Cold enough for you ?


It’s the Netherlands, I have to have a picture of windmills…

Left Amsterdam two days ago. A very pleasant stay it was, although the weather has not been cooperating much (rained almost non stop over the past five days). Things are no better in Egham (UK), where I am staying until next week. Not only is it raining here too (not surprising an occurrence in England, I suppose), it is actually quite chilly…
Altogether, I am attending four different conferences during this European trip (the other three in Wroclaw, Egham and Trieste). The meeting that I attended in Amsterdam is a decent size one by physics standards (approximately 1300 delegates), with presentations and posters spanning different areas of experimental and theoretical research, mostly in condensed matter and atomic physics. Actually, though, all four are centered around the same general broad theme, namely low temperature physics.


Photo taken at conference banquet, Krasnapolski Hotel. Thankfully, dress code is “casual” at physics meetings anything

Among the subjects that are most interesting to me, one that was extensively discussed at the LT25 conference is the elusive Supersolid phase of helium (I have written about it some time ago). There have been a number of interesting experiments over the past few months, but the jury is still out on whether what has been seen in laboratories worldwide over the past five years, is indeed what has been initially claimed, possibly the most fascinating phase of matter ever discovered.
I presented my work on hydrogen clusters, about which maybe I shall write in a future blog post.

“Low” Temperature
What is considered low temperature ? Why is the field of “low temperature physics” interesting ?
For humans, the idea of cold is usually associated with places such as Canada, Alaska or the two Poles, but no temperature that our body can withstand without serious harm, even if only for few minutes, would be regarded as anywhere near “low” by the standards of low temperature physics. This is because, aside from hypothermia, frost bites and slippery sidewalks, in fact even down to temperatures as low as, say, -400 F (-240 oC), at which life as we know it would have long ceased to exist, the physical behavior of matter is not, for the most part, fundamentally different than at room temperature (say, 75 F). Indeed, the laws of Classical Mechanics state that assemblies of large numbers of atoms and molecules (i.e., matter as we know it), will behave in exactly the same way, given specific initial positions and velocities of all particles, regardless of the temperature. The only difference is the time scale at which things will take place, everything happening more slowly, the lower the temperature. Otherwise, the evolution of the system will be the same at any two different temperatures.

Things start getting more interesting at -452 F (-269 oC), when helium, the second most abundant element of the universe (hydrogen being the first) liquefies. This remarkably low temperature, at which all other known substances are frozen, is in many respects the demarcation line (a fuzzy one, to be sure, like pretty much everything in physics) between “high” (classical) and “low” (quantum mechanical) temperature regimes, the latter extending all the way to the theoretical lowest temperature in nature (one that can be approached arbitrarily but not quite reached — also known as absolute zero), namely -460 F (-273 oC).
It was first ingeniously achieved exactly 100 years ago, in the laboratory of Heike Kamerlingh-Onnes at the University of Leiden. After succeeding in making the helium gas transition to a liquid phase, Kamerlingh-Onnes attempted to freeze it, i.e., turn it into a solid, by lowering the temperature even further. Any other liquid substance known to man forms a solid crystal at sufficiently low temperature, under the pressure of its own saturated vapor. Not helium though, it turns out.

Much to Kamerlingh-Onnes’ own, and to the continuing bewilderment and fascination of many of us, the helium liquid did not (and does not) freeze. Indeed, it stays a liquid arbitrarily close to absolute zero, without solidifying [0]. Such a failure of helium to crystallize is a manifestation of Quantum Mechanics on a macroscopic scale (by this I mean big chunks of matter, i.e., a glass of liquid helium, not just individual atoms or molecules) [1].
It is by no means the only, or most remarkable manifestation of quantum behavior that one can detect at these temperatures. The most important is arguably persistent flow, either in helium itself (Superfluidity), or in materials brought to the same temperatures, which become capable of conducting electric current without loss (Superconductivity) [2].

Why does liquid helium not freeze ?
It is possibly best to address this issue by discussing why everything else does, first. Classical mechanics predicts that, in the limit of very low temperature, everything should come to a complete “stop”. As thermal motion vanishes, atoms and molecules of any substance should form regular crystalline arrays (solids), which minimize the energy of mutual interaction among particles. Such an orderly configuration, which takes on a different characteristic particle arrangement (“crystal structure”) for different substances, is referred to as classical ground state; it is the state of lowest total energy, as predicted by classical mechanics. The specific freezing temperature for a given system depends on the interaction among atoms or molecules, but according to Classical mechanics everything should freeze, at some low temperature.

Quantum mechanics introduces an additional ingredient in the above description, namely zero point motion. This means that, even at absolute zero, “motion” as we understand it intuitively does not stop altogether. On the contrary, even atoms and molecules in a solid continue to perform a kind of oscillation around their crystalline equilibrium positions, the amplitude of such oscillations increasing the lower a) the temperature and b) the mass of the particles. For most elements, these quantum oscillations do not qualitatively alter the classical picture, i.e., the solid crystal forms anyway, albeit at lower temperatures than predicted classically. But in the case of helium, whose atoms are relatively light, zero-point motion is large; coupled with the weakness of the interaction among helium atoms, it destabilizes the solid phase at arbitrarily low temperature. In other words, the state of lowest energy is no longer the crystal, but the liquid.

Over the past decade, advances in the field of Laser cooling have taken the notion of “low temperature” to a whole new level. At this point, what was considered low temperature until only fifteen years ago pales compared to what is routinely reached nowadays in laboratories performing experiments on cold atoms. Still, there is something about liquid helium that continues to mesmerize many experimentalists and theorists.

Notes
[0] Helium can be made to crystallize (i.e., solidify), but only on applying pressure (approximately 25 atmospheres in the limit of very low temperature).
[1] Note that many important consequences and effects of Quantum Mechanics do not require that everything be cooled down to almost zero in order to be detected — indeed, they are easily observed at room temperature. More on this here.
[2] Although these phenomena have first been observed at liquid helium temperatures, there is nothing (currently known) that may prevent them in principle from occurring at room temperature as well. While it is difficult to think of a liquid capable of superflowing at room temperature, the search for materials that may turn superconducting at room temperature is ongoing. If discovered, such materials would conceivably have important (likely revolutionary) technological applications.

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7 Responses to “Cold enough for you ?”

  1. Anonymous Says:

    the amplitude of such oscillations increasing the lower a) the temperature and b) the mass of the particles

    The lower the mass makes sense, but the lower the energy leading to increased amplitudes I think is not right, is it? At classical temperature is the opposite, the higher T the larger the amplitude, is there a point where it flips?

    • Massimo (formerly known as Okham) Says:

      the lower the energy leading to increased amplitudes I think is not right, is it?

      The lower the temperature — Yep, that’s exactly how it goes. And yes, it is the opposite trend than predicted classically.
      Keep in mind that, although it is true that the overall energy goes down with T, the kinetic energy does not go down all the way to zero. This is because, even though the thermal contribution vanishes linearly with T, the residual zero-point energy remains, and becomes dominant at low T (think of the quantum mechanical harmonic oscillator, for instance).
      Thus, the kinetic energy of a 4He atom in the equilibrium liquid phase at T = 1 K (1 degree above absolute zero) is slightly over 14 K, as opposed to 1.5 K predicted by classical statistical mechanics. The kinetic energy is the quantity that is most sensitive to quantum effects, and in fact one can use the parameter
      &eta = 1- (Eclass/E)
      to determine how “quantum” the system is (here, Eclass is the classical kinetic energy, E is the actual one). The closer to zero, the more classical the system is. In the highly quantal limit, &eta approaches a value of 1.

      The precise way of expressing what I tried to say in words is that a free particle has a thermal wavelength, given by the square root of &#x210F2/2mT (I hate Boltzmann’s constant, and therefore always set it to one). The thermal wavelength is the characteristic size of the “delocalization cloud” of the particle — i.e., of the quantum “oscillations” I am talking about. As you can see, this quantity diverges in the low temperature limit, i.e., a particle (no matter how massive) will be completely delocalized in that limit (this is just for a free particle, interactions alter this picture by promoting localization).

      Now, quantum corrections can become important even at room temperature, if the system is pressurized, and again, the measurement of the kinetic energy is the most direct indicator (typically by neutron diffraction).

      Thank you for your comment.

      • Anonymous Says:

        I’m confused. I was going to ask why zero-point oscillation amplitudes vary inversely with temperature, but I can’t tell whether you’ve already addressed that in your comment above because I can’t understand a word of it! 🙂

        Keep in mind that, although it is true that the overall energy goes down with T, the kinetic energy does not go down all the way to zero. This is because, even though the thermal contribution vanishes linearly with T, the residual zero-point energy remains, and becomes dominant at low T

        It sounds to me from this statement that:

        1. Kinetic energy is the sum of the contributions of thermal and zero-point energies

        2. As T decreases, thermal energy decreases and therefore kinetic energy decreases

        3. But kinetic energy does not decrease to zero because zero-point energy does not decrease (the “remains” makes it sound like zero-point energy is constant) with T

        4. If zero-point energy increases with decreasing T, then it does not increase as much as thermal energy decreases, otherwise kinetic energy would remain constant

        Is any of this in the right ball-park? And does this mean that there is some higher T at which zero-point energy actually approaches zero? Or can zero-point energy only be detected at zero-point when its “signal” is not swamped by thermal energy?

        Won’t you teach a poor biologist some physics? 🙂

        MH

      • Massimo (formerly known as Okham) Says:

        Kinetic energy is the sum of the contributions of thermal and zero-point energies. As T decreases, thermal energy decreases and therefore kinetic energy decreases

        As they say, “That’s the name of the game” — OK, OK, no more Abba… for today. But fundamentally, that’s the idea (things would have to be stated more precisely — in particular, the expression “thermal energy” [I am the one who is using it, to be sure] can be ambiguous).

        But kinetic energy does not decrease to zero because zero-point energy does not decrease

        Absolutely. The simplest case is that of a particle confined by some external force (e.g., the interaction with other particles) to move inside a finite region of space of size d. In that case, the zero point kinetic energy is approximately given by

        E ~ ℏ2/2md2

        where m is the mass of the particle and ℏ is a physical constant. You can never go below that, no matter how low T is.

        If zero-point energy increases with decreasing T, then it does not increase as much as thermal energy decreases

        The thing that increases with decreasing T is the size of the “quantum oscillations” that the particles perform. The energy associated to these oscillations, i.e. the zero-point energy, depends on the problem, but unless a system is free from interactions (a merely theoretical scenario), it is finite, even at T=0.

        Zero-point energy is essentially the kinetic contribution to the total energy in the ground state, namely the state of lowest total (kinetic plus potential) energy for the system under consideration.

        Or can zero-point energy only be detected at zero-point when its “signal” is not swamped by thermal energy?

        Yes, that’s precisely the idea. Another way to detect zero-point energy is to “make it bigger” by brute force. Take the above example — if the size of the confining region d becomes small, then the zero-point energy can in principle grow without bound. So, if I take liquid helium, or even neon, and compress it, all atoms will be “squeezed”, and their available space before they start running into other atoms (thereby increasing the interaction energy) become tinier.
        As a result, due to their increased spatial confinement, their zero-point energy increases, and if the compression is very high (e.g., 10000 atm or something like that — quite easily reachable in the lab), the “excess kinetic energy” can be measured, even at room temperature.

        Hmmm… why do I think that you do understand what I wrote in my previous reply ? Are you sure you do not know these things already ? 😉

  2. Anonymous Says:

    Nice post! I think I learned something. And I hope that either the UK warms up or you find a nice pub.

    Cath

  3. Anonymous Says:

    For you!

    Cath

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