N-point functions and low-energy scattering by N centres

Vol. 20 (1984)

REPORTS

N-POINT FUNCTIONS

No. 2

ON MATHEMATCALPHYSICS

AND LOW-ENERGY BY N CENTRES

SCATTERING

I. A. EGANOVA Siberian

Branch

of Acad. of Sci., Computer Novosibirsk, USSR

(Received

February

Center,

1, 1980)

The scattering of a low-energy particle by a potential of a small range is known to be described satisfactorily by the s-wave alone. In the present paper we give a method of describing low-energy scattering by N potentials with the aid of N waves. For this purpose, a special system of Laplacian eigenfunctions is suggested. The scattering amplitude depends on only N parameters, irrespective of overlapping of potentials, The physical significance of these parameters fij,, 1 = 1, 2, ,.,, N, is shown by exp(2i6,) = S,, where S, is the eigenvalue of the S matrix. The parameters 6, may be obtained by direct methods and perturbation theory. The low-energy scattering by an arbitrary configuration of N centres is discussed. The differential cross-section is averaged over all orientations of the configuration and radially about the direct beam, giving it as a function of the scattering angle. This formula may be used for the phase shift analysis.

1. Introduction

In many problems, particularly in molecular and nuclear physics. the scattering system includes several centres, two in the simplest case. Various approximate methods of solving these problems are discussed in the monograph [ 11. As to the accurate solutions, “little progress has been made in dealing accurately” with many-centre scattering problems “in general when the fields overlap considerably or indeed with problems involving non-central interactions in general” [l] (see Chap. VIII, Sec. 3). An important contribution to the development of the general approach has been given in [2], where elastic scattering by a non-spherical scatterer was described in terms of the S-representation. In the present paper elastic scattering by many centres, including overlapping ones, is described by a method based on a new special system of Laplacian eigenfunctions. It is known that the scattering of a low-energy particle by a potential of a small range is satisfactorily described by the s-wave alone. The mathematical reason for c2171

I. A. EGANOVA

218

this phenomenon

is that the eigenfunctions

1 g2 = ---d with a definite orbital momen2m turn I, -I< m < i, vanish at r = 0, with the exception off,,, with the momentum 1 = 0 corresponding to the s-wave. The idea to describe low-energy scattering by several centres in the same way, using several waves, was suggested in our earlier work [S]. For this purpose it is enough to construct a special system of eigenfunctions of the operator 12, in which only N ’ functions are non-zero at the N points where the centres are located. Such a complete system of k^’ eigenfunctions we call the ‘N-point functions’. The scalar and vector two-point functions were constructed in [3] (see Appendix B of ref. [4], too), and applied methodically in several subsequent papers [S-7]. By using the vector solenoidal two-point functions, we obtained a solvable model of quantum electrodynamics [S], which generalizes van Kampen’s model [S] for the case of two ‘atoms’ separated by a fixed distance. The description of scattering by two overlapping potentials in terms of the scalar two-point functions [6, 71 allows us to calculate the correction for the so-called zero-range potential approximation, see, e.g., [2], which is used in the theory of elastic scattering by molecules and crystals, and if the phases are small, to comment on the validity of the approximate method of multiple scattering and extend the range of validity of Brueckner’s formula Cl], see (112) on p. 197. In the present paper elastic scattering by an arbitrary configuration which consists of N centres is considered. The corresponding system of N-point functions is constructed in Sec. 2. Using N-point functions we find in Sec. 3 that the scattering amplitude depends on only N parameters 6,, Iz = 1, 2, . . . , N, (the socalled ‘N-point phases’, resembling the S-phase of the scattering by a single centre) independently of the overlap of centres. The physical significance of these phases is that exp(2i6,) = S,, where S, is the eigenvalue of the S matrix, see Sec. 3. In Sec. 3 we also discuss the application of some methods of the approximate determination of the N-point phases by variational method and perturbation theory; the variational principle, resembling Hulthen’s, is formulated, and the formula corresponding to the first Born approximation is given. The expressions obtained for cross-sections can be applied in the problem of elastic scattering by an arbitrary system of N components, where, firstly, the range of each component b, is such that kb, 6 1, and secondly, the intercomponent distances might be thought of as fixed. of the squared

momentum

operator

’ The minimum number of functions of a complete system, which do not vanish at the N points, equals N, see ref. [3], Sec. 2.

N-POINT

FUNCTIONS

AND LOW-ENERGY

SCATTERING

BY N CENTRES

219

In connection with successful application of neutron low-energy scattering to understanding of biological structures, see, e.g., [P-12], in Sec. 4 we give the formula for the differential cross-section of small-angle scattering by an arbitrary configuration of N centres, averaged over all orientations of the configuration and, besides, averaged radially about the direct beam, which gives it as a function of the scattering angle. Finally, in Sec. 4 we briefly discuss a possible phase shift analysis. 2. N-point functions Two-point functions [3] were constructed by means of a unitary transformation % from the known complete and orthonormalized system of the functions (1). The method of constructing N-point functions Fkr(r, 8, cp) is, in principle, the same: Fkr(rr 0, cp) = f

i

I=0

m=

z = 1, 2,...

.fklm(r, 0, cp)@2dl+

(2)

-I

The unitary matrix % must be constructed in such a way that at the N points where the centres are located all the functions Fkr, except for N of them, vanish. As a unitary matrix, X! satisfies the following conditions [13], see Chap. V, 8 2, Sec. 164: (3)

We take the centres as located at d,, d,, . . . , dN. The general scheme of constructirig the first N columns of % (z = 1, 2,. . ., N) is as follows [3]: they should be linear superpositions of the functions f&(dn), n = 1, 2, . . . , N, ort hogonal and normalized : 42~m,r= 4

:

PI=1

z=l,2

4,f&,(4),

,...,

N.

(5)

is the normalization factor. Then at dl, d,, . . . , dN all the N-point functions will be zero, except for the first N functions. In fact, the functions &,,,(d,), in turn, can be represented as linear superpositions of the columns a!:,,,, z = 1, 2, . . ., N, and B,

F,, (d,) = f I=0

vanish according

to (3) if z > N.

i m=-l

.L,(d,)

@lm,r

1. A. EGANOVA

220

Determination of the coefficients a”, and normalization factor determination of the eigenvectors and eigenvalues of the matrix 1 . sin kd,,, Mm, = g5 lid,,. ’ In fact, the columns

or according

(5) must satisfy the condition

Since M is a real symmetric (see. Chap. IX, Sec. 13):

A

n=l

the normalization detIM-111

matrix

0 exists [14]

N

2 2 Therefore

(3), i.e.

matrix, such a real orthogonal N

to

n, n’ = 1, 2, . . . . N.

d,,. = Id,,-dJ,

to (A.2) of Appendix

B, reduces

0; M,,

O,,+f

= i, ii,,,.

n’=l

factors = 0,

B, are determined

by the secular

equation

B, = A, ‘I’,

I,,, = d”,,,

and the coefficients ai, n = 1, 2, . . . , N, are components of the eigenvector of the matrix M, corresponding to its eigenvalue il,. Now upon substituting (5) into (2) and using the formula (A.2) of Appendix A, we find the first N N-point functions

=$, ;

F,cr(d

4

sin k/r- d,,l k,Y_dn,

,

z

=

1,

2,

. . . . N.

(7)

n=l

The remaining columns q!lm,rr z > N, are constructed so as to make all columns and rows of the matrix *& mutually orthogonal and normalized.’ Then the conditions (3) and (4) are fulfilled, and the functions Fkr(r), defined by formula (2), will be N-point ones. Only the N functions (7) are required for describing scattering by N centres. That is why we restrict ourselves to finding them only.

* This can be realized just as for two-point

functions,

see ref. [3], Sec. 2.

N-POINT FUNCTIONS

3. Description 1. Consider

AND LOW-ENERGY

SCATTERING BY N CENTRES

221

of scattering by N centres in terms of N-point functions elastic scattering

by N centres

V”

H = y;+

V(r),

with the hamiltonian

v(r) = ; n=

(8)

U,(Y-t&J. 1

All the potentials U, in question are of range b, so small that kb, < 1. The eigenfunctions tikL(r) of the hamiltonian (8), (9) are all the N-point functions F,,(v), except for the first N. Indeed, they all vanish where the potentials are non-zero. The N remaining eigenfunctions describing the scattering can be obtained as superpositions of the N remaining N-point functions (7) since they form a complete system. The conditions under which such a description is successful, just as for the scattering by a single centre, must be of the form: kh, 4 1, n = 1, 2,. .., N. 2. To solve the scattering problem it is sufficient to know the N eigenfunctions 1, 2, .., N, explicitly for large r: I’ % d, d = max (Id,,\). Beyond the range of potentials, the hamiltonian eigenfunctions are known to expand not only in the normalized functions ,jl(kr) Y,m(O,q), but also in the system of functions ~,j.(~),

1.

=

n,(kr) Ylm(O, q), t+(x) = IV[+ 1,2(x)/,

of ref. [15]. Therefore, N- point functions,

Gkr(r) denotes q(kr)

if

the system

2, independent of the former, see, e.g., Sec. 35 the H eigenfunctions are linear superpositions of the

of functions

obtained

from

the system

q”‘(L), cp) by the same unitary transformation +P in terms functions F,, were constructed from the system j,(kr) I;“(O, q):

of functions of which the

The mathematical form of the functions Gkr(r), T = 1, 2, . . ., N, is given in (A.3) of Appendix A. The constants C;(k) and D;(k) are determined by boundary conditions and normalization of ekl(r). If we denote by A:(k) and 6;(k) the new constants: C;(k) = A’,(k) ‘cos 6 J(k) and 0; (k) = A’,(k) . sin 6; (k), and take into account that Iv-- d,J rr Y- v’d,, for large r, v’ being a unit vector in the direction of scattered wave, the functions $kl will

222

I. A. EGANOVA

become

with 92(-v’)

= -T&

i exp(ikv’d,) n- 1

izi(v’) = -Gn$, exp(

F A;exp(-i&)w,,, r=

-ikv’d,)

1

i

A’,exp(i&)w,,,

I=

1

W, = a:B,.

Generally

speaking,

all we know of the matrix

w is that it satisfies the equation:

i: w:, M,,. w,,,,, = h,,., n,n’=1 see Sec. 2. Therefore

we will assume

that

w, = t

a::B,u,,,

p=l

where u is a unitary matrix. It is known ([16], Sec. 124), that

h, (v’) = C S,,, 91, (4,

where S is the S matrix.

1’

Hence, by the orthogonality

of the matrix

A;exp(id;)

=

i

af: and unitarity

of u,,:

S,,.A’,,exp(-id;,).

I’= 1 Therefore if the subscript i indicates the S-representation, = ~3~~~ exp(2i6,), we get (the unique case where the functions $ the ‘N-point’ subscript r is not considered): 6;(k) = SA(k), z = On the other hand, by normalizing the eigenfunctions tJkl asymptotic expressions as in ref. [16], Sec. 21, we obtain the

cf. [2]: S,,. are numbered by 1, 2, . . . , N. with use of their condition:

N

2 r=

Therefore,

taking

A;* A’,. cos (Pi* - &) =

is,,?.

1

into account

the unitarity

of the S matrix,

we find that

the

N-POINT FUNCTIONS

matrix A; must be unitary.

AND LOW-ENERGY

SCATTERING BY N CENTRES

That is why in the set of systems of N-point

223

functions

Fkr(r) = i: &Jr)nPr jl= 1 one will always be found

$knW=

sin(kr+6,)

B,

” 1 u; cos (kd, cos co,) ‘2nZ”_,

kr

-

cos(kr+6,) kr

with Us,,= A;*. Using this system we get

B, %F,=,

N 1 ai sin (kd, cos CO,,).

r$d,

i=1,2

,...,

N.

(11)

o,, is the angle between d,, and v’. As a matter of fact, expression (11) contains, only the N parameters J,(k), i = 1, 2, . ..) N. We call them the ‘N-point phases’ by analogy to the S-phase of scattering by a single centre. N-point phases are determined by a concrete form of scattering potentials, they will be zero when there is no interaction. The equations for 6,(k) are obtained below. 3. Consider the general solution of the Schrijdinger equation with the hamiltonian (8) corresponding to the state with the energy k2/2m: h&9

= ~bAk)h(r). i.

We have to choose the coefficients scattered wave - i.e.

b,(k)

xb,(k)IC/,,(r)-eiLr 1

(12)

so that

=Sy-’

this does actually

eikr,

represent

r % d.

a

(13)

Expanding the scalar plane wave in terms of the N-point functions (see Appendix A) and substituting instead of t,hkl(r) its asymptotic form (1 l), we determine the coefficients b,(k) so that there are no terms of the type r-l eCikr on the left-hand side of (13): b,(k) = B, exp(i6,) v

5 u;?exp(ikvd,),

II=1

is the unit vector in the direction of the incident The scattering amplitude is then given by f

(v, v’) =

& kilBf (ezisa -

1) i

n,n’=1

wave.

uiu:exp (ik (vd,, -

v’d,.)),

(14)

so that the scattering by N centres is determined by only N parameters 6,, even if centres overlap; this fact was observed in ref. [2] for the 61, 62, ..*, zero-range centres.

224

I. A. EGANOVA

As usual, we can find the total cross-section G(V) = fdv’(f(v,

v’)(?

Using the formula sin kd,;,; ~dv’exp(-ikv’(d,;-d.i))=4”

and the formula

kd

dn;“; =I&;-d,;(,

(6), we find ui1 ai,exp(--ikv(d,,-d,,)).

f “1,“2=

Analogically, wave

,

fliPI;

the total

cross-section

(15)

1

averaged

over all directions

of an incident

N

dvo(v) = 4nkm2 c

0 = 4’, s

sin26,.

(16)

A=1

The formulae (14), (15), and (16) become the well-known ones for scattering by a single centre, if N = 1 or if d -+ 0. 4. The N-point phases 6, (k) can be determined by direct methods with the aid of variational formulae. Just as in ref. [17], see Chap. I, we obtain a variational principle corresponding to that of Hulthen. The N-point phase 6, can be determined by the following procedure. We introduce

where +1 is a trial function with the asymptotic form (1 l), qn is a trial N-point phase. We choose a trial function 4L with the N-point phase vi, containing n adjustable parameters cr, c2, . . . , c,. Then calculating I,, we find it as a function of yapand parameters cl, c2, . . . , c,. The variational equation 6F = 0, which is equivalent to the stationary condition that leads to 4z = $1, and therefore vi = 6,, gives a system of (n+ 1) equations for determination of yap and cl, c2, . . ., c,: IJF ---Z

hA

0,

aF az,

-=z=o, dci

In this way we find the value of 6,. 5. Find the equations for N-point

i = 1) 2,

. . .)

n.

,

phases. Subtracting

the equation

N-POINT FUNCTIONS

AND LOW-ENERGY

SCATTERING BY N CENTRES

225

from (9) both sides of which were multiplied on the left by FtL, and integrating over a volume V, of large enough radius Q B (i, we get

"0

Vo

Carrying out the Ostrogradsky-Gauss find

integration on the left-hand side with the aid of the theorem and using the asymptotic forms of tiki. and F,&, we

-__

’

nkm

sin&

A

s

=

h-F,*, V$,, ,

1”= 1, 2, . .) N.

(17)

“0

In case [nkm(kE./V/kl)l

< 1

where (kl/V/k’il)

= JdrF&(r)

V(r) Fksl(r),

the equation (17) can be solved by means of perturbation theory. This method is given in detail in ref. [6], see Appendix B, for the case N = 2. Just as in ref. [6], we obtain the first approximation of the N-point phases iii: L (ki/V/lk;l)

+ 2m. P

s

dk’kf2

0

m

where

P 1 denotes

the Cauchy

principal

value integral.

b 4. Conclusion In the last few years neutron low-energy scattering by a system of centres has been important for understanding biological structures, see, e.g., [S-12]. Special unique experimental methods [lo] allow us to perform scattering on two fixed centres. The distance between these centres is determined by small-angle scattering data. Then from a set of intercentre distances, the tentative map of the entire configuration of centres is obtained by triangulation. The map is indeterminant only with respect to hand. For example, information about the locations of proteins in the 30s subunit of the ribosome of Escherichicr co/i was obtained in ref.

c121. In connection with studies of this kind, we point out one of the possible methodical applications of the N-point-functions method. We mean of course, for example, cold neutrons, such where the condition of validity of that method is fulfilled.

226

I. A. EGANOVA

Suppose that as a result of investigation of some structure, information about relative spatial position of each of its components is obtained, inter alia, when the structure was in an extraordinary state, e.g., two components were deuterated and the rest of the complex was rich in hydrogen. Cold-neutron scattering by such a configuration as a whole can be one of the methods of controlling results obtained. In experiments of this type, the differential cross-section I(e), averaged over all orientations of the configuration relative to an incident beam and likewise averaged radially about the direct beam, is measured as a function of the scattering angle 8, see Fig. 1.

‘/r‘\

_N

’

,’

_,,’

:

\ \

Y

Incident

beam

Fig. 1

As it is shown

in Appendix

B, if 0 is small,

where d, = d,,l - d,,,, d = d~i - d,+, Q = dl - d, and y, cp, and I/I are the angles between d, and d, -d and Q, dl and Q, respectively. On the other hand, we can solve the problem of the phase shift analysis determination of the N-point phases 6,. According to (18) I(0) is of the form

T,x(~)(~,-1)(~,*.-1),

f &A’=

T,,, = GA,

1

and we can determine the eigenvalues S, of the scattering data Z(Q), solving a system of algebraic equations.

matrix by experimental

N-POINT FUNCTIONS

AND LOW-ENERGY

SCATTERING BY N CENTRES

221

Acknowledgments

The author is grateful to Professor Yu. N. Demkov, Corresponding Member of the Academy of Science M. M. Lavrent’ev, Academician A. B. Migdal, and Dr. M. I. Shirokov for discussions and helpful comments. Appendix

A

1. Calculation

of the sum

1 c fkym(r, 19, cp)fklm(e,

S = f

9, 4).

According

to

I=0 m= -1 (1)

The sum over m is known,

see (4) on p. 140 in [18]:

5, = u is the angle between f

P, (cos co).

r(r, 0, q)

vectors

(A.11

and Q(Q, 9, 4). The sum

(21+ l)j,(kr)j,(kQ)P,(coso)

= ~.si~~~r,“,

I=0 see, e.g., ref. [6]. Thus s

1 sinklr-et

_

2X2. 2. The functions

klr-@I

Gkr(r). On substitution

64.2)

.

of ozdl,,r from (5) in (10) we have

where en and (P,, are polar and azimuthal angles of the vector d.. The sum over m can be calculated by (A.l) and the sum r$0(21+l)j,(kfIJn,(kr)P,(coso.)

II = 1, 2, . . . . N,

=~.co~~~~~‘, r n

see, e.g., ref. [6]. Thus

G,,(r) 3. Expansion is of the form

=&, n=51 a’,cosklr-d,j n k,r_d,

,

r = 1, 2, . . .) N.

of a plane wave in terms of the N-point

exp(ilir) = C {~dvc,,(k)F,,(r)} r

0

functions.

3 Cek,(r)

T

(A.3) This expansion (A-4)

228

I. A. EGANOVA

where c,,(k) = v2jdr’eikr’ F$(r’). which leads through

(A.5)

(7) to

c,(k)

= B,6(v-k)

5

a’, exp (ikd,,),

z=l,2

)...,

N.

n=l

according ekr

(‘+)

=

4

Fk,

b’)

a’,

;

exp (ikd,),

z = 1, 2, . . .) N.

n= 1

Appendix B Calculation of Z(0). The averaging of the differential cross-section over all orientations of a system of scatterers is equivalent to averaging over all directions of an incident beam v, and radial averaging about the direct beam (when a scattering angle 1!3is fixed) is equivalent to averaging over an azimuthal angle cp’ of the vector a, see Fig. 2.

t Y

Fig. 2. Y, v’ are unit vectors ii the directions

of incidence and of scattering,

respectively.

a = VI-V,

a

0

= IP( =

2sinZ. 9 and cp are the polar and azimuthal angles of v; 0’ is the polar angle of a, its azimuthal

angle q’ IS not denoted

in Fig. 2 to avoid overcrowding. We take d, lying in the YZ plane. The vector along the z axis. The .x’:’ plane passes through d. .

d points

Thus 1

Z(O) = Pk$ZIBiB:.(e2i”‘-l)(e~2i6*._1)~

1671~k2 x

i n,.ni.n2,ni=

(B.1) 1

N-POINT

FUNCTIONS

AND LOW-ENERGY

SCATTERING

BY N CENTRES

229

where (B.2;

4 = 4, -4,2,

d

=

d,,; -d,,;.

In -Aew of vd, = d, (cos y cos 9 + sin y sin 9 cos q) and (v + a) d = d (cos 0 cos 9 + +sin 8sin9cos cp’), see Fig. 2, and using the integral representation of the Bessel functions 8.411, 7 in [197. we get on integration over cp’, cp I nlnin2+(@ = 47?jsinScos(p

cos 9) Jo (a sin 9) Jo (/? sin 9) d9

(B-3)

0

where p = k(d,cosy-dcos8), Using the formula

a = kdl sin y,

for multiplication Jo(xsin9)Jo(flsin9)

/I = kdsin8.

of Bessel functions

(B.4)

6.684, 1 in ref. [19]

= x-l yJ,(qsin9)dp 0

where q = (a2+j?2-2a~cosp)‘i2,

the formula 6.688, 2 in ref. [19] n/2 j sinYcos(pcos@J,(qsinY)d$

(W

= (7r/2)‘i2(p2+q2)-1/4Jl,2(L:/p2+q2),

0

where according to (B.4) and = cos y cos 8 + sin y sin 8 cosp, and functions (see, e.g., [18], p. 551)

(B.5) p2 +q2 = k2 (df -t-d’- 2d, d cos o), coso the Gegenbauer addition theorem for Bessel

03.6) where W(P) =

f (2~+W,+1,2(kdN,+1,z(kW,(cos~), m=O

we find I nlnin2ni(R=

TV,d)- 1’2j W(ddp. 0

(B.7)

230

I. A. EGANOVA

By going through an exactly similar analysis to that of Appendix I of ref. [20], we can show that the ex ansion W(p) is a uniformly convergent series. Hence when substituted into (B. 7Q the order of summation and integration may be interchanged. In view of 7 dp P, (cos y cos 8 + sin y sin 6,cos 11)= nP, (cos y) P, (cos O), 0 see

[18], p. 141, we thus have

I “l”j”-p>

(0)=

$=$ ; (2~+1)J,+1,2(k~l)J,+1,2(k~)P,(cosy)P,(cos~).03.8) \ mo 1

In order to verify our calculation, the case N = 2 was considered. Z(0) calculated by the formulae (B.l) and (B.8) and the differential cross-section averaged over all orientations of a system of two centres obtained in ref. [6] coincide. If the scattering

angle 0 is small P,(cosO)

2 1 -+z(m+l)B2,

(B.9)

as follows, e.g., from the integral representation of the Legendre polynomials [19], see 8.822, 1. Then on substitution of (B.9) in (B.8), using (B.6), we obtain for a small scattering angle (B.lO)

I “r”i “2”i (0) ‘v 4x2

The series S = f

m(m+1)(2m+1)J,+,,2(kd,)J,+li2(kd)P,(cos1;)

m=l

may be summed with the aid of the addition theorem for the functions J:mj corresponding to the (n - 1)-dimensional Euclidean space motion group representations of class 1. These functions are a generalization of the Bessel functions, see [21], p. 251. Thus

fl JL 0-AP%o @OSPICl

1

(r2)

N-POINT FUNCTIONS

AND LOW-ENERGY

231

SCATTERING BY N CENTRES

where T, cp, and $ are defined in Fig. 3. According (3) on p. 246 of [21],

to formulae

(3) on p. 248 and

1

(B.12)

1)

Fig. 3

In terms of the Jacobi special functions

polynomials,

by formula

P$&)

=+(l+i”)Pgy(<).

Then in terms of the Legendre [19], we get:

where 5 # -1. Thus, the left-hand

polynomials,

(2) on p. 133 of ref. [18], the

by the functional

relation

8.961, 6 in

side of (B.ll)

(rl r2))312{S+2sin2~.S,

1,

(B.13)

d Sl

=

f s=

m+ 1

l)Js+l,2(rl)Js+l,2(r2)

-----Ps(-COSPL

dcosj?

As in ref. [20], it can be shown for fl=n:-y, rl =kd, and r,=kd,. expansion S, is a uniformly convergent series,

S2= 1 (2s+1)J,+l,2(rl)J,+l,2(r2)~s(-cosB) s=

=

1

r

2r, r2 T-(rf+rZ+2r,

r2cosP)-“4Jl,2(t~r~+rf+2r,r2~~~~),

that

the

232 see

I. A. EGANOVA

(B.6), hence, d

s, = -S2 dcos/? =-

J$rl

r2)3’2(rI+rZ+2rlr2cos~)~3'4J3i2(i~r~+2rl

r,cosa).

(B.14)

The special functions (B.15)

P:$o (cos CC)= cos2 (a/2) and P$,(cosa)

= Pz;b(coscr) J;,l

Jhh

=

On substitution S=2

=Isincc, ,:‘z

see ref. [lS],

(.Y)= 3\,k?.~-~~~ t&/2x(3J,,2

(x) - 2J,!,

of (B.13)-(B.17)

$ ;(kdldp-

J3 2(.~), (x)),

in (B.ll),

pp. 130, 131;

see (B.12);

(B.16)

see [21], p. 246.

(B.17)

we have

’ ) 3’2(k~sin~sin$[lJ,;,(ko)-~JS,2(k~)~+

+J3,2(k@)

COS2~-COS2~‘COS2~

[

*

,

(B.18)

.

if fl=x-7, rl =kd, and r,=kd,, e=ld,-4. Using the addition theorem (B.ll) for /J = TC,one finds that formula (B.18) is fulfilled even if y = 0 (in this case 40 = 0 and $ = 7~). Finally, substituting (B.lO) and (B.18) in (B.l), we obtain the required formula (18). When N = 2 it coincides with the corresponding approximation to the formula (19) in ref. [6]. REFERENCES Cl] Mott, N. F. and H. S. W. Massey: 7‘hr Theory of Atomic Co/lisions, Third ed., Oxford University Press, London, 1965. [2] Demkov, Yu. N. and V. S. Rudakov: Zh. Eksp. i 7kor. Fiz. 59 (1970), 2035. [3] Jeganova, I. A. and M. I. Shirokov: Ann. Phys. 21 (1968), 225. [4] Eganova, 1. A.: Izvesr. Akad. Nauk AzSSR, ser. fiz.-techn. i mat. nauk 1 (1971), 94. [5] Eganova, I. A. and M. I. Shirokov: JINR preprint P2-4645, Dubna, 1969. [6] Eganova, I. A. and M. I. Shirokov: JINR preprint P4-5438, Dubna, 1970. [7] Eganova, I. A.: preprint no. 4, Phys. Inst. Acad. of Sci. AzSSR, Baku, 1971. [S] van Kampen, N. G.: Kg/. Dunske Kdenskah. Selskah. Maf.-jy.s. Medd. 26 (1951) no. 15. [9] Engelman, D. M. and P. B. Moore: Proc. Nar. Acud. Sci. U.S.A. 69 (1972), 1997. [IO1 Fugelman. D. M. and P. R. Moore: Pi. .4mer. 235 (1976) no. 4. 44.

N-POINT [ll] [12] [13] [14] 115) [16] [17] Cl83 [19] [X] [21]

FUNCTIONS

AND LOW-ENERGY

SCATTERING

BY N CENTRES

233

Stuhrmann, H. B., M. H. J. Koch, R. Parfait, J. Haas, K. Ibel, and R. R. Crichton: J. Mol. Biol. 119 (1978), 203. Larger, J. A., D. M. Engelman, and P. B. Moore: J. Mol. Biol. 119 (1978), 463. Smirnov, V. I.: Course in Higher Mathematics, Vol. 5, GIMFL, Moscow, 1959 (in Russian). Gantmacher, F. R.: The Theory of‘ Matrices, Nauka, Moscow, 1967 (in Russian). Davydov, A. S.: Quanrum Mechanics, GIMFL, Moscow, 1963 (in Russian). Landau, L. D. and E. M. Lifshitz: Quanrum Mechanics, GIMFL, Moscow, 1963 (in Russian). Wu, T. Y. and T. Ohmura: Quantum Theory of Scattering, Prentice-Hall Inc., New York, 1962. Wilenkin, N. Ya.: Special Functions and the Theory qf Group Representations, Nauka, Moscow, 1965 (in Russian). Gradshtein, I. S. and I. M. Ryzhik: Tables of Inregrals, Sums, Series, and Producrs, Fifth ed., Nauka, Moscow, 1971 (in Russian). Friedman. B. and J. Russek: Quart. Appl. Marh. 12 (1954). 13. Wilenkin, N. Ya.: Trudy Moskou. Mat. Obshch. 12 (1963), 185.