THE MOLECULAR TACTICS OF
A CRYSTAL
_LORD KELVIN_
London
HENRY FROWDE
OXFORD UNIVERSITY PRESS WAREHOUSE
AMEN CORNER, E.C.
[Illustration]
New York
MACMILLAN & CO., 66 FIFTH AVENUE
THE
MOLECULAR TACTICS OF
A CRYSTAL
BY
LORD KELVIN, P.R.S.
PROFESSOR OF NATURAL PHILOSOPHY IN THE UNIVERSITY OF GLASGOW
AND FELLOW OF PETERHOUSE, CAMBRIDGE
_Being the Second ROBERT BOYLE LECTURE, delivered before
the Oxford University Junior Scientific Club
on Tuesday, May 16, 1893_
WITH TWENTY ILLUSTRATIONS
Oxford
AT THE CLARENDON PRESS
1894
Oxford
PRINTED AT THE CLARENDON PRESS
BY HORACE HART, PRINTER TO THE UNIVERSITY
ON THE MOLECULAR TACTICS OF A CRYSTAL
By LORD KELVIN, P.R.S.
§ 1. My subject this evening is not the physical properties of
crystals, not even their dynamics; it is merely the geometry of the
structure--the arrangement of the molecules in the constitution of a
crystal. Every crystal is a homogeneous assemblage of small bodies
or molecules. The converse proposition is scarcely true, unless in
a very extended sense of the term crystal (§ 20 below). I can best
explain a homogeneous assemblage of molecules by asking you to think
of a homogeneous assemblage of people. To be homogeneous every person
of the assemblage must be equal and similar to every other: they must
be seated in rows or standing in rows in a perfectly similar manner.
Each person, except those on the borders of the assemblage, must have
a neighbour on one side and an equi-distant neighbour on the other: a
neighbour on the left front and an equi-distant neighbour behind on the
right, a neighbour on the right front and an equi-distant neighbour
behind on the left. His two neighbours in front and his two neighbours
behind are members of two rows equal and similar to the rows consisting
of himself and his right-hand and left-hand neighbours, and their
neighbours’ neighbours indefinitely to right and left. In particular
cases the nearest of the front and rear neighbours may be right in
front and right in rear; but we must not confine our attention to the
rectangularly grouped assemblages thus constituted. Now let there be
equal and similar assemblages on floors above and below that which
we have been considering, and let there be any indefinitely great
number of floors at equal distances from one another above and below.
Think of any one person on any intermediate floor and of his nearest
neighbours on the floors above and below. These three persons must be
exactly in one line; this, in virtue of the homogeneousness of the
assemblages on the three floors, will secure that every person on the
intermediate floor is exactly in line with his nearest neighbours above
and below. The same condition of alignment must be fulfilled by every
three consecutive floors, and we thus have a homogeneous assemblage of
people in three dimensions of space. In particular cases every person’s
nearest neighbour in the floor above may be vertically over him, but
we must not confine our attention to assemblages thus rectangularly
grouped in vertical lines.
§ 2. Consider now any particular person _C_ (Fig. 1) on any
intermediate floor, _D_ and _D′_ his nearest neighbours, _E_ and _E′_
his next nearest neighbours all on his own floor. His next next nearest
neighbours on that floor will be in the positions _F_ and _F′_ in the
diagram. Thus we see that each person _C_ is surrounded by six persons,
_DD′_, _EE′_ and _FF′_, being his nearest, his next nearest, and his
next next nearest neighbours on his own floor. Excluding for simplicity
the special cases of rectangular grouping, we see that the angles of
the six equal and similar triangles _CDE_, _CEF_, &c., are all acute:
and because the six triangles are equal and similar we see that the
three pairs of mutually remote sides of the hexagon _DEFD′E′F′_ are
equal and parallel.
[Illustration: FIG. 1]
§ 3. Let now _A_, _A′_, _A″_, &c., denote places of persons of the
homogeneous assemblage on the floor immediately above, and _B_, _B′_,
_B″_, &c. on the floor immediately below, the floor of _C_. In the
diagram let _a_, _a′_, _a″_ be points in which the floor of _CDE_ is
cut by perpendiculars to it through _A_, _A′_, _A″_ of the floor above,
and _b_, _b′_, _b″_ by perpendiculars from _B_, _B′_, _B″_ of the floor
below. Of all the perpendiculars from the floors immediately above
and below, just two, one from each, cut the area of the parallelogram
_CDEF_: and they cut it in points similarly situated in respect to
the oppositely oriented triangles into which it is divided by either
of its diagonals. Hence if _a_ lies in the triangle _CDE_, the other
five triangles of the hexagon must be cut in the corresponding points,
as shown in the diagram. Thus, if we think only of the floor of _C_
and of the floor immediately above it, we have points _A_, _A′_, _A″_
vertically above _a_, _a′_, _a″_. Imagine now a triangular pyramid,
or tetrahedron, standing on the base _CDE_ and having _A_ for vertex:
we see that each of its sides _ACD_, _ADE_, _AEC_, is an acute angled
triangle, because, as we have already seen, _CDE_ is an acute angled
triangle, and because the shortest of the three distances, _CA_, _DA_,
_EA_, is (§ 2) greater than _CE_ (though it may be either greater than
or less than _DE_). Hence the tetrahedron _CDEA_ has all its angles
acute; not only the angles of its triangular faces, but the six angles
between the planes of its four faces. This important theorem regarding
homogeneous assemblages was given by Bravais, to whom we owe the whole
doctrine of homogeneous assemblages in its most perfect simplicity and
complete generality. Similarly we see that we have equal and similar
tetrahedrons on the bases _D′CF_, _E′F′C_; and three other tetrahedrons
below the floor of _C_, having the oppositely oriented triangles
_CD′E′_, &c. for their bases and _B_, _B′_, _B″_ for their vertices.
These three tetrahedrons are equal and heterochirally[1] similar to the
first three. The consideration of these acute angled tetrahedrons, is
of fundamental importance in respect to the engineering of an elastic
solid, or crystal, according to Boscovich. So also is the consideration
of the cluster of thirteen points _C_ and the six neighbours
_DEFD′E′F′_ in the plane of the diagram, and the three neighbours
_AA′A″_ on the floor above, and _BB′B″_ on the floor below.
§ 4. The case in which each of the four faces of each of the
tetrahedrons of § 3 is an equilateral triangle is particularly
interesting. An assemblage fulfilling this condition may conveniently
be called an ‘equilateral homogeneous assemblage,’ or, for brevity, an
‘equilateral assemblage.’ In an equilateral assemblage _C_’s twelve
neighbours are all equi-distant from it. I hold in my hand a cluster of
thirteen little black balls, made up by taking one of them and placing
the twelve others in contact with it (and therefore packed in the
closest possible order), and fixing them all together by fish-glue.
You see it looks, in size, colour, and shape, quite like a mulberry.
The accompanying diagram shows a stereoscopic view of a similar cluster
of balls painted white for the photograph.
[Illustration: FIG. 2.]
§ 5. By adding ball after ball to such a cluster of thirteen, and
always taking care to place each additional ball in some position in
which it is properly in line with others, so as to make the whole
assemblage homogeneous, we can exercise ourselves in a very interesting
manner in the building up of any possible form of crystal of the class
called ‘cubic’ by some writers and ‘octahedral’ by others. You see
before you several examples. I advise any of you who wish to study
crystallography to contract with a wood-turner, or a maker of beads for
furniture tassels or for rosaries, for a thousand wooden balls of about
half an inch diameter each. Holes through them will do no harm and may
even be useful; but make sure that the balls are as nearly equal to one
another, and each as nearly spherical, as possible.
[Illustration: FIG. 3.]
§ 6. You see here before you a large model which I have made to
illustrate a homogeneous assemblage of points, on a plan first given,
I believe, by Mr. William Barlow (_Nature_, December 20 and 27, 1883).
The roof of the model is a lattice-frame (Fig. 3) consisting of two
sets of eight parallel wooden bars crossing one another, and kept
together by pins through the middles of the crossings. As you see, I
can alter it to make parallelograms of all degrees of obliquity till
the bars touch, and again you see I can make them all squares.
§ 7. The joint pivots are (for cheapness of construction) of copper
wire, each bent to make a hook below the lattice frame. On these
sixty-four hooks are hung sixty-four fine cords, firmly stretched by
little lead weights. Each of these cords (Fig. 4) bears eight short
perforated wooden cylinders, which may be slipped up and down to any
desired position[2]. They are at present actually placed at distances
consecutively each equal to the distance from joint to joint of the
lattice frame.
[Illustration: FIG. 4.]
§ 8. The roof of the model is hung by four cords, nearly vertical, of
independently variable lengths, passing over hooks from fixed points
above, and kept stretched by weights, each equal to one quarter of
the weight of roof and pendants. You see now by altering the angles
of the lattice work and placing it horizontal or in any inclined
plane, as I am allowed to do readily by the manner in which it is
hung, I have three independent variables, by varying which I can show
you all varieties of homogeneous assemblages, in which three of the
neighbours of every point are at equal distances from it. You see
here, for example, we have the equilateral assemblage. I have adjusted
the lattice roof to the proper angle, and its plane to the proper
inclination to the vertical, to make a wholly equilateral assemblage
of the little cylinders of wood on the vertical cords, a case, as
we have seen, of special importance. If I vary also the distances
between the little pieces of wood on the cords; and the distances
between the joints of the lattice work (variations easily understood,
though not conveniently producible in one model without more of
mechanical construction than would be worth making), I have three
other independent variables. By properly varying these six independent
variables, three angles and three lengths, we may give any assigned
value to each edge of one of the fundamental tetrahedrons of § 3.
§ 9. Our assemblage of people would not be homogeneous unless its
members were all equal and similar and in precisely similar attitudes,
and were all looking the same way. You understand what a number of
people seated or standing on a floor or plain and looking the same way
means. But the expression ‘looking’ is not conveniently applicable to
things that have no eyes, and we want a more comprehensive mode of
expression. We have it in the words ‘orientation,’ ‘oriented,’ and
(verb) ‘to orient,’ suggested by an extension of the idea involved in
the word ‘orientation,’ first used to signify positions relatively
to east and west of ancient Greek and Egyptian temples and Christian
churches. But for the orientation of a house or temple we have only
one angle, and that angle is called ‘azimuth’ (the name given to an
angle in a horizontal plane). For orientation in three dimensions of
space we must extend our ideas and consider position with reference to
east and west and up and down. A man lying on his side with his head
to the north and looking east, would not be similarly oriented to a
man standing upright and looking east. To provide for the complete
specification of how a body is oriented in space we must have in the
body a plane of reference, and a line of reference in this plane,
belonging to the body and moving with it. We must also have a fixed
plane and a fixed line of reference in it, relatively to which the
orientation of the moveable body is to be specified; as, for example,
a horizontal plane and the east and west horizontal line in it. The
position of a body is completely specified when the angle between the
plane of reference belonging to it, and the fixed plane is given; and
when the angles between the line of intersection of the two planes and
the lines of reference in them are also given. Thus we see that three
angles are necessary and sufficient to specify the orientation of a
moveable body, and we see how the specification is conveniently given
in terms of three angles.
§ 10. To illustrate this take a book lying on the table before you with
its side next the title-page up, and its back to the north. I now lift
the east edge (the top of the book), keeping the bottom edge north and
south on the table till the book is inclined, let us say, 20° to the
table. Next, without altering this angle of 20°, between the side of
the book and the table, I turn the book round a vertical axis, through
45° till the bottom edge lies north-east and south-west. Lastly,
keeping the book in the plane to which it has been thus brought, I turn
it round in this plane through 35°. These three angles of 20°, 45°,
and 35°, specify, with reference to the horizontal plane of the table
and the east and west line in it, the orientation of the book in the
position to which you have seen me bring it, and in which I hold it
before you.
§ 11. In Figs. 5 and 6 you see two assemblages, each of twelve equal
and similar molecules in a plane. Fig. 5, in which the molecules are
all same-ways oriented, is one homogeneous assemblage of twenty-four
molecules. Fig. 6, in which in one set of rows the molecules are
alternately oriented two different ways, may either be regarded as
two homogeneous assemblages, each of twelve single molecules; or one
homogeneous assemblage of twelve pairs of those single molecules.
[Illustration: FIG. 5.]
§ 12. I must now call your attention to a purely geometrical
question[3] of vital interest with respect to homogeneous assemblages
in general, and particularly the homogeneous assemblage of molecules
constituting a crystal:--_what can we take as ‘the’ boundary or ‘a’
boundary enclosing each molecule with whatever portion of space around
it we are at liberty to choose for_ _it, and separating it from
neighbours and their portions of space given to them in homogeneous
fairness?_
[Illustration: FIG. 6.]
§ 13. If we had only mathematical points to consider we should be at
liberty to choose the simple obvious partitioning by three sets of
parallel planes. Even this may be done in an infinite number of ways,
thus:--Beginning with any point _P_ of the assemblage, choose any other
three points _A_, _B_, _C_, far or near, provided only that they are
not in one plane with _P_, and that there is no other point of the
assemblage in the lines _PA_, _PB_, _PC_, or within the volume of the
parallelepiped of which these lines are conterminous edges, or within
the areas of any of the faces of this parallelepiped. There will be
points of the assemblage at each of the corners of this parallelepiped
and at all the corners of the parallelepipeds equal and similar to
it which we find by drawing sets of equi-distant planes parallel to
its three pairs of faces. (A diagram is unnecessary.) Every point of
the assemblage is thus at the intersection of three planes, which is
also the point of meeting of eight neighbouring parallelepipeds. Shift
now any one of the points of the assemblage to a position within the
volume of any one of the eight parallelepipeds, and give equal parallel
motions to all the other points of the assemblage. Thus we have every
point in a parallelepipedal cell of its own, and all the points of the
assemblage are similarly placed in their cells, which are themselves
equal and similar.
§ 14. But now if, instead of a single point for each member of the
assemblage, we have a group of points, or a globe or cube or other
geometrical figure, or an individual of a homogeneous assemblage of
equal, similar, similarly dressed, and similarly oriented ladies,
sitting in rows, or a homogeneous assemblage of trees closely planted
in regular geometrical order on a plane with equal and similar
distributions of molecules, and parallel planes above and below,
we may find that the best conditioned plane-faced parallelepipedal
partitioning which we can choose would cut off portions properly
belonging to one molecule of the assemblage and give them to the cells
of neighbours. To find a cell enclosing all that belongs to each
individual, for example, every part of each lady’s dress, however
complexly it may be folded among portions of the equal and similar
dresses of neighbours; or, every twig, leaf, and rootlet of each one
of the homogeneous assemblage of trees; we must alter the boundary by
give-and-take across the plane faces of the primitive parallelepipedal
cells, so that each cell shall enclose all that belongs to one
molecule, and therefore (because of the homogeneousness of the
partitioning) nothing belonging to any other molecule. The geometrical
problem thus presented, wonderfully complex as it may be in cases
such as some of those which I have suggested, is easily performed for
any possible case if we begin with any particular parallelepipedal
partitioning determined for corresponding points of the assemblage
as explained in § 13, for any homogeneous assemblage of single
points. We may prescribe to ourselves that the corners are to remain
unchanged, but if so they must to begin with either in interfaces of
contact between the individual molecules, or in vacant space among
the molecules. If this condition is fulfilled for one corner it is
fulfilled for all, as the corners are essentially corresponding points
relatively to the assemblage.
§ 15. Begin now with any one of the twelve straight lines between
corners which constitute the twelve edges of the parallelepiped, and
alter it arbitrarily to any curved or crooked line between the same
pair of corners, subject only to the conditions (1) that it does not
penetrate the substance of any member of the assemblage, and (2) that
it is not cut by equal and similar parallel curves[4] between other
pairs of corners.
[Illustration: FIG. 7.]
Considering now the three fours of parallel edges of the
parallelepiped, let the straight lines of one set of four be altered to
equal and similar parallel curves in the manner which I have described;
and proceed by the same rule for the other two sets of four edges. We
thus have three fours of parallel curved edges instead of the three
fours of parallel straight edges of our primitive parallelepiped with
corners (each a point of intersection of three edges) unchanged.
Take now the quadrilateral of four curves substituted for the four
straight edges of one face of the parallelepiped. We may call this
quadrilateral a curvilineal parallelogram, because it is a circuit
composed of two pairs of equal parallel curves. Draw now a curved
surface (an infinitely thin sheet of perfectly extensible india-rubber
if you please to think of it so) bordered by the four edges of our
curvilineal parallelogram, and so shaped as not to cut any of the
substance of any molecule of the assemblage. Do the same thing with
an exactly similar and parallel sheet relatively to the opposite face
of the parallelepiped; and again the same for each of the two other
pairs of parallel faces. We thus have a curved-faced parallelepiped
enclosing the whole of one molecule and no part of any other; and by
similar procedure we find a similar boundary for every other molecule
of the assemblage. Each wall of each of these cells is common to two
neighbouring molecules, and there is no vacant space anywhere between
them or at corners. Fig. 7 illustrates this kind of partitioning by
showing a plane section parallel to one pair of plane faces of the
primitive parallelepiped, for an ideal case. The plane diagram is in
fact a realization of the two-dimensional problem of partitioning the
pine pattern of a Persian carpet by parallelograms about as nearly
rectilinear as we can make them. In the diagram faint straight lines
are drawn to show the primitive parallelogrammatic partitioning.
It will be seen that of all the crossings (marked with dots in the
diagram) every one is similarly situated to every other in respect to
the homogeneously repeated pattern figures: _A_, _B_, _C_, _D_ are four
of them at the corners of one cell.
§ 16. Confining our attention for a short time to the homogeneous
division of a plane, remark that the division into parallelograms
by two sets of crossing parallels is singular in this respect--each
cell is contiguous with three neighbours at every corner. Any
shifting, large or small, of the parallelograms by relative sliding
in one direction or another violates this condition, brings us to
a configuration like that of the faces of regularly hewn stones in
ordinary bonded masonry, and gives a partitioning which fulfils the
condition that at each corner each cell has only two neighbours. Each
cell is now virtually a hexagon, as will be seen by the letters _A_,
_B_, _C_, _D_, _E_, _F_ in the diagram Fig. 8. _A_ and _D_ are to be
reckoned as corners, each with an interior angle of 180°. In this
diagram the continuous heavy lines and the continuous faint lines
crossing them show a primitive parallelogrammatic partition by two sets
of continuous parallel intersecting lines. The interrupted crossing
lines (heavy) show, for the same homogeneous distribution of single
points or molecules, the virtually hexagonal partitioning which we get
by shifting the boundary from each portion of one of the light lines to
the heavy line next it between the same continuous parallels.
[Illustration: FIG. 8.]
Fig. 8 bis represents a further modification of the boundary by
which the 180° angles _A_, _D_, become angles of less than 180°. The
continuous parallel lines (light) and the short light portions of the
crossing lines show the configuration according to Fig. 8, from which
this diagram is derived.
§ 17. In these diagrams (Figs. 8 and 8 bis) the object enclosed
is small enough to be enclosable by a primitive parallelogrammatic
partitioning of two sets of continuous crossing parallel straight
lines, and by the partitioning of ‘bonded’ parallelograms both
represented in Fig. 8, and by the derived hexagonal partitioning
represented in Fig. 8 bis, with faint lines showing the primitive and
the secondary parallelograms. In Fig. 7 the objects enclosed were
too large to be enclosable by any rectilinear parallelogrammatic
or hexagonal partitioning. The two sets of parallel faint lines in
Fig. 7 show a primitive parallelogrammatic partitioning and the
corresponding pairs of parallel curves intersecting at the corners of
these parallelograms, of which _A_,_B_,_C_,_D_ is a specimen, show a
corresponding partitioning by curvilineal parallelograms. Fig. 9 shows
for the same homogeneous distribution of objects a better conditioned
partitioning, by hexagons in each of which one pair of parallel
edges is curved. The sets of intersecting parallel straight lines in
Fig. 9 show the same primitive parallelogrammatic partitioning as
in Fig. 7, and the same slightly shifted to suit points chosen for
well-conditionedness of hexagonal partitioning.
[Illustration: FIG. 8 bis.]
[Illustration: FIG. 9.]
§ 18. For the division of continuous three-dimensional space[5] into
equal, similar, and similarly oriented cells, quite a corresponding
transformation from partitioning by three sets of continuous mutually
intersecting parallel planes to any possible mode of homogeneous
partitioning, may be investigated by working out the three-dimensional
analogue of §§ 16-17. Thus we find that the most general possible
homogeneous partitioning of space with plane interfaces between the
cells gives us fourteen walls to each cell, of which six are three
pairs of equal and parallel parallelograms, and the other eight are
four pairs of equal and parallel hexagons, each hexagon being bounded
by three pairs of equal and parallel straight lines. This figure, being
bounded by fourteen plane faces, is called a tetrakaidekahedron. It
has thirty-six edges of intersection between faces; and twenty-four
corners, in each of which three faces intersect. A particular case
of it, which I call an orthic tetrakaidekahedron, being that in
which the six parallelograms are equal squares, the eight hexagonal
faces are equal equilateral and equiangular hexagons, and the lines
joining corresponding points in the seven pairs of parallel faces
are perpendicular to the planes of the faces, is represented by
a stereoscopic picture in Fig. 10. The thirty-six edges and the
twenty-four corners, which are easily counted in this diagram, occur
in the same relative order in the most general possible partitioning,
whether by plane-faced tetrakaidekahedrons or by the generalized
tetrakaidekahedron described in § 19.
§ 19. The most general homogeneous division of space is not
limited to plane-faced cells; but it still consists essentially of
tetrakaidekahedronal cells, each bounded by three pairs of equal and
parallel quadrilateral faces, and four pairs of equal and parallel
hexagonal faces, neither the quadrilaterals nor the hexagons being
necessarily plane. Each of the thirty-six edges may be straight
or crooked or curved; the pairs of opposite edges, whether of the
quadrilaterals or hexagons, need not be equal and parallel; neither
the four corners of each quadrilateral nor the six corners of each
hexagon need be in one plane. But every pair of corresponding edges of
every pair of parallel corresponding faces, whether quadrilateral or
hexagonal, must be equal and parallel. I have described an interesting
case of partitioning by tetrakaidekahedrons of curved faces with curved
edges in a paper[6] published about seven years ago. In this case each
of the quadrilateral faces is plane. Each hexagonal face is a slightly
curved surface having three rectilineal diagonals through its centre in
one plane.
[Illustration: FIG. 10.]
The six sectors of the face between these diagonals lie alternately
on opposite sides of their plane, and are bordered by six arcs
of plane curves lying on three pairs of parallel planes. This
tetrakaidekahedronal partitioning fulfils the condition that the
angles between three planes meeting in an edge are everywhere each
120°; a condition that cannot be fulfilled in any plane-faced
tetrakaidekahedron. Each hexagonal wall is an anticlastic surface of
equal opposite curvatures at every point, being the surfaces of minimum
area bordered by six curved edges. It is shown easily and beautifully,
and with a fair approach to accuracy, by choosing six little circular
arcs of wire, and soldering them together by their ends in proper
planes for the six edges of the hexagon; and dipping it in soap
solution and taking it out.
§ 20. Returning now to the tactics of a homogeneous assemblage, remark
that the qualities of the assemblage as a whole depend both upon the
character and orientation of each molecule, and on the character of
the homogeneous assemblage formed by corresponding points of the
molecules. After learning the simple mathematics of crystallography,
with its indicial system[7] for defining the faces and edges of a
crystal according to the Bravais rows and nets and tetrahedrons of
molecules in which we think only of a homogeneous assemblage of points,
we are apt to forget that the true crystalline molecule, whatever its
nature may be, has sides, and that generally two opposite sides of each
molecule may be expected to be very different in quality, and we are
almost surprised when mineralogists tell us that two parallel faces on
two sides of a crystal have very different qualities in many natural
crystals. We might almost as well be surprised to find that an army in
battle array, which is a kind of large-grained crystal, presents very
different appearance to any one looking at it from outside, according
as every man in the ranks with his rifle and bayonet faces to the front
or to the rear or to one flank or to the other.
§ 21. Consider, for example, the ideal case of a crystal consisting
of hard equal and similar tetrahedronal solids all same-ways oriented.
A thin plate of crystal cut parallel to any one set of the faces of
the constituent tetrahedrons would have very different properties on
its two sides; as the constituent molecules would all present points
outwards on one side and flat surfaces on the other. We might expect
that the two sides of such a plate of crystal would become oppositely
electrified when rubbed by one and the same rubber; and, remembering
that a piece of glass with part of its surface finely ground but not
polished and other parts polished becomes, when rubbed with white
silk, positively electrified over the polished parts and negatively
electrified over the non-polished parts, we might almost expect that
the side of our supposed crystalline plate towards which flat faces
of the constituent molecules are turned would become positively
electrified, and the opposite side, showing free molecular corners,
would become negatively electrified, when both are rubbed by a rubber
of intermediate electric quality. We might also from elementary
knowledge of the fact of piezo-electricity, that is to say, the
development of opposite electricities on the two sides of a crystal
by pressure, expect that our supposed crystalline plate, if pressed
perpendicularly on its two sides, would become positively electrified
on one of them and negatively on the other.
§ 22. Intimately connected with the subject of enclosing cells for
molecules of given shape, assembled homogeneously, is the homogeneous
packing together of equal and similar molecules of any given shape. In
every possible case of any infinitely great number of similar bodies
the solution is a homogeneous assemblage. But it may be a homogeneous
assemblage of single solids all oriented the same way, or it may be
a homogeneous assemblage of clusters of two or more of them placed
together in different orientations. For example, let the given bodies
be halves (oblique or not oblique) of any parallelepiped on the two
sides of a dividing plane through a pair of parallel edges. The two
halves are homochirally[8] similar; and, being equal, we may make a
homogeneous assemblage of them by orienting them all the same way
and placing them properly in rows. But the closest packing of this
assemblage would necessarily leave vacant spaces between the bodies:
and we get in reality the closest possible packing of the given bodies
by taking them in pairs oppositely oriented and placed together to form
parallelepipeds. These clusters may be packed together so as to leave
no unoccupied space.
Whatever the number of pieces in a cluster in the closest possible
packing of solids may be for any particular shape, we may consider each
cluster as itself a given single body, and thus reduce the problem to
the packing closely together of assemblages of individuals all sameways
oriented; and to this problem therefore it is convenient that we should
now confine our attention.
§ 23. To avoid complexities such as those which we find in the familiar
problem of homogeneous packing of forks or spoons or tea-cups or bowls,
of any ordinary shape, we shall suppose the given body to be of such
shape that no two of them similarly oriented can touch one another
in more than one point. Wholly convex bodies essentially fulfil this
condition; but it may also be fulfilled by bodies not wholly convex, as
is illustrated in Fig. 11.
[Illustration: Fig. 11.]
§ 24. To find close and closest packing of any number of our solids
_S_{1}_, _S_{2}_, _S_{3}_ ... of shape fulfilling the condition of § 23
proceed thus:--
(1) Bring _S_{2}_ to touch _S_{1}_ at any chosen point _p_ of its
surface (Fig. 12).
(2) Bring _S_{3}_ to touch _S_{1}_ and _S_{2}_, at _r_ and _q_
respectively.
(3) Bring _S_{4}_ (not shown in the diagram) to touch _S_{1}_, _S_{2}_,
and _S_{3}_.
(4) Place, any number of the bodies together in three rows continuing
the lines of _S_{1}S_{2}_, _S_{1}S_{3}_, _S_{1}S_{4}_, and in three
sets of equi-distant rows parallel to these. This makes a homogeneous
assemblage. In the assemblage so formed the molecules are necessarily
found to be in three sets of rows parallel respectively to the three
pairs _S_{2}S_{3}_, _S_{3}S_{4}_, _S_{4}S_{2}_. The whole space
occupied by an assemblage of _n_ of our solids thus arranged has
clearly _6n_ times the volume of a tetrahedron of corresponding points
of _S_{1}_, _S_{2}_, _S_{3}_, _S_{4}_. Hence the closest of the
close packings obtained by the operations (1) ... (4) is found if we
perform the operations (1), (2), and (3) as to make the volume of this
tetrahedron least possible.
[Illustration: FIG. 12]
§ 25. It is to be remarked that operations (1) and (2) leave for (3)
no liberty of choice for the place of _S_{4}_, except between two
determinate positions on opposite sides of the group _S_{1}_, _S_{2}_,
_S_{3}_. The volume of the tetrahedron will generally be different for
these two positions of _S_{4}_, and, even if the volume chance to be
equal in any case, we have differently shaped assemblages according as
we choose one or other of the two places for _S_{4}_.
This will be understood by looking at Fig. 12, showing _S_{1}_
and neighbours on each side of it in the rows of _S_{1}S_{2}_,
_S_{1}S_{3}_, and in a row parallel to that of _S_{2}S_{3}_. The plane
of the diagram is parallel to the planes of corresponding points of
these seven bodies, and the diagram is a projection of these bodies
by lines parallel to the intersections of the tangent planes through
_p_ and _r_. If the three tangent planes through _p_, _q_, and _r_,
intersected in parallel lines, _q_ would be seen like _p_ and _r_ as a
point of contact between the outlines of two of the bodies; but this is
only a particular case, and in general _q_ must, as indicated in the
diagram, be concealed by one or other of the two bodies of which it is
the point of contact. Now imagining, to fix our ideas and facilitate
brevity of expression, that the planes of corresponding points of the
seven bodies are horizontal, we see clearly that _S_{4}_ may be brought
into proper position to touch _S_{1}_, _S_{2}_, and _S_{3}_ either from
above or from below; and that there is one determinate place for it if
we bring it into position from above, and another determinate place for
it if we bring it from below.
§ 26. If we look from above at the solids of which Fig. 12 shows the
outline, we see essentially a hollow leading down to a perforation
between _S_{1}_, _S_{2}_, _S_{3}_, and if we look from below we see a
hollow leading upwards to the same perforation: this for brevity we
shall call the perforation _pqr_. The diagram shows around _S_{1}_
six hollows leading down to perforations, of which two are similar
to _pqr_, and the other three, of which _p′q′r′_ indicates one, are
similar one to another but are dissimilar to _pqr_. If we bring _S_{4}_
from above into position to touch _S_{1}_, _S_{2}_, and _S_{3}_, its
place thus found is in the hollow _pqr_, and the places of all the
solids in the layer above that of the diagram are necessarily in the
hollows similar to _pqr_. In this case the solids in the layer below
that of the diagram must lie in the hollows below the perforations
dissimilar to _pqr_, in order to make a single homogeneous assemblage.
In the other case, _S_{4}_ brought up from below finds its place on the
under side of the hollow _pqr_, and all solids of the lower layer find
similar places: while solids in the layer above that of the diagram
find their places in the hollows similar to _p´q´r´_. In the first
case there are no bodies of the upper layer in the hollows above the
perforations _similar_ to _p´q´r´_, and no bodies of the lower layer in
the hollows below the perforations _similar_ to _pqr_. In the second
case there are no bodies of the upper layer in the hollows above the
perforations _similar_ to _pqr_, and none of the under layer in the
hollows below the perforations _similar_ to _p´q´r´_.
§ 27. Going back now to operation (1) of § 23, remark that when the
point of contact _p_ is arbitrarily chosen on one of the two bodies
_S_{1}_, the point of contact on the other will be the point on it
corresponding to the point or one of the points of _S_{1}_, where its
tangent plane is parallel to the tangent plane at _p_. If _S_{1}_
is wholly convex it has only two points at which the tangent planes
are parallel to a given plane, and therefore the operation (1) is
determinate and unambiguous. But if there is any concavity there will
be four or some greater even number of tangent planes parallel to any
one of some planes, while there will be other planes to each of which
only one pair of tangent planes is parallel. Hence, operation (1),
though still determinate, will have a multiplicity of solutions, or
only a single solution, according to the choice made of the position of
_p_.
Henceforth however, to avoid needless complications of ideas, we shall
suppose our solids to be wholly convex; and of some such unsymmetrical
shape as those indicated in Fig. 12 of § 25, and shown by stereoscopic
photograph in Fig. 13 of § 36. With or without this convenient
limitation, operation (1) has two freedoms, as _p_ may be chosen
freely on the surface of _S_{1}_; and operation (2) has clearly just
one freedom after operation (1) has been performed. Thus, for a solid
of any given shape, we have three disposables, or, as commonly called
in mathematics, three ‘independent variables,’ all free for making a
homogeneous assemblage according to the rule of § 22.
§ 28. In the homogeneous assemblage defined in § 24, each solid,
_S_{1}_, is touched at twelve points, being the three points of
contact with _S_{2}_, _S_{3}_, _S_{4}_, and the three 3’s of points
on _S_{1}_ corresponding to the points on _S_{2}_, _S_{3}_, _S_{4}_,
at which these bodies are touched by the others of the quartet. This
statement is somewhat difficult to follow, and we see more clearly
the twelve points of contact by not confining our attention to the
quartet _S_{1}_, _S_{2}_, _S_{3}_, _S_{4}_ (convenient as this is
for some purposes), but completing the assemblage and considering
six neighbours around _S_{1}_ in one plane layer of the solids as
shown in Fig. 12, with their six points _prq″p′r′q″′_ of contact with
_S_{1}_; and the three neighbours of the two adjacent parallel layers
which touch it above and below. This cluster of thirteen, _S_{1}_
and twelve neighbours, is shown for the case of spherical bodies in
the stereoscopic photograph of § 4 above. We might of course, if we
pleased, have begun with the plane layer of which _S_{1}_, _S_{2}_,
_S_{4}_ are members, or with that of which _S_{1}_, _S_{3}_, _S_{4}_
are members, or with the plane layer parallel to the fourth side
_S_{2}_ _S_{3}_ _S_{4}_ of the tetrahedron: and thus we have four
different ways of grouping the twelve points of contact on _S_{1}_ into
one set of six and two sets of three.
§ 29. In this assemblage we have what I call ‘close order’ or ‘close
packing.’ For closest of close packings the volume of the tetrahedron
(§ 24) of corresponding points of _S_{1}_, _S_{2}_, _S_{3}_, and
_S_{4}_ must be a minimum, and the least of minimums if, as generally
will be the case, there are two more different configurations for
each of which the volume is a minimum. There will in general also be
configurations of minimax volume and of maximum volume, subject to
the condition that each body is touched by twelve similarly oriented
neighbours.
§ 30. Pause for a moment to consider the interesting kinematical
and dynamical problems presented by a close homogeneous assemblage
of smooth solid bodies of given convex shape, whether perfectly
frictionless or exerting resistance against mutual sliding according to
the ordinarily stated law of friction between dry hard solid bodies.
First imagine that they are all similarly oriented and each in contact
with twelve neighbours, except outlying individuals (which there must
be at the boundary if the assemblage is finite, and each of which is
touched by some number of neighbours less than twelve). The coherent
assemblage thus defined constitutes a kinematic frame or skeleton
for an elastic solid of very peculiar properties. Instead of the six
freedoms, or disposables, of strain presented by a natural solid it has
only three. Change of shape of the whole can only take place in virtue
of rotation of the constituent parts relatively to any one chosen row
of them, and the plane through it and another chosen row.
§ 31. Suppose first the solids to be not only perfectly smooth but
perfectly frictionless. Let the assemblage be subjected to equal
positive or negative pressure inwards all around its boundary. Every
position of minimum, minimax, or maximum volume will be a position of
equilibrium. If the pressure is positive the equilibrium will be stable
if, and unstable unless, the volume is a minimum. If the pressure
is negative the equilibrium will be stable if, and unstable unless,
the volume is a maximum. Configurations of minimax volume will be
essentially unstable.
§ 32. Consider now the assemblage of § 31 in a position of stable
equilibrium under the influence of a given constant uniform pressure
inwards all round its boundary. It will have rigidity in simple
proportion to the amount of this pressure. If now by the superposition
of non-uniform pressure at the boundary, for example equal and opposite
pressures on two sides of the assemblage, a finite change of shape is
produced: the whole assemblage essentially swells in bulk. This is the
‘dilatancy’ which Osborne Reynolds has described[9] in an exceedingly
interesting manner with reference to a sack of wheat or sand, or an
india-rubber bag tightly filled with sand or even small shot. Consider,
for example, a sack of wheat filled quite full and standing up open. It
is limp and flexible. Now shake it down well, fill it quite full, shake
again, so as to get as much into it as possible, and tie the mouth very
tightly close. The sack becomes almost as stiff as a log of wood of
the same shape. Open the mouth partially, and it becomes again limp,
especially in the upper parts of the bag. In Reynolds’ observations on
india-rubber bags of small shot his ‘dilatancy’ depends, essentially
and wholly, on breaches of some of the contacts which exist between the
molecules in their configuration of minimum volume: and it is possible
that in all his cases the dilatations which he observed are _chiefly_,
if not wholly, due to such breaches of contact.
But it is possible, it almost seems probable, that in bags or boxes
of sand or powder, of some kinds of smooth rounded bodies of any
shape, not spherical or ellipsoidal, subjected persistently to unequal
pressures in different directions, and well shaken, stable positions
of equilibrium are found with almost all the particles each touched by
twelve others.
Here is a curious subject of Natural History through all ages till
1885, when Reynolds brought it into the province of Natural Philosophy
by the following highly interesting statement:--‘A well-marked
phenomenon receives its explanation at once from the existence of
dilatancy in sand. When the falling tide leaves the sand firm, as the
foot falls on it the sand whitens and appears momentarily to dry round
the foot. When this happens the sand is full of water, the surface of
which is kept up to that of the sand by capillary attractions; the
pressure of the foot causing dilatation of the sand more water is
required, which has to be obtained either by depressing the level of
the surface against the capillary attractions, or by drawing water
through the interstices of the surrounding sand. This latter requires
time to accomplish, so that for the moment the capillary forces are
overcome; the surface of the water is lowered below that of the sand,
leaving the latter white or drier until a sufficient supply has been
obtained from below, when the surface rises and wets the sand again.
On raising the foot it is generally seen that the sand under the foot
and around becomes momentarily wet; this is because, on the distorting
forces being removed, the sand again contracts, and the excess of water
finds momentary relief at the surface.’
This proves that the sand under the foot, as well as the surface around
it, must be dry for a short time after the foot is pressed upon it,
though we cannot see it whitened, as the foot is not transparent. That
it is so has been verified by Mr. Alex. Galt, Experimental Instructor
in the Physical Laboratory of Glasgow University, by laying a small
square of plate-glass on wet sand on the sea-shore of Helensburgh, and
suddenly pressing on it by a stout stick with nearly all his weight.
He found the sand, both under the glass and around it in contact with
the air, all became white at the same moment. Of all the two hundred
thousand million men, women, and children who, from the beginning
of the world, have ever walked on wet sand, how many, prior to the
British Association Meeting at Aberdeen in 1885, if asked, ‘Is the sand
compressed under your foot?’ would have answered otherwise than ‘Yes!’?
(Contrast with this the case of walking over a bed of wet sea-weed!)
§ 33. In the case of globes packed together in closest order (and
therefore also in the case of ellipsoids, if all similarly oriented),
our condition of coherent contact between each molecule and twelve
neighbours implies absolute rigidity of form and constancy of bulk.
Hence our convex solid must be neither ellipsoidal nor spherical
in order that there may be the changes of form and changes of bulk
which we have been considering as dependent on three independent
variables specifying the orientation of each solid relatively to rows
of the assemblage. An interesting dynamical problem is presented by
supposing any mutual forces, such as might be produced by springs, to
act between the solid molecules, and investigating configurations of
equilibrium on the supposition of frictionless contacts. The solution
of it of course is that the potential energy of the springs must be a
minimum or a minimax or a maximum for equilibrium, and a minimum for
stable equilibrium. The solution will be a configuration of minimum or
minimax, or maximum, volume, only in the case of pressure equal in all
directions.
§ 34. A purely geometrical question, of no importance in respect to
the molecular tactics of a crystal but of considerable interest in
pure mathematics, is forced on our attention by our having seen (§ 27)
that a homogeneous assemblage of solids of given shape, each touched
by twelve neighbours, has three freedoms which may be conveniently
taken as the three angles specifying the orientation of each molecule
relatively to rows of the assemblage as explained in § 30.