# Diamagnetism

The main constituent part in every material is the atom and atom has electrons which are continuously rotating around the positively charged nuclei. Due to this motion of the electron around the nuclei, a magnetic field is generated and this property of the material is known as Diamagnetism. The magnetic field developed is so weak that it does not exhibit its effect externally.

All the materials in the universe are diamagnetic. When an external magnetic field applied to the materials the orbital motion of electrons changes and a small induced magnetic moment establishes which is opposite to the direction of the external magnetic field. Under the influence of strong external magnetic field diamagnetic materials attract towards the areas where the magnetic field is weak. It is such a weak property of the material that does not affect everyday life.

Diamagnetism was discovered and named in 1845 by Michel Faraday who demonstrated that all the materials exhibit diamagnetism and it is a property of matter.

## Langevin Theory of Diamagnetism

Inside the diamagnetic materials, the orbital motion of negatively charged electrons in the atoms create a tiny current loop which produces a magnetic field. In the absence of external magnetic field, the net magnetic moment is zero because the orientation of electronsâ€™ orbit inside the atom is such that the vector sum of all magnetic moment is zero. But when the external magnetic field has applied the velocity of electrons changes and magnetic moment develops in the opposite direction to the applied magnetic field.

The current loop I produced by an electron e with charge $$-q_{e}$$ on it in per unit time is:

I= $$-\frac{q_{e}}{t}$$

So current of an atom with Z electrons in it,

I= $$-\frac{Ze}{t}$$Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â â€¦â€¦â€¦â€¦â€¦.. (1)

When a magnetic field of intensity B is applied on an electron with charge and mass m, generates an angular frequency $$\omega$$, known as Larmor Frequency and number of revolutions per unit time is $$\omega /2\pi$$. Then current produced per atom is

I= $$-\frac{Ze\times eB}{4\pi m}$$

I= $$-\frac{Ze^{2}B}{4\pi m}$$Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â â€¦â€¦â€¦â€¦â€¦â€¦.(2)

The magnetic moment produced due to this current loop is given by the product of current I loop and the area of the loop. Â Â  If â€˜aâ€™ the average radius of the loop then

$$\mu =I\times \pi \left \langle a^{2} \right \rangle$$Â  Â  Â  Â  Â  Â  Â  Â  Â  Â â€¦â€¦â€¦â€¦â€¦â€¦ (3)

Now from eq. (2) and (3), we get

$$\mu =-\frac{Ze^{2}B}{4\pi m}\times \pi \left \langle a^{2} \right \rangle$$

Simplifying the above expression, we get

$$\mu =-\frac{Ze^{2}B}{4m}\left \langle a^{2} \right \rangle$$ â€¦â€¦â€¦. (4)

If the magnetic field is in z direction and orbit plane in x-y direction then

$$\left \langle a^{2} \right \rangle=\left \langle x^{2} \right \rangle+\left \langle y^{2} \right \rangle$$

The mean square distance $$\left \langle r^{2} \right \rangle$$ of the electron from the nucleus is

$$\left \langle r^{2} \right \rangle=\left \langle x^{2} \right \rangle+\left \langle y^{2} \right \rangle+\left \langle z^{2} \right \rangle$$Â  Â  Â  Â  Â  Â  Â  Â  â€¦â€¦â€¦.Â  (5)

Assuming that the charges are distributed inside the spherical atom symmetrically then

$$\left \langle x^{2} \right \rangle=\left \langle y^{2} \right \rangle=\left \langle z^{2} \right \rangle$$

This way, from eq. (5),

$$\left \langle r^{2} \right \rangle=\frac{3}{2}\left \langle a^{2} \right \rangle$$

Or $$\left \langle a^{2} \right \rangle=\frac{2}{3}\left \langle r^{2} \right \rangle$$

Now replacing this value of $$\left \langle a^{2} \right \rangle$$ in eq. (4) we get

$$\mu =-\frac{Ze^{2}B}{4m}\times \frac{2}{3}\left \langle r^{2} \right \rangle$$

Rearranging the above equation, we get

$$\mu =-\frac{Ze^{2}B}{6m}\left \langle r^{2} \right \rangle$$Â  Â  Â  Â  Â  Â  â€¦â€¦â€¦â€¦ (6)

Now, if N = no. of atoms per volume

Then, M= $$N\times \mu$$Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  â€¦â€¦â€¦â€¦ (7)

Where, M = Magnetization (magnetic moment developed per unit volume in solid)

So, from eq. (6) and (7),

M= $$-\frac{NZe^{2}B}{6m}\left \langle r^{2} \right \rangle$$ â€¦â€¦â€¦â€¦ (8)

Since, B= $$\mu _{0}H$$ and

$$\frac{M}{H}=\chi _{dia}$$ is the diamagnetic susceptibility then

$$\chi _{dia}=-\frac{NZe^{2}\mu _{0}}{6m}\left \langle r^{2} \right \rangle$$Â  Â  Â â€¦.. (9)

From the above equation, it can easily be understood that diamagnetic susceptibility of any material does not dependent on the external magnetic field and the temperature outside.Â  But it is directly proportional to the number of atoms per unit volume. So more are the number of atoms per volume more will be the diamagnetic susceptibility of the material.

### Properties of Diamagnetic Materials

Diamagnetism is an induced magnetism of the materials which is developed when materials are exposed to the strong magnetic field. Though these materials do not exhibit magnetic properties in general when they are put in the strong external magnetic field the atoms in the diamagnetic materials produce a negative magnetization to oppose or repel the external magnetic field. This negative magnetism of the material or diamagnetism produces a repulsive force and therefore diamagnetic materials repel the magnetic field except for paramagnetic and ferromagnetic materials. Because of its repulsive behaviour, the value of diamagnetism of materials bears a negative sign.

The paramagnetic and ferromagnetic materials also have diamagnetism but the paramagnetism and ferromagnetism of these materials are so strong that suppresses the diamagnetic properties of these materials. Diamagnetism of the materials is so weak that sometimes these Materials are referred to as non-magnetic materials. The materials which do not show any kind of strong magnetic properties are diamagnetic materials or diamagnets. In diamagnets, all the electrons are paired and there is no free electron available. Once the external magnetic field arises it influences the path of electrons and realigns it and creates a net magnetic moment in them.

There is a term magnetic susceptibility which measures how much a diamagnetic material can be magnetized when exposed to the magnetic field. Magnetic susceptibility is a dimensionless value as this is a ratio of the internal magnetic field to the applied field. Also, it shows the repulsive behaviour of the material, therefore the value of magnetic susceptibility of materials bears a negative sign. For example, magnetic susceptibility of water is $$-9.05\times 10^{-6}$$. Superconductors are said to be as perfect diamagnets as the magnetic susceptibility for them is -1. These are the materials which expel all the magnetic field.

## FAQs on Diamagnetism

Q.1: What is diamagnetism?

Answer: All the materials are made up of atoms and atoms, in turn, are packed with electrons and protons. These negatively charged electrons when rotating around the nucleus containing positively charged protons a tiny current loop generates. This current loop produces a very weak magnetic field. This property of the material is known as diamagnetism.

Q.2: Give some examples of diamagnetic materials?

Answer: Diamagnetic materials are not a special kind of materials. In fact, all materials are diamagnetic in nature. Diamagnetism is possible in all phases of the materials, i.e., solids, liquids, and gases. Diamagnetic materials are those materials in which all the electrons are paired and no electrons are available freely. For example, wood, copper, gold, bismuth, mercury, silver, lead, neon, water, etc. Superconductors are the perfect diamagnetic materials as they expel all the external magnetic field.

Q.3: Define the term magnetic susceptibility?

Answer: Magnetic susceptibility is the measurement of the diamagnetic nature of the material. Magnetic susceptibility of a material is the ratio of the internal magnetic field to the external magnetic field. As this is a ratio of two magnetic fields therefore magnetic susceptibility is dimensionless.

Q.4: Water is a diamagnetic material. It is boiled at 373 K temperature. What is the change in diamagnetic susceptibility of water?

Answer: Zero. Diamagnetic susceptibility is not dependent on temperature therefore the diamagnetic susceptibility of water will be the same even after boiling. But the steam generated will have different diamagnetic susceptibility as compared to water.

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