COMPOSITION.
Most chemical elements and compounds are capable of assuming the crystalline condition. Crystallization May take place when solid matter separates from solution (e.g. sugar; salt, alum), from a fused mass (e.g. sulphur, bismuth, felspar), or front a vapour (e.g. iodine, camphor, haematite; in the last case by the interaction of ferric chloride and 'steam). Crystalline growth may also take place in solid amorphous matter, for example, in the devitrification of glass, and the slow change in metals when subjected to alternating stresses. Beautiful crystals of many substances may be obtained in the laboratory by one or other of these methods, but the most perfectly developed and largest crystals are those of mineral substances found in nature, where crystallization has continued during long periods of time. For this reason the physical science of crystallography has developed side by side with that of mineralogy. Really, however, there is just the same connexion between crystallography and chemistry as between crystallography and mineralogy, but only in recent years has the importance of determining the crystallographic properties of artificially prepared compounds been recognized.
History.—The word " crystal " is from the Gr. KpI rraXXos, meaning clear ice (Lat. crystallum), a name which was also applied to the clear transparent quartz (" rockcrystal ") from the Alps, under the belief that it had been formed from water by intense cold. It was not until about the 17th century that the word was extended to other bodies, either those found in nature or obtained by the evaporation of a saline solution, which resembled rockcrystal in being bounded by plane surfaces, and often also in their clearness and transparency.
The first important step in the study of crystals was made by Nicolaus Steno, the famous Danish physician, afterwards bishop of Titiopolis, who in his treatise De solido infra solidum naturaliter contento (Florence, 1669; English translation, '671) gave the
results of his observations on crystals of quartz. He found that although the faces of different crystals vary considerably in shape and relative size, yet the angles between similar pairs of faces are always the same. He further pointed out that the crystals must have grown in a liquid by the addition of layers of material upon the faces of a nucleus, this nucleus having the form of a regular sixsided prism terminated at each end by a sixsided pyramid. The thickness of the layers, though the same over each face, was not necessarily the same on different faces, but depended on the position of the faces with respect to the surrounding liquid; hence the faces of the crystal, though variable in shape and size, remained parallel to those of the nucleus, and the angles between them constant. Robert Hooke in his Micrographia (London, 1665) had previously noticed the regularity of the minute quartz crystals found lining the cavities of flints, and had suggested that they were built up of spheroids. About the same time the double refraction and perfect rhomboidal cleavage of crystals of calcite or Icelandspar were studied by Erasmus Bartholinus (Experimenta crystalli Islandici disdiadastici, Copenhagen, 1669) and Christiaan Huygens (Traite de la lumiere, Leiden, 169o); the latter supposed, as did Hooke, that the crystals were built up of spheroids. In 1695 Anton van Leeuwenhoek observed. under the microscope that different forms of crystals grow from the solutions of different salts. Andreas Libavius had indeed much earlier, in 1597, pointed out that the salts present in mineral waters could be ascertained by an examination of the shapes of the crystals left on evaporation of the water; and Domenico Guglielmini (Riflessioni filosofiche dedotte dalle figure de' sali, Padova, 1706) asserted that the crystals of each salt had a shape of their own with the plane angles of the faces always the same.
The earliest treatise on crystallography is the Prodromus Crystallographiae of M. A. Cappeller, published at Lucerne in 1723. Crystals were mentioned in works on mineralogy and chemistry; for instance, C. Linnaeus in his Systema Naturae (1735) described some forty common forms of crystals amongst minerals. It was not, however, until the end of the 18th century that any real advances were made, and the French crystallographers Rome de 1'Isle and the abbe Hauy are rightly considered as the founders of the science. J. B. L. de Rome de l'Isle (Essai de cristallographie, Paris, 1772; Cristallographie, ou description des formes propres a tous les corps du regne mineral, Paris, 1783) made the important discovery that the various shapes of crystals of the same natural or artificial substance are all intimately related to each other; and further, by measuring the angles between the faces of crystals with the goniometer (q.v.), he established the fundamental principle that these angles .are always the same for the same kind of substance and are characteristic of it. Replacing by single planes or groups of planes all the similar edges or solid angles of a figure called the " primitive form he derived other related forms. Six kinds of primitive forms were distinguished, namely, the cube, the regular octahedron, the regular tetrahedron, a rhombohedron, an octahedron with a rhombic base, and a double sixsided pyramid. Only in the last three can there be any variation in the angles: for example, the primitive octahedron of alum, nitre and sugar were determined by Rome de 1'Isle to have angles of Ito°, 126 ° and toe respectively. Rene Just Hauy in his Essai d'une theorie sur la structure des crystaux (Paris, 1784; see also his Treatises on Mineralogy and Crystallography, 18o1, 1822) supported and extended these views, but took for his primitive forms the figures obtained by splitting crystals in their directigns of easy fracture of " cleavage, " which are aways the same in the same kind of substance. Thus he found that all crystals of calcite, whatever their external form (see, for example, figs. 16 in the article CALCITE), could be reduced by cleavage to a rhombohedron with interfacial angles of 95 Further, by stacking together a number ,of small rhombohedra of uniform size he was able, as had been previously done by J. G. Gahn in 1793, to reconstruct the various forms of calcite crystals. Fig. I shows a scalenohedron ,(vkaXrlvbs; uneven) built up in this manner of rhombohedra; and fig. 2 a regular octahedron builtup of cubic elements, such as are given by the cleavage of galena and rocksalt.
The external surfaces of such a structure, with their steplike arrangement, correspond to the plane faces of the crystal, and the bricks may be considered so small as not to be separately visible. By making the steps one, two or three bricks in width and one, two or three bricks in height the various secondary
faces on the crystal are related to the primitive form or " cleavage nucleus by a law of whole numbers, and the angles between them can be arrived at by mathematical calculation. By measuring with the goniometer the inclinations of the secondary faces to those of the primitive form Hauy found that the secondary forms are always related to the primitive form on crystals of numerous substances in the manner indicated, and that the width and the height of a step are always in a simple ratio, rarely exceeding that of I : 6. This laid the foundation of the important " law of rational indices" of the faces of crystals.
The German crystallographer C. S. Weiss (De indagando formarum crystallinarum charactere geometrico principali dissertatio, Leipzig, 1809; Ubersichtliche Darstellung der verschiedenen naturlichen Abtkeilungen der KrystallisationsSysteme, Denkschrift der Berliner Akad. der Wissenst., 18'41815) attacked the problem of crystalline form from a purely geometrical point of view, without reference to primitive forms or any theory of structure. The faces of crystals were considered by their intercepts on coordinate axes, which were drawn joining the opposite corners. of certain forms; and in this way the various primitive forms of Hauy were grouped into four classes, corresponding to the four systems described below under the names cubic, tetragonal, hexagonal and orthorhombic. The same result was arrived at independently by F. Mohs, who further, in 1822, asserted the existence of two additional systems with oblique axes. These two systems (the monoclinic and anorthic) were, however, considered by Weiss to be only hemihedral or tetartohedral modifications of the orthorhombic system, and they were not definitely established until 1835, when the optical characters of the crystals were found to be distinct. A system of notation to express the relation of each face of a crystal to the coordinate axes of reference was devised by Weiss, and other notations were proposed by F. Mohs, A. Levy (1825), C. F. Naumann (1826), and W. H. Miller (Treatise on Crystallography, Cambridge, 1839). For simplicity and utility in calculation the Millerian notation, which was first suggested by W. Whewell in 1825, surpasses all others and is now generally adopted, though those of Levy and Naumann are still in use.
Although the peculiar optical properties of Icelandspar had been much studied ever since 1669, it was not until much later that any connexion was traced between the optical characters of crystals and their external form. In 1818 Sir David Brewster found that crystals could be divided optically into three classes, viz. isotropic, uniaxial and biaxial, and that these classes corresponded with Weiss's four systems (crystals belonging to the cubic system being isotropic, those of the tetragonal and hexagonal being uniaxial, and the orthorhombic being biaxial). Optically biaxial crystals were afterwards shown by J. F. W. Herschel and F. E. Neumann in 1822 and 1835 to be of three kinds, corresponding with the orthorhombic, monoclinic and
up of Rhombohedra.
of Cubes.
anorthic systems. It was, however, noticed by Brewster himself that there are many apparent exceptions, and the " optical anomalies " of crystals have been the subject of much study. The intimate relations existing between various other physical properties of crystals and their external form have subsequently been gradually traced.
The symmetry of crystals, though recognized by Rome de l'Isle and Hauy, in that they replaced all similar edges and corners of their primitive forms by similar secondary planes, was not made use of in defining the six systems of crystallization, which depended solely on the lengths and inclinations of the axes of reference. It was, however, necessary to recognize that in each system there are certain forms which are only partially symmetrical, and these were described as hemihedral and tetartohedral forms (i.e. 'jµ1, halffaced, and rirapros, quarterfaced forms).
As a consequence of Hauy's law of rational intercepts, or, as it is more often called, the law of rational indices, it was proved by J. F. C. Hessel in 183o that thirtytwo types of symmetry are possible in crystals. Hessel's work remained overlooked for sixty years, but the same important result was independently arrived at by the same method by A. Gadolin in 1867. At the present day, crystals are considered as belonging to one or other of thirtytwo classes, corresponding with these thirtytwo types of symmetry, and are grouped in six systems. More recently, theories of crystal structure have attracted attention, and have been studied as purely geometrical problems of the homogeneous partitioning of space.
The historical development of the subject is treated more fully in the article CRYSTALLOGRAPHY in the 9th edition of this work. Reference may also be made to C. M. Marx, Geschichte der Crystallkunde (Karlsruhe and Baden, 1825) ; W. Whewell, History of the Inductive Sciences, vol. iii. (3rd ed., London, 1857); F. von Kobell, Geschichte der Mineralogie von z650–1860 (Munchen, 1864) ; L. Fletcher, An Introduction to the Study of Minerals (British Museum GuideBook) ; L. Fletcher, Recent Progress in Mineralogy and Crystallography [1832–1894] (Brit. Assoc. Rep., 1894).
I. CRYSTALLINE FORM
The fundamental laws governing the form of crystals are:
1. Law of the Constancy of Angle.
2. Law of Symmetry.
3. Law of Rational Intercepts or Indices.
According to the first law, the angles between corresponding faces of all crystals of the same chemical substance are always the same and are characteristic of the substance.
(a) Symmetry of Crystals.
Crystals may, or may not, be symmetrical with respect to a point, a line or axis, and a plane; these " elements of symmetry " are spoken of as a centre of symmetry, an axis of symmetry, and a plane of symmetry respectively.
Centre of Symmetry.—Crystals which are centrosymmetrical have their faces arranged in parallel pairs; and the two parallel faces, situated on opposite sides of the centre (0 in fig. 3) are alike in surface characters, such as lustre, striations, and figures of corrosion. An octahedron (fig. 3) is bounded by four pairs of parallel faces. Crystals belonging to many of the hemihedral and tetartohedral classes of the six systems of crystallization are devoid of a centre of symmetry.
Axes' of Symmetry.—Consider the vertical axis joining the opposite corners a3 and a3 of an octahedron (fig. 3) and passing through its centre. 0: by rotating the crystal about this axis through a right angle (go°) it reaches a position such that the orientation of its faces is the same as before the rotation; the face a1d2a3, for example, coming into the position of ala2a3. During a complete rotation of 36o° (= 90°X 4), the crystal occupies four such interchangeable positions. Such an axis of symmetry is known as a tetrad axis of symmetry. Other tetrad axes of the octahedron are a2d2 and a,a,.
Art ells of symmetry of another kind is that which passing through the centre 0 is normal to a face of the octahedron. Ry rotating the crystal about such an axis Op (fig. 3) through vi' angle of r 20° those faces which are not perpendicular to theaxis occupy interchangeable positions; for example, the face ala3a2 comes into the position of a2alas, and d2ald3 to a3a2a1. During a complete rotation of 36o° (=12o°X3) the crystal occupies similar positions three times. This is a triad axis of symmetry; and there being four pairs of parallel faces on an octahedron, there are four triad axes (only one of which is drawn in the figure).
An axis passing through the centre 0 and the middle points d of two opposite edges of the octahedron (fig. 4), i.e. parallel
Axes and Planes of Symmetry of an Octahedron.
to the edges of the octahedron, is a dyad axis of symmetry. About this axis there may be rotation of 18o°, and only twice in a complete revolution of 36o° (=18o°X 2) is the crystal brought into interchangeable positions. There being six pairs of parallel edges on an octahedron; there are consequently six dyad axes of symmetry.
A regular octahedron thus possesses thirteen axes of symmetry (of three kinds), and there are the same number in the cube. Fig. 5 shows the three tetrad (or tetragonal) axes (aa), four triad (or trigonal) axes (pp), and six dyad (diad or diagonal) axes (dd).
Although not represented in the cubic system, there is still another kind of axis of symmetry possible in crystals. This is the hexad axis or hexagonal axis, for which the angle of rotation is 6o°, or onesixth of 36o°. There can be only one hexad axis of symmetry in any crystal (see figs. 77So).
Planes of Symmetry.—A regular octahedron can be divided into two equal and similar halves by a plane passing through the corners a1a3d1a3 and the
centre 0 (fig. 3). Onehalf is the mirror reflection of the other in this plane, which is called a plane of symmetry. Corresponding planes on either side of a plane of symmetry are inclined to it at equal angles. The octahedron can also be divided by similar planes of symmetry passing through the corners ala2a,az and a2a3a2a3• These three similar planes of symmetry are called the cubic planes of symmetry, since they are parallel to the faces
of the cube (compare figs. 68, showing combinations of the
p
octahedron and the cube).
A regular octahedron can also be divided symmetrically into two equal and similar portions by a plane passing through the corners a3 and a3, the middle points d of the edges alai and d1a2, and the centre 0 (fig. 4). This is called a dodecahedral plane of symmetry, being parallel to the face of the rhombic dodecahedron which truncates the edge a,a2 (compare fig. 14, showing a combination of the octahedron and rhombic dodecahedron). Another similar plane of symmetry is that passing through the corners a3a3 and the middle points of the edges ala2 and a1a2, and altogether there are six dodecahedral planes of symmetry, two through each of the corners al, a2, a3 of the octahedron.
A regular octahedron and a cube are thus each symmetrical with respect to the following elements of symmetry: a centre of symmetry, thirteen axes of symmetry (of three kinds), and nine planes of symmetry (of two kinds). This degree of symmetry, which is the type corresponding to one of the classes of the cubic system, is the highest possible in crystals. As will be pointed out below, it is possible, however, for both the octahedron and the cube to be associated with fewer elements of symmetry than those just enumerated.
(b) Simple Forms and Combinations of Forms.
A single face ala2a3 (figs. 3 and 4) may be repeated by certain of the elements of symmetry to give the whole eight faces of the octahedron. Thus, by rotation about the vertical, tetrad axis aaa3' the four upper faces are obtained; and by rotation of these about one or other of the horizontal tetrad axes the eight faces are derived. Or again, the same repetition of the faces may be arrived at by reflection across the three cubic planes of symmetry. (By reflection across the six dodecahedral planes
of symmetry a tetrahedron only would result, but if this is associated with a centre of symmetry we obtain the octahedron.) Such a set of similar faces, obtained by symmetrical repetition, constitutes a " simple form." An octahedron thus consists of eight similar faces, and a cube is bounded by six faces all of which, have the same surface characters, and parallel to each of which all the properties of the crystal are identical.
Examples of simple forms amongst crystallized substances are octahedra of alum and spinel and cubes of salt and fluorspar. More usually, however, two or more forms are present on a crystal, and we then have a combination of forms, or simply a
combination." Figs. 6, 7 and 8 represent combinations of the octahedron and the cube; in the first the faces of the cube predominate, and in the third those of the octahedron; fig. 7 with the two forms equally developed is called a cubooctahedron. Each of these combined forms has all the elements of symmetry proper to the simple forms.
The simple forms, though referable
to the same type of symmetry and
axes of reference, are quite independent,
and cannot be derived one from the
other by symmetrical repetition, but,
after the manner of Rome de l'Isle,
they may be derived by replacing
edges or corners by a face equally
combination withCube. inclined to the faces forming the edges
or corners; this is known as " truncation " (Lat. truncare, to cut off). Thus in fig. 6 the corners of the cube are symmetrically replaced or truncated by the faces of the octahedron, and in fig. 8 those of the octahedron are truncated by the cube.
(c) Law of Rational Intercepts.
For axes of reference, OX, OY, OZ (fig. 9), take any three edges formed by the intersection of three faces of a crystal. These axes are called the crystallographic axes, and the planes in which they lie the axial planes. A fourth face on the crystal intersecting these three axes in the points A, B, C is taken as the parametral plane, and the lengths OA : OB :OC are the parameters of the crystal. Any other face on the crystal may bereferred to these axes and parameters by the ratio of the intercepts
OA, OB. OC h k l '
Thus for aface parallel to the plane ABe the intercepts are in the ratio OA : OB : Oe, or
OA. OB 0C
I I 2
and for a plane fgC they are Of: Og: OC or
OA OB, 0C
2 3I
Now the important relation existing between the faces of a crystal is that the denominators h, k and l are always rational whole numbers, rarely exceeding 6, and usually o, r, 2 or 3. Written in the form (hkl), h referring to the axis OX, k to OY, and 1 to OZ, they are spoken of as the indices (Millerian indices) of the face. Thus of a face parallel to the plane ABC the indices are (III), of ABe they are (112), and of fgC (231). The indices are thus inversely proportional to the intercepts, and the law of rational intercepts is often spoken of as the "law of rational indices."
The angular position of a face is thus completely fixed by its indices; and knowing the angles between the axial planes and the parametral plane all the angles of a crystal can be calculated when the indices of the faces
are known.
Although any set of edges formed by the intersection of three planes may be chosen for the , crystallographic axes, it is in practice usual to select certain edges related to the symmetry of the crystal, and usually coincident with axes of symmetry; for then the indiceg will be simpler and all faces of the same simple form
will have a similar set of Z
indices. The angles between FIG. 9.Crystallographic axes of
the axes and the ratio of the reference.
lengths of the parameters
OA: OB: OC (usually given as a: b: c) are spoken of as the " elements " of a crystal, and are constant for and characteristic of all crystals of the same substance.
The six systems of crystal forms, to be enumerated below, are defined by the relative inclinations of the crystallographic axes and the lengths of the parameters. In the cubic system, for example, the three crystallographic axes are taken parallel to the three tetrad axes of symmetry, i.e. parallel to the edges of the cube (fig. 5) or joining the opposite corners of the octahedron (fig. 3), and they are therefore all at right angles; the parametral plane (III) is a face of the octahedron, and the parameters are all of equal length. The indices of the eight faces of the octahedron will then be (III), (Iii), (Ili), (III), (Ill), (ill), (III), (III). The symbol {III} indicates all the faces belonging to this simple form. The indices of the six faces of the cube are (too), (olo), (ooi), (Too), (olo), (ool); here each face is parallel to two axes, i.e. intercepts them at infinity, so that the corresponding indices are zero.
(d) Zones.
An important consequence of the law of rational intercepts is the arrangement of the faces of a crystal in zones. All faces, whether they belong to one or more simple forms, which intersect in parallel edges are said to lie in the same zone. A line drawn through the centre 0 of the crystal parallel to these edges is called a zoneaxis, and a plane perpendicular to this axis is called a zoneplane: On a cube, for example, there are three zones each containing four faces, the zoneaxes being coincident with the three tetrad axes of symmetry. In the crystal of zircon (fig. 88) the eight prismfaces a, in, &c. constitute a zone,. denoted
by [a, m, a', &c.], with the vertical tetrad axis of symmetry as zoneaxis. Again the faces [a, x, p, e', p', x"', a"] lie in another zone, as may be seen by the parallel edges of intersection of the faces in figs. 87 and 88; three other similar zones may be traced on the same crystal.
The direction of the line of intersection (i.e. zoneaxis) of any two planes (hkl) and (h,k11,) is given by the zoneindices [uvw], where u=kl1lkl, v=lh,—h11, and w=hk1kh1, these being obtained from the faceindices by cross multiplication as
h k l h k l
x x x
h, kl ll h, k, l,.
Any other face (h2k212) lying in this zone must satisfy the equation
h2u+k,v+12w=o.
This important relation connecting the indices of a face lying in a zone with the zoneindices is known as Weiss's zonelaw, having been first enunciated by C. S. Weiss. It may be pointed out that the indices of a face may be arrived at by adding together the indices of faces on either side of it and in the same zone; thus, (311) in fig. 12 lies at the intersections of the three zones [210, Io1], [201, IIO] and [211, Ioo], and is obtained by adding together each set of indices.
(e) Projection and Drawing of Crystals.
The shapes and relative sizes of the faces of a crystal being as a rule accidental, depending only on the distance of the faces
from the centre of the crystal and not on their angular relations, it is often more convenient to consider only the directions of the normals to the faces. For this purpose projections are drawn, with the aid of which the zonal relations of a crystal are more readily studied and calculations are simplified.
The kind of projection most extensively used is the " stereographic projection." The crystal is considered to be placed inside a sphere from the centre of which normals are drawn to all the faces of the crystal. The points at which these normals intersect the surface of the sphere are called the poles of the faces, and by these poles the positions of the faces are fixed. The poles of all faces in the same zone on the crystal will lie on a great circle of the sphere, which are therefore called zonecircles. The calculation of the angles between the normals of faces and between zonecircles is then performed by the ordinary methods of spherical trigonometry. The stereographic projection, however, represents the poles and zonecircles on a plane surface and not on a spherical surface. This is achieved by drawing lines joining all the poles of the faces with the north or south pole of the sphere and finding their points of intersection with the plane of the equatorial great circle, or primitive circle, of the sphere, the projection being represented on this plane. In fig. Io is shown the stereographic projection, or stereogram, of a573
cubic crystal; al, a2, &c. are the poles of the faces of the cube. ol, 02, &c. those of the octahedron, and d', d2, &c. those of the rhombic dodecahedron. The straight lines and circular arcs are the projections on the equatorial plane of the great circles in which the nine planes of symmetry intersect the sphere. A drawing of a crystal showing a combination of the cube, octahedron and rhombic dodecahedron is shown in fig. II, in which the faces are lettered the same as the corresponding poles in the projection. From the zonecircles in the projection and the parallel edges in the drawing the zonal relations of the faces are readily seen: thus [alold5], [aldla5], [a5old2l, &c. are zones. A stereographic projection of a rhombohedral crystal is given in fig. 72.
Another kind of projection in common use is the " gnomonic projection " (fig. 12). Here the plane of projection is tangent to
the sphere, and normals to all the faces are FIG. II—Clino drawn from the centre of the sphere tographic Drawing of a intersect the plane of projection. In this Cubic Crystal. case all zones are represented by straight
lines. Fig. 12 is the gnomonic projection of a cubic crystal, the plane of projection being tangent to the sphere at the pole of an octahedral face (III), which is therefore in the centre of the projection. The indices of the several poles are given in the figure.
In drawing crystals the simple plans and elevations of descriptive geometry (e.g. the plans in the lower part of figs. 87 and 88) have sometimes the advantage of showing the symmetry of a crystal, but they give no idea of solidity. For instance, a cube would be represented merely by a square, and an octahedron by a square with lines joining the opposite corners. True perspective drawings are never used in the representation of crystals, since for showing the zonal relations it is important to preserve the parallelism of the edges. If, however, the eye, or point of vision, is regarded as being at an infinite distance from the object all the rays will be parallel, and edges which are parallel on the crystal will be represented by parallel lines in the drawing. The plane of the drawing, in which the parallel rays joining the corners of the crystals and the eye intersect, may be either perpendicular or oblique to the rays; in the former case we have an " orthographic " (6p0os, straight; ypa4ew, to draw) drawing, and in the latter a " cinographic " (icXtvew, to incline)
drawing. Clinographic drawings are most frequently used for representing crystals. In representing, for example, a cubic crystal (fig. II) a cube face a5 is first placed parallel to the plane on which the crystal is to be projected and with one set of edges vertical; the crystal is then turned through a small angle about a vertical axis until a second cube face a2 comes into view,
follows:
and the eye is then raised so that a third cube face a' may be seen.
(f) Crystal Systems and Classes.
According to the mutual inclinations of the crystallographic axes of reference and the lengths intercepted on them by the parametral plane, all crystals fall into one or other of six groups or systems, in each of which there are several classes depending on the degree of symmetry. In the brief description which follows of these six systems and thirtytwo. classes of crystals we shall proceed from those in which the symmetry is most complex to those in which it is simplest.
1. CUBIC SYSTEM
(Isometric ; Regular ; Octahedral ; Tesseral).
In this system the three crystallographic axes of reference are all at right angles to each other and are equal in length. They are parallel to the edges of the cube, and in the different classes coincide either with tetrad or dyad axes of symmetry. Five classes are included in this system, in all of which there are, besides other elements of symmetry, four triad axes.
In crystals of this system the angle between any two faces P and Q with the indices (hkl) and (pqr) is given by the equation
Cos PQ= hp+kq+lr
End of Article: COMPOSITION 

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