Magnetism and Magnetic Materials

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Magnetism and Magnetic Materials DTU (10313)
10 ECTS KU 7.5 ECTS
Module 2 04/02/2001 Isolated magnetic moments
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Flashback
What have we learned last time? You tell us
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Intended Learning Outcomes (ILO)
(for todays module)

1. Establish the link between g-factor and measured
   angle in the EdH experiment
2. Verify that magnetism in Fe is mostly from spin
   angular momentum
3. Describe what Diamagnetism is
4. Estimate the diamagnetic response of He
5. Describe what Paramagnetism is
6. Explain why some atoms are diamagnetic and why
   some are paramagnetic
7. List the three Hunds rules and apply them to a
   few atoms

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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 R4 cm, H6 cm, k10-11
Nm r7874 kg/m3, MS1707 A/m
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Hamiltonian of an atom in a magnetic field
The Hamiltonian of an atom in a magnetic field
has two main components the diamagnetic term,
and the paramagnetic term
Spin (for each electron)
OAM
Initial Hamiltonian of the atom
Hamiltonian in presence of a magnetic field
Explicit version of the Hamiltonian
Is the Hamiltonian gauge invariant?
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Magnetic susceptibility
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 (cVm) --mass (c/r)
Focus on the initial magnetization
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Diamagnetism
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?
Theres another contribution Landau diamagnetism
(will be dealt with later)
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Diamagnetic and paramagnetic susceptibilities
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Paramagnetism semiclassical
If J is not zero, then you have paramagnetism
Total angular momentum
Energy of the moment at an angle
What is the expected averaged moment along the
field axis?
Langevin function
n is the volume density of moments
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The J1/2 case
Several similarities
Two spins, J1/2, just two states (parallel or
AP), to average statistically
Estimate the paramagnetic susceptibility
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Generic J and the Brillouin function
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Lande g-value and effective moment
J1/2
J3/2
J5
Curie law cCC/T
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Van Vleck paramagnetism
Another contribution to the paramagnetic
susceptibility (theres one moremobile electrons
Pauli)
If J0, 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
Which is positive (para), and T-independent.
Why is it T-indepenent?? And why was the Langevin
term T-dependent instead?
John H. van Vleck, Nobel prize lecture
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The multi-electron atom and the Hunds rules
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 JL-S if
   the shell is less than half-filled, and JLS is
   the shell is more than half-filled. This third
   rule arises from an attempt to minimize the
   spin-orbit energy.

Find the electronic structure of Fe3, Ni2,
Nd3, Dy3, and determine their spin configuration
2S1LJ
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Sneak peek
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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
• Hunds rules introduction

Next lecture Tuesday February 8, 1315, DTU
(?) Crystal fields (MB)