PHYSICAL REVIEW B VOLUME 3, NUMBER 4 15 FEBRUARY 1971 Theory of Metal Surfaces: Work Function* N. D. Lang IBM Watson Laboratory, Columbia University, New York, New York 10025 and W.Kohn, University of California, San Diego, La Jolla, California 92037 Hebrew University, Jerusalem, Israel (Received 16 October 1970) In a recent paper we presented a contributio to the theory of metal surfaces with emphasis on the shape of the electron-density distribution and the surface energy. The present paper extends this analysis to a consideration of the work function. Some general theoretical rela- tionships are established. Effects of the ions are included using a simple pseudopotential theory, permitting the calculation of the variation of the work function from one crystal face to another. For simple metals(li,na,k,rb,cs,l,pb,zn, and Mg), agreement with available experimental data is good (5-10%); for the noble metals, the computed work func- tions are 15-30% too low. I. INTRODUCTION puted in LK-I, and we take from the available theory of exchange and correlation of a uniform The present paper represents a sequel to one electron gas. This yields the work function f concerned primarily with metal-surface charge this model as a function of the mean bulk density densities and surface energies. In the earlier (or of the Wigner-Seitz radius).4 These results paper we presented, first of all, a theory of the are compared with experiment and with the theo- electronic structure of a model metal surface in retical calculations of Smith, who used a similar which the lattice of approach but did not carry out a fully self-consis- uniform background cha tent calculation. 5 tion effects were include Finally, we incorporate the effect of the actual version of the theory ion cores. We show first that when the difference gas. 23 Following this, the effect of etween the pseudopotentials of the ion cores and structure on the surface the electrostatic potential of the uniform charge count by calculating background is treated as small perturbation(), contribution(similar to the change of the work function of a particular crys- by evaluating the in tal face due to this perturbation is given to first the ion cores using f order by the following rigorous expression: ory. In the present p (1.2) plan in developing a Here the integral is carried out over a slab whose This quantity, denoted surface consists overwhelmingly of the face in ques- in Sec. II in terms mo tion; and() is the change of the electron density, analysis, is equal to the mini e uniform-background model, fol- be done to remove an electro ctron from the system. se localized near We give first a rigorous demo We have calculated the charge =△中一μ, s to that of LK- d to look for self-consis- where is the rise in mean electrostatic potential tent solutions with a zero mean electric field deep potential of the electrons relative to the mean elec- Since n () depends in fact only on the distance trostatic potential in the metal interio surface, it is possible to re- its simple form, this expression i duce(1. 2)to a one-dimensional quadrature.In body effects, in particular, that of the image force. this way we have calculated the total work function For the uniform-bac from the electronic charge density n( ckgroun 虫+ (1.3) 31215
1216 N. D. LANG ANd W. KOHN 3 for the principal faces of nine simple metals -Al Pb. Z metals Cu, Au, and Ag. Experimental data are generally available only for polycrystalline samples Background, n+(x] of unknown surface structure. This makes Electrons, n(x) tailed comparison between theory and experiment mpossible. Nonetheless, we can state the follow ng conclusions imple metals: The measured work functions range over 2.. 3 ev. with the possible exception of Li (where there is considerable uncertainty both in the experimental data and in the pseudopotential) agreement between the full theory and experiment is typically within 5-10%0. The ionic lattice contri- outions 8, which are characteristically of the or- der of 10% of the total work functions, contribute to establish this rather good agreement. Anisotro- evel (u) ies among the different faces are typically also of the order of 10% of the mean work function. In ac cordance with the arguments of Smoluchowski, w find the lowest work function to be associated wit FIG. 1. Schematic representation of (a) density dis- the least densely packed face among those consid- tributions at a metal surface and (b) various energies ered [(110) for icc, (111)for bcc]. relevant to a study of the work function. Noble metals: In view of the success with simple metals, we have applied the same technique to the noble metals to learn about the limits of validity of ative to the mean electrostatic potential there(see our theory. Here the experimental work functions Fig. 1). It is important to know if this expression range over 4.0-5 2 ev, and the calculated values includes properly all many-body effects, in par are 15-30% too low. It may be assumed that the ticular, the work done against the image force in presence of the filled d bands not far from the Fer- removing an electron from the metal. This is in mi level makes our highly simplified theory, based fact the case. As we have not found any rigorous on the inor ous-electron-gas model with demonstration of Eq .(2. 1)in the literature,we small pseudopotential corrections, much less ap propriate for these metals Since we are interested in removing one electron In summary, the theory we have outlined appears from the metal at K, we first develop a simpl to describe well the work functions of simple met- extension of the theory of Hohenberg and Kohn (HK)2 Is. Additional reliable experimental data for this to allow for a variable number of electrons, and class of metals would be highly desirable, partic- then use this theory in establishing the validity of ularly data on the work functions of single-crystal Eq.(2.1) ces. In the case of the noble metals. on the other In hk. it was shown that for a fixed number of hand, where, for metallurgical reasons, the ex- electrons N and arbitrary static external potential perimental data are much more consistent and re- v(r), there exists an energy expression liable, the present theory is less successful, and further theoretical work is needed. There is need also for additional theoretical studies on the transi- tion metals, which are not discussed in this paper. IL, RIGOROUS EXPRESSION FOR WORK FUNCTION +Gn], (2.2) alitative considerations, in the spirit of the with the following properties: (a)Gn] is a unive Sommerfeld electron theory of metals, strongly sal functional of n(r), not explicitly dependent suggest that the work function is given by the v(r), given by G[n]=(Vn, [T+U,) 1 n(r)n (2.1) Here Ao is the change in electrostatic potential where the wave function y, refers to the unique across the dipole layer created by the "spilling electron ground state with density n(r), and T and out"of electrons at the surface, and u is the chem- U are, respectively, the kinetic- and interaction- ical potential of the electrons in the bulk metal rel- energy operators; (b)eIn] is equal to the correct
3 THEORY OF METAL SURFACES: WORK FUNCTION ground-state energy for a given v(r)when the cor- rect density n(r), corresponding to u, is used on 小()≡v()+ the right-hand side of the equation; (c)the first variations of En] about this density, consistent is the total electrostatic potential. Now consider with the restriction first a particle-conserving, but otherwise arbi ≡∫6n()dF=0 (2. 4) trary variation On. Equation(2. 11)then gives vanIs where u 'is some constant, independent of r. 6En]=0 (2. 5) Next, consider a particle nonconserving variation Now consider an ensemble of from(2.11)and (2. 13)we then see that tronic systems at the absolute zero of tempera μ=μ=中()+6G[n]/6n() (2.14) ture, specified by the external potentials v(r)and the chemical potentials u. The subsidiary cond The work function is, by definition tion( 2. 4)will no longer be imposed on the density 更=[中(∞)+El]-E variations of interest. We define R, In]=[n]-ufn(r)dr (2.6) where (oo)is the total electrostatic potential far from the slab considered above and eu is the Clearly, for a first variation of the density n,(F) ground-state energy of the slab with M electrons which does satisfy the condition(2. 4) (but still with N units of positive charge).[Both 中( 613nu]=51En]=0 ( 2. 7) but the combination(2. 15)does not. ]Using the definition of the chemical potential and Eq . (2. 14) Now let nu,(F)and nu+ u, (r)be the correct elec- this can also be written as tron densities corresponding to a given u(r)and respectively,toμandμ+bμ, These two chemi- 中=(∞)-以=[(∞)-列 cal potentials describe two systems whose total mumbers of particles differ by 8N: 6N=∫[n,(式)-n,(6)1d=∫b6n2(F)dF (2.8) 乒=μ-可=(6G[n/6m(F》 (2.17) The corresponding first-order change of &p, u is Here◇ denotes an average over the metal.μis given by the bulk chemical potential relative to the mean interior potential; its independence of this poten 2202, [n]=Emn+6u, u]-Elnm, J]-u6N=0 tial may be verified from the definition (2. 3)of (2.9) GLn]. Equation (2. 16)is equivalent to the postu- where the vanishing follows from the thermody (2.1) namic definition of the chemical potential at T All many-body effects are contained in K. Since an arbitrary small variation on is a change and correlation contributions to u and in (unique)sum of variations of the types 6n, and &n2 their effect on the barrier potential As. In par it follows that, in general, ticular, the image-force effect on p may be re- garded as contained in the disappearance of part (2. 10) of the correlation energy when one electron is We now apply this theory to the work function. moved away from the metal surface We consider a neutral slab of metal. all of whose IlL. UNIFORM-POSITIVE-BACKGROUND MODEL dimensions are macroscopic but whose surface consists overwhelmingly of two parallel faces the We consider in this section a model of a metal ork function of which we wish to consider(the surface in which the positive ions are replaced by physical properties of these two surfaces are taken a uniform positive charge background filling the to be identical). Let n(f)be the correct electron half-space x<0. 10 The electron density in thi density corresponding to the given nuclear poten- model is shown schematically in Fig. 1 (a) tial, the chemical potential u, and a total number We consider first the quantity u in Eq.(2. 1) of electrons N. By(2.6),(2. 2), and(2. 10)w Since deep in the metal interior the electron den have, for a small variation in density on(F) sity has a constant value n, u takes on the simple ∫(,c) 正=k+2 (3.1) where Here kp=(372n1/3 is the bulk Fermi wave numb
N D LANG AND W. KOHN TABLE L. The work function u of the uniform-back- Other, more recently suggested, forms of the cor round model, and its bulk and surface-barrier compo- relation energy give substantially similar results nents. The Wigner-Seitz radius r& characterizes the in terior density.面=△d-园, where u=量h》+Hxe, The bar The quantity H and its two components 2k and rier term Ap is given with a self-consistency of 0. 03 eV uxe are shown in Table I for rs in the metallic or better (this is a somewhat greater self-consistency than that of the preliminary report, Ref. 4) the bulk contribution to , It will be seen from Table I that, for metals of low electron density respectively),-H is much larger than the othe 212.52 9.61 6,803.89 term△φ. The rather good agreement with ex 2.58.01790 11 3.83 3. 72 periment(see Table II)constitutes 5.57 5.92-1.18 2.32 3.50 good confirmation of the expression (3. 3)for 5. 28.83 1.43 3. 26 This is especially meaningful since the theory of -2.31 0.91 3.06 the correlation energy is most difficult at low elec- 0.56 2.00 4.38 2.38 tron densities 2.38 We turn now to the double-layer contribution 1.39 3.76-2.370,042.41 △中. By Poisson' s equation, p=d()-(-)=4mJxm(x)-n,()dx (3.5) nd Are is the exchange and correlation part of with n(x)and n,(x), respectively the electron the chemical potential of an infinite uniform elec and positive-background densities in the uniform tron gas of density n. Axe is given by the relation model. n(x), calculated in LK-I with a self-con sistency of better than 1% of n, was recalculated (3. 2) for the present work to an accuracy of 0. 2% or better. The resulting values of Ap are listed in with Exc the exchange and correlation energy per Table i, as is the total work function in the uni- particle of the uniform gas. In our computations, form model we have used the expression (3.6) ∈x)=-(0.458/ys)-0.44/(+7.8) (3.3) It will be noted that while△φ is negligible for here the wigner-Serls C analysis of the electron gas; nant for high-eledr and 4p separately change by from wigner’ s class metals of low electron density, it becomes domi radius rs is given by on-density metals. It is also trik mr3=1/ 5.3 and 6.8 ev, respectively, over the metallic Theoretical and experimental work functions of nine simple metals. u is the work function for the uni- ound model; ois the first-order pseudopotential correction: =u+o (rounded to the nearest 0.05 eV) potential core radii re are taken from the work of Ashcroft and Langreth (Refs. 18-20). In the cases in which these authors give two possible values of r for a metal, the choice which yields agreement with experiment for were taken from Refs. 25-27(see text for details of selection). [The most densely packed faces for the various struc tures are: fcc (111), hep(0001), bec(110).] Metal Structure rs 4ulev) b币(eV) 西ev) (110) 00) (111) (110)(100) (111)(polycrystalline) 2.073,871,12 0.210 0.193.654.204.05 2.303.801.12 0 0.130.063.803.953.8 2,303,801.270.36for(0001) face 4.15 for(0001) face 0.38 for (0001) face 4,05for(0001)fac 3,283,371.06*0,19-0,050,133.553,303,25 2.00-0.99-0.95-1.052.402.402.30 bcc 393.102.752.65 4.96 2.7 0,01 0 2,402.35 2 2.63 2.61 0.45-0.53-0.602.202.102.05 2.21 0.26-0.32652.352.30 bcc 0.23 0.61 2.14 0.10-0.21
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)
1220 N. D LANG AND W. KOHN 3 Now in the (N-1)-electron ground state, associ ted with Ex, the electric field inside the metal is zero, and outside each of the two parallel faces is given by &=4/2A (directed along the outwar normal), with 2A the total surface area. Let us take the x axis to have its origin in one of th e Iaces and point outwards [see Fig. 1(a). Then n, is a unction of x only. Further, let us denote the y-z average of &0(r)by Bu(x)=j. v(x, y, 2)dv dz/,dy dz (4.7) cE atomic units Then, using (4.6), Eq.(4. 4)reduces to Bulk density corresponds to fs"2 1.00 x io-3 atomic units 6=A.6(x)(x)dx =-∠。6(x)nx)dx/n(x)dx Js 5v(x)no()dx/no(x)dx where a is a length large compared with a scre ing length but small compared with the thickness the slab. In the last step both numerator and de- nominator have been reduced by a factor of 2. Since (4. 8)is homogeneous in no, no(x) can now be taken to be the surface-charge density in the semi-in finite uniform-background model induced by an ar Bulk densify corresponds to rs =4 bitrary weak electric field perpendicular to the s-3.00x 10-4 atomic units) B. Calculations To evaluate (4. 8)we require n(x) and 5v(x). na(x) was determined as the difference between two self consistent densities: one corresponding to a neu tral metal with vanishing field outside, and the other corresponding to a metal with a small surface charge and finite field outside. The method used was the same as that described in LK-I, and n,(x) was evaluated for r between 2 and 6.15 The func tions n,(x)for rs=2, 4, and 6 are shown in Figs. 3(a),(b), and (c), respectively. They are rather wide density distributions, which outside the metal DISTANCE (atomic units) drop to half their maximum value at approximately 2-3 (or 1.0-1.5 A)from the nominal surface. 18 is. 3.00x 10". atomic units The perturbation potential 6v(x), Eq.(4.7),was obtained, as explained in LK-I, Appendix D, using seudopotentials of the form employed by ashcroft FIG. 3. The e n(x) in the electron-density dis and Langreth8-20 tribution nx) of the uniform-background model, induced oy a weak external electric field along the outward nox 0f=(4) along +x). Curves are shown for(a)rs=2,(b) nd (c)r=6, n is the These authors have determined the radius r fo electron density deep in the metal. each metal to give a good description of bulk prop the cubic crystals the calculations were done for the(111),(100), and (110)crystal faces, and for Our calculated results for u, 8p, and =u+6p hep crystals, for the(0001)face. In the are given in Table II for nine simple metals. For which Ashcroft and Langreth give two possible
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
ANG AND W. KOHN and the barrier potential Ao. (Significant out ACKNOWLEDGMENTS ward movement of the last layer of atoms would reduce A and thus further increase the error. 3) One of us (W. K)wishes to thank the Hebrew It is not whether more sophisticated pseudo- University in Jerusalem for their hospitality during potential ry would remove most of the discre a sabbatical leave. We would also like to express ancy or whether these metals, with hybridized s-d our appreciation to S. Berger for assistance with bands, require a different approach a part of the numerical work. *supported in part by the National Science Foundation using an artificial small 6v()localized near the sur- and doing a direct self-cor stent calculation of Ad pResent address: IBM Thomas J. Watson Research and hence of =Ao-F for this case. The result for 64 Center, Yorktown Heights, New York 10598 agreed within a few percent with the perturbation result Permanent addre D. Lang and W. Kohn, Phys. Rev. B 1, 4555 16The surface-charge distributions n,(x)also play an (1970); this paper will be denoted as LK-L. important role in theories of the image force and of 2P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 small-gap condensers, which will be discussed in a forth (1964) coming paper Sw. Kohn and L, J. Sham, Phys. Rev. 140, A1133 IThe case re >ld(d is the spacing of the lattice planes (1965) which did not occur in LK-l, does occur in some of the A preliminary account of this part of the calculation present calculations as been given in N. D. Lang, Solid State Commun N. W. Ashcroft and D. C. Langreth, Phys. Rev. 1047(1969) 682(1967) J. R, Smith, Phys. Rev. 181, 522(1969) IN. W. Ashcroft and D. C.Langreth, Phys. Rev 661(1941) 500(1967) An example of a semiempirical study of the work func 2tof these three faces, the(111) face is the most this paper is given by D. Steiner and E. P. Gyftopoulos, closely packed and the (110)face the least closely packed in Proceedings of the Twenty-Seventh Annual Conference for the fcc structure; for the bcc structure the roles a n Physical Electronics, Massachusetts Institute of reversed. Technology, Cambridge, Mass. 1967(unpublished Al [(111),(100)1: B. Sieroczynska-Wojas, Fiz. Tverd The extension of HK to the grand canonical ensemble Tela 10, 693(1968)[Soviet Phys, Solid State 10,544 at arbitrary temperatures has already been given by 1968); Zn (0001): L. P. Mosteller, T. Huen, and F. Phys,Rev,137,A1441(1965).[ On the Wooten,Phys.Rev.184,364(969);K[(110,(100 ght-hand sides of Eg (9) and the secon numbered C. Richard and P. Saget, J. Phys. Suppl. 31, C1 quation of this paper the term -un(r)dr must be 155(1970). Professor G -A. Boutry (private communi- added. I We present de novo the specially simple case of cation) has described current work on the (110) faces of zero temperature K, Rb, and Cs. Single-crystal data on the noble metals ' e use atomic units, with lelam=h=1. In this Cu, Au, Ag [(111),(100)] is given by P. system the unit of energy is 27.2ev. Coquet, and E. Boursey, Czech. J. Phys 10The first wave-mechanical computation of the work (1969); see also, for Cu, P. Kohler, z Phys function using this model was given by J. Bardeen, Phys 21191(1966) Rev.49,653(1936) 2Note, however, the exceptions to this trend for the For this corrected form, see Eg. (3. 58)of D. Pine 100) faces of Al and Pb. Elementary Excitations in Solids(Benjamin, New York, 24 For example, W, Mo: O. D. Protopopov, E. V. Mi 1963 heeva, B. N. Sheinberg, and G. N. Shuppe, Fiz. Tverd. ingwi, A, Sjolander Tela 8, 1140(1966)[Soviet Phys, Solid State 8, 909 and R. H. Land, Phys. Rev. B 1, 1044(1970) (1966)]; Ni: A, Kashetov and N. A. Gorbatyi, Fiz. Tverd 1 One can easily show that in the high-density limit Tela 10, 2135(1968)[Soviet Phys. Solid State 10, 1673 he leading terms of F and Ao are both I*p; they diverge (1969)] dividually but cancel each other in (3. 6). We conjee- 25A1: J. C. Riviere, Proc. Phys. Soc. (London)B70 ture that in the uniform model the work function tends to 676(1957) erected in Ref. 28) a finite limit (in the vicinity of 4 ev) as the interior ik 158, 607(1960)(as quoted in Ref, 28); Zn: mann and G. Wedler, Z. Angew. Phys. 14, 70 1In connection with the material of this section we (1962); Mg: R. Garron, C. R. Acad. Sci 258, 1458 (1964): Na: R. L. Gerlach and T. N. Rhodin, Surface e double-layer moment has beer i.19,403(1970), and priv nett and c uke, Phys. R 1060(1969) 15 The values of were chosen large enough that n,(x) for Na deposited on three different faces of Ni); K, Rb Cs: G.-A. Bountry and H. Dormount, Philips Tech. Rev. as well above the "noise level" associated with lack of 30,225(1969) tency, and small enough to be in th P.A. Anderson, Phys. Rev. 75, 1205(1949)(as sponse region. These conditions were checked by re- erected peating the calculations with the values of 8 doubled 2A. P O ikov and B. M. Tsarev, Fiz. Tverd. The entire theory of this section was also checked by 9,3512 [Soviet Phys. Solid State 9, 2766
THEORY OF METAL SURFACES: WORK FUNCTION 1223 high-coverage-limit the net importance of (6v)2 terms in d work functions for Li deposited on various faces of Reference 19 gives also a second, much larger, valu and R of r for each of these metals, which leads to somewhat J. C. Riviere, in Solid State Sunface Science, edited less satisfactory results for bulk properties. The work by M. Green(Dekker, New York, 1969), Vol. 1 BC. Herring and M. H. Nichols, Rev. Mod. Phys than those corresponding to the preferred set, giving an 21,185(1949) even greater discrepancy with experiment It might be mentioned that T. Schneider [Phys. Status Cu and Ag: D. E. Eastman, Phys. Rev. B 2, 1 32, 323(1969) has shown second-o 970); Au: E. E. Huber, Appl. Phys. Letters 8, 169 ce-A, the bulk contribution to the work func n, by 0.1-0.2 eV for the alkalis and by 0. 3 and 0. 8ev There is some theoretical evidence suggesting the for Al and Mg, respectively. The second-order effect existence of occupied surface states in the noble metal of 6v on the surface term Ao must be determined, how this could of course also affect Ad. See F. Forstmann ever, before any conclusions can be drawn concerning and J. B. Pendry, Z. Physik 235, 75 (1970) PHYSICAL REVIE W B VOlUME 3, NUMBER 4 15 FEBRUARY 1971 Some Formal Aspects of a Dynamical Theory of Diffusion University of llinois, Urbana, llinois 61801 eceived 2 October 1970) A classical dynamical theory of diffusion is presented in which the reaction coordinate and its to the Kac equation is shown to be exact for classical statistics, and the jump rate is calcu- lated accordingly. The jump rate can be expressed in terms of the vector R and the dynam matrix; this leads to a frequency factor different from that obtained from the reaction-rate theory, and an indication that the system does not jump through the relaxed saddle-point con- iguration. The self-diffusion isotope effect is also considered. It is shown explicitly that the migration energy is mass independent, and that the effect may be expressed simply in terms f the reaction coordinate. In terms of the phonons, the isotope effect depends on a weighted verage of the fraction of the energy carried by the jumping atom in each mode, Quantum corrections are discussed and a connection is made between the isotope effect and thermomi- gration . INTRODUCTION quilibrium configurations The troubling aspect of rate theory is that been increasing interest in recent on prop years in formulating the theory of diffusion jump saddle-point configuration. In a quantum-mechani- rates in a way that avoids some of the conceptual cal theory it would not only be impossible to treat difficulties of the absolute -rate theory. The rate theory was applied to diffusion by wert and Zener positions and velocities independently, but, more significantly, it would be completely inappropriate plicitly how the motions of many atoms are involved as has been pointed out by Flynn and Stoneham, ate and elegantly formulated by vineyard to show ex to speak of the properties of the intermediate in the jump process. This can be seen in Vine Attempts to circumvent these difficulties have yards form for the preexponential or frequency been made by several authors. These"dyi factor cal"theories view the jump process as resulting special kind of fluctuation f and attempt to calculate the frequency of such fluc tuations. In the earlier work, fluctuations were n which v and vl are the normal-mode frequencies considered that carried the system to a definite of the equilibrium and saddle-point configurations configuration, e.g., the relaxed saddle-point con respectively, with the unstable "mode"of the sad figuration. Glydehas shown that, for this case, dle point left out of the product in the denominator. the dynamical and rate theories have the same The many-body nature is also apparent in the iden- formal content. More recently, Flynn has con- tification of the migration energy with the potentia sidered fluctuations in a reaction coordinate, made energy difference of the relaxed saddle-point and up of a linear combination of particle displacements