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On the absorption and dispersion of sound in molecular gases
Gordon, R. G. 1967
Physica 34 398-412
ON THE ABSORPTION AND DISPERSION OF SOUND IN MOLECULAR GASES by R. G. GORDON*) UniversiteLibre de Bruxelles, Bruxelles, Belgique.
synopsis The propagation of ultrasonic waves in dilute molecular gases IS studied theoretically. Explicit relations are obtained between the dynamics of successive binary molecular collisions, and the absorption coefficient and velocity of sound. These relations are especially useful in the study of rotationally inelastic collisions, since they remain valid even when many rotational energy levels are populated. The analysis is applied to rotational absorption in Ds gas, and it is shown that a good account of the experimental frequency dependence may be obtained by taking into account at least two successive binary collisions.
1. Introduction. The absorption and dispersion of sound by molecular gases is an important method in the study of inelastic collisions between moleculesr). As a sound wave passes through a molecular gas, the increased translational energy near a peak of compression is transformed by collisions into internal energy of rotations), vibrations) and chemical reactiona). This transfer of energy lags behind the sound wave itself, thereby contributing to the absorption of the sound. This phase lag may then be related to the probabilities for the various inelastic transitions induced by molecular collisions 4-g). The simplest possible system showing a relaxation effect would be a molecule with only two internal energy states. This model leads to simple formulas for acoustic absorption and dispersion, in terms of a single parameter, related to the collision cross section for the transition between the two states.6) 7). Another simple case is that in which the internal states are represented by a harmonic oscillator which is weakly coupled to the translational and rotational motion. Landau and Tellers) showed that the simple two-state formulas6) 7) still remain valid for an oscillator, if only a small change is made in the definition of the rate constant. The great simplification achieved for an oscillator by Landau and Teller (compare with refs. 5 and 8!) was *) Present address: Department of Chemistry, Harvard University, Cambridge, Mass. -
398 -
ABSORPTION
AND
DISPERSION
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IN MOLECULAR
GASES
399
only possible because of the special values of the energy levels and transition probabilities
in a weakly coupled harmonic
oscillator.
For other cases, such
as rotational energy levels, their simplification is not possible. The general case of many (N) arbitrary internal energy levels appears very complicateds) s), and the results depend in detail on all the &V(N - 1) transition probabilities between the states. This multitude of transition probabilities between many quantum states is certainly szlfficient to describe the acoustic properties in systems of many levels. However, we propose here that in fact such detail is not always necessary. We show that sound absorption by molecules with many excited rotational states are well described by only a few parameters, which are defined in terms of the molecular collisions which occur in a gas. In the first approximation, we obtain again the familiar absorption shape, with a generalized cross section which reduces to Kneser’s result for the two-level casee), and to Landau and Teller’s result for an oscillator a). This first approximation corresponds physically to keeping only the independent effects of single binary collisions. Higher approximations are obtained which correct for the correlated effects of two, three . . . successive binary collisions. These correlations are present because the final rotational state in one collision is the initial state for the next collision. Since the probabilities for energy transfer in a collision depend on the initial state, two successive collisions in general have a different effect than a pair of uncorrelated collisions. The correction terms which are thereby introduced vanish for either a two-level system or for an oscillator. Physically we expect that these correlation effects will rapidly become small as we consider longer sequences of collisions. Thus we expect the correction terms to decrease rapidly in magnitude as we go to higher correlations. These results find some experimental support in recent measurements of ultrasonic absorption in Ds gaslo), which can be represented keeping only the first correction term. This illustrates the advantage of the present method in the analysis of experimental data. The parameters to be determined in the conventional analysis are the transition probabilities between the internal states of the molecules. If there are only two levels significantly populated (for example, the two lowest rotational levels at low temperatures), then one can readily determine a single parameter related to the cross section for excitation. If there are three significant levels, it is already very difficult to determine the three independent transition probabilities from experimentrr). If there are more than three levels, it becomes quite impossible to deduce the +N (N - 1) transition probabilities uniquely from an experiment, even if the precision were much improved over that presently obtained. In contrast to these difficulties, the present approach leads to a rapidly convergent series of approximations, in which only a few welldefined dynamical parameters are needed to describe the entire sound absorption spectrum, even in a many-level system.
400
R.G.GORDON
A second advantage of the present method is related to the correspondence principle between quantum mechanics and classical mechanics. The usual description of sound absorption in terms of transitions between the internal quantum levels has no simple meaning in terms of classical mechanics. However, the description which we use here has the same form, whether classical or quantum mechanics is used to describe the collision dynamics. This point is particularly important for actual calculations. Quantum mechanical transition probabilities in molecular collisions can at present only be calculated with drastic approximationsis-1s) which break down when many levels can be excited by a collision. In contrast, exact classical calculations are possible for molecular collision traj ectoriesi7) is) and various cross sectionsi7) la). Using the results of the present study, such classical calculations may be compared with experiments on the absorption and dispersion of sound. Such comparisons should lead to a better understanding of intermolecular forces and torques, and of molecular collision processes. The general theory of sound absorption and dispersion is extended and clarified in section 2. The case of a dilute molecular gas is analyzed in sections 3 and 4. The results are discussed and applied in section 5 to the analysis of ultrasonic absorption experiments. 2. Oa the general theory of zcltrasonic absor#tion and disfiersio%. A microscopic theory of the absorption and velocity of ultrasonic waves must relate these properties to the molecular motion in the medium. The absorption coefficient 0: is defined by CL= dln A/d1
(2.1)
where A is the amplitude of the sound wave, and I is the distance of travel in the medium. The actual experimental value of u depends somewhat on whether the experiment is done with a steady wave pattern (as in an interferometer), or with a pulse. If the fluid obeys the usual hydrodynamic equations, then the absorption coefficient, for a steady state experiment, has the well-known valueso-2s) ci
1 + (4N2
=
$$ [+q +i-+ (c,’
- c,‘) K]
where 11. c and K are the hydrodynamic coefficients of shear viscosity, bulk viscosity, and thermal conductivity, respectively. C, and C, give the heat capacities at constant volume and pressure, p is the mass density, co the radian frequency, and k the wave vector. The phase velocity vp, which may be measured in a steady-state interferometer, is given by vP = co/k. The analysis
of a pulse experiment
is more complicated.
(2.3) In fact,
if the
ABSORPTION
AND
DISPERSION
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IN MOLECULAR
GASES
401
medium absorbs very strongly, then the shape of a pulse is badly distorted as it progresses, and analysis of the results is very difficult. However, if the absorption per wavelength is only moderate, then a pulse is distorted only gradually, and it travels with the well-known group velocity vg = do/dk.
(2.4)
From the definitions (2.3) and (2.4), we may express vg in terms of vP:
-&(kvp)
Vg =
dv,
= vp + kv, -
do
or (2.5) We wish to point out here that a pulse experiment also measures a somewhat different value of the absorption coefficient than does a steady-state experiment. It is not hard to show, from the basic definition (2.1), that
instead of the classical result (2.2). If the quadratic term in (2.2) is important, then the absorption is so strong that a more elaborate analysis of a pulse experiment would probably be necessary. In general the hydrodynamic coefficients 7, f and K must be considered to be frequency dependent 23-9. Allowing such a frequency-dependence has been criticized as a rather arbitrary device for obtaining agreement between theory and experiment 26)27). However, if the fluid dynamics are in fact well described by hydrodynamical equations with frequencydependent coefficients, then these coefficients may be given a well-defined interpretation in terms of the molecular motion in the fluids5) 28-34). For example, the frequency-dependent bulk viscosity is given by Fourier analysis of the fluctuations of the pressure: co
C(w)=
&
s
dt ~0s wt<(Tzt(o)- %t)(T,,(t)- Tj3)>
(2.7)
0
where TU is the total momentum flux tensor, ?=‘tjis its average value
in which P is the pressure of the system with Hamiltonian H, internal energy U, volume Q, and number of molecules N. Similarly, the frequency-de-
402 pendent
R. G. GORDON
shear viscosity
is
00
“rl(m)=
l
3052/&T s
dt cos UJ~<(3Tzj(O) - BiJ40))(3Tij(t)
-
BiJzz(t)>
(2.9)
0
Finally, analysis
the frequency-dependent heat conductivity of the fluctuations of the heat current J(t),
is given
by a Fourier
m
K(W) =
&
s 0
dt cos C&I(O) *J(t)>
(2.10)
Zwanzigsd) has also taken this point of view, and he demonstrated that the statistical methodas) agrees with the usual relaxation theoryr) in the weak coupling limit. Our next step is to evaluate the frequency-dependence of the velocities (dispersion) from the frequency-dependent transport coefficients. This relation may be found by calculating the response of a system to a weak field which couples to longitudinal waves. This response function will show a sharp resonance when the frequency and wave vector of this applied field match the phase velocity of a sound wave in the system. The response function may be put in the convient forms2) 3s)
co
s
do’D(o’) (o’)s _ cos + iwD(m)
where the dissipative
coefficient
D(o) = &(4 Thus we have a resonant
1-l
(2.11)
D(w) is given by
+ T(o) + +J~(C,~-
C,l)l
p-l
(2.12)
response when
(2.13)
This relation allows us to calculate the phase velocity from the frequencydependent transport coefficients. Finally, the group velocity may be found from the phase velocity, using eqn. (2.5), and hence, through (2.12) and (2.13), from the transport coefficients. These coefficients are, in turn, given in terms of the molecular motion by eqns. (7) to (10). This completes the program of this section, which was to define precisely the relation between the absorption coefficient, the phase velocity, the group velocity, and the molecular motion in a fluid.
ABSORPTION
AND
DISPERSION
OF SOUND
IN MOLECULAR
GASES
403
We conclude this section with as discussion of the limits of validity of these general formulas. The most important assumption is that the ordinary equations of continuum hydrodynamics describe the fluid. This assumption breaks down when the wavelength becomes comparable to the average intermolecular distances. If one attempts to extend the hydrodynamic approach into this region, then the hydrodynamic equations must be modified, and the transport coefficients must be considered nonlocal (i.e. dependent on the wave vectorsg) 3s) 3s). However, it appears doubtful that the hydrodynamic range can be practically extended very far in this way. Systems with long-range correlations (eg. superfluids) require special care because of their singular dependence on the wavevectorss) as). The sound wave must have a small amplitude. Otherwise, the nonlinear response at large amplitudes generates harmonics which distort the waveform from the simple harmonic form we have assumedss). 3. Dilute molecular gases. In a gas at low densities, the molecular motion consists of essentially free flights interrupted by binary collisions between molecules. The purpose of the present section is to relate the absorption and dispersion of sound in gases to explicit properties of these binary collisions. The frequency dependence of the shear viscosity becomes significant in gases only at very high sound frequencies. Then the wavelength has become comparable to the mean free path, and the Navier-Stokes equations of hydrodynamics lose their validity. Therefore it is of little use to make corrections for the frequency-dependence of the shear viscosity, without at the same time systematically studying better approximations to the hydrodynamics. On the other hand, the bulk viscosity often shows dispersion at frequencies well below the collision frequency. This low frequency dispersion arises from the slow exchange of energy between the translational motion and the internal degrees of freedom (rotational, vibrational, electronic). The case of heat conductivity is slightly more complicated. Heat is transported both as translational kinetic energy, and as internal energy. The translational part of the heat conductivity, like the shear viscosity, should show dispersion only in the very high frequency region, where the hydrodynamics breaks down. However, the internal part might be expected to show frequency dependence at lower frequencies, just as the bulk viscosity does, if the internal degrees of freedom are weakly coupled to the translational motion. However, one may estimate that the sound absorption due to this internal contribution to the heat conductivity is smaller than that due to bulk viscosity, by a factor of approximately r& where rV and 75 are the relaxation times of the shear and bulk viscosities, respectively. Thus, if the internal degrees of freedom are weakly coupled (TC3 T,J, dispersion in the heat conductivity may be neglected in calculating sound absorption,
404
R. G. GORDON
at least in the region of dispersion. On the other hand, if the internal degrees of freedom are strongly coupled to translation (7~ = T,J, then the heat conductivity will not show any dispersion in the hydrodynamical region. Thus, in all cases, the heat conductivity, like the shear viscosity, may be taken as constant in the frequency range of interest in the study of energy transfer. Therefore in the following, we need only consider the frequency dependence of the bulk viscosity, which is implicit in eqn. (2.7). In a dilute gas, the only significant contribution to the pressure from the kinetic energy. The trace of the pressure tensor is
Tit = 2Ht, where Ht, is the contribution of the kinetic thermodynamic derivative in (2.8) is
comes
(3.1) energy to the Hamiltonian.
(-g), =(1-g)
The
(3.4
where Ca is the contribution to the total specific heat from the internal degrees of freedom. The number fluctuation does not enter, since we are using a canonical ensemble. Combining these expressions with (2.7) and (2.8) gives the correlation function C(t) which determines the bulk viscosity : c(t)
zi?s&
<(Tii(O)
-
%,)(W)
In this the energy fluctuation &z(t) = C,‘[(C,
-
<&k(O) 4))
(3.3)
- &)I,
(3.4)
&l(t) is given by
C,)(H;(t) - a:) - Ct(H#)
-
where Hj is the internal (eg. rotational) For simplicity we will first consider diluted in a large excess fluctuation of the internal
2533)) = &
energy for the Zth molecule. the special case of a few molecules
of an atomic gas. In this case &l(t) reduces energy of a molecule away from its average, E$) = Hi(t) - a;.
to
(3.5)
Then to compute the bulk viscosity we must study the time dependence of <&t(O)&l(t)>. We use a method which has proved useful in the problems of non-resonant dipolar absorption and nuclear spin relaxationa6). We simply count up the effects of successive binary collisions, and resum the resulting infinite series into a usable form. Let us define a persistence ratio P, for the internal energy, for the nth collision by p
12
=
Q(4)n
<&Z(O) &Z(Q>n-I
collisions collisions
’
(3.6)
ABSORPTION
In general
AND
DISPERSION
OF SOUND
IN MOLECULAR
GASES
405
P, will be a function of the initial state of the Zthmolecule. Then
we may write <&Z(O)&Z(t)>= :
. . . pn
N(%
t)>
(3.7)
where N(%, t) is the probability that the Zthmolecule has suffered n collisions up to time t. Here we introduce the major simplifying assumption: We approximate this counting of collisions by the Poisson distribution
N(n, 4 =
$.( $ >nexp(--t/70)
(3.8)
where TO has the value given by kinetic theory To =
(p&j-l
(3.9)
where p is the number density of atoms, fi is the mean relative velocity (8k~T/np)* and ,u is the reduced mass of the colliding atom molecule pair. This counting of collisions would be exact if all of the relative velocity in a collision came from the atom, rather than the molecule. This would be the case if the atom were light compared to the molecule. It would also be exact if the rotation or vibration is weakly coupled to the translation, since in this case the molecular velocity would have relaxed between any two collisions which changed the internal states. Thus the velocities in these two inelastic collisions could be taken as uncorrelated. Finally, the results would also be exact if the velocity dependence of the inelastic cross sections was of the form v-1, which corresponds to the case of Maxwellian collisions in the kinetic theory of atoms. Since any one of the three above conditions is sz#cient to make our results exact (at low densities), one may expect that the corrections to the Poisson counting of collisions should not be very large in most real cases. Finally, we remark that without this approximation the present results, if applied to a two-level system, would not reduce to Kneser’s result, for that case. In other words, the Poisson counting of collisions is equivalent to the usual assumption that one may write a set of (coupled) first order rate equations for the internal states. The length ba appearing in (3.9) is an upper limit to the range of the intermolecular forces. At the end of the calculations we would like to let be become large. Then, provided the intermolecular forces fall off fast enough at large distances, the results should tend to a well-defined limit. However, such a stable limiting behavior is not shown by the series (3.7), since each term depends strongly on the cutoff ba, although the entire infinite sum does not. We have introduced a resummation proceduress) which results in a series, each of whose terms approaches a finite limit at large 60. The resultae) is C(t) = (3C~/2C,)(K~T)2{exp(--p~u~t)}(l
+ 5 (pdogt)~ fg)/~!} n=2
(3.10)
406
R. G. GORDON
where the cross section Q for bulk viscosity
is
G C5 =
2<(H,I -
a;)s>(&
-
Ci) s
M
27cb db.
(3.11)
0
In this cross section LIE is the total energy transfer from translation to internal energy in a single binary collision with initial relative speed V. The mean square internal energy fluctuation is given by equilibrium as <(H,” - g;)“) = cakBTs
theory
where cg is the internal heat capacity per molecule. series (3.10) have the form of fluctuations:
in the
/'['= 12
For example, f2
=
2
j-l)"+"
n!
k-0
k!(n. -
k)!
f2 is
The coefficients
... fvc>"-k
(3.12)
(3.13)
Pl)>”
given by
{
PlP2X4O)>
-
m2)/<@)(1
-
P1)>2
(3.14)
Examining these results, we see that the first term in the series (3.10) depends only on the dynamics of single binary collisions, through the cross section (3.11). The second term brings in the dynamics of two successive binary collisions, through the coefficient fs in (3.14). In general, the tith term contains information about the dynamics of n successive binary collisions. The above results were written for a molecular gas in a large excess of a monatomic gas. The same arguments may be applied to a pure molecular gas, by generalizing the definitions slightly. The persistence ratio becomes Z (&Z(O)Em(t)>~collisions pn
=
X7:E1(o)
m
‘mCt))n-I
(3.15) oollisions
Similarly, in the formula (3.13) for fn, E:(O) must be replaced by Cm &t(O)~~(0) Then the subsequent formulas remain unchanged. The bulk viscosity may now be calculated by Fourier analysis of C(t). From (2.7) and (3.3), we have 00
s
(3.16)
(3.10) into (3.16) and integrating,
gives
dt cos cot C(t).
0
Substituting
the correlation
c(w) = (2Ci/3C,)(pk~T~)(l
function + &%2)-i
x [t(n+l)l
x
1 +
5 f$)(l + co2.T2)-”
T&=2
,Fo
(-l)
f
2+k ‘>
(co,,,,],
(3.18)
ABSORPTION
AND
where the relaxation
DISPERSION
OF SOUND
IN MOLECULAR
GASES
407
time is (3.19)
7 = (pda&l
and the cross section ag is given by (3.11). Now we may apply this result to the absorption and dispersion of ultrasonic waves. It is convenient to write the steady state absorption coefficient a (2.2) as a sum of a translational part @trails and an internal molecular contribution CQ: 1
+
yajk_2 = atrans +
aint
(3.19)
with atrans =
&
[$j + (c,’ - c,‘) K]
(3.20)
and “int
(3.21) For convenience, follows :
we write out the explicit
form of the first few terms as
To complete the calculation, we must compute the frequency dependence (dispersion) of the phase and group velocities. As in the case of absorption, dispersion due to the shear viscosity and heat conductivity becomes significant only at such high frequencies that the hydrodynamic equations are no longer applicable. Slow transfer of energy to and from the internal degrees of freedom leads to dispersion at lower frequencies,-which we may compute from the bulk viscosity C(U) in (3.18). After substituting into the dispersion relation (2.13), the integral may be done by contour integration, giving
It is easily seen that this expression for the phase velocity approaches the proper limit when the frequency is so high that the internal degrees of freedom can no longer adjust with the sound wave. Finally, from this phase velocity we obtain the group velocity using (2.5),
408
R. G. GORDON
with the result
(3.24) According to this result, the group velocity is always larger than the phase velocity. However, the difference goes to zero in both the high frequency and zero frequency
limits.
4. Connectiort with the inelastic collision c~0s.s sections. It is interesting to express the results of our analysis in terms of the collision cross sections for inelastic transitions between the internal (eg. rotational) energy levels. We define a matrix Q of these cross sections by
where Pfi is the probability that a certain collission, with impact parameter b and initial relative speed V, causes a transition from the molecular energy state i to a state f of different energy. For this purpose, all (degenerate) states of the same energy are counted together. Sf6 is unity if i = f and zero otherwise. The diagonal elements of Q give the total inelastic cross sections for changing the internal ene?gy state of a molecule. The off-diagonal elements give (minus) the partial cross sections for collisions which cause the various specific transitions between states of different energy. To the same accuracy of description as discussed in section 3, we may consider the collisions as occurring at random times, causing the molecules to jump between the energy states at rates ~6s according to a Markov random process. Then the correlation function of the internal energy has the Markoffian formaT) = d.[exp(-@cd)] where the weighting
vector
-p-d,
(4.2)
d is given by
da = (H&
-
a,,)
+.
(4.3)
The population matrix p has as its diagonal elements the equilibrium populations of the molecular energy levels, and zero off-diagonal elements. If we now expand (4.2) about a mean cross section a = d.cr.p.d,
(4.4)
we may obtain (E(O) s(t)> = <.G>{exp(--p%t)){l Comparison
of (4.5) with (3.
+ g (p&f)” d.(o
n=2
-
16)n*p*d/rt!}. (4.5)
IO)gives us the identification ag = a = d.cr.p.d
(4.6)
ABSORPTION
AND
DISPERSION
OF SOUND
IN MOLECULAR
GASES
409
and /f’
= d*(a -
lo)n*p*d/(C?)n.
(4.7)
Equations (4.6) and (4.7) relate the mean cross section ag and the correlation coefficients fla to the inelastic collision cross sections, for collisions between an atom and a molecule with the internal energy levels. These relations may also be established in a more direct but more lengthy manner by expanding the definitions (3.11) and (3.13) in a matrix form. 5. Discussions artd comfiarison with exfierimemf. The results of the previous sections express the absorption and dispersion of ultrasonic waves in terms of explicit properties of successive binary collisions. The expressions are ordered in a series, of which the first term takes into account the energy transfer from translation to internal motion (eg. rotation) in single collisions. The second term corrects for the fact that energy transfer in two szlccessive collisions differs in general from the transfer averaged over two uncorrelated collisions. Physically one expects these higher order correlations to be rather small. Then we may expect to obtain good agreement with experiment by keeping only the first two or three terms in the series. If this is the case, then the present method gives a very efficient representation of the experiments using only two or three well-defined dynamical parameters, instead of the +N(N - 1) transition probabilities required for an analysis with the conventional theory 5)s) for an N-level system. For a system with only two internal energy states, one may show that the present series actually tmncates after the first term, Furthermore, the cross section appearing in the first term becomes, for a two level system,
(5.1) where olcO is the total collision cross’section for excitation from the 0 to 1 level, and CT~+~ is the cross section for de-excitation. The present results then reduce exactly to Kneser’s results for a two-level system 6). This correspondence gives us a second reason for expecting the correlation terms to be small. The relaxation properties of an N-level system often resemble, at least qualitatively, those of a two-level system (eg. the use of a spin temperature in magnetic problemsss). Since the correlation terms vanish identically in a two-level system, one expects them to remain small even in the case of more general energy levels. Finally, of course, the convergence of the series can best be tested by actual calculations of the collision dynamics. Such calculations are in progress, and it is hoped that we will report on these in the future. It may be helpful to note here that in the two-level case our 7, defined by (3.19) and (3.11) corresponds to the relaxation time T” of ref. 1, p. 84. In order to compare these results with experiments on ultrasonic absorp-
410
R. G. GORDON
tion, it is convenient to subtract the “translational” part (3.20) to obtain the absorption directly due to relaxation of the internal degrees of freedom. However, because of the frequency dependence in the phase velocity, even the “translational” part depends to some extent on the relaxation. This coupling appears to have been overlooked in ultrasonic work. However, we may obtain a consistent interpretation by one or two stages of successive approximation. A rough estimate of the relaxation time T is used to calculate the phase velocity, and the “translational” contribution to the absorption. This is subtracted from the experimental absorption coefficient to obtain the “internal” contribution, and a better estimate of T and, say, fs is obtained by comparison with the theoretical formula (3.22). This process can then be repeated until a consistent set of parameters is found. In practice this procedure converges quickly. Recent measurements at Leidenla) of absorption in Ds gas provide a good test case for the present calculations. At low temperatures (5 90°K) essentially only two levels are populated in 0-Ds. In this case the experiments fit a curve for a single relaxation time within experimental error. This agrees with Kneser’s calculation for a two-level system, and with our calculations, for which the first approximation is exact in a two-level system. At room temperature, 5 or 6 rotational levels are populated in Ds, and the experimental absorption curvelo) is definitely broader than the first approximation (single relaxation time). We find that a good fit is obtained by taking fs = 0.2, and neglecting the higher correlation terms. The experimental points are compared with the theoretical curves in fig. 1. The cross section (3.11) becomes, for classical linear molecules, co 5
0.0:’ Fig.
1. The rotational
experimental
points
c+Da(O)
Sluijter, Knaap and Beenakker a cross section q = 0.31 Aa and for
I
5
1
to ultrasonic and
n-De(+)
(reference
I
I
10 frequency / pressure,
contribution for
I
2
50
20
absorption in Da gas at 293°K. The were calculated from the results of 10). The solid curve is calculated we also give a dotted
f2 = 0.2. For comparison
f2 =
0 (single
100
MHz/atm
relaxation
time).
with curve
ABSORPTION
AND
The best fit is obtained
DISPERSION
OF SOUND
IN MOLECULAR
with cry= 0.31 As. The experimental
GASES
411
value for the
rotational heat capacity, also taken from this fit, is about 6% larger than the theoretical value. This difference is, however, probably within the experimental error. Because of the scatter of the experimental points it is not possible to rule out a small contribution of, say, a few per cent from fa. It would be most interesting to repeat these measurements with greater precision. However, these experiments at least suggest that the fz correlation can account for the major correction to the first approximation. The order of magnitude of this fz is also consistent with similar correlation coefficients appearing in the theory of nonresonant absorptions6). Hydrogen and deuterium are especially favorable cases in which to study the detailed shape of the rotational absorption. In these molecules, rotational energy transfer is relatively slow, and thus the entire dispersion curve occurs at frequencies low enough so that hydrodynamics remains valid. By studying the complete shape of the absorption, one can more or less evaluate separately the relaxation time and the correlation coefficients f2, f3 . .. In most other molecules, rotational energy transfer is more efficient, and the region of rotational relaxation becomes mixed up with the breakdown of macroscopic hydrodynamics. In such cases, meaningful comparison with the present theory may only made at lower frequencies. Then the observable quantity is the low frequency limit of eqn. (3.21), which is
cc
co2 St
N-
Thus in the case of strongly
-
3vo ( c, ) coupled
41 +
fnl n=2
T[l + :
rotation
(5.3)
only the combination
5 fnl
n=2
is observable. Physically this result [means that if measurements are reastricted to a low frequency region, one necessarily observes effects of correlated sequences of collisions, and one cannot experimentally separate out the effects of single collisions. This situation is similar to that of nuclear relaxation in gasess*), where for other technical resons measurements are often made in the low frequency limit. The present theory thus gives a well-defined connection between energy transfer in collisions, and ultrasonic absorption and dispersion, which is useful even when many levels are populated. If the dispersion occurs at low frequencies (weak coupling), a more detailed experimental separation of collision effects may be made. However, even if the rotational motion is strongly coupled in collisions, a meaningful interpretation of ultrasonic experiments has been established.
412
ABSORPTION
AND
DISPERSION
OF SOUND
IN MOLECULAR
GASES
Acknowledgements. The author wishes to thank Dr. H. F. P. Knaap and Dr. G. Thomaes for interesting discussions. We also thank Professor Prigogine for his hospitality at the Universit6 Libre de Bruxelles. This work was supported by the Society of Fellows of Harvard University. Received
7-6-66 REFERENCES
1)
For a general review, see Herzfeld, K. F. and Litovitz, of Ultrasonic Waves (Academic Press, New York, 1959).
2)
Jeans, J. J., The Dynamical Theory of Gases (Cambridge University Herzfeld, K. F. and Rice, R. O., Phys. Rev. 31 (1928) 691.
3) 4)
Einstein,
5)
Bourgin, D. G., Nature 112 (1928) ibid. 42 (1932) 721; ibid. (1936) 355.
6)
Kneser,
7)
Rutgers,
A., Sits. der Akad.
Wiss.
Berlin,
Physik
11 (1931)
A. J., Ann.
Physik
16 (1933) 350.
Richards,
9)
Landau,
L. and Teller,
W.T.,
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Sluijter,
C. G., Knaap,
11)
Beyer,
Press, London,
(1920) 380. Phys.
Rev. 34 (1929)
10 (1936) 34.
Sot. Am. 29 (1957) 243.
Arthurs, A. M. and Dalgarno, A. Proc. roy. Sot. A259 K., Suppl. Prog. theor. Phys. 25 (1963) 1. 13) Takayanagi, W. D., Disc. Faraday Sot. 33 (1962) 71. 14) Davidson, 15)
Roberts,
16)
Lawley,
K. P. and Ross,
J., J. them.
17)
Karplus,
M. and Raff,
L. M., J. them.
(1960) 540.
C. S., Phys. Rev. 131 (1963) 209. Phys. 43 (1965) 2930. Phys. 41 (1964)
1267.
D. R., J. them. Phys. 43 (1965) 3530. 18) Cross, Jr. R. J. and Herschbach, R. G., J. them. Phys. 44 (1966) 3083. 19) Gordon, L. D. and Lifshitz, E. M., Fluid Mechanics (Pergamon Press, London, 20) 20) Landau, Sections
49 and 79; S. R. de Groot
Holland Stokes,
Publishing Co., 1962). G., Trans. Cambridge Phil. Sot. 8 (1845) 287.
22)
Kirchhoff,
23)
Tisza,
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Herzfeld, Landau,
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G., Poggendorf’s
L., Phys.
Physics
Non-Equilibrium
Ann. Phys. 134 (1868)
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(North
34 (1958)
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177.
Rev. 61 (1942) 531.
K. F., Ann. Physik L. D. and Lifshitz, - JETP
and P. Mazur
7 (1958)
(6) 3 (1948) 69. E. M., op. cit., section
132; J.E.T.P.
262;
182.
26)
Truesdell,
27) 28)
Markham, Landau,
29) 30)
sections 118-124. McLennan, J. A., Phys. Fluids 3 (1960) 493. Montroll, E. W., Rend. Scuola, Intern. Fis. Enrico
Fermi 13 (1960) 242.
31) 32)
Zwanzig, Kadanoff,
24 (1963) 419.
33)
Martin,
P. C., Proc.
Meixner,
North
C., J. Rational
Mech. Anal. 3 (1953) 643.
J, J., Beyer, R. T. and Lindsay, R. B., Rev. mod. Phys. 23 (1951) 353. L. D. and Lifshitz, E. M., Statistical Physics (Pergamon Press, London,
R. W., Phys. Rev. 124 (1961) 983. L. P. and Martin, P. C., Ann. Physics Intern.
Holland
Sym. Stat. Mech. and Thermodynamics
Publishing
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1964 (Ed. J.
Co., Amsterdam).
35)
Zwanzig, R. W., J. them. Phys. 43 (1965) 714. Fox, F. E. and Wallace, W. A., J. Acoust. Sot. Am. 26 (1954) 994.
36) 37) 38)
Gordon, Gordon, Abragam,
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521:
J. M., Physica 30 (1964) 745; 31 (1965) 915.
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863.
Z. Sowjetunion
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Math. naturw.
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196 1).
Physica 34 398-412
ON THE ABSORPTION AND DISPERSION OF SOUND IN MOLECULAR GASES by R. G. GORDON*) UniversiteLibre de Bruxelles, Bruxelles, Belgique.
synopsis The propagation of ultrasonic waves in dilute molecular gases IS studied theoretically. Explicit relations are obtained between the dynamics of successive binary molecular collisions, and the absorption coefficient and velocity of sound. These relations are especially useful in the study of rotationally inelastic collisions, since they remain valid even when many rotational energy levels are populated. The analysis is applied to rotational absorption in Ds gas, and it is shown that a good account of the experimental frequency dependence may be obtained by taking into account at least two successive binary collisions.
1. Introduction. The absorption and dispersion of sound by molecular gases is an important method in the study of inelastic collisions between moleculesr). As a sound wave passes through a molecular gas, the increased translational energy near a peak of compression is transformed by collisions into internal energy of rotations), vibrations) and chemical reactiona). This transfer of energy lags behind the sound wave itself, thereby contributing to the absorption of the sound. This phase lag may then be related to the probabilities for the various inelastic transitions induced by molecular collisions 4-g). The simplest possible system showing a relaxation effect would be a molecule with only two internal energy states. This model leads to simple formulas for acoustic absorption and dispersion, in terms of a single parameter, related to the collision cross section for the transition between the two states.6) 7). Another simple case is that in which the internal states are represented by a harmonic oscillator which is weakly coupled to the translational and rotational motion. Landau and Tellers) showed that the simple two-state formulas6) 7) still remain valid for an oscillator, if only a small change is made in the definition of the rate constant. The great simplification achieved for an oscillator by Landau and Teller (compare with refs. 5 and 8!) was *) Present address: Department of Chemistry, Harvard University, Cambridge, Mass. -
398 -
ABSORPTION
AND
DISPERSION
OF SOUND
IN MOLECULAR
GASES
399
only possible because of the special values of the energy levels and transition probabilities
in a weakly coupled harmonic
oscillator.
For other cases, such
as rotational energy levels, their simplification is not possible. The general case of many (N) arbitrary internal energy levels appears very complicateds) s), and the results depend in detail on all the &V(N - 1) transition probabilities between the states. This multitude of transition probabilities between many quantum states is certainly szlfficient to describe the acoustic properties in systems of many levels. However, we propose here that in fact such detail is not always necessary. We show that sound absorption by molecules with many excited rotational states are well described by only a few parameters, which are defined in terms of the molecular collisions which occur in a gas. In the first approximation, we obtain again the familiar absorption shape, with a generalized cross section which reduces to Kneser’s result for the two-level casee), and to Landau and Teller’s result for an oscillator a). This first approximation corresponds physically to keeping only the independent effects of single binary collisions. Higher approximations are obtained which correct for the correlated effects of two, three . . . successive binary collisions. These correlations are present because the final rotational state in one collision is the initial state for the next collision. Since the probabilities for energy transfer in a collision depend on the initial state, two successive collisions in general have a different effect than a pair of uncorrelated collisions. The correction terms which are thereby introduced vanish for either a two-level system or for an oscillator. Physically we expect that these correlation effects will rapidly become small as we consider longer sequences of collisions. Thus we expect the correction terms to decrease rapidly in magnitude as we go to higher correlations. These results find some experimental support in recent measurements of ultrasonic absorption in Ds gaslo), which can be represented keeping only the first correction term. This illustrates the advantage of the present method in the analysis of experimental data. The parameters to be determined in the conventional analysis are the transition probabilities between the internal states of the molecules. If there are only two levels significantly populated (for example, the two lowest rotational levels at low temperatures), then one can readily determine a single parameter related to the cross section for excitation. If there are three significant levels, it is already very difficult to determine the three independent transition probabilities from experimentrr). If there are more than three levels, it becomes quite impossible to deduce the +N (N - 1) transition probabilities uniquely from an experiment, even if the precision were much improved over that presently obtained. In contrast to these difficulties, the present approach leads to a rapidly convergent series of approximations, in which only a few welldefined dynamical parameters are needed to describe the entire sound absorption spectrum, even in a many-level system.
400
R.G.GORDON
A second advantage of the present method is related to the correspondence principle between quantum mechanics and classical mechanics. The usual description of sound absorption in terms of transitions between the internal quantum levels has no simple meaning in terms of classical mechanics. However, the description which we use here has the same form, whether classical or quantum mechanics is used to describe the collision dynamics. This point is particularly important for actual calculations. Quantum mechanical transition probabilities in molecular collisions can at present only be calculated with drastic approximationsis-1s) which break down when many levels can be excited by a collision. In contrast, exact classical calculations are possible for molecular collision traj ectoriesi7) is) and various cross sectionsi7) la). Using the results of the present study, such classical calculations may be compared with experiments on the absorption and dispersion of sound. Such comparisons should lead to a better understanding of intermolecular forces and torques, and of molecular collision processes. The general theory of sound absorption and dispersion is extended and clarified in section 2. The case of a dilute molecular gas is analyzed in sections 3 and 4. The results are discussed and applied in section 5 to the analysis of ultrasonic absorption experiments. 2. Oa the general theory of zcltrasonic absor#tion and disfiersio%. A microscopic theory of the absorption and velocity of ultrasonic waves must relate these properties to the molecular motion in the medium. The absorption coefficient 0: is defined by CL= dln A/d1
(2.1)
where A is the amplitude of the sound wave, and I is the distance of travel in the medium. The actual experimental value of u depends somewhat on whether the experiment is done with a steady wave pattern (as in an interferometer), or with a pulse. If the fluid obeys the usual hydrodynamic equations, then the absorption coefficient, for a steady state experiment, has the well-known valueso-2s) ci
1 + (4N2
=
$$ [+q +i-+ (c,’
- c,‘) K]
where 11. c and K are the hydrodynamic coefficients of shear viscosity, bulk viscosity, and thermal conductivity, respectively. C, and C, give the heat capacities at constant volume and pressure, p is the mass density, co the radian frequency, and k the wave vector. The phase velocity vp, which may be measured in a steady-state interferometer, is given by vP = co/k. The analysis
of a pulse experiment
is more complicated.
(2.3) In fact,
if the
ABSORPTION
AND
DISPERSION
OF SOUND
IN MOLECULAR
GASES
401
medium absorbs very strongly, then the shape of a pulse is badly distorted as it progresses, and analysis of the results is very difficult. However, if the absorption per wavelength is only moderate, then a pulse is distorted only gradually, and it travels with the well-known group velocity vg = do/dk.
(2.4)
From the definitions (2.3) and (2.4), we may express vg in terms of vP:
-&(kvp)
Vg =
dv,
= vp + kv, -
do
or (2.5) We wish to point out here that a pulse experiment also measures a somewhat different value of the absorption coefficient than does a steady-state experiment. It is not hard to show, from the basic definition (2.1), that
instead of the classical result (2.2). If the quadratic term in (2.2) is important, then the absorption is so strong that a more elaborate analysis of a pulse experiment would probably be necessary. In general the hydrodynamic coefficients 7, f and K must be considered to be frequency dependent 23-9. Allowing such a frequency-dependence has been criticized as a rather arbitrary device for obtaining agreement between theory and experiment 26)27). However, if the fluid dynamics are in fact well described by hydrodynamical equations with frequencydependent coefficients, then these coefficients may be given a well-defined interpretation in terms of the molecular motion in the fluids5) 28-34). For example, the frequency-dependent bulk viscosity is given by Fourier analysis of the fluctuations of the pressure: co
C(w)=
&
s
dt ~0s wt<(Tzt(o)- %t)(T,,(t)- Tj3)>
(2.7)
0
where TU is the total momentum flux tensor, ?=‘tjis its average value
in which P is the pressure of the system with Hamiltonian H, internal energy U, volume Q, and number of molecules N. Similarly, the frequency-de-
402 pendent
R. G. GORDON
shear viscosity
is
00
“rl(m)=
l
3052/&T s
dt cos UJ~<(3Tzj(O) - BiJ40))(3Tij(t)
-
BiJzz(t)>
(2.9)
0
Finally, analysis
the frequency-dependent heat conductivity of the fluctuations of the heat current J(t),
is given
by a Fourier
m
K(W) =
&
s 0
dt cos C&I(O) *J(t)>
(2.10)
Zwanzigsd) has also taken this point of view, and he demonstrated that the statistical methodas) agrees with the usual relaxation theoryr) in the weak coupling limit. Our next step is to evaluate the frequency-dependence of the velocities (dispersion) from the frequency-dependent transport coefficients. This relation may be found by calculating the response of a system to a weak field which couples to longitudinal waves. This response function will show a sharp resonance when the frequency and wave vector of this applied field match the phase velocity of a sound wave in the system. The response function may be put in the convient forms2) 3s)
co
s
do’D(o’) (o’)s _ cos + iwD(m)
where the dissipative
coefficient
D(o) = &(4 Thus we have a resonant
1-l
(2.11)
D(w) is given by
+ T(o) + +J~(C,~-
C,l)l
p-l
(2.12)
response when
(2.13)
This relation allows us to calculate the phase velocity from the frequencydependent transport coefficients. Finally, the group velocity may be found from the phase velocity, using eqn. (2.5), and hence, through (2.12) and (2.13), from the transport coefficients. These coefficients are, in turn, given in terms of the molecular motion by eqns. (7) to (10). This completes the program of this section, which was to define precisely the relation between the absorption coefficient, the phase velocity, the group velocity, and the molecular motion in a fluid.
ABSORPTION
AND
DISPERSION
OF SOUND
IN MOLECULAR
GASES
403
We conclude this section with as discussion of the limits of validity of these general formulas. The most important assumption is that the ordinary equations of continuum hydrodynamics describe the fluid. This assumption breaks down when the wavelength becomes comparable to the average intermolecular distances. If one attempts to extend the hydrodynamic approach into this region, then the hydrodynamic equations must be modified, and the transport coefficients must be considered nonlocal (i.e. dependent on the wave vectorsg) 3s) 3s). However, it appears doubtful that the hydrodynamic range can be practically extended very far in this way. Systems with long-range correlations (eg. superfluids) require special care because of their singular dependence on the wavevectorss) as). The sound wave must have a small amplitude. Otherwise, the nonlinear response at large amplitudes generates harmonics which distort the waveform from the simple harmonic form we have assumedss). 3. Dilute molecular gases. In a gas at low densities, the molecular motion consists of essentially free flights interrupted by binary collisions between molecules. The purpose of the present section is to relate the absorption and dispersion of sound in gases to explicit properties of these binary collisions. The frequency dependence of the shear viscosity becomes significant in gases only at very high sound frequencies. Then the wavelength has become comparable to the mean free path, and the Navier-Stokes equations of hydrodynamics lose their validity. Therefore it is of little use to make corrections for the frequency-dependence of the shear viscosity, without at the same time systematically studying better approximations to the hydrodynamics. On the other hand, the bulk viscosity often shows dispersion at frequencies well below the collision frequency. This low frequency dispersion arises from the slow exchange of energy between the translational motion and the internal degrees of freedom (rotational, vibrational, electronic). The case of heat conductivity is slightly more complicated. Heat is transported both as translational kinetic energy, and as internal energy. The translational part of the heat conductivity, like the shear viscosity, should show dispersion only in the very high frequency region, where the hydrodynamics breaks down. However, the internal part might be expected to show frequency dependence at lower frequencies, just as the bulk viscosity does, if the internal degrees of freedom are weakly coupled to the translational motion. However, one may estimate that the sound absorption due to this internal contribution to the heat conductivity is smaller than that due to bulk viscosity, by a factor of approximately r& where rV and 75 are the relaxation times of the shear and bulk viscosities, respectively. Thus, if the internal degrees of freedom are weakly coupled (TC3 T,J, dispersion in the heat conductivity may be neglected in calculating sound absorption,
404
R. G. GORDON
at least in the region of dispersion. On the other hand, if the internal degrees of freedom are strongly coupled to translation (7~ = T,J, then the heat conductivity will not show any dispersion in the hydrodynamical region. Thus, in all cases, the heat conductivity, like the shear viscosity, may be taken as constant in the frequency range of interest in the study of energy transfer. Therefore in the following, we need only consider the frequency dependence of the bulk viscosity, which is implicit in eqn. (2.7). In a dilute gas, the only significant contribution to the pressure from the kinetic energy. The trace of the pressure tensor is
Tit = 2Ht, where Ht, is the contribution of the kinetic thermodynamic derivative in (2.8) is
comes
(3.1) energy to the Hamiltonian.
(-g), =(1-g)
The
(3.4
where Ca is the contribution to the total specific heat from the internal degrees of freedom. The number fluctuation does not enter, since we are using a canonical ensemble. Combining these expressions with (2.7) and (2.8) gives the correlation function C(t) which determines the bulk viscosity : c(t)
zi?s&
<(Tii(O)
-
%,)(W)
In this the energy fluctuation &z(t) = C,‘[(C,
-
<&k(O) 4))
(3.3)
- &)I,
(3.4)
&l(t) is given by
C,)(H;(t) - a:) - Ct(H#)
-
where Hj is the internal (eg. rotational) For simplicity we will first consider diluted in a large excess fluctuation of the internal
2533)) = &
energy for the Zth molecule. the special case of a few molecules
of an atomic gas. In this case &l(t) reduces energy of a molecule away from its average, E$) = Hi(t) - a;.
to
(3.5)
Then to compute the bulk viscosity we must study the time dependence of <&t(O)&l(t)>. We use a method which has proved useful in the problems of non-resonant dipolar absorption and nuclear spin relaxationa6). We simply count up the effects of successive binary collisions, and resum the resulting infinite series into a usable form. Let us define a persistence ratio P, for the internal energy, for the nth collision by p
12
=
Q(4)n
<&Z(O) &Z(Q>n-I
collisions collisions
’
(3.6)
ABSORPTION
In general
AND
DISPERSION
OF SOUND
IN MOLECULAR
GASES
405
P, will be a function of the initial state of the Zthmolecule. Then
we may write <&Z(O)&Z(t)>= :
. . . pn
N(%
t)>
(3.7)
where N(%, t) is the probability that the Zthmolecule has suffered n collisions up to time t. Here we introduce the major simplifying assumption: We approximate this counting of collisions by the Poisson distribution
N(n, 4 =
$.( $ >nexp(--t/70)
(3.8)
where TO has the value given by kinetic theory To =
(p&j-l
(3.9)
where p is the number density of atoms, fi is the mean relative velocity (8k~T/np)* and ,u is the reduced mass of the colliding atom molecule pair. This counting of collisions would be exact if all of the relative velocity in a collision came from the atom, rather than the molecule. This would be the case if the atom were light compared to the molecule. It would also be exact if the rotation or vibration is weakly coupled to the translation, since in this case the molecular velocity would have relaxed between any two collisions which changed the internal states. Thus the velocities in these two inelastic collisions could be taken as uncorrelated. Finally, the results would also be exact if the velocity dependence of the inelastic cross sections was of the form v-1, which corresponds to the case of Maxwellian collisions in the kinetic theory of atoms. Since any one of the three above conditions is sz#cient to make our results exact (at low densities), one may expect that the corrections to the Poisson counting of collisions should not be very large in most real cases. Finally, we remark that without this approximation the present results, if applied to a two-level system, would not reduce to Kneser’s result, for that case. In other words, the Poisson counting of collisions is equivalent to the usual assumption that one may write a set of (coupled) first order rate equations for the internal states. The length ba appearing in (3.9) is an upper limit to the range of the intermolecular forces. At the end of the calculations we would like to let be become large. Then, provided the intermolecular forces fall off fast enough at large distances, the results should tend to a well-defined limit. However, such a stable limiting behavior is not shown by the series (3.7), since each term depends strongly on the cutoff ba, although the entire infinite sum does not. We have introduced a resummation proceduress) which results in a series, each of whose terms approaches a finite limit at large 60. The resultae) is C(t) = (3C~/2C,)(K~T)2{exp(--p~u~t)}(l
+ 5 (pdogt)~ fg)/~!} n=2
(3.10)
406
R. G. GORDON
where the cross section Q for bulk viscosity
is
G C5 =
2
a;)s>(&
-
Ci) s
M
27cb db.
(3.11)
0
In this cross section LIE is the total energy transfer from translation to internal energy in a single binary collision with initial relative speed V. The mean square internal energy fluctuation is given by equilibrium as <(H,” - g;)“) = cakBTs
theory
where cg is the internal heat capacity per molecule. series (3.10) have the form of fluctuations:
in the
/'['= 12
For example, f2
=
2
j-l)"+"
n!
k-0
k!(n. -
k)!
f2 is
The coefficients
... fvc>
(3.12)
(3.13)
Pl)>”
given by
{
PlP2X4O)>
-
m2)/<@)(1
-
P1)>2
(3.14)
Examining these results, we see that the first term in the series (3.10) depends only on the dynamics of single binary collisions, through the cross section (3.11). The second term brings in the dynamics of two successive binary collisions, through the coefficient fs in (3.14). In general, the tith term contains information about the dynamics of n successive binary collisions. The above results were written for a molecular gas in a large excess of a monatomic gas. The same arguments may be applied to a pure molecular gas, by generalizing the definitions slightly. The persistence ratio becomes Z (&Z(O)Em(t)>~collisions pn
=
X7:E1(o)
m
‘mCt))n-I
(3.15) oollisions
Similarly, in the formula (3.13) for fn, E:(O) must be replaced by Cm &t(O)~~(0) Then the subsequent formulas remain unchanged. The bulk viscosity may now be calculated by Fourier analysis of C(t). From (2.7) and (3.3), we have 00
s
(3.16)
(3.10) into (3.16) and integrating,
gives
dt cos cot C(t).
0
Substituting
the correlation
c(w) = (2Ci/3C,)(pk~T~)(l
function + &%2)-i
x [t(n+l)l
x
1 +
5 f$)(l + co2.T2)-”
T&=2
,Fo
(-l)
f
2+k ‘>
(co,,,,],
(3.18)
ABSORPTION
AND
where the relaxation
DISPERSION
OF SOUND
IN MOLECULAR
GASES
407
time is (3.19)
7 = (pda&l
and the cross section ag is given by (3.11). Now we may apply this result to the absorption and dispersion of ultrasonic waves. It is convenient to write the steady state absorption coefficient a (2.2) as a sum of a translational part @trails and an internal molecular contribution CQ: 1
+
yajk_2 = atrans +
aint
(3.19)
with atrans =
&
[$j + (c,’ - c,‘) K]
(3.20)
and “int
(3.21) For convenience, follows :
we write out the explicit
form of the first few terms as
To complete the calculation, we must compute the frequency dependence (dispersion) of the phase and group velocities. As in the case of absorption, dispersion due to the shear viscosity and heat conductivity becomes significant only at such high frequencies that the hydrodynamic equations are no longer applicable. Slow transfer of energy to and from the internal degrees of freedom leads to dispersion at lower frequencies,-which we may compute from the bulk viscosity C(U) in (3.18). After substituting into the dispersion relation (2.13), the integral may be done by contour integration, giving
It is easily seen that this expression for the phase velocity approaches the proper limit when the frequency is so high that the internal degrees of freedom can no longer adjust with the sound wave. Finally, from this phase velocity we obtain the group velocity using (2.5),
408
R. G. GORDON
with the result
(3.24) According to this result, the group velocity is always larger than the phase velocity. However, the difference goes to zero in both the high frequency and zero frequency
limits.
4. Connectiort with the inelastic collision c~0s.s sections. It is interesting to express the results of our analysis in terms of the collision cross sections for inelastic transitions between the internal (eg. rotational) energy levels. We define a matrix Q of these cross sections by
where Pfi is the probability that a certain collission, with impact parameter b and initial relative speed V, causes a transition from the molecular energy state i to a state f of different energy. For this purpose, all (degenerate) states of the same energy are counted together. Sf6 is unity if i = f and zero otherwise. The diagonal elements of Q give the total inelastic cross sections for changing the internal ene?gy state of a molecule. The off-diagonal elements give (minus) the partial cross sections for collisions which cause the various specific transitions between states of different energy. To the same accuracy of description as discussed in section 3, we may consider the collisions as occurring at random times, causing the molecules to jump between the energy states at rates ~6s according to a Markov random process. Then the correlation function of the internal energy has the Markoffian formaT)
vector
-p-d,
(4.2)
d is given by
da = (H&
-
a,,)
(4.3)
The population matrix p has as its diagonal elements the equilibrium populations of the molecular energy levels, and zero off-diagonal elements. If we now expand (4.2) about a mean cross section a = d.cr.p.d,
(4.4)
we may obtain (E(O) s(t)> = <.G>{exp(--p%t)){l Comparison
of (4.5) with (3.
+ g (p&f)” d.(o
n=2
-
16)n*p*d/rt!}. (4.5)
IO)gives us the identification ag = a = d.cr.p.d
(4.6)
ABSORPTION
AND
DISPERSION
OF SOUND
IN MOLECULAR
GASES
409
and /f’
= d*(a -
lo)n*p*d/(C?)n.
(4.7)
Equations (4.6) and (4.7) relate the mean cross section ag and the correlation coefficients fla to the inelastic collision cross sections, for collisions between an atom and a molecule with the internal energy levels. These relations may also be established in a more direct but more lengthy manner by expanding the definitions (3.11) and (3.13) in a matrix form. 5. Discussions artd comfiarison with exfierimemf. The results of the previous sections express the absorption and dispersion of ultrasonic waves in terms of explicit properties of successive binary collisions. The expressions are ordered in a series, of which the first term takes into account the energy transfer from translation to internal motion (eg. rotation) in single collisions. The second term corrects for the fact that energy transfer in two szlccessive collisions differs in general from the transfer averaged over two uncorrelated collisions. Physically one expects these higher order correlations to be rather small. Then we may expect to obtain good agreement with experiment by keeping only the first two or three terms in the series. If this is the case, then the present method gives a very efficient representation of the experiments using only two or three well-defined dynamical parameters, instead of the +N(N - 1) transition probabilities required for an analysis with the conventional theory 5)s) for an N-level system. For a system with only two internal energy states, one may show that the present series actually tmncates after the first term, Furthermore, the cross section appearing in the first term becomes, for a two level system,
(5.1) where olcO is the total collision cross’section for excitation from the 0 to 1 level, and CT~+~ is the cross section for de-excitation. The present results then reduce exactly to Kneser’s results for a two-level system 6). This correspondence gives us a second reason for expecting the correlation terms to be small. The relaxation properties of an N-level system often resemble, at least qualitatively, those of a two-level system (eg. the use of a spin temperature in magnetic problemsss). Since the correlation terms vanish identically in a two-level system, one expects them to remain small even in the case of more general energy levels. Finally, of course, the convergence of the series can best be tested by actual calculations of the collision dynamics. Such calculations are in progress, and it is hoped that we will report on these in the future. It may be helpful to note here that in the two-level case our 7, defined by (3.19) and (3.11) corresponds to the relaxation time T” of ref. 1, p. 84. In order to compare these results with experiments on ultrasonic absorp-
410
R. G. GORDON
tion, it is convenient to subtract the “translational” part (3.20) to obtain the absorption directly due to relaxation of the internal degrees of freedom. However, because of the frequency dependence in the phase velocity, even the “translational” part depends to some extent on the relaxation. This coupling appears to have been overlooked in ultrasonic work. However, we may obtain a consistent interpretation by one or two stages of successive approximation. A rough estimate of the relaxation time T is used to calculate the phase velocity, and the “translational” contribution to the absorption. This is subtracted from the experimental absorption coefficient to obtain the “internal” contribution, and a better estimate of T and, say, fs is obtained by comparison with the theoretical formula (3.22). This process can then be repeated until a consistent set of parameters is found. In practice this procedure converges quickly. Recent measurements at Leidenla) of absorption in Ds gas provide a good test case for the present calculations. At low temperatures (5 90°K) essentially only two levels are populated in 0-Ds. In this case the experiments fit a curve for a single relaxation time within experimental error. This agrees with Kneser’s calculation for a two-level system, and with our calculations, for which the first approximation is exact in a two-level system. At room temperature, 5 or 6 rotational levels are populated in Ds, and the experimental absorption curvelo) is definitely broader than the first approximation (single relaxation time). We find that a good fit is obtained by taking fs = 0.2, and neglecting the higher correlation terms. The experimental points are compared with the theoretical curves in fig. 1. The cross section (3.11) becomes, for classical linear molecules, co 5
0.0:’ Fig.
1. The rotational
experimental
points
c+Da(O)
Sluijter, Knaap and Beenakker a cross section q = 0.31 Aa and for
I
5
1
to ultrasonic and
n-De(+)
(reference
I
I
10 frequency / pressure,
contribution for
I
2
50
20
absorption in Da gas at 293°K. The were calculated from the results of 10). The solid curve is calculated we also give a dotted
f2 = 0.2. For comparison
f2 =
0 (single
100
MHz/atm
relaxation
time).
with curve
ABSORPTION
AND
The best fit is obtained
DISPERSION
OF SOUND
IN MOLECULAR
with cry= 0.31 As. The experimental
GASES
411
value for the
rotational heat capacity, also taken from this fit, is about 6% larger than the theoretical value. This difference is, however, probably within the experimental error. Because of the scatter of the experimental points it is not possible to rule out a small contribution of, say, a few per cent from fa. It would be most interesting to repeat these measurements with greater precision. However, these experiments at least suggest that the fz correlation can account for the major correction to the first approximation. The order of magnitude of this fz is also consistent with similar correlation coefficients appearing in the theory of nonresonant absorptions6). Hydrogen and deuterium are especially favorable cases in which to study the detailed shape of the rotational absorption. In these molecules, rotational energy transfer is relatively slow, and thus the entire dispersion curve occurs at frequencies low enough so that hydrodynamics remains valid. By studying the complete shape of the absorption, one can more or less evaluate separately the relaxation time and the correlation coefficients f2, f3 . .. In most other molecules, rotational energy transfer is more efficient, and the region of rotational relaxation becomes mixed up with the breakdown of macroscopic hydrodynamics. In such cases, meaningful comparison with the present theory may only made at lower frequencies. Then the observable quantity is the low frequency limit of eqn. (3.21), which is
cc
co2 St
N-
Thus in the case of strongly
-
3vo ( c, ) coupled
41 +
fnl n=2
T[l + :
rotation
(5.3)
only the combination
5 fnl
n=2
is observable. Physically this result [means that if measurements are reastricted to a low frequency region, one necessarily observes effects of correlated sequences of collisions, and one cannot experimentally separate out the effects of single collisions. This situation is similar to that of nuclear relaxation in gasess*), where for other technical resons measurements are often made in the low frequency limit. The present theory thus gives a well-defined connection between energy transfer in collisions, and ultrasonic absorption and dispersion, which is useful even when many levels are populated. If the dispersion occurs at low frequencies (weak coupling), a more detailed experimental separation of collision effects may be made. However, even if the rotational motion is strongly coupled in collisions, a meaningful interpretation of ultrasonic experiments has been established.
412
ABSORPTION
AND
DISPERSION
OF SOUND
IN MOLECULAR
GASES
Acknowledgements. The author wishes to thank Dr. H. F. P. Knaap and Dr. G. Thomaes for interesting discussions. We also thank Professor Prigogine for his hospitality at the Universit6 Libre de Bruxelles. This work was supported by the Society of Fellows of Harvard University. Received
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