Introduction to Crystallographyby Donald E. Sands
"This is truly a delightful monograph." — Canadian Chemical Education
Designed as a useful, accessible introduction to the logical development of basic crystallographic concepts, this book presents important principles in a clear, concise manner that will enable the nonspecialist to read and comprehend crystallographic literature. Explanations are/i>
"This is truly a delightful monograph." — Canadian Chemical Education
Designed as a useful, accessible introduction to the logical development of basic crystallographic concepts, this book presents important principles in a clear, concise manner that will enable the nonspecialist to read and comprehend crystallographic literature. Explanations are concise and mathematical prerequisites have been kept to a minimum.
In the first four chapters, the author presents the vocabulary of crystallography, with discussions of lattice points, unit cells, symmetry, point groups, crystal systems, space groups, and equivalent positions. The principles of x-ray diffraction and methods of determining crystal structures are summarized in the next two chapters. The final chapter describes various simple structures. Appendixes list the 230 space groups, introduce the reciprocal lattice, and describe the powder method. A well-chosen selection of problems (with solutions) encourages self-study.
Ideal as the basis for a course in crystallography and highly useful as an adjunct to physical chemistry courses, this book will also serve as an excellent reference for practicing chemists, mineralogists, metallurgists, and other workers in the field. 1969 edition.
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Introduction to Crystallography
By Donald E. Sands
Dover Publications, Inc.Copyright © 1975 Donald E. Sands
All rights reserved.
CRYSTALS AND LATTICES
Crystallography is concerned with the structure and properties of the crystalline state. Crystals have been the subject of study and speculation for hundreds of years, and everyone has some familiarity with their properties. We will concentrate on those aspects of the science of crystallography that are of interest to chemists. Our knowledge of chemistry will help us to understand the structures and properties of crystals, and we will see how the study of crystals can provide new chemical information.
1-1 Definition of a crystal
Crystals frequently have characteristic polyhedral shapes, bounded by flat faces, and much of the beauty of crystals is due to this face development. Many of the earliest contributions to crystallography were based on observations of shapes, and the study of morphology is still important for recognizing and identifying specimens. However, faces can be ground away or destroyed, and they are not essential to a modern definition of a crystal. Furthermore, crystals are often too small to be seen without a high-powered microscope, and many substances consist of thousands of tiny crystals (polycrystalline). Metals are crystalline, but the individual crystals are often very small, and faces are not apparent. The following definition provides a more precise criterion for distinguishing crystalline from noncrystalline matter.
A crystal consists of atoms arranged in a pattern that repeats periodically in three dimensions.
The pattern referred to in this definition can consist of a single atom, a group of atoms, a molecule, or a group of molecules. The important feature of a crystal is the periodicity or regularity of the arrangement of these patterns. The atoms in benzene, for example, are arranged in patterns with six carbon atoms at the vertices of a regular hexagon and one hydrogen atom attached to each carbon atom, but in liquid benzene there is no regularity in the arrangement of these patterns.
The fact that benzene is a liquid rather than a gas at room temperature is evidence of the existence of attractive forces between the molecules. In the case of benzene these are relatively weak van der Waals' forces, and thermal agitation keeps the molecules from associating into ordered clusters. If benzene is cooled below its freezing point of 5.5°C, the kinetic energy of the molecules is no longer sufficient to overcome the intermolecular attractions. The molecules assume fixed orientations and positions with respect to each other, and solidification occurs. As each molecule joins the growing solid particle, it is oriented so as to minimize the forces acting upon it. Each molecule entering the solid phase is influenced in almost exactly the same way as the preceding molecule, and the solid particle consists of a three-dimensional ordered array of molecules; that is, it is a crystal.
Another example is afforded by a crystal of sodium chloride. The crystal contains many positive and negative ions held together by electrostatic attractions. The details of the arrangement depend upon the balancing of attractive and repulsive forces, which include both electrostatic and ionic size effects. The structure of sodium chloride is shown in Fig. 1-1, and further discussion will be given in Section 7-8. Each ion is surrounded by six ions of opposite charge, at the vertices of a regular octahedron, and the crystal structure represents an arrangement of these ions that leads to a potential energy minimum.
The point we want to emphasize here is that the formation of a solid particle naturally leads to crystallinity. There is a preferred orientation and position for each molecule to attach to the solid, and if the rate of deposition is slow enough to let the molecules attain this favored arrangement, the structure will fit our definition of a crystal. We have a pattern consisting of atoms or molecules. This pattern may be as simple as a single atom or it many consist of several molecules, each of which may contain many atoms. This entire pattern repeats over and over again, at regularly spaced intervals and with the same orientation, throughout the crystal.
1-2 Lattice points
Suppose we imagine a tiny creature wandering through the interior of a crystal. He stops at some point and closely examines his surroundings, and he carefully notes his position relative to the various atoms that constitute the pattern. He then walks in a straight line to an identical point in an adjacent pattern. For example, he might travel from one Na+ ion to another Na+ ion in sodium chloride or from the center of one ring to the center of another ring in benzene. When he arrives at this second point, there is absolutely nothing in his environment that will enable him to detect that he has moved at all. Furthermore, if he continues his walk without turning, he will come to another identical point when he has covered the same distance. Of course, the surroundings will look different near the surface, but in much of our discussion we will assume that the crystal contains so many repeating patterns that surface effects are quite negligible.
A useful two-dimensional analog of a crystal is an infinite wall covered with paper. The wallpaper pattern can be of any complexity, and the entire pattern repeats periodically in two dimensions. The array is actually periodic in the direction defined by any line connecting two identical points, but all of these directions can be described by taking vector sums of two arbitrary nonparallel base vectors.
Figure 1-2 shows a rather simple wallpaper pattern. To aid our discussions and calculations, it is convenient to choose some points and axes of reference. A system of reference points may be obtained by choosing one point at random (Fig. 1-3). All points identical with this point constitute the set of lattice points (Fig. 1-4). These points all have exactly the same surroundings, and they are identical in position relative to the repeating pattern or motif. This set of identical points in two dimensions constitutes a net. The term lattice or space lattice is frequently reserved for a three-dimensional distribution of points, and in one dimension the proper term is row.
1-3 Unit cells
If we now connect the lattice points by straight lines we can divide our two-dimensional space into parallelograms (Fig. 1-4). In three dimensions the space is divided into parallelepipeds. Repetition of these parallelepipeds by translation from one lattice point to another generates the lattice. The generating parallelepiped is called a unit cell. A unit cell is always a parallelepiped, and it is sort of a template for the whole crystal. If we know the exact arrangement of atoms within one unit cell, then we, in effect, know the atomic arrangement for the whole crystal. The process of determining the structure of a crystal consists, therefore, of locating the atoms within a unit cell.
The size and shape of a unit cell may be specified by means of the lengths a, [??], and c of the three independent edges and the three angles α, β, and γ between these edges. These quantities are shown in Fig. 1-5. The angle α is the angle between [??] and c, β is between a and c, and γ is between a and [??]. These axes define a coordinate system appropriate to the crystal. In some respects it would be simpler to always use a Cartesian coordinate system, in which the three axes are equal in length and mutually perpendicular, but the advantages of a coordinate system based on the lattice vectors outweigh the simplicity of Cartesian geometry.
EXERCISE 1-1 The density of NaCl crystals is 2.16 g/cc. Referring to Fig. 11, calculate the length of the edge of a unit cell of NaCl. (The unit cell in this case has a = [??] = c, α = β = γ = 90°.)
1-4 Fractional coordinates
The location of a point within a unit cell may be specified by means of three fractional coordinates x, [y, and z. The point x, y, z is located by starting at the origin (the point 0,0,0) and moving first a distance xa along the a axis, then a distance y [??] parallel to the [??] axis, and finally a distance zc parallel to the c axis (Fig. 1-6). If one of these coordinates is exactly 1, then the point is all the way across the unit cell, and if one of the coordinates exceeds 1, the point is in the next unit cell. For example, the point (1.30,0.25,0.15) is in the next unit cell to the right in Fig. 16. This point is equivalent to (0.30,0.25,0.15) since all unit cells are identical. It is thus apparent that a crystal structure can be entirely specified with fractional coordinates. One of the advantages of basing our coordinate system on lattice vectors is that two points are equivalent (or identical) if the fractional parts of their coordinates are equal. Also note that (-0.70,0.25,0.15) is equivalent to (0.30,0.25,0.15) since the x coordinates only differ by an integer; that is, equivalent points result when any integer is added to a coordinate.
1-5 Unit cell calculations
Calculations involving oblique coordinate systems are certainly more tedious than they would be if the axes were at right angles to each other, but compensation is provided by features such as the identity of the fractional coordinates of equivalent points in different unit cells. The following formulas will be useful.
The volume V of a unit cell is given by
V = abc(1 - cos2 α - cos2 β - cos2γ + 2 cos α cos β cos [gamma)1/2(1-1)
The distance / between the
points x1, y1, z1, and x2, y2, z2 is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1-2)
You should verify these formulas for the familiar case where α = β = γ = 90°. Derivation of these formulas is accomplished easily by means of vector algebra.
1-6 Primitive and centered cells
As is shown by Figs. 1-4, 1-7, and 1-8, the choice of a unit cell is not unique. Any parallelepiped whose edges connect lattice points is a valid unit cell according to our definition, and there are an infinite number of such possibilities. It is even permissible to have lattice points inside a unit cell (Fig. 1-8). In such cases there is more than one lattice point per unit cell, and the cell is centered. A unit cell with lattice points only at the corners is called primitive. The centered cell in Fig. 1-8 is doubly primitive. A unit cell containing n lattice points has a volume of n times the volume of a primitive cell in the same lattice (see Exercise 1-2).
EXERCISE 1-2 A primitive unit cell has a = 5.00, [??] = 6.00, c = 7.00 [Angstrom]; and α = β = γ = 90°. A new unit cell is chosen with edges defined by the vectors from the origin to the points with coordinates 3,1,0; 1,2,0; and 0,0,1.
(a) Calculate the volume of the original unit cell.
(b) Calculate the lengths of the three edges and the three angles of the new unit cell.
(c) Calculate the volume of the new unit cell.
(d) Calculate the ratio of this new volume to the volume of the original cell. How many lattice points does the new cell contain?
EXERCISE 1-3 A unit cell has dimensions a = 6.00, b = 7.00, c = 8.00 [Angstrom], α = 90°, β = 115.0°, γ = 90°.
(a) Calculate the distance between the points 0.200,0.150,0.333 and 0.300, 0.050, 0.123.
(b) Calculate the distance between the points 0.200,0.150,0.333 and 0.300, 0.050,—0.123.CHAPTER 2
Some of the earliest studies of crystals were motivated by observations of their external symmetry. Snowflakes have hexagonal shapes; sodium chloride forms perfect cubes; and crystals of alum are frequently regular octahedra. Our study of crystallography will show us that these symmetrical shapes are manifestations of the internal structures of crystals. If the individual molecules have symmetry, then it is perhaps reasonable that they should pack together in a symmetrical array. There are some complicating factors, however, in the actual development of a crystal, and the arrangement of molecules is influenced by the availability of space as well as by the symmetry of the intermolecular forces. In the case of the very symmetrical benzene molecule, for example, it may turn out that six additional benzene molecules cannot approach closely from six equivalent directions without interfering with each other. Some of the molecules tilt to achieve a balance of attractive and repulsive forces, and the aggregate has less symmetry than the single molecule. In energy terms, the potential energy minimum is associated with a lower symmetry. It is also possible to have crystalline arrays where the symmetry is higher than that of the individual molecules, and we will encounter examples of this as we proceed. The important point here is that symmetry in crystals is a result of essentially the same factors that make crystallinity the natural state of a solid particle. If we have learned not to be surprised that solid matter is crystalline, then we must also accept the appearance of symmetry as an inevitable consequence of some rather simple laws. We will eventually see that crystallinity itself may be regarded as a special type of symmetry.
The utility of symmetry considerations extends beyond their application to crystals. For example, it is extremely useful to know that all six carbon atoms in a benzene molecule are identical (related by symmetry), and calculations pertaining to molecular vibrations or chemical bonding are vastly simplified by taking such symmetry into account. For this reason, we will devote this chapter to a general discussion of symmetry concepts and nomenclature, especially as applied to molecules. This material is indispensable to our later treatment of crystal geometry and symmetry; it is hoped that it will also provide some background for those who want to learn more about the role of symmetry in chemistry.
2-2 Definition of symmetry
So far we have spoken of symmetry as though everyone knew what is meant by the term. This is probably true, at least qualitatively, but a definition will ensure a common understanding of the symmetry concept. An object or figure is said to have symmetry if some movement of the figure or operation on the figure leaves it in a position indistinguishable from its original position. That is, inspection of the object and its surroundings will not reveal whether or not the operation has been carried out.
2-3 Symmetry operations and elements of symmetry
A molecule of H2O is shown in Fig. 2-1. The H—O—H angle is 104.5°, and the dotted line in the picture is the bisector of this angle. Suppose that the H2O molecule is rotated 180° about the axis represented by the dotted line. The oxygen atom will be rotated 180°, but it will end up looking exactly as it did before the rotation. The two hydrogen atoms will be exchanged, so that now H(2) is on the left and H(1) is on the right. However, there is no possible way of telling the hydrogen atoms apart, since all hydrogens atoms are identical and in this molecule they have identical chemical environments. The 180° rotation has left the molecule in a position indistinguishable from its original position, so our definition of symmetry is satisfied. In H2O the 180° rotation about the bisector of the H—O—H angle is a symmetry operation, and the rotation axis is a symmetry element. This particular symmetry element is designated by the symbol C2 in the Schoenflies notation used extensively by spectroscopists, or simply by the symbol 2 in the Hermann-Mauguin or international notation preferred by crystallographers. Besides designating the symmetry element, the symbol C2 (or 2) also implies the operation of rotation by 180°.
2-4 Rotation axes
A symmetry element for which the operation is a rotation of 360°/n is given the Schoenflies symbol Cn (or the Hermann-Mauguin symbol n). For example, the chloroform molecule (CHCl3) has a C3 axis. In this molecule (Fig. 2-2), the three chlorine atoms form an equilateral triangle, the carbon atom is directly above the center of this triangle, and the hydrogen atom is directly above the carbon atom. If the molecule is rotated about the axis defined by the C—H direction, it reaches identical orientations after every 120° of rotation, and there are a total of three equivalent orientations in a complete 360° turn.
Excerpted from Introduction to Crystallography by Donald E. Sands. Copyright © 1975 Donald E. Sands. Excerpted by permission of Dover Publications, Inc..
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