Magnetism and Magnetic Materials
DTU (10313) – 10 ECTS
KU – 7.5 ECTS
Sub-atomic – pm-nm
Mesoscale – nm-mm
Macro – mm-mm
Module 1 – 01/02/2001 – Introduction
Context
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Magnetism is a physical phenomenon that
intrigued scientists and laymen alike for
centuries.
Some materials attract or repel each other,
depending on their orientation.
Experimentally, it became soon clear that
magnetism was related to the motion of
charges.
But how exactly? And why?
Classical physics gives us a basic
framework, but doesn’t help us much in
developing a coherent and comprehensive
bottom-up picture
The advent of QM provided some answers
Relativity provided some additional answers
How far can we go? Can we understand
what “magnetism” is, and how a magnet
works? Yes we can…
Magnetism:
A tangible macroscopic manifestation
of the quantum world
Meet your teachers
KL
CF
MB
MFH
BMA
JBH
LTK
JOB
Timeline
The basics
Tuesday, Feb 1
Introduction
Advanced topics
13:00-16:45
MB
Tuesday, Mar 8
13:00-16:45
Order and broken symmetry KL
Friday, Feb 4
08:15-12:00
Isolated magnetic moments MB
Tuesday, Feb 8
Crystal fields
13:00-16:45
MB
Friday, Feb 11
Interactions
08:15-12:00
MB
Tuesday, Feb 15
Magnetic order
13:00-16:45
MB
The mesoscale
Friday, Feb 18
Micromagnetism I
08:15-12:00
MB
Tuesday, Feb 22
Micromagnetism II
13:00-16:45
MB
Friday, Feb 25
Macroscopic magnets
08:15-12:00
MB
Experimental methods
and applications
Friday, Mar 11
08:15-12:00
Bulk measurements/dynamics KL
Tuesday, Mar 15
Nanoparticles I
13:00-16:45
CF
Friday, Mar 18
Nanoparticles II
08:15-12:00
CF
Advanced topics
Tuesday, Mar 29
Thermodynamics
GMR and spintronics
13:00-16:45
LTK
JBH
Friday, Apr 1
Magnetism in metals
08:15-12:00
BMA
Assigments deadlines
Feb 18
Mar 11
Mar 26
May 9
A1
A2
A3
PW (DTU only)
Evaluation
Tuesday, Mar 22
13:00-16:45
Magnetization measurements MFH
Hard disks
MFH
30/3 or 4/4
Oral exam (KU students)
Friday, Mar 25
08:15-12:00
Imaging and characterization CF
4/4-9/5
Oral exam (DTU students)
26/5 or 27/5
Oral exam (DTU students)
Workload
Your homework will be:
• Go through what you have learned in each Module,
and be prepared to present a “Flashback” at the
beginning of the next Module
• Carry out home-assignments (3 of them)
• Self-study the additional reading material given
throughout the course
Your group-work will be:
• Follow classroom exercise sessions with Jonas
• DTU only: project work
Your final exam will be:
• Evaluation of the 3 home-assignments
• Oral exam
• DTU only: evaluation of the written report on the
project work
This course will be successful if…
•Macroscopic magnets, how they work (MB)
• In depth (QM) explanation of bound currents (ODJ)
• I know why some things are magnetic (JJ)
• Know more about magnetic monopoles (ODJ)
• Lorentz transformations of B and E (MB)
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•.
•.
•.
Students’ feedback to be gathered in the classroom – 01/02/2011
Intended Learning Outcomes (ILO)
(for today’s module)
1.
2.
3.
4.
5.
6.
7.
8.
Describe the logic and structure of this course, and what will be learned
List the electron’s characteristics: charge, mass, spin, magnetic moment
Predict the main features of electron motion in presence of an applied field
Calculate the expression and values of Larmor and cyclotron frequencies
Define the canonical momentum, and explain its usefulness
Describe the connections between magnetism and i) QM, ii) Relativity
Write down simple spin Hamiltonians, and solve them in simple cases
Manipulate consistently spin states (spinors) with spin operators
Meet the electron
Mass: me=9.10938215(45) 10-31 Kg
Charge: e=-1.602176487(40) 10-19 C
Spin: 1/2
Magnetic moment: ~1 mB
Size: <10-22 m (from scattering)
Classical radius: 2.8 fm (little meaning)
Calculate the classical electron radius
Electron in motion and magnetic moment
m  I  dS
I 

S
L
v


I
ev
2 R
ˆ
m  L  lm B L
 
e
2me
mB 
eh
2me
m

Calculate the classical electron velocity for
some hypothetical l=1 state with R=a0.
Precession
Since magnetic moment is linked with angular momentum…
B, z

q
coil
Ferromagnetic rod
Einstein-De Haas
Barnett
E  m  B
dL
 mB 
dt
dm
 m  B
dt

 the Larmor precession
Calculate
frequency

Electron motion in applied field
The Lorentz force
z
y
-
F  q(E v  B)
x
B
B, z
y
x
Left or right?
Calculate the cyclotron frequency
More in general: canonical momentum
F  q(E v  B)
E  V  t A
dv
F  qV   t A  v    A  m
dt
v    A  (v  A)  (v  )A
d
mv  qA  qV  v  A
dt
p  mv  qA

p  p  qA

p  ih  qA
Classical
Quantum mechanical
To account for the influence of
a magnetic field in the motion
of a point charge, we “just”
need to replace the momentum
with the canonical momentum
in the Hamiltonian
p i  eA i (ri )
p 2i

T 

2m
2m
i
i
2
Connection with Quantum Mechanics
F  qv  B
Perpendicular to velocity
 F  dl   F  vdt  q  (v  B) vdt 0
No work = no change in energy = no magnetization
p i  eA i (ri )

H 
 other terms
No work
A classical system of charges
at thermal equilibrium has no
net magnetization.
The Bohr-van Leeuwen
theorem
2
2m
i
Z
  dr dp exp H ({r , p })
i
i
i
i
1
F   log Z

F 
1  log Z 
M      

B T ,V
  B T ,V
Z independent of B, ergo M=0
i
“It is interesting to realize that essentially everything that we
find in our studies of magnetism is a pure quantum effect. We
may be wondering where is the point where the h=/=0 makes
itself felt; after all, the classical and quantum Hamiltonians look
exactly the same! It can be shown […] that the appearance of a
finite equilibrium value of M can be traced back to the fact that
p and A do not commute. Another essential ingredient is the
electron spin, which is a purely quantum phenomenon.”
P. Fazekas, Lecture notes on electron correlation and magnetism
Connection with Relativity
B and E, two sides of the same
coin. No surprise we always
talk about Electromagnetism
as a single branch of physics.
(a)

(b)
(a)
qm0 v 2
F
zˆ
2 r
(b)
F  0 ??
Hint: Lorentz contraction

From: M. Fowler’s website, U. Virginia
Restore relativity and show that
the force experienced in (a) and
(b) is the same, although in (b)
the force is electric
Quantum mechanics of spin
Quantum numbers: n,l,ml,s,ms
Orbital angular momentum: l,ml;
l(l+1) is the eigenvalue of L2 (in hbar units)
ml is the projection of L along an axis of choice (e.g. Lz)
The resulting magnetic moment is m2=l(l+1)mB and mz=-mlmB
Spin angular momentum: s,ms;
s(s+1) is the eigenvalue of S2 (in hbar units)
ms is the projection of S along an axis of choice (i.e. Sz)
The resulting magnetic moment is m2=gs(s+1)mB and mz=-gmsmB
Zeeman splitting: E=gmsmBB
(remember the Stern-Gerlach experiment)
The g-factor (with a value very close to 2) is one difference between oam and spin
Another difference is that l can only be integer, while s may be half-integer
Also, spin obeys a rather unique algebra (spinors instead of “normal” vectors)
Other than that, they behave similarly.
But there are consequences… [exercise on EdH effect]
Pauli matrices and spin operators
0 1 
0 i 
1 0 
ˆ x  
ˆ y  
ˆ z  

 
 

1 0 
i 0 
0 1
s=1/2
a 
    a   b
b 
1
ˆS  Sˆx , Sˆy , Sˆz  1 
ˆ
ˆ x , 
ˆ y , 
ˆz


2
2
Commutators
Ladder operators
Sˆ , Sˆ  i
i
j
ˆ
ijk Sk
Sˆ  Sˆx  iSˆy

Sˆ 2 s, ms  s(s 1) s, ms

ˆ
Sz s, ms  ms s, ms
1 1
 
 
2 1
Sˆ , Sˆ  0
2
i
 ˆ
ˆ
ˆ
S
,
S
     2 Sz

Generic spin state
Sˆ s, ms  s(s 1)  ms (ms 1) s, ms 1

1 1 
 
 
2 1
Stern Gerlach
What is the final state?
Will the final beam split?
The simplest spin Hamiltonian: coupling of two spins
H  ASˆ a  Sˆ b
Sˆ tot  Sˆ a  Sˆ b
tot 2
a 2
b 2
ˆ
ˆ
ˆ
S   S   S   2Sˆ a  Sˆ b

Combining two s=1/2 particles gives an entity with s=0 or s=1.
The total S2 eigenvalue is then 0 or 2.
Hence, the energy levels are:
Possible basis:
A

2
2
2
A
ASˆ a  Sˆ b  Sˆ tot   Sˆ a   Sˆ b    43A
2
 4


 ,  ,  , 
Consider symmetry of wave function for Fermions

     

,
, 
Eigenstates:  ,
2
2
triplet and singlet
s0
s 1
Sneak peek
Paramagnetism
Diamagnetism
Hund’s rules
Wrapping up
•Magnetic moment
•Electron motion under an applied field
•Precession of magnetic moments
•Magnetism as a quantum-relativistic phenomenon
•Einstein-de Haas effect
•Orbital and spin angular momentum
•Spin behaves strangely
•Stern-Gerlach
•Coupling of spins
Next lecture: Friday February 4, 8:15, KU
Isolated magnetic moments (MB)