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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.