The Identification of Opaque Solids by Selective ... - ACS Publications

THE IDENTIFICATIOK O F OPAQUE SOLIDS BY SELECTIVE IRIDESCENT FILMIKG. I

OPTICS A. ?VI. GAUDIN' M i n e r a l Dressing Laboratories, M o n t a n a School of M i n e s , Butte, Montana Received November 88, 1936

In previous articles (20, 21) the discovery was announced of a means of staining characteristically some sulfide minerals by reaction with a suitable aqueous solution. Further work (22, 47) in this field has shown that the development of characteristic, brilliant colors on normally white, gray, or even on colored sulfide minerals is not peculiar to a few sulfide minerals, but is susceptible of development into a general method for the systematic identification of opaque solids. The method consists in coating a plane surface of each opaque substance with a virtually transparent thin film of such physical properties that interference of the rays reflected from the top of the film and from the bottom of the film takes place for some particular wave lengths of the visible spectrum. Thus, a polished surface consisting of several different phases acquires films that display in white light substantially the same color whereever substance A occurs, while displaying characteristically different colors where substances B, C, D, etc., occur. There is nothing new in the development of uniform iridescent colors a t the surface of solids. The phenomenon has been extensively investigated, in particular by Evans and his associates (12 to 17), Tammann and his associates (48 to 54), and also by others (1, 27,29,30,33,40, 55, 57) in what concerns the action of oxygen, hydrogen sulfide, and halogens on metals. In fact, Tammann has touched on the phenomenon in what concerns the action of oxygen on sulfides at elevated temperatures (50). Use of iridescent filming was proposed by Koerber (28) as a means for the determination of silicon in iron-silicon alloys, the interference color obtained, after reaction for a given time, depending upon the amount of silicon in solid solution in the iron. Observations were also recorded by Leo (32) concerning the natural occurrence of iridescent faces on crystals, and concerning the artificial preparation of interference colors on polished sections of sulfide minerals. 1

Research Professor of Mineral Dressing. 811

812

A. 31, GAUDIN

It would seem that Leo had in mind in 1911 a method analogous to that which has been developed at Montana School of Mines. This is indicated, for example, by the subtitle of Leo’s book. Careful reading of the book, however, shows that he sought a method of producing a permanent endcolor, something which is constitutionally incompatible with interference colors; it appears also that Leo rejected treatments which gave various colors in succession, as surface films become thicker and thicker. In view of the apparently general applicability of the phenomenon of selective iridescent filming to determinative microscopy, a search of the literature was made to find out what had been done in a theoretical way concerning the optics of these films. This search was disappointing. Most recent investigators have been content with a general statement of the problem for a thin film included between layers of the same substance. Older investigators have examined the same problem in some detail, particularly Newton (37), Wertheim (%), and Rollett (43). No one apparently has analyzed the general problem of a thin film between layers of dissimilar substances, although an experimental spectrophotometric study was made by Constable (7, 8,9). The object of this article is partly to fill this gap, and at the same time to pave the way for a clear understanding of the experimental phenomena to be detailed in subsequent articles. The interference between the light reflected from the top of the film and that reflected from the bottom involves both a qualitative and a quantitative requirement. The qualitative requirement is that the rays which are to interfere must differ in phase by some fraction of a wave. The quantitative requirement is that the intensity of the net ray resulting from the interfering components must differ sufficiently as between one visible wave length and another, or else, even if interference takes place qualitatively, it will not be quantitatively perceptible to the eye. Let nA be the index of refraction of the medium adjoining the filmed mineral, nFthat of the film, and nMthat of the mineral. Consider further that nA < nF < n M

(1)

This will be the general case in dealing with polished opaque solids, such as sulfide minerals, viewed by reflected light. INTERFERENCE BY SINGLE REFLECTION

In figure 1 is represented the interference between the ray reflected at the top of the film -4QB (reflection order 0 ) and the ray reflected once a t the bottom of the film, A’M’P’QB, (reflection order 1). The time required to traverse M’T’ and S Q is the same, since M’S and T’Q, drawn perpendicular to AQ and M’P’, are the wave fronts of the incident and refracted light. Hence, the interfering rays differ in path by T‘P’ + P‘Q, which equals

IDENTIFICATION O F OPAQUE SOLIDS

813

T'W' or 2t cos r, in which t is the thickness and r the angle of refraction. If X is the wave length of the light (considered, for the present, as being monochromatic), interference, qualitatively speaking, occurs if 2nF t cos r = m e

"1 -

2

i

in which m is an odd integer. This is correct in view of the assumption nA < nF < nM, as such an assumption (42) necessitates reflection with a phase retardation of Xi2 at

I,

M/'

FIG.1. Zero- and first-order reflections a t a filmed surface

both interfaces. A phase retardation of A, 2 occurs in any event upon reflection at a metallic surface (62). INTERFERENCE BY MULTIPLE REFLECTION

Interference, however, is not limited to the interaction of reflections of the zero and first orders. Consider the reflection of the p order involving light that has been reflected p times a t the film-substratum interface and light that is being reflected for the first time at the air-film interface. Reflection within and at the top of the film does not involve a phase retardation, since nF > n A , but each of the reflections a t the bottom of the film requires a phase retardation of X/2, since n F < n w ;also, the zero-order reflection requires a phase retardation of X/2. The path difference between the reflection of the zero order and that of the p order is p times the path difference between the reflections of zero and first orders or, p.21 cos r

814

A. RI. GAL-DIN

Hence the retardation of the reflection of p order is

x

p*2np*t*cos 7. + ( p - 1) 2

If interference takes place between reflections of the zero and first orders, then

x

2nF.t.cos 1’ = ?n-

2

and the retardation of the reflection of p order as compared to that of zero order is

x + ( p - 1) x2

pin-

2

Since m is odd, the term p ( m be expressed as

+ 1) niust

be even and the retardation can

in which L is an integer. It follows that if interference occurs between the reflections of the zero and first orders, all the other reflections team up with the reflection of the first order, working against the reflection of the zero order. QUASTITATIVE ESTIMATIOK O F INTERFERENCE

I n order to arrive at a quantitative estimation of the extent to which this interference will be perceived, i t is necessary to consider the indices of refraction of the transparent media and the reflectivity of the plane opaque solid. Let (58) the amplitude of the incident ray be unity, b and c be fractions which describe the amplitude of the ray reflected from the upper film surface (zero order) and of the first refracted ray. Let k2 be the reflectivity of the plane opaque solid in contact with the film. As shown in figure 2, the ray reflected at the bottom of the film has amplitude ck. This is based on the well-known relationship that the intensity of illumination with monochromatic light is directly as the square of the amplitude of the light. The ray refracted from the film into the medium A , above it (reflection order l ) ,has the amplitude c f k ; that reflected back into the film, cbk, etc. After substitution of 1 - b2 for cf (46), the ampli-

815


tudes of the various reflections bo, bl, bz. . . . b,, are as follows, in relation to amplitude of unity for the incident ray: bo b1 bz b3 bb b,

= = = = = =

b (1 (1 (1 (1 (1 -

b2)k b2)kbk b2)b2k3 b2)b3k4 b’)bP

It has already been pointed out that if the reflections of the zero arid first orders interfere, the reflections of all other orders team up with the reflection of first order. The amplitude of the net reflection is then .. .. bo - (bl bz b3 bp). On the contrary, if the thickness of the film is such that reflections of the zero and first orders differ by a full wave, bz will lag behind bl by P A ,

+ + +

+

n,

FIG.2. Multiple reflections a t a filmed surface

since its retardation involves not only that due to the thickness of the film, but also one more

X

retardation due to reflection into a rare medium at n 2 dense medium boundary; likewise b, will lag behind bl by 2.3X or 3 X ; br by 3 .$A or PA, etc. The amplitude of the net reflection is then (bo

-

+ bl + b3 +

. . . . ) - (bz

+ + bg + 64

. . . .)

This, of course, represents the largest possible amplitude for a reflection, just as bo - (bl

+ bz + b3 +

....

+ bp)

represents the smallest possible amplitude. Generally, then :

+ +

+

+

bmin. = b - ( 1 - b2)k[1 bk b2k2 b3k3 . . . . + bpkp] b,,. = b + (1 - b2)k[l + b2kZ b4k4 + . . . + b2p’k2p’]( 1 - b2)k[bk b3k3+ . . . , + b2p’+1k2p’+1]j (IT)

+

+

I

816

A. M. GAUDIN AMPLITCDE O F REFLECTION O F ZERO ORDER

The quantity b, or relative amplitude of zero-order reflection, is well known and depends solely upon the indices of refraction of the transparent media, and upon the angle of refraction. I n the case of light at normal incidence b depends only upon the indices of refraction of the transparent media. The relationship, due to Fresnel, is

in which nF and n A are the indices of refraction of the two media. The intensity of the reflected light of zero order, expressed as a fraction of the TABLE 1 I n t e n s i t y o j reJlected light of zero order, expressed as a percentage o,f the i n t e n s i t y of ihe incident light, at air-film and oil-film surjaces, according to Fresnel’s f o r m u l a l o r normal incidence I N D E X OF REFRACTIOX OF FILM

16 1.8 20 2.2 2 4 26 2 8 3 0

I

AT FILM-AIR SURFACE

I

5.3 8 2 11 1 14 1 17 0 19 8 22 5 25 0

~

I

* Index of refraction of oil

=

1.505 (45).

incident light is, of course, measured by b?. Its numerical value for some values of nF,expressed as a percentage of the incident light, is presented in table 1. I n this table data are presented for the two cases corresponding to viewing of the film through air and through oil. INDEX O F REFRACTION O F FILM SUBSThSCES

To date the method of selective iridescent filming has been applied principally, but not exclusively, t o the identification of sulfides, both natural and synthetic. -iccordingly, an inquiry \m,s directed to the ascertainment of the indices of refraction of solids of the type that might be expected to make up thin films at the surface of sulfides. Such solids have rather high indices of refraction, ranging roughly from 1.8 to 2.8 (see, e.g., International Crztical Tables). Attention is called in particular to the high indices of refraction of oxides of antimony, tin, zinc, nickel, lead, copper, iron, and mercury, all of which are higher than 2.0.


817

REFLECTIVITY OF POLISHED SURFACES

Besides the indices of refraction of the film and of the viewing medium, the other important quantity that plays a part in controlling the amplitudes of the reflections of various orders is the reflectivity of the polished surface in contact with the film. KO direct data are available on the reflectivity of polished surfaces in contact with films having various indices of refraction. However, measurements are available of the reflectivity in air and in oil of polished minerals. These measurements, combined with the Fresnel relationship, can be used to calculate the reflectivity ( I C . ~ ~against )~ a medium of index of refraction nF. This treatment is wholly sound if the solid is transparent, but there is some ground for doubt if it displays the property of metallic conduction. Since our knowledge of refraction in metals is far from satisfactory, the lesser evil would seem to be to assume that opaque metallic minerals can be dealt with as if transparent (39). On this assumption,

Solving each for

nM

and equating

Hence

Strictly speaking, these equations apply for normal incidence only. The reflectivity in air of most metals varies from 30 to over 90 per cent. This is shown by table 2, which is abstracted from Wood (59). The average reflectivity in air of sulfide minerals is much lower: it ranges from 15 t o 60 per cent, as may be seen from table 3. This table presents not only the reflectivity of some sulfides in air, but also their reflectivity in oil, as averaged from the data of Schneiderhoehn and Ramdohr (45), and as calculated from the average reflectivities in air with the help of formula T’Ia. Comparison of the calculated with the experimentally observed reflectivities constitutes a test of the validity of equation T’Ia. The agree-

818

A. M. GAUDIN

ment is fair, although a systematic deviation should be noted, the calculated reflectivities being lower than the corresponding observed reflectivities. This may, of course, be ascribable to the inexactitude of the formulas VI. An obvious source of error in the experimental values is the inability to polish any brittle solid so as to obtain a geometrically plane surface. Even a polish apparently perfect is merely a polish in which the defects are small enough in relation to the wave length of the visible light to be imperceptible. But from the standpoint of loss of light by reflection in a direction oblique to the plane of the section these defects may be significant. This is confirmed by Cissarz’s recent observation that polished cleavage surfaces have a lolyer reflectivity than unpolished cleavage surfaces (3, 4, 5). If then the hypothesis is made that all measurements of reflectivity of sulfides made to date fall short, the calculated and observed values of their TABLE 2 R e j e c t i u i t y o j some metals i n air, (k.li.4)2, in p e r cenl R A V E L E N G T H I N A . O. . . . . . . . . . . . . . . . . . . . . . . . . .

’

4200

~

4500

5000

- .1COLOR (TO THE E Y E ) . . .........................

Silver. .............................. Platinum. . . . . . . . . . . . . . . . . . . . . . . . . . . Steel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gold. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Copper. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xickel.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I

~

Violet

Blue

1

5500

,

(44) 6000

~

~

7000

~

Green , Yellow 1 Orange

Red

__

-~--

86 6 90 5 51 8 54 7 52 1 54 29 1 33 33 I 37 56 59

9 1 3 , 927 1926 5 8 4 , 611 642 55 55 55 47 74 84 44 ’ 48 72 , 63 65 55

,

1

’

94 6 69 58 92 83 69

reflectivity in oil (table 3) can be brought in close accord. If, for example, the true reflectivity of millerite is one-third greater than it iq measured to be, the actual observations should have been 54 X 4/3 = 72 per cent and 46 X 4/3 = 61 per cent. Should such an error in observation have been made, the reflectivity in oil calculated from the 72 per cent reflectivity in air would have been exactly 61 per cent. The most recent measurements of the reflectivity of sulfides yield values considerably higher than those abstracted in table 3 from Schneiderhoehn and Ramdohr’s standard text. The latest values are such as to substantiate the hypothesis put forth in this section of this paper. Thus, in the case of galena, viewed in orange light, Schneiderhoehn and Ramdohr give k M a = 37.5 per cent, k llo = 25 per cent; formula T’Ia yields k.llo (calcd.) = 18.6 per cent,-a discrepancy of 26 per cent between the observed and calculated reflectivities in oil. Corresponding figures from Cissarz ( 5 ) for cleaved galena are k,, = 43.2 per cent, k M o (obsd.) = 27.9 per cent, k.wo (calcd.) = 25.1 per cent indicating a discrepancy of only 10 per cent instead

819

IDESTIFICATION O F OPAQUE SOLIDS

of 26 per cent. The comparison is even more startling in the case of stibnite. The corresponding discrepancies between observed and calculated values for orange light, and the mineral viewed in the c-orientation are 40 TABLE 3 ReJEectivityof some sulfide minerals i n air and i n oil* AYERAQE REFLECTIVITY IN A1R

MINERAL

Pyrite. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bismuthinite.. . . . Chalcopyrite. . . . . Cubanite.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

........................ Emplectite.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Galena. . . . . . . . . . . . . . . . . . . . . Zinkenite. . . . . . . . . . . . . . . . . . . Jamesonite. . . . . . . . . Argentite. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.......................... .......................... Bournonite. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miargyrite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pyrargyrite. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stephanite . . . . . . . . . . . . ..... ...........................

.......................... .......................... Famatinite.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tetrahedrite. . . . . . . . . . . . . . . . . . . . Molybdenite. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... ...

~

I i

1

.... , . . . . . . . . . . . . . . . . . . . . . . . . ............ .! Enargite Stannite.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bornite. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ., I Covellite.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

A V E R A G E REFLECTIVITY IN OIL

54 53 51 51 44 40 40 37 37 36 35 34 33 33 33 31 31 30 28 27 27 26 26 25 24 24 24 23 22 21 19 17 15

Observed

Calculated

46 46 44 39 33 32 31 29 24 23 26 26 20 18 24 19 18 18 17 13 15 12 13 13 15 16 8 12 12 10 9 6 8

39 38 36 36 28 25 25 21 21 20 20 19 18 18 18 16 16 15 14 13 13 12 12 12 11 11 11 10 10

9 7 6 4

~

* Arranged

in order of decreasing rcflectivity in air.

per cent (Schneiderhoehn and Ramdohr, 1931) and 9 per cent (Cissarz, 1932). Clearly, then, as better measurements are made the observed and calculated reflectivities in oil approach each other. Such an error due to imperfect polishing as is here postulated would

820

A. 31. GATDIh-

result in a systematic deviation between the observed and calculated values for the reflectivity in oil in the direction in which a systematic deviation is actually obtained. Such an error should be practically lacking in minerals soft enough to flow, as in chalcocite or argentite, but quite marked in the brittle or cleavable minerals, as pyrite, chalcopyrite, and galena. This is precisely what is observed in table 3. I n view of these considerations it may fairly be said that in what concerns substances that are not better electronic conductors than the substances listed in table 3, equations T-I are substantially valid. It is also to the point to observe that equation V is approximately correct in the case of the good conductor, metallic potassium (61). If equation V is valid in that case, then the foundation for equations VI is valid, indeed. AMPLITUDE O F REFLECTIONS O F VARIOUS ORDERS

From the Fresnel formula, and from the formulas describing the amplitude of the reflections of various orders f o r normal incidence there have been calculated the values of bo, bl, 6 2 , bs . . . . b, for (k.v,)z = 0.2, 0.5, 1.0, 2.0, 3.0, 5.0, 10.0, 15.0, 20.0, 30.0, 40.0, 60.0,80.0 per cent and for nF= 1.6, 1.8, 2.0, 2.2, 2.4, 2.6, 2.8, 3.0. The amplitudes of successive reflections were calculated until one was found numerically less than 0.05 per cent (corresponding to an intensity amounting to less than 0.000025 per cent of that of the incident light). This was done for both air immersion and oil immersion (no = 1.505). For the sake of brevity these data are omitted from this paper as they are not of significance by themselves, although their calculation is necessary if bmin. and b,,. are to be determined. I t is of interest to note that if the reflectivity is low the convergence to zero of the b values of successive orders is rapid, but that slow convergence is the rule if the reflectivity is high. Again a low index of refraction of the film connotes rapid convergence and a high index slow convergence. For example : with nF = 1.6, (kMF)*= 0.2 per cent, air immersion, bo = 23.0; bl = 4.2; bz = 0.0 with nF = 3.0, ( I C . , ~ ~ = ) ~ 0.2 per cent, air immersion, bo = 50.0; b, = 3.4; b:! = 0.1; b 3 = 0.0 with nF = 1.6, (k,v,):!= 80.0 per cent, air immersion, bo = 23.0; bl = 84.8; bz = 17.5; ba = 3.6; b, = 0.7; bg = 0.2; bg = 0.0 with nF = 3.0, (k.wF)Z= 80.0 per cent, air immersion, bo = 50.0; bl = 67.2; b:! 30.1; ba = 13.5; bq = 6.1; b j = 2 . 7 ; bg = 1.2; b.i = 0.5; bs = 0.2; bo = 0.1; bio = 0.0 As appears in table 3, the refleeti\-ity of all sulfides in air is less than 60 per cent and in oil i t is less than 50 per cent. Sulfides filmed with sub-

821


TABLE 4 illinimum amplitude of total light rejlected normally b y filmed solids immersed i n a i r (per cent of incident amplitude) f o r definite values of the i n d e x of refraction o j t h e j l m (1.6 to 3.0) a n d of the reflectivity ( k b f ~ at ) ~the film-substratum interface (0.3 to 80 per cent). bmin. = bo - ( b i bz ba ........ b,)

+ + +.

REFLECTIVITY

nF = 1.8

n p = 2.0

n p = 2.2

24 5 22.1 19 2 15 2 12 0 6 8 3 1 11 3 18 5 31 0 42 3 62 9 81 9

29.2 26.9 24.1 20.1 16.9 12.1 1.8 6.4 13.6 26.4 38.1 59.6 80.1

33.7 31 3 28 7 24 7 21 6 16 7 6 8 1 3 8 6 21 7 33 7 56 4 78 2

.+

BF

= 2.4

per cent

0.2 0.5 1 2 3

5 10 15 20 30 40 60 80

18 8 16 2 13 3 9 2 5 9 0 6 9 4 17 3 24 0 36 4 46 9 66 3 83 8

37 5 35 2 32 7 28 9 25 8 20 8 11 3 3 0 4 2 17.4 29 6 53 3 76 6

24 5 15 0 6 9 0 3 134 26 0 50 5 74 8

28 2 18 7 108 34 99 22 4 47 6 73 4

31 1 21 9 14 0 6 7 6 5 19 4 44 8 71 6

TABLE 5 M i n i m u m amplitude of total light rejfected normally b y filmed solids immersed in oil, no = 1.605 (per cent of incident a m p l i t u d e ) , f o r definite values of the i n d e x o j refract i o n of the f i l m (1.6 to 3.0) a n d of the rejfectivity (k,l.iF)2 at the film-substratum interface (0.8 to 80 per cent). bmin. = bo - (bi bZ br f . . . . . . . . . b,)

+ +

REFLECTIVITY

nF = 2.0

np

=

.+

2.2

I

nF = 2.4 nF = 2.6

Z F = 2.8

I

n p = 3.0

p e r cent

0.2 0.5 1

2 3 5 10

15 20 30 40 60 80

1 4 4 0 69 11 1 14 3 19 4 28 8 36 1 42 2 52 5 61 3 76 2 89 0

4 5 1 9 11 5 3 8 6 13 7 23 4 30 8 37 3 48 1 57 4 73 6 87 6

9 7 7.1 42 0 1 3 3 8 5 18 3 26 0 32 8 44.1 54 0 71.1 86.3

14 5 11 9 10 0 4 7 1 6 37 13 6 21 5 28 3 40 2 50 5 68 7 85 0

18 7 16 1 13 2 9 1 5 8 0 5 9.5 17 4 24 2 36 5 47 3 66 5 83 8

22 4 20 0 17 2 13 1 99 4 5 5 4 13 4 20 6 32 9 44 0 64 0 82.6

25.9 23 6 20 7 16 7 13 6 8 3 1 6 9 7 16 9 29 5 41 1 61 6 81 3

~

29.1 26 8 24 0 20 0 16 8 11 8 1 7 6 3 13 5 26 4 38 1 59 7 80 3

822

A. M. GAUDIN

TABLE 6 J l a x i m u m a m p l i t u d e o f total light reflected normally b y $filmed solids immersed in a i r (per cent o f incident amplztude) f o r de$nite mlues of the i n d e x ~ J r e f r a c t i o nof the f i l m (1 6 to 8.0) a n d o j the reffectivzty (k.tfF)Z at the film-subsiraturn interJace (0.2 to 80 per cent). b ~ p f + ~ -) ( b ~ br be bmax. = bo (b, b3 bo b~pr)

+ +

+ + +

REFLECTIVITY

nF = 2.2

n F = 1.6

? F = 2.4

n p = 2.6 I n F

=

2.8

F = 3.0

per cent

0 2 0 5 1

2 3 5 10

15 20 30 40 60 80

27 2 29.6 32 3 36 0 38 7 43 2 51 0 56 7 61 4 69 0 75 3 85 3 93 4

32.7 35 1 37 6 41 2 43 8 48 0 55 3 60 7 65 1 72 2 77 9 86 9 94 1

37 39 41 45 47 51 58 63 67 74 79 88 94

2 5 9 3 9

7 6 6 8 4

7 0 5

41 43 45 49 51 55 62 66

io

76 81 89 95

44 9 47 0 49 3 52.3 54 8 58 4 64 5 69 0 72 6 78.4 83 0 90 1 95 6

3 5 9 1 6 3 0 5 4 7 7 2 2

48 0

50 9 52.8 54 8 57 8 59 9 63 2 68 8 72 8 76 2 81 3 85 2 91 4 96 2

50.0 52 1 55 1

57 4 60 9 66 7 io 9 74 5 79 8 84 0 90 7 95 8

53.3 55 1 57 1 59 9 62 0 65 1 io 4 74 4 77 5 82 3 86 2 91 8 96 4

TABLE 7 J i a x i m u m a m p l i t u d e o f total light re$ected n o r m a l l y b y filmed solids immersed in oil, no = 1.505 (per cent of i n c i d e n f a m p l i t u d e ) , f o r definite values OJ” the i n d e x of refraction o f the f i l m (1.6 to 8.0) a n d of the rejectivity ( k , \ f ~ )at ~ the film-substratum interf a c e (0.2 to 80 per cent) b~p’) h z p f + I ) - (bl br be bm,x = bo (bi b3 b,

+ + + +

+ + + + +

REFLEC-

t~ = 3.0

TIVITY

per cent

0 2 0 5 1 2 3 5 10 15 20 30 40 60 80

7 6 10 2 13 1 17.1 20 3 25 2 34 4 41.3 47 2 56 9 65.1 78.6 90 2

13 3 15 9 17 7 22 7 25 8 30 7 39 4 46.0 51 7 60.7 68 2 80.8 91 2

18 5 20 9 23 8 27 7 30 7 35 3 43 7 50.1 55 4 63 9 71 2 82 5 91 9

23 25 28 32 35 39 47 53 58 66 73 84 92

1 5 2 1 0 5 6 7 7 7 3 1

8

’

I ~

1 j

I

27 29 32 35 38 43 50 56 61 69 75 85 93

1 5 2 9 6 1 9 6 4 1

3 3 2

30.8 33 2 35 8 39 3 42 1 46 5 53 8 59.4 63 8 71.1 77.0 86 4 93 8

’

I

34 1 36 4 38 9 42 5 45 0 49.1 56 4 61 .7 65.9 72 9 78 5 87 2 94 1

37.1 39.4 41.8 45.2 47.8 51.8 58.7 63 8 67.9 74.4 79.9 87.9 94 7

823


stances whose index of refraction lies, say, between 1.8 and 2.8 will have a reflectivity against the film in all cases under 40 per cent. Vnder these circumstances the amplitudes of reflections of the fifth and higher orders are all less than 0.5 per cent and could be neglected. In the calculations for b,in. and b,,,. (tables 4 to 7) this approximation was not made, amplitudes being neglected only if under 0.05 per cent. Had this further approximation been allowed, slightly different values for bmin. and b,,. would have been obtained. The effect of neglecting high-order reflections is mentioned in this connection, as the approximation it connotes is used in some of the calculations presented in this paper. T'alues of the minimum reflected amplitude of filmed solids are presented in tables 4 and 5 for the cases of air immersion and oil immersion, for various values of the reflectivity of the solid against the film substance ranging from 0.2 to 80 per cent, and for various indices of refraction of film substances ranging from 1.6 to 3.0. Tables 6 and 7 present corresponding values for the maximuni reflected amplitude. OPTICAL ANALYSIS TO ASCERTAIK T H E QUALITY O F T H I N FILMS

The expression 1 -

(;E)'can be regarded as measuring the maximum

quantity of interference of light observable in a thin film of suitable thickness.

Tables 8 and 9 give the values of the ratio

(:=,)'

for the various

values of nFand (klNF)2used in tables 4 to 7. It is obvious that high values of this ratio connote the possibility of but poor interference, and that, conversely, low values of this ratio connote the possibility of good interference. Consider the case in which the net intensity of the reflected light (from all reflections taken together) is minimum for wave length XI, and assume furthermore that this represents interference of the first order. It follows that for XZ = $11 the net intensity of the reflected light is maximum (17). If X I is a t 8000 A. U., the extreme red end of the visible spectrum, then X B = 4000 A. C. is at the extreme violet end of the visible spectrum. Suppose that

~

is 25 per cent; it follows that in relation to the abundance

2)::!(

of red and violet in the incident light, the filmed surface reflects four times as much violet light as red light, perhaps three times as much blue light, twice as much green, etc. The impression on the eye will be of a colored film, the color perceived in all probability being some shade of blue.

While it is certain that a low ratio for colored filni-if

(&)*will yield an intensely

a film of the proper thickness is prepared-it

is debatable

824

A . 41. GAUDIN

TdBLE 8 R a t i o (per cent) o j intensitu o j rejected rays whose waue lengths are in the ratio o j 1 to 2* ( a s for 4000 A . U . a n d 8000 A . C . ) i f the r a y of longer wave length is of m i n i m u m amp1itude;jor various values o j t h e i n d e x ofrejraction of the film and of the rejectiuiti/ ( k , ~ p at ) ~the film-air substratum interface; a i r i m m e r s i o n REFLECTIVITY

nF = 1.6

per cent

47.8 30 0 17 0 6 5 2 3 00 3.4 92 15 3 27 9 38 7 60 5 80 5

0.2 0.5 1

2 3 5 10 15 20 30 40 60 80

56.4 39.7 26.1 13 6 7 5 2 0 0 3 2.9 8.1 18 4 29 6 52 5 75 8

61 6 46 4 331 19 7

1,1

5 5 0 1 1 0 12 6

66 7 51 8 391 25 4

~

00

I

45 9 720

i

~

40 0 674

I

* It is, of course, assumed that these rays have equal intensity in the incident light.

595 70 2

5 10

40

60 80

I

200 35 3

5 8 17 6

88 8 70 9 57 6 940 830 743 1 9 7 5 ~ 9 2 38 8 ~5

1

0 9 8 2

47 5 , 6 6 7 840 ~

,

00

395 1609 810 ~

,

0 9

2 9

,

327 274 550 1 4 9 8 776 1 7 4 6

'

5 2

j

0 1

228 460 721


825

where the line should be drawn between conditions indicating the possibility of good interference colors and those indicating the lack of such a possibility. In this connection it is useful to recall the variation with wave length in the reflectivity of the two colored metals, gold and copper (see table 2). I n the case of these metals the ratio in reflectivity for X = 4000 A. U. and 8000,4. U. is as 28:95 and 30:89, respectively, that is, t h e relative intensity of the reflected violet to the reflected red light is 29 and 33 per cent, respectively. L4ccordingly,one might well choose 25 per to (bmaXJ2as the limit delineating the possicent in relative value of

*

rE

V I-

40 K w

2 20301

nF

FIG.3. Relation between index of refraction of film, n p , reflectivity of mineral in contact with film, R.vp, and ratio of intensity of reflected rays whose wave lengths are in ratio of 1 to 2 (as for 0.400 and 0.800~)if the ray of longer wave length is a t minimum intensity,

(L,) bmin.

.

Air immersion.

bility of good from that of fair interference colors. Colors about as intense as those of gold or copper would be described as fair. With less justification 10 per cent has been chosen as the delineation between the possibility for excellent interference colors and t h a t for good colors. Many minerals usually rated as white or gray present variations in intensity of reflected light for various wave lengths such t h a t the faintest reflectivity is but 80 to 90 per cent of the brightest. T o err on the side of conservatism, therefore, 75 per cent for

(&,)'can be regarded as marking

t h e impossibility of producing even a faint color.

826

A. M. GAUDIS

------°0‘ 80

60

P3

40

30 20

15 IO ’ 5. O

L

n, FIG.4 . Relation between index of refraction of film, n p , reflectivity of mineral in contact with film, R.qF, and ratio of intensity of reflected rays whose Rave lengths are in ratio of 1 to 2 (as for 0.400 and 0.8009) if the ray of longer wave length is a t minimum intensity,

(bz,) . oil immersion bmin.

90 80 70

60

F

k

50 40

30 20 10. 5 L

16

1.8

2.0

22 nf

2.4

2.6

ZB

30

FIG.5. Relation between index of refraction of film, nnp, reflectivity in air a t mineral surface (treating mineral as if its reflectivity were due solely t o its index of refraction), R Y A , and possibility of iridescent filming. Air immersion.


827

Judging from some colored minerals, as, for example, covellite, and from the color of the sky, it appears reasonable to take 50 per cent as the delineation between the possibility of producing fair iridescence and that of producing faint (or poor) iridescence. Bearing in mind the conventions just outlined, namely, the set-up of divisional points at 10, 25, 50, and 75 per cent for

(k,)2,

the data of

tables 8 and 9 have been diagrammed as figures 3 and 4. In these figures the coordinates are the index of refraction of the film and the reflectivity of the solid against the film. Lines of equal possibility of producing inter-

90

80 70 60

'

50

40

30

20 10 5 I

1.6

1.8

2.0

22

2.4

26

28

3D

2,

FIG.6. Relation between index of refraction of film, n F , reflectivity in air a t mineral surface (treating mineral as if its reflectivity were due solely to its index of refraction), R . M d ,and possibility of iridescent filming. Oil immersion.

ference colors are drawn. Turning to figure 3, for example, it appears that if R M F = 100 per cent, no interference colors can be obtained; likewise, if R M F= 0 per cent. The first of these results agrees with the finding of Lord Rayleigh (60). The second implies that there is no reflection a t the mineral-film boundary, hence no interference colors. Comparison of figure 4 with figure 3 makes it plain that a considerable difference exists between conditions as they involve air immersion and oil immersion. In figures 3 and 4 the ordinate is the reflectivity of the solid in contact with the film. Since this quantity is generally unknown, and since the

828

A. hl. GAUDIN

reflectivity in contact with air is generally known or readily determinable for most minerals, use has been made of the relations between k,,, k,,, and n F to chart figures 5 and 6 from figures 3 and 4. In these charts the ordinate is the reflectivity of the substratum in contact with air. Again, in this connection, emphasis must be placed on the assumption involved in the preparation of figures 5 and 6, namely, that the refractive index of an opaque (hence absorbing) medium can be obtained from its reflectivity just as if the substance in question were transparent. This in turn implies that the opaque substance is a perfect dielectric, a condition obviously not fulfilled in the case of the metals, and but imperfectly fulfilled in the case of sulfide minerals (but see Wood (61) and Phillips (39)). AGREEMENT O F OPTICAL ANALYSIS K I T H EXPERIMENT I N RESPECT TO POSSIBLE QUALITY O F THIK FILMS

It is interesting to note that considerable evidence is available to corroborate figures 5 and 6 and to give weight to the predictive value of these figures. ( a ) Galena, when treated for a very short time with a solution of chromic trioxide, sulfuric acid, and water analogous to the solution of chromic trioxide, hydrochloric acid, and water described elsewhere (20), gives films brilliant by air immersion, but not so brilliant by oil immersion. One might expect the formation of films of lead chromate or lead sulfate (lead chloride, while possibly insoluble in the staining bath, is soluble in the wash mater). The indices of refraction of these substances average 2.44 and 1.88, respectively, and the average of these average indices is 2.16. Figures 5 and 6 show that excellent iridescent films are possible if viewed in air and good if viewed under oil, since the reflectivity of galena in air is about 43 per cent (5). ( b ) Iridescent films were not obtained on bornite, although the mineral was discolored when using either the chromic trioxide-hydrochloric acidwater solution or analogous solutions with sulfuric or hydrofluoric acids replacing hydrochloric acid, or solutions of iodine in methyl alcohol containing sulfuric or hydrochloric acids. This experimental fact is confirmed by figures 5 and 6, from which it appears that the film should have an index of refraction lower than 2.0 to display “good” iridescence in air and 2.2 in oil. The index of refraction of copper iodide is over 2 . 2 ; cuprous and ferrous chlorides can hardly be expected to form in oxidizing baths; cupric and ferric chlorides and the cupric and ferric sulfates are soluble in the bath. The only other likely film substances are copper and iron chromates; no data are available concerning copper chromate, and chromite, FeCrOr, has an index of refraction of 2.16. Pending direct contradiction the experimental case of bornite can be considered as verifying figures 5 and 6.

IDEKTIFICATION O F OPAQUE SOLIDS

829

It might be argued that good iridescent colors form on bornite exposed to the action of oxygen, this being indeed the most characteristic feature of bornite in hand specimens. I n this case the films are largely copper oxide, the index of refraction of which is very high and changes much with wave length. Constable (7) gives n as ranging froin 3.1 a t X = 4600 A. U. to 2.9 a t X = 6000 A. U. and 2.55 a t X = 7600 A. U. At the same time t h e reflectivity of bornite in air (45) is 18.5 per cent for green light (A = 5200 A. U. i), 19 per cent for orange light (X = 6500 A. U. A),and 21 per cent for “rose” light (X = 7000 A. U. =t).From figure 5 it is seen that interference is impossible for red and orange light, but possible (conditions below critical line) for blue and violet light. This suggests that with thickening film the flesh color of bornite should first darken to a red-brown, then become violet, then blue, these shorter wave-length colors appearing not because of destructive interference at the longer wave-length end of the spectrum, but because of constructive interference a t the shorter wavelength end of the spectrum. (c) After immersion in the chromic trioxide-hydrochloric acid-n-ater bath, pentlandite takes on colors that are brilliant if viewed by air immersion but faint if viewed by oil immersion. Among possible oxidation products are the oxide of nickel and the chromates of iron and nickel. No data are available for nickel chromate, but the indices for NiO and FeCrOl are given as 2.18 and 2.16, respectively, in the International Critical Tables. Constable (8) gives n (for NiO) as ranging from 2.38 to 2.00 with X ranging from 4500 -4. U. to 6800 -4. U. ; these values average at about 2.18. The film on pentlandite can then be regarded as having an index in the range of 2.15 to 2.2. On the other hand, the reflectivity of the mineral is reported by Schneiderhoehn as 51 per cent. At the same time, the reflectivity in oil is given as ranging from 41 per cent in green light to 47 per cent in orange light. Correcting the reflectivities to allow for surface roughness, pits, etc. by means of equations VI, treating the reflectivity in air as unknown, and making use of the ratio of the reflectivities in air and in oil, the corrected reflectivity in air becomes 6G per cent. Figures 5 and 6 show that good interference colors should be obtained in air and fair to poor colors in oil, thus confirming the experimental observations. ( d ) Polybasite, if stained by a solution of iodine in methyl alcohol and sulfuric acid, takes on good colors in air; the colors beconie more brilliant still if viewed under oil. The reflectivity is about 27 per cent; the index of refraction of silver iodide is 2.22 and that of antimony oxide 2.0. Assuming an average index of refraction of the order of 2.15, it appears from figures 5 and 6 that good colors should be obtainable if the mineral is viewed in air and excellent colors if i t is viewed in oil. (e) Steel can be made iridescent b y suitable tempering in air a t a temperature below red heat.

830

A. M. GACDIN

From the reflectivity of steel (about 55 per cent) it is clear that excellent (or at least good) interference colors should be possible whether the oxide formed is ferric oxide ( n in the vicinity of 3.0) or a mixture of FeO and Fez03(8) whose index of refraction varies from 2.26 a t X = 4500 A. U. to 2.00 at X = 5500 A. U. and 1.76 at X = 7000 A . U. The problem is somewhat complicated here by the strong absorption of iron oxide, which makes the interference colors indistinct except for colors of the first order (8). (f) Adherent films of mica on selenium display excellent iridescent colors, as may be seen by referring to the color photographs reproduced by Marcelin (34, 3 5 ) . This agrees with the prediction of figure 5 in the case of a film having an index of refraction in the neighborhood of 1.6 (muscovite, n = 1.58) on a substratum having a reflectivity of the order of 25 per cent (selenium, nM= 20, 25, 35 per cent for X = 7600, 5890, 4900 A. U.). (9) In all fairness attention should be called to the fact that figure 5 predicts failure in regards to the filming of silver with silver iodide, while the experimental fact is to the contrary (e.g., 18, 51). This can be explained away by invoking the high conductivity of silver, hence the failure of formula VIa expressing the unknown reflectivity of the metal in the film in terms of the reflectivity of the metal in air. The high reflectivity of silver is tantamount to the assumption of a very high index of refraction for that metal (of the order of 30 to SO), while modern theory of reflection a t metallic surfaces places this index a t from 0.16 to 0.18 (36). It can also be explained as has already been done by Evans (17) on the basis of absorption in film. In the present state of our knowledge neither “explanation” is wholly satisfying. COLORS OF FILMS I N THE CASE THAT SUBSTANTIALLY PERFECT INTERFERENCE CAN B E PRODUCED

It is of interest to ascertain how the color of filmed minerals varies with increasing thickness of film. This can be done readily as follows: If 4 denotes the retardation of the reflection of first order as compared with the reflection of zero order for a given wave length of light X, the amplitudes of the various reflections being bo, bl, bz, b 3 . . . . b,, it follows that the amplitude of the x and y components of the total reflection are (remembering the X/2 or T retardation caused by reflection from the surface of a denser medium into a rarer medium) :

+ 4+ (24 + + (34 + 2 ~ + ) [n++ ( n - l ) ~ ] (44 + 3 ~ + ) . . . . + b, b, = bo sin 0 + bl sin 4 + b2 sin (24 + + sin (34 + 2 ~ +) bq sin (44 + 3n) + . . . . + b, sin [n++ (n - l ) ~ ]

bz

=

bo

COS

0

bi COS

b2 COS

T)

b3 COS

COS

T)

b3

bq COS

83 1

IDENTIFICATION OF OPAQUE SOLIDS

In the particular example 4

In the particular example 4 bmin,

=

0, b,

=

=

b,

=

7,

0, and b

=

=

0 , and b

bo - b1 - bz - 63 . . . . = bo - (b,

b,

=

=

b,,,,

b , = bmin.

+ bz + b3

. . . .)

These special formulas are those already established for b,,, by a less general method. The general formulas for b, and b, can be simplified to: b, b,

= =

+

and

bmin.

+

bo b1 COS 4 - bz COS ( 2 4 ) b3 COS ( 3 4 ) - bq COS ( 4 4 ) bl sin 4 - bz sin ( 2 4 ) + b3 sin (34) - b4 sin ( 4 4 ) . . . .

cos (n4) and sin (n4)can be expressed as functions in cos 4 and sin 4 of the nth degree. Since it is rarely necessary to be concerned with reflections beyond that of the fourth order, b, and b, become fourth-degree functions in cos 4 and sin 4, the parameters being various summations in bo, bl, bz, b3, and b4: By suitable substitution b takes the form:

+ bz - + (bl - 3b3) 4 + (8b4 - 2bz) cos24 + 4b3 cos3 4 - 8b4.cos4 4 b, = (b1 + 3b3) sin 4 - (2b2 - 4b4) sin 4 cos 4 - 4b3 sin34 - 8b4

bz = (bo

b4)

COS

cos3 4 sin

I

(VIII)

+J

I t is clear also that the retardation, 4, Yaries directly as the frequency of the light, that is, inversely as the wave length in air (provided the speed of light of all wave lengths through the film is the same, that is, providing the film has the same index of refraction for all wave lengths). In order to place this analysis on the concrete basis of a typical example, consider the case in which the reflectivity of the mineral against the film is 10 per cent and the index of refraction of the film is 2.2 (reflectivity against air, 38.3 per cent). In this case bo bl bz

= = =

b3 = bq

=

37.6 27.2 3.2 0.4 0.0

832

A. M. GAUDIN

TABLE 10 Relationship between wave length of light and intensity of illumination jor a film of thickness 910 A . U.(nF = 2.2; ( k , w ~ ) *= 10.0 p e r cent, air immersion) i n relation to intensity os illumination o j the suine surjuce, unfilmed x

h

(b&Y

-I-

A . u. m

96,000 48,000 32 000 24,000 19,200 16,000 13,720 12,000 10,670 9,600 8,730 8,000 7,380 6,860 6,400 6 000 5,650 5,340 5,050 4,800 4 570 4,360 4,170 4,000 3 690 3,460 3,200 3 000 2 830 2,670 2,530 2,400 2,290 2,180 2,090 2 000 1,920 1,850 1,$76 ~

~

~

degrees

A.C.

per cent

0 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225 240 255 270 285 300 315 330 345 360 390 420 450 480 510 540 570 600 630 660 690 720 750 780 810

1,714 1,655 1,600 1,548 1,500 1 454 1,411 1,371 1,333 1 264 1 200 1,143 1,091 1,044 1,000 889 800 727 667 616 572 533 500 471 445 421 400 381 363 348 333 320 308 296 286 276 267 258 250

28 7 8.4 1.2 8 4 28 7 54 8 78 6 94 1 100 0 78 6 28 7 1 2 28 7 78 6 100 0 12 100 0 1 2 100 0 1 2 100 0 1 2 100 0 1 2 100 0 1 2 100 0 1 2 100 0 1 2 100 0 1 2 100 0 1 2 100 0 1 2

~

~

~

32 1 2 32

548 67 4 78 6 87 9 94 1 98 7 100 0 94 1 78 6 54 8 28 7 8 4

'

990 1020 1050 1080 1140 1200 1260

1800 1980 2160 2340 2520 2700 2880 3060 3240 3420 3600 3780 3960 4140 4320 4500 4680 4860 5040 5220 5400

1,

lo(:

;

100 0

833


and the equations describing the amplitude of the reflection for all wave lengths become :

+

+

b, = 40.8 26.0 COS 4 - 6.4 c0s2I#J 1.6 c0s34 b, = 28.4 sin I#J - 6.4 sin 4 cos 4 - 1.6 sin3$

with b,,.

If

I#J

62.0; bu,in, = 6.8;

=

= x =

180'forX

I#J = I#J = I#J =

=

180°, X 360°, X 540°, X

(!e)'

= 1.2 per cent.

8000A. U., then4X = 144 X 104andfor

= = =

X

8000 A. U., b = bmi,,, 6.8; 4000 A. U., b = b,,. = 62.0; 2667 A. U., b = bmin. = 6.8, etc. IN A U

1?560

?80

36.40

l8,20

10

455

22,IS

ll!1

t (FOR h=8000AU)

?SO

IiW

18.20

?IO

"!5

2?75

11!7

569

t (FOR X- 4000AU)

FIG.7. Relative intensity of reflected light of various wave lengths if film is of such thickness that interference of first order occurs for X = 0 . 8 0 0 ~[ n =~ 2.2; ( 1 2 . 1 1 p ) ~ = 10 per cent].

for various values of X are presented in table 10 and are plotted in figure 7 . I n this figure a logarithmic scale was chosen for X; even with that arrangement the squeezing together of successive crests and troughs of intensity of light with decreasing wave length of light is very marked. This would have been even more obvious had an arithmetic scale been chosen to plot X. The data of table 10 and figure 7 can be applied to thicknesses of film other than that which yields $1 = 144 X lo4. This is done by mere multiplication of the scale of abscissae by a suitable factor. Thus the case qbX1 = 144 X lo4 requires (equation 11) a film haIring a thickness tl =

834

A. M. GAUDIN

- 910 A. U. for maximum interference of the first order (first m 2 minimum amplitude). I

100,

0,440

0.500

0.600

0700

0.8QO

A IN U

FIG. 8a. Film thickness, 152 A. U. Film color, gray-white.

0400

0500

0.600

0700

0800

x IN U, FIG.8b. Film thickness, 227 A. U. Film color, yellowish-tan.

MOO

0500

0.600

0,700

am

4, I N p

FIG.8c. Film thickness 455 A. U. Film color, dark orange-red (orangebrown).

‘1 10

o m

0600

05po

A

FIG.8d. Film thickness, 637 A. U. Film color, dark purple.

FIG.8e. Film thickness, 910 A. U. Film color, brilliant blue.

IN

&TOO

0800

p

FIG.8f. Film thickness, 1183 A. E. Film color, greenish-straw.

Likewise, for

and +XZ = 180 X 4000 = 72 X lo4 = $+XI

It is merely sufficient, in order to plot

(&)2

versus X in the case of t z , to

read from table 10 values of X equal to those appearing in the table multi-

835

IDEXTIFICATIOK OF OPAQUE SOLIDS

Xz X1

plied by the ratio - (which in this particular instance is $), and to plot them against

'h I N

(L)'. b,,.

p

FIG.8g. Film thickness, 1820 A. U. Film color, brilliant crimson-purple.

FIG.8h. Film thickness, 2366 A. U. Film color, brilliant green.

FIG.8k. Film thickness, 3640 A. U. Film color, green.

I00

90 80 70 60 50

90

30 20 10 0

0400

0500

0600

'h

a700

OBOO

I N p

FIG. 81. Film thickness, 7280 A. U. Film color, greenish-Khite.

0400

0500

a600

0700

0800

. i IN p

FIG.8m. Film thickness, 14,560 A . U. Film color, gray-white.

FIG.8 (a to m ) , Intensity of reflected light in relation to intensity of light of same wave length in unfilmed surface. Conditions are for possibility of substantially perfect interference (7tF = 2.2; ( k M F ) 2= 0.10; air immersion). U-V = ultra-violet; V = violet; B = blue; G = green; Y = yellow; R = red; I-R = infra-red.

Figures 8a to 8m were drawn for thicknesses of film increasing from 152 A. U. to 14,560 A. U. These figures show many interesting features and are in agreement with observations from films produced experimentally: ( a ) Extremely thinly filmed surfaces behave as if unfilmed and have comparatively high reflectivities. ( b ) The thinnest films that are perceptible color the mineral faintly with a color toward the red end of the spectrum;

836

A . 11. GAUDIN

because of the considerable amount of yellon- in the reflection, their color is some shade of tan. (c) Thicker films are gradually not only darker in color (lower reflectivity) but more intensely colored uiitil a dark purple color is obtained (figure 8d). ( d ) The first blue film is one of the most characteristic (figure 8e). (e) X o film showing a brilliant first-order green color can be produced (figure 8f). On the contrary good greens of second and third order can be prepared (figures 8h, 8k). (f) Films of higher order than third become increasingly near white in color (81, 8m). TABLE 11 Characteristics ojjilms of various thickness (nF = 2.2) formed at the surface of a solid hauing a reflectivity of 10 per cent against the film (38.3 per cent in air),viewed b y air immersion. at n o r m a l incidence

THICXN E 8 8 OF FILM

IF U N F I L M E D , IN R A N Q E OF 4200 T o 7000 A.U.

?1

I

Ih-TENBITY OF LIQHT I S RELAT I O S TO T E A T WHICR W O E L D

LNTENBITY OF COLORATION

1 R,,,

3

-

Minimum

Maxima

Emin. --

ma -

A.U.

152 80

h i

93

1 1 16'

I

0

0

Xone

1

0

0

1

0

1

1

0

1

Very pale Strong Strong

Very bright Bright

I

227 58

84

455 637

43 124 28 123

1 8 1 2

910 6 5 1183 53

~

1 45

98 15 1 1 1 88 1-2

1

1

100

1820

1 2

95 76

2366

1 2

100 183

3640 7280

1 2

100 83 100 8 3 100 183

I

' I

2

1

,

1

'2-4 1+2@+ 4-8 3 2@,4 8-16 7 f 2@;

i

+

Strong Very pale i Very Bright strong I Very Bright strong 1 Strong Medium Medium Pale Medium Xone

1

2@+

1-2

Dark Very dark Medium Bright

2 2 4 8

White Tan Orange violet Blue Greenishstraw Crimsonpurple Green Green Greenish Grav-

I

In view of the fact that the eye is very much less sensitive to light whose wave length is near the ends of the visible spectrum than to that near t h e center (25), i t is desirable in estimating the color perception of films from the intensity versus wave length charts (figures 8a to 8m) to neglect wave lengths shorter than 4200 A. U. and longer than 7000 A . U. The relative visibility of light of 4200 A . U. and 7000 A. U. is but 0.4 and 0.3 per cent, respectively, of light of the most visible wave length, 5560 A. U., and t h e

837


relative visibility of wave lengths decreases very rapidly as the wave length decreases below 4200 A. U. or increases above 7000 A. U. Table 11 summarizes some of the features brought out by figures Sa to 8m. The thickness of film corresponding to incipient coloration can be obtained from figure 7. That figure has been drawn to represent the ratio (&)2

as a function of X for a constant value of the thickness of film, t.

TABLE 12 V a r i a t i o n s in relative i n t e n s i t y of extreme red and extreme violet in light rejlected from a $lmed solid jor nF = 2.2, ( k M F ) 2 = 0.10, w i t h thickness of f i l m in the range of first-order interference

A.U.

per cent

per cent

72.8 145.6 182 202 228 243 280 331 347 364 379 405 417 455 486 520 560 606 662 728

94 2 80 1 69 0 62 5 54 8 507 36.5 21 5 16 5 12 0 8 4 5 2 3.2 1 2 1 2 1 2 1.2 1 2 1 2 1 2

98 8 94 2 92.0 90 5 87 9 86 0 81 5 74 5 72 0 69 0 67 5 62 5 60 548 48 43.5 36.0 28 7 47 70

But since At changes in

=

per cmt

95.4 85 1 75 0 69.1 62 4 59 0 44 8 28 8 22.9 17.4 12 4 8 3 5 3 22 2 5 2 75 3.3 4.2 2 6 1 7

None None None Slight Slight Slight Fair Fair Good Good Good Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent Excellent

constant, a mere change of scale is sufficient to express the

(b&-

with changes in t for X = constant: thus, the first t

scale expresses variations of the ratio

(2.)'

the second for X = XZ = 4000 A. U.

for X = X I = 8000 A. U. and

To get

( b,L,. Y

and

(bk)2 it is

merely necessary to read the ordinates for t corresponding to the two scales of t-abscissae. Results are presented in table 12.

838

A. M. GAUDIN

It appears that the thinnest visible film is of the order of thickness of 190 A. U., or some fifty atomic layers. This conclusion agrees with that of Tammann (48 to 54)) who in one case regards the thinnest visible film of silver iodide as consisting of fortyfive atomic layers. However, in that case Tammann’s estimate of the film thickness is 500 A. U.; this would mean that an atomic layer of silver or iodine is some 11 A. U. thick; this value exceeds considerably the diameter of either the silver or iodine atoms or the silver or iodide ions, as determined by x-ray investigations of crystal structures. The concordance must therefore be regarded as rather of the nature of a coincidence. Table 12 also shows that a t first considerable thickening of the film is necessary to produce some change in brilliancy of coloration, while for thicker films, even small changes in thickness result in considerable changes in brilliancy. This is in general agreement with the statement of Marcelin (34) that the first-order purple is most sensitive to changes in thickness. It is also in partial agreement with his statement (35) that the eye can perceive changes of shade corresponding to changes of filni thickness (for nF = 1.6) of the order of 7 A. U. It would seem that this statement holds only for films of such thickness as to yield the most sensitive colors, namely, in the first order purple to blue; in second- and third-order colors it is probably not approximated, and for first order below the violet as for higher orders than three, it is certainly not valid. In the judgment of the writer it is estimated that changes of color are perceivable without direct comparison (for nF = 2 to 2.5) in the first-order tans if At > 30 to 50 A. U.; in the first order purple to blue if At > 10 to 15 A. E , ;in the first order greenish-white to straw (the “hiatus” of Evans (12 to 19)) if At > 50 to 75 A. C . ; in the second order yellow, red, and green if At > 20 to 30 A. U. COLORS O F FILMS I N T H E CASE THAT IMPERFECT INTERFERENCE ONLY CAN B E PRODUCED

The case so far considered was one corresponding to the possibility of securing practically perfect interference colors. It is of interest to consider also a case in which merely fair interference colors can be obtained a t optimum thickness. The case ( k y F ) 3= 0.30, nF = 2.2, oil immersion, has been chosen as representative of this class of films. In this case:

(!E)’ =

36.4 per cent

839


From equations VI11 b, = (bo

+

- b4) 3. (b, - 3b3) cos + + (8b4 - 2b2) cos2+ + 4b3 cos3+ -

b2

8b4 cos4+ 24.1 51.1 COS - 10.0 cos2+ 2.4 cos3+ - 0.8 cos4+ b, = (bl 3b3) sin - (2b2 - 4b4) sin cos - 4 b sin3+ ~ - 8b4 sin + cos3+ = 54.7 sin - 10.4 sin cos - 2.4 sin3+ - 0.8 sin cos3+ =

+

+

+

+

+

+

+

+

+

+

+

100

100

BO BO

70

60 50

40

40

u1

30

20

20

10

10

0 O W

A

a600

0440

0.'

h

IN ,A4

FIG. 9a. Film thickness, 152 A. U. Color, white

IN A, 4

FIG. 9b. Film thickness, 227 A . U. Color, white

0.600

t 00

:n h

1 N p

FIG. 9c. Film thickness, 455 A. U. Color, tan

100

1~~

50

44

40

30

30

30 20

20 10

10 0

0,440

aaoo

0.600

1I N p

FIG. 9d. Film thickness, 637 A. U. Color, lavender

10 a400

-~

OM0

0.600

moo

A 1NP FIG. 9e. Film thickness, 910 A. U. Color, blue

X0.600 INp

0.BW

FIG. 9f. Film thickness, 1183 A. U. Color, greenish-white FIG.9 (a to f ) . Intensity of reflected light in relation to intensity of light of same wave length in unfilmed surface. Conditions are for possibility of but imperfect interference (np = 2.2; ( k ~ ~ =) 0.30; 2 oil immersion). Intensity of reflected light is expressed in per cent of that reflected by a n unfilmed surface.

From these equations there were calculated values of b, and b,,.. hence of b2 and of

("7

+

for increasing from 0" by 15" increments. Figures 9a bmsx. to 9f have been drawn from these values in the same way as figures 8a to ,

I

840

A. M. GAUDIN

8m were drawn from the values in table 10. Table 13 likewise parallels table 11. These data show that if partial interference only can be made to occur, the colors of the films are paler than those of films obtained in those cases where complete interference is possible, and that these films are visible, but over a shorter range of film thicknesses. A detailed comparison of tables 11 and 13 is instructive. TABLE 13 Characteristics of f i l m s of various thickness ( n F = 2 . 2 ) f o r m e d at the surface of a solid having a rejlectivity of SO per cent against t h e film (44.4 per cent in a i r ) , viewed bii oil i m m e r s i o n

1

INTENSITY O F REFLECTED

1

TIONTOTAAT

1

I LIQHT I N RELA- 1

I

I 'E 1

,

I

LNETy

COLORATION

APPEARANCE

4200 TO 7000

A.

I

c.

152 227 455 637 910 1183 1820 2366 3640

92 68 57 100

77 37 36 36

36

I 1071 1 120 1 1.84~ 1

I

100

I

I

2 7 8 2-4

1

'

7280

I

0 0 0

0

1

and

and 2@3 2 78 8 16 7 and 2Gf

4

i Sone

8

1 Sone

1

14560

36

100

1

I

Bright Pale Pale Bright Very pale Bright

1 ~

White White Tan Lavender Blue Greenish white Old rose Green GreenishJvhite

Very bright Very bright Bright Bright Bright Very bright

Sone 3 Very pale 1 1 Very pale f 1 Pale Very pale 1

I

II Bright Bright

II

'

White

I

E F F E C T O F OBLIQUE ILLUMISATIOK, AS I N A METALLOGRAPHIC MICROSCOPE

In the foregoing analysis it was assumed that the incidence of the light falling on the filmed surface was normal. If it is not normal but oblique, everything is changed: the reflectivity, b2, at the film surface, the reflectivity, IC2, a t the substratum-film interface, and the retardation between the reflections of various orders $1, $2, &, etc., due to the time required to traverse the film, are altered. There are also changes in the retardation due to reflection a t rare medium to dense medium boundaries.

841


This for example, is recalled by Jung (27), who sketches the changes in the relative retardations of the component of the light vector polarized parallel and perpendicular to the plane of incidence with changes in the angle of incidence, using plane polarized incident light a t 45" azimuth. According to Jung, a t normal incidence there is no change of phase between the reflected components, while a t grazing incidence these components differ by half a wave or X/2. This does not mean, of course, that a t normal incidence there is no retardation on reflection; Wood (62), for instance, makes a statement to the contrary. However, Wood and Jung do not agree as to the amount of the phase retardation on reflection at normal

x

incidence, their values being A = - and 2

1

A = arc tan

?%'(I + 2nK K') -1

in which n is the index of refraction, K the absorption coefficient, and X the wave length. Both of these are presented on theoretical grounds. TrlBLE 14 Characteristics of some microscope objectiaes ( B a u s c h a n d L o m b ) MAXIMUM ANQLE

FOCAL LENGTH

OF INCIDENCE

mm.

32 16

8 4 2 (oil)

0 10

5 10 20 25 90

0.25 0 50 0 85 1 30

5"45' 14"30' 30 58"20' 59"40'

Fortunately, the phase difference between the parallel and perpendicular light vectors as determined experimentally is inconsiderable, unless the angle of incidence to the metal exceeds 30". This appears, for example, in figure 5b of the article by Dayton (lo), in which data for iron are presented after Jamin (26). At an angle of incidence to the metal of 25O, A is about

so 5' that is, but 3 degrees of arc.

As will appear presently, even with the

widest-angle lens in use as a dry objective, the incidence to the mineral surface through the film rarely exceeds 25' (table 15). In the case of oil-immersion objectives an incidence possibly as great as 40-45' may be encountered, particularly if the index of refraction of the film is low. In that case the relative retardation on reflection of the two components of polarized incident light, respectively parallel and per1 x pendicular to the plane of incidence, may rise to - - or 15 degrees of arc. 12 2 THE J O U R N l L OF PHYSICAL CHEMISTRY, T'OL.

41,

NO.

6

842

A. hl. GAUDIS

It happens that in the examination of polished surfaces under a reflecting microscope the light impinging upon the object under examination ranges all the way from normally incident light to oblique light forming an angle i, whose sine is the numerical aperture of the objective divided b y the index of refraction of the immersing medium. (Maximum i is about BO".) For various objectives made by Bausch and Lomb, t h e focal length, approximate magnifying power, numerical aperture, and maximum angle of incidence are as given in table 14. This is in part according to W. L. Patterson (38). VARIATION I N AMPLITUDE O F REFLECTIOSS W I T H INCIDENCE O F LIGHT; AIR IMMERSIOK

It is, therefore, of interest to examine first what happens to the aniplitude of the reflectioiis of various orders for angles of incidence up to 60' for both oil and air immersion. This can be done by means of the general Fresnel formula describing the reflectivity of a transparent substance a t oblique incidence,

in n-hich i and r are the angles of incidence and refraction related by na sin i

=

n F sin r

For air immersion n A = 1 so that sin i = nF sin r . To proceed with a similar estimation of the change in reflectivity at polished opaque mineral surfaces with changes in incidence, the assumption is again made that the reflectivity of the polished mineral can be regarded as related to the index of refraction as if the mineral were transparent. In the case of normal incidence, 2 (kMF)

=

n.lf n.u

+

n F

n F

From this relationship nlf can be calculated if n F and (k.vF)? are known. For example, if nF = 2.2 and ( ? c , ~=~0.10, ) ~ n.M = 4.235. Extending the general formula describing the reflectivity a t oblique incidence to polished opaque substances:

R = k = z {'sinsin

(e)?

(T (T

+ -

T')]'

+ [tan tan

+

((Tr - r'> T')]')

From the values of b and k , values of bo, bl, bz, . . . . b,, of

bmi,,,,

(XI b,,.

and

can be calculated as functions of the incidence. Table 15 presents

843


a typical analysis of this sort for n F = 2.2, ( k M F ) *= 10.0 per cent at normal incidence. It is interesting to observe that within the whole range of air-immersion lenses, even for incidences as high as 60", there is no appreciable change in k M F . Also one should note that there is no appreciable change in b provided an incidence lower than about 30 to 35' is used, while the change in b is marked at high obliquity (at 60" incidence an increase in b of over 15 per cent is noted). TABLE 15 V a r i a t i o n w i t h angle of incidence, i, u s i n g a i r i m m e r s i o n , in the case n F = 8.8 i f R J ~ F i s 10 per cent at normal incidence, of the a m p l i t u d e of the zero-order reflection, b; of the a m p l i t u d e of the jirst-order reflection in the jilrn layer, ~ M F of ; the amplitudes 0.f reject i o n in a i r oj" various orders, bo, bl, b l . . . b,; and of the ratio of the i n t e n s i t y oj" minimum i l l u m i n a t i o n to m a x i m u m i l l u m i n a t i o n , Angle of incidence, i, in degrees.. . . . . . . 0 20 30 Angle of refraction (in film), r , in degrees.. 0 8"563' 13" 8' -4ngle of refraction (in substratum), r ' , in degrees.. . . . . . . 0 4'38' 6'48' 37 8 b in per cent . . . . . . . . . . 37.6 37.6 31 6 k.tf~ in per cent.. . . . . . 31.6 31 6 37.8 bo.. . . . . . . . . . . . . . . . . . . . 37.6 37 6 bi.. . . . . . . . . . . . . . . . . . . . 27.2 27 2 27.1 bz.. . . . . . . . . . . . . . . . . . . . 3.2 32 32 ba. . . . . . . . . . . . . . . . . . . . . 0.4 0 4 0 4 b,. . . . . . . . . . . . . . . . . . . . . 0.0 00 00 bmin, . . . . . . . . . . . . . . . . . . 6 . 9 6.9 7 1 bmax,. . . . . . . . . . . . . . . . . . 62.0 62 0 62.1 1.2

1.2

1.3

40

45

50

55

17" 0'

18"45' 20"23' 21'52'

8'44' 38 3 31 6 38 3 27 0 3 3 0 4 0 0 7.6 62 4

9"37' lO"26' 39.8 38.9 31.6 31.6 39 8 38 9 26.8 26 6 3.3 34 0.4 0 4 00 0 1 8 4 9 4 62 8 63 4

1 5

1 8

22

60 23"ll'

11"lO' ll"48' 41 2 43 5 31.6 31.7 41 2 43 5 26 2 25 7 3 4 3.5 0 4 05 0 1 0 1 11 1 13 7 64 3 66 1 3 0

4 3

___ -

Thus the analysis presented in this paper, up to the calculation of the ratio

( F ), for normal incidence, applies to a very high approximation bmin. max.

for the partly oblique illumination obtaining in the case of all objectives weaker than the 8 mm. objective, and including that objective, but it does not apply to the same high degree of approximation in the case of higherpower objectives. It is seen from table 15 that, even if using a 4 mm. lens (maximum incidence near eo"), there is little change in the possible perfection of the interference with change in incidence. In all cases

(2)'

is less than 10 per

844

A. M. GAUDIN

cent, thus indicating that the interference for individual rays of every possible obliquity (considered singly) remains in the zone of possible excellence. This does not mean that this is true of the total effect; indeed, it will later be shown that, strictly speaking, this is not always the case. Two additional cases will be considered briefly with nF = 2.2, one in which k.wF is large, the ,other small, compared to the value chosen for table 15. Specifically 1 per cent and 40 per cent were chosen as type values for ( k M F ) 2 . I n the first case nMis 2.689; in the second, 9.73. I n the case nF = 2.2, k M Fsmall, increasing the angle of incidence makes the iridescence worse and worse. On the other hand, in the case nF = 2.2, kMUF large, increasing the angle of incidence makes the iridescence better, at least up to an angle of incidence of 60". It would seem that perfect TABLE 16 V a r i a t i o n w i t h angle of incidence of the reflectivity in the f i l m and of the m a x i m u m and minimum amplitudes of reflection o j surfaces having reflectivities of 1 per cent (case A ) and 40 per cent (case B ) , respectively, against the jilm; n F = 8.8 Angle of incidence t o film, i, in degrees ............................... Amplitude of reflection from film surface, b, in per cent.. . Amplitude of reflection f substratum, ~ M F in , per cent Maximum amplitude of net reflection, bmax., i n p e r c e n t , . . . Minimum amplitude of net flection, b m i n , , in per cent

(e)',

in per cent.. . . . . . , . . .

iridescence,

(2) =

0, should be obtained in the latter case for an inci-

dence somewhere between 70 and 80°, and that for a still higher incidence the iridescence would become poorer and poorer until grazing incidence, at which position it would be nil since there is no refracted ray in the film a t grazing incidence. This explains, for example, why films of barium stearate on polished chromium formed by the successive transfer of unimolecular layers of barium stearate at the surface of water by the method of Langmuir and of Blodgett (2, 31, 6) are most brilliantly colored when viewed a t some particular, highly oblique incidence.

IDENTIFICATION O F OPAQUE

SOLIDS

845

Table 16 shows that there is very little change in k with change in incidence, whether k2 is 1 per cent or 40 per cent at normal incidence, just as there is very little change in it if k2 is 10 per cent (table 15). Table 16 shows also the variations in the amplitude of reflections of various orders, and in

(py,

with variations in incidence of the light for the two cases

max.

in which the reflectivity in the film, a t normal incidence, is 1 per cent and

(k92

40 per cent, respectively. The change in the trend of - with change in incidence is of interest. For practical purposes the important thing to remember is that even for an incidence of 60" corresponding to a numerical aperture of 0.866 for air-immersion lenses, the change in the quality of the iridescence with changes in incidence of the light is not considerable, so that the analysis of interference effects when using a 4 mm. objective limits itself to a study of the effect of the various retardations caused by the various incidences. VARIATION I N RETARDATION O F REFLECTIONS O F VARIOUS ORDERS WITH VARIATIOS I N INCIDENCE

It is now of interest to find out what change is effected by changing incidence in the retardation of the first-order reflection over the zero-order reflection. Since the effective thickness of the film must be t cos r it is sufficient in figure 7 to change the axis of abscissae by a translation equivalent to cos r. Figures loa, lob, and 1Oc show the relative intensities of illumination at various wave lengths in the two cases of normally incident light and light incident at 60". It will readily appear that when dealing with color shadings due to very thin films (first-order tan) the use of a high-power lens giving rise t o very oblique rays is not very objectionable. All it does is to require somewhat greater film thickness for visibility. On the other hand, in dealing with colors of the second to fourth orders the use of a highpower lens is bound to have a marked damping effect, as shown by figure lOc, since where a maximum occurs due to normal incidence a low intensity is had due to highly oblique incidence, and where a minimum occurs due to normal incidence a substantial intensity due to highly oblique incidence is obtained. Comparison of figures loa, lob, and 1Oc strongly suggests that with increasing thickness of film the height of the successive maxima of total illumination will gradually decrease, reaching some minimum value, while the height of the minima will gradually increase, reaching some maximum value. The net effect of simultaneous incidence at various angles will be t o bleach the film, as compared with the effect that would be obtained if all the incident light struck the film at the same angle.

846

A. M. GAUDIK

VARIATION I S AMPLITUDE O F REFLECTIONS W I T H IKCIDEXCE O F LIGHT; OIL IMMERSION

I n the case of an oil-immersion lens having a numerical aperture of 1.30, the oil having an index of refraction of 1.505, the inclination CY of the most 1100

100,

0.500

a600

0.700

0.800

a FIG.loa. Film thickness, 227 A. C .

XINP

FIQ.lob. Film thickness, 910 A. U.

h ",#A

FIG.1Oc. Film thickness, 3640 A. U. FIG.10 (a to c). Relative intensities of illumination a t various wave lengths for normally incident light and light incident a t i = 60" if n p = 2.2, ( k ~ p ) = * 10 per cent.

oblique ray admitted by the lens is such that sin CY =

1.30 1.505

__ =

0.864, CY =

59"50', or practically 60'. Table 17 parallels table 15, for the case of oil immersion. It is seen that, while b varies considerably with change in incidence, there is practically

847

IDESTIFICATION O F OPAQUE SOLIDS

no change in k.vMF. It is clear that the effect of miscellaneous obliquities of incident rays is much more marked in the case of oil immersion than in the case of air immersion. In figure 11 are plotted the relationship between distance along the face of the objective from center t o edge, and the corresponding values of b,,i,.,

TABLE 17 V a r i a t i o n w i t h angle of incidence, i , u s i n g oil i m m e r s i o n (no = 1.505), in the case n p = 3.9, it" R.11~i s SO per cent at n o r m a l incidence, of the a m p l i t u d e of the zero-order r e j e c t i o n in the oil layer, b; o.f the a m p l i t u d e ot" the Jirst-order r e j e c t i o n in the f i l m layer, k ~ i o ~f t h;e a m p l i t u d e s of r e j e c t i o n in oil of various orders, bo, b l , b ? . . . .b,; a n d of the ratio of the i n t e n s i t y of minimum i l l u m i n a t i o n to maxiniuna i l l u m i n a t i o n ,

Angle of incidence, i , 1 in degrees. . . . . . . .~ 0 Angle of refraction (in film), r , in

bi . . . . . . . . . . . . . . . . . . . . .

b?. . . . . . . . . . . . . . . . . . . . .

I

I

20

I

1

30

~

~

40

~

45

~

50

55

60

, 13'32'

20" 0'

30554' 18.9 54.8 18 9 52 8 5 5 0 6 0.6 0.1 0 1 40 1 66 8

j"443'

(2y

. . . . . . . . . . . . ..~ 36 2

19.1 54.8 19.1 52 8 5 5 0 6 0 1 39 9 66 9

36 1 ~

35 6 ~

26" 6' 28"56'

31'37'

34" 5' 36"20'

7"23' 20 1 54.8 20.1 52 6 5 8 0 6 0.1 39 0 67 4

8" 21 54 21 52 6

8"49' 22.8 54 8 22 8 52 0 6 5 0 8 0 1 36 6 69 0

9"26' 9"58+' 25 2 28 8 54 8 54 9 25 2 28 8 51 4 50 3 79 7.1 1 3 10 0 1 0 2 34.4 30 9 70.4 72 3

33 5

31 2

____

0 0 38 68

8' 1 8 1 3 0 7 1 0 0

28 1 -

23.9 ~

18 3 ___

RELATIONSHIP O F WAVE LENGTH TO AMPLITUDE O F REFLECTED LIGHT; OIL IMMERSION

It is proposed to determine the A-b curve for the total reflected light from all parts of the lens and for a given thickness of film, t. For any one value of A, b is the sum of all the individual values Ab of the elementary rays. It can fairly be assumed that the illumination on the object is uniform, and that each unit area on the lens face contributes equally to this illumination. Hence, the light coming from all the points A on the face of the objective situated at distance z from the center can be regarded as coming from z such points. As a first approximation the

848

A. M. GAUDIN

objective can be regarded as made up of ten concentric annular bodies of average radii equal to 1/20, 3/20, 5/20, 7/20, 9/20, 11/20, 13/20, 15/20, 17/20, 19/20 of the outer radius of the front lens of the objective. The abundance of the light from these bodies is as 1,3, 5 , 7 , 9, 11, 13, 15, 17, 19. J-alues of bmin. and b,, as a function of z are presented in figure 11. Table 18 presents the weighted average value of b for the case t = 910 A. U., for various values of X ranging down from CQ to 500 A. U. These values mere obtained from curves (not presented in this paper) in which individual values of b against X were plotted for various values of z, namely, 1, 3, 5 , 7, 9, 11, 13, 15, 17, and 19. 75 70 55

a

\

50

55 70 I

*=

t5 4r.i

*

**

7 to

;5 TO

TER OF ‘ECTIVE

FIG.11. Values of bmin,, b,,,.,

and

t

EDGE OF OBJECTIVE

for a n oil-immersion lens (no = 1.505,

numerical aperture = 1.30) if n , ~ = 2.2 and k.vF = 30 per cent.

The intensity of the reflection for the various maxima and minima is obtained by squaring the various maxima and minima in ‘b’ which can be read from a charted presentation of the data of table 18. These particular intensities, together with the ratio of successive pairs, are presented in table 19. Figure 12, which corresponds to figure 7 , and which must be compared with it, sums up in graphical form the damping action of a wideangle lens on the brilliancy of iridescent films. In this theoretical estimation of the damping action of wide-angle lenses on iridescence, the tacit assumption has been made that the phase retardation on reflection at the surface of the opaque solid is not a function of the

849


x

b

A . U.

p w cent

68 67 65 63 60 56 49 43 40 39 38 39 41 52 64 66

m

50,000 30,000 20 000 16,000 13,000 10,000 8,625 7,750 7,250 6,750 6,500 6,000 5,000 4,000 3,700 3,500 3,300 3,000 ~

1

5 9 3 4 7 0 6 4

1 7 3 7 0 6 8

1 !

A.U.

1

1 '

63 8

2,500 2,300 2,250 2,200 2,000 1,850 1,750 1,700

1,400 1,350 1,300

~

1

b

A per cent

~

'1

1,200 1,150 1,100 1,000 960

46 41 41 42 53 63 66 66 61 54 46 44 47 54 61 63 61 50 47

per cent

9 6 6 4 9

880 840 800 760 740 720 690 670 650 620 600 580 560 550 520 510 500

I

,

1'

4 1 1 9 2

1

0 8 9 9 4 8 6 3 9

1,

50 59 61 57 51 49 51 57 60 59 53 49 54 60 60 52

0 4 5 8 1 3 1 9 6 3 5 8 8 1 3 3

ii

~

TABLE 19

Intensities of maxima and minima i n light reflected from a filmed mineral viewed with an oil-immersion objective, N . A . = 1.30, if YLF = 2.2, (kdwp)2 = 0.30, t = 910 A . U . ; also ratio of each minimum to the corresponding maximum INTENSITY

RATIO

1

per cent

per cent

14.9 17.2 20.0 22.8 Maximum fourth or .................. Minimum fifth order . . . . . . . . . . . . . . . . . . . . . . . . Maximum fifth order. .................. Minimum sixth order.. . . . . . . . . . . . . . . . . . . . . .

37.9 24.2 36 8 24.8 25 0

I

1 I

32 38 45 56

I

64 67

I

69

850

A. M. GAUDIN

incidence. This, however, is possibly not strictly true (11, 23, 27). Obviously, variations in the phase retardation with incidence will make the various amplitude-wave length curves (e.g., such as shown in figure 11) still more out of step. Unfortunately so little is known concerning t h c changes in phase retardation with incidence that it is impossible to evaluate the effect of this variable on the amplitude of the net reflected light in function of wave length. Table 19 and the consideration just outlined give theoretical justification for the observation otherwise frequently made in the laboratory that iridescent filnis of a high order are bleached when viewed by high-power oil-iininersion objectives, while still fairly well colored when viewed with low-power, dry objectives. Comparing films giving interference colors of high order with first-order films it is clear that their quality is inipaired if vieTTed with low-power h

IN

AU .n4

FIG.12. Relative intensity of reflected light of various Tvave lengths if film is of such thickness that interference of first order occurs for A = 8000 A. U. ( n =~ 2 . 2 ; ( k , ~ p ) *= 30 per cent), using an oil-immersion lens having a numerical aperture of 1.30.

objectives because of the crowding together of maxiina and minima within the risible spectrum; if the viewing is through high-power objectives, a further impairment in the quality of the film arises froin the increasing damping in the maxima and minima of the amplitude of the net reflection n-ith increasing thickness of film. Thiy defect is particularly acute in the case of oil-immersion objectives and adds itielf in this case to a third defect arising from the usually inadequate reflection at the film-oil boundary. I T T E R F E R E X C C EFFECTS CAUSED BY ABSORBIXG FILMS WHICH ABSORB I N KOS-SELECTITE Fa4SHIOK

It ib of importance t o consider the effect of films displaying absorption, first of a non-selectire character, then of a selective character. I n the case of non-selective absorption, let T be the distance that must be traveled by light in the absorbing film in order that the amplitude be

85 1


FIG.13. Multiple reflection a t a filmed surface, filmed by a n absorbing substance TABLE 20 D a m p i n g factor due to absorption in film f o r single (dla3, CY*] e t c . ) : -2

and multiple rejkctions

(CY)

4

CY=^

t T

a

0.010 0 020 0 050 0 100 0 150 0 200 0 300 0 400 0 500 0 600 0 700 0 800 0 900 1 000 1 200 1 500 1 800 2000 2 500 3000 4000

0 9802 0 9608 0 9049 0 8187 0 7408 0 6704 0 5488 0 4494 0 3679 0 3012 0.2466 0 2019 0 1653 0 1353 0 0908 0 0498 0 0274 0 0183 0 0069 0 0025 0 0003

a4

0.961 0 923 0 819 0 669 0 549 0 444 0 301 0 202 0 135 0 092 0 060 0 041 0 027 0 018 0 008 0 002 0 001

0.942 0 887 0 741 0 550 0 407 0 296 0 166 0 091 0 050 0.027 0 015 0 008 0 005 0 002 0 001

0.923 0.852 0.670 0.488 0 301 0.197 0 090 0 041 0.018 0 008 0.004 0 002 0 001

reduced e-fold (e being the base of naperian logarithms). The amplitude of successive reflections from the film surface is as follows:

852

A. M. GAUDIN

in d i i c h p , t , r , b, k have the same significance as previously. This is made obvious by mere inspection of figure 13. TABLE 21 A m p l i t u d e of refiection by filmed solid in a i r ( p e r cent o j incident a m p l i t u d e ) due to multiple reflection at normal incidence for nF = 2 . 2 , (k&fF)* = 10 per cent, f o r various oalues of t/T (T = thickness at w h i c h a m p l i t u d e i s reduced jrom 1 to e-1, a n d t = thickness of f i l m ) 1

br

62

+

T

37 6 37 6 37 6 37 6 37 6 37 6 37 6 37 6 37.6 37 6 37 6 37 6 37 6 37 6 37 6 37 6 37 6 37 6

0 0 050 0 100 0 200 0 300 0 400 0 500 0 600 0 700 0 800 0 900 1 000 1 200 1 500 1 800 2 000 2 500 3 000

1,

27 2 24 6 22 3 18 2 14 9 12.2 10 0 8 2 6 7 5 5 4 5 37 2.5 1 4 0 7 0 5 0 1 00

32 26 2 1 1.4 1 0 0.6 0.4 0 3 0 2 0 1 0 1 0 0 0 0

1 0 0 0 00 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1

I ' 1

At normal incidence cos r = 1, and if a: b, = (1-b?)

Values of

CY,

~2,a 3 , a4,in terms of

bp-1

4 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0

=

e-':,

bmin.

bmax.

6 8 10 1 13 0 17 9 21 6 24 8 27 2 29 1 30 7 32 0 33 0 33.8 35 1 36 2 36 9 37 1 37 5 37 6

62 59 58 54 51 49 47 45 44 43 42 41 40 39 38 38 37 37

0 9 0 5 6 2 2 5 1 0 0 2 1 0 3 1 7 6

(:*.>' max

1 2 2 8 5 0 10 8 17 5 25 4 33 2 40 9 48 4 55 4 61 8 67 3 76 7 86 2 92 9 95 0 99 0 100 0

b, takes the form:

(XI)

k p a:+p

t T are given in table 20.

Consider again the particular case nF = 2.2, (k.wF)2= 10 per cent. Values of bo, bl, b2, b3, for various thicknesses expressed in terms of t / T , at normal incidence are given in table 21.

t

From this tabulation it is abundantly clear that as - exceeds 0.2, 0.4, T


853

0.7, and 1.2, respectively, the optimum quality of the interference colors is reduced from excellent to good, fair, poor, and none. This is portrayed in a most graphic manner by figures 14a to 14j in which are plotted simultaneously the b2-X curves for various thicknesses of film ranging from 200 A. U. to 2400 A. U., first in the case that the absorption is nil ( T = C Q ; dash lines), secondly in the case of rather strong but non-selective absorption ( T = 2000 -4.U.; solid lines), and lastly in the case of selective absorption (dash and dot lines). Attention is now drawn to the first and second sets only. Figures 14a to 14c show that for thin films (in the special case under consideration) there is little difference as between the absorbing and the non-absorbing films. A considerable and constantly increasing difference is manifest as films are made thicker. This is shown by figures 14g to 14j which contrast the brilliant interference colors resulting from non-absorbing films with the dully tinted gray-whites of absorbing films. It is of interest to note that if the reflectivity of the solid is too high, for a given index of refraction of the film, to permit more than the poorest interference (see, e.g., figure 5 , upper part of chart), this defect can be eliminated to some extent if the film is absorbing. I n such a case, as the film thickens the amplitude of the reflections which have traveled through the film are gradually reduced and become of an order of magnitude small enough to interfere, constructively and destructively according to the wave length, with the zero-order reflection. Thin films in such a case fail to display interference colors, but thicker films may begin to display them, so that theoretically at least, good second- and third-order colors should be possible for a surface which fails to give first-order colors, or gives but poor colors of that type. INTERFERENCE EFFECTS CAUSED BY ABSORBING FILMS WHICH ABSORB SELECTIVELY

This problem is somewhat more complicated, but it can be treated just as the foregoing problem, making use of tables 20 and 21 together with the fundamental general equations T’III. Figures 14a to 14j present the relationship between the reflectivities and wave length in such a way that direct comparison is possible between the characteristics of a selectively absorbing film [nF= 2.2; ( k M F ) 2= 10 per cent; air immersion; normal incidence, dash and dot lines], a non-selectively absorbing fiZm [T = 2000 A. U.], and a perfectly transparent film. The selectively absorbing film was chosen arbitrarily to have the X-T relationship summarized in table 22. This relationship is such that by transmitted light the substance will have a strong orange or red color (relative paucity of violet, blue, and green). Figure 14a shows that a solid filmed in such a manner as is here postu-

100 I

FIG. 14a.Film thickness, 200 A. U. - - -, tan; ___ , pale tan; -.-._ , brown.

FIG.14b. Film thickness, 400 A. U. - - -, orange; ___ , orange; _.-., redbrown.

4000

6d00

FIG. 14c. Film thickness, 600 A. U. - - -, dark purple; -, purple; , dark blue.

-.-.-

8ooo

FIG. 14d. Film thickness, 800 A. U. ---, brilliant blue; -, blue; -.-.-, dark bluegreen.

FIG. 14e. Film thickness, 1000 A. U. - --, clear light b l u e ; - - - - - , slate blue; -.-.-, gray-green.

FIG. 14f. Film thickness, 1200 A. U. - - -, pale green; -, steely gray; -*-+--, grayish-yellow.

FIG.lig. Film thickness, 1600 A. U. - - -, orange; ___ ,reddishgray; -.-., grayishred.

F I G . l i h . Film thickness, 2000 A. U. -- -, brilliant violet; -, lavender-gray; -.-.-, grayish-purple.

FIG. 14;. Film thickness, 2400 A. U. - --, brilliant green; -, gray; -.-.I graygreen.

FIG.14. Relative intensities of light reflected by a filmed surface, in relation to the light that would be reflected by the same surface, if unfilmed, in the case n F = 2.2, ( I i . w ~ ) * = 10 per cent, if (a) the film is not absorbing (- - -), (b) the film is absorbing uniformly a t all \Tave lengths according to table 26 for T = 2000 A. U. (---), (e) the film is absorbing more in the short xvave-length than in the long wave-length end of the spectrum, according t o table 26 (--.--.-), 854

~

855


lated acquires a substantially equivalent color a t a lesser thickness than a solid filmed by a non-selectively absorbing substance or by a transparent sub9tance. This advance in sequence of colors with thickness is confirmed by figures 14b to 14f, but i t apparently has disappeared in thicknesses giving second-order colors. By analogy it would seem that a film substance absorbing the red and orange more than the blue, violet, and green would display interference colors only when its thickness is greater than is indicated by the theory for transparent films. I n the case of a film substance absorbing the central part of the visible spectrum more than the edges, the first color might well be some qhade of violet or violet gray followed by red (or brown), gray [first intermediary “hiatus,” to folloiy the terminology of Evans], blue, gray [the “hiatu?” of TABLE 22 A r b i t r a r i l y chosen relationship between absorption and waz’e length of a selectiuely absorbing substance (Ti s the thickness ojfilm substance resulting i n a n e-jold reduction in a m p l i t u d e ) h

T

Z4.r.

A . l.

4000 4500 5000 5500 6000 6500 7000 7500 8000

200 400 1000 2000 4000 8000 10,000 16,000 20,000

!

Evans], grayish-orange, gray-violet, green, etc. The various colors just listed would correspond to the various thicknesses shown in figures 14a to 14j. Thus, selective absorption could cause not only deviations in the color scale-film thickness relationship, but also t h e appearance (at least in extraordinary cases) of exceptional color sequences. I t is of interest to compare these theoretical curves with some experimental curves, figures 15a to 15j. These charts, redrawn from data by Constable ( 7 ) , deal with the interference colors on iron and nickel. It is clear that the quality of the interference decreases materially with thickness, and that this deterioration is of the type pictured by figure 14. The only difference between these figures is that with increasing film thickness the curves (figure 15) show lesser and lesser amplitudes. This may be caused by an irregularity and roughness of the films that increases with

/+’

100

SO.

,

I’

0

4000

(Iron, straw

8000

6000

[ Iron, red-

FIG. l5b. I Iron, brownish-yellow

brown

brown light brown 100-

4004

bo00

80

(Iron,

(Iron, very intense

,I

Nickel, very dark blue

violet

Iron, steely blue-green FIG. 15f. I {Kickel, greenish-

(Iron. red(Iron, FIG.15h. I1 Iron, yellow orange FIG.15g. I1 {Sickel, [ brown yellow FIG.15. Wave length-intensity relationship for the reflection of filmed nickel and iron according to Constable (7a). The metals \yere filmed by oxidation and the intensity of the light reflected from the filmed surfaces was determined spectrophotometrically. 856

\


857

the thickness, resulting in a smaller amplitude of specular reflection and a greater amplitude of diffuse reflection. LIMITATIONS A S D ASSUMPTIOKS I N THIS ANALYSIS

The analysis that is presented in this paper pertains to the condition

nM> nF > nA Five other conditions can be regarded to exist; they are described by permutations of M , F , and A . A study of all these permutations mould cover entirely the field of thin films. This analysis has involved but one assumption which can be questioned, namely, that the reflectivity of an opaque medium can be expressed in terms of its refractive index and of the refractive index of the adjoining medium as if the opaque were a transparent medium. This assumption is probably invalid in those cases in which the opaque medium is a good conductor of electricity, but it is acceptable in the case of sulfide minerals, since these minerals cannot be regarded as possessing metallic conductance (39). Other assumptions have tacitly been’made in the numerical examples. These are that the reflectivity of the polished surface and the index of refraction of the film do not vary with the wave length of the light. These assumptions are convenient as a first approximation, but it must be kept in mind that as a rule indices of refraction vary in regular manner with the wave length (as a first approximation according to the Cauchy formula (63)

B

C

A2

x4

n = A + - + -

in which n is the index of refraction, X the wave length, and A , B, C experimental constants) and that there are many exceptions related to the existence of absorption bands. Treatment of the general problem of color of film versus thickness is possible by equations developed in this paper (by carrying out individual calculations for each wave length and each thickness of film). I n the analysis dealing with oblique incidence, such as is obtained with high-power microscope objectives, it is further implicitly assumed that the film is isotropic. This assumption is necessarily a limitation. Finally, it has been assumed that the light used is truly white light, that is, that the abundance of rays of various wave lengths is such as to include all wave lengths in the proper amounts. Clearly this is merely an approximation more or less approached experimentally with incandescent lights fitted with suitable absorbing filters. In spite of these various limitations and assumptions the analysis pre-

.

858

A. hf. GAUDIN

sented in this paper agrees remarkably with experiment, a t least in so far as experimentation has been carried at present. It gives, furthermore, a sound basis for further experimentation, pointing out the optical conditions which must be secured in order for success to be possible. In another article some of the chemical phases of the production of thin films will be discussed; this will lead to another set of limitations which must he observed in order t o be successful in the production of characteristic iridescent films on minerals (or more generally on solids) which it is desired to identify. I n later articles detailed accounts will be presented of the results obtained with various filming solutions and various minerals and synthetic preparations. REFERESCES (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29) (30) (31) (32)

BEUTEL,ERNST,AND KCTZELSIGG, ARTUR:Z. Elektrochem. 36, 523-8 (1930). BLODGETT, KATHERINE: J. Am. Chem. SOC.67, 1007-22 (1935). CISSARZ,A.: Xeues Jahrb. Mineral. Geol. A64, 137-62 (1931). CISSARZ, A.: Z. Krist. 78, 445-61 (1931). CISSARZ,8.: Z. Krist. 82, 438-50 (1932). CLARK, G. L., STERRETT, R. R., ASD LEPPLA,P. W. J. Am. Chem. Soc 67, 330-1 (1935). CONSTABLE, F. HURN:Proc. Roy. Soc. London A117, 376-87 (1927). CONSTABLE, F. HURN:Proc. Roy. SOC. London A116, 570-88 (1927). F. HURN:Proc. Roy. Soc. London A126, 630-5 (1929). CONSTABLE, DAYTOK-, RESSELL Tt’. : Am. Inst. RIining hIet. Engrs., Tech. Pub. No. 593 (1935). DRUDE,P . : W e d . Ann. 61, 86 (1894). EVANS, U. R.: Trans. Electrochem. Soc. 46,247-82 (1924). E v a ~ s U. , R.: Proc. Roy. SOC.London A107, 228-37 (1925). EVANS,U.R.: S a t u r e 120, 584 (1927). EVANS, C . R.: S a t u r e 123, 16 (1929). EVANS,U.R.: J . Inst. 1Ietals 46,7-24 (1931). EVAXS, U. R.: Kolloid-Z. 69, 129-37 (1931). EVANS, U. R.,AXD BANNISTER, L. C . : Proc. Roy. SOC.London A126, 370-94 (1929). EVANS, U.R., ASD STOCKDALE, JOHN: J. Chem. Soc. 1929,2651-60. A. 31.: Econ. Geol. 30, 552-62 (1935). GAUDIN, GAUDIN,A. 31.: Congr. intern. mines mBt. gCol. appl., Paris, 1935. GAUDIN,A. h l . : Gluck Auf, Montana School of Mines, 1, S o . 5 (1936). GISH, OLIVERH.: Phys. Rev. [2] 3, 353-63 (1914). Handbook of Chemistry and Physics, 17th Edition, p. 1317. Chemical Rubber Publishing Co , Cleveland, Ohio (1932). Reference 21, p. 1198. Jkhim: Ann. Physik 19,296 (1847). JGA-G, GERHARD: Z. physik. Chem. 119,111-12 (1926). KOERBER, FRIEDRICH: Z. anorg. Chem. 164, 267-74 (1926). KOHLSCHCTTER, v., AND KRAEHENBUHL, E. : z. Elektrochem. 29,570-80 (1923). KURODA, h 1 ~ s . 4 ~ 0Sci. : Papers Inst. Phys. Chem. Research Tokyo 14, 145-52 (1930). LANGMUIR, I : J. Franklin Inst. 218, 153 (1934). LEO, ;\Tax: Die Anlauffarben-Eine neue llethode zur Untersuchung opaker Erze und Erzgemenge. Theodor Steinkopf, Berlin (1911).


859

(33) LUCE,L. R.: Ann. phys. [lo] 11, 167-251 (1929). (34) MARCELIS,ANDRE : Solutions Superficielles; fluides a deux dimensions et stratifications monomolecularies, p. 139. Les Presses Universitaires de France (1931). (35) Reference 34, p. 147. (36) XINOR: Ann. Physik 10, 614 (1903); quoted by Wood in Physical Optics, 1934 edition, p. 566. Opticks, Lib. 11, edited by Clarke. Lausannae et Genevae (37) NEWTON,ISAAC: (1740); quoted by Rollett (43). (38) PATTERSON, W.L.: Trans. Am. SOC.Steel Treating 2, Xo. 2 (1921). (39) PHILLIPS, F. COLES:Aheralog. Mag. 23, 458-62 (1933). (40) PILLISG,N. B., AXD BEDWORTH, R. E . : J. Inst. Metals 29, 529-91 (1923). THOMAS: The Theory of Light, 3rd edition, p. 171. The RIacmillan (41) PRESTON, Co., London (1901). (42) REFERENCE 41, p. 172. (43) ROLLETT,ALEXANDER: Sitzber. kaiserl. Akad. Wiss. Math. Katurw. Klasse 77, Part 111, 177-261 (1878). (44) RUBENSAND HAGES: Ann. Physik 1, 352 (1900): 8, 1 (1902); quoted by Wood (58, 61). (45) SCHNEIDERHOEHN, H., AND RAMDOHR, PAUL:Lehrbuch der Erzmikroskopie, Vol. 11, p. 4 (1931). Borntraeger Bros., Berlin (1931). (46) STOKES:Cambridge and Dublin Mathematical Journal, Vol. IV, p. 1 (1849); quoted by Preston (41, p. 178). W. R . : Thesis, Montana School of Mines, 1936. (47) SWILER, (48) TAMhf.4NN) G.: anorg. Chem. 111, 78-89 (1920). (49) T A M M A N N , G., AND BOCHOW, K.: anorg. Chem. 169, 42-50 (1928). (50) TAMMANN, G., A N D BREDEMEIER, H.: Z. anorg. Chem. 136, 337-57 (1924). (51) TAMNANN, G., A N D KOESTER,W.:Z. anorg. Chem. 123, 196-224 (1922). (52) TAMNASS, G., ASD SCHNEIDER, J.: Z. anorg. Chem. 171, 367-71 (1928). (53) T A M M ~ G., N NAND , SCHROEDER, E.: anorg. Chem. 128, 179-206 (1923). (54) TAMMANN, G., A N D SIEBEL,G.: anorg. Chem. 162, 149-59 (1926). (55) VERNON, W,H. J.: Trans. Faraday SOC.19, 839-900 (1924). (56) KERTHEIM:Ann. chim. phys. [3] 10, 156: quoted by Rollett (43). (57) ILKIN INS, FREDERICK JAMES:J. Chem. soc. 1930, 1304-9. (58) WOOD,ROBERTW.:Physical Optics, p. 157. The Macmillan Co., S e w York (1924). (59) Reference 58, p. 466. (60) Reference 58, p. 172. (61) WOOD,ROBERTW.:Physical Optics, p. 563. The Macmillan Co., New York (1934). (62) Reference 61, p. 553. (63) Reference 61, p. 469.

z.

z.

z.

z.

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