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