Widths of the isobaric analog state of 208Pb

Nuclear Physics A462 (1987) 61-71 North-Holland, Amsterdam

WIDTHS

OF THE

ISOBARIC

ANALOG

STATE

OF =Pb

Shizuko ADACHI Theoretische Kemphysik,

Universitiit

Hamburg,

Luruper Chaussee

149, 2000 Hamburg 50, W.-Germany

Shiro YOSHIDA Physics Department,

Tohoku University,

Sendai 980, Japan

Received 17 June 1986

Abstract:Both escape and spreading widths are evaluated microscopically in a consistent framework for the isobaric analog of the “sPb ground state. A TDA Green function is obtained within the space of discretized J = O+ proton-particle neutron-hole configurations using the Skyrme interaction. Couplings of these configurations with continuum and more complicated configurations are included into TDA matrix elements with a form of energy dependent terms. The energy and the widths of the isobaric analog resonance are obtained as a result of the matrix diagonalization. Comparison is made of the results with the former theoretical calculations as well as experimental data.

1. Introduction

Great progress has been made for the last decade in theoretical study of strength distributions of resonance states above the threshold of particle emission ‘*‘). These states are described by a superposition of particle-hole states in microscopic theories. There are three origins which give the width to the excitation function. The first one is a splitting of the strength which results from correlations in the lp-lh space, whose effect is inherent in the framework of the lp-lh Tamm-Dancoff (TDA) or random phase approximation (RPA). The second one is an escape effect which is brought about by escape of a particle directly from a lp-lh configuration. The last one is a spreading effect brought about by the coupling of lp-lh configurations with more complicated configurations. In order to include a full escape effect by using a complete lp-lh space, the coordinate-space response function method ‘) is known to be very powerful when a residual two-body interaction is a delta function type one like Skyrme interactions. This method was adopted in TDA or RPA calculations of giant resonances or isobaric analog resonances within a self-consistent framework 4z5). The advantage of including a complete lp-lh space is, however, achieved at the expense of information on contributions from each p-h component. As for the spreading effect, a lot of numerical calculations have been done in recent years for the strength distribution of giant multipole resonances 6-9). In a microscopic treatment, discretized p-h bases are used to calculate coupling matrix 0375-9474/87/$03.50 @ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

S. Adachi, S. Yoshida / IAR widths

62

elements

between

The importance

lp-lh

of collective

In the case where contributions

and 2p-2h states which appear escape

in the TDA or RPA matrices.

effects in 2p-2h states has been pointed effects are important

from each p-h component

same time, one must look for a method

and

one wants

or to deal with the spreading which is different

out in refs. 6,8). to distinguish width at the

from the above mentioned

coordinate-space response function method in order to work in the p-h configuration representation. In a recent paper lo), we presented an approximation called the “Q-space approximation” where escape effects are taken into account in a response function of the p-h configuration representation. Using a simple separable type interaction, this approximation was shown to reproduce well the results which include a full escape effect provided that the employed p-h space consisting of discretized states is large enough. We adopt this method in the present study to evaluate the widths of isobaric because analog resonance of the *‘*Pb ground state using the Skryme interaction, both escape and spreading widths are of comparable magnitude and the evaluation of partial escape widths is of importance in this case. Formulae for the Green function to describe analog resonances are given in the next section. It is based on the projection operator method of Feshbach ‘l). In sect. 3, we show how to construct subspaces; first the single-particle wave functions in the Q-space are discretized, then the P-space single-particle wave functions are constructed to be orthogonal to the Q-space ones. The practical way of calculating escape widths is given not in this section but in our forthcoming paper ‘*). The last section is devoted to showing the results and discussions.

2. The Green function Many works were done starting with the definition of Piza and Kerman 13). Following their definition, analog states are obtained by operating the isospin lowering operator on physical parent states which are actual eigenstates of the hamiltonian. One then takes account of couplings between the analog and other states by isospin-violating parts of the hamiltonian. Giant isovector monopole resonances and anti-analog states are known to have significant couplings to the analog state. The correlation between different particle-hole states was shown to be important to obtain a reasonable value for widths of the isobaric analog resonances (IAR) [refs. 14,15)]. In the present work we do not adopt this method but the response function method in which the Green function is calculated within a space of J = O+ protonparticle neutron-hole states following Auerbach and Nguyen Van Giai ‘). They performed a TDA calculation of the complete J = O+ proton-particle neutron-hole spectrum using the Skyrme interaction. The coupling of the analog with isovector monopole resonances and anti-analog states is inherent in this approach and correlation effects are taken into account. Since we intend to evaluate the spreading width

63

S. Adachi, S. Yoshida / IAR widths

microscopically and distinguish the partial escape widths in this work, the Green function is obtained in a p-h configuration representation in contrast with their coordinate representation. The whole space is divided into two subspaces Q and P. Both spaces consist of AT, = -1, J = O+p-h states (IpI-lh, 2p-2h and so on). The Q-space includes discretized particle states with positive single-particle energies as well as negative-energy bound states. This space is an extended space of bound p-h states. On the other hand, in the P-space at least one particle is in the continuum state which is constructed to be orthogonal to the bound states in the Q-space. The Q- and P-spaces are further divided into subspaces Qi’s and Pi’s according to their complexity. The subspace with a suffix i consists of ip-ih states. The coupling between the Q, and QZ spaces under the presence of a two-body residual interaction gives spreading effects, while the coupling between the Qi and Pi spaces gives escape effects. The Green function in the Q1 space is

where %‘o, is an effective hamiltonian

in the Q, space

XQ,=QIHQ,+

WT+ WL.

(2)

The coupling terms are W’ = Q1HP,

W’= Q,HQz

(3)

E-Pl~P,+i8p1HQ1’

1 E-Q2HQ2-

W,+iS

QzHQI

.

(4)

In deriving the above expression, we kept direct couplings of Qr with Q2 and P1 spaces only. The coupling term between the Q2 space and the space of more complexity is included in the denominator of WL by W,. In this work the hamiltonian in W’ is replaced by a single-particle hamiltonian H,, W’ + W; = Q, H,,P,

E-P,~,,P,+,plHoQ1~

(5)

This quantity is easily calculated using the single-particle Green function in coordinate representation if the single-particle potential is local. In the case of a separable interaction, this replacement was shown to be a good approximation if a sufficiently large Q1 space is employed lo). In the evaluation of W’, we neglected the residual interaction in the Q2 space, that is, replaced Q2HQ2 by Q2HOQ2. This approximation may result in missing some

64

S. Adachi, S. Yoshida / IAR widths

correlation

effects, whereas

and the Pauli principle

it enables

violation.

us to be free from double-counting

problems

We use

w3+ QlHQ2E-Q,I!ioQ2+ilQ2HQ1' where the coupling term W, is simulated by a constant imaginary value -il. When we use a density-dependent interaction, p-h matrix elements in QIHQ, are obtained as the second derivative of the energy functional with respect to the one-body density matrix. On the other hand, the antisymmetric part of the p-h interaction derived from the original density-dependent interaction is employed in evaluating

matrix

elements

in Q1 HQ,, since 2p-2h states are antisymmetrized. 3. Construction of subspaces

We start with the construction of the Q-space. The single-particle hamiltonian h is generated by the Hartree-Fock (HF) calculation based on the Skyrme interaction. The parameter set SIII [ref. ‘“)I is utilized throughout the calculation. Next this HF hamiltonian is diagonalized within a limited basis of harmonic oscillator wave functions, (h-G$#Uj)=O. The single-particle

states for occupied

(7)

and unoccupied

orbits are given by

Ns.p.-1 19,rj>=

where

]n’2j, V) denotes

a harmonic

“ZO

G&o,

oscillator

4,

wave function

(8) with the node number

n’, orbital angular momentum 1, total angular momentum j and the size parameter V. The quantity N,.,. indicates the number of states used in the diagonalization. All resulting eigenvalues are discrete and some of them are positive. If a single-particle state has a negative energy, its eigenvalue and eigenvector are very close to those obtained by the HF calculation in a full space provided that a sufficiently large N,.,. and a proper Y are used. When a single-particle state with a positive energy corresponds to quasi-bound state, its eigenvalue and eigenvector are close to the corresponding HF results. This fact does not change if N,,,. is further increased. On the other hand, when a positive energy level does not correspond to any resonance or echo states “) due to the HF potential, its energy changes if N,.,. is changed. Among the particle states constructed in this way we choose some particle states to be included into the Q-space. The number of p-h states in the Q1 space which consists of these selected particle states is denoted by Nph. In the present work the maximum difference of number of nodes in the particle and hole wave functions Anmax is used to select particle and hole orbits. The lowest s1/2 orbit is excluded

S. Adachi, S. Yoshida / IAR widths

65

from the hole orbits because its effect is very small. The Q-space is specified with

three parameters v, N,.,. and Nrh (or An,,,). There is no clear way to choose these parameters. We take the parameter Y to be a standard value. A-1’3 fmp2, and the appropriate values of other parameters should be found. We should increase the values of iV_ and Nph until further increase does not give a significant change in the results. After the construction of the Q-space, single-particle wave functions in the P-space wave functions are obtained. The projected continuum wave function vliZ,which is orthogonal to the bound wave functions in the Q-space, is a solution of the equation, (E-PirP)Jvrj,)=O. This is given in terms of the unprojected It&)= I&)-C

n

(9)

continuum wave function v& [ref. ‘)I,

sm#%l&hjl~lvijJ,

(10)

where g(c) is the unperturbed particle Green function and r&‘s are the bound wave functions included in the Q-space. The wave function v$= satisfies (E-~)]v;,)=O.

(11)

This is a generalization of eq. (2.8) of ref.14). Though we do not need to use the projected wave functions explicitly in the evaluation of Wi (see ref. ‘“)) or the partial escape widths (see ref. 12)), it is interesting to see the radial shape of these wave functions. As an example, the radial wave functions of the proton orbit ~312 in 208Pb are shown in fig. 1 for N,,,, = 20. The lowest two states with n = 0 and n = 1 have negative energies and their eigenvalues and eigenfunctions are very close to the corresponding HF results. The next higher state has E = 2.08 MeV which is almost equal to the resonance energy obtained by solving the equation with the HF potential in a whole space. The higher states with n 2 3 have positive energy eigenvalues and their wave functions extend over 20 fm. The lowest of these, that is, the n = 3 wave function has very small values for I c 10 fm. In fig. 1 the real part of the continuum wave function without orthogonalization is plotted for E = 10 MeV. Also shown is the real part of the projected P-space wave functions. The wave function presented by the dot-dashed curve is orthogonalized to 5 bound states used in the Q1 space with Npi,= 43, while the dashed curve is a result of the orthogonalization to 6 bound states in the Q1 space with A$,-,= 85. They have small values for I ~20 fm and become closer to the unprojected one beyond that point. In table 1 we show the eigenvalues calculated for two cases corresponding to N,.,. = 10 and IV,.,. = 20. The positive energy eigenvalues with n 2 3 becomes lower when N,.,. is increased from 10 to 20. In the third column we show the resonance energies. A resonance was found at E, = 2.08 MeV corresponding to n = 2, but other resonances were not found at higher energies. At 38.3 MeV we found a broad echo “) with a width of more than 20 MeV. This does not correspond to any bound states.

66

S. Ada&i,

S. Yoshida f IAR widths

-n=U,E=-36.72MeV ---n = I , E =- I8.66MsV

-ff=4,E=12.38MeV ---n=5, fz~16.77MeV

0.5

0

0

----

Fig. 1. Single-particle

wave functions

Re fVof Re (. VI

rlfm f

f=IOMeV +E=iOMeV Nph -85

in Q and P space for protons

in zOsPb par2 orbit.

The Q-space wave functiofi obtained above may be interpreted in the following way. The harmonic oscillator wave functions make a complete set if we do not restrict the value of N,,. If we restrict the number N_,, we may say in a very rough way that they span a limited spatial subspace. The radius R of this limited subspace may be obtained from the mean square radius of the highest orbit allowed by N,,,,.. So if we increase N_,, R increases. Then it is easily understood that the level distance becomes small with increasing R in analogous way to the case of

67

S. Adachi, S. Yoshida / IAR widths TABLE 1 Single-particle

n

0 1 2 3 4 5 6

energies

of proton

state p3,2 in *“sPb

E [MeV]

E [MeV]

(X, = 10)

E, lMev1 (HF)

( NSP = 20) -36.72 -18.66 2.08 8.61 12.38 16.77 21.89

-36.71 -18.65 2.10 12.75 19.57 28.27 41.95

2.08

(38.3)

The energies are obtained using 10 and 20 harmonic basis, while E, is the resonance energy calculated Hartree-Fock potential.

oscillator using the

wave functions for an infinitely deep square well potential. The Q-space wave function may describe physical phenomena occurring within the limited space and energy.

4. Results and discussions The matrix XQ,(E) is diagonalized in this work to obtain the excitation energy and the width of the IAR instead of drawing the strength distribution of the IAR from the Green Since the matrix orthogonal

function by using an appropriate X,(E) is complex and symmetric,

matrix

T (Z-%+(E)T-‘),,=

We identify

one-body operator as a probe. it is diagonalized with a complex

&,(K-$i~,).

(12)

one state )A) with the IAR, IA)=C

a

(13)

T,Ia),

where ICY)is a J = O+ proton-particle neutron-hole component. The real eigenvalue of that state E, is close to the displacement energy and the overlap with the simple shell model

analog

state is nearly

0.95. The value

E, is taken

for E in the 1.h.s. of

eq. (12). The results of the diagonalizations in the case of N,.,. = 20 is summarized in table 2. The parameter Nph from 22 to 106 represents the number of p-h configurations with Anmax= 1 to 5. The results are shown for various cases: including only W’, only W’ or both; only the imaginary part of W or complex W. The excitation energies are measured with respect to the ground state of *08Pb. All 2p-2h states with unperturbed energies from 17 to 20 MeV are taken into the evaluation of W”. Since we have little information on the magnitude of W,, the quantity I should be thought of as a kind of parameter for numerical calculations. We took

68

S. Adachi, S. Yoshida / ZAR widths TABLE

2

Analog resonance energies and widths for *s8Pb with NSP= 20 Iv Anmar

%.h

E,[MeV] 1

22

2

43

3

64

4

85

5

106

W”

urf+wi

I/C

I C I C I C I C I C

18.71 18.63 18.62 18.52 18.50 18.42 18.48 18.37 18.45 18.34

r’[keV] 7 12 238 231 57 77 66 91 59 86

E,[MeV]

rL [keV]

E,[MeV]

r [keV]

18.71 18.66

178 179

18.71 18.59

185 178

18.48 18.47

159 165

18.50 18.40

219 253

18.43 18.42

175 176

18.45 18.34

232 274

The imaginary part of W(I) or the complex W(C) is included in the TDA matrix.

0.2 MeV for I and the final result remains essentially unchanged when the value of I is changed around that value. As for the spreading widths, the results are rather stable with respect to Nph, and a real part of W’ gives a very small effect. On the other hand, the escape width changes dramatically in the region of small Nph and shows a convergence to a stable value for large Q1 spaces. We see that it is important to include the real part of W’. In the case of N,.,. = 10, if we take both the real and imaginary part of WT into the calculation with Nph = 106, the obtained value for r’ is very close to the one in the N,.,. = 20 case. The relation r = r’ + r” holds for all cases within the difference of 10%. Let us compare our results with the former Van Giai ‘) used the analog spin lowering function and they obtained a narrow peak half maximum of 140 keV. After including

theoretical one. Auerbach and Nguyen operator as a probe for the response around E = 18.9 MeV with a width at the corrections r4) from the Coulomb

exchange term (-350 keV), vacuum polarization (100 keV) and others (nearly zero in the “*Pb case), they finally obtained the analog energy of 18.63 MeV. Since the Coulomb exchange term is approximately included in our calculation, we include other corrections. Then the obtained value 18.44 MeV is smaller than theirs by 190 keV. If we assume that the two definitions of the analog energy do not bring a significant difference in the final results, their results and our results including only W’ should be close to each other provided that the approximation adopted here is good enough. The discrepancy about 190 keV in the excitation energy may result from the numerical accuracy in the HF calculation which gives single-particle energies. On the other hand, the difference of 50 keV in the escape width seems too large. As mentioned before, we put a consistency condition E = E, in solving eq. (12). Since the escape width is very sensitive to the choice of E, it becomes larger by

S. Adachi, S. Yoshida / ZAR widths

69

30 keV if we increase E by 400 keV from E, in the case of N,.,, = 20, Nr,,,= 106.

Therefore it seems that the difference in the calculated excitation energy of the IAR is the main reason for our small escape width. The partial escape widths are presented in table 3 for the case of N,.,.=20, NPh= 106 and complex WT. This partial escape width corresponds to the proton width r,. When a hole state specified by quantum numbers (nh , I, j) remains after the escape of a particle which has the same orbital and total angular momenta, the contribution from that component is (14) where T’ is a coefficient for a discretized p-h component of the IAR (A) with a particle orbit (not, Z,j) and a hole orbit (nh, I,j), and uliE(k) denotes a wave function in the P1 space with an available particle energy E = E,+ &hand a wave number k satisfying E = h2k2/(2m). These available particle energies are also listed in table 3. The wavefunctions are normalized as (u~~(~))o~~(/c’))=(~T)~S(~-~‘). Since we used the single-particle hamiltonian, the sum over (Y is restricted to the p-h components with the same orbital and total angular momenta as the remaining hole state. TABLE 3 Partial escape widths for the ground analog state of “*Pb exp

f

I%18

Theor.

)

Hole E [MeV] PI/Z fW P3/2 i 13,2 fl/2 h9/2 total

11.49 10.92 10.59 9.74 9.15 8.06

r,

lkevl

56h2.8 25.2*3 63.2*4 10.014 4.8&l CO.01 150

EWV]

C, lkevl

11.38 9.99 10.33 8.26 7.25 5.76

47.8 12.5 43.2 0.5 0.1 0.1 104

In table 3 we see that a sum of partial escape widths is large than the total escape width r’ by about 20%. When only the escape width is included, this sum is exactly equal to the total escape width multiplied by a factor N (a sum over a of the absolute square of T_) [ref. 12)]. The sum of partial escape widths should be compared with a sum of experimental proton widths, while the quantity r, which is close to the sum r’ + r’, should be compared with the experimental total width. Looking into each partial escape width, we find that the theory underestimates it compared with the experimental data 14*18)because of smaller available particle energies. This is conspicuous for the f-orbits. Though the Skyrme interaction can

70

S. Adachi, S. Yoshida / IAR widths

reproduce bulk properties of nuclei quite well with a small number of parameters, it sometimes fails in describing the details such as a precise level scheme. If we increase the excitation energy up to the experimental excitation energy 18.87 [ref. ‘*)I, 18.88 [ref. “)I MeV with respect to the ground state of *‘*Pb it makes the escape width r’ larger by about 30 keV as mentioned in the above paragraph. Multiplying that width by the factor N, we could obtain a comparable value to the experimental data for the sum of partial widths. But there remains a problem in each contribution, because the available particle energy of the p-orbit becomes higher than the experimental value, while that of the f-orbit remains still lower with the increase of the excitation energy by 400 keV. If we take account of the above expected increase in the escape width r’, the calculated spreading width is larger by about (30-40)% than the value which reproduces the experimental total width 230* 10 [ref. ‘*)I, 220 f 20 [ref. “)I keV. As for the spreading width, there may be a possibility that the used level density of T = 1, J = 0+2p-2h states is overestimated due to the neglect of the correlation in 2p-2h space. There may be another possibility that the used interaction is not adequate in the present case. The parameter set SIT1 gives reasonable values to the spreading width in some cases 20921),but it is necessary to test it in various cases. If we use another parameter set of the Skyrme interaction, for example the parameter set SGII [ref. “‘)I which reproduces a compressibility well and has better spin properties, we may obtain a better result. Though the HF-TDA framework was used in this work, it should be generalized to the RPA case 23) when one wants to obtain a more consistent picture. In summary, we performed a TDA matrix diagonalization in the space of discretized J = O+proton-particle neutron-hole configurations to obtain the isobaric analog resonances of the ground state of *‘*Pb. The advantage of this approach lies in the fact that the coupling of the analog with isovector monopole and anti-analog states is inherent and correlation effects are taken into account. Both escape and spreading widths are obtained by evaluating microscopically the coupling terms of bound lp-lh configurations with continuum and more complicated configurations in the TDA matrix. The excitation energy of the IAR is lower than the experimental ones by about 400 keV. There remains a problem in numerical values of the widths. Because we have a lot of information about the IAR, the evaluation of the quantities about the IAR could become a good test for both the formalism and the effective interactions. The present work is the first step in a fully microscopic evaluation. Even though there are some limitations and problems at the present, progress in this direction is encouraged. References 1) 2) 3) 4)

G.F. Bertsch, P.F. Bortignon and R.A. Broglia, Rev. Mod. Phys. 55 (1983) 287 S. Yoshida, Prog. Theor. Phys. Suppl. 74-75 (1983) 142 S. Shlomo and G.F. Bertsch, Nucl. Phys. A243 (1975) 507 K.F. Liu and Nguyen Van Giai, Phys. Lett. 65B (1976) 23

S. Adachi, S. Yoshida / IAR widths 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23)

71

N. Auerbach and Nguyen Van Giai, Phys. Lett. 72B (1978) 289 V.G. Soloviev, Ch. Stoyanov and A.I. Vdovin, Nucl. Phys. A288 (1977) 376 S. Adachi and S. Yoshida, Nucl. Phys. A306 (1978) 53 P.F. Bortignon and R.A. Broglia, Nucl. Phys. A371 (1981) 405 B. Schwesinger and J. Wambach, Nucl. Phys. A426 (1984) 253 S. Yoshida and S. Adachi, Nucl. Phys. A457 (1986) 84 H. Feshbach, Ann. of Phys. 19 (1962) 287 S. Yoshida and S. Adachi, Z. Phys. A., to be published A.F.R. de Toledo Piza and A.K. Kerman, Ann. of Phys. 43 (1967) 363 N. Auerbach, J. Hiifner, A.K. Kerman and C. M. Shakin, Rev. Mod. Phys. 44 (1972) 48 N. Auerbach, Phys. Lett. 44B (1973) 241 M. Beiner, H. Flocard, Nguyen Van Giai and P. Quentin, Nucl. Phys. A238 (1975) 29 K. W. McVoy, Ann. of Phys. 43 (1967) 91 P.v. Brentano, H.J. GlBckner, E. Grosse and C.F. Moore, Jahresbericht, Max-Planck Institut, Heidelberg (1968). G.H. Lenz and G.M. Temmer, Nucl. Phys. All2 (1969) 625 S. Adachi, Phys. Lett. 125B (1983) 5 S. Adachi and N. Auerbach, Phys. Lett. 131B (1983) 11 Nguyen Van Giai and H. Sagawa, Phys. Lett. 106B (1981) 379 N. Auerbach, A. Yeverechyahu and Nguyen Van Giai, Proc. of 1980 RCNP Int. Symp. on highly excited states in nuclear reactions, ed. H. Ikegami and M. Muraoka (Osaka, 1980).