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Spin-Orbit Effect

Spin-Orbit Effect. In addition to its motion about the nucleus, an electron also has an intrinsic angular momentum called “ spin ” similar to the earth moving about the sun and spinning on its axis orbital angular momentum L = r x p

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Spin-Orbit Effect

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  1. Spin-Orbit Effect • In addition to its motion about the nucleus, an electron also has an intrinsic angular momentum called “spin” • similar to the earth moving about the sun and spinning on its axis • orbital angular momentum L = r x p • spin angular momentum S cannot be written in terms of coordinates • total angular momentum of the electron is J = L + S • if L // S, then J=L+S • if L antiparallel S, then J=L-S • in quantum mechanics angular momentum is quantized and has magnitude given by • Electrons have spin s=1/2 and are fermions

  2. Spin-Orbit Effect • Consider a state with l=1 and s=1/2 . What are the possible values of j • j= 1-1/2=1/2 and j=1+1/2 = 3/2 • Quantum states with the same values of n and l but different j have small energy differences => fine structure due to spin

  3. Electron moving about a proton with angular momentum L Magnetic field due to apparent motion of charged proton is up When electron spin is up, its magnetic moment is down and energy is higher When electron spin is down, its magnetic moment is up and energy is lower E~ L.S

  4. Fine Structure n=2 l=0,1 => S or P 2S level has j=1/2 2P level has j=1/2 or 3/2 n=1 l=0 => 1S j=1/2 E~ L.S

  5. Periodic Table • Atoms with more than one electron cannot be solved exactly • assume the Z electrons do not interact with one another but rather only see the nucleus with charge +Z • the state of each electron is described by four quantum numbers • n, l ,m and ms • the fourth number, ms = 1/2 is the spin quantum number • l = 0, 1, 2, 3, 4, 5, … correspond to • s p d f g h • Pauli principle: no two electrons can have the same set of values of n, l ,m and ms • eg. Hydrogen (Z=1) has only one electron • lowest energy state has n=1 => l=0 => m=0 and ms = 1/2 • 1s state

  6. Periodic Table • Helium ( Z=2) has two electrons • we can put both in the n=1 energy state with l=0 and m=0 but with opposite spin • hence total spin is zero, total orbital angular momentum is zero, and total (spin + orbital) angular momentum is zero ==> j=0. • Denote as 1s2 • Lithium (Z=3) has three electrons • first two as in He but third must go into n=2 level =>l=0 or 1 • l=0 has lower energy • denote as 1s22s • Beryllium (Z=4) has 1s22s2electron configuration

  7. 1s 2s 2p Ne 3s 3p Periodic Table • An s state (l=0) can hold a maximum of 2 electrons • a p state (l=1) can hold a maximum of 6 electrons • a d state (l=2) can hold a maximum of 10 electrons • in general, 2(2l+1) states • neon (Z=10) has ten electrons • configuration is 1s22s22p6 • Hund’s rules • argon (Z=18) has eighteen electrons • configuration is 1s22s22p63s23p6 E~ L.S

  8. configurations Periodic table

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