Information
Atoms V
Summary of spectroscipy of hydrogen over 80 years from precursors to the Lamb shift, to the proton radius puzzle .
Atoms in external magnetic fields
Fine structure and Landé g-factor
"game of $\vec{L}, \vec{S}, \vec{B}$ "
Add the Zeeman term
$$\begin{align} H_Z = - \vec{\mu} \cdot \vec{B} \end{align}$$to the SOC Hamiltonian
$$\begin{align} H = H_0 + H_{FS} + H_Z = H_0 + A_{FS} \vec{L}\cdot \vec{S} -\frac{\mu_B}{\hbar} (g_s \vec{S} + g_l \vec{L})\cdot \vec{B} \end{align}$$where $|g_s| = 2(1 + \frac{\alpha}{2\pi} + \cdots )$ for electron spin, and $|g_l| = 1$ electon orbital angular momentum. FS term: an internal magnetic field $\sim 1$ Tesla.
direct derivation by using
$$\begin{align} \langle S, L, J, m_J | H_Z | S', L', J', m_J' \rangle \end{align}$$weak field limit $H_{FS} \gg H_{Z}$
vector model: $\vec{L}, \vec{S}$ rapidly process aroud total angular momentum $\vec{J}$ , only projection on $\vec{J}$ .
use $\vec{L}^2 = (\vec{J} - \vec{S})^2 \Rightarrow 2\vec{J}\cdot \vec{S} = \vec{J}^2 + \vec{S }^2 - \vec{L}^2$ and let $g_s \to -2, g_l \to -1$
$$\begin{align} H_Z =& - \left(\vec{\mu}\cdot \frac{ \vec{J}}{|\vec{J}|} \right) \left( \vec{B}\cdot \frac{\vec{J}}{|\vec{J}|} \right) \\ =& \frac{\mu_B}{\hbar} \frac{(\vec{L}+ 2\vec{S})\cdot \vec{J}}{|\vec{J}|^2} B J_Z \\ =& \frac{\mu_B B}{\hbar}J_z \cdot \frac{\vec{J}^2 + \frac{1}{2}(\vec{J}^2 + \vec{S}^{2} - \vec{L}^2)}{\vec{J}^2}\\ =& \mu_B B m_J \left(1 + \frac{J(J + 1) + S(S + 1) - L(L + 1)}{2 J(J + 1)} \right)\\ \equiv & \mu_B B m_J g_J \end{align}$$where $g_J$ is Landé g-factor
limiting case
- $S = 0 \Rightarrow g_J = 1$
- $L = 0 \Rightarrow g_J = 2$
HFS in an Applied Field
Now add nucleus spin $\vec{I}$ :
$$\begin{align} \vec{L} , \vec{S}, \vec{I}, \vec{B} , \quad \vec{I} + \vec{J} = \vec{F} \end{align}$$(No fine structure Hamiltonian?)
$$\begin{align} H = H_0 + a h \vec{I}\cdot \vec{J} - \vec{\mu}_J \cdot B_0 - \vec{\mu}_I \cdot \vec{B}_0 \end{align}$$LOW FIELD $\vec{\mu}_J\cdot \vec{B}_0 \ll a h \vec{I}\cdot \vec{J}$
$$\begin{align} H_Z = -(\vec{\mu}_J + \vec{\mu}_{I})\cdot \vec{B}_0 \end{align}$$treated in a perturbation theory, similar to landé g factor
$$\begin{align} H_Z = -\mu_B ( -|g_J| \vec{J}\cdot \vec{F} + g_I \frac{\mu_N}{\mu_B}\vec{I}\cdot \vec{F}) \frac{\vec{F} \cdot \vec{B}_0}{|\vec{F}|^{2}} \end{align}$$where $\mu_N = \frac{e\hbar}{2m_p}$ (Wikipedia: Nucleon magnetic moment) , and because $m_p \gg m_3$ , so $\mu_N\ll \mu_B$ , we neglect the second term, we define
$$\begin{align} g_F \equiv \frac{\vec{J}\cdot \vec{F}}{|\vec{F}|^2}g_J \end{align}$$thus
$$\begin{align} H_Z = \mu_B g_F B_0 m_F \end{align}$$HIGH FIELD $\vec{\mu}_J\cdot \vec{B} \gg a h \vec{I}\cdot \vec{J}$
Zeeman energy fist comes, $\vec{B}_0$ quantize $\vec{J}$ and $\vec{I}$ along $B_0$
$$\begin{align} H_Z = |g_J| \mu_B m_J B_0 - g_I \mu_N m_I B_0 + a h m_I m_J \end{align}$$GENERAL SOLUTION: EXACT DIAGNOLIZATION
$$\begin{align} H = a h \vec{I}\cdot \vec{J} + (g_J\mu_B m_J - g_I \mu_N m_I) B_0 \end{align}$$ $$\begin{align} m_F = m_I + m_J = m_{\mathrm{TOTAL}} \end{align}$$structure: repulsive and anti-crossings of state with the same $m_F$ , for example: Na $^{87}\mathrm{Rb}$ . $^2S_{1/2}$ ground state $I = \frac{3}{2}$ .