My guess is that some research labs would do this work for a price.
I don't know of any commercial sources for this kind of work.

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P.E. Metallurgy, consulting work welcomed

I figured. NIST has some tools as do a few other research labs, but the typical commercial materials characterization business don't seem to, understandably.

5 Tesla, 5 Kelvin are pretty typical SQUID unit in many university labs, 14-16 T is very rare in Commerical. Florida high magnetic field has all kinds of high fields, DC up to 20T for all purpose, 32T for HTS materials. pulsed field can be over 100T. temp can reach 0.02 K.
curious about what are your "feebly magnetic and some strongly magnetic structural materials"?

The materials I am interested in are various grades of austenitic steels and various nickel alloys. They undergo magnetic transition at temperatures I care about. There exists some scant literature on transition temperatures, and some even have magnetization curves at "typical" test temperatures like liquid helium and liquid nitrogen. I need more than that for robust engineering analysis.

Strain induced martensitic transformation in austenitic stainless causes ferromagnetic. There are also thermal expansion alloys undergo a martensitic phase transformation leading to anomaly CTE. These are pretty well documented, and this often happens above liquid N2 temp. it is interested to see if lower temp, and higher magentic field can lead to anything interesting things.
There are some ASTM specs to test low permeability (1.001) for feebly materials at room temp. Is it practical to go under liquid He temp? i often think that is for theoretical studies.

Changes in saturation behavior with temperature are what I most care about. In strain hardened stainless steels generally the magnetic induction at a given field increases with decreasing temperature. Many studies were unable to fully saturate at these cryogenic fields.

It is not totally impractical to measure below liquid Helium temperatures. Quantum Design builds devices which can test in these conditions. Properties at those temperatures are rarely useful for engineering consideration.

I made a set a samples for someone once, I'll try to recall whom.
We used five or six different austenitic alloys.
It was available material, 304L, AL6-XN, Nit 40, 625, and a couple that I don't remember.
We drew samples to a range of conditions (reductions) for each alloy.
The other party was measuring cryo properties (LHe and LH2), physical and mechanical.
I never saw the data.
At the time I was doing work to control the martensitic transformation in 304 by adjusting the temperature of the draw process.
For most commercial 304L chemistries if you start with material above 135F you can reduce the amount of transformation by >85%.
It really helps to get larger draw reductions.

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P.E. Metallurgy, consulting work welcomed

@OP: are you looking at M-H curves which will never be "saturated" (flatten)? you need to convert to B-H to subtract the external H effect. From what I saw with austenitic stainless, they are easily to be saturated.
magnetic induction tends to increase as temp decreases due to decrease in thermal energy. Blocking temperature and Neel temperature are extreme examples.

@ED, yes, 304, especially 304L, is pretty easier to have martensitic transformation at "high" temperature, while 316 is in a better position to gain less magnetism, and 625 perhaps even less than 316.

@ben, I had converted the M-H to J-H, subtracting the external H effect. This study with measurements at liquid helium temperature I have used for reference. https://doi.org/10.1016/0011-2275(92)90353-C
I care about polarization differences on the order of 10 mT and at fields much higher than in this study. As Ed states the processing history of the materials matters really quite a lot, especially for these cryogenic properties, so I must make the measurements on my own materials.

Ben, Drawing 304 'warm' can nearly eliminate the martensitic transformation.
Even at very heavy cold reductions.
Modern chemistries make this a challenge to do in production.

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P.E. Metallurgy, consulting work welcomed