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Quantum fluidsDavid S. Betts aa School of Mathematical and Physical Sciences, University ofSussex , BrightonPublished online: 20 Aug 2006.

To cite this article: David S. Betts (1969) Quantum fluids, Contemporary Physics, 10:3, 241-262, DOI:10.1080/00107516908224595

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CONTEMP. PHYS., 1969, VOL. 10, NO. 3, 241-262

Quantum Fluids DAVID S. BETTS School of Mathematical and Physical Sciences, University of Sussex, Brighton

SUMMARY. A quantum fluid is one in which energy quantization has a marked effect on behaviour. Sometimes the effect can be dremstic, as with superfluid helium and with the electron gas in superconducting metals. I n this article a simple criterion for distinguishing quantum fluids is obtained, and this is followed by a brief outline of the properties of perfect quantum gasea. These ideas are then used as a unifying frsmework, within which some properties of real quantum fluids sre described and amlysed. ”he approach to the subject is descriptive and physical, with very little dependence on mathematical argument.

1. Introduction: wave character of moving particles

wavelength X which is related to its momentum by de Broglie’s relation. A particle moving with momentum p has, associated with its motion, a

de Broglie’s relation: X = h/p (h = 6.626 x J.5).

There is nothing in our normal human experience to prepare us for this curious fact, but we are obliged to accept it as a result of experiments in which beams of atomic-sized particles can be made to exhibit interference and diffraction effects in much the same way that light waves or sound waves can. Consequently, we have to visualize particles as being rather like ‘ wave-packets ’, extending over distances of the order of A, which might in some circumstances be con- siderably in excess of their actual physical size. If this appears remote and unlikely, it might help to give as an example that a man walking at 5 km/h has an associated wavelength of only about metre which is infinitesimal when compared with his physical size. It follows that when he walks through a doorway one naturally observes no diffraction; broadly this is why de Broglie’s relation seems contrary to our common sense. But it is merely a question of magnitudes. kg), it would find nothing strange about de Broglie’s relation; it would be common electron experience.

If an electron had common sense as well as a tiny mass (

2. The quantum fluid criterion The wave character of a moving particle can only manifest itself when the

particle finds itself near another object or particle, that is, within a distance of about the de Broglie wavelength A. In such situations one should expect diffraction effects. Consider a gas of atoms in equilibrium’ht temperature T, in which, classically, the mean kinetic energy per atom is 3kT/2 (where k is Boltzmann’s constant) and consequently the mean momentum .magnitude is (3mkT)1/2. Thus from de Broglie’s relation, the associated wavelength of each atom in the gas is approximately

Ax h/(3rnkT)1/2. (2.1)

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242 David S. Betts

We can now decide whether or not quantum effects are likely to be important by comparing X with the mean interatomic separation d. This can be estimated by making the rough assumption that the atoms are on a simple cubic lattice, in which case N atoms would occupy a volume Nd3. Therefore

Nd3= V , i.e., d = ( V / N ) 1 / 3 . (2.2)

If h is much greater than d we shall expect strong quantum effects, whereas if X is much less than d we shall expect ordinary classical behsviour. Thus we have arrived ah a simple criterion for identifying quantum fluids, which from (2.1) and (2.2) can be written,in the form:

Q,uantum fluid if T 4 -

In fact, there are few systems in nature which can be made to satisfy this condition, mainly because most materials are unable to remain in a h i d phase to low enough temperatures with high enough number densities. For example, air with a ‘ normal ’ number density would only become a quantum fluid at temperatures below 0.1 K; it is already solid by then. It is clear from the form of (2.3) that the best chance of success will be to try light clements, and it turns out that the only true fluids which fall unmistakably into the quantum fluid class are the two isotopes of helium, 3He and 4He-and mixtures of the two. In addition, the conduction electrons in ordinary metals can be con- sidered as constituting a quantum fluid, and so can an assembly of photons (quanta of electromagnetic radiation). Of course, many more gas and liquid systems show slight quantum effects, but we are only interested here in strong quantum ‘ degeneracy ’. In fact we shall not deal with photons.

3. Multiple occupancy of available energy levels There is another and slightly more fruitful way of deriving the quantum

fluid criterion, hut in order to follow it, it will be necessary to recognize or to take on trust the fact that a particle restricted to move in a cubical box of volume V. is only allowed energies qmn given by

h2(P + m2 + n2) 8mVa13 (3.1) €lmn =

with I , m, n as any positive integers. The particular choice of a cubical box does not affect any conclusion, but it

simplifies the notation a little. Eqa. (3.1) derives directly from the de Broglie relation: a moving particle has an adsociated wavelength, but a wave reBtricted in a box is only allowed certain modes or wavelengths (as, for example, in ordinary stationary waves); therefore a particle, being wave-like, is only allowed certain momenta and energies. Again, the reason why eqn. (3.1) is not common experience is simply that for man-sized objects the energy difference between one allowed level and the next is almost infinitesimally small. Electrons would be quite hmiliar with the result.

There is one further point in relation to eqn. (3.1), namely that there can be different combinations of integral values I , m, n which yield the same value for the cnergy qmn. For example, the combinations 1= 13, m = 3, n = 4 and

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Quantum Fluids 243

l = 5 , m = 5 , n= 12 both give 12+m2+n2= 194. Strictly 1, m, n specify states and there may be several or many states having the same energy. The problem of working out how many states correspond to a particular energy can safely be left to a computer, although there is a neat way of making a direct calcula- tion provided that 1 , m, n are all large numbers. This method can also be used to find out how many states there are with energies somewhere in the range 0 to E (with large E ) . The answer turns out to be (very closely):

47r v 3 h3

number of states (with qmn less than E ) = - - (2mE)3/2 (3 .2)

Thus, for a gas, the number ( S ) of states available for occupation will be found approximately by setting E = Q LT, the average classical thermal energy. Then

47r v 3 h3

S = - - ( 3 m l ~ T ) ~ J ~ . (3 .3)

If there are N atoms spread out among these S states, we can argue that the number of atoms in each state is roughly, and on average

N - (atoms per state) = S

3 A 3

47r d = - (-> (using 2.1 and 2.2) . (3 .4)

But our quantum fluid criterion is ‘ A-much-greater-than4 ’, from which we conclude that in a quantum fluid there is likely to be more than one atom (or other particle) in each available state, that is, states are multiply occupied. There are thus two ways of interpreting the quantum fluid criterion:

(a) The de Broglie wavelength should be larger than the interparticle

( b ) there should be on average, more than one particle in each available state. separations, or

4. Quantum statistics Actually although some kinds of particle are allowed unlimited multiple

occupancy, others are forbidden this liberty by the Pauli exclusion principle. This quantum mechanical principle ensures, for example, that in a many- electron atom only two electrons occupy the first and lowest orbit (or state), eight occupy the next and so on. It is this simple organization of electrons into ‘ shells ’ which is the cause of valency aFd thereforb of much of chemistry. The Pauli exclusion principle is of fundamental importance and has many applications.

From this example, it can be seen that electrons are restricted by the exclusion principle-all but two are excluded from the orbit of lowest energy. However, some other kinds of particles are not so restricted. Fortunately there is a simple rule determining which particles are restricted in this way, and which are not.

All atomic particles, including whole atoms, have an angular momentum or spin which is quantized in units or half-units of h/27r Electrons, neutrons and protons all have a spin angular momentum of h/47r, that is, half-integral spin.

i

It is related to the angular momentum of the particle.

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244 David 8. Betts

Combinations of these basic particles may have integral or half-integral angular momentum, and often there is a tendency for pairs to cancel each other out. For instance, an alpha-particle (two protons plus two neutrons) has zero spin, because two of the four ' nucleons ' spin right-handedly, the other two left- handedly, thus cancelling. A 4He atom has an alpha-particle as nucleus, plus two orbiting electrons. Again the angular momenta of the electrons cancel and the net angular momentum of the atom is zero (which counts as integral). On the other hand a 3He atom has two protons and two electrons, but only one neutron. The pairing is incomplete, and the atom is left with net angular momentum h/4r (half-integral). This cancelling tendency is not a universal principle and many atoms have total angular momenta of 3h/4~ , h/T, 5 h / 4 ~ , 2 h / ~ , etc. We will not be concerned with them here.

Quantum fluids with particles having integral angular momentum (bosons) are called Bose (strictly Bose-Einstein) fluids, those whose particles have half- integral angular momentum (fermions) are called Fermi (strictly Fermi-Dirac) fluids. Fermions (electrons, 3He atoms) are subject to the exclusion principle, bosons (4He atoms, paired 3He atoms, paired electrons, photons) are not. Paired 3He and paired electrons would make Bose fluids because in each case the two half-integral spins would combine to produce an integral spin. We shall return to this important matter of pairing in a later section.

A perfect or ideal gas can be characterized as one in which the constituent particles interact with each other so weakly that the interaction can be ignored, except in so far as it is responsible for establishing thermodynamic equilibrium in the gas. The properties of such a gas can be inferred from this definition by the standard techniques of statistical mechanics, but it will suffice to say here that the familiar properties of a perfect gas (for instance, its equation of state or its specific heat) can be derived from the ' distribution function ', which indicates how many particles there are likely to be in each available state. Thus, the numerical value of the distribution function for a particular energy E is equal to the average number of particles in each state having that energy. Because of the exclusion principle, the distribution function for a Fermi gas is not the same as for a Bose gas. The two cases will be discussed in the following two sections.

5. The perfect F e d gas The ' distribution function ' for an assembly of particles which are dis-

tributed among a number of available states (specified by eqn. (3.1) ) in such a way that they do not violate the Pauli exclusion principle is shown in Figure 1. At T = 0 the graph is a rectangle of height 1 (atom per state) and width equal t o a characteristic energy called the Fermi energy 0.J. The physical meaning of this is that all states with energy less than po are occupied by one particle each, and all states with greater energy are empty. The Fermi energy po can be calculated to be

where the Fermi-Dirac temperature, Ti, is simply defined as Polk. A com- parison of (5.1) with (2.3) shows that the quantum fluid criterion for a Fermi gas can now be written simply as T 4 Ti.

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Quan turn Pluids 245

Referring again to fig. 1, one notices that when T -g Ti (that is, in the quantum fluid region) the rectangular appearance of the graph is retained, but the sharp edge becomes rounded. Physically, there js a band of levels (near po) of which only about half are occupied. The width of this band is about kT. Then at T B Ti the characteristic rectangular appearance of the Fermi distribution is almost entirely lost and the probability of finding any particular state occupied is much less than unity. Classical behaviour is thc result.

T*T,

T: 3Tr

I

0 5 1 1.5

Fig. 1. The average number of particles per state is shown for a perfect Fermi gas, as a function of the energy of the state (in units of kTf) for several different temperatures. The scale of this figure is the same as that of fig. 3, for com- parison.

Fig. 2 shows what the calculated specific heat looks like. The smooth transition from quantum fluid behaviour to classical behaviour is illustrated rather well here, and this smoothness is typical of a Fermi system. The degeneracy temperature Ti simply indicates whereabouts the changeover occurs, and nothing abrupt happens there. At high temperatures (T % Ti) the specific heat approaches the classical limit C/R = 3 /2 ; a t low temperatures ( T g Ti) it approaches the quantum limit C/R= # T ~ ( T / T ~ ) . It is possible to give a rough ,physical argument for the appearance of the factor TIT! as follows. Referring back to fig. 1, and taking note of the energy range kT within which the rounding off of the rectangle takes place, one might argue that only a fraction kT/kTf are truly ‘ free ’, that is, able to shift between states. The remainder can be regarded as ‘ locked ’ in their states, oiic to a cell. SO when heat energy is supplied to a Fermi gas, only the ‘ free ’ atoms are able to accept it. If the ‘ free ’ atoms behaved entirely classically the specific heat would be (3R/2)x(kT/kTf). More precisely, the numerical factor is r2/2 rather than 3 /2 , but one would hardly expect to get exactly the right answer without a little more sophistication. The physical picture is broadly right.

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246 David S . Beth

Other properties of a Fermi gas can also be calculated from the distribution function. For example, the equation of state takes the classical limiting form P V = RT at high temperatures (T B Ti) but approaches a different form, PV = ERTf, at low temperatures (T< Ti). Physically, this means that the pressure approaches a limiting value which is not zero at very low temperatures. This is because the speed of the ' locked ' atoms (the majority) is not affected by temperature, so that collisions with the walls of a containing vessel involve fixed momentum changes, resulting in a fixed pressure, independent of temperature.

Also, atoms may collide with each other and, as in an ordinary gas, one can estimate transport properties (viscosity, thermal conductivity) if one knows thc size of the atoms and something about the forces between them.

Fig. 2. The specific heat of a perfect Fermi gas, showing the approach to linearity at low temperatures and the approach t o clilssical behaviour at high temperatures.

For example, viscosity arises from collisions between atoms in a fluid, because these collisions provide a mechanism which converts ordered flowing motion into disordered thermal motion. Thus ordered kinetic energy degene- rates into heat and this conversion is associated with a frictional or viscous resistive force. According to simple kinetic theory the coefficient of fiscosity is proportional to the number of atoms per unit volume and to their mean free path. In a classical gas of hard spheres, it can be shown that the viscosity is proportional to the square root of the mcan thermal energy, that is, pro- portional to (3kT/2)1f2. Surely, the ' locked ' atoms are unable to exchange any energy or momentum, or, to put it another way, no useful collisions take place between them. Of all the Collisions which take place, the fraction which involve one ' free ' atom is kT/kTf (see fig. 1 ) and the fraction which involve two ' free ' atoms (and are therefore useful) is (kT/kTf)2. That is, the useful number of collisions is proportional to T2, and it follows that the mean free path between useful collisions is propoi.tiona1

But what of a degenerate Fermi gas?

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Quantum Fluids 247

to 1/T2. The lower the temperature, the higher the mean free path, the easier to convert kinetic energy into heat, and consequently the higher the viscosity. Moreover, all the useful collisions are between atoms with energies close to the Fermi energy po. Thus, out of all this it appears that the viscosity of a degenerate Fermi gas is proportional to 1/T2.

In this article we shall repeatedly refer to the properties of specific heat and viscosity, because they are generally well known and are good indicators of the quantum fluid state. The equation of state is not so valuable for most real quantum fluids, with one exception, but it is included in the summary of con- clusions of this section:

Property Classical (high T ) Quantum (low T) Specific

heat QR

Viscosity proportional to proportional to (hard sphere model) p i 2 T-2

Equation of PV=RT PV=$RTi state

6. The perfect Bose gas Fig. 3 shows the distribution function for an assembly of particles which

are distributed among a number of available states, but are now allowed unlimited occupancy of any one state.

A comparison with fig. 1 (for a Fermi gas) shows almost no similarity except a t high temperatures. The most striking and important thitlg t o notice about fig. 3 is that there is a qualitative difference between the distribution function at high and low temperatures. A t high temperatures tho curve approaches the vertical axis with a finite gradient, and gives an intercept which is

Fig. 3. The average number of particles per state is shown for a perfect Bose gas, as a function of the energy of the state (in units of kT,,) for several different temperatures. The scale of %his figure is the same as that of fig. 1, for com- parison.

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248 David 8. Bette

extremely small compared with the total number of particles present. At low temperatures the curve approaches the vertical axis sharply and, in addition, a sizeable fraction of all the particles present have zero energy. This change in character occurs at a very well-defined temperature, the Bose- Einstein degeneracy temperature Tb. It is really impossible to show the abruptness of the change in a simple diagram like fig. 3. By far the best way of showing it would be by a cine film in which the alteration of shape of the distribution function could be shown continuously as the temperature falls. A t high temperatures the ordinate intercept would move upwards, getting higher with falling temperature. Then, at the degeneracy temperature, the magnitude of the zero-energy ordinate would suddenly rush up to a huge value, faster than the eyc would follow, abruptly.

The temperature at which this dramatic change takes place can be calculated. It is given by

h2 N 112 T b = - ( - ) 2 r m k 2.612V .

The reader will notice the familiar appearance of this expression. Apart from a numerical factor, close to unity, it is the same as T i , given in the last section (eqn. (5.1)). Clearly, also, the quantum fluid criterion (eqn. (2.3)) can be written as T-g Tb for a Bose gas.

Physically, what happens at Tb is that large, macroscopic numbers of particles start falling into the ground state ( 6 = 0 ) at the expense of all the other states. In fact, the mathematics of the situation show that the number N , of particles in the ground state at temperature T , below Tb, is given by

N , = N [ l - ( T / T I , ) ~ / ~ ] . (6.2) Thus, at T = 0, all particles are in the ground state. The fraction N,/N falls as T rises, becoming zero at Tb. The particles in the ground state are said to have wndensed, and the phenomenon is referred to as Bose-Einstein condensation. It is important to realize that the word condensation is used here in a special sense : it does not imply solidification, or physical separation of condensed and non-condensed particles.

The onset of condensation in a Bose gas occurs abruptly and it is to be expected that the behaviour of the gas will reflect the change. The specific heat can be calculated from the distribution function and it is shown in fig. 4. There is a cusp in the specific heat a t Tb. At higher temperatures there is an approach to the classical value of 3R/2, whereas at low temperatures the specific heat varies as T3I2. The reason for the T 3 J 2 variation is not hard to find- it is related directly to the fraction of uncondensed particles.

The equation of state at very low temperatures has the form 312

P V =Om4 (g) R T . (6.3)

Actually, since Tb-'I2 is proportional to volume, eqn. (6.3) implies that the pressure is independent of the volume, a somewhat surprising result.

The question of the viscosity of a perfect Bose gas is a rather troublesome one. There is no reason why condensed particles should not have useful collisions; they are not ' locked ' in the sense that some fermions are. So it is

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Quun turn Flu ids 249

5

Fig. 4. The specific heat of a perfect Bose gas showing a cusp at Tb and thc approach to classical behaviour at higher temperatures.

not clear that the viscosity would show any dramatic change at Tb. However, the effect of inter-particle interactions, which in any real quantum fluid cannot be ignored, when combined with the existence of the Bose-Einstein condensate, is to produce superfluidity. A discussion of superfluidity belongs in a later section; a truly perfect Bose gas would probably not exhibit it.

Property Classical (high T ) Quantum (low T )

Specific heat $ R 1.92R( T/Tb)'r2

Viscosity proportional to T1I2 see text (hard sphere model)

Equation of state PV=RT P V = 0.514( T / Tb) ''2R T

Now at last, the real quantum fluids can be introduced, first the Fermi fluids, followed by the Bose fluids.

7. Real Fermi qumtmn fluids Reference to the conclusions of Section 5 will show some kinds of behaviour

which might be expected. All gases, without exception, liquefy or solidify at temperatures which are

considerably higher than their degeneracy temperatures Tb or Ti. Some gases, notably the two helium isotopes, do show effects which are traceable to the beginnings of the onset of degeneracy, but one would like better examples. Surprisingly there are several, although none of them seem on first sight really to be much like gases.

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(a) Dilute solutions of 3He in liquid 4He These are perhaps the best and most recently realized examples of Fermi

degeneracy. One might expect that the presence of the solvent liquid 4He would dominate the beliaviour of such solutions, but this would be to forget that at sufficiently low temperatures nearly all the 4He atoms are condensed (in the Bose-Einstein sense). The result of this is that in some respects the 4He behaves as if it were not really there at all; just a sort of vacuum with mass. This is, of course, an oversimplification, but for our key properties, specific heat, viscosity and equation of state, it is a tolerable one.

The convenience of 3He-4He solutions for testing lies in the fact that degeneracy temperatures Tf can be freely selected by choosing the coims- ponding number density N / V (provided only that the 3He concentration is not made greater than 6yo, above which a complicating phenomenon of phase- separation occurs). It would seem that researchers now have a Fermi ' quasi- gas ' for investigation, whose degeneracy temperature may be preselected anywhere between 0 K and 0-37 K (the value for a 6% 3He concentration). Much work remains to be done on this system, but some aspects are becoming clear.

I I I 2 5 T(in n-3~) 50 75

Fig. 5. The specific heats of two different solutions of 3He in liquid 4He (with molar concentrations of 1.3% and 5.0%) showing the approach to linearity at low temperatures. The shape of the curves is close to pcrfect Fermi gas behaviour.

Some behaviour of such solutions is in fact dominated by the 3He content and it is found to be very close to Fermi gas prediction. For example, the specific heats of 5% and 1.3% solutions of 3He in liquid 4He (see fig. 5 ) have been shown to depend on temperature in a way which is entirely consistent with the Fermi gas results except that the 3He atoms behave as if they had an ' effective ' mass equal to 2.4 times their actual mass. The difference is attributable to the background 4He solvent.

The viscosity also seems to behave as expected, approaching proportionality to 1 / T 2 at low enough temperatures. The only proviso here is that the method

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Quantum Fluids 25 1

of measurement should not be ‘ capillary flow ’ because these solutions are in fact superfluid (see the section on liquid 4He for an explanation of this proviso).

Dilute 3He solutions offer the only real opportunity to see a gas-like equation of state in a Fermi fluid, or for that matter in any quantum fluid. The point is that although the 3He does not exert a true gas pressure, it does exert an osmotic pressure against pure liquid 4He. Fig. 6 shows diagrammatically how the experiment is done.

SUPERLEAK Fig. 6. The osmotic pressure of dilute solutions of 3He in liquid 4He can, in principle,

be obtained by measuring h and making allowance for the difference in density of the two liquid columns.

A perfectly serviceable ‘ semi-permcable membrane ’ in the present case is a superleak, that is, a hole so small that any viscous fluid would be unable to pass through, but which would pass a superfluid (liquid 4He) without trouble. (Again, see below for fuller discussion of superfluidity.) So, in fig. 5 , there is pure 4He on one side of the superleak, and the dilute 3He mixture on the other. The osmotic pressure might be measured by the height difference, although this can be rather large so that other methods are often used. In this way, the temperature dependence of the osmotic pressure has been investigated and, sure enough, the pressure is found to be independent of temperature a t low enough temperatures. In fact the relation P V = QRTi seems to be obeyed, again provided that the ‘ effective mass ’ rather than the real mass of the atom is used.

( b ) Normal metals The second physical approximation to a perfect Fermi gas is the whole

class of metallic solids. It seems unbelievable that a good example of a gas should be a solid, and perhaps it is misleading to say it in quite that way. The crystalline lattice naturally behaves in ways which are unmistakably solid-like (for instance, it is almost incompressible). But metals are distinguished by the property of easy electrical conduction, and one can make a good deal of sense of this property by imagining that each atom in the lattice contributes one or more electrons to a sort of floating pool or gas of free electrons, and is itself left ionized in the process. So a metal is visualized as a lattice of positive ions together with a ‘ gas ’ of electrons. This is by no means a new model; it was first successfully used long ago to illustrate the physical origin of Ohm’s law. However, in the early days there was a rather worrying contradiction which required quantum statistics for its resolution. The problem was, that

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although good electrical conduction strongly suggested the presence of free electrons in the metal, this electron gas did not seem to exhibit any detectable specific heat. This appeared strange because the room-temperature specific heat of solids is generally found to have the constant value of 3R per mole, arising from internal vibrational modes, and gas specific heats are usually 3R/2 per mole due to translational modes. Why, then, do not metals have a com- bined specific heat of [ 3 + (3/2)]R = 9R/2? The answer, it subsequently appeared, was that the electron gas is highly degenerate even at room tem- perature. The high degeneracy temperature follows from the low mass of the electron and the high number density (see eqn. (5 .1)) . Magnitudes are typically about 50 000 K for ordinary metals, so that at room temperature, rather less than 1% of the electrons actually contribute to the specific heat. Th:: remainder are ' locked ' as discussed in Section 5.

This interpretation of metallic specific heats can be verified experimentally in a rather neat way. It can be shown that at low enough temperatures the specific heat of the lattice is no longer constant a t 3R, but approaches zero proportionately to T 3 . So the lattice specific heat approaches zero much more rapidly than the degenerate electron gas specific heat, proportional to T . Thus, although the lattice dominates at room temperature, the position must be reversed at low enough temperatures. By ' low enough ' is usually meant a few degrees above absolute zero. A t such temperatures, then, metallic specific heats should take the form

C = yT + A T 3 (7.1) (electron gas) (lattice)

where we expect y to be determined by eqn. (5 .2 ) , using the correct value of Ti and A is constant which is characteristic of the lattice. Eqn. (7.1) is tested by measuring the specific heat of the metal and then plotting Cexp/T versus T 2 . The graph should be a straight line of gradient A and intercept y. Fig. 7 shows such a line for metallic potassium. The unmistakable presence of a non-zero intercept y is a very strong indication of Fermi degeneracy in the electron gas. The magnitude of y is also very close to the value calculated from eqn. (5.2) for a Fermi gas with the same number density and particle mass as for electrons in potassium.

Fig. 7. The experimental specific heat of potassium, plotted as C/T versus T2 in order to show the electronic contribution.

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Quantum Fluids 253

Metallic specific heats, then, support the model of a degenerate gas of electrons obeying Fermi-Dirac statistics. The viscosity and the equation of state of these electrons cannot be investigated in the same straightforward way, so the matter will not be pursued any further here.

( c ) Liquid helium-3 Again, this is hardly a true gas, although its density is somewhat low

(82 kg/m3) by ordinary liquid standards. 3He liquefies a t 3.2 K and remains a liquid right down to absolute zero unless pressures of a t least 28 atmos- pheres are applied. The two isotopes of helium are the only substances which can remain liquid in this way, partly because their atoms do not interact very strongly (no chemical compounds of helium are known) and partly because of the quantum effect known as ‘zero point motion’ which discourages the atoms from approaching each other too closely.

Liquid 3He is a Fermi liquid rather than a Fermi gas, which means that interactions, though weak, can modify the behaviour. The degeneracy tem- perature Tf, calculated from eqn. (5.1), should be about 5 K. Experimentally it is clear that even at temperatures below &Ti, i.e. 0.5 K, liquid 3He is far removed in behaviour from a Fermi gas. Fig. 8 shows the specific heat as a function of temperature.

I

V I I

1 1 (K)

Fig. 8. The experimental specific heat of liquid 3He compared with that of a perfect The approach to Fermi gas of the same atomic mass and number density.

linearity at sufficiently low temperatures is indicated in both cases.

The specific heat can hardly be said to be in line with Fermi gas predictions, at least, not if the degeneracy temperature is truly 5 K. Actually, the curve does become approximately linear, but not until temperatures more like &Tf, i.e. 0.05K, are reached. Another property, thermal conductivity, which ought to vary as 1/T, actually varies in the wrong direction down to 0.2 K, where it has a minimum before rising

Q C.P.

This is not a t all what was expected.

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254 David S. Betts

as the temperature is lowered further, eventually reaching the 1/T variation below about 0.06 K. Fig. 9 shows the viscosity rising sharply as the tem- perature is lowered.

The behaviour of the viscosity indicates the onset of degeneracy, but the full Fermi prediction of 1/T2 proportionally is not reached until temperatures somewhat below 0.1 K are attained.

I

\ ' APPRMCHING I/lL DEPENDENCE

Figure 9. The experimental viscosity of liquid 3He, showing the approach to Fermi gas l /TZ dependence at low enough temperatures.

So liquid 3He appears to be like a Fermi gas, but only a t much lower tem- peratures than we might have expected. The reason for this is not easy to see, but it has to do with non-negligible interactions between the 3He atoms. These interactions make it impossible to consider the motion of single atoms in the liquid-rather there is a sort of collective motion. In Fermi liquid theory (as opposed to Fermi gas theory) one is obliged to imagine ' quasi- particles ' whose energy is, in some complicated way, a function of the behaviour of all the other quasi-particles in the system. This model gives a way of dis- posing of the troublesome interactions by a mathematical trick, and one is left with a restoration of the Fermi gas of quasi-particles having a mean-free- path proportional to 1/T2 at low enough temperatures. By ' low enough ' is meant ' below 0.1 K '. This is not actually Ti, but rather it is a temperature below which the blurring of the details of the Fermi distribution functions (fig. 1) by interactions becomes small.

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Quantum Fluids 255

Perhaps all this should be summarized thus: liquid 3He does behave very much like a Fermi gas, but only at temperatures which are very low indeed when compared with Ti. Interactions are responsible for this feature, con- verting particles into ' quasi-particles ' and forcing down the temperature range of applicability.

8. Real Bose quantum fluids In Section 6 can be found an outline of the behaviour of a perfect Bose gas.

As with Fermi fluids, the effect of interactions is important, but there are still real Bose fluids showing recognizable Bose traits.

(a) Liquid helium-4 The most startling characteristic of liquid 4He is that it undergoes a phase

transition at the temperature 2.172 K (known as the lambda point because of the shape of the graph of specific heat versus temperature, fig. 10). Fig. 10 shows the specific heat and compares it with the theoretical result for a Bose gas having the same number density as liquid 4He.

2 -

1 -

5

Fig. 10. The specific heat of liqud 4He compared with that of a perfect Bose gas of the same atomic mass and number density. Note that the comparison between C,,, (liquid 4He) and C, (Bose gas), although commonly made, as here, is in fact strictly inadmissible.

It is not difficult to convince oneself that the lambda-point anomaly in liquid 4He is closely related to the phenomenon of ' condensation ' discussed in Section 6. True, the temperatures are not quite the same and neither are the shapes, but then interactions might account for the difference. This does not imply that the comparison makes possible an immediate identification. There are difficulties; for example in liquid 4He the lambda anomaly seems to be of logarithmic character (not so in the perfect Bose gas), and at low enough temperatures below 0.6 K the specific heat varies as T 3 (in a perfect Bose gas it would vary as T3f2) . As a matter of fact the experimental T 3 variation in specific heat is rather revealing because i t suggests the importance of phonons in liquid 4He. Phonons are ' quantized

.

These are not trivial differences.

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256 David S. Betts

sound-waves ’ which are familiar in solids, giving rise to a T 3 specific heat a t temperatures well below the characteristic Debye temperature. However, the presence of phonons is confirmed by other more direct kinds of experiments involving neutron diffraction. These experiments show very clearly the existence of ‘ quantized excitations ’, or packets of energy and momentum, and also give an unambiguous relationship between their momentum arid energy. Most excitations occur either in the region of small energy or near the minimum, and for convenience these two types are given different names: phonons and rotons.

Fig. 11 shows this relationship.

20

- - x . Y ROTONS

I I

0 10 20 p/h (nrn-’

Fig. 11. The relation between energy and momentum (in suitable units) of the quantized excitations in liquid 4He.

One can now approach another aspect of condensation and ask: is there any direct evidence for the presence of a ‘ condensed ’ fraction of liquid He? The specific heat anomaly only hints at it, since very similar lambda anomalies occur in many other substances which certainly have no Bose condensation. However, viscosity measurements strongly indicate condensation a t tem- peratures below the lambda point. There are broadly two ways of measuring viscosity; the capillary flow method and the ball-bearing method. That is, one can either measure the rate of flow of the fluid through a tube due to a pressure gradient or one can measure the resistance to motion of an object through the fluid. In ordinary fluids it does not matter which method is used (except for experimental convenience), the same value for viscosity is obtained. I n liquid 4He also this is true down to the lambda point, but below that tem- perature the capillary method gives a value which is very much lower, by a factor of a t least a million, than the ball-bearing method. Another remarkable observation is that in the capillary experiment the liquid flowing out of the capillary is found to be cozder than the reservoir it came from.

How can one make any sen?? of these extraordinsry results? To do this we must imagine that the helium is mxle cp of an intimate mixture of two$uids,

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Quantum Fluids 257

the normal fluid having an ordinary viscosity and the superfluid hzving no viscosity whatever. The normal fluid may be supposed to consist of the phonons and rotons introduced earlier and they are conceived of as moving about in a background superfluid which is the condensed part of the liquid. This two-fluid model can now explain qualitatively the odd viscosity results as follows. In the capillary method the normal fluid is unable to flow freely because of its viscosity, but the superfluid is able to flow without viscous friction. Moreover, the superfluid is effectively at zero temperature in the sense that its atoms are in the lowest available energy level (the ground state) already. The capillary merely filters out the hot excitations, allowing only the cold frictionless component to flow. On the other hand, a ball-bearing falling through the liquid or a surface moving through it is impeded by the normal fluid and ordinary viscous behaviour is shown.

Fig. 12. The fraction of the density of liquid 4He which, experimentally, is associated with the ‘ normal ’ component in the two-fluid model. For comparison, the fraction of uncondensed atoms in a perfect Bose gas is shown.

It is possible to measure the partial density of the normal and superfluid components by an ingenious method due to Andronikashvili. The instrument consists of a large number of circular discs, mounted on a central axis so that there is a small gap between adjacent discs. This ‘ rotor ’ is suspended in liquid 4He and set into slow rotational oscillation. The measured period of oscillation determines its moment of inertia, which is due partly to its own structure and partly to the normal component of the helium. Only the normal component is dragged round with the rotor; the superfluid, being inviscid, remains motionless. Hence the ratio of normal to total density pnlp is obtained and this is another quantity which can be compared with the prediction for a Bose gas, in which ~ n / p = ( T / T b ) ~ / ~ . Fig. 12 shows the comparison which, qualitatively, is a reasonably satisfying one.

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258 David S. Betts

One is left reinforced in the belief that the phenomenon of Bose-Einstein condensation has a great deal to do with some aspects of the behaviour of liquid 4He. There are many properties of 4He which, in this outline, I must leave untouched. However, the phenomenon of superfluidity is so dramatic as to demand a little more explanation. If we accept that the superfluid com- ponent can in some sense be identified with the Bose-Einstein condensate, what then? Does superfluidity follow? The answer is: not necessarily-in fact the condensed atoms of a perfect Bose-Einstein gas would not constitute a superfluid component. The true origin of superfluidity can be seen in fig. 11. A phonon in liquid 4He travels with a speed z/p equal to the speed of sound. It follows that to create phonons in 4He at absolute zero temperature, one must move an object through the liquid at a speed of at least 200 m/s. It is easier to create a roton, for which only 60 m/s. is necessary. Movements at lower speeds should involve no viscous losses because there is no way of dis- sipating energy by creating excitations, that is, converting kinetic energy into heat. This situation is in extreme contrast with a ' particle-like ' excitation spectrum for which z =p2/2m. With a particle-like excitation spectrum any speed, no matter how small, will be enough to produce an excitation and therefore viscosity. (Note that this argument applies strictly only at absolute zero and has to be extended for higher temperature. However the conclusion remains valid.)

Actually the figure of 60m/s is not supported by experiment, the reason being that even at much lower speeds turbulence produces other kinds of excitations called vortex rings for which the critical velocity is much lower and dependent on capillary size. What I want to emphasize is the way of deciding in advance whether a fluid is super- fluid or not. First find the form of the z versus p graph, making sure that all branches are present. Then draw a straight line through the origin and tangential to the curve. The line of lowest possible gradient then defines the critical velocity. For almost all fluids, including the perfect Bose gas, this critical velocity turns out to be zero, that is, the line of lowest possible gradient is actually the horizontal axis. This is certainly the case for a particle-like excitation spectrum c =p2/2m. So even a perfect Bose gas would not be super- fluid. However, careful analysis of a weakly-interacting Bose gas suggests that it may well have a phonon-like spectrum (rather than particle-like) so that superfluidity would be possible.

Superfluidity, then, is the result when there is a condensation and at the same time interactions produce a suitable excitation spectrum.

This is not the place to pursue this point.

( b ) Superconducting metals Actually, superconductors are rather far removed from the perfect Bose gas,

but there are enough points of similarity to justify their inclusion here. I have described in Section 7 ( b ) how some properties of metals can be inter-

preted in terms of a Fermi gas of electrons. But a number of metals, when cooled to low temperature, exhibit behaviour which finds no parallel in the Fermi gas model. For instance a tin wire behaves quite normally above 3-7 K, but below this critical temperature (To) its electrical resistance vanishes entirely or, to be more precise, it falls by a factor of at least 1015. There are also some

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Quantum Fluids 25 9

new magnetic effects. This sharp transition, taking place within a tempera- ture range of one millikelvin or less in pure metals, is also accompanied by sudden changes of other pr~perties. Fig. 13 shows the specific heat of gallium.

5

Fig. 13. The electronic specific heat of gallium, showing typically Fermi behaviour at higher temperatures, but disrupted catastrophically by the onset of super- conductivity just above 1 K.

Above To the specific heat shows typically Fermi-Dirac linear temperature dependence. Suddenly, as the transition is passed, the variation with tem- perature changes in a way which is very reminiscent of Bose-Einstein condensa- tion. Could it be that condensation can occur in a Fermi system? Some years ago a two-fluid model was applied to superconductors and it was possible to calculate approximate experimental specific heats. The idea was to con- sider the electron gas as if it were two interpenetrating gases with different propcrties: a gas of normal Fermi-Dirac electrons, and a gas of ‘super- electrons ’ with zero entropy and zero flow resistance. The implication is that the ‘ superelectrons ’ can in some way be identified with a Bose-Einstein condensate, even though ordinary electrons obey Fermi-Dirac statistics. Fair, but not fully satisfactory, agreement with experimental specific heats can be achieved if the fraction of superelectrons at temperature T is taken to be 1 - (T/TO)*. True, this is not quite the same as the Bose-Einstein result 1 - (T/T0)3/2 , but the similarity is there. In the same way the BoseEinstein low temperature specific heat proportional to ( T/T0)3/2 can be compared with the experimental values which are in many cases proportional to exp (-1-76T0/T). Not of course the same, but surely, one finds oneself arguing, not tremendously different, since both functions and their derivatives are monotonic and approach zero at IT = 0.

Actually, the exponential form of the variation with temperature of experi- mental specific heats is quite revealing. In other physical systems, such a form results from the presence of an energy gap or a range of forbidden energies. For example, the specific heat of an intrinsic semiconductor contains such a

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term and so does the specific heat of a crystal with two close electronic energy levels (the Schottky anomaly). Perhaps, therefore, we can infer the presence of an energy gap A , such that A = 1-76kT0. The superelectrons are those which are in states lower than those accessible to normal electrons by an amount A. They show no flow resistance because a certain minimum energy A has to be supplied (by collision with other electrons) before they can be dislodged from their state. If such an energy is not available, then there can be no resistance to motion. The situation is rather similar to that in liquid 4He flowing at speeds less than the critical speed. For super- conductors there is a critical current, and higher currents destroy super- conductivity.

As a result of this line of reasoning we accept the presence of an energy gap A . Arather more detailed examination of the facts shows that A = 1*761c11,, is the gap energy for electrons whose energy is close to the Fermi energy kTr. A t lower energies the gap energy is very much lower, approximating to zero. Fig. 14 shows a comparison of the relative density of normal electrons, com- pared with the Bose gas result to show the broad similarity.

This appears to make sense.

" 0 0 5 1 1.5 7/10

Fig. 14. The relative density of normal electrons in a fiuperconductorc omparcd with ' normal fluid ' densities in liquid 4He and in a perfect Bose gas.

It appears that in some metals electrons are able to reduce their energy by a process ofpairing and that electron pairs are the ' superelectrons ', that is they are condensed into a ground state separated from other states by an energy gap. These electrons cannot easily be knocked out of their ground state (because of the gap) and consequently axe not free to give rise to dissipative, that is, resistive mechanisms. Superfluidity, or rather superconductivity, is the result. On first sight this process of pairing is hard to believe. After all, electrons repel other electrons, whereas the existence of a bound pair pre- supposes an attraction between them. The electrons are not really quite like a gas because of the presence of the crystal

The explanation is as follows.

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Quantum Fluids 261

lattice which is positively ionized. the following way. the lattice vibrations because of ordinary electrostatic forces. situation illustrated in fig. 15.

Electrons can interact, via the lattice, in Any movement of an electron in the lattice must affect

Imagine the

\ / PHONON - PHONON CRuTlOW ANNIHILIION

t

Fig. 15. Diagrammatic representation of clectron interaction via the lattice. I t is said that ' a phonon is exchanged '.

Electron 1 jumps spontaneously to a lower energy state and this occurrence is accompanied by the appearance of some extra lattice vibration to make good the energy difference. One speaks of the electron ' emitting a phonon ' when it excites the lattice. Now think of electron 2. As it moves, it meets the phonon and absorbs it, shifting to a new state to maintain energy conservation. Thus the two electrons have a means of interaction whereby the lattice acts as a middleman. Usually such a process would leave the two electrons with as much energy between them, no more or less, as they had to begin with. How- ever, if the phonon is in flight for an extremely short time, T, then the quantum- mechanical uncertainty principle says its energy is ill-defined within a range of about h/r. The implication of this is that the electron pair may have a different energy after exchanging the phonon than before. If this energy is more, then the mechanism yields a repulsive force between the electrons, but if it is less, then we have found the attraction we are looking for.

Thus, in some metals, conditions are such as to produce attraction between electrons strong enough to overcome their electrostatic repulsion. Con- sequently, pairing is favoured energetically, and paired electrons form a superfluid condensate since pairs are not subject to the exclusion principle. Once a current of such frictionless electrons is set up in a superconducting loop it will flow almost indefinitely, certainly for many years.

Finally, in an ordinary metal, the switching on of a magnetic field induces ' eddy currents ' which die away very quickly because of electrical resistance. These eddy currents, which act in such a way as to resist penetration of the field, are not able to persist. But what if there is no electrical resistivity as in perfect conductor? Then the eddy currents will persist and will successfully maintain whatever magnetic field happens to be passing through the metal a t the time of switching. Real superconducting metals do better than this; eddy currents at their surface are persistent and always exclude all magnetic flux from the interior. This property of flux expulsion is not merely a con- sequence of zero resistance; it is more than that, it is a property in its own right and a very important one. However, its explanation is only remotely connected with the other material in this article, and I say no more about it here.

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262 Quantum Fluids

The point should be made that the identification of superconductors with a perfect Bose gas is extremely tentative. The energy gap plays a crucial role in superconductors but not in the Bose gas. Also the electron pairs are not really quite like molecules, and the most one can say is that there is a charac- teristic length in the theory which can be loosely interpreted as the size of a pair.

9. Conclusion To continue the story, the reader should consult the two textbooks given as

references. These will show that there is a very great deal known about the systems which have here been discussed so briefly and with so little real detail. The purpose of this article has been to present the general ideas, and to try to explain them in a non-mathemat,ical way.

REFERENCES LYNTON, E. A., 1962, Superconductivity (John U'iley). W n m , J . , 1967, The Properties of Liquid and Solid Helium (Oxford UniversityPress).

The Author: David S. Betts took his first degree a t Oxford in 1958 and remained at tho Clarondon Laboratory

to work for a D.Phi1. degree under the supervision of Dr. J. Wilks. This was completed in 1962 and he then spent a further two years in Oxford as a Research Fellow. I n 1964 he became Visiting Assistant Professor at Ohio State University before returning to a Lectureship at Sussex University in 1905.

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