THE DISTRIBUTION OF CRYSTAL FACES IN ZONES, AND THE MODE OF CONSTRUCTING
A PLAN OF THE FACES.
It will have been clear from the facts related in the previous chapters
that the salient property possessed by all crystals, when ideal
development is permitted by the circumstances of their growth, and the
substance is not one of unusual softness or liable to ready distortion,
is that the exterior form consists of and is defined by truly plane
faces inclined to each other at angles which are specific and
characteristic for each definite chemical substance; and that these
angles are in accordance with the symmetry of some particular one of the
thirty-two classes of crystals, and are such as cause the indices of the
faces concerned to be rational small numbers.
It will also be clear that, given the presence of any face other than
the three axial planes, the symmetry of the class—supposing the crystal
to exhibit some development of symmetry and not to belong to class 32,
the general case possessing no symmetry—will require the repetition of
this face a definite number of times on other parts of the crystal. Such
a set of faces possessing the same symmetry value we have already learnt
to call a “Form,” and the faces composing it will have the same
Millerian index numbers in their symbols, but differently arranged and
with negative signs over those which relate to the interception of the
back part of the _a_ axis, the left part of the _b_ axis, or the lower
part of the vertical _c_ axis; that is, parts to the front and right,
and above, the centre of intersection of the three crystal axes are
considered as the positive parts of those axes.
A form, if of general character, that is, if composed of faces each of
which is inclined to all three axes, will comprise more faces the higher
the symmetry. Thus, in the cubic system, the form shown in Fig. 21, the
hexakis octahedron, comprises as many as forty-eight faces, all covered
by the form symbol {321}; while in the rhombic system the highest number
of faces in a form is eight, in the monoclinic only four, and in the
triclinic system two. It will also have become clear that the law of
rational indices limits the number of forms possible of any one type.
For instance, very few hexakis octahedra are known, the most frequently
occurring ones besides {321} being {421}, {531}, and {543}. Forms, of
any class, possessing higher indices than these are very rare,
especially in the systems of lower symmetry.
[Illustration:
FIG. 47.—The Spherical Projection.
]
We next come to a further very interesting fact about crystals. Let us
imagine a crystal, on which the faces are fairly evenly developed, to be
placed in the middle of a sphere of jelly, as indicated in Fig. 47
(reproduced from a Memoir by the late Prof. Penfield), so that the
centre or origin of the axial system of the crystal and the centre of
the sphere coincide. Let us now further imagine that long needles are
stuck through the jelly and the crystal, one perpendicular to each
crystal face, and so as to reach the centre. The crystal represented in
Fig. 47 is a combination of the cube _a_, octahedron _o_, and rhombic
dodecahedron _d_. If such a thing as we have imagined were possible, we
should find that the needles would emerge at the surface of the sphere
in points which would lie on great circles, that is, on circles which
represent the intersection of the sphere by planes passing through the
centre. Moreover, the points would be distributed along these circles at
regularly recurring angular positions, corresponding to the symmetry of
the crystal. If the crystal belonged to one of the higher systems of
symmetry, it would happen that four of the points on at least one of
these great circles, and possibly on three of them, would be 90° apart,
that is, would be at the ends of rectangular diameters, which would most
likely be the axes of reference. The other points would be distributed
symmetrically on each side of these four points.
The great circles on which the points are thus symmetrically
distributed—and they may legitimately be taken to represent the faces,
for tangent planes to the sphere at these points would be parallel to
the faces—are known as “zone circles,” and the faces represented by the
points on any one of them form a “zone.” Now a zone of faces has this
practical property, that when the crystal is supported so as to be
rotatable about the zone axis—which is parallel to the edges of
intersection of all the faces composing the zone, and is the normal to
the plane of the great circle representing the zone—and a telescope is
directed towards the crystal perpendicularly to the zone axis, while a
bright object such as an illuminated slit is arranged conveniently so as
to be reflected from any face of the crystal into the telescope, an
image of it being thus visible in the latter, then it will be found that
on rotating the crystal a similar image will be seen reflected in the
telescope from every face of the zone in turn. Moreover, when the
crystal is mounted on a graduated circle, the angle of rotation between
the positions of adjustment to the cross-wires of the telescope of any
two successive images, reflected from adjacent faces of the crystal, is
actually the angle between the two points representing the faces
concerned on the zone circle, and is the supplement of the internal
dihedral angle between the two crystal faces themselves. It is, in fact,
the angle between the normals (perpendiculars) to the two faces, the
angle which is measured on the goniometer.
This is, indeed, the very simple principle of the reflecting goniometer,
invented by Wollaston in the year 1809, and which in its modern improved
form is the all-important principal instrument of the crystallographer’s
laboratory. The work with it consists largely in the measurement of the
angles between the faces in all the principal zones developed on the
crystal. The very fact, however, that crystal faces occur so absolutely
accurately in zones immeasurably lightens the labours of the
crystallographer, and is one of prime importance.
[Illustration:
FIG. 48.—The Reflecting Goniometer.
]
The most accurate and convenient modern form of reflecting goniometer,
reading to half-minutes of arc, and provided with a delicate adjusting
apparatus for the crystal, is shown in Fig. 48. It is constructed by
Fuess of Berlin.
The graduated circle _a_ is horizontal and is divided directly to 15′,
the verniers enabling the readings to be carried further either to
single minutes, which is all that is usually necessary, or to
half-minutes in the cases of very perfect crystals. The divided circle
is rotated by means of the ring _b_ situated below, and the reading of
the verniers is accomplished with the aid of the microscopes _c_. The
circle which carries the verniers is not fixed, except when this is done
deliberately by means of the clamping screw _d_, but rotates with the
telescope _e_ to which it is rigidly attached by means of an arm and a
column _f_. A fine adjustment is provided with the clamping arrangement,
so that the telescope can be adjusted delicately with respect to the
divided circle. Both telescope and collimator are rigidly fixed at about
120° from each other during the actual measurements. The collimator _g_
is carried on a column _h_ definitely fixed to one of the legs (the back
one in Fig. 48) of the main basal tripod of the instrument. The signal
slit of the collimator is carried at the focus of the objective about
the middle of the tube _g_, the outer half of the latter being an
illumination tube carrying a condensing lens to concentrate the rays of
light from the goniometer lamp on the slit. The latter is not of the
usual rectilinear character, but composed of two circular-arc jaws, so
that, while narrow in the middle part like an ordinary spectroscope
slit, it is much broader at the two ends in order to be much more
readily visible; the central part is narrow in order to enable fine
adjustment to the vertical cross-wire of the telescope to be readily and
accurately carried out. The shape of this signal-slit will be gathered
from the images of the slit shown in Fig. 61 (page 126) in Chapter X.
The telescope carries an additional lens _k_ at its inner, objective,
end, in order that when this lens is rotated into position the telescope
may be converted into a low power microscope for viewing the crystal and
thus enabling its adjustment to be readily carried out.
The crystal _l_ is mounted on a little cone of goniometer wax (a mixture
of pitch and beeswax) carried by the crystal holder. The latter fits in
the top of the adjusting movements, which consist of a pair of
rectangularly arranged centring motions, and a pair of cylindrical
adjusting movements; the milled-headed manipulating centring screws of
the former are indicated by the letters _m_ and _n_ in Fig. 48, and
those which move the adjusting segments are marked _o_ and _p_. The top
screw fixes the crystal holder.
The crystal on its adjusting apparatus can be raised or lowered to the
proper height, level with the axes of the telescope and collimator, by
means of a milled head at the base of the instrument, there being an
inner crystal axis moving (vertically only) independently of the circle.
Moreover, a second axis outside this enables the crystal to be rotated
independently of the circle, the conical axis of which is outside this
again. The two can be locked together when desired, however, by a
clamping screw provided with a fine adjustment _q_. Freedom of movement
of the crystal axis, unimpeded by the weight of the circle, is thus
permitted for all adjusting purposes, the circle being only brought into
play when measurement is actually to occur. With this instrument the
most accurate work can be readily carried out, and for ease of
manipulation and general convenience it is the best goniometer yet
constructed.
The idea of regarding the centre of the crystal as the centre of a
sphere, within which the crystal is placed (Fig. 47, page 62), gives
crystallographers a very convenient method of graphically representing a
crystal on paper, by projecting the sphere on to the flat surface of the
paper, the eye being supposed to be placed at either the north or south
pole of the sphere, and the plane of projection to be that of the
equatorial great circle. The faces in the upper hemisphere are
represented by dots which are technically known as the “poles” of the
faces, corresponding to the points where the needles normal to the faces
emerge from the imaginary globe, and all these points or poles lie on a
few arcs of great circles, which appear in the projection either also as
circular arcs terminating at diametrically opposite points on the
circumference of the equatorial circle, which forms the outer boundary
of the figure and is termed the “primitive circle,” or else, when the
planes of the great circles are at right angles to the equatorial
primitive circle, they appear as diametral straight lines passing
through the centre of the primitive circle.
Such a stereographic projection offers a comprehensive plan of the whole
of the crystal faces, which at once informs us of the symmetry in all
cases other than very complicated ones. A typical one, that of the
rhombic crystal of topaz shown in Fig. 22 (page 40), is given in Fig.
49.
It will happen in all cases of higher symmetry, as in that of topaz, for
instance, that the poles in the lower hemisphere will project into the
same points as those representing the faces in the upper hemisphere; but
in cases of lower symmetry, where they are differently situated, they
are usually represented by miniature rings instead of dots. From the
interfacial angles measured on the goniometer the relative lengths and
angular inclinations (if other than 90°) of the crystal axes can readily
be calculated, by means of the simple formulæ of spherical trigonometry;
and the stereographic projection constructed from the measurements as
just described proves an inestimable aid to these calculations, by
affording a comprehensive diagram of all the spherical triangles
required in making the calculations.
[Illustration:
FIG. 49.—Stereographic Projection of Topaz.
]
The relative axial lengths _a_ : _b_ : _c_ (in which _b_ is always
arranged to be = 1), and the axial angles α (between _b_ and _c_), β
(between _a_ and _c_), and γ (between _a_ and _b_), form the “elements”
of a crystal. These, together with a list of the “forms” observed, and a
table of the interfacial angles, define the morphology of the crystal,
and are included in every satisfactory description of a crystallographic
investigation. They are preceded by a statement of the name and chemical
composition and formula of the substance, the system and the class of
symmetry, and the habit or various habits developed by crystals from a
considerable number of crops. An example of the mode of setting out such
a description will be found on pages 157 to 160.
Having thus made ourselves acquainted with the real nature of the
distribution of faces on a crystal, and learnt how the crystallographer
measures the angles between the faces by means of the reflecting
goniometer, plots them out graphically on a stereographic projection,
and calculates therefrom the “elements” of the crystal, it will be
convenient again to take up the historical development of the subject so
far as it relates to crystal forms and angles, and their bearing on the
chemical composition of the substance composing the crystal, by
introducing the reader to the great work of Mitscherlich, whose
influence in the domain of chemical crystallography was as profound as
that of Haüy proved to be as regards structural crystallography.