Transfer reactions on 58Ni and 56Fe induced by polarized
protons of 25 MeV
Polane, J.H.
DOI:
10.6100/IR157606
Published: 01/01/1981
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Citation for published version (APA):
Polane, J. H. (1981). Transfer reactions on 58Ni and 56Fe induced by polarized protons of 25 MeV Eindhoven:
Technische Hogeschool Eindhoven DOI: 10.6100/IR157606
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TRANSFER REACTIONS ON 58Ni AND 56 Fe
INDUCED BY
POLARIZED PROTONS OF 25 MeV
PROEFSCHRIFT
TER VERKRI.JGING VAN D~ GRAAD VAN DOCrOI'< IN DE
TeCHNISCHE WET~NSCHAPPEN AAN 0" TECHNI~CHE
HOGE5CHOOl EOINOHOVEN, OP GEZAG VAN DE
RECTOR MAGNIFICU5, PROF, II'<. J. ERKELENS, VOOI'<
EEN COMMISSIE AANGEWEZEN DOOR HET COllEGE;
VAN DEKANEN IN HET oPEN6AAR TE VEROEDIGEN OP
VRIJDAG 5 JUNI 1981 IE 16.00 UUR
0001'<
JOHANNES HENDRIKUS POLANE
GE90l'-:EN TE AMSTEFlOAM
~188E~r""'I':: O~IJt<,tI;I!RIJ
. .Ibro
""",>AQHI;l T~~~.QQN O~g20-1ml
DIT PROF,!,,,CHRIl"T IS GOEDGllKllURl)
DOOR DE PR(lMO'l'(!Rb:N
1?R01",DH, O.J,
POPPflMA
,:N
1'110,' . DH, B, ,l, VJlRlil\AR
CO-PR()M()'.l'()l~
DR,
P,J. VAN HALL
CONTENTS
CHAPTEI\
I~TRODUCTION
CHAPTEH 2
EXPEl',~MBNTAL
PROCEDURE AND DATA REDUCTION
2.1
Beam
2.2
The telescope system
2.3
Tha elect,onic equipment
p,od~ction
and scattering chamber
8
References
CHAi>'l'ER 3
11
SOME ASPECTS OF THE THEOR¥ OF TRANSFER REAC'nONS
13
3.1
DWBA theory
13
3.2
One-neutron trangier
16
3.3
TW"O-neutron transfer
19
3,4
The optical-model
20
3,5
Finite-range and non-locality
3,6
Adiabatic deuteron
potenti~l
20
correction~
potenti~l
21
23
Ref~ren~es
CHAPTER 4
THE (p,d) ReACT"QN ON SSNi AT 24.6 MeV
25
4.1
Introd~ction
25
4.2
Tha experiment
26
4.3
The reaction model and the calculational procedures 27
4.4
4.3.1
Reaction model
27
4.3.2
Th.e neutron fOJ::'m factor
29
4.3.3
Optical potentials
4.4.1
4.5
CHAPTER 5
TB:E
31
DiScussion
33
The pa,ticl,e-states 3/2
(0.00) , $/2
(O.77J
and 1/2
(1.
11)
33
4.4.2
The 7/2
states
36
4.4.:)
The optical deuteron-potential
39
4.4.4
The 3/2
42
states
4.4.5
1'he states at 3.71 and 5.78 MeV
45
4.4.6
'l'he 1"0 and 1=2 states
46
4.4.7
SUl1}
rules
17
Conclusions
49
Reference~
51
+
(p,d) REl!.CTrON ON 56 Fe AT 24.6 MeV
So
5.1
Introduction
53
5.2
l'he experiment
53
5 ..j
The calculations
[,. :3.
1
:'. :1. 2
5.4
5.5
55
l1.e;oction moclel
55
Optical-model and other DWBA parameters
57
Discus~ion
59
,.4.1
The negar.ive-parity states
59
s.4.2
The posit.ive-parity states
64
:;.4.J
Sum-rule results and compc:t.r$,son with
5eN~ (p, d)
65
5.4.4
The deuteron potencial
67
68
cone h\~ion"
70
RCference$
CHAPTER"
TlH;
(p,r.) REAl":TION ON 58 Ni
6.1
Tnt)::oduction
6.2
Experiment~l
6.3
rJ.4
5~ve r-:r 24.6 MeV
71
procedures
"12
Ca.1Cu.liOlr.ion<>l procedures
73
i<eElction m<O'ch"nisms
n
G.J.2
simultaD<O'OIJS r.wo-neute-on transfer
74
6.3.J
SequentiEll transfer
76
6.3.4
Inelastic sc"ttering
78
6.1.5
Optical potentials
79
6.3.6
Normalization parameters
flO
Discussion
81
6.4.1
Detormination of A and
6.4.2
Tho:
:1
Il~he
6.4.4
2t
al~d
.t-
2~
W
stat_es of o'Fe
+
Oz stOlte of
5~'Fe
DWBA analysis
Conclusions
References
CHAPTER 1
THE (p,~) REACTlO~ ON 58 Ni AT 24.6 MeV
·1.
,SUMMAR";
1
71
1.,.3.1
6.1.
6.5
A\'ID
In trOde<Ci;ion
81
88
92
93
96
97
99
99
7.2
Re~ction
7.3
Disc'L1.ssion
"1.4
condu~ion~
lU4
Heferenc8s
105
model
99
101
107
1(J~
NIIWOO!<D
111
!,IWI':Nf;T,OOP
112
CHAPTER 1
INTRODUCTION
Nuclea~ t.an~fer-r~actions
a small number of nucleons
are
char~cterized
(protons or
n~utrons)
by the transfer of
from one atomic
nucleus to another. In this thesis we present the results of various
types of
reactions induced by
tr~nEfer
foils consisting of
bomb~rding
S6Ni and 56pe nuclei, respectively, with a beam of polarized protons
having an
The
ene~9Y
ta~get
of 24.6 MeV.
foas are w thin that the p;(ob"bil.l,ty of meeting
mOre than one nucleus is very small for the protons.
most of the
p~otons
~onsequently
go through the target foil without feeling
~ny
nuclear interaction at all. Of the protons that do encounter a
nucleus, most are sCattered (in)elastically and leave the target as
a proton
~gain~
In our experimetlts we selected those
~~~ct~ons,
in
""hich the proton "changes" into a deuteron, a t);"iton 0;( an alpha
particle by picking up from the target
n~cleu~ ~
neutron, a pair of
neutrons Or two neutrons plUE one proton, res.,ect.ively. The firll11
nucleus is often left in an excited state by such reactions.
Counting the number of deuterons,
t~itons
em1tted in a certain oirection and mSdsuring
and alpha particles
thei~
called energy-spectra. These 5pectra reveal peaks
are cantered around discrete
levels of the final nuclei.
represent (after proper
Which
value~),
Th~
corr~spond
energ~es
to the ene);"gy
number of counts in these peaks
~ormalization}
the various diffarential cross
sections. In fact, because the protons are
ditf.e;(entia' cross sections fa.
energy yields so(ife+ the
e~ch
polari2~d,
we have two
level: one for polarization up
and one for polarization down. It is convenient to
);"epl~ce
these two
cross sections by two quantities which are Lndependent of the
of
pol~rization
do/o~unpOl
and the
analy~~ng
power A given by
do (6) /dllunpol = 1/2 (do/dilt + do/dllt)
A(B)
d~gree
P. These quantities are the unpo!arized CrOSs section
1
(do/dllt - do/dOf)
-p x
(do/dnt + d%ll+)
(1.1 )
(1.2)
where B is the angle bet""ecn the emitted particle and the proton beam.
Relation
(I.~)
is only
v~11d
for
p~rticle~
emitted in the
ho~izont~l
plane and to the left of the proton beam direction. To be more precise
A(O), dC1(O)/,:m\.lnPOl an') th", di£"fe,ential cross section in the
dtrectiol1 (O,t) aye related by
., +
do(O,tJ!dn ~ (I + P'n A(O)} dC(8)/dOunpol
+
wht;!-re n
~,s
t.he uni t
ve~to;r; co~:t"esponding
1.nJ,tiaJ. ano fInal Ught.-part.tcle momentum
(1.3)
to the vector product of t.he
(B,,~el
convention).
S1)ch a stlJ8y o[ nl)clea"(, reactions yields information On the
reaction mechanisms and on the internal structure of the
volv~d.
Both aspects are investigated
FOr irlstomce
Orle"
ext~nsivoly
nucl~l
in-
in the present
and two-neutron tnlrlsfer rc"ctions prob", the
~ork.
~ingle­
neutron "nd n0utroJI-pairir1g properties, respectively, of the jnitial
and tin,tl. nuclei. cono0rning the reaction mechan.l.sm of e. g. twon<'!ut.ron transfer re<letions one may investigate the relative importance
of two $ucce.ssive transfers of one neutron and
th~
si.multaneous
transfer of " neutron pair. The polarization of the protons orten
prov~s
to be crucial in answering questions about both
~uclear
strUGture and reaction mechanisms.
The choicu of 5BNi il.r'ld 56 pe as target nuclei was xnotivated by
two fucos. First.ly, 5C Ni and 56Fe Itre close to the doubly-magic
nucleu.~ ooNi. whi.ch means that their internOl.l st);ucture is rel<J.tivcly
simple within the conte-xt of the :::;hell . . .moilel .. Secondly, we expectOd
similar.ities o,f the neutron ..... tr,ans[er. data since 58 Ni and S6 F0 oontatn
the same nl,lmber of neutrons.
The experimental set up is described in thG following ohapter,
while we give an outli.ne of the theory in chapter :J. 'l'he experimental
r~sults
are presonted and discussed in the remaining ch"pters. The
r8f.ercnces are given at the erld of each clutptcr.
2
CH~PTER
EXPERIMENTAL
P~OCEDURE ~ND
th~~ the~1s
The experiments described in
of a polarized proton beam
DATA REDUCTION
acceler~ted
have been induced by means
to 24.6 MeV by the AVF cyolotron
of the Eindhoven University of Technology 1). In section 2.1 we give
an outline of the production of this polarized proton beam and we
describe the arrangement of the
chamber, where the nuclear
~catt~ring
reactions take place. In section 2.2 we discuss how the outqoinq
are identified as protons, deuterons, tritons and alpha
p~rt1cles
particles by the telescope device. Finally we
of the electronic circuitry. More
det~ils
~ketch
the main featur8s
on the experimental set UP
and the spectrum analysis can be fOund in the refs. 2 and 3.
2.1
Beam production and scattering chamber
The polarized protons were
~upplieo ~y
a source Of the atomic
b~am type 4,5). which has been developed by van der Heide 6). The
intensity, enerqy and degree of polarization of the proton beam were
2 to 4
~A,
injected
5 kev ano about
r~dially
80~,
respectively. This proton beam was
into the center Of the cyclotron with a trochoidal
injection system, which is a oopy of the one used ~n saclay 7). The
Lorent~
force dcting on the protons during injection was balanced by
an electrostatioal force generated by a system of electrOdes. The
shape ot the electrical field was such that in combination with the
magnetical field a strong focussing was achieved resulting in a
transmission efficiency of about
70~.
only a small fraction of the injected
cu~rent
is actually
accelerated by the cyclotron. During our experiment the
an energy of about 80 keV per.
~evolution,
50
the
revolutions was about 300. The intensity Of the
10 to 20 nA with an energy
~pread
After extraction the beam was
tot~l
proton~
gained
number of
eKt~acteo
beam was
of 60 to 90 keV.
tr~nsported
chamber by a system compQsed of five bending
to the scattering
magnet~,
twenty quadru-
poles and five steering magnets 2,8) oovering a distance of 40 m. In
order. to qet as muOh intensity as possible on the target we used the
doubly achromatic mooe of the beam-guiding system, which means
conservation of the energy profile of the bssm during transport.
We wGre able t.o focus t.he proton beam in a spot of les" than t;wo
mm dtamet;er at t.he tarCJet. Th;, direction of the beam was ched:ed with
probes at. the center of th" Far"day cup, which was situated twe and a
half mete.rs after the target..
The outgOitlg protons r deut.e::rons I tri teIls and alpha
J?arti~le5
were
detected by an array Of four telescopes TI-T4 (fig. 2.1; see also
section 2.2). Two di fferent configurations hav<l becn \lS"d: in the ~eNi
exper1ment the te)."$~opes we);e 10° apart, while in the 56)'e experiment
thb; waS 6
0
The angu.1.ar ,,"cceptance was in both cases about 1°.
•
Two Su.J.COI1 det-<'c\;o£S (Dl, D2) in th<l V<lrtical plane through th<l
bei:lHl "t
con5tan~
sc.,.t.Cering angles of 4S ilnd -45 degrees, re!jpectively.
IlIea..:;Ured. p.:lastj,cally sc.:a.tter.-ed protons in order t.o prov,i(le a relative
normalization of t.he va1:"io\ls runs and a clock signal for reve>:5ing the
polartzation di",->ction. The vertical plane w"" ChoSen because the
"""tt<:ring in tilc,t plane is independent
ot the d8gr<l<l of polarization
(the polarization ",-xi::; of the proton beam being vertical
tOOl
cf.
for.mula 1.3).
The deqree::: of pol.:lriL;<.:...tion of t.he.
~rQton
beam was monitored
"ontitlUously in " secood smaller scattering chamber down"tream (fig.
2.1) i,l the following way: fit"st the beam en<lrgy was degraded to a
mc"n energy of 16 MeV »y an aluminiUm foil; next thc number of protons
t=!1.aDt.ic311y scat.ter,'eti by the carbon in a thin polyct,hyl,ene foil was
m"a~\\red
hy t.wo
~iliccon
c'letectm:s (03, 04) pl"""d in the ho:dzontal
plane at. 52.5 and -52 .. 5 degrees. Since the unalysing PQwer of ~2C is
... /
I''''I~I
T'.'J'.q
"
"
'I
~
./
:>(]hematic, draun:ng oj' the
;~r:.rt7~on
mor'!.",?: tor_
8c:att;$t'-[.1I.(J
dli:ll1iber and
polar'~:­
ratner energy-independent at
tni~
ang"e, one
determine in tnis
~an
way a reliable value for the polarization degree of the be.!!.!".
2.2
The telescope sYijtem
In fig. 2.2 we give a
cross~se~tional
view of the detector block
used in the S6 Fe experiment. Bach of the four telescopes consists of
two silicon surfaOe-barrier detectors: a 200 urn
and a 2 mm E-deteotor
teles~opes
(O~TEC).
sweep away the
target. During the
5B Ni
~E-detector
(Philips)
Permanent magnets in front of the
se~ondary
electrons arriving from the
experiment w~ did not yet have this detector
blOck at Our disposal and we had to mount each
tel~~cope
on
~
separate
holder. Both experiments were performed using the same deteetors. The
telescopes were normalized to each other by comparing meaSUrements
done at the same angle.
Toe
particl~5
are
ident~fied
by their specific energy loss dE/dx
in the ~E-det~ctor, which ~5 apprOximated by the semiempirical
rBlation 9)
_ .... -_.------_ .... ....... --,
,- ,
\
\
\
~
\
\
,
ITS\ \\\S]
tS\\\\\33l
I
\
I
~
I
~f--6.E
/
~~~~~
I
'\, \ 1- :m J: }/-'"''''
\
I
\
I
\
\
.... -
Fig. :1.2
........... _--- .....
Cross-sectional view Of ,the four>-td"'",::op$ d$t$ctor'
block u8~d in the S6 pe experiment.
5
P.ll
whe,t'e M ..
Z and .E!: are ths ffiu.SS n.umber I
cha:r:ge number and energy of the
particles, rcspcutivcly. The constants c and 0.73 dep8nd upon the
c\el;e(;~or
SlU,con
mat_eei"l
(",j
licon) and the energy rarlge (5 to 100 MeV) _ lFe:.:
c = (L045 if 1>
J,$
in MeV and" in wm.
j,g
dete~mine
In order to
che
p~rticle
type irrespective of the
ene,gy a qu"ntity PIC) ()t.a,ticle .!:.dentifier
_~utput)
is cal.cul,ated
according Co "_he algo):itl1m of GO\lloing et a1. 9)
(2.2)
~
where liB and
are the energies deposited by the particles in the IIll-
:e . . detector,
and
respectively. Assuming thEl.t the pttrticle is $topped
in the it-detector and using eq.
PlO
=
(2.1) one co.n derive
1.73 C MO• 73 Z2 d
(2.3)
where d is t:he thickness of the bE-detector. The relatiOn (2.3)
that PTO is
ener~y-indep~ndent
and allows to discriminate
VO-riOllS p"r.ticle types. 1\ct\J"lly the quantity
.hows
b~twC0n
pro d"fined by (2.2)
-----------1
el.protl)n~
n
o
C
,.
10~
Z
!
---l
I
CJ)
, I
\
\.i
\
I
"I
,
.!
!.
.... L
'(>
1"i:7, ;.'.
6
:_1
tr:!~~~
I
M(U,H'
H[.I(-)(:ir'Wn
!)l!;-
and
(I
CHANNELS
ob/:ained
,': rr~n
at;
etab
g-dRteetOl'.
/
1
\
\
i
• • 11
i6o ----f!----------sio-----'-
the
gives rise to " so-called mass spectrum (fig. 2.3), wh<:!r<:! th<:! p<:!aks
<Ire centered "round the values given by the relation (2.3). 'rhe width
of the peaks is caused by several etfects, of
inhomogenities in th<:! thickness of the
wh~ch
~~-detector
electronic noise,
and straggling in
th<:! 6E-detecto~ appear to be the most important ones 10). We Will
discuss th<:! straggling effect in SOme d@tail because it
~ets
an
intrinsic lower limit on the peak widths. Since we only want to make
here a rough estimate of the importance of the
will not use the
Vav~lov-theory
3t~aggling
effect we
11) but we will assume a Gaussidn
energy-distribution 12) of the straggled part~cle5, where the standard
deviation c is given (in units of keV) by
a=4.3Z,fct
(2.4)
with Z the charge number of the part1eles
the (silicon)
deviation
~
~E-detector
is
in
appro~imately
~m,
~nd
The change
with d the
~PIO
thie~ess
of
indue eo oy the
given oy
I~PIOI = 1.73 a ~D.73
(2.5)
combining (2.3), (2.4) and (2.5) we find
8PIO
(2.6)
1 PIO I
where 0 is given in
~m
and E in MeV.
~or
inStanC0, eq. (2.6) gives the
Values 0.07 (protons), 0.03 (deuterons), 0.02 (tritons) and 0.01
(~­
particles) for the ratio Of IIPIO and P,O, when d = 200 vm ana the
energy of the proton oeam is 2S MeV, In order to obtain the FWHM
(full-wLdth-half-maximum) one has to multiply the right-hand-side of
eq. (2.6) with 2.35, A comparison with the experimental mass-spectra
showed that in our case 50 to
70~
of the widths could be uttributed
to straggling. we also tested the relation (2.6) with maSs-spectra
obtained with different thicknesses of the liB-detector (table 2.1)
and found a semiquantitative agree1l'"aent..
From (2.6) it is clear that the resolution of the mass spectrum
gets better when the thickness d increases. On the other hand particles
that are stopped in the liE-detector are not analysed because the
particle type cannot be determined in that case. $0 the liE-detector
must not be too
thic~
in order to allow the detection of low-energy
deuterons and tritons. To meet th& two c.iteria m&ntioned aoove we
selected 200
our
~m
as the appropriate thickness Cf the
6~~detectors
in
experiments~
7
Table
CompariEon of
calc~lated
~.1
and experimental
~elative
peak-widths in the pl(J spectrum
particle type
thickness
of lIE-det
<l")
p
lOa
um
~OO VIn
500 \lm
a)
11~he
calc.
1.38
1.00
0.69
0.38
expo
1.11
1. 00
0.54
0.49
cal".
1.44
1. 00
0.675
0.32
expo
1. 43
1. 00
0.50
0.29
CCllc.
1.14
1. 00
expo
1.46
1. 00
the relative
of the (leuteron peak.
peak~width
u5C':ld to normalize
Wi:tS
a
t
proton peak th"(eatens to
the other
aw~n'\P
);'"",ult~
the deuteron
pe~k
especially
c:tt forwa.rd angles because Q,f the large number of elastically scattered
prt:ltOn:=:.
Th~rE
$C,~1;t<?-r.i.ng
valu<!s by
fore ... since we were not primarily interested in proton
- 1'02 Ghift.,d the elct<,tic purt of th" proton pea};, to lower
choo~itlq
al'\
E-detsct6r which
WilS
too thirt to stop eliJ.sti"ally
scattet'ed prot.ons but. thick enough to stop all the other pilrticles. Orte
can easily check
1;ha~
the expression (2.2) yields lower vllim,,, for PIO
when (,,,,.l,culated for partiel"" that o.re not stopped by the telescope.
In order to avcid broadening of the mass peaks by unequal
Ilmpli~
fic"L.ion of the AE- and E-signa~s we ad~ ... stGd ttl" relative ampli fication
by C'ompar.inC'l the signals i.nduced by the 5,8 Mev alpha partiel"s emittOld
by 24 4 Cm .
2 ..1
Th.e electronic equipment
Ttle main components of the eleC1;rOnj,C cLtcu!, cry used tor eaeh Of
the telescope"
aL'e
d;i,spl"yed. irt fig. 2.4. Both the /lE- and the E-
signals are fed into a
pulse~BhapinlJ
The fast discriminator
produee~
~;.Ignal
A
circui1; and a
fast-~og;i.c
cireui,t.
" lQgiciOIl puls<?- whenever the input
exceed. a p"eBet l€!vel ( .. bo'lt 0.1 mV). AfI:-<?-r conversion frOm
E
tel. bits
E
gate
part. bits
PA
energy
Di~S
preampUj"ier (CANBERRA 970D)
amp~ifi",r
~;S5L)
(LJi'S
FA
fast
FIA,
tiLter amp~ifier (l'HE)
~'tJ
j'aat di{]criminator (LRS 621AL)
,dqpt$~'
LA
level
RU
routing Im/t
MA
main ampUj"iel' (OH1't;C l8b)
(DH8
688r;)
('l'HL')
LGS,
lin.ear gat.;; stywtoh(ty, (OR1'EC
Pl
par tide id$rlUfiiJr- ('('HE)
.J1!'.)
DU
disar-iwinator uni t (THE)
PL
partiele logic (THf;)
MI
mixer (ORTE'C 16:5A)
ADC:
analog digital convertBr (NVCL. CllIC. 27850)
9
NJM
to
'!'ltL
logic this signal is pas.sen on to the routing unit. When-
eve:r- the 't'C\ltin9' unit l':"eceives simultaneously (;t"esolving time 150 ns)
" LlE- and E-signal the gates of the stretchers (LGS) <lre opened.
two telescope bits identify the telesoope in that
O<lSO.
Th~
Then tho E-
and 6E .. outputs of. the st.,retche:rs are fed into a.n analogue particle
identifier developed by Sllliters et
g€fl.0ri:ltes a PIO
al.
13). This particle identifier
(particle identifier output f
tot<ll energy signal E+6E. The
mas~
see
far.Ol1,J,l~
2~2'
and a
peaks of the )?IO spectrum are
separated by (01)r dls<'rilftJ,nat.or levels a, b, c and d in t.he
discr.iminator. unil,. whenever an event in one Of the windOws Of this
unit is 1:"ecQroea, t,he gate Of t.ho ADC is oponed and two bit£:; are:
qene"ated to la.bel t.he part.iclo type.
The IIDC bits t.ogether wah t.he telescope bits, tho p"rtide bits
and the spin bit are sent via <tn ADC controll"r 11nd " CAMAC system to
"- MOS-memory
(18 k,
memory are writ.ten
24 b.l ~~).
"1".0
IIfteI' each run the content.s of t.he MO,S-
floppy diSkS by a PDPl1 computer. FOr details
on t.l\e v<lrious contl:ol emits and the data acquisition We refer to the
):'e£s.
10
3 and 14.
RE:fJ::RENCES
N~F.
Verster. H.L.
HageaoQ~n,
J. Ceel, NUcl.lnstr.
2
J.P.M.G. Mel5sen,
J.
Zwanenburg,
A.~+~+
~~~nken
and
113,19 (1%2) 88.
the$i~,
Eindhoven
Unive~5ity
of Technology,
1971;1.
S.b. Wassenaar, thesis (to be published), Eindhoven University of
T"-<;hnology, 1981.
4
C. Clausnitzer,
~.
F~e~sehmann
and H. Schopper, Z.Phys. 144 (1956)
336.
5
H.F. Glavish, Froe. 3rd Symp. on Polarization Phenomena in Nuclear
Reactions, Madison 1970 (eds. H.H. Barschall and W. Haeberli, univ.
of Wisconsin Press, Madison, Wisc., 1971) p.267.
6
J.a. van der Heide, thesis, Eindhoven University of Technology,
1972.
R. Beurtey and J.M. Durand, Nucl.lnstr. 57 (1967) 313.
8
B.L. Hagedoorn, J.W. BrOer and F. schutte, Nucl.rnstr. 86 (1970)
9
F.B. Goulding, O.A. Landis, J. Cerny and R,H. Pehl, Nucl.lnstr.
~53.
31
10
(1964) 1.
F.S. Gouldtng, Annual Rev.Nucl.So. 25 (1975) 167.
11
P.V. Vavilov. JETP 5 (1957) 749.
12
N. Bohr, Mat.Fys.Medd.Dan.Vid.Selsk. 18,8 (1948).
13
J.E. Sluiters and 5.5. Klein. NuCl.Instr. 120 (1974) 305,
14
A.J. de Raaf, Nucl.lnstr. 163 (1979) 313.
11
transfer reactions are generally
Nu~~~ar
gra~ing
collisions
between nuclei, that is the nuclei do not interpenetrate much and the
effective
inter~ction
is mOrC Or less localized
~t
the nuclsar
sur face'3. 'rhe relative weakness Or this "surf<,cG-interaction" allows
the usc of the so-called distorted-wave Born-approximation (DWBA). In
DWBA the 5urface-irtteraction is treated in
fi~~t
Drde~
only,
the elastic scattering before and after this interaction is
to all OrderS by
di~torted
whe~eas
de~cribed
waves.
In this chapter we will give a small
r8vi~w
of DWBA-theory in
order to prOvide bOrne bilCXground to the following chapters_ More
details can be found in e. g. th@ textbook of AU9tern 1). ·the general
features of the DW6A are treated in s@ction 3.1, while the applications
to one- and two-neutron tran",fer are presented in the sections 3.2 and
3.3, respectively. For details of the t;r:iton
t~an~fer
fOrm<ll.ism, which
i~ applied to the analysis of the 58Ni(p,~) ;r:ea~tion (~hap~er 7), We
refer to the thesis of Smits 2). Finally, the opti~al-mooel pot~ntials,
which
generat~
the distorted waves, are discussed in the sections 3.4,
3.5 and 3.6.
3. 1
DWBA theory
~o
describe the reaction A+a
H
+
B+b we
~ntroduce
the Hamiltonian
(3.1)
g
whe,e HI'.' Ha' HB and Hb represent the internal motion of the target
nucleus A, the projectile a, eh@ reSidual
nucleu~
B and the ejectile b.
The relative motion 1s governed by the kinetic energy T« (Te) and th0
potential Va
(V~).
The state vector
~(E)
satisfies
H~
~ ~o/
and has in our mOdel tne
following form
(3.2)
where
~e
and
~c
are the internal wave functions of the heavy (Cl and
light (c) component of the system and Consequently are eigenfunctions
of He and Ho' The
f~ction
Xy is the disto,ted wave for the reaction
!3
cnan)1el y and
~ep~es"nt6
tho rel<'ttive "lotion. 'rhe sum in (3.2)
qenerally is limited to a small.
n~~lobey
of reaction
r'eITkt.ird nCJ p:coce:'):;',.e:=: aYe hidden in thG imilgir'l~rY
channel~T
w})ile the
U~bs~-..,(ption)
pa.rt of
the optical··model put-unt.ials mentioned below.
It. .1.8
j 1"1.
f.=!iC':t the ?I:=:ymptotic be:h!lviO\.lr
(l~rqe
::;.~par~ticn
between
C and c) of the functions Xy that is rneasur<;:d by t.he det.ector". 'l'he
l ..I:::ansition 6lITlp1.1t-w~e 'r cd~ for' the r:-eaction
X~+)
from
(A,a) ..). (B,b)
io ext.xacted
(r·e-J....W), wher.'e t.:ht;>.: ,. indicates lIoutqoing" boundary condit.ions.
All ttl" distOrt .. d wiiv,,''; in the cxpansio)1 0_2) "-re of the x(+)
except for Xu (elastio cho.'lTIr)Ol), which ttlso cont.ains incomin..-;
type
wav~s
(the proton beam fo1'" instance) ..
NOW,
for
tru~
511pposinq that <11'B'~blljlc:~c> = 0fY (which is only r.igor.ou~ly
iII~lastic
~outtering)
a)1d starting from
(3.3)
where we hd.ve replacE!(J, ';~JB~bl VR I$BWb ';:0 by the opti.c;;3.1 potential
Us (r s)
and where the matrix tll"rntlnt on the right.-hand s:i,d" J." defined by
(3.5)
Loosely
~'peakir"lq w"':'~
Ci..uJ
eay that. Vy -tly represents the
Il
sur fact·-
in l:.erttci:,lon" mcn tioncd above.
~~q:\)Q.t
hl.,lt
Th" "quatio))
(3 _4)
~eo.lly
ton$ a,no by
~olvin9
thi."1.
~l!3o
lJWBA
ze~o,
is 0. system of coupled differential
~ygtem
the surface"'inte:c,::tctions are
not only the elastic soat.tering'
tr~ilt8d
sam" of the term" at the r_h.s. of "q.
whi~h
is
~
to all orders. Tn the
(3.5)
are set; equal to
yeasonable procedure if Glastlc scattering
i~
the
dominant process. We will clarify this procGdure by the following
example, whiGh
ela.stic one
include~
the outgoing chann"ls
~
and y besides the
,[ig-_ ., _ 1). The system of equatioJ\s. COrrespondinr'l to the
coupling scheme Of fig.
3. J ts
,~).
ven by
(3.6a)
(3.6b)
(j.6,,)
14
I--
Pig. S.l
\~'J
'··1
a
(CI,y)
Coupling s("fliem$ with reaotion dlCl,rm.tds
(a+~),
(~+y)
and
(CI~Y)
CI.,
S and y
are now d®scribed by a
order approximation (lIone ... way coupling"),
s~attering
--I
eq. 3, (J)
(("fl'.
The transitions
y
)I
is still
t~eated
whe,=,,~a!3
to all orders. The
fi~st-
the elustic
~equenCe
(CI~G) (~+y)
is
a two-step OWBA transition, while (a+yl is a one-step DWBA
call~d
transition.
If we are nat interested in channel y then the system (3.6)
reduces to (3.6a) and (3.6b) and in that case the transition ampl1tUde
Tu8 can elso be calculated tram the expression
(3.7)
where
Xs
is the
gol~tioo
of
th~
homogeneous
equ~tion
Corresponding to
sq. (3.6b) with incoming boundary oonditions. Displaying
th~
into-
gration over relative coordinates explicitly eq. (3.7) becomes
T~S ~ J
+
f"
dr
f drC(
e
+
I I (+)"
..
Xe(-)*".
(k~,rS)<Bb V Aa>x
Cl (ka,rC\)
(3.8)
with J the Jaoobian fOr the transformation to the relative COOrdinates
and with k
and kC\ the relative momenta.
f
Before di8~~~~ing the matrix element <BbIVIAa> we will f~rst
connect TaB with the analysing power A and the unpolarized cross
ssct~on
da/dn. In expression (3.8) we have not explicitly shown the
angular momenta. In fact TaB depends
a,
II
and
b: M , m , ME
a
A
~on
the spin
Proje~tions
of A,
and mb and the unpolarized cross section is
given by
15
do
(3.9)
-'v
elll
Thf':!
~~na.l
~er.tion~3.
ysi.nq po we r: i$
obt.=:.i..nf':!cl w.i
.'":!
lTIp.<;:I.=;H).;re
fOJ::
the di fference betweer'l the G"rOsG
protDn gpin up
t}""l
(m
Cl
(m
e,
=
1/;>)
and proton spir\ down
= -lh),
(3.10)
A
I t 1,. "ls<>r from
D .10) th"t A 1i"',. between -1 and L
In urcler to have" qu"ntit<>tiv<= indication of t.he
betw~en
r.;o~~
ca leul at-cd
::::~ct:i.on
X~
2
X
A
r
r
we l)st=:!d
N
ii
NI
~91:'"ement
measured (see cha,pte:r 1) anf.l.lysing power and
d.r"ld
foll.owing chi-squa]""e expressions:
tJ~*
) - d 1 (e i )}2
ca
Mexp (e t )
(8 i
""£'
exp (e.)
~
--~
(3.11a)
- Aca l(A.)}Z
~
() _lIb)
(e.)
6A
exp
~
where N is Lh", tot"l. Ilumb"r of eX!?8rim8ntal dat'" eaken at different
5catterinq angles 0 i _ Irhe quanti ties no and ~A are t.he errOr:=. of the
experimental cross s8ction and analysing pow8r.
3.2:
One-Ilc::utron tr.::l.Ilsfer
working CIllt "BblvIAa"
for a (p,d) tr.;>nsition one arrives "t 1),
r
-T
I
-><'Bb IVila>
" vn
<4> (r-> -r> ) ~ I~) IV' ' '(r' -r
»
I
rJ p
n
B'
"P P n ) i(tA(t,r
.
n
(3.12)
wi t.h f:; be 1.119 the in t.e:rnal coor.dinates of nucleus Band n the number of
"act.i,ve" (.1"e. taken explicit.ly iflto account. in the
w~ve
function 1/1)
neutrons in nlKleus A.. The /'JGutc.:ron irltcrnal. wt\ve function
i~
indic~ted
by ¢rl and i5 generillly limited to ttl" L=O component (s-state d<,ut"e:ron",),
which h",:; th" iUl!?Ortant oorollary that the transferred "flgular momentum
i6 equal to that of the neutron picked up from nUGleuS A. The "surface
interaction" V
i.s :regtricted to the potential between the incoming
np
pt:'oton
~nd
the
tr..7,ln.~f'e:C(t;:!(1
approxim~tion on~ t~k~G V
np
16
neut:ron.
In the so ... called
zero-:r:~nge
proportional to the delta function 6(tp-~n)'
~nteraot
i.e. the proton and the neutron only
...
~eplaces ~dVn~
facl one
when they coincide. In
+
by 00 6 (rp-r n ) • where the zero-range constant
DO is calculated frOm
~
DO
-+
fdr
->-
~d(r)V
...
np
(3.1 J)
(r)
with V
a mo~e Or less realistic potential.
np
'the nuclear overlap ("form factor")
<!JIB1wl':
is exp«nded 35
follows;
(3.14)
One usually takes for $jm a singLe-particle shell-ooodel wave function
Woods-Sa~on
calculated with a real
~eLL-dePth
potential (eq. 3.26). of which the
is adjusted to coOOPly with the condition that the binding
energy of the neutron has to be equal to the experimental
energy EB-E
<i)iSlw A>.
A
5epar~tion
in order to guarantee a correct asymptotic behav10ur of
~hi~ is oalled the well-depth p~ocedure (WDP).
'the nuc"ear structure ~nfOrmation is contained in the spectroscopic amplitude~ 5 1/ 2 and they mea~ure th~ am~lltud~s with which the
variou~
single~particle
states appear in the nuclear wave
functio~s.
For example, the 0+ ground state of 58 Ni may be described by the
follo~ing
two-neutrOn
~ave
function:
In this model th~ 56 N! core is considered to be inert (closed shells)
anc\ the two "valence" neutrons are occupying 2p1/2, 2pl/? and lf5/2
orb~ts_
~he
configuration mixing
~s
caused by the residual interaction
(not present in the shell-mOdel potential) between the neutrons. The
spectroscopic amplituc\e sl/2(2p3/~) for the transition of this 0+
state to the 3/2
ground state of 5?Ni is then (using eq. 3.14)
l;)/Z.
So by ,nea;;uring the spectroscOl?ic amplitudes one thus determines the
magnitude of the coefficients of the various COmponents of the 0+ W~ve
fUnction.
Since we want to investigate the interference between one-step
and two-step DW5A transitions. knowledge of the relative signs of the
spectroScopic amplitudes
i~
crucial. Therefore one must be sure that
the pha58 conventions used in calCulating these
model wave
functio~s
~~lit~de5
frOm Shell-
are the Same as thos€ of the computGr code that
caLculates the cr055 sections
~nd
the
analy~ing
powers.
17
F'OY' the coupiecl-c:!tt1.one18 code CHUCK2
sec:tion~
:-1
r
and anillysing pOwer5 in this thesit;
with 1/iihich the CrOSS
r
h~ve
been
cCLlcul~ted,
one
has to \.u~"(~ ,9p~(:t:..tQ$CQpic a(flplitudGs Jefirae("j by
1:1.16)
where p star"Lds fo:r" U"J.e quaflL.Ulrl numbers
::;ymrnet_r.~
7;erl_
(nlj)
an,c:1
11i;1~11
means::: anti-
rn addition the rild.;i.al pa:rt of th€ si("J.gle-pa"(ticle wave
func;tions is po:oit.ive ... t
infinity "nu the an9u1ax- pa"t is
,~",fined
without thc' U,eto!:: i 1.
Summ.::J.rizing we qet t:he Eollowinq expression (or the one-step DWBA
transition .:::I.mplltudc
\'
1/'
L S
l(nlj)
To.ll '" l)O(pd)
Id'~
(-)* -. "r
+
(+) +
1\~1 +
• X~
(k~,r)4nlj(~)X" (kct'T r1
(3,17)
which is .:::I.1so valid for' ot.her one . . . neutron
triJ.nSfer~1
b\.lt with
-1l
di ffere"t UlrO-rilrlge constant I)Q' The maSS nun,ber of the targGt
is indicated by A. The angular momentum traII.sf0r j
h~s
nuc)."u~
to satisfy t.ht;":!;
triilng.le condition implied by the C.lebsch-Gordan ooeffl.c1ent of
cq.
= 0 then only one term survives, j = J
3.14 and if J
A
4)
:B'rench .and M.;3.C [arlane
'
S
have derived :sum -r:ules, which
~onnect
neutron and prot"n occupation numbe"" with th" spectroscopic ampJ.l.t\1des.
Since we apply the::;;.e Sum .tt),le9 to our
experi,rn~ntal
r:esults
(chapte:(s -1
and 0) wO give them he):"e. If the to.rgot "ue)."u" has iso6pin TlI then one
"<lrl
r,~,,(;h viii one-neutron transfer
wl.t.h T
(t
1/2, t
" 1/2 ) final state.
z
T" + II?, The sum rules then ar.e:
-,
I
c 2s
T
(1 j)
"lj
-
<
j]lj
2T
+!
A
(3.18)
1j
c2 S
where
./.1
rr
(lj)
TI
2T l'l
A
is tll" number' of
correspondingly J,lJ
ne\)t~on5
occupying the Lj orbit and
fOr t.be pyotons. C sto.rlUS fOx.: the isospin Glebsch-
Gordan coefficient <TBTllz; Ii? 1/2ITI\TAz' wj.th either TB ~ T ... - 1/2 or
'l'B = T A
11)
+ I /2 •
3.3
Two~neutron
transfer
In case Of a one-step (P,t) transition the explicit form of the
transition oensity is 5) :
(3.19)
In tho Zero-range approximation one takes:
(3.20)
where
...r
-7
-+
+
and
r
p
-,r
~
(3.21 )
2
l?
In this approximation the interact.ion takes place when the proton
coincides with the centGr of mass of the two neutrons. The function
do can be determineo analyt10ally from:
-+
dD(r)
..
Jdp<j>t(r,o)(V
-+
-+-+
pnl
+V
pn2
(J.22)
)
if one takes Gaussian shapes for ¢t and V. The important parameter
entering this evaluation i.s the root-mean-square raoius of the tritOn,
Which usually is chosen
equ~l
to 1.7 fm. In aodition a pure L=O triton
internal wave function is aSsumed.
The nuclear overlap <1jJI'!(~) I1jJA(f;/~L,tZ»
is
related to the
spectroscopic amplitudes in ths following WaY
vn(~-l)
.el/lBll/I '
L SI/2(jlj2'J')<JMJI'!~IJ}\MA>[1>j
=
A
jjj2
()';I)x<l>.
1
{;2))
J2
J
(3.23)
The one-particle wave functions in
tn~s
equation a,e
~etermined
in the
same way as in the case of one-neutron transfer, wnile the binding
energy is usually
t~ken
equal to half the separation energy EB-B
A
of
the two neutrons.
The inverse of eg. 3.23 is given by
(3.24)
In oroer to calculate the transition amp,Utud0 TaS one then proceeds
as follows. Pirst [<p.
products of
Jl
r~dial
(t 1 )
x ~j
2
(;2))
is transformed into a SUm of
and angulur functions expressed in the relative
19
and
c~nter""'of-mas::;
cOordinat.es -; .:.. ·;2-;1 i.\I1d
R=
1/2
(~1+;2)'
respectivGly. Next. one restricts this t..tan.sformed function to the part
with
r",.\~tJ.ve
l~O
(i.e. th" tW'o neut):cns being it. tl1\ s-state relativ8
too each other). This mea1\,~ that the S-t"",1\SfOL' is zer-o and thilt J
,
= L-
+
Then the integri:.lt..l..on OVE'-r .1:' ~nd p is performed resulting in th~ fOrm
J
I'. +
facto( b'O~ (A-2 R), where I'. i" the mass number of the taL'<)et nucleus.
The final result then is
(3.25)
rrhe distorted w.::..ves aloe
c~lc.;ulCtt.e(~
f~Qffi
(:3. 6a)
equations lik.e
\lsinq- an opticill-model potent-ial of the following form:
U(r)
= Uc (r)
- Vf(XO)
-
iw f(xl)
V
-
4iw f'(x l
d .
1
-
(~)2
V
.!:.df(Kz)
meso r
dr
d.-r
-IT
(3.26)
has the WOOds-SOI"cn shape
where f (xi)
- r.A 1 / 3
r
le"i + 1)-1
with
0,
x_
(3.27)
1
is t.lle Coulomb potential g,morat0d by a uniformly charged
and where U
c
sphere oE radius reAl/::..;. For protons .and tritons. ;; :-
deuterons;;
=,
-;;1 where s is the spin
orbital anguJ.n: momentum of the 5catteruc\
terms of U (r)
absorption
3.5
2~' and for
operator. The v~ttor
p":c~l,,,l ..
t
is the
, The two imo.gin"ry
represent tho volu.m.e absorption (w ) and the
v
::;.urfGl,~e
(01 )0
~inite-ranQe
ano non-locality corrections
In sElotion 3.2 wo have discussed the zero-riJ.nge approximation
for one-neutron transfer. It
;i.~,
however, possible to make
Q
first-
0rder cOrr0ctj..on to th;ls Etpproxim,;:.,tion, which is called the local",n""'9Y appro:dmat-.i"" 6)
(",ljA) _ 'l'he LEA con,;ists of multiplyin<) the
[o,-m factor (eC(, 3,14) with the llulthen form
A(d
(I:I.:.!:.
A
r)
+ En - Vn(r)
-
E~
+
US(~))}
_I
D.28)
20
with Vn
th~ bOund-5tat~
potential and R the finite-range parameter.
The optical potential should
shown by Perey 7)
reall~
be a non-local potential. As
distorted waves in the nuclear interior. This
simulated
th~
a non-local potential reduces the amplitude Of
~ppF.oximatel~
Pere~-effect
can be
by multiplying the form factor (eq. 3.14) with
the factors
(3.29)
where 8
is the non-locality param~t8r of projectile, neutron and
i
ejectile , respectively, and Pi the cor~esponding red~~ed ro~ss_
Adiabatic deuteron
3-Q
In the
~eaction
pot~ntial
model implied by
3.2 three-body effects are
e~.
not taken into account. since the deuteron is a loosely-bound particle,
tho> nucle<lr field of the final nucleus can easily sl?lit the deuteron
in a proton and a neutron and thus give rise too the channel (B,p,n).
The adiabatic model of Johnson and Soper 8) provides an
apprc~imate
d~uteron
where
and
treatment of these effect5 by replacing the optical
potentia" by a kind of folding potential,
~h~r~
ii(r)
~
Do
fds...
~
-I
Jd~ (Un (; +
DO
.,.
+
f)
+ U (r
p
+
- t)}
+
Vpn(S) <!>d (s)
+
Vpfl (s)'fld(S)
(3.30)
(3.31 )
the proton and the neutron eaoh
g~t
half of the deuteron
<;onex:gy. The distOrted waves geneJ::ated by the "adinb"tic" potential G
also include unbound (p,n)
states, in which the proton and the neutron
continue to mave together in an
An
S-stat~
approximate expr~5Sion fm:
U(,)
with little relative momentum.
h .. ~ been derived by satchler 9)
f.or the case that U .. nd Up are woods-saxon potent.ials (or first
n
derivativ~s thereof.) with the same geom~try parameters but diffe):"ent
well-depths. We will give
he:r;e
the formulas leading to this approximate
"xJ;lression for U(r) and refer to the study of Sat"hler
If Un and Up have a
WoodE~Saxon
and (Vp,r,a), r0speetiv0ly, then sO has ij with
a~
ij
a + 0.0217
a
fOI;
detailS.
shape with parametel;$ (vn,r,a)
r
= rand
(3.32a)
(3.32b)
21
3.}:Ib differs from the prcscript-~on
wl-1en' R = r ,,1/3. We remark th"t eg.
of Satchler by the faot.Or ~ in front or .(t'2. We think, however, tJl?J.t our
"pproxilllallon is bet.ter. We will not go into det.ails, sinc" the fim'l
:t"~8ult::::
~f
a~:e
hardly influenced by this alteJ:'nat.ive pr-escription.
On and up have the shape of the derivative of a Woods-SaxOn
p"t~nti<'l
with pllramct.cr·s (wn,r,a) and (W ,r,l1), respectively, then
p
U with
= rand
r
the same is true for
a
=
W
=
a.Ole2
" +
(
(3.
a
2
1
2 n
)
1·· O.OlGi (~+
] liT)
(3.Db)
We also had to constrllct pot"ntials W starting
geOmetry
'par~mOl.ers
:33,,)
.f:ro,o different
fOr t.he:.- nent.ron find the pt:'oton potentials. This
was don" by separo.tel y calClll"ting
Wn
ann
Wp
from eq. 3.331:> putting
Wp " 0 and wn " 0, ):'espectively and then adding th",,,,.
REFERENCES
N.
Au~tern,
Direct Nuclear Reaction 'theoJ:."ies (wiley-Interscience.
1970) •
2
J.W&
$m~tg,
thesis,
Rijksuniver5it~it
GrotlingCn, the Notherlands,
1977 .
P.D. Kunz, University of Colorado, unpublished.
4
J.B.
5
tl.w. aaer, J.J. Kraushaar, C.E. MOSS, N.S.P. King, R.E.L. Green,
6
P.J.A. Buttle and L.J.B. Goldfarb, Proc.Phys.Soc. 83 (1964) "l0l.
F~ench
and M,H. Macfarlane,
Nucl.~hy~.
26 (1961) 168.
P.D. Kunz and E. Rost, Ann.Phys. 76 (1973) 437.
7
F.G. P€rey. Direct Interaction and Nuclear Reaction Mechanisms,
eds. E. ClementeL and C. villi, Gordon and Breach, New York, 1963.
B
R,C. John<;:on and P.J.R. soper, Phys.Rev. ci
9
G.R. Satchler, Phys.Rev. C4 (1971) 1485.
(1970) 976.
23
CHAPTER 4
4.1
THE (g,d) REACTION ON 5B Ni AT 24,6 Mev
Introduction
Th~
(p,d)
r~aotion
is a well-known instrumGt,t fOr obtaining
speotrosoopio information like l-values and spectroscopio faotors.
TIl';' elsefulness of this reaction is greatly enh€l.nced by using polarized
protons, becullse the analysing powers show a clear j-dependence 1). As
~
general rule one can say that the sign of the analysing power at the
angle wh~re thG cross section reaches its main maximum, is positive OY
negative depending on whether j
=
1 + 1/2 or
the measured analysing powers with DWBA
j
=
1 - II?
c~lculationB
By comparing
or with the
experimental results tor levels with well-established j-values
(I~pa.tterrt
rscogni tion") one thus obtain~ a r-eli';"Qle ~pj,n ass;i.g-nUlent.
of the final state in the caBe of
~pin-ze~o
target nuolei.
Some (p,d) cross sections da not exhibit the oharacteristic 1dependent diffraction pattern predicted by the
rather flat and featureless shape 2,3)
on~-~t~p
levels showing suoh a behaviour is that they are
wherefore in these cases
two-~tep
DWBA, but &
A COmmOn property of the
wea~ly
excited,
preoesses arc generally thought to
be competing with the one-step pick-elp. such transitions are present
j.n
the 58 Ni(p,d) reaction at 27.5 MeV 3).
Mayer et al. 1) already perfOrmed the 58 Ni(p,d) experiment with
pol~rized
protons at the same energy as we did. These authors
r
however,
only analysed the strongly excited levels. So we decided to extend this
study to the
wea~er
level~
espscially in view of possible two-step
contributions. In addition we wanted to investigate the importance of
two-step prooesses fOr strongly
All OrOBS
section~
e~cited
levels.
and analysing powers hav@ b@@n calculated with
the computer code CHUCK2 4).
Our results are discussed in section 4.4. An outline of. the
reaction model and the ingredients for the calculations is presented
in section 4.3. In section 4.2
WE
have brought together the details
Of the experiment and finally in seotion 4.5 we have
main oonclusions of
su~"arized
the
thi~ ~hapter.
25
The exporiII1e~
1. L
'l'hu
(p, l)
(p,d) .J:·l";8.c:tion
wa.~
5imultctw~Qu::ily
measured
W;l..~:h
t:".he (P,P),
(p, 1),) r".'Iction,,- The outqoinq particle,s we"',, identified using
,~nd
the telescope sySlem dOScI·ibed in chapt.er.- 2.
determin~
In oyde"r" to
absolute cross sli..:.'ctioIlS we compared the elast:ic p:roton .... scatterinq
with th" optic"i-mockl un"lysis performooct before by
yea~tiQn ~HiNj.(pIP) at
the
Our
group 5)
d~ti.1.
for
Th~ o.2Sitimut..:cd ac(:u.:rac:y of the
211.b MeV.
nO.J:'lllaliz.ati.on is :20't.
'The
en"J"-iched
ta.t:~Jet.
consisted of:
(q~J.q~,)
~
gelf .... supporting foil Of .isotopically
51~Ni with a thickness of 1 mg/cmi.. Th-e: intensity and
neq:t:"ee of polarization of the proton
bc~m
r,::spi::c:;tiv~ly. Dutu. wt":.~r~ t:.1.:l.kcn fJ:'om 10
0
e::::I1C.:.'rgy re:.:::=.olutioll wo":t:=:
~bolJ.t.
were about. 10 nA
t.o 80
0
in
:;::b:~ps of
100 keV FWllM. which ilYlposed
iJ.
ann 80%,
SO _
The
limit L.o the
number of lr::-vels th.Qt. could be:' analysed.
Wf;;;: meoJ.surBd
th(~
J.IE.1.1ysing powers anrJ c:;:::rQS!;: sections of eight.een
level", (table 'L d). These '''". bo seen in the speer.rum of fig. 4.1
CD
/
u·,
10 3
()
a
'"I
~.
'"
..
'"
eo
,'+..,
.~
l\'!. .:'\~'. "I'
,~cD\O
.
/'
co,
u")
-0
,
1--'
cr,
"'
(-- L..O
<D
<J)
Ii"]
<>'
-
I \
I \ \
0
,",
0
'"
,,'"'
102
M
I
'"'\
IJl
fZ
;;)
0
U
1
10
,--,--,--,--,--,--,---,---,-,,-J .... ,L,. . L...L...L.-'....JJl..llL--'-.llil-'1lL..llliJIL.J....,.J.JJlllll..l!.l...-Ll.-' 10°
200
320
CHANNELS
FlU,
26
d.1
440
together with levels of contaminant.s and oth0r lo;,vels of 57Ni which
w~re
excited too weakly to yield reliaole angular distributions. Tho;,
excitation energies were taken from the survey of Auble 6) and from the
study of the 58 Ni (T, cd re1lction 1lt 25 Mel! by ~'ortier and ~ales 7).
,he
5a Ni(p,d)
rcaction has already been measured several times
at different energies and with unpoLarized protons
3,8-10)
~
polarized proton beam has been used by Mayer et al. 1) (24.5 MeV) and
by Hosono o;,t al. 11)
4.3
4 3.1
T
The
~eaction
(65 11eV).
model and tho;, calculational procedures
Reaction model
Seven out of the measured eighteen transitions have bBen analysed
according to the reacc10n model that is implied by the coupling scheme
given in fig. 4.2. The remaining transitions were described by oncstep DWBA only. In the two-step prOco;,Ss
inelasti~
proton-scattering to
the first 4+ (1.45 Mev) of ,SNi is followed by the pi~k-up of a P3/2.
Pl/2, £5/2 or f7/2 neutron.
s~atterin?
P~ck-up
followo;,d by inelastic deuteron-
turned out to yteld negligible contributions and tho;,refore
was ignored. Replacing DWBA by CCBA in the proton channel (i,e.
~
two-
way coupling between the 0+ and 2+ states) ct~d not change the ,esults
8~gnific~ntly
and 50 CCBA
Wa~ al~o
left out of consideration_
The one- plus two-step DWBA calculation (fig. 4.2) is complicated
by the fact that five spectroscopic
~mplitudes
may be 1nvolvCd against
one in case of .a simple one-step D'ir'fflA an..alysis. i)'heref.'ot'e we mad~ us0
of shell-model calculations done by KOOps and Glaudemans 12) and
Van Hees et al. 13). Th", J.atter Calculations allow for aile If'll?
Fig. 4.2
Coupling $"heme for the do'-,pt,d-ahanneh eakuZatiO>l8
J"
= 1/2-,6/2-,
5/2-, 7/2
27
rLucl(::!on-h(Jlt:'; in the
')5 Ni
GOre with the partic.1e:8 Qccupyil)g 2p3!'::,
2pl/~
,mel 1 [0/2 orbits i'tt"ld t.hey
(~Dl)
en with a
r"norm"li~",d
cont~ineu.
Cca.l(:I,llations
a.~e
p"rformeo with th" surfa.ce-delta
Kuo~ll"own
i,t1 ref_
(KBI)
re~idual
12) wert) rest'r:icted to
interaction. The
2p.~/:l1
2pl/2 and
1fS/2 pctrticles (i_~. ~Il inert ~GNi core) and the 511rfacc-delta inter.~ction
(SDU). FrOm th", )·""ults of tiles", nuclear structc",e ~t\.ldi"s ttlr",e
s",t.s o( spectr.o"c;opic i'tmpJ.J.t.udes, also indica.t_ed by SOl,
h~v('
been comp1)tec1 14)
0"'" :1/7-,
.~(\(.l 1/2
0/:,
stat" orlly, while SDI and KSI
5p""tl-um ot ne~'Jiltive pal;ity st<lL.cs ri'tnqinq from 1/2
the 9/'
2U)J
qrouIlll ~t·.a.r.*
ll/;':
states can not
l.ead to "- wllOle
Co
ll/Z -. Of these
b~ :reached in Dr'lC ~tep from the.: 0'1-
~iHNi_
\,'J[
:'i/?1 ilr", 1/:'1 stat.es are idenl:l.fied with ttl" 57 t<1 levels
The 3/21,
at 0_00,0.71 and 1.11 MeV,
~·e3pectivc.:.~ly.
rrht'::!se three stutes
only ones which Lurn Qnt. to have .an u.pprecJ.~ble 1/2-
transfer strengtt• .t~orn the 2+ "t<tte Of 511 Ni
.~.tate~ are populateu by
"I/O'
.....
.,
__ . _ - - -
2·t-_:r
J
for onc..'-lIuul,rOJ"1
b~ rCi.\ched
soO
!;()1
...
"
t.(an~f~~t:'
KBI
......-,,_.•. _ . _ - - -
-0_7il
0,')4
1.11
0_B9
1_15
1 (1)
0.40
0_11
0_ 4·/
S (I))
-n.~2
-()_,,7
""U_ 22
5 (:J)
0.3<
o. :~ ~
0_17
(1)
().:!"
0.44
0.17
r, (7)
[LOD
() _ l()
·:0_10
o. :,,,
D_17
~,
the
(table 4.1) _ 1'he ~emaining
:J (J)
,; (S)
J
aI'€:!
b.nd. ?~/'2
~_1
"'~mpl11:\ldf':!s
/JpJ)CL)
Pr.L_'L_·I...'::.'~';
.].!;r
statE::!5 whi(:li ('.:an only
T"bk
,Spl..'!_·Lr.l':...sc_·vj.d.r:
r
t.r.'lnsfer frOm either th" 0' only or the
;,; + on1 y. calculations :::how that_ the
.,.
f)
KBI and SDO,
The set SDD obviously desc:rj,bes transitions to
-0. ,r,:;
, (5)
o. '"
.J (:1)
1. 01
0.7:'
1 _12
Ill)
-0.42
-0.39
-0_11
J Ii)
0.00
0_22
0_11
1 ( ))
-D_,"Ej
-0_11
-0.14
I (1)
0_1:2
0_ 3"/
0.40
.... " .. , - , - , - - - - - - - - , , - , _ .
~'d
).8
t.h~
tT<'1r1st'.::.::rI'Cd
dnyu.] ar momentum B
iD(!i"at_"d by j .
i"
vi" th",
ar .. too weak to be: obs<!>rv",d in our experiment. Some of
these states probably have been found by FOrtier
~nd Gal~5
7)
~nd
m<LY
be identified with a weak-coupling multiplet 2+"7/~- somewhere around
E (2+) + E (7/2-) = 4.0 MeV excitation energy.
x
x
The cross sections and analysing powers of the partiole-states
at 0.00, 0.77 and 1.11 MeV have been
amplitudes from table
~,1.
~alculated
using the spectroscopic
We applied the usua" one-step DWBA analysis
to the other states except for four cases, where we added a two-step
transfer to the one-step transition.
7/2
The
inelasti~
s~attering
has been described macroscopically using
the following form factor
6
~ -':'l.
F2 (r)
15
{
R
0
dU r (r)
J
[R~
2
~~~ + R - - - ~ - ~e2 -;;T 0 (r-R ) + ~ 0 (R -r)
dr
r
dr
~
r
c
~C
C
d[] a (r)
l}
(4, I)
where U and U are the real (volume W-S) and the imaginary parts,
I
o
respectively, of the optical proton-potential (eq. 3.26). The parameters
Ri (i=o,I,C) are defined by ~1 ~ r A1/ 3 , while e is the usual step funci
tion. fO. the deformation parameter 62 we used the value -0.22 found
before by OUr group 5) from the 58Ni(~,p} ~ea~t~on at 24.6 MeV.
Aotually the sign of the deformation parameter oannot be
+
determined by the (p,p') experiment since it depends on the
~elative
phas" of the w;ove f1mctions of the 0+ and 2+ iOtates. The sign of S2
has to be found by
chot~e
tr~al
and error. It was p05sible to make", oefinite
fOr this sign by making use of the interferenoe between the
one-'i'tep and the two-step tran5itions. This is Ulustr"ted in fig. 4.3
(J/~-,
solid ~nd dashed-dotted lines).
The code CHUCK2 does not account for the O-state of the deuteron
~nd
the finite range Of the
interact~On.
50 our oaloulations are
limited to deuterons ill the S-state and the finite range is deScribea
in the
lo~al-ener9Y
approximation (LEA). We used the Hulthen form
(3.29) with the v;olue Of 0.69 fm fOr the finite-range parameter. The
LEA reduces the Cross sections by about 3t
~ffect
on the
sh~pes.
~nd
has no significant
The Cross sections were normalized with the
empirical constant bOCpd) = -122.5 MeV fm 3/ 2 ,
The need for a non-locality correction at the neutron Wdve
fun~tion
has been demonstrated recently for neutron- and proton-
29
7
(]"n",ity calculations 15-1 ). So we
",pp~y
this correelion in
OilY
cas,;:
Following the aut.ho:t:'3 of refs. 15 .... 1·1) we \lsed a Gaus:=;ian fOrm
t.GD.
(3.30)
CrOSS
wit.h the Vi'lille or: U.<l:' fIn fOr t.h", non-locality parameter'. The
~J,ection5
non-loc~lity
increase with about 3S% by the
correction,
wherot';!as tht7! shapes r.emF.l.in virtud.lly unchanqe(-'l..
l)$u.~lly
proc~dUr'"
the neutron form factor is calculat.,d by the well-depth
wi. th bindin<] energy of the ne1)tron e"lual to the o,,-per imental
"epClration en€r<]y (t;El, So: = Sn
,)nC'rgy of 5U N;. UJ.'~.)
i·
Ex with Sn t.he
("1f'~n"utron
separation
"nd 0:" the 8xcitat,i.071 en"''''9Y· In this way the
cor.rect asymptotic behaviour of triG fOrn! fact.or is .assured.
Tn order to explain the observed j-dependence of certain (p,d)
C)":Qg~ section~
Shery et ul.
18) proposed to
):."eplac~
the separation
ener-gies of the pa.rticlG-::;tut.es l)y effective enerqics, wnj,ch
in line with the
:sinql(~-purt.j.r..:.l.t';!:
to give a better.
d~sc:ription
In f.u.ct both
t.~ei'3.tment
",pprQtl,.(.:.:h~s
~t.e
more
Shell . . . . model energios and a,(e sllpposed
in the nuclear interiOr.
(.:.;:.n be reconciled by d mOre sophisticatOd
of. the form fuutOr, whi,ch P..Xl?licitly accounts t'or the
rGsidutl.l j.n,te.ract:iQn of the valerh.2;G nOUt.rOnS by considering the
factor as "
~o.lutior>
We tried to constru(;t..
the foliowinq way:
fOri!!
nf on inho,"o<]eneous S"hr6dinger ""I\lation 19-24)
.:s.pproxim3.te solution to suc.;h an
..:t(l
~qU,;3:tion
in
firl5t by choosing the residual intcract:ion to be
a delta interaction w(~ Sd.mk,,"1 it'ied the inhomogeneou:=: Schr&rJint)er
equution to
[I
(E - T - ul F (rl
j
a .. ,F l ,]
Jj'=1 JJ
J
F. (r)
]
w1 r,h J7: th~ :::Iep~ra.tion enerqy, T the }dn~l.ic enertJ"Y operator, U t.he
sholl-model pot.ential of t.he ~,f'Ni car<, .e,-,d a
jj
, cert",in coefficIents
determined by the spect.roscopic -'lmplitudes; secondly, we,
the solutions
p~?)
J
(0 )
( E - T - u l Fj'
~n
nawly-found
(1~d
(rl
the 't:'iqht'·hand 5idC Of the
numGriC'~llYi
~1Jb.'3titut"d
of t.hp. hQrooqeneous equation::;
fi(~d.1I.y WP.
~alutions
~g:t,l~:::ati(ln!3
(4,3)
(-1.2) and solved l.he.!;':€:!
r:p.peated this proce:durc by
in the right-hand
n.ot chan!Je any more
~ftGr
8id~
$'I),~)~titutinq
t.wO Or thxee i teY{:Itions. Ttw I'rO:=.S
Clnd
however~
lurned out. t·o be virtually the same d,e those obtained
powers
the
Of (4.21. The solutions
So€;!!.,.":t.ton~
)0
~na.lysing
=0
G<11Gu.l~t.ed
wit.h l.hese form
f~ctorst
Table 4.2
Effective
energie~
(1;:1;:)
for the bound-neutron wave function
jTI
I;:I;:-Snll.)
EE
3/r
0.00
12_20
5/r
-0.77
11 _13
1/2-
-1.11
11.09
7/r
2.58 b )
7/F
5.23<:)
17.43
1/2+
5.50
17.78
3/2+
6.01
.\8. ;:1
14.78
a) Sn is the r.ep<lration energy of a nel,tron from the ground
st~te of 58Ni ,
Sn=12.20 Mev
b)J.£ Ex "5.23 MeV
c) i f EK~5. 23 MeV
Since mOre carefully constructQd solutions of the
inhomogeneou~
S~hradinger equation yield result~ 21,22) which deviate significantly
~~Dm
that preScription, we decided to
(SE) with the following
compa~e
effe~t1ve-en~rgy
the
separatian-ene~gy
(Eg) prescription,
EE = Sn ± ~x. Sn is the one-neutron sep<lratian energy of 58Ni in the
ground state and ~x is th€ excitation energy of the final 57 Ni state.
'rhe minus and the l?lUS signs ""e chosen fOr particle (i.e. 1/2-, 3/2-,
5/2-) ~nd holG (7/2-) ~tates, respectively. The effective ~inding­
energies are
al~o
used in a well-depth procedurG, which
homogeneous Schrodinger equation. They are
We took the usual
r = 1.25 fm, a
A
=
gQomet~1
giv~n
iIDJ?l~es
a
in table 4.2.
parameters for the woods-Saxon well,
0.65 fm and the factor multil?lying the Thomas term
25 (i.e. V /V .. 0.138).
so
The optical-model param€ters are given in table 4.3. The prOton
potential has been determined before by our g~oup S). It was ext~acted
from a
globe~
fit to elastic scattering data of polari2ed
p~Dtons
on
several i,on and nickel isotopes at energies ranging from 15 to 25 MeV.
31
Tublc 4.3
Opticill-mod"l
V
N.:IHlC
j.'
t'
'>2 .0
.'l
0
0./6
1.15
Dl
101.1
1.05
0,1)6
D2
10" _4 c, d I 1 _1 ")
0_ -In
111
')U _4
w
V
Wd
2,1J!i
(L
()
bl
o.m
Ll"I
6V!6 I'
__
r
'" SO
0.4'1
5.6
1.04
0.54
1.1]
0.69
7.0
0_75
0.50
10.1 c)
I. 26
O_Gl
11.8
1 _(l1
0.56
9_6 c )
l.n
0.5/
15.2 cl
1.26
(L61
J 1 .8
1 _04
0.56
14 -4(:)
1.32
CL
.'---
-
~)7
.
0_:31:3
0_11B
-(1-6;'6
G)
SO
LJ5
0_290
\:.)
r
60
14.7
D4
v'Wv'Wcj .and V~(J in MuV,
for tho:::
v
"r
1
7
n'.l
,0 )
a)
"Wall:. Ex
x
-_._.
p!J.ri\Hl.(:tCr":::~
pl"ot~..">n
the ot.hp.y parametero in fm; )·C-~·.1.2~5 fin
po!.:t:nl.iij,). 6l.nrl. . . . r:=1 .
Tht~
\l~lIlE:~
("..of
w..... s
L.tt(;~r·
(':!J.r. r..C'(:
t:his parilmr..:-tc.:.'r· 1:.;..;
t.r:·(l by t.h!7!
b'or the grounli :=: t·.,) t·.1":;
Q
ju
r
fiU
:~o
frn for the
):"e}!Qy.t~a
t:hDr.S to
d('l."!I~.0:J!)n
to be O.
[to
pot".enti!lls
l.n t"c-!L.
2(;)
but
0.86.
b~
Ni
d)
This potclntitll i::; ral:.h":r close to the Becchetti-c.:;reenlees 25) potcntii!\J.
""copt. fOL t.he
ditflls",ne$~
of the spin-orbit part.
We tried 6"v"ral deutoron potentials. The first one (Dl)
is the
global Lohr-llaeberli 26) potential obtilined from elast"l.c scatte:ring of
POlariz.e-d
o?nc..~
unPO.l
f.I'(j
ze:f1
deute~·on~
on nuclides with A. ::- 40 ttt cnergie.s
frOm D t.o 1) MOV r wh tch 1& just t.he region
The adi-=tbat.ic deut.eron potential (D2)
t.h-p. Becchetl.l.-Green l~H~!S
;,j)
0
f ou . . . experiment.
has been
construct~d
pot".ent.ials for protons and
Q-.ccordinq to the p.l":escY"iption of Satchler ),7)
from
neutrotl~
(chttpt.er 3 1
s~ct:ion
1.6).
In cOnst.ructing the SPir"l-()ri)it potent.tal woe ro:::pl.acei3. the qeometry
pO-rarneters of ClIO BG poterlt.ial by t.ho8e of
t.:.he li..1ttcr was determineu by a polarj
Out of the pr-I"::!sent study Slros8
potent;alG_ On" o[ th"s" (DJ)
~?!.t_ion
OUr
proton potential, sinGe
exper.irnent..
the need for' two rnOr'e rle1J,teron
", the same asD2 but fOr. u reduction
4.5~
of
of the real well-depth and is lett out of table 4.3. The
othe. one (D4), which gave the best result.s, differs f;("om D2 by a 7,5%
reduction of the real well-depth and by an increase of the imaginary
potenti~l.
5
This increase is 50% for the ground state and about 10% at
excitation GnGrgy. The Curves displayed in the
M~V
f~gs.
4.3-4.11
huve been calculated uSing deuteron potential D4, unless otherwise
mentioned.
Non-locality corrections have been appliBd using ttl" Gaussian
form of the correction factor with
~(p)
0.85 fm and e{d)
~
~
0.54 fm.
The cross sections rise about 8% by these corrections and show slightly
more pronOunced diffraction patterns. The
~ections
4.4
and non-locality
finite~range
corrections fOr thG proton and the deuteron together
r~iEe
the cross
by about 5%.
Discussion
one-neutron pick-up preceded by
yields
G~06S
section~
which
a~e
at
inel~~tic
le~~t
sc~ttering
one oroer of
smaller than those obtained by pick-up only, The,efore
which are strongly
by a one-step
popul~ted
~ransi~ion,
weakly-popula~ed
~ll
by the (p,d) reaction, are
while the tWC-St0P
(fig. 4.2)
m~gnit~oe
tr~nsition
level$
m~inly
reached
may dominate
~
level.
For the strong
particle-s~~tes
ae 0.00, 0.77 and 1.11 MeV thG
addition of the two-step transitiotls introduces a small but significant
change of the calculated oross sections and analysing
solid and dashed lines). All crOss section
cu~ve~
powe~~
but one
~~
(f~g.
4.3,
£ig_ 4.3
were normalized to the first maximum in order to faoilitate the
OOmparison of the shapes.
Cal~ulation5
$D1, Kal)
with different sets of spectroscopic
yield almost identical results except
the cross sections. Only for the 1)2
fo~
state there is
~mplitudes
(SDO.
the magnitudes of
~
slight
difference in shapl;' When KBI is used instead of SDO Or SDl
(fig. 4.3).
The axperimental data are described rather well by the calculations
apart from the CrOsS section of the l/Z
s<owly. This does not
appea~
state which is falling off too
to be a general problem. because in tho
corresponding 56 Fe (P.d)55 Fe (1/Z-) reaction (cn~pter 5) the d~screpan~y
33
·~.::,m
V(m.
',0
, ',0
lO~
, 8.0
'0 '
'i
r
,
, au
'I
00
r'
",.'"
5BN'"P.d I 2' 0 MeV
312- iD.OOJ
- .•.._- 1":ZST, 1)-=0
---1ST
i
---1'251 p.O
I,
fl
I.
1/1
~
[
\
~
~,
I
L
r
I
I
I
r
I
I
I
r
i
I
I
,
I '
I
'
I
I
I
I
I
I
I
I
I
I
I
I
I
i
~
I
I
I
,,,!
I
!SaNi \P,\J) Z4.oM~V
111 \.11)
' .. , 1 .2S1,
_. -- .,.,. 1ST
-
Suu
_. - 1 .~Sl. K8)
o'q
""
Im~/'S1")
0.1-
I ".
f
r
I
0,1,.-
1
I
I
!
I /"
1/
-:r
). u 2S
I
0.01.
/0
III
·.11
~,.
liIi!. './.;1
c...":Omp Z.(;r t~'{-J (.Xl
f;~
,
1'(1
~n
III
C}'ODD C<1U t-i.orw
/;Iw papt'{cle
(,I)
and analysing pOiJel's 1'0}' the
tPGnr;i Uons
~o
o·t,atc3 oj' 5? N·I:. The cor·id Uneo l'epn'8ent the
'/..81..£
1..(.1 i.7.:cJns +
rllor'mq Z. 7.:;:~I:i"/;'l~()n. fo.e 1~(}r~8 (.H'e
disp'l,a.yc:Jri
()nly tor' Uw h/;l- :;taj.:(,. '/'ne dasned-ciot"(;ed ('Ur'WN fOJ' Ole
,'0/2
34
~i.:'.!i;('
I"'.rr)e (.I(;(m !:a7.-(,u7.-(.t(;d W3inCi t;h(, ()pr()~il'o: "'i(fn
measur~d
between the calculated and
~ro~s
sections is considerilbly
sm<lller.
~h~
~n~lusion
o{ the
two-st~p
~an
the fie co the data as
prooesses does not always
imprQv~
be seen from the CrosS section of the 1/2
state and the analysing power of the 5/2~ state, which are better
by the one-step calculations (fig. 4.3).
des~ribed
~he
first maximum of the 3/2
about 10% after adding
spectroscopic factors
th~
and 1/2
states is
dec~eased
by
two-step prOCeSSes, wh;l.ch means th"t the
ext~acted
from a DWBA-nnalysis have to be
increa$ed by about 10%. 'rhe m<lgnituds of this
Oorr~etion
is
rath~r
to the optical deuteron-potential and to the set of
ins~nsitiv~
spectroscopic amplitudes (SDO, SOl, Ka1). There is no need for a
correction of the spectroscopic
facto~
of the 5/2
state. So the
interference between the one-step and the two-St8P
proc~ssCS
fi~st
The intorference
i~ de~t~uotiv8
maximum
5tates~
tor all three
at the
angle ~s abOut 120 0 fOr the 3J?~ and 1/<- states and 100 0 fOr the
5/Z
state.
The separation-energy (SE) Prescription fo. the bound neutron
(section 4.3.2) generally gives minor differences in the results
compared to the effective-energy (EE) prescription. Yet we preferred
the latter,
pe~ause
1/2
a~e
states
in that Case the cross sections of the 5/2
t~lling
otf more r,apidly, which
c~n
be
and
~nde~gtood
from
the fact that the neutron is less bound when using ER. 1.\11 curves dis-
played in
figures of this chapter are calculated with the effective-
th~
energy prescription.
Table 4.4
Spectroscopic factors of the one-step transitions frOm th0 ground
state of 58 Ni to the part~cle st<ltes of 57Ni
experim~ntala)
theoretical
11
SOO
501
3/2 (0.00)
1.23
0.79
1.
32
0.91
0.91
5/2- (0. 77)
0.62
0.88
0.20
0.53
0.68
1/2-(1.11)
2
0.16
0.17
0.22
0.12
D.19
<.01
1_64
1_62
LS6
1_7B
3
(8 )
l(
-
1: C 5
a)
R:61
EE
SE:
determined by using potential D4
35
In t"blc 4.4 w'fl give ehe c:<perime)1t.al spe"troscopic f""tol:'s,
which are corrected as
describ~d
above, together with the p<edictions
Of the )1ucleotr-Slructure caJ.cl1L"tions SDO, SDI and KBl. None of these
cal.c111ations gives the correct m"qnitude of all the experimenta], c 2 s_
factors. Yet we can say that the surface-delta interaction (SDO and
~lo5er
SDI) is generally
to the ma:rk th(:ln the K\lO-D:t'Ovm inte:r8,ction_
'rhe summed strength is best. described by the calculations which allow
fo"( on.,-1101"
4.4.2
stat~s
The 7/2
(sDl
""et
KB1).
states
-------------..,.-
The cross sections of thB ,/2
st~tes
state (fig. 4.3) and the 7/2
(hg9. 4.4 "nd 4.5) show an uflmist"k"bl" j-dependence which is
only s1J.ght.1 y reproduc<ld by the calculations. This j -dependence, which
ha~
al,ready be<il" noticed by Sher, et al. 18), cannot be explained by
two-s~cp
processes, because all investigated 7/2
effeut to the same extent. JohnSQn and Santos 28)
states show this
found the inclusion
of the deuteron D-state to be largely responsible for this effect. I
<llthough Delic and Robson 29) cl"im t.hat t"king int.o <>ccount "exact
flnlt-'~-r"ngclt
effect~
"",,,cels til'.! D-stat-e
"lmost- completely. Jt may
al .. o be that. a mure tl:!reful trecttment Of the neutron fOrm ["ctor 19-24)
a:=l
(3,isc\1~:!=Ied
in section 4.). 2 ~ will s11ed light on 't.he p:t'oblem.
The structure calc1l1i;!.tions SOl and KBI do not predict a sizable
two-step contribut i.ol1 for the 7/1.
DWB~~analygis
step
in
the~e
states, so we only applied a one-
cases and this resulted in a reasonable
fit to the analysing powers. It is obvious from the j ... effect of the
""alysing
powe~9
that the levels at 1·.23, 1.58 and 7.12 MeV are 7/2
5ti;!.te".
'floe speutroscopic f"ctors are given in tab),e 4.5. They show that
!~.he
.:;truCtur0 Cctl(;ulatioBo do
'1'= 1/ '
I~ot.
r0prOduC(~
tho fr<:tgmentation of the
5tat"~.
r'ollowin9 the stuc'ly o( F'ortie,( Cl,nd Gales 7)
strength or the T=1/7
t~~ngitiong
we assume that the
is mainly concent~~ted in the states
n.t 5.2) and 7.12 MeV. Then accDyding to the following sum rule 30)
L
'l"'l'A t I /2
G2SU/~-)
1T(7/2-)
21' +1
(4.4)
A
where 11"(7/2.-) is the av.:;raq.:; number of f7/2 protons and T (=!) i5 the
A
~'.f"T\.
,0
,0
60
r'"'"'T'""T"'T""""T"'"'j""'"'T".,...,.--r"""T""T""""T""
80
D.2S
. ·O.l>
0.1
-:J
t· t
I
Itt
j
!
I
I
f·
+-+-+"1--' j
I
J ,-
'r-'I",-I-"T"
I 'I- ..., ..
,.,_··t·"'1"-"I""1 ~
~ I
I
I
SBNilp,d) 2Ut¥.ii
~2"IS.'l)
15l
•.- - - - - - - - . - 0
f
. I
I
A
- I I
.. 0.25
j. ,
I
1I
I
I
I ·f ; +·1·++·+·+-+
-+-+-!-t-+-+-+-t-,
'·-"-+'1
·'·-'·-1
f
5BNiip.dl 2MM.V
712 "17.12)
- 0.2.
---712-~-
Fig. 4. tj
~'roes
>1,-
sections and anMyt>inq powers for the transi tions to
the 7/2- 3tat~$ at 2.58, 5.12 and 7.12 MeV. The dash~d
aurVes have been catc:u.tatrJd at>t>uming a 5/2- FI;na~ ~to;trJ.
37
,0
, I
II
,
,
1,0
~r,.m
'\'"m
I'
6U
r r ]"r
iU
r ,',
I
SBN.I ['i.d) 2/•. 6Me\l
7/1-'J.2)1
I
I
r ,-,
'
,0
I
r r
r
'D
I
'
r
nn
6D
r",
\
"l
1ST
- D 25
1
~
0.1·
iD
'I
I
'\
rr
-D.2S
. 0
I
I
Itl.l.----i.o
II
"-.----~--
1
'
I
I
I
I
"
I
I
~8NI /~,d 1 21,.5MeV
m-II.. S81
i
I
I
I
I
I
'
is
I
'\,1
I
\
A
'1- .
0 25
-J
,
\'
I
j
I
I
I
I
I
I
I
I
I
I
i
0.15
0.11
f
1
/
\
I
I
I
\
\ II
J
I
f
10
\
\~
I
ON· I
I
I
I
I
I
r,
I
I :
20
I!
I,D
I
I
I
L
SO
I
, I,
60
I
,I,
,0
VI~U'
1, h
I "
1,0
, I
60
.1
I,
80
Cl'~O~~~~ ~~euf..L)"lG (.'au] analy:;;in(J [JoWr-:.r l ,"3
t..,rr.e ?!?
(:I'..I,'r'·r)(?,~~
.1fl
I
-il crn .
:..i'r,r.;
;:;U".J~.ea
aL 6,)·f!..1~
fOp ~h~; c:rl..P'1s,:tioH.:;:
1"',::,, <:rrf..(i, 4.fjfj MeV. ,1'h~;'? datJhod
h,·.toe bc:u?tl ea leu.l.ated
(H)f}ll,rrn:.7'L(j {~/)/2
f1~nc."(!.
ci:ate,
to
Table 4.5
C2$-fa~tors of the one-step tr~nsitiDn~ from the ground st~t~
of 58 Ni to the 7/2
",tate", of 57 Ni
Eoxperimental a )
theoretical.
Tb )
E
3/2
$DI
x
l:C
a)
b)
2
51;:
5.2~
Lee
1.68
2.25
2.25
0.10
0.53
0.56
0.67
1.98
2.21
2.81
2.92
2.58
3.60
3.00
3.0
3.0
3.23
0.56
0,88
0.50
0.54
3.37
0.11
0.12
4.23
0.21
0.26
4.58
0.22
0.28
4_0
4.2
~ (T= 1/2)
det~rmined
EE
7.12
2
LC S(T=3/2)
1/2
KBl
by
4,16
us~ng
3.88
potential D4
isospin of the Hnal state
isospin of the target nucleus, the summed strength should not exceed
the
v~"ue
2.67. We used this limit in determining the potential D4
(section 4.4.3).
We took ",orne trouble in finding an acceptable
deut~ron potenti~l.
In this Bection we describe how this was done.
The Lohr-Haeberli potentia" (Dl) yields crosS sections which fall
off too slowly except for the
by using the
~diab~tic
analysing pOwer Of
$t«rt~ng
th~
7/~
states (fig. 4.6). This is remedied
potential (02), but then the fit co the
7/2
state at 2.58 MeV is quite bad (fig. 4.6).
from the adiabatic potential one
of the re«l parc with 5 to 10% in
orde~
to
h«~
1;.0 reduce the strength
obt~in
an acceptable fit
to this analysing power (fiq. 4.7). Such a reduction has also
b~~n
proposed for other reasOns by $atchler 27) and Blankert 31) and is
39
,0
1~"1""1""
.0
r"T"'!"'! -
50
80
i
,',
, 90,
I
"-\ '----...
I
I
O.J!!i
I
I
~
I
+
............ . . .
\,!./
........ ....
~
I
do
;iii"
(l'T"~bh,r)
, ,I . , , I
1
'"~,:,~"; ,;:,.1
,;.
I·
. I
I
'/
'f
r
I
i .
I
L
J.
l
/,0
C\!I7T[UT'1:80n
,
,
,,
I
I I
i
·
3
I
L
,)1'
I
I ,I
-0
~~
~/
""\
i
\ .............
1
\ f
~
I
,. , ,
I
~I
'---
,
,
I
A
i
---02
;,\
I IJ
,cJ
~
--01
~
~
......
7/]'12.bB1
"1. (i
60
,
--01
'w Ol
v
t I
0.'
1"1:(1.
I
~12-(O.m
. I
[1
,0,
20
I
T'-r'~~'~';';I~,'~';I;'~~~'~~~'
//
.• . . . t.......
J
I
\
,I,I
\.
, I
I
\
,II
\
..--.-- ..............
I
[.0
I
I.
I
I
gO
~
,
,
,
,
?O
au, r.ohl>-IJaebed·i
,
~o
,
I,
~o
(01) and the a(Habo.H" (D2)
deutepon potenti(!/s for' i;lu? tY'(wG-itions to the 5/3- ((J,17)
and ~/:I- I:I.:,<'U 8·t"trJd.
effect~
p050ibly elu" t.O the ne.-,lect of """h"nge
in the construct.tom
of tho .:tdiabat. i c potential.
It "Oems that
thj.~
J::ed\lction h/15 to be acc:ompanied by a
::.:imul tan'::OUE incrci:tse of the absorption potent.ial, be..;a\lse a l:;"educt.ion
of the
re~l
the 5/1
potcnti8.1 alone c,;I.qses a fall-off Of 'the cross section of
sU,te, Whil;h i3 tee "low. This is llemonstrated by the
calculat.ion. lIsin'J the D3 and D4 potentials. The applied reduction is
4.5~,
and
7.5~
fot· 1)3 and 04, r.espectivcly, while in addition fOr the
1)4 potential the abso>:ption potential is
il1crea~ed
by about 35% (fig.
4.7).
J\n increi;'.lse of that size, t"Lowever, causes the spectroscopic
factors of. the ';/:,
states at
~.2] cHid
7.12 MeV
t.O
be so li:lrq'" that
tl,," SL,m-rule limit of 2.67 j,s largely exoeodod. We solved this probleID
40
20
'I:'
, I
1
,0
r
T
i·
00
00
10
"T"","""r-'T
I .~8'~'~' ;'~'.T ~';'~67.~-;1-
"""1'"
40r"T
I
\'
so,"I
80
1'"
&12- W.771
--03
,
---Q4
"
o. 2~
t
J
I
0,1
I
..\"-r..-r •..-- ........ ~ ...... ""-
.,
./
I
........
)
/
do
dll
'." ,I
1mbl!:irl
j
·'1'-t-t-+-"j.-+-I---I---Io-,~·" ..,.. t',,+-,·,·,1
SSNllp,dI24.6M.V
/
-(US
I
A
'.,
712- !2.58)
--OJ
---04
1 ..Ll. 1 , 1
LL---'----'-'"Jo-O.L1-L
,
40
.I. .1. I.
6()
d_J
80
r-
..i
j ·0. 25
.1.-,---,-..,. 20I. ,
1 .. 1
.{)-.;.m.
Fig. 4.1
L
I ••
40
>
I
1._1-.•
I..
60
SO
-:tc,m.
Corrrp=·iso" of the adjusted
deu~B'I'o?1
foT' the tmnlf>itions to ·the [,/2
potmtials D3 and D4
(o.??) and 7/2
(2.58)
8tates.
by assuming a
lin~ar 8n~rqy-dependBrtce
for the percentage of increase
(the hiqher Ex' the lower the percentage;
Eventually we fixed the
par~meters
s~@
table 4.3).
of D4 by
t~king
the sum of
the 5pectroscopic strengths of the states at 5.23 and 7.12 MeV
approximately equal to the
sum-rul~
limit dnd by fitting as well as
possible the shapes of the Cross sGctions and the analysing
the particle states at 0.00, 0.77 and 1.11 MeV
~nd
the hole
powe~s
ot
~tates
at
2.58 and 5.12 MeV.
The
in~re~se
equiv~lent
to
~n
of the surface absorption turned out to be almost
in~reasc
of thG
volum~
absorption with slightly
better fits in the farmer case.
Generally there is not So much
obtained with 03 and D4
(s~e
difte~enCe
between the result5
fOr instance the 7/2
state
~n
fig. 4.7).
41
Table 1.G
futentiuJ
dependc.::.'r'I(;O of t.he c;2s-.factors a)
J'IT (E)
x
D2
OJ
D4
0,91
3/T(0,00)
().8~
0.80
S/2- (0.77)
0.51
0.43
0,53
1/T(1.11 }
0.12
0.11
0.) 2
;; G2 S
1.1S
1. 34
1.%
7/T (?, 58)
:L1
2.8
3.0
'1/2-(5.23)
- - - ,... ........
,
"
..
~~
. 7S
~.
:l
2.25
__.
iJ.) deb"jrmirl.ud u:;:in(j the effecti ve-€nr~r(JY
pre::;criptioD
«:ll)
This is also t:rue for t.he spectroscopic f""tors which vary at most
20~
(tahle 4,6). Unless othe):'wise stilted, ,,11 CurveS displayed in the
fiqur.es h.;ave been calculated using D4.
[o'J,na11y we Itke to remark that the
locality
cor~ection5
extent an increase of the- absorption
incY.'ease is mo"(e urgent
The
4.4,Q
:j/~
fiI~.tt.e-runge
(LEA) and non-
(for proton iJ.ncJ deuteron) rnimi[,: to a. certain
if one
do~s
r
so
t-h~t_
the need :for such un
not appl.y these corrections.
Stilt()S
---------------
Apurt fro:m
t.h~
g.rol~nrl
state we found four
mor(~
:~/'-
st.ate~.
Alt.hough their Cr0SS sections "how the characteristic forward peak,
lll('Y L,ll off much mar'" slowly t.han the
4.8 arId 4.9, daSh<ld line,;)
one~gtep
DWBA curvoios (figs,
fOr which reason Il(lwards et "l. 3) label
these as unassigned levels:.
Fortier.- and Gales 7),
anulysis of the ~8Ni (T ,a)
how8vf~r,
do not. me!.7!t.
t.hi~ ~I:'oblem
in their
reaction leading to t:hese states. SinCE" the
resolution of this experiment. is much bet-t.e.!: (20 keV) than ours (100
keV)
Or
tM' one of Edwards ct a1.
(7~
key) / we checked for possible
unr"solv"d multiplc,ts. Ac"ord.l.ng t.o the results of ref.
Of lhc fOur
~~i:\t(;S
7)
only one
is an unrCSolved dO\lblet, n.'lmely the one at 4,95
MeV whict, is not r,,0;01 VGU from
u,e
7/2
8tate at 4.89 MeV. The fit
to the cr.oss S8ctl.Or~ indeed improves somewhat by including the 7/2
state. whil€;:! the t.wo
42.
c;.;':I.l~l)l.;.l.ted
analysitlg pOwG;rS a.re very
siml,l.~L
and
60
80
T I'
, 'I
0.1
,r,
f
\
!l2.
oR
\
/
\_~~,
1/
o.01t'.
(mO/5.0
\
{
I
.I
.t--+-+ "r
iI
\
I I
I
-1'1,
S8N'i~.dl
"liZ"iqSI
I
"1.. ,1".
2•. 6MeV
~
1"
I
t ~
I
I
:
I
I··'
I
.+_+ J
A
~
~~-I·'SI
---151
!
Fig. 4,8
desc~ibe
! ,
Cross $~atiOn8 and analysing po~er8 for the transition$ to
the 3/2- states at 3.85 and 4.1(/ MeV.
t.he dat.a very well (fig. 4.9).
Cross sections origin&ting from a
be
rath~r
two~step
flat and feat.ureless (e.g. 5/2
transition tend to
state, fig. 4.3). The cross
sections of the remaining three states are too large, however. to be
explained by
two~8te~
processes exclusively and we are fOrced to
alloW for a one-stsp component too.
The only sizable two-step contributions arise from inelastic
excitat~on to the 2t state of 5B~i followed by the pick-up of a 1/2neutron +
~oreover,
leaving aside the three
parti~le~~tatesJ
the
structure calculations SDl and KBl generate only spectroscopic
strength from the 2t state to stat.es of 57Ni through 7/2
tranSfer.
we performed calculations involving such a two-step
transition for the ~tate5 at 3.85, 4.46 and 5.95 MeV. The C2 S-factors
The~efore
43
)0
1,0
'111
l
'60
1
1
1
8U
11
l
111
1111
~m
I
irnI1r';f)
I
I ' ,
I
i
I
!)8 Nifp,rll ~J.,6MI;:\i
312-(0.9,)
---1+25T
,-- --1ST
II
-\!
oIF'.
\
I111I
I
I
It
.. O.lS
~I
I
/' .-............. """t-·~-L
\
\
"..........
(
'",-/
,
'--.,
I
I
0.0"
I, •
I'
,~
.
I
111
10
1
I
\...-'
I
~(~Ji
I
,
I ,
1~
I
I
I
L
I
1.0
I
0.2,
I
I
II.J
I
GO
80
~('.m.
Om'-H~q) (.'{tl.l.·l"I.[H-icm~ ]"0Y'
I:Itat-r.'<' Qt '1.9,"
(1r1d
the
t~ani!it1:0rw
'/;(J
trul J/?
1).95 MeV omnpm'ed tJ1:th a on,,- p1.l.IS
tlvo-s'l:ep ,.",i.r:l.""t;·icm (6.95 M~'V) and a one-step ('ai·(,,"/.c~I,i{)rt
for· the
~pank!'iI'i01!
I.. !) (In
l"nresolved doubLet (4.1l.S M(eV).
~re q:i.ven in tab18 4.1. The C?~.i2 str-ength was chosen to be 18 tim. .~s
larg~r than th~ C~SO strength.
The fits to til'" ~rO"S se(O"tions are ~orlsiderably impr.o\1ed bj the
inclusion Of the t.WD-st€!P processes (fig:;. 4.R and 1.9, dashed l)nes),
while the c'JSo-facto-r-s a . . . e uno.ffccted, oe~all~e the erOSb sections aX'e
completely dOml.l1.atei'"'l. by the onQ..-step
it seems that
j,11
spite of
ttl"
pro~e~ses
nt forward anglfl.::3. So
large c2S2-fa(:tors a pure one-step
~na)ysi~ ~5 sufficient to detarmine C2S0.
Finally, althollgh the analysing
powers
lack statistic", we can tent<ltiv"ly assj.gn a
behaviollr <ltoune'l ISO.
14
are not
J~3/2
well
desc~ibed
villu<!o fl:"Om thE!
or
1.4.5
The states at 3.71 and 5.78 MeV
-------------------------~---~-
From the very flat structure of the cross section at 3.71 MeV
we get no indication about the l-value,
~
result already reported by
Edwards et al. J). The 58Ni(p,d) reaotion at 121 MeV 9)
shows a clear
l~3
however,
behaviour far this state, to which FOrtier and
Gal/)s 7) and Anderson et ai. 9) tentatively assign a J-value of 7/2.
we think, on the contrary, that this value has to be 5/2. In analogy
with the 3/2- states discussed above wG added a strong two-step
process (table 4.7) and this improved the fit to in particular the
analysing power (fig. 4.10). We were not able to Obtain
'I/~
gOOd fit assuming a
~n
equally
final state. The one-ste., DWBA calculations
of the analysing .,ower already seem to indicate a 5/2
value (fig.
4.10) .
The state at 5.78 :1eV is, according to ref. 7), a multiplet of
7/2
st~tes_
This is confirmed by the shape of the analysing power
(fig. 4.10). The shape of the cross section,
howeve~,
dev1ates
state, ~ fact already noticed befo~e 3,9), we
strongly from a 7/2
investigated the possibility of the presence of an unresolved 3/2
state (fig, 4.10, solid line), but fOund no real improvement_
Table 4.7
Spectroscopic ~aGtors of the t~an5itions to the states a)
of 57 Ni with a large weak-coupLing co~ponent
J"II
E
x
a)
3.2
5/2
3.71
0.05
)/2
3.85
0.04
0.77
3/2
4.46
0.02
0.32
3/2
6.95
0.06
1.10
determined using potential D4 and the EE prescription
b) via 7/2
transfer: relative sign of
negative in alL
CS~(O+) and CS~(2+)
ca~es
45
0.1 ..
70
I
1
I
I
40
1
I
1
"T"
! V'""i\'"
I
h
~-,
'\
.it'
\....
- IJ
':....""'
I'
//
sn-
,
20
40
50
80
-T··-r-,·_,·.....,._·T"'·...,.-r-T ·...,.-,..-r
! •
,
\ \.
/
,1~
0.01··
BO
50
11
!; . . . . . . . . . . .
.......
.
1+2ST
50- 1ST
I
I
~
712- I 51
712- hlSl
I
I
~
I
I
~
I
I
I
I
I '
I
I
~
I
I
I
I
I
I ~
I
I
I t · ,...
t
I· .... ~ ..
1\
WNilDfd) 21,6MeV
Ex-fl.7SMcN
0.25
, I ,
BO
.:){m.
C)"Ol,[;
cO OW
"'c.m.
r,ecUonc and ana[ysinl'{ pOwef'$ fOl:' ·the tl:'an"itionl;'
8"0'"8S
at .3.71 MeV (c:ompar'?¥m of 5/2
(mil 7/2-
c·tateD) and at ';'.?II M(;V, for. /,)hLoh ·the l:'esuLts for'
,or,
lml'ecolved (;; /;'\-, ;,/:.l-) (/olJ.bZ,"" at'e a/fiO shown_
4.4.6
~he
1=0 and
1~2
states
--~~------------------
~'" re",~onab1y well described by a.
The 1/2+ :;t<ltc at 5.50 MeV
one··step Dwf3A calculation nnd taking
~tate
at S.67 MeV
n
brinqs even
rnor~
~cco'!).nt_
of the unresolved
3/~
tigreeiner"J.c -with the experimentcJ.l
da ta. (fi9 _ 4 _ 1 1 ) .
The behaviol.1r
~t
forwar.d anqles of tho .::tnalys;inq power of the
1=2 state at 6 . 01 MeV fF.l.vours a j-vi11uG of 3/?. a8 can be seen from
" compartson wit.h t-h"
one-~,t"p
DWBA ca1cu1"tionE (fig. 4 _ 11). The
overa l.l U t. t-o both t.h" en'ss section and th" an!11ysing
though, is rather bad, Includin9 a
46
two~step
powe~,
process vi" the 3
of
,.~.~q'T..,.~..'·=!,·p'-"-.,-...,,.!;6,!!-DT" ~,.3Pl
I 0.5
D.I
,
I,
0.01
Ii
"
1V'··'.. +1-111-+',"",'>-+1-11-+ ..... +···.· ..
A
58~, (ji.d) 2•. W ..V
312 (6.011
---'1:/"
----SQ'
0.15
0·'
'0.25
0.01
!
~LL I
?O
1.. L..J.....!........!.J.......L...LI...
<0
00
~-c,m.
Fig. 4.11
M
,I. l __ L...L ..L_L....L.........~_.~
10
40
J 0'
50
I.c.j
60
"'c.m.
One-step caZoula'tions for' the tmnsitions te) the 1/2+
state at; .5 • .08 MeV and tile 3/2+ Mat", at 6.01 MeV. The
resutt8 fop an unresotv@d (1/3+, 0/2-) doublet QPe also
8ho~m.
5ti Ni Or aSSuming un unresOlved 7/2- or 5/2+ state did not improve
t.his sit.uation.
4.4.7
Sum
);"ules
------_
... In table 4.8 we h~ve summarized the C~S-factors obtained with
the effective-ene.gy
(~B)
~nd
separation-energy (3E) prescription fOr
the bOlJnd neutron. It should be r€:membered that the "previous results"
are not cQ);"rect.ed for two-step contributions.
~he summed C2S-factors are relat8d to the average numbers of
protons and neutrons in the va"ious shell-model orbits through the
47
crabl.e 4.8
R8Slll t.s
._-
~
.
'"
_..
,-,--"
of t.he saN i (p,d) 57 N;.
r~ClC'tion
.. """,.. .... ,-. ..... ........
,
c:\ (0+-+.1")--,"'/"--......--J1T
Ex
Rb
EE
SE
FO a )
NDS o )
0.83
0.00
s/r
\2. "0
0.91
0.'l1
1.l:J
O.n
5/2-
11.43
0 ..;;3
0.68
1.14
0.54
1.11
1/2-
11.09
0.12
0.19
D.22
0.16
2.58
7/2-
H.7B
:l.0
3.0
3.3
2.8
3.23
"1/2-
14.7G
0.50
0.54
0.61
0.39
0.15
3.37
7j'l.-
14. '/13
0.11
[).12
[).1'1
:l. 71
(S/'2")
11.43
0.05
0.09
D. [(;c)
'L8S
(')/2-)
12.20
0.04
0.07
0.09
4.2:,1
7/'[
14."/8
0.)1
[).25
0.34
4.46
(J/[)
\? ;?O
Q.02
0.04
0.03
4.58
'liZ-
14.78
0.22
0.28
0.25
4.95
3!?-
1).20
0.02
0.04
0.03
2./.5
2.25
??9d )
Z.O
0./0
0.60
5.23
3
7!?-
17.43
~)
_ ~'; t~
0
1/2+
l'I.·m
1.0
1.0
,i)
I
7H
.}
17/T)
1"/ .4:3
0.32
0.34
(l.
6.01
on t·)
18.21
0.96
0.96
0.98 d )
6.9S
(Jjr)
1/../.0
o.oe;
0.12
0.11
7.1?
7/2-
17.41
u.S6
0.6)
0.72
0.22
0.23
26c;l.)
(). ')2
0.14
.--,~,,'--.-
0)
r:~~om
b)
(,)
the study of 1"(.Irtic..:,r tind CaltJ5 7)
From the survey of Auble- (,)
ha .• .,d on JT!~7/2-
sum rules of French and M0.cfa.rl.='l.ne 30)
~
which are also g.iven in
chapter J.
[n ta.ble <1 _ 9 we compar-c:- Our resu 1. t.S with the predictions of the
t>!ll and Kill
cillc\Jl<ttior\~ 11) for the ground state of
o8 Ni. The J;;f.
r.esults 2l.rf:::: oomr2!wILi:1t lOwGr than the t.heoretical limits 2.0 and 8.0,
l:)\~t
thUg 1.9 not sut""prisinq,
we t.;>ke the r"mainin(l
since we have not. l{"f\.lnd all levels.
~tren'Jth
from rcf. 7) the E:E
and 7.7, respectively, whictl is vc:;ry
clo~e
If
~esult5 b"comC 1.9
to the lirlli ts indeed.
1'''°1", 4. 9
compi:lrisOII of sUm-rule results
experimentill
shell wodel
------
-~--~--
BE
SD1,KBl
BE
r
2
C S(I/2-,3/2-,5/2-)
2.0
L 75
2 _14
L
C2S(7/2-)~~(7/2-)
8_0
7_2
7_5
L
C2S(3/2-)=~(J/2-)
0.10
0.18
0- 36
T=3/2
The average number of ~rotons in a 3/2
orbit scems to be some-
what underestimated oy the structure calculations.
The determination of C2 S-£actors is hampered by m~ny uncertainties
and although the sum-rule results give us some
confidenc~,
We
e$tim~te
the error of these factors for the strongly-excited states to oe
20~,
Throughout this chapter we applied " non-locality correction to
the neutron form-factor. One hus to raise all C2 S-factors with ~bout
35% if this correction is left out_ It is obvious that the sum-rule
limi ts ar", violated in that case. 'rhi.s remains true if one uses the
D2 and D3 potentials. So the only alternative then is to change these
I?otentials (for
iI'l~tllnce
the ",nerqy dependence) .
rinally we like to ~emark that the summed strength for 7/<
transfer from tho
does not
e~ceed
2t state to 57 NL (table 4.7), which is equal to 5_4,
the thtl>oretical limit of 7.96 given by the structure
calculations.
4.5
Gonclusion~
In this chapter we ~tudied the ~eaction :,BNi(t,d) leading to
eighteen levels of 57 Ni • FO~, of these levels showed a featureless
shape of the
cro~s
section, which could be interpxeted to a large
extent by a strongly competing two-step
of inelast~c proton-scattering to the
pick-up of a 7/2
2t
proces~.
This process consists
state of 58 Ni
fou'owed by the
n8utron,
The description of the
~n"Lysing
powers, which
we~e
also rather
featu,eless, was generally not improved, though, by the i"clusion of
49
the two-Qtop processes.
1t
turnOd out tha.t.. the
l~stlal
the qroUJ\d ,state and the II?
analysis yi.eld values which
spectroscopic fa(:t.0rs
i:'.r~7!:
~~p}?e.::t.red
lO'l!! too low.
to be
the r::;t.rue:ture calculat:ions that we
fragment,at,ion of the 7/7
w~
al~o
dcetermination of the
S-f~ctOr5
Of
st"tc at 1.11 MeV by a one-step DWBII
'l~he
6~tisficd
us~d
...
sum rules of the
rat-her well, while
failp.o to describe the
level".
dete-r:mined some new spin assignments
u~ing
the j-eff"ect.
of the ana lysin,,; power. The problem of the j-effect of the l=J ero,,:;;
sG(;tion$
rerna~ned
Ur>solveQ and was ter\t11tively ascribed to the neqlGct
of tll« D-"latc Of Lhe deut.eron".
F"l nally /
we were forced l.O reduce the ~tY"ength of the real P.;3l:r:t
oC the adiabatic deuteron potcnUi1J. in order to obtain satisfying fits
to
u\e dat.a a'ld we got "0 me indications th<lt this reduc-tion has to be)
.-:.c::ccmpa.l1ied by
50
:~.n
increase of
th~
absorp,tion potential ~
REPEREtJCI::S
B. Mayer, J.
J.L. £$cudie and H.
Go~set,
Kamit~ubo,
Nucl.PhY$.
AI77 (1971) 205.
w.R. Zimmerman,
B.P. Blok,
J,J. Krau5haar and P.A, Batay-CsOrba,
Nucl.Phys. A267 (1977) 156.
3
~.M.
Edwards, J.J. Kraushaar and B.W. Ridley, Nucl.Phys. A199
(1973)
4
P,D~
463.
UniVT of
Kun~,
PrJ. van ijallJ
Color~do,
J.P.M.G~
unpublished.
Mels~enl
$.0.
W~s5enaa~,
5.5. KleJ.n and G.J. tJijgh, Nu<;),.Phys, 1.\.291
6
O.J. POppema,
(1977) 63.
R.L. Auble, Nucl. Data sheets 20 (1977) 327.
S. Portier and S.
Gal~s,
Nucl.Phy". A321 (1979) 137.
8
H. Ohnum<> , T. Suehiro, M. Sekiguchi and S. Yamadl1, J.Phys.Soc.Jap.
9
R. E. Apderson, J. J. g;raush<>ar, J. R. Shepard ano J, R. Con,fort,
36 (1974) 1236.
NUcl,Phy$, A311 (1978) 93.
10
H.!. Kegami, T. Yamazaki, S. Morinobu, I. Katayama, M. Fujiwara,
Y. Pujita, H. Taketani, M. Adachi, T. Matsuzaki, N. Koori and
M. Matoba, Nucl.Phys. A329 (1979) 84.
11
K. flosono, M. Kondo, '1'. Saito, tJ. Mat$uoka, S. tJagamacni, T. tJoro,
H. Shimizu, S. Kato, K. Okada, K. Ogino and Y. Kadota, Nucl.Phys.
A343 (1960) 234.
12
J.E. KOOps and P.W.M. Gl"udGmans, Z.Phys. A280 (197"1) 181,
P,J, BruSSaard and P.W.M.
Gl~UdGmanS,
ShGll-model applicationS
in nuclear spectroscopy, Amsterdam, North-Holland (1977).
13
A.G.M. van Hees, P.W.M. r,laudemans and B.C. Metsch, Z.Phys. A293
14
A.G.M. van
15
F. Malaquti, A. Uguzzoni, E. Verondini and p.e. Hodgson,
16
L.
17
F. Mal<>guti,
(1979) 327,
H~es,
private communication.
Nucl.Phys. A297 (1978) 287.
~<>y
and P.E. Hodgson, Phys.Rev. C20 (1979) 2403.
~.
Ugu~~oni,
E. Verondini and P.E. Hodgson,
Nuovo Cim. 49A (1979) 412.
IB
R. Sherr,
19
N. AU5tern,
~.
Ro~t
and M.E. Rickey, Phys.Rev.Lett. 12 (1964) 420.
Phy~.Rev.
136 (1964) B1743.
20
R. auby and J.L. Hutton, PhYS.LGtt. 19 (1966) 660.
21
W.T. p1nkstOn apJ G,R, satchlCr, Nucl,Phys. 72 (1965) 641,
51
22
~.J.
A119
21L
:n
A,
~rak"sh
24
N.
i'>tI~tc'r",
"nd N.
i'>u~t"rn,
Ann.
i'>29~
Nuc1.Phys.
of
pg77)
Phys. 51
(1c)69)
2S
F.n. neoohetti, Jr:,
,!.M. Loln- and W, llaeberli, Nucl.PhY6. A232
:1.7
G.R. ""tchlel::, Phy".Rev. c1
2!l
E.C.
29
G.
30
J.B. French and M.H. Macfarlane, NUc1.Phye. 26
31
P"J.
~nd
Johnson "no'! F,D.
G.W.
Gr.eenlees, PhY6.nev.
(1:171)
~3"nto~,
418.
lC)O,
26
52
Nucl.~hy5.
Philpott, W.T. Pinkston and G.R. satchler,
(1968)
(19'/4)
HlJ
(1969)
1190.
3131.
I1H5.
~article6
?.nd Nuclei 2 (19'11)
(Jelte and R.A. Robson, Nu"l,Phys. A19, (1972)
Blunkcrt. th"sis, VU Amsterdam, 19'/9.
510.
()961)
168.
2135.
CEA~ER 5
5.1
TEE (P,d) REACTION ON SSpe AT 24.6 Mev
Introduction
since the nuelide 55 pe contains the same nu~er of neutr.ons as
58 Ni , one oan expect a certain ana"ogy ~etween the one-neut~on pickup reactions S5 pe (p,d) and 5S Ni(P,d). consequently our treatment of
the raaction 56 Fe (p,d) will be quite similar to that of 56 Ni (p,dl as
descriQed in the foregoing chapter.
Only for the most prominent transitions to 5Spe there arC
spectroscopic factors available from. the literature 1-3). The values
given for these
casss
2,3)
o~
spectrc~ccplc
la~gely.
factors are plainly conflicting in some
exceed the
Bum~~~le
limits~
U~ing
the
expe~ience
acquired throu9h analysing the reaction 58 Ni (;,d) we hope to remOv~
some of these inconsistencies, the mOre
50
seemed to be fulf,illed quite well in that
as the Sum rules generally
analy~i5.
until how there has been ~eported only one 5~~e{p,d}
in which a polarized-proton beam was used. l·his "xper,iment
expe~iment,
",a~
P""-
formed by Hosono et al. 3) at a much highe. proton energy (65 MeV)
than ours and the analysis was restricted to the transit.ions to the
three lowest levels of 55 pe . So we may be able to make some ~BW spin
assignments for states of S5pe by exploiti~g the j-effect of the
analysing power.
The results of our investigations are discussed in section 5.4,
while the experimental and <;:alcul.ational. details arB given in the
sections 5.2 and 5.3, respectively.
5.2
The expe.iment
we analysed the transitions to nine levels of 55 Fe (t~ble 5.4).
These are shown in the energy spect.rum of fig. 5.1 together with some
other levels, Which were excited too w€akly to give reliable angular
distl:ibutions.
AS can be seen from this speotrum the peak shape is slightly
asymmetrical, there is a small hump on the >:ight flank of the peaks.
Most of the spectra showed this feature but sOlllet:i.mes t.he hump
appei;:l.red on
the left flank or had about the same magnitude as the
53
""' 0_
'"0
."
oo
(j)
.".
-",
I
[-
103
00
N
0
I:I
.".
I
c'
'".....;
::;;:
N
\
+
rN
M
'"
+
~
N
M
10~
IJl
I-Z
-<
N
;:)
N
0
\
U
1
10
~-L-L~~-L-L~~-L~~ . ~~~~Lll-L-LliWLL~~~~100
190
430
310
CHANNELS
Pi-y_ 5.-1
- 24.6 MsV
main peak. We did not investigate the origin of these peouliar peak
shapes, since they hardly affected the results of the analyses of the
spectr,a _
A good definition of the peak shape is important when resolving
doublets. Actl1crlly we met this situation far the state:s at 1. 32
1.41 MeV, since the
ene~gy
~nd
resolution was about 100 keV PWHM. The
pe"k-fitting computer code (see chapter 2) that we used provides
fo~
peak shapes such ,,03 d.escrib"d "bove. We determined the peak paxamete):"s
from the low-lying single
pe~k5
~nd us~d
with precise excitation-energies f):"om the
resolve the stutes at
The target
enriched (99.9%)
1.~2
con~i~ted
56 Fe with
these parameters togethey
literatu~e
j.n
o~der
to
and 1.41 MeV.
of a
It
~elf-supporting
foil of isotopically
thickness of about 1 mg!cm 2 . The intensity
and degree of poladzatton Of the proton beam were about 10 nA and
respectively, The 4at.a were taken at 11°, frOm 15
0
80%,
to 33° in steps of
30 ~nd from 3]° to 81 0 in steps Of 6°.
We did nat Oe.terrnj.ne the
~b.solut8
normalization of the
crOSE
"GctiOI1s independently but interpolated linearly between data obtained
54
at 17.0 4 ), 22.3 5) and 28 MeV 6)
We fe~t confident in using such
an interpolation because these data virtually were lying on a straight
line ..
The reaction 56 pe (p,d) has been measured before 3,4-7) but e~cept
for the experiment of Hosono et al. 3)
(65 MeV proton energy) in all
cases with an unpolarized proton beam.
5.3
TQe calculations
S.J.l
_-
Reaotion model
.... _--------- ....
We applied the Same kind of coupling Bcheme to the reaction
S6 pe (p,d) as we did in the case Of S8 Ni (P,d). i.e. inelastic protonscattering followed by neutrcn pick-up (two-3tep DWBA) in addition to
neutron pick-up from the grOund state of the target nucleus (one-step
DWB~}
t
2t
we 1ncluded two 2+ states in the p~oton chann~l: the
0.85 MeV with E21 = -0.24 and the 2t at 2.66 MeV with $2
(sc~me
2
A. fig. 5.2). The inelastic scattering was described
macroscopically (see formula 4.1).
~he m~gnitud~s
at
-0.10
Of the deformation
A
B
Fig . .5.2
Coupting $eheme8 for the
J"
= 1/2-,
oQupred-~hannels
3/2-, 5/2-, 7/2
aatouZationa
param8t8r~ h1l.ve beilJl determined b0fOre by our group 8) from scattering
of pola:ri~f~d protons hy the nuclc:!us S5 Fe ,
th~
We we:re able to reducO
computing time with a factor of two by
co\nbining both 2+ stutes tn an e(Fectiv8 2+ $t_ate
B,.
fig.
O-t
1.1:' M.,V (scheme
5.2). 'The excitation eW3rgy Of t:".his state was culculat.ed
as~o~ding
f0rmul,a~
to the following
(5.1)
The defoL-m"tion ",,,to-meter hZ was <:hosen to be -0.24. Th" spectroscopic
amplitudes for
r'lOutrOI'l
pick-up from the second 2+ state wer,e multiplied
Table 5.1
SpC;:~Ct.rosc:op'ic
amplitudes
(0),""
one-ILr~utI:on
t:r"ansfer
---~------~------~-------
SDl
o++J
2
+
'J
KBI
-0.42
( 5)
-0.91
(j)
0.65
1. 06
(1)
0.66
0.64
7) II) b)
(J.
73 (7) b)
0.73
bO
s
(5)
:n
-0,1(1
5
(3)
o. )6
0.12
(1)
0.46
0.26
(7)
0.14
0.10
0.02
-0.
IS)
0.14
(.3)
0.;>6
0.'04
II)
-0.18
-0.45
17)
0.17
(S)
0.13
-0.07
D)
0.58
/) (3)
(J.n
D)
0.10
73 (:q
-0.24
73 (7)
0.30
7)
tt)
e)
The t ..t:"an~f€rrf,;3d i:l.ilgulo.r" momentum is indi(:,;:l.teil by j
b) The stutes ,,\"_ L 32 MeV and <.94 MeV: 71 and 73, re"peccively
c)
S6
T.r~r. . siti'Jns
to 7/2
st.:ttC:~
not;. r.'".:.;llculated with SD1
by -0.10, divided by -0.24 and then added to the corresponding spectroscopic amplitudes for the first 2+ state.
Tne results of using the scheme
)l,
Or 8
were compared in several
Apa~t
cases and were found to yield negligible differences.
s~ving
from the
of COmPutex t1me there is another advantage ot the simplified
coupling scheme a,namely that it can be transferred unchanged to the
multistep c~lculations for the 56 Fe (p,t) reaction (chapter 6), whexe
the maximum number of channel couplings allowed by the computer COde
CI,(]CK2
would be largsly exceeded when using scheme A.
The states at 1.41, 4.51, 4.BB and 7.79 Mev were analysed with
one-step DWBA calculations only. The spectroscopic amplitudes for the
one- plus two-step
in table 5.1.
Thes~
txanBit~on~
to the five remaining states are given
amplitudes aXe based
u~on
the
S~me
kind of shell-
model calculations as described in section 4.3.1 Of the foregoing
chllpter.
TWO sets of
spect~ogcopic
amplitudes. indicated by SDI
(Surf~Ge
delta) and KBI (renormalized Kuo-Brown) were computed by Van Sees 9)
from the work of Venn ink and Glaudemans )0)
~sed
The configuration space
for these calculations is described in the following sentences.
The nuclide 56 pe has two protons 185s than 58 Ni and these are
represented by two f7/Z holes in the SG Ni CD"e. The configuration
space is constructed with P3i2, Pli2 and £5/2 particleS and t7!2 holes
with the only
~e~triction
that the maximum number of f7/2 holes is
th~ee fox both 56 Fe and 55~e. Since two of the holes are proton holes
this means that one additional proton or neutron hole is allowed.
The neutron form-factor was oaloulated using the well-depth
procedu,e with effective binding-energies
(s~otion
be found in table 5.4 in the column headed by
4.3.2), which can
The Zero-range
normalization constant Do(~d) was taken equal to -122.5 MeV fm Siz .
We
a~pl1ed
~E.
the same non-locality and finite-range oorrections (table
5,2) as we did :eor the reaotion 5B Ni (p,d)
The optical-model parametcrs are
(chapter 4).
g~ven
in tdble 5.2. The proton
potential has been determined before by our group 8) in the way
de~cX~bed
1n section 4.3.3. We tried the same
optic~l
deuteron-
potentials a~ we did for the ~eaotion SSNi(p,d) with esSentially the
same results.
57
In this chapt'or we there[QJ:"e wi,ll only pres0r\t L'E'sults oot,ained
with the M.dt~l,'bat_i(": potenti81s D2 ana 04 a6 described in chaptel" 4_
ethe potential lJ1 qener"lly qave the !Jest description or the cross
sections and analysirlg powers and
wa~
used in all calculationg
unl~65
st\:1.tcd otherwi.::iL':.
Tahl" 5.2
Optical-model parameters
p
1)2
,.
V
Ni:lil\e
a
0
51.0
b
100.D )
1S
0.76
1.17
0.78
l.
W
v
0
2.50
r
wd
G.17
10.0
9,6
97 _"b)
1)4
0.113
1. 17
t~).
7
15.0
c)
n
b)
b)
b)
b)
J
a)
"1
r
sO
~a
a
sO
1. 34
0.55
S.S
\. 04
0.04
\.26
0.61
11 .13
1.04
0.S6
1.]2
0.57
1. 26
0.61
11.8
1.04
0_56
1. 32
0.57
J,
0.6S
d)
U.60
1.. 25
V
1
Energy dependence of the deuteron potentiill£
l!.V/l!.E
ilWd/lIE"
x
D2
0.313
0.118
D4
O. NO
-0.626
Non-loc.~l j.ty
p
n
0.85
a)
V,
V
so'
wv
}'ini to-range p,;srameter R
p.=:!.rameter
0.85
(p,d)
J
0 .. 51
R
0.69
and Wd in MeV, the other parameters in fm; rC = 1.2~ fm
proton potential and \:: = 1-30 fm for the deuteron
b)
FQr the q~ound stat" of 55 pe
fOr
~he
c) V is foun,j from the w"ll-depi-h procedure
d) Factor multiplying lh" Thm\\as-term is 25
potenti~ls
:;.4
Discussion
The transitions to the particle-states at 0.00, 0.41 anct 0.9,
MeV yield cross-sect1on shapes
are very
s~milar
~nd
analysing powers (fig. 5.3) which
to those obtained for the corresponding
tr~ns~tions
to 57~i. This result can he understood from the fact that in both
a~e
cases the particle-states
stron9ly excited and
conse~uently
are
rsached dominantly by the one-step transfer.
Ths analysing powers of the transitions to the 1/2
states,
though, are somewhat different beyond 4So. Th1s difference 15 mainly
due to the presence of a larger two-step contribut1on to the transition
to the l/~
~tate of 57~i as the OW~A curves show (figs. 4.3 and 5.3).
Although the experimental CrOSS
both 1/2
of the
6ect~ons
~r~nsitions
to
states are almost identical in shape, the best fit is
obtained tn the case of 55Fe. This must be ascribed to the sensitivity
of the
calculat~on
to the deuteron energy (Or the Q-value), which
differs about 1.7 MeV (for the 3/2
a~e
and 5/2
states the difference$
1.0 and 0.85 MeV, respectively). It is not
clea~
if thi5
irnplie~
a weaker energy dependence of the deuteron potential, $ince we do not
know yet the
~xact
influence of neglecting
e~act
finite-range and the
deuteron D-state.
The unrealistic sharp
Of the J/Z- state (at 50
mi~irna
of the crags
peak~
of the oaloulated analysing
power~
and the 1/2~ state (at 30°) coincicte With
0
)
~ection~.
It
~s
obvious that
instabilitie~
of the
calculations will show up first at such places (see also section
5.4.4) .
The inclusiOn of the
two~step
fits to the crOss sections and
~lthough
transitions generally improves the
ana~ysing
pOWers Of the particle-states,
the difference with the one-step oaloulation remains small
(fig. 5.3).
The analysing power of the fLrst 7/2
state (1.32 MeV) clearly
shows a phase shift with respect to that of the s8cond 7/2
state
(1.41 MeV) and since the latter state, being strongly excited, is
mainly reached by a one-step transition, it
tr&nsition to the first 7/<
i~
evident that the
state receives a considerable two-step
contribution (fig. 5.4). This is confirmed by the shell-model
calculations
(tabl~
5.1) and is related to tne
presenc~
of a
lar~e
59
..)o:.:.rr'l
\l<':.ITI
20
'Q
, I'
I
I
r
I
&0
I r
1,0
80
III
ao
6{J
'.T-'.-"' ..r~'··"--~1
S5F. ID,dl 24.SM.V
3R- 10.001
---1>251
·,~.2S
---1ST
i
1
1:::-
.
~
I '
1
'
o.?~
~
'1
• r
1
( I
, J
1,1 IT
;m
'I
~
l'
1
I
I·' : '\1
56~ ISIdr Lt•. €MeV
512- 10.9))
1
I
0,2,
-,
0·2,
,
Q.Oli I
.,
"I
I,
,,,
, I,
to
}O
I
iff'
,1-..1"'6]
L..L_L-diJ"L.
"(r1"\,
V/'u.
D. ,'1
Ci~O~:' . ~ {~I:~(:!... /OHf3 <Tr~d r.~r1.()1..ys·~no pO?Jep,s 1'OP
u
t;h('
1';".lr1t,f.....·re
~'t(.;;te~ of ')··'Fc.
hav(, /")C'on rW,'rnaU""d
60
"(;0
th&
the
I/lhl~~ O(ll.cul.a't-ed
for~"ar·d rfll:p::iml)'lIr.
tI.ran.sitioi"/.,;: to
O}''OG.f;
s8otions
,.
"0
60
60
r.~~:~~.~¥~
•
'r
2D
"0
60
-'---r""-·---'-""T""T""""T-rr-'
8D
f' I' 1
\
7/2 lIn)
"-1
j
---t"2ST
(
o.)~
I
\
r
\.
/
O.2S
"1
---lSI
j
\
\
\./
I
I
:1
_I
'0,25
I
:I
00
'"~~I.,rOI~'H "",·_<·++t+
A
i.
I
D~
. 0,1.
D.l'.
.•. 1.
10
L.L.J. ..1..L. ~ _.. J.
40
I
I
~D
L" ••
L"L.
80
-'-'-'""'2;"O-'-'-'-'to" ,
Pig. 5.
~
do '
"c.m.
." 1i:,1"f\.
Cro088 SedUoYr.$ and
ana~ys{ng
row",!'$ for' the tmnsitioYr.G
to ttl" ?/2- states at 1.32 and 1.11 MeV. Tn8 OM- t)~US
two-et@p C'akmlation
{$
7/~
two-hole component in the first
ba,8IJd an the ampU/;wk$ KB1.
~tate
which can of cOurse only
be reached from the 2+ state of S6~e, whereas the ~econd 7/2- st~te
is dOminated by a three-hole OOmponent (two protons and One neutron).
The direction of the
caloulation~
structure
with the
the
ekperim~ntal
experim~ntal
transition
(f~g.
shift is cDrrectly predioted by the
pha~e
(fig. 5.4, 1.32 MeV), although the
data is obscured by an additional
OCmpa~i50n
~hift
between
and calculated analysing powers of the one-step
5.4, 1.41 MeV).
~he
instability of the calculations
of the analysing pOwer~ near 0° will be discussed in section 5.4.4.
The analysing
~owers
of the transitions to the 7/2
2.94 and 1.41 MeV are almost identical,
90
states at
we conclude that the state
at 2.94 MeV is mainly reached by a one-step transter, contrary to the
predictions of the shell-model
C~lculations
The state at 7.79 MeV probablY is the
the 7/2
(fig. 5.5).
i~obaric
analog (rAS) of
(0.126 Mev) state of 5'Mn. The analysing power indicates that
61
'l
10
11
I
r
1,0
10
1"1"
1
'"
"
,
"
eo
ouI '
I
0,15
do
d([
I
t
1mb/sri
•
~I'
i
0,1
5~~'ID,d) 14SM.v]
.1
10
I
I
I
I
,n
I
I
I
I
m
I
r ~
I
I
/"0 the
3(:'{~t·¥>)"ru., (1"f~d
i/,:r Dtate~
>
I·,"
A
1
L
IlL
40
L.
1. 015
,"j
L
511
BO
.,'}!': m.
a.n(lLysina
pOr.r)r=:'J')f}
foY' the
tl'ar18'it"ir,),r1$
"I" ,~. 94 and I. Ii! M(iV. Till£" one- pI-us
ICllo-step ",a/em i.qt·irm
l'h~
...
V
..~ 1..:.r'T)
Ct'OD.D
't
0.25
~
~
>
·~!lj~O
·1
7~\Sl
~
{8
based on t}w (£mpH I.",d.,,. ](B1.
"')'lin of this state is inde",(! 'l/2 (fig. 5.5).
'l'he two sets of spectros(:opic amplitudes SDI and KBI gener<tred
id8l'1tical shages of the crOsS sect.ions and analysing powers like th.:y
did for the 58 Ni (p,d) transttions. ~)o their relative merits have to
be judged from the 5p<lctro5copic fac:tor~ they predict (table 5.3).
~ne
~~o~n~OWTI
interaction clearly gives the best results. Therefore
<Ill CUrves lhspJ aye,) in this chapter are cal culated using- the set of
~pectroscopic
amplitudes KE1.
The relative strengt.h of the second and first 7/Z
state whBn
analysed with a one-step cri'lnsitiDn only (1ST), is about 4,0. This
lies closely to t.he val \le of 4.4 fOund by Majumdsr et a1.
Il)
from
the reaction 56 Fe (d,t) at 12 M0V.
In table 5.4
WC3
GOmpare ou):" "(esults with those obt:f.l.ined, by other
o.uthers. In 1111 c"ses we found
6,
(sometimes co,\sider<1bly) ,smalle"
Table 5.3
2
Comparison of experimental and theoretical C S-factors
theoretical
JTT
sol
0.00
3/2
0.41
1/2
0.93
experimental
KBI
1ST
1+2ST
0.42
1. 13
0.64
0.69
0.44
0.41
0.25
0.28
5/2
0.94
0.19
0.32
0.33
1. 32
7/2
0.04
0.35
0.58
0.41
1. 41
7/2
3.72
2.66
2.41
2.41
2.94
7/2
0.41
0.54
0.87
0.92
7.79
7/2
<1.17
<1.17
0.82
0.82
E
a)
x
lAS
2
C S extraoted from a one- plus
two~step
a1
DWBA-analysis.
Table 5.4
Resul tS of the 56 Fe (;,d)55 pe reaction
C2S (O+_>J")
J~
E a)
EE a )
TflE b )
pdC)
dtC)
"
3He ,,,-C)
0.00
3/2
11. 20
0.69
1.0
1.02
1. 12
0.41
1/2
10.79
0.28
0.42
0.49
0.42
0.93
5/2
10.n
0.33
0.67
0.54
1. 71
1.32
7/2
12,60
0,41
1.41
7/2
12,60
2.41
7/2
1/2+
12.60
0.87
15.70
0.51
3/2+
16,07
0.97
7/2
18.99
0.82
2.94
3
4.51 d )
0
4. SSd)
7.79 d )
3
6.1
0.69
0.69
3.02
4.65
5.!34
4.88
a)
E and Illl in MeV
b) This el<)?eriment
"
c)
d)
FrOm th<!: survey of Kocher I)
Given as 4.45, 4.82 and 7.73 MeV, respectively, by Kocher 1)
spe(:t:roscopic factors and we t.hjnk thG!s€: to be more realistic because
oC their consistsnOy with t.h~ sum rules
~rtg\\lar
'T'h ..
(li$tri.butions of the
g<lcr:.i.on for the st"t.e at 1.01 M~V
for the l/? + (,,_Sa Mev)
(see section 5.4.3).
~nal.ysing
power and the "eoss
(fig. 5.1;) closely resembl.e those
state of S7 Ni. 'rheeefore a spin-p"d,ty
assignment of lj?+ is obvious. The DWBI'> calculation qivc~
poo~
~ensltive
too
I'ather
It
fit to the data. This i6 partly caused by the calculation being
to the Q-v.=:tlue or tho
d(~ut.et'Qn
by the one-step DWBA "naly,,!,s ifi'). 1.11
~(]
I
energy, since the
l!~+ state of S7 Ni is (lescribed considerably better
tyansition to the
1,0
I
I
~
flO
8iJ.
r
dashed line). Notice that
1,0., ,1',0,
I
50
I
I
SO
I
I
I
'
5GFdp,!'l1 1.4.5 MeV
1J/(~.SI)
-·--lS.T
10
11.5
....--~
0.1.
I
I
0.01.
,Il'
~
d\1
~ i1i.:)
I"'. ~ f
05
1-
I
,
I
,
'J-/
-r
A
I
'56FI1(p.o:1i 2/1 5MoiV
3/2 ·~/I tim
---.JI2-
-----51,·
, 02b
,
---.. ./
01
0.25
,
o oP
Ft·.I).
;). U
I
60
i:)
CCOIJtl
$(l ...'dorr.,;
, I
80
,
10
I,
1,0
and anaiysl.no power'? [(n'
, I
60
~he
I
I .......I .
~o
....J
1:l'ansitio"", to
tfw -1/;/ (4.5.1) and the :if:!.1- (4.88) states of 55 pe . 'j'h&
dr+",wd
Cia'l!et;
finq/, ;;Ude'.
64
hm)!] 1m,"" L'ah'lI~ated ,.<$$um{nq a ::;/f/ (4.88)
the unrealistio peaks of the calculated analysing power coincide
again with minima of the
~rO~~ ~ection.
The shape of the cross section and tha analysing power of the
transition to the state at 4.88 MeV (fig. 5.6) olearly excludes an 1=0
or 1=1 tranSfer. we have to
between 1=2 and 1=3. The
choo~e
ana~ysing
power then limits the ohoioe of the J~ values to 3/2+ and 5/2-.
Assuming a 5/2
state results in a spectro9copic factOr Of 0.47 which
is improbably large. Moreover
Fulme~
and McCarthy
12)
find an 1=2
transfer for the 54~e(d,p) transition to this state. so we feel
confident in ass~gnLng a J~ value of 3/2+ to the state at 4.88 MeV.
The one-step DWBA-analysis gives a fair description of both the cross
section and the analysing power.
1n table 5.5 we oompare the summed C~S-5trength with the ~esults
of the ghell-model calculations (KB1, SD1). The total strength for
the particle-states is 32% lower thzm given by KBl and SDI. '11\iS may
be an indication that our valuas are too low. The more so,
tot~l
si~ce
the
strength, whioh has to be larger than 2.0, seems to be mainly
cono~ntrat~d
The
in the states
str~ngth
~nalysed
by us.
of tho transitions to the 7/2
states is given
rather well by the struoture oaloulations. We found about 60% of the
Table 5.5
Comparison of sum-rule results
Tal
-
1/2-+3/2 +5/2
3/2
1. 30
1. 72
1. 70
2.09
7/2
3/2
3.69
3.55
4.17
6.71
7/2
5/2
0.82
1.17
3/2.5/2
4.51
7.88
7/2
a) i50sp~n ot the tinal state
b) this ~xperiment
cl restricted to the states analysed in this ohapt~r
d) total c 2 s-st.ength for J~
total 7/2
sr.rength. The remaining strength is probably spread over
a labJ* nu]tlbe):' Of weakly-ex\.::ited states.
The ratios of the spectrosc;opic factors of ttl", tro.nsitions to
the particle sLates of '5 Fc and 5 7Ni are reproduced rem~rkably well
oy
the KBI
c,~l(;Cll"t:i.orls
(t"ble 5 .1;) altho\.\gh the e}{perimental
Spec:tT:OS(:0PtC ftl.(;i.:.0rS Q.rt:!
I!l
and 3/2
~m.;..lle):"
than the calculated on0s for the
st~t~.
states and la.ger fa, the 5/2
The total calclllated
"t~"ngth
for transition" to the
part.~cJ.e
states is smalleY' foX' 5~5l-"e than fo,t:' S7 Ni . 'l'his is col'lfirmed by the
experimental resul, t.s (t·.able 5. ~ 1 .
Table 5.6
Comparison between C2~~-factors of the reiictions
58
+
56
+
tH (p,d) and
Fe(p,d)
J"
1!21
Nucleus
55
57
V2!
S!2t
55
expo
J<;Ill
Fe
0.28
0.41
0.44
Ni
0.12
0.17
0.22
SDI
Fe
0.69
I.IJ
0.42
57 Ni ,
(J.91
1. 32
0.79
5\."
0.3.1
O.lS
0.84
O.SJ
0.28
0.B8
57
Ni
C2S(55Fe)/C2S(57Ni)
1/21
2.3
2.4
2.0
3/2j
0.76
0.86
0.53
5!2t
0.62
0.64
0.95
2
C .S
66
I. 30
1.72
1. 70
1. 56
1.<32
1. 84
~
·.-0
~5
-I
--D4
i'
:"+
---02
r"+-!' I ,,...+ -I '
I ....··' . , ·-i·--+- 4
., I I
S6Fe(~.d) ;>i..6M.V
.
J
+-++-+-+-f-l1
711-«·~"
I
1-
Ja.z~
/. . . . .\i
I
M
5Q 0.1:.(MtJ/S'-)
:'
·--··----jO
I~
0.1-
,
10
,
4'0 LI_._16~·L.L...l·8~·I.·
I.I,J
I
J
20
.:tor;.m
F-i-g. 5.7
CorrrpO;!'£$on of
l)QI'1:0U8
I
,0
J
111
,) ~.I'1'1 .
I
;;0
deuteI'on potenUats faY' U-w
transitions to th(~ ?I.~- states at 1.11 and 2.94 MeV. The
aaZauZated
fOI'UXll'd
Cl'OCS secl;ioi1~
hCi;-Ue INen
nO·~maU'led
maximum except foY' thO$'~ of -tlw
to the
tOIJest Pal't 0/
the
figuI'$,
67
In fig. 5.1 we show the e ff.;,ct of using the adiabatic deute);on
pot<)l)t,i"l D2 insteo.d of
th~
adjustGd deuteron potCt'lti.;>l [)4 for t.he
transit.ions t.O t.wo '1/2- ~t.ate:s of 55 pe . It. is cle~r frOm thesr:
examples that.
1)4
is to bG !?L'efe:n:ed to D2 fo~ the rcaction oO"e (;,d)
tao.
It is also clear from this figure that there
calculate'~
of instubility for the
80 degreeS.
~hey
~ppro~imately
coincide
ar~
three regions
analysillg pow,,!:s, around 5, 00 und
with minima of the
~r.056
soc;U,ons. Notice that in the etal:)le reqions the fit to the experimental
d11tl1
i~
quit." ,__ ea~onabl",
The i.nBt~bility around 50 depend~ upon the deuteron energy outsid.;, the "ucle"s (or the, Q-vctlue) and is relatively insensitive to
the dCu.t-e=on potential used .. The energy of t"he deuteron in.!3iOe thO
nucleus E~n ~ Eout - v(d)
so a change o£ E
(or the Q-va.lue) is
Qut
morC Or le55 eqUivalent to a chan,]e of the real well-depth V (d) of
J
the deuteron potential.
The inf\u~nce of E
on the unst~ble regions i~ shown in fig. 5.7
in
by three calcul"tions psrfOrIlled for. £l( = 2.94 MeV usinq potenti,al D2
8~
with Ot,
ClTItI 1,,% reduotion of V(d).
we feel. that the introduction of exaot finite-ran"", and
deuterons may stahilize
"amO
ti~le
th~
OttlCU1ated analysing
~owers
fill up the minima of the 1=1 CrOSS sections.
The use of
th~~
an~
lye.:in(J
pOWEr
in
as~ignin9"
the j-value is net
seriously il.ffect0(i by the instabllit"y I?robleml bGuL\use its
is most evident
maximum
(~
D~state
and at the
~t
j-dependen~e
the angle where the cr,oss sect.ion reaches its main
stable region).
,'inally, we like to >:emar,k that because Of the IIIentioned
instabilities the magnH.udes of the adjustments of the real anti
imaginary
well-rle~th:,;
of the ""ute,ron potential D4 are quite uncertain.
Conclusions
We
inv0st:t.g~~:~d
the one"'neut.ro)"l
l?.t~k"""up
tran::;itiona to niIl€
levels of SG Fe by the :r:eactioIl 56 Fe (tId). All of these transitions
wer;, dOminated by one-step triJ.nSfer. The
r8~ult'"
quite siIllila~ to those foun<l fO'- the reacHon
G8
were in gen,,:ral
se Ni (Prd) .
The shell-model calculations based on the renormalized Kuo-Brown
interaction yielded a better
de~~ri~t~on
of the spectroscopic factors
than those based on the surface-delta interaction.
We added some new spectroscopic factors and spin-parity
assignments. our spectroscopic factors turned out to be (sometimes
considerably) smaller than those obtained by other authors.
Finally, we found the calculated analysing
the transitions to 7/2
~owers
states to be unstable against
the deuteron energy inside and outside the
instabilities
o~~ur
we think that
"se~of)d-order"
at angles
~her9
nu~leus.
of especially
v~riations
of
since these
tha cross sections have minima,
effects like 9",,,,,t finite range or the
D-state of the deuteron may remove this prOblem.
69
D.C. KOch"", N\jcl. Data Sheet.s lb
2
(19'/6)
L.D. Rio:.:.:kert.senr M...I_ Schneiderr ,.1 .•J.
phy3.~ett.
H.
Rudolph,
K.
Ho ..~o'riO, M. Kondo, '1'.
saito
fl_
Shimi?u, 1,. Kato , K,
oJ.; ad" ,
Nucl.Phys. A343
.LC.
L~qg
0.))(\
60B
(lg7S)
r
463-
Krat~shaar,
w.
2imrne:rm~ln
N . . Matsuoka, S. Nagamachi. T.
K.
NOr"O
(19UO) 234_
!'._ R"'jt , I'hys.Rev.
).34
(1~t.4)
Bi52.
(1962)
6
R. SlIerr, B.F. BOIyman, ),;_ Rose, M.!'._ Rj.r:key and C.G. Hoot ,
7
C-A. Witten, Jr,
1J9 (1965)
,74.
1\1~72.
E. Ka:;hy and .!_!'. Schiff"r, Nucl.PhyS. 86
()966)
,07_
e r.J.
S-~;-
9
van [-10311, . .LP.M.G. Melssen,
~_D_
Was::senattr .. O.J.
Klein and G.:I. Nijgh, NU,::l.Phys. A291
~.G.M.
PoppemCl,
(1977) 63.
van Hees, pyivate communication.
vennj,nk and P.W.M. Glaudem"ng, Z_Phys. A294
10
R.
11
/I.R. Majumder "nd H.M. sen G\\pt<,I
12
R.B. Ful.mer <,nel Po.L. McCilrthy, PhyS_Rev.
70
I
c!'Jinb ilnd Y. 1("rJot"I
C.D. Goodman, J.n. 1>,,11 and C,B, Ful'ileL', phys.l<sv. 127
Phys.Rc:v.
and
19.
(1980)
Jndian J.Phys. 47
131
(1963)
241.
(1973)
2Ln.
396.
THE (e,C) REACTION ON 56 Ni and 56 Fe ~T 24.6 MeV
CHAPTER 6
6.1
Int~oduction
the
~n
of one-neutron
~na~ysi~
between different
pos~ibLe
tr~n~fer
proce~ses
~e~ctions
interte~ence
arises only when going beyond
one-step calculations. $0 inte.ference generally is a small effect
~s
we have seen in the two foregoing chapters.
two-neutron tranSfer reactions, however. interference already
~n
shows up in
caLcuLations. The importance of interference in
one-~tep
two-neutron transfer reactions makes the (p,t) prooess a valuable tool
fOr testing calculated nuclear wave functions. For the Wi and Fe
isotopes sophistioated wave functions are available 1,2) and these
were used in our oaloulations.
The signifioanoe of two-step prOcesses like for
instan~e
sequential
crartsfer (two successive one-neutron transferS) for "allowed" 3-6) and
even for "forbidden" 7-g)
(p,t) transitions is still
2l
matter of
discussion. In our case we included the following prOG8SS8S in the
reaction mechanism: simultaneous transter (one"step), sequential
in~l~stic
transfer,
multaneous transfer
scattering eollowing Or preceding
(two-~tep)
or
~equential
e~ther
si-
transfer (three-step).
Since there are indications 9,10) that e8pec~aL~y the analysing
power~
are sensitive to interference between
prOCe3~e~,
one-~tep
and multistep
we focussed Our attenticn on obtaining a good fit to these.
Several authors 11,12) found th~t if they used the "wrong"
relative
s~gn
of the
s~qu~nti~l
and the simultaneous transfers the
agreement with the Gxperimental data wag improved in particular at
forward angles. Therefore we
con9ide~ed
thts possibility in our
analyses too. Because we made the zero-range approximation and neglected
non-orthogonality effects 13) ou" results wilL probably give no final
answer tc this problem.
The calculations ot the cross sections and analysing powers have
4
been performed with the coupled~reactions code CHUCK2 1 ), which had
been modified to raise the maximum number of couplings to twenty one.
The results of our investigations are discussed in section 6.1,
while the
m~in
mOdel and
~ll
conclusions are
numbers
r~levant
~ummari~ed
in section 6.5. The reaction
to the calculations can be found in
section 6.:J. The experl.merttal details
an~
given in the next section_
71
6. 2
~.!l:tUilental.
(p, t)
Tile
measured
proced1lres
re"el.ions invest.i<)ated ;.n this "h<:tptOr hav<:o been
=imultun~'-'lls1y with t.ht;: (~J d)
",hid, huv'i! been
cl~alt
Or) Si.l N1 and 55 f 'e,
1';"eactioT1S
with in t_he chapter,,,, 4 und 5. We refer to these
chapter:; fOr mOre j.tifOr"[Jlet.tion on the experim(!!;I1tttl deL.tlils.
Tn t.be S8 Ni expe:rirnent the data wl2"r~ taken from \5° to HO
o
~n
steps DE 50, while for the ;;r;Ff2: experiment.. the ~ize of the steps was
4° fr.om 11° to 15°, :{' from 150 to 33° "nd 6° from .)_,0 to 81°. The
ener.qy resolution was about 100 kBV FWHM r of whtch most was due to
the ene rqy sp:l':-ead
0
In fiq. 6.1 "'€
f
the proton bettill.
displ~y ~
typioal energy spectrum
to~
re~~t~o~
the
S6t-'e (p,t). We hClve chosen the anqlc '::;0 as to gi,ye ~ 9DOd picture of
the rlependence on the polarization for the ground st",te of 5"Fe. We
analysed the tran~ition3 to the four low~5t states. to w~t the
{qround state)
~ the ~i (1.41 MeV)} tho
MeV). For ths saNi(~,tJ
at.
(2.56 Mev)
~nd the
2t
01
(~.96
reaction we obtctined d"'t:'" only fo):" the qround-
::!ib:lto to qrOuIlu"'state trClI'l.si tiofl.
To obtain absolute crosS sections we used the same normalization
factors as deduced in the an03JYO=:0S of the (PIC) re3ctionsl- The estimated
accuracy of the nOl":'malization is 20% in both
cases~
--_._,.,-,.. ,. _...
$
--
spin dowli
pin up
D
-'
0
0
uo
Cj.
cO
- '"
,-.-,
N
\
m
I
L. JI
-""
''""'
-
<~
I
230
c~
if1
-
'7>
;:;::
"\
/
0
'" '"I"
0
.".
I
\
I
I
I
310
230
I
I
270
310
CHANNELS
2,1. Ii
M(! V
fop bath
There have eeen other (p,t) experiments on 56 Ni IS-IS) and
56Fe 18,19) cefore, but at higher energies ~nd not with a polarized
proton beam ..
6.3
6.3.1
CalOulational
procBdure~
Reaction mechani5ms
-------------~-~---
In order to limit computing time and to comply with the
m~ximum
number of channels and Of channel-couplings in the code CRUCK2 we
to restrict our coupling
~cheme$
h~d
SOmewhat. These restrictions depend
upon the final state and will be justified in section 6.4. We desc<ibed
the ground-state transitions with
th~
COupling 5cheme A (fig. 6.2; the
2+ ~tate of 54Fe is the effective ~tate at 1.12 MeV, see chapter 5),
which consists of the one-step
transit~on
(pt), the
two-~tep
transitions
(pdt) and (pp't) and the three-step transition (pp'dt).
A
B
c
Fig.
G.2
Coupl-i"g 8(,hemes fop th" ooupled-ahatm(d8 aakulaUO>"18
if = 1/2-.
3/2-, 5/2-. 7/2
73
For the "nttly~is of th~ t.ran~J.t_j,ons to the 2+ states Of 5 ' ,["" we
~cF-\tt_erin9
rcplC'LGcd t.he inelast.lc prot-on
Then we have t-he £ollow;."<)
Thes~
(pdt\:').
ar-e
t_ran~i_tion5'
-~ulrml<lri::;~d
in scheme
wi t.h
dlff~ret"l(,;e
not :3iqnit'i.(:ant~
'le:.:i.~
th!7~
than
1>
~:cat_t_ering,
ewo-way coupling t.o eh", trH_,,,,
case the
by inelastic trit.OIl scat.t.e:r:ing.
(pt '),
(pdt '),
but we found that in OUI'
c:al,":ulation tlsing a one ... way' couplirlg was
~~
1,11 ma(]nitude of: the c:t:"oss section).
the cF.l.se of the O~ ~t:ate 'We only investigated ttl!)
t.r,:ul.~i tiO(I:::;.
6.3.2
(ptL ') and
(fig. 6.2), We "pplieCt a
(pt I)
dII.U
11"1
the (pdt I)
C ~ fiq. 6.2;.
(scheme:
Simultaneous two-neutron
transf~r
------------------------~-~------
Since we usud thO "oCt" CHUCK2 we had to make the ::;e);o-1:"<lnge
approxim.o.tion r which means that the
co~.nGide.~
the proton
ne\'tl:'Qn~_
"'ft~r
t-h~$
wi.th the
~enteJ.
ir'lt~raotion
of
m~Sg
is non-zero onlYI when
of the two transferred
simplifi<;:<ltJ.on the t.ran-'Oition amplitude for the
t:r:ansfe:r:- af two ne"L\tron .: ', with a gi\ven ang\lla):" momentum .J becomes:
(6.1)
fJ
PA
is the normalized form factor of the neutron pair piOl<;old-up fl:Qrt,
the shell-moclel
o:ebit_~l~
D ,;I..n.i.l A and coupled to an angular momentum J.
'r'he spectroscopic amplitudes SIjl. (pAiJ)
f\mctions according to f.o1:"mllla
configuration (pAl in eg_
effect~
(:J,
(6.1)
follow from nuolear w~ve
24). The coherent summ1ltion
ove~
the
is the origin of the interference
in th", one-st.ep calcul at.ion as mentioned in section 6.1.
'rhe zero-range normalization constant NO(ptl has been tak"n equal
too -1560 MeV Em 3 ",td.dl corresponds too DO (pt.) = -,00 MeV
The form f"ctors
F~,
have been generat.ed by the
fm 3/2 _
Bayman~Kallio
20)
method with the;, root-mean-squ<lr" r"dius Of the triton equal to 1 _7 fm.
The neLtlrOn bOllIlu-ctatl";;' Wu-V8 fu(1(;t.i.OI"'iS "'ere corastr\).(.tt;!:cl ac..-::ortling to
t.h0 well-depth procedure
USiI~g
half the tWO-I"'ieu.tron sep.::lration energy
of t_he target nuclous as tho binding energy,
We assumed an inert 56 Ni COre in the _>ANi eXl?"r:iment and derived
the sp",ctro'3Gopic ampU tIldes S 1h (p, ;J) from the
Wi'!ve
function6 of
Koops ,'ma Glallaemar\o (given in ref. 1)), which are based on
in.te.t'"f:l.c:t;"j on
Or
t.ht;!:
$u.t'f.."cr:~-del
occupying t,he 2pJ/2,
74
lf5/z and
ta toru). The two
"pl/z shells.
C\
residual
ty.~nsf;e;( . . . ne\Jtrons
are
Since the amount of (lf7/2)-2 configurations ~n the ground state
Of S6 Ni is probab~y small, we think that neglecting the lf7/2 shell
will not change very much.
~he 56 Fe and 54Fe wave functions have been taken from the work
Of Vennink and Glaudemans 2)
They allowed lp-lh excitations of
neutrons and protons with the hole being in the lf7/2 shell.
~ogether
with twc lf7/2 proton holes this results e.g. for 54 Fe in
(f7!ZI-n-2(p9!2,fS/Z,pl/Z)n configurations with n = 0 and n ~ 1. We
have chosen the wave functions computed with the modified KuO-Brown
Table 6.1
spectroscopic amplitudes fOr simultaneous twc-neutron transfer
Process
J;"e
(5,5)
-0.21
(3,3)
-0.58
(1,1)
-0.29
(5,'»
(5,3)
4
(5,5)
-0.55
(3,3)
-0.78
(l , 1)
-0.28
(5,5)
-0.30
-0.09
(5,3)
-0.32
0.16
Ni
(5,1)
0.13
(5, j)
0.36
(3,3)
-0.28
(.),3)
-0.71
(3,1)
-0.29
(3,11
-0.42
(3,7)
-0,16
(5,5)
0,09
(5,5)
~0.O2
(5,3)
-0.01
(5,3)
-0,03
(5,1)
-0,07
(S,I)
0.09
(3,3)
0.21
(:1
,3)
-0.15
(3,1)
0.25
(3,1)
-0.17
(3,7)
0.16
(3,7)
0.43
(3,3)
0.38
1)
0.21
(l,
2
Ni
0,1 )
0.20
(3,7)
0.10
(3,7)
O. !l
Fe
a) we took the effective 2+ stdte of 56 Fe at 1.12 Mev (chapter 5)
b) we on~y cOnsider~d the amplitudes originating from the first
2+ ~tate of 56 F0 at 0.85 MeV
75
residual interaction
.
bettl.:I"
2)
re~t.A1Ls
(KB1), because
th~8
intera~tion gen~r«lly qiv0~
Eor the Fe J..,sotopes th,;:.,n the
~l)rJ:ace .... delta
inter-
action (.5Dl). WG have calculat"J t.he .p~ctr"",copic «mplitudes accordinq
to formula 15.40 Of ref. 1)
Gare
been taken l.O bring Che resul (;ing
ht\~
S'pect.~o",copic
amplitude"
int.o .Une wi t.h the phase convcnUons of CllUC1(2. We refer t.O chapt.er 3
for
detail~_
G.l?~~ct-.t.:'Qgcopic
The
amplitl,l0es are given in table 6.1 as they
ente!eri CHUCK2 (apart from the normal.i.zation c;on>:lt.ant. NO 1 _
Irl our calculation the sequential transfGr proCrl::r.::cJ:=. through th.e
following inte:r,mediate 8tates~
1/21' 3/21' 5/2.1
for ~)S}"e al!30 the th;(ee lowes:.t '1/:':
almost e"hausted by these
.st~te9,
fOl"
S7 Ni ar'Ld in ~ddition
states. The transf~r strerlgth is.
80 we
feel confident in neglecting
high"r stilt.eS,
The one-neutron fOrm f<1ctors were comput_ed wIth the well-depth
p,'ocedure. The
billdi~g
Cllergy of " lle"tron in orbit p is given by
Eb(P) =Sn(A) :I-£p(A-l) forth"
(po) step and Eb(P)
(dt) step. Sn is the one-neutron
for t.he
number of the target
nl.)c:-le~]g
.;:tnd V:p the
Sn(A-l)
~Ep(!'.-l)
5ep~rilt,~on
energy, 1'> the mass
ex~i tat ion
energy _ The minus
and Lhc plus siqn are chosen for particle and hole stat"s, r""pGctively.
We appli"d tt,,, finite-range correction of the interaction in the
local-energy approximation in all one-neutron trdtlsfr:-r-channels. For
this we used the Hllithen form with the value of ll.69 fill for the filliterange parameter.
The one"'nelltYOn t,l"'"ansfer amplitudes have to be normalized with
empirical cOTIstant9 1 becal,se the finite range of the interaction has
not been tre"-t-ed 1 n "n
NO (pdl
= +122.5
to DQ(pd)
= -12~,5
convenience'
"~~ct
way. We took the widely used villues
Mev fm3!~ i'lnd NO (dt_) = -225.0 MeV f m3/2. They cOJ"re"pono
MeV fm 3 / 2 and Do(dt)
= -184.0
MeV fm 31 ?
For
p,ake we have gather.ed all normalization constants in table-
6 _1) _
'l~he
spectroscopic amplitudes for the one-neutron transfers have
been computed 21) wi th
th~ nuolE:.~ar
wave functions of section 6.3.2
""CertC!ed cO u\e lnl~r'ITIeulilt,e nuclides 57 tH and 55 f .... we r"nQrmaH~ea
the
~p ... ct:t'osc(>p:lc
ilITIf'l i tude:> of t_he
(pd) -8t_ep in orde:.: to reproduce
the experimental erO>;>; >;Gction>; of the
4
76
~nd
(p ,d)
reactions (see chapte~5
.5). The rd .. tivc mugnit.udc of the (p'd) and (pd) amplitudes
'!'able 6.2
Spectroscopic
PrOCeSS
0+ +J
}'e
Fe
Fe
:fe
a)
b)
b)
2++J
J+ot
J+2i
am~litudes
2J(2j)a)
for sequential two-neutron
Sl,
transf~r
P'-:OCeS5
2J (2j)
0+ "J
S';!
5 (5)
-0.58
5 (5)
-0.7j
.3 (3)
0.83
3 (3)
0.95
1 (I)
0.53
1 (1)
O_~5
7 (7)
1.53
5 (5)
-0.24
5 (5)
-0.39
5 (3)
0.16
5 (3)
0.30
S (1)
0.36
511 )
0.31
5(7)
0.13
3(5)
0.01
.315 )
0.28
3 (3)
0.42
) Cl)
0.87
3 (1)
-0.35
3 (1)
-0.36
3(7)
0.10
Ni
Ni
2+ "J
1 (5)
-0.06
1 (5)
-0.31
1 (3)
0.48
I (3)
0.3'1
5 (5)
0.70
3m
-0.74
1 (1)
-0.61
5 (5)
-0.28
5 (3)
-0.27
5 (1)
0.37
J (5)
-0.14
3 (3)
0.41
Ni
Fe
J->-O
+
J"'2~
5 (5)
1. 00
3 (:l)
-1.00
1 (1)
-1.00
5 (5)
-0.11
.3 (5)
-0.20
0.10
.3 ())
0.35
3 11)
.3 (7)
0.17
3 (7)
0.56
1 (5)
-0.30
1 (5)
-0.14
1 (3)
-0.64
1 (3)
0.10
7 (3)
0.40
7 (3)
0.28
tM transferred angular momentum is indicated by j
we took the effe~t~ve 2+ state of 56 1"e at 1.12 MeV (chapter r,)
Table 6.3
Zer.ou.y.Q.nge normalizatior'l
Reaction
a)
(;on~taI'lts
a)
DO
NO
+122.5
(p,d)
-122.5
(d,t)
-184.0
-22 5.0
(p, tl
-500.0
-1560.0
used
in t.he code
light-ion
Sp8<:
includes
CHUCK2 f NO
troscopic ampli t.u,~e",
l""ding to the same final state in ttl" d"uteron channel remained unc;hdnged. Since we :pe~forrned calculations with a.nd wit.hout inclusioD
Of ilh"::~la6tic proton !;':cattering we employed two slightly differerit
L'enOrrnali.z;i:ttior'l.6. The corresponding spsctrosoopi0 arnplitud88
~re
g~,ven
J.n tilble 6.2.
w~
r:emark tha L for Lhe grou/),j-statc to
the relation
Ii SL/~(P~fO)
= Slh(p)
pd
ground-st"t~
x Sl/?(\)
dt
t~:a/)st tion~,
is approJ{i,mate1y valid.
This rGla.tion ena.bles one to check the slgns of the spectroscopic
amp1.it.lldcs.
We desc'(ibea the inelastic pr.oton and tritotl.
,s;oatt~ring
u,s;'lng
t.he form f"ctor 'liven by eq. 4.1. We found no determinat.ion of the
deformation p<tr""eLcrs F.'2 from 5"Fe(t,~') experiments in t.he literature_
Th~refor~ Wr:!
used the values obtained in our
0'\0\11'1
Ittboratory from
5"Fe(p,p') experiments 22). The ~2 parameters fOr inelast.ic proton
scattering were taken also from experiments 22)
done before by our
{jI"OUP. FOr tho proton channel the actuOll value6 are
~2
= -0.22 an.d
~~ = -0.24 for ,aNi 0.45 MeV) and 56 Fe (1.12 MeV), respectively (se~
chapter 4
62
2
arHj
5). For the triton channel thG values
~<l
= +0.15 and
= +0.13 have been used. The inelastic lriton ~catter1ng is not
ap]?lJ.""bl" "1;0 the Ni experiment, ))(:Ci:lllSe we
Inea"L1~'ed
ord.y t.he
transj. tion to the grOund st<lte of 55 Ni.
The siqns of the (le formation pa:rameters depend upon the phas86
of the nuclear wave
function~
and can only de determined by a
Gcopic calcul.cJ.tion of the form, factor.
between. the va.rious
proce.s~e:Eil
~icro-
Because of the int8rference
a :r.ever.sal of the sign
o~
132 considerably
changes the results of a
s~mp~y
mult~ste~
calcu!ation. OUr procedure was
to try both Signs and then to cOmpare the results of the
ca!culat~ons
with the experimental data. In this way we were aLway5
able to make an unambiguous choice.
~he optic~l-model
parameters for the triton potentials arc listed
in table 6.4. 'the pr.-atan and deuteron potentials have been taken
the chapterS 4 and 5.
we
f~om
chose the modified adiabatic deuteron
potential 04, becaUSe this potential describes the (p,d) experiments
best.
We tried three di ffe),;ent td ton potentials indicated by
1'F, TE
and T6. TF is the ),;e$ult of a fixed-geometry fit 23) to elasticscattering data of 20 M0V tritons Over a broad range of the mass number
A.
Since there is no TF potential for Sb Ni , we hQve ext-rapolated th~ Tl"
potential Of ~4Fe to 50 Ni, We havs also applied an energy extrapolation
tc the
T~
potentiQls. Doth
extrapol~tions
were done according to the
energy and symmetry dependence of the glob"l triton potential, Of
Becchetti and Gr8enlees 24). Th~ TB pot8nti~1 w~s obtained f~om a
global fit of (t,p)
re~ction
and 1.=2 using a triton energy
data with angular momentum transfers
o~
L~O
12 MeV 25) , which is close to the
Table 6.4
Optical-model and DWBA parameters
Name
TB
TF
Nucleus
56 Ni ,54 Fe
56
54
T6
56
r
V
0
a0
Ii
"
Wd
a)
rr
a
rc
I
155.0
1. 24
0.70
]0.0
0
1. 37
0.70
1. 25
Ni
144.2
1.
24
0.69
34.1
0
1. 43
0.87
1. 2S
Fe
143.6
1. 24
0.69
29.4
a
1.43
0.8)
1. 2S
Ni
137.6
1. 10
0.85
0
32_"
1. 31
0_75
1. 25
Non-locality paramet8r
p
0.85
d
0.54
t
0.25
Finite~range
n
0.85
R
pa~ameter
(p,d)
(d,t)
0.69
0.69
R
a) V, Wv and lid ~n MeV; the other parameters ~n fm
79
v<lluo we nce(j, The Tfi p"tential was teste(l only In the, ,8 Ni experiment.
ThtG potentia.l ha? beo:?:n applied with :some ::;ucceS$ to Ti{P/t)
reac-
tiNts 26) aI.d orig.i.nated fL'am fits of elastiC triton and :lH" scattering
d"t" for cnergie" of 20 MeV te 80 MeV. Sinee TF g8nen.lly
g~ve
the best
rEsult!=.: we us€d thls polentla).. in most of our calculations. We did not
aCCO\.1nt for
W"c.:.~
WOrc:.i.:=.
diffc~'r0i\C~~:i
t),~er::l.
thC:!:
.:=:.~rn~
in tr'iton energy in
Tn the calC'ulation~ where a
final cha.nnel.
two-way
In othr.::r
final state5.
c;Ol.:ipling is prS:5c:H}t, .1,0 t.he
process one must chang\::: tht" iIIl8.9.ln';"'l"Y triton pot8nti.~1 SOtue-
(t,t')
what.
t.h-:::~
t_(itan potential for al]
n).
applied an estimated reductio)1 "f
We
5~
to the imaginary
poten t:"a 1 and we found thi;lt the change Of OrObS sections and analysing
powers were
Tll~
negligiol~.
msrits of th", non-localj,ty cor.reotions have aJ.reaoy been
discussed in chapter 1. All
"al(;UL\t~on$
descr,ibed in this chapt.e:t
were done with the non-locality corrections. The corresponding pa:raare given in table 6.4.
metey~
comparison of exact finite-range and zero"'raI"lge
A
of on,'-step
ang"la~
(pt) cross sections 3, 28)
$how,~
calc'11~t.ion.9
that the shapes of the
nisttibutions are verY similar and that the main difference
lies i.n t.lle ma'lnitude o( the cross sGetion. The exact finite-range
calculat,l..ons often lJndt;:!rl7!!3ti)"YJ.;:tte the 8xpcrirncnta 1
C:1"OS$
sections
consinerably. Although the effect of finite range on the magnitude
of (po) "nd (ot) cross seetiorls is negligible, the same is not t.rue
for the two~5tep pr.ocess (Pdt) as a ~ecent study by Burch e~ al. 3)
showed. Again. the only difference between the finite-runge .;:tnd zero . . .
ruTlge u..::'.i(;"uli:1.t..ior)s t:'5 tht:: magni tude of the cross SCCt-;i.,O().. :r.c W,;:ts a190
fOlmd
:l)
.
l.hat. the ,'elCltlve phase angl" between \-he
(pt) and (pdt)
t-ran:;;..ition noes not change significantly by the intr.oduction of ax act
finite r.ange_
~;o
i.f we renormaliz8 the zero-range
II factor A ann th" zero-range
(pdt.)
{pt)
amplitud~
with
amplitude with a factor u, we C<ln
slIl'lulate an exact finite-range c.::..lculation.
However, we do not know the magnitudes of
w
~
and
~.
Burch et al.
obtained A
0.69 and
27 MeV and
'" 0.60 and U = 0.39 for the satl)e reaction at 40 Mev. Jt
3)
0.65 for the reaction 02' Ni (p,t)6D Ni(OT) at
is important to rea.li:.:::c that these rE:!51)1t-s depend much UPOIl the details
of the assumed internal deuteron ttr1d triton w.;!.ve functions.
80
We decided to determine the ratio of the
A and W from a best fit to th8
to tne grol,lnd
~tate.
.
~ormalization
parameters
power of the (p,t) reaction
Next "- norml11i"ation Of the calculated cross
e~perimental
section to the
analy~ing
these values have also been
one fixed the magnitudes of A and
~sed
in the
c~lculations
~.
Then
for the excited
states of the final nucleus.
6.4
Discussion
The analysing poWers and the cross sections of the (p,t) reactions
leading to the ground st~tes of S6 Ni and 5~Fe have been co~uted from
transition amplitudes of the following form (scheme A, fig. 6.2)
T = A{T(ptl + T(Pp't)} +
~(T(pdt)
+ T(pp'dt) f
(6.2)
The re6ults of the calculat:i()ns with (A,\Jl = (1,0) and
1;0>:
= (0,1)
(A,~)
"OFe and 5 RNi using the potential TF are shown in fig. 6.3. First
we like: to remark that the analysing powe;r.9 do not behave ;Like the
ot::!riyatives of the
crO~8
~e~tion:!;:
at for.-W'ard ,;:I,ngles as they do for
instance in the reaction 116 Sn (;,t} 6). A conspicuous difference
between the "'"perimcnt"l o.nalyslng powers of SSp", "nd S6 Ni is th" ctccpnes~ of the minimum at 25°. This is reproduced qualitatively by the
ca~culations.
we see that the
seotionS than
th~
the cross
simultan~ous
tranSfer
9iv~s
l"~ger
cross
sequential transfer and that for both transfer modes
~ectiorts
are of the same
orde~
of magnitude
~~
the experimental
re:sults.
One now oan determine the ratio of ). and 11 [,om fitting the .. ngul"r
d~strib~tion
to the
of the CrOsS section or from applying the
~naLystng
s~me
procedure
power. We examined both ways. Since we were
inte~ested
in the relative Sign 11) at the simultaneous- and sequential-tr.ansfer
amplit~d~s,
we
a~so
looked into negative values of
chi-square values x~ and
xi
A/~.
Using the
defined by (3.11al and l3.11b) to
indicate the quality of the £its we found in general two minima for
both X~ and X!, one for A/U positive and one for A/u negative. 1n a
consi~tent calculation the positions of the minima of X~ and X~ should
coj,n<;~de. This is !!lOre Or les5 the cas" for S8 N;i., but not for 56 Fe . The
sensitivity to
vari~tions
of A/U is of the same order of magnitude for
the cross section and the analysing power, but the minima are mOre
81
20
I
I
1.0
fill
I
11'
,I)
f1.0
I
I
I
,u
~o
III
I
I
I
.
80I :
-=-5F~(j;'t) Str.Fcdg ~.)
i
l
2HM.V
._ _
11 \
1-
\.
c
\
.
\
\
\
\
0·1-
ptl:D~.I,
____ pd 1+ pp'J {
I
I
1 I
'/
1/
II
/--\
11
\r'fl 11 ,~, r)
I'
I'
'I)
/---~
\
I,
~~ 0.01.
~-'-l
\
\ I
\ I
~!"!Nlq})n
L
~ ~../I'
5.5 N11g
,
.
::--\
~
11
.-
~.)
2&..6 MC!:V
1·-
-u
I
=.~ =--= ~~+t~:·~·t
05
-1
\
\
\
\
0.1.
1-\ . . ' \ /
\
IT....
\. I
T
i' . . . -.
"'-- -....
\
\
\
~
I
\
01.11
.)
J
J
)/o'! ) .. ,,1 .. 1
,
L
I
J•. J\" I
J
(';(1
0
O.S
It
'I
v
f
J
!.
IW
L.I)
J.
)
)0
FlU'
6.;i
Ca r,:n/la-t'(onG oj' ')'hm"Uan,w,-,~
",Yr.lWf"''-'
i,()
aI"
(IT'mlnd Dtatea
n -I)
oI
l
L.
I,
)
(0
~c..rT'1
..)"':.r(j
BO
r~,1d ~f!qwmi;-ial
(~-: 1)
S1 re and ~6Ni.
pronoun<:,,,,j fo~ X2 th"n for X2 . MoreOVer the ""lue of A/~ which 'lives
Il.
0
a minimum x~ depel1d~ upon lhe nQl:'Ih"l.i,zation procedure used for the
CX'09~
section.
~~o
we decfde(l to Ch005E the best fit of
th~
aW:llysing
power u.s thE" critErion for detErmining A/U.
The magnit-ude,;; Of A
~tld
\l hav'~ too be found [yom the ab901ute
cross sections. As a rule one normalizEs the cross section to
th~
ftrsL. mts.xl.rnum. For an 1:.=0 transfer this maximum is situi:l.ti::d -Ltt 0°.
co()'sequerlt.ly. one he:as to
extrapol~te
the
experimeI"lt~l
dttta to this
angle. FOr 58NiJ howe:ver ~ the experimental points at forwa.rd a.r~gl'8::;
urO too ~curce for a reliable determination of the maximum at
0°.
Since we wanted to c0~~re the ~eg~ltg for S8Ni and 56Fe r we decided
to noc-mali;o;" th" cro.;:", 5ect-Lons to the $econd ma)(imum in both caSES.
l\c:tt)ally tQ!:' 5Er'f€ w~ (',:hO!;le: the ~xpE!rin~ental point at 51
.'r,"eference r while
r(";Y'
0
as a
S8Ni we ~pplied a best-fit pY.'oced~lY.'e f:rom. :~OO to
80 0
,
because in the latter case the maximum at 50
0
is less p~ominent.
we will returt'l. to this normulization pr.ob1em later on in thj,~ !:;le~tion ..
The resulting values of A and
there is no minimum of
Xi
~
are given in
6.5. Since
t~Qle
fo>: A/~ posiUve when the triton potential
T6 is Ilsed, .,,'" give the results obtainec) frQm the minimum Of X~
instea<:1.
The ratio of A ~nd u tends to be independent of the triton
potential.
Thi~
property is a oonsequence of the two [allowing facts.
Firstly. the main
l~es
diffe~ence
between the various triton potentials
in the magnitlldes of the
relative
transfer,
no~rnali4ation
whe>:e~s
sections are quite
c~oss
sections with about the same
factors for simultansous and sequential
the analysing powers und the shapes of the cross
Secondly, the interference angle between
simil~r.
the stmultaneous- and seqllential-transfer amplitud0s is rather insensitive to the triton potentia,. This angl'" is
gener~lly ~bo~t
90° when both amplitudes are ~arge.
If we look at thE!
chi~square
values in table 6.5 it is clear
th~t
the best fits are obt"ined with 1'F. But th", fact that TB cOnE.istently
gives cross
gection~
which are too ltirge is even more important for
T"b18 6.5
NormaliZation
parameter~
Nucleus potential
TB
TF
Til
T6
obteined from the ground-stat'"
transition~
2/ a)
Vv
XA N
2
X(J / N
0.76
0.81
1.065
46
440
-0.11
-0.17
\.59
41
718
0.69
0.64
D,9~
39
609
-0.10
-0.13
1. 30
49
771
0,67
0.53
0.78
~5
85
-0.22
-0.22
0.97
13
154
0.67
0.40
0.59
29
99
-0.21
-0.15
0.72
18
187
0.69
o.ss
48
85
-0.20
1.04
16
196
0,81 b )
-0.20
a) N is the number of eKp",rimental pOints
oj
obtained from the absolute m1n~mum of X2
(j
83
'J'able 6.6
Ratio of calcuL"l..ted .,:3.nd experimenta.l
sect.ions
DrosS
for th~ ground-:=-CQ.te t..:r"i;I,nsitions
54
Solution
C
P:r.ocess
+
c
a)
a)
56 N,.
Fe
TF
TR
TP
'l'B
0.45
pclt+pp'dt
0.39
0.4.3
0.15
ptrpp't
0.91
0.97
0.J9
0.54
pt
0.91
0.97
Cl, :00
0,36
pdt+pp'dt
O.ilG
0.84
0.82
0.79
pt.rpp't
0.04
O. D4
0.08
0.09
pt
0.04
0.01
0.06
0.06
~he complete C+ i!lnd C
Calclllations have 1.0 as
ratio
the ~hotcp. 0,£ the tri ton potential.
F'ar example, the 50 Ni Orol::';,s
section calclllated from the (pt) amplitude only (\=1) iG 7..4 times
l~rq(:r
thctn the experimental croSs section (for TF t.h:i.s ractol:" 1.5 1.2).
Th"r"for" "U curves present"d itl the figures have been computed with
th", TF polential.
In table 6.6 we have listed the magnitudes Of the cross sections
of the separate
at 50°
(Fe)
proceg~eg
and 80
0
relative to the
exp8riment~1
.cross
s~otioI1S
(Ni). The sequential trans;fer is the dominant
proceS9 for t.he C.,.",A/p -:: 0)
:!=lolutions, while the 9imultaneous transfer
(;c)!ltributes less than 10*. _ Tn cant't"ast, the sequential- and simultuneous-tl-ansfel-
C1::"O~3
.sections f<..""Jr the C+ O./w > 0)
solutions are of
th.e St\m(~ Qrd~~r of maqni tude. Comparing the results of SoPe and S 8Nt ir)
r.hi,', l.able, He'.' C;Q"oto.",:;y of the 58qu8nti,,1 c;0ntributcior, J.S not,,,,worthy,
],'rom fig. (,.4 one can see that the analysing );lowe);' of 5ijN.t is
fittt:d better by the sol\ltioll with
V)l "
0 (c_) , because the dIp at
2:;0 is de"c~;,bed well l)y c~. Concerning the analysl.ng power Df 561"e
(fig. ".4), it is clettr th"t the m"in difference bet.ween C+ and
o
c~
is
ag~J.n around 2S , but now both fits arC equally good. The CrOSS section
Of S8 Ni cl"arly f~vours the C+ solution, which is r8lated to the f~ct
that the "bsOlute minimum of x~ is readIed fol' 1./)1 positive, Ttle C+
CrOsS sl":?ctior'l. of 5G·r~e i~ too low at forward angles. 'fhis intr.oduces
81
-
Lor, " T,,·~P"·I
10
~\
I
\
I
~
'1
r'i..8p,,;·
r
20
;
,
,
,
80
56Fa(p,tl !)L.F~ Ig.5.)
\.
Ij
~4.6M'V
--c,
'=-
'
0.5
----C
-0.5
~~;
(inh
0
01~;-
r~.rl
[.
t
'·1 ."
";
'J"
I
__-c.
\
I
fit
,;
j
ll,6 M,V
"' , \
IL.
'I
!3-eNi(p,t) 56N1 1g ~
]
----c_
O.G
I
.,i
\.
J
~
\
\
\/
. I
lO
l
I
1.0
I;
-0.5
'/
4
I
I
60
I.!.
80
1
j
11--.1
80
J.
10
~-c,1Y1
Fig.
6,1
The ()ompZ"te caZcuLations (l]QupUng
~'3h(lm(l /I)
with Vl.I
pO$itive (c+) and AI).! negatil)e (C_) for the ground
etate transitions.
an ambiguity in the normalization and hence in the values of A and
The extrapolat~d value of th~ cross section of 56 Fe at 0
and if one
0
~.
is about
6.0
mb/s~
ha~
to be multiplied by the factor 1.7. As mentioned before we can not
normali~e8
s~etiOn
to this value then the c+ CroSs
0
give such a reliable estimate of the ~ross section at 0
in the case
of S8 Ni.
we will
dis~~~s
two reasons why One
maximum when
norm~lizing
second-orde~
effe~ts
prefers the first
Fir~tly,
are thought to be less important for this maximum
than fa. the rema1n1ng
nicely illustrated in
~s~al~y
the calculated cross sections.
pa~t
~ig.
ot the angular distribution,
wh~ch
6.5 by the (pt) and (pt)+(pp't)
is
prOG~sSes.
Now, the relative contributions to the cross sections at 0° are rathe~
similar to the results at 50
0
and 80° (table 6.6) and sinoe these
85
,esult" ShQw that [a, t_ll" C+ solutions <tll proces"es gi.ve c;ompar"ble
contributions the"r:'e is no cleClr-cut fir::;t.-order process .i,n these
CClsr:::s.
S0condly 1 the shu.pe of the "-11gulur dist.rihution be:yono the fih"gt:
i:?; of ton s(u'lsitive to
ma~imurn
solut. .1 Qn~,
triton
ther:::e
liowev~'"<:r r
th(~
::;hap~s
opt.ica1. potentials. Fo:r the c+
~re
.remark,~bly
s:i.m.i.lar for all three
potential~.
l"inally
Ha~hi.moto
I
co,(rection applied Lo a
and Kawai )0)
t_wo-~~tep
::;:howed that u.
IlOI)-orthogon~J i ty
calculation of 1f8 Ca (PI t) 46 Cu
(ot)
at.
40 MeV conside,,(ably r.et"luceti the magnitude of the cross section at 0°,
,,1 thou']h at 27
M~V
t.M.
~
conect.ion is much ",m"lle)"" _ Anyway, we will
m~xim\lm
quote the oon,;eqll0noos Of norrnulizing to the first_
at the
Clppropriato pl<lc8".
Comp.:u-in,] the ma'lnitu,;les
~i.!Jn
same pot_ent.ial and
0
f
~
(table 6 _5) calcul"ted with the
of VII w" find considerably larger values for
~\.iFe than [Or ~1(!.N.i. Thi(,.; 1.-:; connected t.o the fact that for 58 Ni the::
«It) ,"Ol);lli.tudes (tilbl" 6,2) are about
il
fo.ctOr 1. S l"rger t.han fOr
%{'"e_ Ilene" tbe ~equenlii'll ~r"nS£e:r.· wit_h ~~1 <:lives lat"ger
c1:05S
than fOr S6 pe (fig_ 6_ 3), whereas the magnitudes of
secr.tons for S8 Ni
the eXI?e:r;.meotal CrOSS sections are quite close to one another. Sin<;e
the ampU t.udes for
space
(no
"/2
sa Ni
holes)
have been obt"inGd from a simpler configuration
than the one used fer ,oPe (up till one -//2
hole) we th.ink that: t.he values of ~ extJ:"acted (rom the SOFe an"lYEis
are mOre realistic_
Accord.lng to Burch et al.. 3) finj,t;.e-range cor,rections r.esult in
t~ansfeJ:",
a reducej.on Of the CrOSS sect.iOn fo..: bOt.h types of
expect \ and
gives u
~
II
t_C! be "mallei: than one_ The C
1_S9 and the c+ sol.\.ltion
~
,,0 we
sQlution for. .)(iFe (TP)
-, 1.065_ 'rhis iE another point.
in favour of the latter possibility_
'rhe r(,L,tiv<.) C I cross section of the
(p.~+pp'
t)
t ..ci'lns£e):" is about
two times lar'le,\: For S~~'e than for 58 Ni (tabU, 6.6) _ This hi'ts been
ca.used by the different v;:;l:lues of A.
Hecause the C+ solution s~emb
more r~Uabl~ for: SIJNi t.han for 561;"e (oette, fit""
oOinoiding minim" of x~ ilnd X~) we think that the
section is closer to O.
:~9
see table 6_5, "ncl
(pt+pp't) cross
than to 0.91.
Sumrnarizin'] we can say that A/)l is 0_"!±O_2 and O_5±0_2 for 56 Fe
and S8 Ni
(after correction for the
The <lrrOr of 0_ 2
86
WilS
(dt) amplitudes), respectively_
obtained from comparing X2 (min) with 1_ 5 X2 (mi,n).
The~e
reSUhts are rather independent of the triton potential and they
are not affected by the uncertainties in the
normal1~at1on
since they
follow from the analysing power. They should be compared with
A/U = 1.1 obtained from finite-range calculations 3) of the cross
section of 6 2 Ni(p,t)60 Ni(Ot) at 27 MeV. Keeping ~ fi~ed at 1.0 Feix
et al. 6) found V~ = 0.6 for 116 sn (p,t) 114Sn (Ot) at 2S Mev.
We wondered if the fact that the 56Fe data are not described as
well as the Ni data (table 0.5 and Ug. 6.4) Could be connected wj.th
pre~ence
the
of
~
fQur~hole
component in the
wav~
ground state of 54 Fe (because of mixing with the
we added the tranSfer of two 7/7
function of the
oi
stat~). The~efo~e
neutrons with an adjustable strength
to the originul calculation. A new sGArCh was made with A and Wfree.
,C
,. ,- I
eo
-.-.--,-.-.--,,-rn
50
T
58N,lji,t1 56 N,lg 51
«_~
r
20
n- -, 1 '
MeV
--PI
--- -- pH· pp t
,.1
I
\
t
~
00
dii"
(mb/sr)
~
A
I,,~ '~··I··-+-"""'··I·,-t-+-
5~N'(~, t) 56Ni I~
, 1
1<,6MoV
---C.
0.5
- - - - pt+;;lIjt
,11'--M7
-c.S
so
Fig"
6.5
60
Calculations with and ",ithout in~ta6ti(! pl"'oton seat-t81'ing
fol"' the gZ'our!d-state tl"'ansiUon of 56 Ni ; A/ll is positive
in all eases; see caption of fig. 6.4
fOl'
C+_
87
Yet the best fit of the ~naly5ir\g pOw8r was obtained with nQ (7/2):>
transfer ~t. ~ll.
We want to finish this subsection with some remarks about the
coupling scl"'m,, A (fig. 6.2) which we Llscd for the
gI'ound-~tate
transitions. Leaving out the inelastic triton ecattering only changed
th", magni;'.Llde Of the
CrO~S
section w,ith about.
5~,
whi(,h we vey.ified
with the limited cal elll "tion (pt) + (pdtl, In ccmplin'J scheme fl, which
+
is used fo):" the 2
states
l
have dropped the inelastic proton
Wf:!
"""t.t.ering. Tl"i,e):"efDre it is of som", itIWortance tD lODk at the eff;,,,t
D[ thig simpli£jcation on the grDund-sti'ite transition. We perform;,d
a
(ptl + (pc~tl cal,culation for Ni with the same
value~
of ), and
)1
as
for the c+ o;olution (fig. 6.5). Although the fits have be('ome wo/:,se,
tl'l-8 change of the cross section and the y.nalys.ing pOw~r is not. drastic.
So we expect that leaving ont the inelastic proton scattering
i~
a
reilsonable app.roximatlon for the calculation of the 2+ state::;. of
aOursC one could ~lso optimize
I;run(:"Q.l.~(,.~
WDrse \X~ ;
190)
determin~tion
xi
for the ground state wi chin chis
but. in tho!1.t.:. c:a$t;! the fit of the
~ch~mel
CrOgg
section gets
i'ind \ chan<Je~ from 0.5:3 to O. U. Apparently the
of A i9
ve~y
sensitive to the inclusion of
(pp 1 dt) and
(pp' t), W.. fOlAnd that. th", cha"ge 01 \ is chiefly d\le to the omission
proC~88
Of the thr8<'l-sC"p
p~OGB65
(pp' dt). That the import.anee of the (pp' t)
should not be underestimated is clear from fig. 6.5 where we
(,o,"par", the (ptl and the (pt+pp'tl c"leulations using \ = 1. Th"
description Of th"
"r\<lly~ing
power is substant.ially improved af.ter
inolu81on Of the two-st."p process (pp' t) .
6.4.2
!~~_~I_~~~_~i_~~~~==_~f_:~~=
The 2
+
st.at-os have been calcul.ated according to coupling.
B with the norm31:Lzation par.amete.t' values from table 6.S.
potentia15
(TV, TB)
sch8m~
Both tritOfl.
give ve'f:Y similar analysing powers and {absolut.e)
cross sections. Therefore w.. onl.y
pre~ent.
the results obtained with
TF. wo already mentioned th" fllct that. replacing the two-way ('oupling
+
in tl\~ tritorl channel by <lone-way coupling t.o the 2 st.ate leads to
vii.t.ually the same r.esul ts. Consequently, the oalculations are really
bal;ecl on arnplitlAdes of the following form.
'J' = A(')'
(pt. , ) + 1'(ptt')) .. ll{I'(pdt') + T(pdtt')}
(6,3)
In Cur diScusSiOn we \.<Jill disc'drn the sir'l"lult.aneous tr.ansfer (pt'), the
88
o1",--.-.-',",,-0"-r"["O'-,-,---r,::;O'-,,,,,a'i'-n"1-:----- r-~--'
5~f~(p.n 5t.F"e(2i)
-;~, -T-'-'P ,-
50
'1
-
, 80, '
2<_oMeV
025
I
__
".,"~
\/,\ 1
---- pdf,o_es
do
iITf
f---+-+--t-i'---+--+--+---j---1---+-'-i--'--r..,.--t -+
>5peIP,n 54 Fe{'i)
:i!/.. ()Iv19"V
Imb/-sr)
l
------, , I ' ,
l
,
I
i
A
I
I
-'
_ _ pt',O,73
Itl.2'5
- - - - pd;t'~9.1
-D_2S
O_Ot
LJ.~-'-!2"'O.Ll----'-~<O;--'-~C_~_J---'-_J-~OJ---­
l_.l,
20
';cm,
Fig,
6.6
,j,. ,J ........
.J ....... ".l_"I, .J.
40
I,
I
50
80
vern.
CalauZations of simuZtaneous (A=1) and
transfer to the 2+ states
Of
s~qU(lnt'i.a~
()J=1J
54 pe , The oross sections
ape nopmatized with the factors disptayed in the figupe,
se~uential
t~ansfer
(~dt')
and the inelastic transition (ptt'+pdtt')
as the basic procEsses.
Fig. 6.6 shows the Cross sections and analysing powers of two of
these
p~ceS5es calculat~d
with A = 1 and)J = 1. The
c~oss
section
Curves are normalized to the first maximum to allow a better judgement
of the shapes. The experimental analysing powers of both states are
quite different (almost opposite phase). which is connected with their
dif~e~ence in ~tructurel the 2~ being predominantly a two-hole state
+
and the 22 a three-hole st~te. The (pt') t~ansfer oy itself describes
the transition to the
+
2[ erOsS
~eetion.
2i
ort the
~tate very well, ~e first maxim~m of the
~ontrary,
is better fitted by the (pdt')
process. But neither the (pt·) nor the (pdt') transfer produces a
89
UI
1,0
,
2U
, I '
.
(0
51J
11
I
so
I
<>_~~":'COI~:~ CI'F" II;,
i 025
'-~
A
,
no,
I
\\
/1
I"
\,1
c+
~ 1 'I
,
1- 0.2>
I~""'"
.. - C H 1),72
,Ill
dO
imb!:;'"
/
'
,
,
,"
J
\
,
,
u
!
I
\~
,
:l5F~q:;'t) S!:r,'~ (2))
"
)/, F.M<!'V
0.2,
0.01
,
,
"
bU
'.'
'I
V.I
I
10
,
.1_ L ....L. .~
l'h8 ()Oml-'i.ete oakul.aUom; (OOUP!.,;"!!
pOS"l.·.I~·r'J"
•• _
1,0
1
I,
I
(,I)
~O
i'~(.m
8d'Elr~(, fJ)
(,JUh All.!
(17+) I.lrld A/I! n"9utive Ie) POl' the :/ ctate
fJf'<Tn.~'ib.>Jt1(-L
IJI}W (j)'~().~$
1::3{';:(:JL-·t~O"ft~"; Ql~e '1.o~ma"ll;::;ed
w,'!>th the
'l'able 6.7
a
H.atio of cillculated )
~nJ ~xperiment".a.l c:t'o~!:'i
sections
for tho 2+ transit.ions of S4 Fe
Solution
j-o'inal
ptl
pdL I
state
c ·r
c:
a)
the C+
2i
Complete
pdt'
calculation
0.80
1. 29
2,0
0.60
;or
0.91
0.13
0.84
2.0
+
21
0.035
2.9
2.9
1.4
2;
0.04
0.28
0.38
0.7
r~sults
normalizes the
90
pt. '+
haIJ6 to be mul tiplier) by 1.7 if onE
g~ound-state
transition to the maX.
~t
0°
reasonable
anal~sing ~owe~ for the
fD~ the
2i
the (pdt') cross section
2T
state. Note the smallness of
state. Thers is nO Simple reason
th1~.
for
The angular
di~t~ibution and analysing powers of the 2+ states
clearly select the C+
the C
solution~
a.!3 the better ones (fiC/. 6.7), altllough
solutions give cross section normalizations closer to experiment.
However., one should not
wor~y
too much about these
~ormalizatton5
since
they turned out to be rather sensitive tc the magnitude and the sign
of the spectroscopic amplitudes.
In table 6,7 we h~v~ brOll9ht together the cross 6ections of v~rioll5
u
p.;r:;-oc8s5e5 normalized by the experimental cros,s section at 25 _ '1'he
,0
0,1"" , , T ,.. ,...,
e. "".,. . .
r'"
r" ,T
1"
S6Fe(~Tt)S"Fc(2;)
.,.~O,., ""P'r"r'" ~Q.' .. '.. r·?,o'·i·
I
"/
I~
<0
.0
SO
CT'!"
24iJM~V
..."
~"
0.2,
··/··10
"
'J
I
i
'!
~ --c.
-
.. ·021
- - C.'ppT
00
dO
(mb/sr)
·t,-,,····,~,·-+--1-.t--.+--I.-+-
....+-I-t--I-.
A
'OPeIS, I I" F<11\ I
·0, ~~
.L •. l __ L ..L_...L....i.......L....L..L.
20
<0
.0
1..l.....J
~O
~(.m.
Fiq.
fi.R
Tlv,; c:atei.tation -indu.cUng ir!dastic scatterin,J in the
proton c:hcmne ~ (+pp '/; I) and U!e one OJith
e.)
negaUve
" in an cases;
compared with the C+ r&8utt; A/lJ {" po,git-ive
the cross sections have been mu Ztip Z7:ed
,v7: th
I;)';' far~ ~oY'r;
indicated in the figure.
91
prOCCSS~5
multistep
q:i.ve C":onside.rable contributions to t.he tot.:=.tl
~.n­
c,"oss ::::eclton.::;. Notice the (1t.asl,1.(: Char"lg8 of magnit\]de when the
ela5tic tral)silioCl
~s
incl'l~ert.
Th~
j.l1ulastic tran9ition i)1terferCS
dc:=tructively for th(~ fir6t and c.:on~t.l:'uclivC:1:1y for tl1e ~ecGnl~ 2+
w~
~15Q
c:u.lculation~ b9:~e(1
pert'ormed
tunded with an ),=0 (pp't')
t.):'an~fer
on coupling scheme E ex-
(cable 6.11. The chang"i,nt.roduced
by this addition,,-) two·,.,~tep P):",QC~~S J s n~gli9ible fo( the 2; $tat~'! and
disast):,ol15 fOr thw
2t Rtat" (f1q. 6.8). The situation for the 2r state
mlqtLt be improve,j by also i.nclUlling the L=2 and L"1
processe~.
Ipp't".t·.' ) and (pp'dt1".')
ilnd the
Anyhow,
(pp' t ')
thi~
t.ranSfer
""aches uS to
be C'aut.ious when neglect 1 n(",J inelastic p,l":"oton scatte:!ring fOr tra.nsi tions
to ",«::it.eu states in
t.I~~
tdt.On c;himne!. The (:
p"(,"oce!';l.'3 b~cf.t.u:s:e of. the !:',mall value of
~'ina Ily,
w<)
of the
:!t
(pp' t')
A.
show how t.o ,Jistinguist\ the 91.IJn of t.h" defOrmatiOn
0;, (Hg. 6.B).
paruMetet;"
sollltion for the
~ilditj,on
",tat.c, .lS not ch"n'le(l signifioantly by the
Ob","rv" 1".11<" norm<Llization ar the
CrO.~~
sec-lion when f3;~ ,i.e nCgutive_
!~~_~~_!~!~~_~!_~~~!
6.4.J
Since t.he
~h~ll-mOd"l
calculations of Vennink "t al.
dom:i,nantly
cJ.
fOll'(-hol~
8l.~tt;.~,
2) d" not
O~ 5t.ate is thought to be p:r",-
cant";.n f"\lr-holo State5 and t.h"
W~
do not have cal.culat.ed spectroscopic
rtmplitude::: tit our- dispOSCtL. Therefore we computed the analysing powe"("
and the Gross sect.ion wit.h adjustable spectroscopic amplitudes. In
r:?(;l.
combined tho:!: one-step simllltaneo't)9
we.;-
s8quential
t:l:·.~n,sfr:.!"r of
mil~imizing
the (":hi-sc';l'..~.J.re of
two ·I/?
1)(~utr6Ijr5
th~
t:ran~fer
(schsme LI
und the tWO·r9tep
fig_
G,?).
By
analysing power and next nQrll'l.ttl$.7.ing
tJle cross section to t,he maximum at 0° we d~termiIl~d both 9pecttC$'~OriC
amplitlldes. "r11e rosult is given in fig. 6.9.
Although t.he tit of the completo;> calcul<ltion is fat;" t:rom h,.l.ng
perfect r it :i.s certainly
(pt')
~ncj
sect_L:m,
bett.~r
i.h~,J't
the one""step calcultttion. The
(pdl') proc"ss"s separately give
rCsp~c.:;tively.
The s.trengt.h of both
SI/2(7/2 1/2;0) = 0.61
P 8 1 / 2 (7(;:)
pd
,
$lh (7/0)
dt·
2.50
40~
and SSt of the crOss
pro~t7!sses
was found to be
~,
,
I
,
'
'.
'1""T5Z;~~'~~'~}i54~i'~'~" ' 1' '1:''
, I
I>
I' ,
2"'.6M~V
--pl'.pdl'
,,0.\1)
1
----~~'t'
i\ -,
I \
\.
\ 10
, !
I
10
J
J .1
40
I
I
_L
50
, I
1 .1_0..._1.1
, I
"(;.m
(J. (I
III
'0
'c m
10
80
I
I
I.
60
!J,
I.
80
'Ph" (,orrtp~e t" UCI~r:Urat?:Or' (r:O"pUl1g $dhremrl C) fo}' the
tr'ansition to the seuond 0+ otate of ,)1 Fe togdh(!!' lJdh
th,!
If.
rx~r'tial pr'O(,!$ 88 (;' 8 ,
w~ ~ubstitute
amplitud~
(pt')
the values of A from
tabl~
6,5 (TF) wG find for the
0,78 and -3.6 for ,>0 and A<O,
r~spectively.
Apparently
). h<1s to be positive because the theoretical limit of ISl/2 (7/1 7/2;0)
is 1.
Finally, fox- a pure one-step calculation thE! sPectrOScOpic
amplitude i& 0,90 if one
ta~es
A=I.
~~~~_~~~~Z~~~
6.4.4
The multistep c~lculation5 fit the d~t~ of the 0+ state~ and
the
2t
stttte significantly better than the one-step DWBA calculations
(figs. Q,~, 6.10 <1nd 6.11), The
2t
state is alsD well described by a
pure one-step transition (fi9' 6,)0), However, the magnitudes Of the
CrOSs sections are surprisingly
~lose
to the experimental
value~
in
the DWBA analysiS (tnble 6.8).
The wave function of the
2t
state 2) consists prE!domin~ntly of
two proton holes coupled to angular momentum 2. The renson that the
one-~te~
two-neutron tr.ansfer has a large cross section stems from
the
that thex-e is a considerable component in the wave function
fa~t
of the grO~nd state of 56 pe with two proton hOles coupleQ to angul~r
momentum 2. The magnitude of the (pt') CrOss section of the
2r
statE!
(table 6.8) should be cOmpared with the results obtained dt 52 MeV 19)
where the calculated cross section is an order of magnitude too small.
I
~o
10
I
~()
l'
6()
r'
SC-;rt:("~.tJ V'Fe(y ~ )
1,\
..
21+.5M~V
~
..._- ( .
\
\.
----pt
\
~
0.,
0.71
I.
I
\
QI:
;.-..\
,
~1
~
(-\
\
•• ' \
1
~.~.
0 o,~
[~[bl,;)
t·,
I
I
I
,\
I
~C.Nd~S)
'S8 N1 (pi,t)
2'" f.. M~\j
...._- [ .
______ ptx085
: ,
I,
I
,~
(0
L ,.,1
iO
.1..
I.
,.0
I
50
"c,m.
·/'i"ie 0/11'·-:) 1".,1'
OQ
iCI.dation,~
i'~(,~ l(::',~.1. (.~ j,. / On;j j'ol"
~.h:.:
(flUl'mr21.-i.zed) a'1d tlw
C+
Y.Y"Otmd-8 t;(+f.:(~ i. Y.f..I~~b-i. 't:ions.
Table 6.8
Ratio of
llW13l\
ana experimental cross sections for the
ground "tltLo
pctent.ial
a)
uno the 2
\ a)
+
states of
0+
1
51
Fe
+
2~
27
"l"l'
0.84
l.OO
0.82
0.96
TB
0.61'0
1,00
0.S7
1. 02
fOr Ni:
A 0"0,92 (1'F) and>' = 0.61
('1'B)
0' E" . ··1·· T"?~'1"""T-r4·i""T"'T·;g"'T--;-"r-~r.. ·
2U
I
.0
r
,
1 ;
T
60
r
I'
56~lp.!) ~'Fc{2i)
II
H_'Mo"
I 020
,
/
,
I
I
I
0_01
~,./
I
i
itI: -u 25
_ _ C... )[1,7
'!
____ pll). 0,87
do
IT
(mb/~rl
'r
1
+-,_.+-+--1.+.,.,
~
.. I..
T
,
t '
6.6F"eIP,D 6./'F<a/2i,)
ZG,0"10V
_ _ '" C.j..O,io9
- - ....... (:It'!!:
0 7S
0_20
U-----'---'----!;"'OLJ.I-·1.."10· I
r"
~,m
Fig. 6.11
,I, ,,~O ..,L ,..J..
eo
i""L.).."•.r...."I,
1,0
"\.,, 1
,0
I ••
80
-l(,m.
'l'h>J Or1>J-st>JP and the C aakulati-ons foY' ~he 2+ Mute
r
trandti01-tG_ The normati:-:a"tiol1 fa(;to"['$ ar\! given in the
j"igure.
95
6.
~~
COrlel usions
""-~~.----
In
(p ~ t",)
th1~
Chap~cr
t::r.,~n$~, t::l,on:::::.
one'~5tep
with the
'(1"1
we
1:1
otudte~
11
CF.!.~~!=:
£~ve
nBtu~al
m\~l tistep
process. The sbape- of
th~
ana.ly::sinq PQwe rs improve .F:fte1'" .:l.1di tion of
Ther.efo)~e
q(~nOrd.t(;,]
parity
(i.e. "allowod")
procE::':=':::H:)::; compete st ron9J. Y
eroS::; sect-i.ons Q,nc)
t~li.!
mul tist.ep
proct?,s~e~.
the shell-moeJel wave functions 1,2) which WGrr...: uSed 1:,0
the spectroscopic arnpl i tudes seem to bO rCI i.:l.ble.
IPhe ciJ.lculations have been performed with t.he :zero-range codoF:!
CHUCK~"
We have tY'l.ed t,o m1:lke IIp fo:r: the omission of 8ffects
fin,i. t"~ ranrJoF:! by ren.r...h'ma ll!':.lnfJ th* ,e',im1.ll taneous
tl:,',71liS(oF:!):'
;:",):'tl'pli tudes.
By m1.n:i.uit.;~inr']
SUl=~l
a~
seq1J~::nt..ial­
the chi-square of:' the Q,nQ,ly::siilg
powe)", of thF.! qround-f:;tat-e
r,ran.~.itions
norm~li7,.:.It;,j,.(H)
of which ont;':! yi.elded
paralTlE' ters,
and
"T
th~
we [ound two solutions for
f.L
'(everse relative
phase between the sililul to:'lneol).$ and the seql,lential amplitudes. But I
consideYing all yegults, the 90111tion with the natural
pt~~c
clo~rly
is the best one"
'rh<:~
rQ,tio of. 'the normalization parameters;" and
\...l
was found tG
be approximately eqllal t.o 0.7 foOl: ~,IlF8 and 0.5 fOr 5B Ni. SO we had
to :t,"educe the 5imul taneOllS ty.ans f.er two to four times more
seq~ential
tr~n~fet.
th~il
tk'l,e
REFERENCES
P.J. Brussaard and P.w.M. Glaudsmans, Shell model applieatio n $
in nuclear
2
~.
8~ectrosco~y
Vennink and P,W.M.
(North-Holland. Amsterdam, 1977).
Glaudema~5,
J,D. Burch, M.J, Schneidsr
a~d
z.Phy~.
J.J.
(1~80)
A294
241.
NUC1.Phys.
Krau~haar.
A2~9
(1978) 117.
4
N. Hashimoto and M. Kawai, Phys.Latt. 59B (1975) 223,
D.H. Peng, M.A. Nagal:"ajan, M.R. Strayer, M. Vallieres and
W.'j·. Pinkston, Phys.Rev.Lett. 44 (1980) l03}.
6
W.F. Feix, J.B. Polane, P.J. van Hall, O.J. Poppema, J.M.J. van
Oosten, S.S. Klein and G.J. Nijgh, to be published in Nuclear
Physics
M.A.
A~
Nagarajan, M.R. Strays); and M.F. wsrby, Phys.Lett. 68B (1977)
421.
8
Y. Toba, Y. Aoki, H. rida, S. Kunori, K. Nagano
fifth
Inte~national
Phys~cs,
9
M.
~nd
K.
Y~gi.
Symposium on Polarization Phenomena in
Nucle~r
Santa Fe, august 1980, cOntribution 2.110,
Igara~hi
and K.-I, Kubo, Fifth International Symposium on
polari~at~on
phenomena in NucLear physics, santa Fe, aU9ust 1980,
contribution 2.112,
10
K. Yagi, S. Kunori, y, Aoki, K. Nagano,
Phys. Rev. el9
(l. 979)
~.
Tagishi and
11
N.B. de TaKacsy, Nucl.Phys. A231 (1974) 243.
12
P.
Guazzon~,
M.
~.
Toba,
285.
P~9nane""1,
E. colombo and P. Crcscentini,
prsprint.
13
T.
(Jdagawa, H.B. wolte. and
W.R.
coke):"
PhYS.Rsv.Lett. 31 (1973)
1507.
14
P.D. Kunz, Univ. of Colorado, unpublished.
15
H. Nann and W. Bsnenson, Phys.Rev. CIO (1974) 1880.
16
W.G. Davies, J.E. Kitching, W. McLatchie, O.G. Montague,
17
G. Bruge and R.F. Leonard,
18
,. Suehiro, Y.
K. Rarnavatararn and N.S. Chant, Phys,Lett.
~shizaki,
Phy~.Rev.
27~
(1968) 363.
(1970) 2:100.
C2
H. 09ata. J. Kokame. Y. Saji, A. Stricker,
Y. Sugiyama and I. Nonaka. Phys.Lstt. 33B (1970) 468.
19
T. Suehiro, J. Kokame, Y. !shizaki,
~.
Ogata, y. Sugiyama. Y.
S~ji
and I. Nonaka, Nuol.phys, A220 (1974) 461.
20
B.F. Bayman and A. Kallio, phY3.ReV.
1~6
(1967) 1121.
97
21
l\.C.M.
22
P.,J-
V,;>l1 Het"':!!=::r p:r:-ivate communicatioI\.
van ITf.l.l1,
,LP_M_G_ Melssen
S.D. WaS88I)aar, O.J.
l
s.B. Klein and G.J. Nijgh, Nucl.Phys. A291
2:1
Ii:. R.
182
21
F_D_
"hy~.Rev.
Jr and G.W. Greenlees, in Polarization
Ptl~nOmana
Uecchetti~
R~ar.t.i.on,~
Un~,vt"':!.t'sity
01. H. J3iu3chall and W. llaeberli, "ds) p, 6132,
of Wi.SCQnsln P).:'f':!.!;':5
2",
rr.w. nRn" NI.(Cl,Phy;;. lItn
2f..
fLW
~7
r
B.::ter, J.J.
,~n~
Kr-:.tu$.haar,
P.D.
Klll11.
N.K.
Glandenni!lq, D.L.
(19(,8)
.T.(;.
111:1.
in Nilcleat'
The
Beery and A.G. Bittir,
Flynn, D.D. Armstrong,
(1%9)
E. R,'sr.,
\1971).
(1%8) 625.
e.E.
A~n,"hys.
MOSS ..
76
KLS_J?_ Kinc)J R_E . . L . . Green,
(197~)
HCI\driu and O.N.
131.
Takem~l::;.a,
28
T.
2')
T.K. I,j,m,
30
N, UltShimoto <me1 M, Knwtti, preprtflt._
98
poppem,~,
(1977) 63.
'N'-Ic1.Phys. A:??O
Phy~.R,",v.
C7
(1973)
(lq74)
12813.
'31_
4.17.
J8rrj.S,
phyS_T,ett_
26~
THE (P,a) REACTION ON 58Ni AT 24.6 MeV
CHAPTER 7
7.1
Introduction
Although direct (p,a)
r~~Gtio~g h~ve
been studied for guite
~
long while 1), it is only during the l~Et few yea~E that ~ol~rized
proton be~mg have been used 2~7) to induce them. rt wa$ found by
Van Hall et ai. 2) that th~ analysing powers gener~lly show much mOre
structure
th~n
the cross sections and that they
between positive and negative parity
digcrimin~te
clearly
whereas the cross sectIons
st~te5,
u~ually displdY a j-dependence 8) only.
A S8 Ni (p,a) experiment has already been performed by Tagishi et
al. 6) at the proton energy of 22 MeV, which is close to our value of
24.6 MeV.
~he
data of our
those of the 56 Ni (p,d)
expe~iment
were taKen
sim~ltaneou6ly
(chapter 4) ~nd 55 Ni(P,t)
with
(Gh~pter 6) y,eactions
and we thougnt it worthwhile to extend the multi5tep
c~lculation5
applied to these cases to the (t,a) reaction.
We on~y observed the transition to ,he 7/2- 9~ound st~te of 55 co ,
bec~USe the alpha P~rticles, that leave the 55Cc nucleus in an excited
state, arG too slow to pdSS through the 200
telesco~es.
ThG
experiment~l
~m ~E-detectorS
Of the
resuLts are in close agreement with thosB
obtained by Tagishi at al. 6)
7.2
Reaction model
The 7/2
ground state Of 5SCo is assumed to be a pure ~7!' F~oton
ho~e in the SONi care and the (p,a) reaction ~s thought to proceed by
the pick-up Of the two valence neutrons of S8 Ni and one f7!2 proton.
The reactiOn model is summarized in the coupling sCheme of fig. 7.1,
which i5 scheme
~
of fig. 6.2
suP~lemented
with a (t,a) and a (p,a)
step. We did not take into account transitions via excited states of
56 Ni.
The one-step (p,a) transition was treated in a cluster
app~oximation,
a
t~iton
wherein
th~
tran5ition was describeo by the piCK-Up of
bound in a woods-saxon
w~ll
with the binding
ene~gy
equal to
the experimental separation energy of the triton. ThG number oF. nooes
N of the form factor is equal to 3 and
tollow~
from
th~
harmon1c-
99
lor
l:olApl/:nq .coheme
,J Il" :=
7/:.·:-
;';/,~~
j
,
"the ooupl..ed-(:h.annR1..$ c:a1..(;ul.a.{·-((.")r·l:J
il/;~
oscHlat.o,' rU,l.e: f(~nj+l~) = 2N+L, where the n
i
and Ii erE: quantum
numbers of the three con8titl.l.ent nucleons. The geometry parameterS Of
the
potenti~l
well
~re
di ffu3eness parameter
listeQ in table 7.1. The 6mall size of ths
w~s fO~lnd
to be necessary 9,10)
for one-stl!:\p
calculations in or.de);":" to get good fits to the cross sections and is
w1dely a(;ccptc>d now.
~'h"
well-depth procedure resulted in
<l
valuG of
170 Mev for the: triton binding potential.
'l'he
(t r (I.)
step was calculated like the other one-nel.ltron trc:l.TI:=>fGr:=;
2lnd uSlng the zero ... ranCJe normalization constant DO (ted
= -479 MeV fm 3 / 2
corresponding to the CHlJCK normalization constant NO (t,,)
-678 MeV
(m,l:1.
The proton "11d deutur<Jn (D4) potenl,iall:; are discul:;""d in chapter
.:;
(~e~-::t.j.on
<1.3.3, and tl)e tritOn potentials 1':10' und 1'8 in chapter. b
(,.',etion G.:l.:;). We t.yje.-] I:.wo alpha-potent.tals 1\1 and 1\2 (table 7.1).
The first one (Ill) en-i,].i
nate,~ from"
study by Brobowska ",t. a1.
was applied suCCeS8£1),11 y t.o the one-$t.e)? cl.uster
reaction
~'f;Ni(p,(1) "t. 30
A2
was taken
of
~lph"
The
particl~
by Smits and. Siemssen 10). The potential
f:!
I': a 1,
~4.7
Mev On nuclei in the region Of 1\=60 by
12).
nOrt ... locality pa.yaiT),eteY,"5
for the neutr.-on, t,-:-iton antI the alpha
are qiven in tAble 7,1 together with the
meters (LEAl
11) and
of t.he
[rom a global tit to data obtained from elastic scattering
particle5 or
B1.l(}Zrtnow::.ki
MeV
."naly~is
ED,' the
finit~-r1in~e
para-
it,a) t.ran31.tj.Dn. we applied no finite-range
COrr""tion to the triton pJ.ck-up.
WG i.ntrOduced normalization
(t.,a)
~1:.0p,
normalization of the
100
p~rameters
A and
)J
for. the
(p,(I.)
and
respeotivelYr because of the unoertainty of the
(Pr(1.)
tr~nsition
and the neglect of effects such
'rable 7.1
a)
Optical-model and other nWBA parameters
N~me
V
r0
"0
wv
rr
aI
r so
V
so
a
"c
sO
AI
175.3
1.46
0.48
26.0
1.46
0.48
1. 34
1\2
204.7
1.42
0.52
24.0
1. 42
0.52
1. 34
n
l;J)
1.25
0.65
t
b)
1.2S
0.45
1.
c)
O~65
1.25
param~te" ~
Non-locality
n
(t,a)
t
0.85
2,
0.25
0.20
a)
b) V, Wv and Vso in MeV,
t~e
0.70
other parameters in fm
V is found from the well-depth procedure
c) FactOr multiplying the Thomas-term is 25
as exact finite-range. The no.malization parameters were determined
analogously to the procedure of section 6.3.6, namely by minimizing
the
chi~square
va'ue of the fit to the analysing power of the (p,u)
reaction and then
normali~ing
the calculated cross section to the
experimental data.
7.3
Discussion
The resul ts of the procedure for determining A and 1.1 are displayed
in fig. 7.2 for the C+ and C
solutions (see chapt.er 6) using triton
potential TF and alpha potential A2. It is clear that
th~
difference
with the one-step calculation is small but significant (;,specially
for the analysing power) .
'the contribution of the
Ct,a) piilth to the CrOSS Section (at 3{')
is about fOt for both the C+ and the C
C+ solution to
~ive
solution. Since we found the
the correct description of the (p,t)
~e~ction,
we
will fraIl) now On only disouss oalculations based en this solu1;ion. The
(p,a) step alone gives 130% of the c~oss section (at 35°), so the
interference be1;ween the (t,a) und
(p,~)
path" is
dcstr~ctiv'"
(the
101
""C,m
;J'l..1T1
I
10
I
I
I
I
1.0
I
I
I
,
60
:-,
I""
80
'1"1'
.
"T''''j''''
58Nlq~',od .2/•. 6Mr-V_
712"
20
-r""
'0
, • "'.",.
60."
"'1
~"'T"'-T""r
'" eo;--"l
(g,:;,)
--lsrA1
Ij
---ISTAI
I,
I
'0,2,
I
I I
I ~ I I ~
58NI (~,CL) l~,$M ~v
:
I
"1Q- (1 ';)
'''-/"\'~
. :
-
C.. T F
[; .n
\
,
,
('. ,13
,,
,
,
GO
40
CP()~~.s ~?eej.~£()n8
'f:o
(fu:~ 7/2
80
i
I
J
, I ,
20
I
I
J
1,0
60
J
J
I.
e~
I
0c m.
and an.a.lyc"I.:n.a
poweY''[; pOP
the
l:~an81..~ l.-iO"fl
(lY'or,{.ixi et;Lx"tC! of' 65 Co • 'phe eymbo"lc are ex-
pl.ahuki in I;h,,' te:r.t.
102
-
."''-I
\
,
~~,m
VI~UT
~
\
1SI
)0
'
\
interference angle is 130°). ~he 6~ contribution of the (t,~) path
can be translated in a spectroscopio faCtOr
exact
finit~-rang~ ~ff~otS),
value ot
e
based on a
Bimpl~
S(t,~)
(disregarding
= 0,04
whioh is much smaller than th0
e~pected
shell-model. Even if we only try to obtain
the oorreot magnitude of the crOsS seotion, not oaring about the shape
of the OrOSS seotion and the analysing power, we find that S(t,a) can
not exceed the value 0.9 (assuming an interference angle of 130°). 1n
section 7.4 we give some possible explanations of this problem.
In fact if we look at the
values only then the result
chi-s~uare
of fitting the analysing power by var,ying A
a pure one-step description
VIA
~nd ~
(PJ~)
is
~on~L~tent
~eactiQn,
with
namely
value of the cross-section shape sustains this finding'
0.0;1;0.1.
~hat
p~re
of the
-O.ltO.IS. The corresponding result from minimizing the chi-
=
g~uare
ll/l>. =
(~~O)
(p~)
the
(p,~)
and ( •.. t)
path has to be dominant, is also
(t~)
ole~r
from the
calculations (fig. 7.2), whiCh show a much
clo$er agreement with espeoially the
Oro~s-section
data in the
fo~me~
case.
The choice of the triton potential has SOme influence on the
shape at" the crosg ",ectJ.on and the analysing power of the ( .•. ) (t,,)
tran~ition,
potential
is quite
fo~
~B
whi10 the magnitude Of the cross section calculated with
is about twice as
~imilar
la~ge
to the dependence
at that obtained with TF, which
~on
~he
triton potential found
the (p,t) reaction (ohapter 6).
Whereas the a,lpha potenti<tl Ai is
~uite
successful in de,,"oribing
the cross section calculated in the one-step cluster approximation
(fig. 7.2). it Obviously is deficient in reproducing the analysing
power. The global potential A2, on the other hand, gives
one~step)
a
reason~ble
he~e
(also
fit to both the cress section and the analysing
power. For this reason we selected this potential for the calculations
including the multistep transitions.
The ratio Of the eXperimental cross section and the one given by
the code DWUCK4 13) is 40.5 and 16.7 using the potentials A2 and Ai,
respectively, 1'he",e values correspond to
S is the COuPling parameter Of the COde
a~
2600 and
e=
1600 wh",r",
CHUCK2 13). These numbers
contain normalization constants as well as
spect~oscopic
factors.
103
One-step
(PI
cO
cal(:ulatian~
ba~ef.l.
on a mic.:ro!=:copi.c: form factor
generally yield fi loS t.o the .experimental dat.a which are inferior t.o
'~l.u5ler
those obtaine(l witch a
mixed the
"mic~o~cC)pic
form fact.or 11). In our iinalysis we
appr')~ch"
of the
( ... 10) (tetl p"th with the
(pal pat-h and fOurld th"t th" contribution
"cluster tyeatm",nt" Of t.h"
of the f.o"(met.' Pt"O..-;e:SS y.:~1.i~ muCh sm~llor than 0:x:poctGd.
:rea~Dn
the pot.ent.i.,al
th.~
(p,
0:1 reac t.i,on5. Therefor<o it. m"y be th"t mul ti-
SLOp ,jffects are hi(lc]",n in t.h",
I.'*:sul t..s obtained with mj
t.he inclusion of e.q.
clu.~ter
cra~copic
approximation and t.hat. the
form .fact-ors will be improved by
Iptllt-.Ct) trans.;,ti.ons, in which case Our results
to indicate that the inter-fe'renee ha.fo tc he
el<pl~n~tion may be th,;t t.h"
de$truc;tivEly with t.he
destructiv~.
Anot.her
(1', 3He ) (3 8e ,,,) prO(;"Ss ir'ltCrf0rCs
(;p,t) (t.,u)
i.:'loH'I.:tn'::Ul.ue .of the one . . . . ~tep
104
obviou.s
L bi.n(js the triton clu6ter have been determined from
OnG-step analyses 0 f
~eE':!IIl
Or'l.(~
fo.r ·Chi.s rC'.:::;:ult io th8 faOt that the qeometry parameters of
(P, t:d
process resulting in a seeminq
process.
C.B. Fulmer and B.L. Cohen, Pnys.Rev. 112 (1958) 1672.
P.J. van Hall, J.P.M.G. Melssen, O.J. Poppema and J.W. Smits,
Pnys.Lett. 74B (1978) 42.
3
F. Brummer, H. Cech, H.H. Muller, G. Ratel, H. Jasicek and
4
y~
H. Oberhummer, J_PhyB_ G5 (1980) 473.
~a9i~hi,
~f
K~~O~~,
M_
Sa~agage,
M_ Sato and T. MikumO,
PhYS.ReV_Lett. 41 (1978) 16.
5
y.
T~gi~hi,
K_
Katori, M- Sasagase, M. Sato and T. Mikumo,
pnys.Rev. CI9 (1979}2420.
6
Y. T"gishL K. Katori. M. sasagase, M. Sato and T. Mikumo,
Nucl.Phys. A339 (1980) 262.
Y. Tagishi, K. Katori, Y. Toba, M. sasagase, M. sato and
~.
8
MikumD, J.pnys.Soc.Jap. 47 (1979) 1735.
L.~.
Lee, Jr, A. Ma,inoV, C. Mayer-Boricke. J.P. Schiffcr.
R.H. Bassel, R.M. Drisko and G.R. Satchler, Phys_Rev_Lett. 14
(1965) 261.
J.A. Nolen. Jr, C.M.
(1966)
9
10
Glashau~ser
D.L. Dittmer and W.W. Daehnick. Phys.Rev. 187 (1969) 1553.
J.W. Smits, R.H. siemssen,
Nucl.Phys. A319 (1979)
11
and M.E. Rickey, Ph.ys.Lett. 21
705.
s.~.
van der werf and A. van der Woude,
Z~.
A. srobowska et al., rnst. of Nucl.Phys.,
~rakov,
~eport
no. 777PL
(1971), unpublished.
12
A.
Bud~ancwsk1,
A.
Str~alkowsk~,
K.
Grotowsk~,
J_
~uzminski,
S. Sykutowski, J-
S~m1der
ll_ Niewcdniczanski,
and R. Wcllski,
Nucl.Phys. A106 (1967) 21.
13
P.D. Kunz. univ. of
14
W.R. Falk, R. Abegg and S.K. Datta,
colo~ado.
unpublished.
Nucl.Phy~.
A334 (1930) 44:\.
lOS
SUMMARY
Nuclear transfer-reactions are characterized by the transfeh of
a small number of nUCLeons from
thesis deals with
Beve~~l
o~a
atomi~
nucl~u~
to
anoth~r.
Thi~
types of transfer reactions which were
induced by bombarding seNi and 56 pe foils, respectively, with a beam
of polarized protons having an energy of 24.6 MeV. Three
ty~es
of
reactions have been studied, the transfer of one neutron (p,d), the
transfer Of two neutrons (p,t) and the transf.er of two neutrons and
one proton (p,«)
The purpose of this study was twofold, namely the e~tension of
the spectroscopic knowledge of the nuclei 57Ni and S51"e ~ :Oy rneang
of the (p,d) reactions - and the determination of the importance of
multistep processes for all reactions investigated in this
th~sis.
For this the analysing powers, which were available thanks to the
polar~zation
of the proton beam, turned out to be very helpful.
The description of the (p,d) reactions 9snerally improved by
taking account of the two-step prOCess (pp') (p'd), i.e. inelastic
proton scattering (to the 2+ states) followed by one-neutron transfer,
in addition to the dir8ct (pd) process. The
the
expe~imental
re~ulting
corrections of
spectroscopic factors amounted to at most 10%.
In most cases the multistep caLculations were based on transition
amplitudes obtained from
Koo~~
5hell~rnodel
calculations of Glaudemans,
and Vennink. The latter calculations have been p8rformed using
two different types of residual nucleon-nucleon interaction, surfacedelta
(~O)
and KUO-Brown (KB). The shapes of the differential
sections and the analysing
~esidual
powe~s
wer,e
~nsens~tiv@
cros~
to the type of
int@raction, :Out not. however, the magnitudes of the cross
sections. The KB
~nteraction gener~lly rep~oduced
these magnitudes
:Oetter than the SO interaction in the case ot the 55 Fe1P ,d) reaction,
while it was not possible to make a definite choice for the
5B Ni(p,d)
<eactJ.on.
The analysis
o~
the IP,d) reactions yielded some problems with
the adiabatic deuteron potenti~l~ we were fo~ceo to reduce the r8~1
well-depth with about 7% in order eo obtain an accepta:Ole description
of both the CrOss sections
were
~ndications
~nd
the analysing pOwers, In addition there
that the imaginary
(i.e. mOre deuteron
ab~orption)
well-d~pth
has to be
~ncreased
in that case. We guess that these
107
re~nllt~
Glr.~
(:QnsequC'r)(:c::; of the.:: fCl(.;t that exchange effec.·ts \.\fere not_
t.aken tnt.o aCCOt1t1t in tIL", <:Orlotru(;tion Of th" adiab<ltic dGut0ron
PQc~nt.j,id.•
NOt, OVGn tlw "djusted deuteron potentiill D4 could
.~u£ficlent.ly
explain the difference observed between the experimental
CrOSs s~c:.,tion~ with j=7/7. ancJ j=S/2. WP. th;i.nk thQ,t th,is sho('tcoroing
is llue to the neglect of t.he
n-~tat"
of
t.h~
(l"\\t"ron~.
Ttl" sequ""tial t.yansfer of t.wo n",ut.rOIlS (pel) (dt) generctlJ.y
contributed
n~~\.l.t.rOr~
a~
much to the
(p,~,
CtOSS sections as the 01zect two-
transfer (pt). !5ecause or; uncertainties about the norma.lization
Ot bot.h t.he (pel.) (dt) and the (pt) process due to ... among other things ....
tht:: zero-ranqe
.=":!.p'pJ.'o).;:1m,~t.ior"r"
we .Lfl.trOduced two
meters ~ ilnd U. The rati.o of A and
).l
')Orm.:j,liz~t.iOI1
pLl.ra-
was det;ormined by fitting the
(;"u.lcula teO ,?,nal YfJing' powers to the cxparim~ntal on~s. Next the
magni tur1e
"ro~,,,,
0
f " and
follow~d
f~om
In th" case of the 58 N1 (p,"')
t.OO,
the magnitudes of the
expe~imental
s"c:t.ion~.
,'eaction we use(l such param~ter3
From this we fOllnd the on",-sT.",p FroG"S:s
( •• ,t.) (tOL)
beGau::.e of
(1'(>.)
to 0", dominant. The
procesge$ contdbuted much leS$ than eXFectec'l poss;ll;ll 'I
dr=-otruct.i V"e interference with
( . _ . jHe) (3He~r J?roce~.~es I
whlch were le.ft out of the calculCltion.:3.
In yie'W of t_he accumuli1tion of uncG'rtYinti~:s etbout the various
inqre~ient~
of the calculalions fOr the (p,e) and
(p,~)
reactions,
we think it wise to conc"nt.rate - fOt' the time being - on ga.lnJ.ng
more insi<)ht. tnt.o the d0t"-ils Of the (p,d)
108
rCaCti,OD~.
Kernreacties. wdarbij een
Kle~n
aantal nucleonen (protonerr of
neutronen) van de ene atoomkerrr naar de andere overgaat, worden overdrachtreacties genoemd. Dit
van srrige
pro~f5chrift
overdrachtr~actie5,
handelt over de resulCaten
die werden opgewekt door folies van
bijna zuiver 5$Ni respectievelijk 56Fs materi~al to b0bchieten met
gepolariseerde protonen, waarvan de enerqie 24,6 MeV Was ten qovolqe
van versrrelling door het
werdGn drie
typ~s
cyclotron van de T.H.
A.V.~.
re~cties
bestudeerd: overdracht van
Eindhov~n.
6~n
Er
neutron
(p,d), overdracht van twee neutronen (p,t) en overdracht van tWee
neutronen en
~en
proton
(p,~).
Het doel Van dit onderzoek was cweeledig, n.l. uitbreiding van
de spectroscopische informatie over de Kernen S7 Ni en S5 Fe - via de
(p,d) reacties - en bepalinq van hee belanq van meerstapsprocessen
voor al de bestudeerde overdrachtreacties. Hierbij ward met vrucht
gebruik gemaakt van de extra gegevens die experimenten met gepolariseerde protonen opleveren: de analyserende vermogens.
Het in rekening brengen van het twee5tapsproces (pp') (p'd),
d.w.z. in~lasti5ch~ protonverstrooiing (naa~ 2+ niveaulg) gevolgd
door de overdracht van een neutron. naast het direkte (pd)
bleek in het algemeen de bescnr1jving van
betersn. De
hie~doo~
nDodz~kelijk
mentele speotro5copische
~eworden
d~
(p,d)
reacti~s
p~oces
te
v~r-
correctie van de experi-
bedroeg maximaal lOt.
f~ctoren
In de messte gevallen war.en bovengenoemde meerstapsberekeninqcn
gebaseerd op overgangsamplitudes verkregen uit de schillenmodelberekeningen van Glaudemans, Koops en Vennink. Deze
laatst~
bereke-
ningen zijn uitgevoerd voor tweo versohillende vormen Van de residue
nucleon-nucleon wisselwerkirrgo
oppervlakte-delt~
(So) en KUO-Brown
(KB). Oe vorm van de diffsrentiele werkzame doorsnedes en de analy-
serende vermog'ens bleek ongevoelig voor de keus tUS5<1l'1 SD en KB.
scheer niet de grootte van de werkzame doorsnedes. Daarbij was KB
over het algem~err beter dan SD voor: de 56 Fe (p,d) reacHc, tcrwijl bet
nlet mogelijk waS "en voorkeur te bepalen bij de S5 Ni(p.d) reactie.
llij de analyse van de (p ,d)
.r,eactie~
<;!tui tten we op problem",n met
de "di"bat;i.scne deuteronpotentiaal, we wa"(en gedwongen de putdiepte
Van het
re~le
deel mat ongaveer
7~
te
ve~kleinen
om een redelijke be-
5ch rijving van zowel de warkzame door.snedes als de analyserende vCr-
109
mog~l1.s
w;~ren
te ve'(kl:".i jrye?:n. verde!."
van het. inh=':l,qtnai.ce uo.el
d~r'l
eL' aanw.i.j7.inryen nat de putdiepte
t.egelljk venJroot D)Oet: wOr'uen (l'Ileer
(loE;!ut.f:!tTma.bsorptiu). We vermocden dat ueze resultatcn te makcn hebbcn
u~t
met het, [eit.
"i..::xr..::hangC:l:"
r.:sffect~n
rck~nirlg
rliet in
zij71 gebrtich1..
blj (1e cons t.!.'\.IcLil::: Vlill de adiaba.tische deuteronpotentia.al.
.=l.~n9f;!pa.stC
t\.ls,~,f!n
d~uterortpotentiaal
werk~~rnc
verklaren. IN*,
Ook dE:::::
IJtJ kon het experimentele verschil
doorsnedes met j·:l/;':: en die met j . . ·S/2 niet afdoende
d~nk~r'),
dttL dit te wi]tr::O:Tl is
u~rl
het verwaarlozeTl
VuTl
lie D-t0estund van het. deute.t'()n.
Bij
d~
neutyonen
~an
(p,t)
l'.·eacties bleek de sequentiele over.Cl:r.ar.:ht van t.wee
(pd, (ctt) in het alqemeen ongeveer evenveel bij te draf)en
de werk;.,:;tt.rne uoorsnede als de direkte overdracht (pt} , Oezien de
onzekerh<?id over de no,-mel:"ing
\1~n
zowel het (pol (('It-l
~l~
het (j;ltl
pyoces in verband met o_a. de oracht-nul benadering, voerden we tw"e
norrnr::rinqsparamete'l'·s \ en 1.J i.n. De verhouding van). eiJ IJ wOrd bepaald
door. de beJ:"ekende an.:tlyserende vermoqens
2.0
goed moqelijk aan te
passen aan de exp8rim-entele ani11yserende verrnogens. De grootte van
en
IJ
.3nel~e
volgdc:: du.n ui t de
~yootte
van de experj.mentele werkza'ITle dcol':'--
•
ook voc!';' (le ~lar'H {p,CX} r,-":",otic voe.rdBH we dergelijk~ parLlmOter::;
A "n 11 in. Daarbij bleek het_ eenstapspyoces (pa) dominant- t'" zijn.
De
bijllrag" van de (. _ .t) (ta)
proce~sen
was veel kleiner dan verwacht,
mogclij);. ten ryevolrye van (lest,.u~tj_eve ;'nterferent.l.e met ( .. _ J se ) (3 se l))
proC~s5en,
die niet explic1et 1n rekening werden gebracht.
Het lijkt on" verst.anoig v00r).op1g meer aanoacht te schenken "an
het beqrijpen van d" ,j"taU" v"n de (p,o) t"ea<;ties, ge0:1en d., op",,,nstapelinry van on:zekerheden over de verscheidene ondordelen vall de
berekeningen bij
ll()
(,!"
(p,t) en (p,c<l ,'eacties.
NI\WOORD
Het
onderzoe~ Q~t
in dit pr.oefschrift beschreveTI
i~,
werd ver-
richt binnen de onderwe;(pgroep "experimentele kernfysiclt'l Btl mGt de
daadwerkeLijke seeun van alle leden van die graep.
~n ker.nfysi~ch o~z~cht
van
~all,
r~ank
sieb Klein,
Gera~d
kwarn die steun van prof. Poppe rna , Pict
Ni~gh,
"os Melssen, Sietse Wassenaar en
Dautzenberg. Technisch hebben mij bijgestaan Wim Gudden, Gerard
Hamers, Harm Rozema, Peter Teunisse en Jan van de Berg. De laatstG
dank ik in hee bijzonder voor zijn hulp bij het ontwerp en de constructie van de detectorhouders en het telescooJ?blok.
Van essentieel belang was oak de ontwikkeling van het computersysteem door Adri de
Raa~_
Ik denk met plez!.er terug aan de nauwe samenwerking met
ooste" gedurende de ontwikkeling en net te.ten
Vdn
lJan~
v~n
het tele.coop-
SYst(J(Jm.
Heel erkentelijk ben ik pr,of_ Glaudemans en zijn medewerkers
in het bijzonder Fons van Hees - voor het
en spsctroscopische amplitudes van de
ver9ch~ffen
ijze~~
en
n~kkelkern~nT
Van Wolfgnng Peix heb ik - tijdens zijn verblijf
"r."~eo.);"h
~
van golffDnctie9
hie~
als
feU.ow" - veel. gcleerd over de mc«rst<lps!1n1l1yse van (p,t)
reClcties.
V«el dank. oak jegens de cyclotronsbedrijfsgroep voor het leveren
van de gepol"riseerde pratonenbundel.
De uiteind(Jlijke vormgeving van dit praefschrift, tenslotte,
kwam tot stand dank zij de grate inzet vo.n Ruth Gruyters (tek.eningen),
Rian Teurlings en Aafje Smit (typewerk).
111
10 f"vruari
mei
1946
geboren te Amsterdam
1964
8iI~d~x~mur'l
qynmasi um-~ aan het i~.
Lely ..... lyceWl')
te Amsterdam
r.;ept,ember 197'2.
doctoraaJ. ex..::Unr...':n. theoretische natuuY."kunde
(hoge-ener9ie-fy~ic8.)
8.ctn
/'Jq:
Universiteit van
Amsterdam
SGptGmver 1 ~72-
wetcnschappelijk medewerker aan de vrije
auqust\.lS
\ g74
augustus
1974-
leraar. natuurkun<:le aan het MichalHcollGge te
augustu,s
19'/C
zaanc'1arn
a\\gu~tus
197C-
wetenschapl?elijk ambL.ena"r in do. qroep
f~ln'uad
1901
expedlnentele kernfyGica van de T.H. EindhOvel)
112
Universiteit van Bru5sel
beho~end
bij het proefschrift van
J.B. Palane
Eindhoven, 5 juni 1981
In de uitdrukking die Donner I) seKEt vuor de BBnslag-energie van
( . . . eecJ~eltje~exc.itatie.!::i iIl Let RCS-mode.l t client de term
C(u
k1
uk
2
+ v
Vj
k 1 '2
I) W. I)orm('"',
)2 vervange"
(e
wo~deD
door
-2C;u
u
V
v
1< 1 k2 kl k2·
/:,"[r'fuhY'tmg hi. (h(, 'l'hr~oJ'1:e deY' KernBpektr 1(i>l
(Ii. I., Mr.mnheim 1971) ,
p. 2JD
2
l[l hll~ be~prekin~ van
llru>;sadrd
0Il
pararnl2tcrs 8'
de oppervlakte-deltol.-wisselw".,.king m"kc1\
Gl a"demans I)
Ltl\
o1\):"ecl't" Dnderscheid tussen de
en C' enerzijds CI\ d~ p.Ct.rsmet.e.rs II i:::':n C anderzijds.
I) P.J. Bl"'USBam'd 1m P.W.H. GZaudemana, SheU-mode/. r.lpplicatio>la
Iju<,t8a1' cpe<:ttr'(Jf:"'"oPY
f?j
(NoY'th-Ho/.Zand. Amstel"'dam 1017) , p. 111
Effecten van de tw€€de orde
kvon~n
Dok bij voor~aartse hoeken
CeIl
a.anzienlijke bijdr<lg~ geVei\ .".n de differEntlele w~.rkzarne dQ<lrSIlede
V(ln een d"eltje6overdrac:htre(lctie. Het is raadzaam hj"rmee r"kcning
lc houclen 1)1 j h~t norm"rCIl V<1I) ber~kend" dif[erc!ltiele "e"kzame
4
voor het
bepal~n
van dc relatieve bijdragen
v~n
pt'oc('ssen a.an de ov.::::rgungs,-;implltudes vuor (p~t)
goede
;)~1.()p8.~5ing
de verschillende
l':"E=!aC'.tiE!:'
v.crdient f':en
vmt het analyserende vermog€'.'''J de voorkeur boven di e
v.:tn d~ c.1iffer'i.-'.I"J.t.i;?_l~ tuerkzame doorsncd~ •
.5
In e~n ~antal r~~~Iltc popu1aire artik~l~n 1,2) over d~ uniticatie
dH
Qlektromagn~ti.che,
de
ste~ke en
onrechLe tic (J.l()g(.11jkl':teid vdn
1,!(.'11
~)0I1trino met rustm~S8a
g~neg~crd.
1)
~)
~,I. iii !.('.'z~!k.~ :,'('
,.;, (J'~':'o'r'rJ7~!I lin.
!lrm-:r~i,uaJ'1.J d.e(~ember'
Ameriu(zn"
up r''r.: 1..
VCln
de zwakkc wissel werking wordt ten
1980, !.i.
U'ftJI J p . .tJO
au
ongE::!lijk
n\.I~1
Een kerncentrale c;!ie getroffen wordt door
lange
~ijd
in een
veJ;"",o"5t~rtd
c;!oende reden moeten
1) S.A. Fe.Uer en K.
z~jn
~.,n
kel:"nhom, veI:and.::rt yO,,\"
stalingsw"p"n 1). Dit feit zou een af-
om af tc zien van kernenergic.
l'Bipi$, Sa. Amel"iC(ln, apl,a 1981, p. ;;;,
Het verplichte eirtd",xamenprograrnrna natuurkunde voor het HAVO laat te
weinig ruimte oveI: voor projectonderwij •.
8
Tert onrechte wordt de aauleg
veel
hui.~n
V~n
een afvoerputje 1n de
als overbodige luxe beschouwd.
badkame~ van