Dispersion relations for elastic n-d scattering in the Amado model

Volume 48B, number 1

DISPERSION

PHYSICS LETTERS

RELATIONS

FOR ELASTIC n-d SCATTERING

7 January 1974

IN T H E A M A D O M O D E L

A T STELBOVICS and L.R. DODD Department of Mathemattcal Physics. UmverslO' of ddelaMe, AdelaMe, S Austraha 5001

Received 15 November 1973 The formulatmn of rigorous, dispersion relations for on-shell three-body amplitudes m a separable model is reported The results of approximate N/D calculations of s-wave n-d elastic scattering are compared with exact, numerical solutmns of the Faddeev equatmns While calculations of elastic n-d scattering based on on-shell dispersion relations [1,2] have met with some success, it is difficult to assess their rehabthty because of the number of theorettcal approximations and experimental uncertainties Involved. The work reported m this letter was undertaken In order to test the on-shell approach to the calculation of threebody scattering amplitudes within the context o f a well-known separable-potential model [3] where comparison with accurate numerical solutions of the Faddeev equations is possible Rigorous dispersion relations for three-particle scattering anaplatudes have been derived by Rubln, Sugar and Tlktopoulos [4] for a system of three particles interacting via local two-body Yukawa forces It can be shown [5] that their results are essentially unchanged if the Yukawa potential is replaced by a sum of separable Interactions In such a way that the two-body t-matrices have the same bound state poles Except for the possibility of simple poles in the first order terms, all the singularities of the on-shell three-particle amphtudes lie on the real axis of the complex energy plane and consist of the right hand unitarlty cuts, the left hand potential cuts and under certain condmons associated with two particle bound states, anomalous singularities. It follows that the on-shell amplitudes o f the Amado model [3] of n-d scattering, in which the two-body interactions are separable, non-local Yamaguchi potentmls, satisfy dispersion relatmns of the type formulated In ref. [4] * Because accurate solution o f the Faddeev equations * It is shown m ref [5] that the elastic n-d amphtudes have no anomalous slngularlt~es

with separable potentials is straightforward by the method of rotation of contours [6], the Anlado model provides a means of testing the three-body dispersion relations and investigating the utility of various approximations xn their solutions In this note, we present the results of a direct comparison of solutions of the on-shell N / D equations for s-wave elastic n-d scattering with exact solutions of the offshell Faddeev equations for the Amado model The N / D equations m the form in which N is determined by a non-singular integral equation involving only the inelasticity and an input functton B(E), have been used to represent the analytic structure of the quartet and doublet amplitudes. The inelastlctty which was regarded as part of the input information**, was found dlrectl2¢ from the solutions of the Faddeev equatmns The input function B(E), which is defined by an integral ove) the discontinuity of the amplitude T(E) on the left hand cut was calculated from the following formula, 1 B(E) = Re T(E +le) - - ~ P

f lm

r+ T(E le) dE, E' - E

(1)

ER which only requires a knowledge o f the amphtude for energies on the right hand cut In order to check the dispersion-relations, the exact input was computed from the accurate, numerical solutions of the Faddeev equations On solving the N / D equations with this input, the exact amplitude was reproduced both in the quartet and doublet cases. ** If three-particle umtanty is used then a system of coupled dispersion relations revolving elastic, breakup and free particle amphtudes must be solved 13

Volume 48B, number 1

PHYSICS LETTERS

22 20 1 8

16

1 4

12

1 0

I

I

20

20

60

Energy (MeV)

Fig 1 Inelastlcltles for the quartet and doublet channels are shown as a function of centre of mass energy an the Amado model (Inelasticity P = (I/kl TI2)Im T

7 January 1974

In the usual apphcatlons o f the N/D method, the exact input is not known but must be approximated In two-body potential scattering the most convenient approximation is based on the multiple scattering series obtained by iteration o f the L i p p a m n n Schwmger equation. Following this approach we have solved the doublet and quartet dispersion relations using input functions calculated from the first to the fourth order terms of the iteratlve solution of the Faddeev equations. In fig. 1 the melasticlties for the quartet and doublet channels in the Amado model are shown. In the quartet channel the inelasticity increases slowly at energies just above the threshold for breakup and is approximately linear at higher energies. In the doublet channel, the coupling between the neutron and the slnglet antibound state has an important influence; there is an abrupt rise m the inelasticity near threshold but at higher energies the inelasticity is again almost linear. In fig. 2 the approximate input functions calculated from the multiple scattering series are compared with the exact input functions, eq. (1). In the doublet case we note that the approxl-

I I

I

i

O0

0 2

i

I

-

Z

40

60

o21~\\\\.a

I

l

~

--

--0

A -04 A

2,

-o,i///,,, -02

-, I

0

20

I

I

40

60

Energy (MeV)

ok~i~,, 11

0

20

Energy ( M e V )

Fig 2 Inputs labelled I-IV are derwed from the first four orders of the multiple scattering serxes (a) Quartet channel The exact input from the Faddeev equations is represented by the dashed curve (b) Doublet channel. The dashed curve labelled a is the exact input calculated from the Faddeev equations. The dashed curve b represents the exact input with a contribution from the tmton pole substracted 14

Volume 48B, number 1

PHYSICS LETTERS

7 January 1974

Table 1 Real parts of phase shifts and absorption coefficients (r/= e x p ( - 26 i)) for varying centre of mass energies The first four sets are from the solution of the N/D equations with the approximate inputs. The exact set has been obtained by solution of the Faddeev equations For the doublet channel the triton binding energy is shown for all the input QUARTET 1

2

3

E(MeV)

6R

"0

6R

r~

- 2 225 -2 0 5 10 12 20 25 30 40 50 60 70

180 0 151 0 109 3 83.5 73 2 66 7 61 4 58 1 54 3 48 5 43 8 39 8 36 2

1 000 1 000 1 000 0 992 0 957 0 914 0 876 0 838 0.810 0 766 0.735 0 716 0 707

180 0 150 9 107 7 97 6 65 l 55.8 48 6 43 0 38 0 30.0 24 4 19.8 16.3

1.000 1.000 1 000 0 992 0 961 0 930 0 907 0 893 0 886 0 888 0 898 0 913 0 928

4

6R 180 150 107 78 65 55 48 42 37 30 25 21 17

0 5 8 7 0 8 7 8 8 7 1 0 8

~

exact

6R

77

-i.000 1 000 1 000 0 992 0.961 0 930 0 907 0 893 0 887 0 885 0 893 0.904 0 915

180 0 151 2 107 4 78 6 64 8 55 5 48 3 42 4 37 5 29 8 24 2 19 8 16 3

1 000 1 000 1 000 0 992 0 961 0 930 0.908 0 895 0 889 0 890 0 899 0.913 0 921

180 150 107 78 65 55 48 42 38 30 25 21 17

6R 0 7 7 7 2 9 7 9 1 7 2 1 9

1 000 1 000 1 000 0 992 0 962 0 930 0 908 0 893 0 886 0 884 0 892 0 903 0 914

rt

1 000 1 000 1 000 0431 0294 0 369 0473 0 576 0 637 0736 0811 0 872 0 934

180 0 178 0 154 6 1283 1037 77.5 564 47 2 40 8 322 26 2 21 4 17 6

1 000 1 000 1 000 0515 0310 0 278 0364 0 437 0 506 0622 0706 0 780 0 835

180 0 181 0 159 0 1358 115.0 91 5 706 57 0 48 7 382 31 8 27 3 23 9

1 000 1 000 1 000 0595 0358 0 263 0.276 0 333 0 396 0516 0591 0 668 0 707

DOUBLET - 2 225 -2 0 5 10 15 20 25 30 40 50 60 70 Triton binding energy

00 65 6 39 8 192 120 84 6 2 4.6 3.4 17 0.6 0 1 -0 7

1.000 1 000 1.000 0898 0.953 0 975 0986 0 992 0 995 0999 1 000 1 000 1 000

No bound state

180 0 162 7 130 8 898 539 40 3 318 25 5 21 5 152 105 69 32

1 000 1 000 1 000 0344 0442 0 577 0688 0 774 0 826 0902 0948 0 976 0 994

-51

180 0 173 0 146.9 1180 847 59 0 46 3 37 7 33 0 259 205 16 1 12 6 77

94

-110

In table 1 our results for the phase shifts and ab-

m a l e inputs do n o t converge to the e x a c t input. The reason for this b e h a v l o u r IS t h a t t h e m e t h o d used to

s o r p t i o n c o e f f i c i e n t s are listed. In the q u a r t e t c h a n n e l ,

c o m p u t e the e x a c t i n p u t f r o m the F a d d e e v e q u a t i o n s

the a g r e e m e n t o f the s e c o n d , t h i r d and f o u r t h o r d e r

includes a c o n t r i b u t i o n f r o m the t r i t o n pole w h i c h is not p r e s e n t m the m u l t i p l e scattering series B o t h f o r m s o f i n p u t satisfy the

N/D r e p r e s e n t a t i o n

of the

a m p h t u d e , in the case o f the e x a c t i n p u t t h e t r i t o n occurs as a pole o f the N f u n c t i o n r a t h e r t h a n as a zero o f t h e D f u n c t i o n . In fig. 2b the d a s h e d curve b r e p r e s e n t s the e x a c t m p u t w i t h the c o n t r i b u t i o n f r o m the t r i t o n pole s u b t r a c t e d .

a p p r o x i m a t i o n s w i t h the e x a c t values is r e m a r k a b l y g o o d . In the m o r e c o m p l e x d o u b l e t c h a n n e l , the results are less s a t i s f a c t o r y The phase shifts in all orders decrease t o o q m c k l y as the e n e r g y increases, a l t h o u g h t h m r values i m p r o v e w i t h increasing o r d e r o f the Input f u n c t i o n . Similarly the a b s o r p n o n coefficients r e p r o d u c e the & p o f the e x a c t c o e f f i c i e n t b u t are t o o large at low energies and t o o small at 15

Volume 48B, number 1

PHYSICS LETTERS

higher energies. The results do, however, appear to be convergang slowly to the exact values. The binding energy of the triton, which is determined by the posltaon of the zero of the Fredholm denominator, although also approaching the exact model value, falls short of it by 1.6 MeV m the fourth order approximation. It is clear from the details of the calculations that while the solution of the quartet channel N/D equation is relatively lnsensltwe to the input the doublet channel is quite sensltwe to small changes m the input. We conclude that the N/D method is a practical way of umtarlzlng the multiple scattering series and is capable of reproducing the features of the exact amphtudes in the Amado model with reasonable accuracy Although not competing m accuracy with complete solutions of the off-shell Faddeev equations, the on-shell dispersion relations would

16

7 January 1974

appear to provide a useful tool in the phenomenologlcal analysis of n-d scattering.

References [1] G Barton andAC Phllhps, Nucl Phys A132(1969) 97 [2] Y Avlshal, W Ebenhoh and A.S Rmat-Rexner, Annals of Physics 55 (1969) 341 [3] R D Amado, Phys Rev 131 (1963) 485, R Aaron, R D Amado and ¥.Y Yam, Phys Rev. 140 (1965) B1291, R Aaron and R.D Amado, Phys. Rev. 150 (1966) 857 [4] M Rubm, R Sugar and G. Tiktopoulos, Phys Rev 146 (1966) 1130, 159 (1967) 1348, 162 (1967) 1555 [5] A StelbovIcs, Ph D. thesis, University of Adelaide (unpubhshed) [6] J H Hertherlngtom and L H Schlck, Phys. Rev 137 (1965) B935