Chapter 10

Atomic Structure and Atomic Atomic Structure and Atomic SpectraSpectra

Spectra of complex atomsSpectra 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

Multiplicity = (2S + 1)

= (2·1 + 1)

= 3

Triplet state

Spins are partiallyparallel

Three ways to obtain nonzero spin

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

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 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 couplingEnergy 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: magneticInteractions are small

In heavy atoms: magneticinteractions may dominatethe 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

P

D

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

Singlet

Triplet

For several electrons outside the closed shell,For several electrons outside the closed shell,

must consider coupling of all spin and all orbitalmust consider coupling of all spin and all orbital

angular momentaangular 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

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 = j1 + j2

Selection rules for atomic (electronic) transitionsSelection rules for atomic (electronic) transitions

• Transition can be specified using term symbols

• e.g., The 3p1 → 3s1 transitions giving theNa 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 angularmomentum and photon spin of 1 (boson)

Selection rules for atomic (electronic) transitionsSelection 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 mayor may not change: J = L + S