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Lecture 22 Spin-orbit coupling Spin-orbit coupling Spin makes an electron act like a small magnet. An electron orbiting around the nucleus also makes a magnet. These two magnetic moments can interact and, depending on the relative orientations of the two moments, orbital energy can be slightly altered. We use the so-called Na D line as a paradigm. We use the first-order perturbation theory to describe the shifts in orbital energies. The spin-orbit interaction is a relativistic effect and its derivation is beyond the scope of this course. We treat it as a phenomenological effect explained in analogy to two interacting magnets. Na D line The orange color of the sodium lamp is due to the Na 3p→3s emission at ca. 17000 cm−1. A close examination of this transition reveals that the emission band consists of two bands separated by 17 cm−1. Public-domain image from Wikipedia Spin-orbit coupling Spin of an electron makes it a magnet. Orbital motion of the electron also makes it a magnet. These two magnetic moments can interact or “couple” (spinorbit coupling) and cause energy level splitting. N N S S N S S N Sum of angular momenta Each electron has two angular momenta (a dual magnet): orbital angular momentum and spin angular momentum. The total momentum is the most naturally defined as their vector addition. j =l +s Total Orbital Spin s l j Sum of angular momenta j must be (space) quantized. So its total angular momentum quantum number j is either a full or half integer in the range: j = jmin (0 or greater), jmin+1,…, jmax–1, jmax j =l +s jmax = l + s = l + 12 jmin = l - s = l - 12 Examples Identify the levels that may arise from the configurations (a) (3p)1, (b) (3s)1. Examples (a) 3p orbital → l = 1. j = l ± ½ = 3/2 or 1/2. (b) 3s orbital → l = 0. j = 0 + ½ = ½ (j = 0 – ½ is not allowed because j is non negative). Spin-orbit coupling Two magnets are the most stable when they are antiparallel and the least stable when they are parallel. In general, the energy due to the interaction of spin and orbital momenta should be ESO µ s ×l = s × l cosq l θ s Spin-orbit coupling operator The atomic Hamiltonian does not have this: H = 2 Ñ +V 2 2m This is because we do not have a counterpart in the classical energy, from which the Hamiltonian is derived. We add spin-orbit interaction operator: H =- 2 Ñ +V + ( hcA) 2 2m sˆ × lˆ 2 Spin-orbit coupling operator The spin-orbit interaction operator H =- 2 Ñ +V + ( hcA) 2 2m sˆ × lˆ 2 has the spin-orbit coupling constant A. It is in units of cm−1, which is why hc is multiplied. The value of A is extracted from experiment (11.5 cm−1 for Na 3p from the splitting of 17 cm−1) or relativistic quantum mechanics. Homework challenge #6 Study the special theory of relativity. One of the best textbooks is “Special Theory for Relativity for Beginners” by Jürgen Freund. Study Dirac’s theory of relativistic quantum mechanics and explain how it introduces the concepts of spins and positrons from the first principles. Study the work of Pekka Pyykkö on the effect of relativity on chemistry. Spin-orbit coupling operator The spin-orbit interaction operator H =- 2 Ñ +V + ( hcA) 2 2m sˆ × lˆ 2 makes the solution of the Schrödinger equation difficult. Since A is very small (0.001 of 3p-3s energy difference), we use perturbation theory. First-order perturbation theory E (1) SO ( )Y ˆ × lˆ hcA s (0)* = ò Y ls 2 (0) ls dt j × j = (l + s ) ×(l + s ) = l × l + s × s + 2s × l s ×l = E (1) SO 1 2 ( j × j - l ×l - s ×s) ( )Y ˆ × lˆ hcA s (0)* = ò Y ls = 12 hcA ò Y (0)* ls 2 (0) ls dt ˆj 2 - lˆ2 - sˆ2 2 Y (0) dt ls = 12 hcA{ j( j + 1) - l(l + 1) - s(s + 1)} Na D line E3/ 2 E1 / 2 1 2 hcA 23 52 1 2 1 2 hcA 12 23 1 2 1 2 1 2 3 2 3 2 1 2 hcA hcA Na D line 4-fold degenerate 3 2 hcA = 17 cm -1 A = 11.5 cm -1 2-fold degenerate Spin-orbit coupling constants The measured values of A: Li: 0.23 cm–1 Na: 11.5 cm–1 K: 38.5 cm–1 Rb: 158 cm–1 Cs: 370 cm–1 Spin-orbit coupling arises from the special theory of relativity and greater for the heavier elements because the 1s electrons in high-Z elements can go nearly as fast as the speed of light. Consequences of SO coupling An electron in each orbital no longer has a well defined spin (magnetic quantum number, α or β). States are no longer pure spin-singlet, doublet, triplet, etc. Radiative transitions between singlet and triplet, between doublet and quartet, etc. become weakly allowed (phosphorescence). Nonradiative transitions between singlet and triplet, etc. become weakly allowed (intersystem crossing). These are more prominent in heavier elements. Singlet and triplet states The singlet and triplet states have different spin and spin magnetic momenta. They are orthogonal functions if it were not for the SO interaction.. spatial spin spin ˆ TDM = ò Y f zˆY i dt = ò Yspatial z Y d t Y f i ò f Y i dt This is zero when final and initial states have different spin eigenfunctions, e.g., singlet and triplet. Separable because z operator does not act on spin part. Fluorescence & Phosphorescence Fluorescence Fluorescence is an emission of light between the same spin states (e.g., singlet to singlet). Since this is an allowed transition, it is intense and fast. Intersystem crossing Phosphorescence is between different spin states and mediated by SO. It is “forbidden” and it is weak and slow. Both public-domain images from Wikipedia Phosphorescence Summary Spin angular momentum as a magnet and orbital angular momentum as another magnet interact (spin-orbit coupling). Spin-orbit coupling is a relativistic effect and is greater for heavier elements. It causes splitting of subshell states, phosphorescence, and intersystem crossing. The first-order perturbation theory describes the coupling.