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# Magnetic Susceptibility

My students and I are preparing to go up to Bellingham on Thursday to do some work in the paleomagnetic lab up there, so we spent today’s lab meeting getting everyone acquainted with the data they are going to collect. I started explaining something in the lab meeting that I thought could use a demonstration. So here it is.

Rocks might have lots of different magnetic particles in them. They might contain magnetite, an iron oxide that forms in igneous and metamorphic rocks as well as in soils; they might contain titanomagnetite, a common  consitituent of oceanic basalt; they might contain maghemite that formed as magnetite was oxidized — rusted — by weathering, or they might contain maghemite that formed in soils; they might contain hematite or goethite, indicating soil formation in dryer or wetter environments… there are even rock-forming minerals like pyroxenes and micas that are magnetic to a certain extent. In addition to forming in different environmental conditions, all of these minerals have particular quirks in their record of Earth’s magnetic field. So we need to be able to tell the kinds of magnetic minerals apart.

One way we differentiate between magnetic minerals is by their response to weak magnetic fields. So I tried a little experiment. I put a bunch of different materials inside a wire coil. I could send a current through the coil, producing a (weak) magnetic field at the coil’s center. I also set up a magnetometer to measure the magnetic field just outside the coil.

Why might the field inside the coil be different from what I measure with the magnetometer? This is a secret that is not addresed in the physics textbook we use in Physics II: there are two different ways of describing magnetic fields. We call the magnetic field that the magnetometer $\vec{B}$, and we measure it in Tesla. But there are two kinds of things that produce that magnetic field. One is the current through the coil, and the other is whatever is inside the coil (or outside it but close by). So we say that there is an applied magnetic field (applied by the coil to whatever is inside it) as well as a little bit of extra magnetic field due to whatever we put inside the coil. In physics terms, we call the applied magnetic field $\mu_0\vec{H}$, and the bit of extra field from whatever is in the coil $\mu_0\vec{M}$. Both of these are measured in A/m. We say that:

$\vec{B} = \mu_0(\vec{H}+\vec{M})$

There are two ways you can get a little bit of extra field from putting stuff inside the coil. A large number of Earth materials become magnetic when you put them in a magnetic field, but then revert to what most people would call “non-magnetic” when the magnetic field is turned off. For example, if you put an iron-bearing garnet crystal inside an area with zero magnetic field, it wouldn’t attract a compass needle. But as soon as you turn the magnetic field on, the compass needle begins to deflect – ever so slightly – toward the garnet. We call that garnet paramagnetic. Other minerals, like quartz, are diamagnetic: put them in a magnetic field, and the compass needle deflects away from the mineral. For both paramagnetic and diamagnetic materials, the effect on the compass disappears when you shut off the magnetic field you’ve applied. We call this an induced magnetization.

Some materials also have a remanent magnetization – a magnetization that remains after the $\mu_0\vec{H}$ field is switched off. Magnetite is a good example of this. Besides behaving like an induced magnet, magnetite also has induced magnetic behavior.

So: I took pieces of a bunch of different materials – steel, teflon, hematite, and various other minerals – and put them in the middle of the coil to see what would happen to the magnetic field as I increased and decreased the current. I tried the mica in two different directions (with the edge pointed toward the magnetometer, and with the flat face 45° from the magnetometer) to see if there was an effect.

Here is a plot illustrating the response of these materials to the magnetic fields produced in the coil:

The first thing you might notice is that all materials, more or less, make a linear trend on this plot. So the total magnetic field is proportional to the applied field. The biggest effect is in the bar magnets: they are ferromagnetic (the line of points does not intersect the origin, meaning that there is some magnetic field that remains when you turn off the $\mu_0\vec{H}$ field). The rest of the materials have a different slope, varying between low (Teflon) and high-ish (empty sample holder, keys, nail).

The ratio between applied magnetic field and a material’s (induced) magnetization is called magnetic susceptibility. It is given the symbols k, κ, or χ. If you were to measure magnetic susceptibility carefully, you could identify differences between these minerals – perhaps even between the different orientations of the mica. To do that, you need to have a good idea about what the response of your magnetometer would be if your coil were empty. That’s your model for how your measurement device works. It’s just a linear equation here: $y = m x$ (using the variables we have here, $B = \chi_0\mu_0H$). You can then subtract your prediction based on the empty coil model from all of your $\vec{B}$ magnetic field measurements to see whether the stuff you put in the coil is adding to $\vec{B}$ (ferromagnetic, paramagnetic) or decreasing it.

Here is what you get when you subtract out the empty sample holder’s response:

On these graphs, a positive slope indicates a material that behaves as a paramagnet; a negative slope indicates a diamagnet. Most of these materials behave like a combination of the two – not a particularly steep positive slope (except for the bar magnets) and a variety of negative slopes. The Teflon rods have the steepest negative slope because they contain the most diamagnetic material. Because ferromagnetic materials retain a $\vec{M}$ when the applied magnetic field is reduced to zero, their behavior shows up as a vertical offset of the whole graph, as seen in the bar magnets and in the hematite, below:

Here is the R code for the graphs:

…and the data file.

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