Magnetism and Magnetic Materials
DTU (10313) – 10 ECTS
KU – 7.5 ECTS
Module 2
04/02/2001
Isolated magnetic moments
Sub-atomic – pm-nm
Flashback
What have we learned last time? You tell us…
Intended Learning Outcomes (ILO)
(for today’s module)
1.
2.
3.
4.
5.
6.
7.
Establish the link between g-factor and measured angle in the EdH experiment
Verify that magnetism in Fe is mostly from spin angular momentum
Describe what ”Diamagnetism” is
Estimate the diamagnetic response of He
Describe what ”Paramagnetism” is
Explain why some atoms are diamagnetic and why some are paramagnetic
List the three Hund’s rules and apply them to a few atoms
Einstein de Haas
Consider the following scenario:
A soft iron bar magnet placed in a
torsion balance, initially
demagnetized, saturates under the
influence of the magnetic field
generated by a coil.
The angular momentum J associated
with the acquired moment MV sets
the magnet in motion. A laser beam is
used to measure an equilibrium
angle.
Can you measure the g-factor from
this experiment, and assess whether
magnetism in Fe is from spin or oam?
Parameters:
R=4 cm, H=6 cm, k=10-11 Nm
r=7874 kg/m3, MS=1707 A/m
Hamiltonian of an atom in a magnetic field
E  g B Bms   B B
Spin (for each electron)
hL   ri  p i
OAM
The Hamiltonian of an atom in
a magnetic field has two main
components: the
“diamagnetic” term, and the
“paramagnetic” term
i

 pi2
ˆ
H 0   Vi 
2m

i 
Initial Hamiltonian of the atom

p  eA(r )2
i
i
Hˆ   
Vi  g B B  S
2m

i 
Hamiltonian in presence of a magnetic field
2
e
2
Hˆ  Hˆ 0   B (L  gS)  B 
(B

r
)

i
8me i
Explicit version of the Hamiltonian
Is the Hamiltonian gauge invariant?
Magnetic susceptibility
B  0 (M H)  0 (M Ha  Hd )
M  H
How strongly a material
magnetizes in response to an
external field
Note:
-Difference between intrinsic and
experimental susceptibility (demag
factor involved)
-Other definitions of susceptibility:
--molar (Vm)
--mass (/r)
Focus on the initial magnetization
Diamagnetism
(B  ri )2  B2 (xi2  yi2 )
2 2
e2 B 2
e
B
2
2
2
E0 
0
(x

y
)
0

0
r


i
i
i 0
8me i
12me i
1 F
N E0
N e2 B
2
M 


0
r

i 0
V B
V B
V 6me i
N a  0 e2
 
0 ri2 0

Vm 6me i
There’s another contribution:
Landau diamagnetism (will be
dealt with later)
All atoms and ions (except
perhaps H+), have nonzero
diamagnetic responses to an
external field. It is, however,
pretty small, and often masked
by the paramagnetic response
of atoms with a net moment
(unpaired electrons)
Try this out with He. Does it match?
Diamagnetic and paramagnetic susceptibilities
Paramagnetism – semiclassical
J  LS
Total angular momentum
E( )  B cos
Energy of the moment at an angle
What is the expected averaged moment along the field axis?
z
 cos  exp B cos  )sin  d


 exp B cos )sin  d
z
M
1
B

 coth y   L(y), y 
MS

y
k BT
Langevin function
n0 2

3k BT
n is the volume density of moments
If J is not zero, then you have
paramagnetism
The J=1/2 case
Two spins, J=1/2, just two states (parallel or AP), to average statistically
Several similarities
 B B 
 B exp  B B)   B exp  B B)
 z  g B m J 
  B tanh

exp  B B)  exp  B B)
k BT 
 B B   B B
mJ
M

 tanh y)  tanh

MS
J
k BT  k BT
n0 2B

k BT
Estimate the paramagnetic susceptibility
Generic J and the Brillouin function
J
m
mJ 
J
exp m J x)
m J  J
J
 exp m x)
1 Z
gJ  B B

, x
Z x
k BT
J
m J  J
 ln Z
M  ngJ  B m J  nk BT
B
sinh (2J 1) 2x 
Z
sinh 2x 
M  ngJ  B JBJ (y)  M S BJ (y)

2J 1  1
 y 
2J 1
gJ  B JB
BJ (y) 
coth
y coth , y 
 2J  2J
2J 
2J
k BT
Lande’ g-value and effective moment
J 1
BJ (y) 
y
3J
J=1/2
J=5
2
M n0eff


MS
3k BT
eff  gJ  B J(J 1)
3 S(S 1)  L(L 1)
gJ  
2
2J(J 1)
J=3/2
Curie law: =CC/T
Van Vleck paramagnetism
If J=0, in principle there is no paramagnetic term.
However, if we go second-order, and consider the
possibility of excited states (off-diagonal matrix
terms) with nonzero J, then we have:
E0  
n
  20 B2
0  B (L  gS)  B n
E0  En
N

V n
Another contribution to the
paramagnetic susceptibility
(there’s one more…mobile
electrons – Pauli)
2
0 (Lz  gSz ) n
En  E0
2
Which is positive (para), and T-independent.
Why is it T-indepenent?? And
why was the Langevin term Tdependent instead?
John H. van Vleck, Nobel prize lecture
The multi-electron atom and the Hund’s rules
2 
2
 pi2
Ze
Ze
Hˆ   
 
2m 4  0 ri  i j 4  0 | ri  ri |
i 
With many electrons, it gets
messy. How do electrons
“choose” which state to occupy?
(1) Arrange the electronic wave function so as to maximize S. In
this way, the Coulomb energy is minimized because of the Pauli
exclusion principle, which prevents electrons with parallel spins
being in the same place, and this reduces Coulomb repulsion.
(2) The next step is to maximize L. This also minimizes the energy
and can be understood by imagining that electrons in orbits
rotating in the same direction can avoid each other more
effectively.
(3) Finally, the value of J is found using J=|L-S| if the shell is less
than half-filled, and J=L+S is the shell is more than half-filled.
This third rule arises from an attempt to minimize the spinorbit energy.
2S+1L
J
Find the electronic structure of Fe3+,
Ni2+, Nd3+, Dy3+, and determine
their spin configuration
Sneak peek
Wrapping up
•Einstein de Haas: how to distinguish L and S
•Diamagnetism: universal, temperature independent
•Paramagnetism: J.neq.0, temperature dependent
•Curie law and implications
•Lande’ g-factor
•Hund’s rules introduction
Next lecture: Tuesday February 8, 13:15, DTU (?)
Crystal fields (MB)