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Slide 1/16 Where Are We Going…? Week 10: Orbitals and Terms Russell-Saunders coupling of orbital and spin angular momenta Free-ion terms for p 2 Week 11: Terms and ionization energies Free-ion terms for d 2 Ionization energies for 2p and 3d elements Week 12: Terms and levels Spin-orbit coupling Total angular momentum Week 13: Levels and ionization energies j-j coupling Ionization energies for 6p elements

Where Are We Going…?

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Where Are We Going…?. Week 10: Orbitals and Terms Russell-Saunders coupling of orbital and spin angular momenta Free-ion terms for p 2 Week 11: Terms and ionization energies Free-ion terms for d 2 Ionization energies for 2p and 3d elements Week 12: Terms and levels Spin-orbit coupling - PowerPoint PPT Presentation

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Page 1: Where Are We Going…?

Slide 1/16

Where Are We Going…?

• Week 10: Orbitals and TermsRussell-Saunders coupling of orbital and spin angular momentaFree-ion terms for p2

• Week 11: Terms and ionization energiesFree-ion terms for d2 Ionization energies for 2p and 3d elements

• Week 12: Terms and levelsSpin-orbit couplingTotal angular momentum

• Week 13: Levels and ionization energiesj-j couplingIonization energies for 6p elements

Page 2: Where Are We Going…?

Many Electron Atoms

• For any 2 e- atom or ion, the Schrödinger equation cannot be solved for every electron:

• Treatment leads to configurations

for example: He 1s2, C 1s2 2s2 2p2

rij

e2

HH-like = ri

Ze ½ mvi2 +

ii

• Inclusion of interelectron repulsion leads to terms

for example: p2 1D, 3F and 1S

characterized by S and L quantum numbers

energy given by Hund’s 1st and 2nd rules

(2S+1)(2L+1) degenerate

i≠j

Page 3: Where Are We Going…?

Slide 3/16

Magnetism Due To Spin

• Electron(s) with spin angular momentum generate a magnetic field perpendicular to plane of loop

magnitude related to S

direction related to MS

Page 4: Where Are We Going…?

Slide 4/16

Magnetism Due To Orbit

• Electron(s) with orbital angular momentum generate a magnetic field perpendicular to plane of loop

magnitude related to L

direction related to ML

Page 5: Where Are We Going…?

Orbital Magnetism

• Electrons generate magnetism through their orbital motion• This is associated with an ability to rotate an orbital about an axis into

an identical and degenerate orbital.

x

y

z

x

y

z

rotation of a px orbital by 90° gives a py orbital and vice versa: generating magnetism about the z-direction

Page 6: Where Are We Going…?

Slide 6/16

Orbital Magnetism

• To be able to do this: the orbitals involved must have the same energy there must not be an electron in the second orbital with the same

spin as that in the first orbital. If there is, the electron cannot orbit without breaking the Pauli principle.

free orbitals available for

electron to hop into:orbital magnetism

free orbital available for electron to hop

into:orbital magnetism

no free orbital available for electron to hop into:no orbital magnetism

rotation of a px orbital by 90° gives a py orbital and vice versa: generating magnetism about the z-direction

L = 1 L = 1 L = 0

Page 7: Where Are We Going…?

Slide 7/16

Spin Orbit Coupling

• There is a magnetic interaction between the magnetism generated by the spin and orbital motions

results in different values for the total angular momentum, J

orbital magnetism spin magnetism

lowest energy highest energy

Page 8: Where Are We Going…?

Russell – Saunders Coupling

• The magnetic interaction increases with the atomic number

for most of the periodic table, electrostatic >> magnetic

• Treat electrostatic to give terms characterized by L and S

l1 + l2 + … = L, s1 + s2 + … = S

rij

e2

i≠j

H = HH-like + λL.S+

• Then treat spin-orbit second to give levels:

L + S = J

J is the total angular momentum

configurations terms levels

Page 9: Where Are We Going…?

Slide 9/16

Russell – Saunders Coupling

• For each L and S value:

J = L + S, L + S – 1, L + S – 2 …. L – S

Each level, MJ = J, J -1, J - 2, … -J (2J+1 values)

2S+1L

J

Page 10: Where Are We Going…?

Slide 10/16

Hund’s 3rd Rule

• For less than half-filled shells, smallest J lies lowest

p2: ground term is 3P with S = 1 and L = 1

J = 2, 1 and 0

less than half-filled:

3P

3P0

3P1

3P2

Page 11: Where Are We Going…?

Slide 11/16

Hund’s 3rd Rule

• For more than half-filled shells, highest J lies lowest

p4: ground term is 3P with S = 1 and L = 1

J = 2, 1 and 0

more than half-filled:

3P

3P2

3P1

3P0

Page 12: Where Are We Going…?

Slide 12/16

Magnetism

• The magnetic moment is given by:

where g is the Landé splitting factor,

12effμ g[J(J 1)]

[S(S 1) L(L 1)]3g2 2J(J 1)

• p2: ground level is 3P0 with J = 0, S = 1, L = 1

μeff = 0 (p2 is diamagnetic, at least at low temperature)

• p4: ground level is 3P2 with J = 2, S = 1, L = 1

g = 3/2 and μeff = 3.68 B.M. (B.M. = “Bohr Magnetons”)

Page 13: Where Are We Going…?

Slide 13/16

0

5

10

15

20

25

1 2 3 4 5 6pn

ion

iza

tion

en

erg

y (e

V)

2p

3p4p5p6p

Ionization Energies: (iii) Hund’s 3rd Rule

• For 6p, there is a decrease between p2 and p3

No half-filled shell effect!

p-block ionization energies: M M+

5

6

7

8

9

10

11

12

1 2 3 4 5 6pn

ion

iza

tion

en

erg

y (e

V) 6p

Page 14: Where Are We Going…?

Slide 14/16

j-j Coupling

• For very heavy elements, magnetic coupling becomes large

• Then add individual j values together to give J

j1 + j2 + … = J

• Treat spin-orbit first to give spin-orbitals for each electron:

j = l + s each value is (2j+1) degenerate

• For p-electrons, l = 1 and s = 1/2

j = 1/2 and 3/2 with former lowest in energy

j = 1/2

j = 3/2

Page 15: Where Are We Going…?

Slide 15/16

j-j Coupling

• For p-electrons, l = 1 and s = 1/2

j = 1/2 and 3/2 with former lowest in energy

j = 1/2

j = 3/2

• If electrostatic >> magnetic overall increase due to increasing nuclear charge decrease in ionization energy for p4 due to pairing (1st rule)

• If magnetic > electrostatic overall increase due to increasing nuclear charge decrease in ionization energy for p3 due to repulsive magnetic

interaction (3rd rule)

Page 16: Where Are We Going…?

Summary

Spin and orbital magnetism • Electrons have intrinsic magnetism due to spin• Electrons may also have orbital magnetism Spin-orbit coupling• Usually weak magnetic coupling between spin and orbit• Characterized by levels with total angular momentum, J Hund’s 3rd Rule• Lowest J lies lowest for < 1/2 filled shells• Highest J lies lowest for > 1/2 filled shellsConsequences• Magnitude of magnetism due to J, L and S• Stabilization of p1 and p2, destabilization of p4 – p6

Task!• Work out ground levels and magnetism for fn elements