Medium susceptibility

How strongly does this medium magnetise per unit field?

The formula

χ = φ \frac{χi}{1 + χi / 3} + (1 - φ) χc

LaTeX

χ = φ \cdot \frac{χi}{1 + χi / 3} + (1 - φ) \cdot χc

Work it out

c Carrier fluid

What the particulate is suspended in. Contributes its own small susceptibility — negative for every liquid offered.

m Particulate

The magnetic solid. Sets the saturation magnetisation and the single-domain limit.

φ Particulate loading

The solid's share of the volume. The medium's susceptibility and its saturation both scale with it.

d Particle diameter

µm

Decides which picture applies: below the material's single-domain limit, Langevin; above it, a demagnetisation-limited soft sphere.

T Temperature

°C

Only the single-domain susceptibility depends on it — thermal motion is what a superparamagnet fights.

or see a worked example with typical numbers

Compare the particle diameter with the material's single-domain limit — 15 nm for iron, 80 for magnetite, a micron for barium ferrite.
Below it the particle is one moment of Ms·V, and its initial susceptibility is the Langevin slope μ₀Ms²V ÷ 3k_BT. It grows with the cube of the diameter and is already in the tens for a ten-nanometre magnetite grain.
Above it the particle is a soft ferromagnet with an intrinsic initial susceptibility of order a thousand. The exact figure hardly matters, for the next reason.
A sphere demagnetises itself: its apparent susceptibility is χ_i ÷ (1 + χ_i/3), which is 3 for any large χ_i. Micron iron shows 2.99, not 1000.
Weight the particle's figure by the volume fraction and the carrier's by the rest. The carrier's is negative and of order 10⁻⁵ — negligible against any loading, everything at none.
The medium saturates at φ·Ms, and the field that gets it there is that over χ.

Particles that do not feel each other — the dilute limit. Above a few per cent the neighbours' fields raise the effective susceptibility somewhat (a Clausius–Mossotti-style correction), and in chains far more; treat the figure as a floor at heavy loading.
The multi-domain intrinsic susceptibility of 1000 is an order of magnitude for a soft, annealed material. A cold-worked or oxidised powder is lower, but any figure above about 30 gives the same demagnetisation-limited answer of 3 within a tenth, which is why the sphere's shape rather than the material's purity decides the number.
Low field. The susceptibility is the initial slope; the medium follows it until it approaches φ·Ms and then flattens, and the field-to-saturate output is where the two lines cross, not where the curve actually bends.
The temperature enters only the single-domain case. The carrier's susceptibility and the multi-domain figure are taken at room temperature.
Nothing here is Meyer's. The estate material specifies a speed for the medium and no loading, size or material; the calculation exists because the magnetisation inside the drive coil is 3φH for a micron iron slurry and the carrier's own few parts per million without one, and the difference is the whole question.