Van Leeuwen - Bohr Teorema

1. QUANTUM THEORY OF PARAMAGNETISM

2. DIAMAGNESTISM AND PARAMAGNETISM
DIAMAGNESTISM AND PARAMAGNETISM
 Magnetic susceptibility per unit volume
M
 M:magnetic moment per unit volume
(CGS) B:macroscopic magnetic field intensity
H
M


(SI)
H
B
M
0


 In general
B
 In general
paramagnetic material has a positive susceptibility
diamagnetic material has a negative susceptibility
diamagnetic material has a negative susceptibility

3.  Three principal sources for the magnetic moment of a
 Three principal sources for the magnetic moment of a
free atom:
 Electrons spin
 Electrons orbital angular momentum about the
nucleus
 Change in the orbital moment induced by an applied
 Change in the orbital moment induced by an applied
magnetic field
First to sources give paramagnetic contributions.
 First to sources give paramagnetic contributions.
 The last source gives a diamagnetic contribution
.
.

4. Bohr-van Leeuwen Theorem
M = γ L = 0 according to classical statistics.
→ magnetism obeys quantum statistics.
→ magnetism obeys quantum statistics.
c
diamagneti
of
lities
susceptibi
magnetic
stic
characteri
ce
subs
ic
paramagnet
and
c
diamagneti
of
lities
susceptibi
tan
Electronic structure Moment
Electronic structure Moment
H: 1s M  S
He: 1s2 M = 0
unfilled shell M  0
All filled shells M = 0

5. PARAMAGNETISM
PARAMAGNETISM
Occurrence of electronic paramagnetism:
• Atoms, molecules, & lattice defects with odd number of electrons ( S  0 ).
E.g., Free sodium atoms, gaseous NO, F centers in alkali halides,
organic free radicals such as C(C6H5)3.
• Free atoms & ions with partly filled inner shell (free or in solid),
• Free atoms & ions with partly filled inner shell (free or in solid),
E.g., Transition elements, ions isoelectronic with transition
elements,rare earth & actinide elements such as Mn2+, Gd3+, U4+.
• A few compounds with even number of electrons.
E.g., O2, organic biradicals.
• Metals
• Metals

6. QUANTUM THEORY OF PARAMAGNESTISM
QUANTUM THEORY OF PARAMAGNESTISM
 The central postulate of QM is that the energy of a
 The central postulate of QM is that the energy of a
system is discreat.
Improving the quantitative agreement ,between
 Improving the quantitative agreement ,between
theory and experiment.
In quantum case is restricted to certain
 In quantum case is restricted to certain
variables.

 In classical case is contious variables.

7. QUANTUM THEORY OR PARAMAGNETISM
QUANTUM THEORY OR PARAMAGNETISM
Magnetic moment of an atom or ion in
free space is given by
free space is given by
 
1

 J



 the total angular momentum
the sum of the orbital
 
1

 J







L
J
 the sum of the orbital
 the sum of the spin




S
L
 the radio of the magnetic moment to
the angular moment




8. factor
splitting
pic
spectrosco
or
factor
g
g
ratio
ic
magnetogyr
or
ratio
ic
gyromagnet



by
defined
is
it
factor
splitting
pic
spectrosco
or
factor
g
g 
J


 
 
 
3
2






J
g
g
B
B



  J


 
g
spin
electron 0023
.
2

     
    
is
atom
free
for
equation
lande
     
 
1 1 1
1
2 1
J J S S L L
g
J J
    
 

 
4


9.  
CGS
mc
e
magneton
bohr
B
2



  
5

 
SI
m
e
mc
B
2
2



  
5

.
electron
free
a
of
moment
magnetic
spin
the
to
equal
closely
is
It
field
magnetic
the
of
level
energy
The
U   
μ B J B
m g B


, 1, , 1,
J
m J J J J
    

 
6

number
quantum
azimuthal
m j 
field
magnetic
the
of
level
energy
The
, 1, , 1,
J
m J J J J
    

For a free electron, L = 0, S = ½ , g = 2,
→ mJ =  ½ , U =  μB B.

 
   
T
B
T
B
T
B
N
N
/
exp
/
exp
/
exp
1





  
7


   
 
   
T
B
T
B
T
B
N
N
T
B
T
B
N
/
exp
/
exp
/
exp
/
exp
/
exp
2








 

10. 2
1 N
N
N
is
atoms
of
number
total
the

  
8

The upper state along the field
direction is –μ.
2
1 N
N
N 

direction is –μ.
The lower state along the field
The lower state along the field
direction is μ.
The magnetic moment is
proportional to the difference
between the two curves.
between the two curves.

11. B
x
k T


 
   
 
T
B
N
T
B
T
B
T
B
N
N
/
exp
/
exp
/
exp
/
exp
2
1








B
x
k T

x
N e x
N e

 
   
T
B
T
B
T
B
N
N
/
exp
/
exp
/
exp
2





T
K


x
x x
N e
N e e





x
x x
N e
N e e





T
KB


volume
unit
per
atoms
N
for
ion
magnetizat
resulant
the
 
M N N 
 
  

e
e
e
e
e
e
N
N
N
x
x
x
x
x
x


























volume
unit
per
atoms
N
for
ion
magnetizat
resulant
the
x x
x x
e e
N
e e






tanh
N x



N
e
e
e
e
M
e
e
e
e
N
x
x
x
x






 

 
 
9

x x
N
e e
 


tanh
N x

  
9


12. 2
N B
M N x
k T

 










T
k
B
N
x
N
B



High T ( x << 1 ):
B
k T
gJ B

Curie-Brillouin law:
 
B J
M N g J B x


B
B
gJ B
x
k T


  x
x
B J
3

 
 
2 1
2 1 1
2 2 2 2
J
J x
J x
B x ctnh ctnh
J J J J

 
  
 
   
 
 
Brillouin function:
B
gJ
NgJ
M
x
NgJ
M
B
B
3





2 2 2 2
J J J J
 
 
3
1 x x
ctn h x     
High T ( x << 1 ): x

 
T
k
B
g
J
NJ
M
T
k
NgJ
M
B
B
B
B
B
3
1
3
2
2





3 4 5
ctn h x
x
    
  2 2
1
N J J g
M 

High T ( x << 1 ):
3

 
T
k
g
J
NJ
B
M
T
k
B
B
B
3
1
3
2
2



  2 2
1
3
B
B
N J J g
M
B k T



 

13.   2 2
1 B
N J J g
M 


 
2 2
B
N p 

 
3
B
B
M
B k T
  
B
k
n
C
3
2


law
CURIE
3
B
B
N p
k T

 B
k
3
 

C


 
 
10

T
C


 
1
p g N J J
  = effective number of
Bohr magnetons

14. .  Potassium chromium
 Ferric ammonium alum
 Ferric ammonium alum
 Gadolinium sulfate
octahydrate.
octahydrate.
 Over 99.5% magnetic
saturation is achieved at
1.3k and about 50,000
1.3k and about 50,000
gauss.

15.  Plot of for a
gadolinium salt,
T
v s

1
gadolinium salt,

O
H
SO
H
C
Gd 2
3
4
5
2 9
.
)
(
 The straight line is the curie
law.

16. RARE EARTH IONS
Lanthanide
Lanthanide
contraction
1.11A
ri = 0.94A
ri = 0.94A
4f radius ~
0.3A

17. HUND’S RULES
For filled shells, spin orbit couplings do not change order of levels.
Hund’s rule ( L-S coupling scheme ):
Outer shell electrons of an atom in its ground state should assume
1.Maximum value of S allowed by exclusion principle.
For filled shells, spin orbit couplings do not change order of levels.
1.Maximum value of S allowed by exclusion principle.
2.Maximum value of L compatible with (1).
3.J = | L−S | for less than half-filled shells.
J = L + S for more than half-filled shells.
Cases:
1. Parallel spins have lower Coulomb energy.
2. e’s meet less frequently if orbiting in same direction (parallel Ls).

3. Spin orbit coupling lowers energy for LS < 0.
Mn2+: 3d 5 (1) → S = 5/2 exclusion principle → L = 2+1+0−1−2 = 0
Ce3+: 4 f 1 L = 3, S = ½ (3) → J = | 3− ½ | = 5/2
Pr3+: 4 f 2 (1) → S = 1
2
5/2
F
Pr3+: 4 f 2 (1) → S = 1
(2) → L = 3+2 = 5 (3) → J = | 5− 1 | = 4
3
4
H