Magnetism: Part 6 - Diamagnetism

Previous posts: I, II, III, IV, V

Welcome back, folks. In the last post, we took a whistle-stop tour of the quantum mechanical principles that govern spin. Hopefully the 'take-home' message that spin is a description of the intrinsic angular momentum of an electron, was recieved. We saw that spin behaves in no such way that its name would have us expect! This post will build on what we've seen so far, and serve as an introduction to diamagnetism. There are three main types of magnetism; namely, diamagnetism, paramagnetism and ferromagnetism. We will meet the latter two forms of magnetism in later posts. 

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Levitation of small objects is possible with a diamagnet in a magnetic field. In the famous case of the 'levitating frog' (see above), the minimum criterion for levitation is met, given by: B dB/dz = u0 pg/X, where B is magnetic field, dB/dz is the rate of change of magnetic field along vertical axis, u0 is the permeability of free space, p is the density of the object/material, g is the local gravitational acceleration, and X is the magnetic susceptibility. This criterion is equivalent to stating that the force due to the induced diamagnetic field must be at least equal to the weight of the object/material, in order to observe levitation. 

Image source/credit.  

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Every material you can imagine exhibits some degree of diamagnetism. We can think of diamagnetism by considering the magnetic susceptibility, which we saw in part II. It turns out that for every diamagnet, the suspectibility is negative, and in most cases, weak. This type of magnetism is characterised by the induction of a magnetic moment within the material, which is in direct opposition to the applied field that brought it about. The prefix dia means 'against' or 'across' - we can glean some visualisation of the mechanism of diamagnetism from this link! 

It's quite tempting to make the following classical argument:

Consider a diamagnetic material in a magnetic field at thermal equilibrium.

Tthe action of the field on the orbital motion of an electron brings about a back e.m.f. (electromotive force).

This e.m.f opposes the magnetic field via Lenz's law.

This does make sense on an intuitive level, but we should note that by the Bohr-Van Leeuwen theorem, magnetism is not possible in a classical system at thermal equilibrium. Diamagnetism is a purely quantum mechanical phenomenon. We will continue this discussion with our quantum theory hats on!

Lets consider the case of an atom, which has no unfilled electronic shells. If the applied field is parallel to the z-axis, we have

and it is the case that

It's possible to show that the first-order shift in the ground state energy due to the diamangetic term, is given by

where |0> is a ket in Dirac notation, representing the ground state wave function. Now, if we postulate that our atom under consideration is spherically symmetric, then by symmetry we may write

from which we arrive at a restatement of the first-order shift in the ground state energy due to diamagnetism 

Now, lets think somewhat more generally, for a moment, by considering a solid that is comprised of N ions; each with Z electrons of (rest) mass m, enclosed within a volume V, where all electron sub-shells are filled. We can derive the functional form of the magnetisation (at T = 0) by  making use of the Helmholtz free energy, F = U - TS, where U is the internal energy of the system, T is the absolute temperature of the surroundings (modelled as a heat bath), and S is the entropy of the system. 

Interestingly (and rather deeply), the Helmholtz free energy function is a Legendre transformation of the internal energy, U, in which temperature replaces entropy as the independent variable. 

We are now able to write the magnetisation as

which allows us to extract the magnetic susceptibility in the weak-field limit (with chi << 1)

This leads us straightforwardly to the 'master equation' for diamagnetic susceptibility 

This expression has assumed first-order perturbation theory. 

If we increase the temperature of our system above zero, states which are above the ground state become progressively more important in the realisation of the diamagnetic susceptibility, but this effect is a marginal one. Diamagnetic susceptibilities are usually large and temperature independent. 

It is possible to (rather crudely) test this formulation of the diamagnetic susceptibility by plotting experimentally determined diamagnetic molar susceptibilities for a number of different ions against the ordinate Z_(eff) r^2. Here, Z_(eff) is the number of electrons in the outer shell of a given ion and r is the measured ionic radius. The major assumption in this comparative regime is that all the electrons in the outer shell of the ion have roughly the same value of <r_i>^2. It is possible to show that in this regime, we have 

Thanks for reading. If anyone has a question about this post, or magnetism in general, then feel free to leave a comment, and I'll do my best to get back to you. In the next post, we're going to take a step back, and consider the more general (quantum mechanical) picture of a single atom in a magnetic field. With a bit of luck, this exercise will give us a deeper understanding of the origins of magnetism, and indeed the concepts covered in this post. 

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References & Further Reading:

• Magnetism in Condensed Matter, Stephen Blundell, Oxford University Press, New York, 2003.
• Physics of Magnetism,  Sōshin Chikazumi,  Wiley, 1964.