Atomic Structure and Atomic Spectra

Atomic Structure and Atomic Spectra Chapter 10

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

Fig 10.24 Vector model for parallel-spin electrons Three ways to obtain nonzero spin Multiplicity = (2S + 1) = (2·1 + 1) = 3 Triplet state Spins are partially parallel

Fig 10.25 Grotrian diagram for helium Singlet – triplet transitions are forbidden

Fig 10.26 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 10.27 Parallel and opposed magnetic momenta Total angular momentum (j) = orbital (l) + spin (s) e.g., for l = 0 → j = ½ for l = 1 → j = 3/2, ½ Result: For l > 0, spin-orbit coupling splits a configuration into levels

Fig 10.28 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 El,s,j = ½ hcA{ j(j+1) – l (l+1) – s(s+1) }

Fig 10.29 Levels of a 2P term arising from spin-orbit coupling of a 2p electron El,s,j = 1/2hcA{ j(j+1) – l(l+1) – s(s+1) } = 1/2hcA{ 3/2(5/2) – 1(2) – ½(3/2) = 1/2hcA and = 1/2hcA{ 1/2(3/2) – 1(2) – ½(3/2) = -hcA

Fig 10.30 Energy level diagram for sodium D lines Fine structure of the spectrum

Fig 10.31 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 10.32 Total orbital angular momentum (L) of a p and a d electron (p1d1 configuration) L = l1 + l2, l1 + l2 – 1,..., |l1 + l2| = 3, 2, 1 F D P

Fig 10.33 Multiplicity (2S+1) of two electrons each with spin angular momentum = 1/2 S = s1 + s2, s1 + s2 – 1,..., |s1 - s2| = 1, 0 Triplet Singlet

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 10.34 Correlation diagram for some states of a two electron system Russell-Saunders coupling for atoms with low Z, ∴ spin-orbit coupling is weak: J = L+S, L+S-1,..., |L-S| jj-coupling for atoms with high Z, ∴ spin-orbit coupling is strong: J = j1 + j2

Selection rules for atomic (electronic) transitions • Transition can be specified using term symbols • e.g., The 3p1→ 3s1 transitions giving the • Na doublet are: • 2P3/2→ 2S1/2 and 2P1/2→ 2S1/2 • In absorption: 2P3/2← 2S1/2 and 2P1/2← 2S1/2 • 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 Δl = ±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

Atomic Structure and Atomic Spectra