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Relation between the Landau-Migdal interaction and the separable interaction
Nuclear
Physics
@ North-Holland
A438 (1985) 503-S 11 Publishing Company
RELATION
BETWEEN
THE LANDAU-MIGDAL
AND THE SEPARABLE
INTERACTION
INTERACTION
H. TOKI
Department of Physics, Tokyo Metropalitnn Universit.v, Sefuguyu, Tokyo I%, Japan and f-5. EJIRI ~gpurt~~~t of Physics, Osaka ~~iver~~ty, Toy~nak~, Osaka 56U, Japan Received 14 June 1984 (Revised 30 October 1984) Abstract: The Landau-Migdal interaction has been used widely in analyses of the quenching of spinisospin transitions, The separable interaction, on the other hand, has been used for studies of electric giant resonances and low-lying vibrational states, and recently for those of spin-isospin polarizations. The relation between these two interactions in the spin-isospin channel is studied. Quantitative comparison of these interactions in the spin-isospin L= 1 channel shows good correspondence between the nuclear matrix elements of the two types of interactions.
1. Introduction The quenching
phenomena of spin-isospin transitions such as magnetic and Gamow-Teller transitions have drawn much attention in recent years le4). Among many possibilities, the delta-isobar mechanism as well as the second-order core polarization due to the strong tensor correlation are believed to be responsible for the major part of the missing strength s-‘2). In these discussions, our knowledge of the short-range correlations among nucleons is essential for the determination of the magnitudes of these mechanisms. It is commonly assumed that a Landau-Migdal interaction of delta-function form should represent the short-range correlations as long as the physics is limited to small-and medium-size momenta 13). Systematic studies of the positions of the giant Gamow-Teller states and the transition matrix elements in nuclei close to the magic numbers suggest a Landau-Midgal parameter in the spin-isospin channel of g’-0.6 in pion mass units 14). On the other hand, hither-multipo~e magnetic y-transitions (M2, M4) and analogous axial p-transitions in medium-heavy nuclei have been extensively studied by the Osaka group ‘5*‘6). It has been shown that these spin-isospin /3- and ytransitions between low-lying single-quasiparticle states are uniformly quenched by factors of g”“/g - 0.2-0.3 [ref. ‘5>].Ejiri and Fujita ‘2*‘s**6) have analyzed the quenching in terms of the spin-isospin core polarization. This has been interpreted as destructive interference of the spin-isospin giant resonance with the single-particle 503
504
motion.
H. Toki, H. Ejiri / Landau-Migdal
The spin-isospin
core polarization
interaction
and the giant resonances
are collective
features in the spin-isospin channel. Hence, they may be treated by means of separable-form polarization (phonon) interactions in analogy with those used for the El giant resonances, E2 core polarization and other modes I’). Because of the complexity of the nuclear structure in these nuclei, the separable interaction was used to analyze the quenching phenomena of the spin-isospin transitions ‘2*‘5*‘6). They extracted the coupling of the polarization strengh of the transition matrix elements. Defining the spin-isospin tensor operator as FTsU = T’[(+’ x ( I/ b,JLYL] with b,, being the oscillator parameter, the separable interaction is given by V= x&_,F * F. They used the random-phase approximation with a model space of lho ph states and evaluated the coupling strengths x& so as to reproduce the data 12,‘57’6).The derived interaction strengths are xzU = (60-85)/A MeV for AL = 1 channels. They depend a little on whether S = 0 or S = 1 as well as the type of experimental data used for the analysis. In this paper, we relate these two interactions and discuss the consistency of the interactions. Since extensive works have been made on the spin-isospin channels 20) with AL = 0, we discuss mainly the AL = 1 channels in this paper. The GT mode with AL = 0 has recently been analyzed by several authors. In this way, we hope to obtain a general view of the quenching mechanisms for the AL= 1 channels. In sect. 2, we shall introduce the two interactions and relate them explicitly by comparing the harmonic-oscillator matrix elements. We will discuss the relation between the two interactions by assuming that the transition densities are those of the macroscopic model in sect. 3. Furthermore, numerical results will be provided in sect. 4, where M2 matrix elements with L = 1, S = 1, T = 1 are explicitly compared. Sect. 5 will be devoted to the conclusion and a discussion.
2. Landau-Migdal
interaction and the separable interaction
In this section we introduce the Landau-Migdal interaction and the separable interaction in the spin-isospin channel and calculate the particle-hole matrix elements of these interactions. We begin with the Landau-Migdal interaction. The spin-isospin part is written as’
V(LM)=$g’S(r,-r,)rr, Tr
‘uzrl
-12,
(2.1)
where fm is the 7rNN coupling constant and rnrr the pion mass. The combined value is f',/ rnt= 396 MeV * fm3 in common units. g’ is called the Landau parameter and the empirical value is believed to be around 0.6 [ref. ‘“)I. g and T are the spin and isospin Pauli operators, respectively. Note that the Landau-Migdal interaction is constant in the momentum space.
H. Toki, H. Ejiri / Landau-Migdal
In order transform
to take the particle-hole the jj-scheme lj,j,‘;
matrix
wave functions JMTM,)
elements
of the above
= C ((~,9jp(~h~ljh; Jl(~&L(%W;
where the transformation coefficients the angular integral, the particle-hole
interaction,
we
L)@(f$-‘;
S)]‘,lff-‘;
J> TM,),
(2.2)
are the unitary 9-j coefficients r8). After taking matrix elements are reduced to the form
= (jrj,‘;
JMTM,I
=$g’C ?r
L
Z*(j,j,;
X (47r)2
505
to those in the LS scheme:
x[&I,‘;
M(LM)
interaction
I
V( LM)( j; jl;’ ; JMTM,) LJ)Z(jLjh;
Q;&I)Q;W(q)
W)
$3
(2.3)
where
-X&h; JW = ((~&j,(~h~>jh; JI(~,4)UI~)S= 1; J) x2v$21,+ The function function:
1)(21,,+ 1)/4~(2L+
Q is the Fourier-Bessel
transform
1)(1,01,01LO)(-)‘“.
of the particle-hole
(2.4) radial
wave
dr. J ~,(~)G(~Mq~)~2
Q;,,(q)=a3
(2.5)
0
On the other hand,
the separable
interaction
is written
V(EF) = x;~F.
F
as (2.6)
with F ;LJ=Tr[aSX(l/bo)LYL]J, where the oscillator parameter b, is given by b, = G with ho = 41A-“3 MeV. In eq. (2.6), this separable form has been denoted by V(EF) due to the intensive use of the separable interaction for the spin-isospin modes by Ejiri and Fujita 15). strength The spin-isospin interaction strength x zH is related to the dimensionless fsTUasxs=r=fsTuhw/2CiiGii, where G,=(iJJ[~~~~(r/b,)~Y~]~(]j)/fiand the summation is made over all 1 ho ph excitations 12,16). The values for the isospin modes were derived as xALs= 320/A (2.1)2L MeV for L= 0 and L= 1 so as to reproduce the observed quenching factors and the El giant resonance energies. This gives xAoo= 320/A MeV for the L = 0 channel, corresponding to the symmetry energy lOOtT/A MeV, and x;,, - 73/A MeV for the L = 1 channel. The f2corresponding strength ,yt ,, for the spin-isospin to the xAI1 is fT= 1.15 [refs. 12,‘6)]. The interaction
506
H. Toki, H. Ejiri / ~ndau-~jgdal
inrerac~ion
mode with L = 1 is of the same order of magnitude as the xi,, for the isospin mode. However, analysis of the observed giant-resonance energies with T = 1, S = 1, J” = 22, l+, O- in terms of the interaction eq. (2.6) yields a slightly smaller value off, = 0.9, which corresponds to the xi IJ = 60/A MeV[refs. ‘2Z’6)].The ratiof,/f, 0.78 compares well with that for the L = 0 channels. On the other hand, a slightly larger value of fm- 1.35 is derived so as to reproduce both the observed giant resonance energies and the quenching factors in terms of both the lhw ph polarization of eq. (2.6) and the higher excitation ro strength such as AN-’ and the second-order polarization ‘2,‘6). This corresponds to xIti = 85/A MeV [refs. ‘2*16)]. Therefore we evaluate the g’ corresponding to the separable interaction strength of ~~~~=(60-85)/AMeV. We take now the particle-hole matrix elements of this interaction: M(EF) = (j,j,‘;
JMTn/r,l V(EF)l jbjk-;' ;JMTM,)
=,y~uZ*(jpj,,;
L=J-
I, J)Z(jgj;l;
L=J-
1, I) (2.7)
Although the two forms given in eqs. (2.3) and (2.7) look similar, we cannot relate the two interaction strengths analytically because of the integral and the summation involved in eq. (2.3). In the next section, therefore, we shah use the macroscopic wave function in order to compare the two interactions at the matrix-element level. After making this quahtative comparison, we shall calculate explicitly the matrix elements of the two interactions in the harmonic-oscillator basis in order to make a direct comparison between the two interaction forms.
3. Qualitative
comparison in the macroscopic
model
In a discussion of collective states such as giant resonances it is often assumed that those collective states are described in terms of slight oscillations around the nuclear ground-state density. We might be able to assume also that the spin-isospin modes under consideration have similar collectivities and therefore the matrix elements of the two interactions with the collective wave functions have the same value. The transition density in the macroscopic model is “)
G(r) L-, k(r) = ~LT r YLhf($),
(3.1)
which is applicable for L > 1. The constant oL is determined by the sum-rule value for the electric mode. An extension of this form to spin-isospin modes is not straightforward, since the spin wave function with S = 1 couples with L = J - 1 and
507
H. Toki, H. .Ejiri / Landau- Migdal interaction
J+ 1 to provide the total spin J However, we shah assume that the collectivity is
caused by the repulsive correlations at small momentum, where the smaller L-states have dominant roles over the larger L-states. In fact the transition density with L = J + 1 makes a negligible contribution to the electromagnetic and beta decays, which are transitions with q = 0, and the long-wavelength approximation is valid. Hence, the same form as the transition density in eq. (3.1) will be assumed for the spin-isospin modes also, where L should be read as L = J - 1 and the spin-isospin wave functions are added simply to eq. (3.1). In eq. (3.1) above, the ground-state density is well described in the Woods-Saxon form p(r) =
po/[1 + e(r--R)/c] ,
(3.2)
where the constants are provided by electron scattering excperiments is): C=OS7fm.
R=(1.18AL’3-0.48)fm, The matrix element of the Landau-Migdal
(3.3)
interaction is written as 13.4)
On the other hand, the separable interaction gives
[I
mdp 2L+1dr
M(EF) =x&b;*’
0 z
2
3 _
(3.5)
The derivative of p with respect to r is localized at the nuclear surface. Therefore the integrals of eqs. (3.4) and (3.5) may effectively be given by the approximate forms (3.6a) (3.6b) where (dp,/dr),,,
= p,/4C, a - 2.6C and a’- 3.3C. Then, 2 W~~~,,,,,-~d~:
2 (>
?r
RZL2.6C,
(3.7)
2
M(EF),,,,,-X&
R4L+2(3.3C)Zb;2L.
(3.8)
Dropping the common terms, we find the relation
.f’, g’=
,z
7r
The A-dependence
,yLR2L’24.2Cb;zL.
(3.9)
given in eq. (3.2) is effectively given by a simple form of
H. Toki, H. Ejiri / Landau-Migdat
508
interaction
R =
1.1A”3 in the present mass region of A = 90- 150. Then the interaction strength ~1,~ = (60-85)/A MeV for the L= 1 channel leads to .f’, 1212g’==210-290 MeV * fm3. (3.10) 7r This value corresponds to g’ = 0.53-0.73 which is consistant with the value obtained from the analysis of giant Gamow-Teller states 14) although the result should be considered qualitative. Furthermore, if we assume a delta-function-type interaction, the A-dependence obtained by Ejiri and Fujita for the L = 1 (J = 2) mode is consistent with the short-range nature and no A-dependent assumption of g’. 4. Numerical comparison In this section we shall calculate the particle-hole matrix elements explicitly using eqs. (2.3) and (2.7), and obtain the corresponding g’ from the strength ~1,~ for the M2 mode (S = 1, L = 1, J = 2). The experimental data are taken for nuclei around A=90 and 150 by utilizing the fact that the lh,,,* single-particle state is lowered by the spin-orbit force and makes a low-lying 2- state together with the lg,,z orbit TABLE 1
Particle-hole matrix elements of M2 states for the Landau-Migdal and separable interactions in nuclei with A = 90 (“Zr) and A = 146 (la6Gd) 7r.
ph
v
A=90 -2.13 3.61 -1.77 0.82 -0.47 -0.76 0.56 -1.09 1.27 -1.15 0.19 -0.62 0.00 0.08 -0.89 -2.14 -1.91 -0.25 -0.42
C Dh
CWirY,tO)
177
-113 168 -68 59 0 -50 27 -89 96 -67 21 -39 20 0 -74 -133 -85 -53 -79 10800
62 134 22 17 0 12 3 38 44 21 2 7 2 0 26 85 35 14 30
H. Toki, H. Ejiri / Landau- Migdal TABLE
27.
Y
eh
509
interaction
1 (cont.) CO-M) [Meal
C(W 1m,*1
COW,,,, [m21
-0.93 0.15 -0.50 0.00 0.06 -0.72 -1.73 0.82 -1.54 -0.31 0.17 -0.20 -0.75 1.12 0.08 -0.34 -0.55 0.18 -0.18 -0.07 -0.36 0.2 1 -0.61 -0.42 0.00 -1.58
-77 24 -45 22 0 -85 -153 92 -98 0 37 -62 -123 164 32 -91 -67 50 0 0 -78 34 -113 -83 17 -176
25 2 8 2 0 30 98 35 40 0 6 16 63 111 4 34 19 10 0 0 25 5 53 29 1 129
123
15 800
A= 146
w/2 1G/2 24,2 1G/Z 2d,,,2P,,, 2d,,22P,,z 3s,,zlfs,2 3s,,,2~,,, lh 1,/z 1g112 th,,,lg,,, Ih,,,lg,,, lh,,,2d,,, 2f,,, 1g7/2
2f7,2k,,, ‘Z,,W,, 2f,,,2d,,, 2f*,,lg,,, 2fs,, lg,,, 2f,,,2d,,, 2fs,,2d,,, 2fs,,3s,,, 3e,/z1g7/, 3e,,,2ds,, 3e,,,2d,,, 3Ps,,3%,, 3e,,,2d,,, 3er/,2d3/, Ii ,,/zlh,,,,
C C(ehkY,IO) Ph
The particle-hole states shown in the second column are the partner of the lh , 1,21g7,2-’ state for the matrix elements. C(LM) and C(EF) are directly related to the M(LM) and M(EF) of eqs. (2.3) and (2.7). C(EF),i,.a is l((ph)JM(r[cTY,]2(0)(2. The summed values Z,“h’ C(ph(r[aY,]‘jO) are shown at the bottom of each column.
nearby. Hence, we take the nucleus with A = 90, where the Fermi levels are at 2 = 40 and N = 50 and the nucleus with A = 146, where they are at Z = 64 and N = 82. For the case of Z = 64, we assume that the lg7/2 and 2d,,, orbits are occupied in the 50-82 major shell. With this setting, we take all the particle-hole matrix elements with lhw excitations. This model space is consistent with the one taken by Ejiri and Fujita in order to obtain the coupling constant x [ref. “)I. In table 1 we show the particle-hole matrix elements for J =2. In the second column, the values C(LM) = M(LM)/g’ are given and in the next column, the value We give also, in the fourth column, C(EF) = M(EF)/[xSTU beZL((2L+ 1)!!/4~)*].
510
H. Toki, H. Ejiri / Landau- Migdai interaction
the square of the transition matrix elments I((ph)Jnci(r[oYJ2[O)(“. In these numbers, we have not included the r-spin matrix elements, which are trivial and irreievant to our discussion here. Note that the signs of the two matrix elements C(LM) and C(EF) coincide. Hence, all the particle-hole states act coherently in both cases to quench the transition strength of the low-lying state. In order to compare the two interactions, we take the weighted sums of C(LM) and C(EF) as ((ph)JMlr[aYJ*O), which appear in the perturbation calculations assuming all the particle-hole states have the same unpe~urbed energy. The weighted sums are listed at the bottom of the columns for C(LM) and C(EF). We can now find g’, which corresponds to a given ~1,~. When we take the coupling constant of the separable interaction xi I2= (60-85)/A MeV, we find g’ = 0.50-0.73 for A = 90 and 0.55-0.80 for A = 146. These values are quite consistent with the number of 0.53-0.73 obtained by the macroscopic description in the previous section, and with the value g’-0.6 from the giant Gamow-Teller states 14).
5. Conclusion and discussion We have studied the relation between the delta-function-type interaction (LandauMigdal interaction) and the separable interaction at the matrix element level. The macroscopic model provides g’=O.53-0.73 with the use of xl,,== (60-85)/A MeV. Numerical studies of harmonic-oscillator matrix elements for ~1,~ have resulted in g’ = 0.50-0.73 for A = 90 and 0.55 - 0.80 for A = 146. It is very interesting to find that the g’ values evaluated from the macroscopic model are quite consistent with the values evaluated from the comparison of the numerical matrix elements. These values are also in accord with the value g’-0.6 obtained by the systematic study of giant Gamow-Teller states in the medium-heavy nuclei. In fact there are slight differences between these g’ values obtained by various methods. The value g’= 0.75 for ,Y!,~= 85/A MeV, which corresponds to the fm = 1.35, is about 20% larger than the value g’ = 0.60 evaluated from the observed GT energies 14). This difference may partly be due to the delta-isobar effect and the second-order core-polarization effect. The spin-isospin strengths in the higher excitation region push down the spin-isospin giant resonances such as the GT for L = 0, and CR’s for the first-forbidden P-decays and M2 -y-decays for L = 1. These effects are taken into account in evaluating the value frc = 1.35, and not in evaluating the g’ = 0.6 from the observed GT energies. Correction for the delta-isobar effect on the GT energy results in a somewhat larger value of g’. The values g’-0.55 for xi ,J = 60/A MeV, which correspond to fib - 0.9, are about 10% smaller than the g’ = 0.6. The value fTr - 0.9 is evaluated from the GR energies for L= I without including the effects of the higher excitation strengths as mentioned above. Actually the Landau-Migdal interaction with g’ = 0.6 gives the GR energies for L = 1 higher by 15% than the observed GR energies 14). Consequently, adjusting the LandauMigdal interaction so as to reproduce the observed CR energies for the L = 1 channel
511
H. Toki, H. Ejiri / Landau- Migdal interaction
gives g’- 0.57 resulting in better agreement interaction with ,yt ,, =60/A MeV. We thank G.F. Bertsch for suggesting the comparison of the two interactions.
with the g’corresponding
to the separable
us the use of the macroscopic
model
for
References 1) J.I. Fujita and K. Ikeda, Nucl. Phys. 67 (1965) 145; K. Ikeda, S. Fujii and J.I. Fujita, Phys. Lett. 3 (1963) 271; 2 (1962) 169 2) H. Ejiri, K. Ikeda and J.I. Fujita, Phys. Rev. 176 (1968) 1277 3) C. Gaarde, in Proc. Int. Conf. on spin excitations in nuclei, Telluride, 1982, ed. F. Petrovich (Plenum, New York, 1982): Nucl. Phys. A369 (1981) 258. 4. A. Richter, Proc. Int. Symp. on electromagnetic properties of atomic nuclei, Tokyo (1983) 5) M. Ericson er al., Phys. Lett. 45B (1973) 19 6) H. Toki and W. Weise, Phys. Lett. 97B (1980) 12 7) E. Oset and M. Rho, Phys. Rev. Lett. 42 (1979) 47 8) G.E. Brown and M. Rho, Nucl. Phys. A226 (1981) 397 9) A. Bohr and B. Mottelson, Phys. Lett. IOOB (1981) 10 10) K. Shimizu et al., Nucl. Phys. A226 (1974) 282 11) I.S. Towner and F.C. Khanna, Phys. Rev. Lett. 42 (1979) 51 12) H. Ejiri, Nucl. Phys. A396 (1983) 181~ 13) S.-O. Baeckman and W. Weise, Mesons in nuclei, ed. M. Rho and D.H. Wilkinson (North-Holland, Amsterdam, 1979) p. 1097 14) G.F. Bertsch, D. Cha and H. Toki, Phys. Rev. C24 (1981) 533 15) H. Ejiri and J.I. Fujita, Phys. Reports 38C (1978) 85, and references therein 16) H. Ejiri, Phys. Rev. C26 (1982) 2682 17) A. Bohr and B. Mottelson, Nuclear structure, ~01s. 1 and 2 (W.A. Benjamin, Reading, Mass., 1969 and 1975) 18) G.F. Bertsch, The practitioner’s shell model (North-Holland, Amsterdam, 1972) 19) G.F. Bertsch, P.F. Bortignon and R.A. Broglia, Rev. Mod. Phys. 55 (1983) 287 20) K. Nakayama et a/., Phys. Lett. 114B (1982) 217; T. Suzuki, Nucl. Phys. A379 (1982) 110; C. Gaarde ef al., Nucl. Phys. A396 (1983) 127
Physics
@ North-Holland
A438 (1985) 503-S 11 Publishing Company
RELATION
BETWEEN
THE LANDAU-MIGDAL
AND THE SEPARABLE
INTERACTION
INTERACTION
H. TOKI
Department of Physics, Tokyo Metropalitnn Universit.v, Sefuguyu, Tokyo I%, Japan and f-5. EJIRI ~gpurt~~~t of Physics, Osaka ~~iver~~ty, Toy~nak~, Osaka 56U, Japan Received 14 June 1984 (Revised 30 October 1984) Abstract: The Landau-Migdal interaction has been used widely in analyses of the quenching of spinisospin transitions, The separable interaction, on the other hand, has been used for studies of electric giant resonances and low-lying vibrational states, and recently for those of spin-isospin polarizations. The relation between these two interactions in the spin-isospin channel is studied. Quantitative comparison of these interactions in the spin-isospin L= 1 channel shows good correspondence between the nuclear matrix elements of the two types of interactions.
1. Introduction The quenching
phenomena of spin-isospin transitions such as magnetic and Gamow-Teller transitions have drawn much attention in recent years le4). Among many possibilities, the delta-isobar mechanism as well as the second-order core polarization due to the strong tensor correlation are believed to be responsible for the major part of the missing strength s-‘2). In these discussions, our knowledge of the short-range correlations among nucleons is essential for the determination of the magnitudes of these mechanisms. It is commonly assumed that a Landau-Migdal interaction of delta-function form should represent the short-range correlations as long as the physics is limited to small-and medium-size momenta 13). Systematic studies of the positions of the giant Gamow-Teller states and the transition matrix elements in nuclei close to the magic numbers suggest a Landau-Midgal parameter in the spin-isospin channel of g’-0.6 in pion mass units 14). On the other hand, hither-multipo~e magnetic y-transitions (M2, M4) and analogous axial p-transitions in medium-heavy nuclei have been extensively studied by the Osaka group ‘5*‘6). It has been shown that these spin-isospin /3- and ytransitions between low-lying single-quasiparticle states are uniformly quenched by factors of g”“/g - 0.2-0.3 [ref. ‘5>].Ejiri and Fujita ‘2*‘s**6) have analyzed the quenching in terms of the spin-isospin core polarization. This has been interpreted as destructive interference of the spin-isospin giant resonance with the single-particle 503
504
motion.
H. Toki, H. Ejiri / Landau-Migdal
The spin-isospin
core polarization
interaction
and the giant resonances
are collective
features in the spin-isospin channel. Hence, they may be treated by means of separable-form polarization (phonon) interactions in analogy with those used for the El giant resonances, E2 core polarization and other modes I’). Because of the complexity of the nuclear structure in these nuclei, the separable interaction was used to analyze the quenching phenomena of the spin-isospin transitions ‘2*‘5*‘6). They extracted the coupling of the polarization strengh of the transition matrix elements. Defining the spin-isospin tensor operator as FTsU = T’[(+’ x ( I/ b,JLYL] with b,, being the oscillator parameter, the separable interaction is given by V= x&_,F * F. They used the random-phase approximation with a model space of lho ph states and evaluated the coupling strengths x& so as to reproduce the data 12,‘57’6).The derived interaction strengths are xzU = (60-85)/A MeV for AL = 1 channels. They depend a little on whether S = 0 or S = 1 as well as the type of experimental data used for the analysis. In this paper, we relate these two interactions and discuss the consistency of the interactions. Since extensive works have been made on the spin-isospin channels 20) with AL = 0, we discuss mainly the AL = 1 channels in this paper. The GT mode with AL = 0 has recently been analyzed by several authors. In this way, we hope to obtain a general view of the quenching mechanisms for the AL= 1 channels. In sect. 2, we shall introduce the two interactions and relate them explicitly by comparing the harmonic-oscillator matrix elements. We will discuss the relation between the two interactions by assuming that the transition densities are those of the macroscopic model in sect. 3. Furthermore, numerical results will be provided in sect. 4, where M2 matrix elements with L = 1, S = 1, T = 1 are explicitly compared. Sect. 5 will be devoted to the conclusion and a discussion.
2. Landau-Migdal
interaction and the separable interaction
In this section we introduce the Landau-Migdal interaction and the separable interaction in the spin-isospin channel and calculate the particle-hole matrix elements of these interactions. We begin with the Landau-Migdal interaction. The spin-isospin part is written as’
V(LM)=$g’S(r,-r,)rr, Tr
‘uzrl
-12,
(2.1)
where fm is the 7rNN coupling constant and rnrr the pion mass. The combined value is f',/ rnt= 396 MeV * fm3 in common units. g’ is called the Landau parameter and the empirical value is believed to be around 0.6 [ref. ‘“)I. g and T are the spin and isospin Pauli operators, respectively. Note that the Landau-Migdal interaction is constant in the momentum space.
H. Toki, H. Ejiri / Landau-Migdal
In order transform
to take the particle-hole the jj-scheme lj,j,‘;
matrix
wave functions JMTM,)
elements
of the above
= C ((~,9jp(~h~ljh; Jl(~&L(%W;
where the transformation coefficients the angular integral, the particle-hole
interaction,
we
L)@(f$-‘;
S)]‘,lff-‘;
J> TM,),
(2.2)
are the unitary 9-j coefficients r8). After taking matrix elements are reduced to the form
= (jrj,‘;
JMTM,I
=$g’C ?r
L
Z*(j,j,;
X (47r)2
505
to those in the LS scheme:
x[&I,‘;
M(LM)
interaction
I
V( LM)( j; jl;’ ; JMTM,) LJ)Z(jLjh;
Q;&I)Q;W(q)
W)
$3
(2.3)
where
-X&h; JW = ((~&j,(~h~>jh; JI(~,4)UI~)S= 1; J) x2v$21,+ The function function:
1)(21,,+ 1)/4~(2L+
Q is the Fourier-Bessel
transform
1)(1,01,01LO)(-)‘“.
of the particle-hole
(2.4) radial
wave
dr. J ~,(~)G(~Mq~)~2
Q;,,(q)=a3
(2.5)
0
On the other hand,
the separable
interaction
is written
V(EF) = x;~F.
F
as (2.6)
with F ;LJ=Tr[aSX(l/bo)LYL]J, where the oscillator parameter b, is given by b, = G with ho = 41A-“3 MeV. In eq. (2.6), this separable form has been denoted by V(EF) due to the intensive use of the separable interaction for the spin-isospin modes by Ejiri and Fujita 15). strength The spin-isospin interaction strength x zH is related to the dimensionless fsTUasxs=r=fsTuhw/2CiiGii, where G,=(iJJ[~~~~(r/b,)~Y~]~(]j)/fiand the summation is made over all 1 ho ph excitations 12,16). The values for the isospin modes were derived as xALs= 320/A (2.1)2L MeV for L= 0 and L= 1 so as to reproduce the observed quenching factors and the El giant resonance energies. This gives xAoo= 320/A MeV for the L = 0 channel, corresponding to the symmetry energy lOOtT/A MeV, and x;,, - 73/A MeV for the L = 1 channel. The f2corresponding strength ,yt ,, for the spin-isospin to the xAI1 is fT= 1.15 [refs. 12,‘6)]. The interaction
506
H. Toki, H. Ejiri / ~ndau-~jgdal
inrerac~ion
mode with L = 1 is of the same order of magnitude as the xi,, for the isospin mode. However, analysis of the observed giant-resonance energies with T = 1, S = 1, J” = 22, l+, O- in terms of the interaction eq. (2.6) yields a slightly smaller value off, = 0.9, which corresponds to the xi IJ = 60/A MeV[refs. ‘2Z’6)].The ratiof,/f, 0.78 compares well with that for the L = 0 channels. On the other hand, a slightly larger value of fm- 1.35 is derived so as to reproduce both the observed giant resonance energies and the quenching factors in terms of both the lhw ph polarization of eq. (2.6) and the higher excitation ro strength such as AN-’ and the second-order polarization ‘2,‘6). This corresponds to xIti = 85/A MeV [refs. ‘2*16)]. Therefore we evaluate the g’ corresponding to the separable interaction strength of ~~~~=(60-85)/AMeV. We take now the particle-hole matrix elements of this interaction: M(EF) = (j,j,‘;
JMTn/r,l V(EF)l jbjk-;' ;JMTM,)
=,y~uZ*(jpj,,;
L=J-
I, J)Z(jgj;l;
L=J-
1, I) (2.7)
Although the two forms given in eqs. (2.3) and (2.7) look similar, we cannot relate the two interaction strengths analytically because of the integral and the summation involved in eq. (2.3). In the next section, therefore, we shah use the macroscopic wave function in order to compare the two interactions at the matrix-element level. After making this quahtative comparison, we shall calculate explicitly the matrix elements of the two interactions in the harmonic-oscillator basis in order to make a direct comparison between the two interaction forms.
3. Qualitative
comparison in the macroscopic
model
In a discussion of collective states such as giant resonances it is often assumed that those collective states are described in terms of slight oscillations around the nuclear ground-state density. We might be able to assume also that the spin-isospin modes under consideration have similar collectivities and therefore the matrix elements of the two interactions with the collective wave functions have the same value. The transition density in the macroscopic model is “)
G(r) L-, k(r) = ~LT r YLhf($),
(3.1)
which is applicable for L > 1. The constant oL is determined by the sum-rule value for the electric mode. An extension of this form to spin-isospin modes is not straightforward, since the spin wave function with S = 1 couples with L = J - 1 and
507
H. Toki, H. .Ejiri / Landau- Migdal interaction
J+ 1 to provide the total spin J However, we shah assume that the collectivity is
caused by the repulsive correlations at small momentum, where the smaller L-states have dominant roles over the larger L-states. In fact the transition density with L = J + 1 makes a negligible contribution to the electromagnetic and beta decays, which are transitions with q = 0, and the long-wavelength approximation is valid. Hence, the same form as the transition density in eq. (3.1) will be assumed for the spin-isospin modes also, where L should be read as L = J - 1 and the spin-isospin wave functions are added simply to eq. (3.1). In eq. (3.1) above, the ground-state density is well described in the Woods-Saxon form p(r) =
po/[1 + e(r--R)/c] ,
(3.2)
where the constants are provided by electron scattering excperiments is): C=OS7fm.
R=(1.18AL’3-0.48)fm, The matrix element of the Landau-Migdal
(3.3)
interaction is written as 13.4)
On the other hand, the separable interaction gives
[I
mdp 2L+1dr
M(EF) =x&b;*’
0 z
2
3 _
(3.5)
The derivative of p with respect to r is localized at the nuclear surface. Therefore the integrals of eqs. (3.4) and (3.5) may effectively be given by the approximate forms (3.6a) (3.6b) where (dp,/dr),,,
= p,/4C, a - 2.6C and a’- 3.3C. Then, 2 W~~~,,,,,-~d~:
2 (>
?r
RZL2.6C,
(3.7)
2
M(EF),,,,,-X&
R4L+2(3.3C)Zb;2L.
(3.8)
Dropping the common terms, we find the relation
.f’, g’=
,z
7r
The A-dependence
,yLR2L’24.2Cb;zL.
(3.9)
given in eq. (3.2) is effectively given by a simple form of
H. Toki, H. Ejiri / Landau-Migdat
508
interaction
R =
1.1A”3 in the present mass region of A = 90- 150. Then the interaction strength ~1,~ = (60-85)/A MeV for the L= 1 channel leads to .f’, 1212g’==210-290 MeV * fm3. (3.10) 7r This value corresponds to g’ = 0.53-0.73 which is consistant with the value obtained from the analysis of giant Gamow-Teller states 14) although the result should be considered qualitative. Furthermore, if we assume a delta-function-type interaction, the A-dependence obtained by Ejiri and Fujita for the L = 1 (J = 2) mode is consistent with the short-range nature and no A-dependent assumption of g’. 4. Numerical comparison In this section we shall calculate the particle-hole matrix elements explicitly using eqs. (2.3) and (2.7), and obtain the corresponding g’ from the strength ~1,~ for the M2 mode (S = 1, L = 1, J = 2). The experimental data are taken for nuclei around A=90 and 150 by utilizing the fact that the lh,,,* single-particle state is lowered by the spin-orbit force and makes a low-lying 2- state together with the lg,,z orbit TABLE 1
Particle-hole matrix elements of M2 states for the Landau-Migdal and separable interactions in nuclei with A = 90 (“Zr) and A = 146 (la6Gd) 7r.
ph
v
A=90 -2.13 3.61 -1.77 0.82 -0.47 -0.76 0.56 -1.09 1.27 -1.15 0.19 -0.62 0.00 0.08 -0.89 -2.14 -1.91 -0.25 -0.42
C Dh
CWirY,tO)
177
-113 168 -68 59 0 -50 27 -89 96 -67 21 -39 20 0 -74 -133 -85 -53 -79 10800
62 134 22 17 0 12 3 38 44 21 2 7 2 0 26 85 35 14 30
H. Toki, H. Ejiri / Landau- Migdal TABLE
27.
Y
eh
509
interaction
1 (cont.) CO-M) [Meal
C(W 1m,*1
COW,,,, [m21
-0.93 0.15 -0.50 0.00 0.06 -0.72 -1.73 0.82 -1.54 -0.31 0.17 -0.20 -0.75 1.12 0.08 -0.34 -0.55 0.18 -0.18 -0.07 -0.36 0.2 1 -0.61 -0.42 0.00 -1.58
-77 24 -45 22 0 -85 -153 92 -98 0 37 -62 -123 164 32 -91 -67 50 0 0 -78 34 -113 -83 17 -176
25 2 8 2 0 30 98 35 40 0 6 16 63 111 4 34 19 10 0 0 25 5 53 29 1 129
123
15 800
A= 146
w/2 1G/2 24,2 1G/Z 2d,,,2P,,, 2d,,22P,,z 3s,,zlfs,2 3s,,,2~,,, lh 1,/z 1g112 th,,,lg,,, Ih,,,lg,,, lh,,,2d,,, 2f,,, 1g7/2
2f7,2k,,, ‘Z,,W,, 2f,,,2d,,, 2f*,,lg,,, 2fs,, lg,,, 2f,,,2d,,, 2fs,,2d,,, 2fs,,3s,,, 3e,/z1g7/, 3e,,,2ds,, 3e,,,2d,,, 3Ps,,3%,, 3e,,,2d,,, 3er/,2d3/, Ii ,,/zlh,,,,
C C(ehkY,IO) Ph
The particle-hole states shown in the second column are the partner of the lh , 1,21g7,2-’ state for the matrix elements. C(LM) and C(EF) are directly related to the M(LM) and M(EF) of eqs. (2.3) and (2.7). C(EF),i,.a is l((ph)JM(r[cTY,]2(0)(2. The summed values Z,“h’ C(ph(r[aY,]‘jO) are shown at the bottom of each column.
nearby. Hence, we take the nucleus with A = 90, where the Fermi levels are at 2 = 40 and N = 50 and the nucleus with A = 146, where they are at Z = 64 and N = 82. For the case of Z = 64, we assume that the lg7/2 and 2d,,, orbits are occupied in the 50-82 major shell. With this setting, we take all the particle-hole matrix elements with lhw excitations. This model space is consistent with the one taken by Ejiri and Fujita in order to obtain the coupling constant x [ref. “)I. In table 1 we show the particle-hole matrix elements for J =2. In the second column, the values C(LM) = M(LM)/g’ are given and in the next column, the value We give also, in the fourth column, C(EF) = M(EF)/[xSTU beZL((2L+ 1)!!/4~)*].
510
H. Toki, H. Ejiri / Landau- Migdai interaction
the square of the transition matrix elments I((ph)Jnci(r[oYJ2[O)(“. In these numbers, we have not included the r-spin matrix elements, which are trivial and irreievant to our discussion here. Note that the signs of the two matrix elements C(LM) and C(EF) coincide. Hence, all the particle-hole states act coherently in both cases to quench the transition strength of the low-lying state. In order to compare the two interactions, we take the weighted sums of C(LM) and C(EF) as ((ph)JMlr[aYJ*O), which appear in the perturbation calculations assuming all the particle-hole states have the same unpe~urbed energy. The weighted sums are listed at the bottom of the columns for C(LM) and C(EF). We can now find g’, which corresponds to a given ~1,~. When we take the coupling constant of the separable interaction xi I2= (60-85)/A MeV, we find g’ = 0.50-0.73 for A = 90 and 0.55-0.80 for A = 146. These values are quite consistent with the number of 0.53-0.73 obtained by the macroscopic description in the previous section, and with the value g’-0.6 from the giant Gamow-Teller states 14).
5. Conclusion and discussion We have studied the relation between the delta-function-type interaction (LandauMigdal interaction) and the separable interaction at the matrix element level. The macroscopic model provides g’=O.53-0.73 with the use of xl,,== (60-85)/A MeV. Numerical studies of harmonic-oscillator matrix elements for ~1,~ have resulted in g’ = 0.50-0.73 for A = 90 and 0.55 - 0.80 for A = 146. It is very interesting to find that the g’ values evaluated from the macroscopic model are quite consistent with the values evaluated from the comparison of the numerical matrix elements. These values are also in accord with the value g’-0.6 obtained by the systematic study of giant Gamow-Teller states in the medium-heavy nuclei. In fact there are slight differences between these g’ values obtained by various methods. The value g’= 0.75 for ,Y!,~= 85/A MeV, which corresponds to the fm = 1.35, is about 20% larger than the value g’ = 0.60 evaluated from the observed GT energies 14). This difference may partly be due to the delta-isobar effect and the second-order core-polarization effect. The spin-isospin strengths in the higher excitation region push down the spin-isospin giant resonances such as the GT for L = 0, and CR’s for the first-forbidden P-decays and M2 -y-decays for L = 1. These effects are taken into account in evaluating the value frc = 1.35, and not in evaluating the g’ = 0.6 from the observed GT energies. Correction for the delta-isobar effect on the GT energy results in a somewhat larger value of g’. The values g’-0.55 for xi ,J = 60/A MeV, which correspond to fib - 0.9, are about 10% smaller than the g’ = 0.6. The value fTr - 0.9 is evaluated from the GR energies for L= I without including the effects of the higher excitation strengths as mentioned above. Actually the Landau-Migdal interaction with g’ = 0.6 gives the GR energies for L = 1 higher by 15% than the observed GR energies 14). Consequently, adjusting the LandauMigdal interaction so as to reproduce the observed CR energies for the L = 1 channel
511
H. Toki, H. Ejiri / Landau- Migdal interaction
gives g’- 0.57 resulting in better agreement interaction with ,yt ,, =60/A MeV. We thank G.F. Bertsch for suggesting the comparison of the two interactions.
with the g’corresponding
to the separable
us the use of the macroscopic
model
for
References 1) J.I. Fujita and K. Ikeda, Nucl. Phys. 67 (1965) 145; K. Ikeda, S. Fujii and J.I. Fujita, Phys. Lett. 3 (1963) 271; 2 (1962) 169 2) H. Ejiri, K. Ikeda and J.I. Fujita, Phys. Rev. 176 (1968) 1277 3) C. Gaarde, in Proc. Int. Conf. on spin excitations in nuclei, Telluride, 1982, ed. F. Petrovich (Plenum, New York, 1982): Nucl. Phys. A369 (1981) 258. 4. A. Richter, Proc. Int. Symp. on electromagnetic properties of atomic nuclei, Tokyo (1983) 5) M. Ericson er al., Phys. Lett. 45B (1973) 19 6) H. Toki and W. Weise, Phys. Lett. 97B (1980) 12 7) E. Oset and M. Rho, Phys. Rev. Lett. 42 (1979) 47 8) G.E. Brown and M. Rho, Nucl. Phys. A226 (1981) 397 9) A. Bohr and B. Mottelson, Phys. Lett. IOOB (1981) 10 10) K. Shimizu et al., Nucl. Phys. A226 (1974) 282 11) I.S. Towner and F.C. Khanna, Phys. Rev. Lett. 42 (1979) 51 12) H. Ejiri, Nucl. Phys. A396 (1983) 181~ 13) S.-O. Baeckman and W. Weise, Mesons in nuclei, ed. M. Rho and D.H. Wilkinson (North-Holland, Amsterdam, 1979) p. 1097 14) G.F. Bertsch, D. Cha and H. Toki, Phys. Rev. C24 (1981) 533 15) H. Ejiri and J.I. Fujita, Phys. Reports 38C (1978) 85, and references therein 16) H. Ejiri, Phys. Rev. C26 (1982) 2682 17) A. Bohr and B. Mottelson, Nuclear structure, ~01s. 1 and 2 (W.A. Benjamin, Reading, Mass., 1969 and 1975) 18) G.F. Bertsch, The practitioner’s shell model (North-Holland, Amsterdam, 1972) 19) G.F. Bertsch, P.F. Bortignon and R.A. Broglia, Rev. Mod. Phys. 55 (1983) 287 20) K. Nakayama et a/., Phys. Lett. 114B (1982) 217; T. Suzuki, Nucl. Phys. A379 (1982) 110; C. Gaarde ef al., Nucl. Phys. A396 (1983) 127