Chapter 10 Atomic Structure and Atomic Spectra

Spectra of complex atoms Energy levels not solely given by energies of orbitals Electrons interact and make contributions to E Singlet and triplet states Spin-orbit coupling

Fig Vector model for paired-spin electrons Multiplicity = (2S + 1) = (2·0 + 1) = 1 Singlet state Spins are perfectly antiparallel Ground state Excited state

Fig Vector model for parallel-spin electrons Multiplicity = (2S + 1) = (2·1 + 1) = 3 Triplet state Spins are partially parallel Three ways to obtain nonzero spin

Fig Grotrian diagram for helium Singlet – triplet transitions are forbidden

Fig Orbital and spin angular momenta Spin-orbit coupling Magnetogyric ratio

Fig 10.27(a) Parallel magnetic momenta Total angular momentum (j) = orbital ( l ) + spin (s) e.g., for l = 0 → j = ½ for l = 1 → j = 3/2

Fig 10.27(b) Opposed magnetic momenta

Fig Parallel and opposed magnetic momenta Result: For l > 0, spin-orbit coupling splits a configuration into levels e.g., for l = 0 → j = ½ for l = 1 → j = 3/2, ½ Total angular momentum (j) = orbital ( l ) + spin (s)

Fig Spin-orbit coupling of a d-electron ( l = 2) j = l + 1/2 j = l - 1/2

Energy levels due to spin-orbit coupling Strength of spin-orbit coupling depends on relative orientations of spin and orbital angular momenta (= total angular momentum) Total angular momentum described in terms of quantum number j Energy of level with QNs: s, l, and j where A is the spin-orbit coupling constant E l,s,j = ½ hcA{ j(j+1) – l ( l +1) – s(s+1) }

Fig Levels of a 2 P term arising from spin-orbit coupling of a 2p electron E l,s,j = 1/2 hcA{ j(j+1) – l ( l +1) – s(s+1) } = 1/2hcA{ 3/2(5/2) – 1(2) – ½(3/2) = 1/2 hcA and = 1/2hcA{ 1/2(3/2) – 1(2) – ½(3/2) = -hcA

Fig Energy level diagram for sodium D lines Fine structure of the spectrum

Fig Types of interaction for splitting E-levels In light atoms: magnetic Interactions are small In heavy atoms: magnetic interactions may dominate the electrostatic interactions

Fig Total orbital angular momentum (L) of a p and a d electron (p 1 d 1 configuration) L = l 1 + l 2, l 1 + l 2 – 1,..., | l 1 + l 2 | = 3, 2, 1 F P D

Fig Multiplicity (2 S +1) of two electrons each with spin angular momentum = 1/2 S = s 1 + s 2, s 1 + s 2 – 1,..., |s 1 - s 2 | = 1, 0 Singlet Triplet

For several electrons outside the closed shell, must consider coupling of all spin and all orbital angular momenta In lights atoms, use Russell-Saunders coupling In heavy atoms, use jj-coupling

Fig Correlation diagram for some states of a two electron system J = L+S, L+S-1,..., |L-S| Russell-Saunders coupling for atoms with low Z, ∴ spin-orbit coupling is weak: jj-coupling for atoms with high Z, ∴ spin-orbit coupling is strong: J = j 1 + j 2

Selection rules for atomic (electronic) transitions Transition can be specified using term symbols e.g., The 3p 1 → 3s 1 transitions giving the Na doublet are: 2 P 3/2 → 2 S 1/2 and 2 P 1/2 → 2 S 1/2 In absorption: 2 P 3/2 ← 2 S 1/2 and 2 P 1/2 ← 2 S 1/2 Selection rules arise from conservation of angular momentum and photon spin of 1 (boson)

Selection rules for atomic (electronic) transitions ΔS = 0 Light does not affect spin directly Δ l = ±1 Orbital angular momentum must change ΔL = 0, ±1 Overall change in orbital angular momentum depends on coupling ΔJ = 0, ±1 Total angular momentum may or may or may not change: J = L + S