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Chemistry 431 Lecture 9 Spin-orbit coupling NC State University Spin-orbit Hamiltonian The magnitude of spin-orbit coupling is measured spectroscopically as a splitting of spectral lines. The Hamiltonian is: HSO = 1 hcA j( j + 1)) – l(l ( + 1)) – s(s ( + 1)) 2 Where l is the orbital angular momentum quantum number and s is the spin quantum number. The total angular moment is j = l + s and A is the magnitude of the spin-orbit coupling in wavenumbers. HSO = 1 hcA (l +s)(l + s + 1) – l(l + 1) – s(s + 1) 2 = 1 hcA l 2 + s 2 + ls + sl + l + s – l 2 – l – s 2 – s) = hcA l⋅s 2 Spin-orbit p Hamiltonian The magnitude of the spin orbit coupling can be calculated in terms of molecular parameters by the substitution 2 Zα hcA L⋅S= < 13 >L⋅S 2 r where α is the fine structure constant α = e2/hc4πε0, which is a dimensionless constant (α ( = 1/137.037). 1/137 037) L and S are operators. Z is an effective atomic number number. The spin orbit coupling splitting can be calculated from E SO = Ψ H SOΨ dτ = Z 2 2(137) * LS L⋅S Ψ 3 Ψ dτ r * Spin-orbit Hamiltonian This can be evaluated using the above identity that can be recast 2 2 2 1 L⋅S= J –L –S 2 to give an spin-orbit coupling energy in terms of molecular parameters Z 1 ESO = 1 j( j + 1) – l(l + 1) – s(s + 1) 2 2(137) 2 r 3 where 1 = r3 Ψ * 1 Ψdτ r3 Spin-orbit Hamiltonian We can evaluate this integral explicity for a given atomic orbital. For example for Ψ210 we have 3/2 –Zr/2a 0 Ψ 210 = 1 aZ Zr e cos θ a 0 4 2π 0 so that the integral is 1 = 1 Z 32π a 0 32 r3 π 2π 5 dφ 0 cos 2θsinθdθ 0 0 which integrates to 2 a < 13 > = 1 aZ 2π 2 0 32π 0 3 Z r 5 2 or Z3/24 in atomic units. ∞ = 1 Z 24 a 0 3 r 2e Zr/a 0 13 r 2dr r Spin-orbit Hamiltonian Therefore we have Therefore, 3 Z < 13 > = 3 3 r n a ol(l + 1/2)(l + 1) Therefore, in general the spin-orbit splitting is given by 4 j( j + 1) – l(l + 1) – s(s + 1) Z ESO = 2l(l + 1/2)(l + 1) 2(137) 2ao3n 3 Note N t th thatt the th spin-orbit i bit coupling li iincreases as th the fourth power of the effective nuclear charge Z, but only as the third power of the principal quantum number n. This indicates that spin orbit-coupling interactions are significantly g y larger g for atoms that are further down a particular column of the periodic table. The Sodium D Line One notable atomic spectral line of sodium vapor is the so-called D-line, which may be observed directly as the sodium flame-test line and also the major light output of low-pressure low pressure sodium lamps (these produce an unnatural yellow. The D-line is one of the classified Fraunhofer lines. lines Sodium vapor in the upper layers of the sun creates a dark line in the emitted spectrum of electromagnetic g radiation byy absorbing g visible light g in a band of wavelengths around 589.5 nm. This wavelength corresponds to transitions in atomic sodium in which hi h the th valence-electron l l t ttransitions iti ffrom a 3s 3 tto 3p 3 electronic state. The Splitting of the D Line Closer examination e amination of the visible isible spectr spectrum m of atomic sodium reveals that the D-line actually consists of two lines called the D1 and D2 lines at 589 589.6 6 nm and 589.0 nm, respectively. The splitting between These lines arises because of spin-orbit p coupling. p g The constant A is usually given in cm-1. For Na, it is 11.5 cm-1. Na has one unpaired electron (s = ½). If we consider id the th s -> p transition, t iti th then ffor the th excited state, p, we have l = 1. Thus, j = 3/2 or ½. Practical calculations using the Spin-orbit Hamiltonian The two t o energy energ le levels els can be calc calculated lated in terms of the constant A. E3/2 = 1 A 3/2(3/2 + 1) – 1(1 + 1) – 1/2(1/2 + 1) = 1 A 2 2 E1/2 = 1 A 1/2(1/2 + 1) – 1(1 + 1) – 1/2(1/2 + 1) = – A 2 The energy difference Th diff between b t the th lines li iis 3/2A 3/2A. Thus, the energy splitting for Na is 17.3 cm-1.