Proton induced reactions on mass-14 nuclei

1

2.B

Nuclear Physics Al65 (1971) 19-32; @

1

Nortb-~~l~a~d PubZishj~g Co., Amsterdam

Not to be reproduced by photoprint or microfilm without written permission from the publisher

PROTON INDUCED REACTIONS ON MASS-14 NUCLEI T. H. CURTIS

t, H. F. LUTZ,

D. W. HEIKKINEN

and W. BARTOLINI

Lawrence Radiation Laboratory, University of California, Livermore ft Received 14 December 1970 Targets of r4N and t4C were bombarded with 14.5 MeV protons. The r4N was in the form of natural N2 gas, which is 99.6 % i4N, while the r4C was in the form of CO2 with a 14C enrichment of 91.4 %. Particles from the (p, p’), (p, d) and (p, t) reactions on these nuclei were identified and analyzed using conventional solid-state detectors and electronics. The overall energy resolution was 70 keV. Angular distribution were taken from 20” to 150” for the (p, p’) reaction and from 20” to 110” for the (p, d) and (p , t) reactions. For 14N, 18 levels were seen below 10 MeV excitation energy in the (p, p’) reaction and one transition to the ground state of 13N in the (p, d) reaction. The (p, t) reaction is energetically forbidden. For 14C, 5 levels were seen below 8.5 MeV excitation but is was not possible to obtain angular distributions because of masking due to excited states of 160. Also for 14C, 3 levels were seen in the (p, d) reactions and the (p, t) reaction leading to the ground state of l*C. The fp, p’) data were analyzed assuming a microscopic model of the nucleus and a two-body interaction, which included a tensor component for the study of the 2.31 MeV level of 14N. The DWBA code DRC was used for the (p, p’) analyses. The (p, d) data were analyzed using the code DWUCK to extract spectroscopic factors.

Abstract:

E

NUCLEAR REACTIONS -N, ‘“C(p, p), (p, p’), E = 14.5 MeV; measured o(E,,, 8). 14N, 14C(p, d), E = 14.5 MeV; measured a(&, 0). i4C(p, t), E = 14.5 MeV; measured cr(E,, 8). isC, i3N levels deduced S.

1. Introduction In this paper we report on a series of proton-induced reactions, namely (p, p’), (P, t> using 14.5 MeV protons on targets of 14N and 14C. In the analysis of the inelastic proton smattering from 14N we have used a microscopic description [refs. ’ - “)I of the nuclear reaction, which requires nuclear wave functions and the interaction between the target m&eons and the projectiIe in addition to the usual information necessary for a DWBA analysis. We employed the she&model wave functions calculated by True “) and by Freed and Ostrander “). The effective interaction is taken to be the so-called one-fm equivalent of the Kallio-Kolltveit semi-realistic force ‘), which is an even-state force of Yukawa shape and 1 fm range. We have also investigated theeffect of the tensor force for the transition to the2.31 MeV state in 14N Rose et al. “) have found that a tensor component in the interaction is necessary to explain both the gamma- and beta-decay matrix elements for the mass-14 system. (P, d), and

t Present Address: tt Work performed

Bell Telephone Laboratories, Holmdel, New Jersey 07733. under the auspices of the U.S. Atomic Energy Commission. 19

20

T. H. CURTIS

ef af.

For the analysis of this transition we used a version of the computer code DRC modified by Schmittroth “) and used to analyze the (p, n) data of Anderson et al. lo). In this part of the analysis we used the Visscher-Ferrell rl) wave functions, which give the large cancellation in the 14C-14N beta-decay matrix element. In addition to the inelastic studies, we have investigated the (p, d) reactions on 14C and 14N. Single-particle transfer reactions have recently been done on several p-shell nuclei 12) and th e results showed good agreement with the theoretical spectroscopic factors given by Cohen and Kurath 13). We report on two additional single-particle transfer experiments and compare the results with theoretical predictions. The 14C(p, d)13C reaction has been done previousIy by Legg 14) at an energy of 18.5 MeV. The extraction of spectroscopic factors in his analysis was complicated by the use of the plane wave Born approximation and uncertainties in target composition. The 14N(p, d) reaction was investigated by Kozub et al. 1“) at an energy of 30 MeV with good agreement between theory and experiment. In the present experiment we have obtained angular distributions for 14C(p, d)13C (g-s.) and 14N(p, d)13N(g .s. ) reactions. Because of the rather low energy of the outgoing deuterons only forward-angle data were obtained for the 14C(p, d)13C reaction to the 3.09 and 3.68 MeV levels of 13C. The (p, t) reaction is energetically forbidden for 14N . The angular distribution for 14C(p, t)““C(g.s.) is presented. 2, Experimental method A beam of 14.5 MeV protons was produced by the Livermore variable energy cyclotron 16). The beam was momentum-analysed using a 90” bending magnet with a 76.2 cm radius of curvature and focused by a quadrupole triplet to a spot at the center of a 60.9 cm diameter scattering chamber. The target gases were contained in a cylindrically shaped cell, 7.56 cm diameter, with a thin window (2.5 pm thick nickel-cobalt alloy) in the scattering plane that subtended 270”. For 14N we used natural N2 gas, which contains 99.8 % of the mass-14 isotope. The gas cell was filled in position in the scattering chamber. The pressure and temperature of the gas were monitored during bombar~ent. For ‘*C we used CC& gas with the carbon enriched to 91.4 % in 14C. Due to the radioactive decays associated with 14C the gas cell was filled and sealed at a place remote from the scattering chamber.‘For 14N the yield as a function of angle was converted to absolute cross section by the calculation described in detail by Silverstein 17). The 14C cross sections were normalized by comparing the I60 cross section resulting from the oxygen in the CO, mixture to measured cross sections I*). The particles were detected with a counter telescope consisting of a transmissiontype surface barrier dE detector and a lithium-drifted silicon E detector. The LIE detector was 100 pm thick operated with a reverse bias of 350 V. The detectors were cooled to - 25” by thermoelectric coolers to ~nirni~ leakage currents. A collimator

I

I

t

CFIRBCIN- 111

I I

GRD.

(PY P)

I

STATE

I

I

r ,

NITROGEN-lq

I

,

I

(P,P)

GFiO. STAT

Fig. 1. Elastic proton scattering from 14N and 14C. The solid curves are the optical model fits to the experimental data. In the region around 90” the cross section for “C is less than the cross section for r4N . This behavior is accounted for by the increase in the imaginary part of the optical model potential due to its dependence on isospin.

I

h)

T. H. CURTIS ei at.

22

consisting of a circular hole in front of the detectors and a verticai slit close to the gas cell defined the region of the gas target viewed by the detection system. The particle identification system of Goulding et al. ’ “) was used to select the particle type whose spectrum was stored in a 800-channel pulse-height analyzer. The system was run first to accept only protons and then to accept both deuterons and tritons from the (p, d) and (p, t) reactions. So few levels were seen in the latter reactions that they could be identified unambiguously by their energies and kinematic shifts with angle. The spectra were punched on paper tape that was converted to computer cards and analyzed on the CDC 66~ with a program that subtracted backgrounds, fitted the spectra with sums of Gaussian-shaped peaks and calculated cross sections. Angular distributions for levels separated by at least ‘70 keV could be obtained in the present experiment. TABLE1 Optical model parameters used in DWBA calculations Channel

V

(MeV) pl.14N P-f-‘4C d n

W

(MeV)

60.4 63.0 132.0 varied

W’

v*.o.

(MeV)

(MeV) _.

(2)

(&I)

(Tm”,

5.5 9.7 5.8

7.5 7.5 5.8 #I==25

1.13 1.13 0.886 1.1

0.65 0.65 0.95 0.65

1.13 1.13 1.57

(L) 0.51 0.51 0.717

(k) 1.25 1.25 1.3

Non-locality (fm) 0.85 0.85 0.54 0.85

3. Analysis of the data 3.1. ELASTIC

SCATTERING

The elastic scattering angular distributions of protons from r4N and 14C were analyzed using an optical model potential to obtain parameters of a distorting potential to use in the DWBA calculation of the inelastic scattering and particle-transfer reactions. The experimental angular distributions and the fits obtained with the optical model are shown in fig, 1. The parameters obtained from this analysis are listed in table 1. The increase in the values of T’and FF for 14C are due to the isospin dependence of the optical model and yield A V = 18.5

N-Z MeV, A

AW = 29.5LZMeV A

for the geometrical parameters used. The possibility of compound elastic scattering contributing to the elastic cross section was considered. The 3.95 MeV levet of 14N has the same values for J”; T as the ground state so it is reasonable to take its integrated cross section as an upper Iimit

14N, 14C‘CP, P), (P, 0

@>t)

23

for the compound contribution to each state. For the ground state this implies an isotropic angular distribution with a magnitude less than 2 mb * sr-I. This has no effect on the elastic cross section at forward angles, but at the minimum near 60” it could induce 10 y(, changes in the cross section, which could alter the optical model parameters. To get a measure of this effect we analyzed the elastic data several times each time subtracting 0,5 mb - sr-l from the cross section. No particular improvement in the chi-squared test of the fit could be discerned with this procedure. We therefore made no compound-nucleus correction to the data. 3.2. INELASTIC

SCATTERING

The inelastic proton scattering cross sections were analyzed using the computer code DRC. This is a distorted-wave reaction code in which the distorting potential of the incoming and outgoing waves are provided by the optical potential, which we obtained by fitting the elastic scattering data. No spin-orbit terms areincludedin DRC. The interaction potential is taken to be a sum over projectile-target nucleon pairs. In one version of this code the interaction is restricted to be central with an arbitrary spin-isospin mixture. Direct and exchange contributions to the scattering amplitudes are included in the calculation. In another version the interaction may also have a tensor component but no exchange terms. Both versions of the code were used in the analysis of the inelastic data. We point out that, in addition to spin-orbit distortions, the codes employed neglected the effects of core polarization and channel coupling. Inputs to the code consists in specifying the mass, charge and spin of all particles in the reaction under study, optical model potentials for the incoming and outgoing waves, angular momentum and isospin quantum numbers associated with the state, parameters of the Woods-Saxon well in which the shell-model wave functions of the bound state are computed and spectroscopic factors for promoting a nucleon from one bound state to another. The spectroscopic factors are calculated from the shellmodel con~~urations of the initial and final states using the formulae given by Madsen “). The effective interaction used in the direct-plus-exchange code was the one-fm equivalent of the Kallio-Kolltveit semi-realistic force that has F/0 = 36.2 MeV, V, = 6.23 MeV, V, = 17.8 MeV and V,, = 12.1 MeV. This interaction, when used in a phase-shift calculation, duplicates the results of a real force with hard core. The force with a hard core could not, of course, be used in DWBA calculations. The shell-model descriptions of mass-14 nuclei that we used interpret these nuclei as an inert 12C core with two valence nucleons. The valence nucleons can occupy ip+, Id,, 2s+ and Id, orbits. The calculated angular distributions can be the sums of many single-particle transitions allowed by the spectroscopic factors calculated from the model wave functions and allowed by the selection rules on the quantum numbers I, I’, L and AT. The total angular momentum transferred from initial to final state obeys the selection rule IJi-J;I ~ I ~ IJiS-JIJ.

T. H. CURTIS

24

et al.

The spin-flip quantum number may take on the values 0 or 1. The orbital angular momentum transferred from initial to final state obeys the rule 14-M

5 1 S 1~1+4,

where I, and I2 are the initial and final orbital angular momentum of the particle in the nucleus that is excited. The tensor force has two units of angular momentum resulting in the relation z = 1+2. The parity relation is 7ci7cnf = (-1)”

= (-1)“.

3950 keV

1\ + + t+

‘*

t +

+ + t

+

t

.l

+t

5110 k&J

5830 kc?/

7030 keV

I-.t

+t,

t

t

lo C.M.FINGLE I OEGREESI

Fig. 2a

w

t+

+ tt”

+

SO

t

120

160

14N,14C(P, PI, (P. d), (P. t)

25

6q90kff

6620 keV

1

6910 keV 1

-o-lov

120

160

f +tt+t’t

f’ ‘IO

60

120 CJ4.l

E I OEGREES I

Fig. 2. Angular distributions for inelastic proton scattering from 14N. Part a contains the data for excited states from 2310 keV to 7030 keV and part b for excited states from 7970 keV to 10100 keV. The curves are generated with the exchange version of the code DRC using the so-called one-fm equivalent of the Kallio-Kolltveit semi-realistic force. The results are summarized in table 2.

The isospin can change by 0 or 1. The spins, parities and isospins for the mass-14 system have been determined previously, mainly by the work of Warburton et al. [refs. “-*‘)]. These values formed the basis of the shell-model calculation and we also use them. The experimental angular distributions and fits to the data generated with the version of DRC that includes exchange terms are shown in fig. 2. Some of the fits, for example the 5.11 MeV and 5.69 MeV transitions, are rather good but the overall agreement must be termed poor. The fits to the 6.44 MeV, 9.51 MeV and 9.70 MeV transitions miss the trend of the data entirely. Table 2 summarizes the results of the experiment and analysis of the inelastic scattering data. It lists the energy levels and quantum numbers of each level, the total cross

26

T. H. CURTIS

et al.

section for exciting the level in the present experiment,

the dominant

shell-model

con-

figuration, the dominant transition calculated from the shell-model wave functions and the factor by which the theoretical curves were normalized to get the best overall agreement with the experimental data when the calculation is made with the wave functions of True ‘) or Freed and Ostrander “). There are some states that are given a detailed description by one author but not the other. For example True “) lists the 3.95 MeV, the 7.03 MeV and the 9.17 MeV states as merely core-excited states and we could only perform the computations with the wave functions of Freed and Ostrander. This explains the omissions in the last two columns of table 2. TABLET Summary of the results for the 14N(p, p’)14N experiment and analysis

(L”)

cr total

J”: T

(mb)

Dominant configuration

Dominant transition )3 I I’

_ 0.0

1+;0

2.3 1

o+;1

2.9

(P+Y

2

1

1

3.95

1+;0

17.5

(p+Y+c.e.

2

2

0

4.91

o-;o

3.3

p-: s3

5.11

2-; 0

14.0

5.69

1-; 0

8.8

5.83

3-;o

19.1

6.20

1+;0

5.8

6.44

3+;0

6.0

7.03

2+; 0

7.97 8.06

Normalization True 5,

(~+)*+c.e. 2.3

1.7 6.8

1

1

0

3.8

3

3

0

4.4

4.9

1

1

0

4.6

8.2

P& d+

3

3

0

5.1

6.0

(s#

2

2

0

12.7

s+ d;

2

2

0

17.1

23.6

c.e.

2

2

0

2-;o

6.9

p+ d+

1

1

0

0.9

1-; 1

1.1

pt

1

1

0

0.8

8.49

(4-);

8.62

o+; 1

8.91 9.17

pa de P) s+

s3 ?

(s+Y

2

1

1

416.0

3-; 0

11.7

1

2

1

5.3

2+; 1

8.8

p+ d+ c.e.

2

3

1

9.39

2-;o

9.0

?

9.51

2-;

8.1

9.70

1+;0

4.1

pa d* (P#

2+;0

8.4

s3 d*

1

1

5.7 59.7

12.3

10.1

FO 6,

3.3 5.8 69.7

3

3

1

6.3

2

2

0

12.1

4.0

The integrated cross sections were obtained by fitting the experimental angular distributions with a sum of Legendre polynomials. The data for the 10.1 MeV level was assumed symmetric about 90” to obtain an integrated cross section. In a qualitative manner the normalizations can be taken to be an indication of corepolarization or other inadequacies in the wave functions. For example, most of the normalizations are about 10 or under, but the core-excited states at 7.03 MeV and 9.17 MeV have normalizations of 59.7 and 69.7. The ground state wave function of

l“N 14C(P, PI, (P, d), (P, t)

21

Freed and Ostrander “) has a larger (2~~)’ component than True’s “) wave function. This explains the difference in normalizations in the cases of the 6.44 MeV and 8.63 MeV levels for the two sets of wave functions. ,

I

I

I

I

VT-l.3

I t++++t

a-O.71

t

‘r-l\

t+ ‘\ ++

: ttttttt+ tttt+1

t

+

“1

L

t I

I

60

120

I

Q-3.9 a-0.71

\

0

‘/

90

so

120 160 C.M.ANGLE

0 Lfo ( DEGREES 1

1 1Ei0

Fig. 3. Fits to the inelastic scattering data leaving 14N in the 2310 keV state. The calculations were performed with tensor force version of the computer code DRC. The value of V,, in the central part of the interaction was kept at 6.2 MeV while the magnitude of the tensor force vr and the range parameter a were varied. No satisfactory fit was obtained.

As previously noted, in addition to the exchange version of DRC we have used the tensor version to analyze the transition from the J”; T = 1’; 0 ground state to the J”; T = O+; 1 level at 2.31 MeV. Recent work by Crawley et al. ““) done at 24.8 MeV bombarding energy has shown that inclusion of a tensor component in the two-body effective interaction produced a substantial improvement in agreement between experimental and theoretical angular distributions. In addition, analysis of the 14C(p, n) 14N ground state angular distribution showed better agreement when a tensor force was included “). However, the fit to the polarization data in the (p, n) reaction showed no improvement and may have been poorer.

28

T. H. CURTIS

et al.

The form of the tensor force used in the present by Crawley et al. 26), namely,

where fT(rol)

is a regularized f=(r)

spherical

Hankel

= h’,“(icrr)-

(i)3

calculation

is the same as that used

function h’:‘(ijlr).

A range of a = 0.71 fm- ’ was used for the one-pion tensor form factor. The regularizing term had a range /? = 2.0 fm-I. The results of the present work are shown in fig. 3. In these calculations we used a value of 6.2 MeV for V,, in the central part of the interaction. Several values of Vr were tried. As can be seen from fig. 3 the quality of fits to the data is poor for all values of Vr. Changing the value of CIto 1.4 fm- i also did not produce any improvement in the quality of the fits. In general, a comparison of the shapes of the angular distributions tend to indicate a preference for Vr < 3.9 MeV, which is the value found at 24.8 MeV proton bombarding energy. The r4C(p, n)14N analysis for bombarding energies comparable to our energy resulted in values of Vr around 5 MeV.

3.3. TRANSFER

REACTIONS

The (p, d) reactions have been analyzed using the DWBA code, DWUCK ‘“). This code allows both finite-range and non-local effects to be taken into account. The optical model parameters for the proton channel are taken from the present work and are shown in table 1. The deuteron parameters are those used by Schiffer et al. 12) for the analysis of (d, p) results on lp shell nuclei. These parameters are averages of those found in scattering of 11.8 MeV deuterons 2g). Since the deuteron energy in this experiment corresponds to approximately 8 MeV in the (d, p) studies we have increased V, to 132 MeV from 118 MeV. This is in accord with the results of Satchler [ref. 30)] on the analysis of the 12C ( d , d)“C scattering over a wide range of deuteron energies. The deuteron optical model parameters are also shown in table 1. The calculations were performed for both local-zero range (LZR) and non-local finite range (NLFR) cases. For the LZR analysis the calculation was performed with no radial cut-off and a cut-off at 2 fm. The non-local parameters were /? = 0.85 and 0.54 for protons and deuterons. The finite range parameter was taken to be 0.65 fm-‘. The following general remarks can be made concerning the results For the LZR calculations, the shape of the 14C(p, d)i3C(g .s. ) reaction was reproduced better by using a cut-off radius of 2 fm. Inclusion of finite-range and non-local effects did not improve the quality of the fits. The spectroscopic factors obtained differed by approximately 25 %. Using the proton optical model parameters determined by Watson [ref. “‘)I gave essentially the same quality of fiits with 10 % smaller spectroscopic factors.

14W14C(P, P), (P, d),

29

(P, t)

-----r--

r---

14C (p,d) "C

5 a

-

LZR

Rc.0

-

--

LZR

R,-2

--LZR

LZR

R,-0 Rc-2

--- NLFR R,-0

--- NLFR R,-0

.5 s z $1 m w 2

Fig. 4. Neutron pick-up reactions on 14N and r4C proceeding to the ground states of r3N and I’C. The symbols LZR refer to local zero range and NLFR refer to non-local finite range. The computer code DWUCK was used to make the calculations.

Spectroscopic

factors determined

Reaction

-C(p,

d)=C(g.s.)

“‘C(p, d)-C(3.68)

-N(p,

d)r3N(g.s.)

TABLE3 from the DWBA analysis of the (p, d) reactions, studied in the present work

Type of calculation

Cut-off radius (fm)

s present “)

S Watson b,

LZR LZR NLFR LZR LZR NLFR LZR LZR NLFR

0

1.27 1.58 1.41 1.52 1.81 1.97 0.83 1.30 1.11

1.18 1.47 1.32 1.33 1.68 1.76 (c)

2 0 0 2 0 0 2 0

“) Refers to proton optical model parameters from present work. ‘) Refers to proton optical model parameters from Watson er nl. 31). ") Not calculated.

Although sufficient data could not be obtained for an angular distribution for the 3.68 MeV level in “C, a spectroscopic factor was obtained by normalizing the theoretical curve to the experimental data at forward angles. Once again the extracted spectroscopic factor differed by approximately 25 ‘A depending on the cut-off radius and whether it was LZR or NLFR.

30

T. H. CURTIS et al.

The 14N(p, d)t4N(g.s.) reaction has been analyzed in the manner described above. The quality of the fits was in general not as good as those for the 14C(p, d)13C reaction. The same sensitivity to the cut-off radius was noted. These results are summarized in table 3 where the spectroscopic factors are listed and fig. 4 where representative fits are shown. The DWBA fits for the 13C ground state transition were best for a cut-off radius of 2 fm. From table 3 we see that this gives a value of 1.58 for the spectroscopic factor. This is to be compared with the theoretical value of 1.73 predicted by Cohen and Kurath I”). The agreement is quite reasonable. For the transition to the 1~~ state at 3.68 MeV we havea spectroscopic factor of 1.81 compared to the predicted value of 2.03. For the 14N(p, d)13N reaction we have I

t

I

‘% (PA)“c

Fig. 5 Experimental

-

data for the reaction ‘*C(p, t)‘% proceeding to the ground state of W. diffraction pattern typical of an L = 0 transfer is evident.

A

1.30 and 1.38 for the experimental and predicted spectroscopic factors. The 14N(p, d) 13N reaction can, in principIe, proceed by the pick-up of either a lpt. or lp3. neutron. The results of Cohen and Kurath 13), however, show that the transition shouid be mainly lp*. The spectroscopic factors refer to this. The agreement between theory and experiment is quite satisfactory. To the extent that using DWBA “best fits” is the correct procedure for extracting spectroscopic factors the present results support the intermediate-coupling calculation of Cohen and Kurath 13). The results however, are sensitive to the cut-off radius as well as the optical model parameters. It would be more satisfactory to perform the (p, d) reactions at an energy comparable to the proton energy in the (d, p) studies of Schiffer et al. 12). Th is p resumably would allow one to use a self-consistent set of optical parameters and thereby gain a more meaningful comparison of the results.

“N, 14C (P, P), (P, d), (P, t)

For completeness

an angular

distribution

31

of the ’ 4C(p, t)“C

ground

state reaction

was obtained. The results are shown in fig. 5 where a typical L = 0 diffraction is evident. No attempt has been made to fit the data.

pattern

4. Discussion The poor agreement between theory and the inelastic data requires some comments. Three areas present themselves as possible causes of the difficulties: simplifying assumptions made in the DWBA calculation, the model wave functions and compoundnucleus contributions in the inelastic scattering. We have already pointed out that the codes we used neglected spin-orbit, core-polarization and channel-coupling effects. In addition, exchange and tensor effects could not be treated simultaneously. Love et al * 32 ) have shown recently that tensor plus knock-out exchange contributions are important in the 14N(p, n)14C(g.s.) t ransition. Because we are treating light nuclei with relatively few open channels, one might expect that channel-coupling effects could bc important. Schmittroth performed a study ‘) in which he coupled together the ground states of 14C and 14N and the first two excited states of r4N at 2.31 MeV and 3.95 MeV. He concluded that the DWBA calculations were about as accurate as the coupled-channel calculation, and moreover, spin-orbit effects were small in the system he considered. The effects on the DWBA results of the shell-model wave functions and the twonucleon interaction are intimately related. Within a single calculation it is almost impossible to separate the two effects and this makes it very unlikely that one can derive any information from parameter variation. It would be desirable to calculate the shellmodel wave functions in a manner consistent with the reaction calculation: bound in a potential well with the Woods-Saxon shape and using the assumed two-nucleon interaction. The possibility exists that we are not in an energy region for this nucleus that can be described by a direct reaction theory. Our analysis of the elastic data indicated that there should not be too great compound nucleus scattering in that channel. The situation in the inelastic channels might be one described by neither the simple compoundnucleus scattering nor direct reactions. The cross sections could have a fluctuating behavior determined by subtle interference effects. This is a question that can only be answered

by further

experiments

performed

as a function

of energy.

References 1) A. K. Kerman. 2) 3) 4) 5) 6) 7)

V. G. N. W. N. F.

H. McManus and R. M. Thaler, Ann. of Phys. 8 (1959) 551 A. Madsen, Nucl. Phys. 80 (1966) 177 R. Satchler, Nucl. Phys. 77 (1966) 481 K. Glendenning and M. Veneroni, Phys. Rev. 144 (1966) 839 W. True, Phys. Rev. 130 (1963) 1530 Freed and P. Ostrander, Nucl. Phys. All1 (1968) 63 Pctrovich, H. McManus, V. A. Madsen and J. Atkinson, Phys. Rev. Lett.

22 (1969)

895

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T. H. CURTIS et al.

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