## Fermi and Gamow-Teller Transition

The beta decay is caused by the weak interaction. The weak interaction is very short range, because the mediate particles, the $W^{\pm}$ and $Z^0$ bosons are 80 GeV and 91 GeV respectively. The effective range is like $10^{-3}$ fm. So, the interaction can assumed to be a delta function and only the coupling constant matter. The Fermi coupling constant is

$1.17 \times 10^{-11} (\hbar c)^2~ \mathrm{MeV^2}$

The fundamental process of beta decay is the decay of quark.

$\displaystyle u \xrightarrow{W^+} d + e^+ + \nu_e$

Since a pion is made from up and down quark, the decay of pion into position and electron neutrino is also due to weak interaction.

The Hamilton of the beta decay is

$\displaystyle H_w(\beta^{\pm})=G_V \tau_{\mp} + G_A \sigma \tau_{\mp}$

where $G_V$ is the vector coupling constant, the term is called Fermi transition. The $\tau_{\pm}$ is the isospin ladder operator. The beta+ decay changes the isospin from +1/2 (neutron) to -1/2 (proton). The $G_A$ is the axial coupling constant, the term is called Gamow-Teller transition. $\sigma$ is spin operator. Because of this operator, the Gamow-Teller transition did not preserve parity.

The $G_A$ is different from $G_V$, which is caused by the effect of strong interaction. The Goldberger-Trieman relation

$\displaystyle g_A = \frac{G_A}{G_V} = \frac{f_\pi g_{\pi N}}{M_N c^2} = -1.3$

where $f_\pi \sim 93~\textrm{MeV}$ is the pion decay constant. $g_{\pi N} \sim 14 \times 4\pi$ is the coupling constant between pion and nucleon.  This, we can see the effect of the strong interaction, in which pion is the meson for strong nuclear force.

The transition probability can be estimated by Fermi-Golden rule

$\displaystyle W(p_e)=\frac{2\pi}{\hbar}|\left< \psi_f|H|\psi_0\right> |^2 \rho(E_f)$

the final state wavefunction

$\displaystyle \left|\psi_f\right> = \frac{1}{\sqrt{V}} e^{ik_e r} \frac{1}{\sqrt{V}} e^{ik_{\nu}r} \left|j_f m_f\right>$

$\displaystyle e^{ikr} = \sum \limits_{L}\sqrt{4\pi (2L+1)} i^L j_L(kr) Y_{L0}(\theta)$

using long wavelength approximation, the spherical Bessel function can be approximated by the first term.

$\displaystyle j_L(kr) \sim \frac{(kr)^L}{(2L+1)!!}$

$\displaystyle \left| \psi_f\right>=\frac{1}{V}(1 + i \sqrt{\frac{4\pi}{3}} Y_{10} + ...) \left|j_f m_f\right>$

The first term 1, or L=0 is called allowed decay, so that the orbital angular momentum of the decayed nucleus unchanged. The higher order term, in which the weak interaction have longer range has very small probability and called L-th forbidden decay.

The density of state is

$\displaystyle \rho(E_f) = \frac{V}{2\pi^2 \hbar^7 c^3} F(Z,E_e)p_e^2 (E_0-E_e) ( (E_0-E_e)^2-(m_{\nu} c^2)^2)^2$

where the $F(Z, E_e)$ is the Fermi function.

The total transition probability is the integration with respect to the electron momentum.

$\displaystyle W = \int W(p_e) dp_e = \frac{m_e^5 c^4}{2 \pi^3 \hbar^7} f(Z,E_0) |M|^2$

where $f(Z,E_0)$ is the Fermi integral. The half-life

$\displaystyle T_{1/2} = \frac{\ln{2}}{W}$

To focus on the beta decay from the interference of the density of state, the ft-value is

$\displaystyle ft = f(Z,E_0) T_{1/2} =\frac{2\pi^3\hbar^7}{m_e^5 c^4} \frac{\ln{2}}{|M|^2}$

The ft-value could be difference by several order.

There is a super-allowed decay from $0^{+} \rightarrow 0^{0}$ with same isospin, which the GT does not involve. an example is

$\displaystyle ^{14}\mathrm{O} \rightarrow ^{14}\mathrm{N} + e^+ + \nu_e$

The ft-value is 3037.7s, the smallest of known.

Fermi Gamow-Teller
$\Delta S=0$ $\Delta S=1$
$J_f=J_i + L$ $J_f=J_i + L+1$
$T_f=T_i + 1$

transition L $\log_{10} ft_{1/2}$ $\Delta J$ $\Delta T$ $\Delta \pi$
Fermi GT
Super allowed 3.1 ~ 3.6 $0^+ \rightarrow 0^+$ not exist 0 no
allowed 0 2.9 ~ 10 0 (0), 1 0, 1 ; $T_i=0\rightarrow T_f=0$ forbidden no
1st forbidden 1 5 ~ 19 (0),1 0, 1, 2 0,1 yes
2nd forbidden 2 10 ~18 (1), 2 2, 3 no
3rd forbidden 3 17 ~ 22 (2), 3 3, 4 yes
4th forbidden 4 22 ~ 24 (3), 4 4, 5 no

The () means not possible if either initial or final state is zero. i.e $1^{-} \rightarrow 0^+$ is not possible for 1st forbidden.

## Method II (decay)

there is another way to study nuclear physics, which is by observe the decay process.

there are 3 major decays, the alpha, beta and gamma. the alpha decay is an excited nucleus go to a lower energy state by emitting an Helium ( 2 protons and 2 neutrons) nucleus. This process change the nucleus constitution and make it lighter. the beta decay is an excited nucleus go to a lower energy state by emitting an electron ( or positron, or capture an electron). This process also change the nucleus constitution but the change in mass is very small. the gamma decay is an excited nucleus go to a lower energy state by emitting a photon ( or the light particle). this process does not change the nucleus constitution.

there are many other decays, like neutron, proton, and even fission can regard as decay. in general, decay is a process that a nucleus go to a lower energy state and become more stable. But other decays has too short for lifetime, so that after the earth was formed, they are almost gone. and only alpha, beta and gamma has long left time to survive.

Via a decay process, we can know the energy level, lifetime, parity of each level of a nucleus. These informations can help us to build nuclear models and theories to understand and predict nuclear properties, like the strong and weak force.

i will present a series about decay in future posts. watch out!

## symmetry breaking, mass and higg field.

Above curie temperature, the spin of iron is isotropic. The spin can be rotated to any direction without any resistance, they like massless. Below curie temp, the iron has phase transition and all spin now point to a particular direction. And we need some force to rotate the spin direction. The spin has mass now. This is what symmetry breaking in simple manner.

When matter becomes superconduct, the magnetic field inside is decay exponentially,which is similar as Yukawa force. And we said the force carrier particle is massive. The magnetic field decay is due to the copper pair, Which respond to the magnetic field and tend to cancel it. Thus, the direction of copper pair is not isotropic and this is another symmetry breaking due to the external field and low temperature.

At very high temperature, the weak force carriers are massless. And we assign an isotropic field (scaler field) for the force carrier and call it Higg field. The Higg field quata is called Higg boson. It act like the copper pair, which respond with the force carrier. When there is a force carrier, a Higg boson will be induced. And the symmetric breaking in Higg field, the symmetry breaking makes the force carrier has mass. That we need to apply a force to change the motion.

## Objects of Interest

Nuclear Physics is a study on nuclear matter which is fundamental building block of the world.

electron, proton , neutron, deuteron, tritium, etc… those are objects in nuclear, we call them “particle”. the most simple particle in here is electron, proton and neutron.

The different between nuclear and atom is:

Nuclear core (sit in the center) + Electrons (moving around) = Atom

the mass of atom is almost contributed by nuclear. This is because the mass of proton is about 1830 times bigger than electron, and neutron’s mass is only heavier a bit then proton.

There are many properties contained in each particle. there are electric charge, mass, spin, kinetic energy, etc… and the objective of nuclear physics is understand all these properties and how these properties affect the inter-reaction among them. for example, how a proton and neutron form a nuclear core in deuteron? how they attract each other?

these properties, some may say, are ASSIGNED to the particles. Basically, we can only measure the effect or the result from each interaction. we think, there is a FORCE to make particles able to INTER-ACT with each others. simple to say, when an electron meets another electron, they affect each other by ELECTROMAGNETIC force. but when consider an electron meet a neutron, they don’t interact by electromagnetic(EM) force. in order to distinguish these. we assign an electric CHARGE to electron, and no charge for neutron.

so, basically, Nuclear Physics is study the PROPERTIES of particles and the INTERACTION among them.

There are 3 major forces/interactions, Weak force, EM force and Strong force. Until this moment, we only know the weak and EM force and not fully understand the strong. We neglect the gravity in here, because it is very weak and do no observable effect.

## Range

Strong 10,000 10-15m
EM 1000 long
Weak 1 10-18m