THEORY OF SUPERFLUIDITY OF HELIUM II

E. M. Lifshitz

CONTENTS

I. Quantization of the Motion of a Liquid . . . . . . . . . . . . . . . . . . . 512
1. Helium II—a quantum liquid (512). — 2. Quantization of the motion of a liquid (513). — 3. Energy spectrum of a quantum liquid (517). — 4. Energy spectrum of an almost ideal Bose–Einstein gas (521). — 5. Calculation of the thermodynamic quantities of helium II (523).

II. Superfluidity of Helium II . . . . . . . . . . . . . . . . . . . . . . . . . 525
6. Superfluidity of helium II at absolute zero (525). — 7. Helium II at temperatures above absolute zero (529). — 8. Calculation of the ratio $\rho_n/\rho$ (533). — 9. Heat transfer in helium II (536). — 10. Thermomechanical effect in helium II (537). — 11. Behavior of impurity atoms in helium II (539).

III. Macroscopic Hydrodynamics of Helium II . . . . . . . . . . . . . . . . . 540
12. System of hydrodynamic equations of helium II (540). — 13. Hydrodynamic equations for an incompressible liquid (543). — 14. Propagation of sound in helium II (545). — 15. Attenuation of sound in helium II (549). — 16. Scattering of light in helium II (551). — 17. Viscosity of helium II (552). — 18. L. Tisza’s work on the theory of helium II (556).

I. QUANTIZATION OF THE MOTION OF A LIQUID

1. Helium II—a quantum liquid

As is well known, at the temperature $2.19^\circ\mathrm{K}$ liquid helium has the so-called $\lambda$-point (a phase transition of the second kind); at temperatures below this point liquid helium (helium II) possesses a number of remarkable properties connected with the quantum nature of this liquid.

Indeed, as the temperature is lowered, the de Broglie wavelength corresponding to the thermal motion of the atoms increases. In liquid helium the de Broglie wavelength of the helium atoms becomes comparable with interatomic distances at temperatures of the order of $2$–$3^\circ\mathrm{K}$. Accordingly, in this temperature region the properties of liquid helium are connected with quantum phenomena; in this sense one may speak of helium at very low temperatu—

about a “quantum liquid.” In particular, a quantum property is also the fact that helium remains liquid (at ordinary pressure) at all temperatures, down to absolute zero, whereas according to classical mechanics every body at absolute zero must be a solid crystal.

Liquid helium is the only quantum liquid existing in nature; all other liquids solidify considerably before quantum effects become noticeable in them.

The most essential of the special properties of helium II is its superfluidity, discovered by P. L. Kapitsa*). A rational explanation and quantitative theory of this phenomenon were given by L. Landau1, 2, 3, who for the first time constructed a consistent theory of a quantum liquid. This article is devoted mainly to the exposition of this theory.

Among other attempts to construct a theory of superfluidity we mention the works of F. London4, 5, 6 and L. Tisza7, 8 (on Tisza’s later works9, 10 see below, § 18). In these works the properties of a degenerate ideal Bose–Einstein gas are invoked to explain the behavior of helium II, and it is assumed that atoms in the normal state (the state with zero energy) move through the substance without experiencing friction. Such a representation, however, cannot be regarded as satisfactory. First of all, helium II has nothing in common with an ideal gas, and there are no grounds for transferring to it results obtained for a gas. But even in an ideal gas, atoms in the normal state would by no means behave as “superfluid”; on the contrary, nothing would prevent them from colliding with excited atoms, exchanging momentum with them, i.e. in their motion through the substance they would experience friction, and superfluidity would be absent. Thus, such an explanation of superfluidity not only lacks a sufficiently solid basis, but in fact is also in direct contradiction with the initial assumptions.

2. Quantization of the motion of a liquid

An arbitrary system of interacting particles (a liquid) can be described in classical theory by means of the density \(\rho\) and the mass current \(\mathbf{j}\), which are defined in the following way. Let \(\mathbf{R}\) be the radius vector of an arbitrary point of space, and \(\mathbf{r}_a\) the radius vector of a particle of mass \(m_a\). Then the density is defined as

\[ \rho=\sum_a m_a\,\delta(\mathbf{R}-\mathbf{r}_a), \tag{2,1} \]

*） A review of the relevant experimental results is given in the article by E. L. Andronikashvili, U.F.N. 33, issue 4, 469 (1947).

where \(\delta\) is the three-dimensional \(\delta\)-function, and the summation is carried out over all particles in the system. The volume integral \(\int \rho\, d v\) gives the total mass of the system. Similarly, the mass-flow density is defined as

\[ \mathbf{j}=\sum_a m_a \mathbf{v}_a \delta(\mathbf{R}-\mathbf{r}_a) =\sum_a \mathbf{p}_a \delta(\mathbf{R}-\mathbf{r}_a), \]

where \(\mathbf{v}_a, \mathbf{p}_a\) are the velocity and momentum of the particle \(m_a\).

Let us emphasize that with such a description of the liquid no averaging is performed in the sense in which it is performed in statistics. The description proceeds from the microscopic picture, since all particles possess (at a given instant) definite coordinates \(\mathbf{r}_a\) and velocities \(\mathbf{v}_a\).

In passing to quantum theory one must regard \(\rho\) and \(\mathbf{j}\) as certain operators; let us determine the form of these operators. For simplicity of reasoning, suppose that the system consists of just one particle. Then the classical density is \(\rho=m\delta(\mathbf{R}-\mathbf{r})\). The operator \(\rho\) must be defined in such a way that its mathematical expectation \(\int \psi^*(\mathbf{r})\rho\psi(\mathbf{r})\,dV\) (where \(\psi(\mathbf{r})\) is the wave function of the particle) is equal to the mass density at the point \(\mathbf{R}\), i.e. \(m|\psi(\mathbf{R})|^2\). Hence it follows that the operator \(\rho\) must have the same form \(\rho=m\delta(\mathbf{R}-\mathbf{r})\), and in the case of an arbitrary system of particles, correspondingly, the form \((2,1)\).

The classical flow density for one particle is \(\mathbf{j}=\mathbf{p}\,\delta(\mathbf{R}-\mathbf{r})\). It is easy to see that the corresponding quantum operator is

\[ \mathbf{j}=\frac{1}{2}\{\mathbf{p}\delta(\mathbf{R}-\mathbf{r})+\delta(\mathbf{R}-\mathbf{r})\mathbf{p}\}, \]

where \(\mathbf{p}\) is the usual momentum operator

\[ \mathbf{p}=\frac{\hbar}{i}\nabla \]

(\(\nabla\) denotes differentiation with respect to \(\mathbf{r}\)). Indeed, the mathematical expectation of \(\mathbf{j}\)

\[ \int \psi^*(\mathbf{r})\mathbf{j}\psi(\mathbf{r})\,dV = \]

\[ = \frac{\hbar}{2i}\int \psi^*\nabla\delta(\mathbf{R}-\mathbf{r})\psi\,dV +\frac{\hbar}{2i}\int \psi^*\delta(\mathbf{R}-\mathbf{r})\nabla\psi\,dV, \]

or, integrating the first term by parts,

\[ \int \psi^*\mathbf{j}\psi\,dV =\frac{\hbar}{2i}\int\{-\psi\delta(\mathbf{R}-\mathbf{r})\nabla\psi^* +\psi^*\delta(\mathbf{R}-\mathbf{r})\nabla\psi\}\,dV = \]

\[ =\frac{\hbar}{2i}\{\psi^*(\mathbf{R})\nabla\psi(\mathbf{R}) -\psi(\mathbf{R})\nabla\psi^*(\mathbf{R})\}, \]

i.e. precisely what it should have been. For an arbitrary system of particles we have analogously

\[ \mathbf{j}=\frac{1}{2}\sum_a\{\mathbf{p}_a\delta(\mathbf{R}-\mathbf{r}_a)+\delta(\mathbf{R}-\mathbf{r}_a)\mathbf{p}_a\},\qquad \mathbf{p}_a=\frac{\hbar}{i}\nabla . \tag{2,2} \]

Instead of the operator \(\mathbf{j}\), it is more convenient to use the operator of the velocity of motion of the liquid \(\mathbf{v}\), defined in a natural way according to \(\mathbf{j}=\frac{1}{2}(\rho\mathbf{v}+\mathbf{v}\rho)\), or

\[ \mathbf{v}=\frac{1}{2}\left(\frac{1}{\rho}\mathbf{j}+\mathbf{j}\frac{1}{\rho}\right) \tag{2,3} \]

(in the classical theory \(\mathbf{j}=\rho\mathbf{v}\)).

The commutation rules between the operators introduced in this way can be obtained by a direct, though rather lengthy, calculation. Omitting these calculations here, we give the relations obtained as a result. Denoting by the indices 1 and 2 the values of the operators taken, respectively, at the points of space \(\mathbf{R}_1\) and \(\mathbf{R}_2\), we shall have

\[ \rho_1\rho_2-\rho_2\rho_1=0, \tag{2,4} \]

\[ \mathbf{v}_1\rho_2-\rho_2\mathbf{v}_1=-i\hbar\nabla\delta(\mathbf{R}_1-\mathbf{R}_2), \tag{2,5} \]

\[ v_{1i}v_{2k}-v_{2k}v_{1i}=-i\hbar\delta(\mathbf{R}_1-\mathbf{R}_2)\frac{1}{\rho_1}(\operatorname{rot}\mathbf{v})_{ik} \tag{2,6} \]

(in the last relation \(i,k=x,y,z\), and \((\operatorname{rot}\mathbf{v})_{ik}\) denotes the difference

\[ \frac{\partial v_k}{\partial x_i}-\frac{\partial v_i}{\partial x_k} \]
).

If we apply to both sides of equation (2,5) the operation \(\operatorname{rot}\) (with differentiation with respect to the coordinates \(\mathbf{R}\)), we obtain the relation

\[ \operatorname{rot}\mathbf{v}_1\cdot\rho_2-\rho_2\operatorname{rot}\mathbf{v}_1=0. \tag{2,7} \]

It is easy to verify that, when the relations obtained are applied to the macroscopic motion of a liquid, one obtains, as one should, the ordinary hydrodynamic equations written in operator form. The energy per unit volume of a classical liquid, considered macroscopically, is

\[ \frac{\mathbf{v}\rho\mathbf{v}}{2}+\rho E(\rho), \]

where \(E(\rho)\) is the internal energy per unit mass of the liquid. The energy \(E\) is assumed to depend only on the density \(\rho\) of the liquid; in this the macroscopic character of the consideration, connected with statistical averaging, is manifested. In a microscopic consideration such an assumption is, of course, unjustified.

The quantum operator corresponding to the written classical expression is

\[ \frac{\mathbf{v}\rho\mathbf{v}}{2}+\rho E(\rho). \]

The Hamiltonian function \(H\) of the liquid is the volume integral

\[ H=\int \left\{\frac{1}{2}\mathbf v\rho \mathbf v+\rho E(\rho)\right\}dV . \tag{2,8} \]

To determine the time derivative \(\dot\rho\) of the density, we must, in accordance with the general rules of quantum mechanics, compute the result of commuting the operators \(\rho\) and \(H\):

\[ \dot\rho=\frac{i}{\hbar}(H\rho-\rho H). \]

Let us temporarily denote the coordinates of the point at which \(\rho\) is taken by the index 1, and the coordinates of the variable point in the region of integration in (2,8) by the index 2. Then

\[ \dot\rho_1=\frac{i}{\hbar}\int \left\{\frac{1}{2}\left[\mathbf v_2\rho_2\mathbf v_2\rho_1-\rho_1\mathbf v_2\rho_2\mathbf v_2\right] +\left[\rho_2E(\rho_2)\rho_1-\rho_1\rho_2E(\rho_2)\right]\right\}dV . \]

By virtue of (2,4) the second term under the integral sign vanishes, and the first may be written in the form

\[ \frac{1}{2}\left[\mathbf v_2\rho_2(\mathbf v_2\rho_1-\rho_1\mathbf v_2) +(\mathbf v_2\rho_1-\rho_1\mathbf v_2)\rho_1\mathbf v_2\right] \]

or, substituting (2,5):

\[ \frac{\hbar}{2i}\nabla\delta(\mathbf R_2-\mathbf R_1)\cdot (\mathbf v_2\rho_2+\rho_2\mathbf v_2) =-i\hbar\nabla\delta(\mathbf R_2-\mathbf R_1)\cdot\mathbf j_2 . \]

Thus,

\[ \dot\rho=\int \nabla\delta(\mathbf R_2-\mathbf R_1)\mathbf j_2\,dV_2 =-\int \delta(\mathbf R_2-\mathbf R_1)\operatorname{div}\mathbf j_2\,dV_2 =-\operatorname{div}\mathbf j_1, \]

i.e. we arrive at the continuity equation in operator form

\[ \frac{\partial\rho}{\partial t}+\operatorname{div}\frac{\rho\mathbf v+\mathbf v\rho}{2}=0. \tag{2,9} \]

In an analogous way one can compute the derivative

\[ \dot{\mathbf v}=\frac{i}{\hbar}(H\mathbf v-\mathbf vH). \]

The calculation leads to the equation

\[ \frac{\partial v_i}{\partial t} =-\frac{1}{2}\sum_k\left(v_k\frac{\partial v_i}{\partial x_k} +\frac{\partial v_i}{\partial x_k}v_k\right) -\frac{1}{\rho}\frac{\partial}{\partial x_i}\frac{dE}{d\rho}, \tag{2,10} \]

i.e. the Euler equation is obtained in operator form (\(dE/d\rho\) is the pressure \(P\) of the liquid).

Let us emphasize once again that equations (2,9–10) are less general than the commutation conditions (2,4–7), which are applicable also in an exact, microscopic treatment of the liquid.

3. The energy spectrum of a quantum liquid

In the classical hydrodynamics of an ideal liquid it is shown that if at some instant the motion is potential ($\operatorname{rot}\mathbf v = 0$) throughout the entire volume of the liquid, then it will remain potential at all other instants (the so-called Lagrange theorem). It turns out that in quantum hydrodynamics there is an analogue of this theorem.

According to the commutation rule (2,7), $\operatorname{rot}\mathbf v$ always commutes with the density $\rho$. The components of $\operatorname{rot}\mathbf v$, however, do not commute, in general, either with one another or with the components of the velocity $\mathbf v$ (when the operation $\operatorname{rot}$ is applied to equation (2,6), the right-hand side of the equality does not vanish). Therefore $\operatorname{rot}\mathbf v$ does not commute, in general, with the Hamiltonian operator either; as is known from quantum mechanics, this means that $\operatorname{rot}\mathbf v$ is not conserved.

An exception is the case in which, throughout the whole volume of the liquid, $\operatorname{rot}\mathbf v = 0$. In this case the right-hand side of (2,7) is zero, so that $\operatorname{rot}\mathbf v$ commutes both with $\rho$ and with $\mathbf v$, and consequently also with the Hamiltonian operator *).

Thus, a vanishing curl of the velocity is conserved. In other words, a quantum liquid always possesses such stationary states in which $\operatorname{rot}\mathbf v$ is zero throughout the entire volume. Such states may be called, by analogy with classical hydrodynamics, states of potential motion of the liquid.

In connection with this result it is useful to draw an analogy (though a purely formal one) with the angular momentum $\mathbf M$ in quantum mechanics. The commutation of two components of $\mathbf M$ with one another leads to the third component; as a result, the components of the angular momentum commute with one another, and therefore exist simultaneously, only if they are all zero. It is also known that there are no states with arbitrarily small values of the angular momentum—the first nonzero eigenvalues of it are of order $\hbar$. This is a consequence of the inhomogeneity of the commutation relations—their left-hand side is quadratic in $\mathbf M$, while the right-hand side is linear.

An analogous assertion may also be made concerning the curl of the velocity in quantum hydrodynamics. Namely, there cannot exist such states in which the curl of the velocity would be different from zero, but arbitrarily small throughout the whole volume of the liquid.

From this one may draw an essential conclusion about the basic properties of the energy spectrum of the liquid (we emphasize that we are speaking not of levels for individual helium atoms, but of levels corresponding to states of the whole liquid as a whole). Namely, there cannot exist energy levels arbitrarily close to the nor-

*) Moreover, not only with the Hamiltonian (2,8), but also with any other Hamiltonian containing $\rho\mathbf v$ and their derivatives of any order with respect to the coordinates.

small ones, which would correspond to vortex motion of the liquid. In other words, states excited sufficiently close to the normal state must correspond to potential motion of the liquid.

It is necessary to emphasize that in this reasoning it is assumed that the normal state of the liquid is vortex-free. In principle, however, both possibilities are logically admissible—a vortex and a vortex-free normal state. We shall see below that superfluidity is led to by an energy spectrum with a vortex-free normal state; it should therefore be accepted that in liquid helium it is precisely this case that occurs. It must, however, be borne in mind that, since in nature there exists only one quantum liquid—liquid helium—the question of whether such an energy spectrum, and hence also superfluidity, is a general property of quantum liquids cannot be decided experimentally. Theoretically, the existence of non-superfluid quantum liquids is quite admissible.

Let us consider an excited level situated not too high above the beginning of the spectrum (the normal level). Any weakly excited state of a macroscopic body may be regarded in quantum mechanics as an aggregate of separate “elementary excitations.” These “elementary excitations” behave like certain “quasi-particles,” moving in the volume occupied by the body and possessing definite energies and momenta. Thus, for example, the thermal excitation of a solid body (a crystal), in which the atoms perform small oscillations about their equilibrium positions, may, as is known, be regarded as an aggregate of “sound quanta” (phonons) moving in the body.

Let us denote by \(\varepsilon(p)\) the energy of an “elementary excitation” in liquid helium as a function of its momentum \(p\). The form of the dependence of \(\varepsilon\) on \(p\) is the principal characteristic of the energy spectrum of the liquid. On the basis of what was said above it is not difficult to establish the form of this function for sufficiently small values of the momentum (long waves). Indeed, sufficiently small excitation energies correspond to potential motions of the liquid. But potential internal motion is nothing other than longitudinal sound waves. Therefore the corresponding elementary excitations are simply sound quanta, i.e. phonons. The energy of phonons, as is known, is a linear function of their momentum, i.e.

\[ \varepsilon = cp, \tag{3,1} \]

where \(c\) is the velocity of sound. Thus, in its initial part the curve of the dependence of \(\varepsilon\) on \(p\) is rectilinear.

As the momentum \(p\) increases, the curve \(\varepsilon=\varepsilon(p)\) deviates from linearity. However, it seems impossible to determine in general

in the form of a dependence of \(\varepsilon\) on \(p\) for a quantum liquid from theoretical considerations alone, and in order to determine the form of the energy spectrum of helium II it is necessary also to make use of the available experimental data; these are measurements of various thermodynamic quantities of helium II (entropy, heat capacity), as well as of the velocity of propagation of the so-called “second sound” in helium II (see § 14). Analysis of these data shows (L. Landau\(^5\)) that they can be naturally explained if one assumes that the curve \(\varepsilon=\varepsilon(p)\) has the form shown in Fig. 1. After the initial linear segment (phonons), the energy \(\varepsilon\) reaches a maximum, then begins to decrease, and at a certain value of the momentum \(p=p_0\) the function \(\varepsilon(p)\) has a minimum.

Fig. 1. Energy spectrum of helium II.

In thermal equilibrium the elementary excitations present in the liquid are distributed mainly near the minima of the energy, i.e. in the region of small \(\varepsilon\) (the region near \(\varepsilon=0\)) and in the region near the value \(\varepsilon(p_0)\). Therefore precisely these regions are especially important. Near the point \(p=p_0\) the function \(\varepsilon(p)\) can be expanded in a series in powers of \(p-p_0\). Since

\[ \left.\frac{d\varepsilon}{dp}\right|_{p=p_0}=0, \]

the linear term in the expansion is absent, and, to terms of second order, we have

\[ \varepsilon=\Delta+\frac{(p-p_0)^2}{2\mu}, \tag{3,2} \]

where \(\Delta=\varepsilon(p_0)\) and \(\mu\) are constants.

Elementary excitations of this second type correspond, in contrast to phonons, to vortical motions of the liquid (motion with \(\operatorname{rot}\mathbf v\) different from zero). They may be called “rotons,” and the constant \(\mu\) in (3,2) may accordingly be spoken of as the “effective mass” of the roton. It must, however, be borne in mind that, for an energy spectrum of the type described, one cannot, strictly speaking, distinguish phonons and rotons as two qualitatively different types of elementary excitations, since there is a continuous transition from one to the other. It would be more correct simply to speak of long-wavelength (small \(p\)) and short-wavelength (\(p\) near \(p_0\)) quanta of excitation.

Bearing this reservation in mind, we shall nevertheless continue to use the convenient terms—phonons and rotons*).

Knowing the form of the energy spectrum of a liquid, one can calculate all its thermodynamic quantities (§ 5). The formulas obtained in this way, with a suitable choice of the constants $\Delta$, $p_0$, $\mu$, turn out to be in excellent agreement with the available experimental data on the heat capacity of helium II (the measurements of Keesom; cf., however, the footnote on p. 525), its entropy (the measurements of P. L. Kapitza), and the velocity of “second sound” in it, measured by V. P. Peshkov. The values of the constants determining the energy spectrum of helium II are then as follows:

\[ \Delta/k = 9.6^\circ\mathrm{K}, \qquad p_0/\hbar = 1.95 \cdot 10^8\ \mathrm{cm}^{-1}, \qquad \mu = 0.75\,m_{\mathrm{He}} . \tag{3,3} \]

Let us note that the effective mass of the roton turns out to be of the order of magnitude of the mass $m_{\mathrm{He}}$ of the helium atom, while the wavelength $\hbar/p_0$ turns out to be very small, even smaller than atomic dimensions**).

*) In the original version of the theory L. Landau considered an energy spectrum in which the function $\varepsilon(p)$ consists of two branches—the phonon branch (3,1) and the roton branch $\varepsilon=\Delta+\dfrac{p^2}{2\mu}$; both began at $p=0$, so that between the lowest states of potential and vortex motion there was an “energy gap” $\Delta$. Quite apart from the fact that such a spectrum leads to a dissatisfactory agreement with the experimental data, it is intrinsically contradictory. The latter is manifested in the fact that rotons in this spectrum could spontaneously decay into phonons, i.e., would be unstable; thus, for example, a roton with energy $\Delta$ and momentum $p=0$ could decay into two phonons moving in opposite directions, with momenta $p=\Delta/2c$ and energies $\varepsilon=\Delta/2$. We also mention that the energy spectrum of liquid helium was studied by A. Byley^[11], who came to the conclusion that there is an “energy gap” between the normal and all excited states. This result, however, appears very doubtful already because it would mean, in particular, the impossibility of propagation in a liquid of sound waves of small frequency.

**) This, of course, by no means implies that rotons can to any degree be identified with individual helium atoms! Let us emphasize once again that phonons and rotons are not connected with any definite atoms or groups of atoms, but are, in essence, only a means of describing the collective motions of all the atoms of a liquid.

Let us note the following circumstance in this connection. If one introduces the “roton velocity” $v=d\varepsilon/dp$, then from (3,2) we obtain $v=(p-p_0)/\mu$. In particular, for $p=p_0$ one obtains $v=0$, i.e., at a momentum equal to $p_0$ the roton has no velocity. This circumstance, of course, contains no internal contradictions and admits a perfectly useful “classical analogy.” Imagine a wave packet propagating in some medium. The velocity $v$ of its displacement is determined by the dispersion law of the waves $\omega=\omega(k)$ ($\omega$ is the frequency, $k$ the wave vector), according to the well-known formula for the group velocity $v=d\omega/dk$, and for certain values of $k$ it may vanish. This will mean that the whole wave packet as a whole, i.e. the entire picture of motion in the medium, remains stationary in space, which, of course, does not prevent internal motion from taking place in the medium, which may be accompanied by a transfer of mass in some direction.

If the number of rotons and phonons (referred to unit volume of the liquid) is comparatively small, then their aggregate may be regarded as a mixture of two ideal gases—the phonon gas and the roton gas. Such a situation obtains at temperatures not too close to the \(\lambda\)-point.

Both the phonon gas and the roton gas must obey Bose statistics. This can be seen on the basis of the following very general considerations. From the properties of the energy spectrum described above, when the liquid is excited, phonons and rotons can appear singly. On the other hand, the momentum of any quantum-mechanical system (in the present case, of the entire liquid) can change only by an integral number. Therefore the singly arising “elementary excitations” must possess integral momentum and, consequently, obey Bose statistics.

Concerning the roton gas, it should be noted, however, that since its energy always contains the quantity \(\Delta\), which is large in comparison with \(kT\) (at low temperatures, where alone one can speak of a roton gas), in its application to it the Bose distribution may, with sufficient accuracy, be replaced by the Boltzmann distribution.

4. Energy spectrum of an almost ideal Bose–Einstein gas

As has already been noted, the problem of a complete theoretical determination of the energy spectrum of a real liquid may be regarded as hopeless. In view of this, it is of interest to consider some substantially simpler model of a macroscopic body, even if it has no direct relation to liquid helium, in order to clarify how an energy spectrum with properties analogous to those described above can actually arise.

As such a model it is most natural to choose a Bose–Einstein “almost ideal” gas, i.e. a gas with weak interaction between the particles. A consistent quantum-mechanical treatment of such a problem was first carried out by N. N. Bogolyubov\({}^{12}\), who, by an ingenious application of the method of second quantization, succeeded in completely determining the energy spectrum of the weakly excited states of such a gas.

Referring the reader to Bogolyubov’s original work for the detailed execution of the calculations, we shall describe here only the resulting results. It turns out that the weakly excited state of the gas can indeed be described as an aggregate (an “ideal gas”) of mutually noninteracting “elementary excitations,” so that the total energy of the gas is composed of the energy of the ground sta-

the momenta and the sum of the energies of the individual “quasi-particles”—elementary excitations. To each elementary excitation one may assign a certain momentum \(p\), while its energy as a function of the momentum is expressed by the formula

\[ \varepsilon(p)=\frac{1}{2m}\sqrt{4\rho\,\nu(p)\,p^2+p^4}, \tag{4,1} \]

where \(m\) is the mass of an individual particle of the gas, \(\rho\) is its density, and the function \(\nu(p)\) is defined as the integral

\[ \nu(p)=\int U(r)e^{-ipr/\hbar}\,dV, \tag{4,2} \]

taken over all space; here \(U(r)\) is the potential energy of interaction of two particles of the gas (assumed to depend only on the mutual distance \(r\) between the particles).

In order that the expression under the radical in (4,1) always be positive (including for small momenta), it is necessary that

\[ \nu(0)=\int U(r)\,dV>0. \tag{4,3} \]

Otherwise, for sufficiently small \(p\), the energy \(\varepsilon(p)\) would be complex, which would mean instability of the excited states of the gas under consideration. It can be shown that \(\rho\nu(0)/m^2=\partial P/\partial\rho\) (\(P\) is the pressure of the gas), so that condition (4,3) is equivalent to the familiar condition of thermodynamic stability of a gas \((\partial P/\partial\rho>0)\). From the mechanical point of view, condition (4,3) means that repulsive forces act between the particles of the gas. The necessity of this condition in the model under consideration is obvious beforehand, since in the case of attractive forces, at absolute zero temperature and near it (which corresponds to considering weakly excited states), the substance could not exist in the form of a gas and would condense into a liquid. Let us note that this circumstance, in a certain sense, makes the model under consideration more remote from real helium, in which attractive forces act between the atoms.

For small momenta the energy (4,1) takes the form

\[ \varepsilon(p)\simeq \frac{\sqrt{\rho\nu(0)}}{m}\,p = \sqrt{\frac{\partial P}{\partial\rho}}\,p = cp \]

(\(c=\sqrt{\partial P/\partial\rho}\) is the velocity of sound in the gas). Thus, in the initial part of the curve \(\varepsilon=\varepsilon(p)\), we are indeed dealing with phonons. For large momenta the function \(\nu(p)\) tends to zero (since it contains, in the integrand, a rapidly oscillating exponential factor). Therefore, for sufficiently large \(p\) we have

\[ \varepsilon(p)\simeq \frac{p^2}{2m}, \]

i.e. \(\varepsilon\) becomes the kinetic energy of an individual particle of the gas.

THEORY OF SUPERFLUIDITY OF HELIUM II

5. Calculation of the thermodynamic quantities of helium II

The features of the energy spectrum of helium II described in § 3 make it possible to draw definite conclusions about the temperature dependence of its thermodynamic quantities (entropy, heat capacity, etc.). The calculations carried out below refer to temperatures not too close to the $\lambda$-point, when one may speak of a phonon and a roton gas in the liquid. Under these conditions all thermodynamic quantities consist of two parts, associated respectively with each of these gases; we shall refer to them as the phonon and roton parts.

The phonon parts of the thermodynamic quantities are determined directly by the formulas of the well-known Debye theory of the heat capacity of solids (whose thermal excitation is regarded as an aggregate of phonons). Thus, for the free energy (referred to 1 g of liquid) we have

$$ F_{\phi}=-\frac{\pi^{2}k^{4}}{90\hbar^{3}c^{3}\rho}\,T^{4} \tag{5,1} $$

($\rho$ is the density of helium). The phonon part of the entropy is obtained from this by differentiation with respect to temperature,

$$ S_{\phi}=\frac{2\pi^{2}k^{4}}{45\hbar^{3}c^{3}\rho}\,T^{3}, \tag{5,2} $$

and the phonon part of the heat capacity $C_{\phi}=T\dfrac{\partial S_{\phi}}{\partial T}$ is equal to

$$ C_{\phi}=\frac{2\pi^{2}k^{4}}{15\hbar^{3}c^{3}\rho}\,T^{3}. \tag{5,3} $$

Thus, the phonon part of the heat capacity is proportional to the cube of the temperature*).

To calculate the roton part of the thermodynamic quantities we shall use the fact that, at the temperatures under consideration, the roton gas is described with sufficient accuracy by the Boltzmann distribution. In doing so one must remember that the number of particles in the roton gas does not remain unchanged, but is a function of temperature and is determined by the condition of minimum free energy. According to the well-known formula of statistical physics, the free energy of an ideal gas with number of particles $N$ in a volume $V$ is

$$ F=-NkT\ln\frac{eV}{N}\int e^{-\varepsilon/kT}\frac{d\tau_{p}}{(2\pi\hbar)^{3}}, \qquad (d\tau_{p}=dp_{x}\,dp_{y}\,dp_{z}). $$

*) The formulas written down are obtained in Debye theory when only longitudinal sound waves are present, and correspond to temperatures small in comparison with the Debye temperature. The latter condition may be considered fulfilled, since in the present case the Debye temperature is

$$ \Theta=\frac{\hbar c}{k}\left(\frac{18\pi^{2}\rho}{m_{\mathrm{He}}}\right)^{1/3}=30^\circ\mathrm{K}. $$

Equating to zero the derivative \(\partial F/\partial N\), we find for the number of particles in the roton gas the formula

\[ N_{\mathrm p}=\frac{1}{\rho(2\pi\hbar)^3}\int e^{-\varepsilon/kT}\,d\tau_{\mathrm p}; \]

here we have put \(V=1/\rho\), in accordance with the fact that we refer all quantities to unit mass of helium. The corresponding value of the free energy is

\[ F_{\mathrm p}=-\frac{kT}{\rho(2\pi\hbar)^3}\int e^{-\varepsilon/kT}\,d\tau_{\mathrm p}. \tag{5,4} \]

Into these formulas one must substitute expression (3,2) for the energy of a roton. In view of the fact that \(p_0^2\gg \mu kT\), in integrating with respect to \(dp\) one may, with sufficient accuracy, replace \(p\) in the pre-exponential factor by \(p_0\). The integration of the exponential factor with respect to the difference \(p-p_0\) may, owing to its rapid decrease with increasing \(|p-p_0|\), be carried out within the limits from \(-\infty\) to \(+\infty\). Therefore

\[ \int p^2 e^{-(p-p_0)^2/2\mu kT}\,dp \simeq p_0^2\int_{-\infty}^{+\infty} e^{-\xi^2/2\mu kT}\,d\xi = p_0^2\sqrt{2\pi\mu kT}. \]

Thus we obtain the following formulas for the number of rotons and the roton part of the free energy:

\[ N_{\mathrm p}= \frac{2(\mu kT)^{1/2}p_0^2}{(2\pi)^{3/2}\rho\hbar^3}\, e^{-\Delta/kT}, \tag{5,5} \]

\[ F_{\mathrm p}=- \frac{2\mu^{1/2}(kT)^{3/2}p_0^2}{(2\pi)^{3/2}\rho\hbar^3}\, e^{-\Delta/kT}. \tag{5,6} \]

For the roton part of the entropy \(S_{\mathrm p}=-\dfrac{\partial F_{\mathrm p}}{\partial T}\) and of the heat capacity \(C_{\mathrm p}=T\dfrac{\partial S_{\mathrm p}}{\partial T}\), we hence find

\[ S_{\mathrm p}= \frac{2(k\mu)^{1/2}p_0^2\Delta}{(2\pi)^{3/2}\rho T^{1/2}\hbar^3} \left(1+\frac{3kT}{2\Delta}\right) e^{-\Delta/kT}, \tag{5,7} \]

\[ C_{\mathrm p}= \frac{2\mu^{1/2}p_0^2\Delta^2}{(2\pi)^{3/2}\rho k^{1/2}T^{3/2}\hbar^3} \left[ 1+\frac{kT}{\Delta}+\frac{3}{4}\left(\frac{kT}{\Delta}\right)^2 \right] e^{-\Delta/kT}. \tag{5,8} \]

Thus, the temperature dependence of the roton part of the thermodynamic quantities has, in the main, an exponential character \((\sim e^{-\Delta/kT})\). Therefore, at sufficiently low temperatures the roton part of the thermodynamic quantities must become smaller than the phonon part, whereas at higher temperatures the opposite situation occurs and the roton part predominates over the phonon part. Calculation shows that in fact the two parts of the heat capacity and of the entropy become comparable at a temperature of about \(0.8\)—\(0.9^\circ\mathrm K\).

In Figs. 2 and 3 (pp. 526 and 527) are shown the curves of the entropy and heat capacity of helium II, calculated from the formulas obtained (the calculations were carried out by I. Khalatnikov and O. Kramer). For the parameters \(\mu, \Delta, p_0\) the values (3.3) were adopted, the speed of sound was taken to be \(235\ \text{m/sec}\), and the density of helium \(0.145\ \text{g}/\text{cm}^3\). At temperatures close to the \(\lambda\)-point, the theoretical formulas may give a considerable error; therefore the upper parts of the curves (above approximately \(1.8^\circ\text{K}\)) have been extrapolated according to the available experimental data*).

II. SUPERFLUIDITY OF HELIUM II

6. Superfluidity of helium II at absolute zero

We shall now show that the superfluidity of helium II follows from the above-described properties of the energy spectrum. Let us begin with a consideration of liquid helium at absolute zero; at this temperature the liquid is in its normal, unexcited state.

Consider helium flowing through a capillary with constant velocity \(\mathbf{v}\). The presence of viscosity would manifest itself in the fact that, owing to friction against the walls of the tube and friction within the liquid itself, dissipation of the kinetic energy of the liquid and a gradual slowing of the flow would occur.

It will be more convenient for us to consider the flow in a coordinate system moving together with the liquid. In this system the helium is at rest, while the walls of the capillary move with velocity \(-\mathbf{v}\). If viscosity is present, the helium at rest must also begin to move. Physically it is obvious that the entrainment of the liquid by the walls of the tube cannot lead from the very beginning to motion of the liquid as a whole. The appearance of motion must begin with the excitation of internal motions in the layers of liquid close to the wall, i.e. with the appearance in the liquid of excitation quanta.

*) It is necessary to note the following circumstance. A careful study of the results of measurements of the heat capacity (Keesom and Miss Keesom) and of the entropy of helium II (P. Kapitza) reveals that between these data, taken by themselves, there is a noticeable contradiction. This discrepancy, which violates the thermodynamic relation \(C = T(\partial S/\partial T)\), shows up when one attempts a more accurate determination of the parameters entering into the theoretical formulas.

There are grounds for believing that the entropy measurements in Kapitza’s experiments were made with greater accuracy than the heat-capacity measurements of Keesom (above all, measurements of heat capacity, as a quantity of differential character, are in themselves more difficult; it should also be borne in mind that the temperature intervals over which the heat capacity was measured in Keesom’s experiments were comparatively large). Therefore, in comparing theory with experiment, precisely the data on the entropy of helium II were chosen as the basis. The curve for the heat capacity (Fig. 3), however, was then calculated over the whole temperature range from the theoretical formula (with, for temperatures above \(\sim 1.8^\circ\text{K}\), a correction for the nonideality of the roton gas—see § 17).

Suppose that in the liquid a quantum is excited with momentum \(\mathbf p\) and energy \(\varepsilon(p)\). Then the energy \(E_0\) of the liquid (in the reference frame in which it was initially at rest) is made equal to the energy \(\varepsilon\) of this quantum, and its momentum \(\mathbf P_0\) to the momentum \(\mathbf p\) of the quantum. We now pass back to the coordinate system in which the capillary is at rest. According to

Fig. 2. Entropy of helium II.

the well-known formulas for the transformation of energy and momentum in classical mechanics, for the energy \(E\) and momentum \(\mathbf P\) of the liquid in this system we have

\[ E=E_0+\mathbf P_0\mathbf v+\frac{Mv^2}{2}, \qquad \mathbf P=\mathbf P_0-M\mathbf v \]

(\(M\) is the mass of the liquid). Substituting \(\varepsilon,\mathbf p\) for \(E_0,\mathbf P_0\), we write

\[ E=\varepsilon+\mathbf p\mathbf v+\frac{1}{2}Mv^2 . \]

The term \(\dfrac{1}{2}Mv^2\) represents the initial kinetic energy of the flowing helium; the expression \(\varepsilon + \mathbf{p}\mathbf{v}\) is the change of energy due to excitation of a quantum. This change must be negative, since the energy of the moving liquid must decrease

Fig. 3. Heat capacity of helium II.

\[ \varepsilon + \mathbf{p}\mathbf{v} < 0. \]

For a given value of \(p\), the quantity on the left-hand side of the inequality has the least value when \(\mathbf p\) and \(\mathbf v\) are antiparallel; therefore in every case one must have \(\varepsilon-pv<0\), i.e.

\[ v>\varepsilon/p. \]

This inequality must be satisfied for all values of the momentum \(p\) of the excitation quantum. Therefore the final condition for the possibility of the appearance of excitations in a liquid moving through a capillary is obtained by finding the minimum of the quantity \(\varepsilon/p\). The equality

\[ \frac{d}{dp}\left(\frac{\varepsilon}{p}\right) = \frac{1}{p}\frac{d\varepsilon}{dp}-\frac{\varepsilon}{p^2}=0, \]

i.e. \(\dfrac{\varepsilon}{p}=\dfrac{d\varepsilon}{dp}\), determines on the curve \(\varepsilon(p)\) the point at which the straight line drawn from the origin is tangent to the curve. Denoting the corresponding value of \(p\) by \(p_m\), we finally have

\[ v>\left(\frac{d\varepsilon}{dp}\right)_{p=p_m}. \tag{6,1} \]

We see that at flow velocities smaller than a certain limit [determined by the slope of the tangent drawn from the origin to the curve \(\varepsilon=\varepsilon(p)\)], excitations cannot arise in helium. This means that the flow of the liquid will not be slowed down, i.e. helium will exhibit the phenomenon of superfluidity.

At flow velocities satisfying condition (6,1), the superfluidity of helium must in any case disappear. It must be borne in mind, however, that these velocities are very large (of the order of magnitude

\[ \frac{d\varepsilon}{dp_m}\simeq \frac{1}{\mu}\left(\sqrt{2\mu\Delta+p_0^2}-p_0\right) \sim 60\,\frac{\mathrm m}{\mathrm{sec}} \]
)—considerably greater than the actually observed “critical velocities” for the disappearance of superfluidity. Therefore the latter must have some other physical nature.

Let us note that the condition under which the energy spectrum leads to the property of superfluidity consists, according to (6,1), in the derivative appearing on the right-hand side of the inequality having a value different from zero. This condition is essentially reduced to the requirement that the curve \(\varepsilon=\varepsilon(p)\) not be tangent to the abscissa axis at the origin (apart from the unlikely possibility of its tangency to this axis in its subsequent course). Thus, in essence, any spectrum in which sufficiently small excitations reduce to phonons alone leads to superfluidity.

7. Helium II at Temperatures Above Absolute Zero

Let us now consider helium at a temperature different from absolute zero. Then helium is not in the ground state; it contains excitations, of which, at sufficiently low temperature, one may say that they form a gas consisting of excitation quanta. The derivation carried out above remains valid in itself even now, since it did not make direct use of the circumstance that the liquid was initially in the ground state. Therefore, also at temperatures different from \(0^\circ K\), the motion of helium relative to the walls of the tube cannot lead to the appearance in it of excitation quanta. It is necessary, however, to determine in what way the presence of quanta already existing in the liquid itself will manifest itself.

For this purpose let us consider liquid helium contained in an axially symmetric vessel rotating about its axis with a constant angular velocity \(\Omega\). Let us pass to a coordinate system rotating together with the vessel. In this system the vessel is at rest, i.e. the external conditions in which the liquid is found are stationary. Therefore in it the Gibbs distribution obtains, i.e. the probability of finding the liquid in one or another excited state is determined by the formula

\[ \text{const}\cdot e^{-E'/kT}, \]

where \(E'\) is the energy of the excited state of the liquid in the rotating coordinate system. Let us now pass back to the stationary frame of reference. As is known, the energy \(E'\) of a body in a rotating frame of reference is related to the energy \(E\) in the stationary frame by the relation

\[ E' = E - \mathbf{M}\Omega, \]

where \(\mathbf{M}\) is the angular momentum of the body in the given excited state (in the normal state \(\mathbf{M}=0\)). Thus, the Gibbs distribution in the stationary frame of reference is

\[ \text{const}\cdot e^{-(E-\mathbf{M}\Omega)/kT}. \tag{7.1} \]

For clarity, let us assume that the temperature is sufficiently low, so that one may speak of an excited state as of an ideal gas of excitation quanta. Then the energy \(E\) and the angular momentum \(\mathbf{M}\) of the excited state may be written in the form

\[ E=\sum \varepsilon,\qquad \mathbf{M}=\sum \mathbf{m}, \tag{7.2} \]

where \(\varepsilon,\ \mathbf{m}\) are the energy and angular momentum of the individual quanta.

As is known, upon substituting energies in the form \(\sum \varepsilon\) into the Gibbs distribution \(e^{-E/kT}\), one can pass to the distribution for the individual “particles” of the gas—in the present case, to the Bose distribution

\[ \left(e^{\varepsilon/kT}-1\right)^{-1} \]

(incidentally, which particular distribution it is is immaterial for what follows).

Similarly, upon substituting (7,2) into (7,1), we obtain in the same way the distribution for the quanta of excitation in a rotating vessel, with the only difference that instead of \(\varepsilon\) there will stand \(\varepsilon-\Omega m\); thus the Bose distribution takes the form

\[ \left(e^{(\varepsilon-\Omega m)/kT}-1\right)^{-1}. \]

But such a distribution is nothing other than the distribution in a gas rotating as a whole (with angular velocity \(\Omega\)). Thus we arrive at the result that, in a rotating vessel with helium II, a statistical equilibrium is established which differs from the equilibrium in a stationary vessel only in that the gas of excitation quanta rotates together with the vessel, as though entrained by its walls.

If from the written distribution one calculates the angular momentum of helium in a rotating vessel at a given temperature, i.e. the quantity

\[ \bar{M}= \frac{1}{(2\pi\hbar)^3} \int \int \frac{m\,d\tau_p\,dV}{e^{(\varepsilon-\Omega m)/kT}-1}, \tag{7,3} \]

then at absolute zero, i.e. in the complete absence of excitation quanta, we would obviously obtain zero. At higher temperatures the angular momentum will be different from zero, but the moment of inertia (i.e. the coefficient of proportionality between \(\bar{M}\) and \(\Omega\)) will, at sufficiently low temperatures, still be far smaller than the ordinary one (corresponding to rotation together with the vessel of the entire mass of the liquid).

We thus arrive at the fundamental result that, when the walls of the vessel move, only part of the mass of liquid helium is entrained by them, while the other part remains, as it were, motionless. Therefore liquid helium may be vividly regarded as if it were a mixture of two liquids—one superfluid, possessing no viscosity and not entrained by the walls of the vessel, and another normal one, “catching” on the wall as it moves and behaving like a normal liquid. Here it is very important that between these two liquids moving “through one another” there is no “friction,” i.e. no transfer of momentum occurs from one of them to the other. Indeed, the very existence of such mutual motion was obtained by us in considering statistical equilibrium in a uniformly rotating vessel. But if some kind of

relative motion can take place in a state of static equilibrium, this means that it cannot be accompanied by friction.

Let us emphasize that the consideration of helium as a “mixture” of two liquids is no more than a convenient manner of expression, useful for describing the phenomena occurring in helium II. Like any description of quantum phenomena in classical terms, it is not fully adequate. In reality one must say that in a quantum liquid, such as helium II is, two motions can exist simultaneously, each of which is associated with its own “effective mass” (so that the sum of these two masses is equal to the total true mass of the liquid). One of these motions is “normal,” i.e. it has the same properties as the motion of an ordinary liquid; the other is “superfluid.” Both these motions occur without transfer of momentum from one to the other. Let us especially emphasize that here there is no division of the real particles of the liquid into “superfluid” and “normal” ones. In a certain sense one may speak of the “superfluid” and “normal” masses of the liquid, as masses associated with the two simultaneously possible motions; but this by no means signifies the possibility of a real separation of the liquid into two parts.

Bearing in mind all these reservations concerning the true nature of the phenomena occurring in helium II, it is nevertheless convenient to use the terms “superfluid liquid” and “normal liquid” as a convenient way of briefly describing these phenomena.

When a vessel containing helium is rotated, the superfluid part of the liquid, as was indicated, remains at rest. One may say that a superfluid liquid is incapable of rotation. Mathematically this means that the curl of the velocity of the superfluid motion is equal to zero. Thus, the motion of the superfluid liquid is always potential. On the contrary, the normal part of the liquid can execute both potential and vortical motion.

In particular, from this follows the following interesting property of the motion of helium II. As is known from hydrodynamics, in potential flow a liquid exerts no pressure on bodies over which it flows (the so-called d’Alembert paradox). Therefore the superfluid part of the liquid will, in its motion, exert no pressure whatever on a body immersed in helium II; it will experience pressure only from the normal part of the liquid.

The most important parameter determining the properties of helium at each given temperature is the ratio between the masses of the superfluid and normal parts of the liquid. Let us introduce the density \(\rho_n\) of the normal liquid and the density \(\rho_s\) of the superfluid; the sum \(\rho = \rho_n + \rho_s\) is the total true density of the liquid.

At absolute zero the ratio \(\rho_n/\rho\) is equal to zero. As the temperature is raised it increases until it becomes equal to unity, after which, of course, it will remain constant. The temperature at which \(\rho_n/\rho\) becomes unity is the point

of the transition of helium II into helium I. Thus, the phase transition in liquid helium is associated with the disappearance of the superfluid part of the liquid. This disappearance occurs gradually, i.e. \(\rho_n/\rho\) tends to unity continuously, without a jump. Therefore the transition is a phase transition of the second kind (it is not accompanied by the liberation or absorption of latent heat). The presence of a discontinuity in the heat capacity is, as is known, a direct thermodynamic consequence of a phase transition of the second kind.

The ratio \(\rho_n/\rho\) can be measured directly in experiment by determining the moment of inertia \(J\) of a cylindrical vessel filled with helium II and rotating about its axis. The ratio of \(J\) to the moment of inertia \(J_0\), computed on the assumption that the entire mass of helium rotates together with the vessel, gives the ratio \(\rho_n/\rho\) at the given temperature. Similar measurements were carried out by E. Andronikashvili and led to values of \(\rho_n/\rho\) in very good agreement with the theoretical ones (and with the values that can be calculated from the velocity of “second sound”—see § 14).

The ideas developed above make it possible to give a simple explanation of the properties observed experimentally in the flow of helium II. P. L. Kapitsa showed that, when flowing through a capillary or through a thin narrow slit, helium II exhibits no viscosity. From the point of view of the theory presented here, this is explained by the fact that, when helium II flows out of a vessel through a narrow slit, the superfluid “part” of the liquid passes through the slit without displaying friction; the normal “part,” however, is retained in the vessel, flowing through the slit incomparably more slowly, with a velocity corresponding to its viscosity and to the width of the slit.

In an equally natural way one explains the fact that, when the viscosity of helium II is measured from the damping of torsional oscillations of a disk immersed in the liquid, values different from zero are obtained. Indeed, when the disk rotates in a liquid containing both superfluid and normal parts, it will be brought to rest already by friction against the normal liquid. Thus, in experiments on flow through a capillary the presence of the superfluid part of the liquid is revealed, whereas in experiments with the rotation of a disk in helium II its normal part is revealed.

The well-known property of helium II of forming films moving over a solid surface is an effect related to its property of flowing through thin capillaries. The very fact of film formation is not a special property inherent only in helium II. Films are formed by any liquid that wets a solid surface. In ordinary liquids, however, the formation of a film and its spreading over a large area of the surface occur extremely slowly owing to the presence of viscosity; in helium II, by contrast, the formation and motion of the film occur rapidly owing to its superfluidity. (Ya. I. Frenkel\(^{13}\), in this work, gave a quantitative analysis of the film formed on a vertical wall.)

8. Calculation of the ratio \(\rho_n/\rho\)

At sufficiently low temperatures the ratio \(\rho_n/\rho\) can be calculated. As was indicated above, the “ideal gas” of excitation quanta in helium II may be regarded as consisting of “particles” of two kinds—phonons and rotons. Correspondingly, at low temperatures \(\rho_n\) is composed of two independent parts—the effective mass of the ideal phonon gas and the mass of the roton gas.

Let us imagine that the gas of excitation quanta moves as a whole translationally with velocity \(\mathbf v\). As is known, the distribution function for a gas moving as a whole is obtained from the distribution function of a stationary gas simply by replacing the particle energy \(\varepsilon\) by \(\varepsilon-\mathbf p\mathbf v\), where \(\mathbf p\) is the particle momentum. Therefore the total momentum of the gas (referred to unit volume) will be

\[ \mathbf P=\int \mathbf p\, n(\varepsilon-\mathbf p\mathbf v)\,d\tau_p, \]

where by \(n(\varepsilon)\) one must understand the Bose distribution in the case of phonons, or the Boltzmann distribution in the case of rotons.

Suppose that the velocity \(\mathbf v\) is small, and expand the integrand in powers of \(\mathbf p\mathbf v\). The term of zeroth order vanishes upon integration over the directions of the vector \(\mathbf p\), and there remains

\[ \mathbf P=-\int \mathbf p(\mathbf p\mathbf v)\frac{\partial n(\varepsilon)}{\partial\varepsilon}\,d\tau_p. \tag{8,1} \]

Let us first consider the phonon gas. Since for phonons \(\varepsilon=cp\), we have

\[ \frac{cp}{p}\frac{\partial n}{\partial\varepsilon}=\frac{\partial n}{\partial p}, \]

so that one may write

\[ \mathbf P=-\frac{1}{c}\int \mathbf p(\mathbf p\mathbf v)\frac{\partial n}{\partial p}\,d\tau_p. \]

Integrating by parts, we have

\[ \mathbf P=\frac{1}{c}\int\left\{\mathbf p\mathbf v+\frac{\mathbf p}{p}(\mathbf p\mathbf v)\right\}n\,d\tau_p. \]

Let us average the integrand over the directions of the vector \(\mathbf p\). Under such averaging \(\mathbf p(\mathbf p\mathbf v)\) gives \(\frac{1}{3}p^2\mathbf v\), so that we obtain

\[ \mathbf P=\frac{4}{3c}\mathbf v\int pn\,d\tau_p =\frac{4}{3c^2}\mathbf v\int \varepsilon n(\varepsilon)\,d\tau_p. \]

But the integral standing here is nothing other than the energy \(\rho E_\phi\) of a unit volume of the phonon gas, so that

\[ \mathbf P=\frac{4E_\phi}{3c^2}\rho\mathbf v. \]

The coefficient of \(\mathbf v\) is the desired effective mass of a unit volume of the phonon gas, i.e. the phonon part \((\rho_n)_\phi\) of the density \(\rho_n\):

\[ (\rho_n)_\phi=\frac{4E_\phi}{3c^2}\rho=\frac{2\pi^2 k^4}{45\hbar^3 c^5}T^4, \tag{8,2} \]

(for the energy \(E_\phi=F_\phi+TS_\phi\) formulas (5,1—2) have been used). Thus, the phonon part of \(\rho_n\) is proportional to \(T^4\).

To calculate the roton part of \(\rho_n\), we note that for the Boltzmann distribution

\[ \frac{dn(\varepsilon)}{d\varepsilon}=-\frac{n}{kT}; \]

also averaging the integrand in (8,1) over the directions of \(\mathbf p\), we obtain

\[ \mathbf P=\frac{\mathbf v}{3kT}\int p^2 n(\varepsilon)\,d\tau_p =\mathbf v\,\frac{\overline{p^2}}{3kT}\rho N_p, \]

where \(\overline{p^2}\) is the mean value of the square of the roton momentum.

Thus we obtain the formula

\[ (\rho_n)_p=\frac{\overline{p^2}}{3kT}\rho N_p. \tag{8,3} \]

In view of the fact that \(p_0^2\gg \mu kT\), one may with sufficient accuracy put \(\overline{p^2}=p_0^2\), and we arrive at the final formula

\[ (\rho_n)_p=\frac{p_0^2}{3kT}\rho N_p =\frac{2\mu^{1/2}p_0^4}{3(2\pi)^{3/2}(kT)^{1/2}\hbar^3}\,e^{-\Delta/kT}. \tag{8,4} \]

[\(N_p\) from (5,5)]. Thus, the roton part of \(\rho_n\) depends on the temperature exponentially.

The phonon part of \(\rho_n\) becomes substantial only at the very lowest temperatures. Even at \(1^\circ\) K the phonon part is 60 times smaller than the roton part; they become equal at a temperature of about \(0.6^\circ\) K.

The part of the curve \(\rho_n(T)\) close to the \(\lambda\)-point cannot, of course, be calculated exactly. One may, however, expect that, owing to the very rapid growth of \(\rho_n\) according to formula (8,4), the very value of the temperature of the \(\lambda\)-point can be approximately obtained by putting \(\rho_n/\rho=1\) and using formula (8,4) for \(\rho_n\). Such a calculation gives for the temperature of the \(\lambda\)-point the value \(2.38^\circ\)K, in good agreement with the actual value \(2.19^\circ\)K.

In Fig. 4 the curve of the temperature dependence \(\rho_n/\rho\), calculated from the formulas obtained, is given (the calculations were carried out by I. Khalatnikov and O. Kramer). The upper part of the curve was obtained by interpolation with the aid of experimental data.

Fig. 4. Temperature dependence of the ratio \(\rho_n/\rho\).

9. Heat Transfer in Helium II

The entropy of helium II is determined by the statistical distribution of excitation quanta. Therefore, in any motion of the liquid in which the “gas” of excitation quanta remains at rest, no macroscopic transport of entropy arises. We thus arrive at the very important result that, in the flow of a superfluid liquid, no transport of entropy occurs. In other words, a superfluid liquid does not carry heat with it in its motion. Hence it follows, in turn, that the motion of helium II in which only its superfluid part takes part is thermodynamically reversible.

As was already indicated, when helium II flows through thin capillaries or slits, we are dealing precisely with such a flow of the superfluid liquid. Therefore such flow must be reversible (more precisely, the closer to complete reversibility the thinner the capillary is and the less normal liquid penetrates into it). This circumstance was indeed found in the experiments of P. L. Kapitza.

Furthermore, since helium flowing through a thin capillary carries no heat with it, we can draw one more conclusion. Namely, liquid helium flowing out of a vessel through a thin capillary must be at the temperature of absolute zero (more precisely, at a temperature lower than the temperature of the helium in the vessel, and equal to zero only in the ideal case of an infinitely thin capillary).

The transport of heat by the moving normal part of the liquid constitutes the mechanism of heat transfer in helium II. It thus has a peculiar convective character and is fundamentally different from ordinary thermal conduction. Any temperature difference in helium II immediately gives rise in it to internal motions of the normal and superfluid parts “through one another”; at the same time, there may be no real macroscopic flow accompanied by transport of mass in the liquid.

This kind of situation occurs in heat transfer in a capillary, the large magnitude of which was discovered by Keesom and Miss Keesom and which was studied in detail by Kapitza. When helium is heated at one end of a capillary, two oppositely directed currents arise in the helium. From the heated end to the cold end there goes a current of normal liquid carrying heat; the heat thus transported is more than sufficient to explain the experimentally observed large magnitudes of heat transfer. In the opposite direction there goes a current of superfluid liquid. The two currents exactly compensate one another in the amount of mass carried by them, so that no real macroscopic flow in the helium actually arises.

P. L. Kapitza observed the deflection of a leaf suspended in front of the open end of a capillary when helium was heated at its closed end. This phenomenon receives the following natural explanation. The stream of superfluid liquid flowing into the capillary exerts no pressure on the leaf which it washes over (owing to the potential character of the motion—see § 7). On the contrary, the stream of normal liquid flowing out of the capillary exerts on the leaf a pressure that deflects it away from the end of the capillary. Let us note that we arrive at a very peculiar picture: a force may act on a body immersed in helium II, while no motion of the liquid as a whole takes place.

10. The thermomechanical effect in helium II

The so-called thermomechanical effect in helium II consists, as is known, in the fact that when helium flows out of a vessel through a thin capillary, heating is observed in the vessel; conversely, at the point where helium flows from the capillary into the vessel, cooling is observed. The existence of the thermomechanical effect in itself is by no means peculiar to helium alone; what is anomalous in helium II is only the large magnitude of the effect. The thermomechanical effect in ordinary liquids is an irreversible phenomenon of the type of the thermoelectric Peltier effect.

An effect of this kind must also exist in helium II; however, in this case it is masked by another effect, greatly exceeding it, which is specific to helium II and has nothing in common with irreversible phenomena of the Peltier-effect type. Namely, the heating upon outflow occurs simply because the helium flowing out through the capillary carries away no heat, and therefore the heat present in the vessel is distributed over a smaller amount of helium II. When helium flows into the vessel the reverse phenomenon takes place.

It is easy to write down what the amount of heat \(Q\) absorbed is when \(1\ \mathrm{g}\) of helium flows into a vessel through a capillary. Since the inflowing liquid contains no excitation quanta, its entropy is zero. In order that the helium in the vessel remain at its temperature \(T\), it would be necessary to supply to it an amount of heat \(TS\) (\(S\) is the entropy of \(1\ \mathrm{g}\) of helium at temperature \(T\)), so as to compensate for the decrease in the entropy per unit mass due to the introduction of \(1\ \mathrm{g}\) of helium with entropy equal to zero. This means that when \(1\ \mathrm{g}\) of helium flows into a vessel with helium at temperature \(T\), the amount of heat absorbed is

\[ Q = TS. \tag{10,1} \]

Conversely, when \(1\ \mathrm{g}\) of helium flows out of a vessel with helium at temperature \(T\), the amount of heat \(TS\) is liberated.

Let us now consider two vessels with helium II at temperatures \(T_1\) and \(T_2\), the vessels being connected with one another by a thin capillary. Owing to the possibility of the free flow of the superfluid liquid through the capillary, mechanical equilibrium of the helium in the two vessels will quickly be established. Since, however, the superfluid liquid does not carry heat, thermal equilibrium (in which the temperatures of the helium in the two vessels become equal) will be established only much more slowly.

The condition of mechanical equilibrium of the helium is easily written down, using the fact that the establishment of this equilibrium proceeds, in accordance with the foregoing, at constant entropies \(S_1\) and \(S_2\) of the helium in the two vessels. If \(E_1\) and \(E_2\) are the internal energies per unit mass of helium at temperatures \(T_1\) and \(T_2\), then the condition of mechanical equilibrium (i.e. of minimum energy), effected by the transfer of the superfluid liquid, will be

\[ \left(\frac{\partial E_1}{\partial N}\right)_{S_1} = \left(\frac{\partial E_2}{\partial N}\right)_{S_2}, \]

where \(N\) is the number of atoms in \(1\ g\) of helium. But the derivative \((\partial E/\partial N)_S\) is the chemical potential \(\zeta\). We therefore obtain the equilibrium condition in the form

\[ \zeta(P_1,T_1)=\zeta(P_2,T_2) \]

(\(P_1, P_2\) are the pressures in the first and second vessels), or

\[ \Phi(P_1,T_1)=\Phi(P_2,T_2), \]

where \(\Phi\) is the thermodynamic potential of helium II.

If the pressures \(P_1, P_2\) are small, then, expanding in powers of them and remembering that \(\partial \Phi/\partial P\) is the specific volume \(V\), we obtain

\[ V\,\Delta P=\Phi(T_1)-\Phi(T_2)=\int_{T_1}^{T_2} S\,dT \]

(\(\Delta P=P_2-P_1\)). If the temperature difference \(\Delta T=T_2-T_1\) is also small, then, expanding in powers of \(\Delta T\) and noting that \(\partial\Phi/\partial T=-S\), we obtain

\[ \frac{\Delta P}{\Delta T}=\frac{S}{V}. \tag{10,2} \]

Since \(S>0,\ V>0\), it follows that \(\Delta P/\Delta T>0\), in agreement with experiment. Formulas (10,1) and (10,2) are fully confirmed by the experiments of P. L. Kapitza.

Formulas (10,1)–(10,2) were also obtained by G. London\(^{14}\), starting from Tisza’s considerations, the verbal formulation of which in this point coincides with the results of Landau’s theory.

Let us note that the phenomena described can be vividly represented as osmotic phenomena in a “solution” of excitation quanta in liquid helium, the role of the semipermeable partition being played by thin capillaries or slits.

11. Behavior of impurity atoms in helium II

The question of the behavior of atoms of foreign substances dissolved in helium II is of interest; in particular, one may be speaking of atoms of the helium isotope (\(\mathrm{He}^{3}\)) admixed in negligible concentration with the principal isotope (\(\mathrm{He}^{4}\)); the considerations set forth below are due to L. Landau and I. Pomeranchuk\({}^{15}\).

For a theoretical approach to this question one must consider the energy spectrum of the system helium II + impurity atom (the impurity concentration is assumed to be so small that its atoms may be regarded as not interacting with one another). The presence of a foreign atom in the liquid will lead to the appearance of a new branch of the energy spectrum, corresponding to the motion of this atom through the liquid; of course, in view of the strong interaction of the impurity atom with the atoms of the liquid, this motion is in reality a “collective” effect, in which the helium atoms also take part. To this motion one may assign a certain resultant conserved momentum \(p'\).

Thus, in the liquid there will appear “elementary excitations” of a new type (in a number equal to the number of impurity atoms), whose energy \(\varepsilon'\) is a definite function of the momentum \(p'\). In statistical equilibrium these excitations will have energy values concentrated near the smallest of the minima of the function \(\varepsilon'(p')\). Near the minimum the function \(\varepsilon'(p')\) has the form \(\varepsilon'=(p'-p'_0)^2/2\mu'\) (\(\varepsilon'\) being measured from its minimum value), where \(p'_0\) may in particular cases be equal to zero; \(\mu'\) is the effective mass associated with the motion of the impurity atom.

These new “elementary excitations” will interact with phonons and rotons, colliding with them, and thus will enter into the composition of the “normal part” of the liquid. Therefore, for example, when helium II is passed through a thin slit, the impurity atoms will be filtered out together with the entire “normal liquid.”*) We emphasize that this circumstance has no relation whatever to the question of whether the impurity substance itself (in particular, the pure isotope \(\mathrm{He}^{3}\)) would exhibit the property of superfluidity, contrary to the incorrect view expressed in the literature\({}^{6,9}\).

*) Such “filtering out” of the isotope \(\mathrm{He}^{3}\) has indeed recently been observed by Dount, Hiram, and others.

E. M. LIFSHITZ

III. MACROSCOPIC HYDRODYNAMICS OF HELIUM II

12. The system of hydrodynamic equations of helium II

Starting from the ideas developed above concerning the microscopic mechanism of the phenomenon of superfluidity, one can construct a complete system of hydrodynamic equations which would describe helium II in a macroscopic (phenomenological) way.

The initial premise is the fundamental circumstance that in a quantum liquid—helium II—two motions occur simultaneously and, accordingly, the motion of helium II must be described at each point not by one velocity, as in ordinary hydrodynamics, but by two velocities. We shall denote by $\mathbf{v}_s$ and $\mathbf{v}_n$ the velocities of the “superfluid” and “normal” motions, respectively. Since the superfluid motion is always potential, we have

\[ \operatorname{rot}\mathbf{v}_s=0. \tag{12,1} \]

It turns out that the hydrodynamic equations with two velocities $\mathbf{v}_s$ and $\mathbf{v}_n$ can be obtained in an entirely unambiguous way from the single requirement that all necessary conservation laws be satisfied. These equations for the general case of arbitrary velocities are rather complicated, and we shall not present them here, but shall confine ourselves only to a simplified derivation of the equations applicable to motion with not too large velocities $\mathbf{v}_s$ and $\mathbf{v}_n$.

Let us denote by $\mathbf{j}$ the macroscopic mass flux of the liquid. It is a function of both velocities $\mathbf{v}_s$, $\mathbf{v}_n$, and at small velocities may be expanded in powers of them. In the first approximation,

\[ \mathbf{j}=\rho_s\mathbf{v}_s+\rho_n\mathbf{v}_n. \tag{12,2} \]

The coefficients $\rho_s$ and $\rho_n$ are, evidently, what we called the densities of the superfluid and normal “parts” of the liquid. Their sum is equal to the density $\rho$ of helium II:

\[ \rho=\rho_s+\rho_n; \tag{12,3} \]

$\rho_s$ and $\rho_n$ are, of course, functions of temperature. Let us note that the mass flux $\mathbf{j}$ is at the same time nothing other than the momentum density, i.e. the momentum of unit volume of the liquid.

The density $\rho$ and the flux $\mathbf{j}$ must satisfy the continuity equation

\[ \frac{\partial \rho}{\partial t}+\operatorname{div}\mathbf{j}=0. \tag{12,4} \]

For the present we shall consider only such motions of the liquid in which the viscosity of its normal part plays no role. Then the equations of conservation of momentum are written in the form

\[ \frac{\partial j_i}{\partial t}+\sum_{k=1}^{3}\frac{\partial \Pi_{ik}}{\partial x_k}=0, \tag{12,5} \]

where \(\Pi_{ik}\) is the tensor of momentum-flux density, equal to

\[ \Pi_{ik}=P\delta_{ik}+\rho_s v_{si}v_{sk}+\rho_n v_{ni}v_{nk}, \tag{12.6} \]

where \(P\) is the pressure (recall that in ordinary hydrodynamics \(\Pi_{ik}=P\delta_{ik}+\rho v_i v_k\)).

In the absence of viscosity effects, the motion of the normal part of the liquid helium II is reversible. Accordingly, the entropy conservation equation must hold,

\[ \frac{\partial(\rho S)}{\partial t}+\operatorname{div}(\rho S\mathbf{v}_n)=0 \tag{12.7} \]

(\(S\) is the entropy per unit mass of helium II). The “entropy flux” is equal to \(\rho S\mathbf{v}_n\), since entropy is transported only by the normal part of the liquid. The heat flux \(\mathbf{q}\) is, correspondingly, equal to

\[ \mathbf{q}=\rho TS\mathbf{v}_n . \tag{12.8} \]

Finally, we obtain the last equation of the complete system of hydrodynamic equations by equating the acceleration \(d\mathbf{v}_s/dt\) to the force acting on a unit “superfluid” mass. To determine this force, imagine that a unit mass of the liquid is transferred from point 1 to point 2 in such a way that the distribution of excitation quanta in the liquid is not changed. In other words, one may say that in the transfer only the “superfluid liquid” is displaced, while the distribution of the “normal” component remains unchanged. The energy \(E\) of the liquid changes under such a transfer by

\[ \left(\frac{\partial E}{\partial M}\right)_1-\left(\frac{\partial E}{\partial M}\right)_2 \]

(\(M\) is the mass of the liquid). The derivatives here must be taken at constant entropy (since the entropy is associated only with the normal liquid) and at constant momentum of motion of the normal mass of the liquid relative to the superfluid*); in addition, the volume of the liquid is, of course, also considered constant.

From the expression written for the change in energy it is clear that the quantity \(\partial E/\partial M\) may be regarded as the “potential energy” of the superfluid liquid, so that the force acting on it is

\[ -\operatorname{grad}\frac{\partial E}{\partial M}. \]

To compute the derivative \(\partial E/\partial M\), we note that the derivative of the energy at constant entropy and volume is equal to the derivative

\[ \text{*) The motion of the superfluid liquid may be regarded as external conditions in which the excitation quanta move. Therefore the Lagrange function for the motion of the normal part of the liquid depends not simply on its velocity \(\mathbf{v}_n\), but on the difference of the velocities \(\mathbf{v}_n-\mathbf{v}_s\). The conserved momentum is therefore the derivative of the Lagrange function with respect to \(\mathbf{v}_n-\mathbf{v}_s\), i.e., the indicated momentum of relative motion.} \]

of the thermodynamic potential at constant temperature and pressure. The thermodynamic potential of the liquid \(M\Phi\) (\(\Phi\) is the potential per unit mass) can be written as the sum of the thermodynamic potential of a liquid at rest and the kinetic energy \(P^2/2M_n\) of the relative motion of the superfluid and normal “parts”:

\[ M\Phi=M\Phi_0+\frac{P^2}{2M_n}. \]

Here \(P\) is the momentum of motion of the normal mass relative to the superfluid. Differentiating \(M\Phi\) with respect to \(M\) at constant \(P, T, P\) and remembering that the normal mass \(M_n\) is proportional (for given \(P\) and \(T\)) to the total mass \(M\), we obtain

\[ \Phi_0-\frac{P^2}{2M_nM}. \]

Substituting here \(P=M(\mathbf v_n-\mathbf v_s)\) and replacing the ratio of masses by the ratio of densities, we finally find for the required derivative \((\partial E/\partial M)_{S,V,P}\) the expression

\[ \Phi_0-\frac{\rho_n}{2\rho}(\mathbf v_n-\mathbf v_s)^2. \]

Consequently, the hydrodynamic equation sought has the form

\[ \frac{d\mathbf v_s}{dt} = \frac{\partial \mathbf v_s}{\partial t} + (\mathbf v_s\nabla)\mathbf v_s = -\operatorname{grad}\left\{\Phi-\frac{\rho_n}{2\rho}(\mathbf v_n-\mathbf v_s)^2\right\} \]

(the subscript on \(\Phi_0\) is omitted). It can be written otherwise, noting that, since always \(\operatorname{rot}\mathbf v_s=0\), we have

\[ (\mathbf v_s\nabla)\mathbf v_s=\operatorname{grad}\frac{v_s^2}{2}. \]

Thus,

\[ \frac{\partial \mathbf v_s}{\partial t} = -\operatorname{grad}\left\{\Phi+\frac{v_s^2}{2}-\frac{\rho_n}{2\rho}(\mathbf v_n-\mathbf v_s)^2\right\}. \tag{12,9} \]

Equations (12,1–9) constitute the complete system of hydrodynamic equations of helium II. It remains to write the boundary conditions for these equations. First of all, on every (stationary) solid surface the component of the mass flux \(\mathbf j\) perpendicular to this surface must vanish. Further, it must always be remembered that the “normal part” of the liquid is in fact the aggregate of the thermal excitations in it. In motion along a solid surface the excitation quanta interact with it, which must be described macroscopically as “sticking” of the normal part of the liquid to the wall, just as occurs for ordinary viscous liquids. Thus, on a solid surface the tangential component of the velocity \(\mathbf v_n\) must vanish.

As for the component \(\mathbf v_n\) perpendicular to the wall, it must be borne in mind that excitation quanta may be absorbed by the solid—this corresponds simply to heat transfer from the liquid to the solid. Therefore the component of the velocity \(\mathbf v_n\) perpendicular to the wall need not necessarily vanish; the boundary condition requires only continuity of the component, perpendicular to the wall, of the heat flux \(\mathbf q\) (12.8). Finally, the temperature must be continuous.

Thus, the boundary conditions at the surface of an immobile solid body may be written in the following form (choosing a coordinate system with the \(x\)-axis perpendicular to the surface at the given point):

\[ \rho_s v_{sx}+\rho_n v_{nx}=0,\qquad v_{ny}=v_{nz}=0, \tag{12.10} \]

\[ \rho T S v_{nx}=-\varkappa\left(\frac{\partial T}{\partial x}\right)_{\mathrm{тв}},\qquad T_i=T_{\mathrm{тв}}, \tag{12.11} \]

where \(\varkappa\) is the coefficient of thermal conductivity of the solid.

In fact, however, down to the very lowest temperatures, heat transfer in the solid is extremely slow in comparison with heat transfer in helium II, and the effects due to the thermal conductivity of the solid prove to be very small. In these cases one may put \(\varkappa\) equal to zero, and the boundary conditions then take the form

\[ v_{sx}=0,\qquad \mathbf v_n=0. \tag{12.12} \]

In other words, we obtain the usual boundary conditions of an ideal liquid for \(\mathbf v_s\), and of a viscous liquid for \(\mathbf v_n\).

13. Hydrodynamic equations for an incompressible liquid

Let us consider such motions of helium II for which it may be regarded as an incompressible liquid, as is usually the case for all kinds of flow past bodies (L. Landau\(^2\)). In doing so we shall also take into account the viscosity of the normal part of the liquid.

In order to take into account the viscosity of the normal part of the liquid in equation (12.5), one must add to the tensor \(\Pi_{ik}\) terms expressed in the usual way through the coefficient of viscosity and the derivatives of the velocity \(\mathbf v_n\) with respect to the coordinates. For an incompressible liquid,

\[ \Pi_{ik}=P\delta_{ik}+\rho_s v_{si}v_{sk}+\rho_n v_{ni}v_{nk} -\eta\left(\frac{\partial v_{ni}}{\partial x_k}+\frac{\partial v_{nk}}{\partial x_i}\right), \tag{13.1} \]

where \(\eta\) is the coefficient of viscosity of the normal liquid\(^*\). Taking account of visco-

\(^*\) For a compressible liquid, \(\Pi_{ik}\) contains one more term, proportional to \(\operatorname{div}\mathbf v_n\), with the coefficient “second viscosity.” In addition, for a compressible liquid, additional terms proportional to \(\operatorname{div}\mathbf v_n\) must also be added to equation (12.9) (to the expression standing under the sign \(\operatorname{grad}\)). Thus, generally speaking, not one but several different coefficients of “second viscosity” must enter the hydrodynamics of helium II.

also leads to the appearance of additional terms on the right-hand side of equation (12.7), expressing the increase of entropy as a result of irreversible processes of viscous friction. These terms, however, generally turn out to be small quantities of higher order than the additional terms in (12.5), and therefore may still be omitted.

Taking the densities \(\rho_s\), \(\rho_n\) and the entropy \(S\) to be constant, we have from equation (12.7): \(\operatorname{div}\mathbf v_n=0\); and from equation (12.4) \(\operatorname{div}\mathbf j=0\), or, in other words,

\[ \operatorname{div}\mathbf v_n=\operatorname{div}\mathbf v_s=0. \tag{13,2} \]

Bearing these equalities in mind and substituting (13.1) into (12.5), we obtain the equation

\[ \rho_s\frac{\partial\mathbf v_s}{\partial t} +\rho_n\frac{\partial\mathbf v_n}{\partial t} +\rho_s(\mathbf v_s\nabla)\mathbf v_s +\rho_n(\mathbf v_n\nabla)\mathbf v_n = -\nabla P+\eta\Delta\mathbf v_n. \tag{13,3} \]

Equation (12.9), however, remains unchanged.

Since the superfluid motion is potential, one may introduce the velocity potential \(\varphi_s\) of \(\mathbf v_s\) according to

\[ \mathbf v_s=\operatorname{grad}\varphi_s. \tag{13,4} \]

In view of \(\operatorname{div}\mathbf v_s=0\), the potential \(\varphi_s\) must satisfy Laplace’s equation

\[ \Delta\varphi_s=0. \tag{13,5} \]

Introducing \(\varphi_s\) into equation (13.3) and writing
\((\mathbf v_s\nabla)\mathbf v_s=\nabla \dfrac{v_s^2}{2}\), we obtain

\[ \rho_n\frac{\partial\mathbf v_n}{\partial t} +\rho_s\frac{\partial\mathbf v_s}{\partial t} +\rho_n(\mathbf v_n\nabla)\mathbf v_n +\rho_s\,\operatorname{grad}\frac{v_s^2}{2} +\rho_s\,\operatorname{grad}\frac{\partial\varphi}{\partial t} = -\nabla P+\eta\Delta\mathbf v_n. \]

We introduce, as two auxiliary quantities, the “pressures” of the normal and superfluid flows, \(P_n\) and \(P_s\), according to the equation

\[ P=P_0+P_n+P_s, \tag{13,6} \]

where \(P_0\) is the pressure at infinity, while \(P_s\) is defined by the usual formula for an ideal fluid

\[ P_s=-\rho_s\frac{\partial\varphi_s}{\partial t}-\frac{\rho_s v_s^2}{2}. \tag{13,7} \]

The equation of motion for the velocity \(\mathbf v_n\) then assumes the form

\[ \frac{\partial\mathbf v_n}{\partial t} +(\mathbf v_n\nabla)\mathbf v_n = -\frac{1}{\rho_n}\nabla P_n +\frac{\eta}{\rho_n}\Delta\mathbf v_n. \tag{13,8} \]

This equation coincides formally with the Navier–Stokes equation;

for a liquid of density \(\rho_n\) and viscosity \(\eta\) (and, correspondingly, with kinematic viscosity \(\eta/\rho_n\)).

Thus, the problem of the motion of incompressible helium II proves to be reduced to two problems of ordinary hydrodynamics for ideal and viscous fluids. Namely, the distribution of the superfluid velocity \(\mathbf{v}_s\) is determined by the solution of Laplace’s equation (13.5) with a boundary condition for \(\partial\varphi_s/\partial n\), as in the ordinary problem of potential flow of an ideal fluid past a body. Further, the distribution of the normal velocity \(\mathbf{v}_n\) is determined by the solution of the Navier–Stokes equation (13.8) with the same boundary condition for \(\mathbf{v}_n\) as in the ordinary problem of flow past a body by a viscous fluid. The pressure distribution is then determined from formula (13.6).

Finally, let us derive the formula determining the temperature distribution in moving helium II. Writing in equation (12.9) \(\mathbf{v}_s=\nabla\varphi_s\) and integrating, we obtain

\[ \Phi+\frac{v_s^2}{2}-\frac{\rho_n}{2\rho}(\mathbf{v}_n-\mathbf{v}_s)^2+\frac{\partial\varphi_s}{\partial t}=\mathrm{const}. \tag{13.9} \]

The changes of temperature and pressure in an incompressible liquid are small, and therefore the thermodynamic potential may be expanded in powers of \(T-T_0\) and \(P-P_0\) (\(T_0, P_0\) are the temperature and pressure at infinity). To terms of the first order we have

\[ \Phi-\Phi_0=-S(T-T_0)+\frac{1}{\rho}(P-P_0). \]

Substituting this expression into (13.9), we obtain

\[ -S(T-T_0)+\frac{1}{\rho}(P-P_0)+\frac{v_s^2}{2} -\frac{\rho_n}{2\rho}(\mathbf{v}_n-\mathbf{v}_s)^2 +\frac{\partial\varphi_s}{\partial t}=0. \]

Introducing \(P_s\) and \(P_n\), we finally obtain

\[ T-T_0=\frac{\rho_n}{\rho S}\left\{\frac{P_n}{\rho_n}-\frac{P_s}{\rho_s} -\frac{(\mathbf{v}_n-\mathbf{v}_s)^2}{2}\right\}. \tag{13.10} \]

14. Propagation of sound in helium II

Let us apply the equations of hydrodynamics of helium II to the propagation of sound in this liquid. As usual, in a sound wave the velocities of motion are assumed small, while the density, pressure, entropy are almost equal to their constant equilibrium values. Then the system of hydrodynamic equations can be linearized—in (12.6) and in (12.9) we neglect terms quadratic in the velocity, and in equation (12.7) one may take the entropy \(\rho S\) outside the sign in the term \(\operatorname{div}(S\rho\mathbf{v}_n)\) (since this term already contains the small quantity \(\mathbf{v}_n\)).

Thus, the system of hydrodynamic equations assumes the form

\[ \frac{\partial \rho}{\partial t}+\operatorname{div}\mathbf{j}=0, \tag{14,1} \]

\[ \frac{\partial(S\rho)}{\partial t}+S\rho\,\operatorname{div}\mathbf{v}_n=0, \tag{14,2} \]

\[ \frac{\partial\mathbf{j}}{\partial t}+\nabla P=0, \tag{14,3} \]

\[ \frac{\partial\mathbf{v}_s}{\partial t}+\nabla\Phi=0. \tag{14,4} \]

Differentiating (14,1) with respect to time and substituting into (14,3), we obtain

\[ \frac{\partial^2\rho}{\partial t^2}=\Delta P. \tag{14,5} \]

For the thermodynamic potential the relation holds:

\[ d\Phi=-S\,dT+V\,dP=-S\,dT+\frac{1}{\rho}\,dP \]

(\(V\) is the specific volume). Hence we have

\[ \nabla P=\rho S\nabla T+\rho\nabla\Phi, \]

or, substituting \(\nabla P\) from (14,3) and \(\nabla\Phi\) from (14,4),

\[ \rho_n\frac{\partial}{\partial t}(\mathbf{v}_n-\mathbf{v}_s)+\rho S\nabla T=0. \]

We apply to this equation the operation \(\operatorname{div}\), and for \(\operatorname{div}(\mathbf{v}_s-\mathbf{v}_n)\) we substitute the expression

\[ \operatorname{div}(\mathbf{v}_s-\mathbf{v}_n)=-\frac{\rho}{\rho_s S}\frac{\partial S}{\partial t}, \]

which follows from the equality

\[ \frac{\partial S}{\partial t} =\frac{1}{\rho}\frac{\partial(S\rho)}{\partial t} -\frac{S}{\rho}\frac{\partial\rho}{\partial t} =-S\operatorname{div}\mathbf{v}_n+\frac{S}{\rho}\operatorname{div}\mathbf{j} = \]

\[ =\frac{S\rho_s}{\rho}\operatorname{div}(\mathbf{v}_s-\mathbf{v}_n). \]

As a result we obtain the equation

\[ \frac{\partial^2 S}{\partial t^2} =\frac{\rho_s S^2}{\rho_n}\,\Delta T. \tag{14,6} \]

Equations (14,5–6) determine the propagation of sound in helium II. Already from the fact that there are two such equations it is clear that in helium II there must exist two velocities of sound propagation. Let us write \(S\), \(P\), \(\rho\), \(T\) in the form \(S=S_0+S'\), \(P=P_0+P'\), etc., where the primed quantities represent the small changes of the corresponding quantities caused by the sound wave, and the quantities with subscript \(0\) are their constant equilibrium values. Then one may write

\[ \rho'=\frac{\partial\rho}{\partial P}P' +\frac{\partial\rho}{\partial T}T', \qquad S'=\frac{\partial S}{\partial P}P' +\frac{\partial S}{\partial T}T', \]

and equations (14, 5–6) take the form

\[ \frac{\partial \rho}{\partial P}\frac{\partial^2 P'}{\partial t^2} -\Delta P' +\frac{\partial \rho}{\partial T}\frac{\partial^2 T'}{\partial t^2}=0, \]

\[ \frac{\partial S}{\partial P}\frac{\partial^2 P'}{\partial t^2} +\frac{\partial S}{\partial T}\frac{\partial^2 T'}{\partial t^2} -\frac{S^2 \rho_s}{\rho_n}\Delta T'=0. \]

We seek the solution of these equations in the form of a plane wave, in which \(P'\) and \(T'\) are proportional to the factor \(e^{i\omega(t-x/u)}\) (\(u\) is the speed of sound). As the compatibility condition for both equations we obtain the equation

\[ u^4\frac{\partial(S,\rho)}{\partial(T,P)} -u^2\left(\frac{\partial S}{\partial T} +S^2\frac{\rho_s}{\rho_n}\frac{\partial \rho}{\partial P}\right) +\frac{\rho_s S^2}{\rho_n}=0 \]

(where

\[ \frac{\partial(S,\rho)}{\partial(T,P)} \]

denotes the Jacobian of the transformation from \(S,\rho\) to \(T,P\)).

By a simple transformation using thermodynamic relations, this equation may be given the form

\[ u^4-u^2\left[\left(\frac{\partial P}{\partial \rho}\right)_S +\frac{\rho_s T S^2}{\rho_n c_v}\right] +\frac{\rho_s T S^2}{\rho_n c_v} \left(\frac{\partial P}{\partial \rho}\right)_T =0 \tag{14,7} \]

(\(c_v\) is the heat capacity referred to unit mass). This equation, quadratic (in \(u^2\)), determines two velocities of sound propagation in helium II.

For \(\rho_s=0\), i.e. at the \(\lambda\)-point, one of the roots of equation (14,7) becomes zero, and we obtain, as indeed should be the case, only the ordinary sound velocity \(u=\sqrt{(\partial P/\partial \rho)_S}\) (which we denoted above by \(c\)).

In practice, at all temperatures the heat capacities \(c_p\) and \(c_v\) are close to one another. According to a known thermodynamic formula, under these conditions the isothermal and adiabatic compressibilities are also close to one another:

\[ \left(\frac{\partial P}{\partial \rho}\right)_T \simeq \left(\frac{\partial P}{\partial \rho}\right)_S. \]

Denoting the common value of \(c_p\) and \(c_v\) by \(C\), and the common value of \((\partial P/\partial \rho)_T\) and \((\partial P/\partial \rho)_S\) simply by \(\partial P/\partial \rho\), we obtain from equation (14,7) the following two expressions for the speeds of sound:

\[ u_1=\sqrt{\frac{\partial P}{\partial \rho}},\qquad u_2=\sqrt{\frac{T S^2 \rho_s}{C\rho_n}}. \tag{14,8} \]

Thus one of the velocities (\(u_1\)) is almost constant, while the other (\(u_2\)) depends strongly on the temperature, vanishing at the \(\lambda\)-point*).

Near absolute zero, where \((\rho_n)_\phi \gg (\rho_n)_p\), for the heat capacity and entropy one may use their “phonon” expressions (5,3)

*) An elementary derivation of the formula for the velocity \(u\) was proposed by Gorter and Patak \(^{17}\).

and (5.2), while the ratio \(\rho_n/\rho\) is determined by formula (8.2). Substituting these expressions into formula (14.8) for \(u_2\), we find

\[ u_2=\frac{u_1}{\sqrt{3}}. \tag{14.9} \]

Thus, as the temperature tends to zero, the velocity \(u_2\) (like \(u_1\)) tends to a constant limiting value, in such a way that their ratio tends to \(1/\sqrt{3}\).

Fig. 5. Velocity of “second sound” in helium II.

The graph of the temperature dependence of the velocity of “second sound” is shown in Fig. 5.

For a better clarification of the physical nature of both kinds of sound waves in helium II, let us consider a plane sound wave. In such a wave the velocities \(v_s\), \(v_n\) and the variable parts \(T'\), \(P'\) of the temperature and pressure are proportional to one another. Let us introduce the coefficients of proportionality according to

\[ v_n=av_s,\qquad P'=bv_s,\qquad T'=cv_s. \tag{14.10} \]

A simple calculation with the aid of equations (14.1–6), carried out with the requisite degree of accuracy (E. Lifshitz\({}^{18}\)), gives for the “first sound”

\[ a_1=1+\frac{\alpha\rho}{S\rho_s}\frac{u_1^2u_2^2}{(u_1^2-u_2^2)},\qquad b_1=\rho u_1,\qquad c_1=\frac{\alpha T}{C}\frac{u_1^3}{(u_1^2-u_2^2)}, \tag{14.11} \]

and for the “second sound”

\[ a_2=-\frac{\rho_s}{\rho_n}+\frac{\alpha p}{S\rho_n}\frac{u_1^2u_2^2}{(u_1^2-u_2^2)}, \qquad b_1=\frac{\alpha p u_1^2u_2^3}{S(u_1^2-u_2^2)}, \qquad c_2=-\frac{u_2}{S} \tag{14,12} \]

(\(\alpha=\frac{1}{\rho}\frac{\partial \rho}{\partial T}\) is the coefficient of thermal expansion). Quantities containing \(\alpha\) are small in comparison with the corresponding quantities not containing \(\alpha\).

We see that in a sound wave of the first type \(\mathbf v_n \cong \mathbf v_s\), i.e. in such a wave, in each volume element, the liquid oscillates, to a first approximation, as a whole; the normal and superfluid parts move together. Naturally, these waves correspond to ordinary sound waves in ordinary liquids.

In a wave of the second type, however, we have \(\mathbf v_s \cong -\dfrac{\rho_s}{\rho_n}\mathbf v_n\), i.e. the total density of the matter flux

\[ \mathbf j=\rho_s\mathbf v_s+\rho_n\mathbf v_n\cong 0. \]

Thus, in a wave of “second sound” the superfluid and normal parts of the liquid oscillate “toward one another,” so that, to a first approximation, their “center of inertia” in each volume element remains motionless and the total matter flux is absent. It is clear that this type of wave is specific to helium II.

An essential difference between the two types of waves is also that in a second-sound wave the amplitude of pressure oscillations is relatively small, while the amplitude of temperature oscillations is large [as is seen from formulas (14,12)], whereas in an “ordinary sound” wave the opposite situation obtains.

15. Emission of sound in helium II

The question of various methods of exciting sound waves in helium II was considered by E. M. Lifshitz\(^ {18}\). It turned out that the usual mechanical methods of exciting sound (by oscillating solid bodies) are extremely unfavorable for obtaining “second sound,” in the sense that the intensity of the emitted “second sound” is negligible in comparison with the intensity of the ordinary sound emitted simultaneously.

Let us consider, for example, the emission of sound waves by a plane executing oscillations in the direction perpendicular to itself (which we choose as the direction of the \(x\)-axis). We seek the velocities \(\mathbf v_s\) (directed along the \(x\)-axis) in the “first” and “second” emitted waves, respectively, in the form

\[ v_s^{(1)}=A_1\cos\omega\left(t-\frac{x}{u_1}\right), \qquad v_s^{(2)}=A_2\cos\omega\left(t-\frac{x}{u_2}\right). \]

(\(\omega\)—the frequency of oscillation of the plane). The boundary conditions on the surface of the solid require equality of the velocity of the body to the components of the velocities \(v_s, v_n\) perpendicular to its surface (cf. conditions (12,10) on a stationary body). In the present case this gives the equations

\[ A_1 + A_2 = v_0, \qquad A_1 a_1 + A_2 a_2 = v_0 \]

(\(v_0 \cos \omega t\) is the velocity of oscillation of the solid plane), whence

\[ \frac{A_2}{A_1} = -\,\frac{1-a_1}{1-a_2}. \]

As for all small oscillations, the mean (in time) potential energy is equal to the mean kinetic energy. Therefore the total energy density in the sound wave in helium II is equal to

\[ \rho_s \overline{v_s^{\,2}} + \rho_n \overline{v_n^{\,2}} = \frac{1}{2} A^2(\rho_s+\rho_n a^2); \]

the energy flux (intensity) is obtained by multiplying this by the corresponding sound velocity \(u\). With the aid of the values \(a_1, a_2\) from (14,11—12), we thus obtain for the ratio of the intensities of the emitted waves of the “second” and “first” sounds

\[ \frac{I_2}{I_1} \simeq \frac{a^2 T}{C}\,\frac{u_2^3}{u_1} \tag{15,1} \]

(here it has been assumed that \(u_2 \ll u_1\), which is valid down to very low temperatures). At \(2^\circ\mathrm{K}\), for example, this ratio is equal to \(2\cdot 10^{-6}\), i.e. the “second sound” is practically not emitted.

In helium II, however, still other methods of exciting sound waves are possible, methods specific to it and impossible in ordinary liquids. Such are the emission of sound by a plate oscillating in its own plane and thereby “dragging along” the normal part of the liquid*), and also emission from a surface with a temperature varying periodically in time. This second method is especially advantageous for obtaining “second sound.”

Let the temperature of a plane solid surface vary according to the law \(T' = T_0' \cos \omega_0 t\). The conditions of continuity of temperature and of the vanishing of the component perpendicular to the plane of the total matter flux \(\mathbf{j}\) [see (12,10—11)] give

\[ \rho_s(A_1 + A_2) + \rho_n(a_1 A_1 + a_2 A_2)=0, \qquad c_1 A_1 + c_2 A_2 = T_0'. \]

For the ratio \(A_2/A_1\) we have

\[ \left|\frac{A_2}{A_1}\right| = \frac{\rho_n a_1 + \rho_s}{\rho_n a_2 + \rho_s} \simeq \frac{S}{a u_2^{2}}. \]

Hence we find for the ratio of intensities

\[ \frac{I_2}{I_1} = \frac{C}{T a^2 u_1 u_2}. \tag{15,2} \]

*) This method was proposed by P. L. Kapitsa.

At \(2^\circ\mathrm{K}\) this quantity is equal to \(5\cdot 10^3\), and at lower temperatures it is still larger. Thus, as regards intensity, practically only second sound is emitted here. For the ratio of the pressure and temperature amplitudes we have

\[ \frac{P'_2}{P'_1}=\frac{b_2A_2}{b_1A_1},\qquad \frac{T'_2}{T'_1}=\frac{c_2A_2}{c_1A_1}, \]

whence

\[ \frac{P'_2}{P'_1}=\frac{u_2}{u_1},\qquad \frac{T'_2}{T'_1}=\frac{C}{\alpha^2 T u_1u_2}. \tag{15,3} \]

At \(2^\circ\mathrm{K}\), \(P'_2/P'_1=0.1\), so that the conditions for observing “second sound” through the change in pressure are still unfavorable. The conditions for observing “second sound” through the change in temperature, on the other hand, are very favorable.

16. Scattering of light in helium II

The scattering of light in helium II should reveal certain peculiarities in comparison with the scattering of light in ordinary liquids (V. L. Ginzburg\(^{19}\)). Although these phenomena are apparently beyond the limits of experimental possibility, they nevertheless are of definite theoretical interest.

First of all it must be noted that the expectation, expressed in the literature\(^{20,21,22}\), of anomalously strong scattering of light in helium II is completely unfounded. It was connected with the scattering of light by an ideal Bose–Einstein gas, which near the condensation point must scatter light extremely strongly. Since, however, Bose–Einstein condensation has no relation to the properties of helium II, there are no grounds for transferring to helium II the optical properties of such a gas.

The total intensity of light scattering in helium II is determined by the well-known general formula for scattering associated with density fluctuations in any isotropic body (liquid or gas)

\[ I=\frac{\pi^2 V}{2\lambda^4}\left(\rho\frac{\partial\varepsilon}{\partial\rho}\right)^2 \frac{1}{\rho}\left(\frac{\partial\rho}{\partial P}\right)_T kT(1+\cos^2\varphi). \]

Here \(I\) is the ratio of the intensity of the scattered light (calculated per unit solid angle) to the flux density of the incident light, \(V\) is the scattering volume, \(\lambda\) is the wavelength of the light in vacuum, \(\varepsilon=n^2\) is the square of the refractive index, and \(\varphi\) is the scattering angle (the light is assumed to be unpolarized). An estimate by means of this formula shows that for helium near the \(\lambda\)-point one obtains \(I\sim 2\cdot 10^{-8}\) (for \(V=1\,\mathrm{cm}^3\), \(\lambda=4\cdot 10^{-5}\,\mathrm{cm}\)), which is approximately equal to the intensity of light scattering by air at room temperature\(*\).

\[ \text{*) The absence of anomalously large scattering was indeed established experimentally by MacLennan et al.} \]

As is known, density fluctuations can be divided into adiabatic and isobaric*). Fluctuations of the first type propagate in a liquid with the speed of sound. Scattering by these fluctuations leads to a splitting of the scattering line into the so-called Mandelstam–Brillouin doublet, with relative distance between the components

\[ \frac{\Delta \omega}{\omega}\sim \frac{u}{c} \]

(\(c\) is the speed of light, \(u\) is the speed of sound). Isobaric fluctuations (entropy fluctuations) in an ordinary liquid do not propagate and are scattered owing to thermal conductivity. Scattering by these fluctuations does not change the frequency of the light. As a result, the scattering line is a triplet with an unshifted component in the middle.

In helium II the situation changes, since entropy fluctuations also propagate in the liquid with the speed of second sound \(u_2\). Therefore, along with the usual doublet (with relative splitting \(\Delta \omega_1/\omega\sim u_1/c\)) there should be observed one more doublet, lying inside the first, with relative splitting

\[ \frac{\Delta \omega_2}{\omega}\sim \frac{u_2}{c}. \]

Thus, as a result, the scattering line should be a quadruplet. It must be borne in mind, however, that not only is the interval between the components of the “anomalous” doublet very small, but its intensity (as the corresponding estimate shows) is negligibly small in comparison with the intensity of the normal doublet.

17. Viscosity of Helium II

A special problem in the theory of helium II is the question of calculating its viscosity (the viscosity of its “normal part”). It was considered quite recently by L. Landau and I. Khalatnikov. We shall present here briefly the results of their work, omitting the very complicated and lengthy calculations.

In the question of viscosity we encounter a phenomenon of kinetic character, connected with the processes of establishment of equilibrium in the phonon and roton “gas.” The calculation of viscosity requires, first of all, an investigation of the different types of elementary collision processes between the particles of this gas and the calculation of their effective cross sections**).

*) Writing a small change in density in the form

\[ \Delta \rho=(\partial \rho/\partial P)_S \Delta P+(\partial \rho/\partial S)_P \Delta S \]

and remembering that pressure and entropy fluctuations are independent \((\overline{\Delta P \Delta S}=0)\), we find that the mean square fluctuation \((\Delta \rho)^2\) is represented as the sum of the adiabatic pressure fluctuation and the isobaric entropy fluctuation (with the corresponding coefficients).

**) We mention that A. Akhiezer and I. Pomeranchuk\(^{23}\) considered collisions of slow neutrons with rotons and phonons for the purpose of determining the intensity of neutron scattering in helium II (it turned out that helium II is practically “transparent” for slow neutrons).

Momentum transport in helium II (when there is a velocity gradient in it) is carried out both by phonons and by rotons. Accordingly, the viscosity may be conventionally represented in the form of the sum of two parts—the phonon part and the roton part. Let us first consider the roton part of the viscosity.

According to the well-known formula of the kinetic theory of gases, the order of magnitude of the viscosity of a gas is determined by the formula

\[ \eta \sim m v l n, \]

where \(m\) is the mass of a gas molecule, \(v\) is its mean thermal velocity, \(l\) is the mean free path of the molecules, and \(n\) is their number per unit volume. From formula (8.3) it is seen that the role of the particle mass in the roton gas is played by the quantity \(p_0^2/3kT\) (the roton part of the density \(\rho_n\) is obtained by multiplying this quantity by the number of rotons). The mean thermal velocity of a roton is determined from \(\overline{v^2} \sim \overline{(p-p_0)^2}/\mu^2 \sim kT/\mu\). Thus, for the roton viscosity we have

\[ \eta_{\mathrm{p}} \sim \frac{p_0^2}{\sqrt{\mu kT}}\, l n_{\mathrm{p}} . \tag{17.1} \]

To determine the mean free path of rotons, one must consider the various types of collisions experienced by rotons: 1) elastic collisions of rotons with one another, 2) elastic collisions of rotons with phonons, 3) various kinds of inelastic collisions of rotons with rotons, accompanied by the emission (or absorption) of new rotons or phonons. Calculations show that the probability of the various processes of inelastic scattering of rotons by rotons is considerably less than the probability of elastic scattering, as, indeed, could have been expected in advance. As for collisions of rotons with phonons, at not too low temperatures the probability of these processes is considerably less than the probability of elastic scattering of rotons by rotons, owing to the fact that the number of phonons is very small in comparison with the number of rotons. Only at sufficiently low temperatures (beginning approximately from \(0.5^\circ\) K), owing to the sharp decrease in the number of rotons in comparison with the number of phonons, do collisions of rotons with phonons begin to predominate. It must be borne in mind, however, that at temperatures below \(0.5^\circ\) K the usual concept of viscosity practically loses its meaning altogether, since the mean free paths of phonons and rotons become, as calculation shows, of the order of \(1\ \mathrm{cm}\), i.e. comparable with the dimensions of the instruments and vessels by means of which viscosity measurements are usually made.

Leaving aside, therefore, the region of very low temperatures, we may consequently say that the mean free path of rotons is determined mainly by their elastic collisions with one another. Denoting the effective cross-section of these collisions by-

by means of \(\sigma_{pp}\), we have \(l \sim 1/n_p\sigma_{pp}\). The effective cross section \(\sigma_{pp}\) for rotons with mean thermal velocity turns out to be inversely proportional to \(\sqrt{T}\); substituting in formula (17.1), we arrive at the conclusion that the roton part of the viscosity is a temperature-independent constant\(^*\)

\[ \eta_p=\mathrm{const}. \tag{17.2} \]

The calculation of the phonon part of the viscosity is especially complicated. A careful analysis of the various processes of elastic and inelastic collisions of phonons with one another and with rotons leads to the following results.

At temperatures higher than approximately \(0.8^\circ\mathrm{K}\), the principal role in momentum transfer is played by elastic collisions of phonons with rotons. The effective cross section \(\sigma_{\phi p}\) of these collisions (for phonons with mean thermal energy\(^ {**}\)) can be calculated and proves to be proportional to \(T^4\). Qualitatively, this dependence may be obtained if one notes that, because of the large magnitude of the roton momentum in comparison with the mean phonon momentum \((p_0 \gg kT/c)\), the scattering of phonons by rotons must proceed analogously to the scattering of sound waves by a stationary solid sphere. As is known, the effective cross section of such scattering is proportional to the fourth power of the sound frequency. Therefore we may conclude that the effective cross section for scattering of a phonon by a roton is proportional to the fourth power of the wave vector (i.e., momentum) of the phonon, whence (in view of the relation \(p\sim kT/c\)) the temperature dependence indicated above is obtained.

In the gas-kinetic formula \(\eta \sim mvln\), one must now replace the product \(mn\) by the phonon part of the density \(\rho_n\) [formula (8.2)],

\(^*\) For an exact calculation of the probability of collisions of rotons with rotons it would be necessary to know the law of their interaction with one another. The existing theory does not make it possible to draw any definite conclusions on this matter. However, to determine the temperature dependence of the effective cross section one can manage by means of a device known from the theory of neutron scattering, writing the interaction energy \(U\) of two rotons in the form of a \(\delta\)-function of the difference of their coordinates: \(U=U_0\delta(r_2-r_1)\), where \(U_0\) is some constant.

The collision probability of two rotons (located in some volume \(\Omega\)), calculated by perturbation theory, proves to be a constant quantity, expressible only through \(p_0\), \(\mu\), \(U_0\) (and \(h\)). This, incidentally, could have been expected in advance on the basis of the fact that the roton momenta are, in absolute value, close to the constant quantity \(p_0\) (which is also taken into account in the calculation). The effective cross section is obtained by multiplying the resulting probability by \(\Omega/v\), whence the law indicated in the text is obtained.

An exact expression for the viscosity (in terms of the constants \(p_0\), \(\mu\), \(U_0\)) can be obtained by solving the corresponding kinetic equation.

\(^ {**}\) It is essential that the processes of energy exchange of the phonons with one another and the establishment of equilibrium in energy occur, as the investigation shows, very rapidly.

and the velocity \(v\) by the speed of sound \(c\). Writing also for the mean free path \(l \sim 1/\sigma_{\phi p} n_p\), we obtain

\[ \eta_{\phi}\sim \frac{k^4 T^4}{\hbar^3 c^4 \sigma_{\phi p} n_p}. \]

The number of rotons, according to formula (5.5), depends on the temperature according to the law \(\sqrt{T}e^{-\Delta/kT}\). Taking also into account that \(\sigma_{\phi p}\sim T^4\), we find the following law for the temperature dependence of the phonon viscosity (for \(T \gtrsim 0.8^\circ\mathrm{K}\)):

\[ \eta_{\phi}\sim T^{-1/2} e^{\Delta/kT}. \tag{17.3} \]

At temperatures below \(\sim 0.8^\circ\mathrm{K}\), as a consequence of the sharp decrease in the number of rotons, scattering of phonons by phonons begins to play the principal role. In this temperature region the calculations lead to the following temperature dependence of the viscosity:

\[ \eta_{\phi}\sim T^{-5}. \tag{17.4} \]

Thus we see that the phonon part of the viscosity increases with decreasing temperature according to a very rapid law—at first exponentially, and then as \(1/T^5\). It is significant that in both of the indicated temperature regions it proves possible not only to determine the temperature dependence of the phonon viscosity, but also to compute fully its absolute magnitude; moreover, the resulting expressions contain no undetermined constants (such as the constant \(U_0\), characterizing the interaction of two rotons—see the note on p. 554). This circumstance makes it possible to compare the theoretical results with experiment without introducing new parameters. It should be borne in mind, however, that the expression for the viscosity includes not only such quantities as \(\rho\), \(\Delta\), \(p_0\), \(c\), but also their derivatives with respect to the density of the liquid. These derivatives can be determined by means of the available experimental data on the variation of the velocities of first and second sound with pressure; such a determination, however, can at present be made only with comparatively low accuracy.

Systematic measurements of the viscosity of helium II have recently been carried out by E. Andronikashvili. He found that in the region from \(1.9^\circ\mathrm{K}\) to \(1.6^\circ\mathrm{K}\) the viscosity remains almost constant, while with further lowering of the temperature it increases rapidly (measurements were made down to \(1.3^\circ\mathrm{K}\))*). If from the experimental values of the viscosity one subtracts the values \(\eta_{\phi}\), calculated by the theoretical formula,

*) In the region from the \(\lambda\)-point to \(1.9^\circ\mathrm{K}\) the viscosity decreases with decreasing temperature. In this region, however, the concepts of roton and phonon gases already lose their meaning, and the theoretical calculation of the viscosity becomes altogether impossible.

then one obtains an approximately constant quantity, which can be identified with the roton part of the viscosity*). Thus a quite satisfactory agreement between theory and experiment is found.

18. L. Tisza’s works on the theory of helium II

As was already indicated in § 1, in his first works L. Tisza, following F. London, considered helium II as an ideal Bose–Einstein gas. In his later works9, 10 the discussion is of a “quantum Bose–Einstein liquid,” but it must be stated that this distinction in Tisza’s works is rather phraseological than physical in character. It reduces to several contentless statements about the wave function of a many-particle system, after which one continues to speak of atoms (!), “condensed” in the lowest state, and of excited atoms (the excitation energy of which the author calls “translational quanta”). Meanwhile, such a consideration of the motion of individual atoms in a system of strongly interacting particles (a liquid) is in complete contradiction with the basic principles of quantum mechanics. On the contrary, an inevitable consequence of quantum mechanics is, as is known, the possibility of introducing, for any weakly excited macroscopic system, the concept of elementary excitations (lying at the basis of L. Landau’s microscopic theory), describing the “collective” motion of the particles; moreover, to each such excitation there may be assigned a definite energy \(\varepsilon\) and an effective momentum \(p\) (regardless of the specific functional relation \(\varepsilon(p)\), i.e., of the form of the spectrum).

Although, in this way, the microscopic theory is in essence wholly absent in Tisza’s works, it is nevertheless necessary at the same time to note his undoubted merit, consisting in the introduction, independently of L. Landau**), of the idea of a macroscopic description of helium II by separating its density into two parts and introducing two velocity fields. On the basis of these ideas he predicted

*) Comparing the value of the viscosity thus obtained with its theoretical expression, one can determine the constant \(U_0\) characterizing the interaction of rotons. On the other hand, through this same constant one can express the coefficient in the “van der Waals correction,” which can be introduced in order to take into account the nonideality of the roton gas arising when approaching the \(\lambda\)-point (\(\rho_n\) is then written in the form \(\rho_n = (\rho_n)_p[1 + a(\rho_n)_p]\), where \((\rho_n)_p\) is the ideal-gas expression (8.3), and \(a\) is a constant). This gives the possibility of an independent determination of \(U_0\), from data on the ratio \(\rho_n/\rho\) at temperatures about \(1.9\text{–}2.0^\circ\) K. The values of \(U_0\) obtained by both methods turn out to be of the same order of magnitude.

) Tisza’s detailed 1940 paper9 was received in the USSR, owing to wartime conditions, only in 1943, and a short note8 in the Comptes rendus of the Paris Academy of Sciences remained unnoticed at the time.

still in 1938, the existence of a second kind of sound waves in helium II (which he called temperature waves).

However, Tisza did not succeed in constructing a quantitatively correct and consistent hydrodynamic and thermodynamic theory of helium II. His last paper as well is devoted mainly to the macroscopic theory.^10 But reading this paper gives the clear impression that, in writing it, the author was substantially motivated by the desire to ascribe to himself results that did not belong to him. The thermodynamic derivation of the hydrodynamic equations, the boundary conditions for them, the thermodynamic formula for the velocity of second sound, the discussion of experiments on measuring viscosity, etc., are presented by the author without any mention that all this had been done by Landau (whereas these formulas were absent from Tisza’s preceding papers^9).*)

At the same time, Tisza’s paper^10 contains many incorrect assertions; since these assertions may give rise to misunderstandings, we shall discuss them in more detail.

Tisza’s basic error consists in his failure to understand the role of phonons. Tisza excludes phonons from the normal part of the density of helium II, arguing that phonons are “bound to the liquid as a whole,” as opposed to “excited helium atoms in translational states of the Bloch type” (which, in his theory, constitute the normal part of the liquid). This assertion, like his argumentation, is fundamentally wrong. Phonons, like rotons, refer equally to the “collective” motion of the atoms of the liquid (although phonons have a greater wavelength than rotons) and, according to quantum mechanics, can equally well be regarded as certain “quasi-particles” possessing momenta. Quite apart from the fact that the inevitability of the entrainment of phonons (as well as rotons) by moving walls is proved by rigorous thermodynamic arguments in § 7), it is already a priori obvious that, for example, in the case of flow—

*) It is characteristic that Tisza uses all the terminology proposed by Landau, again without any references. Thus, he uses the normal and superfluid densities \(\rho_n\) and \(\rho_s\) (and the corresponding velocities), whereas in the preceding papers the discussion concerned the density of degenerate and excited atoms, \(\rho_{\mathrm{deg}}\) and \(\rho_{\mathrm{exc}}\).

**) Tisza’s remark about the lack of convincingness of these arguments (since they “are used to obtain information about the kinetic coefficient (viscosity), proceeding from considerations of equilibrium”) is a pure misunderstanding. The rotation of the vessel is an equilibrium thermodynamic treatment, and in the arguments of § 7 it is used only to prove the entrainment by a rotating vessel of the liquid as a whole, from which, of course, no conclusions about the magnitude of the viscosity can be, and are, drawn.

Equally a misunderstanding is the remark that rotons should be “associated with a definite mass enclosed in the volume in which the curl of the velocity is nonzero”; the incorrectness of such an assertion is sufficiently clear from what has just been said.

of helium through a narrow slit, the phonons will inevitably be scattered by its walls, i.e., will be delayed by them.

Furthermore, Tisza’s assertion of proportionality between the entropy and the normal density of helium is incorrect (formula (12) of his paper). Such a relation cannot be obtained thermodynamically, and from the formulas of §§ 5, 8 it is clear that it in fact has no place*).

In deriving the hydrodynamic equations Tisza admits a number of inaccuracies, as a result of which only the equations of the first approximation (coinciding with Landau’s equations) turn out to be correct. For the velocity of second sound one naturally obtains a formula coinciding with Landau’s formula (14,8). However, in connection with the above-mentioned interpretation of the role of phonons, Tisza excludes from the entropy \(S\) entering this formula its phonon part. For the temperature dependence of the remaining part of the entropy Tisza postulates the relation

\[ S = S_0 (T/T_0)^r \]

(\(T_0\) is the temperature of the \(\lambda\)-point; the exponent \(r\) is chosen equal to 5.5 in order to obtain agreement with the empirical values of the entropy**). Using also the proportionality, assumed by him, between the entropy and the normal density, Tisza obtains the following formula for the velocity of second sound:

\[ u_2 = 26 \sqrt{\frac{T}{T_0}\left[1-\left(\frac{T}{T_0}\right)^{5.5}\right]}\ \text{m/sec}. \]

As \(T \to 0\) this formula gives \(u_2 \to 0\), i.e., the velocity of second sound tends to zero instead of the finite value \(c/\sqrt{3}\), which follows from Landau’s theory.

Tisza’s article also contains some considerations concerning the viscosity of helium II. However, his arguments on this in fact extremely complex question (see § 17) are distinguished by extreme naïveté. They are limited to remarks on the necessity of distinguishing between viscosity of the “liquid type” and of the “gas type,” as a result of which Tisza arrives at the conclusion that the viscosity falls with decreasing temperature, a conclusion in complete contradiction with the latest measurements of E. Andronikashvili.

*) For accidental reasons, the temperature dependences of the roton parts \(\rho_n\) and of the entropy turn out to be similar [cf. formulas (5,7) and (8,4)]. This is connected with the circumstance that Tisza succeeded in achieving a good fit of the formulas to the experimental values of the entropy and the velocity of second sound in the region of temperatures above approximately \(1.3^\circ\text{K}\).

**) A theoretical calculation of the temperature dependence of the thermodynamic quantities of helium II could not be carried out by Tisza because he lacked a microscopic theory.

References Cited

1. L. Landau, JETP, 11, 592, 1941.
2. L. Landau, JETP, 14, 112, 1944.
3. L. Landau, J. of Physics, 11, 91, 1947.
4. F. London, Nature, 141, 643, 1938.
5. F. London, Phys. Rev., 54, 947, 1938.
6. F. London, J. Phys. Chem., 43, 49, 1940.
7. L. Tisza, Nature, 141, 913, 1938.
8. L. Tisza, C. R. Paris, 207, 1035, 1186, 1938.
9. L. Tisza, J. de Phys. et Rad., 1, 165, 350, 1940.
10. L. Tisza, Phys. Rev., 72, 838, 1947.
11. A. Bijl, Physica, 7, 869, 1940.
12. N. Bogolyubov, Izv. Acad. Sci. USSR, Phys. Ser., 11, 67, 1947.
13. Ya. Frenkel, JETP, 10, 650, 1940.
14. F. London, Proc. Roy. Soc., A, 171, 484, 1939.
15. L. Landau and I. Pomeranchuk, Dokl. Acad. Sci. USSR, 59, 669, 1948.
16. Y. Frank, Phys. Rev., 70, 261, 1946.
17. D. Gogate, P. Pathak, Proc. Phys. Soc., 59, 457, 1947.
18. E. Lifshitz, JETP, 14, 116, 1944.
19. V. Ginzburg, JETP, 13, 213, 1943.
20. A. Galanin, JETP, 10, 1267, 1940.
21. L. Goldstein, Phys. Rev., 57, 241, 457, 1940.
22. L. Schiff, Phys. Rev., 57, 844, 1940.
23. A. Akhiezer and I. Pomeranchuk, JETP, 16, 391, 1936.