Phy331 l3, Essays (university) of Physics

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PHY

Magnetism

Lecture 3

Last week…

• Derived magnetic dipole moment of a

circulating electron.

• Discussed motion of a magnetic dipole

in a constant magnetic field.

• Showed that it precesses with a

frequency called the Larmor

precessional frequency

Langevin’s theory of diamagnetism

• We want to calculate the sample magnetisation M and the diamagnetic susceptibility χ (recall M = χ H)
• Every circulating electron on every atom has a magnetic dipole moment
• T he sum of the magnetic dipole moments on any atom is zero ( equal numbers circulating clockwise and anticlockwise? )
• T he magnetisation M arises from the reaction to the torque Γ due to the applied magnetic field B which creates the Lamor precessional motion at frequency ω L

• i) the angular momentum L of the circulating electron is,
• The angular momentum L p of the precessional motion is, where ω L is the Larmor frequency and < r 2 > is the mean square distance from an axis through the nucleus which is parallel to B € L = mvr = m ω r

€ L

p

= m ω

L

r

So substitute for each quantity, in turn, so that, € M = − N Z e 2 m L p € M = − N Z e 2 m m ω L r 2 € M = − N Z e 2 m m eB 2 m ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ r^ 2

χ =

M

H

= − μ 0

N Z

e

2

4 m

r

2

Now express < r 2

in terms of the mean square radius of the orbit < ρ 2 , where will give € r

= 2 3 ρ

χ = − μ 0 N Z e 2 6 m ρ 2 The result we wanted!! € ρ 2 = x 2

• y 2
• z 2 € r 2 = x 2
• y 2 € x 2 = y 2 = z 2 = ρ 2 / 3

Paramagnetism

• Paramagnets have a small positive magnetisation M (directed parallel the applied field B ).
• Each atom has a permanent magnetic dipole moment.
• Langevin (classical) theory
• The paramagnet consists of an array of permanent magnetic dipoles m
• In a uniform field B they have Potential Energy = - m. B

A dipole parallel to the field has the lowest energy

• BUT , the B field causes precession of m about B.
• However it can’t alter the angle between m and B (as the L z component is constant in the precession equations).
• For the dipole to lower its energy (and become parallel to the field) we need a second mechanism. This is provided by the thermal vibrations
• The magnetic field “would like” the dipoles aligned to lower their energy
• The thermal vibrations “would like” to randomise and disorder the magnetic dipoles

Calculate the magnetisation M of a paramagnet

• M must be the vector sum of all the magnetic dipole moments To do this, use Boltzmann statistics to obtain the number dn of dipoles with energy between E and E + dE where k = Boltzmann’s constant, c = is a constant of the system and T = T measured in Kelvin €

dn = c exp (− E kT ) dE

M =

[1] Resolved component of a dipole in field direction

X

[2] Number of dipoles with this orientation

How to find c?

Integrate over all the energies, which must

give N the total number of dipoles

€ N = c exp (^) (− E kT ) dE 0 ∞ ∫ Next week we will do this to calculate the susceptibility of a paramagnet. The important result we will get will be That susceptibility is temperature dependent with χ = C/T This is Curie’s law.