Spin•Electronic charge in motion - A current loop behaves as a magnetic dipole and has a magnetic moment. - Note the current direction is opposite to the electron velocity, and also the angular momentum direction is opposite to the magnetic moment•Two contributions to the electronic magnetic moment -An orbital magnetic moment due to orbital angular momentum -A spin magnetic moment due to electron spin

Electronic Magnetic Moments -Orbital Contribution

•The orbital motion of an electron around the nucleus may correspond to a current in a loop of wire having no resistance where m=(area of loop) (current)•Note that the angular momentum is continuous (not quantized), indicating a classical treatment of the problem

2)

/)2(()( 22

0

evr

vr

er

erm

rmvprp 0

00 )2

( pm

em Bm

ehm

40

Electronic Magnetic Moments-Spin Contribution

Spin? - Was postulated in 1925 by Paul Dirac in order to explain certain features of optical spectra of hot gases subjected to a magnetic field(Zeeman effect) and later theoretical confirmation in wave mechanics - The root cause of magnetism and an intrinsic property, together with charge and mass, of subatomic particles of fermions (eg.electrons, protons and neutrons) and bosons (photons, pions)

It was found, theoretically and experimentally, that the magnetic moment due to electron spin is equal to,m0

BS m

ehm

4

Electronic Magnetic Moments -m vs. p

For a given angular momentum, the spin gives twice the magnetic moment of orbit

00

0 )2

(

244

pm

ehp

m

eh

m

ehm

SS

S pm

eshp

m

eh

m

ehm )(

244

Electronic Magnetic Moments-Total Moments

The total magnetic moment per electron is the vector sum of the orbital and spin magnetic

moments

SStot pm

ep

m

emmm 2)

2()

2( 00

tottot pm

egm )

2(

The term ‘g’ is called the Lande splitting factor — g=2 for spin only components — g=1 for orbital only components

Electronic Magnetic Moments-Lande’Equation

)1(2

)1()1()1(1

JJ

LLSSJJg

•Orbital is quenching : L=0, J=S g=2•Spin =0 : S=0, J=L g=1

Schrodinger Equation

Schrodinger Equation

m=hml ml= l(l+1)

Schrodinger Equation

Electronic Magnetic Moments-Quantum Mechanical

The orbital angular momentum quantum number (l)

)2

(0 h

lp

The spin angular momentum quantum number (l)

)2

(h

spS

The spin angular momentum quantum number (l)

)2

)(()2

(h

slh

JpJ

Hund’s Rule Empirical rules which determine the occupancy

of the Available electronics within an atom

Used to calculate L, S and J for an unfilled shell 1. Maximum total S=max Sz with Sz=imsi Obeying the Pauli exclusion principle 2. Maximum total L=maxLz with Lz= imli Minimizing the Coulomb interaction energy 3. Spin-orbit interaction: L -S if less than half-filled J= L+S if more than half-filled

Hund’s Rule - Examples

•Sm3+ ion having 5 electrons in its 4f shell (n=4, l=3) S=5/2, L=5, J=L-S=5/2

ml 3 2 1 0 –1 –2 –3 ms

occupancy s •Fe2+ ion having 6 electrons in its 3d shell (n=3, l=2) S=2, L=2, J=L+S=4. Actually, however, S=2, L=0 (quenched) J=S=2 ml 2 1 0 –1 –2 2 1 0 -1 -2 ms

occupancy s

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

12

1

2

1

2

1

2

1

2

1