# Zeeman Effect Preview Report

## Outline

- Purpose
- Theory
- Zeeman Effect
- Fabry-Perot Interferometer
- Equipment
- Method
- Points for attention

## Purpose

Observe both normal and anomalous Zeeman effect with Fabry-Perot interferometer. Calculate Bohr magneton by the effect of different magnetic field to Zeeman effect.

## Theory

### Zeeman Effect

Hamiltonian of an Hydrogen atom under magnetic field: $H = H_0 + H_z$
$$
H_z = - \left( \vec{\mu}_\ell + \vec{\mu}_s\right) \cdot \vec{B}_{ext}
\\
\vec{\mu}_\ell = -\frac{eg_\ell}{2m}\vec{L}, \quad \vec{\mu}_s = - \frac{eg_s}{2m}\vec{S}
$$
Take $g_\ell = 1$, $g_s \approx 2$, after simplifying we get:
$$
H_z = \frac{\mu_B}{\hbar}\left(\vec{L}+2\vec{S}\right)\cdot \vec{B}_{ext}
$$
where $\mu_B = \frac{e\hbar}{2m}$. The state that diagonize the hamiltonians is $\left| n\:l\:j\:m_j \right>$. The energy corrections are:
$$
E_z = \frac{\mu_B}{\hbar} \left< \vec{L}+ 2\vec{S}\right> = \mu_B g m_j B_{ext}
$$
$$
g = 1 + \frac{j(j+1)+s(s+1)-\ell(\ell+1)}{j(j+1)}
$$

#### Normal Zeeman Effect

The states of Cadmium atoms we use are $(L, S, J) = (2, 0, 2) \rightarrow (1, 0, 1)$:


$s=0,\: g=1$, energy splittings are all $\mu_B B_{ext}\rightarrow$ 3 lines!

#### Anomalous Zeeman Effect

$(L, S, J) = (0, 1, 1) \rightarrow (1, 1, 2)$:

Possible values of $\Delta E = \mu_B B_{ext} \Delta (g m_j)$:

### Fabry-Perot Interferometer

Interference condition: $n = \frac{2\mu d \cos \theta_n}{\lambda} = n_0 \cos \theta_n$
Define: $n_1 = n_0 - \epsilon \quad (0 < \epsilon < 1), \quad n_p = n_0 - (p-1)$
The relation between the radius of circles and order of maximum:
$$
\frac{r^2_{p+1}}{r^2_{p+1}-r^2_p} = p + \epsilon
$$
If two beam of light of slightly different wavelength is used, then:
(define $k=\frac{1}{\lambda}$)

$$
\Delta k = \frac{1}{2 \mu d} \frac{\delta}{\Delta}
$$
where
$$
\delta^p_{a,\: b} = r^2_{p+1, \: a} - r^2_{p+1, \: b}
\\
\Delta^{p+1,\: p}_{a} \approx \Delta^{p+1,\: p}_{b} \approx r^2_{p+1, \: a} - r^2_{p, \: a}
$$
Using
$$
E_\nu = hc\Delta k = \frac{hc}{2\mu d} \frac{\delta}{\Delta}
$$

For normal zeeman effect: $E_\nu = \mu_B B_{ext}$
For anomalous Zeeman effect with vertical polarization: $E_\nu = 2\mu_B B_{ext}$
For anomalous Zeeman effect with horizontal polarization: $E_\nu = \frac{1}{2}\mu_B B_{ext}$

## Equipment


## Method

**Exp 1.**
Observe both transversal and longitudinal Zeeman effect, with different polarization

**Exp 2.**
1. Change the distance between the magnets, measure magnetic field with Gauss meter.
2. Calculate the area of the circles under different magnetic field and calculate Bohr magneton

## Points for Attention

1. Make sure the Cd lamp is at the middle of the magnets. Don't break it.
2. Do not touch optical equipments with hands.
3. Do not clean the camera with alcohol.
4. Make sure the Gauss meter doens't touch the Cd lamp.