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Calculation of the photodisintegration cross section of 4He
Volume 27B, number 6
P H Y SIC S L E T T E R S
19 August 1968
CALCULATION OF THE PHOTODISINTEGRATION
CROSS SECTION OF 4He
F. BECK and A. MULLER-ARNKE
Institut flit Theoretische Kernphysik der Technischen Hochschule, Darmstadt, Germany Received 10 June 1968
Older c a l c u l a t i o n s of the @,p) c r o s s section of 4He [1,2] have shown only m o d e r a t e a g r e e m e n t with e x p e r i m e n t [3-5]. A m o r e r e c e n t a t t e m p t [6] to fit the c r o s s s e c t i o n by a t w o - l e v e l r e s o n a n c e , employing r e s u l t s of a p a r t i c l e - h o l e calculation in the o s c i l l a t o r shell model [7] was able to r e produce the e x p e r i m e n t a l l y o b s e r v e d position of the m a x i m u m in the c r o s s section. Such a des c r i p t i o n however cannot p r e d i c t the width. In this c a l c u l a t i o n we t r e a t the p a r t i c l e - h o l e s t a t e s of light nuclei as states of a s y s t e m , cons i s t i n g of t h r e e different p a r t i c l e s , a hole (with negative m a s s ) , a p a r t i c l e , and the core, i n t e r acting via m u t u a l two-body i n t e r a c t i o n s . By i n troducing the k i n e m a t i c s employed by Faddeev [8] with a p p r o p r i a t e changes for the n e g a t i v e m a s s p a r t i c l e , we a r e able to s e p a r a t e the c e n t e r - o f - m a s s motion. F o r the s c a t t e r i n g s t a t e s we have thus a s y s t e m of i n t e g r a l equations in two r e l a t i v e m o m e n t a : the r e l a t i v e m o m e n t u m of the h o l e - c o r e s y s t e m and the m o m e n t u m of the p a r t i c l e r e l a t i v e to the h o l e - c o r e s y s t e m . The fact that the p h y s i c a l l y r e a s o n a b l e h o l e - c o r e s t a t e s a r e bound ones e n a b l e s us to r e s t r i c t our H a m i l t o n i a n to a s u b s p a c e of d i s c r e t e h o l e - c o r e s t a t e s . This p r o j e c t i o n which is p o s s i b l e in a t r a n s l a t i o n a l l y i n v a r i a n t way leads us to a finite s y s t e m of c o u p l e d - c h a n n e l equations in one v a r i a b l e only. It can be shown that the violation of the P a u l i p r i n c i p l e , which has to be expected in such a t r e a t m e n t in g e n e r a l , has no c o n s e q u e n c e s for T = 1 s t a t e s , so the model is well suited for the c a l c u l a t i o n of the final state in the (~, p)-dipole r e a c t i o n of 4He. F u r t h e r m o r e in this p a r t i c u l a r ly s i m p l e case only one hole state e x i s t s , so that j u s t one i n t e g r a l equation for the s c a t t e r i n g a m plitude of the final state has to be solved. The (~, p)-dipole t r a n s i t i o n probability as c a l culated in this model is given by
w h e r e n is the r e d u c e d m a s s of the p-3H s y s t e m , w the energy of the i n c o m i n g photon. ~o(q), ~ h ( q ) a r e the i n i t i a l and final state wave f u n c tions of the p a r t i c l e ejected into the continuum; q is the m o m e n t u m of the p a r t i c l e r e l a t i v e to the h o l e - c o r e s y s t e m . Since each nucleon of 4He can r e p r e s e n t the p a r t i c l e of this model, pf, the density of final s t a t e s , contains an additional factor 4. T h i s can be d e m o n s t r a t e d by c o m p a r i son with the plane wave e x p r e s s i o n for u n d i s t i n guishable p a r t i c l e s . We have c a l c u l a t e d the c r o s s section using two different e x p r e s s i o n s for the d i p o l e - m a t r i x e l e m e n t . Since the equation
Ipl i> = im ~ - - ( E f - E i ) < g j f l r l ¢ i)
(2)
is only c o r r e c t if no exchange f o r c e s a r e p r e s e n t and if ~/i, ~f a r e t r u e eigenfunctions of H, dif.. f e r e n c e s between c r o s s sections c a l c u l a t e d with the right or left side of eq. (2) have to be expected if a p p r o x i m a t e wave functions a r e u s e d or if exchange f o r c e s a r e i m p o r t a n t . Denoting the c r o s s sections c a l c u l a t e d with the right and left side of eq. (2) by ar and ~._ r e s p e c t i v e l y we obtain form eq. (1) and an anPalogous e x p r e s s i o n , employing the right side of eq. (2)
%(~)
-
3(1
2~2 e2 nk~o +¼~2k2r2(k, k) )
x
(3)
V2dk,)2
rl(k', O
27r2e2k ~P@) = 3n¢o(1 +¼~2k2r2(k,
× (k~o(k)-f
k)) ×
(4)
rl(k"k)dd°(k')k,3dk,; o
w=
- ~ ( f d 3 q ~ i ( q ) q z , ~ o ( q ) ) 2 Of
(1)
with k = 42n(¢o -~), where ~ = 19.8 MeV i s the t h r e s h o l d of the (~, p) r e a c t i o n . S y m m e t r i z e d 343
Volume 27B, number 6
PHYSICS LETTERS
19 August 1968
Fig. 1. Photodisintegration cross section of 4He. The points, circles and the dotted line are the compilation of experimental results given in ref. 6. crr and Op are calculated according the eqs. (2) and (3) respectively.
of about 23 MeV and c o n s i s t e n t with the r . m . s . r a d i u s of 4He. It is r e m a r k a b l e that this choice of p a r a m e t e r s , which g i v e s a g r e e m e n t with e x p e r i m e n t , l ead s a l s o to a r a t h e r s m a l l d i f f e r e n c e between a r and ap, as shown in fig. 1. V a r i a t i o n s of the wave function and of the s t r e n g t h of the p a r t i c l e - h o l e potential affect the c r o s s s e c t i o n in a quite d i f f e r e n t way. A wave function which is m o r e c o n c e n t r a t e d in s p a c e l e a d s to a l a r g e r high e n e r g y p a r t , changing only weakly the position of the m a x i m u m , w h e r e a s an i n c r e a s e of the p a r t i c l e - h o l e potential shifts the m a x i m u m d i s t i n c t l y to h i g h e r e n e r g i e s . Thus it is concluded that the r o l e of the p a r t i c l e - h o l e p o t en t i al is e s s e n t i a l in the photo d i s i n t e g r a t i o n of 4He, in a v e r y s i m i l a r way as has been demons t r a t e d f o r h e a v i e r n u cl ei , and t h e r e f o r e a genuine sh el l s t r u c t u r e e f f e c t [6] shows up in this reaction.
boundary conditions have been chosen f o r the i n t e g r a l equation of the s c a t t e r i n g state, r l , the l = 1 component of the H e i t l e r r e a c t i o n m a t r i x , i s then the solution of
References
2.f
t5
16 "
o
115
I 25
nl#
"20
i 30
I 35
i 40
I /,5
E#M.v)
rl(k ,k') Vl(k,k') ?o =
_
V l (k,k")rl (k",k')
(5) k.2dk .
k"2 - k'2
v 1 contains the sum of an a t t r a c t i v e o n e - p a r t i c l e potential and a r e p u l s i v e (since the f i n a l s t a t e has T = 1) p a r t i c l e - h o l e potential. Eq. (5) has been s o l v e d n u m e r i c a l l y . T h e p a r a m e t e r s w e r e chosen as follows. Our one p a r t i c l e potential is such that it g i v e s the o b s e r v e d low e n e r g y p - 4 H e s c a t t e r i n g [9], but with n e g l e c t of the s p i n - o r b i t and Coulomb f o r c e . We employ a p a r t i c l e - h o l e potential of G a u s s i a n shape with a s t r e n g t h of 30 MeV ( r e p u l s iv e ) and a r a n g e of 2.0 fm. F i n a l l y ~ o is taken as the 1 s - e i g e n f u n c tion of a Gaussian potential of 100 MeV depth and a r a n g e of 1.6 fm, c o r r e s p o n d i n g to an e i g e n v a l u e
344
1. J.C.Gunn and J.Irving, Phil. Mag. 42 (1951) 1353. 2. B.H. Bransden, A.C. Douglas and H. H. Robertson, Phil. Mag. 2 (1957) 1211. 3. J . E . P e r r y and S.J.Bame, Phys. Rev. 99 (1955) 1368. 4. D.S.Gemmel and G.A.Jones, Nucl. Phys. 33 (1962) 102. 5. R.G.Clere, J.R.Stewart and R.C.Morrison, Phys. Letters 18 (1965) 316. 6. B.R. Barrett, W.E. Meyerhof and J, D. Walecka, Phys. Letters 22 (1966) 450. 7. A. De-Shalit and J. D. Walecka, Phys. Rev. 147 (1966) 763. 8. L.D. Faddeev, Mathematical aspects of the threebody problem in the quantum scattering theory, Israel Programm for Scientific Translations, Jerusalem 1965. 9. S. Sack, L.C. Biedenharn and G. Breit, Phys. Rev. 93 (1954) 321.
P H Y SIC S L E T T E R S
19 August 1968
CALCULATION OF THE PHOTODISINTEGRATION
CROSS SECTION OF 4He
F. BECK and A. MULLER-ARNKE
Institut flit Theoretische Kernphysik der Technischen Hochschule, Darmstadt, Germany Received 10 June 1968
Older c a l c u l a t i o n s of the @,p) c r o s s section of 4He [1,2] have shown only m o d e r a t e a g r e e m e n t with e x p e r i m e n t [3-5]. A m o r e r e c e n t a t t e m p t [6] to fit the c r o s s s e c t i o n by a t w o - l e v e l r e s o n a n c e , employing r e s u l t s of a p a r t i c l e - h o l e calculation in the o s c i l l a t o r shell model [7] was able to r e produce the e x p e r i m e n t a l l y o b s e r v e d position of the m a x i m u m in the c r o s s section. Such a des c r i p t i o n however cannot p r e d i c t the width. In this c a l c u l a t i o n we t r e a t the p a r t i c l e - h o l e s t a t e s of light nuclei as states of a s y s t e m , cons i s t i n g of t h r e e different p a r t i c l e s , a hole (with negative m a s s ) , a p a r t i c l e , and the core, i n t e r acting via m u t u a l two-body i n t e r a c t i o n s . By i n troducing the k i n e m a t i c s employed by Faddeev [8] with a p p r o p r i a t e changes for the n e g a t i v e m a s s p a r t i c l e , we a r e able to s e p a r a t e the c e n t e r - o f - m a s s motion. F o r the s c a t t e r i n g s t a t e s we have thus a s y s t e m of i n t e g r a l equations in two r e l a t i v e m o m e n t a : the r e l a t i v e m o m e n t u m of the h o l e - c o r e s y s t e m and the m o m e n t u m of the p a r t i c l e r e l a t i v e to the h o l e - c o r e s y s t e m . The fact that the p h y s i c a l l y r e a s o n a b l e h o l e - c o r e s t a t e s a r e bound ones e n a b l e s us to r e s t r i c t our H a m i l t o n i a n to a s u b s p a c e of d i s c r e t e h o l e - c o r e s t a t e s . This p r o j e c t i o n which is p o s s i b l e in a t r a n s l a t i o n a l l y i n v a r i a n t way leads us to a finite s y s t e m of c o u p l e d - c h a n n e l equations in one v a r i a b l e only. It can be shown that the violation of the P a u l i p r i n c i p l e , which has to be expected in such a t r e a t m e n t in g e n e r a l , has no c o n s e q u e n c e s for T = 1 s t a t e s , so the model is well suited for the c a l c u l a t i o n of the final state in the (~, p)-dipole r e a c t i o n of 4He. F u r t h e r m o r e in this p a r t i c u l a r ly s i m p l e case only one hole state e x i s t s , so that j u s t one i n t e g r a l equation for the s c a t t e r i n g a m plitude of the final state has to be solved. The (~, p)-dipole t r a n s i t i o n probability as c a l culated in this model is given by
w h e r e n is the r e d u c e d m a s s of the p-3H s y s t e m , w the energy of the i n c o m i n g photon. ~o(q), ~ h ( q ) a r e the i n i t i a l and final state wave f u n c tions of the p a r t i c l e ejected into the continuum; q is the m o m e n t u m of the p a r t i c l e r e l a t i v e to the h o l e - c o r e s y s t e m . Since each nucleon of 4He can r e p r e s e n t the p a r t i c l e of this model, pf, the density of final s t a t e s , contains an additional factor 4. T h i s can be d e m o n s t r a t e d by c o m p a r i son with the plane wave e x p r e s s i o n for u n d i s t i n guishable p a r t i c l e s . We have c a l c u l a t e d the c r o s s section using two different e x p r e s s i o n s for the d i p o l e - m a t r i x e l e m e n t . Since the equation
Ipl i> = im ~ - - ( E f - E i ) < g j f l r l ¢ i)
(2)
is only c o r r e c t if no exchange f o r c e s a r e p r e s e n t and if ~/i, ~f a r e t r u e eigenfunctions of H, dif.. f e r e n c e s between c r o s s sections c a l c u l a t e d with the right or left side of eq. (2) have to be expected if a p p r o x i m a t e wave functions a r e u s e d or if exchange f o r c e s a r e i m p o r t a n t . Denoting the c r o s s sections c a l c u l a t e d with the right and left side of eq. (2) by ar and ~._ r e s p e c t i v e l y we obtain form eq. (1) and an anPalogous e x p r e s s i o n , employing the right side of eq. (2)
%(~)
-
3(1
2~2 e2 nk~o +¼~2k2r2(k, k) )
x
(3)
V2dk,)2
rl(k', O
27r2e2k ~P@) = 3n¢o(1 +¼~2k2r2(k,
× (k~o(k)-f
k)) ×
(4)
rl(k"k)dd°(k')k,3dk,; o
w=
- ~ ( f d 3 q ~ i ( q ) q z , ~ o ( q ) ) 2 Of
(1)
with k = 42n(¢o -~), where ~ = 19.8 MeV i s the t h r e s h o l d of the (~, p) r e a c t i o n . S y m m e t r i z e d 343
Volume 27B, number 6
PHYSICS LETTERS
19 August 1968
Fig. 1. Photodisintegration cross section of 4He. The points, circles and the dotted line are the compilation of experimental results given in ref. 6. crr and Op are calculated according the eqs. (2) and (3) respectively.
of about 23 MeV and c o n s i s t e n t with the r . m . s . r a d i u s of 4He. It is r e m a r k a b l e that this choice of p a r a m e t e r s , which g i v e s a g r e e m e n t with e x p e r i m e n t , l ead s a l s o to a r a t h e r s m a l l d i f f e r e n c e between a r and ap, as shown in fig. 1. V a r i a t i o n s of the wave function and of the s t r e n g t h of the p a r t i c l e - h o l e potential affect the c r o s s s e c t i o n in a quite d i f f e r e n t way. A wave function which is m o r e c o n c e n t r a t e d in s p a c e l e a d s to a l a r g e r high e n e r g y p a r t , changing only weakly the position of the m a x i m u m , w h e r e a s an i n c r e a s e of the p a r t i c l e - h o l e potential shifts the m a x i m u m d i s t i n c t l y to h i g h e r e n e r g i e s . Thus it is concluded that the r o l e of the p a r t i c l e - h o l e p o t en t i al is e s s e n t i a l in the photo d i s i n t e g r a t i o n of 4He, in a v e r y s i m i l a r way as has been demons t r a t e d f o r h e a v i e r n u cl ei , and t h e r e f o r e a genuine sh el l s t r u c t u r e e f f e c t [6] shows up in this reaction.
boundary conditions have been chosen f o r the i n t e g r a l equation of the s c a t t e r i n g state, r l , the l = 1 component of the H e i t l e r r e a c t i o n m a t r i x , i s then the solution of
References
2.f
t5
16 "
o
115
I 25
nl#
"20
i 30
I 35
i 40
I /,5
E#M.v)
rl(k ,k') Vl(k,k') ?o =
_
V l (k,k")rl (k",k')
(5) k.2dk .
k"2 - k'2
v 1 contains the sum of an a t t r a c t i v e o n e - p a r t i c l e potential and a r e p u l s i v e (since the f i n a l s t a t e has T = 1) p a r t i c l e - h o l e potential. Eq. (5) has been s o l v e d n u m e r i c a l l y . T h e p a r a m e t e r s w e r e chosen as follows. Our one p a r t i c l e potential is such that it g i v e s the o b s e r v e d low e n e r g y p - 4 H e s c a t t e r i n g [9], but with n e g l e c t of the s p i n - o r b i t and Coulomb f o r c e . We employ a p a r t i c l e - h o l e potential of G a u s s i a n shape with a s t r e n g t h of 30 MeV ( r e p u l s iv e ) and a r a n g e of 2.0 fm. F i n a l l y ~ o is taken as the 1 s - e i g e n f u n c tion of a Gaussian potential of 100 MeV depth and a r a n g e of 1.6 fm, c o r r e s p o n d i n g to an e i g e n v a l u e
344
1. J.C.Gunn and J.Irving, Phil. Mag. 42 (1951) 1353. 2. B.H. Bransden, A.C. Douglas and H. H. Robertson, Phil. Mag. 2 (1957) 1211. 3. J . E . P e r r y and S.J.Bame, Phys. Rev. 99 (1955) 1368. 4. D.S.Gemmel and G.A.Jones, Nucl. Phys. 33 (1962) 102. 5. R.G.Clere, J.R.Stewart and R.C.Morrison, Phys. Letters 18 (1965) 316. 6. B.R. Barrett, W.E. Meyerhof and J, D. Walecka, Phys. Letters 22 (1966) 450. 7. A. De-Shalit and J. D. Walecka, Phys. Rev. 147 (1966) 763. 8. L.D. Faddeev, Mathematical aspects of the threebody problem in the quantum scattering theory, Israel Programm for Scientific Translations, Jerusalem 1965. 9. S. Sack, L.C. Biedenharn and G. Breit, Phys. Rev. 93 (1954) 321.