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The temperature dependence of the Cotton-Mouton effect of N2, CO, N2O, CO2, OCS, and CS2 in the gaseous state
~Chemical Physics~(1984)~207-214 North-Hohand; Amsterdam
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T~TE~MPE~ATuREDEPENDEN~E~FTHEC~~O~~MOUTONEFFE~ OF N,, CO,-NzO. H. KLING.and
W.
CO,, -_
OCS, AND Cs,
I-kJT”hIER
Abrerlung Chemisehe Pizpk,
Uttrversttiit
_
. -
:
IN THE G&kOUS-STATE __
Ulm; D -7900
_ Ulm.
West Get-nrutt~
Received 30 Apnl 1984
The magnetic-field-Induced birefringence of the SIY substances Na. CO, N,O. CO,. OCR and CS_ has been measured in the gaseous state, between - 70 and + 120 o C and at pressures up to 1 bar. A quantum correction term. accounting for the influence of the rotation-Induced magnetic moments. is derived to modify the eKpressron from classical orientahon theory describtng the Cotton-Mouton effect of linear molecules. The temperature dependence of the measured molar Cotton-Mouton constants ,C(T) has been analyzed in terms of the hyperpolanzabihty anisotropy parameter A1) and the onentation parameter BnL$. The latter is discussed in view of the polarizabihty and suscepubrlity anisotroptes ha and LI< and of electric quadrupole moments Q,, known by other tecbmques- There IS generally good agreement between results from Rayleigh seattenng. high-resolutton spectroscopy, eltoztric-field-gradient-Induced buefringence. and the present Cotton-Mouton effect results
Introduction We have investigated the magnetic-field-induced birefringence(Cotton-Mouton effect, CME) of gases consisting of the linear molecules N,, CO, N,O, CO,, OCS and C&. at pressures below 1 bar. Molecular orientational correlation effects and local field corrections are disregarded. The ellipsometric phase difference, + = 2nC
/
B’df;
(1)
may then be described in terms of parameters by the relations [ 1,21 C=$V’(N/N*)
m
c,
molecular
(2)
and ,C = 2+V,(
fAq + 2AaA.$/ 4:5kT).
(3)
B = B(I) is the magnetic field strength along the pathlength I; X, NA, N, kT are wavelength, Avogadro’s. number, number density and -thermal energy per molecule, respectively. -A+ is a suitably defined [3] hyperpolarizability parameter describing the magnetic field perturbation of the polariza-
bility tensor and Aa = a,, -a, and Ag = <,, -5 i are the molecular polarizability and susceptibility anisotropies. Kdnig’s definition [4] of the molar Cotton-Mouton constant, ,C, is used. Eq. (3), as a result of classical theory, is valid only for the high-temperature approximation and will have to be slightly modified below. From our recent measurements of the temperature dependence of the CME in gaseous non-polar hydrocarbons [5] we concluded that Aq in eq. (3) must not be neglected if one wishes to maintain the accuracy of the empirical phase differences in evaluating the orientation parameters AaA.$. As almost all CME studies on diamagnetic gases reported in the literature were restricted to constant (room) temperature, it appeared important to -extend the investigation of the temperature depen-_ dence to other substances. The polarizab$ity anisotropies Aa of all of the above molecular species -have been determined from observations of the depolarized Rayleigh scattering by various authors [6-S]. Yet-y accurate susceptibility ahisotropies At are; in addition, available from molecular beam electric_resonance Zeeman measurements for the three polar- species CO,-N,O, and OCS [9-111. Thus, measurement of
0301-0104/84/$03.00 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
the slope of the function ,C(T’-‘) given in eq. (3) provides redundant information for Acu and A< in these three cases which seems a rather welcome situation in view of the widely differing experimental and theoretical concepts with their occaslonal simplifications. For example, undiscnminated intensity contributions from the vibraRayieigh tional R&man effect ma> influence scattering results [6]. and unknown centrifugal distortion effects or vibrational dependencies may restrict the transferrability of A=$ values determined in 10~~rovibrational quantum states_ Spectroscopic&y determined susceptibility anisotroples A.$ are not available for the three nonpolar species N,. CO?. and CS,. In these cases. accurate orientation parameters An.L$ are valuable sources for obtaining A& There exists also a connection bet\%een A.$ and the molecular electric quadrupole moment. Q,, _ as established by RamseJ’s relation (see eq. (8) beIow)_ Q,, of nonpolar molecules can be determined rather directI) by the electric-field-gradient-Induced blrefringence [12]. uhich in turn opens an interesting possibility to chech the consistency of independent experimental and theoretlcai methods.
2. Experimental The measurements have been carried out under conditions described in detctii in ref. [5]. The reievant parameters are X = 632-S nm. /-B’ddl= 1.157(5)x 10” G’=5cm. and QETT;O-? x *o_” ‘i; corre, sponding to C,,,, m --I or - referred to gases at normal conditions - ,C,,,, = 5 x lo-”
G-2
cm’ mo]-’
The maximum coLered temperature range \\ds - 70 to 120 OC. which w;1s shortened in some cases owing to 10~~ vapor pressure below the boiirng point. The minimum and maximum temperatures used are indicated in table 1. Commercial purity specifications are &so given in table 1. Whenever possible. samples were degassed at liquid-nitrogen and higher temperatures to remove traces of foreign gases. first of ail oxygen_ Several mdss spectromerric analyses taken from the cell contents have shown that impurities were negligible. There are. however. two other small perturbing
effects to be considered which are obviously of molecular origin. They vary quantitatively from substance to substance, and sometimes they are well resolved with the present ellipsometric sensitivity. First. equihbration of the sample gas after cell filling may take several hours. which is easily monitored by repeated observation of the sIowly converging magnetic-field-induced birefringence. Drifts as large as 5% have been registered. Second. it is frequently observed that CME measurements carried out at narrowly spaced temperatures show very accurately a straight line T-’ dependence. but produce a slightly incorrect slope X/i3(T-‘). In order to avord correlations of this kind. temperatures are usually vaned in large steps in one direction (typically 10 “. increasing or decreasing). and the missing data points are then filled in on the (decreasing or increasing) way back. subdividmg the temperature axis effectively in intervals of = 5 O_ During the course of this work. we have repeated rarher measurements on N, [13] and found discrepancies somewhat outside the error limits. It IS likely that these were caused by the described effects. The exact redsons of the (rather time consuming) irregularities are not understood_ It seems unlikely that orientational correlation, like in molecular liquids. should play a measureable role at pressures below one bar. The fact that the reproducibdity is best at room temperature points to residual temperature gradients which could in principle give rise to magnetic-fieid-induced molecular alignments via the SenftlebenBeenakker effect [14]_ The variation of temperature in our present cell amounts. however. to not more than lo.
3. Data evaluation and esperimental
uncertainties
The low moments of inertia of some of the molecules concerned require to investigate to what extent quantum corrections of the classical Cotton-Mouton constant will be important. In addltion, it cannot be excluded without detailed analysis that rotation-induced magnetic moments may cause contributions to the observed birefnngence. We have recently developed a rigorous quantum statistical expression for the Cotton-Mouton con-
stant of a general dilute diamagnetic or paramagnetic gas (see eq. (10) of ref. [13]) which we can now specialize to the cases at hand. We can simplify the problem to that of an ensemble of linear rigid rotors since the population of excited vibronic states is too low to influence the results significantly. It is then, at the same time, assumed that centrifugal distortion of the molecular parameters is negligible, which has the effect that the least-squares-fitted- parameters represent actually mean values over the rotational population. With these restrictions, eq. (10) of the quoted paper [13] simplifies considerably and reduces to the expression C = (27iR’/h)(3/2) x[(2/45kT)AcrA~(l - (2/45kT)Aa( x(1
-+r-&u’+
--a+&a”+
___)
g”,p;/2hG) ___)+$A&
which is valid for u = hG/kT -=x 1. hG 1s the rotational energy per unit angular momentum. and glpLn the gyromagnetic ratio (pEcn = nuclear magneton). Summations of expansions in the rotational quantum number [15]. utilized by Buckingham and Pariseau 1161 in a similar treatment of the electric-field-gradient-induced birefringence, were useful in developing eq. (4)_ The first term in the brackets was already given by Corfteld [17] and leads, together with the term m AT, to the known classical result in the hmlt u = 0 (see eq. (3)). The second term describes the orIentationa influence of the rotational magnetic moments. It remains in the classical limit, and IS, in distinction from the corresponding well-known orientation term Aa(p,_/kT)’ in the Kerr effect of dipolar gases. proportional to T-’ only (jl~~ is the electric dipole moment). Thus term may become important for heavy molecules with exceptionally large rotational g-factors. Finally, we note that all terms in the third-order hyperpolarizability tensor [2,13] vanish completely. This is also different from the Kerr effect, where an (electric) moment causes a cross term j.Q/kT, fl being the anisotropy of the (electric) third-order hyperpolarizability [18]. The correction terms are easily taken mto
account by introducing an effective susceptibility anisotropy parameter A,& into eq. (3): ,,C = 27rNA( fAq + 2AaA&/45kT),
(5)
where A&
\
= Ac( 1 - u + Aa’) -(&;/2hG)(l
--~u-~u2).
For the gases investigated here, the second term of A& turns out to be an order of magnitude smaller than the experimental uncertainty, and will therefore be disregarded_ The remaining correction factor, truncated to first order in u, affects the results of N, and CO up to one standard deviation and should therefore be retained. The values of 1 - u at room temperature are listed in table 1. The plots ,C(T) versus (1 - u)/T are given in figs. i and 2. and show the expected linear behaviour. Van der Waals corrections have been applied to obtain the
(14 3
L
T)lT-‘/lQ3K-’ 5
I
mC/lO-“cm3 G-%wJ~-’
Ftg 1. The molar Cotton-Mouton constants of Nz. CO, N,O and co2 \ersus (1 - a)T_ ‘_ see eq (6) The error bars are i one standard deviation, obtamed from a senes of measurements at constant temperature, and added contribut:ons from uncertainties in the number density. A’. The strrtighthnes result from least-squares fitting these data to the slopes and ordinate crossing points which in turn yield the molecular parameters in table 1.
(l-a(T))
T-‘/lOSK-’
O@fl
,,,C/lO-%n3
G-*mol-
Flp 1 The molar Cotton-Mouton wnbtants of OCS and CS2 compare the tc\t beloa fig I Obsenc the \cr3u5 (1 - a)7-1. decreased ordmats sale H hlch maha the error bars unrcpresentable 1x1the case of OCS
number densitres from the pressure and temperature readings. The parameters CIand b uttltzed are gtven in the first column of table 1. together with the commercial sample puritres. The (vertical) error bars in figs. 1 and 2 represent one mean standard deviation to either side of the entries. where each one is the result of a series of about ten single ellipsometric compensations. The errors caused by the uncertainty m the pressure gauge readings (0.005 bar. a conservative estimate) are also included_ They dre comparably small near normal pressure. but contribute signifiin the low-temperature. small-pressure cantly ranges of the coudensable gases. An increased scale would show that doubling the error bars is sufficient to bring the least-squares-fitted straight lures to withm the range of each single entry. We have therefore chosen to assign two standard deviations to the parameters 1111and AcxA& which clre given in table 1 ds the final result of a weighted least-squares procedure for each gas. The rather small multiplicative systematic error contributions arising from /B’dl and the Faraday compensator calibration are also included. Finally. an addittve systematic uncertklty corresponding to a phase difference of f 2 x lo-% is introduced_ The latter covers empty-cell effects which are sometimes ob’ served and well below that signal strength_ The additive error drops out in determining Aa.L$. but increases the error of A2 typically by 7 X 10eA3 cm3 G-‘. compare table 1.
4. Results_ The vahres of AIJ. A.aA& and ,,,C at room temperature of all gases are collected in the fourth to sixth columns of table 1. Those of (AaA<), in column seven are obtained from the room-temperature ,C values under the constraint Aq = O_ Several room-temperature ,C values of the gases considered here have been published previously by different authors [3,17,19]_ We have compiled them in the last column of table 1. The literature ,C values not measured at He-Ne laser wavelength were adJusted to X = 632.8 nm with the aid of the polarizabtlity anisotropies obtained at various wavelengths by Bogaard et al. [6] and by Ahns et al. [7]. Fortunately. there are no dramatic disagreements, though the uncertainty ranges somettmes do not overlap. The (A(rAE),, values are generally not consistent with their counterparts &A.$ obtained from the slope of the (quantum corrected) ,C(T-‘) function, which shows that the temperature dependence has to be surveyed for full exploitation of the experimental accuracy. The differences lie. however, below lo%, while it was previously found in ethyne [5] that AaA& deviated from (A~YAE), by as much ds 30%. Comparison of the data in the fourth column of table 1 shows that introduction of a sulfur atom increases A17 substantially_ A doubling of the hyperpolarizability anisotropy of CSz m comparison to OCS. as indicated in table 1, seems therefore plausible. Unfortunately, the accuracy of the carbon disulfide result is strongly reduced as a consequence of the high boiling point of this substance. It is now interesting to discuss the orientation parameters AaAt in the light of Aa and A,$ values obtained previously with other methods. Accurate spectroscopic molecular-beam measurements are available for the susceptibility anisotropies of the three polar molecules. We have used them to calculate the polarizability anisotropies, and the resulting values are collected in table 2. Recent values determined by depolarized Rayleigh scattering are also included for comparion. As can be seen. the new values are in excellent agreement with the most accurate literature results, and equally accurate-
Table
L
Hyperpolarizabdity and &Id-orientation ptiamctrrs from the ordinate crossing pomts and slopes in figs. 1 and 2 (fourth and fifth column). and firld-orientation parameters from room-temperature ,,,C co%tants_(sixth c@umn) and cq (5). $th AT = 0 (seventh column). The temperature ranges and quantum correcttons (ai room temperature) are Indicated ~ti the second and third columns The errors include two standard deviations from the fitting routme and additional systematic coniributions, see text. All measurements have been carried out with HeNe laser radiation_ The room-temperature ,,,C values from other authors are listed in the last column Substance a)
7m,n:Tmzt
l-lYb)
(“C)
.lp (IO-”
cm3
AC&$ (10es3 cm6)
,c =’ (IC-‘“cm3 G-’ mol-t)
G-‘)
,c d’ (10-t”
(AQA&J (1O-s3 cm6)
G-’
cm3 mol-t)
-75:17-o
0 9903
2 6(20)
- 103(4)
-0 391(20)
- 0 95(5)
-0 -0 -0
- 64;120
0.9906
0 2(16)
- 0 75(2)
- 0.305(20)
- 0 75(4)
- 0 35(7) [3]
N1O (99.5-3.78:4 42)
- 84.92
0 9980
- 4 8(34)
-5 20(11)
- 2 21(4)
- 5 34(Y)
-2
co, (99.995.3 59.4 27)
-56:120
ocs (97 5:3 93,5 82)
-61;70
N, (9Y.YYY;l 39 3 91) co (99 97.1.49.3 99)
39(3) 131 38(2) [17] 50(10) [lY]
-0 29(l) [17] 31(5) (31
- 2 04(4) r171 -2 12(22) [19] 0 9981
- 2 Y(B)
- 2 21(4)
-O-95(2)
- 2.30(6)
- 0 89(5) [3] -0 90(4) P71 - 0 95(14) [19]
5,121
CSI ( > 99 6.116;7 69)
0.9990
- 23 (3)
0 9995
- 47 (29)
-6
33(12)
- 2 91(4)
- 29.1(11)
- 7.04(10)
- 12 6(2)
- 2.58(9) [17] -2 85(26) [IY] - 11 j(2) [17] - 12 2(12) [lY]
- 30 5(j)
il’ The first number in parentheses IS the commercial purity specification in volume%, the second and third are the van der Waats constants LI (lo6 cm3 atm mol-‘) and h (10 cm3 mol-’ ) taken from ref [20] and used to determine N in eq (2). b’ See eq (5). values referred to 21 OC ‘) Referred to 21 “C ” Literature room-temperature results Those of refs [3.17] were obtained at Increased pressure. with the exception of CS_. Values measured at different wavelengths were adJusted lo 632 8 nm usmg the dispersion mformatlon
For the non-polar
species there are no spectra-
scopic At parameters determine
available,
them by reversing
with the aid of the Rayleigh of Bogaard
and, therefore,
AEd’=‘, and using the relation
we
the above procedure, scattering
Acx values
et al. 161. The results are collected
table 3, and compared
for Aa m refs [6.7]
with semi-empirical
= Atdla + (e’/4n@$,c’)l,g~)
(7)
in
values
(e
and
MP are the proton
charge
and mass, and
Amos and others [21-231. These authors have obtained them by ab initio calculating the diamag-
r~z, and c are the electron mass and speed of light)_
netic contribution
contribution
of
of the susceptibility
anisotropy,
I I,
the moment
of inertia, and
to the rotational
g$),
the electron
g-factor,
are accu-
Table 2
Comparison of differently
obtamed polatizability anisotropies Aa. The values in the second row are calculated wtth AaAS in table 1 and At in the first row, ail others are from Rayleigh scattering work. Unless otherwise stated. ail Aa refer to 632.8 nm co A5 (10m30 cm3) Aa (lo-”
cm3)
- 13.72(2)
N,O
ocs
Ref.
- 17.32(10)
- 15 56(2)
[Y-11] this work
0.532(10) =)
3.oo(6) 2.946) a1
4.07(8) 4_077(76) a)
PI
O-53(2)
2 96(6) b’ 3-l(2) =’
4 14(g) 4 O(2)
171 PI
O-543(19)
-
*) We have taken into account 3% uncertainty In the depolarization c, X = 514.5 nm. b)X=647nm.
parameter pO_
212
rately known from Raman effect and molecularbeam magnetic resonance studies. There is generally good agreement between theory and experiment_ Eq. (7) represents d difference of two large quantities. meaning that Atdra was calculated with much higher relative accuracy than evident from table 3. The knowledge of i1.$ can be extended to that of the electric molecular quadrupole moments by using the well-known relation 1341
(8) The results are also collected in table 3, and compared with the values which follow more directly from the method of the field-gradient-induced birefringence [l?]. This experiment is, analogously to the CME. described by an orientation term containing the quadrupole moment. and a temperature-independent hyperpolarizability contribution_ In the case of CO?. Battaglia et al. [31] have investigated the temperature dependence and deduced Q,, = -4.49(15) X 10ez6 esu from the slope of the T-’ plot. This value is almost in agreement
with ours<--4 2(2)x 10mZ6esu) within the experimental uncertainties. For N, and d&, only measurements at Yoom temperature seem to be available [30.31], but Amos has calcul&ed the (rather small) hyperpolarizability contribution [29]. We have combined both to extract Q,, and tind agreement with our own values for the two molecules (see table 3). It is interesting to realize that the quadrupole moment changes sign in going from CO, to CS,. This trend is also followed by the known ab initio values which we have included in table 3. One of them (2.1 x lo-‘” esu of CS,) is far beyond the experimental error limit. The quality requirements for wavefunctions are here obviously higher owing to the near cancellation of nuclear and electronic contributions. Finally. we have listed two quadrupole moments m table 3 which stem from ion scattering investigations [32]. The agreement is encouraging and calls for further improvement of this independent techmque.
-l-able 3 Comparilson ol dIfferentI> obtamrd susceptIbtht> nnwotropies A $ .md electnc quairupolc moments Q,# The ~.du~~ AC m the flrbt rob of catch moleatlc arc calculated =tth .InL< m t.tble 1 .md An from Ra>iielgh scnttcrmg gwcn m the wcond column. the corrc~ponding ~aluez QIz m turn mlth cq (S) usng (C(GHz) g, ) = (59 646 [2-I]: -0 2777(s) [Xl). (II 6Y8 [Xl. -0 055OS(5) [37]) (3 1713(Z)
5,
co,
[lS]
- 0 Olii-J(Z)
0 704(13)
2 11(J)
[17]) for N,.
CO,
.md CS,.
- i-1 6(6)
- 110 h’ - LO j(3)
-YY cs,
947(18)
rc.\pcctl\cI\
h’
- 30 7(13) h’ - ‘9 7 h’ - 777
The ongms
of the other
~.ducs
are e~plamed
m footnotes
- 1 3(-I) - 1
c’
4 Z(Y) 3 61( 1s) .’ 2 10 n 3 :3 n
P31
From Ra>lergh scattcnng soA. X = 632 8 nm [6]. str\ hat t&en mto dc‘coun: 3% unccrtint> m the dcpolanzation parameter p0 From ab motto c&uicltton of At”‘” .md ux of cq (7) From ctertnc-field-grJdisnt-Induced birefnngcncs at room temperature corrected b> calculntcd [lYj hqperpolanzabtlity contnbunon the uncorrected values are -147(Y) and 3 60115) for h’, and Cq. respectr.cl> (10-” eau). From temperature dependence of slectnc-field-gr.tdtmt-induced btrefnngcnce From ion scattenng Hark Ab tmtlo ralculauon
. .
-
5. Conclusion
We have carried oui extensive meas&ements of the magnetic-field-induced bircfringence on a series of diamagnetic gases, consisting of &near molecules. over a large temperature range. The primary data were reduced to molar qzlantities ,,,C and analyzed by an expression proportional to f Al) + 2AaA.&rF/45kT where A&., rep;-sents, apart from a small known quantum correction factor, the molecular susceptibility anisotropy. Leaving this factor off would, for example. decrease the quantity AaAt of N, by 2% Like in our previous CME work on ethane, ethene, and ethyue [5]. it was found that neglecting AT would in most cages falsify AaA.$ well beyond the present error limits. The relative accuracy of most Aq values is rather poor. However. the error limits are conservative estimates. and the signs, with the exception of CO, seem to be determined. Improved data could be obtained by employing higher temperatures and perhaps better insulated sample cells. The AT data now available show, however. already interesting trends (including the exceptionally large value AI = 12.2(9) x 1O-43 cm3 G-’ previously observed in ethyne [5]) which might be worthwhile to be followed by theoretical methods. The parameters AaA< have been further analyzed in tables 2 and 3. In the cases of CO, N,O and OCS, accurate values of A.$ and Aa were already known from high-resolution spectroscopy and Rayleigh scattering work. The CME product parameters AaAs of these molecules may therefore, first of all, serve as consistency checks. The results of table 2 show a beautiful harmony between the three independent experimental procedures_ At the same time, it IS seen that the A.$ values which are measured in low rotational quantum states do not deviate perceptibly from their means over the rotational populationThe accuracy of the experimental A,$ values of N?, CO, and CS, in table 3 is that typically achieved in rotational Zeeman effect work on polar molecules [35]. The relative error increases largely when electric quadrupole moments are calculated with eq. (8). It is certainly more advantageous to determine Q ,, of a non-polar molecule from the
213
.
?I_ hlrng. FE Hiimrer i Cotton - Mouton effect of diunzugnezic gases
-temper&ture dependence of- the electric-field-gradie&induced bjrefnngence. _-
Acknowledgement
The support of the Deutsche Forschungsgemeinschaft and of Fonds der Chemischen Industrie is gratefully acknowledged.
References [l] M. Born, Optlk (Spnnger, Berlin 1933) [2] AD. Buckmgham and J A Pople, Proc. Phys Sot (London) B69 (1956) 1133. [3] AD Buckmgham. W H Pnchard and D H Whffen. Trans Faraday Sot 63 (1967) 1057 [4] H KBmg, Ann. Physlk 31 (1938) 289 [5] H Kling. H Gcschka and W Huttner. Chem Phys. Letters 96 (1983) 631. [6] M P. Bogaard. A D. Buchingham. R.K Pterens and A.H. Wfute. J Chem Sot. Faraday Trans I 74 (1978) 3008 [7] G R Alms, A.K Bumham and W H Flygarc, J Chem. Phys 63 (1975) 3321 [g] E Baa and K D ban den Hout, PhysIca 95A (1979) 597. [9] W L
Mccrts.
F-H
de Lcew
[lo]
Phjs 22 (1977) 319. J M L J. Remartz. W L
[ll]
Phys 31 (1978) F-H de Lccuw
[i2] [13] [14] [15] [16] 1171 [18] [19] [20] (21) [22] [23]
19 and A
and A
Dymanus.
Chem
and A
Djmanus.
Chem
Meerts Dlmanus.
Chcm
Phys
Letters
7
(1970) 258. A D. Buckingham. R L. Disch and D A Dunmur. J Am Chem Sot. 90 (1968) 3104. H Khng. E Dreier and W Hiittner, J. Chcm. Phys 78 (1983) 4309 J J M. Bccnakker. Acta Phys Austriaca Suppl X (1973) 267 HP. Mu!holland. Proc Cambridge Ph11. Sot. 24 (1928) 280. A D. Buckmgham and M Panscau, Trans Faraday Sot. 62 (1966) 1. M G. Corfield. Ph D Thesis, Bnstol University. Bnstol (1969). AD. Buckingham and B J Orr. Proc Roy Sot A305 (1967) 259 H. Geschka. S Pfcrrer. H HSussler and W. Hluttner. Ber. Bunsenges. Physlk Chem 86 (1982) 790 R C Weast. rd. Handbook of Chemist9 and Physics (The Chemical Rubber Co., Cleveland, 1974) R D. Amos, Mol Phys 39 (1980) 1 R D Amos and M R Battaglia. Mol Phys. 36 (197X) 1517 J H. W~lhams and RD. Amos. Chem Phys. Letters 66 (1979) 370.
214
H. Ming. IV. Hii~_pr /
Corton - Xfouton effect of diunmgnetrc gases
(24] 3. Bendtsen. 3. Ran-ran Spectry_ 2 (1974) 133. [25] S I. Chart. M R. Baker and N.F. Ramsey. Phys Rev_ 136X (1964) 1224 [26] AS Pine and G. Guelachrlli. I Mel Spectry. 79 (1980) 84_ [27] J-W Ccderbcrg. C H Anderson and N-F_ Ramsey. Phyb Ret 136A (1964) 960. [1X] W J Walker and A. Weber. J. Mel Spectry 39 (1971) 57. [29] R D -4mos. Chem. Phyb Letters 85 (1981) 123. [30] AD. Buckingham. C Graham and J H Wtlhams. Mol. Phys 49 (1983) 703 (311 hl R. Battag,!,!ta.AD. Buclrmgham. D Xcumarh. R K Puxcns and J.H Wtlhams. Mel Phys 43 (1951) 1015
[32] FE Buchenholzer. EA Gislason. A D. Jorgensen and J.G. Sachs. Chem Phys Letters 47 (1977) 419 [33] W B. England. B J. Rosenberg. P_J_ Fortune and A-C. WahI. J. Chem Phys 65 (1976) 684. (341 N F. Ramsey. Molecular Beams (Clarendon press. Oxford. 1956). [33] K-H Hellwege and A M. Hellnege. eds. in: LandoltB&nstein. Numerical data and funchonal relatronsbips m science and technology. New Senes II/6 and II/14 hfokcular constants from microua\e. molecular beam. and electron spin resonance spectroscopy (Spnnger. Be&n, 1974 and 1982) table Z 9.
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\
-:
T~TE~MPE~ATuREDEPENDEN~E~FTHEC~~O~~MOUTONEFFE~ OF N,, CO,-NzO. H. KLING.and
W.
CO,, -_
OCS, AND Cs,
I-kJT”hIER
Abrerlung Chemisehe Pizpk,
Uttrversttiit
_
. -
:
IN THE G&kOUS-STATE __
Ulm; D -7900
_ Ulm.
West Get-nrutt~
Received 30 Apnl 1984
The magnetic-field-Induced birefringence of the SIY substances Na. CO, N,O. CO,. OCR and CS_ has been measured in the gaseous state, between - 70 and + 120 o C and at pressures up to 1 bar. A quantum correction term. accounting for the influence of the rotation-Induced magnetic moments. is derived to modify the eKpressron from classical orientahon theory describtng the Cotton-Mouton effect of linear molecules. The temperature dependence of the measured molar Cotton-Mouton constants ,C(T) has been analyzed in terms of the hyperpolanzabihty anisotropy parameter A1) and the onentation parameter BnL$. The latter is discussed in view of the polarizabihty and suscepubrlity anisotroptes ha and LI< and of electric quadrupole moments Q,, known by other tecbmques- There IS generally good agreement between results from Rayleigh seattenng. high-resolutton spectroscopy, eltoztric-field-gradient-Induced buefringence. and the present Cotton-Mouton effect results
Introduction We have investigated the magnetic-field-induced birefringence(Cotton-Mouton effect, CME) of gases consisting of the linear molecules N,, CO, N,O, CO,, OCS and C&. at pressures below 1 bar. Molecular orientational correlation effects and local field corrections are disregarded. The ellipsometric phase difference, + = 2nC
/
B’df;
(1)
may then be described in terms of parameters by the relations [ 1,21 C=$V’(N/N*)
m
c,
molecular
(2)
and ,C = 2+V,(
fAq + 2AaA.$/ 4:5kT).
(3)
B = B(I) is the magnetic field strength along the pathlength I; X, NA, N, kT are wavelength, Avogadro’s. number, number density and -thermal energy per molecule, respectively. -A+ is a suitably defined [3] hyperpolarizability parameter describing the magnetic field perturbation of the polariza-
bility tensor and Aa = a,, -a, and Ag = <,, -5 i are the molecular polarizability and susceptibility anisotropies. Kdnig’s definition [4] of the molar Cotton-Mouton constant, ,C, is used. Eq. (3), as a result of classical theory, is valid only for the high-temperature approximation and will have to be slightly modified below. From our recent measurements of the temperature dependence of the CME in gaseous non-polar hydrocarbons [5] we concluded that Aq in eq. (3) must not be neglected if one wishes to maintain the accuracy of the empirical phase differences in evaluating the orientation parameters AaA.$. As almost all CME studies on diamagnetic gases reported in the literature were restricted to constant (room) temperature, it appeared important to -extend the investigation of the temperature depen-_ dence to other substances. The polarizab$ity anisotropies Aa of all of the above molecular species -have been determined from observations of the depolarized Rayleigh scattering by various authors [6-S]. Yet-y accurate susceptibility ahisotropies At are; in addition, available from molecular beam electric_resonance Zeeman measurements for the three polar- species CO,-N,O, and OCS [9-111. Thus, measurement of
0301-0104/84/$03.00 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
the slope of the function ,C(T’-‘) given in eq. (3) provides redundant information for Acu and A< in these three cases which seems a rather welcome situation in view of the widely differing experimental and theoretical concepts with their occaslonal simplifications. For example, undiscnminated intensity contributions from the vibraRayieigh tional R&man effect ma> influence scattering results [6]. and unknown centrifugal distortion effects or vibrational dependencies may restrict the transferrability of A=$ values determined in 10~~rovibrational quantum states_ Spectroscopic&y determined susceptibility anisotroples A.$ are not available for the three nonpolar species N,. CO?. and CS,. In these cases. accurate orientation parameters An.L$ are valuable sources for obtaining A& There exists also a connection bet\%een A.$ and the molecular electric quadrupole moment. Q,, _ as established by RamseJ’s relation (see eq. (8) beIow)_ Q,, of nonpolar molecules can be determined rather directI) by the electric-field-gradient-Induced blrefringence [12]. uhich in turn opens an interesting possibility to chech the consistency of independent experimental and theoretlcai methods.
2. Experimental The measurements have been carried out under conditions described in detctii in ref. [5]. The reievant parameters are X = 632-S nm. /-B’ddl= 1.157(5)x 10” G’=5cm. and QETT;O-? x *o_” ‘i; corre, sponding to C,,,, m --I or - referred to gases at normal conditions - ,C,,,, = 5 x lo-”
G-2
cm’ mo]-’
The maximum coLered temperature range \\ds - 70 to 120 OC. which w;1s shortened in some cases owing to 10~~ vapor pressure below the boiirng point. The minimum and maximum temperatures used are indicated in table 1. Commercial purity specifications are &so given in table 1. Whenever possible. samples were degassed at liquid-nitrogen and higher temperatures to remove traces of foreign gases. first of ail oxygen_ Several mdss spectromerric analyses taken from the cell contents have shown that impurities were negligible. There are. however. two other small perturbing
effects to be considered which are obviously of molecular origin. They vary quantitatively from substance to substance, and sometimes they are well resolved with the present ellipsometric sensitivity. First. equihbration of the sample gas after cell filling may take several hours. which is easily monitored by repeated observation of the sIowly converging magnetic-field-induced birefringence. Drifts as large as 5% have been registered. Second. it is frequently observed that CME measurements carried out at narrowly spaced temperatures show very accurately a straight line T-’ dependence. but produce a slightly incorrect slope X/i3(T-‘). In order to avord correlations of this kind. temperatures are usually vaned in large steps in one direction (typically 10 “. increasing or decreasing). and the missing data points are then filled in on the (decreasing or increasing) way back. subdividmg the temperature axis effectively in intervals of = 5 O_ During the course of this work. we have repeated rarher measurements on N, [13] and found discrepancies somewhat outside the error limits. It IS likely that these were caused by the described effects. The exact redsons of the (rather time consuming) irregularities are not understood_ It seems unlikely that orientational correlation, like in molecular liquids. should play a measureable role at pressures below one bar. The fact that the reproducibdity is best at room temperature points to residual temperature gradients which could in principle give rise to magnetic-fieid-induced molecular alignments via the SenftlebenBeenakker effect [14]_ The variation of temperature in our present cell amounts. however. to not more than lo.
3. Data evaluation and esperimental
uncertainties
The low moments of inertia of some of the molecules concerned require to investigate to what extent quantum corrections of the classical Cotton-Mouton constant will be important. In addltion, it cannot be excluded without detailed analysis that rotation-induced magnetic moments may cause contributions to the observed birefnngence. We have recently developed a rigorous quantum statistical expression for the Cotton-Mouton con-
stant of a general dilute diamagnetic or paramagnetic gas (see eq. (10) of ref. [13]) which we can now specialize to the cases at hand. We can simplify the problem to that of an ensemble of linear rigid rotors since the population of excited vibronic states is too low to influence the results significantly. It is then, at the same time, assumed that centrifugal distortion of the molecular parameters is negligible, which has the effect that the least-squares-fitted- parameters represent actually mean values over the rotational population. With these restrictions, eq. (10) of the quoted paper [13] simplifies considerably and reduces to the expression C = (27iR’/h)(3/2) x[(2/45kT)AcrA~(l - (2/45kT)Aa( x(1
-+r-&u’+
--a+&a”+
___)
g”,p;/2hG) ___)+$A&
which is valid for u = hG/kT -=x 1. hG 1s the rotational energy per unit angular momentum. and glpLn the gyromagnetic ratio (pEcn = nuclear magneton). Summations of expansions in the rotational quantum number [15]. utilized by Buckingham and Pariseau 1161 in a similar treatment of the electric-field-gradient-induced birefringence, were useful in developing eq. (4)_ The first term in the brackets was already given by Corfteld [17] and leads, together with the term m AT, to the known classical result in the hmlt u = 0 (see eq. (3)). The second term describes the orIentationa influence of the rotational magnetic moments. It remains in the classical limit, and IS, in distinction from the corresponding well-known orientation term Aa(p,_/kT)’ in the Kerr effect of dipolar gases. proportional to T-’ only (jl~~ is the electric dipole moment). Thus term may become important for heavy molecules with exceptionally large rotational g-factors. Finally, we note that all terms in the third-order hyperpolarizability tensor [2,13] vanish completely. This is also different from the Kerr effect, where an (electric) moment causes a cross term j.Q/kT, fl being the anisotropy of the (electric) third-order hyperpolarizability [18]. The correction terms are easily taken mto
account by introducing an effective susceptibility anisotropy parameter A,& into eq. (3): ,,C = 27rNA( fAq + 2AaA&/45kT),
(5)
where A&
\
= Ac( 1 - u + Aa’) -(&;/2hG)(l
--~u-~u2).
For the gases investigated here, the second term of A& turns out to be an order of magnitude smaller than the experimental uncertainty, and will therefore be disregarded_ The remaining correction factor, truncated to first order in u, affects the results of N, and CO up to one standard deviation and should therefore be retained. The values of 1 - u at room temperature are listed in table 1. The plots ,C(T) versus (1 - u)/T are given in figs. i and 2. and show the expected linear behaviour. Van der Waals corrections have been applied to obtain the
(14 3
L
T)lT-‘/lQ3K-’ 5
I
mC/lO-“cm3 G-%wJ~-’
Ftg 1. The molar Cotton-Mouton constants of Nz. CO, N,O and co2 \ersus (1 - a)T_ ‘_ see eq (6) The error bars are i one standard deviation, obtamed from a senes of measurements at constant temperature, and added contribut:ons from uncertainties in the number density. A’. The strrtighthnes result from least-squares fitting these data to the slopes and ordinate crossing points which in turn yield the molecular parameters in table 1.
(l-a(T))
T-‘/lOSK-’
O@fl
,,,C/lO-%n3
G-*mol-
Flp 1 The molar Cotton-Mouton wnbtants of OCS and CS2 compare the tc\t beloa fig I Obsenc the \cr3u5 (1 - a)7-1. decreased ordmats sale H hlch maha the error bars unrcpresentable 1x1the case of OCS
number densitres from the pressure and temperature readings. The parameters CIand b uttltzed are gtven in the first column of table 1. together with the commercial sample puritres. The (vertical) error bars in figs. 1 and 2 represent one mean standard deviation to either side of the entries. where each one is the result of a series of about ten single ellipsometric compensations. The errors caused by the uncertainty m the pressure gauge readings (0.005 bar. a conservative estimate) are also included_ They dre comparably small near normal pressure. but contribute signifiin the low-temperature. small-pressure cantly ranges of the coudensable gases. An increased scale would show that doubling the error bars is sufficient to bring the least-squares-fitted straight lures to withm the range of each single entry. We have therefore chosen to assign two standard deviations to the parameters 1111and AcxA& which clre given in table 1 ds the final result of a weighted least-squares procedure for each gas. The rather small multiplicative systematic error contributions arising from /B’dl and the Faraday compensator calibration are also included. Finally. an addittve systematic uncertklty corresponding to a phase difference of f 2 x lo-% is introduced_ The latter covers empty-cell effects which are sometimes ob’ served and well below that signal strength_ The additive error drops out in determining Aa.L$. but increases the error of A2 typically by 7 X 10eA3 cm3 G-‘. compare table 1.
4. Results_ The vahres of AIJ. A.aA& and ,,,C at room temperature of all gases are collected in the fourth to sixth columns of table 1. Those of (AaA<), in column seven are obtained from the room-temperature ,C values under the constraint Aq = O_ Several room-temperature ,C values of the gases considered here have been published previously by different authors [3,17,19]_ We have compiled them in the last column of table 1. The literature ,C values not measured at He-Ne laser wavelength were adJusted to X = 632.8 nm with the aid of the polarizabtlity anisotropies obtained at various wavelengths by Bogaard et al. [6] and by Ahns et al. [7]. Fortunately. there are no dramatic disagreements, though the uncertainty ranges somettmes do not overlap. The (A(rAE),, values are generally not consistent with their counterparts &A.$ obtained from the slope of the (quantum corrected) ,C(T-‘) function, which shows that the temperature dependence has to be surveyed for full exploitation of the experimental accuracy. The differences lie. however, below lo%, while it was previously found in ethyne [5] that AaA& deviated from (A~YAE), by as much ds 30%. Comparison of the data in the fourth column of table 1 shows that introduction of a sulfur atom increases A17 substantially_ A doubling of the hyperpolarizability anisotropy of CSz m comparison to OCS. as indicated in table 1, seems therefore plausible. Unfortunately, the accuracy of the carbon disulfide result is strongly reduced as a consequence of the high boiling point of this substance. It is now interesting to discuss the orientation parameters AaAt in the light of Aa and A,$ values obtained previously with other methods. Accurate spectroscopic molecular-beam measurements are available for the susceptibility anisotropies of the three polar molecules. We have used them to calculate the polarizability anisotropies, and the resulting values are collected in table 2. Recent values determined by depolarized Rayleigh scattering are also included for comparion. As can be seen. the new values are in excellent agreement with the most accurate literature results, and equally accurate-
Table
L
Hyperpolarizabdity and &Id-orientation ptiamctrrs from the ordinate crossing pomts and slopes in figs. 1 and 2 (fourth and fifth column). and firld-orientation parameters from room-temperature ,,,C co%tants_(sixth c@umn) and cq (5). $th AT = 0 (seventh column). The temperature ranges and quantum correcttons (ai room temperature) are Indicated ~ti the second and third columns The errors include two standard deviations from the fitting routme and additional systematic coniributions, see text. All measurements have been carried out with HeNe laser radiation_ The room-temperature ,,,C values from other authors are listed in the last column Substance a)
7m,n:Tmzt
l-lYb)
(“C)
.lp (IO-”
cm3
AC&$ (10es3 cm6)
,c =’ (IC-‘“cm3 G-’ mol-t)
G-‘)
,c d’ (10-t”
(AQA&J (1O-s3 cm6)
G-’
cm3 mol-t)
-75:17-o
0 9903
2 6(20)
- 103(4)
-0 391(20)
- 0 95(5)
-0 -0 -0
- 64;120
0.9906
0 2(16)
- 0 75(2)
- 0.305(20)
- 0 75(4)
- 0 35(7) [3]
N1O (99.5-3.78:4 42)
- 84.92
0 9980
- 4 8(34)
-5 20(11)
- 2 21(4)
- 5 34(Y)
-2
co, (99.995.3 59.4 27)
-56:120
ocs (97 5:3 93,5 82)
-61;70
N, (9Y.YYY;l 39 3 91) co (99 97.1.49.3 99)
39(3) 131 38(2) [17] 50(10) [lY]
-0 29(l) [17] 31(5) (31
- 2 04(4) r171 -2 12(22) [19] 0 9981
- 2 Y(B)
- 2 21(4)
-O-95(2)
- 2.30(6)
- 0 89(5) [3] -0 90(4) P71 - 0 95(14) [19]
5,121
CSI ( > 99 6.116;7 69)
0.9990
- 23 (3)
0 9995
- 47 (29)
-6
33(12)
- 2 91(4)
- 29.1(11)
- 7.04(10)
- 12 6(2)
- 2.58(9) [17] -2 85(26) [IY] - 11 j(2) [17] - 12 2(12) [lY]
- 30 5(j)
il’ The first number in parentheses IS the commercial purity specification in volume%, the second and third are the van der Waats constants LI (lo6 cm3 atm mol-‘) and h (10 cm3 mol-’ ) taken from ref [20] and used to determine N in eq (2). b’ See eq (5). values referred to 21 OC ‘) Referred to 21 “C ” Literature room-temperature results Those of refs [3.17] were obtained at Increased pressure. with the exception of CS_. Values measured at different wavelengths were adJusted lo 632 8 nm usmg the dispersion mformatlon
For the non-polar
species there are no spectra-
scopic At parameters determine
available,
them by reversing
with the aid of the Rayleigh of Bogaard
and, therefore,
AEd’=‘, and using the relation
we
the above procedure, scattering
Acx values
et al. 161. The results are collected
table 3, and compared
for Aa m refs [6.7]
with semi-empirical
= Atdla + (e’/4n@$,c’)l,g~)
(7)
in
values
(e
and
MP are the proton
charge
and mass, and
Amos and others [21-231. These authors have obtained them by ab initio calculating the diamag-
r~z, and c are the electron mass and speed of light)_
netic contribution
contribution
of
of the susceptibility
anisotropy,
I I,
the moment
of inertia, and
to the rotational
g$),
the electron
g-factor,
are accu-
Table 2
Comparison of differently
obtamed polatizability anisotropies Aa. The values in the second row are calculated wtth AaAS in table 1 and At in the first row, ail others are from Rayleigh scattering work. Unless otherwise stated. ail Aa refer to 632.8 nm co A5 (10m30 cm3) Aa (lo-”
cm3)
- 13.72(2)
N,O
ocs
Ref.
- 17.32(10)
- 15 56(2)
[Y-11] this work
0.532(10) =)
3.oo(6) 2.946) a1
4.07(8) 4_077(76) a)
PI
O-53(2)
2 96(6) b’ 3-l(2) =’
4 14(g) 4 O(2)
171 PI
O-543(19)
-
*) We have taken into account 3% uncertainty In the depolarization c, X = 514.5 nm. b)X=647nm.
parameter pO_
212
rately known from Raman effect and molecularbeam magnetic resonance studies. There is generally good agreement between theory and experiment_ Eq. (7) represents d difference of two large quantities. meaning that Atdra was calculated with much higher relative accuracy than evident from table 3. The knowledge of i1.$ can be extended to that of the electric molecular quadrupole moments by using the well-known relation 1341
(8) The results are also collected in table 3, and compared with the values which follow more directly from the method of the field-gradient-induced birefringence [l?]. This experiment is, analogously to the CME. described by an orientation term containing the quadrupole moment. and a temperature-independent hyperpolarizability contribution_ In the case of CO?. Battaglia et al. [31] have investigated the temperature dependence and deduced Q,, = -4.49(15) X 10ez6 esu from the slope of the T-’ plot. This value is almost in agreement
with ours<--4 2(2)x 10mZ6esu) within the experimental uncertainties. For N, and d&, only measurements at Yoom temperature seem to be available [30.31], but Amos has calcul&ed the (rather small) hyperpolarizability contribution [29]. We have combined both to extract Q,, and tind agreement with our own values for the two molecules (see table 3). It is interesting to realize that the quadrupole moment changes sign in going from CO, to CS,. This trend is also followed by the known ab initio values which we have included in table 3. One of them (2.1 x lo-‘” esu of CS,) is far beyond the experimental error limit. The quality requirements for wavefunctions are here obviously higher owing to the near cancellation of nuclear and electronic contributions. Finally. we have listed two quadrupole moments m table 3 which stem from ion scattering investigations [32]. The agreement is encouraging and calls for further improvement of this independent techmque.
-l-able 3 Comparilson ol dIfferentI> obtamrd susceptIbtht> nnwotropies A $ .md electnc quairupolc moments Q,# The ~.du~~ AC m the flrbt rob of catch moleatlc arc calculated =tth .InL< m t.tble 1 .md An from Ra>iielgh scnttcrmg gwcn m the wcond column. the corrc~ponding ~aluez QIz m turn mlth cq (S) usng (C(GHz) g, ) = (59 646 [2-I]: -0 2777(s) [Xl). (II 6Y8 [Xl. -0 055OS(5) [37]) (3 1713(Z)
5,
co,
[lS]
- 0 Olii-J(Z)
0 704(13)
2 11(J)
[17]) for N,.
CO,
.md CS,.
- i-1 6(6)
- 110 h’ - LO j(3)
-YY cs,
947(18)
rc.\pcctl\cI\
h’
- 30 7(13) h’ - ‘9 7 h’ - 777
The ongms
of the other
~.ducs
are e~plamed
m footnotes
- 1 3(-I) - 1
c’
4 Z(Y) 3 61( 1s) .’ 2 10 n 3 :3 n
P31
From Ra>lergh scattcnng soA. X = 632 8 nm [6]. str\ hat t&en mto dc‘coun: 3% unccrtint> m the dcpolanzation parameter p0 From ab motto c&uicltton of At”‘” .md ux of cq (7) From ctertnc-field-grJdisnt-Induced birefnngcncs at room temperature corrected b> calculntcd [lYj hqperpolanzabtlity contnbunon the uncorrected values are -147(Y) and 3 60115) for h’, and Cq. respectr.cl> (10-” eau). From temperature dependence of slectnc-field-gr.tdtmt-induced btrefnngcnce From ion scattenng Hark Ab tmtlo ralculauon
. .
-
5. Conclusion
We have carried oui extensive meas&ements of the magnetic-field-induced bircfringence on a series of diamagnetic gases, consisting of &near molecules. over a large temperature range. The primary data were reduced to molar qzlantities ,,,C and analyzed by an expression proportional to f Al) + 2AaA.&rF/45kT where A&., rep;-sents, apart from a small known quantum correction factor, the molecular susceptibility anisotropy. Leaving this factor off would, for example. decrease the quantity AaAt of N, by 2% Like in our previous CME work on ethane, ethene, and ethyue [5]. it was found that neglecting AT would in most cages falsify AaA.$ well beyond the present error limits. The relative accuracy of most Aq values is rather poor. However. the error limits are conservative estimates. and the signs, with the exception of CO, seem to be determined. Improved data could be obtained by employing higher temperatures and perhaps better insulated sample cells. The AT data now available show, however. already interesting trends (including the exceptionally large value AI = 12.2(9) x 1O-43 cm3 G-’ previously observed in ethyne [5]) which might be worthwhile to be followed by theoretical methods. The parameters AaA< have been further analyzed in tables 2 and 3. In the cases of CO, N,O and OCS, accurate values of A.$ and Aa were already known from high-resolution spectroscopy and Rayleigh scattering work. The CME product parameters AaAs of these molecules may therefore, first of all, serve as consistency checks. The results of table 2 show a beautiful harmony between the three independent experimental procedures_ At the same time, it IS seen that the A.$ values which are measured in low rotational quantum states do not deviate perceptibly from their means over the rotational populationThe accuracy of the experimental A,$ values of N?, CO, and CS, in table 3 is that typically achieved in rotational Zeeman effect work on polar molecules [35]. The relative error increases largely when electric quadrupole moments are calculated with eq. (8). It is certainly more advantageous to determine Q ,, of a non-polar molecule from the
213
.
?I_ hlrng. FE Hiimrer i Cotton - Mouton effect of diunzugnezic gases
-temper&ture dependence of- the electric-field-gradie&induced bjrefnngence. _-
Acknowledgement
The support of the Deutsche Forschungsgemeinschaft and of Fonds der Chemischen Industrie is gratefully acknowledged.
References [l] M. Born, Optlk (Spnnger, Berlin 1933) [2] AD. Buckmgham and J A Pople, Proc. Phys Sot (London) B69 (1956) 1133. [3] AD Buckmgham. W H Pnchard and D H Whffen. Trans Faraday Sot 63 (1967) 1057 [4] H KBmg, Ann. Physlk 31 (1938) 289 [5] H Kling. H Gcschka and W Huttner. Chem Phys. Letters 96 (1983) 631. [6] M P. Bogaard. A D. Buchingham. R.K Pterens and A.H. Wfute. J Chem Sot. Faraday Trans I 74 (1978) 3008 [7] G R Alms, A.K Bumham and W H Flygarc, J Chem. Phys 63 (1975) 3321 [g] E Baa and K D ban den Hout, PhysIca 95A (1979) 597. [9] W L
Mccrts.
F-H
de Lcew
[lo]
Phjs 22 (1977) 319. J M L J. Remartz. W L
[ll]
Phys 31 (1978) F-H de Lccuw
[i2] [13] [14] [15] [16] 1171 [18] [19] [20] (21) [22] [23]
19 and A
and A
Dymanus.
Chem
and A
Djmanus.
Chem
Meerts Dlmanus.
Chcm
Phys
Letters
7
(1970) 258. A D. Buckingham. R L. Disch and D A Dunmur. J Am Chem Sot. 90 (1968) 3104. H Khng. E Dreier and W Hiittner, J. Chcm. Phys 78 (1983) 4309 J J M. Bccnakker. Acta Phys Austriaca Suppl X (1973) 267 HP. Mu!holland. Proc Cambridge Ph11. Sot. 24 (1928) 280. A D. Buckmgham and M Panscau, Trans Faraday Sot. 62 (1966) 1. M G. Corfield. Ph D Thesis, Bnstol University. Bnstol (1969). AD. Buckingham and B J Orr. Proc Roy Sot A305 (1967) 259 H. Geschka. S Pfcrrer. H HSussler and W. Hluttner. Ber. Bunsenges. Physlk Chem 86 (1982) 790 R C Weast. rd. Handbook of Chemist9 and Physics (The Chemical Rubber Co., Cleveland, 1974) R D. Amos, Mol Phys 39 (1980) 1 R D Amos and M R Battaglia. Mol Phys. 36 (197X) 1517 J H. W~lhams and RD. Amos. Chem Phys. Letters 66 (1979) 370.
214
H. Ming. IV. Hii~_pr /
Corton - Xfouton effect of diunmgnetrc gases
(24] 3. Bendtsen. 3. Ran-ran Spectry_ 2 (1974) 133. [25] S I. Chart. M R. Baker and N.F. Ramsey. Phys Rev_ 136X (1964) 1224 [26] AS Pine and G. Guelachrlli. I Mel Spectry. 79 (1980) 84_ [27] J-W Ccderbcrg. C H Anderson and N-F_ Ramsey. Phyb Ret 136A (1964) 960. [1X] W J Walker and A. Weber. J. Mel Spectry 39 (1971) 57. [29] R D -4mos. Chem. Phyb Letters 85 (1981) 123. [30] AD. Buckingham. C Graham and J H Wtlhams. Mol. Phys 49 (1983) 703 (311 hl R. Battag,!,!ta.AD. Buclrmgham. D Xcumarh. R K Puxcns and J.H Wtlhams. Mel Phys 43 (1951) 1015
[32] FE Buchenholzer. EA Gislason. A D. Jorgensen and J.G. Sachs. Chem Phys Letters 47 (1977) 419 [33] W B. England. B J. Rosenberg. P_J_ Fortune and A-C. WahI. J. Chem Phys 65 (1976) 684. (341 N F. Ramsey. Molecular Beams (Clarendon press. Oxford. 1956). [33] K-H Hellwege and A M. Hellnege. eds. in: LandoltB&nstein. Numerical data and funchonal relatronsbips m science and technology. New Senes II/6 and II/14 hfokcular constants from microua\e. molecular beam. and electron spin resonance spectroscopy (Spnnger. Be&n, 1974 and 1982) table Z 9.