(unit6:terms)=
## Unit 6 : Equations & Terms

### Equations
To aid with exam revision below is a list of all equations that you should learn for this unit in the course.

**The time for half the sample to decay (half-life)**

$$T_{1/2} = \tau \textnormal{ln}(2) = 0.693\tau$$

**Mass Deficit**

```{math}
:label: massdeficit
\begin{equation}
Q = M(A,Z)^{2}c^{2} - [M(A-X,Z-Y)c^{2} + M(X,Y)c^{2}] > 0 
\end{equation}
```

**Beta Decay Example**

$$
\bf
 _{83}^{210}Bi \rightarrow _{84}^{210}Po + e^- + \bar{\nu}_e 
$$


**Q Value Beta- Decay**

```{math}
:label: betamqvalue
\begin{equation}
Q_{\beta^{-}} = [ M(A,Z) - M(A,Z-1) ]c^{2}
\end{equation}
```

**Q Value Beta+ Decay or EC**

```{math}
:label: betapqvalue
\begin{equation}
Q_{\beta^{+}} = [ M(A,Z) - M(A,Z-1) - 2m_{e}]c^{2}
\end{equation}
```

**Lambda and Matrix Element**

```{math}
:label: lambdacalc
\begin{equation}
\lambda = \frac{2\pi}{\hbar} ~| M_{if} |^{2} ~\frac{dn_{f}}{dE_{f}}
\end{equation}
```

```{math}
:label: matrixelement
\begin{equation}
|M_{fi}| = \int \psi^{*} \hat{H} \psi dV.
\end{equation}
```

**Beta Flux**

```{math}
:label: densityfinal
\begin{equation}
N(p) = \frac{C}{c^{2}} p^{2} (Q-T_{e})^{2}
\end{equation}
```


**Beta Superallowed Transition**

$$
 \Delta I = 0, \quad \Delta \pi = \textnormal{none}
 $$

**Beta Allowed Transition**

$$
 \Delta I = 0 \text{ or } \pm 1, \quad \Delta \pi = \textnormal{none}
 $$
 except for $0 \rightarrow 0$ transitions, which are forbidden.

**Beta Forbidden Transitions**

$$
 \Delta I = 0, \pm 1, \pm 2, \quad \Delta \pi = 1 
 $$
 
except for $0 \rightarrow 0$ transitions, which are still completely forbidden.

In general, the "n-th forbidden" transition is characterized by:

$$
 \Delta I = n, n \pm 1, n \pm 2, ..., \quad \Delta \pi = (-1)^{n}
 $$
 


**Alpha Geiger Nutall**

$$
\log_{10} t_{1/2} = b_{1} \frac{Z}{Q^{1/2}} + b_{2}
$$

or

$$
\ln\lambda = -a_{1} \frac{Z}{Q^{1/2}} + a_2
$$


**Alpha Tunnelling Transmission Probability**

```{math}
:label: transmissionprob
\begin{equation}
X = Ae^{-αL}
\end{equation}
```


## Terms

- Nuclear Sseperation, Coulomb's Law, Coulomb Potential
- Beta Decay, Three-body processss, mononergetic, neutrino, anti-neutrino
- Daughter Nucleus, Weak Interaction, Positron Emission
- Electron Capture, K or L Capture, Proton Excesss, Positron Annihilation
- Possitron Emission Tomography, Decay Sequence
- Energy LEvel Diagramme, Emnergetics, Fermi, Fermi's Golden Rule
- Phase SPace Factor, Nuclear Matrix Element, Fermi THeory
- Beta Decay Endpoint, Femri-Kurie Plot