III

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III5 Magnetic Properties of Materials
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Main Topics

• Introduction to Magnetic Properties
• Magnetism on the Microscopic Scale.
• Diamagnetism.
• Paramagnetism.
• Ferromagnetism.

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Introduction Into Magnetic Properties I

• Magnetic properties of materials are generally
  more complicated than the electric ones even on
  the macroscopic scale. We had conductors in which
  the electric field was zero and dielectrics
  (either polar or non-polar), in which the field
  was always weakened. Other behaviour is rare.
  More subtle differences can be revealed only by
  studying thermal or frequency properties.

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Introduction Into Magnetic Properties II

• If a material is exposed to an external magnetic
  field is gets magnetized and an internal magnetic
  field appears in. It can be described as
  the density of magnetic dipole moments
• The volume V is small on macroscopic but large on
  the atomic scale.

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Introduction Into Magnetic Properties III

• The total field in the magnetized material can be
  then written as a superposition of the original
  field
• and internal field
• Here, we can shall deal only with linear
  behavior
• The parameter ?m is the magnetic susceptibility
  which can now be greater or less than zero.

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Introduction Into Magnetic Properties IV

• We can combine these equations
• and define the relative permeability Km ,
  usually also written as ?r.
• The absolute permeability is defined as
• ? ?0 ?r ?0 Km
• The internal field of a long solenoid with a core
  can then be written as B ?nI.

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Introduction Into Magnetic Properties V

• Three common types of magnetic behavior exist.
  The external field in materials can be
• weakened (?mlt 0 or Km lt 1) this is called
  diamagnetism
• slightly intensified, (?mgt 0 or Km gt1) this is
  called paramagnetism
• considerably intensified, (?mgtgt 0 or Km gtgt 1)
  this is called ferromagnetism.

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Introduction Into Magnetic Properties VI

• If a material can be ferromagnetic is is a
  dominant behavior which masks other behavior
  (diamagnetism) that is also always present but is
  much weaker.
• But the dominant behavior may disappear with high
  temperature. Ferromagnetism changes to
  paramagnetisms above Couries temperature.

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Magnetism on Microscopic Scale I

• Magnetic behavior of materials is an open field
  of research. But the main types of behavior can
  be illustrated by means of relatively simple
  models. All must start from the microscopic
  picture.
• We know that if we cut a piece of any size and
  shape from a permanent magnet, we get again a
  permanent magnet with both poles.

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Magnetism on Microscopic Scale II

• If we continue to cut a permanent magnet we would
  once get to the atomic scale. The question is
  which elementary particles are responsible for
  magnetic behavior?
• We shall show that elementary magnetic dipole
  moment is proportional to the specific charge so
  electrons are responsible for the dominant
  magnetic properties.
• Experiments exist, however, which are sensitive
  to nucleus magnetic moment (NMR, Neutron Diff.).

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Magnetism on Microscopic Scale III

• Electrons can generate magnetism in three ways
• As moving charges as current.
• Due to their spin.
• Due to their orbital rotation around a core.
• The later two mechanisms add together and the way
  it is done is responsible for magnetic behavior
  in particular material.

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Magnetism on Microscopic Scale IV

• Electrons can be viewed as a tiny spinning
  negative charged particles. The quantum theory
  predicts spin angular momentum s
• s h/4? 5.27 10-35 Js
• Here h 6.63 10-34 Js is the Planck constant
• Since electron is charged it also has a magnetic
  dipole moment due to the spin
• 1 ms eh/4?me 9.27 10-24 J/T

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Magnetism on Microscopic Scale V

• ms mb is called Bohr magneton and it is the
  smallest magnetic dipole moment which can exist
  in Nature. So it serves as a microscopic unit for
  dipole moments.
• We see that magnetic dipole is quantised.
• Spin is a quantum effect not a simple classical
  rotation. Electron would irradiate energy and
  slow down and fall on the core.

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Magnetism on Microscopic Scale VI

• When electrons are bound in atoms they also have
  orbital angular momentum. It also is a quantum
  effect.
• It is illustrative to look at a classical
  planetary model of electron, even if it is not
  realistic, to see where the dependence on the
  specific charge comes from.

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Magnetism on Microscopic Scale VII

• Even in a very small but macroscopic piece of
  material there is enormous number of electrons,
  each having some spin and some angular momentum.
  The total internal magnetic field is a
  superposition of all electron dipole moments.
• The magnetic behavior generally depends on
  whether all the magnetic moments are compensated
  or if some residual magnetic moment remains.

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Diamagnetism I

• Materials, in which all magnetic moments are
  exactly compensated are diamagnetic. Their
  internal induced magnetic field weakens the
  external magnetic field.
• We can explain this behavior on (non-realistic
  but sometimes useful) planetary model of one
  electron orbiting around an atom.

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Diamagnetism II

• Due to an external magnetic field a radial force
  acts on the electron. It points toward or out of
  the center depending on the direction of the
  field. The force cant change the radius but if
  it points toward the center it speeds the
  electron and if out it slows it. This leads to a
  change in the magnetic moment which is always
  opposite to the field. So the field is weakened.

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Paramagnetism I

• Every electron is primarily diamagnetic but if
  atoms have internal rest magnetic dipole moment
  diamagnetism is masked by much stronger effects.
  If the spin and orbital moments in matter are not
  fully compensated, the atoms as a whole have
  magnetic moments and they behave like magnetic
  dipoles. They tend to line up with the external
  field and thereby reinforce it.

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Paramagnetism II

• The measure of organizing of dipoles due to the
  external field depends on its strength and it is
  disturbed by temperature movement.
• For fields and temperatures of reasonable values
  Curies law is valid
• Bm CB/T
• where C is a material parameter.

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Ferromagnetism I

• If we think of magnetism, we usually have in mind
  the strongest effect ferromagnetism.
• In some materials (Fe, Ni, Co, Ga and many
  special alloys) a quantum effect, called
  exchanged coupling leads to rigid parallel
  organizing of atomic magnetic moments in spite of
  the randomizing tendency of thermal motions.

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Ferromagnetism II

• Atomic magnetic moments are rigidly organized in
  domains which are microscopic but at the same
  time large on the atomic scale.
• Their typical volumes are 10-12 10-8 m3 , yet
  they still contain 1017 1021 atoms.
• If the matter is not magnetized the moments of
  domains are random and compensated.

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Ferromagnetism III

• In external magnetic field the domains whose
  moments were originally in the direction of the
  field grow and the magnetic moment of some other
  can collectively switch its direction to that of
  the field.
• This leads to macroscopic magnetization.

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Ferromagnetism IV

• Ferromagnetic magnetization
• Is a strong effect ?r ? 1000!
• Depends on the external field.
• Ends in saturation.
• Has hysteresis and thereby it can be permanent.
• Disappears if T gt TC, Curies temperature.

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Ferromagnetism V

• The internal magnetization is saturated at some
  point. That means it cant be further increased
  by increasing of the external field.
• The alignment at saturation can be of the order
  of 75.
• The Curies temperature for Fe is 1043 K.

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Ferromagnetism VI

• The hysteresis is due the fact that domains cant
  return at low temperatures and in reasonable
  times to their original random configuration. Due
  to this, so called memory effect, some permanent
  magnetization remains.
• This effect is widely used e.g. to store
  information on floppy and hard-drives.

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Homework

• Homework from yesterday is due tomorrow!

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Things to read

• This Lecture Covers
• Chapter 28 7, 8, 9, 10
• Advance Reading
• Chapter 29 1, 2, 3, 5

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Planetary model of a charge I
Lets have a charge q with speed v on orbit of
the radius r and calculate its magnetic dipole
moment m0 IA. The area is simply A ?r2. To
get the current we first have to find the period
of rotation T 2?r/v. Then if we realize that
every T one charge of q passes, the current is I
q/T qv/2?r.
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Planetary model of a charge II
Now the magnetic moment m0 IA rqv/2. On the
other hand the angular momentum is b mvr. If
we put this together, we finally get m0 b
q/2m. This can be generalized into a vector
form If the charge is an electron q -e so
the vectors of the magnetic moment and orbital
momentum have opposite directions.