CRYSTALLOGRAPHY

--Crystallography is the study of crystal shapes--
--in this section we will study crystal symmetry, crystal forms
crystallographic axes, unit cells, axial ratios, face intercepts
and Miller Indices

A. Introduction
-atoms of elements combine in mineral formation to build specific geometric shapes
on the smallest scale (we will study this later in a section dealing with the nature of atomic
shapes)--these shapes combine to form crystal shapes--since each mineral is restricted
by the crystal shape (s) it can have, the study of crystallography can aid in the
determination of the name of a mineral--

-the determination of the symmetry functions present on a crystal is paramount and allows
the crystal to be categorized into one of 32 possible crystal classes

-to become familiar with crystal class determination we will analyze symmetry functions
on many crystals--wooden crystals precut to show symmetry functions will be used for
practice before doing the same for real mineral crystals--

B. Symmetry Functions (Crystal Symmetry)
1. axis of rotation
-is an imaginary line through a crystal around which a crystal is rotated through 360
degrees--the complete occurrence of the original face or set of faces in a number of
multiple appearances during the full rotation determines the kind (type) or fold of axis of
rotation
A2 = (2-fold axis of rotation) a reproduction of original face or faces twice
A3 = (3-fold axis of rotation)  "                        "                         "        three times
A4 = (4-fold axis of rotation)  "                        "                         "        four times
A6 = (6-fold axis of rotation)  "                        "                         "        six times
-a crystal can have more than one type of fold axis and may have a multiple of that
fold type--an example--1A6, 6A2 indicates the crystal has one six fold axis and six two fold
axes-in all but the isometric crystal system additional rotation axes types and their multiple,
if present, occur only in the 90 degree horizontal plane from an axis already found--in the
isometric system, fold axes may also be present at 45 degrees from another axis-

-an axis of rotation can represent only one yAx---in order to have more than one kind of
rotation axis or a multiple of the same rotation axis, you must move your fingers and hold
the crystal in another place to define these other axes

2.  mirror plane (symmetry plane)
-is an imaginary plane that divides a crystal into two equal halves, each a mirror image of
the other--there may be 0-9 different mirror images on a crystal--a crystal which has
6 different mirror images as an example is designated as 6m-
-in all but the isometric system mirror images, if present, occur in the vertical plane of
rotation axes and one possibly in the horizontal plane of an axis--in the isometric
system, mirror planes may be present as above and also can occur at 45 degrees
to the rotation axes-
-in the determination of rotation axes and mirror planes, it is paramount not to
count the same individual axis or mirror plane more than once

3.  center of symmetry
-occurs if the same feature is located on the exact opposite sides of a crystal and both
are equal distance from the center of the crystal--features include points, corners, edges
or faces--
-there is a center of symmetry or none--if there is a center present, there is an infinite
number of places where this analysis will occur and the crystal has an (i) appearing in the
designation of its symmetry functions-

4. axis of roto-inversion
-is present if the original face or faces are reproduced after a crystal is rotated either at
180degrees (A2), 120 degrees (A3), 90 degrees (A4) or 60 degrees (A6) and then inverted-
-if present, a crystal can have only one type of roto-inversion axis and the designation for
this is similar to that of an axis of rotation except the AX includes a bar above the letter
or number
-each roto-inversion A3 is equivalent to an (A3) + i
-each           "            A4             "               A2
-each           "            A6             "              (A3 + m)
-since the roto-inversion axis has equivalents of the other functions of symmetry, we need
only to use the first 3 functions to categorize a crystal in a class--even so you need to
be able to identify a roto-inversion axis on crystals--
If the total symmetry of a crystal is ascertained, the table below can be used to
determine to which crystal class and system the crystal belongs

-the table below includes the equivalent non-rotoinversion symmetry where ap-
propriate

 Crystal System Crystal Class Name of Class Symmetry Content Triclinic 1 Pedial none 1 Pinacoidal i

 Monoclinic 2 Sphenoidal 1A2 m Domatic 1m 2/m Prismatic i, 1A2, 1m

 Orthorhombic 222 Rhombic-disphenoidal 3A2 mm2 Rhombic-pyramidal 1A2, 2m 2/m2/m2/m Rhombic-dipyramidal i, 3A2, 3m

 Tetragonal 4 Tetragonal-pyramidal 1A4 4 Tetragonal-disphenoidal 1A2 4/m Tetragonal-dipyramidal i, 1A4, 1m 422 Teragonal-trapezohedral 1A4, 4A2 4mm Ditetragonal-pyramidal 1A4, 4m 42m Tetragonal-scalenohedral 3A2, 2m 4/m2/m2/m Ditetragonal-dipyramidal i, 1A4, 4A2, 5m

 Hexagonal 3 Trigonal-pyramidal 1A3 Rhombohedral 3 Rhombohedral i, 1A3 Division 32 Trigonal-trapezohedral 1A3, 3A2 3m Ditrigonal-pyramidal 1A3, 3m 32/m Trigonal (Hex.)-scalenohedral i, 1A3, 3A2, 3m Hexagonal 6 Hexagonal-pyramidal 1A6 Hexagonal 6 Trigonal-dipyramidal 1A3, 1m division 6/m Hexagonal-dipyramidal i, 1A6, 1m 622 Hexagonal-trapezohedral 1A6, 6A2 6mm Dihexagonal-pyramidal 1A6, 6m 62m Ditrigonal-dipyramidal 1A3, 3A2, 4m 6/m2/m2/m Dihexagonal-dipyramidal i, 1A6, 6A2, 7m

 Isometric 23 Tetartoidal 3A2, 4A3 2/m3 Diploidal i, 3A2, 3m, 4A3 432 Gyroidal 3A4, 4A3, 6A2 43m Hextetrahedral 4A3, 3A2, 6m 4/m 32/m Hexoctahedral i, 3A4, 4A3, 6A2, 9m

You should note from the table there are certain associated symmetry functions as:
1.  a crystal which has an A6 cannot have an A4 and vice versa
2.  a crystal with only 1A4 is tetragonal, and if more present, has to have 3A4 and
must belong to a specific isometric crystal class
3.  a crystal with an A6 cannot have an A3 and must be in the hexagonal
division of the hexagonal system
4.  a crystal with only 1A3 must be in the hexagonal system, and if more present
there has to be exactly 4A3 and belong in the isometric system
List other similar symmetry functions associations from the table above

Holohedral refers to the highest symmetry class in a crystal system-list those
classes--know them

C.  Crystal Forms

-a crystal form is a group of faces, all of which have the same relation to the functions of
symmetry on the crystal--the symmetry on a crystal has a direct influence on which specific
forms are present
-the forms present on crystals may be classified as (1) non-isometric and (2) isometric--
isometric crystals have essentially a different set of forms than the same in the other
crystal systems
-the faces of a crystal form may intersect faces of another form on a crystal resulting in shapes
being altered not resembling the perfect texbook "full form" shown below, hence
the faces comprising a form must be extended visually to resemble
the picture drawings below--this may be more difficult to do for multiform crystals in some
systems and especially those in the isometric system--practice will solve this problem
-note in the picture drawings, each form is shown related to axes (is) of rotation or roto-
inversion--see page 127 in text if you are not familiar with symbols for these axes

1. non-isometric forms
-CLICK to see drawings of single full form non-isometric forms taken from the textbook
and explained below--important to note is the name of a form with a

specific number of faces can indicate the crystal system or
possibly the crystal class-
-an example is a prism--if comprised of:
3, 6 or 12 faces it must belong in the hexagonal system
4                 "     "   "        "       "   "orthorhombic, tetragonal or monoclinic system
8                 "      "   "       "       "   "tetragonal system
-specific names are given to a form depending on how many faces it has--from the
example above: 3 faces= (trigonal prism); 4 faces= (tetragonal, orthorhombic or monoclinic
prism--depending on which crystal system it belongs based on symmetry);6 faces=
(hexagonal prism or ditrigonal prism--based on symmetry);12 faces (dihexagonal prism)

a. Pedion--a single face comprising a form
b. Pinacoid--an open form comprised of 2 parallel faces
c. Dome--an open form comprised of 2 nonparallel faces symmetrical with respect to
a mirror plane
d. Sphenoid--two nonparallel faces symmetrical with respect to a 2-fold axis
e. Prism--an open form consisting of 3, 4, 6, 8 or 12 faces all parallel to the same axis--
except for some prisms in the monoclinic system, the prisms are parallel
to the vertical or highest fold axis
f.  Pyramid--an open form comprised of 3, 4, 6, 8 or 12 nonparallel faces that meet at a
point at the top of a crystal
g. Dipyramid--a closed form comprised of  6, 8,12, 16, or 24 faces--basically it is a
pyramid appearing at the top and bottom of a crystal with a mirror plane
separating each
h. Trapezohedron--a closed form comprised of 6, 8 or 12 faces with 3, 4, or 6 upper
faces offset from 3, 4, or 6 lower faces--this results in a 3, 4, or 6
fold axis with 3 2-fold perpendicular axes
iScalenohedron--a closed form with 8 or12 grouped in symmetrical pairs
j.  Rombohedron--a closed form comprised of 6 faces of which 3 faces on top are offset
with 3 faces on the bottom each by 60 degrees
k.  Disphenoid--a closed form comprised of 2 upper faces that alternate with 2 lower
faces, offset by 90 degrees

2. isometric forms
-CLICK to see drawings of single isometric forms taken from text book and explained
below:
-many of the forms have their nature based on 3 elementary forms--this triad of forms
include the cube (hexahedron), octahedron and the tetrahedron--many form names
include a prefix with the suffix being one of these 3 basic forms--an example is the
tetrahexahedron having 4 (tetra) faces on each hexahedron face (6) for a total of 24 faces

a. Cube(hexahedron)--6 equal faces intersecting at 90 degrees
b. Octahedron--8 equilateral triangular faces
c. Dodecahedron--12 faces, each rhomb shaped
d. Tetrahexahedron--24 isosceles triangular faces
e. Trapezohedron--24 trapezium shaped faces
f. Trisoctahedron--24 isosceles triangular faces
g. Hexoctahedron--48 triangular faces
h. Tetrahedron--4 equilateral triangular faces
i. Tristetrahedron--12 triangular faces
j. Deltoid dodectahedron--12 faces corresponding to 1/2 of the faces of a trisocta-
hedron
k. Hextetrahedron--24 faces-6 triangular faces formed on each side of the tetrahedron
l.  Diploid--24 faces
m. Pyritohedron--12 pentagonal faces

3.  forms present in the 32 crystal classes
-one or a combination of the forms above can be present on individual crystals based on
crystal symmetry--it is possible to determine the crystal class based on the com-
bination of forms present--
-in some cases it is possible to determine the crystal class if a special form is present on the
crystal since that form can occur only in that class--some examples would be:

-the rhombic dipyramid occurs only in the rhombic dipyramidal class;
the ditrigonal dipyramid occurs only in the ditrigonal dipyramidal class;
the hextetrahedron occurs only in the hextetrahedral class;
the tetrahexahedron occurs in the hextetrahedral class;

-crystal class names are based on the most outstanding form possible in that class-
-form tables below list all possible forms that can be present on crystals in each of the 32 classes-
-in the tables, numbers in parentheses after the form name represent the number of
faces comprising the form-

-CLICK HERE  and page down to the desired system or CLICK below on a crystal system
1. triclinic classes
2. monoclinic classes
3. orthorhombic classes
4. tetragonal classes
5. hexagonal-rhombohedral division classes
6. hexagonal-hexagonal division classes
7. isometric classes

D.  Crystallographic Axes
-are imaginary reference lines constructed through crystals and in most instances
coincide with symmetry axes or normals to symmetry planes
--these axes aid in the
orientation of crystals and help to explain other crystal concepts like unit cells and
Miller Indices which we will discuss below
-crystals in all crystal systems have 3 axes associated with them except for 4 axes in the
Hexagonal System
- these axes are compared by lengths and angles of intersection with each other
axes are designated as a, b and c when unequal in lengths in a crystal or by the same letter,
"a" whenever equal in length--the "c" axis when present is the principal axis and can be
larger or smaller than the other axes associated with it
-the axes have a preferred vertical and horizontal orientation--the c axis when present
always occurs in the vertical--when an a and b axis are present, they are located in the
horizontal plane, the a is in a front-back orientation, and the b in a left to right position
--if there is no b but a c axis present, the a axes are located horizontally in the front-back,
and left-right positions if 2 are present, or if 3 are present  the same exists  and the third lies
between the other 2--
-the following explains relative lengths of axes and angles of intersection of axes with each other
and the association with symmetry functions in each of the crystal systems--CLICK to
see drawings of the crystallographic axes

1. Triclinic
-consists of 3 unequal axes (a, b, c) all intersecting at oblique angles--since there
are no symmetry axes or symmetry planes the a, b, and c axes are not associated
with any symmetry functions

2. Monoclinic
-consists of 3 unequal axes (a, b, c), in which a and c are inclined to each other
at an oblique angle and the third (b axis) perpendicular to the other two--the b
axis corresponds to the 2-fold axis and lies in a perpendicular plane to the m if
either or both are present--the faces of the monoclinic prism (main prism), if
present, parallel the c axis

3.   Orthorhombic
-consists of 3 unequal axes (a, b, c) all mutually perpendicular--the c axis
corresponds to one of the 2-fold axes and is that on which the pyramid,
dipyramid or disphenoid are found if present--the faces of the orthorhombic
prism, if present, parallel the c axis

4.   Tetragonal
-consists of 3 mutually perpendicular axes (a, a, c), two of which are equal in
length (a, a) and the third axis (c) is shorter or longer than the other two--the faces of
most forms in this crystal system form on or parallel the c axis

5.   Hexagonal
-consists of 4 axes, three of which are equal in length, found in the horizontal
plane, and intersect at 60 degrees (a, a ,a); the fourth axis (c) is perpendicular to
and shorter or longer than the other 3-- faces of most all forms in this crystal system
form on or parallel the c axis

6.   Isometric
-consists of 3 mutually perpendicular axes of equal length

E. Unit Cell
-is a 3-dimensional geometric figure constructed in the array of atoms of a mineral so that
there are specific (nodes) atoms occurring at the corners of the figure and possibly at the center
of  face or faces and/or at the very center of the figure--there are additional (nodes) atoms located
within the cell but not at these specific locations

-is the smallest unit of a mineral that retains all of the physical, chemical, and crystallographic
properties of a mineral.  The unit cell comprises the symmetry of each holohedral class
of the crystal system

-there are only 14 unique figures possible for the 6 crystal systems and include the following:
1. primitive
-a cell with (nodes) atoms at all the corners and not at the center of faces nor an atom at the
very center of the cell-

2. non primitive
-a cell with (nodes) atoms at all corners and also an atom on each of 2 opposite faces--if at
top and bottom, C-centered; if on both side faces, B-centered; if on
front and back faces, A-centered; or on all faces, F-or face-centered--the A, B,
and C lattices are symmetrically identical and can be converted to each other by
an appropriate change of the crystallographic axes
-also, a cell with (nodes) atoms at the corners and one at the very center of the unit cell is a
body-centered ( I ) non primitive cell

CLICK to see the 14 types of unit cells (Bravais Lattices)
-although many minerals may have the same type of primitive or non primitive unit
cell, each mineral has a sole combination of crystallographic axes lengths which acts
like a genetic code for that mineral--hence no two minerals will have the same 3 unit
cell axes dimensions--this fact allows X-ray analysis to generate the name of a mineral

CLICK HERE to see how unit cells on an atomic scale pack together to form crystal forms on
a much larger seeable scale (micro or macroscopic)

-The volume of the unit cell is important in Specific Gravity calculations--the table below indicates the
formula to calculate the volume of the unit cell for each crystal system: a, b and c are axes lengths (10-8cm)
and alpha, beta and gamma are axes angles

 CRYSTAL SYSTEM UNIT CELL VOLUME ( V ) Isometric V = a3 Tetragonal V = a2c Hexagonal V = a2csin(60o) Trigonal V = a2csin(60o) Orthorhombic V = abc Monoclinic V = abcsin(beta) Triclinic V= abc(1-cos2alpha-cos2beta-cos2gamma + 2cos alpha cos beta cos gamma)1/2

F.  Axial Ratio
-is a ratio of the unit cell lengths measured in Angstrom units (Å) using the length of the b axis as
the common denominator, or the a axis as the same if the b axis is not present--thus the unit
cell lengths of sulfur with the following axial lengths: a = 10.47 Å;   b=12.88 Å;
c = 24.49 Å will have an axial ratio of a:b:c = 0.8135:1:1.9029--as you can see all unit cells
in the isometric must have an axial ratio of 1:1:1

G.  Face Intercepts (Parameters)
-face(s) of forms (is) are defined by their intercepts or non-intercepts on the crystallographic
axes--an example is, do faces parallel one or two axes and intersect the other?-- and if so,
it must be determined at what relative distance the faces intersect the different axes--this
relative distance is expressed as a multiple or fraction related to the axes lengths or ratios
of the unit cell as measured from the vortex--for instance, if the aforementioned unit cell of
sulfur is used and a face representing a form intercepts the a axis at 21.94 Å, the b axis at
6.44 Å, and the c axis at 24.49 Å, the parameters or face intercepts would be
2a, 1/2b, c--if the face parallels an axis (axes) an infinity sign (looks like an 8 turned
90 degrees) is placed before the axis letter--a minus intercept on an axis is then so
noted with a minus sign above the intercept number

H.  Miller Indices
-
the indices of a face is a series of whole numbers derived from the intercepts by inverting
the whole or fraction numbers and clearing of fractions if present--examples of conversion
from intercepts to indices would be: ( 2a, 1/2b, c) = (1/2, 2/1,1/1) = (1,4,2): ( infinity a, 3b,
2/3c) = (0, 1/3, 3/2) = (0,2,9)--once again the minus sign intercept should be included
above the appropriate number
Now let us consider the nature of some basic mineral chemistry principles--this
includes the nature of mineral formulas, element and element oxide weight
percents, specific gravity determinations and other principles--we will solve problems
to aid in understanding these concepts