Monday, July 27, 2009

Röser's equation

At low temperatures, some materials display a feature known as "super-conductivity" - the total loss of electric resistance. Electrical currents in such a material, once initiated, propagate ideally forever. Super-conductivity sets in at a so-called "jump temperature," below which the material changes its electric properties. "Low" temperature in this case means indeed very low: The phenomenon was first observed on mercury with a jump temperature of only 4.19 Kelvin. Other jump temperatures for metallic compounds reach up to 23 Kelvin in Nb3Ge. Bardeen, Cooper and Schrieffer got a Nobel prize in 1972 for their theoretical explanation of super-conductivity in these materials.

These temperatures seem way too low to ever make super-conductivity useful for daily life. But over the last two decades an increasing amount of substances has been discovered, the so-called "high temperature supra-conductors," (HTS), starting with the discovery of super-conductivity in barium lanthanum copper oxide at 35 Kelvin by Georg Bednorz and K. Alexander Müller in 1986, to the recent class of so-called iron pnictides.

For them, the jump temperature can be much higher, typically above 30 Kelvin, with the current record at 138 Kelvin hold by a ceramic oxide containing thallium, mercury, copper, barium, and calcium. However, this search has up to now been rather erratic with groups of researchers more or less systematically testing promising classes of laboratory-grown crystals. The jump-temperature of a crystal so far could not be predicted.

It is then quite astonishing that Hans-Peter Röser, professor at the Institute of Space Systems at the University of Stuttgart, Germany, found a simple equation relating the geometric structure of a crystal to its jump temperature. Röser's equation for the (critical) jump-temperature Tc reads
    4 π k me(2 x)2 n-2/3 = h2/ Tc

where k is Boltzmann's constant, h is Planck's constant, me is the electron mass, x is the doping distance of the crystal and n is the number of supra-conducting layers in the crystal (it is usually 1,2, or 3). If one plots these quantities for known HTS one obtains the following graph, where the straight line is the prediction of Röser's equation:


(from: A Correlation Between Tc of Fe-based HT Superconductors and the Crystal Super Lattice Constants of the Doping Element Positions by Felix Huber, Hans Peter Roeser, Maria von Schoenermark, Proc. Int. Symp. Fe-Pnictide Superconductors, J. Phys. Soc. Jpn. 77 (2008) Suppl. C, pp. 142-144)

Now, neither Stefan nor I are specialists in super-conductivity, but this relation is quite interesting for it may harbor the possibility to better direct the search for high-temperature supra-conductors. Though one is left to wonder whether there is a way to know if a material will be super-conducting at all, it is intriguing how well the data points fit to the curve. It is however not clear to us whether the above shown curve depicts all known data points or only those that fit nicely, and how big the uncertainty of the quantity x is. In any case, the proposed relation is purely heuristic, it was obtained from accumulated data rather than derived from a theoretical model.

The relation was published in Acta Astronautica 62 733-736 (2008) and J. Phys. Soc. Japan Suppl. C 77 142-144 (2008).




Update: Check out also the continuation of the post, Röser's equation, again

26 comments:

Arun said...

It will be interesting to get a derivation from theory of this phenomenal phenomenological law.

saurabhmadaan said...

interesting... data-patterns were pretty much the driving force behind planck's amazing results. Who knows this may lead to another fundamental understanding about materials!

Uncle Al said...

Tc = Kn^(2/3)/x^2

Making stuff doesn't happen with engineers or physicists. The former diddle things, the latter diddle theory. Stuff is chemistry. Hose down some lab chiggers, stick them in the hole, throw in raw meat, wait for results.

Get the smallest unit cell with the smallest dopant. Somebody look at fluoride-doped lamellar transition metal borides, carbides, nitrides, oxides, sulfides. Zirconium phosphonates are begging to be diddled with redox sites. OLED technology is lamellar. Do something clever - no, dopey! - with h-BN or the 200+ polytypes of silicon carbide,

http://www.ifm.liu.se/matephys/new_page/research/sic/Chapter2.html

Aaron said...

Whoaaaaa! I took a glance at the paper where they propose the relationship, and it strikes me as surprisingly un-fishy. I mean, it's fishy, but not fishy enough! I don't pretend to understand the heuristics they use to define x and get the x dependence, but they look reasonable to me. And it's true that by playing with combinations of physical constants, you can get almost any quantity in any units to pop out... but these people only use a few constants, all of which are clearly relevant, and their left-over dimensionless fudge factor just happens to be almost exactly pi! Excellent.

Zephir said...

/*..it will be interesting to get a derivation from theory of this phenomenal phenomenological law..*/

The finding of empirical formula is different task from simulation of it and the derivation of formal formula from first principles is the other thing. Currently we are able to simulate behavior of electrons, which are attracted and confined to hole stripes like into channels or pipes and to estimate the point, where repulsive forces between electrons would lead into unstable and chaotic motion of electrons comparable to chaotic motion of lattice (which corresponds Tc, as I presume). Such simulation wouldn't be very difficult with current state of knowledge both in quantum mechanic arrangement (3D numerical solution of Schrodinger equation), both in its classical approximation with particle model of electrons - it's just computationally intensive. I do believe, in such way Rossler equation can be verified soon.

The another step is to derive formal equation from this model. This is completely different task: it corresponds the derivation of Reynolds criterion from Navier-Stokes equations, which can be fitted to particle simulation independently.

Another task is to propose new semiconductor structure for high Tc from known materials and the synthesis of such semiconductor is the very last job. All these tasks are highly specialized and all they require a different knowledge and way of thinking.

Zephir said...

I've a certain problem with approach of mainstream science to verification of some fundamental experiments. Where we can read for example about peer-reviewed verification of

1)room temperature superconductivity experiments of J.F. Prins
2)cold fusion experiments of Arata or US Navy research
3)Podkletnov's antigravity experiments

Kea said...

Very interesting, indeed. Brings back memories from my teenage research days: mixing highTc ceramic ingredients with a mortar and pestle, measuring their properties ... lots of fun. Hmmm.

Anonymous said...

Hi Sabine, hi Stefan,

in the theory of HTS one still searches for the elementary excitations which are able to produce HTS at these temperatures. Also of course the crystal symmetries or also broken symmetries are interesting for the theorists.

Greetings

Kay zum Felde

Phil Warnell said...

Hi Bee,

Well I have to hand it to you, for you’ve finally pointed to a paper which I can’t make much sense out of, beginning with what the graph is an indication to demonstrate. I understand that the X axis has something to do with doping distance within a crystal (whatever that is) and the Y axis somehow relates to the point in temperature when superconductivity sets in. However, the temperature component, unless I’m mistaken isn’t straight up temperature. The only thing I do have a grasp of is what a superconductor does and the promise they hold if they can ever be developed to operate at room temperature; not the least of which would be an ability to build machines like the LHC more easily and reliable.

What will be interesting to find out, as you mentioned, is if there is some theory that can be built out of Roser’s equation. That is oft times such things indicate new direction, not just in understanding existing phenomena, yet lead to future understanding and development. I remember reading for instance that even though Maxwell’s theory had indicated the existence of EM waves well beyond the visible spectrum Maxwell himself didn’t place much stock in them ever being of practical value. So one is lead to wonder where something like this may eventually lead; that is beyond levitating skate board competitions:-)

Best,

Phil

Bee said...

Zephir: During the last weeks I've deleted several of your comments and another one in this comment section. Would you please keep in mind that our blog is not the place to promote your own theory of whatever. As to your problem with mainstream science, you possibly don't find a verification because there has been no verification.

Bee said...

Hi Phil,

Well, it certainly hasn't been my intention to confuse you. The x-axis actually shows the inverse of the temperature where superconductivity sets in (multiplied by some constants). I'm not perfectly sure myself what the doping distance is, not even if that's the right word (the German word is "Dotierungsdistance"). It's roughly the distance between impurities in the lattice of the crystal, the impurity also known as 'doping,' which is quite common in superconductivity research. Best,

B.

Bee said...

Oompf, the German word is of course "Dotierungsdistanz," sorry, jetlag.

Zephir said...

/*..during the last weeks I've deleted several of your comments ..*/
It's no secret, for contemporary scientists is more important to read explanation in mainstream press then to read it at all, because they're not interested about explanations at all, but about system, which provides and maintains salary for them. The [1:33 AM, July 28] just correlates this stance at another level.

Phil Warnell said...

Hi Bee,

First, let me say that you never had me to be confused, for if anything your explanation gave me at least what the researcher had accomplished in discovering some sort of mathematical correlation between doping distance and temperature. What I meant to say is if I had simply read the paper I woudn’t have had much of a clue what he had discovered. So I thank you for giving me at least that much.

I also have to apologize as I got my X and Y axes confused. I don’t know what it is for I always want to think of the vertical one as X and the horizontal as Y. I should be ashamed seeing I’m such a big fan of the inventor of Cartesian coordinates being Rene Descartes:-) It then seems I did essentially get the gist of what the graph was displaying, even though I still don’t fully grasp the significance of the measure(s). So the temperature component has been modified by the constants and as for the distance between the impurities I assume this is where the de Broglie wavelength consideration comes into play.


Best,

Phil

Bee said...

Zephir: You miss the point. This is my blog. It's not your advertisement platform. Period. I don't care if you promote Nike Shoes, Dating Hotlines or a handmade GUT - if you leave links to promote your site, I will delete them.

Bee said...

Hi Phil,

I assume this is where the de Broglie wavelength consideration comes into play.

There is no derivation of the equation, it's pretty much hand assembled. The significance of it is that previously you wouldn't have known where the jump temperature of the crystal is before you measured it. Best,

B.

Zephir said...

/*..you miss the point. This is my blog...*/
I've "my blog" too and I'm not prohibiting anybody in promotion of private theories and ideas.

It means, while I respect your private property and decision - I really don't think, it's a true reason of your stance. It's just a (necessary) condition of it.

Bee said...

Well, you're welcome to do on your blog what you want, but if you come here, you play according to my rules, or you don't play.

Peter Orland said...

It is interesting that they only claim this for iron-based high-Tc materials. This suggests that this formula doesn't apply to the cuprates.

Bee said...

Do you happen to have a data point for these that doesn't fit on the curve?

Peter Orland said...

No, I don't know the data well enough. But the papers only concern Fe-based materials. If they had something similar for the cuprates, the authors probably would have said so.

Zephir said...

/*..they only claim this for iron-based high-Tc materials. This suggests that this formula doesn't apply to the cuprates...*/

Nope, they don't:

"...This equation was used to test the correlation for various kinds of HTSCs, including oxygen deficiency doping and cuprates. The Tc used was always the center of the transition by using the mean of the onset temperature and the end of the transition as published in other groups’ papers....

... A correlation was found between the doping positions and Tc of iron based superconductors. This correlation fits well to the condition found in cuprates giving rise to the assumption that there is a uniform mechanism within the various types of superconductors forming a resonance between the lattice and the charge carriers that finally leads to superconductivity."

Peter Orland said...

Yes, you are right. I guess somehow I only noticed that one of the papers concerns Fe, the other Cu

Georg said...

Hello,
The Materials on the lower left
side with Bi, Tl or Yttrium
plus some Number as a name are
Cuprates.
This correlation is much too precise
for not having a "meaning".
Georg

Zephir said...

This correlation simply means, temperature of HT superconductivity transition is driven by pure geometry, i.e. density of electrons, which are compressed mutually in superconductor lattice.

Tumbledried said...

A remarkably interesting relation, and such a precise fit to the known data. I had not seen this before. Thanks for sharing!