Finite-range calculation of two-neutron transfer reactions on rare-earth nuclei

I

2.G

[ I

NuclearPhysics A220 (1974) 31--44; (~) North-HollandPublishiny Co., Amsterdam Not to be reproduced by photoprint or microfilmwithout writtenpermissionfrom the publisher

FINITE-RANGE CALCULATION OF TWO-NEUTRON TRANSFER REACTIONS ON RARE-EARTH NUCLEI TADASHI TAKEMASA Department of Physics, Osaka University, Toyonaka, Osaka 560, Japan and Japan Atomic Eneryy Research Institute, Tokai-mura, Naka-yun, lbaraki-ken, Japan Received 9 November 1973 Abstract: Finite-range DWBA calculations have been done for (p, t) reactions on rare-earth nuclei.

The perturbation interaction acting between the proton and each transferred neutron was assumed to have a Gaussian shape. At low incident energy, very good agreement with the zero-range calculation was found both in the shapes of the angular distributions and in the magnitudes of the cross sections for the 2 + and 4 + ground-band members relative to the ground-state transition. At high incident energy, however, the finite-range calculations produced considerable changes in the shapes of the angular distributions. The better agreement with the experimental data indicates the importance of the finite-range interaction at high incident energies. It was also found that the errors introduced by assuming a transferred pair to be in a relative s-state have suiprisingty little effect on the shapes and the magnitudes of the cross sections.

1. Introduction

U p to now, analyses of t w o - n e u t r o n transfer reactions o n rare-earth nuclei by m e a n s of the distorted-wave B o r n a p p r o x i m a t i o n ( D W B A ) have been d o n e by the zero-range a p p r o x i m a t i o n 1, 2). This assumes that the transfer process takes place at the center of mass (c.m.) of the transferred pair. The a s s u m p t i o n allows for red u c t i o n of a six-dimensional integral to a three-dimensional one in the calculation of the D W B A t r a n s i t i o n amplitude. However this simplication has n o t received a n y physical justification. W h e n the reactions involve large m o m e n t u m transfers, large Q-values, a n d high incident energies, the neglect of finite-range effects m a y be u n realistic. Therefore, it is worthwhile to investigate the effects due to the finite range of the n u c l e o n - n u c l e o n interaction. I n the present paper we report the results of exact finite-range D W B A calculations for the (p, t) reaction on even nuclei in the rare-earth region. The interaction potential which acts between the p r o t o n a n d two individual transferred n e u t r o n s is t a k e n into account. This interaction can deal with the relative m o t i o n of the transferred n e u t r o n s in a n essentially exact m a n n e r . This work was partly motivated by the recent report of Ascuitto et al. 3). They suggested that the effects of multi-step processes which are neglected in a D W B A 31

32

T. TAKEMASA

calculation are large for (p, t) cross sections on deformed nuclei 3, 4). However, some differences between the coupled-channel (CC) calculations taking account of the multi-step processes and the experimental data are still found, in particular concerning in 4 + state transition. These may indicate the need to investigate other features of the calculation, such as the finite range of the nuclear interaction. And the limitation of the DWBA theory will only be apparent after the results of exact finite-range calculations are compared with experimental data. Sect. 2 gives an outline of the derivation of the DWBA cross section. Sect. 3 gives numerical details of the present calculations. In sect. 4, the finite-range DWBA calculations are compared with zero-range DWBA calculations and with the experimental data. Finally in sect. 5, the conclusions drawn from the present study are presented. 2. Derivation of the D W B A cross section

The differential cross section for the A(t, p)B reaction is given in DWBA direct reaction theory 5) by (do) _ 1 m'm* kp ~ [TI2' d-~ (,. p) 2(2IA + 1) (2/1:h2)2 k t MaMa,

(2.1)

.uo.ut

/T = J dz f~-)*(k v , rp)(~,.u,,(B)x~.,,(p)lV17%.u..(A)ckt Z½~,(t)>ft~+)(kt, rt),

(2.2)

where m* and m~ are the reduced masses and k t and kp the wave numbers for the relative motion in the initial and final states. The functions ft c+) and f ~ - ) are the distorted wave functions for the elastic scattering of t from A and p from B, respectively, and 7tl,u.,.(A) and ~h,u,,(B) represent the target and residual nuclear states with spin (z-component) IA(MA) and le(MB), respectively. The functions Z½~,(t) and X½up(P) are the triton and proton spin wave functions and t~t is the triton internal wave function. The evaluation of the matrix element that remains in eq. (2.2) depends upon the nuclear model chosen to describe the nuclear wave functions. We closely follow the

A Fig. 1. Coordinate system for the description of the reaction t+ p -+ p ÷ (A q-2).

TWO-NEUTRON TRANSFER

33

methods developed by Glendenning 6) and Lin and Yoshida 7). The coordinate system of the (t, p) reaction appropriate to the evaluation of the transition matrix T is shown in fig. 1. For the triton wave function q5t a Gaussian form is assumed for simplicity: q~t = Nt exp ( - 6 2 Z r~) = ~ooo(362, r)~kooo(462, ¢),

(2.3)

i>j

where rij = r i - r j and ~,tm( V, r) represents a normalized harmonic oscillator wave function with a number n of radial nodes and a size parameter v [ref. s)]. The residual nuclear wave function ~IBM~,(A+2) is expanded into a core wave function ~'ZAMA(A) and a two-neutron wave function ~ala2;JJ~f in the j-j coupling scheme, ~v,~u~(A+2) = E S(IA; al a2, J; In)[~,,,~;j(xl, X2)~PtAuA(A)],BM~,

(2.4)

al~a2

J

where al and a 2 represent the shell model orbits which are characterized by the quantum numbers (nl lljl) and (n212j2). The quantum number J is the total angular momentum of the two transferred neutrons. The expansion coefficient S(I A; a~ a2, J , Ia) contains all the information on nuclear structure and is usually called the spectroscopic amplitude. For convenience, the two-neutron wave function ~ a ~ ; s~t is separated into the orbital and spin parts by transforming it into the L-S coupling scheme. Further, the orbital part of the wave function for the two neutrons is transformed into a wave function of relative motion ~ z ( r ) and one of c.m. motion ~k~E(R) by means of the Talmi-Moshinsky transformation 9). Here E, and I are the orbital angular momentums of the c.m. and relative motions, while ~ and ~ are their respective principal quantum numbers. Substituting kVh,~rB(A+ 2) recoupled in this manner into the matrix element of eq. (2.2) and integrating over the A-nucleon coordinates, we obtain

<~X~MB(A+ 2)X+.p(p)[ V[ kU,AuA(A)qStZ+u,(t)) = E (IAMAJM[IeMB>(SMsLMLIJM> (AA---~2)N+½Z x B(IA; S37L~TL, J; IB)(Z~up(p)Zsu~(a,, a2) x [ ~b~Z \A+2(2Av'R)Ip~(½v,

r)]LML

VlO0o0(4fi 2, r)0ooo(3~52, ~)2½.,(t)>,

(2.5)

where S and L are the spin and orbital angular momenta of the two transferred neutrons and Zsu~(a,, 42) is the spin wave function of the two neutrons. The factor ((A +2)/.4) ~+~z arises from taking into account the change of the general center of mass 7). The coefficient B(IA; SI~E~TL, J; Ia) is defined by B(IA; SI~,~L, J;/a) = X g(a~ , a2)S(lA; a 1 a2, J; IB) al,a2

× <½l~(j~)½12(J2); d[{½(S)l~/2(L); J)(n~ It n2/2; LI~L~7; L),

(2.6a)

34

T. T A K E M A S A

where

42(1 +6(al,

a2~"

(2.6b)

Here the bracket (½11(jO½12(j2); JI½½(S)/112(L); J ) is the transformation coefficient from the j-j coupling scheme to the L-S coupling scheme and (n, l 1n 2/2 ; L[/~ai7; L) is the Talmi-Moshinsky transformation coefficient. The interaction potential responsible for the two-neutron transfer is assumed to be a spin-dependent, central force with a Gaussian form between the proton and two individual neutrons: V = Vr,,,, + Vpn,, (2.7a) Vp, = (P1 Vx+Ps V3) exp [-a2(rp-r.)2],

(2.7b)

where P1 and Ps are singlet and triplet spin projection operators and r p - r . , = ¢ - k r . Then the spin-part integration in eq. (2.5) can be simply evaluated using the standard method of Racah algebra: (Z~,(P)ZsM~(a~, a2)IP~ 1/1 +e3 V3[z~u,(t))

= (½1.tpSMs[½1A>x/2--~I(-)S[¼v~+¼(3-2S(S+I))Vs].

(2.8)

The integration over the relative coordinate r can also be carried out 1o): D~(¢) =

f

r){exp [-- a2(~ + ½r)2] + exp [ - aZ(¢-- ½r)23} x ~000(462, r)~kooo(362, ¢)dr = I~I(¢)Y7";(~), (2.9a)

in which L-c3~-_J~3~v~661~LV(2"l+2fi+l-)!!-]~v~i(-)iexp(-D¢2)2;-÷r~! J " "

r=o ~ ('½v)~ (r~--tc)!

" 1 Cl+~:-S)! 1 la z '~i+2(r--$) xs~=oSI(x-S), (27+2a:-2S+1)' (C)s ~--C ¢) , (2.9b) where

C - ¼v+¼a2+~6 z,

a4

D -- a2+262- - - . 4c

(2.9c)

The DWBA expression for the differential cross section for the (t, p) reaction is finally given by (da)

2IB+l m'm*

kp ~,

(t, p) = 21A "}- 1 ( 2 . h 2 ) 2 k t JLMz, S ×

x/2L+I J

,J

1/1 + 3 - 2 S ( S + I ) 44

vs 2

TWO-NEUTRON TRANSFER

35

Here the form factor Ft, uL(rt, rp) is defined by

NL

x

¢~,7~\A + 2"

L~tL'

(2.11a)

where rt = R+½~,

rp --

A A+2

(2.11b)

R+~.

The sums over/V, E,, 17and 7 in eq. (2.1 la) are restricted according to: 2n~+l 1 + 2 n 2 + l 2 = 2 N + L + 2 ~ + 7 , 11 + / 2 + L , + 7 = even.

(2.12)

In the zero-range approximation the terms ~ in the distorted waves ft(+)(R+½~) and f~-)((A/(A + 2))R + ¢) are dropped before the integration over ¢. One can then integrate D~(~) over ~ and obtain D~(zero) = f o ~ ( ¢ ) d ¢ = [37zvt~4'~÷2~r(7-2])V(2n~ 1)!! (1

4DC+a2 v)~3(', 0). (2.13)

In the zero-range approximation, the effects of the finite range of the nucleon-nucleon interaction and the size of the triton wave function are included in this approximate manner. In the case of the pick-up reaction B(p, t)A, the differential cross section is simply obtained using the relation 11): dtr (d~)(p pt) = )\kp] (kt~22/A+1(d--~)(t ' 2[ B +, I .

(2.14)

3. S o m e n u m e r i c a l details

The spectroscopic amplitudes are calculated by the method developed in ref. 2), where the nuclear wave functions are obtained by projecting states of good I and M [ref. 12)] from the Nilsson and BCS intrinsic state. The difference between the deformation of a target nucleus and that of a residual one is explicitly taken into account. In the calculation of the spectroscopic amplitude, all the levels in the N --- 3, 4 and 5 oscillator shells for protons and all the levels in the N = 4, 5 and 6 oscillator shells for neutrons are taken into account. The Nilsson single-particle states are generated by a deformed oscillator potential well with a quadrupole deformation plus a hexadecapole deformation and are expanded within the same major oscillator shells.

36

T. TAKEMASA

The well-known parameters x and # o f the Nilsson model is) are those given by ref. 14). The oscillator energy unit is taken to be h & o = 4 1 / A ~ MeV.

(3.1)

The strengths of the pairing force are taken over f r o m ref. 14): Gp = 23.5 A -1 MeV,

G n = 18.0A -1 MeV.

(3.2)

The deformation parameters f12 and f14 are taken f r o m the literature ~5,16) and are listed in table 1. TABLE 1

Deformation parameters Nucleus

15aGd

1SSGd

16OGd

16ZDy

164Dy

f12

0•32

0.33

0•34

0•33

0.34

f14

0.040

0.030

0.032

0.010

0.008

The size parameter 62 o f the triton a n d the oscillator parameter v are reasonably well establised by electron-scattering experiments 17). We shall take 62 = 0.06 fm -2 and v = 0.98 A -~ fm -2. Unfortunately the value o f the parameter a -1, the range o f the effective nucleon-nucleon potential, is less well determined. We shall choose a 2 = 0.3 fm -2 [ref. 7)]. This gives g o o d fits to the nucleon-nucleon effective range and the scattering length for the free interaction is, 19). The transferred n e u t r o n pair is assumed to have S = 0 in the (t, p) and (p, t) reactions. The effect o f the transferred pair with S = 1 is very small in general 20). The effective strength l,reff • = ~V1 1 + ~3V 3 which determined the absolute cross section is chosen to be the standard value - 32.5 MeV [ref. 7)]. The f o r m factor FLu,_(rt, rp) is constructed according to eq. (2.1 l a) except that the oscillator function ~k~£ is replaced by the function ff~Z that is matched outside the TABLE2 Optical-model parameters used in the DWBA calculations V

W

rr

ar

rl

al

rc

protons I tritons I

55.6 168.8

14.5 12.6

1.25 1.16

0.72 0.752

1.25 1.498

0.47 0.817

1.1267 1•1267

protons II tritons II

45.0 170.0

15.5 19.0

1.25 1.15

0.65 0.74

1.25 1.52

0.46 0.76

1.25 1.40

Ref.

3) z) 22) 23)

The symbol I refers to the X6°Gd(p, t)lSSGd reaction and II to the 15SGd(p,t)lS6Gd and 164Dy(p, t)t62Dy reactions• The derivative-type absorption is assumed for the imaginary part of the potential for the protons, while the volume one is assumed for the tritons• All energies are given in MeV, all lengths in fro.

TWO-NEUTRON

TRANSFER

37

nuclear radius onto a spherical Hankel function corresponding to the two-neutron separation energy 6). The differential cross sections are calculated using the finiterange DWBA code DWBA-4 2z), where spin-orbit coupling terms in the optical potentials of the entrance and exit channels are neglected. The calculations are carried out for the (p, t) reactions induced by 18 or 51.9 MeV protons on several even nuclei in the rare-earth region. The optical-model parameters used in the DWBA calculations are taken from the literature 3, 22, 23) and are tabulated in table 2. 4. Results and discussions

First the contributions of the relative angular momentum of a transferred pair 7 ~ 0 to the cross sections have been investigated. Since we assume the S = 0 transfer and focus our attention on the transitions from the g.s. to the members of the g.s. rotational band in the rare-earth even nuclei, the summations over E and 7 terms in eq. (2.11a) are restricted to even values only. The restriction 7 = even comes from the requirement that for identical nucleons the spatial part must be symmetric when S = 0, and the restriction ~ = even results from conservation of parity. In fig. 2, we have plotted 1~i(~) (eq. (2.9b)) as a function of ¢ for several of the smallest ~ and 7 values. It is seen that the values of 1~(~) decrease very rapidly as 7 becomes large. Thus the effect of the 7 ~ 0 terms on the cross section is expected to be small, since the factor I~7(~) determines the magnitudes of the form factor. io °

i

i

,

i

160Gd G

i

/

i

,

i

,

J

( ~',t )tSaGd

16I

"~

,('~,II

'E

t6 z v

',g

t63 J6' o

5D (fro)

0

5.0 (fro)

Fig. 2. T h e b e h a v i o r o f I~7(~e) as a f u n c t i o n o f ~ f o r the leOGd(p ' t) 15 SG d reaction.

In fig. 3, the differential cross sections of the 0+, 2+, and 4 + transitions on the 160Gd (p, t) 15SGd reaction with Ep = 18 MeV are shown, where the individual differential cross sections from the 1 = 0 and 7 ~ 0 components are plotted. The total cross sections are given by the coherent sums of the two amplitudes. As shown in fig. 3, the contributions from the I ~ 0 terms, in which all allowed values of 7 ~ 0 are taken into account, to the cross sections are surprisingly small and give rise to, on average,

38

T. T A K E M A S A

I000

i

i

i

i

i

i

I000

i

I

I

I

16°Gd (p,t)15eGd

I

I

I

I

i

t6°Gd(p,t)lSaGd

Ep=18 MeV

Ep=18 MeV I00

~100 A

.D

I0

o v

x I000

0.1

O•I

50 60 ec.m. (deg.)

I

I

I

0

90

f

I

I

50 60 e c r n . (deg.)

I

I

90

Fig. 3b. Same as fig. 3a, for the 2 + transition.

Fig. 3a. Finite-range D W B A cross sections for the 0 + transition• The individual contributions f r o m the *1= 0 term and all the allowed ~ :~ 0 terms are shown• I0 O0

'6°Gd(p,t)'58Gd Ep=I8MeV 4+

IOO

.d

~.0

I0

~o f

0, l

I

x 200

I

0

I

I

I

I

I

30 60 ec.m• (deg.)

I

I

90

Fig. 3c. Same as fig. 3a, for the 4 + transition•

an increase of about 3 %, an increase of about 1%, and a reduction of about 1 % for the 0 +, 2 +, and 4 + state cross sections, respectively. These results indicate that the finite-range calculation is dominated by the transfer of a neutron pair in a relative s-state. This is mainly due to a Gaussian-shape assumption for the triton internal wave function. In a (p, t) reaction, a neutron pair bound to the core is transferred

TWO-NEUTRON TRANSFER

io"

I

I

I

i

I

i

39

.

le°Gd(pH)lSeGd. Ep= 18 MeV 10 3

B-FR ---ZR

1°2 1

I

0~

g

• •

2 +

I 01

to°l i i i r i i ~ 0

30 60 ec.rn. (de g.)

90

Fig. 4. Comparison of the ~6°Gd(p, t)tSSGd data 24) with the DWBA calculations. The solid and dashed curves are the results of the FR and ZR calculations, respectively. The FR and ZR curves are separately normalized to each transition.

into components of the triton wave function, in which the relative motion of the neutron pair is a 0s state. Thus the overlap integral between the neutron-pair wave functions in the target nucleus and in the triton has the maximum value when fi = 7 = 0, and this determines the magnitudes of the cross section. In addition, the quantum ~r of the c.m. wave function of the transferred pair has the maximum value when the relative state is 0s. The form factor with the larger N value is predominant in determining both the shape and the magnitude of the cross section. Thus, the contribution from terms other than 7 = 0 is small. In a (t, p) reaction, the above qualitative discussion is also valid. The relative s-state approximation saves computing time by a factor of about 2, 5, and 8 for the 0 +, 2 +, and 4 + transitions, respectively. Thus this approximation is very useful for a systematic study of the (t, p) and (p, t) reactions based on the finite-range DWBA theory. In fig. 4, the DWBA results for the 160Gd(p ' t)t 58Gd reaction with Ep = 18 MeV are compared with the experimental data 24). The solid and dashed curves correspond to the finite-range (FR) and zero-range (ZR) DWBA calculations, respectively. The zero-range DWBA calculations have been carried out using the same optical-model

40

T. T A K E M A S A

p a r a m e t e r s as in t h e f i n i t e - r a n g e c a l c u l a t i o n s . T h e finite- a n d z e r o - r a n g e c u r v e s a r e s e p a r a t e l y n o r m a l i z e d t o give t h e best o v e r a l l fits t o t h e e x p e r i m e n t a l d a t a f o r e a c h

tr(O)exp./a(O)rR

transition. The normalization factors N(FR) = of the finite-range c u r v e s a r e 0.68, 5.29, a n d 5.82 f o r t h e 0 +, 2 +, a n d 4 + t r a n s i t i o n s , r e s p e c t i v e l y . T h e corresponding normalization factors N(ZR)

o f t h e z e r o - r a n g e c u r v e s are 0.83, 2.42,

a n d 5.59. T h e r e l a t i v e m a g n i t u d e s o f t h e cross s e c t i o n s b e t w e e n t h e finite- a n d z e r o r a n g e c a l c u l a t i o n s are s u m m a r i z e d in t a b l e s 3 a n d 4, a n d t h e s e will be d i s c u s s e d b e l o w in detail. It is seen f r o m fig. 4 t h a t t h e finite- a n d z e r o - r a n g e c u r v e s a g r e e well w i t h t h e s h a p e s o f t h e a n g u l a r d i s t r i b u t i o n s f o r t h e 0 +, 2 ÷, a n d 4 + t r a n s i t i o n s . T h e f i n i t e - r a n g e TABLE 3 Absolute values of summed cross sections for the g.s. transitions DWBA results

~cr(0)e~. a)

Residual nucleus

(~b)

zero-range ~tr(0) (J~b)

finite-range E~r(0) (/~b)

15 SGd

3972 b)

4766

5815

156Gd

816 c)

1514

696

162Dy

510 ¢)

1127

518

a) Xtr(0) is the sum of the differential cross sections for 5 ° steps in the range 50-80 ° for 16°Gd(p, t)lSSGd and for 5 ° steps in the range 50-45 ° for 15SGd(p, t)lS6Gd and 164Dy(p, t)~62Dy. b) Ref. 24). ¢) Ref. 2s). TABLE 4 Relative summed cross sections for the 2 +, 4 +, and 6 + ground-band members to the g.s. transition DWBAresults Residual nucleus

Fr

Etr(0)c,p a)

zero-range ~.(r(0)

1SaGd

0+ 2+ 4+

100 b) 26.4 6.6

100 9.05 0.98

100 7.34 0.78

156Gd

0+ 2+ 4+ 6+

100 c) 35.9 6.9 9.8

100 12.0 1.65 0.87

100 12.4 2.33 1.56

162Dy

0+ 2+ 4+ 6+

100 c) 34.3 8.4

100 10.6 3.82 0.56

100 11.2 6.32 0.85

a-c) See notes below table 3.

finite-range ~tr(0)

TWO-NEUTRON TRANSFER

41

fit to the experimentally observed angular distribution for the 0 + transition is excellent and the zero-range fit is also very good. However, the DWBA calculations for the 2 + transition in both cases disagree significantly with the experimental data. The finiterange calculation does not offer any significant improvement for this state. The CC calculation for this state 3) shows a good agreement with the experimental angular distribution. The result of the finite-range DWBA calculation indicates the limitation of the single-step DWBA theory for this state. The envelopes of the DWBA curves for the 4 + transition agree fairly well with the e~xperimental data, but it is clear from fig. 4 that a detailed fit to the shape of the angular distribution is lacking. On the other hand, the CC calculation for the 4 + state does not improve the agreement with experiment 3). These results may indicate the need of a more complicated explanation for this state, such as a realistic interaction potential and a more accurate form factor, etc. Recently, Bayman et al. reported exact finite-range calculations for the 4°Ca (t,p)42Ca and 4SCa(t,p)S°Ca reactions with E t = 10.1 MeV and 11.97 MeV, respectively z o). They used a realistic triton wave function and an realistic interaction instead of a Gaussian-shape assumption for a triton wave function and an interaction potential. They found that the zero-range curves agreed not only with the shapes of the finite-range angular distributions for various L values, but also with their relative cross sections. Our results for the (p, t) reactions on deformed nuclei are consistent with the results of Bayman et al. for the (t, p) reactions on non-deformed nuclei. This fact suggests that the finite-range interaction has very little effect on the shape of the angular distribution in (t, p) and (p, t) reactions at low incident energies on other nuclei. In order to investigate the energy dependence of the finite-range effects, the finiterange calculations were carried out for the 158Gd(p ' t)l 56Gd and 164Dy(p ' t)162Dy reactions at 51.9 MeV leading to the 0 +, 2 +, 4 +, and 6 + members of the g.s. rotational band. The calculations were done using the s-state approximation (7 = 0). As mentioned above, the errors caused by this approximation are negligible. The results are shown in fig. 5 and compared with the zero-range DWBA results and also with the experimental data 25). For the 1 SSGd(p ' t)156Gd reaction, the normalization factors N(FR) (N(ZR)) of the finite-range (zero-range) curves are 1.17 (0.54), 3.39 (1.61), 3.46 (2.26), and 7.35 (6.08) for the 0 +, 2 +, 4 +, and 6 + transitions, respectively. The corresponding factors N(FR) (N(ZR)) for the 164Dy(p, t)162Dy reaction are 0.98 (0.45), 3.00 (1.46), and 1.30 (0.99) for the 0 +, 2 +, and 4 + transitions. Unfortunately the experimental data for the ~64Dy(p, t)~62Dy reaction are not available for the 6 + transition to compare with the calculated results. As can be seen from fig. 5, there are considerable differences in the angular shapes between the finite- and zero-range calculations at the high incident energy. This result shows a definite distinction from the result at low incident energy. The finiterange calculations for the 0 + transitions markedly improve the agreement with the experimentally observed data, particularly at forward angles. The poorest DWBA

42

T.

TAKEMASA 10~

"2

0

30 60 Oc.rn" (deg.)

90

0

30 60 Oc.rn. (deg.)

90

Fig. 5. Comparison of the 15aGd(p,t)156Gd and 164Dy(p,t)t62Dy data 2s) with the DWBA calculations. The solid and dashed curves are the results of the F R and ZR calculations, respectively. The F R and ZR curves are separately normalized to each transition.

fit obtained in this study is to the 2 + transition. The finite-range calculations offer no significant improvement. In particular, at very forward angles the calculated cross sections decrease, whereas the experimental cross sections tend to increase. A similar result was also obtained in our previous study 2 6 ) o f the l~4Sm(p, t)lS2Sm reaction at 51.9 MeV. These results indicate that the 2 + transition cannot be described by a single-step pick-up mechanism alone. The same conclusion has also been drawn for low incident energy. For the 4 + transitions, the finite-range calculation fits the data better than the zero-range one for the 15SGd(p,t)156Gd reaction. For the 164Dy(p, t) 162Dy reaction, however, both calculations are in disagreement with the measured cross section. Improved agreement for this 4 + transition probably requires the inclusion of the multi-step processes. The angular shape of the 6 + transition for the 158Gd(p,t)156Gd reaction is reasonably well accounted for by the finite-range calculation. A summary of the absolute values of the summed cross sections predicted by the DWBA calculations and the experimental summed cross sections for the g.s. transi-

TWO-NEUTRON TRANSFER

43

tions is given in table 3. The sums ~,oa(O) are taken for 5 ° steps in the range 50-80 ° for 16°Gd(p,t)lSSGd and for 5 ° steps in the range 50-45 ° for 15SGd(p, t) 156Gd and a64Dy(p, t)16ZDy. The absolute values of the summed cross sections predicted by the zero-range DWBA are always larger than those of the experiments. As with the shapes of the angular distributions, there are also some differences between the DWBA results at low incident energy and at high incident energy. At low incident energy, the finite-range result is larger than the zero-range one by a factor of about 1.2, while at high incident energy, the finite-range results are smaller than the zerorange ones by a factor of about 2.2. And the finite-range results at high incident energy give good agreement with the experimental ones. Our results concerning the dependence of the absolute values of the g.s. 0 + cross section upon the incident energy agree with the result of L i n e t al. 27). They found that the finite-range effects reduce the absolute cross section of the g.s. 0 + state on the 56Fe(p, t)S4Fe reaction with Ep = 51.9 MeV. Finally, table 4 presents the ratios of the summed cross section for the 2 +, 4 +, and 6 + ground-band members relative to a value of 100 for each of the g.s. transitions. The finite- and zero-range calculations give almost similar results except for the 6 + transition in the 15SGd(p ' t)l S6Gd reaction and for the 4 + transition in the 164Dy (p, t) 16ZDy reaction. In these cases, the finite-range results are larger than the zerorange ones by a factor of about 1.8. In other cases, however, the differences between the finite- and zero-range results are within about 40 %. The theoretical ratios are always much smaller than the observed ones at both the low and high incident energies. We reach the conclusion that the finite-range calculations give no significant improvement in the relative magnitudes of the cross sections. 5. Summary and conclusions The finite-range DWBA calculations have been carried out for the (p, t) reactions on several even nuclei in the rare-earth region. It is found that the 7 = 2, 4 . . . . states of the relative motion in a transferred pair contribute negligibly both to the shapes of the angular distributions and to the magnitudes of the cross sections. At low incident energy, the finite-range calculations give no appreciable differences from the zerorange ones in either the angular shapes or the relative magnitudes of the cross sections. At high incident energy, however, the finite-range calculations produce considerable changes in the shapes of angular distributions, but have little effect on the magnitudes of the cross sections for the 2 ÷, 4 +, and 6 + transitions relative to the g.s. transition. Since finite-range effects are expected to bc largest in reactions in which the momentum change of the projectile is largest 28), the conclusions obtained from the present work are not surprising. The absolute summed cross sections of the g.s. transitions predicted by the finiterange DWBA are in good agreement with the experimental data for the reactions at high incident energy. The 2 ÷, 4 +, and 6 + transition strengths relative to the g.s.

44

T. TAKEMASA

t r a n s i t i o n are m u c h smaller t h a n the experimental data at b o t h low a n d high i n c i d e n t energies. P e r h a p s this discrepancy is due to the a s s u m p t i o n that two n e u t r o n s are picked u p f r o m the h a r m o n i c oscillator orbits 3). The fact t h a t the fits to the 2 + a n g u l a r distributions are n o t i m p r o v e d at all by i n t r o d u c t i o n of the finite range of the n u c l e o n - n u c l e o n i n t e r a c t i o n indicates the limi t a t i o n o f the single-step D W B A theory. H o w e v e r the i m p r o v e m e n t s o f the fits of the a n g u l a r distributions to the experimental data, except for the 2 + transition, a n d of the absolute values o f the g.s. 0 ÷ cross section give a n i n d i c a t i o n of the i m p o r t a n c e o f finite-range effects at high incident energies. The a u t h o r would like to t h a n k Dr. H. Y o s h i d a for permission to use his D W B A code D W B A - 4 . He is also grateful to Drs. K. H a r a d a , M. Sano a n d H. Y o s h i d a for their careful reading of the m a n u s c r i p t a n d to Dr. E. Takekoshi for her interest a n d encouragement. References 1) R. A. Broglia, C. Riedel and T. Udagawa, Nucl. Phys. A135 (1969) 561 2) T. Takemasa, M. Sakagami and M. Sano, Phys. Rev. Lett. 29 (1972) 133, and to be published 3) R. J. Ascuitto, N. K. Glendenning and B. Sorensen, Nucl. Phys. A183 (1972) 60; R. J. Ascuitto and B. Sorensen, Nucl. Phys. A190 (1972) 297, 309 4) T. Tamura et al., Phys. Rev. Lett. 25 (1970); 26 (1971) 156 5) N. Austern, Direct nuclear reactions (Wiley-lnterscience, New York, 1970) 6) N. K. Glendenning, Phys. Rev. 137 (1965) B102 7) C. L. Lin and S. Yoshida, Prog. Theor. Phys. 32 (1964) 885; C. L. Lin, Prog. Theor. Phys. 36 (1966) 251 8) A. de-Shalit and I. Talmi, Nuclear shell theory (Academic Press, New York, 1963) 9) I, Talmi, Helv. Phys. Acta 25 (1952) 185; M. Moshinsky, Nucl. Phys. 13 (1959) 104 10) A. Y. Abul-Magd and M. EI-Nadi, Nucl. Phys. 77 (1966) 182 11) R. H. Bassel, R. M. Drisko and G. R. Satchler, Oak Ridge National Laboratory report ORNL3240 (1962), unpublished 12) R. E. Peierls and J. Yoccoz, Proc. Phys. Soc. A70 (1957) 381 13) S. G. Nilsson, Mat. Fys. Medd. Dan. Vid. Selsk. 29 no. 16 (1955) 14) O. Prior, F. Boehm and S. B. Nilsson, Nucl. Phys. All0 (1968) 257 15) D. L. Hendrie et aL, Phys. Lett. 26B (1968) 129; A. A. Aponick, Jr., et al., Nucl. Phys. A157 (1970) 367 16) K. A. Erb et aL, Phys. Rev. Lett. 29 (1972) 1010 17) H. Collard et al., Phys. Rev. Lett. 11 (1963) 132 18) J. M. Blatt and V. F. Weisskopf, Theoretical nuclear physics (Wiley, New York, 1952) p. 55 19) S. A. Afzal and A. Ali, Nucl. Phys. A157 (1970) 363 20) B. F. Bayman, Nucl. Phys. A168 (1971) 1; B. F. Bayman and D. H. Fleng, Nucl. Phys. A205 (1973) 513 21) H. Yoshida, DWBA-code DWBA-4, unpublished 22) F. G. Perey, Phys. Rev. 131 (1963) 745 23) J. B. Ball et al., Phys. Rev. 177 (1969) 1699 24) D. G. Fleming et aL, Phys. Rev. C8 (1973) 806 25) Y. Ishizaki, Proc. Phys. Soc. Jap. 26 (1971) 147; Y. Sugiyama et aL, J. Phys. Soc. Jap. 30 (1971) 602 26) T. Takemasa and H. Yoshida, Phys. Lett. 46B (1973) 313 27) C. L. Lin, S. Yamaji and H. Yoshida, Nucl. Phys. A209 (1973) 135 28) R. M. Drisko and G. R. Satchler, Phys. Lett. 9 (1964) 342