  According to the laws of magnetism, the shape of magnetically soft objects
limits the effective susceptibility. For example, spherical soft magnets cannot
display an effective susceptibility larger than 3. Although true for
macroscopic multi-domain magnetic materials, we show that magnetic
nanoparticles in a single-domain state do not suffer from this limitation. This
is a consequence of the particle moment being forced to saturation by the
predominance of exchange forces, and only allowed to rotate coherently in
response to thermal and/or applied fields. We apply statistical mechanics to
determine the static and dynamic susceptibility of single-domain particles as a
function of size, temperature and material parameters. Our calculations reveal
that spherical single-domain particles with large saturation magnetisation and
small magneto-crystalline anisotropy, e.g. FeNi particles, can have very a
large susceptibility of 200 or more. We further show that susceptibility and
losses can be tuned by particle easy axis alignment with the applied field in
case of uniaxial anisotropy, but not for particles with cubic anisotropy. Our
model is validated experimentally by comparison with measurements on
nanocomposites containing spherical 11$\pm$3 nm $\gamma$-Fe$_2$O$_3$ particles
up to 45 vol% finely dispersed in a polymer matrix. In agreement with the
calculations for this specific material, the measured susceptibility of the
composites is up to 17 ($\gg$3) and depends linearly on the volume fraction of
particles. Based on these results, we predict that nanocomposites of 30 vol% of
superparamagnetic FeNi particles in an insulating non-magnetic matrix can have
a sufficiently large susceptibility to be used as micro-inductor core materials
in the MHz frequency range, while maintaining losses below state-of-the-art
ferrites.
