Cohesive-related energies

There are no absolute values of energies in physics—only energy dif-ferences between two states, one of which can be a reference state that is given by a definition or convention. If U(k —> oo) refers to separated neutral atoms of the elements, [/bind agrees with the normal definition of the cohesive energy. However, if we consider a lattice of NaCl it would be natural to let [/bind refer to infinitely separated ions Na⁺ and Cl~ rather than neutral atoms Na and CI. Some authors would call that quantity the cohesive energy. It is larger than the conventional cohesive energy by the electron ionisation energy of Na atoms minus the electron affinity of CI atoms (cf. the example below).

Another important quantity referring to binding energies is the en-thalpy of formation, e.g. the formation of NaCl from the elements Na and CI in their standard states. The standard reference state of an el-ement is normally chosen to be in the most stable structure of that element at 298.15 K (25 °C), and at a standard pressure. Thus the stan-dard state of Al refers to a solid in an fee lattice, that of Hg to a liquid while that of CI refers to the diatomic gas CI2. The standard pressure was long chosen to be 101,325 Pa (1 atm) but is now recommended to be at the slightly different value of 10⁵ Pa (1 bar). Other reference temperatures than 298.15 K are occasionally used.

Cohesive-related energies 5

Thermodynamic quantities of a standard state are usually identified by the superscript °, for instance H° for the enthalpy and S° for the en-tropy. When one wants to make it clear which temperature the standard state refers to, one may use the notation 7/_{298 15} or //°(298.15), S_{298 15} etc. Other notations for standard states are also used, for instance °H or

H+.

The enthalpy of formation (previously often called heat of forma-tion) of a compound is defined as the enthalpy difference between the compound and the constituent elements, all in their standard states. It is often written A//f° (or, e.g. AfH°). For instance,

A#f°(NaCl) = #°(NaCl) - #°(Na) - ]-H°{C\2). (1.10) This quantity is negative since the NaCl compound is energetically more

stable than the separated constituents. Obviously, the standard enthalpy of formation of an element in its standard state is zero.

The reader is warned that the conventions and reference states cho-sen by some authors for quantities called binding energies, cohesive energies and enthalpies (energies) of formation may not be those that are used by others. In particular, experimental values usually refer to 298.15 K or some other finite temperature, while theoretical results usually refer to 0 K.

Example: cohesive-related energies of NaCl One of the most com-plete tables of thermodynamic data (Barin 1989) gives i/29815(Na)

= #298.i5(^cl2) = 0, //₂°₉₈.₁₅(NaCl) = -411.120 kJ/mol, #₂°₉₈₁₅(C1) = 121.286 kJ/mol and #2°98'15(Na, gas) = 107.300 kJ/mol, where the last two terms are for a monatomic CI and Na gas, respectively. Another source (Tosi 1964) gives the cohesive energy £/COh(NaCl) = 764.0 kJ/mol (relative to separated ions). Furthermore, the ionisation energy for Na -> Na⁺ + e~ is £_ion = 495.8 kJ/mol, and the electron affinity for CI + e" -* CI" is £aff = 348.7 kJ/mol (Emsley 1989). We now write

^bind(NaCl) as the result of a process where solid NaCl is separated into solid Na and a gas of CI2, then further separated into monatomic gases of Na and CI, and a final step with ionisation of the gas atoms.

Thus, C/_bind(NaCl) = -#₂°₉₈.₁₅(NaCl) + #₂°₉₈.₁₅(C1) + #₂°₉₈.₁₅(Na, gas) +

£ion ^— #aff = 786.8 kJ/mol. However, this value assumes that the (infi-nitely dilute) gases of Na⁺ and Cl~ ions have the temperature 298.15 K,

and hence a total enthalpy 2(5#772) = 12.4 kJ/mol, where R is the gas constant. Subtracting this from 786.8 kJ/mol we get [/bind = 774.4 kJ/mol at T = 298 K, in good agreement with the value [/con = 764.0 kJ/mol from Tosi (1964). In fact, the calculation by Tosi follows exactly the steps of this example, but with slightly different data.

In order to predict the actual crystal structure of a solid (at T = 0 K), one has to compare binding energies [/bind (or cohesive energies [/con

etc.) for all conceivable lattice structures, and find the lowest [/bind- In practice, a comparison is often limited to the most likely structures, such as fee, bcc, hep and tetragonal lattices in the case of metals. At finite temperatures one should compare Helmholtz or Gibbs energies (see Chapter 7 for a treatment of temperature induced structural changes).

An additional complication, that is often neglected in calculations, is that of dynamical instability. For instance, a bcc lattice may have a minimum in [/(A) when A corresponds to a certain value of the lattice parameter, but a further lowering of U may occur if the lattice is sheared (Chapter 4, §3). Therefore, one should consider U(k\, A2,.. •, A„), where the parameters A; describe all possible atomic configurations in a unit cell containing any number of atoms. Figure 1.1 corresponds to a minimum when U is a function of only one A,,-, but it does not say if this is a true minimum or, say, a saddle point in the complete A space.

It is instructive to express some characteristic energies in the unit

^B?fus pe^r atom, where 7fus is the melting temperature. Table 1.1 gives values for the cohesive energy [/coh = //(gas) — H(solid), relative to separated neutral atoms (Al, W, GaAs, TiC) or ions (NaCl), and the enthalpy difference AZ/f_us between the liquid and the solid at 7f_us (as an example of the effect of a significant change in the atomic config-uration). It also gives the quantity Ez ~ 9&B#D/8 as an approximate measure of the energy associated with the zero-point lattice vibrations (Chapter 7, §2) where #D ~ #D ^{o r} ^D(O) is a Debye temperature taken from the tables in Appendix I. E_vfo is the vibrational energy if anhar-monic effects are neglected, i.e. the classical value 3kBT per atom at high temperatures. Data are from the JANAF thermochemical tables (1985), Barin (1989) and the above example for NaCl. The large value of AHfus for GaAs reflects the fact the bonding in GaAs changes from covalent in the semiconducting solid state to metallic in the liquid. It should be remarked that A//f_us/ 7f_us is the entropy of fusion (see Chapter 12). The quite small variation among [/con of different materials, when

Simple models of cohesive properties

Table 1.1

Some characteristic energies, expressed in the unit k# Tfus per atom

Al(7fus = 933K)

expressed in k_BTf_us per atom, is a significant feature (cf. Chapter 19,

§10).

4. Simple models of cohesive properties