Magnetic Properties

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Magnetic Properties

• Subhalakshmi Lamba

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Langevins Theory of Diamagnetism
i
e
Magnetic Dipole moment of an electron in an
atomic orbit ??
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Langevins Theory of Diamagnetism
electron
r
Nucleus
d A
d l
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Langevins Theory of Diamagnetism
ANGULAR MOMENTUM
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Langevins Theory of Diamagnetism

• What happens to the magnetic dipole moment of the
  circulating electron in a constant magnetic
  field?
• The field produces a torque ? m ? B on each
  dipole
• Purely precessional motion

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Larmor Precession

• Frequency of precession Larmor frequency ?p
• Angular momentum
• L p m ?p lt r2gt
• (eB/2) lt r2gt

e
L
?peB/2m
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Langevins Theory of Diamagnetism

• The magnetic moment due to the precession is
  mp-(e/2m) Lp.
• Total magnetization of a sample
• (No. of atoms) ? (No. of electrons/atom) ? mp

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Langevins Theory of Diamagnetism

• M - N Z (e/2m) Lp.

• M - N Z (e/2m) (eB/2) lt r2gt B ?0H

• ? M/H - ?0N Z (e2/4m). lt r2gt

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Finding ltr2gt

• lt r2gt is the mean square distance of the
  electron from the axis through the nucleus and
  parallel to the field

Z

• Mean square
• radius of the
• electron orbit

r2x2y2
Y
r
X
?2x2y2z2
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Finding ltr2gt

• ltx2gtlty2gtltz2gt(1/3) lt?2gt for a spherically
  symmetric charge distribution

• ltr2gtltx2gtlty2gt(2/3) lt?2gt

• ? - ?0N Z (e2/6m). lt?2gt

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The diamagnetic susceptibility

• is negative

• does not depend on temperature

• Is very small because lt?2gt is very small

• is a property of all matter

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Merit of the Theory

• Langevins theory explains for all the major
  properties of diamagnetic materials

• The atom as a whole does not have a permanent
  magnetic moment
• Magnetization is produced by the effect of the
  externally applied magnetic field on the
  circulating electron

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PARAMAGNETISM

• Each atom has a permanent magnetic dipole moment

• Each dipole has an energy of m.B
• which is m B cos ??
• Energy is minimum when m is parallel to B

?
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Two mechanisms at work

• Effect of the field is to cause a precession of
  the magnetic moment about the field.
• Precession cannot change the angle between B and
  m

• How do the magnets align to the external field?

Thermal vibrations
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Two mechanisms at work
The magnetic field tends to align the dipole
Thermal vibrations tend to randomize the dipoles

• Statistical Theory

Total Magnetization ?? (component of the dipole
moment in the direction of the field) X (No. of
dipoles which have this orientation wrt the field)
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A classical theory for Paramagnetism
Component of the dipole in the direction of the
field is m cos ?
Energy E -m B cos ?
?
dE m B sin ? d ?
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A classical theory for Paramagnetism
No of dipoles which have an energy between E and
E dE No. of dipoles which have an orientation
between ? and d?
dn c exp(-E/kBT) dE c
exp(mBcos?/kBT)m B sin? d?
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A classical theory for Paramagnetism
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A classical theory for Paramagnetism

• A parameter which is a measure of the ratio of
  the two competing energies

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A classical theory for Paramagnetism
LANGEVIN FUNCTION L(a)
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A classical theory for Paramagnetism
L(a)
a
PAUL LANGEVIN
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A classical theory for Paramagnetism
Magnetization of a paramagnetic substance M N
m L(a) amB/kBT ? M/H
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At the limits

• At small applied fields and high temperatures T
• a is small

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At the limits
CURIES LAW
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At the limits

• Large applied field B compared to the
  temperature T
• a is small

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At the limits
SATURATION maximum possible magnetization Magnet
ization becomes independent of the applied field
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A RECAP

• Classical Theory of Magnetism
• Circulating Electron magnetic dipole of the
  atom
• Diamagnetism due to the effect of the external
  magnetic field on the magnetic dipole moment.
• A weak effect depends on atomic radius, which
  is very small

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A RECAP

• Paramagnetism competitiion between the magnetic
  field and temperature
• Alignment to the field versus randomization
• Statistical Treatment
• Saturation at high fields
• Curies law for low fields

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NEED for a Quantum Theory

• Niels Bohr 1911, and J.H van Leeuwen
  independently in 1919 in her PhD thesis proved a
  famous theorem for classical nonrelativistic
  electrons using Maxwell's equations and
  statistical mechanics
• "At any finite temperature, and in all finite
  applied electrical or magnetic fields, the net
  magnetization of a collection of electrons in
  thermal equilibrium vanishes identically."

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NEED for a Quantum Theory

• Classical Physics cannot give any kind of
  magnetism
• Explanation for ferromagnetism Pauli principle
  that no two electrons could occupy the same
  state. Together with the Coulomb interaction
  between electrons, this leads to a scalar
  isotropic interaction of two spins with a
  positive exchange constant J.
• Physical explanation for the Molecular fields of
  Weiss

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NEED for a Quantum Theory

• Zeeman Effect offered experimental support for
  the quantization of angular momentum

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NEED for a Quantum Theory

• Anomalous Zeeman Effect the magnetic field
  split the lines into four, six, or even more
  lines and some triplets showed wider spacings
  than expected
• Only explained by including electron spin