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Coupling of the pairing vibrations with the fission mode
Volume 161B, number 4,5,6
PHYSICS LETTERS
31 October 1985
C O U P L I N G OF T H E P A I R I N G V I B R A T I O N S W I T H T H E F I S S I O N M O D E ~ A. STASZCZAK, A. BARAN Department of Theoretical Physics, University MCS, Lublin, Poland
K. P O M O R S K I 1 Institute of Theoretical Physics, University of Regensburg, Regensburg, Fed. Rep. Germany
and K. B O N I N G Institute of Nuclear Studies, Warsaw, Poland
Received 24 May 1985; revised manuscript received 15 July 1985 The influence of collectivepairing vibrations on the spontaneous fissionlifetimes are studied. The Nilsson single-particle potential plus the monopolepairing residual interaction are used. The collectivehamiltonian is obtained within the cranking model. The fissionlifetimes are evaluated in the WKB approximationalong the least-action trajectory in the three-dimensional space (e24, Ap, A). When taking into account the mean pairing-field parameters (Ap, An) as dynamical variables, the theoretical fission lifetimesdecrease by a few orders of magnitude.
1. Introduction. T h e majority of the theoretical papers devoted to calculations of spontaneous-fission lifetimes (Tsf) start from the mean field single-particle hamiltonian of Nilsson [1] or W o o d s - S a x o n [2] type. The residual pairing correlations are usually taken in the stationary BCS approximation. The idea of dynamical calculations of Tsr done along the least-action trajectories was proposed in ref. [2] and then practically applied by the Warsaw-Lublin group (see e.g. ref. [3] and papers quoted there). The dynamic programming method [4], based on the two-dimensional grid in deformation space, was used there to get the least-action trajectory in a W K B sense. The least-action trajectories were traced there in deformation space. The Strutinsky renormalisation procedure was applied to calculate
Work supported in parts by Deutsche Forschungsgemeinschaft, Bonn, Fed. Rep. Germany. I On leave of absence from Department of Theoretical Physics, University MCS, Lublin, Poland.
the potential energy surface of nuclei and the mass parameters were evaluated within the cranking model. The aim of the present paper is to extend the least-action trajectory method to the space spanned by the deformation parameters (ex) as well as the collective pairing variables, e.g. the proton (Ap) and neutron (A n) energy gaps. The pairing energy gap (A) and the gauge angle (q0 describe the mean field and can be used as generators of the pairing collective motion [5]. The A parameter is connected with the vibrational series, while gauge angle q~ produces the well-known quasi-rotational b a n d built from the ground states of even-even nuclei. The mass parameters of fissioning nuclei depend strongly on the pairing energy gap (almost proportional to 1/A 2) which suggests that the coupling of the pairing vibrations with the fission m o d e has to be strong. As an example we have chosen F m isotopes for which the experimental fission lifetimes are well known. We have performed the numerical calculations in the
0 3 7 0 - 2 6 9 3 / 8 5 / $ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
227
Volume 161B, number 4,5,6
PHYSICS LETTERS
(E24, Ap, An) space in order to estimate the magnitude of this coupling. Our model is similar to that proposed and discussed schematically in ref. [6], where the pairing gap was introduced as a dynamical variable in the penetration of a fission barrier. But contrary to ref. [6] we take into account two kinds of particles and all components of the mass tensor. Also all quantities appearing in the final formula for Tsf are evaluated microscopically, not only estimated as it was done in ref. [6].
2. The model The mass parameters were evaluated within the cranking model [2,5]: nqq, 2h 2 ~ =
(BfSla_~a~/OqlBCS)
~,,~>0
× (BCSI(O/Oq')a;a+_~IBCS)(E~ + E~)-I,
(1) where q = (e24, A) and a~+ creates a quasiparticle with energy E~ in the state 1~). The potential energy is obtained using the Strutinsky prescription [2]: V = ELD q- EBCs --/~,
(2)
i.e. replacing the smooth part of the BCS energy by the liquid-drop energy. No zero-point correction energy was added to the potential energy surface (confer ref. [7]). The WKB theory gives the following estimates of the spontaneous fission halflives [2]: T~f= l n 2 / n [1 + exp (S)],
(3)
where S is the action integral equal to
s = 2 f~?(2IV- EIB)I/2 ds.
(4)
along the path of fission (s). The turning points (01 , 02) are the lower and upper limits of the integral. E is the energy of the fissioning nucleus and, for spontaneous fission, is equal to the ground-state energy. The number of assaults of a given nucleus on the fission barrier per unit time (n) is assumed to be 7.62 × 10 27 yr -1, which corresponds to a typical frequency of the fission mode hw = 1 MeV. The effective inertia along the path of fission (B) contains also contributions from other than 228
s
=
e24
31 October 1985
components of the mass tensor:
B = B~z4~:4+
Bapap(dAp/de24) 2 " "
+ BAnAn(dAn//de24) 2 + 2Be24An dA n/de24
+ 2 B,2,Ap dAp/de24.
(5)
The first term is leading only when d A i / d e is small. Using the numerical technique described in ref. [3] we have estimated the least-action trajectory in the (e24 , Ap, An) space and with eq. (3) we have evaluated the fission halflives.
3. Results and discussion. The numerical calculations were performed with a standard set of parameters, the same as in ref. [3]. The mass parameters and the potential energy were evaluated along the symmetric path to fission e24 = (e, e4(e)) and for different values of protons and neutrons gaps: Ap and, A n. No corrections for the y non-axiality and the left-right asymmetry are added to the potential energy surface. The even-even isotopes of Fm with N = 134-164 were chosen as examples. The estimations of the fission lifetimes were performed along the static path to fission, i.e. assuming the BCS solutions for Ap and A n and along the dynamic trajectory. The fission barriers, the mass parameters along both paths and the gap parameters (Ap and An) of 252Fm are plotted in fig. 1. It is seen that, when comparing the results for the dynamic and static paths, the fission barrier is increased by 1.4 MeV while the mass parameter B is on the average 2 times smaller. Also the shell fluctuations in the mass parameters along the dynamic trajectory are less pronounced than along the static one. The enhancement of the dynamic inertia (B) around the ground-state deformation is due to the large value of d A / d e when the dynamic path begins (see bottom part of fig. 1 and eq. (5)). Nevertheless, that enhancement of B gives rather small contributions to the path-integral (4) because the facto~ ( V - E) is small when the path begins. The gap parameters minimizing the action integral (4) lie pretty far from the BCS solutions. The spontaneous fission halflife of 252Fm evaluated along the static
Volume 161B, number 4,5,6 i
PHYSICS LETTERS
i
I
i
'
J
tog
X
v
31 October 1985
tyr] /
\\
//
-3
",,"
sfo.f
-6
-6
-12 I
,
,
,
I
,
,
,
I
,
,
,
I
800: ; B ['h~/MeV]
/ ~00
s
,1~\,.JstQt
~,_y
\ ,.--.._/
/
-18
!
~ i
N ~
,
]
I/+0
'.1
i
~
t
J
I
150
k
t
J
k
[
~
i
160
Fig. 2. Theoretical estimates of the spontaneous fission halflives T~f for Fm isotopes compared with the experimental data. I
0 S
,
,
,
I
,
,
,
I
,
,
,
i
,
,
,
A [MeV]
S
1.0 1.5 I 0.5 0
• . -prof. I
0.2
,
,
,
I
O.t,
-- - neul. ,
,
,
I
0.6
,
,
,
I
0.8
,
,
,
E:24
Fig. 1. The fission barrier and the mass parameter for fission mode of 252Fm along the dynamic and static trajectories in (~24, Ap, A ) space. The trajectories are shown in the lower part of the figure.
(traditional) p a t h is 6 orders of magnitude larger than T~f for the dynamical trajectory. These results are in line with the conclusions drawn in ref. [6]. T h e systematic data from F m isotopes are presented in fig. 2. The experimental values of Tsf are taken f r o m the compilation made in ref. [3]. T h e static estimates of T~ lie always above the experimental points, the dynamic results are m u c h closer to the latter. Both these estimates were m a d e without adding any zero-point energy ( E o = 0) to the Strutinsky ground-state energy of the nuclei. Traditionally, an energy E 0 = 0.5 MeV was a d d e d (see e.g. refs. [2,3]) but, as it was shown in ref. [6], such a procedure is inconsistent.
Nevertheless, the dynamical estimates of Tsf with E o -- 0.5 M e V are plotted in fig. 2 for comparison. T a k i n g into account E 0 = 0.5 MeV instead of E o = 0 decreases the fission lifetimes by 3 orders of m a g n i t u d e on the average. T h e rather p o o r agreement of the theoretical estimates with the experimental data is connected with the fact that we have not taken into account the nonaxial (3,) and the left-right (%s) degrees of freedom, also the neck parameter (e4) was not i n d e p e n d e n t on the elongation (e).*x It seems also that the single-particle potential parameters (set " A = 242") f r o m ref. [1]) do not reproduce properly the shell structure of the F m isotopes. O u r calculations have shown that the collective pairing degrees of freedom are strongly coupled with the fission m o d e and can not be omitted in Tsf calculations. The fission barriers along the
*~ The action integral (and consequently T~t) must not necessarily be reduced when more deformation parameters are introduced. It could happen (and it does often) that the new degrees of freedom push down the ground-state energy and the effective fission barrier (and afterwards Tsf) is increased. 229
Volume 161B, number 4,5,6
PHYSICS LETTERS
dynamical trajectories are only by 18% higher than the static (BCS) ones, while the dynamical mass parameters are reduced, on the average, by a factor 2 with respect to the traditional (static) ones. This big reduction of the collective inertia is due to its strong dependence on pairing gaps. C o n s e q u e n t l y the action integral (4) for the dynamical p a t h in (e24, Ap, An) space is smaller by 5 - 1 5 units as c o m p a r e d with the static result. This effect causes the dynamical estimates of the fission lifetimes to lie a few orders of magnitude ( 2 - 6 ) below the static ones, i.e. those evaluated along the lines Ap(e24 ) and An(e24 ) corresponding to the BCS solutions.
230
31 October 1985
References [1] S.G. Nilsson, C.F. Tsang, A. Sobiczewski,Z. Szymahski, S. Wycech, C. Gustafson, I.L. Lamm, P. MSller and B. Nilsson, Nucl. Phys. A131 (1969) 1. [2] M. Brack, J. Damgaard, A.S. Jensen, H.C. Pauli, V.M. Strutinsky and C.Y. Wong, Rev. Mod. Phys. 44 (1972) 320. [3] A. Baran, K. Pomorski, A. Lukasiak and A. Sobiczewski, Nucl. Phys. A36! (1981) 83. [4] R.E. Bellman and R.E. Kalaba, Quasilinearization and nonlinear boundary-value problem (Elsevier, Amsterdam, 1965) Ch. 7. [5] D.R. B~s, R.A. Broglia, R.P.J. Perazzo and K. Kumar, Nucl. Phys. A143 (1970) 1. [6] L.G. Moretto and R.P. Babinet, Phys. Lett. 49B (1974) 147. [7] A. Gr~dL K. Pomorski, M. Brack and E. Werner, Nucl. Phys. A442 (1985) 26.
PHYSICS LETTERS
31 October 1985
C O U P L I N G OF T H E P A I R I N G V I B R A T I O N S W I T H T H E F I S S I O N M O D E ~ A. STASZCZAK, A. BARAN Department of Theoretical Physics, University MCS, Lublin, Poland
K. P O M O R S K I 1 Institute of Theoretical Physics, University of Regensburg, Regensburg, Fed. Rep. Germany
and K. B O N I N G Institute of Nuclear Studies, Warsaw, Poland
Received 24 May 1985; revised manuscript received 15 July 1985 The influence of collectivepairing vibrations on the spontaneous fissionlifetimes are studied. The Nilsson single-particle potential plus the monopolepairing residual interaction are used. The collectivehamiltonian is obtained within the cranking model. The fissionlifetimes are evaluated in the WKB approximationalong the least-action trajectory in the three-dimensional space (e24, Ap, A). When taking into account the mean pairing-field parameters (Ap, An) as dynamical variables, the theoretical fission lifetimesdecrease by a few orders of magnitude.
1. Introduction. T h e majority of the theoretical papers devoted to calculations of spontaneous-fission lifetimes (Tsf) start from the mean field single-particle hamiltonian of Nilsson [1] or W o o d s - S a x o n [2] type. The residual pairing correlations are usually taken in the stationary BCS approximation. The idea of dynamical calculations of Tsr done along the least-action trajectories was proposed in ref. [2] and then practically applied by the Warsaw-Lublin group (see e.g. ref. [3] and papers quoted there). The dynamic programming method [4], based on the two-dimensional grid in deformation space, was used there to get the least-action trajectory in a W K B sense. The least-action trajectories were traced there in deformation space. The Strutinsky renormalisation procedure was applied to calculate
Work supported in parts by Deutsche Forschungsgemeinschaft, Bonn, Fed. Rep. Germany. I On leave of absence from Department of Theoretical Physics, University MCS, Lublin, Poland.
the potential energy surface of nuclei and the mass parameters were evaluated within the cranking model. The aim of the present paper is to extend the least-action trajectory method to the space spanned by the deformation parameters (ex) as well as the collective pairing variables, e.g. the proton (Ap) and neutron (A n) energy gaps. The pairing energy gap (A) and the gauge angle (q0 describe the mean field and can be used as generators of the pairing collective motion [5]. The A parameter is connected with the vibrational series, while gauge angle q~ produces the well-known quasi-rotational b a n d built from the ground states of even-even nuclei. The mass parameters of fissioning nuclei depend strongly on the pairing energy gap (almost proportional to 1/A 2) which suggests that the coupling of the pairing vibrations with the fission m o d e has to be strong. As an example we have chosen F m isotopes for which the experimental fission lifetimes are well known. We have performed the numerical calculations in the
0 3 7 0 - 2 6 9 3 / 8 5 / $ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
227
Volume 161B, number 4,5,6
PHYSICS LETTERS
(E24, Ap, An) space in order to estimate the magnitude of this coupling. Our model is similar to that proposed and discussed schematically in ref. [6], where the pairing gap was introduced as a dynamical variable in the penetration of a fission barrier. But contrary to ref. [6] we take into account two kinds of particles and all components of the mass tensor. Also all quantities appearing in the final formula for Tsf are evaluated microscopically, not only estimated as it was done in ref. [6].
2. The model The mass parameters were evaluated within the cranking model [2,5]: nqq, 2h 2 ~ =
(BfSla_~a~/OqlBCS)
~,,~>0
× (BCSI(O/Oq')a;a+_~IBCS)(E~ + E~)-I,
(1) where q = (e24, A) and a~+ creates a quasiparticle with energy E~ in the state 1~). The potential energy is obtained using the Strutinsky prescription [2]: V = ELD q- EBCs --/~,
(2)
i.e. replacing the smooth part of the BCS energy by the liquid-drop energy. No zero-point correction energy was added to the potential energy surface (confer ref. [7]). The WKB theory gives the following estimates of the spontaneous fission halflives [2]: T~f= l n 2 / n [1 + exp (S)],
(3)
where S is the action integral equal to
s = 2 f~?(2IV- EIB)I/2 ds.
(4)
along the path of fission (s). The turning points (01 , 02) are the lower and upper limits of the integral. E is the energy of the fissioning nucleus and, for spontaneous fission, is equal to the ground-state energy. The number of assaults of a given nucleus on the fission barrier per unit time (n) is assumed to be 7.62 × 10 27 yr -1, which corresponds to a typical frequency of the fission mode hw = 1 MeV. The effective inertia along the path of fission (B) contains also contributions from other than 228
s
=
e24
31 October 1985
components of the mass tensor:
B = B~z4~:4+
Bapap(dAp/de24) 2 " "
+ BAnAn(dAn//de24) 2 + 2Be24An dA n/de24
+ 2 B,2,Ap dAp/de24.
(5)
The first term is leading only when d A i / d e is small. Using the numerical technique described in ref. [3] we have estimated the least-action trajectory in the (e24 , Ap, An) space and with eq. (3) we have evaluated the fission halflives.
3. Results and discussion. The numerical calculations were performed with a standard set of parameters, the same as in ref. [3]. The mass parameters and the potential energy were evaluated along the symmetric path to fission e24 = (e, e4(e)) and for different values of protons and neutrons gaps: Ap and, A n. No corrections for the y non-axiality and the left-right asymmetry are added to the potential energy surface. The even-even isotopes of Fm with N = 134-164 were chosen as examples. The estimations of the fission lifetimes were performed along the static path to fission, i.e. assuming the BCS solutions for Ap and A n and along the dynamic trajectory. The fission barriers, the mass parameters along both paths and the gap parameters (Ap and An) of 252Fm are plotted in fig. 1. It is seen that, when comparing the results for the dynamic and static paths, the fission barrier is increased by 1.4 MeV while the mass parameter B is on the average 2 times smaller. Also the shell fluctuations in the mass parameters along the dynamic trajectory are less pronounced than along the static one. The enhancement of the dynamic inertia (B) around the ground-state deformation is due to the large value of d A / d e when the dynamic path begins (see bottom part of fig. 1 and eq. (5)). Nevertheless, that enhancement of B gives rather small contributions to the path-integral (4) because the facto~ ( V - E) is small when the path begins. The gap parameters minimizing the action integral (4) lie pretty far from the BCS solutions. The spontaneous fission halflife of 252Fm evaluated along the static
Volume 161B, number 4,5,6 i
PHYSICS LETTERS
i
I
i
'
J
tog
X
v
31 October 1985
tyr] /
\\
//
-3
",,"
sfo.f
-6
-6
-12 I
,
,
,
I
,
,
,
I
,
,
,
I
800: ; B ['h~/MeV]
/ ~00
s
,1~\,.JstQt
~,_y
\ ,.--.._/
/
-18
!
~ i
N ~
,
]
I/+0
'.1
i
~
t
J
I
150
k
t
J
k
[
~
i
160
Fig. 2. Theoretical estimates of the spontaneous fission halflives T~f for Fm isotopes compared with the experimental data. I
0 S
,
,
,
I
,
,
,
I
,
,
,
i
,
,
,
A [MeV]
S
1.0 1.5 I 0.5 0
• . -prof. I
0.2
,
,
,
I
O.t,
-- - neul. ,
,
,
I
0.6
,
,
,
I
0.8
,
,
,
E:24
Fig. 1. The fission barrier and the mass parameter for fission mode of 252Fm along the dynamic and static trajectories in (~24, Ap, A ) space. The trajectories are shown in the lower part of the figure.
(traditional) p a t h is 6 orders of magnitude larger than T~f for the dynamical trajectory. These results are in line with the conclusions drawn in ref. [6]. T h e systematic data from F m isotopes are presented in fig. 2. The experimental values of Tsf are taken f r o m the compilation made in ref. [3]. T h e static estimates of T~ lie always above the experimental points, the dynamic results are m u c h closer to the latter. Both these estimates were m a d e without adding any zero-point energy ( E o = 0) to the Strutinsky ground-state energy of the nuclei. Traditionally, an energy E 0 = 0.5 MeV was a d d e d (see e.g. refs. [2,3]) but, as it was shown in ref. [6], such a procedure is inconsistent.
Nevertheless, the dynamical estimates of Tsf with E o -- 0.5 M e V are plotted in fig. 2 for comparison. T a k i n g into account E 0 = 0.5 MeV instead of E o = 0 decreases the fission lifetimes by 3 orders of m a g n i t u d e on the average. T h e rather p o o r agreement of the theoretical estimates with the experimental data is connected with the fact that we have not taken into account the nonaxial (3,) and the left-right (%s) degrees of freedom, also the neck parameter (e4) was not i n d e p e n d e n t on the elongation (e).*x It seems also that the single-particle potential parameters (set " A = 242") f r o m ref. [1]) do not reproduce properly the shell structure of the F m isotopes. O u r calculations have shown that the collective pairing degrees of freedom are strongly coupled with the fission m o d e and can not be omitted in Tsf calculations. The fission barriers along the
*~ The action integral (and consequently T~t) must not necessarily be reduced when more deformation parameters are introduced. It could happen (and it does often) that the new degrees of freedom push down the ground-state energy and the effective fission barrier (and afterwards Tsf) is increased. 229
Volume 161B, number 4,5,6
PHYSICS LETTERS
dynamical trajectories are only by 18% higher than the static (BCS) ones, while the dynamical mass parameters are reduced, on the average, by a factor 2 with respect to the traditional (static) ones. This big reduction of the collective inertia is due to its strong dependence on pairing gaps. C o n s e q u e n t l y the action integral (4) for the dynamical p a t h in (e24, Ap, An) space is smaller by 5 - 1 5 units as c o m p a r e d with the static result. This effect causes the dynamical estimates of the fission lifetimes to lie a few orders of magnitude ( 2 - 6 ) below the static ones, i.e. those evaluated along the lines Ap(e24 ) and An(e24 ) corresponding to the BCS solutions.
230
31 October 1985
References [1] S.G. Nilsson, C.F. Tsang, A. Sobiczewski,Z. Szymahski, S. Wycech, C. Gustafson, I.L. Lamm, P. MSller and B. Nilsson, Nucl. Phys. A131 (1969) 1. [2] M. Brack, J. Damgaard, A.S. Jensen, H.C. Pauli, V.M. Strutinsky and C.Y. Wong, Rev. Mod. Phys. 44 (1972) 320. [3] A. Baran, K. Pomorski, A. Lukasiak and A. Sobiczewski, Nucl. Phys. A36! (1981) 83. [4] R.E. Bellman and R.E. Kalaba, Quasilinearization and nonlinear boundary-value problem (Elsevier, Amsterdam, 1965) Ch. 7. [5] D.R. B~s, R.A. Broglia, R.P.J. Perazzo and K. Kumar, Nucl. Phys. A143 (1970) 1. [6] L.G. Moretto and R.P. Babinet, Phys. Lett. 49B (1974) 147. [7] A. Gr~dL K. Pomorski, M. Brack and E. Werner, Nucl. Phys. A442 (1985) 26.