Ferro Magnetic Materials

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    FERROMAGNETIC MATERIALS

    DKR-JIITN-PH611-MAT-SCI-2010

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    DKR-JIITN-PH611-MAT-SCI-2010

    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)

    DKR-JIITN-PH611-MAT-SCI-2010

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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