Читать книгу Crystallography and Crystal Defects - Anthony Kelly - Страница 24

All of the symmetry operations in a crystal must be mutually consistent. There are no fivefold axes of rotational symmetry because such axes are not consistent with the translational symmetry of the lattice. In Section 1.6 we derived the possible combinations of pure rotational symmetry operations that can pass through a point. These combinations are classified into different crystal systems and we will now investigate the types of space lattice (that is, the regular arrangement of points in three dimensions as defined in Section 1.1) that are consistent with the various combinations of rotation axes. We shall find, as we did for the two‐dimensional lattice (or net) consistent with mirror symmetry (Section 1.5), that more than one arrangement of points is consistent with a given set of rotational symmetry operations. However, the number of essentially different arrangements of points is limited to 14. These are the 14 Bravais, or space, lattices. Our derivation is by no means a rigorous one because we do not show that our solutions are unique. This derivation of the Bravais lattices is introduced to provide a background for a clear understanding of the properties of imperfections studied in Part II of this book. The lattice is the most important symmetry concept for the discussion of dislocations and martensitic transformations.

We start with the planar lattices or nets illustrated in Figure 1.14. To build up a space lattice we stack these nets regularly above one another to form an infinite set of parallel sheets of spacing z. All of the sheets are in identical orientation with respect to an axis of rotation normal to their plane, so that corresponding lattice vectors t₁ and t₂ in the nets are always parallel. The stacking envisaged is shown in Figure 1.18. The vector t₃ joining lattice points in adjacent nets is held constant from net to net. The triplet of vectors t₁, t₂, t₃ defines a unit cell of the Bravais lattice.

Figure 1.18 Stacking of nets to build up a space lattice. The triplet of vectors t₁, t₂, t₃ defines a unit cell of the Bravais lattice

We now consider the net based on a parallelogram (Figure 1.14a). If we stack nets of this form so that the points of intersection of twofold axes in successive nets do not lie vertically above one another then we destroy the twofold symmetry axes normal to the nets. We have a lattice of points showing no rotational symmetry. The unit cell is an arbitrary parallelepiped with edges a, b, c, no two of which are necessarily equal, and where the angles of the unit cell α, β, γ can take any value; the cell is shown in Figure 1.19a. By choosing a, b, c appropriately we can always ensure that the cell is primitive. Although this lattice contains no axis of rotational symmetry, the set of lattice points is of course necessarily centrosymmetric.

Figure 1.19 Unit cells of the 14 Bravais space lattices. (a) Primitive triclinic. (b) Primitive monoclinic. (c) Side‐centred monoclinic – conventionally the twofold axis is taken parallel to y and the (001) face is centred (C‐centred). (d) Primitive orthorhombic. (e) Side‐centred orthorhombic – conventionally centred on (001) (C‐centred). (f) Body‐centred orthorhombic. (g) Face‐centred orthorhombic. (h) Primitive tetragonal. (i) Body‐centred tetragonal. (j) Primitive hexagonal. (k) Primitive rhombohedral. (l) Primitive cubic. (m) Face‐centred cubic. (n) Body‐centred cubic

To preserve twofold symmetry we can proceed in one of two different ways. We can arrange parallelogram nets vertically above one another so that t₃ is normal to the plane of the sheets, as in Figure 1.20a, or we can produce the staggered arrangement shown in plan, viewed perpendicular to the nets, in Figure 1.20b.

Figure 1.20 Lattice points in the net at height zero are marked as dots, those at height z with rings

In the first of these arrangements, in Figure 1.20a, the twofold axes at the corners of the unit parallelogram of the nets all coincide and we produce a lattice of which one unit cell is shown in Figure 1.19b. This has no two sides of the primitive cell necessarily equal, but two of the axial angles are 90°. A frequently used convention is to take α and γ as 90° so that y is normal to x and to z; β is then taken as the obtuse angle between x and z.

The staggered arrangement of the parallelogram nets in Figure 1.20b is such that the twofold axes at the corners of the unit parallelograms of the second net coincide with those at the centres of the sides of the unit parallelogram of those of the first (or zero‐level) net. A lattice is then produced of which a possible unit cell is shown in Figure 1.21. This is multiply primitive, containing two lattice points per unit cell, and the vector t₄ is normal to t₁ and t₂. Such a cell with lattice points at the centres of a pair of opposite faces parallel to the diad axis is also consistent with twofold symmetry. The cell centred on opposite faces shown in Figure 1.21 is chosen to denote the lattice produced from the staggered nets because it is more naturally related to the twofold symmetry than a primitive unit cell would be for this case.

Figure 1.21 A three‐dimensional view of the staggered arrangement of nets in Figure 1.20(b) in which two‐fold symmetry is preserved. In this diagram, the vector t₄ is normal to t₁ and t₂

The staggered arrangement of nets shown in Figures 1.20b and 1.21 could also have shown diad symmetry if we had arranged that the corners of the net at height z had lain not above the midpoints of the side containing t₁ in Figure 1.20b but vertically above either the centre of the unit parallelogram of the first net or above the centre of the side containing t₂ in Figure 1.20b. These two staggered arrangements are not essentially different from the first one, since, as is apparent from Figure 1.22, a new choice of axes in the plane of the nets is all that is needed to make them completely equivalent.

Figure 1.22 Lattice points in the net at height zero are marked with dots. The rings and crosses indicate alternative positions of the lattice points in staggered nets at height z, arranged so as to preserve twofold symmetry. The dotted lines show an alternative choice of unit cell

There are then two lattices consistent with monoclinic symmetry: the primitive one with the unit cell shown in Figure 1.19b and a lattice made up from staggered nets of which the conventional unit cell is centred on a pair of opposite faces. The centred faces are conventionally taken as the faces parallel to the x‐ and y‐axes; that is, (001), with the diad parallel to y (see Figure 1.19c). This lattice is called the monoclinic C lattice. The two lattices in the monoclinic system can be designated P and C, respectively.

The two tetragonal lattices can be rapidly developed. The square net in Figure 1.14b has fourfold symmetry axes arranged at the corners of the squares and also at the centres. This fourfold symmetry may be preserved by placing the second net with a corner of the square at 00z with respect to the first (t₃ normal to t₁ and t₂) or with a corner of the square at with respect to the first. The unit cells of the lattices produced by these two different arrangements are shown in Figures 1.19h and 1.19i, respectively. They can be designated P and I. The symbol I indicates a lattice with an additional lattice point at the centre of the unit cell (German: innenzentrierte). In the tetragonal system the tetrad axis is usually taken parallel to c, so a and b are necessarily equal and all of the axial angles are 90°.

The nets shown in Figures 1.14d and 1.14e are each consistent with the symmetry of a diad axis lying at the intersection of two perpendicular mirror planes. It is shown in Section 2.1 that a mirror plane is completely equivalent to what is called an inverse diad axis: a diad axis involving the operation of rotation plus inversion. This inverse diad axis, given the symbol , lies normal to the mirror plane. The symmetry of a diad axis at the intersection of two perpendicular mirror planes could therefore be described as 2, indicating the existence of three orthogonal axes: one diad and two inverse diads. The lattice consistent with this set of symmetry operations will also be consistent with the arrangement 222 in the orthorhombic crystal system (Table 1.3).¹³ To develop the lattices consistent with orthorhombic symmetry, therefore, the two relevant nets are the rectangular net (Figure 1.14d) and the rhombus net (Figure 1.14e). The positions of diad axes are shown on the right‐hand sides of Figures 1.14d and 1.14e. The rhombus net can also be described as the centred rectangular net.

If we stack rectangular nets vertically above one another so that a corner lattice point of the second net lies vertically above a similar lattice point in the net at zero level (t₃ normal to t₁ and t₂) then we produce the primitive lattice P. The unit cell is shown in Figure 1.19d. It is a rectangular parallelepiped.

If we stack rhombus nets (centred rectangles) vertically above one another, we obtain the lattice shown in Figure 1.23. This is the orthorhombic lattice with centring on one pair of faces.

Figure 1.23 The stacking of rhombus nets vertically above one another to form an orthorhombic lattice with centring on one pair of faces

We can also preserve the symmetry of a diad axis at the intersection of two mirror planes by stacking the rectangular nets in three staggered sequences. These are shown in Figures 1.24a, b and c. The lattices designated A‐centred and B‐centred are not essentially different since they can be transformed into one another by appropriate relabelling of the axes.¹⁴ The staggered sequence shown in Figure 1.24c is described by the unit cell shown in Figure 1.19f. It is the orthorhombic body‐centred lattice, symbol I. There is only one possibility for the staggered stacking of rhombus nets. Careful inspection of the right‐hand side of Figure 1.14e shows that the only places in the net where a diad axis lies at the intersection of two perpendicular mirror planes is at points with coordinates (0, 0) and , of the rhombus primitive cell. We have already dealt with the vertical stacking of the rhombus nets. If we take the only staggered sequence possible, where the second net lies with a lattice point of the rhombus net vertically above the centre of the rhombus in the zero‐level net (so that the end of t₃ has coordinates , , z in the rhombus net) then we produce the arrangement shown in Figure 1.25. This is most conveniently described in terms of a unit cell shown in Figure 1.19g, which is a rectangular parallelepiped with lattice points at the corners and also in the centres of all faces of the parallelepiped. This is the orthorhombic F cell. The symbol F stands for face‐centred, indicating additional lattice points at the centres of all faces of the unit cell.

Figure 1.24 The three possible stacking sequences of rectangular nets. In the three left‐hand diagrams, lattice points in the net at zero level are denoted with dots and those in the net at level z with open circles. The corresponding three‐dimensional views of the arrangements of the nets are shown in the three right‐hand diagrams

Figure 1.25 The staggered stacking of rhombus nets. This form of stacking generates the orthorhombic F Bravais lattice

All of the lattices consistent with 222 – that is, orthorhombic symmetry – are shown in Figures 1.19d, e, f and g. The unit cells are all rectangular parallelepipeds, so that the crystal axes can always be taken at right angles to one another – that is, α = β = γ = 90° – but the cell edges a, b and c may all be different. The primitive lattice P can then be described by a unit cell with lattice points only at the corners, the body‐centred lattice I by a cell with an additional lattice point at its centre and the F lattice by a cell centred on all faces. The A‐, B‐ and C‐centred lattices, shown in Figures 1.24a, 1.24b and 1.23, respectively, are all described by choosing the axes so as to give a cell centred on the (001) face; that is, a C‐centred cell.

We have so far described nine of the Bravais space lattices. All further lattices are based upon the stacking of triequiangular nets of points. The triequiangular net is shown in Figure 1.14c. There are sixfold axes only at the lattice points of the net. To preserve sixfold rotational symmetry in a three‐dimensional lattice, such nets must be stacked vertically above one another so that t₃ is normal to t₁ and t₂. The lattice produced has the unit cell shown in Figure 1.19j. The unique hexagonal axis is taken to lie along the z‐axis so a = b ≠ c, γ = 120° and α and β are both 90°. This lattice (i.e. the array of points) possesses sixfold rotational symmetry and is the only lattice to do so. However, it is also consistent with threefold rotational symmetry about an axis parallel to z. A crystal in which an atomic motif possessing threefold rotational symmetry was associated with each lattice point of this lattice would belong to the trigonal crystal system (Table 1.3).

A lattice consistent with a single threefold rotational axis can be produced by stacking triequiangular nets in a staggered sequence. A unit cell of the triequiangular net of points is shown outlined in Figure 1.26 by the vectors t₁, t₂ along the x‐ and y‐axes. Axes of threefold symmetry pierce the net at the origin of the cell (0, 0) – at points such as A – and also at two positions within the cell with coordinates , and , , respectively, which are labelled B and C, respectively in Figure 1.26. We can preserve the threefold symmetry (while of course destroying the sixfold one) by stacking nets so that the extremity of t₃ has coordinates of either , .or , . The two positions B and C in Figure 1.26 are equivalent to one another in the sense that the same lattice is produced whatever the order in which these two positions are used.

Figure 1.26 The stacking of triequiangular nets of points in a staggered sequence. Lattice points labelled A, B and C belong to nets of lattice points at different heights relative to the plane of the paper

A plan of the lattice produced, viewed along the triad axis, is shown in Figure 1.27, and a sketch of the relationship between the triequiangular nets and the primitive cells of this lattice is shown in Figure 1.28. In Figures 1.27 and 1.28 the stacking sequence of the nets has been set as ABCABCABC… Exactly the same lattice but in a different orientation (rotated 60° clockwise looking down upon the paper in Figure 1.27) would have been produced if the sequence ACBACBACB… had been followed. The primitive cell of the trigonal lattice in Figure 1.28 is shown in Figure 1.19k. It can be given the symbol R. It is a rhombohedron, the edges of the cell being of equal length, each equally inclined to the single threefold axis. To specify the cell we must state a = b = c and the angle α = β = γ < 120°.

Figure 1.27 Lattice points in the net at level zero are marked with a dot, those in the net at height z by an open circle, and those at 2z by a plus sign. The projection of t₃ onto the plane of the nets is shown

Figure 1.28 The relationship between a primitive cell of the trigonal lattice and the triply primitive hexagonal cell

An alternative cell is sometimes used to describe the trigonal lattice R because of the inconvenience in dealing with a lattice of axial angle α, which may take any value between 0 and 120°. The alternative cell is shown in Figure 1.28 and in plan viewed along the triad axis in Figure 1.29. It is a triply primitive cell, three mesh layers high, with internal lattice points at elevations of and of the repeat distance along the triad axis. The cell is of the same shape as the conventional unit cell of the hexagonal Bravais lattice and to specify it we must know a = b ≠ c, α = β = 90° and γ = 120°.

Figure 1.29 Plan view of the alternative triply primitive hexagonal unit cell used to describe the trigonal R lattice

Crystals belonging to the cubic system possess four threefold axes of rotational symmetry. The angles between the four threefold axes are such that these threefold axes lie along the body diagonals of a cube (Figure 1.30), with angles of 70.53° (cos⁻¹(1/3)) between them. Reference to Table 1.2 and Figure 1.17b shows that these threefold axes cannot exist alone in a crystal. They must be accompanied by at least three twofold axes. To indicate how the lattices consistent with this arrangement of threefold axes arise, we start with the R lattice shown in Figure 1.28 and call the separation of nearest‐neighbour lattice points in the triequiangular net s and the vertical separation of the nets along the triad axis h. The positions of the lattice points in the successive layers when all are projected onto the plane perpendicular to the triad axis can be designated ABCABC… as in Figures 1.26 and 1.28. In a trigonal lattice, the spacing of the nets, h, is unrelated to the separation of the lattice points within the nets, s. If we make the spacing of the nets such that h = (= 2s/), the angle α in Figure 1.28 becomes 60° and triangles A₁B₁B₂, A₁B₂B₄, A₁B₁B₄ all become equilateral. Planes such as A₁B₁C₁B₂, A₁B₂C₂B₄, A₁B₁C₃B₄ all contain triequiangular nets of points. Planes parallel to each of these three planes also contain triequiangular nets of points and are also stacked so as to preserve triad symmetry along lines normal to them. When α = 60°, the original trigonal lattice becomes consistent with the possession of four threefold axes. The conventional unit cell of this lattice is shown in Figure 1.19m; it is a cube centred on all faces. The relationship between this cell and the primitive one with α = 60° is shown in Figure 1.31.

Figure 1.30 The four 〈111〉 three‐fold axes with acute angles of 70.53° between one another which together define the minimum symmetry requirements for a crystal to belong to the cubic crystal system

Figure 1.31 The relationship between the primitive unit cell and the conventional cell in the face‐centred cubic lattice

The large non‐primitive unit cell in Figures 1.19m and 1.31 is the face‐centred cubic lattice, which can be designated F. It contains four lattice points. These are at the corners and centres of each of the faces.

When h in Figure 1.28 becomes equal to , the primitive unit cell of the R lattice becomes a cube with α = 90°. This is the cubic primitive lattice P shown in Figure 1.19l, containing lattice points at the corners of the cubic unit cell.

Lastly, if in Figure 1.28 h takes the value , the angle α is equal to 109.47° (= 180° − 70.53°) = cos⁻¹ (−1/3). The lattice formed by such an array of points also contains four threefold axes of symmetry. The conventional unit cell of this lattice is shown in Figure 1.19n. It is a cube with lattice points at the cube corners and one at the centre. It can be designated I, the cubic body‐centred lattice. The relationship between the doubly primitive unit cell shown in Figure 1.19n and the primitive unit cell, which is a rhombohedron with axial angles of 109.47°, is shown in Figure 1.32.

Figure 1.32 The relationship between the primitive unit cell and the conventional cell in the body‐centred cubic lattice

We have just described the three lattices consistent with the possession of four threefold axes of rotational symmetry. They are shown together in Figures 1.19l, m and n. The unit cell of each can be taken as a cube with a = b = c; α = β = γ = 90°. The primitive cell contains one lattice point, the face‐centred cell four, and the body‐centred cell two.

The unit cells of the 14 Bravais space lattices are shown together in Figure 1.19. All crystals possess one or other of these lattices, with an identical atomic motif associated with each lattice point. In some crystals a single spherically symmetric atom is associated with each lattice point. In this case the lattice itself possesses direct physical significance because the lattice and the crystal structure are identical. In other cases the lattice is a very convenient framework for describing the translational symmetry of the crystal. If the lattice is given and the arrangement of the atomic motif about a single lattice point is given, the crystal structure is fully described. The lattice is the most important symmetry aspect for describing the properties of imperfections in crystals.