INTRODUCTION

(Translated from Spanish)

Thin Sections are two dimensional cuts of bodies with crystallographic and optical properties belonging to three

dimensions, so a good knowledge of Solid Geometry, can be of great help in studies of these sections with the

petrographic microscope. These notes are intended to emphasize geometric aspects, taking into account, first

crystallography, 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 random

cuts. Note that a good example would be the volcanic rocks, where the majority of crystals can take their own forms.

1. SIMPLE CUBE

In figure 2, we begin with a section coincident with the frontal face of the cube (100). If we turn the section 90° with

the axis of rotation indicated by the arrows, we will obtain several rectangles and two squares. Two of the sides of

rectangles are the same than the cube’s side (a) and the maximum length that can reach the other two is the diagonal

of 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 of

the cube, one of which is the axis of rotation. Initially we have a rectangle (cut 1) then isosceles trapezoids (cut 2) and

finally an equilateral triangle (cut 3). If we continue turning the cross section in the same direction, we get isosceles

triangles 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 of

the section and parallel to the diagonal of upper face (001). After the initial rectangle, we get a regular hexagon (cut

2), then a rhombus (cut 3) whose major axis is the main diagonal of the cube. If we continue turning in the same

direction 90°, we get rhombus, whose major axis reduces until we get a square.

(100) (010) (001) (101) (011) (110) (111) (210)

1

2 3

1 2 3

a

2

a

1 2 3

1 2 3 3

Now (Figure 5) we have the same initial section than before, but now we move the section in a parallel way. We get

several rectangles and a square section. It should be noted that the length of two sides of the rectangles, is the same

than 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 isosceles

triangles, 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, triangles

and trapezoids. Less likely are hexagons, squares and rhombuses. The rectangles have two sides equal to the

side of the cube. The size of the sides of the triangles may vary from minuscule to the diagonal of one face of

the 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 unit

area) in a perpendicular section. Cut 1 of figure 7 is perpendicular to x, y and z planes. Cut 2 is tilted and the traces of

the planes in this section are thicker and more spaced (less dense).

4

a 3

a

1 2 3

1 2

a

a

5

6

a

With this short introduction, let us consider a cube

than 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 cut

the 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 presence

of two cleavages, one thinner and denser and the other

thicker and less dense. If the objective (x40) is moved

slightly, the thinner cleavage does not seem to move, while

the other shows a neat movement. The thickness and dense

difference is due to section, almost perpendicular for the

thinner and tilted for the thicker one. One section

perpendicular to both cleavages, show them with the same thickness and density. Plane polarized light. x10.

Thin cleavage

Thick cleavage

8

x y z

2

x y z

1

2

x y

z 7

1

The other sections cut all the planes with increasing inclination, therefore the thickness of the traces decreases and

density 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 is

contained in the frontal face of the cube (100) and then parallel to the inner planes. Therefore, the traces of these

planes in all sections will be parallel to this border line. Note also that the direction of rotation is toward the upper

face (001) which is perpendicular to the inner planes. Therefore, the thickness of traces decrease and their density

rise gradually.

2

3

4

5

1

9

1

2

3

4

5

3

10

1

2

1

2

3

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 their

traces have the minimum thickness and the maximum density in all them. For traces of Y planes is the same case than

figure 9. It should be noted that traces of both planes are perpendicular to each other, but for one family, their

thickness and density will vary.

11

y

3

x

1

x

2

3

y

4

1

2

3

4 x y

12

Figure 13 shows exactly the same cut that section 3 of figure 10 or section 3 of figure 3. The cut is an equilateral

triangle, where one side is the diagonal of the front face (100) and then parallel to Y planes. Another side is the

diagonal of (010) face and then parallel to X planes. Therefore the traces of both planes in the section will have the

same thickness and density since the slope of cut is the same for both planes. Note that the angle between both traces

is 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 parallel

each other. The thickness and density will depend of the slope of cut and will be the same only for section 3, but

note that the direction of inclination is opposite.

13

x

y

x

y

1 2 3

4

x y

1

y x y

2

3

y x

4

x

14

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 in

perspective for more clarity. It is clear, that the

probability that a random section cuts the inner cube,

will be low if the size is small, but if its size approach

the size of the external cube, more random sections

will contain both cubes.

We could see an analogy with all these figures and Thin Sections. Inner planes could be cleavage or twin planes. The

cube inside another one is similar to zoning of minerals or the external portion of a crystal altered by some chemical

reaction with his environment. It is important to note that although the thin sections are essentially two dimensional

bodies, their thickness (30 microns) is of great help in finding particular sections of a mineral. When the objective is

displaced slightly (40x would be appropriate), a cleavage or twin plane perpendicular to the thin section, will present

a fine trace and remain static in the field of view, but seems thicker and to move more or less insofar as the slop is

farther away of perpendicular to the thin section.

15

16

2

1 3

1 2

3

2

Plagioclases. Nevado del Ruiz. Colombia. Cross

polarized light. The left crystal shows the traces

of twin planes very thin and their density is high

suggesting that they are perpendicular to the

section. The right crystal instead, shows the twin

traces thicker and less dense. x4.

Plagioclase. Nevado del Ruiz volcano. Colombia. Left image plane

polarized light. Right image cross polarized light. Note both cleavages

almost mutually perpendicular and the trace of albite twin very thin.

These characteristics belong to a section very close to perpendicular to

a axis o [100]. The probability to find this section is very low but very

interesting, because it allows the better determination for composition

in routine methods of Thin Sections. x10.

OPTICS

Although these notes are not intended as a manual of Optical Mineralogy, always is useful to recall some basic

concepts 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 the

ratio between the speed of light in vacuum with respect to its velocity in the medium considered. The light used in

Petrography typically is orthoscopic and then is possible to associate directly a direction vibration of light with a

refractive 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 the

vibration of light associated with each of them, the resulting envelope is an ellipsoid called the indicatrix. Depending

on 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 its

construction, 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 but

rationing is essentially the same if we take the minor axis. A

perpendicular section to the major axis will be a circumference since

all points on the ellipse to rotate, describe a circle perpendicular to the

axis of rotation. A parallel section to major axis will be an ellipse with

the largest eccentricity can be obtained, that will decrease with the

angle of cut. Note that the minor axis is contained in all sections.

SECTIONS OF A ELLIPSOID NOT OF REVOLUTION

In this case the ellipsoid has three axes: large, medium

and small that are mutually perpendicular. Sections

perpendicular to one of these axes, contain the other

two. If we take the section that contains the major and

minor axis of the ellipsoid (figure 20), somewhere in

the ellipse, will give a distance to the center, equal to

the intermediate axis of the ellipsoid. If we continue

with the same procedure for cuts parallel to the axis of

the ellipsoid (Figure 21), we obtain a circular area

whose radius is the length of the intermediate axis of

the ellipsoid. These sections will be isotropic and its

perpendicular is called the optical axis. There will be

two circular sections is an ellipsoid not of revolution.

17

18

20

Np

Ng

Nm

19 great

petty

middle

There are several ways to symbolize the refractive indices. In

an effort to emphasize its size, is used here Ng (g great) for the

major axis of the ellipsoid (Figure 22), Np (p petty) for the

smaller one and Nm for the middle axis (these are equivalent

to the vibration directions Z, X and Y). In the case of an

anisotropic section, we will use n'p and n'g when we only

know the relative size between the two indices.

The birefringence of a mineral, is the difference between its major and minor refractive indices, that is, between

the major and minor axes of the indicatrix. The birefringence of a section is the difference between the major and

minor indices of the section. It is clear then that for a given mineral, the birefringence of the sections will range

from zero (the refractive indices of the section are equal, which corresponds to circular sections of the indicatrix) to

a maximum value that coincides with the nominal value given for the mineral (the section contains then the major

and minor axis of the indicatrix).

Usually the light used in the petrographic microscope is white and normal to the thin section (orthoscopic) being

polarized according to the direction of the Polarizer that is often taken NS; over the thin section, is the Analyzer

whose polarization direction is perpendicular or EW. Polarizer and Analyzer arranged in this way (crossed

polarizers) do not allow the passage of light. If we interpose between the two, another polarizer with its polarization

direction at 45 degrees of both, fwe see that there will be light transmission (Figure 23). This can be seen by vector

decomposition (Figure 24).

Clearly, if the direction of the intermediate polarizer coincides with either the Polarizer or Analyzer, no light is

transmitted.

Ng

Nm

1

1

Np

Nm

Ng

21

Circular

section

Ng

Np

Nm

22

Polarized

direction

23

Polarizer

Analyzer

The sections of the indicatrix, other than circular

sections, can be seen as a polarizer, with two

polarization directions mutually perpendicular. If any

of these directions coincides with the Polarizer of the

microscope, the beam of light coming from it, will

break down in the section into two beams with

mutually 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 and

its thickness. These two factors together, constitute what is called the retardation. It is therefore important to

remember that the observed color (with crossed polarizers), depends not only on the birefringence of the section, but

also its thickness. Thus using the same thickness for the sections (typically 30 microns), the interference colours

varies 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) it

will exists only one direction of vibration in the section parallel to the Polarizer (the other index is 90 degrees to the

Polarizer 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 must

be parallel to Polarizer. This is accomplished by taking the section to extinction (crossed nicols), and remove the

Analyzer. The relief and color observed, belong to the index parallel to Polarizer. It is important to note that for

different positions, relief and color will be intermediate between both indices.

REAL CASE

HYPERSTHENE

Orthorhombic system. Two good cleavages {210} that cut a axis at

half distance and b axis at unity, but a is almost twice b in length,

that means that they are almost perpendicular (88°). Np (X)= 1.712

light rose-brown. Nm(Y)=1.724 pale yellow-green. Ng(Z)=1.727

pale grey-green. Biaxial Negative (the minor index is the bisector

of optical axes). 2V between 50° - 60° (Tröger, 1971). With these

values we can see that the birefringence of Hypersthene (Ng-Np) is

0.015 that is, an interference color of orange first order for 30

microns.

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 Nm

indices. The birefringence of this section (Nm-Np) is 0.012 yellow first order for 30 microns of thickness. The

Polarizer

Analyzer

Middle

Polarizer Vibration

of light

de la luz

24

Polarizer

Anisotropic

Section 25

10

0

(210)

(010)

Ng c

Np

b

Nm

a

Optical axis 26

cleavages are perpendicular to section then their traces are very thin, perpendicular each other and stay static if the

objective (x40) is moved slightly. In plane polarized light, if the major index of the section (Nm) is parallel to

Polarizer, the color of section will be yellowish hue. If we rotate the section 90º (Np will be parallel to Polarizer), we

will 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. The

birefringence is the same than the mineral (Ng – Np) 0.015 orange first order. The cleavages are at 45º to the

section and then is difficult to observe them. The section is pleochroic between a greenish hue (Ng parallel to the

polarizer) 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 same

reason as above, it will be difficult to observe the cleavages. The colors of the indices are greenish, therefore the

pleochroism is not obvious. The section is perpendicular to bisector of optical axis and can therefore be seen the

interference figure well centered.

The probability to obtain strictly these three sections is very small. However the sections close to them show

similar 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 be

between Np and Nm therefore the birefringence is between 0.003

and 0.015. We may consider a white hue of interference more or

less. The color in plane polarized light will be between a greenish

(Ng) and pinkish hue (n’p). We can see only one cleavage with a

trace 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 they

remain parallel to the b axis, therefore, all sections contain the Np index. The major index of the initial section is

Nm 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 the

angle 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 symmetrical

with respect to cleavages.

In the second case, the cuts are directed toward the face (010), remaining parallel to the a axis. Note that the plane

formed by the optical axes and indices Ng and Np is perpendicular to all sections and then the cuts will contain the

index 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 section

where this index is Nm and the section is isothropic or cyclic and perpendicular to one of optical axes. This section is

not pleochroic. The birefringence decrease then progressively from the initial section, to zero (cyclic section), before

rising slightly to a final birefringence of 0.003 (face (010)). As in the previous case the traces of cleavages thicken

progressively, 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 almost

perpendicular to one of the cleavages. The situation is similar to the cuts in Figure 12. The traces of cleavage

perpendicular to the sections, will be fine, keep the same density and remain static when slightly displace the

(001) section (100) section (010) section

Nm

Ng

Np

Ng

Nm

27

Np

Ng

n‘p

28

microscope objective. The traces of the other cleavage will thicken, its density will decrease and appear to move

more strongly as the tilt angle decreases, when slightly move the objective. Note that only the initial section contains

two of the indices of the mineral, while the final contains only one (Ng). Both indices of the other cuts will have

intermediate values. The sections of the indicatrix are not evident in this case, however, the major index of initial

section bisects the traces of cleavages, while in the final one, this index is parallel to the only visible cleavage. It

could be seen that the major index of sections will be progressively close to the fine cleavage and the extinctions are

not parallel which means that extinctions for these sections are neither straight nor symmetric.

Toward (100) Face

n‘g

n‘g

Np

Np

Toward (010) Face

Cyclic section

Np

(001) Section

(001) Nm Nm

n‘g

Nm

Nm

29

Toward (210) Face

n‘g

n‘p

n‘g

n‘p

Hypersthene. Section close to (001) face.

The traces of two cleavages are thin and

almost perpendicular each other. Yellow first

order in crossed polarized light (30 microns).

Left, plane polarized light. Right, crossed

nicols. Andesite of Nevado del Ruiz

volcano.Colombia. x10.

EPILOGUE

From 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 of

their textures. In short, a three-dimensional ‘vision’, can go beyond the simple identification of minerals and is a

necessary starting point in structural studies.

BIBLIOGRAPHY

SHELLEY, 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 clear

and interference colors are grayish. The

section shows the interference figure well

centered.

Hypersthene. Section close to (100) face. This

section does not show cleavages, as they are far

from perpendicular to section. Pleochroism very

clear between greenish and pinkish colors. In

crossed polarized light, the interference color is

orange of first order (30 microns) and is the same

that the mineral (contains the major and minor

indices of mineral).