29. 7. 2003 1

III–5 Magnetic Properties of Materials

29. 7. 2003 2

Main Topics

• Introduction to Magnetic Properties

• Magnetism on the Microscopic Scale.

• Diamagnetism.

• Paramagnetism.

• Ferromagnetism.

29. 7. 2003 3

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.

29. 7. 2003 4

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.

V

mBm

mB

29. 7. 2003 5

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.

mB 0B

mBBB

0

0BB mm

29. 7. 2003 6

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.

00)1( BBB rm

29. 7. 2003 7

Introduction Into Magnetic Properties V

• Three common types of magnetic behavior exist. The external field in materials can be• weakened (m< 0 or Km < 1) this is called

diamagnetism

• slightly intensified, (m> 0 or Km >1) this is called paramagnetism

• considerably intensified, (m>> 0 or Km >> 1) this is called ferromagnetism.

29. 7. 2003 8

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 Courie’s temperature.

29. 7. 2003 9

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.

29. 7. 2003 10

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.).

29. 7. 2003 11

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.

29. 7. 2003 12

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/4me = 9.27 10-24 J/T

29. 7. 2003 13

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.

29. 7. 2003 14

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.

29. 7. 2003 15

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.

29. 7. 2003 16

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.

29. 7. 2003 17

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 can’t 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.

29. 7. 2003 18

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.

29. 7. 2003 19

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 Curie’s law is valid:

Bm = CB/T where C is a material parameter.

29. 7. 2003 20

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.

29. 7. 2003 21

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.

29. 7. 2003 22

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.

29. 7. 2003 23

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 > TC, Curie’s temperature.

29. 7. 2003 24

Ferromagnetism V

• The internal magnetization is saturated at some point. That means it can’t be further increased by increasing of the external field.

• The alignment at saturation can be of the order of 75%.

• The Curie’s temperature for Fe is 1043 K.

29. 7. 2003 25

Ferromagnetism VI

• The hysteresis is due the fact that domains can’t 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.

29. 7. 2003 26

Homework

• Homework from yesterday is due tomorrow!

29. 7. 2003 27

Things to read

• This Lecture Covers

Chapter 28 – 7, 8, 9, 10

• Advance Reading

Chapter 29 – 1, 2, 3, 5

Planetary model of a charge ILet’s 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 = 2r/v.Then if we realize that every T one charge of q passes, the current is: I = q/T = qv/2r.

Planetary model of a charge IINow 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. ^

mqbm 20