Absorption and Pleochroism : Two Niuch-Neglected Optical I'roperties of Crystals*

THE AMERICAN MINERALOGIST, VOL. 44, JANUARY_FEBRUARY, 1959

ABSORPTION AND PLEOCHROISM : TWO NIUCH-NEGLECTED OPTICAL I'ROPERTIES OF CRYSTALS*

Josnen A. l\'IeNlenrNo,t Uniaersityof Michigan, Ann Arbor, Michigon' Assrnlcr Certain fundamental relationships of absorption and pleochroism are presented to stimulate interest in the quantitative expression of these two properties. The absorption coefficient (i) is given preference as the absorption constant, while biabsorption is re- emphasized as a quantitative expression for pleochroism. Methods for the determination of the absorption coefficients are briefly discussed. Two methods for the determination of biabsorption are presentedin detail.

INrnooucrroN With the polarizingmicroscope (probably the most widely usedtool in mineralogytoday), methodshave beenevolved which make possiblethe accuratemeasurement of many optical propertiesof crystals.IIowever, investigatorshave overlookedcertain other optical properties.Absorp- tion and pleochroism,although they are sometimesmentioned in a quali- tative manner,have beenaimost entirely ignored. The purposeof this paper is three-fold.First, it is hoped that interest in absorption and pleochroismwill be stimulated among mineralogists and petrographers.Secondly, certain constantsfor the quantitative ex- pressionof thesetwo propertiesare suggested.Lastly, some methodsfor measuringthese properties are described. The materialpresented in this paper stemsfrom a project sponsoredby the Ofice of Naval Research.Their assistanceis gratefully acknowledged' Thanks are also extendedto ProfessorReynolds n[. Denning of the Uni- versity of l\fichigan,who contributed many suggestionsduring the course of the project. Hrsronv lihe absorption of light in crystals was first investigatedearly in the 19th century, but it did not progressbeyond a rule-of-thumb system until much later. Among the earlier investigators was Babinet (1838), who discoveredthat the greatestabsorption in a crystal generallycoin- cided with the direction of greatest refractive index. He found many exceptionsto this rule, however.Laspeyres (1880) pointed out the exist- enceof absorptionaxes (directionsof least, intermediate, and greatest absorption).He investigatedcertain biaxial crystals and found that the

x Contribution No. 229 from the Department of Mineralogy, University of Michigan, Ann Arbor, Michigan. t Present address: Department of Geology and Geological Engineering, Ntichigan College of Mining and Technology, Houghton, Michigan. 66 J. A. MANDARINO absorption axes, although subject to the symmetry of the crystal, did not necessarilycoincide with the principal directions of the indicatrix. Between 1880 and 1900,Voigt (1885) and Drude (1900) presentedthe fundamental theoriesof absorptionin crystals.During this sameperiod, notable experimental work confirming these theories was carried out by Voigt (1885),Becquerel (1887), Ramsay (1888),and Ehlers (189S). Shortly afterwards, Pockels (1906) presentedan excellentsummary of the work up to thar time. The introduction of photoelectricdevices restimulated activity in this field during the 1930's.Vogel (1934) investigatedabsorption in certain crystalscolored by chromium (emerald,natural and synthetic ruby, and synthetic spinel). Unfortunately, his absorption data were given in terms of galvanometer readings and served only to determine the wave lengths at which absorption maxima and minima occurred.He did not specify the orientation of his crystals; nor, apparently, did he measure their absorption for any particular direction. Tromnau (1934) studied the optical properties of synthetic spinel colored by various amounts of cobalt. Again absorption data were presentedas galvanometerread- ings. Tromnau's paper is interesting, however, for the refractive index data presented.It is one of the few tabulationsof refractiveindices that clearly illustrates the change from normal dispersion to the so-called anomalousdispersion in an inorganicisomorphous series. Berek's many contributions in the field of absorption are summarized in the book by Rinne and Berek (1953). In the last ten years much work has been done on the study of light absorptionin crystals.However, a great deal of this work includes the same objectionablefeatures of some of the earlier studies: especially, inconsistenciesin the choiceof absorption constants.Some modern in- vestigatorsstill use terms such as "optical density" [Cohen (1956)] and "absorption modulus" [Vult6e and Lietz (1956) and Lietz and Nfiinch- berg (1956)1. Bloss (1955) presenteda study of absorption in the seriesNi(NHa)2 (SOD, 6HrO-X'Ig(NH+)r(SOr)r'6HzO,and correlatedabsorption (optical density and molar absorbance)with composition. Pleochroism,although included in the titles of many early papers, has been almost completely omitted from any quantitative investiga- tion. Slawsonand Thibault (1939)expressed the pleochroismof a tourmaline crystal in terms of the ratio I,/I.. Denning and I,Iandarino (1955) suggestedthe term "biabsorption" as a quantitative expressionfor pleochroism.They definedbiahsorption in a uniaxial crystal as the difference between the absorption coefficients for the extraordinary and ordinarv rays. ABSORPTION AN D PLEOCH ROISM 67

AssonprroN TBnMrrqorocv Theory If light having an initial intensity of 16passes through a medium, the initial intensity is redircedby: 1) scattering, 2) reflectionat both sur- facesof the medium, and 3) the inherent absorbingpower of the medium. In order to convert the measuredtransmittances I'/Io to the correct transmittancesI/16, the lossesdue to scatteringand reflectionmust be accountedfor. Lossesdue to scattering can be ignored if the crystal being investigatedis relatively free from inclusionsand imperfections. Losses due to reflection can be rectified by applying the well-known Fresnel corrections.The relation between the true intensity I and the measuredintensity I' is given by I:r,(,:n)/ 1 \2 (r) in which R is the reflectingpower. The value of R may be calculated from the following equation: R: r,\n*l/ i)' Q) in which n is the index of refractionof the crystal for the vibration direction being investigated. Substitution of the value of R in Equation 1 yields -t2 r (n * l\2 (3) I: r'1---_ I Absorption must be expressedin terms of carefully definedconstants. A variety of constants have been employed in the past. All of these constantshave been basedon the following generalequation: I/Is: s-"t (4)

In this equation,Io is the intensity of the incident light, I is the intensity of the emergent light corrected for reflection losses,e is the base of the natural logarithms, a is the absorption constant, and D is a constant dependent on the conditions of measurement. The value of I/16 is determined and the value of a is then calculated from the equation to yield the desired absorption constant. The numerous absorption constants found in the literature are due to various expansions of the ex- ponent 6 in Equation 4. Table 1 is a compilation of terms which have been used to express absorption. Many of these terms over-simplify the process of absorption; others,although suitablefor specialapplications (i.e., colorimetry) are unsuitable for fundamental absorotion studies. A term for the ex- 68 J, A, MANDARINO

Tnrr,r 1. V.qnlous Tbnus UsBo ro Expnnss AssonprroN In the equations, i: optical path length in the medium, c: concentration of the coloring material, M:molecular weight of the coloring material, tr:wave length of the light used, z:refractive index of the medium, Io:the intensity of the incident light, and I:the intensity of the emergent light corrected for reflection losses.

Symbol Term Equation

Transmission : Transmittancv T : I/IO

A Absorption : Absorbancy A:1-T

1) Optical density D : loglo (IolI)

L Extinction E : logro (IolI)

K Extinction coefficient

h Specific extinction - lqrlry!- ct

Specific absorption coefficient a-,t : I/Io

M logro (IolI) e Molecular extinction ctr

tn Absorption modulus r/rs : 2 *t

h Absorption coefficient IfIs : ettr{artlt'l

Absorption index IfIs : s-xrarh111 pression of absorption should be based on the following precepts: 1) an absorption constant, like index of refraction,should be mathe.rtat- ically dimensionless;2) it shouldbe basedon acceptedphysical theories, thus lending itself to the calculation of special constants for special absorption problems; and 3) it should be easily measurableor easily calculable from an easily measurable quantity. An expressionsuch as "absorption modulus," becauseof its dimen- sions (cm-l), gives the erroneousimpression of "absorption per centi- meter." But more important: it has very little significancein the concept ABSORPTIONAND PLEOCHROISM 69 of the complex index of refraction for absorbing media. Simply stated, the complex index is a combination of the refractive index and the absorption (N:z-aft). Consequently,both propertiesshould be expressed as dimensionlessquantities. In analyzingthe terms found in Table 1, it can be seenthat two terms meet the first two requirementsstated above. In addition, they may both be calculatedfrom easily measurablequantities. These terms are: the "absorption index" (X) and the "absorption coefficient" (ft). They are related by the following equation: k:nX (s) in which z is the index of refraction. In this paper the nomenclatureof Berek (1953)is used. Absorption is expressedin terms of the absorption coefficient i. Consequently, only equationsdealing with fr are presentedhere. The basicequation relating the absorption coefficient to the intensities of the incident and emergent light is as follows: IfIn : s-e"ft1x (6)

In this equation,ft is the absorptioncoefficient, Io is the intensity of the incident light, I is the intensity of the transmitted light correctedfor reflectionlosses, I is the path length in the absorbingmedium, e is the baseof the naturai logarithms,and tr is the wave length of the light used. If a plate of a uniaxial crystal cut parallel to the c-axisis usedfor the specimen,the absorptioncoefficients for the ordinary and extraordinary rays are derivedfrom the following equations: rtkat^ I,/I": e (7) l,flo: ;+r*elt (8)

in which I, is the intensity of the transmitted ordinary ray, corrected for reflectionlosses; and I. is the samefor the extraordinary ray. If the specimenis mounted in the usual type of petrographic thin-section, the measured intensities must also be corrected for the reflection lossesdue to the glassand balsam.The absorptioncoefficients for the ordinary and extraordinaryrays are

, I ln (I,/Io) Ra:- (9) ,,-- +tt

I ln (I./Io) h,:, - -=;7t (10)

In like manner, similar equations can be deri'r'ed lor ko, k9, and k, lor biaxial crystals. These absorption coefficients are independent of thickness and can 70 T, A, MANDARIN)

be used to calculateany of the specialabsorption constantspreviously mentioned.

M ethod.sof M easwrentent All absorption measurements,regardless of the constant desired,re- quire a measurementof the ratio If Is.In somemethods of measurement this quantity is measureddirectly (that is, I'/I11is measured,and then corrected for reflection lossesto I/16). However, in many methods, a function of this quantity is measured instead. There are three common spectrophotometricmethods of measuringI/Io or somefunction thereof: visual, electronic,and photographic. Typical instruments for all three types of measurement are fully de- scribedby Gibb (1942),N{ellon et al. (1950),Judd (1952),Brode (1943), and the Committee on Colorimetry of the Optical Society of America (1953).These references also include comprehensivebibliographies of the literature on spectrophotometry. The choiceof the instrument to be used in absorption work usually dependson the instrument available. For this reason, descriptionsof individual instruments will not be presentedhere.

Pr,BocnnorsuTnnlrrNorocy Theory Although the study of absorptionin crystalshas been complicatedby the use of many difierent constants,Iew attempts have been made to apply a quantitative term for the expression of pleochroism. The use of the ratio I./L has been suggested(for uniaxial crystals), but this has the,objectionablefeature of being dependenton the thicknessof the sample. Since pleochroismis the differencebetween the extraordinary absorptionand the ordinary absorptionin a uniaxial crystal, the quantity h,-k. could be used to express pleochroism quantitatively. This quantity, like the individual absorption coeffi.cients,is independent of thickness.It follows, then, that pleochroismcan be calculatedfrom the values of fr. and ft.. This method of calculating pleochroism is analogous to the method of calculating birefringence from the indices of refraction. By using various compensators,birefringence can be measured directly without recourse to the values of the refractive indices; similarly, pleochroismcan be measureddirectly. The equationfor k,-k, can be derived from Equations 7 and 8 for the ordinary and extraordinary rays. Division of the ordinary equation (Eq. 7) by the extraordinary equation (Eq. 8) yields

(k€-ft,) Id/J€ : e(atrllI) (11) ABSORPTIONAND PLEOCHROISM 7r

Solution for fr.-fr" yields x ln (I./I.)

+t h,- k" (r2) T The similarity of this equation to that for birefringence(n,-n,:lf t) led to the expressionol h,-k, as b'iabsorption.Calctiation of biabsorp- fion can be accomplishedby measuringthe ratio I,/ I, ditectly. Just as in the measurementof I/Iq, reflectioncorrections must be applied to the measured valtte I,'f I,'. If the birefringence is relatively small, the correction is negligible.For specimensin ordinary thin-sections'no correction is necessaryfor reflectionlosses due to the glassslide, balsam, and cover glass.

ilIpruoos or MEASUREMENTS There are two generally applicable methods of measuring I,fI, directly. One method uses a standard retardation compensator and a rotatable analyzer. The other method dependson visually matching the brightnessof two imagesformed by a double-imageplate.

Double-ImageMethod This method employs a double-imageplate above or below a pleochroic plate so oriented that the vibration directions of the crystal plate are parallel to the vibration directionsof the double-imageplate. The double-imageplate may be a Wollaston plate, Rochon plate, or simply a calcite cleavageplate. When monochromatic light is passed through both platesand observedby meansof a suitableoptical system, two imagesare seen.The Iight forming both imagesis linearly polarized srch that the vibration direction of one image is normal to the vibration direction of the other image. Thus, one image representsthe intensity of the light transmitted by the ordinary ray of the pleochroic crystal and the other image representsthe intensity of the light transmitted by the extraordinary ray. A linear polar placedbetween the plates and the eye can be rotated until the two imageshave the samebrightness. Figure 1 showsthe relationsof the amplitudesof the two transmitted rays. OA is the amplitude of the ordinary ray and OB is the amplitude of the extraordinary ray. When the polar is rotated to a position such that its vibration direction is OP, the two imageshave the samebrightness' If angle AOP is designatedas a, 94: tu,'': & (13) OB A. Squaringboth sidesof this equation produces 7,) J. A. MANDARINO

Frc. 1. Relationship between the amplitudes of the transmitted ra1's of a pleochroic crystal plate.

I.fIr: 16nzo (14) Thus the ratio of I, to I. can be measured directly. Very often the two images have difierent brightnessesbefore the pleochroicplate is placedin the system.If such is the case,the ratio betweenthese two incident intensitiesmust be determinedand applied as a correctionto the measuredvalue of I,/I.. This correctionis determined by rotating the polar to a position of equal brightnesswhen the pleochroicplate is out of the system.If this angleis designatedas B, the equation lor I,fI,be- comes: I./I,: tan2a cot2P (15) When a calcite cleavageplate is used as a double-imageplate, the value of cot2B may vary considerably from unity due to the extreme difference between the refractive indices. The use of a Wollaston or Rochon plate greatly reducesthe correction. The double-imagemethod was used to determine biabsorption in synthetic ruby and the resulting data will be presentedin a later paper.

Compens ator- Analyzer M ethod. In general,when a non-pleochroic,optically anisotropiccrystal plate is placedbetween crossed linear polarssuch that the vibration directions of the crystal are at 45oto thoseof the polars,elliptically polarizedlight is produced.The axesof the resultant ellipseare parallel to the vibration directionsof the crystal plate and their ratio is a function of the phase- difference between lhe two rays transmitted by the crystal. The solid curve in Fig. 2 representsthe trace of the elliptically polarized light pro- ABSORPTIONAND PLEOCHRO]SM 73 duced by a crystal plate whose two rays were subjected to a phase- difference of thirty degrees(tr/6 linear path difierence). If the crystal is pleochroic, an entirely different ellipseis traced by the resultant elliptically polarizedlight. Not only is the ratio of the axesof the ellipse changed, but the axes are inclined to the vibration directions of the crystal. The dotted curve in Fig. 2 is the trace of the resultant elliptically polarized light from a pleochroic crystal whose two rays are subjectedto the same phase-differenceas before (thirty degrees).

Frc. 2. Traces of the elliptically polarized light produced from crystal plates with a phase-difierence of thirty degrees. Solid curve, non-pleochroic crystal; dotted curve, pleochroic crystal.

The angle through which this second ellipse has been rotated is a function of the ratio of the intensitiesof the ordinary and extraordinary rays. Nleasurement of this angle makes possible the calculation of I.fI, and, subsequently,k,-h.. The problem is very similar to the determination of bireflectiondiscussed by Berek (1953) and Hallimond (1es3). In order to developthe mathematicalrelations upon which this method is based,the dotted ellipseof Fig. 2 has beenredrawn in Fig. 3. The vibration directions of the polarizer and analyzer are representedby CP and CA, respectively.CO is the ordinary vibration direction of the crystal and CE is the extraordinary vibration direction. COo and CEo are the componentsof the incident light resolved along the ordinary and extraordinary vibration directions of the crystal, respectively. COr is the amplitude of the transmitted ordinary ray and CEr is the amplitude of T, A. MANDARINO

Frc. 3. Geometricalrelationships of the ellipticallypolarized light producedby a pleochroiccrystal plate.

the transmitted extraordinary ray.It can be seen that COr/CEr equals the square root of I,/I.. Because COr/CE1 :tan ECS, it is necessary to determine the value of angle ECS. The relationships between angle ECS, angle ECR, and A are given by the following equations from Hallimond (1953). Proofs of these equations are given by Schuster and Nicholson (1924). tan 2 ECR tan 2 ECS (16) cos A tan2(tan-t b/a) tanA: (17) sin 2 ECR Angle ECI{ is the angle between the extraordinary vibration direction of the crystal and the major axis of the ellipse.A is the angular phase- differencebetween the ordinary and extraordinary rays. From E,quations16 and 77, it can be seenthat measurementsoI b/a and angleECR are necessaryfor the determinationol I,f I,. Proceduresfor the measurementof both of these quantities are described by Hallimond. The measurementrequlres a mrcroscopeso con- structed that the ana"lyzermay be rotated. In addition, an accessory slot must be provided at 45o to the analyzer vibration direction and must rotate with the analyzer. If a compensatoris piacedin the accessoryslot, the crystal plate can be compensatedonly when the vibration directionsof the compensator ABSORPTIONAND PLEOCHROISM i5 are parallel to the axes of the ellipse.I'hus, in the caseof a non-pleochroiccrystal, the elliptically polarizedlight can be resolvedinto linearly polarizedlight by adjusting the compensatorwhen the vibration directions of the compensatorare parallel to the vibration directionsof the crystal plate. In the caseof a pleochroiccrystal plate, however,the axes of the ellipseare not parallel to the vibration directionsof the crystal. Therefore, the compensator and analyzet must be rotated until the vibration directions of the compensatorare parallel to the axes of the ellipse.This angle of rotation is equal to the angle ACR which is equal to 45o-ECR. x.,Ianycompensators yield values of retardation in terms of iinear path-difference,f. The angular phase-differenceat 45o to the ellipseis designated6. The relation between{ and I is

o: ?: (18) 'iJ*:l or therigr't emproved' r is reratedto the )i; *iJ,X[rength i10) 6 : 2(tan-t b/o) From Equations 16 and 17 it follows that tan 2(45'- ACR) tan 2 ECS : (20) COS A tan d tan-\: (2r) ii.7Gs- etnr Substitution of the measuredvalues of ACR and 4 in the preceding equationsyields a solution for the value of ECS' The value of I"fI, is determinedfrom the following equation: I./I": tan2 ECS Biabsorptioncan then be calcuiatedin the usual manner from the value

of relatively high retardation.

CouueNrs It can be seen,from the foregoingdiscussion of measurementmethods, that rather expensiveequipment is necessaryfor the determination of absorptioncoefficients. Unless a suitable spectrophotometeris available to interestedworkers, it is doubtful whether the absorption coemcient 76 T. A. MANDARINO

will become a commonly noted constant in minerarogicalliterature. However, the measurementof biabsorption requires very little equip- ment not already presentin most mineralogicalor petrographiclabora- tories. The emphasisin this paper has been on uniaxial crystals. llowever, these same methods, with appropriate changesin equations,may also be applied to biaxial crystals. It is proposed that ko, ke, and.k, refer Io the absorption associatedwith the directions of the refractive indices a, B, and 7, respectively.It should,be noted that k.(ka1k, will not neces_ sarily hold,true.

Rcrrnrltcns

BaorNnr, M. (1838), sur I'absorption dans les milieux colores birefringents: compl. Rend. Acatl Sci. P aris, 7, 832-333. Brcqurnnr,, H. (1887), Sur I'absorption de la lumiere au travers des cristaux: Butrt. soc. Fr an. M in., tO, 120-124. Brnor, (Rr*na-Brnnr) M. (1953), Anleitung zu optischen untersuchungen mit dem Polarisationsmikroskop. Stuttgart. Br,oss, (1955), F. Relationship between light absorption and composition in a solid solution series: ,4r2. Mi.neral., 40, 571-397. Bnooa, (1943), W. Chemical Spectroscopy.John Wiley and Sons, Nerv york. CourN, A. (1956), Color centers in the a-quartz called amethyst: Am. Mineral.,4l, gI4_ 891. cou*rrrnr oN cor,onrunrR.o oF THE orrrcar, Socrrrv or Ar*nrc,q, (1953), The science of Color. Thomas Y. Crowell, New york. DeNNrNc, R., aNl Me*oanrno, J. (1955), pleochroism in synthetic ruby: Am. Mineral., 40, 1055-1061. Dnune, P. (1900), Lehrbuch der Optik. S. Hirzel, Leipzig. Enrrns, J. (1898), Die Absorption des Lichtes in einigen preochroitischen Krystailen: Neues Jahrb., ll, 259117. Grrn, T. (1942), optical Methods of chemical Analysis. McGraw-Hill Book company, Inc., New York. Ila'lruoNo, (1953), pnlarizing A. Manual of the Microscope. Cooke, Troughton, and Simms.York, England. D. (1952), Juno, Color in Business, Science, and Industry. John Wiley and Sons, New York. Lasrnvnrs, H. (1880), Mineralogische Bemerkungen: Zeit. Kryst.,4, 433-467. Lrrrz, J., uxo MuNcnnnnc, w. (1956), Uber symmetrie und pleochroismusvon Amethvst: N euesI ahrb., 88, 217-232. MELL.N, M. (Editor) (1950), Analytical Absorption Spectroscopy.John wiley and sons, New York. Pocrnr,s, F. (1906), Lehrbuch der Kristalloptik. Leipzig_Berlin. Rausa.v, (1888), w. ueber die Absorption des Lichtes im Epidot vom sulzbachthal: zei,t, Kryst., 13, 97-134. Scnusrrn, A., ano Nrcnolsow, I.0924), Theory of Optics. Arnold, London. Sr.ewsow, C., aNl Tnmaurr, N. (1939), euantitatrve measurement of dichroism in tourmaline: Am. Mineral., 24, 492-498, ABSORPTION AND PLNCHROISM 7i

TnolrN.qu, H. (1934), Chemischeund physikalischeuntersucbungen an synthetischen mit Kobalt gefiirbtenSpi4ellen: Neaes f ahrb.,68,349-376' vocu,, P. (1934), Optische Untersuchungenam smaragd und einigen anderen durch Chrom gefarbtenMineralien: Neues f ahrb',68,401-438' Vorct, W. (1885),Erkl:irung der Farbenerscheinungenpleochroitischer Krystalle: Neues tahrb., l, ll9-141. BIau- Vurrfr, J., UNoLrerz, J. (1956),Uber die Rolle desTitans als Ferbungsursachevon und Rosenquarzen',Nanes Jalrrb,88, 49-58.

Monus.cri,ptreceitd April 18, 1958