Spin_Orbit_Interaction_MSc_Level.pptxUltrafast Laser–Electron Interactions in Nanophysics

Ultrafast Laser–Electron Interactions in Nanophysics

1. Spin-Orbit Interaction M.Sc. Physics Level Presentation Prepared by Zahid

2. Introduction • Spin–orbit interaction arises due to coupling between an electron’s spin and its orbital motion. • It is a relativistic effect resulting from the motion of an electron in the electric field of the nucleus. • Leads to fine structure splitting in atomic spectra. • Important in explaining atomic, molecular, and solid-state phenomena.

3. Physical Origin of Spin-Orbit Interaction • In the electron’s rest frame, the nucleus appears to move, creating a magnetic field. • The electron’s magnetic moment due to spin interacts with this magnetic field. • This results in an additional energy term that depends on the relative orientation of spin and orbital angular momentum.

4. Mathematical Expression The spin–orbit interaction energy is given by: H_SO = ξ(r) L · S where: • L = orbital angular momentum • S = spin angular momentum • ξ(r) = (1 / (2m²c²r)) (dV/dr) This term depends on the gradient of the central potential V(r).

5. Spin-Orbit Interaction in Hydrogen Atom • For hydrogen, V(r) = -Ze² / (4πε₀r). • Substituting into ξ(r) gives the fine structure correction term. • The energy correction depends on total angular momentum quantum number j = l ± ½. • Leads to splitting of spectral lines in hydrogen and hydrogen-like ions.

6. Total Angular Momentum • Total angular momentum J = L + S. • The possible values are j = l + ½ and j = l – ½. • The energy shift due to spin-orbit coupling is proportional to ⟨L · S⟩. • ⟨L · S⟩ = ½ [j(j+1) – l(l+1) – s(s+1)].

7. Fine Structure Splitting • Spin–orbit coupling leads to splitting of atomic energy levels into closely spaced sub-levels. • These are observed as fine structure in spectral lines. • Example: 2P₁/₂ and 2P₃/₂ levels in hydrogen. • Splitting increases with atomic number (Z⁴ dependence).

8. Coupling Schemes • LS (Russell–Saunders) Coupling: - Dominant in light atoms. - L = Σl_i and S = Σs_i are coupled first, then J = L + S. • jj Coupling: - Dominant in heavy atoms. - Each electron’s l_i and s_i combine to form j_i; total J = Σj_i.

9. Relativistic Interpretation • The Dirac equation naturally includes spin–orbit coupling as a relativistic correction. • The interaction arises from the Thomas precession correction to the classical magnetic interaction. • Explains the fine structure without ad hoc assumptions.

10. Applications • Explains fine structure in atomic spectra. • Essential in understanding magnetic anisotropy and spintronic devices. • Determines selection rules in atomic transitions. • Important in quantum materials, e.g., topological insulators and Rashba effect.

11. Summary • Spin–orbit interaction couples spin and orbital angular momenta. • It originates from relativistic motion of electrons in atomic electric fields. • Leads to fine structure and affects magnetic and spectroscopic properties. • Its strength increases with atomic number.

12. References 1. J. J. Sakurai – Modern Quantum Mechanics 2. Griffiths – Introduction to Quantum Mechanics 3. Eisberg & Resnick – Quantum Physics 4. C. Cohen-Tannoudji – Quantum Mechanics, Vol. 1