On an inversion procedure for nuclear transition densities

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On an inversion procedure for nuclear transition densities

Citation for published version (APA):

Overveld, van, C. W. A. M. (1985). On an inversion procedure for nuclear transition densities. Technische

Hogeschool Eindhoven. https://doi.org/10.6100/IR108514

DOI:

10.6100/IR108514

Document status and date:

Published: 01/01/1985

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ON AN INVERSION PROCEDURE

FOR

NUCLEAR TRANSITION DENSITIES

PROEFSCHRll1

TER \l!:RKRIJGING VAN DE GRAM VAN DOCTOR IN DE TE:CHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL E IND HOVEN, OP GEZAG VAN DE RECTOR MAGNIFICUS, PROF. OR. S.T.M. ACKERMANS, VOOR £ION COMMISSIE AANGEWEZEN DOOR HET COLLEGE VAN DEKANEN IN HET OPENBAAR TE VEROEOIGHI OP VRIJDAG 29 MAART 1985 TE 16.00 UUR

DOOR

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Dit proe.fscbrift. ,i.$ g(l~clg<:.k~u"J:"d do-or de promotor Prof. Dr. O.J. Poppe11'1;L

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This invcstig.:i.tioa was pa.rt of l';.h~ t"ese:ar~h program Qf the "Stichting voor Fundarn~nte.el Oaderzoek de:r Mate);"i-et' (FOM),. which ~s fin,,noially suppo~te.d by the "Nedcdo.nd$• Organisatie

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Aan m:i.jn vad(3:r>; aan de nagedachtenis van miJn moeder.

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5 Summary

6 Cllapter

6 I-1 Introduction

\5 I-2 Scattcr1nS: th.eory

21 I-3 Inveatisatton of fora hetQr significance

29 Ghapt.,r II

30 II-l A rev1<W of the theory of &pin-orbit

41

defoi:me.Hon

Il-2 The 1mplementaelon of the •pin-orbit coupling

Il-3 Re~utt• of the spin-orbit co...plios

44 Cb4~ter III

45 HI-1 An inversion procedure for " linear p~roll111etr1zation

53 1H-2 the o:ohoice of the b&Sia { siJ

66 111-3 Aco:ouracy, reprodoc1bility, !loiqueneaa

72 III-4 A summary of

the

inversion method

75 III-A Appendix

7S Chapter lV

7S tv-1 the interpretation of th• 5SNi result•

88 IV-2. A mkto~copic interpretation of the iovcte1o"

95 111 117 U2 129 results

IV-3 Rnults fo-r G°"1C OT .. 2+ e"dtations IV-4 The

1nvers~0>1

Qt 1

b

6•

110 cd(p,p')OJ: •

31

IV-S The inversion procedure fQr i""laatic alpha

ecatterin:g

IV-6 The inve"tsion of (p,d) tradaition densities IV-<\ ApperuU"

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135 Chapter

v

135 v-1

Seeond

orQe~

DWBA

141

V-2

Results of th• two step re.s.~tions

159 V-3 An interpretation of the results

164 Chapter Vl

166 VI-1 The computation of clasaic:al trajectories

168 Vl-2 'l'he differential cross section for inelaa~1e ~catterlng

171 VI-3 A cla.asieal inv~r~ion

174 Concluding remarks

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Su111111Sry

The pV.rpose of doing scattel'fng ¢11;p~rimen.ts :ts to increase our

knowledge and undentanding of nuclear structure and ~a~tion

mech.an:l.Slll$· Ttie l;lim. of the cuI'rent Wol'"k 11'1. th:l.6 ¢:0i:ltex.t is to

l'reae"t a ""'thod by llK!ans of vhich

we

can analyse the <esult' from an experiment in a vei:y d:tree.t. w.-y to establish ce1:ta10 pr.;pcrtlet

ot

the nuelear reaction under study. We foeu• our attentio" on the so-called translt1on density for this reaction.

The ~ecessit:y of such a method 19 ex:pl.C-.il"led in chapter I, together with th" .-eacUon tho0Qr_11 involved. 'this <.hapter ends wHh an 1nvestig<>tion of the a@nsi tlvity of ""~tdt> reaeHons

for certaln f~.$.t.ot:"~S of the transition densities.

Chapter II is devoted to the e~tet'ls.iQ'!\ of a comput:er code

for the scattering calculation~ fo oi:der to ll'lclude the spin orbit coupllng. thie chapter can be omitted in a first gfohd

reading.

The method as p'telu.dcd t.o 1(1 c.h.aJ;lter I ls presented in

chapter Ill. Detailed attention is paid to ita U111thc ... t~e<1l and numerical properties.

ln cho\pter Iv the ""'thod 1B applied to BOlllO shople

Qne-step reactions. The resulting tTansition densities a1:e

intel"p .. ~t"~ ~" ter""' of the shell model thoory o~ "uelear

st1:1,1.t.t1,1re+

For a n1.1mbe:r of more complicated t"cELct;.~Q'l\S:io the procedure:

and the rc~\llU are given fo chapter

v.

The validity of the method fQr these reactions is checked by means of a p~e .. do-4~t" >11ethod.

Chapter VI deda With .i.o entirely different approach to tbc ext.-actf.1>n of transition densities from "iq>er~mc~Ul data. Here the poaaibilities of th<! classical 8catteril'ls theory "" a method to solve the p'"oblem ,..., st\l<lled.

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Ch•pter l.

I-1 Introduction

ln order to exp1aitl. the role of the current work as a part of a general reaearc.h program,.. we start with ~ am.all tn.r:tv~y ¢of

the conventional way in .;.;.hi ch the theoretical aaalysis of :3eattering experiments is performed. We will put emphasis on the underlying assumptions. Ai:; an example Orte ~ould thiflk cf low-cn~i:"gy ~c~tt~r-J;n,!:'; of prQC¢n(l frQm an t;!v'eti.-t!:veI~ 11ui:;:leua. where the nucleus is left tn :lt$ ground $t.nit:e (el.a$t.!i:;: &¢4tt.er1l'lg) or iI'I

one of its e><dted states (inelastic scattering), The dat• that a"C"c obtained c-omp~i.ae the diff'er~nt.i~l (!.t'Oi!l:IEI: s~ctiol"I. If crz.e uses polarized protons) as e~g· in the ca~c of th~ E1f!dh<Jveu l'luclear ~hysica R.eoe•rch Group, ol~o the aMlYd"S power can be measured (MEL78, WAS8?).

c;ii;pet~mll?!'tt(ll t:f~r:.~ t~k.e$ pl.iiee tn. ae'leral, more or leas independent steps~

a) Optical Model Anolyoi•·

Of all possible z:-cactlou pt:"-l)C~.$1'.i=i!l:i- the e.lastio:: ac.s.ttering is thought to b~ understood best. !t \$ de~~r1be.d aa predominantly potential seat teri ng f :i;"om Q ¢0mpie.x p.:ite:c"J.t is.l, the optkal model potential (abbreviated "" OMP). 'I:lw ge9mo•r~ca1 nuclear ptOpt:l:"t.:.1~6, 1!h.I.Ch .as o.u<;.lear radius, surf.ace thicknc:S~ etc., enter the model in the fotm <)f ~ $er of poto'1tial parametets. o~e. can hc:ipe to obt.aia the values for tbie:e.c

parameters tram a first-principles theory.

practice of nuclear reaction analysis, hO~~vt:!l."1 they are obtainod ftom <I x2_-ftt_to_{the data.}

Noi.r :t.t ii;i ii;nportant to res.lize that this procedure is h~!l~d on numero1,1$ n~auinpttona.. I/lie JJtE.ntion a few;

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-6-n

l'he exchange effect, @Uenth,lly rton-local, that originates from the 1ndiatinguiahability of the projectile p.irtic.l<!o and th<! target nucle\IS parHelea, ia approximated by a purely loc<1l potential.

ii} The chosen pa"t"amet\"1i:at10ta of the OMP allows only .a

li1:1dt.ed variety of OMP' s; ccrto1n aiubtle atruc.ture dlfferen.c:e('

between nuclei a~e: $1,1.1."~ to reDMlin undetected in s\1.r::;il •~ optical model analyol•·

Ui) Ap•rt fr""" potential scattering, tl\ere "re other proceasea that <:Mtrib.,te to the elaatic. cross section. Some

of them are compound elaatic scatt<>rlng ond other mulUatep proeeaaea. I t ia l<nowu that it> lllBoy C<leee both the shape and the size of tl'le elutic differential cross section ls affected by contdb\IH0>:>6 froro the latter (PET83).

A few words should also be 6dd on the iroag1nary part of

tQ~ OMP. l!ven though recently complex n~cleon-uucleo~ f°Q'l"ees ~re

\l~cd :r.~ the .,_forelDf:;nt:loned flrst-prlnciplee calc1,1latl1:;rn.~1 lt will still remain tmpOS$~b1e to ioterpret the imaginar7 part of

the lJ.MP in terma of nncles"° propei:-t1e• only -Ila .f.t arises

pn:domino.ntl)' fr.;.m oeglected reaction channels. Therefore tho: parameters of the imaginacy part have an @lltUely d1Here"-t

~tatua than those of the re.a;l p.srt. Nevertheless they are.

sometimes fitted together with the latter~ and even correlations

bet'feen the two k1i:i.ds of p~r1;tmeter~ are known to occur.

Usually one doe~ Mt ~eeo.,o:>t fQr the processes 1), 11) and iil) explieit1y, so they will ohseure the interpretation

of

the

re9ulting OMP pa~~mcte~$· Theref~re it ahottld be considered to

p\lt 1o as llh.l(.h physics as pe>ooibl• before 8tartlog the Htti<>g p?;"0¢t:dure.

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The above c.Ot'll'JfdcJ:"ations do not apply to the ftc~1.:i:ig of OMP parameters only; sooa w~ w~l;l, izp1cQ:i,mter other fitting pl'"oblems where the 1.nte!."p'('ttOJ:t.lon of the results also needs a thorovgh I"eflection on which phyi!11¢A;l, '\:>foce:S.s;es have been included and whieh hol!lv-c n.ot.

!,) C"'"P"t<1tlon of inels.stic •c<ltt~~~O.i! <::~l>o$ ••ct ion•.

The second step in the 8.i;'l;t.ly~is, \l.$1J.Qlly involves perfori::11.1t1.g calculations of iaelast:te eroG:e. sections and aaalysif.Lg poljqi;:tS;. The ingrec:H-e:ni:.11.i- th-Elt .ax-e involved are th~ following~

i.) for each channel that c.ottu~!i. 1n1 .l.ilTI OM'f" haB to be provided

for. It. 1$ u~:1-1,1 ... lly assumed that the OM'.P's are ehe .r:i.a.mi:: for all ehAnne-1s1 although in eome c.ases of highly excited statee

it t.8.;i;'I appeaI" to be necessary tc;i C(H,"t:"eei:: th!:! OMP for the energy differenc.e w:f.l~.h thie ground state following e.s:. th~ prescriptions of lleoohetei and Greenlees ( 8EC69).

11) A rea¢t1¢n model: -one has to decide which e)(e.lted tl~.ace$

and ~h1ch couplings between them ar.e to 'be de..,lt. "il'ith

explicitly. Couplings alw;;iy• ,.re ol.l(>wcd to act ln both direo:tioa.s, b1,1t in many ca$C:$ One: of the two directions may be

QmHt~d. '.l"his bappens in the so-called [)istorted Wne Born

Approx.i~t!Od:io ot: DWBA. In fact, the latter case will be our

main item of interest in the following chopters.

Hi) >\ t<1msit1on density ha• to be provided for. THs is a function of the spatial pQ1!11t1on of the l:"e.Q.cting proj~ctile 'lhich specifies the nuclear exdtaHQn prob1>bllity. J:t follows fI"QID a model governing nu~lear ~yndmi¢~ ..,_~~ p~ojectile

kinematics~ as well a.a the interplo.y bec'liieen thcGC· The two

followiag mode-t~ -!li:-c u:!Jcd most frequently:

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-ll--The micro~c<ipie model (GER71)·

Here the nucleus is des-c:ribed .a.$ 6fl inert core to whieh

valence nucllli!::OTu~ .a.rE: c::oupled • Thi:: ene:i;'g)I' .$.r'l.d 41\g:uls.r momcntu.Ill: that ~~e transferred to the ... ·1:1,1ele11a during the Ci:"9nfil;l~ion._{1 are}

1,1,s-ed to I'earrange th~se \1'.i!.lenc:e particles. J:n mO.$t ea.sea there sr-e @!Cver<.il of the.ae ini¢.rl)4¢.opi.:::. transitions that e.Qntribute

to the total t:i::-ansition density. Spe-cf.J.:11 ~4re has to be e.ak=n

for pToce$iilea in which the pl:Ojec.tile takes the place Q( one

of the tarset t\ueleons "hct"M <:>ne of the latter is emitted

ss an eje~tile1 the so called exch~~s~ ti:ie.~h~diam~ In this c~6e

the transition density ca.m1~t be writt.el'l aclyMore ss a functiol');

of the proj@cUle eoorHnatea only (fig. I. l).

ejectile

}----projectile

Pig. I.1. Schema:tfr d·tag'l':f111 of a one-pal'·~icle •:wita.tion. lsft;

dil'Mt; •xaitation, Her'e

t-ne

sjectilB ie the same pa.>'ticZ• as t>he p:t'(!;jectifo. A t>ale11ae pa>'tfrle is e>xdted fr"m orbit>

J

to ,j'. RirJht: e:tchcmge p>'oe:es~. ~'he> pl'o#ctifo takas the pfoce of a val<tnt:e pctl'Ncie i• o"f'bit j ' wheioeas the oPiginai vorlenM pavtiaZe from or>bit ;j iea.ves the nu~Zeus aG the ejectile.

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-The coll.oct ive model (tAl165).

Now, the nucleons are not dlstfogvhhed lndividually, since thG nucleus .ts detl(:r.f.h~..:£ .::i.s a deformed liqt.iiil drop. Jn this case the l'l.1J.cl~~1=" transition det'la..1cy d-¢Ptnd.s: on the pr'ojec.tile CQc)t"dinatea only. It it;: o'bt,i:1,ined by defori:o.i:i'!..& th-Ii!: OMP. The transir:Jon i:l~neity is then11 to first Qtd,~'(1 the derivative of tho optical potential.

The rod~d part of the tr;i.nslt~on density ia sometime$

c.alled the ~orm f.actor. It c:an be viewed .so as to depict the (unno•,.,.H•ed) "xcitation pr<>b'i'bUHy o•n•it:r aa a ftlnction of tile distance bet"""" the prOjcctne and the c"ntre of ""'M of i::'he nt.1cle1.1s~ This f\lt"lt:t!.cin.(t.l cl~pcu,dence of the form factor;- is giv-en hy thi! mii::::::i:oeicopic model il'l. it!:'I $.!mplcst form as the overl~p b~t1i11een the Wave f ul'l(:t I on.s of the excited ouc. lenn J \I !c.s: lniti.al .and final 8t.1H.e. The no~mali2:ation fs.~tor thtit m.ultipl!i";$ tho form factor 1il: ori;!e:i::' to get the correc.t tr)r.(11. transition •ironsth io call"d effedive ¢Mrgc.

I:n thie collec.titJe D10d.el th~ fonn factor is obt~ined "by

deforitll:il:& r;h~ Ol1.P. Here the l'"l.Ormallzstion factor is the de-l:ormation paramt!:ter.

Rotll the "ffectloe charge and th• defor'"'1tiM P<'•-1'.,•tH have to be .d.etertnll'l.ed by i:ueana of a fitting proee(l1,1~e. 1"'his is easily dol'"l.~ in the c:aaiz: of a DWBA calculation where the cross ~e'=-t~ou. c.a.TI be proven to b~ pi::-opo:i:-tlonal to the square of th1$ n.O'l"m.alii:atioa. fal::tcr.

lo. the c~ldl'J: of a more c.omple:ii:. reai::.t 1-0n Scheme~ several exe.itation .strengths are involved. Their l;'i1t;l.1JE1 can be obtoincd either froli:i. direct ~.\.tt;l.n.g to the experiDJ.et"lt Cit' f:i;"Oro nuclear st.ruc::.tu:c-G calculatioaa. A group t:.h~Q(~tlcsl model for eollectlve excitations, the Inter<)ctlag ~oson Model (HIM) of A.rlma and Iachello (AIU76-7$), 11~~ become available to eo .. pute relative e;ii;-e:(e~tion strengths. These etalcula.tions also predic.t the Wf.l.'f .~1;1 ""h;l.-ch th-e excitation atrength.s vary from nucle:uA to

oucl~us within a certain (A,Z) roglon· !he IBM 1n iteelf,

however, ia not. reaction theory, l;it:!:(:..;).1,1$r:: it gives ao p"J;""cscI"iptiona for the $.,JJ.t.:l.al depEndencE: of the Ci."o!Jnsition

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0-10 64

_zn

_(~,p'l

..

20.4 MeV

o+

1 (o. om

+O, 5

rl!rJ ruth.

do'

d!l

(mblsrl

do'

d!l

(mblsrl

0.0

0,

i

-o.'

10 64

_zn

_(t,p'I

20.4

MeV

t\

10.99) +0. 5

-.

. ..

o.

Q

..

0.1

,,

"

.... ,

..

,

....

:·

,,,

"O. 5

64zn (t,p'I

20.4 MeV

i•

2+2 (l.80l +O. 5

O. l

\f•t

•t ....

0. 0

..

11+ 0,01

.

..

..,

.

,.

.

..

"t .,·.

-0. j 0 30 60 'IQ 12'0 150 0 30 60 'IQ 120 150 A A A

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densities. Undor """ "U~mption that the transitiort d~n$Hy

p'tcacrlptiona of the ¢0llect1ve model are applicable:!' 'V~~ Hell

and De....,rteau (~~82) formulated a c.orreaportdence scheme

between IBM trattQitlon strengt:he and norm.aliz~tiO'L"I eo~st.ante

for for~ fti.l!!to~GI· Onell! the exc:its.tlatt str.ength& have been

fixed, there are no more ~dJustable parameters left.

If we compare ehe optiea.l model s.nslyeis for el.s.sti(: 9csttering with the DwBA a"dyUO for inelastic. sc.attH1ng, we

see tha•

they differ

substantially in approaoh.

llhercas

the

OMP

has

a

great variety of adjustable po~ametc~s. making it up to a

certain exten~ Qrt ~mpirlcal quantity, the D~IIA results -~e less

empirical,' This r<;!fle<:tij

1tdelf

among other things in <he ~iffe~e~ce in accuracy in which the e~per1m8ntal cross sections

for elaatie add 1.nelMtlc scattering cao be described. A8 an

example., we preee11t tn f1gut'"c I .. 2 the elastic cross seetiQl"l 0£

6~ Zn(!I', p) and tlfo inelastic cross seetions, foi; •he 2i "-"d the 2; et,,,t'l!::8;1 -reepectlvely ..

At this point, an empirical approaeh to itielunc

sutte~ing appears to be an interestin$ pQnibUitY· ;:his «ill

be the main topic of our study.

<:) ;:he interpretation of the result".

The last step of the reaetion a11alyds eo.,.•teta <:>f th" comparison of the theoretical ob~ervablca with the 0~perimental

qalue~.

This

c<>mparisoo is based oo necessarily s~bjectlvc

4\"',Q"u:mcnt!Ji. Once a 'tE".actlon theory bee been formulated"' it 1$ impossible to

preo;l1et "

prio~:t i u attainable degree of

agreement to t:he experimental data. A c.e:rt:ain. ~o"O.ee-n.l!;tui;i e.«1'1 evolve i-o. the CQIJ.t":f31-e of tlm.e .SbOIJt 14h&t can be expected from such a theoretie.al analysis. In th.e next ph~.1J;e~ (n;:ice ~ e:ti:.Qt-er ex.perience h.as bee-o 'ilbtaln~d, .a .;:.ert;:~f,.:i .e~p~r!,.:11~.~H1t-E1l ~-Cl'J\llt c.aTI

be said t<> ho> ~"¢•pt~¢'10.l h that it h oignificently worse Q~.$-ct:lbcd by the theot"y than on the average, A close study of

the properties of su~h a~ 'exe~ptiona.1 ~ase~ ~46t i~ art

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-12-e11e~ 14ter sta.get lead to ainendments to the theory which starts

the entire c::yele again ... It is important to note, however~ tha.t

currently there is no di re ct back-couplins from the quality of

agreement between theor-y and experiment to the assumptions with!.> the model. More specifically this means that:

l) In case th<!: tbe¢~eC1cd eurvu hf.i'P~"" C¢ P•U thro.,,gh the error bars of t.he expe:time:ntal data, no further steps arc t$ken.

11) In ca<J<: the eheoret1¢$1 C'>rv<O~ devUt<O tro:n. one or more <>f the expeHmental pMnts ln such !I

""1

that th~ erlCOf Mrs

e.011.11.eeted to these points lie entirely outside the curve~ a

decioion 15 made whetl\et tlJU i$ Uill acceptable. If the devi!lt lon ls judged to be lnacceptable, sometimes an 3d hoc

Qlte~atlon in the mod~l a~~~mpt1Qns is made and n~W

theoretical curves .are generated.

From a m.:!:thodological point of view, objections ea.n be ""l~ed to thh p~Oc<:d.,~e.

ad 1) It ls ino1>fficiently known how s<!verely we have teoted the model aoaumptfono, eopeeially eoneerning t:he n"clear

l!;'tr1,1ett,trt;!. In the best eases an error ~r i!J a~~ig-n.ed to the

vslu" of the tr,.ns1tion strength, but e~n th<!n we have no firm

Mo.yt>e for"' factors exist tho.t differ fo sho.pe co,.plHely fro,. the liquid drop model form factors, but yield theorEttl<:al curv<!s that c::annot be dist:lnguiahed within the current experimental

ac.c.urae.y from each other~ A. famous example of thia phenotrienon

"""" be found 1n lo"-ell-ergy <>-~eal:ter1ng "'""~" $'>V<!ral dhtinct group• of OM!''• <1re kMwn to e"ht, yidding almost """'ctly the same cross section (IG058).

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ad ii) Here two objectlons mu•t be mentioned.

First of all~ it l.s. 1,1'(ldesirable to har,;re -only ad-hO(; 1D¢th(><l~ ~t Me'• di•posal to improve the quality of the fit of

the theoretic.al curves to the d.ata.

Furthermot"e~ the d~cf,~:l,ort c::ot1cerning the acc.eptsbllity of

t.h~ deviation is very .arbitt'.sty. Thi1;1 ie again because we haYe n.6 good as no idea of thi:- $,::nv};J.tlvtey of the cross .sections to the details 0£ t:he for111 factors or to othet' !.ngrei;l.1e::n.ts of the <;llkulationa. I t ~ould b~ t\rnt e.5. a •"'-'11 variation in a certain part of a foi::'m factox- Wi 11 yield very large changes Xn the calculated Qb6*1,"vQ.b1el!. 'therefore the resulting Qboerv~hl.es can only be c61icul<1ted. ul:) tei a. certain s.ccuracy-bar:id wtd.c.h Dli,,ght be brol)der than the experimental accur~cy Wnd. Moreover:!' in these ci tcum::i tchtc.e:s effects of rounding ('J~~ e't"l"Ol"9 Ml.ght show themst::l11es in d1ffereJJct3e bct:we.en che aumerical resulte of various algoi:-tehmto or ¢(Hrap11te.r codes+

The :i:M.in goal of the (;Ut'tel"l.t WQrk now can be summa.:-h;ed m!I

follo'IS'

1) To pre&iz:~t a m('H"e ayatetnJ;ltic study on thir: ~H~:l.'l$1t.1Y'ity of the ob6.~('v«bl'*s to the mode 1 ingredie-nt@ Q£ reaction calculat lon.S 1

.especi«lly to t:.h4 forin factor.

2) To derive fro" this study an algorithm for imprnving the quality .,, ehe fits of the theorotical e,,rveo to the data in a

mod.el-ind~p~n~en~ wBy. (Later on we ~ill dlseuss more

thoroughly in 'bO'il' r::~r Q!J.l" approach is really mod-=1-1t1.depende11.L) Th!i;s ls !fl order to pt'Ovide. .·u1 answer to the GecOt'J,Q. obj.e:ctio11 under

ii) .

3) Ai; we hope to get mor~ i;i.et;!iled inform.a.t:ton Qn, the excitation m~c.h~illam of nuc:le:i. under atudy ~ this info!:'meat \¢r'l. he.ari11g a

ruo,,t:J.-:1.nd.ependent and highly- ~iup1t1¢al character, thet;"e w;l.11 be a u.C.a;~ for iflterpretation of thia information in tei:'"m.6 Qf e.urrent theories of m.1cleol:li;" !J~l"ueture.

(18)

-14-1-2 Scattedng 'l'hsorr

X<•

order to quantif)' tho program""' .ae presented in the

previoue :t1eetiQt.1.. w.; ~eed an algorithm to compute croes 5ections

s.rad analysing powers. etarc1ng f1"0~ OHP's ond tra-nsition

d1;:n:81:te:l'.erJ;. lt ie not our intention to present a c01Dpletc

derivation of the form.,ilae (AflS70, H0071). We reetrict ourselves

to the preaentatlon of soma of tho 1ntcrmudiatc ei<preuiOT&a. As our comp\ltatl.one ore done predominantly with the coupled channels code CijOCK.,

we

will etay elo6e to the notation as used in the '1rite-up of thi~ program (K.ON69).

We consider s resctiol'I ~roeeas il'I vhteh the entrance ehannel is denoted by _{c0 and any other channel by} C· In tli1• context, we denote by 'a ch•Mel' s c0111bl.nation 0£ a apeelf1e

Q.QclC'!.11:" $t.4te tQgether with a projectile st.ate+ The nuclear

otatea s.re defined by the quantum numbers l and M, vhich stand

for the nu<:>lear spin and its projection on the direction

perpendicular to the ('t:ta.c'tf.Qn :plc-.ne. r-ct.J:i;:ie4;:t:f.v-c::i.y. To d.ef'trr.e th~

po).re~c.le stat.es, we use the quantumnumber ~. In case we consider inelastic scattering <""ctlono oTily, the lnttinsic opin of the 11\"'Clji;i:,e.U.~ (..i!.nnot change, and therefore cannot serve to dioe~ng...hh. sev<>ral "hannels. Th.e projeHile otate the11 io completely defined by the projection of th" proj<!ctil" intrinsic spin ms on the d1~eetlo" perpe"d~e~l~I;' to the re~etlo~ r>l~~e.

This means that here "' ~qu.als. m~. In the ca1;e 14'e »ant i:o

conudel;' one pi;>rttele tra"afer reaetio"•· " ati;>"da for l:>oth th.e i<ltri.,al.c spin s of the project I.le and its projection m

8•

FQr """h transition between the channel• (I

0,M0,o0) and

(I ,M ,o ) we con define " p"rtid ei:'oee section. Thie e~n be

co co co

seen as @xp<"<>$$ed ;1" the .... ntx ele..i:"u of ~ tr.>.n81n0<1

operator t~

(19)

In ease the several spin directions .,.r.e i;i.Qe df.t;tf,ngu:f..i:;h~d in the exper!~n:i::.1 we 'h9ve to sum over all final states aad

.;.vet.".tige over all initial states. In order to coid.pute the e.rCJsa section with g~ven 0MP1_$ _.ttr;'IQ _{t.J:"Q~$;t.tfon}_{den9itiee.1' one needs to} relate the tl:".l!l.n81t1o:n operator to the wave fone.ti-ona that desc'l'ibe the 8c1J.ttCt"in.g process, and, more prec.iselyt to the asympt<>tic behavi<>ur <>f these "~ve !ud¢r (¢nG. 'rhis Mtmptotk ~ehav(w~ oal'\ be oxpressed in the so-called T....,.,Ui>< ele.oet1U.

These oan be v(e.,ed oils ., $et of transmission and reflection c~ffkient& for projectile wave hh'l<:tions with Well defined <>rbital and t<>Ul Aflg~l"r '"°"""l'ltl.\ ~ ""d j, the so-called partial waves .. Employing the T-""" tri>: eleincttts ""' write the ctosa •i:ct1.on as:

&:> U '{Ml U' Z

d!l • constant •

l

I I

A {e)T

I

{M} ).,). '

(I.2)

Here, ;\, stands for the cot1tbin~ti011 ot .t .al'\d. j. 'tbe BymbQl {M} stands for all relevant spin project.ions in the entrance- .s.nd exit channels ... The coefficient A is not written out explicitly; lt cont&ln8 the angular dependence ln the

form of

an associated Legendre funetiQn

ot

the scat:t.~rlng .s.ngl~ 0 ~l'ld. ft;trthe~ore it.

lncorporat.es all relevant vector codpling ~o~fficients.

ln ord"r t<:> c<:>mpute th" T-matrix elements ../'' '. "le will encounter two different methods... The first one ia exact in so far ae the number of channels eoaside'l'ed deeeribes t.he scattering problem completely. This ls called the coupled-channels method (CC). In the second method the T-m.atrix is obtained in an approximative way by neglecting couplings that only contribute t.o t:he excitation process in second or higher order in the coupling strength ... Thie is named the Disto~t~d Wave BQrn Appro~imi;ltio~ {rni~A)~ already .entioned a few times before ...

(20)

-16-ln or-d.er to ohtain the CC-~alues foI" the T-mat:i::-ix -eleill:e'llts

we have to solve the co ... pled Schrlldinger-like equations for the

particle wave ful'lcC10(j,6 1-0. the eh4nn.els involved. In ord@r CQ d¢ so we tfrite the total w~ve fanct1on for the nuc:1eu9 plu$ the !f.eattered particle as an ex:pa·(u~iOQ. 11'1 ter:i:o:s of t.he set of n1>cle..r

""'l'e

fu<>ctions t ('(Cl):

Here r denott::S the relati11e displac::ement between the

p.art1clt-and the nucleu.8 whet:ea.s a stp.art1clt-ands for the internal tar.get nucle~'( cQQrd,1'1"-s.t:es. We insert the above ic::xpaTIQ1Qn 1n th-e twl) partii:.le

Schr'C!dlnger equation:

(I.3b)

Here Hint accounts for

the degrees

of freedom of the target

nucleuat T is the kinetic e~ergy operator and V stands for the

inte~3cti<:ln between the target nucleus and the prOj~¢tile. Ne~t

"" project bQth members of (I.3b) onto te. Since a p1'rt of the intet:oil'.et.l.::m ope-ratQr V is non-diagonal in the va't10uG. nucle~r

~tates {

<-},

this yields our set Of

¢0upled

partial differential equations for the partlclt:: wavt: £ui;t¢t10tl.S Xe(_!):

(T+u

0

-E

0

)x"·

-1

v¢e'x",

c'*c

In (l.4), we have used the following definitions;

(I.4)

(I.4s.)

(l.4b)

(21)

Note that the l.itl:~r two ~xpx-ea.sicmB are functions of

.!·

Th~y

r~sp€ct1vely acco~nt for th~ d1ato~e10~ of the wavefunction

x

0

"-"d th~ ~~Ci.tot \011 prubability as a function of the relative Qp.et13t por:;itl.oa of the reaction partners. The f'm·•etJ.OT1. Vc.ci

~therefore,. is a transition denslt,Y ns Introduced in section

I.I. As stated there, it. .f.& ObtiJ!ned from eith.er the collective 11H>del o~ tho rnicroacupic model for nuclear e~d t•tion$. In the next section w-~ wfl.l i:itu.dy the s.ensitivity of the cross sec.tio11 and the analysiag power (o~ lts functional form.

In pr1ne:1ple, ~he $-ll:t {I.,.4) is a set of an !11f111it12 11\lmher of coupled equations. In pt.,ctlce~ the set is truncated to those th.$t. c¢('respond to the lowest few excited o.u..c.left.r -~t:.(l~e:ei wh:C.ch ar~ usually the states coupled most str-ongly. FQ:i;" thf.1.:1 i:-e.a.Bon~ the c:o..iple.d c.h.et'lt'lel~ method 1~ ala(! 'tcf0rred to as the close .... coupling method. By neglecting the more weakly C¢1.1f,ll~d chann.els ~

however~ we introduee a loss of probability fl~x· We compensatE

for it by adding an i"'~Biaary P<l!-rt to the d1ag¢nal putent1al energy o~er$.t.Qr. As ~ith a complex refraction index in opt.i<;.&,. its function i• to red,.ce the ~mpHt\>de uf the 'lavefunctions. The teeulting complex potential U¢ io called opticol model potential; it i• the 0111' M l"t\"oduced in section 1.1, As stated thel"t:.. "lt~ pa:t'.ameters are usu;ally obta.11"1.ed fr-om .a £it to the experimental data. In do1..rag r:io, it: :{.r; a~!J1.1med that for the

clast~c channel _{(c•c 0 ) the right haad par• of (I .4) h}

negligible compared w:( th the ~magino.ry port of Uc. In case this auslllllption is not valid, an indepe<>de<>t .;>pei¢al model O.nalrsis cannot be perfor:m.ed. In. t.hei,ii;.e ¢1, t:'r;~11.s ~~IJ.Clt$, the couplings to (some of) the excited states need to be tak~l"l iflt(t a;.c¢¢1.11;1,t. explicitly when performin~ th" O'!P Ht.

(22)

In order to oolve (1;4) numerl<:dly,

ve.

use the well known part1"1 w11ve expansion. It loves lis vith a set of <:ou?led ordi11Sry difhrotnt!.al eqllllti<;>n• for the radial puu

~~(r)

of the functiont Xc(_t), where again for brevity the q.,.0,,tuOl 11umbers ~ and j are replaced by ~, Thi• n~>I *et of equations has

a

u .. u ..

r

·~ru~ture 88 (1.4):

(1.5)

H'

lhe ~~etors V~~' , app~a~ins 1d the swnma.tion~ aTe the radial

part~ (>f the transition densities together with some vector

cQ.,pling coefficients. Th<: <I.et (t.5) h solved nwneric.ally,

).

hereby i"trodueing a matriK of 8olut:t.011e { ~ } ). , because of ~ COp

the fact that the integration starts with ~c •O in 1111 parUsl .,.,,,.. ehsnnels exoept fot the P*rtid V&'1e chstmel (). ',e₀), whereas (1. ',c₀) runs consecutively over all possible partial wave channe\s ~elonging to the entrance ch1h'ntd. tllue f11t1ctiot1s

{~).}

l , are th<it

~o

e11-1led mathematical solutions. The

" "o.

physical solutlons co:>ndat of linear combinations of these, ouch

~hat the asymptotic b¢undaty ¢0dd1tlQ11S are -met. In this way we

obtain the T--..~t~~x ~1~m~nta:

lim I'."""

(I.6)

llere the fone.tions F and C arc ...,gular and .l.rree; ... lar codOd!b

functions~ respectively.

Ha Iring introduced the CC-...: thod fQr eo,.puting the T...,,,,.trix elcmenct, v;; p;11y attention to the DllBA· Now we consider the

case of only two eh4~~elst c•c

0 and c•c1 whereas th@ coupling

(23)

order 1n the coupling otrength Ye

may

omit the ter"' 1ti the right hand aide of (I -4) that allows for the backcoupU>'lg ~tom the exc1t8d state to the ground state channel. The s..t (!. 5)

reduc.e1;1 to t"il'O @qus.tions:

d~ t'(t'+l) A' _{(I, 7a)}

c-

d?'"

+!,!

"-o +

----rr-

-i;; )~ - 0

"o "o

d2 (l +1) ).

-I

\ \ ' I.' (-~ +uel+ r~ -E _{cl cl})( "

v

t

(X. 7b) I.' "0"1

"-o

Agai" ve <:<In proceed alo<1s; tho& above way tQ cQmputo the T-matrix. Note that e~uation (l.7a)

:r..

homogeneous, and thllt "e theref<ore Mn apply Green'• fonct1oos for solving (l. 7). Let

*(+)A ' *(-)~'

~ and ~c be the h<:lmOgsnsoue solution• of (l.7b) with

"-1 1

ingoing end ontgoia.g asymptotic behavionr, i;e8pectively. Then the asymptotic hehaviQur of the full eoluti<on of (I. 7b) can he

shown to be (AUS70):

(I.8)

If we compare this e"J'reesion wttll (l.6)

we

see that fo this approximation (DllBA):

(I. !I)

In other 'fiiorda; the T--ma.t.rl:z. elements depit.nd linearly on the form

fa~tora VI.~'

and therefore OD the tradaition density.

CO Cl

Thie concl~eion will be of ~ltal 1mportanc~ later o~.

(24)

-20-I-3 Investigation of form factor slgnific.a.nce

The first item of 01,1r program as presented 1tt section I-1

announeed there. 'kle will present some: te51.1lt!}; (If c:.ale.ula.tione

that have been done 14 oTder to gain insight ~~ the sensitivity

of cnlc ... bted <.ross s@ct10ns t<) cert"itt details of tbc f<)rm

f.a.c:.tors. As f1r&t. e;ii:umples we take the one-step i:e~etions

56 ...

+

88 ~ + +

Ni(p,p')0

1 ~ 21 at 20.4 MeV and Sr(p,p')01 + 21 at 24.6 MeV. Both r~aetio~e have be.en cdculat<:d I.a ~!IA with OMP's as ~i~en by Melssen (Mli:~78) and Wassenaar (WAS82), where the form

factors have beed taken from the collective .od.el prescription,

th"t is:

V • 6

~U

(I,10)

'"'o

""o

a

It

co

.,here R is the radius pa•ameter fr""' the OMP multipHed w~th Al / g • As usually, the ~ valuM h<tve tie<:oo. obtained by scaling tlie

theoret;1cal .croiJs sections to tb~ ~xpel:~'Q)4dtttl ones. It turns

out

tor

Basr that tbO c<OH ~ectiod bears a gr8at Simila~tty

with tM ,;xp~rimoioo.td data whereas for HN1 oiajo~ diserepandes arise at the backward angle~ <13 cao. be seen from figur" I .3.

Now the <lc>Ct step in our calculations COl'ld~t~ 0£ making p"rtur~at~O"t to the form facto~.

This

ia done by replacing tlie

•

fo= factor V , , by V where•

~~o c.co'

~ere the function S;I. (<") is " cubic spline function "'ith a ¢edtr" r-value of l/<0.625 fm atid " half width of Q.6H fm.

The

height is ta<.en eq ... al to 6% of the maximum hdght of the c.ollective

fonn. factor. With thid perturbed form fa.ctor ve ~gaJ.n eQinpute

tt,e eross section, say : 1.

l'O~

the unperturbed cross section ""

(25)

dO'

dQ

O, l lmblsrl 0,1 30

• _••••

53Ni I~, p'I

20.4 Mev

2•1 0.451

...

_••

_•

.

..

_•

••

_••••

•

&!sr (Jl,p'l 24,6 MeV z+I {L831 60 90 120 150 0 ~

_{c.m_}

+O, 5

o_o

A +0_ 5 0,0 A -0, 5 30 60 90 120 150

Fig. I. J. Cvoss sectionj

and anaty?ing

powere fo>' the 08Ni arid 88

_sr•

_(p';p

_{')2: !'eactiQns.}_The58

Ni figures haw been taken from IMEl?8! and the 88

s-r

fig1<l'f!B am f-rom

(WASS.~).

(26)

the two cross sections by mean• of* qusotity

x

₁2 that '• defined ao:

(l.12)

~bere Che factor 625 is t~trQd1,1¢ed to simulate 4d aveTage dat.a

e:ttot of 4% and N equals the ""'"ber of a"l!l,.;, for which the c~I)$.$ se.::.tions h.ave been e~puted. It hs.s been ver-1f1-ed. th4.t. the

x

~ ~ thus de Hoed is in good appro.:lmati<>n proportional to the

square of th~ perturbation height. By mean8 of theee

x

12 we can a$eig" an uncertalnty width to each radial p<>1nt. This 1B accomplished as follow&· .\&~\l"Qle that with such a 6% perturbation we get •or i•8 (that is at a radius of S fm) a

x

₁2_ value of 4.0. That ..,ans that with a pe:ttu:tbatlon height of 3% we wo'1ld get xi2 eq""l to LO. The latter value 111eans that the

d<I d(!

.ave~age distance between dO and ~ then eq\J.l\ls the average sbt:lstieal error, in otller wo:i:-dt: with the given e:xpe:rlmentd

.(lc.c.11racy this 1.t: About the smallest diffei:--e~ce det.ectable

between the two curves. Therefore to that tadi~l point of S fm 'le assig" an accuracy >11dth of 3% of the maximum.

In figore t .4

we

preaent thes" 1J11cen<11nty wUths. The soU<I eorve represent& the 88 Sr-case, whereas the dHhed c .. ne is for SSN1, the ta<lbl coordinRtes for

SSsr

are •ealed by a factQr ()8/88)113 to account for the dlfference in radl1 of the two nuc.lei.

We obscrv<> two features. 11:1.ratly, not;rithstand:l.ng the great difference in th<:o er<>H sections for the two 1:uctto"'s

c.one1dered11 we observe a. great simil.erit)' bet.we~11 th.e t.wQ

C\U,"Yes. Secondly, we note that there is a relatively larg"

1111cer~aiftty width over the *nt1re rad1al region: it appears to

be of the order of 30% of the value of th"' collective model form

(27)

~

103 :;:;.

c:

-!!!

_....

"'

_Hf

u ~ ;;;} ___:

"'

I

-101

a

2 "

r tfm)

..._

...

6 "

8

10

Fig. I.1. 1JnMl'Minty ba,.da for 58Ni (daahed) and 88sr (fuiiJ cw

obtained from 1-apZin<l penurl:>atioM. 8=20.4 Ml!IV.

What ln feet turM out to """8~ tM9t l.:.~e:e "Men<Hnty value• 10 tile folll)Wlrtg. Aput ft¢ill thoeo ra<IU.1 pudtlon of tM perturbation, its width .appe.ars to be of vital 1mpo:i:-ts.nce for

the ntt.i:iln.-ible: d~t.ec;:.t.lcn 111;cl}'l"~¢y by me:an& o.t a

x

2 cric.~t"fOt'I. In ol'"der to lav~liiltig.at'C:: t1J.i:s;. we 't~pi;;iit the .Q.bQve Pt'"Q(:e:d•,;1-1::'"'&> "bv.t

idste4d of using one splin~ function ~s a perturbation~ we shall

(28)

-24-~

~ ~

.:!!

_{,_}

<l;> u i:::: ::l --'

:.:'.

103

102

101

0

2

4 r (fml

l

spline perturbation

6

8

10

Pig .

.r.

5. As figure I.4 fOP 1- to 5-spUn" pertUPbcwions.

use groups of

2, 3, 4

and ~ adjacent splines with equal heights.

These perturbatlo.n f1,1t1.et;..i,Qni;1- represent 1_jbumpsw

with im:.rcaia.ing;

"ldth•. l'.;.r the radius where we apply """h

*

b\1111p we take its "1ean value. Each &et of pe•~"~b>;1HQtts ... tth s fixed width yields

(29)

In figure I .5 the total e;~t of r:hes~ l:.1).l"v'.t!:ISI 1~ pr-et~nt.~Q:. ;lg~t~n fot' both nuc.lides. The saine sc.aliug presc.riptiou as. defit'led (lbo•< hM \>e.;,n "i'PUod to all of th~ ees~ curvea, not only for the radial coordinates b,.t ~lso far the bump width~. In oa~o xi

from (I· lZ) would be a linear functional of the perturbation function~ we could compute the uncertainty widths corresponding

to the broader perturbo.tlons from the cutvee of flgure

r.5

by

applying the easily derived e~pre$Sion:

(I.13)

Hore f(r;s) ia the uncertdnt~ width o.t r.i.dhs r •~•ultlng fr0m a perturbation S. lt turns out that in fnct the nppUeab1llty of (Ll3) h l1mH•d to thoeo 1eh upper p"n of Hg. I.$. For the practical eomputnt tons of enor bande therefo>"c, we will apply

the ~ct~ of numetically obtained curves as :f.n flg. I.5 rather

than the analytical result of (l.13).

If we apply this figure to the collective model form

f'a.ctor th.at is eo.nQI'l.1)1' Uf1ed. f(lr t.h~ ea1¢'l,llollt~Q?'I (If th~ C::t;."(H;l.$

section of the 56Ni

(Oi

+

2i)

excitation, we end up with an .f.eeu'('.P.~y f.h~t. 1fil ~.,.r bette'(' th.~'tl. J.0%. Itt p:i:"~ctice1 the

CI'I'Or:-bars assigned to the

a

values are in the 5%-r.ange.

Or.1 th-e l)t.he:r: h.a'l;'ld 1 if thct~ wo1,1ld 'b~ ,,. $t('1,.11:::t\1't'e of, say 1 2

fm width round r=-2 fm, this could only be detected with an

«oc~~~cy of ~bo~t 25X of th@ ""'x~mum fonn factor value.

(30)

-26-104

i!3

103

"'

<;;

s

~ 1o2

"

"'

101

I t 1.8 interestll>S to inve~tlgiote to wbac e.:t.eo.t the uncertd11ty widths can b" ced"o:;"d by increasing the bOtob•rding enu·3y of the reaction. This seems to he not unreasonable, &ince. it1.tuit.iwly the radial resoluUo», aad the naoeiated prnjMtil" wave number are thought co be r"lated .q.,,.n~ities. We repeated th.,n:foro; th" abo"" procedure for the 08 111.(p,p'

>ot •

2t

re•cHon at 8'1etg1"• of 30.(> """ 40.8 M<;V. The rnulu ••e given in Ug,.re I.6. We oba.,rve indeed that tl>e curves for 11igher energiU. lay

coJ:\$14erably lower~ this is the ut0st. obvious £0\" t:he na"°l'011J

p<>rturbatlo11s. SurprUhgly anothn ~tg"1ficant d1ffer.,nce conohts of a more oae'1114tory ltehavlout: of the cu!.'V':O.

r lfmf

l'ig.

I.6. l!s

figu:t'e

I.S fol'

58 Ni. Lsft' E=M.6

MeV.

(31)

1) The eensithHy Qf ehe er<>;;e ;;eeti<>ns to details of the form

factors depends very heavily on the locations and widths of

the$e det<lU;;.

2) Despite the great differences 1n structuC"e of the C'C"0$5

E1ec:::.tiod11 f(lr SBN1 .i:ltad SBsr:. the ude~rt.~int::y widche of both nuclides have a very eimlla~ behaviour as demonstrated in figu~~

l.4. and I.~. This l!ICans thet the global x2 _{e.riterion, that}

forms the back.ground of this figure might not be the one best

su.ited for the detection of features of the individual form

factors... We will come back t.o this statement more extensively

id seetiQd III.3.

3) A1th01>$b it hM not been tested for a wid" range of nucl,,i, it seems not unlikely that the applicability of figure i: .4 ill

""t t,"est.1eeed to:> 88

_s,,.

_{attd 5B1'1t.}

4) In order to improve the radial resolution nbout

o

fnctor two:>

QTI :lne-:-e~1;1;e o.t' t:h.e 'bom.b~rd1ng energies towards JO MeV seems to

suffice.

5) Tfte linear behaviour of

X

as a functional of the perturbatioa

allov& os to apply

the

approxit1111tion (I.ll); its eppl1ceb1lity, how.ever, is restric't-ed to ne.i-l'"ow pC:1,"'tu:;tb.a.t1one in the c.e.atrs.l pert of the nucleus.

Finallr it might be \1Geh1l to note that the <esulta of figurn l-5 can d@o be e~pres•ed by an empirkal formula, i.e.:

(l.14)

In tbla (Q'°'"l" t.

1,

the udceruinty width in the fom

fa~t(lr didded by its "'4Ximum value; 1\ h the r~d-"l~l position of

• "bump" in units of ( 56/A)113 fa, whereas II is the width of the

b1,1111p 11'. th~ same units+

(32)

-28-Chapter II.

Ai:. has be-en .et.-1.1ted i:n section ;i: ... 1. wheoever one t'{"i~I}; to

extract the values of i::i:o.e Qr more modal par~-m~ters:io it 11.:1 w~y

important tQ i<ldude n "'8ny kn<>Wtt conttibuUna mechanisms in the '"odel aa possible because an omission of these "ill be kind of

¢ompe.nssted for by erroneous valu.CS of sa-aiie: other model pa.'l"al:l;u:teT~.

Now a ma1n topic of this <1ork will be the r.,coMn ... ctiQo. of

form facto("~ from. c.roae sect10U3 .e.nd analysing powe"tU fot idel.aatic

scatted"&• If, ,,.g,

we

ll-:PPlY the collective model, the fotw factors result from the deforoia,tion of th<: centr<l-l part of tho OM!>.

lt is obvious that then not only the oentr~l part of the Ol!l' •hould be defo...,d, but also tll< e""lo'"b a"d spin o1"bH parts. The defo...,..ttons of the$e puts lead to proceue• that contrilrute,

t.ogie.ther with tbe (:Ctatral form. facto-.; .. tQ the e~c.itatlon proce:S.$

under study. They are known M the coulomb deformatlon and the ~plta-orblt defonmt1011::1 re.spe.ctively.

<he coulomb d~£o~mation haa a relati~ely transparant

strnct:urc; it ¢6.11. be written a.e; ~ form. factot tb4t st~ply adds to

the form fa.etor from the CedHal defomatlon. It baa been included (be i t approxlm11.ti11ely) in the coupled ehannela code CHUCK.

'l'hia ia not the cas~. how~ver, for the spin-orbit deformation.

Sf.nee tbere are, on the. other h.t.i;'ld.1 r.u:·g:ent reasons to 1,1.Se espe..c.ially

this co1,1p1ed chanm>h cod", we devote thio chapter to the dea~dption of ne¢usary modification& ot C!l.tJCI{. Fo• the Mk<!: of

i::ompleteness.1 we give in ecc:::t1.0n l;l--1 a abort aurv-::y Qf the

treattllent of th>io spi,,~orbit defotwn~on as given by She(H£ •nd

co-workers (SHE68-70) + We proi:.$t::nt t.he exact formaliE111111 'b~sed o-n. a

¢QD>pilation of v~~haar (V~R72), which hO$

Bo

far been de•lvcd only

for fltl'Jt order proces8C:$ 1,tm;l w~ also mention 0.TI $ppro-x1ination due

to the Oak Ridge Group (FRI67) which can be aeneralized easily fo( higher order processes. Section 2 dola with the more tcehnieal aspects of the m<>d1Heatlon of CHUCK, "he~eae aec.tion

(33)

1I-L A r~view of l:he theory of spin-orbit d.eformatiol"l

h the opti<41 potential we note an obvious dUferenee between the central pe~t an~ the c¢~lo~b p~r~ on one hand and the spin-orbit p.s.rt ou the other hand. Wher.eas the fir.et two operators have a purely multiplicative nature, not deperu:l.1t1.g oa the. quantumnumbers of the ""ve fonction• they act upon, th" latter depends on the•e q,ull"l.tum c\umbersT Thia difference shotii:s it:Se1f evi:n more et:.r-ongly

i f we deform the OMP. Ih" d<>nvaU<>" of the •pin-orbit coupling accordh'lS to Sheriff and Blair starts by writing down the foH01<1ng

e~preaa1on for the spin-orbit part of the OMf:

n~.1)

lien> p "I' (E_) '"P'""""~" i;:he ""clear density

and

s is the spin opctator a<>tlng on the projeotile spin· Ibis oxptoH!on 11 .. 0 1,.;en given by Brown (Bll057) "ho d'°l:'hed it fo a high-energy limit from the nucleon-n:i,icl.e:=:on int.ernc.tion by :ine.a:ns of a folding procedure.

I t 1.& COl"LV'enient to decompose the matrix elements of VSO 1k6 .a sum over products of nucle:l:l't and pi:ojec.tile parts. In order to make this explicit by ~e~~$ o~ the W!a;ner-Eckart theorem, We exploit the spheric.al tenao'{" ch.8.~~ei::e:r of eai::.h of the component9.. As :i.::i '(,)'ell l<nown thh lo "eeomplishe<;I by writing:

(l'.I.2)

(H.3)

(34)

-JO-M3kidS use of (II-Z) 4dd (II.3) the Vso, up to flrst

ord~r

ld the

d~form.Jt.ion pat".ametie'l;':ti; a.6t , 'becomes:

Here pD H.odd1;1 for the 8phericd term P(r-R). tbh term c.annot

cau.01': t rads it lons be.t1o1 .. en nates wlth diff.,rent "'"8"lar

"'°""'nta.

The other t<!tmll h•ve been given by Shoi:iff (SHl!:70) as:

(ll.S)

ll<!r<!:

t

(f)is the 2t• a opl':r.OtOl' worl<ing on the initial {fl,,,..l) state.

1 - - ~

*

'the Hrst factor (1.e.

Z"t.il

aeta up<HI the nll<:\e.oi:- states wheres& the aeeond factor (1 • .,. y" ) acts upon the .ons~lar part of the

projectile et"te. The third factoi: (1 .e. the. expreaU<>n between

square buckets) acts upon the rll,!ial part of the pl:ojcctilc wave d

dr

operator. The ent:1re expI:e&.oioto is <>ften referr<!d to u the full note the tecotod term herein vhich contains a

fi,znCtiO~i.

Thomas fQt'lll.·

Unfo-rt.tl1111.t.ely:1o this e~~t expression cannot be gcQe~•l1!.~d

readily to .a.e-cond order prac.essest s;uch s& t:h~ dire..:::.t exc.itatloc. of

a tvo phonon st.ate. S\l~h exc.itations a."°e nev•t'tl'Jele~a believed to

cont~(b~~~ substantially to che ~rose sections for t~o phQdOtt

~tates+ Theri::£ore. io. c.aee we 11i'i8b to ¢(1rtl!;l.'er\1¢t fc.m. factors fot these 'Second o:c-der processes'I we need a form.u.lac1on to ~ceount for

seeoo.d order spin-orbit e>Celtation. This "'"' ~ foo.o.d in. t:he so ~alled Oak Ridge expression. F<>r first order excitation•, it

cooaisu of neglecuns the second and thll."d te,,... of the thlrd factor in (II,5) ~~d, in order to k,.,ep the operator hermitian, ~ddi~S the

(35)

he,,..,~t~<>n conjugate, The analogon of (11.~) then reads:

(XI.6)

Altho~gh th15 expression is only #PP~O~iwat~, lt turns out. that the

ero.st.i sections and ana.lys.t(lg pOtiire'l:s tb.at are calculated with it show a great rese-m.hlai:tee wieh the result.a using; the full thou.aSi form ¢f (II.5), except for the m<>H fQ""'""d •m3les (SHE69 ,GLA6'.I),

In order to gener~1:l,~c thill operator to sei:.oo.d or.der pt"oceseea,. we follow ,o. d.e'l:'ivation .analogous to the ol'l.e for l;:h~ eeot~-l fotm. fsctoT. The result.tns e~pteGsion is (assuming d1•2)

l

3~!>

[;.1.·s·l +a•1

;t'j

(2 2 O O

lt>~'O)

r

~ ---:~..--~-~ /(2w)

1>~·-0,2,4

(II. 7)

whereas the spin orbit excitation strength in this case ia given by 2 2

i&QR inate"d 0£

a

80R for the first order term. In case

we

intend to

describe " t"'o quadrupol'" phonon excitatio.-. ata>:th•g (i;.,,. a

o+

ground state, only the tei:.i t~

....

th6 su111111Btiou with 4.1.' equal to the

nuclear spin 0£

the

¢0q&1dcr~d two phonon state suTvivesT

(36)

-32-II-2 The i"'plementation of the spin-orbit coupling

Tvo aepccts •~ to be dolt vith.

1) In the full Thomas expression, three terms act upon the

pi:-ojcctUc v11ve f'1d¢tio"·

tvo

tel;'(!itl Ari': multiplicativ<!:, wt, a&

stated above, one term contains a radial derivative. (This is the

s"cond te.111 from th" third factor of equation (II.5).) Originally,

t!\e lntcgi:&Ho" ""'t!\<:>d .i~ .. sed ~" Cl!l.TC1{ for s<>lvlng th" tadial equations (I.5) works 69 follows. Lot the vector ~ be the set of

'1nknovn

tunct~one

{'tl

at

a glvcn i:-adl"$

r.

ijy lnti:oduc1ng the

"- ~ t(t+l) )

dis.goaal m.at.rix U :io U • -F' TJ + ---:;:r--E , and the non-diagonal

2

f:!:'•

c r c

~t\"'l:;< V 1 V c.ct•

-W

~c•, we write equat:ion (I .. 5) as:

(II,8)

the matrix I being the identity matrix.

No~ •~•"m~ a ~csh r•i~i:-. 1•0,1,2, .•. , and

i

knovn in the

p<>ln1'9 1 and 1+1. Then a value fot ~+

₂

foll<>Ws ftom: (U.9)

~t can be proven easHy that t~h Numsrov-like method bu ordon:· 2 in ehe. 8t"-i>she. (IJAL74). N""' e<:>.,"1d..r the 8ecoo.d tei:m of the foll

Thomas operator. It causes (U.S) to be of the form:

(II.10)

A genet.aligat10n g.f (II.9) could bl;!: based on the consideration that 1t is derived from:

(37)

whereas a ~im:l

ta.t:'

eymmetric for1rrt1la for the fir!lt detivative is ghen byt

(II.12)

Making 11Be Qf (II.12), the equivalent expreui<>n of (U.9) then becomes:

(II.13)

Thia '""a"" th~t fot every x-adial step the matrix l+llW has t<> be

inverted ... Even if !:.. 1.8 !!!;mall enough to insure a Bt"1ble lnversion,

thl~ is a .-at her elaborate proc&du.-.. Therefore we ~~e the approximation (HAN65) foate11d of (II .12):

.!!!lLl_ M Z5f(r1-4Bf(r-t. )+36f(t-2l1)-16f(r-3l1)+3f(r-4t.) + O(A~).

d"l" 1.U

(II,14)

whi¢h ,a:lvetJ an. expression analogous to ('.tt.9). In ca$e 'rril'e do not have en<>ugh precedittg po~ "to a.t our disposal, {II, 14) h"s to b;, replaced by: df llf(r)=l.8f(r-a~9£(r-2d )-2£(>:-3.l.) -l<l(l1J) _(XX.15a)

d7.

df 3f(r)-4f(;c. )+f(r~ ~(aZ) _(II.15b) or di: -df t(r)-Hi:-t.) -t0(<1.

l

_(II.15~)

or

_dr.

_c,

depeno;I~"g "" the number of radial steps already integrated. Thia

approach has been tt:!Slted in the following way ... We c.onei~er the

difft'!:re:n.t!a.1 ~quati(1n fo"t th~ spb-erical 'Bessel func:tio"O.e:io wh1¢h ~.1;1rt

be written in two fo.ri»..s:

or (tI.16)

(38)

Eq11ation (ll.16) ean be solved m.qoerieally by applying " $eheme similar t<:i (II.'1)1 while f<:ir equation (Il,17)

we

w•t

use an dtern<ltive schame employing (II, 14).

~Qth schemes have been used for ~-o ""d t•IO resulting ia,

respectively. a rapidly o•c11latins s<>lution and a. •ol"Uon

m.onotonously increasing fo£ a lsl'"gie: x j;pt~rv•l+ These calculat1o~o

have bl!<m rcpe<lte4 for several &t<:pab<:e: ll•.l, ,OS, .025 • • 0125

a~d .00625. Integration has been done between x•O and 20. For the

latter point t.1'e ~lat.ive diffe'°ance bet"'c~n. t.he aolut.ion for a

give" ll and the C.•.00625 aol.1tion are depk.ted in figure Il .1. Ill! not<: the following,

For

t"°·

both schemes have eic..ctly

the

aame convergence orde~. (The eo1>vergence o..:de~ ts 4ef1ned as the slope of the e"rves in figure U.l for "reasonable values" Of ll.) I t la cleat f .. Om the fisure that for t-0 the •1>i;>Uc~bility of both sche:mu8 h equally good,

For i•lO, the -tOd~r,gence order" c.onsldered :coua.d 4.•, l ~ of both

schemes ls less than it ie in th-c .t..O uee:. MoreoveT;i. thiE

df

approximat10n for

;rr-

ceuae8 a alight deterioration of the ~o.-.versence fo .. ft811 "*luee of ~, ExpresBion (U.14) is exact in caee f 18 a polynomial in x up t<:i the third degree. The aol,,.t1on~ of (II.l6) or (ll.17) however, for ..,.11 x behave like x1• Thia means that a H•tem.atlcal error b introdt,teed 111 the fi..:et pa..-t of the 1ntegrat1on intatval in t~e eaaee where 1)4. Th8 hlgher the 1-vdue, the la.:'gcr thb l'rror will be, Once tl\e ~ ... oetio"a start

osc1118Cltig1 the derivatives will beCQliC l!;lm«ller in absolute v.alue1

aod therefore the el."'rore will become less. On th@ other haod, since we Me th"'t the relative er .. ot in che i .. 10 case for I'. ) 2• lO-~ 18 leas tbart for t•O, We o;\o .-.ot have to worry too m11c1'1 •bout this 1owe:t" co-o.ve:rgenc.e ord6't·

We conclude fro'" this numetical expel."'ime.-.t that expression (U .14) for the first derivative givee sufficiently accurate rce .. l h for Bessel-like differential equations, such ee the radial Schr!ldingar

(39)

L.. 0 L.. L.. (I)

10

0.01 v;

l:

0 A:

l: 10

step size

0.1

Fig. II. l. Conve"""""" fol' t=o and t=10 f!lotwtions at x"30. 5oUd curvei!: /flotu.tiDn of equation II.17. Dashed curves: 1Jotwtion of oquation II.16.

Z} The second aep<rct of the implementati<m of the full Thomas and Oak Ridge opin orblt co..pH<>J!;$, ie the computatlon of the !llBttlx

u•

element• for ~he pa~tt~l wa~e co~pling, that is the Vee' frcm equation (1.5). In the derivation of

(t.,)

~~ o~itted the individual

(40)

-36-t, s btld j-d.epe:ndenct:· Kei:e we h•ve to give full act:Outlt Qf all relevant quantum numhr&. '!'hey are:

I,I' • ... the nvcle~1; spin before and aftgr the l:o.tera.etio-n.;

s,s' ••• -the i"trinaic project He opln before and aftoo< tho interaction~

t,t'•+••the projectile Ofb1t~l ~ng~lar moment.a before artd after

the intc..-action;

j,j~ ••.. the total proje:c.tile angular momenta before and after the

interac'tlon;

J ••••••• the system's total ang .. t;,~ oio,.entum;

1-1 +1 •

.:!.•l'+.!.'

61 .•.••• the transferred o..-b~td ansular m<>mentum; .C.t-£ '-t

As •••••• the transfc(i:eQ .spin;

!!._•!_'-.!.

fl j• +.+++the transferred t¢t~1 ~11.e:ular M.oment.ttm.;

~-1·-1'

~-..!,'-.!.

u•

(41)

multipolarf.ty Id~ the combined :interaction being a scalar~ we have;

(II.lB)

In equation (II.IS)~ P and N refer to the projectile paI"t Of the i"ter<>et1on ~nd the nuclMr p(l.rt, respeetlvely. Nw we wrUe the

reduction (II .3) in an explicit way;

( II.19)

tn gMHal P :ls an oper3tot whloh is able to eh,.nge the spin of the

projectile (e.g. in transfer reactions). In macroacopic oaleulations, however, one usu.ally omHa the ~a·l terms from the transition operator. Thia la justified by the fact that the major

pa~t of tho opinyfUp prob .. MHty stems_ f'l"om the dbgonnl parts of

the spin orbit operator. We therefore restrict ourselves to the case

6.a•O:i s'111111_s_{and l:r.j ..}~t.

No"

P(~O can either be the operator of (11.5), ,.here

its

tedu~ed matri~

element is

w~itte~

as a product of YAt and the term bet,.ecn large p;;renthosM h"I the se¢o"d Ude of (II.5), or it co.a be the operator from (II,6) or (11.7)· fo <'HM< of tli• two :l<'ttor ca.E;C$, iife have co i:::omp\J.tl!= 111,Qtr111; eieul.Cl"lt:EI of the fori:e:

(I!.20)

The: f'1rat t:~rm of this matr:f.J( eleu:aent is easily seen to be

(II .21)

Re('e -y 1.$ ~he 1,1~1,1,jj.l ~hbrev'.loi3.t.1on for the eigenvalue of the 2.t• s. ope-rat or.

(42)

-38-The .eei:.oad term. can be computed by letting .!.'"

!,

operate on the

left oide:

(II.22)

We eooelude that 1n any Ca$e y6tdete...Unes the multipolarlty of the

operator f(61.)_ Therefore the second matrix clement

trcim

(ll.19) can always b.. 11rit ten a-11:

(II.23)

with:

for let ord"r Oak Ridge eoupling:

N- the third factor from (U.5) for f\lll Tllo,.., coupling.

Now with ~ "' I (2lt+l) and

"

...

(1•u.t1t)-<u

too

Jt•

o>/!,.',

(II.24)

~:i..•_.~(-)J+l'+2j+a+.l.'+6t!j' 6t j

J

I.' t.t !

I

v.

"""!. .J { j '

I

(M. I. 0 0 1' O)~

cc tt I I' j e

(lt.2~)

The value of the nuclear ..,,~r1x e~eo>edt, (1•1N(tt)11), d"p<>nda on whether the excit~tion is of first nrd<>r or of second order. We

(43)

l".aave fo"t fi;tst order coupling~

(U.i6B)

(II.26b)

IQ~ q~«drupole _{excitations 2+1+l'•}

r

1

_-o+

2, 2+2 or 4+1•

For secol\d ord~r e""pl~ot. tA~R needs to be replaced by:

2 2 J

~t.~R (2 2 0 0 ~'0)/.'(2'r) (II.27).

We eonelude thh p.ira31'<1ph by noting that, for convenienee,

CHUCK sppUu the co11.o;ention that the coupling strength as entered

into the

progra~

ls ""'ltiplied by

((2Al+l)f(2I'+l))~

This additions.I factor allows one to use the Bame

a

for all e~eit~tiOll.8

2t

~ L+ where L-0~, ₂₂or _41•