Müller I. A history of thermodynamics. The doctrine of energy and entropy (1185104), страница 44
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From January of that year, and for the whole year,he lay in a coma after an automobile accident. They say that he passed awayseveral times but was brought back to life by drastic methods. Lifshitzpresented the award to him in the hospital. Landau survived, but not as anactive physicist.Another peculiarity of He II – apart from second sound – occurs underrotation: Since the super-fluid has no viscosity, it should be impossible toimpart a rotation to it.
Accordingly Landau – faithful to the Landau equations – predicted that the surface of super-fluid helium should remain flat,even if its container sits on a rotating turntable. That, of course, presented achallenge for experimentalists and it was not long before D.V. Osborne30came up with a rotating container of liquid helium. The surface turned outto be a perfect paraboloid, just like for any other incompressible liquid inrigid rotation, – in contradiction to Landau’s expectations.In that case it was Lars Onsager (1903–1976) who proposed an ingenioussolution of the dilemma during a panel discussion of the Osbornephenomenon.
Onsager knew that a homogeneous distribution of potentialvortices 31 mimics a rigid rotation, i.e. has the same velocity field. Thereforehe suggested that Osborne’s rotating helium was a superposition of suchpotential vortices. In this way he saved Landau’s theory and yet explainedOsborne’s experiment. Moreover, sceptics were quickly convinced, becausethe vortices could in fact be made visible when it turned out that an electronbeam could pass through the cores of the vortices and nowhere else.It remains to understand the physical reason for super-fluidity of helium.The usual assumption seems to be that this phenomenon is a case of BoseEinstein condensation, which we shall come to know later in this chapter.Adiabatic DemagnetisationThe wish to study super-conductivity of metals and super-fluidity of heliumhas motivated a drive for lower and lower temperatures. New methods wereneeded to get below 0.5K and they were found.
Peter Joseph WilhelmDebye (1884–1966) and William Francis Giauque (1895–1982) came upindependently with the idea of adiabatic de-magnetization. A magneticsalt – gadolinium sulfate in Giauque’s case – was put under a strongmagnetic field so that the magnetic dipoles of the salt lined up in thedirection of the field, because energetically that is the most favourableposition. That material – still under the magnetic field – was cooled with30D.V.
Osborne: “The rotation of liquid Helium II” Proceedings Physical Society A 63(1950).31 Potential vortices are like the vortex in an emptying bathtub, or like a tornado. Ideally theyare free of dissipation and thus should be able to exist in super-fluid helium.1866 Third Law of Thermodynamicsliquid helium and then adiabatically isolated. Afterwards the field wasslowly switched off, so that the thermal motion of the dipoles couldrandomise their orientation by sending the dipoles uphill, as it were, in theenergetic landscape, against the direction of the remaining field. This meansthat the salt was cooled and the salt in turn cooled the surrounding helium.Giauque reached 0.25K with gadolinium sulfate and later, with other salts,temperatures as low as 0.02K. The technique was refined and eventuallyproduced temperatures as low as 3mK. Further cooling proved to beimpossible in this way, because the dipoles of the electron shells start toalign themselves, so that the magnetic field had no effect, nor doesrandomisation take place.He3 – He4 CryostatsHowever, there are also nuclear dipoles, of copper (say).
In order to alignthem, very strong magnetic fields and sustained small temperatures areneeded and those can be provided by a He3-He4 cryostat. The method formaintaining low temperatures by evaporation of He3 was first conceived byHeinz London (1907–1970) in 1962. Let us consider this.Helium comes in two isotopes He3 and He4. Under natural conditionsthere are about a million times more He4 atoms than He3 atoms. But themixture can be enriched and, when this is done, it turns out that below0.87K – in the liquid phase – a miscibility gap opens up, cf. Fig. 6.8,because the now sluggish thermal motion cannot supply the energy neededto form (He3-He4)-neighbours.
Roughly speaking that gap is bell-shaped inthe (T,X)-diagram, see Fig. 6.8.32 Since He3 is lighter, it floats on top, whereit may be made to evaporate. As always for adiabatic evaporation thetemperature drops – by ǻT – and, since the light constituent is morevolatile, the system loses He3, even though the He3-rich solution on topbecomes even more enriched, cf. Fig. 6.8. Thus more He3 evaporates and soit happens that a low temperature of 10µK can be reached and maintainedfor days. The copper is eventually just as cold and the magnetic field keepsits nuclear dipoles aligned. Afterwards, when demagnetisation occurs, thedipoles randomise and the copper cools to 1.5 µK, the lowest temperaturereached so far.Physicists have so much faith in the third law that a jargon has developedamong them according to which the third law forces miscibility gaps toappear in alloys and mixed isotopes because, after all, the mixture mustshed its entropy of mixing, if the entropy is to go to zero.32I am told that the bell is not quite symmetric and that it does not seems to cover the wholerange 0 < X < 1 when T tends to zero.
For the present consideration this is not important.Entropy of Ideal Gases187Fig. 6.8. Miscibility gap in He3-He4 phase diagram (schematic). Enriching He3 in the He3 -richphase by evaporation. He3-He4 cryostat of the Physikalisch Technische Bundesanstalt inBerlin.33Entropy of Ideal GasesAlthough in this chapter we are dealing with low and lowest temperatures,we have to consider ideal gases for several reasons, but primarily becausewe wish to have further confirmation of the third law.
Also we wish tounderstand super-fluidity, perhaps.We recall that Boltzmann’s extrapolationS33k ln W with WN!· N xc !xcThe photograph is taken from an article by P. Strehlow: “Die Kapitulation der Entropie –100 Jahre III. Hauptsatz der Thermodynamik.” [The capitulation of entropy – 100 years of3rd law of thermodynamics] Physik Journal 4 (12) 2005.1886 Third Law of Thermodynamicswas seriously flawed, cf.
Chap. 4. The basic reason is the way of countingrealizations of the distribution {Nxc}, because Boltzmann believed – aseverybody did in his time – that an interchange between identical particlesin different elements of the (x,c)-space leads to a new realization.According to quantum mechanics of many particles this is not the case.Also Boltzmann could not know about bosons and fermions and de Brogliewaves. So, if we wish to repair Boltzmann’s reasoning, we have to take twoobservations from quantum mechanics into account:x There is no way to distinguish between identical particlesThe classical idea is that we may mark particles, e.g. paint them in different colours.But this is not only impractical, it is incompatible with quantum mechanics, where theparticles are de Broglie waves, as it were.x There are two types of particles, fermions and bosonsNo two fermions may occupy the same state, but there is no such restriction onbosons; they may all pile up in one state.For a unified treatment of fermions and bosons we assume here that eachstate may be occupied by up to d particles.
Of course, d = 1 holds forfermions and d = for bosons and these seem to be the only two cases thatoccur in nature.34The new argument was prompted by Satyendra Nath Bose (1893–1974),who made two important contributions when he improved the derivation ofPlanck’s radiation formula in 1924:x Bose was the first person to take seriously Boltzmann’s cells in x,cspace and to give them a definite volume.35 We recall that Boltzmannhimself had considered those cells as a calculational trick withoutphysical significance, cf. Chap. 4.
Not so Bose; he quantized the phasespace – spanned by coordinates and momenta – into cells of size h3. Heneeded that value in order to arrive at Planck’s formula.36x Also Bose introduced a new way of characterizing realizations anddistributions. He does that without any fanfare as a matter of course,and without commenting on the move, and without showing a sign thathe was aware of revolutionizing statistical mechanics. Bose sent his34The idea of having an occupancy of an arbitrary number d was introduced by G. Gentile:“Osservazioni sopra le statistiche intermedie.” [Observations on intermediate statistics]Nuovo Cimento 17, p. 493–497.35 We shall review Bose’s contribution in detail in the next chapter which deals withradiation. Let it be said here – in anticipation – that Planck’s radiation formula hadresulted from an interpolation between two empirical functions. This was not satisfactory,at least not for Bose.
Einstein had already improved Planck’s derivation by introducingstimulated emission; but he, too, relied on classical thinking when he adopted theBoltzmann factor, cf. Chap. 4, for the relative frequency of atoms in different energeticstates. Again Bose found this unsatisfactory.36 Accordingly the measure factor Y which I have used heretofore will henceforth be chosen33as Y = µ /h .Entropy of Ideal Gases1894 page-paper to Einstein who translated it into German and had itpublished in the Zeitschrift für Physik.37Einstein added a note of the translator saying that Bose’s derivation ofthe Planck formula represents … an important step forward. The methodused [by Bose] furnishes also a quantum theory of the ideal gas as I shallexplain elsewhere.
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