- Email: [email protected]
Average Description of Dipole Gamma Transitions in Hot Atomic Nuclei
Available online at www.sciencedirect.com
Nuclear Data Sheets 118 (2014) 237–239 www.elsevier.com/locate/nds
Average Description of Dipole Gamma Transitions in Hot Atomic Nuclei V.A. Plujko,1, 2, ∗ O.M. Gorbachenko,1 E.P. Rovenskykh,2 and V.A. Zheltonozhskii2 1
Nuclear Physics Department, Taras Shevchenko National University, Kiev, Ukraine 2 Institute for Nuclear Research, NAS of Ukraine, Kiev, Ukraine
A new version of the modified Lorentzian approach for radiative strength function is proposed. It is based on renewed systematics for giant dipole resonance (GDR) parameters. The gamma-decay strength functions are calculated using new GDR parameters and compared with experimental data. It is demonstrated that closed-form approaches with energy-dependent width of the gamma strength, as a rule, provide a reliable simple method for description of gamma-decay processes. I.
INTRODUCTION
The average probabilities of γ-transitions at γ-ray emission and photoabsorption can be described using the radiative (photon) strength functions (RSF) [1]. These functions are involved in calculations of the observed characteristics of most nuclear reactions. They are also used for investigation of nuclear structure (nuclear deformations, energies and widths of the giant dipole resonances, contribution of velocity-dependent force, shapetransitions, etc.) as well as in studies of nuclear reaction mechanisms. Isovector dipole electric γ-transitions (E1) are usually dominant compared to transitions of other multipolarities and types. Therefore reliable model of dipole RSF description is needed and simple expressions for RSF are preferable to decrease computation time. The dipole radiative strength functions of γ-decay and photoabsorption at γ-ray energy Eγ coincide with the spectral function FE1 (Eγ , τ ) at different temperatures τ for the excitation and decay channels. The spectral function is proportional to imaginary part χE1 of response function of a nucleus on dipole electric field with frequency ω = Eγ / [2] FE1 (Eγ , τ ) ∝ χE1 (Eγ ) /[1 − exp(−Eγ /τ )].
(1)
The γ-decay RSF function depends on the final state temperature τ = Tf which is a function of the γ-ray energy in contrast to the initial state temperature, τ = T , for the case of photoabsorption. The scaling factor alongside with χE1 determines the enhancement magnitude of the RSF in heated nuclei as compared to the zero temperature case.
In the approximation of the excitation of one strong collective state (GDR), the function χE1 has Lorentzianlike shape [3] and in spherical nuclei is given by χE1 (Eγ ) ∝
Eγ2
−
Eγ Γ (Eγ , Uτ ) , 2 2 Er + (Γ (Eγ , Uτ ) Eγ )2
where Er is the GDR energy and Γ (Eγ , Uτ ) is an energydependent width at excitation energy Uτ corresponding to the temperature τ . For deformed nuclei, we adopt the approximation of axially deformed nuclei with two Lorentzian-like components Eq. (2) with different values of GDR energy and width for vibration along and perpendicular to the axis of symmetry [1, 4]. The phenomenological models of the RSF are mainly different in the expressions for width Γ (Eγ ), which is governed by the description of the GDR damping. The RSF expressions based on Eqs. (1) and (2) are named here and in Refs. [1, 3, 4] as modified Lorentzian approach (MLO). In this contribution, a new version MLO approach (MLO4) for RSF is proposed with the use of the renewed GDR width systematics[4, 5]. Different Lorentzian-type models of E1 strength functions are tested by comparison of experimental data with theoretical calculations. II.
CALCULATIONS AND DISCUSSION
In Ref. [6], the values of GDR parameters and their uncertainties from Refs. [4, 5] are used to obtain new systematics of GDR parameters. The expression for new systematics for the GDR width has the following form (in units of MeV) Γr,j = a1 · Er,j + a2 · βdyn · Er,j · γj ,
∗
Corresponding author: [email protected]
http://dx.doi.org/10.1016/j.nds.2014.04.046 0090-3752/2014 Published by Elsevier B.V.
(2)
(3)
where a1 , a2 are constants, Er,j and Γr,j are GDR energy and width for vibration along j-axis respectively,
Average Description of Dipole . . . 0.6
167
-3
(MeV )
γj = (R0 /Rj ) , with Rj , R0 for nuclear radii along j axis and for radius of equivalent spherical nuclei. The exponent 0.6 was taken in line with the one-body dissipative model[7] for fragmentation component Γfr,j of the GDR width in deformed nuclei that used Γfr,j ∝ 1.6 ∝ Er,j γj . Quadrupole dynamical deforma(R0 /Rj ) tion parameters βdyn were calculated as [8]: βdyn = 1224A−7/3 /E2+ with E2+ the energy of the first col1
V.A. Plujko et al.
NUCLEAR DATA SHEETS
-7
Er
aver, E1+M1
10
- MLO1; - GFL; - SMLO;
f
1
-8
10
lective 2+ state. The values of constant and their uncertainties a1 = 0.249(20 ), a2 = 0.360(83 ) were obtained by the fitting within SLO model [1]. Based on the systematics Eq. (3) for the GDR width, we take a new expression for description of the energy dependent width Γj (Eγ , U ) = bj a1 (Eγ + U ) + a2 βdyn Er,j γj , (4)
3
- SLO - EGLO - MLO4 5
E (MeV)
FIG. 1. The gamma-decay strength functions within different RSF models for 167 Er. The experimental data are taken from Ref. [11].
where Eγ and U are the energy of the gamma-rays and excitation energy respectively. Parameters bj are normalization factors if experimental data on GDR widths Γr,j are exist. They are found to fit Eq. (4) at U = 0 to the experimental data. In accordance of Eq. (3), the values bj = 1 are taken with no experimental data. The calculations of the MLO model[1] with this expression for energy dependent width are named below as MLO4. The comparison of the calculations of the E1 RSF of different forms with the experimental data is shown on Figs. 1-3. The calculations were performed within the framework of the approaches(see[1] for details): Standard Lorentzian (SLO), Enhanced Generalized Lorentzian (EGLO)[9], Generalized Fermi Liquid (GFL)[10], Modified Lorentzian (MLO1), Simplified Modified Lorentzian (SMLO)and MLO4. Calculations correspond to the sum of the gamma-
f
E1+M1
(MeV-3)
168 -7
10
Er
-8
- MLO1; - GFL; - SMLO;
10
3
E (MeV)
- SLO - EGLO - MLO4 5
FIG. 2. The gamma-decay strength functions within different RSF models for 168 Er: U = Sn . Experimental data are taken from Ref. [13].
←
decay strength functions for E1 and M1 transitions ( f E1 ←
92
Mo
-7
10
- MLO1 - SLO - GFL - EGLO - SMLO - MLO4
f
E1+M1
(MeV-3)
+ f M1 ). The M1 strength functions were calculated by the method described in the RIPL [1]. The Fig.1 shows dipole gamma-decay strength functions for 167 Er within different RSF models in comparison with experimental data from [11]. Due to measurement peculiarities, the experimental data from [11] correspond to that ones averaged on the excitation energy U ← Ui : faver (Eγ ) = 4 m f (Eγ , Ui − Eγ )dUi /(Um − 4) for U ← 1 < Eγ ≤ 4, and Eγm f (Eγ , Ui − Eγ )dUi /)Um − Eγ ) at 4 < Eγ ≤ Um with Um = 8 M eV ≈ Sn , Sn -neutron separation energy. Fig. 2 shows the dipole gamma-decay strength functions for 168 Er within different RSF models in comparison with experimental data from [13]. The calculations were performed for initial excitation energy Ui = Sn . In Fig. 3 results of the calculations for the 92 M o are compared with experimental data from Ref. [15]. The calculations were performed for Ui = Sn . One can seen that RSF models with vanishing values of strength function at zero gamma-ray energy (EGLO, GFL, MLO1, SMLO, MLO4) lead to better description of
-8
10
7
10
E (MeV)
FIG. 3. The gamma-decay strength functions within different RSF models for 92 Mo: Ui = Sn . Experimental data are taken from Ref. [15].
the experimental data in the low-energy region than the SLO model. The results of the calculations of gammadecay RSF within EGLO, GFL, MLO, MLO4 and SMLO 238
Average Description of Dipole . . .
V.A. Plujko et al.
NUCLEAR DATA SHEETS
models are all characterized by a non-zero limit. It can be also noted that different variants of the MLO (SMLO) approach are based on general relations between the RSF and the nuclear response function [2]. Therefore, they can potentially lead to more reliable predictions among different simple models.
action calculated by the use of different RSF models.
´ Pn ` 2 2 TABLE I. The average i=1 χi (model)/χi (SLO) /n ratio of chi-square deviations of the theoretical RSF of γ-decay from experimental data. n-cumulative number of nuclei ([11, 12]: n = 41, [13, 14]: n = 38, [15, 16]: n = 7). Exp.Data
n
EGLO Refs. [11, 12] 41 0.18 Refs. [13, 14] 38 1.22 Refs. [15, 16] 7 2.22
GFL 0.17 0.91 2.11
Model MLO SMLO MLO4 0.11 0.11 0.13 0.98 1.01 0.89 1.16 1.71 1.20
The cross section calculations were performed by the EMPIRE 3.1 Rivoli code [17]. In these calculations, gammadecay widths were normalized on their experimental values at the neutron binding energy. The difference in the calculations of excitation function with the different RSF increases in heavy nuclei. On the whole, the calculations within the RSF models with energy-dependent width agree better with the experimental data for middleweight and heavy nuclei. FIG. 4. The excitation functions of (n, xγ) reaction on nat F e: calculations performed within listed RSF models; experimental data are taken from EXFOR.
III.
CONCLUSIONS
The overall comparison of the calculations within different simple models and experimental data shows that the EGLO and MLO (SMLO) approaches with vanishing values of strength function at zero gamma-ray energy provide a universal and rather reliable simple method for estimation of the dipole RSF over a relatively wide energy interval ranging from zero to slightly above the GDR peak. In general, the new version of the MLO model (MLO4) leads to better description of the experimental data than previous ones as for gamma-decay and for photoexcitation functions. It should be also mentioned that the experimental data from [15, 16] are better described within the SLO approach.
Table I presents the ratio χ2 (model)/χ2 (SLO) of chisquare deviations of the theoretical RSF of gamma-decay from experimental data. The average values of the ratio for approximately 40 nuclei with 25 < A < 200 were obtained. As one can see from this table and figures, asymmetric RSF gives better agreement with the experimental data at least in approximation of axially-deformed nuclei which is adopted in the presented calculations. On the whole, the proposed variant of the MLO model (MLO4) leads to the best description of the experimental data. Fig. 4 shows excitation function of the nat Fe(n,xγ) re-
[1] R. Capote et al., Nucl. Data Sheets 110, 3107 (2009); http://www-nds.iaea.org/RIPL-3/. [2] V.A.Plujko (Plyuiko) Yad. Fiz. 52, 1004 (1990); Sov. J. Nucl.Phys. 52, 639 (1990). [3] V.A. Plujko et al., Intern. J. Mod. Phys. E. 18, 996 (2009). [4] V.A. Plujko et al., At. Data Nucl.Data Tables 97, 567 (2011). [5] V.A. Plujko et al., J. Korean Phys. Soc. 59(2), 1514 (2011). [6] V.A. Plujko et al., Nucl. Phys. Atom. Energy 13, 335 (2012). [7] B. Bush et al., Nucl. Phys. A 531, 27 (1991).
[8] [9] [10] [11] [12] [13]
[14] [15] [16] [17]
239
L. Esser et al., Phys. Rev. C 55, 206 (1997). J. Kopecky et al., Phys. Rev. C47, 312 (1993). S. F. Mughabghab et al., Phys. Lett. B487, 155 (2000). E. Melby et al., Phys. Rev.C 63, 044309 (2001). U. Agvaanluvsan et al., Phys. Rev.C 70, 054611 (2004). A.M. Sukhovoj et al., in Proc. of the XV Int. Seminar on Interaction of Neutrons with Nuclei. (Dubna, May 2007) 92 (2007) A.M. Sukhovoj et al., Izvestiya RAN. Seriya Fiz. 69, 641 (2005). R. Schwengner et al., Phys. Rev. C 81, 034319 (2010). R. Schwengner et al., Phys. Rev. C 78, 064314 (2008). M. Herman et al., Nucl. Data Sheets 108, 2655 (2007); http://www.nndc.bnl.gov/empire/.
Nuclear Data Sheets 118 (2014) 237–239 www.elsevier.com/locate/nds
Average Description of Dipole Gamma Transitions in Hot Atomic Nuclei V.A. Plujko,1, 2, ∗ O.M. Gorbachenko,1 E.P. Rovenskykh,2 and V.A. Zheltonozhskii2 1
Nuclear Physics Department, Taras Shevchenko National University, Kiev, Ukraine 2 Institute for Nuclear Research, NAS of Ukraine, Kiev, Ukraine
A new version of the modified Lorentzian approach for radiative strength function is proposed. It is based on renewed systematics for giant dipole resonance (GDR) parameters. The gamma-decay strength functions are calculated using new GDR parameters and compared with experimental data. It is demonstrated that closed-form approaches with energy-dependent width of the gamma strength, as a rule, provide a reliable simple method for description of gamma-decay processes. I.
INTRODUCTION
The average probabilities of γ-transitions at γ-ray emission and photoabsorption can be described using the radiative (photon) strength functions (RSF) [1]. These functions are involved in calculations of the observed characteristics of most nuclear reactions. They are also used for investigation of nuclear structure (nuclear deformations, energies and widths of the giant dipole resonances, contribution of velocity-dependent force, shapetransitions, etc.) as well as in studies of nuclear reaction mechanisms. Isovector dipole electric γ-transitions (E1) are usually dominant compared to transitions of other multipolarities and types. Therefore reliable model of dipole RSF description is needed and simple expressions for RSF are preferable to decrease computation time. The dipole radiative strength functions of γ-decay and photoabsorption at γ-ray energy Eγ coincide with the spectral function FE1 (Eγ , τ ) at different temperatures τ for the excitation and decay channels. The spectral function is proportional to imaginary part χE1 of response function of a nucleus on dipole electric field with frequency ω = Eγ / [2] FE1 (Eγ , τ ) ∝ χE1 (Eγ ) /[1 − exp(−Eγ /τ )].
(1)
The γ-decay RSF function depends on the final state temperature τ = Tf which is a function of the γ-ray energy in contrast to the initial state temperature, τ = T , for the case of photoabsorption. The scaling factor alongside with χE1 determines the enhancement magnitude of the RSF in heated nuclei as compared to the zero temperature case.
In the approximation of the excitation of one strong collective state (GDR), the function χE1 has Lorentzianlike shape [3] and in spherical nuclei is given by χE1 (Eγ ) ∝
Eγ2
−
Eγ Γ (Eγ , Uτ ) , 2 2 Er + (Γ (Eγ , Uτ ) Eγ )2
where Er is the GDR energy and Γ (Eγ , Uτ ) is an energydependent width at excitation energy Uτ corresponding to the temperature τ . For deformed nuclei, we adopt the approximation of axially deformed nuclei with two Lorentzian-like components Eq. (2) with different values of GDR energy and width for vibration along and perpendicular to the axis of symmetry [1, 4]. The phenomenological models of the RSF are mainly different in the expressions for width Γ (Eγ ), which is governed by the description of the GDR damping. The RSF expressions based on Eqs. (1) and (2) are named here and in Refs. [1, 3, 4] as modified Lorentzian approach (MLO). In this contribution, a new version MLO approach (MLO4) for RSF is proposed with the use of the renewed GDR width systematics[4, 5]. Different Lorentzian-type models of E1 strength functions are tested by comparison of experimental data with theoretical calculations. II.
CALCULATIONS AND DISCUSSION
In Ref. [6], the values of GDR parameters and their uncertainties from Refs. [4, 5] are used to obtain new systematics of GDR parameters. The expression for new systematics for the GDR width has the following form (in units of MeV) Γr,j = a1 · Er,j + a2 · βdyn · Er,j · γj ,
∗
Corresponding author: [email protected]
http://dx.doi.org/10.1016/j.nds.2014.04.046 0090-3752/2014 Published by Elsevier B.V.
(2)
(3)
where a1 , a2 are constants, Er,j and Γr,j are GDR energy and width for vibration along j-axis respectively,
Average Description of Dipole . . . 0.6
167
-3
(MeV )
γj = (R0 /Rj ) , with Rj , R0 for nuclear radii along j axis and for radius of equivalent spherical nuclei. The exponent 0.6 was taken in line with the one-body dissipative model[7] for fragmentation component Γfr,j of the GDR width in deformed nuclei that used Γfr,j ∝ 1.6 ∝ Er,j γj . Quadrupole dynamical deforma(R0 /Rj ) tion parameters βdyn were calculated as [8]: βdyn = 1224A−7/3 /E2+ with E2+ the energy of the first col1
V.A. Plujko et al.
NUCLEAR DATA SHEETS
-7
Er
aver, E1+M1
10
- MLO1; - GFL; - SMLO;
f
1
-8
10
lective 2+ state. The values of constant and their uncertainties a1 = 0.249(20 ), a2 = 0.360(83 ) were obtained by the fitting within SLO model [1]. Based on the systematics Eq. (3) for the GDR width, we take a new expression for description of the energy dependent width Γj (Eγ , U ) = bj a1 (Eγ + U ) + a2 βdyn Er,j γj , (4)
3
- SLO - EGLO - MLO4 5
E (MeV)
FIG. 1. The gamma-decay strength functions within different RSF models for 167 Er. The experimental data are taken from Ref. [11].
where Eγ and U are the energy of the gamma-rays and excitation energy respectively. Parameters bj are normalization factors if experimental data on GDR widths Γr,j are exist. They are found to fit Eq. (4) at U = 0 to the experimental data. In accordance of Eq. (3), the values bj = 1 are taken with no experimental data. The calculations of the MLO model[1] with this expression for energy dependent width are named below as MLO4. The comparison of the calculations of the E1 RSF of different forms with the experimental data is shown on Figs. 1-3. The calculations were performed within the framework of the approaches(see[1] for details): Standard Lorentzian (SLO), Enhanced Generalized Lorentzian (EGLO)[9], Generalized Fermi Liquid (GFL)[10], Modified Lorentzian (MLO1), Simplified Modified Lorentzian (SMLO)and MLO4. Calculations correspond to the sum of the gamma-
f
E1+M1
(MeV-3)
168 -7
10
Er
-8
- MLO1; - GFL; - SMLO;
10
3
E (MeV)
- SLO - EGLO - MLO4 5
FIG. 2. The gamma-decay strength functions within different RSF models for 168 Er: U = Sn . Experimental data are taken from Ref. [13].
←
decay strength functions for E1 and M1 transitions ( f E1 ←
92
Mo
-7
10
- MLO1 - SLO - GFL - EGLO - SMLO - MLO4
f
E1+M1
(MeV-3)
+ f M1 ). The M1 strength functions were calculated by the method described in the RIPL [1]. The Fig.1 shows dipole gamma-decay strength functions for 167 Er within different RSF models in comparison with experimental data from [11]. Due to measurement peculiarities, the experimental data from [11] correspond to that ones averaged on the excitation energy U ← Ui : faver (Eγ ) = 4 m f (Eγ , Ui − Eγ )dUi /(Um − 4) for U ← 1 < Eγ ≤ 4, and Eγm f (Eγ , Ui − Eγ )dUi /)Um − Eγ ) at 4 < Eγ ≤ Um with Um = 8 M eV ≈ Sn , Sn -neutron separation energy. Fig. 2 shows the dipole gamma-decay strength functions for 168 Er within different RSF models in comparison with experimental data from [13]. The calculations were performed for initial excitation energy Ui = Sn . In Fig. 3 results of the calculations for the 92 M o are compared with experimental data from Ref. [15]. The calculations were performed for Ui = Sn . One can seen that RSF models with vanishing values of strength function at zero gamma-ray energy (EGLO, GFL, MLO1, SMLO, MLO4) lead to better description of
-8
10
7
10
E (MeV)
FIG. 3. The gamma-decay strength functions within different RSF models for 92 Mo: Ui = Sn . Experimental data are taken from Ref. [15].
the experimental data in the low-energy region than the SLO model. The results of the calculations of gammadecay RSF within EGLO, GFL, MLO, MLO4 and SMLO 238
Average Description of Dipole . . .
V.A. Plujko et al.
NUCLEAR DATA SHEETS
models are all characterized by a non-zero limit. It can be also noted that different variants of the MLO (SMLO) approach are based on general relations between the RSF and the nuclear response function [2]. Therefore, they can potentially lead to more reliable predictions among different simple models.
action calculated by the use of different RSF models.
´ Pn ` 2 2 TABLE I. The average i=1 χi (model)/χi (SLO) /n ratio of chi-square deviations of the theoretical RSF of γ-decay from experimental data. n-cumulative number of nuclei ([11, 12]: n = 41, [13, 14]: n = 38, [15, 16]: n = 7). Exp.Data
n
EGLO Refs. [11, 12] 41 0.18 Refs. [13, 14] 38 1.22 Refs. [15, 16] 7 2.22
GFL 0.17 0.91 2.11
Model MLO SMLO MLO4 0.11 0.11 0.13 0.98 1.01 0.89 1.16 1.71 1.20
The cross section calculations were performed by the EMPIRE 3.1 Rivoli code [17]. In these calculations, gammadecay widths were normalized on their experimental values at the neutron binding energy. The difference in the calculations of excitation function with the different RSF increases in heavy nuclei. On the whole, the calculations within the RSF models with energy-dependent width agree better with the experimental data for middleweight and heavy nuclei. FIG. 4. The excitation functions of (n, xγ) reaction on nat F e: calculations performed within listed RSF models; experimental data are taken from EXFOR.
III.
CONCLUSIONS
The overall comparison of the calculations within different simple models and experimental data shows that the EGLO and MLO (SMLO) approaches with vanishing values of strength function at zero gamma-ray energy provide a universal and rather reliable simple method for estimation of the dipole RSF over a relatively wide energy interval ranging from zero to slightly above the GDR peak. In general, the new version of the MLO model (MLO4) leads to better description of the experimental data than previous ones as for gamma-decay and for photoexcitation functions. It should be also mentioned that the experimental data from [15, 16] are better described within the SLO approach.
Table I presents the ratio χ2 (model)/χ2 (SLO) of chisquare deviations of the theoretical RSF of gamma-decay from experimental data. The average values of the ratio for approximately 40 nuclei with 25 < A < 200 were obtained. As one can see from this table and figures, asymmetric RSF gives better agreement with the experimental data at least in approximation of axially-deformed nuclei which is adopted in the presented calculations. On the whole, the proposed variant of the MLO model (MLO4) leads to the best description of the experimental data. Fig. 4 shows excitation function of the nat Fe(n,xγ) re-
[1] R. Capote et al., Nucl. Data Sheets 110, 3107 (2009); http://www-nds.iaea.org/RIPL-3/. [2] V.A.Plujko (Plyuiko) Yad. Fiz. 52, 1004 (1990); Sov. J. Nucl.Phys. 52, 639 (1990). [3] V.A. Plujko et al., Intern. J. Mod. Phys. E. 18, 996 (2009). [4] V.A. Plujko et al., At. Data Nucl.Data Tables 97, 567 (2011). [5] V.A. Plujko et al., J. Korean Phys. Soc. 59(2), 1514 (2011). [6] V.A. Plujko et al., Nucl. Phys. Atom. Energy 13, 335 (2012). [7] B. Bush et al., Nucl. Phys. A 531, 27 (1991).
[8] [9] [10] [11] [12] [13]
[14] [15] [16] [17]
239
L. Esser et al., Phys. Rev. C 55, 206 (1997). J. Kopecky et al., Phys. Rev. C47, 312 (1993). S. F. Mughabghab et al., Phys. Lett. B487, 155 (2000). E. Melby et al., Phys. Rev.C 63, 044309 (2001). U. Agvaanluvsan et al., Phys. Rev.C 70, 054611 (2004). A.M. Sukhovoj et al., in Proc. of the XV Int. Seminar on Interaction of Neutrons with Nuclei. (Dubna, May 2007) 92 (2007) A.M. Sukhovoj et al., Izvestiya RAN. Seriya Fiz. 69, 641 (2005). R. Schwengner et al., Phys. Rev. C 81, 034319 (2010). R. Schwengner et al., Phys. Rev. C 78, 064314 (2008). M. Herman et al., Nucl. Data Sheets 108, 2655 (2007); http://www.nndc.bnl.gov/empire/.