7. MagPropSolids

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Magnetization M is a magnetic moment per unit volume (units of A/m or T).

Individual subatomic charges such as protons and electrons can be

thought of as tracing out tiny circuits and behaving as tiny magnetic

moments. They respond to external magnetic fields and give rise to an

induced magnetization.

The relationship between the magnetization induced in a material M and

the external field B is defined as:

M = c·B/μ0 = c·H

The unitless parameter c is the bulk magnetic susceptibility of the material

(per unit volume).

Some materials are more easily “magnetized” than others

c is a function of orientation, temperature, state of stress, time scale of

observation and applied field.

Magnetization and Magnetic Susceptibility



Diamagnetism and Paramagnetism

The magnetic moment of a free

atom has three principal

sources:

(1) the spin of the electron,

(2) their orbital angular

momentum about the nucleus

and

(3) the change in orbital

momentum induced by anexternal magnetic field.

(1) and (2) give rise to

paramagnetism and

(3) gives rise to diamagnetism.

Positive c Paramagnetism, Negative c Diamagnetism



Associated with the tendency of electrical charges partially to shield

the interior of a body from an applied magnetic field

Lenz's law states that the emf induced in an electric circuit always acts

in such a direction that the current it drives around a closed circuit

produces a magnetic field which opposes the change in magnetic flux.

In other words, the magnetic field of the induced current is opposite of the applied field, leading to a diamagnetic moment.

All materials exhibit diamagnetic behavior .

Diamagnetism

cd is fairly weak, but measurableand temperature independent

Examples:

quartz (-0.62 x 10-8 m3/kg) and

calcite (-0.48 x 10-8

m3

/kg)



Langevin Diamagnetism Equation

Larmor theorem:

For e-s around nucleus ,

Motion in presence of B = Motion in absence of B + precession of e-

s

Larmor’s frequency m

eB

2

Application of B will cause finite average current around the nucleus which

is equivalent to a magnetic moment with direction opposite to that of B

1charge revolutions per unit time

2 2

eB I Ze

m

2current area of the loop I r

If N is the number of atoms per unit volume,

0 M

B

c

22

4

Ze Br

m

0 N

B

220

6

NZer

m



Paramagnetism is the partial alignment of permanent atomic magnetic

moments in the direction of an applied field B.

Substances with odd #s of electrons, metals, and atoms with partially filled

inner shells (transition elements) normally exhibit paramagnetism.

Thermal disorder resists the tendency of the field to orient the moments.

Paramagnetism

Magnetization in thermal equilibrium is given by Langevin equation

( ), M N L x

Curie Law

1where / , and the Langevin function ( ) is ( ) ctnh . B x B k T L x L x x

x

2

For 1, we have ( ) / 33 B

N B x L x x M k T

C M B

T

2

where the urie constant .3 B

N

C k

C



,3

2

BT C

T k B N M

B



,3

2

BT

C

T k

B N M

B

22

00

6r

m

NZe

B

N c

)( x L N M

N M

Diamagnetism

Paramagnetism

Langevin Description (summary)



Quantum theory of Paramagnetism

The magnetic moment of an atom or ion in free space is given by

, B g J J

where , J L S

gyromagnetic ratio or magnetogyric ratio

-g factor or spectroscopic splitting factor (~2.00 for e spin)

B g

g

- ( 1) ( 1) ( 1)For a free e with , 1 .

2 ( 1)

J J S S L L L g

J J

The Bohr magneton .2

B

e

m

Energy levels in magnetic field . J B E m g B



Case L = 0

1, and 2

2 J m g

Curie Law

J B B E m g B B

1If , x

x x

B

N B C e x B

k T T N e e

2and . x

x x

N e

N e e

1 2( ) M N N x x

x x

e e N

e e

tanh . N x

For 1, tanh x x x B

B M N

k T



T k

B N M

B

2



General Case

magnetonsBohr of number effectivetheis ,)]1([

.

33

)1(

,453

1ctnh,1For

.2

ctnh2

1

2

)12(ctnh

2

12

bydefinedisfunctionBrillouinthewhere

)/( ),(

2/1

2222

3

J J g p

T

C

T k

Np

T k

J Ng

B

M

x x

x x x

J

x

J J

x J

J

J B

B

T k B gJ x x B NgJ M

B

B

B

B

J

J

B B J B



T

C

T k

Np

B

M

B

B 3

22



Hund Rules to fix configurations



L is quenched

3d shell experiences intense inhomogeneous electric field (Crys tal field )

produced by neighbouring ions.

LS coupling is broken & sublevels split due to crystal field



In a noncentral field the plane of the

orbit will move about and hence the

angular momentum components areno longer constant and may

average to zero.

When Lz averages to zero then L is

said to be quenched.



Ferromagnetism

• M decreases rapidlywith H

• Beyond the Curietemperature it behaveslike a paramagneticsubstance

• Examples : iron, cobalt,nickel

Behaves like a paramagnet

Curie Temperature

TC

M decreases rapidly with H



Antiferromagnetism

•

like paramagnets above a critical temperature

TN called Neél

temperature.

•

Below TN c is small &T -dependence is

different from

paramagnets.

• Example : Cobalt



Ferrimagnetism

• Like ferro magnets, but

the effect tends to be

smaller.

• The 1/c curve is very

close to zero below a

critical temperature,

also called Neél

temperature.

• Examples : magnetite

(Fe3O4) and spinelferrites







Novel Magnetism

• SPIN GLASS : A random orientation of frozen spins.

• CLUSTER GLASS : spins make small clusters with

magnetic order but no order between clusters

• METAMAGNET: Field induced magnetic transition

from a low magnetization state to a relatively muchhigher magnetization state

• SUPERPARAMAGNET – when the size of the

magnetic particle is very small – domains are not

formed . Each magnetic particle behaves a giantparamagnetic ion.

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