On the Theory of Accommodation and Condensation Phenomena

Ya. I. Frenkel, Leningrad

Introduction

When a gas particle strikes the surface of a solid or liquid body, this particle may either adhere to the latter (the phenomenon of “condensation”), or rebound from it. In the latter case the energy of the particle’s translational motion changes, on the average, toward approaching the value corresponding to the temperature of the body (the phenomenon of “accommodation”). These phenomena may be characterized quantitatively by two coefficients, namely the coefficient of adhesion $f$, representing the ratio of the number of adhering gas particles to the number of incident ones (or the coefficient of reflection $r = 1 - f$), and the coefficient of accommodation $\alpha$, which is usually defined for the limiting case of infinitesimal proximity of the gas temperature $T$ and the temperature of the solid (or liquid) body $T_0$ by the formula

\[ \alpha = \frac{T' - T_0}{T - T_0}, \]

where $\frac{3}{2} kT'$ is the mean energy with which the gas particle rebounds from the surface of the body (having struck it with energy $\frac{3}{2} kT$). For $T \simeq T_0$, the coefficient $\alpha$ is a function of one of these temperatures; in the general case, however, it depends on both of them, and in a non-identical manner.

Strictly speaking, there is no fundamental difference between the phenomena of condensation and accommodation. The rebound of a gas particle from the surface of a solid (or liquid) body is preceded by at least a very brief period of its residence on the surface, in any case no shorter than the duration of a collision between two gas molecules, or the period of oscillation of a particle of the solid body about its equilibrium position ($\tau_0 \sim 10^{-13}$ sec.). On the other hand, an adsorbed gas particle, i.e. a particle that has adhered to the surface, may again detach from it after some time $\tau$, equal on the average to $\tau_0 e^{\frac{U}{kT}}$, where $U$ is the energy of evaporation or desorption.

The actual residence time of a particle in the adsorbed state may differ from \(\tau\) in either direction; moreover, the probability that the particle will not detach from the surface during a time \(\geq t\) is expressed by the formula \(e^{-t/\tau}\). Thus adsorption does not exclude the possibility of almost instantaneous evaporation, which is in no way different from ordinary rebounding.

In the case where the adsorption energy \(U\) is very small in comparison with \(kT\), the phenomenon of sticking may practically be ignored altogether, assuming that the interaction of a gas with the surface of a solid (or liquid) body reduces to rebounding, characterized by a definite value of the accommodation coefficient \(\alpha(T,T_0)\). In the opposite case, however, i.e. when \(U \gg kT\), the assumption that every gas particle striking the surface remains bound to it for a time comparable with \(\tau\) proves to be incorrect. The possibility of being adsorbed does not yet imply the necessity of being adsorbed, and some of the incident particles, so to speak neglecting this possibility, rebound from the surface in exactly the same way as in the preceding case.

It would, of course, be possible to regard these rebounding particles as those adsorbed particles for which the lifetime on the surface \(t\) is very small in comparison with \(\tau\) and is of the order of \(\tau_0\). If, however, this point of view were correct, the reflection coefficient \(r\) could be determined from the formula

\[ r = 1 - e^{-\frac{\tau_0}{\tau}} \simeq \frac{\tau_0}{\tau} = e^{-\frac{U}{kT}}. \]

Although the experimental data on this question are still very scanty, they nevertheless apparently do not agree with this result. Thus it is more appropriate to regard the phenomenon of rebounding not as the limiting case of rapid evaporation following sticking (adsorption), but as a phenomenon sui generis, distinct from sticking and capable of competing with it (provided that \(U \gg kT\)).

Let us note that the opposition considered by us between the phenomena of condensation and rebounding of gas particles on the surface of a solid or liquid body is to some extent analogous to the opposition between the absorption and scattering of a light quantum by an atom or molecule.

Strictly speaking, the absorption of light, connected with the excitation of an atom, is always only the first stage of the process of interaction of light with the atom, followed by a second stage—the return of the excited atom to the normal state with the emission of a light quantum. The two stages, taken together, constitute the phenomenon of scattering, differing from ordinary scattering of light only in its resonant character. Despite this circumstance, it is expedient to distinguish ordinary scattering from resonant scattering for the reason that, in the absence of resonance, the scat—

decomposition of the scattering act into absorption and emission turns out to be impossible or, more precisely, possible only when applying quantum perturbation theory in the second approximation (with the introduction of an intermediate stage between the initial and final states), whereas in the case of resonance this decomposition can already be carried out in the first approximation.

The question of the interaction of gas particles with the surface of a solid (or liquid) body, from the point of view of determining the coefficients of sticking and accommodation (in the case of rebound), began to be studied only recently—approximately from 1930, if one does not count several early works (Knudsen and others), in which it was rather posed than solved. The experimental results and theories currently available relate almost exclusively to the question of the accommodation coefficient (when adsorption is impossible, i.e. when \(U\) is small), whereas the question of the sticking coefficient and of the magnitude of the accommodation coefficient in the case where adsorption is possible has so far hardly been considered at all1. This circumstance is explained, on the one hand, by experimental difficulties and, on the other, by fundamental defects in the quantum-theoretical treatment of the question of energy exchange between gas particles and solid bodies by the various authors who have dealt with this question up to the present time. Without pausing here for a critical examination of these treatments (to which we shall return below), let us note that, in view of the relatively large mass of atoms and molecules, the use of quantum mechanics in constructing a theory of their interaction with a solid body cannot be considered obligatory. On the contrary, there is then no need to abandon the methods of classical mechanics, and in the case of high temperatures also those of classical statistics (in the case of low temperatures quantum-statistical relations may be introduced only into the final formulae when passing to mean values).

Of the various theories of accommodation phenomena proposed up to now, only one theory has been constructed on the basis of classical mechanics; its results we shall derive below in a more elementary way than was done in the original, by a method (which moreover gives a better idea of the nature of the neglects made in the calculation); further, we shall attempt to improve this theory and to compare it with the quantum theories of accommodation phenomena. In the concluding part of the article we shall consider the foundations of the classical and quantum theory of sticking phenomena.

§ 1. Model of a Free Atom of a Solid Body

Let us first suppose, for simplicity, that the atoms of the solid body, before the impact of the gas particle against the surface of the latter, are at rest in their positions of equilibrium (which, from the point of view of classical

statistics corresponds to absolute zero temperature \(T_0=0\)). The process occurring upon the impact of a gas particle is then reduced to its imparting to the solid a certain part \(\Delta\varepsilon\) of its initial kinetic energy \(\varepsilon'\). The ratio \(\frac{\Delta\varepsilon}{\varepsilon'}\) may serve as a measure of the accommodation coefficient \(\alpha\) for the “individual” impact under consideration. In the case of interaction with a body at \(T_0=0\) of a (rarefied) gas at temperature \(T\), the mean accommodation coefficient \(\bar{\alpha}\) may be defined as the ratio of the mean value of \(\Delta\varepsilon\) to the mean value of \(\varepsilon'\) (for a Maxwellian distribution of velocities among the gas particles), or else as the mean value of the ratio \(\frac{\Delta\varepsilon}{\varepsilon}\) for individual impacts1.

The energy lost by a gas particle in striking the surface of a solid may be estimated in the crudest way if the solid is replaced by one perfectly free atom and the process is considered as a head-on collision of an elastic sphere representing the gas particle (mass \(m'\), initial velocity \(v'\)) with another elastic sphere (mass \(m\), initial velocity 0) representing an atom of the solid. Denoting the velocities of the gas particle and of the atom after the collision respectively by \(v'-\Delta v'\) and \(v\), and using the laws of conservation of momentum \(m'\Delta v'=mv\) and of energy \(m'[v'^2-(v'-\Delta v')^2]=mv^2\), we obtain

\[ \frac{1}{2}mv^2=\frac{2m'^2mv'^2}{(m+m')^2}, \]

i.e.

\[ \frac{\Delta\varepsilon}{\varepsilon'}=\frac{4mm'}{(m+m')^2}. \tag{1} \]

This expression has a maximum, equal to 1, when \(m'=m\), which corresponds to the interaction of the solid with its own vapor. For \(m'>m\) it can hardly be applied to the problem of interest to us, since in this case the incident particle must, after the impact, move in the same direction as before, i.e. into the body under consideration. Replacing the latter by a single surface atom is in this case clearly inadmissible.

On the contrary, for \(m'\ll m\), formula (1), reducing in this case to

\[ \frac{\Delta\varepsilon}{\varepsilon'}\simeq 4\frac{m'}{m}, \tag{1a} \]

must constitute a sufficiently good approximation to reality. Indeed, when a comparatively light particle falls upon a solid, it will rebound, imparting to the struck (relatively heavy) atom only an insignificant velocity.

Thus the latter will not have time to be displaced from its equilibrium position by any noticeable distance during the impact (i.e., during the intense interaction with the striking particle), and therefore will behave as though no other forces were acting on it. These considerations can be made more precise if one introduces the effective “duration of the collision” \(\tau\) (i.e., the time of strong interaction between the colliding particles). The product of this time by the velocity \(v=\sqrt{\frac{2\Delta\varepsilon}{m}}\), imparted to the atom in the impact, must be small in comparison with its distance \(a\) from the neighboring atom in the equilibrium position.

As for the duration of the collision \(\tau\), in the case of interaction between an atom and a gas particle with potential energy of the form

\[ U(x)=Ae^{-\frac{x}{b}} \tag{2} \]

(where \(x\) denotes the distance between their centers, and \(b\) is a constant of the order \(10^{-8}\)—\(10^{-9}\) cm), this time may be estimated by the formula

\[ \tau=\frac{2b}{v'} \tag{2a} \]

(since the intense interaction is confined to an interval \(\Delta x\) close to \(b\)).

Substituting into \(v=\sqrt{\frac{2\Delta\varepsilon}{m}}\) the expression for \(\Delta\varepsilon\) from (1a), we can rewrite the inequality

\[ v\tau\ll a \]

in the form

\[ 4\sqrt{\frac{m'}{m}}\,\frac{b}{a}\ll 1. \tag{2b} \]

For a sufficiently small ratio \(\frac{m'}{m}\), this inequality may be regarded as unconditionally satisfied.

It is interesting to note that, despite its extreme simplicity and schematic character, the theory set forth (first proposed by Baule\(^1\)) is in satisfactory qualitative agreement with experimental data. The experiments of Spivak and Zakharyin\(^2\) on the accommodation of gas particles on the surface of metals covered with various monomolecular films show that the proportionality of the accommodation coefficient to the ratio of the masses of the incident particle and of the particle receiving the impact (a metal atom or a molecule of the monolayer adsorbed on it) is justified even when these masses are comparable with one another, i.e., when the assumptions of formula (1a) become incorrect.

§ 2. Linear elastic model of a solid

In the preceding paragraph we replaced the solid by a single atom, completely ignoring the forces connecting it with the other atoms of the body and taking into account only its interaction with the incident particle. In order to clarify the influence of the forces binding the atoms of a solid to one another, we shall first consider a one-dimensional model of a solid in the form of a chain of elastically bound atoms. To simplify the problem further, we shall treat this one-dimensional model as a continuum, i.e. as a continuous elastic rod, and characterize it by a definite value of the propagation velocity of elastic deformations, i.e. the speed of sound \(c\) 1.

Fig. 1.

The impact of the incident particle on the end of the rod will produce in it a compression wave which, during the collision time \(\tau\), will propagate over the distance \(l=c\tau\). In this process the front part of the rod, which initially had length \(l\), will be shortened by the amount

\[ \Delta l = v\tau, \]

where \(v\) denotes the velocity acquired by the atoms forming it (Fig. 1). This velocity, transmitted by elastic forces from one atom to the next, may be regarded as the same for all atoms in the segment \(l\).

Therefore the total momentum communicated during the impact to the atoms of the rod may be represented as the product \(mv\) by the number of atoms in the segment \(l\), i.e. by the ratio \(\frac{l}{a}\), where \(a\) is the distance between the equilibrium positions of neighboring atoms. This momentum must be numerically equal to the change in the momentum of the incident particle.

We shall assume that its impact on the rod is “almost” elastic, i.e. that it loses only a small part of its energy \(\left(\frac{\Delta \varepsilon}{\varepsilon} \ll 1\right)\), rebounding with a velocity close in magnitude to the initial one. Under these conditions the law of conservation of momentum may be written in the following form:

\[ 2m'v' = mv\frac{l}{a}. \tag{3} \]

As for the energy of the rod, it consists of the potential energy of compression of the segment \(l\) of the rod by \(\Delta l\),

\[ U=\frac{1}{2}E\frac{(\Delta l)^2}{l}, \]

where \(E\) denotes the modulus of compression, and of their kinetic energy

\[ K=\frac{1}{2}mv^{2}\frac{l}{a}. \]

It is not difficult to see that these two parts of the energy must be equal to one another.

For this purpose let us note, first of all, that the speed of sound \(c\) is related to \(E\) by the relation

\[ c^{2}=\frac{E}{\rho}, \]

where \(\rho=\dfrac{m}{a}\) is the linear density of the rod. Putting \(E=c^{2}\dfrac{m}{a}\), we obtain

\[ U=\frac{1}{2}c^{2}\frac{m}{a}\frac{(\Delta l)^{2}}{l} \]

or, since \(l=c\tau\) and \(\Delta l=v\tau\),

\[ U=\frac{1}{2}\frac{c\tau}{a}mv^{2}=\frac{1}{2}\frac{l}{a}mv^{2}=K. \]

The law of conservation of energy may therefore be expressed by the formula

\[ \Delta \varepsilon=mv^{2}\frac{l}{a}. \tag{4} \]

From (3) it follows that

\[ \varepsilon'=\frac{1}{8}\frac{m^{2}v^{2}}{m'}\left(\frac{l}{a}\right)^{2}. \tag{4a} \]

Thus we obtain the following expression for the accommodation coefficient

\[ \frac{\Delta \varepsilon}{\varepsilon'}=8\,\frac{m'}{m}\frac{a}{l} \tag{4b} \]

or, since

\[ l=c\tau=c\frac{2b}{v'}, \]

\[ \frac{\Delta \varepsilon}{\varepsilon'}=4\,\frac{m'}{m}\frac{a}{b}\frac{v'}{c}. \tag{5} \]

This expression differs from (1a) by the factor \(\dfrac{a}{b}\dfrac{v'}{c}\), which, for average values of \(v'\) at ordinary temperatures, is of the order of unity.

It follows from this that formula (5) is applicable only in the case where the mass of the incident particle \(m'\) is considerably smaller than the mass of the atoms forming the body under consideration, or if the velocity of this particle \(v'\) is very small in comparison with the speed of sound in this body (i.e. at very low gas temperatures).

If these conditions are not fulfilled, i.e. if the loss of energy in the impact \(\Delta \varepsilon\) is significant, then the change in momentum

of the striking particle upon rebounding must be less than \(2m'v'\). In the limiting case of its complete stopping at the surface of the body, the quantity of motion lost is equal to \(m'v'\) (instead of the maximum \(2m'v'\)). Equating it to \(mv\,\dfrac{l}{a}\) and combining this equality with formula (4), we obtain, instead of (5),

\[ \frac{\Delta \varepsilon}{\varepsilon'} = -\,\frac{m'\, [[unclear: obscured symbol]]\, v'}{m\,b\,c}. \]

But in the case under consideration \(\Delta \varepsilon=\varepsilon'\). Therefore the preceding equality can hold only under the condition that its right-hand side is equal to 1. Since there are no a priori grounds for the fulfillment of this condition, it follows that the theory set forth above is applicable only in the case of “weak impacts,” connected with a relatively small loss of energy, i.e. under the condition that \(\dfrac{\Delta \varepsilon}{\varepsilon'}\) is small in comparison with 1.

§ 3. Volumetric elastic model

We shall now pass from the one-dimensional case to the three-dimensional one, i.e. to the fall of a gas particle onto a solid body with a plane surface (Fig. 2). If, as before, this body is treated as an elastic continuum, then the effect produced by the impact of a small sphere representing a gas particle (when the latter falls normally) reduces to the occurrence of a hemispherical compression wave, which during the time of impact propagates in all directions over a distance \(l\). We shall take into account only the longitudinal (radial) displacement of the particles of the body, i.e. in effect regard the latter as a liquid rather than a solid1. In this case the velocity of displacement of the particles along the radius vector \(r\), which makes an angle \(\theta\) with the direction of impact, may be expressed as a function of this angle and of the distance \(r\) by the formula

Fig. 2.

\[ v=\frac{v_0}{r}\cos\theta, \tag{6} \]

where \(v_0\) is a certain constant.

We shall now apply the laws of conservation of momentum and energy to the hemispherical volume under consideration. Then, just as in the one-dimensional case the number of particles encompassed by the motion was equal to \(\dfrac{l}{a}\), in the case under consideration it is equal to

\[ \frac{2\pi}{3}\frac{\beta^3}{a^3} = \frac{2\pi}{3}\frac{c^3}{a^3}\tau^3. \]

We have, therefore (under the assumption \(\Delta \varepsilon \ll \varepsilon\)),

\[ 2m'v'=\frac{2\pi}{3}\frac{l^3}{a^3}\,m\overline{v\cos\theta}, \tag{7} \]

where \(v\cos\theta\) is the projection of the radial velocity of one of the particles onto the direction of impact, and \(\overline{v\cos\theta}\) is its mean value over the entire hemispherical volume, and

\[ \Delta\varepsilon=\frac{2\pi}{3}\frac{l^3}{a^3}\,m\overline{v^2} \tag{8} \]

[cf. formulas (3) and (4)].

But according to (6)

\[ \overline{v\cos\theta} = v_0\cos^2\theta\, \frac{\displaystyle\int_0^l 4\pi\,dr} {\displaystyle\int_0^l 4\pi r^2\,dr} = v_0\cdot\frac{1}{3}\cdot\frac{3}{l^2}\cdot\frac{1}{l} = \frac{v_0}{2l} \]

and

\[ \overline{v^2} = v_0^2\cos^2\theta\, \frac{\displaystyle\int_0^l 4\pi\,dr} {\displaystyle\int_0^l 4\pi r^2\,dr} = v_0^2\cdot\frac{1}{3}\cdot\frac{3}{l^2} = \frac{v_0^2}{l^2}. \]

Thus

\[ 2m'v'=\frac{\pi}{3}\frac{l^2}{a^3}\,mv_0 \]

and

\[ \Delta\varepsilon=\frac{2\pi}{3}\frac{l}{a^3}\,mv_0^2. \]

From the first equality we find

\[ \varepsilon'=\frac{1}{2}m'v'^2 = \frac{1}{8m'}\frac{\pi^2}{9}\frac{l^4}{a^6}\,m^2v_0^2. \]

Hence it follows that

\[ \frac{\Delta\varepsilon}{\varepsilon'} = 4\frac{m'}{m}\cdot\frac{3}{\pi}\frac{a^3}{l^3} \tag{7a} \]

or, since

\[ l=c\tau=c\frac{2b}{v'}, \]

\[ \frac{\Delta\varepsilon}{\varepsilon'} = 4\frac{m'}{m}\frac{3}{8\pi} \left(\frac{a}{b}\right)^3 \left(\frac{v'}{c}\right)^3. \tag{8a} \]

This expression is similar to that which was obtained above for the one-dimensional case, and, like the latter, is applicable only under the condition \(\frac{\Delta\varepsilon}{\varepsilon'}\ll 1\), i.e. either under the condition that the ratio \(\frac{m'}{m}\) is small, or else at not too high temperatures.

In the case of a Maxwellian distribution of velocities among the gas particles at temperature \(T\), the mean value of the ratio \(\frac{\Delta\varepsilon}{\varepsilon'}\)

turns out to be proportional to \(T^{3/2}\). The mean value of the energy \(\Delta E\) given up by the gas per unit time to a solid body, if the temperature of the latter is equal to absolute zero, is determined by the formula

\[ \Delta E=\int_{0}^{\infty}\Delta \varepsilon\, v' f(v')\,dv', \]

where

\[ f(v')\,dv'=A e^{-\frac{m'v'^2}{kT}}\,dv' \]

is the number of particles (per unit volume) whose velocity in the direction normal to the surface is contained in the interval between \(v'\) and \(v'+dv'\).

On the other hand, the total energy carried by these particles to the surface of the body is equal to

\[ \overline{E'}=\int_{0}^{\infty}\varepsilon' v'f(v')\,dv'. \]

Let us note that per one particle this amounts to

\[ \overline{\varepsilon'}= \frac{\displaystyle\int_{0}^{\infty}\varepsilon' v'f(v')\,dv'} {\displaystyle\int_{0}^{\infty}v'f(v')\,dv'}, \]

i.e.

\[ \overline{\varepsilon'}= \frac{\displaystyle \frac{1}{2}m'\int_{0}^{\infty}e^{-\alpha v'^2}v'^3\,dv'} {\displaystyle\int_{0}^{\infty}e^{-\alpha v'^2}v'\,dv'} =\frac{m'}{2\alpha}, \]

where

\[ \alpha=\frac{m'}{2kT}, \]

i.e.

\[ \overline{\varepsilon'}=kT \]

(and not \(2kT\)\(^{1}\)).

In an analogous way, for the mean loss of energy per one gas particle we obtain

\[ \Delta\varepsilon'= \frac{\displaystyle P\int_{0}^{\infty}v'^6 f(v')\,dv'} {\displaystyle\int_{0}^{\infty}v'f(v')\,dv'} = P\frac{15}{8}\alpha^{-\frac{5}{2}} = Q(kT)^{\frac{3}{5}}, \]

\(^{1}\) The value \(2kT\) is obtained if one starts from the distribution of velocities in three dimensions. In this case \(v'f(v')\,dv'\) should be replaced by

\[ v'_x f(v')\,dv'_x\,dv'_y\,dv'_z, \]

putting

\[ v'=\sqrt{v_x'^2+v_y'^2+v_z'^2}. \]

where, according to (8),

\[ P=\frac{3}{4\pi}\left(\frac{a}{b}\right)^3 \frac{m'^2}{m c^3} \]

and

\[ Q=\frac{8}{15}P\left(\frac{2}{m'}\right)^{\frac{5}{2}} . \]

The accommodation coefficient is defined as the limiting value of the ratio of the energy given up by the gas to the body, when the temperature of the latter \(T_0\) is infinitely close to the temperature of the gas, to the difference between the energy brought by the gas particles and that which they would carry away after assuming the temperature of the body. Proceeding from the circumstance that at \(T=T_0\) the energy given by the gas particles to the atoms of the solid must be equal to the energy which they receive from the latter (otherwise the exchange of energy between the gas and the solid would lead to a violation of the equality of their temperatures), it is sometimes assumed that the energy received by the gas from the solid at temperature \(T_0\) is equal to \(Q(kT_0)^{\frac{5}{2}}\). As a result, for the accommodation coefficient at temperature \(T \to T_0\) one obtains the formula

\[ \alpha =Q\lim_{T\to T_0} \frac{(kT)^{\frac{5}{2}}-(kT_0)^{\frac{5}{2}}}{kT-kT_0} =\frac{5}{2}Q(kT)^{\frac{3}{2}}, \]

or, according to the definition of \(Q\),

\[ \alpha=\operatorname{const}\left(\frac{a}{bc}\right)^3 \frac{1}{m\sqrt{m'}}(kT)^{\frac{3}{2}}, \tag{9} \]

where const is a numerical coefficient.

This result cannot be regarded as rigorously established, since it is based on the assumption (not explicitly formulated) that, when the temperatures of the gas and of the solid are unequal, the exchange of energy between them can be calculated by subtracting from one another the energies \(\Delta \varepsilon'\) taken away by the solid at temperature 0 from the gas at temperature \(T\) in one case and \(T_0\) in the other.

This assumption evidently requires special verification (see below).

§ 4. Model of a Quasielastically Bound Atom

(Classical Theory)

According to formula (7a), the relative loss of energy of the striking particle is the greater, the shorter the impact, i.e. the smaller the distance over which the compression wave propagates during the collision. In this case the minimum value of \(l\) is evidently the distance between neighboring atoms, \(a\).

At those velocities \(v'\) which correspond to ordinary tem-

temperatures \((v' \simeq 10^5\ \mathrm{cm/sec})\), the effective duration of the collision (impact) \(\tau\) is approximately \(10^{-13}\ \mathrm{sec}\); in this case the range of propagation of the compression wave \(l\) is of the order of one or several interatomic distances. Under such conditions the replacement of a solid body by an elastic continuum is completely inadmissible.

We must therefore, in order to improve the preceding theory, regard the solid body as a lattice formed by individual atoms. Since, however, the effective range of propagation of the compression wave in a sharp impact (with small effective duration) is measured by one or several atomic distances, it seems quite permissible, for the simplification of calculations in the case of such impacts, to replace the solid body by a single atom only—not a free one, as in § 1, but quasi-elastically bound to a certain equilibrium position determined by all the other atoms (which may be replaced by one immobile atom of infinitely large mass). We thus arrive at the model of a harmonic oscillator, which has already been used by a number of authors for semiclassical (Zener) or consistently quantum-mechanical (Zener, Johnson, Lennard-Jones, and others) calculations of the accommodation coefficient.

Fig. 3.

Incidentally, a similar model was first introduced by Bohr as early as 1915 to describe the behavior of bound electrons when fast charged particles pass through an atom. In this connection Bohr showed that in the case when the effective duration of interaction \(\tau\) of such a particle with the electron under consideration is small in comparison with the period \(\tau_0\) of its proper oscillations, the electron may be regarded as completely free, and the energy taken by it from the passing particle may be calculated without taking account of the forces binding it inside the atom; whereas in the opposite case \((\tau \gg 1)\) it may be regarded as bound infinitely firmly and, accordingly, as altogether incapable of taking energy from the particle flying past.

In Bohr’s theory the effective duration of the collision was determined by a formula of the same form as in the preceding paragraphs, namely
\[ \tau=\frac{2b}{v'}, \]
where by \(b\) was meant the “impact distance” (i.e., the minimum distance of the undeflected trajectory of the particle from the electron).

We shall show, first of all, that Bohr’s considerations remain valid quite independently of the law of interaction between the impacting particle (in the case of interest to us, a gas particle) and the oscillator, i.e., the quasi-elastically bound atom of the solid on which the impact falls. We shall here regard the impact as head-on, just as in § 1 (Fig. 3, where the binding of the atom with all the others is replaced by a spring connecting it with an immobile wall).

Suppose that the force with which the incident particle acts on the atom under consideration is known to us as a function of time, \(mF(t)\), different from zero only during some short time \(\tau\). The equation of motion of this atom may be written as follows:

\[ \frac{d^{2}q}{dt^{2}}+\omega_{0}^{2}q=F(t), \tag{10} \]

where \(q\) denotes its displacement from the equilibrium position, and \(\omega_{0}=\dfrac{2\pi}{\tau_{0}}\).

To solve the problem we expand \(F(t)\) in a Fourier integral

\[ F(t)=\int_{-\infty}^{\infty} F_{\omega}^{0} e^{i\omega t}\,d\omega \]

and seek \(q\) in the form

\[ q(t)=\int_{-\infty}^{+\infty} q_{\omega}\,d\omega, \]

where \(q_{\omega}\) satisfies the equation

\[ \ddot q_{\omega}+\omega_{0}^{2}q=F_{\omega}^{0}e^{i\omega t} \]

with the initial conditions at the moment \(t=0\) (“before the collision,” i.e. when the force \(F(t)\) is still equal to zero): \(q_{\omega}=0\) and \(\dot q_{\omega}=0\). We then obtain

\[ q_{\omega}=\frac{F_{\omega}^{0}}{2\omega_{0}} \left[ \frac{e^{i\omega t}-e^{-i\omega t}}{\omega+\omega_{0}} - \frac{e^{i\omega t}-e^{i\omega_{0}t}}{\omega-\omega_{0}} \right] \tag{11} \]

and, further,

\[ q(t)=-\frac{e^{i\omega_{0}t}}{2\omega_{0}} \int_{-\infty}^{+\infty} F_{\omega}^{0}\, \frac{e^{i(\omega-\omega_{0})t}-1}{\omega-\omega_{0}} \,d\omega + \]

\[ +\frac{e^{-i\omega_{0}t}}{2\omega_{0}} \int_{-\infty}^{+\infty} F_{\omega}^{0}\, \frac{e^{i(\omega+\omega_{0})t}-1}{\omega+\omega_{0}} \,d\omega . \]

Replacing \(\omega\) by \(-\omega\) in the second integral and noting that, in view of the reality of \(F(t)\), \(F_{-\omega}^{0}\) is equal to \(F_{\omega}^{0*}\) (i.e. is equal to the complex conjugate of \(F_{\omega}^{0}\)), we can represent this integral as a quantity complex-conjugate to the first integral.

Since, furthermore, the factor \(\dfrac{e^{i(\omega-\omega_{0})t}-1}{\omega-\omega_{0}}\) in the integrand has a resonant character, i.e. has a sharply pronounced,

On the Theory of the Phenomena of Accommodation and Condensation

maximum at \(\omega=\omega_0\), and for sufficiently large values of \(t\), then in the latter case one may put

\[ \int_{-\infty}^{+\infty} F_\omega^0 \frac{e^{i(\omega-\omega_0)t}-1}{\omega-\omega_0}\,d\omega = F_{\omega_0}^{0} \int_{-\infty}^{+\infty}\frac{e^{i\xi t}}{\xi}\,d\xi \qquad (\xi=\omega-\omega_0), \]

i.e., take outside the integral sign the resonant value \(F_\omega^0\) at the point \(\omega=\omega_0\). Finally, putting \(e^{i\xi t}=\cos \xi t+i\sin \xi t\) and observing that, owing to the oddness of \(\dfrac{\cos \xi t}{\xi}\) as a function of \(\xi\), the integral \(\displaystyle \int_{-\infty}^{+\infty}\frac{\cos \xi t}{\xi}\,d\xi\) vanishes, we obtain

\[ \int_{-\infty}^{+\infty}\frac{e^{i\xi t}}{\xi}\,d\xi = i\int_{-\infty}^{+\infty}\frac{\sin \xi t}{\xi t}\,d(\xi t) = i\pi \]

and, consequently,

\[ q(t)=Ae^{i\omega_0 t}+A^*e^{-i\omega_0 t}, \tag{12} \]

where

\[ A=-\frac{i\pi}{\omega_0}F_{\omega_0}^{0}. \tag{12a} \]

It should be remembered that these formulas are valid not for every value of \(t\), but only for such values of \(t\) for which the difference

\[ \omega-\omega_0=\frac{\varepsilon\pi}{t} \]

(corresponding to the turning of the resonance multiplier into zero) is so small that the difference between the values

\[ F_\omega^0\bigg|_{\omega=\omega_0+\frac{2\pi}{T}} \]

and \(F_{\omega_0}^0\) may be neglected.

Thus formula (12) has an asymptotic character and is justified the more accurately, the larger \(t\) is. It is therefore entirely suitable for describing the state of the oscillator after the end of the collision, i.e., after the force \(F(t)\) has been brought to zero. In this case the oscillator retains an energy \(\Delta\varepsilon\), equal to

\[ \Delta\varepsilon = \frac{1}{2}m\omega_0^2q^2+\frac{1}{2}m\dot q^{\,2} = m\omega_0^2q^2 = 2m\omega_0^2AA^*; \]

i.e.,

\[ \Delta\varepsilon=2\pi^2m\,|F_{\omega_0}^{0}|^2. \tag{13} \]

We shall apply this formula to the case of a force whose dependence on time is expressed in the following way:

\[ \begin{array}{rcl} \text{for } t<t_1, &\quad& F=0,\\[2mm] \text{” } t_1<t<t_2, && F=\dfrac{f}{m}=\text{const},\\[2mm] \text{” } t>t_2, && F=0. \end{array} \]

We then obtain, according to the general formula,

\[ F_{\omega}^{0}=\frac{1}{2\pi}\int_{-\infty}^{+\infty} F(t)e^{-i\omega t}\,dt, \tag{14} \]

\[ F_{\omega_0}^{0}=\frac{f}{\pi m}\,\frac{\sin \dfrac{\omega_0\tau}{2}}{\omega_0}, \tag{14a} \]

where \(\tau=t_2-t_1\) is the duration of the collision.

If this duration is small in comparison with the period of oscillation \(\tau_0=\dfrac{2\pi}{\omega_0}\), i.e. if \(\omega_0\tau\ll 1\), one may put

\[ F_{\omega_0}^{0}=\frac{f\tau}{2\pi m}, \]

which gives

\[ \Delta\varepsilon=\frac{1}{2}\frac{f^2\tau^2}{m}. \]

Here \(f\tau\) represents the impulse of the force with which the striking particle acts on the atom during the collision. Taking the energy \(\Delta\varepsilon\) acquired by the latter to be small in comparison with the initial energy of the particle

\[ \varepsilon'=\frac{1}{2}m'v'^2, \]

one may put

\[ f^2\tau^2=(2m'v')^2=8m'\varepsilon', \]

whence we finally obtain

\[ \frac{\Delta\varepsilon}{\varepsilon'}=\frac{4m'}{m}. \]

This formula coincides exactly with formula (1a) for the energy taken away by a free atom.

If, on the contrary, the duration of the collision \(\tau\) is large in comparison with \(\tau_0\), then one may put

\[ \left|F_{\omega_0}^{0}\right|^2 = \frac{f^2}{\pi^2m^2}\, \frac{\sin^2\dfrac{\omega_0\tau}{2}}{\omega_0^2} = \frac{f^2}{2\pi^2m^2\omega_0^2}, \]

which gives

\[ \Delta\varepsilon=\frac{f^2}{4m\omega_0^2} \simeq 2\frac{m'}{m}\,\varepsilon'\,\frac{1}{\omega_0^2\tau^2}, \]

so that, for \(\omega_0\tau\gg 1\), the relative loss of energy turns out to be much smaller than in the preceding case [by a factor of the order of

\[ \left(\frac{\tau_0}{\tau}\right)^2 \]

].

The results set forth make it possible to calculate the value of the accommodation coefficient when gas particles with temperature \(T\) rebound from a solid body with temperature \(0\) in the following approximate manner.

Let us divide the gas particles into two groups: “slow” ones, for which the duration of the collision \(\tau=\dfrac{2b}{v}\) is greater than \(\tau_0\), and “fast” ones, for which it is less than \(\tau_0\).

The energy given up by the slow particles may be neglected, while the energy given up by the fast ones may be identified with that which corresponds to the absence of a quasi-elastic bond of the struck atom, i.e., to the case of a free atom considered in § 1. The mean value of the energy given up, referred to one gas particle, is then expressed by the formula

\[ \Delta e= \frac{ 4\,\frac{m'}{m}\displaystyle\int_{v_0'}^{\infty}\varepsilon' v' f(v')\,dv' }{ \displaystyle\int_{0}^{\infty} v' f(v')\,dv' }, \tag{15} \]

where \(v_0'\) is the limiting velocity corresponding to the condition \(\tau=\tau_0\), i.e.

\[ v_0'=\frac{2b}{\tau_0}. \]

Noting that

\[ \int_{v_0'}^{\infty} v^3 e^{-\alpha v^2}\,dv = -\frac{1}{2}\frac{\partial}{\partial\alpha} \int_{v_0'}^{\infty} e^{-\alpha v^2}\,d(v^2) = \]

\[ = -\frac{1}{2}\frac{\partial}{\partial\alpha} \frac{e^{-\alpha v_0'^2}}{\alpha} = \frac{1+\alpha v_0'^2}{2\alpha^2}e^{-\alpha v_0'^2}, \]

we obtain

\[ \Delta e = 2\,\frac{m'}{m}\, \frac{1+\alpha v_0'^2}{\alpha} e^{-\alpha v_0'^2} \]

or, since \(\alpha=\dfrac{m'}{2kT}\),

\[ \Delta e = 4\,\frac{m'}{m}\,(kT+\varepsilon_0')\,e^{-\frac{\varepsilon_0'}{kT}}, \tag{16} \]

where

\[ \varepsilon_0'=\frac{1}{2}m'v_0'^2=\frac{2m'b^2}{\tau_0^2} \tag{16a} \]

is the kinetic energy corresponding to the limiting velocity (15a).

Computing from this the accommodation coefficient \(\alpha\) as \(T\to T_0\) by the formula

\[ \alpha= \frac{\overline{\Delta e}(T)-\overline{\Delta e}(T_0)} {kT-kT_0}, \]

i.e.

\[ \alpha= \frac{d\overline{\Delta e}(T)}{d(kT)}, \tag{17} \]

just as in the preceding paragraph, we obtain

\[ \alpha= 4\,\frac{m'}{m} \left[ 1+ \frac{(kT+\varepsilon_0')\varepsilon_0'}{(kT)^2} \right] e^{-\frac{\varepsilon_0'}{kT}}. \tag{17a} \]

We note that for \(b=5\cdot 10^{-9}\) and \(\tau_0=10^{-13}\), \(v_0'=10^5\ \mathrm{cm/sec}\), which corresponds to an energy \(\varepsilon_0\) of the order of \(kT\), at moderate temperatures.

The preceding formulas have meaning only when the ratio \(\dfrac{m'}{m}\) is sufficiently small, i.e., when the gas particles are light in comparison with the atoms of the solid (this condition is fulfilled, for example, in the accommodation of hydrogen or helium on metals; see below, § 7). When \(m'\) is comparable with or greater than \(m\), the ratio \(\dfrac{m'}{m}\) must be replaced by

\[ \frac{m'm}{(m+m')^2} \]

[cf. formula (1)]. It should, however, be borne in mind that for \(m'>m\) the replacement of the solid by a single quasi-elastically bound atom is, strictly speaking, inadmissible. In this case it proves necessary to take into account a larger number of atoms—the larger, the greater the ratio \(\dfrac{m'}{m}\). We shall return to this question below (§ 7). The preceding theory can claim agreement with experiment only under the condition \(\dfrac{m'}{m}\ll 1\).

§ 5. Quantum theory of a quasi-elastically bound atom

It is interesting to note that the preceding results coincide exactly with those to which the quantum-mechanical theory of perturbation of a quasi-elastically bound atom (harmonic oscillator), under the action upon it of a prescribed external force \(mF(t)\), leads.

A force of this kind corresponds to the potential function

\[ U'(q)=-mF(t)q. \tag{18} \]

The motion performed by the oscillator under the influence of this perturbing force is described by a wave function of the form

\[ \psi=\sum C_n(t)\psi_n(q,t), \]

where

\[ \psi_n(q,t)=\varphi_n(q)e^{-i\frac{2\pi}{h}W_n t} \]

is the normalized wave function describing the unperturbed motion of the oscillator with energy

\[ W_n=\left(n+\frac{1}{2}\right)h\nu_0 \quad \left(\nu_0=\frac{\omega_0}{2\pi}\right). \]

The probability coefficients \(C_n\) are determined by the system of equations

\[ -\frac{h}{2\pi i}\frac{dC_n}{dt} = \sum U'_{np}C_p e^{i\omega_{np}t}, \tag{19} \]

where

\[ U_{np} = [[unclear: obscured formula]] \qquad (19a) \]

and, for brevity, it has been put

\[ q_{np}=\int q\psi_n^{*}\psi_p\,dq \]

and

\[ \omega_{np}=2\pi\frac{W_n-W_p}{h}. \]

Suppose that at the initial moment the oscillator is in the state \(p\), so that at \(t=0\), \(C_p=1\), while all the remaining \(C_n=0\). The system of equations (19) then reduces, in the first approximation, to

\[ -\frac{h}{2\pi i}\frac{dC_n}{dt}=U_{np}e^{i\omega_{np}t}\quad (n\ne p). \]

Substituting here expression (19a) and integrating with respect to \(t\) within the limits from \(t=0\) to \(t=\infty\), we obtain

\[ C_n=-\frac{2\pi i}{h}mq_{np}\int_0^\infty F(t)e^{i\omega_{np}t}\,dt. \tag{20} \]

The integral appearing in this formula is, according to (14), nothing other than the quantity \(2\pi F^0_{-\omega_{np}}=2\pi F^0_{\omega_{pn}}\), obtained in expanding \(F(t)\) into a Fourier integral. Thus the preceding formula may be written in the form

\[ C_n=-\frac{4\pi^2}{h}imq_{np}F^0_{\omega_{pn}}. \tag{20a} \]

The square of the modulus of this quantity,

\[ |C_n|^2=\frac{16\pi^4}{h^2}m^2|q_{np}|^2|F^0_{\omega_{pn}}|^2, \tag{20b} \]

is the probability that the oscillator, under the action of the force under consideration, will pass from the initial state \(p\) into the state \(n\).

As is known from the quantum-mechanical theory of the harmonic oscillator, the quantity \((q_{pk})^2\) differs from zero only for \(n=p+1\) or \(n=p-1\), being expressed by the formula

\[ |q_{n,p}|^2=\frac{h}{4\pi m\omega_0}(p+1) \]

in the first case, and

\[ |q_{n,p}|^2=\frac{h}{4\pi m\omega_0}p \]

in the second. Noting that in this case \(\omega_{pn}=\pm\omega_0\), we obtain

\[ |C_{p\pm1}|^2=\frac{4\pi^3 m}{h\omega_0}|F^0_{\omega_0}|^2\left(p+\frac12\pm\frac12\right). \tag{21} \]

The energy which, according to quantum theory, an oscillator receives on the average is, obviously,

\[ \Delta \varepsilon = h\nu_0 \left(\left|C_{p+1}\right|^2-\left|C_{p-1}\right|^2\right), \tag{21a} \]

i.e., consequently,

\[ \Delta \varepsilon = 2\pi^2 m \left|F^0_{\omega_0}\right|^2, \tag{21b} \]

which coincides exactly with formula (13) of the classical theory.

The quantum-mechanical derivation of this formula shows, among other things, that the energy acquired by the oscillator under the action of a given “impact” force does not depend on the initial state of the oscillator, i.e., consequently, on the temperature of the body which it represents.

This result (which, incidentally, can also be easily obtained by the methods of the classical theory, by averaging over all initial phases of the oscillator) appears at first sight to contradict the circumstance that, when the temperature of the solid and of the gas is the same, the particles of the latter, upon collision with the former, on the average lose no energy. This contradiction is explained by the fact that in reality the force \(mF(t)\) cannot be regarded as prescribed, and that the result of the interaction between the gas particle and the atom of the solid body elastically bound in it is determined by the dependence of the force \(mF\) not on the time, but on the distance between them (more precisely, between their centers) \(x=q'-q\). Here \(q'\) denotes the coordinate of the particle, measured from the equilibrium position of the atom.

Denoting the potential energy of their interaction by \(U(x)\), and regarding the displacement \(q\) of the atom from its equilibrium position as small, one may expand \(U\) in a series in powers of \(q\) and put, in the first approximation,

\[ U(q'-q)=U(q')-qU'(q'), \tag{21'} \]

where \(U'(q')=f(q')\) is the force experienced by the atom under consideration from the gas particle at the distance \(q'\) between them [cf. formula (18)].

The energy of the system formed by them is represented in the form

\[ K=\frac{1}{2m}p^2+\frac{1}{2m'}p'^2+U(q'-q)+\frac{1}{2}m\omega_0^2q^2, \tag{21'a} \]

where \(p\) and \(p'\) denote the momenta of the corresponding particles or, in quantum mechanics, the operators \(\dfrac{h}{2\pi i}\dfrac{\partial}{\partial q}\) and \(\dfrac{h}{2\pi i}\dfrac{\partial}{\partial q'}\).

Substituting expression (21) here, we see that the energy \(K\) is composed of the part

\[ H=\frac{1}{2m}p^2+\frac{1}{2}m\omega_0^2q^2, \tag{22} \]

which characterizes the unperturbed motion of the atom of the solid body, and, further, of the part

\[ H'=\frac{1}{2m'}p'^2+U(q'), \tag{22a} \]

defining the motion of the gas particle when this atom is fixed in its equilibrium position, and, finally, from the quantity

\[ S=-qU'(q'), \tag{22b} \]

which may be interpreted as the perturbation energy causing an exchange of energy between the gas particle and the atom of the solid body when they collide with one another.

§ 6. A consistent quantum theory of an elastically bound atom and a gas particle

The method which we used above can be refined somewhat if one determines the dependence of \(t'\), and thereby also \(U'(q')=-mF(t)\), on time by integrating the equation \(H'=\varepsilon'=\mathrm{const}\). We shall not, however, proceed by this “semiclassical” route, but shall consider the question of interest to us from a consistent quantum-mechanical point of view.

Fig. 4.

Let us suppose that the gas particle can move away from the point \(q'=0\) (i.e. from the equilibrium position of the atom of the solid body) only to a finite distance \(L\), after traversing which it is thrown back by an absolutely rigid wall \(B\) (Fig. 4).

Its displacement in the opposite direction, i.e. toward negative values of \(q'\), is in practice limited by a very small quantity depending on the form of the potential function \(U(q'-q)\), or \(U(q')\) for \(q=0\), i.e. for the unperturbed motion of the particle corresponding to the equilibrium position of the atom. Let us note that in the case of a potential function of the form \(U=Ae^{-\alpha q'}=Ae^{-q'/b}\), \(q'\) can in theory vary down to \(-\infty\); however, the potential energy then increases so rapidly that in practice only very small negative values of \(q'\) (of order \(-b\)) are attainable.

The unperturbed motion of the particle (for \(q=0\)) is determined by the quantum-mechanical equation

\[ H'\varphi'=\varepsilon'\varphi' \tag{23} \]

with a discrete spectrum \(\varepsilon'\), which approaches a continuous one as the distance \(L\) is increased. Taking the latter to be sufficiently large, we may determine the quantized values of the energy \(\varepsilon'\) by the same formula

\[ \varepsilon'=\varepsilon'_n=\frac{h^2 n^2}{8m'L^2}, \tag{23a} \]

which corresponds to the free motion of a particle between two absolutely rigid walls separated from one another by a distance \(L\).

The wave functions describing such motion, which vanish at \(q'=0\) and \(q'=L\), have the form

\[ \psi_{n'}(q',t)=\varphi_{n'}(q')e^{-\frac{i2\pi\varepsilon_{n'}t}{h}} = \sqrt{\frac{2}{L}}\sin\frac{\pi n' q'}{L}\, e^{-\frac{i2\pi\varepsilon_{n'}t}{h}}, \tag{23b} \]

where the normalization factor has been chosen so that

\[ \int_{0}^{L}\varphi_{n'}^{2}\,dq'=1. \]

It should be noted that, whereas formula (23a) is a quite sufficient approximation for the true values of the energies determined by equation (23), expression (23b) for the wave functions cannot be regarded as sufficiently accurate, since precisely the region adjoining the point \(q'=0\), where these functions deviate most strongly from the eigenfunctions of equation (23), is the most important in calculating the matrix elements of the perturbation energy (22), by which the probabilities of the various transition processes in the system under consideration are determined.

Nevertheless, in order to simplify the calculations, which in any case cannot claim great accuracy,1 we shall in what follows use the approximate functions (23b).

Let us denote the initial state of our system (before the impact) by the quantum numbers \(p\) (oscillator) and \(p'\) (gas particle). The probability of finding it at the moment \(t>0\) in the state \(n,n'\) is equal to the square of the modulus of the coefficient \(C_{n,n'}(t)\), determined in the first approximation by the equation

\[ -\frac{h}{2\pi i}\frac{d}{dt}C_{n,n'} = S_{n,n';p,p'}e^{\,i\frac{2\pi}{h}(\varepsilon_n+\varepsilon_{n'}-\varepsilon_p-\varepsilon_{p'})t}, \tag{24} \]

where

\[ S_{nn';pp'} = \iint \varphi_n^*\varphi_{n'}^{\prime *}S\varphi_p\varphi_{p'}'\,dq\,dq' = -\,q_{np}U_{n'p'}. \tag{24a} \]

Here \(q_{np}=\int q\varphi_n^*\varphi_p\,dq\) is the matrix element of the coordinate of the quasi-elastically bound atom (cf. § 5), and \(U_{n'p'}\) is the matrix element of the force exerted on it, in the equilibrium position, by the impacting particle.

Integrating the preceding equation under the condition \(C_{nn'}=0\) at \(t=0\) and forming the squared modulus of the coefficient \(C\), we obtain

\[ |C_{nn'}|^2 = \frac{4\pi^2}{h^3}\,|S_{nn';pp'}|^2\, \frac{\sin^2 \dfrac{\Delta\omega_{nn'}t}{2}} {\left(\dfrac{\Delta\omega_{nn'}}{2}\right)^2}, \tag{25} \]

where, for brevity, we have put

\[ \Delta\omega_{nn'} = \frac{2\pi}{h}\left(\varepsilon_n+\varepsilon'_{n'}-\varepsilon_p-\varepsilon'_{p'}\right). \tag{25a} \]

Only those (resonant) transitions are of essential importance for which this quantity is close to zero.

In this case, to each admissible value \(p(=p\pm1)^1\) there corresponds, as \(L\to\infty\), an almost continuous sequence of values of the quantum number \(n'\), approximately satisfying the resonance (or energy-conservation) condition \(\Delta\omega_{nn'}=0\). The probability of the transition \(p\to p\pm1\) for a given value of \(n'\) is expressed, consequently, by the sum of the values \(|C_{nn'}|^2\) over all values \(n'\) close to the resonant one. According to \((23a)\), the number of such values (i.e. of stationary states of a gas particle) in the energy interval between \(\varepsilon'\) and \(\varepsilon'+d\varepsilon'\) is equal to

\[ dn'=\frac{L}{h}\sqrt{\frac{2m'}{\varepsilon'}}\,d\varepsilon' \]

or, since \(d\Delta\omega_{nn'}=\dfrac{2\pi}{h}\,d\varepsilon'\),

\[ dn'=\frac{L}{2\pi}\sqrt{\frac{2m'}{\varepsilon'}}\,d\Delta\omega_{nn'}. \]

Replacing the sum by an integral, we obtain, therefore,

\[ \int |C_{nn'}|^2\,dn' = \frac{2\pi L}{h^2} \int |S_{nn';pp'}|^2 \frac{\sin^2 \dfrac{\Delta\omega_{nn'}}{2}\,t} {\left(\dfrac{\Delta\omega_{nn'}}{2}\right)^2} \sqrt{\frac{2m'}{\varepsilon'}}\,d\Delta\omega_{nn'} . \]

The integration extends over an interval of values of \(\Delta\omega_{nn}\) of very small width, of order \(\dfrac{\pi}{t}\). Without appreciable error it may be widened to \(\pm\infty\), taking outside the integral sign the values of the slowly varying factors \((S^0)^2\) and \(\sqrt{\dfrac{2m'}{\varepsilon'}}\) at the maximum point of the resonance factor, i.e. at \(\varepsilon'=\varepsilon'_0=\varepsilon_p+\varepsilon'_{p'}-\varepsilon_n\). Noting that

\[ \int_{-\infty}^{+\infty}\frac{\sin^2 \xi t}{\xi^2}\,d\xi = t\int_{-\infty}^{+\infty}\frac{\sin^2 \xi t}{(\xi t)^2}\,d(\xi t) = \pi t, \]

\(^1\) For other values the matrix element vanishes.

we find for the probability of the transition under consideration over a time interval \(t\) the expression

\[ \int |C_{nn'}|^2\,dn' = \frac{2\pi^2}{h^2} \sqrt{\frac{2m'}{\varepsilon_0'}} \,L\,|S_{nn';\,pp'}|^2\,t . \]

During this time the gas particle must collide with the oscillator

\[ \frac{V_{p'}t}{2L} \]

times, where \(V_{p'}=\sqrt{2\varepsilon_{p'}'/m'}\) is its initial velocity. Dividing the preceding expression by this quantity, we obtain the probability of the transition under consideration in one collision

\[ \Gamma_{np}(\varepsilon_{p'}') = \frac{4\pi^2}{h^2} \,L^2\, \frac{m'}{\sqrt{\varepsilon_{n'}'\varepsilon_{p'}'}} \,|S_{nn';\,pp'}|^2 \tag{26} \]

(the index \(0\) at \(n'\) we shall omit for brevity).

This expression appears, at first glance, to depend on \(L\), which obviously would have no physical meaning. It is not difficult, however, to verify that the quantity \(|S|^2\) is inversely proportional to \(L^2\), so that this factor in fact cancels.

According to formula (24a), this question reduces to consideration of the matrix element

\[ U_{n'p'}'=\int U'(q')\varphi_{n'}^{*}\varphi_{p'}'\,dq' . \]

Whatever the exact expression of the functions \(\psi'\) may be, they, just like the approximate functions (23b), must be inversely proportional to

\[ \frac{1}{\sqrt{L}} \]

in order that the normalization condition

\[ \int_{-\infty}^{+\infty}|\varphi'|^2\,dq'=1 \]

be satisfied as \(L'\to\infty\). Putting

\[ \varphi_n'=\frac{\psi_n}{\sqrt{L}}, \]

where the functions \(\psi\) retain a finite value as \(L\to\infty\), we obtain

\[ U_{n'p'}' = \frac{1}{L}\int U'\psi_{n'}^{*}\psi_{p'}'\,dq' = \frac{1}{L}\,\overline{U}_{n'p'}' \]

and, consequently,

\[ S_{nn';\,pp'}=q_{np}\overline{U}_{n'p'}'L^{-2}. \tag{26a} \]

Thus we finally find

\[ \Gamma_{np}(\varepsilon_{p'}') = \frac{4\pi^2}{h^2} \frac{m'}{\sqrt{\varepsilon_{p'}'\varepsilon_{n'}'}} \,|q_{np}|^2\,|\overline{U}_{n'p'}'|^2 = \]

\[ = \frac{8\pi^2}{h^2} \frac{1}{v'^2} \,|q_{np}|^2\,|\overline{U}_{n'p'}'|^2, \tag{26b} \]

where \(v'=\sqrt{v_{n'}'v_{p'}'}\) is the geometric mean value of the velocity of the gas particle before and after the impact.

This expression, as is not difficult to see, is very close to expression (20b), derived under the assumption that the elastically bound particle experiences, from the incident force, \(mF(t)\), varying in a prescribed manner with time. Indeed, since by assumption \(mF(t)=U'(q')\), where \(q'\) should be treated as a prescribed function of time, we have

\[ 2\pi mF_{\omega_0} = \int_{-\infty}^{+\infty} U'(q')e^{-i\omega_0 t}\,dt . \]

Putting here \(dt=\frac{dq'}{v'}\) and regarding the velocity as constant at each of the two stages of the collision (\(v'=v_p'\) before the collision, \(v'=v_{n'}'\) after the collision), we obtain

\[ 2\pi mF_{\omega_0} = \int_{-\infty}^{0} U'(q')e^{-\frac{i\omega_0 q'}{v_p'}}\,\frac{dq'}{v_p'} + \int_{0}^{\infty} U'(q')e^{-\frac{i\omega_0 q'}{v_{n'}'}}\,\frac{dq'}{v_{n'}'} , \]

or approximately

\[ 2\pi mF_{\omega_0} = 2\int_{0}^{\infty} U'(q')e^{-\frac{i\omega_0 q'}{v'}}\,\frac{dq'}{v'} , \]

where \(v'\) is some velocity intermediate between \(v_p'\) and \(v_{n'}'\), for example the “geometric-mean” velocity already derived above, \(\sqrt{v_p'v_{n'}'}\).

Thus formulas (20) and (26) differ from one another only in that the expression

\[ 2\int_{0}^{\infty} U'(q')e^{-\frac{i\omega_0 q'}{v'}}\,\frac{dq'}{v'} \tag{27} \]

in the first of them corresponds to the expression

\[ \sqrt{2}\,\frac{1}{v'}\int_{0}^{\infty} U'(q')\varphi_{n'}^{*}\varphi_p\,dq' \tag{27a} \]

in the second.

Putting here

\[ \varphi_{n'}=\sqrt{2}\sin\frac{\pi n' q'}{L'} \]

according to (23) and replacing \(\sin x\) by \(\frac{e^{ix}-e^{-ix}}{2i}\), we can represent (27a) in the form of a sum of four terms containing exponential factors with exponents \(\pm \frac{\pi(n'+p')}{L}q'\) and \(\pm \frac{\pi(n'-p')}{L}q'\). The first two terms, in view of their rapidly oscillating character,

corresponding multipliers may be neglected, whereas the second two give

\[ +\frac{\sqrt{2}}{4}\frac{1}{\bar v'} \left\{ \int_{0}^{\infty} U'(q') e^{\frac{i\pi}{L}(n'-p')q'}\,dq' + \int_{0}^{\infty} U'(q') e^{-\frac{i\pi}{L}(n'-p')q'}\,dq' \right\}. \]

In the case where the difference \((n'-p')\) is small in comparison with \(n'\) and \(p'\), i.e., for a relatively small change of the energy of the incident particle in the collision, one may put, according to (23a),

\[ \varepsilon_{n'}'-\varepsilon_{p'}' = \frac{h^{2}}{8m'L^{2}}(n'+p')(n'-p') \simeq \frac{h}{4m'L}\frac{hp'}{L}(n'-p') \simeq \]

\[ \simeq \frac{h}{4m'L}\frac{hn'}{L}(n'-p') \]

or, since \(\dfrac{hp'}{2L}=m'\bar v'_{p'}\) and \(\dfrac{hn'}{2L}=m'\bar v'_{n'}\),

\[ \varepsilon_{n'}'-\varepsilon_{p'}' = \frac{h}{2L}\bar v'(n'-p'). \]

Replacing here \(\varepsilon_{n'}'-\varepsilon_{p'}'\) by \(-\dfrac{h\omega_{0}}{2\pi}\), we obtain

\[ \frac{\pi}{L}(n'-p')=-\frac{\omega_{0}}{\bar v'}. \]

Hence it is seen that expression (27a) practically coincides with the real part of expression (27).

Without dwelling on further details, we thus convince ourselves that, in the approximation under consideration, the consistent quantum theory of the collision of a gas particle with a harmonic oscillator replacing the solid body leads to the same results as the “semiclassical” theory of the preceding paragraph, where the motion of the gas particle was described classically.

The essential difference between the two theories consists only in the fact that whereas the first leads to a probable increase of the oscillator energy in the collision [according to formula (21a)], the second admits, along with an increase, also a probable decrease of the oscillator energy, depending on the relation between its initial energy and the initial energy of the gas particle.

To determine the mean value of the energy \(\overline{\Delta\varepsilon}\) given by the gas particle to the oscillator or, conversely, taken from the latter (in the case \(\Delta\varepsilon<0\)) in the collision, it is necessary to multiply expression (26b) by \(\varepsilon_{n}-\varepsilon_{p'}'\), sum over \(n\), and average over all initial states of both particles. In this connection the probability that the energy of the oscillator before the collision is equal to \(\varepsilon_{p}=ph\omega_{0}\) is proportional to the expression \(e^{-\frac{\varepsilon_{p}}{kT_{0}}}\), where \(T_{0}\) is the temperature of the solid body, while the probability that the energy of the incident par-

particle, corresponding to the longitudinal component of its velocity (in the direction toward the oscillator), is equal to \(\varepsilon'_{p'}\), is the expression \(e^{-\varepsilon'_p/kT}\), where \(T\) is the temperature of the gas. Taking into account the circumstance that the number of gas particles falling upon an area perpendicular to the direction of their motion per unit time with a given velocity \(v'\) is proportional to this velocity, for the probability of collision with particles whose energy lies in the interval between \(\varepsilon'\) and \(\varepsilon' + d\varepsilon'\), we obtain the expression

\[ \operatorname{const} e^{-\frac{\varepsilon'}{kT}} v'\,dv' \sim \operatorname{const} e^{-\frac{\varepsilon'}{kT}} d\varepsilon' \]

(cf. the end of § 3).

Summation of the product \((\varepsilon_n-\varepsilon_p)\Gamma_{np}\) over \(n\) gives

\[ \sum_n(\varepsilon_n-\varepsilon_p)\Gamma_{np}(\varepsilon') = \frac{h\omega_0}{2\pi}\left(\Gamma_{p+1,p}-\Gamma_{p-1,p}\right) = \]

\[ = \frac{h\omega_0}{2\pi}\cdot\frac{4\pi^2}{h^2}\frac{hm'}{4\pi m\omega_0} \left[ (p+1)V\left(\varepsilon'-\frac{h\omega_0}{2\pi},\varepsilon'\right) - pV\left(\varepsilon'+\frac{h\omega_0}{2\pi},\varepsilon'\right) \right], \]

where \(V\left(\varepsilon'\mp \dfrac{h\omega_0}{2\pi},\varepsilon'\right)\) is the value of

\[ \frac{|U_{n'p'}|^2}{\sqrt{\varepsilon'_{n'}\varepsilon'_{p'}}} \]

respectively in the case of transfer of the energy \(\dfrac{h\omega_0}{2\pi}\) by a gas particle to the oscillator or of its receiving this energy from the latter. Using the approximate functions (23) and replacing in them \(n'/L\) by

\[ \frac{\sqrt{8m'\varepsilon'}}{h}, \]

according to (23a) we have

\[ V\left(\varepsilon'\mp\frac{h\omega_0}{2\pi},\varepsilon'\right) = \]

\[ = \frac{2}{\sqrt{\left(\varepsilon'\mp\frac{h\omega_0}{2\pi}\right)\varepsilon'}} \left| \int U'(q')\sin\frac{2\pi}{h}\sqrt{2m'\left(\varepsilon'\mp\frac{h\omega_0}{2\pi}\right)} \right. \]

\[ \left. {}\times q'\sin\frac{2\pi}{h}\sqrt{2m'\varepsilon'}\,q'\,dq' \right|^2 . \]

The mean value of the energy transferred by a gas particle to the solid is then calculated from the formula

\[ \Delta\varepsilon = \frac{m'}{2m} \left[ (p+1)\overline{V(\varepsilon'-h\nu_0,\varepsilon')} - p'\overline{V(\varepsilon'+h\nu_0,\varepsilon')} \right], \tag{29} \]

where

\[ p = \frac{\displaystyle\sum_0^\infty p\,e^{-p\frac{h\nu_0}{kT_0}}} {\displaystyle\sum_0^\infty e^{-p\frac{h\nu_0}{kT_0}}} = \frac{1}{e^{\frac{h\nu_0}{kT_0}}-1}, \]

\[ \overline{p+1} = \frac{\displaystyle \sum_{0}^{\infty} (p+1)e^{-p\frac{h\nu_{0}}{kT_{0}}}} {\displaystyle \sum_{0}^{\infty} e^{-p\frac{h\nu_{0}}{kT_{0}}}} = p+1 = \frac{1}{1-e^{-\frac{h\nu_{0}}{kT_{0}}}}, \]

\[ V(\varepsilon' - h\nu_{0},\, \varepsilon') = \frac{ \displaystyle \int_{h\nu_{0}}^{\infty} V(\varepsilon' - h\nu_{0},\, \varepsilon') e^{-\frac{\varepsilon'}{kT}}\, d\varepsilon' }{ \displaystyle \int_{0}^{\infty} e^{-\frac{\varepsilon'}{kT}}\, d\varepsilon' } = \]

\[ = kT \int_{h\nu_{0}}^{\infty} V(\varepsilon' - h\nu_{0},\, \varepsilon') e^{-\frac{\varepsilon'}{kT}}\, d\varepsilon', \]

\[ V(\varepsilon' + h\nu_{0},\, \varepsilon') = kT \int_{0}^{\infty} V(\varepsilon' + h\nu_{0},\, \varepsilon') e^{-\frac{\varepsilon'}{kT}}\, d\varepsilon'. \]

We shall not dwell on the actual computation of these expressions. Let us note only that for \(T=T_{0}\) the mean value \(\Delta \varepsilon\) identically vanishes. Indeed, in this case we have

\[ \overline{\Delta \varepsilon} = \frac{m'}{2m}\, \frac{kT}{e^{\frac{h\nu_{0}}{kT}}-1} \left[ e^{\frac{h\nu_{0}}{kT}} \int_{h\nu_{0}}^{\infty} V(\varepsilon' - h\nu_{0},\, \varepsilon') e^{-\frac{\varepsilon'}{kT}} \right. \]

\[ \left. - \int_{0}^{\infty} V(\varepsilon' + h\nu_{0},\, \varepsilon') e^{-\frac{\varepsilon'}{kT}}\, d\varepsilon' \right]. \]

In view of the symmetry of the function \(V(\varepsilon'',\, \varepsilon')\) with respect to both variables \(\varepsilon'\) and \(\varepsilon''=\varepsilon'-h\nu_{0}\), we have

\[ e^{\frac{h\nu_{0}}{kT}} \int_{h\nu_{0}}^{\infty} V(\varepsilon'',\, \varepsilon') e^{-\frac{\varepsilon'}{kT}}\, d\varepsilon' = \int_{0}^{\infty} V(\varepsilon'',\, \varepsilon''+h\nu_{0}) e^{-\frac{\varepsilon''}{kT}}\, d\varepsilon'' = \]

\[ = \int_{0}^{\infty} V(\varepsilon''+h\nu_{0},\, \varepsilon') e^{-\frac{\varepsilon'}{kT}}\, d\varepsilon', \]

whence it follows that \(\overline{\Delta\varepsilon'}=0\) in the case \(T=T_{0}\).

For \(T\ne T_{0}\) one may put

\[ e^{\frac{h\nu_{0}}{kT_{0}}} \int_{h\nu_{0}}^{\infty} V(\varepsilon' - h\nu_{0},\, \varepsilon') e^{-\frac{\varepsilon'}{kT}}\, d\varepsilon' = \]

\[ = e^{\frac{h\nu_{0}}{k}\left(\frac{1}{T_{0}}-\frac{1}{T}\right)} \int_{0}^{\infty} V(\varepsilon'' + h\nu_{0},\, \varepsilon'') e^{-\frac{\varepsilon''}{kT}}\, d\varepsilon''. \]

and, correspondingly, rewrite formula (29) in the form

\[ \overline{\Delta \varepsilon} = \frac{m'}{2m}\, \frac{kT}{e^{\frac{h\nu_0}{kT_0}}-1} \int_0^\infty V(\varepsilon' + h\nu_0,\varepsilon') e^{-\frac{\varepsilon'}{kT}}\,d\varepsilon' \left[ e^{\frac{h\nu_0}{kT}\left(\frac{1}{T_0}-\frac{1}{T}\right)}-1 \right]. \tag{29a} \]

It follows from this that, in the limit as \(T \to T_0\),

\[ \frac{\overline{\Delta \varepsilon}}{k(T-T_0)} = \frac{d\bar{\varepsilon}}{kdT} = \]

\[ = \frac{m'}{2m}\, \frac{h\nu_0}{e^{\frac{h\nu_0}{kT}}-1} \cdot \frac{1}{kT} \cdot \int_0^\infty V(\varepsilon' + h\nu_0,\varepsilon') e^{-\frac{\varepsilon'}{kT}}\,d\varepsilon' . \tag{29b} \]

This formula determines the accommodation coefficient (cf. § 3). Let us note that the expression

\[ \varepsilon = \frac{h\nu_0}{e^{\frac{h\nu_0}{kT}}-1} \]

is nothing other than the mean energy of the thermal motion of a quasi-elastically bound atom (replacing the solid body). At high temperatures \((kT \gg h\nu_0)\) it reduces to the classical expression \(kT\), canceling with the factor \(kT\) in the denominator of (29b). In this case the dependence of the accommodation coefficient on temperature is determined entirely by the integral factor

\[ \int_0^\infty v(\varepsilon' + h\nu_0,\varepsilon') e^{-\frac{\varepsilon'}{kT}}\,d\varepsilon' \]

and has a weakly expressed character. In the case of low temperatures \((kT \ll h\nu_0)\), however, the accommodation coefficient rapidly decreases with decreasing temperature, and this decrease is determined chiefly by the pre-integral factor \(\frac{\varepsilon}{kT}\).

These results are in good agreement with Richards’ experimental data on the accommodation coefficient of hydrogen and helium on the surface of various metallic bodies (see the following paragraph).

§ 7. Quantum theory of the linear and three-dimensional model of a solid body

The theory set forth above can claim to describe reality approximately only in the case when \(m' \ll m\), i.e. in the case of collisions of particles of a light gas, for example hydrogen or helium, with a solid body consisting of significantly heavier particles.

If this condition is not fulfilled, replacement of the solid body by a single quasi-elastically bound atom is impermissible. Better re-

results in this case in a linear model of a solid, i.e. a chain formed by an aggregate of equidistant atoms, which we have already considered in § 3, using the methods of classical mechanics. However, the application of quantum mechanics is complicated in the case of an infinite chain. We shall therefore replace the body under consideration by a chain of a finite number \(N+1\) atoms, and we shall regard the last atom as fixed immovably (for \(N \gg 1\) this circumstance plays no role from the physical point of view).

The motion of such a chain may be represented as a superposition of standing waves with an antinode at one end and a node at the other. Assigning to the fixed atom the number \(r=0\), and to the end atom of the chain the number \(r=N\), and treating the chain as a practically continuous rod of length \(L=N\cdot a\), we may represent the displacement of the atoms in one of the normal vibrations by the formula

\[ q_{rs}=\xi_s \sin \frac{2\pi ar}{\lambda_s}, \tag{30} \]

where \(s=1,2,\ldots\) is the number of the vibration,

\[ \lambda_s=\frac{4L}{2s-1} \tag{30a} \]

is the corresponding wavelength (in the fundamental vibration \(s=1\), the latter must be 4 times the length of the rod, i.e. the distance from the node to the antinode), and \(\xi_s\) is the \(s\)-th normal coordinate. In the case of free vibrations

\[ \xi_s=A_s\cos(2\pi\nu_s t+\varphi_s), \]

where \(A_s\) is the amplitude and \(\nu_s=\dfrac{c}{\lambda_s}\) the frequency of the vibrations (\(c\) is the velocity of their propagation). Taking account of the discrete structure of the chain, this formula is applicable only in the region of comparatively long waves; moreover, the maximum value of \(s\) is equal to \(N\) (i.e. to the number of degrees of freedom).

In the general case of free vibrations of the chain, the displacement of the end (the \(N\)-th) atom may be represented by a sum of expressions (30). Noting that

\[ \frac{Na}{\lambda_s}=\frac{L}{\lambda_s}=\frac{2s-1}{4}, \]

we obtain

\[ q_0=\sum_{s=1}^{n}\sin \frac{\pi}{2}(2s-1)\xi_s \]

or, since \(\sin \dfrac{\pi}{2}(2s-1)=\sin\left(\pi s-\dfrac{\pi}{2}\right)=-\cos\pi s=1\) for odd \(s\) and is equal to \(-1\) for even \(s\),

\[ q_0=\xi_1-\xi_2+\xi_3-\ldots \tag{31} \]

In the quantum-mechanical theory of the free vibrations of a chain, the normal coordinates \(\xi_s\) should be regarded not as harmoni-

...functions of time, but as arguments of the wave functions \(\psi_s(\xi_s)\), characterizing the probability of one or another value of \(\xi_s\) and determined by wave equations of the form

\[ \left(\frac{1}{2M}\frac{d^2}{d\xi_s^2}+\frac{1}{2}M\omega_s^2\xi_s^2\right)\psi_s=W\psi_s, \]

where \(\omega_s=2\pi\nu_s\), and \(M\) is a coefficient having the dimension of mass.

The quantum theory of vibrations of a chain caused by the impact of a gas particle upon it can be constructed as a theory of transitions between free vibrations caused by the perturbing function already considered above,

\[ S=-q_0U'(q'). \tag{31a} \]

Substituting here the value of \(q_0\) from (31), we obtain for the perturbation energy a sum of values corresponding to different normal coordinates, i.e. to different harmonic oscillators, the totality of which is equivalent to the chain under consideration.

Thus we in fact return to the problem already solved by us for a single harmonic oscillator, namely the single quasi-elastically bound atom. The difference consists only in the fact that, in order to estimate the mean energy given up by the gas particle to the solid body, we must now take into account \(N\) different possibilities, corresponding to the acquisition of this energy by one of the \(N\) oscillators in the form of the corresponding quantum \(h\nu_s\). The quantum number \(p\) introduced above, characterizing the initial state of the oscillator, must now be replaced by a set of \(N\) numbers \(p_s\), and in the final state of the body only one of the \(N\) numbers \(n_s\) can differ from \(p\) (and then, as before, only by \(\pm 1\)).

The probability that a ready particle, in collision with a solid body, exchanges energy precisely with the \(s\)-th oscillator can be calculated in exactly the same way as before, i.e. by formula (26 b) (with \(q\) replaced by \(\xi_s\)). Denoting it by \(P_{nsp_s}^{\nu}(\varepsilon')\), we can calculate the mean value of the energy given to the solid body, \(\Delta\varepsilon\), by summing expression (29a), in which \(\nu_0\) is replaced by \(\nu_s\), \(p\) by \(p_s\), and \(m\) by \(M\). In view of the fact that, for large values of \(N\), neighboring values of the frequencies \(\nu_s\) lie very close to one another, summation over \(s\) may be replaced by integration over \(\nu\), taking into account that the number \(ds\) of vibrations with frequencies in the interval between \(\nu\) and \(\nu+d\nu\), according to

\[ \nu_s=\frac{c}{\lambda_s}=\frac{c(2s+1)}{4L}, \]

is equal to

\[ ds=\frac{2L}{c}\,d\nu \]

and that \(\nu_{\min}\simeq 0\), while \(\nu_{\max}\simeq \dfrac{c}{2a}\).

In an analogous way the quantum theory can also be extended to the three-dimensional model corresponding to a real solid body. In this case the number of longitudinal vibrations, whose frequency ...

is contained in the interval \(\nu\) and \(\nu + d\nu\), is expressed, as is known, by the formula

\[ ds=\frac{4\pi v}{c^3}\nu^2 d\nu, \]

where \(v\) is the volume of the body.

For the accommodation coefficient at low temperatures one obtains an expression of the same form

\[ \varkappa=\mathrm{const}\,\frac{\varepsilon}{kT}, \]

as in the case of a single oscillator (if one neglects the dependence of the integral factor on the frequency), where, however, \(\varepsilon\) denotes the mean value of the energy

\[ \frac{h\nu}{e^{\frac{h\nu}{kT}}} \]

for all frequencies from \(\nu_s=0\) to \(\nu_s=\nu_{\max}=\frac{k\theta}{h}\) (\(\theta\) is the “characteristic temperature” of Debye). This mean value, for \(T \ll \theta\), as is known, is directly proportional to the fourth power of the absolute temperature. Thus, in the region of low temperatures, the accommodation coefficient proves to be approximately proportional to \(T^3\), like the heat capacity of solids. Richards’ experimental curves for the accommodation coefficient of hydrogen and helium on metals in fact resemble the heat-capacity curves of solids (Fig. 5). We shall not dwell on the further development of this question, and shall consider some other corrections needed by the theory set forth above.

Fig. 5.

§ 8. Further corrections and modifications of the theory of accommodation

These corrections may be reduced to the following principal points, which have special significance in the case of energetic impacts:

1. Allowance for nonlinear terms in the expression for the forces of interaction between the atoms of a solid as functions of their relative displacements. These nonlinear terms, proportional to the squares and higher powers of the relative displacements of neighboring atoms \((q_{r+1}-q_r)\), give rise, as is known, to relaxation phenomena in solids, expressed in the transfer of energy from some normal coordinates (“oscillators”) to others and, in particular, in the conversion of mechanical energy into heat (“damping” of sound waves, thermal resistance). When a ball rebounds from the surface of the earth, the energy of the ball decreases not only because part of it is propagated into the earth in the form of an elastic wave, but also on account of heating of the surface of the earth and of the ball during their contact with one another (this heating may, however, have a secondary—

...secondary character, being a consequence of the partial conversion of the energy of elastic vibrations caused by the impact into heat, i.e. the energy of incomparably more rapid vibrations of which thermal motion in solids is composed).

In our calculations we have taken no account whatever of this thermal effect. In a number of works concerned primarily with the question of electron emission and cathode sputtering when fast positive ions strike the cathode of a discharge tube (Kapitsa, Morgulis, etc.), the thermal effect, on the contrary, is placed in the foreground, and various secondary phenomena caused by the impact of an ion are derived from the local and short-lived heating which this impact produces. Instead of the equations of the theory of elasticity, one then uses the equations of heat conduction, assuming that practically the entire kinetic energy of the ion is converted into heat. Thus the question of the accommodation coefficient is simply removed. Indeed, in the case of gas particles (ions) possessing energies of the order of several hundred or thousand volts, i.e. exceeding by tens and hundreds of thousands of times the energy of thermal vibrations, the concept of the accommodation coefficient in its exact meaning \((T \to T_0)\) has no sense. This circumstance does not, however, eliminate the question of the mean value of the energy \(\overline{\Delta \varepsilon}\) given up when an ion strikes the cathode surface. Yet any satisfactory solution of this question is impossible without taking into account the nonlinear part of the interatomic forces and the relaxation effects associated with it.

3. Accounting for nonlinear terms in the expression for the interaction energy between a gas particle and an outer (surface) atom of a solid. In the quantum theory of the two preceding sections we restricted ourselves to the first-order term in the expansion of the interaction energy of the gas particle with the atom it strikes in powers of the displacement of the latter, \(q\) (or \(q_0\)), i.e. we set

\[ U(q' - q) = U(q') - qU'(q'). \]

In the case of “strong” impacts it is necessary to take into account higher-order terms in this expansion, i.e. to determine the perturbation energy by the formula

\[ S = -qU'(q') - \frac{q^2}{2} U''(q') - \frac{q^3}{6} U'''(q') - \ldots . \]

If, in doing this, the solid is replaced by a single bound quasi-elastic atom, then alongside the processes considered above, associated with the receipt or surrender by it of a single vibrational quantum, it proves necessary to take into consideration impacts in which its energy changes by 2, 3, and more quanta. The probability of these processes is determined by the matrix elements of the terms of the corresponding order in the preceding expression for \(S\) and, as is not difficult to see, decreases rapidly with increasing “order,” if ...

the energy of the incident particle is small, attaining, in the opposite case, a considerable magnitude.

When a solid is replaced by an aggregate of a large number of harmonic oscillators with different frequencies \(\nu_l\) (the linear and volume model), the terms of the second order correspond, generally speaking, to collisions in which the gas particle gives energy at once to two oscillators—one quantum to each—or receives energy from two of them, or, finally, gives energy to one of them and receives it from the other; instead of giving (or acquiring) one quantum each to two different oscillators, it may, however, give (or acquire) two quanta at once from one of them.

It should be borne in mind, however, that when quadratic terms in the expression for \(S\) are taken into account, it is necessary, in order to remain consistent, also to take into account the nonlinear terms in the expression for the forces of interaction between the atoms of the solid, which excludes the possibility of replacing the latter by a quasi-elastically bound atom or by a system of harmonic oscillators.

3. Transitions of the second and higher orders. The quantum theory of the two preceding sections is approximate not only in that it is restricted to terms of the first order in the expression \(S\) and in the expression for the forces of interaction between the atoms of the solid. It is approximate also in that it is restricted to the first approximation in solving the problem of perturbation theory, i.e., it considers only “simple” or direct transitions of the system gas particle plus solid from the initial state \((p,p')\) to the final \((n,n')\)\(^1\). Solving the same problem in the second approximation, we would obtain, alongside the simple transitions \(p,p' \to n,n'\), double transitions through various intermediate stages \(m,m'\), i.e., transitions of the form

\[ p,p' \to m,m' \to n,n', \]

the probability of which is proportional to the expression

\[ \left| \sum_{m,m'} \frac{S_{pp',\,mm'}\,S_{mm',\,nn'}}{W_{mm'}-W_{pp'}} \right|^2 \]

This probability proves to be different from zero for such transitions in which the vibrational state of the solid changes at once by 2 quanta, even in the case where one restricts oneself to the first (linear in \(q\)) term in the expression of the perturbation energy.

It follows from this that, taking into account terms of the second (and higher) order in this expression as applied to simple transitions, we must, in order to remain consistent, also take into consideration double (triple, etc.) transitions, for which the terms

\(^1\) Mathematically, this corresponds to replacing, on the right-hand side of the equations, the true values of the probability coefficients \(C_n(t)\) at the moment under consideration by their initial values at the moment \(t=0\).

of the first order give values comparable with those which, in the case of simple transitions, are given by terms of the second order.

We thus see that the consideration of more energetic collisions of a gas particle with a solid body—collisions in which the energy of the latter changes by several vibrational quanta—encounters extremely great difficulties.

§ 9. Adhesion (Adsorption) and Reflection

These difficulties appear especially clearly when considering the question of the adhesion of an impinging gas particle to the surface of a solid body. Such adhesion can take place only in the presence of attractive forces, to which there corresponds an adsorption energy. The latter, as is known, is of the order of ten large calories per gram-molecule, i.e. approximately 10–20 times greater than the energy of thermal motion of gas particles at ordinary temperatures. When a particle adheres, the adsorption energy, together with the kinetic energy of thermal motion, is transferred to the solid body in the form of vibrational energy. Taking into account that the energy of a vibrational quantum of maximum frequency \(h\nu_{\max}=k\theta\) is comparable with the energy of thermal motion at ordinary temperatures, we see that when a gas particle adheres the solid body must at once absorb several tens of vibrational quanta.

Using the methods of calculation set forth in the preceding paragraph, in order to describe such a process we would have to carry the expansion of the potential energy \(U(q'-q)\) to terms of the 10th or 20th order with respect to \(q\), or else, retaining only terms of first order, carry perturbation theory to the 10th or 20th order corresponding to transitions through 10–20 intermediate states. It is clear that calculations of this kind are impracticable and useless (even if they were practicable). Therefore, in order to compute the probability of adhesion (or reflection), it is necessary to resort to another, more adequate method.

This method is in principle very simple and consists in the following. When a gas particle adheres to a solid body, it ceases to be a gas particle and becomes a particle of the solid body, differing from the original one by the presence of this additional particle. The latter remains bound to the other particles of the body by forces of the same type as the interaction forces between them, i.e. by forces which may be regarded as quasi-elastic (proportional to relative displacements).

Thus, if the initial state of the system under consideration—the solid body \(A\) and the gas particle \(A'\)—before the impact can be described approximately by the product of two wave functions \(\psi_p\) and \(\psi'_{p'}\), as we did above, then the final state, i.e. the state in which the particle is adsorbed on

surface of the body, must be described by a function \(\Psi'_n\) of the same type as \(\psi_p\), but corresponding to the vibrational motion of the new, “compound” solid body \(AA'\). The latter differs from the original not only by the presence of an extra particle, but, in connection with this, by a completely different vibrational spectrum, i.e. by a different number and character of the normal coordinates, or of the harmonic oscillators equivalent to them. The significance of this difference is especially evident in the case where the body \(A\) itself consists of only a small number of particles. Thus, for example, if it is replaced by just one atom bound quasielastically, then the system \(AA'\), formed by the attachment to it of the atom \(A'\), can be described as a set of two harmonic oscillators having nothing in common with the simple oscillator that is the atom \(A\) taken separately.

As the number of atoms forming the body \(A\) increases, the difference between it and the “compound” body \(AA'\) does not decrease, but only, as it were, is dissolved throughout the entire vibrational spectrum, adding new vibrations to it and shifting the frequencies of the old ones by an insignificant amount.

Without going into the details of this question—which, incidentally, has not so far been solved in any satisfactory way—let us note that, knowing the functions \(\psi_p\) and \(\Psi'_n\), one can calculate the matrix element

\[ H_{n;pp'}=\int \cdots \int \Psi_n^{\prime *}H\psi_p\psi'_{p'}\,dq_1dq_2\ldots dq_Ndq', \]

where \(H\) denotes the complete energy operator of the whole system, and from this the probability of the transition \(p,p'\to n\), i.e. of adhesion of the particle upon its impact with the solid body [in this case the states \((p,p')\) and \(n\) must correspond to approximately the same energy].

In the case of the impact of a particle on a body capable of adsorbing it, it is necessary to introduce adsorbed states as intermediate ones also in calculating the probability of impacts accompanied in fact by rebound of the particle. It may then turn out that the probability of such impacts, determined in the second approximation of perturbation theory, i.e. as a quantity proportional to the expression

\[ \left|\sum_m \frac{H_{pp';m}\,H_{m;nn'}}{W_{p,p'}-W_m}\right|^2, \]

will be greater than the value obtained when calculating it in the first approximation by the method of §§ 6 and 7.

This applies especially to such impacts in which the solid body takes from the particle, or gives to it, several vibrational quanta, i.e. to sharply inelastic impacts.

The scheme set forth has in the last two years found wide application in the theory of collisions of elementary particles (\(L\)-particles, protons, neutrons) with atomic nuclei, especially with nuclei of heavy atoms, which can be treated as ma-

to ordinary solid (or liquid) bodies. In this case, however, the phenomena of resonance, connected with the existence of discrete quantized energy levels in the nucleus and appearing distinctly at small energies of the incident particle \(A'\), are of substantial interest. In the case of strong impacts, corresponding to a large excitation energy of the compound nucleus \(AA'\), resonance phenomena, as well as other quantum effects, recede into the background, and the probability of capture (i.e. adsorption) of the incident particle can be calculated by means of classical mechanics and statistics, as can the various secondary effects accompanying this capture, namely the heating of the nucleus and the “evaporation” of the captured particle, or of some other particle instead of it (which corresponds to nuclear reactions of various kinds).

If the application of classical mechanics proves possible in the region of nuclear collisions of sufficiently great strength, then it should be expected that in the case of collisions of atoms with ordinary solid bodies the classical methods are quite suitable and require quantum corrections only at low temperatures. We have already had occasion to convince ourselves of this in considering the question of the accommodation coefficient. The classical theory of this question, set out in §§ 1, 2, and 3, can easily be generalized to the case of collisions of a gas particle with a body capable of adsorbing it. In this case the interaction energy \(U(q-q')\), along with a term of the form \(Ae^{\alpha(q'-q)}\), corresponding to repulsive forces at small distances, must contain a term of the form \(-Be^{\beta(q'-q)}\), corresponding to attractive forces at somewhat larger distances \((\beta<\alpha)\).

The action of these attractive forces can be taken into account approximately by adding to the initial kinetic energy of the incident particle \(\varepsilon'\) the work done by these forces when the particle reaches the equilibrium position, and equal (approximately) to the adsorption energy \(U\). We may, consequently, calculate the energy \(\Delta\varepsilon\) given up by the particle in the impact by means of the methods set out in §§ 1, 2, and 3 and relating to the case of absence of attractive forces, replacing the initial kinetic energy \(\varepsilon'\) by the effective

\[ \varepsilon'_{\mathrm{eff}}=\varepsilon'+U \]

(at ordinary temperatures \(\varepsilon'_{\mathrm{eff}}\gg\varepsilon'\)).

Thus, for example, if the mass of the incident particle \(m'\) is small in comparison with the mass of the atoms of the solid body, then, replacing the latter by one free atom and regarding the latter as motionless before the impact, we obtain

\[ \Delta\varepsilon=\frac{4m'}{m}\varepsilon'_{\mathrm{eff}}. \]

If the energy \(\Delta\varepsilon\) proves greater than the initial energy \(\varepsilon'\), i.e., if

\[ 1+\frac{U}{\varepsilon'}>\frac{m}{4m'}, \]

then the particle will not be able to rebound again from

body and “stick” to it. The smallest value of the energy \(\varepsilon'\) at which sticking becomes impossible is

\[ \varepsilon'_0=\frac{U}{\dfrac{m}{4m'}-1}. \]

If the energies of the incident particles are distributed according to Maxwell’s law for the temperature \(T\), while the solid is at absolute zero temperature, then for the reflection coefficient one obtains the expression

\[ r=e^{-\frac{\varepsilon'_0}{kT}}, \]

equal to the fraction of the total number of incident particles whose energy is greater than \(\varepsilon'_0\).

Analogous results are obtained when the solid is replaced by a quasi-elastically bound atom or by a chain of similar atoms.

Experimental data on the coefficient of reflection and sticking are as yet extremely scanty. One may mention only the recent work of Sines, who, comparing the rapid evaporation of various liquids under intensive pumping of the vapor with the theoretically calculated rate of vapor condensation in the state of statistical equilibrium at the same temperature, found that in the case of metals the coefficient of reflection (for particles of their own vapor) is practically equal to zero, while for water it is close to 1 (sticking coefficient 0.03). The correctness of the latter result is, however, called into question by the circumstance that the surface of the evaporating water, owing to its relatively low thermal conductivity, could have had a temperature considerably lower than that which was measured in Sines’s experiment.

LITERATURE

1. Baule, Ann. d. Phys., 44, 145, 1914.
2. Zakharyin and Spivak, ZhETF, 6, 1113, 1936.
3. C. Zener, Phys. Rev., 40, 335, 1932.
4. C. Zener, Proc. Cambr. Phil. Soc., 29, 136, 1933.
5. Lennard-Jones and Strachan, Proc. Roy. Soc., 150, 442, 1935.
6. Lennard-Jones and Devonshire, Proc. Roy. Soc., 156, 6, 1936.
7. Lennard-Jones and Devonshire, Proc. Roy. Soc., 158, 253, 1937.
8. Roberts, Proc. Roy. Soc., 152, 445, 464, 1935.
9. Synes, Proc. Roy. Soc., 1937.

1. Since the solid body is replaced by a single atom, and in expanding \(U(q'-q)\) in a series only the first-order term with respect to \(q\) is retained. ↩↩↩↩↩