Crystallography and Optical Mineralogy (L2-6) Flashcards
What was first observed about growth morphology of crystals?
Morphology dominated by faces with the slowest growth rate
Tend to be parallel sets of planes with small hkl values
What are the symmetry elements of crystal shapes?
Rotation axes
Mirror planes
Inversion point
Inversion axes
What is a point group?
A combination of symmetry elements in three dimensions that shows the symmetry of a crystal shape
What are the geometrical constraints of point groups?
Intersection of two symmetry axes automatically leads to the existence of a third
Unless the three symmetry axes have a particular angular relationship, repeated operation of the symmetry elements will generate an infinite number of additional axes
What are the various types of point group?
Single symmetry element
Mirror plane perpendicular to a single major rotation axis
Mirror plane parallel to a single major rotation axis
Two sets of mirror planes parallel to a single major rotation axis
Mirror planes parallel and perpendicular to a single major rotation axis
Two or more rotation axes, no mirror planes
How can point groups in the cubic system be identified?
A ‘3’ in the second position of the point group symbol
i.e. 23, m3, 432
What is a form?
A set of faces all related to one face (hkl) by the point group symmetry
What do the various brackets represent for Miller indices?
(hkl) = face
{hkl} = set of faces
[UVW] = vector
= set of vectors
What is habit?
The overall shape of a crysdtal depending on the relative size of the faces of the various forms present
What is a twin?
Two or more component crystals with a fixed orientational relationship between them
What is a reflection twin?
The operator is a mirror plane called the twin-plane
What is a rotation twin?
The operator is an axis of 180° rotation called the twin-axis
What are the processes by which twins can form?
Growth twins: accident occurs in growth, new crystal added to the face of an existing crystal, twinning occurs if new crystal shares lattice points on the existing crystal’s face but a different orientation
Transformation twins: pre-existing crystal undergoes phase transition from high symmetry to lower symmetry form, T or P change = low symmetry stable, crystal parts arrange in different orientations
Deformation twins: in response to deformation by an external stress
What is the difference between a simple twin and a multiple twin?
Simple twin only has two components
Multiple twin has many components
What is a lattice point?
All the points in a structure which have identical environments w.r.t the distribution of atoms around them
What is a unit cell?
A small unit which can be repeated in three dimensions to reproduce the lattice
What are the various types of unit cell?
Primitive: lattice points only on corners
Body centred: Primitive + lattice point in the centre
Face centred: Primitive + lattice points in the centre of each face
A/B/C face centred: Primitive + lattice points in the centre of the respective faces
Rhombohedral: Primitive but angle between X and Y =/= 90°
Which symbols are used to represent the unit cell types?
Primitive = P Body centred = I Face centred = F A/B/C face centred = A/B/C Rhombohedral = R
What is the shape of a unit cell described by?
Length of its sides (a,b,c)
Angles between the sides (α,β,γ):
b^c = α, a^c = β, a^b = γ
What defines the cubic crystal system?
Minimum symmetry: 4 triads - parallel to <111>
Point groups: 2nd symmetry element = 3
Unit cell shape: cube
Geometrical constraints: a=b=c, α=β=γ=90°
Lattice types: P, I, F
What defines the tetragonal crystal system?
Minimum symmetry: 1 tetrad - parallel to [001]
Point groups: 1st symmetry element = 4
Unit cell shape: Square prism
Geometrical constraints: a=b=/=c, α=β=γ=90°
Lattice types: P, I
What defines the hexagonal crystal system?
Minimum symmetry: 1 hexad - parallel to [001]
Point groups: 1st symmetry element = 6
Unit cell shape: 120° rhombus prism
Geometrical constraints: a=b=/=c, α=β=90°, γ=120°
Lattice types: P
What defines the trigonal crystal system?
Minimum symmetry: 1 triad - parallel to [001]
Point groups: 1st symmetry element = 3
Unit cell shape: 120° rhombus prism
Geometrical constraints: a=b=/=c, α=β=90°, γ=120°
Lattice types: P, R
What defines the orthorhombic crystal system?
Minimum symmetry: 3 diads - parallel to [100], [010] and [001]
Point groups: 222, mm2, mmm
Unit cell shape: Rectangular prism
Geometrical constraints: a=/=b=/=c, α=β=γ=90°
Lattice types: P, C, I, F
What defines the monoclinic system?
Minimum symmetry: 1 diad - parallel to [010]
Point groups: 2, m, 2/m
Unit cell shape: Parallelogram prism
Geometrical constraints: a=b=/=c, α=γ=90°, β=/=90°
Lattice types: P, C
What defines the triclinic system?
Minimum symmetry: none Point groups: (+/-)1 Unit cell shape: PGeneral parallelepiped Geometrical constraints: a=/=b=/=c, α=/=β=/=γ=/=90° Lattice types: P
What is a lattice vector?
Any vector joining two lattice points
t = U a + V b + W c
t = [UVW]
What are Miller indices?
The first lattice plane away from the origin intercepts the x, y, and z axes at distances a/h, b/k, and c/l
If parallel to an axis, index is zero
What are screw axes?
A combination of a rotation and a translation
General symbol n_m
Corresponds to a rotation of 360°/n followed by a translation of m/n of the unit cell repeat along the axis
What are glide planes?
A combination of a reflection and a translation
Translation vector is parallel to the plane of reflection and is half the length between neighbouring lattice point
What are space groups?
Symbols providing the lattice type and minimum defining symmetry
First letter = lattice type
Following three letters refer to mirror planes or glide planes
What is relief?
The ease with which a grain or crystal edge can be seen
Depends on the difference between the RI of the grain and the medium
Low RI diff = Low relief
What is the Becke line test?
Bright concentric line around a grain
When stage is raised it moves into the medium of higher RI
How is the variation in RI with direction in a crystal represented?
In 3D by an optical indicatrix
Shape and orientation w.r.t crystallographic axes is constrained by symmetry
What are the three variants of optical indicatrix?
Isotropic
Uniaxial
Biaxial
What defines isotropic materials?
Which crystal system do they belong to?
What else are usually isotropic?
Same value of RI in all directions
Optical indicatrix is a sphere
Cubic crystal system
Amorphous materials
What defines anisotropic materials?
Lower than cubic symmetry
RI varies with direction
What defines the uniaxial indicatrix?
An ellipsoid of revolution with a single circular section (isotropic section)
Direction perpendicular to the isotropic section is the optic axis
Which crystal systems have a uniaxial indicatrix?
Why?
Tetragonal, trigonal, and hexagonal
Single high symmetry rotation axis parallel to z
View down z, indicatrix shape must be a circle
What defines the two types of uniaxial indicatrix?
RI in isotropic section = ω
RI parallel to optic axis = ε
If ε > ω, rugby ball shape = positive indicatrix
If ε < ω, burger shape = negative indicatrix
Which crystal systems have a biaxial indicatrix?
Why?
Orthorhombic, monoclinic, and triclinic
No symmetry constraints on the relative size of RI parallel to the three semi axes
In a biaxial indicatrix, where are the isotropic sections?
A direction in the α-γ plane is equal to β
The section parallel to this direction and β semi axis is circular
Two of these sections exist
Where are the optic axes in the biaxial indicatrix?
In the α-γ plane
What is the relevance of the angle between the optic axes of a biaxial indicatrix?
Angle between = 2V
2V value characteristic of mineral
2V < 90°, mineral = biaxial positive
2V > 90°, mineral = biaxial negative
What is the symmetry of a biaxial indicatrix?
What is the implication for a crystal?
Diad symmetry parallel to each of the semi axes
If the crystal has a diad in a particular direction, one of the semi axes must lie along that direction
What defines the orientation of the biaxial indicatrix for orthorhombic, monoclinic, and triclinic crystals respectively?
Orthorhombic: α, β, and γ parallel to x, y, and z
Monoclinic: Either α, β, or γ parallel to y
Triclinic: No orientation constraints
Define permitted vibration directions (PVD’s)
A general cross section through the indicatrix yields an ellipse with a major and a minor axis
Light in an anisotropic medium is only permitted to vibrate parallel to these two axes
What is birefringence defined as?
The difference between the two RI’s of the PVD’s
What causes pleochroism?
Polarised light seen may vary with vibration direction for some anisotropic crystals
What causes interference colours?
Polarised light vibrating at an angle to the two PVD's Light splits into two components One slow, one fast Speed difference -> phase difference Colour generated on recombination
What causes extinction?
Crystal rotated such that one PVD is parallel to the polariser
All waves pass through in the E-W orientation
So all waves blocked by analyser
Crystal appears black
What is optical path difference?
Δ = d(n_1 - n_2)
What is the purpose of the sensitive tint plate?
Gives the first-order sensitive tint-colour
Used to distinguish between slow and fast vibration directions in any crystal
What causes addition/compensation when two crystals are superimposed?
Addition occurs when the two slow or fast directions are parallel
Compensation occurs when a slow direction is parallel to a fast direction
What is an extinction angle?
An angle measured between one of the PVD’s and a significant feature such as cleavage, lines of exsolution
What is produced by a convergent-beam interference figure?
A map of directions
Can tell if the indicatrix is uniaxial/biaxial, +ve/-ve and estimate 2V
Explain these terms for convergent-beam interference figures:
Isogyre
Isochromes
Melatope
Isogyre: centred black cross, mapping the directions that are in extinction
Isochromes: concentric coloured rings of constant birefringence
Melatope: centre of the isogyre, marking the optic axis
How can the optic sign be determined from a centred-uniaxial interference figure?
Look for 1st-order grey interference colour close to the centre of the figure
Insert the sensitive tint plate
If the upper right quadrant is blue, optic sign is positive
Negative if yellow
Which two biaxial interference figures are considered useful?
Acute bisectrix figure
Optic axis figure
When is the acute bisectrix figure obtained?
What does this mean for a biaxial crystal?
When the acute bisectrix of the two optic axes is oriented parallel to the axis of the microscope
Positive: γ is vertical
Negative: α is vertical
What are the main features of an acute bisectrix figure?
Black cross at 0° and 90° positions
On rotation breaks into two curved isogyres
At 45° and 135° these are symmetrical hyperbolae and pass through the two optic axis melatopes
Coloured rings around each melatope
How can the optic sign be determined from an acute bisectrix figure?
Blue upper right = positive
How can 2V be estimated from an acute bisectrix figure?
Separation of isogyres at 45° is proportional to 2V
Can be estimated if the angular diameter of the FOV is known (normally ~60°)
Why is it easier to obtain an interference figure along or close to an optic axis for a biaxial crystal?
The thin section shows a low order interference colour
Black if the optic axis is centred
How can the optic sign be determined from an optic axis figure?
If the isogyre can be seen to curve in the 45° position
Then the rest can be imagined
Sign determined using blue upper right rule
If 2V is large (>85°), no detectable curvature so can’t find the sign
2V = 90°, crystal is optically neutral, not +ve or -ve
How can 2V be estimated from an optic axis figure?
From the maximum curvature of the isogyre
