THEORY OF METAL SURFACES: CHARGE DENSITY TABLE III. Cleavage energy constants a where o, is the surface energy in the uniform Lattice Cleavage planes background model and ooa and 8gpa are given by (111) Eqs.(3.3)and(3.4) 0.00325 0.01434 004407 B Computations and Comparison with Experiment 003100 We would like to thank M. Rao for his help with Since there are practically no data available for computations of a the surface energies of simple metals in the soli phase, we have computed values of o which would be most appropriate for comparison with mea negative charges [including the effective ionic surements of the surface tensions of liquid met arges nion G)] absence of a satisfac We recall the definition of surface energy as the the ionic configurations of liquid-metal surfaces, energy required, per unit area of new surface we have calculated o for such ordered lattice struc tures and cleavage planes as, in our view, re the two models, we calculate the electrostatic con- sembled most closely a liquid surtace. Exper g.5) similar to that in the solid state, provides some Step 1. We divide the crystal in two, holding the justification for this approach electron density uniform up to the nominal metal The coordination numbers, in the liquid state, boundary in each half. The contribution to 8o from of the metals considered lie between the coordina- this step is a classical cleavage energy which will tion numbers of the bcc and fcc lattices, which are be denoted by 6o 8 and 12, respectively. Therefore, calculations Step 2. Next we change, in both models, the were carried out for both of these two lattice electron density from its step-function form to its types. The faces selected were those most dense- actual form n(x). The contribution from this step ly packed,(111)for fcc and(110)for bcc.Such to 5g will be called 8o a choice has been considered reasonably repre Step 1 requires no energy in the uniform back sentative of a liquid surface by various authors, o ground model. In the ion lattice model we may, in and we have verified by sample calculations that calculating the energy required for this step, re place the pseudopotentials by point-charge poten orgies and hence would be expected to appear or tials.a dimensional argument shows that, for the surface a given lattice type and a given cleavage plane We have calculated surface energies for the 8 metals listed in Table IV. These include all of 80 =azn, (3. 3) those considered in the bulk pseudopotential calcu- where a is a dimensionless constant. The compi tation of this constant is described in Appendix c The results for the body-centered and face-cen tered cubic lattices with cleavage planes perpen 8v(x)with rc=0 dicular to the [100], [111], and [110] directions n(x}-百 0.2 are given in Table Ill Step 2, as inspection of Fig. 5 shows, contrib utes the following term to 60 COps- 6v()n(x)-n.()]dx Here v(x)is the average, over the y-z plane, of he sum le ionic pseudopotentials of the half- 06 lattice, minus the potential due to the semi-infinite round. The algebraic expres sion for 6v(x)is derived in Appendix D. The two STANCE (Bohr radii factors entering the integrand in (3. 4)are plotted FIG. 6. Factors in the integrand giving dopg [Eq. in Fig. 6 (3.4)1. The case of potassium is shown here. In the The total surface energy in the present model is absence of pseudopotential cancellation, &v(x)is the then given by unction represented by the deeply cusped dashed line (the lattice planes are at the cusps). The presence of substantial oscillations in n(x)-n,)has in general an 0=y+601+b0p, (3. 5) important effect on the value of &