Lattice Translation Vectors

10 Key excerpts on "Lattice Translation Vectors"

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  One-Dimensional Metals

  Conjugated Polymers, Organic Crystals, Carbon Nanotubes and Graphene

  • Siegmar Roth, David Carroll, David L. Carroll(Authors)
  • 2015(Publication Date)
  • Wiley-VCH

    (Publisher)

  translational invariance of the crystal lattice and of the laws governing the physics within a crystal.

  Once the most simple, primitive lattice structure has been determined, it is important to realize how many possibilities exist for distinctively different lattices. Different lattices in this context are described by the different symmetry operations that will take the lattice into itself. This is governed by the ratio of lengths in lattice parameters as well as the angles in the lattice. Therefore, the three-dimensional analog of our rectangular lattice mentioned above, with , is a cubic lattice. This lattice has a number of symmetry operations – rotations, translations, reflections, that when performed on this lattice leaves the lattice unaltered. However, in lattices of lower symmetry, the lengths of the elementary cell might differ, ; also the angles between the axes do not need to be 90°. This means that an entirely different set of symmetry operations leaves the lattice unaltered.

  When the whole set of invariant symmetry operations: rotations and translations on a set of primitive lattice points are considered, they form a mathematical point group. An enumeration of all the possible lattice point groups can be made – which means there are only a finite number of primitive lattice arrangements expressing unique symmetries, that can exist – regardless of the values of the lattice parameters and angles. (Bravais lattices are said to be “equivalent” if they have isomorphic space symmetry groups.) These distinct groups are listed in Ashcroft and Mermin [1] as well as several other solid-state physics texts. In fact, in three-dimensional space, there are 14 possible symmetry groups, or symmetry variations of this most simple lattice. But in two dimensions, there are only 5 such symmetry groups allowed. In zero and one dimension, there is only one type of Bravais or primitive lattice.

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  Introductory Solid State Physics

  • H.P. Myers(Author)
  • 1997(Publication Date)
  • CRC Press

    (Publisher)

  2 Crystallography 2.1 Lattices We are most familiar with condensed matter in the form of solid crystalline substances and, although interest in liquids and amorphous solids has grown considerably in the past few years, the physics of condensed matter is to a very great extent the physics of crystals. To begin, we therefore need to learn how to describe the regular geometrical arrangement of atoms in space that is the essence of crystallinity. Crystals are finite regular arrangements of atoms in space. In practice the atomic arrangement is never perfect, but in crystallography we neglect this aspect and describe crystals by reference to perfect infinite arrays of geometrical points called lattices. A lattice is an infinite array of points in space so arranged that every point has identical surroundings. All lattice points are geometrically equivalent. A lattice therefore exhibits perfect translational symmetry and, relative to an arbitrarily chosen origin, at a lattice point, any other lattice point has the position vector (2.1) The numbers n are necessarily integral and the vectors a, b and c are fundamental units of the translational symmetry; the latter are arbitrary, but a sensible choice is usually that which gives the shortest vectors or the highest symmetry to the unit cell. On the other hand, by definition, the volume associated with a single lattice point is unique, but since there is a choice regarding the vectors a, b and c, it may take a variety of shapes as illustrated for a two-dimensional example in Figs 2.1 and 2.2. The volume associated with a single lattice point is called the primitive cell, and this usually takes one of two forms.

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  Solid State Physics

  • Mircea S. Rogalski, Stuart B. Palmer(Authors)
  • 2000(Publication Date)
  • CRC Press

    (Publisher)

  This implies that we ignore the boundaries of the crystal, assuming that the average size of elementary translation vectors is very small with respect to the crystal dimension. In 52 Lattice Symmetry 53 other words we neglect the surface effects on the bulk properties of the crystal. The primitive vectors are often omitted and the lattice vectors, Eq.(2.1) written by convention [pqs]. It follows that the primitive vectors themselves may be denoted by a = [100], £ = [010], c = [001]. The lattice is generated by repeated arbitrary translation, Eq.(2.1), of either a single lattice point or a primitive cell associated with it, which is illustrated in Figure 2.1 for hypothetical lattices in one, two and three dimensions, which are formed by the centres of a set of spheres. A primitive cell always contains only one lattice point. Either there are only points at the comers of the cell (a crystallographic cell) or one lattice point is in the centre of a cell which is defined by the planes bisecting the lines joining the central point with the adjacent lattice points (a Wigner-Seitz cell). • ' M r -X -(P) (WS) (b ) / / / / / / / WS) Figure 2.1. Primitive and Wigner-Seitz cells in the (a) linear, (b) plane and (c) three- dimensional lattices Figure 2.1 shows that the sides of the Wigner-Seitz cell are determined by the equation: -p -R = ^ R 2 or 2 p R + R 2 =0 i.e. (p + R)2 -p 2 (2.2) 54 Solid State Physics where p is the position vector with respect to the central lattice point. It is apparent that the complete set of polyhedrons, generated from the unit cell by all the translation operations described by Eq.(2.1), will cover all points in space. Lattices are classified into groups or systems according to the shape of the unit cell , which generates the entire lattice by translation. The shape of the unit cell is chosen for geometrical convenience and so may contain more than one lattice point.

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  Introduction to Solid State Physics for Materials Engineers

  • Emil Zolotoyabko(Author)
  • 2021(Publication Date)
  • Wiley-VCH

    (Publisher)

  Auguste Bravais):

  (1.1)

  The nodes,

  r

  s

  , of Bravais lattice are produced by linear combinations of three non-coplanar vectors,

  a

  1

  , a

  2

  , a

  3

  , called translation vectors. The integer numbers in Eq. (1.1) can be positive, negative, or zero. Atomic arrangements within every crystal can be described by the set of analogous Bravais lattices.

  Classification of Bravais lattices is based on the relationships between the lengths of translation vectors, |

  a

  1

  | = a, |

  a

  2

  | = b, |

  a

  3

  | = c and the angles, α, β, γ, between them. In fact, all possible types of Bravais lattices can be attributed to seven symmetry systems:

  Triclinic: a ≠ b ≠ c and α ≠ β ≠ γ;

  Monoclinic: a ≠ b ≠ c and α = β = 90°, γ ≠ 90°; in this setting, angle γ is between translation vectors

  a1

  (|

  a1

  | = a) and

  a2

  (|

  a2

  | = b); whereas the angles α and β are, respectively, between translation vectors

  a2

  ^

  a3

  and

  a1

  ^

  a3

  ;

  Orthorhombic : a ≠ b ≠ c and α = β = γ = 90°;

  Tetragonal: a = b ≠ c and α = β = γ = 90°;

  Cubic: a = b = c and α = β = γ = 90°;

  Rhombohedral: a = b = c and α = β = γ ≠ 90°;

  Hexagonal: a = b ≠ c and α = β = 90°, γ = 120°.

  A parallelepiped built by the aid of vectors

  a1

  ,

  a2

  ,

  a3

  is called a unit cell and is the smallest block, which being duplicated by the translation vectors allows us to densely fill the 3D space without voids.

  Translational symmetry, however, is only a part of the whole symmetry in crystals. Atomic networks, described by Bravais lattices, also possess the so-called local (point) symmetry, which includes lattice inversion with respect to certain lattice points, mirror reflections in some lattice planes, and lattice rotations about certain rotation axes (certain crystallographic directions). After application of all these symmetry elements, the lattice remains invariant. Furthermore, rotation axes are defined by their order, n. The latter, in turn, determines the minimum angle, , after rotation by which the lattice remains indistinguishable with respect to its initial setting (lattice invariance). In regular crystals, the permitted rotation axes, i.e. those matching translational symmetry (see Appendix 1.A ), are twofold (180°-rotation, n = 2), threefold (120°-rotation, n = 3), fourfold (90°-rotation, n = 4), and sixfold (60°-rotation, n = 6). Of course, onefold, i.e. 360°-rotation (n = 1), is a trivial symmetry element existing in every Bravais lattice. The international notations for these symmetry elements are: – for inversion center, m – for mirror plane, and 1, 2, 3, 4, 6 – for respective rotation axes. We see that fivefold rotation axis and axes of the order, higher than n = 6, are incompatible with translational symmetry (see Appendix 1.A

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  Electronic Structure

  Basic Theory and Practical Methods

  • Richard M. Martin(Author)
  • 2020(Publication Date)
  • Cambridge University Press

    (Publisher)

  ● The crystalline order is described by its symmetry operations. The set of translations forms a group because the sum of any two translations is another translation. 1 In addition 1 A group is defined by the condition that the application of any two operations leads to a result that is another operation in the group. We will illustrate this with the translation group. The reader is referred to other sources for the general theory and the specific set of groups possible in crystals, e.g., books on group theory (see [281– 283], and the comprehensive work by Slater [284]. 82 Periodic Solids and Electron Bands there may be other point operations that leave the crystal the same, such as rotations, reflections, and inversions. This can be summarized as Space group = translation group + point group. 2 The Lattice of Translations First we consider translations, since they are intrinsic to all crystals. The set of all translations forms a lattice in space, in which any translation can be written as integral multiples of primitive vectors, T(n) ≡ T(n 1 ,n 2 , . . .) = n 1 a 1 + n 2 a 2 + . . . , (4.1) where a i , i = 1, . . . ,d are the primitive translation vectors and d denotes the dimension of the space. For convenience we write formulas valid in any dimension whenever possible and we define n = (n 1 ,n 2 , . . . ,n d ). In one dimension, the translations are simply multiples of the periodicity length a, T (n) = na, where n can be any integer. The primitive cell can be any cell of length a; however, the most symmetric cell is the one chosen symmetric about the origin (−a/2,a/2) so that each cell centered on lattice point n is the locus of all points closer to that lattice point than to any other point. This is an example of the construction of the Wigner–Seitz cell. The left-hand side of Fig. 4.1 shows a portion of a general lattice in two dimensions. Space is filled by the set of all translations of any of the infinite number of possible choices of the primitive cell.

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  Mechanics And Mathematics Of Crystals: Selected Papers Of J L Ericksen

  Selected Papers of J L Ericksen

  • Jerald L Ericksen, Millard F Beatty, Michael A Hayes(Authors)
  • 2005(Publication Date)
  • World Scientific

    (Publisher)

  Any 1 -lattice has points with position vectors relative to a point in space repre-sentable in the form x K =n a K e a +r, n a K e Z, K = l,2,..., (2) where r is a vector depending on the choice of the point, and the e a , a = 1, 2, 3 are lattice vectors, required to be linearly independent, all of these vectors being independent of position. If there are n lattices describing the crystals, the position vectors are describable in the form XKa =n a K e a +r a , a=l,...,n, (3) where the r a are position vectors locating some point in the ath lattice relative to a point in space. The condition that no two atoms occupy the same position translates to the condition that the differences of these cannot be integral combinations of lattice vectors, or n e Z = > r o -r ^ n a e a when a 4= p. (4) Generally, one makes some particular choice of kinds of atoms and number of lattices n,-containing the ith species, considered to be fixed numbers such that m i = l m being the number of different kinds of atoms. Also, I will make some use of the reciprocal lattice vectors (dual basis) e such that e a -e b = %, e a ®e a = e a ®e a = 1. (6) Those interested in related continuum theories generalize this, regarding e a and pi as vector fields but, for the present study, this is not important. (5) 79 On Groups Occurring in the Theory of Crystal Multi-Lattices 147 2.2. Some Relevant Transformations Certain transformations of e a and r a give values of these vectors describing exactly the same lattices, leave the atoms in each set the same, and change r a by simply replacing the point in the ath lattice by another. These are of the form e a -> e a = m b a e b , m = m b a e G. (7) r a -+r a =r a +n a a e a , < 6 Z. (8) Changing one point from which the position vectors are defined to another gives other transformations of the form fa -* r a + C, C = Const., (9) which could be interpreted as translating the whole configuration, something com-monly regarded as inconsequential.

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  Physics, Optics, and Spectroscopy of Materials

  • Zeev Burshtein(Author)
  • 2022(Publication Date)
  • Wiley

    (Publisher)

  Instead, our following discussions are limited to perfectly periodic crystals. Figure 9.4 Bravais lattices. 248 9 Crystalline Solids Figure 9.4 (Continued) Figure 9.4 (Continued) 9.2 Periodic Crystals 249 Table 9.1 Descriptions of crystalline point group notations according to Shoenflies and Hermann-Mauguin. The alternative notations that are sometimes used are written in parentheses. Point group notation No. in Table 8.2 No. crystal system Schoenflies Hermann-Mauguin 1 Triclinic C 1 1 1 2 C i 1 6 3 Monoclinic C 2 2 2 4 C S m 9 5 C 2h 2/m 15 6 Orthorhombic D 2 (V) 222 19 7 C 2v mm2 (2mm,mm) 10 8 D 2h (V h ) mmm (2/m 2/m 2/m) 26 9 Tetragonal C 4 4 4 10 S 4 4 7 11 C 4h 4/m 17 12 D 4 422 21 13 C 4v 4mm 12 14 D 2d (V d ) 4 2m 23 15 D 4h 4/mmm (4/m 2/m 2/m) 28 16 Trigonal C 3 3 3 17 S 6 (C 3i ) 3 8 18 D 3 32 20 19 C 3v 3m 11 20 D 3d 3m ( 3 2/m) 24 21 Hexagonal C 6 6 5 22 C 3h (S 3 ) 6 (3/m) 16 23 C 6h 6/m 18 24 D 6 622 (62) 22 25 C 6v 6mm 13 26 D 3h 6m2 27 27 D 6h 6/mmm (6/m 2/m 2/m) 30 28 Cubic T 23 32 29 T h m3 (2/m 3) 34 30 O 432 (43) 35 31 T d 4 3m 33 32 O h m 3m (4/m 3 2/m) 36 250 9 Crystalline Solids 9.3 Lattice-Vector and Lattice-Plane Orientations A lattice-vector orientation is the orientation of a lattice vector of the form L = h a + k b + ℓ c, where h, k, and ℓ are specific integers that do not have a common factor; for brevity, the orientation is noted as [h, k, ℓ] (if, at first, the h, k, and ℓ numbers do have common factors, they are divided by the biggest one). A crystal (lattice) plane is a plane that contains lattice points. A fundamental plane is defined by three lattice points on vector tips oriented along fundamental lattice vectors, with the first at p 1 a, second at p 2 b, and third at p 3 c (p 1 , p 2 , and p 3 integers), as demonstrated in Figure 9.5. A family of planes is the set containing all lattice planes parallel to the fundamental plane.

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  Crystallography and Surface Structure

  An Introduction for Surface Scientists and Nanoscientists

  • Klaus Hermann(Author)
  • 2016(Publication Date)
  • Wiley-VCH

    (Publisher)

  Many researchers in the surface science community (and not only there) find it convenient to think in Cartesian coordinates, using orthogonal unit vectors in three-dimensional space. Therefore, they prefer to characterize lattices, if possible, by orthogonal lattice vectors R 1 , R 2 , R 3 even at the expense of having to consider corresponding crystal bases with a larger number of atoms. This will be discussed for face- and body-centered cubic lattices in Section 2.2.2.1. 16 2 Bulk Crystals: Three-Dimensional Lattices Theoretical studies on extended geometric perturbations inside a crystal, such as those originating from periodic imperfections or distortions, often require con- sidering unit cells with lattice vectors R ′ 1 , R ′ 2 , R ′ 3 which are larger than those given by R 1 , R 2 , R 3 of the unperturbed crystal. Here, a direct computational compari- son of results for the perturbed crystal with those for the unperturbed crystal is often facilitated by applying the same (enlarged) lattice vectors R ′ 1 , R ′ 2 , R ′ 3 to both systems. As a result, the unperturbed crystal is described by a lattice with a larger than primitive unit cell and an appropriately increased number of atoms. This is the basic idea behind so-called superlattice methods, which will be discussed in Section 2.2.2.2. Ideal single crystal surfaces that originate from bulk truncation yielding two- dimensional periodicity at the surface, will be treated in great detail in Chapter 4. Here, the analysis of structural properties at the surface can be facilitated greatly by using so-called netplane-adapted lattice vectors R ′ 1 , R ′ 2 , R ′ 3 . These are given by linear transformations of the initial bulk lattice vectors, where the shape of the morphological unit cell may change but not its volume nor the number of atoms inside the cell.

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  Surface Structure Determination by LEED and X-rays

  • Wolfgang Moritz, Michel A. Van Hove(Authors)
  • 2022(Publication Date)
  • Cambridge University Press

    (Publisher)

  The lattice constants are: a, b, c and α, β, γ: The angles are defined such that: a  b ¼ a j j b j j cos γ, b  c ¼ b j j c j j cos α, a  c ¼ a j j c j j cos β: (2.1) The position vector R of each unit cell in the crystal is then given by: R ¼ n 1 a þ n 2 b þ n 3 c, (2.2) where one of the unit cells is taken as origin. The atomic positions inside the unit cell are usually described in relative coordinates, that is, in length units of the translation vectors. Thus, r i ¼ x i a þ y i b þ z i c, (2.3) where 0  x i , y i , z i <1 for positions inside the unit cell. For example, the CsCl lattice is shown in Figure 2.1, with a j j ¼ b j j ¼ c j j ¼ 0:4123 nm ¼ 4:123 Å, α ¼ β ¼ γ ¼ 90 ∘ , x Cs ¼ y Cs ¼ z Cs ¼ 0, 0, 0 ð Þ and x Cl ¼ y Cl ¼ z Cl ¼ 0:5, 0:5, 0:5 ð Þ. The atom coordinates are in general described in an oblique coordinate system, that is, as relative coordinates based on non-rectangular translation vectors, or, for lattices with suitable symmetry, as relative coordinates based on rectangular translation vectors in rectangular coordinates with different unit lengths – except for the cubic system, which has identical unit lengths. The use of oblique coordinate systems is mostly unfamiliar to the physicist who prefers vector calculations in Cartesian coordinates. An oblique system has, however, substantial advantages over the Cartesian coordinates in the diffraction theory so that oblique coordinate systems are used in all diffraction methods. Vector calculations in oblique coordinates are more complex than in rectangular coordinates and require the definition of a reciprocal coordinate system or covariant and contravariant systems: some rules for vector calculations in oblique coordinate systems are given in Appendix C. The periodicity of the crystal allows the description of the positions of all atoms once the positions are known for only the atoms within a single unit cell.

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  Periodic Materials and Interference Lithography

  For Photonics, Phononics and Mechanics

  • Martin Maldovan, Edwin L. Thomas(Authors)
  • 2009(Publication Date)
  • Wiley-VCH

    (Publisher)

  In this book, we choose to represent point lattices with the points located at the corners of the cells. Because all cells in Figure 1.3b are equivalent, we can arbitrarily choose any of those cells as a unit cell that repeats to form the point lattice. In this particular example, the edges of the unit cells have equal lengths. In the general case, however, the unit cell is an arbitrary parallelogram that can be described by the primitive vectors a 1 and a 2 (Figure 1.4), and when repeated by successive translations in space the unit cell forms the corresponding point lattice. The distances a = |a 1 |, b = |a 2 |, and the angle γ are called the lattice constants. The primitive vectors a 1 and a 2 defining the unit cell also construct the entire point lattice by translations. For example, the point lattice can be viewed as an array of regularly spaced points located at the tip of the vector R = n 1 a 1 + n 2 a 2 (1.1) where n 1 and n 2 are arbitrary integer numbers. Equation 1.1 indicates that for each particular set of n 1 and n 2 values there is a corresponding lattice point R in space. Therefore, the vector R fills the plane with regularly spaced points and it is said to generate the point lattice. The number of conceivable two-dimensional periodic structures is infinite. However, when two-dimensional periodic structures are transformed into their Fig. 1.4 A general two-dimensional unit cell defined by two nonorthogonal vectors, a 1 and a 2 . 1.2 Two-dimensional Point Lattices 9 corresponding point lattices, only five distinct point lattices are found. In other words, the existence of an object that repeats regularly in space, together with the requirement that the repeating objects must have identical surroundings, determines the existence of only five types of point lattices. Therefore, we have achieved one of our goals: we can classify the infinite set of two-dimensional periodic structures by their spatial periodicity into five categories.

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