Spin-orbit coupling

Spin-orbit coupling is the interaction between a particle’s spin angular momentum and orbital angular momentum. An electron orbiting around the nucleus ‘sees’ the nucleus circling it, just like a person on earth perceives the sun circling the earth as the latter orbits around the sun.

This apparent nuclear orbit creates a magnetic field B that exerts a torque on the electron’s spin magnetic dipole moment \mu_s, resulting in an additional term of \hat{H}_{so}=\sum_{i=1}^{n}A_i\hat{\boldsymbol{\mathit{S}}}_i\cdot\hat{\boldsymbol{\mathit{L}}}_i (where A_i=\frac{1}{2m_e^{\;2}c^{2}}\frac{1}{r_i}\frac{dV_i}{dr_i}) in the multi-electron Hamiltonian. To derive this term, we consider a 1-electron atom.

Let L=m_erv_{\perp} be the orbital angular momentum of the electron and I be the proton’s current loop, which generates a magnetic field of magnitude B=\frac{\mu_0I}{2r} given by the Biot-Savart law. Since \frac{1}{t}=\frac{v_{\perp}}{2\pi r}, we have

I=\frac{e}{t}=\frac{eL}{2\pi m_er^{2}}\; \; \; \; \; \; \; \; 259

Substitute eq259 in B, we have, B=\frac{\mu_0e}{4\pi m_er^{3}}L or \frac{\boldsymbol{\mathit{B}}}{\hat{\boldsymbol{\mathit{B}}}}=\frac{\mu_0e}{4\pi m_er^{3}}\frac{\boldsymbol{\mathit{L}}}{\hat{\boldsymbol{\mathit{L}}}}, where \hat{\boldsymbol{\mathit{B}}} and \hat{\boldsymbol{\mathit{L}}} are unit vectors. Since \boldsymbol{\mathit{B}} and \boldsymbol{\mathit{L}} point in the same direction, \hat{\boldsymbol{\mathit{B}}}=\hat{\boldsymbol{\mathit{L}}}=\hat{\boldsymbol{\mathit{M}}}. Multiplying both sides of \frac{\boldsymbol{\mathit{B}}}{\hat{\boldsymbol{\mathit{M}}}}=\frac{\mu_0e}{4\pi m_er^{3}}\frac{\boldsymbol{\mathit{L}}}{\hat{\boldsymbol{\mathit{M}}}} by \hat{\boldsymbol{\mathit{M}}}, we have \boldsymbol{\mathit{B}}=\frac{\mu_0e}{4\pi m_er^{3}}\boldsymbol{\mathit{L}}, which we substitute in eq65 (where \boldsymbol{\mathit{\mu}} is the spin analogue of eq61) to give U=-\frac{\gamma_e\mu_0e}{4\pi m_er^{3}}\boldsymbol{\mathit{S}}\cdot\boldsymbol{\mathit{L}}. Substitute eq164 and \frac{1}{c^{2}}=\mu_0\varepsilon_0 in U,

U=\frac{e^{2}}{4\pi\varepsilon_0 m_e^{\;2}c^{2}r^{3}}\boldsymbol{\mathit{S}}\cdot\boldsymbol{\mathit{L}}

For a 1-electron atom, V=-\frac{e^2}{4\pi\varepsilon_0r} and so

U=\frac{1}{ m_e^{\;2}c^{2}r}\frac{dV}{dr}\boldsymbol{\mathit{S}}\cdot\boldsymbol{\mathit{L}}\; \; \; \; \; \; \; \; 260

Eq260 can be written in terms of the Larmor frequency of the electron. From eq149, \omega_L=\frac{eB}{m_e}=\frac{e}{m_e}\frac{\mu_0e}{4\pi m_er^{3}}L=\frac{1}{m_e^{\;2}c^{2}r}\frac{dV}{dr}L. So, U=\frac{\omega_L}{L}\boldsymbol{\mathit{S}}\cdot\boldsymbol{\mathit{L}}. Swapping \omega_L with the Thomas precession rate \omega_T, we obtain the correction term of U_{tp}=-\frac{1}{2 m_e^{\;2}c^{2}r}\frac{dV}{dr}\boldsymbol{\mathit{S}}\cdot\boldsymbol{\mathit{L}}. The total spin-orbit Hamiltonian is

\hat{H}_{so}=U+U_{tp}=U_{tp}=\frac{1}{2 m_e^{\;2}c^{2}r}\frac{dV}{dr}\boldsymbol{\mathit{S}}\cdot\boldsymbol{\mathit{L}}\; \; \; \; \; \; \; \; 261

For a multi-electron atom,

\hat{H}_{so}=\sum_{i=1}^n\frac{1}{2m_e^{\;2}c^2r_i}\frac{dV_i}{dr_i}\boldsymbol{\mathit{S}}_i\cdot\boldsymbol{\mathit{L}}_i\; \; \; \; \; \; \; \; 261a


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