Ferro Magnetic Materials

8/2/2019 Ferro Magnetic Materials

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FERROMAGNETIC MATERIALS

DKR-JIITN-PH611-MAT-SCI-2010


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Ferromagnetic Materials

MHB 00 +=

Certain metallic materials posses permanent magnetic moment inthe absence of an external field, and manifests very large andpermanent magnetization which is termed as spontaneousmagnetization. Example: Fe, Co, Ni and some rare earth metals

such as Gd.

Magnetic susceptibility as high as 106 are possible for ferromagneticmaterials. Therefore, H


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Atomic magnetic moments due to un-cancelled electron spin is

responsible for Spontaneous magnetization. Orbital motion alsocontributes but its contribution is very small in comparison to theelectron spin.

Coupling interaction causes net spin magnetic moments ofadjacent atoms to align with one another even in absence ofexternal field. This mutual spin alignment exists over a relativelylarger volume of the crystal called domain.

The maximum possible magnetizationis called saturation magnetization. Itresults when all magnetic dipoles in a

solid piece are mutually aligned withthe external field. There is also acorresponding saturation flux densityBs.

NMs = (3)



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1. Weiss theory of spontaneous magnetization classical theory

Hypothesis of Weiss theory:

1. A ferromagnetic specimen contains, in general, large number ofsmall regions called domains (dimension ~ 10-2 cm andcontaining as many as 1015 to 1017 atoms) which arespontaneously magnetized. The magnitude of spontaneousmagnetization of the specimen is determined by vector sum ofdipole moment of individual domains.

2. Within each domain spontaneous magnetization is due to theexistence of a molecular field which tends to produce a parallelalignment of the atomic dipoles despite effect of thermal energy.This internal field is equivalent to a magnetic field Hm, which is

proportional to the magnetization M of within a domain i.e.

MHm =

Where is constant independent of temperature, called molecularfield constant or Weiss constant.

(1.1)


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Effective field experienced by each dipole would be then,

MHHe +=

Let us consider a ferromagnetic solid containing N number ofatoms/ m3, then magnetization due to spins can be given as

])(

tanh[]tanh[ 00

kT

MHN

kT

HNM BB

BB

+==

At sufficiently high temperature,

1)(

0


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Where,

kT

MHNM B

)(2

0 +=

kT

MN

kT

HNM BB

2

0

2

0 +=kT

HN

kT

NM BB

2

0

2

0 )1(

=

)1(

2

0

2

0

kT

NkT

N

H

M

B

B

==

)(

2

0

2

0

k

NTk

N

H

M

B

B

==

)(

=T

C

Ck

NkNC BB ===

2

0

2

0 and

Therefore,(1.3)

(1.4)

(1.5)


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T

1/

Ferromagnetic

Paramagnetic

0 Now when H = 0, then

]tanh[])(

tanh[ 00

kT

MN

kT

MHNM B

B

B

B

=

+=

]tanh[ 0

kT

M

M

M

N

M B

sB

==

(1.5)

tanh==sB M

M

N

M

Where, kT

MB

0

=


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)(aBNgJM JB=

Where

and

kT

MHgJ

kT

HgJa BB

)(00 +==

We know that,Bs NgJM =)0( Therefore

)()0()( aBMTMJs

=

)()0(

)(aB

M

TMJ

s

=

Further,kT

MgJ

kT

MHgJa BB 00 )( =+=

J

a

J

a

J

J

J

JJB

2coth

2

1

2

)12(coth

2

12)(

++=

2. Weiss theory of spontaneous magnetization Quantum theory


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kT

MgJa B

0=

BgJ

kTaTM

0

)( =

BBs NgJgJ

kTa

M

TM

1

)0(

)(

0

=

2

0

22)0(

)(

Bs JNg

kTa

M

TM=

)0(

)(

sM

TM

a

)(aBJ

T =T


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)0(

)(

sM

TM

T

1

15.00

)0(sM

T

1

5.0

Ms(0) is spontaneous magnetization at 0K.


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Let us consider Brillouin function (BJ) again,

Ja

JJaJ

JJaBJ

2coth

21

2)12(coth

212)( ++=

For a


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+

+++=

aJ

aJ

aJ

aJJJaBJ 2

22

2

222

12

12

12

)414(12)(

+++=

aJ

aJJaaaJJaBJ 2

2222222

12

124412)(

+=

aJ

JaaJaBJ 2222

12

44)(

+=

aJ

JJaaBJ 2

2

12

)1(4)(

+=J

JaaBJ

3

)1()(

Therefore, from the equation of magnetization

)()0()( aBMTM Js=J

JaMTM s

3

1)0()(

+=

Thus, slope of the curve for a


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2

0

22)0()(

Bs JNgkTa

MTM =

We also know that,

2

0

22)0()(

Bs JNgkT

aMTM =

Thus,

(B)

Comparing (A) and (B) at T = ,

J

J

JNg

k

B3

12

0

22

+=

k

JJNg B

3

)1(2

0

2 += (C)

Equation gives relation between curie temperature, and molecular fieldconstant, .


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Now let us consider the case of T>, a


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)3

)1(1(

3

)1(

20

2

2

0

2

kTJJNg

kT

JJNg

H

M

B

B

+

+

==

)3

)1((

)1(

2

0

2

2

0

2

kJJNgTk

JJNg

H

M

B

B

+

+==

)(

=T

C(C)

Where,k

JJNgC B

)1(2

0

2 +=

and k

JJNg B

3

)1(20

2 +=

(D)

(E)


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3. Exchange interaction in magnetic materials

212 sJsEexch =

2

1

2.2 ZJSSSJEj

jiijexch = =rr

Heisenberg in 1928 gave theoretical explanation for large Weissfield in ferromagnetic materials.

According to his theory, parallel arrangements of spins inferromagnetic materials arises due to exchange interaction inwhich two neighboring spins in the solid are coupled togetherwith an energy given as

Here J is known as exchange integral. Its value depends uponseparation between atoms as well as overlap of electron chargecloud. When J > 0, lower energy is obtained for parallel

configuration of spins, while for J < 0, spins are anti-parallel.

(3.1)

If there are Z nearest neighbors to a central ith spin, theexchange energy for this spin is

(3.2)


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ijJ

02r

rab

Cr

Mn

Fe

Co

Ni

Gd

22ZS

KJ

KZJS 22

This energy must be equal to K as at , ferromagnetic order isdestroyed. Thus,

(rab is inter-atomic distance

and r0 is atomic radius).

0>J

0

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