Any dia- or para-magnetic material will experience a force when immersed in an inhomogeneous magnetic field. The Gouy method for measuring susceptibility χ. avoids the need for measuring the actual field gradient involved by using a sample that extends, with uniform cross-sectional area A, from a uniformly high-field region to a uniformly low-field region.

## Accessories

In our apparatus, the vertical force is measured by seeing it create an apparent mass change, Δ m = Fz / g, as detected on a one-pan electronic balance. If the magnetic field Bbottom can be measured (and if Btop ≈ 0 ) then this result provides an absolute measurement of χ (that is, a result not depending on the use of anyone else’s calibration of some ‘standard sample’).
A curious application of diamagnetism is that diamagnetic materials can be stably, and passively, levitated in suitably inhomogeneous magnetic fields. Straightforward theory predicts that the figure of merit for materials’ levitation is their mass susceptibility or susceptibility per unit density, χ / ρ . This apparatus readily measures the magnetic susceptibility for graphite, which possesses the highest-known room-temperature figure of merit. Our apparatus also includes a magnet array which produces magnetic fields with suitable geometry, and size of gradient, to permit a demonstration of the levitation that can be achieved.

Diamagnetic levitation​:

The Gouy Method

## Specifications

Foundational Magnetic Susceptibility Debut - APS March Meeting, 2015:

Why is it worth measuring?

Balance: 200.000-g range, 0.001-g resolution

Magnet: Field near 0.4 T, between pole faces with 25-mm diameter and 15-mm gap

Field non-uniformity over bottom of sample: 3%; field non-uniformity correction: 1/4 +/- 1/4%

Sample holder: equipped for x-, y-, and z-translations of sample in magnetic field

Samples: fit into cuvette of (10 mm)2 cross-sectional area, of height about 32 mm

Samples included: Cu, Al, Ti, Bi, Co, C (as pyrolytic graphite), Nd chloride, Gd2O3, Er2O3, Mohr’s salt, Iron alum, Copper sulfate, Copper acetate, Manganese oxide, Manganese chloride, Nickel-Zinc ferrite

Sensitivity: balance’s 1-mg resolution limit corresponds to (mks) susceptibility changes of about 1 x 10-6 .

Curie paramagnetism​:

‘Weighing the magnet’ method​

While any number of everyday materials are diamagnetic, students can certainly find special materials that are paramagnetic. Such materials are unambiguously attracted into a high-field region. Students can also discover that paramagnetism is almost always associated with transition-metal or rare-earth content of compounds, and they can measure the susceptibility of such compounds in solid or powdered form. Our Manual leads students through the issues of SI-vs.-cgs units for susceptibility, and also makes clear the various forms (volume, mass, and molar) of the susceptibility often quoted. Quantitative results of considerable precision (uncertainties of a few percent) are obtainable over a wide range of susceptibilities.

Materials possess many properties that can be measured, including their color, conductivity, density, etc. What distinguishes magnetic susceptibility as especially worth measuring? Perhaps the best answer is that magnetic susceptibility can, in many cases, directly indicate of the character of electron spins inside the atoms or molecules comprising the material. Its measurement therefore provides a direct, table-top, non-destructive, and non-invasive test of properties that can be readily (and exclusively) predicted by quantum mechanics.
Magnetic Susceptibility is a measure of the how magnetized a material becomes when it is exposed to an external magnetic field. The definition of susceptibility χ connects the magnetization M of a material (its magnetic moment per unit volume) with the external field H that is magnetizing it. For any but ferromagnetic materials, that response is linear, and is described by M = χ H . Michael Faraday was the first to measure this kind of magnetic response for ‘non-magnetic’ materials, and he invented the terms diamagnetic (for materials with χ < 0 ) and paramagnetic (for materials with χ > 0 ).
Our Foundational Magnetic Susceptibility apparatus comes with 16 prepared samples, and the facilities for students to prepare lots more. The prepared samples include some transition-metal salts, and some rare-earth compounds, so that student can measure quantitatively the susceptibility of these (paramagnetic) compounds. The student Manual will also lead students from raw data (the mass changes read off the balance) to susceptibility, to molar susceptibility, and finally to the ‘effective magnetic moment per atom’, expressed in Bohr magnetons. This is the level at which students can compare their experimental results with quantum-mechanical theory, derived from a count of d- or f-shell electrons in their material.
One of the pedagogically-useful features of the Gouy method used in this apparatus is that the sign of susceptibility is so straightforwardly measured. Despite the tiny susceptibility of liquid water ( χ ≈ -9 × 10-6 ), it is easily shown that water is repelled by a high-field region, establishing its diamagnetism. The apparatus comes with prepared samples of other diamagnetic materials, including copper, bismuth, and graphite; the graphite supplied is crystalline, and can easily display the remarkable anisotropy of susceptibility attributable to its planar crystalline layers.
Magnetic susceptibility is predicted to be zero, at any temperature, for any material governed by classical mechanics (that’s the Bohr-van Leeuwen theorem). So measuring any non-zero value is evidence against classical mechanics. Better still, for some systems, magnetic susceptibility can be predicted from first principles by relatively uncomplicated calculations based on quantum mechanics.

The magnetic susceptibility of inert-gas atoms, for example, can readily be predicted to be negative, and to have a magnitude related to the mean-square radius of atomic orbitals, ∑ < r 2 >. Those predictions can be verified (though not in this apparatus, since the numerical values obtained are so small).

By contrast, the paramagnetic susceptibility of Curie paramagnets can be measured as easily as it can be predicted, and its value provides an indirect measure of the number of spin-unpaired electron per atom. So the prediction of paramagnetic susceptibility is a test of the Aufbau Principle (which describes how electron energy levels fill in multi-electron atoms) and of Hund’s First Rule (which describes the energy-preferred spin content in multi-electron atoms).

The traditional Gouy method uses a heavy fixed electromagnet, and ‘weighs the sample’ suspended in its field, to determine the force Fz computed above. We take advantage of modern rare-earth permanent magnets to turn this method around, choosing instead to ‘weigh the magnet’. We measure Fz by seeing the magnet’s apparent mass change, as the sample in brought from a far-away location to the one illustrated above. Newton’s Third Law connects the force we measure to the force we need – and our method also provides an elegant way to use the same balance to measure, absolutely, the field B in the center of the permanent-magnet structure we use.

## The Instrument

Connecting susceptibility to theory

Extra boxes of (24) cuvettes and (25) caps, priced separately

Optional d.c. power supply (for field calibration), priced separately

# Foundational Magnetic Susceptibility

To establish the needed value of the magnetic field B in the central region of our magnetic-field structure, we arrange its field to be in the (horizontal) y-direction, and in that field we position a surrogate sample which carries an external d.c. electrical current i moving along a fixed and known (horizontal) length Lx in the x-direction. The interaction of the current with the field produces a force Fz = i Lx By , which again shows up as an apparent mass change of the magnet structure we are weighing. Correlating that mass change with the current used, we can establish the field B (whose value is near 0.4 T, or 4000 gauss) to better than 1% precision and accuracy.

What is Magnetic Susceptibility?

Paramagnetism​:

## introduction

Diamagnetism:

For the positioning shown above, the force is maximal, and has the value

$\color{White} F_z = \frac{\chi A}{ 2 \mu_0} \left( B^2_{top} - B^2_{bottom}\right)$

Absolute calibration method: