# Explaining the reason behind the negative susceptibility of diamagnetic material

Susceptibility means how much is a particular substance allowing or tends to allow magnetic field to pass through it. Diamagnetic substance are those which repel magnetic field and henceforth its susceptibility is negative. But for paramagnetic which are weaker than ferromagnetic but still attracting the magnetic field lines into it have positive value of magnetic susceptibility . For paramgnetic susceptibility is small but positive and for ferromagnetic it nearly approaches 1.

Is it by convention that diamagnetic materials have susceptibility as negative or is there any other reason?

In the case of diamagnetism, the presence of an externally applied magnetic field $$\mathbf{B}$$ will induce a magnetisation $$\mathbf{M}$$ that acts opposite to the direction of the field. This is why diamagnetic materials expel magnetic flux from the bulk of the material. Therefore, in order to understand why the susceptibility ($$\chi$$) is negative, it is important to know why the magnetic moments of the atoms ($$\mu$$) anti-align with the field (thus resulting in $$\mathbf{M}$$ antiparallel to $$\mathbf{B}$$).

### Semi-classical intuition: Lenz's law

(Disclaimer: Whilst this provides a rough semi-classical intuition of why $$\chi < 0$$. The true explanation requires quantum mechanics, which I have alluded to at the end) Consider an electron in a circular orbit around the nucleus at radius $$r$$. Now we switch on a magnetic field, so that the field rises from 0 to $$\mathbf{B}$$ in time $$\delta t$$. Using Lenz's law, we can determine the electric field acting on the electron due to the change in magnetic flux: $$$$\oint_{circle} \mathbf{E}.d\mathbf{l} = -\frac{\partial \Phi_{B}}{\partial t}$$$$ With magnetic flux $$\Phi_{B} = B\pi r^2$$ : $$$$E = -\frac{Br}{2\delta t}$$$$ This electric field will exert a torque on the electron which, as you can check, increases the angular momentum by $$\delta L$$, where: $$$$\delta L = \frac{eBr^2}{2}$$$$ The electron is travelling in a circular current loop, so this increase in angular momentum will change the magnetic dipole moment by $$\delta \mu$$. From magnetostatics, we have $$\delta \mu = I (\pi r^2)$$, where $$I$$ is the electric current. Since current is the rate of flow of charge: $$$$I = \frac{-e}{T}$$$$ where $$T$$ is the time-period for one orbit. $$$$T = \frac{2\pi r}{v} = \frac{2\pi m r^2}{\delta L}$$$$ Plugging everything into the expression for $$\delta \mu$$, we find: $$$$\delta \mu = -\frac{e^2 B r^2}{4 m}$$$$ Therefore, the magnetisation $$M$$ (total magnetic moment per unit vol.) is given by: $$$$M = -\frac{\rho e^2 B r^2}{4m}$$$$ where $$\rho$$ is the number of atoms per unit vol. The negative sign is crucial, because when you take the derivative with respect to $$H (= B / \mu_{0})$$, you get the susceptibility $$\chi$$: $$$$\chi = -\frac{\mu_{0} \rho e^2 r^2}{4m}$$$$

### With Quantum Mechanics

Of course, electrons don't travel in circular orbits around the nucleus. Instead, they exist in orbitals/ wavefunctions around the nucleus. This means that we can only meaningfully speak of $$\langle r^2 \rangle$$ in the above expression. Assuming the magnetic field is aligned in the $$z$$-direction, the electron will be moving in the $$xy$$-plane, so we need $$\langle x^2 + y^2 \rangle$$. Assuming the atom is spherically symmetric: $$$$\langle x^2 + y^2 \rangle = \frac{2}{3} \langle r^2 \rangle$$$$ Using this, instead of $$r^2$$ in the expression for $$\chi$$, we get: $$$$\chi = - \frac{\mu_{0} \rho e^2 \langle r^2 \rangle}{6m}$$$$ A proper treatment of the quantum mechanics involves deriving the Hamiltonian for an electron in a magnetic field, before applying 1st order perturbation theory to the diamagnetic term.

There are two points to be made:

1. The prefixes dia and para are used oppositely for the relation between H and B than for that between D and E because for magnetism the associated field H is given as $$H=\frac{1}{\mu}B$$ for historical and hysterical reasons.
2. A simple explanation (perhaps too simple): $$\mu$$ is less than 1 (diamagnetic) when the magnetization M is caused the the action of B on current loops in atoms. Then Lenz's law tends to repel B. $$\mu$$ is greater than 1 (paramagnetic or ferromagnetic) when M is composed of the spin magnetic moments getting aligned along the direction of B.