Some one-neutron transfer reactions with 18O and 19F beams

Nuclear Physics Al72 (1971) 25-32;

i@ ~orth-~o~~~~

P~l~sh~~~ Co., Amsterdam

Not to be reprodueeti by photoprint or mlcroglm without written permission from the publisher

SOME ONEXVEUTRON TRANSFER REACTIONS WITH “0 AND “F BEAMS H. KNOTH, P. H. BARKER, A. HUBER, U. MATTER, P. M. COCKBURN and P. MARMIER Luboratorium fiir Kernphysik. ETH Zurich, Switzerland Received 15 March 1971 (Revised I5 April 1971) Abstract: The differential cross sections of the neutron transfer reactions 13C(z80, r90)rzC, %e (‘*O, lqO)%e, gBe(‘80, 170)10Be, l”B(igF, 18F)kzB and z0B(i80, 170)r1B have heen measured at energies in the region of the Couiomb barrier. Using a DWBA formalism, the products of the involved neutron spectroscopic factors have heen extracted.

E

NUCLEAR REACTIONS i”B(180, IsO), (rgF, 19F), r3C(‘*0, ‘*O), 9Be(‘80, 180), E = 12-24 MeV; 1oB(180, “O), (lgF, i8F), E = 20,24 MeV; gBe(180, i90), E = 12.1, 16 MeV, (i*O, “O), E = 16,20 MeV; t3C(‘80, ‘*O), E = IS, 2424 MeV; measured a(@. s* r”Be, llB, izC deduced neutron spectroscopic factors.

1. Iutroduction

Studies of heavy-ion one-neutron transfer reactions have shown that they can be a valuable tool for the determination of spectroscopic factors. Earlier total cross-section measurements rm3), wh’ICh were made in the energy region below the Coulomb barrier, where the approximation of a pure Coulomb intera~~on could be considered a good one, were fitted well and resulted in reasonable spectroscopic factors. In some recent work, Barker et al. “) examined several one-neutron transfer differential cross sections in the neighbourhood of the Coulomb barrier and analysed them with the DWBA formulation of Buttle and Goldfarb “). At these energies the nuclear absorption was significant. In principle, the nuclear interaction can be described by means of an optical potential, but because the parameters of this cannot be uniquely determined, use was made of the suggestion of Goldfarb and Steed “) to construct the distorted wave eigenfunctions from the scattering phase shifts. The resulting fits to the oneneutron transfer angular distributions were generally good and the spectroscopic factors extracted agreed well with those calculated from theory. The present study is an extension of this earlier work in order to examine further the applicability of the DWBA analysis of heavy-ion transfer reactions to the determination of neutron spectroscopic factors. Angular distributions of the following reactions have been measured at energies below and near the Coulomb barrier; i3C(‘80, 1gO)‘2C, 9Be(‘80, r’O)*Be, 9Be(180, 170)10Be, ‘*B(l*O, l’O)‘lB and l”B(19F, ‘*F)‘rB. In order to determine the phase shifts from which the distorted waves are calculated, the elastic scattering of the same incoming nuclei at the same energies has also been studied. 25

H. KNOTH er al.

26

Using

the DWBA

have been calculated, the involved

theory

of Buttle and Goldfarb

and from these the products

‘), fits to the measured of the spectroscopic

points

factors

of

nuclei determined.

2. Experimental

method

The experimental techniques have already been described elsewhere 4, ‘* *), and therefore the procedure will only be summarised here. The two outgoing nuclei were measured in coincidence using two semiconductor detectors. Targets were lo-20 jcg/cm’ thick, those of “C being self-supporting with an enrichment to 50 % “B being on carbon backings of about 10 pg/cm’ in 13C, and those of 9Be and thickness.

CL-t

0,:

005

45

_--~_ _____ ELab:

t_+.-I

*‘_!

.-.

\

I

__ .A_._...I.__._ 60 100

'L--I i20

140

Tg

Fig. 1. Differential cross sections of the one-neutron transfer reaction 13C(‘B0, 1gO)12C at 15, 20 and 24 MeV.

The coincidence method is very suitable for measuring heavy-ion neutron transfer albeit only in the angular region where they are reaction angular distributions, sufficiently separated from the elastic scattering. In addition, transitions to different states of the same final nucleus may be differentiated from one another. In the reactions to be described here, with the exception of 9Be (180, ’ 90)8Be, only transitions to the ground states of the final nuclei were measured, the cross sections for the transitions to other states being too small to be measurable.

ONE-NEUTRON

TRANSFER

REACTIONS

27

Figs. 1-S show the measured angular d~stribu~ons of the transfer reactions. The curves have been calculated with the DWBA (see sect. 4), and the error bars contain only the statistical uncertainties in the points, there being a further overali error of IO-15 % involved in the normalisation.

EL,b: 2

16MeV

12,lMeV 1 0.5 I

‘.

~

q ‘.

‘.

”

$1

o.05L_-. so

100

Fig. 2. Differential cross sections of the one-neutron trausfer reaction 9Be(*a0,*90+ at 1.47 MeV)sBe at 12.1 and 16 MeV.

150

j_

c++$

Fig. 3. Differential cross sections of the one-neutron transfer reaction 9Be(is0, 170)*oBe at 16 and 20 MeV. ‘“Bf’80.170)‘~B

‘%(

lgFI

‘6F 1 “6

Fig. 4. Differential cross sections of the one-neutron transfer reaction xOB(tvF. “F)“B at 20 and 24 MeV.

Fig. 5. Differential cross sections of the one-neutron transfer reaction l”B(reO, ‘70)x1B at 20 and 24 MeV.

28

H. KNOTH ef al.

In the 13C(180, 190jX2C reaction, only the transition to the ground state of 190 could be measured, whilst the reaction ‘Be(“*O, r ‘0 ) ‘Be led principally to the second excited state of IgO at I.469 MeV. The combined cross section for the transitions to the ground and the first excited states in the latter reaction was at least a factor thirty smaller. These two reactions illustrate the Q-value dependency of the transfer reaction cross sections, which has been described by Barker et al, “). According to this, for t~nsitions to di~erent states of the same final nucleus, the greater probability is for the reaction whose Q-value lies closer to zero, when the influence of the spectro9Be ( ~*o,@o 1%e 04 i

%ab:

1

121MeV

0.5

16MeV

20MeV

1

0.5

13c( 80, l*otl3c 010; t

Eiab:

1

0.2~

I.

*..

20 MeV ..

15MeV

0.5 24MeV

24MeV

0.2 0.1 0.05 50 Fig. 6. Differential

100

150 B(..N

cross sections of the eiastic scattering of the ions in the entrance of the reactions ilfustrated in figs. l-5.

channels

ONE-NEUTRON

TRANSFER

REACTIONS

29

scopic factors has been removed. Since the Q-value for 13C(‘*0, ’ 90)‘2C is -0.99 MeV and that for 9Be(‘80, r90)‘Be is 2.292 MeV it is to be expected that in the first case the ground state of 190 and in the second the excited states would be preferentially populated. The reaction 9Be(180, 1‘O)l ‘Be has a Q-value of - 1.23 MeV, and no transition to an excited state of “Be could be found. The Q-values of the reactions l”B(19F, “F)“B and “B(“O, l’O)lrB are positive, 1.03 and 3.41 MeV respectively, and so one might expect transitions to excited states in these cases. None however were detected, which is probably ascribable to the complicated nature of the lower excited states of rlB. Fig. 6 shows the measured angular distributions of the elastic scattering of the nuclei in the entrance channel of each of the transfer reactions. Wherever the statistical errors are greater than the diameter of the points, then they are given by the bars. The solid curves give the theoretical fits calculated with the smooth cut-off model (see sect. 4). 3. Theory Buttle and Goldfarb ‘) have given the following expression for the differential cross section of the one-neutron transfer reaction (c,+n)+c, 1

+ c,+(c,+n), 2

Here S, and S2 are the spectroscopic factors of the nuclei between which the neutron is exchanged, AI is an integral involving the neutron bound in nucleus one and N2 is the normalisation constant with which a spherical Hankel function is fitted to the outer part of the neutron wave function in nucleus two. The spins of the nuclei one and two are j, and j, and I is the transferred angular momentum, with projection 1; T,,represents the integral containing the elastic scattering wave functions in the entrance and exit channels. At low energies, these latter are simply Coulomb waves, but at energies nearer the Coulomb barrier, they are calculated according to the proposal of Goldfarb and Steed “). At large distances they may be represented using the nuclear phase shifts, 6r, and the regular and irregular Coulomb wave functions, FL and GL: eidL[F,(kr)cos &+G,(kr) sin SJ. At a radius R,, where the above expression is still valid, it is fitted to a linear combination of Coulomb functions, ~1 &(kr)+

a2 G&r),

of which the coefficients are so chosen that the combination than a cut-off radius R,.

vanishes for radii less

30

H. KNOTH et al.

The phase shifts, a,, can be determined from the measured elastic scattering differential cross sections using the smooth cut-off model 9), in which the absorption function in R-space ,

is taken. Using the semi-classical relationship L(L+

the AR can be transformed with AL = exp 2i&.

1) = k2R2 -2qkR,

into AL, from which the phase shifts may be extracted

4. Analysis 4.1. ELASTIC

SCATTERING

In fig. 6 the calculated fits with the smooth cut-off model are shown. The parameters with which the cross sections were calculated are displayed in table 1. From the radius rO and the masses of the nuclei involved, the radius R. is R.

= r,(Mf

+ Mf).

Since only the elastic scattering in the entrance channel of the transfer reaction could be measured, the phase shifts for the scattering in the outgoing channel were determined by using parameters from the entrance channel which were extrapolated to the correct energy in the outgoing channel. TABLE 1

Parameters used in the calculation of the elastic scattering cross sections shown in fig. 6 r0 (fm)

dR (fm)

15 20 24

2.10 1.72 1.66

0.6 0.9 1.1

180-9Be

12.1 16 20

2.36 1.86 1.68

0.4 0.7 1.3

lsO.--‘OB

20 24 30.5

1.81 1.65 1.67

0.6 0.9 1.4

19F_‘OB

20 24

1.98 1.76

0.3 0.5

Scattering ions ~SO--‘3C

4.2. TRANSFER

bb

(MeV)

REACTIONS

In order to determine the form factor A, and the normalisation constant N,, the wave functions of the neutron in both the initial and the final nucleus are calculated.

ONE-NEUTRON

TRANSFER

REACTIONS

31

For this a Woods-Saxon potential is used, which has a Fermi radius cf 1.2 fm and a diffuseness of 0.55 fm. The potential also contains a spin-orbit term of 20 % of the depth of the Woods-Saxon potential. These parameters affect the overall normalisation. If, for example, the Fermi radius or the diffuseness is altered by 0.1 fm then the product Ai Ni , which appears in the expression for the cross section, changes by around 50 % on the average. However, the potential used has normally been successful in describing the neutron bound state wave function, and so the uncertainty introduced into the calculation of the spectroscopic factors should be small. The distances R, and R, are determined by fitting the calculated transfer reaction cross section to the shape of the measured differential cross section. Because the distance between R, and R, has little influence on the calculated shape, it was set arbitrarily at 2 fm. The product of the neutron spectroscopic factors is then just the factor with which the calculated differential cross section is normalised to the measured points. In table 2 the spectroscopic factor products which have been derived from these experiments are displayed with the theoretical values. The values of R, and R,,, which were used to obtain the fits shown in figs. l-5 are also included in table 2. The theoretical spectroscopic factors for beryllium, boron and carbon were taken from Cohen and Kurath lo), and for the heavier nuclei from other sources l l- ’ “). TABLE2 Parameters used in the calculation of the differential cross sections shown in figs. l-5 Reaction

K (fm)

R, (fm)

&SZ (exp.)

SISz (theor.)

15 20 24

8.5

10.5

0.49 *0.05 1.47*0.2 1.43 hO.2

1.2

gBe(l*O, “O*)*Be

12.1 16

8.3

10.3

0.37hO.07 0.44&-0.06

0.58

%epo,

16 20

8.0

10.0

6.6 *1.5 7.7 k1.8

3.9

-cpo,

190)‘fC

170)‘oBg

&.

(MeV)

In the r3C(‘*0, ’ 'O)"C reaction, very good fits were obtained at all three energies. However, the spectroscopic factors shown in table 2 clearly show a discrepancy. While the values at 20 and 24 MeV are in good agreement, and are also compatible with theory, the value at 15 MeV is a factor three lower. The fits for the ‘Be(‘a0, r’O)*Be reaction are also very good (see fig. 2). The spectroscopic factors obtained at the two energies agree among themselves and are only slightly lower than the theoretical value. Fig. 3 contains the curves for the reaction ‘Be(180, “O)l’Be. The agreement with the data is reasonable, and there is fair correspondence between the experimental and theoretical products.

H. KNOTH

32

er al.

In the l”B(19F, 18F)11B, and the “B(180, “O)‘lB reactions it proved impossible to calculate differential cross sections which both fitted the experimental points and gave reasonable values for S, S2. In the former case this is probably ascribable to the complicated nature of the ground state wave function of 19F, which could not be taken into account in the calculation. The Q-value for the latter reaction is very large, +3.41 MeV, which has the effect of shifting considerably the wave functions in the exit channel relative to those in the entrance channel, which makes the integral very sensitive to small changes in the parameters. This has been described more fully elsewhere “). 5. Conclusions Using the DWBA theory of Buttle and Goldfarb 5), most of the measured oneneutron transfer reactions have been well analysed, and although all the reactions were performed at energies at which nuclear interaction is not negligible in the relative motion of the heavy ions, this was successfully taken into account using the method of Goldfarb and Steed “). Furthermore the products of spectroscopic factors from the same reaction at several energies agree well with one another. It did not prove possible to calculate the differential cross section of the reaction i”B ( 180 “O)l’B which is probably due to the large Q-value. The large normalisaa tion constants needed for the i”B(19F, 18F)11B reaction are almost certainly consequence of the complicated structure of the ground state of i9F. In all other cases, the products of spectroscopic factors which are determined differ only slightly from those predicted by theory. These differences are certainly smaller than those which would arise from making small alterations in the neutron bound state wave function parameters. Excellent descriptions of the elastic scattering of the heavy ions in the entrance channels

of the transfer

reactions

were provided

by the smooth

cut-off model.

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15)

J. C. Hiebert, J. A. McIntyre and J. G. Couch, Phys. Rev. 138 (1965) B346 R. M. Gaedke, K. S. Toth and I. R. Williams, Phys. Rev. 141 (1966) 996 J. G. Couch, J. A. McIntyre and J. C. Hieberl, Phys. Rev. 152 (1966) 883 P. H. Barker, U. Matter, A. Gobbi, A. Huber, H. Knoth and P. Marmier, Nucl. Phys. A155 (1970) 401 P. J. A. Buttle and L. J. B. Goldfarb, Nucl. Phys. 78 (1966) 409 L. J. B. Goldfarb and J. W. Steed, Nucl. Phys. All6 (1968) 321 A. Gobbi, U. Matter, J. L. Perrenoud and P. Marmier, Nucl. Phys. All2 (1968) 537 H. Knoth, Dissertation, Zurich (1970) J. A. McIntyre, K. H. Wang and L. C. Becker, Phys. Rev. 117 (1960) 1337 S. Cohen and D. Kurath, Nucl. Phys. Al01 (1967) 1 J. P. Elliott and A. M. Lane, in Handbuch der Physik, ed. S. Fliigge, vol. 39 (SpringerVerlag, Berlin, 1957) M. G. Redlich, Phys. Rev. 110 (1958) 468 N. K. Glendenning, Ann. Rev. Nucl. Sci. 13 (1963) 191 I. Kanestrom and H. Koren, Nucl. Phys. A130 (1969) 527 M. H. Macfarlane and J. B. French, Rev. Mod. Phys. 32 (1960) 567