Magnetism and Magnetic Materials

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Magnetism and Magnetic Materials DTU (10313)
10 ECTS KU 7.5 ECTS
Module 3 08/02/2001 Crystal fields
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Intended Learning Outcomes (ILO)
(for todays module)

1. Explain why paramagnetism is T-dependent whereas
   diamagnetism is not
2. Estimate the value of the Curie constant for a
   given paramagnetic substance
3. Predict the ground state of ions by applying
   Hunds rules
4. Explain the origin of the spin-orbit interaction,
   and describe its main effects
5. Compare Hunds rule predictions with data on 4f
   and 3d elements
6. Describe how crystal fields arise
7. Explain phenomena such as crystal field
   splitting, Jahn-Teller distortions, low/high spin
   states

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Flashback
Einstein de Haas -measure g-factor
Diamagnetism -small -T-independent -Orbital size
Paramagnetism -small -T-dependent ---gt Curie
law -Total angular momentum J
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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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Lande g-value and effective moment
J1/2
J3/2
J5
Curie law cCC/T
Estimate the Curie constant for a paramagnetic
ionic salt with a0.3 nm, JS3/2
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Check where are we?
All atoms and ions are diamagnetic diamagnetism
arises from a perturbation of the ground
state diamagnetism is small and
T-independent Whenever J differs from zero, we
observe a paramagnetic response J can be either
from OAM or from Spin or both paramagnetism is
larger than diamagnetism but still small at
room T The question now is What gives angular
momentum to an atom? Why are some atoms more
magnetic than others? Thats what we focus on
today.
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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, JLS is the
   shell is more than half-filled, JS (L0) if the
   shell is exactly half-filled (obviously). 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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Spin-orbit and the fine structure
For the atomic Hamiltonian weve considered so
far, L and S were good quantum numbers. Problem
is they are not
For multi-electron atoms
Where the sign of Lambda depends on the shell
occupancy.
This is an opportunity to put QFT in action! Try
to re-derive spin-orbit in a fully relativistic
framework.
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Spin-orbit in the multi-electron atom
For multi-electron atoms
Where the sign of the energy depends on the shell
occupancy (see table).
This justifies Hunds third rule, whenever
spin-orbit is a significant perturbation.
If spin-orbit dominates (large atomic number, as
it goes as Z4), the L-S coupling scheme fails.
Alternative j-j coupling.
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Composition of angular momentum
Possibilities JLS, LS-1L-S
How many?
Without spin-orbit, L and S are good quantum
numbers (i.e. L and S are conserved), and J is
not useful.
With spin-orbit, L and S are not good quantum
numbers (i.e. L and S are not conserved, although
L2, S2 and J2 are), and J becomes important.
States are L,S,J,MJgt
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Summary and example
Fine structure of the Co2 ion 3d7 S3/2, L3,
J9/2, gJ5/3
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Data and comparison (4f and 3d)
Hunds rules seem to work well for 4f ions. Not
so for many 3d ions. Why?
How do we measure the effective moment?
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Origin of crystal fields
When an ion is part of a crystal, the
surroundings (the crystal field) play a role in
establishing the actual electronic structure
(energy levels, degeneracy lifting, orbital
shapes etc.).
Not good any longer!
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A new set of orbitals
Octahedral
Tetrahedral
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Crystal field splitting low/high spin states
The crystal field results in a new set of
orbitals where to distribute electrons.
Occupancy, as usual, from the lowest to the
highest energy. But, crystal field acts in
competition with the remaining contributions to
the Hamiltonian. This drives occupancy and may
result in low-spin or high-spin states.
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Orbital quenching
Examine again the 3d ions. We notice a peculiar
trend the measured effective moment seems to be
S-only. L is quenched. This is a consequence of
the crystal field and its symmetry.
Is real. No differential (momentum-related)
operators. Hence, we need real eigenfunctions.
Therefore, we need to combine ml states to yield
real functions. This means, combining plus or
minus ml, which gives zero net angular momentum.
Examples
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Jahn-Teller effect
In some cases, it may be energetically favorable
to shuffle things around than to squeeze
electrons within degenerate levels.
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Sneak peek
m1
m2
Interactions
Ferromagnetism (Weiss)
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Wrapping up

• Temperature dependencies
• Curie Law
• Van Vleck paramagnetism
• Hunds rules
• Spin-orbit
• Crystal field
• Orbital quenching
• Jahn-Teller distortions

Next lecture Friday February 11, 815,
KU Interactions (MB)