CRYSTAL STRUCTURES WITH CUBIC UNIT CELLS

Transcription

1 CRYSTAL STRUCTURES WITH CUBIC UNIT CELLS Crystalline solids are a three dimensional collection of individual atoms, ions, or whole molecules organized in repeating patterns. These atoms, ions, or molecules are called lattice points and are typically visualized as round spheres. The two dimensional layers of a solid are created by packing the lattice point spheres into square or closed packed arrays. (Figure 1). Which packing arrangement makes the most efficient use of space? square array close-packed array Figure 1: Two possible arrangements for identical atoms in a 2-D structure Stacking the two dimensional layers on top of each other creates a three dimensional lattice point arrangement represented by a unit cell. A unit cell is the smallest collection of lattice points that can be repeated to create the crystalline solid. The solid can be envisioned as the result of the stacking a great number of unit cells together. The unit cell of a solid is determined by the type of layer (square or close packed), the way each successive layer is placed on the layer below, and the coordination number for each lattice point (the number of spheres touching the sphere of interest.) Primitive (Simple) Cubic Structure Placing a second square array layer directly over a first square array layer forms a "simple cubic" structure. The simple cube appearance of the resulting unit cell (Figure 3a) is the basis for the name of this three dimensional structure. This packing arrangement is often symbolized as "AA...", the letters refer to the repeating order of the layers, starting with the bottom layer. The coordination number of each lattice point is si. This becomes apparent when inspecting part of an adjacent unit cell (Figure 3b). The unit cell in Figure 3a appears to contain eight corner spheres, however, the total 1

2 number of spheres within the unit cell is 1 (only 1/8 th of each sphere is actually inside the unit cell). The remaining 7/8 ths of each corner sphere resides in 7 adjacent unit cells. Figure 3a: Square Array Layering Figure 3b: Simple Cubic Figure 3c: Space Filling Simple Cubic The considerable space shown between the spheres in Figures 3a is misleading: lattice points in solids touch as shown in Figure 3b. For eample, the distance between the centers of two adjacent metal atoms is equal to the sum of their radii. Refer again to Figure 3a and imagine the adjacent atoms are touching. The edge of the unit cell is then equal to 2r (where r = radius of the atom or ion) and the value of the face diagonal as a function of r can be found by applying Pythagorean s theorem (a 2 + b 2 = c 2 ) to the right triangle created by two edges and a face diagonal (Figure 4a). Reapplication of the theorem to another right triangle created by an edge, a face diagonal, and the body diagonal allows for the determination of the body diagonal as a function of r (Figure 4b). e d g e e d g e edge Figure 4a Figure 4b 2

3 Few metals adopt the simple cubic structure because of inefficient use of space. The density of a crystalline solid is related to its "percent packing efficiency" (1). The packing efficiency of a simple cubic structure is only about 52%. (48% is empty space!) How would you find the volume of a unit cell and the volume of atoms in a unit cell? (1) % packing efficiency = volume of atoms in a unit cell 100 volume of unit cell Body Centered Cubic Structure A more efficiently packed cubic structure is the "body-centered cubic" (bcc). The first layer of a square array is epanded slightly in all directions. Then, the second layer is shifted so its spheres nestle in the spaces of the first layer (Figures 5a, b). This repeating order of the layers is often symbolized as "ABA...". Like Figure 3, the considerable space shown between the spheres in Figures 5 is misleading: spheres are closely packed in bcc solids and touch along the body diagonal. The packing efficiency of the bcc structure is about 68%. The coordination number for an atom in the bcc structure is eight. How many total atoms are there in the unit cell for a bcc structure? Draw a diagonal line connecting the three atoms marked with an "" in Figure 5b. Assuming the atoms marked "" are the same size, tightly packed and touching, what is the value of this body diagonal as a function of r, the radius? Find the edge and volume of the cell as a function of r. Figure 5a: Square Array Layering Figure 5b: Body Ctr d Cubic (bcc) Figure 5c: Space Filling bcc 3

4 Cubic Closest Packed (ccp) A cubic closest packed (ccp) structure is created by layering close packed arrays. The spheres of the second layer nestle in half of the spaces of the first layer. The spheres of the third layer directly overlay the other half of the first layer spaces while nestling in half the spaces of the second layer. The repeating order of the layers is "ABC..." (Figures 6 & 7). The coordination number of an atom in the ccp structure is twelve (si nearest neighbors plus three atoms in layers above and below) and the packing efficiency is 74%. Figure 6: Close packed Array Layering. The 1 st and 3 rd layers are represented by light spheres; the 2 nd layer, dark spheres. The 2 nd layer spheres nestle in the spaces of the 1 st layer marked with an. The 3 rd layer spheres nestle in the spaces of the 2 nd layer that directly overlay the spaces marked with a in the 1 st layer. A C B A Figure 7a & 7b: Two views of the Cubic Close Packed Structure 4

5 If the cubic close packed structure is rotated by 45 the face centered cube (fcc) unit cell can be viewed (Figure 8). The fcc unit cell contains 8 corner atoms and an atom in each face. The face atoms are shared with an adjacent unit cell so each unit cell contains ½ a face atom. Atoms of the face centered cubic (fcc) unit cell touch across the face diagonal (Figure 9). What is the edge, face diagonal, body diagonal, and volume of a face centered cubic unit cell as a function of the radius? A C B 45 o rotation A Figure 8: The face centered cubic unit cell is drawn by cutting a diagonal plane through an ABCA packing arrangement of the ccp structure. The unit cell has 4 atoms ( 1 / 8 of each corner atom and ½ of each face atom). 4r Figure 9a: Space filling model of fcc. Figure 9b: The face of fcc. Face diagonal = 4r. 5

6 Ionic Solids In ionic compounds, the larger ions become the lattice point spheres that are the framework of the unit cell. The smaller ions nestle into the depressions (the holes ) between the larger ions. There are three types of holes: "cubic", "octahedral", and "tetrahedral". Cubic and octahedral holes occur in square array structures; tetrahedral and octahedral holes appear in close-packed array structures (Figure 10). Which is usually the larger ion the cation or the anion? How can the periodic table be used to predict ion size? What is the coordination number of an ion in a tetrahedral hole? an octahedral hole? a cubic hole? Tetrahedral Octahedral Octahedral Cubic Figure 10. Holes in ionic crystals are more like "dimples" or "depressions" between the closely packed ions. Small ions can fit into these holes and are surrounded by larger ions of opposite charge. The type of hole formed in an ionic solid largely depends on the ratio of the smaller ion s radius the larger ion s radius (r smaller /r larger ). (Table 1). Table 1: Radius Ratio & Hole Type Hole Type Radius Ratio Tetrahedral Octahedral Cubic Empirical Formula of an Ionic Solid Two ways to determine the empirical formula of an ionic solid are: 1) from the number of each ion contained within 1 unit cell 2) from the ratio of the coordination numbers of the cations and anions in the solid. 6

7 Table 2: Position of atom and fraction contained in a single unit cell: Position of Atom in the Unit Cell # of Adjacent Cells Sharing Atom Fraction Contained Within a Single Unit Cell Cube Corner 8 1 / 8 Edge 4 ¼ Face 2 ½ Internal 1 1 Eample: Find the empirical formula for the ionic compound shown in Figures 11 & 12. First Method: When using the first method, remember most atoms in a unit cell are shared with other cells. Table 2 lists types of atoms and the fraction contained in the unit cell. The number of each ion in the unit cell is determined: 1/8 of each of the 8 corner X ions and 1/4 of each of the 12 edge Y ions are found within a single unit cell. Therefore, the cell contains 1 X ion (8/8 = 1) for every 3 Y ions (12/4 = 3) giving an empirical formula of XY 3. Which is the cation? anion? When writing the formula of ionic solids, which comes first? X Y Figure 11a: Unit cell of ionic solid XY 3. Figure 11b: Space filling unit cell of ionic solid XY 3. Second Method: The second method is less reliable and requires the eamination of the crystal structure to determine the number of cations surrounding an anion and vice versa. The structure must be epanded to include more unit cells. Figure 12 shows the same solid in Figure 11 epanded to four adjacent unit cells. Eamination of the structure shows that there are 2 X ions coordinated to every Y ion and 6 Y ions surrounding every X ion. (An additional unit cell must be projected in front of the page to see the sith Y ion ). A 2 to 6 ratio gives the same empirical formula, XY 3. 7

8 X Y Figure 12: Four unit cells of an ionic solid with formula XY 3. 8