IL NUOVO CIMENTO

VoL. 28A, N. 1

1 Luglio 1975

The 4°Ar(d, p)~Ar Reaction at Deuteron Energies below the Coulomb Barrier (*). S. CAVALLARO, G. COI~LEO, 2k. CUNSOLO, A. FOTI, S. GIAMBRUNO, M. ~DI TORO and G. I:)APPALAlCDO Istit~tto di Fisica dell'Universitd - Calania Istituto .Yazionale di Fisica Nucleate - Sezione di Catania Centro Siciliano di .Fisica 2¢¢tcleare e di Slruttura della Materia - Catania

(ricevuto il 24 Febbraio 1975)

- - Spectroscopic factors are obtained from stripping reactions at energies below the Coulomb barrier. Compound-nucleus effects are taken into account. The dependence of spectroscopic factors on the energy of the incident particle, on the nuclear optical parameters and on the geometrical parameters of the captured-neutron potential well is investigated in detail.

Summary.

1. -

Introduction.

I t is well k n o w n (~-3) t h a t (d, p) reaction dat~ obtained at deuteron ~nd p r o t o n energies below the Coulomb barrier can provide, in the f r a m e w o r k

(*) To speed up publication, the authors of this paper have agreed to not receive the proofs for correction. (1) L . J . B . GOLDFARB:Nuel. Phys., 72, 537 (1965); K. A. T~R MAlZTmOSYA~: Zurn. Eksp. Teor. Fiz., 2, 620 (1956); R. H. L~M~R: N~tel. Phys., 39, 680 (1962); L. J. B. GOLDFARB and K . K . WONG: Nucl. Phys., 90A, 361 (1967); F. P. GIBSON and A. K. Kv.R~A~: Phys. Rev., 145, 758 (1966); W. D. BARFIELD, B. M. BACONand L. C. BI~D~,NHARN: Phys. Rev., 125, 964 (1962). (2) J. RAPAPOI~Tand A. K. K~MAN: N~el. Phys., 119 A, 641 (1968); L. J. B. GOLDFA~S, J. A. GO~ZAL~S, M. POSN~I~ and K. W. JON]~S: preprint, University of Manchester (1971). (3) W. R. ]-h~RI~G and M. DOST: Nucl. Phys., l l l A , 561 (1968). 12

Trm 4°Ar(d,p)4~Ar REAC't'*O.~AT DEu'r~nOX EN]~RGIES BELOW :ETC.

13

of D W B A , precise spectroscopic information on the single-particle c o m p o n e n t of the nuclear w a v e functions for the following reasons: i) the stripping m e c h a n i s m being mainly due to the k n o w n Coulomb interaction, the relative spectroscopic factors 5' b e c o m e i n d e p e n d e n t of the nuclear optical potential p~r~meters; ii) the compound-nucleus contribution to the expected to be negligible a t such a low energy.

reaction m e c h a n i s m

is

I n the present work we test the a b o v e a p p r o x i m a t i o n s for the 4°Ar(d, p)4~Ar reaction p e r f o r m e d at E d = (1.5--2.5) MeV. Due to the Q value the energies of some p r o t o n groups leading to the low-lying levels of 41Ar do not satisfy the r e q u i r e m e n t of being lower t h a n the Coulomb barrier, and so we p e r f o r m e d a preliminary s t u d y of the reaction m e c h a n i s m t h r o u g h the analysis of the e x p e r i m e n t a l d a t a on angular distributions and excitation functions. W e investigated also the dependence of the spectroscopic factors S on the nuclear optical p a r a m e t e r s and on the geometrical p a r a m e t e r s (ro~ and %) of the c a p t u r e d - n e u t r o n potential well (~).

2. - T h e e x p e r i m e n t .

The deuterons were accelerated b y the 2.6 MeV H V E C Van de Graaff of the CSFI~/SM (*), Catania. The t a r g e t was m a d e up b y a differentially p u m p e d scattering c h a m b e r (4) filled with Ar gas ~t a pressure ranging from 4.0 to 5.0 m m Hg. The outgoing particles were detected with surface barrier silicon detectors. The overall incident-energy resolution was a b o u t 20 keV at 0 = 10°; the angular resolution was ~ 5 °. Figure ] shows the energy s p e c t r u m of the emitted particles. W e measured : i) Differential excitation functions of the elastically scattered deuterons a t E d ranging f r o m 1.5 to 2.3 MeV, in 20 kcV steps and for 0~b = 100 °, 120 °, 140 ° , 160 ° . The overall error in the absolute value of the elastic cross-section is estimated to be ~ 6 % . Figure 2 shows a comparison of the elastic-scattering crosssections with the R u t h e r f o r d calculations; the r.m.s, deviation found is ~ 4 ~o.

(*) Centro Sicili~no di Fisica Nuclcarc c di 8t,ruttur~ della Materia. (4) S. CAVALLARO,A. CUNSOLO, R. •OT131NZA and A. RUBmNO: report INFN/TC-70/16 (1970).

14

s.

CAVALLA-RO, G. C O R L E O , A. C U N S O L O , A. F O T I , S. G I A ~ B R U N O ,

~TC.

p,

de[ x 100

1000

800

-~° 600

~

r o

5

t

4o0

2o0

o

i

3O0

200

lOO

channel, numben

Fig. 1. - Energy spectrum of the emitted particles. The positions of the arrows correspond to the quoted energy values (ref. (~)); 4°Ar(d, p)alAr, Ed=2.36MeV , 01~b=140 °.

E z ~I0 LU

10 ~

1.4

I

,

I

,

I

1.8 Eci(MeV)

,

I

,

2.2

Fig. 2. - Excitation functions for the deuteron elastic scattering by 4OAr. The solid lines represent the results of a pure Coulomb-scattering calculation. (5) E. KASHY, A. 1~{. HOOGENBOOI~Iand W. W. BUECHNER: Phys. Rev., 124, 1917 (1961).

THE

4°Ar(d,p)a~Ar

REACTION

AT

DEUTERON

ENERGIES

BELOW

15

ETC.

ii) Differenti~fl excitation functions of t h e ~°Ar(d, p)4~Ar reaction, at E d r a n g i n g f r o m 1.5 to 2.5 MeV, in 50 keV steps, und for 0~ b ----100 °, 120 °, 140 °, 160 ° . iii) A n g u l a r distributions of sever~l groups of p r o t o n s (see Table I) e m i t t e d in t h e a b o v e - m e n t i o n e d r e a c t i o n at E d ----1.96 MeV, in t h e a n g u l a r r a n g e 0~b - - (15--170) °, in 10 ° steps. T h e ~bsolute value of t h e (d, p) cross-section has been o b t a i n e d b y n o r m a l i z a t i o n w i t h t h e R u t h e r f o r d elastic scattering. TABLE I. -- Spectroscopic ]actors extracted Item DWBA calculations with various sets o] optical parameters. The S values have been obtained by fitting the (da/dY2),~p with the D W B A calculations without taking into account the CN contribution. The 1mcertainty in the S values is less than 10~, and it is duc mainly to the experimental errors. Group

E~ (a)

[ (b)

J~: (b)

(MeV)

,~ FP

MP

PC

FB

£oo

0

3

~-

0.43

0.47

0.44

0.54

P2

0.52

1

~-

0.14

0.16

0.15

0.17

2

~3+

0.03

0.03

0.03

0.04

~3

1.04

P4

1.35

1

~-

0.39

0.44

0.44

0.45

-P6

1.87

0

!+2

0.053

0.054

0.049

0.055

/a s

2.40

1

1

0.20

0.22

0.22

0.22

D e u t e r o n p a r a m e t e r s : (F) FITZ el al. (*); (M) MELKANOFF et al. (~); (C) COSACK et al. (s). P r o t o n p a r a m e t e r s : (P) PEREY (~); (B) BECCttETTI (lo). N e u t r o n well p a r a m e t e r s : roa = 1.25 f~l, a = 0.65 fro, Va.o. = 27.5 MeV. (a) R e f . (~.a). (b) R e f . (').

P r e l i m i n a r y d~t~ on the present investigation h~ve been published in ref. (11).

3. -

The

reaction

mechanism.

T h e resulting angulax distributions a n d differential excitation functions are s h o w n in Fig. 3, 4; for p r o t o n groups c o r r e s p o n d i n g to higher excited levels, t y p i c a l C o u l o m b stripping p a t t e r n s are displayed. (6) W. FITZ, R. JAHR and R. SANTO: Nucl. Phys., l 1 4 A , 392 (1968). (7) M . A . MELKANOFF, T. SAWADAan4 N. CINDRO: Phys. Zett., 2, 98 (1962). (s) M. COSACK, M. K. LEUG, M. T. MCELLISTREM, R. L. SttULTE, M. M. STAUTBERG, J. L. WEIL and J. C. NORMAN: Nucl. Phys., 136A, 532 (1969). (9) F. C. PEREY: Phys. ~ev., 131, 745 (1963). (lo) F. D . BECCH:ETTIjr.: M. S. Thesis, University of Minnesota (1968); F. D. BECCIIETTI and G. W . GREENLESS: Phys. Rev., 182, 1190 (1969). (11) G. CORLEO, A. CUNSOLO, A. FOTI, M. DI TORO and G. PAPPALARDO: Lett. Nuovo Cimento, 1, 169 (1971).

16

8. G I A 1 K B R U N O , :ETC.

s . C A V A L L K R O , G. CORL:EO, A. C U N S O L O , A.. F O T I ,

For a preliminary investigation of the sensitivity of the c o m p u t e d stripping cross-sections to the parameters we performed zero-range D W B A calculations (') using different sets of optical parameters, and we extracted normalization constants S through the relation

(3.])

o.D = ] . 5 8 ~ 2J, + :t ~

~

j.~"

Table I shows the results.

10-1

,

I

'

I

'

J

'

10-2

i

~

i

i

1"

---" . . . . . . . . . . . .

- ....

~c X~ I

,10-1

'L_ xlO 10-20

I

I

i

9O

I

I

i--

180 0

-- [ I I

-- [

I

I

Ill

I

-- ~ i

90 ec.m

I

IiI--"

180 0

I

i

I

•90

I

i

180

Fig. 3. - Angular distributions of the proton groups for the *°Ar(d, p)*lAr reaction. The solid lines represent the IIF q-DWBA results, the dashed lines the HF contribution. Only representative error bars are shown. E a = 1.96 MeV.

We observed that the deduced values of S are rather insensitive to changes in the deuteron optical parameters; however, they depend on the protoa parameters until the proton energy exceeds the Coulomb barrier, allowing in t h a t case a mean variation of S of about 15 %. Figure 5 shows the dependence of S on the geometrical parameters (ron and an) of the captured-neutron well. The observed strong dependence (see also ref. (2))

(*) We used the code J U L I E (i~). (12) R. H. :BASSEL, :R. M. DI~ISKO and G. R. SATCIILER: ORNL-report 3240 (1962) and supplemcnt.

mile S°Ar(d, p)41Ar REACTION AT DEUTERON ENERGIES BELOW EG~C.

P2

10-~ P3

//~60 °

10-'~

~4~r

4

102+

!

++~/¢v'lq~

+/iooo

r

17

/+

'

,~¢~b {t

i/(F/"

4101103-

~ Z , J

z/~.-j4/''

~ 10-2

[ 4/

--

~~,.+-yc,---

{.."

7/ _ / ' ~ ~t 4~o'1o-' / + +.+++,.+~,,t p ,./.-~o: J~o-: t ' -~o',o ~L + . . # ; " I Soo ~ ++~oo I I/ .,/----1

lo-~m +

L / I /

,^-2L4

, ~+-,¢'] +

, ~

.~

bl

,i4

10 ~L'

I

r .~ , 7 /

. . . . .

_i___

~i-!3q

.~40oo'

~

,o ~-'

1°7' i

I

1.5

.

I

t

i

I

I

2.0

i

, ~-,--'~,o

71°1°I<

i l

10-311

1.5

,

~

i

,

,

~

2.0 E~.(MeV)

_,

iF

,v~6()°

,

L~

,

I

2.5

/

,o ~- . d

10<1i

I

1.5

o

.,/f] J

o~, " "

"if'a°

1°I+ i

-

.~10

7 + ,].:,.~-,[

+" ! #

2.5

t

I . , . ,l

10 10 L /

o_,r +/,ooo

++

I

J10-z

l O ° ~

9

. _ , . + - , - + ' 1 'o ,o [+~,"

,f/=

101/l''l

]

l/l ,~:

r )~'~ 140 ° o.~ ~[, g j ' '

]

°

--

1

,

m~,'

. ~I_

.

~--

10-~/

/ (( . . . .

#

,

"~

/J//'lO0

'3"~/

1I 10 3/f7" ......

4; 't~ +# 2F

+L,..tT~/'"

I

+. ~ P6

!

.~

!- +t I

JlUll0 3L

, ....

} lO-'~

i

10-2 , . . 10° t Pz,

~

" "~'~

410 i10-2 t4'. 't . . . .

./'+1ooo

/

_,-4

/

/

.)..,~'

I

/

I

/,~ '1," ~ + +

i

I

_o

-~,u

t

1

100°

'

i

2.0

I

I

I

i

II

2.5

Fig. 4. - Excitation functions of the proton groups for the a°Ar(d, p)alAr reaction. The solid lines show ~he H F + D W B A results. I n ~he P3 transition the strong CN contribution is explicitly indicated (dashed line). 2 -

ll

Nuovo

Cimento

A.

1~

S. CAVALLARO, G. CORL:EO~ A. CUNSOLO~ )r. FOTI~ S. GIAI~IBRUNO, ~ T C .

gives rise to the problem of an absolute determination of S. As expected~ t h e shapes of the angular distributions are insensitive in our case to ro= and a, variations. F o r the following discussion and analysis we choose the F B set (see 1.00 ,5 an=0.5 0.80

0.60

0.40

0.20 I 1.00

I i 1.10

I , 1.20

I 1.30

~r,(fm)

Fig. 5. - Dependence of the spectroscopic factor on the geometrical parameters of the oaptured-neutron well for ~he P4 proton group.

f o o t n o t e of Table I) of optical parameters t h a t give the best fit to the experimental angular distributions~ and employed~ r a t h e r arbitrarily, the values to. = 1.25 fm and a= = 0.65 fro. To estimate the Clq contribution to the cross-sections we used the wellk n o w n H a u s e r - ~ e s h b a c h espression (13)

(3.2)

1 2

(--1) ~'-'

~.~,(0) = ¥~5.~.~ (2~ + ~)(2i ÷

1)]~(1JiJ;

sZ). J

J

•Z(l'Jl'J; s'L)

2~(eos0). a

F o r a discussion of the symbols see ref. (14). The level density p a r a m e t e r s used in the c o m p u t a t i o n are those suggested in ref. (14). E q u a t i o n (3.2) gives the energy-averaged statistical CN cross-section in its absolute value and it is valid when the statistical mechanism is the only one present in all channels; however in some cases eq. (3.2) can give an esti-

(13) It. F~SHBACrT: Nuclear Spectroscopy, Part B, edited by F. AJZ~B:ERG-S~LOY3~ (New York, N.Y., 1960), 13. 1033. (la) ]~. CxADIOLI,I. IORI and L. ZETT~,: Phys. Bey., 174, 1140 (1968).

TIIE

~°Ar(d,p)~Ar

ItEACTION

AT

DEUTERON

ENF, RGI_'ES B E L O ~ V

19

:ETC.

m a t e of t h e statistical c o n t r i b u t i o n to t h e e n e r g y - a v e r a g e d cross-section (~*,~) also if direct processes ure present, as in o u r case. T h e H a u s e r - F e s h b a c h calculations r e p o r t e d in Fig. 3 (dashed lines) i n d i c a t e t h a t t h e CN m e c h a n i s m is negligible for m o s t t r a n s i t i o n s e x c e p t for t h e p r o t o n g r o u p s p~ a n d ps. U n d e r t h e a p p r o x i m u t i o n t h a t direct a b s o r p t i o n is negligible, we a d d e d inc o h e r e n t l y t h e H F a n d D W B A cross-sections, o b t a i n i n g t h e theoretical curves (solid lines) of Fig. 3. T h e spectroscopic factors o b t a i n e d f r o m

are listed in Table I I . I t is e v i d e n t t h a t t h e r e is a large u n c e r t a i n t y in t h e S values for t h e less intense transitions, due to t h e a s s u m e d 3 0 % a m b i g u i t y (~G) in t h e e v a l u a t i o n of t h e statistical c o n t r i b u t i o n . TABLE I I . - Spectroscopic factors obtained from a~gular distributions and excitation functions. The S values have bcen obtained by fitting the (( direct ~>part of the crosssection ((da/d~)~,p--(da/dt'2)~F)with the DWBA calculations. The quoted uncertainties are due to the experimental errors and to the assumed 30% ambiguity in the t I F evaluation. Group

S[a(0)] (a) S[a(0)] (1.96 (1.96 MeV) McV)

S[100 °]

S[120 °]

S[140 °]

Pc

0.54

0.47 ~: =[:0.03

0.46 :[: ±0.02

0.44 ~ ±0.02

T2

0.17

0.128=]= -]=0.006

0.087=]= 0.102~ -j=0.005 ~=0.006

Pa

0.04

0.010:j: -~0.005

p~

0.45

P6 Ps

S[160 °]

S

S[a(0)] (b) (11.6 MeV)

0.45 ~: 0.47 ~: _-4=0.02 :~0.03

0.46

0.49

0.115:J= :[=0.007

0.101 ~= =~0.007

0.101

0.10

0.007~ ~:0.004

0.008± 0.013-[: =]=0.004 :L0.005

0.023 :t: ±0.006

0.013

0.03

0.46 -~ -j:O.O1

0.55 ~ ~0.01

0.47 :~ ~0.01

0.47 :~ ±0.01

0.44 ~: ~_0.01

0.48

0.49

0.055

0.044~: ~0.004

0.038~: ~0.004

0.036 :L ~:0.004

0.035-~ ~:0.003

0.037~ :J 0.004

0.037

0.06

0.22

0.215-]= ±0.009

0.216:L 0.211 :~ :J:O.O09 ±0.008

0.217~: ~0.008

0.208~: ±0.009

0.213

0.18

(a) x,Vithout t a k i n g i n t o a c c o u n t CN c o n t r i b u t i o n s . (b) R e s u l t s o b t a i n e d r e a n a l y s i n g t h e e x p e r i m e n t a l d a t a of ref. (s) u s i n g our set of p a r a m e t e r s .

(15) T. EI~ICSO~;: Ann. of Phys., 23, 390 (1963). (16) G. CORLEO, Cx. I)AP]?ALAI~DOand A. RCBBINO: Nuovo Cimento, 7 0 A , 323 (1970).

2D

S. CAVALLARO, G. COI~LEO, A. CUNSOLO, A. FOTI, S. GIAMBRUNO, ETC.

Finally, we tested the behaviour of S vs. the incident energy E~; the experimental data obtained b y F~Tz et al. (~) at Ed----11.6 MeV were analysed using our set of parameters. A t this energy the Clq contribution t u r n e d out to be negligible ( ~ 1 ~b/sr). The extracted S values are listed in Table I L I f we exclude the d a t a for the p~ transition (for which we have not f o u n d a n y simple interpretation), the spectroscopic factors obtained at Ea----1.96 1V[eV agree reasonably well with those at E a = 11.6 MeV.

4. -

Conclusions.

W e can conclude the present paper b y pointing out the following: i) The ~°Ar(d, p)~lAr reaction at E d ~ 1.9 MeV is a good tool for determining spectroscopic factors as far as relative values are concerned. A strong dependence of S on the geometrical parameters of the c a p t u r e d - n e u t r o n well has been found. ii) W h e n the Clq statistical contribution is taken into ~ecount a n d the d a t a coherently analysed, no significant energy dependence of S is observed.

•

RIASSUNT0

Si sono misurate le sezioni d'ur~o differenziali delle reazioni a°Ar(d,d)4°Ar e 40Ar(d,p)*lAr nell'intervallo energetieo AEa= (1.5--2.5)1V[eV. In queste eondizioni si sono ottenuti i fattori spe~troseopiei per energie dei deutoni inferiori nile barriere eoulombiane, tenendo eonto anehe degli effetti di nueleo eomposto. Si g s~udiata dettaglia~amen~e la dipendenza dei fat~ori spe~roseopiei sin dall'energia delle partieellc ineiden~i ehe dai parametri o~iei dei eanali dl ingresso e, d'usei~a e dai paramc~ri geometrici dell~ buea di potenziale del neutrone eat~urato.

PeaKuua 4°Ar(d, p)4L&r Ha ~efiwpHU npx 3Heprmqx nH~e Kya0noncKoro 6apbepa.

PeamMe (*). - - Ortpe)xeaa~orcg crteKTpOcrormnecrrm ~barrop~i n3 pearmL~ cTprtmmHra IIpr~ artepr~x Hmre KyJIOHOBCKOFO6apbepa. YqHTIalBa/OTC~ 3qbqbeKT/aIKOMIIayH~-~l~ep. 1-Io~po6Ho rtcc~e~yeTca 3aBH¢I~dVlOCTBctIeKTpOCrOr~qeCKrtX d~aKTOpOB OT 3neprrlrt rta~amme~t qaCT~t,I, OT ~epHBLX OIITHqeCKtlX~apaMeTOB HOT reoMeTpyqecKnx HapaMeTpoB IIOTeHuHaHbHO~ flMbI 3axBaqennoro He~TpoHa.

(*) Ilepeee3eno pe3amluef~.