(unit3:terms)=
## Unit 3 : 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.

**MAGIC NUMBERS**
```{math}
:label: shellmodelnuclei
\begin{equation}
\bf \textnormal{Magic Nuclei -}~ Z~\textnormal{or}~N=2, 8, 20, 28, 50, 82, 126...
\end{equation}
```

**BINDING ENERGY PARABOLA ESTIMATE**
```{math}
:label: BindingEnergyParabola
\begin{equation}
M(A,Z) = aA + bZ + cZ^{2}
\end{equation}
```

**BINDING ENERGY SOLVING**

```{math}
:label: BindingEnergyParabolaDif
\begin{equation}
\frac{dM(A,Z)}{dZ} \rightarrow 0\\
\end{equation}
```

**PARABOLA DIFFERENCE**
```{math}
:label: BindingEnergyParabolaDif
\begin{equation}
\frac{-\hbar^{2}}{2m} \nabla^{2} \psi(r,\theta,\phi) + V(r,\theta,\phi) \psi(r,\theta,\phi) = E \psi(r,\theta,\phi)
\end{equation}
```

**SHELL MODEL POTENTIAL**
```{math}
:label: simpledropcombined_extraterms2
\begin{equation}
V(r) = \frac{-V_{0}}{1+\textnormal{exp}(\frac{r-b}{a})}
\end{equation}
```


**SHELL QUANTUM NUMBERS AND PROJECTIONS**
```{math}
:label: shellmdoelam
\begin{equation}
\langle l^{2} \rangle  = \hbar^{2}l(l+1)
\end{equation}
```

```{math}
:label: shellmodelamproj
\begin{equation}
\langle l_{z} \rangle = \hbar m_{z} ~~~~\textnormal{where}~~~~ m_z=0,\pm1,\pm2,...,\pm l
\end{equation}
```


```{math}
:label: shellmodelspin
\begin{equation}
\langle s^{2} \rangle = \hbar^{2} s(s+1)
\end{equation}
```

```{math}
:label: shellmodelproj
\begin{equation}
\langle s_z\rangle  = \hbar m_{s} ~~~~\textnormal{where}~~~~ m_{s} = \pm \frac{1}{2}.
\end{equation}
```

**SHELL MODEL SIMPLE STATES**

$$
n_{\textnormal{states}} = 2(2l+1)
$$


**SPIN ORBIT COUPLING POTENTIAL OPERATOR**

$$
V=V(r) \hat{L} \cdot \hat{S}.
$$


## Terms

The following represent the key terms you should be able to explain after reviewing this module.

- SEMF, Binding Energy Equation, Binding Energy Curve
- Nuclear Spin, Parity, Magnetic Moment, Nuclear Material
- Collective Body, Energy Levels, Collective Body Model
- Individual Particle Model, Symmetry Term in SEMF, Pairing Term in SEMF
- Magic Number Nuclei, Concentric Shells in Shell Model
- Nuclear Potential Well, Harmonic Oscillator
- Nuclear Decay, Mirror Nuclei, Relative Abundance
- Neutron Binding Energy, Neutron Capture Energy, Neutron Seperation Energy
- Neutron Shielding, Pauli Blocking, Reduced Quadrupole Moment
- Nuclear Excitation Energy
- Ground State, First Excited State, Doubly Magic Nuclei
- Double Closed Shell Nucleus, Decay Chain Sequences
- Principal Radioactive Series, Decay Chain
- Total Angular MOmentum, Nuclear Spin, Angular Momentum Substates
- Spin-Orbit Correction, Spin Up, Spin Down, Degeneracy
- Unpaired Nucleon, Parity, Parity Operator, Collective Models