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The atomic cascade of kaonic and pionic hydrogen
Volume 248, number 3,4
PHYSICSLETTERSB
4 October 1990
The atomic cascade of kaonic and pionic hydrogen G. R e i f e n r & h e r a n d E. K l e m p t Institut fur Physik der Universitdt Mainz, D-6500 Mainz, FRG
Received 5 June 1990;revised manuscript received 20 July 1990
The X-rayintensities ofkaonic and pionic hydrogenatomsare calculatedas functionsof the hydrogendensityand are compared to experimental data. Data on kaonichydrogenare statistically not very significant, the comparisonwith the cascaderesults allows to test their intrinsic consistency.X-rayintensities from pionic hydrogenagree reasonablywith the results of the cascade model.
The interest in the low-energy kaon-proton interaction has recently been restimulated by the fact that intense kaon beams of high purity and low momenta may become available at the planned KAON factory in Vancouver [1,2]. Particularly interesting in this field are measurements of the strong interaction shift and the broadening of the 1S ground state of kaonic hydrogen atoms. These strong interaction parameters allow to determine the kaon-proton scattering length. The scattering length can also be determined from kaon-nucleon scattering data. It is strongly influenced by the A (1405). A measurement of the Xray energies of K - p atoms with high precision would therefore allow a consistency check and may contribute to a better understanding of the A(1405) resonance. Experiments searching for X-rays from kaonic hydrogen atoms meet, however, formidable difficulties; the low production rate ofkaons at low momenta and their short lifetime require the use of short beam lines. Yet short beam lines do not allow a clean kaon/pion separation, and the overall background is high. On the other hand, low momenta are essential to stop kaons in a low-density Ha target in order to get areaTable 1 The strong interaction parameters whichwere used. Set
/~kEIs[eVl
F~s [eV]
F2p [meV]
A B C
270 270 270
560 300 300
3 3 8
250
sonably high X-ray yield per kaon stop. Recent experiments at LEAR on antiprotonic hydrogen atoms have shown that X-ray yields are much larger when antiprotons are stopped in H2 gas at low densities [ 3 ]. The purposes of this letter are the following: we will compare the results of a cascade model with experimental data from three experiments on K - p atoms in order to check the consistency of the - statistically rather poor - data with cascade calculations; we will present theoretical X-ray yields as function of the H2 density to facilitate planning of future experiments; and we will present results on the cascade of pionic hydrogen. Measurements of strong interaction effects o f n - p and n - d atoms are in progress at the Paul Scherrer Institute (formerly SIN), and cascade calculations may be useful as a guide to optimizing the target density. So far, three experiments on kaonic hydrogen have been carried out by stopping kaons in a liquid H2 target. X-ray line patterns were reported which possibly originated from kaonic hydrogen, but the evidence for this assignment was only weak. In the first experiment by Davies et al. [4] a 2a (two standard deviation) p e a k at an energy of 6.52 + 0.06 keV and with a yield of (11 + 6 ) × I 0-4 per stopped kaon was observed. The expected K~ (2P-o IS) transition energy is 6.48 keV, hence the measured energy corresponds to a strong interaction shift of 40 + 60 eV. No line broadening was observed, / ' t s < 2 3 0 eV. The 3 P ~ I S transition was not observed, an upper limit of 9 × 10-4 was given. Izycki et al. [ 5 ] found a pattern of three lines at
0370-2693/90/$ 03.50 © 1990- Elsevier SciencePublishers B.V. ( North-Holland)
Volume 248, number 3,4
PHYSICS LETTERS B
4 October 1990
I0 2
.+
+;;..
10
10-i
÷ ÷ ÷ + ÷ ÷
10-~
Z
Ktot
IO 3
Ktot
Kp
o~ #
.'~ 10-~
i
i
i
i
i
¢~_~ 102
10 X
//
10 -~
to _~10 2
0 13£ ÷÷ ..~. ÷ ,...~....
1
i
X
÷ +÷+
10-1
I0
x ~x
+ ÷ ÷ ÷ .
10-2 Ltot
k~
÷+
hot
10 -3
Lp
~
10-1 i 10-~ 10-3 10-210 -I I
i
l h
10 102 103
10 - 3 1 0 - 2 1 0 -1
i
i
1
10
102
103
Fig. 1. X-ray yields of K-p atoms as a function of H2 density for different strong interaction parameters: Set A corresponds with +, set B with ..., and set C with -. Data points O are from ref. [ 4 ] and/x from ref. [ 5 ].
" . . ~
60-
1%-3
10-2
10' -1 Pressure
1'
10 '
102 T,. ÷.÷~~-.03
lotto]
Fig. 2. F r a c t i o n o f P c a p t u r e in K - p
10-3 10-210 -I I
I0 102 103
10-310 -210 -I I
10 102 103
Pressure [atml
Pressure [atml
a t o m s as a f u n c t i o n o f hy-
d r o g e n density. Strong interaction parameters as in fig. [ 1 ].
6 . 9 6 + 0 . 0 9 , 7 . 9 9 + 0 . 0 7 and 8 . 6 4 + 0 . 1 0 keV with yields o f 2.1, 2.5 and 1.5X 10 -4, respectively. The statistical significance o f the lines was 2tr, 3tr, and 2tr. Since the interpretation o f these lines as I ~ line, Kp
Fig. 3. X-ray yield of pionic hydrogen atoms as a function of hydrogen density. Data points O are from ref. [ 10], [] from ref. [ 12 ], and/x from ref. [ 13 ]. line and K~ line is not unambiguous, the authors conservatively quote 8 X 10 -4 as upper limit for the total X-ray yield to the ground state. If the lines are assumed to originate from K - p atoms, a strong interaction shift and width o f A E + ½iF= ( 2 7 0 + 80) + i ( 2 8 0 + 130) eV can be deduced. Bird et al. [ 6 ] reported four lines at energies compatible with K~, Ka, Kr, I ~ transitions of K - p atoms, and with yields o f 8, 33, 4, and 2 X 10 -4, respectively. The K B line at 7.872 + 0.060 keV was observed as 3a effect, the other yields were compatible with zero. Clearly, the experimental data are at the limit o f being statistically significant, and the results o f the three experiments are not compatible with each other. We will compare the results with results o f our cascade model which we have developed recently with emphasis on antiprotonic hydrogen [ 7 ]. A large variety o f experimental data exists for pp atoms, the Mainz cascade model was capable of describing these
251
Volume 248, number 3,4
PHYSICS LETTERSB
data without need for any adjustable parameter. Previous cascade calculations for the K - p atom [ 8 ] had to use free parameters. In absence of solid experimental input data, the predictive power of these calculations was limited. The Mainz cascade model is described in detail in ref. [ 7 ], and therefore we present directly the results of the model. The energy levels of kaonic hydrogen without strong interaction effects were taken from ref. [ 9 ]. The strong interaction parameters that were used are displayed in table 1. The results of the cascade calculations are shown in fig. 1. The strong interaction parameters (A) lead to intensities shown by plusses. The width F2p= 3 meV had been chosen to match the total intensity of ref. [ 5 ]. This set of parameters predicts a very low yield of K~ X-rays. The I ~ X-ray yield can be increased by decreasing the width of the 1S level. The value of F~s = 300 eV chosen for set (B) is compatible with the experimental result o f F , s= 560 +- 260 eV. The calculated X-ray yields are represented by dots in fig. 1. The total intensity is now too high, reasonable consistency can be obtained by choosing set (C). For this choice of strong interaction parameters the predicted intensities for K~, Kp, Kr, and Ktot are 1.5, 0.5, 1.8, and 10 × 10-4, respectively. Transitions from high-nP orbits to the 1S state contribute significantly, but these are not observed in refs. [ 4-6 ]. The high yield for emission of K~ lines observed in ref. [4] and the high yield of the Kp line in ref. [6] are clearly in conflict with the upper limit of the total intensity reported in ref. [ 5 ]. Nevertheless we notice that the strongest line in refs. [ 5,6 ] is the Kp line. In addition one may argue [ 2 ] that the efficiency of detecting low-energy X-rays may have been overestimated in ref. [ 5 ], so that the two data sets may not necessarily be in conflict. We do not think that the statistical significance of the data allows a "fine-tuning" of the strong interaction parameters. Instead we conclude from the variation of the calculated X-ray yields with changes of strong interaction parameters, that the results of ref. [ 5 ] are likely to be compatible with the cascade model while the strong dominance of one X-ray line, K~ or Kp, as found in refs. [4,6], is not supported by our cascade model. Nevertheless, one should stress that the compatibility of the results of ref. [ 5 ] with cascade calculations does not guarantee that the identi252
4 October 1990
fication of the experimental structure with X-rays from K - p atoms was correct. Fig. 1 also shows that considerable gain can be achieved if experiments are carried out by stopping kaons in gaseous H2 at low density. In this case one has to struggle against the time which elapsed between kaons entering the target and the emission of a X-ray. This time interval was determined to 30 ns for antiprotons stopping in Hz gas at normal temperature and pressure after a path in H E of 0.5 m [ 10]. This interval leads to a kaon decay probability of 90%. Due to the finite width of the 2P level, capture of K from P states is also possible. The fraction of P state capture as a function of H2 density is shown in fig. 2. Results on the cascade time from atomic capture to nuclear capture are also obtained. They are nearly identical to the cascade time of the pp atoms [ 7], and therefore not shown here. Pionic hydrogen atoms were first observed in a pioneering experiment by Bailey et al. [ 11 ]. In a H2 gas target at 4 atm a ratio K~/Ktot of 0.53+_0.04 was found. The absolute yield per pion captured on a proton was determined to 0.40_+ 0.04. The energy of K~ X-rays from rc-p atoms was measured in two experiments [ 12,13 ] with a graphite crystal spectrometer by stopping pions in a 2.7 arm (3 atm) cooled saturated hydrogen gas. A strong interaction shift AEls= - (4.9 +-0.5 ) eV was derived. The yield of I ~ X-rays was 0.023 +0.010 at a density equivalent to 20 atm gas at normal temperature [12] and 0.04+-0.02 at 23 atm [ 13]. Capture ofpions in n - p atoms is improbable, pion capture is a process which requires presence of two nucleons. Only the processes ~ - p - , n ° n or "/n are known. They result in a strong interaction width of about 0.5 eV [ 14,15 ]. The results of the cascade calculations with these strong interaction parameters are compared to the experimental data in fig. 3. The pressure dependence of the X-ray intensities suggests the use of gas targets at a few atmospheres. In the case of pionic hydrogen, the gain in intensity when going to lower pressures is rather small. In the case of kaonic hydrogen the kaon decay losses will certainly not compensate the increased X-ray yields if targets with pressures of or below 1 arm are used. We would like to thank Dr. Ch. Batty and Professor Dr. L. Tauscher for useful comments and a criti-
Volume 248, number 3,4
PHYSICS LETTERS B
cal r e a d i n g o f t h e m a n u s c r i p t , a n d P r o f e s s o r H. Pilk u h n for p r o v i d i n g c a l c u l a t i o n s o f e l e c t r o m a g n e t i c energy levels p r i o r to p u b l i c a t i o n . T h e w o r k was supp o r t e d by the B M F T , G e r m a n y , u n d e r c o n t r a c t n u m b e r 06 M Z 223.
References [ 1 ] C.J. Batty, Nucl. Phys. A 508 (1990) 89c. [ 2 ] C.J. Batty, Exotic atoms and the kaon-nucleon interaction, talk at the Workshop on Intense hadron facilities and antriproton physics (Turin, 1989 ).
4 October 1990
[ 3 ] C.J. Batty, Rep. Prog. Phys. 52 ( 1989 ) 1165. [4] J.D. Davies et al., Phys. Lett. B 83 (1979) 55. [5] M. Izycki et al., Z. Phys. A 297 (1980) 11. [6] P.M. Bird et al., Nucl. Phys. A 404 (1983) 482. [7] G. ReifenrSther and E. Klempt, Nucl. Phys. A 503 (1989) 885. [ 8 ] E. Borie and M. Leon, Phys. Rev. A 21 (1980) 1460. [9] H. Pilkuhn and H.G. Schaile, Z. Phys. D 15 (1990) 321. [10] G. Reifenr~ither et al., Phys. Lett. B 214 (1988 ) 325. [ 11 ] J.M. Bailey et al., Phys. Lett. B 33 (1970) 369. [ 12] A. Forster et al., Phys. Rev. C 28 (1983) 2374. [13] E. Bovet et al., Phys. Lett. B 153 (1985) 231. [ 14] G. Rosche and W.S. Woolcock, Nucl. Phys. A 381 (1982) 405. [ 15 ] W.B. Kaufmann and W.R. Gibbs, Phys. Rev. C 35 (1987) 838.
253
PHYSICSLETTERSB
4 October 1990
The atomic cascade of kaonic and pionic hydrogen G. R e i f e n r & h e r a n d E. K l e m p t Institut fur Physik der Universitdt Mainz, D-6500 Mainz, FRG
Received 5 June 1990;revised manuscript received 20 July 1990
The X-rayintensities ofkaonic and pionic hydrogenatomsare calculatedas functionsof the hydrogendensityand are compared to experimental data. Data on kaonichydrogenare statistically not very significant, the comparisonwith the cascaderesults allows to test their intrinsic consistency.X-rayintensities from pionic hydrogenagree reasonablywith the results of the cascade model.
The interest in the low-energy kaon-proton interaction has recently been restimulated by the fact that intense kaon beams of high purity and low momenta may become available at the planned KAON factory in Vancouver [1,2]. Particularly interesting in this field are measurements of the strong interaction shift and the broadening of the 1S ground state of kaonic hydrogen atoms. These strong interaction parameters allow to determine the kaon-proton scattering length. The scattering length can also be determined from kaon-nucleon scattering data. It is strongly influenced by the A (1405). A measurement of the Xray energies of K - p atoms with high precision would therefore allow a consistency check and may contribute to a better understanding of the A(1405) resonance. Experiments searching for X-rays from kaonic hydrogen atoms meet, however, formidable difficulties; the low production rate ofkaons at low momenta and their short lifetime require the use of short beam lines. Yet short beam lines do not allow a clean kaon/pion separation, and the overall background is high. On the other hand, low momenta are essential to stop kaons in a low-density Ha target in order to get areaTable 1 The strong interaction parameters whichwere used. Set
/~kEIs[eVl
F~s [eV]
F2p [meV]
A B C
270 270 270
560 300 300
3 3 8
250
sonably high X-ray yield per kaon stop. Recent experiments at LEAR on antiprotonic hydrogen atoms have shown that X-ray yields are much larger when antiprotons are stopped in H2 gas at low densities [ 3 ]. The purposes of this letter are the following: we will compare the results of a cascade model with experimental data from three experiments on K - p atoms in order to check the consistency of the - statistically rather poor - data with cascade calculations; we will present theoretical X-ray yields as function of the H2 density to facilitate planning of future experiments; and we will present results on the cascade of pionic hydrogen. Measurements of strong interaction effects o f n - p and n - d atoms are in progress at the Paul Scherrer Institute (formerly SIN), and cascade calculations may be useful as a guide to optimizing the target density. So far, three experiments on kaonic hydrogen have been carried out by stopping kaons in a liquid H2 target. X-ray line patterns were reported which possibly originated from kaonic hydrogen, but the evidence for this assignment was only weak. In the first experiment by Davies et al. [4] a 2a (two standard deviation) p e a k at an energy of 6.52 + 0.06 keV and with a yield of (11 + 6 ) × I 0-4 per stopped kaon was observed. The expected K~ (2P-o IS) transition energy is 6.48 keV, hence the measured energy corresponds to a strong interaction shift of 40 + 60 eV. No line broadening was observed, / ' t s < 2 3 0 eV. The 3 P ~ I S transition was not observed, an upper limit of 9 × 10-4 was given. Izycki et al. [ 5 ] found a pattern of three lines at
0370-2693/90/$ 03.50 © 1990- Elsevier SciencePublishers B.V. ( North-Holland)
Volume 248, number 3,4
PHYSICS LETTERS B
4 October 1990
I0 2
.+
+;;..
10
10-i
÷ ÷ ÷ + ÷ ÷
10-~
Z
Ktot
IO 3
Ktot
Kp
o~ #
.'~ 10-~
i
i
i
i
i
¢~_~ 102
10 X
//
10 -~
to _~10 2
0 13£ ÷÷ ..~. ÷ ,...~....
1
i
X
÷ +÷+
10-1
I0
x ~x
+ ÷ ÷ ÷ .
10-2 Ltot
k~
÷+
hot
10 -3
Lp
~
10-1 i 10-~ 10-3 10-210 -I I
i
l h
10 102 103
10 - 3 1 0 - 2 1 0 -1
i
i
1
10
102
103
Fig. 1. X-ray yields of K-p atoms as a function of H2 density for different strong interaction parameters: Set A corresponds with +, set B with ..., and set C with -. Data points O are from ref. [ 4 ] and/x from ref. [ 5 ].
" . . ~
60-
1%-3
10-2
10' -1 Pressure
1'
10 '
102 T,. ÷.÷~~-.03
lotto]
Fig. 2. F r a c t i o n o f P c a p t u r e in K - p
10-3 10-210 -I I
I0 102 103
10-310 -210 -I I
10 102 103
Pressure [atml
Pressure [atml
a t o m s as a f u n c t i o n o f hy-
d r o g e n density. Strong interaction parameters as in fig. [ 1 ].
6 . 9 6 + 0 . 0 9 , 7 . 9 9 + 0 . 0 7 and 8 . 6 4 + 0 . 1 0 keV with yields o f 2.1, 2.5 and 1.5X 10 -4, respectively. The statistical significance o f the lines was 2tr, 3tr, and 2tr. Since the interpretation o f these lines as I ~ line, Kp
Fig. 3. X-ray yield of pionic hydrogen atoms as a function of hydrogen density. Data points O are from ref. [ 10], [] from ref. [ 12 ], and/x from ref. [ 13 ]. line and K~ line is not unambiguous, the authors conservatively quote 8 X 10 -4 as upper limit for the total X-ray yield to the ground state. If the lines are assumed to originate from K - p atoms, a strong interaction shift and width o f A E + ½iF= ( 2 7 0 + 80) + i ( 2 8 0 + 130) eV can be deduced. Bird et al. [ 6 ] reported four lines at energies compatible with K~, Ka, Kr, I ~ transitions of K - p atoms, and with yields o f 8, 33, 4, and 2 X 10 -4, respectively. The K B line at 7.872 + 0.060 keV was observed as 3a effect, the other yields were compatible with zero. Clearly, the experimental data are at the limit o f being statistically significant, and the results o f the three experiments are not compatible with each other. We will compare the results with results o f our cascade model which we have developed recently with emphasis on antiprotonic hydrogen [ 7 ]. A large variety o f experimental data exists for pp atoms, the Mainz cascade model was capable of describing these
251
Volume 248, number 3,4
PHYSICS LETTERSB
data without need for any adjustable parameter. Previous cascade calculations for the K - p atom [ 8 ] had to use free parameters. In absence of solid experimental input data, the predictive power of these calculations was limited. The Mainz cascade model is described in detail in ref. [ 7 ], and therefore we present directly the results of the model. The energy levels of kaonic hydrogen without strong interaction effects were taken from ref. [ 9 ]. The strong interaction parameters that were used are displayed in table 1. The results of the cascade calculations are shown in fig. 1. The strong interaction parameters (A) lead to intensities shown by plusses. The width F2p= 3 meV had been chosen to match the total intensity of ref. [ 5 ]. This set of parameters predicts a very low yield of K~ X-rays. The I ~ X-ray yield can be increased by decreasing the width of the 1S level. The value of F~s = 300 eV chosen for set (B) is compatible with the experimental result o f F , s= 560 +- 260 eV. The calculated X-ray yields are represented by dots in fig. 1. The total intensity is now too high, reasonable consistency can be obtained by choosing set (C). For this choice of strong interaction parameters the predicted intensities for K~, Kp, Kr, and Ktot are 1.5, 0.5, 1.8, and 10 × 10-4, respectively. Transitions from high-nP orbits to the 1S state contribute significantly, but these are not observed in refs. [ 4-6 ]. The high yield for emission of K~ lines observed in ref. [4] and the high yield of the Kp line in ref. [6] are clearly in conflict with the upper limit of the total intensity reported in ref. [ 5 ]. Nevertheless we notice that the strongest line in refs. [ 5,6 ] is the Kp line. In addition one may argue [ 2 ] that the efficiency of detecting low-energy X-rays may have been overestimated in ref. [ 5 ], so that the two data sets may not necessarily be in conflict. We do not think that the statistical significance of the data allows a "fine-tuning" of the strong interaction parameters. Instead we conclude from the variation of the calculated X-ray yields with changes of strong interaction parameters, that the results of ref. [ 5 ] are likely to be compatible with the cascade model while the strong dominance of one X-ray line, K~ or Kp, as found in refs. [4,6], is not supported by our cascade model. Nevertheless, one should stress that the compatibility of the results of ref. [ 5 ] with cascade calculations does not guarantee that the identi252
4 October 1990
fication of the experimental structure with X-rays from K - p atoms was correct. Fig. 1 also shows that considerable gain can be achieved if experiments are carried out by stopping kaons in gaseous H2 at low density. In this case one has to struggle against the time which elapsed between kaons entering the target and the emission of a X-ray. This time interval was determined to 30 ns for antiprotons stopping in Hz gas at normal temperature and pressure after a path in H E of 0.5 m [ 10]. This interval leads to a kaon decay probability of 90%. Due to the finite width of the 2P level, capture of K from P states is also possible. The fraction of P state capture as a function of H2 density is shown in fig. 2. Results on the cascade time from atomic capture to nuclear capture are also obtained. They are nearly identical to the cascade time of the pp atoms [ 7], and therefore not shown here. Pionic hydrogen atoms were first observed in a pioneering experiment by Bailey et al. [ 11 ]. In a H2 gas target at 4 atm a ratio K~/Ktot of 0.53+_0.04 was found. The absolute yield per pion captured on a proton was determined to 0.40_+ 0.04. The energy of K~ X-rays from rc-p atoms was measured in two experiments [ 12,13 ] with a graphite crystal spectrometer by stopping pions in a 2.7 arm (3 atm) cooled saturated hydrogen gas. A strong interaction shift AEls= - (4.9 +-0.5 ) eV was derived. The yield of I ~ X-rays was 0.023 +0.010 at a density equivalent to 20 atm gas at normal temperature [12] and 0.04+-0.02 at 23 atm [ 13]. Capture ofpions in n - p atoms is improbable, pion capture is a process which requires presence of two nucleons. Only the processes ~ - p - , n ° n or "/n are known. They result in a strong interaction width of about 0.5 eV [ 14,15 ]. The results of the cascade calculations with these strong interaction parameters are compared to the experimental data in fig. 3. The pressure dependence of the X-ray intensities suggests the use of gas targets at a few atmospheres. In the case of pionic hydrogen, the gain in intensity when going to lower pressures is rather small. In the case of kaonic hydrogen the kaon decay losses will certainly not compensate the increased X-ray yields if targets with pressures of or below 1 arm are used. We would like to thank Dr. Ch. Batty and Professor Dr. L. Tauscher for useful comments and a criti-
Volume 248, number 3,4
PHYSICS LETTERS B
cal r e a d i n g o f t h e m a n u s c r i p t , a n d P r o f e s s o r H. Pilk u h n for p r o v i d i n g c a l c u l a t i o n s o f e l e c t r o m a g n e t i c energy levels p r i o r to p u b l i c a t i o n . T h e w o r k was supp o r t e d by the B M F T , G e r m a n y , u n d e r c o n t r a c t n u m b e r 06 M Z 223.
References [ 1 ] C.J. Batty, Nucl. Phys. A 508 (1990) 89c. [ 2 ] C.J. Batty, Exotic atoms and the kaon-nucleon interaction, talk at the Workshop on Intense hadron facilities and antriproton physics (Turin, 1989 ).
4 October 1990
[ 3 ] C.J. Batty, Rep. Prog. Phys. 52 ( 1989 ) 1165. [4] J.D. Davies et al., Phys. Lett. B 83 (1979) 55. [5] M. Izycki et al., Z. Phys. A 297 (1980) 11. [6] P.M. Bird et al., Nucl. Phys. A 404 (1983) 482. [7] G. ReifenrSther and E. Klempt, Nucl. Phys. A 503 (1989) 885. [ 8 ] E. Borie and M. Leon, Phys. Rev. A 21 (1980) 1460. [9] H. Pilkuhn and H.G. Schaile, Z. Phys. D 15 (1990) 321. [10] G. Reifenr~ither et al., Phys. Lett. B 214 (1988 ) 325. [ 11 ] J.M. Bailey et al., Phys. Lett. B 33 (1970) 369. [ 12] A. Forster et al., Phys. Rev. C 28 (1983) 2374. [13] E. Bovet et al., Phys. Lett. B 153 (1985) 231. [ 14] G. Rosche and W.S. Woolcock, Nucl. Phys. A 381 (1982) 405. [ 15 ] W.B. Kaufmann and W.R. Gibbs, Phys. Rev. C 35 (1987) 838.
253