lecture 4_ para magnetism

These materials don't possess permanent dipole and hence the magnetic effects are smallSince the induced magnetic moment always oppose the applied field, the diamagnetic susceptibility is negative.

It is always temperature independent.

Diamagnetic materials usually repel the magnetic line of force.

Summary Diamagnetism

e.g.: Bi, H2O, CO2, Ge, Si etc.

Material Χ = (µr -1)

Cu -9.5x10-6

Al2O3 -5.0x10-6

Au -3.7x10-5

Ge -0.8x10-5

Si -0.3x10-5

Se -1.7x10-5

He -0.5x10-5

Para-magnetism

• B>B0 , is positive, μr>1, .

• The permanent magnetic moment results from the following contributions:

1. The spin or intrinsic moments of the electrons.

2. The orbital motion of the electrons.

3. The spin magnetic moment of the nucleus.

A form of magnetism which occurs only in the presence of externally applied magnetic field and materials are attracted to magnetic field.

Paramagnetism

• Paramagnetism is observed in:1. Metals2. Atoms and molecules possessing an odd number of

electrons, that is free sodium atoms, gaseous nitric oxide etc.

3. A few compounds having an even number of electrons (example Oxygen molecule)

4. Free atoms or ions having a partially filled inner shell e.g. rare earth and actinide elements, ions of some transition elements such as Mn2+ Manganese, platinum, tungsten, some members of rare earth group and ions formed by removing and adding electrons to basic atoms there by creating unpaired spins.

Basic assumptions of Langevin’s classical theory

• The theory considers the paramagnetic solids in terms of paramagnetic gas, in which each particle is assumed to bear a permanent magnetic moment .

• Mutual interaction between dipoles is assumed to be negligible.

• Orientation of permanent magnetic dipole moment.

Classical Theory of Paramagnetism

T

CCurie’s law :

)(

T

CCurie-Weiss law :

. Langavin’s analysis

. Temperature dependence of paramagnetism

- Paramagnetism has net magnetic moments:

• No field : M=0 • Field is applied, low Temp.

B• Field is applied, and High Temp.

B

T

Classical Theory of Paramagnetism

:μ(vector)

cosBEP

• The Magnetization(M) =The total magnetic moments per unit volume

0cos

0cos

sin

cossin

de

dena

a

n dnM 0 cos n a deKdnn 0 0

cos sin2 0cos cossin2 deK a

0

cossin

2d

nK

ea

B

. Langavin Function

• Let dn be the number of moments inclined at an angle between & +d•When no field is applied dn dA (solid angle)•When field is applied dn is proportional to dA x Boltzmann factor

.)sin(cos

exp2exp dkT

BK

kT

EKdAdn p

aan

aee

een

dxe

dxexn

aa

aa

ax

ax

1coth

1

11

11

ddx

x

sin

cos

aa

M

M 1coth

0

nM 0

Langavin function :

945

2

453)(

53 aaaaL

T

Ba

• Conclusions from Langavin function: 1. Saturation will occur if a is large enough. ⇒ Large B or low T is necessary 2. At small a, the magnetization M varies linearly with B.

. Relationship between Langavin theory and Curie’s lawFor small a, L(a)=a/3=B/3kT, M=M0 L(a)

.

.3

,

32

0

200

kTBforvalid

constCuriek

nCwhere

T

C

kT

n

B

M

HM

χ is called paramagnetic volume susceptibility, n is number density.

Mass susceptibility :

kgNumber

mNumber

N

N

mm /

/,

3

Molecular Susceptibility :

N should be replaced by NA

Understand some more formulae from S.O.Pillai – for Numerical

Note:

. Curie-Weiss law; Wiess Theory of Paramagnetism:

Lengevin’s theory failed to explain the complicated type of dependence of susceptibility upon temperature exhibited by many paramagnetic substances

e.g. Compressed & cooled gases

Solid salts & crystals etc.

Moreover, this theory does not throw light on the intimate relationship between para & ferromagnetic materials

To overcome these problems, the concept of intermolecular field was introduced by Weiss, on assuming the mutual influence of magnetic moments.

Molecular field (Hi) : The interaction of elementary moments with one another

Let this internal molecular field Hi be represented in terms of its own magnetization

Hi = λ M

λ - molecular field coefficient

Quantum Theory of Paramagnetism

. Quantum theory and Classical theory cosBEP

• Classical theory : The energy of a system is varied continuously. =All values of angle are possible. •Quantum theory : The change of energy is discrete i.e is discrete

321 ,,

L

ml=+l

ml = - l

B

• The component of in the direction of the applied field eff

BJH MgMJ : Quantum number associated with J J, J-1, J-2, ……-(J-2), -(J-1), -J ※J : Integer or half-integer of 1/2 ∞∼

• The effective moments( )eff

OeergJJmc

ehg

eff/)1(

4

BJJg )1(

Quantum Theory of Paramagnetism

. Brillouin function- The procedure of derivation is the same as Langavin, except that.

(1)

(2)

BJH Mg cos

J

JJMd )(0

HMgE BJP

∴ Boltzmann factor = ee kTHMgkTE BJP //

• M = (n atoms per unit volume) ×(The average magnetic moment resolved in the direction of the field

J

J

kTHg

J

J

kTHg

BJ

e

eM

BJ

BJ

M

Mgn

/

/

J

a

JJ

J

J

JngM aBJ 2

`coth

2

1

2

12coth

2

12 `

kT

H

kT

HgMa HBJ

`,

A special case

• If

kT

JJ

kTHgM

B

BJ

3

)1(gthen

If22

Χ = M/H

μ0

Quantum Theory of Paramagnetism

• When a` is small,

`),(3

)1`(`),(

aJBngJMJ

JaaJB

B

kT

Hn eff

3

2

T

C

AkT

N

H

M eff 3

2

Ak

NC eff

3

2X =?

For detail expressions follow S.O. Pillai book

Quantum Theory of Paramagnetism

J

a

Ja

J

J

J

J

M

M

2

`coth

2

1

2

12coth

2

12 `

0

Brillouin function, B(J,a)

1) If J = infinite, B(J,a`) = `

1`coth

aa

)(),( aLaJB 2) If J=1/2(only spin contribution),

`tanh a`)B(J,0

aM

M

. Paramagnetic materials ⇒atoms or ions which have a net magnetic moment because of noncancellation of the spin and orbital component.

• Salts of the transition elements - Incomplete inner shells - Magnetic moments due almost to spin(g 2) - This metal salts obey the Curie or Curie-Weiss law with a small Θ

• Salts and oxides of the rare earths• Rare-earth elements