---
title: "Optical Mineralogy"
book: "University Earth and Space Sciences — Year 1"
subject: earth
language: en
chapter: 8
exercises: 12
source: https://one-course.com/books/earth/3/en/chapter/8-optical-mineralogy
license: CC-BY-NC-SA-4.0
credit: "One Earth and Space Sciences Book, One Course (one-course.com)"
---

# Chapter 8 — Optical Mineralogy

Between [crossed polars](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-microscope) a [thin section](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-thin-section) of gabbro lights up like stained glass. Long grey and white laths, striped like a barcode, cross the field; between them, grains glow blue, yellow and magenta; a few stay black whatever the angle. Each of these effects has a cause that can be calculated, and each tells which [mineral](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) is there. This chapter explains how light crosses a crystal, why a [mineral](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) between [crossed polars](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-microscope) is coloured, how the colour measures a property of the [mineral](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral), and how a petrologist names the grains of a [thin section](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-thin-section) with nothing but a microscope. It ends with an unknown section whose [minerals](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral), and whose [true thickness](https://one-course.com/books/earth/3/en/chapter/5-cross-sections#def-b1-cross-sections-thickness), are found from what it shows.

**You already know.**

A [thin section](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-thin-section) is a slice of rock about $30\,\text{µ}\mathrm{m}$ thick on a glass slide; the [polarising microscope](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-microscope) looks at it in [plane-polarised light](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-microscope) (one polariser) and between [crossed polars](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-microscope) ([Chapter 6](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#ch-b1-field-lab-methods-1)). The refractive index $n$ of a medium is the ratio of the speed of light in vacuum to its speed in the medium, and refraction at an interface obeys Snell’s law $n_1\sin i_1 = n_2\sin i_2$ (physics, University Year 1). Light is a transverse wave; a polariser passes only the component of the electric field along its direction, so that an analyser at angle $\alpha$ to it passes the fraction $\cos^2\alpha$ of the intensity (Malus’s law, physics).

## 8.1 Light in crystals

In a glass, a liquid or a cubic crystal, light travels at the same speed in every direction and whatever its vibration direction. In most [minerals](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) it does not: the arrangement of atoms differs along different directions, and so does the response of the electrons to the electric field of the wave.

**Definition 8.1 (Isotropic and anisotropic minerals).**

A [mineral](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) is an *isotropic mineral* when its refractive index is the same for every direction of propagation and of vibration of light; cubic [minerals](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) ([garnet](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups), halite, spinel) and glasses are isotropic. It is an *anisotropic mineral* when its refractive index depends on the vibration direction; all [minerals](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) of the other crystal systems are anisotropic.

A plate cut from an anisotropic crystal does not let light through as a single wave. We admit the following result of crystal optics (physics, University Year 2): for a given direction of propagation, a plate of an [anisotropic mineral](#def-b1-optical-mineralogy-anisotropic) transmits two waves that vibrate in two perpendicular directions fixed in the plate, its *vibration directions*, and travel with two different refractive indices $n_1$ and $n_2$. Light vibrating along one of these directions crosses the plate unchanged in polarisation; light vibrating in any other direction is split into its two components along them. The two indices depend on the direction of propagation, and there are particular directions in which they are equal.

**Definition 8.2 (Optic axis, uniaxial and biaxial minerals).**

An *optic axis* of an [anisotropic mineral](#def-b1-optical-mineralogy-anisotropic) is a direction of propagation along which light travels with a single refractive index, whatever its vibration direction, as in an isotropic medium. A [mineral](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) is *uniaxial* when it has one optic axis ([minerals](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) of the trigonal, tetragonal and hexagonal systems, such as quartz and calcite) and *biaxial* when it has two (orthorhombic, monoclinic and triclinic [minerals](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral), such as [olivine](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups), [pyroxenes](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups), [amphiboles](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups), [micas](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups) and [feldspars](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups)).

The geometry is summarised by a surface, the indicatrix: an ellipsoid whose radius in each direction is the refractive index for light vibrating in that direction. The section of the ellipsoid perpendicular to the direction of propagation is an ellipse whose axes are the two vibration directions and whose semi-axes are $n_1$ and $n_2$. For a [uniaxial](#def-b1-optical-mineralogy-optic-axis) [mineral](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) the ellipsoid is one of revolution about the [optic axis](#def-b1-optical-mineralogy-optic-axis), with two principal indices $n_\omega$ and $n_\varepsilon$: quartz has $n_\omega = 1.544$ and $n_\varepsilon = 1.553$, calcite $1.658$ and $1.486$.

A [biaxial](#def-b1-optical-mineralogy-optic-axis) [mineral](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) has three principal indices $n_\alpha < n_\beta < n_\gamma$ and two circular sections, perpendicular to its two optic axes: forsterite has $1.635$, $1.651$ and $1.670$.

**Definition 8.3 (Birefringence (of a mineral)).**

The *birefringence* of a [mineral](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) is the difference $\delta = n_{\max} - n_{\min}$ between its largest and its smallest principal refractive indices: $n_\varepsilon - n_\omega$ in absolute value for a [uniaxial](#def-b1-optical-mineralogy-optic-axis) [mineral](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral), $n_\gamma - n_\alpha$ for a [biaxial](#def-b1-optical-mineralogy-optic-axis) one. In a grain cut in any other direction, the difference $|n_1 - n_2|$ lies between 0 (a section perpendicular to an [optic axis](#def-b1-optical-mineralogy-optic-axis)) and $\delta$ (a section containing the directions of $n_{\max}$ and $n_{\min}$).

The [birefringence](#def-b1-optical-mineralogy-birefringence) is a property of the [mineral](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral), like its density: quartz $0.009$, calcite $0.172$, forsterite $0.035$, fayalite $0.052$. A [thin section](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-thin-section) cuts the grains of a [mineral](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) in every orientation, so its grains show every value from 0 to $\delta$; only the highest is characteristic.

## 8.2 Observations in plane-polarised light

With the lower polariser only, the microscope shows the shape of the grains, their colour, their cleavages and their [relief](#def-b1-optical-mineralogy-relief).

**Definition 8.4 (Pleochroism).**

A [mineral](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) is *pleochroic* when its colour, seen in [plane-polarised light](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-microscope), changes as the stage is rotated; the *pleochroism* is this change. It arises because the absorption of light depends on its vibration direction in the crystal.

Biotite is the textbook case: a grain cut across its cleavage passes from pale yellow-brown to dark brown as it turns, darkest when the cleavage traces lie along the polariser. Hornblende changes from yellow to green. Most other rock-forming silicates (quartz, [feldspars](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups), [olivine](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups), diopside, muscovite) are colourless in [thin section](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-thin-section), and their colourless grains are told apart by other means.

**Definition 8.5 (Relief, Becke line).**

The *relief* of a grain in [thin section](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-thin-section) is how strongly its outline, cracks and surface stand out against the mounting medium; it is high when the refractive indices of the grain and of the medium differ much, low when they are close. The *Becke line* is the thin bright line that borders a grain whose index differs from that of its surroundings when the microscope is slightly out of focus.

**Proposition 8.6 (Relief and refractive index).**

A ray crossing at incidence $i$ a boundary between a mounting medium of index $n_m$ and a grain of index $n$ close to $n_m$ is deviated by

$$
\varepsilon \approx \frac{|n - n_m|}{n_m}\,\tan i
$$

(in radians), to first order in $(n - n_m)/n_m$. The deviations at the inclined edges of a grain, and so its [relief](#def-b1-optical-mineralogy-relief), grow in proportion to $|n - n_m|$.

**Proof.** Snell’s law gives $n_m \sin i = n \sin r$. Write $n = n_m(1 + \eta)$ with $\eta$ small and $r = i - \varepsilon$ with $\varepsilon$ small. Then $\sin i = (1 + \eta)(\sin i - \varepsilon\cos i) + O(\eta^2, \varepsilon^2)$, so $\eta \sin i = \varepsilon\cos i$ to first order, that is $\varepsilon = \eta\tan i$, with $|\eta| = |n - n_m|/n_m$. ∎

A [thin section](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-thin-section) is mounted in a resin of index about $1.54$.

Quartz ($1.544$–$1.553$) and the [feldspars](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups) ($1.52$–$1.59$) are nearly invisible against it: low [relief](#def-b1-optical-mineralogy-relief), the grain boundaries hard to see. [Olivine](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups) ($1.64$–$1.88$), [pyroxenes](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups) and [amphiboles](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups) stand out with dark rims and a rough-looking surface: high [relief](#def-b1-optical-mineralogy-relief). [Garnet](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups) ($1.83$) looks almost black-rimmed. Calcite, whose index runs from $1.486$ to $1.658$ with the vibration direction, changes [relief](#def-b1-optical-mineralogy-relief) as the stage turns, an effect called twinkling.

The [relief](#def-b1-optical-mineralogy-relief) tells only how far $n$ is from $n_m$, not on which side; the [Becke line](#def-b1-optical-mineralogy-relief) tells the side.

**Method 8.7 (The Becke line test).**

Focus on the edge of a grain in [plane-polarised light](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-microscope), close the aperture diaphragm a little, and raise the focus slightly (move the objective away from the section). The bright [Becke line](#def-b1-optical-mineralogy-relief) moves into the medium of higher refractive index: if it moves into the grain, the grain has the higher index (positive [relief](#def-b1-optical-mineralogy-relief)); if it moves out into the mount or the neighbouring grain, the grain has the lower index (negative [relief](#def-b1-optical-mineralogy-relief)).

The method works because a grain boundary is never exactly vertical: at an inclined boundary, light is refracted towards the side of higher index, and beyond the critical angle it is totally reflected on that side, so that the light crossing the boundary is concentrated above the higher-index medium. Raising the focus looks at a plane above the section, where that concentrated light lies inside the higher-index material.

## 8.3 Between crossed polars

With the analyser inserted, at right angles to the polariser, an isotropic grain is black in every position: light reaches it vibrating along the polariser and leaves it vibrating the same way, and the analyser stops it entirely. An anisotropic grain is generally bright and coloured. The reason is the delay between its two waves.

**Definition 8.8 (Retardation (in a crystal plate)).**

The *retardation* $\Delta$ of a plate of an [anisotropic mineral](#def-b1-optical-mineralogy-anisotropic) is the difference between the optical paths of its two waves on leaving the plate, the slower wave lagging behind the faster.

**Proposition 8.9 (Retardation and birefringence).**

A plate of thickness $d$ whose two waves have indices $n_1 > n_2$ has the [retardation](#def-b1-optical-mineralogy-retardation)

$$
\Delta = d\,(n_1 - n_2),
$$

and its two waves leave it with a phase difference $\varphi = 2\pi\Delta/\lambda$ for light of wavelength $\lambda$ in vacuum. In a [thin section](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-thin-section), the largest [retardation](#def-b1-optical-mineralogy-retardation) shown by grains of one [mineral](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) is $d\,\delta$.

**Proof.** A wave crossing a thickness $d$ at speed $c/n$ takes the time $nd/c$; its optical path, the distance light would cover in vacuum in that time, is $nd$. The two waves enter together; on leaving, the slower one is behind by $n_1 d - n_2 d = d(n_1 - n_2)$, a delay that is the fraction $\Delta/\lambda$ of a period, that is a phase $2\pi\Delta/\lambda$. The largest value of $n_1 - n_2$ among all orientations is $\delta$ ([Definition 8.3](#def-b1-optical-mineralogy-birefringence)). ∎

A quartz grain of the most favourable orientation in a section $30\,\text{µ}\mathrm{m}$ thick has $\Delta = 30\,000\,\mathrm{nm}\times0.009 =
270\,\mathrm{nm}$; a calcite grain $30\,000\,\mathrm{nm}\times0.172 = 5160\,\mathrm{nm}$.

![Light through a crystal plate between crossed polars. The polariser passes the east–west vibration (a); the plate splits it into two waves along its vibration directions, at and + 90 from the polariser (b); the analyser, north–south, passes the north–south component of each (c). The two components have equal sizes and opposite signs: without the delay they cancel, and the plate looks dark; wave 2 lags behind wave 1 by the phase = 2π /, and the sum no longer vanishes.](https://one-course.com/images/onecourse/chapters/earth-3/b1-optical-mineralogy/fig-c49f1f793e71.svg)

*Light through a crystal plate between [crossed polars](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-microscope). The polariser passes the east–west vibration (a); the plate splits it into two waves along its vibration directions, at $\theta$ and $\theta + 90{}^{\circ}$ from the polariser (b); the analyser, north–south, passes the north–south component of each (c). The two components have equal sizes and opposite signs: without the delay they cancel, and the plate looks dark; wave 2 lags behind wave 1 by the phase $\varphi = 2\pi\Delta/\lambda$, and the sum no longer vanishes.*

**Theorem 8.10 (Intensity between crossed polars).**

A plate of [retardation](#def-b1-optical-mineralogy-retardation) $\Delta$ whose vibration directions make the angle $\theta$ with the polariser, between [crossed polars](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-microscope), transmits at the wavelength $\lambda$ the intensity

$$
I = I_0\,\sin^2 2\theta\,\sin^2\frac{\pi\Delta}{\lambda},
$$

where $I_0$ is the intensity that leaves the polariser.

**Proof.** Take the polariser along $x$ and the analyser along $y$, and write the field leaving the polariser as the real part of $E_0 e^{i\omega t}$ along $x$, with $I_0 = E_0^2$ up to a constant. The plate’s vibration direction 1 is the unit vector $(\cos\theta, \sin\theta)$, direction 2 is $(-\sin\theta, \cos\theta)$. The incident field has the components $E_0\cos\theta$ along 1 and $-E_0\sin\theta$ along 2. The plate leaves wave 1 unchanged, up to a common phase, and delays wave 2 by $\varphi = 2\pi\Delta/\lambda$ ([Proposition 8.9](#prop-b1-optical-mineralogy-retardation)); in complex amplitudes the waves leaving the plate are $E_0\cos\theta$ along 1 and $-E_0\sin\theta\,e^{-i\varphi}$ along 2. Their components along $y$ are $E_0\cos\theta\sin\theta$ and $-E_0\sin\theta\cos\theta\,e^{-i\varphi}$; the analyser passes their sum,

$$
A = E_0\sin\theta\cos\theta\,(1 - e^{-i\varphi}) .
$$

Since $|1 - e^{-i\varphi}|^2 = 2 - 2\cos\varphi = 4\sin^2(\varphi/2)$ and $\sin\theta\cos\theta = \frac12\sin2\theta$, the intensity is $|A|^2 = E_0^2\,\sin^2 2\theta\,\sin^2(\varphi/2)$, that is $I = I_0\sin^2 2\theta\sin^2(\pi\Delta/\lambda)$. If wave 1 is the slower one, $\varphi$ changes sign and $I$ is unchanged. ∎

The theorem contains the whole of what the microscope shows between [crossed polars](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-microscope). Its first factor depends only on the orientation of the grain on the stage; the second only on the grain’s [retardation](#def-b1-optical-mineralogy-retardation) and on the colour of the light.

**Definition 8.11 (Extinction position, extinction angle).**

An anisotropic grain between [crossed polars](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-microscope) is in an *extinction position* when it is dark because its vibration directions are parallel to the polariser and the analyser. The *extinction angle* of a grain is the angle between a direction visible in it (a cleavage trace, an elongation, a [twin](#def-b1-optical-mineralogy-twin) plane) and the nearest extinction position, read on the graduated stage.

The factor $\sin^2 2\theta$ vanishes at $\theta = 0$, $90{}^{\circ}$, $180{}^{\circ}$ and $270{}^{\circ}$: a grain goes dark four times in a full turn of the stage, and is brightest midway, at $45{}^{\circ}$ from extinction (figure below). A grain with $\Delta = 0$ — an [isotropic mineral](#def-b1-optical-mineralogy-anisotropic), or a section perpendicular to an [optic axis](#def-b1-optical-mineralogy-optic-axis) — stays dark throughout. The [extinction angle](#def-b1-optical-mineralogy-extinction) depends on the symmetry of the crystal: a [mineral](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) whose vibration directions are tied to its cleavages, as in orthorhombic [minerals](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral), extinguishes parallel to them (straight extinction); a monoclinic or triclinic one generally does not (inclined extinction). The largest [extinction angle](#def-b1-optical-mineralogy-extinction) between the cleavage traces and the slow vibration direction is about $38{}^{\circ}$ in diopside and $15{}^{\circ}$–$19{}^{\circ}$ in hornblende, in sections parallel to $(010)$: one of the ways of telling a [pyroxene](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups) from an [amphibole](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups) in [thin section](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-thin-section).

![The intensity passed by a grain as the stage turns, at the wavelength 550\, nm, for three retardations (). Four extinctions per turn, four maxima at 45 from them; the retardation sets the height of the maxima, which is nearly zero when is close to a whole number of wavelengths.](https://one-course.com/images/onecourse/chapters/earth-3/b1-optical-mineralogy/fig-398f1e71a98e.svg)

*The intensity passed by a grain as the stage turns, at the wavelength $550\,\mathrm{nm}$, for three [retardations](#def-b1-optical-mineralogy-retardation) ([Theorem 8.10](#thm-b1-optical-mineralogy-intensity)). Four extinctions per turn, four maxima at $45{}^{\circ}$ from them; the [retardation](#def-b1-optical-mineralogy-retardation) sets the height of the maxima, which is nearly zero when $\Delta$ is close to a whole number of wavelengths.*

### 8.3.1 Interference colours

In white light, the second factor $\sin^2(\pi\Delta/\lambda)$ acts as a filter that depends on the wavelength. A grain with $\Delta = 270\,\mathrm{nm}$ passes almost all the green and yellow ($\lambda$ near $2\Delta = 540$ nm, where the factor is 1) and somewhat less of the violet and red: it looks white. A grain with $\Delta = 550\,\mathrm{nm}$ suppresses the yellow-green ($\lambda = \Delta$) and passes red and blue: it looks magenta. As $\Delta$ grows, the suppressed wavelengths move across the spectrum and new ones appear, and the colour runs through a fixed sequence.

**Definition 8.12 (Interference colour, order, Michel-Lévy chart).**

The *interference colour* of an anisotropic grain is its colour between [crossed polars](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-microscope) in white light, at $45{}^{\circ}$ from extinction; it depends only on its [retardation](#def-b1-optical-mineralogy-retardation) $\Delta$. The colours repeat in bands of about $550\,\mathrm{nm}$ of [retardation](#def-b1-optical-mineralogy-retardation), each ending with a red-violet tint; each band is an *order* (first, second, third…). The *Michel-Lévy chart* shows the interference colours against the [retardation](#def-b1-optical-mineralogy-retardation), with the lines $\Delta = d\,\delta$ that relate the thickness of a section, the [birefringence](#def-b1-optical-mineralogy-birefringence) of a [mineral](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) and its colour.

![A Michel-Lévy chart computed from first principles: for each retardation, the colour of white light (CIE standard illuminant D65) through crossed polars and a plate at 45 from extinction, from the CIE 1931 colour-matching functions (data © CIE, CC BY-SA 4.0) and the sRGB transform, normalised to the light passed by parallel polars. The black lines give = d\, for the birefringence of each mineral; the dashed line is the standard thickness of 30\, µ m. The red-violet tints that end the orders fall at 542, 1074 and 1605\, nm in this computation.](https://one-course.com/images/onecourse/chapters/earth-3/b1-optical-mineralogy/fig-bcbbfe7a6add.svg)

*A [Michel-Lévy chart](#def-b1-optical-mineralogy-interference-colour) computed from first principles: for each [retardation](#def-b1-optical-mineralogy-retardation), the colour of white light (CIE standard illuminant D65) through [crossed polars](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-microscope) and a plate at $45{}^{\circ}$ from extinction, from the CIE 1931 colour-matching functions (data © CIE, CC BY-SA 4.0) and the sRGB transform, normalised to the light passed by parallel polars. The black lines give $\Delta = d\,\delta$ for the [birefringence](#def-b1-optical-mineralogy-birefringence) of each [mineral](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral); the dashed line is the standard thickness of $30\,\text{µ}\mathrm{m}$. The red-violet tints that end the [orders](#def-b1-optical-mineralogy-interference-colour) fall at 542, 1074 and $1605\,\mathrm{nm}$ in this computation.*

The chart is computed exactly as the theorem says. For each [retardation](#def-b1-optical-mineralogy-retardation), each wavelength of the light is weighted by $\sin^2(\pi\Delta/\lambda)$; the transmitted spectrum is then turned into a colour as the eye sees it, by the standard colour-matching functions of colorimetry, and into the red, green and blue of a screen or a printer by the sRGB transform. Three features are worth noticing. The first [order](#def-b1-optical-mineralogy-interference-colour) starts black, passes through grey and white (the only white of the chart is near $270\,\mathrm{nm}$, where the spectrum is passed almost entirely), then yellow, orange and red. The second and third [orders](#def-b1-optical-mineralogy-interference-colour) are the brightest, with vivid blue, green, yellow and magenta. Beyond the fourth [order](#def-b1-optical-mineralogy-interference-colour) the colours fade: when $\Delta$ is many wavelengths, the filter $\sin^2(\pi\Delta/\lambda)$ oscillates several times across the visible spectrum, passes every colour in turn and, on average, half the light: the grain looks pale pinkish or greenish grey, the [high-order](#def-b1-optical-mineralogy-interference-colour) white of calcite.

Reading the chart is the core of optical mineralogy. In a section of known thickness, the highest colour shown by a [mineral](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) gives its [birefringence](#def-b1-optical-mineralogy-birefringence): follow the thickness across to the colour, then the line through that point back to its [birefringence](#def-b1-optical-mineralogy-birefringence). Conversely, a [mineral](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) of known [birefringence](#def-b1-optical-mineralogy-birefringence) gives the thickness: in a good $30\,\text{µ}\mathrm{m}$ section quartz never shows more than a [first-order](#def-b1-optical-mineralogy-interference-colour) white or the palest yellow, and a section in which quartz shows yellow or orange is too thick.

**History — Auguste Michel-Lévy.**

![](https://one-course.com/images/onecourse/chapters/earth-3/b1-optical-mineralogy/img-da5c8106aa0c.jpg)

*Michel-Lévy as a student, about 1862. Public domain.*

Auguste Michel-Lévy (1844–1911), who became director of the Geological Survey of France, was among the first to identify [minerals](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) in rocks systematically with the [polarising microscope](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-microscope). Before him, the colours of [minerals](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) between [crossed polars](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-microscope) were used empirically; he showed that what matters is the maximum [birefringence](#def-b1-optical-mineralogy-birefringence) of each [mineral](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral), which can be measured by matching the colours of its grains against a wedge of quartz whose [retardation](#def-b1-optical-mineralogy-retardation) grows steadily along its length. With his student Alfred Lacroix he published in 1888 *Les minéraux des roches*, a handbook of the optical properties of rock-forming [minerals](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral). The chart that bears his name, colours against [retardation](#def-b1-optical-mineralogy-retardation) with lines of equal [birefringence](#def-b1-optical-mineralogy-birefringence), is still the way [interference colours](#def-b1-optical-mineralogy-interference-colour) are read.

Two waves leaving a grain can be given an extra, known [retardation](#def-b1-optical-mineralogy-retardation) by an accessory plate inserted above the section at $45{}^{\circ}$ to the polars. The commonest one is cut to show the [first-order](#def-b1-optical-mineralogy-interference-colour) red, a [retardation](#def-b1-optical-mineralogy-retardation) of about $540\,\mathrm{nm}$. If the slow vibration direction of the grain is parallel to the slow direction of the plate, the delays add and the grain’s colour rises by about one [order](#def-b1-optical-mineralogy-interference-colour); if it is parallel to the fast direction, they subtract and the colour falls. A [first-order](#def-b1-optical-mineralogy-interference-colour) grey grain of $100\,\mathrm{nm}$ becomes [second-order](#def-b1-optical-mineralogy-interference-colour) blue ($640\,\mathrm{nm}$) in one orientation and [first-order](#def-b1-optical-mineralogy-interference-colour) orange ($440\,\mathrm{nm}$) in the other; the test shows which of the grain’s two vibration directions is the slow one, a property used to identify elongated [minerals](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral).

## 8.4 Identifying minerals in thin section

**Definition 8.13 (Twin, polysynthetic twinning).**

A *twin* is an intergrowth of two or more crystals of the same [mineral](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) whose lattices are related by a symmetry that the crystal itself does not have, such as a reflection across a plane (the twin plane). *Polysynthetic twinning* is repeated twinning on parallel planes, which divides a crystal into thin parallel lamellae whose orientations alternate.

Between [crossed polars](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-microscope) the lamellae of a polysynthetic [twin](#def-b1-optical-mineralogy-twin) extinguish alternately: half of them are dark when the other half are bright, and the crystal looks striped. [Plagioclase](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups) almost always shows these stripes, which distinguish it at once from quartz and from [alkali feldspar](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups); its other feature, low [relief](#def-b1-optical-mineralogy-relief) and [first-order](#def-b1-optical-mineralogy-interference-colour) grey to white colours, it shares with them. The [extinction angles](#def-b1-optical-mineralogy-extinction) of the two sets of lamellae, measured either side of the [twin](#def-b1-optical-mineralogy-twin) plane, depend on the composition of the [plagioclase](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups), and charts of those angles give its anorthite content.

**Method 8.14 (Identifying a mineral in thin section).**

1. In [plane-polarised light](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-microscope) : note whether the grain is opaque; its colour and [pleochroism](#def-b1-optical-mineralogy-pleochroism) (rotate the stage); its [relief](#def-b1-optical-mineralogy-relief) , and on which side of $1.54$ its index lies ( [Becke line](#def-b1-optical-mineralogy-relief) ); its shape and cleavages, and the angle between two cleavages.
2. Between [crossed polars](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-microscope) : check whether it stays dark (isotropic, or a section perpendicular to an [optic axis](#def-b1-optical-mineralogy-optic-axis) ); among the grains of the same [mineral](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) , find the highest [interference colour](#def-b1-optical-mineralogy-interference-colour) and read the [birefringence](#def-b1-optical-mineralogy-birefringence) on the [Michel-Lévy chart](#def-b1-optical-mineralogy-interference-colour) for the thickness of the section.
3. Measure the [extinction angle](#def-b1-optical-mineralogy-extinction) on grains showing a cleavage or an elongation; look for twinning (polysynthetic stripes, simple [twins](#def-b1-optical-mineralogy-twin) ).
4. Compare with a table of optical properties ( [Table 8.1](#tab-b1-optical-mineralogy-key) ) and with the [minerals](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) the rock can contain ( [Chapter 7](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#ch-b1-rock-forming-minerals) ).

| [mineral](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) | colour (PPL) | indices | [birefringence](#def-b1-optical-mineralogy-birefringence) | highest colour at $30\,\text{µ}\mathrm{m}$ | other features |
| --- | --- | --- | --- | --- | --- |
| quartz | colourless | 1.544–1.553 | 0.009 | [first-order](#def-b1-optical-mineralogy-interference-colour) white | no cleavage; no [twins](#def-b1-optical-mineralogy-twin) visible; [uniaxial](#def-b1-optical-mineralogy-optic-axis) |
| orthoclase | colourless, often cloudy | 1.518–1.525 | 0.005 | [first-order](#def-b1-optical-mineralogy-interference-colour) grey | negative [relief](#def-b1-optical-mineralogy-relief); simple [twins](#def-b1-optical-mineralogy-twin) |
| [plagioclase](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups) | colourless | 1.526–1.590 | 0.008–0.013 | [first-order](#def-b1-optical-mineralogy-interference-colour) white to pale yellow | [polysynthetic twinning](#def-b1-optical-mineralogy-twin) |
| muscovite | colourless | 1.552–1.618 | 0.035–0.042 | bright second- to [third-order](#def-b1-optical-mineralogy-interference-colour) | one perfect cleavage; straight extinction |
| biotite | brown, green | 1.565–1.696 | 0.04–0.07 | masked by its own colour | strong [pleochroism](#def-b1-optical-mineralogy-pleochroism); one perfect cleavage |
| hornblende | green, brown | 1.675–1.695 | 0.020 | [second-order](#def-b1-optical-mineralogy-interference-colour) blue | [pleochroic](#def-b1-optical-mineralogy-pleochroism); cleavages at $56{}^{\circ}$/$124{}^{\circ}$; extinction $15{}^{\circ}$–$19{}^{\circ}$ |
| diopside | colourless | 1.664–1.694 | 0.030 | [second-order](#def-b1-optical-mineralogy-interference-colour) yellow | cleavages at about $87{}^{\circ}$/$93{}^{\circ}$; extinction about $38{}^{\circ}$ |
| [olivine](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups) | colourless (Mg) to pale yellow (Fe) | 1.635–1.879 | 0.035–0.052 | end of second [order](#def-b1-optical-mineralogy-interference-colour) to third [order](#def-b1-optical-mineralogy-interference-colour) | high [relief](#def-b1-optical-mineralogy-relief); irregular cracks, no good cleavage |
| [garnet](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups) | colourless to pale pink | 1.830 (almandine) | isotropic | black | very high [relief](#def-b1-optical-mineralogy-relief); six- or eight-sided outlines |
| calcite | colourless | 1.486–1.658 | 0.172 | [high-order](#def-b1-optical-mineralogy-interference-colour) white | [relief](#def-b1-optical-mineralogy-relief) changes on rotation; rhombohedral cleavage |
| magnetite | opaque | — | — | — | black in both lights |

***Table 8.1.** Optical properties of common rock-forming [minerals](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) in [thin section](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-thin-section). PPL: [plane-polarised light](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-microscope). The highest colour is read on the [Michel-Lévy chart](#def-b1-optical-mineralogy-interference-colour) at $\Delta = 30\,000\,\mathrm{nm}\times\delta$.*

![(a) plane-polarised light](https://one-course.com/images/onecourse/chapters/earth-3/b1-optical-mineralogy/img-e88b8668f4d4.jpg)

![(b) crossed polars](https://one-course.com/images/onecourse/chapters/earth-3/b1-optical-mineralogy/img-3e410af05130.jpg)

*(a) [plane-polarised light](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-microscope)*

*(b) [crossed polars](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-microscope)*

*The same field of a gabbro in [thin section](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-thin-section). In [plane-polarised light](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-microscope) (a), [plagioclase](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups) is colourless with low [relief](#def-b1-optical-mineralogy-relief), the [pyroxene](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups) and [olivine](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups) grains are pale brown-grey with high [relief](#def-b1-optical-mineralogy-relief) and dense cracks, and the opaque ore is black; the scale is a micrometer in the eyepiece. Between [crossed polars](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-microscope) (b), the [plagioclase](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups) laths are striped by [polysynthetic twinning](#def-b1-optical-mineralogy-twin), the [pyroxene](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups) and [olivine](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups) grains shine in [second-order](#def-b1-optical-mineralogy-interference-colour) colours, and the ore stays black. Photographs Chmee2, CC BY-SA 3.0 (Wikimedia Commons).*

**In the lab — A session at the microscope.**

![](https://one-course.com/images/onecourse/chapters/earth-3/b1-optical-mineralogy/img-7b74980cdd77.jpg)

*A student at a [polarising microscope](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-microscope) in a teaching laboratory (AI-generated illustration).*

A petrologist describing a new [thin section](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-thin-section) works through it at low magnification first, to see the texture and the main [minerals](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral), then grain by grain. For each [mineral](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) she records the observations of the method above, estimates its proportion in the field, and notes how the grains touch: which crystallised first, which fill the spaces left, which replace others. Light measurements are made with the stage graduated in degrees and an accessory plate; compositions, when they matter, are measured later with a microprobe on the same section. A teaching laboratory works the same way, one microscope per student and a box of reference sections of known [minerals](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) beside it.

## 8.5 Exercises

**Exercise 8.1 ★.**

Compute the largest [retardation](#def-b1-optical-mineralogy-retardation) of quartz ($\delta = 0.009$) and of calcite ($\delta = 0.172$) in a section $30\,\text{µ}\mathrm{m}$ thick. Using the [Michel-Lévy chart](#def-b1-optical-mineralogy-interference-colour), describe the colour of each.

**Solution of Exercise 8.1.**

Quartz: $30\,000\,\mathrm{nm}\times0.009 = 270\,\mathrm{nm}$, a [first-order](#def-b1-optical-mineralogy-interference-colour) white. Calcite: $30\,000\,\mathrm{nm}\times0.172 = 5160\,\mathrm{nm}$, far beyond the chart: a pale [high-order](#def-b1-optical-mineralogy-interference-colour) white, faintly pink or green.

**Exercise 8.2 ★.**

A grain is turned on the stage between [crossed polars](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-microscope). Compute $I/I_0$, as a fraction of its maximum, at $\theta = 0$, $22.5{}^{\circ}$, $45{}^{\circ}$, $67.5{}^{\circ}$ and $90{}^{\circ}$ from an [extinction position](#def-b1-optical-mineralogy-extinction).

**Solution of Exercise 8.2.**

$I/I_{\max} = \sin^2 2\theta$: $0$ at $0{}^{\circ}$; $\sin^245{}^{\circ} = 0.5$ at $22.5{}^{\circ}$; $1$ at $45{}^{\circ}$; $0.5$ at $67.5{}^{\circ}$; $0$ at $90{}^{\circ}$.

**Exercise 8.3 ★.**

Explain why a [garnet](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups) stays black between [crossed polars](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-microscope) whatever the position of the stage, and why a quartz grain may do the same. How would you tell them apart in [plane-polarised light](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-microscope)?

**Solution of Exercise 8.3.**

[Garnet](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups) is cubic, hence isotropic: light leaves it vibrating as it entered, along the polariser, and the analyser stops it; $\Delta = 0$ in every position. A quartz grain cut perpendicular to its [optic axis](#def-b1-optical-mineralogy-optic-axis) also has $n_1 = n_2$ for light travelling along that axis, so $\Delta = 0$: it too stays dark. In [plane-polarised light](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-microscope) the [garnet](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups) has very high [relief](#def-b1-optical-mineralogy-relief) ($1.83$ against $1.54$), the quartz almost none.

**Exercise 8.4 ★.**

Rank by [relief](#def-b1-optical-mineralogy-relief), in a mount of index $1.54$, orthoclase ($n$ about $1.52$), quartz ($1.55$), forsterite ($1.65$) and almandine ($1.83$). Which has negative [relief](#def-b1-optical-mineralogy-relief)? In which direction does its [Becke line](#def-b1-optical-mineralogy-relief) move when the focus is raised?

**Solution of Exercise 8.4.**

$|n - 1.54|$: almandine $0.29$, forsterite $0.11$, orthoclase $0.02$, quartz $0.01$; [relief](#def-b1-optical-mineralogy-relief) in that order, from highest to lowest. Orthoclase has the lower index: negative [relief](#def-b1-optical-mineralogy-relief), and its [Becke line](#def-b1-optical-mineralogy-relief) moves out of the grain, into the mount, when the focus is raised.

**Exercise 8.5 ★★.**

In a $30\,\text{µ}\mathrm{m}$ section, the grains of a green [pleochroic](#def-b1-optical-mineralogy-pleochroism) [mineral](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) with two cleavages at $56{}^{\circ}$ and $124{}^{\circ}$ show at most a [second-order](#def-b1-optical-mineralogy-interference-colour) blue, read on the chart as $\Delta = 650\,\mathrm{nm}$. Compute its [birefringence](#def-b1-optical-mineralogy-birefringence) and name the [mineral](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) with [Table 8.1](#tab-b1-optical-mineralogy-key).

**Solution of Exercise 8.5.**

$\delta = 650/30000 = 0.022$. A green [pleochroic](#def-b1-optical-mineralogy-pleochroism) [mineral](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) with cleavages at $56{}^{\circ}$ and $124{}^{\circ}$ and a [birefringence](#def-b1-optical-mineralogy-birefringence) near $0.020$: hornblende.

**Exercise 8.6 ★★.**

Using [Proposition 8.6](#prop-b1-optical-mineralogy-relief) with $n_m = 1.54$ and $i = 30{}^{\circ}$, compute the deviation at the edge of a quartz grain ($n = 1.553$) and of a forsterite grain ($n = 1.651$), in degrees, and the ratio of the two. Check the second with Snell’s law exactly.

**Solution of Exercise 8.6.**

$\tan30{}^{\circ} = 0.5774$. Quartz: $(0.013/1.54)\times0.5774 = 0.004\,87\,\mathrm{rad}
= 0.28{}^{\circ}$. Forsterite: $(0.111/1.54)\times0.5774 = 0.0416\,\mathrm{rad} =
2.38{}^{\circ}$. Ratio $0.111/0.013 = 8.5$. Exactly: $\sin r = 1.54\sin30{}^{\circ}/1.651
= 0.4664$, $r = 27.80{}^{\circ}$, a deviation of $2.20{}^{\circ}$: the first-order formula overestimates by $8\,\%$ for so large an index difference.

**Exercise 8.7 ★★.**

A [first-order](#def-b1-optical-mineralogy-interference-colour) red plate has $\Delta = 540\,\mathrm{nm}$. A quartz grain of $\Delta = 270\,\mathrm{nm}$ is placed at $45{}^{\circ}$ with the plate inserted. What are the two possible total [retardations](#def-b1-optical-mineralogy-retardation), and the colours they give on the chart? Which orientation puts the two slow directions parallel?

**Solution of Exercise 8.7.**

$270 + 540 = 810\,\mathrm{nm}$, a [second-order](#def-b1-optical-mineralogy-interference-colour) green, when the slow directions of the grain and the plate are parallel (the delays add); $|270 - 540| =
270\,\mathrm{nm}$, [first-order](#def-b1-optical-mineralogy-interference-colour) white again, when the slow direction of the grain is parallel to the fast direction of the plate.

**Exercise 8.8 ★★.**

At its tapering edge, a grain of fayalite ($\delta = 0.052$) thins from $30\,\text{µ}\mathrm{m}$ to zero. How many red-violet tints, which end the [orders](#def-b1-optical-mineralogy-interference-colour) at 542, 1074 and $1605\,\mathrm{nm}$, appear as coloured fringes between the edge and the full-thickness part of the grain?

**Solution of Exercise 8.8.**

At full thickness $\Delta = 30\,000\,\mathrm{nm}\times0.052 = 1560\,\mathrm{nm}$. From the edge ($\Delta = 0$) inwards, the [retardation](#def-b1-optical-mineralogy-retardation) passes 542 and $1074\,\mathrm{nm}$ but not $1605\,\mathrm{nm}$: two red-violet fringes.

**Exercise 8.9 ★★.**

On several elongated grains with one good cleavage, a geologist measures [extinction angles](#def-b1-optical-mineralogy-extinction) of $0{}^{\circ}$, $21{}^{\circ}$, $36{}^{\circ}$ and $37{}^{\circ}$ between the cleavage trace and the slow vibration direction. Why do the values differ? Is the [mineral](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) more likely diopside or hornblende?

**Solution of Exercise 8.9.**

The [extinction angle](#def-b1-optical-mineralogy-extinction) measured on a grain depends on how the grain is cut: only sections in a particular orientation show the largest angle, and others show less, down to zero. The largest value, about $37{}^{\circ}$, is the one to compare: it matches diopside (about $38{}^{\circ}$), not hornblende (at most $19{}^{\circ}$).

**Exercise 8.10 ★★★.**

Redo the proof of [Theorem 8.10](#thm-b1-optical-mineralogy-intensity) with the analyser parallel to the polariser, and show that $I_\parallel = I_0\bigl(1 - \sin^2 2\theta\sin^2(\pi\Delta/\lambda)\bigr)$. What colour does a grain of $\Delta = 550\,\mathrm{nm}$ show at $45{}^{\circ}$ between parallel polars?

**Solution of Exercise 8.10.**

With the analyser along $x$, the $x$ components of the two waves are $E_0\cos\theta\cdot\cos\theta$ and $-E_0\sin\theta\,e^{-i\varphi}\cdot(-\sin\theta)$, so $A = E_0(\cos^2\theta + \sin^2\theta\,e^{-i\varphi})$ and

$$
|A|^2 = E_0^2\bigl(\cos^4\theta + \sin^4\theta + 2\sin^2\theta\cos^2\theta\cos\varphi\bigr)
  = E_0^2\bigl(1 - 2\sin^2\theta\cos^2\theta\,(1 - \cos\varphi)\bigr),
$$

using $\cos^4\theta + \sin^4\theta = 1 - 2\sin^2\theta\cos^2\theta$. Since $2\sin^2\theta\cos^2\theta = \frac12\sin^2 2\theta$ and $1 - \cos\varphi =
2\sin^2(\varphi/2)$, $I_\parallel = I_0(1 - \sin^2 2\theta\sin^2(\pi\Delta/\lambda))$. At each wavelength, $I_\parallel + I_\perp = I_0$: the colours between parallel polars are the complements of those between [crossed polars](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-microscope). A grain of $550\,\mathrm{nm}$, magenta between [crossed polars](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-microscope), is pale yellowish green between parallel polars.

**Exercise 8.11 ★★★.**

For a calcite grain with $\Delta = 5000\,\mathrm{nm}$, the factor $\sin^2(\pi\Delta/\lambda)$ vanishes whenever $\Delta/\lambda$ is a whole number. (a) How many times does it vanish between $400$ and $700\,\mathrm{nm}$? (b) Show that near $\lambda = 550\,\mathrm{nm}$ two successive zeros are about $\lambda^2/\Delta$ apart, and compute it. (c) Explain why the grain looks nearly white.

**Solution of Exercise 8.11.**

(a) $\Delta/\lambda$ runs from $5000/700 = 7.14$ to $5000/400 = 12.5$; it passes the whole numbers 8 to 12: five zeros. (b) The zeros are at $\lambda_k = \Delta/k$; two successive ones are $\Delta/k - \Delta/(k + 1) =
\Delta/(k(k + 1)) \approx \lambda^2/\Delta$ apart, that is $550^2/5000 =
60.5\,\mathrm{nm}$ near $550\,\mathrm{nm}$. (c) Five narrow dark bands spread across the spectrum remove a little of every colour, and the rest passes with an average weight of one half: the eye sees a pale, nearly neutral tint.

**Exercise 8.12 ★★★.**

Show that, averaged over a full turn of the stage, a grain between [crossed polars](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-microscope) passes $\frac12 I_0\sin^2(\pi\Delta/\lambda)$. A geologist estimates [birefringences](#def-b1-optical-mineralogy-birefringence) from the highest colour of each [mineral](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral), assuming $30\,\text{µ}\mathrm{m}$, in a section that is in fact $25\,\text{µ}\mathrm{m}$ thick. By what factor are all her values wrong, and in which direction?

**Solution of Exercise 8.12.**

The mean of $\sin^2 2\theta$ over a turn is $\frac1{2\pi}\int_0^{2\pi}\sin^2 2\theta\,
\mathrm d\theta = \frac12$, so the mean intensity is $\frac12 I_0\sin^2(\pi\Delta/\lambda)$. She computes $\delta = \Delta/30000$ instead of $\Delta/25000$ (in nm): all her values are $25/30 = 0.83$ of the true ones, too low by $17\,\%$.

## 8.6 Problem: Identifying the minerals of an unknown thin section

**Problem 8.1.**

Weekend problem — observing relief, pleochroism and interference colours, measuring the thickness of a section with quartz, and naming its minerals

A [thin section](https://one-course.com/books/earth/3/en/chapter/6-field-and-lab-methods-i#def-b1-field-lab-methods-1-thin-section) of an unknown igneous rock, mounted in a resin of index $1.540$, is said to be $30\,\text{µ}\mathrm{m}$ thick. It contains five [minerals](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral), labelled A to E. Use the table of optical properties of the chapter, the [Michel-Lévy chart](#def-b1-optical-mineralogy-interference-colour), and the following values: quartz $n_\omega = 1.544$, $n_\varepsilon = 1.553$; diopside $n_\alpha = 1.664$, $n_\beta = 1.672$, $n_\gamma = 1.694$.

**Part I — Plane-polarised light.**

1. [Mineral](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) A is colourless, with grain boundaries hard to see. [Mineral](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) B is colourless, with strong dark rims and two cleavages meeting at about $87{}^{\circ}$ . Which has the higher [relief](#def-b1-optical-mineralogy-relief) ?
2. For A, taken as quartz, and B, taken as diopside, compute $|n - n_m|$ using the mean of the principal indices of each.
3. Using [Proposition 8.6](#prop-b1-optical-mineralogy-relief) , compute the ratio of the deviations at their edges for the same inclination of the boundary.
4. On raising the focus, the [Becke line](#def-b1-optical-mineralogy-relief) of B moves into the grain. What does this show?
5. [Mineral](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) C turns from pale yellow to dark brown as the stage rotates, darkest when its single cleavage lies along the polariser. Name the property and the [mineral](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) .
6. [Mineral](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) D is colourless, of low [relief](#def-b1-optical-mineralogy-relief) , in long laths; E is green to yellow and [pleochroic](#def-b1-optical-mineralogy-pleochroism) , with two cleavages at $124{}^{\circ}$ . What does the angle suggest for E?

**Part II — The thickness of the section.**

7. What is the largest [retardation](#def-b1-optical-mineralogy-retardation) of quartz in a section of exactly $30\,\text{µ}\mathrm{m}$ ? What colour is it?
8. Why must the largest [retardation](#def-b1-optical-mineralogy-retardation) be sought among many grains of A, rather than measured on any one of them?
9. The highest colour shown by the grains of A is a [first-order](#def-b1-optical-mineralogy-interference-colour) pale yellow; a quartz-wedge comparator gives its [retardation](#def-b1-optical-mineralogy-retardation) as $342\,\mathrm{nm}$ . Compute the real thickness of the section.
10. By how many per cent is the section thicker than it should be?
11. At $\lambda = 550\,\mathrm{nm}$ and $45{}^{\circ}$ from extinction, compute $I/I_0$ for this quartz grain, and for the same grain in a section of $30\,\text{µ}\mathrm{m}$ .
12. Compute the range of [retardations](#def-b1-optical-mineralogy-retardation) of D, taken as [plagioclase](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups) ( $\delta = 0.008$ to $0.013$ ), in this section, and say whether its colours remain in the first [order](#def-b1-optical-mineralogy-interference-colour) .

**Part III — The birefringence of B.**

13. The highest colour of B is a blue read as $\Delta = 1140\,\mathrm{nm}$ . In which [order](#def-b1-optical-mineralogy-interference-colour) is it?
14. Compute the [birefringence](#def-b1-optical-mineralogy-birefringence) of B with the thickness found in question 9.
15. What [birefringence](#def-b1-optical-mineralogy-birefringence) would one have found by assuming $30\,\text{µ}\mathrm{m}$ ? Which colourless [mineral](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) of high [relief](#def-b1-optical-mineralogy-relief) would that have suggested, and why is it out of place in a rock with quartz (consider $\ce{Mg2SiO4 + SiO2 -> 2 MgSiO3}$ )?
16. Compare the result of question 14 with the table, together with the [relief](#def-b1-optical-mineralogy-relief) and the cleavages of B, and conclude.
17. What colour would B show at most in a correct $30\,\text{µ}\mathrm{m}$ section?
18. The section is ground down to $30\,\text{µ}\mathrm{m}$ . By how much must it be thinned, and what is then the highest colour of quartz?

**Part IV — Extinction and twinning.**

19. The laths of D are divided into parallel stripes that extinguish alternately. Name the feature and the [mineral](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) .
20. The stripes of one lath extinguish at $27{}^{\circ}$ and $28{}^{\circ}$ either side of the [twin](#def-b1-optical-mineralogy-twin) plane. What do such angles, measured on many laths, give when compared with published charts?
21. The highest [extinction angle](#def-b1-optical-mineralogy-extinction) between the cleavages of E and its slow direction is $17{}^{\circ}$ . Name E.
22. Compute the largest [retardation](#def-b1-optical-mineralogy-retardation) of E in this section and the colour it shows.
23. A grain of B stays dark during a full rotation. Explain how it can be the same [mineral](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) as the bright grains of B.
24. List the [minerals](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-mineral) A to E, and say what kind of rock they make.
25. What is the real thickness of the section, in micrometres?

**Solution of Problem 8.1.**

**1.** B: dark rims mean high [relief](#def-b1-optical-mineralogy-relief); A’s barely visible boundaries mean low [relief](#def-b1-optical-mineralogy-relief).

**2.** A: $(1.544 + 1.553)/2 = 1.5485$, $|n - n_m| = 0.0085$. B: $(1.664 + 1.672 + 1.694)/3 = 1.6767$, $|n - n_m| = 0.1367$.

**3.** The deviations are proportional to $|n - n_m|$: $0.1367/0.0085 =
16.1$.

**4.** B has a higher index than the mount: positive [relief](#def-b1-optical-mineralogy-relief).

**5.** [Pleochroism](#def-b1-optical-mineralogy-pleochroism); biotite.

**6.** Two cleavages at $56{}^{\circ}$ and $124{}^{\circ}$: an [amphibole](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups).

**7.** $30\,000\,\mathrm{nm}\times0.009 = 270\,\mathrm{nm}$: [first-order](#def-b1-optical-mineralogy-interference-colour) white.

**8.** Each grain is cut in its own orientation and shows a value of $|n_1 - n_2|$ between 0 and $\delta$; only grains cut parallel to the [optic axis](#def-b1-optical-mineralogy-optic-axis) show $\delta$ itself. The highest colour among many grains is the one to use.

**9.** $d = 342/0.009 = 38\,000\,\mathrm{nm} = 38\,\text{µ}\mathrm{m}$.

**10.** $(38 - 30)/30 = 26.7\,\%$.

**11.** $\sin^2(\pi\times342/550) = 0.861$; at $30\,\text{µ}\mathrm{m}$, $\sin^2(\pi\times270/550) = 0.999$.

**12.** $38\,000\,\mathrm{nm}\times0.008 = 304\,\mathrm{nm}$ to $38\,000\,\mathrm{nm}\times0.013 = 494\,\mathrm{nm}$: white to orange-red, all below the end of the first [order](#def-b1-optical-mineralogy-interference-colour) at $542\,\mathrm{nm}$.

**13.** Between 1074 and $1605\,\mathrm{nm}$: third [order](#def-b1-optical-mineralogy-interference-colour).

**14.** $\delta = 1140/38000 = 0.030$.

**15.** $1140/30000 = 0.038$, which would suggest an [olivine](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups). A magnesian [olivine](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups) and quartz react to give [pyroxene](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups), $\ce{Mg2SiO4 + SiO2 -> 2 MgSiO3}$; a rock that has crystallised with quartz should not keep both.

**16.** $0.030$ is the [birefringence](#def-b1-optical-mineralogy-birefringence) of diopside; its high [relief](#def-b1-optical-mineralogy-relief), its lack of colour and its two cleavages near $87{}^{\circ}$ agree: B is a clinopyroxene, diopside.

**17.** $30\,000\,\mathrm{nm}\times0.030 = 900\,\mathrm{nm}$: [second-order](#def-b1-optical-mineralogy-interference-colour) yellow.

**18.** By $8\,\text{µ}\mathrm{m}$; quartz then shows at most $270\,\mathrm{nm}$, [first-order](#def-b1-optical-mineralogy-interference-colour) white.

**19.** [Polysynthetic twinning](#def-b1-optical-mineralogy-twin): D is [plagioclase](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups).

**20.** The composition of the [plagioclase](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups) (its anorthite content).

**21.** An [amphibole](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups) with an [extinction angle](#def-b1-optical-mineralogy-extinction) of $17{}^{\circ}$: hornblende ($15{}^{\circ}$–$19{}^{\circ}$), not a [pyroxene](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups) ($38{}^{\circ}$ for diopside).

**22.** $38\,000\,\mathrm{nm}\times0.020 = 760\,\mathrm{nm}$: a [second-order](#def-b1-optical-mineralogy-interference-colour) green.

**23.** It is cut perpendicular to an [optic axis](#def-b1-optical-mineralogy-optic-axis): its two indices are equal for light travelling along the axis, $\Delta = 0$, and it stays dark like an isotropic grain.

**24.** A quartz, B diopside, C biotite, D [plagioclase](https://one-course.com/books/earth/3/en/chapter/7-rock-forming-minerals#def-b1-rock-forming-minerals-groups), E hornblende: an intermediate igneous rock with some quartz, such as a quartz diorite.

**25.** The section is **$38\,\text{µ}\mathrm{m}$** thick.