Basic elements of crystallography 2nd Edition by Nevill Gonzalez Szwack, Teresa Szwacka ISBN 9814613576 9789814613576 by Gonzalez Szwacki, Nevill; Szwacka, Teresa 9789814613583, 9814613584 instant download after payment.

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ISBN 10: 9814613576 
ISBN 13: 9789814613576
Author: Nevill Gonzalez Szwack, Teresa Szwacka

This textbook is a complete and clear introduction to the field of crystallography. It includes an extensive discussion on the 14 Bravais lattices and their reciprocals, the basic concepts of point- and space-group symmetry, the crystal structure of elements and binary compounds, and much more.The purpose of this textbook is to illustrate rather th

Basic elements of crystallography 2nd Table of contents:

Chapter 1 One- and Two-Dimensional Crystal Lattices
1.1 Introduction
1.2 One-Dimensional Crystal Structures
Figure 1.1 Three different one-dimensional crystal structures: (a) periodic repetition of identical atoms, (b) periodic repetition of a building block composed of two different atoms, and (c) periodic repetition of a building block composed of two identical atoms.
Figure 1.2 Two different arrangements of lattice points with respect to atoms of each of the crystal structures shown in Fig. 1.1. In all cases, the atomic arrangement looks the same from any point of the lattice. The lattice basis vector defines its primitive cell.
Figure 1.3 Crystal structure from Fig. 1.1a with two choices of lattice points (different from those shown in Fig. 1.2a) with respect to atoms of the structure. In this case, the basis is composed of two atoms.
Figure 1.4 Crystal structure from Fig. 1.1a with graphical symbols (small circles) that represent reflection points. The upper drawing shows the reflection points that overlap with the centers of atoms, and the lower drawing shows the set of reflection points that are equidistant from the centers of atoms.
1.3 Two-Dimensional Crystal Structures
Figure 1.5 Two-dimensional crystal structure. The lattice points overlap with the centers of atoms. In addition, we show different positions of twofold rotation points using graphical symbols of such symmetry elements.
Figure 1.6 Two-dimensional crystal structure. The lattice points overlap with the centers of atoms. Cells I and II are examples of two unit cells that can reproduce the lattice.
Figure 1.7 Two-dimensional crystal structure. The highlighted atom (lattice point) belongs to four cells which are marked from 1 to 4, therefore only a fraction of this atom (lattice point) belongs to the highlighted cell 1.
Figure 1.8 Two-dimensional crystal structure. The lattice points overlap with the centers of atoms. The cell shown in the figure is not primitive since contains 3 lattice points.
Figure 1.9 A two-dimensional crystal structure obtained from the structure from Fig. 1.6 by placing an additional atom at the center of each unit cell of type I. The lattice points of the resulting structure overlap with the centers of atoms. Vectors define a primitive unit cell which contains one atom. The figure shows also the diagram of symmetry elements (twofold rotation points) of the structure.
Figure 1.10 A two-dimensional crystal structure made up of two types of atoms. The unit cell, defined by vectors has 2 atoms. The diagram of symmetry elements (twofold rotation points) of the structure is supplied.
Figure 1.11 A two-dimensional crystal structure that is a superposition of two identical substructures (structures from Fig. 1.6). The lattice points overlap with the centers of atoms of one of the substructures. The primitive cell of the lattice is defined by vectors and has 2 atoms. Each of the basis atoms belongs to a different substructure.
Figure 1.12 Two-dimensional crystal structure. The lattice points overlap with twofold rotation points of the structure, so the unit cell defined by vectors is conventional. The diagram of symmetry elements of the structure is provided.
Figure 1.13 A two-dimensional crystal structure made up of two types of atoms. The structure is a superposition of two substructures. Each of them is made up of one type of atoms. The lattice points overlap with the centers of atoms of one of the substructures. The unit cell defined by vectors is conventional.
Figure 1.14 The honeycomb structure. In the figure are shown two positions of the origin of the unit cell with respect to the atoms of the structure.
Figure 1.15 A hexagonal lattice for the honeycomb structure shown in Fig. 1.14. The basis vectors are defined in Fig. 1.14. The lattice points are sixfold rotation points and the geometric centers of the equilateral triangles overlap with threefold rotation points of the lattice. In the figure, we also show the graphical symbols for the threefold and sixfold rotation points.
Figure 1.16 A two-dimensional hexagonal structure. The primitive unit cell defined by vectors is a conventional cell.
Figure 1.17 A construction made using the basis vector and the opposite to it, to show that there are only one-, two-, three-, four-, and sixfold rotation points in a two-dimensional crystal structure or lattice.
Figure 1.18 Rotation points that overlap with the geometric centers of the following plane figures: (a) parallelogram, (b) rectangle, (c) square, and (d) regular hexagon. The graphical symbols of the rotation points are shown.
Figure 1.19 Conventional unit cells for the five lattices existing in two dimensions: (a) oblique, (b) rectangular, (c) centered rectangular, (d) square, and (e) hexagonal. In cases (a–d) the n-fold rotation point shown in the figure corresponds to both the unit cell and the lattice, while in case (e) it corresponds only to the lattice.
1.4 Crystallographic Point and Space Groups in Two Dimensions
Figure 1.20 Lattice direction in a two-dimensional lattice. Vector is the shortest one in this direction.
Table 1.1 Printed symbols for symmetry elements and for the corresponding symmetry operations in two dimensions
Figure 1.21 Reflection lines orthogonal to directions [10] and [01] in a centered rectangular lattice.
Figure 1.22 Glide lines a orthogonal to direction [01] and glide lines b orthogonal to direction [10] in a centered rectangular lattice.
Table 1.2 Specification of the possible symmetry elements in the primary, secondary, and tertiary positions in the set of characters indicating symmetry elements in the Hermann–Mauguin symbol of a plane group. Mirror and glide lines are orthogonal to lattice symmetry directions specified in the last two columns
Table 1.3 Classification of the crystallographic point and plane groups in crystal systems
1.5 Wyckoff Position
Figure 1.23 Two-dimensional structure of B2C. A conventional unit cell (a) and the diagram of symmetry elements (b) are shown. The structure has been obtained using computational simulations. The calculated lattice parameters are a = 10.727 Å and b = 4.814 Å.
Figure 1.24 Atoms (points) in different positions within the unit cell of the oblique lattice. (a) Each of the atoms (points) located at the vertices contribute with a fraction to the unit cell (draw to the left). The sum of these fractions represents an entire atom (point), which is located at the origin of the cell (draw to the right). (b) In each case shown to the right, an entire atom (point) represents the sum of two fractions of atoms (points) shown to the left, and is located at the position which is closer to the origin (draw to the right). The basis vector is pointing downwards and vector to the right. The coordinates of the entire atoms (points) are given in units of a and b.
Figure 1.25 (a) Diagram of symmetry elements for the p2mg plane group. (b) An example of symmetrically equivalent points. The basis vector is pointing downwards and vector to the right. The coordinates of points are given in units of a and b.
Figure 1.26 The p2 plane group. (a) Diagram of symmetry elements. (b) Diagram of the general position. (c) Coordinates of points shown in (b). (d) Wyckoff positions. In (a), (b), and (c) the basis vector is pointing downwards and vector to the right.
Figure 1.27 The p1m1 plane group. (a) Diagram of symmetry elements. (b) Diagram of the general position. (c) Coordinates of points shown in (b). (d) Wyckoff positions. In (a), (b), and (c), the basis vector is pointing downwards and the vector to the right.
Figure 1.28 The p2mg plane group. (a) Diagram of symmetry elements. (b) Diagram of the general position. (c) Coordinates of points shown in (b). (d) Wyckoff positions. In (a), (b), and (c) the basis vector is pointing downwards and vector to the right.
Problems
Figure 1.29 One-dimensional crystal structure composed of three types of atoms.
Figure 1.30 A hexagonal lattice.
Figure 1.31 A centered rectangular lattice.
Figure 1.32 Hexagons that have the same point symmetry as the two-dimensional infinite structures composed of two types of atoms.
Figure 1.33 Unit cell for a two-dimensional lattice.
Figure 1.34 Two-dimensional B3C structure. A conventional unit cell is shown. The structure has been obtained using computational simulations. The calculated lattice parameters are a = 2.827 Å and b = 7.750 Å.
Figure 1.35 The ideal (001) surface of the NaCl crystal structure. Note that usually atoms at the crystal surfaces assume a different structure than that of the bulk atoms; however, this was not taken into account in this figure. Vectors and are primitive translation vectors of the three-dimensional NaCl structure. The experimental lattice constant is a = b = 5.6401 Å.
Figure 1.36 A two-dimensional boron structure with its conventional unit cell. One reflection line and one glide line are also specified in the draw. The structure has been obtained using computational simulations. The calculated lattice parameters are a = b = 3.228 Å.
Figure 1.37 Another example (see Fig. 1.36) of a two-dimensional boron structure. The structure has been obtained using computational simulations. The calculated lattice parameters are a = 2.882 Å and b = 3.328 Å.
Table 1.4 Wyckoff positions and the coordinates of the corresponding boron (first block) and carbon (second block) atoms located within the unit cell shown in Fig. 1.23a. The coordinates are given in units of a and b
Table 1.5 Wyckoff positions and the corresponding coordinates of Na and Cl ions located within the unit cell of the NaCl(001) surface (plane group p4mm). The coordinates are given in units of the length of surface basis vectors
Table 1.6 Wyckoff position and the corresponding coordinates of boron atoms located within the unit cell shown in Fig. 1.36 (plane group p4gm). The coordinates are given in units of a
Table 1.7 Wyckoff positions and the corresponding coordinates of boron atoms located within the unit cell shown Fig. 1.37 (plane group p2mm). The coordinates are given in units of a and b
Figure 1.38 One-dimensional crystal structure composed of two types of atoms, labeled A and B. A conventional unit cell is also shown.
Chapter 2 Three-Dimensional Crystal Lattices
2.1 Introduction
Figure 2.1 A primitive unit cell of a three-dimensional lattice.
2.2 Examples of Symmetry Axes of Three-Dimensional Figures
Figure 2.2 Some rotation axes of three solid figures: (a) rectangular prism, (b) square prism, and (c) cube.
Figure 2.3 A regular tetrahedron (a) and a regular octahedron (b) inscribed in a cube.
Figure 2.4 Symmetry points of a superposition of plane figures: (a) two equilateral triangles and (b) two squares.
2.3 Rotation Axes of a Cube
Figure 2.5 Six twofold rotation axes of a cube.
Figure 2.6 Each diagonal of a cube represents one of its threefold rotation axis.
2.4 Rotation Axes of a Set of Points
Figure 2.7 The 13 rotation axes of a cube.
Figure 2.8 The system of 14 points placed at the vertices and in the geometric centers of the faces of a cube have the same threefold rotation axes as the cube.
Figure 2.9 Axial view of one of the triangles from Fig. 2.8.
Figure 2.10 Fourfold rotation axis of a system consisting of 14 points located at the vertices and centers of the faces of a cube. On the right is shown the projection of the 14 points on the cube bottom face.
Figure 2.11 Three systems consisting of (a) 8 points at the vertices of a cube, (b) 9 points at the vertices and the geometric center of a cube, and (c) 14 points at the vertices and face centers of a cube. Each set of points has the same 13 rotation axes as the cube.
2.5 Crystal Systems
Figure 2.12 Solid figures with point symmetries of lattices belonging to the 7 crystal systems in three dimensions: (a) triclinic, (b) monoclinic, (c) orthorhombic, (d) tetragonal, (e) trigonal, (f) hexagonal, and (g) cubic. The solid figures are defined by the basis vectors and parallel (in most cases) to the main symmetry axes, if there are any in a lattice belonging to a given crystal system.
Table 2.1 Restrictions on conventional cell parameters for each crystal system. The following abbreviations are used:
2.6 The Rhombohedron and Hexagonal Prism
Figure 2.13 Rhombohedron constructed inside a hexagonal prism.
Figure 2.14 A 1/8 of the point placed in each vertex of a cubic cell belongs to this cell.
2.7 The 14 Bravais Lattices
2.7.1 Introduction
2.7.2 The Triclinic System
2.7.3 The Monoclinic System
Figure 2.15 (a) Primitive monoclinic unit cell. In the figure, we have also drawn the monoclinic cells centered in three different ways: (b) C-face centered, (c) body centered, and (d) all-face centered. The c-axis setting is assumed.
Figure 2.16 The monoclinic cells centered in two different ways: (a) A-face centered and (b) B-face centered. The c-axis setting is assumed.
Table 2.2 Symbols for the centering types of the cells shown in Figs. 2.15 and 2.16
2.7.4 The Orthorhombic System
Figure 2.17 A primitive monoclinic unit cell located inside the monoclinic lattice from Fig. 2.15b. The c-axis setting is assumed.
Figure 2.18 A body centered monoclinic unit cell placed inside the monoclinic lattice from Fig. 2.15d. The c-axis setting is assumed.
Figure 2.19 A body centered monoclinic unit cell located inside the monoclinic lattice from Fig. 2.16a. The c-axis setting is assumed.
Figure 2.20 (a) Primitive orthorhombic unit cell. In the figure, we have also drawn the orthorhombic cells centered in three different ways: (b) C-face centered, (c) body centered, and (d) all-face centered. Figure (e) shows two bases for two adjacent cells from (b) and (d). The base of a primitive (or body centered) cell of the lattice shown in (b) (or (d)) is highlighted in (e).
Figure 2.21 Two bases (labeled as I) of two adjacent C-face centered or all-face centered tetragonal unit cells. The base labeled as II corresponds in one case to a primitive tetragonal unit cell and in the other case to the body centered tetragonal cell.
2.7.5 The Tetragonal System
2.7.6 The Cubic System
Figure 2.22 Two bases (labeled as I) of two adjacent all-face centered cubic unit cells. The base labeled as II corresponds to a non-cubic body centered unit cell.
2.7.7 The Trigonal and Hexagonal Systems
2.7.8 Symbols for the Bravais Lattices
Table 2.3 Symbols for the 14 Bravais lattices
2.7.9 Final Remarks
Figure 2.23 The 14 Bravais lattices.
Figure 2.24 The NNs, NNNs, and TNNs of a lattice point in a sc lattice. The 6 NNs of a lattice point placed in the center of the large cube are at the vertices of the regular octahedron. The 12 NNNs are in the middle of the large cube edges and the 8 TNNs are in its vertices.
2.8 Coordination Number
2.9 Body Centered Cubic Lattice
Figure 2.25 Three sets of primitive translation vectors of the bcc lattice.
Figure 2.26 A primitive rhombohedral unit cell of the bcc lattice.
Figure 2.27 Demonstration of the equivalence of the two lattice points within the cubic unit cell of the bcc lattice.
Figure 2.28 The lattice points from the vertices of the cubic unit cell of the bcc lattice represent the NNs of the lattice point that is in the center of the cell.
Figure 2.29 (a) Conventional unit cell of the bcc lattice. Each of the lattice points located at the vertices contributes with 1/8 to the unit cell so the cell contains 2 lattice points. (b) The coordinate triplets of the two lattice points within the cubic cell. The point, which is a sum of eight fractions, is placed at the origin of the cell. The coordinates are expressed in units of a.
2.10 Face Centered Cubic Lattice
Figure 2.30 (a) Face centered cubic unit cell of the fcc lattice. (b) Coordinate triplets of the four lattice points within the cubic cell. The coordinates are expressed in units of a, which is the length of the cube edge.
Figure 2.31 A primitive rhombohedral unit cell of the fcc lattice.
Figure 2.32 Demonstration of the equivalence of all lattice points in the cubic cell of the fcc lattice.
Figure 2.33 Nearest neighbors of a lattice point in the fcc lattice.
2.11 Rhombohedral Unit Cell in a Cubic Lattice
2.11.1 Introduction
2.11.2 A Rhombohedral Unit Cell of the sc Lattice
Figure 2.34 A body centered rhombohedral unit cell of the sc lattice.
Table 2.4 The characteristics of the two unit cells, shown in Fig. 2.34, of the sc lattice
Figure 2.35 Simple cubic crystal structure. Face centered cubic lattice is used to describe this structure. The cubic F and the rhombohedral P unit cells of this lattice, defined by the basis vectors and respectively, are shown. The cubic F unit cell contains 8 atoms, while the rhombohedral P unit cell contains 2 atoms.
2.11.3 Simple Cubic Crystal Structure
Figure 2.36 The fcc lattice used to describe the sc structure. The figure shows a primitive rhombohedral unit cell of the fcc lattice (defined by vectors in Fig. 2.35) with two atoms in it. (a) The cell origin overlaps the center of an atom. (b) An alternative choice of the cell origin.
2.11.4 Interpretation of Data for As, Sb, Bi, and Hg
Table 2.5 Experimental lattice parameters obtained under normal conditions for arsenic, antimony, and bismuth. It is also given the value for mercury at normal pressure and temperature 227 K. The conventional basis vectors and define a rhombohedral P unit cell that in the case of As, Sb, and Bi contains two atoms. The data about the location of basis atoms are given with respect to the conventional cell origin, whose location is similar to that from Fig. 2.36b. In the case of Hg the basis is composed of one atom
2.12 Rhombohedral Lattice
Figure 2.37 Primitive rhombohedral and a R-centered hexagonal unit cells of a rhombohedral lattice. The centering points, within the hexagonal cell, reduce the sixfold rotation axis of the hexagonal prism to a threefold rotation axis.
Figure 2.38 (a) Projections of the centering points of the triple hexagonal cell from Fig. 2.37 on the base of the cell. The coordinates of these points are given in terms of the basis vectors and (b) Projections of the 6 points that are inside the hexagonal prism on its base. The fractions and near the lattice point projections are coordinates of these points in terms of the basis vector The hexagonal prism base translated by a translation vector is also shown.
Figure 2.39 The reverse setting of a triple hexagonal cell in relation to the primitive rhombohedral cell.
Figure 2.40 Coordinate triplets of the three points within the triple hexagonal R unit cell in obverse setting (a) and in reverse setting (b). The coordinates are expressed in units of a and c.
2.13 Triple Hexagonal R Cell in the Cubic Lattice
Figure 2.41 Three types of unit cells of the bcc lattice. Each of the three triple hexagonal R cells shown in the figure is defined by basis vectors and or their linear combinations. Inside the hexagonal prism there is a rhombohedral P unit cell defined by basis vectors and Besides that, there is a cubic I cell of the bcc lattice defined by vectors and All three unit cells have the same origin O.
Table 2.6 Basic information about three types of unit cells of the bcc lattice
Table 2.7 Basic information about three types of unit cells of the fcc lattice
2.14 Wigner–Seitz Cell
2.14.1 Introduction
2.14.2 Construction of the Wigner–Seitz Cell
Figure 2.42 The Wigner–Seitz cell of the bcc lattice.
2.14.3 The Wigner–Seitz Cell of the bcc Lattice
Figure 2.43 In (a) is outlined a cross section of two cubic F cells of the fcc lattice. In (b) the cross section is used to demonstration that in the construction of the Wigner–Seitz cell participate only the NNs of the lattice point belonging to this cell.
2.14.4 The Wigner–Seitz Cell of the fcc Lattice
Figure 2.44 (a) The Wigner–Seitz cell of the fcc lattice. (b) A face of the dodecahedron shown in (a).
Problems
Figure 2.45 (a) A regular tetrahedron and (b) a regular octahedron.
Figure 2.46 A tetradecahedron inscribed in a cube of edge length a. This tetradecahedron may be seen as a regular octahedron truncated by 6 faces of the smaller cube. The regular octahedron is inscribed in a cube of edge length (3/2)a and has the same geometric center as the cube of edge length a.
Figure 2.47 The Wigner–Seitz cell of the fcc lattice.
Figure 2.48 A set of 27 points located at the vertices of the 8 small cubes.
Figure 2.49 Monoclinic cells centered in two different ways: (a) A-face centered and (b) C-face centered. The b-axis setting is assumed.
Figure 2.50 All the NNs, some of the NNNs, and also some of the TNNs of a lattice point located at the center of the displayed fcc lattice.
Figure 2.51 A two-dimensional square lattice of lattice constant a.
Figure 2.52 A rhombohedron constructed inside a hexagonal prism.
Figure 2.53 (a) A rhombohedron constructed inside a hexagonal prism of side a and height c. (b) The highlighted face of the rhombohedron from (a). (c) The right triangle highlighted in (a).
Figure 2.54 Body centered rhombohedral unit cell. The coordinates of the two lattice points are expressed in units of
Figure 2.55 A centered rhombohedral unit cell for the fcc lattice.
Chapter 3 Crystal Structures of Elements
3.1 Introduction
3.2 Pearson Notation and Prototype Structure
Table 3.1 Pearson symbols corresponding to 14 Bravais lattices. In the symbols, n expresses the number of atoms per conventional unit cell. The last column gives examples of Pearson symbols, which together with the prototype structures correspond to crystal structures of elements
3.3 Filling Factor
3.4 Simple Cubic Structure
Figure 3.1 The plane that contains a face of the cubic P unit cell of the sc structure with the cross sections of atoms considered as hard spheres.
3.5 Body Centered Cubic Structure
Figure 3.2 (a) Cubic I unit cell of the bcc structure. (b) Cross sections of atoms considered as hard spheres in a plane defined by two body diagonals of the cube shown in (a). In this plane, there are the points of contact between the central atom and its NNs.
Table 3.2 Lattice constants of elements that crystallize in the bcc structure at normal pressure. The data is provided at room temperature, unless otherwise specified
Figure 3.3 (a) Cubic F unit cell of the fcc structure. (b) Cross sections of 5 atoms considered as hard spheres from the front face of the cube from (a). The points of contact between the NNs are found in this plane.
3.6 Face Centered Cubic Structure
Table 3.3 Lattice constants of elements that crystallize in the fcc structure at normal pressure. The data is given at room temperature, unless otherwise specified
3.7 Close-Packed Structures
Figure 3.4 A close-packed layer of equal spheres that is a two-dimensional close-packed hexagonal structure.
Figure 3.5 The centers of spheres of the first layer and the projection of the centers of spheres of the second layer in a close-packed arrangement of equal spheres.
Figure 3.6 Draws (a) and (b) show two close-packed arrangements of equal spheres. The case described in (a) differs from that one in (b) in the positions of spheres of the third layer with respect to the spheres of the first and second layers.
Figure 3.7 The fcc structure viewed as a close-packed structure (ccp). Three consecutive layers of this structure are marked as A, B, and C.
Figure 3.8 The ABAB ... stacking sequence of atomic layers in the hcp structure.
3.8 Double Hexagonal Close-Packed Structure
Table 3.4 Lattice parameters of lanthanides that crystallize in the dhcp structure under normal conditions. The data for δ-Sm correspond to room temperature and 4.0 GPa
Table 3.5 Lattice parameters of actinides that crystallize in the dhcp structure under normal conditions
Figure 3.9 Double hexagonal close-packed structure. The ABACABAC ... sequence of layers is shown.
3.9 Samarium Close-Packed Structure
Figure 3.10 Hexagonal prisms for three of the four close-packed structures considered by us: hcp, dhcp, and Sm-type. In each case, the sequence of the two-dimensional hcp layers is shown. The hexagonal prism for the fourth close-packed structure (ccp) is displayed in Fig. 3.11.
Figure 3.11 Hexagonal prism for the ccp (fcc) structure (the cell parameters ratio is see Exercise 2.21). The sequence of layers A, B, and C is shown.
3.10 Symmetry Axes
Table 3.6 Rotoinversion axes and lower symmetry axes included in them
Table 3.7 Rotation axes with center of symmetry and lower symmetry axes included in them
Table 3.8 Two- and threefold screw axes and the corresponding screw vectors of a right-handed screw rotation
Table 3.9 Fourfold screw axes with the corresponding screw vectors of a right-handed screw rotation and lower symmetry axis included in each case
Table 3.10 Basic information about sixfold screw axes
Table 3.11 Basic information about screw axes with center of symmetry
Figure 3.12 (a) Threefold screw axis 31 in the infinite selenium structure. The fractions and near the atom projections, in the drawing to the right, are coordinates of these atoms given in terms of the basis vector (b) Sixfold screw axis with center of symmetry, 63/m, in the infinite graphite structure. The dashed line with arrows indicates changes in the positions of atoms after the right-handed rotation of 60° around the axis accompanied by a fractional translation by screw vector The vector pointing in the direction of the screw, is the shortest lattice translation vector parallel to the axis.
3.11 Hexagonal Close-Packed Structure
Figure 3.13 (a) A hexagonal prism for the hcp structure. Shown are the rotoinversion axis and the threefold rotation axis included in it. The -inversion point is located on layer B (in an infinite volume the -inversion points are located on layers A and B). (b) Hexagonal prism for the hcp structure shifted horizontally with respect to that from (a) in the way shown in (c). The symmetry axis 63/m and also the lower symmetry axes 3 and 21 that are sub-elements of it are shown. The centers of symmetry are equidistant from layers A and B. (c) Top view of the set of atoms within the hexagonal prism from (a) and the projection of the atoms within the infinite volume specified in (b) on the base of the prism.
Figure 3.14 The hexagonal P unit cell of the hcp structure defined by the basis vectors and In (a) the origin O of the cell coincides with an -inversion point, whereas in (b) the origin O′ coincides with a center of symmetry placed on the 63/m axis.
Figure 3.15 Coordinate triplets of the two atoms that are within the hexagonal P unit cell of the hcp structure: (a) from Fig. 3.14a and (b) from Fig. 3.14b. The coordinates are expressed in units of a and c.
Figure 3.16 Six of twelve NNs (labeled 1–3 and 5–7) of the atom labeled as 4. The 6 NNs are located at the vertices of a hexagonal P unit cell. Vector gives the position of the atom marked as 4.
Figure 3.17 The base of the hexagonal P unit cell shown in Fig. 3.16.
Table 3.12 Lattice parameters of metals that crystallize in the hcp structure. The data is given at room temperature and normal pressure, unless otherwise specified. Values for helium (3He and 4He) are also included
Table 3.13 Crystal structures of all metals that crystallize in dense-packed structures (fcc, hcp, dhcp, Sm-type, and bcc) under normal conditions. The structures of noble gases at low temperatures are also included
3.12 Interstices in Close-Packed Structures
Figure 3.18 Centers of spheres of the A layer and projection of the centers of spheres of the B layer. (a) Highlighted are the top views of two tetrahedrons with a different spatial orientation, one with three vertices and one vertex projection and the other one with one vertex and three vertex projections. (b) Top view of octahedron bases lying on A and B layers.
Figure 3.19 Three tetrahedral interstices inside a hexagonal prism for the hcp structure. The sequence of layers A and B is shown.
Figure 3.20 Two tetrahedral (a) and one octahedral (b) interstices in a cubic F unit cell of the fcc (ccp) structure. The sequence of layers A, B, and C, which are orthogonal to a body diagonal of the cube, is shown.
Figure 3.21 Distribution of the octahedral interstices within the cubic F unit cell of the fcc structure. The interstices that have their centers placed on cube edges belong only partially to the cube.
Figure 3.22 (a) Cubic F cell of the fcc structure. (b) Eight cubic I cells of the bcc structure.
Figure 3.23 (a) Hexagonal prism for the fcc structure shown in Fig. 3.22a in relation to the cubic F unit cell. (b) Four hexagonal prisms for the bcc structure shown in Fig. 3.22b. The large prism (of height 4cbcc) is related to the large cube from Fig. 3.22b.
3.13 Diamond Structure
Figure 3.24 Two-dimensional schematic diagram of covalent bonds in the diamond structure.
Figure 3.25 (a) A tetrahedron defined by the NNs of an atom in the diamond structure. (b) The tetrahedron from (a) inscribed in a cube. (c) Three-dimensional schematic representation of covalent bonds between an atom and its 4 NNs.
Figure 3.26 Two small cubes from Fig. 3.25b placed in two of the four possible positions inside a cubic F unit cell of the diamond structure.
Figure 3.27 Relative positions of atoms belonging to the diamond structure. The 4 atoms that are inside the cubic F unit cell are distributed in two vertical planes defined by the body diagonals of the cube.
Figure 3.28 (a) Comparison of the distribution of atoms from vertices and faces of the cubic F unit cells with those from their body diagonals in the diamond structure. (b) Cross sections of atoms (considered as hard spheres) from (a) are shown. In this figure, the equivalency between the relative distributions of atoms of each type (those from vertices and faces of the cube and those from its diagonals) is visualized.
Figure 3.29 Left part of Fig. 3.28b with the cross sections of the A, B, and C layer planes added to it. In this figure, the sequence of layers is easily seen.
Figure 3.30 Cubic F and rhombohedral P unit cells of the diamond structure (left). In the figure, is also shown the location of the two atoms belonging to the rhombohedral P unit cell (right).
3.14 Arsenic Structure
Figure 3.31 (a) Coordinate triplets of the eight atoms within the cubic F unit cell of the diamond structure. The coordinates are expressed in units of a. (b) Projection of atoms on the cell base. The fraction nearby the projection of an atom represents its coordinate in units of c = a.
Figure 3.32 (a) Hexagonal prism for the rhombohedral Bravais lattice hR of the normal phase of arsenic. To the right, the atomic basis is shown. (b) Hexagonal prism for the fcc lattice proposed to describe the sc structure of arsenic at room temperature and pressure of 25 GPa. The atomic basis is shown to the right. (c) Crystal structure of arsenic at normal conditions. (d) Crystal structure of arsenic at room temperature and pressure of 25 GPa. Both figures to the right were rescaled to keep a′ = a.
3.15 Selenium Structure
Figure 3.33 (a) Base of the hexagonal P unit cell with a conventional origin O for the structure of selenium at normal conditions. The projection of the three atoms within the unit cell on its base is shown. The fraction nearby the projection of each atom represents its fractional coordinate along the axis. (b) Structure of selenium at normal conditions. In the upper part is shown the projection, on the prism base, of atoms from the hexagonal prism shown in the lower part of the figure. The second hexagon corresponds to the projection of a hexagonal prism (not shown) shifted with respect to that shown in the figure. (c) Structure of selenium at room temperature and pressure of 46 GPa. In the upper part is shown the projection, on the prism base, of atoms located in the hexagonal prism shown in the lower part of the figure.
3.16 Graphite Structure
Figure 3.34 (a) Relative position of the projections of the honeycomb A, B layers in the graphite structure. (b) Hexagonal prism for the graphite structure. The rotoinversion axis is shown. The -inversion point is located on layer B (in an infinite volume, the -inversion points are located on layers A and B). (c) Hexagonal prism shifted horizontally with respect to that from (b) in the way shown in (a). The symmetry axis 63/m is shown. The centers of symmetry are equidistant from layers A and B.
Table 3.14 Sets of symmetry directions in a three-dimensional hexagonal lattice corresponding to a given position in the Hermann–Mauguin space-group symbol
3.17 Atomic Radius
Table 3.15 Lattice constants of elements that crystallize in the diamond structure under normal conditions. In addition, the NN distances, d, and the covalent radii, rcov, are given
Table 3.16 Nearest neighbor interatomic distances (in Angstroms) of metals that crystallize in dense-packed structures under normal conditions. The values have been calculated using the data from Tables 3.2–3.5 and 3.12. In the case of metals that crystallize in the hcp, dhcp, and Sm-type structures, we report two interatomic distances: the distance to the 6 NNs from the same layers and the distance to the 6 NNs from adjacent layers
Table 3.17 Metallic radii (in Angstroms) of metallic elements that under normal conditions crystallize in dense-packed structures (fcc, hcp, dhcp, Sm-type, and bcc). In the case of elements that crystallize in the hcp, dhcp, and Sm-type structures, the radius is half of the average value for the NN interatomic distance
Problems
Figure 3.35 Two hexagonal prisms for a dhcp structure. One is shifted with respect to the other by 1/4 of the prism height.
Figure 3.36 Hexagonal prism for the α-Sm structure.
Figure 3.37 Cross section in the middle of a hexagonal prism of side 2ah and height 2ch, where ah and ch are parameters of a triple hexagonal unit cell of the fcc structure. The gray colored hexagon corresponds to the bottom base of the hexagonal prism of side ah and height ch. The atoms located in the cross sectional plane and the projections of atoms within the top half of the large prism are shown. The fraction nearby the projection of an atom represents its coordinate in units of ch.
Chapter 4 Crystal Structures of Important Binary Compounds
4.1 Introduction
4.2 The Ionic Radius Ratio and the Coordination Number
Figure 4.1 Zinc blende structure. (a) Regular tetrahedron defined by Zn cations with the S anion in its center. (b) Nearest neighbors of the Zn cation at the vertices of a regular tetrahedron.
Figure 4.2 Cubic F unit cell for the zinc blende structure. A plane defined by two body diagonals of the cube is shown.
Figure 4.3 Cross section from Fig. 4.2 of the cubic F unit cell for silicon carbide in the β phase. Larger circles correspond to the cross sections of Si atoms and the smaller ones to the cross sections of C atoms.
Figure 4.4 A plane defined by two body diagonals of the cubic F unit cell for the zinc blende structure shown in Fig. 4.2. In the figure, we show the limiting case in which the anions, represented by larger circles, touch one another and are in contact with the cations (smaller circles).
Table 4.1 Expected radius ratio ranges for different cation coordination numbers. The crystal structures from the last column of the table will be fully described in this chapter
Table 4.2 Ionic radii for Na+, K+, and Ca2+, for different coordination numbers. For comparison, we have also listed the metallic radii for Na, K, and Ca taken from Table 3.17
4.3 Zinc Blende Structure
Figure 4.5 (a) Two cubic F unit cells for the zinc blende structure of ZnS: one with Zn cations and the other one with S anions at the vertices. (b) A rhombohedral P unit cell for the zinc blende structure with the two ions belonging to it.
Table 4.3 Lattice constants (in Angstroms) obtained under normal conditions for III–V compounds that crystallize in the zinc blende structure
Table 4.4 Lattice constants (in Angstroms) obtained under normal conditions for compounds of Be-VI and TM-VI type that crystallize in the zinc blende structure
Table 4.5 Lattice constants (in Angstroms) obtained under normal conditions for compounds of TM-VII type that crystallize in the zinc blende structure
4.4 Fluorite and Anti-Fluorite Structures
4.4.1 Fluorite Structure
Figure 4.6 Cubic F unit cells for the CaF2 structure. In the cube vertices are placed Ca2+ cations in (a) and F− anions in (b).
Figure 4.7 (a) Regular tetrahedron defined by the NNs of the F− anion in CaF2. (b) Cube defined by the NNs of the Ca2+ cation in CaF2.
Table 4.6 Lattice constants (in Angstroms) obtained under normal conditions for II–VII compounds and hydrides, silicides, oxides, and fluorides of some TMs, all of them crystallizing in the fluorite structure. In addition, the data for β-PbF2 and α-PoO2 are included
Figure 4.8 Cubic F unit cells of Li2O which crystallizes in the anti-fluorite structure. In the cube vertices are placed O2− anions in (a) and Li+ cations in (b).
4.4.2 Anti-Fluorite Structure
Figure 4.9 (a) Regular tetrahedron defined by the NNs of the Li+ cation in Li2O. (b) Cube defined by the NNs of the O2− anion in Li2O.
Table 4.7 Lattice constants (in Angstroms) obtained under normal conditions for I–VI compounds that crystallize in the anti-fluorite structure
Table 4.8 Lattice constants obtained under normal conditions for some II–III and II–IV compounds, and also phosphides, all of them crystallizing in the anti-fluorite structure
4.5 Wurtzite Structure
Figure 4.10 Hexagonal prism for ZnS in the wurtzite structure, with Zn cations at the vertices (a) and with S anions at the vertices (b).
Figure 4.11 Two hexagonal P unit cells for ZnS in the wurtzite structure: in (a) with a Zn cation at the origin and in (b) with a S anion at the origin.
Figure 4.12 (a) and (b) give the coordinate triplets of ions belonging to the unit cells from Figs. 4.11a and 4.11b, respectively. The coordinates are expressed in units of a and c.
Figure 4.13 (a) A tetrahedron defined by the NNs of the S anion in α-ZnS. (b) A tetrahedron defined by the NNs of the Zn cation in α-ZnS. We can envision the tetrahedrons from (a) and (b) in both cells from Fig. 4.11.
Figure 4.14 (a) A regular tetrahedron, defined by Zn cations, located inside a hexagonal P unit cell for the wurtzite structure of ZnS. (b) A vertical cross section of the tetrahedron shown in (a). See text for detailed explanation.
Figure 4.15 Hexagonal layers AS, AZn, BS, and BZn in the wurtzite structure of ZnS. The distances between the consecutive layers for the ideal case are shown.
Table 4.9 Lattice parameters, obtained under normal conditions, of binary compounds that crystallize in the wurtzite structure
Table 4.10 Comparison between NN distances for zinc blende and wurtzite structures of some binary compounds. The values were obtained from the lattice constants listed in Tables 4.3, 4.4, and 4.9
4.6 Nickel Arsenide Related Structures
4.6.1 NiAs Structure
Figure 4.16 (a) Hexagonal prisms for the prototypical NiAs. (b) The NiAs structure for the ideal case when
Figure 4.17 Octahedrons defined by anions that are the NNs of a cation (a) in the NiAs compound and (b) in the ideal NiAs structure. In this figure we also show the two nearest cations to the cation placed in the center of each octahedron.
Figure 4.18 (a) Two hexagonal prisms for the NiAs structure: one with anions at the vertices and another one with cations at the vertices. (b) Projection of the centers of ions belonging to each hexagonal prism on the hexagonal base. We can observe that the triangles defined by the ions from the Ba layer have in each case from (a) different orientations with respect to the hexagonal prism base.
Figure 4.19 Octahedron defined by the NNs of a cation (a) and the trigonal prism defined by the NNs of an anion (b) in the NiAs structure.
Figure 4.20 (a) Hexagonal prism for an ideal NiAs structure with the cations at the vertices. The regular octahedron defined by 6 anions located inside this prism is also shown. In addition, this octahedron is inscribed in a cube. (b) The cube defined in (a). The longitude of a body diagonal of the cube is expressed as a function of the lattice constant c. (c) One of the triangles shown in (b). In this figure we show the relation between the lattice constant a and the cube edge ac.
Figure 4.21 Two hexagonal P unit cells for the NiAs structure: (a) with an anion at the origin and (b) with a cation at the origin. Neither of the two choices for the origin is considered standard. In (a) the origin coincides with an -inversion point, whereas in (b) it coincides with a center of symmetry placed on the axis 63/m (see Section 3.11 for the hcp structure and Fig. A.31 in Appendix).
Figure 4.22 (a) and (b) show the coordinate triplets of ions belonging to the unit cells from Figs. 4.21a and 4.21b, respectively. The coordinates are expressed in units of a and c.
Figure 4.23 Anti-NiAs structure shown on the example of the VP compound.
Table 4.11 Lattice parameters, obtained under normal conditions, of PtB and NiTl that crystallize in the anti-NiAs and NiAs structures, respectively
Table 4.12 Lattice parameters obtained under normal conditions for compounds of TM-IV type that crystallize in the NiAs, anti-NiAs, or TiAs structures
Table 4.13 Lattice parameters obtained under normal conditions for compounds of TM-V type that crystallize in the NiAs, anti-NiAs, or TiAs structures
Table 4.14 Lattice parameters obtained under normal conditions for compounds of TM-VI type that crystallize in the NiAs or anti-NiAs structures. The values for MgPo are also included in the table
Figure 4.24 Hexagonal prisms: (a) for the NiAs structure and (b) for the TiAs structure. The lattice parameters, a and c, are shown in both cases.
4.6.2 TiAs Structure
4.7 Sodium Chloride Structure
Figure 4.25 The structure of NaCl. (a) Regular octahedron defined by Na+ cations with the Cl− anion in its center. (b) Nearest neighbors of a Na+ cation at the vertices of a regular octahedron.
Figure 4.26 (a) Two conventional cubic F unit cells for the structure of sodium chloride: one with Na+ cations at the vertices and the other one with Cl− anions at the vertices. (b) A rhombohedral P unit cell with two ions (one anion and one cation), which is the smallest unit cell that reproduces the NaCl structure.
Figure 4.27 The sequence of two-dimensional hexagonal layers ANa, BNa, CNa, ACl, BCl, and CCl in the structure of NaCl.
Figure 4.28 Coordinate triplets of ions belonging to the cubic F unit cell of the NaCl structure shown in Fig. 4.26a to the left. The coordinates are expressed in units of a. The cell is conventional.
Table 4.15 Lattice constants (in Angstroms), obtained under normal conditions, of alkali metal halides and silver halides that crystallize in the NaCl structure. In the table, it is also indicated which of the considered compounds crystallize in the CsCl or zinc blende structures
Table 4.16 Lattice constants (in Angstroms), obtained under normal conditions, of II-VI and also some IV-VI and V-VI compounds that crystallize in the NaCl structure. In the table, it is also indicated which of the considered compounds crystallize in the wurtzite or NiAs structures
Table 4.17 Lattice constants (in Angstroms) obtained under normal conditions for compounds of the TM-VI type that crystallize in the NaCl structure. In the table, it is also indicated which of the considered compounds crystallize in the NiAs or zinc blende structures
Table 4.18 Lattice constants (in Angstroms) obtained under normal conditions for the compounds of the TM-V type that crystallize in the NaCl structure. The data for some tin pnictides are also included. In addition, we indicate in the table which of the considered compounds crystallize in the NiAs or TiAs structures
Figure 4.29 The plane of a face of the cubic F unit cell of the NaCl structure with the cross sections of 9 ions considered hard spheres. The large ion, located in the center of the face, makes contact with its NNs (small spheres) and also with the NNNs (large spheres). The NN distance d is equal to the sum of the ionic radii r− + r+.
Figure 4.30 The same plane as in Fig. 4.29, but now the large ion, located in the center of the cube face, makes contact only with the NNs (small spheres). The NN distance, d, is equal to r− + r+.
Figure 4.31 The same plane as in Figs. 4.29 and 4.30, but now the smaller ion is too small to make contact with larger ions and as a consequence the ion located in the center of the cube face makes contact only with its NNNs (large spheres). The NN distance, is defined only by the radius of the larger ion.
Table 4.19 Several values for alkali halides: (a) cation and anion radii (to the right of the ion symbols), (b) ionic radius ratios (r+/r− or r−/r+), (c) sums of the ionic radii (r− + r+) in cases when r+/r− > 0.414 or in cases when r+/r− < 0.414, and (d) experimental values for the NN distances, d = a/2, where the lattice constants, a, are given in Table 4.15
4.8 Cesium Chloride Structure
Figure 4.32 A conventional unit cell for cesium chloride with two ions in it.
Figure 4.33 A plane defined by two body diagonals of the cube shown in Fig. 4.32. In this plane, there are the points of contact between the cation and its four NNs.
Table 4.20 Lattice constants, obtained under normal conditions, of cesium and thallium halides that crystallize in the CsCl structure
Table 4.21 Lattice constants (in Angstroms) obtained under normal conditions for intermetallic compounds of the RE-Mg or RE-III type that crystallize in the CsCl structure. The elements in the compound symbols are listed alphabetically
Table 4.22 Lattice constants (in Angstroms) obtained under normal conditions for intermetallic compounds of the RE-TM type that crystallize in the CsCl structure. The elements in the compound symbols are listed alphabetically
Table 4.23 Lattice constants obtained under normal conditions for intermetallic compounds that crystallize in the CsCl structure. The elements are listed alphabetically in those compounds where at least one of the elements is a TM
Problems
Chapter 5 Reciprocal Lattice
5.1 Introduction
5.2 The Concept of the Reciprocal Lattice
Figure 5.1 A two-dimensional crystal lattice. The points at and have equivalent positions in the lattice.
Figure 5.2 The relation between the components of the vector in the orthogonal and non-orthogonal coordinate systems.
Figure 5.3 The unit cell of a reciprocal lattice defined by the primitive translation vectors and
5.3 Examples of Reciprocal Lattices
5.3.1 Introduction
5.3.2 Reciprocal of the Triclinic Lattice
Figure 5.4 Primitive unit cell of the reciprocal of the triclinic lattice. In the figure, we show also the primitive unit cell for the direct lattice.
Figure 5.5 Cubic P unit cells of the sc lattice and its reciprocal lattice.
5.3.3 Reciprocal of the Simple Cubic Lattice
Figure 5.6 Cubic F unit cell of the fcc lattice and the primitive translation vectors that define a rhombohedral P unit cell for this lattice.
5.3.4 Reciprocal of the Face Centered Cubic Lattice
Figure 5.7 Rhombohedral P unit cell for the bcc reciprocal lattice. The cell is defined by primitive translation vectors with components given in the figure.
Table 5.1 Correspondence between types of centering in direct and their reciprocal lattices
Problems
Figure 5.8 Vectors and that generate the two-dimensional direct and reciprocal lattices, respectively.
Chapter 6 Direct and Reciprocal Lattices
6.1 Introduction
6.2 Miller Indices
Figure 6.1 Four lattice planes with different orientations in the sc lattice.
Figure 6.2 Three-dimensional crystal lattice generated by the primitive translation vectors The lattice plane shown in the figure intersects the axes ξ1, ξ2, and ξ3 in the lattice points.
Figure 6.3 A complementary figure to Fig. 6.1d. In this figure is indicated the lattice point where the () plane crosses the z-axis defined by the translation vector
Figure 6.4 The (111) and () planes in the sc lattice.
Figure 6.5 Three lattice planes in the sc lattice that are equivalent by symmetry of the lattice.
Figure 6.6 A lattice direction. The vector is the shortest one in this direction.
6.3 Application of Miller Indices
Property 1
Figure 6.7 Three directions equivalent by lattice symmetry in the sc lattice.
Figure 6.8 Two (133) lattice planes in a lattice generated by the primitive translation vectors
Property 2
Figure 6.9 Two (623) lattice planes in a lattice generated by the primitive translation vectors
Figure 6.10 The direction given by the vector in the reciprocal lattice is orthogonal to the family of (hkl) direct lattice planes.
Figure 6.11 Two consecutive (hkl) planes. The direction orthogonal to these planes, defined by unit vector ňhkl, is shown. The distance, dhkl, between these planes is also indicated.
Property 3
Figure 6.12 The (100) planes which include two cube faces that are parallel to each other in the direct sc lattice. The cubic P unit cell defined by primitive translation vectors of the reciprocal lattice is also shown. The translation vector is orthogonal to the (100) planes.
Figure 6.13 The cubic P unit cells of the sc direct lattice and the reciprocal to it. The vector of the reciprocal lattice is orthogonal to the (111) planes in the direct lattice.
Problems
Figure 6.14 A simple cubic lattice, of lattice constant a, generated by the translation vectors
Figure 6.15 Two-dimensional crystal lattice generated by the primitive translation vectors In the figure four consecutive (41) lattice planes are shown.
Chapter 7 X-Ray Diffraction
7.1 Introduction
7.2 Laue Equations
Figure 7.1 X-ray scattering by two atoms within a crystal structure with one-atom basis. One atom is located at the origin O and another one at a lattice point of position given by The unit vectors ň and ň′ are parallel to the directions of the incident and scattered X-ray beams, respectively.
Figure 7.2 Geometric illustration of Eq. 7.9.
7.3 Ewald Construction and the Bragg Diffraction Formula
Figure 7.3 Graphical demonstration of the fact that vectors and are orthogonal to each other.
Figure 7.4 Ewald construction shown in a two-dimensional reciprocal lattice.
Figure 7.5 X-ray reflection from two consecutive (hkl) lattice planes.
7.4 Atomic Structure Factor
Figure 7.6 Scattering of X-rays by a single electron. The scattered ray propagates in any ň′ direction.
Figure 7.7 X-ray beam scattered in the ň′ direction by electrons placed at and at the origin O. The scattering amplitudes are given.
Figure 7.8 Scattering of the X-ray beam by an electron placed at As a reference, the scattering by an electron placed at the origin is also considered. The z-axis is parallel to vector defined in the draw.
Figure 7.9 An isolated spherical atom with electron density The infinitesimal electron charge within the volume d3r at a point produces an infinitesimal electrostatic potential at a point
Figure 7.10 X-ray scattered elastically in the ň′ direction by an infinitesimal charge placed at As a reference, the scattering by an electron placed at the origin O is also considered.
Figure 7.11 Definition of the spherical coordinates r, ϑ, and φ with respect to vector defined in Fig. 7.8.
Figure 7.12 A two-dimensional crystal structure. Basis vectors and define the unit cell. The atomic basis is composed of two atoms at positions and
7.5 Structure Factor
Figure 7.13 (a) Conventional unit cell of a two-dimensional crystal structure defined by lattice vectors and The reciprocal lattice is defined by vectors The unit cell contains two atoms of different types. (b) X-ray scattering by the atom at position in (a) and the same atom placed at the origin. The vector is orthogonal to the family of (11) planes.
Figure 7.14 Conventional unit cell cubic I for the bcc structure. This cell contains two atoms located at and
Table 7.1 Integral reflection conditions for centering cells
Problems

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