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- 1. INTRODUCTION(Translated from Spanish)Thin Sections are two dimensional cuts of bodies with crystallographic and optical properties belonging to threedimensions, so a good knowledge of Solid Geometry, can be of great help in studies of these sections with thepetrographic microscope. These notes are intended to emphasize geometric aspects, taking into account, firstcrystallography, then optics, and finally an integration of both. We use the Miller indices only for simple planes:CRYSTALLOGRAPHIC SECTIONS.The idea is to do many imaginary cuts to a cube, in order to obtain an approximation to the most likely real randomcuts. Note that a good example would be the volcanic rocks, where the majority of crystals can take their own forms.1. SIMPLE CUBEIn figure 2, we begin with a section coincident with the frontal face of the cube (100). If we turn the section 90° withthe axis of rotation indicated by the arrows, we will obtain several rectangles and two squares. Two of the sides ofrectangles are the same than the cube’s side (a) and the maximum length that can reach the other two is the diagonalof a face of the cube (section 3).Now (figure 3) we begin with section 3 of the previous case: a plane containing the diagonals of two opposite faces ofthe cube, one of which is the axis of rotation. Initially we have a rectangle (cut 1) then isosceles trapezoids (cut 2) andfinally an equilateral triangle (cut 3). If we continue turning the cross section in the same direction, we get isoscelestriangles up to a square.In figure 4, we begin with the same type of section that the previous case, but the axis of rotation is now bisector ofthe section and parallel to the diagonal of upper face (001). After the initial rectangle, we get a regular hexagon (cut2), then a rhombus (cut 3) whose major axis is the main diagonal of the cube. If we continue turning in the samedirection 90°, we get rhombus, whose major axis reduces until we get a square.(100) (010) (001) (101) (011) (110) (111) (210)12312 3a2a1 231 2 33
- 2. Now (Figure 5) we have the same initial section than before, but now we move the section in a parallel way. We getseveral rectangles and a square section. It should be noted that the length of two sides of the rectangles, is the samethan the side of the cube.The initial section in figure 6, is an equilateral triangle (the section of a tip of the cube), then we get isoscelestriangles, trapezoids (not shown) and finally we obtain a rectangle.So far, we can see that most likely sections that we can obtain of random cuts of a cube are rectangles, trianglesand trapezoids. Less likely are hexagons, squares and rhombuses. The rectangles have two sides equal to theside of the cube. The size of the sides of the triangles may vary from minuscule to the diagonal of one face ofthe cube.2. CUBE WITH INNER PLANES PARALLEL TO A FACE.A set of equally spaced parallel planes, will show the smaller thickness and higher density (amount of traces per unitarea) in a perpendicular section. Cut 1 of figure 7 is perpendicular to x, y and z planes. Cut 2 is tilted and the traces ofthe planes in this section are thicker and more spaced (less dense).4a 3a1 2 312aa56a
- 3. With this short introduction, let us consider a cubethan contains a set of planes parallel to one face(figure 8).Figure 9 shows the same type of cuts that figure 2. Section 1 is the frontal side of the cube (001) and does not cutthe inner planes. Section 2 cut only one plane with a weak slope. The trace of the plane in this section is then thick.Amphibole of Sotará volcano, Colombia. Note the presenceof two cleavages, one thinner and denser and the otherthicker and less dense. If the objective (x40) is movedslightly, the thinner cleavage does not seem to move, whilethe other shows a neat movement. The thickness and densedifference is due to section, almost perpendicular for thethinner and tilted for the thicker one. One sectionperpendicular to both cleavages, show them with the samethickness and density. Plane polarized light. x10.Thin cleavageThick cleavage8x y z2x y z12x yz71
- 4. The other sections cut all the planes with increasing inclination, therefore the thickness of the traces decreases anddensity increases.Figure 10 shows the same cuts than figure 3, but the axis of rotation is now the diagonal of the top face of the cube(001). In order to obtain the different sections it should be noted that in all cases, one of the edges of the sections iscontained in the frontal face of the cube (100) and then parallel to the inner planes. Therefore, the traces of theseplanes in all sections will be parallel to this border line. Note also that the direction of rotation is toward the upperface (001) which is perpendicular to the inner planes. Therefore, the thickness of traces decrease and their densityrise gradually.2345191234531012123
- 5. 3. CUBE WITH TWO INNER PLANES MUTUALLY PERPENDICULAR.(Figure 11)Figure 12 shows the same sections than figures 2 and 9. Note that cuts are perpendicular to X planes therefore theirtraces have the minimum thickness and the maximum density in all them. For traces of Y planes is the same case thanfigure 9. It should be noted that traces of both planes are perpendicular to each other, but for one family, theirthickness and density will vary.11y3x1x23y41234x y12
- 6. Figure 13 shows exactly the same cut that section 3 of figure 10 or section 3 of figure 3. The cut is an equilateraltriangle, where one side is the diagonal of the front face (100) and then parallel to Y planes. Another side is thediagonal of (010) face and then parallel to X planes. Therefore the traces of both planes in the section will have thesame thickness and density since the slope of cut is the same for both planes. Note that the angle between both tracesis 60º because is an equilateral triangle.Note in figure 14 that vertical sides of cuts are parallel to both inner planes, therefore their traces will be paralleleach other. The thickness and density will depend of the slope of cut and will be the same only for section 3, butnote that the direction of inclination is opposite.13xyxy12 34x y1y xy23y x4x14
- 7. 4. CUBE INSIDE ANOTHER CUBE.Figure 15 shows a cube included in the center of another, twice its size.Figure 16 shows three sections, the second one inperspective for more clarity. It is clear, that theprobability that a random section cuts the inner cube,will be low if the size is small, but if its size approachthe size of the external cube, more random sectionswill contain both cubes.We could see an analogy with all these figures and Thin Sections. Inner planes could be cleavage or twin planes. Thecube inside another one is similar to zoning of minerals or the external portion of a crystal altered by some chemicalreaction with his environment. It is important to note that although the thin sections are essentially two dimensionalbodies, their thickness (30 microns) is of great help in finding particular sections of a mineral. When the objective isdisplaced slightly (40x would be appropriate), a cleavage or twin plane perpendicular to the thin section, will presenta fine trace and remain static in the field of view, but seems thicker and to move more or less insofar as the slop isfarther away of perpendicular to the thin section.151621 31 232Plagioclases. Nevado del Ruiz. Colombia. Crosspolarized light. The left crystal shows the tracesof twin planes very thin and their density is highsuggesting that they are perpendicular to thesection. The right crystal instead, shows the twintraces thicker and less dense. x4.Plagioclase. Nevado del Ruiz volcano. Colombia. Left image planepolarized light. Right image cross polarized light. Note both cleavagesalmost mutually perpendicular and the trace of albite twin very thin.These characteristics belong to a section very close to perpendicular toa axis o [100]. The probability to find this section is very low but veryinteresting, because it allows the better determination for compositionin routine methods of Thin Sections. x10.
- 8. OPTICSAlthough these notes are not intended as a manual of Optical Mineralogy, always is useful to recall some basicconcepts that are handled in this discipline.In anisotropic crystals, the speed of light can vary according to its direction of vibration. The refractive index is theratio between the speed of light in vacuum with respect to its velocity in the medium considered. The light used inPetrography typically is orthoscopic and then is possible to associate directly a direction vibration of light with arefractive index and in that way simplify the reasoning used in the determination of anisotropic sections.If the refractive indices of a crystal are put all together in a point with the same direction in space that have thevibration of light associated with each of them, the resulting envelope is an ellipsoid called the indicatrix. Dependingon the crystal symmetry this will be an ellipsoid of revolution (Tetragonal and Hexagonal systems) or not(Orthorhombic, Monoclinic and Triclinic systems). It is useful to remember that the indicatrix is an artifact and by itsconstruction, its sections must necessarily pass through its center.SECTIONS OF A REVOLUTION ELLIPSOID (17 and 18)We take as the axis of rotation, the major axis of the ellipsoid butrationing is essentially the same if we take the minor axis. Aperpendicular section to the major axis will be a circumference sinceall points on the ellipse to rotate, describe a circle perpendicular to theaxis of rotation. A parallel section to major axis will be an ellipse withthe largest eccentricity can be obtained, that will decrease with theangle of cut. Note that the minor axis is contained in all sections.SECTIONS OF A ELLIPSOID NOT OF REVOLUTIONIn this case the ellipsoid has three axes: large, mediumand small that are mutually perpendicular. Sectionsperpendicular to one of these axes, contain the othertwo. If we take the section that contains the major andminor axis of the ellipsoid (figure 20), somewhere inthe ellipse, will give a distance to the center, equal tothe intermediate axis of the ellipsoid. If we continuewith the same procedure for cuts parallel to the axis ofthe ellipsoid (Figure 21), we obtain a circular areawhose radius is the length of the intermediate axis ofthe ellipsoid. These sections will be isotropic and itsperpendicular is called the optical axis. There will betwo circular sections is an ellipsoid not of revolution.171820NpNgNm19 greatpettymiddle
- 9. There are several ways to symbolize the refractive indices. Inan effort to emphasize its size, is used here Ng (g great) for themajor axis of the ellipsoid (Figure 22), Np (p petty) for thesmaller one and Nm for the middle axis (these are equivalentto the vibration directions Z, X and Y). In the case of ananisotropic section, we will use np and ng when we onlyknow the relative size between the two indices.The birefringence of a mineral, is the difference between its major and minor refractive indices, that is, betweenthe major and minor axes of the indicatrix. The birefringence of a section is the difference between the major andminor indices of the section. It is clear then that for a given mineral, the birefringence of the sections will rangefrom zero (the refractive indices of the section are equal, which corresponds to circular sections of the indicatrix) toa maximum value that coincides with the nominal value given for the mineral (the section contains then the majorand minor axis of the indicatrix).Usually the light used in the petrographic microscope is white and normal to the thin section (orthoscopic) beingpolarized according to the direction of the Polarizer that is often taken NS; over the thin section, is the Analyzerwhose polarization direction is perpendicular or EW. Polarizer and Analyzer arranged in this way (crossedpolarizers) do not allow the passage of light. If we interpose between the two, another polarizer with its polarizationdirection at 45 degrees of both, fwe see that there will be light transmission (Figure 23). This can be seen by vectordecomposition (Figure 24).Clearly, if the direction of the intermediate polarizer coincides with either the Polarizer or Analyzer, no light istransmitted.NgNm11NpNmNg21CircularsectionNgNpNm22Polarizeddirection23PolarizerAnalyzer
- 10. The sections of the indicatrix, other than circularsections, can be seen as a polarizer, with twopolarization directions mutually perpendicular. If anyof these directions coincides with the Polarizer of themicroscope, the beam of light coming from it, willbreak down in the section into two beams withmutually perpendicular vibration with different speeds(different refractive indices).If the analyzer is crossed, the two rays from the section are decomposed vectorially, interfering with each other,resulting in a characteristic color (interference colours), which will depend on the birefringence of the section andits thickness. These two factors together, constitute what is called the retardation. It is therefore important toremember that the observed color (with crossed polarizers), depends not only on the birefringence of the section, butalso its thickness. Thus using the same thickness for the sections (typically 30 microns), the interference coloursvaries only with the birefringence of the section.In the event that one of the indices of the anisotropic section, match with the direction of the Polarizer (Figure 25) itwill exists only one direction of vibration in the section parallel to the Polarizer (the other index is 90 degrees to thePolarizer and can not provide components). With crossed nicols, there is no light transmission (extinction position).In order to observe the optical characteristics of one index of a section in natural light (without the Analizer) it mustbe parallel to Polarizer. This is accomplished by taking the section to extinction (crossed nicols), and remove theAnalyzer. The relief and color observed, belong to the index parallel to Polarizer. It is important to note that fordifferent positions, relief and color will be intermediate between both indices.REAL CASEHYPERSTHENEOrthorhombic system. Two good cleavages {210} that cut a axis athalf distance and b axis at unity, but a is almost twice b in length,that means that they are almost perpendicular (88°). Np (X)= 1.712light rose-brown. Nm(Y)=1.724 pale yellow-green. Ng(Z)=1.727pale grey-green. Biaxial Negative (the minor index is the bisectorof optical axes). 2V between 50° - 60° (Tröger, 1971). With thesevalues we can see that the birefringence of Hypersthene (Ng-Np) is0.015 that is, an interference color of orange first order for 30microns.Figure 26 is a schematic perspective view of hypersthene. The refractive indices coincide with crystallographic axes.The section (001), that is perpendicular to c axis (Orthorhombic system) contains a and b axis and then Np and Nmindices. The birefringence of this section (Nm-Np) is 0.012 yellow first order for 30 microns of thickness. ThePolarizerAnalyzerMiddlePolarizerVibrationof lightde la luz24PolarizerAnisotropicSection25100(210)(010)Ng cNpbNmaOptical axis26
- 11. cleavages are perpendicular to section then their traces are very thin, perpendicular each other and stay static if theobjective (x40) is moved slightly. In plane polarized light, if the major index of the section (Nm) is parallel toPolarizer, the color of section will be yellowish hue. If we rotate the section 90º (Np will be parallel to Polarizer), wewill see a pinkish hue. In crossed polarized light, the extinction will be symmetric with respect to traces of cleavages.Section (100) is perpendicular to a axis and then contains b and c axis and therefore Np and Ng indices. Thebirefringence is the same than the mineral (Ng – Np) 0.015 orange first order. The cleavages are at 45º to thesection and then is difficult to observe them. The section is pleochroic between a greenish hue (Ng parallel to thepolarizer) and pink (Np aligned with the polarizer).The section (010) contains Ng and Nm indices. The birefringence will be (Ng – Nm) 0.003 dark gray. For the samereason as above, it will be difficult to observe the cleavages. The colors of the indices are greenish, therefore thepleochroism is not obvious. The section is perpendicular to bisector of optical axis and can therefore be seen theinterference figure well centered.The probability to obtain strictly these three sections is very small. However the sections close to them showsimilar characteristics.A section (210) that is parallel to one of the cleavages (figure 28)contains Ng index while the minor axis of the section, will bebetween Np and Nm therefore the birefringence is between 0.003and 0.015. We may consider a white hue of interference more orless. The color in plane polarized light will be between a greenish(Ng) and pinkish hue (n’p). We can see only one cleavage with atrace very thin because is almost perpendicular to section.In figure 29 we start with a (001) section. In the first case, the cuts are directed toward the face (100) and theyremain parallel to the b axis, therefore, all sections contain the Np index. The major index of the initial section isNm and Ng for the final one, then the major index of the section (n’g) will have a value between these two indices.The interference colors will be between yellow and orange. The traces of cleavages will become thicker and theangle between them gradually decreases. If the objective is moved slightly the traces will move to opposite sides,because although the angle is the same, the direction of inclination is opposite. Extinctions remain symmetricalwith respect to cleavages.In the second case, the cuts are directed toward the face (010), remaining parallel to the a axis. Note that the planeformed by the optical axes and indices Ng and Np is perpendicular to all sections and then the cuts will contain theindex Nm of the mineral. The other index in the initial section is Np and in the final one is Ng, so it will be a sectionwhere this index is Nm and the section is isothropic or cyclic and perpendicular to one of optical axes. This section isnot pleochroic. The birefringence decrease then progressively from the initial section, to zero (cyclic section), beforerising slightly to a final birefringence of 0.003 (face (010)). As in the previous case the traces of cleavages thickenprogressively, the angle between them diminishes and extinctions remain symmetrical.Starting from the same previous cut (001) towards (210), it can be seen that in this case, the sections will be almostperpendicular to one of the cleavages. The situation is similar to the cuts in Figure 12. The traces of cleavageperpendicular to the sections, will be fine, keep the same density and remain static when slightly displace the(001) section (100) section (010) sectionNmNgNpNgNm27NpNgn‘p28
- 12. microscope objective. The traces of the other cleavage will thicken, its density will decrease and appear to movemore strongly as the tilt angle decreases, when slightly move the objective. Note that only the initial section containstwo of the indices of the mineral, while the final contains only one (Ng). Both indices of the other cuts will haveintermediate values. The sections of the indicatrix are not evident in this case, however, the major index of initialsection bisects the traces of cleavages, while in the final one, this index is parallel to the only visible cleavage. Itcould be seen that the major index of sections will be progressively close to the fine cleavage and the extinctions arenot parallel which means that extinctions for these sections are neither straight nor symmetric.Toward (100) Facen‘gn‘gNpNpToward (010) FaceCyclic sectionNp(001) Section(001)NmNmn‘gNmNm29Toward (210) Facen‘gn‘pn‘gn‘pHypersthene. Section close to (001) face.The traces of two cleavages are thin andalmost perpendicular each other. Yellow firstorder in crossed polarized light (30 microns).Left, plane polarized light. Right, crossednicols. Andesite of Nevado del Ruizvolcano.Colombia. x10.
- 13. EPILOGUEFrom all these examples, it can be seen the importance of the Solid Geometry. A good knowledge of the ellipsoids,together with the crystalline forms, will allow more reliable identification of the crystals and a better understanding oftheir textures. In short, a three-dimensional ‘vision’, can go beyond the simple identification of minerals and is anecessary starting point in structural studies.BIBLIOGRAPHYSHELLEY, David. Manual of Optical Mineralogy. 1975. Elsevier.STOIBER, Richard; MORSE, Stearns. Crystal identification with the polarizing microscope. 1994. Chapman & Hall.TRÖGER, W.E.. Optische Bestimmung der gesteinsbildenden Minerale. 1971.Hypersthene. Section close to (010) face.Cleavages are no visible. Pleochroism is clearand interference colors are grayish. Thesection shows the interference figure wellcentered.Hypersthene. Section close to (100) face. Thissection does not show cleavages, as they are farfrom perpendicular to section. Pleochroism veryclear between greenish and pinkish colors. Incrossed polarized light, the interference color isorange of first order (30 microns) and is the samethat the mineral (contains the major and minorindices of mineral).

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