The measured noise spectrum is close to (but not exactly) 1/f.

**The disparity of time/energy scales.**

What is the relationship (if any) between the noise (which is sometimes measured on time scales as long as one thousand seconds (mHz)) and microscopics (which one might calculate with quantum chemistry and/or Hubbard models, but typically involves energies larger than meV or frequencies that can be ten orders of magnitude larger)?

**Obscure trends.**

If one looks at the actual exponent alpha of the noise, 1/f^alpha. It varies in a non-monotonic way as the temperature T varies. This looks rather "random" to me (i.e. I found it hard to believe there was any systematics involved).

However, Jens Muller and collaborators have used a model due to Dutta, Dimon, and Horn (DDH) to nicely elucidate what is going on in a series of papers such as this one.

Origin of the glass-like dynamics in molecular metals κ-(BEDT-TTF)2X: implications from fluctuation spectroscopy and

*ab initio*calculations

Jens Müller, Benedikt Hartmann, Robert Rommel, Jens Brandenburg, Stephen M Winter, and John A Schlueter

Here are the basic ideas of the DDH model.

There is a distribution of relaxation times tau, which arise because there are a distribution of activation energies E for relaxation.

tau0 is a typical "attempt frequency"/molecular vibration frequency for something like a conformational change of a molecule.

One assumes that for a specific tau that the noise is simply Lorentzian. But one then averages over D(E), the distribution of activation energies.

One can then show that at a given temperature the noise has a 1/f^alpha form with an exponent given by,

A specific consistency test of the model is to then compare the measured alpha to that calculated from the above expression using the observed temperature dependence of the noise spectrum. This comparison is shown in the figure above.

One can also invert the equation above to extract D(E), giving the result in the figure below.

These two points give a better understanding of where the temperature dependence of alpha comes from; it has a reasonable explanation in terms of the distribution of activation energies.

Furthermore, the origin of the low frequency noise is the relatively large value of the activation energies. This leads to conformational transitions being extremely rare. In particular,

**I find it amazing that the noise at the Hz scale is detecting the fact that in the macroscopic crystal about every one second a single molecule (yes, just one undergoes a conformational change)!**Note that the activation energy distribution D(E) is peaked around 230 meV. This is the same energy that is deduced from studies of the activation energy for the glassy behaviour seen in NMR, specific heat, and thermal expansion. Moreover it is also the energy barrier calculated from quantum chemistry for the transition between the two conformations of the ethylene end groups (staggered vs. eclipsed) that I discussed in a recent post.

The reference given above also gives an explanation using

*ab initio*calculations as to why the presence of the glass transition depends on the chemical identity of the anion X in kappa-(BEDT-TTF)2X. It relates to the relative strength of the bonding between X and the ethylene end groups of the BEDT-TTF molecules.

One thing that is not clear is what determines the width of the distribution D(E).

There are subtleties that I have glossed over here and other interesting things but the aim of this post is to focus on the big picture and some of my basic puzzles.

I thank Jens Muller for a very helpful discussion about his work.

Interesting indeed.

ReplyDeleteA remark/a word of caution:

Browsing (not properly reading yet) the paper by Mueller et al, I get that the relaxation involves cooperative movement of molecules (or molecular groups).

Then one has to be careful interpreting the data in terms of a distribution of (independent) activation energies: the activation energies of a system where cooperative (correlated, or maybe very specific: hierarchical) events happen may not reflect the activation energies of different individual steps.

(As with so many things: ) This was recognized by Phil Anderson in 1984.

Palmer et al, PRL 53, 958 (1984).

The correspondence between the maximum of the extracted activation energy distribution and the calculated activation energy barrier for conformal changes may (...) therefore be fortuitous,

Though in a system where part of the events are indeed independent (in Anderson's terminology: parallel) and part are in series (hierarchical), this would depend on the relative fraction of independent and correlated events.