# Is there a theoretical size limit by ferromagnetic materials?

Hemoglobin contains 4 iron atoms and oxidized hemoglobin is diamagnetic.1 On the other hand iron-oxide nano particles can be ferromagnetic.2 I did not manage to find any reference about the size limit where this property disappears but obviously at some point it does. By nanoparticles my impression that there is a size limit due to the production technology and not because the particle size reaches a theoretical limit. Can somebody explain why this property disappears and in theory what is the minimum size where it is present? A hemoglobin contains 4 iron atoms one in each heme ring. Would it be ferromagnetic if these iron atoms were closer to each other?

Many magnetic nano particles (MNPs) are actually superparamagnetic, not ferromagnetic. This is often a desired quality in many applications. So some MNPs that are 1 um large are actually made out of 50 nm beads, giving the entire particle a super paramagnetic property. It is certainly possible to manufacture beads this small.

A ferromagnetic material starts acting superparamagnetic once it is small enough to contain only one magnetic domain. This occurs below the Curie temperature of the material. How big this is depends on the temperature. So "small enough" is typically given by this expression:

$$T_B = \frac{KV}{k_B \log{ \frac{\tau_m}{\tau_0}} }$$

Where K is the nanoparticle’s magnetic anisotropy energy density, V its volume, $k_B$ the Boltzmann constant, $\tau_0$ the "attempt time" which is material dependent (usually 10^-9 seconds), and $\tau_m$ is the measurement time. So by setting the blocking temperature to room temperature, you can solve for V for a particular material and estimate the minimum size to be superparamagnetic.

Physically, what this means is the particle has enough thermal energy that it "flips" many times during the measurement, thus the material's magnetization averages to zero, as opposed to having a fixed remnant magnetization, like a ferromagnetic material.

Iron is not ferromagnetic in oxyhemoglobin because of its bonding which ends up giving it a filled shell. This has nothing to do with number of iron atoms.

You can see ferromagnetism in very small nanoparticles of iron, but eventually if you look at a small enough nanoparticle you start to see the iron sit in a superposition of spin up and spin down due to quantum/thermal finite size effects.

• Thanks! Can you elaborate your answer on how big the "small enough" is? – inf3rno Nov 12 '17 at 18:22
• Take a look at Kthaxt's answer for that – KF Gauss Nov 13 '17 at 0:33
• Iron atoms do not have a filled shell. And as I remember, in at least one of its forms (oxygenated or deoxygenated, I do not remember now), the heme group has a magnetic moment. Pauling measured it long ago. – Pieter Nov 13 '17 at 7:20
• Yes the atoms on their own do not have filled shells of course. You are right that unoxygenated hemoglobin have magnetic moments, but oxyhemoglobin as mentioned in the question does not. The point is that just having iron atoms does not give you ferromagnetism. – KF Gauss Nov 13 '17 at 9:02

In tiny particles of ferromagnetic substances, the magnetization direction will fluctuate spontaneously under the action of thermal energies. There is a transition temperature to a blocked state (a frozen state, permanent moment). At high temperatures, the substance will display superparamagnetism.

It is difficult to say something about sizes and/or temperatures. It depends primarily on the magnetocrystalline anisotropy, etc.