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Crystal faces are the result of a solid
substance growth by adding atoms in a completely orderly,
repetitive, 3-dimensional array called its atomic structure.
It is crystalline if it has this orderly atomic structure,
even if it lacks the regular faces (which are dependent
on the growth environment and the history since formation).
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The
unit cell.
The unit cell of a mineral is the smallest divisible unit
of a mineral that possesses the symmetry and properties of
the mineral. It is a small group of atoms, from four to as
many as 1000, that have a fixed geometry relative to one another.
The atoms are arranged in a "box" with parallel
sides called the unit cell which is repeated by simple translations
to make up the crystal. The atoms may be at the corners, on
the edges, on the faces, or wholly enclosed in the box, and
each cell in the crystal is identical. |
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This
is what was meant by an "ordered internal arrangement"
in the definition of a mineral. It is the reason why crystals
have such nice faces, cleavages, and regular properties.
The box of the unit cell is, in general, a parallel-piped
with no constraints on the lengths of the axes or the angles
between the axes. The box is defined by three axes or cell
edges, termed a, b, and c and three inter-axial angles alpha,
beta, and gamma, such that alpha is the angle between b and
c, beta between a and c, and gamma between a and b.
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The
Seven Crystal Systems |
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Triclinic
The triclinic unit cell is defined by three axes, a, b, and c,
of unequal lengths. None of the angles, alpha, beta, and gamma,
between these axes are exactly 90°. The standard convention
for assigning axes is c < a < b. The angle alpha lies between
the b and c axes; beta lies between a and c, and gamma lies between
a and b. The cell is usually chosen so that alpha and beta are
obtuse (between 90º and 180º).
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Orthorhombic
The orthorhombic unit cell is defined by three axes, a, b, and c,
of unequal lengths. The angles between all axes are exactly 90°.
The axes are chosen to correspond to 2-fold axes of rotational symmetry
axis or to be perpendicular to mirror symmetry planes. The standard
convention is that c < a < b.
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Hexagonal
The hexagonal unit cell can be defined by three axes. Two of them,
labeled a, are equal in length and at an angle of 120º to one
another. The third axis, labeled c, is perpendicular to the a axes
and of a different length. The c axis corresponds to a 3-fold or
6-fold symmetry axis.
To highlight the presence of 3-fold or 6-fold symmetry, usual practice
is to include a third a axis at 120° to the other two, and correspondingly
to use a redundant 4th integer in the Miller index. (The extra integer
is placed in the 3rd position and equals the negative of the sum
of the first two).
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Trigonal
A subgroup of the hexagonal crystal system characterized
by one three-fold symmetry axis. The remaining crystal systems in
the hexagonal crystal system have a six-fold symmetry axis
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| Monoclinic
The monoclinic
unit cell is defined by three axes, a, b, and c, of unequal lengths.
The angles between the a and b axes and between the c and b axes
are exactly 90°. The b axis is chosen to correspond to a 2-fold
axis of rotational symmetry axis or to be perpendicular to a mirror
symmetry plane. The standard convention for assigning the other
axes is c < a. The unit cell is also chosen so that the angle
beta, lying between the a and c axes, is obtuse (between 90º
and 180º).
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Tetragonal
The tetragonal
unit cell is defined by three axes. Two of them, labeled a, are
equal in length; and the c axis is of a different length. All angles
between axes are 90°.
The c axis corresponds to a symmetry axis of either 4-fold rotation
or 4-fold rotation inversion. The c axis can be either longer or
shorter than the a axes.
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Isometric,
Cubic
The isometric
(=cubic) unit cell is defined by three axes of equal length, all
labeled a. All angles between axes are 90°. Because of the equivalence
of all axes and angles The isometric system can contain combinations
of many different types of symmetry elements: 2-, 3-, and 4-fold
rotation axes, 3- and 4-fold rotation-inversion axes, mirror planes,
and centers of symmetry
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| These
basic features (Unit Cell Axial Constraints,
and Allowed Symmetry Operations) of the Seven Crystal Systems |
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basic features (Unit Cell Axial Constraints, and Allowed
Symmetry Operations) of the Seven Crystal Systems are outlined in
the following table :
System Constraints Operations
Triclinic None 1, -1
Monoclinic alpha = gamma = 90 1, -1, 2, -2(m)
Orthorhombic alpha = beta = gamma = 90 1, -1, 2, -2(m)
Trigonal alpha = beta = 90 gamma = 120 a = b 1, -1, 2, -2(m), 3,
-3
Hexagonal alpha = beta = 90 gamma = 120 a = b 1, -1, 2, -2(m), 3,
-3, 6, -6
Tetragonal alpha = beta = gamma = 90 a = b 1. -1, 2, -2(m), 4, -4
Cubic alpha = beta = gamma = 90 a = b = c 1. -1, 2, -2(m), 3, -3,
4, -4
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Symmetry
Operations
Symmetry is an important property of mineral crystals. The symmetry
we see in the external shape of a crystal results from the symmetry
of the mineral's atomic structure. The symmetry of the crystal may
not be obvious because of irregular growth; however, the angles
between the crystal's faces will always be related by the true symmetry
of the mineral.
A symmetry operation is a transposition of an object These may be
of three distinct types: rotations, inversions (including roto-inversions
i.e. improper rotations), or translations, or combinations thereof.
Symmetry groups made up of rotation and inversion operations, are
called the point groups, each of which is one of the 32 crystal
classes.
Groups made up from all three types of operation give rise to the
230 space groups.
Rotations
Permissible
rotations - Proper
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1-fold 360 º I Identity
2-fold 180 º 2
3-fold 120 º 3
4-fold 90 º 4
6-fold 60 º 6
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Permissible rotations - Improper (result in enantiomorphs).
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1-fold 360 º + i i
2-fold 180 º + i -2 = m
3-fold 120 º + i -3
4-fold 90 º + i -4
6-fold 60 º + i -6
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Translations
Permissible translations are unit cell translations or fractions
thereof that are consistent with the rotational symmetry (e.g. 1/2,
1/3, 1/4, and 1/6), plus combinations |
Allowable
Rotations.
Illustrated
below are stereographic projections of general crystal forms that
have the allowable rotation operations that are consistent with
translation symmetry.
Each of these ten allowable rotations generates, by itself, a unique
point group. In addition, there are 22 possible combinations of
rotation operations, giving a total of 32 possible 3-dimensional
point groups. Each point group corresponds to different crystal
class. Each crystal class places constraints on the axial geometry
such that each of these 32 classes may be associated into one of
the 7 crystal systems, each having different constraints on the
axial lengths and inter-axial angles.
In determining a point group, one must have diagnostic faces such
as the general form. For example, if you have a cube, it can occur
in several point groups as a special form. Thus there is no way
to uniquely determine the point group.
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Each of the 10 allowed proper and improper rotations is, by itself,
one of the 32 point groups, and we have seen stereographic projections
of each of these. The additional 22 point groups are generated by
combinations of these 10 symmetry operations. These are illustrated
below
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The
32 Crystal Classes
The symbols used to represent the symmetry elements are combined to
represent each of the crystal classes. The rules for interpreting
these combined symbols are:
(1) if a mirror plane symbol "m" immediately follows a rotational
axis symbol, the rotational axis lies in the mirror plane;
(2) if a slash "/" separates the rotational axis and the
mirror symbol, the mirror plane is perpendicular to the axis;
(3) if two or three mirror plane symbols or two or three rotational
axis symbols immediately follow one another, they are perpendicular
to one another.
These conventions are not rigorously followed for the cubic system
in which the symmetry relationships are more complex.
Following the rules of groups, there is a limited number of ways in
which the 10 proper and im proper rotations can be combined to form
groups, that is, there are 32 possible combinations to form groups.
These are the 32 3-dimensional point groups which correspond to the
32 Crystal Classes. Each of the 32 crystal classes can be ascribed
to one of the 6(7) crystal systems. Triclinic:
1, B1
Monoclinic: 2, m, 2/m
Orthorhombic: 2/m 2/m 2/m, 222, mm2
Tetragonal: 4, B4, 4/m, 4mm, B42m, 422, 4/m 2/m 2/m
Hexagonal: 3, B3, 3m, B3 2/m, 32, 6, B6, 6/m, 6mm, B6m2, 622, 6/m
2/m 2/m
Isometric or cubic: 23, 2/m B3, 4/m B3 2/m, B43m, 432 |
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Morphology (Crystal shape) |
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| Crystal
Morphology (Crystal shape)
The formalism of crystal morphology (shapes). The morphology of
a perfect crystal (i.e., our wooden blocks), in general, reflects
the maximum symmetry that a crystal can have. That is, there may
be portions of the crystal structure that violate some of the apparent
symmetry, but if high-symmetry forms (crystal faces) are present,
the crystal is likely to have high symmetry. (e. g., if the crystal
is a cube, it is most probably isometric.)
A crystal form is a crystal face plus its symmetric equivalents.
For example, a cube is a crystal form made up of six symmetrically
equivalent faces.
A special form is a crystal form that is repeated by the symmetry
operations onto itself so that there are fewer faces than the order
of the point group. The projections of special forms or special
faces will lie on symmetry operations in our stereographic projections.
A general form is one that is not repeated onto itself by the symmetry
operations so that it has the same number of faces as the order
of the group.
Forms are either general or special. In addition to being special
or general, forms may also be open or closed.
A closed form is one that encloses a volume; (e.g., a cube, tetrahedron,
octahedron, etc). A closed form may then be the only form present
on a perfect crystal.
An open form is one that does not enclose a volume; (e.g., prism,
pinacoid, etc.). A crystal that has an open form must have more
than one form present.
An illustration
of several crystal shapes is given in this chapter.
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Groups
A set of elements (operations) is a group if the following properties
hold:
1. Closure: combining any two elements of the group gives a third
element of the group.
2. Association: For any three elements of the group (ab)c = a(bc).
Note: not necessarily commutative (ab = ba). If it is true for all
members of the group, the group is called Abelian.
3. Identity: There is an element of the group, I, such that aI = Ia
= a for each element of the group.
4. Inverses: For each element, a, there is another element, b, such
that ab = I = ba.
The order of the group is the number of elements of the group. We
will first consider groups made up of all allowable combinations of
rotation and inversion operations to make up the point groups in two
dimensions and in three dimensions. There are ten possible 2-dimensional
point groups and 32 possible 3-dimensional point groups. Each of these
32 3-D point groups corresponds to one of the crystal classes. We
will then combine these with the possible translation operations to
form the 17 2-dimensional space groups and 230 3-D space groups.
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Miller
Indices
Three
integers (sometimes four in the hexagonal crystal system) used to
indicate the orientation of a plane or direction in a crystal such
as those corresponding to a crystal face or cleavage. The three
numbers are related to the three (or four) axes that define the
unit cell. The three numbers are enclosed in parentheses, as (111),
to indicate a single face or plane. They are enclosed in braces,
as {111}, to indicate a crystal form (set of planes related by symmetry).
They are enclosed in brackets, as [111], to indicate a direction.
Planes
A crystal face (or plane) cuts the crystallographic axes
at , 2, and 1. These intersections are called intercepts. Because
symbols are cumbersome, these intercepts are inverted and all fractions
are cleared, as shown below. , 1/2, 1/1 = 0, 1/2, 1 = (0 1 2)
These operations give us the Miller indices of any plane. These
planes may be a cleavage plane, a crystal face, or any diffracting
X-ray plane. Thus, a cube face is (0 0 1), the octahedron (1 1 1),
and a dodecahedron (1 1 0). There may also be negative (0 0 -1)
Miller indices. Miller indices are always in relation to the crystallographic
axes, not any orthogonal system of convenience. The general form
for Miller indices is (h k l).
For hexagonal axes the general form is (h k i l).
In general,
crystal faces, diffracting X-ray planes,and cleavages will be denoted
with simple parentheses, e.g. (2 1 0). However, a crystal form (a
face plus its symmetric equivalents will be denoted with curly brackets,
e.g. {2 1 0}. Hence the cube, {1 0 0} is made up of faces (1 0 0),
(0 1 0), (0 0 1), (-1 0 0),(
0 -1 0), and (0 0 -1).
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Directions
Directions in a crystal are merely the vector components with respect
to the crystallographic axes that have been reduced to the smallest
whole numbers. These are given in square brackets [1 3 0], [0 1
0], etc. In general, the [1 1 1] is not normal to the (1 1 1), except
for isometric (cubic) crystals. |
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