Symmetry

Question 1

Space group

Answer 1

Space groups represent the various ways motifs (such as atoms in xtals) can be arranged in space in a homogeneous array.

Each motif is equivilant to every other motif in the pattern

230 possible combinations of symmetry operations and translation operations in space

1. Based on 1 of the 14 Bravais lattices
2. Isogonal w/ 1 of the 32 point groups

Question 2

Point group

Answer 2

The 32 possible combinations of rotational symmetry operations - both proper and improper

Point indicates that the symmetry operations are fixed around an “unmoved” center axis

• 21 without a center of symmetry
• 11 with a center of symmetry

Identical to the 32 crystal classes that are assigned based on morphology

The translation-free “residue” of a family of possible isogonal space groups

Question 3

Isogonal

Answer 3

The rotation and screw axes have the same rotational angle

ex: 60° in a six-fold rotation (6) of six-fold screw axis (6₁)

Question 4

Crystallography

Answer 4

The study of the external form and internal atomic arrangement of crystalline solids and the principles that govern their growth, external shape, and internal structure.

Question 5

Unit Cell

Answer 5

The smallest unit of structure (or pattern) that can be infinitely repeated to generate the entire structure

Maintains the diagnostic properties of the mineral

Question 6

Lattice

Answer 6

An imaginary pattern of points (or nodes) in which every point/node has an environment that is identical to that of every other point/node in the pattern

No specific origin as it can be shifted parallel to itself

Question 7

Law of Bravais

Answer 7

Faces are most likely to develop parallel to lattice planes that have a high density of lattice points

The larger the number of points/nodes, the more common the face

Question 8

Steno’s Law of the constancy of interfacial angles

Answer 8

The angles between equivilant faces of crystals of the same substance, measured at the same T, are constant

Question 9

Symmetry Operations

Answer 9

The processes of rotation about an axis, of refection by a mirror, and of inversion about a central point

Question 10

Rotation

Answer 10

Rotation is established by rotating a motif about an imaginary axis that intersects the crystal center

When rotational symmetry is present, a specific face on the crystal is repeated n times in a complete rotation of the crystal

Question 11

Reflection (mirror)

Answer 11

A mirror plane (m) is an imaginary plane that divides a crystal into two halves, each of which, in a perfectly developed crystal, is the mirror image of the other

A reflection across a mirror plane produces a mirror image

Question 12

Inversion

Answer 12

A center of symmetry (i) is present in a crystal if an imaginary line can be passed from any point on its surface thru its center and the same point is found on the line at an equal distance beyond the center of the opposite side

Tested by confirming that for each face of the crystal there is some parallel face

Produces an enantiomorphic pair

Question 13

Rotoinversion

Answer 13

1. First rotate the motif thru α (the rotation angle)
2. Then invert the motif thru the center

Best seen and evaluated in 3D objects

Question 14

Hermann-Maugin (HM) notation

Answer 14

The symbols for the overall symmetry content of a crystal

example:

i, 3A₄, 4A₃, 6A₂, 9m

i = center of symmetry

3A₄ = three four-fold rotations

4A₃ = four three-fold rotations

6A₂ = six two-fold rotations

9m = nine mirror planes

Question 15

Miller Indices

Answer 15

The Miller indices of a face consist of a series of whole numbers that have been derived from the intercepts by inverting and, if necessary, the subsequent clearing of fractions.

Question 16

Morphology

Answer 16

Morphology is commonly used to indicate the general outward appearance or shape of crystals

Question 17

Crystal Habit

Answer 17

In crystallography, external shape is denoted by the term, habit whereas form is used in a special and restricted sense

Question 18

Crystal Form

Answer 18

Crystal habit = external shape

Crystal Form is used in a special and restricted sense

A form consists of a group of like crystal faces, all of which have the same relation to the elements of symmetry

Question 19

Translation

Answer 19

An ordered pattern can be generated by taking a motif and repeating it or translating it a specific distance t₁

The external form of a crystal, although an expression of its internal structure, lacks any expresion of translation

Question 20

Space Lattices

Answer 20

When the five planar nets are stacked in various ways along a third direction (z), periodic arrays in three-dimensions results

Question 21

Primitive

Answer 21

A primitive space lattice is a parallelpiped with lattice points only at the corners

Question 22

Nodes may be centered on a pair of opposite faces of the unit cell

Answer 22

When nodes are centered on a pair of opposite faces of the unit cell

A-centered = along the a (x) axial direction

B-centered = along the b (y) direction

C-centered = along the c (z) direction

Question 23

F-centered

Answer 23

When nodes are centered on all faces of the unit cell

Question 24

I-centered

Answer 24

When the node is present in the center of the unit cell, it is refered to as body-centered

Question 25

Bravais Lattices

Answer 25

There are 14 lattices that are compatible with the 32 point groups

They represent the only possible wys in which nodes can be arranged in three dimensions

The varios lattices can be arranged among the seven crystal systems

Question 26

Screw Operation

Answer 26

The combination of a rotarional operation with a translation (t) parallel to the axis of rotation

Any additional motifs are generated from the original motif in a 3-D, helical path - as in a screw motion

possible screws: 2, 2₁, 3, 3₁, 3₂, 4, 4₁, …

When the ratio of the subscript to the number of rotation is less than 1/2 the screw is right-handed

When the ratio of the subscript to the number of rotation is greater than 1/2 the screw is left-handed

Question 27

Glide Operation

Answer 27

The combination of a mirror reflection with a translation component (t/2 or t/4) parallel to the mirror

also known as glide reflection

It relates motifs on either side of the line with a periodicity of hald the lattice translation

Question 28

Glide Planes

Answer 28

3-D glide movements expressed in terms of the crystallographic axes: a, b, and c

Example: If it is parallel to the a-axis, then it is an a-glide. Similarly, if it is parallel to the b-axis, a b-glide, and a along the c-axis, a c-glide

diagonal (n) glides and diamond (d) glides exist

Question 29

Right-handed

Answer 29

When the ratio of the subscript to the number of a rotation axis is less than 1/2

Defined as one that advances away from the observer when rotated clockwise

Question 30

Left-handed

Answer 30

When the ratio of the subscript to the number of the rotation axis is greater than 1/2

Defined as one that advances toward the observer when rotated ?

Question 31

R-centered

Answer 31

Contains lattice points at 1/3 and 2/3 along the body diagonal