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Cluster radioactivities of odd-mass nuclei.

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Ann. Physik 3 (1994) 107- 117
Annalen
der Physik
@ Johann Ambrosius Barth 1994
Cluster radioactivities of odd-mass nuclei
D. N. Poenaru
'I*
*, E. Hourani ', and W. Greiner'
Institut de Physique Nuclkaire, F-91406 Orsay, France
'Institut fur Theoretische Physik der Universittit, Postfach 11 1932, D-60054 Frankfurt/M.,
Germany
Received 22 November 1993, revised version 17 January 1994, accepted 3 February 1994
Abstract. The partial half-lives of the hypothetical even-even equivalent of an odd-mass nucleus for
cluster transitions toward various excited states of the daughter, used as a reference to find the hindrance factor, can be calculated within analytical superasymmetric fission model, by taking into
account the angular momentum of the emitted cluster. Detailed tables are presented for I4C radio22JAc;24Ne radioactivity of 233U,23'Pa, and 23Fdecay of 23'Pa, showing
activity of 221Fr,221.223Ra,
that, except for "*Ac, the existing experimental evidences, do not exclude (moderate) hindered transitions to the ground states of the daugther nuclei.
Keywords:
Cluster radioactivities; Transitions to excited states; Fission model.
1 Introduction
A rich variety of nuclear decay modes have been discovered and intensively studied during the last decade [l]. Among them we would like to mention cluster (or '%?exotic")
radioactivities - intermediate phenomena between fission and alpha decay, in which
a parent nucleus AZ is splitted into two fragments: the emitted cluster *cZe and the
daughter
The spontaneously emitted light fragment 'tZe is a small nucleus (like
'.4C, 24Ne, etc.) heavier than a particle, but lighter than the lightest fission fragment.
The hadronic numbers are conserved: A = A,+Ad and Z = Z e + Z d .
A growing interest for these new processes has been manifested since 1984, when the
first experiment has been reported, confirming our predictions of 1980 that 14C should
be the most probable emitted cluster from Ra isotopes with mass numbers A = 222 and
224. The related theoretical models [2,3] and experimental results [4,5] have been
recently reviewed. The predicted halflives within analytical superasymmetric fission
model (ASAFM) have been experimentally confirmed.
Like for spontaneous fission or alpha decay, the basic explanation relies on the
quantum mechanical tunnelling through a potential barrier. Theoretically any parent
nucleus for which the released energy is a positive quantity, Q > 0, can be a cluster emit-
*
Permanent address: Prof. Dr. D. N. Poenaru, Institute of Atomic Physics, P.O.Box MG-6, RO-76900
Bucharest, Romania.
108
Ann. Physik 3 (1994)
ter, but practically there is a severe selection imposed by the available techniques,
requesting a short-enough halflife and a large enough branching ratio. We shall give
below some examples of technical achievements.
The decay constant L = In 2/T characterising the well known exponential law e - I r
of variation in time, t, of the number of parent nuclei, can be expressed as a product
of three (model dependent) quantities
where v is the frequency of assaults on the barrier per second (related to the zero-point
vibration energy E, = h v/2, with h the Planck constant), Po,is the preformation probability of the cluster at the nuclear surface, strongly dependent on the nuclear structure,
and P, is the quantum penetrability of the potential barrier. Usually P, is calculated by
using the semiclassical Wentzel-Kramers-Brillouin (WKB) approximation.
Up to now there are successful measurements on 14C, "0, 23F 24-26Ne, 28*30Mg,
and 32*34Si cluster decays of the following trans-francium nuclei: i2'Fr, 221-224*226Ra,
225Ac,228*230Th
, 231P
a, 232-234U7
236,238Pu,
and (preliminary) 242Cm.The shorter halflife, T= 10" s, corresponds to the 14Cradioactivity of 222Raparent nucleus, leading to
a double magic daughter 'O'Pb.
There is a strong competition with 01 decay. The largest branching ratio with respect
to a-decay, b = Ta/T is about 10-9.2 and has been observed in I4C radioactivity of
223Ra.With the high sensitivity of solid state track detectors, it was possible to measure
a branching ratio as low as 10-'6.25for Mg emission from 238Pu.The longer partial
s, is that of 231Panucleus against 23Femishalf-life determined up to now, of =
sion.
Usually only one kind of emitted cluster (the most probable) could be experimentally observed, for the next one the emission rate being smaller by some orders of
magnitude. Nevertheiess, there are few exceptions: both F and Ne emitted by 231Pa;Ne
and Mg from 234U,and Mg+Si from 238Pu.
The main difficulty of such an experiment comes from necessity to select few rare
events from an enormous background of alpha particles. Either solid state track detectors (which are not sensitive to alpha particles), or magnetic spectrometers (in which
alpha particles are deflected by a strong magnetic field), have been used to overcome
this difficulty. The superconducting spectrometer SOLENO, at I.P.N. Orsay has been
employed since 1984 to detect and identify the 14Cclusters spontaneously emitted from
222i223*224*226Ra
parent nuclei. Moreover, its good energy resolution has been exploited
in 1989 to discover 161 a 'ifine structure" in the kinetic energy spectrum of 14C emitted
by 223Ra.Cluster emission leading to excited states of the final fragments have been
considered for the first time [7] in 1986.
When the fine structure of 14C radioactivity of 223Rahas been discovered it was
shown that the transition toward the first excited state of the daughter nucleus is
stronger than that to the ground state. In other words, like in spontaneous fission of
odd-mass nuclei [8], or in fine structure of a-decay, one has [9] a hindered and a
favoured transition, respectively. Spontaneous fission rates of odd-A nuclei are slower
than that of e - e nuclei, indicating a higher fission barrier, due to shell effects. In both
kind of theories the physical explanation relies on the single-particle spectra of neutrons
or/and protons. If the uncoupled nucleon is left in the same state both in parent and
heavy fragment, the transition is favoured. Otherwise the difference in structure leads
to a large hindrance
D. N. Poenaru et al., Cluster radioactivities of odd-mass nuclei
109
H = T eX P/Te- ,
where TeXPis the measured partial half-life for a given transition, and Te-.eis the corresponding quantity for a hypothetical even-even equivalent, estimated either from a
systematics (log T versus Q-"* for example) or from a model. The released energy
during the process, Q, is given by the difference between the initial (parent) mass M a n d
the final one (the sum of fragment masses M e f M d ) expressed in units of energy (that
is - multiplied by the light velocity c squared): Q = [M- (Me+ M d ) ]c2. A transition
is favoured if H = 1, and it is hindered if H%-1.
Particularly interesting are the transitions in which at least one of the three partners
possess an odd number of nucleons, because of the spectroscopic informations one can
get. Unlike in a-decay, where the initial and final states of the parent and daughter are
not so far one from the other, in cluster radioactivities of odd-rnass nuclides, one has
a unique possibility to study a transition from a well deformed parent nucleus with complex configuration mixing, to a spherical nucleus with a pure shell model wave functions. In this way, one can get direct spectroscopic information on spherical components of deformed states. In 1986, after taking properly into account the even-odd
effect, we have improved our earlier predictions. The purpose of the present paper is
to learn about some possible hindered transitions by analyzing the still existing
discrepancies. Before displaying the results we shall compare two different kinds of
systematics of experimental data of even-even nuclei, which could be used to determine
Te-e from Eq. (2).
2 Two kinds of systematics
The main quantities experimentally determined are the partial halflife,
and the
kinetic energy of the emitted cluster Ek = QAd/A. This equation is a direct consequence of the "cold" character of this decay mode - the total kinetic energy of the
two fragments practically exhausts the released energy Q, which is shared between the
two fragments.
Intuitively one can imagine that the preformation probability is a decreasing function of the cluster mass number A,; e.g. the chance to find 14 nucleons close together
would be smaller than that of only 4 of them. One may assume an empirical linear
dependence of the logarithm of this quantity in the range of emitted clusters already
measured:
log Po, = (Ae-l)logP:u
3
(3)
where the preformation probability of a particle P:" can be determined by fit with experimental data.
On this basis we found [lo] a single universal curve log T =f(log P,), for each kind
of cluster radioactivity of even-even parent nuclei. This has been done by making a further assumption that the frequency of assaults v , which enters into the decay constant
relationship as was shown above, is independent on the emitted cluster and the daughter
= constant. From a fit of experimental data we obtained
nucleus v(Ae,Ze,Ad,Zd)
s-', leading to
Pzu = 0.0160694 and v =
110
Ann. Physik 3 (1994)
log T = -log P, -22.169 +0.598 ( A ,- 1)
(4)
which is a straight line for a given A,, with a slope equal to unity.
For all cases of practical importance the daughter nucleus has a spherical shape (a
doubly magic 208Pbor one of its neighbours). When the cluster is preformed, by
neglecting in the first approximation its deformation, we can assume a touching point
configuration of two spherical nuclei with a separation distance R = R, between
centers expressed in fm, given by the Myers-Swiatecki liquid drop model:
R, = 1.2249(Ai’3+A:/3). For any combination of fragments AeZe,AdZd one can
calculate easily the WKB penetrability of the external (Coulomb) part of the potential
barrier E ( R ) = Z,Zde2/R, between the classical turning points R, and R b . The electron charge e = 1.43998 MeV-fm and Rb is defined by E(R,) = Q. The nuclear inertia
B ( R ) at the touching point equals the reduced mass p = mpA = mAdAe/A, where rn
is the nucleon mass.
P, = exp (-K,) ; K, =
2
1v2p[E(R)-Q]dR
Rb
R,
in which K, is the action integral. By expressing the length in units of fm we get
where r = Rt/Rb,Rb = 1.43998z d z e / Q .
The advantage of such a handy relationship is evident. The up to now 14 even-even
half-life measurements are well reproduced (within a ratio 3.86, or rms = 0.587 orders
of magnitude). A similar result is obtained with the “hand-pocket” formula presented
in Ref. 3 . Another argument of its universality is illustrated in Fig. 1. Instead of having
different lines for various parent nuclei, like in the “classical” systematics
log T = f ( Q - ’ l 2 ) ,one get practically only one line when we plot log T =f(log P,).
We had to use calculated (ASAFM) values for the half-lives in both kind of
systematics presented in Fig. 1, but in principle a linear least-square method through experimental points (if available), would give similar results. None of these two
systematics can take into account the angular momentum of the emitted cluster, which
can play a role for odd-mass nuclei.
Fig. 1 Comparison of a “classical” systematics (left hand side) with universal curve-like systematics
(right hand side), calculated within ASAFM for 14C radioactivity of even-even Ra isotopes (full line)
and of Pb isotopes (dashed line).
D. N. Poenaru et al., Cluster radioactivities of odd-mass nuclei
111
3 Analytical superasymmetric fission model
The above mentioned universal curve has been recently developed and it is certainly of
limited use in comparison with the more general ASAFM. Since 1980, in order to make
a systematic search for new cluster decay modes we had to take into account a large
number of combinations parent-emitted nuclei (at least 2000x250 = 5 . lo’). The corresponding large amount of computations can be performed in a reasonable time by
using an analytical relationship for the halflife. Such a formula has been obtained on
the basis of the Myers-Swiatecki liquid drop model adjusted with a phenomenological
shell correction in the spirit of Strutinsky method.
Within ASAFM we can easily simulate the even-even assumption. We also can take
into account the allowed angular momenta, I, determined from the spin (1)and parity
(n)conservation:
The half-life
T = [(hIn 2)/(2E,)]exp K
(8)
of a parent nucleus A 2 against the split into a cluster AeZe and a daughter A d z d is
calculated by using the WKB approximation, according to which the action integral is
given by
with B ( R ) = p - the reduced mass defined above, and E ( R ) replaced by
[E(R)-E,,,]-Q. In fact the action integral contains two terms K = K,,+K,, the
former corresponding to the overlapping stage of the fragments (the limits of the integral R,,R,), and the latter - to separated fragments (limits R,,Rb). E,,, is a
phenomenological correction energy similar to the Strutinsky shell correction, also taking into account the fact that Myers-Swiatecki’s liquid drop model overestimates fission
barrier heights, and the effective inertia in the overlapping region is different from the
reduced mass.
In order to avoid a lengthy numerical computation of the deformation energy of two
overlapping spheres from a separation distance R = R, to the touching point R = R,,
and to obtain an analytical relationship for K O , , we are using a parabolic approximation of the potential barrier:
E ( R ) = Q+(Ej-Q)[(R-Ri)/(R,-Rj)J2 ; R S R ,
(10)
where the interaction energy at the top of the barrier, in the presence of a nonnegligible
angular momentum, I A , is given by Coulomb and centrifugal terms:
Ei = E , + E , = e 2 Z e Z d / R , + h 2 1 ( 1 + 1 ) / ( 2 p R ~.)
(11)
The touching point separation distance R, = Re +Rd represents the distance between
the centers of the two tangent spheres. Rj = rOAj”(j = O,e,d; ro = 1.2249 fm) are the
112
AM. Physik 3 (1994)
radii of parent, emitted and daughter nuclei. Initially, one has an internal touching
point, when the separation distance is Rj = Ro -R e . During the deformation process
from one parent nucleus to two final fragments we presume that the radius of the small
sphere, representing the emitted cluster, remains constant R2 = Re, and that of the
daughter decreases from R t = Ro to R , = R d , allowing to conserve the total volume a condition imposed by the incompressibility of the nuclear matter. After substitution
into the above integral we obtain a closed formula
where the parameter b 2 = (Ecor+E*)/EEappears as a consequence of the following
expression of the first turning point (defined by E(R,) = Q ) :
R, = Rj + (Rt -Rj) [(E,,, +E *)/E:]‘I2
(1 3)
in which E* is the excitation energy concentrated in the separation degree of freedom,
and E t = Ej- Q is the barrier height before correction.
For separated spherical fragments in the Coulomb-plus-centrifugal potential
E ( R ) = EcR,/R+E,Rj/R2 ; R > R t
(14)
it is not necessary to make any approximation; we get
K, = 0.4392[(Q+EC,,+E*)A,A,j/A]‘/2RbJ,
(15 )
with
in which r = Rt/Rb and c = rEc/(Q+E,,, + E * ) . When the angular momentum I = 0,
one has c = 1 and the above equations take a simpler form. The external turning point
(defined as E ( R b )= Q ) is given by
In order to reduce the number of fitting parameters, we took E, = E,,, though it is
evident that, owing to the exponential dependence, any small variation of E,,, induces
a large change of T, and thus plays a more important role compared to the preexponential factor variation due to E, (see Eq. (8)). Both shell and pairing effects are included
in E,,, = ai(Ae)Q(i = 1,2,,3,4 for even-even, odd-even, even-odd, and odd-odd parent
nuclei). Pairing effects are clearly seen: for a given cluster radioactivity we have four
values of the coefficients ai, the largest for even-even parent and the smallest for the
D. N. Poenaru et al., Cluster radioactivities of odd-mass nuclei
113
odd-odd one [14]. The shell effects for every cluster radioactivity is implicitly contained
in the correction energy due to its proportionality with the Q value, which is maximum
when the daughter nucleus has a magic number of neutrons and protons. Another contribution of shell effects, which is always present, comes from the fact that we subtract
from the deformation energy (with corrections) the true value of the released energy Q,
obtained as a difference of experimentally determined atomic masses.
4 Hindrance factors
In the following we shall present the results of calculations within ASAFM, concerning
cluster emission from the ground state of some odd mass number parent nuclei toward
ground state and first excited states of the corresponding daughters.
As can be seen from the Table 1, the spin and parity of the (initial) ground state of
223Rais JP = 3/2+. In the final states of 2wPb one has J7f = 9/2+ (gs), 11/2+ (first
excited state, at 0.779 MeV), and 1512- (second excited state, at 1.423 MeV). Consequently the allowed angular momenta are 4 and 6 units of li in the first two cases, and
7 and 9 in the last one.
The favoured transitions are explained within the present version of ASAFM (one
should allow an accuracy of + I order of magnitude of any theoretical calculations),
but the hindered one needs a larger action integral, K. Such an increase can be obtained
Table 1 Hindrance factors, H = TeXP/Tee
and log Tee for I4C transitions from 223Ra(gs 3/2+) to
various excited states of 209Pbcalculated from a systematics (S) log T = j ’ ( Q - ” 2 ) of 3 even-even
neighbours (linear least-sq.) and1 within analptical superasymmetric fission model (ASA). T is given
in seconds.
~~
E*
keV
~
J?
~
Q
log Texp
I
log Tfe
log T f A
HS
NASA
0
4
6
0
4
6
0
7
9
0
2
4
0
2
0
13.21
13.36
13.60
13.86
14.90
15.14
15.40
16.22
16.89
17.29
16.52
16.59
16.76
17.51
17.58
18.47
18.54
18.71
18.58
18.65
640.3
456.4
264.9
145.8
2.5
1.4
0.8
2.4a
MeV
0
9/2+
31.852
16.02
778
11/2+
31.073
15.29
1422
15/2-
30.429
16.60a
1567
5/2+
30.285
2032
1/2+
29.820
2470
7/2+
29.382
2520
3/2+
a preliminary.
29.332
4
0
2
14.79
16.13
16.44
17.45
18.41
18.53
3.2
2.9a
0.5a
0.2a
114
Ann. Physik 3 (1994)
Table 2 log Teeand H = Texp/Tee
for 14C transitions from Z2‘Ra(gs??) to various excited states of
207Pbcalculated from a systematics (S) log T =f(Q-’’*)of 3 even-even neighbours (linear least-sq.)
and within analytical superasymmetric fission model (ASA). T is given in seconds. log TexP= 13.39.
E*
keV
J3
Q
I
0
3
0
t
0
HS
HASA
17.4
8.9
6.5
0.7
0.7
0.2
0.2
MeV
0
1/2-
32.394
570
512-
32.824
898
312-
3 1.496
12.15
13.27
13.93
1
12.44
12.58
13.53
13.55
14.17
14.19
1.3
0.3
Table3 log Teeand H = Texp/T,,for “Ne transitions from 233U(gs 5/2 *) to various excited states
of ‘”b calculated from a systematics (S)log T = f ( Q - ” 2 ) of 3 even-even neighbours (linear leastsq.) and within analytical superasymmetric fission model (ASA). T is given in seconds.
log Texp= 24.84.
E*
keV
J7
Q
1
log T:e
log T”,””
HS
HASA
23.51
23.55
23.65
23.80
24.76
24.89
25.05
25.25
25.81
26.01
26.19
26.43
26.04
26.08
26.18
44.0
21.3
19.4
15.6
11.0
MeV
0
9/2+
60.503
0
2
4
6
23.20
778
11/2+
59.724
24.44
1422
1512-
59.080
0
4
6
8
0
25.49
5
1567
5/2+
58.937
7
9
0
2
4
25.73
2.5
0.2
1.2
0.9
0.6
0.4
0.1
0.1
0.0
3.1
0.0
0.1
0.1
0.0
with a larger potential barrier, due to the so-called “specializationenergy” [I 11, in a
similar way to what has been done for spontaneous fission of odd-mass nuclei 112, 131.
This energy arises from the conservation of spin and parity of the odd particle during
the fission. With an increase of deformation (distance between centers within the singleparticle two center shell model) the odd nucleon may not be transferred on a low energy
at a level crossing, if in this way it cannot conserve the spin and parity. The corresponding barrier becomes higher and wider, compared to that of the even-even neighbour. The
examples given in [I 1- 131 are showing how the barrier height and width become larger
due to this effect. If we introduce the correction, this would produce a larger area of
the barrier E ( R )-E,,, integrated over the range of separation distances from the first
to the second turning point, (Ra,Rb). We need a difference of about 15.76 MeV * fm, in
order to get an hindrance factor of 400.
D. N. Poenaru et al., Cluster radioactivities of odd-mass nuclei
115
Table 4 log Teeand H = Tcxp/Tee
for I4C transitions from 221Fr(gs 5 / 2 - ) to various excited states
of 207Tl
calculated within analytical superasymmetric fission model (ASA). T is given in seconds.
log Texp= 14.53.
E*
keV
J?
0
Q
I
log T$tA
HASA
MeV
1/2+
31.295
0
3
13.60
13.74
8.5
6.2
35 1
3/2'
30.944
0
1
3
2.0
1.6
1.2
1340
11/2-
29.955
0
4
6
8
14.29
14.32
14.44
16.32
16.57
16.83
17.19
1670
(5/2)+
29.625
0
1
3
5
17.03
17.05
17.17
17.39
Table 5 log Teeand H = Texp/Tee
for I4C transitions from 2 2 5 A(gs
~ (3/2-)) to various excited states
of 211Bicalculated within analytical superasymmetric fission model (ASA). T is given in seconds.
log Texp= 17.16.
E*
keV
J?
Q
I
log T$A
HASA
MeV
0
(9/2)-
30.479
0
4
6
17.00
17.24
17.50
1.4
0.8
0.5
405
(7/2)-
30.074
0
2
4
17.86
17.94
18-10
0.2
0.2
0.1
766
29.818
0
18.42
832
?
(9/2)-
29.752
0
4
6
18.56
18.80
19.06
95 1
?
29.633
0
18.83
We have performed similar calculations for other odd-mass parent nuclei (see Tables
2-7). The influence of the angular momentum is not very large, as can be seen by comparing the results obtained for I = 0 and for the allowed values of 1. The spin and parity
of ''Ra in its ground state (see Table 2) and of some excited states of 2"Bi (Table 5 ) are
not known.
The Tfe values obtained from the systematics of even-even neighbours (when the
measurements are available, like in Tables 1- 3), are not essentially different from the
calculated ones T$A within ASAFM.
116
Ann. Physik 3 (1994)
Table 6 log Teeand €2 = TCXp/Tee
for 24Netransitions from 23'Pa(gs 3/2-) to various excited states
of 207Tlcalculated within analytical superasymmetric fission model (ASA). T is given in seconds.
log TexP= 22.89.
~
J?
E*
keV
0
1/2+
Q
1
log Tf:A
HASA
0
22.37
22.39
22.93
22.94
23.01
24.51
24.72
24.90
3.3
3.2
0.9
0.9
0.8
MeV
60.421
1
351
3/2+
60.070
0
59.081
3
0
1
1340
11/2-
5
7
Table 7 log Teeand H = TexP/Te,
for 23Ftransitions from 231Pa(gs 3/2-) to various excited states
of 2osPb calculated within analytical superasymmetric fission model (ASA). T is given in seconds.
log TexP= 26.02.
J?
E*
keV
Q
log T$""
I
~
O+
0
HASA
MeV
51.844
0
1
0
2
4
2615
3-
49.229
3198
5-
48.646
0
4
6
~~
24.93
24.94
29.74
29.79
29.89
30.87
31.03
31.19
~~
12.4
12.0
We got a large hindrance factor (H = 44) for "Ne decay of 233U(gs 5/2+) to the
gs (912') of 209Pb;the best fit with TeXP
= 1024.84
s is obtained for the transition to the
= 1024.89.
first excited state (11/2+ at 778 keV) with 1 = 4, TAsAFM
Other hindered transitions could be: 23Fdecay of 231Pa(gs 3/2-) to the O+ gs of
208Pb ( H = 12); I4C decay of "'Fr (gs 5/2-) to the 1/2+ gs of 207Tl(H = 8.5); I4C
decay of 221Rato the 1/2- gs of 207Pb(H= 9), and 24Ne decay of 23'Pa to the gs of
207~1.
The I4C transition from '"Ac (gs(3/2-)) to the (9/2-) gs of "'Bi seems to be not
hindered. An experimental search for the fine structure in this decay mode will be performed soon.
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