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Susceptibility measurement setup

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Enthalpy:
Hello everybody!

Because it is small, like 10-5, measuring the para- or diamagnetic susceptibility (χ = µr-1) of materials is uneasy.
   Overview of Methods for Magnetic Susceptibility Measurement
   by P. Marcon and K. Ostanina
describes several setups.

The Gouy balance and the Evans balance measure a force, often for liquids
https://en.wikipedia.org/wiki/Gouy_balance
https://en.wikipedia.org/wiki/Evans_balance
http://www.iiserkol.ac.in/~ph324/ExptManuals/quincke%27s%20manual.pdf
An other method uses a Squid, which is sensitive and differential by nature, but demands cold. The less expected setup measures the tiny change of coil (or transformer) inductance due to the material, with an identical but unloaded coil for differential measurement.

I propose instead to use a constant excitation magnetic field immobile versus the measurement coil and to move the sample in and out the sensitive volume.

Then the induced voltage results from the susceptibility of the sample and nacelle. The electronics can easily integrate the voltage over time to obtain the flux variation in the measurement coils. Adequate arrangement of magnets and measurement coils, for instance as by Helmholtz, define a measurement volume where both the excitation field and the measurement sensitivity are uniform.

In the sketch, permanent magnets make the excitation field, and a rotation passes the sample in and out the sensitive volume. Other possibilities exist. Figures should follow.

Marc Schaefer, aka Enthalpy

Enthalpy:
On the previous sketch, permanent magnets create the excitation field without iron. Rare-earth and ferrite magnets have a small permeability, like 1.05, which I neglect. They are equivalent to a current sheath I = e×H flowing at their rim. This lets arrange them as Helmholtz coils
https://en.wikipedia.org/wiki/Helmholtz_coil

10mm thick Nd-Fe-B magnets with H=1MA/m are as strong as 10kA×turn, or 100A in 100 turns, wow. With D=60mm, the maximally flat induction distribution would result from 30mm spacing between the magnets if they were thin. This creates B=0.30T at the centre, dropping by 1% at +-5mm axial distance and a similar radial one. A tiny induction dip at the centre would slightly widen the good volume, but the maximally flat condition is algebraically simple:
dB/dx = d2B/dx2 = d3B/dx3
where the odd derivatives vanish by symmetry and the second by adjustment.

The magnets' thickness e spreads the equivalent current by the amount e. It's equivalent to a convolution of the current distribution by a square function of length e, and this square function is the integral of two opposed Dirac spaced by e. The convolution at the created induction incorporates some d2B/dx2 from x around the centre. For reasonable e, this can be compensated by increasing a bit the spacing. I expect an algebraic solution by this Diracs convolution method, writing d2B/dx2 with the degree change due to the integral, and solving for the maximally flat condition. I didn't check if this method is new, and don't plan to detail it. It applies to coils too.

Permanent magnets are strong but dangerous. Electromagnets could replace them, with cores to achieve an interesting induction. The cores must allow easy flux variations. A uniform induction would result from FEM optimization and CNC machining of the poles. The measurement coils can surround the exciting coils; I'd keep them distinct to reduce noises.

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The measurement coils too are arranged according to ol' Hermann. A sine current flowing in them would create a uniform induction in the measurement volume, hence induce a voltage in a small loop independent of the position there. Because the mutual induction is the same from object A to B and B to A, the same sine current in the small loop would induce the same voltage in the measurement coils from all positions in the measurement volume. This way, all points of the sample within this volume contribute equally to the measurement signal.

To compute the signal induced by the sample, of which all points are equivalent, one can integrate over the measurement coils the vector potential A created by a small magnetic dipole, which has an algebraic expression. Instead, I compute here the mutual induction from the measurement coils in the sample. 1A×t in each R=50mm coil create 18µT so a 1cm2×1cm sample receives 1.8nWb and the mutual induction is 1.8nH, from the sample to the measurement coils too. If the 1cm2×1cm sample has a susceptibility χ=10-5 in 0.30T, it's equivalent to a dipole of 24mA and 2.4µA×m2 that induces 43pV×s per turn in the coil pair, for instance 14nV over 3ms transitions of the sample in and out the excitation field.

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If each coil has 2000 turns, the signal is 86nV×s or 29µV over 3ms. D=0.29mm enameled copper make 340Ω resistance and about as much reactance at 40Hz. Integrated over 40Hz and with 2dB noise by the differential amplifier, the thermal background is 19nV, so the measure can be accurate even for smaller samples. Good amplifiers can cope with a smaller source resistance, that is, fewer turns.

The measurement coils (and their feed cables) demand shielding against slow electric fields, especially at 50Hz and 60Hz. A nonmagnetic metal sleeve can surround each coil if the torus it makes is not closed, but its ends can overlap if insulated. It's better symmetric starting from the feed cable and connected to the feeder's shield. The situation is much easier than for electrocardiograms.

100mA×2mm×0.5m at 1m distance and 50Hz induce 0.2µV in the coil pair, so a decently calm environment needs no magnetic shielding.

Averaging successive measures would further filter out interferences by the mains. The sample's cycle can be desynchronized from the mains or have a smart frequency ratio with it.

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An electric motor can rotate the sample if it's shielded by construction or is far enough. An adjustable counterweight is useful. Or let the sample oscillate as a pendulum.

The nacelle, or bottle for liquids and powders, should contribute little signal. Thin construction of polymer fibres is one logical choice. Its contribution can be measured separately and subtracted by computation.

The parts holding the magnets and coils must be strong and also stiff. A 2000 turns coil moving by 1nm versus the magnet gets roughly 100nV×s, as big as the signal, so this shall not happen at the measurement timescale. Not very difficult, but needs computational attention. The nacelle's movements must be isolated from the magnets and coils.

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If a sample keeps a permanent magnetization, it should be measured without the excitation magnets.

Metals conduct too much, but most electrolytes fit.

Marc Schaefer, aka Enthalpy

Enthalpy:
Some crystals are anisotropic and their susceptibility is a tensor. By orientation of the measurement coils, my apparatus measures the nondiagonal terms of the susceptibility.

Here a continuous rotation can't insert the sample in the measurement volume and extract it. A mechanical oscillation seems better, with enough amplitude to go over the centre or even exit the measurement volume in both directions, and can serve with the previous orientation too. The oscillatory pumps over oil wells may give inspiration. A spring can let exceed 1g if desireable. The sample's orientation is paramount, so parallel wires as sketched don't suffice: it needs at least a truss.

Marc Schaefer, aka Enthalpy

Enthalpy:
It is well known, but not by everybody, so here's a sketch of an electrostatically shielded coil.

A closed loop would allow current induced in the shield to reduce the voltage induced in the windings, so the shield is interrupted.

At such low frequencies, the ends of the shield can overlap if something prevents an electric contact, and the shield needs not be symmetric around the feed point. Nor is a symmetric cable vital here.

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I have forgotten "=0" in the conditions of a maximally flat induction at a Helmholtz coil.

Enthalpy:
"One can integrate over the measurement coils the vector potential A created by a small magnetic dipole" to compute the signal induced by the sample, but on 07/22/18 03:11 PM I took an other route. Here is the standard one, with most computations in the drawing.

The A potential by a magnetic dipole has a know algebraic expression computed from Biot and Savart
https://de.wikipedia.org/wiki/Biot-Savart-Gesetz
https://en.wikipedia.org/wiki/Magnetic_dipole

If I inject the former m=2.4µA*m2 and R=50mm I get 22pV*s per turn in the coil pair, precisely half as much as previously. I trust today's computation. The former had possibly a logic flaw because of the two coils.

The setup has much noise margin anyway.

Marc Schaefer, aka Enthalpy

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