(unit4:terms)=
## Unit 4 : 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.

**PARITY COMBINATION**

```{math}
:label: paritycombination
\begin{equation}
\pi_{total}=\pi_{1}\pi_{2}\pi_{3}\pi_{4} = \prod_{A}(-1)^{l} = + (\textnormal{even})~~\textnormal{or}~~-(\textnormal{odd})
\end{equation}
```


- if two neutrons (or protons) occupy a level with the same $j$ and $l$, then their total angular momenta couple to give zero contribution.


**Magnetic Moment Free Nucleon**
```{math}
:label: giprojection
\begin{equation}
\mu = g_{I} \frac{e\hbar}{2m_{p}} I = g_{I} I \mu_{N}
\end{equation}
```

**Nuclear Magneton**
```{math}
:label: nuclearmagneton
\begin{equation}
\mu_{N} = \frac{e\hbar}{2m_{p}} = 5.05\times 10^{-27}\textnormal{J/T} = 3.25 \times 10^{-8} \textnormal{eV/T}
\end{equation}
```

**Magnetic Moment and Projection**
```{math}
:label: magneticmoment
\begin{equation}
\mu = g_{j}~j~\mu_{N}
\end{equation}
```

```{math}
:label: magneticmoment_obs
\begin{equation}
\mu_{obs} = g_j j_z \mu_N
\end{equation}
```



**Magneton Estimates**


```{math}
:label: gyromnagneticratio
\begin{equation}
g = g_{l} \frac{j(j+1)+l(l+1)-s(s+1)}{2j(j+1)} + g_s \frac{j(j+1)+s(s+1)-l(l+1)}{2j(j+1)}
\end{equation}
```

We arrive at only two possible scenarios for $g$:


```{math}
:label: magnetonestimate
\begin{equation}
\frac{\mu}{\mu_N} = (j-\frac{1}{2})g_l + \frac{1}{2}g_s \rightarrow (stretched)
\end{equation}
```

```{math}
:label: jackknifeestimate
\begin{equation}
\frac{\mu}{\mu_N} = ((j+\frac{3}{2})g_l - \frac{1}{2}g_s))\frac{j}{j+1} \rightarrow (jackknife)
\end{equation}
```


**G Factors**
```{math}
:label: gyromagneticfactors
\begin{equation}
g_l = 1~~\textnormal{(proton)} ~~~~~~~ g_l = 0 ~~\textnormal{(neutron)}\\
g_{s} = 5.85 ~~\textnormal{(proton)} ~~~~~~ g_{s} = -3.826 ~~\textnormal{(neutron)}
\end{equation}
```


**Quadrupole Operations**

```{math}
:label: quadrupoleoperation
\begin{equation}
eQ_{EQM} = e\int \psi^{*} (3z^{2} - r^{2})\psi dV
\end{equation}
```

Calculations to find the expectation value $< \psi >^{2}$ end up giving as values for $Q$ for single particle missing/addition states of:

```{math}
:label: quadrupolecalculation
\begin{equation}
Q_{EQM} = - < r^{2} > \frac{2j-1}{2(j+1)}
\end{equation}
```


For unfilled shells with more than one particle missing:

```{math}
:label: quadrupoleunfilled
\begin{equation}
< Q_{EQM} > = < Q_{sp} > \left [ 1-2\frac{n-1}{2j-1} \right ]
\end{equation}
```


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## Terms

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

- Nuclear Notation
- Conserved Quantum Number
- Electric Quadrupole Moment
- Nuclear Magnetic Moments
- Unpaired Nucleons, Nucleon Degeneracy
- Spherical Shell Model, Zero Moment
- Nucleus Odd Parity, Nucleus Even Parity
- Pairing Term, Volume Term, Bulk Nucleus, Collective Treatment
- Bulk Nuclear Material, Rotations, Vibrations, Magnetic Moments
- Magnetic Dipole Moment, Torque, Electric Current, Quarks
- Internal Charge Structure, g Factor, Unpaired Nucleon
- Nuclear Magneton, Bohr Magneton, Intrinsic Magnetic Moment
- Observed Magnetic Moment, Expectation Value
- Magnetic Moment Operator, Gyromagnetic Ratio, Stretched, Jack-nife
- Schmidt Lines, Monopole, Dipole, Quadrupole
- Angular Momentum Quantum  Number, Oblate, Prolate, Nuclear Force
- Tide of Nuclear Material, Electron-Barn, Single Particle Value
- Permanent Distortions, Ground State Spins, Bulk Deformations
- Excited States, Central Nuclear Potential, Core Potential
- Collective Mode of Excitation, Shell Theory Potential, Core Potential
- Collective Mode of Excitation, Shell Theory Potential, Saxon-Woods
- Average Potential, Schrodinger Equation, Top-Hat Potential
- Centrifugal Potential, Laplace SPherical Harmonics, Rigid Rotation
- Eigenvalues, Eigen-functions, First Excited State, Incompressible
- Standing Waves, First Harmonic, Phonon Couplings -->