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 shapessince 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 crystalswooden
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
degreesthe 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 = (2fold axis of rotation) a
reproduction of original face or faces twice
A3 =
(3fold axis of rotation) "
"
" three times
A4 = (4fold axis of rotation)
"
"
" four times
A6 = (6fold axis of rotation)
"
"
" six times
a crystal
can have more than one type of fold axis and may have a multiple of that
fold
typean example1A6, 6A2 indicates the crystal has one six fold axis
and six two fold
axesin 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
foundin the
isometric system, fold axes may also be present at 45 degrees from another
axis
an
axis of rotation can represent only one yA_{x}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
otherthere may be 09 different mirror images on a crystala 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 axisin 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 crystalfeatures include points,
corners, edges
or
faces
there is a
center of symmetry or noneif 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 rotoinversion
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 rotoinversion 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 rotoinversion A3
is equivalent to an (A3) + i
each "
A4
"
A2
each "
A6
"
(A3 + m)
since the
rotoinversion axis has equivalents of the other
functions of symmetry, we need
only to use the first 3 functions to categorize a crystal in a classeven
so you need to
be able to identify a rotoinversion 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 nonrotoinversion 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

Rhombicdisphenoidal

3A2


mm2

Rhombicpyramidal

1A2, 2m


2/m2/m2/m

Rhombicdipyramidal

i, 3A2, 3m

Tetragonal

4

Tetragonalpyramidal

1A4


4

Tetragonaldisphenoidal

1A2


4/m

Tetragonaldipyramidal

i, 1A4, 1m


422

Teragonaltrapezohedral

1A4, 4A2


4mm

Ditetragonalpyramidal

1A4, 4m


42m

Tetragonalscalenohedral

3A2, 2m


4/m2/m2/m

Ditetragonaldipyramidal

i, 1A4, 4A2, 5m

Hexagonal

3

Trigonalpyramidal

1A3

Rhombohedral

3

Rhombohedral

i, 1A3

Division

32

Trigonaltrapezohedral

1A3, 3A2


3m

Ditrigonalpyramidal

1A3, 3m


32/m

Trigonal (Hex.)scalenohedral

i, 1A3, 3A2, 3m





Hexagonal

6

Hexagonalpyramidal

1A6

Hexagonal

6

Trigonaldipyramidal

1A3, 1m

division

6/m

Hexagonaldipyramidal

i, 1A6, 1m


622

Hexagonaltrapezohedral

1A6, 6A2


6mm

Dihexagonalpyramidal

1A6, 6m


62m

Ditrigonaldipyramidal

1A3, 3A2, 4m


6/m2/m2/m

Dihexagonaldipyramidal

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
systemlist those
classesknow 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
crystalthe symmetry on a crystal has a direct influence on which specific
forms are present
the forms present on crystals may
be classified as (1) nonisometric 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
belowthis may be more difficult to do for multiform crystals in some
systems and especially
those in the isometric systempractice will solve this problem
note in the picture drawings,
each form is shown related to axes (is) of rotation or roto
inversionsee page
127 in text if you are not familiar with symbols for these axes
1. nonisometric forms
CLICK to see drawings of single full form nonisometric forms
taken from the textbook
and
explained belowimportant to note is the
name of a form with a
specific number of faces can indicate the crystal system or
possibly
the crystal classan
example is a prismif 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 hasfrom the
example
above: 3 faces= (trigonal prism); 4 faces=
(tetragonal, orthorhombic or monoclinic
prismdepending
on which crystal system it belongs based on symmetry);6 faces=
(hexagonal prism
or ditrigonal prismbased on symmetry);12
faces (dihexagonal prism)
a. Pediona single face comprising a form
b. Pinacoidan open form comprised of 2 parallel faces
c.
Domean open form comprised of 2 nonparallel faces symmetrical with respect
to
a mirror plane
d. Sphenoidtwo
nonparallel faces symmetrical with respect to a 2fold axis
e.
Prisman 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.
Pyramidan open form comprised of 3, 4, 6, 8 or 12 nonparallel faces that
meet at a
point at the top of a crystal
g. Dipyramida closed form comprised of 6, 8,12, 16,
or 24 facesbasically it is a
pyramid appearing at the top and bottom of a crystal with a mirror plane
separating each
h. Trapezohedrona closed form comprised of 6, 8 or 12
faces with 3, 4, or 6 upper
faces offset from 3, 4, or 6 lower facesthis results in a 3, 4, or 6
fold axis with 3 2fold perpendicular axes
i. Scalenohedrona
closed form with 8 or12 grouped in symmetrical pairs
j. Rombohedrona 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. Disphenoida 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 formsthis triad of forms
include the cube
(hexahedron), octahedron and the tetrahedronmany form names
include a prefix
with the suffix being one of these 3 basic formsan 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.
Octahedron8 equilateral triangular faces
c.
Dodecahedron12 faces, each rhomb shaped
d. Tetrahexahedron24 isosceles triangular faces
e. Trapezohedron24 trapezium shaped faces
f. Trisoctahedron24 isosceles triangular faces
g. Hexoctahedron48 triangular faces
h.
Tetrahedron4 equilateral triangular faces
i. Tristetrahedron12
triangular faces
j. Deltoid
dodectahedron12 faces corresponding to 1/2 of the
faces of a trisocta
hedron
k. Hextetrahedron24 faces6 triangular faces formed on each
side of the tetrahedron
l.
Diploid24 faces
m. Pyritohedron12 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
symmetryit 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 classsome 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. hexagonalrhombohedral division classes
6. hexagonalhexagonal
division classes
7. isometric classes
CLICK HERE
FOR MORE ON CRYSTAL FORMS AND CRYSTAL CLASSES
CLICK HERE FOR CRYSTAL FORMS AND
JAVA ROTATING CRYSTALS
D. Crystallographic
Axes
are imaginary reference lines constructed through crystals
and in most instances
coincide with
symmetry axes or normals to symmetry planesthese 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 lengththe "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 orientationthe c axis when present
always occurs in the
verticalwhen an a and b axis are present, they are located in the
horizontal plane,
the a is in a frontback 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 frontback,
and leftright
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 systemsCLICK to
see
drawings of the crystallographic axes
1. Triclinic
consists of 3 unequal axes (a, b, c) all intersecting at oblique
anglessince 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 twothe b
axis corresponds to the 2fold axis and lies in a perpendicular
plane to the m if
either or both are presentthe 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 perpendicularthe
c axis
corresponds to one of the 2fold axes and is that on which the
pyramid,
dipyramid or disphenoid
are found if presentthe 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 twothe 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 3dimensional 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 figurethere 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 facesif at
top and bottom, Ccentered; if on both side faces, Bcentered;
if on
front and back faces, Acentered; or on all faces, For facecenteredthe
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
bodycentered ( 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 mineralhence no two minerals will
have the same 3 unit
cell axes dimensionsthis fact allows Xray 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)
CLICK
HERE to see Bravais Lattices that rotate
The volume of the unit cell is important in Specific
Gravity calculationsthe 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^{8}cm)
and alpha, beta and gamma are axes
angles
CRYSTAL SYSTEM

UNIT CELL VOLUME ( V )

Isometric

V
= a^{3}

Tetragonal

V
= a^{2}c

Hexagonal

V
= a^{2}csin(60^{o})

Trigonal

V
= a^{2}csin(60^{o})

Orthorhombic

V
= abc

Monoclinic

V
= abcsin(beta)

Triclinic

V=
abc(1cos^{2}alphacos^{2}betacos^{2}gamma
+ 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 presentthus 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 = 0.8135:1:1.9029as
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 nonintercepts on the crystallographic
axesan 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 axesthis
relative distance is
expressed as a multiple or fraction related to the axes lengths or ratios
of the unit cell as measured
from the vortexfor 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, cif the
face parallels an axis (axes) an infinity sign (looks like an 8 turned
90 degrees) is placed
before the axis lettera minus intercept on an axis is then so
noted with a minus
sign above the intercept number
CLICK HERE to see other examples of
face intercepts
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 presentexamples 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 principlesthis
includes the nature of mineral
formulas, element and element oxide weight
percents, specific gravity determinations
and other principleswe will solve problems
to aid in understanding these concepts
