1036 M I CHAEL P. GREEN E et al. Comparison of (C25) and (C30) gives 2'! d (n. o. ) = — (4srP f')'"Q, (X, X'). (C31) Our result can be written s (X) It-(n. o. ) = — 1+3ys Q rt»'. (C34) g) res+72 Insertion of (C31) into (C9) yields Now we wish to express this in terms of Xp=|Ipr/to, instead of X. Using st=1 — 8s and s (X)=s (Xp) — Xps '(X) ra8s, we obtain a "nonoscillating" con- tribution 8 8 Qf 2 Ir" = — 6 '— — rt'" 2 — Qa(X, X') I g-p" (C32) pr' 8$ 8$' ~ as+y' s.(xp) We introduce the Cohen-Harrison-Harrison" nota- I, **(„o)= — 1+sobs+3&s tion ~ ~s+7s s. (X) = and note that P '(X sin8) j' sin'8d8, (C33) 00 8 3s +Xp- s(Xp) . (C35) a=~ crs+p' 8Xp 2 P-'(X)3'=s Combination of (C22) and (C35) gives Eq. (3. 45) of the text. PH YS ICAL REVIEW VOLUME 177, NUMBER 3 15 JAN VARY 1969 Size Effects on the Diamagnetic Susceptibility of a Free-Electron Gas* D. CHIZ, DERS DepartrrIent of Physics, University of California, los Angeles, California P. Pzxcvs Douglas Adpanced Research Laboratory, Pnntengton Beach, California, Institlto Venesolano de Ineestigaciones Cientzficas, Caracas, Venezuela, Department of Physics, Uneeersity of California, Los Angeles, California 900Z4 (Received 3 August 1968) W'e consider a noninteracting electron gas constrained in one dimension by a harmonic-oscillator potential as a model of a metallic Glm with specular surfaces. The diamagnetic response of this system to an applied Geld is investigated to study the effect of sample size on both the de Haas-van Alphen (dHvA) oscillations and the steady susceptibility. When the diameters of the dominant cyclotron orbits become comparable to the effective sample "thickness, " we Gnd large departures from the familiar II ' law for the dHvA oscilla- tions. However, the steady part of the susceptibility appears to remain independent of size. I. INTRODUCTION 'HIS paper is devoted to an investigation of a model for size eQ'ects on the diamagnetic susceptibility of a degenerate free-electron gas. In particular, we study the 61m geometry shown in Fig. 1, in which the mag- netic 6eld is taken to be parallel to the surface of a 61m of thickness L. The problem is to elucidate the correc- tions to both (a) the steady part of the diamagnetic susceptibility an. d (b) the oscillatory de Haas — van Alphen (dHvA) structure which arise when the ratio 1 = 2R,/L is nonzero. (R, is the cyclotron radius of an electron at the Fermi surface travelling in a plane per- pendicular to the applied 6eld. ) We shall restrict our * Work supported in part by the National Science Foundation and the„U. S. OKce of Naval Research Contract No. 233(88). attention to a model (described below) which is es- sentially equivalent to specular re6ection from the surfaces of the sample. This assumption raises the question of the applicability of such a calculation to real systems, where diffuse reQection will certainly play a role. However, recent experiments by Koch et al. ' indi- cate that a specular boundary may be obtainable, at least for certain groups of electrons. The physical effect of the boundaries is to lift the degeneracy associated with a particular Landau level2 which arises from cyclotron orbits centered at different positions across ' F. Koch (private communication) has shorn that in several experiments on the transport properties of metals such as Sn and In the results are consistent only with highly specular surface scattering. ~ C. Kittel, Qeantlm Theory of Solids (John Wiley R Sons, Inc. , New York, 1963).