_{h}=\|D_{\mu }h\|^{2}-\lambda \left(\|h\|^{2}-{\frac {v^{2}}{2}}\right)^{2}\,\!$ ${\mathcal {L}}_{y}=-y_{u\,ij}\epsilon ^{ab}\,h_{b}^{\dagger }\,{\overline {Q}}_{ia}u_{j}^{c}-y_{d\,ij}\,h\,{\overline {Q}}_{i}d_{j}^{c}-y_{e\,ij}\,h\,{\overline {L}}_{i}e_{j}^{c}+h.c.\,\!$
Quantum electrodynamics	${\mathcal {L}}={\bar {\psi }}(i\gamma ^{\mu }D_{\mu }-m)\psi -{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }\;,\,\!$

Footnotes

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Sources

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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.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
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.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
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.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
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.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
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.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
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.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
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.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
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.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

Farrell–Markushevich theorem

Contents

Definitions