CRYSTAL DESCRIPTION AND CALCULATION A. INTRODUCTION This exercise develops the framework necessary for describing a crystal. In essence we shall discuss how we fix the position of any crystallographic plane in space, and how such a plane may be given a convenient but meaningful name. Among the things we shall address are: defining the crystallographic axes by system; parameters and parametral plane; axial ratio and other interaxial relations; angular coordinates, intercepts, and miller indices of a plane; zone and zone axis. B. CONCEPTS AND DEFINITIONS The following concepts and definitions are essential to aid the students to become familiar with the terms which would be constantly used in this exercise as well as in the following set of exercises.. Crystallographic Axes These compose a set of reference axes similar to the cartesian system used to describe the elements of a crystal. It differs from the cartesian system in the sense that the crystallographic axes are made to fit a crystal's symmetry, more precisely its unit cell or lattice. In general, each crystal system has its own characteristic crystallographic axes. But, in detail, even individual crystals belonging to the same crystal system or class each has its unique set of crystallographic axes. Crystallographic axes are defined such that they are either parallel or perpendicular to prominent symmetry elements. This practice insures that the crystallographic axes are parallel to principal lattice lines, and that they reflect the symmetry of the crystal they correspond. Furthermore the crystallographic axes also reflect the properties of the crystal lattice. 2. Choice of Crystallographic Axes The choice of the crystallographic axes is dictated by the crystal symmetry. Consequently, these axes are chosen differently and have different properties and relations for each crystal system.
2.. Isometric system GEOL. 40 In this system the crystallographic axes are taken as the lines parallel to the axes of four-fold symmetry, A 4 (A 2 if no A 4 is present). Since one cannot distinguish between one A 4 from another, the crystallographic axes are usually chosen as follows: a - axis facing the observer; a 2 - axis horizontal and parallel to the plane of the observer; and a 3 - axis oriented vertically and parallel to the plane of the observer The following interaxial relations also follow. For the angle between the crystallographic axes we have the following: A = a 2 ^ a 3 = 90 degrees B = a ^ a 3 = 90 degrees C = a ^ a 2 = 90 degrees For the linear relations of the axes we have a = a 2 = a 3 (relation of the unit lengths) 2.2. Hexagonal System This system has a unique symmetry axis, either A 6 or A 3, which is made to coincide with the c- axis. Perpendicular to the A 6 or A 3 are six or three A 2 's. The horizontal axes, a, a 2, and a 3, are made to coincide with the three horizontal A 2 axes, which are separated by an angle of 20 degrees and are equal in unit lengths. The following relations characterize the system. For the interaxial angles, we have A = a 2 ^ c = B = a ^ c = 90 degrees C = a ^ a 3 = a ^ a 2 = a 2 ^ a 3 = 20 degrees 2.3. Tetragonal System For this system, the c-axis is taken as coincident with the 4-fold axis of symmetry, A 4, while the two other axes perpendicular to it are referred to 2-fold axes, A 2. In several classes every pair of crystallographic axes, (a, a 2 ); (a, c); (a 2, c), is contained within a mirror plane. 2
A feature that distinguishes it from other systems is the equality of the unit lengths of the orthogonally arranged a and b axes such that they are designated, respectively, as a and a 2. The c-axis is chosen as the one which is the shortest or the longest of the three axes. In summary the interaxial relationships characteristic of the tetragonal system are: a - front-back horizontal position a 2 - left-right horizontal position c - vertical position The above axes have the following relationship a = a 2 > or < c The interaxial angles are A = c ^ a 2 = B = c ^ a = 90 degrees C = a ^ a 2 = 90 degrees 2.4. Orthorhombic system All of the three crystallographic axes are made parallel to the 2-fold axes of symmetry, A 2. The c-axis is taken to be parallel to the direction of elongation or perpendicular to a prominent pinacoid form or cleavage of the crystal. In orienting an orthorhombic crystal, the c-axis is referred to the vertical position. The two horizontal axes, one in front-back and one in left-right position, represent the a and b crystallographic axes respectively. Crystallographic dimensions and interaxial angles characteristic of the system are as follows: a =/= b =/= c =/= a b > c > a (relative lengths) A = B = C = 90 degrees 3
2.5 Monoclinic System GEOL. 40 In this system the b-axis is usually referred to a 2-fold symmetry axis, A 2. An orientation preferred is one with the c-axis vertical, b-axis horizontal and to the right, a-axis inclined towards the observer. With this orientation the c- and a- axes and the angle, B, between them are contained in a vertical (usually a mirror) plane. The unit lengths and the angular relationships for the system are defined as follows: 2.6 Triclinic System a =/= b =/= c =/= a A = b ^ c = 90 degrees C = a ^ b = 90 degrees B = a ^ c > 90 degrees, where a and c refer to the positive ends of the crystallographic axes For this system, the c-axis is taken as the axis of the most prominent vertical zone. A distinctive feature is that, the three crystallographic axes are obliquely inclined with each other such that no two interaxial angles are equal. The following crystallographic constants distinguish the system from the others. (Note that a, b, c, refer to the positive ends of the crystallographic axes). A = b ^ c > 90 degrees B = a ^ c > 90 degrees C = b ^ a < or > 90 degrees b > a > c The following figure illustrates the crystallographic axes and the interaxial relationship of each of the crystal systems, where A, B, C, represent the interaxial angles; a,b,c, represent the crystallographic axes; and a, b, c represent the unit lengths in the three corresponding axes. The axes have positive and negative ends similar to the three-dimensional coordinate system. 4
Figure 2. The Crystallographic Constants of the Six Crystal Systems. Isometric System 4. Orthorhombic System 2. Hexagonal System 5. Monoclinic System 3. Tetragonal System 6. Triclinic System 5
3. Fixing the Position of a Crystal Plane GEOL. 40 The position of a crystal plane, given the crystallographic axes can be fixed in three ways. These are described in the succeeding discussions. 3. By Intercept Method (or Miller Indices) If the lengths of the intercepts that a plane makes with respect to the crystallographic axes are known, then the position of the plane is known. For instance, if a plane intersecting all the axes has an equation 6(x/a) + 4(y/b) + 3(z/c) = 2, where a,b,c are unit lengths, then, its intercepts can be determined as: x = OA = 2a, y = OB = 3b, and z = OC = 4c Thus, the figure can be drawn as in Figure 2.2. Figure 2.2 Intercept Method of Fixing the Position of a Crystal Plane Crystallographers have a shorter way of designating the position of the same plane by Miller Indices instead of equation. By definition, the Miller Indices are the reciprocals of the intercepts of the plane, provided the intercepts are expressed in terms of the unit lengths on the axes. That is, the Miller Indices of a plane can be defined as follows: MI = (h k l) = OA / a, OB / b, OC / c,where, 6
h = OA / a = a OA k = l = OB / b OC / c = = b OB c OC For the given example, the resulting Miller Indices are: h = a 2a = 2 k = b 3b = 3 l = c 4c = 4 After clearing off fraction we get the following: (h k l ) = 2, 3, 4 = (6 4 3) as the Miller Indices of the given plane. Conversely, if the miller indices of a plane are given, the intercepts can be calculated and the desired plane can be drawn. 3.2. Two-Circle Method Imagine that the crystal is inscribed within a sphere as in Figure 2.3. From the common center, a line normal to a given crystal face can be drawn and extended until it intersects the inscribing sphere. The intersection of the sphere and the line normal to the crystal face is the pole of the face. This pole can be given spherical coordinates called the phi and rho coordinates. The rho angle is the angle that the line normal to the face makes with the vertical. It is analogous to the co-latitude of a point on the globe. The phi angle is the angle between the great circle passing through the intersection of the b-axis and the sphere and the great circle passing through the given point, measured along the arc of the equatorial circle. It is the analogue of the longitude of a point on the globe. 7
Together, the two angles phi and rho of a crystal plane (actually of its pole) fixes its position on the sphere, in the same way that the longitude and latitude of a place, say Manila, fix its position on the globe. Figure 2.3 Two-Circle Method The planes of an octahedral crystal inscribed within a sphere can be represented by normal lines originating from the center of the sphere. The spherical coordinates, phi and rho, of the intersection of these lines with the sphere (called poles) fix their exact positions. 3.3 Single-Circle Method A plane's position can also be fixed by measuring the angle it makes with some planes whose positions are known. The angle, which a crystal plane makes with another, is called the interfacial angle, which is designated as I in Figure 2.4. 8
Figure 2.4 Single-Circle Method Coordinates of face AC (drawn normal to the plane of the paper) is angle I, the interfacial angle between AC and BC. This means that the position of a plane can be fixed by measuring its interfacial angle with respect to some known plane, which is contained in the same great circle. 4. Parameters and Parametral Plane The intercepts of a crystal plane with the crystallographic axes are known also as its parameters. Specifically, a unit plane, which intersects all the crystallographic axes at unit lengths, has the parameters a, b, and c. However, the unit planes of the crystals belonging to different crystal systems are not equivalent despite similar parameters and indices. For instance, the () face in the isometric system is very different from the corresponding face in the orthorhombic system. This is due to the fact that each crystal system has its own distinct crystallographic constants. A parametral plane, (hkl), is one whose Miller indices would permit the calculation of the axial ratio of a crystal. It is also used to define the parameters or unit lengths of the axes. Generally, the chosen parametral plane intersects all the axes, is the most prominent and well developed face, and has simple (low) and rational indices. 9
5. Axial Ratio The axial ratio is an expression of the relative lengths or dimensions of the unit lengths on the crystallographic axes. It is derived by dividing the unit length of each axis by the unit length of the b-axis. Thus, a b : b b : c is the general form of the b axial ratio and can be expressed as P: : Q in the triclinic, monoclinic, and orthorhombic systems. In the tetragonal and hexagonal systems, where the horizontal axes are equal, the axial ratio takes a different form and can be expressed as :Q, where Q is the ratio of the unit length on the vertical axis to any of the horizontal axes. For the isometric system the axial ratio is : : since the unit lengths of its crystallographic axes are equal. The axial ratio for any system can also be calculated if the exact relationship of the crystallographic axes with the phi and rho coordinates and the indices of a crystal plane are known. This relationship is expressed in mathematical formula and is different for each system. However, the derivation is generally the same for all systems. As a general case, the derivation of the mathematical relationship among the axial ratio, phi and rho coordinates and indices of a crystal plane for the orthorhombic system is shown as an example below. The same general method applies for all the other systems, though quite tedious for the monoclinic and triclinic systems. For our purpose the orthorhombic case is sufficient to illustrate the pertinent relationships among the angular coordinates, Miller indices, and axial ratio in any given crystal. 0
Consider a plane ABC having intercepts OA, OB, and OC along a, b, and c axes and miller indices () as illustrated in Figure 2.5. Figure 2.5 Illustrations of the trigonometric relationships of the parametral plane () with the crystallographic axes. We know that rho = <COQ and phi = <BOD from the previous discussions. We also have the following relationships from figure 2.5: cos P = OD / OB tan R = OC / OD tan P = OB / OA (b-) (c-) (b-2) Rearranging, b- becomes OD = OB cos P, and substituting this equation in c-, we have: tan R = OC OBcosP (c-2) Using the definition of intercepts, c-2 and b-2 become tan R = c / l ( b / k)cosp and
tan P = b / k a / h To obtain expressions for the axial ratio, we rearrange the above equations into: tan R cos P = (c/b) (k/l) and cot P = (a/b) (k/h) For the plane being considered, we need only to know its phi and rho coordinates to get the axial ratio of the crystal it represents since (k/l) and (k/h) are both equal to unity. 6. Zone and Zone Axis A zone is composed of a group of faces that are parallel to a common line of reference called the zone axis (fig. 2.6). The zone is designated by zone symbols, [uvw], which is derived from any two of the crystal faces belonging to the zone. Figure 2.6 Illustration of a zone composed of four faces (00), (20), (20) (00) and the zone axis [20]. 2
The derivation of the zone symbol is comparable to the method of determinants wherein the variables consisting of two sets are cross-multiplied and the products are algebraically added to give the desired result. The variables in this case are derived from the indices of any two-crystal faces lying in the same zone. For example, we want to know the zone symbol for the zone composed of the faces (00) and (20) shown in Figure 2.6. In general, what we do are as follows: (a) choose any two representative faces; (b) write the indices of the chosen faces twice, one set of indices directly above or below the other; (c) perform the cross multiplication. Suppose we choose (00) and (20), then, by performing the next two steps we have: and 0 0 0 0 2 0 2 0 u = (0)(0) - ()() = - v = ()(2) - (0)(0) = 2 w = (0)() - (2)(0) = 0 the zone symbol, therefore is: [uvw] = [20]. The student can try to derive the zone symbol using any other combination of the faces. The result should be the same for the faces lying on the same zone. 3
E. REFERENCES Berry, L. G. and B. Mason (959) Mineralogy: concepts, descriptions, determinations. San Francisco: W. H. Freeman and Company, pp. 25-35, 43-50. Bishop, A. C. (967) An outline of crystal morphology. London: Hutchinson and Co. (Publishers) Ltd., pp.76-82. Buerger, M. J. (956) Elementary Crystallography. New York: John Wiley and Sons, Inc., pp. 99-. Hurlbut, C. S. and C. Klein (977) Manual of Mineralogy. 9th ed.. New York: John Wiley and Sons, Inc., pp. 58-98. Philips, F. C. (949) An introduction to crystallography. st ed.. Great Britain: Robert Maclehouse and Co. Ltd., pp. 40-43, 8-23. Wade, F. A. and R. B. Mattox (960) Elements of crystallography and mineralogy. New York: Harpers and Brothers Publishers, pp. 27-35. 0/4/98: lab-ex2.doc 4