Chapter08 Atomic Structure and the Periodic Table General Bibliography 1) Various wikipedia, as specified 2) Thornton-Rex, Modern Physics for Scientists & Eng, as indicated

Outline 8.1 Atomic Structure and the Periodic Table 8.2 Total Angular Momentum

H-atom Energy Level Scheme 1s 2s2p 3s3p3d 4s4p4d4f

8.2 Multi-Electron Atoms Z

n = 1

n = 2

n = 3

2s 3s 2p 4s 1s 3p 3d 4p orbitals get sucked down the most Crossings occur for the upper orbitals 1s sucked off bottom of page

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Two kinds of Angular Momentum Classical Angular Momentum –L = r x p – r vector, p vector  L vector –L obeys vector math –Any L possible, no contraints on L x L y L z Quantum –Quantum Mechanical Angular Momentum –L = r x p L – r vector, p vector operator  L 3 component operator –L –L obeys …… got to be careful –L –L described by two labels l, m –L and L z can be known, L x and L y cannot

Bohr Model of Ang Momentum Note: s-states (l=0) have no Bohr model picture Eisberg & Resnick: Fig 7-11 Classical or Semi-classical description

Vector Model of QM Ang. Momentum quantum numbers E&R Fig 7-12

pg 19: “We might imagine the vector moving in an unobservable way about the z-axis...” Edmonds “A.M. in QM” pg 29: “The QM probability density, not being time dependent, gives us no information about the motion of the particle in it’s orbit.”  *(r,t)  (r,t)  (r,t)=  (r) e  i  t

Morrison, Estle, Lane “Understanding More QM”, Prentice-Hall, 1991

Addition of Orbital Angular Momentum for two electrons Problem: Two objects each travel in a p-orbit ( l=1 ). What are the allowed values of l tot, m tot ? L1L1 L2L2 L tot = L 1 + L 2 L1L1 L2L2 L tot

L tot = L 1 + L 2

Addition of Angular Momentum aligned configuration jack-knife configuration “aligned” does not mean straight “jack-knife” does not mean antiparallel

Addition of Intrinsic Spin Angular Momentum S tot = S 1 + S 2 S2S2 S1S1 Because objects are all spin-1/2 s tot = 0, 1 singlet triplet WARNING: CAN’T MAKE TRIPLET IF BOTH ELECTRONS ARE IN THE SAME ORBIT BECAUSE THEY WOULD BE IN THE SAME m s SUBSTATE. i.e. m s = + ½ and + ½

Total Angular Momentum for a single electron J tot = L 1 + S 1 S1S1 L1L1

Add It All Up for Two Electrons What is the “total” total angular momentum, J tot ? L1L1 L2L2 S1S1 S2S2 LS Coupling JJ Coupling

Two Ways to Add LS Coupling (aka Russell-Sanders Coupling) –L tot = L 1 + L 2 and S tot = S 1 + S 2 –J tot = L tot + S tot –Preferred for lighter atoms, Z<30 JJ Coupling –J 1 = L 1 + S 1 and J 2 = L 2 + S 2 –J tot = J 1 + J 2 –Preferred for heavier atoms, where the nucleus has a lot of charge and creates big internal magnetic fields from the point of view of the electrons.

Hund’s Rules Hund observed in lighter atoms that –The S 1 – S 2 orientation energy is very strong –Bigger S tot have lower energy –The L 1 – L 2 orientation energy is not as strong –Bigger L tot have lower energy –If shells are less than half full, smaller J tot have lower energy L1L1 L2L2 S1S1 S2S2 4 February March 1997

4p-4d Example

QUANTUM NUMBERS principal: n l tot, s tot j tot. S tot = S 1 + S 2 + … L tot = L 1 + L 2 + … J tot = L tot + S tot Multi-e Spectroscopic Notation s tot = 1, l tot =0, j tot =1 3 S 1

Two Kinds of Spectroscopic Notation Where an individ electron is at n l j –1s 1/2 –2s 1/2 –2p 1/2 –2p 3/2 A.M. for whole atom 2S tot +1 L tot J tot – 1 S 0 – 3 S 1 – 3 P 0, 3 P 1, 3 P 2