Theory of Electron Diffraction by Crystals I. GREEN'S Function and Integral Equation
KYOZABURO KAMBE
Fritz-Haber-Institut der Max-Planck-Gesellschaft, Berlin-Dahlem *
(Z. Naturforschg. 22 a, 422—131 [1967] ; received 17 December 1966)
A general theory of electron diffraction by crystals is developed. The crystals are assumed to be infinitely extended in two dimensions and finite in the third dimension. For the scattering problem by this structure two-dimensionally expanded forms of GREEN'S function and integral equation are at first derived, and combined in single three-dimensional forms. EWALD'S method is applied to sum up the series for GREEN'S function.
The fundamental theory of electron diffraction We formulate the problem at first as a boundary- by crystals is BETHE'S dynamical theory1. In this value problem for the SCHRÖDINGER equation, but go theory the wave field inside the crystal is represent- over at once to the equivalent integral equation. The ed by a superposition of BLOCH waves and continued integral equation is convenient for mathematical into the vacuum by a superposition of plane waves. analysis and renders the exact solution of the prob- The main problem is to solve the eigenvalue prob- lem. The integral equation is also a convenient tool lem for the wave vector (propagation vector) of for investigating various existing theories of elec- BLOCH waves. In one dimension this is represented tron diffraction (including BETHE'S theory) and by HILL'S determinantal equation (WHITTAKER— provides an unifying view over these theories. WATSON2, p. 415). In three dimensions, however, the solubility of the corresponding determinantal equation is somewhat problematic. In contrast to the § 1. General Survey of Green's Functions one-dimensional case, we cannot secure the absolute convergence of the infinite determinant. In fact, Before going into our problem let us consider at first some related problems and corresponding BETHE argues that the problem is not exactly soluble (1. c.1, p. 73, § 4) and confines himself to approxi- GREEN'S functions to clarify our point of view. mate solutions. The most general form of the integral equation for potential scattering may be Apart from this difficulty in fundamental prin- ciple, BETHE'S assumption of geometrically plane v(r) =v(0)(r) + fG(r\r') V(r') ip(r') dr', (1) surfaces will not be valid especially in practical where ip{r) is a solution of SCHRÖDINGER'S equation application to low-energy electron diffraction. V2v + (*2_F(r))v; = 05 (2) In the present paper we propose an alternative formulation of the problem which appears to be I//°)(r) is the incident wave, and GREEN'S function more fundamental and more widely applicable than BETHE'S. We consider, as BETHE, non-relativistic elastic scattering and an infinitely extended crystal c(rii')-- (3) slab, but assume that the potential is periodic only is constructed to satisfy in two dimensions parallel to the surface. The varia- \7;G + x2 G= +d(r-r') (4) tion of the potential in the third dimension is at first almost arbitrary. This assumption allows us to apply and the condition of outgoing waves. This GREEN'S the theory to monatomic layers and to crystals with function is valid only for scattering objects which regularly distributed adsorbed atoms or steps on are finite in three dimensions. Therefore, it is ques- their surface. These cases are important for low- tionable to use this GREEN'S function in our problem energy electron diffraction. of infinitely extended slabs.
* Abteilung Piof. Dr. K. MOLIERE. 2 E. T. WHITTAKER and G. N. WATSON, A Course of Modern 1 H. BETHE, Ann. Phys. (4) 87, 55 [1928]. Analysis, Cambridge University Press 1927 (2nd ed.).
Dieses Werk wurde im Jahr 2013 vom Verlag Zeitschrift für Naturforschung This work has been digitalized and published in 2013 by Verlag Zeitschrift in Zusammenarbeit mit der Max-Planck-Gesellschaft zur Förderung der für Naturforschung in cooperation with the Max Planck Society for the Wissenschaften e.V. digitalisiert und unter folgender Lizenz veröffentlicht: Advancement of Science under a Creative Commons Attribution Creative Commons Namensnennung 4.0 Lizenz. 4.0 International License. Another extreme case of crystals infinite in three § 2. Expansion in Two Dimensions dimensions appears in the theory of BLOCH functions The two-dimensional periodicity of the potential (for example, KOHN and ROSTOKER 3). GREEN'S func- tion for the integral equation of BLOCH function is allows us to expand the solution in a kind of FOU- RIER series. This process has been applied already exp {i(k+ Bg) • (cf. § 2). In the third direction perpendicular to the pendicular to CLX and do, the non-zero region of surfaces the solution should have the property of V(r) is assumed to be confined to a finite range outgoing waves as required for (3). Thus, it is clear zj FUJIWARA 4 who investigated the same problem as ours, used the spectral representation for GREEN'S function 2 exp {ik• (i—r') } G(r * 2 dfc, (6) V) = [2 a)- / and put it in the iterative solution of the integral equation (1). This GREEN'S function appears to be effectively that what we want. However, the form (6) is not very convenient to be used in further mathematical investigations, because it contains a somewhat indefinite contour integral. Although FUJI- WARA'S treatment appears to be satisfactory so far as the iterative solution is concerned, he could not prove its convergence. His solution appears hence Fig. 1. The column of reference. to be a formal one. In any case his result is identical with the iterative solution of TOURNARIE'S system of This restriction is introduced only to simplify the integral equations 5, which is discussed in § 2, and mathematics, and can be removed afterwards by consequently should be identical also with our exact the process of shifting z\ and ze to — oo and + oo solution if the iteration is convergent. respectively. 3 W. KOHN and N. ROSTOKER, Phys. Rev. 94, 1111 [1954], 7 R. W. ATTREE and J. S. PLASKETT, Phil. Mag. (8) 1. 885 4 K. FUJIWARA, J. Phys. Soc. Japan 14, 1513 [1959]. [1956]. 5 N.TOURNARIE, J. Phys. Soc. Japan 17, Suppl. B-II, 98 [1962]. S K. HIRABAYASHI and Y.TAKEISHI, Surface Sei. 4, 150 [1966]. 6 M. VON LAUE, Phys. Rev. 37, 53 [1931]. We call the planes z = z; and 2 = ze the surfaces, where Kmt = Kot + Bmt. (16) the space z; = exp {i K0 • (/i! «! + n2 d2)} xp{r). 11 Vm(z) = Tm exp {i rm z) + ^^exp { - i Fm z}. (21 b) We can put therefore XFm , Rm, Tm , and are constants and can be in- terpreted, if r.m is real, as the amplitudes of plane xp(r) = exp {iKot-rt} yy>m(z) exp {*'Bwt-rt} , (11) m waves going upwards or downwards, and, if Tm is imaginary, as the amplitudes of "evanescent where Kot and rt are the components of K0 and V waves" 9 which increase or decrease exponentially. tangential to the surfaces. Bmt are given by The condition of outgoing waves can be adapted Bmt = m1Blt + m2B2t, (12) to the present problem in the form where the index m stands for the pair of integers m1 m = (22 a) and m2 . Blt and Bot are the Jico-dimensional reci- and procal basis vectors which are determined from dx and a2 to satisfy 0m=O for all m. (22 b) arBjt = 27i3ij (i,/ = 1,2). (13) From the continuity of ip and dy/dz on the surfaces it is found at once that the solution xp inside the Blt and Bot may not be identical with the conven- crystal must satisfy the boundary conditions tional basis vectors of the three-dimensional reci- procal lattice. Vm + • > d!fW ) = 2 <5omi dornt exp {iro Zi} lim /z=zi Putting (11) into the SCHRÖDINGER equation (2) (23 a) we obtain and 1 drpjr 4t Vm (2) + r,n Wm («) ~ Z Vm - n U) Wn U) = 0 , = 0 for all TO. (23 b) i r„ (14) d z where Tm2 is given by This is an explicit form of the condition of outgoing |2 10 r, K ml i 5 (15) waves (SOMMERFELD , p. 179). 9 Cf. for example M. BORN and K.WOLF, Principles of Optics, 10 A. SOMMERFELD, Vorlesungen über theoretische Physik, Bd. Pergamon Press, London 1959, p. 560. — J. A. RATCLIFFE, 6, Partielle Differentialgleichungen der Physik, Akadem. Rep. Progr. Phys. 19. 188 [1956]. Verlagsges. Leipzig 1958 (4. Aufl.). Thus we have a boundary value problem for an These GREEN'S functions satisfy the equations [the infinite system of ordinary differential equations main part12 of (14) ] (14) in the finite region z\ ^ z ^ ze with the bound- d2 ary conditions (23 a, b). Our aim is to find from the Gm(z | z') + -Tm2 Gm(z | z') = + d(z — z) (25) solution of this problem the amplitudes of the waves going out from the crystal, i. e. Rm and Tm of Eqs. and the homogeneous boundary conditions corre- al a, b). sponding to (23a, b), that is, if zj MERFELD10, p. 180, MORSE-FESHBACH11, p. 1071) it Gm{z\\z') + Yr~ dz Gm{z\z ) I =0, (26 a) is easy to find the system of GREEN'S functions ("GREEN matrix" of TOURNARIE 5) and Gm (z I z) =-- Yfr~ exp RM\z-z'\} • (24) Gm (ze I z) - T^— — Gm (z | z) = 0. (26 b) The system of integral equations is ZE Wrn (2) = \mi <5o in 2 exp {i roz} +fGm(z\z')£V m _ n (z) V« (2') d z, (27) or with (24) 2 V-'m(z) = <50mi <50w,a exp {i ro z} + -p- exp {1 rm z) / exp { - i rm z'} 2 Vm_n{z) \pn{z) dz 11 1 m J n Ze +—p—exp{-irmz} / exp{irtnz'}2Vm_n{z) ipn{z) d z. Z I I M. J N If we have obtained the solution of this system we can calculatz e the amplitudes of the outgoing waves by Zq Rm= 9, r / exp {irmz'}1ym_n(z') y>n(z') dz, (29 a) Z I 1 M J N "e Tm = <3om i a + 2 m J exp { - i Tm z'} lLVm_n (z) ipn (Z ) dz . (29 b) § 3. Recombination in Three Dimensions tion as well be shown in a separate paper12a. In fact, the forms of the systems in § 2 suggest that these The infinite system derived in § 2 have a satis- can be obtained by expansion of single three-dimen- factorily explicit and simple form to be used for fur- sional equations. ther mathematical analysis or for practical calcula- For the beginning, it is obvious that the system tions. For example, VON LAUE'S theory 6 of electron of differential equations (14) is an expanded form diffraction by monatomic layers is nothing but the of (2). The periodicity (10) shows that the three- first iterated solution of (28). In many cases, how- dimensional boundary-value problem may be con- ever, it is more convenient to have one integral fined to a column parallel to the z-axis ("column of equation in three dimensions in the form (1). This reference") (Fig. 1). Its cross-section is a unit cell is particularly true for low-energy electron diffrac- of the two-dimensional lattice, that is, a parallelo- 11 P. M. MORSE and H. FESHBACH, Methods of Theoretical where the constant ßm can be chosen arbitrarily. We ar- Physics, 2 Vols., McGraw-Hill, New York 1953. rive in this way at various forms of existing theories. This 12 It should be noted that the choice of the main part is some- will be shown in a later paper. what arbitrary. Thus, we can write 12a K. KAMBE, Z. Naturforschg. 22 a, 322 [1967]. d2 -j^ Gm(z\z') + (/V— ßm) Gm(z\z')=8(z — z'), gramm of the basis vectors dt and a.,. The top and and on the bottom surface z = ze bottom of the column are parts of the surfaces of incidence and exit respectively. y(rt,*e) - jA(rt-rt') yz(rt',ze) d/ = 0, (31b) On the side planes of the column of reference the where boundary conditions are given, in accordance with (10), by Vz(rt,z) = -^-y>(rt,z), (32) v(1*81 +Ol) =exp{« Kot-aJ y(r81), (30 a) and and A(rt- rt') =/im 4 lim2 T^T- exp {i Kmt • (rt - rt')}, T/.'(rs2 + a2) = exp {{Kot'ßo} y;(rs2), (30 b) (33) where rsl and rs2 are points on one of the pairs of where A is the area of the cross-section of the col- side planes. The points rsl + dj and I*s2 + Cl2 are umn, that is, the two-dimensional unit cell. The sur- then situated on the opposite side planes. face integral / ds' of (31a, b) The boundary conditions on the top and bottom are derived from (23 a, b). We get, by multiplying is taken for Tt over with exp {i Kmt • rt} , summing over m, writing the top and bottom surfaces respectively. xp(r) =yj(Tt, z) and applying at first formally (31 a, b) are the explicit forms of the conditions PARSEVAL'S theorem, on the top surface Z = Z[ of outgoing waves which has been stated by FUJI- WARA 4 somewhat vaguely in words. They are, how- v(rt,«i) + fA(rt-rt') y>a(rt',zi) &s' ever, at first purely formal, since we do not know (0) ( } V t i = 2 ' (r,z), whether the double series (33) converges. We obtain GREEN'S function by recombining the system (24) in a similar manner as G(r\r') = I I—~exP{irm\z-z'\ +iKmt-(rt-rt')} (34) fi m £ i i m to satisfy the Eq. (4) and the boundary conditions on the side planes, that is, if the point V lies inside the column G(rsl + ax | r) =exp {iKot-aJ G(rsl|r), (35 a) and =exp {iKot'ßo} G(rs21 (35 b) G(rs2 + a21 r') r'). Further, if the point I*! = (rtl,zj) lies inside the column, the boundary conditions on the top and bottom surfaces are G{rt,zi\ rn,zx) + f A (rt — rt') Gz (rt\ z\ | rtl, zt) d/ = 0, (36 a) G(rt,ze | rtl,zt) - J A(rt- rt') Gz(rt\ ze | rtl, d/ = 0, (36 b) where Gz = 3G(rt\ z | rtl, zt)/dz . The form (34) shows that it is really a mixed type between (3) and (5) as was expected in § 1. Thus (34) is hermitic just as (5) for the interchange of variables rt and rt', and complex symmetric just as (3) for the interchange of variables z and z . Obviously the double series (34) is absolutely and uniformly convergent if z + z , but for z = z we do not know whether it converges. We note that we have the relation A(rt-rt')=2[G(r\r')]z=z', (37) where z may be arbitrarily chosen as zero. The integral equation having the GREEN'S function (34) is obtained from the system (28) in the recom- bined form (1), where (**) is given by (9) and the range of integration is extended over the "column of reference". If we have found the solution of the integral equation the amplitudes of the outgoing waves can be cal- culated as J fexp{-iKmt-rt}G(rt,Zi\r') V(r') w{r) dr'ds, (38a) f Tm= + ^ exp { - r^ ze} J j exp { — i Kmt • rt} G (rt, ze | r) V{r )y{r) dr'ds, (38b) where the surface integral / ds is taken for rt over the cross-section of the column. It remains to investigate the convergence of the series (34) forz = z'. § 4. Green's Function in an Explicit Form To evaluate the sum of (34), in particular for the case z = z', we apply EWALD'S method 13 for summing the series (5). The method is essentially a generalization of RIEMANN'S method (WHITTAKER—WATSON 2, p. 273) for evaluating zeta functions. Its main feature is to derive the unknown sum of a series by ana- lytic continuation of another series whose sum is known to be an analytic function of a parameter. The proposed series for (34) is (see Appendix 2) its generalized form Cs(R) = 27r(f)* H%2{rm\Z\) exp {t Kmt • Rt} , (39) where R = r — r', Rt — rt, Z = z — z, s is a complex parameter, and //-«/2 j Z j) is a HANKEL = rt function. This double series is absolutely and uniformly convergent (with respect to s, Rt, and Z) if 9te(.s) >2, so that it is an analytic function of s in that domain. By means of an integral representation of HANKEL functions we obtain, on applying RIEMANN'S method, the analytic continuation of it into the domain 9ic(s) 2 , and particularly G(R) for 5 = 1. In an explicit form it can be written ooexp{i G <»> = - jhr (2I «p J ^ «p ( t (r- f •- f") I CO oo - 412 (t? I exP { -Kot • °»t> /r 4 exP {- T R + a>* I2 £ - T) 1 ' (40) 11 to where cpm = JI — 2 arg R,N , ant = nla1 + n2 tt2 stands for a pair of integers n1 and n2), and co is a com- plex number which can be chosen arbitrarily in the domain ^(co) >0, | co | < oo. These integrals can be expressed after EWALD 13 by error functions. (4) can be modified to (cf. Appendix 2) oo C(R).i2exp{,-K„,-Rt} C^'^nb/^-P\W>- ^V' 1/to oo -—(^LEXPL-iKof^T} friex p { - ± (L R + ant |21 - ^) [ dl, (41) \lto where oj > 0 and the integrals are taken on the real It can be proved directly (Appendix 3) that G(R) axis. A comparison with (34) shows that to each given by (40) or (41) satisfies the equation (4) term of (34) the second term in the square bracket and the boundary conditions (35 a) — (36 b). There- is added to make the series convergent. These terms fore it is the required GREEN'S function. are compensated by the second series. A(Rt) given by (33) can be obtained from (40) 13 P. P. EWALD, Ann. Phys. (4) 64, 253 [1921]. - 0. EMERS- according to the relation (37). We find easily that LEBEN, Phys. Z. 24, 73 [1923]. — The author is greatly in- the integrals of (31 a, b) exist if xpz is a continuous debted to Prof. Dr. K. MOLIERE for calling his attention to these papers. function of J*t'. In this way it is proved that the three-dimensional the FREDHOLM'S series is more complicated than the form of § 3 is consistent quite independent of its iterated (NEUMANN'S) series, the convergence of the derivation from the form of § 2 14. series is guaranteed for a wide range of potential The formal expressions (33) and (34) are the functions V (r). The exact solution is no doubt a limit of (40) for | co j —^ 0 (Appendix 2). Another valuable help to prove various approximation me- formal expression can be obtained (Appendix 2) in thods. the limit | co | —> oo . Thus The rapid convergence of our representation (40) is significant particularly for low-energy electron G(R) = - ? 4 n (R+ a,it) (42) diffraction, for which we can apply the cellular me- thod of the band theory of metals (e. g. KOHN—Ro- X exp {t x\R + ant J -£ Kot-Oat}. STOKER3), in virtue of the fact that the partial wave This is apparently a two-dimensional lattice of unit expansion of the single-atom scattering is rapidly sources of spherical waves. One source lies within convergent for slow electrons 12a. the column of reference at the point !*', that is, j R ] = 0. The other sources lie outside the column The author wishes to thank Prof. Dr. K. MOLIERE at the points r — dnt which are generated from r' by for his encouragement to this work. repeated lattice translations ax and d2 . 2 Finally it is to be noted that the case Fm = 0, Appendix 1. Periodicity of Solution which is excluded in the present treatment, can be taken into account by means of a modified GREEN'S If K0 is perpendicular to the surface we see at function (MORSE-FESHBACH u, p. 822). once that the solution should be two-dimensionally periodic in the whole space with the same periods § 5. Discusssion as those of the two-dimensional lattice. For, if we observe two points V and V = T + nx a1 + n2 Cl2 with The form of GREEN'S function (40) is only an arbitrarily chosen integers n1 and n2, then the geo- alternative of the expanded form (24) of TOURNA- metry of the problem is completely identical for r RIE 5 and the spectral representation (6) of FUJI- and r'. Thus the solution has the form (11) with WARA 4. Our form has, however, the advantage of ! Kot! = o. being explicit and relatively rapidly convergent If K0 is not perpendicular to the surface, then the (particularly in the case of low-energy electrons). geometry is not quite the same for T and 1*'. The The main disadvantage is that it contains somewhat only difference is, however, that the primary wave cumbersome integrals. For analytical calculations it has different phases at V and r', the phase differ- is often more convenient to use the function GS(R) ence being equal to exp {£ K0 • (n1 a1 + n2 d2) } • ^ given by (39) at first, and to put s = 1 in the result we shift the phase of the primary wave by this 14 if analytic continuation is allowed (cf. ). amount, then the geometrical condition for V is The explicit form of our GREEN'S function allows again identical to the geometrical condition for I*' us, in virtue of the finite integration range (that is, before the phase shift. As we have a time-stationary the column of reference), to derive the exact solu- problem, the phase shift of the primary wave should tion of our problem, namely, the FREDHOLM'S series. only change the phase of the solution by the same This will be shown in a succeeding paper. Although amount. Thus (10) follows. 14 The question which arises now is whether the form of § 2 panding at first the function GS(R), which is quadratically can be derived from the form of § 3. The difficulty lies in integrable for 9\C(s)>2, applying PARSEVAL'S theorem, the fact that PARSEVAL'S theorem cannot be applied directly and then deriving from the expanded expression the form in the expansion in two-dimensions, because the functions for s = l by analytic continuation. In this way the form in G(R) for Z = 0 and ^l(/?t) are not quadratically inte- § 2 proves to be really equivalent to the form in § 3. grate for the surface integral / ds (HOBSON 15, Vol. 2, p. 15 E. W. HOBSON, Functions of a Real Variable, 2 Vols., Cam- 718). We can, however, circumvent this difficulty by ex- bridge University Press 1926-1927. Appendix 2. Derivation of Green's Function The expression (34) can be modified to the series of HANKEL functions I Z\\i C(R) = 2 H We consider the generalized series (39). The power (\ Z \/rm)^s is made definite by the convention that (WHITTAKER—WATSON 2, p. 589) exp LOF - — i arg r„ (A 2) r- r and the argument of Tm takes the value 0 or n\2 according to (17 a, b). The series (39) can be easily seen to be an analytic function of 5 for all values of R = (Rt,Z) if 9ie(s) >2, because then the double series S | Tm \~s is convergent (WHITTAKER—WATSON 2, p. 51). We want to find the analytic continuation of m (39) for a domain containing s = l to obtain the sum of (A 1), that is, of (34). From an integral representation of HANKEL functions (WATSON16, p. 178) we obtain the expression ooexp{i (Fm\z\) = iV/^expfj (rmH (A3) 0 + where cpm = n-2 argTm . (A4) The lower limit 0 + indicates that the contour integral starts from 0 in the direction of the positive real axis. On putting (A3) into (39) we have 2 1 GS(R) 2 exp{iKmt-Rt} J' Cis_1exp — r m ts dC 2 n A \ 2 m 0+ 1 z Y exp {i Kmt-Rt} f P'-iexp IrjC- dC. (A 5) 2 ti A \ 2 0+ I 2 We note that the first sum has only a limited number of terms so that the order of integration and sum- mation can be inverted. We apply now the theta-transformation formula in two dimensions (KRAZER and PRYM 17, p. 88; KRAZER 18, p. 108; EPSTEIN19, p. 624) which is valid for 3le(0 >0 2 exp ;o I Kot 1+ "m Bmti ; !2 '+ " iv--—u (Kout '+ —Bmimt/) —"• Ri t j =o 2 exp j - y1, \Rt + aR Kot' flnt (A 6) 2. ) z 71 t, n where a,lt = nt ax + n2 do {n stands for the pair of indices nt and n2). From this formula we find (in a similar manner as described in WHITTAKER—WATSON 2, p. 273) that the order of integration and summation of the second sum of (A 5) can be inverted if (5) >4 and if always >0 on the contour. After the inversion we split the integrals in two parts by taking a point co somewhere in the domain 9ie(a>) >0, ! to j < 00 (WHITTAKER—WATSON 2: co=l). On applying (A 6) for the parts of the integrals from 0 to OJ, and putting into this part 1 /£ instead of C, we obtain 7V>0 1 GS(R) = - 1 2~7t~A J t^- 2 exp 2 (rj, + i Kmt' Rt j d; rm' G.N.WATSON, Theory of BESSEL Functions, Cambridge Uni- 18 A. KRAZER, Lehrbuch der Thetafunktionen, Teubner, Leip- versity Press 1944 (2ND ed.). zig 1903 (referred to by EWALD 13). A. KRAZER and F. PRYM, Neue Grundlagen einer Theorie 19 P. EPSTEIN, Math. Ann. 56, 615 [1903], der allgemeinen Thetafunktionen, Teubner, Leipzig 1892 (referred to by EPSTEIN 19). This expression can be proved (again in a similar to + oo and (A 9) from the part co to 0. It follows manner as WHITTAKER—WATSON 2, p. 273) to be an then (41). analytic function of s for all values of s if R + 0. To derive the form (42) we assume, just as It represents the analytic continuation of (39) for EWALD 13 did, that x has a small positive imaginary 3ie(s) ^2. On putting 5 = 1 and inverting again part. We put then in the expression (40) the order of summation and integration (since it OJ oc exp {i cp) , can be proved to be allowed) we obtain (40). The where restriction 3te(s)>0 can then be removed. arg^<9?< . (A 10) Then the contours of the first integrals can be re- presented by segments of an infinitely large circle from oc exp {199} to oc exp {i cpm} . By applying JORDAN'S lemma one can prove that these integrals vanish. The second integrals of (40) can be trans- formed to the form (EWALD 13) exp {i x j R + a,it | } V2rt (All) ri>o IR + o«t I so that (42) follows. 0 u) Appendix 3. rm<0 Proof of the Properties of Green's Function Fig. 2. The paths of integrals for Eq. (41). If Z = z — z 4= 0 the termwise differentiation of We take OJ on the real axis (O>>0) and the con- (34) is allowed, and it follows at once that G(R) tours shown in Fig. 2 for the integrals of (40). We satisfies consider at first the case JH,W2> 0. On applying JOR- 2 DAN'S lemma (WHITTAKER-WATSON 2, p. 115) to the V|G(R) +x G(R) =0. (A 12) infinitely large and infinitely small quadrants of If Z = 0 and 3ie(s)>2 we find that the termwise circles we find that the integrals on these vanish differentiation of (39) is allowed. On applying the (WATSON16, p. 180). We obtain from the part 0 to properties of BESSEL functions (WATSON 16, p. 74) i oc on the imaginary axis we obtain, if R | =f= 0 , exp {i rm\Z\} - V2.1 (A 8) GS(R) = -x2 GS(R) - — 1)GS_2(R). (A13) i T m VI (5 The right-hand side is an analytic function of s, so and from the part OJ to 0 on the real axis that (A 12) follows for s = 1 if R | 4= 0. oo The expression (A 7) shows that G6(R) is a — j" t~3/l exp f(iZ|»«- rf')U. (A9) smooth function of R if R =f= 0. In the neighbour- I/O) hood of R = 0 the term with ant | = 0 in the last In the case 1 m~ Thus G(R) -> d t !«-> 0 4«. VI/ /r*exp(-J-(!Rp( I/o, \R]'-/2L 4 .T I r r This equation shows, combined with (A 12), valid for R + 0, that G(R) satisfies (4) (MORSE-FESH- BACH11, p. 809). The function G(R) satisfies the boundary conditions on the side planes (35 a, b) as it can be seen directly from (40). The boundary conditions (36 a, b) are also satisfied. This can be proved as follows. For example, for (36 a) we consider at first the integral /s = G„(rt',2i | rn,zx) d/, (A15) 2jGs(rt,0|rt\0) where Gsz=dGs/dz. If 9ie(s)>2 we obtain from the expression (39), using the known properties of HANKEL functions, /, = - .A 21*-2 exp { i sf J r (-f) 2 ! H-\s+i (rm \Z[-Zl !) exp {i Kmt• (rt - rtl) } . (A 16) Since it is assumed that Z[ 4= zx this expression is an analytic function of s except at the poles of T 5). Hence putting s = 1 we have = 2 J G(rt, 0 J rt', 0) Gz(rt', Zi\rtl,Zl) ds = - G(rt, Zi j rtl, zx) . (A17) This proves (36 a). (36 b) can be proved in the same way. Gravitationsinstabilitäten eines Plasmas bei differentieller Rotation R. EBERT Institut für Theoretische Physik der Universität Frankfurt a. M. (Z. Naturforschg. 22 a, 431—437 [1967] ; eingegangen am 14. Oktober 1966) Herrn Prof. Dr. L. BIERMANN zum 60. Geburtstag gewidmet In this paper an instability calculation is given for an axially symmetric gas distribution which has a differential rotation and in which a magnetic field is present. It is a generalization of similar calculations given by CHANDRASEKHAR and BEL and SCHATZMAN. The generalization becomes necessary for the study of problems of the formation of planetary systems, and star formation. The instability conditions and the critical wave lengths are calculated for plane-wave-like dis- turbances. For disturbances running perpendicularly to the axis of rotation instability can occur only if the gas density exceeds a critical value which depends on the differential rotation at the considered distance only as long as pressure gradients and gradients of the magnetic field strength are negligible. If the gas density exceeds this critical value the shortest unstable wave length is proportional to the square root of vt2+vj$2, where i/p means the velocity of sound and t>B the ALFVEN-velocity. For disturbances running parallel to the axis of rotation in addition to the JEANS instability a new type of instability occurs due to the simultaneous action of the magnetic field and the differen- tial rotation; for rigid rotation this instability vanishes. In vielen astrophysikalischen Untersuchungen SEKHAR und FERMI 2, FRICKE 3, CHANDRASEKHAR 4, BEL spielt die Frage nach den Gravitationsinstabilitäten und SCHATZMAN 5, SAFRONOV 6 und GLIDDON 7 ausge- eines unter seiner Eigengravitation stehenden Gases führt worden. Die hier durchgeführte ist eine Ver- eine wichtige Rolle. JEANS 1 führte als erster In- allgemeinerung der von CHANDRASEKHAR 4 und BEL stabilitätsrechnungen dieser Art aus. Erweiterungen und SCHATZMAN 5. Sie geht aus neueren Untersuchun- auf kompliziertere Gaskonfigurationen ohne und mit gen zur Entstehung von Sternen und Planetensyste- Magnetfeld sind in unserer Zeit u. a. von CHANDRA- men hervor 8. 1 J. H. JEANS, Phil. Trans. Roy. Soc. London 199, 1 [1902]. 6 V. SAFRONOV, Ann. Astrophys. 23, 979 [I960]. 2 S. CHANDRASEKHAR U. E. FERMI, Astrophys. J. 118, 116 [1953]. 7 J. E. C. GLIDDON, Astrophys. J. 145, 583 [1966], 3 W. FRICKE, Astrophys. J. 120, 356 [1954]. 8 R. EBERT, Zur Theorie der Entstehung von Planetensyste- 4 S. CHANDRASEKHAR, Vistas in Astronomy, Pergamon Press, men, Habilitationsschrift, Universität Frankfurt a. M. 1964; London 1955, p. 344. Magnetohydrodynamical Model of the Formation of Pla- 5 N. BEL U. E. SCHATZMAN, Rev. Mod. Phys. 30. 1015 [1958]. netary Systems, Astrophys. J., in Vorbereitung.