Direct nuclear reactions and interaction in the initial and final states

12.----~ q

N~,clearPhysics 48 (1963) 375--384; (~) North-Holland Publishing Co., Amsterdam

~

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

DIRECT NUCLEAR REACTIONS

AND INTERACTION

IN THE INITIAL AND FINAL STATES v. A. KAMINSKY and Yu. V. ORLOV Nuclear Research Institute, Moscow State University, Moscow, USSR Received 25 February 1963 Abstract: A formal scheme is developed to take into account the interactions in the initial and final states in direct nuclear reactions. The unitarity of the S-matrix and analytical properties of the amplitude of the reaction as function of energy are used. The solution of the singular integral equation for the partial amplitude of the reaction is found. The result is not affected if the amplitude has an essential singularity of the type e-ikR connected with the nuclear size. The solution of the problem is factorized into two factors, one of which (p,,(E) = pz(E)pe(g)) contains all singularities connected with the interaction in the initial and final states, while the singularities of the other factor are determined by the mechanism of the reaction. The function pc(E) for a rectangular well is calculated. The problem of the bound states of the system is considered. The solution found for the partial amplitude of the reaction satisfies the physical requirements and has the correct behaviour near the threshold. The solution also takes into account the resonance effects of the compound nucleus in a direct nuclear reaction. A method of boundary problem equivalent to the dispersion method but more easily vizualizableis presented. 1. Introduction I n the t h e o r y o f direct nuclear reactions 1 - 3 ) a new a p p r o a c h using the u n i t a r i t y o f the S - m a t r i x a n d analytical p r o p e r t i e s o f the a m p l i t u d e s o f direct processes is being developed. T h e i n t e r a c t i o n in the initial a n d final states which is usually dealt with b y the d i s t o r t e d wave m e t h o d is in fact c o n n e c t e d with the a m p l i t u d e singularities with respect to energy. T h e d i s t o r t e d wave m e t h o d has n o t been sufficiently well justified. I n this m e t h o d the wave functions o f the optical m o d e l a r e used, which, strictly speaking, give only a g o o d d e s c r i p t i o n o f the scattering phases, i.e. the a s y m p t o t i c b e h a v i o u r o f the wave function. Besides, the d i s t o r t e d wave m e t h o d is b a s e d on the a p p l i c a t i o n o f p e r t u r b a t i o n theory, which c a n h a r d l y be justified since, first, nuclear interactions are n o t small and, secondly, the wave functions o f the initial a n d final states for the c o m p l e x p o t e n t i a l are n o t o r t h o g o n a l . I n t h e dispersion a p p r o a c h the basic a s s u m p t i o n s are f o r m u l a t e d m o r e distinctly a n d the n u m e r i c a l calculations seem to be simpler t h a n in the d i s t o r t e d Jiwave m e t h o d . I n a p r e v i o u s p a p e r 3) the s o l u t i o n o f the O m n ~ s - M u s k h e l i s h v i l i singular integral e q u a t i o n was c o n s i d e r e d for the case t h a t the i n t e r a c t i o n in the final state is t a k e n into account. W i t h the help o f the r e p r e s e n t a t i o n o f the p h a s e 61(E) in the f o r m tg 6~(E) = x/F.Q ( E ) / P ( E ) ( Q ( E ) a n d P ( E ) are a r b i t r a r y p o l y n o m i a l s ) the s o l u t i o n was r e d u c e d to one q u a d r a t u r e d e t e r m i n e d b y the m e c h a n i s m o f the r e a c t i o n a n d the scattering phase. The i n t e r a c t i o n b o t h in the initial a n d the final states is c o n s i d e r e d in this p a p e r . 875

376

V. A. KAMINSKY AND Yu. V. ORLOV

The role of an essential singularity of the type e -ikR ( k is the wave number and R is the nuclear radius) in the amplitudes of the reaction is investigated. The function

p,(E) exp

-

~,(E')dE'

I

f f (E,_E,)(E,_e,),

(1)

is calculated for the case when scattering is described with the help of a rectangular well. The problem of bound states of the system is considered. A method of boundary problem equivalent to the dispersion method but more easily vizualizable is proposed. 2. Statement of the Problem

Consider the amplitude of the reaction A + x ~ B + y as a function of non-relativistic variables s, t and u of which only two are independent. Let us isolate from the amplitude M the part M (°) containing all pole terms as well as the terms having branch points only with respect to t and u. Let us go over to the variables E and z, where E is the kinetic energy of the outgoing particle in the c.m.s., z = cos 0, and 0 is the angle between the m o m e n t a of the particles x and y in the same system. Let us now write the dispersion relation with respect to energy on the sheet I m x/E > 0 for a fixed z for the function M ' ( E , z ) = M(E,z)--M(°)(E,z)which has only a cut from E o to oo: E° =

Q, 0,

Q<0 Q >0

Q = m x + m A - - m y - - m B.

Expressing the amplitude through its discontinuity on the cut in the conventional way, we obtain

I rE'o EA(E', z) dE'. '-E-ir 1

M'(E,z) = ~

For the sake of simplicity we assume that subtractions need not be made. Let us perform the expansion in partial waves and use the unitarity relation, keeping only the terms corresponding to elastic scattering of x on A and y on B. We obtain the following singular integral equation for the partial amplitudes t Mw(E)(I, l' are the quantum numbers of the angular m o m e n t u m of the relative motion at the beginning and at the end; their difference is only due to the mechanism of the reaction if the approximation in which the elastic scattering amplitude is diagonal with respect to l is considered):

1 fe ~ Mu,(E')hT,,(E') + M~,(E')fI(E') dE', where

h, = {;"' sin 6,,

E>0 E < 0

[ e~*' sin qh, fz = [0,

E-Q > 0 E - Q < 0;

t The kinematical factor A/El' (E--Q)I

is i s o l a t e d s o t h a t M w ( E )

(2)

(2a)

--¢-const, w h e n E ~ 0.

NUCLEAR REACTIONS

377

got(E- Q) and 6~(E) are the partial scattering phases at the beginning and at the end. The problem thus reduces to the solution of eq. (2). In a more general case when in the elastic scattering amplitude there remain terms not diagonal with respect to orbital momentum we have a set of integral equations. However, keeping only the diagonal terms in the elastic scattering amplitude appears to be a sufficiently good approximation. 3. An Essential Amplitude Singularity at Infinity In the general case, it may be possible that the direct process amplitude as a function of energy have at infinity an essential singularity of the type e-aR connected with the size of the nucleus R. Let us use an example when the interaction is considered only in the final state (generalization for interactions at the beginning and the end introduces no difficulties, nor does it change the result) to show that this amplitude singularity does not affect the solution of the unhomogeneous problem, i.e. the integral over a large circle can still be discarded. Let us multiply the amplitude Ml(E) by eiX(x = kR). As the factor we can take e~kn'(R' > R). However, the final result does not depend on R'. The function Ml(E)e i~ decreases when IE[---} oo(Imx/E > 0) and the dispersion relation can be written in the form

M t(E)e~:"

~,f~o),,i~. 1 f ° M,(E')e'~'H*(E ' ) - M~°)(E ') sin x' dE', E'--E-irl

,'-t ~ -- -~Jo

(3)

where

H?(E) = e-'~[h?(E)e- 'x + sin x].

(4)

Eq. (3) reduces to a boundary problem if a Cauchy-type integral is considered (see ref. 4), for example)

1 ~° M,(E')e'X'H*(E ' ) - M~°)(E') sin x' dE', F(E) = ~ ) o E'--E

(5)

where E is complex. The discontinuity of the function F(E) on the real axis is

F(E +) - F(E_) = M,(E)e'XH?(E)- M~°)(E) sin x;

(6)

from eq. (3) we have

M,(E) = M~°)(E)+e-iX2iF(E+ ).

(7)

Substituting eq. (7) into eq. (6) we obtain

F(E +)[1 - 2in?(E)] - F(E_ ) = Ml°)(E)h'~(E)e-'X.

(8)

Using the equation

1--2iH~(E) = e -2'('''+x)

(9)

we obtain the solution in the form

Mt(E)e '~ = M}°)eiX+ ff'(E*) f °° M~°)(E')h*(E')e-iX'dE' +O(E)fft(E+), .Io p,(E')(E'-E-itl)

(10)

378

V . A. K A M 1 N S K Y A N D

Yu.

V. ORLOV

where O(E)~t(E+)is a solution of the homogeneous equation; on which restrictions will be considered further on, and ~,(E) = exp

IE-E1 f~o .

O(E)is

the function

6*(E')+ x' dE'}, (E'-Ex)(E'-e)

(11)

where E is complex. Using the equation (El, E are real and positive)

f[ (E'-E,)(E'-E)

= 0,

(12)

we obtain

~,(E~) = pt(E+)e±,x,

(13)

where Pl is given by the relation (1). Substituting eq. (13) into eq. (10) and cancelling the common factor e ~x we obtain the solution

M,(E)= MI°)(E)+ P'(E+)f °° ~

.Io .,(E_)(E-V.-i,1)

dE'+O(E)p,(E+),

(14)

which does not differ from the solution of the problem for the case when the amplitude has no essential singularity at infinity. 4. Choice of the Solution

Let us now interpret the function O(E).This function must belong to a class of functions without singularities in the final part of the plane, i.e. it must be an entire function. It has been shown above that the integral over a large circle can be discarded even when an essential singularity of the type e -~kR is allowed in the amplitude at infinity. Consequently, the function O(E) may also have an essential singularity of such a type. The integral equation has an infinite manifold of solutions out of which one solution must be chosen. To isolate the solution let us use the following requirements: (1) M l -* M[ °) when 6j --* 0 and (2) M t ~ 0 when Mlt°) -* 0. Both requirements are physically sensible. Indeed, letting 6~ tend to zero for all energies means that the scattering of the given partial wave can be neglected, i.e. the amplitude must tend to M[ °). On the other hand, the amplitude Mz must tend to zero if M~ °) ~ 0 since M[ °) is connected with the very mechanism of the reaction. The solution with O(E) = 0 was taken in ref. a). Such a choice satisfies the above requirements if M~ ~ 0 at infinity. In the general case the solution of the homogeneous equation must be taken into account, while O(E)should be determined by the above requirements. 5. Interaction in the Initial and Final States

Let us now consider eq. (2) in which the interaction in the initial as well as final states is taken into account. Let 2i~(E+)denote the integral term of eq. (2) and the

379

NUCLEAR REACTIONS

equation be reduced to a boundary problem: • (E+) I1 -

2 i f l + h ~ ] - ~ ( E _ ) - f l + h ~ M~O). 1 + 2ift/ l+2iA

(15)

Here the following relation, valid in the two-particle approximation of elastic scattering, is used: Im Mu, = M , , h e*+ M u , * f l .

(16)

1 --2i ft + h~, _ e-2t(6,,*+,,),

(17)

Since 1+2~ eq. (15) can be rewritten as (E > 0, E - Q

> O)

O(E+)e -2tO'''+*')- ~(E_) = e -`~6'''+*') sin (t$~,+ tpz).

(18)

The case differs from that in which the interaction only in the final state is taken into account in that the phase t$~, is replaced by the sum of the phases t$~,+tp~. Thus, the solution of the problem has the same form (14), the product of the functions Pt and Pr being taken as Pt (the functions refer to the interaction in the initial and final states, respectively). It will be recalled (see eqs. (2a)) that in the integrals from fi~, and from tpz the lower limits are 0 and Q, respectively. 6. Calculation of the Function p~(E) In contrast to the distorted wave method, the method under study does not require knowledge of the wave functions. The interaction in the initial and final states is expressed only through the partial scattering phases. These can be found from experiment by the phase shift analysis. The latter, however, is not feasible at high energies because of the large number of partial waves. In particular cases it was shown that the set of optical model phases describing the experiment in the best manner is quite close to the minimizing values of the phase analysis 5). Thus, there is a hope to obtain the "experimental" phases 3z(E) by the optical model. Accordingly, pt(E) should be calculated for the scattering on the optical model potential. The simplest case is that of a rectangular well for which we have St = e 2i6'-- - YJ'+l(Y)h[2)(x)-xh~2+)l(x)j'(Y)

(19)

YJr + l(Y)h~l)(x) - xh~l+)l(x)jt(Y)'

where JL, hiE1), and htL2) are the Bessel and Hankel spherical functions, x = k R , y = ~/2g(E+ U)R/h, # is the reduced mass, U is the depth of the well (Re U > 0), R is the radius of the well; S t can conveniently be represented as St = e-tx[f(E) + 2ixQ(E)'] . etX[p(E)_ 2 i x a ( E ) ]

'

(20)

380

V(E)=ot,(E)

V. A. KAMINSKY A N D Y u . V. ORLOV

E(½(l + X))

[ 2 ( I + 1 -- n)] !

,,=o

(l+l-2n)l(2n)!

Z

(-1)"

(2x) 2" E(½0 [ 2 ( 1 - n)]! -2~,(E) y' ( - 1 ) " " (2x)2"; ,,=o (l-2n)!(Zn)!

E(½1)

(21+ 1 --2n)t

.=o

(l-2n)!(Zn+ 1)!

Q(E) = ~,(E) E ( - 1 ) "

(2x) 2"

E(½(t - 1))

--2~/(E)

Z n=0

Q(E) = %(E),

(-1)"

(21 - 1 - 2n) ! (2x) 2", ( l - 2n - 1)t(2n + 1)!

1 > O,

l = O.

In the upper limits of the sums E(Q denotes the integral part of ~. For the rectangular well we have

oq(E) = j,(y)/2 '+ ly,;

y,(E) = j,+ t(y)/2 '+ ly,-1

(21)

From eqs. (20) and (21) it follows that for E ~ oo (on the real axis) we have y ~ x and St --* 1. From quasi-classical considerations it can be found that 61(E ) ~ 0 for E - * oo and therefore we assume that 31(oo) = 0. To calculate pl(E) (see eq. (1)) we represent 6~ as

6,(E) = 1__In e-'X[p(E) + 2ixQ(E)]

2i

(22)

e'X[P(E)- 2ixQ(E)]

and consider the integral over the contour C on the sheet Im x/E > 0 of the function ['2xi(E' - E)(E'- Et)] -1 In {e'X'[P(E ') -2ix'Q(E')]}. The contour C encloses all singularities of the function, including the poles at the points E and Ex on the real axis and branch points of the logarithm given by the equation

P(E)-2ixQ.(E) = O.

(23)

All roots Ev of eq. (23) on the sheet Im ~/E > 0 are the poles S~ corresponding to the bound states. The integral over the last circle can be discarded since e~x[P(E) -2ixQ(E)] --, 1 for IEI ~ oo on the sheet Im~/E > 0 for a rectangular well. Note that the function e-~X[P(E)+2ixQ(E)] increases on the large circle a s e - 2 i x with the radius, i.e. S z has an essential singularity at infinity for a rectangular well. We obtain

pt(E) =v01 ( 1 - ~ ) / e ' X [ P ( E ) - 2 i x Q ( E ) ] ,

(24)

where n is the number of bound states with the given l. The expression for the function

Pz(E) was found in ref. a) under the assumption that 3t(0 ) = 0 and the phase is described in the form tg3t(E) = x/F.Q(E)/P(E),

(25)

NUCLEAR REACTIONS

381

where Q(E) and P(E) are arbitrary polynomials. The rectangular well case (22) does not differ essentially from the model (25) and therefore we keep the notations Q(E) and P(E). In the given case they are entire functions of E, which, however, does not affect the method of calculation. The expression found for pt(E) makes it possible to give a simpler proof of the Levinson theorem 6) for a rectangular well (61(0)-6t(m) = nT~, n is the number of bound states with a given/) since gl(0) can easily be found with the aid of eq. (24). Indeed, from eqs. (20), (21) and (24) it is clear that for E -, 0

pt(E) cc E-".

(26)

On the other hand, the estimate of the behaviour of the Cauchy-type integral close to the left end of the integration line a) leads to the expression

p,(E) oc E -''~°~/'~.

(27)

Comparing eqs. (26) and (27) we obtain 6,(0) = n• and hence 6~(0)-6t(0o ) = nn (we choose a branch of 61(E) for which 6~(o0) = 0). In the general case we can write

p,(E)= f i t (1-- - ~ ) / f ( - k ) ,

(28)

where f(k) is the Jost function (for the definition of this function see ref. 7)). Note that the function pt(E) is normalized so that p~(E) ~ 1 when E ~ or. 7. Boundary Problem Method The consideration of the interaction in the initial and final states is connected with a singular integral equation which is solved by reducing it to a boundary problem for a certain auxiliary function. It is possible to formulate the boundary problem directly for the amplitude M(E) (the subscript l is omitted here and in the following) if its singularities are known. Let us note that it is necessary to know the singularities to be able to write the dispersion relation as well. For the sake of simplicity let us consider only the case of the interaction in the final state. The amplitude has the righthand cut L o running along the positive real axis connected with the interaction in the final state, and singularities of the function M c°~(E) determined by the mechanism of the process. The boundary problem is

M(E+ ) -- M(E_) = 2i2(E),

(29)

where M(E±) is the value of the function M(E) on the positive and negative side of the cut, respectively, if the beginning and end of each line are indicated, and 2(E) is the discontinuity of M(E), assuming a definite value on any cut. We solve the problem (29) under the assumption that 2(E) contains the wanted amplitude M(E) only on L o where one can write

(E) = M(E+ )h* (E) + ~ (E), ~U(E) representing the other terms in the unitarity relation.

(30)

382

v.A.

KAMINSKY

AND

Yu.

V. ORLOV

It is convenient to rewrite eq. (29) as

M(E+)G(E)- M(E_) = 2ix(E),

(31)

where the functions x(E) and G(E) are taken to be known, while

x(e) =

G(E)

1

e_2ia.(E)j on Lo,

x(E) = x(E)/ on the other cuts. G(E) 1 J

(32)

The solution of the problem (31) is of the form M(e+) =

The line L includes all cuts. The two-particle approximation of elastic scattering implies neglecting in eq. (33) the integral over Lo as compared with the integrals over all other cuts. The equivalence of eqs. (14) and (33) can readily be shown. F o r this purpose we should consider that h*(E)/p(E_) is the discontinuity of the function - [ 2 i p ( E ) ] - 1 on L o. Let us now consider the integral over the contour enclosing all cuts L r of the function M(°)(E')/Lo(E')(E'-E)]. It is easy to express the integral over part of the contour enclosing the cut L o through the integral over the contour enclosing the left cuts Lx(K ~ 0). Then eqs. (14) and (33) will coincide if

AM(E) = M(°)(E+)-M(°)(E_) = 2ix(E ) on LK

(K ~ 0).

(34)

Eq. (33) shows that the solution is factorized into two factors, one of which is connected with the interaction in the final state and the other has only singularities connected with the mechanism of the reaction. A similar result was obtained by Bosco s) who considered the analytical properties of the amplitude in the distorted wave method. In the particular case of M (°) = const, the solution obtained coincides with the solution of the homogeneous problem. For this simple case Jackson 9) considered the connection between the distorted wave method and the dispersion method. 8. Bound States and the Behaviour Near the Threshold Let us now discuss the solution obtained in connection with the behaviour of the amplitude near the threshold of the reaction. I f the system has n bound states for a given l then pt(E) oc E-n when E ~ 0. Since the amplitude of the reaction must be finite at zero energy there arise additional restrictions on M (°) or on the factor of pt(E) in eq. (33). Let M(E) ~ 0 on the large circle. Then we have

M(e) = P(E)r . 7C JL p(E_)(E --E)

(35)

NUCLEAR

383

REACTIONS

If there are bound states formally incorporated in the integral (35) (near the pole E = Ev one has x(E) = rtrv tS(E-Ev) we can rewrite eq. (35), making the pole terms over E explicit:

= p ( E )F l f

p( z!E')dE'E)

"

r,

],

(36)

where L' does not contain poles any longer. The finiteness of the amplitude at E = 0 means that the following relations hold: 1f

;~(E')dE'

~

=,=

rv

= 0,

K = 1, 2, .. n.

(37)

Eqs. (37) connect the discontinuities on the cuts and the residues r, and are the generalization of analogous conditions obtained by Bosco 1o) for the elastic scattering amplitude h(E)/k. Bosco considered the dispersion relation for the scattering amplitude decreasing at infinity. However, for the potentials vanishing at r > R (R is the potential radius) the scattering amplitude in the final part of the plane has no singularities except the fight-hand cut and poles due to bound states, and has an essential singularity at infinity. In this case the Cauchy integral of the function eU'rh(E)/kp(E) (R' > 2R) should be considered, which leads to the following expression for the residue of the elastic scattering amplitude at the pole for E = E o (for simplicity we consider the case when there is only one bound state): r =

e-'k°R'EoP(Eo)f ~°h(e') sin k'R' -

.

Jo

dE'.

(38)

k'E'p(E+)

9. Conclusion The formal scheme of taking into account the interaction in the initial and final states in direct nuclear reactions was considered in this paper. The unitarity of the S-matrix and analytical properties of the amplitude of the reaction as a function of energy are used. The problem of constructing the amplitude of the reaction by its singularities was formally divided into two parts: (1) singularities with respect to t and u determine the mechanism of the reaction described by the function M (°) which has definite discontinuities on the left-hand cuts in the complex plane of energy; and (2) singularities with respect to s other than the poles are connected with the interaction in the initial and final states. Only the second part of the problem was investigated in this paper. The mechanism of the process determining the function M (°) was not considered. The function M (°) and its discontinuities on the left-hand cuts were assumed to be known. It has been assumed that the interaction in the final state does not add new singularities in the unphysical energy region. This can be justified in the case of shortrar, ge forces in the final state when additional singularities connected with the

384

V . A. K A M I N S K Y

AND

Yu.

V. ORLOV

interaction in the final state are situated far from the physical region and are therefore inessential. I f we have long-range forces (i.e., Coulomb interaction) the above singularities cannot be neglected. The amplitude in the form (14) and (33) has all singularities required, according to the condition of the problem, in the complex plane E, satisfies the physical requirements isolating the unique solution and has the correct behaviour near the threshold of the reaction. It should be noted that in the problem under study the amplitude of the reaction was constructed by its singularities on the physical sheet, i.e. the explicitly complex poles of the amplitudes corresponding to the unstable states and lying on the unphysical sheet were not taken into account. However, the presence of complex poles essentially affects the scattering phase which enters the solution. As is clear from eq. (33) the amplitude of the reaction has complex poles corresponding to the scattering matrix resonance contained in the function p(E). Thus, the solution found takes into account the resonance effects of the compound nucleus in direct processes. In conclusion we wish to express our deep gratitude to Professor I. S. Shapiro for his attention to this work and stimulating discussions and to L. D. Blokhintsev and E. I. Dolinsky for discussions.

References I) 2) 3) 4) 5) 6) 7) 8) 9)

I. S. Shapiro, JETP 41 (1961) 1616; Nuclear Physics 28 (1961) 244 L. D. Blokhintsev et aL, Nuclear Physics 40 (1963) 117; JETP 43 (1962) 1914 V. A. Kaminsky and Yu. V. Orlov, JETP 43 (1962) 2146; Nuclear Physics 43 (1963) 236 N. I. Muskhelishvili, Singular integral equations (Fizmatgiz, Moscow, 1962) D. S. Saxon, Proc. of Int. Conf. on Nuclear Structure, Kingston, Canada, 1960 N. Levinson, Mat. Fys. Medd. Dan. Vid. Selsk. 25, No 9 (1949) R. G. Newton, J. Math. Phys. 1 (1960) 319 B. Bosco, Phys. Rev. 123 (1961) 1038 J. D. Jackson, Nuovo Cim. 26 (1962) 342