Parametrization of the deuteron wave function of the paris n-n potential

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Parametrization of the deuteron wave function of the

paris n-n potential

M. Lacombe, B. Loiseau, R. Vinh Mau, J. Cote, P. Pires, R. de Tourreil

To cite this version:

M. Lacombe, B. Loiseau, R. Vinh Mau, J. Cote, P. Pires, et al.. Parametrization of the deuteron wave

function of the paris n-n potential. Physics Letters B, Elsevier, 1981, 101, pp.139-140.

�10.1016/0370-2693(81)90659-6�. �in2p3-00017487�

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Volume 101B, number 3 PHYSICS LETTERS 7 May 1981

PARAMETRIZATION OF THE DEUTERON WAVE FUNCTION OF THE PARIS N - N POTENTIAL

M. LACOMBE 1, B. LOISEAU and R. VINH MAU

Division de Physique Thdorique 2, lnstitut de Physique Nucl&ire, Orsay 91406, France

and LPTPE, Universitd Pierre et Marie Curie, Paris 75230, France

J. COT/~ and P. PIRI~S

Division de Physique Thdorique, lnstitut de Physique Nucl&ire, Orsay 91406, France

and

R. de TOURRE1L

Laboratoire GANIL, F-14201 Caen, France

Received 22 January 1981

We present a convenient analytical parametrization, in both configuration and momentum spaces, of the deuteron wavefunction calculated with the Paris potential.

The deuteron wavefunction of the recently pub- lished Paris potential [1 ] reproduces quite well the known low energy properties and the electromagnetic form factor A ( q 2) o f the deuteron. In particular the predicted value of the asymptotic D- to S-wave ratio ~, 0.02608, is in excellent agreement with the recent high precision measurements o f ref. [2], 0.0259 -+ 0.0007, and ref. [3], 0.02649 + 0.00043. It has been shown furthermore by Klarsfeld et al. [4] that for internucleon distances r ~ 0.6 fro, this wavefunction satisfies rigorous bounds imposed by the experimental deuteron data and now improved by the new experimental values of ~ [2,3].

In view of the use of this wavefunction in low and medium energy nuclear reaction problems, numerical values have been given in table VI of ref. [1]. We have also found that a cubic spline interpolation of this table reproduces very accurately the original wavefunction. The numerical values and the cubic spline interpolation subroutine can be obtained on request. However, since in some cases an analytical form is use- 1 Facult~ des Sciences de Reims, 51-Reims, France.

Laboratoke associ~ au CNRS.

ful, we present here an algebraic parametrization of the Paris-potential deuteron wavefunction, in both configuration and momentum spaces.

It is appropriate to parametrize the radial wavefunc. tion components

U(r)/r

and

W(r)/r

as a discrete super- position of Yukawa-type terms since this was done for the potential itself. Therefore

U(r)

and

W(r)

of table VI of ref. [1 ] are parametrized as follows

n Ua(r ) = ~

C j e x p ( - m j r ) ,

J = l

(1)

n Wa(r ) = ~

D j e x p ( - m j r ) ( 1 + 3~m j r + 3/m2r2),

J=l

where the subscript a refers to "analytical".

The corresponding m o m e n t u m space wavefunction components are n

Ua(P)/p

= (2/701/2 ~

Cs/(p 2 + m~),

J = l n (2)

Wa(p)/p

= (2/1r)1/2 ~ .

D j / ( p 2 + m2).

J=l

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Volume 101B, number 3 PHYSICS LETTERS 7 May 1981 T h e b o u n d a r y c o n d i t i o n s U a ( r ) ~ r a n d Wa(r ) -+ r 3 as r -+ 0 i m p l y 3 n ~ n n

,m2--o

J = l J = l =

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I n o r d e r t o satisfy t h e s e c o n s t r a i n t s w i t h a c c u r a c y t h e last c o e f f i c i e n t o f Ua(r ) a n d t h e last t h r e e o f Wa(r) are t o b e c o m p u t e d w i t h t h e f o l l o w i n g f o r m u l a e n - 1

C n = - C C j ,

J = l O n _ 2 =

_(.,2

m~_2

2

2 _ . , 2 _ 2 )

n - mn-2)(mn-1

n_3D

2 2 ~ ~'J+

2 +m 2)

X -mn-lmn

(mn-1

J= 1 m 2

n-3

× C

D j -

C

Djm 2)

J = l J = l (4) a n d t w o o t h e r r e l a t i o n s d e d u c e d b y c i r c u l a r p e r m u t a - t i o n o f n - 2, n - 1 a n d n. T h e masses

mj

are c h o s e n t o b e

mj = o~ + (J -

1 ) m 0 , w i t h m 0 = 1 f m - 1 a n d

a = (2mRIEDI)I/2/h

= 0 . 2 3 1 6 2 4 6 1 f m - 1 , w h e r e m R a n d E D are t h e n e u t r o n p r o t o n r e d u c e d mass a n d t h e d e u t e r o n b i n d i n g e n e r g y o f ref. [ 1 ] . This c h o i c e e n s u r e s t h e c o r r e c t a s y m p t o t i c b e h a v i o r . T h e values o f t h e p a r a m e t e r s C j a n d D j are l i s t e d in t a b l e 1. T h e a c c u r a c y o f t h i s p a r a m e t r i z a t i o n is i l l u s t r a t e d b y t h e m a g n i t u d e o f

,s (0 I

=

drI,, ,

-

r,12)"2

= 3 . 5 X 1 0 - 4 Table 1

Coefficients of the parametrized deuteron wavefunction components. The last

Cj

and the last three

Dj

are to be computed from eq. (4). The notation a -+ n stands for a X 10 +-n.

Cj

(fm -1/2)

Dj(fm -1/2)

0.88688076 + 00 0.23135193 - 01 -0.34717093 + 00 - 0 . 8 5 6 0 4 5 7 2 + 00 - 0 . 3 0 5 0 2 3 8 0 + 01 0.56068193 + 01 0.56207766 + 02 - 0 . 6 9 4 6 2 9 2 2 + 02 - 0 . 7 4 9 5 7 3 3 4 + 03 0.41631118 + 03 0.53365279 + 04 -0.12546621 + 04 -0.22706863 + 05 0.12387830 + 04 0.60434469 + 05 0.33739172 + 04 - 0 . 1 0 2 9 2 0 5 8 + 06 -0.13041151 + 05 0.11223357 + 06 0.19512524 + 05 -0.75925226 + 05 see eq. (4) 0.29059715 + 05 see eq. (4) see eq. (4) see eq. (4) a n d

ID = ( ? dr[W(r)- Wa(r)]2)l/2=4.0 X 10 -4.

We h a v e also c h e c k e d t h a t this p a r a m e t r i z a t i o n re- p r o d u c e s v e r y well t h e d e u t e r o n p a r a m e t e r s a n d f o r m f a c t o r o b t a i n e d in ref. [ 1 ].

References

[I ] M. Lacombe et al., Phys. Rev. C21 (1980) 861. [2] W. Grfiebler, V. K6nig, P.A. Schmelzbach, B. Jenny and

F. Sperisen, Phys. Lett. 92B (1980) 279.

[3] K. Stephenson and W. Haeberli, Phys. Rev. Lett. 45 (1980) 520.

[4] S. Klarsfeld, J. Martorell and D.W.L. Sprung, Nucl. Phys. A352 (1981) 113;

S. Klarsfeld, private communication.