Solid State Communications, Vol. 21, pp. 257-261,1977. Pergamon Press Printed in Great Britain DIRECT-TRANSITION INTERPRETATION OF ANGULAR-DEPENDENT VALENCE-BAND PHOTOEMISSION FROM SINGLE-CRYSTAL COPPER IN THE ENERGY RANGE 40 < hv < 200 eV L.F. Wagner, Z. Hussain and C.S. Fadley* Department of Chemistry, University of Hawaii, Honolulu, Hawaii 96822, U.S.A. (Received 13 September 1976 by L. Hedin) Angular-dependent photoemission data from copper single crystals in the energy range 40 < hv < 200 eV obtained recently by Stohr et al. is com- pared with theoretical calculations based upon a bulk direct-transition model with free-electron final states and complete neglect of matrix- element effects. Very good agreement is found between experiment and theory. The addition of matrix elements based upon plane-wave final states is by contrast found to yield very poor results. 1. INTRODUCTION ANGULAR-DEPENDENT photoemission studies of the valence bands of single crystals of the noble metals cop- per and gold have previously been performed with pho- ton energies in the conventional UPS range from approxi- mately IO-2 1 eV,lw4 as well as at the often-used XPS energy of 1.487 keV. 5-7 With few exceptions, the spec- tral changes noted with both angle and photon energy have been successfully explained in terms of a bulk direct-transition model requiring rigorous conservation of energy and wave vector and neglecting matrix-element variations.1-6 In analyzing the XPS data, a free-electron final state has been assumed, and, within this approxi- mation, it has also been suggested that angular-dependent matrix elements may play a signifant role in producing such effects.7*8 The potential importance of additional final-state complexities such as those producing emission in secondary directions9 has also been discussed.1-7 Very recently, Stohr et al. lo have used synchrotron radiation to obtain angular-dependent photoemission data from copper and gold single crystals in the inter- mediate photon energy range from 32 to 200 eV. In particular, copper valence spectra were obtained for photoelectron emission along the [OOl] and [ 11 l] directions and significant changes were noted with both direction and photon energy, as indicated by the rep- resentative spectra shown in Fig. 1. For the data shown, the angle between the electron emission direction and the electric field vector of the polarized source was fixed at 27.5”, and emission was normal to the crystal surface. As no quantitative theoretical interpretation of this data has as yet been attempted, it is thus of interest to deter- mine the extent to which a straightforward application of the direct-transition model can be used to predict the * Alfred P. Sloan Foundation Research Fellow. observed effects. In this paper, such calculations are carried out and found to compare favorably with experi- ment. Also, the effect of including plane-wave matrix elements is investigated and found to yield significantly decreased agreement with experiment. 2. THEORETICAL MODEL The direct-transition model used in these calcu- lations is identical to that applied previously to the pre- diction of angular-dependent XPS spectra.5*6 The photo- electron energy distribution N(_K, hv) is given by N(E, hv) a c I d3 kf I@‘(r)lA * Vk$‘(r))1’ i k’ obs. x 6(kf - k’ - khv -g)s(Ef-E’-hv)G(E-E’) in which @f(r) and @j(r) are the final and initial electron wave functions, kfand k’ are the final and initial electron wave vectors, A-V represents the perturbation due to radiation, khv is the photon wave vector (with khv = 2nv/c), and g is a reciprocal lattice vector. ki is thus taken to be in the first Brillouin zone and k’is expressed in an extended zone scheme. The delta functions rep- resent wave vector and energy conservation, with that for wave vector resulting from a partial evaluation of the transition matrix element. The integral is over observed kf values, whose directions span a rectangular solid angle with overall dimensions of 8” x 12” for the experimental geometry utilized.” (Summing over a square grid of 400 directions within this rectangular solid angle was found to yield accurate results as judged by increasing the number of directions.) The sum on i is over all occupied initial states. In view of the present absence of band structure calculations at high energies for copper, it is further assumed that the magnitude of k’ can be estimated with sufficient accuracy by using 2.57