c Pleiades Publishing, Ltd., 2012. ISSN 1063-7788, Physics of Atomic Nuclei, 2012, Vol. 75, No. 11, pp. 1324–1330. c I.N. Izosimov, V.G. Kalinnikov, A.A. Solnyshkin, 2012, published in Yadernaya Fizika, 2012, Vol. 75, No. 11, pp. 1400–1405. Original Russian Text
NUCLEI Experiment
Resonance Structure of Strength Functions for First-Forbidden β + /EC Transitions I. N. Izosimov* , V. G. Kalinnikov, and А. А. Solnyshkin Joint Institute for Nuclear Research, Dubna, Moscow oblast, 141980 Russia Received January 16, 2012
Abstract—Experimental data obtained by measuring the fine structure of the strength function Sβ (E) in spherical and deformed nuclei were analyzed. The use of modern nuclear-spectroscopy methods made it possible to reveal the nuclear-deformation-induced splitting of peaks in Sβ (E) for transitions of the Gamow–Teller type. For first-forbidden transitions, the resonance nature of Sβ (E) was proven experimentally both for spherical and for deformed nuclei. It is shown that, at some values of the excitation energy, the intensity of first-forbidden transitions in nuclei can be commensurate with the intensity of Gamow–Teller transitions. DOI: 10.1134/S1063778812110099
1. INTRODUCTION The probability for a β transition is proportional to the product of its lepton part described by the Fermi function f (Qβ − E) and the nucleon part described by the strength function for β transitions, Sβ (E). The strength function Sβ (E) is one of the most important features of nuclei [1–3] and is the distribution of the squares of moduli of matrix elements of the β-decay type with respect to excitation energies E of the nucleus being considered. Until recently, total-absorption γ-ray spectrometers and methods of total-absorption spectroscopy, which are characterized by a low energy resolution, were used to study the structure of the strength function Sβ (E) experimentally. By means of these methods, it could be proven experimentally that the strength function Sβ (E) for Gamow–Teller β transitions has a resonance structure [2, 3]. However, the methods in questions have a number of drawbacks associated with a low energy resolution of NaI-based spectrometers. The development of experimental techniques made it possible to apply, in studying the fine structure of Sβ (E), nuclear-spectroscopy methods boasting a high energy resolution. In the present study, such methods are used to obtain data on the structure of Sβ (E). For Gamow–Teller transitions, the strength function Sβ (E) has a pronounced resonance structure [1–3]. Until recently, the question of a resonance *
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character of the strength function Sβ (E) for firstforbidden β transitions remained open. In [4], we showed for the first time that the strength function Sβ (E) for the first-forbidden β + /EC decay of the 160m Ho deformed nucleus has a resonance character. In the present study, we show that the strength function Sβ (E) for the first-forbidden β + /EC decays of the 160g Ho deformed and 147g Tb spherical nuclei also have a resonance nature. We also show that, for a number of energy values, the intensities of firstforbidden β + /EC transitions are commensurate with the intensities of Gamow–Teller transitions. In addition, we discuss special features of the resonance structure of Sβ (E) for first-forbidden transitions.
2. STRENGTH FUNCTION Sβ (E) FOR β DECAY The strength function Sβ (E) determines the energy (E being the energy of the nucleus being considered) distribution of elementary charge-exchange excitations and their combinations belonging to the proton particle–neutron hole [(πp)–(νh)] and neutron particle–proton hole [(νp)–(πh)] types: [πp ⊗ νh]J π and [νp ⊗ πh]J π , respectively, where J π is the total angular momentum of a combination. The strength function of Fermi-type β transitions takes into account [πp ⊗ νh]0+ or [νp ⊗ πh]0+ excitations. Since the isospin is a rather good quantum number, the strength of Fermi-type transitions is concentrated in the region of the isobaric analog resonance. The
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strength function for β transitions of the Gamow– Teller type describes [πp ⊗ νh]1+ or [νp ⊗ πh]1+ excitations. For first-forbidden β transitions in the Coulomb approximation, the [πp ⊗ νh]1− 0− or [νp ⊗ πh]1− 0− configurations are significant. The residual interaction can cause the collectivization of these configurations and the appearance of resonances in Sβ (E). The positions and intensities of resonances in Sβ (E) are calculated on the basis of various microscopic models [1, 5–9]. For Gamow–Teller β transitions, first-forbidden transitions in the Coulomb approximation, and firstforbidden unique transitions, the reduced probabilities B(GT), [B(λπ = 0− ) + B(λπ = 1− )], and [B(λπ = 2− )]; the half-lives T1/2 ; the populations of levels, I(E); the strength function Sβ (E); and the quantities f t are related by the equations [4] d (I(E)) = Sβ (E)T1/2 f (Q − E), dE 1 = Sβ (E)f (Q − E)dE, T1/2 Sβ (E)dE =
(1) (2)
1 , ft
(3)
gV2 , 4π · f t
(4)
ΔE
ΔE
B(GT, E) = D
2 g2 D 4πA If T± (k)σμ (k)Ii , (5) B(GT, E) = 2Ii + 1 [B(λπ = 2− )] =
g2 3 D V , 4 4π · f t
[B(λπ = 0− ) + B(λπ = 1− )] = D
gV2 , 4π · f t
(6)
(7)
where Q is the total β-decay energy, f (Q − E) is the Fermi function, t is the partial half-life with respect to β decay to the level whose excitation energy is E,T is the isospin of the nucleus, If T± (k)σμ (k)Ii is the reduced nuclear matrix element for the Gamow– Teller transition, Ii is the spin of the parent nucleus, If is the spin of the daughter nucleus in the excited level being considered, and D = 6147 ± 7 s. By measuring the populations of levels in β decay, one can determine the reduced probabilities and the strength function for β decay. These probabilities are proportional to the squares of the nuclear matrix elements and reflect the fine structure of the strength functions for β decay. PHYSICS OF ATOMIC NUCLEI
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The scheme of levels that are significant for analyzing the strength functions for the Gamow–Тeller transitions is shown in Fig. 1. In the β + /EC decay of N > Z nuclei, there is only one isospin value of T0 + 1 in coupling the the isospin (τ = 1, μτ = +1) of the [νp × πh]1+ configuration to the neutron-excess isospin T0 . The most collective state formed by [νp ⊗ πh]1+ excitations characterized by the isospin τ = 1 and the isospin projection μτ = +1 is also referred to [2] as a μτ = +1 Gamow–Teller resonance. For the β − decay of N > Z nuclei, the Gamow–Teller resonance (τ = 1, μτ = −1) lies (Fig. 1) at excitation energies in excess of Qβ and is energetically inaccessible to population in β − decay, but the μτ = +1 Gamow–Teller resonance can be populated by β + /EC decay [2]. At the present time, there is no theory that would describe adequately the strength functions for the β decay of deformed nuclei. Only for spherical and weakly deformed nuclei does the existing theory allow us to calculate quite correctly the positions and relative intensities of the peaks in the strength functions for Gamow–Teller transitions [1, 2, 5–9]. Even for spherical nuclei, the deviation of the calculated absolute intensities of the strengthfunction peaks from their experimental counterparts is as large as a few tens of percent to a few hundred percent, the theory predicting more intense peaks than those observed experimentally [2, 3, 5]. For first-forbidden β + /EC transitions in the Coulomb approximation, [νp ⊗ πh]0− ,1− configurations are significant. The question of whether the resonance structure is (or is not) present in the strength function for first-forbidden β − or β + /EC transitions remained open until recently. Information about the structure of the strength function Sβ (E) is of importance for many fields of nuclear physics [1–4]. It is necessary to have reliable experimental data on the structure of Sβ (E) in order to predict half-lives of nuclei far from the stability band; to verify the completeness of decay schemes; to calculate the energy deposition in the decay of fission products in nuclear reactors; to calculate the spectra of delayed particles; to calculate the probability of delayed fission; to evaluate the fission barriers for nuclei far from the β-stability band; to study the production of various isotopes in astrophysical processes; and to develop microscopic models for calculating strength functions Sβ (E), especially in deformed nuclei. 3. INVESTIGATION OF THE STRENGTH FUNCTION Sβ (E) BY MEANS OF HIGH-RESOLUTION NUCLEAR SPECTROSCOPY The application of methods of high-resolution nuclear spectroscopy makes it possible to obtain de-
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(b) GTR(T>)
(a)
GTR(T
