Surface energy for crystalline solids

The surface is created by cutting the crystal along a line at an angle of 9 to the horizontal. The number of bonds directed vertically per unit projected length is 

The number of bonds directed horizontally per unit projected length is 

Thus the total number of broken bonds per unit projected length will depend on 0 as 

Note that the factor of 2 arises in L 6) because two surfaces are being created by this process. We have also 

The three-dimensional WuhT plot is cumbersome to use. A more convenient approach is to use a stereographic projection. If a crystal is located at the center of a sphere, the normal to each kkl plane will intersect the sphere at a particular location or pole. The angles 

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In this chapter the consequences of the orientation-dependent surface energies for crystalline sohds have been described. The effects on equilibrium crystal shape and the thermodynamics of grain-boundary behavior and faceting have been used as examples. Two common techniques for measuring y for solid surfaces, namely zero creep and multiphase equilibrium, have been described. 

We turn now to the interaction energy e2/r12 between electrons and consider first its effect on the Fermi surface. The theory outlined until this point has been based on the Hartree-Fock approximation in which each electron moves in the average field of all the other electrons. A striking feature of this theory is that all states are full up to a limiting value of the energy denoted by F and called the Fermi energy. This is true for non-crystalline as well as for crystalline solids for the latter, in addition, occupied states in fc-space are separated from unoccupied states by the "Fermi surface . Both of these features of the simple model, in which the interaction between electrons is neglected, are exact properties of the many-electron wave function the Fermi surface is a real physical quantity, which can be determined experimentally in several ways. 

The energy required by an electron to escape from the surface of a crystalline solid is called the work function (9) of the material. It is a characteristic parameter for its electron emission behavior. The work function of metals is in the range of 2 to 6eV and correlates mainly with the electronegativity of the element (Table 1.23). 

The illustrative data presented in Table VII-3 indicate that the total surface energy may amount to a few tenths of a calorie per gram for particles on the order of 1 /xm in size. When the solid interface is destroyed, as by dissolving, the surface energy appears as an extra heat of solution, and with accurate calorimetry it is possible to measure the small difference between the heat of solution of coarse and of finely crystalline material. 

In contrast to this, for a crystalline solid, the free movement of the adsorbed molecules is very much restricted by the periodic nature of the force fields at such surfaces. Then the activation energy to move from one place of adsorption to another (Alf ) becomes very important. Little is known, theoretically or experimentally, about the size of AH relative to AH, but it is generally considered H5-47) thafAH is about AH/2 or less. Movement of molecules along this surface takes place by a series of jumps from site to site rather than by a sliding motion. 

Besides this experimental method, there is a semitheoretical method by which y can be estimated. Assuming Equation 3 to be valid for any value of s and assuming that y is independent of s, we may outline this method as follows. One mole of a coarse crystalline solid consisting of ions A+ and B" is immersed in an aqueous solution. When the large crystal is cut into small pieces, the interface s increases, and the system gains Gibbs energy. This pulverization finally yields the hydrated ions Aaq+ and Ba( ", and the molar surface reaches a value of ... 

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