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Vibrational energy transfer in hydrogen liquid and its isotopes
Chemical Physics 34 (1978) 205-217 0 North-Holland Publishing Company
VIBRATIONALENERGYTRANSFERiNHYDROGENLIQUIDANDITSISOTOPES G.M. GALE and C. DELALANDE Laboratoire d’Optique Quantique du Centre National de la Reelzerclte Scientifique, Ecole Polytechnique, 91 I28 Palaisean Cedex, France Received 7 August 1978
The transfer of vibrational energy (V-V) from H2 to isotopic impurities (HD or D2) has been studied in the liquid state, between 15 and 30 K. The subsequent relaxation (V-T) of the excited impurity by the HZ liquid host has also been measured and contrasted with the vibrational relaxation behaviour of pure Hz and DZ liquids. The isothermal density dependence of both V-V and V-T transfer has been investigated in the fluid state at 30 K. High density relaxation rates are also
compared to our data in the pure gases and to other available gas phase results. Measurements in the solid, near the triplepoint temperature, are equally reported for each process studied.
1. Introduction Laser spectroscopic techniques have been applied to the study of energy transfer in liquids since 1972. These experiments employed either stimulated Raman scattering or infrared resonant absorption to excite a well-defined vibrational level of the milieu. The decay of the excited state population was monitored either by spontaneous anti-Stokes scattering, infrared double resonance, or by direct fluorescence from infrared active molecules (see, for example, ref. El]). These methods are superior to the previously employed ultrasonic technique, being selective in excitation, sensitive to decay mode, and applicable to a wide range of relaxation times. In the majority of the experiments reported to date the molecules studied were organic polyatomics with decay times in the picosecond domain and with complex vibrational mode structure and relaxation channels. Recently, however, some observations of V-T transfer in simpler liquids, such as N2 [2,3], H2 [4], CO [S], and 0, [6] have been made using spectroscopic techniques. For these diatomics the relaxation times in the pure liquid are very long (lOds to 10 S) and, except for H2 and perhaps 02, are dominated by radiative processes_ Vibrational energy transfer (V-V) from excited N2 liquid to a number of small impurity molecules has also been studied in some detail albeit
at a futed temperature and density of the host liquid [2] _ Because of the complexity of many body interactions, even in simple liquids, and the lack of complete experimental data, theory has been slow to develop. Although the results to date are not inconsistent with an isolated binary collision model [7], a deeper understanding of vibrational energy relaxation processes in liquids evidently requires a more sophisticated theoretical analysis. In a previous detailed study [4] the vibrational relaxation behaviour of pure normal hydrogen was shown to exhibit three important features: (i) There is no apparent discontinuity in the relaxation process across the gas/liquid phase transition. That is, the inverse relaxation time T-I, remains linear in density N, up to liquid densities of about 1.7 X 10z2 cme3, as it is in a very low density gas where the deexcitation is due to isolated binary collisions. (ii) In the pressurised liquid, a small non-linearity appears which is well described by a simple cage law: 7 a (N-I/3 - cr), where N-II3 is the intermolecular distance in a cubic system (varying from 3.6 to 4.2 a in our experiment), and (Tis an effective collision diameter which is found to be close to the value of the Lennard-Jones parameter for Hz (2.93 a). (iii) As in the gas phase, the relaxation time in the liquid at constant density varies very slowly with temperature.
206
GM. G&T, C. Delalandef Vibrational energy transfer in Hz and its isotopeS
We wish to report here our observations of vibrational energy transfer in hydrogenic isotopic mixtures, specifically the vibration-vibration transfer of energy from excited H, liquid to dissolved impurities in the form of HD aniDa, and subsequent vibration-translation relaxation of the transferred energy by the host liquid. Forcompleteness we also present measurements of vibrational relaxation in pure “D2 liquid. The study of these isotopic mixtures and the pure isotopes is particularly interesting as the intermolecular potential remains essentially unchanged on isotopic substitution, but large variations in mass and vibrational quantum are obtained. The aim of this paper is to compare, as far as possible, the relaxation behaviour for the above processes with that of pure hydrogen, paying particular attention to the three features previousIy underlined: the gas/liquid phase transition, the isothermal density variation in the liquid, and the temperature dependence. In addition, we have also studied the liquid/solid phase transition, near the triple point, in all cases.
2. Experimental Excitation of the u = 1) J = 1 level of normal (75% ortho) liquid HZ, either pure, or containing smali quantities of HD or D7, was achieved by stimulated Raman scattering (SRS) of a powerful ruby laser pulse in the medium. The ruby laser system consisted of a single mode oscillator, Q-switched by an RG 685 Schott glass filter, producing a 2 MW, 30 ns pulse, from which a 2.5 ns segment was chopped out using standard Pockels’ cell/pulse generator techniques. This short pulse was subsequently amplified to a typical power level of 100 MW and slightly focussed to a 5 mm spot diameter at the 30 mm long cruxiform liquid cell (fig. l), giving a typical excitation ratio of the order of 2 X 10m5. The J= 1 level of orthohydrogen and the J = 0 level of parahydrogen are populated in the 3 : 1 ratio at the end of the laser pulse because of rapid thermalisation between the rotational levels [S]. When pure nD2 liquid was excited by SRS some evidence for V-V transfer to higher vibration levels was observed*. Thus, it was necessary to reduce the excitation ratio (to abdut 2 X 10m6). Stable, low level, * For footnote see next column.
X.YTable Fig. I. Schematic diagram of the experimental arrangement. P photodiode, S synchronisation, ADC analogue-to-digital converter, W sapphire window, L calcium fluorite lens, Flow or high psss filter, G germanium f&s, FL sapphire tieid lens, D indium antimonide detector cooled to 77 K.
excitation is difficult to produce by SRS and hence we empIoyed infrared absorption in the Q + S overtone band at 1.64 pm [9], which promotes two molecules into the u = 1 state and is quite distinct from the direct transition of a single molecule to u = 2, due to the large anharmonicity of the system. The absorption coefficient is of the order of 3 X IO-” cm-l_ Generation of 1.64pm radiation with about 30% photon efficiency from a ruby laser pulse is achieved by stimulated second Stokes scattering in a hydrogen gas cell at room temperature. The decay of the various excited populations (designated Hr, HD* and D;) was monitored by detect-
ing directly the infrared collision-induced fluorescence from the excited vibrational level of these species. The fluorescent lifetime evaluated from absorption coefficients [9] in the liquid is several seconds, very much * At high levels of SRS excitation the Dz fluorescence signal was found to be super-exponential at relatively short times. When the excitation was decreased this rapid signal disappeared. This effect is probably due to the Treanor pumping of highervibrational levels of the D2 manifold and would seem to indicate that the V-V/V-T transfer ratio is much larger for pure D2 than for pure Hz_ Spectroscopic evidence for this type of process has been obtained at high excitation levels, where the observed relaxation time is no longer inde pendent of emission wavelength, being = l/2 of its low level value at a wavelength corresponding to the v = 2 + v = 1 transition and continuing to decrease with increasing fIttorescence wavelength.
GM. Gale, C, Delahde/VibrationaI
207
energy transfer in Hz and its isotopes
PURE H, LIQUID 21.0 “K
I
1
0
50
Time
100 (ps)
Fig. 2. A typical infrared fluorescence signal obtained from pure “Hz liquid at 27.8 K. Trace height in millimeters time in microseconds. Also shown is the natural logarithm of the trace height (straight line), over 3.5 7.
longer than the collisional relaxation times involved (2 ps -2 ms), and the fluorescent decay signal thus gives the relaxation time directly. The emission from the excited population (in the 2-4 pm range) passes through a sapphire side-window and is focussed by an f(2) calcium fluorite collecting lens onto a cylindrical sapphire field lens placed just in front of a 2 mm diameter InSb infrared detector cooled to 77 K (fig. 1). The first and second Stokes waves generated by SRS in the cell are blocked by a 13 mm thick germanium filter. The electrical output from the suitably polarised and shunted detector is amplified and converted to digital form. Signal to noise ratios were improved (where necessary) by digital adding of shots. Final results were plotted on a X- Y recorder. Fig. 2 shows a typical curve for pure “Hz at 27.8 K. Due to the large isotope effect in hydrogen, the fluorescent emissions from the three species involved (Hz, HD*, D;) were readily distinguished by the use of appropriate high pass or low pass Filters. We verified that the observed infrared signals do indeed correspond to the species involved by recording their (time resolved) spectra with anf(3.5) grating monochromator. Fig. 3 shows the fluorescent spectra near
is graphed against
20 K of pure ‘rH2, a 0.5% solution of HD in Hz (where HD is excited by a rapid V-V transfer from Hz) and of pure lzD2 liquids. The hydrogen was supplied by Air-Liquide and had a stated general purity of 99.9999% and an isotopic impurity, in the form of D?, of less than 1.2 ppm. Deuterium was obtained from the Centre d’Enegie Atomique and contained less than 0.25% HD. A small quantity of HD was procured from Merkx and contained approximately 0.5% DzFL.Solutions of a known concentration of HD or D2 in Hz liquid were prepared as follows: a 1 II stainless steel container was tilled to a given pressure (normally between 1 and 100 torrs) with the impurity gas. A continuous stream of pure Hz was then admitted to the container until a final desired pressure, typically about 10 atm, was reached, the recipient was closed, and mixing allowed to proceed. This gas sample was liquified rapidly in the experimental cell at the required temperature and the cell was isolated. * It can be shown from our present measurements
that OUT experimental results were unaffected by these small amounts of isotopic admixture in the dopants, taking account of the low concentrations of impurity used (2 1%).
208
GM. Gale. C. Delnlande~lfibratioml
z;= --
;-
=;; __
;_
z= I-
HD
“H*
;; I
Iz
nD2 n
energy transfer in Hz and-its isotopes
and hence we shall subsequently refer only to the u= 1 level of hydrogen (denoted by Hz), treating the two species of different nuclear moment globally. There are two main channels available for the vibrational deexcitation of the hydrogen molecule (see fig. 4): (1) Vibration-translation transfer via the reaction:
This relaxation time is a function of temperature T and of total hydrogen liquid density N. (2) Because the isotopic impurity XY has a lower vibrational quantum than H,, vibration-vibration transfer can occur: 26
2.3
3.2
3.6
h(v)
Fig. 3. Fluorescence spectra, near 20 K, of pure nHz (full line), a 0.5% solution of HD in nHa (dashed), and pure “D? (dotdashed). Although translational splitting is not resolved the main bands are identified in the figure using the notation Qlo(J’) fora rransicion from u= l;J=J’ to II= O,J=J*, and Olo(J’) for a transition from u = 1, J = J’ to u = 0, J = J’ + 2. Notice that the Qto(O) line of HD in ‘IHz, absent from the absoqtion spectrum of pure HD because of a cancellation effect [9], is seen here due to the presence of orthohydrogen (J= 1) molecules. This band was distinguished from the Olo (1) line of the excited nHz by time resolving the spectrum.
‘H;+XY 2 +XY*
H*2 +XY -H
+ A.!?(=
530 cm-l
=1170 cm-l followed
by the vibration-translation
for D2) ,
deexcitation
of
XY*: rXY*:H2
XY*+H
2 ------,H
2 +XY + AE(z3630
cm-l
k2990 cm-l The temperature of the experimental cell was variable in steps of 0.1 K (in the range S-300 K) and was measured by an incorporated H2 and D, vapour pressure thermometer. Densities were cakulated from P and Tvalues using refs. [IO-131.
for HD and
Tag*: H, depends only on the temperature
for HD and for D2). T and the
3. The population equation for H2-HD and H2-D2 mixtures We studied very low concentrations of isotopic impurities (between 0.01 and 1%) and only the u = 1, J= 1 level of orthohydrogen is initially populated by SRS. We have already pointed out that a rapid thernalisation occurs, during the excitation, with the u= 1, J= 0 level of the 25% parahydrogen molecules present. This thermalisation is fast because of the quasi-resonant nature of the reaction oHz(u=1,J=1)+~H2(~=0,J=0)=OH2(~=0,J=1) +PH2(v=
l,J=O)
+ dE(%6 cm-l),
v=o
i
HZ
I
yzo
XY
Fig. 4. Energy level diagram representing V--V and V-T tramfer processes for VibrationaLly excited hydrogen containing a dissolved &atomic dopant XY (HD or D,).
GM Gale. C. DefuiandefVibrationatenergy transfer in Hz and its isotopes
total density of the host N(at very low impurity concentration), but the V-V transfer time T~H*_~~ depends also on the concentration of the im&ity, x. In the above WChave neglected two V-T transfer reactions, Hz : XY and XY* : XY. This omission is justified in view of the low concentrations of impurity involved and, as will be seen later from the results, because of a general mass effect observed at low temperature, viz; the heavier the (isotopic) collision partners, the less efficient the V-T transfer_ Hence, the Hz : XY process should be slower than H; : Hz for example, and can be ignored at feeble dopant levels. We can also neglect all reverse endothermic vibrational reactions because of the low temperature (/CT < 20 cm-l). It should be noted that the initial concentration of excited molecules is always much less than the impurity concentration. This avoids complications due to possible saturation of a V-V channel. With these conditions, it can easily be shown that the excited species populations [H;] and [XY*], and their corresponding fluorescence, evolve according to the following equations: -1
1. [Hal@)= IH~l@)exe(--t~,bs
where ~;t, = ~i$. H + T$_x~ is the observed inverse fluorescen?deciy time of the excited hydrogen and gives the V-V transfer rate for a known impurity concentration [XY*](t) = exp(-rr&:
Hz) - exp(-rT;l,)
,
which, under the condition realised in our experiments behaves as exp(--tr&*: HZ) at large t, yielding rxy*: H, directly.
TXY*: H, ~,‘olw
209
is also inapplicable to l*Dz vapour as the very long relaxation time is comparable to the time of thermal diffusion out of the excited region. We have, however, been able to measure I- values in deuterium vapour near the critical point using the infrared excitation and detection technique. Fig. 5 compares our measurements of the rate constant k [defined by X-= (r&)-l s-l cm31 for “HZ and nD2, to earlier data at higher temperatures [15,16]. The agreement is seen to be excellent. This figure also shows jzD; : nH~ relaxation constants above 55 K [ 161, to which we will refer later. 4.2. Gas-liquid phase tramitim We have verified the linearity of 7-l in density, across the phase transition, for ‘*Dz near the critical temperature (fig. 6). This linearity is maintained up to densities of about 1.6 X 10” cmm3, and so it would seem that for ‘*D?, as for IzH2, there is no abrupt change in the relaxation process as one passes from gas to liquid. 4.3. Liquid demity depetrdence The published density values of compressed liquid ‘ID7 are not so precise as those for ‘*H? +, and we measured only the density dependence of r>*,: Hz and HD* : H, V-T transfer, near 30 K, in the same range of intermolecular dis?ance studied for pure IzH2. The agreement with the cell law, T m (iV-1/3 - a), is also very good in these two cases and we find, within experimental error, the same value of u as for “Hz liquid (see fig. 7). 4.4. Temperature dependence and isotopic ratio
4. Vibration-translation
transfer
We begin by considering the V-T transfer measurements compiled in table 1. 4.1. Gas phase The Schlieren detection method [14], used for nH2 gas, detects the global transfer of energy from vibration to translation. The interpretation of observed signals in the case of mixtures, where several relaxation times are involved, is then very difficult. This method
As we have seen, the V-T transfer density dependence in the liquid phase is the same for pure hydrogen and for isotopic mixtures. Thus it seems reasonable to compare the renzperafwe dependence of the mixture relaxation rate to that of H, by plotting the ratio ~~,t: H, /Tag* I H2, at the same saturated liquid den* Our data for nDp liquid indicates some deviation from the simple gage law at high densities (quantum effects are smaller for D2 than for Ef2 and hence D2 liquid is denser). We prefer, . however, to reserve judgement on this point until our measurements of both T and N for “D2 liquid, over an increased
density range, are completed.
210
G.M. Gale, C. Delalondef Vibrational energ$ransfet- in H2 ahi its isotopes
Table 1 ‘Ihe ensemble of relaxationtime measurements 7 in the liquid state as a function of temperature T, density Iv, and impurity concentration x. Some low density (gas phase) results in pure Da are also shown V-T
transfer
V-V transfer
T-
N
T
W)
(10z2 cmd3)
ols)
HD*: H,
14.8 18.6 21.9 24.7 27.5 29.8 27 27 21 27 27
2.26 2.18 2.06 1.95 1.8 1.6 1.84 1.96 2.06 2.14 2.19
19.4 21.3 22.3 26.3 28-4 31.9 29 25_1 23.5 21.5 19.3
D;: H,
14.3 15.0 21.9 24.7 28.1 30.1 29.8 29.8 29.8 29.8 29.8 29.8 29.8 29.8 29.8
2.27 2.25 2.06 1.94 1.75 1.60 1.64 1.69 1.75 1.78 1.83 1.88 1.93 1.97 2.05
23.6 24.7 28.1 30.5 33.8 37.7 36.4 34.0 32.9 31.8 30.9 29.3 28.5 27.6 25.6
19.3 23.5 25.0 27.8 30.0 32.2 33 33.6 35 36 37 37.4 37.4 38 33 35 37 38
2.57 2.45 2.39 2.27 2.15 2.01 1.95 1.9 1.78 1.69 1.54 1.43 1.30 1.25 0.21 0.33 0.50 0.83
0.96 x L-96 x 1.12 x 1.22 x 1.32 x 1.41 x LA7 x 1.46 x 1.56 X 1.50 x 1.69 x 1.8 x 1.85 X 2.08 x 17.25 x 8.6 x 5.38 x 3.1 x
D;: D,
T (K)
IO3 IO3 LO3 103 IO3 103 103 lo3 LO3 103 lo3 lo3 lo3 10s 103 IO3 lo3 103
Hz-HD
14.8 21.9 26.0 29.9 14.8 21.9 26.0 29.8 14.8 21.9 26.0 29.8 29.8 29.8 29.8 29.8 29.8
HZ-D,
14.8 14.8 14.8 14.8 14.8 21.4 21.4 21.4 21.4 24.1 24.7 24.7 24.7 24.1 28.2 28.2 28.2 28.2 28-2 30.1 30.1 30.1 30.1 30.1 29.8 29.8 29.8 29.8 29.8
>/ooI. 1.44 1.44 1.44 1.44 0.91 0.91 0.91 0.91 0.45 0.45 0.45 0.45 1.49 1.49 1.49 1.49 1.49 0.0 2.8 5.4 8.1 10.8 0.0 2.0 4.0 6.7 0.0 2.9 5.5 8.2 10.8 0.0 3.0 5.8 8.4 11.1 0.0 2.8 5.5 8.2 11.0 5.6 5.6 5.6 5.6 5.6
N (1O22 cmm3)
5
2.26 2.06 1.87 1.63 2.26 2.06 1.87 1.63 2.26 2.06 1.81 1.63 1.64 1.74 1.84 1.93 2.02
3.4 3.9 4.7 5.5 4.6 5.4 6.4 7.4 6.8 7;9 9.9 11.4 5.5 4.8 4.5 4.3 3.8
2.26 2.26 2.26 2.26 2.26 2.08 2.08 2.08 2.08 1.95 1.95 1.95 1.95 1.95 1.75 1.75 1.75 1.75 1.75 1:61 1.61 I.61 1.61 1.61 1.64 1.74 1.84 1.92 2.02
12.2 6.8 4.7 3.5 2.8 14.7 9.5 6.5 4.7 16.3 9.8 6.6 5.1 3.4 18.2 10.5 7.1 5.6 4.3 21.0 11.9 7.5 6.1 4.6 7.4 7.1 6.4 5.5 4.9
Gus)
G.M. Gale, C. DclatandejVibmtional energy transfer in Hz and its isotopes
211
-16-
I
-18-
50
3.5
4.0
Fig_ 7. The relaxation time as a function of mean intermolecular distance in the liquid a ~~‘f-~‘~, for the V-T transfer process D;: Hz.
-19-
T 50
100
“K
150
Fig. 5. The logarithm of gas phase rate constants li as a function of temperature. The low temperature measurements of this wvorkandref. [4] are represented by open symbols. Earlier measurements (refs_ [ 15,161) at higher temperatures are shown as full points.
,
-i-
.’
_-’
Fig. 6. The behaviour of the inverse relaxation time 6’ of pure nD, as a function of density Nacross the phase transition and in the pressurised liquid at 38 K. The point on the lower left
represents the saturated vapour.
sity, as a function of temperature (fig. 8). This ratio is essentially constant in the 14-30 K temperature range for HD’ : H, indicating that, as in “H,, the deexcitation rate is almost independent of temperature. For
D; : Hz the ratio increases by about 10% between i 5 and 30 K. We have also plotted in fig. 8 the ratio of the slightly extrapolated rate constant in the gas for Dg : Hz (see fig. 5) to that of pure nH2. The result is perfectly consistent with the liquid phase data which demonstrates that the isotopic ratio is the same in rhe gas and the liquid. For a given intermolecular
potential the vibrational
energy transfer rate depends essentially on two parameters; the reduced mass of the collision pair and the enerov gap. Table 2 gives the isotopic ratios of the transfer rates at 30 K, as well as the reduced mass and the vibrational quantum, for the different reactions studied. One notes that in every case the reduced mass effect predominates; i.e. the larger the reduced mass the smaller the rate. The large “pure” influence of reduced mass can be seen by comparing D; : D2 to Dt : H,, where the vibrational quantum is the same, but the reaction rate continues to increase as one passes from Df : Hz via HD* : H2 to HD;: H2 even though the vibrational gap is also increasing, which would tend to decrease the rate. However, it is known that the deexcitation channel is not necessarily a direct V-T transfer, but that relaxation can occur via intermediate rotational states of the fun-
212
G.M. Gale, C. DelalondelVibrational
energy
0.7 -
tramferin If2 and its isotopes
HD
15
20
25
Temperature Fig. 8. The isotopic ratio of the V-T transfer times in the saturated HD or Dz). Also shown is the ratio of the slightly extrapolated
30
T OK
liquid 7~;:
rate constant
H2/~xy* : H2 as a function of temperature
(XY =
for Dz: H, (see fig_ 5) to that in pure “H, gas, near
25 and 30 K (open triangles).
darnental vibrational level [1’7,18], and so the exact value of the energy gap is uncertain. A detaiied inter-
pretation of the isotopic ratios at these low temperatures would require a complete quantum mechanical calculation including all V-R decay channels. Conversely, these ratios should provide a sensitive test of the accuracy of any such calculation. 4.5. Theories of V-T transfer in simple liquids Experimental information on the collisional relaxation of vibrational population in liquids composed of small molecules is very scarce. In fact, for several of the diatomic liquids studied to date the predominant Table 2 The efficiency E of the V-T relaxation process at 30 K compared to pore nH2, for various isotopic interactions_ The reduced mass of the collision partners p, and the vibrational energy gap AE are also indicated V-T transfer H;:H2
HD* : H,
D;:H,
E P (au)
1.0 1.0
0.63 1.2
0.57
0.01
AE(cm-L)
4155
3630
1.33 2990
2.0 2990
D;:D,
relax? iion Frocess seems to be radiative. This paucity of data has retarded the advance of the theory of energy transfer in simple liquids. On the other hand, a relatively large number of organic molecules have been studied in the liquid state. In this case however, a different theoretical approach may be required due to the internal complexity of these systems. The testing of any given theory by comparison of relaxation times in the gas and liquid phase must be made with care. Any extrapolation of high temperature results to liquid temperatures can be dangerous, as illustrated in fig..5 where, even on isotopic substitution, the dependence of the gas relaxation rate on temperature varies enormously. It is obviously preferable to compare measurements made in the same temperature range and, if possible, on the same sample. The only previous complete study of the relaxation process in the gas and liquid at the same temperature is that of Madigosky and Litovitz [7] in CO2 and CS?, employing global acoustic techniques. The behaviour of these systems is very similar to that observed in H2 and its isotopes; i.e. (i) 7-1 remains linear in density across the base tran-2 sition up to densities of the order of(0.4 XaL J), where aL, is he Lennard-Jones parameter of the molecule involved.
G.M. Gale. C. DelalandejVibrarional energy tran.@r in Hz and its isotopes
(ii) The isothermal density dependence of 7 in the liquid phase is well described by a cage law, even though the absolute temperature and u values were significantly different in the two studies. (iii) The temperature dependence of T is found to be the same in the gas and liquid phase, where comparison is possible. (iv) For hydrogen, the isotopic ratios are the same in the gas and the liquid. The last two points are, of course, closely related to the first one. In the light of these results it seems not unreasonabIe that a theory of relaxation in the liquid could be linked to low-density, two-body processes_ Litovitz [7] has proposed a classical theory of relaxation in liquids which supposes that deexcitation is due to isolated binary collisions with a probability p* per collision, independent of density, but with a frequency Z of elastic collisions at high density given by a simple cage model. Hence one can write,T-l =p*Z at all densities. The cage model, for the calculation of the frequency of collisions in the liquid, gives 2 = v/(N-1/3 - 0). where v is the average velocity, and thus,
This semi-empirical formula describes well the observed N dependence of 7 at liquid densities. Another value forp* V can be calculated from low density gas k data, using x_= (_m?-1 = 21/z 3 j? = 21/2,*J
cv,
where J and J, are the suitably averaged inelastic and elastic cross sections respectively, and taking somewhat arbitrarily, in a classical hard-sphere approximation, J, = ga2. This p* v value is found to be in fair agreement with the experimental value obtained in the liquid from the slopes of cage law plots at high density [4,71. This theory is attractive in view of its simplicity and empirical accuracy. Nevertheless, it is difficult to understand the notion of an elastic collision at densities such that the intermolecular distance is of the order of the molecular “diameter” and neighbouring particles are always in interaction_ Davis and Oppenheim have formalised the theory of vibrational transfer in dense media [ 19,201 and have
been able to show, under certain conditionst,
213 that the
liquid relaxation time can be correlated to a gas relaxation time, using a linear extrapolation and a correction factorg(T, IV, R*)/g(T, 0, R*), where g(T, N, R*) is the pair distribution function at a characteristic classical distance of closest approach, responsible for deexcitation,R*. The correction factor is then the pair distribution function, for this value of intermolecular distance, at liquid densities, normjlised to its zero density limitg(T, 0, R*). Experimentally this correction factor is found to be not far from unity, which would require a value of R* close to a (see, for example, ref. [31]), invalidating one of the conditions (R* 2 0.9 o)t used in the derivation of the above expression. Also, for hydrogenic systems, quantum mechanical calculations [22] have shown that even low energy collisions deexcite efficiently and hence it is difficult to define a specific effective R* as required in the approximation of Davis and Oppenheim. Both of the above theories attempt to correlate the relaxation time in the liquid to its supposedly known value in the low density gas. Although the process of collisional deexcitation at low density is we!l understood, an a priori calculation is generally difficult. Recently, complete quantum-mechanical calculations have been undertaken for H, [23] but as yet these calculations have not been extended to liquid temperatures even for the simplest system (Hz : He). Another approach has been made by Diestler [24] who has tried to calculate directly the relaxation rate in simple liquids by a quantum cell model. In view of the difficulty of the calculation in the gas, the simplifications made in this theory, particularly in the evaluation of the interaction potential, lead to erroneous results. 5. Vibration-vibration
transfer
Fig. 9 shows the variation of the inverse observed relaxation time of H;, T;&, previously defined, versus isotopic impurity concentration, at various saturated liquid temperatures. 7iks is found to be linear in impurity concentration, as expected at these low dopant t In particular, ifE* is a characteristic kinetic energy of dee.ucitinerencounters IE* = VCR*).where V(R) is the inter-
molecular potential], and fiw is the vibrational quantum; kTQE*--StiwandR*<,0.9a_
214
&W Gale. C. Delalandef Vibrational energy transfer in Hz and its isotopes
13JF’K
(a)
21.95’K
26.0SaK 29.63’K
I
I 2.e
l.ldJ
Concentration
X
levels and for a non-diffusion-limited process_ A cage law isothermal host density dependence is also observed_ From the slopes of the plots in fig. 9 we can deduce a pure V-V relaxation time r”;_,~u(T), at an impurity concentration taken equal to unity. This enables us to compare directly the efficiency of the V-V transfer processes to that of V-T transfer in pure hydrogen_ The ratio of the V-T relaxation time in H2 to a given V-V time, at the saturated liquid density, was found to be independent of temperature in the 15-30 K range. The V-V transfer mechanisms are much more efficient than V-T transfer as can be seen in table 3. Comparison with the only other existing measurement (Hz -+ D2 at 300 K [25]) shows that the V-T/V-V ratio is larger by a factor of three at low temperature. This type of behaviour is quite frequ&tly remarked; i.e. the temperature variation of a high probability process is less than that of a low probability one, and can lead to experimenta difficulties in the measure of pure V-T transfer times at low temperatures [2,5]-
Ag. 9. The inverse observed relasation time of Hz, T& as a function of impurity concentration at various saturnted liquid host temperatures and in the solid, for the dopants (a) HD and 01) D,.
I 0
o.5-lD-2
1.d
1.5.lE
G.M. Gale.C. Delalandc/Vibrational energy transferin H2 and itsisotopes Table 3 The efficiency of V-V relaxation at 30 K and 300 K cornpared to pure V-T relaxation in “H2, for the two processes H;-HDandH;-tD,
Hz-D2 H;+HD
300 K
15-30 K
90
280 1850
In our V-T experiment in hydrogen, for example, the concentration of HD must be less than 10 ppm to be negligible. The isotopic ratio (rH;_HD/TH;_-D, = 7) is greater than that for V-T transfer (Tag* I H /rD; : H, = 1.2) as the energy gap and reduced mass e‘ffects here operate in the same direction, and ofcourse the change in Ihe energy to be transferred to translation is much greater here for V-V, than for the corresponding V-T
processes. Finally, supposing that the linearity of T-l in N up to liquid densities also holds for V-V transfer we can estimate gas rate constants near 30 EC.These are k = 8.7. X lo-l6 cm3 s-l for H; + D2 and k = 5.6 X lo-l5 cm3 s-l for H; --f HD.
215
6. Liquid/solid phase transition Up to now, we have compared the processes of vibrational relaxation in the liquid and gas phase, considering the liquid as a high density gas. Alternatively, not far from the melting point, a liquid could be regarded as a quasi-crystalline solid without long-range order, and it is interesting to study the behaviour of the relaxation time across the liquid/solid phase transition. We have been able to measure relaxation times in the solid for all six of the cases previously discussed. The results in the solid close to the triple point temperature are presented in table 4. For comparison the corresponding relaxation times in the liquid just above the melting point are also shown. In the third column of this table the ratio of Tsolid to Tliquid is indicated.
It can be seen that there is no striking variation in the deexcitation times as one passes across the phase transition. In contrast to the gas/liquid transition however, a small change of the isotopic ratios between the liquid and the solid is observed. There is, of course, a substantial density increase on solidification, and we can
Table 4 Liquid-solid transition (near 13.9 K for nH1 and 18.7 K for “D,). The V-V rela..ation times are calculated at a supposed impurity concentration of unity. The third column shows the ratio of the experimental relaxation times, and the fourth column is the corrected ratio (see text) Liquid
Solid
2.27 x 1022 (cm-j)
2.68 x 1O22 (cmm3)
TH; : H, rHD* : H,
12.0 ps 19.6 fis
8.5 PS 16.25 /IS
0.71 0.83
1.02 1.19
‘D; : H,
23.6 PS
24.7 US
1.05
1.50
7H,+HD * 7H; -+ D,
6.9 ns
6.9 ns
1.00
1.43
39.5 ns
30.3 ns
0.77
1.10
251 x 10” (cmJ)
3.07 x 10” (cm-a)
970 ps
850 fis
0.88
1.46
density
V-T
v-v (x = 1) density V-T
‘D; : D,
attempt to correct for this by assuming that the liquid cage 1~ holds right down to characteristic solid intermolecular distances. Ths last column of table 4 shows Fe estimated ratios rs&/rEq(Nn = Ns). The corrected ratio is always greater than unity; i.e. deexcitation appears to be somewhat less efficient in the solid. Another point to note is that it seems to make little difference whether one is dealing with deexcitation in the pure solid or of an impurity in a matrix. (The change of the ratio r&e in going from HT : H2 to HD* : H2 is of the same order of magnitude as that between HD* : H2 and D; : H2.) There are only a few experimental results on this phase transition in simple systems [5,36] and the main concern of theoretical and experimental studies of vibrational relaxation in solids is with impurities in monoatomic matrices [27-29]_ From these studies it would appear, however, that the local environment of the excited species plays a dominant role in deexcitation, and hence a dramatic change in relaxation time on solidification might not be expected. Direct measurements of relaxation times at high density in the liquid would ahnost certainly be helpful for the detailed understanding of the isotopic variation in the solid and the solid/liquid transition.
7. Conclusion
Our resuIts show that vibrational relaxation in the liquid phase is highly correlated to low density gas phase processes_ The semi-empirical isolated binary collision theory of Litovitz gives good order of magnitude scaling and reproduces well the isothermal density dependence in the liquid. For a better comprehension of the mechanisms underiying the phase transition behaviour and the effects of density and temperature in the liquid, more fo,mal and sophisticated theories are required. It is to be hoped that the development of such theories will be stimulated by the increasing fund of experimental data on simple systems.
References [l] WV. Kaiser and A. Laubereau, in: Tunable lasers andap plications, Proceedings of the Loen Conference, eds. A. ~Mooradian, T. Jaeger and P. Stoksetb (Springer, Berlin,
1976). [2] W.F_Calaway and GE. Ewing, J. Chem. Phys. 63 (1975) 2842. [3] S.R.J. Brueckand R.M. Osgood, Chem. Phys. Letters 39 (1976) 568. [4] C. Delalande and GM. Gale, Chem. Phys. Letters 50 (1977)
339.
(51 N.
Legay-Sommaire and F. Legay, Chem. Phys. Letters 52 (1977) 213. [6] C. Renner and hi. Maier, Chem. Phys. Letters 35 (1975) 226. [ 7] W.M. Madigosky and T.A. Litovitz, J. Chen. Phys. 34 (1961) 489. [8] C.G. Sluijter, H.F.P. Knaap and J.J.M. Beenakker, Physica 30 (1964) 745. [9] E.J. Alhi, H.P. Gush, W.F.J. Hare, J.L. Hunt and H.L. Welsh, in: Coltoque sur les prop&t& optiques et acoustiques des fluides cornprimes et les interactions molbcul&es, 1-6 juiuet 1957, Bellevue (France) (CNRS, 1959) p. 21-39. PO1 A. van Itterbeek, 0. Verbeke. F. Theeuwes and V. Jansoone. physica 32 (1966) 1591. [Ill H.J. Hoge and R.D. Arnold, J. Research NBS 47 (19.51)
[I21
63. H.J. Hoge and
WI Wl
505. P-K. Davis, J. Chem. Phys. 57 (1972) 517. J-0. Hirschfelder, CF. Curtiss and R-B. Bird, in: Mo-
J.W. Lassiter, J. Research NBS 47 (1951) 75. [131 J.D. Rogers and F.G. Brickwedde, J. Chem. Physics 42 (1965) 2822. 1141 .I. Ducuinp. C Joffrin and 3.P. Coffmet, Opt. Commun. 2 (1970) 245. 1151 MM. Audibert, C. Joffrin and J. Ducuing, Chem. Phys. Letters 25 (1974) 1.58. [161 I. Lukasik and J. Ducuing, Chem. Phys. Letters 27 (1974) 203. 1171 M.M. Audibert, R. Vilaseca, J. Lukasik and J. Ducuing, Chem. Phys. Letters 31 (1975) 232. ]18] M.H. Alexander, J. Chem. Physics 61 (1974) 516. 1191 P.K. Davis and I. Oppenheim, I. Chem. Phys. 57 (1972)
lecular theory of gases and liquids (Wiley, New York, 19.54): L. Verlet, Phys. Rev. 165 (1968) 201; F. Mandel, R.J. B-an Phys. 52 (1970) 3315.
Acknowledgement We would like to thank Professor J. Ducuing for his continuing support, Dr. C. Flytzanis for helpful discussion, and Mr. J. Debrie for technical assistance.
and M.Y. Bearman. 5. Chem.
[22] J. Lukasik, Ph.D. Thesis, University of Paris-Sud, France (1975). [23] R. Ramaswamy and H. Rabitz, J. Chem. Phys. 66 (1977) 152. [24] D.J. DiestIer, Chem. Phys. 7 (1975) 349. .1251 J.F. Bott. J. Chem. Phys. 65 (1976) 3921.
GM. Gale, C. Delaiande~Vibmtionnl energy transfer in Hz avd
[261
IL Dressier, 0. Oehler and D.A. Smith, Phys. Rev. Letters 34 (1975) 1364. 127) F. Legay, in: Chemical and biochemical applications of laser, Vol. 3, ed. G.B. Moore (Academic Press, New York, 1977).
its isotopes
217
[281 J. Goodman and L.E. Brus, J. Chem. Physics 65 (1976) 3146. 1291 K.F. Freed and H. Metiu, Chem. Phys. Letters 48 (1977) 262.
VIBRATIONALENERGYTRANSFERiNHYDROGENLIQUIDANDITSISOTOPES G.M. GALE and C. DELALANDE Laboratoire d’Optique Quantique du Centre National de la Reelzerclte Scientifique, Ecole Polytechnique, 91 I28 Palaisean Cedex, France Received 7 August 1978
The transfer of vibrational energy (V-V) from H2 to isotopic impurities (HD or D2) has been studied in the liquid state, between 15 and 30 K. The subsequent relaxation (V-T) of the excited impurity by the HZ liquid host has also been measured and contrasted with the vibrational relaxation behaviour of pure Hz and DZ liquids. The isothermal density dependence of both V-V and V-T transfer has been investigated in the fluid state at 30 K. High density relaxation rates are also
compared to our data in the pure gases and to other available gas phase results. Measurements in the solid, near the triplepoint temperature, are equally reported for each process studied.
1. Introduction Laser spectroscopic techniques have been applied to the study of energy transfer in liquids since 1972. These experiments employed either stimulated Raman scattering or infrared resonant absorption to excite a well-defined vibrational level of the milieu. The decay of the excited state population was monitored either by spontaneous anti-Stokes scattering, infrared double resonance, or by direct fluorescence from infrared active molecules (see, for example, ref. El]). These methods are superior to the previously employed ultrasonic technique, being selective in excitation, sensitive to decay mode, and applicable to a wide range of relaxation times. In the majority of the experiments reported to date the molecules studied were organic polyatomics with decay times in the picosecond domain and with complex vibrational mode structure and relaxation channels. Recently, however, some observations of V-T transfer in simpler liquids, such as N2 [2,3], H2 [4], CO [S], and 0, [6] have been made using spectroscopic techniques. For these diatomics the relaxation times in the pure liquid are very long (lOds to 10 S) and, except for H2 and perhaps 02, are dominated by radiative processes_ Vibrational energy transfer (V-V) from excited N2 liquid to a number of small impurity molecules has also been studied in some detail albeit
at a futed temperature and density of the host liquid [2] _ Because of the complexity of many body interactions, even in simple liquids, and the lack of complete experimental data, theory has been slow to develop. Although the results to date are not inconsistent with an isolated binary collision model [7], a deeper understanding of vibrational energy relaxation processes in liquids evidently requires a more sophisticated theoretical analysis. In a previous detailed study [4] the vibrational relaxation behaviour of pure normal hydrogen was shown to exhibit three important features: (i) There is no apparent discontinuity in the relaxation process across the gas/liquid phase transition. That is, the inverse relaxation time T-I, remains linear in density N, up to liquid densities of about 1.7 X 10z2 cme3, as it is in a very low density gas where the deexcitation is due to isolated binary collisions. (ii) In the pressurised liquid, a small non-linearity appears which is well described by a simple cage law: 7 a (N-I/3 - cr), where N-II3 is the intermolecular distance in a cubic system (varying from 3.6 to 4.2 a in our experiment), and (Tis an effective collision diameter which is found to be close to the value of the Lennard-Jones parameter for Hz (2.93 a). (iii) As in the gas phase, the relaxation time in the liquid at constant density varies very slowly with temperature.
206
GM. G&T, C. Delalandef Vibrational energy transfer in Hz and its isotopeS
We wish to report here our observations of vibrational energy transfer in hydrogenic isotopic mixtures, specifically the vibration-vibration transfer of energy from excited H, liquid to dissolved impurities in the form of HD aniDa, and subsequent vibration-translation relaxation of the transferred energy by the host liquid. Forcompleteness we also present measurements of vibrational relaxation in pure “D2 liquid. The study of these isotopic mixtures and the pure isotopes is particularly interesting as the intermolecular potential remains essentially unchanged on isotopic substitution, but large variations in mass and vibrational quantum are obtained. The aim of this paper is to compare, as far as possible, the relaxation behaviour for the above processes with that of pure hydrogen, paying particular attention to the three features previousIy underlined: the gas/liquid phase transition, the isothermal density variation in the liquid, and the temperature dependence. In addition, we have also studied the liquid/solid phase transition, near the triple point, in all cases.
2. Experimental Excitation of the u = 1) J = 1 level of normal (75% ortho) liquid HZ, either pure, or containing smali quantities of HD or D7, was achieved by stimulated Raman scattering (SRS) of a powerful ruby laser pulse in the medium. The ruby laser system consisted of a single mode oscillator, Q-switched by an RG 685 Schott glass filter, producing a 2 MW, 30 ns pulse, from which a 2.5 ns segment was chopped out using standard Pockels’ cell/pulse generator techniques. This short pulse was subsequently amplified to a typical power level of 100 MW and slightly focussed to a 5 mm spot diameter at the 30 mm long cruxiform liquid cell (fig. l), giving a typical excitation ratio of the order of 2 X 10m5. The J= 1 level of orthohydrogen and the J = 0 level of parahydrogen are populated in the 3 : 1 ratio at the end of the laser pulse because of rapid thermalisation between the rotational levels [S]. When pure nD2 liquid was excited by SRS some evidence for V-V transfer to higher vibration levels was observed*. Thus, it was necessary to reduce the excitation ratio (to abdut 2 X 10m6). Stable, low level, * For footnote see next column.
X.YTable Fig. I. Schematic diagram of the experimental arrangement. P photodiode, S synchronisation, ADC analogue-to-digital converter, W sapphire window, L calcium fluorite lens, Flow or high psss filter, G germanium f&s, FL sapphire tieid lens, D indium antimonide detector cooled to 77 K.
excitation is difficult to produce by SRS and hence we empIoyed infrared absorption in the Q + S overtone band at 1.64 pm [9], which promotes two molecules into the u = 1 state and is quite distinct from the direct transition of a single molecule to u = 2, due to the large anharmonicity of the system. The absorption coefficient is of the order of 3 X IO-” cm-l_ Generation of 1.64pm radiation with about 30% photon efficiency from a ruby laser pulse is achieved by stimulated second Stokes scattering in a hydrogen gas cell at room temperature. The decay of the various excited populations (designated Hr, HD* and D;) was monitored by detect-
ing directly the infrared collision-induced fluorescence from the excited vibrational level of these species. The fluorescent lifetime evaluated from absorption coefficients [9] in the liquid is several seconds, very much * At high levels of SRS excitation the Dz fluorescence signal was found to be super-exponential at relatively short times. When the excitation was decreased this rapid signal disappeared. This effect is probably due to the Treanor pumping of highervibrational levels of the D2 manifold and would seem to indicate that the V-V/V-T transfer ratio is much larger for pure D2 than for pure Hz_ Spectroscopic evidence for this type of process has been obtained at high excitation levels, where the observed relaxation time is no longer inde pendent of emission wavelength, being = l/2 of its low level value at a wavelength corresponding to the v = 2 + v = 1 transition and continuing to decrease with increasing fIttorescence wavelength.
GM. Gale, C, Delahde/VibrationaI
207
energy transfer in Hz and its isotopes
PURE H, LIQUID 21.0 “K
I
1
0
50
Time
100 (ps)
Fig. 2. A typical infrared fluorescence signal obtained from pure “Hz liquid at 27.8 K. Trace height in millimeters time in microseconds. Also shown is the natural logarithm of the trace height (straight line), over 3.5 7.
longer than the collisional relaxation times involved (2 ps -2 ms), and the fluorescent decay signal thus gives the relaxation time directly. The emission from the excited population (in the 2-4 pm range) passes through a sapphire side-window and is focussed by an f(2) calcium fluorite collecting lens onto a cylindrical sapphire field lens placed just in front of a 2 mm diameter InSb infrared detector cooled to 77 K (fig. 1). The first and second Stokes waves generated by SRS in the cell are blocked by a 13 mm thick germanium filter. The electrical output from the suitably polarised and shunted detector is amplified and converted to digital form. Signal to noise ratios were improved (where necessary) by digital adding of shots. Final results were plotted on a X- Y recorder. Fig. 2 shows a typical curve for pure “Hz at 27.8 K. Due to the large isotope effect in hydrogen, the fluorescent emissions from the three species involved (Hz, HD*, D;) were readily distinguished by the use of appropriate high pass or low pass Filters. We verified that the observed infrared signals do indeed correspond to the species involved by recording their (time resolved) spectra with anf(3.5) grating monochromator. Fig. 3 shows the fluorescent spectra near
is graphed against
20 K of pure ‘rH2, a 0.5% solution of HD in Hz (where HD is excited by a rapid V-V transfer from Hz) and of pure lzD2 liquids. The hydrogen was supplied by Air-Liquide and had a stated general purity of 99.9999% and an isotopic impurity, in the form of D?, of less than 1.2 ppm. Deuterium was obtained from the Centre d’Enegie Atomique and contained less than 0.25% HD. A small quantity of HD was procured from Merkx and contained approximately 0.5% DzFL.Solutions of a known concentration of HD or D2 in Hz liquid were prepared as follows: a 1 II stainless steel container was tilled to a given pressure (normally between 1 and 100 torrs) with the impurity gas. A continuous stream of pure Hz was then admitted to the container until a final desired pressure, typically about 10 atm, was reached, the recipient was closed, and mixing allowed to proceed. This gas sample was liquified rapidly in the experimental cell at the required temperature and the cell was isolated. * It can be shown from our present measurements
that OUT experimental results were unaffected by these small amounts of isotopic admixture in the dopants, taking account of the low concentrations of impurity used (2 1%).
208
GM. Gale. C. Delnlande~lfibratioml
z;= --
;-
=;; __
;_
z= I-
HD
“H*
;; I
Iz
nD2 n
energy transfer in Hz and-its isotopes
and hence we shall subsequently refer only to the u= 1 level of hydrogen (denoted by Hz), treating the two species of different nuclear moment globally. There are two main channels available for the vibrational deexcitation of the hydrogen molecule (see fig. 4): (1) Vibration-translation transfer via the reaction:
This relaxation time is a function of temperature T and of total hydrogen liquid density N. (2) Because the isotopic impurity XY has a lower vibrational quantum than H,, vibration-vibration transfer can occur: 26
2.3
3.2
3.6
h(v)
Fig. 3. Fluorescence spectra, near 20 K, of pure nHz (full line), a 0.5% solution of HD in nHa (dashed), and pure “D? (dotdashed). Although translational splitting is not resolved the main bands are identified in the figure using the notation Qlo(J’) fora rransicion from u= l;J=J’ to II= O,J=J*, and Olo(J’) for a transition from u = 1, J = J’ to u = 0, J = J’ + 2. Notice that the Qto(O) line of HD in ‘IHz, absent from the absoqtion spectrum of pure HD because of a cancellation effect [9], is seen here due to the presence of orthohydrogen (J= 1) molecules. This band was distinguished from the Olo (1) line of the excited nHz by time resolving the spectrum.
‘H;+XY 2 +XY*
H*2 +XY -H
+ A.!?(=
530 cm-l
=1170 cm-l followed
by the vibration-translation
for D2) ,
deexcitation
of
XY*: rXY*:H2
XY*+H
2 ------,H
2 +XY + AE(z3630
cm-l
k2990 cm-l The temperature of the experimental cell was variable in steps of 0.1 K (in the range S-300 K) and was measured by an incorporated H2 and D, vapour pressure thermometer. Densities were cakulated from P and Tvalues using refs. [IO-131.
for HD and
Tag*: H, depends only on the temperature
for HD and for D2). T and the
3. The population equation for H2-HD and H2-D2 mixtures We studied very low concentrations of isotopic impurities (between 0.01 and 1%) and only the u = 1, J= 1 level of orthohydrogen is initially populated by SRS. We have already pointed out that a rapid thernalisation occurs, during the excitation, with the u= 1, J= 0 level of the 25% parahydrogen molecules present. This thermalisation is fast because of the quasi-resonant nature of the reaction oHz(u=1,J=1)+~H2(~=0,J=0)=OH2(~=0,J=1) +PH2(v=
l,J=O)
+ dE(%6 cm-l),
v=o
i
HZ
I
yzo
XY
Fig. 4. Energy level diagram representing V--V and V-T tramfer processes for VibrationaLly excited hydrogen containing a dissolved &atomic dopant XY (HD or D,).
GM Gale. C. DefuiandefVibrationatenergy transfer in Hz and its isotopes
total density of the host N(at very low impurity concentration), but the V-V transfer time T~H*_~~ depends also on the concentration of the im&ity, x. In the above WChave neglected two V-T transfer reactions, Hz : XY and XY* : XY. This omission is justified in view of the low concentrations of impurity involved and, as will be seen later from the results, because of a general mass effect observed at low temperature, viz; the heavier the (isotopic) collision partners, the less efficient the V-T transfer_ Hence, the Hz : XY process should be slower than H; : Hz for example, and can be ignored at feeble dopant levels. We can also neglect all reverse endothermic vibrational reactions because of the low temperature (/CT < 20 cm-l). It should be noted that the initial concentration of excited molecules is always much less than the impurity concentration. This avoids complications due to possible saturation of a V-V channel. With these conditions, it can easily be shown that the excited species populations [H;] and [XY*], and their corresponding fluorescence, evolve according to the following equations: -1
1. [Hal@)= IH~l@)exe(--t~,bs
where ~;t, = ~i$. H + T$_x~ is the observed inverse fluorescen?deciy time of the excited hydrogen and gives the V-V transfer rate for a known impurity concentration [XY*](t) = exp(-rr&:
Hz) - exp(-rT;l,)
,
which, under the condition realised in our experiments behaves as exp(--tr&*: HZ) at large t, yielding rxy*: H, directly.
TXY*: H, ~,‘olw
209
is also inapplicable to l*Dz vapour as the very long relaxation time is comparable to the time of thermal diffusion out of the excited region. We have, however, been able to measure I- values in deuterium vapour near the critical point using the infrared excitation and detection technique. Fig. 5 compares our measurements of the rate constant k [defined by X-= (r&)-l s-l cm31 for “HZ and nD2, to earlier data at higher temperatures [15,16]. The agreement is seen to be excellent. This figure also shows jzD; : nH~ relaxation constants above 55 K [ 161, to which we will refer later. 4.2. Gas-liquid phase tramitim We have verified the linearity of 7-l in density, across the phase transition, for ‘*Dz near the critical temperature (fig. 6). This linearity is maintained up to densities of about 1.6 X 10” cmm3, and so it would seem that for ‘*D?, as for IzH2, there is no abrupt change in the relaxation process as one passes from gas to liquid. 4.3. Liquid demity depetrdence The published density values of compressed liquid ‘ID7 are not so precise as those for ‘*H? +, and we measured only the density dependence of r>*,: Hz and HD* : H, V-T transfer, near 30 K, in the same range of intermolecular dis?ance studied for pure IzH2. The agreement with the cell law, T m (iV-1/3 - a), is also very good in these two cases and we find, within experimental error, the same value of u as for “Hz liquid (see fig. 7). 4.4. Temperature dependence and isotopic ratio
4. Vibration-translation
transfer
We begin by considering the V-T transfer measurements compiled in table 1. 4.1. Gas phase The Schlieren detection method [14], used for nH2 gas, detects the global transfer of energy from vibration to translation. The interpretation of observed signals in the case of mixtures, where several relaxation times are involved, is then very difficult. This method
As we have seen, the V-T transfer density dependence in the liquid phase is the same for pure hydrogen and for isotopic mixtures. Thus it seems reasonable to compare the renzperafwe dependence of the mixture relaxation rate to that of H, by plotting the ratio ~~,t: H, /Tag* I H2, at the same saturated liquid den* Our data for nDp liquid indicates some deviation from the simple gage law at high densities (quantum effects are smaller for D2 than for Ef2 and hence D2 liquid is denser). We prefer, . however, to reserve judgement on this point until our measurements of both T and N for “D2 liquid, over an increased
density range, are completed.
210
G.M. Gale, C. Delalondef Vibrational energ$ransfet- in H2 ahi its isotopes
Table 1 ‘Ihe ensemble of relaxationtime measurements 7 in the liquid state as a function of temperature T, density Iv, and impurity concentration x. Some low density (gas phase) results in pure Da are also shown V-T
transfer
V-V transfer
T-
N
T
W)
(10z2 cmd3)
ols)
HD*: H,
14.8 18.6 21.9 24.7 27.5 29.8 27 27 21 27 27
2.26 2.18 2.06 1.95 1.8 1.6 1.84 1.96 2.06 2.14 2.19
19.4 21.3 22.3 26.3 28-4 31.9 29 25_1 23.5 21.5 19.3
D;: H,
14.3 15.0 21.9 24.7 28.1 30.1 29.8 29.8 29.8 29.8 29.8 29.8 29.8 29.8 29.8
2.27 2.25 2.06 1.94 1.75 1.60 1.64 1.69 1.75 1.78 1.83 1.88 1.93 1.97 2.05
23.6 24.7 28.1 30.5 33.8 37.7 36.4 34.0 32.9 31.8 30.9 29.3 28.5 27.6 25.6
19.3 23.5 25.0 27.8 30.0 32.2 33 33.6 35 36 37 37.4 37.4 38 33 35 37 38
2.57 2.45 2.39 2.27 2.15 2.01 1.95 1.9 1.78 1.69 1.54 1.43 1.30 1.25 0.21 0.33 0.50 0.83
0.96 x L-96 x 1.12 x 1.22 x 1.32 x 1.41 x LA7 x 1.46 x 1.56 X 1.50 x 1.69 x 1.8 x 1.85 X 2.08 x 17.25 x 8.6 x 5.38 x 3.1 x
D;: D,
T (K)
IO3 IO3 LO3 103 IO3 103 103 lo3 LO3 103 lo3 lo3 lo3 10s 103 IO3 lo3 103
Hz-HD
14.8 21.9 26.0 29.9 14.8 21.9 26.0 29.8 14.8 21.9 26.0 29.8 29.8 29.8 29.8 29.8 29.8
HZ-D,
14.8 14.8 14.8 14.8 14.8 21.4 21.4 21.4 21.4 24.1 24.7 24.7 24.7 24.1 28.2 28.2 28.2 28.2 28-2 30.1 30.1 30.1 30.1 30.1 29.8 29.8 29.8 29.8 29.8
>/ooI. 1.44 1.44 1.44 1.44 0.91 0.91 0.91 0.91 0.45 0.45 0.45 0.45 1.49 1.49 1.49 1.49 1.49 0.0 2.8 5.4 8.1 10.8 0.0 2.0 4.0 6.7 0.0 2.9 5.5 8.2 10.8 0.0 3.0 5.8 8.4 11.1 0.0 2.8 5.5 8.2 11.0 5.6 5.6 5.6 5.6 5.6
N (1O22 cmm3)
5
2.26 2.06 1.87 1.63 2.26 2.06 1.87 1.63 2.26 2.06 1.81 1.63 1.64 1.74 1.84 1.93 2.02
3.4 3.9 4.7 5.5 4.6 5.4 6.4 7.4 6.8 7;9 9.9 11.4 5.5 4.8 4.5 4.3 3.8
2.26 2.26 2.26 2.26 2.26 2.08 2.08 2.08 2.08 1.95 1.95 1.95 1.95 1.95 1.75 1.75 1.75 1.75 1.75 1:61 1.61 I.61 1.61 1.61 1.64 1.74 1.84 1.92 2.02
12.2 6.8 4.7 3.5 2.8 14.7 9.5 6.5 4.7 16.3 9.8 6.6 5.1 3.4 18.2 10.5 7.1 5.6 4.3 21.0 11.9 7.5 6.1 4.6 7.4 7.1 6.4 5.5 4.9
Gus)
G.M. Gale, C. DclatandejVibmtional energy transfer in Hz and its isotopes
211
-16-
I
-18-
50
3.5
4.0
Fig_ 7. The relaxation time as a function of mean intermolecular distance in the liquid a ~~‘f-~‘~, for the V-T transfer process D;: Hz.
-19-
T 50
100
“K
150
Fig. 5. The logarithm of gas phase rate constants li as a function of temperature. The low temperature measurements of this wvorkandref. [4] are represented by open symbols. Earlier measurements (refs_ [ 15,161) at higher temperatures are shown as full points.
,
-i-
.’
_-’
Fig. 6. The behaviour of the inverse relaxation time 6’ of pure nD, as a function of density Nacross the phase transition and in the pressurised liquid at 38 K. The point on the lower left
represents the saturated vapour.
sity, as a function of temperature (fig. 8). This ratio is essentially constant in the 14-30 K temperature range for HD’ : H, indicating that, as in “H,, the deexcitation rate is almost independent of temperature. For
D; : Hz the ratio increases by about 10% between i 5 and 30 K. We have also plotted in fig. 8 the ratio of the slightly extrapolated rate constant in the gas for Dg : Hz (see fig. 5) to that of pure nH2. The result is perfectly consistent with the liquid phase data which demonstrates that the isotopic ratio is the same in rhe gas and the liquid. For a given intermolecular
potential the vibrational
energy transfer rate depends essentially on two parameters; the reduced mass of the collision pair and the enerov gap. Table 2 gives the isotopic ratios of the transfer rates at 30 K, as well as the reduced mass and the vibrational quantum, for the different reactions studied. One notes that in every case the reduced mass effect predominates; i.e. the larger the reduced mass the smaller the rate. The large “pure” influence of reduced mass can be seen by comparing D; : D2 to Dt : H,, where the vibrational quantum is the same, but the reaction rate continues to increase as one passes from Df : Hz via HD* : H2 to HD;: H2 even though the vibrational gap is also increasing, which would tend to decrease the rate. However, it is known that the deexcitation channel is not necessarily a direct V-T transfer, but that relaxation can occur via intermediate rotational states of the fun-
212
G.M. Gale, C. DelalondelVibrational
energy
0.7 -
tramferin If2 and its isotopes
HD
15
20
25
Temperature Fig. 8. The isotopic ratio of the V-T transfer times in the saturated HD or Dz). Also shown is the ratio of the slightly extrapolated
30
T OK
liquid 7~;:
rate constant
H2/~xy* : H2 as a function of temperature
(XY =
for Dz: H, (see fig_ 5) to that in pure “H, gas, near
25 and 30 K (open triangles).
darnental vibrational level [1’7,18], and so the exact value of the energy gap is uncertain. A detaiied inter-
pretation of the isotopic ratios at these low temperatures would require a complete quantum mechanical calculation including all V-R decay channels. Conversely, these ratios should provide a sensitive test of the accuracy of any such calculation. 4.5. Theories of V-T transfer in simple liquids Experimental information on the collisional relaxation of vibrational population in liquids composed of small molecules is very scarce. In fact, for several of the diatomic liquids studied to date the predominant Table 2 The efficiency E of the V-T relaxation process at 30 K compared to pore nH2, for various isotopic interactions_ The reduced mass of the collision partners p, and the vibrational energy gap AE are also indicated V-T transfer H;:H2
HD* : H,
D;:H,
E P (au)
1.0 1.0
0.63 1.2
0.57
0.01
AE(cm-L)
4155
3630
1.33 2990
2.0 2990
D;:D,
relax? iion Frocess seems to be radiative. This paucity of data has retarded the advance of the theory of energy transfer in simple liquids. On the other hand, a relatively large number of organic molecules have been studied in the liquid state. In this case however, a different theoretical approach may be required due to the internal complexity of these systems. The testing of any given theory by comparison of relaxation times in the gas and liquid phase must be made with care. Any extrapolation of high temperature results to liquid temperatures can be dangerous, as illustrated in fig..5 where, even on isotopic substitution, the dependence of the gas relaxation rate on temperature varies enormously. It is obviously preferable to compare measurements made in the same temperature range and, if possible, on the same sample. The only previous complete study of the relaxation process in the gas and liquid at the same temperature is that of Madigosky and Litovitz [7] in CO2 and CS?, employing global acoustic techniques. The behaviour of these systems is very similar to that observed in H2 and its isotopes; i.e. (i) 7-1 remains linear in density across the base tran-2 sition up to densities of the order of(0.4 XaL J), where aL, is he Lennard-Jones parameter of the molecule involved.
G.M. Gale. C. DelalandejVibrarional energy tran.@r in Hz and its isotopes
(ii) The isothermal density dependence of 7 in the liquid phase is well described by a cage law, even though the absolute temperature and u values were significantly different in the two studies. (iii) The temperature dependence of T is found to be the same in the gas and liquid phase, where comparison is possible. (iv) For hydrogen, the isotopic ratios are the same in the gas and the liquid. The last two points are, of course, closely related to the first one. In the light of these results it seems not unreasonabIe that a theory of relaxation in the liquid could be linked to low-density, two-body processes_ Litovitz [7] has proposed a classical theory of relaxation in liquids which supposes that deexcitation is due to isolated binary collisions with a probability p* per collision, independent of density, but with a frequency Z of elastic collisions at high density given by a simple cage model. Hence one can write,T-l =p*Z at all densities. The cage model, for the calculation of the frequency of collisions in the liquid, gives 2 = v/(N-1/3 - 0). where v is the average velocity, and thus,
This semi-empirical formula describes well the observed N dependence of 7 at liquid densities. Another value forp* V can be calculated from low density gas k data, using x_= (_m?-1 = 21/z 3 j? = 21/2,*J
cv,
where J and J, are the suitably averaged inelastic and elastic cross sections respectively, and taking somewhat arbitrarily, in a classical hard-sphere approximation, J, = ga2. This p* v value is found to be in fair agreement with the experimental value obtained in the liquid from the slopes of cage law plots at high density [4,71. This theory is attractive in view of its simplicity and empirical accuracy. Nevertheless, it is difficult to understand the notion of an elastic collision at densities such that the intermolecular distance is of the order of the molecular “diameter” and neighbouring particles are always in interaction_ Davis and Oppenheim have formalised the theory of vibrational transfer in dense media [ 19,201 and have
been able to show, under certain conditionst,
213 that the
liquid relaxation time can be correlated to a gas relaxation time, using a linear extrapolation and a correction factorg(T, IV, R*)/g(T, 0, R*), where g(T, N, R*) is the pair distribution function at a characteristic classical distance of closest approach, responsible for deexcitation,R*. The correction factor is then the pair distribution function, for this value of intermolecular distance, at liquid densities, normjlised to its zero density limitg(T, 0, R*). Experimentally this correction factor is found to be not far from unity, which would require a value of R* close to a (see, for example, ref. [31]), invalidating one of the conditions (R* 2 0.9 o)t used in the derivation of the above expression. Also, for hydrogenic systems, quantum mechanical calculations [22] have shown that even low energy collisions deexcite efficiently and hence it is difficult to define a specific effective R* as required in the approximation of Davis and Oppenheim. Both of the above theories attempt to correlate the relaxation time in the liquid to its supposedly known value in the low density gas. Although the process of collisional deexcitation at low density is we!l understood, an a priori calculation is generally difficult. Recently, complete quantum-mechanical calculations have been undertaken for H, [23] but as yet these calculations have not been extended to liquid temperatures even for the simplest system (Hz : He). Another approach has been made by Diestler [24] who has tried to calculate directly the relaxation rate in simple liquids by a quantum cell model. In view of the difficulty of the calculation in the gas, the simplifications made in this theory, particularly in the evaluation of the interaction potential, lead to erroneous results. 5. Vibration-vibration
transfer
Fig. 9 shows the variation of the inverse observed relaxation time of H;, T;&, previously defined, versus isotopic impurity concentration, at various saturated liquid temperatures. 7iks is found to be linear in impurity concentration, as expected at these low dopant t In particular, ifE* is a characteristic kinetic energy of dee.ucitinerencounters IE* = VCR*).where V(R) is the inter-
molecular potential], and fiw is the vibrational quantum; kTQE*--StiwandR*<,0.9a_
214
&W Gale. C. Delalandef Vibrational energy transfer in Hz and its isotopes
13JF’K
(a)
21.95’K
26.0SaK 29.63’K
I
I 2.e
l.ldJ
Concentration
X
levels and for a non-diffusion-limited process_ A cage law isothermal host density dependence is also observed_ From the slopes of the plots in fig. 9 we can deduce a pure V-V relaxation time r”;_,~u(T), at an impurity concentration taken equal to unity. This enables us to compare directly the efficiency of the V-V transfer processes to that of V-T transfer in pure hydrogen_ The ratio of the V-T relaxation time in H2 to a given V-V time, at the saturated liquid density, was found to be independent of temperature in the 15-30 K range. The V-V transfer mechanisms are much more efficient than V-T transfer as can be seen in table 3. Comparison with the only other existing measurement (Hz -+ D2 at 300 K [25]) shows that the V-T/V-V ratio is larger by a factor of three at low temperature. This type of behaviour is quite frequ&tly remarked; i.e. the temperature variation of a high probability process is less than that of a low probability one, and can lead to experimenta difficulties in the measure of pure V-T transfer times at low temperatures [2,5]-
Ag. 9. The inverse observed relasation time of Hz, T& as a function of impurity concentration at various saturnted liquid host temperatures and in the solid, for the dopants (a) HD and 01) D,.
I 0
o.5-lD-2
1.d
1.5.lE
G.M. Gale.C. Delalandc/Vibrational energy transferin H2 and itsisotopes Table 3 The efficiency of V-V relaxation at 30 K and 300 K cornpared to pure V-T relaxation in “H2, for the two processes H;-HDandH;-tD,
Hz-D2 H;+HD
300 K
15-30 K
90
280 1850
In our V-T experiment in hydrogen, for example, the concentration of HD must be less than 10 ppm to be negligible. The isotopic ratio (rH;_HD/TH;_-D, = 7) is greater than that for V-T transfer (Tag* I H /rD; : H, = 1.2) as the energy gap and reduced mass e‘ffects here operate in the same direction, and ofcourse the change in Ihe energy to be transferred to translation is much greater here for V-V, than for the corresponding V-T
processes. Finally, supposing that the linearity of T-l in N up to liquid densities also holds for V-V transfer we can estimate gas rate constants near 30 EC.These are k = 8.7. X lo-l6 cm3 s-l for H; + D2 and k = 5.6 X lo-l5 cm3 s-l for H; --f HD.
215
6. Liquid/solid phase transition Up to now, we have compared the processes of vibrational relaxation in the liquid and gas phase, considering the liquid as a high density gas. Alternatively, not far from the melting point, a liquid could be regarded as a quasi-crystalline solid without long-range order, and it is interesting to study the behaviour of the relaxation time across the liquid/solid phase transition. We have been able to measure relaxation times in the solid for all six of the cases previously discussed. The results in the solid close to the triple point temperature are presented in table 4. For comparison the corresponding relaxation times in the liquid just above the melting point are also shown. In the third column of this table the ratio of Tsolid to Tliquid is indicated.
It can be seen that there is no striking variation in the deexcitation times as one passes across the phase transition. In contrast to the gas/liquid transition however, a small change of the isotopic ratios between the liquid and the solid is observed. There is, of course, a substantial density increase on solidification, and we can
Table 4 Liquid-solid transition (near 13.9 K for nH1 and 18.7 K for “D,). The V-V rela..ation times are calculated at a supposed impurity concentration of unity. The third column shows the ratio of the experimental relaxation times, and the fourth column is the corrected ratio (see text) Liquid
Solid
2.27 x 1022 (cm-j)
2.68 x 1O22 (cmm3)
TH; : H, rHD* : H,
12.0 ps 19.6 fis
8.5 PS 16.25 /IS
0.71 0.83
1.02 1.19
‘D; : H,
23.6 PS
24.7 US
1.05
1.50
7H,+HD * 7H; -+ D,
6.9 ns
6.9 ns
1.00
1.43
39.5 ns
30.3 ns
0.77
1.10
251 x 10” (cmJ)
3.07 x 10” (cm-a)
970 ps
850 fis
0.88
1.46
density
V-T
v-v (x = 1) density V-T
‘D; : D,
attempt to correct for this by assuming that the liquid cage 1~ holds right down to characteristic solid intermolecular distances. Ths last column of table 4 shows Fe estimated ratios rs&/rEq(Nn = Ns). The corrected ratio is always greater than unity; i.e. deexcitation appears to be somewhat less efficient in the solid. Another point to note is that it seems to make little difference whether one is dealing with deexcitation in the pure solid or of an impurity in a matrix. (The change of the ratio r&e in going from HT : H2 to HD* : H2 is of the same order of magnitude as that between HD* : H2 and D; : H2.) There are only a few experimental results on this phase transition in simple systems [5,36] and the main concern of theoretical and experimental studies of vibrational relaxation in solids is with impurities in monoatomic matrices [27-29]_ From these studies it would appear, however, that the local environment of the excited species plays a dominant role in deexcitation, and hence a dramatic change in relaxation time on solidification might not be expected. Direct measurements of relaxation times at high density in the liquid would ahnost certainly be helpful for the detailed understanding of the isotopic variation in the solid and the solid/liquid transition.
7. Conclusion
Our resuIts show that vibrational relaxation in the liquid phase is highly correlated to low density gas phase processes_ The semi-empirical isolated binary collision theory of Litovitz gives good order of magnitude scaling and reproduces well the isothermal density dependence in the liquid. For a better comprehension of the mechanisms underiying the phase transition behaviour and the effects of density and temperature in the liquid, more fo,mal and sophisticated theories are required. It is to be hoped that the development of such theories will be stimulated by the increasing fund of experimental data on simple systems.
References [l] WV. Kaiser and A. Laubereau, in: Tunable lasers andap plications, Proceedings of the Loen Conference, eds. A. ~Mooradian, T. Jaeger and P. Stoksetb (Springer, Berlin,
1976). [2] W.F_Calaway and GE. Ewing, J. Chem. Phys. 63 (1975) 2842. [3] S.R.J. Brueckand R.M. Osgood, Chem. Phys. Letters 39 (1976) 568. [4] C. Delalande and GM. Gale, Chem. Phys. Letters 50 (1977)
339.
(51 N.
Legay-Sommaire and F. Legay, Chem. Phys. Letters 52 (1977) 213. [6] C. Renner and hi. Maier, Chem. Phys. Letters 35 (1975) 226. [ 7] W.M. Madigosky and T.A. Litovitz, J. Chen. Phys. 34 (1961) 489. [8] C.G. Sluijter, H.F.P. Knaap and J.J.M. Beenakker, Physica 30 (1964) 745. [9] E.J. Alhi, H.P. Gush, W.F.J. Hare, J.L. Hunt and H.L. Welsh, in: Coltoque sur les prop&t& optiques et acoustiques des fluides cornprimes et les interactions molbcul&es, 1-6 juiuet 1957, Bellevue (France) (CNRS, 1959) p. 21-39. PO1 A. van Itterbeek, 0. Verbeke. F. Theeuwes and V. Jansoone. physica 32 (1966) 1591. [Ill H.J. Hoge and R.D. Arnold, J. Research NBS 47 (19.51)
[I21
63. H.J. Hoge and
WI Wl
505. P-K. Davis, J. Chem. Phys. 57 (1972) 517. J-0. Hirschfelder, CF. Curtiss and R-B. Bird, in: Mo-
J.W. Lassiter, J. Research NBS 47 (1951) 75. [131 J.D. Rogers and F.G. Brickwedde, J. Chem. Physics 42 (1965) 2822. 1141 .I. Ducuinp. C Joffrin and 3.P. Coffmet, Opt. Commun. 2 (1970) 245. 1151 MM. Audibert, C. Joffrin and J. Ducuing, Chem. Phys. Letters 25 (1974) 1.58. [161 I. Lukasik and J. Ducuing, Chem. Phys. Letters 27 (1974) 203. 1171 M.M. Audibert, R. Vilaseca, J. Lukasik and J. Ducuing, Chem. Phys. Letters 31 (1975) 232. ]18] M.H. Alexander, J. Chem. Physics 61 (1974) 516. 1191 P.K. Davis and I. Oppenheim, I. Chem. Phys. 57 (1972)
lecular theory of gases and liquids (Wiley, New York, 19.54): L. Verlet, Phys. Rev. 165 (1968) 201; F. Mandel, R.J. B-an Phys. 52 (1970) 3315.
Acknowledgement We would like to thank Professor J. Ducuing for his continuing support, Dr. C. Flytzanis for helpful discussion, and Mr. J. Debrie for technical assistance.
and M.Y. Bearman. 5. Chem.
[22] J. Lukasik, Ph.D. Thesis, University of Paris-Sud, France (1975). [23] R. Ramaswamy and H. Rabitz, J. Chem. Phys. 66 (1977) 152. [24] D.J. DiestIer, Chem. Phys. 7 (1975) 349. .1251 J.F. Bott. J. Chem. Phys. 65 (1976) 3921.
GM. Gale, C. Delaiande~Vibmtionnl energy transfer in Hz avd
[261
IL Dressier, 0. Oehler and D.A. Smith, Phys. Rev. Letters 34 (1975) 1364. 127) F. Legay, in: Chemical and biochemical applications of laser, Vol. 3, ed. G.B. Moore (Academic Press, New York, 1977).
its isotopes
217
[281 J. Goodman and L.E. Brus, J. Chem. Physics 65 (1976) 3146. 1291 K.F. Freed and H. Metiu, Chem. Phys. Letters 48 (1977) 262.