JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Refractive Index and Thickness of Ultrathin Sections of Coals andGraphite by Interferometry

JAMES T. MCCARTNEY* AND SABRI ERGUN*(Received March 10, 1961)

The development of techniques for preparing ultrathin sections of coals now permits the direct measure-ment with an interference microscope of the refractive index of coal components. The change in phase ofpolarized monochromatic light (5460 A) upon passage through an absorbing ultrathin section is a functionof refractive index, thickness, and extinction coefficient. A graphical procedure has been developed forcalculating these parameters from phase change and independent absorption measurements. The effects ofmultiple reflections on the latter data were carefully considered. The ultrathin sections were usually be-tween 500 and 1000 A thick. The refractive indices obtained agreed well with those previously derived fromreflectance measurements. Values of 2.20 and 0.51 for indices of refraction and absorption of graphite at5460 A were obtained by these direct measurements as compared to 2.15 and 0.66 by reflectancemeasurements.

REFRACTIVE indices of coals have thus far beendetermined somewhat indirectly from reflectance

measurements.'- 3 Although the relation between re-flectance, refractive index, and absorptance is theo-retically well established, questions arise as to the rela-tion of the properties of the polished surface layer ofthe coal to those of the bulk material. Development ofan interference microscope for transmission work hasnow afforded a means of directly measuring the re-fractive index of small coal components, providedsections thin enough to transmit sufficient light forthe measurement can be made. Simultaneously thethickness of the section can be determined accurately,and this is of prime importance for other optical meas-urements on these coal sections.

The AO Baker interference microscope used for thiswork is a polarizing microscope modified into a two-beam interferometer. One beam passes through asupporting quartz plate and then the specimen, andthe reference beam passes through the quartz only.The difference in phase of the two polarized beams isdetermined by passage through a quarter-wave plateand an analyzer.

For high-rank coals and graphite, absorption issignificant and must be considered in the treatment ofresults. The phase change 5 of a plane monochromaticwave upon transmission through an absorbing film ona transparent substrate' is expressed as:

2rnt -k nikZ=- ± tan-' ± tan-' + tan-'

X n,+n n2+nni+k 2

exp (47rkt/X) sin (4rnt/X) PcPi sin (c¢+±bci)- (1)

exp(47rkl/X) cos(47rnt/X) +PcPi cosJ(D+i)

* Physicist and Project Coordinator, respectively, PittsburghCoal Research Center, Bureau of Mines, U. S. Department ofthe Interior, Pittsburgh 13, Pennsylvania.

' F. J. Huntjens and D. W. Van Krevelen, Fuel (London) 33,88-103 (1954).

2 S. R. Broadbent and A. J. Shaw, Fuel (London) 34, 385-402(1955).

3 J. T. McCartney and S. Ergun, Fuel (London) 37, 272-282(1958).

4 Max Born and Emil Wolf, Principles of Optics, (PergamonPress, London, 1959), p. 624.

where(n-nc)2+k2

P 2= -

(n+n,)2+k2

2nckDc= tan-'

n2- lt2+k2

(n- n,)2+k2

(n+ni)2+k2

2nikC = tan' l-

n2_ni1+k 2

In these equations, n, n, and n are the refractiveindices of the absorbing film, the substrate, and themedium covering the film, and k and t are the extinc-tion coefficient and thickness of the film. The extinctioncoefficient k is defined by k = X ln(Io/I)/47rt when theintensity of light of wavelength X drops from Io to Iin traversing the film.

Refractive indices of the specimens used in thisstudy were higher than those of the quartz substrateand the immersion media. Therefore the amplitude ofthe reflected light is negative at the quartz-specimeninterface and positive at the specimen-immersionmedium interface. This may be seen from the equationsfor the reflectances at these interfaces:

n- (n+ik); ~~r = pei=

n,+ (n+ik)

(n+ik) - iri= piei4'i-

(n+ik) +ni

Therefore the negative root of p 2 and the positive rootof p,

2 must be used in Eq. (1) and succeeding equations.The phase change of the comparison beam passing

by the film in the interference microscope and travers-ing instead the same thickness t of the immersionmedium is

bi= 27rnit/X. (2)

The difference in phase of these two beams, that is,5-3i, is measured experimentally. If k is negligible,Eq. (1) takes the form

1-pcpi 27rnttana= tan

1+pcPi X

(3)

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J. T. IcCARTNEY AND S. ERGUN V

and the phase difference -5i is obtained from Eqs.(3) and (2). A further simplification can be made ifmultiple reflections are neglected;

a - a = 2t (n -1 ij/X. (4)

For practical purposes, when k<0.1, as it is for coalsbelow high volatile A rank, the simpler equation maybe used with little error. For high-rank coals andgraphite, the three parameters , , and k can be ob-tained by measuring phase differences in two immersionmedia and determining the product kR by separateabsorption measurements.

Extinction coefficients obtained by normal trans-mission should be corrected for reflection. In trans-mission measurements the thin section is usuallymounted on a quartz plate and the monochromaticlight passes first through the quartz, then throughthe sample. Reference light passes through the quartzplate alone. At normal incidence the reflectance R ofa quartz plate in air is given by

2r12- 2r1

2 cos(47r1nt/X)(5)

1+rl - 2ri2 cos(47rnct,/X)

where r 2, the reflectance of the quartz-air interface, isdefined by

rll= (l-stc)2/(+jjC)2, (5a)

sit is the refractive index of the quartz and I, is itsthickness.

The above equation applies when t, is uniform withina small fraction of wavelength X of the monochromaticlight and when angular spread of the incident beam isnegligible. In this case the reflectance depends uponthe thickness of the plate and becomes zero whenn)t=0, X/2, 2(X/2), etc., and assumes a maximumvalue of [2rl/(1+r2)]2 when nBt,=X/4, 34X, 5/4X,etc. For a relatively thick fused quartz plate, about0.5 mm, nt, cannot be regarded as uniform within asmall fraction of X and slight angular spread of theincident beam prevents a coherent combination ofmultiple reflections. For this case Eq. (5) takes theform

Rl=2r12/ (I+r12 ) : 2r,2 . (6)

When the quartz plate is backed by an absorbingmaterial, for example, a thin section of coal, the re-flectance R2 of the system is given by

ri2+r2 2- 2r12r22

R2= r+r 22, (7)

1-rl r2

where 22 is the reflectance of the quartz-coal interfaceand is defined by

pC2 exp(47rk/X)+pi 2 exp(-47rkt/X)+2pcpi cos(@i-4?c+47rnt/X)r22=

exp(47rkt/X) +pc2Pi2 exp(-4rkt/X) +2pCpi cos(e±i+4+47rnt/X)'

where Pc, pi, 1,, and hi are as defined following Eq.Equations (6) and (7) permit calculation of irected for reflection:

I io X 1-R2kl=- n I-n 47r I ob. 47r 1-Ri

The last term of Eq. (8) is the correction term.

EXPERIMENTAL PROCEDURE

Thin sections of vitrinite from coals of differrank were prepared by three methods. At firstsections were made by grinding by the usual teniques,5 except that the final abrasion was careftcontinued until the section was much thinner t1usual. During this treatment the thickness becaquite variable and parts of the section were torn The section was examined under the microscope the thicker parts were cut away and discarded. very thin small areas remaining were freed frommounting cement by immersion in a solvent. TIwere then picked up on slides, dried, and mounted va drop of immersion oil and cover glass.

6 R. Thiessen, G. C. Sprunk, and H. J. O'Donnell, U. S. Burof Mines I. C. 7021 (1938).

(1). Because of checking and cracking in sections of low-or- rank coals (lignite and sub-bituminous), caused by

wetting and drying in the grinding process, cutting ofthese sections by conventional microtomy with a steel

8 knife was tried and yielded very small uniform sections) a micron or two thick. The coals were kept moist

throughout this process, and the cut fragments werestored in water. This method was not suited to theharder, higher rank vitrinites.

Later a new technique of cutting microtome sectionsof the harder coals with a diamond knife was developed.

ent These sections were much thinner (500 to 1000 A) andthe more uniform than the abraded sections, and werech- much more satisfactory for interferometric measure-Illy ments because of their greater transmittance. A de-han tailed description of the technique of preparing theseme sections will be given elsewhere.ut. Very thin flakes of graphite were peeled from naturalLnd crystal masses by means of cellophane tape. The'he thinnest fragments were selected under a microscope,the freed from the tape in a solvent, and then picked upiey on glass slides or quartz plates.ith Immersion oils of suitable index of refraction were

selected for each section by consideration of the prob-eau able index of that coal. To avoid ambiguities intro-

duced when the path difference is greater than one

198 Vol. 52

ULTRATHIN SECTIONS OF COALS AND GRAPHITE

wavelength, the oils were selected so that (n- ni)t< 5460A. Oils available were methylene iodide, ni= 1. 74, anda series of microscope immersion oils of index 1.46 to1.64. For the ultrathin fragments cut with the diamondknife, measurements could be made in air as one of theimmersion media. The oil index values used in thecalculations are for sodium light at 5890 A, whereasthe mercury green line at 5460 A was used in themeasurements. However, the order of accuracy attaineddoes not justify correction for this difference.

To check the performance of the interference micro-scope, measurements were made on Mylar film nomi-nally 0.0005 in. (12.7 u) thick and having a reportedindex of refraction of 1.64. Calculated values of n and from determinations in three immersion liquids rangedfrom 1.622 to 1.632 for the former and 10.2 to 14.9 forthe latter. This appears to be reasonably good per-formance for a specimen thickness that is rather largefor best application of this method.

Values of n, , and k are obtained from the inter-ferometric and transmission measurements by a graphi-cal procedure. Approximate values are first calculatedfrom Eq. (4) and the kt product. Values of n and above and below the approximate are assigned, anda-6i is calculated for each combination of values forthe two immersion media. The phase difference a-ai

is then plotted as a function of t for each value of n.Corresponding values of it and t are then obtainedfrom the plot for the experimentally determined valuesof 5-Si. A plot of n versus t for the two media will thengive two curves whose intersection yields the desiredvalues of n and t and the resulting k. Usually one as-sumed value of n and t above and below the approxi-mate gives sufficient accuracy.

The above procedure is illustrated in Fig. 1 using ananthracite sample that showed phase differences, S-5i,of 490 in air and 18033' in an immersion oil having arefractive index 1.64. Substitution of these values intoEq. (4) yielded n= 2.03 and = 722 A. The value of ktobtained from absorption measurements was 452 A.

TABLE I. Chemical analyses of vitrinites.

Ultimate analysis, percent m.a.f.aCoal (Vitrinite) H C N 0 S

Lignite, North Dakota 4.8 72.8 0.7 21.0 0.7Sub-bituminous, Wyoming 5.1 74.4 0.8 19.2 0.SHigh volatile bituminous, 5.0 78.7 1.3 12.2 2.8

Illinois No. 6 bed, Ill.High volatile bituminous, 5.1 81.9 1.9 10.5 0.6

Illinois No. 5 bed, Ill.High volatile bituminous, 5.3 84.1 1.6 8.0 1.0

Pittsburgh bed, Pa.Medium volatile bituminous, 4.9 89.5 0.8 4.3 0.5

Sewell bed, W. Va.Low volatile bituminous, 4.6 90.8 0.7 3.3 0.6

Pocahontas No. 3 bed, W. Va.Semianthracite, W. Va. 3.8 92.4 0.5 2.3 1.0Anthracite, Pa. 2.6 94.1 0.4 2.2 0.7

a Moisture- and ash-free.

Air 2.1

Oil2.0

60

> ~~~~~~~~~~~n=2

30-

Oil (ni=1.64) n=2.1

= ~~~~~~~~~~ ~~n=2.0

10llI700 750 800 850 900

THICKNESS, t, angstroms

FIG. 1. Example of graphical derivation of real values of refractiveindex and thickness of ultrathin sections of absorbing materials.

The above values of n, , and ki would result in a re-flectance loss of 17.3% for an air-quartz-anthracite-airsystem as compared to 7.0% for an air-quartz-air sys-tem, requiring an approximate correction for kt from452 to 401 A, cf. Eq. (8). Values of -6i calculatedfrom Eqs. (1) and (2) for two assigned values ofn (2.00 and 2.10) are shown in Fig. 1 as a function oft (at 700, 775, and 850 A) for the two immersion media.Assigned values of n are plotted (top of Fig. 1) againstthicknesses that correspond to the observed values of6-3i. The intersection of the plots for air and oil im-mersion corresponds to n=2.06, =780 A, and, hencek=0.51. The accuracy of this graphical procedure waschecked by calculating the corresponding phase changes[from Eqs. (1) and (2)] and comparing with the ob-served changes. The reflectance was also recalculated[from Eqs. (6) and (7)] to see if the approximatecorrection made to the observed value of kt was accuratewithin the limits of the accuracy of kt measurements.

RESULTS AND DISCUSSION

Chemical analyses of the vitrinites studied are shownin Table I. Refractive indices, extinction coefficients,and the thicknesses of the fragments with which theywere obtained are listed in Table II. Measurementswere made on conventional microtome sections of thelignite and sub-bituminous vitrinites, ground sectionsof the two Illinois and Pennsylvania high volatilebituminous samples, both ground and ultramicrotomesections of the medium and low volatile West Virginiavitrinites, and ultramicrotome sections of the WestVirginia semianthracite and Pennsylvania anthracite.

199Februar'y 1962

J. T. McCARTNEY AND S. ERGUN

TABLE II. Refractive indices, thicknesses, and extinction coe-fficients of coal thin sections by interferometry and absorptionmeasurements at 5460 A.

Index of Refraction Extinc-By Average tion

Interfer- By thickness, coeffi-Coal (Vitrinite) H/C ometry Reflectance microns cient

Lig, N. D. 0.79 1.70 1.72a 1.2 0.066Sub, Wyo. 0.82 1.71 1.733 1.1 0.093Hvb, Ill. No. 6 0.76 1.80b 0.12

Component 1 1.82 1.1Component 2 1.74 1.3

Hvb, Ill. No. 5 0.75 1.81b 0.14Component 1 1.84 1.3Component 2 1.80 1.2

Hvb, Pa. 0.76 1.80 1.78c 1.9 0.18Mvb, W. Va. 0.66 f1 .90 1.85-1.88bd 0. 7 0.24

Lvb, W. Va. 0.61 J1.85 1.88-1 91b d 1.1 0.22185) 0.067

Sa, W. Va. 0.49 2.05 1.94-1.98d 0.080 0.48An, Pa. 0.33 2.06 1.90-2.00bd 0.078 0.51Graphite ... 2.20 2.15 0.077 1.12

a Interpolated from a relation between refractive index and carboncontent.

b Interpolated from a relation between refractive index and H/C ratio.e Determined specifically for this vitrinite.d Range from minimum to maximum indices.

The thicknesses of the former two types are 1-2,u, andthose of the ultramicrotome sections are 500 to 800 A.

From observations during the measurement of op-tical path differences, it was concluded that all of thevitrinites except the two Illinois specimens consistedof one major component. The two lowest rank vitrinitesobviously contained minor amounts of other compo-nents. The Illinois vitrinites appeared to contain nearlyequivalent amounts of two different, randomly-inter-mixed components. Independent measurements of thesewere quite difficult to make but the separate indicesfound for the No. 6 vitrinite are believed to be quitereliable while those for the No. 5 are somewhat lessreliable.

The refractive indices found by interferometry arecompared in Table II with those deduced from re-flectance measurements.' Only the Pennsylvania bi-tuminous and West Virginia semianthracite specimenswere measured by both methods. The indices of theothers (by reflectance) were interpolated approximatelyfrom a relation between refractive index and hydrogen/carbon ratio,3 except for the two lowest rank vitrinitesfor which an index-carbon content curve was used be-cause H/C ratio appears to be an unreliable measureof rank in this range. Furthermore, for the high-rankcoals in which anisotropy becomes significant, theorientation of the ultrathin sections is undeterminedso that the index found is probably intermediate be-tween the maximum and minimum indices. The ground

sections are parallel to the bedding; the determinedindex is therefore the maximum because coal is uni-axially negative. Considering these modifying factors,the indices determined by the two methods are in goodagreement. The largest deviations are for the threehighest-rank coals.

The comparable indices found for the sections of themedium and low volatile coals of greatly differentthickness indicates that these optical properties arenot a function of thickness. Apparently there is nosignificant "skin" effect that might appreciably affectthe characteristics of the ultrathin sections.

In a study of optical density and thickness of evapo-rated carbon films and graphite, Cosslett and Cosslett6

used an equation similar to Eq. (4) in which complexindex N(=n-ik) was substituted for n. Since theirequation involves no other complex quantity, it ispresumed here that they substituted the modulus ofN, i.e., N I = ( 2+k2 )1 , in place of n in the equation.In our calculations we have found that this assumptionis not rigorous. For high-rank coals the index valuesobtained by Eq. (4) were not much different from thereal values derived from Eq. (1), but the thicknesseswere appreciably lower. The real index of graphite,2.20, was also not much higher than that obtainedfrom Eq. (4), 2.14.

Cosslett and Cosslett apparently obtained 2.11 fromEq. (4) and then used k as 0.55 to calculate the realindex from N=n-ik to be 2.04. This value of k theyobtained from absorption measurements on evaporatedcarbon films which they believed to be highly graphi-tized. [They erroneously stated that their extinctioncoefficient (k =0.55) agrees with Huntjens and VanKrevelen's' absorption index of 0.56.] Our study showedk to be 1.12, corresponding to an absorption index of0.51. If we had calculated the real index from N= n-ikas Cosslett and Cosslett did, using our values of N=2.14 and k=1.12, we would have obtained n=1.82,a value much lower than any yet reported for graphite.Our present values for graphite, n= 2.20 and absorptionindex=0.51 may be compared with Cosslett andCosslett's values of 2.04 and 0.27, Huntjens and VanKrevelen's 2.01 and 0.56, Mukherjee's 7 2.03 and 0.55,and McCartney and Ergun's3 2.15 and 0.66. All ofthese are for green light approximating the 5460-Amercury line and are maximum values because theywere obtained for surfaces or sections perpendicular tothe optic axis of a uniaxial negative substance. Thelatter three sets of values were obtained from reflectancemeasurements.

6 A. Cosslett and V. E. Cosslett, Brit. J. Appl. Phys., 8, 374-376(1957).

' B. C. Mukherjee, Fuel (London) 31, 153-158 (1952).

200 Vol. 52