Langvein Theory of Diamagnetism and Magnetic Properties of Materials, Study notes of Materials science

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Langvein Theory of Diamagnetism

Classifications of Materials Based on Magnetic field

Diamagnetic Materials

Diamagnetic materials exhibit induced magnetic fields opposite to the applied magnetic field.

They have a small negative magnetic susceptibility (χ < 0) and examples include bismuth,

copper, and gold. Diamagnetic properties are typically temperature-independent. More simply,

Diamagnetic materials are the materials in which there is repulsion in presence of magnetic field.

Paramagnetic Materials

Paramagnetic materials are faintly attracted by magnetic fields, have small positive susceptibility

(χ > 0), and examples include aluminum and platinum. Their susceptibility decreases with

increasing temperature according to Curie's law.

Ferromagnetic Materials

Ferromagnetic materials strongly attract magnetic fields and retain their magnetization after the

field is removed. Examples include iron and cobalt. They exhibit large positive magnetic

susceptibility and lose their magnetic properties above the Curie temperature.

Antiferromagnetic Materials

Antiferromagnetic materials have atomic magnetic moments oriented opposite to each other,

resulting in no net magnetization. Examples include manganese oxide and iron oxide. They

exhibit antiferromagnetic ordering below the Neel temperature.

Ferromagnetic Materials

Ferromagnetic materials have magnetic moments aligned in opposite directions but of unequal

magnitude, leading to net magnetization. Examples include magnetite and ferrites. They have

large positive susceptibility and a Curie temperature above which they become paramagnetic.

Overall, the Langevin Theory of Diamagnetism explains diamagnetic material behavior,

while paramagnetic, ferromagnetic, antiferromagnetic, and ferromagnetic materials

exhibit distinct magnetic properties. Understanding these classifications is crucial for

various applications in electronics, magnetics, and materials science.

Figure 1 .We see that the current I is moving along clock wise direction as the movement of the

electron (e) is along anti-click wise. The electrons revolve in a definite path called Orbital with

same angular velocity but in the opposite direction f the direction of the current. Therefore, these

electrons in their orbital motion behave as a tiny current loop with magnetic moment (M) and are

directed towards the opposite direction. The magnetic momentum is generally written as M = I A.

In the Absence of any External Magnetic Field

In the absence of an external magnetic field, diamagnetic materials exhibit a net magnetic

moment of zero. It happens because the magnetic moments of electrons in these materials get

effectively canceled due to mutual cancellation. Diamagnetic materials are those for which

electrons occur in pairs, each electron having a magnetic moment caused by its intrinsic spin.

That is to say, in a pair with opposite spins, the magnetic moment of one electron would be

canceled out by that of the other, causing the net spin-related magnetic moments to be zero. In

diamagnetic materials, the orbital magnetic moments of electrons, which arise due to the motion

of electrons around the nucleus, also get canceled. This again happens because of the symmetric

filling of atomic orbitals. In these configurations, the magnetic moments associated with orbital

motion get balanced in such a manner that their vector sum is zero; hence, there is no net orbital

contribution to magnetization. The Pauli Exclusion Principle asserts that no two electrons

belonging to an atom can share the same set of quantum numbers. This principle requires

electron pairing within orbitals, with each electron having its magnetic moment opposed by that

of its paired counterpart. For completely filled electron shells, the pairing leads to symmetry in

the arrangement and hence complete cancellation of magnetic moments. Consequently,

diamagnetic materials do not exhibit spontaneous magnetization in the absence of an external

magnetic field. This can be attributed to the orbital angular motion that cancels the electron spin,

thereby having a zero net magnetic moment.

In the Presence of an External Magnetic Field

Contrary to no external magnetic field, where in fact diamagnetic materials do not indicate any

net magnetic moment, the presence of an external magnetic field introduces the following

remarkable changes:

Changing Magnetic Field:

As far as an external magnetic field impinges on a diamagnetic material, it introduces a change

in magnetic flux through the material. This alteration directly affects the behavior of the

electrons within the material.

Induced Electromotive Force, emf:

The Faraday's Law of Induction says that a time-varying magnetic flux produces an

electromotive force, emf, in the material. This is due to the variation of magnetic fields.

Induced Electric Field:

This induced emf produces an electric field in the material. That may drive the currents, called

the induced currents. The currents flow in response to the external applied magnetic field.

Induced Magnetic Moment:

It is this induced electric field which will give rise to the induction of an extra magnetic moment

in the material. The induced magnetic moment comes into existence because the induced

currents, in turn, will respond to the external magnetic field change.

Opposition to External Field:

According to Lenz's Law, an induced magnetic moment opposes the direction of any change in

an applied external magnetic field. This means that the material's response generates a magnetic

moment that counteracts the change in the external field.

Δm: change in the magnetic moment

The application of the external magnetic field causes a change in the magnetic moment, Δm,

oriented opposite to the applied change in the magnetic field.

In the presence of an external magnetic field, diamagnetic materials respond such that an

electromotive force, emf, is induced as a result of a change in magnetic flux. The latter will then

give rise to an electric field that will drive induced currents and finally end with an induced

magnetic moment. The induced magnetic moment is opposite to the external field change; hence,

the change, Δm, in magnetic moment oriented opposite to the direction of applied field change.

Total Magnetic Moment

Let

B

is applied perpendicular to the plane of the paper then magnetic flux associated with

current loop

¿ BA

Where T is the time period of one rotation and d is the distance so the magnetic moment μ will

μ =

− π r

2

2 πr

( ∆ v )

μ =

− e ∆ v r

Equation number 4 gives the value one electron revolving in one orbital only. In general an atom

may have a large number of electrons and orbitals

So in general magnetic moment is μ =

− e r

2

B

4 m

Since the atoms have many different orbitals with different radii. So the mean radius is

r

2

= x

2

• y

2

• z

2

But for the spherical symmetry we have x

2

= y

2

= z

2

r

2

=¿ 3 x

2

and x

2

r

2

Equation (6) is the general case of the atom but in the case of confinement when magnetic field

is applied. Let the magnetic field is applied along (say) z axis then the atom gets confined in XY

plane as shown in the Figure 3

Figure 3

Therefore the mean radius in case of confinement and magnetic field is applied will be

R

2

= x

2

• y

2

i.e. R

2

= x

2

• x

2

= 2 x

2

R

2

2 r

2

Hence the equation (5) becomes after replacing r

2

by R

2

we have the magnetic moment (μ)

μ =

− e

2

R

2

B

4 m

As the atom with the atomic number Z will have the magnetic moment will be

μ =

− Z e

2

R

2

B

4 m

And the total magnetic moment when the diamagnetic material is placed in external magnetic

field is given by

Magnetization

The magnetization is defined as the net magnetic moment per unit of volume of a material denoted by M.

it is the representation of the degree up to which the material (dielectric in this case) is magnetized. So

mathematically;

M =

μ

total

V

Where,

μ

total

is the net magnetic movement of all molecules or atoms in the material and V is the volume

of the same material. If there are N number of atom/molecules in the given material then the

magnetization of the diamagnetic material will be

M =

− N e

2

R

2

B

6 m

Magnetization in diamagnetic substances

The diamagnetic materials when exposed to external magnetic field, the induced magnetic moment get

oriented in the opposite direction to the field applied. As a consequence of it the magnetization M

becomes negative as induced magnetic moment opposes the applied field. Such produced negative

magnetization (M= negative) is a key characteristic of all diamagnetic substances.

The magnetization in later is time independent. The reason for it is the magnetic moments of electrons

cancel out due to symmetric configuration within the material, which results in no net magnetic moment

in absence of the external magnetic field. While there is no slight change in magnetization even in the

presence of external magnetic field as induced magnetic moment does not significantly change with

temperature

μ

total

− e

2

Z R

2

B

4 m

𝜇 0

: Permeability of free space, a constant that describes the relationship between magnetic field and

magnetic force.

𝑚 : Mass of the electron.

In general, the magnetic susceptibility (χ) of diamagnetic materials is always negative, and its

value is typically very small. The range for χ dia

in diamagnetic materials generally falls between:

−

to −

−

This means that when a diamagnetic material is placed in an external magnetic field, it will be

weakly repelled by the field. This property is due to the induced magnetic moments that are

opposite to the direction of the applied magnetic field. The magnitude of the susceptibility

depends on the specific material, but it remains small and negative across all diamagnetic

substances.