PHY331 Magnetism

Lecture 6

Last week…

Learned how to calculate the magnetic dipole moment of an atom.

Introduced the “Landé g-factor”. Saw that it compensates for the different contributions from the orbital motion and the electron spin.

Compare calculated values with measured data

This week…. •  Will derive a quantum theory of Paramagnetism. Will

obtain a new expression for paramagnetic susceptibility that we will compare with the classical value. Will see that it confirms Curie’s Law.

•  Will introduce some basic concepts relating to Ferromagentism.

Quantum theory of paramagnetism What are the changes from Langevin’s classical theory? The

magnetic moment value is,

and the resolved component of the moment along the field direction is, where

the potential energy of the dipole in the field is then,

€

µ J = g J J +1( ) µB

€

MJ gµB

€

MJ = J , J −1( ) , .....,− J −1( ),−J

€

−MJ gµB B

so this gives all we need to substitute into,

€

M =∑ [1] Resolved component of a dipole in field direction X [2] Number of dipoles with

this orientation

we now have a rather unusual result: the dipoles no longer have an infinite set of orientations in space,they have a finite set of orientations described by MJ, this means that, the integral over dθ that was used in Langevin’s treatment

€

mm

=

cosθ ⋅ cmB sinθ ⋅ exp mB cosθ kT( ) dθ0

π

∫

cmB sinθ ⋅ exp mB cosθ kT( ) dθ0

π

∫

can be replaced by a summation over the MJ

€

µJ

µJ=

MJgµB ⋅ exp MJgµB B /kBT( )−J

J

∑

exp MJgµB B /kBT( )−J

J

∑

whose solution is:

€

µJ

µ J=2J +12J

coth 2J +12J

⋅y⎛

⎝ ⎜

⎞

⎠ ⎟ −

12Jcoth 1

2J⋅y

⎛

⎝ ⎜

⎞

⎠ ⎟

This is the Brillouin function BJ(y). It is plotted as a function of

€

y =µJ BkT

€

coth x =1

tanh x=1x

+x3−x 3

45+ ...

(1)

B/T ~ y

€

µJ//

µB

The magnetic susceptibility χ Just as with the classical theory, we can derive an

expression for the magnetic susceptibility χ in the limit of small applied fields and high temperatures,

that is Use

Substitute into equation (1) to show €

y =µJ BkT

<<1

€

BJ y( ) ≈J +13J

y + ......

€

BJ y( ) ≈J +13J

µJ BkT

€

coth x =1x

+x3

or (2)

but, or

and

so that substituting (3) into (4) gives,

so that,

€

µJ

µ J= BJ y( )

€

µJ = µJ BJ y( )

€

M = N µ J

€

M = N µ J = Nµ J BJ y( )

€

M = NµJJ +13J

µJ BkT

= N J +1J

µJ2 µ0H3kT

€

χ =MH

= µ0 NJ +1J

µ J2 13kT

(3)

(4)

(5)

Using (1)

Finally using, gives,

which is Curie’s Law (again).

So we compare with the classical result,

which indicates that,

€

µJ = JgµB

€

χ =µ0N g2 J J +1( ) µB

2

3kT=

CT

€

χ =MH

=µ0 Nm

2

3kT=

CT

€

Jeffective = g J J +1( )

which is consistent with the earlier definitions,

and our understanding of what the “Landé g factor” is meant to do.

Experimental evidence of Curie’s Law behaviour

€

µL = − L L +1( ) µB

€

µS = − 2 S S +1( ) µB

Gadolinium salt Gd(C2H3SO4)3.9H2O

€

1χ∝T

T

€

1χ

Paramagnetism..

•  Calculated paramagetic susceptability via both quantum and classical approaches similar.

•  Both confirm statistical competition between magnetic alignment and thermal effects.

•  Now consider systems where have a strong, spontaneous alignment between dipoles even in the absence of a magnetising field.

Ferromagnetism The bulk properties of a ferromagnet are: The magnetisation M is large and positive

It is contribution to the total B field is significant χ >> 0 and µr >> 1

The magnetisation M is a complex function of the applied magnetic field H

The magnetisation M also depends on the past history of the sample

A graph of M against the applied magnetic field H gives the characteristic magnetisation curve of the ferromagnet

The curve divides naturally into three regions:

A linear region at small values of H in which the initial susceptibility can be defined

An intermediate region, in which the slope rises to a maximum

A region where decreases

to show saturation of the magnetisation

€

χ = M H

€

dM dH

€

dM dH

The magnetisation M is a function of the past history of the sample

A permanent magnet and an iron nail can both lie in the same Earth’s magnetic field but, the

magnetisation M inside each is quite different

The permanent magnet has has already been exposed to a much larger “magnetising field”

On demagnetising M lags behind H. It requires a reverse field to demagnetise the sample

Important definitions: RM = Remanent Magnetisation HC = Coercive Field Msat = Saturation magnetisation

€

Msat = NpµB

N: atoms m-3

pµB: Magnetic dipole per atom

Msat

Summary

•  Derived a quantum theory of Paramagnetism.

•  Introduced ferromagentism. Saw magnetization was a function of previous history.

•  Next week will look at this phenomena in more detail using the domain theory of ferromagnetism.

€

χ =µ0N g2 J J +1( ) µB

2

3kT=

CT