Volume 125, number 1
19 May 1983
SPREADING WIDTH OF GAMOW-TELLER RESONANCES IN 208Bi Shizuko ADACHI
Institute for Nuclear Study, University of Tokyo, Tanashi, Tokyo 188, Japan Received 15 September 1982
The spreading width of the Gamow-Teller resonance in 2°8Bi is studied in the self-consistent TDA with the Skyrme interaction. Coupling between lplh states and 2p2h states are included in the TDA linear response function.
Our knowledge on spin and spin-isospin excitation modes has been much extended by recent (p, n)  experiments with interi,lediate energy protons as well as (e, e')  and (p, p')  experiments on many nuclei. These experiments show us that the observed strengths of Gamow Teller (GT) resonances and M1 states are much smaller than those predicted by simple shell models. The mixing of A particle-nucleon hole components in these states can be one of the reasons for this quenching . Another reason is nuclear structure effect. The strength spreads out by coupling of l p l h states with more complicated states . In this letter, I show the results of microscopic calculations on the spreading of the GT strength due to the nuclear structure effects for 208Bi. A self-consistent Hartree-Fock (HF) + T a m m - D a n c o f f approximation (TDA) formalism  is used, which includes coupling between l p l h states and 2p2h states. Due to this coupling, the main GT resonance peak has a spreading width and the GT strength is shifted into other energy regions. Although these two effects of this coupling cannot be clearly separated, I concentrate mainly on the spreading width of the GT resonance in the present work. Bertsch and Hamamoto  aimed to show that a large part of GT strength is shifted into the higher energy region by mixing with 2p2h configurations at high excitation energies. On the other hand, the aim of the present work is to show the spreading width of the GT resonance by coupling with 2p2h states near the resonance. First I carry out HF calculations for 208pb with the Skyrme interaction using a code by Negele and 0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
Vautherin . Once the HF single-particle hamiltonJan is obtained, this hamiltonian is diagonalized in a large space of harmonic oscillators. The oscillator parameter u(= mco/~) is chosen to be 0.1688 fm - 2 , and the dimension of harmonic oscillator space MAXN to be 10 or 20 for each (l,]) pair. Thus I construct the bound wave functions both in the negative energy region and in the positive energy region. These bound wave functions are used for 2p2h states as well as l p l h states in the present calculation. I will show the results with using the parameter set Sill. The configuration space used in the l p l h TDA diagonalization is restricted to 0 h w proton-particleneutron-hole states for isobaric analog resonances (IAR) and GT resonances in 208Bi. Enlarging the configuration space gives only little effect on excitation energies and transition strengths. In the case of 208Bi, the TDA calculation gives very similar results to the RPA calculation, because RPA correlations in the ground state are prevented by the Pauli principle . The results of the TDA diagonalization show that only one state carries almost all the isobaric analog strength, while the GT strength splits mainly into two states. The calculated GT strengths in 208Bi are presented in fig. 1. The excitation energies are relative to the ground state of 2°8pb. The black bars show the GT strengths of unperturbed particle-hole states, while the white bars show those of the GT resonances by the TDA diagonalization. The position of the IAR is shown by an arrow in fig. 1. The highest GT resonance has roughly 65% of all the GT strength. The lower state has about one quarter of the GT strength of the 5
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PHYSICS LETTERS I
taken into account. A method for including this effect was proposed by S. Yoshida and the author  some years ago, but in this letter I restrict myself to the spreading width. We know that about a half of the total width comes from coupling to continuum in the case of the IAR , while the escape width of the GT resonance is very small and the observed width is almost entirely spreading width . The matrix elements of the effective hamiltonian in the configuration space are given by
GT STRENGTH 8060-
C~ php 'h ,(CO) = ~ pp ,~ h h ' [ep - e h ] + (phi Vph Ip'h')
Fig. 1. The calculated GT strengths (in units ofg}t/4cr) in
2°8Bi. The black bars show the GT strengths of unperturbed particle-hole states, while the white bars show those of the GT resonances by the TDA diagonalization. The excitation energies are relative to the ground state of 2°8pb in all figures.
highest state. As for the excitation energies, the relative position of GT resonances to the IAR is well reproduced in the case of Sill. The strength function b as a function of the excitation energy co is given by 1/n times the imaginary part of the linear response function: b(co) = (l/n) Im(01M+(~(co) - co - i $ ) - l M I 0 ) ,
where M is a column vector for the transition operator which corresponds to the excitation considered and trig(co) is an effective hamiltonian. In the case of IAR, the isospin-lowering operator is used for the transition operator: M = ~t_(i). i
On the other hand, the following transition operator is used for the GT resonances:
g2A ~i. ou(i)t_(i),
19 May 1983
where gA is the axial-vector coupling constant. As the Green function is evaluated in l p l h configuration space constructed with borund wave functions, c;g (co) is the effective hamiltonian in this space. In this letter, coupling between bound l p l h states and bound 2p2h states are included in evaluating the effective hamiltonian. As the IAR and GT resonances are in the continuum region, the effect of coupling between bound l p l h states and continuum l p l h states must be
- iYph p,h,(co)/2 .
The first term is the single-particle energy and the second term consists of the usual effective particle-hole matrix elements of density-dependent interactions. The usual TDA matrix elements consist of these two terms. The last term represents contributions from couplings of l p l h states to 2p2h states: (phi V'ph12p2h)(2p2h IV'ph Ip'h') Pphp'h' = 2 Im 2p2h ~ ' e2p2h -- co _ ii, ave/2
(s) Only the imaginary part of the coupling term is included in the present calculations. The real part gives shifts of the resonance energy due to couplings of l p l h states to more complicated states, which are neglected in the present work. In the above equation, e2p2h represents tha sum of single-particle energies of 2p2h states. The averaging width Pave represents the width of 2p2h states arising from the coupling of these states to more complicated states. In evaluating matrix elements between l p l h states and 2p2h states, the antisymmetrized form of the effective particle-hole interaction is used . The sum in eq. (5) is taken over all the antisymmetrized 2p2h states near the energy co. The correlations in the 2p2h states are not included in the present calculation, Although it is most desirable to used correlated 2p2h states obtained by the diagonalization in the space of bound 2p2h states, it is practically almost impossible. The advantage of the present method is that the Pauli principle is exactly taken into account in the 2p2h states. For the IAR, the off-diagonal part of Fphp, h, (ph 4: p'h') plays an important role in obtaining the width of the IAR. By including the off-diagonal part fully, the decay to 2p2h states with lower isospin is almost
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I GT STRENGTH 208Bi
19 May 1983
2o1~ I-'pho 11/2i ~~/2)
10 - I
I 20 Ex (MeV)
Fig. 2. The strength function of GT resonances (in units of
Ex (MeV) Fig. 3. The energy dependence of the spreading width Fph for the l p l h state [~ .u-1 ]z+ tll 1/2113/2
prohibited and this gives a very narrow width to the IAR. If only the diagonal part of P is included, we cannot obtain a sharp peak. For parameters MANN = 10 and Pave = 400 keV, I get 170 keV for the spreading width of the IAR. This value is almost unchanged when I change the parameters MAXN and PaveThe strength function of GT resonances in 208Bi is depicted in fig. 2. The used parameters are MAXN = 10 and Pave = 800 keV. The highest peak has a width of about 2.5 MeV. The lower two peaks have widths less than 1 MeV. The GT strength in fig. 2 is integrated within some energy intervals. The obtained GT strengths in units ofg2/4~r are as follows: 15.1 for E x = 10.0-12.3 MeV, 28.4 f o r E x = 1 2 . 3 - 1 5 . 0 MeV, 20.6 f o r E x = 15.0 17.5 MeV and 58.3 f o r E x = 17.5 - 2 3 . 0 MeV. These values show that almost all the GT strength remains in the energy range presented in fig. 2. One can compare the result of the TDA diagonalization in fig. 1 with the strength function in fig. 2. The lower two peaks have corresponding TDA states. The highest peak and the plateau which lies between 15.0 MeV and 17.0 MeV correspond to the highest TDA state. In this plateau there is about a quarter of the GT strength in the highest TDA state. This plateau comes from strong coupling between l p l h states and 2p2h states. In constrast with the case of the IAR, the strength function is not so affected by neglecting the off-diagonal part of l"php,h, in the case of GT resonances. Therefore we show in fig. 3 the energy dependence of -Tr .V--I Vph (ph = p'h') for the l p l h state [111/2113/2 ] 1 +. This l p l h state has a large GT transition matrix ele-
ment and it has a large coupling strength to 2p2h states near E x = 15 MeV. This situation is similar for other l p l h states which have large GT transition strength. This may be the reason why the GT strength is found between the higher and lower GT resonances although the original TDA calculation does not give strength there. I finally comment on the parameters used in the present calculations. If we use a smaller value for Pave, we have more fine structures but the overall structure is not changed. The calculations were done with MANN = 20 as well as MAXN = 10. In the case of MAXN = 20, the energies of particle states lower and as a result the level density of 2p2h states becomes higher than that for MANN = 10. In spite of this fact, the resultant strength function is surprisingly almost the same for MAXN = 10 and MAXN = 20. This might be due to the result of some kind of averaging effect in treating many 2p2h states. In summary, the spreading width of the GT resonance in 208Bi is studied in the present work within the self-consistent TDA using the Skyrme III interaction. The imaginary part of the coupling between l p l h states and 2p2h states is included in the TDA linear response function. The uncorrelated 2p2h states are used in the present calculation and the Pauli principle is exactly taken into account in these 2p2h states. The calculated spreading width of the main GT peak is about 2.5 MeV. A shift of GT strength into the energy region between the lower GT states and the highest GT state is observed. It will be interesting to compare the present results with the results of spreading width calculations by using a p a r t i c l e - p h o n o n coupling picture.
g2A/4e MeV-1) in 2°8Bi.
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I am very grateful to Professor Shiro Yoshida for helpful discussions and careful reading o f the manuscript. I thank m e m b e r s o f the t h e o r y division in the Institute for Nuclear S t u d y for w a r m e n c o u r a g e m e n t . The numerical calculations were p e r f o r m e d at the c o m p u t e r r o o m o f the Institute for Nuclear S t u d y by using the F A C O M M 180II-AD c o m p u t e r .
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