The calcium isotopes and N = 28 isotones and the optical model

1 2.B:Z.E

(

Nuclear Physics Al92

(1972) 641-650;

@

North-Holland

Publishing

Co., Amsterdam

Not to be reproduced by photoprint or microfilm withont written permission from the publisher

THE

CALCIUM ISOTOPES AND N = 28 ISOTONES AND THE OPTICAL MODEL

J. C. LOMBARDIT, Department

of Physics,

R. N. BOYDtf, Rutgers

R. ARKINGttt

University,

and A. B. ROBBINS

New Brunswick,

New Jersey 08903 f

Received 30 December 1971 (Revised 29 May 1972) Abstract: Differential cross-section and analyzing-power angular distributions have been measured for protons elastically scattered from 40* 44* 48Ca, s0Ti, 51V, %Zr and 54Fe. The rms radii of the real-central potentials were determined from the optical-model analyses of the angular distributions. From these results and the rms radii of the charge distributions, the rms radii of the neutron distributions were calculated and found in this analysis to be larger than the rms radii of the proton distributions in the isotones 48Ca, 50Ti, 51V, 5ZCr and 54Fe. This analysis yields an rms radius for the neutron distribution in 48Ca which is larger than that determined from Coulomb energy-difference measurements.

E

NUCLEAR REACTIONS 5oTi, slV, 52Cr, 54Fe(p, p), E = 10.8-15.4 MeV; measured o(E, 0) and A(E, 0): deduced optical model parameters. “Ti, 51V, %r, 54Fe deduced rms radii. Enriched taraets.

1. Introduction In a previous paper ‘) we reported an investigation which entailed the measurement of differential cross-section and analyzing-power angular distributions for protons elastically scattered from ‘03 44S48Ca, and the optical-model analyses of these data. The present paper reports on an extension of our investigation to the nuclei ’ 'Ti, ‘IV 52Cr and 54Fe. These nuclei along with 4”Ca are the stable members of the isotonic sequence with neutron number N = 28. The high quality of the cross-section and analyzing-power data resulted in accurate determinations of the optical-model parameters, and hence the mean square radii of the real-central term of the optical-model potential
details

The experimental details are essentially the same as in ref. ‘). The experiment was carried out at the Rutgers-Bell tandem Van de Graaff accelerator. Differential crosssection and analyzing-power angular distributions were measured for protons elastict Presented by J. C. L. in partial fulfilment of the requirements of Rutgers University. Present address: Department of Physics, Pennsylvania. tt Present address: Department of Physics, Stanford University, tit Present address: Department of Nuclear .Physics, Weizmann Israel. 1:Supported in part by the National Science -Foundation. 641

the Doctor of Philosophy degree, Allegehny College, Meadville, Stanford, Institute

California. of Science,

Rehovot,

642

9. C. LOMBARDI

et al.

ally scattered from 5‘Ti, ‘IV, 52Cr and 54Fe. For the analyzing-power measurements a Lamb-shift polarized ion source “) was used which produced typically 1 nA of 55 % polarized beam on target. The angular distributions were measured at two energies in the range 10.80 to 15.35 MeV. Excitation functions were examined in the energy region of each angular distribution to determine that the angular distributions were not influenced by prominent resonances. The 51V, 52Cr and 54Fe targets had isotopic purities of 99.8, 99.9 and 97.1 % respectively. The 50Ti target had an isotopic purity of 76.4 % with a 18.7 % “‘Ti impurity plus other titanium impurities. Because of these target impurities it was necessary to use natural titanium cross-section and analyzing-power data to correct the 5oTi data. The uncertainty in the detector placement was determined to be A” or less by comparison of yields on the right and left sides of the chamber. The statistical uncertainties in the cross sections are typically 1 %. However, relative errors of 2 ‘A were assigned for optical-model analyses to account for angular misaligrrments. The errors assigned to the analyzing-power data are the statistical uncertainties. Typical errors for these data are 2 %. 3. Optical-model analysis Optical-model fits to the elastic scattering data for each nucleus and energy were made using an optical-model potential with the following form:

U,(r) = u&Q+ U&“)+~Ul(~)+Us(r),

(1)

where g

WJ”) =

[3-

1

(&)“I

$ __ r

U,(r) = 4a,

for

r 4 Rc

for

r>&,

w, Y!!@ )

dr

A, is the Compton wavelength of the pion and A is the atomic mass number. The parameters r, and a, are the radius and diffuseness parameters associated with the form factor fx(r). The Coulomb potential is calculated from a uniform charge

49.1

18.60 “)

1.22

1.13

1.24

1.33 1.19

1.21

1.26

1.19 1.19

0.70

0.72

0.65

0.57 0.67

$69

0.62

0.68 0.67

r1

1.34

1.25

1.34

1.32 1.26

1.29

1.28

1.28 1.24

(fm)

0.53

0.71

0.34

0.39 0.69

0.61

0.46

0.68 0.65

data taken. from refs. ii. 12).

8.3

6.9

14.21

13.7 7.5

8.6

11.1

8-G 8.9

WD (MeV)

QR

(fm)

“) Cross-section and analyzing-power

50.3

55.9

10.80

s4Fe

45.8 51.6

15.13

10.77 15.3s

50.3

52Cr

49.9

I$81

15.20

“IV

52.9 52.2

14.15 15.35

(iE.?V) (M?V) (fit)

s OTi

TABLE I

5.78

6.49

5.22

6.67 5.70

6.57

6.64

6.38 6.84

1.07

0.95

0.88

1.06 1.02

1.01

1.12

0.99 0.98

‘0.46

0.58

0.50

0.47 0.45

0.64

0.42

0.71 0.66

(ii, 1.21

1.29

1.29

1.29

1.27 i.27

1.27

19.49

18.20

19.03

19.30 18.09

18.56

18.55

17.79 17.75

{fm) 1.26 1.26

(fm2)

&IA*

4.57

434

473

507 444

456

423

454 453

(MeV

JNA * fm3)

Optical-model parameters for the best fits to the differential cross-section and analyzing-power data

50.8

48.0

3.2

8.6 19.2

30.9

1.4

6.7 10.1

x:

18.1

25.8

8.5

8.6 14.5

13.0

9.4

1.3 5.6

x:

8.4

7.0 7.8

x$

31.9

36.9

5.7

8.6 17.3

22.0

--

% E

s s

g

2 ,,

644

J. C, LOMBARD1 et al.

15.35 MeV

QCM (deg.)

Fig. 3. Angular distributions of the differential cross sections and analyzing powers for the elastic scattering of protons from 52Cr at 10.77 and 15.35 MeV.

1.6 1.4 1.2

20

I 1

40

I

40

60

I

60

I

I

120

80 100 120 @CM(deg.)

I

80 100 %Mkkg.)

140

I

140

160

I

160

.

Fig. 4. Angular distributions of the differential cross sections and analyzing powers for the elastic scattering of protons from s4Fe at 10.80and 15.13 MeV.

-0.81

20

1.61.2-

646

J. C. LOMBARD1 et al.

distribution “) with its radius given by Rc = &(r&)“, (2) where %is the charge-distribution rms radius. The values of * were taken from ref. “). The optical-model parameters I’,, W,, V,, r,, a,, rI, a,, r, and as were varied to find a parameter set which minimized the quantity

(3) wherebth (ei), 4h(b), ~erp(4)and 4, (6 > are the theoretical; and experimental differential cross sections and analyzing powers at the c.m. angle Bi, and Ad and AA are the respective experimental uncertainties. Figs. l-4 show the cross-section and analyzing-power angular distributions and their optical-model fits for 50Ti, 51V, 52Cr and 54Fe. Table 1 gives the optical-model parameters obtained for each of these isotones and at each energy, the corresponding values of the mean square radius of the real-central potential <&> and the volume integral per nucleon of the real-central potential J/A. 4. The optical model and nuclear sizes In this section we report the results of assuming a simple relationship between the quantity & and the rms radius of the nuclear-matter distribution (PA)*. The uncertainties assigned to various quantities are discussed in sect. 5. Makofske et al. “) have shown that the reformulated model of Greenlees “) gives values of (r&)* which are larger at 16 MeV than they are at 30.3 MeV. However, a comparison of values of * determined with a standard optical model for energies of 16 MeV [ref. ‘)I and 39.6 MeV [ref. ‘)I sh ows very little energy dependence of “. We have assumed the energy independence of <&>* determined with the standard optical model. We have assigned an uncertainty to has been derived “) in terms of <&> and the mean square radius of the proton distribution
=

-

s

~i(~)Voje-K~(ly-~l)2d~.

Here, - VOje-kJr2 is the nucleon-nucleon potential; i = p for protons and n for neutrons; j = t for spin-triplet interactions and s for spin-singlet interactions; & = 0.415 fme2, KS = 0.292 fmm2, F,, = 66.92 MeV and V,, = 29.05 MeV. The

N = 28 ISOTONES

647

parameters&I .&, Vat and V,, were d~t~~n~

’ “) from an analysis of 01-gscattering data, and were shown to yield the experimental n-p triplet and p-p singlet scattering 1,engths and effective ranges. Eq. (4) comes from folding the nucleon-nucleon triplet and singlet potentials with the neutron and proton distributions pa and p,, . The mean square radius of the real-central potential can be written as

(5) From eqs. (4) and (5) it follows that

where 2 aud N are the proton and neutron numbers,

(Y;> = <$)+

(rj”> = ($}+3/2&,

and (rj2> is the mean square radius of the singlet or triplet potential. Eq. (6) can be rearranged to give @,“> = +O.3938 $ [-(r,2}-5.137]-3.914.

(7)

This expression was used to calculate * for each nucleus. The values of (&> used in eq. (7) were determined for ?I%, “V and s2Cr by averaging the results of the two optical-model &s for each nucleus. For these nuclei the energies at which the angular distributions were made are well above the (pJ n) threshold. However, for 54Fe the data at 10.80 MeV were taken only slightly above the (pt n) threshold of 9.78 MeV. The results of the fit to these data may be affected therefore by the presence of compound elastic scattering. For this reason an additional optical-model fit to cross-section 11) and analyzing-power 12) data at 18.60 MeV was made. The value of{&> determined at 10.80 MeV was found to f&l1between the values determined at 15.13 and 18.60 MeV. An average of the three values of (P&> for 54Fe was used in the subs~uent analysis. The values of (I,“> used in eq. (7) were obtained from the results of electron scattering experiments “) which determine the mean square radii of the charge distributions in nuclei. In order to account for the kite size of the proton I3314) we subtracted 0.64 fm2 from the experimental values uf (r&J to determine ($3. The mean. square radii of the. matter distributions
J. C. LOMBARD1

648

et al.

TABLE2 Experimental values of quantities defined in the text Nucleus

40Ca “) 44Ca “) 48Ca *) 5oTi J’V 52Cr 54Fe

3.9s 3.98 4.22 4.22 4.31 4.32 4.35

o&> W)

*

(fm)

* (fm)

4.28 4.18 4.74 4.59 4.81 4.71 4.68

3.40 3.41 3.62 3.63 3.71 3.74 3.77

3.40 3.40 3.78 3.74 3.88 3.88 3.90

0.01 -0.02 0‘39 0.26 0.39 0.31 0.27

The absolute uncertainties in 3, -k and d are kO.07, 10.10, &to.18 and kO.19 fm, while the relative uncertainties are &0.06, 10.06, 10.13 and f0.15 fm respectively. “) Results are taken from ref. ‘).

RMS RADII

3.9

00

x MATTER 3.8

*

t

= ?J

”

0

3.6

Fig. 5. The rms radii of the proton, neutron and matter distributions in 40*44S48Ca, 50Ti, 51V, 52Cr and ?4Fe. The solid line represents the fit * = 0.99 A+ fm. Relative uncertainties in the matter radii are estimated to be ho.06 fm. The dashed line represents the fit
determined from the following relationship: 0-3

= &-

+

&-

09

(8)

The values of *, <&+, * uncertainties indicated are relative uncertainties which are estimated to be kO.06 fm. The solid line represents the function C, A&where C, has been varied in order to fit

N = 28 ISOTONES

649

the matter radii; C, was found to be 0.99 fm. This function is seen to give a reasonable representation of our data. If one assumes a Fermi distribution with a diffuseness of 0.5 fm, then an rms radius of 0.99 A* fm corresponds to a half-way radius of approximately 1.1 A” fm. The dashed line represents the function CPZa where C, has been varied in order to fit the proton radii; C, was found to be 1.24 fm. 5. Discussion

of uncertainties

Two contributions are included in the uncertainties of the values of (r&-J”. The first comes from extracting <&> from the data; the second from the effects of coupling between the ground state and inelastic states. For the nuclei investigated a variation in (r&>* of f0.06 fm resulted in an increase of approximately 50 % in x2 from its minimum; this is regarded ‘) as a significant difference in the quality of the fit. The first contribution to the uncertainty in (Y&)* is therefore taken to be t-O.06 fm. We have investigated the effect of coupling between the ground state and strongly excited 2+ and 3- states, using the coupled-channels code of Raynal 16). Deformation parameters for these states in 54Fe have been measured by Eccles et al. ‘l). It is concluded that the effect on the value of (P&)” is less than +0.04 fm. These two sources of uncertainty result in a total uncertainty in (&>” of +0.07 fm. In order to assign an absolute uncertainty to (r-i)* the uncertainty in for each nucleus from the relationship I’) = <&) - . (9) The values of (r&> from ref. ‘) and the present investigation have been listed in table 2, and are in agreement with the values of refs. 17-20). From this we feel that a reasonable uncertainty in the value of ” are calculated from the uncertainties in (r&)” and and are found to be &O. 10 fm. The uncertainties in values of ’ and the uncertainties in (ri)* which are assumed to be 2 %_ The absolute uncertainties in (r,f)” and A are +0.18 and 1-0.19 fm respectively. To get an estimate for the relative uncertainties we take only the uncertamty in 4 and the uncertainty which comes from extracting ” from the data. This assumes that the uncertainty in the nucleon-nucleon interaction and the uncertainty due to inelastic coupling affect only the absolute uncertainties. The relative uncertainties in (ri)*, ” and A are found to be +O.OS, kO.13 and i-O.15 fm respectively.

G50

J. C. LOMBARD1

et aE.

Investigations of Owen and Satchler “‘) indicate that exchange effects decrease the measured value of * is consistent with an A” variation through the calcium isotopes and N = 28 isotones. Such a variation has been observed for the radius of the optical potential for the reactions 40344348Ca(160, .160) [see ref. ““)I. An A3 variation for (ri)+ was not seen in the calcium isotopes in Conlomb energy difference experiments “). It is seen from fig.‘5 that a .Z* variation gives a reasonable representation of the proton distributions in the calcium isotopes and N F 28 isotones. The present results, the results of ref. I), and the results of electron scattering 4, ’ “) suggest that there is a difference in the variation of the matter and proton distributions in this mass region. The results of Coulomb energy difference experiments “) do not yield as large a difference between the matter ,and proton distributions. References ‘1’) J. C!: Lombardi, R. N. Boyd, R. Arking and A. B. Robbins, Nucl. Phys. A188 (1972) 103 2): R. N.‘Boyd, J. C. Lombardi,

A. B. Robbins and D. E. Schechter, Nucl. Instr. 81 (1970) 149 3) F. D. Becchetti, Jr.,,and G. W. Greenlees, Phys. Rev. 182 (1969) 1190 4) H. R. Collard, L. R. B. Elton and R. Hofstadter, Landolt-Bornstein, new series 12 (1967) 30 5) W. Makofske, G. W. Greenlees, H. S. Liers and G. J. Pyle, Phys. Rev. C5 (1972) 780 6) G. W. Greenlees, G. J. Pyle and Y. C. Tang, Phys. Rev. 171 (1968) 1115 7) H. S. Liers, R. N. Boyd, C. H. Poppe, J. A. Sievers and D. L. Watson, Phys. Rev. C2 (1970) 1399 8) J. P. Schiffer, J.-A. Nolen, Jr., and N. Williams, Phys. Lett. 29B (1969) 399 9) G. W. Greenlees, V. Hnizdo, 0. Karban, L. Jowe and W, Makofske, Phys. Rev. C2 (1970) 1063 10) I. Reichstein and Y. C. Tang, Nucl. Phys. Al39 (1969) 144 11) S. F. Eccles, H. F. Lutz and V. A. Madsen, Phys. Rev. 141 (1966) 1067 12) P. Kossanyi-Demay, R. de Swiniarski andC.‘Glashausser, Nucl. Phys. A94 (1967) 513 13) L. R. B. Elton, Phys. Rev. 158 (1967) 970 14) .E. E. Chambers and R. Hofstadter, Phys. Rev. 103 (1956) 1454 15) R. F. Frosch, R. Hofstadter, J. S. McCarthy, G. K. Noldeke, K. J. van Oostrum, M. R. Yearian, B. C. Clark, R. Herman and D. G. Ravenhall, Phys. Rev. 174 (1968) 1380 16) J. Raynal, ECIS, unpublished 17) D. Slania and H. McManus, Nucl. Phys. All6 (1968) 271 18) G. W. Greenlees, W. Makofske and.G. J. Pyle, Phys. Rev. Cl ,(1970) 1145 19) G. L. Thomas and B. C. Sinha, Phys. Rev. Lett. 26 (1971) 325 20) R. N. Boyd, J. C. Lombardi, R. Mohan, R. Arking and A. B. Robbins, Nucl. Phys. A182 (1972) 571 21) L. W. Owen and G. R. Satchler, Phys. Rev. Lett. 25 (1970) 1720 22) H. R. Kidwai and J. R: Rook, Nucl. Phys. Al69 (1971) 417 23) M. C. Bertin, S. L. Tabor, B. A. Watson, Y. Eisen and G. Goldring, Nucl. Phys. Al67 (1971) 216