J . Phys. Chem. 1989, 93, 3035-3038
3035
Second Hyperpolarizability and Static Polarizability Anisotropy of Carbon Dioxide Ian R. Gentle,lPDerek R. Laver,'" and Geoffrey L. D. Ritchie*?lb School of Chemistry, University of Sydney, New South Wales 2006, Australia, and Department of Chemistry, University of New England, New South Wales 2351, Australia (Received: August 3, 1988; In Final Form: September 20, 1988)
Improved equipment for measurements of the electrooptical Kerr effect in gases is described, and a study of carbon dioxide over a range of temperatures (2299-490 K) and pressures (~100-1OOOP a ) is reported. Analysis of the temperature dependence of the effect yields definitive values for the second Kerr hyperpolarizability (1060yK/C m4 V3= 0.125 i 0.032) and the static polarizability anisotropy ( 1040Acu0/Cm2 V 1= 2.41 i 0.08) of the carbon dioxide molecule, and these are compared with discordant results from other sources.
Introduction The polarizability and hyperpolarizability tensors of the carbon dioxide molecule have been extensively investigated by means of a variety of optical and electrooptical techniques. Apparently reliable values of combinations of elements of the second hyperpolarizability, y, at visible-region wavelengths have been obtained from studies of electric-field-induced second-harmonic generation,2 third-harmonic g e n e r a t i ~ nand , ~ four-wave m i ~ i n g . ~ However, such results are at variance with the value of the closely related second hyperpolarizability, yK, derived from an earlier studys of the temperature dependence of the Kerr effect, even when appropriate allowance is made for dispersion and vibrational contributions to the latter quantity. In addition, the static polarizability anisotropy, AcuO, of carbon dioxide as deduced by molecular-beam laser Stakk spectroscopy6 is significantly higher than the result from the same Kerr effect study. In view of the importance of these fundamental properties of carbon dioxide, and because of the doubt that has arisen in relation to the electric birefringence measurements, we have used improved equipment to reexamine the effect, with increased accuracy and precision, over an extended temperature range. The results that are reported here serve to resolve the two discrepancies mentioned above and to provide definitive values of both yKand AaO. Theory The definition of the molar Kerr constant, ,K, is7
+
+
= 6nV,[(n2 2)'(er 2)2]-'[(nn- n,)FZ]F=O (1) where n and cr are the refractive index and relative permittivity of the medium in the absence of the field; rill - n, is the fieldinduced refractive index difference for light polarized parallel and perpendicular to the uniform electric field, F ; and V, is the molar volume. To take account of molecular interactions, the Kerr constant can be expressed in terms of the molar volume as K , = AK BKV,-' + . . . (2) in which A K and BKare the first and second Kerr virial coefficients, respectively. In the case of a linear, nondipolar molecule (for which the internuclear axis is labeled z ) , the theoretical expression7 for AK, in SI units,s is A K = (N~/405to)[5y~ (~T)-'ALIAC~O] (3) K ,
where yK= [3y&aas(yw;w,0,0) - yaolae(-w;w,O,O)]/10 is the second Kerr hyperpolarizability; and Acu (=a,, - a X xand ) Aao (=a,: - ax:) are the anisotropies in the optical frequency and static polarizabilities, respectively. There are, therefore, two contributions to the observed effect: first, a relatively small temperature-independent term that originates in distortion of the electronic structure.by the field; and second, a temperature-dependent, and typically dominant, term that arises from molecular orientation. If Aa is known, as it is for carbon dioxide? then measurements of AK over a suitable range of temperatures serve to define both yK and AaO, the two quantities sought for this molecule. Experimental Section The apparatus, shown in Figure 1, which was constructed at the University of Sydney to measure the temperature dependence of electric birefringence in gases and vapors, is an improved version of earlier arrangements;l0 the particular configuration described here was chosen to provide the maximum possible accuracy and precision of results, without unnecessary complications. Two Kerr cells are used; one contains the gas under examination, and the other, a calibrated reference cell, contains liquid cyclohexane. A variable ac voltage is applied to one electrode of the gas cell, and a fixed dc voltage is applied to the other electrode; the same ac voltage is also applied to one electrode of the reference cell, which has a variable dc voltage on its opposite number. For each cell, the relative retardance of light polarized at r/4 to the direction of the electric field is cp
= k( V
+ Vo sin
(4)
where Vis the dc voltage, Vosin wt is the ac voltage, and k is a characteristic constant. Application of Mueller calculus" shows that the intensity of light modulated at the fundamental frequency w will be zero when
+
+
(1) (a) University of Sydney. (b) University of New England. (2) (a) Ward, J. F.; Miller, C. K. Phys. Rev. A 1979, 19, 826-833. (b) Shelton, D. P.; Buckingham, A. D. Phys. Reo. A 1982,25, 2787-2798. (c) Shelton, D. P. J . Chem. Phys. 1986, 85, 4234-4239. (3) Ward, J. F.; New, G. H. C. Phys. Rev. 1969, 185, 51-72. (4) (a) Rado, W. G. Appl. Phys. Lett. 1967.11, 123-125. (b) Lundeen, T.; Hou, S.-Y.;Nibler, J . W. J . Chem. Phys. 1983, 79, 6301-6305. (5) Buckingham, A. D.; Bogaard, M. P.; Dunmur, D. A,; Hobbs, C. P.; Orr, B. J. Trans. Faraday SOC.1970, 66, 1548-1553. (6) Gough, T. E.; Orr, B. J.; Scoles, G. J . Mol. Spectrosc. 1983, 99, 143-158. (7) Buckingham, A. D.; Pople, J. A. Proc. Phys. Soc. A 1955,68,905-909. (8) For conversion factors and some discussion of SI units see (a) Buckingham, A. D.; Orr, B. J. Trans. Faraday Soc. 1%9,65,673-681. (b) Lukins, P. B.; Laver, D. R.; Buckingham, A. D.; Ritchie, G. L. D. J . Phys. Chem. 1985, 89, 1309-1312.
0022-3654f 89 f 2093-3035$01.50/0
in which the subscripts 1 and 2 refer to the gas cell and the reference cell, respectively. Such a technique is particularly convenient as the null condition depends only on the two dc voltages and the characteristic constants for the cells, and not on the ac voltage; the need for accurate measurement of the ac voltage, using voltage dividers, is therefore obviated. Additionally, the use of frequency doublers and phase shifters is avoided, so that possible errors due to mismatch of phases are eliminated. The pressure vessel consists of a brass tube, of 102-mm external diameter and 3-mm wall thickness, with brass end-flanges. Brass end-caps support stainless steel tubes on which the window (9) Bogaard, M. P.; Buckingham, A. D.; Pierens, R. K.; White, A. H. J . Chem. SOC.,Faraday Trans. 1 1978, 74, 3008-3015. (10) See, for example (a) Boyle, L. L.; Buckingham, A. D.; Disch, R. L.; Dunmur, D. A. J . Chem. Phys. 1966,45, 1318-1323. (b) Buckingham, A. D.; Sutter, H. J . Chem. Phys. 1976, 64, 364-369. (c) Burnham, A. K.; Buxton, L. W.; Flygare, W. H. J . Chem. Phys. 1977, 67, 4990-4995. (11) Shurcliff, W. A. Polarized Light; Harvard University Press: Cambridge, MA, 1962.
0 1989 American Chemical Society
3036
L
The Journal of Physical Chemistry, Vol. 93, No. 8, 1989
P
RK
I
GK
I
Q
A
LF
PM
I
Figure 1. Optical system for measurements of electric birefringence: L, 2-mW He-Ne laser (632.8 nm); P and A, Glan-Taylorcalcite polarizers; RK, reference Kerr cell, filled with liquid cyclohexane; OP, bipolar operational power supply; PA, power amplifier; GK, gas Kerr cell; CP, calibrated power supply; Q, quarter-wave plate; LF, laser line filter; PM, photomultiplier tube detector; F, active band-pass filter; CRO, oscilloscope; 0, oscillator; LIA, lock-in amplifier; I, integrator; AD, analogto-digital converter; C, microcomputer.
housings are mounted. Sealing between the end-caps and flanges is effected either by O-rings or by 0.5-mm gold-wire seals, depending on the gas under investigation and the temperature. The windows are Schott BK7 optical glass of IO-mm thickness. Pressure measurements are facilitated by the use of an MKS Baratron 22 1AHS pressure transducer, which gives results accurate to * O S % of reading and a range of 0-1330 kPa. Each of the electrical feedthroughs consists of a solid alumina ceramic cylinder with a length of nichrome wire sealed axially through the center and stainless steel rods soldered to its ends. Stainless steel connectors of novel design make contact with the electrodes under spring pressure, with minimal risk of corona discharge. A West Opus 72 microprocessor-based temperature controller, electrical heating tape, and extensive insulation are used to maintain a constant cell temperature to within f0.2 K in the range 300-500 K. The actual temperature is determined by using close-tolerance T-type thermocouples in stainless steel probe tubes and an Omega 412B digital thermometer. Temperatures are measured at evenly spaced intervals along the region between the electrodes to account for small nonuniformities due to unavoidable heat losses and unevenness in the heater winding. The accuracy of temperatures measured in this way is better than f l K over the full range. Independent control of the temperature of the windows is made possible by separate heating tape powered by an autotransformer. The gas cell electrodes were fashioned from SC23 special alloy steel which, when correctly heat-treated, offers a high resistance to distortion at elevated temperatures. In order that the effective length, I, be accurately known, the ends of each electrode were shaped to a profile determined by the method of Rogowski12 for ideal plane electrodes; the value of I for the assembly is (0.500 f 0.001) m. Each electrode was annealed, fine-ground to a flatness of 5 X 10" m, and nickel-plated for corrosion resistance. The electrodes are supported by alumina ceramic mounts and stainless steel spacers; the gap, d, was measured by using a hole gauge and micrometer to give a mean value for the required quantity &/1 of (5.236 f 0.015) X m. Considerable care was taken to ensure that the value of & / 1 is accurate and, furthermore, that it remains constant over the entire temperature range of the experiments. To this end, a pair of small brass electrodes was constructed to be used with glass spacers of accurately known length, so that & / I is precisely known for this assembly; the effective length of the assembly is 41.71 mm and the gap 6.89 mm. Because of the small size and relatively large spacing of these electrodes, it was assumed that only thermal ( 1 2) Cobine, J. D. Gaseous Conductors: Theory and Engineering Applications; Dover: New York, 1958.
Gentle et al. expansion needed to be considered at high temperatures and that the effects of distortion were negligible. By alternate measurements of the birefringence of a sample of carbon dioxide at 298 K with each electrode assembly, the value of & / l for the gas cell electrodes was confirmed to be correct to within *OS%. Repetition of the experiment at 413 K showed that the effect of distortion at that temperature was less than 1.5%. Light from a 2-mW plane-polarized He-Ne laser traverses a Clan-Taylor prism with its transmission axis oriented at ir/4 with respect to the electrodes, and the ellipticity induced in the beam by the birefringent gas is nulled by the reference Kerr cell, whose electrodes are parallel to those of the gas cell. Any residual ellipticity is converted to a rotation by a quarter-wave plate, passes the analyzer, and is detected by the photomultiplier tube. An active band-pass filter and an Ithaco Dynatrac 393 lock-in amplifier detect only the fundamental signal; the output is integrated and amplified by a Kepco BOP-IOOOM operational power supply which, in turn, supplies the dc bias voltage for the reference cell. This feedback system therefore maintains a null irrespective of the size of the bias voltage applied to the gas cell electrodes, so that any subjectivity is removed and long averaging times can be used. The voltage required to maintain the null is monitored and analyzed by an Apple IIe microcomputer fitted with a Data Translation DT2834 analog input module. A signal generator, 150-W power amplifier, and transformer provide the ac voltage of up to 10 kV at 512 Hz, while the dc bias voltage for the gas cell is derived from a Fluke 410B calibrated supply. The precision of results obtained after averaging for several minutes is f 2 0 nrad. In order to optimize the performance of the optical system and to quantify potential sources of error, an analysis of the system was made with Mueller calculus. The major optical errors arise from relative misalignment of the Kerr cells and the retardances of the cell windows. If the Kerr cells are misaligned by an angle a , the relative error in the null condition is 2 sinZa, and careful alignment reduces the error from this source to less than 0.06%. A stray static retardance 6 between the birefringent element and the nulling retardance gives rise to a maximum error of 2 sin2 (6/2) when the azimuth is parallel to that of the polarization direction of the beam. The use of low-birefringence windows ensures that this error is less than 0.05%. Accurate calibration of the reference cell is extremely important, and great care was taken to minimize uncertainties associated with this procedure. After evaluation of several procedures, the optimum method was found to be the application of a static voltage to one electrode with the other grounded, modulation of the intensity emerging from the quarter-wave plate with a Faraday coil, and nulling with a second, dc Faraday coil, which was, in turn, calibrated against an analyzer/micrometer arrangement. Repeated calibrations over a period of several months demonstrated that the cell is extremely stable and that the calibration constant is reproducible to better than f l % . The overall accuracy of measurements of A K is conservatively estimated at f2% over the full temperature range. The sample of carbon dioxide (Commonwealth Industrial Gases, food grade, >99.8%) was used without further purification. At each temperature many observations were made over a range of pressures. Measured nulling voltages were used to calculate retardances and hence birefringences, from which values of ,KO = (2/27)(nll- n , ) V m F 2 were derived and fitted to the relation
in which A , and A , are the low-density molar dielectric polarization and refraction, respectively. Density virial coefficient~l~ were used to obtain molar volumes, V,, from the measured gas pressures and temperatures; values of A, and A , were calculated from the p ~ b l i s h e d ~static , ' ~ polarizability (3.241 X 1040 C m2 V-I) and optical frequency polarizability (2.933 X lo4 C m2 VI). (13) Dymond, J. H.; Smith, E. B. The Virial Coefficients of Pure Gases and Mixtures; Clarendon Press: Oxford, U.K., 1980. (14) Newell, A . C.; Baird, R. C. J. Appl. Phys. 1965, 36, 3751-3759
The Journal of Physical Chemistry, Vol. 93, No. 8, 1989 3037
Temperature Dependence of the Kerr Effect of COz
TABLE I: Temperature Dependence of the Kerr Effect of Carbon Dioxide at 632.8 nm
T /K 489.5 455.8 422.8 394.5 370.9 348.8 330.9 3 14.9 299.2
1037-l/~-'
no. of pressures
p/kPa
2.043 2.194 2.366 2.535 2.697 2.867 3.023 3.176 3.343
9 9 8 9 12 7 9 9 7
347-663 243-792 242-737 240-774 245-781 232-874 270-7 25 216-790 150-1040
106s/m3mol-' -32.2 -41.1 -51.8 -62.9 -74.0 -86.3 -97.9 -1 10.0 -123.5
1 0 2 7 ~ K / mv-2 S moP 1.508 f 0.004 1.613 f 0.006 1.740 f 0.003 1.846 f 0.004 1.979 f 0.005 2.063 f 0.004 2.219 f 0.005 2.292 f 0.003 2.404 f 0.003
1032~K/m8 v-* moI-1 -5 f 3 -3 f 4 -3 f 2 2f2 -6 f 2 2fl 4f2
Of1 6fl
2.46
2.44
2.42
2.40
2.38
3 0
100
200 -1
V,/mol
300
400
500
-3
m
Figure 2. Density dependence of the Kerr effect of carbon dioxide at
0
. 0.0
1.o
299.2 K.
Figure 2 shows, as a typical example, the dependence of ,,,KO on V,-l at 299.2 K. The results are summarized in Table I, where the uncertainties quoted are based on the standard deviations derived from the least-squares fitting of straight lines to the density-dependence data, together with appropriate allowance for systematic errors. Unfortunately, the values of BK are too imprecise to draw any firm conclusions; more reliable values could be obtained from measurements at much higher pressures, but the equipment described here was designed specifically for studies -of gases and vapors at relatively low pressures.
Results and Discussion From eq 3 it can be seen that the zero-density molar Kerr constant should exhibit a linear dependence on the reciprocal of the absolute temperature. Figure 3 shows two plots of AKagainst TI; one represents the data from ref 5 and the other the results of the present investigation. Table I1 contains the intercept and slope of our weighted-fit least-squares straight line, together with the new values of y K and Aa0 derived from these quantities. It is immediately apparent that the magnitude of yK(expressed here as 10myK/C m4 V3)as deduced from the measurements now reported (0.125 f 0.032) is much smaller than that from the original temperature-dependence study (0.56 f 0.~;K )! and, in fact, the term (NA/81c0)yK contributes only 4.4%,rather than 17%,5 of the value of AK at 300 K. As noted by Shelton,2cthe most likely reason for the disagreement lies in the uncertainty limits attributed to the earlier observations and, in particular, in the error in the T I = 0 intercept in the plot of AK against TI. A reexamination of the original data indicates that the intercept has an uncertainty, expressed as the standard deviation, of 75%
0 3.0
2.0 3
7 4.0
-1
10 T /K-' Figure 3. Temperature dependence of the zero-density molar Kerr con-
stant of carbon dioxide: upper line, data from ref 5, 5% uncertainties; lower line, data from Table 11, 2% uncertainties. TABLE II: Analysis of the Temperature Dependence of the Zero-Density Molar Kerr Constant of Carbon Dioxide
property 102*(intercept)/msV2m o P 1o6OyK/Cm4 V-3, 102s(slope)/m5 K mol-l
value 1.05 f 0.27'
0.125 f 0.032 6.89f 0.10" ~ o ~ ~ A m4 ~ v-2 A ~ ~ / c ~5.66 f 0.08 2.35 f 0.07b 1040Aa/Cm2 V-I 1 0 4 0 ~ a om2 / ~ v-1 2.41 f 0.08
'From weighted least-squares straight line fitted to data in Table 1. uncertainty assumed.
'Referehce 9;X = 632.8nm, 3%
rather than the quoted value of 21%; furthermore, if the lowest temperature point (103T' = 3.97) is omitted, the value of yK changes from +0.56 f 0.42 to -0.06 f 0.18. It is worthy of emphasis here that the determination, with acceptable precision, of the temperature-independent contribution to the Kerr constant is an extremely demanding objective, because a long and therefore I = 0 is required. In the somewhat uncertain extrapolation to T earlier study the extrapolation ratio R = Tmi,,/(T,,, - T,,,J was 3.0, whereas in consequence of the wider and higher temperature range, the value in the present investigation is only 1.6, a much more favorable situation. In contradistinction to the earlier result, the new value of yK (0.125 f 0.032) is much closer to the prediction made by Shelton
J. Phys. Chem. 1989, 93, 3038-3041
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(0.073 f 0.002) on the basis of results from his study of gas-phase electric-field-induced second-harmonic generation.2c The procedure used by Shelton involved allowance for the frequency dependence of the electronic contribution and inclusion of a vibrational contribution; however, on the basis of our findings it now seems that the latter correction could be larger than was thought to be the case. By contrast the value of Aao (expressed here as 10@Aa0/Cm2 VI)that emerges from our measurements (2.41 f 0.08) is, fortuitously, the same as the result of the original investigation (2.39 f 0.18); however, the uncertainty in the latter quantity appears, once again, to have been underestimated. Both are lower than the value derived by Cough et aL6 by laser Stark spectroscopy (2.89 f 0.20), which now must be considered to be about 20% too high. In the same study,6 Aao was also independently determined from considerations of the frequency dependence of the gas-phase Rayleigh depolarization ratio and the intensities of the infrared-active vibrational bands. The result (2.57 f 0.06) is in better agreement with the definitive value now available from the Kerr effect and serves to reinforce the view that the estimate from the spectroscopic investigation is in error.
Solid-state ''C
Summary The present study of the temperature dependence of the electrooptical Kerr effect of carbon dioxide has provided reliable measurements of the second Kerr hyperpolarizability and the static polarizability anisotropy, two fundamental molecular properties that have previously attracted much interest and some controversy. Our value for the hyperpolarizability is much smaller than that found in the earlier Kerr effect investigation but in reasonable agreement with a recent estimate based on observations of electric-field-induced second-harmonic generation; the result for the static polarizability anisotropy has served to emphasize a deficiency, as yet unidentified, in the laser Stark spectroscopic study of this property.
Acknowledgment. A Commonwealth Postgraduate Research Award (to I.R.G.), financial support from the Australian Research Council (to G.L.D.R.), technical assistance from Mr. J. Zyimans (University of Sydney), and helpful conversations with Dr. M. P. Bogaard (University of New South Wales) and Professor B. J. Orr (Macquarie University) are gratefully acknowledged. Registry No. C 0 2 , 124-38-9.
M R Studies of the Structural Transition in (TMTSF),(Nl(tds),)
A. K. Whittaker, P. C. Stein, P. Bernier,* Groupe de Dynamique des Phases Condensges, UniuersitP des Sciences et Techniques du Languedoc. Place E. Bataillon, F 34060 Montpellier Cedex, France
W. B. Heuer, and B. M. Hoffman* Department of Chemistry and Materials Research Center, Northwestern University, Euanston, Illinois 60208 (Received: August 5, 1988; In Final Form: October 25, 1988)
The I3C NMR spectra of the conducting salt (TMTSF),(Ni(tds),) have been recorded across the metal-metal transition at 272 K. The changes in the NMR spectra are related to the changes in the symmetry of the system and to variations in the Knight shifts. Below the transition there is a decrease in symmetry, in agreement with previously published X-ray data. The NMR data demonstrate that the structural modification associated with the transition extends over a much wider temperature range than suggested by ESR and conductivity measurements. A comparison with the spectra of the Bechgaard salts reveals the sensitivity of the technique to variations in the distribution of conduction electrons.
Introduction Organic conductors exhibit a large number of different phases and transitions between those phases.' The phase transitions are often of structural origin and can lead to changes in the lattice periodicity that open a gap at the Fermi surface leading to an insulating low-temperature phase. The properties of these materials depend strongly on the size and symmetry of the anion. For example, at ambient pressure (TMTSF)2C104(TMTSF = tetramethyltetraselenafulvalenium) exhibits a superconducting ground state,2whereas the isostructural (TMTSF),Re04 exhibits a semiconducting ground state due to ordering of the anion^.^,^ Heuer et a1.s*6recently have synthesized a series of isostructural TMTSF salts with the anions M(tds),- (tds = bis(trifluor0methy1)ethylenediselenato-, (TMTSF),(Ni(tds),) = bis(tetra( I ) JBrome, D.;.Schulz, H. J. Ado. Phys. 1982, 31, 299. (2) Ribault. M.; Jlrome, D.; Tuchendler, J.; Weyl, C.; Bechgaard, K. J. Phys. Lett. 1983, 44, L953. (3) Tomic, S.Thesis, UniversitB Paris Sud, Orsay, 1986. (4) Moret, R.; Pouget, J . P. In Crystals Chemistry and Properties of Materials with Quasi One Dimensional Structure; Rouxel, J., Ed. Reidel: Dordrecht, 1987: p 87. ( 5 ) Heuer, W. B.; Hoffman, B. M. J. Chem. Soc., Chem. Commun. 1986, 174.
(6) Heuer, W. B.; Squattrito, P. J.; Hoffman, B. M.; Ibers, J. A. submitted for publication in J . Am. Chem. SOC.
0022-3654/89/2093-3038$01.50/0
methyltetraslenafulvalene) [ bis(trifluoromethyl)ethylene]diselenonickel), where M is Ni, Pt, or Cu. The chemical structures of the TMTSF and the Ni(tds), molecules are given in Figure 1. The Ni and Pt derivatives differ from the other TMTSF systems in that the anion carries a localized spin and that these salts have a novel first-order metal transition at a transition temperature (T,) of 272 and 240 K, respectively. The conductivity at lower temperature is enhanced. The X-ray diffraction analysis of the structure of the Ni derivative above and below T, showed that this transition is of structural origin. The structure of these materials in the high-temperature phase has been determind6 The anions occupy the corners of the unit cell. In contrast to the Bechgaard salts,'+ the TMTSF molecules occupy a center of inversion symmetry at ambient temperature, and therefore there are two distinct pairs of halfTMTSF molecules within the unit cell. The anions reorient in the low-temperture phase, and the stacks of the TMTSF molecules and the anions are randomly translated in the c direction by distances of f 1/8 c, such that the TMTSF molecules no longer occupy centers of inversion symmetry. The higher conductivity at lower tem(7) Rindorf, G.; Soling, H.; Thorup, N. Acta Crystallogr. 1982, B38,2805. (8) Thorup, N.; Rindorf, G.; Soling, H.; Bechgaard, K. Acta Crystallogr. 1981, B37, 1236. (9) Wudl, F. J. Am. Chem. SOC.1979, 103, 7065.
0 1989 American Chemical Society
