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 elastically isotropic.

Who is wrong? Actually, we believe both approaches are equivalent, in disguise. There are three seemingly compelling questions that need to be addressed:

  1. Is it the oxygens, or is it the aluminums?
  2. 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.

  3. Are the elastic constants nearly isotropic, or strongly anisotropic?
  4. 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.

  5. If random oxygens do as well, why make up such a fancy theory?
  6. 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.
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.

Thanks to Simon Moss for a useful discussion leading to this note.

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Last modified: January 2, 1995

James P. Sethna,

Statistical Mechanics: Entropy, Order Parameters, and Complexity, now available at Oxford University Press (USA, Europe).