The Experimentalists Fit Their Own Peak: a Caveat
The comparison between our theory and the experiment looks fantastic,
until you find out that the experimentalists find a similar fit from
a model of elastic deformations around randomly arranged oxygens.
(Jiang et al., PRL 67, 2167 (1991))
We attribute the funny shape of the peaks to a nonlinear response of the
material to random distribution of aluminums substituted into the
square copper lattice: the disorder changes the desire for elastic
distortion (via a quadratic coupling), which together with the elastic
anisotropy leads to the peaks. They attribute it to a linear response of the
material to the random placements of oxygens, which sit on the bonds
between the coppers: indeed, in their model the material is roughly
Who is wrong? Actually, we believe both approaches are equivalent,
in disguise. There are three seemingly compelling questions
that need to be addressed:
Of course, the real thing we're proud of is not that we can
quantitatively model the shape of a peak in the X-ray scattering (even
if the experimentalists didn't already to a good job). It's that we've
connected the whole phenomenon with spin
glasses! If the tweed were a result of random oxygen deformations,
why would it exhibit
hysteresis? That is, on heating the tweed needs a higher
temperature than on cooling: something that happens in glasses but not
usually in elastic responses to random impurities. The mapping to a
spin glass has inspired us to make a prediction, too: the hitherto
mysterious two-way shape memory effect
we think is due to the glassy memory stored in the tweed configurations.
- Is it the oxygens, or is it the aluminums?
For the experimentalists, the oxygens are the most important degrees of
freedom. After all, the shape transition occurs when the oxygens decide
to line up into chains! (This oxygen diffusion makes the high temperature
superconductors slightly different from mainstream martensites, in a way
that doesn't matter for us here.) On the other hand, we think of the oxygens
as being very much like the elastic deformations: because they are
mobile, they relax into a configuration which minimizes the free energy.
So we lump the oxygens with the phonons as part of our order parameter,
and think of the response of the order parameter to the (static) random
arrangements of aluminums.
- Are the elastic constants nearly isotropic, or strongly
The experimentalists measure the elastic deformations either by ultrasound
or by neutron scattering: i.e. they stretch and squeeze the material
rather fast: the oxygens don't have time to rearrange in response to the
measurement. This is the right measurement to do for their model, but
when we (theorists) ask about elastic constants we need the ones where
the oxygens also come to equilibrium: if you squeeze the material in a vise
and wait several oxygen rearrangement times, how much will it deform?
There is no doubt that this will give a larger anisotropy; we've not
checked whether it's enough for our model.
- If random oxygens do as well, why make up such a fancy theory?
Actually, to get really good fits to their data, the experimentalists need to
put some correlations between oxygens. The tails of the diffraction peaks
fall off too fast for the oxygens to be random, and the experimentalists
say that the oxygens start forming into rows and the rows are broken
up by the aluminums. Our theory implicitly is a model of just this
kind of effect: if aluminum concentration raises the transition temperature
then the ordering will take place in regions far from the aluminums ...
In fact, our model has the tails of the diffraction peaks falling off
faster than the data, which (as mentioned by the experimentalists) falls
off faster than random oxygens would imply.
Thanks to Simon Moss for a useful discussion leading to this note.
Sethna's Research 90-94
Entertaining Science done at
Last modified: January 2, 1995
James P. Sethna, email@example.com
Statistical Mechanics: Entropy, Order Parameters, and Complexity,
now available at
Oxford University Press