Transfer reactions on 58Ni and 56Fe induced by polarized protons of 25 MeV

Transfer reactions on 58Ni and 56Fe induced by polarized

protons of 25 MeV

Citation for published version (APA):

Polane, J. H. (1981). Transfer reactions on 58Ni and 56Fe induced by polarized protons of 25 MeV. Technische

Hogeschool Eindhoven. https://doi.org/10.6100/IR157606

DOI:

10.6100/IR157606

Document status and date:

Published: 01/01/1981

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TRANSFER REACTIONS ON

58Ni

AND

56

Fe

INDUCED BY

POLARIZED PROTONS OF 25 MeV

PROEFSCHRIFT

TER VERKRI.JGING VAN D~ GRAAD VAN DOCrOI'< IN DE TeCHNISCHE WET~NSCHAPPEN AAN

0"

TECHNI~CHE HOGE5CHOOl EOINOHOVEN, OP GEZAG VAN DE RECTOR MAGNIFICU5, PROF, II'<. J. ERKELENS, VOOI'< EEN COMMISSIE AANGEWEZEN DOOR HET COllEGE; VAN DEKANEN IN HET oPEN6AAR TE VEROEDIGEN OP

VRIJDAG 5 JUNI 1981 IE 16.00 UUR

0001'<

JOHANNES HENDRIKUS POLANE

GE90l'-:EN TE AMSTEFlOAM

DIT PROF,!,,,CHRIl"T IS GOEDGllKllURl) DOOR DE PR(lMO'l'(!Rb:N

1?R01",DH, O.J, POPPflMA ,:N

1'110,' . DH, B, ,l, VJlRlil\AR

CONTENTS

CHAPTEI\ I~TRODUCTION

CHAPTEH 2 EXPEl',~MBNTAL PROCEDURE AND DATA REDUCTION 2.1 Beam p,od~ction and scattering chamber 2.2 The telescope system

2.3 Tha elect,onic equipment References

CHAi>'l'ER 3 SOME ASPECTS OF THE THEOR¥ OF TRANSFER REAC'nONS 3.1 DWBA theory

3.2 One-neutron trangier 3.3 TW"O-neutron transfer 3,4 The optical-model potenti~l

3,5 Finite-range and non-locality correction~ 3,6 Adiabatic deuteron potenti~l

Ref~ren~es

CHAPTER 4 THE (p,d) ReACT"QN ON SSNi AT 24.6 MeV 4.1 Introd~ction 4.2 Tha experiment 8 11 13 13 16 19 20 20 21 23 25 25 26 4.3 The reaction model and the calculational procedures 27

4.3.1 Reaction model 27

4.3.2 Th.e neutron fOJ::'m factor 29

4.3.3 Optical potentials 31

4.4 DiScussion 33

4.4.1 The pa,ticl,e-states 3/2 (0.00) , $/2 (O.77J

and 1/2 (1. 11) 33

4.4.2 The 7/2 states 36

4.4.:) The optical deuteron-potential 39

4.4.4 The 3/2 states 42

4.4.5 1'he states at 3.71 and 5.78 MeV 45

4.4.6 'l'he 1"0 and 1=2 states 46

4.4.7 SUl1} rules 17

4.5 Conclusions 49

Reference~ 51

CHAPTER 5 TB:E (p,d) + REl!.CTrON ON 56Fe AT 24.6 MeV

So

5.1 Introduction 53

5 .. j The calculations 55

[,. :3. 1 l1.e;oction moclel 55

:'. :1. 2 Optical-model and other DWBA parameters 57

5.4 Discus~ion

,.4.1 The negar.ive-parity states

s.4.2 The posit.ive-parity states

:;.4.J Sum-rule results and compc:t.r$,son with

5eN~ (p, d)

5.4.4 The deuteron potencial 5.5 cone h\~ion"

RCference$

CHAPTER" TlH; (p,r.) REAl":TION ON 58 Ni A\'ID 5~ve r-:r 24.6 MeV 6.1 Tnt)::oduction

6.2 Experiment~l procedures 6.3 Ca.1Cu.liOlr.ion<>l procedures

1.,.3.1 i<eElction m<O'ch"nisms

G.J.2 simultaD<O'OIJS r.wo-neute-on transfer 6.3.J SequentiEll transfer 6.3.4 Inelastic sc"ttering 6.1.5 Optical potentials 6.3.6 Normalization parameters rJ.4 Discussion 6.4.1 Detormination of A and W

6.4.2 Tho:

2t

al~d 2~ .t- stat_es of

6.1.

:1

Il~he _Oz + _{stOlte of} 5~'Fe 6.4.4 DWBA analysis

6.5 Conclusions

References

o'Fe

CHAPTER 1 THE (p,~) REACTlO~ ON 58 Ni AT 24.6 MeV

·1. 1 In trOde<Ci;ion ,SUMMAR"; NIIWOO!<D !,IWI':Nf;T,OOP 7.2 Re~ction model 7.3 Disc'L1.ssion "1.4 condu~ion~ Heferenc8s 59 59 64 65 67 68 70 71 71 "12 73

n

74 76 78 79 flO 81 81 88 92 93 96 97 99 99 99 101 lU4 105 107 1(J~ 111 112

CHAPTER 1 INTRODUCTION

Nuclea~ t.an~fer-r~actions are char~cterized by the transfer of

a small number of nucleons (protons or n~utrons) from one atomic

nucleus to another. In this thesis we present the results of various types of tr~nEfer reactions induced by bomb~rding foils consisting of S6Ni and 56pe nuclei, respectively, with a beam of polarized protons having an ene~9Y of 24.6 MeV.

The ta~get foas are w thin that the p;(ob"bil.l,ty of meeting mOre than one nucleus is very small for the protons. ~onsequently most of the p~otons go through the target foil without feeling ~ny nuclear interaction at all. Of the protons that do encounter a nucleus, most are sCattered (in)elastically and leave the target as

a proton ~gain~ In our experimetlts we selected those ~~~ct~ons, in

""hich the proton "changes" into a deuteron, a t);"iton 0;( an alpha particle by picking up from the target n~cleu~ ~ neutron, a pair of neutrons Or two neutrons plUE one proton, res.,ect.ively. The firll11 nucleus is often left in an excited state by such reactions.

Counting the number of deuterons, t~itons and alpha particles em1tted in a certain oirection and mSdsuring thei~ energy yields

so-called energy-spectra. These 5pectra reveal peaks (ife+ the energ~es

are cantered around discrete value~), Which corr~spond to the ene);"gy levels of the final nuclei. Th~ number of counts in these peaks

represent (after proper ~ormalization} the various diffarential cross

sections. In fact, because the protons are polari2~d, we have two ditf.e;(entia' cross sections fa. e~ch level: one for polarization up and one for polarization down. It is convenient to );"epl~ce these two cross sections by two quantities which are Lndependent of the d~gree of pol~rization P. These quantities are the unpo!arized CrOSs section do/o~unpOl and the analy~~ng power A given by

do (6) /dllunpol = 1/2 (do/dilt + do/dllt)

A(B) -_p 1 x (do/dllt - do/dOf) (do/dnt + d%ll+)

(1.1 )

(1.2)

where B is the angle bet""ecn the emitted particle and the proton beam. Relation (I.~) is only v~11d for p~rticle~ emitted in the ho~izont~l plane and to the left of the proton beam direction. To be more precise

A(O), dC1(O)/,:m\.lnPOl an') th", di£"fe,ential cross section in the dtrectiol1 (O,t) aye related by

., +

do(O,tJ!dn ~ (I + P'n A(O)} dC(8)/dOunpol +

(1.3)

wht;!-re n ~,s t.he uni t ve~to;r; co~:t"esponding to the vector product of t.he

1.nJ,tiaJ. ano fInal Ught.-part.tcle momentum (B,,~el convention).

S1)ch a stlJ8y o[ nl)clea"(, reactions yields information On the

reaction mechanisms and on the internal structure of the nucl~l

in-volv~d. Both aspects are investigated ext~nsivoly in the present ~ork. FOr irlstomce Orle" and two-neutron tnlrlsfer rc"ctions prob", the ~ingle­

neutron "nd n0utroJI-pairir1g properties, respectively, of the jnitial and tin,tl. nuclei. cono0rning the reaction mechan.l.sm of e. g. two-n<'!ut.ron transfer re<letions one may investigate the relative importance

of two $ucce.ssive transfers of one neutron and th~ si.multaneous

transfer of " neutron pair. The polarization of the protons orten prov~s to be crucial in answering questions about both ~uclear strUGture and reaction mechanisms.

The choicu of 5BNi il.r'ld 56 pe as target nuclei was xnotivated by

two fucos. First.ly, 5C Ni and 56Fe Itre close to the doubly-magic nucleu.~ ooNi. whi.ch means that their internOl.l st);ucture is rel<J.tivcly

simple within the conte-xt of the :::;hell ... moilel .. Secondly, we expectOd similar.ities o,f the neutron ... tr,ans[er. data since 58 Ni and S6 F0 oontatn

the same nl,lmber of neutrons.

The experimental set up is described in thG following ohapter, while we give an outli.ne of the theory in chapter :J. 'l'he experimental r~sults are presonted and discussed in the remaining ch"pters. The

CH~PTER 2 EXPERIMENTAL P~OCEDURE ~ND DATA REDUCTION

The experiments described in th~~ the~1s have been induced by means of a polarized proton beam acceler~ted to 24.6 MeV by the AVF cyolotron of the Eindhoven University of Technology 1). In section 2.1 we give an outline of the production of this polarized proton beam and we describe the arrangement of the ~catt~ring chamber, where the nuclear reactions take place. In section 2.2 we discuss how the outqoinq p~rt1cles are identified as protons, deuterons, tritons and alpha particles by the telescope device. Finally we ~ketch the main featur8s of the electronic circuitry. More det~ils on the experimental set UP and the spectrum analysis can be fOund in the refs. 2 and 3.

2.1 Beam production and scattering chamber

The polarized protons were ~upplieo ~y a source Of the atomic

b~am

type 4,5). which has been developed by van der Heide 6). The intensity, enerqy and degree of polarization of the proton beam were 2 to 4 ~A, 5 kev ano about 80~, respectively. This proton beam was injected r~dially into the center Of the cyclotron with a trochoidal injection system, which is a oopy of the one used

~n

saclay 7). The Lorent~ force dcting on the protons during injection was balanced by an electrostatioal force generated by a system of electrOdes. The shape ot the electrical field was such that in combination with the magnetical field a strong focussing was achieved resulting in a transmission efficiency of about 70~.

only a small fraction of the injected cu~rent is actually accelerated by the cyclotron. During our experiment the proton~ gained an energy of about 80 keV per. ~evolution, 50 the tot~l number of revolutions was about 300. The intensity Of the eKt~acteo beam was 10 to 20 nA with an energy ~pread of 60 to 90 keV.

After extraction the beam was tr~nsported to the scattering chamber by a system compQsed of five bending magnet~, twenty quadru-poles and five steering magnets 2,8) oovering a distance of 40 m. In order. to qet as muOh intensity as possible on the target we used the doubly achromatic mooe of the beam-guiding system, which means conservation of the energy profile of the bssm during transport.

We wGre able t.o focus t.he proton beam in a spot of les" than t;wo mm dtamet;er at t.he tarCJet. Th;, direction of the beam was ched:ed with probes at. the center of th" Far"day cup, which was situated twe and a

half mete.rs after the target..

The outgOitlg protons r deut.e::rons I tri teIls and alpha J?arti~le5 were

detected by an array Of four telescopes TI-T4 (fig. 2.1; see also section 2.2). Two di fferent configurations hav<l becn \lS"d: in the ~eNi exper1ment the te)."$~opes we);e 10° apart, while in the 56)'e experiment thb; waS 60• The angu.1.ar ,,"cceptance was in both cases about 1°.

Two Su.J.COI1 det-<'c\;o£S (Dl, D2) in th<l V<lrtical plane through th<l bei:lHl "t con5tan~ sc.,.t.Cering angles of 4S ilnd -45 degrees, re!jpectively.

IlIea..:;Ured. p.:lastj,cally sc.:a.tter.-ed protons in order t.o prov,i(le a relative

normalization of t.he va1:"io\ls runs and a clock signal for reve>:5ing the polartzation di",->ction. The vertical plane w"" ChoSen because the """tt<:ring in tilc,t plane is independent

ot

the d8gr<l<l of polarization

(the polarization ",-xi::; of the proton beam being vertical tOOl cf.

for.mula 1.3).

The deqree::: of pol.:lriL;<.:...tion of t.he. ~rQton beam was monitored

"ontitlUously in " secood smaller scattering chamber down"tream (fig. 2.1) i,l the following way: fit"st the beam en<lrgy was degraded to a mc"n energy of 16 MeV »y an aluminiUm foil; next thc number of protons

t=!1.aDt.ic311y scat.ter,'eti by the carbon in a thin polyct,hyl,ene foil was m"a~\\red hy t.wo ~iliccon c'letectm:s (03, 04) pl"""d in the ho:dzontal

plane at. 52.5 and -52 .. 5 degrees. Since the unalysing PQwer of ~2C is

...

/

I''''I~I ./ T'.'J'.q

"

"

'I ~

:>(]hematic, draun:ng oj' the 8c:att;$t'-[.1I.(J dli:ll1iber and polar'~:­ ;~r:.rt7~on mor'!.",?:

tor_

ratner energy-independent at tni~ ang"e, one ~an determine in tnis way a reliable value for the polarization degree of the be.!!.!".

2.2 The telescope sYijtem

In fig. 2.2 we give a cross~se~tional view of the detector block used in the S6Fe experiment. Bach of the four telescopes consists of two silicon surfaOe-barrier detectors: a 200 urn ~E-detector (Philips) and a 2 mm E-deteotor (O~TEC). Permanent magnets in front of the

teles~opes sweep away the se~ondary electrons arriving from the target. During the 5BNi experiment w~ did not yet have this detector blOck at Our disposal and we had to mount each tel~~cope on ~ separate holder. Both experiments were performed using the same deteetors. The telescopes were normalized to each other by comparing meaSUrements done at the same angle.

Toe particl~5 are ident~fied by their specific energy loss dE/dx in the ~E-det~ctor, which ~5 apprOximated by the semiempirical rBlation 9)

Fig. :1.2

,-

-\

_,

\ \ \ \

,

I \ _ ....

-_.---_

.... --,

...

-~

ITS\

\\\S] I tS\\\\\33l ~ I ~f--6.E ~~~~~

/

I

'\, \ 1-

\

:m

J: }/-'"''''

I \ I \ \ .... -

...

_---

...

Cross-sectional view

Of

,the

four>-td"'",::op$

d$t$ctor'

P.ll

whe,t'e M .. Z and .E!: are ths ffiu.SS n.umber I cha:r:ge number and energy of the

particles, rcspcutivcly. The constants c and 0.73 dep8nd upon the

c\el;e(;~or mat_eei"l (",j licon) and the energy rarlge (5 to 100 MeV) _ lFe:.: SlU,con J,$ c = (L045 if 1> j,g in MeV and" in wm.

In order to dete~mine che p~rticle type irrespective of the ene,gy a qu"ntity PIC) ()t.a,ticle .!:.dentifier _~utput) is cal.cul,ated according Co "_he algo):itl1m of GO\lloing et a1. 9)

(2.2) where liB and ~ are the energies deposited by the particles in the

IIll-and

:e ....

detector, respectively. Assuming thEl.t the pttrticle is $topped

in the it-detector and using eq. (2.1) one co.n derive

PlO

=

1.73 C MO• 73 Z2 d (2.3)

where d is t:he thickness of the bE-detector. The relatiOn (2.3) .hows

that PTO is ener~y-indep~ndent and allows to discriminate b~twC0n the

VO-riOllS p"r.ticle types. 1\ct\J"lly the quantity pro d"fined by (2.2)

n

o

C 10~ Z ---l CJ)

"I

I

1"i:7, ;.'. :_1 6 el.protl)n~

, I

\.i

,.

!

I .... L '(> \

\

I

.!

,

!.

CHANNELS

M(U,H' H[.I(-)(:ir'Wn ob/:ained at;

e

tab !)l!;- and (I ,': rr~n g-dRteetOl'.

---1

tr:!~~~ 1 \

/

\

i

• • 1 1

i6

_o

----f!---sio---'-gives rise to " so-called mass spectrum (fig. 2.3), wh<:!r<:! th<:! p<:!aks <Ire centered "round the values given by the relation (2.3). 'rhe width of the peaks is caused by several etfects, of wh~ch electronic noise, inhomogenities in th<:! thickness of the ~~-detector and straggling in th<:!

6E-detecto~

appear to be the most important ones 10). We Will discuss th<:! straggling effect in SOme d@tail because it ~ets an intrinsic lower limit on the peak widths. Since we only want to make here a rough estimate of the importance of the 3t~aggling effect we will not use the

Vav~lov-theory

11) but we will assume a Gaussidn energy-distribution 12) of the straggled

part~cle5,

where the standard deviation c is given (in units of keV) by

a=4.3Z,fct (2.4)

with Z the charge number of the part1eles ~nd with d the thie~ess of the (silicon) ~E-detector in ~m, The change ~PIO indue eo

oy

the deviation ~ is appro~imately given oy

I~PIOI = 1.73 a ~D.73

combining (2.3), (2.4) and (2.5) we find

1 8PIO I PIO (2.5) (2.6)

where 0 is given in ~m and E in MeV. ~or inStanC0, eq. (2.6) gives the Values 0.07 (protons), 0.03 (deuterons), 0.02 (tritons) and 0.01 (~­ particles) for the ratio Of IIPIO and P,O, when d = 200 vm

ana

the energy of the proton oeam is 2S MeV, In order to obtain the FWHM

(full-wLdth-half-maximum) one has to multiply the right-hand-side of eq. (2.6) with 2.35, A comparison with the experimental mass-spectra showed that in our case 50 to 70~ of the widths could be uttributed to straggling. we also tested the relation (2.6) with maSs-spectra obtained with different thicknesses of the liB-detector (table 2.1)

and found a semiquantitative agree1l'"aent..

From (2.6) it is clear that the resolution of the mass spectrum gets better when the thickness d increases. On the other hand particles that are stopped in the liE-detector are not analysed because the particle type cannot be determined in that case. $0 the liE-detector must not be too thic~ in order to allow the detection of low-energy deuterons and tritons. To meet th& two c.iteria m&ntioned aoove we selected 200 ~m as the appropriate thickness Cf the 6~~detectors in

Table ~.1

CompariEon of calc~lated and experimental ~elative

peak-widths in the pl(J spectrum

thickness particle type

of lIE-det

<l")

p t a

lOa

um

calc. 1.38 1.00 0.69 0.38 expo 1.11 1. 00 0.54 0.49 ~OO VIn cal". 1.44 1. 00 0.675 0.32

expo 1. 43 1. 00 0.50 0.29

500 \lm CCllc. 1.14 1. 00

expo 1.46 1. 00 a)

the relative peak~width of the (leuteron peak.

Wi:tS u5C':ld to normalize the other );'"",ult~

11~he proton peak th"(eatens to aw~n'\P the deuteron pe~k especially

c:tt forwa.rd angles because Q,f the large number of elastically scattered

prt:ltOn:=:. Th~rE fore ... since we were not primarily interested in proton

$C,~1;t<?-r.i.ng - 1'02 Ghift.,d the elct<,tic purt of th" proton pea};, to lower valu<!s by choo~itlq al'\ E-detsct6r which WilS too thirt to stop eliJ.sti"ally scattet'ed prot.ons but. thick enough to stop all the other pilrticles. Orte can easily check 1;ha~ the expression (2.2) yields lower vllim,,, for PIO when (,,,,.l,culated for partiel"" that o.re not stopped by the telescope.

In order to avcid broadening of the mass peaks by unequal Ilmpli~

fic"L.ion of the AE- and E-signa~s we ad~ ... stGd ttl" relative ampli fication by C'ompar.inC'l the signals i.nduced by the 5,8 Mev alpha partiel"s emittOld by 24 4Cm .

2 .. 1 Th.e electronic equipment

Ttle main components of the eleC1;rOnj,C cLtcu!,

cry

used tor eaeh Of the telescope" aL'e d;i,spl"yed. irt fig. 2.4. Both the /lE- and the E-signals are fed into a pulse~BhapinlJ circui1; and a fast-~og;i.c cireui,t. The fast discriminator produee~ " lQgiciOIl puls<?- whenever the input

~;.Ignal exceed. a p"eBet l€!vel ( .. bo'lt 0.1 mV). AfI:-<?-r conversion frOm

E tel. bits E gate part. bits PA preampUj"ier (CANBERRA 970D)

FA fast amp~ifi",r

(LJi'S

~;S5L)

FIA, tiLter amp~ifier (l'HE)

~'tJ j'aat di{]criminator (LRS 621AL) LA level ,dqpt$~' (DH8 688r;)

RU routing Im/t ('l'HL')

MA main ampUj"iel' (OH1't;C l8b)

LGS, lin.ear gat.;; stywtoh(ty, (OR1'EC .J1!'.) Pl par tide id$rlUfiiJr- ('('HE)

DU disar-iwinator uni

t

(THE)

PL partiele logic (THf;)

MI mixer (ORTE'C 16:5A)

energy Di~S

NJM to '!'ltL logic this signal is pas.sen on to the routing unit. When-eve:r- the 't'C\ltin9' unit l':"eceives simultaneously (;t"esolving time 150 ns)

" LlE- and E-signal the gates of the stretchers (LGS) <lre opened. Th~

two telescope bits identify the telesoope in that O<lSO. Then tho

E-and 6E .. outputs of. the st.,retche:rs are fed into a.n analogue particle

identifier developed by Sllliters et al. 13). This particle identifier

g€fl.0ri:ltes a PIO (particle identifier outputf see far.Ol1,J,l~ 2~2' and a tot<ll energy signal E+6E. The mas~ peaks of the )?IO spectrum are separated by (01)r dls<'rilftJ,nat.or levels a, b, c and d in t.he

discr.iminator. unil,. whenever an event in one Of the windOws Of this

unit is 1:"ecQroea, t,he gate Of t.ho ADC is oponed and two bit£:; are:

qene"ated to la.bel t.he part.iclo type.

The IIDC bits t.ogether wah t.he telescope bits, tho p"rtide bits

and the spin bit are sent via <tn ADC controll"r 11nd " CAMAC system to

"- MOS-memory (18 k, 24 b.l ~~). IIfteI' each run the content.s of t.he

MO,S-memory are writ.ten "1".0 floppy diSkS by a PDPl1 computer. FOr details

on t.l\e v<lrious contl:ol emits and the data acquisition We refer to the ):'e£s. 3 and 14.

RE:fJ::RENCES

N~F. Verster. H.L. HageaoQ~n, J. Zwanenburg, A.~+~+ ~~~nken and

J. Ceel, NUcl.lnstr. 113,19 (1%2) 88.

2 J.P.M.G. Mel5sen, the$i~, Eindhoven Unive~5ity of Technology, 1971;1.

S.b. Wassenaar, thesis (to be published), Eindhoven University of T"-<;hnology, 1981.

4 C. Clausnitzer, ~. F~e~sehmann and H. Schopper, Z.Phys. 144 (1956) 336.

5 H.F. Glavish, Froe. 3rd Symp. on Polarization Phenomena in Nuclear Reactions, Madison 1970 (eds. H.H. Barschall and W. Haeberli, univ. of Wisconsin Press, Madison, Wisc., 1971) p.267.

6 J.a. van der Heide, thesis, Eindhoven University of Technology, 1972.

R. Beurtey and J.M. Durand, Nucl.lnstr. 57 (1967) 313.

8 B.L. Hagedoorn, J.W. BrOer and F. schutte, Nucl.rnstr. 86 (1970) ~53.

9 F.B. Goulding, O.A. Landis,

J.

Cerny and R,H. Pehl, Nucl.lnstr. 31 (1964) 1.

10 F.S. Gouldtng, Annual Rev.Nucl.So. 25 (1975) 167. 11 P.V. Vavilov. JETP 5 (1957) 749.

12 N. Bohr, Mat.Fys.Medd.Dan.Vid.Selsk. 18,8 (1948). 13 J.E. Sluiters and 5.5. Klein. NuCl.Instr. 120 (1974) 305, 14 A.J. de Raaf, Nucl.lnstr. 163 (1979) 313.

Nu~~~ar transfer reactions are generally gra~ing collisions

between nuclei, that is the nuclei do not interpenetrate much and the

effective inter~ction is mOrC Or less localized ~t the nuclsar

sur

face'3. 'rhe relative weakness Or this "surf<,cG-interaction" allows the usc of the so-called distorted-wave Born-approximation (DWBA). In

DWBA the 5urface-irtteraction is treated in fi~~t Drde~ only, whe~eas

the elastic scattering before and after this interaction is de~cribed to all OrderS by di~torted waves.

In this chapter we will give a small r8vi~w of DWBA-theory in

order to prOvide bOrne bilCXground to the following chapters_ More

details can be found in e. g. th@ textbook of AU9tern 1). ·the general features of the DW6A are treated in s@ction 3.1, while the applications to one- and two-neutron tran",fer are presented in the sections 3.2 and 3.3, respectively. For details of the t;r:iton t~an~fer fOrm<ll.ism, which

i~ applied to the analysis of the 58Ni(p,~) ;r:ea~tion (~hap~er 7), We refer to the thesis of Smits 2). Finally, the

opti~al-mooel pot~ntials,

which generat~ the distorted waves, are discussed in the sections 3.4,

3.5 and 3.6.

3 . 1 DWBA theory

~o describe the reaction A+a + B+b

we

~ntroduce the Hamiltonian

H g _(3.1)

whe,e HI'.' Ha' HB and Hb represent the internal motion of the target nucleus A, the projectile a, eh@ reSidual nucleu~ B and the ejectile b. The relative motion 1s governed by the kinetic energy T« (Te) and th0 potential Va (V~).

The state vector ~(E) satisfies H~ ~ ~o/ and has in our mOdel tne following form

(3.2)

where ~e and ~c are the internal wave functions of the heavy (Cl and light (c) component of the system and Consequently are eigenfunctions of He and Ho' The f~ction Xy is the disto,ted wave for the reaction

cnan)1el y and ~ep~es"nt6 tho rel<'ttive "lotion. 'rhe sum in (3.2)

qenerally is limited to a small. n~~lobey of reaction channel~T w})ile the

r'eITkt.ird nCJ p:coce:'):;',.e:=: aYe hidden in thG imilgir'l~rY U~bs~-..,(ption) pa.rt of

the optical··model put-unt.ials mentioned below.

It . . 1.8 j 1"1. f.=!iC':t the ?I:=:ymptotic be:h!lviO\.lr (l~rqe ::;.~par~ticn between

C and c) of the functions Xy that is rneasur<;:d by t.he det.ector". 'l'he l . .I:::ansition 6lITlp1.1t-w~e 'r cd~ for' the r:-eaction (A,a) .. ). (B,b) io ext.xacted from

X~+)

(r·e-J....W), wher.'e t.:ht;>.: ,. indicates lIoutqoing" boundary condit.ions. All ttl" distOrt .. d wiiv,,''; in the cxpansio)1 0_2) "-re of the x(+) type except for Xu (elastio cho.'lTIr)Ol), which ttlso cont.ains incomin..-; wav~s

(the proton beam fo1'" instance) ..

NOW, 511pposinq that <11'B'~blljlc:~c> = 0fY (which is only r.igor.ou~ly

tru~ for iII~lastic ~outtering) a)1d starting from

(3.3)

where we hd.ve replacE!(J, ';~JB~bl V R I$BWb ';:0 by the opti.c;;3.1 potential

Us

(r s) and where the matrix tll"rntlnt on the right.-hand s:i,d" J." defined by

Loosely ~'peakir"lq w"':'~ Ci..uJ eay that. Vy -tly represents the Ilsur

fact·-in l:.erttci:,lon" mcn tioncd above.

(3.5)

Th" "quatio)) (3 _ 4) ~eo.lly is 0. system of coupled differential

~~q:\)Q.t ton$ a,no by ~olvin9 thi."1. ~ygtem not only the elastic soat.tering' hl.,lt ~l!3o the surface"'inte:c,::tctions are tr~ilt8d to all orders. Tn the lJWBA sam" of the term" at the r_h.s. of

"q.

(3.5) are set; equal to

ze~o, whi~h is ~ yeasonable procedure if Glastlc scattering i~ the dominant process. We will clarify this procGdure by the following example, whiGh include~ the outgoing chann"ls ~ and y besides the ela.stic one ,[ig-_ ., _ 1). The system of equatioJ\s. COrrespondinr'l to the coupling scheme Of fig. 3. J ts ,~). ven by

(3.6a)

(3.6b)

I--Pig. S.l

a

'··1

\~'J

y

(CI,y)

) I

Coupling s("fliem$ with reaotion dlCl,rm.tds CI., S and y

(("fl'.

eq.

3, (J)

--I

The transitions (a+~), (~+y) and (CI~Y) are now d®scribed by a fi~st-order approximation (lIone ... way coupling"), whe,=,,~a!3 the elustic s~attering is still t~eated to all orders. The ~equenCe (CI~G) (~+y) is call~d a two-step OWBA transition, while (a+yl is a one-step DWBA

transition.

If we are nat interested in channel y then the system (3.6)

reduces to (3.6a) and (3.6b) and in that case the transition ampl1tUde

Tu8

can elso be calculated

tram

the expression

(3.7)

where

Xs

is the gol~tioo of th~ homogeneous equ~tion Corresponding to

sq. (3.6b) with incoming boundary oonditions. Displaying th~ into-gration over relative coordinates explicitly eq. (3.7) becomes

f

+

f"

(-)*". +

I I

( + ) " ..

T~S ~ J drC( dr

_e

Xe

(k~,rS)<Bb

V

Aa>x_Cl (ka,rC\) (3.8)

with J the Jaoobian fOr the transformation to the relative COOrdinates and with k

f and kC\ the relative momenta.

Before di8~~~~ing the matrix element <BbIVIAa> we will f~rst connect TaB with the analysing power A and the unpolarized cross

ssct~on da/dn. In expression (3.8) we have not explicitly shown the angular momenta. In fact TaB depends ~on the spin Proje~tions of A,

a,

II and b: M A,

given by

m ,

a ME and mb and the unpolarized cross section is

do - ' v

elll (3.9)

Thf':! ~~na.l ysi.nq po we r: i$ .'":! lTIp.<;:I.=;H).;re fOJ:: the di fference betweer'l the G"rOsG

~er.tion~3. obt.=:.i..nf':!cl w.i t}""l protDn gpin up (m

=

1/;>) and proton spir\ down

Cl

(m

_e,

= - l h ) ,

A

I t 1,. "ls<>r from D .10) th"t A 1i"',. between -1 and L

(3.10)

In urcler to have" qu"ntit<>tiv<= indication of t.he ~91:'"ement betw~en ca leul at-cd d.r"ld measured (see cha,pte:r 1) anf.l.lysing power and r.;o~~ ::::~ct:i.on we l)st=:!d tJ~* foll.owing chi-squa]""e expressions:

r

(8_{i ) -} d 1 (e_i)}2

X~

i

""£'

ca _(3.11a) N _{Mexp (e} t ) i 2 X A

N

I

r

(e.) - A l(A.)}Z exp ~ ca ~ () _lIb) --~ 6A (e.) exp ~

where N is Lh", tot"l. Ilumb"r of eX!?8rim8ntal dat'" eaken at different 5catterinq angles 0 i _ Irhe quanti ties no and ~A are t.he errOr:=. of the

experimental cross s8ction and analysing pow8r.

3.2: One-Ilc::utron tr.::l.Ilsfer

working CIllt "BblvIAa" for a (p,d) tr.;>nsition one arrives "t 1), I I r -> > I ' ' ' ' -T I

->-<'Bb Vila> " vn <4> (r -r ) ~ I~) V (r -r ) i(tA(t,r »

rJ p n B '

"P P

n . n (3.12)

wi t.h f:; be 1.119 the in t.e:rnal coor.dinates of nucleus Band n the number of

"act.i,ve" (.1"e. taken explicit.ly iflto account. in the w~ve function 1/1)

neutrons in nlKleus A.. The /'JGutc.:ron irltcrnal. wt\ve function i~ indic~ted

by ¢rl and i5 generillly limited to ttl" L=O component (s-state d<,ut"e:ron",), which h",:; th" iUl!?Ortant oorollary that the transferred "flgular momentum

i6 equal to that of the neutron picked up from nUGleuS A. The "surface

interaction" V i.s :regtricted to the potential between the incoming

np

pt:'oton ~nd the tr..7,ln.~f'e:C(t;:!(1 neut:ron. In the so ... called zero-:r:~nge

i.e. the proton and the neutron only ~nteraot when they coincide. In ... +

facl one ~eplaces ~dVn~ by 00 6 (rp-r_n) • where the zero-range constant DO is calculated frOm

f

-+ ->- ...

DO ~ dr ~d(r)V (r)

np (3.1 J)

with V

np a mo~e Or less realistic potential.

'the nuclear overlap ("form factor")

<!JIB1wl':

is exp«nded 35 follows;

(3.14)

One usually takes for $jm a singLe-particle shell-ooodel wave function calculated with a real Woods-Sa~on potential (eq. 3.26). of which the ~eLL-dePth is adjusted to coOOPly with the condition that the binding

energy of the neutron has to be equal to the experimental 5epar~tion

energy EB-E

A in order to guarantee a correct asymptotic behav10ur of

<i)iSlw

A>. ~hi~ is oalled the well-depth p~ocedure (WDP).

'the nuc"ear structure ~nfOrmation is contained in the spectro-scopic amplitude~ 5 1/ 2 and they mea~ure th~ am~lltud~s with which the variou~ single~particle states appear in the nuclear wave functio~s. For example, the 0+ ground state of 58 Ni may be described by the follo~ing two-neutrOn ~ave function:

In this model th~ 56 N! core is considered to be inert (closed shells) anc\ the two "valence" neutrons are occupying 2p1/2, 2pl/? and lf5/2 orb~ts_ ~he configuration mixing ~s caused by the residual interaction

(not present in the shell-mOdel potential) between the neutrons. The spectroscopic amplituc\e sl/2(2p3/~) for the transition of this 0+ state to the 3/2 ground state of 5?Ni is then (using eq. 3.14)

l;)/Z.

So by ,nea;;uring the spectroscOl?ic amplitudes one thus determines the magnitude of the coefficients of the various COmponents of the 0+ W~ve fUnction.

Since we want to investigate the interference between one-step and two-step DW5A transitions. knowledge of the relative signs of the spectroScopic amplitudes i~ crucial. Therefore one must be sure that the pha58 conventions used in calCulating these ~~lit~de5 frOm Shell-model wave functio~s are the Same as thos€ of the computGr code that caLculates the cr055 sections ~nd the analy~ing powers.

F'OY' the coupiecl-c:!tt1.one18 code CHUCK2 :-1 r r with 1/iihich the CrOSS

sec:tion~ and anillysing pOwer5 in this thesit; h~ve been cCLlcul~ted, one has to \.u~"(~ ,9p~(:t:..tQ$CQpic a(flplitudGs Jefirae("j by

1:1.16)

where p star"Lds fo:r" U"J.e quaflL.Ulrl numbers (nlj) an,c:1 11i;1~11 means::: anti-::;ymrnet_r.~ 7;erl_ rn addition the rild.;i.al pa:rt of th€ si("J.gle-pa"(ticle wave

func;tions is po:oit.ive ... t infinity "nu the an9u1ax- pa"t is ,~",fined

without thc' U,eto!:: i 1.

Summ.::J.rizing we qet t:he Eollowinq expression (or the one-step DWBA transition .:::I.mplltudc

\' 1/' Id'~ (-)* -. "r + (+) + 1\~1 + To.ll '" l)O(pd) L S l(nlj) • X~ (k~,r)4nlj(~)X" ( k c t ' T r1

(3,17) which is .:::I.1so valid for' ot.her one ... neutron triJ.nSfer~1 b\.lt with -1l

di ffere"t UlrO-rilrlge constant I)Q' The maSS nun,ber of the targGt nuc)."u~ is indicated by A. The angular momentum traII.sf0r j h~s to satisfy t.ht;":!;

triilng.le condition implied by the C.lebsch-Gordan ooeffl.c1ent of cq. 3.14 and if J = 0 then only one term survives, j = J

S'

A 4)

:B'rench .and M.;3.C [arlane have derived :sum -r:ules, which ~onnect

neutron and prot"n occupation numbe"" with th" spectroscopic ampJ.l.t\1des.

Since we apply the::;;.e Sum .tt),le9 to our experi,rn~ntal r:esults (chapte:(s -1 and 0) wO give them he):"e. If the to.rgot "ue)."u" has iso6pin TlI then one

"<lrl r,~,,(;h viii one-neutron transfer (t 1/2, t

z " 1/2 ) final state. wl.t.h T

_-,

T" + II?, The sum rules then ar.e:

I

c

2

_s

_{(1 j)} "lj

_-

j]lj T 2T A

+!

< (3.18) c2 S_rr (lj) TI 1j 2T Al'l

where

./.1

is tll" number' of

ne\)t~on5

occupying the Lj orbit and

correspondingly J,lJ fOr t.be pyotons. C sto.rlUS fOx.: the isospin

Glebsch-Gordan coefficient <TBTllz; Ii? 1/2ITI\TAz' wj.th either TB ~ T ... - 1/2 or 'l'B = T A + I /2 •

3.3 Two~neutron transfer

In case Of a one-step (P,t) transition the explicit form of the transition oensity is 5) :

In tho Zero-range approximation one takes:

where

...

r and + p -+ r l? (3.19) (3.20) -7 -,r ~ 2 (3.21 )

In this approximation the interact.ion takes place when the proton coincides with the centGr of mass of the two neutrons. The function do can be determineo analyt10ally from:

-+

J

-+ -+-+

dD(r) .. dp<j>t(r,o)(V +V )

pnl pn2 (J.22)

if one takes Gaussian shapes for ¢t and V. The important parameter entering this evaluation i.s the root-mean-square raoius of the tritOn, Which usually is chosen equ~l to 1.7 fm. In aodition a pure L=O triton internal wave function is aSsumed.

The nuclear overlap <1jJI'!(~) I1jJA(f;/~L,tZ» is related to the spectroscopic amplitudes in ths following WaY

vn(~-l)

.el/lBll/I

A' =

L

SI/2(jlj2'J')<JMJI'!~IJ}\MA>[1>j

()';I)x<l>.

{;2))

jjj2 1 J2 J

(3.23) The one-particle wave functions in tn~s equation a,e ~etermined in the same way as in the case of one-neutron transfer, wnile the binding energy is usually t~ken equal to half the separation energy EB-B

A

of

the two neutrons.

The inverse of eg. 3.23 is given by

(3.24)

In oroer to as follows. products of

calculate the transition amp,Utud0 TaS one then proceeds Pirst [<p.

(t

1 ) x ~j

(;2))

is transformed into

a SUm of

Jl 2

r~dial and angulur functions expressed in the relative

and c~nter""'of-mas::; cOordinat.es -; .:.. ·;2-;1 i.\I1d

R

= 1/2 (~1+;2)'

respectivGly. Next. one restricts this t..tan.sformed function to the part

with r",.\~tJ.ve l~O (i.e. th" tW'o neut):cns being it. tl1\ s-state relativ8 too each other). This mea1\,~ that the S-t"",1\SfOL' is zer-o and thilt J =

L-, +

Then the integri:.lt..l..on OVE'-r .1:' ~nd p is performed resulting in th~ fOrm

J I'. +

facto( b'O~ (A-2 R), where I'. i" the mass number of the taL'<)et nucleus. The final result then is

(3.25)

rrhe distorted w.::..ves aloe c~lc.;ulCtt.e(~ f~Qffi equations lik.e (:3. 6a)

\lsinq- an opticill-model potent-ial of the following form:

U(r)

=

U (r) -

_c

Vf(XO) - iw f(xl) 4iw f'(x l (~)2 V .!:.df(Kz)

d.-r

V - d . 1 - m e s o r dr

where f (xi) has the WOOds-SOI"cn shape le"i + 1)-1 _{with x_} 1 r - r.A 1/ 3 0, -IT (3.26) (3.27) and where U

c is t.lle Coulomb potential g,morat0d by a uniformly charged

sphere oE radius reAl/::..;. For protons .and tritons. ;; :- 2~' and for

deuterons;; =,

-;;1

where s is the spin operator. The v~ttor

t

is the

orbital anguJ.n: momentum of the 5catteruc\ p":c~l,,,l .. , The two imo.gin"ry

terms of U (r) represent tho volu.m.e absorption (w ) and the ::;.urfGl,~e v

absorption (01 0

)-3.5 ~inite-ranQe ano non-locality corrections

In sElotion 3.2 wo have discussed the zero-riJ.nge approximation

for one-neutron transfer. It ;i.~, however, possible to make Q

first-0rder cOrr0ctj..on to th;ls Etpproxim,;:.,tion, which is called the

local-",n""'9Y appro:dmat-.i"" 6) (",ljA) _ 'l'he LEA con,;ists of multiplyin<) the [o,-m factor (eC(, 3,14) with the llulthen form

A(d

(I:I.:.!:.

r)

A

_I

+ En - Vn(r) -

E~

+

US(~))}

D.28)

with Vn th~ bOund-5tat~ potential and R the finite-range parameter. The optical potential should reall~ be a non-local potential. As

shown by Perey 7) a non-local potential reduces the amplitude Of

th~

distorted waves in the nuclear interior. This Pere~-effect can be simulated ~ppF.oximatel~ by multiplying the form factor (eq. 3.14) with the factors

(3.29) where 8

i is the non-locality param~t8r of projectile, neutron and

ejectile , respectively, and Pi the cor~esponding red~~ed ro~ss_

3-Q Adiabatic deuteron pot~ntial

In the ~eaction model implied by e~. 3.2 three-body effects are not taken into account. since the deuteron is a loosely-bound particle, tho> nucle<lr field of the final nucleus can easily sl?lit the deuteron in a proton and a neutron and thus give rise too the channel (B,p,n).

The adiabatic model of Johnson and Soper 8) provides an apprc~imate treatment of these effect5 by replacing the optical d~uteron potentia" by a kind of folding potential,

-I +

.,.

+ +

ii(r) ~ DO

Jd~

(Un (; +

f)

+ U (r

- t)}

Vpn(S) <!>d (s) (3.30) p

where Do ~

f ...

ds V (s)'fld(S) + (3.31 )

pfl

and ~h~r~ the proton and the neutron eaoh g~t half of the deuteron <;onex:gy. The distOrted waves geneJ::ated by the "adinb"tic" potential

G

also include unbound (p,n) states, in which the proton and the neutron

continue to mave together in an S-stat~ with little relative momentum. An approximate expr~5Sion fm: U(,) h .. ~ been derived by satchler 9) f.or the case that U

n .. nd Up are woods-saxon potent.ials (or first derivativ~s thereof.) with the same geom~try parameters but diffe):"ent well-depths. We will give he:r;e _{the formulas leading to this approximate}

"xJ;lression for U(r) and refer to the study of Sat"hler fOI; detailS. If Un and Up have a WoodE~Saxon shape with parametel;$ (vn,r,a) and (Vp,r,a), r0speetiv0ly, then sO has ij with

r

= rand

a

~ a + 0.0217

a (3.32a)

ij (3.32b)

wl-1en' R = r ,,1/3. We remark th"t eg. 3.}:Ib differs from the prcscript-~on

of Satchler by the faot.Or ~ in front or .(t'2. We think, however, tJl?J.t our

"pproxilllallon is bet.ter. We will not go into det.ails, sinc" the fim'l

:t"~8ult:::: a~:e hardly influenced by this alteJ:'nat.ive pr-escription. ~f On and up have the shape of the derivative of a Woods-SaxOn

p"t~nti<'l with pllramct.cr·s (wn,r,a) and (W

p,r,l1), respectively, then

the same is true for

U

with

r

=

rand

a

=

" + a.Ole2 _a (3. :33,,)

W- ( 1 2 n

2 )

= 1·· O.OlGi (~+ ]

liT)

(3.Db)

We also had to constrllct pot"ntials W starting .f:ro,o different

geOmetry 'par~mOl.ers fOr t.he:.- nent.ron find the pt:'oton potentials. This

was don" by separo.tel y calClll"ting

Wn

ann

Wp

from eq. 3.331:> putting Wp " 0 and wn " 0, ):'espectively and then adding th",,,,.

REFERENCES

N. Au~tern, Direct Nuclear Reaction 'theoJ:."ies (wiley-Interscience.

1970) •

2 J.W& $m~tg, thesis, Rijksuniver5it~it GrotlingCn, the Notherlands,

1977 .

P.D. Kunz, University of Colorado, unpublished.

4 J.B. F~ench and M,H. Macfarlane, Nucl.~hy~. 26 (1961) 168. 5 tl.w. aaer, J.J. Kraushaar, C.E. MOSS, N.S.P. King, R.E.L. Green,

P.D. Kunz and E. Rost, Ann.Phys. 76 (1973) 437.

6 P.J.A. Buttle and L.J.B. Goldfarb, Proc.Phys.Soc. 83 (1964) "l0l.

7 F.G. P€rey. Direct Interaction and Nuclear Reaction Mechanisms,

eds. E. ClementeL and C. villi, Gordon and Breach, New York, 1963. B R,C. John<;:on and P.J.R. soper, Phys.Rev. ci (1970) 976.

9 G.R. Satchler, Phys.Rev. C4 (1971) 1485.

CHAPTER 4 THE (g,d) REACTION ON 5B Ni AT 24,6 Mev

4.1 Introduction

Th~ (p,d) r~aotion is a well-known instrumGt,t fOr obtaining speotrosoopio information like l-values and spectroscopio faotors. TIl';' elsefulness of this reaction is greatly enh€l.nced by using polarized

protons, becullse the analysing powers show a clear j-dependence 1). As

~ general rule one can say that the sign of the analysing power at the

angle wh~re thG cross section reaches its main maximum, is positive OY

negative depending on whether j = 1 + 1/2 or j = 1 - II? By comparing the measured analysing powers with DWBA c~lculationB or with the experimental results tor levels with well-established j-values

(I~pa.tterrt rscogni tion") one thus obtain~ a r-eli';"Qle ~pj,n ass;i.g-nUlent.

of the final state in the caBe of ~pin-ze~o target nuolei.

Some (p,d) cross sections da not exhibit the oharacteristic 1-dependent diffraction pattern predicted by the on~-~t~p DWBA, but & rather flat and featureless shape 2,3) A COmmOn property of the levels showing suoh a behaviour is that they are wea~ly excited, wherefore in these cases two-~tep preoesses arc generally thought to be competing with the one-step pick-elp. such transitions are present

j.n the 58 Ni(p,d) reaction at 27.5 MeV 3).

Mayer et al. 1) already perfOrmed the 58Ni(p,d) experiment with pol~rized protons at the same energy as we did. These authors_r however,

only analysed the strongly excited levels. So we decided to extend this study to the wea~er level~ espscially in view of possible two-step contributions. In addition we wanted to investigate the importance of two-step prooesses fOr strongly e~cited levels.

All OrOBS section~ and analysing powers hav@ b@@n calculated with the computer code

CHUCK2

4).

Our results are discussed in section 4.4. An outline of. the reaction model and the ingredients for the calculations is presented in section 4.3. In section 4.2 WE have brought together the details Of the experiment and finally in seotion 4.5 we have su~"arized the main oonclusions of thi~ ~hapter.

1. L The exporiII1e~

'l'hu (p,d) .J:·l";8.c:tion wa.~ measured 5imultctw~Qu::ily W;l..~:h t:".he (P,P),

(p, l) ,~nd (p, 1),) r".'Iction,,- The outqoinq particle,s we"',, identified using the telescope sySlem dOScI·ibed in chapt.er.- 2. In oyde"r" to determin~

absolute cross sli..:.'ctioIlS we compared the elast:ic p:roton .... scatterinq d~ti.1.

with th" optic"i-mockl un"lysis performooct before by Our group 5) for the yea~tiQn ~HiNj.(pIP) at 211.b MeV. Th~ o.2Sitimut..:cd ac(:u.:rac:y of the nO.J:'lllaliz.ati.on is :20't.

'The ta.t:~Jet. consisted of: ~ gelf .... supporting foil Of .isotopically en"J"-iched (q~J.q~,) 51~Ni with a thickness of 1 mg/cmi.. Th-e: intensity and neq:t:"ee of polarization of the proton bc~m were about. 10 nA ann 80%, r,::spi::c:;tiv~ly. Dutu. wt":.~r~ t:.1.:l.kcn fJ:'om 100 t.o 800 in :;::b:~ps of SO _ The

e::::I1C.:.'rgy re:.:::=.olutioll wo":t:=: ~bolJ.t. 100 keV FWllM. which ilYlposed iJ. limit L.o the

number of lr::-vels th.Qt. could be:' analysed.

Wf;;;: meoJ.surBd th(~ J.IE.1.1ysing powers anrJ c:;:::rQS!;: sections of eight.een level", (table 'L d). These '''". bo seen in the speer.rum of fig. 4.1

CD / 103 u·, ()

'"

a

I

0 ~.

'"

,-

",

eo

'"

+ 0 .~

,'..,

'"

"'

102

..

/'

,,'

M

"'

I

cr,

'"'

<>' \ IJl f-Z ;;)

,~cD\O

l\'! . :'\'

~

"I'

-

.

co u") -0 1--' ,

.

,

.

(-- L..O <D <J) Ii"]

I \

I \ \

0 U 101 ,--,--,--,--,--,--,---,---,-,,-J .... ,L,. . L . ..L...L.-'....JJl..llL--'-.llil-'1lL..llliJIL.J....,.J.JJlllll..l!.l...-Ll.-' 10°

200

FlU, d.1 26

320

CHANNELS

440

together with levels of contaminant.s and oth0r lo;,vels of 57Ni which w~re excited too weakly to yield reliaole angular distributions. Tho;, excitation energies were taken from the survey of Auble 6) and from the study of the 58 Ni (T, cd re1lction 1lt 25 Mel! by

~'ortier

and

~ales

7).

,he

5a

Ni(p,d) rcaction has already been measured several times at different energies and with unpoLarized protons 3,8-10)

~

polarized proton beam has been used by Mayer et al. 1) (24.5 MeV) and by Hosono o;,t al. 11) (65 11eV).

4.3 The ~eaction model and tho;, calculational procedures

4T3.1 Reaction model

Seven out of the measured eighteen transitions have bBen analysed according to the reacc10n model that is implied by the coupling scheme given in fig. 4.2. The remaining transitions were described by onc-step DWBA only. In the two-onc-step prOco;,Ss inelasti~ proton-scattering to the first 4+ (1.45 Mev) of ,SNi is followed by the

pi~k-up

of a P3/2. Pl/2, £5/2 or f7/2 neutron. P~ck-up followo;,d by inelastic deuteron-s~atterin? turned out to yteld negligible contributions and tho;,refore was ignored. Replacing DWBA by CCBA in the proton channel (i,e. ~ two-way coupling between the 0+ and 2+ states) ct~d not change the ,esults 8~gnific~ntly and 50 CCBA Wa~ al~o left out of consideration_

The one- plus two-step DWBA calculation (fig. 4.2) is complicated by the fact that five spectroscopic ~mplitudes may be 1nvolvCd against

one in case of .a simple one-step D'ir'fflA an..alysis. i)'heref.'ot'e we mad~ us0

of shell-model calculations done by KOOps and Glaudemans 12) and Van Hees et al. 13). Th", J.atter Calculations allow for aile If'll?

Fig. 4.2 Coupling $"heme for the do'-,pt,d-ahanneh eakuZatiO>l8

rLucl(::!on-h(Jlt:'; in the ')5 Ni GOre with the partic.1e:8 Qccupyil)g 2p3!'::, 2pl/~ ,mel 1 [0/2 orbits i'tt"ld t.hey a.~e p"rformeo with th" surfa.ce-delta

(~Dl) en with a r"norm"li~",d Kuo~ll"own (KBI) re~idual interaction. The

Cca.l(:I,llations

cont~ineu.

i,t1 ref_ 12) wert) rest'r:icted to

2p.~/:l1

2pl/2 and

1fS/2 pctrticles (i_~. ~Il inert ~GNi core) and the 511rfacc-delta

inter-.~ction (SDU). FrOm th", )·""ults of tiles", nuclear structc",e ~t\.ldi"s ttlr",e

s",t.s o( spectr.o"c;opic i'tmpJ.J.t.udes, also indica.t_ed by SOl, KBI and SDO,

h~v('

been comp1)tec1 14) The set SDD obviously desc:rj,bes transitions to

0"'" :1/7-,

0/:,

.~(\(.l 1/2 stat" orlly, while SDI and KSI l.ead to "- wllOle

5p""tl-um ot ne~'Jiltive pal;ity st<lL.cs ri'tnqinq from 1/2 Co ll/Z -. Of these

the 9/' 2U)J ll/;': states can not b~ :reached in Dr'lC ~tep from the.:

0'1-qrouIlll ~t·.a.r.* \,'J[ ~iHNi_

The 3/21, :'i/?1 ilr",

1/:'1

stat.es are idenl:l.fied with ttl" 57t<1 levels

at 0_00,0.71 and 1.11 MeV, ~·e3pectivc.:.~ly. rrht'::!se three stutes aI'€:! the only ones which Lurn Qnt. to have .an u.pprecJ.~ble 1/2-r .].!;r b.nd. ?~/'2

transfer strengtt • .t~orn the 2+ "t<tte Of 511 Ni (table 4.1) _ 1'he ~emaining .~.tate~ are populateu by "I/O' t.r.'lnsfer frOm either th" 0' only or the ;,; + on1 y. calculations :::how that_ the statE::!5 whi(:li ('.:an only b~ rCi.\ched

T"bk ~_1

,Spl..'!_·Lr.l':...sc_·vj.d.r: "'~mpl11:\ldf':!s for onc..'-lIuul,rOJ"1 t.(an~f~~t:'

Pr.L_'L_·I...'::.'~'; /JpJ)CL)

_soO

!;()1 _KBI . . . __ . _ - - -

...

" ... -,,_ .•. _ . _

-.,.

f)

.,

J ,; (S) -0_7il 0,')4 -0. ,r,:; :J (J) 1.11 0_B9 1_15 1 (1) 0.40 0_11 0_ 4·/ 2·t-_:r J S (I)) -n.~2 -()_,,7 ""U_ 22 5 (:J) 0.3<

o.

:~ ~ 0_17 ~, ( 1 ) ().:!" 0.44 0.17 r, (7) [LOD () _ l() _{·:0_10} , (5)

o. '"

o. :,,,

D_17 .J (:1) 1. 01 0.7:' 1 _12 Ill) -0.42 -0.39 -0_11 J Ii) 0.00 0_22 0_11 1 ( )) -D_,"Ej -0_11 -0.14 I (1) 0_1:2 0_ 3"/ 0.40 .... " .. , - , - , - - - , , - , _ .

~'d t.h~ tT<'1r1st'.::.::rI'Cd dnyu.] ar momentum B iD(!i"at_"d by j .

vi" th",

i"

ar .. too weak to be: obs<!>rv",d in our experiment. Some of these states probably have been found by FOrtier

~nd Gal~5

7)

~nd

m<LY be identified with a weak-coupling multiplet 2+"7/~- somewhere around E (2+) + E (7/2-) = 4.0 MeV excitation energy.

x x

The cross sections and analysing powers of the partiole-states at 0.00, 0.77 and 1.11 MeV have been ~alculated using the spectroscopic amplitudes from table ~,1. We applied the usua" one-step DWBA analysis to the other states except for four cases, where we added a two-step

7/2 transfer to the one-step transition.

The inelasti~ s~attering has been described macroscopically using the following form factor

6 { d[] a (r) dUr (r) J

[R~

2

l}

F2 (r) ~ -':'l. R ~~~ + R - - - ~ - ~e2 -;;T 0 (r-R ) + ~ 0 (R -r) (4, I)

15

0 dr r dr ~ r

c

~C C

where U

o and UI are the real (volume W-S) and the imaginary parts,

respectively, of the optical proton-potential (eq. 3.26). The parameters Ri (i=o,I,C) are defined by ~1 ~ r

i A1/ 3 , while

e

is the usual step func-tion. fO. the deformation parameter 62 we used the value -0.22 found before by OUr group 5) from the

58Ni(~,p} ~ea~t~on

at 24.6 MeV.

Aotually the sign of the deformation parameter oannot be +

determined by the (p,p') experiment since it depends on the ~elative phas" of the w;ove f1mctions of the 0+ and 2+ iOtates. The sign of S2 has to be found by tr~al and error. It was p05sible to make", oefinite chot~e fOr this sign by making use of the interferenoe between the one-'i'tep and the two-step tran5itions. This is Ulustr"ted in fig. 4.3

(J/~-, solid ~nd dashed-dotted lines).

The code CHUCK2 does not account for the O-state of the deuteron ~nd the finite range Of the interact~On. 50 our oaloulations are limited to deuterons ill the S-state and the finite range is deScribea in the lo~al-ener9Y approximation (LEA). We used the Hulthen form

(3.29) with the v;olue Of 0.69 fm fOr the finite-range parameter. The LEA reduces the Cross sections by about 3t ~nd has no significant ~ffect on the sh~pes. The Cross sections were normalized with the

empirical constant bOCpd) = -122.5 MeV fm 3/ 2 ,

The need for a non-locality correction at the neutron Wdve

proton-(]"n",ity calculations 15-17). So we

",pp~y

this correelion in OilY cas,;: t.GD. Following the aut.ho:t:'3 of refs. 15 .... 1·1) we \lsed a Gaus:=;ian fOrm

(3.30) wit.h the Vi'lille or: U.<l:' fIn fOr t.h", non-locality parameter'. The CrOSS ~J,ection5 increase with about 3S% by the non-loc~lity correction, wherot';!as tht7! shapes r.emF.l.in virtud.lly unchanqe(-'l..

l)$u.~lly the neutron form factor is calculat.,d by the well-depth

proc~dUr'" wi. th bindin<] energy of the ne1)tron e"lual to the o,,-per imental "epClration en€r<]y (t;El, So: = Sn i· Ex with Sn t.he ("1f'~n"utron separation ,)nC'rgy of 5U N;. UJ.'~.) "nd 0:" the 8xcitat,i.071 en"''''9Y· In this way the cor.rect asymptotic behaviour of triG fOrn! fact.or is .assured.

Tn order to explain the observed j-dependence of certain (p,d) C)":Qg~ section~ Shery et ul. 18) proposed to ):."eplac~ the separation ener-gies of the pa.rticlG-::;tut.es l)y effective enerqics, wnj,ch ~t.e more in line with the :sinql(~-purt.j.r..:.l.t';!: Shell ... model energios and a,(e sllpposed to give a better. d~sc:ription in the nuclear interiOr.

In f.u.ct both ",pprQtl,.(.:.:h~s (.:.;:.n be reconciled by d mOre sophisticatOd

t.~ei'3.tment of. the form fuutOr, whi,ch P..Xl?licitly accounts t'or the

rGsidutl.l j.n,te.ract:iQn of the valerh.2;G nOUt.rOnS by considering the fOri!! factor as "

~o.lutior>

nf on inho,"o<]eneous S"hr6dinger ""I\lation 19-24) We tried to constru(;t.. ..:t(l .:s.pproxim3.te solution to suc.;h an ~qU,;3:tion in the foliowinq way: firl5t by choosing the residual intcract:ion to be

a delta interaction w(~ Sd.mk,,"1 it'ied the inhomogeneou:=: Schr&rJint)er equution to

(E - T - ul F

j (rl

[ I

_{Jj'=1 JJ} a .. ,Fl ,] F. _{J ]} (r)

w1 r,h J7: th~ :::Iep~ra.tion enerqy, T the }dn~l.ic enertJ"Y operator, U t.he

sholl-model pot.ential of t.he ~,f'Ni car<, .e,-,d a

j j, cert",in coefficIents determined by the spect.roscopic -'lmplitudes; secondly, we, ~1Jb.'3titut"d

the solutions

p~?)

of t.hp. hQrooqeneous equation::; J

(0 )

( E - T - u l Fj' (rl =0 (4,3)

~n the 't:'iqht'·hand 5idC Of the ~g:t,l~:::ati(ln!3 (-1.2) and solved l.he.!;':€:!

numGriC'~llYi fi(~d.1I.y WP. r:p.peated this proce:durc by $'I),~)~titutinq the nawly-found ~alutions in the right-hand 8id~ Of (4.21. The solutions

(1~d n.ot chan!Je any more ~ftGr t.wO Or thxee i teY{:Itions. Ttw I'rO:=.S

So€;!!.,.":t.ton~ Clnd ~na.lysing powers G<11Gu.l~t.ed wit.h l.hese form f~ctorst however~ lurned out. t·o be virtually the same d,e those obtained

Table 4.2

Effective energie~ (1;:1;:) for the bound-neutron wave function

jTI I;:I;:-Snll.) EE 3/r

0.00

12_20

5/r

-0.77 11 _13 1/2- -1.11 11.09 7/r 2.58b) 14.78 7/F 5.23<:) 17.43 1/2+ _{5.50 17.78} 3/2+ 6.01 .\8. ;:1

a) Sn is the r.ep<lration energy of a nel,tron from the ground

st~te

of 58Ni , Sn=12.20 Mev

b)J.£ E_x"5.23 MeV c) _{i f EK~5.} 23 _MeV

Since mOre carefully constructQd solutions of the inhomogeneou~

S~hradinger

equation yield

result~

21,22) which deviate significantly ~~Dm that preScription, we decided to compa~e the separatian-ene~gy

(SE) with the following effe~t1ve-en~rgy (Eg) prescription,

EE = Sn ± ~x. Sn is the one-neutron sep<lratian energy of 58Ni in the ground state and ~x is th€ excitation energy of the final 57 Ni state. 'rhe minus and the l?lUS signs ""e chosen fOr particle (i.e. 1/2-, 3/2-, 5/2-) ~nd holG (7/2-) ~tates, respectively. The effective ~inding­ energies are al~o used in a well-depth procedurG, which iIDJ?l~es a homogeneous Schrodinger equation. They are giv~n in table 4.2.

We took the usual gQomet~1 parameters for the woods-Saxon well, r = 1.25 fm, a = 0.65 fm and the factor multil?lying the Thomas term A 25 (i.e. V /V .. 0.138).

so

The optical-model param€ters are given in table 4.3. The prOton potential has been determined before by our g~oup S). It was ext~acted from a globe~ fit to elastic scattering data of polari2ed p~Dtons on several i,on and nickel isotopes at energies ranging from 15 to 25 MeV.

Tublc 4.3 Opticill-mod"l p!J.ri\Hl.(:tCr":::~ a) N.:IHlC V t' .'l w _Wd r 1

"r

v r

'"

0 () V 60 SO SO j.' '>2 .0 1.15 0./6 2,1J!i (L 7 LJ5 0.4'1 5.6 1.04 0.54 Dl 101.1 1.05 0,1)6bl 14.7 1.1] 0.69 7.0 0_75 0.50 D2 10" _ 4 c, d I 1 _ 1 ") 0_ -In 10.1 c) I. 26 O_Gl 11.8 1 _(l1 0.56 9_6c )

_l.n

_0.5/ 111 ')U _ 4 Ll"I

o

.m

15.2cl 1.26 (L61 J 1 .8 1 _04 0.56 14 -4(:) _{1.32 CL} ~)7 6V!6 I' x "Wall:. Ex -_._.

__

. ' - - -- . n'.l 0_:31:3 0_11B D4 0_290 -(1-6;'6 ,0 )

v'Wv'Wcj .and V~(J in MuV, the ot.hp.y parametero in fm; )·C-~·.1.2~5 fin for tho::: pl"ot~..">n po!.:t:nl.iij,). 6l.nrl. ... r:=1 . :~o frn for the d('l."!I~.0:J!)n pot".enti!lls

\:.) 2(;)

Tht~ \l~lIlE:~ ("..of t:his parilmr..:-tc.:.'r· 1:.;..; ):"e}!Qy.t~a to be O.

[to

l.n t"c-!L. but w ... s L.tt(;~r· (':!J.r. r..C'(: t.r:·(l by t.h!7! fiU t:hDr.S to b~ 0.86.

G) ju

b'or the grounli :=: t·.,) t·.1":; Q r Ni

d)

This potclntitll i::; ral:.h":r close to the Becchetti-c.:;reenlees 25) potcntii!\J. ""copt. fOL t.he ditflls",ne$~ of the spin-orbit part.

We tried 6"v"ral deutoron potentials. The first one (Dl) is the global Lohr-llaeberli 26) potential obtilined from elast"l.c scatte:ring of POlariz.e-d o?nc..~ unPO.l f.I'(j ze:f1 deute~·on~ on nuclides with A. ::- 40 ttt cnergie.s frOm D t.o 1) MOV r wh tch 1& just t.he region 0 f ou ... experiment.

The adi-=tbat.ic deut.eron potential (D2) has been construct~d from

;,j)

t.h-p. Becchetl.l.-Green l~H~!S pot".ent.ials for protons and neutrotl~

Q-.ccordinq to the p.l":escY"iption of Satchler ),7) (chttpt.er 31 s~ct:ion 1.6). In cOnst.ructing the SPir"l-()ri)it potent.tal woe ro:::pl.acei3. the qeometry pO-rarneters of ClIO BG poterlt.ial by t.ho8e of OUr proton potential, sinGe t.:.he li..1ttcr was determineu by a polarj ~?!.t_ion exper.irnent ..

Out of the pr-I"::!sent study Slros8 the need for' two rnOr'e rle1J,teron potent;alG_ On" o[ th"s" (DJ) ", the same asD2 but fOr. u reduction

of 4.5~ of the real well-depth and is lett out of table 4.3. The othe. one (D4), which gave the best result.s, differs f;("om D2 by a 7,5% reduction of the real well-depth and by an increase of the imaginary potenti~l. This increase is 50% for the ground state and about 10% at 5 M~V excitation GnGrgy. The Curves displayed in the f~gs. 4.3-4.11 huve been calculated uSing deuteron potential D4, unless otherwise mentioned.

Non-locality corrections have been appliBd using ttl" Gaussian form of the correction factor with ~(p) ~ 0.85 fm and e{d) ~ 0.54 fm.

The cross sections rise about 8% by these corrections and show slightly

more pronOunced diffraction patterns. The finite~range and non-locality

corrections fOr thG proton and the deuteron together r~iEe the cross

~ections by about 5%.

4.4 Discussion

one-neutron pick-up preceded by inel~~tic sc~ttering (fig. 4.2)

yields G~06S section~ which a~e at le~~t one oroer of m~gnit~oe

smaller than those obtained by pick-up only, The,efore ~ll level$ which are strongly popul~ted by the (p,d) reaction, are m~inly reached by a one-step ~ransi~ion, while the tWC-St0P tr~nsition may dominate ~ weakly-popula~ed level.

For the strong particle-s~~tes ae 0.00, 0.77 and 1.11 MeV thG

addition of the two-step transitiotls introduces a small but significant

change of the calculated oross sections and analysing powe~~ (f~g. 4.3, solid and dashed lines). All crOss section cu~ve~ but one ~~ £ig_ 4.3 were normalized to the first maximum in order to faoilitate the OOmparison of the shapes.

Cal~ulation5 with different sets of spectroscopic ~mplitudes (SDO. $D1, Kal) yield almost identical results except fo~ the magnitudes of the cross sections. Only for the 1)2 state there is ~ slight

difference in shapl;' When KBI is used instead of SDO Or SDl (fig. 4.3). The axperimental data are described rather well by the calculations apart from the CrOsS section of the l/Z state which is falling off too s<owly. This does not appea~ to be a general problem. because in tho corresponding 56 Fe (P.d)55 Fe (1/Z-) reaction (cn~pter 5) the d~screpan~y