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Dalitz diagrams and decay of 12C states
Volume 19, number 4
PHYSICS LETTERS
DALITZ
DIAGRAMS
AND DECAY
1 November 1965
OF
12C
STATES
N. MACDONALD D e p a r t m e n t o f Natural Philosophy, The University, Glasgow
Received 24 September 1965
The s y m m e t r i c Dalitz d i a g r a m [1] has been used [2] to study the r e a c t i o n p + l i b ~ 12C* --, 3 a . However c e r t a i n g e n e r a l f e a t u r e s of this and r e l a t e d r e a c t i o n s , which can be exhibited in t e r m s of these d i a g r a m s , do not s e e m to have been pointed out. They a r e s i m i l a r to f e a t u r e s of the decay o~ ~ 3~ u s e d [3] to d e t e r m i n e the spin and p a r i t y of the w meson. In d i s c u s s i n g the k i n e m a t i c s of this s y s t e m we work in the 12C r e s t f r a m e so that m o m e n t a a r e coplanar. We denote k i n e t i c e n e r g i e s by T1, T 2 and T3, and t h e i r s u m by Q. We m a y d i s t i n guish between s t a n d a r d Dalitz d i a g r a m s which plot T 1 a g a i n s t T2, and s y m m e t r i c Dalitz diag r a m s which plot T1, T2, T 3 along axes making a n g l e s of 120 ° . The f i r s t p r o p e r t y we d i s c u s s u s e s the facts that a - p a r t i c l e s have z e r o spin and positive p a r i t y . The b o u n d a r y of the d i a g r a m c o r r e s p o n d s to c o l l i n e a r m o m e n t a [4]. C o n s e quently we m a y take the axis of q u a n t i s a t i o n along t h e i r c o m m o n d i r e c t i o n . We have J = 11 + ! 2 + 13 where the I i a r e o r b i t a l angul.ar m o m e n t a , so that m i = 0 a n d ~ = (-) I1+12+13. By c o n s i d e r i n g the coupling I 1 + 12 = 1, ! + 13 = J, and u s i n g the r e s u l t that the v e c t o r coupling coefficient (Jl O J2 O [ J3 O) = 0 if J l + J2 + J3 is odd, one s e e s that 11 + l~ + l~ + J is even and so the p a r i t y is (-)J. So any state of 12C with u n n a t u r a l p a r i t y decays so a s to avoid this r e g i o n of phase space. The second p r o p e r t y m a k e s u s e of the s y m m e t r y of the w a v e - f u n c t i o n u n d e r exchange of the a - p a r t i c l e s . Let us c o n s i d e r the situation in which all t h r e e m o m e n t a a r e of equal magnitude and at a n g l e s of 120 ° to each other. This c o r r e sponds to the c e n t r e point of the s y m m e t r i c d i a g r a m , or to the point -~Q, ~Q in the s t a n d a r d d i a g r a m . It is r e a d i l y shown that s t a t e s with a n g u l a r m o m e n t u m add p a r i t y 1- and 2- in 12C decay so as to avoid this r e g i o n in p h a s e space ~. The a r g u m e n t is a s follows. Take the Z axis p e r p e n d i c u l a r to the p l a n e of the m o m e n t a . The Inspection of the tables in [5] shows that 0- and 1+ states can not decay directly to 3a states.
o r i g i n a l configuration is r e s t o r e d ff one r o t a t e s through {~ about the Z axis and i n t e r c h a n g e s two p a i r s of p a r t i c l e s . This i m p l i e s exp({~ mi) = = 1. So, if J < 3, m = 0 . Now r e f l e c t the x - y p l a n e in the o r i g i n and rotate through ~ about the Z axis. T h i s r e s t o r e s the o r i g i n a l configuration and so r e q u i r e s that the p a r i t y should be positive. A r g u m e n t s b a s e d on s y m m e t r y of a 3a state should be t r e a t e d with caution when an i n t e r m e d i a t e 8Be state may be involved. If such a s i t u a t i o n gave a s h a r p r e s o n a n c e c o n t a i n i n g a l m o s t all the events, the p r e s e n t r e s u l t s could t e l l us nothing new, since whatever the 12C spin and p a r i t y the r e s o n a n c e would not fall at the c e n t r e or on the edge of the Dalitz d i a g r a m . However, in p r a c t i c e one d i s c u s s e s [6, 7] the m o d e s of decay on the b a s i s of j u d g e m e n t s r e g a r d i n g the p r e s e n c e of r e s o n a n c e s a g a i n s t a " d i r e c t " 12C* ~ 3a background. It is c l e a r l y n e c e s s a r y to a p p r e c i a t e any special f e a t u r e s of the b a c k g r o u n d which m a y affect these judgements. In d i s c u s s i n g the actual e x p e r i m e n t a l situation one c o m e s up a g a i n s t the fact that both T = 0 and T = 1 a r e found [6-8] to b r e a k up into 3 a ' s . While we have made no explicit u s e of isotopic spin p u r i t y , one m a y have doubts about the v a l i dity of the s i m p l e p r o c e s s , going through one w e l l - d e f i n e d state of 12C, where in p r a c t i c e t h e r e m a y be two or m o r e o v e r l a p p i n g s t a t e s with m i x e d isotopic spin c h a r a c t e r . However it m a y be that the method will have some use in i n d i c a t i n g how f a r the s i m p l e p r o c e s s is valid. The e x p e r i m e n t a l r e s u l t s [8] which have b e e n displayed in this way r e f e r to a 2+ state. T h e r e i s a f a i r l y high d e n s i t y of points at the c e n t r e . T h e r e i s a m a r k e d fall in d e n s i t y of p o i n t s tow a r d s the c i r c u m f e r e n c e in some r e g i o n s but an a p p r e c i a b l e n u m b e r of points on it. It m a y be noted that the c o n t o u r plot of those r e s u l t s in [9] does not b r i n g out all the d e t a i l s of the b o u n d a r y region. It would be of g r e a t i n t e r e s t to have the data on the 2-, T = 1 state at 16.57 MeV d i s played in this way, with the complete Dalitz d i a 293
Volume 19, number 4
PHYSICS LETTERS
g r a m covered. The 1- state at 17,23 MeV should also be examined. C l e a r l y our a r g u m e n t s can be extended to 160* ~ 4 a dcays, although t h i s may be s o m e what f r i v o l o u s in view of the e x p e r i m e n t a l c o m p l e x i t i e s . The a p p r o p r i a t e way to plot data i s in the f o r m of IPl - P21 a g a i n s t - P41, where Pi is m o m e n t u m in the 160 c e n t r e of m a s s f r a m e . R e g i o n s of p a r t i c u l a r s y m m e t r y , i n v o l v ing r e s t r i c t i o n s on the decay p o s s i b i l i t e s , c o r r e s p o n d to c o l l i n e a r m o m e n t a , m o m e n t a of equal magnitude d i r e c t e d t o w a r d s v e r t i c e s of a r e g u l a r t e t r a h e d r o n , and c o p l a n a r m o m e n t a of equal magnitude making a n g l e s of 900 with each other.
[P3
1 November 1965
1. R.H.Dalitz, Phil. Mag. 44 (1963) 1068. 2. D.Dehnhard, D.Kamke and P. Kramer, Physic s Letters 3 (1962) 52; 12 (1964) 113; Annalen der Physik 14 (1964) 201. 3. M.L.Stevenson, L.W.Alvarez, B.C.Maglic and A. H.Rosenfeld, Phys.Rev. 125 (1962) 687. 4. See for example, O.Skjeggestad, CERN 64-13, Notes on phase space, for a discussion of these diagrams and the ~ meson decay. See also A. E. Litherland, Can. J. Phys. 39 (1961)1245. 5. Z.Koba, Acta Physica Pol0nica 22 (1962) Supplement, 103. 6. G.C. Phillips, Revs. Mod. Phys. 37 (1965) 409. 7. D.Delmhard, Revs.Mod. Phys.37 (1965) 450 8. D.Kamke, J.Krug, F-W.Richter, Revs.Mod.Phys. 37 (1965) 453. 9. P. Kramer, Revs. Mod.Phys. 37 (1965) 346; Annalen der Physik 15 (1965) 278.
AN INTER-NUCLEON POTENTIAL DEDUCED FROM SHELL MODEL CALCULATIONS J. M. CLARK and J. P. E L L I O T T
School of Physical Sciences, University of Sussex, Brighton, Sussex Received 25 September 1965
T h e r e a r e s t i l l m a n y p r o b l e m s to be solved c o n c e r n i n g the foundation of the shell model, and the r e l a t i o n b e t w e e n any effective i n t e r - n u c l e o n shell model potential and the r e a l potential as deduced f r o m n u c l e o n - n u c l e o n s c a t t e r i n g . In this l e t t e r we p r e s e n t the r e s u l t s of an a t t e m p t to deduce an effective potential f r o m s h e l l model c a l c u l a t i o n s t h e m s e l v e s . T h i s potential is then c o m p a r e d with a r e a l i s t i c one. We have a s s u m e d , as a f i r s t and somewhat naive g u e s s , that the potential i s m a s s - i n d e p e n dent and has the s a m e G a u s s i a n shape and range, a = 1.8 fm, for all c o m p o n e n t s , i.e.
~j =
I A l 3 p 13 + A31p 31 + A l l p I I + A33p33+
+ (BlZP 13 + B 3 3 p 3 3 ) ~ j +
(1)
÷ ( c l 3 p 13 + C33p33)S.L} exp(-?-~'/a 2) J
Here p 2 T + l , ZS+I is a p r o j e c t i o n o p e r a t o r in the spin and i s o s p i n of the p a i r i, j and the n o n - c e n t r a l t e r m s have t h e i r u s u a l d e f i n i t i o n s :
L =rxpwtth 294
r=r i
rj, P = ~ t P i - P j ) . W e h a v e
used o s c i l l a t o r wave functions, adjusted to the n u c l e a r size. The eight l i n e a r p a r a m e t e r s A, B, C a r e then d e t e r m i n e d by a l e a s t s q u a r e s fit to a collection of 48 shell model e n e r g y m a t r i x e l e m e n t s , deduced f r o m e x p e r i m e n t a l l y known s p e c t r a of c e r t a i n nuclei. The data were t a k e n only f r o m n u c l e i which a r e b e l i e v e d to have a r e a s o n a b l y p u r e j - j conf i g u r a t i o n or a m i x t u r e of just two configurations. The configurations and m a s s r a n g e s c o n s i d e r e d a r e indicated in table 1. In m o s t c a s e s the m a t r i x e l e m e n t s were t a k e n f r o m p r e v i o u s work, e.g. T a l m i and T h i e b e r g e r [1], in which t h e i r o p t i m u m v a l u e s had b e e n d e t e r m i n e d by the b e s t fit to a n u m b e r of e n e r g y l e v e l s in that m a s s region. Only for A = 210 have we deduced each m a t r i x e l e m e n t f r o m a single e n e r g y level. The v a l u e s for the eight potential s t r e n g t h s which e m e r g e f r o m our l e a s t s q u a r e s fit a r e given in table 2, together with a s t a t i s t i c a l e s t i m a t e for the s t a n d a r d deviation as a m e a s u r e of the r e l i ability of each value. As might be expected, t h e s e deviations a r e a p p r e c i a b l e but hot o v e r whelming. With the aid of a s t a t i s t i c a l t e s t , the F - t e s t [10], we have b e e n able to show that the
PHYSICS LETTERS
DALITZ
DIAGRAMS
AND DECAY
1 November 1965
OF
12C
STATES
N. MACDONALD D e p a r t m e n t o f Natural Philosophy, The University, Glasgow
Received 24 September 1965
The s y m m e t r i c Dalitz d i a g r a m [1] has been used [2] to study the r e a c t i o n p + l i b ~ 12C* --, 3 a . However c e r t a i n g e n e r a l f e a t u r e s of this and r e l a t e d r e a c t i o n s , which can be exhibited in t e r m s of these d i a g r a m s , do not s e e m to have been pointed out. They a r e s i m i l a r to f e a t u r e s of the decay o~ ~ 3~ u s e d [3] to d e t e r m i n e the spin and p a r i t y of the w meson. In d i s c u s s i n g the k i n e m a t i c s of this s y s t e m we work in the 12C r e s t f r a m e so that m o m e n t a a r e coplanar. We denote k i n e t i c e n e r g i e s by T1, T 2 and T3, and t h e i r s u m by Q. We m a y d i s t i n guish between s t a n d a r d Dalitz d i a g r a m s which plot T 1 a g a i n s t T2, and s y m m e t r i c Dalitz diag r a m s which plot T1, T2, T 3 along axes making a n g l e s of 120 ° . The f i r s t p r o p e r t y we d i s c u s s u s e s the facts that a - p a r t i c l e s have z e r o spin and positive p a r i t y . The b o u n d a r y of the d i a g r a m c o r r e s p o n d s to c o l l i n e a r m o m e n t a [4]. C o n s e quently we m a y take the axis of q u a n t i s a t i o n along t h e i r c o m m o n d i r e c t i o n . We have J = 11 + ! 2 + 13 where the I i a r e o r b i t a l angul.ar m o m e n t a , so that m i = 0 a n d ~ = (-) I1+12+13. By c o n s i d e r i n g the coupling I 1 + 12 = 1, ! + 13 = J, and u s i n g the r e s u l t that the v e c t o r coupling coefficient (Jl O J2 O [ J3 O) = 0 if J l + J2 + J3 is odd, one s e e s that 11 + l~ + l~ + J is even and so the p a r i t y is (-)J. So any state of 12C with u n n a t u r a l p a r i t y decays so a s to avoid this r e g i o n of phase space. The second p r o p e r t y m a k e s u s e of the s y m m e t r y of the w a v e - f u n c t i o n u n d e r exchange of the a - p a r t i c l e s . Let us c o n s i d e r the situation in which all t h r e e m o m e n t a a r e of equal magnitude and at a n g l e s of 120 ° to each other. This c o r r e sponds to the c e n t r e point of the s y m m e t r i c d i a g r a m , or to the point -~Q, ~Q in the s t a n d a r d d i a g r a m . It is r e a d i l y shown that s t a t e s with a n g u l a r m o m e n t u m add p a r i t y 1- and 2- in 12C decay so as to avoid this r e g i o n in p h a s e space ~. The a r g u m e n t is a s follows. Take the Z axis p e r p e n d i c u l a r to the p l a n e of the m o m e n t a . The Inspection of the tables in [5] shows that 0- and 1+ states can not decay directly to 3a states.
o r i g i n a l configuration is r e s t o r e d ff one r o t a t e s through {~ about the Z axis and i n t e r c h a n g e s two p a i r s of p a r t i c l e s . This i m p l i e s exp({~ mi) = = 1. So, if J < 3, m = 0 . Now r e f l e c t the x - y p l a n e in the o r i g i n and rotate through ~ about the Z axis. T h i s r e s t o r e s the o r i g i n a l configuration and so r e q u i r e s that the p a r i t y should be positive. A r g u m e n t s b a s e d on s y m m e t r y of a 3a state should be t r e a t e d with caution when an i n t e r m e d i a t e 8Be state may be involved. If such a s i t u a t i o n gave a s h a r p r e s o n a n c e c o n t a i n i n g a l m o s t all the events, the p r e s e n t r e s u l t s could t e l l us nothing new, since whatever the 12C spin and p a r i t y the r e s o n a n c e would not fall at the c e n t r e or on the edge of the Dalitz d i a g r a m . However, in p r a c t i c e one d i s c u s s e s [6, 7] the m o d e s of decay on the b a s i s of j u d g e m e n t s r e g a r d i n g the p r e s e n c e of r e s o n a n c e s a g a i n s t a " d i r e c t " 12C* ~ 3a background. It is c l e a r l y n e c e s s a r y to a p p r e c i a t e any special f e a t u r e s of the b a c k g r o u n d which m a y affect these judgements. In d i s c u s s i n g the actual e x p e r i m e n t a l situation one c o m e s up a g a i n s t the fact that both T = 0 and T = 1 a r e found [6-8] to b r e a k up into 3 a ' s . While we have made no explicit u s e of isotopic spin p u r i t y , one m a y have doubts about the v a l i dity of the s i m p l e p r o c e s s , going through one w e l l - d e f i n e d state of 12C, where in p r a c t i c e t h e r e m a y be two or m o r e o v e r l a p p i n g s t a t e s with m i x e d isotopic spin c h a r a c t e r . However it m a y be that the method will have some use in i n d i c a t i n g how f a r the s i m p l e p r o c e s s is valid. The e x p e r i m e n t a l r e s u l t s [8] which have b e e n displayed in this way r e f e r to a 2+ state. T h e r e i s a f a i r l y high d e n s i t y of points at the c e n t r e . T h e r e i s a m a r k e d fall in d e n s i t y of p o i n t s tow a r d s the c i r c u m f e r e n c e in some r e g i o n s but an a p p r e c i a b l e n u m b e r of points on it. It m a y be noted that the c o n t o u r plot of those r e s u l t s in [9] does not b r i n g out all the d e t a i l s of the b o u n d a r y region. It would be of g r e a t i n t e r e s t to have the data on the 2-, T = 1 state at 16.57 MeV d i s played in this way, with the complete Dalitz d i a 293
Volume 19, number 4
PHYSICS LETTERS
g r a m covered. The 1- state at 17,23 MeV should also be examined. C l e a r l y our a r g u m e n t s can be extended to 160* ~ 4 a dcays, although t h i s may be s o m e what f r i v o l o u s in view of the e x p e r i m e n t a l c o m p l e x i t i e s . The a p p r o p r i a t e way to plot data i s in the f o r m of IPl - P21 a g a i n s t - P41, where Pi is m o m e n t u m in the 160 c e n t r e of m a s s f r a m e . R e g i o n s of p a r t i c u l a r s y m m e t r y , i n v o l v ing r e s t r i c t i o n s on the decay p o s s i b i l i t e s , c o r r e s p o n d to c o l l i n e a r m o m e n t a , m o m e n t a of equal magnitude d i r e c t e d t o w a r d s v e r t i c e s of a r e g u l a r t e t r a h e d r o n , and c o p l a n a r m o m e n t a of equal magnitude making a n g l e s of 900 with each other.
[P3
1 November 1965
1. R.H.Dalitz, Phil. Mag. 44 (1963) 1068. 2. D.Dehnhard, D.Kamke and P. Kramer, Physic s Letters 3 (1962) 52; 12 (1964) 113; Annalen der Physik 14 (1964) 201. 3. M.L.Stevenson, L.W.Alvarez, B.C.Maglic and A. H.Rosenfeld, Phys.Rev. 125 (1962) 687. 4. See for example, O.Skjeggestad, CERN 64-13, Notes on phase space, for a discussion of these diagrams and the ~ meson decay. See also A. E. Litherland, Can. J. Phys. 39 (1961)1245. 5. Z.Koba, Acta Physica Pol0nica 22 (1962) Supplement, 103. 6. G.C. Phillips, Revs. Mod. Phys. 37 (1965) 409. 7. D.Delmhard, Revs.Mod. Phys.37 (1965) 450 8. D.Kamke, J.Krug, F-W.Richter, Revs.Mod.Phys. 37 (1965) 453. 9. P. Kramer, Revs. Mod.Phys. 37 (1965) 346; Annalen der Physik 15 (1965) 278.
AN INTER-NUCLEON POTENTIAL DEDUCED FROM SHELL MODEL CALCULATIONS J. M. CLARK and J. P. E L L I O T T
School of Physical Sciences, University of Sussex, Brighton, Sussex Received 25 September 1965
T h e r e a r e s t i l l m a n y p r o b l e m s to be solved c o n c e r n i n g the foundation of the shell model, and the r e l a t i o n b e t w e e n any effective i n t e r - n u c l e o n shell model potential and the r e a l potential as deduced f r o m n u c l e o n - n u c l e o n s c a t t e r i n g . In this l e t t e r we p r e s e n t the r e s u l t s of an a t t e m p t to deduce an effective potential f r o m s h e l l model c a l c u l a t i o n s t h e m s e l v e s . T h i s potential is then c o m p a r e d with a r e a l i s t i c one. We have a s s u m e d , as a f i r s t and somewhat naive g u e s s , that the potential i s m a s s - i n d e p e n dent and has the s a m e G a u s s i a n shape and range, a = 1.8 fm, for all c o m p o n e n t s , i.e.
~j =
I A l 3 p 13 + A31p 31 + A l l p I I + A33p33+
+ (BlZP 13 + B 3 3 p 3 3 ) ~ j +
(1)
÷ ( c l 3 p 13 + C33p33)S.L} exp(-?-~'/a 2) J
Here p 2 T + l , ZS+I is a p r o j e c t i o n o p e r a t o r in the spin and i s o s p i n of the p a i r i, j and the n o n - c e n t r a l t e r m s have t h e i r u s u a l d e f i n i t i o n s :
L =rxpwtth 294
r=r i
rj, P = ~ t P i - P j ) . W e h a v e
used o s c i l l a t o r wave functions, adjusted to the n u c l e a r size. The eight l i n e a r p a r a m e t e r s A, B, C a r e then d e t e r m i n e d by a l e a s t s q u a r e s fit to a collection of 48 shell model e n e r g y m a t r i x e l e m e n t s , deduced f r o m e x p e r i m e n t a l l y known s p e c t r a of c e r t a i n nuclei. The data were t a k e n only f r o m n u c l e i which a r e b e l i e v e d to have a r e a s o n a b l y p u r e j - j conf i g u r a t i o n or a m i x t u r e of just two configurations. The configurations and m a s s r a n g e s c o n s i d e r e d a r e indicated in table 1. In m o s t c a s e s the m a t r i x e l e m e n t s were t a k e n f r o m p r e v i o u s work, e.g. T a l m i and T h i e b e r g e r [1], in which t h e i r o p t i m u m v a l u e s had b e e n d e t e r m i n e d by the b e s t fit to a n u m b e r of e n e r g y l e v e l s in that m a s s region. Only for A = 210 have we deduced each m a t r i x e l e m e n t f r o m a single e n e r g y level. The v a l u e s for the eight potential s t r e n g t h s which e m e r g e f r o m our l e a s t s q u a r e s fit a r e given in table 2, together with a s t a t i s t i c a l e s t i m a t e for the s t a n d a r d deviation as a m e a s u r e of the r e l i ability of each value. As might be expected, t h e s e deviations a r e a p p r e c i a b l e but hot o v e r whelming. With the aid of a s t a t i s t i c a l t e s t , the F - t e s t [10], we have b e e n able to show that the