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Simultaneous phase shift analysis of 210 MeV nucleon-nucleon scattering
~
Nuclear Physics B4 (1968) 527-533. North-Holland Publ. Comp., Amsterdam
SIMULTANEOUS
PHASE
SHIFT
NUCLEON-NUCLEON Z. JANOUT
ANALYSIS
OF
210
MeV
SCATTERING
*, Yu. M. KAZARINOV, F. LEHAR *, P. WINTERNITZ Joint Institute f o r Nuclear Research Dubna, USSR
**
Received 2 January 1968
A b s t r a c t : P h a s e - s h i f t analysis at 210 MeV has been performed taking /max - 5. New data on the differential c r o s s section, polarization and triple scattering p a r a m e t e r s have been used and the procedure leads to a unique solution. The angular dependences of experimental quantities a r e shown in d i a g r a m s .
i . INTRODUCTION
T h e p h a s e - s h i f t a n a l y s i s of n u c l e o n - n u c l e o n s c a t t e r i n g d a t a a t 210 M e V h a s p r e v i o u s l y b e e n p e r f o r m e d in r e f s . [1-4]. S i n c e m a n y n e w e x p e r i m e n t a l d a t a h a v e b e e n o b t a i n e d r e c e n t l y , i t i s n e c e s s a r y to t a k e t h e s e into a c count and find more accurate phase-shifts at this energy. T h e p u r p o s e of t h i s i n v e s t i g a t i o n w a s , on one h a n d , to s p e c i f y the p r e v i o u s l y e x i s t i n g s e t of p h a s e - s h i f t s f o r / m a x = 3 a n d / m - x = 4. On t h e o t h e r h a n d t h e v a l u e / m a x , up to w h i c h the p h a s e s a r e v a r i e d ~ o r l > / m a x t h e y a r e c a l c u l a t e d u s i n g the o n e - p i o n e x c h a n g e a p p r o x i m a t i o n ) i s d e t e r m i n e d u s i n g v a r i o u s s t a t i s t i c a l c r i t e r i a a n d d e p e n d s on t h e a m o u n t a n d a c c u r a c y of t h e e x p e r i m e n t a l d a t a . T h u s t h e i n c l u s i o n of n e w d a t a u s u a l l y s h i f t s / m a x to h i g h e r v a l u e s , w h i c h c a n c a u s e t h e a p p e a r a n c e of new p h a s e - s h i f t s e t s . In the c o n s i d e r e d c a s e i t p r o v e d n e c e s s a r y to i n c r e a s e t h e a n g u l a r m o m e n t u m v a l u e to / m a x = 5. T h e a m b i g u i t y of t h e p h a s e - s h i f t a n a l y s i s f o r h i g h e r /max has not been investigated previously.
2. EXPERIMENTAL
DATA
AND
PHASE-SHIFT
ANALYSIS
T h e d a t a u s e d in t h e p h a s e - s h i f t a n a l y s i s a t 210 M e V [5-16] a r e g i v e n in t a b l e 1. T h e new d a t a on t h e d i f f e r e n t i a l c r o s s s e c t i o n f o r s m a l l s c a t t e r i n g a n g l e s [6], t h e p o l a r i z a t i o n in p , p [6] a n d n , p s c a t t e r i n g [13] a n d t h e t r i p l e s c a t t e r i n g p a r a m e t e r s D t ( r e f . [15]) a n d R t ( r e f . [16]) f o r t h e r e c o i l p a r t i c l e s * On leave of absence from the Faculty of Technical and Nuclear Physics, Technical University, Prague, Czechoslovakia. ** On leave of absence from the Nuclear R e s e a r c h Institute of the Czechoslovak A c a demy of Sciences, f~e~.
Z. JANOUT et al.
528
Table 1 The e x p e r i m e n t a l data, u s e d in the p h a s e - s h i f t a n a l y s i s at 210 MeV
Energy (MeV)
Angular r a n g e in degrees (c .m .s .)
Number
Measured quantity
ffpp
213
30-90
7
2.39
213
9-39
13
13.65
210
30-90
7
3.88
210
13-113
14
213
9-39
Ppp
Autho:rs
Refs
Konradi (1961)
[5]
M a r s h a l l et al. (1966)
[6]
Tinlot, W a r n e r (1961)
[7]
12.12
B a s k i r et al. (1957)
[8]
13
13.88
M a r s h a l l et al. (1966)
[6]
of points
AX2,
C oe nmt m a
b
217
60-120
7
6.42
Tinlot, Warner (1961)
[7]
Rpp
213
30-90
7
2.74
England et al. (1961)
[9]
App
213
30-90
7
5.51
c
England et al. (1961)
[9]
R' PP
213
30-90
5
3.36
d
G o t o w ,Lobkowicz (1964)
[10]
Dpp
213
30-90
7
4.80
Gotow et al. (1962)
[11]
~np
200
6-180
21
28.25
K a z a r i n o v , Simonov (1962)
[12]
P
199
77-158
7
7.20
Thorndike (1967)
[13]
Ppn
215
40-120
9
10.94
Tinlot, Warner (:1961)
[7]
Dpn
212
13-39
5
3.01
Warner, Tinlot (1962)
[14]
Dt
197
127-148
3
2.05
Spalding et al. (1967)
[15]
Rt
203
139-179
5
5.38
Reay et al. (1966)
[16]
(Ttnp
200
1
0.12
Kazarinov, Simonov (1962)
[12]
e
np
* The contribution to X 2 f o r the 1 - s t set.
f
PHASE SHIFT ANALYSIS
529
C o m m e n t s to t a b l e 1 a) Their data were multiplied by the correction factor 0.986 according to ref. [4]. b) The polarization data from ref. [8], which do not agree with the more accurate values obtained at 210 MeV (ref. [7]) in the angular region 50o-70 ° were not included in the phase-shift analysis. c) The values of the p a r a m e t e r Apo at 800 and 90 ° given in the original paper (see ref. [9]) were excluded since th6y differ by more than 3 standard deviations from the calculated values. Instead of these points we used the new data, as given by Thorndike (ref. [13]). In this case the contributions to X2 are equal to 1.58 and 0.06 for 80 ° and 90 ° , respectively. d) The values at the angles 60 ° and 70° had to be excluded since their contributions to X2 are equal to 12.9 and 16.3, respectively. The number of points in table 1 are shown without the excluded points. e) The values at the angle 86.6 ° had to be excluded because they differ by more than 3 standard deviations from the calculated values. f) The quantities P n - and R t were m e a s u r e d in the quasi-elastic scattering of protons on neutrons in deuteron nuclei. The measured values were corrected later by Koehler et al. (ref. [17]) taking the final S-state interaction into account. The corrected values of R t are in worse agreement with the calculated value, than the uncorrected data. This may be due to a systematical e r r o r of about 14% contained in the corrections (ref. [13]).
i n n , p s c a t t e r i n g , w e r e p r e v i o u s l y n o t i n c l u d e d i n the p h a s e - s h i f t a n a l y s i s . T h e t r i p l e s c a t t e r i n g p a r a m e t e r s f o r the r e c o i l p a r t i c l e a r e d i s c u s s e d i n d e t a i l i n r e f . [18] w h e r e a l l the f o r m u l a e u s e d i n t h i s p a p e r a r e g i v e n , taking r e l a t i v i s t i c effects into account. A l l the e x p e r i m e n t a l d a t a a r e c o n c e n t r a t e d i n the e n e r g y r e g i o n 1 9 7 - 2 1 7 MeV. T h e m e a s u r e d v a l u e s w e r e not n o r m a l i z e d to the m e a n e n e r g y 210 MeV. T h e p h a s e - s h i f t a n a l y s i s w a s c a r r i e d out a c c o r d i n g to the p r o g r a m m e , d e s c r i b e d i n d e t a i l i n r e f . [1]. Hundred searches from random initial parameters were made, from w h i c h t h r e e s o l u t i o n s with ×2 = 127.3, 159.8 a n d 190.4 w e r e o b t a i n e d i n the X2 --< 2X-2 r e g i o n ( t a b l e s 2 a n d 3). It s h o u l d be n o t e d t h a t a l l the o b t a i n e d s o l u t i o n s c o r r e s p o n d to the p h a s e - s h i f t s s e t s o b t a i n e d p r e v i o u s l y i n the f i r s t a n a l y s i s of p , p a n d n , p d a t a a t t h i s e n e r g y [1] ( s o l u t i o n s b, b I a n d c, r e s p e c t i v e l y ) . T h e s e a r c h f o r the s o l u t i o n s f r o m r a n d o m i n i t i a l p a r a m e t e r s w a s p e r f o r m e d u s i n g a l l the a v a i l a b l e d a t a a n d the p o i n t s m e n t i o n e d i n the c o m m e n t s to t a b l e 1 w e r e e x c l u d e d only d u r i n g the s p e c i f i c a t i o n of the o b t a i n e d s o l u t i o n s . T h e s e p o i n t s did n o t a g r e e with a n y of the t h r e e s o l u t i o n s i n the r e g i o n . S o l u t i o n 3 i n t a b l e 2 (×2 = 190.4) c a n b e r e j e c t e d a c c o r d i n g to the ×2 c r i t e r i o n m a k i n g a m e a n s q u a r e e r r o r s m a l l e r t h a n 10-3%. T h e e r r o r due to the r e j e c t i o n of the f i r s t a n d s e c o n d s o l u t i o n i s P(X 2 = 127.3) = 14% a n d p(×2 >I 159.8) = 0.14%, r e s p e c t i v e l y . It s h o u l d b e n o t e d , t h a t i n r e j e c t i n g
530
Z. JANOUT e t al. Table 2 The p h a s e - s h i f t s in d e g r e e s (the Stapp e t al. [19] p a r a m e t r i z a t i o n ) f o r 210 MeV nucleon-nucleon scattering P r e s e n t solution Phase shifts ± A5
Solution f r o m r e f . [2]
Solution f r o m r e f . [3]
5 ± A5
5 ± A5
1S 0
4.99
0.46
5.14
0.49
5.18
0.57
3S 1
15.44
1.84
17.11
2.87
18.23
3.10
3P 0
-1.34
0.50
-1.02
0.51
-0.79
0.61
lP 1
-23.99
2.56
-30.34
1.80
-23.37
8.01
3P 1
-22.46
0.20
-22.04
0.15
-21.59
0.60
3P 2
15.73
0.14
16.14
0.12
15.89
0.27
5.77
0.81
1.22
1.88
3.13
2.87
E3 3D 1
-19.84
1.64
-22.40
2.58
-22.98
4.04
1D 2
6.96
0.24
7.10
0.18
7.02
0.32
3D 2
27.09
2.06
22.47
3.58
23.39
4.33
3D 3
4.20
0.91
2.36
1.36
2.82
1.66
E2
-2.62
0.12
-2.76
0.11
-2.78
0.19
3F 2
1.32
0.21
1.56
0.25
1.58
0.34
1F 3
-4.72
1.35
-2.98
1.42
-5.53
2.78
3F 3
-2.62
0.17
-2.44
0.17
-2.58
0.21
3F 4
2.19
0.11
2.14
0.16
2.32
0.20
~3 3G 3
6.36
0.83
5.87
1.04
7.09
0.96
-1.66
0.95
0.13
1.11
-0.42
1.50
1G 4
1.10
0.09
1.08
0.13
1.04
0.16
3G 4
5.48
1.13
4.08
1.66
4.40
2.60
3G 5
-0.12
0.67
1.55
1.15
0.00
1.50
c4
-0.87
0.08
-0.94
0.09
3H 4
-0.06
0.21
0.47
0.36
1H 5
-0.54
0.66
-
-
3H 5
-0.96
0.17
-0.64
0.22
3H 6
0.00
0.13
0.41
0.27
f2
0.063
0.008
0.071
fix.
0.064
0.006
PHASE S H I F T ANALYSIS
531
Table 3 T h e p h a s e - s h i f t s in d e g r e e s (the Stapp e t al. [19] p a r a m e t r i z a t i o n ) f o r 210 MeV nucleon-nucleon scattering The rejected solutions Phase shifts
2nd s e t 5 ~: A 5
1S 0
5.41
0.50
3S 1
14.20
31) 0
3.13
3rd set 5 + A5 -15.27
0.71
2.15
8.77
4.17
0.54
-25.71
0.65
11) 1
-19.92
2.34
-0.02
2.06
31) 1
-20.26
0.19
-2.40
0.41
31) 2
17.51
0.15
18.83
0.36
6.78
0.82
29.65
1.46
E1 3D 1
-17.47
1.31
-9.73
2.73
1D 2
7.95
0.20
3.96
0.18
3D 2
29.26
1.63
16.32
1.22
3D 3
0.92
1.20
0.91
1.22
-1.83
0.15
-6.65
0.20
3F 2
-1.67
0.19
0.58
0.47
1F 3
-6.08
1.43
2.81
1.27
3F 3
-2.01
0.16
-1.64
0.16
3F 4
-0.02
0.14
1.25
0.36
E2
c3
6.33
0.79
10.08
0.72
3G 3
-4.13
0.72
-2.44
0.81
1G 4
0.86
0.09
1.24
0.17
3G 4
6.62
0.89
4.57
0.56
0.08
0.73
-0.47
0.53
-1.07
0.08
-0.78
0.12
3G 5 ~4 3H 4
0.33
0.15
-0.21
0.16
1H 5
-0.44
0.60
-2.00
0.81
3H 5
-0.64
0.15
-0.62
0.09
3H 6
0.51
0.10
-0.22
0.15
f2
0.077
0.008
0.067
0.010
o
~I .JJ; ~ ~J:_--~:.-* -~
,'/," ~ , , N
O
PHASE SHIFT ANALYSIS
533
s o l u t i o n 2 i t w o u l d b e m o r e c o r r e c t to u s e a d i f f e r e n t c r i t e r i o n [20]. It follows from this discussion that for /max = 5 the phase-shift analysis a t 210 M e V r e m a i n s u n a m b i g u o u s .
3. R E S U L T S T h e s e t of p h a s e - s h i f t s w i t h ×2 = 127.3 i s g i v e n in t a b l e 2. T h e p r e v i o u s l y k n o w n s e t s of K a z a r i n o v e t al. [2] a n d A r n d t a n d M c G r e g o r [3] a r e g i v e n in t h e s a m e t a b l e f o r c o m p a r i s o n . It m a y b e s e e n f r o m t h e t a b l e , t h a t the m e a n v a l u e s of t h e p h a s e - s h i f t s h a v e n o t c h a n g e d v e r y m u c h , but t h e i r e r r o r s h a v e decreased considerably. On t h e b a s i s of p h a s e - s h i f t s e t 1 t h e a n g u l a r d e p e n d e n c e s of the e x p e r i m e n t a l v a l u e s w e r e c a l c u l a t e d t o g e t h e r w i t h t h e i r c o r r i d o r of e r r o r s . T h e c o r r i d o r of e r r o r s r e p r e s e n t s t h e m e a n s q u a r e e r r o r a t e a c h p o i n t of the c a l c u l a t e d c u r v e . T h e m o s t i n t e r e s t i n g e x p e r i m e n t a l v a l u e s u s e d in t h e p r e s e n t p h a s e - s h i f t a n a l y s i s (D t a n d Rt) a r e g i v e n in t h e g r a p h s on f i g . 1. F o r t h e r e m a i n i n g q u a n t i t i e s c o n f r o n t r e f . [18]. In c o n c l u s i o n t h e a u t h o r s e x p r e s s t h e i r d e e p g r a t i t u d e to S. M. B i l e n k y , L. I. L a p i d u s , A. P a z m a n , R. M. R y n d i n a n d J. A. S m o r o d i n s k y f o r h e l p f u l d i s c u s s i o n s , F. L o b k o w i c z f o r a p r i v a t e c o m m u n i c a t i o n a n d to E. D u d o v a , J. F i n g e r o v a a n d N. V. V o l c h k o v a f o r h e l p in t h e w o r k .
RE FERENCES
[1] Yu. M. Kazarinov, I. N. Silin, J E T P 43 (1962) 692. [2] Yu. M. Kazarinov, V.S. Kiselev, Yu. N. Simonov, P r e p r i n t JINR P-2241, Dubna 1965. [3] R. A. Arndt, M.H. MacGregor, Phys. Rev. 141 {1966) 873. [4] P . S i g n e l l , N . R . Y o d e r , J. E. Matos, Phys. Rev. 135 (1964) Bl128. [5] A. Konradi, T h e s i s , University of Rochester, 1961 {unpublished) [6] J. F. Marshall, C.N. Brown, F . L . Lobkowicz, Phys. Rev. 150 (1966} 1119. [7] J. H. Tinlot, R. E. W a r n e r , Phys. Rev. 124 (1961) 890. [8] E. B a s k i r , E. M. Hafner, A. Roberts, J. H. Tinlot, Phys. Rev. 106 {1957) 564. [91 A. C. England, W.A. Gibson, K. Gotow, E. Heer, J. H. Tinlot, Phys. Rev. 124 (1961) 561. [10] K. Gotow, F. Lobkowicz, Phys. Rev. 136 (1964) B1345; F. Lobkowicz, p r i v a t e communication. [11] K. Gotow, F. Lobkovicz, E. Heer, Phys. Rev. 127 {1962} 2206. [12] Yu. M. Kazarinov, Yu. N. Simonov, J E T P 43 (1962} 35. [13] E. H. Thorndike, Rev. Mod. Phys. 39 (1967) 513. [14] R. E. W a r n e r , J . H . Tinlot, Phys. Rev. 125 (1962) 1028. [15] D. Spalding, A. Thomas, E. Thorndike, Phys. Rev. 158 (1967} 1338. [16] N. W. Reay, E.H. Thorndike, D. Spalding, A.R. Thomas, Phys. Rev. 150 {1966} 801. [17] P. Koehler, E . T h o r n d i k e , A. C r o m e r , Phys. Rev. 134 (1964)B1030. [18] Z. Janout, Yu. M. Kazarinov, F. L e h a r , P. Winternitz, p r e p r i n t JINR E1-3345, Dubna 1967. [19] H. P. Stapp, T. Ypsilantis, N. Metropolis, Phys. Rev. 105 {1957) 302. [20] Yu. M. Kazarinov, V. S. Kiselev, A. M. Rozanova, I. N. Silin, P r e p r i n t JINR E13268, Dubna 1967.
Nuclear Physics B4 (1968) 527-533. North-Holland Publ. Comp., Amsterdam
SIMULTANEOUS
PHASE
SHIFT
NUCLEON-NUCLEON Z. JANOUT
ANALYSIS
OF
210
MeV
SCATTERING
*, Yu. M. KAZARINOV, F. LEHAR *, P. WINTERNITZ Joint Institute f o r Nuclear Research Dubna, USSR
**
Received 2 January 1968
A b s t r a c t : P h a s e - s h i f t analysis at 210 MeV has been performed taking /max - 5. New data on the differential c r o s s section, polarization and triple scattering p a r a m e t e r s have been used and the procedure leads to a unique solution. The angular dependences of experimental quantities a r e shown in d i a g r a m s .
i . INTRODUCTION
T h e p h a s e - s h i f t a n a l y s i s of n u c l e o n - n u c l e o n s c a t t e r i n g d a t a a t 210 M e V h a s p r e v i o u s l y b e e n p e r f o r m e d in r e f s . [1-4]. S i n c e m a n y n e w e x p e r i m e n t a l d a t a h a v e b e e n o b t a i n e d r e c e n t l y , i t i s n e c e s s a r y to t a k e t h e s e into a c count and find more accurate phase-shifts at this energy. T h e p u r p o s e of t h i s i n v e s t i g a t i o n w a s , on one h a n d , to s p e c i f y the p r e v i o u s l y e x i s t i n g s e t of p h a s e - s h i f t s f o r / m a x = 3 a n d / m - x = 4. On t h e o t h e r h a n d t h e v a l u e / m a x , up to w h i c h the p h a s e s a r e v a r i e d ~ o r l > / m a x t h e y a r e c a l c u l a t e d u s i n g the o n e - p i o n e x c h a n g e a p p r o x i m a t i o n ) i s d e t e r m i n e d u s i n g v a r i o u s s t a t i s t i c a l c r i t e r i a a n d d e p e n d s on t h e a m o u n t a n d a c c u r a c y of t h e e x p e r i m e n t a l d a t a . T h u s t h e i n c l u s i o n of n e w d a t a u s u a l l y s h i f t s / m a x to h i g h e r v a l u e s , w h i c h c a n c a u s e t h e a p p e a r a n c e of new p h a s e - s h i f t s e t s . In the c o n s i d e r e d c a s e i t p r o v e d n e c e s s a r y to i n c r e a s e t h e a n g u l a r m o m e n t u m v a l u e to / m a x = 5. T h e a m b i g u i t y of t h e p h a s e - s h i f t a n a l y s i s f o r h i g h e r /max has not been investigated previously.
2. EXPERIMENTAL
DATA
AND
PHASE-SHIFT
ANALYSIS
T h e d a t a u s e d in t h e p h a s e - s h i f t a n a l y s i s a t 210 M e V [5-16] a r e g i v e n in t a b l e 1. T h e new d a t a on t h e d i f f e r e n t i a l c r o s s s e c t i o n f o r s m a l l s c a t t e r i n g a n g l e s [6], t h e p o l a r i z a t i o n in p , p [6] a n d n , p s c a t t e r i n g [13] a n d t h e t r i p l e s c a t t e r i n g p a r a m e t e r s D t ( r e f . [15]) a n d R t ( r e f . [16]) f o r t h e r e c o i l p a r t i c l e s * On leave of absence from the Faculty of Technical and Nuclear Physics, Technical University, Prague, Czechoslovakia. ** On leave of absence from the Nuclear R e s e a r c h Institute of the Czechoslovak A c a demy of Sciences, f~e~.
Z. JANOUT et al.
528
Table 1 The e x p e r i m e n t a l data, u s e d in the p h a s e - s h i f t a n a l y s i s at 210 MeV
Energy (MeV)
Angular r a n g e in degrees (c .m .s .)
Number
Measured quantity
ffpp
213
30-90
7
2.39
213
9-39
13
13.65
210
30-90
7
3.88
210
13-113
14
213
9-39
Ppp
Autho:rs
Refs
Konradi (1961)
[5]
M a r s h a l l et al. (1966)
[6]
Tinlot, W a r n e r (1961)
[7]
12.12
B a s k i r et al. (1957)
[8]
13
13.88
M a r s h a l l et al. (1966)
[6]
of points
AX2,
C oe nmt m a
b
217
60-120
7
6.42
Tinlot, Warner (1961)
[7]
Rpp
213
30-90
7
2.74
England et al. (1961)
[9]
App
213
30-90
7
5.51
c
England et al. (1961)
[9]
R' PP
213
30-90
5
3.36
d
G o t o w ,Lobkowicz (1964)
[10]
Dpp
213
30-90
7
4.80
Gotow et al. (1962)
[11]
~np
200
6-180
21
28.25
K a z a r i n o v , Simonov (1962)
[12]
P
199
77-158
7
7.20
Thorndike (1967)
[13]
Ppn
215
40-120
9
10.94
Tinlot, Warner (:1961)
[7]
Dpn
212
13-39
5
3.01
Warner, Tinlot (1962)
[14]
Dt
197
127-148
3
2.05
Spalding et al. (1967)
[15]
Rt
203
139-179
5
5.38
Reay et al. (1966)
[16]
(Ttnp
200
1
0.12
Kazarinov, Simonov (1962)
[12]
e
np
* The contribution to X 2 f o r the 1 - s t set.
f
PHASE SHIFT ANALYSIS
529
C o m m e n t s to t a b l e 1 a) Their data were multiplied by the correction factor 0.986 according to ref. [4]. b) The polarization data from ref. [8], which do not agree with the more accurate values obtained at 210 MeV (ref. [7]) in the angular region 50o-70 ° were not included in the phase-shift analysis. c) The values of the p a r a m e t e r Apo at 800 and 90 ° given in the original paper (see ref. [9]) were excluded since th6y differ by more than 3 standard deviations from the calculated values. Instead of these points we used the new data, as given by Thorndike (ref. [13]). In this case the contributions to X2 are equal to 1.58 and 0.06 for 80 ° and 90 ° , respectively. d) The values at the angles 60 ° and 70° had to be excluded since their contributions to X2 are equal to 12.9 and 16.3, respectively. The number of points in table 1 are shown without the excluded points. e) The values at the angle 86.6 ° had to be excluded because they differ by more than 3 standard deviations from the calculated values. f) The quantities P n - and R t were m e a s u r e d in the quasi-elastic scattering of protons on neutrons in deuteron nuclei. The measured values were corrected later by Koehler et al. (ref. [17]) taking the final S-state interaction into account. The corrected values of R t are in worse agreement with the calculated value, than the uncorrected data. This may be due to a systematical e r r o r of about 14% contained in the corrections (ref. [13]).
i n n , p s c a t t e r i n g , w e r e p r e v i o u s l y n o t i n c l u d e d i n the p h a s e - s h i f t a n a l y s i s . T h e t r i p l e s c a t t e r i n g p a r a m e t e r s f o r the r e c o i l p a r t i c l e a r e d i s c u s s e d i n d e t a i l i n r e f . [18] w h e r e a l l the f o r m u l a e u s e d i n t h i s p a p e r a r e g i v e n , taking r e l a t i v i s t i c effects into account. A l l the e x p e r i m e n t a l d a t a a r e c o n c e n t r a t e d i n the e n e r g y r e g i o n 1 9 7 - 2 1 7 MeV. T h e m e a s u r e d v a l u e s w e r e not n o r m a l i z e d to the m e a n e n e r g y 210 MeV. T h e p h a s e - s h i f t a n a l y s i s w a s c a r r i e d out a c c o r d i n g to the p r o g r a m m e , d e s c r i b e d i n d e t a i l i n r e f . [1]. Hundred searches from random initial parameters were made, from w h i c h t h r e e s o l u t i o n s with ×2 = 127.3, 159.8 a n d 190.4 w e r e o b t a i n e d i n the X2 --< 2X-2 r e g i o n ( t a b l e s 2 a n d 3). It s h o u l d be n o t e d t h a t a l l the o b t a i n e d s o l u t i o n s c o r r e s p o n d to the p h a s e - s h i f t s s e t s o b t a i n e d p r e v i o u s l y i n the f i r s t a n a l y s i s of p , p a n d n , p d a t a a t t h i s e n e r g y [1] ( s o l u t i o n s b, b I a n d c, r e s p e c t i v e l y ) . T h e s e a r c h f o r the s o l u t i o n s f r o m r a n d o m i n i t i a l p a r a m e t e r s w a s p e r f o r m e d u s i n g a l l the a v a i l a b l e d a t a a n d the p o i n t s m e n t i o n e d i n the c o m m e n t s to t a b l e 1 w e r e e x c l u d e d only d u r i n g the s p e c i f i c a t i o n of the o b t a i n e d s o l u t i o n s . T h e s e p o i n t s did n o t a g r e e with a n y of the t h r e e s o l u t i o n s i n the r e g i o n . S o l u t i o n 3 i n t a b l e 2 (×2 = 190.4) c a n b e r e j e c t e d a c c o r d i n g to the ×2 c r i t e r i o n m a k i n g a m e a n s q u a r e e r r o r s m a l l e r t h a n 10-3%. T h e e r r o r due to the r e j e c t i o n of the f i r s t a n d s e c o n d s o l u t i o n i s P(X 2 = 127.3) = 14% a n d p(×2 >I 159.8) = 0.14%, r e s p e c t i v e l y . It s h o u l d b e n o t e d , t h a t i n r e j e c t i n g
530
Z. JANOUT e t al. Table 2 The p h a s e - s h i f t s in d e g r e e s (the Stapp e t al. [19] p a r a m e t r i z a t i o n ) f o r 210 MeV nucleon-nucleon scattering P r e s e n t solution Phase shifts ± A5
Solution f r o m r e f . [2]
Solution f r o m r e f . [3]
5 ± A5
5 ± A5
1S 0
4.99
0.46
5.14
0.49
5.18
0.57
3S 1
15.44
1.84
17.11
2.87
18.23
3.10
3P 0
-1.34
0.50
-1.02
0.51
-0.79
0.61
lP 1
-23.99
2.56
-30.34
1.80
-23.37
8.01
3P 1
-22.46
0.20
-22.04
0.15
-21.59
0.60
3P 2
15.73
0.14
16.14
0.12
15.89
0.27
5.77
0.81
1.22
1.88
3.13
2.87
E3 3D 1
-19.84
1.64
-22.40
2.58
-22.98
4.04
1D 2
6.96
0.24
7.10
0.18
7.02
0.32
3D 2
27.09
2.06
22.47
3.58
23.39
4.33
3D 3
4.20
0.91
2.36
1.36
2.82
1.66
E2
-2.62
0.12
-2.76
0.11
-2.78
0.19
3F 2
1.32
0.21
1.56
0.25
1.58
0.34
1F 3
-4.72
1.35
-2.98
1.42
-5.53
2.78
3F 3
-2.62
0.17
-2.44
0.17
-2.58
0.21
3F 4
2.19
0.11
2.14
0.16
2.32
0.20
~3 3G 3
6.36
0.83
5.87
1.04
7.09
0.96
-1.66
0.95
0.13
1.11
-0.42
1.50
1G 4
1.10
0.09
1.08
0.13
1.04
0.16
3G 4
5.48
1.13
4.08
1.66
4.40
2.60
3G 5
-0.12
0.67
1.55
1.15
0.00
1.50
c4
-0.87
0.08
-0.94
0.09
3H 4
-0.06
0.21
0.47
0.36
1H 5
-0.54
0.66
-
-
3H 5
-0.96
0.17
-0.64
0.22
3H 6
0.00
0.13
0.41
0.27
f2
0.063
0.008
0.071
fix.
0.064
0.006
PHASE S H I F T ANALYSIS
531
Table 3 T h e p h a s e - s h i f t s in d e g r e e s (the Stapp e t al. [19] p a r a m e t r i z a t i o n ) f o r 210 MeV nucleon-nucleon scattering The rejected solutions Phase shifts
2nd s e t 5 ~: A 5
1S 0
5.41
0.50
3S 1
14.20
31) 0
3.13
3rd set 5 + A5 -15.27
0.71
2.15
8.77
4.17
0.54
-25.71
0.65
11) 1
-19.92
2.34
-0.02
2.06
31) 1
-20.26
0.19
-2.40
0.41
31) 2
17.51
0.15
18.83
0.36
6.78
0.82
29.65
1.46
E1 3D 1
-17.47
1.31
-9.73
2.73
1D 2
7.95
0.20
3.96
0.18
3D 2
29.26
1.63
16.32
1.22
3D 3
0.92
1.20
0.91
1.22
-1.83
0.15
-6.65
0.20
3F 2
-1.67
0.19
0.58
0.47
1F 3
-6.08
1.43
2.81
1.27
3F 3
-2.01
0.16
-1.64
0.16
3F 4
-0.02
0.14
1.25
0.36
E2
c3
6.33
0.79
10.08
0.72
3G 3
-4.13
0.72
-2.44
0.81
1G 4
0.86
0.09
1.24
0.17
3G 4
6.62
0.89
4.57
0.56
0.08
0.73
-0.47
0.53
-1.07
0.08
-0.78
0.12
3G 5 ~4 3H 4
0.33
0.15
-0.21
0.16
1H 5
-0.44
0.60
-2.00
0.81
3H 5
-0.64
0.15
-0.62
0.09
3H 6
0.51
0.10
-0.22
0.15
f2
0.077
0.008
0.067
0.010
o
~I .JJ; ~ ~J:_--~:.-* -~
,'/," ~ , , N
O
PHASE SHIFT ANALYSIS
533
s o l u t i o n 2 i t w o u l d b e m o r e c o r r e c t to u s e a d i f f e r e n t c r i t e r i o n [20]. It follows from this discussion that for /max = 5 the phase-shift analysis a t 210 M e V r e m a i n s u n a m b i g u o u s .
3. R E S U L T S T h e s e t of p h a s e - s h i f t s w i t h ×2 = 127.3 i s g i v e n in t a b l e 2. T h e p r e v i o u s l y k n o w n s e t s of K a z a r i n o v e t al. [2] a n d A r n d t a n d M c G r e g o r [3] a r e g i v e n in t h e s a m e t a b l e f o r c o m p a r i s o n . It m a y b e s e e n f r o m t h e t a b l e , t h a t the m e a n v a l u e s of t h e p h a s e - s h i f t s h a v e n o t c h a n g e d v e r y m u c h , but t h e i r e r r o r s h a v e decreased considerably. On t h e b a s i s of p h a s e - s h i f t s e t 1 t h e a n g u l a r d e p e n d e n c e s of the e x p e r i m e n t a l v a l u e s w e r e c a l c u l a t e d t o g e t h e r w i t h t h e i r c o r r i d o r of e r r o r s . T h e c o r r i d o r of e r r o r s r e p r e s e n t s t h e m e a n s q u a r e e r r o r a t e a c h p o i n t of the c a l c u l a t e d c u r v e . T h e m o s t i n t e r e s t i n g e x p e r i m e n t a l v a l u e s u s e d in t h e p r e s e n t p h a s e - s h i f t a n a l y s i s (D t a n d Rt) a r e g i v e n in t h e g r a p h s on f i g . 1. F o r t h e r e m a i n i n g q u a n t i t i e s c o n f r o n t r e f . [18]. In c o n c l u s i o n t h e a u t h o r s e x p r e s s t h e i r d e e p g r a t i t u d e to S. M. B i l e n k y , L. I. L a p i d u s , A. P a z m a n , R. M. R y n d i n a n d J. A. S m o r o d i n s k y f o r h e l p f u l d i s c u s s i o n s , F. L o b k o w i c z f o r a p r i v a t e c o m m u n i c a t i o n a n d to E. D u d o v a , J. F i n g e r o v a a n d N. V. V o l c h k o v a f o r h e l p in t h e w o r k .
RE FERENCES
[1] Yu. M. Kazarinov, I. N. Silin, J E T P 43 (1962) 692. [2] Yu. M. Kazarinov, V.S. Kiselev, Yu. N. Simonov, P r e p r i n t JINR P-2241, Dubna 1965. [3] R. A. Arndt, M.H. MacGregor, Phys. Rev. 141 {1966) 873. [4] P . S i g n e l l , N . R . Y o d e r , J. E. Matos, Phys. Rev. 135 (1964) Bl128. [5] A. Konradi, T h e s i s , University of Rochester, 1961 {unpublished) [6] J. F. Marshall, C.N. Brown, F . L . Lobkowicz, Phys. Rev. 150 (1966} 1119. [7] J. H. Tinlot, R. E. W a r n e r , Phys. Rev. 124 (1961) 890. [8] E. B a s k i r , E. M. Hafner, A. Roberts, J. H. Tinlot, Phys. Rev. 106 {1957) 564. [91 A. C. England, W.A. Gibson, K. Gotow, E. Heer, J. H. Tinlot, Phys. Rev. 124 (1961) 561. [10] K. Gotow, F. Lobkowicz, Phys. Rev. 136 (1964) B1345; F. Lobkowicz, p r i v a t e communication. [11] K. Gotow, F. Lobkovicz, E. Heer, Phys. Rev. 127 {1962} 2206. [12] Yu. M. Kazarinov, Yu. N. Simonov, J E T P 43 (1962} 35. [13] E. H. Thorndike, Rev. Mod. Phys. 39 (1967) 513. [14] R. E. W a r n e r , J . H . Tinlot, Phys. Rev. 125 (1962) 1028. [15] D. Spalding, A. Thomas, E. Thorndike, Phys. Rev. 158 (1967} 1338. [16] N. W. Reay, E.H. Thorndike, D. Spalding, A.R. Thomas, Phys. Rev. 150 {1966} 801. [17] P. Koehler, E . T h o r n d i k e , A. C r o m e r , Phys. Rev. 134 (1964)B1030. [18] Z. Janout, Yu. M. Kazarinov, F. L e h a r , P. Winternitz, p r e p r i n t JINR E1-3345, Dubna 1967. [19] H. P. Stapp, T. Ypsilantis, N. Metropolis, Phys. Rev. 105 {1957) 302. [20] Yu. M. Kazarinov, V. S. Kiselev, A. M. Rozanova, I. N. Silin, P r e p r i n t JINR E13268, Dubna 1967.