Transcript Spin-Orbit Effect - University of Manitoba

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
J
j( j 1)
where j  l  s, l  s 1,... l  s
for l  0, j  s
•Electrons have spin s=1/2 and are fermions
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
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
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
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
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
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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
1s22s2 electron configuration
Periodic Table
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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
1s
2s
2p
• argon (Z=18) has eighteen electrons
• configuration is 1s22s22p63s23p6
Ne
3s
E~ L.S
3p
configurations
Periodic
table