# 物理代写|核物理代写nuclear physics代考|PHYS161

## 物理代写|核物理代写nuclear physics代考|Decay Rate

Finally, we must introduce a factor, called the “Fermi factor”, $F\left(Z_D, T_e\right)$, which accounts for the Coulomb attraction between the daughter nucleus and the emitted electron (or the Coulomb repulsion between the daughter nucleus and the emitted positron). For electrons this function is given by
$$F\left(Z_D, T_e\right)=\frac{2 \pi \eta}{1-\exp (-2 \pi \eta)},$$
where
$$\eta=Z_D \alpha \sqrt{\frac{m_e c^2}{T_e}}$$
We note that for small electron energies this is a large factor, indicating that the electron intensity is larger for small kinetic energies, because the electrons are slowed down by the Coulomb attraction.

For the case of positron and neutrino emission, the Fermi function, which now accounts for the Coulomb repulsion between the daughter nucleus and the positron, is given by
$$F\left(Z_D, T_e\right)=\frac{2 \pi \eta}{\exp (2 \pi \eta)-1} .$$
It is small for small positron energy as the Coulomb repulsion tends to speed up the positrons.

Piecing everything together by inserting (7.26), (7.25), and (7.21) into (7.14) and including the Fermi factor we end up with the final expression for $\lambda\left(T_e\right)$ :
$$\lambda\left(T_e\right)=\frac{G_F^2 c}{2 \pi^3 \hbar}\left|\mathcal{M}_{D P}\right|^2\left(Q-T_e\right)^2\left(T_e+m_e c^2\right) p_e F\left(Z_D, T_e\right)$$

## 物理代写|核物理代写nuclear physics代考|Selection Rules

Since the electron and antineutrino emitted in $\beta$-decay are both spin- $\frac{1}{2}$ particles. their total spin is either zero or one. Decays in which the total lepton spin is zero are called “Fermi transitions”, whereas decays in which the total lepton spin is one are called “Gamow-Teller transitions”.

The spins of the parent nuclide, $I_P$, and daughter nuclide, $I_D$, differ by $S$ where
$$\left|I_P-I_D\right| \leq S \leq I_P+I_D .$$
If both $I_P$ and $I_D$ are zero, then only Fermi transitions can occur, since $S$ then also has to be zero. If $I_P=1$ and $I_D=0$ or vice versa, then only Gamow-Teller transitions can occur since $S$ then has to be one. If $I_D$ and $I_P$ are both non-zero and differ by one, then $S$ can be either zero or one and both Fermi transitions and Gamow-Teller transitions are permitted. Such transitions are known as “mixed transitions”.

If $I_P$ and $I_D$ differ by more than one, then the decay is called a “forbidden transition”. Such decays are not really forbidden, but only suppressed. Angular momentum can still be conserved provided that the leptons have one or more units of orbital angular momentum. A $\beta$-transition that requires orbital angular momentum $\ell$ is called an ” $\ell$ th forbidden transition”.

Transitions with non-zero orbital angular momentum are only possible if we go beyond the approximation (7.24) for the wavefunctions of the leptons and expand them as
$$\Psi_e(\boldsymbol{r})=\frac{1}{\sqrt{V}}\left[1+i \frac{\boldsymbol{p}_e \cdot \boldsymbol{r}}{\hbar}+\cdots\right]$$
and similarly for the wavefunction of the (anti)neutrino.

## 物理代写|核物理代写nuclear physics代考|Decay Rate

$$F\left(Z_D, T_e\right)=\frac{2 \pi \eta}{1-\exp (-2 \pi \eta)},$$

$$\eta=Z_D \alpha \sqrt{\frac{m_e c^2}{T_e}}$$

$$F\left(Z_D, T_e\right)=\frac{2 \pi \eta}{\exp (2 \pi \eta)-1} .$$

$$\lambda\left(T_e\right)=\frac{G_F^2 c}{2 \pi^3 \hbar}\left|\mathcal{M}_{D P}\right|^2\left(Q-T_e\right)^2\left(T_e+m_e c^2\right) p_e F\left(Z_D, T_e\right)$$

## 物理代写|核物理代写nuclear physics代考|Selection Rules

$$\left|I_P-I_D\right| \leq S \leq I_P+I_D$$

$$\Psi_e(\boldsymbol{r})=\frac{1}{\sqrt{V}}\left[1+i \frac{\boldsymbol{p}_e \cdot \boldsymbol{r}}{\hbar}+\cdots\right]$$

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