3 THEORY OF METAL SURFACES: WORK FUNCTION choices for the pseudopotential radius re, we have surface energies than the others, and therefore carried out the computation for both. In the should be preferentially present. These surfaces stances, the r. value marked with an asterisk in have the highest work functions. Thus, in the cases the table is generally found to yield agreement with of Al and Pb, for which the work-function measure experiment for a wider range of bulk properties ments were made on evaporated films, the appro than the other. 4 There exist at present practical- priate theoretical estimates are probably somewhat ly no experimental data on the work functions of higher than the crosses. For Zn and Mg whose Is listed. Of ew recent data which we could find, " almost none using evaporated films) the crosses already cor- were determined by more than one group of workers respond to the most closely packed faces, which and some appear to disagree with accepted results were the only ones calculated. or polycrystals. We therefore felt that comparison It should also be mentioned that photoemission of our results with single-crystal measurements measurements weight the low-p faces most strong was premature. We remark however that the gen- ly. This effect is probably of greatest importanc eral trend of a decre ase of the calculated with for the lower-density alkali metals whose quoted decreasing density of ions in the lattice plane 23is work functions were obtained by this method and in keeping with the arguments of Smoluchowski and for which the surface-energy anisotropy is small with the extensive body of single-crystal work on Hence, for these metals, the appropriate theoreti metals such as W, Mo, and Ni. For the polycrys- cal estimates are probably somewhat lower than talline data we have, so far as possible, used recent the crosses results which agreed substantially with earlier mea- In view of the uncertainties concerning the cor surements. With the exception of Li, we have used rect crystal-face average for polycry stalline ma- only experimental data obtained by the photoemission terials, it is not possible to state just how much or Kelvin contact-potential-difference methods if at all, the ion-lattice corrections &d to the These methods are in wide use, they yield the work form model improve the agreement with experi function in a direct way, and they avoid high tem- ment. However, we note (see Fig. 2)that for the peratures. For Li, however, the most frequently alkalis, for which b, >fept, the average 8p is quoted result was obtained over 20 years ago, by a negative, as it should be, and of the right order of type of contact-potential-difference method, " and magnitude, while for Al, Pb, and Zn, p <p est, disagrees seriously with that of a very recent field- and the average 8 is properly positive and agai emission experiment by Ovchinnikov and Tsarev. of the right order. (For Mg, u agrees exactly with In this case we have included both experimental re- boxt, and the addition of 8 produces a 10% error sults. The reader is referred to the article by Thus it appears that, all in all, the inclusion of the Riviere for a general review of the experimental ion potential in the theory improves the agreemen situation with experiment to within 5-10%. The measured work function of a polycrystalline ample represents a certain average over various D. Results and Comparison with Experiment-Noble Metals crystal faces, which depends on the technique used, Our calculated results for the noble metals Cu, the conditions of the measurement(e. g, tempera- Ag, and Au are given in Table III. Again we have ture), and the relative proportion of each face on used local pseudopotentials of the simple form the surface. 29 For qualitative orientation the simple (4. 9), with values of rs taken from Ref. 19. 3The e arithmetic average of the t values computed for calculated work functions are seen to be 15-30% the several faces of each metal is shown in Fig. 2 oo low compared with experiment, 32 the absolute as a cross(there are two crosses for the cases in discrepancies being in the vicinity of 1 ev. For which there were two rs values these metals the experimental data are consistent However, it should be remarked that for higher- and appear to be reliable, so that this error is al- density metals, theory predicts that the most certainly due to the theory. We do not know closely packed faces will have substantially lower at present how this error is apportioned between TABLE III. Theoretical and experimental work functions of the three noble metals. See Table II for explanation of symbols, Experimental values for polycrystalline samples Metal Structure rs 100)(11)(10)(100)(111) polycrystalline) 0. 22 fcc3.023.491.04-0.150,070,19 4.0