Dispersion relations and low-energy elastic scattering of charged particles from nuclei

12~. L J

Nuclear Physics A206 (1973) 498--512; (~) North-Holland Publishing Co., Amsterdam

~

Not to be reproduced by photoprint or microfilm without written permission from the publisher

DISPERSION RELATIONS AND LOW-ENERGY ELASTIC SCATTERING OF CHARGED PARTICLES F R O M NUCLEI (I).

Theory

R. D. VIOLLIER, G. R. PLATTNER, D. TRAUTMANN and K. ALDER Physics Department, University of Basel, Switzerland ¢

Received 24 January 1973 Dispersion relations for the elastic scattering of charged particles from nuclei are deduced for the case of potential scattering. The inclusion of the Coulomb interaction leads to a simple modification of the formalism developed so far for uncharged projectiles only.

Abstract:

1. I n t r o d u c t i o n

During the last decade dispersion relations have become a powerful tool for interpreting experimental data for scattering processes. In particular, they provide a natural framework to investigate the importance of exchange contributions and to study the influence of nuclear bound states 1- 7). So far, the application of forward dispersion relations in low-energy nuclear physics has been restricted to elastic scattering of uncharged particles, i.e. neutrons 3-5). The reason for this lies mainly in the formal divergence of the forward scattering amplitude for charged projectiles, which is caused by the long range of the Coulomb force. In addition, the non-divergent part of the forward scattering amplitude, which remains after subtraction of the Coulomb amplitude, has completely different analytic properties than the corresponding amplitude for uncharged particles. In high-energy physics (e.g. re-nucleus scattering) Coulomb effects play a minor role and can be considered as corrections. At low energies this is no longer true, and the influence of Coulomb forces on the nuclear part of the scattering amplitude must be fully considered. If successful, this opens up new possibilities, since composite particles are invariably charged. Even in nucleon scattering, the scattering amplitude for protons can be experimentally determined with much higher accuracy than for neutrons. Thus, the generalization of the dispersion formalism to include Coulomb effects at low energies promises to be very rewarding. This paper is the first part of a study concerning the application of dispersion relations to the scattering of charged particles. We intend to show how, for potential t Work supported in part by the Swiss National Science Foundation. 498

D I S P E R S I O N R E L A T I O N S (I)

499

scattering, the usual formalism must be modified to take into account Coulomb effects. In the following paper (II) we apply our results to experimental data on the elastic scattering of nucleons from 4He. 2. T h e n u c l e a r

T-matrix

It is instructive to study the analytic properties of the nuclear T-matrix element for the scattering of a spinless charged particle from a spherically-symmetric local potential V(r) = Vo(r)+ Vl(r), (1) where

Vo(r)

Z1 Z2 e 2

-- - -

/.

e -a'

(2 > 0 )

(2)

is the screened Coulomb potential and Vx(r) a short-range nuclear potential. The total T-matrix element T~#(E) = (q~n~[VI~k(E~)), (3) can be separated into a Coulomb part

T~)(E) = <~E2Volx~)),

(4)

T2~)(E) =
(5)

and a nuclear part by means of the two-potential formula 9)

T~,#(E) = T~)(E) + T~)(E).

(6)

The notation q~, is used to denote a plane wave ~bE,(r) = e ' * ' ' ' ,

(7)

' t + ) -~tancts " " for a Coulomb distorted wave with asymptotic wave vector and the sy m b o l ,~.e~

L~

X(±)

~

=

q~E~+

1

E - Ho - Vo-t- i8

Vo 4'~,

(8)

or explicitly for 2 --} 0 in terms of the confluent hypergeometric function X~)(r) = C o ( t / ) e ' k " ' i F l ( T- it/; 1; ___i(kr~k~, r)).

(9)

For elastic scattering, we have

Ik=l =

Ik#l

= k,

E = h2k212m, where m is the reduced mass. The Coulomb parameter t/is given by

(10)

(11)

500

R . D . VIOLLIER ,1 =

et al.

Uk,

(12)

with -- Z 1 z2 e2m h2

(13)

and the Coulomb penetration factor (or Gamow factor) by 1/ 2nr/ Co(r/) = l e - ~ F ( 1 + #/)l = V e 2 n ~ - 1 "

(14)

Finally, ~,~) is a state distorted by the total potential V(r) ~(+)

~(+)+.

EF

~E,S

-~

1

E-H+_i~

V1 A~(E p+ ),

(15)

with the total Hamilton operator (16)

H = Ho + Vo + V~.

Combining eqs. (5) and (15), we get for the nuclear T-matrix element 1 V -(+)\ T~(~) ~(+)\+/~(-) , (E ) = (X~:)IV,zE,¢/ \An, VXE_H+_i8 1 /;E/J / ,

(17)

or

T~(E)

(-) ITllxB~ (+' >, =
(is)

where the Lippmann-Schwinger expression for the nuclear T-matrix T 1 is given by T1

= Vl 3V Vl

1

E-H+

ie

Vl.

(19)

Let us now introduce a complete set of intermediate states ~. (discrete) and ~,~'~) (continuous) of the Hamiltonian H which satisfy the completeness relation E I~,)<¢,1 + . 1 . f d O r k'2dk'kb~+r))(~b~+r)l -- 1. .

(20)

(2~)aJ

Inserting these into the second term of eq. (17), the nuclear T-matrix element can be written as r¢~)(E) =

\~E~/~(-)'VI,,~E.' , +()\/+

Z .

(¢"lV'lz~+P)> E-E.

'"(+)\/'"(+)V1 ,~EB J + ' \/ + __~1 (dOrk,2dk,_
(21)

Eq. (21) represents an exact expression for the nuclear part of the T-matrix element with no assumptions of approximations involved other than those mentioned at the beginning of this section.

D I S P E R S I O N R E L A ' I I O N S (I)

501

3. Singularities arising from the Coulomb interaction To derive a dispersion relation for the nuclear T-matrix element we must know its singularities. The Coulomb penetration factor Co(r/), which occurs in all the terms o f the nuclear T-matrix element, has an essential singularity at r/ = oo (i.e. k = E = 0) and an infinite number of poles at r~ = +_in

(i.e. k = _+i (--, E = - - h2~2 ] \ n 2 m n 2]

n = 1, 2 . . . . .

These Coulomb singularities are of no interest in the present context, since we are concerned with the nuclear amplitude proper. Let us therefore introduce a "reduced" nuclear amplitude 1, 6) ~ ) ( E ) -- T~(;)(E) C2(q ) ,

(22)

and a "reduced" Coulomb wave function ~(+)

~ ( + ) _ ,~E~ '~= Co(~)"

(23)

With these definitions we get

= +

.

E-E.

+ ----:!lfd~2rk'2dk' (2n) ~J

~(-)

t+) (+) E - E' + is

"¢+)

(24)

4. The analytic structure of the "reduced" amplitude In order to study the analytic structure of this amplitude we shall now investigate the terms of eq. (24) separately. 4.1. T H E D W B A A M P L I T U D E

We begin with the first term of eq. (24), the DWBA amplitude

Y,p = <2~;)1v112~)>.

(25)

As an example, we initially assume the nuclear potential V t ( r ) to be a Yukawa-type potential e-t~r

1"1(r) = 9 ' -

/,.

(26)

Then the DWBA amplitude is given in terms of the hypergeometric function by (see appendix A)

502

R.D. V1OLLIER

et al.

]~'=~-~[I + (4k2/# l-2,k/# 12'"[l+4k2sin2 l-' 2) sin 2 ½0_1 ~-y ½0j x2F~

(

--it/, -it/;1;

- - 4k2 -~-

sin2 ½ 0 ) (, 2 7: )

where 0 is the c.m. scattering angle. Introducing the squared momentum transfer h2z, = 2k2(1-cos 0) = 4k 2 sin 2 ½0,

(28)

we get for this amplitude

L(k,

,) =

[1+ ~]-' BF, ( - i ~¢,# - i ~,(" 1;- ~") .

~ - L 1+---#-~-~ ~j

(29)

The physical region is restricted by the condition z =< 4k 2 _

8mE h2

(30)

In the limit k ~ oo we get the Born matrix element lim

f~p(k, z) - 41rg

k--, oo

(31)

"¢ + / . t 2 "

Let us now consider the DWBA amplitude as a function of the complex variable k for a fixed physical value of z. The following statements are evident: (i) (ii)

f~p(k, z) is analytic in the upper half plane Im k > 0, f~p(k, z) satisfies the crossing symmetry f~(k, "0 = l~,,(-k*, "0,

(32)

if time-reversal symmetry exists, i.e.

V~(r) = V*(r),

(33)

which we will assume in the following. The same statements are true for a very general class of potentials which can be expressed as a superposition of Yukawa-type potentials N

V,(r) =

e-~jr

/'co

j=t

e-/~r

r d,.

aj

(34)

r

4.2. T H E P O L E T E R M

The second term in eq. (24), the pole term, is given by ~,~

=

V, IZu .

E

--E n

>

(35)

DISPERSION RELATIONS (1)

503

By means o f the Schr6dinger equations

v~0. = ( ~ . - H o - Vo)q,., (Ho+ V o ) ~ ~

(36)

~±>

(37)

we get (38) and thus (39) n

For a zero-range nuclear potential Vt (r) the bound-state wave function ~. is given by a Whittaker function

~O.(r) = N. W_,., ,,+÷(2x. r) Yz.,..(P),

(40)

T

where N. is a normalization constant and the Coulomb parameter q. is given by q. = - - . x.

(41)

The quantity x. is related to the binding energy by

B.

her"2

= --

2m

> 0.

(42)

For ~/. = 0 the Whittaker function reduces to a spherical Hankel function of the first kind: 141o,,.+½(2x. r) = K. rlh~)(ir., r)l. (43) We can easily evaluate the contribution of the pole term (see appendix B) and get 21rh2--. ~,p(k,~) = ~ ( 2 1 . + l ) ( k 2 + x

" 2+ ( • ) 2) 1-] i(/k N2~2p~. 1 =x 2 - i ( / k ~ '

(44)

where the radial integral ~,. is given by

~. =

fo 'dr W_,.,~.+~(2x.r)P~.(n,

kr),

(45)

and the "reduced" regular Coulomb function by In

( 9 ]¢.l,,~ln + 1 ~ - ikr

ff,.(r l, kr) = I-112+iql~Fi(l,+l-iq; =~

2 / , + 2 ; 2ikr)'-"-" . 2F(2 l~+ 2)

(46)

In eq. (44) we have introduced the relations z. 1+i~l exp (2ioh.(q)) = l I , ~=x 2--iq

(47)

504

R . D . V 1 O L L I E R et al.

Pt.(cosO)= 4 ~ E Y~.,..(E.)Yz*,..(Ep), 21.+1 ,..

cos0= 1-~

(48) (49)

2k2 •

The radial integral (~. is evaluated as (see appendix B) In

(~"

=

_ _

k2+x.2

.

(50)

= F(~.+ l ~q.) 2 N2P`" 1 - 2 k ~5 k 2 +I x .2 .

(51)

,~,

~=1

F(/.+l+q.)

Finally, we get for the pole-term contribution: In

~.,(k, v)

27rh2~" . (21"+ ll

Now, let us consider the pole term eq. (51) as a function of the complex variable k. The following statements are true: (i) 0.p(k, z) is analytic in the upper half-plane I m k > 0 except for N poles at k = + i t ¢ . (n = 1. . . . . N), (ii) 0~p(k, ~) satisfies the crossing relation 0*a(k, T) = 0,p(-k*, ~).

(52)

We now give up the zero-range approximation. Then the bound-state wave function O, is given by ft.(r) = u.(r) Yz..,.(~),

(53)

T

where u.(r) can be separated:

u.(r) = N. W_,.,,.+½(2x. r ) - o . ( r ) .

(54)

This separation of the bound-state wave function is possible for a short-range nuclear potential, eq. (34), since then lim ,~®

o"(r) W_..,~.+~(2~c.

= 0,

(55)

r)

or

u.(r) ,,, N.W_,.,,.+~(2K.r)

for

r-

oo.

(56)

In this case the radial integral eq. (45) must be replaced by

O_ -_fodr [W_..,,.+.(2x.r)-O"(r)q ~,.(q,k O. Nn A

(57)

As for the scattering of uncharged particles by a spherically symmetric potential 9), one can show that the two statements eq. (52) are satisfied provided

DISPERSION RELATIONS (I) <

4/~2,

505 (58)

where # - ' is the exponential range of the potential eq. (34), p = min(/zj)

j = 0, 1. . . . . N.

(59)

4.3. THE INTEGRAL OVER THE PKYSICAL CUT Similar considerations can be made for the third term of eq. (24) /~,p=

1 [dfJrk,2dk,<;~(~)l v, Iee,, (+) > (2r0 s ,/ E - E' + ie '

(60)

the integral over the physical cut. The only difference between this term and the pole term lies in the fact that here the bound-state wave functions are replaced by the free ones. This does not alter the validity of the following statements: (i) ~p(k, z) is analytic in the upper half-plane Im k > 0 for T < 4/.t 2, (ii) /~B(k, x) satisfies the crossing symmetry /~*,(k, T)

=/~,p(-k*,z).

(61)

Summing up our investigation of the analytic properties of eq. (24), we find that (i) the "reduced" nuclear T-matrix element is analytic in the upper half plane except for a finite number of poles on the imaginary axis at k = ix,, provided T < 4# 2 and, (ii) the crossing symmetry T(~)(k, T)* = T(~)(-k *, z),

(62)

holds. 5. The dispersion relation for fixed momentum transfer

With the results derived in sects. 2-4 we can now easily formulate a dispersion relation. 5.1. THE UNSUBTRACTED DISPERSION RELATION We use Cauchy's integral formula T(~)(k, z) = 1 p fr "~P rO):t'' ~'~' ~) dk , +pole terms, rt~ k'- k

(63)

where k lies on the contour F (fig. 1). Since we are interested in the analytic properties of the "reduced" nuclear T-matrix element as a function of E, let us apply Cauchy's integral formula in the k 2 plane. The transformation

(o = k 2

(64)

maps the upper k half-plane onto the whole to-plane, leaving a cut on the positive

506

R.D. VIOLLIER et al.

real co-axis. The poles k = i x j ( j = 1 , . . . , N ) on the positive imaginary k-axis are mapped onto the negative real co-axis and appear at

0.) = --/~j2 < 0.

(65)

The amplitude

¢(co, ~)=

,.,=# r - ~ .t,,~, . ,),

(66)

is analytic in the whole cut co-plane except for a finite number o f poles at co = co~ ( j = 1. . . . . N). Let us now apply Cauchy's integral formula to the amplitude ~b(co, z) using the integration c o n t o u r F shown in fig. 2. We obtain

L

*(co'*) = ~

N

co'-co

j=, co~----o"

~'

(~

~.,

,

I

a~k'

Fig. 1. The integration contour/' and the singularities in the k" plane.

Im o /

~ '

Fig. 2. The integration contour/' and the singularities in the fa" plane.

(67)

DISPERSION RELATIONS (I) The residue

r](z) of ~b(09, z) at the pole 09 =

507

09] is given by

ri(~ ) = - lim (k 2 +

x2)~p(k, ~).

(68)

k-+ igj

Substituting ~p(k, z) as defined in eq. (51), this yields

rj(z)=

2rch2N2(2lj+l)(-1)'s ( bx2j) m F(1 + ¢/x])2 Ptj 1 + .

(69)

This is also true if a finite-range expression [eqs. (55) and (57)] is used, as can be easily checked. If lim ~b(09, z) = 0, (70) go--~ oo

the integral over the infinite circle needed to close the contour vanishes. From eq. (62) we deduce

=

(71)

This equation, known as the mirror principle of Schwartz, permits the analytic continuation across the cut. We now get, evaluating eq. (67), a dispersion relation of the form Re ~b(09, z) = _1 p

Im q~(09', z) d09'+ y'

7"~

09v

09

j=l

r](,)

(72)

09]--09

5.2. THE SUBTRACTION OF THE DISPERSION RELATION However, if as in our case of potential scattering lim 1~b(09,z)l = const. > 0,

(73)

OJ"~ CO

then we have to subtract the dispersion relation at to = 09o in the following manner: Re Eq~(09,z ) - tk(090, z)] = 1__(09 - 090)P

Im $(09', z)

d09'

(09'- 09)(09'-09o) N

rj(.c)(09_090)

+ JE=I (09J-- 09)(09j -- 090)'

(74)

in order to ensure that the integral over the infinite circle (fig. 2) still vanishes. 5.3. DISPERSION RELATION FOR THE "REDUCED" NUCLEAR SCATTERING AMPLITUDE Let us now formulate this subtracted dispersion relation in terms of the "reduced" nuclear scattering amplitude f l (E, z) defined by fx(E, z) = -

m ~ ) ( k , z), 2nh 2

(75)

508

R.D. VIOLL1ER

et aL

i.e. fl(E, z) is the total scattering amplitude minus the Coulomb amplitude divided by Co2(q). This gives us as final result

Re [f,(E, z)_ fl(Eo, T)] = I ( E _ E o ) P i

o

lmf,(E',z)

dE,+~(E, Eo,~) '

(76)

(e'-E)(E'-Eo) where we have introduced the

discrepancyfunction A(E, Eo, z) given by N R

A(E, Eo, ~) = E

j =1

The dimensionless

(0hc(E-eo)

(Ej -- E ) ( E j - go)

•

(77)

residue R i (z) is given by Rj(T) =

h

2(--I)IJ(2/j-F1)

2me N j

Y ( 1 --F-¢/Kj) 2

( elj

2~j ) 1 -F-

.

(78)

To conclude, we interpret the normalization constant in terms of an effective range of the interaction in the bound state ~k,. The effective range reft is defined by reff= In addition,

2fodr(W2_,.,(2x.r)--(ru"(r))2] , ~22 ) "

(79)

Un(r)must be normalized according to

fo'u2(r)r2dr = 1.

(80)

reff= 2 {foW2_~.,~(2x.r)- -~2}.

(81)

Therefore, we have

In appendix C we show that 1 1 ~ q~ . 2r. F(1 +q.)2 s=O (th+S)2(q.+s+ 1) 2

(82)

With this, eq. (8 l) becomes

1 ~ reff - x.F(l+t/.)2s=o~

q2

2

(q.+s)2(rh +s+ l) 2 - N-~"

This equation implicitly relates the pole residue R. to the effective range nuclear interaction in the bound state ¢..

(83)

r.n of the

Appendix A In this appendix we derive an explicit expression for the DWBA matrix element defined by f,p --
DISPERSION

RELATIONS

(I)

509

where X~ ) and 3f~,+#) are Coulomb distorted waves. Using a Yukawa-type potential for VI e - ~r

V,(r)

= a

,

y

(A.2)

we get

f,~ = a fd~,x(~:'*(,) e-", X~+2(,)

(A.3)

r

The same integral appears in the theory of (d, p) reactions below the Coulomb barrier, if the orbital angular momentum transfer in such a reaction is zero. Following ref. 11), this integral can be solved using integration methods developed by Sommerfeld, leading to

FL(e2""'- ".", l)(e2""' - I)_Il' L~~d

~

2F,(-itl~, -irl,; I; -z)

( k , - ka) 2 +/~2

1+ z

'.-

x L ( k , - ka) 2 + #2J

'"' ,

(A.4)

with z --

4k~ kp sin 2 ½0, (k~- kp) 2 + #2

(A.5)

where 0 is the scattering angle in the c.m. system. Since we are interested in elastic scattering E = E',

k, =

kp

=

k,

(A.6)

rG = r/a = r/, we get the final result

2.. I [i f" -- ~ff FLe~.]

I -2~k/.

l ~'.

+ (4k% ~) sin~ ½6J

x [I+ ~sin 2 ½0]-'zF, (-it/,-it/; I;--

4k_2sin2 ½0).

(A.7)

Appendix B

We want to calculate the matrix element I = <~f~)l¢n> = where ~b. is given by

darx~-~)*(r)d/.(r),

(n.0

R . D . V1OLLIER et al.

510

~b.(r) = N. W-""'t"+~r(2t¢" r) Ylnmn(~)"

(B.2)

/,

Let us expand X~)(r) into partial waves: Xte:)(r) = 4n E ite-'~'(n)F,(rl, kr)Yl*m(~)Ylm(P), kr lm

(B.3)

where F t (r/, kr) is the regular Coulomb wave function

F,(~I, kr) = e-~'"lF(/+ 1 + i-r/)[tFl(1 + 1 - it/; 21 + 2; 2ikr)(2kr) '+ %-!k'.

1 2F(2/+2)'

(B.4) and oh.(q ) is the Coulomb phase shift given by In

(a.5)

co,.(t/) = E arctan r/. A=I

We perform the integration in eq. (B. 1) and obtain

I = 4re__i_l.e~O~,.(~)yl.,..(l~)N.Q. ' k

(B.6)

with the radial integral

Q. =

dr W_,.,,.+~(2x. r)Ft.(q, kr).

(B.7)

For the calculation of the radial integral Q., we write the Coulomb wave function as a regular Whittaker function

Fz.(~l. kr)

=

(B.8)

- iCt(r/).~',,.,+~(2ikr),

with

C,(~I) = ½(- i)'e-~"lF(l

(B.9)

+ 1 + in)].

With this we get

Q. = - iCz.(q

dr W_,.,,.+~(2x. r)..¢[i., t.+½(2ikr).

(B.10)

This integral can be solved using the general integral formulae given by Buchholz x 2) f l a 2 - - a l 2 + l(,a - 1- _ ~ a 2

I

4

z

1/2 --]22 /

+ ~z21

(1)

(2)

P~.~.(aa z)P~,½~(a2 z)dz =

DO) [~ 1 z) --~, ½t,~'

d 0(1) t~

Px,(2)~},(a2 z)

d p(a2)v(a2z ) ,

(B.H)

D I S P E R S I O N R E L A T I O N S (1)

511

where P is any kind of Whittaker function. In our case we get

Q, = - iC,.(rl)

~1

1

d

W_~.,,.+~(2x.r) w_,..,.+~(2~.0

d

dl,~,,.+½(2ikr)l°°

~ ~,,.,.+~(2ikr) o

(B.12)

For the calculation of the determinant we have to use the asymptotic forms of the Whittaker functions xa). From this we find that the value of the determinant vanishes at the upper limit so that we get a contribution only from the lower limit

k

Q =e_~, .

(k)'"lF(l.+l+#l)l

kZ + x-----] ,7.,

(B.13)

/-(/.+ 1 +q.)

Inserting eq. (B.13) into (B.6) we get the desired integral L

Appendix C We want to normalize the Whittaker function case I. = 0, i.e. we demand that

N=.fo°(W_

rt., ~:(2 tCn r

r -1 W_,., ~.+½(2x.r)

for the special

))2dr = 1.

(c.1)

To get the normalization constant N. it is necessary to evaluate the integral I =

fo o(W_,., ~(2x. r)) 2 d r .

(C.2)

To do this we use the following integral representation for the Whittaker function xa) W_,, ½(2xr) = with

dx'g(r/, x, x')e -~'',

L, 1

2x

(x'-x~".

(C.3)

(C.4)

Therefore we have

I= =

dr

dx'g(q, x, x')e-""

d~'0(~. ~, ~')

dx"g(rl,x, x")e-'"

d~"O(~,x, ¢ 3

1

~ ' + t¢" '

(C.5)

where we have done the integration over r. Inserting eq. (C.4) into eq. (C.5) and making some substitutions we get 1

I --

1

2xe(q)2 f °

duu"-~(1-U)fodvv'~-l(l-v)(1-uv) -~. ,

(C.6)

512

R.D. VIOLLIER

et aL

One of these integrals is equivalent to a hypergeometric function

fldvv't-l(1-v)(1-uv)-I

F(~)

r(2+n) 2

2F1(1, ~/, 2+t/, u).

(C7)

Since 0 < u < 1, we can use the series expansion of the hypergeometric function. Integrating term by term this leads to the result

I-

1 1 ~ [ F(tl+s) '~2 :K r(~y ,=o \ r ( 2 + ~ s i / '

(C.8)

1 1 ~ (( ~÷ ))2. 2K F(I+F/)2,=o ~+s)( s+X

(C.9)

or

i_

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13)

D. Y. Wong and H. P. Noyes, Phys. Rev. 126 (1962) 1866 H. J. Schnitzer, Rev. Mod. Phys. 37 (1965) 666 T. E. O. Ericson and M. P. Locher, Nucl. Phys. A148 (1970) 1 M. P. Locher, Nucl. Phys. B23 (1970) 116 M. P. Locher, Nucl. Phys. B36 (1972) 634 R. D. Viollier, G. R. Plattner, D. Trautmarm and K. Alder, Phys. Lett. 40B (1972) 625 L. S. Kisslinger, Phys. Rev. Lett. 29 (1972) 505 G. R. Plattner, R. D. Viollier, D. Trautmann and K. Alder, Nucl. Phys. A206 (1973) 513 A. Messiah, Quantum mechanics (North-Holland, Amsterdam, 1961) ch. 19 M. L. Goldberger and K. M. Watson, Collision theory (Wiley, New York, 1964) ch. 10 K. A. Ter-Martirosyan, JETP (Sov. Phys.) 2 (1956) 620 H. Buchholz, Die konfluente hypergeometrische Funktion (Springer-Verlag, Berlin, 1953) A. Erd61yi et aL, Higher transcendental functions (McGraw-Hill, New York, 1953)