Topics in Magnetism
I. Definitions and Atomic Sources
Anne Reilly
Department of Physics
College of William and Mary
After reviewing this lecture, you should be familiar with:
1. Definitions of M, H and B
2. Magnetic units
3. Atomic sources of magnetism
4. Paramagnetic and Diamagnetic responses
Material from this lecture is taken from Physics of Magnetism by Chikazumi
and Solid State Physics by Ashcroft and Mermin (Chp. 31)
Fundamental Definitions
Magnets have two poles (north and south)
Poles exert a force on each other
N
+m1
S
-m1
N
S
+m2
-m2
Definition: magnetic pole m (SI units:Weber, Wb=m2kg/s2A)
Magnetic force (N):
m1m2
F
2
40 r
0=4 x 10-7 H/m
Fundamental Definitions
Electric current in wire exerts a force on a magnetic pole
I
H
N
S
+m
-m
Definition: magnetic field H (SI units: A/m)
Magnetic force (N):
F  mH
Field from Solenoid:
H  nI
n = # turns/m, I = current
Fundamental Definitions
What happens to a magnet in a magnetic field?
+mH
N
+mH
-mH
l

S
S
H
N
H
-mH
Magnetic torque:
  mlH sin 
Translational force ONLY if there is non-uniform H (gradient):
H
Fx  ml
x
Fundamental Definitions
+mH
N
l
S

H
-mH
Definition: magnetic moment M = ml (SI units: Wb m)
(dipole moment)
 
Magnetic torque:   M  H
Magnetic Energy:
 
U  M  H
Fundamental Definitions
Magnetic materials have a density of magnetic moments
Definition: Magnetization M=NM
(SI units: Wb/m2 or Tesla T)
N
M
N=moments per unit volume
S
Fundamental Definitions
To measure magnetization, use induction (vibrating sample magnetometry)
H
M
B
V
dB
Induced Voltage V   NA
dt
Definition: magnetic flux density B (SI units: Tesla T)
 

B  M  0 H
Fundamental Definitions
Magnetization in materials is proportional to applied field H
 

B  M  0 H


M  H
Definition: magnetic susceptibility  (SI units: H/m)


B    0 H  H
Definition: magnetic permeability  (SI units: H/m)
Fundamental Definitions:Review
H
(externally applied)
M
B
 

B  M  0 H
M = magnetization (T)
H = magnetic field strength (A/m)
B = magnetic flux density (also called field) (T)
Fundamental Definitions:Review
Note that sometimes magnetic flux density is defined as:



B  0 M   0 H
In this case, the units of M are A/m
Gaussian System of Units:
Common system prior to 1980’s. Defined by magnetic poles.

 
B  4M  H
Oersted (Oe)
CGS unit of magnetic field (H). The Oersted is defined to be the field strength
in a vacuum at a distance 1 cm from a unit magnetic pole.
Gauss (G)
CGS unit of magnetic flux density (B). A field of one Gauss exerts a force on a
conductor of 0.1 dyne/A cm.
Electromagnetic Unit (emu)
CGS unit of magnetic dipole moment (M) equal to 1.256637 x 10-5 Oe.
emu/cm3 or emu/cc
CGS unit of magnetization (M) In SI units, one emu/cm3 can be interpreted
either as 1.256637 mT as a unit of excess magnetic induction, or as 1000
A/m as a unit of magnetic dipole moment per unit volume.
Unit Conversion:
Gaussian unit
(cgs-emu)
Conversion
(SI/cgs)
SI unit
B
Gauss (G)
x 10-4 =
T or Wb/m2
H
Oersted (Oe)
x 103/4 
A/m
M
emu/cm3
x 103 =
x 4/10 
A/m
mT
Note: In free space (M=0), 1 G = 1 Oe
Source of Magnetic Moment: Moving Electric Charge (Current)
Atomic Magnetism arises from electron angular momentum and spin
ML
I
ML= orbital magnetic moment
= IA=1/2 eL/me
r
L
S
v
e-
Source of Magnetic Moment: Moving Electric Charge (Current)
Atomic Magnetism arises from electron angular momentum and spin
ML
I
r
S
L
Atomic magnetic moment:
Bohr magneton
e
B 
2m
ML= orbital magnetic moment
= IA=1/2 eL/me
v
e-



M   B L  g e  B S
Angular momentum vector
Spin vector
Gyromagnetic ratio ge~ 2
Source of Magnetic Moment: Moving Electric Charge (Current)
Multi-electron atoms: total magnetic moment determined
by total J, L and S
Hund’s rules: electrons fill shells such that
1. Largest total S is achieved
2. Largest total L is achieved
3. J=|L-S| (minimum) in shells less than half full and J=|L+S| (maximum) in shells
more than half full.
Example:
m = -2 -1 0
1
3d
n =4
Iron (Fe)
26
2 (lz)
3p
n =3
4s
3s
2p
n =2
2s
n =1
1s
Maximum values:
L=2+2+1+0-1-2=2
S=4/2 = 2
J=4
Source of Magnetic Moment: Quantum Derivation
(for multi-atom systems)
Magnetization Defined to be:
1 E ( H )
M (H )  
(at T=0)
V H
 E n / k BT
M
(
H
)
e
n n
(at T>0)
M (H ,T ) 
 E n / k BT
e

where
n
1 En ( H )
M n (H )  
V H
Source of Magnetic Moment: Quantum Derivation
(for multi-atom systems)
In terms of Helmholtz free energy F:
1 F
M (H )  
V H
M
N 2F


H
V H 2
To calculate magnetic properites, consider Hamiltonian
in magnetic field and find energy
Source of Magnetic Moment: Quantum Derivation
Write Hamiltonian for atomic electrons in a Magnetic Field (ignore Vatom)
 
 e   2
1
H
( pi  Ari )  g e  B H  S

2m i
c

1 
A   ri  H = Vector potential
2
Consider first term:


2
2



2 e 
 e 2
 e  
 pi  Ari   pi  pi  A  A  pi    A
c
c


c
Source of Magnetic Moment: Quantum Derivation
With
Consider first term:


H  Hzˆ

2
2



2 e 
 e 2
 e  
 pi  Ari   pi  pi  A  A  pi    A
c
c


c
e   
e2   2
ri  p  H
r H
Using these relationships:
c
2 i
   
c
A B  B  A
2
  
  
e
2
2
A  ( B  C )  ( A  B)  C
x

y
i
i H
 
2
4c
L   ri  pi
 
e

i
LH
c


2
2





1
e
1
e


2
2
2
2


p

A
r

p


L

H

H
x

y
 i



i 
i
B
i
i
2
2m i 
c
2
m
8
mc

i
i


Hamiltonian in a Magnetic Field
 
 

1
e2
2
2
2
H
pi  B L  H 
H  xi  yi  ge B H  S

2
2m i
8mc
i

H0

H’
Magnetic field dependent terms considered as perturbation:
2
 

e
2
2
2
H'   B ( L  g e S )  H 
H
(
x

y

i
i )
2
8mc
i
Magnetic field as a perturbation:
E  En  En
Energy (En is ground state energy)
En  n H' n  
n ' n
n H' n'
En  En '
2
Magnetic field as a perturbation:
E  En  En
Energy (En is ground state energy)
En  n H' n  
2
n H' n'
En  En '
n ' n
paramagnetism



En   B H  n L  g e S n  
n ' n
e2
2
2
2
H
n
(
x

y

i
i ) n
2
8mc
i
diamagnetism



 
n  B H  L  g e S n'
En  En '
2

In ground state atoms or ions with closed (filled) shells:
J 0 L0 S 0 0
0



En   B H  n L  g e S n  

0


 
n  B H  L  g e S n'
n ' n
2
En  En '

e2
2
2
2
H
n
(
x

y

i
i ) n
2
8mc
i
Larmor Diamagnetism is only response:
e2
2
2
E0 
H
0
r

i 0
2
12mc
i
let x  y  z 
N  2 E0
e2 N
2
 


0
r

i 0
2
2
V H
6mc V
i
r
3
Summary of magnetic responses:
Diamagnetic
(by Lenz’s Law,
opposes H)
paramagnetic
(aligns with H)
M
M
HB
HBB
<<1, negative
<<1, positive
M=magnetization
from http://www.geo.umn.edu
Summary of magnetic responses:
diamagnetic
(opposes H)
paramagnetic
(aligns with H)
ferromagnetic
(even without H!)
M
M
M
HB
HBB
BB
<<1, negative
<<1, positive
>1, positive