I IRLF i <> - A ^ LIBRARY OF THE UNIVERSITY OF CALIFORNIA GIFT OF A METHOD OF PETROGRAPHIC ANALYSIS BASED UPON Chromatic Interference with Thin Sections of Doubly -Refracting Crystals in Parallel Polarized Light. Presented to the Faculty of Philosophy of the University of Penn- sylvania in partial fulfillment of the requirements for the degree of Doctor of Philosophy. BY HOMER MUNRO DERR, A. M, The Randal Morgan Laboratory of Physics. 1903. Plate 1 155477 PLATE 1. Thin Section showing Free Gold in Metamorphic Diorite FROM ENCAMPMENT, WYOMING. Magnified 50 diameters : /, feldspar, ( oligoclase, albite ) ; q, quartz h, hornblende ; ra, magnetite ; g, gold. Crossed nicols ; thickness of section, 56 microns. Plate I. BEHAVIOR OF ISOMETRIC CRYSTALS. Isometric crystals on account of their isotropic character ex- hibit no special phenomena in polarized light. If a thin plate of such a mineral be placed in the path of a polarized beam of light between the polarizer and the analyzer, the beam will ex- perience no alteration of its plane of vibration, no matter in what direction the plate was cut from the crystal, nor in what position it lies between the polarizer and the analyzer ; for the elasticity of the ether is the same in all directions in such a case, and rotating the mineral about any axis whatever will effect no change of the plane of vibration of the polarized light. If the section is colorless, it will not alter the color or the brightness of the field of view in a polarization microscope, except for the small amount of absorption which a ray of light will experience in passing through any medium. If the section is colored, the field of view will appear slightly different in color from that of the mineral ; however, this color does not change by altering the position of the plate in any way. Furthermore, when the nicols are crossed, a section on the stage of the microscope appears dark, and revolving it in any plane produces no effect. It appears light, and experiences no change in any position be- tween parallel nicols. Sections of transparent isometric crys- tals may always be recognized as such by the fact that they behave as an amorphous substance in polarized light. There are, however, some optical anomalies in this connection. Sec- tions of amorphous or isometric minerals sometimes appear light when viewed between crossed nicols. These results are due to internal strains caused either by inclusions of gases or fluids which exert a pressure on their surroundings, or by con- traction of adjacent parts. These phenomena may usually be distinguished from ordinary double refraction, since the appear- ance is not generally uniform throughout the section. INTERFERENCE. Fresnel and Arago investigated completely the conditions of interference of two rays polarized at right angles to each other after they had been brought back to the same plane of polari- zation by passing them through a crystal of calc-spar whose principal section made an angle of 45 with the planes of polari- zation of each of the two rays. They found, (1) that two rays polarized at right angles to each other, which have come from an unpolarized ray, do not interfere even when they are brought into the same plane of polarization ; (2) that two rays polarized at right angles, which have come from a polarized ray, interfere when they are brought back to the same plane of polarization. Let a ray of polarized monochromatic light be incident at the lower surface of a plane-parallel doubly-refracting plate at any angle, Fig. 1, at the point A. It is separated into two rays AD and AE, vibrating in planes at right angles to each other and following different paths in the plate. On emergence they follow paths parallel to their direction at inci- dence, but are not coinci- dent, and do not produce interference. But other parallel rays from the same - 1 source and incident at B and C, will emerge from the upper surface, so that from all points E and F, the ordinary component of one ray and the extraordinary of an- other follow the same path. These rays will have travelled over slightly different paths in the plate, and with different velocities ; then on emergence one ray must have advanced a certain number of wave-lengths, or fractions thereof, ahead of the other. The waves are therefore in different phases and retain this difference of phase while they travel in the same medium. The plane of vibration of the extraordinary ray is parallel to the principal optic section of the plate, and that of the ordinary ray is at right angles to the same. If an analyzer be placed in the path of these rays, each ray will be by it resolved into components the vibrations of which are in and at right angles to the principal section of the polarizer, and only the former will be transmitted. That is to say, there will emerge two rays advancing in the same line and with parallel vibrations ; hence they are capable of interfering, since light- rays can completely interfere only when their vibration are in the same plane. If these vibrations are in the same phase, the intensity of the resultant wave will be proportional to the square of the sum of their amplitudes ; but if in opposite phases, the intensity will be proportional to the square of the difference. WITH PARALLEL MONOCHROMATIC LIGHT AND CROSSED NICOLS. In sections cut perpendicular to the optic axis, interference phenomena are impossible, and the field remains dark through- out a rotation of 360, since the light from the polarizer traverses the section in the direction of the optic axis, and therefore without change. In all other sections there is double refraction, and consequently interference. The field is dark four times during the entire rotation of the stage at intervals of 90 ; that is, when the planes of vibration of the rays pro- duced in the section coincide with the principal sections of either nicol. For all other positions the field is illuminated by the components of the rays which penetrate the analyzer, and this brightening is most intense in the diagonal positions. The rays pursuing the same path are brought into one plane of vibration by the analyzer and there interfere. The kind of interference is determined by the formula, in which A is the difference in the retardations in microns which the two rays have undergone. h is the thickness of the section in microns. A, is the wave-length in microns. i is the angle of incidence. nj is the index of refraction for the slower ray. n 2 is the index of refraction for the faster ray. For normal incidence, which is usually the case in practice, A = h (HI n 2 ). 7 FIG. 2. When A = A., 2 A, 3 A., etc., the field is dark during an entire revolution of the section on the stage. From Fig. 2, it will be seen that the A components o f PPi , on emer- gence from the plate with vibra- tion directions OOx and EE 1} must be of the same phase; that is, the sim- ultaneously dis- placing forces acting upon any ether particle C are Co and Ce, which when reduced to the plane of the analyzer are Ca and Cai, equal and in opposite directions. When A =1 A., f A, f A., etc. , the light will be at its brightest be- cause the com- ponents of PPi must then be of opposite phase A on emergence from the plate. From Fig. 3, the simultaneous- ly displacing forces acting on any ether parti- cle C are Co and Ce, which re- duced by the analyzer to its plane are Ca and Ca 1? equal and in the same direction. Let a thin wedge of doubly-refracting crystal, such as quartz, be cut so that its planes of vibration are parallel to the length and breadth, and then examined between crossed nicols, using perpendicularly incident monochromatic light. Upon rotation, it will appear dark when in the "normal positions," but in all others it will show a series of dark and light bands which are most marked in the diagonal position. When the nicols are made parallel the portions formerly light become dark, and vice versa. The distance between the dark bands varies with light of different wave-lengths. The formula for intensity of the emerging light is I = a 2 sin 2 2 IF sin 2 in which a the amplitude of the incident wave, A. = the wave- length, A = the retardation, V = the angle between the vibra- tion plane of the polarizer and the slower wave. I will be a minimum : (a) when sin 2 2 V = 0, or 2 W = 0, 180, 360, 540, etc., or W = 0, 90, 180, 270, etc. ; that is, four times in a revolution of the section, or whenever the planes of vibration of the plate coincide with those of the polarizer and the analyzer. () when sin 3 ^ = 0, which will be whenever y = 1, 2, 3, etc. ; that is, whenever the phase difference A is a multiple of A, for then sin 2 -^ becomes sin 2 180, or a mutiple there- l A J of. This is independent of W ; hence with this condition, the section will appear dark throughout an entire revolution. I will be a maximum : When sin 2 2^ = 1 or 2W = 90, 270, etc., and W = 45, 135, 225, etc. When y = |, f, |, etc., for then sin 2 {^-] = sin 2 90, sin 2 270, etc., =1. * See Derivation of Formulae, page 21. WITH PARALLEL WHITE LIGHT AND CROSSED .NICOLS. In sections cut normal to the optic axis, the field appears dark throughout a rotation of 360, the optic axis being the same for all colors. In all other sections there is extinction every 90, and greatest brightness in the diagonal positions ; but, since A may be at the same time approximately an even multiple of i A, and an odd multiple of A. 1 , light of one wave- length may be greatly weakened while that of another wave- length is practically undimmed ; that is, there will result a tint due to the partial extinction of some colors. For example, suppose that in a polarization microscope, parallel white light passes upward through the polarizer, whose principal section is represented by PP^ in Fig. 2. Next let this light, which is polarized in a single plane PPi , pass through a thin section of gypsum. It will there be separated into two waves vibrating in planes at right angles to each other, OOi and EEj. The two waves travel through the section with unequal velocity, and on emerging one is retarded a certain number of wave-lengths, or fractions thereof, as compared with the other. Now let these light waves pass through the analyzer, with its vibration-plane AAi at right angles to PPi. Then each of the two sets of vibrations will have a component in the direction AAi, and these will emerge polarized in the same plane, and are therefore capable of interfering. Light corre- sponding to the other components in the direction PP X will be extinguished. As previously stated, one of the emergent waves is slightly retarded as compared with the other. The amount of this retardation, on which the interference-color of the sec- tion depends, is proportional to the difference of the indices, and also to the thickness of the section. If in the examination of the plate of gypsum above, one of the nicols had been rotated 90, that is to say, if the principal sections of the two nicols had been parallel, interference would have taken place between the emerging rays ; but the color resulting in each case would have been exactly complementary to that obtained at first when the nicols were crossed. For instance, in Fig 2, if W = o, the principal sections of the plate coincide with those of polarizer and analyzer, and complete ex- tinction occurs ; but if A AI be made parallel to PPi , evidently the section would appear white. 10 If we examine in the same way a very thin and gradually- tapering wedge of some doubly-refracting crystal (quartz for example), cut so that its planes of vibration are parallel to the length and breadth, the successive interference-colors of the first order, beginning at the thin end, pass from an iron-gray through bluish-gray to white, yellow, and red ; then follow violet, indigo, blue, green, yellow, orange, and red of the second order; then the similar but paler series of colors of the third order, and finally the very pale shades of green and red of the fourth order. Beyond this the colors are not very distinct, and white of a higher order finally results from the interference. A min- eraj of very strong double refraction, such as calcite, shows only the white of the higher orders unless extremely thin. In general, a thin section of a doubly -refracting crystal, ex- amined between crossed nicols, is not dark except when placed in certain definite positions. In any other position it does not completely extinguish the light, but its effect, in conjunction with the nicols, is partially to suppress the several components of the white light in different degrees, so that in the emergent beam these components are no longer in the proportions to give white light. In this way arise interference-tints, which may be definitely classed according to Newton's color-scale. (See page 15.) The several tints, though graduating into one another, are distinguished by names and divided into several orders. The precise position in the scale of a given tint observed be- tween crossed nicols can be fixed by means of a quartz-wedge or other contrivance for compensating or neutralizing the effect of double refraction of the section. The interference-tints given by a crystal section depend (1) on the amount of double refraction of the mineral (nj n 2 ), which is a specific character ; (2) on the direction of the section relatively to the ellipsoid of optic elasticity, the tint being highest for a section parallel to the greatest and least axes of the ellipsoid ; (3) on the thickness of the section. The last two are disturbing factors, which must be eliminated before we can use the interference-tints as an index of the amount of double refraction of the crystal, and as a useful criterion in identifying the mineral. The fact that the interference-tints depend in part on the direction of the section through the crystal will give little difficulty in estimating approximately the amount of double refraction of the mineral. If several crystals of the 11 same mineral are contained in a rock-section, it is sufficient to have regard to the ones which give the highest interference- tints. Even a single crystal will in the majority of cases give tints not so far below those proper to the mineral as to occasion error, but the possibility of the section having an unfavorable direction must be borne in mind. In making a determination, the manner of proceedure is briefly as follows : Select a crystal, revolve it on the stage of the microscope until it is in the position of maximum intensity (i. e., when the greatest and least axes of elasticity make an angle of about 45 with the principal sections of polarizer and analyzer) ; introduce the quartz-wedge between polarizer and analyzer (in slot provided for the purpose, also making an angle of 45 with principal sections of the two nicols), until a position of maximum darkness of the crystal is reached. If the inter- ference-tint of the crystal cannot in this position, or when revolved through an angle of 90, be reduced through the suc- cessive colors of the scale, in descending order, back to dark- ness, the section is an unfavorable one, i. e. , it is not cut ap- proxmiately parallel to the plane of the axes of greatest and least elasticity of the crystal. When a favorable section is found, after having fixed the quartz-wedge in the position giving maximum darkness to the crystal, first remove the sec- tion from the stage of the microscope. Now the quartz- wedge, as viewed through the microscope, will be of the same color and order in Newton's scale as that of the crystal-section under the same conditions. Carefully withdraw the wedge, at the same time noting the succession of tints in descending order. In this fashion, one can estimate the order of the interference-color in the scale ; and by applying the formula A = h (n x n 2 ), in connection with the tables given on pages 15 and 16, many of the commoner rock-forming minerals may be identified. The following twenty-four examples will illustrate : 12 Rock Section. Thick- ness in Mi- crons. Crystals. Interference Color between Crossed Nicols. Interference Color between Parallel Nicols. Order of Color. QUARTZ- PORPHYBY . . . RHYOLITE . . 26.5 32 8 Quartz Orthoclase Greenish white Gray Straw - yellow Brown Brownish yellow I I 23 6 Orthoclase Biotite Hornblende Almost pure white Light greenish gray Light green Gray Light red Grayish red Carmine -red I IV II I PEGMATITE 21 4 Orthoclase Plagioclase Microcline Biotite Muscovite Hornblende Augite Hypersthene Tourmaline Zircon Apatite Bluish gray Bluish gray Bluish gray Greenish yellow Bright orange- red Purple Reddish orange Pale straw -yellow Brownish yellow Greenish yellow Lavender-gray Brownish white Brownish white Brownish white Grayish blue Greenish blue Light green Bluish green Dark reddish brown Gray-blue Grayish blue Yellowish white I I I III II II I I I III I I' TRACHYTE 27 2 Feldspar Biotite Muscovite Tourmaline Beryl Grayish blue Greenish blue Pure yellow Brownish yellow Grayish blue Brownish white Flesh-colored Indigo Gray-blue Brownish white I III II I I PHONOLiITE 33 1 Biotite Hornblende Augite Dull purple Sky-blue Indigo Dull sea-green Orange Golden yellow III II II TlNGUAITE 18 4 Augite Elaeolite Greenish blue Grayish blue Brownish orange Brownish white II I SYENITE . . 24 7 Biotite Hornblende Augite Leucite Elaeolite Orthoclase Dark violet-red Bright yellow Brownish yellow Black La vender -gray Bluish gray Green Blue Gray-blue Bright white Yellowish white Brownish yellow "ll I I I I I NEPHELINE-SYENITE . ANDESITE . 19.6 25 3 Augite Hornblende Biotite Orthoclase Nepheline Hornblende Augite Biotite Red Indigo Flesh -colored Grayish blue Lavender-gray Reddish orange Brownish yellow Indigo Pale green Golden yellow Sea-green Brownish white Yellowish white Bluish green Gray-blue Impure yellow I II III I I I I III I Augite Biotite Hornblende Deep red Carmine-red Indigo Yellowish green Green Golden yellow I III II 13 Rock Section. Thick- ness in Mi- crons. Crystals. Interference Color between Crossed Nicols. Interference Color between Parallel Nicols. Order of Color. DACITE 33 7 Quartz Light yellow Indigo I DIOBITE 29.1 Plagioclase Biotite Augite Hornblende Plagioclase Yellowish white Whitish gray Green Light green Greenish white Carmine-red Bluish gray Light carmine-red Purplish red Brown I IV II II I BASALT 38 9 Augite Biotite Hornblende Plagioclase Sky-blue Bluish green Greenish blue Orange Lilac Brownish orange Indigo JII IV II I OLIVINE- BASALT 41.7 Augite Plagioclase Greenish yellow Bright yellow Violet Blue II I HORNBLENDE-BASALT . NEPHELINE-BASALT . . . LEUCITE-BASALT 46.3 43.6 45 9 Augite Olivine Plagioclase Augite Hornblende Nepheline Augite Leucite Pure yellow Flesh colored Bright yellow Bright orange-red Dark violet red Gray Orange Indigo Sea-green Blue Greenish blue Green Brownish yellow Dark blue White II III I II II I II I MELILITE-BASALT DIABASE 51.3 49 2 Augite Melilite Augite Bright orange-red Straw-yellow Light bluish violet Greenish blue Deep violet Yellowish green Gray-blue II I III I GABBBO 46 5 Augite Dark violet red Green Blue II I NOBITE 41'3 Augite Bright orange-red Greenish blue Blue II I TBOCTOLITE 55 4 Hypersthene Reddish orange Bluish green Gray- blue I I THEBALITE 33.4 Olivine Augite Whitish gray Bluish gray Brownish orange IV PEBIDOTITE 29.8 Biotite Plagioclase Nepheline Apatite Olivine Whitish gray Yellowish white Grayish blue Grayish blue Dark violet-red Bluish gray Carmine-red Brownish white Brownish white Green II Augite Diopside Hornblende Biotite Sky-blue Bright orange-red Greenish blue Light green Orange Greenish blue Brownish orange Carmine II II II IV 14 COLOR-SCALE ACCORDING TO NEWTON. A IN MICRONS. INTERFERENCE COLOR BETWEEN CROSSED NICOLS. INTERFERENCE COLOR BETWEEN PARALLEL NICOLS. 0.000 Black Bright white > 0.040 Iron-gray White 0.097 Lavender-gray Yellowish white 0.158 Grayish blue Brownish white 0.218 Clearer gray Brownish yellow 0.234 Greenish white Brown 0.259 Almost pure white Light red a M H 0.267 Yellowish white Carmine-red 3 0.275 Pale straw-yellow Dark reddish brown o 9 0.281 Straw-yellow Deep violet U 0.306 Light yellow Indigo a 0.332 Bright yellow Blue 0.430 Brownish yellow Gray-blue 0.505 Reddish orange Bluish green 0.536 Red Pale green 0.551 Deep red Yellowish green 0.565 Purple Lighter green 0.575 Violet Greenish yellow 0.589 Indigo Golden yellow 0.664 Sky-blue Orange 0.728 Greenish blue Brownish orange G0 i 0.747 Green Light carmine-red o a 0.826 Lighter green Purplish red . b 0.843 Yellowish green Violet-purple s 0.866 Greenish yellow Violet 1 0.910 Pure yellow Indigo 0.948 Orange Dark blue 0.998 Bright orange-red Greenish blue 1.101 Dark violet-red Green 1.128 Light bluish violet Yellowish green ] 1.151 Indigo Impure yellow 1.258 Greenish blue Flesh-colored H 1.334 Sea-green Brownish red H M g 1.376 Brilliant green Violet U o 1.426 Greenish yellow Grayish blue a a 1.495 Flesh-colored Sea-green H 33 1.534 Carmine-red Green 1.621 Dull purple Dull sea-green 1.652 Violet-gray Yellowish green -> 1.682 Grayish blue Greenish yellow o 1.711 Dull sea-green Yellowish gray 1.744 Bluish green Lilac - 1.811 Light green Carmine 1.927 Light greenish gray Grayish red ^ 2.007 Whitish gray Bluish gray 15 DOUBLY-REFRACTING POWER OF VARIOUS MINERALS. CRYSTAL. D! D 2 CRYSTAL. n x n 2 Proustite . . . . 0.2953 Phenacite . . . . 0.0157 Sulphur . . . . 0.2900 Anorthite . . . . 0.0130 Rutile .... 0.2871 Natrolite . . . 0.0119 Dolomite . . . . 0.1791 Hyphersthene . . 0.0115 Calcite 0.1722 Andalusite . . 0.0110 Brookite . . . . 0.1580 Gypsum . . . . 0.0100 Aragonite . . . . 0.1558 Bronzite . . . . 0.0100 Cassiterite . . . 0.0968 Corundum . . 0.0092 Zircon 0618 Quartz . 0091 Biotite .... 0.0600 Topaz . . . 0090 Epidote . . . . 0.0545 Cordierite . . . . 0.0090 Anhydrite . . . 0.0430 Axinite . . 0.0090 Muscovite . . . 0.0420 Orthoclase . . 0.0080 Olivine 0.0360 Albite . . . 0.0080 Scapolite . 0.0360 Beryl . . 0.0073 Diopside . . . . 0.0335 Plagioclase . . 0.0071 Cancrinite . . 0.0289 Zoisite . 0.0060 Tremolite . . . . 0.0265 Melilite . . . . 0.0058 Hornblende . . . 0.0240 Nephelite . . . . 0.0049 Augite .... . 0.0220 Elaeolite . . . . 0.0047 Sillimanite . . . 0.0220 Apatite . . . . 0.0042 Glaucophane . . . 0.0216 Vesuvianite . . . 0.0020 Brucite . . . . 0.0210 Apophyllite . . . 0.0019 Tourmaline . . . 0.0189 Leucite . . . . 0.0010 Chlorite . . . . 0.0010 16 Plate 2. PLATE 2. (Crossed Nicols.) Chalcedony, Section cut at right angles to the length of a stalactite ; X 100 diameters. Central sections through the concretions give a spherulitic interference cross. Radial fibers appear to be uniaxial, since extinction is complete in fibers parallel to planes of vibration of polarizer and analyzer, i. e. , when y o ; the illumination is at a maximum when W = 45. Extinction is also complete where fibers are cut at right angles, for the light travels in the direction of their optic axes. The crystal system of this mineral has not yet been definitely determined ; however, it appears to be optically uniaxial. Plate II. \ DKRR DERIVATION OF FORMULAE. Let there be a Y crystalline plate having OX and OY for axes of elasticity, OP the direction of a pol- arizer on this sec- tion (fig. 5) ; a ray of light polarized by OP vibrating in the principal sec- tion of the polari- zer is divided, in the plate, into two rays, which may be regarded as superposed and l which have at emergence a difference of phase : 9 = ^ - (n e n ). A- cos r Let us call x and y the vibrations of these rays which give as a resultant an elliptical movement, and let us receive the corres- ponding doubly-refracted ray on an analyzer having OA for the direction on the plane of the plate ; the angle between the polarizer and the axis OX of the plate is POX. Put Z POX = ft. The angle between the analyzer and the same axis is AOX, and we put Z AOX = a. The components be- longing to OX and OY are : x = a cos 2 cos y = a cos 2 n _ . sin ft. They give rise to two rays which emerge from the plate with a difference of phase

\ = a cos 2 TT r (cos a cos ft -f- sin ar sin ft cos % ft q>) -f- a sin 2 TT sin a sin /? sin 2 n (p. According to Fresnel, let us put a (cos ot cos ft -\- sin a sin ft cos 2 n qt) = E cos A ? a sin <* sin ft sin 2 ?r

= cos 2 a cos 2 /? -f sin 2 a sin 2 y# -f- 2 sin sin ft cos <* cos /? - 4 sin or sin ft cos or cos /? sin 2 rt cp t Replacing cos 2 n cp by 1 2 sin 2 TT 9?, E 2 (cos cos ft -f- sin sin ft)* sin 2 sin 2 /? sin 2 TT q> a = cos 2 ( ft] sin 2 a sin 2 /? sin 2 n cp ..... (1) DISCUSSION OF THE FORMULA. We have that q> = - (n e n ). A cos r h represents the thickness of the section ; r the angle between the rays, which we regard as superposed, and the normal to the faces of the section ; n e , n the indices of refraction of the two rays. When white light, composed of an infinite number of lights of very different wave-lengths, is employed, the second term corresponds to the sum of the intensities of these lights, and the general formula may then be written : ^! = cos 2 (a ft} sin 2 a sin 2 ft Ssin 2 n ^ (n e n ) . - (2) a 2 *- A. cos r 20 THE DIFFERENT CASES. I. a ft = r , case where the principal sections of the po- 2i larizer and analyzer are at right angles to each other. Since ft = a - , formula (2) becomes : f* = cos 2 -^ - sin 2 a sin (2 a TT) ^> sin 2 n cp, or a PARTICULAR CASES. = sin 2 2 a 5 sin 2 TT

(4) a II. a = /?, the polarizer and analyzer are parallel. TT 2 2 = 1 sin 2 2 a ^> sin 2 ?r