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Susceptibility measurement setup
Enthalpy:
What is the consequence of a conductive sample on a measure?
The drawing computes it for a cylindrical sample, but shape details matter little. The excitation induction is supposed essentially unaffected by the material since the main measured effect is about 10-5.
The magnetic moment due to conductivity results from the excitation induction and must be smaller than the susceptibility effect. To achieve 0.01× the previous 2.4µA×m2 from 0.3T/3ms in 1cm2×1cm sample, the material must conduct less than 60S/m = 16mΩ×m or 600mS/cm in the unit of the table:
Electrical Conductivity of Aqueous Solutions, in recent CRC Handbook of Chemistry and Physics
this allows all electrolytes listed there.
Metals conduct much more, up to 106× the previous limit, but can still be measured if the sample consists of insulated bands, wires or coarse powder grains up to 103× smaller as needed: D=10µm. Maybe native or grown oxide suffices to insulate the grains, but I'd trust instead a suspension in a liquid or some solid wax. Apply a law of mixtures.
Marc Schaefer, aka Enthalpy
Enthalpy:
Conductive powder can also be mixed with enough non-conductive powder of known susceptibility.
Enthalpy:
The D=60mm e=10mm Nd magnets that create the 0.3T excitation field are seriously dangerous. Copper electromagnets can be switched off, but they draw much power, even with an iron return path. Superconductors and their cooling are expensive.
Electromagnets with chilled aluminium coils are a decent compromise. Already suggested and described there, with links to data
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The mean D=60mm coils of 10mm×10mm section shall host 1600 turns of D=0.20mm aluminium wire lacquered to D=0.24mm, possibly made on purpose, that carries 6.25A.
At 20K and no field, the resistivity of not-so-pure aluminium (RRR=2000) drops to 24pΩ×m. 10kA×turn in the coil create 0.4T at its edges, where the magnetoresistance doubles the resistivity, but not everywhere. 1.3× the zero-field resistivity as a mean, or 31pΩ×m, gives each coil 0.73Ω. Together they dissipate 23W at 20K. If 0.3× as efficient as Carnot's limit, the chiller consumes 1.2kW. Locate the heat dump in the lab in winter, outside in summer. Warm copper coils would have consumed 8kW, needing active cooling too.
Helium doesn't have to flow between the turns to remove 12W. The coils, helium pipes, wires need vacuum insulation and probably multilayer insulation. Polymer yarn and straps can hold many parts firmly.
Chilled aluminium or copper sensing coils would reduce the background noise. Then the preamplifier is usefully cooled too.
Marc Schaefer, aka Enthalpy
Enthalpy:
I didn't need to show off with distributions and convolutions here on 22 Jul 2018
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because the solution for a maximally flat induction by two magnets or long coils goes without. One derivative gets rid of the dumb integral, see the illustration.
* a = b is the uninteresting I = B = 0.
* All useful solutions spread with a ≤ 0.5 ≤ b where nearly 0.5 is the Helmholtz arrangement.
* The magnets can be long. a = 0 and b = ∞ is the long coil with uniform B.
* So magnets achieve a strong induction, and coils consume less power.I didn't find immediately an algebraic solution for b versus a.
* α ≡ 1+a2 and β ≡ 1+b2 simplify little.
* Then factoring away (β-α) reduces to a clumsy 5th grade equation, oh good.
* Introducing the sum and difference doesn't seem magic.Solving by software will be trivial.
* The function of a or b is maximum at 0.5, it decreases towards 0, slower towards infinity.
* Dichotomy or other method.Marc Schaefer, aka Enthalpy
Enthalpy:
My appended command-line software computes pairs of long coils and magnets with on-axis fields and dimensions for maximally flat induction. Rename .txt to .zip, expand the archive.
========== Use
Cmd.exe.lnk launches windows 2000's console ...\system32\cmd.exe and must be adjusted to the computer. LongCoils.cpp is the Ansi C source that Borland's C++ 5.5.1 compiles to LongCoils.exe. Cmd.exe.lnk sits usefully in the folder of LongCoils.cpp.
When the user types a, b or e, LongCoils recomputes both others for d2B/dx2=0 at x=0, as on 30 Jan 2023 here. e<<R defines Helmholtz coils while e>>R makes a long solenoid, and LongCoils has given sensible outputs on both.
LongCoils supposes μr=1. This needs experimental adjustment with Nd magnets (μr~1.06), ferrite (μr 1.05 to 1.3) but is very wrong for AlNiCo.
Dimensions are in meters but scaling them all at identical H keeps B everywhere, so the user can pretend that dimensions use a different unit, or choose R=1 to use relative units. a, b, e are stored relative to R and scale when the user changes R.
B is in tesla = 104 gauss. H is the magnetic material's bigger coercive force that includes losses through the magnet; in coils it's the current×turns per length unit. Its unit is A/m = 0.012566 oersted in vacuum.
The command t displays B at some predefined positions, the command d displays positions where B drops to predefined values relative to the center. Positions within a coil should be correct. A position very close to 2×b would be wrong.
d2B/dx2 and d4B/dx4 are represented by the change of B they contribute at x=0.5×R. These figures are independent of simultaneous scaling of a, b, e, R.
My integration algorithm in sum () may serve elsewhere. Its steps vary locally with the needs, so LongCoils manages r=5mm and e=1m with accuracy and speed. The last step is a Simpson one, sooner accurate than segments are at Romberg. The big sum is a tree, it accumulates less rounding error.
========== Results
The induction by long coils is more uniform than from thin Helmholtz coils but this isn't spectacular. No magic e/R value makes d4B/dx4=0 at x=0, only long solenoids approach it.
Increased e/R reduce a and the access. LongCoils seems credible for big e/R where it computes the end effect.
Permanent Nd magnets can give a strong uniform induction since thickness is allowed. 2R=40mm e=10mm a=5.7mm (×2) provide 0.45T uniform to 0.5% up to ±5.6mm. For 2R=40mm e=20mm a=3.0mm (×2): 0.79T uniform to 0.1% up to ±4.4mm (collision at 3.0mm). Coils too are easier and consume less if a bit longer.
Marc Schaefer, aka Enthalpy
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