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

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

n=1 l=0 => 1S j=1/2 n=2 l=0,1 => S or P 2S level has j=1/2 2P level has j=1/2 or 3/2 Fine Structure 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 m s the fourth number, m s =  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 m s eg. Hydrogen (Z=1) has only one electron lowest energy state has n=1 => l=0 => m=0 and m s =  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 1s 2 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 1s 2 2s Beryllium (Z=4) has 1s 2 2s 2 electron configuration

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 1s 2 2s 2 2p 6 Hund’s rules argon (Z=18) has eighteen electrons configuration is 1s 2 2s 2 2p 6 3s 2 3p 6 1s2s2p Ne3s3p E~ L.S

Periodic table configurations