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Proton radioactivity and the proton drip line
Nuclear Physics A 752 (2005) 223c–226c
Proton radioactivity and the proton drip line L´idia S. Ferreira a∗ and Enrico Maglioneb a
Centro de F´ısica das Interac¸c˜oes Fundamentais, and Departamento de F´ısica, Instituto Superior T´ecnico, Av. Rovisco Pais, P1049-001 Lisbon, Portugal. b
Dipartimento di Fisica “G. Galilei”, Via Marzolo 8, I-35131 Padova, Italy and Istituto Nazionale di Fisica Nucleare, Padova, Italy. Proton radioactivity from deformed drip–line nuclei is discussed. It is shown that all experimental data currently available on decay from ground and isomeric states, and fine structure of odd-even and odd–odd nuclei, can be interpreted within a consistent theoretical approach, the non–adiabatic quasi-particle model. 1. Introduction Recent studies with exotic nuclei lead to the discovery of a new form of radioactivity[1] in nuclei lying beyond the proton drip-line with 50 < Z <83, where the Coulomb barrier is very high and protons can be trapped in a resonance state. Spherical and deformed emitters were observed, decaying mainly from the ground state, but decay from isomeric excited states and fine structure were also seen. Proton radioactivity is a very important phenomena in nuclear physics, since measurements of the separation energy of the proton in these exotic nuclei, provide a unique way of testing mass formulae, and map the proton drip-line. The use of tagging techniques with the proton, can also determine the decay spectrum of the nucleus, impossible to obtain otherwise. The escape energy of the emitted proton is very small, corresponding to a single particle resonance very low in the continuum. Due to this fact, and the large potential barrier, a simple WKB calculations can already give a good estimate of the experimental data[2] of spherical nuclei. The purpose of this work is to show that decay from deformed nuclei can be studied as decay from a Nilsson resonance[3,4]. As a first approach, the parent nucleus is treated in the strong coupling limit of the particle-rotor model[5]. The inclusion[6] of Coriolis mixing, requires a proper treatment of the pairing residual interaction, leading to a unified interpretation of all available data on these emitters. 2. Decay widths for deformed proton emitters: Adiabatic approach
Considering proton radioactivity from deformed nuclei with an even number of neutrons and odd Z. The states close to the Fermi surface are the most probable ones for decay to occur and the corresponding wave function of the decaying proton, can then be ∗
Work supported by Funda¸c˜ao de Ciˆencia e Tecnologia (Portugal), Grant N. 36575/99 and FEDER.
0375-9474/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2005.02.037
224c
L.S. Ferreira, E. Maglione / Nuclear Physics A 752 (2005) 223c–226c
obtained from the exact solution of the Schr¨odinger equation in a deformed mean field with deformed spin–orbit, imposing outgoing wave boundary conditions, as discussed in Ref.[7,8]. The simplest approach to determine the wave function of the parent nucleus is to impose the strong coupling limit[5], where the nucleus is described by a particle plus rotor model, with infinite moment of inertia. Within these assumptions, if decay occurs to the ground state, only the component of the s.p. wave function with the same angular momentum as the ground state contributes, and the decay width can be determined from, Γlp jp (r) =
|ulp jp (r)|2 h ¯ 2k u2 , µ(jp + 1/2) |Glp (kr) + iFlp (kr)|2 Ki
(1)
where F and G are the regular and irregular Coulomb functions, respectively, and u lp jp the component of the wave function with momentum j p , equal to the spin of the decaying nucleus. The quantity u2Ki is the probability that the single particle level in the daughter nucleus is empty, evaluated in the BCS approach. The decay width obtained from Eq. 1 depends on deformation, and is very sensitive to the wave function of the decaying state. Therefore, will give clear information on the deformation and properties of the decaying state. Decay to excited states, allow few combinations for lp jp according to angular momentum coupling rules, and consequently different components of the parent wave function are tested. Table 1 Total angular momentum and deformation that reproduce the experimental half–lives for the measured deformed odd–even proton emitters compared with the predictions of Ref. [9]. The theoretical results are from Refs. [3,10,11]. Proton decay J 109
I Cs 117 La 117m La 131 Eu 141 Ho 141m Ho 151 Lu 151m Lu 113
1/2+ 3/2+ 3/2+ 9/2+ 3/2+ 7/2– 1/2+ 5/2– 3/2+
β 0.14 0.15 ÷ 0.20 0.20 ÷ 0.30 0.25 ÷ 0.35 0.27 ÷ 0.34 0.30 ÷ 0.40 0.30 ÷ 0.40 –0.18 ÷–0.14 –0.18 ÷–0.14
M¨oller–Nix J
β
1/2+ 3/2+ 3/2+
0.16 0.21 0.29
3/2+ 7/2–
0.33 0.29
5/2–
–0.16
We have applied[3,4,10,11] our model to all measured deformed proton emitters including isomeric decays. The results are shown in Table 1. The experimental half-lives are
L.S. Ferreira, E. Maglione / Nuclear Physics A 752 (2005) 223c–226c
225c
perfectly reproduced by a specific state, with defined quantum numbers and deformation, thus leading to unambiguous assignments of the angular momentum of the decaying states. Extra experimental information provided by isomeric decay observed in 117 La, 141 Ho and 151 Lu, and fine structure in 131 Eu can also be successfully accounted by the model. The experimental half–lives for decay from the excited states were reproduced in a consistent way with the same deformation that describes ground state emission. Similar calculations were done within the coupled channel Green’s function model[12]. Emission from deformed systems with an odd number of protons and neutrons can be discussed in a similar fashion[4]. However, in contrast with decay to ground state of odd-even nuclei where the proton is forced to escape with a specific angular momentum, many channels will be open due to the angular momentum coupling of the proton and daughter nucleus, which has angular momentum equal to the one of the neutron. The unpaired neutron, will contribute significantly with its angular momentum to decay, and at the same time, its Nilsson state can be determined. As in the case of odd-even nuclei, Table 2 shows a perfect description of experimental decay rates for deformations in agreement with predictions of Ref. [9]. The same Nilsson state of the odd proton is used in the calculation of odd–odd and neighbour odd–even nuclei. Similar deformations were also found for the odd-odd and nearby odd-even nuclei. This represents a further consistency check of the model. Table 2 As in Table 1 for the measured odd–odd proton emitters, taken from Ref. [4]. The quantities Kp and Kn are the magnetic quantum numbers of the proton and neutron Nilsson wave functions, J the total angular momentum of the parent nucleus, and β and βM the deformation coming from the calculation and predictions of Ref. [9], respectively.
112
Cs Ho 150 Lu 150m Lu 140
Kp
Kn
J
β
βM
3/2+ 7/2– 5/2– 3/2+
3/2+ 9/2–, 7/2+ 1/2– 1/2–
0+, 3+ 8+, 0– 2+ 1–, 2–
0.12 ÷ 0.22 0.26 ÷ 0.34 –0.15 ÷–0.17 –0.22 ÷ 0.00
0.21 0.30 –0.16
3. The non–adiabatic quasi-particle approach: contributions from Coriolis mixing and pairing residual interaction As we have seen, calculations within the strong coupling limit were able to reproduce the experimental results. Including the rotational spectrum of the daughter nucleus, the Coriolis force has to be taken into account. This effect, was studied within the nonadiabatic coupled channel[13], and coupled-channel Green’s function[14] methods, but the agreement with experiment found in the adiabatic context was lost, a surprising result, since calculations with Coriolis mixing should be better. Decay rates in deformed
226c
L.S. Ferreira, E. Maglione / Nuclear Physics A 752 (2005) 223c–226c
nuclei, are extremely sensitive to small components of the wave function. The Coriolis interaction mixes different Nilsson states, and can be responsible for strong changes in the decay widths. However, the residual pairing interaction can modify this mixing of states, an effect not considered in the calculations of Ref.[13,14]. We have included[6] beside the Coriolis mixing, the pairing residual interaction in the BCS approach. The mixing of states is modified by the residual interaction and transformed trough a Bogoliubov transformation into a mixing between quasi-particle states. The wave function that describes the decaying nucleus corresponds, in the particle case, to the highest state in energy, while in the quasi-particles to the lowest one, as it should be. This inversion implies a change of sign of the wave function components, leading to interference in the calculation of the decay width, destructive for particles obliging the width to decrease, and constructive for quasi-particles. The width is enhanced, and the adiabatic results are recovered. Such non-adiabatic treatment of the Coriolis coupling, brings back the perfect agreement with data observed in the strong coupling limit. 4. Conclusions We have presented a unified model to describe proton radioactivity from deformed nuclei. Decay is understood as decay from single particle Nilsson resonances, evaluated exactly from potentials that fit large sets of data on nuclear properties. The rotational spectra of the daughter nucleus, and the pairing residual interaction in the BCS approach, are taken into account, leading to a treatment of the Coriolis coupling in terms of quasi– particles. All available experimental data on even–odd and odd–odd deformed proton emitters from the ground and isomeric states and fine structure, are accurately and consistently reproduced by the model. The angular momentum of the decaying nucleus, and the state of the unpaired neutron in decay from odd–odd nuclei, are identified without ambiguity. REFERENCES 1. A. A. Sonzogni, Nuclear Data Sheets 95 (2002) 1. ˚berg, P. B. Semmes and W. Nazarewicz, Phys. Rev. C56 (1997) 1762. 2. S. A 3. E. Maglione, L. S. Ferreira and R. J. Liotta, Phys. Rev. Lett. 81 (1998) 538; Phys. Rev. C59 (1999) R589. 4. L. S. Ferreira and E. Maglione, Phys. Rev. Lett. 86 (2001) 1721. 5. D. D. Bogdanov, V. P. Bugrov, S. G. Kadmenskii, Sov. J. Nucl. Phys. 52 (1990) 229. 6. G. Fiorin, E. Maglione and L. S. Ferreira, Phys. Rev. C67 (2003) 054302. 7. L. S. Ferreira, E. Maglione and R. J. Liotta, Phys. Rev. Lett. 78 (1997) 1640. 8. L. S. Ferreira, E. Maglione, and D. Fernandes, Phys. Rev. C65 (2002) 024323. 9. P. M¨oller, et al., At. Data Nucl. Data Tab. 59 (1995) 185; ibd. 66 (1997) 131. 10. L. S. Ferreira and E. Maglione, Phys. Rev. C61 (2000) 021304(R). 11. E. Maglione and L.S. Ferreira, Phys. Rev. C61 (2000) 47307. 12. C. N. Davids, et al., Phys. Rev. Lett. 80 (1998) 1849. 13. A. T. Kruppa, et al., Phys. Rev. Lett. 84 (2000) 4549; B. Barmore, et al., Phys. Rev. C62 (2000) 054315; W. Kr´olas, et al., Phys. Rev. C65 (2002) 031303(R). 14. H. Esbensen and C. N. Davids, Phys. Rev. C63 (2001) 014315.
Proton radioactivity and the proton drip line L´idia S. Ferreira a∗ and Enrico Maglioneb a
Centro de F´ısica das Interac¸c˜oes Fundamentais, and Departamento de F´ısica, Instituto Superior T´ecnico, Av. Rovisco Pais, P1049-001 Lisbon, Portugal. b
Dipartimento di Fisica “G. Galilei”, Via Marzolo 8, I-35131 Padova, Italy and Istituto Nazionale di Fisica Nucleare, Padova, Italy. Proton radioactivity from deformed drip–line nuclei is discussed. It is shown that all experimental data currently available on decay from ground and isomeric states, and fine structure of odd-even and odd–odd nuclei, can be interpreted within a consistent theoretical approach, the non–adiabatic quasi-particle model. 1. Introduction Recent studies with exotic nuclei lead to the discovery of a new form of radioactivity[1] in nuclei lying beyond the proton drip-line with 50 < Z <83, where the Coulomb barrier is very high and protons can be trapped in a resonance state. Spherical and deformed emitters were observed, decaying mainly from the ground state, but decay from isomeric excited states and fine structure were also seen. Proton radioactivity is a very important phenomena in nuclear physics, since measurements of the separation energy of the proton in these exotic nuclei, provide a unique way of testing mass formulae, and map the proton drip-line. The use of tagging techniques with the proton, can also determine the decay spectrum of the nucleus, impossible to obtain otherwise. The escape energy of the emitted proton is very small, corresponding to a single particle resonance very low in the continuum. Due to this fact, and the large potential barrier, a simple WKB calculations can already give a good estimate of the experimental data[2] of spherical nuclei. The purpose of this work is to show that decay from deformed nuclei can be studied as decay from a Nilsson resonance[3,4]. As a first approach, the parent nucleus is treated in the strong coupling limit of the particle-rotor model[5]. The inclusion[6] of Coriolis mixing, requires a proper treatment of the pairing residual interaction, leading to a unified interpretation of all available data on these emitters. 2. Decay widths for deformed proton emitters: Adiabatic approach
Considering proton radioactivity from deformed nuclei with an even number of neutrons and odd Z. The states close to the Fermi surface are the most probable ones for decay to occur and the corresponding wave function of the decaying proton, can then be ∗
Work supported by Funda¸c˜ao de Ciˆencia e Tecnologia (Portugal), Grant N. 36575/99 and FEDER.
0375-9474/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2005.02.037
224c
L.S. Ferreira, E. Maglione / Nuclear Physics A 752 (2005) 223c–226c
obtained from the exact solution of the Schr¨odinger equation in a deformed mean field with deformed spin–orbit, imposing outgoing wave boundary conditions, as discussed in Ref.[7,8]. The simplest approach to determine the wave function of the parent nucleus is to impose the strong coupling limit[5], where the nucleus is described by a particle plus rotor model, with infinite moment of inertia. Within these assumptions, if decay occurs to the ground state, only the component of the s.p. wave function with the same angular momentum as the ground state contributes, and the decay width can be determined from, Γlp jp (r) =
|ulp jp (r)|2 h ¯ 2k u2 , µ(jp + 1/2) |Glp (kr) + iFlp (kr)|2 Ki
(1)
where F and G are the regular and irregular Coulomb functions, respectively, and u lp jp the component of the wave function with momentum j p , equal to the spin of the decaying nucleus. The quantity u2Ki is the probability that the single particle level in the daughter nucleus is empty, evaluated in the BCS approach. The decay width obtained from Eq. 1 depends on deformation, and is very sensitive to the wave function of the decaying state. Therefore, will give clear information on the deformation and properties of the decaying state. Decay to excited states, allow few combinations for lp jp according to angular momentum coupling rules, and consequently different components of the parent wave function are tested. Table 1 Total angular momentum and deformation that reproduce the experimental half–lives for the measured deformed odd–even proton emitters compared with the predictions of Ref. [9]. The theoretical results are from Refs. [3,10,11]. Proton decay J 109
I Cs 117 La 117m La 131 Eu 141 Ho 141m Ho 151 Lu 151m Lu 113
1/2+ 3/2+ 3/2+ 9/2+ 3/2+ 7/2– 1/2+ 5/2– 3/2+
β 0.14 0.15 ÷ 0.20 0.20 ÷ 0.30 0.25 ÷ 0.35 0.27 ÷ 0.34 0.30 ÷ 0.40 0.30 ÷ 0.40 –0.18 ÷–0.14 –0.18 ÷–0.14
M¨oller–Nix J
β
1/2+ 3/2+ 3/2+
0.16 0.21 0.29
3/2+ 7/2–
0.33 0.29
5/2–
–0.16
We have applied[3,4,10,11] our model to all measured deformed proton emitters including isomeric decays. The results are shown in Table 1. The experimental half-lives are
L.S. Ferreira, E. Maglione / Nuclear Physics A 752 (2005) 223c–226c
225c
perfectly reproduced by a specific state, with defined quantum numbers and deformation, thus leading to unambiguous assignments of the angular momentum of the decaying states. Extra experimental information provided by isomeric decay observed in 117 La, 141 Ho and 151 Lu, and fine structure in 131 Eu can also be successfully accounted by the model. The experimental half–lives for decay from the excited states were reproduced in a consistent way with the same deformation that describes ground state emission. Similar calculations were done within the coupled channel Green’s function model[12]. Emission from deformed systems with an odd number of protons and neutrons can be discussed in a similar fashion[4]. However, in contrast with decay to ground state of odd-even nuclei where the proton is forced to escape with a specific angular momentum, many channels will be open due to the angular momentum coupling of the proton and daughter nucleus, which has angular momentum equal to the one of the neutron. The unpaired neutron, will contribute significantly with its angular momentum to decay, and at the same time, its Nilsson state can be determined. As in the case of odd-even nuclei, Table 2 shows a perfect description of experimental decay rates for deformations in agreement with predictions of Ref. [9]. The same Nilsson state of the odd proton is used in the calculation of odd–odd and neighbour odd–even nuclei. Similar deformations were also found for the odd-odd and nearby odd-even nuclei. This represents a further consistency check of the model. Table 2 As in Table 1 for the measured odd–odd proton emitters, taken from Ref. [4]. The quantities Kp and Kn are the magnetic quantum numbers of the proton and neutron Nilsson wave functions, J the total angular momentum of the parent nucleus, and β and βM the deformation coming from the calculation and predictions of Ref. [9], respectively.
112
Cs Ho 150 Lu 150m Lu 140
Kp
Kn
J
β
βM
3/2+ 7/2– 5/2– 3/2+
3/2+ 9/2–, 7/2+ 1/2– 1/2–
0+, 3+ 8+, 0– 2+ 1–, 2–
0.12 ÷ 0.22 0.26 ÷ 0.34 –0.15 ÷–0.17 –0.22 ÷ 0.00
0.21 0.30 –0.16
3. The non–adiabatic quasi-particle approach: contributions from Coriolis mixing and pairing residual interaction As we have seen, calculations within the strong coupling limit were able to reproduce the experimental results. Including the rotational spectrum of the daughter nucleus, the Coriolis force has to be taken into account. This effect, was studied within the nonadiabatic coupled channel[13], and coupled-channel Green’s function[14] methods, but the agreement with experiment found in the adiabatic context was lost, a surprising result, since calculations with Coriolis mixing should be better. Decay rates in deformed
226c
L.S. Ferreira, E. Maglione / Nuclear Physics A 752 (2005) 223c–226c
nuclei, are extremely sensitive to small components of the wave function. The Coriolis interaction mixes different Nilsson states, and can be responsible for strong changes in the decay widths. However, the residual pairing interaction can modify this mixing of states, an effect not considered in the calculations of Ref.[13,14]. We have included[6] beside the Coriolis mixing, the pairing residual interaction in the BCS approach. The mixing of states is modified by the residual interaction and transformed trough a Bogoliubov transformation into a mixing between quasi-particle states. The wave function that describes the decaying nucleus corresponds, in the particle case, to the highest state in energy, while in the quasi-particles to the lowest one, as it should be. This inversion implies a change of sign of the wave function components, leading to interference in the calculation of the decay width, destructive for particles obliging the width to decrease, and constructive for quasi-particles. The width is enhanced, and the adiabatic results are recovered. Such non-adiabatic treatment of the Coriolis coupling, brings back the perfect agreement with data observed in the strong coupling limit. 4. Conclusions We have presented a unified model to describe proton radioactivity from deformed nuclei. Decay is understood as decay from single particle Nilsson resonances, evaluated exactly from potentials that fit large sets of data on nuclear properties. The rotational spectra of the daughter nucleus, and the pairing residual interaction in the BCS approach, are taken into account, leading to a treatment of the Coriolis coupling in terms of quasi– particles. All available experimental data on even–odd and odd–odd deformed proton emitters from the ground and isomeric states and fine structure, are accurately and consistently reproduced by the model. The angular momentum of the decaying nucleus, and the state of the unpaired neutron in decay from odd–odd nuclei, are identified without ambiguity. REFERENCES 1. A. A. Sonzogni, Nuclear Data Sheets 95 (2002) 1. ˚berg, P. B. Semmes and W. Nazarewicz, Phys. Rev. C56 (1997) 1762. 2. S. A 3. E. Maglione, L. S. Ferreira and R. J. Liotta, Phys. Rev. Lett. 81 (1998) 538; Phys. Rev. C59 (1999) R589. 4. L. S. Ferreira and E. Maglione, Phys. Rev. Lett. 86 (2001) 1721. 5. D. D. Bogdanov, V. P. Bugrov, S. G. Kadmenskii, Sov. J. Nucl. Phys. 52 (1990) 229. 6. G. Fiorin, E. Maglione and L. S. Ferreira, Phys. Rev. C67 (2003) 054302. 7. L. S. Ferreira, E. Maglione and R. J. Liotta, Phys. Rev. Lett. 78 (1997) 1640. 8. L. S. Ferreira, E. Maglione, and D. Fernandes, Phys. Rev. C65 (2002) 024323. 9. P. M¨oller, et al., At. Data Nucl. Data Tab. 59 (1995) 185; ibd. 66 (1997) 131. 10. L. S. Ferreira and E. Maglione, Phys. Rev. C61 (2000) 021304(R). 11. E. Maglione and L.S. Ferreira, Phys. Rev. C61 (2000) 47307. 12. C. N. Davids, et al., Phys. Rev. Lett. 80 (1998) 1849. 13. A. T. Kruppa, et al., Phys. Rev. Lett. 84 (2000) 4549; B. Barmore, et al., Phys. Rev. C62 (2000) 054315; W. Kr´olas, et al., Phys. Rev. C65 (2002) 031303(R). 14. H. Esbensen and C. N. Davids, Phys. Rev. C63 (2001) 014315.