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Where Are We Going…?

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

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Where Are We Going…?

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  1. Where Are We Going…? • Week 10: Orbitals and Terms • Russell-Saunders coupling of orbital and spin angular momenta • Free-ion terms for p2 • Week 11: Terms and ionization energies • Free-ion terms for d2 • 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

  2. Ze - ri Many Electron Atoms • For any 2 e- atom or ion, the Schrödinger equation cannot be solved for every electron: S S HH-like = ½ mvi2 + i i • Treatment leads to configurations • for example: He 1s2, C 1s2 2s2 2p2 • Inclusion of interelectron repulsion leads to terms • for example: p21D, 3F and 1S • characterized by S and L quantum numbers • energy given by Hund’s 1st and 2nd rules • (2S+1)(2L+1) degenerate e2 S - rij i≠j

  3. 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

  4. 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

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

  6. Orbital Magnetism rotation of a px orbital by 90° gives a py orbital and vice versa: generating magnetism about the z-direction • 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 L = 1 L = 0 L = 1

  7. 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 spin magnetism orbital magnetism lowest energy highest energy

  8. e2 S - rij i≠j 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 • Then treat spin-orbit second to give levels: • L + S = J • J is the total angular momentum + H = + HH-like λL.S configurations terms levels

  9. 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+1 L J

  10. 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: 3P2 3P 3P1 3P0

  11. 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: 3P0 3P1 3P 3P2

  12. Magnetism • The magnetic moment is given by: • where g is the Landé splitting factor, • 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”)

  13. Ionization Energies: (iii) Hund’s 3rd Rule p-block ionization energies: M  M+ • For 6p, there is a decrease between p2 and p3 • No half-filled shell effect!

  14. j-j Coupling • For very heavy elements, magnetic coupling becomes large • Treat spin-orbit first to give spin-orbitals for each electron: • j = l+ s each value is (2j+1) degenerate • Then add individual j values together to give J • j1 + j2 + … = J • For p-electrons, l = 1 and s = 1/2 • j = 1/2 and 3/2 with former lowest in energy j = 3/2 j = 1/2

  15. j-j Coupling • For p-electrons, l = 1 and s = 1/2 • j = 1/2 and 3/2 with former lowest in energy j = 3/2 j = 1/2 • 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)

  16. 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 shells • Consequences • 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

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