Nucleon-alpha elastic scattering analyses

2 . - ~

Nuclear Physics A209 (1973) 4 2 9 - - 4 4 6 ; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

NUCLEON-ALPHA

ELASTIC SCATTERING ANALYSES

(I). Low-energy n-= and p-~ analyses RICHARD A. ARNDT, DALE D. LONG and L. DAVID ROPER Department of Physics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061 t Received 2 October 1972 (Revised 30 November 1972)

Abstract: Energy-dependent phase-shift analyses of low-energy n-~ and p-= elastic scattering data were performed. Effective range parameters were obtained using the least number of parameters, and data up to the lowest energy necessary to give good fits. Fits to data up to 5 MeV for p-~t scattering and up to 3 MeV for n
1. Introduction Previous e n e r g y - d e p e n d e n t analyses have been p e r f o r m e d by us for n-~ elastic scattering u p to 16 M e V [ref. 1)] a n d p-~ elastic scattering u p to 23 M e V [ref. 2)]. A 0-21 M e V n-ct analysis is p r e s e n t e d in p a p e r II. A l l available total cross section, differential cross section, a n d p o l a r i z a t i o n m e a s u r e m e n t s were used in these p a r t i a l wave analyses. Phase shifts were o b t a i n e d for all p a r t i a l waves necessary to give r e a s o n a b l e fits to the data. T h e p u r p o s e o f the present w o r k was to fit the very low-energy data, using d a t a o n l y u p to the lowest energy for which we c o u l d get g o o d fits using only two p a r t i a l waves a n d two o r three p a r a m e t e r s p e r p a r t i a l wave. Such a p a r a m e t e r i z a t i o n allows theorists to m a k e a direct c o m p a r i s o n with theoretical calculations b a s e d on p a r t i c u l a r potentials. I n such calculations theorists can use a p p r o x i m a t i o n s t h a t a p p l y only at l o w energies to calculate the effective-range p a r a m e t e r s , a n d can then c o m p a r e the c a l c u l a t e d p a r a m e t e r s with o u r results. (See sect. 4.) The n u m b e r o f d a t a at m o s t o f these low energies is n o t sufficient to d o single-energy analyses; a n d e n e r g y - d e p e n d e n t analyses over a large energy range c a n n o t be t r u s t e d to give accurate values for effective-range p a r a m e t e r s at the low-energy end. The p a r a m e t e r i z a t i o n used in the analyses is t h a t o f the effective-range theory. I n the case o f p - ~ scattering, the o p t i m u m fit was chosen to be t h a t to all the d a t a u p t o 5.011 M e V with two p a r a m e t e r s r e t a i n e d in the e x p a n s i o n for the s½ a n d p~ p a r t i a l waves a n d three p a r a m e t e r s for the p~ expansion. F o r n-~ scattering the chosen o p t i m u m t Work supported in part by a grant from the National Science Foundation. 429

430

R.A. ARNDT et al.

fit was that to the data up to 3.042 MeV, with the same seven parameters as for the p-0t analysis. The values and error matrices are given for the parameters in both cases. 2. Method of analysis

The data used were th0se'described in 'refs. 1' 2)i where only the low-energy portions were used in each case. These data include total cross sections, differential cross sections, and polarization measurements. The effective-range parameterization used for the phase shift ~ in the n-0~ analysis is p ctg 5~ = - (1/a~) +,½r~ k 2 , ¼ P ~ k ¢ + . . . . where 1 = orbital angular m o m e n t u m , p = k2~+!, k = non-relativistic relative m o meritum i n fro-1, a~ = scattering length in fro, r t = effective range in fro, and P~ = shape parameter in fm?. : F o r the p-~ fits, the Coulomb corrections are added s), s o t h a t the parameterization becomes pC2(~l) Fctg 6,+ 2r/h(r/)l = - 1 + ½ r t k 2 _ ¼ P , k¢ . . . . L C~(r/)-] a, where ~/ = 2 Mp M~

e2

Mp+M~ h 2 k

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C~(,7) = 2,~,1/(e ~ " " - I), C2(r/) = C2-1(~/) ( 1 + rl~) , oo 1 h(r/) = r / ~1'= s(s2+r/2 ) - I n r / - y ,

= 0.577216 (Euler's constant). The equations relating the phase shifts to the observables are given in ref. 4). The least-squares search procedure is given in ref. s). Starting parameters for the search were taken from the fits of refs. 1, 2), with the higher partial waves neglected. Fits were obtained using all the data available up to some maximum energy, where this maximum energy was assigned various values between 1 and 12 MeV for p-~ scattering and between 1 and 7.1 MeV for n-~ scatterrag. The interval between maximum energies for the various fits was approximately either 0.5, I or 2 MeV, with the smaller energy intervals used in the lower energy range.

TABLE 1 Values o f Z 2 per data point and renormalization for each experiment in the o p t i m u m p-~ solution Tlab (MeV) 0.222 0.300 0.500 0.515 0.940 O.950 1.140 1.350 1.490 1.560 1.700 1.765 1.970 1.997 2.020 2.180 2.220 2.320 2.380 2.510 2.530 2.590 2.610 2.890 3.000 3.006 3.200 3.220 3.470 3.510 3.540 3.580 3.650 3.840 3.910 4.006 4.040 4.150 4.220 4.280 4.460 4.500 4.500 4.560 4.580 4.665 4.757 4.770 4.780 5.011

Expt.

NO. o f data

Ref.

Date

Z z per data point

P P P P P a P P

1 1 1 1 12 10 12 12 11 12 11 11 11 24 13 12 12 1 16 1 15 3 15 14 12 26 12 15 1 15 16 1 1 16 1 25 17 15 1 1 13 15 18 1 15 16 15 2 1 28

P19 PI9 PI9 P19 P17 D2 P17 P17 D2 P17 D2 P17 P17 DI9 D2 P17 D2 P14 PI6 PI4 D2 P17 Pl6 P16 P17 D19 P17 PI6 P14 D16 Pt6 P5 P10 PI6 Pt4 D I9 P13 PI6 PI0 PI4 P16 DI6 PI3 PI0 P23 Pl3 P13 PI0 PI0 D19

1967 1967 1967 1967 1967 1949 1967 1967 1949 1967 1949 1967 1967 1964 1949 1967 1949 1965 1966 1965 1949 1967 1966 1966 1967 1964 1967 1966 1965 1966 1966 1958 1963 1966 1965 1967 1964 1966 1963 1965 1966 1966 1964 1963 1971 1964 1964 1963 1963 1967

0.55 0.76 1.22 0.70 1.02 2.53 0.88 0.77 1.61 1.66 1.91 •.94 0.85 2.26 1.27 0.56 0.74 0.00 1.43 0.37 0.80 0.36 1.86 1.66 1.60 0.80 0.70 2.39 0.31 0.36 0.63 3.08 0.41 0.77 0.17 1.68 2.41 0.96 0.42 2.89 1.84 0.33 0.56 0.36 0.52 3.40 1.69 0.52 0.63 0.98

P a P P a a P a P P P a P P P P a P P P a P P P P P a P P P P P a P P P P P P P a

Renormalization 1.000 1.000 1.000 1.000 1.008 1.045 1.022 1.009 1.003 1.002 0.962 0.998 1.000 0.982 0.968 0.996 0.976 1.000 1.017 1.000 1.000 0.964 0.965 1.004 1.014 0.995 0.995 1.040 i.000 0.974 0.948 1.000 1.000 1.029 1.000 0.997 1.011 0.999 1.000 1.000 0.975 1.008 0.960 1.000 0.988 0.930 0.970 0.972 1.000 1.017

The experiments are: a-differential cross section, P-polarization. [The renormalization n u m b e r given is to be divided into the experimental data points. Reference n u m b e r s are given in the appendix o f ref. 2).]

et al.

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3. Results 3.1. T H E p-u A N A L Y S I S

Fig. 1 is a plot of X2 per data point versus maximum proton kinetic energy, Ep, for the 6-, 7-, and 9-parameter fits. Each value of X2 per data point corresponds to a fit to all the available data up to that particular value of Ep. Six-parameter fits include scattering length and effective-range parameters for the s½, p~, and p~_ partial waves. Seven-parameter fits add the shape parameter to the p~ partial wave. For the nineparameter fits, the shape parameter was added in the s~ and p~ partial waves. A prominent feature of this figure is the plateau in X2 per data point of approximately 1.3 in the vicinity of 5 MeV. The value of g 2 per data point continues to rise with energy for the 6-parameter fits, and nine parameters continue to give a reasonable fit well up in energy. Fig. 2 shows the various parameters as a function of maximum proton kinetic energy for each of the three partial waves. The values for both the six- and sevenTAnLE 2 Parameters obtained from fit o f p-~ data up to 5 MeV

s~_ p~ p~

No. data

z2/data point

a (fro) ~)

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p (fm 3) ¢)

531

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1.295 dz0.082 0.349~0.021 --0.365 ~0.013

--2.394-0.15

a) Scattering length. b) Effective range. c) Shape parameter. a) The error is the change in the parameter that changes Z 2 by 1 when all other parameters are searched.

parameter fits are shown. Where the error bars are not shown, they are smaller than the size of the circle. For energies lower than those for which values are shown, the error is so large that the number is meaningless. In each case, the values of the parameters have become quite well defined and in most cases nearly constant by approximately 5 MeV. The plateau in Z2 per data point at about 5 MeV, the large error bars on the parameters below this energy, and the large g 2 above this energy have led us to choose the seven parameter solution with data up to 5.011 MeV as the optimum low-energy solution for p-ct scattering. Table 1 gives the X2 per data point and the renormalization for each experiment from which data were used in this optimum solution. The number of data points and the type of experiment at each energy are indicated on the figure. The references for these experiments are given in the appendix of ref. 2). Table 2 gives the values of the parameters obtained from the fit.

435

N - x E L A S T I C S C A T T E R I N G (I) 3.2. T H E n - a A N A L Y S I S

Fig. 3 is a plot o f X2 per data point versus maximum neutron kinetic energy for 7-, 8- and 9-parameter fits. N o fits with reasonable values of X2 were obtainable with six parameter s. The 7- and 9-parameter fits used the same parameters as for the p-~ analyses. The 8-parameter fits, which were tried for only a few energies, use three parameters for both of the p-waves. In the seven-parameter fits, a plateau of approximately 1.45 is reached for Xz per data point in the vicinity of 3 MeV. ----1

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Fig. 3. Plot o f Z 2 per data p o i n t as a f u n c t i o n o f m a x i m u m n e u t r o n kinetic energy in n-~ scattering.

Fig. 4 presents plots of the various parameters in the 7-parameter fits as a function of maximum neutron kinetic energy for the three partial waves. As before, the error bars are very large for data up to energies lower than those shown, and where no error bars are shown, they are smaller than the circles. In most cases the values of the parameters are essentially constant within the error bars between 2.5 and 4 MeV. The greater fluctuation in the n-~ parameters than in the p-~ ones is indicative of the fact that the neutron data are less consistent than th~ proton data,

436

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e t al.

TABLE 3 Values o f Z 2 per d a t a p o i n t a n d r e n o r m a l i z a t i o n for e a c h e x p e r i m e n t in t he o p t i m u m n-~t s o l u t i o n T, ab (MeV) 0.1871 × 10 -6 0.3643 × 10 -6 0.5834 × 10 - 6 0.8350 × 10 - 6 0.9805 × 1 0 - 6 1.0745 × 10 -6 1.2263 / l 0 -6 1.2965 × 10 - 6 2.5904 X l 0 -6 6.1946 × 10 - 6 0.122 0.152 0.181 0.202 0.262 0.303 0.362 0.372 0.402 0.483 0.501 0.545 0.590 0.599 0.602 0.704 0.723 0.799 0.840 0.844 0.899 0.907 0.925 0.940 0.993 1.008 1.015 1.023 1.023 1.030 1.055 1.091 1.101 1.106 1.132 1.144 1.175 1.207 1.215

Expt.

aT aT aT aT aT aT aT aT aT aT aT aT aT a P a aT aT a aT a a aT a aT a aT a a aT a aT aT aT aT a P aT aT aT aT aT aT a aT aT aT a aT

No. o f data 1 1 1 1 1 1 1 1 1 l 1 1 1 12 1 23 1 1 28 1 39 14 1 32 1 52 1 36 18 1 39 1 1 1 1 40 11 1 1 1 1 1 1 39 1 I 1 39 1

Ref.

T10 a) T10 TI0 T10 TI0 T10 Tl0 TI0 T10 T10 T5 T5 T5 D9 P5 DI4 T5 T5 D14 T5 " D14 D15 T5 D14 T5 D14 T5 D14 D15 T5 DI4 T5 T5 T4 T5 D14 P7 T5 T5 T5 T5 T5 T5 D14 T5 T5 T5 DI4 T5

Date

Zz per data point

1969 1969 1969 1969 1969 1969 1969 1969 1969 1969 1960 1960 1960 1966 1966 1968 1960 1960 1968 1960 1968 1968 1960 1968 1960 1968 1960 1968 1968 1960 1968 1960 1960 1959 1960 1968 1968 1960 1960 1960 1960 1960 1960 1968 1960 1960 1960 1968 1960

9.98 1.27 1.73 0.23 5.92 0.26 2.60 0.42 4.84 0.00 1.05 0.12 0.51 3.04 2.26 1.56 5.03 0.44 0.60 9.55 0.96 1.92 4.89 2.52 0.06 1.60 4.07 1.52 1.32 3.82 0.80 0.10 5.20 0.43 2.94 1.28 0.80 0.07 4.39 0.06 0.31 0.26 0.00 1.12 1.69 7.16 4.85 0.96 1.28

Renormalization

1.056 1.014

0.973 1.025 1.039 1.068 1.035 0.986 0.996 1.003

1.007 1.001

1.001

1.000

N-ct E L A S T I C S C A T T E R I N G

(I)

437

TABLE 3 (continued) Tlab (MeV)

Expt.

1.306 1.331 1.382 1.462 1.500 1.541 1.620 1.700 1.701 1.780 1.860 1.942 1.961 1.980 1.994 2.000 2.020 2.200 2.259 2.299 2.332 2.381 2.394 2.406 2.440 2.453 2.454 2.476 2.490 2.495 2.836 2.980 2.990 3.020 3.042

cr ax aT trx or err trx a Or ar ar trx a ax trx P trT tr ~T trx trx o'r fiT ~x P crx tr trx ~rx ~rx a.r ~ ax a aT

No. o f data

Ref.

Date

Z z per data point

35 1 1 1 1 l 1 57 1 1 1 1 36 1 1 10 1 63 1 1 1 1 1 1 12 I 40 1 1 1 1 58 1 21 I

D14 T5 T5 T5 T4 T5 T5 DI4 T5 T5 T5 T5 D14 T4 T5 P2 T5 DI 4 T5 T5 T5 T5 T5 T5 P7 T5 D14 T5 T4 T5 T5 D14 T4 D13 T5

1968 1960 1960 1960 1959 1960 1960 1968 1960 1960 1960 1960 1968 1959 1960 1963 1960 1968 1960 1960 1960 1960 1960 1960 1968 1960 1968 1960 1959 1960 1960 1968 1959 1966 1960

1.04 3.93 1.23 2.17 1.59 0.29 2.53 2.12 0.17 0.45 1.25 0.01 1.08 0.14 0.62 0.48 0.00 1.72 0.58 0.25 0.36 0.02 0.05 0.00 1.44 2.30 1.08 0.93 0.00 0.90 0.00 1.48 0.04 1.84 1.05

Renormalization 1.001

1.033

1.006

1.022 1.031

1.020 1.009

1.002 0.993

T h e e x p e r i m e n t s are: aT-total cross section, a-differential cross section, P-polarization. [The r e n o r m a l i z a t i o n n u m b e r given is to be divided into the experimental d a t a points, Reference n u m b e r s are given in t h e a p p e n d i x o f ref. 1).] ~) T10 is n o t listed irt t h e a p p e n d i x o f ref. ~). It is ref. 6) o f the p r e s e n t paper. TABLE 4 P a r a m e t e r s o b t a i n e d f r o m fit o f n-~ d a t a up to 3 M e V

s$ p~ P~

No. d a t a

z2/data point

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1.45

a (fm) a) 2.4641 4-0.0037 d) --13.821 4-t=0.068 --62.951 -b0.003

r (fm) b) 1.385 =]=0.041 --0.419 4-0.016 --0.88194-0.0011

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a) Scattering length, b) Effective range, c) Shape parameter. a) T h e error is t h e change in t h e p a r a m e t e r t h a t c h a n g e s X2 by I w h e n a l l o t h e r p a r a m e t e r s are searched.

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TABLE 5 Values of the phase shifts for our low-energy n-ct and p-ct solutions En (MeV) 0.122 0.202 0.303 0.402 0.501 0.545 0.599 0.704 0.799 0.840 0.899 1.008 1.015 1.207 1.306 1.700 1.961 2.000 2.200 2.440 2.454 2.980 3.020

n-,¢ phases s~. p{ -- 8.642 --11.111 --13.595 --15.644 --17.447 --18.190 --19.059 --20.641 --21.969 --22.516 --23.280 --24.624 --24.708 --26.892 --27.946 --31.758 --34.019 --34.343 --35.946 --37.765 --37.868 --41.508 --41.769

0.184 0.395 0.733 1.131 1.587 1.808 2.094 2.694 3.286 3.555 3.957 4.745 4.798 6.330 7.187 11.036 13.953 14.412 16.860 19.984 20.172 27.611 28.200

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0.926 2.132 4.352 7.440 11.685 14.030 17.365 25.533 35.090 39.851 47.264 61.876 62.816 85.745 94.501 113.247 118.364 118.883 120.895 122.300 122.357 123.321 123.326

0.222 0.300 0.500 0.940 1.140 1.350 1.490 1.560 1.700 1.765 1.970 1.997 2.180 2.220 2.380 2.530 2.610 2.890 3.000 3.200 3.220 3.540 3.840 4.006 4.150 4.460 4.580 4.665 4.757 5.011

-- 1.356 -- 2.405 -- 5.334 --11.393 --13.848 --16.248 --17.759 --18.490 --19.907 --20.546 --22.478 --22.945 --24.372 --24.720 --26.079 --27.310 --27.950 --30.107 --30.922 --32.360 --32.501 --34.688 --36.663 --37.669 --38.547 --40.372 --41.057 --41.535 --42.006 --43.424

0.056 0.127 0.442 1.736 2.563 3.574 4.324 4.720 5.555 5.961 7.314 7.661 8.810 9.106 10.329 11.526 12.183 14.578 15.556 17.381 17.567 20.604 23.544 25.197 26.641 29.765 30.974 31.829 32.752 35.282

pk 0.142 0.331 1.258 6.191 10.255 16.250 21.452 24.456 31.299 34.839 47.189 50.374 60.563 63.044 72.374 79.990 83.553 93.464 96.419 100.737 101.105 105.766 108.613 109.742 110.529 111.750 112.086 112.286 112.471 112.834

T h e l o w v a l u e o f •2 p e r d a t a p o i n t , a n d t h e c o n s t a n c y o f b o t h ~2 p e r d a t a p o i n t a n d t h e p a r a m e t e r s b e t w e e n 2.5 a n d 5 M e V , h a v e led us to c h o o s e t h e d a t a u p t o 3.042 M e V as t h e o p t i m u m l o w - e n e r g y s o l u t i o n f o r n - a scattering. A f t e r t h e a b o v e c a l c u l a t i o n s w e r e c o m p l e t e d , it was called t o o u r a t t e n t i o n t h a t we h a d left o u t s o m e v e r y l o w - e n e r g y (0.1871 to 6.1946 e V ) t o t a l c r o s s s e c t i o n d a t a 6). W e t h e n r e d i d t h e 0 - 3 . 0 4 2 M e V analysis w i t h t h e s e d a t a i n c l u d e d . T h i s analysis is t h e one given below. T a b l e 3 gives t h e X2 p e r d a t a p o i n t a n d t h e r e n o r m a l i z a t i o n f o r e a c h e x p e r i m e n t i n c l u d e d in this o p t i m u m s o l u t i o n . T h e r e f e r e n c e s f o r t h e s e e x p e r i m e n t s are g i v e n in t h e a p p e n d i x o f ref. ' ) . T a b l e 4 gives t h e v a l u e s o f the p a r a m e t e r s o b t a i n e d f r o m this fit. 3.3. PHASE SHIFTS T h e p h a s e shifts at l o w e n e r g i e s as c a l c u l a t e d f r o m t h e o p t i m u m s o l u t i o n s f o r t h e p-o¢ a n d n-:~ a n a l y s e s are p l o t t e d in figs. 5 a - c f o r t h e s~, p~, a n d p~ p a r t i a l waves, re-

440

R . A . A R N D T et aL

spectively. In each case the phase shifts are calculated up to approximately 1 MeV higher in energy than for the respective optimum solutions. Table 5 contains values of the phase shifts at selected energies. 0

'1

I

I

I

I

[

I

I 4.50

I 5.25

7 $1/2

14

21

28-=0

p'a

42

49 56 0

I 0.75

I L,50

I 2.25

I 3.o0 E(MeV)

I 3.75

6.o0

Fig. 5a. P h a s e shifts calculated f r o m t h e o p t i m u m s o l u t i o n s for s~ p a r t i a l waves. l

I

56 L

I

I

I

[

I

3.(30 E(aev)

5.75

4.50

5.25

"---1

P~/z

48

40

~ 32 24 16

8

0

0

0.75

1.50

2,25

FiB. 5b. S a m e as fig. 5a f o r p½ partial waves,

6,00

N-~ 155

I

ELASTIC

I

SCATTERING

I

I

I

(1)

441

I

I

4.50

,5.25

I

120

105

90 no 75

oo

p-a 6O

45

30

15

0

0

0.7'5

1.50

~'25

5.00 E(MeV)

575

6.00

F i g . 5c. Same as fig. 5a for P~r partial waves.

The phases we obtain here are similar to those obtained in previous energy-dependent analyses t, 2. 7), differing from them by less than one degree. As stated in the introduction, one does not expect an energy-dependent analysis over a large energy range to accurately represent the data over a smaller energy range within the large range. 4. Representation of data by inverse error matrices For completeness, both the error matrices, M, and their inverse, M - 1, are presented in tables 6-9 for the solutions given. The matrix M is useful, primarily, for obtaining errors on all quantities which can be calculated from the model parameters (e.g. phase shifts, cross sections, scattering amplitudes, etc.). If the quantity to be determined is q(Pr) then the error Aq is:

aq = [ E

J,K

where

PK = Kth parameter of the representation (e.g. a(sl ), r(sl ), a(pa)), and p0 = parameters for the given solution.

--0.19364E 0.34959E

--0.44831E --0.36491E --0.23924E

a r

a r P

P~

02 00 01

02 00

01 01

0.22310E--01 0.82037E--03 0.10123E--01

--0.26792E--01 0.10395E--02

0.15436E--01 --0196029E--02

a

0.13639E--01 --0.52687E--03

0.67590E--02

--0.44831E 02 --0.36491E--00 --0.23924E 01

a r P

pk

0.12769E 0.13480E 0.77235E

0.34732E 0.83115E

0.51521E 0.77006E

a

TABLE6

p½

TABLI~ 7

--0.54958E--03 --0.19476E--03 0.54193E 03

0.24608E 00 '0.10059E--01

a

0.52160E--03 --0.60279E--05 --0.19698E--03

0.42954E--03

r

04 06 04

03 04

04 04

r

--0.16647E --0.19162E 0.11747E

~0.55224E 0.14025E

0.12256E

S~ .

04 06 05

03 05

05

--0.65860E --0.80247E 0.52572E

0.12780E 0.30379E

a

02 04 03

03 04

P~}

--0.14916E --0.19639E 0.13514E

0.75809E

r

04 06 05

05

a

0.52567E 0.45369E 0.21715E

a

03 05 04

0.25794E 00 --0.62525E 02 0.68321E--01

Inverse error matrix M - 1 for 0-5 MeV p-g solution, Z 2 ~ 697

Values of a and r are given in fm; P is in fm 3.

02 00

--0.19364E 0.34959E

a r

P~

01 01

0.49723E 0.12953E

a r

s~

pO

r

0.16232E--01 --0.59267E 03 --0.80399E--02

s½

Values of a and r are given in fm; P is in fm 3.

p~_

0.49723E 0.12953E

a r

s~

pO

° Errol matrix M for 0-5 MeV p-~ solution; Z 2 = 697

0.43582E --0.23050E

P~ r

07 06

0.16182E--03 0.18502E--02

r

p~

0.13265E

P

05

0.22448E--01

P

Z

--0.62951E --0.88193E --0.30023E

a r P

02 00 01

02 00

01 01

s~

--0.43696E--07 0.76472E--06 0.25002E--04

--0.88239E--05 0.30407E--05

0.14004E--04 --0.69700E--04

a

--0.97400E--05 --0.13956E--04 --0.13679E--02

0.38982E--03 --0.33073E--03

--0.62951E --0.88193E --0.30023E

a r P

02 00 01

02 00

01 01

--0.15631E --0.30038E 0.78121E

0.18151E 0.11503E

0.12620E 0.10789E

a

04 06 04

04 05

06 05

P~

TABLE 9

0.38120E--04 0.36969E--04 --0.22767E 02

0.46007E--02 --0.40877E--03

t/

--0.13254E--04 0.43468E--05 0.28166E--03

0.26176E--03

r

r

0.21053E --0.46432E 0.15812E

0.47235E 0.24482E

0.22738E

s~

04 05 04

03 04

04

--0.38182E --0.56913E 0.56594E

0.50514E 0.65677E

¢1

04 04 03

03 03

r

0.66602E --0.52019E 0.12901E

0.76184E

P~

04 05 04

04

a

0.16293E --0.22896E --0.61660E

a

06 05 04

0.10260E -04 0.11263E--05 0.70176E--04

Inverse error matrix M - 1 for 0-3 MeV n-~ solution; Z z = 1182

Values of a and r are given in fm; P is in fm a.

--0.13821E --0.41884E

a r

P~

pk

0.24641E 0.13851E

a r

s~

pO

r

0.17029E--02

Values of a and r are given in fro; P is in fm 3.

--0.13821E --0.41884E

a r

Pk

pk

0.24641E 0.13851E

a r

s~

pO

TABLE 8

Error matrix M for 0-3 MeV n-~ solution, %2 __ 1182

0.36530E --0.68648E

Pk r

07 05

0.12472E--05 0.57290E--04

P~k r

0.21639E

P

04

0.38164E--02

P

4~

,.-] .q

R. A. A R N D T et al.

444 1.00

1

I

~'~

[

/ \

I

I

I

I

n-Q

0.87 l 0.5 MeV 0.75 ~-

Z _O

0.62 --

N 0.50

-

-

a_ 0.57

0.25 -

02S o

p-n, -=

i 20

r f 80 100 120 140 160 180 8c.M.(deg-) Fig. 6a. Polarization for p-g and n-= scattering at 0.5 MeV as calculated f r o m the inverse e r r o r matrices o f the o p t i m u m solutions. The errors are o f the order o f the size o f the line width. 1.00 I I I I I I I I

0

[ ~ 40

60

0.87 1.0 MeV

0.75

n-t)

O.62 Z

o N O,50 .J

o

11.

0.37 p-o

0.25

0.12

or o

I

I

I

I

f

I

1

40

60

8@

I00

120

140

Oc.M.(deg.l Fig. 6b, Same as fig. 6a at 1.0 MoV.

I 160

I 180

N
I

]

I

I

-~T-

445

(I)

I

.I

I

0.87

1.5 MeV

p-a

0.75

Z 0

0.62

Q

~ O.5O < ._J 0 0.37

0.25

0.12

0 0

I 20

I 40

I 60

I 80

i 100

l 120

I 140

I 160

180

OC.M,(deg.)

Fig. 6c. S a m e as fig. 6a at 1.5 MeV.

We remind the reader that the error, Aq, on q corresponds to a maximum distortion of the solution which is consistent with a X2 increase of 1. The errors on the model parameters themselves are simply the square roots of the diagonal elements of M. The inverse matrix, M - 1, can be used to replace the data if, for instance, one has a low-energy model from which theoretical predictions for a, r and P can be calculated. The Z2 which such a model would obtain if compared to the data is approximately 2+

X2 ~ Zb

0

-1

0

½~, (P.I-PJ)M.oc(PK--PK), J,K

where X~ is the X2 obtained in fitting the data to obtain the pO, and the other quantities are as defined before. The model parameters can then be adjusted to minimize the above expression for X2 with a reasonable certainty that, by so doing, a best fit to the data is actually being obtained. This should be of some use to modelists since the complications of fitting and renormalizing to hundreds of scattering data have been removed. As examples of observables which can be calculated using these inverse error matrices, we present the polarization for the p-ct and n-~t scattering at 0.5, 1.0, and 1.5 MeV. These are shown in figs. 6a-c respectively. Any data available at the particular energies are shown on the graphs. Other observables as well as this one could be calculated for any desired low energy.

446

R . A . A R N D T et al.

References 1) 2) 3) 4)

R. A. Arndt and L. D. Roper, Phys. Rev. C1 (1970) 903 R. A. Arndt, L. D. Roper and R. L. Shotwell, Phys. Rev. C3 (1971) 2100 M. A. Preston, Physics of the nucleus (Addison-Wesley, Reading, Massachusetts, 1962) appx. B L. D. Roper, R. M. Wright and B. T. Feld, Phys. Rev. 138 (1965) B190; L. D. Roper and R. M. Wright, Phys. Rev. 138 (1965) B921 5) R. A. Arndt and M. H. MacGregor, Meth. Comp. Phys. 6 0966) 253 6) D. C. Rorer, B. M. Ecker and R. 0 . Akyiiz, Nucl. Phys. A133 (1969) 410 7) P. Schwandt, T. B. Clegg and W. Haeberli, Univ. of Wisconsin preprint, 1971