- Email: [email protected]
Trinucleon binding energies calculated with rank six separable potentials
Volume 32B, number 2
PHYSICS L E T T E R S
TRINUCLEON
BINDING ENERGIES SIX SEPARABLE
T. BRADY
8 June 1970
CALCULATED POTENTIALS
WITH
RANK
~t
Department of Physics and Astronomy, Rensselaer Polytechnic Institute, Troy. Neu, York 12181. USA Received 12 March 1970
A number of separable potentials which include both attraction and repulsion in all partial waves and which provide good fits to the nucleon-nucleon phase parameters are used in calculations of the trinucleon binding energy.
It has b e c o m e c o m m o n p r a c t i c e to employ s e p a r a b l e p o t e n t i a l s in t h r e e nucleon c a l c u l a t i o n s b e c a u s e of the m a t h e m a t i c a l s i m p l i f i c a t i o n s which they p r o d u c e . C a l c u l a t i o n s to date have g e n e r a l l y been done with p o t e n t i a l s having one s e p a r a b l e t e r m in each of the two nucleon p a r t i a l waves. The d i s a d v a n t a g e of using such a one t e r m pote nt i a l is that it can p r o d u c e only a t t r a c t i o n or r e p u l s i o n # but not both. This difficulty can be o v e r c o m e by the use of rank two p o t e n t i a l s , which a r e the sum of two s e p a r a b l e t e r m s , one a t t r a c tive, the o t h e r r e p u l s i v e . A n u m b e r of a u t h o r s [2, 3] have included r e p u l s i o n in the 1S 0 state in this way but have, in g e n e r a l , n e g l e c t e d r e p u l sion in o t h er p a r t i a l waves° It is the p u r p o s e of this l e t t e r to r e p o r t on d e t e r m i n a t i o n s of the t h r e e nucleon binding e n e r g y , BE3, using rank two s e p a r a b l e potent i a l s in the 1S 0 and coupled 381 + 3D 1 p a r t i a l waves° We n e g l e c t the c o n t r ib u t io n s due to higher p a r t i a l w a v e s . C a l c u l a t i o n s [4] e m p l o y i n g F u d a ' s t - m a t r i x p e r t u r b a t i o n t h e o r y [5] have shown that this is a good a p p r o x i m a t i o n . The coulomb f o r c e is a l s o ignored. We can w r i t e the i n t e r a c t i o n in m o m e n t u m s p ace in the f o r m :
* National Defense Eduacation Act Fellow. Present address: Linear Accelerator Laboratory. University of Saskatchewan, Saskatoon. Saskatchewan, Canada. This work constitutes part of the author's Ph.D. thesis. Rensselaer Polytechnic Institute. August 1969. #Tabakin [1] has proposed one term separable potentials which are claimed to produce both attraction and repulsion, however there seem to be physical reasons for reieeting these potentials [e. g. 131.
v(p,p') =
~ y M (~)%S~, ¢~ ~ , ) v * M (]),) L.L' LSJ w , r , . L , SJ
(1)
J.S.M
L , L ' = 0,2
J , S = 0,1
The YL~," ~ ) a r e the n o r m a l i z e d eigenfunctions of j2, L~, M (= Jz) and $2o The a s s u m p t i o n of rank two s e p a r a b i l i t y for the potential m e a n s that: =
SJ,
SJ,
,
_ hSJ
SJ
,
(2) This f o r m contains both a t t r a c t i o n and r e p u l s i o n ; it is not c o m p l e x since I L - L ' I = 0 or 2. An e a r l y potential of this f o r m is due to Tabakin [61]; m o r e r e c e n t l y Mongan [7, 8] has published a number of fits to the two nucleon data using p o t e n t i a l s of the type (2) with a v a r i e t y of functional f o r m s for the gSLJ and h~ 1. Unfortunately T a b a k i n ' s p o t en t i al does not give the c o r r e c t d e u t e r o n binding e n e r g y , but i n st ead a value of 1.18 MeVo We have a t t e m p t e d to c o r r e c t this by r e a d j u s t i n g the p a r a m e t e r s of the a t t r a c t i v e p a r t s of his t r i p l e t p o t en t i al in suc] a way as to r e p r o d u c e the e x p e r i m e n t a l v a l u e s [10] of the d e u t e r o n binding e n e r g y , quadrupole m o m e n t (Q), and the t r i p l e t s c a t t e r i n g length (at) and e f f e c t i v e range (rt)o In this way we have m a n u f a c t u r e d " m o d i f i e d Tabakin" p o t e n t i a l s having d e u t e r o n p e r c e n t D s t a t e s (PD) of 2 and 4% [9]. No higher p e r c e n t D s t a t e s w e r e cons i d e r e d , si n ce f o r i n c r e a s i n g P D the fits to the 3D 1 e i g e n p h a s e and mixing p a r a m e t e r rapidly d e g e n e r a t e d [9]. 85
Volume 32B. number 2
PHYSICS LETTERS
AS a m a t t e r of n o m e n c l a t u r e we will r e f e r to the potentials of ref. [7] as "old" and those of ref. [8] as hew. F o r the old potentials Mongan i n t r o d u c e s what he t e r m s " s p e c i a l r e p u l s i v e " f o r m s for the functions gSrJ ; the s p e c i a l r e p u l s i v e f o r m f a c t o r s a r e those w~lch do not have the b e haviour:
gLSL(p) ~ pL
p~ 0
fn (a) = (,6 2 +a2) -n/2 L o o s e l y speaking a -1 is the range of the potential in c o o r d i n a t e space. The notation is g~0,~ h00 = 11
.11
o
T .T
g ,n 'gL 'nL = g L ' n L "
t)
Case I old s p e c i a l r e p u l s i v e :
gS(p) = CRP~.f a@R )
hS(p) = CA fl(aA)
hTL(P) = cAL pL f.~ +~.(aA)L New:
gS(p)
=
CRfl(a
)
Case II old s p e c i a l r e p u l s i v e :
gS(p) = CRP2f4(a R )
hS(p)
= CAf2(aA)
New:
gS(p) = CRf2(aR) C a s e IV : The s i n g l e t potential is the s a m e as c a s e II new.
C~p f2L+2(aL), The p a r a m e t e r s for these p o t e n t i a l s can be found in refso [71 and [8]; the Case III have not been given s i n ce no c a l c u l a t i o n s w e r e done with these potentials. The c o r r e s p o n d i n g functions for the modified 86
Tabakin p o t e n t i a l s have the f o r m :
fT (d~ a)
= [(p +d) 2 +a 2 ] - i f ( p - d ) 2 +a2] -1
gT (l, ) = hT(p) =
(3)
The h S / m u s t s at i s fy eq. ( 3 ) i n o r d e r to obtain the c o r r e c t t h r e s h o l d b e h a v i o u r for the two body t r a n s i t i o n amplitudes. Mongan c o n s i d e r s four d i f f e r e n t functional f o r m s for eq. (2) which he labels c a s e I, II, etc. T h e s e a r e given below for the old p o t en t i al s , along with any f o r m f a c t o r s that a r e changed for the new potentials° F o r convenience in writting we define:
S . S
8 June 1970
The modified Tabakin potentials w e r e u s e d with Mongan's c a s e II new singlet potentials. T a b a k i n ' s o r i g i n a l p o t en t i al [6] g i v es a value of 7.02 MeV for the t h r e e body binding energy° This a n o m a l o u s l y s m a l l value is p r e s u m a b l y a r e sult of the underbinding of the deuteron° Table 1 shows the low e n e r g y and deuteron p a r a m e t e r s given by the v a r i o u s p o t e n t i a l s (a S and r S a r e , r e s p e c t i v e l y , the singlet s c a t t e r i n g length and e f f e c t i v e range). It should be noted that all of Mongan's p o t e n t i a l s give quite s m a l l PD o Although this quantity is not w e l l d e t e r m i n e d at p r e s e n t , m o s t p h e n o m e n o l o g i c a l and s e m i p h e n o m e n o l o g i c a l fits to the n u c l e o n - n u c l e o n s c a t t e r i n g data favour a PD in the range of 4 to 7%. In this r e s p e c t , then, Mongan's t r i p l e t potentials a r e somewhat unphysical. However, even with this drawback it was c o n s i d e r e d worthwhile to use them in a t r i n u c l e o n c a l c u l a tion for the following r e a s o n s : i) They p r o v i d e the best fit to the phase p a r a m . e t e r s yet obtained with s e p a r a b l e potentials. ii) The v a r i o u s fits a r e in good a g r e e m e n t with each o t h er on the e n e r g y shell. T h e r e f o r e we can gain i n f o r m a t i o n about the s e n s i t i v i t y of BE 3 to d i f f e r e n c e s in the off e n e r g y sh el l b e haviour. The c a l c u l a t i o n s w e r e done using the t h r e e body F ad d eev equations as s p e c i a l i z e d by Stagut [11] to the c a s e of rank N s e p a r a b l e potentials. The r e s u l t s a r e shown in table 1 w h e r e we have included for c o m p a r i s o n binding e n e r gies c a l c u l a t e s with o n e - t e r m Yamaguchi [12] p o t e n t i a l s which lack repulsion, i . e .
gSLJ(P) -~ 0 The h~ J (D) have the s a m e f o r m as the Mongan Case 1I new fit. It can be seen f r o m the table that the old Mongan potentials give c o n s i s t e n t l y s m a l l e r binding e n e r g i e s than the new potentials, Since the fit to the phase p a r a m e t e r s and low e n e r g y data a r e c o m p a r a b l e in both groups, this p r e sumably r e s u l t s f r o m the use of s p e c i a l r e p u l s i v e f o r m s in the old p o t e n t i a l s The f o r m of the 1S 0 potential is e s p e c i a l l y i m p o r t a n t in this respect°
Volume 32B, n u m b e r 2
PHYSICS
LETTERS
8 June 1970
Table 1 Low energy p a r a m e t e r s and three nucleon binding e n e r g i e s for the various potentials as (fro)
rs (fro)
at (fro)
rt (fro)
Q (fin)
PD
BE3
(~)
(MeV)
Case I old (SR) Case II old (SR)
-23.68 -23.67
2.73 2.73
5.48 5.39
1,83 1.72
0.276 0.278
1.0 1.1
8.94 8.76
Case I new Case II new Case IV new
-23.68 -23.68 -23.68
2.73 2.73 2.73
5.65 5.38 5.38
2.04 1.71 1.70
0.277 0.276 0.274
0.7 1.i 1.4
9.36 10.16 10.05
Potential Singlet
T riplet
Mogan Case I] new
Modified Tabakin Modified Tabakin
-23,68 -23.68
2.73 2.73
5.38 5.38
1.71 1.71
0.276 0.276
2.0 4.0
9.76 8.85
Yamaguehi a) Yamaguchi a) Yamaguehi a)
Yamaguchi Yamaguchi Yamaguchi
-20.34 -20.34 -20.34
2.70 2.70 2.70
5.39 5.39 5.39
2.70 2.70 2.70
0.282 0.282 0.282
1.0 2.0 4.0
10.45 9.97 9.01
Case II old (SR) Case II new
Yamaguchi Yamaguchi
-23.67 -23.68
2.73 2.73
5.39 5.39
2.70 2.70
0.282 0.282
1.0 1.0
8.86 10/33
-23.71
2.70
5.43
1.75
0.282
-
8.48
Experiment a) Singlet potential from ref. [14].
L o o s e l y s p e a k i n g , t h e u s e of s p e c i a l r e p u l s i v e f o r m s h a s t h e e f f e c t of i n t r o d u c i n g a s t r o n g , l o n g r a n g e d r e p u l s i o n ; f o r e x a m p l e , f o r t h e c a s e II f i t s we f i n d f r o m r e f s . [7] a n d [8] t h a t CR/C A (old) = 96, CR/CA(new ) = 11, a R1 (old) = 0,258 f m a n d a ~ l ( n e w ) = 0. 16 fm. The binding energies, for the most part, follow t h e t r e n d f o u n d in c a l c u l a t i o n s w i t h o n e t e r m p o t e n t i a l s [13, 14] of t h e s m a l l e r P D g i v i n g t h e l a r g e r b i n d i n g e n r g y . T h e c a s e I new p o t e n t i a l i s a n e x c e p t i o n ; h o w e v e r , t h e f i t to a t a n d r t i s s o m e w h a t p o o r a n d not c o n s i s t e n t w i t h o t h e r potentials. Also, the case I potential has a pathol o g i c a l b e h a v i o u r in t h e s e n s e t h a t w h e n i t i s a p p l i e d to a s q u a r e i n t e g r a b l e f u n c t i o n t h e r e s u l t i n g f u n c t i o n i s not s q u a r e i n t e g r a b l e . T h e e f f e c t of a d d i n g r e p u l s i o n i n t h e v a r i o u s partial waves can be seen by comparing the binding energies given by the Yamaguchi potential having PD = 1% with those given by the various M o n g a n p o t e n t i a l s . T h e c a s e II f i t s a r e t h e m o s t r e l e v a n t , s i n c e t h e s e h a v e t h e s a m e h/~J a s t h e Y a m a g u c h i . T h e r e p l a c e m e n t of t h e Y a m a g u c h i 1S 0 p o t e n t i a l w i t h M o n g a n ' s d e c r e a s e s t h e b i n d i n g e n e r g y b y 1.59 a n d 0.22 M e V f o r t h e o l d a n d new p o t e n t i a l s r e s p e c t i v e l y . It i s w o r t h e m p h a s i z i n g t h a t t h e 1.37 M e V d i f f e r e n c e b e t w e e n t h e s e two r e d u c t i o n s m u s t b e a t t r i b u t e d to off e n e r g y s h e l l e f f e c t s , s i n c e t h e o l d a n d new p o t e n t i a l s give essentially the same phase parameters. In p a r t i c u l a r , t h e u s e of SR f o r m f a c t o r s l o w e r s the e n e r g y c o n s i d e r a b l y . R e s u l t s s i m i l a r to t h e s e
h a v e b e e n o b t a i n e d b y o t h e r a u t h o r s /15] f o r t h e c a s e of p u r e l y a t t r a c t i v e p o t e n t i a l s . T h e a d d i t i o n of r e p u l s i o n in the t r i p l e t s t a t e c a u s e s f u r t h e r d e c r e a s e s in B E 3 of 0.10 a n d 0.17 MeV. This is a rather small effect, and indicates t h a t t h e a d d i t i o n of r e p u l s i o n in t h e t r i p l e t s t a t e i s of m i n o r i m p o r t a n c e a s f a r a s the t h r e e n u cleon system is concerned. Malfliet and Tjon [16] h a v e o b t a i n e d l i k e r e s u l t s in c a l c u l a t i o n s with spin-dependent local potentials~ The binding energies found using the modified Tabakin potentials in the triplet state show a d e p e n d e n c e on P D w h i c h i s v e r y l i k e t h a t f o u n d i n c a l c u l a t i o n s [13, 14] u s i n g one t e r m p o t e n t i a l s . T h e s e c a l c u l a t i o n s i n d i c a t e t h a t the d e p e n d e n c e of B E 3 on P D i s a p p r o x i m a t e l y l i n e a r o v e r the r a n g e P D = 2 to 7 % w i t h a s l o p e A B E 3 / A P D of a b o u t - 0 . 4 MeV. A l i n e a r e x t r a p o l a t i o n of t h e modified Tabakin potential binding energies gives a fit to t h e e x p e r i m e n t a l v a l u e of B E 3 a t P D ~ 5 % w h i c h i s i n a g r e e m e n t w i t h f i t s to t h e two n u c l e o n data using local potentials. Doing a similar ext r a p o l a t i o n w i t h M o n g a n ' s p o t e n t i a l s we f i n d t h a t f o r the s p e c i a l r e p u l s i v e f i t s a c h a n g e i n P D of a b o u t 1% w i l l g i v e t h e r i g h t b i n d i n g e n e r g y , w h i l e f o r t h e new p o t e n t i a l s a c h a n g e i n t h e n e i g h b o r h o o d of 4 % i s r e q u i r e d . T h i s p o i n t s to t h e n e w e r f i t s [8] a s b e i n g t h e m o r e r e a l i s t i c ; h o w e v e r , t h i s c o n c l u s i o n i s d e p e n d e n t not only on t h e v a l i d i t y of t h e e x t r a p o l a t i o n b u t a l s o on t h e m a g n i t u d e of (unknown) c o r r e c t i o n s to B E 3 due to r e l a t i v istic effects and possible three body forces. 87
Volume 32B. number 2
PHYSICS
It s e e m s e v i d e n t f r o m t h e s e c a l c u l a t i o n s t h a t d i f f e r e n t p o t e n t i a l s w h i c h fit t h e s a m e two n u c l e on d a t a c a n g i v e r a t h e r d i f f e r e n t t h r e e b o d y b i n d ing e n e r g i e s . Of c o u r s e t h i s i s not a new c o n c l u s i o n ; m a n y a u t h o r s h a v e p o i n t e d out the i m p o r t a n c e of off e n e r g y s h e l l e f f e c t s in the t h r e e nucleon problem. However, previous caculations h a v e , in g e n e r a l , u s e d p o t e n t i a l s w h i c h did not p r o v i d e a c o m p l e t e fit to the two n u c l e o n d a t a . T h e v a l u e of the w o r k p r e s e n t e d h e r e l i e s in s h o w i n g t h a t t h i s s t r o n g off s h e l l d e p e n d e n c e p e r s i s t s f o r p o t e n t i a l s t h a t a r e r e a l i s t i c , in the s e n s e t h a t t h e y fit t h e p h a s e p a r a m e t e r s w e l l . In a d d i t i o n the e n e r g i e s o b t a i n e d w i t h t h e m o dified Tabakin potentials strongly suggest that the t r e n d s found in w o r k w i t h p u r e l y a t t r a c t i v e s e p a r a b l e p o t e n t i a l s w i l l c a r r y o v e r to r a n k two o r m o r e p o t e n t i a l s , in p a r t i c u l a r the d e p e n d e n c e of BE 3 on P D s h o u l d be s i m i l a r . T h e a u t h o r w o u l d like to e x p r e s s h i s g r a t i tude to Profo J. So L e v i n g e r f o r m a n y v a l u a b l e s u g g e s t i o n s and stimulating d i s c u s s i o n s .
88
LETTERS
8 June 1970
References [1] F. Tabakin. Phys. Rev. 174 {1964) 1208. [2] G . L . S c h r e n k and A . N . M i t r a . Phys. Rev. L e t t e r s 19 (1967) 530. [3] J. Dabrowski and M. Dworzeeka. Phys. L e t t e r s 28B {1968) 4. [4] A. H. Lu. thesis. R e n s s e l a e r Polytechnic Institute 1969. (unpublished). [5] M.G. Fuda. Phys. Rev. 166 (1968) 1064. [61 F.Tabakin. Ann. Phys. 30 (1964) 51. [7] T.R. Mongan. Phys. Rex,. 175 (1968) 1260. [81 T . R . Mongan. Phys. Rev. 178 (1969) 1597. [9] T. Brady thesis. R e s s e l a e r Polytechnic Institute 1966, (unpublished). [10] T. L. Houk and R. Wilson. Rev. Mod. Phys. 39 (1967) 546:40 (1968} 672. [11] R.W. Stagat, Nuel. Phys. A125 (19C9) 654. [12] Y. Yamaguchi. Phys. Rev. 95 (1954) 1635. [13] T. Brady. M. Fuda. E. Harms. J . S . Levinger and R. Stagat. Phys. Rev. 186 {1969} 1069. [14] A.C. Phillips. Nucl. Phys. A107 (1968) 209. [15] V . F . K h a r c h e n k o . N.M. Petrov and S.A. 8torozhenko. Nucl. Phys. A106 {1968) 464. [16] R.A.Malfliet a n d J . A. Tjon. Nucl. Phys. A127 (1969) 161.
PHYSICS L E T T E R S
TRINUCLEON
BINDING ENERGIES SIX SEPARABLE
T. BRADY
8 June 1970
CALCULATED POTENTIALS
WITH
RANK
~t
Department of Physics and Astronomy, Rensselaer Polytechnic Institute, Troy. Neu, York 12181. USA Received 12 March 1970
A number of separable potentials which include both attraction and repulsion in all partial waves and which provide good fits to the nucleon-nucleon phase parameters are used in calculations of the trinucleon binding energy.
It has b e c o m e c o m m o n p r a c t i c e to employ s e p a r a b l e p o t e n t i a l s in t h r e e nucleon c a l c u l a t i o n s b e c a u s e of the m a t h e m a t i c a l s i m p l i f i c a t i o n s which they p r o d u c e . C a l c u l a t i o n s to date have g e n e r a l l y been done with p o t e n t i a l s having one s e p a r a b l e t e r m in each of the two nucleon p a r t i a l waves. The d i s a d v a n t a g e of using such a one t e r m pote nt i a l is that it can p r o d u c e only a t t r a c t i o n or r e p u l s i o n # but not both. This difficulty can be o v e r c o m e by the use of rank two p o t e n t i a l s , which a r e the sum of two s e p a r a b l e t e r m s , one a t t r a c tive, the o t h e r r e p u l s i v e . A n u m b e r of a u t h o r s [2, 3] have included r e p u l s i o n in the 1S 0 state in this way but have, in g e n e r a l , n e g l e c t e d r e p u l sion in o t h er p a r t i a l waves° It is the p u r p o s e of this l e t t e r to r e p o r t on d e t e r m i n a t i o n s of the t h r e e nucleon binding e n e r g y , BE3, using rank two s e p a r a b l e potent i a l s in the 1S 0 and coupled 381 + 3D 1 p a r t i a l waves° We n e g l e c t the c o n t r ib u t io n s due to higher p a r t i a l w a v e s . C a l c u l a t i o n s [4] e m p l o y i n g F u d a ' s t - m a t r i x p e r t u r b a t i o n t h e o r y [5] have shown that this is a good a p p r o x i m a t i o n . The coulomb f o r c e is a l s o ignored. We can w r i t e the i n t e r a c t i o n in m o m e n t u m s p ace in the f o r m :
* National Defense Eduacation Act Fellow. Present address: Linear Accelerator Laboratory. University of Saskatchewan, Saskatoon. Saskatchewan, Canada. This work constitutes part of the author's Ph.D. thesis. Rensselaer Polytechnic Institute. August 1969. #Tabakin [1] has proposed one term separable potentials which are claimed to produce both attraction and repulsion, however there seem to be physical reasons for reieeting these potentials [e. g. 131.
v(p,p') =
~ y M (~)%S~, ¢~ ~ , ) v * M (]),) L.L' LSJ w , r , . L , SJ
(1)
J.S.M
L , L ' = 0,2
J , S = 0,1
The YL~," ~ ) a r e the n o r m a l i z e d eigenfunctions of j2, L~, M (= Jz) and $2o The a s s u m p t i o n of rank two s e p a r a b i l i t y for the potential m e a n s that: =
SJ,
SJ,
,
_ hSJ
SJ
,
(2) This f o r m contains both a t t r a c t i o n and r e p u l s i o n ; it is not c o m p l e x since I L - L ' I = 0 or 2. An e a r l y potential of this f o r m is due to Tabakin [61]; m o r e r e c e n t l y Mongan [7, 8] has published a number of fits to the two nucleon data using p o t e n t i a l s of the type (2) with a v a r i e t y of functional f o r m s for the gSLJ and h~ 1. Unfortunately T a b a k i n ' s p o t en t i al does not give the c o r r e c t d e u t e r o n binding e n e r g y , but i n st ead a value of 1.18 MeVo We have a t t e m p t e d to c o r r e c t this by r e a d j u s t i n g the p a r a m e t e r s of the a t t r a c t i v e p a r t s of his t r i p l e t p o t en t i al in suc] a way as to r e p r o d u c e the e x p e r i m e n t a l v a l u e s [10] of the d e u t e r o n binding e n e r g y , quadrupole m o m e n t (Q), and the t r i p l e t s c a t t e r i n g length (at) and e f f e c t i v e range (rt)o In this way we have m a n u f a c t u r e d " m o d i f i e d Tabakin" p o t e n t i a l s having d e u t e r o n p e r c e n t D s t a t e s (PD) of 2 and 4% [9]. No higher p e r c e n t D s t a t e s w e r e cons i d e r e d , si n ce f o r i n c r e a s i n g P D the fits to the 3D 1 e i g e n p h a s e and mixing p a r a m e t e r rapidly d e g e n e r a t e d [9]. 85
Volume 32B. number 2
PHYSICS LETTERS
AS a m a t t e r of n o m e n c l a t u r e we will r e f e r to the potentials of ref. [7] as "old" and those of ref. [8] as hew. F o r the old potentials Mongan i n t r o d u c e s what he t e r m s " s p e c i a l r e p u l s i v e " f o r m s for the functions gSrJ ; the s p e c i a l r e p u l s i v e f o r m f a c t o r s a r e those w~lch do not have the b e haviour:
gLSL(p) ~ pL
p~ 0
fn (a) = (,6 2 +a2) -n/2 L o o s e l y speaking a -1 is the range of the potential in c o o r d i n a t e space. The notation is g~0,~ h00 = 11
.11
o
T .T
g ,n 'gL 'nL = g L ' n L "
t)
Case I old s p e c i a l r e p u l s i v e :
gS(p) = CRP~.f a@R )
hS(p) = CA fl(aA)
hTL(P) = cAL pL f.~ +~.(aA)L New:
gS(p)
=
CRfl(a
)
Case II old s p e c i a l r e p u l s i v e :
gS(p) = CRP2f4(a R )
hS(p)
= CAf2(aA)
New:
gS(p) = CRf2(aR) C a s e IV : The s i n g l e t potential is the s a m e as c a s e II new.
C~p f2L+2(aL), The p a r a m e t e r s for these p o t e n t i a l s can be found in refso [71 and [8]; the Case III have not been given s i n ce no c a l c u l a t i o n s w e r e done with these potentials. The c o r r e s p o n d i n g functions for the modified 86
Tabakin p o t e n t i a l s have the f o r m :
fT (d~ a)
= [(p +d) 2 +a 2 ] - i f ( p - d ) 2 +a2] -1
gT (l, ) = hT(p) =
(3)
The h S / m u s t s at i s fy eq. ( 3 ) i n o r d e r to obtain the c o r r e c t t h r e s h o l d b e h a v i o u r for the two body t r a n s i t i o n amplitudes. Mongan c o n s i d e r s four d i f f e r e n t functional f o r m s for eq. (2) which he labels c a s e I, II, etc. T h e s e a r e given below for the old p o t en t i al s , along with any f o r m f a c t o r s that a r e changed for the new potentials° F o r convenience in writting we define:
S . S
8 June 1970
The modified Tabakin potentials w e r e u s e d with Mongan's c a s e II new singlet potentials. T a b a k i n ' s o r i g i n a l p o t en t i al [6] g i v es a value of 7.02 MeV for the t h r e e body binding energy° This a n o m a l o u s l y s m a l l value is p r e s u m a b l y a r e sult of the underbinding of the deuteron° Table 1 shows the low e n e r g y and deuteron p a r a m e t e r s given by the v a r i o u s p o t e n t i a l s (a S and r S a r e , r e s p e c t i v e l y , the singlet s c a t t e r i n g length and e f f e c t i v e range). It should be noted that all of Mongan's p o t e n t i a l s give quite s m a l l PD o Although this quantity is not w e l l d e t e r m i n e d at p r e s e n t , m o s t p h e n o m e n o l o g i c a l and s e m i p h e n o m e n o l o g i c a l fits to the n u c l e o n - n u c l e o n s c a t t e r i n g data favour a PD in the range of 4 to 7%. In this r e s p e c t , then, Mongan's t r i p l e t potentials a r e somewhat unphysical. However, even with this drawback it was c o n s i d e r e d worthwhile to use them in a t r i n u c l e o n c a l c u l a tion for the following r e a s o n s : i) They p r o v i d e the best fit to the phase p a r a m . e t e r s yet obtained with s e p a r a b l e potentials. ii) The v a r i o u s fits a r e in good a g r e e m e n t with each o t h er on the e n e r g y shell. T h e r e f o r e we can gain i n f o r m a t i o n about the s e n s i t i v i t y of BE 3 to d i f f e r e n c e s in the off e n e r g y sh el l b e haviour. The c a l c u l a t i o n s w e r e done using the t h r e e body F ad d eev equations as s p e c i a l i z e d by Stagut [11] to the c a s e of rank N s e p a r a b l e potentials. The r e s u l t s a r e shown in table 1 w h e r e we have included for c o m p a r i s o n binding e n e r gies c a l c u l a t e s with o n e - t e r m Yamaguchi [12] p o t e n t i a l s which lack repulsion, i . e .
gSLJ(P) -~ 0 The h~ J (D) have the s a m e f o r m as the Mongan Case 1I new fit. It can be seen f r o m the table that the old Mongan potentials give c o n s i s t e n t l y s m a l l e r binding e n e r g i e s than the new potentials, Since the fit to the phase p a r a m e t e r s and low e n e r g y data a r e c o m p a r a b l e in both groups, this p r e sumably r e s u l t s f r o m the use of s p e c i a l r e p u l s i v e f o r m s in the old p o t e n t i a l s The f o r m of the 1S 0 potential is e s p e c i a l l y i m p o r t a n t in this respect°
Volume 32B, n u m b e r 2
PHYSICS
LETTERS
8 June 1970
Table 1 Low energy p a r a m e t e r s and three nucleon binding e n e r g i e s for the various potentials as (fro)
rs (fro)
at (fro)
rt (fro)
Q (fin)
PD
BE3
(~)
(MeV)
Case I old (SR) Case II old (SR)
-23.68 -23.67
2.73 2.73
5.48 5.39
1,83 1.72
0.276 0.278
1.0 1.1
8.94 8.76
Case I new Case II new Case IV new
-23.68 -23.68 -23.68
2.73 2.73 2.73
5.65 5.38 5.38
2.04 1.71 1.70
0.277 0.276 0.274
0.7 1.i 1.4
9.36 10.16 10.05
Potential Singlet
T riplet
Mogan Case I] new
Modified Tabakin Modified Tabakin
-23,68 -23.68
2.73 2.73
5.38 5.38
1.71 1.71
0.276 0.276
2.0 4.0
9.76 8.85
Yamaguehi a) Yamaguchi a) Yamaguehi a)
Yamaguchi Yamaguchi Yamaguchi
-20.34 -20.34 -20.34
2.70 2.70 2.70
5.39 5.39 5.39
2.70 2.70 2.70
0.282 0.282 0.282
1.0 2.0 4.0
10.45 9.97 9.01
Case II old (SR) Case II new
Yamaguchi Yamaguchi
-23.67 -23.68
2.73 2.73
5.39 5.39
2.70 2.70
0.282 0.282
1.0 1.0
8.86 10/33
-23.71
2.70
5.43
1.75
0.282
-
8.48
Experiment a) Singlet potential from ref. [14].
L o o s e l y s p e a k i n g , t h e u s e of s p e c i a l r e p u l s i v e f o r m s h a s t h e e f f e c t of i n t r o d u c i n g a s t r o n g , l o n g r a n g e d r e p u l s i o n ; f o r e x a m p l e , f o r t h e c a s e II f i t s we f i n d f r o m r e f s . [7] a n d [8] t h a t CR/C A (old) = 96, CR/CA(new ) = 11, a R1 (old) = 0,258 f m a n d a ~ l ( n e w ) = 0. 16 fm. The binding energies, for the most part, follow t h e t r e n d f o u n d in c a l c u l a t i o n s w i t h o n e t e r m p o t e n t i a l s [13, 14] of t h e s m a l l e r P D g i v i n g t h e l a r g e r b i n d i n g e n r g y . T h e c a s e I new p o t e n t i a l i s a n e x c e p t i o n ; h o w e v e r , t h e f i t to a t a n d r t i s s o m e w h a t p o o r a n d not c o n s i s t e n t w i t h o t h e r potentials. Also, the case I potential has a pathol o g i c a l b e h a v i o u r in t h e s e n s e t h a t w h e n i t i s a p p l i e d to a s q u a r e i n t e g r a b l e f u n c t i o n t h e r e s u l t i n g f u n c t i o n i s not s q u a r e i n t e g r a b l e . T h e e f f e c t of a d d i n g r e p u l s i o n i n t h e v a r i o u s partial waves can be seen by comparing the binding energies given by the Yamaguchi potential having PD = 1% with those given by the various M o n g a n p o t e n t i a l s . T h e c a s e II f i t s a r e t h e m o s t r e l e v a n t , s i n c e t h e s e h a v e t h e s a m e h/~J a s t h e Y a m a g u c h i . T h e r e p l a c e m e n t of t h e Y a m a g u c h i 1S 0 p o t e n t i a l w i t h M o n g a n ' s d e c r e a s e s t h e b i n d i n g e n e r g y b y 1.59 a n d 0.22 M e V f o r t h e o l d a n d new p o t e n t i a l s r e s p e c t i v e l y . It i s w o r t h e m p h a s i z i n g t h a t t h e 1.37 M e V d i f f e r e n c e b e t w e e n t h e s e two r e d u c t i o n s m u s t b e a t t r i b u t e d to off e n e r g y s h e l l e f f e c t s , s i n c e t h e o l d a n d new p o t e n t i a l s give essentially the same phase parameters. In p a r t i c u l a r , t h e u s e of SR f o r m f a c t o r s l o w e r s the e n e r g y c o n s i d e r a b l y . R e s u l t s s i m i l a r to t h e s e
h a v e b e e n o b t a i n e d b y o t h e r a u t h o r s /15] f o r t h e c a s e of p u r e l y a t t r a c t i v e p o t e n t i a l s . T h e a d d i t i o n of r e p u l s i o n in the t r i p l e t s t a t e c a u s e s f u r t h e r d e c r e a s e s in B E 3 of 0.10 a n d 0.17 MeV. This is a rather small effect, and indicates t h a t t h e a d d i t i o n of r e p u l s i o n in t h e t r i p l e t s t a t e i s of m i n o r i m p o r t a n c e a s f a r a s the t h r e e n u cleon system is concerned. Malfliet and Tjon [16] h a v e o b t a i n e d l i k e r e s u l t s in c a l c u l a t i o n s with spin-dependent local potentials~ The binding energies found using the modified Tabakin potentials in the triplet state show a d e p e n d e n c e on P D w h i c h i s v e r y l i k e t h a t f o u n d i n c a l c u l a t i o n s [13, 14] u s i n g one t e r m p o t e n t i a l s . T h e s e c a l c u l a t i o n s i n d i c a t e t h a t the d e p e n d e n c e of B E 3 on P D i s a p p r o x i m a t e l y l i n e a r o v e r the r a n g e P D = 2 to 7 % w i t h a s l o p e A B E 3 / A P D of a b o u t - 0 . 4 MeV. A l i n e a r e x t r a p o l a t i o n of t h e modified Tabakin potential binding energies gives a fit to t h e e x p e r i m e n t a l v a l u e of B E 3 a t P D ~ 5 % w h i c h i s i n a g r e e m e n t w i t h f i t s to t h e two n u c l e o n data using local potentials. Doing a similar ext r a p o l a t i o n w i t h M o n g a n ' s p o t e n t i a l s we f i n d t h a t f o r the s p e c i a l r e p u l s i v e f i t s a c h a n g e i n P D of a b o u t 1% w i l l g i v e t h e r i g h t b i n d i n g e n e r g y , w h i l e f o r t h e new p o t e n t i a l s a c h a n g e i n t h e n e i g h b o r h o o d of 4 % i s r e q u i r e d . T h i s p o i n t s to t h e n e w e r f i t s [8] a s b e i n g t h e m o r e r e a l i s t i c ; h o w e v e r , t h i s c o n c l u s i o n i s d e p e n d e n t not only on t h e v a l i d i t y of t h e e x t r a p o l a t i o n b u t a l s o on t h e m a g n i t u d e of (unknown) c o r r e c t i o n s to B E 3 due to r e l a t i v istic effects and possible three body forces. 87
Volume 32B. number 2
PHYSICS
It s e e m s e v i d e n t f r o m t h e s e c a l c u l a t i o n s t h a t d i f f e r e n t p o t e n t i a l s w h i c h fit t h e s a m e two n u c l e on d a t a c a n g i v e r a t h e r d i f f e r e n t t h r e e b o d y b i n d ing e n e r g i e s . Of c o u r s e t h i s i s not a new c o n c l u s i o n ; m a n y a u t h o r s h a v e p o i n t e d out the i m p o r t a n c e of off e n e r g y s h e l l e f f e c t s in the t h r e e nucleon problem. However, previous caculations h a v e , in g e n e r a l , u s e d p o t e n t i a l s w h i c h did not p r o v i d e a c o m p l e t e fit to the two n u c l e o n d a t a . T h e v a l u e of the w o r k p r e s e n t e d h e r e l i e s in s h o w i n g t h a t t h i s s t r o n g off s h e l l d e p e n d e n c e p e r s i s t s f o r p o t e n t i a l s t h a t a r e r e a l i s t i c , in the s e n s e t h a t t h e y fit t h e p h a s e p a r a m e t e r s w e l l . In a d d i t i o n the e n e r g i e s o b t a i n e d w i t h t h e m o dified Tabakin potentials strongly suggest that the t r e n d s found in w o r k w i t h p u r e l y a t t r a c t i v e s e p a r a b l e p o t e n t i a l s w i l l c a r r y o v e r to r a n k two o r m o r e p o t e n t i a l s , in p a r t i c u l a r the d e p e n d e n c e of BE 3 on P D s h o u l d be s i m i l a r . T h e a u t h o r w o u l d like to e x p r e s s h i s g r a t i tude to Profo J. So L e v i n g e r f o r m a n y v a l u a b l e s u g g e s t i o n s and stimulating d i s c u s s i o n s .
88
LETTERS
8 June 1970
References [1] F. Tabakin. Phys. Rev. 174 {1964) 1208. [2] G . L . S c h r e n k and A . N . M i t r a . Phys. Rev. L e t t e r s 19 (1967) 530. [3] J. Dabrowski and M. Dworzeeka. Phys. L e t t e r s 28B {1968) 4. [4] A. H. Lu. thesis. R e n s s e l a e r Polytechnic Institute 1969. (unpublished). [5] M.G. Fuda. Phys. Rev. 166 (1968) 1064. [61 F.Tabakin. Ann. Phys. 30 (1964) 51. [7] T.R. Mongan. Phys. Rex,. 175 (1968) 1260. [81 T . R . Mongan. Phys. Rev. 178 (1969) 1597. [9] T. Brady thesis. R e s s e l a e r Polytechnic Institute 1966, (unpublished). [10] T. L. Houk and R. Wilson. Rev. Mod. Phys. 39 (1967) 546:40 (1968} 672. [11] R.W. Stagat, Nuel. Phys. A125 (19C9) 654. [12] Y. Yamaguchi. Phys. Rev. 95 (1954) 1635. [13] T. Brady. M. Fuda. E. Harms. J . S . Levinger and R. Stagat. Phys. Rev. 186 {1969} 1069. [14] A.C. Phillips. Nucl. Phys. A107 (1968) 209. [15] V . F . K h a r c h e n k o . N.M. Petrov and S.A. 8torozhenko. Nucl. Phys. A106 {1968) 464. [16] R.A.Malfliet a n d J . A. Tjon. Nucl. Phys. A127 (1969) 161.