Monday, December 03, 2007

Magnetism and the Ising Model

Some materials have the curious property of being magnetic under normal everyday conditions - for example, they stick to the metallic door of your fridge. Technically speaking, they show a spontaneous magnetisation at room temperature, and are called ferromagnetic, for the Latin name of iron, which is the prototype of a material with these properties. It comes out, the state of being magnetic is a phase, similar to being solid or fluid, and indeed, one can study phase diagrams for magnetic materials. For example, if the temperature of a magnetic chunk of iron is raised above a certain, specific temperature, the magnetisation is lost. This temperature is called the Curie temperature for Marie's husband Pierre, a pioneer of solid-state physics. The Curie temperature of iron is at 1043 K. The appearance (or disappearance) of spontaneous magnetisation at the Curie temperature is not only technologically relevant, it is also very useful for geologists: if ferromagnetic minerals in volcanic lava cool down from red-hot molten rock to below the Curie point, they "freeze in" the orientation of the Earth's magnetic field at that very moment. This allows to reconstruct the orientation and strength of the Earth's magnetic field over history.

One goal of physicists in the early years of the 20th century was to understand how spontaneous magnetisation comes about, and to find a quantitative description of the magnetisation as a function of temperature. To this end, they made simplified assumptions, for example, that atoms behave like miniature compass needles which interact just with their neighbours. One of these models was proposed by the German physicist Wilhelm Lenz in 1920, and then analysed in more detail by his student Ernst Ising - it's the famous Ising model (Ising was born in Cologne, Germany, hence the pronunciation of the name is "eeh-sing", not "eye-sing").

In the Ising model, one assumes that the magnetic moments of atoms can have only two orientations, and that it is energetically favourable if the magnetic moments of neighbouring atoms are oriented in parallel - it costs an energy J to flip one magnetic moment with respect to its neighbour. Then, one applies the rules of statistical mechanics and tries to calculate the magnetisation - the average orientation of the magnetic moments. As it comes out, there is indeed a spontaneous magnetisation below a certain temperature - one of the most elementary examples of spontaneous symmetry breaking. And, even more spectacular from the theorist's point of view, in the special case of a restriction to just two dimensions, Onsager and later Yang (the Yang of parity violation and Yang-Mills theories) could derive an exact formula for the magnetisation M as a function of temperature. It looks pretty complicated,

but the interesting thing is that there is only one free parameter in the formula, the Curie temperature TC, which depends on the energy J necessary to flip a magnetic moment. Essentially, the magnetisation is 1 at zero temperature (meaning that all magnetic moments point in the same direction), and drops to zero as the eighth root when the temperature approaches the Curie point.

As nice as it may be to have such a formula, it would be interesting to check in an experiment if it is correct. However, there is a drawback: It's valid only in two dimensions, i.e. for planar layers just one atom thick, and it works only for magnetic moments which can only be parallel or antiparallel to one fixed direction.

Fortunately, progress in materials science in the 1990s has made it possible to produce thin ferromagnetic films only a few atomic layers thick, with magnetic moments which show indeed the restricted orientation with respect to an axis as described in the Ising model. So, these films should behave like the Ising model, and one can try to measure the magnetisation as a function of temperature. This is what is shown in this plot by C. Rau, P. Mahavadi, and M. Lu:

Figure taken from C. Rau, P. Mahavadi, and M. Lu: Magnetic order and critical behavior at surfaces of ultrathin Fe(100)p(1×1) films on Pd(100) substrates, J. Appl. Phys. 73 No. 10 (1993) 6757-6759 (DOI: 10.1063/1.352476).

It is, unfortunately, not possible to measure magnetisation directly, so one has to rely on other effects which are directly dependent on magnetisation - in this case, one uses a method called Electron capture spectroscopy (ECS): A beam of ions is shot on the film, the ions capture electrons from the surface, and emit light which can be detected. If the surface is magnetised, the light is polarised, and thus, the polarisation of the emitted light is a measure of magnetisation. This is what is plotted on the vertical axis: the polarisation P, normalised to the polarisation P0 at low temperatures. For, as it comes out in the experiment, the polarisation - and hence, the magnetisation of the film - is nearly constant at low temperatures, and drops sharply to zero when approaching a specific temperature, to be identified as the Curie temperature TC. In the figure, normalised polarisation is shown as a function of temperature T, where temperature has been normalised to the Curie temperature. Now, one can compare with the theoretical prediction for the magnetisation of the Ising model as a function of temperature. This is the solid black curve. There are no more free parameters, and, as it comes out, the agreement with experimental data is perfect.

Here is an intriguing circle from experiment to theory back to experiment: Experimental data of ferromagnets measured more than 100 years ago show the appearance of spontaneous magnetisation as temperature drops below the Curie point. Models are constructed to try to understand this, and for a simplified model restricted to two dimensions, an exact formula for the magnetisation can be derived. Finally, real materials show up which correspond to the idealisations and simplifications made in the model, the magnetisation can be measured... and it works!

This post is part of our 2007 advent calendar A Plottl A Day.


  1. Hi Stefan and Bee --

    Thanks for bringing up the subject of spontaneous magnetization at the Curie transition. It is certainly interesting physics, and also it gives me a chance to ask for your answer(s) to an old puzzler.

    Suppose we propose the following idea for a goofy kind of heat engine. Take a piece of ferromagnetic material and surround it with an coil of wire connected to some kind of (non-polarized) load. Start with the ferromagnetic piece at high temperature and no magnetic field. Connect the piece to a heat sink and cool it off to below the Curie temperature, and when the magnetic field appears it will momentarily cut through the coil and push a current pulse through the load. Now connect the piece to a heat source and raise its temperature above the Curie temperature, at which point the magnetization vanishes and the collapsing field will put generate another current pulse through the load. Repeat around the cycle and you've turned some amount of heat energy into "useful" electrical potential energy.

    This kind of engine can't produce very much power per weight of material, and so you would never be tempted to build one practically. The puzzle, though, comes when we ask about the efficiency. In principle -- at first glance, at least -- the heat source and heat sink can have an arbitrarily small temperature difference as long as one is above and the other below the Curie temperature. So the Carnot limit on the conversion efficiency (T_source-T_sink)/T_source can be set arbitrarily low. Since the engine produces a fixed amount of useful work on each cycle, this implies that the amount of heat energy that must be pulled from(sunk into) the source(sink) to drive the piece up(down) through the transition becomes arbitrarily large as the source(sink) temperature(s) approach the Curie temperature. That sounds more than a little strange to me; does it make sense to you? if not, where is the error in the puzzle?



  2. Hi Paul,

    that's a nice gadget you are proposing, and an interesting question.

    One thing is not quite clear to me: What do you want to do with the current induced in the coil? Charge a battery or something like that? The direction of the current will have opposite directions when magntisation emerges and vanishes within one cycle, and the direction of the current when magnetisation sets in will be at random from cycle to cycle, since magnetisation is spontaneous. So you will need some ratchet, in the form of a diode?

    For the analysis of the thermodynamics, one probalby should have a close look at the MdH contribution to the internal energy, the energy on the magnetic field... hm... too long ago that I've thought about these kinds of problems... Maybe someone else has an idea?

    Best, Stefan

  3. Dear Stefan,

    I think one can make a Paul-type gadget around many phase transitions.

    E.g., a solid close to its melting point at one atmosphere that expands on melting. You let the solid melt, and it does work = (one atmosphere)*(volume difference of liquid and solid). You then cool down the liquid to freeze it and repeat the cycle.

    The work per cycle is fixed; the source and sink temperatures can be arbitrarily close and above and below the melting point, and the Carnot efficiency can be arbitrarily small, and therefore the amount of heat required in this cycle seems to grow indefinitely.

    However actually, per cycle one is using up the latent heat of phase transition and in return getting the fixed P ΔV work.

    Need to think some more about this to resolve the paradox.


  4. Thanks for another great post. If I recall correctly from stat mech., originally Ising created his model assuming only 1 dimension, whhc has an exact solution. Later L. Onsager extended the model to 2D for which there is also an exact solution (as your equation there implies). But as I recall, there is no exact solution to the 3D extension of the Ising model?

  5. Hi Stefan and Bee,

    1 - There may be a means [beyond my ability] to extend this to 3D.

    Perhaps for Onsager 2D to 3D, someone may be able use to techniques that extended John von Neumann 2D to 3D [and more?] structures.

    RD MacPherson, DJ Srolovitz, “The von Neumann relation generalized to coarsening of three-dimensional microstructures”. [p1053]
    Editor’s Summary

    There was also a concise synopsis of this paper with one figure [1] on a Princeton IAS site “News Briefs”: ‘Materials Science Problem Solved with Geometry’;

    but now all I can find is this pdf summary:
    NY Times Summary

    2 - Is this thermomagnetism related in some manner to electromagnetism?
    IEEE uses CP Steinmetz "phasor" equations based upon Grassmann Algebra, that were shown in 1945 by Gabriel Kron to be related to Shroedinger's Equation.

    PhysRev v67n1-2 Jan 1945

  6. Doug,

    A lot is known about 3D Ising approximately - computer simulations, high temp expansions, low temp expansions, expansions in the number of dimensions (4-epsilon or 2+epsilon), etc. As for exact results in 3D: well, if you find some, you will be rich and famous.

  7. Hi Thomas Larsson,

    Could you provide a web reference discussing the 3D Ising?

    I would be interested in comparing this to the work of RD MacPherson and DJ Srolovitz.

    I sincerely doubt that I "will be rich and famous", but I am curious.

  8. Hi Arun, Paul,

    I am still not sure if I have understood the idea correctly.

    As I see it, this is not related to latent heat - many magnetic transitions are second order anyhow, or have only a small latent heat. And, I mean, high latent heat does not imply that you can extract large amounts of work from the cycle?

    What I am more puzzled about is this question if Paul's apparatus needs some rectifier to extract work - in this case, the machine does not work reversibly, and Carnot's argument doesn't apply anyway?

    Best, Stefan

  9. Hi Changcho,

    If I recall correctly from stat mech., originally Ising created his model assuming only 1 dimension, which has an exact solution.

    yes, you recall nearly correctly ;-) - the model had been formulated by Lenz, who was Ising's PhD thesis advisor. In his thesis, Ising calculated the partition function for the restriction to 1D (that's an quite easy application of what we now call the transfer matrix, if I remember correctly) and could show that the 1D model has no spontaneous magnetisation at non-zero temperature.

    It seems that the name "Ising model" was coined by Peierls, when he described his arguments with domain walls to show that the 2D model has spontaneous magnetisation. The 2D transition temperature was then calculated by Cramers and Wannier using their famous duality relation between the high-T/low-T expansions. And the exact solution in 2D was then obtained by Onsager...

    Best, Stefan

  10. Hi Doug,

    thank you for the reference - I am not so familiar with this kind of stuff, it may be relevant to the Ising model, but I do not see it immediately.

    But I am quite sure that you will be famous if you find an exact solution to the 3D Ising model ;-), at least among the physics geeks. I mean, exact solution says that you can write down a formula for the partition function, the correlation functions, the critical exponents, the magnetisation similar to the formula by Yang in the case of 2D, and so on. Hundreds of brilliant physicists have tried hard to find such an exact solution, so far without success. There is this story of that student at CalTech who was looking for a PhD project or so, and he asked Feynman who hinted him the Hamiltonian of the 3D Ising model in a corner of his blackboard. The student went on to the next office and asked Gell-Mann, who suggested... the 3D Ising model.

    I'm not sure, but such a solution might be important way beyond the Ising model, because the technique may work for other problems as well.

    As for all the other approximative techniques on the market, unfortunately I don't know of any recent review article. Maybe someone of our readers knows one?

    But you can get a quite good impression about what it is going on by checking out the titles of the results of an arXiv search for "3D Ising".

    Best, Stefan

  11. Dear Stefan,

    The paradox is that around a phase transition one can build an almost isothermal engine. However, per cycle, this engine seems to do a fixed amount of work (even if small), even as the source and sink temperatures become arbitrarily close.

    Paul's device needs no diode or rachet. For example, it can be used for electrolysis of water. Yes, the H2 and 02 will be mixed; but you can see that you can accumulate a lot of free energy - a nice explosive mixture :)

  12. Dear Arun,

    thank you for your comment! I'll have to give a second thought to Carnot cycles operating around a phase transition with latent heat... it's probably a typical textbook problem...

    However, I am still confused about Paul's machine. As said before, if you look, for example, at the current induced in the coil when spontaneous magnetisation sets in when crossing the Curie point from higher to lower temperature, the direction of this current will be fluctuating at random from cylce to cycle, since magnetisation is spontaneous. Thus, you cannot accumulate work, for example by electrolysis. Or am I missing something?

    One might add a small external magnetic field which breaks symmetry explicitly, but I am not so sure if the usual Carnot argument then still is valid - solid-state physicists sometimes call such fields time-symmetry breaking, for a reason. So this might by tricky? OK, I should take a break and look in some decent textbook and think about it ;-)

    Best, Stefan.

  13. Hi Stefan --

    I'm glad that you've found the heat engine question interesting. I have to say that I don't think you'll resolve anything focussing on the question of whether the current/voltage needs to be rectified. Even if you assume a polarized DC load that can only do "useful" electrical work with the right polarity, this can easily be accommodated: (1) don't connect the coil to the load or otherwise try to extract energy when the field is being formed, ie on the cooling leg; (2) one the field is established, do a small measurement to determine its direction [I think you can do this at very little energy cost]; then (3) Install a coil in the orientation you want to get the EMF polarity you want, and then collapse the field on the heating leg. (I think you can install a coil, or change one's direction, without an energy cost as long as it's not part of a closed circuit; but if you don't buy that, then just imagine surrounding the piece with a number of fixed coils in different orientations, and then just connect the load to the one that will give you the right polarity on any given heating leg.)

    I'm not 100% sure I agree with Arun's generalization to many types of phase transitions, but that's just because I'm slow and have to think about it. Of course, the whole subject of Carnot efficiencies was derived thinking about steam engines, which certainly do work around a phase transition! so I would guess that the analogous and appropriate logic is in the classical treatment somewhere.

    Lastly, I'll mention that there are such things as magnetic refrigerators; are they in some way related to this proposed heat engine?



  14. Like Stefan, I don't have any references handy, but one place to start at is Wikipedia.

    One can only compute the partition function exactly in 1D and 2D, but some exact information is known in higher dimensions. In particular, the critical exponents when d >= 4 are correctly given by mean field theory. This follows from the Wilson-Fisher renormalization group, which gave Ken Wilson the Nobel prize in 1982. d=4 is the critical dimension because the Ising model at criticality is described by phi^4 theory (a scalar field with a quartic self-interaction), and this theory is renormalizable exactly in 4D.

    On prononciation: Ernst Ising was a german jew who emigrated to the US after WWII, so he probably pronounced his name the american way during most of his long life (almost 100). He left physics directly after his PhD, disappointed that he failed his thesis project, which was to solve this simple model in 3D.

  15. Hi Thomas,

    thank you for collecting the links! It's impressive what is there already... and

    Hi Paul,

    about magnetic cooling, I have just found the Wikipedia entry on Magnetic refrigeration... now, that's some stuff to start with... I'll have to digest all that a bit...

    Best, Stefan

  16. I have managed to confuse myself thoroughly. Suppose we try drawing the cycle of Paul's engine on a temperature-entropy plot, what would it look like?

    The fixed work per cycle means even as T(source) tends to T(sink) the area in this cycle remains fixed.

    See Wiki diagram

  17. Hi Stefan and Thomas Larsson,

    1 - Thanks for the 70 arXiv references to the “3D Ising”
    I have only been able to read the first 7, but hope to read the rest over the next year.

    I have noticed two ideas:
    a - Ref_7 S. Perez Gaviro, et al, ’Study of the phase transition in the 3d Ising spin glass from out of equilibrium numerical simulations’ suggests game theory to me. I think this theme is present in other papers as well.
    b - Ref_4 D. Ivaneyko, et al, ‘On the universality class of the 3d Ising model with long-range-correlated disorder’ may be a link to the paper I referenced above: “non-magnetic impurities”, “linear dislocations, planar grain boundaries, three dimensional cavities”.

    2 - The wiki ’Phase Transition’ page is very interesting.
    a - I wonder if plasma from a star/sun can phase directly into a solid [from the diagram]?
    b - Pressure is also important.
    c- Phase changes seem to be consistent with both concepts of continuous transformation and coupling.

    3 - MRI and NMR imaging in medicine can form 3D images from “magnetic resonance” which would seem only a step or two away from 3D Ising?
    a - A Critical History of Computer Graphics and Animation Section 18:
    Scientific Visualization
    OSU example.
    b - Bioinformatics and Brain Imaging: Recent Advances and Neuroscience Applications
    UCLA example.

  18. Hi doug,

    I have only been able to read the first 7, but hope to read the rest over the next year.

    Sorry, the suggestion was serious, but I didn't want you to read all the papers, but maybe just have a look at the abstracts to get an impression about the issues that are looked at in connection with the 3D Ising model. I don't know anything about your background, so please don't mind if this reference to the arXiv was not useful to you... As a practical remark, since I am not sure about the reproduciblity of hit lists for arXiv searches, it's a good idea in general if you give the arXive numbers, say "cond-mat/0234512", for the references you quote.

    MRI and NMR imaging in medicine can form 3D images from “magnetic resonance” which would seem only a step or two away from 3D Ising?

    Sincerely, I do not see any connection. The magnetic moments of say hydrogen that are used to produce these nice images in MRI are not coupled to each other - as far as I know, they are completely independent. Moreover, they do not from a lattice, but a glass at best.

    Best, Stefan

  19. Doug,

    An Ising spin glass is related to, but not the same as, the Ising model. There is a huge literature on spin glasses, but the little I once knew about it I have long forgotten.

    But if you seriously want to study phase transition theory, the following Phys. Rep. probably contains everything you want to know (and a lot more):

    Title: Critical Phenomena and Renormalization-Group Theory
    Authors: Andrea Pelissetto, Ettore Vicari

  20. Hi Thomas,

    thanks for pointing out this review paper, I didn't know that. It's from after I've switched fields from magnetic phase transitions to heavy ions ;-)

  21. Hi Stefan and Thomas,

    Thanks for the new links [Pelissetto et al] and updates RE Ising glass, lattice and models.

    This Ising concept and phase transitions are really interesting.
    They may be a type of information transition, rather than loss of magnetic information.
    There may be some link to the Bekenstein and Hawking discussions of black hole information ~ a phase transition of information transformation?

    The 70 paper arXiv reference was helpful since I am uneducated with respect to Ising. I do read the abstracts, but gain more from scanning the paper especially if there are accompanying diagrams.

    The arXiv numbers for:
    a - S Perez Gaviro, et al, is arXiv:cond-mat/0603266
    b - D Ivaneyko, et al, is arXiv:cond-mat/0611568

    I have only a math BA with an MD [sort of like have an MS in all Nobel categories if they were preceded by bio-. I have keen interest in “bio-mathematics” which seems to be multidisciplinary with significant contributions from engineering, physics, mathematics, chemistry and economics. Once upon a time I was a naval gunnery officer with basic knowledge of ballistics with related mechanical engineering and fire control radar with related electrical engineering.


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