INTRODUCTION Many structures are based upon the close-packing of atoms or ions into hexagonal layers. In a close-packed structure, each atom or ion is surrounded by six others, resulting in very efficient packing. These hexagonal layers, in turn, may be packed in two different ways, giving rise either to a hexagonal close-packed structure, or a cubic close packed structure. THE CUBIC CLOSE-PACKED STRUCTURE In one type of structure the hexagonal layers are stacked in an ABC fashion, so that the fourth layers lies immediately over the first, the fifth layer lies immediately over the second, and so forth.. This is emphasized by the vertical lines, which connect the first and fourth layers. The ABC stacking of hexagonal layers gives rise to the face-centered cubic structure, and is discussed more thoroughly in a separate section THE HEXAGONAL CLOSE-PACKED STRUCTURE Hexagonal layers may also stack in an ABAB fashion, as shown in this illustration. Note that the two red layers lie immediately above each other, as do the two blue layers. The longer vertical lines connecting the layers are shown for clarity.Whereas the satking of layers in an ABC gives rise to a face-centered cubic structure, the stacking of layers in an AB fashion gives rise to a hexagonal structure. The hexagonal -close packed structure arises when close-packed layers are stacked in an AB fashion so that the third layer lies over the first, the fourth layers lies over the second, and so forth. The hexagonal close-packed structure is adopted by many different metals, including Be, Mg, Sc, Ti, Co, and Zn. In addition, there are several ionic structures in which one set of ions forms a hexagonal close-packed structure and the other set of ions fits into the octahedral or tetrahedral holes. This picture shows a top view of the previous model. Note that one one red layer is shown since one lies immediately on top of the other. The same applies to the blue layers. The holess in the center structure are octahedral holes, formed by three atoms within a single layer and three atoms in the layer above or below. These holes exist in the ABC-stacked structure as well. However, in the case of the AB-stacked layers the holes are distributed through the structure in such a way as it is possible to see all the way through the structure. (hexapic) OCTAHEDRAL HOLES This model shows the location of all the tetrahedral and octahedral holes within a series of ABAB hexagonal layers. The yellow spheres represent the tetrahedral holes; the green spheres represent the octahedral holes (octa) This picture shows the octahedral holes in AB-stacked hexagonal layers as seen from the top. Compare this to the top view show above. Can you find the location of the tetrahedral holes?(octahole) TETRAHEDRAL HOLES This illsutration shows the location of the tetrahedral holes within a a series of AB-stacked hexagonal layers. The actual locations of the tetrahedral holes are much clearer seen from the top. (tetrahole) This image the tetrahedral holes within a series of AB-stacked hexagoal layers. This an isometric view and therefore lacks depth perspective, but the location of the octahedral and tetrahedral holes can clearly be seen. The three larger holes though the center of the structure are the octahedral holes. (tettra) THE FACE-CENTERED CUBIC LATTICE THE BASIC LATTICE The face-centered cubic lattice is one of the most common types of crystal lattices. In addition to the eight atoms located at the corners of the cube, the face-centered cell contains an additional atom on each face of the cube. The face-centered cubic lattice is adopted by many elements, including Ca, Sr,Al, Ni, Cu, Rh, Pd, Ag, Ir, Pt, Au, and Pb. This illustration shows two unit cells of the face-centered cubic lattice. (basic Lattice) OCTAHEDRAL HOLES There exists two different types of spaces or holes within a series of stacked hexagonal layers. This illustration shows in green the location of two octahedral holes. The term octahedral is derived from the fact that the resulting space has eight sides. T Many ionic compounds adopt a structure in which one set of ions forms a face-centered cubic array and the other set of ions reside within the octahedral holes. The rock salt (halite) structure is one example TETRAHEDRAL HOLES The second type of space that can exist between stacked hexagonal layers is called a tetrahedral hole. This illustration shows in green the location of four tetrahedral holes in the two unit cells shown.. A tetrahedral hole is formed by three atoms in one hexagonal layer and a single atom in the layer above or beneath. Many ionic compounds adopt a structure in which one type of ion forms a face-centered cubic array and the other set of ions resides in the octahedral holes. There are 2N tetrahedral holes for every N atoms, so this structure works well for compounds that have 1:2 stoichiometry. THE FLORITE STRUCTURE This illustration shows two unit cells of the calcium fluoride or fluorite structure. The structure can be viewed as a face-centered cubic array of caclium ions, represented by the white spheres, with the fluoride ions residing in the tetrahedral holes. Consdier the stoichiometry of single unit cell. Each of the corner calcium ions is 1/8 inside the cell; since there are eight corners these add up to one ion inside the cell. There are six faces to a sigle cell, each with a calcium ion one-half inside the cell. Therefore a single cell contains four four calcium ions. A single cell also contans eight fluoride ions, each one located entirely within the unit cell. Since there four calcium ions and eight fluoride ions inside the cell, the 1:2 stoichiometry is maintained. (FL pic) OCTAHEDRAL HOLES IN THE FLUORITE STRUCTURE In the fluorite structure, the fluorude ions reside within the tetrahedral holes formed by the face-centered cubic array of calcium ions, and the octahedral holes are vacant. In this illustration the green cylinders outline eight of the vacant octahedral holes. (FL oct) TETRAHEDRAL HOLES IN THE FLUORITE STRUCTURE This illustration shows the location of the tetrahedral holes in the fluorite structure. Consdier why the fluroide ions would reside in the tetrahedral holes rather than the octahedral holes. The most obvious answer to this question is, of course, stoichiometry. There are two fluoride ions for every one calcium ion, and since an array of N atoms results in the formation of N octahedral holes, there would simply not be enough spaces for all fluoride ions. If the ions were reversed, with the fluoride ions forming the face-centered cubic array, there would be enough calcium ions to fill only 1/4 of the tetrahedral holes or 1/2 of the octahedral holes; this would be terribly inefficient. (fl tet) This illustration shows the same model as the previous illustration, seem from the top. Technically, the descriptions of the fluoriute structure given above are inaccurate in the sense that becasue the fluoride ions are in fact larger than the calcium ions, they therefore do not "fit inside" the tetrahedral holes. As can bee seen here, the calcium ions form a sort of "expanded" face-centered cubic structure and do not physically touch each other. Nevertheless this does represent the most efficient packing arrangement. (arrange fl) THE ZINC BLENDE STRUCTURE Zinc sulfide exists as two forms, zinc blende and wurtzite. Zinc blende is cubic zinc structure in which the sulfide ions forn a face-centered cubic array and the zinc ions fill one-half of the tetrahedral holes. This illustration shows a side view of the zinc blende structure; two unit cells are shown. The white spheres represent the sufide ions and the red spheres represent the zinc ions. The sulfide ions are quite large (atomic radius184 pm) relative to the size of the zinc ions (74 pm). (zb) This model outlines the occupied tetrahedral holes in the structure. Recall above that in the zinc blende structure only half of the tetrahedral holes are occupied. The unoccupied tetrahedral holes are not shown. Can you find them? (tetzb) Tetrahedral and octahedral sites in closest packing can be occupied by other atoms or ions in crystal structures of salts and alloys. Thus, recognizing their existence and their geometrical constrains help the study and interpretation of crystal chemistry. The packing of spheres and the formation of tetrahedral and octahedral sites or holes are shown below. (2layers) Solution The octahedral hole is located at the center of any four spheres that form a square. If we represent the radius of a ball fitting in the octahedral holes by r, and the radius of the sphere as R, then we have the relationship: r + R = (1/Ö2) (2 R) r / R = Ö2 - 1 = 0.414 The implication: Pure geometric consideration shows that only small balls fit in the tetrahedral holes of packed spheres. However, if the radii of cations are smaller than 0.225 R, the structure of having ions in the tetrahedral site is unstable. The anions may be pushed apart slightly to reduce the repulsion by fitting a cation in the tetrahedral site. For ionic crystal structure consideration, the cations are usually smaller than anions. Cations fitting into the tetrahedral sites cannot be smaller than 0.225 R. Usually, most ions are slightly larger than 0.225 R, but smaller than 0.414 R. In such cases, the cation coordination is tetrahedral, and a typical structure is ZnS, although covalent bonding is also involved in ZnS. The animated diagram is a model of ZnS structure. When the cation radii are greater or equal to 0.414 R, but less than 0.732 R, the cations occupy the octahedral sites. Sodium chloride is one such structure, and it serves as an important structure type. If the cations are large such that r > 0.732 R, the cation will have a cubic coordination of 8. The strcture is typified by CsCl Crystal Structure of Wurtzite Zinc sulfide crystallizes in two different forms: Wurtzite and Zinc Blende. The ionic radius of the zinc(II) ion is 0.74 angstroms and that of the sulfide ion is 1.70 angstroms. The ratio of radii for the cation and anion is thus r+/r- = 0.74/1.70 = 0.44. With a radius ratio of 0.44, one might expect the zinc(II) ions to occupy octahedral holes; however, the value of 0.44 is only slightly larger than rhole/r = 0.414 for an octahedral hole. In this case, the zinc(II) ions occupy tetrahedral holes. If the sulfide ions originally adopt a hexagonal closest-packed structure, the ZnS crystal is Wurtzite. If the sulfide ions originally adopt a cubic closest-packed structure, the ZnS crystal is Zinc Blende. The images below depict the structure of Wurtzite. The yellow spheres represent the sulfide ions and the blue spheres represent the zinc(II) ions. Examine the images and take note of the following points: 1. The sulfide ions lie in a hexagonal closest-packed arrangement. 2. The zinc(II) ions are much smaller than the sulfide ions. 3. The insertion of zinc(II) ions into the tetrahedral holes causes the structure to expand so that the sulfide ions are not in contact with each other. 4. Only one half of the tetrahedral holes are occupied by zinc(II) ions. The ionic solid is electrically neutral and the unit cell itself must also be electrically neutral. Because the sulfife ions adopt a hcp structure, there are two sulfide ions in the unit cell. Consequently there must also be two zinc(II) ions in the unit cell. Examine the unit cell and verify this fact. 5. Wurtzite has (4,4)-coordination. Zinc Blende has (4,4)-coordination Crystal Structure of Cesium Chloride The ionic radius of the cesium ion is 1.88 angstroms and that of the chloride ion is 1.67 angstroms. In this case the cation is the larger ion, and the ratio of radii for the anion and cation is r-/r+ = 1.67/1.88 = 0.888. With a radius ratio of 0.888, the smaller ion is expected to prefer a cubic hole. The images below depict the structure of CsCl. The green spheres represent the chloride ions and the red spheres represent the cesium ions. Examine the images and take note of the following points: 1. The chloride ions lie in a simple cubic arrangement. Owing to the symmetry of the structure, it does not matter whether one regards the chloride ions as adopting a cubic structure with cesium ions inserted into cubic holes or cesium ions adopting a cubic structure with chloride ions inserted into cubic holes. 2. The cesium and chloride ions are comparable in size. 3. The insertion of chloride ions into the octahedral holes causes the structure to expand so that the cesium ions are not in contact with each other. (The unit cell actually shows chloride ions surrounding a cesium ion. Envision several adjoining unit cells, which produce cesium ions surrounding a chloride ion.) 4. All of the octahedral holes are occupied by cesium ions. The ionic solid is electrically neutral and the unit cell itself must also be electrically neutral. Examine the unit cell and verify that it contains equal numbers of cesium and chloride ions. 5. CsCl has (8,8)-coordination. Crystal Structure of Fluorite Calcium fluoride occurs naturally as the mineral fluorite. The ionic radius of the calcium ion is 1.26 angstroms and that of the fluoride ion is 1.17 angstroms. In this case the cation is the larger ion, and the ratio of radii for the anion and cation is r-/r+ = 1.17/1.26 = 0.929. The two ions are essentially the same size; thus it makes little difference whether one talks about inserting fluoride ions into holes in a calcium ion lattice or vice versa. With a radius ratio of 0.929 (essentially unity), the smaller ion is expected to prefer a cubic hole. The images below depict the structure of Fluoride. The light blue spheres represent the fluoride ions and the red spheres represent the calcium ions. The top two images illustrate the cubic holes occupied by calcium ions; however, these two images do not show the unit cell. The bottom two images show the fluorite unit cell. Examine the images and take note of the following points: 1. The fluoride ions lie in a cubic arrangement. 2. The calcium and fluoride ions are comparable in size. 3. The insertion of calcium ions into the cubic holes causes the structure to expand so that the fluoride ions are not in contact with each other. 4. The top two images, which show a simple cubic cell, do NOT depict the unit cell for fluorite. Because calcium fluoride contains two fluoride ions for every calcium ion, it is not possible to fill all cubic holes with a calcium ion, as this would violate electroneutrality for the structure. In Fluorite, half of the cubic holes contain a calcium ion and the other half are empty. The bottom two images show the unit cell for Fluorite. Examine the bottom two images to verify the 2:1 fluoride ion:calcium ion stoichiometry. 5. The Fluorite structure actually contains calcium ions in a cubic closest-packed structure with fluoride ions in a tetrahedral environment. In the ccp structure, the unit cell contains four atoms and there are eight tetrahedral holes, all of which are filled by fluoride ions in Fluorite. Examine the bottom two images and verify this structure. 6. Fluorite has (8,4)-coordination. Crystal Structure of Sodium Chloride The ionic radius of the sodium ion is 1.16 angstroms and that of the chloride ion is 1.67 angstroms. The ratio of radii for the cation and anion is thus r+/r- = 1.16/1.67 = 0.695. With a radius ratio of 0.695, the cubic holes are too large (rhole/r = 0.732) to be suitable. The sodium ions will prefer to occupy octahedral holes in a closest-packed structure. As it happens, the chloride ions in NaCl pack in a cubic closest-packed structure. The images below depict the structure of NaCl. The green spheres represent the chloride ions and the red spheres represent the sodium ions. Examine the images and take note of the following points: 1. The chloride ions lie in a cubic closest-packed arrangement. 2. The sodium ions are smaller than the chloride ions. 3. The insertion of sodium ions into the octahedral holes causes the structure to expand so that the chloride ions are not in contact with each other. 4. All of the octahedral holes are occupied by sodium ions. The ionic solid is electrically neutral and the unit cell itself must also be electrically neutral. Because the chloride ions adopt a ccp structure, there are four chloride ions in the unit cell. Consequently there must also be four sodium ions in the unit cell. Examine the unit cell and verify this fact. 5. NaCl has (6,6)-coordination. Unit Cells: The Simplest Repeating Unit in a Crystal The structure of solids can be described as if they were three-dimensional analogs of a piece of wallpaper. Wallpaper has a regular repeating design that extends from one edge to the other. Crystals have a similar repeating design, but in this case the design extends in three dimensions from one edge of the solid to the other. We can unambiguously describe a piece of wallpaper by specifying the size, shape, and contents of the simplest repeating unit in the design. We can describe a three-dimensional crystal by specifying the size, shape, and contents of the simplest repeating unit and the way these repeating units stack to form the crystal. The simplest repeating unit in a crystal is called a unit cell. Each unit cell is defined in terms of lattice pointsthe points in space about which the particles are free to vibrate in a crystal. The structures of the unit cell for a variety Unit Cells: NaCl and ZnS NaCl should crystallize in a cubic closest-packed array of Cl- ions with Na+ ions in the octahedral holes between planes of Cl- ions. We can translate this information into a unit-cell model for NaCl by remembering that the face-centered cubic unit cell is the simplest repeating unit in a cubic closest-packed structure. There are four unique positions in a face-centered cubic unit cell. These positions are defined by the coordinates: 0,0,0; 0,1/2,1/2; 1/2,0,1/2; and 1/2,1/2,0. The presence of an particle at one corner of the unit cell (0,0,0) requires the presence of an equivalent particle on each of the eight corners of the unit cell. Because the unit-cell edge connects equivalent points, the presence of a particle in the center of the bottom face (0,1/2,1/2) implies the presence of an equivalent particle in the center of the top face (1,1/2,1/2). Similarly, the presence of particles in the center of the 1/2,0,1/2 and 1/2,1/2,0 faces of the unit cell implies equivalent particles in the centers of the 1/2,1,1/2 and 1/2,1/2,1 faces. The figure below shows that there is an octahedral hole in the center of a face-centered cubic unit cell, at the coordinates 1/2,1/2,1/2. Any particle at this point touches the particles in the centers of the six faces of the unit cell. We can therefore describe the structure of NaCl in terms of the following information. NaCl crystallizes in a cubic unit cell. The cell-edge length is 0.5641 nm. There are Cl- ions at the positions 0,0,0; 1/2,1/2,0; 1/2,0,1/2; and 0,1/2,1/2. There are Na+ ions at the positions 1/2,1/2,1/2; 1/2,0,0; 0,1/2,0; and 0,0,1/2. Placing a Cl- ion at these four positions implies the presence of a Cl- ion on each of the 14 lattice points that define a face-centered cubic unit. Placing a Na+ ion in the center of the unit cell (1/2,1/2,1/2) and on the three unique edges of the unit cell (1/2,0,0; 0,1/2,0; and 0,0,1/2) requires an equivalent Na+ ion in every octahedral hole in the unit cell. CYCLOHEXANE The three important conformations of cyclohexane are chair, twist chair and boat The most stable conformations of cyclohexane two equivalent chair conformations. Their stability arises from the absence of eclipsing interactions Substituents on the chair are either in the plane of the ring (equatorial) or above and below the plane (axial). Chair conformers readily interconvert via the twist chair. This conformational inversion causes substituents that were originally equatorial to become axial and vice versa. Substitution on the cyclohexane ring renders the two chair conformers energetically inequivalent because axial substituents experience unfavourable 1,3–diaxial non–bonding interactions. Consequently the lower energy chair conformer will have bulky substituents in equatorial environments and will be the dominant constituent of the dynamic equilibrium. Note: This is a conformational bias and both conformers are present. Reaction may proceed via either conformer as removal of either will be compensated by rapid re–equilibration. The boat conformer of cyclohexane lies higher in energy than the previous conformations. The energetic disadvantage arises from two eclipsing interactions.