case of more complex atoms. In any case, we are interested in understanding the physics, not just doing the maths of simple systems. In what follows we shall first outline the physics of the electron’s spin magnetic moment μ interacting with the magnetic field, B , due to its motion in the central field (nucleus plus inner shell electrons). The interaction energy is found to be μ · B so the perturbation to the energy, Δ E , will be the expectation value of the corresponding operator ˆ μ · B . We then use perturbation theory to find Δ E . We will not, however, be able simply to use our zero order wavefunctions ψ 0 ( n, l, m l , m s ) derived from our Central Field Approximation, since they are degenerate in m l , m s . We then have to use degenerate perturbation theory, DPT, to solve the problem. We won’t have to actually do any complicated maths because it turns out that we can use a helpful model – the Vector Model, that guides us to the solution, and gives some insight into the physics of what DPT is doing. 4.1 The Physics of Spin-Orbit Interaction What happens to a magnetic dipole in a magnetic field? A negatively charged object having a moment of inertia I, rotating with angular velocity ω , has angular momentum, I ω = λ . The energy is then E = 1 2 I ω 2 We suppose the angular momentum vector λ is at an angle θ to the z -axis. The rotating charge has a magnetic moment μ = - γλ (44) The sign is negative as we have a negative charge. γ is known as the gyromagnetic ratio. (Classically γ = 1 for an orbiting charge, and γ = 2 for a spinning charge). If a constant magnetic field B is applied along the z -axis the moving charge experiences a force – a torque acts on the body producing an extra rotational motion around the z -axis. The axis of rotation of λ precesses around the direction of B with angular velocity ω . The angular motion of our rotating change is changed by this additional precession from ω to ω + ω cos( θ ). If the angular momentum λ was in the opposite direction then the new angular velocity would be ω - ω cos( θ ). 17
Atomic Physics, P. Ewart 4 Corrections to the Central Field: Spin-Orbit interaction Figure 11: Illustration of the precession (Larmor precession) caused by the torque on the magnetic moment μ by a magnetic field B . The new energy is, then E = 1 2 I( ω ± ω cos θ ) 2 (45) = 1 2 I ω 2 + 1 2 I( ω cos θ ) 2 ± I ωω cos θ (46) We now assume the precessional motion ω to be slow compared to the original angular velocity ω . ω << ω so ( ω cos θ ) 2 << ω 2 and we neglect the second term on the r.h.s. The energy change Δ E = E - E is then Δ E = I ωω cos θ (47) = λ ω cos θ (48) Now the precessional rate ω is given by Larmor’s Theorem ω = - γB (49) So Δ E = - γλ B cos θ (50) Hence Δ E = - μ · B (51) So - μ · B is just the energy of the precessional motion of μ in the B -field.
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