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Phase-shift analysis of p-He4 elastic scattering at around 40 MeV
Volume 5, number 5
P HY S I C S L E T T E R S
PHASE-SHIFT
1 August 1963
ANALYSIS OF p-He 4 ELASTIC AT A R O U N D 40 MeV *
SCATTERING
S. SUWA and A. YOKOSAWA Argonne National Laboratory, Argonne, 17linois Received 21 June 1963
The polarization of the protons s c a t t e r e d by He 4 quires a prohibitingly large amount of computer at 38.4 MeV was recently m e a s u r e d 1), and the extime. T h e r e f o r e , we selected a s e a r c h range based perimental r e s u l t s showed an appreciable d i s c r e p upon the theoretical predictions 2, 3). Thus, our ancy f r o m the theoretical ,~redictions of Gammel and starting sets of the phase-shift p a r a m e t e r s were Thaler 2). Kanada et al. 3) derived a non-local random combinations of 5o=1.0 , 1.5; 51+=1.0 , 1.5; p-He 4 potential f r o m the nucleon-nucleon i n t e r a c 51-=0.5 , 1.0; 52+=0 , 0.5; 5 ~ - = 0 , 0 . 5 ; 5 3 + = 0 , 0 . 5 ; tion ; their calculated polarization with this potential and 63- =0 or 0.5, with all b l =1.0, 0.95, 0.90 or is close to the experimental data, but there is still 0.80. F u r t h e r m o r e , as our p r i m a r y interest is to some disagreement. As Gammel and Thaler already investigate the effect of the inelastic p a r a m e t e r s , pointed out, their neglect of the opening of inelastic the p r o g r a m was modified such that at f i r s t the channels is not justified at this energy. T h e r e f o r e , computer v a r i e s the values of only 6/+ by keeping it is hoped that both the differential c r o s s section bl + constant until X2 was minimized and then values data 4) and the polarization data can be fitted simul- of both 5/" and b/+ were varied to f u r t h e r minimize taneously by introducing inelastic p a r a m e t e r s in the ×2. p h a s e - s h i f t analysis, that is, by using complex It was found that, in most c a s e s , phase shifts phase shifts. converge to a single solution (solution A in table 1) The p r o c e d u r e of the analysis is to find the minwhich yields X2 = 176 ; otherwise, they converge imum of X2, which is given by to several sets with values of ×2 higher than 1000. X2
=
£ i [~exp(ei)
i
(ycalc(0/) -2
~(0i)
]
+r'~[pexp(°/)-~p(oj)(i+~) ecalc(e~)]2 ( 2 + (h-7) , where ~exp and P e x p a r e the experimental values of the d i f f e r e n t i a / c r o s s section and the polarization, r e s p e c t i v e l y , and h~ and A P r e p r e s e n t the experimental e r r o r s . The p a r a m e t e r ¢ was introduced to take into account the uncertainty of the incident beam polarization, the fractional e r r o r of which is given by A ¢. F o r each scattering angle, ~calc and P c a l c a r e given as functions of 5l +, the real part of total (nuclear and Coulomb) phase-sRifts, and of inelastic p a r a m e t e r s bl + (ref. 5)). The s u p e r s c i p t s + a r e used in expressing the partial waves of total angular momentum l ± ½. A g e n e r a l l e a s t - s q u a r e s - f i t t i n g p r o c e d u r e 6) p r o g r a m m e d for the IBM 704 computer was adopted for the analysis. The p r o g r a m uses the NewionRaphson method of interaction to determine the set of p a r a m e t e r s which minimizes the X2. All waves up to the F - w a v e s (/max = 3) were taken into account, and the higher waves were ignored, that is, 15 p a r a m e t e r s were considered in the analysis. In this case, an extensive random s e a r c h r e -
Table 1 Total phase shifts, 6l ± (in radian), and inelastic parameters, bl +. X2
50 bo 1.260 A 176 0.839 1.271 B 117 0.839
51+ bl + 1.340 0.727 1.363 0.712
51 b 10.905 0.955 0.911 0.928
52+ b2+ 0.444 0.895 0.446 0.883
62 b2-
53+ b3 +
63b3-
¢
0.275 0.273 0.250 0.013 0.756 0.887 0.878 0.270 0.266 0.245 0.012 0.752 0.880 0.865
In our analysis we had 71 pieces of experimental information to be fitted in t e r m s of 15 p a r a m e t e r s . An attempt was also made to p e r f o r m the analysis by discarding the data of m e a s u r e m e n t of the differential c r o s s section of the f i r s t four smallest angles because such data a r e likely to be subject to systematic e r r o r s f r o m the experimental point of view. In this case, the minimum ×2 b e c a m e 117. Details a r e shown in table 1 and in figs. 1 and 2. The solutions A and B in table 1 a r e r e g a r d e d as the same. Typical c a s e s in which only values of 5l~ were varied by keeping b l + constant a r e shown in table 2. * Work performed under the auspices of the U. S. Atomic Energy Commission.
351
Volume 5, number 5
PHYSICS
LE TTERS
1 August 1963
I00
The resulting value of ×2 changes d r a s t i c a l l ~ with the a s s u m e d values of bl~. The m i n i m u m × = o b tained in our s e a r c h when the inelastic channels a r e ignored (all bl~ = 1.0) was about 5000.
50
51" and bl ± for initial guessed values : 5 , = 1.5, 51 + =1.5,
Table 2 51-=1.0 ' 52+=0.5,-52-=0.5 ' 53+=0.5,'~3-=0.5.
~0
51 +
51 -
52 +
52 -
53 +
53 -
b
0
bl +
b 1-
b2 +
b2 -
b3 +
b3 -
1319
1.124 0.900
1.287 0.900
0.772 0.900
0.491 0.900
0.202 0.900
0.316 0.900
0.217 0.900
3213
1.079 0.950
1.228 0.950
0.759 0.950
0.501 0.950
0.204 0.950
0.345 0.950
0.217 0.950
6084
1.036 1.000
1.167 1.000
0.745 1.000
0.502 1.000
0.198 1.000
0.372 1.000
0.218 1.000
X2
m
E
~,o
I
I 20
0
I
I 40
I
; 60
;
I 80
I
I I00
I
I 120
I
I 140
~¢m (DEG) Fig. 1. D i f f e r e n t i a l elastic scattering cross section. The experimental points are clue to Brussel and W i l ] i a ~ s 4). The solid cmrve is obtained f r o m the solution B.
I.G
/'\ 0.~
l
i
d J I ! Z 0
l I ,'
..
t
\
N n,, ,J
The c a s e o f / m a x = 2 was also run by using the values ~ v e n in table 1 as a starting set: High values of X2, above 10 000, w e r e obtained even though an inelastic channel was taken into cunsideration. In conclusion, within the limit of our analysis by incorporating the inelastic p a r a m e t e r s , one good fit was obtained without considering phase shifts of the G - and the higher waves. Furtherm o r e , the solution A or B is very likely unique. Although more experimental data for the inelastic channels are needed to confirm our results, the values of bl + in our solution axe not unreasonable when compared with the existing data at 40 MeV 7). It will be interesting to perform an analysis by using either conventional or non-local complex optical potential.
-\
s
We a r e grateful to R. George who modified the p r o g r a m of r e f e r e n c e 6 so that the phase-shift a n a l y s i s with 15 p a r a m e t e r s was p o s s i b l e and to his continuous i n t e r e s t throughout this analysis. We would also like to acknowledge the help of Dr. E. C r o s b i e on understanding his original p r o g r a m .
-0.5
-I.0
t
t
I
60
I
i
t
t
t
I00 140 180 ~cm (DEG) Fig. 2. Polarization. The e x p e r i m e n t a l points a r e due to Hwang et al. 1) The dashed c u r v e is due to G a m m e l and T h a l e r 2). The solid curve is obtained f r o m the solution B.
352
20
1) C. F. Hwang et a l . , P h y s . R e v . L e t t e r s 9 (1962) 104. 2) J. L. G a m m e l and R. H. T h a l e r , P h y s . R e v . 109 (1958) 2041. 3) H.Kanada, S.Nagota, S.Otsuki and Y.Sumi, to be published. 4) M . K . B r u s s e l and J . H . W i l l i a m s , P h y s . R e v . 106 (1957) 286. 5) J. H. Foote et a l . , Phys. Rev. L e t t e r s 122 (1961) 959. 6) E . A . C r o s b i e and J . E . M o n a h a n , AN E 208, A r ~ n n e National L a b o r a t o r y . 7) R . M . E i s b e r g , P h y s . R e v . 102 (1956) 1104.
P HY S I C S L E T T E R S
PHASE-SHIFT
1 August 1963
ANALYSIS OF p-He 4 ELASTIC AT A R O U N D 40 MeV *
SCATTERING
S. SUWA and A. YOKOSAWA Argonne National Laboratory, Argonne, 17linois Received 21 June 1963
The polarization of the protons s c a t t e r e d by He 4 quires a prohibitingly large amount of computer at 38.4 MeV was recently m e a s u r e d 1), and the extime. T h e r e f o r e , we selected a s e a r c h range based perimental r e s u l t s showed an appreciable d i s c r e p upon the theoretical predictions 2, 3). Thus, our ancy f r o m the theoretical ,~redictions of Gammel and starting sets of the phase-shift p a r a m e t e r s were Thaler 2). Kanada et al. 3) derived a non-local random combinations of 5o=1.0 , 1.5; 51+=1.0 , 1.5; p-He 4 potential f r o m the nucleon-nucleon i n t e r a c 51-=0.5 , 1.0; 52+=0 , 0.5; 5 ~ - = 0 , 0 . 5 ; 5 3 + = 0 , 0 . 5 ; tion ; their calculated polarization with this potential and 63- =0 or 0.5, with all b l =1.0, 0.95, 0.90 or is close to the experimental data, but there is still 0.80. F u r t h e r m o r e , as our p r i m a r y interest is to some disagreement. As Gammel and Thaler already investigate the effect of the inelastic p a r a m e t e r s , pointed out, their neglect of the opening of inelastic the p r o g r a m was modified such that at f i r s t the channels is not justified at this energy. T h e r e f o r e , computer v a r i e s the values of only 6/+ by keeping it is hoped that both the differential c r o s s section bl + constant until X2 was minimized and then values data 4) and the polarization data can be fitted simul- of both 5/" and b/+ were varied to f u r t h e r minimize taneously by introducing inelastic p a r a m e t e r s in the ×2. p h a s e - s h i f t analysis, that is, by using complex It was found that, in most c a s e s , phase shifts phase shifts. converge to a single solution (solution A in table 1) The p r o c e d u r e of the analysis is to find the minwhich yields X2 = 176 ; otherwise, they converge imum of X2, which is given by to several sets with values of ×2 higher than 1000. X2
=
£ i [~exp(ei)
i
(ycalc(0/) -2
~(0i)
]
+r'~[pexp(°/)-~p(oj)(i+~) ecalc(e~)]2 ( 2 + (h-7) , where ~exp and P e x p a r e the experimental values of the d i f f e r e n t i a / c r o s s section and the polarization, r e s p e c t i v e l y , and h~ and A P r e p r e s e n t the experimental e r r o r s . The p a r a m e t e r ¢ was introduced to take into account the uncertainty of the incident beam polarization, the fractional e r r o r of which is given by A ¢. F o r each scattering angle, ~calc and P c a l c a r e given as functions of 5l +, the real part of total (nuclear and Coulomb) phase-sRifts, and of inelastic p a r a m e t e r s bl + (ref. 5)). The s u p e r s c i p t s + a r e used in expressing the partial waves of total angular momentum l ± ½. A g e n e r a l l e a s t - s q u a r e s - f i t t i n g p r o c e d u r e 6) p r o g r a m m e d for the IBM 704 computer was adopted for the analysis. The p r o g r a m uses the NewionRaphson method of interaction to determine the set of p a r a m e t e r s which minimizes the X2. All waves up to the F - w a v e s (/max = 3) were taken into account, and the higher waves were ignored, that is, 15 p a r a m e t e r s were considered in the analysis. In this case, an extensive random s e a r c h r e -
Table 1 Total phase shifts, 6l ± (in radian), and inelastic parameters, bl +. X2
50 bo 1.260 A 176 0.839 1.271 B 117 0.839
51+ bl + 1.340 0.727 1.363 0.712
51 b 10.905 0.955 0.911 0.928
52+ b2+ 0.444 0.895 0.446 0.883
62 b2-
53+ b3 +
63b3-
¢
0.275 0.273 0.250 0.013 0.756 0.887 0.878 0.270 0.266 0.245 0.012 0.752 0.880 0.865
In our analysis we had 71 pieces of experimental information to be fitted in t e r m s of 15 p a r a m e t e r s . An attempt was also made to p e r f o r m the analysis by discarding the data of m e a s u r e m e n t of the differential c r o s s section of the f i r s t four smallest angles because such data a r e likely to be subject to systematic e r r o r s f r o m the experimental point of view. In this case, the minimum ×2 b e c a m e 117. Details a r e shown in table 1 and in figs. 1 and 2. The solutions A and B in table 1 a r e r e g a r d e d as the same. Typical c a s e s in which only values of 5l~ were varied by keeping b l + constant a r e shown in table 2. * Work performed under the auspices of the U. S. Atomic Energy Commission.
351
Volume 5, number 5
PHYSICS
LE TTERS
1 August 1963
I00
The resulting value of ×2 changes d r a s t i c a l l ~ with the a s s u m e d values of bl~. The m i n i m u m × = o b tained in our s e a r c h when the inelastic channels a r e ignored (all bl~ = 1.0) was about 5000.
50
51" and bl ± for initial guessed values : 5 , = 1.5, 51 + =1.5,
Table 2 51-=1.0 ' 52+=0.5,-52-=0.5 ' 53+=0.5,'~3-=0.5.
~0
51 +
51 -
52 +
52 -
53 +
53 -
b
0
bl +
b 1-
b2 +
b2 -
b3 +
b3 -
1319
1.124 0.900
1.287 0.900
0.772 0.900
0.491 0.900
0.202 0.900
0.316 0.900
0.217 0.900
3213
1.079 0.950
1.228 0.950
0.759 0.950
0.501 0.950
0.204 0.950
0.345 0.950
0.217 0.950
6084
1.036 1.000
1.167 1.000
0.745 1.000
0.502 1.000
0.198 1.000
0.372 1.000
0.218 1.000
X2
m
E
~,o
I
I 20
0
I
I 40
I
; 60
;
I 80
I
I I00
I
I 120
I
I 140
~¢m (DEG) Fig. 1. D i f f e r e n t i a l elastic scattering cross section. The experimental points are clue to Brussel and W i l ] i a ~ s 4). The solid cmrve is obtained f r o m the solution B.
I.G
/'\ 0.~
l
i
d J I ! Z 0
l I ,'
..
t
\
N n,, ,J
The c a s e o f / m a x = 2 was also run by using the values ~ v e n in table 1 as a starting set: High values of X2, above 10 000, w e r e obtained even though an inelastic channel was taken into cunsideration. In conclusion, within the limit of our analysis by incorporating the inelastic p a r a m e t e r s , one good fit was obtained without considering phase shifts of the G - and the higher waves. Furtherm o r e , the solution A or B is very likely unique. Although more experimental data for the inelastic channels are needed to confirm our results, the values of bl + in our solution axe not unreasonable when compared with the existing data at 40 MeV 7). It will be interesting to perform an analysis by using either conventional or non-local complex optical potential.
-\
s
We a r e grateful to R. George who modified the p r o g r a m of r e f e r e n c e 6 so that the phase-shift a n a l y s i s with 15 p a r a m e t e r s was p o s s i b l e and to his continuous i n t e r e s t throughout this analysis. We would also like to acknowledge the help of Dr. E. C r o s b i e on understanding his original p r o g r a m .
-0.5
-I.0
t
t
I
60
I
i
t
t
t
I00 140 180 ~cm (DEG) Fig. 2. Polarization. The e x p e r i m e n t a l points a r e due to Hwang et al. 1) The dashed c u r v e is due to G a m m e l and T h a l e r 2). The solid curve is obtained f r o m the solution B.
352
20
1) C. F. Hwang et a l . , P h y s . R e v . L e t t e r s 9 (1962) 104. 2) J. L. G a m m e l and R. H. T h a l e r , P h y s . R e v . 109 (1958) 2041. 3) H.Kanada, S.Nagota, S.Otsuki and Y.Sumi, to be published. 4) M . K . B r u s s e l and J . H . W i l l i a m s , P h y s . R e v . 106 (1957) 286. 5) J. H. Foote et a l . , Phys. Rev. L e t t e r s 122 (1961) 959. 6) E . A . C r o s b i e and J . E . M o n a h a n , AN E 208, A r ~ n n e National L a b o r a t o r y . 7) R . M . E i s b e r g , P h y s . R e v . 102 (1956) 1104.