1.5
0.5
hbar^2
5.952084e-8
eV
1.5
0.5
hbar^2
5.952084e-8
eV
The Spin-Orbit Coupling Calculator computes the energy splitting of atomic levels due to the interaction between an electron's orbital angular momentum (L) and its spin angular momentum (S). This relativistic quantum mechanical effect, known as spin-orbit coupling or fine structure, causes spectral lines to split into closely spaced doublets or multiplets.
Physically, spin-orbit coupling arises because in the electron's rest frame, the orbiting nucleus creates a magnetic field that interacts with the electron's spin magnetic moment. This interaction energy is proportional to L·S, the dot product of orbital and spin angular momentum vectors. The coupling is described by the Hamiltonian H_so = xi(r) * L·S, where xi(r) is the spin-orbit coupling constant.
The total angular momentum J = L + S takes values |L - S| to L + S in integer steps. For a single electron with s = 1/2, the possible j values are l + 1/2 and l - 1/2 (for l > 0). The L·S value for a given j state is (1/2)[j(j+1) - l(l+1) - s(s+1)] in units of hbar^2.
The fine structure of hydrogen is responsible for the famous sodium D-line doublet at 589.0 nm and 589.6 nm — two closely spaced yellow lines from the 3p level split into j=3/2 (2P_3/2) and j=1/2 (2P_1/2) states by spin-orbit coupling. The splitting grows as Z^4/n^3 for hydrogen-like atoms, making it especially significant for heavy elements.
Spin-orbit coupling is crucial in modern physics: it governs magnetic anisotropy in materials, spin Hall effects in spintronics, topological insulators, and the design of spin-based electronic devices. In chemistry it affects molecular spectra and intersystem crossing in photochemistry.
The spin-orbit energy shift is E_so = A * (L·S) / hbar^2, where L·S = (hbar^2/2)[j(j+1) - l(l+1) - s(s+1)] with s=1/2. The coupling constant A for hydrogen-like atoms scales as Z^4 * hbar^2/(2*me^2*c^2*a0^3*n^3*l(l+1/2)(l+1)). This calculator uses the hydrogen-like approximation.
The j = l+1/2 state has positive L·S and higher energy; j = l-1/2 has negative L·S and lower energy. The splitting delta_E = E(j=l+1/2) - E(j=l-1/2) = (2l+1)*A. For hydrogen 2p: splitting ~ 0.000045 eV (giving the 0.6 nm sodium D-line separation). For heavier atoms the splitting grows rapidly as Z^4.
Inputs
Results
The 2p_3/2 state (j=3/2) of hydrogen has a small positive spin-orbit energy shift of ~11 microeV, contributing to the Lamb shift and fine structure of the hydrogen spectrum.
Inputs
Results
With Z_eff ~ 6 for the 3p electrons of sodium, the spin-orbit splitting is ~2 meV, matching the observed 0.6 nm splitting of the sodium D doublet at 589 nm.
The interaction between an electron's orbital motion and its spin. In the electron's rest frame, the nuclear orbit produces a magnetic field that interacts with the spin magnetic moment, splitting energy levels with the same l but different j.
Fine structure refers to the splitting of atomic spectral lines due to relativistic effects and spin-orbit coupling. It is smaller than the main energy level spacing by a factor of approximately alpha^2 ~ 10^-4, where alpha = 1/137 is the fine structure constant.
j = l + s where l is orbital and s = 1/2 is spin. For l > 0, j takes two values: j = l + 1/2 and j = l - 1/2. For s orbitals (l=0), spin-orbit coupling is zero (no orbital angular momentum to couple with).
The coupling constant A scales as Z^4 because the electric field (and hence the magnetic field in the electron's frame) grows as Z, and the probability density at small radii (where the field is strongest) grows as Z^3 for ns states.
The famous yellow sodium emission lines at 589.0 nm and 589.6 nm arise from spin-orbit splitting of the 3p level into 3p_1/2 and 3p_3/2 states. Both transitions go to the 3s ground state, but with slightly different energies due to the spin-orbit splitting.
For heavy elements, spin-orbit coupling is so strong it changes orbital energies and electron configurations. It causes gold to be yellow (shifting UV absorption into visible range), mercury to be liquid at room temperature, and affects the chemistry of all heavy elements.
In light atoms, orbital angular momenta of all electrons couple first (L = sum of l_i), then spins couple (S = sum of s_i), giving LS (Russell-Saunders) coupling. In heavy atoms, spin-orbit coupling within each electron is stronger, giving jj coupling.
The Lande g-factor determines the magnetic moment of a state with quantum numbers j, l, s: g_J = 1 + [j(j+1) + s(s+1) - l(l+1)]/(2j(j+1)). It governs the anomalous Zeeman effect.
In solid-state physics, spin-orbit coupling lifts spin degeneracy of electronic bands, enables the spin Hall effect, contributes to magnetic anisotropy, and is essential for topological insulators and Weyl semimetals.
Yes. Spintronics devices use spin-orbit coupling to convert charge currents to spin currents (spin Hall effect). It enables Rashba and Dresselhaus effects in semiconductor heterostructures used in spin transistors and quantum computing qubits.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
How helpful was this calculator?
Be the first to rate!
Electron Configuration Calculator
Atomic & Molecular Physics Calculators
Ionization Energy Calculator
Atomic & Molecular Physics Calculators
Electron Affinity Calculator
Atomic & Molecular Physics Calculators
Electronegativity Calculator
Atomic & Molecular Physics Calculators
De Broglie Wavelength Calculator
Atomic & Molecular Physics Calculators
Heisenberg Uncertainty Calculator
Atomic & Molecular Physics Calculators
