We show that the nuclear supersymmetry model (n-susy) in its extended
version, predicts correlations in the nuclear structure matrix elements which
characterize transfer reactions between nuclei that belong to the same
supermultiplet. These correlations are related to the fermionic generators of
the superalgebra and if verified experimentally can provide a direct test of
the model.Comment: Invited talk at "Nuclear Physics: Large and Small", April 19-22,
  2004, Hacienda Cocoyoc, Mexic

Barea, J.

Bijker, R.

Frank, A.

English

arXiv

J. Barea

Crossref

New correlations induced by nuclear supersymmetry

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New correlations induced by nuclear
supersymmetry
J. Barea∗, R. Bijker∗ and A. Frank∗
∗ICN-UNAM, A.P. 70-543, 04510 México, D.F., México
Abstract. We show that the nuclear supersymmetry model (n-susy) in itsextended version, predicts
correlations in the nuclear structure matrix elements which characterize transfer reactions between
nuclei that belong to the same supermultiplet. These correlations are related to the fermionic
generators of the superalgebra and if verified experimentally c n provide a direct test of the model.
The extended n-susy model [1] correlates spectroscopic properties of adjacent nuclei.
It has been particularly sucessful in describing the quartet formed by the isotopes
194Pt, 195Pt, 195Au and196Au using theSpin(6) limit of the dynamical supersymmetry
Uν(6/12)⊗Uπ(6/4) [1, 2] in which the odd-proton is allowed to occupy theπd3/2
orbit, the odd-neutron occupies theν p1/2, ν p3/2 andν f5/2 orbits, even-even nuclei are
described by theSO(6) limit of the interacting boson model.
In this framework, all states belong to the same irreduciblerepresentation (irrep)
of the initial product of supergroups and can be labeled by different irreps of the
subgroups present in the chain of groups. The states in the quartet are said to form a
supermultiplet and the excitation spectra is obtained using the same hamiltonian. In this
way the excitation energies are correlated. Transitions probabilities and moments are
also correlated through the use of the same operators.
Both the hamiltonian and the electromagnetic transition operators are based on
the bosonic generators of the superalgebra, which transform bosons into bosons and
fermions into fermions, inducing transitions inside each nucleus. In this work we ex-
plore the fermionic generators, which transform bosons into fermions and viceversa.
They are thus associated to transitions among different nuclei, which we can associate
to transfer reactions.
The purpose of this contribution is to show that n-susy establi hes correlations be-
tween different transfer reactions. We start with one-nucleon transfer reactions and then
we present the first results, to our knowledge, of a susy analysis of the two-nucleon
transfer reaction198Hg(~d,α)196Au, in which the wave function correlations of proton-
neutron clusters in the target can be tested.
CORRELATIONS IN ONE-NUCLEON TRANSFER REACTIONS
The nuclear structure information which can be extracted from one-nucleon transfer
reactions is contained in the so called spectroscopic intensity, which is the modulus
square of the reduced matrix element of the transfer operator T(J) between the ground
state of the target nucleus in the reaction, and the final state of the residual nucleus
I = |〈α f Jf ||T(J)||αi Ji〉|2. (1)
It is possible to use a transfer operator deduced from microscopic asumptions to calcu-
late these quantities, but here we use an alternative methodbased on symmetry consid-
erations. In this case, the transfer operator can be writtenas a tensor operator under the
subgroups that appear in the group chain of the dynamical super ymmetry and has the
advantage of giving rise to selection rules and closed expressions for the spectroscopic
intensities [3, 4, 5].
For the one-proton transfer reactions, the transfer tensoroperators read [5]
T
〈 12 , 12 ,− 12〉( 12 , 12) 32
1,π = −
√
1
6
(
s̃π ×a†π, 32
)(32)
+
√
5
6
(
d̃π ×a†π, 32
)( 32)
(2)
T
〈 32 , 12 , 12〉( 12 , 12) 32
2,π =
√
5
6
(
s̃π ×a†π, 32
)( 32)
+
√
1
6
(
d̃π ×a†π, 32
)(32)
, (3)
where the upper indices specify the tensorial properties under Spin(6), Spin(5) and
Spin(3). The tensorial character underSpin(6) of T1,π implies that it only excites the
ground state of the odd-even nucleus from the ground state ofthe even-even nucleus.
However,T2,π allows the transfer to an excited state in the odd-even nucleus. The ratio
of the intensities for each transfer operator is given by [5]
R1(ee→ oe) =
Igs→exc
Igs→gs
= 0, (4)
R2(ee→ oe) =
Igs→exc
Igs→gs
=
9(N+1)(N+5)
4(N+6)2
, (5)
whereN is taken as the number of bosons in the odd-odd nucleus of the quartet and
ee and oe refer to even-even and odd-even respectively. In the case of the one-proton
transfer194Pt→195 Au, the second ratio isR2 = 1.12 (N = 5), but the relatively small
strength to excitedJ = 32 states suggests that the operatorT1,π can be used to describe
the data.
Due to theF-spin symmetry structure of the wave functions it is possible to establish
the following correlations with the reaction which involves the even-odd (eo) and odd-
odd(oo) nuclei:
R1(ee→ oe) = R1(eo→ oo) = 0, (6)
R2(ee→ oe) = R2(eo→ oo) =
9(N+1)(N+5)
4(N+6)2
. (7)
A different way to understand this result is through the use of a tensor operator which
transforms as a scalar in the pseudo-l degree of freedom [6] (upper and lower indices
specify the tensorial properties underSpin(6) andSpin(5), Spin(3) andSU(2) respec-
tively)
P〈0,0,0〉
(0,0)0, 12
=
(
s̃ν ×a†ν, 12
)( 12)
−
√
2
(
d̃ν ×a†ν, 32
)( 12)
+
√
3
(
d̃ν ×a†ν, 52
)( 12)
. (8)
This operator links the wave functions of the even-odd and odd- d nuclei in the quartet
to the wave functions of the even-even and odd-even nuclei, respectively. The use of this
property and the fact thatP conmutes with the one-proton transfer operatorsT1,π and
T2,π , allows to write the equations (6) and (7) [7, 8].
We thus find a direct correlation between two different one-proton reactions, which for
the case of195Pt→196 Au can be tested experimentally. Very recently this reaction has
been measured [9] and the experimental data confirms thatT1,π is capable of describing
it, given that the strength to the ground state of196Au is relatively strong and the excited
state whichT2,π can excite is not seen populated in the measurements. This fact confirms
also that the correlation seems to apply.
For the one neutron transfer reaction there is a similar correlation. Let us consider the
following transfer tensor operator (labels like in (8)) [6]
T(2,0,0)
(1,0)2, j ,ν =
√
1
2
(
s̃ν ×a†ν, j
)( j)
−
√
1
2
(
d̃ν ×a†ν, 12
)( j)
j =
3
2
,
5
2
. (9)
The same argument about the link between the wave functions of the even-even and the
odd-even nuclei permits us to correlate the one neutron transfer reaction even-even→
odd-even with the inverse reaction odd-even→ even-even [8]. Taking the following
ratios
R(ee→ oe) =
∣
∣
∣
〈
oe;[N1,N2]〈σ1,σ2,σ3〉(1,0)2;J
∥
∥
∥
T(2,0,0)
(1,0)2, j ,ν
∥
∥
∥
ee; g.s.
〉
∣
∣
∣
2
∣
∣
∣
〈
oe;[N+1,1]〈N+1,1,0〉(1,0)2;J
∥
∥
∥
T(2,0,0)
(1,0)2, j ,ν
∥
∥
∥
ee; g.s.
〉
∣
∣
∣
2 , (10)
R(oe→ ee) =
∣
∣
∣
〈
ee;[N1,N2]〈σ1,σ2,σ3〉(1,0)2
∥
∥
∥
T̃(2,0,0)
(1,0)2, j ,ν
∥
∥
∥
oe; g.s.
〉
∣
∣
∣
2
∣
∣
∣
〈
ee;[N+1,1]〈N+1,1,0〉(1,0)2
∥
∥
∥
T̃(2,0,0)(1,0)2, j ,ν
∥
∥
∥
oe; g.s.
〉
∣
∣
∣
2 , (11)
we find the following relations
R(oe→ ee) = R(ee→ oe) for N2 = 1,
R(oe→ ee) = R(ee→ oe)(N+1)(Nν +1)
(Nπ +1)
for N2 = 0,
In table 1 we show these ratios and quote the experimental values [10] and the
calculated ones for the isotopes194Pt and195Pt. We can observe a consistence between
the experimental and calculated values, but clearly more exp rimental work is needed to
confirm if these correlations hold.
Table 1. Intensity ratios for one-neutron transfer reactions.
[N1,N2]〈σ1,σ2,σ3〉 Ree→oe
194Pt→ 195Pt
calc. exp.
j = 32 j =
5
2
195Pt→ 194Pt
calc. exp.
[N+2]〈N+2,0,0〉 2(N+4)
(N+1)(N+3)(N+6) 0.034 0.264 0.052 0.511 ——-
[N+2]〈N,0,0〉 N(N+4)(N+5)
2(N+2)(N+3)(N+6)2 0.033 ——- ——- 0.498 ——-
[N+1,1]〈N,0,0〉 N
2(N+5)
2(N+2)(N+6)2 0.148 0.087 ——- 0.148 ——-
CORRELATIONS IN TWO-NUCLEON TRANSFER REACTIONS
In contrast to the one-nucleon transfer reactions in which the single particle structure
of nuclear states is examinated, two-nucleon transfer reactions are very sensitive to the
correlation between the transfered nucleons. As a consequence, this kind of transfer
reactions supply a stringent test for the nuclear wave functio s.
Two factors determine the strength of a two nucleon transferreaction, which are
related to the two-nucleon fractional parentage coefficients. On the one hand it depends
on how similar the state ofA+ 2 nucleons is to the state ofA plus two additional
nucleons, and on the other, if the correlation of these two nucleons in theA+2 state
is similar to the correlation in the state of the light nucleus. The nuclear structure
information that can be extracted from a model is related to bo h factors. We shall follow
the formalism developed by Glendenning [11] and briefly sketch the main ingredients
as applied to the reaction198Hg(~d,α)196Au, which has been measured recently to study
the odd-odd nucleus196Au [9].
The nuclear structure information is contained in the spectros opic strengthsGLJ.
This quantities are written as
GLJ =
∣
∣
∣
∣
∣
∑
N jν jπ
β jν jπ G
jν jπ
NLJ
∣
∣
∣
∣
∣
2
, (12)
where
β jν jπ =
〈
196Au;J
∥
∥
∥
T(J)jν jπ
∥
∥
∥
198Hg;0gs
〉
(13)
G jν jπNLJ =
√
3L̂ ĵν ĵπ






lν
1
2 jν
lπ 12 jπ
L 1 J






〈n0N L|nν lν nπ lπ ;L〉 . (14)
For the two nucleon transfer operatorT(J)jν jπ we have chosen the simplest possible form:
T(J)jν jπ = α jν
(
a†jν ×a
†
jπ
)(J)
. (15)
0
-
1
-
2
-
3
-
4
-
J
π
0
1
RLJ
0
1
RLJ
0
1
RLJ
L = 1
L = 3
L = 5
0
-
1
-
2
-
3
-
4
-
J
π
0
-
1
-
2
-
3
-
4
-
J
π
0
1
RLJ
0
1
RLJ
0
1
RLJ
L = 1
L = 3
L = 5
0
-
1
-
2
-
3
-
4
-
J
π
Experimental Calculated
Figure 1. Ratios of spectroscopic strengths. The first column in each frame correspond to states
with
(
3
2,
1
2
)
Spin(5) labels and the second with
(
1
2,
1
2
)
labels, respectively. Each row from the bottom
to the top corresponds to states with labels[6,0]〈6,0〉(13/2,1/2,1/2), [5,1]〈5,1〉(11/2,1/2,1/2) and
[5,1]〈5,1〉(11/2,3/2,1/2) for the groupsUBFνπ (6), SOBFνπ (6) andSpin(6).
The parametersα jν are determined by using a least square fit of the spectroscopic
strengths to the experimental values. To compare the experimental and calculated spec-
troscopic strengths we have choosen seven states of reference, ach one corresponding
to each of the seven possibleLJ transfers that the n-susy model allows, and we have
calculated for eachLJ the transfer ratio
RLJ =
GLJ
Gre fLJ
, (16)
whereGre fLJ is the spectroscopic strength for the reference state for a pa ticularLJ tranfer.
In figure 1 we show the experimental and calculated ratiosRLJ. We observe from this
figure that the comparison is quite reasonable if we take intoacc unt the simple form we
adopted for the two nucleon transfer operator. In concluding this section we can say that
the n-susy model can reproduce remarkably well the main featur s of the neutron-proton
correlations which are present in the nuclear states involved.
SUMMARY
We have found new correlations predicted by extended n-susy. These correlations relate
spectroscopic strengths of different one-nucleon transfer reactions between the nuclei
which are described by the model. We focused on the supermultiplet formed by194Pt,
195Pt, 195Au and196Au whose spectroscopic properties have been previously describ d
in terms of this model. We have shown that these correlationsare partially fulfilled with
the availaible experimental data, but clearly more experimntal work is necessary to test
their full validity.
We have calculated the spectroscopic strengths associatedto the two nucleon transfer
reaction198Hg(~d,α)196Au, which was recently measured. The comparison between the
calculated and experimental data shows good agreement consideri g the simple form
adopted for the two nucleon transfer operator.
We plan to search for other experimental examples to which extended n-susy and
its correlations can be applied, eventually relaxing the constraints set by dynamical
symmetry [12, 13]. We wish to emfasize that nuclear susy may be a model whose range
of applicability is wider than was previously realized and which may lay the foundations
of a new and unifying point of view in nuclear structure.
ACKNOWLEDGMENTS
This work was supported by CONACyT. We are grateful to G. Grawfor sharing the new
experimental data on the transfer reactions195Pt(3He,d)196Au and 198Hg(~d,α)196Au
prior to publication. Many enlighting discussions with J. Gómez-Camacho and P. Van
Isacker are acknowledged.
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