NOTES
2367
The Rotatory Dispersion of Quartz1
by Leonard I, Katzin Chemistry Division,Argonne National Laboratory, Argonne, Illinois (Received M a r c h $0, 1964)
Over a period of some years, Lowry and co-workers accumulated a body of data on the rotatory dispersion of quartz213wliich, for precision and spectral span, is unmatched in the literature on this or any other substance. These data were fitted x\ith an equation which described the degrees of rotation per mm. thickness of quartz, with wave length in microns
RL
=
9.5639/(X2 - 0.0127493)
-
2.3113/(X2 - 0.000974) - 0.1905
to more modern values. The purpose of the investigation was to test the match to the data for which he had apparently already developed a good parametric fit, rather than to make the ultimate adjustment possible in the parameters. In the case of the rotations taken with the lines of the copper spectrum, Lowry preferred the wave length values of Huppers over those of Hasbach.2 Our data fitting showed that the Huppers values actually gave poorer fits. We omitted the copper spectrum results completely in the final computations, to avoid this problem. The least-squares fitting program found to operate satisfactorily in earlier ~ o r kwas ~ , used, ~ with the IBM 704 computer.
Results (1)
The equation was first developed to fit the visible range but apparently agreed with observations over the whole wave length span to the order of thousandths of a degree. Actual rfftation values varied from about 187" per mm. a t 2327 A. to 0.52" a t above 3 p . Inasmuch as the Full original data were available (thanks to the publishing policies of the day), it wab considered useful to investigate whether with modern computer-based fitting methods4J it was really necessary to invoke a fifth parameter, and what the final parameters might show. A further goal of such an analysis would be l,o see how the parameters for optical activity dependent on crystal dissymmetry might relate to the parameters familiar from experience with asymmetric small molecules.
Procedure At some point in the protracted course of Lowry and his co-workers) measurements on a 0.5-m. column of quartz, the sample was reground. On the basis of correspondence a t selected points (the visible), the earlier data were corrected to match to the data obtained with the reground specimen. To avoid such questions as this procedure might raise, the analysis in this report is confiiied to the data obtained with the reground specimen directly. For the first paperj2 this means some 250 points taken at from about 67072327 A. All the infrared data of the Lowry and Snow report3were obi ained with the reground specimen. Some errors in the computation of the original tabulation of resultsZwere detected, generally because they gave bad errors in Lowry's results, and were corrected, thanks again to the data being .+vailable in extenso. KOattempt was made, however, td correct or alter the wave length values given in Lowry's tables
With 250 points from 6707.846 to 2327.49 8., and rotations from 16.535 to 187.247", a two-term Drude equation can be fitted to the data with a root-meansquare deviation of *0.017", and no single deviation as high as 0.100". However, there are unmistakeable signs of systematic misfit to the data. Confining tttention to the 150 points between 4820 and 3073 A., these can be fitted by the equation
RA
=
122.1035/(X2 - 0.00669422) 114.9261/(X2 - 0.00599502)
(2)
to a root-mean-square deviation of *0.0038". Only one point deviates as much as 0.010". A point-bypoint comparison with the deviations from Lowry's equation indicates that, unrecognized by him, there existed a systematic preponderance of negative deviations (observed minus calculated) which our automated statistical fitting eliminates. This aniounts to an approximately 0.004" shift, in the positive direction, of the individual deviations. The correspondence to these 150 points, by eq. 2, is as good as that using the five-parameter equation of Lowry and Coode-Adams. Use of the eq. 2 parameters outside of the range of these points, however, predicts a rotation 0.065" too high at the 6707 A. lithium line, and 0.925" too low at the 232'7 A. line. These are discrepancies of 0.39 and 0.5%, respectively. The 66 shortest wave length points, from 2327 t o ~~
(1) Based on work performed under the auspices of the U. S. Atomic
Energy Commission. (2) T. M.Lowry and W. R C. Coode-Adams, Phil. Trans., Roy. Sac. (London), A226, 391 (1927). (3) T. M.Lowry and C. P. Snow, Proc. R o y . Soc. (London), A127, 271 (1930). (4) L. I. Katzin and E. Gulyas, J . Phys. Chem., 6 6 , 494 (1962). ( 6 ) L I. Katzin and E. Gulyas. J . Am. Chem. Soc., 8 6 , 1685 (1964).
V o l u m e 68, AV?;umher 8 A u g u s t , 1964
NOTES
2368
2887 i., with rotations 107-187', the equation
Rx
=
are duplicated by
134.4039/(X2 - 0.00931987) 127.1814/(X2- 0.00890141) (3)
to a root-mean-square deviation of *0.006'. A group of 27 long wave length points from 6162 to 7741 A, (data on the quartz before regrinding) fit the relation
Rx
=
374.7112/(X2 - 0.0022092) 367.6544/(X2 - 0.00178683) (4)
to &0.010', which is about the experimental reliability claimed for these data. It is easiest to compare and understand the iniplications of these relations if they are converted from the two-term Drude equation to the algebraically equivalent form4,5
R x -- [ ( A - B)X2 -
(AXb2
- BXa2)]/(X4 -
(haz =
+
Xb2)X2
-k
Xa2Xb2)
(5)
[ ( A - B)(X2- L ) - C]/[(X2 - L)' - DI
+
where L = (A,' Xb2)/2. The significant parameters,' ( A - B ) and L , for the several equations, are sunimarized in Table I, for easy comparison. The ini-
Table I : Comparison of Principal Rotatory Dispersion Parameters Spectral span
Full spectrum 6162-7741 8. 3073-4820 8. 2327-2887 b.
contribution, with a negative sign, from a transition in the far-infrared. This is in fact the premise that Lowry had found necessary, though Drude had considered it negligible, and which was expressed as his fifth parameter. Our analysis has shown that it is not possible at all to represent the full quartz data accurately without such a term. The infrared data of Lowry and Snowa were combined with the Lowry and Coode-Adanis2 data to give 332 points ranging from 187.247' rotation a t 2327.49 8. to 0.520' at 32,100 11. These were then analyzed according to the three-term Drude relation. A fit to a root-mean-square deviation of =k0.00376' is given by
R -
-
A2
127.02476 119.77145 - 0.009584310 X2 - 0.009169645
+
3.413 X lo4 (6) X2 - 1.8163 X lo5 The actual significance of the last term is that, for data through 3-4 p in wave length, it represents a constant, -0.1879. This is because the critical wave length comes a t some value sufficiently large that a X2 of 10 does not affect the value of the third tern1 in (6) in the third decimal place ( i e . , to O.OOlo). It can be seen that both parameters of this third term could be smaller by a factor of 10, or perhaps more, without altering the fit to 0.001'. Therefore, one may say that this term may represent an absorption somewhere in the region of 100 p (100 cni.-'), perhaps as low as 50 p or as high as several hundred p .
Discussion (A
- B)
7.1984 7.0568 7.1774 7,2225
L
0.008351 0.001998 0.006345 0,009111
portance of these parameters may be seen by considering that the dispersion curve for a two-term Drude relation is given approximately by the single Drude term, ( A - B)/(X2- L ) , with the corrections to this representation given by the parameters C and D, as eq. 5 suggests. It is seen that there is a steady transition, with wave length, of the best parameters for a fit. Because of this, an attempt to niatch the full wave length span of the data with only four parameters results in the systeinatic mismatch already noted. The fact that the parameters of the shortest wave length section overestimate the rotation a t the longer wave lengths, as do those of the central section also, necessitates consideration of an additional rotatory The Journal of Physical Chemistry
The unique body of data on the optical rotatory dispersion of quartz, from 2327 8. through 32,100 b., left by Lowry and his co-workers, leaves no room for question that a Drude equation of three terms is required to describe the rotation. Because the data terminate a t 3.2 p, the parameters of the third (infrared) Drude term are known only as their ratio, leaving us an equation with five parameters. It may be pointed out that the two terms originating in the ultraviolet may, a t 3 p, be effectively represented, to 0.001', by a single Drude term, ( A - B)/(X2- L ) . Similarly, the tern1 originating in the far-infrared may also be composed of two Drude terms. Data a t much longer wave lengths would be required to make this resolution, and it would be highly desirable that such measurements be made. The five parameters in Lowry's equation (eq. 1) (6) Hereafter the parameters of eq. 5 mill be called polynominal parameters, to distinguish them from the parameters of the Drude equation as normally written.
NOTES
2369
bear little direct resemblance to those in eq. 6, other than the values for the infrared term (-0.1905 and - 0.1879, respectively). Taking the ultraviolet terms in the polynorninal form, however, (6) becomes 72533X2 - 0.016845 R --_A - X4 - 0.018754X2 8.789 ;
