knowledge of the distribution of the phase. That is to say these relationships
are correct for all grain sizes, shapes and orientations, provided representative
sampling is done (Underwood, 1970, Exner, 2004). It is not even necessary to
separate individual crystals or grains. These values are termed stereologically
exact, unbiased or accessible parameters. They are the equivalent of mean
values, in that they only describe a limited aspect of the texture. Other textural
parameters are termed biased or inaccessible, but they are still measurable in
some materials, if assumptions are made about the crystal shape and fabric,
and will be described in later chapters (Underwood, 1970, Howard & Reed,
1998, Brandon & Kaplan, 1999).
The most widely used global parameter is the volumetric fraction of the
phase, V
V
. The subscript indicates that the volume of phase, V, is normalised
to unit volume, V. Delesse (1847) showed that the total area of a phase, A,ina
representative section normalised to unit area, A
A
is equal to V
V
. It may seem
counter-intuitive, but the fabric of the rock and the orientation of the section
are not important – any plane will give the same area fraction, which is equal to
V
V
. Crystal intersection areas are commonly measured at the same time as the
intersection length and width of the crystals and they can be used to calculate
the volumetric phase abundance. In addition, the fraction of a random line
that intersects the phase, L
L
is also equal to V
V
. However, many lines must be
measured to give adequate precision. Finally, the fraction of random or
equally spaced points that lie on the phase, P
P
is also equal to V
V
. This is the
basis of point counting. Hence
V
V
¼ A
A
¼ L
L
¼ P
P
Another common global parameter is the interface density, which is
the total surface area of grains, S, in a unit volume, S
V
. A test line or circle
is drawn on the section and the number of intersections with the surface,
P, counted (Figure 2.7). This value is divided by the test line length to
give P
L
.
Alternatively the number of grains intercepted, N, may be counted. This is
also divided by the length of the test line to give N
L
. Then
S
V
¼ 2P
L
¼ 4N
L
If there are several phases in a material, then the ‘contiguity’ of phase a, C
aa
is
the proportion of the interface area shared between like grains. It is determined
from the number of test points with aa boundaries, P
L(aa)
C
aa
¼ P
LðaaÞ
=P
LðTotalÞ
34 General analytical methods