THEORY OF KINETIC PHENOMENA IN HELIUM II

I. M. Khalatnikov

I. INTRODUCTION

Liquid helium in the temperature range lying below the λ-point*) (helium II) possesses a number of remarkable properties. The chief of these properties—the ability to flow without friction through narrow capillaries—was discovered by P. L. Kapitza\(^{1,2,3}\) and was called “superfluidity.” A simple calculation shows that the de Broglie wavelength of helium atoms in the temperature region of the order of 1–2° K proves to be comparable with interatomic distances. It follows directly from this that helium II is essentially a quantum object. A consistent explanation of the properties of helium II from the standpoint of its quantum nature is due to L. D. Landau.\(^{4,5}\)

The temperature of absolute zero corresponds to the normal state of helium II. At a temperature different from zero one may speak of an excited state of helium II. Each excited state of helium II is a collection of elementary excitations, as is the case, for example, in a solid crystal, in which the thermal vibrations of the atoms at the lattice sites can be represented as a collection of phonons.

1. Energy spectrum of helium II

Elementary excitations in helium II are characterized by a definite dependence of the energy \(\varepsilon\) on the momentum \(p\). It can be shown that states sufficiently close to the normal excited states correspond to potential motion of the liquid.**) Potential motion in a liquid, as is known, consists of longitudinal sound waves. The corresponding elementary excitations are sound quanta—phonons. Thus, excitations in helium II in the initial region of the energy spectrum are repre-

) \(T = 2.19^\circ\) K at atmospheric pressure.
*) This result is obtained under the assumption that the normal state is potential.

are phonons. An analysis of the experimental material on the basic thermodynamic quantities characterizing helium II shows that their temperature dependence can be satisfactorily explained if one proceeds from an energy spectrum for excitations in helium II of the form shown in Fig. 1a. Small values of the momentum correspond to long-wavelength excitations—

Fig. 1a. Energy spectrum of excitations of helium II.
Fig. 1b. Structural factor \(S(k)\).

phonons, for which the energy is proportional to the magnitude of the momentum:

\[ \varepsilon = cp \qquad (c\text{ is the speed of sound}). \tag{1.1} \]

After the linear portion, the curve \(\varepsilon\) bends, reaches a minimum equal to \(\Delta\) at the momentum value \(p=P_0\), and then rises again. Near the minimum the energy \(\varepsilon\) can be expanded in powers of \((p-P_0)\), and this expansion begins with quadratic terms in \((p-P_0)\). Thus, in this range of momenta, we have

\[ \varepsilon=\Delta+\frac{(p-P_0)^2}{2\mu}. \tag{1.2} \]

The long-wavelength excitations corresponding to this range of momenta are called rotons. The coefficient \(\mu\) has the meaning of the effective mass of the roton. The parameters of the theory \(\Delta\), \(P_0\), and \(\mu\) were calculated by us from the experimental values of the heat capacity and the velocity of second sound in helium II (see below). They turned out to be equal to

\[ \Delta=8.9^\circ\mathrm{K},\qquad P_0=2.1\cdot10^{-19}\ \mathrm{g\,cm/sec},\qquad \mu=1.72\cdot10^{-24}\ \mathrm{g}. \]

A rigorous theoretical derivation of the energy spectrum for a real liquid, such as helium II, is, of course, impossible. N. N. Bogolyubov[^6] considered the idealized problem of a dilute Bose gas. The original method developed by N. N. Bogolyubov made it possible to determine completely the energy spectrum of weakly excited states of such a system. It turned out that a weakly excited state is a collection of elementary excitations having an energy spectrum similar to that given above for excitations in helium II.

It can be shown that the energy spectrum of excitations of the type indicated above leads to the property of superfluidity*). In helium II two motions can exist simultaneously—a superfluid one with velocity $\mathbf{v}_s$ and a normal one with velocity $\mathbf{v}_n$. To each of these motions there corresponds its own effective mass. The sum of the superfluid and normal masses is equal to the total mass of the liquid. Both motions occur independently (at least in the region of small values of the velocities $\mathbf{v}_s$ and $\mathbf{v}_n$, not exceeding certain critical values), so that transfer of momentum from one to the other is impossible. The momentum of a unit volume of helium II, $\mathbf{j}$, is thus composed of two parts

\[ \mathbf{j}=\rho_s \mathbf{v}_s+\rho_n \mathbf{v}_n . \tag{1.3} \]

The coefficient $\rho_n$ is called the density of the normal part of the liquid, and $\rho_s$ the density of the superfluid part of the liquid. The sum of $\rho_n$ and $\rho_s$ is equal to the density of the liquid,

\[ \rho=\rho_n+\rho_s . \tag{1.4} \]

The ratio $\dfrac{\rho_n}{\rho}$ at the $\lambda$-point is equal to unity; with decreasing temperature it diminishes. At a sufficient distance from the $\lambda$-point, the aggregate of elementary excitations in helium II may be regarded as a certain ideal gas of excitations. In this case the phonons, evidently, obey Bose statistics. As for the rotons, in this case Boltzmann statistics may be applied, since the term $\Delta$ contained in the energy of a roton considerably exceeds the value $kT$.

Let two translational motions with velocities $\mathbf{v}_n$ and $\mathbf{v}_s$ occur simultaneously in helium. The energy of an elementary excitation in a stationary frame of reference, $E(p)$, is expressed in terms of the energy $\varepsilon(p)$ in the frame of reference in which the superfluid part of the liquid is at rest by means of the relation2

\[ E(p)=\varepsilon(p)+\mathbf{p}\mathbf{v}_s . \tag{1.5} \]

*) The property of superfluidity is also produced by a spectrum of any other type satisfying the condition

\[ \frac{\partial \varepsilon}{\partial p}\ne 0 \quad \text{for } p=0 . \]

Normal motion of the liquid is associated with the translational motion of the gas of excitations, occurring with velocity \(\mathbf v_n\). The distribution function \(n\) for elementary excitations depends on the energy of the relative motion

\[ E' = E - \mathbf p \mathbf v_n = \varepsilon(p) + \mathbf p \mathbf v_s - \mathbf p \mathbf v_n . \tag{1.6} \]

Thus, in moving helium II the distribution of phonons over energies is determined by the Planck function

\[ n=\left[e^{\frac{\varepsilon+\mathbf p\mathbf v_s-\mathbf p\mathbf v_n}{kT}}-1\right]^{-1} \tag{1.7} \]

with the function \(\varepsilon\) (1.1).

The distribution of rotons over energies is determined by the Boltzmann function

\[ n=e^{-\frac{\varepsilon+\mathbf p\mathbf v_s-\mathbf p\mathbf v_n}{kT}} \tag{1.8} \]

with the function \(\varepsilon\) (1.2).

2. Relation of the energy spectrum of excitations to the structure factor of liquid helium II

R. Feynman\(^8\) showed that there is a close connection between the form of the energy spectrum and the structure factor of a liquid. Feynman considers a certain excited state of the liquid and constructs the \(\psi\)-function of this state. Analyzing the various possible configurations of atoms in the liquid, he adduces a number of considerations in favor of the fact that the \(\psi\)-function of such a state can be represented in the form of the symmetric sum

\[ \sum_a f(\mathbf r_a)\Phi, \tag{2.1} \]

taken over all atoms of the system. Here \(f(\mathbf r_a)\) is some function of the radius vector of atom number \(a\), while \(\Phi\) is the wave function of the ground state, depending on the coordinates of all atoms. A function of this type was first proposed by Bijl\(^9\). For small values of the wave vector \(\mathbf k\), Bijl wrote the wave function of the excited state in the form \(\sum_a \exp(i\mathbf k\mathbf r_a)\Phi\). Feynman gave arguments in favor of the fact that such a function is a good approximation to the \(\psi\)-function also for large values of the wave vector \(\mathbf k\). To find the function \(f(\mathbf r_a)\), we apply the variational method.

Let us write the Hamiltonian of the system as

\[ H=-\frac{h^2}{2m}\sum_a \nabla_a^2+U-E_0, \tag{2.2} \]

where \(U\) is the potential energy of the system; the energy is measured relative to the ground state \(E_0\). The wave function of the ground state therefore satisfies the equation

\[ H\Phi=0. \tag{2.3} \]

Next we introduce the function \(F\)

\[ \psi=F\Phi. \tag{2.4} \]

Acting on the function \(\psi\), taken in this form, with the Hamiltonian (2.2) and taking (2.3) into account, we obtain

\[ H\psi=H(F\Phi)=-\left(h^2/2m\right)\sum_a\left(\Phi\nabla_a^2F+2\nabla_a\Phi\nabla_aF\right)= \]

\[ =\Phi^{-1}\left(-h^2/2m\right)\sum_a\nabla_a(\rho_N\nabla_aF). \tag{2.5} \]

Here \(\rho_N=\Phi^2\) is the probability density for the ground state; it determines the probability of one or another configuration \(\mathbf r^N\) (\(\mathbf r^N\) denotes the totality of the radius vectors of all \(N\) atoms of the system).

The energy of the system is found as the minimum of the expression

\[ \mathcal E=\int \psi^*H\psi\,d^Nr= \frac{h^2}{2m}\sum_a\int \nabla_aF^*\Delta_aF\rho_N\,d^Nr \tag{2.6} \]

(\(d^Nr\) denotes integration over the coordinates of all atoms) under the additional condition that the normalization integral

\[ I=\int \psi^*\psi\,d^Nr=\int F^*F\rho_N\,d^Nr \tag{2.7} \]

has a fixed value. The energy \(E\) is equal to \(\mathcal E/I\). According to (2.1), the function \(F\) can be written in the form of a sum

\[ F=\sum_a f_a(\mathbf r_a) \tag{2.8} \]

over all atoms of the system. Substituting this expression into the normalization integral,

\[ I=\int \sum_a\sum_b f^*(\mathbf r_b)f(\mathbf r_a)\rho_N\,d^Nr, \tag{2.9} \]

we fix two arbitrary values of the atom numbers \(a\) and \(b\) and integrate in (2.9) over the coordinates of all the remaining atoms; in this way we obtain

\[ I=\int f^*(\mathbf r_1)f(\mathbf r_2)\rho_2(\mathbf r_1,\mathbf r_2)\,d^3r_1d^3r_2, \tag{2.10} \]

where \(\rho_2\) is the probability of finding one atom at \(\mathbf r_1\), and another at \(\mathbf r_2\). In an analogous manner, by integrating \(\rho_N\) over the coordinates of all atoms except one, one can obtain the function \(\rho_1(\mathbf r_1)\), which gives the probability of finding an atom at \(\mathbf r_1\) in a liquid in the ground state. This quantity, obviously, does not depend on \(\mathbf r\) and is equal to some number \(\rho_0\). As for the function

\(\rho_2(\mathbf r_1,\mathbf r_2)\), then it may be written in the form \(\rho_2=\rho_0\rho(\mathbf r_1-\mathbf r_2)\), where the function \(\rho\) depends only on the mutual distance of the two atoms, so that the normalized integral (2.10) is equal to

\[ I=\rho_1\int f^*(\mathbf r_1)f(\mathbf r_2)\rho(\mathbf r_1-\mathbf r_2)\,d^3r_1d^3r_2 . \tag{2.11} \]

The integral (2.6) for the energy \(\mathcal E\), after substituting \(F\) in the form (2.8) and integrating over the coordinates of all atoms except one, takes the form

\[ \mathcal E=\frac{h^2}{2m}\sum_a\int \nabla_a f^*(\mathbf r_a)\nabla_a f(\mathbf r_a)\rho_N d^N r = \]

\[ =\rho_0\frac{h^2}{2m}\int \nabla f^*(\mathbf r)\nabla f(\mathbf r)d^3r . \tag{2.12} \]

We now choose the function \(f(\mathbf r)\) in such a way that the ratio \(\mathcal E/I\) has the minimum value. Varying the ratio \(\mathcal E/I\) with respect to \(f^*\), with the aid of (2.11) and (2.12), we find the equation

\[ E\int \rho(\mathbf r_1-\mathbf r_2)f(\mathbf r_2)\,d^3r_2 = -\frac{h^2}{2m}\nabla^2 f(\mathbf r_1). \tag{2.13} \]

The equation obtained has the solution

\[ f(\mathbf r)=\exp i\mathbf k\mathbf r . \tag{2.14} \]

To this solution there corresponds an energy value \(E(\mathbf k)\) equal to

\[ E(\mathbf k)=\frac{h^2k^2}{2mS(\mathbf k)}, \tag{2.15} \]

where \(S(\mathbf k)\) is the Fourier component of the correlation function \(\rho(\mathbf r)\),

\[ S(\mathbf k)=\int \rho(\mathbf r)\exp i\mathbf k\mathbf r\,d^3r . \tag{2.16} \]

The function \(S(\mathbf k)\), as is clear from its relation to the correlation function \(\rho(\mathbf r)\), is the structure factor of the liquid, determining, for example, the scattering of neutrons at absolute zero temperature.

The structure factor \(S(\mathbf k)\) is well known from experiment. The most characteristic properties of this function can be obtained from elementary considerations. For large \(k\) the function \(S(\mathbf k)\) tends to unity; in this case the correlation function turns into a \(\delta\)-function. The function \(S(\mathbf k)\) has a maximum for values of \(k\) of order \(\frac{2\pi}{a}\), where \(a\) is the interatomic distance; this corresponds to a maximum of the function \(\rho(\mathbf r)\) at \(r\) of order \(a\), when two atoms are at a distance equal to the mean interatomic distance. Finally, for small values of \(k\) the function \(S(\mathbf k)\) tends to zero linearly-

normal law, in accordance with the fact that at large distances the correlation disappears. Figure 1b shows the form of the function \(S(k)\) and the corresponding form of the function \(E(k)\) following from it. The initial linear segment of \(E(k)\) corresponds to the phonon part of the spectrum. Near the minimum on the curve we represent the function \(E(k)\) in the form

\[ E(k)=\Delta+\frac{h^2}{2\mu}(k-k_0)^2. \]

This part of the energy spectrum corresponds to the roton part of the spectrum. On the basis of the data of Hurst and Henshaw \(^{10}\) on neutron scattering in liquid helium at \(T=4.2^\circ\mathrm{K}\), Feynman obtained for the minimum roton energy the value \(18^\circ\mathrm{K}\), whereas according to thermodynamic data \(\Delta=8.9^\circ\mathrm{K}\). The discrepancy obtained should not be regarded as large, since Feynman used a rather crude approximation for the wave function of the state of the liquid.

3. Thermodynamic functions of helium II

Not very close to the \(\lambda\)-point, the densities of the phonon and roton gases are not large and, as was already said, they may be regarded as ideal gases. In this case all thermodynamic functions are composed of two parts—the part associated with phonons and the part associated with rotons.

In calculating the thermodynamic functions we shall use the distribution functions of phonons and rotons (1.7) and (1.8). In doing so, the dependence of the distribution functions on the relative velocity \(\mathbf{v}_n-\mathbf{v}_s\) may be neglected in the first approximation. The indicated dependence appears in the region of values of \(\mathbf{v}_n-\mathbf{v}_s\) where, under ordinary conditions, superfluidity is violated. Only in the propagation of second sound of large amplitude in helium II is it possible to attain appreciable values of the relative velocity \(\mathbf{v}_n-\mathbf{v}_s\). In this case there arises the need to take account, in the thermodynamic functions, of terms quadratic with respect to the difference \(\mathbf{v}_n-\mathbf{v}_s\).

Let us first calculate the thermodynamic functions for helium II at rest.

The thermodynamic functions of helium II can be obtained by means of the formulas of Bose statistics. Phonons, as is known, obey Bose statistics; the roton distribution is independent of the type of statistics, owing to the presence in the roton energy of the large constant term \(\Delta\gg kT\).

The free energy of the Bose gas is equal to

\[ F=-kT\int \ln(1+n)\,d\tau_p \tag{3.1} \]

\[ \left(d\tau_p=\frac{p^2\,dp\,do}{(2\pi\hbar)^3}\ \text{— element of volume in \(p\)-space, } do\ \text{— element of solid angle}\right). \]

After a single integration by parts

parts in (3.1) we obtain the formula

\[ F=-\frac{1}{3}\int np\,\frac{\partial\varepsilon}{\partial p}\,d\tau_{\mathbf p}, \tag{3.2} \]

which makes it possible to calculate the free energy of the gas of excitations. The entropy of the excitations is found by differentiating the free energy (3.2) with respect to temperature

\[ S=-\frac{\partial F}{\partial T} =-\frac{1}{3kT^2}\int n'\varepsilon\left(p\frac{\partial\varepsilon}{\partial p}\right)d\tau_{\mathbf p} \tag{3.3} \]

(\(n'\) is the derivative of the distribution function with respect to its argument).

Free energy, entropy, and heat capacity of the phonon gas. Carrying out in (3.2) the integration with the distribution function (1.7), we find the free energy of the phonon gas (\(\varepsilon=cp\))

\[ F_{\phi}=-\frac{1}{3}\int\left(e^{\frac{\varepsilon}{kT}}-1\right)^{-1} p\,\frac{\partial\varepsilon}{\partial p}\,d\tau_p =-\frac{1}{3}E_{\phi}. \tag{3.4} \]

The energy \(E_{\phi}\) of phonons per unit volume of helium II is equal to

\[ E_{\phi}=\frac{4\pi^5}{15}\left(\frac{kT}{2\pi hc}\right)^3 kT \simeq \frac{\pi^4}{36}\,kT\,N_{\phi} \tag{3.5} \]

\[ \left( N_{\phi}\simeq 2.4\cdot4\pi\left(\frac{kT}{2\pi hc}\right)^3 \text{ is the number of phonons per unit volume of helium II} \right). \]

The entropy of the phonon gas is found either from the general formula (3.3), or directly by differentiating the obtained expression (3.4) for the free energy; in this way we find

\[ S_{\phi}=-\frac{\partial F_{\phi}}{\partial T} =\frac{16\pi^5}{45}\,k\left(\frac{kT}{2\pi hc}\right)^3. \tag{3.6} \]

Next we calculate the heat capacity of the phonon gas

\[ C_{\phi}=T\frac{\partial S_{\phi}}{\partial T} =\frac{16\pi^5}{15}\,k\left(\frac{kT}{2\pi hc}\right)^3. \tag{3.7} \]

Free energy, entropy, and heat capacity of the roton gas. The free energy of the roton gas per unit volume of helium II is found from the general formula (3.2) with the distribution function (1.8). In performing the integration it is necessary to take into account the circumstance that, according to the energy spectrum, the momenta of the rotons are, in magnitude, close to \(P_0\).

In this way we find

\[ F_p=-kTN_p, \tag{3.8} \]

where \(N_p\) is the number of rotons per unit volume of helium

\[ N_p=\int n\,d\tau_{\mathbf p} =\frac{2P_0^2(\mu kT)^{1/2}e^{-\Delta/T}}{(2\pi)^{3/2}\hbar^3}. \tag{3.8'} \]

By differentiating relation (3.8), we find the entropy of the roton gas

\[ S_{\mathrm p}=-\frac{\partial F_{\mathrm p}}{\partial T} = kN_{\mathrm p}\left(\frac{\Delta}{T}+\frac{3}{2}\right). \tag{3.9} \]

Next we calculate the heat capacity of the roton gas

\[ C_{\mathrm p}=T\frac{\partial S_{\mathrm p}}{\partial T} = kN_{\mathrm p}\left(\frac{\Delta^{2}}{T^{2}}+\frac{\Delta}{T}+\frac{3}{4}\right). \tag{3.10} \]

Summing the results (3.6), (3.7), (3.9), and (3.10), we find expressions for the entropy and heat capacity of a unit volume of helium II

\[ S=S_{\mathrm p}+S_{\phi} = kN_{\mathrm p}\left(\frac{\Delta}{T}+\frac{3}{2}\right) +\frac{16\pi^{5}k}{45}\left(\frac{kT}{2\pi\hbar c}\right)^{3}, \tag{3.11} \]

\[ C=C_{\mathrm p}+C_{\phi} = kN_{\mathrm p}\left(\frac{\Delta^{2}}{T^{2}}+\frac{\Delta}{T}+\frac{3}{4}\right) +\frac{16\pi^{5}k}{15}\left(\frac{kT}{2\pi\hbar c}\right)^{3}. \tag{3.12} \]

Calculations analogous to those given above can be performed for the case of a nonzero relative velocity \(\mathbf w=\mathbf v_n-\mathbf v_s\) \(^{29}\). Without dwelling on the straightforward calculations, we give the final expressions for the thermodynamic functions obtained in this case:

\[ \overline{F}_{\phi}=F_{\phi}(1-w^{2}/c^{2})^{-2}, \tag{3.13} \]

\[ \overline{F}_{\mathrm p} =F_{\mathrm p}\frac{kT}{P_{0}w}\,\operatorname{sh}\frac{P_{0}w}{kT}, \tag{3.14} \]

\[ \overline{S}_{\phi}=S_{\phi}(1-w^{2}/c^{2})^{-2}, \tag{3.15} \]

\[ \overline{C}_{\phi}=C_{\phi}(1-w^{2}/c^{2})^{-2}, \tag{3.16} \]

\[ \overline{S}_{\mathrm p} =S_{\mathrm p}\frac{kT}{P_{0}w}\operatorname{sh}\frac{P_{0}w}{kT} +F_{\mathrm p}\frac{1}{T} \left[ \frac{kT}{P_{0}w}\operatorname{sh}\frac{P_{0}w}{kT} -\operatorname{ch}\frac{P_{0}w}{kT} \right], \tag{3.17} \]

\[ \overline{C}_{\mathrm p} =C_{\mathrm p}\frac{kT}{P_{0}w}\operatorname{sh}\frac{P_{0}w}{kT} +F_{\mathrm p}\frac{1}{T}\frac{P_{0}w}{kT}\operatorname{sh}\frac{P_{0}w}{kT}. \tag{3.18} \]

Here \(w=|\mathbf v_n-\mathbf v_s|\) is the relative velocity of the normal and superfluid motions. Letters without a bar denote the value of the thermodynamic functions in helium II at rest.

4. Normal Density

Helium II is characterized by one more very important function—the normal density \(\rho_n\). The momentum of a unit volume of helium II in a reference frame moving together with the superfluid part is, according to (1.3), equal to

\[ \mathbf p_{\mathrm{отн}}=\mathbf j-\rho\mathbf v_s=\rho_n(\mathbf v_n-\mathbf v_s). \tag{4.1} \]

On the other hand, this momentum can be represented in the form of the integral

\[ \int \mathbf{p} n(\varepsilon+\mathbf{p}\mathbf{v}_s-\mathbf{p}\mathbf{v}_n)\,d\tau_p, \tag{4.2} \]

taken over all elementary excitations. Comparing (4.1) and (4.2), we obtain the relation determining the value of the normal density

\[ \rho_n(\mathbf{v}_n-\mathbf{v}_s)=\int \mathbf{p} n(\varepsilon+\mathbf{p}\mathbf{v}_s-\mathbf{p}\mathbf{v}_n)\,d\tau_p. \tag{4.3} \]

In the case of small values of the difference \((\mathbf{v}_n-\mathbf{v}_s)\), the distribution function \(n\) may be expanded in a series in powers of this difference. The zero term of the expansion makes the integral on the right-hand side equal to zero. The first term of the expansion gives

\[ \rho_n=-\frac{1}{3kT}\int p^2 n'\,d\tau_p. \tag{4.4} \]

Formula (4.4) is valid for small values of the difference \((\mathbf{v}_n-\mathbf{v}_s)\).

Let us first calculate the phonon part of the normal density. Substituting the distribution function (1.7) into (4.3),

\[ \rho_{n\phi}(\mathbf{v}_n-\mathbf{v}_s)=\int \mathbf{p}\left[e^{\frac{\varepsilon+\mathbf{p}\mathbf{v}_s-\mathbf{p}\mathbf{v}_n}{kT}}-1\right]^{-1}d\tau_p. \tag{4.5} \]

Carrying out a simple integration and omitting the common factor \((\mathbf{v}_n-\mathbf{v}_s)\) in the left- and right-hand sides of (4.5), we find

\[ \rho_{n\phi}=\frac{4}{3}\frac{E_\phi}{c^2}(1-w^2/c^2)^{-3}. \tag{4.6} \]

Let us perform analogous calculations for rotons. We have

\[ \begin{aligned} \rho_{np}(\mathbf{v}_n-\mathbf{v}_s) &=\int \mathbf{p} e^{-\frac{\varepsilon+\mathbf{p}\mathbf{v}_s-\mathbf{p}\mathbf{v}_n}{kT}}\,d\tau_p \\ &=(\mathbf{v}_n-\mathbf{v}_s)N_p\frac{kT}{w^2} \left(\operatorname{ch}\frac{P_0w}{kT}-\frac{kT}{P_0w}\operatorname{sh}\frac{P_0w}{kT}\right). \end{aligned} \tag{4.7} \]

Hence we easily find

\[ \rho_{np}=N_p\frac{kT}{w^2} \left(\operatorname{ch}\frac{P_0w}{kT}-\frac{kT}{P_0w}\operatorname{sh}\frac{P_0w}{kT}\right). \tag{4.8} \]

The normal density of helium \(\rho_n\) is equal to the sum of \(\rho_{n\phi}\) and \(\rho_{np}\). According to (4.6) and (4.8), we have

\[ \rho_n=\rho_{np}+\rho_{n\phi} =N_p\frac{kT}{w^2} \left(\operatorname{ch}\frac{P_0w}{kT}-\frac{kT}{P_0w}\operatorname{sh}\frac{P_0w}{kT}\right) +\frac{4}{3}\frac{E_\phi}{c^2}(1-w^2/c^2)^{-3}. \tag{4.9} \]

In the case of small values of the velocity of relative motion of the normal and superfluid parts, the dependence of \(\rho_{n\phi}\) and \(\rho_{np}\) on \(w = |\mathbf{v}_n - \mathbf{v}_s|\) may be neglected. In this case formulas (4.6) and (4.8) give

\[ \rho_{n\phi}=\frac{4}{3}\,\frac{E_\phi}{c^2}, \tag{4.10} \]

\[ \rho_{np}=\frac{P_0^2}{3kT}\,N_p, \tag{4.11} \]

\[ \rho_n=\frac{4}{3}\,\frac{E_\phi}{c^2}+\frac{P_0^2}{3kT}\,N_p . \tag{4.12} \]

In the temperature region above \(0.8\text{--}1^\circ\mathrm{K}\), rotons play the principal role in all thermodynamic functions. However, the relative contribution of rotons to the value of the indicated functions rapidly decreases as the temperature is lowered. This is explained by the fact that the roton parts of the functions decrease exponentially with temperature, whereas the phonon parts decrease according to the power law \(T^3\). Practically, as was already noted, in all problems one may disregard the dependence of thermodynamic quantities on the velocity of relative motion

\[ \mathbf{w}=\mathbf{v}_n-\mathbf{v}_s . \]

This is explained by the fact that in the range of velocities where the phenomenon of superfluidity is observed, the ratios \(\dfrac{w}{c}\) and \(\dfrac{wP_0}{kT}\) are very small. However, when considering the problem of propagation of sound of large amplitude, this dependence can no longer be neglected. Finally, we note that since \(c>\dfrac{kT}{P_0}\), the dependence on \(w\) is manifested to a greater extent for roton quantities than for phonon quantities.

5. Comparison of experimental and theoretical values of thermodynamic quantities

Before proceeding to a direct comparison of the experimental and theoretical values of thermodynamic quantities, let us dwell on the calculation of the basic parameters of the theory that characterize the energy spectrum of helium II \((\Delta, P_0, \mu)\). To find these parameters we shall use the data of Hall, Wilkinson, and Wilks\(^{11}\) on the heat capacity of helium and the data of Peshkov\(^{12}\) on the velocity of second sound. For this purpose the most convenient data are those in the temperature region far from the \(\lambda\)-point. In the immediate vicinity of the \(\lambda\)-point the phonon-roton gas cannot be regarded as ideal, and therefore the formulas obtained above for an ideal gas cannot be applied. It does not seem possible to take the nonideality of the gas of excitations into account sufficiently accurately. A different situation obtains

takes place at low temperatures (below \(1.2^\circ\) K), when the ratio \(\frac{\rho_n}{\rho}\) becomes negligibly small and the gas of excitations may be regarded as ideal.

The calculation of the parameters of the theory was carried out as follows:

1. By integrating the heat capacity\(^{11}\) in the temperature interval \(0.6\)—\(1.2^\circ\) K, the entropy of helium was found.

2. From the value of the velocity of second sound \((u_2)\)\(^{12}\), the heat capacity \((C)\)\(^{11}\), and the obtained values of the entropy \(S\), the ratio \(\frac{\rho_n}{\rho}\) was found according to the formula

\[ u_2=\sqrt{\frac{\rho_s}{\rho_n}\,T\,\frac{S^2}{\rho C}}. \]

1. The phonon part of the ratio \(\left(\frac{\rho_n}{\rho}\right)_{\phi}\) was calculated from formula (4.10). The sound velocity in the temperature interval considered was taken to be \(237\ \text{m/sec}\)\(^{13}\). Next, by subtracting from the experimental values of the ratio \(\frac{\rho_n}{\rho}\) the calculated values of \(\left(\frac{\rho_n}{\rho}\right)_{\phi}\), the roton part of the ratio \(\left(\frac{\rho_n}{\rho}\right)_{p}\) was obtained.

2. Then a graph of the dependence of \(\lg T^{1/2}\left(\frac{\rho_n}{\rho}\right)_{p}\) on \(\frac{1}{T}\) was constructed (cf. (4.12)). This graph was a straight line. Its slope, according to formula (4.12), is directly equal to \(\Delta\).

In this way the value of \(\Delta\) was obtained, equal to

\[ \Delta=(8.9\pm0.2)^\circ\text{K}. \]

This value is approximately half a degree smaller than that previously obtained\(^{14}\) from high-temperature data, \((9.6\pm0.5)^\circ\) K.

The method described for obtaining the value of the parameter \(\Delta\) is the most advantageous when using the available experimental data on the heat capacity and the velocity of second sound in helium II. It was not possible to use the heat-capacity data directly for determining \(\Delta\). The point is that the heat capacity of helium in the low-temperature region has been measured with a rather noticeable error \((5\%\) or \(3\cdot 10^{-4}\ \text{cal}/\text{g}\cdot\text{degree})\).

In order to find the roton part of the heat capacity, it would be necessary to subtract from the measured values of the heat capacity the calculated phonon part, which in the low-temperature region constitutes a noticeable fraction of the total heat capacity.

With such a treatment of the data, the error made in measuring the heat capacity falls entirely on the roton part, so that the relative error in determining the roton part of the heat capacity turns out to be already greater than \(5\%\). From the obtained values of the roton part of the heat capacity, the magnitude of the parameter \(\Delta\) is therefore found with much more

error. When using data on the velocity of second sound, which are known with very high accuracy (error less than 1%), the quantity \(\rho_n/\rho\) is found with the same accuracy with which the heat capacity of helium II is known. (The error in the value of the ratio \(\left(S^2/C\right)\) is equal to the error in the determination of \(C\) or \(S\).) The phonon part of the ratio \(\rho_n/\rho\) is negligibly small; therefore the roton part of the ratio \(\left(\rho_n/\rho\right)_p\) is known practically with the same accuracy as \(\rho_n/\rho\). Thus, in the method used, only the error in the heat-capacity measurements is essential.

To find the parameter \(P_0\), the following procedure was used:

1. The roton part of the entropy \(S_p\) was calculated. For this purpose, from the entropy values obtained by integrating the heat capacity, the phonon part, calculated by formula (3.6), was subtracted.

2. The quantity \(P_0^2\) was found from the obtained values of \(\left(\rho_n/\rho\right)_p\) and \(S_p\), according to the relation

\[ P_0^2=\left(\frac{\rho_n}{\rho}\right)_p \rho \frac{k^2T}{S_p}\left(\frac{\Delta}{T}+\frac{3}{2}\right), \]

which is obtained by combining formulas (3.9) and (4.11).

In this way the value was found

\[ P_0=(2.1\pm0.05)\cdot10^{-19}\ \mathrm{g\,cm\,sec^{-1}}, \]

which differs little from the value \((2.06\pm0.02)\cdot10^{-19}\) obtained earlier.

The value of the effective mass of the roton \(\mu\) was found from the values of the roton part of the entropy, using the accepted values of the parameters \(\Delta\) and \(P_0\). It turned out to be equal to

\[ \mu=(1.72\pm0.68)\cdot10^{-24}\ \mathrm{g}. \]

The large error in the determination of \(\mu\) is caused by the error in the determination of the parameter \(\Delta\), which appears in the exponent. Small changes in the value of the parameter \(\Delta\) cause substantial changes in the value of \(\mu\). The value given for the effective mass of the roton is almost three times smaller than that obtained earlier from high-temperature data.

The obtained values of the parameters \(\Delta\), \(P_0\), and \(\mu\) make it possible to calculate the roton parts of the heat capacity, entropy, and the ratio \(\rho_n/\rho\). To calculate these thermodynamic quantities we use formulas (3.9), (3.10), and (4.11). These formulas, derived under the assumption of complete ideality of the roton gas, are valid only for

at temperatures below \(1.2^\circ\)K \(\left(\dfrac{\rho_n}{\rho}=0.03\right)\). In the region of higher temperatures it is necessary to take into account the nonideality of the roton gas. It is easy to show that, with allowance for the nonideality, the roton parts of the entropy and heat capacity of helium are written in the following form (for more detail see \(^{15}\)):

\[ S_p=S_{p,\mathrm{id}}\left(1+\alpha\frac{\rho_n}{\rho}\right), \tag{5.1} \]

\[ C_p=C_{p,\mathrm{id}}\left(1+2\alpha\frac{\rho_n}{\rho}\right). \tag{5.2} \]

Here \(S_{p,\mathrm{id}}\) and \(C_{p,\mathrm{id}}\) are the values calculated on the assumption of complete ideality from formulas (3.9) and (3.10), respectively. \(\alpha\) is a nonideality parameter—a certain coefficient independent of temperature. The magnitude of the parameter \(\alpha\) depends essentially on the function of interaction of the rotons with one another.

The \(\delta\)-shaped law of interaction between rotons adopted by us in calculating the kinetic coefficients (see below) does not make it possible to use the available data on the viscosity of helium II for an accurate calculation of the coefficient \(\alpha\). Comparing the values of the roton heat capacity calculated by formula (3.10) with those obtained from the high-temperature heat-capacity values measured by Keesom and Westmijze \(^{16}\), one can obtain the value of the coefficient \(\alpha\). It turns out to be

\[ \alpha=0.5. \]

As for the calculation of the ratio \(\left(\dfrac{\rho_n}{\rho}\right)_p\), in this case one can likewise write a relation connecting the true value \(\left(\dfrac{\rho_n}{\rho}\right)_p\) with that calculated by formula for an ideal gas (4.11):

\[ \left(\frac{\rho_n}{\rho}\right)_p = \left(\frac{\rho_n}{\rho}\right)_{p,\mathrm{id}} \left(1+\beta\frac{\rho_n}{\rho}\right). \tag{5.3} \]

The coefficient \(\beta\) cannot be simply expressed in terms of the coefficient \(\alpha\), since the quantity \(\dfrac{\rho_n}{\rho}\) cannot be found by simple differentiation of the free energy, as is the case in the calculation of the entropy and heat capacity of rotons.

Comparing the values \(\left(\dfrac{\rho_n}{\rho}\right)_p\) found in the experiments of E. Andronikashvili \(^{17}\) with the calculations by formula (4.11), we find the value of the coefficient \(\beta\)

\[ \beta=0.25. \]

Measurements of the heat capacity were carried out by Keesom and Miss Keesom \(^{18}\), by Keesom and Westmijze \(^{16}\) in the temperature range above \(1^\circ\)K, by Hollom;

THEORY OF KINETIC PHENOMENA IN HELIUM II

by Wilkinson and Wilks \(^{11}\) and by Kramers, Gorter, and Wasscher \(^{19}\) below \(1^\circ\mathrm{K}\).

It should be noted that heat-capacity measurements have low accuracy, since heat capacity is a differential quantity. In Fig. 2 are shown the results of heat-capacity measurements \(^{11,16,19,18}\), as well as the theoretical values (solid curve). The theoretical values were obtained from formula (3.12), which, after substitution of all constants, takes the following form:

\[ C=1.69\left(\frac{\Delta^{2}}{T^{2}}+\frac{\Delta}{T}+\frac{3}{4}\right)e^{-\frac{\Delta}{T}}+ \]

\[ +\,5.1\cdot10^{-3}T^{3}\ \text{(cal/g degree)}, \tag{5.3'} \]

Above \(1.2^\circ\mathrm{K}\), in calculating the roton part of the heat capacity, a correction for nonideality was introduced according to formula (5.2). The theoretical curve is in good agreement with the data below \(1.2^\circ\mathrm{K}\) and with the data \(^{16}\) above \(1.2^\circ\mathrm{K}\). In comparing the experimental data indicated above, it must be borne in mind that the authors of work \(^{11}\) do not vouch for their high-temperature points. The heat-capacity values for temperatures \(1.5\) and \(1.6^\circ\mathrm{K}\) obtained in this work are apparently too low. These values disagree not only with the heat-capacity measurements of other authors \(^{16,18}\), but also with the entropy data obtained by P. L. Kapitsa \(^{3}\).

For measuring the entropy of helium, P. L. Kapitsa \(^{3}\) used a thermomechanical effect, consisting in the fact that when helium flows out of a vessel through a thin capillary, heating is observed in the vessel, since the helium flowing out through the thin capillary carries no heat away.

Fig. 2. Temperature dependence of the heat capacity of helium II:
● — experimental values of Hull et al. \(^{19}\), ○ — experimental values of Keesom and Westmijze \(^{16}\), —— theoretical values, △ — experimental values of Kramers et al. \(^{19}\).

In Fig. 3 are shown the experimental values of the entropy \(^{3}\) and those calculated from formula (3.11), which after substitution of the values

for all parameters takes the form

\[ \sigma=\frac{S}{\rho}=1.69\left(\frac{\Delta}{T}+\frac{3}{2}\right)e^{-\frac{\Delta}{T}}+ \]
\[ +\,1.7\cdot 10^{-3}T^3\quad (\mathrm{cal}/\mathrm{g}\,\mathrm{degree}). \tag{5.4} \]

Above \(1.2^\circ\) K the roton part of the entropy was calculated with a correction for nonideality according to formula (5.1). In Fig. 3 are also given the entropy values obtained by Kramers et al. by integrating smoothed values of the heat capacity.

Direct measurements of the value of the normal density \(\frac{\rho_n}{\rho}\) were carried out by E. Andronikashvili \(^{17}\). The ratio \(\frac{\rho_n}{\rho}\) in these experiments was determined as the ratio of the moments of inertia of a cylindrical vessel,

Fig. 3. Temperature dependence of the entropy of helium II:

— theoretical values, ○ — experimental values of P. Kapitsa \(^{3}\), \(\Delta\) — values obtained by Kramers et al. \(^{19}\).

Fig. 4. Temperature dependence of the ratio \(\frac{\rho_n}{\rho}\):

— theoretical values, ○ — experimental values of Andronikashvili \(^{17}\).

filled with helium II and with helium I. The moment of inertia was measured from the period of torsional oscillations of such a vessel. In Fig. 4 a comparison is made of the experimental values of \(\frac{\rho_n}{\rho}\) and the theoretical ones,

calculated by formula (4.12). After substitution of all the parameters, this formula takes the form

\[ \frac{\rho_n}{\rho}=55\cdot T^{-1/2} e^{-\Delta/T}+1.28\cdot 10^{-4}T^4 . \tag{5.5} \]

In the same figure are given the values of \(\frac{\rho_n}{\rho}\), calculated from the data on the velocity of second sound (for details see \({}^{22}\)).

II. INTERACTION OF ELEMENTARY EXCITATIONS

In the present section we shall consider the interaction of elementary excitations in helium II with one another. The expressions obtained for the effective cross sections of various processes of interaction of excitations will be used later in calculating the kinetic coefficients of helium II.

6. Dispersion of the phonon part of the energy spectrum\({}^{20}\)

Expansion of the phonon field in a Fourier series; dependence of the parameters characterizing the form of the energy spectrum on pressure. We shall start from the energy spectrum for excitations of the kind indicated in § 1. At the beginning of the spectrum the energy \(\varepsilon\) depends linearly on the magnitude of the momentum \(p\). However, for phonons of large energy the effect of dispersion, i.e. deviations from the linear law, already becomes noticeable. Taking dispersion into account proves to be essentially necessary in calculating the effective cross sections of processes in which only phonons participate.

Since the exact form of the energy spectrum is unknown, the problem of dispersion cannot be solved exactly. However, from the available data on the energy curve for small values of the momentum and near the minimum, one may try to calculate the coefficients in the expansion of the energy in powers of the momentum \(p\). For this purpose we construct an interpolating four-term formula with as yet undetermined coefficients

\[ \varepsilon^2=A_1p^2+A_2p^4+A_3p^6+A_4p^8 . \tag{6.1} \]

Expressions (1.1) and (1.2) give four conditions:

\[ \left(\frac{\partial \varepsilon}{\partial p}\right)_{p=0}=c,\quad \left(\frac{\partial \varepsilon}{\partial p}\right)_{p=p_0}=0,\quad \left(\frac{\partial^2 \varepsilon}{\partial p^2}\right)_{p=p_0}=\frac{1}{\mu},\quad \varepsilon(p_0)=\Delta, \tag{6.2} \]

which make it possible to find the four coefficients \(A_1, A_2, A_3\), and \(A_4\) in the for-

module (6.1). The values of these coefficients turn out to be equal to

\[ \begin{gathered} A_1=c^2,\qquad A_2=\frac{1}{4P_0^2}\left(24\frac{\Delta^2}{P_0^2}+\frac{\Delta}{\mu}-12c^2\right),\\ A_3=\frac{1}{P_0^4}\left(3c^2-8\frac{\Delta^2}{P_0^2}-\frac{\Delta}{2\mu}\right),\\ A_4=-\frac{1}{4P_0^6}\left(4c^2-12\frac{\Delta^2}{P_0^2}-\frac{\Delta}{\mu}\right). \end{gathered} \tag{6.3} \]

Taking the square root of expression (6.1), and retaining only the first two terms, we obtain

\[ \varepsilon=c(p-\gamma p^3), \tag{6.4} \]

where \(\gamma=-A_2/2c^2\). The value of \(\gamma\), calculated with the aid of the known values of the parameters \(\Delta\), \(\mu\), and \(P_0\), turns out to be

\[ \gamma \simeq 2.8\cdot 10^{37}\ \text{erg}^{-2}\,\text{cm}^{-2}\,\text{sec}^2. \]

To calculate the effective cross sections for phonon scattering, we shall use the expansion of the phonon field in a Fourier series. In doing so, we shall proceed from the analogy between the phonon field and the radiation field. We represent the density of helium \(\rho(\mathbf r)\) and the velocity \(\mathbf v(\mathbf r)\) in the form of a Fourier series\(^*\)

\[ \begin{gathered} \rho(\mathbf r)=\rho_0+\Omega^{-1/2}\left\{\sum_{\mathbf p}\rho_{\mathbf p}e^{i\mathbf p\mathbf r/h}+\rho_{\mathbf p}^*e^{-i\mathbf p\mathbf r/h}\right\},\\ \mathbf v(\mathbf r)=\Omega^{-1/2}\left\{\sum_{\mathbf p}\mathbf v_{\mathbf p}e^{i\mathbf p\mathbf r/h}+\mathbf v_{\mathbf p}^*e^{-i\mathbf p\mathbf r/h}\right\}. \end{gathered} \tag{6.5} \]

Here \(\rho_0\) is the equilibrium density in the absence of phonons, and \(\mathbf p\) is the phonon momentum, related to the frequency \(\omega\) by the relation

\[ \omega=cp/h. \tag{6.6} \]

For traveling waves the summation in (6.5) is carried out over both positive and negative values of the momentum.

From the commutation relation\(^4\) between \(\rho(\mathbf r)\) and \(\mathbf v(\mathbf r)\):

\[ \rho(\mathbf r_1)\mathbf v(\mathbf r_2)-\mathbf v(\mathbf r_2)\rho(\mathbf r_1)=(h/i)\nabla\delta(\mathbf r_1-\mathbf r_2) \tag{6.7} \]

one can obtain the commutation relations for the Fourier components. To this end we express \(\rho(\mathbf r_1)\) and \(\mathbf v(\mathbf r)\) by means of (6.5) and substitute the indicated expressions into the left-hand side of (6.7). The term on the right-hand side

\[ {}^*)\ \text{Here and below by } h \text{ we denote Planck’s constant divided by } 2\pi. \]

from expression (6.7) we represent the \(\delta\)-function in the form

\[ \delta(\mathbf r_1-\mathbf r_2)=({}^{1}/_{2})\Omega^{-1} \left\{ \sum_{\mathbf p} e^{i(\mathbf p,\mathbf r_1-\mathbf r_2)/\hbar} + e^{-i(\mathbf p,\mathbf r_1-\mathbf r_2)/\hbar} \right\} \tag{6.8} \]

and compare the coefficients of identical exponentials on the left- and right-hand sides. This gives relations valid for \(\operatorname{rot}\mathbf v_s=0\)

\[ \rho_{\mathbf p}\rho_s^*-\rho_s^*\rho_{\mathbf p} = \frac{\rho_0 p}{2c}\delta_{\mathbf{ps}}, \qquad \mathbf v_{\mathbf p} = \frac{c\mathbf p}{\rho_0 p}\rho_{\mathbf p}, \qquad \delta_{\mathbf{ps}}= \begin{cases} 1 & \mathbf p=\mathbf s,\\ 0 & \mathbf p\ne \mathbf s. \end{cases} \tag{6.9} \]

The total Hamiltonian of the volume of helium under consideration is

\[ \int\{\,{}^{1}/_{2}\rho \mathbf v\rho \mathbf v+\rho\varepsilon(\rho)\,\}\,d\Omega \]

(\(\varepsilon(\rho)\) is the internal energy per unit mass of the liquid), and, in Fourier components, up to terms cubic in \(\rho_{\mathbf p}\), is expressed as follows:

\[ H_0=(c^2/\rho_0)\sum(\rho_{\mathbf p}\rho_{\mathbf p}^*+\rho_{\mathbf p}^*\rho_{\mathbf p}) = \sum(n_{\mathbf p}+{}^{1}/_{2})\hbar\omega . \tag{6.10} \]

Here \(n_{\mathbf p}\) is the number of phonons with momentum \(\mathbf p\).

From relations (6.9) and (6.10) we obtain the nonzero matrix elements of the Fourier components:

\[ (\rho_{\mathbf p})_{n_{\mathbf p},\,n_{\mathbf p}+1} = \sqrt{(p\rho_0/2c)(n_{\mathbf p}+1)}\,e^{-i\omega t}, \tag{6.11} \]

\[ (\rho_{\mathbf p}^*)_{n_{\mathbf p},\,n_{\mathbf p}-1} = \sqrt{(p\rho_0/2c)n_{\mathbf p}}\,e^{-i\omega t}. \tag{6.12} \]

In what follows we shall need the derivatives with respect to density of the parameters \(\Delta\), \(P_0\), \(\mu\), and the sound velocity \(c\), which characterize the shape of the energy curve. They can be calculated from the available data on the velocities of first and second sound under pressure, using data on the coefficient of thermal expansion of helium II.

The derivative of the velocity of first sound \(c\) with respect to the density \(\rho\) is calculated from the data of Findlay et al.\(^{24}\) on the dependence of the sound velocity in helium on pressure, and also from the data of Keesom and Miss Keesom\(^{25}\) on the dependence of the density of helium II on pressure. Both sets of data give

\[ \frac{\rho}{c}\frac{\partial c}{\partial \rho}=1.8, \]

and, consequently,

\[ \left. \begin{aligned} \frac{\rho}{N_\phi}\frac{\partial N_\phi}{\partial \rho} &= -\frac{3\rho}{c}\frac{\partial c}{\partial \rho} \simeq 5.4 \\ \left(N_\phi=2.4\,\frac{4\pi(kT)^3}{(2\pi\hbar c)^3}\right). \end{aligned} \right\} \tag{6.13} \]

The data of Peshkov and Zinov’eva\(^{26}\) on the velocity of second sound under pressure make it possible to calculate a certain linear combination of

of the derivatives $\dfrac{\partial P_0}{\partial \rho}$, $\dfrac{\partial N_{\mathrm p}}{\partial \rho}$, $\dfrac{\partial N_{\phi}}{\partial \rho}$. Since the value of the derivative $\dfrac{\partial N_{\phi}}{\partial \rho}$ is already known to us, the indicated data give one relation between the derivatives $\dfrac{\partial P_0}{\partial \rho}$ and $\dfrac{\partial N_{\mathrm p}}{\partial \rho}$.

To find the second relation we shall use the value of the coefficient of thermal expansion $\dfrac{1}{\rho}\dfrac{\partial \rho}{\partial T}$[^25], which, according to the relation

\[ \frac{1}{\rho^2}\frac{\partial \rho}{\partial T} = \frac{\partial \sigma}{\partial \rho} \tag{6.14} \]

is expressed through the derivatives $\dfrac{\partial N_{\mathrm p}}{\partial \rho}$ and $\dfrac{\partial N_{\phi}}{\partial \rho}$. From the two relations we find the value of the combination $\dfrac{\rho}{P_0}\dfrac{\partial P_0}{\partial \rho}$

\[ \frac{\rho}{P_0}\frac{\partial P_0}{\partial \rho}\simeq \frac{1}{3}. \tag{6.15} \]

Knowing the magnitude of the coefficient of thermal expansion $\dfrac{\partial \rho}{\partial T}$ for several temperature values, we compute the values of the derivative $\dfrac{\partial N_{\mathrm p}}{\partial \rho}$ for several temperature points. Relying on two such points and taking into account the above value of the derivative $\dfrac{\partial P_0}{\partial \rho}$, we find the derivative $\dfrac{\partial \Delta}{\partial \rho}$ (see also[^23]):

\[ \frac{\partial \Delta}{\partial \rho}\frac{\rho}{\Delta}\simeq -\frac{1}{3}. \tag{6.16} \]

As for the derivative $\dfrac{\partial \mu}{\partial \rho}$, no satisfactory value of it can be obtained from the available very crude data for $\dfrac{\partial \rho}{\partial T}$. Within the accuracy of all available experimental data, the indicated derivative may be taken equal to zero.

The second derivative $\dfrac{\partial^2 \Delta}{\partial \rho^2}$, which determines the magnitude of the effective cross section for scattering of a phonon by a roton (§ 8), can be calculated from the temperature dependence of the velocity of first sound in helium II[^13]. The latter data[^13] give for the indicated derivative the value

\[ \frac{\partial^2 \Delta}{\partial \rho^2}\frac{\rho^2}{\Delta}=-5. \tag{6.17} \]

It should be noted that the values of the density derivatives of the parameters characterizing the form of the energy curve for excitations, calculated in this way, are very rough.

This is explained by the extreme scarcity and crudeness of the data on the properties of helium II under pressure. The appearance of new experimental data may substantially change the indicated values.

7. Scattering of a Phonon by a Phonon\(^ {20}\)

We shall calculate the effective cross section for the scattering of a phonon by a phonon by the method of perturbation theory. According to\(^4\), the Hamiltonian of a unit volume of helium II is

\[ H=\frac{1}{2}\mathbf{v}\rho\mathbf{v}+\varepsilon(\rho). \tag{7.1} \]

The scattering process under consideration is a four-phonon effect. The nonvanishing matrix elements of the perturbation energy for transitions of two phonons into two other phonons are obtained from the cubic-in-\(\rho'\) terms in the energy in the second approximation of perturbation theory and from the fourth-degree terms in \(\rho'\) in the first approximation of perturbation theory; \(\rho'\) is the deviation of the density from its value in the stationary liquid, i.e. in the absence of phonons.

Restricting ourselves to terms of fourth order in \(\rho'\), we write the Hamiltonian (7.1) in the form

\[ H=H_0+V_3+V_4, \tag{7.2} \]

where \(H_0\) is the usual density of the sound energy, containing terms of second order with respect to \(\rho'\),

\[ H_0=\frac{\rho_0 v^2}{2}+\frac{\rho'^2 c^2}{2\rho_0}. \tag{7.3} \]

\(V_3\) contains the terms of third order

\[ V_3=\frac{v\rho'v}{2}+\frac{1}{3!}\frac{\partial}{\partial\rho}\left(\frac{c^2}{\rho}\right)\rho'^3 \tag{7.4} \]

and, finally, \(V_4\) contains a term of fourth order:

\[ V_4=\frac{1}{4!}\frac{\partial^2}{\partial\rho^2}\left(\frac{c^2}{\rho}\right)\rho'^4. \tag{7.5} \]

In the scattering of phonons with momenta \(\mathbf{p}\) and \(\mathbf{p}_1\), leading to the appearance of phonons \(\mathbf{p}'\) and \(\mathbf{p}'_1\), six intermediate states I—VI are possible, in which the phonons have momenta:

\[ \begin{aligned} &\text{I. } \mathbf{p}+\mathbf{p}_1; \quad \text{II. } \mathbf{p}-\mathbf{p}',\ \mathbf{p}_1,\ \mathbf{p}'; \quad \text{III. } \mathbf{p}-\mathbf{p}'_1,\ \mathbf{p}_1,\ \mathbf{p}'_1; \quad \text{IV. } \mathbf{p}_1-\mathbf{p}'_1,\ \mathbf{p},\ \mathbf{p}'_1;\\ &\text{V. } \mathbf{p}_1-\mathbf{p}',\ \mathbf{p},\ \mathbf{p}'; \quad \text{VI. } \mathbf{p},\ \mathbf{p}_1,\ \mathbf{p}',\ \mathbf{p}'_1,\ -(\mathbf{p}'+\mathbf{p}'_1). \end{aligned} \]

The matrix element of the transition from the initial state \((A)\) to the final state \((F)\) in the second approximation of perturbation theory

is equal to

\[ H'_{AF}=\sum_{i=1}^{\mathrm{VI}}\frac{(V_3)_{Ai}(V_3)_{iF}}{E_A-E_i}+(V_4)_{AF}. \tag{7.6} \]

We shall express the energy differences in the initial and intermediate states that enter the denominators of (7.6) in terms of the phonon energy:

\[ \left. \begin{aligned} E_A-E_{\mathrm I} &= \varepsilon(p)+\varepsilon(p_1)-\varepsilon(|\mathbf p+\mathbf p_1|),\\ E_A-E_{\mathrm {II}} &= \varepsilon(p)-\varepsilon(p')-\varepsilon(|\mathbf p-\mathbf p'|),\\ E_A-E_{\mathrm {III}} &= \varepsilon(p)-\varepsilon(p'_1)-\varepsilon(|\mathbf p-\mathbf p'_1|),\\ E_A-E_{\mathrm {IV}} &= \varepsilon(p_1)-\varepsilon(p'_1)-\varepsilon(|\mathbf p_1-\mathbf p'_1|),\\ E_A-E_{\mathrm V} &= \varepsilon(p_1)-\varepsilon(p')-\varepsilon(|\mathbf p_1-\mathbf p'|),\\ E_A-E_{\mathrm {VI}} &= -\varepsilon(p')-\varepsilon(p'_1)-\varepsilon(|\mathbf p'+\mathbf p'_1|). \end{aligned} \right\} \tag{7.7} \]

From expressions (7.7) it is easy to see that, if dispersion is neglected, the denominators of the first five terms of expression (7.6) can vanish, leading to a substantial divergence of the matrix element. Indeed, for a linear dependence of the phonon energy on momentum, the denominators vanish whenever the angle between the momenta of the colliding phonons is zero, i.e. when \(|\mathbf p+\mathbf p_1|=p+p_1\). Therefore, for the phonon energy we shall use expression (1.6), which contains, in addition to linear terms, also cubic terms in the momenta.

The perturbation matrix elements \(V_3\) and \(V_4\) entering (7.6) are calculated with the aid of expressions (6.11)—(6.12) for the nonvanishing matrix elements of the density*). The resulting expressions essentially contain two dimensionless parameters:

\[ u=\frac{\rho_0}{c^2}\frac{\partial c^2}{\partial \rho}, \qquad z=\frac{\rho_0^3}{c^2}\frac{\partial^2}{\partial \rho^2}\left(\frac{c^2}{\rho}\right). \tag{7.8} \]

The differential effective cross section for the process under consideration is represented by the relation

\[ d\sigma(\mathbf p,\mathbf p_1,\mathbf p',\mathbf p'_1) =(2\pi/hc)|H'_{AF}|^2\delta(\varepsilon+\varepsilon_1-\varepsilon'-\varepsilon'_1)(2\pi\hbar)^{-3}\,d\mathbf p', \tag{7.9} \]

\[ d\mathbf p'=dp'_x\,dp'_y\,dp'_z . \]

The calculation of the total effective cross section for the scattering of a phonon with momentum \(\mathbf p_1\) by a phonon with momentum \(\mathbf p\) leads to very cumbersome expressions. We shall consider the particular case in which one of the momenta of the colliding phonons is small, namely, when \(p\ll p_1\).

*) In calculating the matrix elements one must take into account all possible permutations of the phonons. To this end it is necessary to multiply the terms containing \(\rho'^2\) by 2, those containing \(\rho'^3\) by 6, and, finally, those containing \(\rho'^4\) by 24.

As for the momenta \(p'\) and \(p'_1\) of the scattered phonons, we shall assume them to be comparable in magnitude with \(p_1\).

In this case, of the three terms in the transition matrix element \(H'_{AF}\) that have resonance denominators and correspond to the intermediate states I, IV, and V, only the term

\[ \frac{(V_3)_{AI}(V_3)_{IF}}{\varepsilon(p)+\varepsilon(p_1)-\varepsilon(|\mathbf p+\mathbf p_1|)} \]

will contain a small quantity in the denominator, unlike the other two, which are proportional respectively to \((p_1-p')^2/p_1p'\) and \((p_1-p'_1)^2/p_1p'_1\); this term proves to be of order \(p_1/p\), and consequently it alone is essential. The indicated circumstance simplifies expression (7.9) for \(d\sigma\), which after simple transformations takes the form:

\[ d\sigma= \frac{(u+2)^4 p_1^2 p'_1 p'}{(16\pi h^2\rho_0)^2 Cp} \cdot \frac{\delta[\varepsilon(p)+\varepsilon(p_1)-\varepsilon(p')-\varepsilon(p'_1)]} {(1-\mathbf n\mathbf n_1+3\gamma p_1^2)^2} \,p'^2\,dp'_1\,dp'. \tag{7.10} \]

The fact that expression (7.10) has a sharp maximum at small angles between the momenta of the colliding phonons allows us to carry out the necessary integration. Let us choose as the polar axis of the spherical coordinate system the direction formed by the sum of the momenta \(\mathbf p+\mathbf p_1\). Let in this system the vectors \(\mathbf p,\mathbf p_1,\mathbf p'\), and \(\mathbf p'_1\) have polar angles respectively \(\theta,\theta_1,\theta'\), and \(\theta'_1\). We now transform to a form convenient for integration the \(\delta\)-function containing the energy difference; the latter, with the aid of (6.6) and the law of conservation of momentum at small angles, is reduced to the form:

\[ c\{(p/2p_1)(p+p_1)\theta^2-(p'/2p'_1)(p'+p'_1)\theta'^2+ \]

\[ +3\gamma(p_1-p')(pp'-p_1p'_1)\}. \]

We now integrate expression (7.10) over the phase volume and average over all angles formed by the momenta \(\mathbf p\) and \(\mathbf p_1\). As a result, for the complete effective cross section for scattering of a phonon with momentum \(p\) by a phonon with momentum \(p_1\), we obtain the following expression*):

\[ \sigma(p,p_1)=\frac{\pi(u+2)^4p_1^4}{(96\pi h^2\rho_0 c)^2\gamma}\quad (p\ll p_1). \tag{7.11} \]

The value of the dimensionless expression \(u\) entering (7.11) can be calculated from Keesom’s data\(^{27}\) for the dependence of the density of helium II on pressure. From these data one obtains a value equal to

*) To take into account the symmetry with respect to the pair of scattered phonons with momenta \(\mathbf p\) and \(\mathbf p_1\), expression (7.11) contains the additional factor \(1/2\).

3.6*). Let us draw attention to the curious fact that the effective scattering cross section (7.11) does not depend on the momentum \(p\) of a phonon of low energy.

Substituting into (7.11) the numerical values of all the parameters, we find

\[ \sigma(p,p_1)=6\cdot 10^{-19}(xT)^4 . \tag{7.12} \]

Here \(x\) is the momentum expressed in units of \(kT/c\), so that \(p_1=x(kT/c)\).

As was already noted, the effective cross section for the scattering of a phonon by a phonon reaches its greatest value at small angles between the momenta of the colliding and scattered phonons. It follows from this that such a process leads mainly to a rapid exchange of energies between phonons and does not lead to any substantial change in the directions of the momenta of the colliding phonons.

8. Scattering of a phonon by a roton\(^{20}\)

In the present section we shall calculate the effective cross section for the scattering of a phonon by a roton. The Hamiltonian of the phonon–roton system may be written in the form

\[ H=H_{\phi}+H_{\rho}+V, \tag{8.1} \]

where \(H_{\phi}\) is the energy of the phonon, \(H_{\rho}\) is the energy of the roton, and \(V\) is the interaction function of the phonon with the roton.

Since the character of the interaction of a phonon with a roton is unknown, we shall regard the roton as some particle situated in the phonon field. In this treatment the internal structure of the roton is immaterial.

The presence of a phonon is equivalent to small oscillations of the density of the medium and to its motion with some velocity \(\mathbf{v}\). In the stationary reference frame the energy of a roton with momentum \(\mathbf{P}\) is expressed as follows:

\[ H'_{\rho}=H_{\rho_0}-\mathbf{P}\mathbf{v} \tag{8.2} \]

(\(H_{\rho_0}\) is the energy of the roton in the reference frame moving together with the medium). From (8.2) it follows that the interaction energy of a phonon and a roton contains a term \(\mathbf{P}\mathbf{v}\) depending on the velocity. The quantum Hermitian operator in the perturbation energy corresponding to this term is

\[ -\frac{1}{2}\,(\mathbf{P}\mathbf{v}+\mathbf{v}\mathbf{P}), \tag{8.3} \]

where \(\mathbf{P}=-i\hbar\nabla\) is the momentum operator.

Let us determine the dependence of the interaction energy \(V\) on the density of the liquid.

* We note that for the value of the parameter \(z\) (see (7.8)) the same data give approximately 20.

To this end let us expand the roton energy in powers of the deviation of the density \(\rho\) from its equilibrium value and retain only terms up to second order in \(\rho\), inclusive. (Here and below, the deviation of the liquid density from its equilibrium value will be denoted by \(\rho\), without a prime.) The terms of first order in \(\rho\) will give the transitions, needed for our problem, in the second approximation of perturbation theory, while the terms of second order in \(\rho\) will give the same transitions, but in the first approximation of perturbation theory. We shall use expression (6.2) for the roton energy and carry out the indicated expansion:

\[ H_p = H_{p_0}+\frac{\partial \Delta}{\partial \rho}\,\rho+ \frac{1}{2}\left[\frac{\partial^2\Delta}{\partial \rho^2} +\frac{1}{\mu}\left(\frac{\partial P_0}{\partial \rho}\right)^2\right]\rho^2 . \tag{8.4} \]

Here \(H_{p_0}\) is the roton energy in the absence of a phonon. Since it follows from the form of the energy spectrum that most rotons possess momenta close to \(P_0\), we shall neglect the terms in the expansion of \(H_p\) containing the difference \((P-P_0)\), and shall not write them out. We shall also neglect the term in (8.4) containing the derivative, whose magnitude, according to (6.16), is extremely small\(^*\).

Thus, finally, with the aid of (8.3) and (8.4) we have for the phonon–roton interaction energy:

\[ V=-\frac{1}{2}(\mathbf{P}\mathbf{v}+\mathbf{v}\mathbf{P}) +\frac{1}{2}\left[\frac{\partial^2\Delta}{\partial \rho^2} +\frac{1}{\mu}\left(\frac{\partial P_0}{\partial \rho}\right)^2\right]\rho^2 . \tag{8.4'} \]

Because of the smallness of the velocity of oscillations \(\mathbf{v}\) and of the deviation of the density from the equilibrium value \(\rho\), the interaction energy may be regarded as a small perturbation in the expression for the Hamiltonian (8.1) of the phonon–roton system. We shall therefore calculate the probability of scattering of a phonon by a roton according to the scheme of perturbation theory.

Since the scattering effect under consideration is a two-phonon one, the terms linear in \(\rho\) or \(\mathbf{v}\) in the interaction energy will give the required transitions in the second approximation of perturbation theory. However, the fact that the hydrodynamic equations are nonlinear somewhat changes the usual picture at this point. Indeed, we shall solve the hydrodynamic equations by the method of successive approximations.

In the first approximation the density \(\rho\) (or the velocity \(\mathbf{v}\)) will be a superposition of plane waves. The terms of the second approximation in the density \(\rho\) (or \(\mathbf{v}\)) will contain pair products of different Fourier components. The matrix elements from the terms of the second approximation in the density \(\rho\) will likewise contain pair products of phonon amplitudes.

\(^*\) As simple calculations show, the term with the derivative \(\partial\Delta/\partial\rho\) in (8.4) may be neglected in the case where the condition

\[ (1/\mu c^2)(\partial\Delta/\partial\rho)^2 \ll \partial^2\Delta/\partial\rho^2 . \]

is satisfied. According to (6.16) and (6.17), this condition is satisfied with a large margin.

Among the pair products there will also be those which correspond exactly to the scattering of a phonon in the process under consideration, i.e., contain the amplitudes of both the incident and the scattered phonons. Therefore already in the first approximation of perturbation theory the terms linear in \(\rho\) (or \(\mathbf v\)) in the perturbation energy will give the transitions needed for our problem. In connection with what has been said, let us solve the hydrodynamic equations taking into account terms of second order in \(\rho\).

In view of the smallness of the changes in density and pressure, we write the variables in the form \(p_0+p,\ \rho_0+\rho\), where \(p_0\) and \(\rho_0\) are the constant equilibrium pressure and density. After neglecting, in Euler’s equation, small quantities of third order we obtain

\[ \frac{\partial \mathbf v}{\partial t}+(\mathbf v\nabla)\mathbf v = -\frac{c^2}{\rho_0}\nabla\rho + \frac{\rho\nabla\rho}{\rho_0} \left( \frac{c^2}{\rho_0} - \frac{\partial c^2}{\partial \rho} \right). \tag{8.5} \]

In the same approximation the equation of continuity reduces to

\[ \frac{\partial\rho}{\partial t}+\rho_0\,\operatorname{div}\mathbf v = -\operatorname{div}\rho\mathbf v. \tag{8.6} \]

We apply the operation \(\operatorname{div}\) to equation (8.5), differentiate equation (8.6) with respect to \(t\), and then subtract the second from the first. As a result we obtain a wave equation with allowance for second-order terms, which we group on the right-hand side:

\[ \frac{\partial^2\rho}{\partial t^2}-c^2\nabla\rho = -\operatorname{div} \left\{ \frac{\partial\rho\mathbf v}{\partial t} - \rho_0(\mathbf v\nabla)\mathbf v + \left( \frac{c^2}{\rho_0} - \frac{\partial c^2}{\partial \rho} \right)\rho\nabla\rho \right\}. \tag{8.7} \]

In the first approximation for the incident and scattered waves we have

\[ \mathbf v=\mathbf v_1 e^{(i/\hbar)(\mathbf p\mathbf r-cpt)},\qquad \rho=\rho_1 e^{(i/\hbar)(\mathbf p\mathbf r-cpt)}, \tag{8.8a} \]

\[ \mathbf v'=\mathbf v'_1 e^{(i/\hbar)(\mathbf p'\mathbf r-cp't)},\qquad \rho'=\rho'_1 e^{(i/\hbar)(\mathbf p'\mathbf r-cp't)}, \tag{8.8b} \]

where quantities without primes refer to the incident wave, and those with primes to the scattered wave. In the second approximation we shall seek only those terms in which the incident and scattered waves are interwoven. The remaining terms do not give energy conservation and vanish in the matrix elements of the perturbation energy.

Let us denote the unit vector in the direction of the incident wave by \(\mathbf n\), and in the direction of the scattered wave by \(\mathbf n'\). Using the method of successive approximations, we substitute (8.8a) and (8.8b) into the right-hand side of (8.7). Then for the density in the second approximation we obtain

\[ \rho_{2\,\mathbf p\mathbf p'} = \frac{\rho_1\rho'_1}{c^2pp'(1-\mathbf n\mathbf n')} \left\{ [p-p'(\mathbf n\mathbf n')] [p'-p(\mathbf n\mathbf n')] \frac{c^2}{\rho_0} - \frac{1}{2}(p-p')^2 \frac{\partial c^2}{\partial\rho} \right\} e^{(i/\hbar)\{(\mathbf p-\mathbf p',\mathbf r)-c(p-p')t\}}. \tag{8.9} \]

For the velocity in the same approximation, from (8.5) we find:

\[ \mathbf{v}_{2\mathrm{pr}}= \frac{\rho\rho'(p-p')(p'-p)}{\rho_0 c p p'(1-nn')} \left\{ (nn')\frac{c^2}{\rho_0} + \frac{1}{2}\frac{\partial c^2}{\partial\rho} \right\} e^{(i/\hbar)[(p-p',r)-c(p-p')t]}. \tag{8.10} \]

The problem now consists in determining the probability of transition of a roton from a state with momentum \(\mathbf P\) to a state with momentum \(\mathbf P'\), in which a phonon \(\mathbf p\) is absorbed and a phonon \(\mathbf p'\) is emitted. Two intermediate states I and II are possible.

I) A roton in state \(A\) with momentum \(\mathbf P\) absorbs a phonon \(\mathbf p\) and passes into the intermediate state \(\mathbf P+\mathbf p\), after which it emits a phonon \(\mathbf p'\) and passes into the final state \(F\), having momentum \(\mathbf P'=\mathbf P+\mathbf p-\mathbf p'\).

II) A roton in state \(A\) with momentum \(\mathbf P\) emits a phonon \(\mathbf p'\) and passes into the intermediate state \(\mathbf P-\mathbf p'\), after which it absorbs a phonon \(\mathbf p\) and passes into the final state \(F\) with momentum \(\mathbf P'=\mathbf P+\mathbf p-\mathbf p'\).

The scattering of a phonon by a roton is in a certain sense analogous to the scattering of a light particle by a heavy one. This analogy is connected with the fact that the phonon has a momentum whose magnitude is considerably smaller than the momentum of the roton. It follows from the distribution function for rotons that the magnitude of the roton momentum is approximately equal to \(P_0\). Taking into account the law of conservation of momentum, the law of conservation of energy for the process under consideration is written, according to (1.1) and (1.2), in the form

\[ cp+\frac{(P-P_0)^2}{2\mu} = cp' + \frac{(|\mathbf P+\mathbf p-\mathbf p'|-P_0)^2}{2\mu}, \tag{8.11} \]

whence, after simple transformations that take into account the smallness of the phonon momenta (\(p\) and \(p'\ll P_0\)), we obtain

\[ c(p-p')=(\mathbf P+\mathbf p-\mathbf p')^2/2\mu P_0^2. \tag{8.12} \]

We shall now use the circumstance that the phonon energy \(\varepsilon=cp\) is considerably smaller than the quantity \(\mu c^2\), approximately equal to \(20^\circ\mathrm K\). This makes it possible to conclude that the law of conservation of energy in the process under consideration simply reduces to equality of the magnitudes of the momenta of the incident and scattered phonons, \(p=p'\). This result confirms the analogy indicated above with the scattering of light particles by heavy ones, where light particles, upon scattering, change only the direction of their momentum, without changing its magnitude.

The energy of the perturbation \(V\), according to (8.4′), contains the quantity \(\mathbf v_{2\mathrm{pr}}\), determined by expression (8.10). However, the quantity \(\mathbf v_{2\mathrm{pr}}\) essentially contains the factor \((p-p')\). Therefore, taking into account the law of conservation of energy, one may omit in the perturbation energy the second-approximation terms \(\mathbf v_{2\mathrm{pr}}\).

The matrix element of the transition \(H'_{AF}\) in the second approximation of perturbation theory, according to \((8.4')\), is equal to

\[ H'_{AF}= \frac{(\mathbf{P}\mathbf{v}+\mathbf{v}\mathbf{P})_{AI} (\mathbf{P}\mathbf{v'}+\mathbf{v'}\mathbf{P})_{IF}} {4(E_A-E_I+pc)} + \frac{(\mathbf{P}\mathbf{v'}+\mathbf{v'}\mathbf{P})_{AI} (\mathbf{P}\mathbf{v}+\mathbf{v}\mathbf{P})_{IF}} {4(E_A-E_{II}-p'c)} + \]

\[ +\,(I/2)\left[ \frac{\partial^2\Delta}{\partial \rho^2} +\frac{1}{\mu}\left(\frac{\partial P_0}{\partial \rho}\right)^2 \right]\rho^2 . \tag{8.13} \]

The changes of energy in transitions to the intermediate states are equal to

\[ \left. \begin{aligned} E_A-E_I &=-\frac{(|\mathbf{P}+\mathbf{p}|-P_0)^2}{2\mu} =-\frac{(\mathbf{P}\mathbf{p})^2}{2\mu P_0^2},\\ E_A-E_{II} &=-\frac{(|\mathbf{P}-\mathbf{p'}|-P_0)^2}{2\mu} =-\frac{(\mathbf{P}\mathbf{p'})^2}{2\mu P_0^2}. \end{aligned} \right\} \tag{8.14} \]

On the same grounds as in \((8.12)\), one may assert that

\[ E_A-E_I\ll pc,\qquad E_A-E_{II}\ll p'c . \]

However, one cannot simply neglect in the denominators of the first two terms in \((8.13)\) the energy differences \(E_A-E_I\) and \(E_A-E_{II}\), since the expressions of zeroth order in the phonon momenta that are then obtained mutually cancel. Therefore we first reduce the first two terms in \((8.13)\) to a common denominator, and then neglect in the latter the changes of the roton energy. Thus we obtain

\[ H'_{AF} = -\frac{vv'}{p^2p'^2c} \left\{ \left(\mathbf{P}+\frac12\mathbf{p},\mathbf{p}\right) \left(\mathbf{P'}+\frac12\mathbf{p'},\mathbf{p'}\right) \left[ p'+\frac{(\mathbf{P}\mathbf{p'})^2}{2\mu cP_0^2} \right] \right. \]

\[ \left. - \left(\mathbf{P}-\frac12\mathbf{p'},\mathbf{p'}\right) \left(\mathbf{P'}-\frac12\mathbf{p},\mathbf{p}\right) \left[ p-\frac{(\mathbf{P}\mathbf{p})^2}{2\mu cP_0^2} \right] \right\} + \]

\[ + \left[ \frac{\partial^2\Delta}{\partial\rho^2} +\frac{1}{\mu} \left(\frac{\partial P_0}{\partial\rho}\right)^2 \right]\rho\rho' . \tag{8.15} \]

Here, in expanding the expression \((\rho^2)_{AF}\), one must take into account the possible interchanges of \(\rho\) and \(\rho'\), leading to a doubling of the corresponding term.

Using the conservation laws, we transform the expression in braces in \((8.15)\) to the form

\[ P_0p^3 \left\{ (\mathbf{n}+\mathbf{n'};\mathbf{m})(\mathbf{n}\mathbf{n'}) + \frac{P_0}{\mu c} (\mathbf{n}\mathbf{m})^2(\mathbf{n'}\mathbf{m})^2 \right\}. \tag{8.16} \]

Here \(\mathbf{m}\) is the unit vector in the direction of the roton momentum \(\mathbf{P}\).*)

\[ \text{*) From the law of conservation of momentum it follows that the direction of the roton momentum } \mathbf{P} \text{ is conserved.} \]

Finally, the transition matrix element (8.13), with the aid of (8.16), takes the form:

\[ \left. \begin{aligned} H'_{AF} &= \frac{P_0 p}{2\rho_0} \left\{(\mathbf n+\mathbf n',\mathbf m)(\mathbf n\mathbf n') +\frac{P_0}{\mu c}(\mathbf n\mathbf m)^2(\mathbf n'\mathbf m)^2 +A\right\},\\[3pt] A&=\frac{\rho_0^2}{P_0 c} \left[ \frac{d^2\Delta}{d\rho^2} +\frac{1}{\mu}\left(\frac{dP_0}{d\rho}\right)^2 \right]. \end{aligned} \right\} \tag{8.17} \]

The desired differential effective cross section for scattering of a phonon by a roton is equal to

\[ d\sigma=(2\pi/\hbar c)|H'_{AF}|^2 \delta(E_A+pc-E_F-p'c)(2\pi\hbar)^{-3}p'^2\,dp'\,do'. \tag{8.18} \]

Substitute (8.17) into (8.18) and carry out the integration over the momenta of the scattered phonon; as a result we have

\[ d\sigma= \left(\frac{P_0p^2}{4\pi\hbar^2\rho_0 c}\right)^2 \left\{ (\mathbf n+\mathbf n',\mathbf m)(\mathbf n\mathbf n') + \frac{P_0}{\mu c}(\mathbf n\mathbf m)^2(\mathbf n'\mathbf m)^2 +A \right\}^2do'. \tag{8.19} \]

In solving the problem of the viscosity of helium II we shall need an expression for the probability of scattering of a phonon by rotons in which the direction of the phonon momentum \(\mathbf p\) changes through a given angle \(\psi\). Therefore we average expression (8.19) over the angles of the incident and scattered rotons. As a result, after simple calculations we obtain

\[ d\sigma(p,\psi)= \left(\frac{P_0p^2}{4\pi\hbar^2\rho_0 c}\right)^2 \left\{ \frac{2}{3}(1+\cos\psi)\cos^2\psi +\frac{1}{105}\left(1+8\cos^2\psi+\frac{8}{3}\cos^4\psi\right) +\frac{2A}{15}\left(\frac{P_0}{\mu c}\right)(1+2\cos^2\psi) +A^2 \right\}do'. \tag{8.20} \]

The angle \(\psi\) appearing in (8.20) is formed by the directions of the incident and scattered phonons. Integrating (8.20) over all scattering angles, we find the total effective cross section for scattering of a phonon with momentum \(p\) by a roton:

\[ \sigma_{\phi\rho} = \frac{1}{4\pi} \left(\frac{P_0p^2}{\hbar^2\rho_0 c}\right)^2 \left[ \frac{2}{9} +\frac{1}{25}\left(\frac{P_0}{\mu c}\right)^2 +\frac{2A}{9}\frac{P_0}{\mu c} +A^2 \right]. \tag{8.21} \]

The value of the parameter \(A\), computed using the values of the derivatives with respect to the parameters \(\Delta\) and \(P_0\) given above in § 6, turns out to be equal to \(-0.66\). It should, of course, be noted that the accuracy of the values of the indicated derivatives and of the parameter \(A\) is comparatively low.

Using the numerical values of all the parameters entering into (8.21), we obtain for \(\sigma_{\phi\rho}\):

\[ \sigma_{\phi\rho}=2.2\cdot10^{-19}(xT)^4,\qquad p=x(kT/c). \tag{8.22} \]

9. Scattering of a Roton by a Roton1

The theory gives no indication as to the character of the interaction of a roton with a roton; therefore, naturally, the problem of the scattering of a roton by a roton cannot be solved exactly. However, since our aim is to calculate only the temperature dependence of the viscosity of helium II, such an exact solution is not necessary.

To find the indicated temperature law it is sufficient to know the probability of scattering of a roton by a roton as a function of temperature to within a constant factor; such a probability is not sensitive to the choice of the roton–roton interaction function.

In calculating the probability of scattering of a roton by a roton we shall use the method of perturbation theory, taking the interaction energy of a roton with a roton to be a \(\delta\)-function of the distance between them. It is known that, in the application of perturbation theory, the choice of the interaction energy in the form of a \(\delta\)-function does not lead to a divergence of the expressions obtained for the probability.

Let the interaction energy of two rotons be equal to

\[ V = V_{0}\,\delta(\mathbf r_{1}-\mathbf r), \tag{9.1} \]

where \(\mathbf r_{1}\) and \(\mathbf r\) are the radius vectors of the rotons, and \(V_{0}\) is a certain constant, whose value can be determined from the values of the viscosity of helium II. We shall denote the energies and momenta of the rotons by \(E\) and \(\mathbf P\), respectively, for the incident rotons without a prime and for the scattered ones with one prime. The probability of transition of the rotons from the state \(A\) with momenta \(\mathbf P\) and \(\mathbf P_{1}\) to the state \(F\) with momenta \(\mathbf P'\) and \(\mathbf P_{1}'\) is determined by the formula of perturbation theory

\[ dw = (2\pi/h)\,|V_{AF}|^{2}\, \delta(E+E_{1}-E'-E_{1}')\, d\mathbf P'\,d\mathbf P_{1}'/(2\pi h)^{2}. \tag{9.2} \]

As the wave functions of the rotons we choose plane waves, symmetrized with respect to the pairs of colliding and scattered rotons. Thus, for the incident rotons the wave function is written in the form

\[ \Psi(\mathbf P,\mathbf P_{1}) = \frac{1}{\sqrt{2}} \left\{ e^{(i/h)(\mathbf P\mathbf r+\mathbf P_{1}\mathbf r_{1})} + e^{(i/h)(\mathbf P\mathbf r_{1}+\mathbf P_{1}\mathbf r)} \right\}. \]

Analogously, the wave function of the scattered rotons is written

\[ \Psi(\mathbf P',\mathbf P_{1}') = \frac{1}{\sqrt{2}} \left\{ e^{(i/h)(\mathbf P'\mathbf r+\mathbf P_{1}'\mathbf r_{1})} + e^{(i/h)(\mathbf P'\mathbf r_{1}+\mathbf P_{1}'\mathbf r)} \right\}. \]

With the aid of wave functions of this form we calculate the matrix element of the transition \(V_{AF}\):

\[ V_{AF} = V_{0}\Omega^{-1} \int \Psi^{*}(\mathbf P,\mathbf P_{1})\, \delta(\mathbf r-\mathbf r_{1})\, \Psi(\mathbf P',\mathbf P_{1}')\, d\Omega\,d\Omega_{1} = \]

\[ = 2V_{0}\Omega^{-1} \int e^{\frac{i}{h}\,[\,\mathbf P'+\mathbf P_{1}'-\mathbf P-\mathbf P_{1},\mathbf r\,]}\, d\Omega. \tag{9.3} \]

The square of the modulus of the matrix element can easily be integrated over the momenta of one of the scattered rotons

\[ \int |V_{AF}|^{2}\,\frac{dP_1'}{(2\pi h)^3}=|V_0|^2 . \tag{9.4} \]

This relation makes it possible to rewrite the transition probability \(dw\) in the form

\[ dw=(8\pi/h)|V_0|^2\delta(E+E_1-E'-E_1')\,dP'(2\pi h)^{-3}. \tag{9.5} \]

The probability defined by expression (9.5) has the dimension \(\text{cm}^3\,\text{sec}^{-1}\). To obtain the effective differential scattering cross section, this probability must be divided by the relative velocity of motion of the colliding rotons, equal to

\[ v=\left|\frac{\partial E}{\partial P}-\frac{\partial E_1}{\partial P_1}\right|. \]

It follows from the form of the energy spectrum that the majority of rotons will have momenta whose absolute values are close to \(P_0\). Consequently, the changes in the momenta of the rotons upon scattering will be considerably smaller in magnitude than \(P_0\).

Fig. 5.

Let the momenta of the rotons \(\mathbf P\) and \(\mathbf P_1\) before the collision form an angle \(\theta\). Then from Fig. 5 it is easy to see that the momenta of the rotons after the collision, introducing the variable \(\mathbf f\), can be represented in the following form:

\[ \begin{aligned} P'&=P_0+f_x\cos\frac{\theta}{2}+f_y\sin\frac{\theta}{2},\\ P_1'&=P_0+f_x\cos\frac{\theta}{2}-f_y\sin\frac{\theta}{2}, \end{aligned} \tag{9.6} \]

where \(|\mathbf f|\ll P_0\).

In the new variables the phase-volume element reduces to

\[ dP'=2\pi P_0\sin\frac{\theta}{2}\,df_x\,df_y . \tag{9.7} \]

and the law of conservation of energy to

\[ f_x^2 \cos^2 \frac{\theta}{2} + f_y^2 \sin^2 \frac{\theta}{2} = \frac{1}{2}(P-P_0)^2 + \frac{1}{2}(P_1-P_0)^2. \tag{9.8} \]

To calculate the total effective cross section \(\sigma\) for roton–roton scattering, it is necessary in (9.5) to carry out an integration over the phase volume of the scattered particle. In expression (9.5), among the coordinates in phase space only the \(\delta\)-function, containing the law of conservation of energy, depends on them.

If, for convenience of integration, one introduces an auxiliary variable \(g\) by means of the relation

\[ g^2=f_x^2\cos^2\frac{\theta}{2}+f_y^2\sin^2\frac{\theta}{2}, \]

then, taking (9.7) and (9.8) into account, the indicated integration of the \(\delta\)-function over the phase volume of the scattered particle is performed quite simply, namely:

\[ \begin{aligned} \int \delta(E+E_1-E'-E_1')\,dP' &= \int \delta\left[g^2-\frac{1}{2}(P-P_0)^2-\frac{1}{2}(P_1-P_0)^2\right] \frac{2\pi P_0\mu}{\cos\frac{\theta}{2}}\,2\pi g\,dg \\ &= \frac{2\pi^2 P_0\mu}{\cos\frac{\theta}{2}} . \end{aligned} \tag{9.9} \]

Using now relations (9.5) and (9.9), we write the expression for the total effective cross section \(\sigma\) for roton–roton scattering:

\[ \sigma = \frac{2P_0\mu |V_0|^2} {\left|\frac{\partial E_1}{\partial P_1}-\frac{\partial E}{\partial P}\right| h^4 \cos\frac{\theta}{2}} . \tag{9.10} \]

The reciprocal of the mean time \(t_p\) between two collisions of a roton is obtained from (9.10) by multiplying by the total flux of rotons with subsequent averaging over all angles formed by the momenta of the colliding rotons. Thus we find\(^*\)

\[ \frac{1}{t_p} = \sigma\, \overline{\left|\frac{\partial E_1}{\partial P_1}-\frac{\partial E}{\partial P}\right|} N_p = \frac{4P_0\mu |V_0|^2 N_p}{h^4}. \tag{9.11} \]

Here \(N_p\) is the number of rotons per unit volume, equal to

\[ N_p= \frac{2P_0^2(\mu kT)^{1/2}e^{-\Delta/kT}} {(2\pi)^{1/2}h^3}. \tag{9.12} \]

The value of the constant \(|V_0|^2\) entering formulas (9.10) and (9.11) can be calculated from experimental values of the coefficient

\(^*\) A bar denotes the operation of averaging over angles.

viscosity of helium II (see § 16). It turns out to be, in order of magnitude, equal to \(1.25\cdot 10^{-76}\,(\text{erg}\,\text{cm}^{3})^{2}\).

In conclusion, let us draw attention to the circumstance that expression (9.10) for the effective cross section \(\sigma\) diverges for angles between the momentum directions of the colliding rotons close to \(\pi\). For rotons possessing the mean velocity

\[ \bar v=(2kT/\pi\mu)^{1/2}, \tag{9.13} \]

formula (9.10), after substitution of the numerical values of the parameters, gives

\[ \sigma \simeq 5\cdot 10^{-15}T^{-1/2}. \tag{9.14} \]

10. Absorption and emission of phonons and rotons\(^ {21}\)

In collisions of rotons and phonons, processes are possible as a result of which the total number of rotons and phonons may change. All such processes in helium II may be divided into three types:

a) processes in which phonons are emitted (or absorbed);

b) processes in which rotons are emitted or absorbed;

c) processes in which rotons are converted into phonons (or conversely).

Let us consider the most probable processes of each type. Only the fastest of the indicated processes will be essential in what follows.

Emission and absorption of phonons. In the collision of two phonons, processes are possible which lead to a change in the total number of phonons. The simplest of such processes—the three-phonon process—is forbidden because of the impossibility of simultaneous fulfillment of the two conservation laws: the law of conservation of momentum and the law of conservation of energy. Therefore we shall consider a five-phonon process, consisting in the absorption (or emission) of a third phonon in the collision of two phonons.

We shall start from the Hamiltonian function of a quantum liquid\(^4\)

\[ H=-\frac{1}{2}\mathbf{v}\rho\mathbf{v}+\varepsilon(\rho). \tag{10.1} \]

The nonvanishing matrix elements for transitions of three phonons into two are obtained from the terms cubic in \(\rho'\) in the energy in the third approximation of perturbation theory, from the cubic and quartic terms in the second approximation, and from the terms of fifth order in the first approximation of perturbation theory (\(\rho'\) is the deviation of the density from its value in the liquid at rest).

Let us represent the Hamiltonian (10.1) in the form of the sum

\[ H=H_0+V_3+V_4+V_5, \tag{10.2} \]

where \(H_0\) is the density of the sound energy, containing terms quadratic in \(\rho'\), while \(V_3\), \(V_4\), and \(V_5\) contain, respectively, terms of the third, fourth, and fifth order with respect to \(\rho'\),

\[ H_{AF}=\sum_{\mathrm{I,II}} \frac{(V_3)_{A\mathrm{I}}(V_3)_{\mathrm{I\,II}}(V_3)_{\mathrm{II}F}} {(E_A-E_{\mathrm{I}})(E_A-E_{\mathrm{II}})} + \sum_{\mathrm{I}} \frac{(V_4)_{A\mathrm{I}}(V_3)_{\mathrm{I}F}} {E_A-E_{\mathrm{I}}} +(V_5)_{AF}. \tag{10.3} \]

The resonance-type denominators that arise when dispersion at small angles between the momenta of the colliding phonons is neglected lead to a divergence of the first two sums in (10.3). Taking into account the dispersion of the phonon part of the energy spectrum (§ 6) eliminates the divergence of the indicated terms. It is easy to see that only the transitions described by the first sum in (10.3) will be essential, some terms of which contain in the denominators products of two expressions that simultaneously vanish when dispersion is neglected \((E_A-E_{\mathrm{I}}\) and \(E_A-E_{\mathrm{II}})\). Therefore in the transition matrix element we shall retain only the first sum, which depends on the terms \(V_3\) of third order in \(\rho'\):

\[ V_3=-\frac{v\rho'v}{2} + \frac{1}{3!}\frac{\partial}{\partial\rho} \left(\frac{c^2}{\rho}\right)\rho'^3 . \tag{10.4} \]

Let there be, in the initial state \(A\), three phonons with momenta \(\mathbf{p}_1,\mathbf{p}_2,\mathbf{p}_3\), and in the final state \(F\), two phonons with momenta \(\mathbf{p}_4\) and \(\mathbf{p}_5\). Altogether, several tens of intermediate configurations (I and II) are possible, through which the transition from state \(A\) to state \(F\) is carried out. However, as was already noted, only those intermediate states will be essential for which, at small angles between the momenta of the colliding phonons, there simultaneously takes place*):

\[ E_A-E_{\mathrm{I}}\to 0,\qquad E_A-E_{\mathrm{II}}\to 0. \tag{10.5} \]

From what has been said it is perfectly clear that calculation of the probability of the transition under consideration \((3\to 2)\)

\[ dw=\frac{2\pi}{h}|H_{AF}|^2 \delta(\varepsilon_1+\varepsilon_2+\varepsilon_3-\varepsilon_4-\varepsilon_5) \frac{d\mathbf{p}_4}{(2\pi\hbar)^3} \tag{10.6} \]

is a very cumbersome operation. Such a calculation hardly makes sense at present. The point is that the main parameters of the theory—the derivatives of the first three orders of the sound velocity \(c\) with respect to density and, especially, the dispersion parameter \(\gamma\), are known with very

\[ \text{*)} \]
Altogether there turn out to be 27 such intermediate states, but, owing to the laws of conservation of momentum and energy, only 15 intermediate states simultaneously satisfy condition (11.5); moreover, which ones in particular depends on the relation between the magnitudes of the momenta.

with little accuracy. Therefore we shall confine ourselves only to establishing the dependence between the transition probability and the energy of the colliding phonons. This will completely determine the temperature behavior of the quantities of interest to us.

To establish the temperature behavior of some average, over all phonons, probability of the transition (2.6) under consideration, there is no need to identify the colliding phonons. Therefore, for the phonon momenta we shall omit the indices. Thus, according to (10.4) and (6.11), we have

\[ V_3 \sim p^{3/2}. \tag{10.7} \]

The energy difference \(E_A-E_I\), for example, for the concrete case in which in the intermediate state I there are two phonons with momenta \(\mathbf p_1+\mathbf p_2\) and \(\mathbf p_3\), may be written in the form

\[ \varepsilon(p_1)+\varepsilon(p_2)-\varepsilon(|\mathbf p_1+\mathbf p_2|)= \]

\[ = \frac{c p_1 p_2}{p_1+p_2}\left[1-\mathbf n_1\mathbf n_2+3\gamma (p_1+p_2)^2\right], \tag{10.8} \]

where \(\mathbf n_1\) and \(\mathbf n_2\) are unit vectors in the direction of the corresponding momenta. The matrix element of the transition will contain in its denominators products of two expressions of type (10.8).

In the present context we are, naturally, not interested in the angular distribution of the phonons. Therefore we average expression (10.6) for the probability \(d w\) over the angles formed by the momenta of the colliding phonons, and integrate over the phase volume of the momentum of the scattered phonon. The integration over angles required here is performed quite simply owing to the smallness of the terms appearing in the denominators of the expression for \(H_{AF}\) and taking account of the dispersion.

Without dwelling on the complicated calculations, we give the final result obtained for the angular-averaged probability:

\[ \overline{w}\sim p'^{\,2}/(\gamma p^2)^2\cdot p^5 \sim p^3 . \tag{10.9} \]

Let the total number of phonons per unit volume (in general not equal to the equilibrium value) be \(N_\phi\). The rate of change of the number of phonons due to the five-phonon process may be written in the form

\[ \dot N_\phi = -\int\!\!\int\!\!\int\!\!\int \{n_1 n_2 n_3 (n_4+1)(n_5+1) - \]

\[ -(n_1+1)(n_2+1)(n_3+1)n_4 n_5\}\, d w\, \frac{d\mathbf p_1 d\mathbf p_3 d\mathbf p_3}{(2\pi\hbar)^9}. \tag{10.10} \]

If the total number of phonons is not equal to the equilibrium value, this means that the distribution functions \(n\) contain a nonzero chemical potential \(\mu_\phi\),

\[ n=\left[e^{[\varepsilon(p)-\mu_\phi]/kT}-1\right]^{-1}. \]

For small deviations from equilibrium we expand the function \(n\) in powers of \(\mu_\phi\), retaining terms linear in \(\mu_\phi\); we have

\[ n-n_0=n_0(n_0+1)\mu_\phi/kT . \tag{10.11} \]

The function \(n\) with subscript zero corresponds to the equilibrium distribution functions for phonons \((\mu_\phi=0)\). After simple transformations, with the aid of (10.11), relation (2.10) takes the form

\[ \dot N_\phi=-\iiint\!\!\int n_{10}n_{20}n_{30}(n_{40}+1)(n_{50}+1)\,dw\, \frac{d\mathbf p_1\,d\mathbf p_2\,d\mathbf p_3}{(2\pi\hbar)^9}\, \frac{\mu_\phi}{kT}. \tag{10.10′} \]

Let us denote the coefficient in the equality relating the rate of change of \(N_\phi\) to the quantity \(\mu_\phi\) by \(\Gamma_\phi\); according to (10.10) we have

\[ \Gamma_\phi=\frac{1}{kT}\iiint\!\!\int n_{10}n_{20}n_{30}(n_{40}+1)(n_{50}+1)\,dw\, \frac{d\mathbf p_1\,d\mathbf p_2\,d\mathbf p_3}{(2\pi\hbar)^9}. \tag{10.12} \]

Without significant loss of accuracy, in the integrand in (10.12) one may neglect the distribution functions \(n_{40}\) and \(n_{50}\) in comparison with unity. Then the integration over the phase volumes of the momenta of the three colliding phonons \((\mathbf p_1,\mathbf p_2,\text{ and }\mathbf p_3)\) is carried out independently, which makes it possible to replace \(dw\) by \(\overline w\), while omitting one integration. Thus we obtain

\[ \Gamma_\phi\simeq \frac{1}{kT}\iiint n_{10}n_{20}n_{30}\,\overline w\, \frac{d\mathbf p_1\,d\mathbf p_2\,d\mathbf p_3}{(2\pi\hbar)^9}. \tag{10.12′} \]

The integrand in (10.12′) is proportional to \(p^{12}\). After integration in (10.12′) over the phase volumes of the colliding phonons, we obtain the temperature law for \(\Gamma_\phi\):

\[ \Gamma_\phi=aT^{11}, \tag{10.13} \]

where \(a\) is a temperature-independent coefficient.

The relation obtained, (10.13), establishes the temperature dependence of precisely that quantity which will enter essentially into the subsequent calculations.

Emission and absorption of rotons. In the collision of two rotons, emission or absorption of a third roton is possible. An exact calculation of the probability of the indicated process does not appear possible, since the specific form of the interaction function of two rotons is unknown.

From the law of conservation of energy it follows that, in the collision of two rotons with momenta \(\mathbf P_1\) and \(\mathbf P_2\), a third roton can be emitted only in the case when the condition is satisfied:

\[ \frac{(P_1-P_0)^2}{2\mu}+\frac{(P_2-P_0)^2}{2\mu}\geq \Delta . \tag{10.14} \]

From the form of the energy spectrum one may conclude that the majority of rotons have momenta close in magnitude to \(P_0\). Therefore condition (10.14) is very stringent and, as the corresponding estimates show, the probability of the five-roton process proves to be negligibly small.

If \(N_p\) is the number of rotons per unit volume, not equal to its equilibrium value, then, by analogy with the preceding case, the rate \(\dot N_p\) of approach to the equilibrium state due to the five-roton process may be written in the form

\[ \dot N_p=-\Gamma_p \mu_p , \tag{10.15} \]

where \(\mu_p\) is the chemical potential of the roton gas.

From the assertion made above concerning the smallness of the probability of the five-roton process there follows the smallness of the quantity \(\Gamma_p\). We shall therefore assume that

\[ \Gamma_p \ll \Gamma_\phi . \tag{10.16} \]

at all temperatures below the \(\lambda\)-point.

Processes of conversion of phonons into rotons. The law of conservation of momentum forbids the process of conversion of a roton into a phonon, since the momentum \(p\) of a phonon satisfies the condition \(p \ll P_0\), whereas the momentum of a roton is approximately equal to \(P_0\). This prohibition occurs in the collision of a roton with a roton if the directions of the momenta of the colliding rotons form small angles; therefore in this case the conversion of the two colliding rotons into scattered rotons and a phonon is impossible. However, if the angle between the directions of the momenta of the colliding rotons is not very small, then the indicated prohibition no longer applies, and such collisions of two rotons are possible in which they are converted into a phonon and a roton and even into two phonons (when the angles between the momenta of the colliding rotons are close to \(\pi\)). For the same reason as in the preceding subsection, calculation of the probability of such processes is impossible. It appears possible only to make the corresponding estimates, the essence of which is as follows.

For convenience let us consider the process inverse to that mentioned above, i.e., let a phonon with momentum \(\mathbf p_1\) collide with a roton with momentum \(\mathbf P_2\), and as a result let two scattered rotons with momenta \(\mathbf P_3\) and \(\mathbf P_4\) be formed. From the law of conservation of energy it follows that the energy of the phonon under consideration must be not less than \(\Delta\). Thus we are dealing with a very energetic phonon. We shall assume that such energetic phonons differ from rotons only in the dependence of the energy on the magnitude of the momentum. They interact with rotons in the same way as rotons interact with one another. Therefore we may carry out all calculations completely analogously to those that were performed in considering scattering

roton with a roton (§ 9). In accordance with this, the interaction energy \(V\) phonon—roton for phonons of high energy (greater than \(\Delta\)) can be represented in a form proportional to the \(\delta\)-function of the distance between the phonon and the roton,

\[ V = V_0 \delta(\mathbf r_1 - \mathbf r_2) \tag{10.17} \]

(\(\mathbf r_1\) and \(\mathbf r_2\) are the radius vectors, respectively, of the phonon and the roton). As was already noted (§ 9), choosing the interaction function in this form makes it possible to determine the temperature law of the probability of the corresponding process. Similarly to the preceding case, for the rate of approach of the number of phonons and rotons to the equilibrium state in a nonequilibrium phonon-roton gas, if this approach is caused by the process of conversion of rotons into phonons and conversely, one may write

\[ \dot N_{\mathrm p} = -\Gamma_{\phi\mathrm p}(\mu_{\mathrm p} - \mu_\phi), \qquad \dot N_\phi = \Gamma_{\phi\mathrm p}(\mu_{\mathrm p} - \mu_\phi). \tag{10.18} \]

The coefficient \(\Gamma_{\phi\mathrm p}\) is determined by the corresponding complete collision integral. Without dwelling on the intermediate calculations, we give the final result obtained with the aid of the interaction function (10.17):

\[ \Gamma_{\phi\mathrm p} = |V_0|^2 \Delta^2 N_{\mathrm p}^2 \pi h^4 c^3 kT . \tag{10.19} \]

The magnitude of the amplitude \(V_0\) for the process under consideration is unknown to us. For purposes of estimation one may use the value of \(V_0\) obtained for the process of interaction of a roton with a roton from the experimental values of the viscosity coefficient of helium II. We note that the estimate made with the value of \(V_0\) chosen in this way gives an upper limit for the values of the coefficient \(\Gamma_{\phi\mathrm p}\). This follows from the fact that such an estimate is equivalent to the assumption of conversion of a phonon into a roton every time that the phonon has sufficient energy and the angle between the momenta of the phonon and the roton is such that the momentum conservation law is satisfied. However, it is quite obvious that these conditions are in reality necessary but not sufficient: when the indicated conditions are satisfied, simple scattering of a phonon by a roton is also possible.

Substituting into the right-hand side of relation (10.19) the values of all parameters and the value of \(V_0\) chosen in the indicated way (\(1 \cdot 12 \cdot 10^{-38}\ \mathrm{erg}\ \mathrm{cm}^3\)), we find

\[ \Gamma_{\phi\mathrm p} < 5 \cdot 10^{50} e^{-\frac{2\Delta}{T}} . \tag{10.20} \]

The five-phonon process must naturally be less probable than the four-phonon one. But for the four-phonon process we have earlier made rather accurate estimates (§ 8), which, thus, give us an upper limit for the value of \(\Gamma_\phi\). Comparison of both upper limits (for \(\Gamma_{\phi\mathrm p}\) and \(\Gamma_\phi\)) shows that they

are quantities of the same order over a wide range of temperatures. Consequently, it is not excluded that the coefficients themselves, \(\Gamma_{\mathrm{fph}}\) and \(\Gamma_{\mathrm{ph}}\), may turn out to be quantities of the same order.

In what follows we shall denote the unknown numerical coefficient in the expression for \(\Gamma_{\mathrm{fph}}\) by \(b\), and shall write, in accordance with (10.19),

\[ \Gamma_{\mathrm{fph}}=b e^{-\frac{2\Delta}{T}} . \tag{10.21} \]

The values of the numerical coefficients \(a\) in formula (10.13) and \(b\) in formula (10.21) can be calculated from the experimental values of the coefficient of absorption of first sound in helium II \(^{28}\). It then turns out that the coefficients \(\Gamma_{\mathrm{ph}}\) and \(\Gamma_{\mathrm{fph}}\) do indeed have the same order of magnitude*). The data \(^{32}\) give the following values for the indicated coefficients:

\[ \Gamma_{\mathrm{ph}}=1\cdot10^{43}T^{11};\qquad \Gamma_{\mathrm{fph}}=4\cdot10^{49}e^{-\frac{2\Delta}{T}} . \tag{10.22} \]

11. Establishment of equilibrium in the phonon gas

A change in the number of phonons in an element of phase volume may occur in the following ways: a) scattering of phonons by phonons (§ 7); b) scattering of phonons by rotons (§ 8); c) absorption and emission of phonons in inelastic collisions of rotons or phonons with one another (§ 10).

Let us calculate the relaxation times (or the corresponding mean free paths) which will characterize the establishment of equilibrium in the phonon gas.

In § 7 it was shown that the effective cross section for phonon scattering by a phonon reaches its greatest value at small angles between the momenta of the colliding phonons. From the conservation laws of momentum and energy it follows that such a scattering process does not lead to a substantial change in the directions of the momenta of the colliding phonons and, consequently, scattering of phonons by phonons mainly leads to a rapid exchange of energies between phonons. However, an exact calculation of the relaxation time characterizing the establishment of energetic equilibrium in the phonon gas cannot be carried out for the simple reason that the problem itself cannot be formulated exactly. Nevertheless, since the time of establishment of energetic equilibrium in the phonon gas is a very important characteristic of our system, we shall attempt to approach its determination from two limiting cases.

*) The calculated effect proves to be considerably less probable than scattering of a phonon by a roton. Comparison of the value of the coefficient \(\Gamma\), obtained from the collision integral for the process of scattering of a phonon by a roton, with the value of the upper limit for \(\Gamma_{\mathrm{fph}}\) convinces us of this.

In the first case we shall assume that, in the phonon gas, the number of phonons possessing small energies (smaller than the mean phonon energy) has in some way been changed, so that the distribution function in the region of small energies is not equal to the equilibrium function; we shall then calculate the relaxation time characterizing the establishment of equilibrium in such a gas.

In the second limiting case we shall assume that, in the phonon gas, the number of phonons possessing large energies (larger than the mean phonon energy) has been changed, so that the distribution function in the region of large energies is not equal to the equilibrium function. In this case we shall calculate the relaxation time characterizing the establishment of equilibrium in such a gas. Comparing this time with the time characterizing transport processes, we shall show that the process of establishing energy equilibrium is faster than the transport processes in a phonon gas.

We shall begin with the first case, i.e., consider the scattering of phonons of small energy by phonons themselves. In this case one may assume that the momentum \(p\) of the phonon under consideration is much smaller than the momentum \(p_1\) of the phonon with which this phonon collides. Such an assumption, as was shown in § 7, considerably simplifies the expression for the angularly averaged probability of scattering of a phonon by a phonon. In the case under consideration, the angularly averaged effective cross section for scattering of a phonon by a phonon, according to (7.11), is equal to

\[ \sigma(p,p_1)=\frac{\pi(u+2)^4 p_1^4}{(96\pi\hbar^2\rho c)^2 T}. \]

Let us consider the kinetic equation for phonons:

\[ \frac{\partial n}{\partial t}+\mathbf{v}\nabla n=I(n), \]

where the collision integral \(I(n)\) for the process of scattering of a phonon by a phonon is equal to

\[ I(n)=-(2\pi\hbar)^{-3}\int c\sigma(p,p_1)\{nn_1(n'+1)(n_1'+1)- \]

\[ -n'n_1'(n+1)(n_1+1)\}\,d\mathbf{p}_1. \tag{11.1} \]

We are interested in the relaxation of phonons of small energy, whose distribution function is equal to \(n\), for a given equilibrium distribution of the remaining phonons. Therefore the distribution functions \(n_1\), \(n'\), and \(n_1'\) in (11.1) will be regarded as equilibrium ones, while the deviation of the distribution function \(n\) from the equilibrium value \(n_0\) will be assumed small and equal to \(\delta n\). Using the known properties of equilibrium distribution functions, we transform the expression standing in braces

of the bracket in (11.1)

\[ \begin{gathered} \{(n_0+\delta n)(n'+1)(n_1'+1)n_1-n_1'n_1'(n_1+1)(n_0+1+\delta n)\}=\\ =\delta n\{n_1(n'+1)(n_1'+1)-n'n_1'(n_1+1)\}=\\ =\delta n n'n_1'(n_1+1)/n. \end{gathered} \tag{11.2} \]

Substituting the expression obtained into the collision integral (11.1), we have

\[ I(n)=-(2\pi\hbar)^{-3}\delta n\int n^{-1} n'n_1'(n_1+1)c\sigma(p,p_1)\,dp_1. \tag{11.3} \]

Thus, the relaxation time for phonons of small energy is determined by the relation

\[ \frac{1}{t_M}=(2\pi\hbar)^{-3}\int n^{-1}n'n_1'(n_1+1)c\sigma(p,p_1)\,dp_1. \tag{11.4} \]

In order to simplify the integration in (11.4), we replace the functions \(n_1\), \(n'\), and \(n_1'\) by Wien functions and take into account the fact that the momenta \(p\) and \(p_1\) satisfy the inequality \(p\ll p_1\). In this case the combination of distribution functions entering expression (11.4) takes the form: \(n_1(n_1+1)pc/kT\). Using this and substituting into (11.4) expression (7.11) for \(\sigma(p,p_1)\), we have*)

\[ \frac{1}{t_M}=(2\pi\hbar)^{-3}\frac{\pi(u+2)^4}{(96\pi\hbar^2\rho_0)^2c\gamma} \int n_1(n_1+1)p_1^4\,dp_1\,\frac{pc}{kT}, \tag{11.5} \]

or, after elementary integration,

\[ \frac{1}{t_M}=\frac{(u+2)^4 6!}{(48\hbar^2\rho_0)^2c\gamma(2\pi\hbar)^3} \left(\frac{kT}{c}\right)^6 p. \tag{11.6} \]

The fact that phonons with momentum \(p_1\) have high energy is taken into account in (11.5) automatically, since in the integration over \(dp_1\) the essential role is played by phonons with energy of order \(6kT\).

Let us turn to the case of relaxation of phonons of high energy. In this case the assumption that the phonon momenta are connected by the relation

\[ p_1\ll p, \tag{11.7} \]

allows one to simplify the expression for the scattering probability of a phonon by a phonon, averaged over angles. Replacing \(p_1\) by \(p\) in expression (7.11), we obtain

\[ \sigma(p,p_1)=\frac{\pi(u+2)^4p^4}{(96\pi\hbar^2\rho_0 c)^2\gamma}. \tag{11.8} \]

*) Integration over \(dp_1\) with a zero lower limit leads to a certain error, which, however, is not significant, since our results have an approximate character.

I. M. KHALATNIKOV

Quite analogously to the case of phonons with low energy, we find for the relaxation time \(t_B\) of phonons with high energy the relation

\[ \frac{1}{t_B}=(2\pi\hbar)^{-3}\int n^{-1}(n_1+1)n'n'_1\sigma(p,p_1)\,d\mathbf{p}_1, \tag{11.9} \]

which coincides with expression (11.4). Such a coincidence is natural, since in deriving relation (11.4) no assumption was made anywhere about the magnitude of the energy of the relaxing phonons. Let us now use the assumption of small momentum and, taking the phonon distribution functions \(n\), \(n'\), and \(n'_1\) to be small compared with unity, simplify the combination of distribution functions entering expression (11.9). We have\(^*\)

\[ n^{-1}n'n'_1(n_1+1)\simeq n_1(n_1+1). \tag{11.10} \]

Substituting (11.8) and (11.10) into expression (11.9), we have

\[ \frac{1}{t_B}=(2\pi\hbar)^{-3}p^4 \frac{\pi(u+2)^4}{(96\pi\hbar^2\rho_0)^2 c\gamma} \int n_1(n_1+1)\,d\mathbf{p}_1. \tag{11.11} \]

Finally, after elementary integration we find

\[ \frac{1}{t_B}= \frac{(u+2)^4(kT/c)^3(\pi^3/3)p^4} {(48\hbar^2\rho_0)^2c\gamma(2\pi\hbar)^3}. \tag{11.12} \]

The fact that the momentum \(\mathbf{p}_1\) is small is taken into account automatically in expression (11.11), since in the integral occurring there the essential role is played by momenta of order \(2kT\). Using the numerical values of the parameters entering formulas (11.6) and (11.12), we calculate the coefficients in these expressions. As a result, for the relaxation time of phonons with low energy we obtain

\[ \frac{1}{t_M}\simeq 2\cdot 10^7T^7x \tag{11.13} \]

and for phonons with high energy:

\[ \frac{1}{t_B}\simeq 10^5\cdot T^7x^4. \tag{11.14} \]

Here the quantity \(x\) is related to the phonon energy by the relation \(\varepsilon=xkT\).

To determine the relaxation time of phonons possessing arbitrary energy, one can choose an interpolation formula which, in both limiting cases, would pass into the obtained formulas (11.13) and (11.14). The following formula satisfies this condition:

\[ \frac{1}{t_\Phi}=10^5\cdot T^7\cdot x(x+6)^3. \tag{11.15} \]

\(^*\) The assumption that the functions \(n\), \(n'\), \(n'_1\) are small is equivalent to the condition \(p,p',p'_1 \gg kT/c\).

Phonon scattering by phonons does not lead to a substantial change in the momenta of the scattered phonons, but this process, as has already been noted, ensures the establishment of equilibrium with respect to energies in the phonon gas. The calculated relaxation time \(1/t_\phi\) characterizes the rate at which such equilibrium is established in the phonon gas.

III. KINETIC COEFFICIENTS OF HELIUM II

A detailed analysis of the kinetic equation for elementary excitations in helium II made it possible to clarify the question of the kinetic coefficients of helium II. It was found that helium II is characterized by a coefficient of first viscosity and by three coefficients of second viscosity. In addition, in helium II there is still another kinetic coefficient, analogous to the coefficient of thermal conductivity for ordinary bodies. This coefficient relates the temperature gradient to the heat flux.

We shall begin our consideration of the question with an analysis of the kinetic equation for elementary excitations in helium II. Such an analysis will allow us to obtain the hydrodynamic equations of helium II and general expressions for the fluxes of energy and momentum.

12. Kinetic equation for elementary excitations in helium II\(^{29}\)

The kinetic equation determining the distribution function \(n\) of elementary excitations in helium II can be written in the form of a continuity equation in the space of coordinates \(\mathbf r\) and momenta \(\mathbf p\)

\[ \frac{\partial n}{\partial t}+\frac{\partial n\dot{\mathbf r}}{\partial \mathbf r}+\frac{\partial n\dot{\mathbf p}}{\partial \mathbf p}=I(n). \tag{12.1} \]

\(I(n)\) is the collision integral, whose concrete form is of no interest to us in the present case.

According to the Hamilton–Jacobi equation, the velocity \(\dot{\mathbf r}\) and force \(\dot{\mathbf p}\) are expressed through the Hamiltonian \(H\) of an elementary excitation:

\[ \dot{\mathbf r}=\frac{\partial H}{\partial \mathbf p},\qquad \dot{\mathbf p}=-\frac{\partial H}{\partial \mathbf r}. \tag{12.2} \]

Substituting (12.2) into (12.1), and expanding the derivatives of the products \(n\dot{\mathbf r}\) and \(n\dot{\mathbf p}\), we rewrite (12.1) in the form

\[ \frac{\partial n}{\partial t}+\frac{\partial n}{\partial \mathbf r}\frac{\partial H}{\partial \mathbf p} -\frac{\partial n}{\partial \mathbf p}\frac{\partial H}{\partial \mathbf r}=I(n). \tag{12.3} \]

Suppose that in helium II there is superfluid motion with velocity \(\mathbf v_s\). The Hamiltonian \(H\) is then equal to

\[ H=\varepsilon(p)+\mathbf p\mathbf v_s, \tag{12.4} \]

where \(\varepsilon(p)\) is the excitation energy in a frame of reference moving with velocity \(\mathbf v_s\).

Equations of motion of helium II. Multiply the left- and right-hand sides of equation (12.3) by the \(i\)-th component of the momentum \(p_i\) and integrate them over all \(p\)-space. By virtue of the law of conservation of momentum for the whole system, the integral of the right-hand side is equal to zero,

\[ \int p_i I(n)\,d\tau_p=0. \tag{12.5} \]

Thus we have

\[ \frac{\partial}{\partial t}\int np_i\,d\tau_p+ \int p_i\frac{\partial n}{\partial r_k}\frac{\partial H}{\partial p_k}\,d\tau_p - \int p_i\frac{\partial n}{\partial p_k}\frac{\partial H}{\partial r_k}\,d\tau_p=0. \]

Next, by a simple transformation we find:

\[ \frac{\partial}{\partial t}\int np_i\,d\tau_p+ \frac{\partial}{\partial r_k}\int np_i\frac{\partial H}{\partial p_k}\,d\tau_p + \int n\frac{\partial H}{\partial r_i}\,d\tau_p=0. \tag{12.6} \]

Substituting here expression (12.4) for \(H\), and, for brevity, denoting integration over \(p\)-space by a bar over the corresponding expressions, we have

\[ \frac{\partial}{\partial t}\overline{np_i} + \frac{\partial}{\partial r_k}\overline{np_i\left(\frac{\partial\varepsilon}{\partial p_k}+v_{sk}\right)} + n\left(\frac{\partial\varepsilon}{\partial r_i}+p_k\frac{\partial v_{sk}}{\partial r_i}\right)=0. \tag{12.7} \]

Equation (12.7) is one of the equations of motion of liquid helium II. It determines the acceleration of the relative motion of the normal and superfluid parts of helium II. This follows directly from the fact that \(\overline{np}\) is the total momentum of the excitations in the frame of reference in which there is no superfluid motion (the superfluid part of the liquid is at rest).

The total momentum per unit volume of helium II is expressed in terms of the excitation momentum \(\overline{np}\) in a frame of reference moving with velocity \(\mathbf v_s\), with the aid of the relation \(\mathbf j=\overline{np}+\rho\mathbf v_s\). The time derivative of the total momentum, as follows from the general law of conservation of energy and momentum, is equal to the divergence of a certain symmetric tensor \(\Pi_{ik}\). We shall express the momentum-flux tensor \(\Pi_{ik}\) in the fixed frame of reference in terms of its value \(\pi_{ik}\) in the frame of reference moving with velocity \(\mathbf v_s\)*)

\[ \Pi_{ik}=\pi_{ik}+v_{sk}\overline{np_i}+v_{si}\overline{np_k}+\rho v_{si}v_{sk}; \tag{12.8} \]

*) The relation connecting the values of the momentum-flux tensor in the fixed frame of reference \((\Pi_{ik})\) and in a frame moving with some velocity \(\mathbf v\) \((\pi_{ik})\) can easily be obtained in the case of classical hydrodynamics, where the form of the tensor \(\Pi_{ik}\) is known. The tensor in classical hydrodynamics is written in the form

\[ \Pi_{ik}=\rho u_i u_k+\delta_{ik}p \]

(\(\mathbf u\) is the velocity of the liquid, \(p\) the pressure). Express the velocity of the liquid in the fixed frame of reference in terms of the velocity of the liquid \(\mathbf u'\) in a frame of reference moving with velocity \(\mathbf v\),

\[ \mathbf u=\mathbf u'+\mathbf v. \]

Thus,

\[ \frac{\partial}{\partial t}\overline{n p_i} + \frac{\partial}{\partial t}\rho v_{si} + \frac{\partial}{\partial r_k} \left( \pi_{ik} + v_{sk}\overline{n p_i} + v_{si}\overline{n p_k} + \rho v_{si}v_{sk} \right) =0 . \tag{12.9} \]

Subtract equation (12.7) from (12.9) and use the continuity equation

\[ \frac{\partial \rho}{\partial t} + \frac{\partial}{\partial r_k} \left( \overline{n p_k} + \rho v_{sk} \right) =0 . \tag{12.10} \]

As a result we obtain

\[ \rho \frac{\partial v_{si}}{\partial t} + \rho v_{sk}\frac{\partial v_{si}}{\partial r_k} + \frac{\partial}{\partial r_k}\pi_{ik} - \overline{n\frac{\partial \varepsilon}{\partial r_i}} - \frac{\partial}{\partial r_k}\overline{n p_i\frac{\partial \varepsilon}{\partial p_k}} =0 . \tag{12.11} \]

In the absence of an external field the derivative \(\dfrac{\partial \varepsilon}{\partial r_i}\) is equal to \(\dfrac{\partial \varepsilon}{\partial \rho}\dfrac{\partial \rho}{\partial r_i}\). From the condition \(\operatorname{rot}\mathbf v_s=0\) there follows the requirement that the sum of the last three terms in (12.11) be the product of the density \(\rho\) by the gradient of some function. Moreover, it is quite obvious that the tensor \(\pi_{ik}\) at absolute zero, in the absence of excitations, must be equal to \(p_0\delta_{ik}\) \((p_0\) is the pressure of liquid helium II without excitations). These requirements determine uniquely the form of the tensor \(\pi_{ik}\):

\[ \pi_{ik} = n p_i\overline{\frac{\partial \varepsilon}{\partial p_k}} + \delta_{ik}n\overline{\frac{\partial \varepsilon}{\partial \rho}}\rho + \delta_{ik}p_0 . \tag{12.12} \]

The pressure \(p_0\) is expressed in terms of the energy \(E_0\) and the potential \(\mu_0(T=0)\) by

\[ p_0=-E_0+\rho\mu_0 . \]

Hence it follows that

\[ \frac{\partial p_0}{\partial r_i} = \rho\frac{\partial \mu_0}{\partial r_i}. \tag{12.13} \]

Substituting expression (12.12) for \(\pi_{ik}\) into (12.11), and taking (12.13) into account, we obtain

\[ \rho \frac{\partial v_{si}}{\partial t} + \rho v_{sk}\frac{\partial v_{sk}}{\partial r_i} + \rho\frac{\partial}{\partial r_i}n\overline{\frac{\partial \varepsilon}{\partial \rho}} + \rho\frac{\partial \mu_0}{\partial r_i} =0 . \tag{12.14} \]

After substituting this expression for \(\mathbf u\) into the formula given above for \(\Pi_{ik}\), we obtain

\[ \Pi_{ik} = \rho v_i v_k + \rho u_i'v_k + \rho u_k'v_i + \rho\left(u_i'u_k'+p\delta_{ik}\right). \]

Denoting the momentum per unit volume of the liquid in the moving frame of reference by the letter \(\mathbf p\), we finally find the desired relation

\[ \Pi_{ik} = \rho v_i v_k + p_i v_k + p_k v_i + \pi_{ik}. \]

Naturally, the transformation formula obtained for the tensor \(\Pi_{ik}\) retains its form also in the hydrodynamics of helium II.

Dropping the common factor \(\rho\) in (12.14), we finally find

\[ \frac{\partial \mathbf v_s}{\partial t} +\nabla\left\{\mu_0+n\,\frac{\overline{\partial \varepsilon}}{\partial \rho} +\frac{v_s^2}{2}\right\}=0 . \tag{12.15} \]

The equation obtained is the equation of superfluid motion. The full thermodynamic potential \(\mu\), according to (12.15), is equal to

\[ \mu=\mu_0+\int n\,\frac{\overline{\partial \varepsilon}}{\partial \rho}\,d\tau_p . \tag{12.16} \]

The second term in (12.16) is due to the contribution of the excitations to the thermodynamic potential \(\mu\).

Instead of equation (12.7), it is sometimes convenient to use the equation of motion (12.9), which, after substitution of expression (12.12) for the tensor \(\tau_{ik}\), takes the form

\[ \frac{\partial}{\partial t}\left(\overline{np_i}+\rho v_{si}\right) +\frac{\partial}{\partial r_k} \left\{ n p_i\frac{\overline{\partial \varepsilon}}{\partial p_k} +\delta_{ik}n\,\frac{\overline{\partial \varepsilon}}{\partial \rho}\,\rho +\delta_{ik}p_0 +n\overline{p_i v_{sk}} +n\overline{p_k v_{si}} +\rho v_{si}v_{sk} \right\}=0 . \tag{12.9′} \]

The law of conservation of energy. The energy per unit volume of liquid helium II, \(E\), in a fixed frame of reference, according to the known transformation formulas, is related to the internal energy \(E_{\mathrm v}\) in a frame of reference moving with velocity \(\mathbf v_s\) by the relation

\[ E=\rho\,\frac{v_s^2}{2}+\mathbf v_s\,\overline{n\mathbf p}+E_{\mathrm v}. \tag{12.17} \]

Let us differentiate with respect to time the first two terms in (12.17), which are the kinetic energy of helium (with the energy of the relative motion of the normal and superfluid parts subtracted), and use the equations of motion (12.7), (12.15) and the continuity equation (12.10). We have

\[ \frac{\partial}{\partial t}E_k = \frac{\partial}{\partial t}\left(\rho\,\frac{v_s^2}{2}+\mathbf v_s\overline{n\mathbf p}\right) = \]

\[ =-(\rho v_{si}+\overline{np_i}) \frac{\partial}{\partial r_i} \left( \mu_0+n\,\frac{\overline{\partial \varepsilon}}{\partial \rho} +\frac{v_s^2}{2} \right) -\frac{v_s^2}{2}\, \frac{\partial}{\partial r_i} \left(\rho v_{si}+\overline{np_i}\right) - \]

\[ -\,v_{si}\frac{\partial}{\partial r_k}\, \overline{np_i} \left( n\,\frac{\partial \varepsilon}{\partial p_k} +v_{sk} \right) - v_{si}\, \overline{\left( n\,\frac{\partial \varepsilon}{\partial r_i} +p_k\frac{\partial v_{sk}}{\partial r_i} \right)} . \tag{12.18} \]

We shall now find the time derivative of the internal energy of helium, \(E_{\mathrm v}=\overline{n\varepsilon}\). For this purpose we multiply the left- and right-hand sides of the kinetic equation (12.3) by \(\varepsilon\) and integrate them over all \(p\)-space. The integral of the right-hand side \(\int I(n)\,\varepsilon\,d\tau_p\) is then equal to zero according to the law of conservation of energy. As a result we obtain

\[ \varepsilon\,\frac{\partial \overline n}{\partial t} +\varepsilon\,\frac{\overline{\partial n}}{\partial r_k}\frac{\partial H}{\partial p_k} -\varepsilon\,\frac{\overline{\partial n}}{\partial p_k}\frac{\partial H}{\partial r_k} =0 . \tag{12.19} \]

Next, by simple transformations of the equation obtained, using the Gauss—Ostrogradsky theorem, we find

\[ \frac{\partial}{\partial t}\,\overline{(n\varepsilon)} + \frac{\partial}{\partial r_k}\,\varepsilon n \left( \overline{\frac{\partial \varepsilon}{\partial p_k}+v_{sk}} \right) - v_{sk}n\,\overline{\frac{\partial \varepsilon}{\partial r_k}} + n\,\overline{\frac{\partial \varepsilon}{\partial p_k}}\,p_i \frac{\partial v_{si}}{\partial r_k} - n\,\overline{\frac{\partial \varepsilon}{\partial \rho}}\, \frac{\partial \rho}{\partial t} =0. \tag{12.20} \]

Thus we have:

\[ \frac{\partial}{\partial t}E_{\mathrm{B}} = - \frac{\partial}{\partial r_k}\, \varepsilon n \left( \overline{\frac{\partial \varepsilon}{\partial p_k}+v_{sk}} \right) + v_{sk}n\,\overline{\frac{\partial \varepsilon}{\partial r_k}} - \]

\[ - n\,\overline{\frac{\partial \varepsilon}{\partial p_k}}\,p_i \frac{\partial v_{si}}{\partial r_k} + \frac{\partial \rho}{\partial t}\, n\,\overline{\frac{\partial \varepsilon}{\partial \rho}} . \tag{12.21} \]

The derivative of the sum of the kinetic and internal energies of helium II according to (12.13) and (12.21) is equal to

\[ \frac{\partial}{\partial t}(E_k+E_{\mathrm{B}}) = - \frac{v_s^2}{2}\, \frac{\partial}{\partial r_k} (\overline{np_k}+\rho v_{sk}) - v_{si} \frac{\partial v_{sk}}{\partial r_i}\,\overline{np_k} - \]

\[ - \rho v_{sk}\frac{\partial}{\partial r_k}\frac{v_s^2}{2} - \overline{np_k}\, \frac{\partial}{\partial r_k} \left( n\,\overline{\frac{\partial \varepsilon}{\partial \rho}}+\mu_0 \right) - \rho v_{sk}\frac{\partial}{\partial r_k} \left( n\,\overline{\frac{\partial \varepsilon}{\partial \rho}}+\mu_0 \right) - \]

\[ - v_{sk}\frac{\partial}{\partial r_i}\, np_k \left( \frac{\partial \varepsilon}{\partial p_i}+v_{si} \right) - v_{sk}\frac{\partial v_{si}}{\partial r_k}\,\overline{np_i} - n\,\overline{\frac{\partial \varepsilon}{\partial r_k}}\,v_{sk} + \]

\[ + n\,\overline{\frac{\partial \varepsilon}{\partial r_k}}\,v_{sk} - \frac{\partial}{\partial r_k}\, \varepsilon n \left( \overline{\frac{\partial \varepsilon}{\partial p_k}+v_{sk}} \right) + \frac{\partial v_{si}}{\partial r_k}\, n\,\overline{\frac{\partial \varepsilon}{\partial p_k}}\,p_i - \frac{\partial \rho}{\partial t}\, n\,\overline{\frac{\partial \varepsilon}{\partial \rho}} . \]

Hence, after transformations using equation (12.10), we obtain the law of conservation of energy, written in the form of a continuity equation for the total energy,

\[ \frac{\partial E}{\partial t} = \frac{\partial E_k}{\partial t} + \frac{\partial E_{\mathrm{B}}}{\partial t} + \mu_0\frac{\partial \rho}{\partial t} = \]

\[ = - \frac{\partial}{\partial r_k} \left\{ (\overline{np_k}+\rho v_{sk}) \left( \mu_0+n\,\overline{\frac{\partial \varepsilon}{\partial \rho}} +\frac{v_s^2}{2} \right) + nH\,\overline{\frac{\partial}{\partial p_k}H} \right\}. \tag{12.22} \]

The third term on the left-hand side of (12.22) characterizes the change in the energy of liquid helium II without excitations. The energy flux \(\mathbf Q\) in helium II according to (12.22) is equal to

\[ \mathbf Q = (\overline{n\mathbf p}+\rho\mathbf v_s) \left( \mu_0+n\,\overline{\frac{\partial \varepsilon}{\partial \rho}} +\frac{v_s^2}{2} \right) + nH\,\overline{\frac{\partial}{\partial p_k}H} \tag{12.23} \]

\[ (\mathbf j=\overline{n\mathbf p}+\rho\mathbf v_s). \]

Equations (12.7), (12.10), (12.15), together with equation (12.22), constitute the complete system of hydrodynamic equations for helium II. Usually, instead of the energy continuity equation (12.22), one writes the entropy continuity equation, which follows from (12.23) and the thermodynamic identities.

13. Kinetic coefficients2

The equilibrium distribution function for excitations in helium II, in which uniform normal motion with velocity \(\mathbf{v}_n\) and superfluid motion with velocity \(\mathbf{v}_s\) take place, has the form (Bose statistics)

\[ n_0=\left[e^{\frac{H-\mathbf{p}\mathbf{v}_n}{kT}}-1\right]^{-1} = \left[e^{\frac{\varepsilon+\mathbf{p}\mathbf{v}_s-\mathbf{p}\mathbf{v}_n}{kT}}-1\right]^{-1}. \tag{13.1} \]

The equilibrium distribution function (13.1) satisfies the kinetic equation, turning its right-hand side into zero. In the case of a deviation of the system of excitations in helium II from equilibrium, the distribution of excitations is determined by another function, whose form in the general case cannot be specified, since the kinetic equation cannot be solved in general form. The problem is simplified for the case when the spatial derivatives of the quantities determining the state of the system are small, i.e., when these quantities themselves change little over distances characterizing the dimensions of the system. In this case the distribution function differs little from the equilibrium function \(n_0\). This circumstance makes it possible to linearize the kinetic equation. To this end, in the left-hand side of the kinetic equation we substitute the equilibrium function \(n_0\), retaining in it after differentiation only terms linear in the spatial derivatives. The right-hand side of the kinetic equation (the collision integral) is expanded in a series in the difference \((n-n_0)\). Thus one obtains a linear integral equation determining the difference \((n-n_0)\) in a form proportional to the spatial derivatives of the quantities characterizing the state. The values of \(n-n_0\) thus found make it possible to calculate the energy dissipation in the nonequilibrium system and to determine the form of the kinetic coefficients.

Let us determine in general form the properties of the kinetic coefficients in helium II.

Substitute the distribution function \(n_0\) into the left-hand side of the kinetic equation. In doing so, along with the smallness of the spatial derivatives, we shall assume small the difference of the velocities \(\mathbf{v}_n-\mathbf{v}_s\). The latter circumstance does not in the least restrict the problem, since the velocity difference \((\mathbf{v}_n-\mathbf{v}_s)\) must be small in comparison with the speed of sound in helium II. As is known, superfluidity is violated considerably earlier than this limit can be reached. Let us write separately each of the terms contained in the left-hand side of the kinetic equation*):

\[ \frac{\partial n_0}{\partial t} = n'\left\{ \frac{1}{kT} \left( \frac{\partial \varepsilon}{\partial t} -\mathbf{p}\frac{\partial \mathbf{v}_n}{\partial t} +\mathbf{p}\frac{\partial \mathbf{v}_s}{\partial t} \right) - \frac{1}{kT^2} \left( \varepsilon-\mathbf{p}\mathbf{v}_n+\mathbf{p}\mathbf{v}_s \right) \frac{\partial T}{\partial t} \right\}. \tag{13.2} \]

We choose as the independent variables the density \(\rho\) and the entropy \(S\).

According to the equations of hydrodynamics, the time derivatives are expressed through the spatial derivatives in the linear approximation as follows:

\[ \frac{\partial \rho}{\partial t}+\operatorname{div}\mathbf{j}=0,\qquad \frac{\partial S}{\partial t}+\operatorname{div}S\mathbf{v}_n=0, \tag{13.3} \]

\[ \rho_n\frac{\partial}{\partial t}(\mathbf{v}_n-\mathbf{v}_s)=-S\nabla T \tag{13.4} \]

(\(\mathbf{j}=\rho_n\mathbf{v}_n+\rho_s\mathbf{v}_s\) is the flux of helium II).

With the aid of (13.4) we transform (13.2) to the form

\[ \frac{\partial n_0}{\partial t} = \frac{n'}{kT} \left\{ \operatorname{div}(\mathbf{j}-\rho\mathbf{v}_n) \left[ \frac{1}{T}\frac{\partial T}{\partial \rho} (\varepsilon-\mathbf{p}\mathbf{v}_n+\mathbf{p}\mathbf{v}_s) - \frac{\partial\varepsilon}{\partial\rho} \right] \right. \]

\[ \left. +\operatorname{div}\mathbf{v}_n \left[ \frac{1}{T} \left( \frac{\partial T}{\partial\rho}\rho+ \frac{\partial T}{\partial S}S \right) (\varepsilon-\mathbf{p}\mathbf{v}_n+\mathbf{p}\mathbf{v}_s) - \frac{\partial\varepsilon}{\partial\rho}\rho \right] \right. \]

\[ \left. +\frac{ST}{\rho_n}\mathbf{p}\frac{\nabla T}{T} -(\mathbf{v}_n\nabla\varepsilon) +\frac{1}{T}(\mathbf{v}_n\nabla T) (\varepsilon-\mathbf{p}\mathbf{v}_n+\mathbf{p}\mathbf{v}_s) \right\}. \tag{13.5} \]

Further, we have

\[ \frac{\partial n}{\partial r}\frac{\partial H}{\partial \rho} - \frac{\partial n}{\partial \rho}\frac{\partial H}{\partial r} = \frac{n'}{kT} \left\{ - \left( \frac{\partial\varepsilon}{\partial\rho}+\mathbf{v}_s \right) \nabla(\mathbf{p}\mathbf{v}_n) \right. \]

\[ \left. -\frac{\nabla T}{T} (\varepsilon-\mathbf{p}\mathbf{v}_n+\mathbf{p}\mathbf{v}_s) \left( \frac{\partial\varepsilon}{\partial\rho}-\mathbf{v}_n+\mathbf{v}_s \right) +\mathbf{v}_n\nabla(\varepsilon+\mathbf{p}\mathbf{v}_s) \right\}. \tag{13.6} \]

Substitute (13.5) and (13.6) into the left-hand side of the kinetic equation (12.1); collecting like terms, we obtain

\[ \frac{n'}{kT} \left\{ \left[ \frac{1}{T}\frac{\partial T}{\partial\rho} (\varepsilon-\mathbf{p}\mathbf{v}_n+\mathbf{p}\mathbf{v}_s) - \frac{\partial\varepsilon}{\partial\rho} \right] \operatorname{div}(\mathbf{j}-\rho\mathbf{v}_n) \right. \]

\[ +\left[ \frac{1}{T} \left( \frac{\partial T}{\partial\rho}\rho+ \frac{\partial T}{\partial S}S \right) (\varepsilon-\mathbf{p}\mathbf{v}_n+\mathbf{p}\mathbf{v}_s) - \frac{\partial\varepsilon}{\partial\rho}\rho \right] \operatorname{div}\mathbf{v}_n \]

\[ +\frac{\nabla T}{T} \left[ \mathbf{p}\frac{ST}{\rho_n} - (\varepsilon-\mathbf{p}\mathbf{v}_n+\mathbf{p}\mathbf{v}_s) \left( \frac{\partial\varepsilon}{\partial\rho} -\mathbf{v}_n+\mathbf{v}_s \right) \right] \]

\[ \left. -\frac{\partial\varepsilon}{\partial\rho}\nabla(\mathbf{p}\mathbf{v}_n) -\mathbf{v}_s\nabla(\mathbf{p}\mathbf{v}_s) +\mathbf{v}_n\nabla(\mathbf{p}\mathbf{v}_s) \right\} = I(n). \tag{13.7} \]

The last two terms in the curly bracket in (13.7) may be neglected, since their difference, as is easily seen, is of order \((\mathbf{v}_n-\mathbf{v}_s)^2\). Of the same order \((\mathbf{v}_n-\mathbf{v}_s)^2\) are all terms containing the factor \(\mathbf{p}\mathbf{v}_n-\mathbf{p}\mathbf{v}_s\). We shall also neglect them. Finally we have

\[ \frac{n'}{kT} \left\{ \left( \frac{\varepsilon}{T}\frac{\partial T}{\partial\rho} - \frac{\partial\varepsilon}{\partial\rho} \right) \operatorname{div}(\mathbf{j}-\rho\mathbf{v}_n) + \left[ \frac{\varepsilon}{T} \left( \frac{\partial T}{\partial\rho}\rho+ \frac{\partial T}{\partial S}S \right) - \frac{\partial\varepsilon}{\partial\rho}\rho \right] \operatorname{div}\mathbf{v}_n \right. \]

\[ \left. +\frac{\nabla T}{T} \left[ \mathbf{p}\frac{ST}{\rho_n} - \varepsilon\frac{\partial\varepsilon}{\partial\rho} \right] - \frac{\partial\varepsilon}{\partial\rho}\nabla(\mathbf{p}\mathbf{v}_n) \right\} = I(n). \tag{13.8} \]

The solution of equation (13.8) may be represented in the form

\[ \begin{aligned} n-n_0={n'\over kT}\Bigg\{& A\left[{1\over T}{\partial T\over \partial \rho}\,\varepsilon -{\partial\varepsilon\over\partial\rho}\right]\operatorname{div}(\mathbf j-\rho\mathbf v_n)+ \\ &+A\left[{\varepsilon\over T}\left({\partial T\over\partial\rho}\rho +{\partial T\over\partial S}S\right) -{1\over 3}{\partial\varepsilon\over\partial p}\,p -{\partial\varepsilon\over\partial\rho}\rho\right]\operatorname{div}\mathbf v_n+ \\ &+B\,{\nabla T\over T}\left[\mathbf p\,{ST\over\rho_n} -\varepsilon{\partial\varepsilon\over\partial p}\right]+ \\ &+C\left[{\partial\varepsilon\over\partial p_i}p_k -{1\over 3}\delta_{ik}{\partial\varepsilon\over\partial p}\,p\right] \left({\partial v_{nk}\over\partial r_i} +{\partial v_{ki}\over\partial r_k} -{2\over 3}\delta_{ik}{\partial v_{kl}\over\partial r_l}\right) \Bigg\} \end{aligned} \tag{13.9} \]

where \(A\), \(B\), and \(C\) are certain functions of state, whose form is determined by the character of the interaction of excitations*). The coefficient \(A\) is common to all terms of scalar type, the coefficient \(B\) to vector terms (terms proportional to \(\mathbf p\)), and, finally, the coefficient \(C\) to tensor terms. From the last term in the curly bracket of (13.8) \({\partial\varepsilon\over\partial p}\nabla(\rho\mathbf v_n)\), we extract the terms of the form \(\operatorname{div}\mathbf v_n\) contained in it, and symmetrize the remaining part.

The result (13.9) allows us to draw preliminary conclusions about the types of kinetic coefficients in helium II. The terms in (13.9) proportional to \(\operatorname{div}(\mathbf j-\rho\mathbf v_n)\) and \(\operatorname{div}\mathbf v_n\) will be responsible for the second viscosity; the terms containing \(\nabla T\) determine a coefficient analogous to thermal conductivity and, finally, the terms containing

\[ {\partial v_{nk}\over\partial r_i}+{\partial v_{ki}\over\partial r_k} -{2\over 3}\delta_{ik}{\partial v_{kl}\over\partial r_l}, \]

determine the first viscosity associated with normal motion. We draw attention to the curious fact that there is no first viscosity associated with the superfluid flow, although the logical possibility of such an effect is allowed, since the condition of superfluidity consists only in the requirement \(\operatorname{rot}\mathbf v_s=0\).

Let us determine the form of the additional terms in the hydrodynamic equations due to nonequilibrium conditions. In doing so one should keep in mind the conditions

\[ \int \mathbf p\,(n-n_0)\,d\tau_p=0,\qquad \int \varepsilon\,(n-n_0)\,d\tau_p=0, \tag{13.10} \]

which follow from the requirement that the nonequilibrium

*) The functions \(A\), \(B\), and \(C\), generally speaking, depend on the momentum \(\mathbf p\) of the elementary excitation. However, in all cases in which the kinetic equation for rotons is solved, the indicated coefficients are constants or pulses in quantities. This is explained as follows. In calculating the collision integrals for rotons, only the process of roton-roton scattering proves essential, the probability of which, according to (§ 9), does not depend on the energy of the colliding rotons. Therefore, in this case the collision integral \(I(n)\) is ultimately written in the form \(\mathrm{const}\,(n-n_0)\).

function \(n\) must lead to the same values of the total energy and total momentum as the equilibrium function gives.

In the general case, the first of conditions (13.10) imposes a restriction on the coefficient \(B\) occurring in the solution of the kinetic equation. It is satisfied automatically if the value of the coefficient does not depend on the momentum \(\mathbf p\). Then, according to (3.4), (4.4), and (13.9), we have

\[ B\int \frac{\nabla T}{T}\left(\mathbf p\,\frac{ST}{\rho_n}-\varepsilon\,\frac{\partial \varepsilon}{\partial p}\right) \frac{n'}{kT}\,\mathbf p\,d\tau_{\mathbf p} = \]

\[ =\frac{B\nabla T}{3kT^2} \left\{ \frac{ST}{\rho_n}\int p^2 n'\,d\tau_{\mathbf p} - \int n'\varepsilon\left(\frac{\partial\varepsilon}{\partial p}p\right)d\tau_{\mathbf p} \right\}=0. \tag{13.11} \]

Since, according to (13.10), the value of the momentum does not change, no additional terms arise in the continuity equation (2.10).

In the equation of motion (12.9), additional terms arise in the first approximation:

\[ \frac{\partial}{\partial r_k} \left\{ \int (n-n_0)p_i\frac{\partial\varepsilon}{\partial p_k}\,d\tau_{\mathbf p} + \delta_{ik}\int (n-n_0)\frac{\partial\varepsilon}{\partial \rho}p\,d\tau_{\mathbf p} \right\}. \tag{13.12} \]

Substituting (13.9) into the integrals (13.12), after simple transformations, taking into account the symmetry properties of the indicated integrals, we obtain

\[ \frac{\partial}{\partial r_i} \left\{ \operatorname{div}(\mathbf j-\rho\mathbf v_n) \int A\,\frac{n'}{kT} \left( \frac{\varepsilon}{T}\frac{\partial T}{\partial \rho} -\frac{\partial\varepsilon}{\partial \rho} \right) \left[ \frac{\partial\varepsilon}{\partial \rho}\rho +\frac{1}{3}\frac{\partial\varepsilon}{\partial p}\,p \right]d\tau_{\mathbf p} +\right. \]

\[ \left. +\operatorname{div}\mathbf v_n \int A\,\frac{n'}{kT} \left[ \frac{\varepsilon}{T} \left( \frac{\partial T}{\partial \rho}\rho +\frac{\partial T}{\partial S}S \right) - \frac{\partial\varepsilon}{\partial \rho}\rho -\frac{1}{3}\frac{\partial\varepsilon}{\partial p}p \right] \left[ \frac{\partial\varepsilon}{\partial \rho}\rho +\frac{1}{3}\frac{\partial\varepsilon}{\partial p}p \right] d\tau_{\mathbf p} \right\} + \]

\[ +\frac{\partial}{\partial r_k} \left\{ \frac{1}{2} \left( \frac{\partial v_{ni}}{\partial r_k} + \frac{\partial v_{nk}}{\partial r_i} - \frac{2}{3}\delta_{ik}\frac{\partial v_{nl}}{\partial r_l} \right) \int C\,\frac{n'}{5kT} \left[ \left(\frac{\partial\varepsilon}{\partial p}\right)^2p^2 - \frac{1}{3} \left(\frac{\partial\varepsilon}{\partial p}p\right)^2 \right]d\tau_{\mathbf p} \right\}. \tag{13.13} \]

Equation (12.9) thus takes the form

\[ \frac{\partial j_i}{\partial t} + \frac{\partial}{\partial r_k} \left\{ -\delta_{ik}\zeta_1\,\operatorname{div}(\mathbf j-\rho\mathbf v_n) -\delta_{ik}\zeta_2\,\operatorname{div}\mathbf v_n +\delta_{ik}p_0 +\right. \]

\[ \left. +n_0p_i\frac{\overline{\partial\varepsilon}}{\partial p_k} +\overline{n_kp_i}\,v_{si} -\eta \left( \frac{\partial v_{ki}}{\partial r_k} + \frac{\partial v_{nk}}{\partial r_i} - \frac{2}{3}\delta_{ik}\frac{\partial v_{kl}}{\partial r_l} \right) + \right. \]

\[ \left. +\delta_{ik}n_0\frac{\overline{\partial\varepsilon}}{\partial\rho}\rho +\overline{v_{sk}n_0p_i} +\rho v_{si}v_{sk} \right\} =0. \tag{13.14} \]

The kinetic coefficients \(\zeta_1\) and \(\zeta_2\), which have the meaning of coefficients

the second viscosity, according to (13.13), are equal to

\[ \zeta_1=-\int A\frac{n'}{kT}\left(\frac{\varepsilon}{T}\frac{\partial T}{\partial \rho}-\frac{\partial \varepsilon}{\partial \rho}\right) \left(\frac{\partial \varepsilon}{\partial \rho}\rho+\frac{1}{3}\frac{\partial \varepsilon}{\partial \rho}p\right)d\tau_p, \tag{13.15} \]

\[ \zeta_2=-\int A\frac{n'}{kT}\left[ \frac{\varepsilon}{T}\left(\frac{\partial T}{\partial \rho}\rho+\frac{\partial T}{\partial S}S\right) -\frac{\partial \varepsilon}{\partial \rho}\rho-\frac{1}{3}\frac{\partial \varepsilon}{\partial \rho}p \right]\times \]

\[ \times\left[\frac{\partial \varepsilon}{\partial \rho}\rho+\frac{1}{3}\frac{\partial \varepsilon}{\partial \rho}p\right]d\tau_p. \tag{13.16} \]

The coefficient \(\eta\) is the coefficient of first viscosity; according to (13.13) it is equal to

\[ \eta=-\frac{1}{2}\int C\frac{n'}{5kT}\left[ \left(\frac{\partial \varepsilon}{\partial \rho}\right)^2p^2 -\frac{1}{3}\left(\frac{\partial \varepsilon}{\partial \rho}p\right)^2 \right]d\tau_p= \]

\[ =-\frac{1}{5kT}\int Cn'\left(\frac{\partial \varepsilon}{\partial \rho}\right)^2p^2d\tau_p. \tag{13.17} \]

In the equation of superfluid motion (12.15), in the nonequilibrium case there arises an additional term

\[ \frac{\partial}{\partial r_i}\int (n-n_0)\frac{\partial \varepsilon}{\partial \rho}\,d\tau_p. \tag{13.18} \]

Substitution into this integral of expression (13.9) for the difference \((n-n_0)\), similarly to (13.13), gives

\[ \frac{\partial}{\partial r_i}\left\{ \operatorname{div}(\mathbf j-\rho\mathbf v_n) \int A\frac{n'}{kT}\left(\frac{\varepsilon}{T}\frac{\partial T}{\partial \rho} -\frac{\partial \varepsilon}{\partial \rho}\right) \frac{\partial \varepsilon}{\partial \rho}\,d\tau_p+ \right. \]

\[ \left. +\operatorname{div}\mathbf v_n \int A\frac{n'}{kT}\left[ \frac{\varepsilon}{T}\left(\frac{\partial T}{\partial \rho}\rho+ \frac{\partial T}{\partial S}S\right) -\frac{\partial \varepsilon}{\partial \rho}\rho -\frac{1}{3}\frac{\partial \varepsilon}{\partial \rho}p \right] \frac{\partial \varepsilon}{\partial \rho}\,d\tau_p \right\}. \tag{13.19} \]

Thus the equation of superfluid motion in the presence of nonequilibrium takes the form

\[ \frac{d\mathbf v_s}{dt}+\nabla\left\{\frac{v_s^2}{2}+\mu_0+n\frac{\overline{\partial \varepsilon}}{\partial \rho} -\zeta_3\operatorname{div}(\mathbf j-\rho\mathbf v_n)-\zeta_4\operatorname{div}\mathbf v_n\right\}=0. \tag{13.20} \]

The coefficients \(\zeta_3\) and \(\zeta_4\) have the meaning of coefficients of second viscosity; according to (13.19) they are equal to

\[ \zeta_3=\int -A\frac{n'}{3kT}\left(\frac{\varepsilon}{T}\frac{\partial T}{\partial \rho} -\frac{\partial \varepsilon}{\partial \rho}\right) \frac{\partial \varepsilon}{\partial \rho}\,d\tau_p, \tag{13.21} \]

\[ \zeta_4=-\int A\frac{n'}{3kT}\left( \frac{\varepsilon}{T}\frac{\partial T}{\partial \rho}\rho+ \frac{\varepsilon}{T}\frac{\partial T}{\partial S}S -\frac{\partial \varepsilon}{\partial \rho}\rho -\frac{1}{3}\frac{\partial \varepsilon}{\partial \rho}p \right) \frac{\partial \varepsilon}{\partial \rho}\,d\tau_p. \tag{13.22} \]

The coefficients \(\zeta_1, \zeta_2, \zeta_3\), and \(\zeta_4\) are not independent. According to Onsager’s principle, they are connected by one relation, \(\zeta_1=\zeta_4\). The validity of the indicated relation can be seen most simply in the case when \(A\) does not depend on the momentum \(p\). According to (13.15) and (13.22) we have

\[ \begin{aligned} \zeta_1-\zeta_4 &= -A\int \frac{n'}{kT}\left\{ \frac{1}{3}\frac{\partial\varepsilon}{\partial p}\,p \left(\frac{\varepsilon}{T}\frac{\partial T}{\partial p} -\frac{\partial\varepsilon}{\partial p}\right) +\frac{\partial\varepsilon}{\partial p}\,p \left(\frac{\varepsilon}{T}\frac{\partial T}{\partial p} -\frac{\partial\varepsilon}{\partial p}\right)\right.\\ &\qquad\left. -\frac{\varepsilon}{T}\frac{\partial T}{\partial p} \frac{\partial\varepsilon}{\partial p}\rho -\frac{\varepsilon}{T}\frac{\partial T}{\partial S}S \frac{\partial\varepsilon}{\partial p}\rho +\frac{\partial\varepsilon}{\partial p}\frac{\partial\varepsilon}{\partial\rho}\rho +\frac{1}{3}\left(\frac{\partial\varepsilon}{\partial p}p\right) \frac{\partial\varepsilon}{\partial\rho} \right\}\,d\tau_p\\ &= A\int \frac{n'}{kT}\frac{S}{T} \frac{\partial T}{\partial S} \frac{\partial\varepsilon}{\partial\rho}\,\varepsilon\,d\tau_p + A\int \frac{n'}{3kT} \left(\frac{\partial\varepsilon}{\partial p}p\right) \frac{\varepsilon}{T} \frac{\partial T}{\partial\rho}\,d\tau_p . \end{aligned} \tag{13.23} \]

According to (3.4), the second integral in (13.23) is equal to \(AS\,\dfrac{\partial T}{\partial\rho}\). As a result of a single integration by parts, relation (3.4) takes the form

\[ S=\frac{1}{3T}\int n\,\frac{\partial}{\partial p}(\varepsilon p^3)\, \frac{1}{p^2}\,d\tau_p . \tag{13.24} \]

Differentiating this relation on the left and on the right with respect to \(\rho\) at constant temperature \(T\), we have

\[ \frac{\partial S}{\partial\rho} = \frac{1}{3T}\int \left\{ n'\frac{1}{kT}\frac{\partial\varepsilon}{\partial\rho} \frac{\partial}{\partial p}(\varepsilon p^3) + n\frac{\partial}{\partial p} \left(\frac{\partial\varepsilon}{\partial\rho}p^3\right) \right\} \frac{1}{p^2}\,d\tau_p . \]

Integrating the second term by parts, we find

\[ \frac{\partial S}{\partial\rho} = \frac{1}{kT^2}\int n'\frac{\partial\varepsilon}{\partial\rho}\, \varepsilon\,d\tau_p . \tag{13.25} \]

The first integral in (13.23), according to (13.25), is equal to

\[ A\int \frac{n'}{kT}\,\varepsilon\, \frac{\partial\varepsilon}{\partial\rho} \left(\frac{\partial T}{\partial S}-\frac{S}{T}\right)d\tau_p = -AS\left(\frac{\partial T}{\partial S}\right)_\rho \left(\frac{\partial S}{\partial\rho}\right)_T = AS\left(\frac{\partial T}{\partial\rho}\right)_S . \]

Thus the sum of the first and second integrals in (13.23) is equal to zero; consequently

\[ \zeta_1=\zeta_4 . \tag{13.26} \]

Finally, let us clarify the form of the additional terms in the energy flux (13.17) that arise when equilibrium is disturbed. We first transform the second term in expression (12.23) for the flux \(Q\), the term \(nH\,\dfrac{\partial}{\partial\rho}H\). Expand the distribution function \(n\) in a series in powers of \((n-n_0)\). The zero-order term, as already indicated, is then reduced to zero owing to the symmetry of the subintegral expression in \(p\)-space. The first-order term, to which we shall restrict ourselves,

is equal to

\[ -\frac{1}{kT}\int n'\left(\frac{H}{kT}\right)({\bf p}{\bf v}_n)H\frac{\partial}{\partial p}H\,d\tau_{\bf p}. \]

Further, after integration by parts we obtain

\[ \overline{nH\frac{\partial}{\partial p}H} = \int n\left[Hv_n+({\bf p}{\bf v}_n)\frac{\partial H}{\partial p}\right]d\tau_{\bf p}. \tag{13.27} \]

In the desired first approximation the additional terms in the energy flux, according to (12.23) and (13.27), are equal to

\[ \overline{(np+\rho v_s)}\int (n-n_0)\frac{\partial\varepsilon}{\partial\rho}\,d\tau_{\bf p} + \int (n-n_0)\left[Hv_n+({\bf p}{\bf v}_n)\frac{\partial H}{\partial p}\right]d\tau_{\bf p}. \]

Substituting here expression (13.9) for \((n-n_0)\); taking into account the symmetry properties of the integrals obtained, as before we find

\[ \begin{aligned} &\overline{(n_0p+\rho v_s)} \left\{ \operatorname{div}({\bf j}-\rho{\bf v}_n) \int A\left(\frac{\varepsilon}{T}\frac{\partial T}{\partial\rho} -\frac{\partial\varepsilon}{\partial\rho}\right) \frac{\partial\varepsilon}{\partial\rho}\frac{n'}{kT}\,d\tau_{\bf p} \right.\\ &\quad \left. +\operatorname{div}{\bf v}_n \int A\left(\frac{\varepsilon}{T}\frac{\partial T}{\partial\rho}\rho +\frac{\varepsilon}{T}\frac{\partial T}{\partial S}S -\frac{\partial\varepsilon}{\partial\rho}\rho -\frac{1}{3}\frac{\partial\varepsilon}{\partial p}p\right) \frac{\partial\varepsilon}{\partial\rho}\frac{n'}{kT}\,d\tau_{\bf p} \right\}\\ &\quad +{\bf v}_n \left\{ \operatorname{div}({\bf j}-\rho{\bf v}_n) \int A\left(\frac{\varepsilon}{T}\frac{\partial T}{\partial\rho} -\frac{\partial\varepsilon}{\partial\rho}\right) \left(\varepsilon+\frac{1}{3}p\frac{\partial\varepsilon}{\partial p}\right) \frac{n'}{kT}\,d\tau_{\bf p} \right.\\ &\quad +\operatorname{div}{\bf v}_n \int A\left(\frac{\varepsilon}{T}\frac{\partial T}{\partial\rho}\rho +\frac{\partial T}{\partial S}\frac{\varepsilon}{T}S -\frac{\partial\varepsilon}{\partial\rho}\rho -\frac{1}{3}\frac{\partial\varepsilon}{\partial p}p\right) \frac{n'}{kT} \left(\varepsilon+\frac{1}{3}\frac{\partial\varepsilon}{\partial p}p\right)d\tau_{\bf p}\\ &\quad +\frac{\nabla T}{3T} \int B\varepsilon\frac{\partial\varepsilon}{\partial p} \left(p\frac{ST}{\rho n}-\varepsilon\frac{\partial\varepsilon}{\partial p}\right) \frac{n'}{kT}\,d\tau_{\bf p}\\ &\quad \left. +\frac{1}{2}\int C({\bf p}{\bf v}_n)\frac{\partial H}{\partial p} \left(p_k\frac{\partial\varepsilon}{\partial p_i} -\frac{1}{3}\delta_{ik}\frac{\partial\varepsilon}{\partial p}p\right) \frac{n'}{kT}\,d\tau_{\bf p} \right\}\\ &\quad\times \left(\frac{\partial v_{nk}}{\partial r_i} +\frac{\partial v_{ni}}{\partial r_k} -\frac{2}{3}\delta_{ik}\frac{\partial v_{kl}}{\partial r_l}\right). \end{aligned} \tag{13.28} \]

Let us now take into account the condition \(\int \varepsilon(n-n_0)d\tau_{\bf p}=0\) and the definition of the coefficients of second viscosity (relations (13.15), (13.16), (13.21) and (13.22)) and of the coefficient of first viscosity (13.17). Finally we obtain the expression for the additional nonequilibrium terms in the \(i\)-component of the energy flux in the following form:

\[ (j_i-\rho v_{ni})[\zeta_3\operatorname{div}({\bf j}-\rho{\bf v}_n)+\zeta_4\operatorname{div}{\bf v}_n] +v_{ni}[\zeta_1\operatorname{div}({\bf j}-\rho{\bf v}_n)+\zeta_2\operatorname{div}{\bf v}_n] +\eta v_{nk}\left(\frac{\partial v_{nk}}{\partial r_i} +\frac{\partial v_{ni}}{\partial r_k} -\frac{2}{3}\delta_{ik}\frac{\partial v_{kl}}{\partial r_l}\right) +\chi\frac{\partial T}{\partial r_i}. \tag{13.29} \]

The first three terms in expression (13.29) are due to first and second viscosities. The fourth term, however, determines the additional energy flux in helium II arising in the presence of a tempera-

temperature \(\nabla T\). The coefficient \(\varkappa\)—the analogue of the coefficient of thermal conductivity—is, according to (13.28), equal to

\[ \varkappa=-\frac{1}{3kT^2}\int B n' \varepsilon \frac{\partial \varepsilon}{\partial p} \left(\mathbf p\,\frac{ST}{\rho_n}-\varepsilon\,\frac{\partial \varepsilon}{\partial p}\right)d\tau_{\mathbf p}. \tag{13.30} \]

14. Thermal Conductivity of Helium II\(^{30,28}\)

In the presence of a temperature gradient in helium II, macroscopic motion arises. The velocities of the normal and superfluid motions that arise in this case are determined by the equations of motion. The normal motion in helium II is nothing other than a macroscopic heat flow. Thus, a temperature gradient in helium II produces a macroscopic heat flow. However, this is not the whole matter. According to (13.29), in this case there also arises a certain additional flux, the magnitude of which is proportional to \(\nabla T\). This phenomenon is analogous to thermal conductivity in ordinary liquids. It should be noted that, although in outward form the expression for the flux (13.29) coincides with the usual expression for the heat flux due to thermal conductivity, there is nevertheless an essential difference between these phenomena. It is easy to see that in a purely phonon gas the coefficient \(\varkappa\), defined by relation (13.30), is identically equal to zero. Indeed, in this case

\[ S_{\mathrm{ф}}=\frac{4}{3}\,\frac{E_{\mathrm{ф}}}{T},\qquad \rho_{n\mathrm{ф}}=\frac{4}{3}\,\frac{E_{\mathrm{ф}}}{c^2},\qquad \varepsilon=cp. \]

Substitution of these values into (13.30) makes the factor under the integral,

\[ \mathbf p\,\frac{ST}{\rho_n}-\varepsilon\,\frac{\partial \varepsilon}{\partial p}, \]

equal to zero. Only the presence in helium II of excitations of another type—rotons—ensures a nonzero value of the coefficient \(\varkappa\).

This circumstance indicates the distinctive character of the effect under consideration, which can be called thermal conductivity only conditionally. The coefficient \(\varkappa\), which we shall call the coefficient of thermal conductivity, is composed of two parts: a part due to rotons, \(\varkappa_p\), and a part due to phonons, \(\varkappa_{\mathrm{ф}}\):

\[ \varkappa=\varkappa_p+\varkappa_{\mathrm{ф}}. \tag{14.1} \]

Roton part of the coefficient of thermal conductivity \((\varkappa_p)\)

The change in the number of rotons in a certain phase volume, if the processes of their emission and absorption (§ 10) are neglected, will occur by two routes: elastic scattering of rotons by rotons (§ 9); scattering of rotons by phonons (§ 8).

However, as simple calculations show, the process of roton scattering by phonons will be insignificant in the thermal conductivity in comparison with the elastic scattering of rotons by rotons, up to temperatures of the order of \(0.6\)—\(0.7^\circ\mathrm{K}\).

If one also takes into account the circumstance that the roton part of the thermal-conductivity coefficient at temperatures below \(1^\circ\mathrm{K}\) becomes negligibly small in comparison with the phonon part, then we arrive at the conclusion that in the kinetic equation it is necessary to take into account only the scattering of rotons by rotons. The collision integral \(I(n)\) for rotons cannot be calculated exactly, since the character of the interaction of rotons with one another is unknown. If, however, we confine ourselves to determining only the temperature dependence of the coefficient \(\chi_p\), then the problem is simplified. In this case the collision integral reduces to

\[ -\frac{n-n_0}{t}, \tag{14.2} \]

where \(t\) is some time which, in order of magnitude, is equal to the mean time \(t_p\) between roton collisions. The time \(t_p\) was calculated by us earlier under the assumption that the interaction of rotons has a \(\delta\)-function character (§ 9):

\[ t_p=\frac{h^4}{4P_0^4|V_0|^2N_p}. \tag{14.3} \]

The magnitude of the amplitude \(v_0\), which determines the interaction of rotons \(\bigl(V=V_0\delta(\mathbf r_1-\mathbf r_2)\bigr)\), can be found from experimental data on the viscosity of helium II. According to § 9,

\[ V_0=1.1\cdot10^{-38}\ \mathrm{erg}\,\mathrm{cm}^3. \tag{14.4} \]

More precisely, the magnitude of the time \(t\) can be found from experimental data. If this time is identified with the collision time of two rotons \(t_p\), then we find the above-mentioned value \(V_0\). Such an identification is possible in the case where the probability of scattering of rotons by rotons has a sharp maximum for scattering through small angles. We shall assume that this circumstance indeed occurs. (In what follows, we shall see that this assumption does not have an essential influence on the final result.) Then not only \(t=t_p\), but also the collision integral \(I(n)\) in the problem under consideration coincides with the collision integral in the problem of the viscosity of helium II. Thus,

\[ I(n)=-\frac{n-n_0}{t_p}. \tag{14.5} \]

Comparing (14.5) with (13.9), we note that the coefficient \(B\) introduced there is simply equal to \(-t_p\). Taking this into account, according to (13.30) we have

\[ \chi_p=t_p\,\frac{1}{3kT^2}\int n'\varepsilon\frac{\partial\varepsilon}{\partial p}\left(p\,\frac{ST}{\rho_n}-\varepsilon\frac{\partial\varepsilon}{\partial p}\right)\,d\tau_p. \tag{14.6} \]

The roton distribution is determined by the Boltzmann function

\[ n=e^{-\frac{\varepsilon}{kT}}; \tag{14.7} \]

the energy of a roton \(\varepsilon\) is equal to

\[ \varepsilon=\Delta+\frac{(p-P_0)^2}{2\mu}. \tag{14.8} \]

Integration of the first term in (14.6), taking (2.4) into account, gives

\[ -\frac{t_p}{3kT^2}\int n'\varepsilon^2\left(\frac{\partial \varepsilon}{\partial p}\right)p\,\frac{ST}{\rho_n}\,d\tau_p = \frac{STS_p}{\rho_n}\,t_p . \tag{14.9} \]

(\(S_p\) is the roton part of the entropy).

Integration of the second term in (14.6) with the distribution function (14.7) gives

\[ -\frac{t_p}{3kT^2}\int n'\varepsilon^2\left(\frac{\partial \varepsilon}{\partial p}\right)^2 d\tau_p = -\frac{\Delta^2 t_p N_p}{3\mu T}. \tag{14.10} \]

Thus, finally we have

\[ \chi_p= \frac{\Delta^2 t_p N_p}{3\mu T} \left\{ 1-\frac{3ST^2S_p\mu}{\rho_nN_p} \right\}. \tag{14.11} \]

Substitution of numerical values of the parameters into (14.9) and (14.10) shows that the second term in (14.6) appreciably exceeds the first at all temperatures. Therefore, with a sufficient degree of accuracy we have

\[ \chi_p= -t_p\frac{1}{3kT^2} \int n'\varepsilon^2\left(\frac{\partial \varepsilon}{\partial p}\right)^2d\tau_p = \frac{\Delta^2 t_p N_p}{3\mu T}. \tag{14.12} \]

The roton part of the coefficient of first viscosity, according to (§ 16), is equal to

\[ \eta_p=\frac{P_0^2t_pN_p}{15\mu}. \tag{14.13} \]

Comparing (14.12) with (14.13), we find a simple relation between \(\chi_p\) and \(\eta_p\)

\[ \chi_p=\frac{5\Delta^2}{P_0^2T}\,\eta_p. \tag{14.14} \]

The validity of relation (14.14) can also easily be verified by comparing (13.17) with (14.12).

The phonon part of the coefficient of thermal conductivity. The kinetic equation determining the phonon distribution function \(n\) in the presence of a temperature gradient \(\nabla T\), according to (13.7), has the form

\[ \frac{n'}{kT}\cos\vartheta\,\frac{\partial T}{\partial x} \frac{1}{T} \left[ p\frac{ST}{\rho_n} -\varepsilon\frac{\partial \varepsilon}{\partial p} \right] = I(n). \tag{14.15} \]

For simplicity we assume that the direction of \(\nabla T\) coincides with the \(x\)-axis. The angle \(\vartheta\) is formed by the momentum \(p\) of the phonon with the indicated axis.

The basic regularities characterizing the scattering of phonons in helium II were elucidated in Chapter II. It is necessary here to consider two temperature regions.

1. The temperature region above \(0.9^\circ\mathrm{K}\). In this temperature region, phonon scattering occurs mainly on rotons. Scattering of phonons by phonons with a change in the direction of the momenta of the colliding phonons has a small probability in comparison with the indicated process. Collisions of phonons with nearly parallel directions of momenta, however, have a large probability, but they are not accompanied by a change in the directions of the momenta of the colliding phonons. This process ensures the establishment of energy equilibrium between phonons moving in a given direction in times shorter than those characterizing the scattering of phonons by rotons. Thus phonons moving in each given direction may be characterized by their own temperature \(T'\), in general not equal to the mean temperature of the phonon gas. Finally, one must also take into account possible processes of inelastic scattering of phonons by phonons, accompanied by a change in their number. The greatest probability among processes of this type is possessed by the process of conversion of two phonons into three (the five-phonon process), characterized by times of the same order as those which determine the scattering of phonons by rotons. Taking all this into account, we come to the conclusion that the distribution function of phonons moving in a given direction can be represented in the form

\[ n=\left[e^{\alpha' + pc/kT'}-1\right]^{-1}, \tag{14.16} \]

where \(\alpha'\) is a certain function of the direction of the phonon momentum, analogous in some sense to the chemical potential.

The deviation of the value of the distribution function \(n\) from the equilibrium value \(n_0\), according to (14.15), is equal to

\[ n-n_0=-n_0(n_0+1)\left\{\alpha' - \frac{pc}{kT}\frac{T'-T}{T}\right\} \tag{14.17} \]

(\(T\) is the mean temperature of the phonon gas).

The collision integral \(I(n)\) in the right-hand side of (14.15) is composed of four parts: \(I_{\mathrm{I}}(n)\), due to the scattering of phonons by rotons (§ 8), \(I_{\mathrm{II}}(n)\), due to the scattering of phonons by phonons (§ 7), \(I_{\mathrm{II}}(n)\), due to the specific process of scattering of phonons by phonons through small angles (§ 11), and, finally, \(I_{\mathrm{IV}}(n)\), characterizing the change in the number of phonons occurring owing to the five-phonon process (§ 10).

The total energy of phonons moving in a given direction, and their number, are conserved in scattering through small angles in the four-phonon process. The five-phonon process, which likewise occurs mainly at small angles between the momenta of the colliding-

ing phonons does not change the energy of phonons moving in a given direction, but substantially changes their number.

Let us integrate the left- and right-hand sides of equation (14.15) over all phonons and over all possible energies. Since, according to what has been said, the integrals

\[ \int I_{\mathrm{III}}(n)\,p^{2}dp,\qquad \int I_{\mathrm{III}}(n)\,\varepsilon p^{2}dp \quad\text{and}\quad \int I_{\mathrm{IV}}(n)\,\varepsilon p^{2}dp \]

are equal to zero, we have

\[ \cos \vartheta\,\frac{\partial T}{\partial x} -\frac{1}{kT^{2}}\int n'\left(p\,\frac{ST}{\rho_n}-\varepsilon\,\frac{\partial\varepsilon}{\partial p}\right)p^{2}dp = \int (I_{\mathrm I}+I_{\mathrm{II}})\,p^{2}dp+ \int I_{\mathrm{IV}}p^{2}dp, \tag{14.18} \]

\[ \cos \vartheta\,\frac{\partial T}{\partial x} -\frac{1}{kT^{2}}\int n'\left(p\,\frac{ST}{\rho_n}-\varepsilon\,\frac{\partial\varepsilon}{\partial p}\right)\varepsilon p^{2}dp = \int (I_{\mathrm I}+I_{\mathrm{II}})\,\varepsilon p^{2}dp. \tag{14.19} \]

In the temperature range under consideration we neglect the magnitude of the integral \(I_{\mathrm{II}}(n)\) in comparison with \(I_{\mathrm I}(n)\). The value of the integral \(\int I_{\mathrm{IV}}p^{2}dp\) was calculated in (§ 10). According to (10.12) and (10.22) we have

\[ \frac{1}{kT}\int I_{\mathrm{IV}}(n)\,d\tau_p=\alpha\Gamma_\phi,\qquad \Gamma_\phi=1.0\cdot10^{43}\cdot T^{11}. \]

The left-hand side of the kinetic equation contains an angular function of the form \(\cos\vartheta\); therefore it is natural to represent the dependence of the parameters \(\alpha\) and \(\dfrac{T'-T}{T}\) on the angle in the same form

\[ \alpha'=\alpha\cos\vartheta,\qquad \frac{T'-T}{T}=\beta\cos\vartheta . \tag{14.20} \]

2. The temperature region below \(0.9^\circ\mathrm K\). In this temperature region, along with the scattering of phonons by rotons, the scattering of phonons by phonons becomes significant. In this case the five-phonon process turns out to be faster than the indicated scattering processes. Thus, phonons moving in a given direction are in equilibrium and therefore may be described simply by the Planck distribution function, depending on the temperature of the phonons in this direction,

\[ n=\left[e^{pc/kT}-1\right]^{-1}. \tag{14.21} \]

The deviation of the distribution function \(n\) from the equilibrium value in this case is determined by a single parameter \(T'-T\):

\[ n-n_0=n_0(n_0+1)\frac{pc}{kT}\frac{T'-T}{T}. \tag{14.21′} \]

The value of this parameter is found with the aid of equation (14.19). Equation (14.18) in this case becomes trivial and reduces simply to the condition \(\alpha = 0\). We shall now show that the integral \(I_{11}(n)\) is in fact also equal to zero. Indeed, we have

\[ I_{11}(n)=\int d\sigma \left(\mathbf p,\mathbf p_1,\mathbf p',\mathbf p'_1\right)\times \]

\[ \times \left\{ n n_1 (n' + 1)(n'_1 + 1)-n'n'_1(n+1)(n_1+1)\right\}\,d\tau_{\mathbf p_1}. \tag{14.22} \]

Here: \(\mathbf p,\mathbf p_1\) are the phonon momenta before scattering, \(\mathbf p',\mathbf p'_1\) are the phonon momenta after scattering; \(n,n_1,n',n'_1\) are the corresponding distribution functions; \(d\sigma\) is the differential effective cross section for the scattering of a phonon with momentum \(\mathbf p_1\) by a phonon with momentum \(\mathbf p\). Substitute into (14.22) the distribution functions expressed in the form of sums of equilibrium values and additions (14.21′). Since the left-hand side of the kinetic equation (14.15) contains the angular function only in the form of the factor \(\cos\vartheta\), it is natural that the quantity \(T' - T\) in a given direction depends on the angle \(\vartheta\), formed by the direction of motion of the phonon with the \(x\)-axis, in the same way, i.e.,

\[ T' - T = \beta \cos \vartheta . \tag{14.23} \]

Taking this into account and retaining only terms linear in \(\beta\), we obtain

\[ I_{11}(n)=\int d\sigma n_0 n_{01}(n'_0+1)(n'_{01}+1)\frac{\beta c}{kT}\times \]

\[ \times \left\{ p\cos\vartheta + p\cos\vartheta_1 - p'\cos\vartheta' - p'_1\cos\vartheta'_1 \right\} d\tau_{\mathbf p_1}. \tag{14.24} \]

The subscript \(0\) denotes equilibrium distribution functions. The angles \(\vartheta,\vartheta_1,\vartheta',\vartheta'_1\) are formed by the corresponding phonon momenta with the \(x\)-axis. The expression standing in braces in (14.24), according to the law of conservation of momentum, is equal to zero. Thus \(I_{11}(n)=0\), and the assertion is proved.

Recalling what was said earlier about the relative magnitude of \(I_{11}\) and \(I_1\) in the temperature region above \(0.9^\circ\mathrm K\), we come to the conclusion that at all temperatures the thermal conductivity is determined by the process of scattering of phonons by rotons.

After all that has been said, let us proceed to calculate the collision integral \(I_1(n)\); we have:

\[ I_1(n)=(2\pi\hbar)^3 \iint c\,d\sigma(p,\psi)\left\{Nn(n'+1)-n'N'(n+1)\right\}d\mathbf P . \tag{14.25} \]

Here \(N\) and \(N'\) are the roton distribution functions before and after scattering; \(n\) and \(n'\) are the phonon distribution functions before and after scattering; \(\mathbf P\) is the momentum of the incident roton; \(d\sigma(p,\psi)\) is the differential effective cross section for the scattering of a phonon with momentum \(p\) by a roton,

for which the direction of motion of the phonon changes by an angle \(\psi\), equal, according to (8.20), to

\[ d\sigma= \left(\frac{P_0p^3}{4\pi h^2\rho_0c}\right)^2 \left\{ \frac{2}{3}(1+\cos\psi)\cos^2\psi+ \frac{1}{105}\left(\frac{P_0}{\mu c}\right)^2 \left(1+8\cos^2\psi+\frac{8}{3}\cos^4\psi\right) +\frac{2A}{15}\frac{P_0}{\mu c}(1+2\cos^2\psi)+A^2 \right\}d o, \tag{14.26} \]

\(d o\) is an element of solid angle,

\[ A=\frac{\rho^2}{P_0c} \left[ \frac{\partial^2\Delta}{\partial \rho^2} +\frac{1}{\mu}\left(\frac{dP_0}{d\rho}\right)^2 \right]. \]

In writing the curly bracket in (14.25) we have taken into account the circumstance that the rotons obey classical statistics, while the phonons obey Bose statistics. The distribution of phonons and rotons in the presence of the temperature gradient \(\nabla T\) is characterized by nonequilibrium functions. However, owing to the relative rapidity of the scattering process of a roton by a roton, the deviations of their distribution functions from the equilibrium values will be considerably smaller than the corresponding deviations for phonons. Therefore in the collision integral \(I_1(n)\) the distribution functions of the rotons may be replaced by their equilibrium values. For the phonons, however, one must substitute in (14.25) the values of the distribution functions (14.27) or (14.21). Let us first consider the case \(T>0.9^\circ\mathrm{K}\). After substitution into the functions (14.17) and (14.25) and integration over the phase volume of the rotons, we obtain

\[ I_1(n)= \]

\[ =N_\rho\int n_0(n_0+1)c\,d\sigma(p,\psi) \left(\alpha-\frac{\beta}{kT}pc\right) (\cos\vartheta-\cos\vartheta') \tag{14.27} \]

(\(N_\rho\) is the number of rotons per unit volume). We express the angle \(\vartheta'\) in terms of the angle \(\vartheta\) and the angle \(\psi\):

\[ \cos\vartheta'=\cos\vartheta\cos\psi+\sin\vartheta\sin\psi\cos\varphi . \]

Further, in an elementary way we find

\[ \int f(\psi)\cos\vartheta'\,d o_\psi = \cos\vartheta\int \cos\psi f(\psi)\,d o_\psi \tag{14.27'} \]

(\(f(\psi)\) is some function of the angle \(\psi\)).

The integral (14.27) contains integration over the solid angle. Taking (14.27′) into account, we rewrite \(I_1(n)\) in the form

\[ I_1(n)= \cos\vartheta\,N_\rho c \int d\sigma(p,\psi)(1-\cos\psi)\times \]

\[ \times \left(\alpha-\beta\frac{pc}{kT}\right)n_0(n_0+1). \tag{14.28} \]

Hence, after integration over the solid angle \(d o_\psi\), we finally find

\[ I_1(n)=n_0(n_0+1)\left(\alpha-\beta\frac{pc}{kT}\right)\left(\frac{pc}{kT}\right)^4\frac{1}{6\theta}\cos\vartheta, \tag{14.29} \]

where \(\theta\) is a quantity having the dimension of time,

\[ \frac{1}{\theta} = \frac{6!N_{\mathrm p}}{4\pi c} \left[ \frac{P_0(kT/c)^2}{h^2\rho} \right]^2 \left\{ \frac{4}{45} + \frac{1}{25}\left(\frac{P_0}{\mu c}\right)^2 + \frac{4}{9}\frac{P_0}{\mu c}A + A^2 \right\}. \tag{14.30} \]

Substituting (14.29) into equations (14.18) and (14.19), after a simple integration we obtain two linear equations

\[ \frac{\partial T}{\partial x}\frac{4\pi^4}{15} \left(c^2-\frac{ST}{\rho_n}\right) = \frac{7}{\theta}(\alpha-8\beta), \]

\[ \frac{\partial T}{\partial x}\frac{36}{5} \left(c^2-\frac{ST}{\rho_n}\right) = \frac{1}{\theta}(\alpha-7\beta)+\frac{1}{\theta_\phi}\alpha. \tag{14.31} \]

The quantity \(\theta_\phi\), having the dimension of time, characterizes the rate of change of the number of phonons occurring because of the five-phonon process. The time \(\theta_\phi\) is related to the coefficient \(\Gamma_{\mathrm p}\), calculated in § 10, by the relation

\[ \frac{1}{\theta_\phi} = \frac{(2\pi\hbar)^3}{4\pi(kT/c)^3}\,kT\Gamma_\phi = 1.95\cdot10^8T^9. \tag{14.31'} \]

Solving equations (14.31) with respect to \(\alpha\) and \(\beta\) and substituting the values obtained into (14.17), we find

\[ n-n_0= \]

\[ = n'\cos\vartheta\,\frac{1}{T}\frac{\partial T}{\partial x}\, \theta\left(c^2-\frac{ST}{\rho_n}\right) \frac{31.2-\dfrac{pc}{kT}\left(3.5-3.7\theta/\theta_\phi\right)} {1+8\theta/\theta_\phi}. \tag{14.32} \]

With the aid of the expression obtained for the nonequilibrium function, we calculate the heat flux that arises,

\[ Q=\int\frac{\partial\varepsilon}{\partial p}\,p(n-n_0)\cos\vartheta\,d\tau_{\mathrm p} = \]

\[ = -19\,\frac{\partial T}{\partial x}\, \theta N_\phi k \left(c^2-\frac{ST}{\rho_n}\right) \frac{1+0.75\theta/\theta_\phi}{1+8\theta/\theta_\phi}. \tag{14.33} \]

Finally, we find the required expression for the thermal-conductivity coefficient

\[ \varkappa_\phi = 19\left(c^2-\frac{ST}{\rho_n}\right) \theta N_\phi k \frac{1+0.75\theta/\theta_\phi}{1+8\theta/\theta_\phi}. \tag{14.34} \]

The number of phonons per unit volume of the phonon gas \(N_\phi\) is equal to

\[ N_\phi\simeq 2.4\cdot4\pi\left(\frac{kT}{2\pi\hbar c}\right)^3. \tag{14.35} \]

As a result of analogous calculations for the temperature range \(T<0.9^\circ\mathrm K\), we find an equation determining the value of the single-

… in this case of the parameter \(\beta\)

\[ \frac{1}{T}\frac{\partial T}{\partial x}\frac{4\pi^4}{15}\left(c^2-\frac{ST}{\rho_n}\right)=\frac{56}{\theta}\,\beta . \tag{14.36} \]

Next we find the expression for the phonon distribution function

\[ n-n_0=n'\frac{1}{T}\frac{\partial T}{\partial x}\cos\vartheta\,\frac{pc}{kT}\, \frac{4\pi^4}{15\cdot 56}\left(c^2-\frac{ST}{\rho_n}\right). \tag{14.37} \]

With the aid of (14.37), we finally obtain for the thermal-conductivity coefficient the expression

\[ \chi_{\phi}=1.8\,\theta N_{\phi}k\left(c^2-\frac{ST}{\rho_n}\right). \tag{14.38} \]

Substitution of the numerical values of the parameters shows that the term \(\dfrac{ST}{\rho_n}\) in the parentheses in (14.34) and (14.38) may be neglected at all temperatures above \(0.8^\circ\mathrm{K}\). Summing the results (14.12), (14.34), and (14.38), for the total thermal-conductivity coefficient we have

\[ \chi= \frac{h^4\Delta^2}{12P_{0t}h\,|V_0|^2} + \theta N_{\phi}k\left(c^2-\frac{ST}{\rho_n}\right) \begin{cases} \displaystyle 19\,\frac{1+0.750/\theta_{\phi}}{1+8\theta/\theta_{\phi}} & \text{(for } T>0.9^\circ\mathrm{K}),\\[1.2ex] 1.8 & \text{(for } T<0.9^\circ\mathrm{K}). \end{cases} \tag{14.39} \]

After substituting the numerical values of the parameters into (14.39), we find

\[ \chi= 2\cdot 10^3\frac{1}{T} + T^{-3/2}e^{\Delta/T} \left(1-\frac{ST}{\rho_n c^2}\right) \begin{cases} \displaystyle 84\,\frac{1+0.750/\theta_{\phi}}{1+8\theta/\theta_{\phi}} & (T>0.9^\circ\mathrm{K}),\\[1.2ex] 7.8 & (T<0.9^\circ\mathrm{K}). \end{cases} \tag{14.40} \]

Thus, the expression for the thermal-conductivity coefficient of helium II consists of two parts: one due to rotons and increasing as the temperature decreases according to the law \(\dfrac{1}{T}\), and a second, due to phonons and increasing as the temperature decreases according to the stronger exponential law \(e^{\Delta/T}\). The phonon and roton parts of the coefficient are approximately equal in magnitude near the \(\lambda\)-point, and then, as the temperature is lowered, \(\chi_{\phi}\) noticeably exceeds \(\chi_p\) (below \(1.4^\circ\mathrm{K}\)). The temperature dependence of the thermal-conductivity coefficient \(\chi\) is shown graphically in Fig. 6. In the same figure are given the experimental values obtained by K. Zinov’eva\({}^{37}\) from data on the absorption of second sound in helium II.

As the temperature is lowered, the time \(\theta\), determined by formula (4.30), rapidly increases owing to the exponential decrease of the number

rotons \(N\). This circumstance nevertheless does not manifest itself strongly in formula (14.39), since the factor contained in the same formula,

\[ c^2-\frac{ST}{\rho_n}, \]

for small values of the number of rotons turns out to be proportional to this number \(N_p\). Expanding the expression

\[ c^2-\frac{ST}{\rho_n} \]

in a series for small values of \(N_p\), we obtain

\[ c^2-\frac{ST}{\rho_n} = c^2\left(\frac{S_p}{S_\phi}-\frac{\rho_{np}}{\rho_{n\phi}}\right) \simeq \frac{P_0^2 c^4 N_p}{4kT E_\phi}. \tag{14.41} \]

With the aid of (14.41) we find the limiting value of \(\chi\) for small \(T\):

\[ \chi=\frac{2\pi^5(\varkappa c^2)^2(hc)^4}{3\cdot 15\cdot 7!(kT)^5TD}, \tag{14.42} \]

\[ D=\frac{4}{45}+\frac{1}{25}\left(\frac{P_0}{\mu c}\right)^2 +\frac{4}{9}\left(\frac{P_0}{\mu c}\right)A+A^2\simeq 0.84. \]

Fig. 6. Temperature dependence of the thermal-conductivity coefficient of helium II \(\chi\) \((\mathrm{erg}/\mathrm{cm}\cdot\mathrm{sec}\cdot\mathrm{degree})\):
— theoretical values, ○ — experimental values of Zinov'eva\({}^{37}\).

An interesting result has been obtained. It was indicated above that in a purely phonon gas the coefficient \(\chi\) is identically equal to zero. It is enough, however, to have a small number of rotons in order to make its value nonzero.

Let us note that the formula determining the value of \(\chi\) for \(T>0.9^\circ\mathrm{K}\) goes over into the formula determining the value of \(\chi\) in the region \(T<0.9^\circ\mathrm{K}\), if the ratio \(\theta/\theta_\phi\) is made to tend to infinity. This circumstance is in complete agreement with the assumption made about the extreme rapidity of the five-phonon process in the temperature region below \(0.9^\circ\mathrm{K}\).

In the calculations we use the following values of the parameters:

\[ \Delta=8.9^\circ\mathrm{K},\qquad P_0=2.1\cdot 10^{-19}\ \mathrm{g}\cdot\mathrm{cm}\,\mathrm{sec}^{-1},\qquad \mu=1.72\cdot 10^{-24}\ \mathrm{g}, \]

\[ A=-0.66. \]

It should be said here that, owing to the inadequacy of the experimental data from which they were obtained, the indicated parameter values are extremely rough. It is therefore necessary to bear in mind the possibility of a noticeable change of these values in connection

with the appearance of new experimental data and, consequently, the possibility of changing the calculated values of the coefficient \(\chi\). In this connection, the value of the ratio of the phonon parts of the viscosity coefficient \(\eta_\phi\) and the thermal-conductivity coefficient, which depends only weakly on the choice of parameters, is of interest. The phonon part of the viscosity coefficient \(\eta_\phi\), with allowance for the five-phonon process, is equal to (§ 16)

\[ \eta_\phi = 3.8 N_\phi kT\bar{\theta}\, \frac{1+0.750/\theta_\phi}{1+8\theta/\theta_\phi}. \]

The time \(\bar{\theta}\) is then determined by the formula

\[ \frac{1}{\bar{\theta}} = \frac{N_\rho b!}{4\pi c} \left[ \frac{P_0(kT/c)^2T^2}{h^3\rho} \right] \left\{ \frac{2}{15} + \frac{33}{35^2} \left( \frac{P_0}{\mu c} \right)^2 + \frac{14}{75} \left( \frac{P_0}{\mu c} \right) A + A^2 \right\}. \]

The indicated value of the time \(\bar{\theta}\) differs little from the time \(\theta\) determined by formula (3.10). Therefore the approximate relation holds (\(T > 0.9^\circ\mathrm{K}\)):

\[ \chi_\phi/\eta_\phi \simeq 5c^2/T. \]

\[ \left( \text{The quantity } \frac{ST}{\rho n} \text{ in the temperature range under consideration may be neglected in comparison with } c^2. \right) \]

One should also allow for the possibility of changing the value of the numerical coefficient in the formula determining the magnitude of \(\theta_\phi\), since the quoted value of this coefficient was found from the experimental values of the coefficient of sound absorption in helium II in a very small temperature range, close to the \(\lambda\)-point, where all phenomena may be complicated by the proximity of the \(\lambda\)-point. Moreover, in calculating this coefficient, very rough values of the derivatives \(\partial P_0/\partial \rho\), \(\partial \lambda/\partial \rho\), and \(\partial c/\partial \rho\) are used. A change in the value of this coefficient substantially affects the value of \(\chi_\phi\).

The mean free path of phonons characterizing thermal conductivity. Let us calculate the mean free path characterizing thermal conductivity. The parameters \(\alpha\) and \(\beta\), which determine the deviation of the distribution function from its equilibrium value, substantially characterize the value of the thermal-conductivity coefficient. If the values of these parameters are changed simultaneously, the thermal-conductivity coefficient changes by the same factor. We shall regard the parameters \(\alpha\) and \(\beta\) as depending on the time \(t\). Then, completely analogously to equations (14.31), one can obtain two equations:

\[ \frac{\pi^2}{3}\frac{d\alpha}{dt} - \frac{36}{5}\frac{d\beta}{dt} = -\frac{1}{\theta}(\alpha-7\beta) - \frac{1}{\theta_\phi}\alpha, \]

\[ \frac{36}{5}\frac{d\alpha}{dt} - \frac{4\pi^4}{15}\frac{d\beta}{dt} = -\frac{7}{\theta}(\alpha-8\beta), \tag{14.43} \]

determining the law of variation of the indicated parameters in the temperature range above \(0.9^\circ\) K. We seek the solution of this system of linear differential equations in a form proportional to \(e^{-t/\tau}\). It is natural to choose the quantity \(\tau\) as the characteristic time determining the thermal conductivity. The corresponding phonon mean free path \(\lambda_{\phi}\) is obtained by multiplying \(\tau\) by the phonon velocity \(c\). To determine \(\tau\) from system (14.43) we obtain a quadratic characteristic equation, whose roots are equal to

\[ \tau \simeq 16\theta\,\frac{1+\theta/4\theta_{\phi}}{1+8\theta/\theta_{\phi}} \quad \text{or} \quad \frac{\theta}{3}\,\frac{1}{1+8\theta/\theta_{\phi}} . \tag{14.44} \]

For purposes of comparison in a macroscopic consideration of the phenomenon of thermal conductivity, one should take the larger of these roots. Taking the larger of the obtained values of \(\tau\), for the phonon mean free path we obtain, with the aid of (14.44) and (14.30), the expression

\[ \frac{1}{\lambda_{\phi}} = \frac{1}{c\tau} = \frac{45N_p}{4\pi c^2} \left[ \frac{P_0(kT/c)^3}{h^2\rho} \right]^2 \times \]

\[ \times \left\{ \frac{4}{45} + \frac{1}{25}\left(\frac{P_0}{\mu c}\right)^2 + \frac{4}{9}\frac{P_0}{\mu c}\,A + A^2 \right\} \frac{1+8\theta/\theta_{\phi}}{1+\theta/4\theta_{\phi}} . \tag{14.45} \]

To determine the characteristic time \(\tau\) in the temperature range below \(0.9^\circ\) K, analogously to (14.43) we obtain the linear differential equation

\[ \frac{4\pi^4}{15}\,\frac{d\beta}{dt} = -\beta\,\frac{56}{\theta}. \tag{14.46} \]

In this case \(\lambda_{\phi}\) is determined by the expression

\[ \frac{1}{\lambda_{\phi}} = \frac{15\cdot 56}{4\pi^4 c}\cdot \frac{1}{\theta} \simeq 1500\,\frac{N_p}{4\pi c^2} \left[ \frac{P_0(kT/c)^3}{h^2\rho} \right]^2 \times \]

\[ \times \left\{ \frac{4}{45} + \frac{1}{25}\left(\frac{P_0}{\mu c}\right)^2 + \frac{4}{9}\left(\frac{P_0}{\mu c}\right)A + A^2 \right\}. \tag{14.47} \]

Substituting into formulas (14.45) and (14.47) the numerical values of all the parameters, we finally obtain

\[ \frac{1}{\lambda_{\phi}} = 7.85\cdot 10^5 T^9 e^{-\Delta/T} \begin{cases} \dfrac{1+8\theta/\theta_{\phi}}{1+\theta/4\theta_{\phi}},\\[6pt] 32. \end{cases} \tag{14.48} \]

In Fig. 7 the dependence of \(1/\lambda_{\phi}\) on temperature is represented graphically. From (14.43) it is easy to see that the phonon path which characterizes the value of the thermal-conductivity coefficient is determined by phonons with energy of the order of \(7\)–\(8\,kT\). In § 11 the time \(t_{\phi}\) was calculated, characterizing the establishment of energy equilibrium in the phonon gas. Comparing the times \(t_{\phi}\) (formula (11.15)) and \(\tau\) for phonons possessing such energy, we are convinced that at all tem-

at temperatures below the \(\lambda\)-point the establishment of energy equilibrium occurs faster than the phonon-scattering processes that determine the thermal conductivity.

15. Coefficients of the second viscosity of helium II\(^ {21}\)

The equations of motion of helium II, taking account of second viscosity, according to (13.14) and (13.20), are written in the form

\[ \frac{\partial \mathbf{j}}{\partial t} +\nabla p +\mathbf{v}_s \operatorname{div}\mathbf{j} +\left(\mathbf{j}-\rho\mathbf{v}_s\right)\operatorname{div}\mathbf{v}_n +\left(\mathbf{j}\nabla\right)\mathbf{v}_s +\left(\mathbf{v}_n\nabla\right)\left(\mathbf{j}-\rho\mathbf{v}_n\right) = \nabla\{\zeta_1\operatorname{div}(\mathbf{j}-\rho\mathbf{v}_n)+\zeta_2\operatorname{div}\mathbf{v}_n\}, \tag{15.1} \]

\[ \frac{\partial\mathbf{v}_s}{\partial t} +\nabla\left(\mu+\frac{v_s^2}{2}\right) = \nabla\{\zeta_3\operatorname{div}(\mathbf{j}-\rho\mathbf{v}_n)+\zeta_4\operatorname{div}\mathbf{v}_n\}. \tag{15.2} \]

The unusual character of the hydrodynamics of helium II leads to the appearance of four second viscosity coefficients. According to the Onsager principle, the coefficients \(\zeta_1\) and \(\zeta_4\) are equal. Let us determine the dependence of the second viscosity coefficients on temperature and on other thermodynamic quantities. In doing so, we proceed from the fact that the second viscosity in helium II is caused by processes connected with changes in the total number of phonons \(N_{\phi}\) and rotons \(N_p\).

Let \(N_p\) and \(N_{\phi}\) be the numbers, respectively, of rotons and phonons in a unit volume of helium II, and let \(\mu_p\) and \(\mu_{\phi}\) be their chemical potentials. In the equilibrium state, when \(\mu_p\) and \(\mu_{\phi}\) are equal to zero, the numbers of rotons and phonons are functions of the density \(\rho\) and the entropy \(S\) of helium and are equal, respectively, to \(N_{p0}\) and \(N_{\phi0}\). In a system removed from the equilibrium state, the numbers \(N_p\) and \(N_{\phi}\) will change with time, tending to assume the equilibrium values \(N_{p0}\) and \(N_{\phi0}\). Consider small deviations from the equilibrium state, for which the density and entropy differ little from the constant equilibrium values. Without loss of generality we shall also assume small velo-

Fig. 7. Temperature dependence of the mean free path of phonons.

velocities \(\mathbf{v}_n\) and \(\mathbf{v}_s\). The equations characterizing the approach of the system to the equilibrium state can be written by expanding the rates of change of the numbers of particles \(\dot N_p\) and \(\dot N_\phi\) in powers of the chemical potentials. If one restricts oneself to terms linear in \(\mu_p\) and \(\mu_\phi\), then the indicated equations take the form

\[ \dot N_p+\operatorname{div} N_p\mathbf{v}_n=-\gamma_{pp}\mu_p+\gamma_{p\phi}\mu_\phi, \tag{15.3} \]

\[ \dot N_\phi+\operatorname{div} N_\phi\mathbf{v}_n=\gamma_{\phi p}\mu_p-\gamma_{\phi\phi}\mu_\phi, \tag{15.4} \]

where \(\gamma_{pp}\), \(\gamma_{p\phi}\), \(\gamma_{\phi p}\), and \(\gamma_{\phi\phi}\) are kinetic coefficients, symmetric with respect to the indices \(p\) and \(\phi\). The terms of the form \(\operatorname{div} N\mathbf{v}_n\) in equations (15.3) and (15.4) take into account the fact that phonons and rotons participate in the motion of the normal part of helium II with velocity \(\mathbf{v}_n\). Neglecting quadratic effects in equations (15.3) and (15.4), we obtain

\[ \dot N_p+N_p\operatorname{div}\mathbf{v}_n=-\gamma_{pp}\mu_p+\gamma_{p\phi}\mu_\phi, \tag{15.5} \]

\[ \dot N_\phi+N_\phi\operatorname{div}\mathbf{v}_n=\gamma_{\phi p}\mu_p-\gamma_{\phi\phi}\mu_\phi. \tag{15.6} \]

The change of the thermodynamic quantities \(\rho\), \(S\) with time is determined by the continuity equations for density and entropy, which in the case under consideration, similarly to equations (15.5) and (15.6), after linearization acquire the form

\[ \left. \begin{aligned} \dot\rho+\operatorname{div}\mathbf{j}&=0,\\ \dot S+S\operatorname{div}\mathbf{v}_n&=0. \end{aligned} \right\} \tag{15.7} \]

The numbers of rotons \(N_p\) and phonons \(N_\phi\) depend on three variables—\(\rho\), \(S\), \(\mu_p\), or \(\mu_\phi\). For sufficiently slow processes, the deviations from the equilibrium state of the indicated numbers of rotons and phonons have time to follow the change of the thermodynamic quantities. In this case the derivatives \(\dot N_p\) and \(\dot N_\phi\) are expressed through the derivatives \(\dot\rho\) and \(\dot S\)

\[ \left. \begin{aligned} \dot N_p&=\frac{\partial N_p}{\partial\rho}\dot\rho+\frac{\partial N_p}{\partial S}\dot S =-\frac{\partial N_p}{\partial\rho}\operatorname{div}\mathbf{j} -\frac{\partial N_p}{\partial S}S\operatorname{div}\mathbf{v}_n,\\ \dot N_\phi&=\frac{\partial N_\phi}{\partial\rho}\dot\rho+\frac{\partial N_\phi}{\partial S}\dot S =-\frac{\partial N_\phi}{\partial\rho}\operatorname{div}\mathbf{j} -\frac{\partial N_\phi}{\partial S}S\operatorname{div}\mathbf{v}_n. \end{aligned} \right\} \tag{15.8} \]

Taking into account (15.7) and (15.8), we rewrite equations (15.5) and (15.6) in the form

\[ -\frac{\partial N_p}{\partial\rho}\operatorname{div}(\mathbf{j}-\rho\mathbf{v}_n) +\left( N_p-\frac{\partial N_p}{\partial S}S-\frac{\partial N_p}{\partial\rho}\rho \right)\operatorname{div}\mathbf{v}_n = -\gamma_{pp}\mu_p+\gamma_{p\phi}\mu_\phi, \tag{15.9} \]

\[ -\frac{\partial N_\phi}{\partial\rho}\operatorname{div}(\mathbf{j}-\rho\mathbf{v}_n) +\left( N_\phi-\frac{\partial N_\phi}{\partial S}S-\frac{\partial N_\phi}{\partial\rho}\rho \right)\operatorname{div}\mathbf{v}_n = \gamma_{\phi p}\mu_p-\gamma_{\phi\phi}\mu_\phi. \tag{15.10} \]

The obtained equations (15.9) and (15.10) determine the values of the chemical potentials \(\mu_{\mathrm p}\) and \(\mu_{\phi}\) through \(\operatorname{div}(\mathbf j-\rho \mathbf v_n)\) and \(\operatorname{div}\mathbf v_n\) for the case of slow nonequilibrium processes in helium

\[ \mu_{\mathrm p} = \frac{1}{\gamma_{\phi\mathrm p}^{2}-\gamma_{\phi\phi}\gamma_{\mathrm{pp}}} \left\{ -\operatorname{div}(\mathbf j-\rho \mathbf v_n) \left[ \frac{\partial N_{\mathrm p}}{\partial \rho}\gamma_{\phi\phi} + \frac{\partial N_{\phi}}{\partial \rho}\gamma_{\phi\mathrm p} \right] \right. \]
\[ \left. + \operatorname{div}\mathbf v_n \left[ N_{\mathrm p} - \frac{\partial N_{\mathrm p}}{\partial S}S - \frac{\partial N_{\mathrm p}}{\partial \rho}\rho \right]\gamma_{\phi\phi} + \left( N_{\phi} - \frac{\partial N_{\phi}}{\partial S}S - \frac{\partial N_{\phi}}{\partial \rho}\rho \right)\gamma_{\phi\mathrm p} \right\}, \tag{15.11} \]

\[ \mu_{\phi} = \frac{1}{\gamma_{\phi\mathrm p}^{2}-\gamma_{\phi\phi}\gamma_{\mathrm{pp}}} \left\{ -\operatorname{div}(\mathbf j-\rho \mathbf v_n) \left[ \frac{\partial N_{\mathrm p}}{\partial \rho}\gamma_{\phi\mathrm p} + \frac{\partial N_{\phi}}{\partial \rho}\gamma_{\mathrm{pp}} \right] \right. \]
\[ \left. + \operatorname{div}\mathbf v_n \left[ \left( N_{\mathrm p} - \frac{\partial N_{\mathrm p}}{\partial \rho}\rho - \frac{\partial N_{\mathrm p}}{\partial S}S \right)\gamma_{\phi\mathrm p} + \left( N_{\phi} - \frac{\partial N_{\phi}}{\partial S}S - \frac{\partial N_{\phi}}{\partial \rho}\rho \right)\gamma_{\mathrm{pp}} \right] \right\}. \tag{15.12} \]

The kinetic coefficients \(\gamma\) occurring in the equations of the present paragraph are simply expressed in terms of the coefficients \(\Gamma\), calculated in § 10 for particular processes. Comparing (15.5) and (15.6) with (10.15) and (10.18) and taking into account the definition of \(\Gamma\), we find

\[ \gamma_{\mathrm{pp}}=\Gamma_{\mathrm p}+\Gamma_{\phi\mathrm p}, \qquad \gamma_{\mathrm p\phi}=\gamma_{\phi\mathrm p}=\Gamma_{\phi\mathrm p}, \qquad \gamma_{\phi\phi}=\Gamma_{\phi}+\Gamma_{\phi\mathrm p}. \tag{15.13} \]

We shall take into account the relative smallness of the coefficient \(\Gamma_{\mathrm p}\) in comparison with \(\Gamma_{\phi\mathrm p}\) and \(\Gamma_{\phi}\). Relations (15.11) and (15.12) then become

\[ \mu_{\mathrm p} = \frac{1}{\Gamma_{\phi}} \left\{ \frac{\partial N}{\partial \rho}\operatorname{div}(\mathbf j-\rho \mathbf v_n) - \left( N - \frac{\partial N}{\partial S}S - \frac{\partial N_{\mathrm p}}{\partial \rho}\rho \right)\operatorname{div}\mathbf v_n \right\} + \]
\[ + \frac{1}{\Gamma_{\phi\mathrm p}} \left\{ \frac{\partial N_{\mathrm p}}{\partial \rho}\operatorname{div}(\mathbf j-\rho \mathbf v_n) - \left( N_{\mathrm p} - \frac{\partial N_{\mathrm p}}{\partial S}S - \frac{\partial N_{\mathrm p}}{\partial \rho}\rho \right)\operatorname{div}\mathbf v_n \right\}, \tag{15.14} \]

\[ \mu_{\phi} = \frac{1}{\Gamma_{\phi}} \left\{ \frac{\partial N}{\partial \rho}\operatorname{div}(\mathbf j-\rho \mathbf v_n) - \left( N - \frac{\partial N}{\partial S}S - \frac{\partial N}{\partial \rho}\rho \right)\operatorname{div}\mathbf v_n \right\}. \tag{15.15} \]

The additional terms in the equations of motion (15.1) and (15.2) are due to processes connected with a change in the total number of rotons \(N_{\mathrm p}\) and phonons \(N_{\phi}\) \((\mu_{\mathrm p}\ne 0,\ \mu_{\phi}\ne 0)\). Thus, the indicated terms are related to the dependence of the pressure \(p\) and the potential \(\mu\) on the chemical potentials \(\mu_{\mathrm p}\) and \(\mu_{\phi}\), and are respectively equal to

\[ \nabla\left( \frac{\partial p}{\partial \mu_{\mathrm p}}\mu_{\mathrm p} + \frac{\partial p}{\partial \mu_{\phi}}\mu_{\phi} \right), \qquad \nabla\left( \frac{\partial \mu}{\partial \mu_{\mathrm p}}\mu_{\mathrm p} + \frac{\partial \mu}{\partial \mu_{\phi}}\mu_{\phi} \right). \]

Equating the right-hand sides of equations (15.1) and (15.2) to the indicated expres-

Substituting the expressions (15.14) and (15.15) for \(\mu_\rho\) and \(\mu_\phi\), we obtain two equalities

\[ \zeta_1 \operatorname{div}(\mathbf j-\rho \mathbf v_n)+\zeta_2 \operatorname{div}\mathbf v_n = \]

\[ = \frac{1}{\Gamma_\phi}\left\{\left(N-\frac{\partial N}{\partial S}S-\frac{\partial N}{\partial \rho}\rho\right)\operatorname{div}\mathbf v_n -\frac{\partial N}{\partial \rho}\operatorname{div}(\mathbf j-\rho \mathbf v_n)\right\}\times \]

\[ \times\left(\frac{\partial \rho}{\partial \mu_\rho}+\frac{\partial \rho}{\partial \mu_\phi}\right) +\frac{1}{\Gamma_{\phi\rho}}\times \]

\[ \times\left\{\left(N_\rho-\frac{\partial N_\rho}{\partial S}S-\frac{\partial N_\rho}{\partial \rho}\rho\right)\operatorname{div}\mathbf v_n -\frac{\partial N_\rho}{\partial \rho}\operatorname{div}(\mathbf j-\rho \mathbf v_n)\right\} \frac{\partial \rho}{\partial \mu_\rho}, \]

\[ \zeta_3 \operatorname{div}(\mathbf j-\rho \mathbf v_n)+\zeta_4 \operatorname{div}\mathbf v_n = \]

\[ = \frac{1}{\Gamma_\phi}\left\{\left(N-\frac{\partial N}{\partial S}S-\frac{\partial N}{\partial \rho}\rho\right)\operatorname{div}\mathbf v_n -\frac{\partial N}{\partial \rho}\operatorname{div}(\mathbf j-\rho \mathbf v_n)\right\}\times \]

\[ \times\left(\frac{\partial \mu}{\partial \mu_\rho}+\frac{\partial \mu}{\partial \mu_\phi}\right) +\frac{1}{\Gamma_{\phi\rho}}\times \]

\[ \times\left\{\left(N_\rho-\frac{\partial N_\rho}{\partial S}S-\frac{\partial N_\rho}{\partial \rho}\rho\right)\operatorname{div}\mathbf v_n -\frac{\partial N_\rho}{\partial \rho}\operatorname{div}(\mathbf j-\rho \mathbf v_n)\right\} \frac{\partial \mu}{\partial \mu_\rho}. \]

Since the equalities obtained must be satisfied for arbitrary values of \(\operatorname{div}\mathbf v_n\) and \(\operatorname{div}(\mathbf j-\rho \mathbf v_n)\), it follows that

\[ \zeta_1 = -\frac{1}{\Gamma_\phi}\left(\frac{\partial \rho}{\partial \mu_\rho} +\frac{\partial \rho}{\partial \mu_\phi}\right)\frac{\partial N}{\partial \rho} -\frac{1}{\Gamma_{\phi\rho}}\frac{\partial \rho}{\partial \mu_\rho} \frac{\partial N_\rho}{\partial \rho}, \tag{15.16} \]

\[ \zeta_2 = \frac{1}{\Gamma_\phi}\left(\frac{\partial \rho}{\partial \mu_\rho} +\frac{\partial \rho}{\partial \mu_\phi}\right) \left(N-\frac{\partial N}{\partial S}S-\frac{\partial N}{\partial \rho}\rho\right) + \]

\[ +\frac{1}{\Gamma_{\phi\rho}}\frac{\partial \rho}{\partial \mu_\rho} \left(N_\rho-\frac{\partial N_\rho}{\partial S}S-\frac{\partial N_\rho}{\partial \rho}\rho\right), \tag{15.17} \]

\[ \zeta_3 = -\frac{1}{\Gamma_\phi}\left(\frac{\partial \mu}{\partial \mu_\rho} +\frac{\partial \mu}{\partial \mu_\phi}\right)\frac{\partial N}{\partial \rho} -\frac{1}{\Gamma_{\phi\rho}}\frac{\partial \mu}{\partial \mu_\rho} \frac{\partial N_\rho}{\partial \rho}, \tag{15.18} \]

\[ \zeta_4 = \frac{1}{\Gamma_\phi}\left(\frac{\partial \mu}{\partial \mu_\rho} +\frac{\partial \mu}{\partial \mu_\phi}\right) \left(N-\frac{\partial N}{\partial S}S-\frac{\partial N}{\partial \rho}\rho\right) + \]

\[ +\frac{1}{\Gamma_{\phi\rho}}\frac{\partial \mu}{\partial \mu_\rho} \left(N_\rho-\frac{\partial N_\rho}{\partial S}S-\frac{\partial N_\rho}{\partial \rho}\rho\right). \tag{15.19} \]

Let us define, with the aid of the following identity, the function

\[ d\varepsilon = T\,dS+\mu\,d\rho-N_\rho\,d\mu_\rho-N_\phi\,d\mu_\phi . \tag{15.20} \]

The function \(\varepsilon\) is an analogue of the energy and, for constant chemical potentials, becomes the energy of a unit volume of liquid. The pressure \(p\) is expressed in terms of the function \(\varepsilon\) by means of the relation:

\[ p=-\varepsilon+ST+\mu\rho . \tag{15.21} \]

THEORY OF KINETIC PHENOMENA IN HELIUM II

From (15.20) and (15.21) there follow expressions for the derivatives \(\dfrac{\partial \rho}{\partial \mu_p}\) and \(\dfrac{\partial \mu}{\partial \mu_p}\)

\[ \frac{\partial \rho}{\partial \mu_p} = N_p-S\frac{\partial N_p}{\partial S} -\rho\frac{\partial N_p}{\partial \rho}, \]

\[ \frac{\partial \rho}{\partial \mu_\phi} = N_\phi-S\frac{\partial N_\phi}{\partial S} -\rho\frac{\partial N_\phi}{\partial \rho}, \tag{15.22} \]

\[ \frac{\partial \mu}{\partial \mu_p} = -\frac{\partial N_p}{\partial \rho}, \qquad \frac{\partial \mu}{\partial \mu_\phi} = -\frac{\partial N_\phi}{\partial \rho}. \tag{15.23} \]

Using relations (15.22) and (15.23), let us rewrite the expressions for the coefficients of second viscosity of helium II in the following form:

\[ \zeta_1 = -\frac{1}{\Gamma_\phi}\frac{\partial N}{\partial \rho} \left( N-\frac{\partial N}{\partial S}S-\frac{\partial N}{\partial \rho}\rho \right) - \frac{1}{\Gamma_{\phi p}}\frac{\partial N_p}{\partial \rho} \left( N_p-\frac{\partial N_p}{\partial S}S-\frac{\partial N_p}{\partial \rho}\rho \right), \tag{15.24} \]

\[ \zeta_2 = \frac{1}{\Gamma_p} \left( N-\frac{\partial N}{\partial S}S-\frac{\partial N}{\partial \rho}\rho \right)^2 + \frac{1}{\Gamma_{\phi p}} \left( N_p-\frac{\partial N_p}{\partial S}S-\frac{\partial N_p}{\partial \rho}\rho \right)^2, \tag{15.25} \]

\[ \zeta_3 = \frac{1}{\Gamma_\phi} \left( \frac{\partial N}{\partial \rho} \right)^2 + \frac{1}{\Gamma_{\phi p}} \left( \frac{\partial N_p}{\partial \rho} \right)^2, \tag{15.26} \]

\[ \zeta_4 = -\frac{1}{\Gamma_\phi} \left( N-\frac{\partial N}{\partial S}S-\frac{\partial N}{\partial \rho}\rho \right) \frac{\partial N}{\partial \rho} - \frac{1}{\Gamma_{\phi p}}\frac{\partial N_p}{\partial \rho} \left( N_p-\frac{\partial N_p}{\partial S}S-\frac{\partial N_p}{\partial \rho}\rho \right). \tag{15.27} \]

The coefficients \(\zeta_1\) and \(\zeta_4\) turned out to be equal, as was to be expected from the symmetry principle for the kinetic coefficients. The derivatives entering the expressions for the coefficients of second viscosity are calculated with the aid of the known relations for the number of rotons and phonons and for the entropy of helium II, (2.5), (2.8′), and (2.11). As a result of a simple differentiation, passing to the variables \(\rho\) and \(T\), we find:

\[ \left( N_p-\frac{\partial N_p}{\partial S}S-\frac{\partial N_p}{\partial \rho}\rho \right) = \]

\[ = \left[ N_p-\rho\left(\frac{\partial N_p}{\partial \rho}\right)_T \right] - \left(\frac{\partial N_p}{\partial T}\right)_\rho \left(\frac{\partial T}{\partial S}\right)_\rho \left[ S-\rho\left(\frac{\partial S}{\partial \rho}\right)_T \right], \tag{15.28} \]

\[ \left( N_\phi-\frac{\partial N_\phi}{\partial S}S-\frac{\partial N_\phi}{\partial \rho}\rho \right) = \]

\[ = \left[ N_\phi-\rho\left(\frac{\partial N_\phi}{\partial \rho}\right)_T \right] - \left(\frac{\partial N_\phi}{\partial T}\right)_\rho \left(\frac{\partial T}{\partial S}\right)_\rho \left[ S-\rho\left(\frac{\partial S}{\partial \rho}\right)_T \right]. \tag{15.29} \]

The derivatives of the parameters \(\Delta, P_0, \mu\) with respect to density, determining the quantity \(\dfrac{\partial N_p}{\partial \rho}\), were calculated in § 6. Using the values of the derivatives given there, according to (2.8′) we find

\[ \left(1-\frac{\rho}{N_p}\frac{\partial N_p}{\partial \rho}\right)\simeq \frac{1}{3}\left(1-\frac{\Delta}{T}\right). \tag{15.30} \]

The derivative of the velocity of first sound \(c\) with respect to the density \(\rho\) can be calculated from Keesom’s data\(^{27}\) on the dependence of the density of helium II on pressure. These data give \(\dfrac{\rho}{c}\dfrac{\partial c}{\partial \rho}\simeq 1.8\), and, consequently,

\[ \left(1-\frac{\rho}{N_\phi}\frac{\partial N_\phi}{\partial \rho}\right) = \left(1+\frac{3\rho}{c}\frac{\partial c}{\partial \rho}\right) \simeq 6.4. \]

Using the indicated values of the derivatives, we write the final expressions

\[ \left(N_p-\frac{\partial N_p}{\partial \rho}\rho-\frac{\partial N_p}{\partial S}S\right) \simeq -\,N_p\,\frac{C_\phi}{3C}\times \]

\[ \times\left\{\left(\frac{\Delta}{T}-1\right) +\left(\frac{\Delta}{T}+\frac{1}{2}\right) \left(1+\frac{3\rho}{c}\frac{\partial c}{\partial \rho}\right)\right\}, \tag{15.31} \]

\[ \left(N_\phi-\frac{\partial N_\phi}{\partial \rho}\rho-\frac{\partial N_\phi}{\partial S}S\right) \simeq N_\phi\,\frac{C_p}{C}\left\{2+\frac{3\rho}{c}\frac{\partial c}{\partial \rho}\right\}. \tag{15.32} \]

The absorption of first sound in helium II is determined by the value of the coefficient \(\zeta_2\) (see § 17). The experimental values of the coefficient of absorption of first sound in helium II for the temperature range \(1.57\text{–}2.0^\circ\mathrm{K}\) make it possible, with a satisfactory degree of accuracy, to determine the unknown coefficients \(a\) and \(b\) in formulas (10.13) and (10.21) for \(\Gamma_\phi\) and \(\Gamma_{\phi p}\). Thus we find\(^*\)

\[ \Gamma_\phi = 1\cdot 10^{43}T^{11},\qquad \Gamma_{\phi p}=4\cdot 10^{49}e^{2\Delta/T}. \tag{15.33} \]

The value obtained for \(\Gamma_{\phi p}\) naturally turned out to be smaller than the upper limit given by formula (10.20).

16. Coefficient of the first viscosity of helium II\(^{31,30}\)

The question of the viscosity of helium II was considered in detail by us earlier\(^{15}\). Recently it has become possible to refine somewhat the results obtained earlier by a more accurate allowance for the five-phonon process. In addition, it has proved possible somewhat

\[ \text{*} \]

\(^*\) The indicated values of the coefficients \(a\) and \(b\) differ from the values given in \(^{38}\). This change is due to a refinement of the value of the derivative \(\partial N_p/\partial \rho\), determining the quantity \(\zeta_2\), which has now been made on the basis of new experimental data.

clarify the values of the derivatives of the parameters \(\Delta\) and \(P_0\) with respect to the density, which determine the magnitude of the effective cross section for scattering of a phonon by a roton. For completeness in the question of kinetic coefficients in helium II, we give here briefly the results obtained earlier, with the refinements made recently.

The viscosity coefficient \(\eta\) is composed of two parts—one due to rotons and the other due to phonons,

\[ \eta=\eta_{\mathrm{p}}+\eta_{\phi}. \tag{16.1} \]

Roton viscosity. The collision integral for rotons, analogously to the way this is done in calculating \(\varkappa_{\mathrm{p}}\), is represented in the form

\[ -\frac{n-n_0}{t}, \tag{16.2} \]

where \(t\) is a quantity differing from the mean time between collisions of two rotons \((t_{\mathrm{p}})\) by a factor of order unity. Since expression (9.11) for \(t_{\mathrm{p}}\) contains an unknown constant factor \(V_0\), we shall not distinguish between \(t\) and \(t_{\mathrm{p}}\), including the factor of order unity in the quantity \(V\). In this case the factor in formula (13.17) determining the magnitude of the viscosity coefficient simply coincides with \(t_{\mathrm{p}}\), and we have

\[ \eta_{\mathrm{p}}=-\frac{t_{\mathrm{p}}}{15kT}\int n'\left(\frac{\partial\varepsilon}{\partial p}\right)^2 p^2\,d\tau_{\mathrm{p}}. \tag{16.3} \]

Carrying out the necessary integration over \(d\tau_{\mathrm{p}}\), we obtain

\[ \eta_{\mathrm{p}}=\frac{P_0^2 t_{\mathrm{p}} N_{\mathrm{p}}}{15\mu}. \tag{16.4} \]

According to (9.13), the quantity \(t_{\mathrm{p}}N_{\mathrm{p}}\) does not depend on temperature and is equal to

\[ t_{\mathrm{p}}N_{\mathrm{p}}=\frac{h^4}{P_0\mu |V_0|^2}. \tag{16.5} \]

Substituting (16.5) into (16.4), we finally find

\[ \eta_{\mathrm{p}}=\frac{h^4P_0}{15\mu^2|V_0|^2}. \tag{16.6} \]

Expression (16.6) contains only factors independent of temperature; therefore the roton part of the viscosity coefficient \(\eta_{\mathrm{p}}\) proves to be constant, independent of temperature.

Phonon part of the viscosity coefficient \(\eta_{\phi}\). Consider helium II in which there is a small gradient of the normal velocity \(\mathbf{v}_n\), directed along the \(x\)-axis. For convenience we choose the velocity \(\mathbf{v}_n\) to be directed along the \(z\)-axis. The direction of motion of a phonon is characterized by two angles \(\vartheta\) and \(\varphi\) in a polar coordinate system with polar axis \(z\). The kinetic equation for phonons then

is written in the form

\[ n(n+1)\frac{cp}{kT}\frac{\partial v_n}{\partial x}\cos\vartheta\sin\vartheta\cos\varphi = I_{\mathrm I}(n)+I_{\mathrm{II}}(n)+I_{\mathrm{III}}(n)+I_{\mathrm{IV}}(n), \tag{16.7} \]

where \(I_{\mathrm I}\) and \(I_{\mathrm{II}}\) are collision integrals due respectively to the scattering of phonons by rotons and of phonons by phonons; \(I_{\mathrm{III}}\) is the collision integral associated with phonon scattering through small angles; finally, \(I_{\mathrm{IV}}\) characterizes the change in the number of phonons that occurs due to the five-phonon process.

In calculating the phonon part of the viscosity coefficient \(\eta_{\mathrm f}\), we proceed from an assumption—whose validity we shall ultimately verify—namely that the process of establishing energy equilibrium in the phonon gas occurs noticeably faster than the scattering of phonons by rotons and by phonons. With respect to processes associated with changes in the total number of phonons, here, just as in the calculation of \(\chi_{\mathrm f}\), two temperature regions must be considered. At temperatures above \(0.9^\circ\mathrm K\), the establishment of equilibrium in the number of phonons (the five-phonon process) is characterized by times comparable with those that determine the scattering of phonons by rotons.

Conversely, at temperatures below \(0.9^\circ\mathrm K\), the establishment of equilibrium proceeds faster than the phonon-scattering processes that characterize the viscosity.

Temperature region above \(0.9^\circ\mathrm K\). In this region \((T>0.9^\circ\mathrm K)\), the kinetic equation (16.7), after integration over all phonons and over all phonon energies, gives two equations

\[ \frac{c}{kT}\frac{\partial v_n}{\partial x}\cos\vartheta\sin\vartheta\cos\varphi \int n(n+1)\varepsilon p^3\,dp = \int (I_{\mathrm I}+I_{\mathrm{II}})\varepsilon p^2\,dp, \]

\[ \frac{c}{kT}\frac{\partial v_n}{\partial x}\cos\vartheta\sin\vartheta\cos\varphi \int n(n+1)p^3\,dp = \int (I_{\mathrm I}+I_{\mathrm{II}})p^2\,dp+\int I_{\mathrm{IV}}p^2\,dp. \tag{16.8} \]

The phonon distribution function depends on two parameters, \(\alpha'\) and \(T'\), which depend on the direction of motion of the phonons,

\[ n=\left[e^{\alpha'+pc/kT'}-1\right]^{-1} \tag{16.9} \]

(\(T'\) is the temperature of the phonons moving in the given direction). The deviation of the distribution function \(n\) from the equilibrium value \(n_0\), according to (16.9), is written in the form

\[ n-n_0=-n(n+1)\left\{\alpha'-\frac{pc}{kT}\frac{T'-T}{T}\right\}. \tag{16.10} \]

Taking into account the symmetry of the problem, we represent the temperature deviation of the phonons moving in the given direction from the equilibrium ...

values of \(T\) and the quantity \(\alpha'\), in the form

\[ \frac{T'-T}{T}=\beta P_2(\cos\vartheta),\qquad \alpha'=aP_2(\cos\vartheta) \]

(\(P_2(\cos\vartheta)\) is a spherical function).

At temperatures above \(0.7^\circ\mathrm{K}\) the integral \(I_{II}\) may be neglected in comparison with \(I_I\). As a result of simple transformations, using the Legendre-function addition theorem, the collision integral \(I_I\), after substitution of the distribution function (16.9), reduces to

\[ I_I(n)=P_2(\cos\vartheta)N_p\int c\,d\sigma(p,\psi)\,[1-P_2(\cos\theta)]\times \]

\[ {}\times\left(\alpha-\beta\frac{pc}{kT}\right)n(n+1). \tag{16.11} \]

Here \(d\sigma(p,\psi)\) is the differential effective cross section for scattering of a phonon by a roton through the angle \(\psi\), defined by expression (8.20). After substituting (8.20) into (16.11) and integrating over the solid angle \(d\omega_\psi\), we obtain

\[ I_I(n)=\frac{1}{\bar{\theta}}P_2(\cos\vartheta)n(n+1)\left(\alpha-\beta\frac{pc}{kT}\right)\left(\frac{pc}{kT}\right)^4, \tag{16.12} \]

where the notation has been introduced

\[ \frac{1}{\bar{\theta}}= \frac{N_p6!}{4\pi c} \left[ \frac{P_0(kT/c)^2}{h^2\rho} \right]^2 \left\{ \frac{2}{15} +\frac{33}{35^2}\left(\frac{p_0}{\mu c}\right)^2 +\frac{144}{75}\frac{P_0}{\mu c} +A^2 \right\}. \tag{16.13} \]

The quantity \(\bar{\theta}\), having the dimension of time, is of the same order of magnitude as the time \(\bar{\theta}\) calculated in the problem on thermal conductivity (14.30). With the aid of (16.12) and (16.8) we obtain two equations determining the parameters \(\alpha\) and \(\beta\):

\[ \frac{\partial v_n}{\partial x}\frac{4\pi^4}{15} = \frac{7}{\bar{\theta}}(\alpha-8\beta), \qquad \frac{\partial v_n}{\partial x}\frac{36}{5} = \frac{1}{\bar{\theta}}(\alpha-7\beta)+\frac{1}{\theta_{\mathrm{ph}}}\alpha. \tag{16.14} \]

Temperature region below \(0.9^\circ\mathrm{K}\). In this region only the first equation (16.8) is meaningful; the parameter \(\alpha'\) is equal to zero. The integral \(I_{II}\), due to scattering of phonons by phonons, has an appreciable magnitude and cannot be neglected. This integral proves to be convergent without taking account of phonon dispersion. As a result of rather lengthy calculations with a distribution function of the form (16.9), we find

\[ \int I_{II}(n)\,\varepsilon p^2dp = -P_2(\cos\vartheta)\beta kT\left(\frac{kT}{c}\right)^3\frac{1}{\tau_{\mathrm{ph}}}, \tag{16.15} \]

where \(\tau_{\mathrm{ph}}\) is a quantity of the dimension of time, defined by the relation

\[ \frac{1}{\tau_{\mathrm{ph}}} = \frac{3.13!(u+2)^4}{5\cdot 2^{19}(2\pi)^3 h^7\rho^2c} \left(\frac{kT}{c}\right)^9 \tag{16.16} \]

\[ \left( u=\frac{\rho}{c^2}\frac{\partial c^2}{\partial\rho} \right). \]

The equation determining the magnitude of the parameter \(\beta\) in this case becomes

\[ \frac{4\pi^4}{15}\frac{\partial v_n}{\partial x} = -\beta\left(\frac{56}{\bar{\theta}}+\frac{1}{\tau_\phi}\right). \tag{16.17} \]

The mean time between phonon collisions characterizing the viscosity. Analogously to (14.13), to determine the mean time between phonon collisions that characterizes the viscosity, we obtain the equations

\[ \begin{aligned} \frac{\pi^3}{3}\frac{d\alpha}{dt} -\frac{36}{5}\frac{d\beta}{dt} &= -(\alpha-7\beta)\frac{1}{\bar{\theta}} -\alpha\frac{1}{\theta_\phi}, \\ \frac{36}{5}\frac{d\alpha}{dt} -\frac{4\pi^4}{15}\frac{d\beta}{dt} &= -7(\alpha-8\beta)\frac{1}{\bar{\theta}}, \end{aligned} \qquad T>0.9^\circ\mathrm{K}, \tag{16.18} \]

\[ \frac{4\pi^4}{15}\frac{d\beta}{dt} = -\beta\left(\frac{56}{\bar{\theta}}+\frac{1}{\tau_\phi}\right), \qquad T<0.9^\circ\mathrm{K}. \tag{16.19} \]

We seek the solution of the system of equations (16.18) and of equation (16.19) in a form proportional to \(e^{-t/\tau}\). Thus, from (16.18) we find two values

\[ \tau = 16\bar{\theta}\, \frac{1+\bar{\theta}/4\theta_\phi}{1+8\bar{\theta}/\theta_\phi} \quad\text{or}\quad \frac{1}{3}\bar{\theta}\, \frac{1}{1+8\bar{\theta}/\theta_\phi}. \tag{16.20} \]

For purposes of comparison we choose the larger of these roots. From equation (16.19) we find

\[ \tau = \frac{4\pi^4}{15\cdot 56}\, \bar{\theta}\left(1+\bar{\theta}/56\tau_\phi\right)^{-1}. \tag{16.21} \]

Substituting the numerical values of the parameters into (16.20) and (16.21), we finally obtain

\[ \frac{1}{\tau} = \begin{cases} 7.5\cdot 10^{10}T^{3/2}e^{-\Delta/T}\, \dfrac{1+8\bar{\theta}/\theta_\phi}{1+\bar{\theta}/4\theta_\phi}, & T>0.9^\circ\mathrm{K}, \\[6pt] 4.75\cdot 10^{11}T^{9/2}e^{-\Delta/T}, & T<0.9^\circ\mathrm{K}. \end{cases} \tag{16.22} \]

Comparing the obtained values of the mean time between phonon collisions \(\tau\) with the time \(t_\phi\) characterizing the establishment of energy equilibrium, we see that at all temperatures below the \(\lambda\)-point \(t_\phi<\tau\). Thus, the assumption made at the beginning concerning the rate of establishment of energy equilibrium in the phonon gas proves to be valid. In Fig. 7 the temperature dependence is given of the phonon mean free path \(\lambda_\phi=c\tau\), characterizing the viscosity.

Temperature dependence of the phonon part of the coefficient of viscosity. Let us solve the system of equations (16.14) with respect to \(\alpha\) and \(\beta\), and substitute the obtained values into expression (16.16) for the deviation \(n-n_0\) of the distribution function from its equilibrium value.

In this way we obtain

\[ n-n_0=-n_0(n_0+1)\cos\vartheta\sin\vartheta\cos\varphi\,\bar{\theta}\times \frac{\left\{31.2-\frac{Pc}{kT}\left(3.5-3.7\bar{\theta}/\theta_\Phi\right)\right\}\dfrac{\partial v_n}{\partial x}} {\left\{1+8\bar{\theta}/\theta_\Phi\right\}} . \tag{16.23} \]

Here we replace the function \(P_2(\cos\vartheta)\) by \(P_{21}(\cos\vartheta)\cos\varphi\), since in the left-hand side of the kinetic equation there is in fact contained the function of the angles \(P_{21}(\cos\vartheta)\cos\varphi\). In an analogous way, from (16.17) and (16.10), we find

\[ n-n_0=-n_0(n_0+1)\cos\vartheta\sin\vartheta\cos\varphi\times \]

\[ \times \frac{4\pi^4}{15\cdot 56}\, \bar{\theta}\left(1+\bar{\theta}/56\tau_\Phi\right)^{-1} \frac{pc}{kT}\frac{\partial v_n}{\partial x}. \tag{16.24} \]

The \(xz\) component of the stress tensor, which is nonzero in our case, is equal to

\[ \sigma_{xz}=\eta_\Phi\frac{\partial v_n}{\partial x}= \]

\[ =-(2\pi\hbar)^{-3}\int pc\,(n-n_0)\cos\vartheta\sin\vartheta\cos\varphi\,p^2\,dp\,do . \tag{16.25} \]

From (16.25), with the aid of (16.23) and (16.24), we find the expression for

\[ \eta_\Phi=3.8\,N_\Phi kT\bar{\theta}\, \frac{1+0.75\bar{\theta}/\theta_\Phi}{1+8\bar{\theta}/\theta_\Phi} \quad \text{for } T>0.9^\circ\mathrm{K}, \]

\[ \eta_\Phi=0.36\,N_\Phi kT\bar{\theta}\left(1+\bar{\theta}/56\tau_\Phi\right)^{-1} \quad \text{for } T<0.9^\circ\mathrm{K}. \tag{16.26} \]

Substituting into (16.26) the numerical values of all the parameters, we finally obtain

\[ \eta_\Phi= \begin{cases} 3.75\cdot 10^{-8}T^{-1/2}e^{\Delta/T} \dfrac{1+0.75\bar{\theta}/\theta_\Phi}{1+8\bar{\theta}/\theta_\Phi}, & \text{for } T>0.9^\circ\mathrm{K},\\[6pt] 3.50\cdot 10^{-9}T^{-1/2}e^{\Delta/T} \left(1+2.15\cdot 10^{-5}T^{9/2}e^{\Delta/T}\right)^{-1}, & \text{for } T<0.9^\circ\mathrm{K}. \end{cases} \tag{16.27} \]

The phonon part of the viscosity coefficient increases as the temperature decreases according to the law \(e^{\Delta/T}\). At temperatures below \(0.7^\circ\mathrm{K}\), however, when only the effect of phonon scattering by phonons is significant, this law is replaced by the law \(T^{-5}\).

Temperature dependence of the viscosity coefficient of helium II.
The roton part of the viscosity coefficient cannot be calculated accurately; however, it can be found from experiment. Subtracting from the experimental values of the viscosity coefficient \(^{18}\) the phonon part calculated by formula (16.27), we find in the temperature interval \(1.9^\circ\mathrm{K}-1.4^\circ\mathrm{K}\) an almost constant quantity equal to

\(1.15 \cdot 10^{-5}\) poise, which we identify with the roton part of the viscosity coefficient. Thus we have

\[ \eta_p=\frac{h^4 P_0}{15\mu^2 |V_0|^2}=1.15\cdot 10^{-5}\ \text{poise}. \tag{16.28} \]

From (16.28) we find the value of the constant \(V_0\):

\[ V=1.1\cdot 10^{-38}\ \text{erg cm}; \]

the value of the viscosity coefficient of helium II is obtained by summing the roton (16.28) and phonon (16.27) parts:

\[ \eta\cdot 10^5 = 1.15+ \begin{cases} 3.75\cdot 10^{-3}T^{1/2}e^{\Delta/T}\dfrac{1+0.75\cdot \bar{\theta}/\theta_\phi}{1+\bar{\theta}/\theta_\phi},\\[6pt] 3.5\cdot 10^{-4}T^{1/2}e^{\Delta/T} \left(1+2.15\cdot 10^{-5}T^{1/2}e^{\Delta/T}\right)^{-1}. \end{cases} \tag{16.29} \]

The ratio of the times characterizing the scattering of phonons by rotons \((\bar{\theta})\) and the five-phonon process \(\theta_\phi\) is calculated by means of formulas (16.13) and (14.31′). Numerical values of this ratio for various temperatures are given in Table I.

Table I

\(T^\circ\mathrm{K}\)	2.0	1.8	1.6	1.4	1.2	1.0	0.8	0.7
\(\bar{\theta}/\theta_\phi\)	1.6	1.41	1.78	2.18	3.10	6.05	20.5	5.6

In the detailed work\(^{31}\) devoted to the consideration of the viscosity of helium II, the calculation of the phonon part of the viscosity coefficient was carried out for two limiting cases. In the first limiting case, \(T>1.0^\circ\mathrm{K}\), the ratio \(\bar{\theta}/\theta_\phi\) was assumed to be so small that equilibrium in the number of phonons did not have time to be established during the times characterizing the scattering of a phonon by a roton. In such a case, the collision integral due to the five-phonon process could be entirely neglected. Conversely, in the second limiting case (\(T<0.8^\circ\mathrm{K}\)) the indicated ratio was assumed to be so large that equilibrium in the number of phonons had time to be established during the same times as characterize the scattering of a phonon by a phonon. In connection with the obtaining of more accurate values of the time \(\theta_\phi\), which it proved possible to calculate on the basis of experimental values of the absorption coefficient of first sound\(^{32}\), it became possible to refine the theory at this point. In solving the kinetic equation for phonons with allowance for the five-phonon process, in the corresponding formulas determining the magnitude of the phonon part of the viscosity coefficient at \(T>0.9^\circ\mathrm{K}\), there appears an additional

the factor

\[ \frac{1+0.75\,\overline{\theta}/\theta_{\phi}} {1+8\,\overline{\theta}/\theta_{\phi}} . \]

This factor differs from unity even for small values of the ratio \(\overline{\theta}/\theta_{\phi}\), owing to the presence in its denominator of the large coefficient 8 multiplying the ratio \(\overline{\theta}/\theta_{\phi}\). This circumstance appreciably changes the magnitude of the phonon part of the viscosity coefficient.

In the temperature region above \(1.4\text{–}1.5^\circ\mathrm{K}\), where the phonon part of the viscosity coefficient is considerably smaller than the roton part, the value of the total viscosity coefficient \(\eta\), under such a refinement, naturally changes practically not at all. At lower temperatures, especially near \(1^\circ\mathrm{K}\), where previously the values of \(\eta_{\phi}\) were obtained by interpolation \(\left(\overline{\theta}/\theta\right.\) of the order of unity\()\), formula (16.27), which takes the five-phonon process into account more accurately, gives values of \(\eta_{\phi}\) smaller than those obtained by interpolation. The results obtained earlier for the temperature region below \(0.8^\circ\mathrm{K}\) thereby remain valid. The correction factor at \(T=0.8^\circ\mathrm{K}\) differs by only \(5\%\) from the limiting value \(3/32\), obtained in the limiting case of an extremely rapid five-phonon process \((\theta_{\phi}\ll \overline{\theta})\). The values of \(\eta_{\phi}\) calculated from formula (16.27) are given in Table II.

Table II

\(T^\circ\mathrm{K}\)	2.0	1.8	1.6	1.4	1.2
\(\eta_{\phi}\) (poise)	\(3.1\cdot10^{-7}\)	\(6.5\cdot10^{-7}\)	\(1.5\cdot10^{-6}\)	\(2.55\cdot10^{-6}\)	\(7.2\cdot10^{-6}\)
\(T^\circ\mathrm{K}\)	1.0	0.8	0.7	0.6
\(\eta_{\phi}\) (poise)	\(3.0\cdot10^{-5}\)	\(17.3\cdot10^{-5}\)	\(56\cdot10^{-5}\)	\(175\cdot10^{-5}\)

Figure 8 shows the temperature dependence of the viscosity coefficient \(\eta=\eta_{\phi}+\eta_{r}\) \((\eta_{r}=1.15\cdot10^{-5}\,n\ \text{poise})\). There, for comparison, are also given the values of the viscosity coefficient measured by E. Andronikashvili\({}^{33}\) from the damping of torsional oscillations of a disk immersed in helium II. These experiments were repeated by a group of authors\({}^{34}\), who fully confirmed E. Andronikashvili’s results. It should be noted that the authors mentioned\({}^{34}\) made an error in calculating the correction for edge effects. In the correct formula, given in work\({}^{33}\), they replaced \(\rho_{n}\) by \(\rho\) in one place.

If, from the values of the damping decrement obtained by these authors, one calculates the values of the viscosity coefficient using the correct formula for the corresponding correction, then one obtains values of \(\eta\) that are in general in good agreement with the data of E. Andro-

Andronikashvili[^33], but at low temperatures they are approximately 10–15% smaller than in [^33].

Recently a paper by Hollis-Hallett[^35][^36] appeared, in which the author also repeated the experiments of E. Andronikashvili. The values of the viscosity coefficient obtained in this work prove to be somewhat lower than those given by E. Andronikashvili, with the exception of one temperature point \((T = 1.255^\circ\mathrm{K})\), for which the author obtained an unnaturally large value of \(\eta\). On the whole, the results of [^35] agree with other measurements of the viscosity coefficient of helium II.

Fig. 8. Temperature dependence of the coefficient of first viscosity:
\(\circ\)—Zinov’eva[^37], \(\triangle\)—Heikkila and Hollis-Hallett[^36], \(\times\)—Hollis-Hallett[^35], \(+\)—de Troyer and Itterbeek[^34], — — —Andronikashvili[^33], — theoretical values[^30].

The general conclusion that can be drawn from a comparison of all the experiments mentioned is that the predictions of the theory concerning the temperature dependence of the viscosity coefficient are fully confirmed.

For all authors, at high temperatures there is observed a temperature-independent constant segment on the curve of the temperature dependence of \(\eta\). As the temperature is lowered, however, the viscosity of helium II increases (the phonon part of the viscosity). The theoretical values of the viscosity coefficient, calculated from formula (16.29), agree, within the accuracy of the experiment, with the measurements in [^33], [^34], [^35], [^36].

The viscosity coefficient of helium II can also be obtained from observations of the damping of second sound. Such a method was used by K. Zinov’eva[^37], who measured the absorption of second sound in a cylindrical vessel. That part of the absorption which is determined by scattering of sound at the walls of the vessel depends mainly on the viscosity of helium II and has a characteristic frequency dependence (the absorption coefficient is proportional to the square root of the sound frequency). Having separated this part from the total absorption, K. Zinov’eva obtained values of the viscosity coefficient down to a temperature of \(0.8^\circ\mathrm{K}\). The values of \(\eta\) obtained in this work agree with the theoretical values calculated earlier.

References

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1. Reference marker as printed in the section heading. ↩

2. *) A prime denotes differentiation of the function with respect to the argument
   \[ \frac{\varepsilon-\mathbf{p}\mathbf{v}_n+\mathbf{p}\mathbf{v}_s}{kT}; \qquad n'=-n(n+1). \] ↩↩