Diamagnetism and Paramagnetism

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Diamagnetism and Paramagnetism

• Physics 355

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• Free atoms
• The property of magnetism can have three origins
• Intrinsic angular momentum (Spin)
• Orbital angular momentum about the nucleus
• Change in the dipole moment due to an applied
  field

In most atoms, electrons occur in pairs.
Electrons in a pair spin in opposite directions.
So, when electrons are paired together, their
opposite spins cause their magnetic fields to
cancel each other. Therefore, no net magnetic
field exists. Alternately, materials with some
unpaired electrons will have a net magnetic field
and will react more to an external field.
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Diamagnetism Classical Approach
Consider a single closed-shell atom in a magnetic
field. Spins are all paired and electrons are
distributed spherically around the atom. There
is no total angular momentum.
nucleus
r
electron
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Diamagnetism Larmor Precession
nucleus
r
v, w0
electron
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Diamagnetism Quantum Approach
starting point
Quantum mechanics makes some useful corrections.
The components of L and S are replaced by their
corresponding values for the electron state and
r2 is replaced by the average square of the
projection of the electron position vector on the
plane perpendicular to B, which yields where R
is the new radius of the sphere.
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Diamagnetism Quantum Approach
If B is in the z direction
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Diamagnetism Quantum Approach

• The atomic orbitals are used to estimate ltx2
  y2gt.
• If the probability density ? ? for a state is
  spherically symmetric ltx2gt lty2gt ltz2gt and ltx2
  y2gt2/3ltr2gt.
• If an atom contains Z electrons in its closed
  shells, then

Consider a single closed-shell atom in a magnetic
field. Spins are all paired and electrons are
distributed spherically around the atom. There
is no total angular momentum.

• The B is the local field at the atoms location.
  We need an expression that connects the local
  field to the applied field. It can be shown that
  it is

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Diamagnetism
Core Electron Contribution

• Diamagnetic susceptibilities are nearly
  independent of temperature. The only variation
  arises from changes in atomic concentration that
  accompany thermal expansion.

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Diamagnetism Example
Estimate the susceptibility of solid argon.
Argon has atomic number 18 and at 4 K, its
concentration is 2.66 x 1028 atoms/m3. Take the
root mean square distance of an electron from the
nearest nucleus to be 0.62 Å. Also, calculate
the magnetization of solid argon in a 2.0 T
induction field.
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Diamagnetism Example
Estimate the susceptibility of solid argon.
Argon has atomic number 18 and at 4 K, its
concentration is 2.66 x 1028 atoms/m3. Take the
root mean square distance of an electron from the
nearest nucleus to be 0.62 Å. Also, calculate
the magnetization of solid argon in a 2.0 T
induction field.
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Paramagnetism
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Core Paramagnetism
If ltLzgt and ltSzgt do not both vanish for an atom,
the atom has a permanent magnetic dipole moment
and is paramagnetic.
Some examples are rare earth and transition metal
salts, such as GdCl3 and FeF2. The magnetic ions
are far enough apart that orbitals associated
with partially filled shells do not overlap
appreciably. Therefore, each magnetic ion has a
localized magnetic moment.
Suppose an ion has total angular momentum L,
total spin angular momentum S, and total angular
momentum J L S.
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Core Paramagnetism
Landé g factor
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Hunds Rules

• For rare earth and transition metal ions, except
  Eu and Sm, excited states are separated from the
  ground state by large energy differences and
  are thus, generally vacant.
• So, we are mostly interested in the ground state.
• Hunds Rules provide a way to determine J, L,
  and S.

• Rule 1 Each electron, up to one-half of the
  states in the shell, contributes ½ to S.
  Electrons beyond this contribute ? ½ to S. The
  spin will be the maximum value consistent with
  the Pauli exclusion principle.

Frederick Hund 1896-1997
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Hunds Rules

• Each d shell electron can contribute either ?2,
  ?1, 0, 1, or 2 to L.
• Each f shell electron can contribute either ?3,
  ?2, ?1, 0, 1, 2, or 3 to L.
• Two electrons with the same spin cannot make the
  same contribution.

• Rule 2 L will have the largest possible value
  consistent with rule 1.

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Hunds Rules

• Rule 3

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Hunds Rules Example
Find the Landé g factor for the ground state of a
praseodymium (Pr) ion with two f electrons and
for the ground state of an erbium (Er) ion with
11 f electrons.

• Pr
• the electrons are both spin 1/2, per rule 1, so
  S 1
• per rule 2, the largest value of L occurs if one
  electron is
• 3 and the other 2, so L 5
• now, from rule 3, since the shell is less than
  half full,

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Hunds Rules Example
Find the Landé g factor for the ground state of a
praseodymium (Pr) ion with two f electrons and
for the ground state of an erbium (Er) ion with
11 f electrons.

• Er
• per rule 1, we have 7(1/2) and 4(?1/2), so S
  3/2
• per rule 2, we have 2(3), 2(2), 2(1), 2(0),
  1(?1), 1(?2), and 1(?3), so L 6
• now, from rule 3, since the shell is more than
  half full,
• J L S 15/2

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Paramagnetism

• Consider a solid in which all of the magnetic
  ions are identical, having the same value of J
  (appropriate for the ground state).
• Every value of Jz is equally likely, so the
  average value of the ionic dipole moment is zero.
• When a field is applied in the positive z
  direction, states of differing values of Jz will
  have differing energies and differing
  probabilities of occupation.
• The z component of the moment is given by
• and its energy is

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As a result of these probabilities, the average
dipole moment is given by
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Brillouin Function
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Paramagnetism
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Paramagnetism
Curie Law
The Curie constant can be rewritten as
where p is the effective number of Bohr magnetons
per ion.
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