Chapter 5 Overview of Crystallographic Concepts
I. From symmetry operations to space groups
a. See the cd-rom
II. Minerals as Crystalline solids
a. Crystals are homogeneous solid possessing long range, three-dimensional internal order.
i. Term crystalline is used to denote the ordered arrangement of atoms in the crystal structure
ii. Microcrystalline needs a microscope
iii. Cryptocrystalline needs an xrd.
b. Internal order
i. Motif or groups of atoms repeated on a lattice
1. Lattice expresses the translation component of an internal order
2. Motif has a certain symmetry that may be reflected in the crystals external shape.
3. The three-dimensional internal order of a crystal can be considered as the periodic repetition of motif.
III. Symmetry
a. Symmetry elements without translation
i. Rotation through an angle about an imaginary axis. (point or line) Can produce another or several motifs
ii. Expressed by any whole number from 1 to infinity. n.
iii. The types of rotation found in internally ordered crystals, are one-fold, two fold, three fold, four fold and six fold. Others are not possible.
1. constrained by the translational symmetry of a lattice
2. see page 177
b. Reflection or mirror
i. Across a mirror plane m. or symmetry plane
ii. Enantiomorphic pair are motifs that are related by a mirror and cannot be superimposed on each other.
c. Center of Symmetry or inversion, i, produces and inverted object through an inversion center.
i. Similar to mirror.
ii. A center of symmetry is present in a crystal if an imaginary line can be passed from any point on its surface through its center and a similar point is found on the line at an equal distance beyond the center.
d. Rotation with inversion
i. Rotoinversion operations (read as bar 1)
e. Symbols used to describe symmetry
i. See page 180
f. Combinations of rotations
i. Single axis of rotation or retroinversion
ii. Combine various axes of rotation
1. All must intersect at a single point.
2. axes must be put together at 90 or 54 degrees 44 minutes
3. minus sign indications the position below the page.
g. Combinations of rotation axes and mirrors
i. Mirror planes within crystals ar eeighter perpendicular to or paralles to any rotation axes.
ii. 4/m four over m is a fourold rotation axis surrounded by four motifs above and below a mirror plane.
h. 32 possible symmetry elements and combinations fo symmetry elements see page 186
i. Hermann-Mauguin notation
i. 32 point groups and crystal classes
i. can be grouped inter 6 crystal systems
1. 21 without a center of symmetry
2. 11 with
3. See page 187
IV. Crystal Morphology
a. External form mimic the internal structure
b. The crystal faces have a direct relationship to the internal structure, the faces have a relationship with each other.
c. Faces are often parallel to the crystal lattice.
d. Stenos law the angles between equivalent faces of crystals of the same substance, measure at the same temperature, are constant. Quartz is a great example.
V. Crystallographic axes
a. Triclinic three unequal axes all intersecting at oblique angles
b. Monoclinic three unequal axes, two of which are nclined to each other at an obliques angle and the third perpendicular to the plane of the other two.
c. Orthorhombid Three mutually perpendicular axes all of different lengths
d. Tetragonal Three mutually perpendicular ases, two of which (the horizontal axes) are of equal length but the vertical axis is shorter or longer than the other two.
e. Hexagonal referred to four crystallographic axes, three equal horizontal axes intersect at angles of 120 and the forth is of a different length and perpendicular to the plane of the other three.
f. Isometric three mutually perpendicular axes of equal lengths
VI. Crystallographic notation for planes
a. Miller indices
i. Method of describing a crystal face in its relationship the intersections of axes.
ii. Whole numbers were fractions are cleared
iii. Example, 2, 2, 2/3 is inverted to ½, ½, 3/2 and then multiply by two gives you 113
iv. A bar over the number indicates the negative side of the axis. A comma only separates numbers if there is a two digit number involved.
v. Parenthesis identifies a single face while braces are used for all faces or forms. Generally the most positive digits are used to describe form.
b. Forms
i. 48 kinds of forms, 32 are general forms and becomes the descriptive name of each 32 crystal classes.
ii. See page 204-207
VII. Internal order
a. Three dimensional periodicity of a mineral or the atomic structure.
b. Repetition of motif
i. Unit cell is the smallest units of repeat in thes lattices outlined.
ii. 14 possible space lattice types, (Bravais lattices)
c. Translation Directions and distances
i. Homogenous if the angles and distances from one motif to surrounding motifs in one location of the pattern are the same in all parts of the pattern.
d. One dimensional order or tows
i. Magnitude is the spacing of the unit translation
e. Two dimensional order or plane lattices
i. Result of regular translations in two different directions.
ii. Patterns can be described by vectors, angles and units of the translation or distance
iii. Only five possible and distinct plane lattices.
iv. Parallelogram, two types of rectangles, a diamond and a rhombus and square.
v. Tesselations are like bricks using two translation directions
vi. Rotation angle Restrictions
1. Shows why you cannot have 5 fold
f. Symmetry Content of Planar Motifs
1. p planer point groups
2. 10 types 5 original and 5 mixed
g. Symmetry content of Plane lattices
i. Two dimensional point groups and space groups
ii. See page 226
h. 3 dimensional lattices
i. Primitive
ii. Centered face
iii. Centered body
iv. Rhombohedra
v. See page 231
i. See page 232
i. Screw Axes and Glide Planes See page 236