Study of nuclei in the vicinity of the "Island of Inversion" through fusion-evaporation reaction
ABSTRACT We report the first observation of high-spin states in nuclei in the vicinity of the "island of inversion", populated via the 18O+18O fusion reaction at an incident beam energy of 34 MeV. The fusion reaction mechanism circumvents the limitations of non-equilibrated reactions used to populate these nuclei. Detailed spin-parity measurements in these difficult to populate nuclei have been possible from the observed coincidence anisotropy and the linear polarization measurements. The spectroscopy of 33,34P and 33S is presented in detail along with the results of calculations within the shell model framework.
arXiv:0904.4777v1 [nucl-ex] 30 Apr 2009
Study of nuclei in the vicinity of the “Island of Inversion”
through fusion-evaporation reaction
R. Chakrabarti,1S. Mukhopadhyay,1, ∗Krishichayan,1, †A. Chakraborty,1, ‡
A. Ghosh,1S. Ray,1S.S. Ghugre,1A.K. Sinha,1L. Chaturvedi,2
A.Y. Deo,3I. Mazumdar,3P.K. Joshi,3R. Palit,3Z. Naik,3S. Kumar,4
N. Madhavan,5R.P. Singh,5S. Muralithar,5B.K. Yogi,6and U. Garg7
1UGC-DAE Consortium for Scientific Research,
Kolkata Centre, Kolkata 700098, INDIA
2Department of Physics, Pt. Ravishankar University, Raipur 492010, INDIA
3Tata Institute of Fundamental Research, Mumbai 400005, INDIA
4Department of Physics and Astrophysics,
University of Delhi, Delhi 110007, INDIA
5Inter University Accelerator Centre,
Aruna Asaf Ali Marg, New Delhi 110067, INDIA
6Department of Physics, Govt. College, Kota 324009, INDIA
7Department of Physics, University of Notre Dame, Notre Dame, IN 46556, USA
(Dated: April 30, 2009)
We report the first observation of high-spin states in nuclei in the vicinity of the “island of
inversion”, populated via the18O+18O fusion reaction at an incident beam energy of 34 MeV.
The fusion reaction mechanism circumvents the limitations of non-equilibrated reactions used to
populate these nuclei. Detailed spin-parity measurements in these difficult to populate nuclei have
been possible from the observed coincidence anisotropy and the linear polarization measurements.
The spectroscopy of33,34P and33S is presented in detail along with the results of calculations
within the shell model framework.
PACS numbers: 21.20.Lv, 23.20.En, 23.20.Gq, 21.60.Cs, 27.30.+t
∗Current affiliation: Department of Physics, Mississippi State University, Mississippi State, MS 39762, USA.
†Current affiliation: Cyclotron Institute, Texas A&M University, College Station, Texas 77843, USA
‡Current affiliation: Department of Physics, Krishnath College, Behrampore 742101, INDIA.
Neutron-rich nuclei are currently of great interest as they exhibit structural properties
very different from nuclei near the β-stability line. Evolving shell gaps and disappearance
of magic numbers seen in neutron-rich nuclei challenge the conventional shell model theory.
The “island of inversion” comprised of neutron-rich isotopes of Mg, Na, and Ne with N∼20 is
one of the best examples of such unexpected structure changes observed in nuclei with large
neutron-proton asymmetry. Investigations into the extent of this island and the transition
region around it will lead to a greater understanding of the evolution of the structure of the
Nuclei in and around the “island of inversion” have in general been studied using trans-
fer/deep inelastic reaction [1–5], or β-decay  or heavy ion inelastic scattering or deuteron
inelastic scattering . However such non-equilibrated reactions have certain limitations, like
contamination from dominant fusion-evaporation channels, low production cross-sections,
low spin population and coincident emissions from the binary partner. The above limita-
tions can be circumvented to a very large degree by using fusion-evaporation reactions with
a neutron-rich target and a neutron-rich projectile. In this paper we present the results of
a spectroscopic investigation of nuclei in the vicinity of the “island of inversion”(33,34P and
33S), populated via the fusion-evaporation reaction.
II. EXPERIMENTAL METHOD
33,34P and33S nuclei were populated utilizing the18O+18O reaction. The18O beam at
an incident energy of 34 MeV was provided by the 14 UD BARC-TIFR Pelletron facility at
TIFR, Mumbai. The choice of the incident energy was determined by the earlier reported
excitation function measurements  which indicate a considerable cross-section for these
nuclei at this incident energy. The neutron-rich18O target was prepared by heating a 50
mg/cm2-thick Ta foil in an atmosphere of enriched Oxygen to form Ta2O5. The total18O
equivalent thickness was estimated to be 1.6 mg/cm2on both sides of the Ta foil. The de-
exciting γ rays were detected by an array of 7 Compton-suppressed Clover detectors placed
at ∼30◦, ∼60◦, ∼90◦, ∼120◦and ∼150◦with respect to the beam direction in the median
plane. An event was recorded when at least 2 Clovers fired in coincidence. A total of ∼1 bil-
lion such γ-γ coincidences were recorded. The data were recorded using CAMAC based data
acquisition system LAMPS  and analyzed using IUCSORT [10–12] and RADWARE 
software packages. The data were pre-sorted to correct for any on-line drifts to ensure that
there were relatively no gain changes between any two list mode data sets within the exper-
iment and, then, were precisely gain matched to ensure that data from each detector had a
constant energy dispersion. The energy calibration was performed using radioactive sources
152Eu and133Ba and beam-off radioactivity data. The data were sorted into symmetric and
asymmetric γ-γ matrices. The genetic correlation between the de-exciting γ rays was estab-
lished from the symmetric γ-γ matrix after background subtraction and efficiency correction.
The asymmetric matrices were used to assign the spin and parity for the observed levels on
the basis of angular correlation and the linear polarization measurements, as described in
the next section.
A.Determination of spins and parities
The multipolarity assignments have been performed from the observed coincidence angu-
lar correlations. Assuming pure (stretched) transition, the coincidence intensity anisotropy
can be used to distinguish between ∆J = 1 and ∆J = 2 transitions. A qualitative assign-
ment for the multipolarity of the γ-transition from the angular correlation measurements
is obtained following the procedure detailed in Ref. . The experimental RDCO in the
present work is defined as:
RDCO=Iγ1(at θ gated by γ2at 900)
Iγ1(at 900gated by γ2at θ)
where θ is 300and 1500. When the gating transition (γ2) is a stretched quadrupole transi-
tion, RDCO∼1 for a pure quadrupole (γ1) and ∼0.5 for stretched pure dipole transition(γ1).
Similarly, a gate on a dipole transition would result in RDCO ∼2 for a pure quadrupole
and ∼1 for a pure dipole transition. These intensity ratios were obtained from the angle-
dependent γ-γ matrices assuming stretched transition for the gates and after incorporating
necessary efficiency corrections. The experimental RDCOvalues determined from quadrupole
and dipole gates have been plotted for several transitions belonging to33,34P,33,34S in Fig. 1
and Fig. 2 respectively. The present statistics did not permit us to extend these measure-
ments to the weak transitions. As seen from the figures, it is possible to distinguish between
∆J = 1, and ∆J = 2 transition following the above procedure. For mixed transitions the plot
essentially provides a qualitative way of determining the dominant multipolarity considering
the proximity of the RDCOvalue to the ∆J = 1 or the ∆J = 2 line.
The angular correlation measurement is not sensitive to the electric or magnetic character
of the radiation. The information on this was obtained from the linear polarization mea-
surements. Clover detectors have an advantage over conventional single crystal detectors as
they allow such measurements to be made.
The angular distribution of linearly polarized gamma rays from an axially oriented en-
semble of nuclei is given by [15, 16]
AλPλ(cosθ) + 2Aλ2P(2)
λ2(cosθ) cos 2ψ
where Bλare orientation tensors describing the degree of orientation of the parent nucleus
and Uλare deorientation coefficients. Pλare the ordinary Legendre polynomials and P(2)
the unnormalized associated Legendre polynomials. Aλare angular distribution coefficients
which depend on the spin of the initial and final state and the multipolarity of the γ-
transition. The coefficients Aλ2depend on the electromagnetic character of the radiation .
θ is the angle that the electric vector of the emitted quanta makes with the orientation axis
and ψ is the angle between the electric vector E of the emitted quanta and the reaction
plane (Fig. 3).
The degree of linear polarization Pθ of a γ ray is defined as the difference between the
intensities of the radiation presenting an electric vector parallel to the reaction plane (ψ =
00) and that with an electric vector perpendicular to the plane (ψ = 900) [17–19]:
P(θ) =W(θ,ψ = 0) − W(θ,ψ = π/2)
W(θ,ψ = 0) + W(θ,ψ = π/2)
where the normalization is such that -1 ≤ P(θ) ≤ +1.
P(θ)= 0 for an unpolarized γ-ray and has a maximum value at θ = 900,
Pcal(900) = ±3a2H2− 7.5a4H4
2 − a2+ 0.75a4
where a2and a4are the angular distribution coefficients and the H2and H4coefficients
depend on the initial and final spin and the mixing ratio, δ [20, 21].
Experimentally linear polarization of gamma rays was detected and measured through
Compton scattering . The differential Compton scattering cross-section is given by 
− 2 sin2ν cos2χ
where r0is the classical electron radius, ν is the Compton scattering angle with respect
to the direction of the incident γ ray, and χ is the angle between the electric vector E of the
primary radiation and the scattering plane defined by the direction of the incident and the
scattered photons (Fig. 3). This cross-section is relatively high and polarization sensitive
for a wide photon energy range. Maximum scattering occurs at χ = 900.
The Clover detectors used in the experiment acted effectively as Compton polarimeters.
The detectors placed at ∼900were particularly useful since polarization is maximum in that
direction. Each crystal of a Clover detector acts as a scatterer and the two adjacent crystals
act as the absorbers. The asymmetry between the perpendicular and parallel scattering
with respect to the reaction plane distinguishes between electric and magnetic transitions.
The experimental asymmetry or ∆IPDCO (IPDCO stands for “Integrated Polarizational-
Directional Correlation from oriented nuclei”) at 900between perpendicular and parallel
coincidence rates is defined  as
where N⊥and N? are the number of photons with a given energy scattered along the
direction perpendicular and parallel to the reaction plane, respectively, in the detectors
placed at ∼900and in coincidence with another photon detected in at least one other detector
in the array. This is called an integrated PDCO because the polarization of one γ quantum
is measured and the information is integrated over all the possible emission directions of the
accompanying coincident radiation.
“a” denotes the correction due to the asymmetry in response of the clover segments. This
factor is energy dependent (a = a0+a1Eγ), and is determined using a radioactive source
(having no spin alignment) under similar conditions. This correction is defined as [18, 19]
The values for a0and a1for the present experimental setup were 1.00007(0.00698) and
∆IPDCOvalues were evaluated from asymmetric γ-γ matrices whose one axis corresponds
to the perpendicular or parallel scattered events in the clovers at ∼900and the other axis
corresponds to the total energy recorded in any of the other detectors. Gates were put on the
full energy peaks of the perpendicular and parallel matrices to obtain spectra representing
either perpendicular or parallel scattering respectively, and from these, the N⊥ and N?
values were obtained for each transition. Fig. 4 is a representative background subtracted
difference spectrum of perpendicular and parallel gates. The positive peaks indicate electric
transitions whereas negative peaks indicate magnetic transitions.
The linear polarization is related to the asymmetry by the polarization sensitivity Q(Eγ)
as [18, 19]
∆IPDCO= PQ(Eγ) (8)
Q(Eγ) is dependent on the incident gamma-ray energy and the geometry of the polarime-
ter and its values, were obtained for a similar setup as reported by Palit et al. . The
theoretical polarizations were determined using eq.(4) by calculating the angular distribu-
tion coefficients a2, a4and the H2, H4functions for each value of the multipole mixing ratio δ
using the formalism given in Refs. [20, 21, 23]. The theoretical ∆IPDCOvalues were obtained
from the theoretical polarization values using eq.(8), and are plotted along with the experi-
mentally obtained ∆IPDCOvalues as a function of Eγin Fig. 5 for several strong transitions
in34S,33P and33S. The results show a good agreement between the theoretical and experi-
mental values. At a given energy, a positive value of the asymmetry parameter indicates an
electric transition, a negative value indicates a magnetic transition and a near-zero value is
indicative of an admixture.
The calculation of theoretical polarization requires two inputs, viz., the width of the
m-state distribution and the mixing ratios. Polarization depends significantly on the distri-
bution of the nuclear state over its magnetic substates. When the alignment is partial, the
angular distribution coefficients for complete alignment  have to be multiplied with the
attenuation coefficients as formulated by Der Mateosian and Sunyar , which depend on
the factor σ/J where σ is the width of the distribution of the m-states, assuming a Gaus-
sian distribution . The choice of σ/J was made following the simultaneous analysis of
the RDCOand ∆IPDCOvalues for several transitions of known spin and parity. Fig. 6 and
Fig. 7 depict the determination of σ/J for one such transition, the 1066-keV [5−→ 3−, δ
= 0] belonging to34S. This nucleus has been populated with substantial cross-section in
the present experiment and its level scheme has been extensively reported earlier by Mason
et al. . The theoretical RDCO values were obtained using the code ANGCOR  as
a function of σ/J. Fig. 6 represents the comparison of the theoretical RDCO values with
the experimentally obtained values. As seen from the figure, a value of σ/J ∼0.35 - 0.45
appears to be reasonable. A similar plot has been made for the ∆IPDCOvalues of the same
transition in Fig. 7. Here also the theoretical values agree with the experimental ones in
the same range. This exercise was repeated for other transitions in34S of known mixing
ratios . It was observed that a value of σ/J = 0.4 consistently reproduced the observed
RDCOand ∆IPDCOvalues and hence this value was used in all our calculations.
Fig. 8 shows the theoretical asymmetry values for a pure E2 transition (Jπ= 2+→ 0+)
as a function of mixing ratio at different values of σ/J. The shaded area gives the range
of observed ∆IPDCOvalues for the known 2127-keV, E2 transition (2+→ 0+) in34S. An
increase in σ/J decreases polarization. At σ/J = 0.6 the theoretical values are not consistent
with the experimental values. The theoretical asymmetry value at δ = 0 for σ/J = 0.4 lies
well within the experimentally observed range. This also justifies our choice of σ/J.
The calculated and observed RDCO and ∆IPDCO values for several strong transitions
in34S,33P and33S are given in Table I. The RDCO and ∆IPDCO values are consistently
reproduced within error bars in each case.
Wherever quantitative measurements were not possible due to insufficient statistics, par-
ity assignment was done qualitatively from the corresponding gated perpendicular and par-
allel spectra. When gates were put on coincident gamma transitions and more counts were
observed in the perpendicular gated spectra than in the parallel, the observed γ-ray was
assigned an electric nature. The reverse was true for assignment of a magnetic nature. Thus
the previously reported spin-parities of the levels in33,34S and33,34P that were observed in
the present experiment, have been confirmed either quantitatively or qualitatively.
The nuclei populated in the experiment as determined from the projection spectra and
supported by the beam-off radioactivity data were34S,33S,34P,33P,32P and30Si. Fig. 9
depicts the projection spectrum of the symmetric γ-γ matrix. The use of fusion evapora-
tion reaction to populate the above-mentioned nuclei has clearly enhanced their production
compared to deep-inelastic/transfer reactions  as is evident from Fig. 10. The comparison
also shows a much cleaner and contamination free spectrum obtained in the present exper-
iment. Moreover, higher spin states have become accessible as a result of utilizing fusion
evaporation reaction mechanism.
The level schemes of these nuclei have been extended with the addition of several new
transitions. Figs. 11, 12 and 13 are spectra obtained by gating on the symmetric γ- γ matrix
by the 429-keV, 186-keV, and 968-keV γ-rays belonging to34P,33P, and33S, respectively.
The deduced level schemes are shown in Figs. 14, 15 and 16. The energies, relative intensities
and assigned multipolarities of the observed transitions, the assigned excitation energies and
spin-parities of the levels, and the γ-ray branching ratios for decay of those levels are listed
in Table II.
A quantitative measure of RDCO and/or linear polarization has not been possible for
some transitions. In those cases, a qualitative assignment has been made as explained in the
previous section. The multipolarities listed in TABLE II for such transitions are essentially
the dominant multipolarity; the extent of mixing could not be determined. The present
setup did not permit us to obtain the mixing ratios experimentally. However mixing ratio
range has been deduced from the RDCOand ∆IPDCOvalues for some transitions in34P, as
explained later. The comparison between theoretical and experimental RDCOand ∆IPDCO
values in34S,33P, and33S is based on previously-reported mixing ratios (TABLE I).
Previous investigations of the level structure of34P employed non-equilibrated reactions
to populate this difficult to access N = 19 nucleus. These studies reported different subsets
of the excited levels in34P. Ajzenberg-Selove et al.  were the first to report the excited
states of34P at 423, 1605, 2225, 2309 and (3345) keV(±10 keV) using the34S(t,3He)34P
reaction. On the other hand, 429-, 1608- and 1178-keV gamma rays were identified by both
Nathan et al.  and Pritychenko et al.  in β-decay and intermediate-energy Coulomb
excitation measurements respectively. Pritychenko et al.  also observed a new 627-
keV transition de-exciting the (2+) level at 2225 keV. However the 1608-keV and 627-keV
transition were not observed in any of the subsequent investigations using transfer and deep-
inelastic reactions [1, 3, 5]. All the excited states of34P reported by Ollier et al. , except
the level at 4723 keV were observed in the present work. Several other strong transitions,
of 679, 1444, 1607, 1638, 1646, 2325 and 3932 keV belonging to34P have been identified
and placed in the level scheme by coincidence and intensity arguments. The placements of
the new transitions were also facilitated to a large extent by the observation of cross-over
transitions. The 1607-keV transition was found to be in coincidence with the 429- and 1876-
keV transitions and hence is different from the 1608-keV transition reported by Nathan et
al.  and Pritychenko et al. . The 1046-keV transition reported by Ollier et al.  but
not observed by Krishichayan et al.  has been observed in the present experiment, but
with an energy of 1048-keV. Further, this transition exhibited a shape asymmetry at forward
and backward angles, which is indicative of a lifetime of the order of a few pico-second for
the 3353 keV level.
One of the main motivations of the present experiment was to undertake polarization
and coincidence angular correlation measurements following fusion reaction to confirm the
spin-parity assignment of the 2305-keV level in34P. In Ref. , we had assigned Jπ= 4+
to this level, whereas it had been assigned 4−by earlier workers [2, 27]. The 429-keV was
established as a magnetic dipole transition from the DCO and polarization analysis method
described in the previous section. The 1876-keV γ ray de-exciting the 2305-keV level has a
RDCOvalue 1.62(26) (Fig. 2) and linear polarization measurements yielded a near-zero value
for ∆IPDCO, establishing it as a highly mixed (L = 2, L
′= 3) transition for the first time
(Fig. 19). This is also evident from the difference between the 429-keV gated perpendicular
and parallel scattering spectra (Fig. 17), where the number of counts under 1876 keV peak
is nearly zero. In such cases, the polarization measurements cannot distinguish between
M2/E3 and E2/M3 mixing. Fig. 18 depicts the theoretical asymmetry curves for 2+→
1+M1/E2 radiation as a function of mixing ratio, at different values of σ/J. The shaded
area represents the experimental dispersion in ∆IPDCO of the 429-keV γ ray. At σ/J =
0.4, the theoretical values are consistent with experimental ∆IPDCOvalues over the range
-80≤ arctan(δ) ≤ 00and 490≤ arctan(δ) ≤ 540. However, as seen from Fig. 18, the mixing
ratio predicted by the shell model (as detailed in the subsequent section and indicated by
the vertical dotted line) limits the value to the former range. We have similarly plotted the
theoretical asymmetry values for a M2/E3 mixing 4−→ 2+as a function of mixing ratio,
at different values of σ/J (Fig. 19). The shaded area on this graph represents experimental
range of ∆IPDCO for the 1876-keV transition. The theoretical values are consistent with
the experimental values over the range -460≤ arctan(δ) ≤ -150and 500≤ arctan(δ) ≤ 760
for σ/J = 0.4. The latter range of very large mixing ratios being physically unreasonable,
has not been considered. When we repeated this exercise considering a E2/M3 mixing, we
obtained almost the same ranges of mixing ratios. Thus an unambiguous identification of
the 1876-keV transition as M2/E3 or E2/M3 is not possible following this procedure. The
∆IPDCOvalue at σ/J = 0.4 corresponding to the mixing ratio predicted by the shell model
calculations (Section IV) does not lie within the aforementioned ranges in both cases. This
mismatch has been discussed in detail in the subsequent section.
Fig. 20 shows the variation of theoretical RDCOfor the 1876-keV γ-ray as a function of its
mixing ratio when the gate is on the 429-keV, the ground state transition. The three plots
correspond to the three values of δ429[ -0.14 ≤ -0.07 ≤ 0.0] that were determined earlier from
Fig. 18. The theoretical RDCOvalues were computed using ANGCOR . The horizontal
lines mark the experimental range of RDCO values for the 1876-keV transition, while the
vertical lines indicate the range of mixing ratios for this transition as obtained from Fig. 19.
As is evident from the graph, the mixing ratio range obtained from the analysis of the linear
polarization measurements is also consistent with the angular correlation measurements.
Thus, both these measurements are indicative of -1.03≤ δ1876≤ -0.27.
Asai et al.  have reported the lifetime of the level at Ex= 2.305 MeV as 0.3 ns ≤ t1/2
≤ 2.5 ns. Combining this lifetime measurement with the mixing ratio range, that we have
obtained for 1876-keV, we have calculated the experimental reduced transition probabilities
assuming both M2/E3 and E2/M3 mixing. The calculations are presented in TABLE III. As
is evident from the table, the lifetime measurements lead to unacceptable M3 strengths .
This supports an M2/E3 assignment for the 1876-keV transition and Jπ= 4(−)to the 2305
keV level. This needs to be confirmed with precise lifetime measurements, however.
The qualitative linear polarization measurements for the 1607-, 1646-, 679- and 2325-keV
transitions indicate an electric nature for them. The spin-parity assignments for the levels
de-exciting via these transitions are based on the assumption of Jπ= 4(−)for the 2305-keV
This work reports the first polarization measurement for33P populated in a heavy-ion
fusion reaction. The earlier light-ion induced reaction investigations had established the
level structure up to a spin of Jπ= 11/2−and Ex∼5.6 MeV . We have been able to
extend the yrast sequence up to Jπ= 17/2(+)and Ex ∼8 MeV due to the observation of
two new transitions of energy 1298 keV (E2) and 1028 keV (M1). The multipolarities of
these two transitions were assigned as quadrupole and dipole, respectively, on the basis of
the observed RDCOvalues. The parity measurements have been shown as tentative since
only qualitative measurements were possible. The 1028-keV transition was identified as a
magnetic transition due to its preferential scattering in the parallel direction as observed
in 1298-keV gated perpendicular and parallel spectra. On the other hand, the 1298-keV
transition was assigned an electric nature due to its preferential scattering in perpendicular
direction. A full Doppler shift has been observed in the 1298-keV transition and hence, the
lifetime of the 6938 keV level is expected to be much less as compared to the stopping time
(∼pico-second). Apart from these, several other new transitions, with energies (in ascending
order) of 237, 247, 980, 994, 1008, 1312, 1825 (D), 2142 (Q) and 3605 keV were observed
and placed in the decay scheme. The present statistics did not permit us to observe the
weak transition to the Ex∼5221-keV from the level at 5454 keV. We have also observed
1170- and 880-keV transitions which could not be placed in the decay scheme. It is worth
mentioning that the single- (1868 keV) and double-escape (1358 keV) peaks corresponding
to the 2379-keV transition were observed and the intensity of this γ-ray reported in TABLE
II was obtained from the sum of the counts under the full photopeak and the two escape
peaks. This was done as our efficiency measurements (performed with a152Eu source) did
not have data points in this energy region where escape contribution becomes significant.
Prior to this experiment33S has been studied via light ion reactions . In this ex-
periment, the level scheme of33S was extended with the addition of 6 new transitions of
energy 597, 597, 603, 845 (E1), 1015 (E1) and 1931 keV respectively. The presence of a
597-keV transition in the 597-keV gated spectrum is indicative of a doublet. As a result it
was not possible to determine their individual intensities and the spin parity of the excited
states that de-excite via these two transitions. The qualitative polarization and angular
correlation measurements for the 845-keV and the 1015-keV transitions indicate that these
are dipoles and electric in nature. Such sequences of electric transitions have been reported
in neighbouring nuclei like32P . The 1931-keV transition exhibits the fully Doppler
shifted peak indicating a short lifetime (≪ pico-second, the stopping time) for the level at
∼4867 keV. The present polarization and angular correlation measurements confirmed the
previously assigned spin-parity of the 1968- and 2936-keV levels. The reported mixing ratios
are consistent with the results of the present measurement (TABLE I).
IV. THEORETICAL RESULTS
Shell model calculations using the code NuShell@MSU  were performed to interpret
the observed level structures of33,34P and33S. The valence space consisted of the 1d5/2,
1d3/2, 2s1/2, 1f7/2, 1f5/2, 2p3/2 and 2p1/2 orbitals outside a16O core. The “sdpfmw” in-
teraction, taken from the Warburton, Becker, Millener, and Brown (WBMB) sd-pf shell
Hamiltonian , was used.
In34P the positive-parity states 1+and 2+, which are expected to be dominated by the
pure sd configurations, are well reproduced within the full sd-space shell model calculations
(0¯ hω) (using the sdpfmw interaction) and are consistent with the sd calculation of Brown .
The predicted binding energy of the ground state is -191.971 MeV, which matches very well
with the experimental value -192.04 MeV . The mixing ratio of the 429-keV transition
predicted by shell model is -0.0024, which is also within the range determined from our
Excitations of nucleons from sd shell into fp shell are essential to explain the negative-
parity states (minimum 1 particle in the fp shell) as well as the high-spin, positive-parity
states (minimum 2 particles in the fp shell). Due to computational limitations, unrestricted
calculations were not possible and only one particle could be excited to the pf shell (1¯ hω).
It has been reported by several authors that there is an overestimation of the sd-pf gap
in the corresponding interaction which required the lowering of the single-particle energies
of the f and p orbitals [24, 35]. No such attempt was made in the present calculation.
Fig. 21 shows a the comparison between the calculated and the experimental levels in34P.
The 7+state predicted by the shell model is at a very high excitation energy (11366 keV)
and hence has not been included in the figure. As seen from Fig. 21, the high-spin positive-
parity states are much higher in excitation energy than the corresponding experimental
levels. This is likely due to our inability to excite more than one particle into the fp shell.
There is a reasonable agreement in excitation energy between the Jπ= 4−, 5−, 6−levels
predicted by shell model and the observed 2305-, 3353- and 4630- keV levels, respectively.
Thus, the theory corroborates our spin-parity assignments at least for the negative-parity
states. However, the above shell model calculations failed to predict the mixed nature of
the 1876-keV transition established from our polarization measurements. The shell model
predicts an almost pure M2 nature for this transition [δ = -0.034]. The B(M2) and B(E3)
values obtained from shell model are 0.1816 W.u. and 0.2167 W.u., respectively, and, as
seen from Table II, the B(E3) values are heavily under predicted, clearly reflecting this
mismatch. We have also performed similar shell model calculations for the neighbouring
N = 19 isotones viz.,
37Ar and35S where similar M2/E3 mixed transitions are reported
(1611 keV (Jπ= 7/2−→ 3/2−in37Ar) and 1911 keV (Jπ= 7/2−→ 3/2−in35S) .
The results are summarized in TABLE IV. In all cases, the calculations predict very little
mixing, unlike the experimental observations. The E3 transition strengths are several orders
of magnitude higher than the corresponding shell model predictions. Clearly, there is a need
to perform these calculations with a better Hamiltonian encompassing a realistic cross-shell
interaction, and/or with a more complete wave function incorporating configurations arising
from multi-particle excitations into the fp orbitals.
The 0¯ hω calculations for33P and33S reproduces the low-spin positive-parity states. How-
ever, the 1¯ hω calculations fail to generate the first experimentally-observed negative- parity
state, 7/2−, in both nuclei. The predicted energies of the high-spin, negative-parity states
are higher than the experimental values by several MeV.
The level structure of the generally difficult to access nuclei
33,34P and33S has been
investigated using heavy-ion fusion reaction which has resulted in a substantial enhancement
in their production cross-sections. The level schemes of these nuclei have been considerably
extended. Spin-parity assignments have been made following a consistent analysis of both
the coincidence angular correlation and linear polarization data. The results indicate that
the 1876-keV transition de-exciting the 2305-keV level in34P is a mixed transition and
plausibly has a M2/E3 admixture; however precise lifetime measurements would be required
to confirm this assignment unambiguously. The shell model calculations emphasize the need
for detailed microscopic calculations to understand the observed level sequences and mixing
ratios. The deformed shell model could provide an insight into the observed level structures
due to the occupation of deformation-driving orbitals such as f7/2.
The authors would like to thank all the participants who have helped set up the Clover
array at TIFR. The help and co-operation received from Mr Kaushik Basu of UGC-DAE
CSR during the experiment is gratefully acknowledged. We would like to thank the BARC-
TIFR Pelletron staff for their excellent support during the experiment. We are thankful
to Mr. J. P. Greene, ANL, U.S.A, for the18O target. Thanks are also due to Dr. W. P.
Tan and Dr. Larry Lamm, Univ. of Notre Dame, U.S.A, for providing us the enriched18O
cathode. Special thanks to Prof. Alex Brown for the indepth discussions and his views and
comments on Shell model calculations.
 B. Fornal et al., Phys. Rev. C 49, 2413 (1994).
 M. Asai, T. Ishii, A. Makishima, M. Ogawa, and M. Matsuda, Proceedings of the Third Inter-
national Conference on Fission and Properties of Neutron-Rich Nuclei, edited by J. H. Hamil-
ton, A. V. Ramayya, H. K. Carter (World Scientific, Singapore, 2002) pp. 295-297.
 J. Ollier et al., Phys. Rev. C 71, 034316 (2005).
 R. Broda, J. Phys. G: Nucl. Part. Phys. 32, R151 (2006).
 Krishichayan et al., Eur. Phys. J. A 29, 151 (2006).
 A. M. Nathan and D. E. Alburger, Phys. Rev. C 15, 1448 (1977).
 I. Iwasa et al., Phys. Rev. C 67, 064315 (2003).
 Y. Eyal and I. Dastrovosky, Nucl. Phys. A 179, 594 (1972).
 LAMPS, http://www.tifr.res.in/∼pell/lamps.html#.