1、American Mineralogist, Volume 72, pages 1011-1013, 1987Procedures and computer programs to refine the double variation methodSnu-CHuN Su,* F. DoN.Lr,o Bloss. Mrcxnv GuNrnnDepartment of Geological Sbiences, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061, U.S.A.Ansrn-l
2、crCalibration procedures, statistical treatment of experimental data, and supporting com-puter programs were developed for the double variation method of measuring refractiveindices to attain precision and accuracy of about +0.0001.INrnooucrroNThe precision and accuracy with which the refractiveindi
3、ces of a solid can be determined by the immersionmethod depend upon the precision and accuracy withwhich the refractive indices of the immersion oil areknown. If the double variation method is to be used, thechange of the oils refractive index (n) with wavelength(dn/dx) and with temperature (dn/dt)
4、must also be knownwith comparable precision and accuracy. These data, al-though sometimes supplied by the manufacturer, maychange as the oil ages. Indeed, sometimes after a manu-facturers supply of a given immersion oil has been ex-hausted, the refractive index of the replacement oil mayprecisely ma
5、tch that of its predecessor only at the wave-length 589 nm. Consequently, its printed label, if that ofits predecessor, may indicate refractive indices that nolonger pertain at the other wavelengths. For precise andaccurate determination of the refractive indices of solidsby the double variation met
6、hod, therefore, it is necessaryto calibrate the immersion oils to be used.This paper describes the techniques and supportingcomputer programs used by the writers to calibrate im-mersion media and to suppress random errors when de-termining the refractive indices of solids by the doublevariation meth
7、od. It also describes procedures for reduc-ing systematic errors through use of glass standards whoserefractive indices are accurately known to the fifth or sixthdecimal place. Only touched upon is the use of a spindlestage (Bloss, l98l), mounted on a polarizing microscope,so as to orient anisotropi
8、c crystals so that their principalrefractive indices can be measured without significanterror from misorientation.EqurnrvrnNrHigh-accuracy refractometerA high-accuracy Abbe refractometer, model 60/HRfrom Bellingham and Stanley, Ltd., has proven ideal forimmersion-oil calibration. It differs from con
9、ventionaltypes of Abbe refractometers in that (l) it does not havea built-in prism to compensate for dispersion so that itcan only be used with monochromatic light sources and(2) its read-out scale is graduated in degrees and, withhelp of a micrometer drum, can be read to one-thou-sandth ofa degree.
10、 The manufacturer provides four con-version tables whereby a critical-angle reading may beconverted into its corresponding refractive index at eachof the four wavelengths, 435.84, 546.07, 589.60, and643.85 nm, respectively. For other wavelengths, as wellas the foregoing, the conversion ofcritical an
11、gle a to re-fractive index .A/, can be calculated fromNL : 0.866025 + ll2p - sin(a - 29.5)lv“+ 0.5 sin(a - 29.5), (1)where,r,: 1.860682 + 1.91832 x l0-z x l+ 1.029446 x tQ_+ y 1_+. (2)While the refractometer is in use, the temperature ofthe oil between its prisms must be monitored to within0.1“C. Co
12、nsideingthatdn/dt for Cargille oils in the 1.50G1.700 index range may equal as much as -0.0006/C, atemperature error of 0.2t would cause an error of ap-proximately 0.0001 in the measured refractive index. Tomaintain a stable temperature for the prism box of therefractometer, room-temperature water f
13、rom a reservoiris circulated through the prism box. The temperature ofthe immersion oil being calibrated is monitored by achromel-alumel thermocouple inserted into a channelhollowed out in the cement surrounding the refractorne-ters illuminating prism.Light sourceRather than using a monochromator to
14、 illuminate therefractometer, we use a set of three metal-vapor spectralbulbs that provide high-purity and high-intensity light atwavelengths 435.84 nm (Hg bulb with blue filter), 546.07nm (Hg bulb with green filter), 589.60 nm (Na bulb), and643.85 nm (Cd bulb with red filter). A heat filter placedb
15、etween the bulb and the refractometer greatly reducesheat transfer to the refractometer by infrared radiationfrom the bulb.Monochromator and oil cell* Permanent address: Institute of Geology, Chinese Academy When employing the double variation (tr, 7) methodof Sciences, Beijing, China. to measure th
16、e refractive indices of solid unknowns, we000304xl87/0910-101 t$02.00 1011tor2control the wavelength of illumination by means of aSchott wedge-interference filter as made by Leitz. Alter-natively, any continuously variable monochromator or aseries of narrow-band-pass filters with peak wavelengthsat
17、5-nm intervals can be used. A heatable oil cell (Bloss,1981, p. 139) changes the oil temperature and is moni-tored by a built-in thermocouple.Spindle stageA spindle stage, mounted on a polarizing microscope,plus the techniques of Bloss (1981) will serve to orientanisotropic crystals so that their pr
18、incipal refractive in-dices can be measured without significant error due tomisorientation. A spindle stage is even valuable if theunknown grain is isotropic. By its means one can orientthe grain so that it best shows a Becke line or an obliqueshadow when comparing the refractive indices of the grai
19、nand oil.CA.r,rnn lrroN PRocEDURESImmersion oilsTo calibrate the oil, we make at least l0 critical-anglemeasurements at each of four wavelengths (643.85,589.60, 546.07, and 435.84 nm). The measurements at643.85 and 435.84 nm are difficult because of the eyesreduced sensitivity to these wavelengths.
20、Consequently,it is necessary to work in a darkened room and to preventstray light from the light source from entering the ob-servers eyes. For each critical-angle measurement, theoils temperature (t) is also determined. The ronrneNcomputer program orr- (available in ronrn-as rv for bothIBM main fram
21、e computer and IBM PC), written byS.C.S., converts each measured critical angle to its cor-responding refractive index (n,). orr- then corrects this in-dex to its value at 25“C (nrr,) by inserting the values of Ia;nd n, plus dn/dt of the oil into the equationnzs: tx,- (t - 25)(dn/dt). (3)Using a lea
22、st-squares method, orr, next fits to the nrrval-ues thus obtained for each wavelength (in nanometers)the Cauchy dispersion equation:,zx:cr*3*.9 (4)After the intercept cr and the regression coefrcients c, andca are obtained, the calibrated oils refractive index forany wavelength and temperature (n,)
23、can be calculatedfrom “ +? + Q - 25)(dn/dt). (5)l?,.1 :C| +p A.If the Cargille oils used in our laboratory have beenkept tightly closed and protected from light between eachuse, their Cauchy constants (c, cr, cr) have remained un-changed, within the experimental error, for at least a year.SU ET AL.:
24、 DOUBLE VARIATION METHODMeasuring systemThe precision and accuracy with which a solids refrac-tive indices can be measured depend upon (l) the sensi-tivity of the criterion of match between the indices ofgrain and oil and (2) knowledge of the exact refractiveindex of the oil at the conditions of mat
25、ch. Louisnathanet al. (1978) addressed item 1. ltem2 depends upon howprecisely and accurately one knows (a) the temperatureand wavelength of match and (b) dn/ dt of the oil, and itsCauchy constants (cr, cr, cr) so that Equations 3 and 4can serve to calculate the refractive index of the oil forthe te
26、mperature and wavelength of match. Although sta-tistical analysis of the data may reduce the efect of ran-dom experimental errors, systematic effors may also af-fect the determinations of the temperature and thewavelength of match. Moreover, if only a few oils areused during the double variation pro
27、cedures, errors indn/dt and the Cauchy constants for these oils will them-selves become systematic errors. In our laboratory, ourmeasuring system is calibrated by means of standard op-tical glasses whose refractive indices at various wave-lengths are accurately known to the sixth decimal place.To ca
28、librate our measuring system so as to reduce sys-tematic errors, we use 13 highly homogeneous opticalglasses kindly supplied to our laboratory by the CorningGlass Company. For these 13 glasses, whose indices rangefrom l.5l to 1.80, the refractive indices were measuredto the sixth decimal at waveleng
29、ths 435.8, 546.1, 587.6,and 632.8 nm by C. J. Parker and Al Werner (pers. comm.)of Corning through the use of the minimum deviationmethod and a specially designed spectrometer.In determining the refractive indices of an unknownsolid, we first select that Corning glass whose refractiveindex ro is clo
30、sest to the refractive index to be measuredfor the solid. We next select the three calibrated oils whoserefractive indices are slightly higher, almost equal andslightly lower than that for the Corning glass selected.Using these oils, successively, we measure the Corningglass by the double variation
31、method. For each matchbetween glass and oil, essentially determined as discussedby Louisnathan et al. (1978), the wavelength (I) and tem-perature (7) is recorded. Approximately 100 to I 50 read-ings, evenly distributed over the visible range (486-656nm), are made within the 20-30“C temperature range
32、.These (tr, t) data are then processed by the ronrnl,N pro-gram solrD (available in ronrneN rv for both IBM mainframe computer and IBM PC), which, through least-squares regression methods, determines the values for theCauchy constants c, cr, and c. by fitting Equation 4 tothe data. sor-ro similarly
33、determines values for the con-stants ao and a, in the linearized Sellmeier equation(Louisnathan et al., 197 8; Bloss, 198 1),! : ao + atx, (6)where x equals tr-2, y equals (n? - l)-, and nequals therefractive index at wavelength tr. The resultant constantswe call observed Cauchy and Sellmeier consta
34、nts andSU ET AL.: DOUBLE VARIATION METHOD 1013symbolize them as cr,ob., c2,ob, ca,ob“and as 40obsand al,ob“,respectively. Similarly, we fit the refractive indices mea-sured at the four wavelengths by Parker and Werner toEquations 4 and 6 to obtain constants that we symbolize?S c1.qa1 c2,cat, c3,cat
35、eo,“ r, and al,“lwhere the subscript “cal“indicates that the constant was calculated from the min-imum deviation data of Parker and Werner. Equation 4and the two sets ofCauchy constants for the Corning glassstandard- cl,obs, c2,obsr cr,oo, and c r,ut, c2,“t, ca,cal -permit r?obsand n.“,to be calcula
36、ted at any wavelength . The correc-tion value Ancan thus be determined for any wavelengthtr sinceLn: nor“ - n“ur. Q)Similarly, for this Corning glass standard, Equation 6 andthe two sets of linearized Sellmeier constants-co.ots, 4r.otsand ao,“r,4r.“r-also permit nob.and n*,to be determinedfor any wa
37、velength . The correction value An can thusbe again calculated for any wavelength l, by use ofEqua-tion 7.When an unknown solid is measured, then nrr, its re-fractive index at 25C for the wavelength at which a matchoccurred, can be calculated from n, the refractive indexof the oil for the wavelength
38、 of match, and I, the tem-perature of match. Thus by inserting n, t, and dn/dt intoEquation 3, we obtain nr,. (It should be pointed out that,since the temperature range for the data collection is heldwithin t5C of 25“C, the effect of temperature variationon the refractive indices of the solid measur
39、ed can beignored.) Systematic error in this value is reduced by sub-tracting from it An as calculated (Eq. 7) for the samewavelength as that of the match. Thus, fl2s,cooectd: rtzs -An. Each match observed by the double variation meth-od within the 20-30C range for the unknown thus resultsin a wavele
40、ngth of match and a corresponding correctedrefractive index of match ilzso*“,a. Typically, 30 or moresuch matches for a principal refractive index are obtainedover the wavelength range 486 to 656 nm. These pairs ofdataare then analyzed by the computer, and either Equa-tion 4 or 6 is fitted to the da
41、ta pairs by the method ofleast squares. In actuality, to the program soLID we mere-ly submit the tr and I values for the 30 or so matches, aswell as dn/dt andn, for the oil. The program then appliesthe temperature correction (Eq. 3) and the correction forsystematic error based on the appropriate Cor
42、ning glassstandard (Eq. 7). sorro then fits a Cauchy equation (Eq.4) and a linearized Sellmeier equation (Eq. 6) to the (,zzso-“.,“a) data pairs to obtain the constants cy cy c3 lrrdao, a, that will permit the unknowns refractive index tobe calculated for any wavelength within the visible range(and
43、perhaps slightly beyond).RnrsnrNcnsBloss, F.D. (1981) The spindle stage: Principles and practice. CambridgeUniversity Press, Cambridge, England, 340 pLouisnathan, S.J., Bloss, F.D., and Korda, E.J. (1978) Measurement ofrefractive indices and their dispersion. American Mineralogist, 63, 39,t-400.M.lruscnryr RECEIvED Apnn 2, 1986Mnrr.rscmrr AccEprED Mav 28. 1987
