PHY 712 Electrodynamics
10-10:50 AM MWF Olin 107
(Make up lecture 2/17/2014 at 9 AM)
Plan for Lecture 13:
Continue reading Chapter 5
1. Hyperfine interaction
2. Macroscopic magnetization density M
3. H field and its relation to B
4. Magnetic boundary values
02/17/2014
PHY 712 Spring 2014 -- Lecture 13
1
02/17/2014
PHY 712 Spring 2014 -- Lecture 13
2
Hyperfine interaction energy:
Interactions between magnetic dipoles
Sources of magnetic dipoles and other sources of
magnetism in an atom:
• Intrinsic magnetic moment of a nucleus  N
• Intrinsic magnetic moment of an electron e
• Magnetic field due to electron current J e (r )
Interaction energy between a magnetic dipole m and a
r
magnetic field B:
E  m  B
int
e
N
Eint   e  B  N   N  B Je (0)

0  3rˆ (μ N  rˆ )  μ N 8
3
Bμ (r ) 

μ N  (r ) 

3
N
4 
r
3

In this case:
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PHY 712 Spring 2014 -- Lecture 13
3
Hyperfine interaction energy: -- continued
Eint   e  B  N   N  B Je (0)
Evaluation of the magnetic field at the nucleus due to the
electron current density:
The vector potential associated with an electron in a
bound state of an atom as described by a quantum
mechanical wavefunction  nlml (r ) can be written:
2
 
ˆ
0 e
z

r
nlm (r )
3 
A J (r )  
ml  d r
4 me
| r  r  | r  2 sin 2  
We want to evaluate the magnetic field B    A

l
e
in the vicinity of the nucleus (r  0).
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PHY 712 Spring 2014 -- Lecture 13
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Hyperfine interaction energy: -- continued
B J e (0)    A J e
2
0 e
zˆ  r  nlm (r )
3 

ml  d r  
4 me
| r  r  | r  2 sin 2  


l
r 0
2
ˆ  r )  nlm (r )
0 e
3  (r  r )  ( z
Bo (r ) 
ml  d r
4 me
| r  r  |3
r  2 sin 2  



r 0
l
r 0
2
ˆ  r )  nlm (r )
0 e
3  r  (z
B o ( 0)  
ml  d r
4 me
r 3
r  2 sin 2  



l
rˆ   (zˆ  rˆ  )  zˆ (1  cos 2 θ )  xˆ cos   sin   cos    yˆ cos   sin   sin   ).
2
 nlm (r )
ˆ r sin   nlm (r )
0 e
0 e
3  z
3 
B o ( 0)  
ml  d r

ml zˆ  d r
3
2
2 
4 me
r
r sin 
4 me
r 3
2
2


l

l
0 e
1
ˆ
=
ml z 3
4 me
r'
02/17/2014
PHY 712 Spring 2014 -- Lecture 13
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2
Hyperfine interaction energy: -- continued
Eint  H HF   e  B  N   N  B Je (0)
Putting all of the terms together:
H HF
0  3(μ N  rˆ )(μ e  rˆ )  μ N  μ e 8
e L  μN 
3


μ N  μ e (r ) 

.
3
3
4 
r
3
me
r

In this expression the brackets
indicate evaluating
the expectation value relative to the electronic state.
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PHY 712 Spring 2014 -- Lecture 13
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Macroscopic dipolar effects -Magnetic dipole moment
1 3
m   d r r  J r 
2
Note that the intrinsic spin of elementary particles is
associated with a magnetic dipole moment, but we often
do not have a detailed knowledge of J(r).
Vector potential for magnetic dipole moment
0 m  r
Ar  
3
4 r
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PHY 712 Spring 2014 -- Lecture 13
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Macroscopic magnetization
3
i
i
M r    m  r  ri 
Vector potential due to “free” current Jfree(r) and
macroscopic magnetization M(r). Note: the designation
Jfree(r) implies that this current does not also contribute
to the magnetization density.




J
r
'




0
M
r
'

r

r
'
free
3 

Ar  
d
r
'

3



4
r

r
'
r

r
'


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PHY 712 Spring 2014 -- Lecture 13
8
Vector potential contributions from macroscopic
magnetization -- continued
 J free r'  M r'  r  r ' 
0
3 


'
r
d
Ar  
3

 r  r'
4 
'
r

r


Note that :
M r'  r  r '
r  r'
3
1
 M r'  '
r  r'
 M r'   'M r' 

 '

r  r'
'
r

r


J free r'   'M r' 
0
3
d r'
 Ar  

r  r'
4
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PHY 712 Spring 2014 -- Lecture 13
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Vector potential contributions from macroscopic
magnetization -- continued
J free r'   'M r' 
0
3
Ar  
d r'

4
r  r'
Note that for the case that   A  0 :
  Br       Ar    2 Ar 
0
3
3
r  r 'J free r'   'Mr' 


d
r
'
4


4
  0 J free r     M r 
   Br    0 M r    0 J free r 
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PHY 712 Spring 2014 -- Lecture 13
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Magnetic field contributions
  Br    0 M r    0 J free r 
Define the magnetic flux density :
 0 H (r )  Br    0 M r 
   H (r ) J free r 
  Br    0 M r    0 J free r 
Note that Br   the magnetic flux density
Define H (r )  the magnetic field
 0 H (r )  Br    0 M r 
   H (r )  J free r 
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PHY 712 Spring 2014 -- Lecture 13
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Summary of equations of magnetostatics :
  Br    0 J total r 
  H (r ) J free r 
Br    0 H (r )  M r 
  Br   0
For the case that J
r   0 :
free
  H (r )  0
n̂
2
  B r   0
02/17/2014
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PHY 712 Spring 2014 -- Lecture 13
12
For the case that J free r  :
  H (r )  0
  Br   0
02/17/2014
At boundary :
H1  nˆ  H 2  nˆ
1
B1  nˆ  B 2  nˆ
2
PHY 712 Spring 2014 -- Lecture 13
n̂
13
Example magnetostatic boundary value problem
M0
M 0 zˆ r  a
M (r )  
ra
 0
  H (r )  0
 H (r )  ΦH (r )
Br    0 H (r )  M r 
  Br   0   0  H (r )  M r 
  2ΦH (r )    M r 
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PHY 712 Spring 2014 -- Lecture 13
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Example magnetostatic boundary value problem -- continued
M 0 zˆ r  a
M (r )  
M0
ra
 0
 2ΦH (r )    M r 
1
3  'M r '
 ΦH (r )  
d r'

4
r  r'
  M r ' 

1
1
3

  M r '  '

d
r
'

'






4
r  r ' 
  r  r ' 
1
3 M r '

   d r'
4
r  r'
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PHY 712 Spring 2014 -- Lecture 13
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Example magnetostatic boundary value problem -- continued
M0
M 0 zˆ r  a
1
3 M r '
M (r )  
ΦH (r )  
   d r'
ra
4
r  r'
 0
For this example:
a
M0  
1
2
ΦH (r )  
 4  r ' dr ' 
4 z  0
r 
For r  a:
  a2 r 2  M 0 z
ΦH (r )   M 0    
z  2 6 
3
For r  a:
  a3  M 0 a3 z
ΦH (r )   M 0   
z  3r 
3r 3
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PHY 712 Spring 2014 -- Lecture 13
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Example magnetostatic boundary value problem -- continued
M0
M 0 zˆ r  a
M (r )  
ra
 0
M0z
For r  a : ΦH (r ) 
3
M 0a3 z
For r  a : ΦH (r ) 
3r 3
B(r )   0 H (r )  M (r ) 
For r  a :
For r  a :
02/17/2014
M0
H (r )  ΦH (r )  
zˆ
3
M 0 a 3  zˆ 3 zr 
H (r )  ΦH (r )  
 3 5 
3 r
r 
M 0 zˆ
2 M 0 zˆ
H (r )  
B(r )   0
3
3
M 0 a 3  zˆ 3 zr 
H (r )  
 3 5 
3 r
r 
M 0 a 3  zˆ 3 zr 
B(r )    0
 3 5 
3 r
r 
PHY 712 Spring 2014 -- Lecture 13
17
Check boundary values:
For r  a :
For r  a :
For r  a :
For r  a :
M 0 zˆ
M0
ˆ
ˆ
H (r )  
H ( ar )  r  
zˆ  rˆ
3
3
M 0 a 3  zˆ 3 zr 
H (r )  
 3 5 
3 r
r 
M 0 a 3 zˆ  rˆ
H (arˆ )  rˆ  
3
a3
2 M 0 zˆ
2M 0
ˆ
ˆ
B(r )   0
B ( ar )  r   0
zˆ  rˆ
3
3
M 0 a 3  zˆ 3 zr 
B(r )    0
 3 5 
3 r
r 
 1 3a 2 
M 0a3
B(arˆ )  rˆ   μ0
zˆ  rˆ  3  5 
3
a 
a
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PHY 712 Spring 2014 -- Lecture 13
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Variation; magnetic sphere plus external field B0
M0
By superposition :
B0
M 0
M (r )  
 0
ra
ra
For r  a :
2
Br   B 0   0 M 0
3
1
1
H r  
B0  M 0
0
3
Br   2  0 H r   3B 0
For an isotropic " paramagnetic" material, Br   H r 
3    0 

B 0
M0 
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 0    2  0 PHY 712 Spring 2014 -- Lecture 13
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Summary of equations of magnetostatics :
  Br    0 J total r 
  H (r ) J free r 
Br    0 H (r )  M r 
  Br   0
For the case that J free r  :
  H (r )  0
  Br   0
02/17/2014
At boundary :
H1  nˆ  H 2  nˆ
1
B1  nˆ  B 2  nˆ
2
PHY 712 Spring 2014 -- Lecture 13
n̂
20
Magnetism in materials
Br    0 H (r )  M r 
For materials with linear magnetism :
B  H
   0  paramagnetic material
   0  diamagnetic material
For ferromagnetic, antiferromagnetic materials
B  f H  (with hysteresis)
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PHY 712 Spring 2014 -- Lecture 13
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Example: permalloy, mumetal /0 ~ 104
B0
Spherical shell a < r < b :

a
0
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b
PHY 712 Spring 2014 -- Lecture 13
0
22
Example: permalloy, mumetal /0 ~ 104 -- continued
B0

a
0 b
For this case :
  H (r )  0
  Br   0
0
Br   H (r )
Continuity at boundaries :
H  nˆ  continuous
B  nˆ  continuous
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PHY 712 Spring 2014 -- Lecture 13
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Example: permalloy, mumetal /0 ~ 104 -- continued
B0
Let :

a
0 b
H (r )   H (r )
  Br   0
For 0  r  a
0
  2  H (r )  0
 H (r )    l r l Pl cos  
l
For a  r  b
For r  b
02/17/2014
 l l 
 H (r )     l r  l 1  Pl cos  
r 
l 
B0
l
 H (r )   r cos    l 1 Pl cos  
0
l r
PHY 712 Spring 2014 -- Lecture 13
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Example: permalloy, mumetal /0 ~ 104 -- continued
Applying boundary conditions
(only l  1 terms contribute) :
B0

a
0 b
At r  a
 
1 
1   1  2 3 
0 
a 
At r  b
γ1
aδ1  aβ1  2
a
B0
 
1 
1
 1  2 3     2 3
0 
b 
0
b
B0 α1
γ1
bβ1  2  b  2
b
μ0 b
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PHY 712 Spring 2014 -- Lecture 13
0
25
Example: permalloy, mumetal /0 ~ 104 -- continued
B0

a
0 b
When the dust clears :
0

 B0
 9 / 0

1  
3
2 
 2  /  0  1 /  0  2   2a / b   /  0  1   0
1   9 / 2 B0 



3
 /  0  1  a / b   0 

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
PHY 712 Spring 2014 -- Lecture 13
26
Energy associated with magnetic fields
Note : We previously used without proof - the force on a magnetic dipole m in an external B field is :
F  m  B 
This implies that energy associated with aligning a
magnetic dipole m in an external B field is given by :
U  m  B
Macroscopic energies - 1 3
It can be shown that : WB   d r Br   H r 
2
1 3
In analogy to :
WE   d r Er   Dr 
2
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