Magnetic Property

The response of the materials to external magnetic field degree of response varies, which is measured in terms of their magnetization (strong or weak)

The parameters used to study the magnetic behaviors of the materials are as follows:

1.Magnetic dipoles & magnetic moment:

Magnetic dipoles are analogous to electric dipoles ;consists of a north pole and a south pole of strength‘m’ each separated by a small distance ‘2l’

Magnetic moment = m x 2l

A circular current loop is equivalent to a magnetic dipole, magnetic moment

μM = I x A ( amp. m2)

Where ‘I’ is the current in the loop

and ‘A’ is the area of the loop

Torque on the dipole τ = μM X B

N S

2l

μM

I

Right handScrew rule

2. Magnetisation = dipole moment / volumeM = μ / V ( amp. / met.)

3.Magnetic susceptibility = magnetization / mag. field strengthχ = M / H (no unit)

4.Magnetic permeability = magnetic induction / mag. field strengthμ = B / H (Wb / amp. met. = H/met)

μ =μ0 μr

5.Relative permeability μr = μ / μ0,

μ0 = absolute permeability = 4πX 10 -7 H/m

6.Relation between H, B & M isB = μ0 (M+H) = μ0 (χH + H) = μ0 (1 + χ) H

7. Relative permeability & Susceptibility B = μ H = μ0 μr H so μr =(1 + χ)

Origin of magnetic moment

• The three sources of magnetic moment in an atom of any material are

[1] Orbital motion of the electron

[2] Spin motion of the electron

[3] Nuclear spin

• If the vector sum of all the contribution is zero then net magnetization is zero :- material is nonmagnetic.

1. Orbital motion of the electron

• Motion of the electron (charged particle)around the nucleus in a circular orbit (orbital motion) is equivalent to a circular current and behaves as a magnetic dipole.

• Associated magnetic moment is

μM = Ix A ( amp. m2)

• if ‘T’ is the time period for one rotation

& ‘v’ is the velocity of the electron

in the orbit , then T = 2 π r / v

v

r+

Orbital magneticmoment

Electronmotion

Current

• Current I = - q/T

• magnetic moment μM = (- q v / 2 π r ) (A) = - ( q v / 2 π r ) (π r2 )

• qvr/2 = - q/ 2m ( mvr)

• = - (q/ 2m ) L

• Orbital magnetic moment

Quantum no. associated with ‘L’ is √* l(l+1) ħ+ ,

‘l’ is the orbital quantum no.

• l=0---s shell, l=1---p shell,

• l=2-----d shell, l=3------f shell

2. Spin motion of the electron

Similarly, the spin motion of the electron around their own axis give rise to spin magnetic moment

where γ is called spin gyromagnetic ratio. (γ=2)

μorb = - (q/ 2m ) L

μspin = -γ (q/ 2m ) S

μorb = -√l(l+1) (qћ / 2m )

Quantum no. associated with ‘S’ is ± ħ/2

3. Nuclear magnetic moment Due to the spin of nucleus ( protons & neutrons) , a magnetic moment is

associated which is very small as compared to the electroniccontribution as heavy mass is involved (10-3 times) is masked byelectronic mag. mom.

Total magnetic moment is due to electron motion inside the atomμM = - g (q/ 2m ) (L+S) = - g (q/ 2m ) J,

‘J’ is the total angular momentum varying from |L+S| to |L- S| If magnetic field is applied along z-direction , the component of the

total magnetic moment in that direction is, μM = - g (qħ/2m) mj

Where ‘mj ‘is the magnetic quantum no. varying from (J to -J) g is called Lande’s g-factor, for spin moment g=2

• g =

• Calculation rules

1.If electrons are in the s-orbit, orbital magnetic moment is zero

(l=0)

2. For completely filled shell, orbital magnetic moment is zero (l=0)

As ml = l to –l (s shell=0, p shell, l=1, d shell, l=2 and f shell, l=3..)

3.If all electrons are paired, spin magnetic momentum is zero

4. Only partially filled p, d and f shells contribute to orbital

magnetic moment

)1(2

)1()1()1(1

JJ

LLSSJJ

Bohr magnetron

• If there is a single electron it will have spin magnetic moment only

• μs = -2 (q ħ / 2m) ±1/2 , as g=2 and mj= ±1/2

• ‘μB‘ = = 9.27x 10-24 Am2

• This is the fundamental magnetic moment, called Bohr magnetron ‘μB‘—it is the magnetic moment of an isolated single electron or the magnetic moment of Hydrogen atom

• Usually magnetic moments of materials are expressed in terms of ‘μB‘

• Spin magnetic moment can be expressed as μ = g μBS

• Hydrogen atom:- One electron in s shell. So l = 0 and s= +1/2 or -1/2 implies that orbital magnetic mom. is zero and spin mag. Mom. is same as the Bohr magnetron

2. Helium atom ?...calculate

m

q

2

• Hund’s rule:

• 1. spins of electrons remain parallel to each other to the max. Extent

• 2. max . Value of L is consistent with the spin S

• 3. if shell is less than half filled J= |L- S| ,

if more than half filled J= |L+ S| &

if exactly half filled then L=0 and J=S

Classification of magnetic materials

• Diamagnetic material:- Have even no. of electrons, so no permanentmoment. When placed in external magnetic field get slightlymagnetised, in a direction opposite to the applied magnetic field.

• Paramagnetic materials:- Have net moment in the absence of ext.magnetic field(partially filled p,d,f orbitals and unpaired electrons).When placed in external magnetic field are magnetised in thedirection of the external magnetic field applied.

• Ferromagnetic materials:- Exhibit spontaneous magnetisation due toan internal field arising due to mutual interaction between thedomains. When placed in ext. magnetic field acquire very large andpermanent magnetisation in the direction of the field.

Antiferromagnetic materials:-Individual magnetic dipoles havemagnetic moment, but due to antiparallel arrangement netmagnetisation become zero.

Ferrimagnetic materials:-Individual magnetic moments areantiparallel, but having different magnitude do not cancel outcompletely.

Diamagnetism Paramagnetism Ferromagnetism

1.Normally referred as non-magnetic as the response isvery weak2.In ext. magnetic fieldmagnetic moment inducedin a direction opposite toapplied field– repelled bythe field

H=0, M=0 H=H→M= - M←

3. Permeability μ<1 4. susceptibility χ <05. Susceptibility does not depend on temperature5. Ex; Cu, Ag, Hg, Au, Zn, SC

1.Normally referred as non-magnetic as the response isweak2. Posses permanent magneticmoments, which are randomlyoriented in the absence of ext.magnetic field., so netmagnetization is zero. When afield is applied dipoles getaligned in the field direction,giving positive magnetization3. Permeability μ>1

H=0, M=0 H=H→ M= M4. susceptibility χ is positive,small and temp. dependant

χ = C/ T →Curie law5.Ex; Al, Cr, Na, Ti, Zr

1.Referred as magnetic asresponse is strong( due toexchange coupling)2. Posses permanent dipoles3.Show spontaneous magnetization—even in the absence of ext. field, magnetization shown is high, when field is applied, M increases4.Permeability μ>1 5.susceptibility χ is positive,

large and temp. dependant χ = C/ T-θ →Curie – Weiss law

6. ferromagnetic domainshow

spontaneous magnetisation

7. Show hysteresis 8. Ex; Fe, Co, Ni

Classification of magnetic materials

Ferromagnetic theory ( Weiss Theory)

• Weiss predicted that, in ferromagnetic materials Spontaneousmagnetization is observed, which is due to a strong internal fieldarising from an exchange interaction between the magneticmoments in the neighborhood domains

• exchange interaction

between two atoms ‘I’ and ‘j’

• = U= -2 J SiSj

‘J’ is called the exchange integral

• H int = λM , where ‘λ’ is called Weiss constant

• H tot= H appl + Hint

• H tot= H appl + λM

• Ferromagnetic domains( 1-100 μm) :-

Domains in a favorable direction grow in size at the expense ofother domains till saturation is reached; there is only onedomain. Energy involved in the orientation is obtained fromthe hysteresis curve.

B

Domain wall ≈ 10-2 μm

Ferromagnetic hysteresis

Ms(0)

TTc

Ms= saturation magnetisation

Mr = remanent magnetisation

is the measure of the strength of the ferromagnetic material as a permanent magnet (is the magnetisation in the absence of the external field).

Hc= coercive field

Temp. dependence of Ms

Ms

Easy direction:- is the crystallographic direction, along which when magnetic fieldis applied, a ferromagnetic single crystal is easily magnetised.(for fairly low field).

SOFT & HARD FERROMAGNETIC HYSTERESIS LOOPS

Soft and Hard Magnetic materials

Soft ferromagnetic Hard ferromagnetic

1. Can be easily magnetized or demagnetized

2. Thin and long hysteresis loop3. High permeability and low coercive

field4. Large susceptibility & low

remanent mag.5. As area of the loop is small,

magnetic energy loss per volume is less during magnetisation and demagnetisation

6. Application: electromagnet, in motors, generators, dynamos and switching circuits

7. Ex: Fe-Si alloy , Fe-Co-Mn alloy and Fe-Ni alloy

1. Can not be magnetised or demagnetised easily

2. Wide hysteresis loop3. Low permeability and high

coercive field4. small susceptibility & high

remanent mag.5. Large area of the loop indicates,

magnetic energy loss per volume is high during magnetisation and demagnetisation

6. For permanent magnet in speakers, clocks

7. Rare earth alloys with Mn, Fe, Co, Ni

Antiferromagnetism:

• when exchange interaction between adjacent or neighboring domains give rise to ordered antiparallel spin arrangement, below a temp. called Neel temp. ex.- MnO,MnS,FeCl2, Co O. Net moment or magnetisation is Zero

χ= C / (T +TN )

TTN

χ

Ferrimagnetic & Antiferromagnetic materials

• Ferrimagnetic material are special class of ferromagnetic material called ‘ ferrites’ with high permeability, saturation magnetisation and they show hysteresis (square loop)

• They are different from the ferromagnetic materials only in the way the spin magnetic moments are arranged in them.

• Formula: Me2+O Fe23+O3 :- Me is a divalent atom

(Fe, Mn, Zn,Cd,Cu,Ni,Co,Mg )

Crystal structure: Inverse spinel

cubic cell has ‘8’ molecules.

In the unit cell, 32 O-2 ions ,

16 Fe3+ ions ,

8 Me2+ ions.

Ferrites :- Me2+O Fe3+2O3

Me= Fe, Mn, Co, Ni, Cu, Mg, Zn, Cd

8 Me2+ ions and 8- Fe 3+, are surrounded by 6 oxygen ions :-Octahedral

8 Fe 3+ ions are surrounded by 4 oxygen ions :- Tetrahedral

• Octahedral site :- 8 Me2+ ions and 8- Fe 3+, are surrounded by 6 oxygen ions and have parallel spins.

• Tetrahedral site :- 8 Fe 3+ ions are surrounded by 4 oxygen ionsand spins antiparallel.

So net magnetic moment of Fe 3+ ions cancel ( 8 up spin and 8 down spin)Only, 8 Me2+ ions contribute to magnetic moment.

Fe3+

Me2+

S= 5/2S = 8 x [μm of one Me 2+]

octahedralFe3+

S= 5/2

tetrahedral

Spin

mo

me

nts

in f

err

ite

s

Magnetisation of a ferrite

• Spin magnetic moment of one Me2+ atom μM = g μBs

• There are 8 Me2+ atoms in a unit cell ,

total moment in unit cell = 8 x μM

• Magnetisation

M = total moment per volume

• So, M = ( 8 x μM ) / a3

where ‘a’ is lattice parameter

Mn2+= 3d5

So,μ = g μBS = 2 x 5/2 x μB = 5μB

Fe2+ = 3d6 μ = 4 μB

Co2+= 3d7 μ = 3 μB

Ni 2+= 3d8 μ = 2μB

Cu2+= 3d9 μ = 1μB

H

M+Ms

-Ms

Hysteresis curve of ferrites

Find the spin magnetic moment of Ni3+

• Applications: resistivity of ferrites are very high so suitablyapplied for high frequency application (eddy current energyloss less) and in special magnetic devices.

• Ferrites have square hysteresis loop. So used for digitalstorage device ( two values of magnetisation +Ms & - Ms; so1 or 0 )

• Soft ferrites are used for high freq. Transformer core,computer memory, hard disc, floppy disk audio videocassette, recorder head

• Hard ferrites are used for permanent magnets in generator,motor, loud speaker, telephone

• Non-volatile memory called magnetic bubbles(magneticdomains in thin films)

• Mixed ferrites are produced by combining two differentdivalent ions in suitable ratios, to obtain a specificmagnetisation desired.

Magnetic Anisotropy and Magnetostriction