2023 Author: Bryan Walter | [email protected]. Last modified: 2023-05-21 22:24
The book of physicists Andrey Varlamov, Attilio Rigamonti and Jacques Villein “Physics of Everyday Life. From soap bubbles to quantum technologies”(“Alpina non-fiction”), translated into Russian by Maria Prilutskaya, is addressed to those who want to understand the amazing natural phenomena that surround a person. The authors consider the natural manifestations of physics that we encounter on a daily basis, how some inventions work, and answer questions related to gastronomy. Finally, the final part of the book discusses how quantum mechanics is changing the way we view nature. N + 1 invites its readers to read a fragment that tells about the history of the study and the unusual properties of helium.
The second element of Mendeleev's periodic table, helium, is perhaps the most interesting to researchers due to its unusual properties. And although it brought insomnia and headaches to scientists, they were rewarded with the beauty of the mechanisms that explain its characteristics.
Liquefaction of helium
The attentive reader already knows from previous chapters that helium becomes liquid only at very low temperatures and does not solidify at atmospheric pressure (see p. 244). Instead, at an even lower temperature, it becomes superfluid, that is, devoid of viscosity (see p. 261). Helium was first liquefied by Kamerlingh Onnes in his Leiden laboratory on July 10, 1908 (Fig. 1). For several months, competition continued with other researchers trying in vain to turn this gas into a liquid. Helium, the only one of all the elements, stubbornly remained gaseous … Kamerling-Onnes was sure that he not only liquefied helium, but also received it in the solid phase back in March 1907. Indeed, immediately after a rapid decrease in pressure, he observed the formation of a whitish cloud in gaseous helium and, without much hesitation, found it solid. Delighted, he telegraphed to his colleague Sir James (Scottish physicist and chemist James Dewar (1842-1923), the first to liquefy hydrogen): "Received solid helium." This achievement has been widely reported in the international press. Alas, the whitish cloud turned out to be composed of hydrogen droplets, which treacherously penetrated into helium! Poor Kamerling-Onnes was ridiculed by his compatriots: they ironically pointed out that instead of solid helium, he found only hal ﬁ um (the word half means “half” in Dutch, while heel means “whole”). Moral: 1) great people are wrong; 2) great scientists make premature conclusions, and you should not imitate them! However, the liquid obtained on July 10, 1908 was indeed helium.
1. Kamerlingh Onnes at the Leiden Laboratory, where he and his colleagues were the first in the world to obtain liquid helium. In 1913, the scientist received the Nobel Prize in Physics for studying the properties of matter at low temperatures.
This major technological advance opened up entirely new experimental avenues for researchers. By cooling devices with liquid helium, they finally gained the ability to conduct experiments at very low temperatures, close to absolute zero. In particular, Kamerlingh Onnes at temperatures below 4 K discovered the phenomenon of superconductivity of mercury (see Chapter 24, p. 255). And now we will tell a much less known story - about the unusual mechanisms of transfer of electric charges in liquid helium.
Electric charges in liquid helium
In the liquids we know, electric charges are always present, and they are relatively mobile. Thus, in water at room temperature, a significant percentage of H2O molecules dissociate into OH– and H + ions (in practice, the latter combines with a water molecule to form an H3O + ion). In liquid helium, such dissociation is completely absent and there are no "free" electric charges. With a very high probability, all atoms are in their lowest energy quantum state (see p. 238). In order for a helium atom to pass from the ground state to an excited one, it is necessary to spend an energy E of about 20 eV (1 eV = 1.6 ∙ 10–19 Joules). According to the Gibbs - Boltzmann formula, the probability that an atom at temperature T is in an excited state with energy E is exp [–E / (kBT)], where kB is the Boltzmann constant (see p. 83). However, liquid helium exists at normal pressure at temperatures below 4.2 K. At this temperature, E / (kBT) = 58,000, therefore, the probability of detecting an excited helium atom is e – 58,000, which is practically equal to 0. Even at room temperature, as the reader can easily be convinced that the probability of finding an excited helium atom is negligible. The probability of encountering an ion (for example, He +) is all the more negligible.
But at the same time, various charge carriers can be artificially introduced into liquid helium in order to measure very low currents. For example, helium nuclei He2 + carrying a positive charge are injected into it using α-rays (see p. 143). Subjecting the surface of helium to bombardment with β-rays, electrons carrying a negative charge are introduced into it. The question arises: why violate the neutrality of the unfortunate helium at all? It turns out that this venture initially leads to unexpected results, intriguing physicists, and then pleases them with unexpected explanations. To begin with, let's talk about the nature of positive charge carriers, whose riddle was solved first, and then about even more unusual negative charge carriers.
Structure and effective mass of a positive carrier
Physicists began to take an interest in these questions at the end of the 50s of the last century. They measured the mass of charge carriers in liquid helium by studying their trajectories in a constant magnetic field. A charged particle moving in a magnetic field with a certain initial velocity describes a spiral, the radius of which depends on the mass of the particle. The measurement results turned out to be quite unexpected: in liquid helium, the mass of carriers of both negative and positive charges was tens of thousands of times greater than the mass of a free electron! Another surprising discovery concerned the mobility of He + ions in liquid helium, that is, the ratio of their velocity to the force moving them. The mobility of atoms of the helium isotope 3He in the most common isotope 4He was already known at that time, and it was expected that the mobility of He + ions would be of the same order of magnitude. However, it was found that for He + ions this value is about 100 times less. How do you explain this new helium quirk?
The solution was found by the American physicist Kenneth Robert Atkins and described in his article in 1959. According to his theory, the presence of the He + ion creates a disturbance in the surrounding helium atoms. This positive ion attracts their electrons to itself and at the same time repels their nuclei (this phenomenon is called polarization of atoms). Due to the small difference in distances, attraction prevails over repulsion, so atoms approach the He + ion: their concentration increases as they approach the He + ion, and the pressure around it increases. As already mentioned, at low temperatures and pressures of 25 atmospheres *, helium solidifies. The calculation shows that such a pressure is reached at a distance r0 = 0.7 nm from the He + ion (Fig. 2) (in order to have an idea of the scale: the radius of the helium atom is 0.13 nm). Thus, a kind of snowball grows around the ion - a ball of solid helium with an ion in the center! When a potential difference is created in the liquid, this snowball, having a charge in the center, begins to move in the direction of the electric field. In his movement he is not alone: he carries along with him a "retinue" of polarized atoms of liquid helium.
2. Local pressure as a function of the distance r to the He + ion at zero pressure (solid curve) and at an external pressure P0 equal to 20 atm (dashed line). The dotted curve is obtained by vertically displacing the solid curve.
This model was able to explain well the available experimental results, including the mass of the carrier of a positive charge exceeding by tens of times in comparison with the mass of the He + ion. According to Atkins, this mass, in addition to the mass of the solid helium snowball itself, includes two additional terms. First, to the mass of the snowball, one should also add the mass of the "suite" - the liquid carried by it into motion. Calculation shows that the latter is 28m0, where m0, = 6, 7 · 10–27 kg is the mass of the He4 atom. Secondly, when moving in a liquid, the body pushes the layers of liquid around itself, which requires energy. Therefore, to impart a certain acceleration to a body when it moves in a liquid, some additional force is required in comparison with that which would be necessary when it is accelerated in a vacuum. Thus, an object in a liquid behaves as if it had a mass m + δm greater than its actual mass m. The excess δm is the “added mass” that we talked about back in Chapter 15 (p. 171) when discussing the motion of bubbles in water. For our snowball moving in helium, the corresponding correction turns out to be 15m0. Finally, the mass of the snowball itself is the product of its volume and the density of solid helium **, which gives 32m0.
Thus, summing up all three terms, we find that the mass of a positive charge moving in liquid helium is 75m0 - a value approximately equal to the value found in the analysis of experiments.
In the above reasoning, we used the concepts of classical physics, which easily describe the motion of positive charges. However, for negative charges, everything turns out to be much more complicated …
* Recall that the atmosphere is a unit of pressure measurement, 1 atm = 100 kPa.
** This density is about 1800 kg ∙ m – 3 at a pressure of 7 million pascals (70 atm). This value is about 14 times higher than at ordinary pressure (1 atm), at which the density of liquid helium is 125 kg ∙ m – 3.
And how is the carrier of a negative charge arranged?
We have already said that liquid helium in an equilibrium state does not contain free charges. If an electron is forced into it, then the latter will cause local shocks. To talk about this, let's digress and talk about the electronic structure of atoms. There is an important law in the quantum world: this is the Pauli exclusion principle, which does not allow two electrons to be in the same quantum state at once (see p. 263). For example, a helium atom has two different states with the same minimum energy, which are occupied by two electrons. There are other energy states for electrons, but they correspond to much higher energies (at least 20 eV), and they remain unfilled. Thus, it is impossible to create an He– ion by adding a third electron to the helium atom. And yet, being accelerated to relatively modest energies of 0.5 eV, electrons penetrate into the thickness of liquid helium!
Three Italian physicists, J. Careri, U. Fasoli, and FS Gaeta, suggested that when an electron penetrates into a volume of liquid helium, the latter does not at all try to “settle down” to a free energy level in one of the atoms, “paying” for this 20 eV. No, he simply remains himself, and pushes the surrounding helium atoms apart, creating a cavity for himself and spending only 0.5 eV for this (ill. 3). The resulting "bubble" is the carrier of a negative charge.
3. Due to the laws of the quantum world, the electron cannot get too close to the helium atoms. Therefore, he "disperses" them around him.
What is the radius of this bubble? Its size is due to the balance between the forces of surface tension and the pressure of an electron on the surface. On the one hand, the formation of a bubble requires the expenditure of energy E1, which is the higher, the larger the volume of the bubble (surface energy, see p. 64). On the other hand, the electron in the bubble is continuously moving and has a kinetic energy E2, which, due to the uncertainty principle, is the higher, the smaller the bubble itself. The bubble radius R will be such that it minimizes the total energy E1 + E2. It is easy to estimate the energies E1 and E2. The first value is E1 = 4πσR2, where σ is the known surface tension of liquid helium. The energy E2 can be found from the uncertainty principle (see p. 234): according to it, the electron momentum p = mev is approximately h / R, so the kinetic energy E2 = mev2 / 2 turns out to be of the order of h2 / (2meR2), where h is Planck's constant, me is the mass of an electron and v is its speed. By minimizing the total energy E1 + E2, one can find that in the equilibrium state R4 = h2 / (me / σ). An accurate calculation gives the value R = 2 nm for the bubble radius. It practically does not have its own mass, because the electron mass is negligible compared to the added mass (see p. 171: δm = (2/3) πρR3, where ρ is the density of liquid helium at normal pressure). It should be noted here that the electron, like the He + ion, also polarizes the helium atoms around the bubble, therefore, the mass of the "suite" accompanying the bubble as it moves in an electric field should be added to δm. However, due to its large radius compared to a snowball, the effect of polarization of the surrounding helium is weak and the corresponding mass turns out to be negligible compared to the added one δm = 245m0, which determines the effective mass of the negative charge carrier in liquid helium.
Influence of pressure
What happens if liquid helium is subjected to external pressure? First of all, we are interested in carriers of positive charges, our famous "snowballs". The higher the external pressure P0, the faster the pressure of 25 atm is reached near the He + ion (Fig. 2). As a result, the size of the "snowball" becomes larger and larger with increasing external pressure (Fig. 4, red curve).
Change in the radius r of the carrier of a positive charge ("snowball") and the radius R of the carrier of a negative charge ("bubble") in liquid helium depending on the external pressure P0.
What, at this time, with an increase in external pressure, happens to the bubble - the carrier of a negative charge? Like any other bubble, it contracts with increasing external pressure (Fig. 4, blue curve). When P0 reaches about 20 atm, the bubble radius R becomes equal to the radius of the "snowball" (1.2 nm). One might think that with a further increase in pressure, the bubble will continue to contract, R will decrease. But not at all! The point is that the total pressure on the bubble surface actually turns out to be higher than the external P0, since it is necessary to add the induced pressure to it due to the attraction by the electron of the liquid helium atoms polarized by it from its "suite". It turns out that at an external pressure of 20 atm, the pressure on the bubble surface reaches those 25 atm, which are necessary for the solidification of helium. Thus, the bubble surrounds itself with a shell of solid helium and becomes a kind of ice “nut”, inside which an electron rushes randomly! A further increase in external pressure leads to a thickening of the "shell" outside, up to the complete solidification of liquid helium. The inner radius of the "nut" practically does not change when the pressure rises above 20 atm. Thus, charged bubbles in liquid helium are the centers of its freezing as the external pressure approaches the critical 25 atm. Recall how the steam bubbles in the kettle serve as the nucleus for boiling.
Let's say a few more words about what happens at pressures above 25 atm with charge carriers in solid helium. They remain the same: bubbles with a negative charge, inside which an electron rushes, and He + ions, whose "snowballs" now become infinitely large. It is clear that the mobility of charge carriers in solid helium turns out to be much lower than in its liquid phase.
Could you imagine that helium has such amazing properties? As Lev Landau noted, the quirks of helium open a window to the quantum world for us.