JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Anomalous Dispersion of g-Carotene Solutions*

WALLACE B. ALLENt AND PAUL LATIMER1Biophysical Laboratory, Department of Physics, Vanderbilt University, Nashville, Tennessee

(Received June 30, 1961)

Refractive indices of liquids were determined by measuring reflectance at a glass-liquid interface, measur-ing transmittance of the liquids, and calculating their relative refractive indices from a Fresnel reflectionequation of Simon. The refractive index of the glass was measured independently and used to calculate nvalues of the liquids relative to vacuum. The spectral region investigated, 400-589 mgi, includes four absorp-tion bands of 0-carotene. Dispersion curves of two solution-solvent pairs were measured. The differences,Alsoluite=ntsolution-nsolvent=f(X), were compared with the predictions of quantum dispersion equationsof Davydov.

INTRODUCTION

B ECAUSE of interest in anomalous dispersion, bothin optical studies of biological materials', 2 and in

infrared spectroscopy, 3' 4 we measured dispersion curves[n= f(X)] of fl-carotene solutions and of the parentsolvents near and within absorption bands. Curves ofAllsoIute, Aflsolute=1solution-11solvent, were calculated from

the experimental results and also from Davydov'squantum dispersion relations for solutions.5' 6 Thisappears to be the first quantitative comparison of certainpredictions of these equations with experimental curvesfor electronic excitation bands of solutions.

The refractive index nt* of an absorbing medium is acomplex number: n*=n-iK, where n governs phaseand K is proportional to absorbance. If K is negligible,n can be readily measured by critical-angle refrac-tometry. However, if the medium is strongly absorbing,large corrections based on independently measuredK values are required (e.g., see reference 3). Both n andK govern reflection and may be evaluated from twodifferent reflectances. Simon7 gives Fresnel equationswhich permit the evaluation of n and K values fromexperimental reflectances for two different angles ofincidence. Kiseleva and Pribytkova8 recently used other

* This investigation was supported by a USPHS research grantand in part by a NSF grant. Parts of this work are from a thesis tobe submitted by Wallace B. Allen in partial fulfillment of therequirements for the M.S. degree in the Graduate School ofVanderbilt University, Nashville, Tennessee.

t Presently Major, U.S.A.F. (M.S.C.), Health Physicist, 7112Central Medical Group, APO 332, New York, New York.

$ Investigator, Howard Hughes Medical Institute at VanderbiltUniversity, Nashville, Tennessee.

P P. Latimer and E. Rabinowitch, Arch. Biochem. Biophys. 84,428 (1959); See also P. Latimer, Plant Physiol. 34, 193 (1959);and R. A. MacRae, J. A. McClure, and P. Latimer, J. Opt. Soc.Am. 51, 1366 (1961).

2 E. Charney and F. S. Brackett, Arch. Biochem. Biophys. 92,1 (1961); See also F. S. Brackett and E. Charney, J. Opt. Soc. Am.50, 811 (1960).

3 J. H. Jaffe, H. Goldring, and U. Oppenheim, J. Opt. Soc. Am.49, 1199 (1959).

" T. S. Robinson and W. C. Price, Proc. Phys. Soc. (London)B66, 969 (1953).

5 P. Latimer, J. Opt. Soc. Am. 51, 116 (1961).'A. S. Davydov, Izvest. Akad. Nauk S.S.S.R., Ser. Fiz. 5, 523

(1953). (English translation by H. W. Milner available from Trans-lation Center, John Crerar Library, Chicago, Illinois.)

I. Simon, J. Opt. Soc. Am. 41, 336 (1951).8 N. K. Kiseleva and N. N. Pribytkova, Optika i Spektroskopiya

10, 133 (1961).

forms of these equations to determine n and K fromsample reflectances at one angle using two differentboundary media. Refractive index n can also be deter-mined from a single reflectance (using Simon's equa-tions) if K is determined from transmittance. We usedthis latter method.

EXPERIMENTAL APPARATUS AND PROCEDURE

To a first approximation, reflectance R at a boundarybetween an absorbing and a nonabsorbing medium isgoverned by n and K:

R- (n- 1)2+K2, (1)

where n is now the refractive index of the absorbingmedium relative to the adjacent medium at the boun-dary. Since Ansoiute will normally be small comparedwith n, the contribution of the solute to reflectance canbe readily measured only if (nhiquid- 1)2<<1. We meas-ured reflectance at glass-liquid interfaces which fulfilledthis condition. For one solution-solvent pair, it- 1>0;in the other case n-1<0.

The experimental apparatus is shown schematicallyin Fig. 1. A collimated incident beam from a B &L grat-ing monochromator was directed through the glassprism to the glass-liquid boundary and out to a photo-multiplier tube (1P28). A galvanometer (sensitivity:2 -109 amp/mm) measured the dc photomultipliersignal. A prism was used for the boundary medium inpart to facilitate isolation of the reflected light fromstray incident light. The angle of incidence at the bound-ary, 4'= 35.4', was determined by removing the vessel,measuring the angle of refraction of light (X=589 mpu)passing out of the prism into air, and calculating theangle of incidence from nPrism with Snell's Law. Inthis case, effects of prism dispersion were neglected.

The sample liquid was contained in a vessel with anopen top and bottom and ground lower edges. Thevessel was pressed against the prism so that leakagewas minimized by surface tension and by solute residuein the case of the solutions. The vessel was shaped likea Wood's horn to minimize scattering of light trans-mitted by the boundary back into the beam.

To determine incident flux (reflectance= reflectedflux/incident flux), we removed the vessel and measured

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VOLUME 52, NUMBER 2 FEBRUARY, 1962

Februaryl962 ANOMALOUS DISPERSION OF

light reflected at the glass-air boundary. From thecalculated reflectance of this boundary7 the measuredreflected flux was corrected to obtain total incidentflux. Of course the measured light fluxes were functionsof the characteristics of the optical system. However,since these characteristics remained constant-the onlychanges in the system during measurements being theinsertion or removal of the liquid vessel and the chang-ing of the detector sensitivity-the experimental ratiosof light fluxes are equal to reflectances.

To determine spectral-reflectance curves, we meas-ured separate spectral curves of detector response toincident and reflected light and relied on the stability ofthe light source and reproducibility of the setting of themonochromator wavelength drum. The reproducibilityof the data gave adequate evidence that the necessaryconditions had been fulfilled.

To permit calculation of refractive indices of liquidsrelative to vacuum from reflectances at a glass-liquidboundary, the refractive index of the glass prism wasmeasured in the region of 400-589 mA by the method ofangle of minimum deviation. To check one value,Hprism: 1.5369 at 589 mAt, and also the reliability of thereflectance apparatus, the apparent reflectances ofseveral solvents of different refractive indices (obtainedwith a water-calibrated Abbe refractometer) weremeasured. In some cases, a trace of pigment was addedto the solvent to absorb any light which might passthrough the boundary in such a way as to be scatteredto the light detector. The results indicated a minimumapparent reflectance of 4.00-10-5 for sovent=1.5199.

This result indicated the existence of two sources ofexperimental error. (1) Ideally, as n-il (solventc

nprism), R-HO. The residual R value was attributed toreflection at defects in the glass surface or to failures inlight baffling within the apparatus, and 4.00- 10-5 wassubtracted from all experimental reflectances. (2) Sincethe minimum apparent reflectance should obtain whennsolvent= lprism (the principle on which the immersiontechnique is based), we assumed the surface of theprism to have a refractive index of 1.5199 at 589 mA andsubtracted 0.0170 from all points on the above-meas-ured dispersion curve of the prism. The difference

PM TUBE

FIG. 1. Apparatus-schematic diagram.

t ~vE AU

4! ~~~~~~~~~~~Solution

1000

400 450 500 550 600

FIG. 2. Reflectance of solution I (-carotene in benzene±CS2)and solvent I for glass-liquid boundary, and absorbance of solu-tion I.(ni~dnz 8 )

between the two n values was attributed to differencesbetween the bulk and surface properties of the glass.

The accuracy of the above experiment was clearlysufficient to necessitate both corrections at 589 myi. Ifthe assumption of wavelength invariance of eithercorrection was not appropriate, errors in the absolutemagnitudes of the nliquid values at wavelengths awayfrom 589 mgu were introduced. However, the effects ofthese errors on An values would be second order only.

A final check of the method and apparatus was ob-tamned by comparing n values calculated from reflec-tance measurements with n values obtained with anAbbe refractometer. Measurements~ were made at 589mt4, where all K values were negligible. The results areshown in Table I.

Since it was necessary to use a wide monochromatorslitwidth (band half-width = 13 mgA), a correction wasmade for its effects. Each observed reflectance wasassumed to be made up of reflectances at five wave-lengths including the nominal one. Thus, at the mthwavelength,

(Observed Reletn = A Sm,2Rm2+ BSm~iRm~

±CSmRm±DS+Rm+i+ES+2Rm,+2, (2)

where Rm,, is the actual reflectance at wavelength m, Sis a weighting factor which is proportional to phototuberesponse to monochromator output (Sm =1, S-i, etc.,were evaluated for each m value), and A, B, C, D, and Ewere determined from the band half-width (0.010,0.1773, 0.6254, 0.1773, and 0.010, respectively, forXm-Xml=10 m). Using four estimated R values forwavelengths outside the spectral region studied, simul-taneous equations of the above form for twenty X values

0-CAROTENE SOLUTIONS 193

19W . B. ALLEN AND P. LATIMER

2000 Absorbance

Soltion

1000

400 450 500 550 600A (ovu)

M-FIG. 3. Reflectance of solution II (-carotene in heptane+benzene) and solvent II for glass-liquid boundary, and absorbanceof solution II. (l&a->nliquid).

were set up and solved (on IBM 650 digital computer)to obtain corrected reflectances.

Working solutions of 3-carotene (from the CaliforniaFoundation for Biochemical Research) were preparedby adding an excess of the crystal to mixtures of reagentgrade solvents (solvent I: benzene+CS 2 ; solvent II:,i-heptane+benzene) and filtering with Whatman No.50 filter paper. K values of the working solutions wereobtained by diluting the solutions by a factor of 103,measuring aC (in logIolo/I= aCt, where t= 1 cm) with aBeckman DU spectrophotometer, multiplying theseaC values by 101, and using C= 2Kw/cLne1O (Seereference 5). The present product aC is equal to a inthis reference. a is a characteristic of the solute, C isconcentration of solute, w is angular frequency, and c isvelocity of light in vacuum. If the extinction coefficientsof /3-carotene in our solvent mixtures and in heptane 9

are the same, the concentrations of the working solu-tions were 7-8 g/liter.

RESULTS

Figures 2 and 3 show spectral reflectance curves oftwo different solution-solvent pairs together with

TABLE I. Comparison of n values measuredby independent methods.

Material it it Difference(Abbe

(Reflection) refractometer)

Solvent I 1.5541 1.5520 -0.0021Solution I 1.5563 1.5578 +0.0015SolvQt 11 t,4114 1 a8on +00,nQ3Solution II 1.4873 1.4888 +0.0015

9 F. P. Zscheile, J. W. White, B. W. Beadle, and J. R. Roach,Plant Physiol. 17, 331 (1942).

absorbance curves of solutions on the same wavelengthscale. Note that the solution reflectance curve in Fig. 3resembles a reverse image of that in Fig. 2. This reversalis predicted by dispersion and reflection theory asdiscussed by Wood."0 nmax values should occur on thelong wavelength sides of absorption bands. Then accord-ing to Eq. (1), the reflectance on the long wavelengthside of an absorption peak will have a maximum valueif n> 1, a minimum value if n< 1.

To obtain refractive indices from reflectances, weused Simon's Eq. (7) which is of the form: R=f(n,K,and k), where 0 is the angle of incidence. This equationwhich is for light polarized parallel to the boundaryplane was sufficiently accurate since more than 95% ofour phototube response was caused by this component.

Simon's equations cannot be solved explicitly forit, e.g., t= f (R,K, and 0), and he suggested (in 1951)the use of nomographs to obtain n values from experi-mental R values. We used the IBM 650 computer pro-grammed to obtain approximate n values from his equa-tion. For each reflectance, the actual values of K and kwere used together with a trial n value to calculate a trialreflectance. The program compared the trial reflectancewith the actual one and added (or subtracted) incre-ments of 0.001 to the trial n value until the actual valuewas bracketed. It then carried out a linear extrapolationon the basis of reflectances to obtain a final n value ofthe liquid. Errors in the extrapolation correspond toerrors in reflectance of <0.1%. The n value at eachwavelength was multiplied by the refractive index ofthe prism to obtain an n value relative to vacuum (air).The resulting solution and solvent dispersion curves andthe final prism dispersion curve are shown in Fig. 4.

500

(P) -

Fio. 4. Refractive indiceg of golutiong, solvents a(l the glamprism (see Fig. 1). Refractive indices of solutions and solventswere calculated from observed reflectances (see Figs. 2 and 3).

10 R. W. Wood, Physical Optics (The Macmillan Company,New York, 1934), 3rd ed., p. 511.

194 Vol. 52

FebrUary1962 ANOMALOUS DISPERSION OF 3-CAROTENE SOLUTIONS

THEORETICAL PREDICTIONS

Within a limited spectral region, dispersion theorypredicts the refractive index of an absorbing medium tobe a constant plus wavelength-dependent terms whichare produced by nearby absorption bands. If a stronglyabsorbing solute is added to a nonabsorbing solvent, theabove mentioned constant will be approximately therefractive index of the solvent, and the wavelength,dependent terms will be governed by the absorptionbands of the solute. Thus experimental curves ofAflsolute(= soIution-soivent), should be very nearly pre-dicted by dispersion theory.

We assumed the contribution to An of all absorptionbands of -carotene outside this spectral region to benegligible. The effects of a failure of this assumptionwould be to add a constant increment to all An valueswithin a curve.

As described elsewhere,5' 6 the absorption and Afsoluteterms of a solution can be expressed as functions of theparameters position, height, and half-width of all nearbyabsorption bands. These parameters can be evaluatedfrom an absorption spectrum [K= f(X)] by curvefitting and used to calculate Ansolute for comparison withexperimental values.

Davydov theoretically treats absorption, dispersion,and transmission of light by solutions (See references4 and 5, and Appendix I). He obtains equations forabsorption and refractive index expressed as functionsof the same parameters for each of two limiting condi-tions. In one case [See Eqs. (4) and (5) in reference 5]the absorption band would be Lorentzian in shape,while in the other [See Eqs. (6) and (7) in reference 5]it would be Gaussian.

It is not clear that either set of conditions postulatedby Davydov prevails for A-carotene in solution or thathe considered all factors relevant to this case. Let us

., .500

A () -

FIG. 5. Solution and solvent I-contribution of solute (a-carotene)to refractive index of solution, and absorption of solution.

400 450 500 550 600

FIG. 6. Solution and solvent II-contribution of solute (-caro-tene) to refractive index of solution, and absorption of solution.

suppose, however, that the general wave equationswhich would describe -carotene in solution were toinclude terms representing processes neglected byDavydov. These equations could be expected to lead toa relation between absorption and refractive indexsimilar to those of that author. On this assumption weused both types of his absorption equations to fit ourempirical absorption spectra of 1-carotene solutions andthereby evaluated absorption band parameters forcalculations of dispersion curves. In each of Figs. 5 and6 is shown the experimental K= f(X) curve with the twotheoretical absorption curves fitted to it, and the experi-mental An= f(X) curve with the two correspondingtheoretical curves.

DISCUSSION

The agreement between the theoretical and experi-mental An curves is reasonably good. The predictedpositions and approximate magnitudes of the four peakswere clearly observed. Note that both experimental Ancurves indicate the presence of four overlapping absorp-tion bands in this spectral region. In contrast, thepresence of only three bands is suggested by our experi-mental absorption spectra of solutions, which are sub-ject to less experimental error than are the An curves.However, other absorption spectra9 of pure 1-carotenein solution do indicate that 1-carotene has four absorp-tion bands in this region, and a four-band model wasassumed in our absorption curve fitting. The presentresults therefore support previous suggestions4 that insome cases absorption spectroscopy may better beaccomplished by measuring reflectance than by meas-uring transmittance.

There are several sources of error which may accountfor discrepancies between theory and experiment. Theexperimental method at present includes several as-

195

W. B. ALLEN AND P. LATIMER

sumptions and corrections, each of which introduces anerror. While some of the corrections are of the order ofmagnitude of the results in Figs. 5 and 6 themselves,errors caused by uncertainties in these corrections arelargely cancelled out in the A'11soiute=f(X) curves. Onthe other hand, the accuracy of the theoretical curves islimited by errors in measuring the experimental absorp-tion curves and probable slight contamination of thesesamples by absorbing degradation products of the,3-carotene, and by the finite patience of the absorption-curve fitter. Nevertheless, we believe the theoreticaldispersion curves to be more accurate than the experi-mental ones.

It is interesting to compare the Gaussian and Lorentz-ian type absorption and dispersion curves in an effortto decide which, if either, distribution is appropriate forabsorption bands of A-carotene in solution. The absorp-tion curves clearly indicate the Gaussian shape (see fitsat 550 myi). On the other hand, the experimental dis-persion curves agree more closely with the Lorentziantype curves. A similar difference seems to occur5 in thecase of the 4.0596-/i band of CS 2. This apparent contra-diction may be related to the fact that our absorptioncurves were obtained from dilute solutions [assumingaen f(C)] while concentrated solutions were used tomeasure the dispersion surves.

The quality of the agreement between experimentand theory in Figs. 5 and 6 is comparable with thatwhich we demonstrated' for the infrared dispersioncurves of Jaffe et al.3 Their curves were determined bycritical-angle refractometry instead of by reflectivity.We believe the reflection method for measuring refrac-tive index within strong absorption bands to be at leastas reliable as is critical-angle refractometry.

ACKNOWLEDGMENTS

We wish to thank J. M. McCue for carrying out someof the exploratory experiments, Dr. W. G. Holladay andDr. T. W. Martin for helpful advice, and Dr. D. C.Bardwell for constructing the sample vessel.

APPENDIX I

Davydov's paper from which we obtained equationsused to calculate theoretical dispersion curves waskindly translated from the Russian for us by Mr. HaroldMilner of the Department of Plant Biology, CarnegieInstitution of Washington. In view of the limitedaccessibility of this translation, and at a reviewer'ssuggestion, we outline below parts of this paper.

Davydov considers light-absorbing solute molecules"amixed" or dissolved in a nonabsorbing solute. Hecomments that absorption studies of such solutions arenormally carried out at low temperatures. In general,interaction between a solute molecule and the surround-ing vibrational lattice of solvent molecules can have twoeffects on an absorption band of the solute. It willincrease the resonance frequency of the absorption band,and it may modify the shape of the band. If the inter-action is sufficiently strong, the solute molecules will bedisplaced to new equilibrium positions during the life-time of excitation of the solute.

Assuming the interaction between solute and solventto be not too strong, Davydov writes the wave functionof this system as the product of the separate wavefunctions of the solute and solvent. The latter includesa term proportional to the displacement of the solutemolecules upon excitation. He then considers an electro-magnetic wave incident on the solution and by first-order perturbation theory obtains a wave function ofthe entire system. From this he obtains general integralexpressions for absorption, refractive index, and trans-mittance of solutions for K<<1, a condition which isnormally obeyed even for concentrated solutions.

Davydov then considers the two limiting cases ofweak and strong interaction between solute and solventmolecules. While interaction always shifts the resonancefrequency, strong interaction will widen an absorptionband and change its shape from the natural Lorentzianform, which is governed by the properties of the solutemolecule only, to a Gaussian form, which is governed bysolute-solvent interaction.

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