HOW CRYSTALS ARE DESCRIBED. THE SIMPLE LAW LIMITING THE NUMBER OF
POSSIBLE FORMS.
The most wonderful of all the laws relating to crystals is the one
already briefly referred to which limits and regulates the possible
positions of faces, within the lines of symmetry which have been
indicated in the last chapter. Having laid down the rules of symmetry,
it might be thought that any planes which obey these laws, as regards
their mode of repetition about the planes and axes of symmetry, would be
possible. But as a matter of fact this is not so, only a very few planes
inclined at certain definite angles, repeated in accordance with the
symmetry, being ever found actually developed. The reason for this is of
far-reaching importance, for it reveals to us the certainty that a
crystal is a homogeneous structure composed of definite structural units
of tangible size, probably the chemical molecules, built up on the plan
of one of the fourteen space-lattices made known to us by Bravais, and
to be referred to more fully in Chapter VIII. In order to render this
fundamental law comprehensible, it will be essential to explain in a few
simple words how the crystallographer identifies and labels the numerous
faces on a crystal, in short, how he describes a crystal, in a manner
which shall be understood immediately by everybody who has studied the
very simple rules of the convention.
It is a matter of common knowledge that the mathematical geometrician
defines the position of any point in space with reference to three
planes, which in the simplest case are all mutually at right angles to
each other like the faces of a cube, and which intersect in three
rectangular axes _a_, _b_, _c_, the third _c_ being the vertical axis,
_b_ the lateral one, and _a_ the front-and-back axis. The distances of
the point from the three reference planes, as measured by the lengths of
the three lines drawn from the point to the planes parallel to the three
axes of intersection, at once gives him what he calls the “co-ordinates”
of the point, which absolutely define its position. In the same way we
can imagine three axes drawn within the crystal, by which not only the
position of any point on any face of the crystal may be located, but
which may be used more simply still to fix the position of the face
itself. The directions chosen as those of the three axes are the edges
of intersection of three of the best developed faces.
If there are three such faces inclined at right angles they would be
chosen in preference to all others, as they would certainly prove to be
faces of prime significance as regards the symmetry of the crystal. If
there are no such rectangularly inclined faces developed on the crystal,
then the three best developed faces nearest to 90° to each other are
chosen, the two factors of nearness to rectangularity and excellence of
development being simultaneously borne in mind in making the choice of
axial planes, and discretion used.
[Illustration:
FIG. 38.—The Cube and its Three Equal Rectangular Axes.
]
[Illustration:
FIG. 39.—Tetragonal Prism and its three Rectangular Axes.
]
If the crystal belong to the cubic, tetragonal, or rhombic systems, for
instance, three faces meeting each other rectangularly are possible
planes on the crystal, and will very frequently be found actually
developed; such would obviously be chosen as the axial planes. The edges
of the cube, or of the tetragonal or rectangular rhombic prism, will be
the directions of the crystallographic axes in this case, and we can
imagine them moved parallel to themselves until the common centre of
intersection, the “origin” of the analytical geometrician, will occupy
the centre of the crystal, and the faces of the latter be built up
symmetrically about it. When the crystal is cubic, the three axes will
be of equal length as shown in Fig. 38; if tetragonal, the two
horizontal axes will be equal, but will differ in length from the
vertical axis, as represented in Fig. 39. If the crystal be rhombic, all
three axes will be of different lengths, as indicated in Fig. 40, which
represents the axes and axial planes of an actual rhombic substance,
topaz, for which the lateral axis _b_ and vertical axis _c_ are nearly
but not quite equal, while the front-and-back axis _a_ is very
different.
When the crystal is of monoclinic symmetry, as in Fig. 41, three axes
will similarly be found as the intersection of three principal parallel
pairs of faces, but two of them will be inclined at an angle other than
90° to each other, while the third, the lateral one in Fig. 41, will be
at right angles to those first two and to the plane containing them;
moreover, all three are unequal in length. In the case of a triclinic
crystal, shown in Fig. 42, however, there can be no right angles, and
the intersections of three important faces meeting each other at angles
as near 90° as possible are chosen as the axes, regard being had to both
factors of approximation to rectangularity and importance of
development. These triclinic axes are the most general type of crystal
axes, for not only are the angles not right angles, but the lengths of
the axes are also unequal.
[Illustration:
FIG. 40.—Axial Planes of a Rhombic Crystal.
]
[Illustration:
FIG. 41.—Axial Planes of a Monoclinic Crystal.
]
[Illustration:
FIG. 42.—Axial Planes of a Triclinic Crystal.
]
[Illustration:
FIG. 43.—Hexagonal Prism of the First Order and its Four Axes.
]
[Illustration:
FIG. 44.—Hexagonal Prism of the Second Order.
]
[Illustration:
FIG. 45.—The Rhombohedron and its Three Equal Axes.
]
The cases of the hexagonal and trigonal systems are somewhat special.
The hexagonal has four such axes, as represented in Fig. 43, the lines
of intersection of the faces of the hexagonal prism closed by a pair of
perpendicular terminal planes, namely, one vertical axis parallel to the
vertical edges, and three horizontal axes inclined at 120° to each
other, and parallel to the pair of basal plane faces, equal to each
other in length, but different from the length of the vertical axis. The
hexagonal axial-plane prism shown in Fig. 43 is known as one of the
first order. The hexagonal prism corresponding to the tetragonal one of
Fig. 39, in which the axes emerge in the centres of the faces, is said
to be of the second order, and is shown in Fig. 44. The trigonal system
of crystals is best described with reference to three equal but not
rectangular axes, parallel to the faces of the rhombohedron, one of the
principal forms of the system, so well seen in Iceland spar, and
illustrated in Fig. 45. The rhombohedron may be regarded as a cube
resting on one of its corners (solid angles), with the diagonal line
joining this to the opposite corner vertical, and the cube then deformed
by flattening or elongating it along the direction of this diagonal. The
edges meeting at the ends of this vertical diagonal are then the
directions of the three trigonal crystallographic axes.
In this last illustration the vertical direction of the altered diagonal
is that of the trigonal axis of symmetry, for the rhombohedron is
brought into apparent coincidence with itself again if rotated for 120°
round this direction. But although a symmetry axis, this is not a
crystallographic axis of reference. It is not shown in Fig. 45,
therefore, but is given in Fig. 19. On the other hand, the singular
vertical axis of reference of the tetragonal and hexagonal systems is
identical with the tetragonal or hexagonal axis of symmetry of these
systems, and the three crystallographic axes of reference of the cube
are identical with the three tetragonal axes of symmetry of the cubic
system. In the rhombic system also, the three rectangular axes of
reference are identical with the three digonal axes of symmetry, and in
the monoclinic system the one axis of reference which is normal to the
plane of the two inclined axes is the unique digonal axis of symmetry of
that system.
Having thus evolved a scientific scheme of reference axes for the faces
of a crystal, it is necessary in all the systems other than the cubic
and trigonal, in which the axes are of equal lengths, to devise a mode
of arriving at the relative lengths of the axes; for on this depends the
mode of determining the positions of the various faces, other than the
three parallel pairs (or four in the case of the hexagonal system)
chosen as the axial planes. This is very simply done by choosing a
fourth important face inclined to all three axes, when one of this
character is developed, as very frequently happens, as the determinative
face or plane fixing the unit lengths of the axes. When no such face is
present on the crystal, two others can usually be found, each of which
is inclined to two different axes, so that between them all three axial
lengths are determined. The faces of the octahedron, of the primary
tetragonal pyramid and the primary rhombic pyramid, and of the
corresponding forms of the other systems, are such determinative planes,
fixing the lengths of the axes. This fact will be clear from the typical
illustration of the most general of these primary or “parametral” forms,
the triclinic equivalent of the octahedron, given in Fig. 46, the faces
being obviously obtained by joining the points marking unit lengths of
the three axes.
[Illustration:
FIG. 46.—Triclinic Equivalent of the Octahedron.
]
Having thus settled the directions of the crystallographic axes and
their lengths, it is the next step which reveals the remarkable law to
which reference was made at the opening of this chapter. For we find
that all other faces on the crystal, however complicated and rich in
faces it may be, cut off lengths from the axes which are represented by
low whole numbers, that is, either 2, 3, 4, or possibly 5, and very
rarely more than 6 unit lengths. By far the greater number of faces do
not cut off more than three unit lengths from any axis. Prof. Miller of
Cambridge, in the year 1839, gave us a most valuable mode of labelling
and distinguishing the various faces by a symbol involving these three
values, employed, however, not directly but in an indirect yet very
simple manner. If _m_, _n_, _r_ be the three numbers expressing the
intercepts cut off by a face on the three axes, _a_, _b_, _c_
respectively, and if the Millerian index numbers be represented by _h_,
_k_, _l_, then—
_m_ = _a_/_h_, _n_ = _b_/_k_, _r_ = _c_/_l_,
or, _h_ = _a_/_m_, _k_ = _b_/_n_, _l_ = _c_/_r_.
Each figure or “index” of the Millerian symbol is thus inversely
proportional to the length of the intercept on the axis concerned. The
intercepts themselves are used as symbols in another mode of labelling
crystal faces, suggested by Weiss, but this method proves too cumbersome
in practice.
The Millerian symbol of a face is always placed within ordinary curved
brackets ( ), but if the symbol is to stand for the whole set of faces
composing the form, the brackets are of the type { }. Thus the
Millerian symbol of the fourth face (that in the top-right front
octant), determinative of the unit axial lengths, is (111), as shown in
Fig. 46, the face in question being marked with this symbol; while the
symbol {111} indicates the set of faces of the whole or such part of the
double pyramid as composes the unit form. In the triclinic system this
form only consists of the face (111) and the parallel one (̄1̄1̄1), but
in the case of the regular octahedron of the cubic system it embraces
all the eight faces. The triclinic octahedron, Fig. 46, is thus made up
of four forms of two faces each. A negative sign over an index indicates
interception on the axis _a_ behind the centre, on the axis _b_ to the
left of the centre, or on the vertical axis _c_ below the centre.
To take an actual example, suppose a face other than the primary one to
make the intercepts on the axes 4, 2, 1; in this case _h_ = _a_/4, _k_ =
_b_/2, and _l_ = _c_/1, that is, when referred to the fundamental
primary form for which _a_, _b_, _c_ are each unity, _h_ = ¼, _k_ = ½,
_l_ = 1, or, bringing them to whole numbers by multiplying by 4, _h_ =
1, _k_ = 2, _c_ = 4, and the symbol in Millerian notation is (124).
Again, suppose we wish to find the intercepts on the three cubic axes
made by the face (321) of the hexakis octahedron shown in Fig. 21. To
get each intercept we multiply together the two other Millerian indices,
and if necessary afterwards reduce the three figures obtained to their
simplest relative values. For the face (321) we obtain 2, 3, 6. This
means that the face (321) in the top-right-front octant of the hexakis
octahedron cuts off two unit lengths of axis _a_, three unit lengths of
axis _b_, and six unit lengths of axis _c_. No fractional parts thus
ever enter into the relations of the axial lengths intercepted by any
face on a crystal, and the whole numbers representing these relations
are always small, the number 6 being the usual limit.
This important law is known as the “Law of Rational Indices,” and is the
corner-stone of crystallography. A forecast of it was given in Chapter
III., in describing how it was first discovered by Haüy, and it was
shown how impressed Haüy was with its obvious significance as an
indication of the brick-like nature of the crystal structure. What the
“bricks” were, Haüy was not in a position to ascertain with certainty,
as chemistry was in its infancy, and Dalton’s atomic theory had not then
been proposed.
That Haüy had a shrewd idea, however, that the structural units were the
chemical molecules, and that while the main lines of symmetry were
determined by the arrangement of the molecules its details were settled
by the arrangement of the atoms in the molecules, is clear to any one
who reads his 1784 “Essai” and 1801 “Traité,” and interprets his
_molécules intégrantes_ and _élémentaires_ in the light of our knowledge
of to-day.
Before we pass on, however, to consider the modern development of the
real meaning of the law of rational indices, as revealed by recent work
on the internal structure of crystals, it will be well to consider
first, in the next chapter, a few more essential facts as to crystal
symmetry, and the current mode of constructing a comprehensive, yet
simple, plan of the faces present on a crystal.