Handbook of Modern Coating Technologies

Milestones Crystalline materials

Like any other branch of science, X-ray techniques too have benefitted from developments in other streams of science and technology, ranging from materials to computing. At the forefront of technological development, this relationship has been symbiotic in nature, as the developments in X-ray techniques have in-turn provided technological push to allied fields, particularly materials science. The versatility of X-ray techniques makes it difficult to tabulate the related developments at one place. A partial timeline depicting major developments in various fields directly concerned with X-rays and allied techniques is provided in Table 3—1.

The associated Nobel awards serve as testimony to the importance and versatility of the developments concerning X-ray techniques. In addition to the developments provided in Table 3—1, many more Nobel Laureates have otherwise benefitted from these developments by using the techniques in their respective areas.

• Crystalline materials

Before the interaction of X-rays with crystalline materials can be understood, it is important to understand certain aspects regarding the crystalline materials. An understanding of crystal struc­ture not only helps classify the vast spectrum of crystalline materials into a finite number of fun­damental structures, but also helps visualize the outcome of their interaction with X-rays.

Solid materials are broadly classified as crystalline and amorphous. The crystalline solids are characterized by a periodic arrangement of atoms. The structure of crystalline solids can either be in the form of a single crystal (namely, a diamond crystal or a calcite crystal) or as an assemblage of multiple crystals with distinct boundaries (namely, grains in metals). This section deals with the concepts of crystal geometry, which are essential for dealing with the subject of XRD by crystals.

• Space lattice and unit cell

The study of crystal structure had started decades before the advent of X-rays or their applica­tion in materials science in the form of diffraction analysis (refer Table 3—1). The structure of crystals is based on some periodic arrangement of atoms according to some particular frame­work, known as space lattice. The space lattice comprises of a three-dimensional arrangement of points, in which the surroundings of every point are identical. For the convenience of draw­ing on two-dimensional sheet of paper, consider the planar array of points shown in Fig. 3—4, which is formed by repeated translation of lattice points along two translation (basis) vectors a and b, making an interaxial angle a, while the third vector c points outward from the plane of paper. The parameters a, b and a together govern the geometry of the lattice. Since by defi­nition of lattice, this array is considered to extend infinitely along both the directions, we can

work-out generalized expressions to calculate the distances to various lattice points surround­ing any given lattice point (refer inset in Fig. 3—4, e.g.).

The space lattice can be expressed in terms of a unit cell—the smallest building block, which when repeated indefinitely, will generate the framework of the corresponding space lattice. The unit cell for a given space lattice can be selected in various convenient ways as shown by the shaded parallelogram toward bottom of Fig. 3—4. Translation of the first unit cell by vectors 6 a/2 and 6 b/2 would center it on a lattice point, similar to the choice of

second unit cell shown in the figure. It should be noted that in either of the cases, the effec­tive number of lattice points per unit cell obviously remains one. Such unit cells, which con­tain only one lattice point, are termed as primitive cell.

Extending the discussion to three dimensions, we arrive at a three-dimensional space lat­tice, an example of which is shown in Fig. 3—5. It is defined by three basis vectors a, b, and c (refer Fig. 3—6) which are designated in the right-handed sense for uniqueness, that is, if the index finger of right hand points in the direction of a, the middle finger and thumb would point in the directions of vectors b and c respectively. The standard notation regarding angles between these vectors designates the angle between b and c as a, between c and a as в and that between a and b as 7. Sometimes, it becomes more convenient to designate the three basis vectors a, b, and c as a_x, a₂, and a₃, respectively.

• The Bravais lattices and crystal structure

Changing the relative lengths of lattice basis vectors a_x and interaxial angles a, в, and 7 (refer Fig. 3—6) leads to formation of differently shaped unit cells and hence, different types of lattice systems. In 1848 French crystallographer Bravais discovered that only fourteen types of lattice systems are possible [31]. In his honor, the term Bravais lattice is often used

in place of space lattice. The fourteen Bravais lattices are based on seven types of unit cells, corresponding to seven types of crystal systems, into which all possible crystalline materials can be classified. Each crystal system is characterized by its compliance to a particular com­bination of macroscopic symmetry transformations of rotation, reflection, inversion and rotoinversion. An n-fold rotational symmetry about an axis calls for self-coincidence for every 360 degrees/n degrees of rotation. It is not possible to have a fivefold or higher than sixfold rotational symmetry, as it will lead to gaps when such unit cells are made to fill space. A detailed discussion on lattice systems and associated symmetries is beyond the scope of this chapter and the interested reader can refer [78] for details.

Rotational symmetry of a given lattice about different axes gives rise to different sets of equivalent planes. Known as planes of a form or family of planes, these are represented col­lectively by enclosing the Miller indices (refer Section 3.5.3 for details) in curl brackets {hkl}. The planes of a form are generally characterized by different Miller indices [30], but same d-spacing. For example, all six faces of a cubic lattice are related by symmetry and have the same d-spacing. Likewise, they can be collectively represented as {100} planes. For a tetrago­nal unit cell, the four side faces belong to one family {100}, while the top and bottom faces form another family {001}. The interplanar spacing, d_hki for parallel planes belonging to the same set (hkl), depends on the indices (hkl) as well as lattice parameters, namely, a, b, c, a, в, and 7. It should be noted that for cubic lattice, the direction [hkl] is normal to the set of planes (hkl) but may not be obviously so for other types of lattice systems. The procedure for designating lattice directions is explained in Section 3.5.3.

A crystal is formed through association of a basis with each lattice point. The bases for dif­ferent crystals range from a single atom to groups of hundreds of atoms. The atomic arrange­ment is governed by combinations of lattice type and symmetry elements, to form 230 unique space groups. The number of nearest neighbors in the bulk of a crystal lattice is known as bulk coordination number. As an example, the coordination number is six for sim­ple cubic lattice; but for complex crystals, the coordination number is subject to the way it is calculated. For simplest examples of monoatomic crystals such as Cu, where each atom occupies a face centered cubic (FCC) lattice point, the lattice parameter and packing fraction can be straightaway calculated from coordination number and atomic radius, since the atoms are supposed to be touching each other (refer Figs. 3—10—3—13). Diatomic ionic solids such as NaCl have two atoms associated with each FCC lattice point. It should be noted that for ionic solids, the atomic radii are affected by degree of ionization—with the negative ions registering an increase in size and vice versa. Furthermore owing to periodicity and symmetry of lattice, there is no way to tell as to which Na^[1] ion is bonded to which Cl^_ atom and so, the term "molecule” becomes moot in the context of crystal structure. The unit cells of proteins may contain thousands of atoms each.

з.   5.3 Designation of points, lines, and planes

Direction vectors are generally required for expressing relative orientation of certain lattice points with respect to one another. Choosing an arbitrary lattice point as origin, the coordi­nates of any other lattice point can be arrived at by a position vector r expressed in terms of integral multiples u, v, and w respectively of the basis vectors a, b, and c, as follows:

r = ua 1 vb 1 wc                                           (3.3)

As an example, taking origin at the tail of vector p shown in Fig. 3—4, the components

и,  v, and w of p along a, b, and c are found to be 1, 2, and 0, respectively. Corresponding Miller indices (named after the English crystallographer W.H. Miller) for the crystal direction p are then written as [120] in square brackets. In case the indices happen to contain double digit figures, it is advisable to separate them with commas to avoid confusion, namely [1,12,3]. Describing the direction indices with reference to the framework provided by basis vectors is quite simple as it eliminates the need to account for lengths of basis vectors as well as their interaxial angles. The parameters of length or interaxial angles are required only when one is interested in finding actual length of the direction vectors or distances between certain lattice points of interest (refer inset in Fig. 3—4).

The Miller indices are expressed as smallest integers, while negative component of a direction is denoted by an overbar. So, the indices for a direction such as 2, 23, and 1 would be expressed as [231], while indices for a direction 4, —6, and 2 should be written as 2[231]. All possible combinations of the direction indices, including both positive and negative values, represent various directions, which are related by symmetry. These are termed as directions of a form or a family of directions and are represented using pointed brackets < > around Miller indices. For example, the directions [231], [213], [231], [312], [321], [123], etc. can be collectively represented in the form of a family, as <231 >. Similarly the families <100>, < 110>, and < 111 > represent respectively the 6 edges, 12 face diagonals, and 4 body diagonals of a cubic unit cell. It should be noted that crystal directions of a family are not necessarily parallel and as the reader would be able to observe in the discussion con­cerning crystal planes, the same rule remains valid for planes as well.

For locations other than the lattice points in primitive unit cells (or lattice points in nonpri­mitive unit cells or for that matter any other location within the unit cell), the vector r is expressed by adding fractions n, p, and q, respectively, to u, v, and w, as r = (u 1 n)a 1 (v 1 p) b 1 (w 1 q)c. Rearranging the terms, we get r = (ua 1 vb 1 wc) 1 (na 1 pb 1 qc), that is, a vector from origin up to a lattice point of the unit cell containing the location of interest, plus another vector from that lattice point to the location of interest within the unit cell.

A plane can be uniquely defined in terms of its intercepts along three noncoplanar direc­tion vectors. The lattice basis vectors form a convenient frame of reference for this purpose. However, if a plane is parallel to any of the vector(s), its corresponding intercept will work out to be infinity. To avoid this situation, Miller introduced the concept of taking reciprocals of the intercepts, thereby converting infinity to zero. Thus the Miller indices of a plane are its frac­tional intercepts with corresponding crystallographic directions. To make a distinction between notations used for directions and planes, the Miller indices for planes are written in parenthe­ses as (hkl). A plane having indices (hkl) would intersect the three axes a, b, and c at distances of a/h, b/k, and c/l, respectively, from the origin. The procedure for writing Miller indices is explained in Table 3—2 with reference to the (112) plane shown in Figs. 3—7 and 3—8.

Some examples for writing Miller indices of planes are shown in Fig. 3—9 for a simple cubic lattice (a = b = c, a = в = у = 90 degrees). In general, the Miller indices for a set of plane are expressed in terms of the plane that is closest to the origin. However, as shown for (022) and (410) planes in Fig. 3—9, the Miller indices may be attributed to an y plane within the set, or even the whole set taken together. Also note how the origin has to be shifted men­tally to find intercepts on the negative side.

The interplanar spacing d_hkl between adjacent planes having Miller indices (hkl) is defined as the distance between first such plane from a parallel plane passing through the origin. Interplanar spacing can be visualized in the examples shown in Fig. 3—9 in the form of a perpendicular dropped from origin (or shifted origin wherever applicable) to the nearest plane. It is apparent that the planes with large indices have small spacing between them and

Table 3-2 Procedure for determination of Miller indices. ¹

FIGURE 3-9 Examples of some Miller indices: (a): (010), (b): (020), (c): (111), (d): (012), (e): (110), (f): (142), (g): (022) and (h): (410)

pass through fewer lattice points. On the other hand, planes with low indices have large spacing and pass through large number of lattice points. As an aid to visualization, Figs. 3—10 and 3—11 show sections along (110) and (111) planes respectively for a monoa­tomic simple cubic crystal, while sections through same planes for a monoatomic FCC crys­tal are shown in Figs. 3—12 and 3—13.

Owing to periodicity of lattice, corresponding to every plane (hkl), there exists a whole set of equidistant parallel planes. It must be obvious by now that planes having Miller indices in

c

b

FIGURE 3-11 Monoatomic simple cubic crystal sectioned along (111) plane.

multiples of each other should also be parallel. For example, the (010) planes are parallel to (020) planes and consecutive planes from the (010) set coincide with alternate planes from the (020) set. Planes, whose indices are negative of each other, namely, (110) and (110), are also parallel but lie on opposite sides of the origin. Regardless of their indices, the set of planes that are parallel to a line, are called planes of a zone and exhibit this relationship in Laue diffraction method discussed in Section 3.8.1.

b

FIGURE 3-12 Monoatomic FCC crystal sectioned along (110) plane. FCC, Face centered cubic.

c

b

FIGURE 3-13 Monoatomic FCC crystal sectioned along (111) plane showing close-packing along this plane. FCC,

Face centered cubic.

• Reciprocal lattice

We have seen how the orientation and location of various lattice planes can be described using Miller indices. However, since the lattice is theoretically assumed to have infinite dimensions along all directions and since actual crystals too are much larger in comparison to atomic dimensions, the absolute location of a plane is not of much interest from the
perspective of crystallography. Simply stated, we are generally interested more in orientation and periodicity (i.e., spacing) of a set of planes rather than location of an individual crystal plane.

As is apparent from discussion so far, there exists an indefinite number of orientations (hkl) in a crystal and corresponding to each orientation, there exists a whole set of parallel planes. It seems very convenient if we could somehow develop a system for collectively representing the planes belonging to each set. This objective was achieved through the con­cept of reciprocal lattice, as introduced by J. Willard Gibbs and subsequently developed by P.P. Ewald. As we shall see later, since the spacing between lattice planes d_hkl is inversely proportional to the spacing between corresponding diffraction patterns, the reciprocal of a real (direct) lattice is naturally associated to its diffraction pattern.

Rather than specifying three points for describing the orientation of a plane, one can specify three components of a vector, which is normal to the plane. Essentially this vector would be normal to the entire set of parallel planes described by the family of Miller indices. For convenience, we shall adopt the notation of a_x, a₂, and a₃ in place of a, b, and c for lat­tice basis vectors.

Fig. 3—14 shows a unit cell OADBCEFG described by three noncoplanar vectors a_x, a₂, and a₃ where a_x = a₃ = a₂/2. Line OR is normal the OADB while R is coplanar with CEFG. It should be noted that the reciprocal vector b₃ points in the direction of OR, while its tip does not necessarily coincide with point R. Proceeding further, OQ is normal to OCEA while Q is coplanar with BGFD. Finally OP is normal to OBGC and P is coplanar with ADFE. In other

words, OP, OQ, and OR are heights of the unit cell normal to the three planes, or simply put, the d-spacings between opposite faces of the unit cell. Again, the points Q and P do not necessarily coincide with the tips of corresponding reciprocal vectors b₂ and b_x.

Mathematically the area of OADB is given by vector (cross) product as a₁ X a₂ and the area vector would point along OR (or along — OR for a₂ X a₁). Furthermore the volume of the parallelepiped in the form of unit cell is given by, V = a₁- a₂ X a₃. The reciprocal space vectors bi are formed by dividing the area (a vector quantity) of corresponding normal plane by volume (a scalar quantity) of the unit cell as under:

a₂ X a₃ ar a2 X a3 a₃ X a_t a2- a3 X a1 a_t X a₂ a3- a1 X a2

The length of reciprocal vectors is expressed in nm^—1 or A^—1. To develop physical under­standing of the concept, we consider the volume of unit cell as the product of its base area and height, that is:

V = Area(OCEA) X (Height of cell) = (a₃ X a₁ )OQ _ (as X a0 _ 1 ² V OQ Similarly, b₁ = 1/OP and b₃ = 1/OR

Thus the magnitude of reciprocal basis vectors equals the inverse of the d-spacing of the planes to which they are normal. So, the tip of vector bi in Fig. 3—14 represents the entire set of planes corresponding to ADFE face—that is (100) planes, while their spacing d₁₀₀ equals reciprocal of the length of b₁. Similarly the vectors b₂ and b₃ signify orientation and d-spacing of the sets of (010) and (001) planes respectively. In this manner, the transforma­tion from direct space to reciprocal space maps the set of direct space planes (hkl) to a single point having coordinates h, k, l in reciprocal space. Note that the indices in reciprocal space are specified without parentheses.

Extending the discussion further, it can be proved that a vector H_hki = hb₁ + kb₂ + lb₃ drawn from the origin to a point hkl in reciprocal space is normal to the plane having Miller indices (hkl) in direct space, while its length is reciprocal of the spacing d_hkl between these planes in real space.

As an aid to visualize this concept, an example of a two-dimensional direct space lattice along with the corresponding reciprocal space lattice is shown in Fig. 3—15. It is to reiterate that the reciprocal vectors b_i are aligned normal to respective pairs of vectors appearing in cross-product in the numerator of Eq. (3.4). As can be observed, the vectors in reciprocal
lattice are normal to their corresponding planes in direct space. Furthermore the planes with higher indices appear closer to the origin in direct space, while they are farther away from origin in reciprocal space. These properties of the reciprocal lattice make it an indispensable tool in crystallographic diffraction studies.

The volume of the reciprocal lattice (V*) works out to be reciprocal of the volume (V) of direct space lattice (refer Eq. 3.6).

V * = b₃- b₁ X b₂

a2 X a3     a3 X a1

X

ar a2 X a3 a2' a3 X a1

(a1 X a₂) [(a₂ X a₃) X (a₃ X aQ]
(a3- a1 X a2)³

(a1 X a2> [{ (a2 X a3) aj a3 - {(a2 X a3) a^ a^
V ³

(a1 X a2)- [Va3 - 0]
V ³

(a_t X a₂)- a₃ =

V²         V

The direct lattice is reciprocal of its own reciprocal lattice, that is, a₁ = (b₂ X b₃)/V*, etc., thereby revealing Pontryagin duality of their respective vector spaces. It also follows from Fig. 3—14 that for simple cubic lattice, since a = в = Y = 90 degrees, bi would be drawn par­allel to a_i and have a magnitude of 1/a (since a_x = a₂ = a₃). Thus the reciprocal of a simple cubic lattice is again a simple cubic with a lattice parameter of 1/a. Apparently a simple cubic lattice of unit dimensions would be its own reciprocal.

[1]    Write the intercepts of the plane in terms of multiples of the basis vectors a, b, and c

2. Take reciprocals of the intercepts
3. Maintaining their ratios, convert them into smallest integers (multiplying by 4 in the present case) and enclose in parentheses