nuclear spectroscopy of ca and sc isotopes from …
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U N I V E R S I T E T E T I B E R G E N
NUCLEAR SPECTROSCOPY OF
Ca AND Sc ISOTOPES
FROM INELASTIC SCATTERING AND
ONE-NUCLEON TRANSFER REACTIONS
ON A RADIOACTIVE 4ICa TARGET
by
Per B. Void
With due permission of the Senateof the University of Bergen to be publiclydiscussed on IS. September 1978 for the
degree of Doctor philosophise
BERGEN 1978
NUCLEAR SPECTROSCOPY OF Ca and Sc
ISOTOPES FROM INELASTIC SCATTERING
AND ONE-NUCLEON TRANSFER REACTIONS
ON A RADIOACTIVE 41Ca TARGET
Per B. Void
Thesis submitted to the University of Bergen
April 1978
C O N T E N T S
Page
1. Qualitative overview of collective statesand coexistence 1
2. Discussion of the results from the one-nucleon transfer data 8
2.1 Spectroscopic factors and sum ruleanalysis 8
2.2 Effective diagonal two-body matrixelements 14
3. PAPERS I - V
I. PARTICLE-HOLE MULTIPLETS IN "°CaOBSERVED IN THE ^ C a ^ H e , ^ REACTION
D. Cline, M.J.A. de Voigt, P.B. Void,0. Hansen, 0. Nathan and D. Sinclair,
Nucl. Phys. A233(1974)91 24
II. NUCLEAR STRUCTURE OF Ca FROMINELASTIC PROTON SCATTERING . . . .
P.B. Void, D. Cline, M.J.A. de Voigtand A. Sperduto,
Nucl. Phys. A292,(1977) 107 39
III. THE EFFECTIVE T=l TWO-PARTICLEMATRIX ELEMENTS IN THE fp SHELL
P.B. Void, D. Cline, R.N. Boyd,H. Clement, W.P. Alford andJ.A. Kuehner,
Phys. Lett. 7ZB(1978)311 58
IV. NUCLEAR SPECTROSCOPY OF THE (f 7/ 2)2,
f7/2P3/2 AND fs/2Pj/2 MULTIPLETSFROM THE ^Ca(d,p) "2Ca REACTION
P.B. Void, D. Cline, R.N. Boyd,H. Clement, W.P. Alford andJ.A. Kuehner,
Nucl. Phys. (1978), in press . . . . 63
Page
V. A STUDY OF THE TWO-PARTICLE STATES"2Sc FROM THE *xCa(3He,d) **2ScIN
REACTION
P.B. Void, D. C.line, M.J.A. deVoig t , O. Hansen and O. Nathan,
to be published
4. Summary and conclusions
112
160
P R E F A C E
The present thesis consists of five papers written
during the period 1974 - 1978. The nuclear structure of
Ca- and Sc- nuclei has been studied using light-ion
41induced reactions on a Ca target. The simple structure
41of the Ca target (consisting largely of a single
40neutron in an f? ,„ orbit outside the doubled closed Ca
nucleus) makes one-nucleon transfer reaction studies on
41
Ca particular interesting because of their properties as
a sensitive probrt of the low-lying one particle-one hole
and two-particle shell model states in mass 4 0 and 4 2
nuclei. Such studies represent the major part of the pre-
sent work and the purpose has been to locate the distri-
bution of these states in order to obtain information on
the effective two-body matrix elements of the residual two-
body part of the nuclear Hamiltonian for (fp) and (fd~ )
shell model states.
The five papers are contained in section 3 of the
thesis and constitute the body of the present work. In
addition, a review of some systematic features of oxygen and
calcium nuclei together with an overview and discussion of
the present results, are given in sections 1 and 2, while
section 4 presents a short summary of the results of this
work.
The experimental part of this work was started in
197 2 when the first experiments were performed at the
Nuclear Structure Research Laboratory, Rochester, USA,
and it was finished in 1976 with the polarized deuteron
beam experiment, which was carried out at the Tandem
Laboratory of McMaster University, Hamilton, Canada. I
wish to express my gratitude to Professors Harry Gove and
John Kuehner for the excellent working conditions offered
me during the performance of the experiments.
The extensive work with the data redr.cLj.on and analysis
has been done partly at Universitetet i Bergen, and partly
at Rochester, and I am indepted to my colleagues in the
nuclear physics groups at these two institutions for
inspiring discussions and their interest in the present
work.
Many thanks are due to my co-workers. I am especially
grateful to Professor Douglas Cline for his invaluable con-
tributions; without his collaboration and support this
work would not have been possible. I would also like to
thank my collaborators from the Niels Bohr Institute, Pro-
fessors Ole Hansen and Ove Nathan for fruitful and stimu-
lating cooperation.
The assistance of Mrs. J. Asphaug in the preparation
of this and other manuscripts is highly appreciated.
Financial support from Norges almenvitenskapelige
forskningsrad and from Universitetet i Bergen is acknowledged.
- 1 -
1. Qualitative overview of collective states and coexistence.
In the exploration of nuclear spectra the interplay of
individual particle motion and collective motion has played
a v.ltal role. The basic features of single particle ex-
citations have been most dominant in the region of closed
shell nuclei, while collective phenomena have been most easily
seen in nuclear systems with several particles outside closed
shells. However, it was early known that the first excited
states of the closed shell nuclei 0 and Ca have positive
parity, while the low lying single particle excitations are
restricted to negative parity, implying that these states
involve excitation of several particles. The positive pari-
ty states were not easy to explain in terms of the shell
model and it was suggested by Morinaga (Mo 56) that they might
be associated with collective quadrupole deformations.
Further indications that the shell model was inadequate
in explaining those states was the observation of strongly
enhanced E2-transitions in 160(Go 63). The most striking re-
sult that these states are strongly deformed was the finding
of several positive parity states observed in inelastic a-
scattering by Carter et al. (Ca 64) in 1 60 and Bauer et al.
40(Ba 65) in Ca, which showed characteristic J(J+1) rotational
band structure. This coexistence of spherical configurations
and deformed low-lying rotational bands provoked considerable
interest in understanding the structure of these nuclei. The
origin of these states was attributed to strongly deformed
multiparticle-multihole (mp-nh) configurations (Br 66, Ge 67).
- 2 -
As can be seen in the Nilsson diagram of fig. 1, it costs
for a large deformation relative little energy to excite
two and four particles from
the sd-shell into the fp-
shell. Somewhat surprising-
ly the energy of the 4p-4h
configuration was found to
be equal or below that of
the 2p-2h state (Za 65,
Ba 65a, Ha66). The lowest
rotational band in these
nuclei (Br 66, Ge 67) were
thus interpreted to be pre-
Fig. 1
Nilsson diagram for the 1^3/2and If 7/2 shells showing two-particlfe two-hole state,• - particle, o - hole.
dominantly a 4p-4h configuration with a large prolate defor-
mation. The strongly enhanced B(E2) values in the adjacent
two-neutron nuclei 0 and Ca suggest that coexistence of
low-lying two-neutron spherical configurations and deformed
bands similar to those in 0 and Ca is occurring. More-
42 +over, the presence in Ca of a second 0 state at 1.84 MeV and
a second 2 state at 2.4 2 MeV additional to the sequence of2
(f_/2) states,which extensive shell model calculations
(Me 70, En 66) cannot explain,is further evidence for co-
existence. However, the absence of any characteristic
rotational energy spacing in Ca indicated that the two
set of levels are not strictly coexistent, but must be
mixed. In fact, it was demonstrated in neutron pick-up
from 43Ca (Bj 64) that the presumed (f7/2 state at 1.52
MeV is strongly mixed with the 2* state at 2.42 MeV.
- 3 -
In addition to inelastic scattering data electromagnetic
transition data are particularly sensitive to the collec-
tive properties of nuclear states. Especially valuable
pieces of electromagnetic data for probing coexistence are
static electric quadrupole moments. They are diagonal
matrix elements depending' on the wave function of only one
state and would thus yield both the sign and the magnitude
of the matrix element. These features have been used to
determine the shape of the deformed bands in the Ca-region
by measuring the static electric quadrupole moment Q_ of the
+ 42 44
first excited 2 states in Ca and Ca. While Gerace and
Green postulated the deformed core-excited states to be of
prolate shape, Towsley, Cline and Horoshko (To 73) used the
experimental E2 matrix elements bo determine the admixture
2 42of (f_ ._) and core-excited states in Ca. In particular,the Q- value of -19±8 efm for the first excited 2 + state in42 2
Ca differing from the shell model prediction by 20 efm
required a prolate shape of the deformed core-excited compo-
nent.
In fig. 2 is shown the unperturbed energies of the core-
excited states deduced by Towsley et al. from their coexis-
tence model wave function and the experimental excitation42
energies in Ca, plotted versus J(J+1) to illustrate theircharacteristic rotational band behaviour. The lowest rota-
40tional band in Ca,also shown in fig.2, has a moment of
42inertia almost the same as that of Ca. Moreover, it was -
found that the intrinsic quadrupole moments derived from the
- 4 -
B (E2) values are also
very similar for the two
bands which also indicates
that the two bands are
about the same and thus
of prolate shape (To 73).
In a very recent study of
the lowest rotational
40band in Ca using the
36Ar(6Li,d)40Ca reaction,
Betts et al. {Be 77)
arrived at the same con-
clusion regarding the
of the band.
Coexistence model
MeV
•01 2 3 4
Fig. 2Energy levels in l*0Ca and lt2Caplotted vs J(J+1) to illustratethe rotational bands in thesenuclei. The lowest band in "*2Cacorresponds to the unperturbedenergies of the complex states.The low-lying levels in lt2Ca con-nected by the dashed line are theunperturbed (fp)2 spectrum (To 73).
calculations similar to those of Towsley et al. have recently
18been applied to 0 by Lawson, Serduke and Fortune (La 76).
The admixture coefficient of the core-excited configuration
+ 18for the 2, in 0 is estimated to only 12%, which yields a
maximum change in the static quadrupole moment Q 2 of 2 efm
2 2from the pure (sd) shell model value of -3 efm (En 76).
Therefore, a measurement of Q 2 in18,0 does not appear as useful
for a determination of the shape of the lowest rotational
bands in ' 0 as in Ca. However, in a recent measurement
of Q 2 in180 (Kl 75) an extraordinarily large valuu of
(-19±2)efm was reported. This measured value was strongly in-
consistent with the predictions from all current nuclear
- 5 -
18structure models of 0 (La 76, En 76, Er 77), which account
18reasonable well for other experimental data available for 0.
Another measurement of the static quadrupole moment of the
+ 18
2, state in 0 was therefore of great interest. Two new
measurements have recently been reported with Q- values of
(Vo 77) and (-4.7+2.7) efm2 * (Fl 77). These
results which are much smaller than the value previously
reported agree very well with the current ideas of the struc-18
ture of 0.
In addition to the collective core-excited states in the
mass 40 region, the strongly enhanced H=3 and 1=5 transitions
observed in inelastic proton and alpha scattering to the— — 4 0
lowest 3 and 5 state in Ca indicate that these states also
contain admixture from collective degrees of freedom. This
41suggests that inelastic scattering to states in Ca should
be a sensitive tool for probing the configurations formed by
the weak-coupling of the f.-. ,, valence neutron to the 3 and
5~ collective states of the Ca core; that is the (3~ x f 7 / 2 )J
and (5~x f /9) septuplets. In table 1 we have summarized the
This value has been corrected for Coulomb-nuclear interferenceusing the prescription, outlined in the paper by Void et al.(Vo 77). All the quoted values for Q2 in
l 8O arebssed on cal-culations assuming negative interference via the second 2+ state.
** The H=3 strength correspond to 27 Weisskopf units (Wu) (1 Wu isthe transition strength estimate for a pure single particlestate) and the £=5 strength to 17 Wu (Gr 72).
- 6 -
Summed t-3 and 1-5 inelastic transition strengths in Ca
J *
8(p ,p ' J 3. J11 * j g . s . ) a )
Btp.p* t 3~ * 0 g . s . )
a i f . p ' l 5, j " - j 9 .J . I a l
8(p',p'l 5" •• 0* 9.1.1
1/2*
1.0
3/2*
0 . 8
5/3*
0 . 5
V2*
0 . 8
9/2*
0 .9
11/2*
0.9
b)1.2
13/3*
c0.6(0.7
co.s
15/2*
0 . 9
17/2*
0 . 5
a) The '--e«k-coupling model predict! that B(p,p'j t, J + 7/2" 9,6.) • 8<p,p'i t * 0* g.s.) foreacti 1. Uvldual spin member of tha multlpleta.
b) This val.u« deduced from the 3973 keV level transition may b« avereatimated {see table I inpuper I) because In a recent 38Ar(a,irr)41Ca experiment it w*i ihown that there are actually twostates at 3974.2*0.5 kaV and 3976.0i0.7 keV with spLns of J1-!5/3,7/2)+ and J*-ll/2* respect-ively (Li 77).
c) Values obtained by attuning the 13/2* state at 4S2Q kev to proceed by 1-3 or 1*5 respectively.
£=3 and 1=5 transition strength results obtained in the
41Ca(p,p') experiment (paper II). These results show that
41the summed A=3 strength in Ca is in reasonable accordance
with a simple weak coupling interpretation in terms of the
(3 x f7/2^ multiplet. However, since the £.=3 strength to
most of the spin members of the multiplet is distributed
over several levels , it implies that the individual states
41in Ca also contain contributions from other configurations.
For example, the 9/2 strength is fragmented into two states
indicating admixture from the -L g/? s i n 9 l e particle state.
Only the 11/2 strength is concentrated in a single level at
— T 1 / O +
3369 keV and is determined to be of almost pure (3 x f7/-2^
structure. It is interesting to note that Lister et al.
(Li 77) have arrived at the same conclusion regarding the
structure of this state in a recent study of the transition
strength in y-decay.
- 7 -
Although the low spin members have not been identified
for 1=5 transitions, the strength distribution of the high
spin members with the exception of the 17/2 state, suggests
a similar weak-coupling picture in terms of the (5 e f7/2)
configuration. The 17/2 strength yields, however, only
about 50% of the expected value. This reduction of the 17/2
strength is probably due to the fact that neutron excitations
to the f7y2 or l a i t a r e blocked by the Pa'ili principle. This
is illustrated schematically in fig. 3. While both proton
and neutron promotion
from the d,/2 to the A0Ca(p,p')to5~ A'ca(p,p) to 17/2*
f 7 / 2 orbit are allowed P n P n p n
for the transition to '
- 40the 5 state in Ca,
only proton excitation
7/2
d3/2 mto the f ., orbit will Fig. 3
' Schematic illustration of blockinggive the maximum align- for Z=5 excitation to the 17/2+
state in ltlCa; • - particle, o - hole,ment of the two f_ ,„
particles which are required to form the 17/2 state. Since the
one-nucleon pick-up strength to the 5~ state indicates almost
pure (f7/2d3/2^ structure '.paper I, Be 75; see also discus-
sion in sect. 2.2) , one should in fact expect a strength re-
duction of about 50% for the 17/2 transition.
It may appear inconsistent to claim a rather pure
(f_._d. ,_) configuration for the 5 state considering the/ / * •i/ i
enhancement factor observed in the inelastic transition
strength to this level. However, the calculations by Gerace
- 8 -
and Green (Ge 68), which suggest that the 5~ state consists of
9 5% (£7/2^3/2 ^ a n d 5 % admixture of deformed 3p-3h core-
exited states, predict an inelastic transition strength in
very good agreement with the observed value.
2. Discussion of the results from the one-nucleon transfer data.
In the previous section we focused the attention to expe-
riments which probed the collective properties of nuclear
states. However, the major goal of the present work has been
to provide information on the simple shell model states in
41mass 4 0-42 nuclei. Since the Ca (g.s.) wave function is
largely of pure f_ ,~ single particle structure one-nucleon
41transfer on Ca should primarily excite the low-lying
two-particle and lp-lh shell model configurations in mass2
42 and 40 nuclei respectively; that is the (f7/2) '
f7/2P3/2' £',/2Pl/2' f7/2d?"j a n d f7/2Sl/2 »ultiplets. The
detailed spectroscopic information derived from these data
is discussed in papers I, III - V experiment for experiment.
In the following sections of the present review some syste-
matic features of the spectroscopic data are discussed.
2.1 Spectroscopic factors and sum rule analysis.
The monopole sum rules of the spectroscopic strength for
a given j orbit are summarized in table 2. Neutron pick-up
41and proton stripping on Ca may proceed to both T=0 and T=l
- 9 -
final states and thus yield information on the degree of
emptiness (stripping) and filledness (pick-up) for both neutrons
and protons in the target ground state. The T=l states of mass
4 2 and 40 nuclei can also be reached with neutron stripping
Table ;
Monopole sum rul«s for one-nucleon transfer within a given ] orbit
(Neutron holes). : t G.(J,,T.•j,} * J,T )] L J I A A •
(proton holes)
(All holes). = T ^ r ^ T G,(J.,TA + j,i -,T,T)3 iJT • lX L l " ft
* J.T+)
Pick up:
(Protons)- =
(Neutrons). =
^ * '•T.1
- J,T.)
T,»j,i - J,T)
Notation: ( ). = number of nucleon or holes in the target in a given orbit J;
J A , T A and J,T are spin, isospin of target and final state, respectively; Tt=TAi
not* thit both the speetroscopic strength G and the spectroscopic facor S are
eiven in the isospin formalism and G • I ; J * " S
(2J.+1)
and proton pick-up, respectively. In table 3 and 4 is given
the sum rule results for all one-nucleon transfer data from
41~eactions on Ca. The results of the simultaneous analysis
40for the same reactions on Ca obtained due to 18% admixture
40 41 41
of Ca in the Ca target, are also quoted. Since the Ca
target spin is different from zero, we obtain in addition to
the overall sum rule separate sum rules for each of the
possible (J,T) values allowed for a given j transfer. For
example, f7/2 transfer in ( He,d) yields eight partial sum
rules. These results are also quoted in table 3 and 4 for
- 10 -
stripping and pick-up, respectively.
In the (d,p) and (3He,d) reactions the summed f7/9, p,,,,
and P l^ 2 strength are expected to be equal to the shell model
prediction if most of the strength has been identified. The
present summed strengths for these orbitals represent only about
(65-80)% of the shell model value. However, this result is not
Table 3
Monopole strength sumg for stripping on
Target 3ff J
*lCa 7/2 ' 01
2
3
•4
5
6
7
Al
Al
Al
3'2" !
314
5
Al
3
1/2' Al
"°Ca 7/2" 7/i
3/2" 3/i
" T
* 1
' 0
* 1
' 0
1
0
' 1
0
0
1
A l l
1
1
1
1
1
0
1
1/2
1/2
GtJA
(3i:a,d)
0.200.67
O.BU
1.50
1.52
2.18
2.172.U2
7 . i 4 0 b
».66
12.06
0.U2
0.78
0.980.96
3.14
0.60
6 . 1
3.6
• VI.1/2
C3,P>
0.15
0.78
1.33
2.1H
U.66
0.37
O.»7
0.714
0.86
2.81
1.214
5.7
3 . 1
<d
Q
0
1
2
5
e.3 .
p . a )
16
91
<.9
iO
06
6
3
S / S s h . l l
(3Ho,<l>
0.79
0.89
0.67
0.86
0.67
0.80
O.ES
0.65
0.82
0.67
0.7S
0.66
0.89
0.880.70
0.79
0.70
0.80
0.90
•odel
(3,p)
0.61
0.63
0.59
0.66
0.63
0 .
0 .
0 .0 .
0 .
0 .
0 .
B9
iu
36
3
0 b l
2
1
0.78
It
0
0
0
0
0
0
0
, p ) a )
. 65
.73
.66
.78
.72
83
83
reft Ha 7Mj b) Includes strength from statas of unknown spin.
considered to be particularly significant as regards to
whether the total strength has been seen because absolute
spectroscopic factors may have large systematic errors.
(More detail on this in paper IV and V). This is not the
case for relative spectroscopic factors which are insensi-
tive to the detailed DWBA procedure employed and therefore
believed to be appreciable more reliable. It thus seems rea-
sonable to conclude from the present summed strength results
- 11 -
of the various fp orbitals shown in table 3, which are seen
to represent for each of the different stripping reactions
separately, about the same fraction of the shell model pre-
diction for transfer to both mass 41 and 42 nuclei, that the
main fragments for these orbitals have been identified. It
is noteworthy that the partial sum rules of mass 42 nuclei for
each individual spin member of the different fp configurations
exhibit the same feature within the experimental uncertainties.
This implies that only the f7/2 orbital is making an appreci-
41able contribution to the ground state properties of Ca as
expected within the simple shell model description of this
state. A renormalization of the spectroscopic factors to
bring them into agreement with the simple shell model
prediction would thus appear reasonable. However, this
requires that the amplitudes of the core-excited components
41in the Ca (g.s.) are small.
There are several experimental methods for determining
40 41the ground state core admixture in ' Ca if one assumes a
40 41coexistence model description of the ' Ca (g.s.). In par-
ticular, the £=2 transition strength in inelastic proton
(paper II) and alpha (Vo 74) scattering experiments on40,41
Ca,
which is primarily sensitive to the collective quadrupole
deformationfcan be used to derive the excited-core admixture
coefficients in the ground state wave functions. Another
method is to use the monopole sum rules for stripping and pick-
40 41up on ' Ca to determine the number of holes in the 2s1>2 and
ld,/_ orbitals and the number of particles in the excited
2 p 3/2lf7/2 s t a t e s >
- 12 -
These various types of data are all consistent with
admixture of core-excited configurations of only about 10%
40 41in ' Ca. With regard to these results we stress especially
4 1 3 40the results of the Ca( He,a) Ca reaction where it was found
that 99% of the f? ,„ strength proceed to the ground state and
1% to the first excited 0 state. This is considered as the
41 40most convincing evidence that the Ca core and the Ca core
are identical. Moreover, the total absence of any detectable
40
Jl=3 and A=l strengths to the rotational bands in Ca is addi-
tional support for that admixture of core-excited configura-41
tions in the Ca (g.s.) is small.
All this evidence suggests that a renormalization of the
( He,d) and (d,p) spectroscopic factors to bring them into
agreement with the shell model predictions appears reasonable.
This results in renormalization factors of 1.33, 1.43 and 1.33
for the ( He,d), (3,p) and (d,p) spectroscopic factors, res-
pectively. These are believed to be accurate to ±15%.
We emphasize that the magnitude of these renormalization
factors is closely related to the specific DWBA analysis em-
ployed in the different reactions.
41Regarding the pick-up experiments on Ca the summed
spectroscopic factors for Z=2 transitions for the (d,t) and
(d,x) data of Betts et al. (Be 75) are in excellent agree-
ment with the shell model predictions. Except for the 3~,
T=0 strength which exceeds the values expected by almost a
factor of two, the individual states of the (£7/9^3/2 ^
multiplet exhibit the same good agreement between the measured
- 13 -
and predicted strengths. The additional £=2 strength is
interpreted as due to <*,-/2 pick-up. The considerable frac-
tionation of the 3~, T=0 strength together with the collec-
tive nature of the lowest 3~, T=0 state provide further
evidence that these states contain fragments of the dc
strength.
Target j"
"lCa 1/2'
3/2*
UDCa 3,2*
1/2*
j "
0*
2"
3"
u"e ~
1"
u"
2"
s"All
All3/2*
1/2*
T
0
0
0
0
0
1
1
1
1
0
1
1/2
1/2
(3He,a)
1.0
0.39
1.0
0.89
x
L.I3.9
,9
.9
.1
.U
. 7
9.0
3.8
. V J,W,T>
Id.tj4' (a,if"
0.B1
0.28
0.72
0.50
0.58
1 J' J J
" • " 3 i)2.1 3-5J2.16.1 6.7
3.8 u.d
1.6 2.U
3H«
1
1
2
1
3
2
2
2
1221
hel
• « >
.0
.3
.3
.6
.6
.0
.3
.0
.5
.7
.5
. 3
.9
. model
<d,t,j) Ci.,."
0.81
0.90
1.65
0.S9
O.B<.
t
'"
0.77
1.3 1-1
1.0
1.1 1.1
0.35 1.1
o.a i.2
a)
The summed strength results of the ( He,a) experiments
agree with the simple shell model description for 1=3,
whereas the £=2 spectroscopic sums exceed the sum-rule limit
by a factor of about 2. Moreover, the strength ratio between
T=l and T=0 states is about 1.5 instead of unity for d-.,
transfer. This latter effect is due to the standard separa-
tion energy procedure used in the DWBA analysis of the (3He,a)
data represents an incorrect treatment of the form factors for
transitions to isobaric analogue states (St 66) . Stock and
- 14 -
Tamura (St 66) have shown that this problem is resolved if
one uses the solution of the Lane coupled-equations as the
form factor in DWBA.
In view of the consistent results of the (d,t) sum rules,
the strong ^-dependence of the expected values for ( He,a)
spectroscopic factors probably reflects a lack of under-
standing of the ( He,a) reaction mechanism.
2.2 Effective diagonal two-body matrix elements.
In theoretical studies of nuclear spectra, knowledge of
the effective nucleon - nucleon interaction and its two-body
matrix elements, is of great interest. The usual procedure
for experimentally determining two-body matrix elements is to
perform shell model calculations in which the two-body matrix
elements are unknown. This procedure has been used to deter-
mine (f7/2' an<* f7/2d3/2 e f f e c t i v e two-particle matrix ele-
ments from a variety of data in the mass 40 region by McGrory,
Erne and Dieperink et al. (Me 73, Er 66, Di 68). However,
the best experimental technique is to deduce the two-body
matrix elements directly from the centroid energies of the
spectroscopic strength observed in one-nucleon transfer reac-
tions. In the present work it has been possible to determine
the energy centroids for all individual spin members of the
2 -1(f_,_) , (£7/2^3/2' a n d ^f7/2d3/2' m u l t iP l e t s« While the
measured energy centroids are given as excitation energies
- 15 -
relative to the ground state, the effective two-body inter-
action energy is measured with respect to a reference energy,
at which a degenerate two-body multiplet would appear in the
absence of a residual two-body interaction.
Greater insight in understanding the effective two-body
matrix elements is obtained if the matrix elements are plotted
versus the analogue of the classical angle between the angular
momentum vectors j, and j_ of the two particles which is
defined by
cos 012
The cos9,2 dependence of the two-body matrix elements
clearly exhibits the range of the two-body force. For
example, a long range force will depend weakly on the dif-
ferent overlap of the two orbits for varying angle 6-i2"
Thus for a long-range force the two-body matrix elements will
be constant, independent of 9-,2" O n t^le o t h e r hand, a short-
range force will have the largest matrix elements when the
wave functions have the greatest overlap which is when the
orbits are coplanar; that is, cos 6,2 = +1. In fig. 4 is
plotted the (f7/->) an<^ ^7/2^3/2^ two-body matrix elements
The energy centroids are defined as
M j t JT) = E G. e./ I G.i * 1 i x
where G, and e. are the spectroscopic strength and excitationenergy if the i^h state of spin JT, which is excited by jtransfer. We note that the centroid energies depend only onthe relative spectroscopic strengths.
- 16 -
derived from the present data as a function of 6-|?. For the2
^ 7/22
' T =^ matrix elements, the behaviour is characteristic
of a short-range attractive force, which becomes large when
912 aPP r o a c h es 0° and 180°. For the T=l matrix elements this
is true for = 180°, but "hot for the small values of 6,-.
TWO-BODY MATRIX ELEMENTS IN 4?Cd
180
;MeV)
576(MsV)
•V 565
-3,15*3? '.' 481
-0,1028
42,Ca
51
-3,1-4*1
m—s, 22—6-,
22 3;0
196 5*0
13? ,*„
TWO BODY MATRIX ELEMENTS IN
(MeV)
- 0
0° G0° 120° 180°
Fig. 4
Values of energy centroids and two-body matrix elements deter-mined from present work. The two-body matrix elements areplotted versus the angle 6 to illustrate the physicalproperties of the force as discussed in text. The relationbetween the angle 912 and the corresponding single particleorbits j a and j 2 with ~3i + 32 - ^
a r e shown to the left inthe figure. The value of the reference energy Erefm for thedifferent multiplets is also indicated.
In the latter case the two particles form approximately a
spatially antisymmetric state. Hence the relative distance
between them is always large and the matrix element approaches
zero with a short-range force.
The matrix element for the minimum J-value of the
- 17 -
configuration is seen to be fairly attractive,
whereas the overall centroid of these matrix elements is
slightly repulsive. Qualitatively, this may be interpreted
as an effective interaction that on the average is slightly
repulsive but has a short-range attractive component which
dominates when the overlap between the two wave functions is
good.
Prior to the present results of (f7/2) two-body matrix
elements there has been some controversy regarding what data
yield the appropriate values. In mass 42 nuclei one assumed
that the lowest states of each spin represent the (f_._)2
multiplet. These values disagreed significantly with those
derived from a similar analysis of the Sc level spectrum,
Fig. 5
Comparison of (f 7/ 2)2 two-
particle matrix elements.The "8Sc particle-holematrix elements have beentransformed into particle-particle matrix elementswith the Pandya transform-ation (Pa 56, Sc 71).
180
- 18 -
which was assumed to represent the (f7/2f7/2^ configuration.
The present work has shown that the assumption made, regarding
2the (^7/2' multiplet, was wrong. For most spin members of
the multiplet, the f7/2 strength was found to fractionate over
several states yielding energy centroids deviating significantly
from the energy of the lowest state of appropriate spin. As42 2
can be seen from fig. 5, the Sc(f7 ,~) matrix elements derived
3 48from the ( He,d) data are in good agreement with the Sc
results (Sc 71). This indicates that the (f7/of_/o) particle-'•/Z 1/4.
hole configuration in Sc probably is less fractionated than
in mass 42 nuclei, which seems reasonable, because Ca is a
40better closed core nucleus than Ca.
— 1 41 3The f_,_d, -_ matrix elements derived from the Ca( He,a)
41data are compared with the corresponding Ca(d,t) results in
table 5. The two data sets yield identical results for the
4~ and 5~, T=0 states as well
as for all members of the T=l
states. This is due to the
fact that the d3>2 strength
to these spin states is
contained in a single level.
The value of the 3~, T=0
matrix element is uncertain.
As discussed, the A=2 tran-
sitions to the 3~, T=0 states
contain admixture from the
Effective (f7,2d3>2~ ) particle-hole matrix •laments
derived from on«-nucleon pick-up experiment! on CA
2~
3"
4~
5"
2*
3'
4~
5"
T
0
0
0
0
1
1
1
1
E(£ 7 / 2 ol3/j~1] (HeV)
(3He,a)
-0.83
-1.78
-1.65
-2.77
1.16
0.43
0.40
1.29
!d,t) "
-0.77
-1.38
-1.65
-2.77
1.16
D.43
0.40
1.29
(d,I) *'
1.36
0.59
0.56
1.46
d ,„ orbit, which most likely
- 19 -
has lead to an upward shift of the (£7/2^3/9) - multiplet
and thus made the 3 ,T=0 matrix element too little attractive.
The (f_/od, ,,) two-particle multiplet in mass 34 nucleiI/ c, if£•
has been located in one-proton and one-neutron stripping on
S and a sum rule analysis of these data yields another esti-
two-body interaction (Cr 72, Er 71).
The particle-hole transform of the mass 34 matrix elements,
as shown in fig. 6, agrees with our values to within a few
hundred keV for the T=l states and the 2 , 3 and 4 , T=0
states. This is within the experimental uncertainties.2r
mate of the f_/9d/ / /
( f 7 / 2 d 3 / 2 ) MATRIX ELEMENTS
A ' M S D Ia^Kuo&Brown•:Erne
T=1
T=0 -
Fig. 6 In the left-hand figure the (fr^cUTj) particle-holematrix elements obtained from the present f5fle,a) data arecompared with the corresponding matrix elements derived frommass 34 nuclei. The (f7/2d3/2) two-particle matrix elementsof mass 34 nuclei were converted into particle-hole matrixelements by means of the Pandya transformation. When trans-forming the (f7/2
d3/2^ matrix elements:, we used a value of0.13 MeV for the 2 ,T=1 state and not the value of 0.39 MeVgiven by Crozier (Cr 72); the new value is a result of new spinassignments in 3<*S (En 74) . This lead to a significant improve-ment of the differences between the two data sets. In particu-lar the discrepancy of the (f7/2d372)s-,T=O matrix element wasreduced from 1.5 MeV to 1 MeV. (See also discussion in text.)The right-hand figure shows a comparison of the ^7/2^3/2^ two-particle matrix elements obtained from present work and fromvarious calculations.
- 20 -
However, the mass 34 data correspond to a 5~, T=0 matrix element
40in Ca, which is about 1 MeV less attractive than our value.
One possible explanation for this difference is that the energy
- 40
of 5 , T=0 centroid in Ca is too low. For example, if the
SL=2 transition to the 8.483 keV (see table 1 in paper 1) is
assigned 5 , T=0, the centroid energy would be shifted upwards
by about 0.6 MeV. In view of the E5 enhancement observed in
inelastic scattering to the 5~, T=0 state admixture of deformed
core-excited states with the (^7/0^3/2^ - configuration, which
would cause a fragmentation of the 5 , T=0 strength, is not
unreasonable. In fact, Gerace and Green (Ge 68) predict a
second 5 , T=0 state at about 8.5 MeV excitation energy. On32
the other hand, it is possible that the non-closure of the S40core, which is considerably larger than in Ca, may cause the
discrepancy.
It is interesting to compare the present data with existing
calculations of the effective two-body interaction between
f7y? and d.,,, nucleons. Kuo-Brown (Ku 68) derived these
matrix elements from the Hamada - Johnston nucleon - nucleon
potential renormalized to account for core polarization.
Dieper.ink et al. (Di 68) calculated the matrix elements from40
a fit to levels in Ca using the phenomenological modified
surface delta interaction (MSDI). And Ernfi (Er 66) obtained
the matrix elements from shell model calculations fitted to
a varity of sd nuclei using an empirical interaction. As can
be seen from fig. 6, the.present data are in very good agree-
ment with the MSDI results. The Kuo-Brown interaction gives
a fairly good description of the J-dependence, but underestimates
- 21 -
the isospin monopole splitting by about 1.5 MeV. However,
Osnes and Kuo (Os 73) have shown that this shortcoming of
the bare Kuo-Brown interaction is greatly reduced when second-
order processes in the Kuo-Brown interaction are taken into
account. This increased the T=l to T=0 splitting to about
80% of the experimental value.
We conclude this section by noting that the (f7/5^3/2^T=0
matrix elements also exhibit a J-dependence characteristic of a
short-range attractive force while the splitting between the
two isospin states, which is considerably larger than the
difference in the J-dependence for the two isospins, indicates
the presence of a long-range isospin dependent interaction.
- 22 -
R E F E R E N C E S
Ba 65 R.W. Bauer, A.M. Bernstein, G. Heymann, E.P. Lippincottand N.S. Wall, Phys. Lett. 14_(1965)129.
Ba 65a W.H. Bassichis and G. Ripka, Phys.Lett, ljj(1965)320.
Be 7 5 R.R. Betts, C. Gaarde, O. Hansen, J.S. Larsen andS.Y. van der Werf Nucl. Phys. A253(1975)380.
Be 77 R.R. Betts, H.T. Fortune, J.N. Bishop, M.N.I. Al-Jadirand R. Middleton, Nucl. Phys. A292(1977)281.
Bj 64 J.H. Bjerregaard, H.R. Blieden, 0. Hansen, G. Sideniusand G.R. Satchler, Phys. Rev. 136(1964)B1348.
Br 66 G.E. Brown and A.M. Green, Nucl. Phys. 75(1966) 401.
Ca 64 E.B. Carter, G.E. Mitchell and R.H. Davis, Phys. Rev.133(1964)B1421.
Cr 72 D.J. Crozier, Nucl. Phys. A198(1972)209.
Di 68 A.E.L. Dieperink, H.P. Leenhouts and P.J. Brussard,Nucl. Phys. A116(1968)556.
En 66 T. Engeland and E. Osnes, Phys.Lett. 20(1966)424.
En 76 T. Engeland and P.J. Ellis, Phys. Rev. Lett. 3_6(1976)994.
Er 66 F.C. Ernë, Nucl. Phys. 84(1966)91.
Er 71 J.P. Erskine, D.J. Crozier, J.P. Schiffer and W.P.Alford, Phys. Rev. Ç3(1971)1976.
Er 77 T. Erikson and G.E. Brown, Nucl. Phys. A277(1977)1.
Fl 77 C.Flaum, J. Barrette, M.J. LeVine and C E . Thorn,Phys. Rev. Lett. 3_9 (1977) 446.
Ge 67 W.J. Gerace and A.M. Green, Nucl. Phys. A93(1967)110.
Ge 68 W.J. Gerace and A.M. Green, Nucl. Phys. A113(1968)641.
Go 63 S. Gorodetzky, P. Mennrath, W. Benenson, P. Chevallierand F; Scheibling, J. Phys. Radium 24(1963)887.
Gr 72 C R . Gruhn, T.Y.T. Kuo, C.J. Maggiore, H. McManus,F. Petrovich and B.M. Preedom, Phys. Rev. £6(1972)915.
- 23 -
Ha 66 J. Hayward, Nucl. Phys. 81(1966)193.
Ha 74 O. Hansen, J.R. Lien, O. Nathan, A. Sperduto andP.O. Tj0m, Nucl. Phys. A243(1975)100.
KI 75 A.M. Kleinfeld, K.P. Lieb, D. Werdecker and V. Smi-lansky, Phys. Rev. Lett. 3_5 (1975) 1329.
Ku 68 T.T.S. Küo and G.E. Brown, Nucl. Phys. A114(1968)241.
La 76 R.D. Lawson, F.J.D. Serduke and H.T. Fortune, Phys.Rev. 014(1976)1245.
Li 77 C.J. Lister, A.M. Al-Naser, A.H. Behbehani, L.L.Green, A.N. James, P.J. Nolan and J.F. Sharpey-Schafer, J. Phys. G: Nucl. Phys. 3J 1 9 7 7) L 7 5»
Mc 70 J.B. McGrory, B.H. Wildenthal and E.C. Halbert, Phys.Rev. £2(1970)186.
Mc 73 J.B. McGrory, Phys. Rev. C8(1973)693.
Mo 56 H. Morinaga, Phys. Rev. 101(1956)254.
Os 73 E. Osnes and T.T.S. Kuo, Phys. Lett. 47B(1973)430.
Pa 56 S.P. Pandya, Phys. Rev. 103(1956)956.
Sc 71 J.P. Schiffer, in Proc. of the Symp. on The Two-BodyForce -in Nuclei, Michigan, 1971, edited by S.M. Austinand G.M. Crowley, p. 205.
St 66 R. Stock and T. Tamura, Phys. Lett. 22(1966)304.
To 73 C.W. Towsley, D. Cline and R.N. Horoshko, Phys. Rev.Lett. .28(1972)368; Nucl. Phys. A204 (1973) 574.
Vo 74 M.J.A. de Voigt, D. Cline and R.N. Horoshko, Phys.Rev. C1O(1974)1798.
Vo 77 P.B. Void, D. Cline, P. Russo, J.K. Sprinkle, R.P.Scarenberg and R.J. Mitchell, Phys. Rev. Lett. 39(1977)325.
Za 65 L. Zamick, Phys. Lett, lj} (1965) 580.
- 24 -
P A P E R I
PARTICLE-HOLE MULTIPLETS IN 4°Ca
OBSERVED IN THE 41Ca(3He,a) REACTION
- 25 -
I.E.I:2.G
Af«r/rari>/vj/«A233(1974)91-104; (£) North-Holland Publishing Co., Amsterdam
Not to be reproduced by photoprint or microfilm without written permission from the publisher
PARTICLE-HOLE MULT1PLETS IN 40CaOBSERVED IN THE 41Ca(t, a) REACTION
D. CLINE and M. J. A. DE VOIGTNuclear Structure Research Laboratory, University of Rochester,
Rochester, NY, 14627, USA*
and
P. B. VOLDInstitute of Physics, Uniienity of Bergen, 5000 Bergen, Norway
and
OLE HANSEN, O. NATHAN and D. SINCLAIRThe Niels Bohr institute, University of Copenhagen, 2100 Copenhagen O, Denmark f '
Received 27 March 1974
Abstract: Energy levels in *"Ca up (o 10.2 MeV have been studied in the neutron pickup reaction41Ca(r, a)40Ca with 20 MeV bombarding energy. Thirty excited states have been identifiedand angular distributions have been measured in the interval from 5° lo 40° by means of asplit-pole magnetic spectrometer. The angular distributions together with DW calculationshave been used lo extract /„ values and spectroscopic factors. The /„ = 2 strength distributionfor the f jd |~ ' particle-hole levels is compared to the /p = 3 strength distribution from protonstripping data.
NUCLEAR REACTIONS 41Ca(r, a), £ = 2 0 MeV; measured <r(£a, 6). "Cadeduced levels, J, n, I, T, spectroscopic factors. Enriched, radioactive target.
1. Introduction
The energy levels of 40Ca below about 8 MeV comprise two classes, namely thespherical "shell-model" states and the deformed multi-particle, multi-hole levels [seee.g. ref. ') and references quoted there]. The shell-model states consist of the groundstate and the negative parity Ip-lh states while the deformed states are the positiveparity levels plus a sequence of negative parity states forming a K" = I" band. Thetwo classes of levels mix to some extent').
The Ip-lh odd parity states can be studied experimentally in at least three distinctlydifferent ways: through direct inelastic scattering, through proton capture from 39Kand through neutron pickup from 4 lCa. The inelastic scattering experiments aresensitive to the coherence properties of the 40Ca levels 2) while the proton stripping
' Research supported by a grant from the National Science Foundation.t' Work supported in part by the Jupan World Exhibition Commemoraiive Association, Tokyo,
Japan.
91
- 26 -
92 D. CL1NE el al.
experiments determine the relative strengths of the (d t)~ l (fp)+l configuration com-ponents and the neutron pickup processes yield information on the (sd)"1 (fj)*'configurations.
Detailed studies of 40Cu utilizing inelastic scattering processes and proton-strippingreactions haveappcared in the literature J~7)whiledata from neutron pickup on 4 lCahave not been published previously. The present paper reports on the 4lCa(T, a)reaction; an enriched 4 lCa target was bombarded with 20 MeV 3He ions and levelsin 40Ca were studied up to an excitation energy of 10 MeV. The values of transferredangular momentum /„, and the corresponding transition strengths were determinedfrom a DWBA analysis of the measured differential cross sections and they are dis-cussed in sects. 2 and 3.
A detailed comparison with other data on *°Ca is presented in sect. 4, and thespectroscopic strength sums for /„ = 2 and /„ = 3 arc discussed in sect. 5.
2. Experimental procedures and results
2.1. TARGET
The target of the radioactive isotope 4JCa, employed in the present measurements,was the same as used in a previous (d, p) experiment 8). The Ca material was enrichedto 81.8 % in 41Ca with an 18.1 % 40Ca residue. The backing was 30fig/cm2 carbonand the Ca thickness was about 25 /<g/cm2. The details of the isotope production andtarget preparation are given in refs. 8- *). The enriched material was purchased fromOak Ridge National Laboratory, Isotopes Service. A 40Ca target was also used foridentification of transitions leading to single-hole stales in 39Ca.
2.2. EXPERIMENTAL PROCEDURE
The 41Ca and 40Ca targets were bombarded by 20 MeV 3He particles from theRochester MP tandem accelerator and the reaction products were momentum ana-lyzed in a split-pole magnetic spectrometer10) and detected in Ilford K.0, 50;imphotographic emulsions placed along the focal plane. Track discrimination wasfacilitated by placing suitable Al foils directly in front of the emulsions.
Spectra were obtained from 41Ca at laboratory angles from 5° to 30° in 5" stepsand at 40°, while control runs on a 40Ca target were made at 5°, 10°, 20°, 30° and 40°.The relative intensity normalizations were made by monitoring the elastic scatteringat 45° by means of a solid state detector.
Absolute cross sections were measured by observing the elastic scattering yields atangles from 10° to 30° in 5° steps using a position sensitive solid state detector in thefocal plane of the magnetic spectrometer and by normalizing to optical model predic-tions. The 41Ca(r, a) absolute cross-section scale is believed to have a systematicerror of less than ± 15 %.
The ratio between 40Ca(r, a) and 41Ca(T, x) yields was obtained in two ways: byrelative measurements on targets of the two isotopes, as described above, and by
- 27 -
41Ca(T,o) 93
500,
EXCITATION ENERGV (MeV)
4 3 2
400
£ 300 i
200
100 P
*1Co ( 3 He,a) '°Ca
E = 20.0 MeV
CM
JU.20
- ~ _ . ^ . . , - J L . . .15 10
DISTANCE ALONG PLATE (cm)
500
EXCITATION ENERGY (MeV)10
o
zr
i 0 0 '
300
200
100-
45
K
J
41Co ( 3 He.a) <
E = 20.0 MeV
5°
DISTANCE ALONG PLATE (cm)
Fig. I. Spectrum of x-particles at 5° laboratory angle. The group numbers refer to table 1.
- 28 -
94 D. CLINE etal.
TABUS 1a°Ca level energies and 4 lCa(r, a)40Ca spcctroscopic strengths
Group £", (kcV)no.
012
3
4
56
7
89
10II12
1.1
14
15
this work(±5kcV)
0(3350)3732
4488
S6I0
(5901)6029
6288
65836748
693069507112
7531
7656
7693
(kcV)rcf. 13)
03352.93736.83904.44491.55212.25249.05278.95614.35627.95902.56025.26029.06285.16509654465816750.56909.869.1069517113.57240728073007.19974267447.17467.175J2.67561.87625.77658 876767695.87768781178677928797280168092fill?8133818682758.1218.157
J";Tr c f l 3 ) ' l
0*0*
a-2*5"0*T +
4*4 -2*("•> -
J*3"4*4*3"2~2*
1-
0*
2 *
<I-J>-
4 " ; T =1
J - ; T = 1
J . b ,
4 * " )
2 ' ' )
if-3)- ")
(0-3)"
4 " )
J"thiswork
-I"')
C(J-•/•!-/ ,)")
/ „, 3 ; = o / « 2
(0 98]10.01]
0.57
( I I ]
0.89
: 0.003(0.17]
0.05
0 04 0.21|022]
(0.07)0.06
0.13 0.20
0.03
1.1
1.3
(dff/dfl)m,, Commentsfmb/srt
1.650.030.59 poor DW fit
1.20
1.12
5 0.005 Id j transfer0.26
0.09
0.410.31
0.(0 poor data0.13 Id] transfer0.66
0.04
1.93
1.88
- 29 -
•"Ca(i, a)
TABLE 1 (continued)
95
Group £•„ (keV) E, J"; T J"no. this work (keV) ref. ")') this
(±5keV) ref.13) work
G(i - * ; + / , ) " ) (do/dQ)mi, Comments— . (mb/sr)
/ = 3 / = o / = 2
16 837417 842318 8483
19 8551
2021
23
903590809145
9222
24
25
26
27
28
29
30
31
9435
9460
9559
9605
9647
9673
10055
10214
83718424847485358551 5"; T «8578 2* ")8626866487438757880588488904893189778993 2* ')902890759136 3" "••)915891719197922892679404 2"; T = 119429194329454 (2,3)-;r-=19535(95369602
J960496391966719668
11004211005110212
(0.08) 0.09 / uncertain|0.62] 0.82(0.21) 0.25 /=--2 probable
117] 2.14
1"
1-
(3 ,4)-
(3,4)"
(3, 4)"
0.33(0.06)
(0.05)
0.98
0.18
(0.11)
(0.05)
(0.31)
(^0.1)
1.470.290.13
0.08
0.22
0.21
0.38
0.58
i 0.10
0.89
6.07
0.76
/ = 0 probable1=1 probable
/ uncertain
/ uncertain
/ uncertain
/ uncertain
•) (d, n) and (r, d) frefs. *•')] have !„ - 1 and J" (0-3)~. Present data have J" =--= 3" or 4". Thusy = 3-.
*) Deduced from ref. " ) .') The lowest 2"; T — 1 level is erroneously quoted in ref. l 3 ) at E, = 8474 keV.d) Spectroscopic factors given in square brackets indicate the /-value is fixed by angular momentum
considerations.c) r = 0 is preferred for this level because there is no 4 0 K parent state to match it ") and because
it was missed in the (p, y) work l4> which otherwise excited all T = 1 levels.') Probably the same state as the 8366 keV state of ref. " ) .
- 30 -
96 D. CLINE el al.
simultaneous observation of a-particle groups originating from 40Ca and 41Ca in theenriched 4 lCa target and using the isotopic analysis provided by Oak Ridge. Thetwo sets of data agreed to within the experimental uncertainties and determined the40Ca(r, a) and 41Ca(T, a) cross-section ratios to better than ± 10 "„.
2.3. RESULTS
A spectrum from the 4lCa(r, a) reaction is shown in fig. 1; it includes 32 at-pariiclegroups corresponding to states in 40Ca up to an excitation energy of 10.21 MeV, Inthis range of excitation more than 100 levels are known 3 " 7 ' " " 1 4 ) . In table 1 wepresent information for all known levels up to £', = 9.0 MeV and for u selection oflevels between 9.0 MeV and 10.2 MeV. The 40Ca(j, a)31)Ca contaminant groupswere used as a corrective secondary energy standard; uncertainties in the excitationenergies amount to +5 keV. The energy resolution was 20 keV FWHM.
TABLE 2
Optical-model parameters
Particle
3He4Hcn
('
165211.6
')
<o
1.14
1.14
i.:o
0.7230.790.65
I f
20.028.8
0
II '
0
0
0
r'o
1.601.1420
a'
0.810.750
Thomas
00
25
Ref.
" )
Well depths are in MeV, geometrical parameters in fm.') Adjusted to give a binding CLJII.II to the experimental separation energy.
TABLE 340Ca(T, a)3*Ca spectroscopic factors
Levelno.
0
1
J"
V
E,(keV)
02470
C(0*
1 0
3.8
•••J,)
1 — 2
9.0
(dtj/d#)mJI
(mb/sr)
12.520.5
3. DWBA analysis
The analysis of (T, a) cross sections near mass number 40 in terms of DW methodsis somewhat problematic l s " 1 8 ) . The angular momentum mismatch at the nuclearsurface is large for (T, <X) transitions to low-lying stales with angular momentumtransfers of 0 and 2 [see e.g. ref. I S)]. Another complicating feature of the DWBAanalysis is the anomalous x-pariicle scattering for targets ncai 40Ca. The net result ofthese effects is a dependence on the incident energy and unphysically large16)extracted spectroscopic factors, especially for /„ = 2, d} pickup. DWBA calculationsfor isobaric analogue slates in this mass region systematically yield too small (T, X)
- 31 -
4lCa(T, a) 97
- \
101(=3
'•\ I =2
(1)U 3
10?
(2)1 = 2
\
15) • (G) • . (7) ,-01 l»2 . io U 2 - : ,01 1=2 -|
e
a
1-0+2
"", ' 0) (10)
' \ 1=2 , I--2• 01 .. H
U2 !
001 . I -i
- v \ . 02) • - (13) . . (141•01 \rj. 01 i=? i o \ 1=2
"w-0+2- -_ • \
- U2 V • - \
(is) :
', J
10 30 50 10 30 50 10 30 50 10 30 50
9c.mFig. 2. Experimental angular distributions and DW predictions in the cm. system. The groupnumbers refer to table 1. The normalization of the DW curves corresponds to the experimental
spectroscopic factors of table I •
98
t>
- 32 -
D. CL1NE el at.
. - , _ , - , -(16) J(1.2) .
(17)
.10 1.2
. . -i_J I-.. . I _ . _ l - i -
(18)
01 ', 1-2 ..(19)
I
\ (20)
• \ 0 1 - 0 -(211 . (22) \
U2(23) .U-2) -
(24)
I I !
(25) (26) .
001 ,
(27)
(31)
10 30 50 10 30 50
10 30 50 10 30 50
6 cm.Fig. 3. See caption to fig. 2.
cross sections and thus too large spectroscopic factors ts'il). The authors of refs. * ' • ' 7 )conclude that this latter effect is caused by the incorrect treatment of the form factorsand not by finite range or non-locality effects.
The DWBA code DWUCK by P. D. Kunz was used together with the optical
- 33 -
41Ca(T,a) 99
model parameters cited in table 2; they are average parameters fitted to elasticscattering for the fj shell region 19~2'). The calculations were made in the zero-range and local approximations, and the radial integrations had a lower boundaryof 0.1 fm. The spectroscopic factors quoted in tables 1 and 3 for the +1Ca(T, a)40Caand 40Ca(t, a)39Ca reactions, respectively, were extracted from the experimentaldata via the relation
where j designates the total transferred angular momentum, V, is the target spin andJt is the final-state spin. The term G™ equals CZS where C is an isospin Clebsch-Gordan coefficient and S the spectroscopic factor defined Jn an isospin 9) formalism.
More than one value of j may contribute, incoherently, to a transition sinceJt = i for the 41Ca g.s. In particular both dj. and d} transfer may contribute to/„ = 2 transfers for Js = 2-5. We assume that the dj. contribution is small for allstates studied with the possible exception of the lowest, collective 3~ state. Both d$and s.j transfer is allowed for •/* = 3~ and 4". Only in two cases, however, have wefound it necessary to fit the angular distributions by a sum of /„ = 2 and /„ = 0(groups no. 8 and 12); in all other cases, the quoted spectroscopic strengths werederived from a fit with one value for the transferred angular momentum. It shouldbe noted that at low excitation energies the angular momentum mismatch makes the(T, a) reaction a poor tool for detecting /„ = 0 strength. Also, the DW /„ = 0 predic-tions in this region are of doubtful value. It is entirely possible that parts of the low-lying 3~ and 4~ strength, recorded here as due to /„ = 2 transfer, may be of /„ = 0character. The DWBA predictions are compared to the experimental cross sectionsin figs. 2 and 3.
4. Comparison with other experiments
The 40Ca(T, a)39Ca(g.s.) forward angle cross section of 12.5±2.0 mb/sr (table 3)may be compared to the value of 10 mb/sr measured at 18 MeV by Bock el al. 21).DWBA calculations predict an increase by 17 % in the cross section when going from18 MeV to 20 MeV (the energy used here). Thus the two cross-section results are inagreement within the experimental uncertainties.
The spectroscopic strength given in ref. 16) for 40Ca(r, a)"Ca(g.s.) at 20 MeV is 8,in good agreement with our value of 9 (table 3). Our value of 3.8 for 4 °CU(T, a)3 9Ca(^+,2.47 MeV) disagrees with the value of 2 quoted in ref. 1 6). On the other hand, theearlier Heidelberg paper " ) on the same reactions quotes a spectroscopic factor forthe 2.47 MeV state which is 2.7 times smaller than the ground-state S-value. Thiscompares quite well with our value of 2.4 for the same ratio. The reasons for thediscrepancy in 5[3 'Ca(g.s.)]/S[39Ca(|+)j between the two Heidelberg papers is notclear.
- 34 -
100 D. CUNE etal.
The comparison of the present results to the proton stripping reactions 3 9 K ( T , d)and 39K(d, n) is of particular interest. The 39K.(g.s.) is mainly a proton d} single-holestate relative to 40Ca(g.s.), while the 4lCa(g.s.) is mainly an f} single-particle staterelative to 40Ca(g.s.). We should therefore expect those lp-lh states in 40Ca thathave dj^fj components to be excited in both proton capture and neutron pickupexperiments. Another set of 40Ca states expected to be common to both experimentsare states having core configurations which are present as components in both39K(g.s.) and 4lCa(g.s.).
The stripping 4> 7) reactions and the pickup reaction both have large cross sectionsto the 40Ca ground state. The proton d} stripping gives G"1 = 0.8 whereas G" = 1.0for fj neutron pickup. For stripping we use C"ripp'"« = (2/f + ])(2/,+ l )~ l
S(J,+j -* Jt), where S is the spectroscopic factor in a non-isospin formalism. Thevalue of G" is obtained with the normalization factor 23 given in eq. (1). The40Ca(d, p)41Ca(g.s.) reaction " ) spectroscopic factor Sdp = 0.8 is in fair agree-ment with the (T, a) result.
Only one positive parity state in addition to the 0+ ground state, was excited in theproton transfer reactions, namely the 3.35 MeV 0+ multi-particle, multi-hole level.In the 39K(i, d) reaction the spectroscopic strength of the 0+ state was 8 % of thatof the ground state and an upper limit of a similar magnitude was quoted for the39K(d, n) reaction 1). This same state is observed in the 41Ca(t, a) reaction with aneven smaller spectroscopic factor of only 1 % of that of the ground-state transition,and again it is the only positive parity excited state definitely observed. The differenceof almost an order of magnitude between the stripping and pickup intensities forexciting the 3.35 MeV 0+ level is believed to reflect a difference in the detailed corestructure of the two target ground states. The non-observation of neutron pickup toany other positive parity state suggests that the 4 lCa ground state looks very muchlike an fj neutron coupled to a 40Ca(g.s.) core.
Those d j 'fj states which are populated in both the proton stripping and the neutronpickup reaction are those states which have proton as well as neutron components.If isospin is a good quantum number, states with T = 0 or T = 1 can be reached andthese states will have equal amounts of proton and neutron configurations. Theobserved /p = 3 spectroscopic strengths C'J for proton stripping are compared, infig. 4, with our 4 = 2 and /„ = 0 pickup spectroscopic strengths. The strippingstrengths are an average of the Seth el al. 4) 3 9 K(T, d) and Fuchs el at. 7) 39K(d, n)results.
Almost all strong f} stripping transitions have strong 41Ca(r, a) /„ = 2 counter-parts, whereas the reverse statement does not hold true. In several cases an *'Ca(T, a)/„ = 2 transition has no /p = 3 3 9 K(T, d) or39K(d, n) counterpart, but rather corre-sponds to an /p = 1 stripping transition. This is not considered to reflect an importantdiscrepancy since both of the stripping experiments inherently favour pt transferstrongly over f$ transfer and thus may contain appreciable undetected /p = 3 strength.
Excitation of 1" levels in the (r, a) reaction can proceed only via pickup of dj
- 35 -
•"Ca(r,<x) 101
neutrons, dj pickup being forbidden by the selection rules. The 6.95 MeV 1" state
is weakly excited in both stripping and pickup, thus demonstrating the presence of
small admixtures of both the fyij ' and d p p j components in this state.
STRENGTH IN '"Co- . 1 . 3 _ - l . J
PROTON CAPTURE M«V NEUTRON PICK UP
J —
-0
20 IS
IJ5
OS os w « 20
Fig, 4. Experimental spectroscopic strengths for proton stripping and neutron pick-up, respectively,to states of *°Ca. The pick-up data are from this work and the stripping data jre average values fromrefs. "•'). The ordinate is excitation energy in 40Ca. The figure is further discussed in the text *.
All the transfer experiments excite the 8.42 MeV state strongly indicating that this
level rather than the 8.48 MeV state l J ) is the T = I, 2~ state. The transfer reaction
assignment has been assumed to be the correct one.
' Note added in proof: At 7.7 MeV excitation a small amount of/,, = 1 (r, d) strength is shown. Inaddition an /„ = 3 strength of 1.4 should have been indicated.
- 36 -
102 D. CLINE el al.
5. The monopole sum rule
The neutron spectroscopic strengths should satisfy the relation 24)
= >•(/), (2)
where G" designates the neutron spectroscopic strength for the transition of the ithstate of spin J( and v(j) is the number of/-neulrons in the target.
If isospin is a good quantum number, one further has2*)
(3)
where the summation extends only over isobaric analogue states and n(j) is thenumber ofy'-protons in the target state. If the fj proton orbits are empty in 41Ca(g.s.)then eq. (3) implies that neutron fj pickup leads exclusively to Tf = 0 states. On theother hand, the d j , s and deeper lying neutron strength split evenly over 7*f = 1 and0 final states, provided the core of 41Ca(g.s.) has T = 0. The latter result applies foreach final state spin value J, separately.
The pertinent strength sums are displayed in table 4. The /„ = 3 summed strengthis unity as predicted by the simplest shell model description. Both the 40Ca and 4 lCa/„ = 2 spectroscopic sums exceed the ldj neutron sum-rule limit of 4 by a factor ofabout 2. The 4 lCa sum for the dj spectroscopic factors is about 90 % of the 40Cavalue. The 40Ca s4 sum likewise exceeds the sum-rule limit of 2 by a factor of about 2.The s summed strength observed in the 4lCa(T, a)40Ca reaction is roughly half thetotal strength observed in the *°Ca(t, «J39Ca reaction. This is as expected if only theT = 0 part of the fjS^' strength has been located in 40Ca.
TABLE 4
(r, a) spectroscopic strength sums
Target
41Ca
40Ca
j "
} -
V
Vi*1 +
J,"
0*2"2"3"3"4"4~5"5~allallall
3-,4"3*
r,00i0i010101all
0,1it
SC" (S
1.00.390.621.01.30.891.31.11.73.44.98.31.79.03.8
B..)'-/(SO»)T-
1.6
1.3
1.5
1.5
1.4
- 37 -
4lCa(r,a) 103
The ratios of [G]T * 'l[G]T " ° are about 1.3-1.6 rather than unity for the dj. pick-up suggesting that the present DWBA calculations do not properly reproduce the Q-dependence of the / = 2 cross sections. This result is parallel to conclusions made inprevious work on (T, a) reactions, in particular in refs. *5l *7).
The unphysically large /„ = 0 and /„ = 2 spectroscopic strengths and incorrectisospin splitting of the strengths both reflect a lack of understanding of the (r, a) reac-tion mechanism. It would be tempting at this point to use the monopole sum rules asguidelines for a renormalization of the spectroscopic strengths. Such renormalii'dstrengths could be used for a further discussion of the dipole and quadrupole sumrules. In view of the mentioned difficulties in the analysis of mixed /„ = 0+2 transi-tions we have found this procedure unjustified.
6. Summary
The 41Ca(T, a)40Ca reaction below Ex = 10.2 MeV is found to populate primarilythe 0+ ground state and 20 negative parity states. These negative parity states appearto contain the major fraction of the d^ 1fi particle-hole multiplet and the T = 0 partof the 2sj *f} multiplet.
The spectroscopic factor for f$ neutron pickup to the 40Ca ground state is approx-imately unity in agreement with the value of 0.8 measured in the 40Ca(d, p)41Careaction " ) . The spectroscopic factors for /„ = 2 and /„ = 0 neutron pickup deter-mined with targets of 40Ca and 41Ca were measured to be a factor of two too large.This is a feature observed in other studies of the (T, a) reaction on nuclei near mass40. In addition the summed strength to the T = 1 states is 40 % greater than to theT = 0 states indicating further problems with the interpretation of the 4lCa(r, a)40Careaction.
The authors want to express their graditudc to Drs. H. E. Gove and P. Stelson fortheir assistance in arranging the 41Ca production in 1965. Discussions with Drs.W. P. Alford, P. Goode, J. P. Schiffer, B. Mottelson and I. Hamamoto are muchappreciated. D. Cline also wishes to thank the Niels Bohr Institute for the hospitalityand financial support given him during his stay in Copenhagen.
Reference
1) W. J. Gerace and A. M. Green, Nucl. Phys. A113 (1968) 6412) A. M. Bernstein, Advances in nuclear physics, vol. 3, ed. M. Baranger and E. Vogt (Plenum
Press, New York, 1969) p. 3253) J. R. Erskine, Phys. Rev. 149 (1966) 8544) K. K. Seth, J. A. Biggerstaff, P. D. Miller and G. R. Satchler, Phys. Rev. 164 (1967) 14505) J. S. Forster, K, Bewpark, J. L. Hutton and I. F. Sharpcy-Schafer, Nucl. Phys. AI50 (1970) 306) M. E. Cage, R. R. tohnson, P. D. Kunz and D. A. Lind, Nucl. Phys. A162 (1971) 6577) H. Fuchs, K. Grabisch and G. Roschert, Nucl. Phys. A129 (1969) 5458) C. EHegaard, J. R. Lien, O. Nathan, G. Sletlen, F. Ingebretsen, E. Osnes, P. O. Tjom, O. Hanson
and R. Slock, Phys. Lett. 40B (1972) 641
104
- 3 8 - -
D. CLINE ttal.
9) J. A. Smith, E. J. Herrelly. C. H. Ice and H. F. Alter, Trans. Am. Nucl. Soc. 8 (1965) 5410) H. A. Enge, Nuci. Instr. 28 (1964) 119;
J. E. Spencer and H. A. Enge, Nucl. Instr. 49 (1967) 18111) M. A. Grace and A. R. Poleiti, Nucl. Phys. 78 (1966) 27312) C. R. Gruhn tl ai, Phys. Rev. C6 (1972) 91513) P. M. Endi and C. van der Leun, Nucl. Phys. A214 (1973) I14) R. J. de Meijer, A. A. Sieders, H. A. A. Landman and C. de Roos, Nucl. Phys. A155 (1970) 10915) R. Stock, R. Bock, P. David, H. H. Duhm and T. Tamura, Nucl. Phys. AI04 (1967) 13616) U. Lynen, R. Sumo, D. Schmitt and R. Stock, Phys. Lett. 27B (1968) 7617) R. Stock and T. Tamura, Phys. Lett, a (1966) 30418) J. Rapaport, W. Dorenbusch and T. Belote, Nucl. Phys. A177 (1971) 30719) R. Bock, P. David, H. Duhm, H. Hefcle, U. Lynen and R. Stock, Nucl. Phys. A92 (1967) 53920) O. Hansen, T. 1. Mulligan and D. J. Pullen, Nucl. Phys. A167 (1971) 121) L. McFadden and G. R. Satchler, Nucl. Phys. 84 (1966) 17722) R. Bock, H. Duhm and R. Stock, Phys. Lett. 18 (1965) 6123) K. Sclh, J. Picard and G. R. Satchler, Nucl. Phys. A140 (1970) 57724) J. B. French and M. H. MacFarlanc, Nucl. Phys. 26 (1961) 168
- 39 -
P A P E R II
NUCLEAR STRUCTURE OF 41Ca FROM
INELASTIC PROTON SCATTERING
- 40 -
Nuclear Phvsiis A292 (1977) 107-124: © North-Holland Publishing Co., Amsterdam
Not lo be reproduced by phutoprinl or microfilm without written permission from the publisher
NUCLEAR STRUCTURE OF 4lCaFROM INELASTIC PROTON SCA1TERING
P. B. VOLD \ D. CLING and M. J A. DE VOIGT "
Nuclear Structure Research Lalwratary Uiuccisilv «l Rochester, Rochester, NY 14627 '"
and
A. SPRRDUTO
Massachusetts Institute of Teihnofagy, Ctwihriilije. Maw. 02139
Received 26 April 1977(Revised 19 July 1977*
Abstract: Angular distributions have been measured for inelastic and clastic scattering of 19 Me V protonson 40 4lCa. A total or 89 levels were identified below 6.4 MeV in "Ca with an energy resolutionof 12 kcV. Inelastic transition strengths have been extracted using DWBA theory with a vibrationalmodel form factor. These transition strengths correlate well with inelastic ^-scattering and electro-magnetic values. The quadrupole strengths arc interpreted in terms of the coexistence model andimply lhat the excited-core admixtures in the ground slates of both 4nCa and 4lCa are ft 5°'o. Theoclupolc strength in 4>Ca exhibits features characteristic of the weak coupling of an f1i2 neutron tothe lowest ) Male in '"C.\ The / - 5 strength exhibits a similar weak-coupling behaviour. In bothcases Ihe microscopic structure appreciably reduces the transition strength for ihc highest spinmembers of the weak-coupling nuiltiplcts.
NUCLEAR REACTIONS 4n 41Ca(p. p). f = 19 Me V; measured a( F.p . l() 4lCa deducedlevels, J, n. inelastic transition strengths. /. 40-4'C"a deduced cxcilcd-corc admixtures.
Coexistence model, weak-coupling model, DWBA, coupled-channel analysis Unriched targets.
I. Introduction
The coexistence of quadrupole deformed core-excited configurations and (If2p)"configurations in the structure of the low-lying states in calcium nuclei is a wellknown phenomenon ' " 3 ) . The structure of the low-lying stales of *'Ca should beparticularly simple if these status are formed hy coupling the odd neutron to thelow-lying stales of the 40Ca core. Thus in addition to the If and 2p single-particlestates there may occur weak-coupling multiplcls formed by coupling a lf} or 2pjneutron to the collective quadrupole band or the 3" and 5" collective states of the'40Ca core.
' Present address, Institute of Physics, University of Bergen, 5000 Bergen, Norway." Present address, KVI. University of Crontngcn, the Netherlands.
" ' Supported by the National Science Foundation.
107
- 41 -
IOX P B. V O L D <•/ <il
Considerable experimental evidence *) exists on the energy level scheme for 41Ca.Recently, the 21AI(I(1O, pn)*'Ca and the 2"Mg(1BO, 3n)*'Ca reactions56) havebeen used to study the yrast high-spin states some of which could be members ofthese weakly coupled multiplcts. In addition, the 4nK(1He, d)4'Ca and the3OK(3(, d)4lCa reactions 7-8<)) have been used to study the 2p-lh nature of thesehigh spin states.
Inelastic scattering is sensitive to the collective properties ofexcited states and thusis ideally suited to investigations ofthe weak coupling of single-particle and collectivemodes in nuclei. The 4lCa(«, a') reaction has been studied 10) previously and theweak-coupling structure in 4lCa was investigated. The present paper described asimilar investigation using the 4lCa(p, p ) reaction. Although these two reactionsshould give similar information, there are several notable differences.
The (z. *') reaction selectively excites collective slates and gives excellent I-discrimination for a bombarding energy of 28.5 McV. The (a, a ) transition operatoris isoscalar and the inelastic transition density for the strongly absorbed oc-particle issimilar to that for the corresponding electromagnetic operator. Thus the inelastica-lran.sition strengths, derived from a conventional distorted wave (DW) analysisarc closely similar to the isoscalar part of the electromagnetic fl(EA) strength l 0) .In contrast, the (p. p') for 19 MeV protons, gives poor /-discrimination. In addition,the selectivity ofthe (p, p ) reaction is low, presumably due to the existence of spindependent components in the (p. p') transition operator, i.e.. three times as manystates were excited in the 4lCa(p, p') reaction compared with the 4lCa(a, a')reaction. However, the (p, p') reaction is more sensitive to low /-transfer and theenergy resolution achieved in the present work, 12 keV, FWHM, is better than the20 kcV obtained using the (a, «') reaction. The inelastic proton transition operatormay be quite complicated and thus the inelastic proton transition strengths coulddiffer appreciably from the corresponding electromagnetic strength. Jn addition,there may be an appreciable indirect contribution to the reaction mechanism for19 McV protons. Thus it is necessary to check carefully that DW analysis of in-elastic proton scattering leading to well known levels produces reasonable transitionstrengths before drawing conclusions for other excited states.
The purpose of the present work is to compare the transition strengths extractedfrom inelastic proton and j-scattering and then to study the implications of the(p. p') data concerning the structure of 4lCa. The experimental procedure andresults are presented in sect. 2. The analysis of the data using DWBA and coupled-channel calculations is described in sect. 3. A comparison of the results of thepresent and previous work is given in sect. 4, while the implications of the presentresults on the structure of 41Ca is given in sect. 5.
42 -
'Ca 109
2. Experimental procedure and results
The target consisted of 25 /ig/cm2 metallic calcium evaporated onto a 30 /Jg/cm1
carbon backing. The target was enriched to 81.8 % 4lCa with an 18.8 % 4UCu residue.The details of the isotope production and the target preparation are described inrefs. " • 1 2 ) . A s;mi)ar target of''"Ca also was bombarded to facilitate identificationof 40Ca levels excited using the 41Ca target.
The 41Ca(p, p') reaction was studied using 19 MeV protons from the University ofRochester MP tandem Van de Graaff accelerator. The outgoing protons weremomentum analyzed using an Enge split-pole magnetic spectrometer and detected
eoo
600
o 400o
200
"Co(p.p)"CoE"l90MeV8c 90°
2000 3000Ex(keV)
4000
800h
600
50001
6000Ex(keV)
Fig. I. Prolon spectrum at 90' lab, recorded with photographic plates The number of counts are givenper 0 25 mm of the plates.
TAHLI 1
Comparison of Ihe *'Ca(a. a') and Ihe present 4 ICa(p. p') results
£, (keV)
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- 45 -
112 P. B. VOLD el ul.
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') Above 5 MeV spin assignments are only given for levels of special interest to the present work.b) Strength for assumed E3 transitions.') Strength for assumed E2 transitions.d) Strength for assumed E5 transitions.') Probably a doublet.') Energies for levels above taken from Belote et al.2').•) Ref. 1 0).") This work. ') Ref. "). k) See text. ") Ref. 5). °) Ref. " ) .*) Experimental error only. See sect. 3 for additional uncertainty.
en
T
- 47 -
114 P. B. VOLD el at.
in the focal plane using Kodak NTB SO /<m photographic plates. Spectra wereobtained at angles from 30' lo 90" in 7.5' steps, while control runs on the 40Catarget were made at 45 and 75". The relative intensity normalization from angle toangle was established by monitoring (he elastic scattering with a scintillator at«ub = 90'.
The absolute cross sections were obtained by observing the elastic scattering andthe two strongest inelastic groups using position-sensitive solid-state detectors inthe focal plane of the magnetic spectrometer and by normalizing the elastic yieldsto optical model predictions. The absolute cross-section scale is believed to be accurateto within 20 %.
A typical spectrum obtained at 0)ab = 90" is displayed in fig. I, A total of 89 levelshave been identified below 6.4 McV excitation energy and these are listed in table I.The energy resolution is 12 keV FWHM. Positions and yields of peaks in the spectrawere obtained using a peak fitting programme which fitted a standard shape takenfrom clearly resolved stro/ig peaks in the spectrum. The background subtractionwas performed manually using the background on either side of the peak. Excitationenergies from the present work are considered accurate to ± 5 keV. The excitationenergies obtained from this work are compared with results of previous work intable I. The level energies, as compiled by Endt and Van der Leun *) andsupplemented with the results of Tabor el al. ' 3 ) , are listed in the first column oftable I, and the energies obtained using the (a, a') reaction are listed in column 2.There is excellent agreement between the accurately known energies and the energiesmeasured in the present work. Thus the present work should give a consistent energyscale for levels in the energy region from 5 to 6.4 MeV. It is notable that the (p, p')reaction excites almost all known levels up to 6.4 MeV in excitation energy.
3. Distorted wave analysis
The inelastic scattering data were analyzed using the DWBA code DWUCK '*).A collective vibrational model form factor was used in the calculations. The inelastictransition strengths were extracted following the ideas of Bernstein '5) as outlined inref. 1 0 ) ; that is, the inelastic transition strength fl(p, p'; /, Jt -» Jf) is defined as
B(p, p';/,./,• » - * 4n (1)
where J, and 7, are the spins of the residual and target nuclei, respectively; R is thereal optical model radius ; / ? „ = ! .2/4* is the radius for a uniform charge distribu-tion; C, is a correction factor to account for (he more reasonable Fermi chargedistribution.
The shape of the angular distribution for inelastic scattering is weakly dependenton the angular momentum transfer in the present experiment. For example, the/ = 3 and / = 5 distributions are almost identical in the angular range studied.
- 48 -
4lCa 115
Therefore, it is necessary to select carefully the optical model parameters in order toreproduce this small /-dependence. Four optical model parameter sets were tried incalculations to see which set best fit data for elastic scattering on 41Ca and inelasticscattering to the strongly excited 3.737 MeV 3" and 4.492 MeV 5" states in 40Ca.
TAIII I 2
Optical model parameters
Set
ABCD
Ref- <MeV)
'*) 49.57") 49.03'•) 50.82'") 48.92
(fm)
1.211,2331.171.16
(fm)
0.660.5690.750.75
W(MeV)
0000
W(MeV)
30.229.3629.3616.28
r,(fm)
1.2011.1381.321.37
(fm)
0.5470.5590.5270.63
(MeV)
27.415.8824.824.16
r, „(fm)
1.0161.011.011.064
" . 0(fmj
0.3510.33!0.750.738
The four parameter sets, listed in table 2, are taken from previous work. Set A isbased on a fit to angulur distribution and polarization data for elastic scatteringof 17.5 to 21.7 MeV protons on 40Ca by Dicello el ai 16). Set B, due to VanOers l 7) , is based on a fit of elastic scattering of 10 to 180 MeV protons on 40Ca,while set C was obtained by Becchetti and Greenlees '8) from a fit to elastic scatteringfor ^ 50 MeV protons on A ^ 40 nuclei. Set D is due to Gruhn el al. 19) and wasobtained from a fit of elastic scattering of 25 to 30 MeV protons on 40Ca. All fourparameter sets fairly well reproduce the elastic scattering data for 4lCa. The qualityofthe fit is illustrated in fig. 2 for three ofthe parameter sets. Unfortunately DWBAcalculations using these four parameter sets did not simultaneously reproduce the/ = 3 and / = 5 angular distributions for inelastic scattering in '"'Ca as illustrated
1.0
0.1
4'Ca(p,p),E=l9MeV
V— DICELLO— VAN OERS— BECCHETTI
20 40 60 60
8cm100
Fig. 2. Elastic scattering data for "Ca wilh optical model predictions '*• "•'"). The cross sections arcgiven relative to the Rutherford cross section.
- 49 -
116 P. B. VOLD ei at.
100
90
Fig. 3. Experimental angular distributions of inelastic scattered protons from 40Ca along with DWBApredictions. The solid lines represent ihe DWBA results with the parameter ol'ref. '") and the dashed lines
represent those ol ref. '*).
in fig. 3. Parameter sets A and B both provided a good fit to the data for / = 3 butgave / = 5 angular distributions which were flatter than the data. On the otherhand, parameter sets C and O produced satisfactory agreement for / = 5 but notfor / = 3. The transition strengths, 5(p, p'; /, J, -» J,) for the lowest states in 40Ca,extracted using parameter set A, are compared with values from inelastic a-scatter-ing 1 0) , inelastic proton 19) scattering and electromagnetic measurements 30) intable 3. These transition strengths were obtained by fitting the calculated distributionsto the data at angles from 0 = 30' to 90'. Parameter set B gives results similar toset A, while sets C and D result in a 15% to 25% increase in the extracted transitionstrengths. The electromagnetic, (a, a') and (p, p') transition strengths are in goodagreement for the 3" states and moderate agreement for the 2 + state. However, thepresent work gives a significantly larger transition strength to the 5" state. In this
(MeV)
3.743.904.49
Jm
Y2*5'
TAUH 3
inelastic transition strengths in "Ca '
DWBA
18- I01
1204.4x10'
cc
17x10'943.4x10'
(p.p'n25 MeV
19.0 xlO1
785.2 x 10'
')
This work')DWBA
I 9 . 3 X 1 0 J
61")7.0x10"
Electro-magnetic ')
(2O±l.3)xlO3
8S±8
') B(/;0* - J') in e1 • fmJl.») Ref. '"). ') Ref. ") .d) Deduced average value from the cross sections al two angles.•) Ref. " ) .') Experimental error ±20%. See sect. 3 for additional uncertainty.
- 50 -
•'Ca 117
E,>2460
30 10 50 60 70 BO 90
0 5
10
01 (b)
30
t
40
•
SO 60
L
70
E.-J20I
£KOJ69 'l"J
""** s. •
•3 6133
80 90
30 40 SO 60 70 80 90"cm
(d)
I
-—1>
V1
E.<4OI4L'2
\
1-4279•2
\ \]
30 40 50 60 70 80 90
Fig. 4. Experimental angular distribulions of inelastic scattered protons form "Ca. The solid linesrepresent DWBA results with the parameters ofrer. ")•
- 51 -
118 P. B. VOLD nul.
connection, it is interesting to note that Gruhn et til. " ) found that the proton in-elastic transition strengths for / = 5 were reduced by almost a factor of two whenthe bombarding energy was increased from 25 to 40 McV.
Coupled-channels calculations were performed using the code CHUCK '*) inorder to test the influence of channel coupling. Calculations were performed for thelowest 3" and 5" stales in 4nCa using a B( E2; 3" -• 5) taken from rcf. -°). The shapesof the angular distributions were indistinguishable from the DW predictions. Thedirect two way coupling between the elastic and inelastic channels increased theextracted transitions strength by about 10 % for / = 3 and about 30','o for / = 5,whereas the change due to the indirect transition via the intermediate state wasnegligible. Since the corresponding / = 3 and / = 5 distributions in 41Ca arefractionated in reasonable accordance with (he simple weak-coupling picture as(3 x f,) and (5" x f,) multiplcls (see seel. 5), the cross section to the individual statesin *'Ca are appreciably reduced as compared to •*°Ca. Ihe influence of channelcoupling is therefore expected to be smaller for the 4lCa(p, p') reaction.
The comparison shown in table 3 suggests that the fi(p, p'; /. J, -* J,) derivedfrom the present data are significantly in that they agree with the electromagnetic and(a, a') values within the ±30",', uncertainty resulting from (he somewhat modeldependent procedure used to analyse the data. The relative transition strengths areprobably less sensitive to Ihe DW analysis procedure. A caveat is appropriate in thatthis argument may fail for weak transition strengths since the (p, p') transitionoperator is complicated; i.e. the form factor may be sensitive to the microscopicstructure. In addition, contributions from nondircct processes may be non-negligible.
The 41Ca(p, p') data were analyzed using optical model parameter set A in theDW calculations. Several states were observed as pure 1=2 transitions with goodfits to the data. Four of the observed transitions to known negative-parity statesexhibit distributions distinctly different from pure/ = 2, thus indicating an admixtureof / = 4 (see fig. 4). These transitions therefore were assumed to be due to mixed/with/ = 2 + 4, and the relative intensities of the two contributions were determinedby filling the DW predictions for / = 2 and / = 4 to the data using a least-squareprocedure. This procedure is somewhat uncertain in that no experimental pure 1=4transition was observed to lest the / = 4 calculations. Thus a ± 100 % uncertaintyis assigned to the transition strengths extracted for these mixed-/lransiiions. Since the1=1 and 1=5 distributions were nearly indistinguishable, it was not possible tosearch for/ = 3 + 5 mixtures. The (a, a') showed no cases of/ = 3 + 5 mixing, and the/-assignments from the (a, a') data were used in the analyses for the weak / = 3 and/ = 5 transitions. Examples of the experimental angular distributions and DWpredic for the 4lCa(p, p')4'Ca reaction are displayed in fig. 4. The peak crosssection, corresponding angle and the extracted fl(p, p' ; t\\'-*J) obtained via ax' fit of the DW predictions to the data, are listed in the right-hand column of table I.
- 52 -
4 1Ca
4. Comparison with other work in 4lCa
I f )
The spin assignments listed in column 2 of table I are taken from rcf. 4) updatedusing the results of rel's. 5 '"• " • 2 a ) . Some additional spin-parity assignments canbe made on the basis of the 3vK(a, il) [rel's. "•")]. 4"K('Hi;, d) and J"K(d, n)[rcf. 7)] reactions and the present results. A few cases of special interest regardingspin assignments arc commented upon briefly.
The upper member of the 3613.0 and 3613.5 kcV doublet has been assignedJ" = J" from p-)' correlation measurements11). Thus, the measured / = 3distribution in the (2, 2') and (p, p') experiments implies excitation of the lowermember. The inelastic scattering data coupled with the spin and parity limitationsof ref. n ) leads to J" = | f -•]+ for the 3613.0 MeV state.
The 3973 keV level has been assigned J" = (5. J)+ by ref. '3) , while the4lCa(2, 2') suggested J" = !
25+ from the observation of pure / = 5. However, ihc
distinct 1 = 6 distribution observed in the 31)K(a, d) reaction *•'') rules out spinvalues smaller than ','*. Recent 2B)y-correlation work determines the spin to be l-r +.This level is also seen with /p = I + 3 in (3He, d) and (d, n) reactions supporting theassignment ol' J" = v,1*. The same argument applies to the 6067 keV state whereagain the combined results of the (2, d)and (3He, d) reactions give J" = '•} *.
Qualitatively, the 4I Ca(p, p') reaction contains approximately the same informationas the 4lCa(2, 2') reaction for the strongly populated positive-parity states whereasmore detailed and comprehensive results are obtained for the weakly excited negative-parity states. The inelastic transition strengths show very good general agreementbetween the two reactions for / = 3 and / = 5 distributions. However, there are afew exceptions; i.e. the / = 5 and / = 3 strengths for the 3973 and 4327 keV states,respectively, are about twice as large in the (p, p') reaction, whereas for the 4730,4813 and 5281 keV state the results are two or three times larger in the (2, a')reaction. The / = 2 transition strength derived from the (2, a') reaction are poorbecause this reaction is unfavoured for! = 2. The agreement for the 1 = 2 transition
7ABI.I 4
Comparison of transition strengths for " C J
Theory ")BIF.I, ] - J")U'* fm")
work J) electromagnetic *)
W422(109246229593370
23223
JriV
29
061.8
2HInO
S 2923
3900
19 + 2• 7 C . 4 !'•> IS
< 0013
3600 ±600')
•) Rcf. " ) ") Rcf ') ') Ref. ! S )") Experimental error ± 2 0 %. See sect. 3 for additional uncertainties.
- 53 -
120 P. B VO1.D el al.
strengths between the (p, p')and (a, a') reactions is excellent for the strong 1.944 MeVstate, but the (p, p') strengths are consistently smaller than the (a, a') strengths for theweaker / = 2 transitions.
Table 4 displays a comparison between inelastic transition strengths of this workand electromagnetic transilions6i 13) in 4lCa. Theoretical calculations as predictedby the coexistence model of Gerace and Green ') are also given. The electromagneticand (p, p') transition strengths agree within the experimental uncertainties for thetransitions to the 1942, 2009, 2959 and 3369 keV stales, whereas there appears to bea discrepancy between the two results for the 2462 keV transition. The (p, p') datafavoured a mixture of / = 2+4 but a pure / = 4 angular distribution is not in-consistent with the data. (See also discussion in sect. 3.)
5. Nuclear structure of 4lCa
The inelastic transition strengths are summarized in Tig. 5, while the summedstrengths for the different multipole transitions are cited in table 5. The summed
\/1eV
)
LLT
6.0
5 0
4 0
3 0
2.0
5/2'
L=2
—
.1/2*— 9'5/2-
-5/2"
- 5 / 2 '
- V
lll/2'l
•
(13/2*1
jl/2'
\V2'
•—' (ll/2*)
•
I
L=3
B(pp'iL)t(!0Le2fm2L)Fig. 5. The energy distributions of the inelastic transition strength in 4lCa are displayed by solid
horizontal lines. The spins and panties indicated for some of the transitions, are taken from table 1.
- 54 -
TAULI 5
Summed proton inelastic transition strengths in "Ca
121
fm")')
/ = 2/ = .1/ = 4/ = 5
8817 2 x I0 1
I 6 x 10-"3'Jx 10°
') The sum includes levels up lo 5.5 MeV.
quadrupole strength for the (p, p') reaction is about half the summed strengthobtaining using the (a, a') reaction. However, the (a, a') reaction value is less reliablesince the (a, a') reaction is less sensitive to I = 2 excitation and thus only upperlimits were assigned for weak 1=2 transitions seen with the (a, <x') reaction.
The summed quadrupole strength can be used to obtain information on theamount of core-excited admixture in the 4lCa(g.s.) wave function. Thus, if weassume that the *'Ca(g.s.) wave function can be written |4'Ca> = [(1—a)2]*|fj> + 2|core>, the core admixture coefficient a for the *'Ca ground state can beevaluated using the procedure outlined in the (a, a') paper 10). This procedure isbased on the assumption that all the quadrupole strength to the ground state has beenincluded in the sum and that the proton inelastic and electromagnetic transitionstrengths are equal. These assumptions may be questionable in that additional/ = 2 strength may lie above 5.5 MeV and the (p, p') and electromagnetic transitionstrengths may differ appreciably for weak transitions in spite of the excellentcorrespondence obtained for the strong transitions listed in tables 3 and 4. Theresulting value for the excited-core admixture (a2) in the *'Ca is (5.5 ±2) %, whichresults in a ground-state static quadrupole moment of — 12 e • fm2. A similar analysisof the summed / = 2 strength for the 40Ca(p, p ) reaction data gives a core-excited admixture of (4.5 ±2)% for the ground state of 40Ca. These values forcore admixtures in 4 0 '4 'Ca are half of the values given by the less reliable (a, a')reaction and about 0.3 times the theoretical values ')• A 4lCa ground-state coreadmixture of £ 10 % is suggested by magnetic moment messurements " ) in 41 • *2Ca.It is interesting that the almost identical core admixture in the ground states of40- 41Ca agrees with the results obtained from one neutron pickup2 4 '") in 41Ca; thatis, the structure of the 4lCa ground state looks very much like a If j neutron coupledto the 40Ca ground state.
The / = 3 transition strength for the 41Ca(p, p ) reaction has a centroid at3.76 MeV and summed strength which is 89 % of the / = 3 strength to the 3.74 MeV3" state seen in the 40Ca(p, p') reaction. This suggests a weak-coupling interpreta-tion in terms of Ihe (3" x f j) J configuration for these positive-parity states in *'Ca.The weak-coupling picture predicts that the transition strengths for the individualmembers of the septuplet should obey the relation that fl(p, p'; 3, J -» J'g.s.) =
P. B. VOLD n ,(/
p, p' ; 3, 3" -»0 + ). Although there are more than seven positive-parity levels'Ca excited by / = 3, it is possible to sum the strength lor each multiplet spin^and compare these with the / = 3 strength to the 3", 3.74 MeV state in 40Ca.
vn in table 6, only the I * and y + summed strengths fall appreciably below 4
" m'. • 2.009 T M3 049 ™1525 "•
i • 4 (»44X135 2K1
(4 327) •')
'_!' i 2014 9W
V 19141*317)')
J) Adopted spin on the bitslsh) Results I'rom inel.islic t-svii*) The /-value not determined
TAIIH ft
Di.slnbunon o ) ' / = 3 sirtn^i
flip. P . . / " - : ».s.l
flip. p \ 3 - ( ) ' g.s )
0 15'1
0.51
| k 0 3(>
^ ^ ^ ^ 1 0 H i " )
0 2 4 ^ ^ ^ ^ ^ K
0 94 J^^^^M
ih.it ' : ' ^ ^ H ^ J evhausi the
h* <<>^^RiFnl data, .issunici
hm J1C\t
(entroid (MeV) ',
U7 JU
jKS) "I^ ^ 3 47
W 172
K. ' ' 7
H k 4 22
MJf
^ 1 (12
0 7,
10 791 h)
(179
0 S 7
0 94
0 ?(i
the weak-coupling val i^H^vcvcr. it is for the 5' states l ^ | | H ^ a') and (p, p )transition strengths ci^jH^he most. The (i. »') data would inct.'.wp|[v,. total fractionfor the V s tates^Bp?;and not appreciably change the values^fc^tbrother spinmembers of ihoJj^F M. The V' strength is weaker than either th^ft^i^.couplingor the [f d r j ^ j p i d c l s predict. This is not fully understood a It hoW>.,-tynpletealignment <i/Jp>(l;) neutron and some components, other than (f|dji^t^)»» thecollective^Skatc may be blocked by the Puuh principle. The fraction
sepluplel and the 1.5 MeV splitting between the centroidmembers indicates that the weak-coupling picture is only a >.'j^
<^?'iiation to the truth.jut 75 "„ of the 1=5 strength to the 4.49 MeV 5 state in 4DCa was observed
'Ca. However, application of the weak-coupling model to the / = 5 strength ist feasible because the low-spin members of the multiplet have not been identified.
of the
- 55 -
i : : P H vo t .n ,-iui
W(p. p'; 3. 3 ->()'). Although there are more than seven positive-parity levelsin ""Ca excited by / = 3, it is possible to sum the strength for each multiple! spinvalue and compare these with the / = 3 strength to the 3 . 3.74 McV state in 4"Ca.As shown in table 6, only the \ * and ' / ' summed strengths fall appreciably below
•'
>'
'•'
1 1 1
V
2 66N
5 046
2 00')
1 IW)
1 *25
4 O')4
4 S I 1
2 NS2
(4 127)')
1 201
4 ')(<<)
1 16')
|s U 7 ) ' |
I Mil 1
Disirihiitum ol' / '
«lp. p . . / ' - • :
«(p. p . .' - 0
(1 15'1
0 51
II .16
n 12 '1
0 52
021 (0211
I) 10 (0 H I
0 |s (11 27|
0 47
(I 32
1164
0 24
(I'M
0 44
0 12
6
> slienyth
[ ! S |
' g s |
'')' • )
in •"(;!
(Vnlrnul iMcV'l
,S7
1 24
4 66
(4 7X1'•)
147
.1 72
.1 17
4 22
Summi'il strength
1 112
0 75
(0 741")
0.79
0X7
(I'M
056
'I V l n p t i x l s p i n nil I hi1 K I M S ih.it '! .mil ',' ' I : \ I I . H M Ilic C \ | X T ! C I I s t i c n g l h
") R e s u l t s l i o m nu' l . l s lk 7-si ' i i i tcring '")
M T h e / - \ t i l u e mi l i l e l c i i m n c i l h \ I he prcMMii d,U;i. . i ssutnci l i n he / - 1
the weak-coupling values. However, it is for the § * stales that the (j . a') and (p. p')transition strengths deviate the most. The (2. 7.') data would increase the total fractionfor the S * states to 79 "„ and not appreciably change the values for the other spinmembers of the multiplet. The ' / * strength is weaker than either the weak-couplingor the [f;dj ' j ^ ' models predict. This is not fully understood although completealignment of the (f.) neutron and some components, other than (f'd[ ')3 , in thecollective 3" stale may be blocked by the Pauli principle. The fractionation of themembers ol'the septuple! and the 1.5 McV splitting between the centroid energiesof the septuple! members indicates that the weak-coupling picture is only a crudeapproximation to the truth.
About 75 "„ of the / = 5 strength to the 4.49 MeV 5 state in 4"Ca was observedin 4lC'a. However, application of the weak-coupling model (o the / = 5 strength isnot feasible because the low-spin members of the multiple! have no! been identified.
- 56 -
'Ca 123
It is interesting to note, however, that the ^ + and ^ + state are consistent with apure weak-coupling fragmentation whereas the l
2-* state represents only about 50 %of the predicted weak-coupling strength. It is believed "•2") that this is a consequenceof the blocking of the complete alignment of the f( neutron and the almost
pure24. 2 5 ) (f,dj ' )5 structure of (he 4.49 MeV 5" states in 40Ca.
'6. Summary
Angular distributions have been measured for elastic and inelastic scattering of19 MeV protons on 40 ' 4lCa. A total of 89 levels was identified below 6.4 MeV in 4lCawith an energy resolution of 12 keV. Inelastic transition strengths have been extractedusing the ideas of Bernstein 15)and these correlate well with inelastic a-scattering l0)and electromagnetic values 20).
A model dependent analysis of the quudrupole transition strengths suggests thatthe ground states of 40Ca and 41Ca have identical core admixed components ofabout 5"/o. The / = 3 strength in 4lCa exhibits features characteristic of weakcoupling of an f, neutron to the lowest 3 " state in 40Ca. Similarly, the less completedata for / = 5 excitation suggests a weak-coupling interpretation in terms of a f i
neutron and the lowest 5" state in 40Ca. However, in both cases, the highest spinmember of the muitiplets have reduced transiiion strengths presumably due to theinfluence of blocking in the microscopic structure of these states.
The authors wish to thank Professor Ole Hansen for instigating the present workand for his involvement in the early stages of the experiment.
References1) W. J Gerace and A. M. Green, Nuel. Phys A93 (1967) 1102) B H Flowers anJ L. D. Skouras, Nucl. Phys. AI36 (1969) .153i) C. W Towskey, IX Clinc and R. N. Horoshko, I'llys Rev. Loll. 2S (1972) .168; Nucl. Phys. A204
(1973)5744) P. M Endl and C. van dcr Leun, Nucl Phys. A2I4 (1973) 15) K. P. Lieb, M. Ohrmacher, F. Dauk and A. M. Kleinl'eld, Nucl. Phys. A223 (19741 4456) P. Gorodelsky, J. J. Kolula, J. W. Olness, A. R. Polem and E. K. Warhurlon, Phys. Rev Lell. 31
(1973)10677) W. Bohne, H. Fuchs, K. Grabisch, H. Kluge, H. Morgcnslern, H. Oeschlerand W Schlegel, Nucl.
Phys. A24O(I975) 1718) C. E. Thorn. I Fiwhmiin, A. M. Bcrnslem and D Cline. Bull Am. Phys. Soc. 16 (1971) 1432; and
unpublished9) H Nann. W. S. Chien, A. Saha and B. H. Wildenlhal, Phys. Rev. CI2 (1975) 1524
10) M. J. A de Voigl, D Cline and R. N. Horoshko. Phys. Rev CIO (1974) 179811) C. Ellegaard, J R. Lien, O. Nalhan, G. Sleltcn, F. Ingebrelsen, E. Osnes. P. O. Tj0m, O. Hansen
and R. Slock, Phys. Lell. 40B (1972) 64112) J. A. Smilh, E. J. Hennelly, C. H. Ice and H. F. Allen, Trans. Am Nucl Soc. 8 (1975) 5413) S. L. Tabor, K. C. Young, Jr., D P. Balamulhund R W. Zurmiihle, Phys. Rev C'I2(1975) 121214) P. D Kunz, University of Colorado, unpublished15) A. M. Bernstein, in Advances in nuclear physics, vol. 3, ed. M Barangcr and E. Vogi (Plenum, NY.
1969) p. 325
- 57 -
124 P. B. VOLD ei al.
16) J, F. Dicclto. G. Igo, W. T. Leiand and K G Percy, Phys. Rev. C4 (1971) 113017) W. T. H. van Ocrs, Phys. Rev. Ci ( I97I ) 155018) F. D. Beccheiti and G. W Greenlees. Phys. Rev. 182 (1969) 119019) C. R. Gruhn, T. Y. T. Kuo, C. i. Maggione, H. McMunui, F. Pcirovich and B. M. Prcedom,
Phys. Rev. C6( 1972) 91520) P. M. End! und C. van der Leun. Nucl. Data AI3 (1974) 6721) K. K. Sclh, A. Sana, W. Stewart, W. Benenson, W. A. Lanford, H. Nann and B. I I Wildenlhal,
Phys. Leu. 4»B(I974) 15722) K. K. Sclh and S. G. Iverscn, Phys. Leu. S3B (1974) 17123) 1. E. Young, R Brenn. S. K. llhauacherjee, O. B. Fossun and G. D. Sprouse. Phys. Rev. Led. 35
(1975)49724) D. Cline, M. J, A. de Voigi, P. B. Void, O. Hunscn. O. Nathan and D. Sinclair, Nucl. Phys. A233
(1974)9125) R. R. Belts. C. Gaarde, O Humen. J. S. LarsenandS. Y valider Werf, Ni»-I. Phys. A2S3( 1975)38026) P. R. Goode and R. N. Boyd, Pbys. Rev CH (1976) 37927) T. A. Beloie, A. Sperdulo and W. A. Buechner, Phys. Rev. 139 (1955) 3802«) C. J. Lihlcr. A. M. A I - N U M T , A. H. liehbehani, L. L. Green, I' J Nolan and J F. Sharpey-Schafer,
Proc. Florence Conf. un pliyMci ot' mcdmin-liglil nuclei, June 1977. p. 36
- 58 -
P A P E R III
THE EFFECTIVE T=l TWO-PARTICLE
MATRIX ELEMENTS IN THE fp SHELL
- 59 -
Volume 72B, number 3 PHYSICS LETTERS 2 January 1978
THE EFFECTIVE f = 1 TWO-PARTICLE MATRIX ELEMENTS IN THE fp SHELL
P.B. VOLD ', D. CLING, R.N. BOYD and H. CLEMENT2
Nuclear Structure Research Laboratory , University of Rochester, Rochester, N. Y. 1462 7, USA
W.P. ALFORDPhysics Department, University of Western Ontario, London, Canada 4
and
J.A. KUEHNHRMcMastcr University, Hamilton, Canada "
Received 28 October 1977
The "'c.ifi!, p)42Ca reaction has been used to locate the strength distributions for p3 / 2 and p1 / 2 transfer in 42Ca.Iftective (l'7/2)2. f-j/2P3,'2 and f7/2Pi/2 two-body matrix elements arc derived from the data and compared 1otheoretical predictions.
Knowledge of the two-body part of the nuclearHamiltonian is required in order to understand com-plex spectra in terms of the nuclear shell model. Ex-perimentally, such information can be obtained bydetermining the effective two-particle matrix elementsof the residual two-nuclcon interaction. These tmlrixelements arc closely related to energy spectra of nucleiwith two nuclcons outside closed shells. However, ifthe shell closure is not perfect, the individual spincomponents of the two-particle multiplcts are oftenfractionated. Therefore, the energy centroids of thefractionated two-particic configurations need to bedetermined to extract the two-body matrix elements.This requires identification of the major componentsof the two-particle configuration of (he relevant /-orbitals, which can only be obtained unambiguouslyfrom one-nucleon transfer reaction data.
The recent production of a 4 lCa target has madepossible such studies of the two-particle spectra usingthe 4 lCa(3lle,d)42Sc and 41Ca(d,p)42Ca reactions
Supported in part by rVorgcs Alrncnvitcnskapcligc l-'urs-VningsrSd. Present address: Institute of Physics, Universityof Bergen, 5000 Bergen, Norway.Present address: Scktion Physik, University of Miinchcn,8046 Garchinp, West Germany.
3 Supported by a grant from the National Science Founda-tion.
* Supported by the National Research Council of Canada.
(1 .2 | . Single-nuclcon transfer on a non-/.ero spinlarpcl may proceed by a mixture of several /• and/-valucs. The shape of the differential cross section uni-quely determines the transferred /, but is insensitive tothe transferred /-value. The above work indicated largefractionation of the / = I strength and suggested con-siderable mixing of P3/2 and p | / 2 transfer. The highsensitivity of the vector analyzing power (i7"|]) to thetransferred/ provides an excellent method for deter-mining the spcctroscopic strength for the individual/transfers for a mixed transition. The present41 Ca(d.p)42Ca measurement was undertaken to util-ize this feature to locate both the fractionated f7/2P3/2and (-112P112 configurations in 42Ca.
The 4 I Ca(d,p)42Ca reaction was studied using 11MeV polari/ed declerons from the McMaster Univer-sity PN tandem Van M Graaff accelerator. The targetconsisted of 81.8^ 41C'aand 18.2%40Ca and had athickness of 25 /ug/ci>,-. The reaction protons weremomentum analyzed in an Huge split-pole magneticspectrometci and detected in the focal plane usingKodak NTH 50 im photographic emulsions. Spectrawere obtained at each angle from successive runs hav-ing spin-up, spin-down vnd unpolarizcd dcuterons andwere laken at angles froi 1 10° to 50° in 5° steps and50° to 70" in 10° steps with an energy resolution of15 keV FW1IM. The rel itive normalization for thedifferenl polarizations and from angle-to-angle was
311
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Volume 721). number 3 PHYSICS LETTERS 2 January 1978
established by monitoring the elastically scatteredparticles with a solid state detector at 30° as well aswith beam cliargc intcgiation. The quench ratio tech-nique was used to determine the vector polarisation ofthe beam, which was typically 0.7. The absolute crosssection scale WJS established by normalizing the crosssections of the 5211 keV level at 15° to the 4 lCa(d.p)v.\luc of ref. | 2 | . In doing this, the 1 2 MeV data ofrcf. \2\ weie extrapolated to 11 MeV using the DWcalculations.
The present data were analyzed using the codeDWIK'K [3). Several optical model parameter sets wereinvestigated in order to test which could best reproduceboth ibe differential cross section and the analyzingpower. It was found thai Ihe dculcron set obtained byLolu and Haeberli [4| with a minor modification ofthe real well depth to 108 MeV, combined willi theproton set of Dicello ct al. | 5 | gave the best simul-taneous fit. Standard finite range and non-locality cor-rections [3] were included in all channels.
The spcctroscopic strengths were extracted from aleast-squares fit analysis of the DW curves to the crosssection and vector analyzing power data. Examplesof the DW fits to the data are displayed in fig. 1. Ex-cept for the fit of the vector analyzing power for theP\/t transition, the predicted curves are in good agree-ment with the data. Since extraction of reliable spec-troscopic factors for the scparale //-values in a mixedtransition requires good fits for (he pure transitions,the experimental analyzing power leading to the 1/2"state in 41Ca. shown as a dashed curve in fig. 1, wasused in the subsequent analysis.
While the /= I strength is fractionated overall encr-g> range of 6 MeV. almost no mixing of PJM and p\nwas observed. About 90" of the p ^ strength is ob-served in the energy region 4.5 6.5 McV. while thewhole pj/2 strength occurs between 6.6 and 7.8 McV.The / = 3 strength below 4 MeV is confirmed by thepresent work to be due to f7/2 transfer as assumed inrcf. (2| . However, the present analyzing power dataindicate that (here is no detectable admixture of/ = 3transfer mixed in with the strongly excited / = I tran-sitions in the 4.5 6.5 McV region in contradictionwith the results of rcf. [2 | .
The summed spcctroscopic strengths observed inthe present experiment for the ff7y2 )2- ^7/2P3/2 a n ^hjlPMl niultiplets arc shown in table 1. It is seenthat these strengths represent about 65% of the shell
a
§15
I ,-3"3i«e.
' , - S i " >e.
A, \\r
60 40 50
lip. 1. Angular distributions of the cross section and analyzingpower for pure f7;3, P3,j and Pj/2 transitions to states in **Ca.The solid lines represent DW results, calculated as described inthe text. The dashed line is based on the analyzing power dataof the 1/2" state at 3943 keV in 41Ca (see text for details).
model limit. The presence of [8%40Ca in the 41Catarget made a simultaneous analysis of transitionswith the same/ in the 4<)Ca(d.p)41 Ca reaction possibleallowing ;i direct comparison of the relative spectro-scopic factor sums for these orbitals in the two reac-tions. The summed spcctroscopic factors for f7y2.p3/2 and P|/2 transfer to 4 1 Ca are 0.71,0.78 and0.65. respectively, of the shell-model value. Moreover,the summed spcctroscopic strengths for these sametransfers to each spin of both the ((j^)2 a r |d f7/2P3/2multiplcts in 4 lCa(d,p) arc within 10% of the same
312
- 61 -
Volume 72 B, number 3 PHYSICS LHTT1-RS 2 January 1978
•> 1
3*
r~ - , - '
2 IF''
U'
•<,i
S ' 1 ' 5 J i G - 8 I ? 3 4 1 2 3 4
' < ' >i ' r ' ' ,
I ip ? Tin1 encr^v dtstrltninnns .>f lltv *pr* bu.si npu "ilrcnplhin 4 3 ( i lor 'he titl fcrcn' fp ufr-it.iK uhscrvrd in Ihc presentrxpcrinH-nl
frai I K H I " I : lu ' -111-11 HKHICI V I I I IC 1 Iicsc icsulls nul l -
U t e l l i . ; ! I ' »e m.?|(>r t o in ponen Is n f I he spectroscopie
sin-nci ' is lor these orbit.ils. h.ur been ident i f ied A
lenorm. i l i / . i l ion l .K l iu ni I 0 ( IS ic<|iiircd In lump
these vi nu: IK- ' I urc i ipths in In .igrcenicnl w i th I he shell-
mt t j t ' l pK'dk Moris is no! unre.ison.iblc Lonsit lcni ig I he
well-Known u iu i rt.unlies .issoLialcd vvilh HW analyses.
The spix i rosLu|-k <nrciipths for il ie individual spin
members " | l!n' l ^ ' i p i o cnn l igur . i lmn ;ire biised tin
an assumed spin value of 3+ for (lie 5211 keV level indisagreement with 2+ from (lie 40Ca(l,p)42Ca reac-(ion |6]. The 2+ assignment would lead to an"V/2P.V2h* s t r e n g | ' 1 which exceeds the average spec-troscopie factor for the ( ^ ^ 3 / 2 ) / configurations bytOO1? and would imply that no (f7^2P3/2^3+ strengthis identified; the unknown p3y^ transitions representfar too little strength to account for (hat missingstrength. However, identification of two close-lyinglevels at 5208 kcV and 5215 keV via the 39K(a.p7)reaction [7| suggests that the (d,p) and (t,p) reactionsexcite different levels.
Identification of the single-particle strengths of the(•JI2. P3/2 und P|/2 orbitals allows determination ofthe energy centroids ef/'j/^./) for these two-particlemultiplets which are listed in table I. The correspond-ing T= I effective two-body matrix elementsA'(/l/2./) can be derived from these centroids bytaking1 the energies with respect to the referenceenergy at which the multiplet would occur if therewere no residual two-body interaction. The referenceenergies used arc 3109 keV, 5291 keV and 7174 keVfor the (f-7/2)2. ^7/2^3/2 ""^ *7/2Pl/2 multiplets,respectively. The latter two were obtained from theenergy centroids for the p3/2 and p t n single-particleconfigurations in 41Ca given by the 4°Ca(d,p)41Careaction |8| .
About 10°; ofthep3/2 transfer strength leads tostates bclwecn 6.3 MeV and 7.1 MeV for which thespin assignments are unknown. This unassignedstrength rfroduccs the largest uncertainty in the ex-tracted matrix elements. One plausible distribution ofthis unassigned strength would be to assign 3+ to the6.82 MeV state and to distribute the remaining 60%of the strength equally among the other spin values.This would result in a summed strength of 0.70 of theshell-model value for each spin and in values of/•.(f7/,.p3/2../) which are 0.35,0.40, 0.13 and 0.06MeV higher than quoted for7= 2. 3.4 and 5, respec-tively.
The data suggest !hat the major fragments of (he( l 7 p) 2 and I7/2P multiplets have been located. How-ever, unobserved minor fragments at high excitationcnerps could result m an appreciable shift in theextracted ccnlroids The spreading of the strengthdistribution due to the residual two-body interactionwas estimated by calculating the theoretical spectro-scopie strength distributions in Ihc fp space using the
313
1
- 62 -
Volume 72B, number 3
hh
PHYSICS LETTERS
Table 1The fp sircngtli and effective two-particle matrix elements in 42Ca.
2 Jwiuuy 1978
J"
0*2*4+
6*aU
2*3+
4+5*unknownaU
all
<27F+1)S
1.226.27
10.6117.1035.20
2.963.785.936.882.91
22.46
9.94
Sshellmodel
0.610.630.590.660.63
0.590.540.660.63
0.70
0.62
'(JtilJ)(McV)
0.582.122.863.192.79
4.435.215.055.78
5.42
7.39
£i/i/i/)(Nexp.
-2.53-0.99-0.25
0.08-0.32
-0.86-0.08-0.25
0.49
0.13
0.21
leV)exp.corrected a )
-2.06-0.94-0.20
0.17-0.28
-0.72-0.08-0.19
0.49
0.17
0.24
Kuo-Brown [9]McGrory (10)
-2.22-1.1$-0.36+0.29-0.27
-0.86-0.03-0.05+0.15
-0.10
-0.14
a ' Corrected for the strength, predicted by the (fp)J shell model calculations, to lie above 7.9 MeV in excitation energy and hencewould be missed by the present measurement.
modified Kuo-Brown {9] matrix elements of McGrory(10]. This calculation indicates that the strength lostat high energy (above 7.9 McV) is extremely small(<3%). The extracted two-body matrix elements cor-rected for omission of this higli-lying strength areIjsted in column 7 of table 1. This correction is insig-nificant for all except the 0+ centroid. Although themajor fractionation is due to mixing with core-excited configurations, the interaction matrix ele-ments are not expected to be large enough to push asignificant fraction of the two-particle configurationsbeyond 7.9 MeV in excitation energy.
Column 8 in table 1 lists the rcnormalized f;/2Pmatrix elements derived from the Hamada-Johnstonnucleon-nucleon potential by Kuoand Brown [9],The (f7/2>2 matrix elements result from a least-squares fit by McGrory [10| to the excitation ener-gies of 29 states and seven binding energies for nucleiwith/) = 42-44. The overall agreement between ourexperimental two-body matrix elements and the val-ues inferred from previous work is good. It is worth
noting, however, that the average contributions to theenergy of the two-body interaction for the f7nP3/2and fy/2Pi/2 configurations are measured to be slight-ly repulsive, whereas Kuo and Brown [9] predict asmall attractive monopole contribution.
References
11 ] P.I). Void ct al., to be published.[2 j O. Ilansen et al., Nucl. Phys. A243 (1975) 100.[3 | P.D. Kunz, University of Colorado, unpublished.|4] J.M. Lohr and W. Haeberli, Nucl. Phys. A232 (1974)
381.[5 | J.P. Dicelloctal., Phys. Rev. C4 (1971) 1130.(6J J.H. Bjcrrcgiiard el al., Nucl. Phys. A103 (1967) 33.(7) E. Bitterwolf, thesis, University of Freiburg, unpub-
lished.[8| D.C. Kochcr and W. Hacberli. Nucl. Phys. A196 (1972)
225.|9) T.T.S. Kuo and G.E. Brown, Nucl. Phys. Al 14 (1968)
241.f 101 JD- McGrory, Phys. Rev. C8 (1973)693.
314
t
- 63 -
P A P E R IV
NUCLEAR SPECTROSCOPY OF THE (f?/2) ,
FROM THE 41Ca(3,p)42Ca REACTION
- b4 -NUCLEAR STECTROSCOl '" ' ' OF TIIH ( f 7/2)2' f7/2P3/2
AND f7/2P1/2 '-lUI/I'IPLETS FROM THE
"'CafSjp) 1<2Ca REACTION
P.B. VOLD,*t, D." CLINE, R.N. BOYD AND H. CLEMENT**
Nuclear Structure Research LaboratoryttUniversity of Rochester, Rochester, NY 14627
W.I?. ALFORD
Physics Department, University of Western Ontario, London, Canada
J.A. KUEHNER
McMaster University, Hamilton, Canadattt
Abstract: Angular distributions of the cross section and the
vector analyzing power hav'e been measured for the
4 *Ca (d,p)'' 2Ca reaction using 11 MeV deuterons. The vector
analyzing power data have been used to determine the total
angular momentum j of the captured neutron for 42 positive
parity levels. Spectroscopic strengths were extracted
separately for each of the allowed j-values which con-
tributed to the cross section and identification of the
major components of the f?/T'PT/O ancJ pi/2 strengfch
distributions have been obtained. An almost total absence
of mixing of the P-wo and Pi/? strength was observed. Of 32
states populated by pure 1=1 transfer, only two consist of
a mixture of P3/5 a n d P 1 / T Tlie e f f e c t i v e ' ^
(f7/2p3/2! and "7/2Fl/2) two-particle matrix elements
are compared with the modified Kuo-Brown matrix elements
•Present address: Institute of Physics, University of Bergen,N-5014 Bergen-U, Norway
**Present address: Sektion Physik , University of Munchen,8046 Garching, West Germany
tSupported in part by Norges Almenvitenskapelige Forskningsrad
ftSupported by a grant from the National Science Foundation
ttt3upported by the National Research Council of Canada
- 65 -
of McGrory. They agree to within a few hundred keV. The
monopole energy centroid of the ^1/2^2/2) a n d (f7/2pl/2>
multiplets are measured to be slightly repulsive, whereas
Kuo-Brown predict a small attractive monopole contribution.
NUCLEAR REACTIONS " ° "* xCa (d,p) , E = 11 MeV, measured a(Ep,6)
and iT,.,(9). "Ca deduced levels, £,j,7r,J, spectroscopic
factors. Enriched, radioactive target.
1. introduction
The basic features of the nuclear shell model are most easily
seen in fairly simple nuclear systems, e.g., those near closed
shells. In particular, the effects of the two-body part of
the nuclear Hamiltonian are especially simple in systems having
a closed shell ±2 nucleons. Imperfect shell closures can cause
fragmentation of the two-particle configurations in such nuclei.
In spite of this, measurement of the energy centroids of the
two-particle configuration in such nuclei allows for a direct
determination of the two-body matrix elements.
The structure of low-lying levels of 42Ca is expected to
be composed primarily of the (f7/2'2 two-neutron shell model
configurations, although higher lying levels will arise from
(f7 ,_P3 ._) and ^-j/2^1/2^ two-neutron configurations. The
apparent existence1) of 4p-2h core-excited states results in
fractionation of the basic two-neutron shell model structure.
Such effects are clearly demonstrated by experimental studies2}
of the E2 properties of low-lying states in **2Ca as well as
by the single-nucleon transfer reaction data3"6) such as
- 66 -
djtj'^Ca, ^3Ca(3He,a)"2Ca and ** ]Ca (d,p) 1>2Ca.
The large fractionation of the I = 1 (& is the transferred
orbital angular momentum) strength observed in the " xCa(d,p) l*2Ca
reaction5) . indicates a complex (fp) spectrum, and a consider-
able mixing of the P3/2 a n^ P1/2 t r a n s ^ e r strength would appear
likely. Because the shape of the differential cross section
depends almost completely on I, a separate identification of
the P0/2 ar>|3 P1/2 s t r e n 9 t n is not possible from the previous
work5) The pronounced differences in the angular distribu-
tions of the vector analyzing powers in a (d,p) reaction initiated
with vector polarized deuterons, 'however, allows such an identi-
fication once the H has been identified from the differential
cross section data.
The present "* 'cafSjp) " aCa reaction study was undertaken
primarily to use this characteristic feature of the analyzing
power to study the (f^/oPT/?' ant^ ^7/2pl/2^ configurations in
42Ca. The experimental procedure and results are described in
section 2. The DWBA analysis of the data and a discussion of
£j sensitivity in a mixed transition for both the cross section
and the analyzing power are presented in section 3. A short
discussion is given in section 4 of spin assignments together
with the spectroscopic information extracted from the present
measurement. In particular, the (f_, ?)2, (f_..p. ,_) and
{f_ ,.p. ,_) two-particle energy centroids and the corresponding
effective two-body matrix elementa are discussed. (An abbre-
viated version of part of this work has been published previously7).
- 67 -
2. Experimental Procedure and Results
The 4 *Ca (d,p) "*2Ca reaction was studied using 11 MeV polar-
ized deuterons from the McMaster University FN tandem Van de
Graaff accelerator. The polarized beam was produced by a Lamb-
shift -source8) and had an intensity on target ranging
from 50-100 nA. • The quench ratio technique9) was used to
determine the vector polarization of the beam and a value of 0.7
was typically found.
The target consisted of 81.8% M1Ca and 18.2% ^ C a and.had
a thickness of 25 ug/cm2. it was vacuum evaporated onto a 30
ug/cm2 carbon backing. The deta'ils of the isotope production
and the target preparation are given in refs. &• and 10. The
presence of the **°Ca contamination allowed for a simultaneous
study of the u °Ca{3,p) "• xCa reaction.
The reaction protons were momentum analyzed in the focal
plane of a magnetic spectrograph using Kodak NTB 50 um photo-
graphic emulsions. Spectra were obtained at each angle for
successive runs having spin up, spin down and unpolarized incident
deuterons (to eliminate any effects from tensor polarized admix-
tures in the beam), and were taken at angles from 10° to 50°in 5°
steps and 50° to 70° in 10° steps. The energy resolution was
15 keV FWHM. The relative normalization for the different.polari-
zations and from angle to angle was established by monitoring
the elastically scattered particles with a solid state detector
at 30°.
The analyzing power of the elastically scattered deuterons
at 30° was obtained.by using the charge collected in a Faraday
cup as normalization. The resulting analyzing power of -0.028±0.012
- 68 -
is in good agreement with the prior result of -0.033±0.004 for
"*°Ca obtained at the same energy and angle11). The former
value was used in obtaining the relative spin-up/spin-down
intensity ratio from the monitor spectra.
The absolute cross section scale was established by normal-
izing the cross section of the 5211 keV level at 15° to the
l*1Ca(d/p) value of ref. 5. In doing this, the 12 MeV data of
ref. 5 were extrapolated to 11 MeV using the DWBA calculations.
Proton spectra obtained from spin-up and spin-down deuteron
beams are compared in fig. 1 to indicate the different yields in
the two cases. Almost all of the 94 levels identified in the
previous study of ^ !Ca (d,p) **2Ca reaction5) were observed in the
present work. However, our analyzing power data for the weakly
excited negative parity states were statistically inadequate
for a detailed analysis.
3. DWBA Analysis
Reaction calculations used to analyze the present data were
performed with the DWBA code DWUCK12). The ^ C a ground state
spin of 7/2 means that an 1 = 1 transition to 3 + and 4 + states in
l*2Ca can proceed via a mixture of p . and p 1 / ? transfer.
An accurate determination of the relative P-,/2 anc^ Pi/? strengths
requires that good fits be obtained for the pure transitions.
The pure transitions for the "°Ca(d,p)"'Ca reaction data, ob-
tained simultaneously with the ^CaCdjp) data, were compared
with DWBA predictions using several optical model parameter sets.
The present study was designed to improve the fits to the pure
p., ._ and p1 ._ transition data compared with those obtained in
previous work by Haeberli et al.l 3) and Seth et al. l u).
- 69 -
The parameter sets investigated are listed in table 2.
The deuteron parameters of set A are those obtained by Lohr and
Haeberli15), but with the real well depth increased from 105
MeV to 108 MeV. This change improved the fit to the P3/2 anc*
pl/2 a n aly zi n9 power data but had little effect on the fit to
the elastic scattering data of Schwandt and Haeberli11). Similar
small variations of the imaginary and spin-orbit potentials
performed independently in both the deuteron and proton channel
produced no appreciable improvement in the fit to the reaction
data. The proton potentials of set A are based on a fit• •
to differential cross section and polarization data for elastic
scattering of 17.5 to 21.7 MeV protons in l|0Ca by Dicello et
a!..",.
As table 2 indicates, calculations using set A include
finite range (FR) and nonlocality (NL) corrections in all
channels. The effects of these corrections for the different
transitions are shown in fig. 2. It is seen that the introduc-
tion of NL in the entrance and exit channels leads to the most
significant improvement of the fits to the data. While the FR
and bound state NL corrections produce little change in the
shaps of the cross sections and the analyzing powers, they
both contribute to the absolute value of the cross section.
The FR correction produces a decrease of 17% for I = 3 while
I =1 remains unchanged. The NL correction in the bound state
leads to an increase of the calculated cross section of about
20%, while inclusion of NL in the incident and exit channels
decreases the peak cross section by about 20%. .
- 70 -
Set B and set C are the potentials used by Hansen et al.5)
in the analysis of the angular distributions in the 41Ca(d,p)
reaction. The deuteron parameters of set B were obtained by
Schwand-t and Haeberli11) by fitting elastic scattering and
analyzing power data on ''"Ca at different energies while those
of the protons are the modified Becchetti and Greenlees17) set.
of ref. l ^ ) . Set C is from ref. 18) and is close to the system-
atic average potentials for the Ca region by Perey 1 9). The
procedure employed with set B is similar to that used with set
A except that set B omits NL corrections in the bound state.
For set C the calculations are performed in zero range and local
approximations.
The results of the three different sets are compared in
fig. 3. Set A obviously provides a better fit to both differen-
tial cross section and analyzing power data for the p, ,_ and pif £.
transfers than do sets B and C. Although the fit to the ampli-
tude of the analyzing power data for p, >2 transfer is not good
with any of the parameter sets tried, set A gives a much better
representation of the data between 30° and 60° than
do the other two sets. The difference in the proton potential
produces the major improvement between sets A and B. A similar
analysis of inelastic scattering20) on ^ C a led to a similar
preference for proton parameters of set A.
The differential cross section cannot discriminate between
the two j-values for a given £-value. The j-value, however,
can be obtained from the measured analyzing power. The analysis
employed for mixed £,j transfer is based on the results of
- 71 -
Satchler for the cross section and the vector analyzing power21)
The cross section is given by
(2J +1) o D W U C K
(1)
where J is the final state spin and S^. is the spectroscopic
factor. In the present analysis of positive parity states,
only the two £-values, Z =1 and 3, contribute to the cross
section.
The analogous expression for the analyzing power follows
from the result that the cross section for an incident polarized
beam depends linearly upon the product of the unpolarized cross
section and the vector analyzing power, so that
(2)
where T,^(0) is the analyzing power for each separate £j-value
D W U C Kand 0 £ j - o^ (8) (2j + l,.
The spectroscopic factors S. . for mixed transitions were*• J
obtained from a least squares analysis of both the angular distribu-
tion of the cross section and analyzing power data. It was not neces-
sary to assume a mixture of more than two transfer (£,j) components
for any state. If one spectroscopic factor came out negative,
or zero within statistics, then the data were reanalyzed using
the single £,j values.
The DWBA predictions were used for the pure transitions,
except for that involving p. ,_ transfer. Since the predicted
curves were not in good agreement with the data for P2/2 t r a n s f e r
- 72 -
the curves used for the P-. /? ' ransition were based on the
iT*| data for the l/2~ state at 3943 keV in ^Ca. Previous
v/ork22'23) using even-A targets has shown that the analyzing
power is essentially identical for transitions of the same
j-value provided that the incident energy, the target mass
and Q-value are approximately the same.
Because of the difference in phase of the differential
cross sections for I = 1 and 3 transfer, and of the amplitude •
of the analyzing power for different j transfer, the two
independent data sets provided by the analyzing power and
the differential cross section'for each level proved to be
invaluable in determining the existence of small admixtures.
The results of the analysis are listed in table 1. A direct
comparison between the summed strengths between 1*0Ca(2,p)
and l*1Ca(cf/p) was made using the data obtained simultaneously
with the mixed t*°>i<1Ca target. The spectroscopic information
for the ilOCa(3/p) reaction for the f?/2'p3/2 a n d pl/2 t r a n s i ~
tions are listed in table 3, and the angular distributions
of the cross sections and analyzing powers, together with
the DWBA predictions, are displayed in fig. 5.
4. Discussion
4.1 J* ASSIGNMENTS
The final experimental results derived from the present
analysis are given in table 1. Although almost all of the 94
levels presented in ref. 5 were also seen in the present work
(see fig. 1) the poor statistics for the weakly excited negative
- 73 -
parity states made it impossible to obtain sufficient accuracy
in the analyzing power data to allow for a determination of
j. Therefore, the negative parity state data were not sub-
jected to any detailed analysis and are omitted from table 1.
The compilation by Endt and Van der Leun21*) lists the spins
and parities of many levels in l<2Ca. Recently some additional
spin and parity assignments have been made mostly for negative
parity states on the basis of the l#3Ca(d,t), 1*1K{3He/d),
U2Ca(a,a') and '•'Cafd/p) reactions25'5). The transferred j
determined by the present v;ork sets limits on the spin of the
final states and thus provides' a check on spin assignments.
In particular, this is the case for the many 2 states excited
in the low energy region below 5 MeV with an & = 1 or a
mixture of I = 1 + 3 where the £ = 1 component is restricted
to j = 3/2. All such spin assignments were confirmed by the
present measurement. It is particularly interesting to note
the extreme sensitivity to the weak P3/2 admixture in the
1524 keV and 2752 keV states, especially at the most forward
angles. For states of unknown spin in the upper part of the
l42Ca spectrum, the present identification of the many j = 1/2
transitions have set the possible final spin values to either
3 or 4 for these states. Below we comment briefly upon a
few cases of special interest.
The 5211 key Level
This is assigned 2 from the *• °Ca (t,p) 1(2Ca reaction26).
Because we observe a pure P3/2 transition, a 2 assignment is
- 74 -
possible. However, the spect>oscopic strength indicates a
2 assignment to be very unlikely. A 2 assignment for this
state would load to a (fT/oP^/?^* strength which exceeds
the average spectroscopic factor for the (f7/2P3/2' configura-
tion by 100%. Moreover, a spin value of 2 would imply that
no (fn/pPi/?'3+ strength is identified and the P3/2 transi-
tions to states of unknown spin represent far too little
strength to account for the missing 3 strength. In contrast,
a very reasonable distribution of spectroscopic strength is
achieved for the Z-i/jP-*/? roultiplet if a 3 assignment is
made. It is noteworthy that this level was rather weakly
excited in the (t,p) reaction and, more importantly, it does
not exhibit the characteristic deep minimum at 50° as the
strongly excited 2 states do. If the (t,p) and (d,p) reactions
are exciting the same level, the anomaly may indicate that the
(t,p) reactions proceed via a two-step process. However,
recently the 39K{a ,py) "*2Ca reaction32) has identified two
closely lying levels in this energy region at 5208 keV and
5215 keV, respectively, suggesting that the (d,p) and (t,p)
reactions may be exciting different levels.
The 6910 keV Level
The j = 1/2 transfer seen in the present data limits the
final spin to 3 or 4 . The (t,p) reaction strongly excites
a level at G920 keV and assigns it 2 . However, a comparison
of the excitation energies resulting from these two reactions
indicate that the level seen in the t,p) reaction probably
- 75 -
which is weakly excited incoi ro-.p -ill;; t a : h" 69 39 keV J-i'
the (d,p) r e a c t i o n 5 ) .
The 7160 kcV Level
The (t,p) reaction again assigns 2 to a level at 7180
keV, while our data imply 3 or 4 . These are also believed
to be different levels on the basis of the same arguments as
for the preceding level.
The 7398 keV Level
The 2 level at 7385 keV, which is strongly excited in
the (t,p) reaction, is probably not the same level as the one
excited by the (d,p) reaction since it is inconsistent with
the derived spin value of 3 or 4 from our data.
4.2 NUCLEAR SPECTROSCOPY OF £)2Ca AND SUM-RULE ANALYSIS
The spectroscopic information for states in l*2Ca derived
from the present data is displayed in table 1 and fig. 6.
The u 1Ca (d,p) **2Ca results of ref. 5 are quoted in table 1
for comparison. All eight states belov; 3.7 MeV in excitation
energy, which proceed via an I = 3 transfer, were found in
the present work to be pure f7/2 transfer, as assumed in
ref. 5. It is difficult to detect weak admixtures of SL = 3
transfer in the presence of strong I = 1 transfer using
differential cross section data. Analyzing power data are
more sensitive to weak admixtures of f7/2 transfer in the
presence of strong P3/5 transfer and weak fc/2 transfer admix-
ture in the presence of strong p, .„ transfer. Reference 5
- 76 -
reported i. - 3 admixture for 'light of the states between 6.1
MeV and 7.8 MeV, which are strongly excited by Z =1'transfer.
Our analyzing power data show that there is a detectable
admixture of I = 3 transfer for only one of these eight states.
This one state at 7.024 MeV proceeds by fc >-, transfer in the
presence of a strong P 1 / 2 transfer. Our data thus suggest
that; the main fragments of the f7 ,? strength are localized
in the energy region below 3.7 MeV, while only one weak frag-
went of the fr/2 transfer strength is seen below 7.8 MeV.
This Is sonewhat in contradiction with the conclusions of
ref. 5.
The I = 1 strength is distributed over a large number of
s'cates in the energy region between 1.5 and 7.8 MeV excitation.
Eighteen transitions were found to be pure P3/2 transfer and
twelve to be pure p. ,_. Mixing of P-,/2 an<^ 1/2 t r a n s f e r w a s
observed for only two rather weakly excited states. The
pronounced j sensitivity of the analyzing power for £ = 1
transfer allows for detection of admixtures as small as 10-15%
for the weaker of the two I = 1 transfer strengths. The
almost total absence of P3y2 an<^ Pl/2 mi-x:*-n9 is surprising
considering the fact that the H = 1 strength is fractionated
over an energy range of 6-7 MeV. However, about 90% of the
p, ,_ strength is observed in the energy region 4.5 -6.5 MeV,
while the Pi/2 strength occurs between 6.6 and 7.8 MeV.
The sum of the spectroscopic strength over all states
of the same spin, i.e.,£ (2J +1)S. ., and the ratio of this£j
summed strength to the simplest possible shell model value;
77 -
that is, assuming a shell closure at l<0Ca, are listed in
table 3 for the "^Ca (d,p)4 *Ca reaction and table 4 for the
k!Ca(d,p)U2Ca reaction. The spectroscopic strength distribu-
tion for the (^T/OPT/P^ multiplet is based on an assumed spin
value of 3+ for the 5211 keV state in U2Ca. The sum of all
spectroscopic strengths leading to l*zCa are 63%, 70% and 62%
of the shell model value for f 7/ 2' P3/2 a n d pl/2 t r a n s f e r '
respectively. The corresponding values for transfer to '•'ca
are 71%, 78% and 67% for ' P3/2 l/2 transfer, respec-
tively. Moreover, the summed spectroscopic strength for
transfer to each spin of both the (f7 ,_) 2 and ^-1/2^2/2^
multiplets in 42Ca are within 10% of the same fraction of the
shell model value. Absolute spectroscopic factors derived
from DW analyses of single-nucleon transfer data are well
known to be uncertain to at least ±30%/ whereas relative
spectroscopic factors are considered to be appreciably more
reliable. The remarkable constancy of the ratio of the
spectroscopic strength to shell model estimates for transfer
to both "^Ca and 'tZCa is strong evidence that the major com-
ponents of the spectroscopic strength for these orbitals has
been identified.
A renormalization of the spectroscopic strengths by a
factor of (1/0.70) would bring the spectroscopic strengths
listed in tables 3 and 4 into very good agreement with the
simple shell model values. The summed spcctroscopic strength
for a given j transfer directly determines the number of
neutron holes in the j orbit of the target. The ground states
- 78 -
ol* both l|(1Ca and '' ' Ca h.tve been shown27) to require about
104 admixture of (2s-ld) dorc-oxcitcd configurations in order
to explain inelastic proton20) and alpha scattering28), the
weakness for I «* 1 pickup29) and £ = 0 and I = 2 stripping 5 ) on
'''Ca, £ = 3 neutron pickup29) to the ground and first excited
0 states to l<0Ca and the magnetic moment of the '''Ca ground
state31)* The above renormalization factor results in orbit
occupancies consistent with a 10% excited-core component. The
coexistence model2) successfully reproduces the electromagnetic
properties of the low-lying states in lt2Ca. Spectroscopic
factors for f7/2 transfer wereicalculated using the coexistence
model wave functions2'27) in g2Ca derived from the E2 properties
and the '''ca ground state wave function deduced above neglecting
the 10% core-excited component. These calculated
spectroscopic factors are compared with the renormalized
experimental values in table 5. The present transfer data
probes the (fp)z part of the coexistence model wave functions
whereas the E2 properties are sensitive to the core-excited
components. The excellent agreement shown in table 5 is strong
support for a coexistence model description of the structure of
these states. Two solutions were obtained in ref. 2 for the
wave functions of the 6 states because only the lowest 6
state was known. Recently the second 6 level was identified
at 4.715 MeV using the 39K (a,pY) <t2Ca reaction37). The 1« yel
energies and spcctroscopic factors both support the coexistence
model solution having almost no mixing of the (fp)2 and core-
excited 6 states. The other solution having slightly more
mixing is quoted in refs. 2 and 27. All this evidence strongly
- 79 -
implies that the above ronomviiization of the spectroscopic
strength is required which, in turn, provides additional
evidence that the major components of the f7,_,p-»_ and p, /,
transfer strength have been located for the (5,p) reaction
on both "°Ca and ulCa.
It is interesting to ascertain to what extent the p, ,_
and P-i /? transfer strength leading to lt2Ca can be reproduced
within the (fp)2 shell model. The (fp)2 shell model calcu-
lations were performed using the Rochester-Oak Ridge shell
model code33) with two-body matrix elements of Kuo and Brown31*)
as modified by McGrory35) to better fit the (f7 , _ ) n spectra.
The P,/2 and p, ,~ spectroscopic strength calculated using these
wave functions are compared with the data in fig. 7. Clearly
the (fp)2 shell model is unable to reproduce the extensive
fractionation of the P-,/2 an<^ pi /? strengths. For example,
the 4 member of the (f.,,p. ,,) configuration is distributed
with significant strength over four states, whereas the model
predicts more than 90% of the strength in one level. The
situation is similar for the p, ,- strength which is spread
rather evenly over more than ten states, whereas theory again
predicts the main strength in one level of each spin. The
discrepancy between the pure (fp)2 calculation and the results
for states excited by f7/2 transfer, as listed in table 5, and
the states excited by P3/2 a n^ PI/T transfer clearly demon-
strates the need for a larger configuration space.
4.3 EFFECTIVE TWO-PARTICLE MATRIX ELEMENTS IN i*2Ca
The energy centroids, e(JiJ2J), for the different spin
members of the (f?/2)2 , (f 7 / 2p 3 / 2, a n d ( f ^ ^ t w o _ p a r t i c l e
- 80 -
multiplets can be obtained frc in the f?/2, P3/2 a n d P-wo transfer
strengths if it is assumed that all the appropriate transfer
strength has been located. The energy centroids e(f.7/ojj)
are given by the relation
J(2J+l)Si(J)ei(J)e(f7/2j,j) = £
where (2J+1)S.(J) and e.(J) are the spectroscopic strength and
excitation energy of the i state of spin J in "*2Ca, which is
excited by j transfer. The experimental energy centroids for
the individual spin members of each two-particle multiplet, in
addition to the average energy.obtained by summing the spectro-
scopic strength over all J values in a multiplet, are presented
in table 6. The present (f-,/•)) 2 energy centroids are in good
agreement with the results of ref. 5 as expected in view of the
fact that the £ = 3 transfer assumed to be f7/-, transfer in ref.
5 was confirmed to be correct by the present work.
Effective T = l two-body matrix element E(jiJ2J) can be
derived by taking the energy centroids with respect to the refer-
ence energy, E D, at which the multiplet would occur if there
was no residual two-body interaction. The reference energy
En(f_,.2) is given by the ground state binding energies.
E R(f ? / 22) = B('t2Ca) +B('*0Ca) -2B(1(1Ca) = 3.109 MeV.
The energy centroids of the p, ,., and P1/2 si n9^ e particle
configurations in 41Ca obtained using the *• °Ca (d,p)'' !Ca reac-
tion22) , were used to determine reference energies of 5291
keV and 7174 keV for the {£7/2^1/2* a n d ^f7/2Pl/2^ c o n f i 9 u ~
rations, respectively. The effective two-body matrix
- 81 -
elements derived from the energy centroids and reference
energies are listed in column 5 of table 6.
The data suggest that the major fragments cf the (f_ , _ ) 2
and(f_,_p) multiplets have been located. However, the
observed transfer strength to states of unknown spin and the
possible existence of unobserved minor fragments at high
excitation energy both lead to uncertainties in the extracted
energy centroids and corresponding two-body matrix elements.
About 10% of the P->/? transfer strength leads to states between
6.3 MeV and 7.1 MeV for which .the spin assignments are unknown.
This unassigned strength produces the largest uncertainty in the
extracted (f7/2p3/2' m a t r i x elements. One plausible distribution
of this unassigned strength would be to assign 3 + to the 6.82
MeV state and to distribute the remaining 60% of the unassigned
strength equally among the other spin values. This would result
in a summed strength of 0.7 0 of the shell model value for each
spin and in values of E(f7 ,,,p, ,^) which are 0.35, 0.40, 0.13
and 0.06 MeV higher than quoted for J = 2, 3, 4 and 5, respectively.
The possible existence of minor fragments of unobserved
strength at high excitation energies was estimated by calcu-
lating the spreading of the strength distribution due to the
residual two-body interaction. The Kuo-Brown34) matrix element
as modified by McGrory35) were used in the (fp)2 shell model
space to calculate the theoretical distribution of spectro-
scopic strength. This calculation indicates that the strength
lost at high energy (above 7.9 MeV) is extremely small (<3%)•
- 82 -
The extracted two-body matrix elements corrected for omission
of this high-lying strength arc listed in column 6 of table 6.
This correction is insignificant for all except the 0 centroid.
Since the experimental and Kuo-Brown-McGrory matrix elements
are very similar, it would appear that this estimate of the
spreading of the strength due to the residual interaction
within the (fp) space is reasonable. The major fractionation
is due to mixing with excited-core configurations. However,
the interaction matrix elements with the core-excited configura-
tion does not appear to be large enough to push a significant
fraction of the two-particle configurations beyond 7.9 MeV in
excitation energy. The narrow width of the P-,/2 s t r e n9th,
together with the absence of p^ ,_ strength above 7 MeV, both
suggest that the spreading due to the core-excited configura-
tions will not push much two-particle strength out of the energy
region studied.
The two-body matrix element given in column 6 of table €
clearly must represent lower limits since unobserved higher
lying strength would raise these energies. The individual
spin (f_/_)2 two-body matrix elements and the averages of
the <f7/2P3/2^
a n d ^f7/2pl/2^ niultiplets are probably accurate
to ~<0.1 MeV, whereas the individual spin values of the ^-1/2
multiplet could be up to 0.4 MeV higher when the unassigned
p., ,_ transfer strength is included.
Column 7 in table 6 lists the renormalized (f_ ,_p) matrix
elements derived from the Hamada-Johnston nucleon-nucleon
potential by Kuo and Brown31*). The (f.,^)2 matrix elements
- 83 -
result from a least squares lit by McGrory35) to the excitation
energies of 29 states and seven binding energies for nuclei
with 42 < A < 44.
The excellent agreement between the (f_ - _ ) 2 matrix
elements derived from the present stripping data and the
previous fit to level spectra implies that use of our effective
(f^y_) 2 two-body matrix elements in shell model calculations
will reproduce those spectral properties used by McGrory for
nuclei with 42 < A < 44.
The overall agreement between our experimental two-body
matrix elements for the (f7 ,.p. ,_) and (f?/2Pi/2^ multiplets
and the predictions of Kuo and Brown is good. It is worth
noting, however, that the average contributions to the energy
of the two-body interaction for the (f7/2P3/2^ a n d ^7/2Pl/2^
configurations are measured to be slightly repulsive, whereas
Kuo and Brown34) predict a small attractive monopole contribu-
tion. The Kuo-Brown prediction for the (ld5/2^si/2^ two-body
matrix elements also are about 0.5 MeV more attractive than
observed experimentally36).
SUMMARY
Angular distributions of the cross section and the vector
analyzing power have been measured for the lf °Ca (3,p) 4 !Ca and
4 afcLp) 1<2Ca reactions using 11 MeV deuterons. The vector '
analyzing power data have been used with DWBA theory to determine
- 84 -
the total angular momentum j of the captured neutron for 42
positive parity levels in 1|2Ca. Spectroscopic strengths were
extracted separately for each of the allowed j-values contri-
buting to the cross section to each state resulting in identi-
fication of the f7/2' P-i/p an& Pi/T transfer strength distri-
butions. The main constituents of the f7/2 strength are found
to be below 4 MeV in excitation energy, whereas no significant
fs/2 strength is identified below 7.8 MeV. An almost total
absence of P-,/? and p. .. admixture is observed. Of 32 states
populated by pure 1 = 1 transitions,only two rather weakly
excited states consist of a mixture of p, ,. and p, ,_. Shell
model calculations in the (fp)2 space predict the observed weak
mixing of the (fT/oPi/?) a n d ^7/2^1/2^ multiplets, but they
cannot reproduce the extensive fractionation of P,/2 ar*d Pi/->
transfer strength observed.
The summed f7/-ji P-w? an& Pi/-> spectroscopic strengths
in **2Ca are found to represent the same fraction of the shell
model limit as are obtained for the same orbitals in the
simultaneous analysis of the *• °Ca (d,p) ** *Ca reaction. Moreover,
the same characteristic feature is obtained for each spin member
of the (£-]/2^2 an<^ ^7/2^3/2^ multiplets. These results have
been used as a criterion for showing that the major components
of the spectroscopic strengths for these orbitals have been
identified. A renormalization of all the spectroscopic strengths
by a constant factor produces results for both the t*°Ca (S,p) *• 1Ca
and ^ C a (cl,p) "* 2Ca reactions in excellent agreement with the cp-
existence model wave functions for the ground state of ^ C a and
low-lying states in 'tZCa deduced from electromagnetic properties,
- 85 -
inelastic scattering and sinqLe-nucleon transfer reactions on
^Ca. All these data are consistent with an admixture of
core-excited configurations of about 10% in the ground state of "tlCe
The effective ( f . ^ ) 2 , (f7/2P3/2
) a n d f7/2pl/2 t w ° - P a r t i c l e
matrix elements were compared with the modified Kuo-Brown
matrix elements of McGrory. They agree to within 0.4 MeV, or
better. However, the average energy contribution of the two-body
interaction for the f7/2P3/2 a n d f7/2pl/2 c o n f i 9 u r a t i o n s a r e
measured to be slightly repulsive, whereas Kuo-Brown predict
a small attractive monopole contribution.
- 86 -
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- 8 9 -
F i g u r e C a p t i o n s .
F i g . 1 P r o t o n s p e c t r a a t 10 l a b o r a t o r y a n g l e f r o m s p i n - u p
a n d s p i n - d o w n d e u t e r o n b e a m s a r e c o m p a r e d to i n d i c a t e
t h e d i f f e r e n t y i e l d s . T h e g r o u p n u m b e r s r e f e r t o
t a b l e 1 .
F i g . 2 E x a m p l e s o f e x p e r i m e n t a l a n g u l a r d i s t r i b u t i o n s
for t h e l l l C a ( d \ p ) " 2 C a r e a c t i o n o f t h e
c r o s s s e c t i o n a n d v e c t o r a n a l y z i n g p o w e r w i t h D W D A
p r e d i c t i o n s f o r p u r e i j - t r a n s i t i o n s u s i n g s e t A .
T h e e f f e c t s o f i n c l u d i n g t h e c o r r e c t i o n s f o r f i n i t e
r a n g e ( F R ) a n d n o n - l o c a l i t y ( N L ) o f t h e d e u t e r o n ,
p r o t o n a n d n e u t r o n p o t e n t i a l s a r e c o m p a r e d to t h e z e r o -
r a n g e ( Z R ) a p p r o x i m a t i o n w i t h l o c a l ( L ) o p t i c a l p o t e n -
t i a l s . T h e F R / L B c u r v e s i n c l u d e n o n - l o c a l i t y in t h e
d e u t e r o n a n d p r o t o n c h a n n e l s o n l y . T h e n o r m a l i z a t i o n
o f t h e F R / N L c u r v e s c o r r e s p o n d s to t h e e x p e r i m e n t a l
s p e c t r o s c o p i c s t r e n g t h s o f t a b l e 1. T h e o t h e r D W B A
c u r v e s a r e n o r m a l i z e d t o t h e F R / N L c u r v e s a t
F i g . 3 D W B A f i t s to t h e a n g u l a r d i s t r i b u t i o n o f t h e c r o s s
s e c t i o n a n d t h e v e c t o r a n a l y z i n g p o w e r d a t a f o r p u r e4 2
£ j - t r a n s i t i o n in Ca a r e c o m p a r e d f o r t h r e e d i f f e -
r e n t o p t i c a l m o d e l p a r a m e t e r s e t s . T h e s e t s a r e g i v e n
in t a b l e 2 .
F i g . 4 a , b , c , d, e , f. E x p e r i m e n t a l a n g u l a r d i s t r i b u t i o n
o f t h e c r o s s s e c t i o n a n d v e c t o r a n a l y z i n g p o w e r a l o n g
- 90 -
w i t h D W B A p r e d i c t i o n s i n t h e c . m . s y s t e m f o r4 1 ->• 4 2
t h e C a ( d , p ) C a r e a c t i o n . T h e s o l i d l i n e s a r e
t h e D W B A r e s u l t s w i t h p a r a m e t e r s e t A . T h e d a s h e d
l i n e s a r e b a s e d o n u s i n g t h e e x p e r i m e n t a l d a t a o f
t h e 3 9 4 3 k e V 1 / 2 " s t a t e in 4 1 C a t o r e p r e s e n t t h e
p u r e iT-11 c u r v e s . T h e n o r m a l i z a t i o n o f t h e D W B A
c u r v e s to t h e e x p e r i m e n t a l c r o s s s e c t i o n c o r r e s p o n d s
to t h e s p e c t r o s c o p i c s t r e n g t h s o f t a b l e 1 .
F i g . 5 E x p e r i m e n t a l a n g u l a ' r d i s t r i b u t i o n o f t h e c r o s s s e c t i o n *
a n d v e c t o r a n a l y z i n g p o w e r a n d D W B A p r e d i c t i o n s f o r
4 0 -* 41t r a n s i t i o n s o b s e r v e d in t h e C a ( d , p ) C a r e a c t i o n .
T h e s o l i d l i n e s a r e t h e D W B A r e s u l t s w i t h p a r a m e t e r s e t
A . T h e n o r m a l i z a t i o n o f t h e D W B A c u r v e s t o t h e
e x p e r i m e n t a l c r o s s s e c t i o n c o r r e s p o n d s t o t h e s p e c t r o -
s c o p i c s t r e n g t h s o f t a b l e 3 .
F i g . 6 T h e e n e r g y d i s t r i b u t i o n s o f t h e s p e c t r o s c o p i c s t r e n g t h4 ~*
in ''Ca f o r t h e d i f f e r e n t f p o r b i t a l s o b s e r v e d in
t h e p r e s e n t e x p e r i m e n t .
F i g . 7 T h e e x p e r i m e n t a l p , , 2 anc' P i / 2 s t r e n g t h d i s t r i -
b u t i o n s a r e c o m p a r e d w i t h t h e p r e d i c t i o n s o f a p u r e
( f p ) s h e l l m o d e l c a l c u l a t i o n . T h e e x p e r i m e n t a l
s t r e n g t h s a r e r e n o r m a l i z e d b y 1 / 0 . 7 0 .
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Jo.
76
78
79
82
83
84
88
91
92
93
94
Ex
present
7024
7122
7160
7267
7345
7398
7519
7641
7706
7756
7789
(keV)
c
refr
7025
7123
7160
7270
7348
7401
7520
7643
7709
7760
7793
J71 cmmax
(pb/sr)
lab £=1 2=3
present
1=' I--.
ref.'
+3+,4
572
702
823
2184
1579
1534
2649
1272
1589
1309
4628
15
15
15
15
10
15
15
10
15
15
15
1/2+5/2 0.19
1/2+3/2 0.22+0.14
1/2
1/2
1/2
1/2
1/2
l/2f)
1/2
1/2)
1/2
0
0
0
0
.1
0
0
0
1
.38
.91
.69
.72
.06
.62
.75
.54
.78
0.37 0.
0.
0.
0.
0.
0,
1
0
0
0
1
2
31
35
98
,74
.96
.1
.45
.74
.47
.6
0.
0
0
0
1
33
i
.65 i
.39
.94
.4
a) Doublet. From the cross section data it was deduced that the spectroscopic strength of1.54 is split with a strength of 1.0 for the 4 and 0.54 for the 2 level.
b) From present j-value.
e) From ref. 5.
c) From ref. 24. d) From present work; see text.
f) Observed at the three most forward angles only whichindicate pure Pi/? transfer.
i
TABLE 2
Optical model parameters. '
Set Particle W W' V s - ° rso aso NL FR
d
P
n
d
P
n
d
P
108 1.05 0.86
62.2-0.664E 1.21 0.66
b) 1.20 0.65
106
51 .2
b)
106
53.0
b)
1
1
1
1
1
1
.05
.17
.20
.05
.20
.20
0
0
0
0
0
0
.85
.75
.65
.85
.65
.65
0.3
36.
30.
44
24
76
2
.6
1
1
1
1
.61
.201
.59
.32
0
0
0
0
.62
.547
.566
.51
14.
27.
14.
20
0
4
0
0
1
1
0
1
.75
.016
.20
.9
.01
0
0
0
0
0
.5
.351
.65 25
.6
.50
0.
0.
0.
0.
0.
38
85
35
38
85
46.8
44.0
1.496 0.63
1.25 i.47
1.20 0.65 25
13 0.9 0.6
32.0 1.20 0.65
1.20 0.65 25
0.62
0. 52
All lengths in fm and all potential energies in MeV . *) The optical model potential was of
the form V(r) = VC0(jl (r) - V [ 1 + exp( (r-r A1 / 3 )/a) ] " ] - i W[l + exp x 1 ] " 1 + i W d/dx'[l+exp x 1 ] " 1
+ Vso(l/r) d/dr[l + exp((r - r ^ A 1 / 3 ) / a S Q ] " ' • t • t , where x = (r - r^ A 1 / 3)/a'.
b) Adjusted to give a binding energy of Q + 2.23 MeV .
1 —
- 95 -
E
1
2
3
x(MeV)
0
.94
.46
.94
f7/2
P3/2
P3/2
Pl/2
"°Ca(d,
this
5
2
0
0
TABLE
P>"Ca
(2JB + 1
work
.7
.3
.80
.90
i
3
results
)S
6
2
0
1
a)
.6
.5
.82
.0
S
S Shell
0.
0.45
Model
71
78
(0.67)b)
a)
b)
Ref. 5
Including correction for the 33% of Pi/2 strength
leading to higher lying states22'23)
- 96 -
TABLE 4
The summed fp strength for the 1*1Ca(d,p) l*zCa reaction
Configu-ration
( f 7 / 2 ) 2
f7/2 P3/2
J
0+
2+
4 +
6 +
ALL
2+
3+
4 +
5+
Unknown
ALL
(2JB + 1)S
1.22
6.27
10.61
17.1
35.20
2.96
3.75
5.93
6.88
2.91
22.46
S
S Shell Model
0.61
0.63
0.59
0.66
0.63
0.59
0.54
0.66
0.63
0.70
f7/2 Pl/2 ALL 9.94 0.62
- 97 -
TABLE 5
Comparison of the experimental values and the coexistence model
predictions for the I = 3 spectroscopic factors to the low-lying
states seen in the *• *Ca (d,p) 4 2Ca reaction.
Coexistence Modela) Present Experiment
1.20
0.44
0.69
0.94
1.37 .
0.33
1.76
<0.01
1.3
0.3
0.77
0.90
1.31
0.37
1.89
<0.07
Spectroscopic factors calculated using the coexistence model
wave functions of ref. 2 in addition to assuming that the ground
state of ^ C a is 90% f^/o neutron coupled to a closed tf0Ca core.
Spectroscopic factors assuming renormalization factors of 1/0.70.
- 98 -
TAHLE 6
The effective two-particle matrix elements in
Config-
uration
(f7/2>
f7/2P3/2
J*
0+
2+
4+
6+
ALL
2 +
3+
4+
5+
ALL
(MeV)
0.58
2.12
2.86
3.19
2.79
A.43
5.21
5.05
5.78
5.42
(MeV)
0.52
2.17
2.85
3.19
•
exp.
-2.53
-0.99
-0.25
0.08
-0.32
-0.86
-0.08
-0.25
0.49
0.13
E(JiJ2J) (HeV)
exp.
corrected )
-2.06
-0.94
-0.20
0.17
-0.23
-0.72
-0.08
-0.19
0.49
0.17
Kuo-Brown
McGrory )
-2.22
-1.15
-0.36
+0.29
-0.27
-0.86
-0.03
-0.05
+0.15
-0.10
f?/2P1/2 ALL 7.39 0.21 0.24 -0.14
b)
Ref. 34
Ref. 35
Corrected for the strength, predicted by the (fp)^ shell modelcalculations, to lie above 7.9 MeV in excitation energy and hencewould be missed by the present measurement
d)Ref. 5
- 99 -
'ON 13NNVHD «3d SINHOO
Fig. 1
7 I
OO3rt-
gPu
1txa
5.0
EXCITATION ENERGY (MeV)S3 60 6.5J ! L
7.0 7.5
41Ca(d,p)42CaSPIN UPEd=11.0 MeV
jm
• 1/3
o
3°
n34 !S3'
3433!'
; 1
ji!l
j
]i
1K
.1
1100 1200 1300 U0O 1500 1600
CHANNEL NO.
EXCITATION ENERGY (MeV)45 5.0 5.5 60 & 5
l : 1 1 •_
1700 1900 2000
7.0 75I
1200 1500 1500
CHANNEL NO.1900 2000
oo
1 . ,
- 101 -
10
FR/NLZR/LFR/LFR/LB
- i
10 "L-
10
lo-V
10'r:
10" V
Ex = 3191 keVr7/2
Ex=5211 keV \
3/2
FR/NL—-ZR/L
FR/L-—FR/LB
0.2
0.1
0
~0.1
-0.2
-0.3
Ex=319lkeV ' * - '
Ex=5211keV
jy.
# <P*\
Y/
^ \F \
Ex=7345 keV
P./2
I I I I I I
0.5
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-—
—
—
—
—
-I .
Tj\\
7/i
Ex=P,«
W
AT T1
7345
I
' \ }
V\V
keV \ y
1r>
RIO
20 40 60Fig. 2
20 40 60
- 102 -
E
a
10° b
10'
Set ASetBSetC
;="= s
10o
Ex=3191keV
Ex=5211 keV
3/2
0.2
0.1
0
-0.1
-0.2
Set A—-- Set B- - - S e t C
/< *
/ / Ex=3191kevV^
Ex-~5211 keV
10°
io-'
-
—-
-
I
\
\ — •
Ex=7345keV
R/2
I I I I
0.5
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
—
—
——
—
1T—\V"
\ j
1
\\I,//~y>
I\
h \\\
/- V \
r- \'T : \ \
/ ; * \
t
Ex=7345 keV
P,/2
1 1
V
\\\\V1
\ i\ ^\ \\ \\\\
1
\\\\\\\
1(*)in
HU
1
20 40 60Fig. 3 e 20 40 60
CM.
- 103 -
10°
'- '10"
10°
p.6
0 . 4
0 .
^ 0.0
- 0 . 2
- 0 . 4
-0 .6
10' r
a 10°CDX
solO" 1
1 0 " '
0 . 6
0-4
0 . 2
E=C1 KEV
10°
HI"'
1 0-3_ A . l ^ - l ^ . l ^ 1 1-1.. U
20 40 6(1- - - , 0.6
'-0 KEV
0 . 4
0-2
0 - 0
- 0 - 2
- 0 - .
20 40 60. 10'
t. = i Li 2 4 K E V
10"
10 -7
Tr. 1835KEVL T
/I
^ . U - l . I., 1 , I.. IJ | C l " . I..J.._l....J._Li_l_i.
20 10 60 20 40 600 , 6
0 . 4
0 . 2
0 . 0
- o . ;
- 0 . 'E:1524KEV
20 JO 60
V 7
E;1835KEVJ--7/?
0-6
0.4
0.2
-0.2
-0-4
\
EJ
20 40 60
I
=2422KEV= 7/2
E=2752KEVL-J *3
20 40 60
£ 0.0
- 0 . 4
-0.6
E;2752KEVJ=3/2+7/2
• ' • ' - ' - • • • • i • •
20 40 60
Fig. 4a
10°
10"
10"
0-6
0.4
0.2
0.0
- 0 . 2
- 0 . 4
- 0 . 6
E = 3 1 9 1 K E V
10"
10"
20 40 60 " 20 40 6010*'- - - - - - -
20 40 60- 0 . 6
0.6
0.4
0.2
0.0
-0.2
-0.4
- n .fi
20 40 60
•TE=3400KEV
>-i • i . i > i • i . i
•
20 40 60
C .M .flNGLE20 40 60
L
- 104 -
.flNTrLE20 40 60
- 105 -
- 0 . 620 40 60
F i g . Ac
2 0 4D 61)
n , M „?0 4 0 6(1 20 40 60
- 106 -
HI" , , M"
0
0
0
0
- 0
-0
_n
.6
.4
.2
.0
.2
. 4
20 40 60
f * •/ ^
\ if \
E;6818KEV
. *i i . i . i . i . i .
20 40 60
,-J .
-o.z
- 0 . 4
20 40 60
Pig. 4d
? 0 1U f - l ! 20 40 600 .6
C . M . Fl N G L E20 40 60
- 107 -
20 40 60
Fig. 4e
- 0 - 620 40 60
C . M .- 0 . 6
20 40 60
G 1 3- 0 . 6
20 40 60
L
- 108 -
- 0 - 4
-0 .620 40 60
C. MANGLE
Fig. 4f
- 109 -
. I I i i i i . p.r-TT-i- 1 —« JJ 1 1 1-
ya
;
- • -
13K
EV
encn —i -II II
LU -J -
CM
OoO
- ID
° O
* —
s
•eie
LU "
cn•^•CM -
N mm -^ II II
/ LOT -
CD
O
CM
O
O
o
CM
OI
o
CM
oo
o
CM
oI
o
oCD
. o
oCM
CD
OI
LU
CMCD CM -
csjm -II II
oCD
oUJ
CD
oi
r
1 •©<
( a
^I©I
\ ,
-
LU
CM
cn \ •— c-i-ii II -
LJ-) -
oI
? cr
CDUD °
' CJ
oCM
OI
j a
\
>LU
CM -
1—o-H II II•-57 LU ~) -
CM O
O O
u
CM
OI
OI
oID
. o
OCM
CD
OI
1!
Fig. 5
- 110 -
>
X
U
8 -
7 —
6 —
5 —
-
3 —
2 —
i
n
2+
+
•
-2*
'7/2
I
l 1 l
—
0
2+
2+
£
2*
p3 / z
1 1 1
>ii h
it.. •3V
PV2
1onr
1 |
I I
5 1 0 1 5 2 4 6 8 1 2 3 4 1 2 3 4
2 3 - f + 1 ) S
Fig. 6
- Ill -
Q)
UJ
P3'2THEORY
3*
?• . _.
— 2*
— V.
- 3 '
2*
-5*
P3/2 Pl/2EXPERIMENT THEORY
Pl/2EXPERIMENT
12 9 6 3 0 3 6 9 12 6 ^ . 2 0 2 ^ . 6
Pig. 7
- 112 -
P A P E R V
A STUDY OF THE TWO-PARTICLE STATES
42 4 1 3 42IN Sc FROM THE Ca( He,d) Sc REACTION
- 113 -
A STUDY OF THK TWO- PARTICLE STATES IN
U2Sc FHOM THE *• JCa { 3IIe ,d) ** 2Sc REACTION
P.B. VOLD*
Institute of Physics, University of Bergen, Bergen, Norway
and
D. CLINE and M.J.A. de VOIGT*
Nuclear Structure Research Laboratory''"''University of Rochester, Rochester, N.Y., U.S.A.
and
OLE HANSENt+t and 0. NATHAN
The Niels Bohr Institute, University of Copenhagen2100 Copenhagen 0, Denmark
Abstract: Energy levels in U2Sc have been studied using the
proton capture reaction "* *Ca (3He,d) "• 2Sc at a bombarding
energy of 20 MeV. A total of 89 levels were identified
below 6 MeV. Angular distributions have been measured and
used together with DWBA calculations to determine & values
2
and spectroscopic strengths. The T=l states of the (^7/0'
and (f_/2P-J/->) configurations were identified from a com-
parison of spectroscopic factors obtained using the ** *Ca (3EIe,d)4 1 "*•
and Ca(d,p) reaction. Deexcitation gamma rays were observed
in coincidence with the emitted deuterons to determine the
individual spectroscopic strengths for the 0.62 MeV 1+, 7 +
unresolved doublet.
* Present address: K.V.I., University of Groningen, TheNetherlands.
t Supported in part by Norges Almenvitenskapelige Forskningsrfid.ft Supported by a grant from the National Science Foundation
ttt Supported by Statens Naturvitenskabelige ForskningsrSd, Denmark
- 114 -
The major components of the f7/2 a n^ ^-ne T =^ P a r t °f the P3/5
strength have been identified and were used as a criterion to
renormalize the strengths by 1/0.75 to bring the summed strengths
into agreement with the shell model predictions. Effective2
(f_ ,„) and (f_ /pP-, /•p)rp=i matrix elements have been determined
and compared to the corresponding matrix elements of Sc and
42Ca. The three sets agreed to within a few hundred keV.
E NUCLEAR REACTIONS 4 0 ' 41Ca (3He ,d) , 41Ca ( 3He ,dr ,
E=20 MeV; measured ad (E ,0), deduced levels, I,
•n,J,T, spectroscopic factors. Enriched targets.
- 115 -
1. Introduction
The existence of deformed multi-particle, multi-hole
configurations additional to the sequence of (f_,_)2 two -
particle shell model states in the low energy spectrum of
mass-42 nuclei is a well known phenomenon. ' ' ). In "42Sc,
about twenty-five levels are observed below 3 MeV excitation ' )
which far exceeds the eight levels expected from the (£7/3)2
configuration. This multitude of states indicates the impor-
tance of core-excited configurations which leads to consider-
able fractionation of the two-particle configurations. Effec-
tive two-body matrix elements of the residual two-body part
of the nuclear Hamiltonian for the (lf-2p) shell can be
determined directly from the strength distribution of the
two-particle configurations in **2Sc. One-nucleon transfer
reaction data provide the only viable method for locating the
fractionated two-particle strength in mass-42 nuclei.
The recent production of a ^Ca target made possible
studies of the two-particle states in mass-4 2 nuclei. The
11 !Ca (d,p) "*2Ca reaction using unpolarized ' ) and polarized
8 9
deuterons ' ) has been reported previously and the T=l effec-
tive two-body matrix elements were determined. The present
paper reports on the l( JCa (3He,d) k 2Sc reaction. This reaction
should strongly excite both the T=0 and T=l members of the
low-lying (f7 ,2)2conf iguration in addition to the higher lying
(f7/2p3/2)' (f7/2pl/2* a n d (f7/2f5/2) two-particle configura-
tions. The *• 1Ca (3He,d) !<2Sc reaction is expected to weakly
- 116 -
excice core-excited configurations in t>2Sc because the ground
.state of " tfa has been shown to have at most about a 10% ad-
9 10mixture of excited core configurations ' ).
The experimental procedure and results for the ^ C a
(3He,d)i'2Sc reaction are discussed in section 2. In addition
this section contains a description of the *• *Ca (3He,dY)'*2Sc
coincidence experiment which was performed to resolve the
transfer strength of the 1+ state at 611 keV from the strongly
excited 7 + state at 618 keV observed as a doublet in (3He,d).
The DWBA analysis of all these data is discussed in Sect. 3.
The J11 and T values for the 1(2Sc states derived from a com-
parison of the present data with the ** !Ca (d, p) k2Ca. results
and other data are given in Sect. 4. The spectroscopic in-
formation extracted from the present results and their impli-
cations regarding the ground state properties of 1<1Ca are
presented in Sect. 5. The energy centroids of the (f 7/ 2)2 a n d
^7/2^3/2^ configurations are discussed in Sect. 6, and the
extracted effective two-particle matrix elements are compared
with the corresponding data from i'8Sc and "2Ca. A comparison
with the coexistence model of Flowers and Skouras ) is
presented in Sect. 7.
- 117 -
2. Experimental Method
2.1 The ** JCa(3He,d)M2Sc reaction
The target consisted of 81.8% hiCa and 18.1% l>0Ca and
had a thickness of about 25 pg/cm2. It was vacuum evaporated
on to a 30 pg/cm2 carbon backing. Details are given in Ref.7.
In addition, a '•Oca target was used in order to facilitate the
identification of transitions to single particle states in
^ S c , excited by reactions on the '•''Ca contamination in the
^ C a target.
The (3He,d) data were obtained using a 20 MeV 3He beam
from the Rochester MP tandem accelerator. The outgoing
deuterons were analyzed in a split-pole magnetic spectrometer
and detected using Kodak NTB, 50 pm photographic emulsions
placed along the focal plane. Suitable Al foils were placed
in front of the emulsions to stop heavier particles scattered
from the target.
Resulting spectra of hzSc were measured in steps of
2.5° or 5 at laboratory angles from 5 to 55 with an energy
resolution of 17 keV. The measurements at 12.5 , 22.5 , 32.5
and 40 were made with a smaller solid angle in order to obtain
better resolution, i.e. 10 keV. Control runs on the l|0Ca target
were performed from 10° to 50 in 10 steps.
The absolute cross sections were obtained by observing
elastic scattering yields using position sensitive detectors
in the focal plane of the magnetic spectrometer and by normal-
izing to optical model predictions. The absolute cross section
scale is believed to be accurate to within 15%.
- 118 -
The ratio between " °Ca( 3He,d) and ^Ca^Hejd) yields
was obtained in two ways: By relative measurements using sep-
arate targets of the two isotopes, as described above, and by
simultaneous observation of deuteron groups originating from the
mixture of Ca and Ca in the enriched Ca target. The two
data sets agreed to within the experimental uncertainties and
determined the Ca( He,d) and Ca( He, d) cross section ratios
to better than ± 10%.
41 3 42A deuteron spectrum from the Ca( He,d) Sc reaction at
12.5° is presented in fig. 1. A total of 89 deuteron groups
42
corresponding to levels in Sc was observed below an excita-
tion energy of 6 MeV. In table 1, the excitation energies
corresponding toLthe groups shown in figure 1 are given. The
excitation energies reported are accurate to within 5 keV and
they are in excellent agreement with the accurate energies
obtained from y-decay studies given in column 2 of table 1.
2.2 The 41Ca(3He,dY) 2Sc Coincidence Experiment
The transfer strength leading to the 611 keV 1 state was
buried under the strongly excited 618 keV 7 + group in the
41Ca(3He,d) Sc reaction data. Since the 7 state is isomeric
it was possible to determine the transfer strength to the 1 +
member of this unresolved doublet by observing the ejectile in
coincidence with the gamma ray de-exciting the 1 + state.
7 41
The 25Mgm/cm Ca target described earlier was bom-
barded with 20 MeV He ions and the deuterons scattered at 8=15°
- 119 -
were detected by position sensitive silicon detectors placed in
the focal plane of the split-pole magnetic spectrometer. The
gamma rays were detected by a 7.6 x 7.6 cm Nal scintillator loc-
cate'd at 9=-80° in the scattering plane and 2 cm from the target.
Both particle singles and coincidence spectra were recorded event
by event on magnetic tape and analyzed off line. Gamma ray spectra
were projected from the coincidence data with windows set on the
time, particle energy and position signals corresponding to the
1 + (611 keV), the 3 + + 5 + (1491 + 1511 keV) and the 2 + (1586 keV)
groups. The spectra were corrected for random coincidences. The
time to random ratio was better than 10 to 1. The gamma spectra
were corrected for summing effects in the Nal detector.
- 120 -
DWBA .i
3.1 The l|1Ca(3He,d)l|ISc Reaction Data
The distorted wave (DWBA) computer code DWUCK by
P.D. Kunz was employed to calculate the differential cross-
sections. The calculations were made using the zero range
(ZR) and local (L) approximations. The optical model para-
meters used in the present analysis are given in table 2; they
are average parameters, fitted to elastic scattering for the
fy/2 s n e 1 1 region * *°) . The spectroscopic strengths G(JA+j-»-J)
listed in table 1 and 3 for the M1Ca(3He,d) and l»0Ca(3He#d)
data respectively, were extracted from the data by normalizing
the DWBA cross section to the experimental cross section via
the relation
exp
.(do/dn)DWUCK (1)
A1
where G(JA+j->-J) is related to the spectroscopic factor given
ir; an isospin formalism by
(2)
C2 is the isospin Clebsch-Gordan coefficient, j and t designate
the transferred spin and isospin, J., T. and J,T are the spin,
isospin for the target and final nucleus respectively. Since
T. = 1/2 and Tft = 1/2 we populate both T = 0 and T =1 states
in l|ZSc. For either choice, C2= 1/2, which merely expresses
equal probabilities that an excited nuclcon in the final state
is a neutron or a proton.
- 121 -
DWBA calculations which included both non-locality (NL)
in the incident and exit channels and finite range (FR) cor-
rections were also performed to check the sensitivity of the
cross sections to these effects. The shape of the angular
distribution was found to be essentially unchanged whereas
the magnitude increased by 20% and 30% for £=1 and i=3, respec-
tively. This increase is similar to the results of other work
on the 3He,d) and (d,3He) reactions 1 4 # 1 5 ) in the fp-shell.
The (ZR-L) procedure of refs. 14, 15) yielded spectroscopic
factors consistent with that expected from the shell model,
while the more complete (FR-NL) calculation underestimated
the spectroscopic factors by about 40%. T*hese results indicate
that the uncertainty associated with absolute spectroscopic
factors derived from the (3He,d) reaction, using the factor
4.42 in Eq. 2, may be as high as + 40%, whereas relative spec-
troscopic factors are considered to be appreciably more reliable
(see Sect. 5 for a further discussion).
The ground state spin of J = 7/2 implies that a transition
may proceed with mixed 2-values? for example £=1+3 are allowed
for final state spins of 2 + £ j" - 5+. In order to obtain
reliable spectroscopic strengths for each of the two i-values
contributing to the cross section in a mixed trancition, it is
necessary that the DWBA curves provide good fits for transitions
involving a single Jl-value. Although the overall quality of the fit to
pure £=3 and pure 1=1 transitions are both very good (see figs. 2
and 3), the fit to the forward angle data for a pure 1=3 transition
- 122 -
such as the 6 *" state at 324'3 koV, allows up to a 2% admix-
ture of a fictitious 1. = 1 transfer whereas small deficiencies
for the I= 1 fit in the 25 - 35° region lead invariably to
a significant fictitious £=3 contribution. For example,
the best fit to the data for the 3/2" state at 1720 !;eV in
^ S c , which must be pure K. = 1 transfer, is obtained with a
fictitious 18% admixture of £ = 3 transfer. It is concluded
from this that an St. = 3 admixture smaller than 25% of the
total strength in a predominantly I = 1 distribution cannot
be determined from the present data, and thus is not quoted
in table 1. In the energy region below 4 MeV, where A =3 pre-
dominates, many transitions were observed to proceed with
i = 1+3, and the strong sensitivity to a =1+3 at small angles
made it possible to determine l =1 admixtures as small as 5%.
The spectroscopic strengths for mixed transitions were obtained
from a least square analysis of the cross section data and
the statistical uncertainty in the spectroscopic strengths
do maxis equal or less than 15% for states with -jrr ^6.1 mb/sr and
believed to be within 25% for weaker transitions.
The form factors were calculated using lf7/2 t r a n s f e r
for all 1 = 3 transitions. All I =1 form factors were assumed
to be 2p,,_ transfer. An.assumption of 2p, .„ transfer would
yield spectroscopic strengths about 10% larger. For i, - 0
and I =2 transitions, we have used 2s. ,» and Id.,/- form fac-
tors. Excited states above 4.2 MeV in excitation energy are
unbound. The spectroscopic strength for the unbound states
was obtained by keeping the proton bound by 100 keV and
- 123 -
using the correct kinematics in the DWBA calculations.
A procedure using extrapolation of the DWBA curves for i, = 1
to the unbound region would have given spectroscopic strengths
15% smaller than those given in table 1 at 5 MeV excitation
increasing to 25% at 6 MeV.
The presence of 18% l*0Ca in the *4lCa target made possible a
simultaneous analysis of the f7/»2 a n d Pi/? transitions in the
lfOCa(3He,d)'*1Sc reaction allowing a direct comparison
of the relative spectroscopic strengths for these orbitals
in the two reactions. The l|OCa(3He,d)'•'Sc results are compared
with those of R. Bock et al. ) in table 3. Our results for
this spectroscopic strength are in good agreement with those
of Ref. 16). Our experimental data together with the DWBA
predictions for the l<oCa(3He,d) "* 1Sc reaction are displayed in
fig. 3.
3.2 Analysis of the " *Ca(3He,dy)"2Sc Coincidence Data
The angular correlation of the deexcitation gammas was
calculated using a version of the coupled-channel code CHUCK )
which had been modified to calculate the d-y correlation for
a 7/2" target spin. The analysis was performed using the
same DWBA procedure and parameter described earlier. Only
one gamma transition between the 0 +, 1 +, 2 +, 3 +, 5 + and 7 +
states is expected to occur with mixed multipole radiation,
that is, the 2 + -»• 1 + transition. The multipole mixing ratio
1 8from this transition has been measured ) to be 6( E2/M1) =
-0.01 + 0.06 and thus this transition was assumed to be a
- 124 -
pure dipole. The y-ray correlation in the scattering plane
was calculated to be, at most, a 27% effect for the transi-
tions studied. However, the Nal detector angle and solid
angle were selected to ensure that the y~ray correlation
produced less than a IS effect.
The d-y coincidence data give relative cross sections for
exciting the 5+, 3+ and 2 states in excellent agreement with
the photographic plate data. The absolute spectroscopic
factor for the (611 keV)l+ state was obtained by normalizing
the relative d-y coincidence data to the cross sections for
the 5+, 3+ and 2+ states measured using photographic plates.
This analysis resulted in a spectroscopic factor for the
(611 keV)l+ state of C2S1+ = 0.66 + 0.07, that is, a G.^ =
0.25 ± 0.03. Thus the remaining strength measured for the
unresolved 1+ - 7+ doublet must lead to the 7+ state, i.e.
G?+ = 1.21.
4. The Jw and T assignments to 42Sc states.
The spin and isospin assignments listed in column 4
of table 1 are taken from the compilation of Endt and
1Q
Van der Leun ). The present results have made it possible
to derive many new assignments by combining them with those
of Ref. 19) and the recent spectroscopic information of the
T = l analog levels in *'2Ca as observed in the ** *Ca (d,p) **2Ca6 7 8 9reaction ' ' ' ). The adopted values are presented in column
5 of table 1. A discussion of how these assignments were
obtained is given below.
- 125 -
The T = 1 states: The identification of the T =1 states is
based on a comparison of the spectroscopic factors in the
l|1Ca(3He,d) M2Sc and *• ]Ca (d*,p) u zCa reaction. The spectroscopic
factors for transitions to the T = l analogue states in It2Ca
and l|2Sc are expected to be equal since isospin is a good
quantum number. The measured spectroscopic factors for the
analogue states are compared in table 4. Very good agreement
is obtained for the low lying analogue states populated pre-
dominantly by f7/2 transfer. Only the unknown analogue of
the 3295 keV 0 + state in l'2Ca was not located, which is as
expected since the peak cross section for the '• a (3He,d)k 2Sc
reaction to this 0 + state should be only 17pb/sr. The spectro-
scopic factors to the higher lying analogue states excited
strongly by £=1 transfer also are in good agreement.
The Coulomb energy displacement energies, AE, listed
in the last column of table 4, fall into two groups, that is,
with AE - + 50 keV for the low lying states excited mainly
by 51=3 and - 50 to - 100 keV for the unbound states excited
mainly by £=1. However, the strongly excited analogue pairs,
4548 and 4448 keV and 5084 and 5020 keV have markedly different
Coulomb displacement energies for the adjacent states. The
levels at 4227 keV and 4827 keV, assumed to be analogues
of the 2 + states at 4760 keV and 4869 keV in "2Ca would give
- 126 -
better agreement for the spectroscopic factors if the states
were interchanged. However, this would result in Coulomb
energy displacement energies which differ by 200 keV which
would appear to be unlikely.
The comparison made in table 4 indicates that all the
T = l analogue states in 'l2Sc with large spectroscopic factors
for {, - 1 anu 2, = 3. transfer have been located. The largest
discrepancy is for the suggested T = l analogue at 3754 keV
of the 36 50 keV state in **2Ca.
The T = 0 States
The spin of the final states for the *• ]Ca (3He ,d) 4 2Sc
reaction are restricted to j ' = 0 - 7 for i. - 3 transfer
and J71 = 2 + - 5+ for I = 1 transfer. However, i, = 3 transfer
to T = 0 states containing the (f 7/ 2)2 configuration is limited
to final state spins of Jv= 1 +, 3 +, 5 , 7 + to ensure anti-
symmetrization. The recent study of the " 1Ca (d,p) "* 2Ca
8 9reaction ' ) showed that no fj. ,~ transfer strength was observed
below 7 MeV in excitation energy and a similar situation should
occur in h2S>c. Thus strongly excited 1-3 transitions to
states below 4 MeV in excitation energy can be safely assumed
to proceed via the (f_. ?)2 configuration while weakly excited
I - 3 transfer may proceed via the deformed core-excited com-
ponents in the "^Ca ground state wave function. A core-excited
admixture in the ^Ca (g.s.) of about 10% (see Ref. 10 and dis-
cussion j p. Sect. 5) could yield an I = 3 spectroscopic strength
- 127 -
of up to about 0.06. We have thus only restricted the final
state spin values to 1+, 3 +, 5+, 7 + for states with I = 3
strength larger than 0.06. It is seen from table 1 that
many.strongly excited T= 0 states below 4 MeV proceed by
I - 1+3 which implies J1T= 3 + or 5+. These results combined
with those of Ref. 19) have resulted in unique Jv = 3 + assign-
ment for the 1490, 3390 and 3930 keV states and Jr = 5 for
the 1510 and 3088 keV states.
The I» = 2 distribution observed for the 2388 keV level in
the 40Ca(a/d)'l2Sc reaction 2 0) ' limits the J* values to 1 +,
2 + or 3+. The present (3He,d) data yield 3 + or 5 +, hence
J1" = 3+ for this level. The present data indicate that a
second level must occur at 2220 keV in addition to the 1 +
level at 2223 keV because the t = 1+3 transfer in (3He,d)
implies J1T= 2 + - 5+. This is consistent with the lifetime
measurements of Roberson and Van Middelkoop ) who suggested
a second level within 5 keV of the known 1 + level.,
5. Discussion and sum rule analysis.
The spectroscopic strengths extracted from the present
data are listed in table 1 and displayed in figs. 4 and 5.
The T = l strength is compared with the l*1ea(d,p) strength in
fig. 4, while Fig. 5 displays the excitation energy distri-
bution of the T= 0 spectroscopic strength.
A striking feature of the present data is that the number
of T = 0 states in the low lying (ZJ/J)2 region exceeds the
T =1 number by about three to one; twenty six T= 0 positive
- 128 -
parity states are seen below 4 MeV and 8 T=l states. However,
a major fraction of the f?/2 strength (250%) is concentrated in the
lowest level of each spin for all spin members of the multiplet for
both the T=0 and T=l states. The extensive fragmentation of the T=0
states indicates that T=0 core excited configurations produce a lar-
ger effect on the low-lying states than the T=l core excited con-
figurations,- it is also interesting to note that the f7y2
and P3/2 transfer strengths appear to mix more strongly for
T= 0 than for T = l states. This is clearly demonstrated for
each of the two strongest 3 + (1490 and 3390 keV) and 5 + (1510
and 3089 keV) transitions; . all of which receive a significant
fraction of both f~ ,_ and P3/5 strength. (See also discussion
in Sect. 7.)
The Si - 1 strength is distributed over forty three states
in the energy region between 1.5 and 6 MeV excitation; nine
of which were found to be T = l states. The recent results of
the polarized *• 2Ca (d,p) l|2Ca reaction ' ) to analogue states
shows that the T = l states are excited by P3/2 transfer below
6.5 MeV and P w 2 transfer at higher excitation energies. More-
over, almost no mixing of P3/2 and Pi/? transfer was observed1,2
in the ^Cafcijp) Ca reaction data. It is interesting to note
that the Jl= 1 transfer strength function for the T= 0 states
exhibits a rather broad minimum around 5 MeV excitation which
may indicate a similar division of the P3/5 anc* P1/2 strength.
The sum of all the proton stripping strength for a given
transferred j determines the number of proton holes in the orbit
j for the garget ground state (J ,T_) i.e.
- 129 -
• > « ?i A
(3)
where TJ(J) is the number of protons in the j orbital in the
target ground state. If isospin is conserved then the sum
of the (x,d) strength to the T = T + 1/2 states determines
the number of j neutron holes in the target,
(4)
2TA
where v(j) is the number of j neutrons in the target.
Application of equation (3) to the present data implies
that there are 6*0 proton holes in the f7/2 orbital in l*1Ca(g.s.)
that is, 75% of the simple shell model sum rule limit. The
summed f7/2 transfer strength of 2.33 for all T = l states
gives a similar result for the number of neutron holes, namely
67% of the shell model value. Moreover, the summed strength
for each spin member of the (f, , , ) 2 multiplet exhibits the
same fraction of the shell model value within the experimental
uncertainties as illustrated in table 5. The (2J+1) propor-
tionality of the summed f? <» strength to each spin member of
the ( f7/ 2)
2 multiplet implies that only the f7/2 n e u t r°n
orbital is making an appreciable contribution to the ground
state properties of '''ca, as expected within the simplest
shell model description of this state. The f?/2 transfer
- 130 -
strength for the l>oCa{3He,d)'''Sc reaction, also studied in
the present work, is concentrated in the ground state of
and this strength was observed to correspond to 6.4 proton
holes in the f7/2 shell , i.e. 80% of the shell model limit.
All these data for both 42Sc and '•1Sc are consistent with
the absolute (3He,d) spectroscopic factors for f--- transfer
being about 75% of the true value. As discussed in Sect. 3,
such an error in the absolute normalization
of spectroscopic strength, derived from an DWBA analysis is
not unexpected and thus a renormalization of the f_,_ strength
by 1/0.75 appears reasonable.
All the T=l states observed via 1= 1 transfer in the
present work are expected to be excited only by P3/2 transfer
as discussed earlier. The summed i. =1 transfer strength to
the T= 1 states is 79% of the sum rule limit for P3/2 transfer.
A 5122 keV T=0 state was excited in the 1*t)Ca (ot,d) l|2Sc reac-
tion. Excitation of this state by I =1 transfer would be
unresolved from transfer to the 5120 keV 3+ T= 1 level which
then could explain the large value for the summed Jl = 1 strength
to the 3+ T = 1 states. Although the ^CafcUp) results imply
that about 10% of the P3/2 strength is above the energy region
covered by the present work, a renormalization by the same
amount as indicated for the f7/2 strength would also bring
the P3/7 strength into good agreement with the shell model
prediction.
A renormalization of the spectroscopic strengths to the
shell morl«l value is only a reasonable procedure if the
- 131 -
amplitudes of the core-excited components in the Ca(g.s.) are
small. As discussed previously, the results of other experiments
22, 23,on wlCa, i.e. inelastic proton''*1) and alpha scattering*""), the
weakness of H=l pickup ' ) , the f?y2 neutron pickup ' )
strength ratio of hundred to one for the ground and first
excited 0 states to "*°Ca are all consistent with admixtures
of (2s-ld) core-excited configurations of only about 10% in
the "'Cafg.s.). The smallness of the admixtures of core-excited
configurations in the M1Ca(g.s.) suggested by all these data
supports the renormalization of the spectroscopic strengths.
Similar renormalizations were also required for the ItlCa(d,p)
data6-9
It is also interesting to note that in the °Ca(a,d) Sc
reaction a second 1=6 distribution was observed for a level
at 3607 keV which could be a second 7 state. We probably
observe the same level at 3601 keV with a pure £=3 distribution
with a strength of only 2% of the 7 1 strength. If this is the
second 7 state the present ( He,d) data thus suggests that2
there is almost no mixing between the (f_ ,-) and core-excited
7 states. Moreover, the recent identification of a second
6+ state in ^Ca by the 39K(a,py) reaction26) lead to a similar+ 9 26
conclusion for the 6 states ' ) . The spectroscopic factors
of the first 6 and 7 states represent thus further evidence
for the above renormalization factor.
- 132 -
The data for l-l transfer to T=0 states indicate that
the major components of the (f7/2p3/2*3+ a n d a considerable
fraction of the (f7/2p3/2*5+ c o n f ig u r a t i°ns have been identified.
The 4469 keV state is consistent with being the 4 + T=0 member
of the (f7/2P3/2^ configuration because the £<=1 transfer strength
is 89% of the shell model value for P3/5 transfer. However, at
this excitation energy an admixture of p, ,» transfer cannot be
excluded. About half of the unassigned £=1 transfer strength
could correspond to P^/2 t r a n s f e r t o T = o states. The remainder
could include some P w 2 t r a n s f e r strength since most of the p 3/ 2
transfer strength to the T=l state is already accounted for.
6. Effective two-particle matrix elements in l<2Sc.
The previous discussion suggests that all the major
,T=1 T=0components of the (f_,~) 2, (f7/2P3/2' a n d t h e
spectrum in 42Sc appear to have,been located. The energy cen-
troids, e(j,j?J), for the different spin members of the
( f 7 / ? )2 and (f7,pP^y~) two-particle multiplets can be obtained
from the f7/? and P T / 2 t r a n s ^ e r strengths if it is assumed
that all the appropriate transfer strength has been located.
The energy centroids c(j,j^JT) are given by the relation
e(j j JT) = E G. (J +j e.(JT)/ E G.(J.+j-»-J) (5)
where G.(J,+j •* J) and e.(J) are the spectroscopic strength
and excitation energy of the i state of spin J,T in " zSc
which is excited by j transfer. The experimental energy
centroids for the individual spin members of each two-particle .
multiplet, in addition to the average energy obtained by
summing the spectroscopic strength over all J values for the
- 133 -
multiplct, are presented in table 6
Effective two-body matrix element E(j,j-JT) can be
derived by taking the energy centroids with respect to the
reference energy, ER{j-j_), at which the multiplet would
occur if there was no residual two-body interaction. The
reference energy ER(f7y22) ^ s given by the ground state
binding energies
B("°Ca) - - B(U1Ca) = 3.17M MeV
In order to obtain the reference ER(f7/2P3/2' ' w e usec^
B(41Sc) the binding energy of the centroid of the P3/0 t r a n s ~
fer strength in '•'Sc derived from our "*°Ca(3He,d) data,
yielding ^^}^i /2^z/l) ~ ^-964 MeV. The effective two-body matrix
elements derived from the experimental energy centroids are given
in column 5 of table 6. The corresponding l*8Sc matrix elements
C\ ft 0
anrl thn l*2C.a r^nnlts ' ' ) for the T=l matrix elements are also
presented in table 6. The effective particle-particle two-body
matrix elements for ueSc were obtained by making a hole-
particle transformation of the particle-hole matrix elements
derived from the ^Sc data 2 7' 2 8).
The "zSc and **2Ca data result in excellent agreement for2
the T=l members of the (f7/2) multiplet. The somewhat more
attractive value for the 0 state in lt2Sc is due the fact that
the third 0 state was not identified in "2Sc. The "2Ca valuematrix element should therefore be used for2
for the
mass 42 nuclei. The overall agreement between the
- 134 -
matrix elements of u2Sc and lt8Sc is also satisfactory; the
differences are of 300 keV or less. However, it is noteworthy
that the T=l matrix elements of 1|2Sc are systematically for
each J-value about 300 keV more attractive than the corresponding
ll8Sc results. This small downward shift of the two body matrix
elements in mass-42 nuclei may be caused by the 10% core-
excited components in '''Cafg.s.). It is also worth noting that
the possible existence of minor fragments of the f7/2 strength
at high excitation energy which would lead to an upward shift
of the mass-42 values, was estimated in the ltlCa(d,ip) work of2
ref. 9 by calculating (fp) shell model wave functions using
the modified Kuo-Brown )matrix elements of McGrory ). The
upward shift was found to be insignificant for all except the
0+ state which was shifted by about 0.5 MeV.
As regards to the (f7/2p3/2^T=l c o nfi9 u r a t i o nr t n e "l2Sc
matrix elements are about 200 keV more repulsive than the 't2Ca
values. This difference probably reflects the fact that the
reference energy ER(f7/2P3/2^ i s a b o u t 300 keV larger in lt2Ca
than in ll2Sc, a result which is caused by different energies
for the P3y2 single particle states in ^Ca and u S c .
The (f7y2p )^+1and (f7/2P3/2)3*° matrix elements are
related to the corresponding pn matrix element by
EP ? = ^(E^?0 + E^? 1), which gives E|9 = -0.77 MeV in ^Sc.
The same matrix element has also been deduced from 50Sc )
data yielding a value of -0.57 MeV ) in good agreement with the
"2Sc results.
- 135 -
The strength V of the symmetry potential V_ =V(T • t)/Asym • c\
can in the absence of pairing forces be determined directlyfrom the splitting of the e(T=TA+l/2) and e(T=TA~l/2) energy
2
centroids. However, when pairing is present as for the {£-. ,~)
configuration the isospin dependence of the two-body interaction
is slightly more complicated. From the formula of de.Shalit and
32 2Talmi ) for the average two-body energy in a j configurationof good seniority, and reduced isospin, we deduce that
V
A
- 2E(j201)
T=l 2where E J > Q = -0.26 MeV for (f7/2)
a n d i s calculated for J>0.
Together with the values of -2.89 MeV for E(f^ 2 01) and -1.51
T=0MeV for E A T T this results in a value of 80 MeV for the strength
A1JLI
of the symmetry potential. This value is close to the value of
80 and 89 MeV derived from ( He,d) experiments in the Ca-Ni
region ) and the "8Sc data but is somewhat smaller than the
value of 101 MeV obtained from neutron pick-up ) on ulCa.
7. Comparison with the Coexistence Model.
It was shown in references 7 and 9 that the spectro-
scopic factors for f7/2 transfer to the lowest 7 T=l states,
seen in the "*'ca (d,p) l(2Ca reaction, are well reproduced by the
coexistence model. The coexistence model calculation by Flowers2
and Skouras ) for T=0 states have recently been modified by
34Thomas and Skouras ). Spectroscopic factors for the low lying
T=0 states were calculated using the (fp) part of these wave-
functions assuming that the ground state of "* lCa has a structure
which is 90% of a t~. ,- neutron coupled to a closed "0Ca core.
- 136 -
The core-excited components are expected to make only a
small contribution to the transfer strength for states with
large spectroscopic factors for f7y2 o r P3/2 t r a n s^ e r- T n e
calculated spectroscopic factors for the strongly excited
T=0 states are compared with the renormalized experimental values
in table 7. The coexistence model predicts that the admixture of
core-excited configurations is small for the states listed in
2table 7, i.e. they are predominately either ( 7/9 or a mixture
2of (f7/o' anc* ^7/2^3/2^ configurations. The agreement between
the coexistence model and experimental spectroscopic factors
for these low lying T=0 states is very good for the 5 and
7 states and reasonably good for the 1 and 3 states. We can
thus conclude that the coexistence model also gives a fairly
good description of the strongly excited low-lying T=0 states
in mass 42 nuclei.
- 137 -
REFERENCES
1) W.J. Gerace and A.M. Green, Nucl. Phys. ASH (1967) 110.
2) B.H. Flowers and L.D. Skouras, Nucl. Phys. A136 (1969) 353.
3) C.W. Towsley, D. Cline and R.N. Horoshko, Phys. Rev. Lett.
££ (1972) 368; Nucl. Phys. A204 (1973 574.
4) R. Sherr, T.S. Bhatia, D. Cline and J.J. Schwartz, Ann.
of Phys. Vol. 615 (1971) 548.
5) F. Puhlhofer, Nucl. Phys. A116 (1968) 516.
6) 0. Hansen, J.R. Lien, 0. Nathan, A. Sperduto and P.O. Tjjzfm,
Nucl. Phys. A243 (1975) 100.
7) C. Ellegaard, J.R. Lien, O. Nathan, F. Ingebretsen, E. Osnes,
P.O. Tj0m, 0. Hansen and R. Stock, Phys. Lett. 40B (1972) 641.
8) P.B. Void, D. Cline, R.N. Boyd, H. Clement, W.P. Alford
and J.A. Kuehner, Phys. Lett. 72B (1978) 311.
9) P.B. Void, D. Cline, R.N. Boyd, H. Clement, W.P. Alford
and J.A. Kuehner, Nucl. Phys. (1978).
10) D. Cline, Proc. of the International Conference on the
Physics of Medium-Light Nuclei, Ed. Blasi, Florence, Italy
(1977).
11) R. Bock, P. David, H. Duhm, II. Hetele, V. Lynen and R. Stock,
Nucl. Phys. A9_2 (1967) 539.
12) O. Ilansen, T.J. Mulligan and D.J. Pullen, Nucl. Phys. A16 7
(1971) 1.
13) G.R. Satchler, D. Armstrong, A. Blair, E. Flynn, R. Philpott
and W. Pinkston, Phys. Rev. 182 (1969) 1141.
14) R.H. Bassel, Phys. Rev. 149 (1966) 791.
- 138 -
15) J.N. Craig, N.S. Wall and R.H. Bassel, Phys. Rev. Lett.
36, (1976) 656.
16) R. Bock, H. Duhm and R. Stock, Phys. Lett. L8 (1965) 61.
17) P.D. Kunz, University of Colorado/ unpublished.
18) R. Hartmann, H. Grawe and K. Handler, Nucl. Phys. A203
(1973) 401.
19) P.M. Endt and C. Van der Leun, Nucl. Phys. A214 (1973) 1.
20) H. Nann, W.S. Chien, A. Saha and B.H. Wildenthal, Nucl.
phys. A292 (1977) 195.
21) N.R. Roberson and G. Van Middelkoop, Nucl. Phys. A176 (1971)
577.
22) P.B. Void, D. Cline, M.J.A. de Voigt and A. Sperduto, Nucl.
Phys. A292 (1977) 107.
23) M.J.A. de Voigt, D. Cline and R.N. Horoshko, Phys. Rev.
CIO (1974) 1798.
24) D. Cline, M.J.A. de Voigt, P.B. Void, O. Hansen, O. Nathan
and D. Sinclair, Nucl. Phys. A233 (1974) 91.
25) R.R. Betts, C. Gaarde, O. Hansen, J.S. Larsen and S.Y. Van
der Werf, Nucl. Phys. A253 (1975) 380.
26) E. Bitterwolf, H. R^pke and P. Betz, J. Phys. G: Nucl.
27) H. Ohnuma, J.R. Erskine, J.A. Nolen Jr., J,P. Schiffer
and N. Williams, Phys. Rev. Cl (1970) 496
28) J.P. Schiffer, In Proc. Top. Conf. on the Structure of
lf7y2 nuclei, Padova, 1971, Ed. Ricci, p. 37 ? J.P. Schiffer,
in Proc. of the Symp. on the Two-Body Force in Nuclei,
Michigan, 1971, edited by S.M. Austin and G.M. Crowley,
p. 205.
- 139
29) T.T.S. Kuo and G.E. Brown, Nucl. Phys. A114 (1968) 241.
30) J.B. McGrory, Phys. Rev. C8 (1973) 693.
31) H. Ohnuma, J.R. Erskine, J.A. Nolen Jr., J.P. Schiffer
and R.G. Roos, Phys. Rev. r77_ (1969) 1965.
32) A. de Shalit and I. Talmi, Nuclear Shell Theory, (Academic
Press, New York, 1963).
33) 0. Hansen and 0. Nathan, Phys. Rev. Lett. 22 (1971) 1810.
34) M.F. Thomas and L.D. Skouras, J. Phys. A: Math., Nucl. Gen.,
Vol. 6 (1973) 1763.
h
TABLE 1. Experimental results from the41 3 4?
Ca( He,d) reaction.
Group
0
12
3
45
6
7
8
9
10
11
12
13
141516
17
Ex<
present )
0
615
1490
1510
1585
1843
1872
1886
2185
2220
2265
2293
2388C)
2433
2452
2486
(2533)
2648
keV)
ref.
0
611.
617.
1490.
1511 .
1586.
1846
1873.
1888.
2187.
2222.
2270.
2297
2389
2455
2488.
2535
2586.
2650.
2669
IS)
2±0
5+1
7+0
1±1
4 + 0
±2
6±0
9±0
9±0
6+0
0+0
+ 2
±5
±2
2+1
±2
8+1
3 + 1
+ 5
.2
.1
.4
.0
.3
.8
.6
.6
.6
.8
.0
.7
ref.IS)
0+;l1 +
7 +
<2+,3+)
(5+,>6)
2+;l
S21(1-3)
1 +
(^2,3+)
T
adopted )
0+;l
1 +
7 +
3 +
5+
2+;l
0+;l1 +
2 + ,3 +
( 2 - 5 ) +
1 +
( l - 7 ) +
d.3 + >
2+-5 +
2 + ;l
/ d a \ M a x
vcTn/C M
(mb/sr)
0.17
0.56
2.72
2.33
5.29
1.19
0.024
0.046
0.059
0.51
0.20
0.058
0.10
0.27
0.21
0.045
0.70
0.017
0.13
SL-,1
111
11
(1(1
+
3
3
3+
+
+
3
3+
+
(3)+
) +
1
3
*2
3
3
3
3
3
3)3
G( J1 A
0
0
0
00
0
0
0
f i -+J)
.049
.14
.026
.016
.003
.0017
.0023
.007
Gl J
0
0
1000
0000
000
0
+ 1 -4A J2
.084
.25
.21
.43
.66
.21
.015
.021
.029
.032
.016
.019
.066
.20
•J)
f )
f )
O
t
- 1 4 1 -
XrdSD•o
•c
o
1-5+csj
•n+
o
r-"•n+<
G(J
CM
+r-H
2* ^^• MO 0)
•M
0
id
en.
q-(US-
• * — *
en«
0)i-
+JG0)u)
pre
dZ
CMCPiO•o
Ho•
o
ro+rH
VOVO
O
in
+
rocr\cCM
COr-t
i-l
VO•o
CM
O•
O
ro
+i-H
COr-t
ro
•-I
+
rAll
in 3"• •
r—i 1—1
+1 +1in en
VO CMi-1 roCO COCM CM
,—1COCM
en
o
oo
ro»+1in
1^^*COCM**""*
VO^ 1
COCM
OCM
o["*.
oo
•r—\
+1H
Or-lCMCM
,_!i-H
enCM
CM
in,—1+i
inkOo>CM
00
O•
©
CM
i—|
roo
l~if)1
1CM
CMenCM
CMCM
00rHO•o
o
in
o
1
1ro
HCM
ro
roCM
voCM»o
CMCM•
O
ro+
00
VD
+in
IOAH+in
vo•
+1renCOoro
00COoro
• < *
<N
rHVDoo
VO
,—tro
inCM
CM
o•o
oot
o
ro+H
VOroo
vovoi—4
ro
voCM
voo«H
ro
*ro
H
+vo
+]
O
CMro
in
CMro
CM
voino»o
ro
CM
O
+r»I•-I
in
CM00CMro
00CMro
ooCM
ini-i•
o
CO
oo•o
ro+
voooo
H
+
in+i
CM
roro
CMCMroro
O\<N
enHo
in in
+l +l
o vo^* VOro roro ro
oCOco
o
ro
00•-I»
O
inH•o
ro+H
Oin
r
ro
VII
r-l•
rH
r~CMenroro
oenroro
^
ro
O
o«o
"H
Q.cso
^H
CM
VO^ t
^ *
ro
CMro
ro
o•o
HO
o•o
CM
+o
• > *
O
l"
1ro
CMrH+1
CO
"3*ro
00vo*3*
ro
roro
voO
o•o
I-l
CO•-•
o
*—*inICM
roen*ro
I1
ro
oo
CMr-linro
inro
r-oo
+r-IrH
inH+1
inCMinro
enCMinro
voro
i
o
oo r co N H<N VO f-> r-t 00O O O O O
O O O O O
inr-o
ro
m
©
tc
o
ro CMo ooo o
oo
00
o o
in<N roo o
voHoo
+r—
ro ro
+ + + 1-1
ro
+
VO
O rH
o o
CO H CTl OCO rH H ^
O O O
in
CM
O O
O O
r- CM
en coroin
o o O O r-t
00
H rH o ro
o o o o
(U+)a.o•s
++ «I + +
r-l rg ro
(U
+Ln inI ICN CM
in
CM
a)u
COVII
a)
H-l
o+100
00VOCO
"3*+1
O\f-r-ro
m.f]
r-o00ro
in+i
o0000ro
rH+1in
roroa\ro
<a0)Q)Ua
r »
inro
oovoro
VD00VOro
in «» CMi~H Lrt O ^r** t*** r**ro ro ro
inin00ro
voooro
voo00ro
oro
ro
C M •"3'CM •>«•O O
•*r in CM vo <no •*»• vo r- ooCM CM CM CM CM
3O
u oI— ooco ro
O\ O r-4 CM
CM••->
•"3+
r~3
- 143 -
ooo
"* <n in oo ui r—I D O ^ C V I l D C O C M r — LDO * J - r — O O O O O
o o o o o o o o
oCOo
r-
o
o o o
O O C M i — C )O S U) CM
00
O O O O C V I r -
m <3" i— CO « * CM CM
O O O O O O O H O O O O O i - H - t f
-a01
+->Q.o-a
mi
CM
UJi
CM
• •
CM
i ni
CM
LT)1
CM
0)
O• J 3 " * 0 0 O l O O C M U 3 0 0 O C M
r0)s-a.
O •S- O
Group
No.
7778
79
80
81
82
83
84
85
86
87
88
89
Ex(keV)
present a)
5326
5352
5370
5380
5434
5473
5520
5572
5633
5651
ref.19)
5771
5865
5964
5820 ±15
J\T
ref.19)
0+;l
adopted )
ir=+
5+;l
(da)\dttJC.M.(mb/sr)
0.45
0.67
0.33
0.69
0.46
0.18
0.35
0.76
1.69
4.18
0.47
0.82
0.77
(1)1
1
(1)
(1)
(1)
0.08
0.21
0.48
0.075
0.16
0.15
a) All + 5 keV.
b) The uncommented assignments of column 5 are based on the results of ref.19 and the present work; seetext for details.
c) Doublet.d) See text.e) Ref.5.
f) From 4 1Ca( 3He,dy); see sect.3.
Table
Optical-model parameters
Part-icle
3He
d
P
V(MeV)
165
106
a)
1
1
1
.14
.05
.20
Fm)1
i 0
0
; 0i
a
.723
.85
.65
W
20.
0
0
2
w1
0
46.8
! o
1
1
1
r'
.16
.496
0
a'
0.
0.
0
o
81
63
Vso
0
13.0
0
r
0
0
0
so
.9
a
0
0
0
so
.6
Thomas
0
0
25
Ref.
11,12
13
a) Adjusted to give a binding equal to the experimental separationenergy.For unbound states the depth was calculated using abinding of 100 keV.
Table
40 3 41Ca( He,d) Sc speotroscopic strengths
• 0
i 1
2
3
7/2
3/2"
(3/2,5/2)+
(3/2,5/2)"
1720
2096
2417
1
2
1
3
0
0
.6
.5 a )
. 4 b )
38
3
3
.2
.0
.0
3
0
0
.6
.4
.4
a) 30% statistical uncertainty.
b) Poor DW fit.
- 146 -
Table *t
4 9 •* no
Analogue states in Ca and Sc
42Sc "CaE (keV)X
0
1587
1872
2486
2814
3245
3322
3446
3754
4548
4727
4827
5084
5120
5651
Ex(keV)
Ref .6
0
1525
1836
2423
2752
3187
3251
3295
3389
3650
44484456
4760
4869
5020
52105474 h)
5778
J ; T
Ref.19)
o+;i
2 +;l 0
Q +;l
2 + ;i
4+;l 0
6+;l
4 +;l 0
0+;l '
2 +;l 0
2 +;l 0
2 +;l 0
2+;l 0
4+;l 0
3+;lc) 0
s+;id) o
V1
.084
.074
.014
.018
.007
.24 a )
.16
.25
.33
.89
.70
£P
1
0
0
0
1
1
0
0
= 3
.3
.66
.24
.64
.1
.3
.26
.04
AE
(keV)
0.05
0.04
0.003
0.01
0.054
.21a)
0.22
0.17
0.29
0.60
0.18
0.80
1.0
0.62
0.23
0.68
1.1
l.S
0.27
0.08
0.01
0.14
(0.10)
(0,16)
(0.11)
e)
e)
e)
0
62
36
63
62
58
71
57
10497
-33
-42
+ 64
-90
-96-127
9 T +1
a) The strength defined as 2J +l S ^
42,b) If £p=l is assumed for the 5380 keV level in Sc,
a spectroscopic factor of 0.2 0 is obtained.
c) Refs. 8 and 9.
d) Refs. 6, 8 and 9.
e) A bracket signifies that £ =3 is possible but not required.
- 147 -
Table 5
42The summed fp strength in Sc
Configu- J77
ration
(f 7 / 2)
f7/2P3/2
f 7 / 2 P
•
2 +
1 +
2 +
3 +
4 +
5 +
6 +
7 +
unknown
All
All
All
2 +
3 +
4+
5 +
All
3 +
5 +
unknown
T
1
0
1
0
1
0
1
0
0
0
1
All
1
1
1
1
1
0
0
0
G
0
0
0
0
0
0
1
1
0
3
2
6
0.
0.
0.
0.
1.
0.
0.
1.
(JA+j->
.10
.34
.42
.7.5
.76
.92(1.
.06
21
..39a)(
62
33
03 •
21 C )
39
49c)
48
57
30
38
50
J) S(J. T,+j W T )
1.58
1.79
1.34
1.64
1.34
09)b1.34 (1.59) b )
1.30
1.30
0.22) a' b )
1.61
1.34
1.50
0.66
0.89
0.88
0.70
0.79
0.70
0.56
S/Sshell model
0.79
0.89
0.67
0.86
0.67
0.67(0.80)b)
0.65
0.65
0.80
0.67
0.75
0.66
0.89
0.88
0.70
0.79
0.70
0.56
a) I - 3 strength up to 4 MeV is included.
b) Values obtained assuming 5 for the 2793 and 3792 keV states.
c) Spectroscopic strength calculated assuming the strength of the 4548 keV
level split with 67% to 4+ and 3 3% to 2+ as obtained for the 42Cao
analogue states " ) .
- 148 -
Table- 6.
Effective two particle matrix elements in
Sc compared to those of Ca and Sc.
Configu-
rationT e(J1J2JT)
(MeV) " So
E(J1J2JT)(MeV)
4 2Ca C )
(f7/2)
f7/2P3/2
3
4 +
5 +
7 +
All
All
2 +
3 +
1
0
1
0
1
0
1
0
0
1
1
1
1
0
0.28
1.39
2.07
2.27
2.91
-2.89
-1.78
-1.10
-0.9
-0.26
-2.13
-2.11
-0.81
-1.04
0.06
) d )1.96(2.16)"' -1.22 (-1.01)d)-0.87
3.25
0.62
1.66
2.80
4.29
5.12
4.81
5.65
3.23
e )
0.07
-2.56
-0.37
-0.68
0-0
0
- 1
.16
.15
. 6 9
.74
0.28
-2.28
-1.59
-0.07
-2.59
-0.94
-0.26
0.08
-0.31
-2.57
-0.99
-0.25
0.08
-0.32
-0.86
-0.08
-0.25
0.49
a) Ref. 28
b) Ref. 6
c) Ref. g
(1) Assumed 5+ for the 2793 and 3792 kcV states.
c) T=0 states of unknown spins were also incorboratcd in the calculation of
these energies.
- 149 -
Table 7.
Comparison of experimental and calculated spectroscopic factors
41 3 42for low-lying T=0 states strongly excited in the Ca( He,d) Sc
reaction.
S(7/2,j+l/2-»-JO)
Theorya)
•7/2P3/2
Experimentb)
'7/2 p3/2
0 .92
1.0
0 . 4 3
1.16
0 .76
1.71
0
0
0
0
.35
. 2 1
.36
.58
1.76
1 .31
0 .55
1.28
0 . 5 1
1.61
0 .15
0 .46
0 .27
0 .43
a) Spectroscopic factors calculated using the coexistence model
41wave functions of ref.34 in addition to assuming that Ca(g.s)
40is a 90% f.7/0 neutron coupled to a closed Ca core.
b) Spectroscopic factors are renormalized by 1/0.75.
- 150 -
Figure captions.
Fig. 1. Deuteron spectrum at 12.5 laboratory angle. The group
numbers refer to table 1.
Fig. 2.a.b.c,d. Experimental angular distributions and DWBA
predictions in the c m . system. The normalization of the DWBA
curves corresponds to the experimental spectroscopic strengths of
table 1. It is worth noting that some of the weak transitions
and angular distributions with few data points are shown with
DWBA curves in the figure, although no Jl-assignments have been
extracted for the transitions. These DWBA curves signify a
possible fit to the data but it is not unambiguous, and the
indicated A-values are therefore not quoted in table 1.
Fig. 3. Experimental angular distributions and DWBA predictions
for the Ca( He,d) Sc reaction. The normalization of the DWBA
curves corresponds to the experimental spectroscopic strengths of
table 3.
Fig. U. Comparison of spectroscopic strengths for stripping to
4-1 3 U?the T=l states observed in the present Ca( He,d) Sc reaction and
in the 41Ca(d,p)I+2Ca reaction.6).
Fig- 5. Experimental spectroscopic strengths for proton stripping
to T=0 states in 2Sc. The ordinate is excitation energy in42Sc.
- 151 -
a.
o
< JD
to
ff
O ii 35 UJCD
S
- 8
o
n o jo
d3d SMDVdi JO
Fig. 1
o
EXCITATION ENERGY (MeV)
400,3.5 55
350H-
300
a 250ma
CKS
S g^ 200
a.H O
a 150— UIO m
•§ I•£ 2 loo
50
E=20.0 MeV
50
2 5 2 6 2 7 2 8 2 9 3 0 3 1 3 2 3 3 3 4 3 5 3 6 7 ?
DISTANCE ALONG PLATE (cm)
38 39 40 41 42 43 44
mto
- 153 -
10 - I I I i I -
I—1—1—I—1Ev-2293keV_
o1
d2
- i i
=~ i i
i i 1 -
E,=28«6keV I
V/1 1
-
10 30 50 10 30 50 10 30 50 10 30 50G C M (degrees)
Pig. 2. a
- 154
10 50
- 155 -
I/I
.a
8
10 50
r- 156 -
.0°
o-
1
- 1
1
* * }
1
1 1 1
Ex = 5O28keV ;
- I p = f 3 z
\
1 l'
-_
1 -
. 1
•
1
\J
I
i i
E, = 5K1^=1
A
i i
i .
-
i
l -
n
1 I i rE« = 5434keVIp = 1
id1
IOU
I I I J J.
10 30 50 10 30 50 10 30
e C M ( degrees)
50
Fig. 2d
- 157 -
006-3353
.Q
I
0 10 20 30 40 50 60 70
0c.m. (degrees)
10
40Ca(3He,d)*ScE,"l720HeV
0 10 20 30 40 50 60 70
0cm. (degrees)
|
cj
1
4OCo(3He.d)4lSc
Ex - 2096 KeV
CJ
1 0.1
40Ca{3He.d)4lSc
E, ' 2417 keV
£?=\
0 . 10 20 30 40 50 60 70
^trn. (degrees)
10 20 30 40 50 60 70
0cm (degrees)
Fig. 3
- 158 -
> ^
O
G(7/2,l/2 + j ,
— pO enr
w+.
OJ .
I
~ * +
* +
ro
5
.ro
+
+
•-1**+Oi
++
f>0
ro"CL
p _ OJ ro _en 'o b #o b o b b
i i i i'U ,
ro-
r \ > -
I I
ro
. r o
-.ro
I IOJ
fD"d.
O -
I I
OJ
o
+
Oi
en00
Fig. 4
- 159 -
G(7/2,l/2+j,l/2-*J,0)ooOf(it
m><OJ
f\>
rv) ben en
ii
O
8 2 en b1 I
en
cn
+ T3II
OJ
Fig. 5
- 160 -
4. Summary and conclusions.
40 42 42The structure of energy levels in ' Ca and Sc
has been studied using inelastic proton scattering and
one-nucleon stripping and pick-up transfer reactions on
a 41Ca target. Application of the monopole sum rules to these
data hass given the following information on the proper-
41ties of the Ca ground state wave function} i) the
41Ca (g.s.) looks very much like an f7y2 neutron coupled
40to the Ca (g.s.) core, ii) The core-excited component
41of the Ca (g.s.) is determined to be 10% or less.
41Excited posity parity states in Ca were found to
exhibit similar weak-coupling properties in that the summed
41i=3 and £=5 strengths in inelastic proton scattering on Ca
are consistent with a coupling of the f7/2 neutron to the
- - 40
collective 3 and 5 states of the Ca core. However, in
both cases, the highest spin member of the multiplets have
reduced transition strengths, which were interpreted as due
to the influence of blocking in the microscopic structure of
these states.
It was inferred from the partial and total monopole42
sum rule results of the one-nucleon stripping data to Ca42and Sc that the main constituents of the spectroscopic
strength leading to the (f_ ,„) , (f7/2' •7/2P3/2> and
(f configurations have been identified. This was
used to deduce the effective two-particle matrix elements
42 42for these configurations. The Sc and Ca data result in
excellent agreement for the T=l members of the (f7/2 multi-
- 161 -
plet while the (f_,,,p,/0) matrix elements derived from
42 1
the Sc data are about 0.2 MeV more repulsive than those42
obtained from the Ca data. The modified Kuo-Brown
matrix elements of McGrory agree with the present values to
within a few hundred keV.
The ^7/2^3/2^ matrix elements derived from the present3( He,a) data were compared to the corresponding values
obtained from one-nucleon stripping to mass 34 nuclei. With
the exception of the 5 ,T=0 effective matrix element where
the disagreement between the results is significant, the two
sets of matrix elements are in very good agreement. The
40Ca values are also well reproduced by calculations using
the modified surface delta interaction.
The experimental spectroscopic factors to both the
2T=0 and T=l states of the (f7/2* multiplet are in
remarkable good agreement with the predicted values of
the coexistence model considering the simplicity of this
model.