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Theory of Electron Diffraction by Crystals I. G R E E N ' S Function and Integral Equation

K Y O Z A B U R O K A M B E

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 by crystals is B E T H E ' S dynamical theory1 . In this theory the wave field inside the crystal is represent-ed by a superposition of B L O C H waves and continued into the vacuum by a superposition of plane waves. The main problem is to solve the eigenvalue prob-lem for the wave vector (propagation vector) of B L O C H waves. In one dimension this is represented by H I L L ' S determinantal equation ( W H I T T A K E R —

W A T S O N 2 , p. 4 1 5 ) . In three dimensions, however, the solubility of the corresponding determinantal equation is somewhat problematic. In contrast to the one-dimensional case, we cannot secure the absolute convergence of the infinite determinant. In fact, B E T H E argues that the problem is not exactly soluble (1. c . 1 , p. 73, § 4) and confines himself to approxi-mate solutions.

Apart from this difficulty in fundamental prin-ciple, B E T H E ' S assumption of geometrically plane surfaces will not be valid especially in practical application to low-energy electron diffraction.

In the present paper we propose an alternative formulation of the problem which appears to be more fundamental and more widely applicable than B E T H E ' S . We consider, as B E T H E , non-relativistic elastic scattering and an infinitely extended crystal slab, but assume that the potential is periodic only in two dimensions parallel to the surface. The varia-tion of the potential in the third dimension is at first almost arbitrary. This assumption allows us to apply the theory to monatomic layers and to crystals with regularly distributed adsorbed atoms or steps on their surface. These cases are important for low-energy electron diffraction.

* Abteilung Piof. Dr. K. MOLIERE. 1 H. BETHE, Ann. Phys. (4) 87, 55 [1928].

We formulate the problem at first as a boundary-value problem for the S C H R Ö D I N G E R equation, but go over at once to the equivalent integral equation. The integral equation is convenient for mathematical analysis and renders the exact solution of the prob-lem. The integral equation is also a convenient tool for investigating various existing theories of elec-tron diffraction (including B E T H E ' S theory) and provides an unifying view over these theories.

§ 1. General Survey of Green's Functions

Before going into our problem let us consider at first some related problems and corresponding G R E E N ' S functions to clarify our point of view.

The most general form of the integral equation for potential scattering may be

v(r) =v ( 0 )(r) + fG(r\r') V(r') ip(r') dr', (1) where ip{r) is a solution of S C H R Ö D I N G E R ' S equation

V 2 v + ( * 2 _ F ( r ) ) v ; = 0 5 ( 2 )

I / / ° ) (r ) is the incident wave, and G R E E N ' S function

c ( r i i ' ) - - (3) is constructed to satisfy

\7;G + x2 G= +d(r-r') ( 4 )

and the condition of outgoing waves. This G R E E N ' S

function is valid only for scattering objects which are finite in three dimensions. Therefore, it is ques-tionable to use this G R E E N ' S function in our problem of infinitely extended slabs.

2 E. T. WHITTAKER and G. N. WATSON, A Course of Modern Analysis, Cambridge University Press 1927 (2nd ed.).

Another extreme case of crystals infinite in three dimensions appears in the theory of B L O C H functions (for example, K O H N and R O S T O K E R 3 ) . G R E E N ' S func-tion for the integral equation of B L O C H function is

G(r | r ' ) = ~ Z exp {i(k+ Bg) • <r-r') } *2-(* + B(Jy- (5)

where T is the volume of the unit cell, k is the pro-pagation vector, and B g are the reciprocal lattice vectors. This G R E E N ' S function is constructed to sa-tisfy (4) and to have the periodic property of B L O C H

functions. Owing to this periodicity the range of integration in the integral equation [which has the form (1) with yjW = 0] is confined to one unit cell or to one W I G N E R - S E I T Z cell.

In the present problem with two-dimensionally in-finite crystals the geometry requires that the solu-tion should have a periodic property like a B L O C H

function in two directions parallel to the surfaces (cf. § 2 ) . In the third direction perpendicular to the surfaces the solution should have the property of outgoing waves as required for (3) . Thus, it is clear that the G R E E N ' S function should have a mixed type between the above two forms (3) and (5 ) .

F U J I W A R A 4 who investigated the same problem as ours, used the spectral representation for G R E E N ' S

function

G(r V) = [2 a)- / exp {ik• ( i — r ' ) } * 2 dfc, (6)

and put it in the iterative solution of the integral equation (1) . This G R E E N ' 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 F U J I -

W A R A ' S treatment appears to be satisfactory so far as the iterative solution is concerned, he could not prove its convergence. His solution appears hence to be a formal one. In any case his result is identical with the iterative solution of T O U R N A R I E ' S system of integral equations 5, which is discussed in § 2, and consequently should be identical also with our exact solution if the iteration is convergent.

§ 2. Expansion in Two Dimensions

The two-dimensional periodicity of the potential allows us to expand the solution in a kind of FOU-RIER series. This process has been applied already in several cases (von L A U E 6 , A T T R E E and P L A S K E T T 7 ,

T O U R N A R I E 5 , H I R A B A Y A S H I and T A K E I S H I 8). Although we would prefer to work with the three-dimensional form (1) , it seems to be convenient to start with the two-dimensional expansion because we can write down relatively easily an explicit form of the bound-ary conditions and corresponding G R E E N ' S function.

In the S C H R Ö D I N G E R equation (2) the "potential" V(r) is now periodic in two dimensions

V(r + n1a1 + n2a2)=V(r), (7)

where and d 2 are the two-dimensional basis vec-tors, and nt and n2 are arbitrary integers. In the third dimension, for which we take the z-axis per-pendicular to CLX and do , the non-zero region of V(r) is assumed to be confined to a finite range z j < z < z e . Thus

V(r) = 0 if or (8)

2

Fig. 1. The column of reference.

This restriction is introduced only to simplify the mathematics, and can be removed afterwards by the process of shifting z\ and ze to — oo and + oo respectively.

3 W . KOHN a n d N . ROSTOKER, P h y s . R e v . 9 4 , 1 1 1 1 [ 1 9 5 4 ] , 4 K. FUJIWARA, J. Phys. Soc. Japan 14, 1513 [1959]. 5 N.TOURNARIE, J. Phys. Soc. Japan 17, Suppl. B-II, 98 [1962]. 6 M. VON LAUE, Phys. Rev. 37, 53 [1931].

7 R . W . ATTREE a n d J . S . PLASKETT, P h i l . M a g . ( 8 ) 1 . 8 8 5 [1956].

S K. HIRABAYASHI and Y.TAKEISHI, Surface Sei. 4, 150 [1966].

We call the planes z = z; and 2 = ze the surfaces, the space z ; < z < z e the crystal, the spaces z < z ; and z > z e the vacuum.

The potential V (T) needs not necessarily be peri-odic as a function of z in the crystal. If V(T) should represent a perfect crystal, however, the above as-sumption requires nothing but that the surfaces are parallel to one of the netplanes of the crystal. The vectors a t and d 2 can be properly chosen in this netplane and may not be identical to the conventio-nal basis vectors of the crystal itself.

We assume that a plane-parallel electron wave falls on the crystal from the "upper" vacuum z < z j with the wave vector K 0 . We call it the primary wave and write

V<°>(r) = e x P { £ K 0 T } , (9)

where K0 has a positive z-component. The surfaces z = zj and z = ze are called then the surface of inci-dence and exit respectively.

It is shown in Appendix 1 that the solution xp has the periodicity like a B L O C H function in two dimen-sions

^ ( r + ^ O i + naOa) = exp {i K0 • (/i! «! + n2 d2)} xp{r). 1 1

We can put therefore

xp(r) = exp { i K o t - r t } yy>m(z) exp { * ' B w t - r t } , (11) m

where Kot and r t are the components of K0 and V tangential to the surfaces. Bmt are given by

Bmt = m1Blt + m2B2t, (12)

where the index m stands for the pair of integers m1

and m2 . Blt and Bot are the Jico-dimensional reci-procal basis vectors which are determined from d x

and a 2 to satisfy

arBjt = 27i3ij (i,/ = 1 , 2 ) . (13)

B l t and Bo t may not be identical with the conven-tional basis vectors of the three-dimensional reci-procal lattice.

Putting ( 1 1 ) into the S C H R Ö D I N G E R equation ( 2 )

we obtain

4 t Vm (2) + r ,n Wm («) ~ Z Vm - n U) Wn U) = 0 , ( 1 4 )

where T m 2 is given by

r , K |2 ml i 5 (15)

9 Cf. for example M. BORN and K.WOLF, Principles of Optics, Pergamon Press, London 1959, p. 560. — J. A. RATCLIFFE, Rep. Progr. Phys. 19. 188 [1956].

where Kmt = Kot + Bmt. (16)

For the later purposes Tm is made unique by the convention

Tm = + y*2 - ! Kmt !2 if * 2 > | Kmt I2, (17 a) and

rm = + « V\ Kmt I2 - if X2 < I Kmt I2 (17 b)

We assume at first

x 4= I Kmt I for all m . (18)

Vm(z) in (14) is the two-dimensional expansion coefficient of the potential V (r) according to the periodicity ( 7 ) . Thus

F ( r ) = ZVm(z)exp{iBmt-rt}. (19) m

In the vacuum z < z; and z > ze we have

W + * 2 V = 0 . (20)

Therefore we can write for z < z\

V'm(z) = Xpm exp {irm z) + Rm exp { - i Fm z } , (21a)

and for z > z e

Vm(z) = Tm exp {i rm z) + ^ ^ e x p { - i Fm z } . (21 b) XFm , Rm, Tm , and are constants and can be in-terpreted, if r .m is real, as the amplitudes of plane waves going upwards or downwards, and, if T m

is imaginary, as the amplitudes of "evanescent waves" 9 which increase or decrease exponentially. The condition of outgoing waves can be adapted to the present problem in the form

m = (22 a) and

0 m = O for all m . (22 b)

From the continuity of ip and dy/dz on the surfaces it is found at once that the solution xp inside the crystal must satisfy the boundary conditions

Vm + • > d ! f W ) = 2 <5omi dornt exp { i r o Zi} lim /z=zi (23 a)

and 1

i r„ drpjr

d z = 0 for all TO. (23 b)

This is an explicit form of the condition of outgoing waves ( S O M M E R F E L D 10, p. 179 ) .

10 A. SOMMERFELD, Vorlesungen über theoretische Physik, Bd. 6, Partielle Differentialgleichungen der Physik, Akadem. Verlagsges. Leipzig 1958 (4. Aufl.).

Thus we have a boundary value problem for an These G R E E N ' 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)

and sponding to (23a, b ) , that is, if z j < z ' < , z e ,

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-a l a, b ) .

In analogy to the one-dimensional problem (SOM-M E R F E L D 1 0 , p. 180, M O R S E - F E S H B A C H 1 1 , p. 1071) it Gm{z\\z') + Yr~ dz Gm{z\z ) I = 0 , is easy to find the system of G R E E N ' S functions ( " G R E E N matrix" of T O U R N A R I E 5 ) and

Gm (z I z) =-- Yfr~ e x p RM\z-z'\} • ( 2 4 ) Gm (ze I z) - T^— — Gm (z | z) = 0.

The system of integral equations is ZE

Wrn (2) = \ m i <5o in 2 exp {i roz} +fGm(z\z')£V m _ n (z) V« (2') d z,

(26 a)

(26 b)

(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 z If we have obtained the solution of this system we can calculate 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, V O N L A U E ' S theory 6 of electron of differential equations ( 1 4 ) 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 Physics, 2 Vols., McGraw-Hill, New York 1953.

12 It should be noted that the choice of the main part is some-what arbitrary. Thus, we can write

d2 - j ^ Gm(z\z') + ( / V — ßm) Gm(z\z')=8(z — z'),

where the constant ßm can be chosen arbitrarily. We ar-rive in this way at various forms of existing theories. This will be shown in a later paper.

12a K. KAMBE, Z. Naturforschg. 22 a, 322 [1967].

gramm of the basis vectors d t and a.,. The top and bottom of the column are parts of the surfaces of incidence and exit respectively.

On the side planes of the column of reference the boundary conditions are given, in accordance with (10) , by

v(1*81 + O l ) = e x p { « K o t - a J y ( r 8 1 ) , (30 a) and

T/.'(rs2 + a 2 ) = exp { { K o t ' ß o } y;(rs2), (30 b)

where r s l and rs2 are points on one of the pairs of side planes. The points r s l + d j and I*s2 + Cl2 are then situated on the opposite side planes.

The boundary conditions on the top and bottom are derived from (23 a, b ) . We get, by multiplying with exp {i Kmt • r t } , summing over m, writing xp(r) =yj(Tt, z) and applying at first formally P A R S E V A L ' S theorem, on the top surface Z = Z[

v ( r t , « i ) + fA(rt-rt') y>a(rt',zi) &s' = 2 V ' ( 0 ) (r t ,z i ), ( }

and on the bottom surface z = ze

y ( r t , * e ) - jA(rt-rt') yz(rt',ze) d / = 0 , ( 31b )

where

Vz(rt,z) = -^-y>(rt,z), (32)

and

A(rt- rt') = 4 2 T̂T- exp {i Kmt • (rt - r t')}, /im lim (33)

where A is the area of the cross-section of the col-umn, that is, the two-dimensional unit cell. The sur-face integral / ds' of (31a, b)

is taken for Tt over the top and bottom surfaces respectively.

(31 a, b) are the explicit forms of the conditions of outgoing waves which has been stated by F U J I -

W A R A 4 somewhat vaguely in words. They are, how-ever, at first purely formal, since we do not know whether the double series (33) converges.

We obtain G R E E N ' S function by recombining the system ( 2 4 ) 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 + a x | r) =exp {iKot-aJ G(r s l |r) , (35 a)

and G(rs2 + a21 r') =exp { i K o t ' ß o } G(rs21 r'). (35 b)

Further, if the point I*! = ( r t l , z j ) lies inside the column, the boundary conditions on the top and bottom surfaces are

G{rt,zi\ rn,zx) + f A (rt — rt') Gz ( r t \ z\ | r t l , zt) d / = 0 , (36 a)

G(rt,ze | rtl,zt) - J A(rt- r t ' ) Gz(rt\ ze | r t l , d / = 0 , (36 b)

where Gz = 3G(rt\ z | r t l , 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 r t and r t ' , 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', ( 3 7 )

where z may be arbitrarily chosen as zero. The integral equation having the G R E E N ' S function ( 3 4 ) is obtained from the system ( 2 8 ) 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 ) d r ' d s , ( 3 8 a )

Tm= + ^ exp { - r ^ z e } J j exp { — i Kmt • rt} G ( r t , ze | r) V{rf)y{r) d r ' d s , ( 3 8 b )

where the surface integral / ds is taken for r t over the cross-section of the column. It remains to investigate the convergence of the series (34) f o r z = z'.

§ 4. Green's Function in an Explicit Form

To evaluate the sum of ( 3 4 ) , in particular for the case z = z', we apply E W A L D ' S method 13 for summing the series ( 5 ) . The method is essentially a generalization of R I E M A N N ' S method ( W H I T T A K E R — W A T S O N 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

C s ( R ) = 2 7 r ( f ) * H%2{rm\Z\) exp {t Kmt • R t } , (39)

where R = r — r', Rt = r t — r t , Z = z — z, s is a complex parameter, and / / - « / 2 j Z j) is a H A N K E L

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 H A N K E L functions we obtain, on applying R I E M A N N ' 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<j jm }

G <»> = - j h r ( 2 I « p J ^ « p ( t ( r - f •- f " ) I CO

oo

- 41 2 ( t ? I e x P { - K o t • °»t> / r 4 e x P { - T R + a>* I2 £ - T ) 1 ' ( 4 0 ) 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 ^ ( c o ) > 0 , | co | < oo. These integrals can be expressed after E W A L D 13 by error functions.

(4) can be modified to (cf. Appendix 2) oo

C ( R ) . i 2 e x p { , - K „ , - R t } C ^ ' ^ n b / ^ - P \ W > - ^ V ' 1/to

oo

- — ( ^ L E X P L - i K o f ^ T } friex p { - ± (L R + ant |21 - ^ ) [ d l , ( 4 1 ) \lto

where oj > 0 and the integrals are taken on the real axis. A comparison with (34) shows that to each term of (34) the second term in the square bracket is added to make the series convergent. These terms are compensated by the second series.

1 3 P . P . EWALD, A n n . P h y s . ( 4 ) 6 4 , 2 5 3 [ 1 9 2 1 ] . - 0 . EMERS-LEBEN, P h y s . Z . 2 4 , 7 3 [ 1 9 2 3 ] . — T h e a u t h o r is g r e a t l y in-debted to Prof. Dr. K. MOLIERE for calling his attention to these papers.

It can be proved directly (Appendix 3) that G ( R ) given by (40) or (41) satisfies the equation (4) and the boundary conditions (35 a) — (36 b). There-fore it is the required G R E E N ' S function.

A(Rt) given by (33) can be obtained from (40) according to the relation (37 ) . We find easily that the integrals of (31 a, b) exist if xpz is a continuous function of J*t'.

In this way it is proved that the three-dimensional form of § 3 is consistent quite independent of its derivation from the form of § 2 14.

The formal expressions (33) and (34) are the limit of (40) for | co j —^ 0 (Appendix 2 ) . Another formal expression can be obtained (Appendix 2) in the limit | co | —> oo . Thus

G ( R ) = - ? 4 n ( R + a,it) (42)

X exp {t x\R + a n t J - £ K o t -Oat } .

This is apparently a two-dimensional lattice of unit sources of spherical waves. One source lies within the column of reference at the point !*', that is, j R ] = 0. The other sources lie outside the column at the points r — d n t which are generated from r ' by repeated lattice translations a x and d 2 .

Finally it is to be noted that the case F m 2 = 0, which is excluded in the present treatment, can be taken into account by means of a modified G R E E N ' S

function ( M O R S E - F E S H B A C H u , p. 8 2 2 ) .

§ 5. Discusssion

The form of G R E E N ' S function ( 4 0 ) is only an alternative of the expanded form ( 2 4 ) of T O U R N A -

RIE 5 and the spectral representation (6) of F U J I -

W A R A 4. Our form has, however, the advantage of being explicit and relatively rapidly convergent (particularly in the case of low-energy electrons). The main disadvantage is that it contains somewhat cumbersome integrals. For analytical calculations it is often more convenient to use the function GS(R) given by (39) at first, and to put s = 1 in the result if analytic continuation is allowed (cf. 1 4 ) .

The explicit form of our G R E E N ' S function allows us, in virtue of the finite integration range (that is, the column of reference), to derive the exact solu-tion of our problem, namely, the F R E D H O L M ' S series. This will be shown in a succeeding paper. Although

14 The question which arises now is whether the form of § 2 can be derived from the form of § 3. The difficulty lies in the fact that PARSEVAL'S theorem cannot be applied directly in the expansion in two-dimensions, because the functions G(R) for Z = 0 and ^l ( / ? t ) are not quadratically inte-g r a t e for the surface integral / ds (HOBSON 15, Vol. 2, p. 718). We can, however, circumvent this difficulty by ex-

the F R E D H O L M ' S series is more complicated than the iterated ( N E U M A N N ' S ) series, the convergence of the series is guaranteed for a wide range of potential functions V (r). The exact solution is no doubt a valuable help to prove various approximation me-thods.

The rapid convergence of our representation (40) is significant particularly for low-energy electron diffraction, for which we can apply the cellular me-thod of the band theory of metals (e. g. KOHN—Ro-STOKER3), in virtue of the fact that the partial wave expansion of the single-atom scattering is rapidly convergent for slow electrons 12a.

The author wishes to thank Prof. Dr. K . MOLIERE for his encouragement to this work.

Appendix 1. Periodicity of Solution

If K0 is perpendicular to the surface we see at once that the solution should be two-dimensionally periodic in the whole space with the same periods as those of the two-dimensional lattice. For, if we observe two points V and V = T + nx a1 + n2 Cl2 with arbitrarily chosen integers n1 and n2, then the geo-metry of the problem is completely identical for r and r ' . Thus the solution has the form (11) with ! Kot! = o.

If K 0 is not perpendicular to the surface, then the geometry is not quite the same for T and 1*'. The only difference is, however, that the primary wave has different phases at V and r ' , the phase differ-ence being equal to exp {£ K 0 • (n1 a1 + n2 d2) } • ^ we shift the phase of the primary wave by this amount, then the geometrical condition for V is again identical to the geometrical condition for I*' before the phase shift. As we have a time-stationary problem, the phase shift of the primary wave should only change the phase of the solution by the same amount. Thus (10) follows.

panding at first the function GS(R), which is quadratically integrable for 9 \ C ( s ) > 2 , applying PARSEVAL'S theorem, and then deriving from the expanded expression the form for s = l by analytic continuation. In this way the form in § 2 proves to be really equivalent to the form in § 3.

15 E. W. HOBSON, Functions of a Real Variable, 2 Vols., Cam-bridge University Press 1 9 2 6 - 1 9 2 7 .

Appendix 2. Derivation of Green's Function

The expression (34) can be modified to the series of H A N K E L functions

C ( R ) = 2iA 2 I Z\\i A H<lh(rm\Z\) exp {iKmt-Rt}. (A l )

We consider the generalized series (39) . The power (\ Z \/rm)^s is made definite by the convention that ( W H I T T A K E R — W A T S O N 2 , p . 5 8 9 )

r -exp LOF r - — i arg r„ (A 2)

and the argument of T m 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 ( W H I T T A K E R — W A T S O N 2, p. 5 1 ) . 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 H A N K E L functions ( W A T S O N 1 6 , p. 178) we obtain the expression ooexp{i<pm}

(Fm\z\) = i V / ^ e x p f j (rmH ( A 3 )

0 + where cpm = n-2 a r g T m . ( A 4 ) The lower limit 0 + indicates that the contour integral starts from 0 in the direction of the positive real axis. On putting ( A 3 ) into (39) we have

GS(R) 2 n A \ 2

2 ti A \ 2

2 e x p { i K m t - R t } J' C i s _ 1 e x p — m 0+ 1 z

r 2t 1 m s

Y exp {i Kmt-Rt} f P'-iexp I r j C -

dC

dC. (A 5) 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 ( K R A Z E R and P R Y M 17, p. 88 ; K R A Z E R 1 8 ,

p. 108; EPSTEIN19, p. 624) which is valid for 3 le (0 > 0

2 exp ; I Kot + Bmt !2 + i (Kot + Bmt) • R t j = 2 exp j - y 1 , \Rt + aR Kot' flnt (A 6) o 1 "mi ; ' " v--—uu ' —mi/ —"i o 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 W H I T T A K E R — W A T S O N 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 < 0 0 ( W H I T T A K E R — W A T S O N 2 : c o = l ) . On applying (A 6) for the parts of the integrals from 0 to O J ,

and putting into this part 1 /£ instead of C, we obtain

GS(R) = -1

2~7t~A

7V>0 J t ^ - 1 2 exp ( r j ,

+ J t ^ 1 CO

Xe (C)>0

2

rm '<o 2 exp 1 (r 2

2 v

+ i Kmt' Rt j d;

z2

K m t ' Rt 1 d ,

4 TT2 \ 2

\ 1 ' °° 1 1 / f 3 J C~i> 2 exp - (i R + ant

1/ 3?e(C)>0

-I- -iKot-a„t 1 ds ( A 7 )

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 a n d F . PRYM, N e u e G r u n d l a g e n e i n e r T h e o r i e 1 9 P . EPSTEIN, M a t h . A n n . 5 6 , 6 1 5 [ 1 9 0 3 ] , der allgemeinen Thetafunktionen, Teubner, Leipzig 1892 ( r e f e r r e d to b y EPSTEIN 1 9 ) .

This expression can be proved (again in a similar manner as W H I T T A K E R — W A T S O N 2, p. 273) to be an analytic function of s for all values of s if R + 0. It represents the analytic continuation of (39) for 3ie(s) ^ 2 . On putting 5 = 1 and inverting again the order of summation and integration (since it can be proved to be allowed) we obtain (40 ) . The restriction 3 t e ( s ) > 0 can then be removed.

ri>o

0 u)

r m < 0

Fig. 2. The paths of integrals for Eq. (41).

We take OJ on the real axis (O>>0) and the con-tours shown in Fig. 2 for the integrals of ( 40 ) . We consider at first the case J H , W 2 > 0 . On applying J O R -

D A N ' S lemma ( W H I T T A K E R - W A T S O N 2, p. 1 1 5 ) to the infinitely large and infinitely small quadrants of circles we find that the integrals on these vanish (WATSON16, p. 180 ) . We obtain from the part 0 to i oc on the imaginary axis

- V2.1 exp {i rm\Z\}

i T m (A 8)

and from the part OJ to 0 on the real axis oo

— j" t~3/l exp I/O)

In the case 1 m~ <C 0 we obtain (A 8) from the part 0

f ( i Z | » « - r f ' ) U . ( A 9 )

to + oo and (A 9) from the part co to 0. It follows then ( 4 1 ) .

To derive the form (42) we assume, just as E W A L D 13 did, that x has a small positive imaginary part. We put then in the expression (40)

OJ oc exp { i cp) , where

a r g ^ < 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 J O R D A N ' S lemma one can prove that these integrals vanish. The second integrals of (40) can be trans-formed to the form ( E W A L D 1 3 )

exp {i x j R + a,it | } IR + o«t I

V2rt

so that (42) follows.

( A l l )

Appendix 3.

Proof of the Properties of Green's Function

If Z = z — z 4= 0 the termwise differentiation of (34) is allowed, and it follows at once that G ( R ) satisfies

V | G ( R ) +x2G(R) = 0 . (A 12)

If Z = 0 and 3 i e ( s ) > 2 we find that the termwise differentiation of (39) is allowed. On applying the properties of B E S S E L functions ( W A T S O N 16, p. 74) we obtain, if R | =f= 0 ,

V I G S (R) = -x2 GS(R) - (5 — 1 ) G S _ 2 ( R ) . ( A 1 3 )

The right-hand side is an analytic function of s, so that (A 12) follows for s = 1 if R | 4= 0.

The expression (A 7) shows that G 6 ( R ) is a smooth function of R if R =f= 0. In the neighbour-hood of R = 0 the term with ant | = 0 in the last series of (A 7) becomes predominant if 5 = 1.

Thus

G(R) -> ! « - > 0 4 « . V I / / r * e x p ( - J - ( ! R p (

I/o,

d t

\R]'-/2L

4 .T I r r

This equat ion s h o w s , c o m b i n e d with ( A 1 2 ) , va l id f o r R + 0 , that G(R) satisfies ( 4 ) ( M O R S E - F E S H -

B A C H 1 1 , p . 8 0 9 ) .

T h e f u n c t i o n G(R) satisfies the b o u n d a r y c o n d i t i o n s o n the side planes ( 3 5 a, b ) as it can be seen direct ly f r o m ( 4 0 ) .

T h e b o u n d a r y c o n d i t i o n s ( 3 6 a, b ) are a lso satisf ied. Th i s can b e p r o v e d as f o l l o w s . F o r e x a m p l e , f o r ( 3 6 a ) we c o n s i d e r at first the integral

/ s = 2jG s ( r t ,0 | r t\0) G „ ( r t ' , 2 i | rn,zx) d / , ( A 1 5 )

where Gsz=dGs/dz. If 9 i e ( s ) > 2 we ob ta in f r o m the express ion ( 3 9 ) , us ing the k n o w n proper t i es o f

H A N K E L func t i ons ,

/ , = - . 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 e x p r e s s i o n is an analyt ic func t i on of s except at the po l es o f T 5 ) . H e n c e putt ing s = 1 we have

= 2 J G(r t , 0 J rt', 0) Gz(rt', Zi\rtl,Zl) ds = - G(rt, Zi j r t l , zx) . (A17) This p r o v e s ( 3 6 a ) . ( 3 6 b ) can b e p r o v e d in the same w a y .

Gravitationsinstabilitäten eines Plasmas bei differentieller Rotation R . E B E R T

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-veloc ity .

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 astrophys ikal i schen Untersuchungen spielt die F r a g e nach den Gravitat ionsinstabi l i täten eines unter seiner E igengrav i ta t i on s tehenden Gases eine wicht ige R o l l e . J E A N S 1 führ te als erster In-stabi l i tätsrechnungen dieser A r t aus. E r w e i t e r u n g e n auf kompl iz ier tere G a s k o n f i g u r a t i o n e n o h n e u n d mit M a g n e t f e l d s ind in unserer Zei t u. a. v o n C H A N D R A -

1 J. H. JEANS, Phil. Trans. Roy. Soc. London 199, 1 [1902]. 2 S . CHANDRASEKHAR U. E . FERMI, A s t r o p h y s . J . 1 1 8 , 1 1 6 [ 1 9 5 3 ] . 3 W. FRICKE, Astrophys. J. 120, 356 [1954]. 4 S. CHANDRASEKHAR, Vistas in Astronomy, Pergamon Press,

London 1955, p. 344. 5 N. BEL U. E. SCHATZMAN, Rev. Mod. Phys. 30. 1015 [1958].

SEKHAR u n d F E R M I 2 , F R I C K E 3 , C H A N D R A S E K H A R 4 , B E L

u n d S C H A T Z M A N 5 , S A F R O N O V 6 u n d G L I D D O N 7 ausge-führt w o r d e n . D i e hier durchge führte ist eine V e r -a l lgemeinerung der v o n C H A N D R A S E K H A R 4 u n d B E L

und S C H A T Z M A N 5 . Sie geht aus neueren Untersuchun-gen zur Entstehung v o n Sternen u n d Planetensyste-men h e r v o r 8 .

6 V. SAFRONOV, Ann. Astrophys. 23, 979 [I960]. 7 J. E. C. GLIDDON, Astrophys. J. 145, 583 [1966], 8 R. EBERT, Zur Theorie der Entstehung von Planetensyste-

men, Habilitationsschrift, Universität Frankfurt a. M. 1964; Magnetohydrodynamical Model of the Formation of Pla-netary Systems, Astrophys. J., in Vorbereitung.