**II.4 The indicatrix
of uniaxial crystals**

The direction-dependent
properties of light in minerals can best be imagined by a three-dimensional
geometric surface corresponding to the numeric values of a particular physical property in all
directions (in mathematic terms we call these vectors). An example of this we
have already seen in the ray velocity surface. Inversely related to the velocity
is the refractive index and the corresponding three-dimensional surface is known
as the indicatrix. This indicatrix has the shape of an ellipsoid. Every mineral has
its own characteristic indicatrix that qua shape and
orientation is determined by the symmetry of the crystal class to which the
mineral belongs.

In general we can distinguish
three main groups

1)
the indicatrix is a sphere. The refractive index is the same in
all vibrational directions. Such a crystal is called optically isotropic and is
only found in minerals belonging to the isotropic or cubic crystal
class.

2)
The indicatrix is a biaxial ellipsoid.

3)
The indicatrix is a triaxial ellipsoid

**Uniaxial crystals**

Uniaxial crystals belongs to the
second group, where the indicatrix has the shape of an
ellipsoid characterised by two perpendicular axes. In the equatorial plane the
refractive indices are the same for all vibrational directions in that plane.
This plane has the shape of a circle and is called the circular section. The
normal to the circular section is known as the optical axis. Because the
ellipsoid has only one such a circular section, minerals in this group are known
as optically uniaxial minerals. The refractive index corresponding to the
circular section is called ω (corresponding to the ordinary ray), the refractive
index parallel to the optical axis is called ε (corresponding to the
extraordinary ray, see calcite example in II.3).

When ε > ω the mineral is
called optically positive. The indicatrix has the
shape of a citrus (prolate) (Fig. 2.6). An example of this is the mineral
quartz

When ω > ε the mineral is
called optically negative. The indicatrix has the
shape of a mandarin (oblate) (Fig. 2.6). An example of this is the mineral
calcite that we have seen before (II.3)

**Fig. 2.6 The indicatrix for optically positive and optically negative
minerals**

Sometime other symbols are used
for the refractive index ε; for example n_{ε},
N_{ε}, E, n_{e} and N_{e}.
Similarly for ω; n_{ω}, N_{ω}, O, n_{o},
N_{o}

We can prove that the indicatrix can be determined from and has the same shape as
the ray velocity surface. Let’s have a look at the situation in Fig. 2.7, which
shows an section through the extraordinary ray wave surface XRY with the optical
axis given as OX.

**Fig. 2.7 Section through a biaxial indicatrix, giving the ellipse
X’R’Y’**

Here we can see
that:

OY = v_{o} =c/ω (1) ↔ ω = c/v_{o}

OX = v_{e} = c/ε (2) ↔ ε = c/v_{e}

Where vo ≥ ve i.e. ω ≤ ε (uniaxial,
positive)

The refractive index in the ellipse is represented by OX’ = 1/OY and OY’ = 1/OX. This ellipse has the same shape as the corresponding section through the ray velocity surface since, based on (1) and (2) we can say:

OY/ε = c/ω x 1/ε and OX/ω = c/ε x
1/ω

So: OY/ε = OX/ω = c/εω = constant !!!!!!

To the ray OR belongs a front
normal ON. Now we need to prove that ε' =c/ON, where ON = v_{ε'}.

For an ellipse we can state that the product ON' x OD = constant = ε x w

OD _{
┴} ON; OD = ε'; ON' x ε' =
ε x w ® ε' = εw/ON'
® ON' = εw/ε' (4)

cos RON = ON/OR and

cos R'ON' = ON'/OR' so ON/OR= ON'/OR' ® ON/ON' = OR/OR' (5)

substitute (3) in (5): ON/ON' = c/εw (6)

so ON' = εw/c x ON (7)

and ε' = εw/ON' = εw x c/(ON x εw) = c/ON