Proton scattering calculations from 23Na and 25Mg

I.E.4:2.L I

Nuclear Physics A152 (,1970) 570 -578; (~) North-HollandPublishing Co., Amsterdam Not to be reproduced by photoprint or microfilmwithout written permission from the publisher

PROTON SCATTERING CALCULATIONS

FROM 23Na AND ~SMg P. L. OTTAVIANI and L. ZUFFI Comitato Nazionale Eneroia Nucleate, Centro di Calcolo, Boloyna, Italy

Received 6 April 1970

Abstract: The scattering cross sections of 17.5 MeV protons from five low-lying states in 23Na and five low-lying states in 2SMg have been analysed with the coupled-channel method. Excitations of rotational bands built on different single-particle states have been taken into account. The interaction potential used contains a macroscopic non-spherical optical part, which gives rise to the collective effects, and a microscopic effective part, which allows the single-particle excitations.

1. Introduction In the last few years several cross-section calculations have been p e r f o r m e d with m i c r o s c o p i c m o d e l s which a d o p t m a n y - b o d y target wave functions a n d assume the interaction which gives rise to the inelastic transitions as a sum o f t w o - b o d y interactions between the projectile a n d the target nucleons 1-4). It is o u r aim to analyse the recent a n g u l a r d i s t r i b u t i o n m e a s u r e m e n t s o f p r o t o n s o f 17.5 M e V energy scattered from 23Na and 25Mg [ref. 5)]. These nuclei can be interpreted with the model o f a r o t a t i n g particle-core system. In fact, in the low-lying level scheme, r o t a t i o n a l bands are f o u n d due to different o d d nucleon excitations 6- 16). W e treat the single-particle excitations in the f r a m e w o r k o f a microscopic scattering description by i n t r o d u c i n g a nucleon-nucleon interaction into the nucleon-nucleus p o t e n t i a l a n d t a k i n g into a c c o u n t the collective effects by means o f a usual nonspherical optical p o t e n t i a l 17, i s). We m a k e use o f the c o u p l e d - c h a n n e l m e t h o d which is the m o r e suitable for the two nuclei considered is). Space exchange effects between the projectile a n d the o d d - t a r g e t nucleon are ignored. In sect. 2 we describe the assumed target nucleus m o d e l a n d in sect. 3, the scattering formalism. 1,1 sect. 4 we present the p e r f o r m e d analyses c o m p a r i n g them with the experimental d a t a and tinally in sect. 5 we include the discussion o f our results a n d the conclusions. 570

23Na, 25Mg PROTON SCATTERING

571

2. The nuclear target model The low-lying levels of 23Na and 25Mg nuclei can be described in terms of a rotational collective model with strong prolatc deformation. For 23Na the last odd proton, for the lower intrinsic configuration, can be considered in the K ~ : - 2 3 + Nilsson orbit [ref. 1,)] no. 7. Excited configurations 6-x4) can be obtained by raising this proton to the Nilsson orbit no. 9 (K '~ = ~-), no. 5 (K s ~ ), no. I I (K ~ = ;+), no. 8 (K ~ = ~ 3+ ) o r b y raising a proton from the K ~ = ½+ orbit no. 6 to no. 7, leaving a hole in orbit no. 6. There is evidence of a strong mixing a m o n g different bands mostly attributed to the RPC 2o). =

The low-lying 25Mg levels have a sirnpler interpretation 6 ~o. is, ~6). The last odd neutron is in the K~ = zs+ Nilsson orbit no. 5, for the ground state conliguration, and can be excitcd to the Nilsson orbits no. 9 (K ~ = ½+) and no. II (K ~ = J_,+). For 25Mg there is less evidence for band mixing. In agreement with the above interpretation for the level schemes o f the two nuclei considered the target wave functions can be written as a linear combination of the form =

K,v

~CbMr' ~,

( 1)

with

= r

l

,,

16~2 [DMr(Z~/31')(PK'"+( - )

,,

(2)

where I is the total angular m o m e n t u m and M its projection along the space z-axis. The wave functions ~tM K , v describe states of a rigid axially symmetric rotor with K being the projection o f the total angular m o m e n t u m 1 along the nuclear symmetry axis and v labelling different intrinsic states with the same K-valuc. Finally, DMr(~xfl?) are the symmetric-top eigenfunctions 2~) and q~g,,, are intrinsic-state wave functions obtained by adding a particle or hole to an axially symmetric doubly cven core (~:,,. --- R2cPK,,, whcre R2 denotes a rotation through 180 ~ about the intrinsic y-axis).

3. Scattering formalism The presence o f single-particle transitions a m o n g the low-lying levels we considcr, does not allow a pure collcctive macroscopic treatment of the scattering. We add therefore to the usual non-sphcrical optical-modcl potential a two-body effective potential between the projcctile and the odd-target nucleon. Then we assume for the interaction potential

v = V x ( r ) + v_,0",

(3)

57z

v.L. orrAvAm ANDL. zurH

The macroscopic term VI(r) is Vl(r ) = - ( V + i W )

! 1+exp[(r-R)/a] -

-4iW o

exp [(r-- R)/~ ] {I + e x p [ ( r - . ~ ) / ~ ] ~

Vs.°. (a" l)~ 2 1 exp[(r-R)/a] + Vcout, ar { 1 + exp [ ( r - R)/a ]} 2

(4)

where R = Ro(l = o(1

(5)

The notations used in (4) and (5) are those of Tamura 18). In our calculations we assume that the deformation parameter fl has the same value for all the rotational bands involved. For the two-body term V2(r, rl) we use the standard form i)

V2(r, ri) = ( Vo + Vl~r " ai)g(lr-ri]),

(6)

where subscript i stands for the odd target nucleon. We use for g ( l r - r i f ) a Gaussian or Yukawa shape. We solve the Schr6dinger equation with the coupled-channel method, which is particularly suitable in our case because we deal both with strongly collective excitations ~8) and with weak excitations for which multiple transitions (via intermediate states) may compete with the direct transitions from the ground state 2). In the coupled equations the potential (4) is expanded in Legendre polynomials P~(cos ,9') up to ;o = 4. The matrix elements of the microscopic potential (6) between two states of the target nucleus belonging to the same intrinsic configuration are put equal to zero. We assume that their contributions are already included in the deformed optical potential 2). 4. Results 4.1. THE 25Mg(p,p') REACTION

We show in fig. 1 the analyses of the 17.5 MeV proton measured angular distributions s) corresponding to the lowcr five levels of ZSMg. These levels are assumed to be members of two rotational bands 6,1o. 15, 16). Namely the levcls at 0. MeV (~-+) and at 1.611 MeV (~v+) belong to the K" = 2s+ band based on Nilsson orbit no. 5, and the 0.584 MeV (½+), 0.976 MeV (3_+), 1.962 MeV (s,÷) levels belong to the K" = ½+ band based on Nilsson orbit no. 9. The results shown in fig. 1 were obtained, by coupling together all the five levels considered, in the three different cases: (i) The Nilsson orbit wave functions, corresponding to the Nilsson deformation parameter r/ = 4, and no band mixing were used.

23Na, 25Mg PROTON SCATTERING

104

~

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573

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"

; 16

•

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-2"

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. , , I , , 0 40

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Oc.m.

101_:

0

~ :___J.~

40

~._...._~. .... , ,_~_..i_~.~

80

120

160

ec.m.

101 _ I

3~E b'd "W°

25Mg

Q:-1.962MeV

10°i ! k.. i l;~.-~4.t=t** ! 16~i_ r t! L 10-2.L;

5/,~ ÷ / --

.. - ".i . ...~ ~ - _ ~ /

E I 1()31 i , , 0

! , , 40

I :, !, 80 120 Oc.m.

=

~, 160

Fig. I. Differential cross sections for scattering o f 17.5 M e V p r o t o n s f r o m 25Mg. (i) Solid lines - results o b t a i n e d w i t h o u t b a n d mixing a n d with the N i l s s o n o r b i t wave f u n c t i o n s (Po --135 MeV, Vl = 0 MeV). (i{) D o t t e d lines - results o b t a i n e d with the N i l s s o n orbit w a v e f u n c t i o n s a n d a b a n d mixing between the t w o K tr = ½+ b a n d based o n Nilsson orbit no. 9 a n d no. ll (Vo = - - I 0 4 MeV, V1 := 0 MeV). (iii) D o t - a n d - d a s h lines - results o b t a i n e d with the wave f u n c t i o n s o f ref. i o) ( Vo = -- 190 MeV, VI = 0 MeV). F o r Q = 0.0 M e V (~+) a n d Q = - 1 . 6 1 1 M e V (3 +) the curves (i), (ii) a n d (iii) nearly coincide. F o r Q . . . . 0.584 (½+) the curves (i) a n d (iii) nearly coincide. T h e e x p e r i m e n t a l p o i n t s are f r o m ref. 5).

574

~'. L o'rIAvAYl AnD I. zvl:H

(ii) The a b o v e - m e n t i o n e d Nilsson orbit wave functions were used, but the b a n d mixing between the two K = = 2~ + b a n d s based on Nilsson orbits no. 9 and no. 11, as suggested for Z'~AI by Litherland et al. 6), was taken into account. (iii) Here the wave functions were the K e l s o n a n d Levinson lo) ones which have no b a n d mixing, but are r a t h e r different from those o f case (i). The p a r a m e t e r values for the m a c r o s c o p i c p o t e n t i a l were: V = 47 MeV, W = 0 MeV, W D = 7.5 MeV, Ws.o. = 3 MeV, a = 0.65 fm, t3 = 0.50 fm, r o = r o

= 1.20 fm, fl = 0.43.

The value o f the d e f o r m a t i o n p a r a m e t e r [~ is in agreement with the m e a s u r e d quad r u p o l e m o m e n t 22). The C o u l o m b potential used was that o f a uniformly c h a r g e d sphere o f radius 1.2()A ~ fm. F o r the m i c r o s c o p i c term wc used a G a u s s i a n form with the range p a r a m e t e r o f 1.85 fm and (i)

Vo = - 1 3 5 MeV,

V l = 0 MeV,

(ii)

V o = - 1 0 4 MeV,

V 1 = 0 MeV,

(iii)

I: o = - 1 9 0 MeV,

V1 = 0 MeV.

H o w e v e r , [Vx[ ~ 41Vol c o u l d be used without a p p r e c i a b l e effects on the a n g u l a r distributions. A further increase in I Vll makes the fits worse. F o r the levels at 0. MeV (s +) and at 1.611 MeV (-~ + ) the three calculations do not give a p p r e c i a b l e differences as is expected for levels o f a p u r e g r o u n d state band. The results for them are shown in tig. 1 for case (i) only. F o r the o t h e r levels a c o m p a r i s o n a m o n g the curves (i), (ii) and (iii) shows that these different choices for the target wave functions have the main effects o f renorrealizing the strength o f the effectivc microscopic interaction. In the a b o v e three calculations thcre is no b a n d mixing between the K" = *z+ g r o u n d state b a n d a n d the K '~ = 2~+ lirst excited band. H o w e v c r a little mixing between these bands, which c a n n o t be excluded ,3. ~,~) may result in a significant decrease in thc rather high strength used for the effective m i c r o s c o p i c interaction. "l"his effect will be seen in the following for the 23Na nucleus. 4.2. THE 23Na(p, p') REACTION The analyses o f the experimental d a t a o f ref. 5) for the first four excited levels a n d for the level at 2.98 MeV o f 23Na are shown in tig. 2. The first three levels at 0. MeV, 0.439 MeV a n d 2.08 MeV are interpreted as the ~+, ~+ a n d ~,+ m e m b e r s o f the g r o u n d state r o t a t i o n a l band based on the K ~ -- 23+ Nilsson orbit no. 7. The level at 2.391 M e V is considcrcd as the lowest m e m b e r o f the K ~" = ½+ band based on the Nilsson orbit no. 9. F o r the levcl at 2.98 M c V there

23N~1, 2 5 M g

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=

; ...... 10-~ _

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80 ec.m.

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F i g . "~ -- Differential cross sections for scattcring o f 17.5 M e V p r o t o n s f r o m 2 3 N a . T h e r e s u l t s shown are obtained ~ith the target wave funct i o n s o f ref. 1o). D a s h e d l i n e s - i n t e r p r e t a t i o n ~+ f o r t h e level a t 2 . 9 8 M e V (Vo -- - - 2 1 M e V , Vl - 7 M e V ) . S o l i d lines - i n t e r p r e t a t i o n .~" f o r t h e l e v e l a t 2 . 9 8 M e V ( Vo ' : -- 33, V~ =-= 0 ) . T h e e x p e r i m e n t a l p o i n t s a r e f r o m ref. s).

576

p.L.

O T T A V A N I A N D L. Z U F F i

is strong enough evidence for the assignment 3+, considering it as the second member of the above mentioned K" = ½+ rotational band. However the assignment ~+, interpreting it as the lowest member of the K ~ = ~+ rotational band based on the Nilsson orbit no. 5, cannot be excluded 6-14). We remark that the results of ref. lo) permit both the interpretations for the 2.98 MeV level. In fig. 2 calculations are shown performed with the target wave functions of ref. 10). The microscopic interaction used throughout for 23Na was, as for 25Mg, a Gaussian with a range parameter of 1.85 fm. The dashed curves were obtained with the interpretation 3 + of the level at 2.98 MeV, and with Vo = - 2 1 McV and V1 = 7 MeV in the microscopic interaction. It was not possible to obtain good tits for both the levels at 2.391 MeV and 2.98 MeV for any value of the ratio [Vt/Vo[. The solid curves correspond to the interpretation 5+ for the level at 2.98 MeV and Vo = --33 MeV and V1 = 0 MeV were used. A positive value of V1 to [VI/Vo[ <~_l.s gives results still acceptable. Calculations obtained by interpreting the level at 2.98 MeV as g)+ and using the Kelson and Levinson ao) orbit wave functions, but neglecting the RPC band mixing, are shown in fig. 3. In the microscopic interaction the values Vo = - 6 0 MeV and V1 -- 0 MeV were used. This calculation shows that the main effect of the band mixing is a renormalization of the strength of the effective microscopic interaction. In all three calculations shown the parameter values of the macroscopic interaction potential were V = 49 MeV, IV = 0 MeV, WD = 6.5 MeV, Vs.o. = 3 MeV, a = 0.65 fm, ~ = 0.50 fro, r o = r o - -

1.20 fro, fl = 0 . 4 8 8 .

The value of the deformation parameter [/is in agreement with the measured quadrupolc moment 2z). The Coulomb potential was the same as in 25Mg calculations. Both for 25Mg and Z3Na we performed calculations with different values for the Gaussian range parameter and with a Yukawa form for the microscopic interaction, without appreciable improvement for the results given above.

5. Discussion and conclusions

The general agreement between the experimental data and the theoretical results shows that the models used for the description of the targets and of the reaction mechanism work well enough in spite of the approximations involved. For z 5 Mg the results are particularly satisfactory. This may be duc to the fact that the simple model of an odd particle plus an even core is suitable for this nucleus. The strength of the effective interaction is however too high compared with that of the force that accounts for the free nucleon-nucleon scattering. This is a general trend

23Na, 25Mg PROTON SCATTERING 104 ,

103~. 23Na

"= " E

+

Q : o.o MeV 3,/~

103;i

= r '=

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Q : -o.,39 MeV

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i:

l/.

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Q : - 2.08 MeV

7//2 '

23Na

2

Q: -2.391 MeV 1 / " ,.'2

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"~' E

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Oc.m.

103:

101

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~ "o'm

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,~

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Fig. 3. Differential crogs sections for s c a t t e r i n g o f 17.5 McV p r o t o n s r r o m -~Na. ' r h c results s h o ~ n arc obtained with the i n t c r p r c t a t i o n ~ + for the level at 2.98 McV and b~,' using the o r b i t ~ a v c f u n c t i o n s o f ref. ~o), but by ncglccting the RI)C b a n d mixing (Vo 60 McV, Vt -- 0 McV). The cxpcrimcnta] points arc f r o m rcf. 5).

578

P.L. OTTAVANIAND L. ZUFFI

when exchange effects are omitted 2 4 - 2 6 ) . Furthermore, such a high value may be due, in part, to ignoring the possible band mixing between the K" = ~ + ground state band and the first excited K~ = 21+ band. For 23Na the results are not equally satisfactory. For this nucleus, in fact, the model of a proton outside a core may be too approximate. The state at 2.391 MeV can also be contributed to by excitations of a neutron ta, z7) from the Nilsson orbit no. 7 to no. 9. Our calculations seem to be more consistent with an interpretation ~+ of the level at 2.98 MeV. However, the involved approximations and principally the neglect of exchange effects do not make our calculations too suitable for level spin assignments. In fact the exchange effects are found to be state dependent 24-26). In our case they might depend in particular on the involved single-particle transitions. So, for each of them, a suitable value of the strength of the microscopic effective interaction would be required. This fact could affect the 23Na results, because here several single-particle transitions are present. Such an argument can in part justify the better fits for 25 Mg, tbr which there is only a singlc-particle transition. The results might be improved also by using a different value of the deformation parameter fl for each rotational band.

References 1) N. K. Glendenning and M. Veneroni, Phys. Rev. 144 (1966) 839 2) N. K. Glendenning, in Proc. Int. School of Physics "Enrico Fermi", Course XL, 1967 (Academic Press, New York, 1969) p. 332; Nucl. Phys. A l l 7 (1968) 49 3) G. R. Satchler, Nucl. Phys. 77 (1966) 481 4) W. G. Love, Nucl. Phys. A127 (1969) 129 and refs. here listed 5) G. M. Crawley and G. T. Garvey, Phys. Rev. 167 (1968) 1070; Phys. Lett. 19 (1965) 228 6) A. E. Litherland et al., Can. J. Phys. 36 (1958) 378 7) A. B. Paul and J. H. Montague, Nucl. Phys. 8 (1958) 61 8) A. B. Clegg and K. J. Foley, Phil. Mag. 7 (1962) 247 9) W. Glockle, Z. Phys. 178 (1964) 53 10) I. Kelson and C. A. Levinson, Phys. Rev. 134B (1964) 269 1 I) J. Dubois, Nucl. Phys. A104 (1967) 657; A l l 6 (1968) 489 12) B. D. Soverby et al., Nucl. Phys. AI21 (1968) 181 13) A. R. Poletti et al., Phys. Rev. 184 (1969) 1130 14) J. R. Priest, Phys. Lett. 27B (1968) 497 15) B. D. Soverby and G. L. McCallum, Nucl. Phys. Al12 (1968) 453 16) B. D. Soverby et al., Nucl. Phys. A135 (1969) 177 17) B. Buck, Phys. Rev. 127 (1962) 940 18) T. Tamura, Rev. Mod. Phys. 37 (1965) 679 19) S. G. Nilsson, Mat. Fys. Medd. Dan. Vid. Selsk. 29, no. 16 (1955) 20) A. K. Kerman, Mat. Fys. Medd. Dan. Vid. Selsk. 30, no. 15 (1956) 21) D. M. Brink and G. R. Satchler, Angular momentum (Oxford University Press, Oxford, 1962) 22) Nucl. Data A5 (1969) 433 23) A. Bottega et al., Nucl. Phys. A136 (1969) 265 24) K. A. Amos, Nucl. Phys. 103 (1967) 657 25) R. SchaelI'er, Nucl. Phys. A135 (1969) 231 26) W. G. Love et al., Phys. Lett. 29B (1969) 478 27) D. Pelte, Phys. Lett. 22 (1966) 448