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THEORY OF METAL SURFACES: WORK FUNCTION 1219 density range, the total work function, given by ment somewhat and also permits calculation, for their difference, varies only from 2. 4 to 3. 9eV a given metal, of the anisotropy of i.e., the The values of fu given in Table I are similar to variation of from one cry stal face to another. those obtained by Smith, although Smith did not clude in his calculations the Friedel density os Ⅳv., ION-LATTICE MODEL cillations near the surface. The reason for this is the followi The contribution -u to u is identical in both calculations. For high electro When we pass from the idealized uniform-back densities, Ap is substantial, but since the Friedel ground model to a more realistic model in which oscillations at these densities are rather small, 1 the effect of each metal ion on the conduction elec Smiths calculations give similar results to ours trons is represented by a pseudopotential,a For low electron densities, the Friedel oscilla- traightforward attempt to calculate 4 from eq tions are important and our A is substantially (2. 1)would involve the prohibitively difficult task maller than that of Smith (by about a factor of 2 of solving self-consistently a system of equations for Cs), but at these densities A is negligible which no longer separate, but which are truly compared with-μ three-dimensional, To avoid this problem we Figure 2 compares the computed values of use the fact that the replacement of the uniforn Shall with recent experimental data on the work func background by the ion pseudopotentials represents tions of polycrystalline simple metals. There is a small perturbation 8v(F). To first order in 6v rather good agreement between theory and experi- the change of the work function 6 will be shown ment for these metals, to which the uniform-back to be given by the expression ground model would be expected to be best appli =∫6v()mn()在 (4.1) cable. A more realistic treatment of the positive ions, presented in Sec. IV, improves this agree hich has already been explained in Sec. I LEq (1.2)], and which avoids the solution of three dimensional wave To derive (4. 1)we begin with the definition (2.15)of =[中(∞)+E]-Ex=E-EN, where the sy stem in question is a large metal slab entirely of tw crystal axes. En is the ground-state energy of the neutral slab, containing N el is the energy of an excited state of the N-electron Unifor system in which (N-1)electrons reside in the low- est possible state in the metal, while one electron is at rest at infinity. The first-order change of 由 due to 5υ is then, by standard ory ∫6v(mF)d-J6v()m(),(4.3) nn and nN are the electron densities in the FIG Comparison of theoretical values of the samples(open circles).(The reason for the presence of wo experimental points for Li is discussed in the text. electron does not contribute to the first integral s the work function in the uniform-background model and we may rewrite(4. 3) heΦva lattice model were computed for the(110),(100), and 师=J6v(Fmn() (4.4) (111) faces of the cubic metals and the(0001) face where the hep metals(Zn and Mg). For qualitative he simple arithmetic average of these values for each (4.5) metal is indicated by a cross(two crosses are shown for the cases in which there were two possible pseudo- with nN, the density distribution of the (N-1)elec- potential radil). The experimental and theoretical points trons in their ground state. Clearly the density for Zn should be at rs=2. 30; they have been shifted deficiency n, satisfies the normalization slightly on the graph to avoid confusion with the data for ∫n()=-1 (4.6)
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