Physics
Miller Indices
Miller indices are a notation system used to describe the orientation of planes and directions in a crystal lattice. They are represented by a set of three integers (hkl) that denote the intercepts of the plane with the crystallographic axes. Miller indices are crucial for understanding the structure and properties of crystalline materials.
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11 Key excerpts on "Miller Indices"
- James A. Newell(Author)
- 2012(Publication Date)
- Wiley(Publisher)
2.6 Miller Indices I n crystals, while directions are important, we are often more concerned with their planes. Any three lattice sites in a crystal can be used to define a plane, so another system of indices was developed to clarify which plane is being discussed. The Miller Indices for a plane passing through any three points in a lattice are determined using a four-step procedure that is similar to that used to find the indices of a direction. The procedure for determining the Miller Indices of a plane is: 1. Identify where the plane intercepts the x-, y-, and z-coordinate lines in terms of number of lattice parameters. 2. Take the reciprocal of these three points. 3. Clear fractions but do not reduce the results. 4. Enclose the results in parentheses. Note: If the plane never crosses an axis, it is taken to intercept at infinity. 2.6 | Miller Indices 43 | Miller Indices | A numerical system used to represent specific planes in a lattice. Example 2-4 Determine the Miller Indices for the planes shown in the figure. (a) (b) 1 2 2 x y z 3 x y z SOLUTION A: Plane intercepts coordinate axes at 1 1,2,2 2 . Reciprocals are 1, 1 2 , 1 2 . Multiply by 2 to clear fractions 2,1,1. 1 2 1 1 2 are the Miller Indices of the plane. B: Plane intercepts axes at 1 ,3, 2 . Reciprocals are 0, 1 3 ,0. Multiply by 3 to clear fractions: 3 * 0, 1 3 , 0, S 0, 1, 0 1 0 1 0 2 are the Miller Indices of the plane. 44 Chapter 2 | Structure in Materials Example 2-5 Determine the Miller Indices of the following plane: SOLUTION The plane passes through the origin, so it intercepts the x- and y-axes everywhere and the z-axis at zero, but a zero intercept would result in an infinite z-index. So we move the origin down by one lattice parameter in the z-direction: Now the z-intercept is at one, while the x- and y-intercepts become infinity. Intercepts at 1 , ,1 2 . The reciprocals are 0,0,1. No fractions to clear. 1 0 0 1 2 are the Miller Indices of the plane. 3 2 x y z 3 2 1 x y z
- Marc J. Madou(Author)
- 2011(Publication Date)
- CRC Press(Publisher)
TherulesfordeterminingtheMillerindicesofa directionoranorientationinacrystalareasfollows: translatetheorientationtotheoriginoftheunitcell, andtakethenormalizedcoordinatesofitsotherver-tex.Forexample,thebodydiagonalinacubiclattice asshownintheright-mostpanelin Figure2.12 is1 a , 1 a ,and1 a orthe[111]direction.TheMillerindices foradirectionarethusestablishedusingthesame procedure for finding the components of a vector. Bracketsorcaretsspecifydirections. Directions[100],[010],and[001]areallcrystal-lographicallyequivalentandarejointlyreferredto asthefamily,form,orgroupof<100>directions.A form,group,orfamilyoffacesthatbearlikerelation-shipstothecrystallographicaxes—e.g.,theplanes ( 001), (100), (010), (001), (100), and (010) —are all equivalent, and they are marked as {100} planes (see Figure 2.13 ). A summary of the typical repre-sentation for Miller Indices is shown in Table 2.2 . Theorientationofaplaneisdefinedbythedirection C/3 C/3 a/2 a/2 a g b 2C/3 height 0,C 3-fold axes 120 ° 120 ° C Mirror Glide (a) (b) FIGURE2.11 Example of a screw axis and a glide plane. (a) N -fold screw axes C : a combination of a rotation of 360°/ n around C and a translation by an integer of C / n . (b) Glide plane: a translation parallel to the glide plane g by a /2. z y x [100] (100) plane z y x [110] (110) plane z y x [111] (111) plane FIGURE2.12 Miller Indices for planes and directions in an SC cubic crystal. Shaded planes are from left to right (100), (110), and (111). (Drawing by Mr. Chengwu Deng.) 46 Solid-State Physics, Fluidics, and Analytical Techniques in Micro- and Nanotechnology of a normal to the plane or the vector product ( A × B = C ).Foracubiccrystal(suchassiliconor galliumarsenide),theplane( hkl )isperpendicular tothedirection[ hkl ].Inotherwords,theindicesofa planearethesamenumbersusedtospecifythenor-maltotheplane.Usingasimplecubiclatticeasan example,youcancheckthatcrystalplaneswiththe smallestMillerindices,suchas{100},{110},{111}, have the largest density of atoms.
- Wole Soboyejo(Author)
- 2002(Publication Date)
- CRC Press(Publisher)
This is illustrated schematically in Figs 2.1 and 2.2. The reciprocals of the inter-cepts are then multiplied by appropriate scaling factors to ensure that all the resulting numbers are integer values corresponding to the least common factors. The least common factors are used to represent the Miller Indices of a plane. Any negative numbers are represented by bars over them. A single plane is denoted by (x y z) and a family of planes is usually repre-sented as {x y z}. Similarly, atomic directions may be specified using Miller Indices. These are vectors with integer values that represent the particular atomic direction [u v w], as illustrated in Fig. 2.3(a). The square brackets are gen-erally used to denote single directions, while angular brackets are used to represent families of directions. An example of the (111) family of directions is given in Fig. 2.3(b). The Miller Indices of planes and directions in cubic crystals may be used to determine the unit vectors of the direction and the plane normal, respectively. Unit vectors are given simply by the direction cosines [/ m n] to be n -/ i + AT7 j + nk (2.1) F igure 2.1 Determination of Miller Indices for crystal planes. (Adapted from Shackleford, 1996. Reprinted with permission from Prentice-Hall.) Defect Structure and Mechanical Properties 25 ( 111 ) F igure 2.2 Examples of crystal planes. (Adapted from Shackleford, 1996. Reprinted with permission from Prentice-Hall.) In the case of a direction, dx, described by unit vector [xx z j, the direction cosines are given by c/i = * ii + Ki j + 2 + y} + A ( 2 . 2 ) In the case of a plane with a plane normal with a unit vector, n 2, that has components ( uv wx), the unit vector, h, is given by a, i + Vi + k 1 + vf + w f (2.3) (a) (b) Figure 2.3 Determination of crystal directions: (a) single [111] directions; (b) family of (111) directions. (Adapted from Shackleford, 1996. Reprinted with permission from Prentice-Hall.)
eBook - PDFUnderstanding Solid State Physics
Problems and Solutions
- Jacques Cazaux(Author)
- 2016(Publication Date)
- Jenny Stanford Publishing(Publisher)
3. Lattice Rows and Miller Indices To describe the crystal structure it is suf ficient to state the choice of lattice vectors, and the nature and position of atoms that make up the basis. These positions are expressed with the help of the vectors a b c , , , considered as the unit vectors: r a b c j j j j u v w = + + . The lattice points are arranged along various rows and planes. When the rows and the planes are parallel and equidistant to each other they are equivalent and they are represented with the same symbols. A series of parallel rows is represented by ( m , n , p ) where r = + + m n p a b c when the row is parallel to the line that connects the origin to the lattice point m , n , p . A series of parallel planes can be represented all by Miller Indices ( h , k , l ), which describe the equation of the form hx + ky + lz = 1 of plane nearest the origin and using a , b , c units (see Ex. 8). This definition means that the intersections of the plane ( h , k , l ) with axes x , y , and z are 1/ h , 1/ k , 1/ l , respectively. The atoms chosen to define a plane must not be collinear. 4. Point Symmetry In nearly all crystals one or several directions are equivalent. This orientation or point symmetry of a crystal can be represented by the symmetry of the figure formed by the group of half-lines which, emanating from the same point 0, are parallel to the directions from which all the properties of the crystal are identical. The point symmetries that are encountered include the rotations of order n 3 Course Summary around an axis (the angle of rotation is 2 p / n with n = 1, 2, 3, 4, 6) and the rotation-inversions, written as n ( 1 ∫ inversion with respect to 0, 2 ∫ � m: mirror symmetry, 3 4 6 , , ). Several symmetry elements can be associated around a point but the number of distinct combinations and possibilities is limited to 32. That is, there are 32 symmetry point groups which result in a classification of 32 crystal classes.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Academic Studio(Publisher)
The crystal structure has a three dimensional shape. The unit cell is given by its lattice parameters, the length of the cell edges and the angles between them, while the positions of the atoms inside the unit cell are described by the set of atomic positions ( x i , y i , z i ) measured from a lattice point. Simple cubic (P) Body-centered cubic (I) Face-centered cubic (F) ________________________ WORLD TECHNOLOGIES ________________________ Miller Indices Planes with different Miller Indices in cubic crystals Vectors and atomic planes in a crystal lattice can be described by a three-value Miller index notation (ℓmn). The ℓ , m and n directional indices are separated by 90°, and are thus orthogonal. In fact, the ℓ component is mutually perpendicular to the m and n indices. By definition, (ℓmn) denotes a plane that intercepts the three points a 1 /ℓ, a 2 /m, and a 3 /n, or some multiple thereof. That is, the Miller Indices are proportional to the inverses of the intercepts of the plane with the unit cell (in the basis of the lattice vectors). If one or more ________________________ WORLD TECHNOLOGIES ________________________ of the indices is zero, it simply means that the planes do not intersect that axis (i.e. the intercept is at infinity). Considering only (ℓmn) planes intersecting one or more lattice points (the lattice planes ), the perpendicular distance d between adjacent lattice planes is related to the (shortest) reciprocal lattice vector orthogonal to the planes by the formula: Planes and directions The crystallographic directions are fictitious lines linking nodes (atoms, ions or molecules) of a crystal. Similarly, the crystallographic planes are fictitious planes linking nodes. Some directions and planes have a higher density of nodes. These high density planes have an influence on the behavior of the crystal as follows: • Optical properties: Refractive index is directly related to density (or periodic density fluctuations).
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Academic Studio(Publisher)
The crystal structure has a three dimensional shape. The unit cell is given by its lattice parameters, the length of the cell edges and the angles between them, while the positions of the atoms inside the unit cell are described by the set of atomic positions ( x i , y i , z i ) measured from a lattice point. Simple cubic (P) Body-centered cubic (I) Face-centered cubic (F) ____________________ WORLD TECHNOLOGIES ____________________ Miller Indices Planes with different Miller Indices in cubic crystals Vectors and atomic planes in a crystal lattice can be described by a three-value Miller index notation (ℓmn). The ℓ , m and n directional indices are separated by 90°, and are thus orthogonal. In fact, the ℓ component is mutually perpendicular to the m and n indices. By definition, (ℓmn) denotes a plane that intercepts the three points a 1 /ℓ, a 2 /m, and a 3 /n, or some multiple thereof. That is, the Miller Indices are proportional to the inverses of the intercepts of the plane with the unit cell (in the basis of the lattice vectors). If one or more ____________________ WORLD TECHNOLOGIES ____________________ of the indices is zero, it simply means that the planes do not intersect that axis (i.e. the intercept is at infinity). Considering only (ℓmn) planes intersecting one or more lattice points (the lattice planes ), the perpendicular distance d between adjacent lattice planes is related to the (shortest) reciprocal lattice vector orthogonal to the planes by the formula: Planes and directions The crystallographic directions are fictitious lines linking nodes (atoms, ions or molecules) of a crystal. Similarly, the crystallographic planes are fictitious planes linking nodes. Some directions and planes have a higher density of nodes. These high density planes have an influence on the behavior of the crystal as follows: • Optical properties: Refractive index is directly related to density (or periodic density fluctuations).
eBook - ePub- R. E. Smallman, A.H.W. Ngan(Authors)
- 2013(Publication Date)
- Butterworth-Heinemann(Publisher)
Figure 1.9(b) , the procedural steps for indexing the plane ABC areThe Miller Indices for the planes DEFG and BCHI are and (1 1 0), respectively. Often it is necessary to ignore individual planar orientations and to specify all planes of a given crystallographic type, such as the planes parallel to the six faces of a cube. These planes constitute a crystal form and have the same atomic configurations; they are said to be equivalent and can be represented by a single group of indices enclosed in curly brackets, or braces. Thus, {1 0 0} represents a form of six planar orientations, i.e. (1 0 0), (0 1 0), (0 0 1), and . Returning to the (1 1 1) plane ABC of Figure 1.9(b) , it is instructive to derive the other seven equivalent planes, centring on the origin O, which comprise {1 1 1}. It will then be seen why materials belonging to the cubic system often crystallize in an octahedral form in which octahedral {1 1 1} planes are prominent.It should be borne in mind that the general purpose of the Miller procedure is to define the orientation of a family of parallel equidistant planes; the selection of a convenient representative plane is a means to this end. For this reason, it is permissible to shift the origin provided that the relative disposition of a , b and c is maintained. Miller Indices are commonly written in the symbolic form (hkl ). Rationalization of indices, either to reduce them to smaller numbers with the same ratio or to eliminate fractions, is unnecessary. This often-recommended step discards information; after all, there is a real difference between the two families of planes (1 0 0) and (2 0 0).As mentioned previously, it is sometimes convenient to choose a non-primitive cell. The hexagonal structure cell is an important illustrative example. For reasons which will be explained, it is also appropriate to use a four-axis Miller–Bravais notation (hkil ) for hexagonal crystals, instead of the three-axis Miller notation (hkl ). In this alternative method, three axes (a 1 , a 2 , a 3 ) are arranged at 120° to each other in a basal plane and the fourth axis (c ) is perpendicular to this plane (Figure 1.10(a) ). Hexagonal structures are often compared in terms of the axial ratio c /a
eBook - PDFCrystallography and Surface Structure
An Introduction for Surface Scientists and Nanoscientists
- Klaus Hermann(Author)
- 2016(Publication Date)
- Wiley-VCH(Publisher)
indices of the cubic lattices are given in sc notation by symmetry-related families {h k l}; (m), with m denoting the number of family members, see below. Miller Indices of the hcp lattice refer to the obtuse representation (i.e., ∠(R o1 , R o2 ) = 120 ∘ , see Section 2.2.2.1) and are given in generic three-index notation. Note that for the hcp lattice the second, fourth and seventh densest 96 3 Crystal Layers: Two-Dimensional Lattices monolayers, denoted by asterisks ( * ) in Table 3.1, contain two atoms in their morphological unit cells, that is, n (h k l) = 2 while for all others n (h k l) = 1. If a lattice exhibits, in addition to translation symmetry, also point symme- try then geometrically identical netplanes may appear for different Miller index values. These equivalent netplanes are often grouped into families, where each family is characterized by Miller Indices {h k l} written inside curly brackets. An example is given by the simple cubic lattice with the six equivalent netplanes, denoted by Miller Indices (±1 0 0), (0 ±1 0), (0 0 ±1), forming a family described as {1 0 0}. This notation is also used in Table 3.1 for monolayers of cubic lattices. In a generalization of Eq. (3.4), directions inside a lattice may also be defined by Miller Indices h, k, l, which are, in general, non-integer- or integer-valued. These directions are usually written as [h k l] inside square brackets. In addi- tion, lattices with point symmetry allow symmetry equivalent directions where the corresponding direction families are written as inside pointed brack- ets. As an example, the simple cubic lattice includes eight equivalent directions ±[1 1 1], ±[−1 1 1], ±[1 −1 1], ±[1 1 −1], which form a direction family <1 1 1>.
eBook - PDF- H.P. Myers(Author)
- 1997(Publication Date)
- CRC Press(Publisher)
Planes making negative intercepts on the axes are treated in similar fashion and the negative sign appears above the Miller index, e.g. (pronounced bar h). The following notation is customary: a particular plane or set of parallel planes (hkl); a set of symmetrically equivalent planes {hkl} (including negative indices). The use of Miller Indices is best appreciated from Fig. 2.8. The positions of lattice or basis points are denoted by their coordinates expressed in terms of base vectors, thus the body-centre position is expressed as . Directions in the lattice are specified by the coordinates of the lattice point that is nearest to the origin in the chosen direction. A direction is written [uvz] and a set of equivalent directions . In cubic lattices (and only in these) one can define directions in terms of the normals to the lattice planes; thus [hkl] indicates the direction normal to the plane (hkl). In structures that possess a centre of symmetry many of the planes containing the same indices are equivalent. Thus in the cubic systems the planes (123), (213), (321), (132), (231) and (312) have the same density of packing and the same interplanar spacing. They are therefore equivalent from these points of view, which are the relevant ones in X-ray diffraction. Remembering that each index may have a negative sign, we find that the indices 1, 2 and 3 may be arranged in 48 different ways. On the other hand the plane (111) has only one distinguishable arrangement of three similar indices; each index, however, may independently take a plus or minus sign. So there are eight equivalent (111) planes in the cubic lattices. The total number of equivalent planes for given numerical values of the indices is called the multiplicity p. Crystallography 35 Figure 2.8 The arbitrary line ll’ belongs to a family of similar lines that may be indexed as (35) in the chosen unit cell (in this case a primitive cell).
- Wolfgang Moritz, Michel A. Van Hove(Authors)
- 2022(Publication Date)
- Cambridge University Press(Publisher)
Miller Indices: it also depends on the material, temperature or further conditions, due to different truncations for structures with multiple atoms in the 3-D unit cell, reconstructions, atomic position relaxations, adsorbates, etc. The morphology of the surface can be described in a more intuitive manner than with high-index Miller Indices, namely in terms of easily visualised terraces and steps; this can be done quantitatively, so as to gauge the terrace width and step density, as well as the step structure (e.g., kinked or not). The general case, for an arbitrary underlying 3-D crystal structure, is analysed thoroughly in the book by K. Hermann [2.6]. We consider here some examples of stepped surfaces of mono-atomic lattices, since steps have been investigated mostly on metal surfaces, while only relatively fewer studies exist on stepped surfaces of semiconductors or compounds. For steps on the simpler mono-atomic lattices, common for metal surfaces, two different notations have been proposed. The first is the ‘step notation’, which counts the atomic rows on a terrace between two steps [2.11], described by: hkl ð Þ ¼ n t h t , k t , l t ð Þ h s , k s , l s ð Þ: (2.25) Here, h t , k t , l t are the Miller Indices of the low-index terrace plane, while h s , k s , l s give the direction of the low-index plane formed by the atoms in the step edge which may be seen as an inclined microfacet, and n t is the number of atomic rows in the terrace. This notation is applicable to all lattices; it is, however, not very practical for surfaces with kinked steps. In order to define the morphology more precisely, a second notation has been proposed: the ‘microfacet notation’ [2.12]. This is based on the fact that Miller Indices are vectors in reciprocal space and can be decomposed into two or three vectors of low- index planes. This decomposition is not unique, and the specific choice characterises the morphology.
eBook - PDF- David Ball(Author)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 766 Chapter 21 | The Solid State: Crystals Unless otherwise noted, all art on this page is © Cengage Learning 2014. unit cells (see Table 21.3). For crystals that are composed of different atoms having similar scattering factors, there may be accidental destructive interferences that can dramatically reduce the intensity of an expected diffraction. We finish this section on Miller Indices by introducing a convenient use of Miller Indices. We like to define a solid crystal as an infinite, regular array of atoms or mol-ecules. In reality, however, we know that the array is not infinite; the crystal stops at some point. It stops at the surface of the crystal. In many cases, the surface of a crystal is not just some random arrangement of atoms or molecules making a microscopically rough boundary. For many crystals, over a large surface area (that is, on a scale of square nanometers or micrometers) the surface corresponds to a particular plane of atoms or molecules that can be described by a particular set of Miller Indices. Figure 21.26 shows examples of some crystalline surfaces. With care, large crystals with specific surface planes can be prepared, and the chemistry that occurs in the presence of each plane can be specific to that surface. We will consider surfaces in the next chapter, and Miller Indices will reappear as a way to specify the arrangement of atoms on a surface.
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