Using the Miller indices

Strategy For the first part, we simply substitute the information into eqn 11.10. For the second part, instead of repeating the calculation, we should examine how d in eqn 11.10 changes when all three Miller indices are multiplied by 2 (or by a more general factor, n). 

It follows that d = 0.22 nm. When the indices are all increased by a factor of 2, the separation becomes 

for these planes d = 0.11 nm. In general, increasing the indices uniformly by a factor n decreases the separation of the planes by n. 

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In general, the lattice points forming a three-dimensional space lattice should be visualized as occupying various sets of parallel planes. With reference to the axes of the unit cell (Fig. 16.2), each set of planes has a particular orientation. To specify the orientation, it is customary to use the Miller indices. Those are defined in the following manner Assume that a particular plane of a given set has intercepts p, q, and r... 

Knowing the unit cell constants and the orientation of the crystal in the x-ray beam, each spot or reflection observed in the diffraction pattern can be defined using the Miller indices corresponding to the set of lattice planes that produced the reflection. There is a reciprocal relationship between the diffraction angle and the spacing between the lattice planes df, in the crystal. Thus, crystals with larger unit cells, such as proteins, will produce a more dense diffraction pattern than that of salt crystals or other small molecules with much smaller unit cells. 

In this case, the x-ray beam enters the sphere enters from the left and encounters a lattice plane, L. It is then diffracted by the angle 20 to the point on the sphere, P, where it is registered as a diffraction point on the reciprocal lattice. The distance between planes in the reciprocal lattice is given as 1/dhkl which is readily obtained from the diagram. It is for these reasons, we can use the Miller Indices to indicate planes in the real lattice, based upon the reciprocal lattice. 

As stated in an earlier section, a crystal is a polyhedral solid, bounded by a number of planar faces that are normally identified using the Miller indices. The arrangement of these faces is termed the habit of the crystal, and the crystal is built up through the repetition of the unit cell. The three-dimensional basic pattern of molecules in a solid... 

Miller uses the Miller indices to designates the crystal faces. 

It is convenient to characterize the various planes in a crystal by using the Miller indices h, k, 1. They are determined by first finding the intercepts of the plane with the basis axis in terms of the lattice constants, and then taking the reciprocals of... 

An account of the use of Miller indices to describe crystal planes and lattice directions is beyond the sco[>e of this article a very adequate treatment of this topic is, however, given in Reference 1. 

Figure 9-7. Elastic electron-diffraction pattern of a highly textured hcxaphenyl film. The Miller indices arc assigned using the intcrplauar spacings calculated in Kef. 11371. Inset Intensity of the f020) peak as a function of the angle between momentum transfer and the Teflon rubbing direction (see text) - taken from Ref. 138. ...

Thus, these intercepts are given in terms of the actual unit-cell length found for the specific structure, and not the lattice itself. The Miller Indices are thus the indices of a stack of planes within the lattice. Planes are important in solids because, as we will see, they are used to locate atom positions within the lattice structure. 

The Miller indices of planes in crystals with a hexagonal unit cell can be ambiguous. In order to eliminate this ambiguity, four indices, (hkil), are often used. These are called Miller-Bravais indices and are only used in the hexagonal system. The index i is given by... 

A better solution is to use a four-axis, four-index system for hep solids. A few examples are depicted in Fig. 4.10. The Miller indices are found as before by taking the reciprocals of the intercepts of the plane with the four axes. Using this system, the equivalent planes discussed above become (1100) and (1010). Now the six equivalent planes resulting from the sixfold axis of symmetry can be identified as the 1100 family of planes. Using four axes, the [hkil] direction is normal to the (hkil) plane, in the same way it was for cubic solids using the three-axis system. ... 

Person 2 Derive a relationship in simplest terms for the rf-spacing of hydroxyapatite in terms of the Miller indices only h, k, and /). Use the cell parameters in nm. 

Using the Hull-Davey chart (Fig. 21.7) determine the Miller indices of the reflections, the ratio b/a, and the corresponding ratio a. 

Lattice Type. From the angles at which X rays are diffracted by a crystal, it is possible to deduce the interplanar distances d using Eq. (3). To determine the lattice type and compute the unit-cell dimensions, it is necessary to deduce the Miller indices of the planes that show these distances. In the case of a powder specimen (where all information concerning orientations of crystal axes has been lost), the only available information regarding Miller indices is that obtainable by application of Eqs. (5) and (6). 

For a simple (primitive) cubic lattice P, there are no restrictions on the Miller indices and therefore none on KF except as noted above. In the case of a body-centered cubic lattice, only those values of KF can be allowed that arise from Miller indices whose sum is even. This has the effect of requiring KF to be even, as seen in the first of the two columns under /.We can then divide them by 2 and thereby reduce them to a relatively prime sequence, shown in the second column under /, for comparison with the sequence obtained from the ldf values. We note immediately that the relatively prime sequence obtained differs from that for a primitive cubic lattice in having gaps at different places. By use of this fact, it is almost always possible to distinguish between a primitive cubic lattice and a... 

Note that a Miller index of zero implies that the plane is parallel to that axis, since it is assumed that the plane will intersect the axis at 1 /oo. A complete set of equivalent planes is denoted by enclosing the Miller indices in curly brackets as hkl. For example, in cubic systems (1 0 0), (1 0 0), (0 1 0), (0 1 0), (001), and (0 01) are equivalent and the set is denoted in braces as 1 0 0. The maximum possible number of (h k 1) combinations that are equivalent occurs for cubic symmetry and is equal to 48. In hexagonal cells, four indices are sometimes used, h k i 1), where the relation i = —(h + k) always holds. The value of the i index is the reciprocal of the fractional intercept of the plane on the axis, as illustrated in Figure 1.10. It is derived in exactly the same way as the others. Sometimes, hexagonal indices are written with the i index as a dot and, in other cases, it is omitted entirely. 

FIGURE 5.17 Cellulose diffraction patterns. Top left synchrotron radiation x-ray diffraction pattern for cotton fiber bundle. The fiber was vertical and the white circle and line correspond to a shadow from the main beam catcher and its support. (Credit to Zakhia Ford.) Top right electron diffraction pattern of fragments of cotton secondary wall. The much shorter arcs in the top right figure are due to the good alignment and small number of crystallites in the electron beam. (Credit to Richard J. Schmidt.) Bottom a synthesized powder pattern for cellulose, based on the unit cell dimensions and crystalline coordinates of Nishiyama et al. [209]. (Credit to Zakhia Ford.) Also shown are the hkl values for the Miller indices. The 2-theta values are for molybdenum radiation instead of the more commonly used copper radiation. 

The convention of Miller indices is used to describe the planes within a unit cell. Miller indices are defined as the reciprocals of the intercept, which the plane makes with each of the three crystal axes. Each plane is denoted by three parameters h k and Planes which are parallel to a crystal axis are given the Miller index of 0 while planes formed in the negative direction are written with a bar over the number in the Miller index. The Miller index of a single specific plane is written within parentheses (hkl) whereas the Miller indices describing a whole family of faces are written with braces hkl. The faces that exist and define the crystal morphology are termed morphologically important and are commonly identified by the Miller indices of the planes represented by those faces. 

First to recall the nomenclature used in naming the faces of, or planes within, crystals. This is done by Miller indices (1839) which arc the reciprocals of the intercepts the plane makes with suitably chosen axes. The idea is most easily understood for the cubic system. The plane shaded in Fig. Ill is a 100 plane, the intercepts made on the OA, OB and OC axes by the plane being 1, oo and oo respectively so that the Miller indices are... 

It is sometimes necessary to determine the Miller indices of a given pole on a crystal projection, for example the pole A in Fig. 2-39(a), which applies to a cubic crystal. If a detailed standard projection is available, the projection with the unknown ])ole can be superimposed on it and its indices will be disclosed by its coincidence with one of the known poles on the standard. Alternatively, the method illustrated in Fig. 2-39 may be used. The pole A defines a direction in space, normal to the plane Qikl) whose indices are required, and this direction makes angles p, <7, t with the coordinate axes a, b, c. These angles are measured on the projection as shown in (a). Let the perpendicular distance between the origin and the hkl) plane nearest the origin be d [Fig. 2-39(b)], and let the... 

Index these lines (i.e., determine the Miller indices of each reflection by the use of Eq. (3-10) and Appendix 10) and calculate their relative integrated intensities. 

The scattering of X-rays from a set of planes defined by the Miller indices h,k,l is shown in Figure 3.5. In order to observe useful data from the X-ray experiment the scattered X-ray beam from the points X and Z must produce diffracted beams which are -. This is only possible if the... 

As one generally uses a vector normal to a lattice plane to specify its orientation, one can as well use a reciprocal lattice vector. This allows to define the Miller indices of a lattice plane as the coordinates of the shortest reciprocal lattice vector normal to that plane, with respect to a specified set of direct lattice vectors. These indices are integers with no common factor other than 1. A plane with Miller indices h, k, l is thus normal to the reciprocal lattice vector G = hb + kb > + lb->. and it is contained in a continuous plane G.r = constant. This plane intersects the primitive vectors a of the direct lattice at the points of coordinates xiai, X2a2 and X3a3, where the Xi must satisfy separately G.Xjai = constant. Since G.ai, G.a2 and G.as are equal to h, k and /, respectively, the Xi are inversely proportional to the Miller indices of the plane. When the plane is parallel to a given axis, the corresponding x value is taken for infinity and the corresponding Miller index taken equal to zero. 

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