ABSOLUTE INTENSITIES AND PRESSURE EFFECTS METHANE … MV 1969.pdf · molecular forces on their...

ABSOLUTE INTENSITIES AND PRESSURE EFFECTS IN THE INFRARED FUNDAMENTAL BANDS OF METHANE AND ITS DEUTERATED SPECIES

by

V. MICHAEL COWAN

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy

in the University of Toronto

Department of Physics October, 1969

(c) V. Michael Cowan

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TABLE OF CONTENTS

PageABSTRACT ..................................... iCHAPTER I : Introduction .............. • * • • • 1CHAPTER II : Apparatus and Experimental Procedure • 6CHAPTER III : Reduction of the Data • • • • • • • • 14CHAPTER IV s Experimental Results • • • • • • • • • 24CHAPTER V : Absolute Intensities and Bond

Properties • • • • • • • • • • • • • 49CHAPTER VI : Pressure Effects in the Spectra of

the Methanes ................... 62 *APPENDIX I : Units for Absorption Coefficient . . . 8l APPENDIX II : The Boltzmann Relation in the Inten

sity Distribution of Rotation-Vibration Bands ............. 83

APPENDIX III: Energy Levels and Line Intensities • • 86APPENDIX IV : Publication Reprint . . . . . . . . . 92BIBLIOGRAPHY ............................. . . . . . 94ACKNOWLEDGEMENTS


ABSTRACT

The infrared fundamental bands of methane and its deuterated species have been examined in dilute mixtures with helium, at densities from ~'100 to ^1000 amagat. There are a total of 24 active fundamentals for CH^, CH3D, CH2D2, CD3H, and CD^, although in some cases only groups of overlapping bands can be observed. For the thirteen separate bands or groups of hands, the density variation of the spectral profiles and the integrated intensities has been studied. The contours have been corrected for slit-width effects by an iterative procedure. There is progressive disappearance of the rotational structure with increasing density, until at the highest pressures only broad, almost featureless bands remain. For the stretching modes, the integrated intensity P* decreases linearly with helium density} the relative decrease is the same for bands at ''-'3000 cra"l and ^2200 cm”l, and equal to 1.53 x 10~4 / amagat. For the deformation modes, the initial decrease in P at low densities is followed by a region of constant or slightly increasing intensity.

Extrapolations of the curves of P 1 vs.yO jje to zero foreign-gas density yield accurate values of the free molecule absolute intensities. The C-H stretching mode intensities are observed to lie in the approximate ratio 4 : 3 i 2 : 1 : 0 for CH^, CH^D, CH2D2, CD^H, and CD^; for

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the C-D stretching modes, the ratio is 0 : 1 : 2 : 3 : 4>The thirteen observed intensities are interpreted in terms of a two-parameter bond-dipole model for infrared absorption. The best-fit values of the C-H (or C-D) bond dipole and its rate of change with bond length are found to be

/ rQ ■ + O.3167 debye / A, ( / 9 r)c «= + O.66O0 debye/S.The overall RMS deviation between the best-fit intensities and experiment is 5»5%* It is observed that all the experimental C-H stretching intensities are higher than the corresponding best-fit values, whereas the C-D intensities are lower. This is ascribed to the neglect of mechanical and electrical anharmonicity in the two-parameter model.

Values of dispersion half-width are obtained by fitting broadened line spectra to the observed profiles.The broadening coefficients range from 0.036 cm--*- / amagat for CH4 to 0.024 cm--*- / amagat for CD4. The density variation in the profiles is interpreted using a rotational autocorrelation function approach. Calculations are presented of the correlation functions for free spherical and symmetric rotors. The effects of collisions are then calculated in an impact approximation. The models of m- and j-diffusion for rotational motion, developed by Gordon, are generalized to the case where either m- or j-diffusion may occur in a given collision. The results, while quantitatively incorrect,

_ ii -


display many of the features of the observed autocorrelation functions. The density variation of P withyO is discussed from the point of view of interference between the free- molecule transition dipole and the moment induced by the overlap interaction in close collisions.

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CHAPTER I INTRODUCTION

Infrared absorption spectroscopy offers a powerful tool for the study of intra- and inter-molecular forces. In the gas phase at low density, analysis of absorption-line frequencies and intensities can yield detailed information on the symmetry and structure of the active molecule, on the forces which bind its atoms, and on the changes in electron distribution which accompany distortions of the molecule during vibration. At higher densities, or with the addition of foreign gas, the absorption lines broaden and may begin to overlap; study of the line widths and frequency shifts helps us to understand the potential fields between the absorber and its environment. With increasing density, the structure due to rotational transitions gradually disappears until only a rotational envelope remains. Further changes in the envelope may be interpreted in terras of simple broadening of the underlying rotational structure, but finally changes in the nature of the rotational motion must be invoked at the highest densities.

A great deal of information has been accumulated in this way for polar diatomic molecules such as HCi. and CO, and in the Raman effect for homonuclear diatomics such as

' 2* particularly In addition, symmetry-forbidden

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2

spectra have been observed in this laboratory and elsewhere, in which the transitions are made active by distortions due to the environment in high density gases, in liquids, and in solids; this process is known as pressure- or collision- induced absorption•

Polyatomic molecules are however more complex systems and only in rare cases have the effects of intermolecular forces on their spectra been analyzed with the same degree of completeness* For study of a polyatomic molecule, it is desirable that the molecule have the following properties: l) low moment of inertia, so the rotational levels will be widely spaced and the broadening observable at reasonable densities with reasonable spectrometer resolution; 2) small number of atoms and high molecular symmetry, to reduce the number of active fundamentals and thus avoid overlapping bands; 3) strong chemical binding, both to avoid the possibility of decomposition and to keep the vibrational frequencies in the near infrared, the range most easily studied; 4) strong absorption, so that dilute mixtures may be used in a cell of moderate length; 5) minimal interactions between rotation and vibration, so that the spectra are as symmetric as possible about the transition frequency.

The methane molecule CH4 satisfies the above requirements admirably. This thesis consists of a study of


all the active fundamentals of methane and its four deuterated species CII3D, CH2^2» CD3H, and CD4 at foreign gas densities from ^100 to '■*'1000 amagat. The choice of perturbing gas was restricted to noble gas atoms to avoid the possibility of pressure-induced absorption in the perturber itself. Of the noble gases, the simplest is helium; its low polari'z- ability ensures that multipole induction effects in the intermolecular forces are small. In the work to be described, only helium was used as a perturbing gas.

Previous work on pressure effects in the spectrum of methane has been carried out by Welsh, Pashler, and Dunn (1951) and Welsh and Sandiford (1952) who studied the and bands in foreign gas mixtures, and developed the extrapolation technique for measuring absolute intensities which is employed in the present investigation. Armstrong (1961) repeated their measurements at higher dispersion for foreign gas pressures from 75 to 3000 atmj in addition he studied the V 3 band of CD4, Certain anomalies in the density variation of the integrated intensities, first obsei*ved by Armstrong (1961), prompted the present investigation.

Chapters XI and III of this work consist of descriptions of the experimental apparatus and of certain mathematical techniques which were found useful in the reduction of the data to physically meaningful form. Chapter IV contains a presentation of the observed band contours with their


assignments; data on the variation of integrated intensity with helium density and comparison with the results of previous workers is also given.

As mentioned above, extrapolation of the integrated intensity to zero foreign gas density yields an accurate value for the free-molecule absolute intensity. Since the deuterated methanes have a total of 24 active bands, at frequencies between 1000 and ~ 3000 cm"^, a great deal of data can be collected on the absolute intensities of infrared bands for such polyatomic molecules. Moreover, since the electrical properties of the deuterated methanes are expected to be identical, the absolute intensities should be closely related to each other. In Chapter V the intensities are analyzed using a model which can in principle give the intensities of all twenty-four bands using only two parameters, the C-H (or C-D) bond dipole moment and its rate of change with bond length ,The spectra of the methanes constitute a sensitive test of such a model; small systematic deviations of the theory from experiment are presented and discussed.

Pressure effects on the contours and integrated intensities are discussed in Chapter VI, The rotational autocorrelation function approach, developed principally by Gordon (1966a, 1966b) offers in principle a way of


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analyzing the effect of collisions on band profiles. In this thesis certain simple models of rotational diffusion are generalized to apply to polyatomic molecules and to cases in which two modes of rotational relaxation may both occur. The results are compared with autocorrelation functions derived from experimental contours. A mechanism for the changes in integrated intensity with helium density is also presented.


CHAPTER II

APPARATUS AND EXPERIMENTAL PROCEDURE

The fundamental absorption bands of the methanes lie in the three general spectral regions 3300 to 2750 cm"1 (corresponding to C-H bond stretching vibrations), 2450 to 1950 cm-1 (C-D bond stretching), and 1600 to 850 cm-3- (C-H and C-D bending). Our overall aim was to examine the spectra of all the active bands in dilute mixtures with helium at densities up to 1000 amagat at room temperature. The experimental apparatus required for. such a program was the following: i) a transmission cell of small volume and accuratelymeasurable path length, capable of sealing repeatedly at pressures up to 3000 atmj ii) stable sources of infrared radiation, a spectrometer of moderate (6 cm-*-*-) resolution, an optical matching system, and pressure windows for the absorption cell, all of which would operate efficiently over the wavelength range from 3 to l2yU. in an enclosure capable of being flushed with dry nitrogenj iii) gas handling systems for preparing dilute methane-helium mixtures to specified concentrations, compressing the mixtures to high density and measuring the pressure and temperature during each experi ment,

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a) The Absorption Cell

The 3000 atm high pressure cell, shown in Fig.II-1, was constructed of Speedcut steel in three sections (A, B, A) held together by six. socket-head cap screws (C) at each end. The central section contained the gas inlet (D) which was threaded to accept a standard Aminco high pressure fitting.The end pieces (A) terminated in one-inch diameter polished surfaces, flat to within several wavelengths of NaD light, to which the high pressure windows (E), also optically flat, were sealed with a thin film of Canada balsam. Because of the 0.25 in.diameter aperture in the steel flats, the windows were self-sealing under pressure, but required Teflon washers (F) and steel caps to prevent movement and leakage during evacuation of the cell. The pressure seals between the end pieces and the central section of the cell were effected by hardened steel lens rings (rl) of trapezoidal cross section, with the sealing surfaces ground to 60° angles with the ring axis.With this arrangement, the pressure could be recycled several times up to 3000 atm before retightening became necessary. Moreover, reassembly of the cell could be quickly performed and gave path lengths repeatable to better them 1%, The free volume of the cell was reduced by a brass filler to allow higher ultimate pressures to be reached. Path lengths were measured by subtracting from the total length of the assembled cell the combined lengths of the cell end pieces, measured


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ner. Further

reproduction prohibited

without

permission.

NWWWWWWWWwJ

Fig.II-1. High pressure infrared absorption cell. A, end piece;B, cell body; C, socket-head' cap screw; D, gas inlet;E, window; F, Teflon washer; G, retaining cap; H, trapezoidal lens ring.

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with vernier calipers and a micrometer depth gauge using a special jig. To simplify optical alignment, the cell was mounted in a rigid cradle which could be moved horizontally across the optical path, and rotated about a vertical axis with a worm gear.

b) The Optical System

For the spectral region from 3400 to 1900 cm”1, the radiation source was a tungsten projection-lamp filament in a water-cooled housing with a sapphire window. The source was run at 350 W, well belo\* its rated power of 750 W; this prolonged its life far beyond the nominal twenty- five hours at full power, thus minimizing the drift in output which accompanies gradual evaporation of the tungsten filament.

For the region from 1650 to 830 cm”1, a Nernst filament was chosen for its high emissivity at these frequencies. Unfortunately, this type of source has a large negative temperature coefficient of resistance; the resulting fluctuations in output were minimized by running a tungsten filament lamp, of positive temperature coefficient, as a ballast resistance in series. However, the Nernst filament remained sensitive to room temperature variations, and to changes in the flow rate of dry nitrogen used for flushing. For this reason the section of the flushing housing around


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the source was made of heavy brass and several hours were allowed before each run for the system to come to equilibrium.

The optical matching system is shown schematically in Fig.IX-2. It consisted of two f/4 rocksalt lenses (L) used at unit magnification, and aluminized 45° plane mirrors (M) which could be adjusted from outside the flushing housing (H). A shutter (s) of polished aluminum could be rotated into the light path to measure the zero intensity level. The absorption cell windows (W) were 0.25 in. thick discs of IR- TRAN 2, supplied by the Eastman Kodak Co. This material transmits efficiently from the deep red to about 12J*- in the middle infrared. Its almost complete absorption of visible light made focusing difficult, but this was outweighed by its high strength, lack of cleavage planes, and resistance to attack by water vapour and organic solvents. An added advantage was the presence in its absorption spectrum of relatively sharp lines at 2349 cm"1 and 2080 cm"1 which were used for frequency calibration purposes.

The spectrometer (P) was a Perkin-Elmer model 112 single-beam double-pass prism instrument, with a LiF prism for the region from 3400 to 1900 cm"1, and an NaCl prism for the 1650 to 830 cm"1 region. Dispersion curves were obtained from the absorption bands of CII , HC1, IIBr, and CO for the LiF prism, and from 1^0, C02 and NII fundamentals for the NaCl prism.


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r

j

Fig.II-2. Optical system schematic. L, rocksalt lens; M, adjustable mirror; H, flushing housing; S, shutter; W, IRTRAN 2 window P, spectrometer; Bt insulated enclosure; R, infrared source.

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In all cases.the data of Plyler et al.(l960) were used for frequency assignments. A string slit—drive was employed to keep the background intensity level and the spectral slit width approximately constant over the frequency range of interest. The greatest demands on the slit-drive were made by the 1650 to 830 cm“ - region, where overlapping bands dictated that the entire region be scanned in one trace. For the desired 6 - 7 cm“^ spectral slit width, this necessitated an inincrease in actual slit width from 110J** to 700y* across the band. After some trial and error, it was decided that 30-lb - test braided nylon fishing line gave the best compromise of strength, flexibility and resistance to stretching for use in the slit-drive. This arrangement allowed a slit-width reproducibility of better than 0.5y“- in successive scans, and no detectable gradual changes even after many cycles.

A final difficulty with the spectrometer was the sensitivity of the output level to changes of room temperature, particularly at low frequencies and with wide slits. Since the radiation is chopped between the first and second passes, it is possible for the detector to llseen the walls of the spectrometer housing at intermediate positions of the rotating chopper mirror. An increase din wall temperature will result din an dLncrease of signal on the portion of the cycle where the entrance slit image is not incident on the exit slit. Since the AC signal is synchronously rectified, this


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will be interpreted as a net decrease in radiation level at the detector'. Indeed it was observed that even with the entrance slit blocked off, there was at 850 cm“^with 700^ slits a reproducible decrease in output level of ~5% of full scale as the room temperature increased from 20.5°C to 23.5°C. Moreover, the decrease varied approximately linearly with slit width, and was much less pronounced at higher frequencies. Although nominally thermostatted at 32°C, the spectrometer evidently did not have sufficient thermal contact between the upper housing and the base to which the heaters were attached. The problem was minimized by enclosing the entire spectrometer in a heavy plywood box (B) insulated top and bottom with aluminum foil and rigid urethane foam, and fitted inside with well distributed Pyrotenax heating cable. The heating was controlled by a Bristol’s electronic temperature regulator using a General Electric Triac solid-state switch; conventional magnetic relays were found unsuitable because of the largp amount of electrical noise generated by current surges on making and breaking contact.

c) The Gas Mixing System

The ordinary methane used in the experiments was Matheson C.P. grade; the deuterated species CH3D, CH2D2,CD^H, and CD^ were supplied by Merck, Sharpe, and Dohme at 98$ minimum isotopic purity. A schematic diagram of the gas handling system is given in Fig.II-3. The methane gas (A)


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- &K

r®-<8>

D

L

■<3

1 (

j

A

B

H

Fig.II-3. Gas mixing system; A, methane cylinder; B, compression chamber; C, Hart oil press; D, mercury monometer; E, He cylinder; F, molecular sieve trap; G, Hart pressure balance; H, mercury piston; J , oil chamber; K, absorption cell; L, vacuum pump.

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was introduced into the gas compressor chamber (B) attached to a Hart oil press (C), The pressure, which ranged from 10 to 30 cm Hg, was read to 0.01 cm Hg on a mercury manometer (D) which had been calibrated using a Wallace and Tiernan precision manometer; the temperature was read to 0.1°C on a thermometer in good thermal contact with the compressor.The helium gas (E), of standard grade supplied by Linde, was introduced at 100 to 180 atm into two large traps (F) containing Linde molecular sieve which removed any residual water vapour and carbon dioxide. The helium was then allowed into the compressor chamber as rapidly as possible to minimize back-diffusion of the methane through the inlet orifice.When the temperature had reached equilibrium, the pressure was measured on a Hart pressure balance (G) nominally accurate to 0.01$. Isotherms for helium below 1000 atm were taken from the data of Wiebe, Gaddy, and Heins (1931); for higher pressures, the less accurate data of Hare (1955) were used. Ideal gas isotherms were assumed for methane.

its density at NTP. For each gas mixture, a mixture ratio

amagat densities of the mixture and absorber, respectively.The value of jDm was computed from the helium isotherms alone; the validity of this approximation is good because of the high


The amagat density of a gas is defined as the ratio of its density at the temperature and pressure of interest to

was calculated, where andy>tt are the

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mixture ratios employed (r^300).During an experiment the mixture was admitted to

the absorption cell and recorder traces were taken for a series of total pressures such that the total absorption and the maximum value of In (I0/I) lay in a convenient range*The total pressures were measured on the Hart balance, and for each pressure the density of absorbing gas was calculated from =j 3 m / ( r + l ) • At each density two or three traceswere taken; these were then superimposed on a light-table and their visual average was traced onto a separate sheet of chart paper. The ordinates of this average curve, measured at regular frequency intervals using a Ruscom Logics chart reducer, formed the raw data for further calculations. Only in rare cases did the individual traces differ from the average by more than .0.5% of full-scale deflection.


CHAPTER III

REDUCTION OF THE DATA

In this chapter we shall describe a number of general techniques which were used to extract the maximum amount of physical information from the raw data. These include i) a method to compensate for the effects of finite spectral resolution on the intensity profiles, ii) a numerical technique of matching background and absorption contours to compensate for instrumental drift, iii) the separation of two overlapping bands without detailed previous knowledge of either profile, and iv) the determination of rotational line shapes and widths from bands in which no individual lines are resolved.

a) Correction for spectral slit width

All of the experiments were performed at a spectral slit width (full width at half height of the slit function) of 6 - 7 cm*"1. At low foreign gas densities, and particularly for bands with sharp and intense features, such resolution appreciably distorted the profiles and gave a decrease of the observed integrated intensity from its true value. Depending on the fraction of the total intensity contributed by the sharp features, the line widths relative to the spectral slit width, and the values of the absorption coefficient fin (IQ/ I)

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15

at the peaks, the decrease in integrated intensity typically amounted to one or two per cent at low helium densities for bands with a sharp Q branch,.The change in the band profile caused by finite resolution was even more serious; however; it was felt that any meaningful discussion of pressure effects on the band envelopes demanded some correction for instrument broadening.

The first method tried was a Fourier transform technique, described by Seshadri and Jones (1963). We define the instrument slit function S ( y ,.-J7 ) as the spectrometer response at nominal frequency V* to light of unit intensity at frequency . The observed profile will be a convolution,I C V 1) = | l ° ( V ) S (y» -l?) d y , where I°(V ) is the light incident on the entrance slit. Now it can be shown that the convolution of the two functions in frequency space is mathematically equivalent to the product of their Fourier transforms in the transformed space. Thus if X(t) and S(t) are the complex Fourier transforms of and S(l7f -17 ), thenthe Fourier transform of I (t) = i("t) / S(t) gives the intensity profile I°(l?)« In practice, I(l7!) is evaluated only at finite frequency intervals; thus the Fourier integrals must be approximated by Fourier series. Since the experiments were performed, at energy-limited resolution, S(V 1 **17) was assumed basically triangular, with a slight Gaussian rounding of the peak and the wings to allow for imperfect optical


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alignment. By trial and error, it was discovered that the efficacy of the overall method depends rather critically on the number of points evaluated per spectral width, and on the range of time and time intervals at which the Fourier series are calculated. For proper choice of these parameters, however, the amount of computing time required was prohibitive for our spectra, which usually consisted of broad bands with sharp features superimposed. Instead an iterative method, described below, was employed.

The observed intensity profile I(y») was taken as a first approximation to the '‘true11 profile I°(y), and the convolution of X ( P f) with S(y 1 -X?) was obtained. Then I ^ y ) = I(»)[l(P ) / j K P 1) S(yt - P ) d y t]X, for X some positive number, formed the next estimate of I°(y ).Still higher approximations could then be made by using In + l(P) = In (^) [l(V) /[ In (V*)S(y» - V ) d»*]X until the correction factor in square br*ackets was as close to unity as desired. The whole process had a tendency to diverge about features, due to noise in the observed profile, which were narrower than the spectral"slit width. It was thus desirable to keep the number of iterations as low as possible. The speed of convergence could be increased by increasing the constant Xj moreover, for X greater than 1, the process was relatively more efficient for features where the correction factor in square brackets was significantly different from


17

unity* The best compromise between meaningful correction of sharp spectral features and introduction of spurious noise was achieved for X <>'1,25* In this case the process converged so rapidly that only two iterations were needed* It was felt that the presence of sharp lines in the background spectrum, due to residual water vapour and carbon dioxide in the light path and to absorption by the windows* made worthwhile the doubling of computer time necessary to include background traces in the "deconvolution1* • Even so, the total time required was a small fraction of that demanded by the Fourier transform technique* All contours and integrated intensities obtained in this study have been calculated using data treated by the iterative "deconvolution1* described above* Effects of the procedure on part of the 3000 cm"l region of ^2^2 are s*lown Fig* III-l. The band has a very sharp and intense Q branch; thus at low foreign gas density, and at high absorber concentration, it constitutes a "worst case"* As expected, the effects of the correction are negligible for regions with no sharp absorption features*

b) Intensity matching procedures

Systematic matching procedures were made necessary by the random intensity drifts observed during the experiments. Such drifts were due to changes in source temperature, instability of the spectrometer and recorder amplifiers, and


In (I0 /I)

i P He

f 91 amagat0.5

30002950 3050FREQUENCY (cmH )

Fig.III-1, Example of correction for finite slit width - the 3000 cm“^ region of CH2D2. The dashed curve is uncorrected.


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changes in the flow rate of flushing gas which determined the amount of atmospheric absorption. In addition, as described above, variations in the temperature of the spectrometer housing could affect the infinite absorption level for the 1600 to 850 cm"^ region,' . Over the period of several hours which elapsed between the taking of background and absorption traces, these random drifts could amount to a mismatch of two or three per cent, which was not necessarily constant across the band. There were in addition systematic shifts which increased \*ith increasing mixture density. These were mainly due to decreasing reflection losses at the window-gas interfaces as the refractive index of the mixture increased, and to progressive narrowing of the light cone inside the cell, A simple calculation showed that anomalous dispersion of the mixture could be neglected for the moderate absorption coefficients used. Thus the fractional intensity change due to increasing refractive index was a monotonic function of frequency, and could be treated in the same way as the slow random drifts.

The time-honoured practise of shifting or tilting the background trace to match the absorption trace in the region of no absorption was rejected for several reasons.Such a procedure destroys the validity of the infinite absorption level, it spoils the accuracy of the frequency calibration, and in the general case where the background spectrum


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is not linear, no single shift can accurately match both high and low frequency regions.

The procedure adopted was to obtain values of o( ( V ) = In [lG(V ) / I(y )J using data to which the only correction applied was the finite slit width "deconvolution*' described above. At frequencies above and below the region of detectable absorption, o( ( V ) generally approached small constant values, o( + and o(_, respectively. Since no other information was available, it was assumed that the value of the intensity shift o(g()? ) varied linearly between + and o(_ across the band. Interpolated values of o<g(V) were then subtracted from the measured o(,(v), resulting in a profile which was zero outside the region of detectable absorption. Another advantage of this method is that the values of o(.+ and o(_ gave a numerical estimate of the amount of drift in the experiment.

c) The determination of rotational line shapes

If a pressure-broadened rotation-vibration spectrum results from a set of rotational lines of known frequency and intensity, each having a shape determined by the collision statistics, the line shapes can be found from experimental envelopes by making certain more or less reasonable assumptions. There are two traditional methods for doing this, each with characteristic advantages and disadvantages.


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In the first method, line shapes are assumed involving parameters which may describe, for example, line width and its variations with rotational quantum number. Each transition in the stick spectrum is thus broadened, and the contributions from all lines are summed at regular frequency intervals. The resulting envelope is compared with experimental contours, and the line shape parameters systematically varied to give a best fit (but see below) between theory and experiment•

In the course of this work it was observed that in the region of the best fit, the sum of squares of deviations between trial contours and observed profiles varied approximately quadratically with assumed line width. By evaluating the constants of the parabola, the best-fit line-width could be found to ~ 0.2 cm“^ whereas the original grid of line- width values contained only five points at intervals of 2.0 cm”'*'.

There are two principal disadvantages to this method. One is the difficulty of finding an unbiased criterion for goodness of fit. If a simple sum of squares of deviation is used, the fit in the broad wings is emphasized- at the expense of any sharp features in the band. Intensity-weighted deviations may be used, but then the question of the weighting scheme to use arises. The second disadvantage is the necessity for assuming some simple form for the line shape. The more


complicated the analytic function, the more parameters must be evaluated in the least-squares fit, and the process quickly becomes unmanageable.

There is a second method to evaluate the broadening function, which requires no assumption about the line shape except that it be the same for all rotation lines.This technique, described by Shapiro (i960), is based on the same mathematical theorem as the Fourier transform slit-width correction. Thus if J*.(t) and .©^(t) are the complex Fourier transforms of the observed spectrum oC( V ) and the free- molecule stick spectrum 0^ ( 1?), then the Fourier transform of J2i(t) “ .(t) / oCQ(t) gives the line-shape function o'(P). Preliminary results with this method for the 2992 cm"^ band of CD3H at various He densities indicated a line shape of dispersion form in the core, but going increasingly negative in the wings for increasing density. This implies that the very concept of a single line shape begins to break down for bands, with severely overlapping transitions. The rotational auto-correlation function approach, to be described in Chapter VX, uses techniques somewhat similar to that above, but in addition offers a physical picture of the mechanisms of pressure broadening.

d) The separation of overlapping bands

The presence of numerous overlapping bands in the spectra of the methanes made it desirable to have some general


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technique for separating them. This was especially useful where expressions for the line frequencies and intensities were not easily obtainable, as in the asymmetric rotor CH2D2* The technique is based on the assumption that the Boltzmann relation holds between intensities equally spaced on either side of the vibrational frequency, i.e., oCg(A}? ) = o(P( A V )eh c A V /kT. A discussion of the validity of this assumption is given in Appendix II. Unless the Boltzmann relation holds strictly for both the overlapping bands, such a method cannot give self-consistent results. In practise it was necessary to assume that the relation held strictly for one of the bands and approximately for the other. The choice of which band to treat strictly was determined by trial and error; a poor choice gave physically unrealistic separated contours. Ve shall give a typical calculation for a case where the overlapping is severe.

Let the band centres be and 1?2 where V 2 - s £ > Oj let the observed sura profile be cK (V) and the desired separated profiles and ol' 2 ( . ' )? ) , Thus atfrequency V ** V 2 + > we have

c i ( V 2 + A “ °î^ V 2+ ^ + A ) + °^2^ 2 + A )We initially assume that the Boltzmann relation holds forboth bands; then for A > S' we have

<* C V 2 - A ) - x< + A - £ >®"( A ■ } 1 E

+ U 2 { V 2 + a )e“ A I E (2)


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where E « kT/hc. We now assume that the contribution of the low frequency band is negligible at sufficiently high frequencies, say A > A 0. Thus, defining A* 2 A - A Q, we can write

l* J ( ’’2 + A o t A ,) » « ( > ;2 + A 0 + A ' ) (3)Substituting equation (3) in (2) and using (l) we finally obtain

oC2(V2 + A0- 2 ST + A,)«o<.(V2 +Ao - 2 S + A • )+ * (v2 + A 0 + A ' ) e - '• / B - < * ( y 2 - - A*) Xe(Ao - S +A») / E (4)

for all A* > (2 £ - A©). The remainder of the V 2 band is found by using the Boltzmann relation, and the band is then obtained by subtraction from the experimental sum profile* The calculations for bands in which the overlapping is slight are more straightforward and will not be given here*

Typical results for the 3000 cm"-*- complex in CH2D2 are shown as dotted curves in Fig. IV-21. The profile is the sum of a C-type contour (see Appendix III) at 3013 cm--*- and a less intense B-type band at 2976 cm"-*-, An evaluation of such separations in terms of intensity ratios will be given in Chapter V.


CHAPTER IV

EXPERIMENTAL RESULTS

a) The spherical rotors: CH^ and CD^

Methane and methane-d^ are spherical top molecules of tetrahedral symmetry belonging to the point group Td. They have four normal modes, one each of totally symmetric A^ species and of doubly degenerate E species, and two of triply degenerate ?2 species. Although all modes are active in the Raman effect, only the F2 vibrations are directly infrared active. In both CH^ and CD^ however, the doubly degenerate mode absorbs weakly due to Coriolis interaction with the nearby active F2 mode.

1) Methane

The following table gives the correlation between observed infrared and Raman bands of methane and the vibrational modes. In this and the corresponding tables for the deuterated methanes, the frequencies given are the band origins obtained from rotational analyses of high resolution spectra, whenever these are available. If the true band origin is unknown, the frequency of the band maximum (or minimum in the case of some perpendicular and asymmetric rotor B-type bands) will be quoted. The notation used throughout is that of Herzberg (1945)*

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TABLE IV-1. Vibrational frequencies of CH^

Assignment Mode Frequency (cm"l) Reference

V 3<F2>^(Ai)

)>2<E>> V P2>

asymmetric C-H stretch symmetric C-H stretch CH2 - CH2 twist H - C - H bending

3019.5 2916.71533.6 1305.9

Herranz and Stoicheff (1963) Thomas and Welsh (i960) Thomas and Welsh (i960) Herranz and Stoicheff (1963)

TABLE IV-2. Experiments on CH4

Region (cm“l)

RunNumber

Path Length (cm)

Mixtureratio

Density range (amagat)

Number of traces

106 2.758 856 86 - 133 53270 - 2690107 n 2846 156 - 874 17105 n 856 133 - 241 6

1530 - 1110108 n 2846 326 - 873 10

25

A .summary of the CH4 experiments is given in Table XV-2* All spectra were taken at ambient temperature, which ranged from 297 to 300°K. No attempt was made to measure the intensity of the ^ 2 band; any improvement on the value of Armstrong and Welsh (i960) would require a more elaborate flushing system and extremely careful purification and handling of the methane to eliminate absorption by traces of water vapour and ethane.

i) The 3019.5 cm”-*- band. - Absorption contours of the V 3band at representative helium densities are given in Fig.XV-1. In this and all similar figures, the ordinates ofspectra at different densities are shifted for greater clarity. At low densities the band fyas pronounced P, Q, and R branches (see Appendix III) which gradually merge until only shoulders on the centred, maximum remain at high densities. There is appreciable rotational structure evident within the P and R branches even at moderate perturber densities.

As the band broadens, the peak absorption coefficient gradually decreases; the area under the contour is not constant however. The variation of integrated intensity Z"7 *3 ) d V with helium density is shown in Fig,IV-2.Within the experimental scatter, the decrease appears linear over the entire range from 100 to 900 amagat. A "least-squares/; straight line of the form P =* ) was fitted to


Reproduced

with perm

ission of the

copyright ow

ner. Further


without

permission.

He DENSITY (amagat)60

873EoV.a>Q. 540

40<DOE

CVJEo 288

155I-l___________________ I____________________ I__2900 3000 3100

(crrf1)Fig.IV-1. The 3000 cm--*- region of CH4 at representative helium densities*

Ordinates have been shifted for clarity of presentation.

oo>oEj d<D

c l

6OoooCO4)

5UO>»+>*«S•o•P

•3>»■P•H«8■p•3•aa>-p(0uto0)■pa•H • •<*<p as o od «n o o•Hs a•H'u H00

o Oo Or o CVJCVJ CVJ

(9|0W/ gUiO ) J

<0 ® > u

O£

bO•HCe*


26

and yielded the constants P Q « 2321 cm2 / mole, x 10“4 / amagat, with an RMS deviation of 18 cm2/ mole

Armstrong (1961) has also observed a decrease of for this band* Although he observed an apparent variation with density, the total decrease in F amagat (~ 10#) was comparable with that observed

here ( ~13#). The possibility was explored that the difference in density dependence was due to the slit-i/idth correction employed in the present work, which would have its greatest effect at low densities* Reducing the spectra without the correction maintained the trend within each run but shifted the low-density run downward with respect to the high-density run. The resulting plot was somewhat similar to that observed by Armstrong, but with higher scatter. In any case, the agreement of the extrapolated absolute intensities was within the combined error estimates. In Table IV-3 is shown a selection of absolute intensity measurements from the literature. It should be remarked that many previous workers have quoted values of AQ = J cK ( V ) d rather than n o = oc ( ) d >> where < * ■ ( ? ) «= * ( ** ) / 'O (see Appendix I).To convert to the units employed here requires a knowledge of the ratio of A © / ^ J i*1 all the bands studied here,the measured was constant to within a small fraction of 1%

(typically ^ 0.1#). In converting the results of other

the data, Tf - 1.33 or** 0.8#.

P with p

quadratic over 1000


I 27

investigators, therefore, our own values of V have been used where necessary, with confidence that no significant error has been introduced*

TABLE XV-3* Experimental absolute intensities for the band of CH^

To(cm^ / mole)

Thorndike (1947) 2229Heicklen (1961) 2310This work 2321Rollefson and Havens (1940) 2406Armstrong and Welsh (i960) 2423Welsh, Pashler, and Dunn (1951) 2662

Armstrong and Welsh (i960) have ascribed the low value of Thorndike (1947) to insufficient pressure broadening, and the high value of Welsh et .al*(l95l) to an impurity absorption* The remaining four values all lie within a range of 4.7/S, which is believed consistent with variations in gas purity, matching difficulties, and slit-width effects* No attempt has been made in the present work to correct for loss of absorption intensity in the wings due to spectrometer insensitivity* Such a correction demands a knowledge of the rotational line-shape functionj at high foreign gas densities this information is still incomplete. Moreover, it is believed


28

that the method of numerical matching described above makes the entire procedure unnecessary. We shall defer to Chapter V a discussion of errors in absolute intensities.

ii) The 1305.9 cm~^ band. - Examination of the contours of Fig. XV-3 for the V 4 bending fundamental of CH^ discloses that it is much narrower than the V 3 fundamental. Teller (1934) explained this in terms of the large value of the Coriolis £ constant for the degenerate V 4 vibration. In addition, Coriolis interaction between the p 4 band and the

twisting mode at 1533*6 cm“^ makes the latter weakly active and distorts the symmetry of the former; this effect has been treated din detail by Jahn (1938)* Given the basic . shape of the band, however, the effects of increasing He density are qualitatively similar to those for the p 3 band: gradual obliteration of the rotational structure coupled with a decrease in oC ( V ) near the peak.

The similarity does not, however, extend to the variation of P w i t h , shown in Fig. IV-4. There is evidently a large discrepancy, amounting to ^ between the results of the only two experiments performed for this band. Examination of the intensity profiles for the two runs reveals no obvious differences ascribable to impurity absorption or bad matching. Moreover, the two runs for the p 3 band, performed using the same two mixtures, agreed well with each other; the possibility remains, however, that the low


(v)

(cm2 m

ole'1 p

er cm

“r) 60 He DENSITY

k (amagat)

87 340

55 8

32 6

155

14001300 z/(cm“ 1)

1200

Fig.IV-3« The 1300 cm”-*' I’eglon of CH^.


— o

LO

o

n o oX

oCDCVJ

olOOJocvj

(0 |O U J/2 UJO) J


29

density mixture was not homogeneous* Armstrong (1961) has pointed out the difficulty of mixing gases in the compressor chamber* In the absence of further information, P 0 was obtained by extrapolating the two runs separately to zero density and averaging the results, giving p Q «= 2616 cm2/mole. In Table IV—4 is given a selection of absolute intensities by other workers; the total variation of 6 ,5 % is not unreasonable*

<*

TABLE IV-4. Experimental absolute intensities for the V 4band of CH4.

Reference • To(cm^ ! mole)

Rollefson and Havens (1944) 2536Heicklen (1961) 2554Thorndike (1947) 2570This work 2616Welsh and Sandiford (1952) 2696Armstrong and Welsh (i960) 2706

Even allowing for the experimental discrepancy in the present work, it is apparent that the integrated intensity of the P 4 band displays very different density behaviour from that of the V 3 band. From 325 to 875 amagat, the decrease is only 57 cm2 / mole or 2*2^, compared to the 7*3%


expected if the slope were the same as that of V 3. Armstrong (1961) has also observed the variation of withyO j he found an apparently quadratic increase with density of ~ 25/6 in 1000 amagat, rather than the reduced decrease observed in this work« The difference may be due to a rather subtle experimental effect* As mentioned in Chapter XI, the detector can "see” the spectrometer walls on part of the chopper cycle when the entrance slit is not imaged on the exit slit* This leads to a decrease in the zero level as the exit slit is opened; if in addition radiation is incident on the entrance slit, the quadratic increase in signal with slit width will be superimposed on the linear decrease of the zero level* In both experiments, the Perkin- Elmer spectrometer was fitted with a slit string-drive to maintain the spectral slit width and background level approxi mately constant. If the zero level were taken as constant at its initial (highest) value, the resulting experimental intensities would tend to be too high, particularly towards the peak* Moreover, as a given mixture was compressed, the discrepancy would increase* In the absence of detailed knowledge about the actual slit widths and zero levels employed by Armstrong, this discussion must, of course, remain a quali tative guess*


30

expected if the slope were the same as that of Armstrong (1961) has also observed the variation of with jo ;he found an apparently quadratic increase with density of 'v 25% in 1000 amagat, rather than the reduced decrease observed in this work* The difference may be due to a rather subtle experimental effect. As mentioned in Chapter II, the detector can “see” the spectrometer walls on part of the chopper cycle when the entrance slit is not imaged on the exit slit* This leads to a decrease in the zero level as the exit slit is opened; if in addition radiation is incident on the entrance slit, the quadratic increase in

1 t

signal with slit width will be superimposed on the linear decrease of the zero level* In both experiments, the Perkin- Elmer spectrometer was fitted with a slit string-drive to maintain the spectral slit width and background level approximately constant. If the zero level were taken as constant at its initial (highest) value, the resulting experimental intensities would tend to be too high, particularly toxyards the peak* Moreover, as a given mixture was compressed, the discrepancy would increase. In the absence of detailed knowledge about the actual slit widths and zero levels employed by Armstrong, this discussion must, of course, remain a qualitative guess*


31

2) Methane-d^

The vibration frequencies of CD4 and the experimental conditions under which it was studied are given in Tables IV-5 and IV-6.

i) The 2259-3 cm"1 band. - This band, shown in Fig.IV-5, is similar in appearance to the corresponding transition in

the principal difference being the compressed rotational structure due to the higher moment of inertia of CD4. The behaviour of the integrated intensity too is similar. The data shown in Fig. IV-6 yield the constants P0 = 1303 cm2/mole,

— 1.58 x 10 ^ / amagat, with an RMS deviation from the best straight line of 8 cm2 / mole or ^ 0.6/C. Comparison with previous results is given in Table IV-7*

TABLE IV-7* Experimental absolute intensities for the V 3 band of CD^

Reference Po(cm'* / mole)

This work 1303Armstrong (1961) 1444Heicklen (196I) 1466

The large (~'11/C) difference between the present value and previous ones is so far unexplained. That it is


with perm

ission of the

copyright ow

ner. Further


without

CD"O— 5oQ.CoCDQ.TABLE IV-5* Vibrational frequencies of CD4

Assignment Mode Frequency (cm“^) Reference

asymmetric C-D stretch symmetric C-D stretch

P 2(E) - CD2 twist174(^2 , D - C - D bending

2259.3'2 1 1 7

1091.9996.0

Olafson, Thomas and Welsh(1961)n

Kaylor and Nielsen (1955)

TABLE XV-6. Experiments on CD4

Region Run Path Length Mixture Density range Number of(cm"-*-) Number (cm) ratio (amagat) traces

T3CD(/)(/)

2440 - 2090

1200 - 840

102103 101104

2.758n

11tt

7892560789

2560

124 - 178 220 - 872 132 - 360 436 -1003

615911

He DENSITY ~ (amagat) > £ 7 2 -

60

eok_<DCl

i£ 40 oE

538

OJEo

326

156

23002200

Fig.IV-5. The 2260 cm-1 region of CD4.


1300

<DOEM' 1200E

t—i

Run 102 o

103 □

8 0 0yCj.|e(amagot)

Fig.lV-6. Variation of integrated intensity with density for the 2260 cm"1 region of CD4.


32

due to a slit width effect is unlikely; the peak intensities in the spectra of Fig.IV-5 are, relatively, higher than those of Armstrong (1961), Earlier work by the present author, although at lower resolution and without the slit-width correction, also indicated a value of P Q much lower than the results of Armstrong and Heicklen £cowan (I964)j.

ii) The 996.0 cm’"^ band. - The increased asymmetry of this band (Fig.XV-7) is due to the Coriolis interaction with the 1091*9 cm"^ mode, designated V 2(^)5 i*1 addition the band is narrower than ")? because of the higher vibrational angular momentum* Otherwise, the behaviour of the profile with He density appears typical.

The change of P with jO (Fig.IV-8) is again different from that observed in the stretching mode* Extrapolation of the eight low-density points yields f"1© — 2194 cm2 / mole; comparison with the only other measurement known to the author is shown in Table IV-8.

TABLE IV-8. Experimental absolute intensities for the 1? 4band of CD4

Reference r »(cm / mole)

Heicklen (1961) 1906This iirork 2194


1010

o

UJO Jad , 0|OUJ _ UJD) I/I) TOI "" O /\

oo

ooo

TfQ •oCMo

^ i•HE to d)o u

BOooo

oO<7>

oH

I>H

b*D•H


2200

oE3a>

(9|0U1/2 UJ0) J


Fig.

IV-8

* Variation

of integrated

intensity

with

density

for

the 100

0 cm

”

aa

The present result is based on too few points toallow any meaningful comment on the size of the discrepancy;the extrapolated value of f""*0 may be somewhat high because of the large value obtained for the low-density slope (7f « 2.75 x 10“4 j amagat).

b) The symmetric tops CH^D and CD^H

Methane-d^ and methane-d^ are, respectively, prolateand oblate symmetric rotors belonging to the point group C^v.•Bach has six vibrational modes, all infrared active, giving three parallel bands of species and three doubly-degenerate perpendicular bands of E species. The lowered symmetry of these molecules, with the consequent threefold increase in the number of active fundamentals, increases the probability of band overlapping, and of Fermi and Coriolis resonance between bands.

•l) Methane-di

Table IV-9 gives the assignments of Wilmshurst and Bernstein (1957) for CH^D, but more accurate transition frequencies have been quoted where available. In addition, the species of 2~)?2 has been corrected from that quoted in the above reference. Table IV—10 summarizes the experiments with CH3D.


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with perm

ission of the

copyright ow

ner. Further


without

permission.

TABLE XV-9• Vibrational frequencies of CH~D.

Assignment Mode Frequency(cm-1)

V jB(B) asymmetric C-H stretch 3016.6

V 3a(Al) symmetric C-H stretch 2969.72 P 2(A 1 + E) overtone 29142 V 4b(A1+ E) overtone 2315.7VitAj) symmetric C-D stretch 2200.0V2(e) asymmetric CH3 bending 1471V4a(Al) symmetric CH3 bending 1300

% < B> C-D bending 1155

Reference

Richardson, Broderson, Krause andWelsh (1962)tt

Wilmshurst and Bernstein (1957) Richardson et al. (1962)

n

Wilmshurst and Bernstein (1957)n

n

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ission of the

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ner. Further


without

permission.

TABLE IV-10. Experiments on CH^D,

Region Run Path length(cm-1) Number (cm)

3270 - 2730 68 2.77470 2.77476 2.76992 2.75893 2.758

2530 - 2000 31 2.75869 2.77488 2.75891 2.758

1650 - 960 67 2.77490 2.75894 2.758

Mixture Density range Number ofratio (amagat) traces

493 109 - 156 5H 7 0 221-433 7

N00H 156 - 433 91879 753 -1029 51879 425 - 872 9

369 204 - 392 5493 199 - 586 8468 621 -1096 8468 621 - 957 8

493468

1879

156 - 399 9232 - 629 8424 - 929 10

34

*■) 3016,6 cm ^ band. — Spectra of the C-H stretching region are given in Fig.IV-9 at representative He densities.Host of the intensity comes from the V.. transition. The twin peaks in the low frequency wing of this band are due to Fermi resonance between 2 V 2 *= 2942(AX + E) and the weak fundamental V 3*^^) for which Wilmshurst and Bernstein (1957) took 2945 cm“* as an unperturbed frequency. A plot of integrated intensity against perturber density is shown in Fig.IV-10• Again the decrease is linear over the entire range, with best-fit constants of » 1773 cm2 / mole, =1.560 x 10“4 ! amagatj RMS deviation was 8.8 cm2 / mole or ^ 0.5/6* Values of H o for this complex of bands are compared in Table IV-11.

TABLE IV-11. Experimental absolute intensities for theTâ and bands of CH3D

Reference P o(cm^ / mole)

Hiller and Straley (i960) 1630This work 1773Heicklen (1961) 1791

ii) The 2200.0 cn'^ band. - This band, shown in Fig.IV-11, consists of the parallel transition due


Reproduced

with perm

ission of the

copyright ow

ner. Further


without

permission.

3 a

He DENSITY v amagat

Eo 20CDQ.

881ijdoE

CMEo529

327

155

31002900 3000( cm 1)

Fig.IV-9. The 3000 cm"^ region of CH^D. The dashed curvevhas been separated using the Boltzmann intensxty relation.

r (cm2/ mole)

1700

Run 68 o 70 □

1600— 76 v92 ■

93 •

800400yOHe(amagat)

Fig,IV-10. Variation of integrated intensity withdensity for the 3000 cm“ bands of CII3D.


35

stretching at 2200*0 cm ^ overlapped on the high frequency wing by an anomalously intense overtone centred on 2317 cra“l. Short of a complete analysis considering both mechanical and electrical anharmonicity, the overtone intensity cannot be explained; a possible contribution is Fermi resonance between2^4b<Al + E) 311(1 ^ l ( A2>*

The variation of P with is shown in Fig.IV-12.Once more a linear decrease is observed, giving the constants

*» 304*2 cm^ / mole, if =* 1*696 x 10“^ / amagat, with an RMS deviation of 3*9 cm2 / mole or ~1.3#. It is evident from the graph that the scatter within a single run is much less than that between runs* This is due partly to the difficulty of obtaining accurate mixture ratios and homogeneous mixtures in the compressor cylinder, and partly to the usual difficulties in matching and in measuring the zero level* Although the discrepancies between runs are here larger than usual, the effect is common to all our results. Table IV-12 gives a comparison of measured intensities.

TABLB IV-12. Experimental absolute intensities for the "V -y

band of CH3D

PReference 2 °(cm / mole)

Heicklen (1961)Hiller and Straley (i960)This work

269281304


( u uuo J0d ,__0 | o i u 2UJ0 ) ( / Z ) D


300

CMEo

Run 31 o

6 9 □2508 8 v

8004000^ (amagat)

Fig*IV-12. Variation of integrated intensity with He density for the band of CH^D*


36

The low values of previous workers may be due to a failure to measure the intensity sufficiently far into the high-frequency wing.

iii) The 1471» 1300» and 1155 cm"1 bands. - Spectra of these three bending modes, shown in Fig.IV-13, are evidently more complicated than any bands discussed so far in this work. Although the Q-branch of the ^4a(î) fundamental stands out clearly, the P- and R-branches are seriously overlapped by perpendicular bands at 1155 and 1471 cm“*. The change in profile with perturber density is qualitatively similar to that for the other bands. Once again, however, the plot of I"”1 vs.yO (Fig.IV-14) differs from that for the stretching modes. The absolute intensity was obtained by extrapolating only data below 500 amagat, giving n 0 = 2501 cm^ / mole,% *= 1.17 x 10"5 / amagat, with an RMS deviation of the points from the straight line of 19*8 cm^ / mole or 0.8^. The high relative width of this band complex ( A V / V ^ O . 5 5 ) imposes great difficulties on the accurate measurement of integrated intensities; this is reflected in the widely scattered values in Table IV-13 below.


aCOXo.oao•Hho©U

aoo.to0H1ototo

©

•COH

K*Hbo•HDu

( ^ o io jad ,_9|ouu 2w o) ( /Z )D


CD <J> <T>

acno£oo

4)A■p

UO

CM

►»■P•r|(Q

s*0

■lrH4)X!

jC-P•rl

>»+>•H10

8-p

mQ«n

4) S•P O<4

CMbO 04)-P dCJ 0•rl •H

bO4)

0 U

CJ HO 1•H fi■P y(4•rl oU o(4 co> H

•

sH•

hO•rl(U

( 9 |0 U J / 2 U J 0 ) J


37

TABLE TV-13. Experimental absolute intensities for the 2 *

_____________ and ^4b bands of CH^D.

Reference r .(cm^ / mole)

Hiller and Straley ( i 960) 2342

This work 2501

Heicklen ( 1961) 2700

2) Methane-d^.

The vibrational frequencies of CD^H are given in TABLE IV-14; although there is some disagreement in the literature, it is believed that the assignments of Wilmshurst and Bernstein (1957) are correct. This will be discussed further in section iii) below.

There is slight but measurable absorption in the entire region between the two low—frequency modes and the band at 1291 cm-1. Thus although the profiles and intensities may be separated for theoretical purposes, proper matching demands that all three bands be observed in one continuous scan. The experimental conditions are summarized in Table XV-15*


Reproduced

with perm

ission of the

copyright ow

ner. Further


without

permission.

TABLE XV-14. Vibrational frequencies of CD^H.

Assignment Mode Frequency Reference(cm-1)

1 — — ' "■ ■ i ii ■ ■ — - . i ■ — ■ ■ ■■ — — mi L i

Vâ(A±) symmetric C-H stretch 2992.3 Rea and Thompson (1 95 6 )

V 3 b ( E ) asymmetric C-H stretch 2250.9 Lepard (1 96 4 )

V i ( A ^ ) symmetric C-D stretch 2142.6 IT

v 2 ( e ) CH bending 1290 Rea and Thompson (1 95 6 )

asymmetric CD^ bending 1037 it

H a ( A l > symmetric CD^ bending 1000 u

X«uQOa0w

ps1&axM•to

HIftW«

H

O ® © (4 O © © x> ua p

©to§ *“N(4 P

*5►> to P eg

a

©tj od h p pX (93 fa

X"6S<-V* 8Xpega

Iu©x

&

O'** OH •H |to a © oC4'-

00 vO to tr> to m oo O' oH

CO X to

o o CO M 04 00 CO co CO 00 00 H v t o O' H"tt CO tv T t 00 t v t v CO CO CO CO vO T t « t vto VO 00 04 04 to OO 00 vO to to o co 04 vO 001 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

o OO t v t v VO O' t v 04 o O' © O' 00 N OO Hto oo oo to ««t 00 CO o O' 04 T t t v co H- MN N to H H 04 to vO C4 H 04 N H H 04 vO

04 to to to to 04 04 to to O' 04 to to O' C4OO 00 00 Tt OO OO 00 vO OO OO H- vO © ©tv O' O' t v t v tv t v O' t v to t v O' tv to O' O'

H

04 O' O' O O 04 04 O' O OO 04 O' O' 00 OO 0004 vO VO vO vO N 04 vO vO tr> 04 VO vO tn to t nO' t v t v t v t v O' O' t v t v t v O' t v t v t v t v t v• • • • • - • • • • • • • • • • •

M 04 04 04 04 04 04 04 04 04 04 04 04 04 04 04

COto

0NON04

1ooCOCO

coo •n-VO

ooo COoo O'to

oO'O'H

ooto04

oVO

tov©

Hoo ooO'

t vto

oto00

otv*0-H

VOV©

04oo tvO'

O'O'

ooH


38

O ,2.992*.3 -£m Jj,_.ban.dt. - Of all the intermediate methanes CH^D through CD^H, this band is least perturbed by overlapping with nearby fundamentals or intense overtones. It is ironic then that the high-frequency region beyond the band contains an almost continuous series of weak overtones and combination bands which extend for ^400 cm”-*- and make matching somewhat uncertain. Typical contours are shown in Fig.IV-15j the behaviour with pressure is qualitatively similar to that in bands previously discussed, i.e. small frequency shifts, disappearance of rotational structure, no appreciable increase in width. The plot of P vs.y0 , shown in Fig.IV-l6, gave the parameters P Q « 628 cm^ / mole, if *= 1.709 x 10”4 / amagat j RMS scatter was 4*7 cm^ / mole

or ^ 0.7£. Results obtained by other workers are compared in Table IV-16.

TABLE IV-16. Experimental absolute intensities for the V 33band of CD3H.

Reference (cm / mole)

Hiller and Straley (i960) Heicklen (1961)This work 628

522581


20 He DENSITY (amagat)

873iEot .a>o.iooE 398

c\jEa

155

310030002900V (crrT1)

Fig.IV-15. The 3000 cm-1 region of CD II.


r (c

m2/mole)

6 0 0

Run 58 o63 □6 4 v 80 a 83 •

550

8 0 04 0 0p (amagat) • H6

Fig.IV-16. Variation of integrated intensity with density for the 3000 cm"1 region of CD3II.


39

The large differences between these results (no two values lie within the combined error estimates) may be due to two factors: l) the weak overtones mentioned above, which, if not allowed for in the matching procedure, give too low a value of / 1 2) the high proportion of the intensity (theoretically O.363) in the sharp Q branch, which makes the result particularly sensitive to slit-width effects. In view of the fact that the agreement between intensity theory (see Chapter V) and our experiments is worst for this band, the value of given here may be somewhat too large; the best consistency would be obtained with a value of P Q between the present one and that of Heicklen (1961).

ii) The 2250.9 and 2142.6 cm~^ bands. - This region, shown in Fig.IV-17, contains a sharp Q branch at 2142 cm”* due to Vj(Ai) and a broad irregular perpendicular band, its central doublet apparently centred on 2268 cm”"*', due to V 3i,(E).The peak at 2330 cm-*, assigned to V 2 + V^b *= 2327(Aj+A2+E), may derive its large intensity from Fermi resonance with the two fundamentals.

The plot of P against He density (Fig.IV-l8) gives the constants P ^ “ 967 cra / mole, ^ *= 1.374 x 10 ^ / amagat, with an RMS deviation of 6.9 cm^ / mole or '"''0.7$* This is compared with the results of other workers in Table IV—17•


5 (Z

/) (c

m2

mo

le’

1 pe

r cm

l)%

81— V\ and (CD3H)

.873

3 6 3

200

2 4 0 02 3 0 02200

The 2200 cm'1 region of CDjH.,


( cm

2 /m

ole

)

950

900

Run.59o 60 □ 65 v81 n

98*

850 800400yOHe(amagat)

Fig.IV-18. Variation of integrated intensity with density for the 2200 cm-1 region of CDjH.


40

TABLE IV-17• Experimental absolute intensities for the V

and ")?2b bands of CD^H.

Reference Po(cm^ / mole)

Heicklen (1961) 00 Wl

Hiller and Stra.ley (i960) 927This work 967

Again the present result is significantly greater than previous values although•the discrepancies are much smaller than those for the ^ 3 3 band.

iii) The 1290. 1037 and 1000 cm"-*- bands. - There is still uncertainty in the literature as to the assignments for these bands, which are illustrated in Fig.IV-19. Rea and Thompson (1956) attributed the sharp peak at 1037 cm"1 to the parallel transition "P 4a(Al)j the somewhat broader 1000 cm"1 they ascribed to a series of Q branches in the degenerate "^^(E) mode. For however, taking A ss 3*28 eraB ~ 2.63 cm"1, and £ « -0.30 obtained from the £ -sum rule,the Q-branch spacing, given by - 2 £b(1 - £ ) - aJ , wouldbe '*■'0.3 cm-1. Thus the total width of the central peak,ignoring the J—structure of the sub-band Q. branches, would be only a few cm"1 for this combination of A, B, and •


\4T

O

o

jQ

CVJ

(l„UJO J 0 d | -0 |O IU 2 LUO) (/Z) D

aCOQO<Moao•H<30<0uH'aoo0O'1o4)

£O*H

£bo

•Hfc.



41

This accidental narrowness of the band will result for any value of t from - 0.10 to - 0.40. For this band then, we cannot make assignments on the basis of the Q-branch width alone. On the other hand, both the normal coordinate calculations of Jones and McDowell (1959) and the intensity calculations of the present work (Chapter V) support the assignments given in Table IV-14 above, namely that the long wavelength peak is the parallel transition V ^ a(Ax).

The integrated intensity for the three bands is shown in Fig.IV-20; again the initial decrease is not maintained to higher densities. Least-squares analysis was performed on both the total intensity and on that above and below 1180 cm“A separately, for densities less than 400 amagat. The results are given in Table IV-18.

TABLE IV-18. Absolute intensities for the 1?2> ^4a> anc*V4b bands of CD3H.

T o t RMS scatterBand (cm2/mole) (10"4 amagat-1) (cm2/mole) (% )

) )2 , 606 0.817 13 2.2»4a and P 4b 1722 1.160 23 1.4V 2 , }>4a,* y 4b «36 1.241 24 l.o


ro

O

O©

□ O ©0 □

C©1 © O

□o

in

OoCO

oC Po

o e2 J

a>o f

<DX■P

U0> .

-P•rl(0

§•o

0-

X-P•rl

>»-P•H •01 X

8 COQ

•p O

tu0

•o4) G

•P O(0 • rlU how ©o u

-pa r l•H 1

&o

Oo

CS oo O '

•rl•P 1ca

•H oU o(0

> r l

•od1

>H

•bO

•rlU.


42

In general, of course, weaker bands are subject to the same relative error as more intense ones since the optical path may be adjusted to bring the percent trans-

i.mission into the desired ..range for best accuracy. In this case, however, the criterion for CD3H density is the peak absorption in the band; the high scatter in the weakerV 2 band is a reflection of the resulting low absorbing gas density. Table IV-19 gives a comparison of the present results with those of other workers.

TABLB IV-19• Experimental absolute intensities for the V 2,^4a» and i^4b bands of CD^H.

Reference r© (cm2 / mole)V i V4a & Z>4b Vi* Z>4a&U4b

Hiller and Straley(l96o) 516 1483 1999Heicklen (I96I) 580 1450 2030This work 606 1722 2336

Once again the present results are appreciably larger than those of previous investigators. It is diffi— cult to conceive of any combination of impurities which could give greater absorption in all three spectral regions without a single patently spurious absorption peak. We note in addition that the discrepancy is much greater for


43

bands which, contain sharp and intense features (V $ a , 1 45) than for broader bands ( 1^2* ^3b^*

c) Methane-d2

The CH2D2 molecule is an asymmetric top belonging to the point group C2V and has nine non-degenerate normal modes: four of species Ai, one of A2, and two each of Bi and B2* All are active in infrared absorption except the A2 vibration, which corresponds to a twisting of the H-C-H and D-C-D planes in opposite directions about the C2 axis* In reality the ^2b^2^ m°de is made weakly active by Coriolis coupling with V 2a(^l) anc* following table (IV-20)gives the observed fundamental frequencies as assigned by Wilmshurst and Bernstein (1957)* For such an asymmetric rotor, the absorption bands are designated as type A, B, or C according to the orientation of the transition dipole (see Appendix III)*

As for the other methanes, the spectrum divides naturally into three non—overlapping regionsj the experiments are outlined in Table IV-21.

'I


TABLE XV-20. Vibrational frequencies of CH2D2*

Assignment Mode Frequency (cm“l) Band Type

^ 3b<Bl) asymmetric C-H stretch 3013 C

V3«<a i > symmetric C-H stretch 2976 B

^3c^b2^ asymmetric C-D stretch 2234 Asymmetric C-D stretch 2202 B

^2a(Al) symmetric CHj bending 1436 B

1 2bÂ2) H-C-H D-C-D twist 1329^4c (B2) CD2 rocking _L H-C-H plane 1234 A

^4b(Bl) CH2 rocking JL D-C-D plane 1090 C

^4a(Al) symmetric CD2 bending 1033 B

TABLE

IV-21.

Expe

rime

nts

CtQNaoao

two w o u oo flj £> Us +>

©to8^ U -P

<9►» M +> <0CO

8 'Q

to

■P(0

It.0)A

G '~'OH

•H Ibfl g (D O 04'-'

NO 0 0 oH 00 NO O NO H

00 o CO CO co H co NO NO N H ON CO ts CO CO tv co O n CO co co NH M NO 00 T t •Nfr 00 © co © o ■N* 00H H H

1 1 1 l 1 1 1 1 1 1 1 1 1•c f H o H NO 00 O ' N NO M H M inS O ' in in o CO tv O ' O ' CO CO

N M H H H NO H •NSf

M m © t v N m m o m o t v NO. t vOO o 0 0 H - 0 0 c t o oo o OO O ' t vT f m H t V •cf CO m H to H t v ■** NO

H H H H H

M m c-t O ' OO m M m c t O ' OO 00t v t v NO NO t v NO t v NO t v N NO m » nt v t v t v t v t v t v t v t v t v O ' t v t v t v

• • • • • • • • • • • • •c* N c< C* C4 <N N M

NO«<*

oCOt vC4I

OOCOCO

H in oo

oCNJ0N1

©N<<*C*

t v coin mm

Ctin

©©ONI

oNNO

NOm

moo inO '

NOO '


44

i) The 3013 and 2976 cm"1 bands. - This region for CH2I>2is illustrated in Fig.IV-21. It consists of an intense C-type band at 3013 cm"-*- and a weaker B-type band with a doublet peak centred at 2976 cm**-*-, corresponding to asymmetric and symmetric C-H stretching vibrations, respectively. At high foreign-gas densities, the overall appearance is very similar to V ^ a * 3b CH^D. At low densities, the central peak at 3013 cm” - is very sharp; indeed, Wilmshurst and Bernstein (1957) appear to have underestimated its intensity by a factor of ~'1.7 because of inadequate resolution.Indirect confirmation of the sharpness of this peak comes from the work of Hiller and Straley (i960) who measured the absolute intensity using the Wilson-Wells technique. They found that this band alone of all the methane fundamentals gave a non-linear plot of integrated intensity against optical path, implying that finite instrumental resolution was playing a significant role even at 3 cm“-** spectral slit width.

Our observed variation of /”* with He density is shown in Fig.IV-22. The linear constants obtained were 1 o 13 1196 cm2 / mole and i = 1.386* X 10“4 / amagat, and RMS deviation was 10 cm2 / mole or 0.8#. Comparison with the only other intensity work on CH2D2 known to the author is given in Table IV—22; agreement is within 2.9#*


2 Xv)

(cm2 mole"1

per c

m"1)

He DENSITY (amagat)

2900 3000 l / t c r r T 1)

3100

Fig.IV-21. The 3000 cm”1 region of CH2 D2


( ©| OUJ / gUJO) J


45

TABLE IV-22. Experimental absolute intensities for the^30 and V 3b bands of CH2D2.

Reference T o(cm^ / mole)

Hiller and Straley (i960) 1162This worke e , . . . » B t u a g r f f M

1196

ii) The 2234 and 2202 cm"-1- bands. - The C-D stretching region in CH2D2, shown in Fig.IV-23, is complex and difficult to interpret, Wilmshurst and Bernstein (1957) assigned the doublet at 2196 and 2208 cm”1 to the B-type mode Vl(Ai), and the peak at 2234 cm”1 to the A-type band of ^ 3 C^B2 *The band at 2275 cm”1 they assigned to the combination tone V 4a + I )4c *= 2267 (B2), without reference to its anomalously large intensity, Jones and McDowell (1959) calculated harmonic frequencies for the deuterated and tritiated methanes and correlated the results with observed frequencies. For the P 3c(B2) bands of CH2T2 and CD2T2 , they found

£ E (“'harm - "obs> /" L r m *» be X 10‘S 1<41 X 10'5 Cmrespectivelyj using the value of 2234 cm ^ f°r CH2D2» however,they found 1,82 x 10” cm. On the assumption that anharmonic constants connecting two modes are proportional to the product of the harmonic frequencies, with the same constant of proportionality for isotopic molecules of the same symmetry,


/Eov_cuCL

a>oEHe DENSITY (amagat)

CJEo 1036

364

1552400230022002100

z/(cm ')Fig.XV-23. The 2250 cm”1 region of CH2D2


46

then f is expected to be constant for all three isotopes. Assuming 1.49 sc 10-5 ? i.41 x 10-5 oa> „e find anexpected range for V>3c of 2252 - 2256 car*. It thus seems clear that the apparent shift of and the large intensityof the peak at 2275 cm”1 are both due to Fermi resonance between V 3c(B2) at - 2255 cm"1 and V 4a + ^ 4c(B2) at -2267 cm"1. In addition to the normal high-frequency branches of V ^ c and V 4a + V 4c, the lobe at — 2340 cm"1 may contain a Fermi resonance between V 4a(A1) + ^ 2b(A2) “2362 (A2) and ^ 4b(Bi) + V 4c(B2) = 2324 (A2), made infrared active by Coriolis interaction with the ^ 3C(B2) and ^ 4a + ^ 4c(b2) pair.

It is evident that the lowered symmetry of CH2D2 and the consequent complexity of its spectrum makes any detailed profile analysis impossible in a work of this scope. We shall be content with a simple integrated intensity over the whole region, the behaviour of which is shown in Fig,IV-24* Analysis yields the parameters *= 615 cm^/mole, ^ *= 1.564 x. 10“4 / amagat, with an RMS deviation of

A4»4 cm / mole or ^0.7^«

TABLE IV-23. Experimental absolute intensities for the and bands of • CH2D2

Reference r 0 < cm2 / mole)

Hiller and Straley 615This work 615


F (c

m2/

mol

e)

3 c

600

550Run 44 o

47 □

53V 55 a

800400p (amagat)

Fig.IV-24. Variation of integrated intensity with He density for the 2250 cm"1 region of

CM 2^29


47

iii) The 1436, 1329. 1234. 1090 and 1033 cm"1 bands. - The deformation mode region of CH2D2 , shown in Fig.IV-25, consists of five overlapping fundamentals with absorption extending from ~1550 to ~950 cm"1. Because of the absence of overtone and combination bands, the spectrum is nonetheless more straightforward than that of the 2250 cm”1 region. The only uncertainty arises from the single peak at 1033 cm"1 ascribed to ^ 43(^1)* a B-type mode. Since both normal coordinate and intensity calculations (see Chapter V) support this assignment, we must assume there is sufficient rotation- vibration interaction within the band, and Coriolis interaction between it and to remove the expected doubletstructure. In any case, the peak is weaker and more shaded than that at 1090 cm"1; this is consistent with the latterbeing a type C band.

The variation of integrated intensity is given in Fig.IV-26. Once again the linear decrease observed throughout the density range for the stretching modes is present only at lower densities for the deformation modes. The discrepancy of ~ 5 % between the two loiir-density runs is probably due to a combination of imperfect mixture homogeneity and matching difficulties. To obtain f”"10, least-squares straight lines were fitted to the two runs separately and the results averaged, giving f~~ = 2563 cm^ / mole, and if 63 1.31 x 10 amagat for the region below 400 amagat; the average RMS


o>

ininro(VI

■oX )

x>CM

(| jijo jod |j>|ouu g luo) (/z) jg


CO

-O

f*4o Q

A*

lOCDOJ

so

oo>oE3

Q)

( 0 | O O 1 / 2 UJO) j


Fig.

IV-2

6• Variation

of integrated

intensity

with

He density

for

the 16

00-9

00

region

of

48

deviation was ^ 0 . 6 % . Hiller and Straley (i960), in measuring the intensity of this band complex, have omitted the higher frequency modes at 1436 and 1329 cm"1, making comparison very difficult.


CHAPTER V

ABSOLUTE INTENSITIES AND BOND PROPERTIES

It has long been known that the fundamental vibration frequencies of polyatomic molecules are often characteristic of the stretching of specific types of bonds. Thus C-H bonds within a molecule give rise to bands at ''■■'3000 cm"^(C—D bonds to ^ 2200 cm"^- transitions. Similar regularities are observed for the absolute integrated intensities of such bands. For organic molecules of high molecular weight and unknown composition, the total intensity in the 3000 cm“^ region has been used to estimate the density of C-H bonds. Indeed, as we shall see below, the intensities in the 3000 cm“l region for the series CH4, CH3D, C^Dj* CD3H, CD4 lie in the approximate ratio 4 * 3 s 2 t 1 : 0, whereas for the 2200 cm"^ region the ratio i s O : 1 : 2 : 3 *4. These intensity patterns imply that each bond of an equivalent set contributes a definite amount to the total intensity in the frequency region characteristic of stretching of the bond.One is thus led naturally to a model for the transition dipole moment as a vector sum of independent, localized moments.There have been numerous attempts to relate observed absolute intensities to the properties of individual bonds, both for . series of isotopically substituted molecules such as the deuterated methanes £Hiller and Straley (i960), Heicklen (1961),

- 49 -


50

Sverdlov (1961)] and for molecules differing more fundamentally [Hornig and McKean (1955), Coulson (1959)J.

Before examining the validity of such bond-dipole models, it is well to recall the approximations implicit in the usual formulas employed* As mentioned above, the experimental definition of integrated intensity used in this work is

H k = (22414 / />aL)( <1 / V ) i n[l0(U) / I(D)]di> (1) The net absorption intensity between vibrational levels E^ and Ej, is given by

P = (N08 T 3 / 3hc) |<i|/ | f> | 2[l - exp (- hcv>fi / kT)]X (1 / Zv) exp (- Ei / kT) (2)

where N0 is Avogadro*s number, J i the dipole moment operator, he 2 (E.f - E^) and Zv is the vibrational partition function* To relate equation (2) to more specific molecular properties, one must assume a form for the vibrational wave functions and for the behaviour of J i • Using harmonic oscillator wave functions, we write Ej[ «= (v^^ + ^) h e w h e r e state iis given by the set of quantum numbers v^^* We further ignore electrical anharmonicity , i.e., we neglect higher terms in the expansion y * . * * + where is the normal coordinate associated with harmonic frequency Wj^*We thus find the only non-zero transition dipole moment to be (vji) ... (vji) +1) ...// |vjtx) ... ... > =

(ffjo (h I » » * “ k«> <vk1} + 1 > *• (3)


51

Summing the contributions from all possible states EA such that the transition frequency lies in the range under examination, and recalling that for the harmonic oscillator *v=7TzVi, one finally obtains

r \ “ <No T / 3c2) (dk / £0 k) (||j03 (4)where d^ is the degeneracy of the normal mode with harmonic frequency CO jc# We emphasize that equation (4) is Correct only to the extent that electrical and mechanical anharmonicity may be neglected*

Crawford (1958) has suggested thatAk " (5)

be the equivalent of (4) when the experimental definition ofintensity is « (22414 / p a1*) f [ ^ o ( ^ ) / * ( P ) ] dV instead of (l) above* In fact however, both and P^ may be measured experimentally; from our data, it was invariably found that — A^ / P^ lay between V ^ and CO

In any case, the difference between the two frequencies is 5% o r less for all bands of the methanes except the 2945 cm“i doublet of CH3D (see Chapter IH-b). Thus if a correction isrequired to equation (5)* it might be expected to have thesame order of magnitude as the uncertainty in the experimental values of P •


52

a) The Calculation of Absolute Intensities

We shall now examine the various bond-moment models for infrared absolute intensities. The simplest and most common theory, the so-called ”zero*th approximation”, has been summarized by Hornig and McKean (1955) as follows: i) in the course of molecular vibration, the stretching of a bond by £ r leads to a dipole moment in the direction of the bond, of magnitude ( ' "j Q ^r» bending of abond through an angle c)c{ produces a dipole in the bending plane and perpendicular to the bond, of magnitude

; iii) the distortion of any one bond has no effect on neighbouring bonds. These assumptions allow us to write the dipole moment induced by an arbitrary distortion of the molecule as a linear function of the various displacements £r£, ^ i * constants involved will be just the bondparameters and ") , plus geometrical factors. If1 o V. d r * owe can now express the normal coordinate Qk in terms of the ( then ( ^ can be found and the problemis in principle solved. The usual procedure is to write the secular equation in terms of symmetry coordinates Sj, which are linear combinations of the ( & r ± t î)* each chosen to belong to one of the symmetry species of the molecular point group. The coordinate transformation matrix Ljk, given by Sj m LjkQk, can then be found by solving the secular equation


53

If the atomic masses, the force constants, and the molecular geometry are known*

Jones and McDowell (1959) have obtained the force constants and harmonic frequencies for methane using observed fundamental, overtone and combination frequencies for CH4 and CD4, and assuming the forces to be unchanged by isotopic substitution* Using their results, Sverdlov (1961) has calculated the vibration form coefficients Ljk for the deuterated and tritiated methanes* However the discovery of a typographical error in one of his quoted force constants made it advisable to recalculate the form factors directly from the data of Jones and McDowell (1959) for use in the present investigation* The computer program was written in double precision throughout to avoid any possibility of error introduced in the matrix arithmetic. The results were generally close to those of Sverdlov (1961) with the exception of the coefficient relating the E-species C-II bending coordinate of CD^H to pjb mode at 2263 cm-lj the quoted value was - 1.1379 x 10-^gm"^whereas we obtained - 0*13769 x 10-^gm- z. Since the V 35 mode is primarily asymmetric C-D stretching, and since the principal normalized form coefficient for any mode is generally of the order of 10^^ gm 2, we are confident that Sverdlov*s value is a typographical error, The Ljk were combined with geometrical factors relating the dipole moment to the symmetry coordinates to give,


54

for each active band, an expression of the formHe " Ak (df)o + Bk (t o + Ck(n?o)2* FiQally,these expressions were grouped for overlapping bands, givinga set of thirteen equations, two each for CH4 and CD4, and three each for CH^D, CH2D2, and CD^H. Assumed values of

( I f L and were systematically varied to give aleast-squares best fit between observed and calculated absolute intensities# The ambiguity of sign in ^results in two possible choices for the parameters, designated ++ and +- in Table V-l below# The best-fit values were (+ 0.491q, + 0.294o) debye / A and (+ -#660q, +0,3167) debye / % respectively#

For every band without exception, the +- choiceis to be preferred; we may take as settled that in the rangeof distortions involved here, the absolute value of the bond dipole decreases with increasing separation# This agrees with the results of Hiller and Straley (i960), of Heicklen (1961) and of Sverdlov (1962), although the evidence here is more complete and convincing# The result also agrees with the theoretical conclusions of Mills (1958)#

With,the +- choice, the RMS deviation of theory from experiment is 5*5#* compared to 7«6# f°r Hiller and Straley (i960), and 10.9# for Heicklen (1961). Although the parameters and ^ —J used by Sverdlov (1962) are not


TABLE

V-l,

Observed and calculated intensities

It,+~.o **U W U fd

OCO

O' •** wo CO NO ■«* O' H co o• • • • • • • • • •H wo vO N •<* HH CO

+ I + + 1 + 1 I + +

O'

4)iHO141L.+^8

NO NO 00 O ' OOCO NO M HC4 NO NO CO WO r lC4 N H H

HOvOtx OO O ' o H NOCO wo O ' CO 'd - wor fC4 wo O ' -N* CO

.HHN

U 'o **f t W U

«

O' WO wo H e* NO wo CO co O' H M• • • • • • • • • • • • •OO O' Wo © wo O' t>» 00 OO oo-N M H M N M H H1 + 1 + + 1 + 1 1 + 1 + 1

L*oHON

CO H CO O' H N M ■Nf OO O' TfCO 00 NO N N ■N* N CO M CO H

NO 00 N CO NO OO ts ** H H W> OH N H N N H H d

L'

4)i—I0) Ox> e oN8

H NO CO •<* H NO wo CO 00 NO CON r l © O O ' H NO C4 NO CO o O 'CO NO N co wo H NO wo NO O ' CO co HN N H C4 H N N H N

ao H •H I» 8 & *—

© O O o O O o O o o O© O © o © O o O o o oo CO O N o N ca o N MCO H co C4 H CO H CO C-J H

ooM

Ooo

4)

■3o4)HO

S

'=}■WO

e *coao

QNao

acoQO

Oo


55

so different from those obtained in this work, his fit of theory to experiment contains several deviations of 30-40# which may represent an error. Taking the theoretical conclusions of Mills (1958) and Coulson (1942) on the direction of the C-H dipole, i.e., from H to C+ , we present in Table V-2 a comparison of calculated bond moment parameters.

TABLE V-2. Bond dipole parameters.

Reference > 0 ( SyU- / c) r)0(debye / X) (debye / X)

Hiller and Straley (i960) - 0.33 + 0.61Sverdlov (1962) - 0.3133 + 0.6994This work - O.3167 + 0.660q

The good overall agreement between observed and calculated intensities in the present investigation implies that the simple two—parameter model is almost as accurate as the experiments themselves.

Further examination of the results in Table V-l reveals that the total intensity in the 3000 cm and 2200 cm“^ regions is closely proportional to the number of

v

C-H and C-D bonds respectively. The situation is illustrated in Fig.V-l where we see that the observed intensities,


Reproduced

with perm

ission of the

copyright ow

ner. Further


without

permission.

ABSOLUTE I N TE N SI T I E S

FOR THE

S T R E T C H I N G MODES

2000 -

observed

c a lcu la ted

ca lcu la ted

E 1 0 0 0

o b s e rv e d

Number of BondsFig.V-l. Absolute intensities in the C-H and C-D

stretching modes as a function of the number of C-H and C-D bonds.

I

56

plotted against the number of C-H or C-D bonds, form good straight lines passing almost through the origin. This result is given numerically in Table V-3; we conclude that, for the methane stretching modes at least, the concept of an absorption intensity per bond is certainly useful within

TABLE V-3 • Absorption intensities per bond.

Molecule r c- H t C-H (cm^/mole) ^C -D /"C-Dobserved calculated observed calculated

ch4 580 559 -ch3d 591 559 304 327

ch2d2 598 559 307 330

chd3 628 558 322 333

cd4 - mm 326 337

b) Limitations of the Modelt

Sverdlov (1961) attempted to improve the intensity theory by allowing for interactions between bonds, i.e., contributions to the bond dipole due to stretching and bending of adjacent bonds. The symmetry of the deuterated methanes is such, however, that the resulting equations for

] all reduce to expressions in two parameters, each a linear combination of the ^


57

Moreover, the coefficients of the two parameters are identical to those for the nzero*th order1* model. Thus the first order approximation cannot improve the fit given in Table V-l, but rather warns us that the two parameters are characteristics of a group of interacting bonds rather than a single bond. Such a warning is particularly apt when we attempt to transfer bond properties from one molecule to another [ c f . Hornig and McKean (1955)J.

Closer examination of the results in Table V-3 reveals two trends which are apparently larger than the experimental error of + 2% in the values of rj, for the stretching modes, i) There is an increase in absorption per bond in both 3000 cm"-*- and 2200 cm"* bands with increasing deuteration of the methane. This effect is so far unexplained, although it may be connected with the decrease in ground state C-H and C-D bond lengths with increasing deuteration, tabulated by Lepard (1964)• AA) The C-H stretching modes have intensities higher than predicted by theory by an average of 6 . 7 whereas the C-D stretching modes are an average of 5»'3% lower. This interesting pattern suggests that mechanical anharmonicity may be responsible for a significant part of the deviations. In support of this is the fact that the bending modes, which would be expected to show lower anharmonicity, have lower deviations


58

which are apparently random, and comparable with the experimental uncertainties of +

A rough estimate of the contribution of mechanical anharmonicity to the intensities, obtained from the classical amplitude of motion at 3000 and 2200 cm*"1 for a harmonic and a Morse oscillator gave a correction of ~ 1% to the ratio .^3000 t ^2200 * experimental discrepancy with the har

monic theory is ^ 14% •t

Evidently mechanical anharmonicity alone is insufficient to explain the observed trends. A more complete treatment, considering higher order terms in both the dipole moment and potential energy expansions, has been done for diatomic molecules by Crawford and Dinsmore (1950) and others.An indication of the difficulty of such a treatment even for the simplest polyatomic case here, namely CH^ and CD^, can be seen from the folloiying outline.

We write an anharmonic vibrational wave function for, say, the mode as a linear combination of harmonic oscillator wave functions each of which has the same symmetry species F2 as the principal term (0010). Thus ^3 *= 380010 (0010) +3a0001 iHoOOl) + 3a1010 (101°) + 3*1001^ (1001) + 3a2010 (2010)+ 2 a 2 0 0 1 ^ (2001) + ••• We now exPress the diPole moment in a Taylor series in the active normal coordinates, since, by symmetry, terms involving the inactive coordinates and Q2 cannot


59

contribute; + + ( i£-) B4

#,) 2i!♦ Lik-)zl + (Jh3 / 0 2! \ dfi3 dfi4/0 ( ^

Writing (j) (0000) = ^ 0 and using tfie selection rules forthe Hermite polynomials, we obtain for this simple case,( f o \ l ^ 3 > ” 3a0010 (f'(oooo) I a 3 I j ) ( 0 0 1 0 ) ) ( j £ - j

+ 3a0001 (0000) J fi4 | ^ (0001)) as the transition

dipole moment. The integrals ^0fQ^|l^ are of course known.The first-order derivatives g'g'." j have been calculated in

1 osolving the harmonic approximation to the intensity problem,whereas derivatives of higher order (- \ . which appear\ d 2 j /0for C^v and C2V molecules, can be expressed in terms of bonddipole anharmonicity parameters. The evaluation of theiavjVzV3V4 coefficients, however, in terms of experimentallymeasurable quantities, i.e., the anharmonicity constantsis a difficult problem, demanding in general a solution of the secular equation for a polyatomic molecule with anharmonicforce constants.

c) The separation of overlapping bands

In a number of cases it has been found possible to separate the absorption contours of overlapping bands, using the methods of Chapter III. In this section we shall compare


60

intensity ratios for several such bands with values predicted by the harmonic intensity theory of section a) above.

i) Methane-d1# the 3000 cm"1 region.

The total intensity of the pair 2 V 2 and in Fermi resonance (see Fig.IV-5) is expected to be the same as that of the unperturbed fundamental, or very slightly higher to allow for the "natural1* intensity of 2 V2* The obviously low intensity of the pair relative to V 3b allows us to assume a relatively unperturbed high frequency wing for the 3021 cm"1 band. Then, by assuming the Boltzmann relationfor intensities (cf. Appendix IX), an estimate of the contribution of V 35 alone to the band below 3021 cm”1 may be obtained. The resulting ratio of absolute intensities

Pv3b / P p3a + 2 was 9*98, compared to a theoretical prediction of 7.28. The disappointingly pooi' agreement is characteristic of cases where one band is much more intense than the other. Thus a slight asymmetry in the strong V ^

band produces a disproportionately large error in the estimated contour of the weak band. From the sign of the discrepancy between the two ratios, we conclude that rotation—vibration interaction in the V 3b transition tends to increase the intensity at higher frequencies.


61

ii) Methane-d2, the 3000 cm"*1 region.

The C-H stretching region was separated into a B-type contour centred on 2976 cm-1 and a C-type band at 3013 era”*-. One component is shown by the dotted curve in

were obtained and averaged, giving I.87 compared to the theoretical value of 1.877* In view of the wide scatter in the experimental results, such close agreement is largely fortuitous.

iii) Methane-d^, the 1400-900 cm”I region.

As described in Chapter XV; the bending mode region of CD^H was separated by dividing it, somewhat arbitrarily, at ll80 cm*"I. The experimental intensity ratio I”1 / f~\>2 obtained from the extrapolated absol

ute intensities was 2*84^ } the corresponding theoreticalIratio is 2.890. Again the agreement is good considering the experimental scatter.

Fig.IV-21. Intensity ratios /


CHAPTER VIPRESSURE EFFECTS IN THE SPECTRA OF THE METHANES

The influence of intermolecular collisions on line and band contours has long been an active field of research. Particularly for polar molecules, accurate data has been collected on line shapes and variations of line width with density and with rotational quantum number. More recently, microwave studies have begun to probe in detail the collision- induced transition probabilities between various rotational states. Such studies are performed at densities sufficiently low that the rotational lines are well separated. In the present work, however, even at the lowest densities employed, only a vestige of the free-molecule fine structure remains.It has been common practice to fit such a spectrum by assuming a line shape and simply adding the contributions from each line in the band. Armstrong and Welsh (i960) studied the fundamentals of CH4 din this way; they found good agreement between dispersion-broadened stick-spectra and the experimental profiles at He densities below ^300 amagat, although the fit for was inferior to that for V 3? The best-fit line width A varied linearly with density and extrapolated to somewhat less than the spectral slit width at zero He density. Within experimental error, the slopes d A / dyO were the same for V z and although the A values differed.

- 62 -


63

We have repeated the calculation incorporating the spectral slit-width correction described above, with stick spectra set up as in Appendix III. The results for the V 3 and bands of CH4 and CD4, and for the P ^ a band of CD3H are summarized in Table VI-1. Half-width A denotes the full width of the rotation line at the half-intensity level.

TABLE VI-1. Variation of half width with He density

Molecule Transition d A / dyO (cm"1 / amagat) />(/>_! o)(cm ± )

ch4 »3 0.036 1.4V4 0.033 1.5

cd4 »3 0.024 0.70.027 3.3

CD3H 0.030 1.1

The low but non-zero line widths at zero density may reflect either imperfect correction for spectral slit width or a real non-linearity in the curve at intermediate densities. The broadening coefficients for the three molecules appear significantly different, but within each molecule, different modes yield the same constant. The values obtained are comparable with those for the broadening of


64

the rotational Raman lines of oxygen and nitrogen by helium j JammUj St.John, and Welsh (1966)J • The fits obtained with dispersion lines are not good, however. As Armstrong (1961) observed, with increasing He density, the experimental profiles show an increasing intensity in the Q branch over the best-fit theoretical contours.

Gerritsen and Heller (1965) have measured the pressure-broadened line widths of a P(7) transition in the

band of CH^, using a timed laser spectrometer. With helium as the perturbing gas at very low pressures, they found d A / dp = 12.0 x 10**5 cm”^ / torr or, assuming the ideal gas law, d A / «= 0.0847 cm“* / amagat. This issignificantly larger than our results for CH4 which, however, agree with those of Armstrong and Welsh (i960) within the estimated error of + 0.003 cra" / amagat. The discrepancy cannot be ascribed to the averaging process inherent in our method of finding the line width, since in the non-polar and non-resonant CH^-He system, the change in A with rotational quantum number should be minimal compared, to that observed for self—broadened polar molecules. We conclude that for high foreign gas densities and closely overlapping transitions, the line width giving a best fit to experiment is not simply the extrapolated value derived from the low-density broadening coefficient.


65

a) The rotational autocorrelation function.

The rotational autocorrelation function approach developed by Gordon (1966a, 1966b, 1968) and others offered the hope of interpreting pressure effects on the band profiles in an intuitively satisfying way. The method is particularly simple when the pressure-induced line shifts are small, as is the case for methane-helium £cf. Goldring,SzBke, Zamir, and Ben-Reuven (1968)J.

We define the rotational autocorrelation function by C(t) ■» </(0) (t)^ where J i (t) is the Heisenbergoperator for the direction of the transition dipole at time t for the ensemble of absorbing systems, and the brack- . ets denote an average over a Boltzmann distribution of initial states. In the absence of coupling between neighbouring absorbers, an assumption certainly valid for our non-polar dilute mixtures, cross terms involving the transition moments of different molecules average to zero, and we may consider yi (t) as operating on a single active molecule. Gordon has shown that for isotropic systems interacting weakly with the radiation field, i.e., no non-linear or population—inversion effects, C(t) is just the Fourier transform about a frequency origin of the normalized spectral intensity I( V ) *= I(^) / J ^ w h e r e I(P) c i ( V ) / (l - e**hc V I kT). For the spectra considered here,


66

the induced emission factor in brackets is negligibly, different from unity• In the absence of rotation—vibration interaction, C(t) is a real symmetric function of the time, whereas I( V 0 + S’ ) and I( V>0 - £ ) differ by the factor e-hc / kT# is thus convenient to define a Boltzmann- symmetrized intensity I( V )e“^c( ^ “ ô) /2kT# imag..inary component in the Fourier transform will thus reflect the presence of rotation-vibration interaction [ Armstrong, Blumenfeld, and Gray (1968) J o r of pressure-induced line shifts* The spectra studied in this section have, therefore, been specifically chosen to have small asymmetry about the band origin and only the real part of C(t) has been considered. Gordon (1963) has shown how the effects of intensity asymmetry may be partially allowed for by the use of a shifted band origin V 6 instead of V Q in the Fourier transform*

Before attempting to interpret experimental envelopes in terms of autocorrelation functions, it is well to have a clear understanding of the form of the curves for free rotors* Calculations of C(t) have been made for linear [Gordon (1968)J and spherical rotors [steele ( 1 9 & 3 ) ]

in a classical approximation. Appendix IV consists of a paper in which it is shown that the presence of discrete rotational quantum numbers and the associated nuclear spin statistics have an appreciable effect on the autocorrelation


67

function (and on the spectrum), A simple calculation is presented of the form of C(t) for the active transitions of CH4 and CD4# The vibrational angular momentum associated with these degenerate vibrations is approximately allowed for by using B(l - fe ) instead of B as the rotational constant, where £ gives the magnitude of the mean vibrational angular momentum along the J direction in units of*h •

In addition, we have calculated C(t) for parallel and perpendicular bands of free symmetric rotors. The selection rules are introduced by appropriately choosing the direction of the transition dipole moment J * . in molecule- fixed axes. The cosine of the angle between this direction at time 0 and at time t is then calculated, the molecular rotation being described by quantum numbers (j,K), Finally, the result is averaged over a Boltzmann distribution of ground state rotational energy levels, allowing for nuclear spin statistics. The method may be clarified by reference to Fig.VI-1,

Fig,VI-1, The geometry of a symmetric top.


6 8

For a parallel band, the transition dipole Jix is parallel to K, the figure axis direction; for a perpendicular band, y*<.2_L K. The molecular motion is a combination of rotation about K at frequency U = 4TTc(Bw - B JlCfc and nutation

-A —xof K about the space-fixed J direction at frequency CO j - »4 TT cBx Y j T T T T J • Here B n and Bx are the rotational constants about axes parallel and perpendicular to the figure axis as in Appendix 111* The angle 0 between K and J is given by cos 0 ** K / l / j ( j + 1),

For a parallel band yt^ has a component *= cos 0-Xwhich is unchanged by rotation about J and gives rise to the

Q branch, and a component y^x “ sin 0 which rotates at angular frequency Ct)j. Thus C(t)s ^/t (0) •yM. (t)^ *=O V o ) y * (, (t) + =((ju& t j ) + f1 - cos *»■ aparallel band of a symmetric rotor. The average implied by the angle brackets is over the ground state distribution n(J,K)given in Appendix III.

For a perpendicular band, rotates about K atjangular frequency CO as K simultaneously precesses about J

at frequency ( O j. By defining ©j - ^jt, © k “ andapplying spherical trigonemtry, it can be shown after a good deal of manipulation that C(t) *=^cos © j c o s © K - sin © j sin 0 K cos 0 + £(1 - cos 0 j)Xcos 0 K sin2 q \ where, as before, cos 0 « K / yj(J + l).


69

Again, since the perpendicular bands are (doubly) degenerate, there is internal angular momentum which lies along the_x

K axis. In a fundamental band, for which i. « + 1, we allow for this approximately by taking = 4itc[b(|(1 - £ ) - bJk*.

Pig.VI—2 shows the resulting curves for perpendicular and parallel bands of CH3D and CD3H. In the approximation used here, all non-degenerate bands of a given molecule have iden txcal spectra and autocorrelation functions5 degenerate bands may differ through the Coriolis constant £ . Superimposed on the above figures are dashed curves obtained from the Fourier transforms of stick spectra. In all cases, the factor e*“kc( ^ " ô) / 2kT has been employed to remove most of the intensity asymmetry before Fourier transformation. It can be shown that the long-time behaviour of C(t) is primarily determined by the sharp Q branch near the frequency origin. The principal difference between the semiclassical C(t) and the stick-spectrum cosine transform is the decrease of the latter towards zero at long times. We conclude that this reflects the broadening of the Q. branch through rotation- vibration interaction; in the approximation of the semiclassical calculation, the Q branch is a r -function at the origin, giving rise to a constant in the Fourier transform.

We are now ready to examine the effect of pressure on the free-rotor curves shown above. Fig.VI—3 contains a selection of autocorrelation functions obtained from


Reproduced

with perm

ission of the

copyright ow

ner. Further


without

permission.

PERPENDICULAR BANDSPARALLEL BANDS

1.00

3 a.75

.0010 155 50 10 15

Fig.VI—2. Rotational autocorrelation functions for some parallel and perpendicular bands of CD3 H and CH3 D. The dashed curves are the Fourier transforms of stick spectra. The results for CD3 H have been shifted for clarity.

70

experimental profiles. The long-time behaviour is approximately exponential [cf. Armstrong, Blumenfeld, and Gray

than linearly with increasing density, and extrapolates at zero density to a value consistent with the free rotor Q-branch width. Interpreted in terms of an equivalent Lorentz broadening coefficient, the low-density slope of the decay constant

to the best-fit value quoted above. Extrapolated to t *= 0, the exponential decay has an intercept of ^ 0.35j the relative intensity of the Q branch is ^ 0.33 for a spherical rotor. No consistent effects of the increase in Q-branch intensity with density are observed, however, due to the sensitivity of the extrapolation to errors in the decay constant.

One observes that the short-time peak in C(t) is very slightly broadened with density. If a dispersion- broadened line spectrum truly described the pressure effects, one would expect a correlation function of the form C (t) = e-t /T CQ(t) where CQ(t) is the free rotor function and 1 / T = T c A . The correlation function would thus be everywhere smaller than the free rotor value. For short times less than ~'2.5 x 10“ 3 sec, this is simply not the case.

We shall now consider a generalization of the simple models of rotational diffusion developed by Gordon (1966a)

with a decay constant which increases slightly faster

is 0.0376 cm**1 / amagat for the band of CH^, very close


wo

1.0

0.5

, ' I 1 1"..

1 . C H 4 z/3 \ CD4 Z/3

’ 1 He D E N S IT Y 1 He D E N S IT Y1 ( a m a g a t ) \ ( a m a g a t )

\ — 2 2 0 I — 2 2 0

- \ — 43 7 V " ~ 4 3 1

I / ^ >Sss^ ^ — 8 74 \ — 872V / ...........

sy _

U l* X x ^ .V' / ' x

10Fig.vi-3.

10t(I0”,3sec)

Observed autocorrelation functions for the *22, bands of CH. and CD..J 4 4

10

71

and show how some of the features of observed autocorrelation functions may be qualitatively explained.

b) The ci-. .1-. and m.i-diffusion models

The treatment is basically an impact approximation, which should be valid for such short-range forces as those between methane and helium; in addition both molecules are of low mass, so the thermal velocities are high. We assume free rotation of the absorbing molecules in between instantaneous collisions which reorient the angular momentum direction; the collisions may or may not change the magnitude of J and K. Let the effective collision rate be 1 f"Xt considered constant for all rotational states; then e“^ !

gives the relative number of molecules which suffer no collisions for a time interval t, during which they rotate freely. The corresponding contribution to the rotational autocorrelation function is simply C(t) *» e“^ C0(t),that is, the spectrum is the convolution of the free rotor spectrum with a Lorentzian line of width 1 / "ff c T.

However, the contribution toÂ(O) . J i ( t)> from molecules which suffer one or more collisions in time t is not negligible at high collision rates. We consider a unit transition dipole /t(t) fixed in the molecule and denote the spherical angle between the directions J l (0) and J i (t) by


72

Thus for a spherical rotor, cos 0 jK(t) »T/3 "*■ (2/3) cos CO jtj for a parallel transition of a symmetric top, cos 0 JK(t) *=K2 / J(J + l) +£l ~ K2 / J(J + l) J cos 60 jtj for a perpendicular transition,C°s 9 jK(t) - cos 60 jt cos co Kt - sin 60 jt sin 60 Rt XK / /j(J + l) + £(l - cos 60 jt) cos £0 Kt £l - K2 / J(J f l)J.The angle 0jK(t) may now be treated identically to that fora diatomic molecule as considered by Gordon (1966a). Thus ifcollisions randomize the J direction, one obtains a contribution

A1 / t j cos 0 JlKl(t ) cos 9 j 2K2(t - tx) dtjL (1)

for molecules which rotate in state (Jj.Kx) up to an arbitrarytime tj[ < t, and then rotate in state (J2K2) from tj until t.Using a Poisson distribution for the probability of n collisionsin time t. one obtains a generalized form of (l), namely

A ■

C(t) = <e**t/T r T ”nJdtn cos 0 Jn+1Kn+1 (t - tn )An n 0Jdtn_i cos 0 jnKn (tn - tn_i)...

^ 2dtx cos 0 j2k2 (fc2 " *l) cos 0 JiKiC^l) (2)as the complete autocorrelation function.

In reality, collisions will induce transitions between rotational states with'a definite average probability for each pair of states (JjK)~>(J + A J, K + A K). These probabilities are as yet unknown, however; Gordon has suggested two limiting cases for the relationship between successive states. In the m-diffusion model, all collisions are


73

rotationally elastic, so that (•*!%) *= (J2K2) = ...«=(Jn+lKn+l) an<* we may write C(t) = e“t/'t'£^Cr-n Cjn n(t).a *In the j-diffusion model, all memory of the state is erased by each collision; for this case we may average over each factor in the convolution integral (2) separately to obtain an expression involving the free rotor autocorrelation functions directly. Thus equation (2) becomesC(t) ** e £dtn CQ(t — tjj) ^ dtn_2 “ ^n-l)»*»

= e“t/T Z T “n Cj,n (t) ° (3)for the j-diffusion case.

A slightly more realistic case is that in whicheach collision either leaves the state (JK) unchanged orrandomizes it completely, with probabilities Pm and Pj forthe two cases. A molecule may suffer N collisions in 2^ ways,each with a probability Pm**13 where Nm + Nj *= N. For eacharrangement of elastic and inelastic collisions, we can writethe contribution to the autocorrelation function in terms ofpure m- and pure j-diffusion contributions Cm>n(t) and Cj#n(t).For example, for N «= 3 and (JK)l *= (JK)2 (JK)^ 5

2 /tthe contribution to Craj j(t) is C 0(t-t3)dt3 •JqOne then simply adds up the contributions from all possible arrangements, each with appropriate probability, to obtain a mixed raj—diffusion rotational autocorrelation function.

In performing the calculations, use was made of the fact that the geometrical angle @jk is essentially equivalent to 2 TTc(Vjk - P D)t, where )JjK is the line frequency


74

originating from state (JK). Thus, by using the Fouriertransform of a stick spectrum, with the intensity oL ( y rv.)

JK'

replacing the population of the initial state nJK, one loses the geometrical significance of the model but gains accuracy and generality in that asymmetries in the band due to rotation-vibration interaction may be allowed for. Collisions in this way can be thought of as transferring intensity between lines in the band, rather than simply causing transitions between rotational states. The three cases of m-, mj-, and j-diffusion have been calculated for up to 4 collisions and for various values of the mean time between collisions; as Gordon (1966a) has mentioned, when evaluated numerically, the convolution integrals reduce to a form of matrix multiplication. The results for the band of CH^ are shown in Fig. VI-4 with

= P. ■= 0.5 for the mixed diffusion curves, m JThe calculated curves show several promising features.

The j-diffusion model gives an exponential long-time behaviour frora 5 to 15 x 10”*3 sec although the mj— and especially the m-diffusion decay is more nearly linear at very long times.The decay constants increase approximately linearly with 1 /'t'Q. Interpreting the low—density broadening coefficient of Gerrit- sen and Heller (1965) as a nominal collision rate 0.930 x 1010 sec”1 / amagat at room temperature, we can relate the value of 1 / T 0 to the foreign gas density. We find the rate of change of the long-time decay constant with density


Reproduced

with perm

ission of the

copyright ow

ner. Further


without

permission.

1.0

CH

T0= 5.0 x I0 “,3sec

0.5

O

10t (10 ” sec)

Fig.VI-4. Calculated correlation functions for ^3 of CH4 in the m-, mj-, and j-diffusion approximations at two values of mean time between collisions T0.

75

is the same for all three models, and for four collisions is equal to -"0.61 x 1010 sec”1 / amagat. Since the j-diffusion model gave the best fit to experiment, calculations were also performed for N «= 9 in this approximateion (Fig.VI-5).The rate of change of the decay rate decreased to 0,58 x 1010 sec"* / amagat, still much larger than the experimental value of 0.356 x 10 -® sec“^ / amagat, but appreciably smaller than that expected from the nominal collision rate and simple dispersion lines. The curves shown in Fig.VI-5 are simply the best-ft calculations, disregarding the expected relation between "X0 and p via the nominal collision rate.

1 / T for the m-j model are intermediate between the values for pure m- and pure j-diffusion j in the range 200-500 amagat* the mj model lies close to the observed values but rapidly diverges at higher densities due to the high value of the broadening coefficient* We note in addition that the value extrapolated from the exponential region to t *= 0 for all the models increases rapidly with densityj a slight increase would be consistent with the observed increase in the Q

All three models give a broadening of the short- time peak in C(t) with increasing density, but the broadening

It is also observed that the actual decay rates

branch at the highest densities £cf. Armstrong


obs. 220amagat

' obs. 7l8amagaf

calc. r0=7.0xlO",3sec\+;o

.05

calc. T0=2.0xl0",3sec.02

Fig.VI-5. Comparison of observed and calculated autocorrelation functions for the Vj band of CH^ using the pure j-diffusion model with N *= 9«


76

is larger than that observed experimentally. The principal failing of the calculated functions is in the region of the "dip*1 near 2.5 x 10-13 sec, where all three models are "filled in" much more than the experimental curves. It is difficult to know how to remedy this; the inclusion of higher order terms will probably make the discrepancy worse.

We conclude that the rotational diffusion models developed by Gordon and generalized in this thesis explain at least qualitatively many of the effects observed in experimental autocorrelation functions. The detailed knowledge of collision-induced rotational transitions desirable for significant improvement in the theory may be difficult to acquire, however, since the methanes have no dipole moment and hence no free molecule rotational spectrum to be studied using double-resonance techniques.

c) The variation of integrated intensity with density

We finally consider the interesting behaviour of the integrated intensity with increasing helium density. As is now apparent, all the stretching mode fundamentals show a linear decrease of I""7 withJO , of the same order of magnitude. The average TT observed for the 3000 cm"-*- bands is


77

1.50 + 0.15 x 10~4 / amagat j for the 2200 cm"1 bands, the average is 1*55 0«12 x 10 4 / amagat. We conclude thatthere is no significant difference between the variation of I""1 with p in the two stretching frequency regions.Close examination of the data plotted in Chapter IV reveals that the decrease in I""1 is slightly more rapid at low densities than at high.

Although the experimental scatter is larger for the bending modes, these too show a pattern in the variation of r* withp : the initial decrease at low densities is followed by a region in which n either remains constant or increases slightly.

In dilute methane-helium mixtures, it should be possible to express the integrated intensity in a power series in yOjje. The constant term is just P ,, the free molecule absolute intensity. The second term in the expansion represents induced effects which are linear in the perturber density. Apparent induced absorption arising from the interaction of the transition dipole with dipoles induced in the foreign gas by the. radiation field is just such a linear induced effect. Because of the low polariz— ability of helium, the linear increase^ calculated according to Van Kranendonlc (1957), is only ~ 1 % in 1000 amagat; this is negligible compared to the variations observed here. For both stretching and bending modes, the observed variation of


78

I""*withyO implies the existence of small positive terms quadratic in the density, with those for the deformation modes somewhat larger.

Although there is no detailed theory for these effects, we will here suggest a possible line of attack, namely, an interference effect between the dipole induced during collisions and that of the free active molecule.In symmetry-forbidden bands, such as the fundamentals of homonuclear diatomic molecules, the only non-zero term in the transition moment is the dipole induced by an asymmetry in the environment; such an induced moment leads to an increase in the intensity, which is given by the square of the transition moment. For an active band, however, the total dipole is the vector sum of the free-molecule moment and the induced moment. Due to the absence of low-order multipole moments for methane and helium, and the small polarizability of the latter, it seems reasonable that the principal induced dipole will be that due to overlap forces. As the active molecule vibrates, the periodic distortion of its electronic structure must result in a modulation of the induced moment; the modulation, moreover, must be in phase with the vibrational motion, i.e., must reach a maximum on each cycle as the vibrational distortion reaches a maximum. Thus, taking the position of the He nucleus as r in a coordinate system fixed in the methane centre of mass,


79

we can write the induced overlap moment as AL (R.O.) *= K o ( K - ) + ( Qi / ^ i + •** wêre the subscript o denotes the methane equilibrium configuration. The first term is modulated by the translational motion only and will not contribute to the vibrational transition moment. The second term will add vectorially to the free-molecule contribution (’ggjj &i> we find that the intensity is proportional to

Now each normal mode of the active molecule belongs to a definite symmetry species of the molecular point group; the modulated component of the induced dipole, averaged over

• Aall orientations of R, will reflect this same symmetry species. Thus the induced factor in the second term above will not average to zero, but by symmetry will lie parallel or anti-

and the properties of the overlap-induced dipole. Wien we further average over all separations R, using the pair distribution function, the second and third terms above will give, to first order, contributions linear in the helium density. Since the total effect observed is small ( ^ 1 5 %

in 1000 amagat), we expect that the squared term in the induced moment is negligible compared to the first-order term. From the fact that all stretching modes display


depending on the form of the vibration

80

approximately the same percent change in intensity with

density may arise.from ternary collisions or from the behaviour of the pair distribution function at small separations and high densities. It is, of course, precisely the behaviour at small separations which determines the effect of the overlap dipole.

density, we conclude that the averaged values of ( ' R I •*-s ProP°rtional to (m£rm'>) for these i\— 5Q • / ‘*’S ProPor^ôna^ to ôr these modes.Terms in the expansion of f""1 which are quadratic in the


80

approximately the same percent change in intensity withdensity, we conclude that the averaged values of

if

( is ProP°rtional to (“If:— J for these modes, i/o

Terms in the expansion of f""1 which are quadratic in the density may arise.from ternary collisions or from the behaviour of the pair distribution function at small separations and high densities. It is, of course, precisely the behaviour at small separations which determines the effect of the overlap dipole.


APPENDIX I

UNITS FOR ABSORPTION COEFFICIENT

It has been remarked [Crawford (1958) ] that theintegrated intensity has the dimensions of a cross section(cm2 mole-* ) when expressed by the equation

P = ( 2 2 4 1 4 / L 1/V) Hn [I0(P ) / I(y )] dVfor L in cm, y0 in amagat (dimensionless) units, and in cm”*-. The corresponding absorption coefficient

* ( V ) = (22414/LyO) ( l / y P ) £n [l0(V) / I(V)]5 ^ has then by analogy the dimensions of a differential cross- section cm2 mole”*- per cm*”*-. for absorption of radiation in the frequency range ( V, V + dV ), When divided by Planck*s constant h, cL (V ) gives directly the molar transition rate per unit radiation energy density at frequency , that is the Einstein coefficient for absorption or stimulatedemission. This is easily shown as follows.

Let the incident radiation energy flux be I( V ) erg cm™2 6ec“^ per cm“^# After passage through a length dX of absorbing medium, the flux is reduced by dI(P) I(P )){ (V) dx where K ( P ) is the absorption coefficient per unit path length. Thus the energy absorbed in unit time in a disc of unit area and thickness dx via transitions with energy difference Ef - E± «- hcV must be just dl(P); the net upward

- 81 -


82

transition rate per unit incident flux per unit volume is then K ( P ) / hep . Now the number of moles of absorbing medium per unit volume is p / 22414# and the radiation energy density is 1 / C per unit incident flux* Thus the net transition rate per mole per unit energy density is K ( v ) 22414 /^°h, But assuming Lambert’s law over the entire path length L, X ( V ) =<1 / t ) Hn [la(V) / ! ( » ) ] . Taking account of induced emissionfrom the upper state of statistical weight gf^ relative to theground state, we obtain(22414 /yOL)(l t V ) In [lQ(p) / I(P)] = 3. (P ) =hB^(l - ^ k'*' )* At room temperature, kT / he 200 cnT-*-*in the spectra of the methanes, there is no significant absorption below ^ 900 cm”-*-. Thus the correction factor in brackets differs from unity by a maximum of ^ 1% and will be neglected.We have then finally d ( P ) » hBfi, the desired result.


APPENDIX IITHE BOLTZMANN RELATION IN THE INTENSITY DISTRIBUTION

OF ROTATION-VIBRATION BANDS

We give here a discussion of the validity of theBoltzmann relation for intensities, used in Chapter III toseparate overlapping bands.

Consider two vibrational levels v *= 0 and v = 1,each of which consists of a set of rotational states (J, K).In the absence of rotation-vibration interaction, the. spacing of the (J, K) levels will be identical in the upper and lower vibrational states; thus we can write E(v, J, K) / he *= G(v) + F(J, K) where F is a function independent of v. The frequency of a transition (l, J + A J, K + A K) (0, J, K) will be given by 1 0 + ^ J> K + AK) - F(J> K) S r whereV Q *=» 6(1) - G(0); the transition (1, J, K)<--(0, J + AJ,K + A K) will occur at "1?0 + F(J, K) — F(J + A«J, K + A IC) *■* ^p. Obviously - V Q S V *= ~ ^ p •

Ignoring induced emission from the upper vibrationalstate, the absorption intensity for the transition is pro

portional to JÔ 1^ I 1>I 2 fj + A J, K + A K^-J, K^J, K where the first factor is the transition dipole moment, the second is the line strength, and the third is the population of the initial rotational state. For the V p transition, the

corresponding expression is

|<0 |/ I 1>| 2 J + A J, K + A K n j + A J, K + A K.

- 83 -


84

The first factor is evidently the same for all transitions of a given band, and does not figure in their relative intensities. The line strengths are squares of direction-cosine matrix elements between rotational states, and therefore

fJ + A J, K + A K «- J,K " fJ,K<- J + A J, K + A K»Thus the only difference between the intensities of V R and

)?p comes from the different populations of their respective initial states.

Now rij# £ is the product of a nuclear spin degeneracy gj, j£, an orientational degeneracy dj^ and a Boltzmann factor e F0(JK)hc/kT^ Both gj^ and dj^ ^ are irregular

functions of J and K. In the classical limit however, as J and K become large, the distribution gj^ ^ dj^ K becomes a smooth, and more important, a slowly varying function of J and K compared to the Boltzmann factor. To the approximation

that gJ# K djf k & gj + A J, k + A IC dJ + A J, K + A K* wethen have IP / IR *= fij + a J, K + A K /n J, K **

-hc[>o(J + A J, K + A K) - F0(J> K) J / kT B e-hc y? / lcT,

which is the Boltzmann relation for intensities*For the experiments described in this work, finite

spectrometer resolution coupled with strong foreign—gas broadening ensured that only a rotational envelope was ob

served. Thus abrupt changes in £jf k ^J, K ôr ne:*-Shboring lines were effectively averaged over a frequency range containing several transitions. To assess the validity of the


85

Boltzmann relation in the intensity distribution, the following procedure was adopted. Stick spectra were set up for the

bands of CH^ and CD^, using the frequency and intensity formulas of Appendix III and constants derived from high resolution experiments. The stick spectra were broadened with a dispersion line shape of width 10 cm”1; the resulting P-branch profiles <^p( & V ) were then multiplied by the factor ehc A3? /kT^ where A'i? is the frequency shift from the band maximum, and compared with the R-branch profiles The use of the Q-branch maximum instead of the transition frequency partially allowed for the effects of rotation-vibration interaction. The results are shown in Fig. AII-lj agreement is seen to be reasonable.

We conclude that for upper and lower state B-values differing by 1$, the Boltzmann relation is valid to ^10% in the region of the P— and R—branch maxima. Although the percentage error increases with increasing A!? , due to progressively larger differences between + ^ andAVj + jl <_ j,the effect on the profile is small since the

actual intensities are rapidly decreasing. Resonances with nearby bands and large rotation—vibration interactions may make the Boltzmann intensity relation a progressively poorer

approximation.



APPENDIX III

ENERGY LEVELS AND LINE INTENSITIES

In this section we present a summary of energy level expressions, selection rules, and line intensity formulas for spherical and symmetric rotors, and a more general discussion of the symmetric rotor.

a) Methane and methane-d^

Tetrahedral molecules have three equal principal moments of inertia and are termied spherical rotors. Ground state rotational energies are given by

F0(J) = B0J(J + 1) - D0J2(J + l)2 where D0 is a correction for centrifugal stretching and B0 is, apai't from constants, the invei’se moment of inertia in the ground state. In a general vibrational level v = (vi V2.»»

•*.v3N-6) one can write Bv = Be “ °<i(vi whereis the degeneracy of normal mode 11 i11 and Be is the rotationalconst nt in the equilibrium configuration.

In CH4 and CD4 the active vibrational modes are triply degenerate; in consequence of Coriolis interaction between rotation and vibration, each degenerate state splits into three sublevels. One component remains unchanged; the other two form linear combinations with components of angular momentum + along the J direction. The energy levels are

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87

F°(J) = BVJ(J + 1) - Dv J2(J +1)2 F+(J) = F°(J) + 2Bv £v (J + 1)F“ (J) = F°(J) - 2By£v J

The rotational selection rules areAj = 0, + 1, provided that AJ = +1, 0, and -1 transitions terminate on F“, F°, and F+ levels respectively. The spectrum consists of three'series of lines, the 11, Q, and P branches; the spacing in the P andR branches is ~2B(1 -§ ). Jahn (1938) has considered in detail the further splitting of the V 4 rotational levels due to Coriolis interaction with the neighboring V2 (E) mode.

Each line intensity is the product of two factors, one giving the population and degeiaeracy of the initial rotational state, the other involving squares of direction-cosine matrix elements giving the transformation from molecule-fixed to space-fixed axes. For tetrahedral molecules, the first factor is (2J + l)^ (naa * nee * n^f)e Fo(J)l c/ . ^QV . ident

ical nuclei of spin I = a = 5* e = 2, f = 3j for 1 = 1, the corresponding values are a = 15> e = 12, f = 18. The level symmetry species naA + nÊ + njpF is given, for instance in Herzberg (1945). The direction cosine factors are

(2J + 3) / (2J + 1), 1, and (2J - l) / (2J + 1)for the R, Q, and P branches, where J is, as always, the quantum number of the initial rotational state.


88

b) Methane-dj and methane-d^

These two molecules, of C^y symmetry, are respectively prolate and oblate symmetric rotors with principal moments of inertia Ia < Ib = Ic, and Ia = Ib < Ic. The ground state energy levels are F0 (JIC) = Bo x J(j + 1) + (B0 || - B0 j_ ) X

- DqjJ^(J + 1)^ - Dqjj^J(J + 1)K^ - D0j^where IC is the com-

ponent of J along the symmetry axis and K = o, +1, + 2,...+ Jj each level for /kJ >0 is doubly degenerate. For the prolate rotor, Bj.= B, B n = Aj for* the oblate rotor, Bj_ = 3, B n — C. There ar*e three non-degenerate vibr*ations of species, and three doubly degenerate E-type modes, all infrared active.In the foi'mer case, only the component ofy“ IIK contributes to the intensity; in the latter case, only the component of

/iK.For parallel bands, the selection rules are (A J = 0,

+ 1,AK = 0, IC 7 0) and ( AJ = + 1, A K = 0, IC = 0). The energy levels for the singly excited non-degenerate state are precisely similar* in form to those above but with the upper state rotational constants By j_ , By a , etc. Thus the spectrum again consists of P, Q, and R branches, with the Q branch (A J = 0) closely concentrated around Vo and the P and R lines with approximately equal separation 2Bj_ . The line intensities are prod- ucts of a factor gK(2 -£K)0><2J + l)c-Fo(JK)hc/lcT a direo_tion cosine factor. For the spin of the identical nuclei I —


89

the nuclear spin degeneracy gK = 2 for 1C = 3n + 1, 4 for IC = 3nj for 1 = 1, gK = 8 for IC = 3n + 1, 11 for K = 3n.The second factor is, for parallel transitions, (J + K + 1) ><(J - IC + 1) / (J + 1)(2J + 1) for A J = + 1, K2 / J(J + 1) for Aj = 0, and (J + IC) (J - IC) / J(2J + 1) for AJ = — 1.

The degenerate E levels are split by Coriolis interaction into two linear combinations having angular momentum ± ^ parallel to K. The energy levels are Fy±(JK) = BVJJ(J + 1)+ (Bv || - Bvj_)K2+ By „ £v K - DvjJ2(J + l)2 - DvjkJ(J + 1)IC2- DyjJC . For transitions from the ground state the selection rules are AlC = + 1, A J = 0, + 1; transitions with AIC = + 1

4- —terminate on Fv levels, A IC = - 1 transitions on Fv levels.For each value of K and AIC, there are three branches A j =•f 1, 0, - 1. The frequencies of the sub-band centers are

V 0 Sub = V o + [ ,1 ( l - 2'fev)- B vx J ± 2. [ B v i i ( I - £ v ) - B vj. J K

+ [ ( P v I I - Bo,. ) “ (B v 4 . - Box ) ] } C f o r AIC = ± 1 .

D e p e n d in g on t h e v a lu e s o f B j _ , B |j , and £ , ' t h e r e s u l t i n g

s p e c tr u m m ay v a r y f ro m a s i n g l e s h a i’p c e n t r a l p e a k w i t h tw o

w in g s , c l o s e l y r e s e m b l in g a p a r a l l e l t r a n s i t i o n , t o a b r o a d

b a n d w i t h a c e n t r a l m in im um # T h e l i n e i n t e n s i t i e s a r e p r o d u c ts

o f gK(2J + l ) e “ F o ( J K ) h c / k T a ild t h e x i n e s t r e n g t h f a c t o r s

(J + IC + 1)(J ± IC + 2) / (J + 1)(2J + 1) f o r A J = + 1,A K = + 1; (J * 1C) (J + IC + 1) / J(J + 1) for A J = 0, AIC = + lj

and (J + IC)(J + IC - 1) / J(2J -I- 1) for AJ = -1, AK = ± 1.


90

c) Methane -d£

Calculation of the frequencies and intensities of asymmetric rotor spectra is a more complex problem than in the case of symmetric and spherical tops. Detailed treatments may be found in Ilerzberg (1945) and Allen and Cross (I963). -The only true quantum number is that for the total angular momentum J. In terms of the rotational constants A, B, and C, there are no simple closed-form expressions for the energy levels; there are however* tabulated values of E x (x ) for rigid rotors where X is an asymmetry parameter (2B - A - C) -r (A - C) and X is a running index from - J to + J. It can be shown that E (A, B, C) = ^ J(J + l) + — ET (X )j corrections for non-rigidity are analogous to those for symmetric rotors although more difficult to apply.

The rotational selection rule is A J = 0, + 1. Spectra may be classified as A, B, or C type according as the ti’ansitiondipole lies along the axis of least, intermediate, or greatestmoment of inertia. In CII2D2, B-type bands arise from transitionsto A^ species vibrations. (J*- parallel to C2 axis); C-typebands from Bjl vibrations {J* perpendicular to D-C-D plane), and A—type bands from Bg vibrations (y^- perpendicular to H-C-li plane). Badger and Zumwalt (1938) have derived band contours based on the integration of approximate intensity formulas. They show that B-type bands consist of a doublet branch


91

centred on }?0 with a broader weaker branch on either side. The A- and C-type bands have a narrow intense central peak (a S' function in their approximation), a weak branch on either side which is probably not resolved, and broader branches similar to P and R branches in appearance.


Reprinted from T in : J o u r n a l o f C h e m ic a l P h y s ic s , Vol. 49, No. 6, 2871-2872, 15 September 1968Prlnlcd In U. S. A.

Effect of Nuclear-Spin Statistics on the Rotational Correlation Functions of

Compressed Gaseous Methane*V. M . C ow an an d R. L. A rm s tro n g

Department of Physics, University of Toronto, Toronto, Canada

(Received 6 M ay 1963)

In a recent paper1 rotational correlation functions where i i jA, i i jh:, i t j f' are the nuclear spin weights for C(/) for CIIi in CH4“Hc, CIi|-Ar, and CH4-Nj.mi.x- , rotational levels of A, E, E symmetry, respectively.3 tures were deduced from infrared i/3 band profiles ob- For nuclear spin / = }-,« -5 , c = 2 ,/=3; for / = 1, a=15, tained at 295°K. For short times the behavior of C(l) agrees with that of a classical freely rotating spherical top.3 For long times C(t) decays exponentially to zero due to the action of intcrmolccular torques. The decay rate increases with increasing density and for a fixed density depends on the broadening gas. Superimposed on the general exponential decay is a pattern of small amplitude oscillations. Correlation functions for CD4 in CD4-IIe mixtures were also obtained. They behaved in a similar manner but did not exhibit any detectable oscillations. This oscillatory behavior can be explained by taking into account the nuclcar-spin statistics of CH4 and CD4.

For a quantum-mechanical freely rotating spherical top it is easily shown that

m = M { c o s i j ( j + i ) y i - h t / i ) , ■

where J is the angular momentum quantum number and / the moment of inertia of the top. The average { ) is over the thermal distribution of rotation-vibration states. Ignoring the small populations in excited vibrational levels, we have for a molecule belonging to point group I'd

Y . j (a„jA+ciuK+/njF) (27+1) cos I fJ ( J + D J W / I } exp [ - 7 (7 + 1W/21 kT] < ( a t iS + o i jK + J n j ^ V J + l ) e x p [-7 (7 + 1)fiV2^ ]


Fig. 1. Theoretical rotational correlation functions for freely rotating Cl I« ami CD , molecules at 295°K. The curves have been displaced vertically for clarity of presentation.

28/2 L E T T E R S T O T H E E D IT O R J. C H E M . P H Y S ., VOL. 49, 1968

T a d l k I . Theoretical and experimental times of the oscillation maxima and minima of C(t) i n CH«.

Positions 1 2 3 4 5

• Theoretical times (10 -" sec)

9 .5 11.9 14.9 17.1 20 .0

Experimental times (10-'* see)

1 0 .2 ± 0 .3 1 1 .9 ± 0 .3 1 3 .8 ± 0 .3 17.3db0.6 1 8 .8 ± 0 .6

e~ 12, / = 18. The functions C(t) evaluated for CH« and CDi at 295°K, for / values up to 35, are shown in I'ig. 1. Both C(i) functions show distinct oscillations; however, the oscillations of C(f) for CH* arc larger in amplitude and begin at a smaller value of /. I t is interesting to note that the amplitude of the oscillations is determined by the nuclear spin whereas the positions in time at which the oscillations occur are determined by the molecular moment of inertia. The first detectable oscillation maximum predicted for CD« occurs at f~20X10~ls sec. However, due to the action of intermolecular torques, the experimental C(l) functions in Ref. 1 are essentially zero for times of this magnitude and no observable oscillations arc predicted.

For He densities <400 amagat and for Ar and N* densities <250 amagat, the experimental CH« band profiles as measured in mixtures with He, Ar, and N5 can be deduced from the free CH4 spectrum assuming suitably broadened rotation-vibration lines.4 A study of the oscillations in the corresponding C(f) functions revealed that the times at which the maxima and

minima occur arc independent of the density and of the broadening gas. Average values of these times are given in Table I as well as the’ corresponding times as deduced from the theoretical correlation function. Theory and experiment arc in good agreement.

For higher densities the molecular rotations are interrupted so rapidly that the rotation-vibration energy states of the perturbed CH4 molecule can no longer be approximated by those of the free molecule. The perturbed energy states no longer possess such distinctive symmetry properties and the oscillations gradually die out with increasing density. This is clearly shown for CH»-Ni correlation functions in Fig. 5 of Ref. 1.

* Research supported in part by a grant from the National Research Council of Canada.

* R . L . Armstrong, S. M . Blumcnfcld, and C. G. Gray, Can. J. Phys. 46, 1331 (1968).

* \V . A. Steele, J. Clicm. Phys. 38, 2411 (1963).»W . A. Childs and I I . A . Jahn, Proc. Roy. Soc. (London) A169,

451 (1939).< R . L . Armstrong and I I . I.. Welsh, Spectrochim. Acta 16, 840

(1960).


if

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Allen, H. C., Jr., and Cross, P. C. 1963* MolecularVib-Rotors. (John Wiley and Sons Inc.,New York and London).

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Acta 16. 840.Badger, R. M,, and Zumwalt, L. R. 1938. J. Chem. Phys. j5, 711 Coulson, C. A. 1942. Trans, Far. Soc. j$j8, 433.

1959* Spectrochim, Acta 14. l6l.Cowan, V. M. 1964. M. A. Thesis, University of Toronto. Crawford, B. L. 1958. J. Chem. Phys. 2&, 1042.Gerritsen, E. J«, and Heller, M. E. 1965. Appl. Opt.

Suppl. 2 f 73*Goldring, H., SzOke, A., Zamir, E., and Ben-Reuven, A. 1968.

J, Chem. Phys. 49« 4253*Gordon, R. G. 1963. J. Chem. Phys. 12., 2|88*

1966a. J. Chem, Phys. .44. 1830.1966b. J. Chem. Phys. 41, 1649.1968. “Correlation Functions for Molecular

Motion11 in Advances in Magnetic Resonance 1, ed. J. S. Waugh (Academic Press, New York and London) p. 1.

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Hare, W. F. J. 1955* Ph.D. Thesis, University of Toronto. Heicklen, J. 1961. Spectrochim. Acta 17. 201.Herranz, J., and Stoicheff, B. P. 1963. J. Mol. Spectros

copy 10. 448.Herzberg, G. 1945. Infrared and Raman Spectra of Poly

atomic Molecules. (D. Van Nostrand Co. Inc., Princeton, N. J.),

Hiller, R. E,, and Straley, J. W, I960. J. Mol, Spectroscopy I t 24.

Hornig, D. F., and McKean, D. C. 1955. J. Phys. Chem.i£, 1133.

Jahn, H. A. 1938. Proc. Roy. Soc. (London) A 168, 495. Jammu, K. S., St.John, G. B., and Welsh, H. L. 1966.

Can. J. Phys. 44* 797.Jones, L. H., and McDowell, R. S. 1959. J. Mol. Spectros

copy JL, 632.Kaylor, H. M., and Nielsen, A. H. 1955. J. Chem, Phys.

21, 2139.Lepard, D. W. I964. Ph.D. Thesis, University of Toronto. Mills, I. M. 1958. Mol. Phys. 1, 107.Olafson, R. A., Thomas, M. A., and Welsh, H.'L. 1961.

Can. J. Phys. 12.» 419.Plyler, B. K., Danti, A., Blaine, L. R., and Tidwell, E. D.

I960. J. Research N. B. S, 64A, 29.Rea, D. G., and Thompson, H. W. 1956. Trans. Far. Soc.

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Richardson, E. H., Broderson, S., Krause, L. and Welsh, H. L.1962. J. Mol* Spectroscopy ,8, 406.

Rollefson, R., and Havens, R. 1940. Phys. Rev. J£2» 710. Seshadri, K. S., and Jones, R, N. 1963* Spectrochim.

Acta 12., 1013.Shapiro, M, M. I960. M. A. Thesis, University of Toronto. Steele, W. A. 1963. J. Chem. Phys. jJJS, 2411*Sverdlov, L. M, 1961. Opt. and Spectroscopy JLO, 17*

1962. Opt. and Spectroscopy .12, 66.Teller, E. 1934* Hand- und Jahrb, d. Chem. Phys. vol.9,

II, 437.Thomas, M. A., and Welsh, H. L. i960. Can. J. Phys.

J 8, 1291.Thorndike, A. M. 1947. J. Chem. Phys. JL£, 868.Van Kranendonk, J. 1957. Physica .23., 825.Welsh, H. L., Pashler, P. E., and Dunn, A. F. 1951.

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Chem. Soc. 23.* 1721*Wilmshurst, J. K., and Bernstein, II. J. 1957. Can. J.

Chem. 32* 226.

IReproduced with permission of the copyright owner. Further reproduction prohibited without permission.

ACKNOWLEDGEMENTS

It is a pleasure to acknowledge the patient help and encouragement of my supervisor, Professor H. L. Welsh, during the course of this work. In addition the interest and assistance of Professors R. L. Armstrong and J, C.Stryland are much appreciated.

Several discussions with Roland Timsit, which led to the slit-width correction technique employed here, have been very helpful.

I am grateful for the co-operation of Bert Owen,Jack Legge and their workshop staffs in the design and construction of apparatus. A special word of thanks is due to the Physics Terminal computer operators, Shirley Thompson and Wanda Slawinski for their cheerful assistance in the later hectic stages of this work.

The receipt of financial assistance from the National Research Council and the Province of Ontario is

also gratefully acknowledged.

Finally, I thank my wife, Dolores, for her boundless

patience through the years.


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