Crystallographic Notation
(Intercepts,
Parameters and Indices)
a) Intercepts:
In
describing crystal faces, it is important to indicate the crystallographic axis
is intersected or not intersected by the faces.
Intercepts
are where crystallographic axes are met and intersected by crystal faces.
The
point of [1]intersection
of the crystallographic axis is called the origin and is given the value
0. Reference is usually taken from that
point. Based on this idea, intercepts
refer to the distance of a crystallographic axis from the origin to a crystal
face, within a unit cell. Usually, the
intercepts of a crystallographic axis in a unit form is always 1 but there are
a few variations.
For example in the figure below, the face (labeled 1)
intersects the a crystallographic
axis at a unit length (OS) and is parallel to the b and c crystallographic
axes. The intercepts of this face will
be 1a, ∞b, ∞c.
Figure
17: Face 1,
intercepting the a crystallographic axis and parallel to the other two,
(1a, ∞b, ∞c)
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The
next figure shows two faces; 2 and 3 which both intersect the a crystallographic axis and are parallel
to both the b and c crystallographic axes. Face 3 intersects the a crystallographic axis at a distance, OS1, which is further
than the distance at which face 1 intersects the same axis. Its distance is considered to be twice the
unit length and its intercepts will therefore be 2a, ∞b, ∞c, relative to
face 1 and the intercepts of face 1 will be 1a, ∞b, ∞c.
Figure
18: A comparison
between 2 faces, 2 and 3 which intersect the same crystallographic axis at
different lengths.
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The
next figure demonstrates a face, 4, which intersects both the a and b crystallographic axes at a unit length (OP and OP1 respectively) and
is parallel to the c crystallographic
axis.
Figure 19:
Face 4 intersects both the a and b crystallographic axes at a unit
length and is parallel to the c
crystallographic axis.
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Its intercept value can be given as 1a, 1b,
∞c.
In some cases, a face can intersect all three
crystallographic axes at unit length, in Figure
20 below, face 5
intercepts all crystallographic axes at unity (OK, OL and OM) giving it the
intercepts 1a, 1b, 1c.
The next demonstration shows face 6 also
intersecting all crystallographic axes but at unequal lengths compared to face
5. It intersects the a and c crystallographic axes at further lengths and the b
crystallographic axis at the same length as face 5.
For
face 5, the intercepts are 1a, 1b, 1c.
For
face 6, the intercepts are 2a, 1b, 2c
Figure
21: A comparison
between 2 faces, 5 and 6 which intersect all crystallographic axes at
different lengths.
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The
most widely used systems of notation are the Parameter system of [2]Weiss
and the Index System of [3]Miller.
a) Parameters
(The Parameter system of Weiss):
Parameters are defined as the ratios
of distances from the origin at which crystal faces cut crystallographic axes
(for short, Parameters are ratios of intercepts).
Weiss considered the crystallographic
axis to be labeled as a, b, c, for
unequal axes, a, a, c for 2 equal and
1 unequal axis and a, a, a, for all 3
axes being equal in length. (See Figure
13, Figure
14 and Figure
15).
The
intercept values that a face makes with the crystallographic axes are written
before the axes.
For
example: na, mb, pc, where n, m, p are
the parameter values when compared to the corresponding lengths cut off by the
unit form (III).
NB:
Unit
form is the form whose face intersects the crystallographic axes at unit
lengths (III) where reference is
taken for the intercepts of the other crystal forms on the same axes.
In
order to represent infinity or a face that is parallel to an axis (does not intersect
an axis), Weiss used the sign ∞.
Thus
following the Weiss parameters;
i.
a,
2b,
∞c, means that the face:
-
Cuts the a crystallographic axis at a distance of
1 unit (the same unit as the unit form),
-
Cuts the b crystallographic axis at twice the
unit of the unit form and is parallel to the c crystallographic axis.
ii.
a,
∞b,
∞c, means that the face:
-
Cuts the a
crystallographic axis at a distance of 1 unit (the same unit as the unit form)
and is parallel to the b and c crystallographic axes.
b) Indices
(The Index System of Miller or Miller Indices):
Miller
indices are a series of whole numbers used to indicate the precise positions or
relative distance at which crystal faces intersect different crystallographic
axes in space.
In this system of notation, the
indices or reciprocals of the parameters are used. They are written in the order of
crystallographic axes (that is the first number stands for the a crystallographic axis, the second
number stands for the b
crystallographic axis and the third number stands for the a crystallographic axis)
NB: In crystal systems that
have 4 crystallographic axes, the first three numbers stand for the horizontal
axes; a1, a2, a3 and the fourth number stands
for the vertical axis which is the c axis.
The whole numbers are taken from
parameters and are always given in their most simple form by inversion and clearing off fractions.
For
example, consider a face with Weiss parameters a, 2b, ∞c
-
The reciprocals of the
parameters will be
,
,
or
a,
b,
c
-
By inversion, we will
have
,
,
-
Simplifying it gives
the Miller indices as 2 1 0, read as two, one, naught (zero) and NOT two hundred and ten.
Similarly,
a face which intersects the a crystallographic
axis and is parallel to the b and c crystallographic axes, notated in
Weiss parameters as a,
∞b,
∞c, will have the Miller index I00, read as one,
zero, zero.
A
Weiss Parameter of
, 1,
, simplified into Miller Indices will be
(213); Two, one, three.
Miller indices are very important
because they show the relationship between a crystal face and a
crystallographic axis by giving the relative length of each crystallographic
axis from the origin (0) to the face in question. The larger the index, the smaller the
corresponding intercepts and the closer the face is to the origin. On the other
hand, the smaller the value, the farther the face is from the origin.
Take
for instance (2 1 3) means that the face in question intersects:
-
The a crystallographic axis at
the unit distance,
-
The b crystallographic axis at a unit
distance and
-
The c crystallographic axis at
the unit length.
Since crystallographic
studies at this level do not deal with measurements, most faces that cut all
three crystallographic axes are considered to be at unity (III).
Any
face which cuts the negative end of a crystallographic axis is indicated by
putting a bar (-) above the index of
that axis. For example (IĪI),
read as one, bar one, one, indicates that the face intersects
-
The a crystallographic axis at the positive
end,
-
The b crystallographic axis at the negative
end and
-
The c crystallographic axis at the positive
end.
The
notation of crystal form is a summary of the crystal faces and written in braces,
that is { } to differentiate them from Miller indices which are written in parenthesis,
that is ( ).
NB:
The number of faces in a Form depends on
the symmetry of the crystal (that is the level of regular repetition of the
same unit cell).
For example:
-
( I
I I ) represents a crystal face that cuts all three
crystallographic axes (a, b, c) at
their positive ends (known as a Pyramidal face),
-
While { I
I I } represents a crystal form with 8 faces with each
face cutting all three crystallographic axis (known as an Octahedron).
Notations
of crystal forms are called Form symbols.
Form
symbols have no bars on them. For
example { I Ī I
} is wrong!
[2]Christian Samuel Weiss (Feb. 26, 1780 – October
1, 1856) was a German Mineralogist
[3]William Hallowes Miller (Apr.
6, 1801 – May 20, 1880)
