Читать книгу Solid State Chemistry and its Applications - Anthony R. West - Страница 28

The unit cell, by definition, must contain at least one formula unit, whether it be an atom, ion pair, molecule, etc. In centred cells, the unit cell contains more than one formula unit and more than one lattice point. A simple relation exists between cell volume, the number of formula units in the cell, the formula weight (FW) and the bulk crystal density (D):

where N is Avogadro's number. If the unit cell, of volume V, contains Z formula units, then

Therefore,

(1.3)

V is usually expressed in ų and must be multiplied by 10^–24 to convert V to cm^–3 and to give densities in units of g cm^–3. Substituting for N, the equation reduces to

(1.4)

and, if V is in ų, the units of D are g cm⁻³. This simple equation has a number of uses, as shown by the following examples:

1 It can be used to check that a given set of crystal data is consistent and that, for example, an erroneous formula weight has not been assumed.

2 It can be used to determine any of the four variables if the other three are known. This is most common for Z (which must be a whole number) but is also used to determine FW and D.

3 By comparison of D obs (the experimental density) and D calc (calculated from the above equation), information may be obtained on the presence of crystal defects such as vacancies or interstitials, the mechanisms of solid solution formation and the porosity of ceramic pieces.

Considerable confusion often arises over the value of the contents, Z, of a unit cell. This is because atoms or ions that lie on corners, edges or faces are also shared between adjacent cells; this must be taken into consideration in calculating effective cell contents. For example, α‐Fe [Fig. 1.11(e)] has Z = 2. The corner Fe atoms, of which there are eight, are each shared between eight neighbouring unit cells. Effectively, each contributes only 1/8 to the particular cell in question, giving 8 × 1/8 = 1 Fe atom for the corners. The body centre Fe lies entirely inside the unit cell and counts as one. Hence Z = 2.

For Cu metal, Fig. 1.11(c), which is fcc, Z = 4. The corner Cu again counts as one. The face centre Cu atoms, of which there are six, count as 1/2 each, giving a total of 1 + (6 × 1/2) = 4 Cu in the unit cell.

NaCl is also fcc and has Z = 4. Assuming the origin is at Na (Fig. 1.2) the arrangement of Na is the same as that of Cu in Fig. 1.11(c) and therefore, the unit cell contains 4 Na. Cl occupies edge centre positions of which there are 12; each counts as 1/4, which, together with Cl at the body centre, gives a total of (12 × 1/4) + 1 = 4 Cl. Hence the unit cell contains, effectively, 4 NaCl. If unit cell contents are not counted in this way, but instead all corner, edge‐and face‐centre atoms are simply counted as one each, then the ludicrous answer is obtained that the unit cell has 14 Na and 13 Cl!