Lecture IV – Crystallographic Axes and Crystal Systems
Axes - serve as a reference frame
1)
2)
3)
4)
3 axes - except in hexagonal system (4)
mutually perpendicular if possible
coincide with symmetry axes of crystal or normal to symmetry planes
if crystals lack sufficient symmetry axes or planes, the axes are
selected to coincide with the intersection of faces with the largest
area
Typical Axes
a-axis is intermediate in length and points toward observer
c-axis is the longest and points up
b-axis is the shortest and points sideways
, ,  are the angles between the axes
Characteristics of different Crystal Systems
Triclinic
Monoclinic
a≠b≠c
 ≠  ≠  ≠ 90°
a≠b≠c
 ≠ 90°
 =  = 90°
- a-axis is inclined
- b-axis – the 2-fold axis
- c-axis is the elongated direction
Orthorhombic
a≠b≠c
 =  =  = 90°
- a-axis and b-axis are arbitrary
- c-axis is the long axis
Tetragonal
a1 = b ≠ c
ll
a2
 =  =  = 90°
- a-axis and b-axis are 2-fold axes of rotation
- c-axis is 4-fold axis of rotation
Hexagonal
a1 = a2 = a3 ≠ c
- a1, a2, a3 all 120° apart and  to c-axis
- American – hexagonal and trigonal subsystems
Isometric (cubic)
a1 = a2 = a3
-  =  =  = 90°
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all crystals have four 3-fold or 3-fold symmetry axes
Unit Cell –
Miller Indices
Miller Indices
- most common method used to express intercepts of crystal faces
upon crystal axes
- Miller indices of a face consist of a series of whole numbers that
have been derived from the intercepts by their inversion and if
necessary, the clearing of the fractions
- Usually has 3 numbers a, b and c axes (hexagonal system has 4
numbers)
- we omit letters and signs for brevity
example: for figure immediately above on previous page (Fig. 5.29 from
Klein – p. 198)
The intercepts for the shaded face are:
1a,1b,1c  1/1, 1/1, 1/1 inverted = 1/1, 1/1, 1/1 = (111)
The intercepts for the smaller face above the shaded one are:
2a,2b, 2/3c  2/1, 2/1, 2/3 invert and clear fractions = 1/2, 1/2, 3/2 = (113)
- use commas in Miller Indexes only when there are 2-digit numbers
(e.g. (1,14,3)
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- use bar over the number (111) for faces intersecting the negative end of
an axis (see Fig. 5.31)
Symbols used
( ) – refers to specific face
{ } – refers to forms (see Forms below)
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(1) ≠ 1 rotoinversion axis
If we don’t know the actual intercepts we use (hkl) – h,k and l are the
reciprocals of rational but undefined intercepts along a, b and c.
(hkl) – cuts all three axes
(h0l); (hk0); (0kl) all cut only 2 axes
there are no (h00) etc. faces as h is taken to be 1,  (100); (010); (001)
Hexagonal crystals have 4 axes (a1, a2, a3, and c) - Bravais-Miller index
- we use (hkil) in hexagonal if we don’t know intercept lengths. The sum
of h, k and i is always equal to zero (h+k+I=0)
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e.g. 1010 = 1+0+1 = 0 (see Fig. 5.32 on page 200)
Form
A group of crystal faces, all of which have the same relation to the
elements of symmetry (see Fig. 5.38, pg. 204) and have the same chemical
and physical properties because all are underlain by like atoms in the same
geometrical arrangement.
- it is important to understand the relation between form and the symmetry
elements of a crystal
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- Triclinic 1
a (111) face will repeat through the center of
symmetry to form the two faces of a pinacoid or parallelohedron
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- Isometric 4/m 3 2/m a (111) face will replicate 7 times to produce
an octahedral form or an octahedron
* The generation of the octahedron from a single (111) face demonstrates
that the number of faces that belong to a form is determined by the
symmetry of the crystal class.
The group of similar crystal faces or form can be indicated using Miller
Indices enclosed in curly brackets {111} and is usually written using
positive numbers if at all possible.
Form names are not always used as the Miller Indices and Point group
symmetry uniquely describe a crystal. That said, some crystal forms such
as the pyritohedron are more commonly used than the corresponding Miller
Indices.
General Form – a form in each crystal class in which the faces intersect all
the crystallographic axes at different lengths {hkl}.
- There are 48 forms in all of which 32 general forms occur in the 32
crystal classes
The other 16 forms are special forms
- 10 of these are closed forms of the isometric system
- 6 of these are open prisms of the hexagonal and tetragonal
system
Rules about General Forms
- in the triclinic, monoclinic and orthorhombic systems the general form is
{111} because the distance along each axis is different.
- in higher symmetry systems with the same unit distance on 2 or more
axes (e.g. tetragonal), a general form must intersect like axes at different
multiples of the unit length.
- {121} tetragonal
- {123} isometric
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- in the example of the {111} face for the 1 and 4/m 3 2/m symmetry, we
developed a 2-face and 8-face form.
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- the 2-face {111} for the 1 is an open form because it does not enclose
space. In order to completely describe a crystal, there must be at least 2
open forms.
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- the 8-faces {111} 4/m 3 2/m is a closed form because it encloses space.
A crystal can be completely described by a single closed form, or the
combination of a closed form and one or more open forms.
Brief Description of Forms
Pedion (monohedron) – a single face comprising a form
Pinacoid (parallelohedron) – an open form made of 2 parallel faces
Dome (dihedron) – an open form of 2 nonparallel faces that are symmetric
wrt a mirror plane
Sphenoid ( dihedron) – 2 nonparallel faces symmetric wrt a 2-fold axis
Prism – open form of 3, 4, 6, 8, or 12 faces parallel to the same axis
(except for monoclinic system, this axis is a crystallographic axis)
Pyramid – open form of 3, 4, 6, 8, or 12 nonparallel faces meeting at a
point
Dipyramid – closed form of 6, 8, 12, 16 or 24 faces (the same as 2
pyramids separated by a horizontal mirror)
Trapezohedron – closed form of 6, 8 or 12 faces with 3, 4 or 6 upper faces
offset from the 3, 4 or 6 lower faces. These are the result of 3-, 4-, or 6fold axis combined with a perpendicular 2-fold axis. (also a 24 face
isometric form)
Scalenohedron – closed form with 8 or 12 faces is symmetric pairs
(tetragonal and hexagonal system) related by a 3- or 4-fold inversion axis.
Rhombohedron – closed form of 6 faces (3 on top and 3 on the bottom)
that are offset by 60°.
Disphenoid – (rhombic or tetragonal tetrahedron) closed form of 2 upper
and 2 lower faces offset by 90°.
Plus – specialized forms in the isometric system (cube, octahedron,
dodecahedron, tetrahedron, pyritohedron, etc.)
Zones – a group of faces with parallel intersecting edges
- a line through the center of a crystal that is parallel to the lines of face
intersections is called the zone axis. It is indicated by [001].
- you can determine the zone axis of 2 intersecting faces (110 and 010) as
follows
110110 (110 written 2 times)
010010 (010 written 2 times)
multiply 2 by three below etc.
0-0;0-0;1-0
results in [001] zone axis