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. Eindhoven: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

C. W. A. M. VAN OVERVELD

GEBOREN IE ROOSENDAAL.

Dit proe.fscbrift. ,i.$ g(l~clg<:.k~u"J:"d

do-or de promotor Prof. Dr. O.J. Poppe11'1;L

co-~ro1'm::ltO\" Dr. 'F.J. Van Hall.

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

vo-o-r ~1,1iVet:' We.tenschappelijk Onderzoek 1' (ZWO).

Aan m:i.jn vad(3:r>;

aan de nagedachtenis

van miJn moeder.

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 1b6

•110cd(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"

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

176 Literature

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.

-5-

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;

-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. It 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.

-7-

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. 1n 1 .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-1s 1 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:

-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~tile 1 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.

-i..)-

-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 then 11 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

-I 0-

10 64zn (~,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

64zn (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

Fig. J. 2. Comparison b~tweeri th~ theorctfoai ~scription~ cf expe­

r·lmentai cross se~tiqns and anoJyaing powe)"? faro the + + + 64

01

, 21

and 2 2 ctates of Zii.

A

A

A

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 iu 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

-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).

-13-

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. It ~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.

-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 m8

•

FQr """h transition between the channel• (I0

,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~

~"o+c) ·l(r0

M0o

0jtlI

0 M

0 "c) Ii (I. l)

0 0 0

-15-

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 0MP 1 $ .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 ...

-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 stands 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+u0

-E0)x"· -1 v¢e'x",

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

-17-

(I.4)

(I.4s.)

(l.4b)

(l.4.;)

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•c0 ) 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.

-lB~

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•c0 and c•c1

whereas th@ coupling

potential may be regatded ~• ""~~- Th1a ,...ano that up to first

-19-

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 )( " v t (X. 7b)

cl cl 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~.

-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')01 ~ 2

1 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 ""

"'ill omH the ind"x i. NCKt we exp~ess the difference between

-21-

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 08

Ni arid 88sr• (p';p ')2: !'eactiQns. The 58Ni figures haw been

taken from IMEl?8! and the 88s-r fig1<l'f!B am f-rom (WASS.~).

the two cross sections by mean• of* qusotity x1

2 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 x1

2 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 x1

2_

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.one1dered 11 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

factor ln the ~eighho~rhood of its maximum.

-23-

~ 103

:;:;. c: -!!! .... "' Hf u ~ ;;;}

___:

"' I-

101

a 2 " r tfm)

..._ ...

6

"

8 10

Fig. I.1. 1JnMl'Minty ba,.da for 58

Ni (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 x2 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

-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 1jbumpsw 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

two curves 49. 11."I f:tsT.Ire I.4. Cona.ee.utive curves~ that is c:urvee

01.tch perNrbationa with increUlP.g "1<1th•, He eo.,siderably

low~r.

-25~

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:"~ctice 1 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.

-26-

104

i!3 103

"' <;; s ~ 1o2 " "'

101

It 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' 58Ni. Lsft' E=M.6 MeV.

Rlght: E"40.8 /lleV.

The uppe:i' ~uz'!lea ao,,r>espona oJith a 1-spline p@,.turbation

succes?ivo io.,.,~ auz'!les ao~,.espond with broade~ per>tu~bations.

-27-

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 x 2 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+

-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 it 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 formaliE11111 1 '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

this chapter with the results of some test calculation•·

-29-

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

if 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.

It 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)

-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

~31-

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£ a80

R 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

-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:

(II.11)

-33-

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)

(II,17)

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.) It 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

osc1118Cltig 1 the derivatives will beCQliC l!;lm«ller in absolute v.alue 1

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 .. lh for

Bessel-like differential equations, such ee the radial Schr!ldingar

equ'11-tion • Therafor-e it h-~ bee11 l:>nilt into the cOmp11to!:1:" .:.ode CHIJCK.

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

-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• In the derivatlOi> o:>t Vee', we start by writing tM co-<>bld<'d nuelear

and pr¢jeetile int.,raction as a product of the nuelear interaction

and the projcct(le interaction. Assuming that both ha~e a

-37-

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' 111111s 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.

-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(6

1.)_ 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

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~, 22 or 41•

-40-

dO' dil (mblsrl

IO

n-3 a.ioult• of the spin-orblt coupling

In o•d"" to check the p~eceding modificst~Q"' ~n Ch" code

CHUCK as described .l)biOve, we have perf-Qt:'MCI Q few test c.alt::.1,ll~t:lOI\G.

Urst of all the DWBA w~culaeions of Melue~ (W::J.78) fo..- the

S" li'.,(lf, p' )O! +2! ~e .. ction st 17 .2 MeV b.,,,., been repeated. Id Hsu•c

II .2 the solid curves ~or-re:s~ond with our e.o11.l.::.\llt1tione: whereas the

dashed e .. i-v<I$ cor..-espond with the calculations 0£ lleh~e.i.

54Fe (Ir, p'l 17.2 MeV 2+1 0.4ll +D.)

o_o A

-0.)

liO 90 120 15Q 30 60 90 120 l~O

a c;,m,

Fig. II.fl. CMM aoation a11d a~aiysing pow@l"' fol"' ths -,.saatit;1'1 .54 -.!A +

Fe (p,p 'J3 1, E'ul.Z t'.'Urve; our aafoutation

d.ai;hed ""1"Vs: <:?afouZatlon from (Mi'L78)

dot-dashed curv<': aalcutation with ,•e­

duced spin--ar>Oit coupling strsngth. (MEL?8!

-41-

The only $l.$o.lfie.,o.t diHe<enee con be seen in the 0-15°

range. This differen.,e, however, h ,..;.et U\\dy n<>t du• to the spin

orhi• coupling, since it la known that differen.,es of this kind may

o"cur betwoio.;.o. th<:o roul u <>f the ll\lllA code llllBA/THE, which has been

u1;<ed l>y !Wlss<m, end CllUCK when used as a DW!IA code. In o>rder to

reproduce Melsset'l 11!1 r~sutt;:t 1 ll-0Wevcf 1 we 'have t'O adopt a different

spin orb~t coupling strength convention:

e (CllUCK) R ~ (DWBA/THE)/a s,o+ s.o. $.Q.

(II .211)

Also in figure (II.2}, Melsaen's edc,,.lstloo. for a smaller value of

S s .. ;._IBee.,tral (l.O instead of 3.0) la gheo., repi:-e~e,..ted by th"

dot-dbahed ¢u~v~. Fram this we observe the ms.in effect of the spift­

orbit eo,.pH"g <ICCQrd~ne; to the full Thomas description: the

an8ly~1ng poweT of the entire forw~rd region ia 'pushed up' 1 whe~e~G

the infl11ence on t;:h.e Cl:'OSS sect.ion ia leas systematic ..

Th" "o"'p"ris<>d bet..,eeo. £.,U ThQma~ dose<~1;>ti<>n o.nd Oak Ridge

description has been ~e~~ed ~moqg others by Glssshsuser (GLA67). In

order to pro'li'lde ~ fu.('tbCJ;" check of oor m.odificat ions~ we have

-i:r:p1:oduced his calculations for the reaction 51jFe('jf1 p~ )O! +2! at.

18.6 MeV. Again "" found a" accurate correspondence between his

cross sectione snd a.nalysin;g: power~ and ourti;I.

The "'Odif1cations in C"UCK that d"al vith the second order Oak

R~dge for,.,.~i~m are harder to check against well-established

~eaults, since it has been rarely appli~d id the liter4~ur~. I~ m(l~t

eases second Qrd,e!," i:::•levl«tione at"e pe't"formed lri'ith the code ECIS

(RAY79) which e~ploys the full Tho.,..a for~ fo~ both f~~•t ~Q~ oecond

order tr~n~1tlon~. Unfo~tunately the latter program has a somewhat

dtffc<ent approach to reaction calculations than CHUCK in the senae

that e.g. second order DWBA calculations with only a limited set of

allowed couplings can only be perfo..,.ed by lllQre or lesa 'cheating'

the program. Therefore the reliability of a compa~1$Qn betwee~ ~he

results of ECIS and CHUCK ahould not be overestimated.

-42-

...

In figure II .3.a, we present croi1e ee-e.tion and analy!Jing power cutvC$

for the G~zn(t,p')Oi ->-2! direct exciht!on with and "'Hho;out spin

orbit coupling (the solid and dashed curves, re9peeeively) as

computed with l:OC1$ in a one itc:r8t1on approxiJDBtiO:Oo\ to a full

eoupled ch8nnels approach· <his ta believed to correspond

no.tmerieally to e eoeoo4 order DllllA cal<:ulaeton. The major difference

between the two ill seen in the baekwnd region of the analysing

power. Thb same difhrvnee nOll ;,.,curs in figure II.% where the

reaults of CHU(:!{ for th"' Mme reaction sre given. 1'i:om the fact tholt

this behaviour 18 reprodueed accurately we conclude th•t our

impleme<>eaUon of the secodd order Oak Ri<le;e coupling oervM u a

re•sonable approximation for the s"cond or<;ler full Thomas e<wpli!tg•

-43-

(arb. units)

do' c1n

lmblsrl

0.1

O. l

0 30 60 90

1>4zn Wop'!

20.4 Mev

2~ IL 801

64zn li\',p'I

20.4 MeV

2~ 11.801

b: CHUCK

"

120 )~0 30

~ <'.'.,m,

,,

60 90 120 150

e·tg. f} T ::;_ Comp(.t.:t>"i$On fa?tWel3n. CHUCK and CCl"fl for the r-~aa·tion

C4 ~ ,+ b b . db Zn (p}f! ')2 2

. n-~e curve8 hav6 sen o tai.ne y

a1•plying a 8<f!t:and. l)Y"der> DWBA reaction paMi o>tl!J.

Top' EC.TS, bottom: CHUCK. The full aW'V<l8 l"'e8urt

j'Y"om cczl.culations that i>idUM the 8~oond order

spin-or·bit coupling. The d.a8heJ. o1"Y'V'3s ""'" without

l'ip-1:,1-orllH om.tpiing.

+O.;

O.o A

-0. ~

+D, ~

0. 0 A

-o. 5

Chaptc>" Ul

Now -we •~rive at the second item of Qv.r progta1IDIN!: •e outlined

1n <:l'l4pter I. In thi~ ehapt<u- we "'Ul concentrate Qd the font f<letor

a& "" idgredi<!nt of theotet1Ml ea.leulatione to be fitted to the

data· We will limit ouraelve$ to the DWBA ..ethod for the

¢.omput.at1on of t.he obsc!rr"..ables. In this ch.o.p.t.e.r we conside"" 0.11.e-step

procetae• only. The first limit<ltioo we see 6$ e~sential. Only io

the !!WllA a lineal:' paramettizaUO" of the form f•etor will yield a

litae~r expression i:n the l'--mat:rii'( elements· 'Ihl!SI !s neceasa'ty fQ1."

doing q"">'Htative 9t•te...,nts com:tt1>lt1S the mathematical properties

of the in...,rs(Qn procedure. The Hcond li11titatlon U leso

essential: in chapter V we will eneO~tlter multi-step reactions that

atUl are manage~l:>le in term& of a atepwhe U>'eior procedur<=.

The firet @eetion of tbl.s chapter d<0"18 vith the imrereion

procedu>:e in geneol terms only. The method 1.s ,!.erived for ""

a!."b:ltrary llne(l;t parametl'iz.a.tloii. of the- fo~ f"1,etor+ The cross

sectl911~ a4d analysins powers arg still quadratic ex~ressiona in the

para.neters. The~etore we have to devise a oOd-linear method for the

p~:i;.-~ter .ae.arcli;.

In section 111-2 the .ac.tual parametrizetio" will be

present4d.· It ia pa'°t.lt based on th-e requirement. th~t at. least the

tr.i.i~foroiati<>n from the p~rametera into the T-matJ:ii< ""'ot be

properl7 1.nqertihle.

ln the third section we dUeusa the method. We pay attent~O"

'tO :possible erroi: sonrcli!ili 8.l'ld we show how o:n.e of the tlOurce.s of

e.,bigulties Ce>' he removed.

A h~ word& have t<;> be aa1d •bo11t the not•Ho"· Since

anal7Bing power& and eroaa sections ean be combined tQ partial cro:>H

aect;ion.a ~ we ;i:efer to ct:"o~u;1 l!;'ec.tions

~~over, wa write cJ as. an ~bbre.viation fof

-44-

ooly dor dli"

in this e.,.pter.

111-1 An invei:'s ioa pr1Jcedure for' a line.Qt patamett"l~atlon

In this aeeUon "" start by "rl.Uns dow" the eroee section as

expressed in the DWBA T-matrix elements (equations 1.z and I.9)~

(lll'..1)

The index k distingt.di!;i-hel!;I bt;!:tween th~ angle~ .i;lt whieh the c.roas

$<Ot1o;." 1a oo:>mput.;d; °' ot'-'M• fo>: th.; coHectl<:n'> of ~$""U¢

qu.antum. nu1Ubers that govern the sc..13.ttering process;. 9.. stands for the

Md ~' <>£ cq.,<1txoll (I .9). Ji'or the p<'rpooe of e!il.s $<:Ct1on,

howe"ler, we i:::s.l"r. iot.erpret t0. $1"1.d .t siu:i:p1y aa ruth'l.ifl,g itt..dlcesT It

should b" m>ted tMt aloo the funotions ~ and v depend on the :lnd"" tm

t ~ The constants Al<. also inclt1de the angular dependence4 If we

demand ~k to equal the measured cross section~ equation (III.l)

becomes a non linear integral equation in the unknown function V • cca

Therefore we cannot hope to solve this equation .analytically. The

"'~tlJod "" "PJ>ly 1<1ote1>d io the follo>11ng.

We expand the function V into a basis of known functions Si ""o

with unknown complex coefficients a1

+ We demand that the dimension

of this basts is BD14\ller th.$.t.1: t.he nur.uber of a'\l'ailable e:t.perimental

cross sections, {o~xp~}. Question.a concerning the nature gf the

basis, its completeness et.c4 ~ will be dealt "W"ith late't on 11\ thl&.

chapter. ln o~dct to avoid complex matti~ at1thmetic~ We ~ill t~~~t

t~e ~e*l ~~d i~Qg~~~ry pa~e~ o~ the eoeff1¢1ents a1

as independent

real pa.ramet.ers .. This means that the dimension of the vector a is

""lee <Iii l.,~s• ~$ the dt...,o.uo~ of tho M8U { s1). wnh the us-: of

this expansion we write (III.1) as:

(111.2)

Here the complex numbers Tit are the appropriate elements of the

DWijA T-mattix ~• def lned in (I.9) with Cho f~nc~~Ono s1 •orv!n$ as

foAm facto~· Thee@ csn be c~lcul~t~d o~c~ th~ $Ct {si} h~~ been

-45-

defined. ln the summation Q~er i we also :to.clu.d.e the T-..tr-i~ea.,

obtd""d from the ~pi .. -orbit ""d c.<>ul<>11b "°"PU"t•• tl)r reasons Qf programming ¢(111.Venlenc.e, 'fe p-refer t:o oall!: th~

set ( DU'"l (ltl,11169), in•tud <>f geou'&t1;ig the oet {r11J. The

elemenU of the &et {DU.'") ean be foull.d by applying • U<>ear one-to­

one tran•~orm.o.tion to. the: T--matt1:s: ~lementB~ so th:f.s. cont11in.t

~><f.etly the sa..e 1dfo~tion about the aeattgring proeeaa as the T­

Mtl:':l)C: does+

'tor the de$eripUon of the search algot!th,., we adopt u

ob..,i<>u.s vector no:>t$ti<>n. lie ""Pren our invetUOQ problem sa 'find a

paramntcr "'eetor !. ouch tlu1.t + ~) is ..tni'""l', where

(III.J)

In the next •ec.tioo we """"' t>aek to this ~ho:f.ce for +, Then ""' vill

also ehov how we "*" t.._pose additional eonetraint• 011 the form

fnetor by addtr:.s n proper non-.1egntive function of.! to+•

lie will solve the non-linear optimization problem (III.J) by

me~d~ of an itet*tive approscb. TQ this aim 'l&Ve co~•truct a sequence

of solution vectors {~} 1n au~h " way that H!.p+1 )<+(!p). We aUrt

'th:f.e seqnence 8.G fQllovs+ We t:«~e 'the central pa.rt of the collect1'7e

'"o:>del form factor and expand 1t tdto the fonctlono s1• lie now d<1U\>e

the <n:etoc a for p..O to con&Ut of these "xp•n81o.> e~fficients.

Since the u:t { s1}hae to be an independent bae:ia in the •f'•ee of

dist1nguUMble form faet<:>r~, the "xpansio11 is uniqll<ll, Mo~eoyer,

the "roes ""ctions aa computed vi th the coll«cthe form factor <l"lld \' i .

as computed with i"o ,S as " form f•ctor vill then •aree :I, '~

sat:iefactg~l1y~ Note that ~ Odly parame~rl~c th~ form. facto~ 0£ th~

central pert of the deformed OM!'. Therefore <>nly thU pj!.rt will I><;

vai::ied. The CO\llomb part and the spin-orbit p"rt 4re left uneh•'-'•$"-d.

Strictly apes~i~S. this is not cot~eet. We assume, bawevec~ that ~he

effeeta of ~orrectiona of the eO\Llomh form f11ctor a"d of the •Pi"-

-46-

orbit form factor on the cross sec:tion are sdl.3.11 enous;h compared

with co~rections of the central form factor, so that we only make a

su:aall el:'ror if w~ neg:lec::t theta. This ie. eoI).s!stet').t with the f.ai:.t

that:r as far aa the coulomb coupling is coneerned~ one commonly uses

3 very rough ~ppro~l,...tlon {Qr lta 8?*tl*l form. For the spin-orbit

coi.Jpling some evidence a.lea exists that the exact functional form of

tho for.. £.."tor U not of to.;. gr,,;ic lmponon¢e for the erooo

oeetiona and analysing powers (SE!E70).

In order to derive an algorith• for producing the sequence of

ootutlon vector• .;, , we could think. dong the fotlo.,lng Une. The

moet straightforward method ma'k.es use of the recursive definition!

with (III.4)

'!-lere ).. C$~ be fo~nd frOD1. a one-p~rameter search. The moat codlDlOnly p

used c:hoiee for d iB d•-V ' • ~ - - p

inethod, however 1 has as a major

This so called steepest descent

disadvantag;e that it can be very

$l<)w. Thh ~opp~n~ p._rt10:..,1arly 1f th" d1re<otiM of the locol

steepest descent differs mu~h from the direction of ~~~, where _!f

ot,.nd$ (o~ the VC¢t<:>r !_ whe~e ~ r'lo .. ¢he• iU absolute mini,.um, In

most case• this can be seen by inopocting the Hcnian 0£ ~. Thh 1$

thol matrh o{ .ill ~econ\! 4erivatitH .;.f t with respect to the

v-.r1..-.b1e$ { ~i}: :t~ i,$ ~ mea~ut;"e for the local c11rvature of ~~). In

ouch caoeo aa dooctibod above, tt 1$ then.f-r froa diagonal, A good

alternative iB the inethod of Newton~ that makes '!Xplicit 1,;1.:&e of r:.he

Ho•$Un, ""d that ~an be proved to converge quadratically (ST080).

It works as follows.

We app~QX1ma~o the £un¢tion ~ ._roudd .!.pbY the firot three

terms of ita Taylor cxpart$~Ou•

~ (~) • H.;,l + (!_-~)·'!_ +P+ \(!,-;,)•Gp(!-~)

~2~ dtb th" HcGshn (Gp)ij • ~-

~a1hj

-47-

(III.5)

In case G 18 poeittve definite, the "ICctor _! that givl>G the •!>solute

.. 1n1mum of tb:ta •f'f>roximation of 4> ia:

(IIl.6)

Th.J.~ a.e:thod is knQtm to work well in maJ.'l.Y -eases. In ou't i:.llrJe the

dimension of the vector ~ '""'"u • epedfk probl"m• Typically, the

di..e'1s1on of ~ U ro..t11d 20, Thie ""'"'"" that thei:" are 210

independent component$ ot G to be computed, Which would cost of th,.

order of 620 seconds for c«eh .f.ter,1J.t ion on out" B7900 eomputer.

'l"herefore it apJ)caU advantageous to H.;16 ._., approximation f-:or G

tllet Mt& be c.omputed with mueh leea effort, Stal."ti"S from:

{Ill.7)

we can app:toxlmate this, if lo-qc~I la aufficiently small, b:i-

(Ill.8)

(III.9)

(III.10)

This io ktaown as the Gauss-Newton approximation (ST080). ~. see by

using it• dcfitaitton formula (lll,10) that the matrix J io -:obta~"ed

in a """ber of calculst10<1e that is linearly p>:oportiMal to the

di..en"1o'1 of a. A.gain for the caee thn the dimension of ~ 18 20,

this takes al>O\lt 37 seconds oomputins time. Now equati<>n OH.6)

-48-

-4D -~[JTJr1 ! 4(~m.e,,>

[JTJr' JT•(o(a-a )-<1exp.) --9 -

(III.11)

So~ ha.vit1S c.0111.puted the ~trl~ J, we can ¢¢0).p~•tt!: tht: ve¢.to-r- 4 . T¢

avoid the need of an explicit •~pression for [JtJ]-1 , w;1'write

(III.11) ao:

(III.12)

Tho set (IU .12) con only be •olvcd in a l""ellable way if •mall

vaLiations 6J1

Q in th€ ~ight hand part do not cause too large

v.:ar:l.~Lt1(Jrn~ 6.!!_ in the ci::n1'putcd vector d. The t'"a.tio:::

is called the condition number of the matrix JTJ. It can be proven

(STO.SO) always to be larger than unity. Moreover it is the square of T

the condition number of J . Therefore, a more stable way of solving

(III.lZ) can be obtained, if we interpret It to be the Gaussian

Tiotmal cquatlon fot ~ least squarcY problem ~nd di~cctly solve the

rectangular least squares problem without left-multiplying J by 1ee

t::ran&pO$e.

The equivalent least-squares problem is easily aeen to be;

(lll.14)

ln solving this linear least squares problem, an additional aspect

has to be considered; it can occur that J is rank deficient or

nearly so~ 'n!~refore our least square5 blgorithm must ~ompute the

~•nk kJ of J (kJ can be •~aller th•n or equal to the smallest

dimension of J) 4nd solve the problem 1" a vector8p8ee with

dimenl'JiOn kJ in or:d~r to get a u.n1.qu~ sol'lltion.

'l:hU ...,qdrement '""' be easily -t if we apply the so-called

Si"a»lar Value An~lyaia method (SVA) when -9¢lvfog (III.14). This

apprnach 18 described e1<t•Mively in the book of 1-<'"$Qn and HanMll

(LAll74). Therefo!,"e, her" w" will give only a brld O<itline of the

method,

Let the met !,"b J have m rows aod n columM • Then squa!,"e

matrices IJ (""'m) and V (nxn) can be ¢Odstructed (J..A.W74) such that

J•IJTS V (III .15)

with U ""m, orthogo:ottal

V TI)( n, orthogonal

W!n db;.go:ital abova (J'l;-n)xtt :i:eroes.

By proper ~earranging the rows and cOlt>•n•s of U and v d.,,,.tng

conatl"uCtloo., S c.an be ~de to have 1 ts elements 111 fl.Ql"J,-l'(lereaaing

orc;ler from loft .,pper to right tower.

Next~ let

(III .16a)

and

(Ul.16b)

then d ~nd ,•xp_~(a ) are expressed in a base 1n which J is -p - - -p

<liogonal.

-50-

In thla spac• we can wr:l.te th• cq\latlon (III.14) as follows:

sl

s2

(n)

0

0 0

0 0

0 0

(m-n) 0

... o 0

0

s3

•4 o ... an

0 ...

0

.•• o 0 0

>;l

x2

xJ

xn

~

bl

b2

b3

bn+l

bn+2

(lll.17)

We oh$e:i:"ve that c)nly the firot n equs.tl(lr'l!!O Qf S.!,. • E_ .i::.an be

sat Is Hod; tht;!; (m-n) remaining elei:z1ent:g of !'; (;.'¢l'ltJ;"lbute to ~he

reaid,.nl """m of th" <><iginal set (III.14). 'l"he reaid.,nl norm is

glven by

(III.18)

whe.,, ~ is ad uhbteviation fo,- ~exp·~(~) ond usage 13 1118de of tho

orthononnality of u.

-51-

ln an sna.10$¢1,1.e way we c•n Wl"lte down the nQrm of the solutlo-n.:

Id TI 2

• In (t;._fSkk)2 (IH.19) p k•l

1t i.8 easily seen th.i>t the solut1M ,;rector ! is given by:

°N¢w> let us have tl mQre detaiH:d look .at the c.ase where J is

ol~s~lar, say havil\$ rank kJ{n. :i:his m¢ads that for k>kJ, the Skk•s

vanish+ !h~~efore the Cq~aeioas for t>kJ cR~~ot be satisfl~d. t~

pra.ctf.i;e,, however, J Cd.fl only be cQ111~1,le~d witb limited aee1.1racy,.

therefore the i!l'!Iiitlleat ele~e1H.s of S will not 'be exactly zeta ..

Neverthe1CS$, the rank of Ir.3

can be establ19hed by examining nipn 2

wt.ere the number Qf terms of (Ul.19) increases ~i:'Q'" zero up'1srd~.

At flrst n d n 2 will inct•Ue gradually, but wiled the number of te"""'

--p 2 exceeds kJ" n~~' rt&.akes a s1,1.d~~~ jump towards (much) larger val~e$.

The 0Mu\"re<1ee of such I.\ j\i'"P is therefore a clear 1nd1cat101\ of the

U<I~ <>f the orig1n*l "'8.trlx J. The ele.oents "k. of~ w1th. k>kJ ue

~ndetermined, b~t because ~e wa~t the norm of our solution to be ~~

.!Jmsl1 as possible" 1c ls best to ms.1.::-c them zero+ This modifies our

S¢l1,,1t.iod into:

(III.21)

-52-

irr-i The c.hoi~• <>f the basi~ I s 1)

the method as pte~cnted above is valid to~ any line~~

p~ram~t~izat!on, independent of th~ fotm of the f~nctlons {s1}. In

orde:t" to establish ..;a suitable cho1ee ~Q't our basis, it might be

hoJ.pful to htn>• " fo<>k at the parametrhatioo. aa "$~d in the field

of ine La.st le -cliec.tron seat teting ainc.e 1u;:i;"i; the problem Qf form

i'ac;:tl):i;' inversion has been aolv~i::I· (SIC74,DRE74 and references

therein).

"t'he' reaction theory as given ~n section !"'2 also holdi;i fo"J:"

ele:c.trc11'l. $catter1ns~ ali::hough instead of distorted w.ave:S one c..a.tt

uoe pbne waves. Th.\$ is juotif 1•d \>y the fact th"'t the ele<;tr<)q

waves ai-e hardly affeeted by the t"IU.C.l~~t potential. Thit equation~

~nalogoua to (!.9) then read$:

lk ·r -1k ·r f(~) • fe ""'f -Pf~_)~ -i - dr

la (III.22), PJ. is the c.harse t~•naition deasi~y, responsible fo~

the coulomb excitation of ~ H"te with wultipolarity lq .1 b ~he U'

transfer~cd momentum~ NQtC that in (1.9) 1 1 is only a n~mber

wherea9 in (III .22) f ie ~ function of the ~unsferred mo.,ene~m and

therefore of the scattering angle ... If the .aogvt~t dependetice of P~.!._)

is writte.-. ln terms of spherical ha.rmol'l.ic~. \Ire i::::~n perform th~

~ng~lar i11tesratloa of (III.22) <'n!'lytically and,,. Hodo

(III.23)

fr1. eq11~t:lon (III.23), \ is a spheric.al Bit!':P~el func.tioI'I and R ta the

-53-

t1\lelear radii.I&. From (n(.23) "" learn that it is •dv4<ltageous to

parametrin p). (r) as follows:

p*r) • L"'i,.A, (q,.,.rl (Ill.Z4)

" with '\v ~h<>Sen so that ql.VR ~8 the v-th &ero of j~. Then, by using

SO'llke res.ult:ei of funct1Qti. theory, a~v..:.~-n. be expt-G!ll~ed. directly f.\I

terms of th" eroou seeHon for tho mod!entum transfer q~" (Dtu;74). We

m1&ht say therefore tho.t the parametou ~ ... give the a,.plitud" of

the form f"etor in mo1110nt11m ·apace, We not" in passing that thU

p11ra.metr1zation ia only modd independent .,p to a limited extent

since the value o~ Tl plays a cruclal rnle 111 the inversion algorithm

(Neuhauscn, (NEU77))·

NO<.! ,,nfortunatdy there are two reasons for which w<:

i;..,_rtnot apply th.is methQd tHtsmendcd to stTongly 1n.teractlng

l'artieles. Hrotly, th<: ph.,aes of the fonetiona f(.'i_} ti:om (III,23)

are k-n.own, Sg th$t, apaTt ~~Qm a simple Coulomb correction. one cs.n

just teke the sq,,.,,re root of the c.roaa ao;ctlon, wher"89 the u·

eo-efficientl'.I' t have "~~~ovn phases. St::C<h"ldly, the 1nveri;i.1on. of

(II1.Z3) 1$ easily obtained in v~rtue of some well-known

orthogonality properties of tesselfunceione 0" finite intervals

(HAN65, £Qrill<1lae ll.4 .5 and 10.1.1). '?he diatoi:-ted wavea f:rom

e<[dation (I.!I) unfortum~tely (lo not obey similar '.l"ebtions.

ThUefore, as two facilitatl"S: aspects of the parametti&at10<1 in

tetll18 of Bessdfl>.,etione do not """"' to J>.. appHe~ble to other ea.see

than electN" scatte'.l"l"g, H makea l'IO sense to adopt thia

paramett1i:.t.t1oa here ..

A eonsiderat1on that can lead to an alternatlve choice is the

physical qae:aaing of t~~ array of p$rameters a. An ~t~ra~tive

proposition ia to *••isn the- ampl1rn<le of the fo~ .. fa.otor fo.- "

rad1~$ ~·rl to th~ para~eter a11 that is~ to adopt$ f1dite-element

method. Id an extreme e~se ~ne coQld u$C ~ O~fdnction cxp~nsion:

V ( r) ~ l a ~ ( r-r ) ""o i I 1

where ;11

~ V (r1

) cco

(Ill.25)

-54-

We have chosen ~ow:- this ~thod 1 but in order:" to reduc:.e the n,1.1mber

of paramete ... s w" repl,,<::o nu ,ZS) by:

(HI.26)

where {sif are cubic $plina functi<)n$ (VEL77) of whieh ~e ~ill ha~e to specify th;, nodes in (I!I.26). Al~o the ~ange of th• ~1>mm.ation

in (III. 26), and thus the radial into ... val to whieh ~h" form fact Qr

is ¢<)~Haed, has to be defi<led· ConcernidS: the spline w~dth, it

•••~s logical at first sight from (lgure (1.5) o~ equation (I.14) to

1,16e .a decreasing value whe11 gt:tf.Og from the tent:re of the t1.1,11:::J..~ue.

01,1t~aI"ds .. Thia ~t"sumentJ' hawevet:-~ is based on x 2 considerat1()n$; but

a x2 crit•rlon is only v<ry sl<>bal concerning sensitivity M "'0 "111

show 1n the next ~eetion.. Moreove~. we IDtlBt CQ~$~de~ more pre~i$¢ly

wha.t exactly we c;:"1n learn from figure (1.5). We observe tha.t only

broad .&t:r:ucturea ia t.he nuclear 111.tet"lor contrib1,1~e tO!l&iderably tQ

the total xi.. Thie does not ID~.Pr,,~ however; that narrow split1.i!l;l l'll

the io.iiel:' tegion nee-e~i·u.rily yield (almost) depend-ent: columns for

eh" J matrix. In order to it111.,$Hgato the minimnl spline width th~t

can he distinguished (th" radial ~esolution) as a foaction of th"

radius para1"otcr, we ha11e h<::en led by the following considHH~oos·

Schemntie..lly we can dep1>0t the path, ~eoding from para .. etct

vector.!!. to cross sectiQM, as follows (compare equation (UL2);

(I) (U)

'W'e 11ow i;l:u;)ose to have th-: iiridths of our spJ. h~elj dependaat on the

propertiea of the linear ttsnsformatio<> (l). This line., ...

-55-

tr.ansfQl"~t.lon can b~ e:iepress@d by th-e matrix K c.>h1¢h is defined

bY•

' ~ '* t i Kji • f ~ (r) ~ (r)S (r)dt

c ¢0 (lII.Z7)

whe.re j nuni.era.tea. all oc.;:1,1rring c.ombinatl01;u~ {i;i'} +

NQ>I let us d<:H~ " rotation "ngle " between t"<> ¢ol1±01<>s i 1 and ~2 of K as~

(III.28)

It is obvio~~ from the forto of (III,28) th"-C • distance ~r12 b"'tweea

il 12 splinu s and S the.t is much om<11le r than the pcdod of the

oscillsti<>n in the fuuct1ono c will yield • v~lue of o1

ith•t is i 2

close to zero and therefore: a in.at.1:'1$. K that ls ne~rl)" dependent &

'l;'hia means th~c.. 11.'l order to :lnvert the. tranSf¢t:~t.ion (I), we h~v.e

to dam..nd a minimal dUC'1tlCe betwean two "'djacent apl1nes. lie

illustt8-te thia by m~ns of an e"8mple. lie tal<e the re .. ction

56 1"e(l\',p')Oi _,_ 2t a.t 24,6 MeV <l$ studied by Melnen (MEL78). First

we comp.,te the function~ E by numericoH)" aolving the appropdate

radial Schrtld1nger eq .... t~ons (I. 7) whete "-S"ia for reasons of

o1mpHcity the $pin dependem:e is omitted. Next we oompute tile

mattf.x K. where for the fu11.c.tion:a. S "'e t::~~ke spline funct1.ons of equal

widths and at equal distancu o<>fh of .5 fm. tn Ug,,.re III.l the

fonetio<t"' from (llI.28) h Si~e" as a functH10 of 12 for th~e~

diffa<ene values of i1

. t" all these plou we observe a UmU$r V­

.s:h,ctped structu-te. The intel"p~.et~tion of this ts. a:s follows& As the

-56-

l 0 0 11 '.; 2. 5 fm

so

a 100 i1 ,. 7. 5 fm

50

4 8 12 16 '2

Fig. III. l.

The rotatiori angie a. i ""1 ~

in d.•:5-gr·~as bf:.t1.;ac11i

two coi..UJlms of' the matr•iJJ K.

lefl upper.- i 7 oal"responda to 2. o fm.

Right upper·_· i1

cor'r•a&pcmdc to 5.0 fm­Left .- i

1 Ml"responda tr> 7. I! fm.

or...... .............................................................. --1

4 8 12

12 16

radial d.11,t.ance between t.<Jo columns i'Ll.~re"'$e6 ~ BO doe a their mutual

.s.ngle1 Siving rise to t!1e steep edse6 1'eat the minita?,1.m of the tV'.

From the f'1et tl\&t all V-$h"p•• frnm flg11~e IU .1 h"-ve "-J>J>~oximately

the Bame w!,dth we conc.l1.1de that the rota~J,on angle for 1.;1 given

radial dl~t-'1nce betw-eea C:"fito aplinea doeiili not depend Ofl tlJ~ actual

lo.c:.acion. of these ~pline-e. Thi$ in turn inean.$ r;'hat the t1..od.1..11

-57-

distance does not necessarily depedd on the radius. From the fa<:C

thollt eve.Ty 1V1 h.f.'l~ a. w:f.dth of ab(n,,1.t 1.2-1 ... J fm we ea'l(e for our

spH>'lea a width of l-2~ fm unless stat"d othervi&e.

\lith ttaPect to the rM..,lU ld fig<1re (I.4). ..., "-Qte that this

spline "idth, although feasible frnm a "'ave functions' point of

view, might sttl~ stve rise to unphya1cal oscillator1 •tructures in

the Ndfident vector !.· It is not possible to detect this a pr(ori

becauu o~ the non-l1near1tY of the transformation (ll). It could

n•verthel""" be that the latter introducca an additional (near-)

dependence in tM nw tri:>< J.

In order to stabilize tM &oluU0"-8 of the problem (H(.3), a

device often ~•e4 is damping. Thia meads that a positive f~"-etion of

the pai:.-.. eeer vector !. 1& added to the ~ in ;,q.,~t:to<> (III.3). The

latter function 13 ealled penalty fun¢tl.O"-; it is chos"n $0 $S ~o

expr@aa a<ldltional conatraines to the solution.

TWg obvious ~boice$ ~o~ su.eh penal~y funct10de are~

(IU.29$)

(Leveaberg-Marquard M'"pfog)

(III .29b)

(d<IOr1,,,.tive damping)

In (Ill.29s,b), ti equals twice the d1mer>~~O"- of I si).

-58-

Equat!(ln (III.29a) expJ:C-ssee the requt.t."-e'IDent that the n¢rT11 of

the s<>lutfon should be 3mall whereas (~11.29b) states that the

•<>lQt(oq should behave smooth\y. The !"lr•~ete~ controls the

strensth of the constroint in both ¢a•e&. ~ote in (lll.29b) that the

tenn with 1-n/2+1 should b., left out a~ "'" cannot de,,,..nd ~hat the

real :Pbt:t of the form t'«r::tor fits ewoothly to the i?Qaginary partT

With some a.lg;eb\"« it can be seen that th~ cone.trs.lnts s.s

e"!'~eooed by the pe<1alty fu11ctlot1G (IU.29a, b) a~e equivalent to o.

modification of the matrix J in the seMe that it lo extended by //,

matr1~ Q that is plae~d below, •Y"~Ollcally:

(Hl.30)

The penalty fu~ction fro~ (111.2~a) correaponds to Q beit1S th.,

identity matrix where~& in the caae of the penalty funeeton from

(iII.29b) the ma<rix Q is sivet1 by:

1 -1 o a o -1 a 0 0 -1

-1 (Hi.31)

-1

l>:> both csses, the •iabe hand part of (UI.14) should be e><tend~d

with n ~CJ;'"O@S.a

Fo'(' the dl.atri~ei:. .J th9t occur 1tJ. Ou.'t inversion probl~m,.

ext.ensiv~ 1'1,1~er\.i:."l tests a.haw th.l).t th41 application of thee.e two

damping devicce for practic.al purposes is equivalent to reducing the

number of s1nsul~~ values fro~ the &U111...,tion (111.20), llhi~h ~. le••

~labora.te

-59-

and thernfo~e ioreferable. Fot the pra.otieal application· of the

inversion p~oeed1J.re we will thei."e.for-e. not use the 4er~"fQ.t.ive .... or

I..evenl>eri;:-~rquard damphS. There is, however, • third damping

c::u~(:hs.niatt1: which indeed 1;1ho'W's to be advau.tag;eo1,1s io. some cases ... 'thie;

has to do "'1th the following.

A.El Wa!J stated in the be.g!.l'ining of this section, we- not only

have to establish the spline width in our parametrization, but al~o

th~ r~dial interval o~ "h~eh we allow the fo~m t8~eor to vary ... This

ingredient ~~ eqni'1alent to the val1,1e <>f R for the invcrUO" of

elei;:;trof\ scatte'('1.t;1,g erosa sec.ti one~ Again ...,e $t4rt with an

11,'lve~tigation of the 11-n.ear transfOC"1J1S.tion bct\il'een the vector a e.nd

the T--'llUltri~. If this tranafonnat1oo has to have an inverse, it i~

es;;,.'IU<11 that there are no "dependent columns. "r.f w<o p1ot the

funcUQn a1 1

as defined 11"1 equation (III.28) in oqch a way that it l 2

depicts the ~oueio<> angle betweel"I d\ pa.irs of adjacent <:o1uDJ'16 as

.$. function of the i-ad1ti~ 1 however~ we a~c th~e there is a minimum

si:oun4 ~s fm (fig. UX.2). Thia seems t<> •uggo$t that ab<>ut this

doci i -12 d r

2 (rac!/fml

0

Pig. Ill.2. The >'Otation angle pel' un.it lBnr;th (rod/fm!

betLJJsan a/.Z pa1~~ "! ad.fria•"t udWTms of th<'

-~o-

10

radiui; there 11:i,-1; linearly depe't'ldent columns. We can expl.t1I't this

phene>menon by h~vh'I~ a. closer look. at the nature Qe the functions

~~.

For e\7ery P.. value 1 exc.ept. f.ot .i=O) there (!:x:l9ta .a c:ertttin '!'."­

value fo~ which che square of the associate wavenumber, k2 ,

.... anf$hes. Thia 1;1:11;'""1;"esponds to th.; classic:s.1 tl,1.tn:t.ng point. Fot ['­

value$ 'liitb@re Jr..2 is negative,. the .t;'T,1.nc.tione /;,.£ h.$v.e; a. more or le!}:t)I

exponent~Ql behaviour~ Whc~eae f~r P0$1tlve ~2 they O$c1llate. Now

If ""e picttJrie: the .t values .;ti a functio11 0£ r:.hese clasa1¢~1 turning

radii (figure III.3). we observe r;hat e1g+ ifl tl'le 5~Fe case there

e:dsts for i•5 ~ i;tl6tiea.u, ra.aging from 4 fm to 7 fm.+ Thls :iueans

that f.or t:adll less tha:n 3.5 fm we h~ve .an incretuUng number of

osc.ill•t1"8 functions i;:t, whereas b"e"een 4 fm and 7 fm their nud!b(!t

is c.onst.i)(l;t. This iE"t. tutn i.s due to the 'behavioiJr of the l'"eal pa.rt

13

6

,.---~ __ ....,.,,,/ ..-----· -·- ,_ ... .........

2

a 10

r lfml

V·ig. III. ;5, the c1.assica?, turning l"'<l"dii. Fuli aur>v':?; around

stat.i? r".!harme i, drJt-dr:A.~".hrtd cu:rve: a:r.a·t·t&.d chcrrmeL _

-61-

af the OMi' tog•ther with the centrlf,.g&l potential t(t+l)/r2 lo the

nuclu.r aurf&<:e region. It1 order to eh.,ek whether this o0Heet really

<:euoes the eolumM of tho 0!4trlx K to b<oeo'"" HM$rl7 d<!pco><l."nt

arouod 4,5 fm, io whi~h ease th• 9""'""'tion rat1ge of (111-26) should

be J;~fie.rict:ed tQ, eay ~ 7 t~r-ms, 11ft!I •hQUld not oi:..ly look o11t the

rotat\Od angles but 4lso e~ ~h~ inn8r prod~cts bct~~ed the colu11:111a

the""!el...,9. (fig. U.1.4). Her<: we obseitv.; th.st the Mrm$ for thoo

col,.mns at brger 1 are very much lacg..r than for "'""Her 1. The

AAt effect is that th" lnnar prQd,.et b8twee>.\ two edja<:eot eolwooo

around 5 fm le by fe'C larser than f: •8• around 3 h• and that

therefore a possible dependenc:r o~ K cannot b<o ca1>&ed by the

pbenomit:i.'\Od eonsideTed above ...

inner­product

1 10

f

-1 t 10

-3 10

0

f j

2 4 6 8 10 r (fm)

Fi,;. II1.4. Th~ inner produatB of aii pai~s of ad;Jacsnt

coZW11718 of t/ie matri:IJ K.

-62~

We conel11d" eh"t the ranse of the """'""'Hon (IILZ6) 1$ not

limited by tnt~inai~ nu~r1cal reasot1~. On the other h~nd. since for

11 $lven k the values of facrk/t.o. 1 1 do ti(loC tend to de.::.rease for

incres.ait1,g: 1'" we hav~ to llll:pose an artificial ..:;:.1,1t-off Ol"I :th.,;

s"""""Uon range of (III.26). We ean use fo>:" this th" J;>llysical

.arguMent that nueleo8.i: exc.itatio11 ptocesses (':$n, only take (Jlace

within nuclear 1114tter- The relat"d cut-off "-"-" be pttfor""'d id ewo

Wbfs· The ei•plest way ia by just li•tt(~g t~e number 0£ uplines at

11118

x•R/0.625 whe~e a equals the nuelear r~d1us and 0.625 i~ half the

spline w(i;!th· In cases where the COl;"l,"ftction to the .fotm factor .i,$

not too large this yields sacisfaetory result• in the ae~•• that the

corr-ection f1,1(1.Ctlon near the nuc.lea.r surface go.eiji to zero for

incre.$4:l:a,g i. Ira eo.3"& where th~ correction need.s to be l~l"S:oet .. 'fe

find that os~lllations oceur near 1 . The reaso~ fQt this is mQl<

thousht to be the folll)Wing. SuppQ~o that the experimental crooo

$e~tion is s~h tbat there e~l8t6 no adequate local form f ~eto~ that

can deaer~be it e"8.ctty. Then there ~ill always relll8in signifloan~

diff@renc:.es between the theoretical ¢roi&;.& section and the data+

In applylug an inveralon step it 1$ likely that these will diaplaY

th41m8elvee in the few most pe~~pheral spline ~mplitudest since there

tbe J llllltri• haa it& largeat e1e.,.,nts, Moreover, ""' find more e1gn

chansea 1n the rishtmoat columns than e.g. in the middle "'""• which

means ehat larger "'ilpllt\ldes for the pe~1pheral splines loose their

major influence+ ~11e: th~ possible ¢3ll5es for the non-e;ii;:ll!iltence of

a purely loelll form factor, ""' cMcked the effect of the bael.:w<1t"d

¢Oupllng. We did thh for the rea¢Uon 58Ni(p',p• )Ot .. 2j" since thi6

reaction ha." • rather large value for the defo'('mation para~~~r:

whieh means that b~ekvsrd couplins to tbe or channel might play an

impo~tant role. No~ e8 ve drop the *~$vmptlon of a one-way couplingt

the llW!A looses its s.ppH~llbility and wteh 1t the inversiQo method

of this chapter. ln order to 4o ~u lnve~slon lo a coupled chann"l~

coutext ther.,fo~e the amo11nt of co111putational ef~o~t is tremendously

...,ch latger than 10 11 DWBA context. We could only p~rfo(w one single

iteration .eittp. since for every change in i:m.e of the ps.ramett:::a."8

~ 1 the entire a~t of coupled equs.~~Q~~ (I~~) needeJ to b~ ~ntegrated

-63-

again. The i:-esuHL"S fQr"' factor h g1ve" 111 figui:-e ut.5. Although

ve a.pply the abi:"»pt <.'1t-off of the radial intervd, we do not

observe o.ecillationtl ~.i:i:r the surface. l"O; r:he same figute: the form

h.ctQr is glven that \"Hults from one 1teraHon step by 111<«ln6 of the

method of tbi9 chaptei:-, We Ob$ervoeo that after tb~« first at"P tbere

is altu.4y the onset of ad oadllatlon !It 5 .625 fm (the \"&d1'.ta of

the '"oat perlpherd varied splfo.,).

1.5

l.0

.5

-.5

-1.0

-u

,· , .~· ...

j... ....... ~2~--~~~ .... ~~~~"""""110

r (fml

Fig. ;III.[;. RecuU of 11 c:oupied ahannsls inverGion (fuil cun;eJ

aompared with r1 on~-itemtion inv~l'sicm itt DWBA (~t­

daeMd l)urve). The d:i8lwd curo<r rtp:l'<'Mnts the col.iea­

tive mode i form f(lator.

-64-

Of course a li:IO"l;"e thorough 1~.,e~tigatlon of fot"m t'actor invert11¢nS in

coupl"d oho<mela ach.,,.es lo needed to '°"ke firm atate'!lent~ on this

point, it oMms not unUkely that the neglect of the b<1ckward

c0"1>ling is indeed panly respoda~bk for the problems near the e .. t­

off ~adiua.

In ordet to guarant:~~ 'W-ell-behaving s:olu:tions near the cut­

off i:-u.dius also il.".I. OU'( OW'BA-baaed lnvt:t"sion algor1 th~;p Iii@ applied

the following mechanis., for tho <'9dial cut-<>ff. We employ a pen~hy

funr.:.tion g1.,en by:

Rct"e p (i) stal.".l.d.e. ~or the nucle.(lt density at r1

:1o ap~~oximated a.a a

r-e~•l. tiloods Sa:ico11 fi,n:'l,ction with llPP\"Opri"ate parametei;$. Expression

(III.32) toke~ account of the fact that the •mcl•ar surfa<:o has a

diffuse <:J1,.Ncter, an effect "hich is ne..;o.,~$;l.rlly neglected by an

~bi:upt c.ut-off. We $t;teas that fQ-1,"' a ~idtao range of values for the

psram.,ter X tlic influence of (III .32) is re~encted to the per:lfetaI

ree~on of the M<:leus.

~6S-

III-l Accuracy, reprod11eihfltt.y:1o 111.'liquene••

Before we can apply the '"4'tliod of this chapter to pracUcd

caeee> there are some queatio1.'ls ¢Ql'4¢itt'n.1ng accuracyt reprod:1;1c1b111ty

~~d l'l'llqvenu .. that .. hould be. answered u ... $t. Thh is partly done by

me.ans Of no.merlcal experimen.t.ifi.. i;:>artly by theoreti~a.1

conaide:rat!ons4

Whe4 we ~on~cruct a series of ve~tQr' ~~~ eonverging

(hope.fully) to a cert•11'1 ve~tot ~ 1 we have to re$l1~t that this

ve~to~ !., is loaded with some. error band. This error band is due to

the follow11.'l& i;::a111Je.s:: expC:t1mental errors:1o Qbf.g'.'.11.C1C.9 1n the OHP

parameters and 1otridS!¢ u.ncertatnties due to th~ 11a.thtmaticsf An

-ti:ilt1matt: of the ["elative imports.nee Qf the first: two error so ... n:~e8

is readily obtained. The influence o{ tl\e expl:timental errors ia

computed by QQ\viTig (III.14) where the right hand part is ~ep1aeed

by the absolute el<perime'1t"1 t~rots, The influence of th<: OHP

ambigr.d.tie1;1; l;~.$ been @Stimated by doio.g t::wo 1.nve"l."slons, one with .a

beet fit OMP and """ehor (lne "'ith a fixed geometry OMP (MEL78),

Cale,,taH<:in~ o{ eheH t'<o effects for the rucUon 56Ni(J\',p')Ol ... 2!

(OVE83) show that in the neighbo,,rh<)(ld of the moat signifle•nt

changes to the form fa.e.t:or t:hey .are a.mall. We w111 dlt"sct our

attention therefore to the mathellllltical method.

A$ 1$ l,l.t;ual foi; non-linear par~:i:Kter eeat"c.h problems:i the

tinal rei1>lt for the parameter vector .!!:. depends on the choice of the

initi~l vcctot !!.;)• This has been chee~ed by varying !obetween 0.5

and 2,0 times the colle<:ti'le model form factor and id e<\<':li case

pc~fo"tlning a complete inver~iQ~. The resulting for~ fa~tor~, again

for the ~eaction HNi(p,p')Oi + 2t. &11 lie between the two f,,.11

curves of figure III .6. °We ID8ntion two c.ausee. fo~ chis lack of

i:-epcod1>cibility.

-66-

0 4 0 4 8

rlfm)

Fig. :IJI_ 6. Unaartainty bar1d Msuiting f>'om mathematiaai

ambiguiei<Js (bstweBn full c:ur-v~s). The daehed """""

l>tcmda fo>' th,; a0Zle1Jtive model form facto>'.

Firotly, W<i tiave to for11tt114te 4 -~~te(1on for terminating the

leeration. Thia i~ do~e. a& usually, in terid& of fClatlve changes of

th" functlon t fro,. equ•t~on (tll,5). In pra<!t1¢e, wh~" ~"" (elat1ve

difference in ~ between tw<> iter.,Uol\ nep• becomes a""'ller th<'o 1%,

the iteration pro¢edu~e le &topped, ••Y at +•top' ~ow oince thia

+ •H<:ip 1'111 """ally not be r:he 11b,olute 10inlmum of H_!), there lo ~

continuum of ve~tOf' .!. connected to the ~81uce Of ~stop, and it

depends on the eh~i~e ~£ t~c vector .!o which will eventually result.

-67-

A possible; ("emed;r foI" th ls seems t.o be to apply overshoot ;1o that is

to replaco tho <>Orre<:tton vvctor for tho p-th tuni, ~· deliberately

by (l-+<: )).~ fo~ a small fixed e. In thia way one can hope to

generate an alternating aeq\lence in the co .. pOllMC& of th<: vectOI"s d

by which means convergen~e le easier d.et.eet.ed.. l:c. OQ.\" ca.rJ.e 1 °hO'f'@Vll!!:r ~

H turna O\lt to be ulle..,.;ceuhl b<':Mull8 of the difference in

di'tect10n b8tween the vecto"°s ~ and !!p+i. Secondly, a related eauae of uocertdnty comea t:i:om che

obscI"v.ation that the overall phase of the t.ra:L'tl!;l!tiQti. den1;1f.ty u~~ot

be detected; for any real B the tranaitio~ denaittea ei6

v y~eld the

aam.e cross sectioni provided th•t the ep1ta otb1c and coulomb

tt'.!f.tt&1c1on densities a.re I"otated over the same phase angle:. For the

1,,1.ll1que:n.eca of ou.r -results 1t is of great. imp-ort.ance t.o know under

which aaaumptlooa the opposite is abo true, chat is; under which

.t.SS\1.mpt10Jlt; do all form factors, yielding the same cross section..

differ only in one overall phaee hctor? We pay &ttentlon co this

question 11> the a.ppe>:t<liic to tl\t3 chapter.

Since ln our method we do not s.llow the spin-orbit

deformation form fac.t.or to 111'.ry, we ex:pei=t the phase of the

corrected form factor to he defined. Neverthel.;,$$, "• th.;, "pin-orbit

cou~Un$ h rdatively small oOlllpared wlth the central form fa.ctor,

1 t could turn ""t id praetiee th.,t, due to the limited numerical

accuracy, the phaa.e of the corre~ted form :follcto\" t.s T.lnSt.able, thus

givtn.,g rise to •>:t •dd1tioM1 broadening of the uncertainty band.

Such phase d<:viBtions, how"v"r, can be computed ano.lyi:icelly *6

shown in the appendix, •o.d if neee~S<lr)' "'" '""' <:ottect for it, In

most of the cases we studied, the phase dlfferencu turned out to be

smaller than about 5ljlo. As a c.onclus!ott. we i;..i;.-p. (lt-E1t-: tba.t: the

u1>eerta.ifttte•, ~ri~tns fro~ the dcpcnd@nc" of the solutions o~ the

1n1Ual vc:ccors ~ app@ar to be by far the largest ones.

It h intereatinJl; t<> •H>te that at the position of the

1ntot:r1or maKimum, at around 2 ,2fm, t:he uncectafot)' width due to Che

numerical ambiguity (0,20fm-2

) correspoad• r~the~ ~ell ~ith the

"a.;oertd~ty th~t is pred1CC<!>d by formula (I.14) which give• a vnlue

gf 0.22 fm-2

• It should be remembered that the latter for~..,1~ ~­baaed on a sensitivity 1nveat1gst1on by means of a global x2

-68-

c:r1tef10n, rather t1um on .ti d~tailed iJ.t:i.idy of our inversion

procedure. In contrast 'Wfth this 1 t:he uncertainty t:"ound 5,.6 .fw 1s

11;1u.ch larger th.an follows .('ro111 (I tl-4) .. Hot-eover, as ~.i;tt.d before 1 the

form of the lattH $Ct"cture is affected by ~he way fo Wh1ch the

t.ut"""off takes place where~'3 the str11et1,1fe t"ound 2 .. 2 ~111 is not.

It l!;'eeDJ.s appropriate to conel1,;1de this se¢i;:f,Qn with 11

discussion o~ two aspect~ of the inv~f8lon method that have nQt been

conaide-:ed up to now. One taf.ght .say, 11': ~thematic.al tetlns~ th4t

they ¢~~~e~n the '~n!queness' al'l.d tbe •existence', respee~tv@ly, of

the method and of the 1~tended form ~~ctor corre~t~oTIS•

10 $t9rt with the onlquenesa Q~ the method, we muBt note t~~t

at two places We have made "1tl31.1mptiona t::l)d¢ti::I:ning the IQ!t:t'tic of

space of Oh$.c"t"vables. The first il'lrH:: . .:mi::e is, whr:i!:tC we base the

definf,ef.on of the: mat~i:x: J on ~'he F;uclide.a.11 d:!.;'1t.snce function of

equation (HI.)). Thia dlstance fun.,t1o:i.., "nlgna equ;!ll weights tQ

the c.ross eecttone as tae8..!J1,1t'.e;d .and CO~ftJ,Itt;d at all artglc:g.L At flri;,;t

sight <:n>e could object to this d"f~M.tlon th .. t, •ince the dat« <'re

taken 1I1 !Su.ch a Wft.Y "1$ to have 8:lmotit equal l:"el.ative error:!)

throughout the whole angular rs11i;:e, at the foi:ward angles one has

larg~~ absolute erro~ (lags th.al'I. at the ba.ckw~rd ~ngles. the~~tore

the weight :factors for thcee forward. cI"ofis sec:tir>:n.1!1 !S!hould be

si:m1ler t.ha11 the weight f.aeto'rs for the C:.\"Q:S& sections at. backward

onglea. fo eomD1on practke (li.OS5J) one appl:f.co a x2 deH<iltion ld

which t.he difference 'b~tween the theo~~t1cal and e"7.pe'("f.mental c.ro1;1;t.

se<tlo" is divided by the exp.,rime<1tal error, a• ~n equation (r.12).

N.CV"i::rtheleaa thete are two '('ca.sons for 1,18 still to pt:.;fer the

euclidea;i metric for equation (HL3). The first tooaoon is, th"t

apart. l~om the e:icper1mental errorfl,. tliel'e is sls-o a contributiol'l to

the error b$;id that at ems from theor~tk1tl '"'cutainties. This is

predQminantly due ti> the uneert.i;t.J,-P.tie.s Jn the pal."ameters. In. Qtlt'

case the~e la a strong tendency th~t the cross sections at eh• h~i:::::kward a.ng:le~ are DJ~h mQr1; .s-ensiti\.'e: to the value~ of the

parameters th~n those for the forward angles,. It ~ould be

.il:'Pl)l"Opriate to ;;.a.1 that thie b.ockward theor~t.~c~l an.glea hop:vc a

larger erro...- fJ,.,.g thatl th~ {Q~1:u'd ones.

-69-

The second ~eaoon for prefen1ug tile Euclidean distance has to

do with the Q\)$ervation Qf Amad<) and SparrQW (AMAB3) that the

exclta.tion streagth il'l !>I/BA calc1,1lat10M ~hould be obtained by

<;lems.ndi"S the nut maximu,. of the theoretical and expui.,eontal

uoos sections to cO~ncide, in other wotdo, to ae.sisn a la.:-g<ioi:'

relative weight to the fo'l;"tiJard angles. Moreov@r, w~ perform.8d same

tei!i't calcul"'-t.ione ~htr~ we compared the two 'Ille.tries 3nd it tutn~d

out thttt d.iffercni;ea in the 1,"'eaulting fo\"~ factors. •re conslde\"*bly

o1'!8ller than the amMg'1ity band in flgur" I(l.6.

"" establiehing the p~ra~eter; that ""1ltipli~• the co.:-.:-~ctlon

\rector _! in ord@r tQ obtain "'" optimal Ht to the <;I-ta aa welt as

obtaining the O:Pti""'l m.••b•r of Slngutar valueo (k) frow the

s1.0ll..-tion (HI.21) we dso apply a x2 procedui:~. The "vide""-e that

e:.c:lsts t°Qr a la'l:'S:~ theoretJ.e"'l uncerta.1nty at the btu:.kwsrd &ns;les

~Qch as due to the Qneertaintics la th€ pB~~w~ters ai" however, Wa$

~ot seeTI for tbt: para:oteters A 3M k.. Therefore "i'a h.l)"le chosen for

the xZ definition ln acco(d~t1ee with Reise for tha <>ptimhat~Qn

problem (1\0553). In thla cue "" also pcr(or~d test ¢~lo_ulat1oM

and the difference was seen tQ be even s....,ller th<lu it1 the prcv~oua

c~se ..

A~ far o.s the 1 ~xl,,stenc.e' of form factQI:' e.orrec.tio11-.ij; ie

eonce"J;"t:J.~4. we must l.').Ote that. i:be pr::ocedur-= does not .a.lwa}"'il lead to a

signlfleant ~~ reduc.tlon. An example of this is found in the

SSsr(jl",p')0\+2-t-1 reaction. \lhat happens there h t~ following. If

we inctease the ni.imber of sinS'J.lar value~~ starting from zero, the

ooJ.\ltion notm front equatl,Qt1 (III,19) 1.,..,dlately narta to !nCi:'eaae

drastically whotc~e for the Hr~t few 01TI5.,1ar valu•o the residual

no;:,, (III.18) $11>0st ma~nMiilB ito ma><lmum value. fo reduce the

latter aignific(ktJ.C:,ly, we h.o.ve to take $Q D'la.l.').y singul1;1.r values 1rit.O

account, th~t the solut~on norm has beeome mueh too large to fulfil

the line•rization <:ot1dition of (Ill.8), Thill """M that e'l'en though

we ft,.4lly can reduce the residual notm of the l:!iu;ariz"d set With •

c~~ts.in wct<:>r .2_, th<': ~esidual M'"111 of the orlgltoal set (llI.2) is

not reduced with this vecto( d. Thi• """ifests it~e1f in •

m.ult.Lplic:ation ¢oefficlent .l.. that is near to tcro

nega.tivit·

-70-

or even

This ~eyq.at"Jc; ,g.a:ln!lil in interest if 1.i1c remember the global

.s:cneitivity for the tt:'lln.Sltion densities as el!;lt°"bJ.~~hr:i:d for the 58 Ni

and aesr ~etu;::t!QT'I.$ in section. I--3. There we have seen that.

o1;:-on<::el;"·i::rlng the x2. cr1te:r1¢n~ the two reactions be.hau-i;; !i'i.11111.a't'ly~ We

MY ther-efore co11clm;lt:!: th.i:tt the total xZ is 01.1.ly a yC;~y incomplete

tool ~ot pred1ction8 ~o~eetoing at~ainable i~pro~e.ment$; it appears

t.hat the bebav'-o\ll." i;;i.~ the residual norm of the linearized set~

together with the 60lut10n norm, as a f1.1l"let~(lr'1. (>f the num.ber of

incorPQ•~ted •ingular valuea, is more reliabl• in thio fld<:\.

lie Mte Hnally that a ais,,ifle,,.,t ~mprovement of tho 88 $r,

2+1 c:ro!::ls section ..£!!!, he ,gai.11ed .. howevert if we use a transition

de.,ait:y ao rnsulting fro"' foetutic electron a<oatteri"S (S<;i!.83).

'l'bh t>:<IMit!on charge de<>nlty has a •tructuro thot 1@ @omewhat

similar to the one found for the 2+1 state ln 68iu. Not 0<1ly h thla

tronUt1(1Q de'1o1ty for BBsr undeteet~blo "1th the pr.,ae~t ""'thod, lt

h413; al$0 aot been fo11n<l with "1 d'-J;"ect ~ unbiased ateef1e.8t del5'~t=:nt

method+

-71-

Hl-4 A S'-'"""'ry of the inversion method

In this section "'" aum ... rt.c the MO"mptio~o on wh:leh the

inversion method is based together with thE actual st.gp& that &tg to

be taken in order to perform an inven:l.on.

The .astn.impt1ons arc~

1) The rea.ctlon that le•d• to the eonalde,,-ed ""cte~r •t<'te is a

p1>te oM-Hei;i \"e .. etior. with<;>ut b4ekw"'r~ .;o<;n;pUr.s to the e"trance

channel·

local+

U!)The api" orbit eoo.lplins "" for..ubted 1n either the foll Thomas

fotm or the Oak Ridge form U correct.

iv) I/her" approptiato:, the co1lect1vo! model can M chosen as a

~Urt. Cori:ectio"s are assumed t<;> be small.

v) The f:°Qrin fo11.ctor 1s ~hosen to have non-vanishing values only

<>Hh1n the ""de ... s.

A.doi;.titJ.g these aasutnptionst the inversion metborl comprises of

the following steps:

l) The spline "idth <ind the ro•Hai hterv<1l for the !"'ve\"slo'< are

established· Th<: fotmet £ol1o..G ,h'(HQ 1nveH1gat1oM oO<lcl!:.:n:lng th•

:a1.1,itual dependency of adjacent c.oltli»ns of the m.a.trbc: of the linear

transformation between the p.a:ram.etet'"s ..! and the transition matrix ...

Tbe latter ls, accor-ding to model a.seumption iv), equal to the

radial 1"terval on which the OM? has oi;m-Wlohhing vahes.

-72-

1:) ~or all splines of 1I\tere$1;:., tht: T-matrices are ,generated.

Moreover, the T-matrices for the spin-orbit eJi:it;:ir:'1t1on .P.nd fo't the

Coulomb ex.citation are computed.

3) '!'he coefficlent vector !o 1• obt<llt\ed by expandlng the collecti~•

model tf6Mlcton density into s oerteo of oi>Uoes of ~pp<op<iate

width.

4) The ci:-oss section is computed re.peatedly f¢r .Q Set:l'.ies of all

vectors ~j "'hr:i:"i: .!j equals .!o except for the j ..... z:.h ~<;iompQ11.1,mt. The

latter c:.om.pottel'\t i!:l l.n~re.r,i.S.i;d by a fixed amount. This is done bQt.h

for the r~.r.11. ~n4' the :f.ll!agins.ry pa.rt of .!j· ln th1$ t..\lay ..te Rla~e an

Mt\mate of the matrix J in equ(ltlo:>n (~~ ~. \ \). This can be said to

corr~$po~d with spproximatiag the x 2 surf~ct ~Q~Ud ~-!!o by a

paraboloid.

5) r'roPJ 1.nv.et"ting the matrix J we C.Ql}Eltr\lc::.t a cQ:ttection vector .!! that poloto eo:>"'"'d• the obsolute minimum of the abo~e p~~abolold.

If ~.e:c~$$Qt')', the matrix J is extefldei;I by 1;1n appropriate square

matrix to account fr.)r th~ dt.ru);it;r dependent damping of th~

¢1Y~t"~~tf.On; vei::::tors+ The invera.1011 t.:!kes p'lllt.e hy means of the

sing:ul.!.r value aaa.lys1s. The number of a!n.gular values (k.) thotlt. i..9

taken 1.nto account as well .is the: \l'~h.i~ c:if ~h~ ~.o.\ctot that is need-ed

co:> qrultiply _!! to obtain an o~tt""'l Ht to th• d*t" ().), ue obtained

by means of s ~rid s~ar~h o~ ~h~~~ ~wo p~tamete~s- This is done by

c;omp\l~~'l;'li,g the x 2 between the experi:irient.a.l al'l.d. the(li.-etJ.ca.l ct"oal'l

a.ec:tions for a well-.t;:.hoset1: tt:!t;. of cQn;ablnationa { k~A} ... The

c<>mlol~at 1011 (I<,;\ ) that yields the l<>WUt x2 LS uSod. •n this "a'/ ""

end up "~th veCtQ'.I;" i and therefore with a new vector .!.i •!.o +A 1~.

-73-

6) By meane of (IIIA.11) we eompute the phae" angle 6, If ~ dHfeto -i6

elgnlflcantly from zero,. ve .apply the tTansforta.;Atil)Q .!p•u ~·

(~<>t,. thillC thie tranefol'Ulation abo- affects the strength& <>{ th,.

oo,,.lomb and Che spin-orbit couplin~s.)

7) If the d1f£erotllc"' 1n x2 between the <0ro•• seet~Otl• <:r(~-l) and

a(.<;,> 1o larger than a eertdn re lac 1ve valu.,, the steps 4-7 are

l:epieated,

-74-

III-A, "'ppend1x

Here we $tudy two aapee~& of the inversion p(ocedure that a~e

both related to the phase of eh" fo•m factor. Ftut we w:tll

investia:ate ~nder whieh ar;s.r;=1,1111pt:lona the o::.omp;l.t;:;,,. ph~ae ambiguity 111

the 011.ly 011~. After that. we present the t.Qm;:puts.tion of s. phd.$11!:

cO'('t'ection whieh In pI"inciple e.oul<:i bt?: u.;ed to reduce th~ width of

the unceruiney ~nd of th" fo...,. factor.

Consider two solution• of (III.2). M)' ~ and !· Thio '""""$:

(IIIA.l)

.. 1 \' lm..11 If we ddine Pk • l '\. .,. , then (IIIA.t) becomes:

l

(tHA.2)

Now we note that

with (IUA.3)

Therefore (i~rA,.2) yield•:

(IliA.4)

Now aesuine th~~ the number of k.-v.t.1ueti. exceeds the aquart;! Qe th" number of p'i'Ometers. Since k dHt»•8u1shes the ocatte<ing

angles, thie Maumption eall al""l'a be folfilled, If we interpret Q

&6 a matrix, we have ~as column index where~8 the combination (1,j)

forms the row hdex. w., assume th<>t the columns of the "1at~lx Q

.span the entire m:at!:"t.x &pace. Then 1t (:411: 'be Seen that the matrix Q

-75-

~oueueo a right inverse Q' such tho.t

~Qi'jQ,1',j' 5 5 tk k -(1,i')(j,j') (IIIA.5)

~Tid tlleI""efore

'tt we f:J.'('8t examine the case .:l .. ji th.C:rt 'We See that

(IIIA· 7)

and next it follows th~t

oo, with (IIIA. 7):

6 - 6 :J I (IIIA.8)

We learn from (IIIA.S) that the ~dCOhCtent summation over ~

does not 1ntrod1,1e-e •dd:t.t1on.al ambiguities 1~ ~· This means, among

others thfogo, that there exist M e .. o form fac.tora th4t yield

exactly the ~ c:t:"ciee sections but diffe:rent analysing powers. In

this uenoe, the analyai"S poWci:' U not an independedt obo;,rvable.

Our c.onc.lusion therefore m;u.is;t be that, apaTt frol:d the On-tr: Overall

phaa~:io thei:-e iP.:tc no more s.mbig:uitier;i in ehc £onD. factor+

-76-

Now .:J.fl an additionul ae.aumptio11 ~ state th.1;1t from all

p¢$1J~ble 6 ""lues, the "'"'tO<' !~(!) with l:he omallost Euclidean

!!:U$tsnce to ~ 1.$ the most fJ;i.vQur.a.ble one. Let _i be the 11ecto:r- auc.h

that _!-~~· Then this ~uclidea<1 dUt•rnce is g1,.en b)';

(HIA.9)

In ord.er to find std.t!Qm~ry value!} of i5 we ha'q'e t.o $Qlve

-o (IIIA.10)

yield1n3

r ~m(•)(l+Re(z~) J d • -~ aret~o (l+Rd•))Z" Im (z) + ""• n-0,l,2, ... ;

whcte;

'We note that there a:re two sol:i,it1Qr\S fen: il,. one 1'.;:.0\"li'.'0tiponding w1t.'h

the minimal ~2 .add oru; correspoad1ng with the ma}(.iI1JQl e:: 2 • For i::: 2 We

find:

which is $e~n easily tQ hav0 a low~r bo~~a of

v•<ue that is reached when~ a<1d !o are pa~allel.

-77-

(!LLA.12)

Chapter IV

In this <0h•pter, the method of "h;l{'ter III ls applied to "'

""'°ber of one step rea<:tt"""· In section IV-1 we eheck whether th<1

form. factor fou::nd fQr 58N1 is Hreal... tl:io.t :l~:1 whethec it 1e not

due to shortcomings Of the D\IBA aeheme used. Section IV-Z le

devot:ed to 111i¢ro&eopic. calcul.at.'-0~0 Q:e the reactions S'-te1-

58 Nl (p, p • )O! _,. zt. It will be ah""" here that the foi:'l!I factors ae

fou:od. for these two t:e~et:lone probably et~m :f\"Oi:a * trderosc:::opic

origiti. EmplOylnS this mi"roocoplc intsrprceat1on, we study a

number of other Ot -> zt nactiO<l8 in se,,tion IV-3. The ruults o:>f

inversions fQt """"' 3i states are presented 1n section lV-4, We end

thiil ch,.pter with • study of t\>e ;ippllcablllty of our invCr&lO(I

procedure to inelastic dph<' •cauerin& and to single-particle pick­

up reactions.

IV-1 The 1nterpretatlon of the SSNi rc~~lts

AB is shown in figure III ,6,

invet-rUQ'il show a seve"°e de:v1•tio:o. from the. collective mod4l

transition density. In order to e~plain it, ~hree possibilities are

auggest:ed.::

l) Effects due to insdeq,.<lte OMP ~rsmetrhation;

2) Non-negligLbl,. co:>dtrit;..,tiods from tl"ansfe.- ...., • .,e~ous;

3) Microscopk effects;

The flrlit t"o show the111selves as form factor correctiot1e !o an

indi~ect way Qoly; the resulting fotin factor co~~ectiOdS are merely

seCoddary effects. In this ~eetiod we study whether taking them into

a.c:.i::.ount prope?'ly 0:¥1'1. elim.inate the need fot a form f.a.Gtot:

"orrectlon. The thtrd possibility indeed should give rioc to foi:.11

f;!Oetor corrections; tbh i8 st"d~ed lft ~eetlon IV-2.

-78-

1) Effects of inadequate OMt para~etrization.

A~ mcl'lt.1on.ed Jn $~¢t1on r-1 1 ;)11 phy.=-1t.h.1 eff.ec:;.ts. that a.re 1."1.Ciit

aooounted for expli~Hlt wlll itiflueM• the reoult• of a model­

independent c.alc.ulation of the transition densities. Here 111e study

the 1t1flu•t1ce of d!HereMM bet..,cen the OMP aa it is d••cribed by

means of the commonly used Saxon-Woods parametrigation and a moi;-e

realistic potential. Therefore we have to provide for tile followingo

.a) real:l.8tic OMP 's, not pia.rametrl~ed in tet"ms of th-e Saxon-WoodGI

formulai

b) a relation between given OMP difference and the cotc'"csponding

fot"m. factor difference such that we obtain the same inel.a.stic ct"oas

see.tiofi1

either with the origin.al Saxon-lri'oods OM'f' and the corrected

form. factor~ er Yith the res.listie OMP and the callec.tive model fonn

f~cto?;"".

Ad a)

We apply two dift'erll?nt methQd& ~Qo'(' obt.i;r.tn!ng: a re.a.11etic. OHP ..

'!'hey diff€!r mainly from a mHhodological point of vhw.

The firet method starts frotti proton and neutron d1stt"ibutionG

p (~) '"'~ p (r) "~ o<>mputod fro., a self-consistent Rortree-Vocl< p "

odcuhti<>t1 by Woro~uier et al. (WARSZ), Th@ proto~ ~H~r~\,,,e1<>n in

the interior part of the nuc.leua :e.hOW=:i. ~ cQrt$:1,i;i~r.)ble de\'l'iation from

a flot, Sa><<>n-Woods Uke for., where.as the ne11tron dhtr!but!o,., i~

TQ.tt¢h fl.attel". f°:t,"om these deneity functions we compute the ~.e.;;.1 po!lrt

of an OMP function by folding them with tho «Ml port of a density

depeftdent nu~lean-nucleon force VC(r;~) ~9 de~~ved hy Von Geramb

(C)ER7l):

(IV .1)

-79-

These integrals are ecnnput.ed uumerically by a atralghtforvar(l

th1"6efold au,.111<1H<:n>. ~n (IV -1), ~ atandt (oi: th<: fel1D1 wave number,

computed aa:

(IV .2)

wllere pn-tp at:ands for the total deMHy fol'l<:t1on, ~n-H>p'

For s:l,"1pl1u1ty we omit the e>:cbange contrib(l.tiodt. The

imagil>.o.ry part of the OMP cannot be derhed in a similar "'*""",,. by

using the 1'CQ.llginary part of the ~\lcleon-nucleoe1. 14te1"4~t1on, as the.

latter stands among others thingG for all neglected 1ne1Mtic

channels. Frequently the imaginary par!: o( " phenomenolosied Ola'

a used.

As a result: of the fQlding procedure:; all internal atruct:ure

present in the proton distrib111:ion, 111 smeared out OOIOpletely and

therefore a totally n .. t innei: region arise• in the potential. The

real part Qf the usulting fold1".!; OMP thenfore bears a very large

slmilarlty to ~'h.e or~gir.ial Saxon--.w'oodo pQt.c:ntial, although th~

radius turns 01,,1.t to bg ebout 7% small.er. The cross section thst ta

<:Olllputed from this folo:Uns: Om> give& a somewhat poorer description

of the data t:h.o.n the original OMP.

llhcrea& this method t:o obtain a realisti('. 0!1l' 1o ti pure f1rat­

pr~nc1ples method, at leut M far as the red pan is concerned,

a ""-"°"d ..et:hQd "e appll' is much ""'r"- emp1'1cal. It 1a baaed ori the

observation th<lt the inversion procedure of chapter III e«MOt be

formulated for inelastic &Mtterlng only l)...t abo for elastie

ecAtter1ng.

Condder !lgaln equation (I. 711) for the partial W<1v~ 1n the

elaetio channel;

(IV.3)

-ao-

Similar to (1.9) we can define a. T-matrix for this elaatic channel:

(IV .4)

NQ\oi' $uppo~e th.at we trattsfortti the central par-t of the potential from

(IV.3):

(IV.6)

lly oomo elcmcntar)'" algcbi:a lt fol1ow1; thot up to Hret order in 6:

(tV. 7)

where the functlon ~ ls the $olut1oo of (IV.3). By fooertlrtS (IV.7)

into (Iv.5), together with a suitable puametrlzation for ~U, we

agaln can apply tM h1V<orUM pro¢ed1>re of chapter III. The only

difference la, however, that no more thOEJ.TI !)JJ.i: 1.tei:-~t:lon ~tt:p ¢.otl\1 br;:

t.;).ke:t'I; & l"lew step woiJld imply again a c.alc.ulatlon of the wave

func:tione whereas in the case of inversion fo"t" in~laetlc :i:-c~ttetJ.ng

tbe same distorted waves c.a.n be u:aed throu.ghou.t.

In the above way we can 1mp«>V¢ th~ d<'$c<lpUon 0£ the

experi01entol dato reoulting in a x~ «d"c~l<:>n Q~ ,,\,o.,t 15%. the

OMP indeed 8ihQws; scnne ~t?;'"\lct1.11'oe; f,n t:.l'le :1.o.ner reg:l.t:)IJ. 111 contrast to

'the difference.~

hQwev~?'." 1 W-it:h the ~L"1$1'141 Saxon w-ciods OMJ:> is nowhere larger than

3% <>f lta rnax(mum V9l'-'C·

l;D tti.'* t."'*mal,t'u~e\" uf this subsection we will concentrate on the

latter, model independent~ OJMP a-9 an ~x:~mpte of .i;i. r~a1t.!lt:i<: OMP.

-Bl-

Ad b}.

Now Ille arrive •t the question whether the differ6u¢ee between

$U.:h e realisUc 01«' and the st•n4ard Saxo>t-Woods 01-!P c.an ghe rise

to the foi:m factor differences in the invei:&lon proccd.,i;e of the

ki4d that wei:-e observ<1d· In oth"r W<>rds: h it pouible to ol:itain

th• $•me ci:-ou aect10>t by applying an 11naltered trans1t1ol."I density

but ineted a rodhtic OlW that differs no.,here ..,roe. than 3% from

the orig1""l OMP TJ?

We solve this pr<:>blem in two atepe. Fii:-~tly we rd•te the

dlffs"teta.ce between the two OMP' I)• say QU:io to ~ wave fu:a.etioo

transform•tion. Next ~oeo constroet th<eo T-matri~ with these

transfor111.ed """" function•• lly demanding that it equals tM T-matrix

ras1>ltil'lg from the original "'•vefunctlOM and the <0orrccted form

factor we obtain the expi:-cu1on for the fon11 1"¢tor co>:reeHon 6V •

Finally we shaU show tha.t with the oU given above we get a ~"lue

foi:- ~ V ..,eh smaller than the coneetion "" obtained by the inversion

prot:edure ..

The doeorivation of the wave f~netion transform 1nvolves

"ppi:-oxim3Hng e solutiot1 of a no<t~linear diffenmtid equaUod.

Thh is possible o<>l'f if the funct1ot<e invol'f"ed behave folloothly

.;.nough to allow lineerho.Hon. for example, H we would apply tl\e

ob~i<>11s pei:-a,.etritetion for the ttan&fo.....,tion:

(IV .Sa)

and <IOUld try to find " relat~oo. up to Uret order in 6 between

the ~ori:-ection function 6S(r} and 5U, we encountoeor expresa1.oos that

involve the hh>ctions ~ and l:lleir derivf.thes to noeogative p~ere.

Th~• makes every linearba1:ioo in the neighboo.rhood of ~eroes of ~

or I;' impossible. The reason f<•r thi.e 18, that the effect of W

consists mainly in ~tretchins or compr<>ssing the ~ and therefor"

shifting ite ~eroes, This efh:~t eannot be panmetrized by "

multiplicative tran$forma.tlon. Instead w~ better use a ..!£!!!.. t r~I\8format.1on 1r

~(:t;!J) + Hr(l+li5'(r));U+liU) (IV.Sb)

We must tak~ car~ aot e~ ~PPly any Taylor expansions of ( around r.

<he oimplest way to pr<>eeed i~ <aS follow•, We define the local

equi~ale<>e waven1'mber k•k(r) and the l<>e.ol scde t~onMonnation

fl.J.IJiCtion Sr•S'(r) S.B~

~ (r;U)•sin(rk); (IV.9a)

~ (~; U1"5U) •sin( r( 1-kl S ')k) (IV .9b)

When inserting E .ln t.he ;i:11dial Schrt\.dinger equ.e.tioa, we fiad

c.omplic:ated no11-litJ.~i).l." ~qlJQt ions for: the complex functions k a.ad S~

For slowly varyiag pQteot1~~~ U and BU~ b0"'7ever) trie can take k eq~dl

to (E-u(~))\. Combining (IV.9) and (IV.3), we find:

1 1 &U(r) 6S (r) • - 2 kZ(r") (IV,10)

NQw we see that 01.1r appro:>e11'.11:8t1on is ro.thet cl"udc indeed as it gives

11 ~anUh1ng aeymptotic phase shift differen~e d .. e to the potenehl

perturbation 6D. Thb clln 1><: "nderstood easily if we. re.alize th"t dk

the approximation ia based on the ~ss.,mpt~on that '1"r is negligible

compared to k(r), wh1eh ts valid only for small values of r. We will

cOTI.c.ent'tate only on the c.orrec.tion to t'he it1ner p•re Qf ~he fo?;""i::a dk

fneto~; here the error due to aegle~tiag ~ 1~ of the order ~f

20%.

For the T-fllt\~r1x ele11el'.lt.Eil tliat c.ot"r113-6pond to the transformed

OMP we cs.tataot $imply k'rite down an equation analogous to (I.9)+

Instead, we have to proe<'ed. 1;,y giving t~e foll solution of the

1"ho0>0Q<eodeOua oqu8tion (I.Jb) by means of the method of Green. In

the expression below all ftlnc.tiona !;; ~re ul"l~et:"~tood tQ re$ul~ f'l;"Qm

~adlal Schrtldinger-like equations with transformed OHP''s:

(lV.ll)

Here the t.ilde -.rk.s the sr;i.lutio:n.s 0£ the homerigeneQIJ.~ eq1,.1~et:Oi;i.$;

p<(>) h the smaller (larger) of p and p' whereas p•(l+liS')r.

-83-

U' :>. The T-matrh element$ T ate obtained by cvd11a.ting ~c(p) at a

fixed, lnrge valu" ¢( 9. Here tl\e ~ifference between ~(p<) and ~(r>)

o1h\ be only a phase facto"<, •aY ""P(i~H ,)

NQ'W we can equate the 1~trlx elemedts from the ~wo

d1£forent approaehea, yieldtns:

... ..... ).'It A I l~' .. "'"P(i~'\ ,)fdr'~ ~··)~ (r')V (r')(l+liV)

A 0

e c0 ce0 (111.12)

Befor~ ev.o.lua.ting the ii.i.tegral l'l;'\ the left hand p.8:rt:io we notl!

that the above ~q11ation cannot hold exactly ae the function ils' (and

tberefo't~ p) is l, -dc:p.end.ent whet"eol:'tG the function l)v ie not, 'l'h:tiJ,

howeV8t° I to.is.ht be S, f!l;hc)'f'tComing of Ql,I.!' paramet'(j,,Z!:,tJ.tiont 80 1.:t 118

.S$Bume for the: to.011te:nt th~t 6 V a.lao d-upe'l'ld.6 on .l.. and l.' . M.oreove:t" 1

in order to "vahtate th<> left hand pi;te, ""- have to neglect the

differenee in ~ --<lependence of p' between ~ ano \I (which is Ht1>i ted

by the tran3(erred angular momentum, ~"Y""Y·)

When we apply the fo~lowing r<':lati<>ns:

d:t'•(l-<IS')dp' (IV.13a)

V(r' )•V( p' )-~ S '!!tp') (lV.13b)

which are of first o>:der in ~, and. when we .a-a&ume that the funcUona

~ """" eufficiently orthogonal~ we get:

(HS(l+f !~)) - n+i1vJ, or:

ilv • \0...;_ dV) ~\I

(IV.14) v Tr kl'

Inserting 1;.he .a.ppropr:lollt.e numerical v"'luee, we get i:tt: r•2.2 fm :for

ilV a value whioh is abo .. t a factor o~ 30 smaller ch•n the invero!on

result, ~t¢tD this we can. ~onclude th.at the correct:lOt"l to the fo~

factOX' eannot be >1'"1'1"1ned by the aaaumpt101' of an inadc~\late OMP

p3rametrfaat1on.

The second sus:g,estion that we want to investigate here; is

whether the forU'l. factor correct ioa mi,ght be due to the neglect of

~:c.rong tr,,n~ft:!:r ¢.ll~:rl:r'l~l,$ th.flt ,;_ir~ ¢'11,1pl.ed t:o ch~ or+ :.rr tnrns;ltion

and that we ought to take: into account explicitly .. As candidates for

these tr•<\ofer rea<tl<>"• woo tak., the ¢<>.,p11'1S• '110 the (pd)(dp)

cMoElinnels "i'it"h the lO'feSit two statll!s of 57 Ni· 1'hesll! .ar~ s.ss\lmed to be

'1;"3thez:- p1.1'(~ one: p.cl~t1c1~ atates so that ...,e can h.opt to ~e.&c~i.be them

adequately by $imp le one <>eutr<:>n tr$.,ofer ¢d4ul.,.tions. It< the

subsequent study we assume that there .are no proton components in

the •ctlv"' &tote~.

The coupling echeme ii;s .i;t~ fQllowi!i'~

~------o+

In the above scheinet the transitions leading to the 2+ state c.an

tal<e place by mean• <>{ •eve<al c<>lllb1nationo ¢{ .!>~, h <rn~ l>j. We

will consider the combinations~

(~~.~ •• ~j) ~ (1,1/1,1/2)

(1,1/Z,3/~l

(3,1/2,5/2)

(1,1/Z,1/Z)

(1,1/2,3/2)

(3,l/2,5/2)

Eve<y single (h ,~s,l>j) tTanoitll'.rn ha~ lu Wn fo(m fo¢tM. If we

adopt the ~•~<> ""'"'S• <l.J)proidmatlon, theee tra<\sit:io" densities are

gi'li'en by the wa'le f\.lnction of the transferred neutron, bound to

-85-

the S7Ni core. The transit·ion 5'C:~cn.gth8 then a..:-e: ~~s.en~1.$l1y .g:t-.,en

by the product of th<: 111f.,'(08<>opl<> •111pl!. t....iu of thes" v"ve fo!lctio!ls

i'-' the o+ o'( 2+ st.,tes, ""'Hip11ed by the usual zero range

no'(malliatiot> codsta"Gt n0

(•122.5). Sueh transition strengths ar6

referred to as the spect:roecopic amplitudes S ::-

(IV.15)

Here the factor t deals vith tl'ie pi:-o,.ei:- 11"Gthnmet,,.ioetio>:1· It <:"" b<!: eomplltcd ~ .ae•>:1e o{ the methQd as de•ci:l bed by Bi:.,ee.it.iti:-d and

Glaudemans (BRU77), The factors o and e are the amplitudes of thfl

(lfte l.'!..ettt:ron components in the states of int:erest. Note tbat elt:her o.

(piek up). Tbe mier<:>aeopic

structures of the at and 2! utates have been taken from the

compilatlon of u~ .. ~u•rd. ""d Gb,,.de.,..,.,a cltfo$ t:ll<: e..ic.,1 .. ttot>a of

Koope and Glaudemane (BRU77, K0077):

lo+> - -.2ajp1/2>~ -.1sjpJ/2>i -.sslH/2>2

\2+> • -.421p3/2xpl/2> -.7llp3/2>2 - • .'IS1fS/2><pl/Z> +.321f5/2'<p3/2>

-.301£5/2>2

A.ftei:' c<1mbereQme 111atehillg of eonvent!ode we ai:-,,.lve at the

apectroec.opic amplitudes for the a possible transfer channels as

listed fo table J;V .1.

Table IV.I: opectroscopic amplitudes for the (pd) and (dp)

transitions ..

o+ 3/r (l>j•3/2) s- +135.l

o+ + s12- (A j•5/:l) s- +95.3

3/2- • 2+ (l>j•l/2) s- -51.45 3/2- + 2+ (A j,.J/2) s- -86.98

312- + 2+ (aj•5/2) s- +39.20 5/2- .. 2+ (4 j•lf2) s- -42-88

5/2~ ... 2+ (A j~3/2) S- +39.20

5/2- + 2+ (A j•5/2) 5• -36.75

-6~-

The £1J.·~t two smplitudeB have been reduced to 100.3 and 72.43~

respei::t1vr::ly 1 in o'tdcJ:' to give a correct representation of the

experimental pick up c~<>•• oeetiO"-$ le<ld~"8 to the )/r s.nd 5/2-

st:atea. For the further details of the calculations we refet:" to the

theoi$ of Pol~n~ (POL8i).

It turned out that the ohape o:>! ~he different~<'~ er<:>~~ oeet1o'l

Q~ th~ 2! state from these sequential proces~~~ bears a reaso~able

similarity with the one from. the c:ollect1"'1e process. The abaolute

val~e, hQwevet 1 1@ roughly a factor of 1000 smaller~ The analysiag

power diffcl"s cow.plctcly from a .collective model e.a.lculatio11.. NolJ w~

have to investigate whether the (pd)(<lp) e<>1>PU"-f!; e"n ln<leed e«'.'Sc

the atrong effeeta that we fwnd ln applyh1g to<> inverolon

proee:d1,1re. Thi~ ls Ol'ccOmpllsbed as follows.

Once th<> collective cross sections and the (pd)(dp) cross

sections ~re kTIQwTI 1 th-e only mh1:;.lng ingredient in order to compute

the t~oss section corresponding to the eombined process~ is an angle

~epe\'ldent ph,.$e f~ctot. T..et <1>1

(0) a<1d <t>2

{i;J) be t"o real function•

of 0. lie auo., tho oxper1ment11l d"t" to be $cUM by ~ hcto( "·

Then 'llfe h.ave;

(IV.l~)

'l"ho 1tbov" o~u,.t1on ha6 to hold for all o.ne;lea 0. w .. .;oodaid"r th"

eroe1;1 sect.ions at: 0 .. 30° add a.t 130° t that is, around the absolute

maximum. and m.inirt1.um of all three cross sections. We then have as

numerical values (mbarn/srad):

30•: C><p( (X<t>, (30• l)•0.97+c.:p(~+ioo• l)•0.00078•0.97<> (I~.17)

From 1V-l7 we h<1ve a.9992~1"1~1.0008 whereo.s fr<>m IY.18 " should

obey 1·92()(1~1'1.925, which clearly leads to a contradiction.

If we would increase the (pd)(dp) <>~<>•$ $ooti<>n o.rt1Hei"tll'

-!$7-

with .o faetor ~, 1C 15 Mon that c has to be *t lent about 750

in order to have equation !V.16 holding for 30' .md 132°. thU ll>!!ans

that 1(1. O\"'det: to explain the observ~d C~Ollilo@ $il?:ct1on differences

between the experi111entd eros~ section and the <cross seeti<>n from

the eolleotivc: <110dcl in tellDB of neglected (pd)(dp) couplings, we

need epectroecopic strengths a f.,etor of approximately 27 tim<:o~

larger than predicted theoretfoally. It is interest!"& tQ "Oto t'1<1t

this is parttdly due to the fact that the Gev<o>:al (pd)(dp)

c<>upllngs interfere destrueti'.'ely! i£ we take e.g. the o+ .. 3/Z­

(l> j•3/Z) and the 3/2--.2+ (l>j•l/2) transitiQ'-"~ 0'1ly 'le find a minimal

v.olue for the ~ as defined above of abo'1t 2SO.

IV-2 A mior<;>aeopie 111te.:-pretat1on of the foverdon results

In the P"evlwll- scctiou 'le have shown th11t the form factor

corrections are not due to SOM obl'i0119 omissions of the >ll<!:thod <>&ed

to i:ompute thEi cross ae~tiona .. We thercfote adopt the in.terpret•t:ti:in

that indeed we e9d Le•~n ~<> ... thing relevant abo'1t th.: transition

densities t'~o~ the corrected form. f.ai:::tors. We now consider the

n•actiona 56Ni(Jt,p')0\+2\ and 5•l"e(1!',p')0+1+z+1 at zo.4 and 24.6

MeV. t"eGpGct1vely. The nucleus 58N1 pOij;$<i93es only two ne.11tTOdi:li

outside a closed ..eutron shell configuratiQrt where"$ the number of

protons is m.agk, Thia mell"8 tho.t lt 1e doubtful whether S8Ni can be

da$c..-ibed by the collective m.;idel. Similarl)' the nucle~a 5 "~e h of

interest b~es.use :f.t ~on•1sts of a c.loaed neutroq eh.ell a.nd only two

proton holes.

For the SSNi(p,p')O+i'•2+1 re•et1on at 27 MeV, orleros~opl<:

croee section calculations hillVe been published by Blok. (;61.062). \le

¢,j.l'lMt,. howaver,. compa'["e his form. fac:;:tQriil wlth our inversion rea1,1lt::.s

aince hia ea.Leulationo include the exchange ef{eets e.:actly. In th la

coatext. a local form fae.tor e~nnQt 'be diafined+ W'e: there('Qrc hav-c to

perfor., .,~er<:>~eoplc calculations which ,..ke 1.;.e<ll <>pprQl<imationa to

the e>:ch&ng;, contributions (RAL71). ?n tl\e appendix to this cha9te~.

the DWBA e):press!Qn fo't' ~ PJ1croscop1c. tratl.11it,1i;n;'I. :!;$ given. In the

c~~e of or . 2t excitation~, there 1$ a number of miero~cop1c

-as-

the 2p3/2' lfS/2 and 2pl/2 orbits only. This is justified by the

~ t:.~t:.~m¢1'1:1;:: of Waro,:p.1:1et (WAR82) tho.t 80% of the occupation

prnbabi llty of the low lying states c.s.n be obtained without admixing

of 3p-lh. 4p-2h or hi sher '=OI}tr1b\lt:.1oI).s. For 5 1tp~ it 1• ti~.::.e~S.d.l'"Y'

to .f.nclude the f1/2 neutron hole state:e.& The model spe.ce seems to be

larse e<><>ugh 1" bo:.t:h co.au for 1't: lei!.H " ochemiiti~ caleulat:1o"·

The transition densities that c:.orrespond to the various

mierO$COp1c:: Ct'$11:$lt.1on.~ ;~re CQIIlp~ted by t:hc code CHUCK and they ~I"c

added coherently in order to arrive at a total transition density.

We only have to pI"ovide for the appropriate spectroscopic.

amplitudes. For saNi these have been computed by meana of the method

of the appendix IVA .. We used the amplitudes of the components of the

«•levant nuclear 1'ave functions from {BRU77 ,K00?7), In the case of

Sr+pe) we had the apec.troac.opie. atti.plitu.des as coinp-uted by Amoe

(l\.H078) at our disposal. Unfortunately these have been obtained

mdng conventiona that differ from those in CHUCK.. By combining

these reaulca of (AH078) "1th co.lcuhtlono of v.,n Hees (HEE80), ""

have succeeded in extracting the proper phases and ampli tudea. The

resulting apectroaeopic. amplitudes are listed in tables IV .z and

IV .3 foi;- ~aNi and ~ 4 1'e, respectively.

Table IV. 2:: coupling strengths for the m.icr-oscopic transi tlona In

SeNi~ using the CHUCK conYentions.

2pl/Z + Zp3/2 -0 .117 (neutron pair break-up)

2pl/2 • lf5/2 -1).099 id.

Zp3/Z + Zpl/Z -0.326 id.

2p3/2 + 2p3/2 0.560 (iie:i.1tron ps.i r reco'1plins)

2p3/2 .. tf5/2 0.25l (neutt:-On p;U~ break-.,p)

1!5/2 .. 2p1/2 -0.197 ~"· lfS/l + 2p3/2 o.178 id.

lf5/i + lf5/2 0.166 (neut-.:-on pai" rnooupling)

-89-

table IV .3: eo<.1plins strengths for the microscopic traMitiCM in

5~ Fe, using the CHUCK conventi<lM·

lf7 /Z + lf7 Ji -o.74l ( ..-eco.,pune; o( p..-otou holes)

lf7 /2 .. 2p3/2 o.2so (p>:ototi p"rt:1¢le-hole e><dt4t1oa)

lf7 /2 + 2p3/2 o.427 (neutron particls-h<lle eJ<<:HilltlO>.\)

2p3/2 .. lf7 /2 0.134 (protot1 p4rtkle-hole recoupling)

2p3/2 + lf7/2 0.301 (mo1>t ton i"'·'Ucle-hole ..-ecoo.plbs)

Apart from th" •1>ecnoocopic ._,.1>l1Nde•, o. ne.:ceoo.o.ry idgredie'1t for

computing the micro~cop1C 't-'11at>,"1;< de...,nts eouS1$U of the nueleot1~

nucleon force. "-• stat"d t" the <IM>e'-'dh, it c<:>mpriae~ ""vero.l

co.ipOl'letlU. We have used th" r4ndharipsnde force (PAN69):

v • 350.9 ~ G(r) (MeV) oe

v tB

~ 526.3 x G(r) (MeV)

v •-438.6 ~ G(r) (MeV) ~o

v to ·0 (11CV)

where G(r) • Bl<p (-l ,6t)/(l 06t") (lV-19)

with r in fm,

Witb tbe above "•1Sbt £.t.ctou wot ha.;e <'""'i'"ted the real p4ru of the

microscopic fotm factots fot the or +z~ transitions in seN1 .and

>•Fe. We ""'~t '""11~e that, d.,e to the 11~~ted d""'ber of ~ieroeeopie

~"<¢1t.,t1Qna helnd.,,d, the total ""dtation strength will be

uDderesti=ted. ln pro.ctice. th!.& h ln p~act:lce M.,i;oensatcd for by

applying additional multiplication fact<>rs, the a<>-called effective

charge•, whlch <:>~ten •~e "\Lowed to be d1f~erent f<:or the pro~on ad(!

i;he tu;!:tltrol1. parts. This approach~ however,. assumes the rs.dial

dependenc.e of the form factor a.a computed in the limited space: to be

conect, In out view thiS 1& not very i:cdhtic- It i• mo-Ce 111.;ely

that the. m:f..i;taing attengt.h should come from a large number of Small

c<>ntributions, all having their own radial dependence. Therefore

the resulting tro.neition d.;neity of these neglected transitions is

-im-

likel,y to h .. ve a ,.ore or le•~ coUectlve shO.P<'· we therefore

follo,,~d the so-oUled c<>re 1>ol<"r~•aUo" r~e$c~11>t~on ~" <>~de~ ~¢

obtain the correct total transition strength (LOV67). Thia means

th-"'t we .add. .a collective form factoL to the mic:roscopic J:orm factor.

The strength ~ is adjusted to reprodu¢e the volu:a:i.e: integral of the

collective model form factor:

(IV-20)

We H<>d vpl..,_.,$ for ~ of 0.50 m>d 0.74 fo< tM 5~Fc ""d l:h" 581U,

respectively~ The resulting microsc.oplc plus core polarization form

factors for the two "'1¢lides are given fo figures IV.l and IV.2,

rc9pectively. as the dot-dashed curve9. As th~ mo~t st~~~~ng

fe~tures "" dote the dearly dQubl•-peak.,d atr,,¢ture id the 58i<1

cMe. p>,"edOU1inpntly '*'"' t<> the 2pV2 .,. 2p3/2, 2p3/2 .,. 2pl/2 and

2p3/Z + 2p3/2 excitations, all interfering conatrucdvely. In the

5'*Fe case, to the contrary, we bav.e stt"Qng 2-p "' lf ~~(!! l~ .,. 2p

excitations, mainly in the neutron o~bits~ whicb IJIB~~fe$~ ~h~m$e~v~$

a• ~ clea~ 1 TI~~at1ve-pQ~1tive'- etructu~e. It 1~ e~~ourasin& to note

that for both n11clei the form factor.a from tbe inv~t:sto~. ti$

r<:preacne .. d by the Mild Hnu, ab(ljl th~ s11111e tre.,da a& the

microscopic onee 1 be it that the iexact locationEI of the!: tt:lo\lt:.l.;v~

lM)C:f.U so>d inJ.ni..a 4re shifted a little, which might be due to ths

ran8e of the force employed ... Even the absolute val.,e of the

correct1om11,. i:::omp.a'.l;"ed. 'fil'1.t.b the eQ:tleet.i"V"e •11)de1 appears to be of the

right magnitude,

With """l''°"t tQ th~; i~$t p<:>int, how~"er, it is necessary to

~omment that apart from. the Pandha~lpande £o~ce thete e~1st numerous

other effective nucleon-nuoleon ll'lt<1t•><'C~On~, b<:>th "1th l<>nger and

with eho~ter ranges. A cOmmOn feat~~~ of the i:ma.Jority of the forces

that ue en vogue to that thdr stte.,gth ~8 density depeadent. A

(li~pl.e way to account for this ie to multiply the 61d$1et eV'en a11d

triplet even cOT!lpOnente -wtt:h a. de4Bity .dependent factor which

"~""Uy ha$ the form:

(IV .21)

-91-

1.0

_5

v,0c1

lfrn-21 / ,0 r-'-"-'--------~---1

- .5

-1, 0

-u

Fi(!- IV_ 1-

l.O

.5

-2 VG,C lfm I

" l ,0

- .5

-LO

0 -i

r lfml

Compa-,.ison b&ttJ.;en th,; aoUMtive model. fm'f11 factor

(dashed (!Ul"W) and ths microscopic model for-m f<ld9>'

(dot-dashBd cuPVe! foi' 58

/1/Up,p'!2+1' The fuU """"° 'is the inv.ersion rGault.

o~ ...... ...,~.._._._'-"-'-l4:--..... -ts~~~~~11a

I (fm)

Fig. IV. 2. As fiu~n IV. 1 fOI' 51 Fe'

1.5

1.0

. 5

- .5

-). 0

D .. 6 6 10

t Cfm)

Fig. IV. S. 1'he effect of a den,,Hy dependent m.cl.son-nucliwn

fo~aq. DaBh,,d: Pesult fr'Om the de"''Hy Jepend.onr.

f'o=s, dot-da:;hed; i>eeult from tha d"nsity i"'"°'P''"­dP.nt ]'oms. The full aui'W ""l'"M""t" tho imJoPsiori

Pe.:.~t .. d'.r. .. Aii fo"Pm fa•:!to:tis al"e fo'T' 58Nit'P,p 1 )2+r

l.5

1.0

". 5

-l. 0

-u 2

-9J-

~ s r ltml

8 10

ller.eo the n.,o;:lear den~Hy is deMted by ~ 1'b1le A, II "''d C are

paramete(t that glther aI"c t1-d.j1,1sted or derived fro'IYI. fir9t

l>~id<iples. We tool< C•2/3 and ll•l .46, thus to.k.i"g the """l:'"S" for

the v8

@ 4nd ~he vte for th~ ttrongest de~Bity dependcdce from Cr~c~

(G\ll!6/), "!;he parametoeor A has bee" •djusted 88 to yield th<' same

vol"""' integral as without the density correction. Tbe rcaglta a.re

given in figurn l\1.3 and IV.4 as the dOt·diiahed curves. As could be

expected, W<: observe that by applying this correetion the St'"'1cture

in tM inner rcs:ion is washed out wh.,reas the main qualitatlve

d.iff-ereQc;.~a with the collect1ve toodel fo\""la fact:or- 'tC:m.d.io.. We

like to su:cmol;lrlze our cQ11elusions .as ~(ll,lows:

i) wr results are consUtedt with the microecoplc P>Qdel fono

h1o:otor. They show tM correct quaU hU ve behaviour for both th<'

SSNi ~dd. the >4 re reactions.

ii) The relative maximum H• the interior of se1u ~ee<M t<> be

sblfted with respect to the "'icroecop1c 1110del predlct10'1S•

Ui) tf we "98\lllle the mlcleon--cnucl<ton for<0e to be density

dependent, the ~gnitude of the correctio~• in tb• ini>er region 10

too lat$•·

-94-

In fi_gQ.re.e; I1J'.5 ~o IV.16 ve present the cross sections and

analysing powers together with the corresponding form factors for

t.btt ot • 2t reactions studied ... Fo'l' the used OMP*s as. well as for

the experimental points ~ refer to Melssen (MBL78), Wassenaar

(WAS82) and Moonen (M0084). The error bands stem from the

numerically i::.om.puted uncertainty band~ as presented previously in

figure I.4 where a see.le transformation with s factor of 9{(58/A)

both for the rad~us add for the tn,i.op wtdth h•• been appli"d• A"

e~t1 ... te tor the «ppro~rtate bump width wi that should be applied at

a given radius r~ was obtained as follows:

(IV.22)

Here fJ. 16 the spline width; the constant C has been chosen such as

tQ y1e1d the eorreet value for Y1

for a one-spline bump+ Moreover~

it :l.8 important to note that the e;<perimcntal errors in figures I.3

~nd 1.4 ~on~ist of both statistic and syatematlc errors~ The latter

m6n!f~et tbcmselve~ llllii!~~ly ~~ ~ ~Qrmali~~tion uncertainty and

therefore cannot contribute: to the unc.ertafnty of the shape of tbs

form facto~. We ehere~Qre Q"~l' h<>ve ~i> consider the statistical

errors which were estimated to be 2%.

For the improvcme~t• etteido4 ~Q ~Ll of the studied reactions

voe refer t:o table IV +4.

Fig . .rv.s to LV. l/J. Reeutts trom inversion aafoutations for the

r>eaations of T(I_bte .rv.4.

Left «Pl'll"-' form fac'tO:t", :t"eal pal"'r ;

Right uppel': fa""' fa.otol', imaginary pa>'t.

The aoUootive modd fa= faatol"' is repI'<lcen­

ted by the daeh<1d dul"Ves 1iheroas the j'l<ti

dlU'1>6S depiat the uncertainty band oOl"'l"SB­

ponding to the inverted fo't'm faotOl'8.

-95-

LO

,5

- . 5

-l. 0

dd dil (mb/~rl

0, l

0 6 8

54re frean

0\-2+1

beta·O. 16

S<lre I~, p'l 17.2 MeV

2+1 fl.41}

r (fml

2 4

$<\Fe Oma9_ l

0'1-2\ bela-Q.16

~O 60 90 12tl 150 0 30 60 90 lZO 150

fr c,m_

F'IG. IV. S.

-96-

+0_ 5

o_o A

-o. 5

u

1,0

. 5

".5

•1,0 I

do' dQ. (mblsrl

0.1

r1frnl

56re lreall

56Fe (!), P'I

17,2 MeV

2\ I0.85l

56re limag. I

u\-2\ beta·0. 26

~o 60 9Q 120 150 o :i,o 60 9Q 120 150 ,,. c.rn.

FIG. IV. G.

-97~

+O, 5

0.0 A

-0, 5

r (fml

2 6

l.5

56F~ (reall '>li~e Uma9. l l.O 0•1-2·1 o\-2\

beta·0.26 beta·O. 26

. s I ·2 vcoc1 1111 '

.0

•• 5

-1.0

10

56fe fl)", p')

20.4 MW

clo' +O. S

d.Q ' - ' lmblsrl

0. 0 A

0, l

·0.5

0 30 60 l)(J 120 150 0 30 60 90 lW ISO

~ c.m.

PIG. IV.?.

-98-

r (fml

0

u 56;e (real! 56Fe limag. I

LO 0\-2\ o•c2t beta•O. 24 beta·O. 24

_5

Vcocl ffm-21 _o

-. 5

-1.0

10

56Fe (Jl', p'I

24.6 M~V

dci 2\ m.as1 +O. 5

d.Q lmb/srl

0.0 A

0.1

·0.5

30 6(1 90 12\J 150 0 ,30 60 90 !20 150

a c.m.

FTG. JV. 8.

-99-

r llml

2

u 58Ni (real! 58Ni (ima9. J

l.O O\-z"1 o\_2•1 beta·O_ 24 be\<l•0.24

.~

·2 vcoci (Im l .0

- • 5

-1.0

10

58Ni {fi',p'l '

20.4 MeV I'

dO' 2\ (1.4~1 +O. 5

d.Q lmblsrl

0, 0 A

0.1

-0. ~

30 60 90 120 150 0 $0 60 90 120 150

" c.m.

FIG. IV. P.

-100-

r(fml

l.5

60Ni (re<il) 60Ni (imag. I LO 0+1-2\ D'J-2+1

t>eta•O. 26 beta•O. 26

.5

Vcoc1 r1m·21 . 0

- .5

-1.0

10

60Ni ()t,p'I

20.4 MeV ,, do

z\ (J.331 +O. 5

dTI (mblsrl

0. 0 A

' ' 0.1 . ... ·0. 5

60 90 120 150 0 30 60 90 120 150

a c.m.

FIG. l'V.10.

-101-

( (fMl

0 8 2 1.5

62NI (real) 62Ni !imig.l LO 0+1_2+1 0•1-21

beta--0.2~ beta-0.25

.5

I -z veoe1 fm l .0

- .5

-!, 0

lO

62Ni qr, p')

20 .4 MeV

do" •0.S

d.Q !mblsrJ

0.0 A

0.1

-o. 5

0 30 60 w 120 150 0 :ro 60 90 120 150

a-c.m.

F'IG. IV.1L

-102-

r \!ml

0

L5 64 Ni (reall 64Ni Om<i9.)

LO 0\-2\ neta-0. 22

·' Vcoc1 ifm-21

.0

- _5

·LO

10

64Ni ilf,p'I

Z0.4 MeV

do' z\ rL 341 •·O. 5

dil lm~lsri .. o_ o A

0.1 +:+ -o. ~

O 30 60 90 120 150 0 30 60 90 120 !SO

ti" c.m.

FIG. IV.12.

-10~-

r lfml

4

u

LO

.5

VC()Cl (fm-21 -.0

".5

-!. 0

10

64zn (Jl", p'I 20.4 MeV 2\ (l.&JI

,, •O. 5 dt'l' dll

imnisf)

' 0.0 A . -. .. 0.1

-0.1

30 60 90 120 110 0 30 60 90 120 150

{t c.m.

/!'IC. IV. 1J.

-104-

r (fm)

4.

1.5

66zn (reilll 66zn (imag.) 1. 0 o+1_ 2\ 01-2+1

beta-0. 25 beta·O. 2~ .5

Vcoc1 (fm-21 .0

- . 5

-LO

lO

66zn (lt, p'l

Z0.4 MeV +O. 5

dct z\ n.041 dll

(rnblsrl

o.o A

0.1

-0. 5

30 60 90 120 150 0 30 60 90 120 150

a c.m.

l"lG. IV.11.

- I Vi-

'\fml

6 a 0 4 8

LS 68zn \(eall 68zn \imag.)

LO 0)-2\ 0\-2i. beta-0.2S bela·O" 25

"~

vcoc1 \rm·2i .0

• • 5

-1.0

10

68 Zn !Jl',p'l 20.4 MeV

dd 2i, U.081 '1 +0.5

dQ \mbl&rl

0.0 A

0.1

-0. s

60 90 120 150 0 ;II) 60 90 120 )~

~ c"m.

FIG. IV. 15.

-106-

r lfml

2 0 u ·r···-ri-·,

70in lreall 70zn rima9. l LO Oi-Oz 0•1-2•1

b~ta·O. 017 beta-0. I 9

.5

vc c lfm"21 o 1 _0

-.5

·LO

10

70zn {il, p'I 20-4 Mey

+0_5 dd 2•1 m. 881 t+... dil (mb/srl

0, 0 A

O_ l

-0, 5

30 60 90 lW 150 0 30 liO 90 120 150

a c.m.

FIG_ IV.16.

-107-

Table IV .4: xz valuea for the o+ .. 1

2+ l

reactions.

Nu.cleua Ref. Energy E x4 )(2 axe 0 ~

(Mc:V) (MoV)

5~ Fe a) 24.6 l-41 0-16 17. 7 a.14 56 Fe a) 17.2 o.85 0.26 13.9 3.32

';G Fe a) 20.4 0.85 0.26 ~1.7 n.6 sl>Fe a) 24.6 0-8~ 0.24 19.9 6.96

58?li a) 20.4 1.45 0.24 92.0 25.2 60J!li \,) 20.4 1.;u 0.26 458. 169.

62Ni b) 20.4 l.17 o.2s 43.3 u.6 ~4 Ni b) 20.4 1,34 0.22 41.4 23 .1

6~zn c) 20.4 Q.99 0.28 388. 114.

~~ :<:u c) 20.4 1.04 0.25 356. 130.

68zn <:) 20.4 1.os 0.22 657. g3 .1

70zn e) 20.4 0-88 0.19 33,4 21.S

x:; • x2 CQ>">"e9ponding to collective m<>del

x.1 • xJ:!: upon completion 0£ the tter~tJ.o~ ei::.tl-eme

n • the number of it;.er-iit.lo~s

F - x~ Ix~

n F

1*> 2 .17

3 4 .19

2 1.63

2.66

3-65

2.11

3 3,73

1.80

3.40

2 2. 75

6 7.06

3 1.53

*) an 1 .. prove,.,<1t t<)'1arde 7 .60 could be gai<1ed ~" Q!l.e more

iteration, but tbia resulted in an QSCill&ting structure on the

ea.tire radial 1'1.terval.

References: a: (MllI.78)

b: (WAS82)

c: (H0084)

1) Ihe S6Fe(p',p')Of + 2t teact1on bas been me~$11red ~Y Melssen

(MEL78) for se,,.H;tl '°"e'gi<:e. We have performed inverstono for the

17.2, 20.4 Md 24.6 MeV cases. In eontrO!l~t to the Ni and Zn

isotopes, we co11ld leave out tho physicsl damp fog CM$t<o~l>t "'itlwut

-108-

gettii'I& into probletti$ in tlie periph.er.at :region. As ca.n h~ 1:uli:::n fro111

figures IV .5 ti) l;V ·] ~ the. lm~tQvements of t:.he c\"oss sec.tions fc)!:;' tit@

17. 2 and the 24. 6 ¢EU<>• are ~oosid~~able, whereas f <:>~ ZO .4 MeV we

have ~he vnusu.al eff~(::r;: that the analp:1ing power itll.~rovGS mostT

Con.cer-nin,g the eQttections, lt ls not surprising: thlit the. gent;!t:"lll

t~ndencies ar~ the same a& with 58 Ni~ ~~nc~ the neutron

configul';;i.t.loll of 56.11~ 1~ the aam.e. We o'b~etve ths.t the mn,e:nf.tude of

the Jnnt;i: s.tru.c.tur~ of the form fn.-ctOt' might decr~~.$c llfith the

energy. This effec.r. lG opposite: to what one wou),i;l. expect from ch~

vat'iation w:f,~h energy of ~he density depcnd-e-n~e of the m.i<:leon­

nucleon fore.". This r~m.eir~. however:. bc.a'{"3 a tentat.:tv.e character

sitl~e the effects of a giai'lt Qctupole resQt.il).nc~ are be¢Qm:J.ng

~"'po<tant at ~hcoe lo.,er eo.,tgies (HALBO).

11) For the Ni isotope$, ve observe fot ~ONi and 62ni toughly the

satll.e s~ rvcture itl th~ .:e:al part Q~ U1e form facto»:- a.a for 56Ni,

whereas for 6 ~1'11 ~h<>re is a dra .... tic change. WhUc the lighter Nl.­

ii!ilQtopcs show a sti:-uctural im.prQV@ment of the t::.1;'¢$$ $ections,. thet"e

i.s h..ardly olitty lmprovement Jn the deseriptlQ'l;'I: of the c.ros$ .$ection

and aaaly.(';f,i;i;g power f(l:r- this isotope. 1'h1s differen¢~ :l.:si also

demonstro.tc::d in the form factor..., both !rt the interior l:(ll"('cction aad

11.1 the dip at 5.625 fQI. As. stated !n r;:hc previous chap~~'('. however~

we <lWbt the signiHcance of peripheral structures. If "" adopt the

micro$c;Opic interpret{ltion as presett~ed in the precedit.i.g: &ec~ion,,. we

tend to o:.0<1clude that for G01u and for 62 N1, aa for 56Ni, the

e:ii:tJ.tation pro¢~.$~ is domin$.tei;I; by excitat1'1t.i.6 and recouplii;'l.$$

within tlle Zp3/2 ond 2plf2 subsho;,lls whereas for •~Ni the

tronsitions bel:.,ecn the lf5/2 Md 2pl/2, 1/2 ee .. tc• may play a tl><)..-o

impo~tant role. o,,., might even ~"1"1' <>f lf7 /Z p..-oton compMedt$

ifhich a.re: gJ;"adually bec.omii;i,g importa11t; 1 nd.ced, because ¢~ the

relatively large neut.r¢TJ exi::.e:ss {!.l'l.4 the strong proton-neutro11

f.ziteractioE'l,, the closed protQn eorc might break 1,1.p. UnforttJni;1.tely

these cunclusi<>ns are Ql:>•c'-'red by the d~p "' 5-625 fm which we hold

to be ~h1¢ to neglec.ti~g the tto11-lo¢•l effects rather th.an to

feature• of the tocal fonn f~cl:ot·

-109-

As to the re:sulti11.g cr-oss Bec:tiotr.s. we Bee s.n. iinpressive

lmp»:"ovcrn~nt for the reproduc.tion of the over.all elope, The. minimum

at 120", however, 18 not deep "'"°"Sh· So it seems that still an

essential ingredient is missing in the es3umcd <:>0oltaUon $ch<o-.

F-or the: a.Mlyai.:>.g powers we can be far lee.s optimistic.+ :tn the

°t'l;,Si.e:m liiC:tW'l;~I'I 90~ at1d 110"" serious deteriorations occur for all of

the Ni isotope• and this is not completely 0C>111penuted !ot by the

eUsht ~,.pr.:;iv.;o'"e"t of the fit at angles larger than 120°.

Apparently the profit of imp~oving the CtQ$6 'eet~<:m more than

compeMatee tor t~e ~eterloriti<X> ot the a"dysio.s power.

iii) For the lightut two Zn t$Ote>peo, G~zn -1\d G6zn, the ~••e

re.,a!."ka hold as for se ,60 ,6.!.Ni. Due to the additional two protons

extra tranaitionB are allowed, but apparently these do not alter the

pltt.1,1.t'"e: tc>o DD,1-Ch:i- DUlybe a.gai11. bee.a.use of the rela.tively wes.k. proton­

proton forc.e. The: moet 8Ct'"1klng ~hl/::nl)J:Qel'l,8. £-oi: the Zn 1$.0tope:(I r....,ki::

plac.e for the 68~n and 70~n Oj + 2t excitations. To begin with the

la.tter 1 we: -observe that both the cross sec:tion and the analysing

powor are '1<1~dl1 hproved loy the in~ere!on. ""d the s01all

c.orrec.tiona that reeult ghe M drMt!c c~<'1'1!:e8 !<' ehe eotlect!"e

it;holi.pe l)f the form factor. We therefore conclude that in 70zn either

there are no J?<lrticulatly dC><>1tt<>t!I"$ t111crQ6eop!c exe!t.,,UQtt6 Qi:' the

d¢im1n.,.titi.g 11).iieroecopii::. transitions i:.orreapond to form factors

without nodes. An argument for the latter possibility can be found

in the. faet that s11bstantial j•9/Z strength has been seen in

70 ~n(p,d) rcacHQM (NUC7\I).

rhe v~Qble~ ehat e~t6ts with 70 zn also af fec.ts otlr

underotandins Qf Sezn. we e>h6erve that the el>ar.;tcter <;£ the f<;rt11

fai::.tor differs completely from that of the other .Zn isotopes whereas

the 1=J;"Qlli$ $ee~lQn .aru:t. ans.lysi'l.'lg power do not differ so -m.ueh. ~e

will try to u<1<lei:-et<>n<I thie e><eeptional 1oeh<>v~o'1t !n eerta6 of

nur;.leo.r ~trueture ans.lyaia. When applying the Interacting Boson.

Model (UM), attempts ar" mad" either to describe &8~n as a

c.ombination of one co'.t"rehted proton pair (pre>te><1 boson) and six

neutron b<>$(mS, o~ as Nie ~~Oto<> boao" i"~ o""' ne'1tN>:I hole l:oo~o<>·

-110-

In the latter case N•40 is .asaiJme:d to he mag:ie. rn this conc.e~t: Van

Hall and Bhat (HAL84) 40ted th4t the foternal •tru<ture <>f th•

neutrou bosons, com.pa.red with. the ~·36 olild 11ghttr Zn 1sotopes 1

might ~~~pl~y ~ ~~dden change. This l; in 90 far consistent ~1th our

findings that the differenc:e betwee11 the 68zt'). s.11d 66z.rr. fortll: f~ctori:i:

io essentially larger than that betw•e" the 66zn and th" light.r Zn

or Mi isotopes.

IV-4 The inversion of l06, ll0cd(lt,p•)o+1+r1

the eon.i::lt,isioo.s ~nd int~t"pr.et~t(Ot.1..!,'1 pre.s;ent:ed f.p the pt:'evi0\1!J

ae<ctio<> are 14 !<>~t <tU Meed on tho mo\"c o< 1¢$$ dctdled

01icroocopk cakulatlo4 of the sallli and S4re(p,p' )Of + 2t rea<otion~.

The inversiofl i:oethod c.a.11 also be aeet'I az3. a iegf.l:lmfil.l:e n.1tern~clve ln

(;!a.sea wb.er.e l)l;uch m.f.c'1;"0$t;.Qp:f.c '('oC;$1.,Jlt$ (Lt¢ J;J;Ot .EJ.va:l.l.a'ble· WE" thin\:: of

the c:.a.ae of hes.vier niJt::lei with MllY V41.eace p.a.rtir.::les outside

cloaed 1;1hella.

defQ''""~ in <><4cr to allow fo~ one otep l)llSA calculations. l'or

theae nue.lei the excitation of the first 2+ state generally consists

o~ iso mai:iy c¢hcreflt m1c:i:"o6.::0pic compon~nts th.at an inte"t"ior

otmcturo> i• n<>t liko>ly tQ ho> P""'"""t in tho> fQr"' fa~tQr. Hie:ho>r

excited positive parity i;1t~tea ~ght .i;1how ~ le1;11;1 coherent. form.

facto~~ but the pre:setl~e 0£ more il'.'lt~rf.e:t"i11g e:~eitatiofl paths ~kes

"'"" metbod ,,.,..,., ha<de~ to "P~ly, M w:lU be oho1'n :l~ ¢lm1>t<" v.

f.U;ireCJv~r-.. l::hfi!; ,f,nt.e't"p~eta[1Qn oE c.he~~ m¢or~ ¢-~ ).e!ls wb.1m$1¢a.l

urue~ure; <'$"-1" ~~"-" f1,>< ~heU m<><;lel ¢<'~C,.lH~On$.

An interesting alternative might be found in the excitation of

low lying negative parity states such aa 31. lt1. th.e r.::olle.e:tive model

they "'°' :lnterp<1~.ted "8 oct1,1pole deformntlone. They "re oHen

~eaerl'beQ r"1ther wi:l1 t>y one step O'WBA empJ,Qying a c.o11eetlv.e fe.rm

fac::tor. On the other hand,. the number of contributing single

p;ii:-Hde cxeU•t:loM o te<luced by the eon6t~"~"~ o~ the p•rHy

chMg.,. 1herdore, wh.:n dolug ~" 1,-,veutol\, we "'1-!;ht hope to flitd

$Ome tueedo~ ~tr,.ctuYe. fo this aeetio.i wo s1~e a deReriptfon of

-111-

L5

1.0

.)

Ve c lfm-21 0 l .0

- • 5

-LO

dO' dll

lmblsrl

0.1

r lfml

z 4 0

l06cd lreall 106cc Ii mag. l

0+1-l)

beta•0.18

106Cd lt,p'I

22. 3 MeY lj !U7l +0.5

0.0 A

-o. 5

30 60 90 120 150 0 JO 60 90 120 l~O

~ ~.m.

/!if! • .i'.V. J!-IV.18. Resuits from ·i;per•!J·ion calcutations for• thd

l'dactions l06' 110cd(p,p ').>-. Conv<Jntions as

i'' fi!Juree zv. 5-fV.16.

- I 12-

do' dfl.

lmblsrl

LO

.5

- . 5

-1.0

0, I

0

..

r lfml

110c~ IP, p'I

22.:JMeV

fj 12.081 ..

JlDcd. limag. I

o\-3i beta•0.16

••

•'

0 30 60 90 120 150 0 30 60 90 120 150

,'j-c.m.

FllJ. I~.111.

-113-

+D. ~

0. 0 A

-0. j

r Uml

4 6 4

1.5 llOcd freall llOcd. Omag, I

I_ 0 0+1-li

beta·O. l6

.5

Vcoc1 (fm-21 .0

- .5

-l. 0

llOcd 1p: p'I 2UMeV

dO' 3·1 (2_081 .. +0_ 5

d.Q •' lmblsri

o_ 1 o_o A

•• -0.5

30 60 QO 120 150 0 30 60 QO 120 150

,j-c.m.

Fig. IV. lf!. Jle in figure IV.18 with a I'<!duced damping faCJtol'.

-114-

the inver9lon of the form f~ctor for the reoctlons \0 6 • 110c~ (p, P' )Or ·> 3l as meoo<>n:d in our ln~tltute by l'etit (PET85).

In figure IV. 17 and IV .18 we proocnt the oroaa sections and

analy$ing powera ~Q~ l06.cd s.nd llOcd that resulted ~i:-o-m our

imrerSi;(ons ... Again w~ u~opt the conv-ettC::!;-OTI that the full c1,1rvc stands

for th.e .::ort:"C:cted result: whei:i=a.e the result £-(Orn the collective

mode1 calculation is Sivel'I by the dashed ¢ ... ~.-e. In order to treat

both isotopes .{\l'J unbiased as P.061$:f.ble, we applied the 5ame damping

o•,,ength; ("'qu$tlon III.32) in h<>tl> cases. A signif1~o,.ny lower xz for l lOcd, howe,,..,r, Muld be obtained by applying a smaller damping

.r::t::t:"cngth+ This r~.tiulted in a e~;:tl1 i;\cgative swing l,n t.lic inner

reg~<)n (fig. IV.19), but still this i8 f~.,.~d to be less th,.., the

neg.>tiv"' ~a~t of the for,. focto-r for 106cd. Tho 'elev-ant X2 's and X2

reduotion factors "re listed ~n table IV .5.

table IV.~: X~ 1 9 of the 1rtvt;t"S;:lons for the Cd r st:$1:eo. I

I~otopc 1- (<;1.0,,,p~~S) x2 x2 )> 0 f

\ OGcd 0-005 96.1 31.J 3.1)8

l IOCd 0.005 3a.2 37. 7 1.01

llOcd 0 .00001 38.2 ll. 7 3.26

In o:tder to give .a.n ¢$~:f,m.at-e of the ac.c.u't"a(:y we npply the same

method a~ fot the 2t states in the Pr•v~Q"o oection. UnfortuMtoly

we chen obse["ve that the collective model for111 fo.cto{"a fall within

the err<>r bands for both ""clci.

It may neverth~lei3$ be int-eresting to pre,se(lt out ideas on an

interpretation in ter010 of th• shell model. The Cd isotopos consist

of. 'two proton bole:a 1rt th~ Z•50 shell and 8 eir 12 neu.trone outside

l:he N~~o shell r<!apecnvcl)". For simplicity we <>6&1.1me that the

ot'l"engtha of the l'f:ro exeJ.t:-E1t:lo:o.9 are the .s.a.DJ:e: for both nuclei+

The only difiet;'cni:t?: tb.en consists of et·.11E! Qc~~ps.t1on of the 2d5/2.

lg7 /2 .odd pOOdbly the lhll/2 and Jsl/2 neutro-n states.

-115-

We note that 0++3- Cxli:.f.ta.t1omJ among these states can be

eoMt>:.,cted hom the follo«'lng tranaiti<>na:

12d5/2>2 .. f 2d:./2> x I lhll/2>

f 1117 /2>2 ... l1a:7/2>• llhll/2> 0

I lhll/2>2 _,. I lhll/2>x l3al/2>

l3al/2)Z _,_ 1381/2>" I lbll/2>

The it1te,ger numbers at the risht h<lo.d ~tde d<onQte tbc number of

nodtBs. in the direct pa.rt of the considered transitioii. detisity,

o.ss"'"1"8 a ~ufhdet1t1y ~hQl;'t i:-~t1go:.d >:tucleon-llucleon iilte..-actioll,

He.re "" exclude both the nodes at the origin and at infinity. Now if

"'° >:ed~ze tMt "'I.th '"' 11'10>:<"1s1ng number of neutrons the Fermi

level shifts 1'> the dire<otio" from th;, 2d5/2 to the lg7 /2 single

particle state, we indeed 1118Y expect the to,,.. factor !or 1 Q6Cd to

NUU$ a .. or" l'l"'Sative/po$1t1ve ot~uctunl (cf. S~Fe or .. zt) than

in the case of llOcd,

-116-

IV-5 the 11"1ver$f.On. procedure for .inelastlc alpha acatter!ng

?:he l)WSA fonnalism .aa pteoented in section I-2 U "ot only

valid for 11"1el"1.&t"1c proton scattering, but: ~l:r:io holds for !ael.aat.tc

"lph~ ocatt•ring- Similarly~• in (p,p') reaction•, eh~ ~orm factor

for the DWBA T-m.atC"ix elet'tlente ·~$ i:.on:mionly derived from l:he

colkctive model. It 16 1ntetest1ng to cheek whHho~ th<: fo<m factor

corre.c.tio(l~ wr 11 look similar to those lq t:he ~'.t'oton case·

In this s•ction w~ ot1,1dy the possibility «> "l'Pl;r our

inveraioa ~th~d t.IJ the inelastic alpha .e:oe.Q.ttiz:t":lng. We: have i::.ho;aen

as s.a exampJ,e the :i:-ee.ction 6 6 Zn(a ,oi: r )Or Y. 2! as me-aaured by MJJ..i:1Ai

(MAA82) at an ind.dent <tlpo" .,n.,rg;r of 25 MeV. TM' tnco<etical

c::13.le.ut\it.:l,¢q$ QVe:reat lma:te the eJi:pel:'imental data itl th!! h~1.¢~1iil:;ii('d

regJ,0n 1 the .s.ame as. in the prot(ll"I. ta.se-. Moreover, the ol!;l:e:l,.ll~tot'y

stnieture be.:.omoo eUghtly out of phase wlth tnoressing angle.

Before we: pt"e.$-=nt the results of the sct\l~l inversion,

'1c>wover, ~ Illl.iSt pay $(1tl'.le atttZntion to the rel~v';)rtt di£'fe['encea

betweeti Pl:'Ot:.Ul"l and alpha scattering. Fl~$t '(oj'~ have the fae.t th~t

"Hhla the nucleus the effeet1ve W'svenumber for ali:>h•'$ 1$ ~bout 3

to 4 u .. es M l~tge •• for protons. At Hot sight one misht be

inclined to conclude fr"" th ls, that the radial r••ol'Jt~OJ\ (ltiould be

at least a.s ,good or even better than for protc)tu). 4.e we observed ia

.r:;ect:lon III-2, how~ver, it is the mutual independ-ency of the •>. ). •

i>roducts of the ?<trti"1 "ave functions ~ ~ that pfo.y$ the dominant

't"Ol~ ::tn determiuil"l.,g the act.ol'liu.ablc ["adial resolutioa .' N-ow .:;!.QT'l;C~"J:"ning

this !!'l!:lpe.;t, ::1.t :l.s of .grea.t import.~ti.¢.e t.o note that alpha's .are

.i:1.'b!llorbe(l m1.1<::h roore ;atron8:1y within the nuclear 1ater1or. Tht::

conaequenc:e of this ~~ thQt thie amplitude of t:.he nlph(l. wave

functions :l,rt the :oucle.ar interior rels.tive t..g the l,J'eripher.al regiotl

is much allialler that\ !Qr {lt"QtonEl+ If we comp.a.re the: lnncJ;" products

of all pail"$ oe Qdjacent coli.Jrti.l'l.S of the u:tp.C.\"l;i1; K ftom section III-2

fo-r alpha is and for prototll);, w-c i::=iln ace the effect of the volume

'-'~$Q~~t\o~ of the alpha ~artklu dt<l~ly domonatrated (figure

tv .20). fo th\• figure the dashed eurve StQnds for the protonB, the

solid Olli!!: for the: .Plpha particlea.

-117-

Ae ex>'lUnvd in section III-1, th~ inversion procedure

d.~.P.Cruis strong.ly (11;1 the aing1,1l~t." value dei::.omp.O&ition i.e. ve only

can handl~ ~la;i.oQt singular sets of equ.;if:i,ons by eliminating th-e

smQllest s1dg~1~t values. fr¢~ inspecting the matri~ VT from

equation (III.15). ..., find that t11'• colu"'-"" <:o~i:"espond1"s to the

sm.a.llest sing~l-~ values have by far thel~ largest ele~ents for the

ampHtudo:s of the '"03C central opUnes. Therefor., one can any that

thefie s~lle~t: :91ngular V4lue& corl'espoll.d to the ep11nes In the

innert110Gt region. No11 let ue aeeum• thet the. de<li:-depe.nde.ncy of the

set (III.14) is entirely due to the .tear-dependency of the cotumne

~h~t correspond to the innei:- splines and ·their direct neighbours,

thet is to the small values of the innel:" products 11'1 the left-~&nd

part of Hgure IV.20. Than we can aeaign a quantity UJ to the

eingular v.ilue 10ethod wlt:h the follo"1"8 Meaning:

For radii 9uch that the Inner ~~oduct between two

:o.ef,ghbouring ¢.Qlgnms is 1.esl'} than l,c) t.!,me6 the maxim.al

occurring it\q~t:- product 1 the correspo!J.ding siii.g:ula't values

have to 'be eliminated. '

:Fo\" the give-o. rdagular vall,le t11ethod we dQ not ka.ow 1 of c:ourae 1 the

numerical value for w. f~om figgre lV.20~ however, we cat\ tlnd for

'"' arbitrary value of w tha minimal radU such Chet the abo""

<;ond~Uon la met, in other words: the s111t1ll0i!t radU for whi<:h .,.,

<:.an expect aoy detail ta oi::cur in the eOt'C'ected .fQ:li'ln factor .. The

<lata from table VI.6 can be to!ad froot Ugure. IV.20.

'.table IV .6: mini.,"1 radii M a functio.t of "'.

., r min, proto"a (fm) r >11in.i< (io1)

. l Z.6 6 .6

.01 1.8 s.s

.001 1.0 3.8

.oooi 0.7 1. 7

-U6-

10~

i ~ ner­prod uct 1 10

-1 10

-3 10

0

I

• t

2 4 r (fm)

6

... ....

8 10

Pig. TV.30, :f7ie irmer pwduct~ of at/. po:if'B of' ad,ictc'3nt columns of

ths matriT K. DaBh•d' pMton•, full,; aipl1a 'i!-

1o4

~ 103

"" "

I Hf " :::

101

2 6 B

r (fml

P·f.y_ IV.21. V~amotainty b,1ndS for alphe< fOIWI factors, cOrr>e8pond·inc1

to J- to S-:;pl·~rio? pBrt-u~br2ti'.0~!!1.

-11 9-

Wher.1 ~e applied. 01,1i:i :t.aviersion prQ¢-edure to pi:iotoa. scatterios~

we foun<;I $1gnlficant dh.;t$ for radii as small H 2 .z fm whieh a~e

believed to be reliable from comparf.oon with ~1.;~oscopic

calculat:f,.Ol'I;.$& If we look. at figure :t-4. however. It seedl.S 'rle'l:y

unlikely that "e can learo anything fr<:>,. radii less than about

l. 5 fm. According co tho table above, thb Vl)uld correspond for

alpha's to about 5 fm M the lld.nimal radius wi,,.,., signific<lat

•Hect& can be e.:pcctad to show up.

The above reuodf.dg appears to be rather i»dl,,..ect and at ff.,,..St

sight needlessly complicated, a:tM" the O.C¢1.1tscy attainable for form

factor ,i,ti.v~~$ion:s c.an be derived d.1~~ctly from a perturbation

approaeh Cl.IJI ln section r.-3. We .a.re a. hit ~elucts.t.t.t 1 however, to uue

the x2 critedon aeeording to equation (I.12) because of the

atrJ.king differe-nee in stru¢t:ure between th.e protoa .s.nd the alpha

¢.ri;u)!J. , sections. The latter ~ea ~ much more oscillatory -'tt:'ucture"'

which eauseG small ang>,1l1;1.r o1lifts to give large contr~butions to x 2 .

Aa long as the differ""""" between th.c theorette<ll curves and

the data man~fcst themsel~es as a systemat1¢ over- or

underestlmating for 1;1. l~rger angular region, we do not have ""'ch of

a problem and we can judge the function$ ln equatiod (UI.3) to be

a reUobl•f penalty {,.<ieti<>n. The procedure 0£ ooction I-3 that

con.Sista of q~.J.ng perturba.tlon.s to the £0-rm factor lirtd tecording

the x2 differences bet.V"eoen the c.orre.eponding ¢1;"¢&6 aectio1ts:, is

likely to yield an un.cert~.f.i;'.l;ty width t:h.c..t is too 1"1.oel.l:';i;"OW bee.a.use .of

the angular ~'hi.fte., Indeed 1 :1,f we compute: the unce:rt~:f.uty band fQr

the alpha-reaction appty~"S the method of sect1Qn i:~3, we end "I'

with eri:ora in the foterior regHin t'hat are hardly larg<>r Clum for

etie proton <:ase. (figure Iv .n). Finally we pre$ent the rU'-'lto for the (a ,a') inv.,rHOn· The

cross section of the e1>Hectf.ve model, together with the i111pr<ned

cross .&cction are given in figure tv.22. Here al&o the re"l !Ind the

imaginary parts of the form fact<:>r& a<e gf, ~e... The error band is

o>btained fro,. figure IV .21 by applying the pre•cnption (IV .22),

but, as stated <Lbove, this esHmate is probobly too opel.mistic. We

m1,.1.iJt therefore ¢0-nclu.de that~ t'hough the t;:l;"¢itiS section 1s improved

suhH~"tidly, there ls no aignifieant struct,,fc in the ""deor

~'l;ltet'iOC" whi~h can he thought: 'J;"caponsib1~. On the othet" hand> it

-120-

2, 5

-n

-5.0

10

0.1

0•1 2\

beta-0. 20

r lfml

66zn lalpha,•lpno'I

25,0 MW

2•1 ll. 041

.. Jo oo 90 1io 150

0 c.m.

66zn lim•g. I

u•1 2+1

beta-0. 20

l"if/· 1 V. 22. &cul is {"Mm in.~ePefon "'''"''·fotions fo>' the ·tnd<wbfr

alpha acatter1in~7. ComJent·ion;.; a$ in figi(f'r?e .TV, 5-IV.16.

-Lil-

seems to be signHie"n~ that the sha>i• of the fono ~ .. etor should be

neuwo0r than c<:>Heeetvo model fo.,.,. f•<!tor, Now it 15 ~"-tereating to

note that .studies of p:c'OtO'L"I ~nd neutro-n dC:l'ltities:io such EL8 per~ormed

by Stdng"U ($'tll82), ahOT< th"t one uaudly fi<>da the largcot

di££e:t"e:t1c.es betwe-::n 1."J.eutron and ptotora densities at: tbe nuclear

surface. Utls seems to cont~adict ouL findiQg&, eiuce~ due to ~t$

laoac.alar cb~rl!Leter:1o alpha's ~..ita excite botb ne11,1.tt:oo.a and protons

eq~tly strong. Therefore one woulQ ie:xpec:;:t a form fei:to...- whie.b is

mo-re Q'I;" less smea'tGd l)\it towards th@ -cc~t\";iil part. of the nuc.lel,liJ.

IV-6 The inversion Qf (p',d) transition deM~Uea

The tt:"fl!.lla!t:ton denstcl~t) as c.onsidered up to 1."J.Q'41 all represent

1ncl11GU<:o scattering re.,etions. The 1ll!SA fo,,,...11.am, howevot, h"s

also been ~:P:P:t.ied suc.cesfully t-o transfer teactiOne;. This means the.t

o"t i<wersion proee<l.,,r• might also be ~P>ilieable in th8S<> eue~. In

tbii; llJ-e.i:::.r.1ott we coni:.1Q.er the one. nc1,.1.t~Qn r.:ra.dsfer re.actiol19

58 Ni(p,d)57N~ (3/r, 5/2-) o~ ""'asured e.t ~-i4 .6 MoV by Polanc

(POl.81).

Itt the tMoreue4l descriptl<>r< of (V,d) reactions, t1'<>

as.sumptionn are comm.only med~. Firstly, the dot\ltetoo. wave functio-ll

is described as a pure S-•tate. Seeondly, the range of the proton­

ne.,.tron force, together witb the size of the dE>uteron h

sufficieqHy ""1;11.11 compa>:ed with the wavelength of the proton- and

d<:.,tetott distorted waves to allow for " iei:o->:~<'>Se approximation.

We then have for the form factor (SAT63):

(Iv .23)

~ere the Gubserlpt ZR stands (or zo0ro range. The factor DO;ZI\ is

chosen such thbt the volume intcsr~1~ of V and of the fl<tite "0"'1

range fotm f~ctor are equal. ~n(r) stands for the radi~l ~art of

-122-

th~ W~rv'~fun<;:tiOLl. Of the tr.ansfe'(''('t;d :i;ie1,1tl"Qll 1 which in this C.0$~ Of l)

o+ e.:srget has A~,. .6.s .ati.d 6 j ~s quantumnumbe:rs. Fo't tht:: t:Wc)

tr<'t16Ui<>t1s <onsidered we ti;;ve (A~,As,Aj) (l,l/2,3/2) .. ,...a (3,1/2,5/2), corrospondiTI~ tQ ~he t1<0 lowest stote$ <)f 57NL

Com~G(t;c;l \iil"j.ch the transition den!El:l:.t1.e$ for 1nel.aat1c scatte:rin.,g 1

this hss the follo"1ng ~ .. Pu"~~1Mo:

1) t.h~ £0~ fa.c.tor11 es..11. be chosen Tcnl.

2) Y.e do not expe:c.t exchange corttl"ibutlon11.

3) DWBA usu8ll)'" oppears to suffice whereas fo~ o..,mer<>Ua inelastic

e-;oec.itat1011s the coupled channel!$. niethod is necessary.

4) no addlt:lon.ut <;.ore polarization~ Coulomb deformation or .spin

orbit dcfo...,ation has to be tal<en into M¢<)utlt.

Apart from these apparent adv.ai:i~l,lge&. there a.re s<:i1:iH~

~ire.um.stances that roakoe 1~ lea.a profitable t;:Q ~pply the inversion

procedure to trar.i.e~ct '((::~<;.r.lons ..

1) St~Qn,g1y e:Jl'..c.it~d inelastic -channels can be :l.m~~a·r;:~nr. (Pol.atl.e~

(POLBl)).

2) OeL1teron brcak~up ~ho~ld be incotpOfated explicitly, .,,.3. by

means of an sdiab~~i~ ~pproximation fQr the deuteron hrenk-~~

chnm1e1 (J0ll70).

J) The ~adlai depe'1den<e of the £Qrm factor is l<nwn in priuc.iple

on-ce the angular mom~nt.v.JD r;:ra11a.fer has br;¢n established. la

practice~ the mi!3:sin.s i:n.grEdient th(!r-efore is not the $hnp-E!: of the

form factQr but ~~t.he=:t the spectroacoptc amplitude and the. la.t:.t~r

can be ohr..!.i11ed without .an 1..nv~~eicn\ approach.

Th~ Bubsequient ~e11ults therefo\"e 4re presentoed. predominantly

to ~rovide an odditional ""Pl<>ration of the 1ttversion method.

FiJ:$t 1 we pe:C"form.cd. the usual 1i:ivera.ion calCYl~tion using the

experimenta:I. pick-up C'C'058 iJections ~ As ~t•rtitl,g fo'C'm fa1;:tori!il we

took the approp~~~te neutron Wave fuu~tions~ ~P Qrder to check the

s.saumpti<>P Qf the ree.li tY Qf the form foot<:>rs, we sllove<I also the

illlBSl"-""Y part to w;.ry. The re;,,,1u are depicted 1" figures i:v .23

and IV .24. the correspond it>& xZ-valueo "'" listed in toJ>~e IV, 7.

Concei:-nt(ls the .a.s:;umpti,on of the :reality of the fo'l;'w, factort we

observe that the >:e•ulting illl8ginory part for ~M ~12- state has ~

shape mote or less slmil$~ to the real pn~~. This me.ans that i~deed

a phase factor exiats su.c.h that the fotm ~(l.~eor can b@ cho.131.en near

to :real. Thia is not the es.Be• however, f<nt the 3/2- st13;tc.

The resulte !"l."QtO. a more sevei:'e test of the 1n11eraion method

are pl'c$eo.ted in figure~ tV.25 and IV.26. Here we dropped the

a.saump-tion thi:tt the re&\J.lt.~ns f()rm factoi;- 6hOu1d be rel.att?:~ r.o

<:Hher a 2p3/2 or 1£5/2 neutroo "'.'ve foactiot" Ao I'- starting form

.fti.etor for botti. 'l;'ea.c;:tions, we tQQk as the radfQl p8rt the average -Qf

the two wave funct:t.<;:i.ni;i;. It turned out that in an invtl;"~f.Od et11.ploying

the experime~t~l c:rosa sectf.Ql"l..S; we could not reproduce tbs co~re-e.t

form faotou. Thio apPe.,red to be due t<> the limited ~ttgular

i:nterval of th.i;: ~xperim.ental C-:"Q$B sections.. E.vCI\ whe11 we. genet'-6-t.e.:;l

tbeoretl¢al (:rose s.cctJ.(u:'J.s that span the entire r.ang~ £ro01 O;;i to

180° .l):ru:i applied thes.e as texperiment~l d.s.ta. i, we Q'(:ll)" c:ould manage

to perform. inv>:r~~OM with s poorer quality tllM equ1valent ('p'; p')

1nvel's1<>0&. This io believed to be due to the foll0"'1"&· The

oentral pa,;t of the form hetor is aamplN 1,y the products of the l. ;I.'

p.&.l:"t'l.~1 wave funct~one. !ict¢.,. Now 1'1\ the c.ase of (f.:P') scattering

for low Q Vo.1'1<!8 (-Q{(f;-Q) ehe ilU .. ber Of dlfferent product$ la

reduced due to the propetty:

(!v.24)

-124-

dd dil

(mbl!r)

1.0

,5

- '5

-]. 0

-1 10

-2 JU

30

' '

r {fml

58Ni (p,dl57Ni !mall 0+

1-312-

1 beta • 100_ 3

58Nilp,di57Ni

24.6 MeV 312- 10. 001

-· ,'

58NI IP, di 57Ni (imag, J

0+1-312-1

beta - 100, 3

60 90 JZO !SO 0 30 60 90 120 150

a c.m_

f·fo . .TV .. 2~-IV.24. Rsm..1..lt:.; f'I'Or:r ·&r~ver-s1~on m:ztm~Zations j'r.:xr· bhtJ

rea<•l·~onB 68Ni('f,dJ57

Ni .~/2-, 5/2-, 1'he f1,u "1<"""" deri••t tho inver•ted form f(wtorn and tl1ri ao-r>r•o•s­

ponding c:t'O::i::i .aeot.·i:onB and (,rt1(1."lysing pouie.i·.a. 17-ie. dot­

&t$hed f0'(111 faotm•s m•o tho r>ea!. part3 of lhe 1>01"wi-

+O, 5

0. 0 A

-0_ 5

c l-(d~e 11..=.?1.dPrm wav& function.:.; for' tht~ f.-"('(1nnfr3r>:t'sd n.sutron.

Tha dot-daiihi.:~d c;r'O(;l."; fJrif(~!.'.i1:in and a.na"lysi.ng po'l.iJer u·1.;::r'-

i)<.:-.R flr>f:: ~Ompldc3d Witf1 th<:!:;e (l'(; fOY'm factors.

-12 ,_

drJ d.\1

]_ 0

.)

-.5

·LO

' I

·'

r {fm\

4

~.:.:-'_· -------"~==-r<-· - ' .. ·-· -· ........... ~--.=-""---,

5SNi (p, cti57 NI (reall

o\-512-l b~ta·12 .4

5SNilp',dl 57NI

24.6 Mev SIT !O. 771

58Ni Ip, d157 NI (imag. I

0+1-512-1

beta•n.4

!mblsrl

. \ ·. '. .......... -· .... _

' , ·-

-3,.___. _ _._~_,__-'------'----'----'-------''---_...~_._-.._____. 10

O 30 60 90 120 ISO 0 30 60 90 120 ISO

~ c.m.

Fig. IV.24.

-126-

+O, s

0.0 A

·0_ 5

LU

. 5

V -2 co• (fm l l .Q

- _ 5

·l.0

\ ... ,/

' '

58Ni (Po dl51Ni (rean

o\-3IZ-1 be1a • 100. 3

4 g

58Ni (p,dJ57Ni Omag. l

o• 1-312" I beta • 100. 3

4

r (lml

F1'.iJ. IV. 85-IV. Uf _ Re8ulta f>'om the irwer~ion .::ak;ukitions for>

the r>eactions 58Ni(p,dJ 57Ni 3/2-, 5/2- ""spectively.

'the <:I'OSG Mations and anatyaintt rower>s tha·t G<?roed

Fi(}-

as '•.t:p••r>imantat' data ware Mmputed from th" dot-dashed

/ot'm /cwtc>'t:'S. 'f'h() daeh"d fo>'m factor>z war« u::;ed as

ztar>ting for>m factoN;. i'he /1,iz ~u»?>~s depict the re-

suitintt fo:mi faator>s, IV.26.

LO

.5

-'

- . 5

-LO

I

I

' .\ '·' ,_'

58Ni $,di57Ni Omog.l o• 1-512-1 beta • 12. 4

4

-127-

58Ni (p, dl57Ni (real)

o•1-s1r1 b~ta • 72, 4

r (fml

A aimiLu relation does not hold for (p',d) reactions.

Moreover, the number of product functions in the latter case is

larger due to the greater number of (.t, a, j) combinations for

deuterona+ For these reasons sign-cancellations in the integrals

n.9) •re .. ore Ukely to occur for (p,d) reactlone than for (p,p')

reactions+

1) U the bpectroseopy of a ~eaet1on 1& kno.,n, the lnverslon

proeed..,re ""Y be used to find mi nor corrections to the foX"IO

.factor. Corrections in the inner part of the fori:a. fae.tor should not

be truated too much.

2) In oder to tind useful results ln COl6e of " complctdy unknown

fotm facto-("~ at least a laI"ge angular 1nte.rval for the e.rosa

section ia oec.eaaary.

Table IV, 7: :r;Z reductions from (p,d) inversions.

)(~

" 59.0

255

-12a-

13-6

76.7

4,33

J.JJ

n

3

J

rv-A Appendix

For the sake of c¢mplcteaeas, we will g~ve hci:-e .a shoc-t ShJ.rvey of

the mieroscopie IJWIM. formaliso» Exee.i~hc tHerature can be found

e.g. in the paper hy VOii Con1mb (GBR71) and references therein.

We $t~n by writing ~o"n tho UWBA expression of the

antisymm.etrized T i:QS.trix element:

I I - l s;n{ <xt<.!'..) J<t ~ ,.(!.' )x'I' cl ff VNN(_!'._-!.') I[ 4> ,.(!_')•'!'cl> 1 lxr (r£ m,n c

ovii..ii

where it 1G undcx-a:tood that in the s.eeo:i;id teX"m not only the Bp$t1~1

coordinates bT,,It .i:s15o the spin and isosp1l"I cc:11:11;"dlnates are

1oterch8nged· In (IVA.l), S 1$ the "mpl:\t1.>de of the cootrlbut~o" to

t.h~ cxc:ltstion by the trane:ltion of a parttc.le fro~ st(lte n to 9tste

m· We have introdueed the: rt\lcJ.e~r state; I[ $mx'¥ C) I)' t wh!<;.h 10 to be

1nt~rptct~d as a single neutrQn ~ave function + th.lit occupies an

orbit with J-j~; and the wave function 0£ ehe core, ~C' ccupl~d

together t.:i ~ e¢t.sl angular momentum I.

The ingredieot~ fH tli~ c¢mputat1on of (IVA.l) "~"'

- the v;).l,1,1e of the apie.<:troso::.op1¢ ~mpt1.t1.Lde S;

- an apprQ;ic;im~t:'-OIJ of the exo::.h.a11.ge te:rm;

~129-

fo order to comp'J.te the e•redgth S for each of the trane:ttf.oaa

id (IV .22) for tlie S8Ni caae, we apply the general DM!tl'lod o~

Brussaard and Cla"d"""'"8 (BRU77). To this aim, the c<:nob~n$tiod 0£

the ti:1.1-cleoo-nuc.leon fotce and the distorted waves is seen .a& an

operat<)~ OA with a multipolatity A eq1.1al to tl'le traaaferred angular

momedtum. We write for this <>~cl."$tOr (BRll77, 13.24):

where a+ and a "re ereatlon and annihilation operators,

tespeetl~ely, This matrix cle1Ue"t o~ OA ean be evaluated by meaM of

the method of second q,1.1antize.t1on. The atetea of seNi ate dCG<:r~bed

.;)i!J two neutrons outs1.d:c ~ 56 N1 c.ore. This means tba.t. tha~e .tiitst~s

can be written as atste opcr.ato:i:e work.irig oa the s6Ni core t d-enoted

by t.he: vac:.uum state t 07:..

(IVA.3}

We can interpret thi• a• a <:<)1Ub~"~U<)" <>f t"<> ~<l.tro'1a,

together fo~mi~g o~r states of interest, The o+ ~ 2+ excitation i$

now deacribed by c1Che1: reeo.,.p1ing within the same orbit or by

promoting gne of th~ deutroos to a different orbit· We h.ave t.O d..t:.i:tl

therefore with two ¢~bit ~trix elements+ Su~h a two orbit matrix

de..o.-.t <O[aj'•aj,]Ifl[a;"aJ'\f[a}a; Jl 1o> ~s •<><:l ... ced and fac.tor1zed a b a b

out as a product of two 810.gle orbit matrix. elements;

Here ~ stands for I (2ic+t).

-l30-

Tbe q~antities appearing ~n (~VA-4) have the follow1ng meaning:

z I> the state of (IVA.J)

F the total Dn,I.lti-o~bit t~ansition operator in se~Qnd

quanti~ation

W an intes~t that depends on the oe~~P~~lon, n,~mber~

"1,rwolved

y l ( ~) the total Sd&~l4r momentum after the excit~~ion p~o~e$s

of the i>artideo thnt occup)'" the orbit labeled 1(2)

th~ tOt.al angular l'.d.Ui:tatl"lt\im b-cfor-e tbe exc.itat!o-11 proeeSS:

of the particl•6 that occupr the orbit l>:>bele<I 1 (2)

the '1=".SI't of the operoltQI:" F that works on the partie:le~

1n orbit 1(2).

Ihe symmetric expreM~on {IVA.4) reduces considerab\y b"c'""'"

of ~i ~o and I £ 2 2. In (BRl,177) " class1ficstio11 sch•"'• i• pruented

for the re'!Jl.Eli:f.r'l'.J.Tig reduced ma.tri:ic element.Q. In our case we have to

consider thr~e Q.f.f:icJ:eo.t ca.teg:ories ~ depend!n,s: ¢n, the .st"C"ucture of

the opera.tot'" f~. l!.v.i'l lu~t.lon of the relevant expre1Hi1.t;ma yie.tQi; fot"

the value of the ...._trix el.,111ento of the a~a6 operator from ( IVA.2):

<a /a+.,•~ IP • ah(2~a6 -t) I: ii

(IVA • .'>)

!!ere • ond b are the am~1itud"6 of t11e ~elevant components in the

wave £~~ctlons of interest. the evaluat~on of «iloAfB> ftom (1VA.2)

is just a matt.er Qf RlE!i:a1.h-slgebra ~ The result iiJ ~

(ct Ii I~>-(£¢<' " o++A' exc1tat1on:)

(IVA.6)

A J)Qt't of this expression le already eoni:.t1i.t1.ed :f.n the code CHUCK·

Thie means that th~ 1lt.t"cngth parameters to be provided for are the

-131-

pl:'<>duct of expression• (tV$,.5) and (IVA.6) divided by th<o h<0tor ill

<;KllCK. this fee.tor tllerefore haa to be rec<lllotructod (Nm the

.compvte-: eode. The final -,;i=&ul t.$ for the strength parametet"i; h..t;lvr:

been l;l.!SleeQ. in aee.t!on IV-2.

Next we muat pay attention to tM al)p«:»<1...,Hon of the

"!xo:;h.i:i:.:p,ge !.erm in terms of a l0¢~1 £ortD. faetor. We adopted the

approximation giveo. by Van Hall (llAL73) >rhl<:h h b'1aed on the local

energy approxl,.,.tlon (LEA) of Perey and Saxon (P~R64). The exchange

matrix eleDJent. then reads:

(tVA. 7)

~~th the excllange correctlon FE(_r) as the Fourier transform of the

nucleow::i.-m.u;::leo-n force that is appro~1m.ated in LEA+ For a YT,Ik.awa

sha.p~ we get:

(IVA.8)

Hare, q2 is the 'local' energy. A ""'-nageahle approximation to q2

whlc.h works well in pr.a.ctlc.e reads:

CIVA-9)

<he oeenrring ~~oper~tors ~re evaluated by meaoa of the

corresponding Schi:'IXl:l.nger equations. Now th¢ ~;.t~gral (IVA. 7) takes

~lgebraically tlle fol."m of ~ direct term. For tho pi:'aetlc~l

realiz.ation of the above scheme we h.p.Q. eo m.pke: soi:ae additional

changes in the code CHUC!(. We started from th<1 <<1<"0-r""S" kMek<:>ut

fori;Q fa.et.or, which is jutit tbt:: pl"'od,1,1<;.t i:;if two nuc.lear wave

functions, and lilllltlpUed it by the function FE ftom (~VA.8).

the eheck of the no<:eM~~y modifications could only be done 1.,

3n indirect way, Binee ~ost authors use exact fintt~ ~$nge

cakul<l~~oo.e, that yield aomawh<llt d:l.fferel"<t cross sections.

Cale~tations by van Hall and Mcl&eeo. (llAL73) however, that al$0 uee

th1G lOC(l.l energy approximation, could be reproduced whereas also

the ld'...-dependence of the tQto11l c.rosa sections as presented b,Y" Lo1/e

al\d $<>eehle.r (LOV67) r<1sulted eQr .. eeUy from our calculations.

-132-

1'he final ingrediel"l.I:: of the microscopic re~enou th-eory is the

nucleon-ttucte:on pot:ential+ For the $eve'l;"al components tha.t o.;<:;v(' :ln

the T-~att"lx we .ei.riply the usu.al J)4t"~meti:t~ation in terUJ.s oJ? a!ng:t.ct

."!n.d t.t"f.~let, even s.00 eii;ld fo'{"cea. We h.ave for the 1.i:i.tlllrai::tions of

Interest (PEf70):

Direct~ AS•O' v ~(v ae +3v Ce tt so +3v to)/8 pn

v •(v se +3vto)/4 !'P

Exchange i ti s .. o: v •(v se +3v te -" "" -3v to)/8 pn

v •(v -3v J/4 PP se to

Direct, i>S•l; v •(•v +v -v +<> )/8 I'" $e te so to

v ~(-v +v )/4 pp i::e to

l!:xchonS'-'• i>S•l: v I'"

•(-v +v 'i-v -v &e te so to

)/8

v •(-v &e -vto)/4 (WA.<0) pp

Tl"\et.i:: cx:ista a. great 11a.r1ei::y of interactions"' both bai;~d on first

~~~<lclplea or on pheMmenology. We apply the pben<>menologkal

PQ1>~haripande force (PAN69). Ile finds for the v00

,vec'v•0

and vto

v " 350.9 G(r) (McV)

"" ,, - 526.3 G(r) (M~V)

~· ., •-438.6 G(r) (MeV)

$(>

v to

-o. wherl!:

G(r)- •~E (-1.6r) (IVA.11) l.6<

-133-

We see that as far .a.s the p:coton•rieutfOt.1 fo:i:ce is concerned,

the l>S•l transitions are neglig~hle eomp'1.>:ed to thMe 'l'ith AS•O,

Moreover, c1nee the proton-proton force <>nly applies to two of the

weaker S~Fe transitions, we skipped the A$•1 ~raMitloM <19 li whole·

The Pandharipande force differs from other forceo llHh reopeet to

the ••nge o~ the "so' whieh nor...,Uy i8 taken much shorter, In that

case it ie allowed to neglect the odd components aa the direet: ati.d

ex.change con.t:rib11tlona interfere d.estr1,1.r;.~!,vely. We de.;.i.dcd to do the

oame ao we had oome do1'hh on the range of ~hla component,

-134-

ChaptH V

I,Jp to now we e:xc:lueively ¢.¢t16,;Qt'l."~i::l li'"eactions where it wa.s

suffie.ie:nt: ~(I t€L~e only one exc.1 tal:iol'I J);)l:b into .,.ccOunt. This ls,

ot' course, s. very 11..:0.u::ed. cJ.•~1:1)9 Qf resctionst The excitation to the

vast majority of 1'11,1¢1¢~~ !Sit.ates proceeds by uu.DJ.erous simult:al"ICOU~

couplings. Fortunately, :f.t ~~ w;iQt always necessary to it1c.lude .;11

these in a coupled channels sc:.hem.~ t.o !'~prc:i.~1.1cc the molilt .salient

In sec:.tioa V-l we

will encounter a l:llethod to .analyse cross aee.tioo.s atld ~rtn.J.y$'l~ir,

pot.\r¢'t"S i(I the case of two concurring reai::.t:f.0"1;1 J?ath:e. 'by me.ans of s.

DWBA-like praoedure. We modified our inveroia~ pra¢•rl~r• tQ h~ *blc

to handle these cases. :8ol;!;(.t1on V-2 di::.als '"fith the .accurac.y of Ollr

method fo~ t.het'!lc more complicat!!d (.Oupl11"1$ :$c;hit:me'11& 1'he i11v0rslon + + + 64 66 68 re$~lt$ {or the o2

, 22

and 41

•ratM of ' ' Zn will be

preeented here. Ft MUy, in ocction V-3 the interpretation of some

f'e.at1,1i:cl'} of th.e resulting: forttt f11.¢tOf.& fl)l;" these cases in terms of

electrofl sl'.::.atter1ua; r;.e1.:11.1J.ts Ernd IBM calc.u1atioaa is gi\l'el"I..

V~l Second order DWBA

Ln l!e<:.tion I-2. we Bhowed how a t.wo channel mQdt;!l eoll-n he u@ed

to derive a DWBA e:icprese1on fO(' the T-m.atth; elements in the c.aee of

ol). one $tCp c:xc1tation. The l"eetrlction for its 'l.J}.11d:l,l;.y WL1-.tl1 t.h._.i.t

the coupling le we.ik compat'ed to the distox-tion potential~ since it

11'3 only tak.en into acco,,.mt up to firet ord~r .. Rere we w:tll. apply .@;

three ehbl'lnel model wh0reas we will derive the T matrix. elementa for

the third e'li.annel 1,1pt.o i:i.econd order in the coupling, strength+

-l35-

Analogous to the ci<proooi<ms (J;.7<1,b) ""have:

(V .!)

These equations correspond to the following coupling ocheme ~<>( (Ju(

eh.a.nni;!l of interest c.2

;

• Cz

Note th.at strictly speaking we should .also incorpor.a.te thie 'back..,.at"d

cq~pl~~$ eer~ v ~nd V in order to ""'ke the entire approach 0 1 <-o 0 2°0

eorrec:t upto second order in the. couplings... The former tel'"m>

howevei::'> doe:~ not 1df1ue:1'.ee e.ha.ll.ael ¢.2

IllOre strcuigly ths.11. upto third

order+ Moreover the co["te8['ond1n,g tI".Qnsition dtanea:lty b.Q& been

-136-

calc.ulated in ~ DWijA context desc.ribe.d prev101,1~ly l,rt .$r;i;:;t.lon IV-J.t

Therefore it would be f.mpI:"i;;i.pcr to ue.~ it in .s. CC scheme. As t-o the

neglec•<~g o< the backooupling from ¢ 2 to o0 , "" 1'1U8t cealioe that

in th€ collective model the ~r:rength:9 o~ the t1="au.sit1ons are related

to the defor1n..::"1r.ion. Th~ i:::0

+c1

and i::1+<:

2 tra1tsit1q.n ei~l,"-cngth-6 .Qt"-e of

first order ~hcrcaa the atretlgth of th~ ¢0•c

2 tt~n9ltion is of

second order ia rhe d~f'.o'l;"mationt Therefore for aot. r;:o9 J.;:i:i:-ge

defotmatlona ~ the latter 1.a ab¢1.1!. iln l)J,"d-c~ o{ magnitude smaller th.s.a

the others. Thereftn•e ~£ W"e omit all backward c.oupl11'1S$i W-c obsc'{"ve

ths.t the T~matrix for c2 ~~n b~ w~1~ten ~a a coherent sum o~er tw¢

ter1J1.B .. We can immediately '"'t~te dotm thE term correspo:i'ldi11g to th=

c0+c

2 coupling (cf. I.9);

(V .2)

The other term ¢.:ln be deJ:"ived by applying Gree11 ra theorem t:\f:lce.

The result is

The function r;}" (< ,'r."') is the solutloa of the SchrMinger equat1<>" "1

fo£" the sec.ond channel whet~ the inhomogeneous 1:).9.rt h~s b~r:n

replo.ced b;r 6 (c-r' ). Unfortun4tely, it ¢4MOt 1>~efull;r be factorized

out 111.to fun.c.tio(l& w:i;t.h, lc;.nown a:siymptotic behaviour· as 1~ the case

~Ot:' the one step T-matri:ic: elem.eat. This mcUn$ that it is not

advantageo'1s to apply equ;<C:l,¢q v .3 fQ< prncticol comp'1tation of the

T-m~trix e1ementr:i;. 'rhl'l& they are co1Dp1Jted by u:ie41\G of 01.101e(icslly

$.Olv:{ng th~ coupled Si:.hrljd1n,ge:r-11ke equ.at1oai:. 1tt£.":cend. Tti:c t:otal T­

m.oti;-ix is given by;

(V.4)

-137-

A• """ to 1,., ""pected, th<l partilll T-,...trieea still depend linea<ly

gn the form factors V ar::i.d V •oc2 "1 c2

For the applicatl<>n of the itlversi<>n method to the case of

parallel excitation po:thr;i;~ l6'e. will confine ourselves tQ th~ .eo­

called two phonon triplet of the 64

•66

•68

zn l~otopes where

diffkulties in the description within the collective model ooour

(~0084). ln the collective model, these $to.t;,$ $r" described as two

¢0.,pled quadrupole surface Mcillntl.on quanta. This can result in a

o+. :t.+ O:t:' 4+ state. Thia mean5 that 8 $p.Ct.iflc pt-¢blem .ariae:s:io as

compared with the Qne $t:ep ease. lf we consider an excitation fi;-om

8. o+ ground state.,. the m1,1.lt!polarities Of the transition densities

v ;)n~ V are equal to the ... ·11,1.::lea:r spins of the states .:::2

and "oc2 "0"1

sever.al multipolai:~t~e.ei: ~re allowed, each of them corresponding tg

its OWl"J. form facto'(". The (l~ty l!i!:x~eptil)il 1g t:he e.s.se rrr mQ+ whe\"e the c2

m"H~polarit iea of V and V a.re equal. Now for ou• inversion COCl GlC2

procedure th1.s u;1e~11s that fotlr form factors would have to be varied

simultJJ.'(leQ\11~ly 1 causing the number of vax'iablc=:.i in (I(!.14) to

¢K<:eed g<eo.tly the number of equatioM. An esMpe from thl~ problem

is to adopt ~n additional assumption e.g. one fx'Qm the

;i.fore10entioned hannon1e vibra~or .. odel. There 1 the nuclear

cxeit~tiotl operator that corre5ponda to V 1~ 1nte~p~etcd as $ cl "-2

~114drupole phonon creation oper~tor. This mMM th~t the ~trl"

eleme.nt.s of t;hi: JP:\llti.PQl~ ~l>ll'.l:POtleo.ti;i::io othel" that1 ttt•2, will v1!Htiah.

In case the nucliSu.9 \1~d.e~ .\ilt:~y i$ not a po.'t"e hal"incn.1~ vibrator the

A~~0.4 Qr 5 components should also be taken into account, but even

then their .::01;1ti::I. but.1Qn ~i;'l t;h!i;i. eot'-te:>!'.t. is ustlally a:n order of

..,.gtlitu<;I" S""'-ller than the t.i•2 tom. for tllc curr.,.,t !lndysh of

the Zn isotope~ we thel"efore restrict ourselves to the lnc.luaion of

t~c M ~2 t""" only.

-l33-

Oni:;!e: Che sec.end .order :OW.BA expression for the T-matrix has

hee'1 form1,1lated,, wwr; readily c.an generalize the. inversion proc.ed.ure

from chapter l!l so as to co(tect both the V and the quad~upolc coci

part of the V transition deno1tieo. It suffice• here to Hot clc2

n;ut:(Cly the d;lfferenci;::a 111ith the one-step vel'"aion of tb.e m'G:thod,

1) The coefficient V"ectoC" ..! from equation III+) now consist.a of

the amplitude.a of the splines of both form factors. If we uae an n­

spline t!Xpal."lsion for e.aeh .of Che forgi: f.o.ct.ots, ..::, 1$ Qrgunf.zed as

follows;

coefficients l, .. + .n: real parts of the spline. .amplitudes of V cocz

n+l •••• 2n: rc4l p~reo -------------------- v clcZ

2n+1 •••• )n: 1mng~nnry pa.rts -------------------- V 0 0<2

:ln+l .... 4n; imaginuy put• -------------------- V clc2

ii) The stru.:oture of the ..,_tri>< J in equatioa (III.ll), including

the phyai<.al da,.plng constraint, ls as follows:

&p-1n up • . .

t------+-1 -·-----+--- ---+-1-

>Q

-139-

The ~qu<'re m.at<~i< Q co<<t$pond• to the penalty function ln equation

(III .33). In prh¢1pL"' th"' d~"p1ns factor ~ for the several ""'tr1ee&

Q ¢"" be eh<>oen 1ndo>peud .. ntly. In practice we preferred l:o use the

$oiJ~e value fo-,:: l, for .all four C.dBeiS in orde:i:- to aC"I"ive at a

straightforward algQrith"1.

iii) l!t pri.,ciple, the •rline width could also differ for the

to;.ro form fa.ctors. Far V we: ¢a1'. 1mi:i:tei;l1a.tely use the results of an ¢0¢2

investigation as in section III-2. Th1o is not the case for V Cl Ci

becauM <>f tha more complicat"d atr11et..tre of T01

,12

• In practice we

took. the &pHne "idtha for the two form fador$ b<:>th eq"d to l fm.

This some,,hat larger width than ua"d pre"1'>""1Y r .. duces the total

number of amplitudes foI' one fQri:q fttetor in such a way ths.t it i!Ji

stUl ~ignifl.cantly- leas than the n111•ber of <1~glc;.

iv) The error width for the two form fa"-tors fa Qbt<>i"e<l ~o.

C:xtl~tly the same way as foC" the one step e~~e. tb€1rt t& by means. of

the global method of section I-3. We performed the error b;>nd

calcula.tio4s for t~e ot, 22 and 4! states of 66zn; the value& for

the other two isotopes were obtdned by scaling with a factor

01 (66/A). 11: turo.ed out th$t senerally- the crosa aec.tfons are ""'ch

mote 9enBitive to the details of the V form factor than for 00°2

those of the V • Th:J.~ re~1,1lts;; ;le, relatively narrow error b,Ji.tt.di;. "-1 °2

for the former compared with the latter.

-140-

l, 5

LO

.5

- . 5

-1. 0

-1.5

V-2 Re:r.ul c.s cf the tW.(.) etep reactions

i) Error anal)"aia.

Evt)I\ t:.houg:h forui:ally the inversion algorithm f-ot t\lfo step

reactions does not diffc~ f:r:om th.i:t.t for one st~p reac.tioas~ details

<)f the re&ulti"S form factors ore likely to be loo• ~"1hblc CM"

for the one step cases. Thia is due to th-= £~~:c t:.hd.t it1 th.e i:M.tr1J( J

.s1,1'btle. c-.orrelstions between e.oh.tmns of the V part Emd of the <0"2

V part rn.e.y r:x~sit. On the othel" h~nd, to S\l;;i.rat\tee a.1.1 uil'bia.se.d <1<2

solut:loo~ it is .ud.vti:..;ible to npply the singular value aaalys1s to

the ll:latrix J as a whole inEltie.g,d of t't"~~ti.o.~ the v-i-r1(H).$ Qblong

~uo°"'tricoo oop4ratoly. We c.aa &H an idea of the reliabillt:r of the

. :..

4

58Ni lreall

o• i-2• l

beta • O, 24

~---!--~- .• ·-,

r llml

' I

I I ;·..._, I /

'\ ..... ,r 'I,,,..\ ...

S8Ni (i mag,)

o• i-2'' 1

beta • 0. 24

"'-.

. . . ,. se . r'* '! • ,. f"·i'.'J. V. 7.. H~~:.;ul.ts fr>om 12 psGudo-dQl-r::t i.nve:r'e-:wn O'rl' Ni i.'1P " 1

•

The d11shr3d cu.~1h:l~ stand for• the. star•ting form fact-or t'

th.t:: d()t-cla.ah~,i r,;ur'PeO dep1:.c t, (,11~? forim facto:t' hy means o }'

whi.ch thi8 psL~udo data. UJl..?l"r3 ~JBl'is::t"'Gtsd and ths j'u."l l cur'iJ~~.·)'

a·1:iir; Pie t'~~F.:'tJ.!.t of '(;h6! { .. Tt'l)lf:''t'GiOt~.

-141-

, - '

0

-I ~~z.n lreall 66zn limag. I 2\_0•2 2\-0'2 beta·0_ 36 beta•O. )6

66zn lreall 66zn llma9. I

o·1-02 0•1-0'2

.is beta •O. 014 beta•O. 014

-.2:l

a 8

r (frlH

' ,,_ 66 (-" ') • • + Fig. V.2-V.4. Rasuota from p~eucw data inversions fol' Zn p,p 0 2,2 3 ,~

1•

The {!OtWBntion~ are as in Fig. V. 1.

9¢~utioM by means Of what "" shall call the pse ... do-dsta method. It

cOMhtQ of tM following. Suppose that the lnvet'U<>n procedure

yields form. factors V' .s.l\d vr !ilt.i:J.t:tlng from the collective 00¢2 °1"2

-14Z-

-I

-2

.25

~l lfm"21

-.25

·" ,1

.1

,-,., ..... ~,,. ...

,1, I

66zn \reall

2\-2•2 b~la•O. 36

66zn (rean

0•1-2'2 beta·O, 014

66zn limag_ I

2•,_22 beta •O, ~

66z~ (ima9. I 0•1-2'12 l ~eta•O. 014 j

-;

l l

--1 l J j J

r (tml

mod~l for~ faccots V and V , ~e6pectivelyf As a s~¢¢nd $tep Co Ci "1°2

we treat the coller::t~ve model cross se¢.t10TI:6 a=s texperi:ments.l 1 d.«t~,

so call~ct peiieudo data, atu;l We try to reprod1,1-ce t:he&e 'by me.ans of the

inver~l<)TI; pt"OC@dure .S.l:)plylrt$ the primed fOI'DJ. faetO'i:'IJ .tl.9 6 Start. rf

-14J-

0

-I

.2'5

-.25

I I

4

66zn lreall

•\-4\ beta·O. 36

66zn lreall

Oj-4j beta·O. 014

66zn (imag'. { 2\_4+ l

bela•O. 36

66zn limag. l Di-4+1

be1a •O. Ol4

r Uml

4

,, I

0"'1J: "tUetliod were e~ct~ 0£ e.011rse the e.ollective model form factors

wonlrl eome out. Applying this pseudo-data mBthod to one otc1>

'('eiJ.et.J,QI'l.$ 1 results as in figtlre V .. l typically can be obtained. It is

tu)t .!;11,1t-prising that this fin.al error is less as the diffe:renc.e

betweeta tb.e for;in factors V and V' is sm.a.ller.

-144-

We presel"r.t the reaL1lt:s of some f:!Seudc) data 1nver$:1.0!le in

fiS<.<!:~$ v.z to V.4. We note ti'iaL though the gone~al teodeno·l .. are

col'1llI!ouly r=produced~ t:h.ert; are still considerable i;!f,:ff.erences ."1.t

4 m.imbex- af vnluea.. Ii;: ii:. interesting to note th«t the

vQl.ume i11tegr.~1s of the fcrm :{actot"a .are: rep:i;-odueed mu~h h~tter~ .i:3:nd

therefcr~ ~i;i.i;i, be use:d foT:' furth~r !mre~tiga.tion. Ic ta with th~r;;e

resut~$ i.n mind th1t.t t:he actual Eora f.actor c.orr'l:ctiona sho'.Ild b12:

j"<lg<>d·

H) a .. sult•.

We perfoC"m.ed the inversioll~ Eot the o+1 ~ z+~ ~n~ 4+1 st.ateB of 64

•66

•68

zn. Fo< overy o+2 w~ ~Pplied the •aoue damping; f9r th• z+2 and 4+) st~tes the d.am.p:l,.ri.g factorli differed,,. but l.!g.,..tn were the l'.limna

for all isotope~. The resultiQg x<•s are li•tcd in table v.l.

table V.lt x2 X'C~ults for th1; two phonoll states fO[" 64166168

zn.

1~Qt.opc .state E ""c 04 Zn o+ 2 t.9\

611 Zi, ~+ 2 1. ao

64 Zn 4+ l 2.31

Hz,, o+ 2 2.37

••za z+ 2 1.87

66zn 4+ l 2 .45

69 Ztt o+ 2 1.66

~e Zn ::·~ 2 \ .88

Se zo 4+ l 2 .42

xz 0

x~ f

113 43 .6

609 %.3

364 17 .2

224 74. 5

331 160

~16 16·6

130 40.2

209 n.z 95.2 41.l

,. 2. 59

io .a 21.2

3.00

2.07

31. l

3.23

2.67

2 .32

n

3

3

2

3

Ml variable• have the ••me me~~lQgB as in tabl• rv.4.

Fiq. V • • S-.(1. 7J. 'l'hB r·esiUts of inve.,-.r;r1:orr calcu.lat·z'.on.s for-

ths two ph"n"n MMec uf" 64 •66

•68 ?.n. The fl<IC our~es give th"

fo"(':rl fo.otor·!3 and or>os? ~eo1,iorts c:md anal.ysing pow~:P~

c~.C 'f'i:."!(;i~l.f,iTi;) f'r"Om ·M1e inVBI':.:;im-1; the d(J.t.hi;:d. 0·4,1·o'.J6.L°7 $/:c..m,d

for ihe ocll. /.c<atiue mod<'/. re<tuito.

-14.S-

r lfml

2 4

. 2

.l

v -2 C()CJ (fm I 0

-.1 I 64Zn !imag.I \ 1 64zn (rea I}

•• 2 0+1_0+2 0\-02

beta-0. 017 beta·O. 017

\

64zn lreall. 64zn (imag_ l

I 2\-02 z +1-02 I be!B-0, 40 beta•OAO

vtot1 ffm·21

0

-l

4

'

~'. 64z~ Ip, p'l I

, I

0 20.4 MeV I

I

do' 10 '-, t +O. 5 dil

o+ 2 n. 911

lmblsr)

0.0 A . -2 ... t ti + .. !O

• t • \I -0. 5

-~ 10

' ' Fig. v. 5.

30 60 90 120 150 30 60 90 120 ]5()

;;tern.

_ 2

_l

·,l

66zn rimag. I

0+1-02 0\-02 beta-o_ 014 beta•0_ 014

._ 2

66:zn lreall 66zn limag_ I , 2+1_0+2 2)-0+2 ,

beta •O. 36 beta•O. >6

-1

-2

00zn ()1, p'I

20.4 MaV

do' 02 (2-37) +O, 5

dn -1 (mblsrl 10

0, 0 A \ ......... \

-2 ' 10

-0,5

Fig_ v.e. 0 30 (){) 90 120 150 30 60 90 120 150

~ Lrn. ~147-

4

• 2

.I

·.I

-. 2

-I

-2

do'

dSl. -I (mblsrl la

·2 w

-3 lO

F'irJ. v. ?_

30 60 90

r(frnl

68zn (real/

o\-o-z beta-0. JO

68zn lreall

2j_o2 beta•O. 31

68zn !P, p'}

20.4 M~v 0+2 (L66)

l20 ISO

ft

68zn (imag. J

0)-0•2 beta-0.10

6Bzn !ima9.}

2j_o+2 beta•O. 31

'• ,,

30 60 QO

+0.5

0. 0 A

·0. s

120 150

-2

_j

-, l

-, 2

-I

-2

-I \

10 I

dd .,.

d_R lmbisrl

-2 IO

Fig. ~-8.

30 60

4

\

I

90

• 0+1-<'z beta•O. 017

64zn lreall 2+,_2•2

bela•O. 40

64zn llf. p'I 20,4 MW

2•, (J_ 80}

. - ,

120 150 0

~,_[1"\_ -149-

64zn Hmag_ 1 oj_z•2

beta-Q_ 017

Mzn limag, 1 2+ 1-2+2

be1a·O_ 40

'I

'• I

I I

'--~I I I j

30 60 QO 120

i' +0.5 •• •

0, 0 A

'I I

I I

, ·0. 5

150

g 4

. 2

.!

-2 vcoc1 (tm I

0

-.I

66zn (lma9. I -.2 0•1_2+

2 o•i-2•2 beta•O. 0\4 beta-0. 014

-I eezn (real) 66zn Hmag. I

2•1-22 2•1-2·2 betl·O. 36 beta ·O. 36

-2

66zn (ll;p'I

20,4 MeV

2+2 U.871 +Q_ 5 -I

10

do' dQ 0. 0 A

lmbl5rl -2

IQ

~+ -0. 5

Fig. V.9. -3

0 JO 60 90 l20 150 0 ~o eo QO 120 150

1\.lf'. -lSO-

_ 2

,I

-_ l

-_ 2

-1

68zn (real!

0·1-•:: bet•·O, 10

rlfml

68zn (imag_ J

0·1 ~2"2 belo·O. 10

68zn lirnag. 1

l\_ 22 beta•O, Jl

. 2

.!

·.l

·.2

-l 10

dO" dQ

(mblsrl ..

-2 . lo t

Fig. V.11.

30

I I

...

r lfml

- o+,_4+1 Deta•O. 017

64zn (real! 2+1-4-)

beta ·0. 40

64zn W,p')

20.4 Mev

4+1 G', 311

60 90 120 !50 0

6-

64zn limag.l 0+1_4+,

bem-0.011

2j-4j betll•0.40

30 60 90 !20 150

+O,:;

0. 0 A

-0, 5

dO" d!1

fmblsrl

.2

,1

-.l

-, <

0

-)

-2

-I ' 10

-2 lO

fig. V.12.

JO

...

r lfml

Mzn freall o\_4•1 bela·0.014

Mzn lre<ill

2+1- 4+1

beta•0,36

66zn [p,p'l

ZQ.4 MBV

4+1 (2, 451

.. .

60 90 120 l50

C'e_m_

66zn (im~g. I 0)_4+1

be1a•O, 014

30

66zn Ii mag_ l

<'1-4•] beta•O, 36

60 90 120 l50

+O, 5

0.0 A

·O_ ~

dll' dfi

(mblsr)

'2

.I

-.1

-. 2

-1

-I lO

10-2

Fig. V. lJ.

0

'• '

30

·,.

60 90

6Szn (reall

2•1_4+1

b~ta·O.}l

6Sz_n qt, P 'I

20.4 MeY

4\ (2.42l

t. . . . , ., 120 150 D

II-c,m.

-154~

68zn Umag_ l

0•1-4\

beta-<l.10

68z11 <imag. 1

2'1- 4+1 beta•0_ 3!

30 60 Qj)

+O,S

o_o A

-o. 5

120 150

We observe that th• ic 2 >:~4ucUon ~o of '"" •<>me wdo.- of

magnitude as for the 2! ~t~te$ 1 ~xc~pt fot the 2! and 4! states of 6 " Zo and the 4r of 06 Zn which are impro ed aubatantially more. The

re:eiulte of these fnversietn~ are 9rese:fl.ted 11"1 f1,sui:es v.5 to v.13.

In the re.,.,inder of this 6ect1on ""' give a doscription of the

j(;t\ll~nt features of these results.

a) The O! state$.

Due ta •~perlmcntnl circumatnnces, the erooo aect!ono for O!

e;.-;;C.ltat ions are generally harder to measure than those of other

~t.ate~. Thi~ c;A~1$el'il th~ l.at:"ge t::'.t"'fO'.t" fl-Rgs ~or these cross !Slec.t:1.on.e

and a11alysit.1.g. pow~rs 4i:r; wi:;11 .al!'I t:he: i;;iomewh~t: ln¢¢het:ent l)ppe.tir~n.c-e:

Qf the ~~Zn data, which, moreover~ should be considered as

preliminary (M0084), The whimokal behavl<>ur of the <oorreotod oroos

sections. .and analysing. powers together w1th the oV"er.all re:du¢tioa of

the cro&e sections, clearly points towards a delicate ue.gative

i"terh~"'nce bet,.,~en the one-step and two-~tep l''""euM. ln the

collecti~e model description this 1a much less apparent. We can

understand this difference. if we look at the nature of the resulting

form factor. 'For 61:i.zn for instance, we see that the correction

essentially coa~iets of red~e1n$ the 2t ~ O! strength where~s the

at ~ O! form factor lo left nearly unalta.-cd. Apparently, in the

collective model dea~r1pt1on the 2f + o; ie ina.inly responsible for

tho shape and size of the or cros& section, wherea6 88 a result of

th.e cQL"'(t::cted £g.ra £'1Jcto("e 'both Pollth5 .$.pp~ar to be lmp(lrtanc. tor

6 c,, Zn a similar mechanism can be observed. although bet'~ &om

peri~her.a.l struc.ture ia th-e Of + 0~ fQrDJ. fact.or seell;).B to appear~

The cross Siecti(ll'l. ~ad e.spe~ially the ads.lysing, i;;iower of 64zn

are reproduced badly and we doubt the relievancta of thla feature.

J;n 66z~, th,. cOH<0c\'.ho mo4<io~ DWBA d~s.;or1pt1<>n uf the oross

section shows a defect in reproducing the angu1"r !'h<'~"-· tt h<1~ bee"

sh(Mn by Van Rall (llAL80) tha\'. U11G O!I" b<io J;"~p~1r~4 by appiying a

mot"c complete coupleQ chnn:aels ~che:me. Th1$ means that our fori::u.

faetor correction~ which also a.-cema tQ 'boe G"blr:: t;:Q i.mprov.e the

description of the da\'.a impl"c•dvely, c<:>.,14 almuh.te the local DWBA

-155-

"<i"1val<:at of ~uch coupled ehar.nel8 effects. As;dr. this seetll(I to be

effectuated by s shift from the zt + O! domlnated process as

represer.ted by the colleetive model toward6 s. deseription in which

both proccucs arc more or leu eq,.ally impot'tant. In thb ease,

however, this 1• not 1;1eeompllshed m1;11nly \>y a red.,ct ton of the real

part of the 2r + or for .. fa.:otor but rather by ""' inere .. sed it1LS&i11ary

term in thi~ excitation path.

A general effect observed for all three isotopes is that the

improvement ie manifest predominantly for the cross sections~

although for 6.&tzn this improvement consists of n.oth:lt1.g more than an

overall redu.~tion of th.e. c.rosa aec.tion without lll'l)" .s:.1-m::llarlty with

the more detailed stru.c.ture of tbe data+

•t ~., comp*•e <he re_sul<• from the O! invc~Glono vteh <he

other states, we have here the major advantage that the form factors

are not lil;ell' to be polluc"d by the efhcto of omitted couplings,

other thail bac.kw.a.rd couplings. 'When we compare the results from our

m@thod with mote fundamental results from nuclear &tructut"e

calculations in section V-2, this aspect will receive special

attention.

b) The 2z states.

Apart f~om any at~Qctu£al effects~ a general tendency of all

eolleetive model descriptions seei:aa to be the overesti:ination of the

cross aec.tioo. da.t.a, be it somewhat leas than for the o+2 etateei ..

The de$<:r~pU.;n .;£ the cl."QU oect~on of tbc 2+2 f>t<>t~ fo~ 0"zu

is greatly lmproved by appl)'ing the corrected form factors. The

<)nly "'~"""ining ""'jor defe.:ot 1& observed !lt 55• wt.ere the da•<'

dispts.y .a sharp interference-like minimum whleh is n<>t &een il'l the

theo•ette•l c~rve. ~he """-1Ysi«$ p<)wer i~pl."Oveo le$o ""'"i~eotly; ~"

ob'l'iOu6 deteriore.tion takes place t<>und 110•. lf we l."estriet

ourselves to the cross eect1ons 1 the improvements gradually become

H:so vh<:n go11'1S from ~~z.., coward~ ~8 ~n. Note tMc tl'IU bol'IRviout u

j"•I: <>ppooite to the case of the 0+2 states. D,.c to the wor9e f1c9

to the analysing powers'" these tendenc.iea not always reflect

themselves id the x2 reductions.

-156-

Concel'ni11g the n~t'.ul;"e oe tt;.e ~0'.l;"m fRCt(l'('g, ~ C.STI observe: that

the fQrt11 h¢Nr <>f the dHoct excitation is again very similar to

the collective moael ~rei~~1pt1oa. The cO~tections to the zt + 2t for., foct<>r~ ~·~ o(>DJo .. 1iat loss <eliable. Thia ia especially true

for th~ pe:dphctal s.tt"uctures for 6i.+:tn and 155zt1.. w-e h~'lt;! 1!la.1a,ee11,,;t:112:d

in the case of one step il"l.version thiti.t: :c:hc occurrence of such a

stru¢.Curi: mi,e;ht be du~ to the omission of some effects that ~annot

be expre!l!A~d 1t'.t tC;i;"ms of corrections to the form factor. B:el"e1

an

¢mHt<td dfoct could be the 2j + 2; coupling by mean8 of t.~mO or

U-4 ~"'"'"itions.

Th~ temaining general tendency is a shif ~ from a 2i + 2; dominated excitation sehem~ f.QWJJ.t<h1 $ $cheme ;1,n '°'hich the one step

(a1Scond order-like) path acquires Hl:Ol"e imp.ort~n¢~ 1 ju5t ~s '(..)"as f'Guad

ift th~ c~~~ of the 02 states. These shifta are~ however~ le~s

drametk than in the latter case.

o) The 4i St3tM.

Of all states atu~1ed, the 4j ~~3teG Q~ 04 zn and 06 in are

those t:har. improve mo$t. Here even tha analysing pmJe:ra shaw 1111.

~~~¢u~agiug imp~ovement. Like for the 2! of ~~Zn 1 it is interesr.1~8

r.o n(Jt.~ tli..f.\t ag~tn the remaining effect in the cross e:ectiot.1.

c:oneia:ts of a ill1t1l.p;i,1,1m roui:td 60° in the data which is not well

reprodJJc~d. Fol;'" 66-z.n we observe an impressive improve.:i:tent of r.he

cro-e-1). $ect:f.¢r.i .. :tt is amazing that here the corrected fort11. fac.tor

backward-peaked cross section whieh ~em!nde to

exc.hang:e-domioPt.t)d t~actions. For an angtJlar momentuin. transfer as

high as ~~mi,, this should not be a p~!oti impossible. The

lmp~Ovemcnt t~~ the 4t of 6Szn is m~ch less spectacular and it

follows the teodenoy of the gradMlly dec~eMing improvgment of the

cJ:"oea sections of the hea..,ier isotopes as found £Qr the 2t $tP.te~. As to the form factor corre.c.tiona, we ob~erve tha.t the gcnetal

beh~viou~ of the 2t and o~ states persists in the aenae thar. ~he

or+ 4! form factor ia herdly affected by toe tnver&ion whereas the

2t + 4i £i:,'("m fp.ctot" shows a clear reduction for r > 4fm.

-157-

We ean s1.1111mnrhc our findings as follows:

l) The collective model prescription oyote111.1Ltks.lly 01>'ereni111&tce

the cross sections for all et.ates wh-ei:e.-e. tb1e overestimation is

less for t:he he.,. vier Zn isotope&.

Z) The conected fom. factor generally yidda s o1gnif1can1:ly

better deec:tiption of the cross see.tiQo. whet:e.Jl.G tlle analysing

power la less ~learly improved. !xeept.f.on.& to this statement are

the 4! states of 64zn and 66zn. The improvements of the croso

sections are leas convincing for the l:t,ghtet: f.Sotopes in the case of

the ot states and fo~ the heavier isotopes in the case of the other

two atatei:;i;.

3) The rssult1ng form factors for the d:l.reet e~<>H<ltion& sre ln good

agree,.,,nt with seco'1.d ord•r c<>Uectlvc model prescriptions, both for

~he real ""d :I.magi nary parts,

4) Ths form factors for the 2t .. t+ (X•02 .Z2 or 41) generally ohw s.

reduction io the peripheral region. Here the O! •l:ate of 6 ~in

fi;>'1lls an exc.0ptlon. In some cases o:o.e O:.OQ .also observe .s. sin.ell

shift of the ma"1mum of the fom factor toward• """'11"~ r.du.

-156-

V-3 A1;1 interpretation of the: results

From a mict:'1;,HJ1;.Qpie: point of vie"(.I, the tr.anEil;f.tf..Q'll densities

for the excit~t~On fro~ o+1 ot~tes to zt ~t~e~s are generally

deee:ri'bed with the 5.U.w.e Dt1c.roscop1c <::0111poo.e"Ota aa the i:ni:c1t$tiot1s to

2t states+ 1h~ $~e~troacopic amplit~dea~ however, dtf£er in such a

way tl;u1t the la.tter tI"ansltion dedaitiea 't'es\.llt £ram a much more

coherent •~Pe•poaition. One would expect this di££~<ed~e to show ~r

in the at+ 21 ~dd 2r + 21 form f~otO•B. Although th16 is, strictly'

spoked, not conttgd:loeed by our findings, ii: should be <>t t .. ast

$tlrpr!aing thQt the rad14l form of the 1eae coherent auperpoaitione

ahows a $tr1kia,g rea@mblanc4!'.: r.o the eollectivc Ii:IOdel fo"t1D f,c1,¢r.()ra.

for the direct t:i:-~.n'li'<1t1ot1s. It is not very likely;t on the other

hand, that our ine:thQd sbould be ~d4ble to produce corrections tq ~he

or~ zi form factot&, siace the ~~mer!eal values of b~th the

element.9 Qf the v-ector !. .and the appropriatl3 ~1~m~t1ts of the mat. ri:ic:

J for the two form factors. a-roe of the same t.'Jtder of ms.gn:lttJ.de. We

find~ moreov¢'t"> th4t in pseudo data inversfQn~ the or+ zt fotm

fattots are more acc~~~Cely reconattucted than tb~ ~t ~ 2! form

factors. In tM$ ¢Q'1text it is inte•eating to tcfo~ to the work of

Neuhaua~n (NEU77), ~hcto ~n inelastic elcct•oo scattering tQ the 2!

states of 64

•66 zn fo<wf~otors have been found that qav~ a rather

c.olle..::tive .api;;u;~l,".l)ne.e.

A inoI"e gener-~l .,_nd qu.e.ntit.at:f.Vf!. way to coni.pD.'L"e our re.1Ji.,1.lt~

with ehe re.suits from wel1 established methods can b-e f¢u11d in the

In.teracting Boo<:n> Model (IBA) Q~ Adma and l<1Ghello (ARI76-7S). We

w111 g1'1e a vie~y $hort and simplified pointwii:J~ 1:1a1mw:i.ry of the

c<,.<1141 iasredient~ Q~ this model.

-15~-

- The phy$lcol idea <>f the 111<1del U the ~phasie on cor~<:lated

pr.citQ'(.1- and ne\lt:ton pa!.rt:i~ Thea.c: pairl;I .a.re 4s1;h,1.med to h.Qve j•O or

j•2, looeely speaking abooe s- and d-bosMQ. The totol n""'ber <>f

prot()1' .and neutron boaot.119 is f:t,.;ed and consists of ~h,e valence

p.artic:le.s o"" holeB.

-The 1'1,1,1.clear h4.mlltonian exists o~ term.a 1n which .$.t moat t:wo boso~$

1.nterat:!.e.

-As a ba.$e for expanding: Qu'[' wa.\"'e.EunctiQ:n..e; tfe c:.hooea a set of stat~

vectors that all ~«ve the number Qf d b¢~ons~ cd~ es a proper

quantumnum.ber. The one- and ewQ phonon atatieiJ from the ha~onic

vib~ator model cor~espond to states having nd•l and nd•i,

respectively. In a IQOte gen~t:al cont.ext, st~tea. c:a.it be written .as

e.,g . .::

(V .5)

-The amplit~des ai ~g (V.5) «<e oht«ined by diago~alizing the

nuekar hamHt<>nian. The para.,.,ters of this h~milto.,ibn usually are

taken from fits to t.he experi1;11ental -e~crgy level ac.heme of the

:D.ti¢.leus of interest.

~Nudea• tra"$1tion probabllitte& can he c<>mputed aD.Blytlcally once

the amplit.,des {a) MEI know.:.. This b acc.ompUshed hy expresUng the

trana.1t1¢1'1 opero11tor in eec-ond. quantiz.atiod. It is 4SBUmed ehent that

o"1y one-boson operators are hlvolved. We th1>s have the first order

opera.to.t"s a+d+d.+s and d+d. Here the neut:\"-on a.nd protoo. p~rts can be

"<Oighted difforedtl:r to express a~ec.ifie properUes of the relevant

operator (e.g. electromagnet~c. iso~~alar).

-160-

In order to compare OU[" ["Ct:.11.111;:~ with the !SM. 1 we have to

provide for tho following:

l) An IBM hamiltonian, '11Jle to reproduce the relevant ener,\lY levels

of thQ Zi;i isotopes under .study. The pa'l'.'.J.meters of thiG hotltn~ lton!an

have ooe" obtaiaed by Maao {MAA82) by a fittitig procedure.

2) Th~ n\leiear tranaitloo pr<Jb.abilities th.P.'- res11lt froN the IBM S["e

mere 1"11.1.inbera ~ without any radial c;lepeddeace+ We hottve to extract

the~~ ~r.aaaition p~oOObilitles therefot,'".i: £?'1)111 ot1r form fnetorsT An

often U&i!d approa.eh is to a~pl)" the expectQt ~on V'alue for the:

ct~etrom.ag,netic t!;"ansition operatO't for a given chnrg.e transition

denoity p (r):

(V .6)

Osterfeld (OST79) has studled the corrupondence between this

electromagn<tle transition operator and tM hadrouic "xcitatlon

opet;:i.tor. From num-et:f.e.(11, e){-periments 1.t n.~peared that fQrtuaately

the o;:;h.srge tt.o!J.'l;'lr)itio11. density and the nuc.lear ti:-a.n$1tion density are

sa.mplti!:d it"L 13. more Q'(' t~ss c.o:mparativ8 W8Y by means oe t.hia approach ..

The cff~c::.t of the iz,.;change eontribut:t.on ~(lo the nucleo51't t.r~nsition In

this cor,.tex~, however, ie nc.'Jt yet c.omplett:J.)" understood.

For the pr.ac::.tics.l compElria.ot>. we pro~e~ded as follow$. W~

computed th• ~ncegul8 from tl:!o right hand p~H fro.; (V ,6) 1>l:!e~e we

took fot" p e1ther our i::or:re~r.ed form factQri:i: or the '!;13:.o!ll P4t"t'S of

the collective mod¢i fQr,,. factors. w., will refot t<:> these integt"l.$

~s C(A). The qu<)tient of C(A)inv dlld C(A)coll is compaxeo W1th the

quotient of the B(eil) Y<'lues &9 oo~ained from the l!IM and

ao oo~ained with the collective model preecr1ption of Tam1>(~

('1;AM65). Note th<'t in the collectlve mode~ description of the

harmonic v:f. 'bra tor as in out 1I'LV"ers1on proc.ed.1,,1.rt!: we only hnve Ji. .. 2 for

-161-

the first ordet trsns!.tl0"8· M to the accuracy of the volume

iotegl'"als we c,;ii.;0. sta.t.e tbst WEI fl)IJ.1'!.d. e-:'rQr~ of a few percent: wltb

the exceptlon Q£ the t:ransition frQm the 2t state tQ the oi aad 4t $'t(l.t.e~,, vhere errors a'["ound 30% oc.011~.

'the result of these ealeulatlon• lo given in table v.i.

64 Zn o+l .. o+2 0.27 o.n 64 Zn 2+1_,.0+2 0.29 2.04 1)

6"zn o\+2+2 l .17 0.89

6•z., z+1 .. 2+2 0.82 o.oa 6• Zn cr\+4+1 0.90 0.89

64 Zn 2+1 .. 4\ 1.12 0.11

66zn o+1 .. o+2 0.24 0.58

6~zn 2+l .. 0+2 0.32 o.os t)

66 Zn 0+1_,.2+2 1.38 o.95

66zn 2+!_,.2+2 0.82 0.38

6Gzn o+1+4+l o.82 o.a6 66zn 2+1_,.4+, l.16 0.53 '>

68 Zn 0+1 .. 0+2 0.26 o.79

GS Zn 2+1+0+2 o.34 0.47 68 Zd o+, .. 2+2 l .73 o.94 68 Zn z\ .. 2+2 0.85 1.80

68 z,. o+, ~4+, 0.87 o.99 GB Z.n 2\+4+1 1.19 0./5

I ) For the O! sta~e3 Q~ 64zn and 6~z .. and the •t state of 66zn we

£"""" a net sign difference bet.,een the UM-.iatrb element:• and the

inversion results.

-162-

1 ..... .

64

the. resu).r:~ 1n tM;l,s t.able are .also present:ed e:'("a.phically in

figure v .. 14~ As to the 4Ce.\lr-~¢y Q~ the .above results_. we rec.all chnt

the C(A) values at"i; t:.$1'.fentis.lly better reprod1,1t.ed l:.h.t:tn the :several

dotails of the form fo.e~ots themselves when doi"& ~ p$cudo data

1a"erG:l,oi;:i, ..

~r6m f1$~~~ V.\4 we learn that th~t"e 1$ &Ome agreement between

the ~BM res~lts and the inversion results fot" the nt -1r r+ t(ansitions where I - o,, 22 <>r 41 • The ""1-jor dise~el>"<lcy fot tho

trallSitiOUd 2t -Ir z; Bild;i 80111eWh8.t teS.$ 1 2t T 4t1

might be d1Je tO the

negle~ted M - 0, 4 and 6 contributions. Indeed, H "c lool< at the

transitiOI'IG 2t -1r 0! 1 we observe for 66zn an~ 68zn ~ ~O~~what better

agreemenr;: between the two re:auJ.tl!i:. O{ CQl,.1.I".S(!: this c:.ould not be

expected from -6.1tzn 'Whei:-c the theoretic.al cross section 1).pd ;)nEJ.ly:i;:lng

power give <'- vc~r poor description of the da[i;.

6B 66

Fig, v_ 11_ Thor Y'~!)Ul-ts from tabtcE V • .2. F1.1.7.7., i:?'l4r1"Vr;?R,' im;ori-eion PR'i;J.J.7.t;p,I

da:;h<:d cur·veo: IBM re~ult~.

-l63-

Chapter VJ; *)

Ae far "-$ "'e know, the o;,.ly correct "'SY to d"aci:ibe phyaieal

phenQD;i.8Ra on the ~uh-atomic level,. is by me.ll:m!I of qus.ntu.111 II!.echanlcs.

<bat ie why the scatter!~$ Cheoi:y fro~ chapter I and the inver8>on

procedure from chapter 1.1.I have heed based oo the q.v.«ntulllllei::.h.att.1cal

cotl¢.eJ?t of the ex.pect:ation v.a.lu~ of an opera to('• i.e. th~ t:('.ansition

op<>rato>• <>f equation ( X .1).

In th18 che.pt.er we 1J1;t\ldy in h..:.~ far a clai;i-e~cal .approx.l.~tlon

of eq'1atfon (l-1) can teacb us so°"'thlng about n~clear traadtfon

deru}f.tles. The.re b.~a two m.otiv3c1one for this study:

--..we ¥~<= curiot1e to the meTits of the c.1~1}61cal approx,.m.stion in

the ca.se of ine:laat1e $catter1ng.. M.o:r:-eover,. it; i:aight incre.i;:r..se our

general und.el"s;tandina of the sc.atter.:f,qg procea1;1.

-in cs.1;1c: the class'1¢*1 approxillMl.t.ton W¢1,1.lQ: turn out to he

auffl.::1en'tly s.ccur.tate, we could. fcplae.e 01,lt," non-linear i~vers!on

by a linear Qi;'.!C, because the clasaical, .analogue t(I i:qu.etion (I. l)

will resule in a linear exp~essionr

llith re$pect to the former p<>i<le, '"' note the follo"~"S· lt ia well

k.n(Kfn that the claaaical .ttPProxims.t.io:o to ela.st.ie ~cattering. ~t"o:ss

oeeHona is <>nly a vety roush one. As io ah....,, in f!g..,re VI.1, <mly

the a.verag@ value of the croz;al!;I .s;ection is 'teprodueed eo:rrec.tly,

alth.,,,gh tbe general effect of ehe imaginary P&•t <>f the potential

corrt:!:spodds nicely to the quao.t.:i,tmme:chanii:::al case. 'thl-6 is not

aurpJ::":f.~ln,g if we: t'C:alige that: the int.eri'~r~-nce struet1,1.:re, clearly

visih1~ f.TI the quattt1,1.mmechanie.a1 dc.sc.ript1¢t.t of the elo..8tic c.ross

1!H!:.i+~f.on orig!oat~.r:; f["om the £tlCt that t;:h~ wave functions of the

pr<>j"¢eU.. and ejoctlle have the sa""' absolute v<'lue of th~

w~v@number k! the dese.r!p~lon of the :r.uterference of two ¢.OhC~ent

W.ckv-cs of e.01,1l."'8e needs .s. w.,_vemt!chanic.al approach.

*) Part of tois ""rk hH$ MO'n do"" ~" collaboro.tfon with M~ F. Meas

as .~:n undergraduate research pi:oject at 01,1.r instittlt;:e.

-164~

10

r:f Id ruth.

O. l

\0

rf Id ruth.

0.1

·1 .,,.,. .

"1 .,,.,. .

30

..

..

,. . : .

·~ : .

66zn (p. pl 20.4MW W • Womp

• I ' . .. ..

•' ·,..

66zn Ip, pl 20.4 Mev W • 2Womp

60 'Ill 120 l~O

a cm.

-165-

..

. . . . • '. . .. .:·

\•

66zn (p, pl 20.4MeV W •O

30 60 'Ill 120 150

Fig. Vl.1-

Cl.ass{t?a.t. ct'0?;3S sea tion r::<'.+ (,~u­

/.atio.is for th" sfo,,~-fo pro~on l:i6

sch."J:tti~rin(J to :i:n at [-1()_ 1 UeV.

Loft. ~ppsr•.- etrew;th uf tho

irriagina:t'y part of tho pot1mtia /,

(W) taken i-ri aaaot>daiiae wUh

the q11'1n·tumrne"harrGoaZ OMI'.

Rir;iit i+pper .- im1:1qinary pai•& ut

to i'.:l~JIO,

I"~:ft l.oi,,J.el' ~ i.:nag-i.nc.z:r•y ~tr-e.n({th

twioi:? as iarge ae; rhe OMP ~Jahu;:.

In the case of 1ndMUc 4e4Herfo.g;, the wavenumber• of the

ptQjeetile and eje.ct1le differ. this deeteoe of eoherence. displays

itself in s mucil leas oseillatory atrncture of the cross $cct1on. Wv

eov.ld. hope therefoI"c.,. th$t " classical app'E'ox1m.et1on of in@laetic

scatterins yields more acceptable results thsn 11'1 the e•se of

elastic scatt.i:(1t.ig.

In this cilspter, the following subjects are dealt Witil:

TTS.ection Vl-L give:.11 the proc.ed"t,t.re for the calculation of

trajecto-riic::8.,. 1.e. t::he classical picture that is .analogous to

the distorted waves. Moreover, attention 13 p~id to the

i<>elua!on of the imaginary part of the potential •

-section VI-2 presents til<> computaUoo. 0£ the d:\Hete;ttl.;o1 eross

section.

-Section VI-3 de8ls "'Hli the .;olasai4al i!iverdon.

lie consider the scatte~it\8 !'rocen in the cel'ltre of •lMs

syoteO!. From el"menury mch•nics we then take the relal:ion

(AL071);

r m

(VI,l)

rm - the largest pole of the integrand, i.e. the classical

t1>r111ng po1nt;

V ( r)~ the real part of the optical model potential.

-166-

The cross oectlon is given by:

- l L

L I d0 1-l ~ dL

(VI.2)

We ac:eo11~t: f!cr t:~~ .absorptiont due to t:h~ lmaein.QI:")'" part W of

tl:lc poteotial, by ""'lUplying each term in the rlght l:land side of

equation (~t.2) by a reduction factor d1

. We compute this factor by

applying the quantu~mecoanical continuity equatlon for the

probublli ty density

(VI· 3)

The w:~.eult is:

(VL4)

The integrals (VJ.\) and (VI.4) have been oomputmd

numer:l.e~lly. Spcci.al care had to be ts.kr;::n in c:hc neighbourhood of

the pole rm; here a Ga~~tl~n qvadrature formula has be~n u~ed

(HAN6)). ihe deviation foaotton 01,, ~hows o $lngularity when the

deno:idn.i!.~(lt" t>f the integrand bec.ol:l'le:s statJot1.a!:"y. The L-value where

th~s happens is. called the: orbit1n,s .(.n'lttula{" nmmentum. The orbiting

doe~ not reflect itself fo the <orooo oe¢Hon Uno< I ~~ I become•

1.-.flnite too. Moreover, the dampi:ng :fo.4+~Qr <IL vaJ:"1::f..~I,e:i:i eor thll! :i;ame

value of L.

Ati: eif'il!.::::.t that does show up ia the c.ro~li $t!:CC1Qrt ..;.., is the d0

~aniehing of dr• In the CQGe of ~laatic scattering we fl~d Ii

$htgula:dty at the corresponding scattering an,g1e. Th.'-$ ~t:t:ci::t iEl

called rainbow sc:atter!ng. D1,,1(~ to i!in additional integration in the

¢.it.6.e c)f 1nel;:a5tic scattering, lt :ts lees manifest there.

-167~

VI-2 Th• dHforent1al "rose section for 1nelastie scattering

'fh.r: problem of c.las.sical and eem.1-elaaa!e.al approaches to both

elastic:: and 1n~l.a&t1C .iile~t:tering ha.a received ample attent1Qt1 d..iring

tli,. l<t't few decades. Th• 81.'rvcy o' Bernstein (BER79) reveal• so'""

of the techniq .. e~ e .. nently used in this field. s1.-.ce our study

beat.a G J:Jrelii:a.~\'kar .and e:duc.at:lonol.\l ehar.,eter~ ~ use a vG'l:'f

simplified rHction mo:>dd here.

We *dopt the following ass..mptlQ~a:

i) tlit eJ<citatioo. 1e local and tal<es pt.Ice in o.-.e step.

ii) effecco •foe to th" projectile spin can be .-.egletoted.

1U.) we IJ;l,@rely e.onsider excitationu Wil:.h L'•Ll;:I where I equals. the

nuclear spin itJ. the excited atate.

ad O The same assumption undc~Hea the IllolllA approach for cQll<letlve

excitatf.On$.. 1.~ the classical ca.:ti;C lt ha.s been implemented .as

follows. We assume a give.-. eet of radii r1

with cQrrcopo~di.-.g

excitation proooblHeies a1

• The latter co<~cepond t<) the a1"plitude

vector!! of ch*Pter III. For each i the lntcgrals (VI.l) and (VI.4)

are compgt~d up to r•r 1

• Herc 111'e .abrl.l-Pt.l.y eh.ange the L and E such as

to acc.allnt foI" the ttt:H'!.$.l;'er of angular momentum and cnctgy. Next we

continue the il"ltegration. Note,. hgwevei;- 1 tha:t the matching of the

two • .,g,.Qnt~ of the trajectory is not .. n1q.,.. ln fact there exist

fotlr c.sees for t.'he Bat!l.e L and E leading to d.1f£~te1\t s.::.a.ttering

~ngles. These are illu~trated in figure VI .z, The •olid c.,~.,e~

..;.Qrreapond to the gtOUJ;J.Q ~t::ate. trajectory whereas the da..th-c~ ¢t,tr'\t~i:i.

depict t1le tojeetories after the eirnitat1Qn. N<)te that the dashed

t!:'<'jeetory has the Mme shape in all cases; it has me~ely been

rotated around the .oca:ttering centre.+ We Observe tha.t 1c. c.e.ses a) dr

and b) the sign 0£ dt is conserved, whereas in ca&co o) 4nd d) the

motion change!9 its dlrec.tion. A schematic.al c1;'.n11putatio11: of

reflection and t'('&n.smit=td.on c.oefficients f01::: the cot"tc:l'.i.porul11"1&: OI).e-

-168-

' \ \

, -~·- ~1 __ ___ ,.

I I

I

' , ' ..a~--' ~

l"ir;. VI.?. l•'our pos!.:J11.:b1.13 01.?mb{nctti.on~ of ent:r'ant?E?"""i:?.X·~t mcl'la'hin.g in computing

eLar;r:;·~(ml tra,i'3ctorii•?P. fo:r ·tnela::;tie scattr?.ring. The full. C'U:t'Ve

derJi.Dt~ Pie U!'Ound stat& t>'(J:j@<JtOI'& ""d the dashed aurv~ th<f!

or1e c:or'~~srondMr.9 t.o th.& ex.cited stat.q., tl:rr.oow.s df:'1'Wt.:.~ th~ entrianc.s­

and. ~\'ti~, ii-igrrr~:nts. Th+! i::.:t·oit.aL1~on i~ mr.lrked by th'° dr;:ii;; on .:;;i:oit.at1'.on

foI' a gz'.ven l'\. oan ·tok..&. place on t.W d-1:fft:'!Y'0;.?t/!,, l-ocat·tons on thi:::

ci'f"'(')Zc w1:rh rodiu.::J r'=it .•

' dimension.al qmmtu!'Dmechauic.s.l sc.s.tte.ring probleto shows t:.hi!.t the

proc.~!!i1!u:~.(O c;) and d) mi;ty ~e ~glected.

o.d 11) Prelimloary studie~ hoV'e ~ho"1:1 ehoe the i\O.ol\.u)~(>" Qf " $p.!n

depend12nt forc112: for elastic scattering does not improve: the quality

<>f the dassks.l deocript1¢rt. $~nee iU 1ndv6~0n in t~c eMc of

inelastic. scattering; is rather a problem, we haiJe 11egleeted the

proj~c~1J.c l'ilpi,n&

ad U:l) fo. the quantummech~n1cd des<:ript1on of inelastic

s~atteri(lg from o+ to r+ by 1J10.a.ns Of DWBA 1 all dy-ada of partial

wa11eo {<t,~t'} that oufHee 1~-Il<~'<l~+II contribute ta the crass

-169-

dO' dQ !mbl~rl

10

0.1

30

•·. ; • .. ... .. ..

66zn !p.p'l 20.4 MaV 2• 1 (1. 041

-: •

611 90 12\l 150

(t c.m.

30

66zn (p,p'I 20.4 MeV

3-1 !2.S3l

. .... . . . 60 90 120 150

4ectfon. Thelr rehtive weights "'" """""'Ulli> gi'1e<> by the square

0£ the Glebsch-Gordan <:o~ff1¢ient (~ 2 • 0 0 Ii 0). At first s~ght, thia seei:ras to be in cQ~t~~~t with the classical ca&e 1 whe~e we have

L' ~ L ± I as the only comblnatio<>s of L and L 1 , The question arUe~

which of the two prescriptions sh<>11ld b~ follow~~. In the classical

limit >te hne h+O. In order to have ffoite angular momenta, it 1$

therefore. neceee..ary t'h$t: the corresponding qua:acum ~u111;bers bee.ome

i<>finite. Now in the 1~m~t for Infinitely large L-v.ol,,eo it cad be

ahown that:

(L UF2 0 oir 0) « (L LH 0 011 0) (VI.5)

Thus lt ~• j..,_stified to limit out'. <:d<:,,lati<>ds to the trsnsit!.ono

L .. L' (•L±I).

-170-

Including the abov~ t:i;i.gl'oedients, we computed the croaa $r;Ctlons,

OTIC for ea~h e~~~tatiQn radius r1

• lf we compare ea¢h One of them

with r.he q\l.m"ltummech.an!c.al cQm.putat ions for t.he corC"espondidfj

sc:iparate tr~t'11$lt.lon d€nsit1ea~ 'IN'~ do not observe ~tty tee;0mblanc.e

whatso~v~~. Adding them togcthe~, however, ~~~htcd with th~

~~plitudes {a1

\ o9<~coponding to the collecti\'e mod•1, tive• a

reasoaable: 1Hmilat"ity with the quantummechan.:teat '('C:iji\.llts (figure:

VI .3). Three main 1;1..Elpccta are repro41,1ci::::d CQrr.ec.tly:

1) the absolute magnltude of the crou section wieh<>"t auy

addlUonel scalias;

2) the location of che first ~ximum, both fo~ I•2 and I~3

e:ic~1tatfona;

3) the ove('all alope.

'Thes-= rersults encouraged u,e, t:Q perform an ii'lver1UQU..

Vt-3 A, classical in\'eruon

Sl .. H<'dy ae in the quantum...,chenical case, th" 1n<!la@Uc

.9ca.ttering is de.serlbed in first order pel:."t1,1.t"bat:f.on wh.ereaa the

elasttc 9Mttering is treated exactly. Thie causes the tuv"to:lon

procedure 1rt tlrn inelaat1e $.C.S.ttet:'!.TIS to be essentially le$$

C:QmpJ..:f..c~ted th.an in the el"-at1c case~ For an i11trod1,1ct1on to the

inverse si::..a.tterlng p:i;-oblcm for c1asa1e. e1t1$t~-c $C~tter1ng we refer

to the w~rk. "~ r.a.-rnby (LAZ80).

Fo~ QU~ inelastic scattering p~Oblem the expression ana1o~o~$

tQ n~~.n reads:

(VI,6)

~171-

Thl6 set of equatio~6 ead b• solved without additlonol

linearizatlon by melu'JS Qf the singular value method .. A,e, ~ praet.1eal

ease «e took aS,ai<> the reaction se1'!l.(p,p 1 )Ot .. zt- It i6 idteresting

to t'iote. that we enc:oo"ter soine problems with the m.oet fl'it~1..phe-:•l

part: of t:he form factor, juGt <'6 tn the q114ntu,.0>e<hanieal case. ln

Q~~er to arrive at a ~ell-b-eb..a.ved .tlQl1,1tion we applied the &a.me

phyolcd daqping recipe as uMd ill the prev10d8 ,,),.,pters. Unlike the

quantummechanica.1 ¢.aee, we .a.lso found a large ampllt1,1.de for the moat

centr•l ai. The phyoical req~ire,.ent th4t the transition density

ElhOuld vati.1sh fo'C r + 0 is le.sa atl"lne;en.t t.ho!lr'l the vanishing outside

the nucleus. Nevertheless, w1 th the ehoUcol ides. of quadrupole

defonnatiOn iti ID.11'1.d~ one can justify th.at, if ni:Cel}~~ry, a.leo a

central damping could be <tl'i>Hed. When doing so, the results no

depicted in Hgure Vl.4 show up. Thia form factor ehould be eQ,.pared

with the quant1>mmaoh.,_..,~¢d re•>llt dso in the figure, It. qud~t'1t~ve

rc.embb.toee between these io the t.,i,..-peo.ked otrueture; the

locations of the p~~ks; however, differ~ Sine~ .. ieo the

:i:-e:pro-i;lu.<::tioa of the expitt'imen,~.p.:t, dttt..l;l h.l;ls not been improved

aub0tal\H.,.l1y .;oa..pared with the colteettve .. o<!el des<:'ription, we

h:a11e to conclude th.at the e1;;1.1;u;lic.1;1.1 deseriptlon lac.kB accuracy fot

pr,•¢tio:oa1 iaveraion calculat~ons.

-l7Z-

l. 5

LO

.5

- .5

-LO

-1.5

a

66zn (rea11

o• 1-z+ 1 beta •0.2>

10

1.0

0.1

b

4 6

0 30 60

66zn lrealJ

o·f i·2• I

b~ta • O. 25

r(frnl

90

..&c.rn.

66zn (p, p') 20.4 MeV

2\ (1. 04)

120 150 180

'';,:gT V l. 4. U.;:.~u (_ ·{;~ f'Y'Om (f ~ l,..(1$e·if?t:i l .,:n·l)r:?'J"i;;·i.or1. ~?<'.+ ?-o·f.1'.lo:tion for> the reac-tion 66 znlp,p ')2+1' Ths j'u/.1. auross i'.n VI.4a r•eprnG<mt the ·i.,.·wet'ted

f'orm. j'actor:J. Left: wi"t:lw1-'t ~C!n.t:r<al. damping~ r1:aht; 1,.r£t.-~~ (N;'t!.l;r<:1l.,

In e~per1mel"lt:at QU<:J.t~t' physics it h.e..:i; been ~~on pract.ice to

per~ot'W. a theoretic:s.l ana.lys!a of the results of i"O.el..f.fitf..c

ooatt.,ring eKp.,ri..,,.,ts by <;1¢~1'5 modd cakulation• o.1ly. In this

work we h.a11e de'V'elo.,-ed an .alternative method :fot these analyses

whi<.h, alJ.o<>s the exp<>rimentalist n<:>~ only to compare e);periiR<ital

c'tOfiB sections and an.a.lyi;i;f.tri.g powers with theorei:.1cal results,. but to

perform th1& comparison also on the leV<!l of rwelear traMition

d"nsities. This ""'thod has been derived 1n a VWBA conte>:~ Wllere th"

inverse inelastie seattcring problem r.,d.,ce9 to the solution of a

set o~ '1"ild~at1<: equations. The latter is aeeompl!.sh"d in an

Herative "'"7 by ""'ans oe 11 $!ngular value aaalys[a. In order t<:>

s11ar.)t1~ee $table solutions~ ~tt.Cnt1on had tQ 'be paid to th~

developm~nt of feasible damJ>1<18 devices,

The method ha• be~n applied to several ol ~t excitations in the region of A • 54 - 70. as well as to or• 3i: ""dtauo,,_~ for

106cd and llOcd. In all cases '"' 1ioprovement of the o;!e3cripdon of

the data could be atUit>ed; tile corresponding x2 reduction faet.:ir

was typiediy of th• order of 3. The re$ulting tranait1<>" densities

have been int.r;;t:'p'('et-ed in tertl1!3; u£ t.hr:: wJ.croac.opic. model. 'Io this aim

6Chema.t1c mfcros~op:t(,! e"'t-culatiODB for t:hl!: t:ta11eition del"lsit:ie$ Qf

tlic reactions 54 Fe(]'l',p'nt and 58 NiCt,J1')2! have been perfonnad. A

satisfyi11:.S 4S'='C::Cment between theij:~ .and the inversion. .results ls

ob~ervcd.

l" o<de£ to get an 1de<' of th• applico.bi11ty Of the metho,! fQi:'

other types of react1on3, "" performed Ql)me pilot st\ld1es of

1ne18stie alphn scattering an.d. O'IJe neutron tra,(18fotl' reaction;i!J. It

appeared that in both eases the performance of tM inversion method

was somewhat less successful.

Next we gener.:!lJ.t:ed the method so S.6 to deal with se.eQ~d order

DWBA cou!'llng sch.,m..s. W" i>erfo(med inversion <:alculations for the

2-phonon trip~ets of 64

•66

•68

zn. A seneral tendency of tM ~esultlng t<ans1tion densities h, that th" 2r + 1+ o-o~, 2,, or 41)

tr.i:u1,sltlo-na are overeiJ.tlmatcd by the c:ollecc1ve model.

-174-

Samel"l.votttting

Hct doel van vel"strooiit1.gs~xperlmentert ~s het verg:r9ten

v~n de ket'l.nis en her:: bcgrip vail ke:rnstructt1t,1)," en reacti.e­

ra.ec::.h.~aism.en .. aee doel Vall. het onderhaV't,g..:: "'l'erk ill de2'.t COIJtext

is het aandra.g~~ van eetl methode dool;' middel waarv.an de

:tC19.ultatea Yan een ex:~el:"1ment op ee~ .teer dire(.t-t;!: wijze kunnen

war.den geanaly$r.::otrd om. zo :;io;ekere eigen.e;chappen vai;:a de b0Bcho11wde

kernr<:actie vast te stellen. lliarbij wordt de aandacht geveetisd

op d" ~ogenaamde ovetgans~dtehtheid (tra<1eition den~Hy) van

de-te res.~eie:.

D~ nooh~.-1< van zotn methode: woNt uitee11.ge.;c:et in

hoof,!3t.,~ I, 64men. met de ben<>d1gde "(C:actietheorie. \let

hoofdstuk efod1st met een onderzoek "~~r d<> gevoeligtidd van

bepaalde reactieiJ VOOI' zek~ ),"C eigenechi:tppcn van de

overganssd1chtheden.

Hoofdstuk II lo gewijd aa<I de u1tbre~~ing van een

teM;.enprogra.mma voor Vt;;t"1)t.('oolitl.gsb.erekeningen om de .Qpf.n ..... bsan

l;o-ppelin.g t.e ku.nnen ve.rd131:l.!'i1contere.1:1T D1t hoofdstuk kan evew;'lt:"IAeel

worden overgealag~n in ee11. eel;"~te globs.le le21ng.

De methode ~~~ aangekoad13d in hcQf4't~~ wordt

gepreeentee<d in hoof dot~~ III, Er wo>:"dt aandacht beote~d "'m de

relevante ra.a.thema.tische en numerieke cigenschappea v~n de

"'dliode.

1.n hoofdstuk :r.v wo"'°dt de met.hQ(!c toegepast op ~nkele

eenvoudise E~nataps l:'eacties. <eaulterende

overg:.r,l.n.gsdichthede11 Wol:den geinterpre:ccerd in tet:i:Q.en van bet

kernfys1a~he schillenmO<lel.

Voor e11.kor::le meer gec.::iri;i.pliceerde reac.t1ca worden de

procedufo en de reeuH<>ten gegeven i" hoo£dstul< V, De geldlghdd

van de methode VOQ-:' de1;e reactiee 15 nagegaan met beh1,.1lp van een

zogeo•~m4e pseudo .... d4to. bereken1ng.

Hoofdot~~ VI beschrijf t ~en volledlg a~dc<~ oenaderins v~"

het problee~ v~n de extractie van overgangsd1¢htheQen u1t

expei:-lmentele ge$eV4=:"1J.&+ Hier vordt oaderzoc.ht in. h.Q-everre de

klaaaieke: verstr-ooiin&r;U:hr;oi:-1..e kan dienen OJcl. hct probleem op t~

loasea.

-175-

Literat1,1.rn

'1.t.071 Alonao,l'!. and F11111,11.J., Fu.Wamental Ua1vers.1ty Fl;iyalca

Vol- 1- Mechanto•, Add hon Wesley, 1967. (dut¢h

translation 1971)

AMA84 Amado,R,I>. and Sparrow,O,A., i'hys. Rev. C~ (1984) 932

AM078 -"'mos,K,, Fae:5s1er,A .. ,. 'Morlsson,r .. ,. Smith,A. . .and

MUther,ll., !fuel. Phys. A.304 (1978) 191

Af\I76 Ar1,.~.A. all<i ~ac'1ello,F., Ann. Phy~.~ (1976) 253

ARI77 Ar1i;aa~A·~ Otsl,]k.~,l·~ lach~llo,F+ and Talmi~I-~ ~hys ...

Lett. B~ (l 977) 205

A.RI78 Ari~,'1.· and I.aehello,F., Ann· Phys • .J:.!.!. (1978) 201

AUS70 Auetern,N., 01rMt Nueleu· Reaction Theorie~. Wiley,

~""' York, l'FO

UC69 lleeeh~tti, F.D. <Ind Greenlee.i.,G.W.,

Phys. Rev. J!!. (1969) 1190

SJ::R79 Berr.~to:in,R.11., Atom-Molee\ile Collision Theory, Plenum

f'ras.ss.t New !fo-.k, 1979

BL082. :Slok,R .. P+:. Kraushaar~J.J., B.at.a.7-Cfmrbs.,P.A,,. and

CecU,F .<;., Nucl. !'bys. A386 (l98Z) 61

BllU7 Briev.,,F. and Rook,J., Nucl. Phys. ~ (1977) 317

SR057 llrow"4,G·E•, Proc. Phys. $¢c, A70 (1957) JtH

BR077 Bruasaard,P ,J. <lnd Glaudemans,P.W.M., Shell Hodd

Applies.Hons In N~otear SpectrO$copy, No,th Holla"~•

A.meterdam, 1977

-~76-

DAH74 Uahlqui.st,G .. and Bjifrek,A.,. NumeI"ical Methods, Prentice

H<1ll, 1974

DRF,74 ntch<><,"·· Frtcd<kh,.J., M"~le,i:., Rothh:w8,!i. and

LUhr$,G •• N,.cl. Phy$. 11.n~ (1?74) 219

F'flI67 Fricke~M.P.,. Gross ~E. and Zucker~A. 1

Phys. Rev. 163 (1967) ll53

FR08l Frnis, B., l'lucl. Phys, A358 (l 981) 383 C

GER7l Von Ge<amb,H. and Amoa,K., Nucl. Phys. Al63 (1971) 337

GER80 Von Ge<amb 1H· 1 Internal Report Hamburg University 1980

GLA67 Glr!shausaer~C., De Swiniar.ski,R., Thir1on,J. and

\Hll,A.o., ~hya. R•"· _!.§.!. (1967) 1437

HAL73 Vl}.n na.11,P.J., Proeeedio.gs ol the r.nternational

Conference on Nu~lear Physics, MUnchen 197), p.417

RAL80 Van Hall ,P .J., Proceedings of the fifth 1atc'("na.tionel

Sytnpos1um on Polariz~t1on Phenol'lle;na in Nuclear Physics,

Santa Ff (Conference Proceedings nr. 69)~ New York \981

-p ... 508: Giant resonance analysis of the teaction

54Fe(p,p' )•

-p.514: t~ai~e[Jn~0 of polarized p.otona from

' ' ' Zn·

HAL82 Vei;n H.i:t1.~.~-~J • .P.nd D¢~.P.t:'~e~u .. M. 1 t-nt:el"o.al 8:.eport E.U.T. 7

1982

HAL84 V~n Ha11,P.J • .lind Bhai:~P.C. 1 priV'ate iC:.Ortli!l:U.Ilic.ation~

l9B4

-177-

HAN&5 H'a~dbook of Mathem.,tical Functions, with FQrmula.a 1

Graphs and Mathematical Tables, edited by Abramowitz,M,

~"d Stea:"'" ! .A., !>Qver 1%5

HEE.SO Van Heea ,'A .. G .M., Private communication of Polane,.J .H .. ,

Thea1s 1 E.u.~. 1 1981

H0D71 Hodg:a-oti.,P.~ .,, Nucl~ar i~4(':C.iont.i .i:tnd N1,I.cle"1t S.t~1,1ctu~e::i­

Cluen~o" 1'ress, Oxford, 1971

IG058 lgo,G., Phys. Rev. Lett. 1 (1956) 72

JOH70 Johnson,R·C· and Soper,P.J.R., Phya. Rev. Cl (1970) ~76

K:0077 lCoops,J.~. and Ghudemans,P.w.M.,

z. Poy$. A280 (1977) u;n

Kl,!N99 K.,u,P.D., u"~ver$~ty of Color.do, 1969, unpublished

LAW74 Lawaon""C.L. and Haason,R.J. 1 Solvi:ng Lea.s~ S~u"'9.t;"el$

Probleml$:11 Pre"'-tiee H.,;ill, New Jc'('Sey, 1974

LA.Z80 Lozenby,J., Am. J. Phys.~ (1980) 432

LOV67 Love~W+G+ and Satcbler 1G .. R.,

Nucl. Phys. A101 (1967) 424

LOV70 Love ,W .G. and Satchler,G.R ••

Nuel. Phys. A.!12_ (1970) L

MAAS2 ·Maas,F.S.M., Internal Report E.U.T., 1982

MAC82 Mackinfosh,R.S. and Koboa 1A.M.,

Phys. Lett. Bll6 (19&:>). 95

MEL78 Melssen,J.J>,M.G., Thesh E.tl.T. 1978

~178-

NJ>ll77 Neuhausen,ll., lluel. Phys. A1£ (1977) 125

ITT1C78 Nuclear !lat"- $beets ~ ( 1918) 20

OVE63 Van Overveld,C.w.A.M. and Van Hall,~.J.,

J. Phy$. 0; Nucl, Phys., 10 (1984) 193

OST79 ostet""feld~F., Wambac'b 1 J. 1 l..ensk:e.,a. and Speth,J .. ,

Nuel. Phys. AB! (1979) 45

PAN69 Pa!ldharip,.nde,v.lt., Nuel. Phys. Al3:; (1%9) 419

l'li:l!.64 Perey,F.G. a<ld S<'><O,.,,o.s., Phys, Lett. !£. (1964) 107

PET70 Peero~tch,F.L., Thesis Mieh18an Stnte llniveroity, l970

P&T85 Schuller-Petit,R.M.A.L., The~i• ~.u.r., to be

pul;>lUhe<I, 1985

?OLSl Pol~nc,J.8., Thesis E.U.T. 19al

l!.AY79 ltaynal,J., Ser~tee de Physique Theorique, CEN-Saelay

9U91, Gif-sur-Yvette, CEDEX, 1979

ROS53 Ito•• ,M. E.. Phys. Rev. !!:.. ( 1.953) 610

SAT63 Satchler,G.R., Nud. Phy •. ~ (1963)

SCH8l Sch~,,;ntke"'(" 1 Q.,. DawsontJ•t l.obh,J.~ ltei1Jen'bet.",g 1 J.,

J .. lchtcnstadttJ•t :Papan!cotas~C.N'.,. W:;l,A;e,J. 1

M¢.Cartney 1 J.s., Van Oer 8ijl,L~T1 .and SloktH.f'.,

Phys. Rev. Lett. 2Q. (1983) l5

-179-

5Hl!68 She.riff,!!. and De. S«1niarsk1,R.,

Phys. Lett. B~ (1968) 96

$HE69 $herlff,H., NuoJ.. l'tiys. f._!E_ (1969) 532

$Hf70 Sh~riff,K. «nd SlaH,J .s., Nucl. Phys. Al40 (1970) 33

$lC74 Skk.,l., Nucl. Phys. Al!!. (1974) 509

S~N73 Sintia,R., Nucl. Phys. A203 (1973) 473

ST080 Stoer,J. and Bulirsch,R., lntroductiot1 to N.,,.eriod

At1alysis, Springer, New York, 1960

STRa2 s''~"s"'~·~· •rnd J.ipparini,E·,

Phys. Lett. Bl!Z_ (1982) 141

!.ABS~ Tsbet,J.p., Bonin"'B~~ Boudard~A.,, Chs.ul'll:eaux,A.~

CQ\l.vt:::re"'~ •• 'f,;fi1cud1~~J·L+.,. Fa.rvaquetL+t Garc.on,M .. ,

Legrand, D ·, Lugol ,J .c •, Nefkens, B. 1 Saudl.rtos 1 J. and

'l:e>:not1,Y., Nud. Phr•• A386 (1982) 552

n.M65 Tamu\"a,'J:., Rav. Mod. Phys., 37 (1965) 679

V!l77 Veltkam~,,G.w. ~ ~nd Ceurts~A .. J., Numerieke Methoden

(dutch), Lecture Note• E.U.T., 1977

VER72 Vcrlinar, ! .J., Hennans ,W .c. and Oberoki,J.,

Nuel. Phyo. Afil (1972) 379

WAR82 Waroquier,M.E.L., Theolo, Ge.it St.,te T.Jniversity,

1982 (dutcb)

UAS82 llaosenMr,s.n., Thc$U, F,.u.r., 1982

-180-

Lt:ven5loop

8 november l957 ge'bq.rei;i, tc ~oolSlend.a..al

mei 1975 einde~a~en ~thenoum John F· Kennedy

te D<:>ngen

februari 1981

april 1981-

maart 1983

1n,e:eulcut"sexamen technische 11.a.tuurk'.ln.:;1.e

{met lof) aan de T .H. Eindho~eto.

wete~~ch~pp@lijk medewerker in FOM~diense

in de groep experimentele kerdfy~l~a v~n de

T.H. Eindhoven~

-181-

STELLINGEN

behorend bij het proefschrift van

C.W.A.M. v. Overveld

Eindhoven, 29 maart 1985

St~Uingeo, b•h•H•rn;!., l>lj het prnefs<hrift vno c.w.A.H. van Over•,,id.

l) Hct ie mogelijk om u1c. gcm.eten differenti~le wet'kz.alJle <loorsneden

en a'1alyser=l'ld~ vennogena van inel~Gt~Gche protoa veratr~(J.UQg

1nfQtJ:11.atie te verkrijgeo'I ¢v¢r het inwendige gedeelte "€1-JJ ~c nuc.leaire

overgangs.di..::htheld 1 7-elfs al bij energie~n tond 20 MeV+

(Oit proefs<hr1f t, hoofdstuk I, III en Iv)

2) Het h1 niet mog.elijk Om eC11 nue.lealre overg~ng$ oFeJ;"atol" te

.s.asac::ll!'fen met eien v-orinfaetor :i:ot.tdct daarbij rekeri.11"1.,g t~ h¢a.uicr.i mE-t de

l;"l.limti:li_jke struc:tuur w;.l"I. de k.eJ:"ntoestanden w;::i.artt,1.$$e11 de!2'.:e operator

werkt..

(JJit proefs<hrifc, hoofdstuk III en IV)

3) fo de klass!eke timlot (h nadert naar aul) io hct ter<>cht OOl bij

inel.a.1!1t18(:hc vei:-6tr-ooiing van een d.eettjc.sb.e.an met iinpulsmc::ui:i.ent 9.. n.sar

"°"" de,.ltjeabaan met 1mpulsmomnt ~' uitslu1tead de bljdragen ""t

t ~-f.!). mm?-e te '1emet1 Wl!arb1.j .\ het over,gedragen itt1p1,.1lsmoment is~

([)it l'N~holirlft, hoofdatuk VI)

4) In de lokale ener,g~e be11ad-ering voor exchai'l,ge wo~dt aangenomen dat

de V" '"' "a ~p~ntoren !a de u1t4r~kk~pg v<>o< q< verdw!jaenl), De

{1rg.umentst le hiervoo.:- eta[ de gemiddelde iinpuls v~n de gabonden

~o~<;Unden gelijk ia ... o 0 ~s <>ntoueikend.

1 )Van Hall 11 procet:::df.ngs of th-e lnte;roationa.1 confeI:"enc-e on

auclea,- pbplcs, M~achen 1973, pag. 417.

5) voor het beproeven van kernmodellen ~ijn !l'>e1;)..i.t1aohe

deeltjesverstrooiinga. e1'p1;t:l,m.eRtt:l"I i;,1.ci:n ~1; bt;-ve1e:t1. bov.eo (nt"f)

res.c.tiesf

5) De argumentatie van Wu ieri. OblD\ll".\I voot: hei;: fet~ d.~t bij het beataan

van m dUorHe ""er11;~e e111;e<>'1;).,i.rde.i in een potentiaal met ko..-te

drac.ht" deze potentiaal a.li:ic.hts tot op eien ""'illei.:;e\1r1se ;ut1.ea.1re

<:oml>1ftBtie V"" m lineair onaJ'ho.nkelijke baaiaf.unctiea na bepaald kan

wordeni ls onjuist T

(Wu .aTI<I Obm1,l.l."~, Q1,1o1'.1\.tum· "theory .of Se..et:l:'.er!ng, Prent!c.e HSll)

TJnUe<l Kiftgdoto., 1"962, PBS· 106)

7) De na.t1m t$1tJ.g;te shell Dl.l).tr1~ eleme11t' uit bet boek vau Brussaard

en Gl.av.d~1:iurn,.$ k;J.~ bci;:~l," 1:1Jn: • $1,ng~.t; ¢!'b~t m~r.r1;11; ~l.em~l'-1:: 1•

(Brussaard a"d Gle.udeto.a<>s, Shell Model Applications in Nuclear

Spectroooopy, North llolland ~ubliahing Company, Amsterdam,

1977)

8) lie ineerpretatie van de waarge"o"-"" roodversehuiving in bet lieht

van verafgelegen $terrcn lrt i:cr-men van ~etl 1,1itdi.je1."1d heel.al

ve~onQe~$telt de afwezighe1~ v~n niet-linea!re termen in de

vergel!1kingen van Maxwe lL Deze afwezlgheid ls niet expel'imenteel

aang•HOOTIO•

9) n., ... 1j~" .,aarop de th.,orct~&chc problemen e .... ~. de

oorloga.neurosen uit de -eer.a;te 'fereldoorlog In de doofpot ~ijn ,getitopt

""<l>t VBn1'lt "'"eeMchO.p$ .. oChQd(ll(l$Ucli &t1>ndp""e """ tV1j£e1MhC1S

lioht: op de totsu,.dk<>,.ing van de dleptepsyohol<>gie io. bet begin v.e.n

de t:w1nt11,i:$tc "'"'"'·

(J .K. v,,n de Bera, O!eptep$ycho1og1e, 6e dr"k, C.e.1lenb.e.oh,

Nijkerk, pag. 216-217)

10) U1t h"t foit dat volgens de rekMk.unMge axioma 'a geldt dat

\00.000.000 + 100.000.000 ~ 200.000.GOO volgt, ~•l<enlng houdende Cllet

de resultoteo ""-" <lo cd•tlv1te!tstheorie, n!et dot !00.000.000m/s +

lQQ.QOO.OOOm/a a 200.000.000m/s. Daarom is het olet o P<lo<~ <I!Ogelijk

om te zeggen hoe""el 100.000.()00 achapen + 100 .000 .000 scllapon ls.

l l) In discussies OY'!!!:r de toepa.sbaarheid van "pergol"J.~1" coinp1..1tf!:'.t'.S

versus mainframe& (,gn;ll;c computel"s) !a t.e~hn!$ch""'wetenschsppelijke

probl-em.t?n vci:-diia:nt het .a..atlbevellr'tS. hct. spreekwoord "wie het k:t~ine

nie~ ee'l."t .f..si hiet grote 111.et w~i:=:t'Q" :l.n acht te nemen ..