Aerogel is a very porous material where, for example, 98% of the volume can be empty. The figure represents some strands of aerogel. The dimension of the figure corresponds to roughly 100 nm. More about aerogels: Los Alamos and Berkeley.

We are interested in the case that liquid ^{3}He fills the aerogel
at very low temperatures. Pure ^{3}He
goes into superfluid state
at temperatures 1 ... 3 mK, depending on pressure. When ^{3}He
is inside aerogel, the superfluid transition
is suppressed.
Experiments studying superfluid ^{3}He in aerogel
are made recently in
Cornell University,
Northwestern University,
University of Manchester, and
Stanford University.

The main effect of the aerogel on ^{3}He arises because the quasiparticles of ^{3}He are scattered from the aerogel strands. The simplest "homogeneous scattering model" replaces the strands with a uniform scattering medium. In other words, the probability for a quasiparticle to be scattered is the same at any location. This is the standard model when considering impurities in superconducting metals (Abrikosov and Gorkov, 1961). We find that this model gives results with the right tendency, but quantitatively it is quite insufficient.
See here for computer code to calculate the parameters of this model.

From the failure of the homogeneous scattering model we infer that the inhomogeneity of the scattering is crucial. One possible model of inhomogeneous scattering is to consider ^{3}He between parallel planes. This gives a better fit to the experiments, but there is a problem that this model is not consistent with the observed isotropy of the aerogel.

Apparently, a
right model should allow some random voids in the scattering medium.
This would be rather tedious to calculate. Therefore we prefer to study
a periodic lattice of voids. Moreover, we approximate the unit cell of
this lattice by a sphere. This "isotropic inhomogeneous scattering model"
can rather well account for the measured suppression of the transition
temperature. It is also superior to the homogeneous model in explaining the
measured pairing amplitude and superfluid density.
We find that aerogel is full of voids whose radius is on the order of 100 nm.
This is roughly consistent with
other measurements of the structure of aerogel.

The aerogel is
anisotropic on a short length scale. The anisotropy forms effectively a
random field on the order parameter of the superfluid.
Depending on parameters, the anisotropy can strongly affect the NMR properties.
The sudden change in the NMR frequency shift at tipping angles near 50 degrees
might be interpreted as a transition from an ordered to a disordered texture.

- E.V. Thuneberg, M. Fogelström, S. Yip, and J.A. Sauls: Localized vs. delocalized scattering in superfluid
^{3}He-aerogel, Czechoslovak J. Phys.**46**Suppl., 113 (1996). - G.E. Volovik, Glass state of superfluid
^{3}He-A in an aerogel, Pis'ma Zh. Eksp. Teor. Fiz.**63**, 281 (1996). - E.V. Thuneberg, S.K. Yip, M. Fogelström and J.A. Sauls,
Models for superfluid
^{3}He in aerogel, Phys. Rev. Lett.**80**, 2861 (1998), cond-mat 9601148. - E.V. Thuneberg: Ginzburg-Landau theory for impure superfluid
^{3}He, cond-mat 9802044, in "Quasiclassical methods in superconductivity and superfluidity, Verditz 96", eds. D. Rainer and J.A. Sauls (1998) p. 53. - R. Hänninen, T. Setälä, and E.V. Thuneberg:
Homogeneous scattering model for impure superfluid
^{3}He, Physica B**255**, 11 (1998), pdf-eprint. - R. Hänninen and E.V. Thuneberg:
Superfluid density for
^{3}He in aerogel assuming inhomogeneous scattering, Physica B**284**, 303 (2000), Proceedings of LT22. - R. Hänninen and E.V. Thuneberg:
Model of inhomogeneous impurity distribution in Fermi superfluids,
Phys. Rev. B
**67**, 214507 (2003), cond-mat 0301154. - Transparencies of a talk: Inhomogeneous impurity distribution in Fermi superfluids (4.1.2003, E. Thuneberg, pdf file, 319 kb).
- L.T.
Kurki and E.V. Thuneberg: Boundary Conditions at the Bulk-Aerogel Interface of
Superfluid
^{3}He-B, J. Low Temp. Phys.**146**, 59 (2007).

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22.4.2007, Erkki Thuneberg, Email