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Giant multipole resonances
,
i
Nuclear Physics A265 (1976) 385-415; ~ ) North.Holland Publishing Co., Arasterdara Not to be reproduced by photoprint or microfilm without written permission from the publisher
GIANT M U L T I P O L E R E S O N A N C E S t K. F. LIU tt
State University of New York at Stony Brook, Stony Brook, N Y 11794 and G. E. BROWN
Nordita,
Copenhaoen, and State University of New York, Stony Brook Received 25 August 1975 (Revised 1 December 1975) Abstract: The positions and the transition strengths of the giant multipole resonances of the spherical nuclei (160, 4°Ca, 9°Zr and 2°spb) are calculated with large configuration spaces in the random phase approximation based on the Hartree-Fock ground states of a newly developed Skyrme interaction. Positions of many giant resonances are reproduced within 5 % of the experimental values. Most of the energy-weighted sum rules calculated from the double commutators are exhausted to within 13 ~ by the contribution from the RPA response function.
1. Introduction
Recently, attention has been focused on the nuclear excitation region above the particle emission threshold. Data from various photonuclear reaction experiments and hadron and electron inelastic scattering experiments seem to indicate the existence of new giant multipole resonances besides the giant dipole resonance. Unlike the resolved discrete low-lying states which depend on the detailed shell structure of the nuclei, the giant resonances exhibit structure governed by the average nuclear properties. It is the aim of this paper to study the systematic locations of the electric multipole resonances and their contributions to their respective energyweighted sum rules within the model of random phase approximation based on the Hartree-Fock ground states of a new Skyrme interaction. This paper is organized as follows. We present in sect. 2 a linear response theory with the random phase approximation formulated in coordinate space which allows us to include a large number of configurations in a simple way numerically. In sect. 3, we present a new Skyrme interaction to be used in the calculation. It has the properties of bringing the single-particle level densities near the Fermi levels in 2°apb close to their experimental values. Sect. 4 is devoted to some of the numerical details of the present t Supported in part by USERDA contract no. AT(I 1-1)-3001. ,t Present address: Service de Physique Th6orique, CEN-Saclay, BP no. 2, 91190 Gif-sur-Yvette, France. 385
386
K.F. LIU AND G. E. BROWN
calculations and we compare, in sect. 5, our results with the experimental values and the results of other theoretical works. Finally, it is concluded, in sect. 6, that it is essential to include a large number of configurations in order to have the experimentally established giant resonances in heavy nuclei reproduced well and that self-consistency of the model is justified through the energy-weighted sum rules. 2. Linear response theory and R P A in coordinate space
Nuclei have excited states that can be detected with experimental probes which couple to the density or the current of the system; for example, the scattering of electromagnetic waves or electrons from nuclei. These probes interact weakly with the nuclei and therefore can be treated in the Born approximation. As a specific example, consider the perturbing Hamiltonian which is coupled to the density of the system in the form
f d3xp.(x,t)f(x, t) ....
H'H(t) =
(1)
Then the change of the density, to first order in H~(t), is related to the densitydensity correlation function and is defined as the linear response,
R[f], = ~(p(x, t)) = ih-1
dt'
dax'f(x ', t')(Ool[p~(x', t'),pn(x, t)][00),
(2)
t.o
where 100) is the ground state of the unperturbed Hamiltonian. Most of the physical information is contained in the overlap of the linear response with the probe in the spectral representation f
R[f f]=
V (~°lFIq~")
daxf(x)R[f]o,=,~
°9-(En-Eo)+i~l
(q%lfl~O")(0"lflq/°)
-
~o+(E _Eo)+i~l
,
(3)
where F = S~b+(x)f(x)~(x)dax is hermitian and q is put in to insure proper causality. This expression, eq. (3), has simple poles at the excitation energies and their residues are the transition probabilities between the ground state and the excited states : Im R [ f f ] (e~ > 0)_~on ~ [(~nlFl~o)126(m-(E,,-Eo)). q
(4)
n
Conventionally, one solves the RPA equation in the configuration space where the dimension of the RPA matrix goes linearly as the number of the partical-hole configurations involved, which might be as large as several hundred for the case of heavier nuclei. Blomqvist 1) and Bertsch 2) have developed a method by inverting matrices in coordinate space instead. It is restricted to zero range interactions and is numerically
GIANT
MULTIPOLE
RESONANCES
387
simpler provided the number of the mesh points used for numerical integration is smaller than the dimension of the particle-hole configuration. Suppose the delta-function interaction has the general form in the multipole expansion Vph
*L L = L ~(rl--r2)"L(l' v~, ~r 1+r2)) 2 L + l M~+(1)DK,M(~I)DK~M(~2)M~(2), a,L,M rlr2 4~
(5)
where {M,} is the set of operators M = {1, a, r, V2,~V, etc.} and K~ is the angular momentum projection of the M~ operator on the body-fixed axis of the particle-hole state. We can, therefore, define the Lindhard function as phL c,hpL(rl)ChpL(r2) atO~L¢~ rE, o9) = ~ C~*phL (rl)CP (rE) + v a i l ~-'1'
(6)
, , z r )M~+ f f 2 L4----~h + l D*LM(f2). cPahL(r)= f dO (~Op~Oh)M(
(7)
ph
•p - - 8h - - 0) - -
it/
8p - -
eh + co-- it/
where
If we replace the integration over the coordinate r by a finite summation over some mesh points, then the random phase approximation to eq. (3) becomes a geometrical series of matrices:
RrtVArffq=F~ ( 1 "Jr-G~°'L )ijFj.L to L J ' J - I ~(O)L
(8)
The indices of the matrices are simply the direct product of the mesh points of the coordinate r and the set of operators {M~}. If we take, say, ten mesh points and three operators, we have typically a 30 × 30 matrix to invert instead of solving a several hundred by several hundred RPA matrix.
3. Particle-hole interaction
Landau's theory of Fermi liquids involves the following assumptions which are generalizations of the usual Hartree-Fock description of the ground state. (a) There is a one-to-one correspondence between the bare particles and the quasiparticles. (b) Quasiparticles on the Fermi surface interact weakly so that they can be described by a self-consistent field. The quasiparticle energy is then obtained by varying the energy with respect to the quasiparticle occupation number, and the effective particle-hole interaction can be identified as the second derivative of the energy functional with respect to the quasiparticle distribution.
388
K . F . L I U A N D G. E. B R O W N
The Skyrme interactions developed by Vautherin and Brink 3), V = Z to(1 +ZoPo)c~(rl-r2)+½tl ~(k'2~(rl - r 2 ) + ~ ( r l - r 2 ) k2) 1,2
1,2
+t 2~k'.6(rl-r2)k+t 1,2
3 ~ 6 ( r l - r 2 ) 6 ( r 2 - r 3 ) + W0 ~ ( a 1+32). k'x 6(r~-r2)k , (9) 1,2,3
1,2
where k and k' are the relative momenta, fit the ground-state energies of spherical nuclei very well, when used in the Hartree-Fock approximation. We take the view that this procedure produces the energy well as a functional of the quasipartide occupation number and shall use the single-particle energies and the particle-hole interaction obtained by functional differentiation of this energy. That is, Vph :
~(rx-r 2) ~ ~(l+(-)~-~'ol"a2)~(l+(-)'-~'r1" x2) ~ , st, s't"
(10)
OPstOPs't'
where s and t are the third components of the spin and isospin labels of the density. All the existing Skyrme forces of the form in eq. (9) have essentially fitted the binding energies and the radii of the spherical nuclei equally well, yet they differ in their HF single-particle spectra and their predictions of the vibrational states, for example, the Skyrme force I [-ref. 3)] yields the dipole state of 2°Spb somewhat lower 4) (12 MeV) than the experimental value (13.5 MeV), while the Skyrme force II which has a stronger velocity dependence yields it somewhat higher. As can be seen from fig. 1, the single-particle levels given by the Skyrme I and II forces are too widely spaced at the Fermi surface. It was noticed long ago 5s) that levels at the Fermi surface are significantly more compressed than predicted by microscopic theories in Hartree-Fock approximation. This can be characterized by saying 53) that m*/m > 1 empirically, at least near the Fermi surface, whereas Hartree-Fock calculations tend to give m*/m < 1, or, it can be characterized by having a wider surface. The empirical compression of levels has since been understood in terms of coupling of the single particles to vibrations 54). Our first objective was to construct a Skyrme-type interaction which had m*/m > 1 in the nuclear surface region and/or a wider surface to see whether we could mock up this effect within the framework of zero-range forces. This is achieved by adding a new term t ~. ½t13(k'26 (r t - r j)8(r j - rk) + ~(r i - r i)6(r j - rk)k2), ijk
to the interaction in eq. (9). It is not at all clear that the level compression described by the single-particle coupling to vibrations can be described in this way, because in doing so we are * We are grateful to D. Vautherin for suggesting such a term; from the formulae for m*/m and the surface thickness in ref. 3), one can see that such a term can be used to increase m* in the nuclear surface and/or to increase the surface thickness.
GIANT MULTIPOLE RESONANCES
389
PROTONS I
II
BLV 1
EXP
0
2f 512 3p 3/2
2f 712 2f 7/2 Li 13/2
-2
_ _
Li 13/2
~ .
2f 712 lh 912
lh 912
-4
2f 712 ~Li1312 lh 912
l h 912 -6 3s 112 X
2d 312
-8
-10
2 d ~1h1112
512
~3s112 ~ 2 d ~ , 3 s
1/2
l h 1 1 / 2 ~ 2 d 3/2
~'-2d ~ l g
-12
lg 712
~ , 2 d
3s 112 312 lh11/2 5/2
2d 312
lh 1112 2d 512
712
5/2
lg 7/2 -11,
NEUTRONS I
II
0
3d 5/2
-2
Li 1112 ,2g 912 "L] 1512
~'/.s
- -
2g 912 Li 1112
~ 2 g ~ .
-4
>
-6
:E UJ
-8 -10 -t2 -14
~ 3 p
112
~ 2 f
512 ~3p312
~Li13/2 ~2f 712 lh 9/2
BLV1
EXP
~ 3 p
3p 112 312 2f 512 Li13/2
~.2f712
,3d 312 7/2 "~s112 :~d512 L|1512 LIl112 2g 912
3p 112 Rf 512 ~3p312 ~Li13/2 ~ 2 f 7/2 ~ l h
912
,.?.g712 1/2 3d 512
- ~ ~
Li 1112 Lj 1512 2g 912
~ 3 p 3 p
112 3/2 2f 5/2
Li1312 2f 7/2 l h 9/2
l h 9/2
Fig. 1, Fitted single-particle spectra near the Fermi levels in 2°Spb by different Skyrme interactions and the experimental results.
390
K. F. LIU AND G. E. BROWN TABLE 1 Parameters of the Skyrme forces
Force
to
tl
t2
I
- 1057.3 - 1169.9 - 1086.2
235.9 585.6 341.14
- 100 - 27.1 - 120
II
BLV1
t3
1446315 9331.1 13114
tl 3
Xo
Wo
1982.9
0.56 0.34 0.52
120 105 130
essentially trying to describe an energy-dependent effect by m o m e n t u m dependence. It is at least interesting to try, and to some extent the degree to which we reproduce the spectrum o f highly excited states gives us some answer, because zero-range forces o f the S k y r m e type are so convenient to use. The fitted p a r a m e t e r s are listed in table 1 together with those o f Skyrme I and II. All the following results are obtained by using the interaction S k y r m e BLV 1. Following the f o r m a l i s m o f Bertsch and Tsai 40), the particle-hole interaction derived f r o m eq. (10) is Vph = 6 ( r , - r2)[a -- b(V 2 + V 2 + V 2, + V2,) + c(V, - Va ,)" (V 2 - V2, ) +d(V 1 + V r ) " (V2 + V2,)],
(11)
with a
=
(to+½tsp+1txa(4z_3V2p))(3_14_~,
. ,t.2 __~all . O.2__4~ 1 . O.2,~1 .T2 ) +ltozo(trl-
a- 2 --~-1 • "¢2),
b = 3~(23tl + 5t 2 + tlaP)+a~(2t2 - t I - ~ t l a P ) ( a 1 • a 2 + ~ 1 " ~2 + a l " trExx" x2), c = 3~(23tl + 5t 2 + ½t 13P) + a~2t2 - ta - ~tl aP)(al " a2 + T~" x2 + a l " aErl" x2), d = 3~(23tl _ 15t2 + ~1t l a p ) - - ~ ( 21t l + 3 t 2 + ~ t l a P ) ( a
1 . a 2 -[- Tl " "['2 "4- O"1 . ~r2"C1 . T2).
(12)
Here the indices 1, l' refer to the'particle-hole coordinates to the left o f the operator, 2, 2' are those to the right. The CP~hL coefficients which a p p e a r in eqs. (6) and (7) will be the same as those given in ref. 40).
4.
Numerical
details
In order to calculate the C~'~hL coefficients, we need to k n o w the single-particle wave functions and energies. The single-particle c o n t i n u u m is discretized by putting a wall at a distance o f a b o u t 2.5 times the nuclear radius. T h e numerical solution o f the radial Schr6dinger equation is carried out by p e r f o r m i n g the R u n g e - K u t t a integration starting f r o m the origin and m a t c h i n g up its logarithmic derivative at a m a t c h i n g radius R M with that obtained by integrating in f r o m the wall. The search
GIANT MULTIPOLE RESONANCES
391
for eigenvalues is carried out by means of the Newton-Raphson method using the condition provided by the matching. An artificial imaginary part ( ~ 0.5 MeV) is introduced to the particle-hole propagator to give the excited states finite widths. Both the wall distance and the particle-hole space cut-off have been varied such that the positions of the excitation are convergent to within 3 ~o and the strengths to within l0 ~ (except in a few cases like the 0 + excitation of 160). The mesh sizes, the particle-hole space truncations, the highest numbers of configurations and the wall distances of the four spherical nuclei are listed in table 2. TABLE 2 Numerical approximation Nucleus
Mesh size (fm)
x60 4°Ca 9°Zr 2°apb
0.6 0.7 0.8 1.0
Highest number of configuration
Emax 120 80 55 40
Wall distance (fm)
86 177 241 398
10 12 12 15
Truncation is such that ep-/~n < Emax; the operator Vr is also neglected in this calculation. It has been pointed out 4s-50) that the spin densities derived from the Skyrme forces are not fitted in the HF ground-state calculations of the self-conjugate nuclei and this may lead to spin instability (i.e. Landau parameter G o < -1). There are two remedies. One is suggested in the paper of B~ickman et al. 50) by the introduction of two additional terms, ~ k 2 + k'2)tl)~lPaC~(rl r2) and I2z2P~k' " kc~(r 1 Y2)" This requires refitting of all the parameters. Another one is proposed and carried out by Blaizot 43) whose idea is to replace the three-body t e r m taC$(r 1 - r 2 ) 6 ( r 2 - r 3 ) by a density dependent term ~tap(1 +Pa)t~(rx-r2), since by doing so, it leaves all the previous HF results unchanged and yet has an effect of removing the spin instability (i.e. GO > - 1). As the matter now stands, it seems there exists some degree of ambiguity in the spin-dependent part (S = 1 in the particle-hole channel) of the residual interaction, which should be pinned down in the self-consistent RPA calculations. On the other hand, we know from experimental data what its strength should be, at least qualitatively, as compared to the spin independent part (S = 0 in the particle-hole channel) of the interaction. The argument is presented as the following: In view of the fact that the unnatural parity states, where the spin part of the interaction dominates, are far less collective than the natural parity states (e.g. they are pushed away from the unperturbed values of the particle-hole gaps by typically 1 or 2 MeV only and the transition rates are not as large as those of the natural parity states in general), one would, therefore, expect that the strength of the spin -
-
-
-
392
K . F . L I U A N D G. E. B R O W N
part of a reasonable interaction should be much smaller than that of the spinindependent part of the same interaction on the nuclear surface where nuclear excitations take place. In addition, we see from table 2 and eq. (32) in ref. 40) that the contribution from the spin part of the interaction is further cut down due to its geometry (e.g. C~_hL does not contribute to the natural parity states and for some configurations Ct~,~Lare either small or vanishing). We have indeed tested for several cases of the giant resonances where many configurations contribute and found that the results are fairly insensitive to the spin part of the interaction even though it may be unphysical (GO > - 1 in the interior of the nucleus). Therefore, in this paper where we are mainly concerned with the giant resonances, the spin part of the interaction is dropped in the RPA calculations together with the spin-orbit interaction and the Coulomb interaction, as was done in ref. 40). The operator Vp+Vh 2 2 is related to other quantities with unit operator through the Hartree-Fock single-particle equation O" O
2
2
Vp +V h =
~
Wq
2u~+ -- [jp(jp+ r
1)-/p(/p+ l)+jh(jh+ 1)--/h(/h+ 1)--~]
2
\2m*]
2~m~*' (13)
where Uq, wq and mq* are respectively the s.p. potential, the spin-orbit potential and the effective mass of neutron or proton. Replacements of this sort help cut down the dimension of the bare Green function ~o~L
5. Comparison with experiments 5.1. SUM RULES
One way of identifying a collective state is by looking at the energy-weighted sum rule: the transition rate of a collective mode should exhaust a fair fraction of the sum rule. Now the linearly energy-weighted sum rule is conserved by the random phase approximation s), i.e.
1 --
[do~ ~oIm (FGRPAF) ½(~o[[F, [H, f]][~o)
7~ d
~
°..,
(14)
where [~bo) is the HF ground state. This has provided us with a check for our numerical work. We found that, for most of the cases, the energy-weighted strengths calculated from the RPA exhaust over 87 % of the sum rule calculated from the double commutator (tables 8-10 and 14-19). The remaining strength is presumably at higher energy in the continuum.
GIANT MULTIPOLE RESONANCES
393
In the actual evaluation of the sum rules, it is noticed that the double commutator of the Skyrme interaction in eq. (14) vanishes, unless the one-body operator F carries degrees of freedom like spin or isospin. We illustrate this point by taking F as the electric multipole operator which acts only on protons: F = e • 12-(1- Z3,)rLyL(p,).
The only contributing terms from the Skyrme interaction are the velocity dependent ones whose commutator with the isospin-independent part of F is proportional to E (V r L YL(~i) -- V r L y L (~j)~(ri _ r j)" (Vi -- V j), ij
but it vanishes because of the zero-range delta interaction. The commutator with the isospin-dependent part of F is a different story; it is proportional to
X (*3,v yL
%Vr r L
r). (v,- v).
ij
This does not vanish as long as i and j are not identical particles. Note that since we use Hartree-Fock ground states calculated from the same force as used in our particle-hole interaction, we should have consistency between the sum rules calculated in the RPA and by the double commutator ss). 5.1.1. lsoscalar multipole sum rule. In view of the fact that the percentage of energyweighted sum rules exhausted by the 2L pole excitation of bound states in spherical nuclei is in general < 20 ~o (except L = 3 in some nuclei where the percentage may be 30%-40%), Nathan and Nilsson ,2) argued that the T ~ Ttransitions are largely dominated by collective motions such that neutron and proton matter move together. Therefore, the electric operator F(EL) = ½E (1 - z a , ) ~ Y o ~ , i
is effectively replaced by F ( E L , T = O) = ½ ~ ( 1 - ( z a , ) ) ~ Y L . ,
(15)
i
where the average <%i> is taken to be ( N = Z ) / A . Therefore, the remainder, -- E I ( ' c 3 i - - <"C3i>)rLYLra, i
(16)
is assumed to act on [d T[ = 1 transitions. As a result, the energy-weighted sum rules for the isoscalar multipoles are multi-
394
K.F. LIU AND G. E. BROWN
plied b y a f a c t o r (l - - ( T 3 1 ) ) 2 = 4Z2/A 2, a n d b e c o m e
S~v( F(EL, T = 0)) =
h2 L ( 2 L + 1) Z 2
2m
4re
A
-
(17)
( r2L - 2 ) .
T h e n u m e r i c a l values o f S Ew L° for the H a r t r e e - F o c k g r o u n d state a r e given in tables 11, 14, 16 a n d 18 a n d are c o m p a r e d with those o b t a i n e d f r o m the r e p l a c e m e n t ,
S~ -
3hZL Z z
R2L- 2,
(18)
8mrc A for a u n i f o r m m a s s d i s t r i b u t i o n o f r a d i u s R in t a b l e 3 a n d fig. 2. W e see t h a t the sum rules c a l c u l a t e d with u n i f o r m m a s s d i s t r i b u t i o n a r e o n l y ~ 20 ~o- ,-~ 70 ~ o f those c a l c u l a t e d with H a r t r e e - F o c k g r o u n d states for higher m u l t i p o l e s . T h e r e f o r e the - -
3
016
Zrg o
0.5
I
I
I
2
4
6
Fig. 2. Comparison of the sum rules calculated with uniform mass distribution and those calculated with Hartree-Fock ground states. TABLE 3
Radii R of the uniform mass distributions and their energy-weighted sum rules as compared to the ones from the Hartree-Fockground states (L = ½(2+ 2))
R'~/~2 + 3) (r~)HF = Nucleus 160 *°Ca 9°Zr 2°Spb
S[°w(uniform distribution) S~° (Hartree-Fock)
R (fm) 2.95 4.01 5.25 6.94
2 = 2 ( L = 2)
2 = 4 ( L = 3)
0.759 0.849 0.910 0.935
0.46 0.62 0.756 0.822
2 = 6(L=4) 0.216 0.385 0.570 0.675
expression in eq. (18) s h o u l d be c o n s i d e r e d i n a d e q u a t e as s o m e t i m e s used in e x p e r i m e n t a l analysis. 5.1.2. Dipole sum rule. It is c o n v e n t i o n a l to express the electric d i p o l e s u m rule in t e r m s o f the classical s u m rule. W h e n the c.m. is t a k e n into a c c o u n t the electric d i p o l e
GIANT MULTIPOLE RESONANCES
395
operator is Dz = ½ e ~ ( l _ z 3 i ) z i -
Ze~
i
h
i " zD
= e Z AN rpYol(~p)_eE. AZr.Y~(~.)"
(19)
P
Then, the dipole sum rule becomes h2 N Z 3 I(OIOzln)lE(g.-go)
=
e 2
n
(20)
__
2m A 4re (1+~)'
where x = ~[(tl+tE)(~(1-zairsj)~ij)+tl3( i
~ i
(1--ZaiZaj)fij~jk)]
h NZ 2m A '
(21)
is called the enhancement factor due to the nucleon correlation. The updated photonuclear experiments 6) show that the integrated absorption cross section to 140 MeV is approximately twice the classical dipole sum rule. This makes the enhancement factor x about two times larger than those from previous experiments and theoretical calculations 7) (and also our results !). One explanation of this increase, offered by Weng, K u o and Brown 8), is largely attributed to the short-range tensor correlation. The additional contributions which they find would presumably come at energies well above that of the giant dipole resonance, and would not be expected to be reproduced by our force, which has no tensor component. TABLE 4
Enhancement factor x for E1 x (Sky. BLV1)
Cut-off
Sky. I
Sky. II
Nucleus (doublecommutator) (responsefun.) 160 4°Ca 9°Zr 2°apb
0.29 0.33 0.37 0.39
0.28 0.29 0.34 0.39
(MeV) 55 55 40 40
(double commutator) 0.16 0.19 0.20 0.21
0.57 0.7l 0.73 0.79
Both results from the double commutator and from the RPA are listed in table 4. The velocity dependence o f the Skyrme BLV1 is intermediate between those of the Skyrme I and II; therefore it places the weighted dipole strength in between those o f Skyrme I and II as we have discussed earlier. Experimentally9),when the cut-off is taken to be at some energy above the giant dipole (30 MeV for 160 and 4°Ca), the sum rule in units of 0.06 N Z / A is 0.63 for 160, 0.73 for 4°Ca and 1.3 _0.1 for 2°apb, and in the present model the values are 0.76, 0.80 and 1.3. Because the sum rule is more or less reproduced up to 30 MeV and the positions
396
K . F . L I U A N D G. E. B R O W N
of the giant resonances are consistent with the experimental results, which we shall show later, it seems plausible that a cut-off at some energy above the giant resonances is appropriate for this model. 5.1.3. Isovector multipole sum rule. Unlike the isoscalar case, the velocity dependent parts of the Skyrme force do contribute to the energy-weighted sum rule. When we take the isovector operator to be the remainder, eq. (16), it introduces a factor 1 - ( ( N - Z ) / A ) 2 = 4 N Z / A 2 therefore the isovector sum rule is h2 L ( 2 L + I ) N Z 4~ A
S ~ ( F ( E L , T = 1 ) ) - 2m
(22)
(r2L_2)(lq-x),
where ~[(t 1 -I- t2)( ½~ c5ij(1- zai'c3j)r 2L- 2> -t- t 13(-~ ff~, C~ijC~jk(1 -- "C3iZ3j)r2Lij ijk X = h 2 NZ _
_
_
_
2m A
27]
(23)
(~>
The x-values for L = 2, 3 and 4 are listed in table 5. The non-vanishing enhancement factor ~c implies that the T = 1 states obtained from the Skyrme interaction either TABLE 5 Enhancement factor r for the isovector s u m rules Nucleus
L = 2
L = 3
t60 4°Ca 90Zr 2°8pb
0.18 0.24 0.29 0.32
0.11 0.16 0.22 0.26
L=4 0.06 0.10 0.16 0.20
will lie higher in energy or will acquire more strength than those obtained from the velocity-independent forces. 5.1.4. Monopole sum rules. Transitions between two nuclear states, both with spin J = 0, are strictly forbidden for electromagnetic radiation. However, the process can take place by the ejection of electrons in the x-shell. Unlike the case of internal conversion where we consider the wave function of the electron being outside the nucleus, here we have to consider the multipole expansion of the Coulomb field interacting with electron wave functions inside the nucleus only. Therefore we shall use the operator F(E0) = ~ = 12!(1 -z3i)r 2 for the monopole excitations and the sum rule. With the proper readjustment condition included, i.e. defining the operator F = ~ f~-(fi), i
such that (qSolFlq50) = 0,
(24)
GIANT MULTIPOLE RESONANCES
397
the respective isoscalar and isovector sum rules are
S~.~(F(E0))
=
Z2
h22 m
__
-A -'
(25)
SL~v(F(E0)) - h2m2NZ <~->(1 + x),
(26)
where
~[(tx + t2)<½ Z 3U(1 - ~3,G)r2> + tl 3<~ Z ~o~ik( 1 - %,%)r2>] r =
u
h 2 NZ
_ _
<~>
(27)
ijk
2m A The values of x are listed in table 6 for x60, *°Ca, 9°Zr and 2°Spb. TABLE 6
E n h a n c e m e n t factor x for the isovector monopole s u m rules
5.2.
Nucleus
t60
4°Ca
9°Zr
2°spb
x
0.18
0.24
0.29
0.32
DISTRIBUTION OF MULTIPOLE TRANSITION RATE
The strength function due to G (°), the l p - l h excitation, is redistributed by G ra'A in such a fashion that the isoscalar modes are pushed down in energy due to the
1t
ISOVECTORDIPOLE
/
1t !
[
1I/
_c
[
/
V
~/
1o
I/
, n,
~
/,1 10 ~
,
. # " , ' - - ~
JL-~..~_,
~ 20
, : ~ z : ~
p
10
30
2o
20 MeV
Co ~
:--
,
30
,
30
Fig, 3. The calculated isovector dipole strength o f four spherical nuclei in single-particle units are plotted in solid lines. The dashed lines are the experimental results taken from refs. 6, s t).
398
K.F. LIU AND G. E. BROWN TABLE7 Positions of giant dipole resonances
Nucleus
Experiment (MeV)
x60 4°Ca 9°Zr 2°spb
Krewald et al. ")
22.3, 24 19 16,7 13.5
21, 22 19, 19.2 16.2, 16.8 %11.2
Present calculation (MeV) 19.5, 23.7 17, 19, 22 15.5, 16.8, 19 13.1, 17.5
~) Refs. 3s,11). TABLE 8 Distribution of isovector dipole strength EWSR (MeV. fm2) double commutator)
Nucleus
State
s.p.u.
% EWSR
160
19.2 23.5 25-30 30-55
0.7 0.7 0.5 0.5
11.4 25.5 23.6 (+) 34.7 95.2 a)
2.55 x 101
4°Ca
0-15 17 19 22 25-55
0.3 0.9 0.8 0.9 0.6
3.9 18.4 17.4 22.4 (+) 21.5 83.6 a)
6.59 x 10l
9°Zr
0-14 15.9 16.8 19 22-30
0.6 1.5 1.1 1.9 1.0
5.6 20.2 15.5 30.9 (+) 20.8 93.0 ")
1.5 x 102
2°spb
13.1 17.6 22-38
5.7 3.6 0.5
49.5 40.0 (+) 9.0 98.5 a)
3.41 x 102
•) The last rows are the sums of the % EWSR exhausted in the energy region as indicated. attractiveness o f the isoscalar interaction, yet the t r a n s i t i o n strength is e n h a n c e d in order to conserve the energy-weighted s u m rule. O n the other h a n d , the isovector m o d e s are raised higher in energy due to the repulsiveness o f the isovector interaction. 5.2.1. Giant dipole resonances. T h e d i s t r i b u t i o n o f the dipole strength is s h o w n in fig. 3 a n d table 8 a n d the peaks are listed in table 7 together with those f r o m experim e n t s a n d f r o m a n o t h e r c a l c u l a t i o n with d e n s i t y - d e p e n d e n t i n t e r a c t i o n by K r e w a l d a n d Speth 38), a n d R i n g a n d Speth 11).
GIANT MULTIPOLE RESONANCES
399
In the present calculation, the splitting of dipole states seems a common feature for all the nuclei considered. While the splitting is observed in light nuclei is attributed to the spin-orbit potential in the theoretical work 10), the second peak has not been observed experimentally in 2°spb. It is probably due to the exaggerated velocity dependence of the Skyrme force in this region, since Skyrme forces are supposed to be parametrized for small momentum transfers, and therefore contain only the first non-zero term in the expansion in momenta. Indeed if we simply switch off the velocity dependence by setting t 1 = t 2 - t13 -- 0, the second peak in 2°Spb is practically gone, while the first one remains essentially at the same position and the splitting in 160 is not much affected. Experimentally, the 160 strength has two main peaks at 22.3 and 24 MeV, and in addition there is a broad absorption extending up to 30 MeV. While the second peak and the broad absorption are reproduced by the present model the first peak is off by about 3 MeV. This is attributed to the fact that our calculated particle-hole gap between d~ and p~ ~ orbits is smaller than the experimental gap by 4.3 MeV for neutrons and 5.5 MeV for protons. If one uses the experimental energies for p~- 1 holes and d~, d~ particles, the two peaks are shifted to 22 and 25 MeV. This shows how important a role a single configuration may play in the giant dipole resonances in light nuclei. It is quite a different story in heavy nuclei like 2°Spb. It is necessary to have a model space containing 200 configurations (this corresponds, in this case, to a 20 MeV cut-off of the particle-hole space with the particle continuum discretized by a wall at a distance of 15 fm from the center of the nucleus) to push the dipole state ISOSCALAR OUADRUPOLE
.4I 4
II
~ _ i
0li
It ~
~
Cal i
g61
Zri0 --
a
20
3O
[wO~
10 t4eV
Fig. 4. The calculated isoscalar quadrupole strength are plotted in solid lines. The experimental results (dashed lines) are taken from ref. 26, 52)for160, ref. 15) for 4OCa, ref " I 7 )for 9 0 Zr, andref. 21 )andref. 19) for 2O8 Pb. The arrows indicate the experimental positions of the low-lying states.
400
K . F . LIU A N D G. E. B R O W N ISOVECTOR QUADRUPOt.E
1
2
, ~ 10
I
o". 20
~30
/,0
LkVk
10
50
60
I I
20
_ :"
30
t,O
50
60
~V Fig. 5. Isovector quadrupole strength. Experimental results (dashed lines) are taken from ref. 17) for 9°Zr; refs. 21, 25) for 2°spb.
to 13.1 MeV. If one takes 100 configurations, the dipole state is pushed up to 12 MeV only. This is probably part of the reason that other calculations with densitydependent forces 11) and realistic force 12) dfd not push the dipole state in 2°spb TABL~ 9 Isoscalar quadrupole excitation RPA calculation
Experiments
Nucleus
E x (MeV)
s.p.u.
~o EWSR
*°Ca
16
12.1
83
9°Zr
13.5
11.0
70
2°spb
11
12.4
57
E x (MeV)
~o EWSR
reaction
ref.
17 10-25 18 17
75 66 32
(p, p,) (e, e') (¢t, ct') (3He ' 3He, )
la) t4) 15) 16)
14 14 14
75
(e, e') (3He ' SHe. ) (*t, et')
17) 16) 2*)
(p, p,) (p, p,) (e, e') (e, e') (3He ' 3He, ) (3He ' 3He, ) (~, ~,)
is) 19) 20) 21) 22)
9.2-11.2 10.9 8.9-11.2 8.9-11.2 11 11 11
54
33 47
16) 22)
It is worthwhile noticing that in the experimental analysis, the sum rule expressions for a uniform mass distribution are sometimes adopted, which are quite different from the H F sum rules as we have pointed out in subsect. 5.1,1.
GIANT MULTIPOLE RESONANCES
401
high enough. In addition, use of self-consistent orbitals keeps the neutron rms radius 2 ½ 2 ½ close to the proton one ((r),,~t,on = 5.61 fro, (r)proto,= 5.49 fm for 2°Spb) and this increases the matrix elements as compared with calculations like those of ref. 12) using harmonic oscillator orbitals 47). 5.2.2. Giant quadrupole resonances. The distributions of quadrupole strength are shown in tables 11 and 12 (figs. 4 and 5). From tables 9 and 10, we found that the calculated values from the Skyrme BLV1 do agree with experimental results reasonably well. TABLE 10 Isovector quadrupole excitation RPA calculation
Experiments
nucleus
E x (MeV)
~ EWSR
E (MeV)
9°Zr
28
26
28.0
23.5
46
22 23.7 22.5
2°Spb
~ EWSR
60 85
reaction
ref.
(e, e')
aT)
(e, e') (p, y) (e,e')
21) 23) 25)
They are also consistent with the predictions given by the collective model (60 A- ~ for T = 0, 125 A - ~ for T = 1). In comparing with the work of Krewald and Speth 3a), we see that the results for 9°Zr and 2°apb agree quite well, but their 2 ÷ in a°Ca has a structure between 22 and 25 MeV which is not present in our calculation. The case with 160 is somewhat complicated. In their recent analysis of the 160(7, no) 150 reaction, Wang and Shakin 45) have assumed a broad E2 resonance in the giant dipole region in order to explain the angular distribution and the polarization data. A study of 12C(~, ~,o)160 by Snover et al. 26) revealed a T = 0, E2 strength in the region E x = 12 to 28 MeV which constitutes ~ 17 ~ of the energy-weighted sum rule in the % channel. Other measurements on the polarized proton capture reaction lSN(p, ~0)160 by Hanna et al. 26) indicate a broad E2 resonance which lies above the giant E1 resonance and the E2 radiation for the (7, Po) channel of 160 exhausted ~ 30 ~ of the isovector quadrupole sum rule which is only a half of the prediction of the neutron data analysis. On the other hand, measurements by Hotta, Itoh and Saito 26) on the electroexcitation of 160 show that in the region of 20--30 MeV excitation the strength exhausts approximately 20 ~ of the sum rule if one assumes an E2 resonance, while the strength in the region below 20 MeV exhausts 43 ~ of the same sum rule. Our calculation shows a single peak around 20 MeV excitation (if experimental single-particle energies are used, thepeak is shifted to 21.5 MeV) which exhausts ,~ 65 ~ of the isoscalar E2 sum rule and this is a common feature for all the existing Skyrme forces with different velocity and density dependence 40). This agrees fairly well with the recent inelastic 0t-scattering experiment
402
TABLE 11 Distribution ofisoscalar quadrupole strength
Nucleus
State
s.p.u.
% EWSR
EWSR (MeV -fm 4) (double commutator)
160
19.5 25-40
6.1 t.2
61.6 ( + ) 19.3 80.9
4.55 x 102
4°Ca
16 21-30
12.2 1.1
83.6 ( + ) 12.1 95.7
1.87 x 103
9°Zr
5 13.5 20-30
3.1 11.0 0.9
7.8 70.0 ( + ) 10.3 88.1~
5.33 x 103
2°SPb
5.6 11 15-25
9 12.4 1.3
25.2 56.7 ( + ) 10.4 92.3
1.65 x 10"
TABLE 12 Distribution of isovector quadrupole strength % EWSR
EWSR (MeV. fm 4) (double commutator)
Nucleus
State
s.p.u.
160
15-24 26.5 31 33 37 39.5-65
0.5 0.8 0.8 0.9 1.4 1.5
5.0 9.0 10.8 13.0 23.3 ( + ) 30.5 91.6
5.38 x 102
*°Ca
0-30.5 33 36-55
2.9 3.7 1.8
25.4 43.5 ( + ) 27.7 96.6
2.31 x 103
9°Zr
0-25 25-30 31.5 34-50
2.4 3.8 3.5 2.1
13.4 26.2 34.3 ( + ) 21.3 95.2
8.56 x 103
2°spb
0-20 23.5 26.5 28-50
0.9 7 3.5 2.5
12.6 46.0 22.6 ( + ) 17.7 98.9
3.34 x 10"
403
GIANT MULTIPOLE RESONANCES
by Breuer et al. 52) in which they find that the giant isoscalar E2 resonance centered at 20.7 MeV with a half-width o f 7.5 + 1 MeV carries ~ 65 ~ of the EWRS. It has been known for a long time that quadrupole transitions in 160 to low-lying states, now interpreted as mainly 2p-2h, 4p-4h in composition, exhaust a good fraction o f the isoscalar sum rule. These states, whose presence at low energies is intimately connected with the easy deformability of 160 in excited states, lie outside of the framework o f the present calculation, so we would not expect detailed agreement between our theory and experiment in light nuclei.
f~
ISOSCALAR OCTLPOLE
Fig. 6. Isoscalar octupole strength. Experimentalresults of ref. 21) are used for 2°SPb (dashed line). The arrows indicate the experimental positions of the low-lying states.
]SOVECTOROCTUPOLE >.
e "
21- /
,1- ~ 1 - /
lo ,o
~o ~o
3o
,o
~o/I,o
so
,o
,o -
-
,I-
,o
10
20
30
40
,o
50
60
60
I~liV
Fig. 7. Isovector octupole strength.
so
7~
,o
404
K. F. LIU AND G. E. BROWN TABLE 13 Isoscalar octupole RPA
nucleus 4°Ca 2°aPb
E x (MeV)
Experiments s.p.u.
~ EWSR
E x (MeV)
10-25
13
25
10-25 13.5
20
11
41
19
s.p.u.
10
~o EWSR
reaction
ref.
37 30
(e, e') (p, p,)
14) 13)
44
(e, e')
2a)
TABLE 14 Distribution of isoscalar octupole strength ~o EWSR
EWSR (MeV- fm 6) (double commutator)
Nucleus
State
s.p.u.
160
6.3 12.5 21 24-30 31.5 33 37 42-62
4.0 2.2 3.3 1.6 2.2 1.2 4.4 1.6
3.9 4.5 10.8 6.6 10.4 6.3 25.7 ( + ) 6.2 74.4
9.87 x l0 a
4°Ca
3.5 10 14 16-27 31 35-60
21 7.5 2.3 2.9 7.5 3.3
11.0 11.3 4.8 8.6 35.5 ( + ) 20.3 91.5
6.18 x 104
9°Zr
1.35 7-8 12.3 15-22 26 30-40
27 14.2 2.6 3.1 8 2.5
6.6 17.7 5.8 10.3 36.9 ( + ) 15.1 92.4
2.66 x 105
2°sPb
2.8 7 10-17 20 25-35
37.6 7.3 5.3 10.6 1.7
18.1 8.9 13.2 41.0 ( + ) 9.1 90.3
1.36 x 106
GIANT MULTIPOLE RESONANCES
405
5.2.3. Octupole resonances. Examining the systematics of the 3 - , T = 0 strength in fig. 6, we find that the strength is more spread out for light nuclei and starts to gain some collectivity in the lowest mode for heavier nuclei. The results in table 14 would support the speculation 27) that about one-half of the energy-weighted sum rule is exhausted by the low-lying 3 - states and the lhco group o f states, leaving the remaining one-half for the 3hco group o f states. In fact, 3 - continuum strength in 4°Ca and 2°Spb has been reportedly seen 13,14,17). The experimental results and the calculated counterparts are listed in table 13. TABLE15 Distribution of isovector octupole strength Nucleus
State
s.p.u.
~ EWSR
EWSR (MeV • fm6) (double commutator)
160
0-10 10-17 17-29 29-35 35-45 45-80
0.6 0.7 2.4 1.6 6.4 4.5
0.6 1.4 7.7 7.4 36.8 (+) 35.0 88.9
1.09 x 104
4°Ca
0-8 8-28 28-47 4%70
0.6 4.7 11.2 3.7
0.5 11.1 58.4 (+) 27.1 97.1
7.17 x 104
9°Zr
0-5 5-27 27-46 46450
0.2 5.3 15.7 2.7
0.1 10.3 70.4 (+) 16.5 97.3
4.04 x 105
2°SPb
0-19 19-29 29-38 38-60
6.9 7.6 17.4 3.4
7.4 19.1 58.0 (+) 13.7 98.2
2.62 x 106
While the 3 - , T = 1 giant resonances have not been identified so far, it is quite possible that the excited states at 33 MeV in 2°apb and 33.5 MeV in 197Au in the (e, e') experiment 25) could, in fact, be the isovector 3 - states 46) (fig. 7 and table 15). 5.2.4. Hexadecapole resonances. According to the shell model, hexadecapole resonances should occur at excitations of 0hco, 2hco and 4hco. As is shown in fig. 8, very little strength is allocated to the bound states (0hog), the energy-weighted sum rule for T = 0 is almost evenly divided between the 2hc~ and 4hco group of states. These
406
K . F . LIU AND G. E. BROWN
ISOSCALAR HEXADE~,POLE
,t-- ~1-/Jl'°/I
I1'°
rlf~,o
,o
1(] . .
,:
IJ
,o
2 -/ v" \ . . 10
~o
rl/~l~^
/5, 20
NI
,o
J-I
,o
,o
,o
c,~O
,o
60
p~,o,
30
40
50
M,V Fig. 8. Isoscalar hexadecapole strength.
I
.
~
~POt£
,2
.
h
AA
>
^
o'~
Ca
_ 20
"
.
I1"
Fo2oe
40 60 HW Fig. 9. Isovector hexadecapole strength.
are consistent with calculations by others 28). The distributions of the strength are presented in tables 16 and 17 and figs. 8 and 9. 5.2.5. Monopole modes. The transition density of the breathing mode,
pno(r) = ( ~l,I ~, ~(r- ri)[~l o) i
....
(28)
resembles that from the collective model of Tassie29). It is mostly localized on the surface where the interaction is very different from that of the interior.
GIANT MULTIPOLE RESONANCES
407
TABLE16 Distribution of isoscalar hexadecapole strength EWSR (MeV • fms) (double commutator)
Nucleus
State
~o EWSR
160
0-34 34-40 40-75
24.1 10.9 (+) 54.8 89.8
2.44 x 105
4°Ca
0-33 33-60
40.6 (+) 49.4 90.0
2.13 x 106
9°Zr
0-25 25-60
35.7 (+) 48.6 84.3
1.30 x 107
2°apb
0-9 9-20 20-50
12.7 29.1 ( + ) 44.2 86.0
1.06 x 108
TABLE 17 Distribution of isovector hexadecapole strength EWSR (MeV • fms) (double commutator)
Nucleus
State
~ EWSR
160
~ 40-80
19.5 (+) 63.8 83.3
2.57 x l0 s
4°Ca
0-30 30-70
13.6 ( + ) 76.6 90.2
2.34 x 106
9°Zr
0-33 33-60
19.3 (+) 58.4 77.7
1.87 x 107
2°apb
0-25 25-70
15.7 (+) 84.2 99.9
1.96 x 108
Recently, b o t h M a r t y et al. 56) in their inelastic d e u t e r o n scattering e x p e r i m e n t a n d T o r i z u k a et al. 56) in their inelastic electron scattering e x p e r i m e n t have tentatively assigned the r e s o n a n c e at ~ 13.2 M e V in 2°Spb to be the b r e a t h i n g mode. O u r m o n o p o l e r e s o n a n c e occurs at m u c h high energy (18.2 M e V in 2°apb), as c a n be seen f r o m fig. 10 (table 18). This is "mainly due to the fact that o u r i n t e r a c t i o n yields a n
408
K.F. LIU AND G. E. BROWN ~S(~CJU.AR M0XOPO~ (BREATH~iO ~ )
,j
'
~
~
o.
~"~--
fV,, Zr w
I ~ , O
/~,o
10
,o
20
30
,o 40
M,,V
Fig. 101 Isoscalar monopole strength (arbitrary unit).
ISOVECTOR MONOPOLE
I
60
/ 10
20
30
40
50
60
MeV
Fig. 11. Isovector monopole strength (arbitrary unit).
extremely high compression modulus in nuclear matter (378 MeV). If we assume a quadratic relation between the monopole energy and the compression modulus in nuclear matter [it is found by Blaizot et al. 44) in their analysis that this relation indeed holds very well for the various interactions], we would find a 200 MeV compression modulus in nuclear matter corresponding to the experimental findings in finite nuclei, a value close to the upper bound o f those generated by the realistic
.forces 57).
GIANT MULTIPOLE RESONANCES
409
TABLE 18 Distribution of isoscalar monopoles strength Nucleus
State
Strength (fm 4)
% EWSR
EWSR (MeV. fm 4) (double commutator)
160
15-20 20-25.5 25.5-29 29-35 35-50
16.6 23.2 12.8 22.5 5.8
12.7 23.5 15.0 31.2 ( + ) 10.7 93.1
2.29 x 103
4°Ca
20 22-30.5 30.5-35 35--40
63.5 238.8 33.7 3.6
12.7 67.2 11.5 ( + ) 1.4 92.8
9.42 x 103
9°Zr
19.5 20.5-31 31~,0
260.2 812 40.9
18.5 71.3 ( + ) 5.2 95.0
2.68 x 104
2616.5 1329.6
54.4 ( + ) 33.4
8.3 x 104
2°aPb
18.2 20.2
87.8
It has been pointed out 30, 31) that at least part of the Nolen-Schiffer anomaly is attributable to the isovector monopole mode coupled to the valence particle. The relation can be established by assuming the Coulomb potential to come from a square distribution of charge Ze2~ 1
Vc = ~ - ~
~(1 - %,)(3 - r2/R2).
(29)
The z-dependent part of Vc gives the isospin breaking in density Pn-Pp, which is the T = 1 mode. The change in energy from isc;spin breaking of the 4°Ca core is dE,,ca =
2('*' Ca[ V~I~b~.%° ) ( ~b~.%°[Vc['*' C a ) E o - E~
(30)
This is shown diagrammatically in fig. 12 where the particle-hole Green function G carries zero energy, i.e. G(0), because the interaction with the valence particle is instantaneous, and where V~ is the isovector part of the interaction. The isotope shift concerns the change in the expectation value of 2
1
1
rch = ~ ~ ( 1
- zs,)r 2.
(31)
410
K . F . LIU AND G. E. BROWN
'",~~.~)-~--C) /
V,
/ Fig. 12. Core polarization.
The contribution from the isovector mode to the change in r~h,2 upon addition of a neutronto 4°Ca, is 2<41Ca1~ 1 2 s=0 J=o 41 ~%ir/I@r=l><@r=llV~[ Ca> 2 i (32) Z(6rcn),,ca = E o - E¢,
Comparing (30) and (32), one finds
(AE),lca--(AE),.~o
Z 2 e 2 (6rcn),'sc 2 2 - (6r~h), ,ca -
2R
R2
(33)
Some preliminary calculations show the results to be sensitive to the position of the node in the transition density of the isovector mode. The isovector monopole transition density in the Green function formalism is defined as
= F1, L= Im ('s+n G(r, r, E ) d E 1~,
(34)
and is sketched in figs. 13 and 14. 1
~T=I
no x
p~l
r
r (fro)
Fig. 13. Theisovectormonopoletransitiondensity and the root-mean-square radius of the valence particle and hole.
x
r2
i I I
t t
Fig. 14. See caption to fig. 13.
r(fm]
GIANT MULTIPOLE RESONANCES
411
The ideal case would be like in fig. 13, where the node lies nearer to the rms radius of the valence hole than that of the valence particle. This will cut down the anomaly of the particle mirror nuclei to the same size as that of the hole mirror nuclei with the remaining anomaly attributable to other mechanisms, for example, charge symmetry breaking. On the contrary, we find that the result is too small to resolve the anomalies for the 41Ca -41Sc and the 170_17 F pair but too large for the anomalies of the 39Ka9fa and the 15N_150 pair. This is simply because the rms radii of the f~ (in 41Ca-41Sc) and the d~ (170_i 7F) valence particle lie nearer to the node of the calculated transition density than those of the d~ (39K-39Ca) and the p½ (15N_150) valence hole (fig. 14). Therefore, it seems plausible to suggest that the combined velocity and densitydependent symmetry potential is too repulsive and asa result, the isovector monopole vibration is localized too far out on the surface. Yet on the other hand, we find that the excitation energy gives a 162 A -~ (table 19 and fig. 11) dependence on the mass number which is smaller than the prediction of the hydrodynamical model (169 A -~) and much smaller than the recent experimental findings by Pitthan et al. 25) where they tentatively established 195 A -~ systematics. Therefore, it is hard to establish a conclusion at this stage. 5.2.6. Low-lying states. Listed in table 20 are the positions of low-lying states located by tracing the poles of the Green function. TABLE 19 Distribution of isovector monopole strength ~ EWSR
EWSR (MeV- fm 4) (double commutator)
Nucleus
State
Strength (fm 4)
160
15-30 30-38 38--60
26.4 26.7 13.4
22.7 34.6 ( + ) 22.3 79.6
2.7 x 103
4°Ca
14-22.5 22.5-33.5 33.5-38.5 38.5-60
52.4 189.4 96.5 55.3
8.5 26.9 29.5 ( + ) 21.7 86.6
1.16 x 104
9°Zr
13-23.5 23.5-30 30-38 38~0
225.6 206.6 602 14.5
10.7 13.3 47.2 ( + ) 15.4 86.6
4.31 x 104
2°sPb
7-17.5 17.5-24 24-34.5 34.5
646.5 1227.1 3061.2 577.3
5.1 15.7 52.7 ( + ) 13.0 86.5
1.68 x l0 s
K. F. LIU A N D G. E. B R O W N
412
TABLE 20 Positions of low-lying states (MeV) L ~ (T = 0)
Nucleus
Exp.
Skyrme BLV1
2+ 33334+ 55-
2°apb 160 #°Ca
4.07 6.13 3.74 2.74 2.61 4.31 4.48 3.2
5.6 6.3 3.5 1.35 2.8 6.4 5.0 3.4
9°Zr 2°spb 2°apb 4°Ca 2°spb
TABLE 21 Transition rate in Weisskopf units
") Ref. 32).
L" (T = 0)
Nucleus
Exp.
2+ 33334+ 5-
2°aPb 160 4°Ca 9°Zr 2°spb 2°spb 2°apb
8 ") 14 b) 25 c) 32 d) 32 a) 15 e) 14 ")
b) Ref. a3).
¢) Ref. 34).
d) Ref. 35).
Skyrme BLV1 9 4 21 27 38 16 14 ") Ref. 36).
Low-lying 2 + states in nuclei with low-lying 0 ÷ states are not seen in this calculation based on the ground state, because these states will not be given by the lp-lh calculation. Even-parity states are reproduced less well than the odd-parity states. Using empirical single-particle energies will lower the states to about the experimental values. But the discrepancies are still larger than the case of odd-parity states. The electromagnetic transition rates for the low-lying states are listed in table 21. The units are the single-particle Weisskopf units. B ( E L , 0 --* L ) = e 2S w
2 L + 1 ( ~ + 1)2(1.2A~)2L"
(35)
The small transition rate of 3- in 160 is even smaller than that of an earlier calculation by Gillet and Vinh Mau aT) which is about 8. It is suggested that the alphaparticle model with some J = 3 deformation in the ground state of 160 could probably resolve part of this anomaly 40).
GIANT MULTIPOLE RESONANCES
413
6. Conclusion
We have learned that the effective particle-hole interaction extracted from a reasonable energy density of a finite Fermi system does reproduce the qualitative excitation properties, and that the response function technique in coordinate space proves very handy to use, especially in heavier nuclei where large configuration space is needed to push the giant dipole states to sufficiently high energies. We have shown that part of the long-standing discrepancy between theoretical and experimental positions of the giant dipole resonance in 2°spb arose from neglect of the continuum in the early theoretical calculations. As a result, most of the giant resonances are reproduced within 5 ~o agreement with experiments. The transition rates of the low-lying states are in general within 12 ~o of the experimental values except for the 3- state in 160. Most of the energyweighted sum rules calculated from the double commutators are exhausted to within 13 ~ by calculated contribution from the response function with proper cutOffs. This also provides a check for self-consistency. The agreement we achieve in detail with experiment is, in many cases, not as good as that in calculations which employ empirical particle and hole energies. We believe there to be advantages, however, in carrying out calculations in a consistent way, in which single-particle energies and residual interactions are obtained from the same underlying force. The complexity of the calculations has necessitated the use of a highly schematical, Skyrme-type interaction. On the whole, this interaction does well for the vibrations of natural parity which we calculated. We achieve reasonable agreement with results from the collective model, which is not surprising, since these could also be derived from knowledge of the ground-state energy as a functional of neutron and proton occupation numbers. The agreement is interesting, however, because collective models use somewhat different spatial boundary conditions. It should be noted that our calculations were for natural-parity states, and there is no expectation that our Skyrme force would be good for unnatural-parity states. In general, our results are comparable with other calculations by Krewald and Speth as) and by Krewald et al. 39) where a RPA model with a density dependent interaction and single-particle wave function from a Woods-Saxon well are employed. Our results for the giant resonances are also consistent with other self-consistent RPA calculations by Bertsch and Tsai 40) using the Skyrme force I and II, except for the giant dipole states. We would like to thank Dominique Vautherin for the original suggestions motivating this work, and for use of the Hartree-Fock programs. Thanks are due to G. Bertsch and S. F. Tsai for kindly lending their computer programs and their stimulating discussions. One of the authors (K.F.L.) wishes to extend acknowledgements to G. Ripka and M. Rho for their encouragement and hospitality during his stay in Saclay where part of the work was carried out.
414
K . F . LIU AND G. E. BROWN References
1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26)
27) 28) 29) 30) 31) 32) 33) 34) 35) 36) 37) 38) 39) 40) 41) 42) 43) 44) 45) 46) 47) 48)
J. Blomqvist, Phys. Lett. 28 (1968) 22 G. F. Bertsch, Phys. Rev. Lett. 31 (1973) 121 D. Vautherin and D. M. Brink, Phys. Rev. C5 (1972) 626 G. F. Bertsch and S. F. Tsai, Phys. Lett. 50B (1974) 319 D. J. Thouless, Nucl. Phys. 22 (1960) 78 J. Ahrens et all, Proc. Int. Conf. o~ nuclear structure, Sendai, Japan (1972) J. S. Levinger and H. A. Bethe, Phys. Rev, 78 (1950) 115 W. T. Weng, T. T. S. Kuo and G. E. Brown, Phys. Lett. 46B (1973) 329 M. Danos and E. G. Fuller, Ann. Rev. Nucl. Sci. 15 (1973) 73 J. P. Elliott and B. H. Flowers, Proc. Roy. Soc. A242 (1957) 57 P. Ring and J. Speth, Phys. Lett. 44B (1973) 477 T. T. S. Kuo and G. E. Brown, Phys. Lett. 31B (1970) 93 M. B. Lewis, Phys. Rev. Lett. 29 (1972) 1257 Y. Torizuka et al., ref. 6) L. C. Rutledge and J. C. Hiebert, Phys. Rev. Lett. 32 (1974) 551 A. Moalen, W. Benenson and G. M. Crawley, Phys. Rev. Lett. 31 (1973) 482 S. Fukuda and Y. Torizuka, Phys. Rev. Lett. 29 (1972) 1109 M. B. Lewis, F. E. Bertrand and D. J. Horen, Phys. Rev. C8 (1973) 398 N. Marry et al., Contribution to Munich Conf., 1973 F. R. Buskirk et al., Phys. Lett. 42B (1972) 194 M. Nagao and Y. Torizuka, Phys. Rev. Lett. 30 (1973) 1068 M. B. Lewis, Phys. Rev. C7 (1973) 2041 K. A. Snover, K. Ebisawa, D. R. Brown and P. Paul, Phys. Rev. Lett. 32 (1974) 317 J. M. Moss et al., Phys. Lett. 53B (1974) 51 R. Pitthan et al., Phys. Rev. Lett. 33 (1974) 849 K. A. Snover, E. G. Adelberger and D. R. Brown, Phys. Rev. Lett. 32 (1974) 1061; S. S. Hanna et aL, Phys. Rev. Lett. 32 (1974) 114; A. Hotta, K. Itoh and Y. Saito, Phys. Rev. Lett. 33 (1974) 790 G. R. Satchler, Phys. Reports 14C (1974) 99 I. Hamamoto, Nucl. Phys. A196 (1972) 101 ; A. Goswan and L. Lin, Phys. Lett. 42B (1972) 310 L. J. Tassie, Austral. J. Phys. 9 (1956) 407 C. W. Wong, Nucl. Phys. A151 (1970) 323 G. E. Brown, V. Horsfjord and K. F. Liu, Nucl. Phys. A205 (1973) 73 A. Barnett et al., Phys. Rev. 186 (1969) 1205; O. H/iusser et al., Nucl. Phys. A194 (1972) 113; E. Grosse et al., Nucl. Phys. A174 (1971) 525 T. Alexander and K. Allen, Can. J. Phys. 43 (1965) 1563 J. Aeisenberg et al., Nucl. Phys. A164 (1971) 353 J. Bellicard et aL, Nucl. Phys. A143 (1970) 213 M. Nagao and Y. Torizuka, Phys. Lett. 37B (1971) 383 V. Gillet and N. Vinh Mau, Nucl. Phys. 54 (1964) 321 S. Krewald and J. Speth, Phys. Lett. 52B (1974) 295 S. Krewald, J. Birkholz, A. Faessler and J. Speth, Phys. Rev. Lett. 33 (1974) 1386 G. Bertsch and S. F. Tsai, Phys. Reports 18C (1975) 125 K. F. Liu, Phys. Lett. 60B (1975) 9 O. Nathan and S. G. Nilsson, Alpha-, beta- and gamma-ray spectroscopy, ed. K. Siegbahn (NorthHolland, Amsterdam, 1965) J. P. Blaizot, Phys. l.,¢tt. 60B (1976) 435 J. P.' Blaizot, D. Gogny and B. Grammaticos, Nucl. Phys. A265 (1976) 315 W. L. Wang and C. M. Shakin, Phys. Rev. Lett. 30 (1973) 301 K. F. Liu, Ph.D. Thesis, unpublished G. E. Brown, Proc. Int. Conf. on photonuclear reactions and applications, Asilomar, 1973 G. Saunier et al., Can. J. Phys. 51 (1973) 629
GIANT MULTIPOLE RESONANCES 49) 50) 51) 52) 53) 54)
B. D. Chang, Phys. Lett. 56B (1975) 205 S.-O. B~ckman, A. D. Jackson and J. Speth, Phys. Lett. 56B (1975) 209 B. L. Berman and F. C. Fultz, Rev. Mod. Phys. 47 (1975) 713 H. Breuer et al., Int. Symp. on highly excited states in nuclei, Jiilich (1975) G. E. Brown, J. H. Gunn and P. Gould, Nucl. Phys. 46 (1963) 598 G. Bertsch and T. T. S. Kuo, Nucl. Phys. 112 (1968) 204; I. Hamamoto and P. Siemens, to be published 55) G. E. Brown, Theory of nuclear models (North-Holland, Amsterdam) 56) N. Marty, M. Mortet, A. Willis, V. Comparat, R. Frascaria and J. Kalline, preprint (Orsay); M. Sasao and Y. Torizuka, preprint (Sendal) 57) H. A. Betbe, Ann. Rev. Nuci. Sci. 21 (1971) 93
415
i
Nuclear Physics A265 (1976) 385-415; ~ ) North.Holland Publishing Co., Arasterdara Not to be reproduced by photoprint or microfilm without written permission from the publisher
GIANT M U L T I P O L E R E S O N A N C E S t K. F. LIU tt
State University of New York at Stony Brook, Stony Brook, N Y 11794 and G. E. BROWN
Nordita,
Copenhaoen, and State University of New York, Stony Brook Received 25 August 1975 (Revised 1 December 1975) Abstract: The positions and the transition strengths of the giant multipole resonances of the spherical nuclei (160, 4°Ca, 9°Zr and 2°spb) are calculated with large configuration spaces in the random phase approximation based on the Hartree-Fock ground states of a newly developed Skyrme interaction. Positions of many giant resonances are reproduced within 5 % of the experimental values. Most of the energy-weighted sum rules calculated from the double commutators are exhausted to within 13 ~ by the contribution from the RPA response function.
1. Introduction
Recently, attention has been focused on the nuclear excitation region above the particle emission threshold. Data from various photonuclear reaction experiments and hadron and electron inelastic scattering experiments seem to indicate the existence of new giant multipole resonances besides the giant dipole resonance. Unlike the resolved discrete low-lying states which depend on the detailed shell structure of the nuclei, the giant resonances exhibit structure governed by the average nuclear properties. It is the aim of this paper to study the systematic locations of the electric multipole resonances and their contributions to their respective energyweighted sum rules within the model of random phase approximation based on the Hartree-Fock ground states of a new Skyrme interaction. This paper is organized as follows. We present in sect. 2 a linear response theory with the random phase approximation formulated in coordinate space which allows us to include a large number of configurations in a simple way numerically. In sect. 3, we present a new Skyrme interaction to be used in the calculation. It has the properties of bringing the single-particle level densities near the Fermi levels in 2°apb close to their experimental values. Sect. 4 is devoted to some of the numerical details of the present t Supported in part by USERDA contract no. AT(I 1-1)-3001. ,t Present address: Service de Physique Th6orique, CEN-Saclay, BP no. 2, 91190 Gif-sur-Yvette, France. 385
386
K.F. LIU AND G. E. BROWN
calculations and we compare, in sect. 5, our results with the experimental values and the results of other theoretical works. Finally, it is concluded, in sect. 6, that it is essential to include a large number of configurations in order to have the experimentally established giant resonances in heavy nuclei reproduced well and that self-consistency of the model is justified through the energy-weighted sum rules. 2. Linear response theory and R P A in coordinate space
Nuclei have excited states that can be detected with experimental probes which couple to the density or the current of the system; for example, the scattering of electromagnetic waves or electrons from nuclei. These probes interact weakly with the nuclei and therefore can be treated in the Born approximation. As a specific example, consider the perturbing Hamiltonian which is coupled to the density of the system in the form
f d3xp.(x,t)f(x, t) ....
H'H(t) =
(1)
Then the change of the density, to first order in H~(t), is related to the densitydensity correlation function and is defined as the linear response,
R[f], = ~(p(x, t)) = ih-1
dt'
dax'f(x ', t')(Ool[p~(x', t'),pn(x, t)][00),
(2)
t.o
where 100) is the ground state of the unperturbed Hamiltonian. Most of the physical information is contained in the overlap of the linear response with the probe in the spectral representation f
R[f f]=
V (~°lFIq~")
daxf(x)R[f]o,=,~
°9-(En-Eo)+i~l
(q%lfl~O")(0"lflq/°)
-
~o+(E _Eo)+i~l
,
(3)
where F = S~b+(x)f(x)~(x)dax is hermitian and q is put in to insure proper causality. This expression, eq. (3), has simple poles at the excitation energies and their residues are the transition probabilities between the ground state and the excited states : Im R [ f f ] (e~ > 0)_~on ~ [(~nlFl~o)126(m-(E,,-Eo)). q
(4)
n
Conventionally, one solves the RPA equation in the configuration space where the dimension of the RPA matrix goes linearly as the number of the partical-hole configurations involved, which might be as large as several hundred for the case of heavier nuclei. Blomqvist 1) and Bertsch 2) have developed a method by inverting matrices in coordinate space instead. It is restricted to zero range interactions and is numerically
GIANT
MULTIPOLE
RESONANCES
387
simpler provided the number of the mesh points used for numerical integration is smaller than the dimension of the particle-hole configuration. Suppose the delta-function interaction has the general form in the multipole expansion Vph
*L L = L ~(rl--r2)"L(l' v~, ~r 1+r2)) 2 L + l M~+(1)DK,M(~I)DK~M(~2)M~(2), a,L,M rlr2 4~
(5)
where {M,} is the set of operators M = {1, a, r, V2,~V, etc.} and K~ is the angular momentum projection of the M~ operator on the body-fixed axis of the particle-hole state. We can, therefore, define the Lindhard function as phL c,hpL(rl)ChpL(r2) atO~L¢~ rE, o9) = ~ C~*phL (rl)CP (rE) + v a i l ~-'1'
(6)
, , z r )M~+ f f 2 L4----~h + l D*LM(f2). cPahL(r)= f dO (~Op~Oh)M(
(7)
ph
•p - - 8h - - 0) - -
it/
8p - -
eh + co-- it/
where
If we replace the integration over the coordinate r by a finite summation over some mesh points, then the random phase approximation to eq. (3) becomes a geometrical series of matrices:
RrtVArffq=F~ ( 1 "Jr-G~°'L )ijFj.L to L J ' J - I ~(O)L
(8)
The indices of the matrices are simply the direct product of the mesh points of the coordinate r and the set of operators {M~}. If we take, say, ten mesh points and three operators, we have typically a 30 × 30 matrix to invert instead of solving a several hundred by several hundred RPA matrix.
3. Particle-hole interaction
Landau's theory of Fermi liquids involves the following assumptions which are generalizations of the usual Hartree-Fock description of the ground state. (a) There is a one-to-one correspondence between the bare particles and the quasiparticles. (b) Quasiparticles on the Fermi surface interact weakly so that they can be described by a self-consistent field. The quasiparticle energy is then obtained by varying the energy with respect to the quasiparticle occupation number, and the effective particle-hole interaction can be identified as the second derivative of the energy functional with respect to the quasiparticle distribution.
388
K . F . L I U A N D G. E. B R O W N
The Skyrme interactions developed by Vautherin and Brink 3), V = Z to(1 +ZoPo)c~(rl-r2)+½tl ~(k'2~(rl - r 2 ) + ~ ( r l - r 2 ) k2) 1,2
1,2
+t 2~k'.6(rl-r2)k+t 1,2
3 ~ 6 ( r l - r 2 ) 6 ( r 2 - r 3 ) + W0 ~ ( a 1+32). k'x 6(r~-r2)k , (9) 1,2,3
1,2
where k and k' are the relative momenta, fit the ground-state energies of spherical nuclei very well, when used in the Hartree-Fock approximation. We take the view that this procedure produces the energy well as a functional of the quasipartide occupation number and shall use the single-particle energies and the particle-hole interaction obtained by functional differentiation of this energy. That is, Vph :
~(rx-r 2) ~ ~(l+(-)~-~'ol"a2)~(l+(-)'-~'r1" x2) ~ , st, s't"
(10)
OPstOPs't'
where s and t are the third components of the spin and isospin labels of the density. All the existing Skyrme forces of the form in eq. (9) have essentially fitted the binding energies and the radii of the spherical nuclei equally well, yet they differ in their HF single-particle spectra and their predictions of the vibrational states, for example, the Skyrme force I [-ref. 3)] yields the dipole state of 2°Spb somewhat lower 4) (12 MeV) than the experimental value (13.5 MeV), while the Skyrme force II which has a stronger velocity dependence yields it somewhat higher. As can be seen from fig. 1, the single-particle levels given by the Skyrme I and II forces are too widely spaced at the Fermi surface. It was noticed long ago 5s) that levels at the Fermi surface are significantly more compressed than predicted by microscopic theories in Hartree-Fock approximation. This can be characterized by saying 53) that m*/m > 1 empirically, at least near the Fermi surface, whereas Hartree-Fock calculations tend to give m*/m < 1, or, it can be characterized by having a wider surface. The empirical compression of levels has since been understood in terms of coupling of the single particles to vibrations 54). Our first objective was to construct a Skyrme-type interaction which had m*/m > 1 in the nuclear surface region and/or a wider surface to see whether we could mock up this effect within the framework of zero-range forces. This is achieved by adding a new term t ~. ½t13(k'26 (r t - r j)8(r j - rk) + ~(r i - r i)6(r j - rk)k2), ijk
to the interaction in eq. (9). It is not at all clear that the level compression described by the single-particle coupling to vibrations can be described in this way, because in doing so we are * We are grateful to D. Vautherin for suggesting such a term; from the formulae for m*/m and the surface thickness in ref. 3), one can see that such a term can be used to increase m* in the nuclear surface and/or to increase the surface thickness.
GIANT MULTIPOLE RESONANCES
389
PROTONS I
II
BLV 1
EXP
0
2f 512 3p 3/2
2f 712 2f 7/2 Li 13/2
-2
_ _
Li 13/2
~ .
2f 712 lh 912
lh 912
-4
2f 712 ~Li1312 lh 912
l h 912 -6 3s 112 X
2d 312
-8
-10
2 d ~1h1112
512
~3s112 ~ 2 d ~ , 3 s
1/2
l h 1 1 / 2 ~ 2 d 3/2
~'-2d ~ l g
-12
lg 712
~ , 2 d
3s 112 312 lh11/2 5/2
2d 312
lh 1112 2d 512
712
5/2
lg 7/2 -11,
NEUTRONS I
II
0
3d 5/2
-2
Li 1112 ,2g 912 "L] 1512
~'/.s
- -
2g 912 Li 1112
~ 2 g ~ .
-4
>
-6
:E UJ
-8 -10 -t2 -14
~ 3 p
112
~ 2 f
512 ~3p312
~Li13/2 ~2f 712 lh 9/2
BLV1
EXP
~ 3 p
3p 112 312 2f 512 Li13/2
~.2f712
,3d 312 7/2 "~s112 :~d512 L|1512 LIl112 2g 912
3p 112 Rf 512 ~3p312 ~Li13/2 ~ 2 f 7/2 ~ l h
912
,.?.g712 1/2 3d 512
- ~ ~
Li 1112 Lj 1512 2g 912
~ 3 p 3 p
112 3/2 2f 5/2
Li1312 2f 7/2 l h 9/2
l h 9/2
Fig. 1, Fitted single-particle spectra near the Fermi levels in 2°Spb by different Skyrme interactions and the experimental results.
390
K. F. LIU AND G. E. BROWN TABLE 1 Parameters of the Skyrme forces
Force
to
tl
t2
I
- 1057.3 - 1169.9 - 1086.2
235.9 585.6 341.14
- 100 - 27.1 - 120
II
BLV1
t3
1446315 9331.1 13114
tl 3
Xo
Wo
1982.9
0.56 0.34 0.52
120 105 130
essentially trying to describe an energy-dependent effect by m o m e n t u m dependence. It is at least interesting to try, and to some extent the degree to which we reproduce the spectrum o f highly excited states gives us some answer, because zero-range forces o f the S k y r m e type are so convenient to use. The fitted p a r a m e t e r s are listed in table 1 together with those o f Skyrme I and II. All the following results are obtained by using the interaction S k y r m e BLV 1. Following the f o r m a l i s m o f Bertsch and Tsai 40), the particle-hole interaction derived f r o m eq. (10) is Vph = 6 ( r , - r2)[a -- b(V 2 + V 2 + V 2, + V2,) + c(V, - Va ,)" (V 2 - V2, ) +d(V 1 + V r ) " (V2 + V2,)],
(11)
with a
=
(to+½tsp+1txa(4z_3V2p))(3_14_~,
. ,t.2 __~all . O.2__4~ 1 . O.2,~1 .T2 ) +ltozo(trl-
a- 2 --~-1 • "¢2),
b = 3~(23tl + 5t 2 + tlaP)+a~(2t2 - t I - ~ t l a P ) ( a 1 • a 2 + ~ 1 " ~2 + a l " trExx" x2), c = 3~(23tl + 5t 2 + ½t 13P) + a~2t2 - ta - ~tl aP)(al " a2 + T~" x2 + a l " aErl" x2), d = 3~(23tl _ 15t2 + ~1t l a p ) - - ~ ( 21t l + 3 t 2 + ~ t l a P ) ( a
1 . a 2 -[- Tl " "['2 "4- O"1 . ~r2"C1 . T2).
(12)
Here the indices 1, l' refer to the'particle-hole coordinates to the left o f the operator, 2, 2' are those to the right. The CP~hL coefficients which a p p e a r in eqs. (6) and (7) will be the same as those given in ref. 40).
4.
Numerical
details
In order to calculate the C~'~hL coefficients, we need to k n o w the single-particle wave functions and energies. The single-particle c o n t i n u u m is discretized by putting a wall at a distance o f a b o u t 2.5 times the nuclear radius. T h e numerical solution o f the radial Schr6dinger equation is carried out by p e r f o r m i n g the R u n g e - K u t t a integration starting f r o m the origin and m a t c h i n g up its logarithmic derivative at a m a t c h i n g radius R M with that obtained by integrating in f r o m the wall. The search
GIANT MULTIPOLE RESONANCES
391
for eigenvalues is carried out by means of the Newton-Raphson method using the condition provided by the matching. An artificial imaginary part ( ~ 0.5 MeV) is introduced to the particle-hole propagator to give the excited states finite widths. Both the wall distance and the particle-hole space cut-off have been varied such that the positions of the excitation are convergent to within 3 ~o and the strengths to within l0 ~ (except in a few cases like the 0 + excitation of 160). The mesh sizes, the particle-hole space truncations, the highest numbers of configurations and the wall distances of the four spherical nuclei are listed in table 2. TABLE 2 Numerical approximation Nucleus
Mesh size (fm)
x60 4°Ca 9°Zr 2°apb
0.6 0.7 0.8 1.0
Highest number of configuration
Emax 120 80 55 40
Wall distance (fm)
86 177 241 398
10 12 12 15
Truncation is such that ep-/~n < Emax; the operator Vr is also neglected in this calculation. It has been pointed out 4s-50) that the spin densities derived from the Skyrme forces are not fitted in the HF ground-state calculations of the self-conjugate nuclei and this may lead to spin instability (i.e. Landau parameter G o < -1). There are two remedies. One is suggested in the paper of B~ickman et al. 50) by the introduction of two additional terms, ~ k 2 + k'2)tl)~lPaC~(rl r2) and I2z2P~k' " kc~(r 1 Y2)" This requires refitting of all the parameters. Another one is proposed and carried out by Blaizot 43) whose idea is to replace the three-body t e r m taC$(r 1 - r 2 ) 6 ( r 2 - r 3 ) by a density dependent term ~tap(1 +Pa)t~(rx-r2), since by doing so, it leaves all the previous HF results unchanged and yet has an effect of removing the spin instability (i.e. GO > - 1). As the matter now stands, it seems there exists some degree of ambiguity in the spin-dependent part (S = 1 in the particle-hole channel) of the residual interaction, which should be pinned down in the self-consistent RPA calculations. On the other hand, we know from experimental data what its strength should be, at least qualitatively, as compared to the spin independent part (S = 0 in the particle-hole channel) of the interaction. The argument is presented as the following: In view of the fact that the unnatural parity states, where the spin part of the interaction dominates, are far less collective than the natural parity states (e.g. they are pushed away from the unperturbed values of the particle-hole gaps by typically 1 or 2 MeV only and the transition rates are not as large as those of the natural parity states in general), one would, therefore, expect that the strength of the spin -
-
-
-
392
K . F . L I U A N D G. E. B R O W N
part of a reasonable interaction should be much smaller than that of the spinindependent part of the same interaction on the nuclear surface where nuclear excitations take place. In addition, we see from table 2 and eq. (32) in ref. 40) that the contribution from the spin part of the interaction is further cut down due to its geometry (e.g. C~_hL does not contribute to the natural parity states and for some configurations Ct~,~Lare either small or vanishing). We have indeed tested for several cases of the giant resonances where many configurations contribute and found that the results are fairly insensitive to the spin part of the interaction even though it may be unphysical (GO > - 1 in the interior of the nucleus). Therefore, in this paper where we are mainly concerned with the giant resonances, the spin part of the interaction is dropped in the RPA calculations together with the spin-orbit interaction and the Coulomb interaction, as was done in ref. 40). The operator Vp+Vh 2 2 is related to other quantities with unit operator through the Hartree-Fock single-particle equation O" O
2
2
Vp +V h =
~
Wq
2u~+ -- [jp(jp+ r
1)-/p(/p+ l)+jh(jh+ 1)--/h(/h+ 1)--~]
2
\2m*]
2~m~*' (13)
where Uq, wq and mq* are respectively the s.p. potential, the spin-orbit potential and the effective mass of neutron or proton. Replacements of this sort help cut down the dimension of the bare Green function ~o~L
5. Comparison with experiments 5.1. SUM RULES
One way of identifying a collective state is by looking at the energy-weighted sum rule: the transition rate of a collective mode should exhaust a fair fraction of the sum rule. Now the linearly energy-weighted sum rule is conserved by the random phase approximation s), i.e.
1 --
[do~ ~oIm (FGRPAF) ½(~o[[F, [H, f]][~o)
7~ d
~
°..,
(14)
where [~bo) is the HF ground state. This has provided us with a check for our numerical work. We found that, for most of the cases, the energy-weighted strengths calculated from the RPA exhaust over 87 % of the sum rule calculated from the double commutator (tables 8-10 and 14-19). The remaining strength is presumably at higher energy in the continuum.
GIANT MULTIPOLE RESONANCES
393
In the actual evaluation of the sum rules, it is noticed that the double commutator of the Skyrme interaction in eq. (14) vanishes, unless the one-body operator F carries degrees of freedom like spin or isospin. We illustrate this point by taking F as the electric multipole operator which acts only on protons: F = e • 12-(1- Z3,)rLyL(p,).
The only contributing terms from the Skyrme interaction are the velocity dependent ones whose commutator with the isospin-independent part of F is proportional to E (V r L YL(~i) -- V r L y L (~j)~(ri _ r j)" (Vi -- V j), ij
but it vanishes because of the zero-range delta interaction. The commutator with the isospin-dependent part of F is a different story; it is proportional to
X (*3,v yL
%Vr r L
r). (v,- v).
ij
This does not vanish as long as i and j are not identical particles. Note that since we use Hartree-Fock ground states calculated from the same force as used in our particle-hole interaction, we should have consistency between the sum rules calculated in the RPA and by the double commutator ss). 5.1.1. lsoscalar multipole sum rule. In view of the fact that the percentage of energyweighted sum rules exhausted by the 2L pole excitation of bound states in spherical nuclei is in general < 20 ~o (except L = 3 in some nuclei where the percentage may be 30%-40%), Nathan and Nilsson ,2) argued that the T ~ Ttransitions are largely dominated by collective motions such that neutron and proton matter move together. Therefore, the electric operator F(EL) = ½E (1 - z a , ) ~ Y o ~ , i
is effectively replaced by F ( E L , T = O) = ½ ~ ( 1 - ( z a , ) ) ~ Y L . ,
(15)
i
where the average <%i> is taken to be ( N = Z ) / A . Therefore, the remainder, -- E I ( ' c 3 i - - <"C3i>)rLYLra, i
(16)
is assumed to act on [d T[ = 1 transitions. As a result, the energy-weighted sum rules for the isoscalar multipoles are multi-
394
K.F. LIU AND G. E. BROWN
plied b y a f a c t o r (l - - ( T 3 1 ) ) 2 = 4Z2/A 2, a n d b e c o m e
S~v( F(EL, T = 0)) =
h2 L ( 2 L + 1) Z 2
2m
4re
A
-
(17)
( r2L - 2 ) .
T h e n u m e r i c a l values o f S Ew L° for the H a r t r e e - F o c k g r o u n d state a r e given in tables 11, 14, 16 a n d 18 a n d are c o m p a r e d with those o b t a i n e d f r o m the r e p l a c e m e n t ,
S~ -
3hZL Z z
R2L- 2,
(18)
8mrc A for a u n i f o r m m a s s d i s t r i b u t i o n o f r a d i u s R in t a b l e 3 a n d fig. 2. W e see t h a t the sum rules c a l c u l a t e d with u n i f o r m m a s s d i s t r i b u t i o n a r e o n l y ~ 20 ~o- ,-~ 70 ~ o f those c a l c u l a t e d with H a r t r e e - F o c k g r o u n d states for higher m u l t i p o l e s . T h e r e f o r e the - -
3
016
Zrg o
0.5
I
I
I
2
4
6
Fig. 2. Comparison of the sum rules calculated with uniform mass distribution and those calculated with Hartree-Fock ground states. TABLE 3
Radii R of the uniform mass distributions and their energy-weighted sum rules as compared to the ones from the Hartree-Fockground states (L = ½(2+ 2))
R'~/~2 + 3) (r~)HF = Nucleus 160 *°Ca 9°Zr 2°Spb
S[°w(uniform distribution) S~° (Hartree-Fock)
R (fm) 2.95 4.01 5.25 6.94
2 = 2 ( L = 2)
2 = 4 ( L = 3)
0.759 0.849 0.910 0.935
0.46 0.62 0.756 0.822
2 = 6(L=4) 0.216 0.385 0.570 0.675
expression in eq. (18) s h o u l d be c o n s i d e r e d i n a d e q u a t e as s o m e t i m e s used in e x p e r i m e n t a l analysis. 5.1.2. Dipole sum rule. It is c o n v e n t i o n a l to express the electric d i p o l e s u m rule in t e r m s o f the classical s u m rule. W h e n the c.m. is t a k e n into a c c o u n t the electric d i p o l e
GIANT MULTIPOLE RESONANCES
395
operator is Dz = ½ e ~ ( l _ z 3 i ) z i -
Ze~
i
h
i " zD
= e Z AN rpYol(~p)_eE. AZr.Y~(~.)"
(19)
P
Then, the dipole sum rule becomes h2 N Z 3 I(OIOzln)lE(g.-go)
=
e 2
n
(20)
__
2m A 4re (1+~)'
where x = ~[(tl+tE)(~(1-zairsj)~ij)+tl3( i
~ i
(1--ZaiZaj)fij~jk)]
h NZ 2m A '
(21)
is called the enhancement factor due to the nucleon correlation. The updated photonuclear experiments 6) show that the integrated absorption cross section to 140 MeV is approximately twice the classical dipole sum rule. This makes the enhancement factor x about two times larger than those from previous experiments and theoretical calculations 7) (and also our results !). One explanation of this increase, offered by Weng, K u o and Brown 8), is largely attributed to the short-range tensor correlation. The additional contributions which they find would presumably come at energies well above that of the giant dipole resonance, and would not be expected to be reproduced by our force, which has no tensor component. TABLE 4
Enhancement factor x for E1 x (Sky. BLV1)
Cut-off
Sky. I
Sky. II
Nucleus (doublecommutator) (responsefun.) 160 4°Ca 9°Zr 2°apb
0.29 0.33 0.37 0.39
0.28 0.29 0.34 0.39
(MeV) 55 55 40 40
(double commutator) 0.16 0.19 0.20 0.21
0.57 0.7l 0.73 0.79
Both results from the double commutator and from the RPA are listed in table 4. The velocity dependence o f the Skyrme BLV1 is intermediate between those of the Skyrme I and II; therefore it places the weighted dipole strength in between those o f Skyrme I and II as we have discussed earlier. Experimentally9),when the cut-off is taken to be at some energy above the giant dipole (30 MeV for 160 and 4°Ca), the sum rule in units of 0.06 N Z / A is 0.63 for 160, 0.73 for 4°Ca and 1.3 _0.1 for 2°apb, and in the present model the values are 0.76, 0.80 and 1.3. Because the sum rule is more or less reproduced up to 30 MeV and the positions
396
K . F . L I U A N D G. E. B R O W N
of the giant resonances are consistent with the experimental results, which we shall show later, it seems plausible that a cut-off at some energy above the giant resonances is appropriate for this model. 5.1.3. Isovector multipole sum rule. Unlike the isoscalar case, the velocity dependent parts of the Skyrme force do contribute to the energy-weighted sum rule. When we take the isovector operator to be the remainder, eq. (16), it introduces a factor 1 - ( ( N - Z ) / A ) 2 = 4 N Z / A 2 therefore the isovector sum rule is h2 L ( 2 L + I ) N Z 4~ A
S ~ ( F ( E L , T = 1 ) ) - 2m
(22)
(r2L_2)(lq-x),
where ~[(t 1 -I- t2)( ½~ c5ij(1- zai'c3j)r 2L- 2> -t- t 13(-~ ff~, C~ijC~jk(1 -- "C3iZ3j)r2Lij ijk X = h 2 NZ _
_
_
_
2m A
27]
(23)
(~>
The x-values for L = 2, 3 and 4 are listed in table 5. The non-vanishing enhancement factor ~c implies that the T = 1 states obtained from the Skyrme interaction either TABLE 5 Enhancement factor r for the isovector s u m rules Nucleus
L = 2
L = 3
t60 4°Ca 90Zr 2°8pb
0.18 0.24 0.29 0.32
0.11 0.16 0.22 0.26
L=4 0.06 0.10 0.16 0.20
will lie higher in energy or will acquire more strength than those obtained from the velocity-independent forces. 5.1.4. Monopole sum rules. Transitions between two nuclear states, both with spin J = 0, are strictly forbidden for electromagnetic radiation. However, the process can take place by the ejection of electrons in the x-shell. Unlike the case of internal conversion where we consider the wave function of the electron being outside the nucleus, here we have to consider the multipole expansion of the Coulomb field interacting with electron wave functions inside the nucleus only. Therefore we shall use the operator F(E0) = ~ = 12!(1 -z3i)r 2 for the monopole excitations and the sum rule. With the proper readjustment condition included, i.e. defining the operator F = ~ f~-(fi), i
such that (qSolFlq50) = 0,
(24)
GIANT MULTIPOLE RESONANCES
397
the respective isoscalar and isovector sum rules are
S~.~(F(E0))
=
Z2
h22 m
__
-A -
(25)
SL~v(F(E0)) - h2m2NZ <~->(1 + x),
(26)
where
~[(tx + t2)<½ Z 3U(1 - ~3,G)r2> + tl 3<~ Z ~o~ik( 1 - %,%)r2>] r =
u
h 2 NZ
_ _
<~>
(27)
ijk
2m A The values of x are listed in table 6 for x60, *°Ca, 9°Zr and 2°Spb. TABLE 6
E n h a n c e m e n t factor x for the isovector monopole s u m rules
5.2.
Nucleus
t60
4°Ca
9°Zr
2°spb
x
0.18
0.24
0.29
0.32
DISTRIBUTION OF MULTIPOLE TRANSITION RATE
The strength function due to G (°), the l p - l h excitation, is redistributed by G ra'A in such a fashion that the isoscalar modes are pushed down in energy due to the
1t
ISOVECTORDIPOLE
/
1t !
[
1I/
_c
[
/
V
~/
1o
I/
, n,
~
/,1 10 ~
,
. # " , ' - - ~
JL-~..~_,
~ 20
, : ~ z : ~
p
10
30
2o
20 MeV
Co ~
:--
,
30
,
30
Fig, 3. The calculated isovector dipole strength o f four spherical nuclei in single-particle units are plotted in solid lines. The dashed lines are the experimental results taken from refs. 6, s t).
398
K.F. LIU AND G. E. BROWN TABLE7 Positions of giant dipole resonances
Nucleus
Experiment (MeV)
x60 4°Ca 9°Zr 2°spb
Krewald et al. ")
22.3, 24 19 16,7 13.5
21, 22 19, 19.2 16.2, 16.8 %11.2
Present calculation (MeV) 19.5, 23.7 17, 19, 22 15.5, 16.8, 19 13.1, 17.5
~) Refs. 3s,11). TABLE 8 Distribution of isovector dipole strength EWSR (MeV. fm2) double commutator)
Nucleus
State
s.p.u.
% EWSR
160
19.2 23.5 25-30 30-55
0.7 0.7 0.5 0.5
11.4 25.5 23.6 (+) 34.7 95.2 a)
2.55 x 101
4°Ca
0-15 17 19 22 25-55
0.3 0.9 0.8 0.9 0.6
3.9 18.4 17.4 22.4 (+) 21.5 83.6 a)
6.59 x 10l
9°Zr
0-14 15.9 16.8 19 22-30
0.6 1.5 1.1 1.9 1.0
5.6 20.2 15.5 30.9 (+) 20.8 93.0 ")
1.5 x 102
2°spb
13.1 17.6 22-38
5.7 3.6 0.5
49.5 40.0 (+) 9.0 98.5 a)
3.41 x 102
•) The last rows are the sums of the % EWSR exhausted in the energy region as indicated. attractiveness o f the isoscalar interaction, yet the t r a n s i t i o n strength is e n h a n c e d in order to conserve the energy-weighted s u m rule. O n the other h a n d , the isovector m o d e s are raised higher in energy due to the repulsiveness o f the isovector interaction. 5.2.1. Giant dipole resonances. T h e d i s t r i b u t i o n o f the dipole strength is s h o w n in fig. 3 a n d table 8 a n d the peaks are listed in table 7 together with those f r o m experim e n t s a n d f r o m a n o t h e r c a l c u l a t i o n with d e n s i t y - d e p e n d e n t i n t e r a c t i o n by K r e w a l d a n d Speth 38), a n d R i n g a n d Speth 11).
GIANT MULTIPOLE RESONANCES
399
In the present calculation, the splitting of dipole states seems a common feature for all the nuclei considered. While the splitting is observed in light nuclei is attributed to the spin-orbit potential in the theoretical work 10), the second peak has not been observed experimentally in 2°spb. It is probably due to the exaggerated velocity dependence of the Skyrme force in this region, since Skyrme forces are supposed to be parametrized for small momentum transfers, and therefore contain only the first non-zero term in the expansion in momenta. Indeed if we simply switch off the velocity dependence by setting t 1 = t 2 - t13 -- 0, the second peak in 2°Spb is practically gone, while the first one remains essentially at the same position and the splitting in 160 is not much affected. Experimentally, the 160 strength has two main peaks at 22.3 and 24 MeV, and in addition there is a broad absorption extending up to 30 MeV. While the second peak and the broad absorption are reproduced by the present model the first peak is off by about 3 MeV. This is attributed to the fact that our calculated particle-hole gap between d~ and p~ ~ orbits is smaller than the experimental gap by 4.3 MeV for neutrons and 5.5 MeV for protons. If one uses the experimental energies for p~- 1 holes and d~, d~ particles, the two peaks are shifted to 22 and 25 MeV. This shows how important a role a single configuration may play in the giant dipole resonances in light nuclei. It is quite a different story in heavy nuclei like 2°Spb. It is necessary to have a model space containing 200 configurations (this corresponds, in this case, to a 20 MeV cut-off of the particle-hole space with the particle continuum discretized by a wall at a distance of 15 fm from the center of the nucleus) to push the dipole state ISOSCALAR OUADRUPOLE
.4I 4
II
~ _ i
0li
It ~
~
Cal i
g61
Zri0 --
a
20
3O
[wO~
10 t4eV
Fig. 4. The calculated isoscalar quadrupole strength are plotted in solid lines. The experimental results (dashed lines) are taken from ref. 26, 52)for160, ref. 15) for 4OCa, ref " I 7 )for 9 0 Zr, andref. 21 )andref. 19) for 2O8 Pb. The arrows indicate the experimental positions of the low-lying states.
400
K . F . LIU A N D G. E. B R O W N ISOVECTOR QUADRUPOt.E
1
2
, ~ 10
I
o". 20
~30
/,0
LkVk
10
50
60
I I
20
_ :"
30
t,O
50
60
~V Fig. 5. Isovector quadrupole strength. Experimental results (dashed lines) are taken from ref. 17) for 9°Zr; refs. 21, 25) for 2°spb.
to 13.1 MeV. If one takes 100 configurations, the dipole state is pushed up to 12 MeV only. This is probably part of the reason that other calculations with densitydependent forces 11) and realistic force 12) dfd not push the dipole state in 2°spb TABL~ 9 Isoscalar quadrupole excitation RPA calculation
Experiments
Nucleus
E x (MeV)
s.p.u.
~o EWSR
*°Ca
16
12.1
83
9°Zr
13.5
11.0
70
2°spb
11
12.4
57
E x (MeV)
~o EWSR
reaction
ref.
17 10-25 18 17
75 66 32
(p, p,) (e, e') (¢t, ct') (3He ' 3He, )
la) t4) 15) 16)
14 14 14
75
(e, e') (3He ' SHe. ) (*t, et')
17) 16) 2*)
(p, p,) (p, p,) (e, e') (e, e') (3He ' 3He, ) (3He ' 3He, ) (~, ~,)
is) 19) 20) 21) 22)
9.2-11.2 10.9 8.9-11.2 8.9-11.2 11 11 11
54
33 47
16) 22)
It is worthwhile noticing that in the experimental analysis, the sum rule expressions for a uniform mass distribution are sometimes adopted, which are quite different from the H F sum rules as we have pointed out in subsect. 5.1,1.
GIANT MULTIPOLE RESONANCES
401
high enough. In addition, use of self-consistent orbitals keeps the neutron rms radius 2 ½ 2 ½ close to the proton one ((r),,~t,on = 5.61 fro, (r)proto,= 5.49 fm for 2°Spb) and this increases the matrix elements as compared with calculations like those of ref. 12) using harmonic oscillator orbitals 47). 5.2.2. Giant quadrupole resonances. The distributions of quadrupole strength are shown in tables 11 and 12 (figs. 4 and 5). From tables 9 and 10, we found that the calculated values from the Skyrme BLV1 do agree with experimental results reasonably well. TABLE 10 Isovector quadrupole excitation RPA calculation
Experiments
nucleus
E x (MeV)
~ EWSR
E (MeV)
9°Zr
28
26
28.0
23.5
46
22 23.7 22.5
2°Spb
~ EWSR
60 85
reaction
ref.
(e, e')
aT)
(e, e') (p, y) (e,e')
21) 23) 25)
They are also consistent with the predictions given by the collective model (60 A- ~ for T = 0, 125 A - ~ for T = 1). In comparing with the work of Krewald and Speth 3a), we see that the results for 9°Zr and 2°apb agree quite well, but their 2 ÷ in a°Ca has a structure between 22 and 25 MeV which is not present in our calculation. The case with 160 is somewhat complicated. In their recent analysis of the 160(7, no) 150 reaction, Wang and Shakin 45) have assumed a broad E2 resonance in the giant dipole region in order to explain the angular distribution and the polarization data. A study of 12C(~, ~,o)160 by Snover et al. 26) revealed a T = 0, E2 strength in the region E x = 12 to 28 MeV which constitutes ~ 17 ~ of the energy-weighted sum rule in the % channel. Other measurements on the polarized proton capture reaction lSN(p, ~0)160 by Hanna et al. 26) indicate a broad E2 resonance which lies above the giant E1 resonance and the E2 radiation for the (7, Po) channel of 160 exhausted ~ 30 ~ of the isovector quadrupole sum rule which is only a half of the prediction of the neutron data analysis. On the other hand, measurements by Hotta, Itoh and Saito 26) on the electroexcitation of 160 show that in the region of 20--30 MeV excitation the strength exhausts approximately 20 ~ of the sum rule if one assumes an E2 resonance, while the strength in the region below 20 MeV exhausts 43 ~ of the same sum rule. Our calculation shows a single peak around 20 MeV excitation (if experimental single-particle energies are used, thepeak is shifted to 21.5 MeV) which exhausts ,~ 65 ~ of the isoscalar E2 sum rule and this is a common feature for all the existing Skyrme forces with different velocity and density dependence 40). This agrees fairly well with the recent inelastic 0t-scattering experiment
402
TABLE 11 Distribution ofisoscalar quadrupole strength
Nucleus
State
s.p.u.
% EWSR
EWSR (MeV -fm 4) (double commutator)
160
19.5 25-40
6.1 t.2
61.6 ( + ) 19.3 80.9
4.55 x 102
4°Ca
16 21-30
12.2 1.1
83.6 ( + ) 12.1 95.7
1.87 x 103
9°Zr
5 13.5 20-30
3.1 11.0 0.9
7.8 70.0 ( + ) 10.3 88.1~
5.33 x 103
2°SPb
5.6 11 15-25
9 12.4 1.3
25.2 56.7 ( + ) 10.4 92.3
1.65 x 10"
TABLE 12 Distribution of isovector quadrupole strength % EWSR
EWSR (MeV. fm 4) (double commutator)
Nucleus
State
s.p.u.
160
15-24 26.5 31 33 37 39.5-65
0.5 0.8 0.8 0.9 1.4 1.5
5.0 9.0 10.8 13.0 23.3 ( + ) 30.5 91.6
5.38 x 102
*°Ca
0-30.5 33 36-55
2.9 3.7 1.8
25.4 43.5 ( + ) 27.7 96.6
2.31 x 103
9°Zr
0-25 25-30 31.5 34-50
2.4 3.8 3.5 2.1
13.4 26.2 34.3 ( + ) 21.3 95.2
8.56 x 103
2°spb
0-20 23.5 26.5 28-50
0.9 7 3.5 2.5
12.6 46.0 22.6 ( + ) 17.7 98.9
3.34 x 10"
403
GIANT MULTIPOLE RESONANCES
by Breuer et al. 52) in which they find that the giant isoscalar E2 resonance centered at 20.7 MeV with a half-width o f 7.5 + 1 MeV carries ~ 65 ~ of the EWRS. It has been known for a long time that quadrupole transitions in 160 to low-lying states, now interpreted as mainly 2p-2h, 4p-4h in composition, exhaust a good fraction o f the isoscalar sum rule. These states, whose presence at low energies is intimately connected with the easy deformability of 160 in excited states, lie outside of the framework o f the present calculation, so we would not expect detailed agreement between our theory and experiment in light nuclei.
f~
ISOSCALAR OCTLPOLE
Fig. 6. Isoscalar octupole strength. Experimentalresults of ref. 21) are used for 2°SPb (dashed line). The arrows indicate the experimental positions of the low-lying states.
]SOVECTOROCTUPOLE >.
e "
21- /
,1- ~ 1 - /
lo ,o
~o ~o
3o
,o
~o/I,o
so
,o
,o -
-
,I-
,o
10
20
30
40
,o
50
60
60
I~liV
Fig. 7. Isovector octupole strength.
so
7~
,o
404
K. F. LIU AND G. E. BROWN TABLE 13 Isoscalar octupole RPA
nucleus 4°Ca 2°aPb
E x (MeV)
Experiments s.p.u.
~ EWSR
E x (MeV)
10-25
13
25
10-25 13.5
20
11
41
19
s.p.u.
10
~o EWSR
reaction
ref.
37 30
(e, e') (p, p,)
14) 13)
44
(e, e')
2a)
TABLE 14 Distribution of isoscalar octupole strength ~o EWSR
EWSR (MeV- fm 6) (double commutator)
Nucleus
State
s.p.u.
160
6.3 12.5 21 24-30 31.5 33 37 42-62
4.0 2.2 3.3 1.6 2.2 1.2 4.4 1.6
3.9 4.5 10.8 6.6 10.4 6.3 25.7 ( + ) 6.2 74.4
9.87 x l0 a
4°Ca
3.5 10 14 16-27 31 35-60
21 7.5 2.3 2.9 7.5 3.3
11.0 11.3 4.8 8.6 35.5 ( + ) 20.3 91.5
6.18 x 104
9°Zr
1.35 7-8 12.3 15-22 26 30-40
27 14.2 2.6 3.1 8 2.5
6.6 17.7 5.8 10.3 36.9 ( + ) 15.1 92.4
2.66 x 105
2°sPb
2.8 7 10-17 20 25-35
37.6 7.3 5.3 10.6 1.7
18.1 8.9 13.2 41.0 ( + ) 9.1 90.3
1.36 x 106
GIANT MULTIPOLE RESONANCES
405
5.2.3. Octupole resonances. Examining the systematics of the 3 - , T = 0 strength in fig. 6, we find that the strength is more spread out for light nuclei and starts to gain some collectivity in the lowest mode for heavier nuclei. The results in table 14 would support the speculation 27) that about one-half of the energy-weighted sum rule is exhausted by the low-lying 3 - states and the lhco group o f states, leaving the remaining one-half for the 3hco group o f states. In fact, 3 - continuum strength in 4°Ca and 2°Spb has been reportedly seen 13,14,17). The experimental results and the calculated counterparts are listed in table 13. TABLE15 Distribution of isovector octupole strength Nucleus
State
s.p.u.
~ EWSR
EWSR (MeV • fm6) (double commutator)
160
0-10 10-17 17-29 29-35 35-45 45-80
0.6 0.7 2.4 1.6 6.4 4.5
0.6 1.4 7.7 7.4 36.8 (+) 35.0 88.9
1.09 x 104
4°Ca
0-8 8-28 28-47 4%70
0.6 4.7 11.2 3.7
0.5 11.1 58.4 (+) 27.1 97.1
7.17 x 104
9°Zr
0-5 5-27 27-46 46450
0.2 5.3 15.7 2.7
0.1 10.3 70.4 (+) 16.5 97.3
4.04 x 105
2°SPb
0-19 19-29 29-38 38-60
6.9 7.6 17.4 3.4
7.4 19.1 58.0 (+) 13.7 98.2
2.62 x 106
While the 3 - , T = 1 giant resonances have not been identified so far, it is quite possible that the excited states at 33 MeV in 2°apb and 33.5 MeV in 197Au in the (e, e') experiment 25) could, in fact, be the isovector 3 - states 46) (fig. 7 and table 15). 5.2.4. Hexadecapole resonances. According to the shell model, hexadecapole resonances should occur at excitations of 0hco, 2hco and 4hco. As is shown in fig. 8, very little strength is allocated to the bound states (0hog), the energy-weighted sum rule for T = 0 is almost evenly divided between the 2hc~ and 4hco group of states. These
406
K . F . LIU AND G. E. BROWN
ISOSCALAR HEXADE~,POLE
,t-- ~1-/Jl'°/I
I1'°
rlf~,o
,o
1(] . .
,:
IJ
,o
2 -/ v" \ . . 10
~o
rl/~l~^
/5, 20
NI
,o
J-I
,o
,o
,o
c,~O
,o
60
p~,o,
30
40
50
M,V Fig. 8. Isoscalar hexadecapole strength.
I
.
~
~POt£
,2
.
h
AA
>
^
o'~
Ca
_ 20
"
.
I1"
Fo2oe
40 60 HW Fig. 9. Isovector hexadecapole strength.
are consistent with calculations by others 28). The distributions of the strength are presented in tables 16 and 17 and figs. 8 and 9. 5.2.5. Monopole modes. The transition density of the breathing mode,
pno(r) = ( ~l,I ~, ~(r- ri)[~l o) i
....
(28)
resembles that from the collective model of Tassie29). It is mostly localized on the surface where the interaction is very different from that of the interior.
GIANT MULTIPOLE RESONANCES
407
TABLE16 Distribution of isoscalar hexadecapole strength EWSR (MeV • fms) (double commutator)
Nucleus
State
~o EWSR
160
0-34 34-40 40-75
24.1 10.9 (+) 54.8 89.8
2.44 x 105
4°Ca
0-33 33-60
40.6 (+) 49.4 90.0
2.13 x 106
9°Zr
0-25 25-60
35.7 (+) 48.6 84.3
1.30 x 107
2°apb
0-9 9-20 20-50
12.7 29.1 ( + ) 44.2 86.0
1.06 x 108
TABLE 17 Distribution of isovector hexadecapole strength EWSR (MeV • fms) (double commutator)
Nucleus
State
~ EWSR
160
~ 40-80
19.5 (+) 63.8 83.3
2.57 x l0 s
4°Ca
0-30 30-70
13.6 ( + ) 76.6 90.2
2.34 x 106
9°Zr
0-33 33-60
19.3 (+) 58.4 77.7
1.87 x 107
2°apb
0-25 25-70
15.7 (+) 84.2 99.9
1.96 x 108
Recently, b o t h M a r t y et al. 56) in their inelastic d e u t e r o n scattering e x p e r i m e n t a n d T o r i z u k a et al. 56) in their inelastic electron scattering e x p e r i m e n t have tentatively assigned the r e s o n a n c e at ~ 13.2 M e V in 2°Spb to be the b r e a t h i n g mode. O u r m o n o p o l e r e s o n a n c e occurs at m u c h high energy (18.2 M e V in 2°apb), as c a n be seen f r o m fig. 10 (table 18). This is "mainly due to the fact that o u r i n t e r a c t i o n yields a n
408
K.F. LIU AND G. E. BROWN ~S(~CJU.AR M0XOPO~ (BREATH~iO ~ )
,j
'
~
~
o.
~"~--
fV,, Zr w
I ~ , O
/~,o
10
,o
20
30
,o 40
M,,V
Fig. 101 Isoscalar monopole strength (arbitrary unit).
ISOVECTOR MONOPOLE
I
60
/ 10
20
30
40
50
60
MeV
Fig. 11. Isovector monopole strength (arbitrary unit).
extremely high compression modulus in nuclear matter (378 MeV). If we assume a quadratic relation between the monopole energy and the compression modulus in nuclear matter [it is found by Blaizot et al. 44) in their analysis that this relation indeed holds very well for the various interactions], we would find a 200 MeV compression modulus in nuclear matter corresponding to the experimental findings in finite nuclei, a value close to the upper bound o f those generated by the realistic
.forces 57).
GIANT MULTIPOLE RESONANCES
409
TABLE 18 Distribution of isoscalar monopoles strength Nucleus
State
Strength (fm 4)
% EWSR
EWSR (MeV. fm 4) (double commutator)
160
15-20 20-25.5 25.5-29 29-35 35-50
16.6 23.2 12.8 22.5 5.8
12.7 23.5 15.0 31.2 ( + ) 10.7 93.1
2.29 x 103
4°Ca
20 22-30.5 30.5-35 35--40
63.5 238.8 33.7 3.6
12.7 67.2 11.5 ( + ) 1.4 92.8
9.42 x 103
9°Zr
19.5 20.5-31 31~,0
260.2 812 40.9
18.5 71.3 ( + ) 5.2 95.0
2.68 x 104
2616.5 1329.6
54.4 ( + ) 33.4
8.3 x 104
2°aPb
18.2 20.2
87.8
It has been pointed out 30, 31) that at least part of the Nolen-Schiffer anomaly is attributable to the isovector monopole mode coupled to the valence particle. The relation can be established by assuming the Coulomb potential to come from a square distribution of charge Ze2~ 1
Vc = ~ - ~
~(1 - %,)(3 - r2/R2).
(29)
The z-dependent part of Vc gives the isospin breaking in density Pn-Pp, which is the T = 1 mode. The change in energy from isc;spin breaking of the 4°Ca core is dE,,ca =
2('*' Ca[ V~I~b~.%° ) ( ~b~.%°[Vc['*' C a ) E o - E~
(30)
This is shown diagrammatically in fig. 12 where the particle-hole Green function G carries zero energy, i.e. G(0), because the interaction with the valence particle is instantaneous, and where V~ is the isovector part of the interaction. The isotope shift concerns the change in the expectation value of 2
1
1
rch = ~ ~ ( 1
- zs,)r 2.
(31)
410
K . F . LIU AND G. E. BROWN
'",~~.~)-~--C) /
V,
/ Fig. 12. Core polarization.
The contribution from the isovector mode to the change in r~h,2 upon addition of a neutronto 4°Ca, is 2<41Ca1~ 1 2 s=0 J=o 41 ~%ir/I@r=l><@r=llV~[ Ca> 2 i (32) Z(6rcn),,ca = E o - E¢,
Comparing (30) and (32), one finds
(AE),lca--(AE),.~o
Z 2 e 2 (6rcn),'sc 2 2 - (6r~h), ,ca -
2R
R2
(33)
Some preliminary calculations show the results to be sensitive to the position of the node in the transition density of the isovector mode. The isovector monopole transition density in the Green function formalism is defined as
= F1, L= Im ('s+n G(r, r, E ) d E 1~,
(34)
and is sketched in figs. 13 and 14. 1
~T=I
no x
p~l
r
r (fro)
Fig. 13. Theisovectormonopoletransitiondensity and the root-mean-square radius of the valence particle and hole.
x
r2
i I I
t t
Fig. 14. See caption to fig. 13.
r(fm]
GIANT MULTIPOLE RESONANCES
411
The ideal case would be like in fig. 13, where the node lies nearer to the rms radius of the valence hole than that of the valence particle. This will cut down the anomaly of the particle mirror nuclei to the same size as that of the hole mirror nuclei with the remaining anomaly attributable to other mechanisms, for example, charge symmetry breaking. On the contrary, we find that the result is too small to resolve the anomalies for the 41Ca -41Sc and the 170_17 F pair but too large for the anomalies of the 39Ka9fa and the 15N_150 pair. This is simply because the rms radii of the f~ (in 41Ca-41Sc) and the d~ (170_i 7F) valence particle lie nearer to the node of the calculated transition density than those of the d~ (39K-39Ca) and the p½ (15N_150) valence hole (fig. 14). Therefore, it seems plausible to suggest that the combined velocity and densitydependent symmetry potential is too repulsive and asa result, the isovector monopole vibration is localized too far out on the surface. Yet on the other hand, we find that the excitation energy gives a 162 A -~ (table 19 and fig. 11) dependence on the mass number which is smaller than the prediction of the hydrodynamical model (169 A -~) and much smaller than the recent experimental findings by Pitthan et al. 25) where they tentatively established 195 A -~ systematics. Therefore, it is hard to establish a conclusion at this stage. 5.2.6. Low-lying states. Listed in table 20 are the positions of low-lying states located by tracing the poles of the Green function. TABLE 19 Distribution of isovector monopole strength ~ EWSR
EWSR (MeV- fm 4) (double commutator)
Nucleus
State
Strength (fm 4)
160
15-30 30-38 38--60
26.4 26.7 13.4
22.7 34.6 ( + ) 22.3 79.6
2.7 x 103
4°Ca
14-22.5 22.5-33.5 33.5-38.5 38.5-60
52.4 189.4 96.5 55.3
8.5 26.9 29.5 ( + ) 21.7 86.6
1.16 x 104
9°Zr
13-23.5 23.5-30 30-38 38~0
225.6 206.6 602 14.5
10.7 13.3 47.2 ( + ) 15.4 86.6
4.31 x 104
2°sPb
7-17.5 17.5-24 24-34.5 34.5
646.5 1227.1 3061.2 577.3
5.1 15.7 52.7 ( + ) 13.0 86.5
1.68 x l0 s
K. F. LIU A N D G. E. B R O W N
412
TABLE 20 Positions of low-lying states (MeV) L ~ (T = 0)
Nucleus
Exp.
Skyrme BLV1
2+ 33334+ 55-
2°apb 160 #°Ca
4.07 6.13 3.74 2.74 2.61 4.31 4.48 3.2
5.6 6.3 3.5 1.35 2.8 6.4 5.0 3.4
9°Zr 2°spb 2°apb 4°Ca 2°spb
TABLE 21 Transition rate in Weisskopf units
") Ref. 32).
L" (T = 0)
Nucleus
Exp.
2+ 33334+ 5-
2°aPb 160 4°Ca 9°Zr 2°spb 2°spb 2°apb
8 ") 14 b) 25 c) 32 d) 32 a) 15 e) 14 ")
b) Ref. a3).
¢) Ref. 34).
d) Ref. 35).
Skyrme BLV1 9 4 21 27 38 16 14 ") Ref. 36).
Low-lying 2 + states in nuclei with low-lying 0 ÷ states are not seen in this calculation based on the ground state, because these states will not be given by the lp-lh calculation. Even-parity states are reproduced less well than the odd-parity states. Using empirical single-particle energies will lower the states to about the experimental values. But the discrepancies are still larger than the case of odd-parity states. The electromagnetic transition rates for the low-lying states are listed in table 21. The units are the single-particle Weisskopf units. B ( E L , 0 --* L ) = e 2S w
2 L + 1 ( ~ + 1)2(1.2A~)2L"
(35)
The small transition rate of 3- in 160 is even smaller than that of an earlier calculation by Gillet and Vinh Mau aT) which is about 8. It is suggested that the alphaparticle model with some J = 3 deformation in the ground state of 160 could probably resolve part of this anomaly 40).
GIANT MULTIPOLE RESONANCES
413
6. Conclusion
We have learned that the effective particle-hole interaction extracted from a reasonable energy density of a finite Fermi system does reproduce the qualitative excitation properties, and that the response function technique in coordinate space proves very handy to use, especially in heavier nuclei where large configuration space is needed to push the giant dipole states to sufficiently high energies. We have shown that part of the long-standing discrepancy between theoretical and experimental positions of the giant dipole resonance in 2°spb arose from neglect of the continuum in the early theoretical calculations. As a result, most of the giant resonances are reproduced within 5 ~o agreement with experiments. The transition rates of the low-lying states are in general within 12 ~o of the experimental values except for the 3- state in 160. Most of the energyweighted sum rules calculated from the double commutators are exhausted to within 13 ~ by calculated contribution from the response function with proper cutOffs. This also provides a check for self-consistency. The agreement we achieve in detail with experiment is, in many cases, not as good as that in calculations which employ empirical particle and hole energies. We believe there to be advantages, however, in carrying out calculations in a consistent way, in which single-particle energies and residual interactions are obtained from the same underlying force. The complexity of the calculations has necessitated the use of a highly schematical, Skyrme-type interaction. On the whole, this interaction does well for the vibrations of natural parity which we calculated. We achieve reasonable agreement with results from the collective model, which is not surprising, since these could also be derived from knowledge of the ground-state energy as a functional of neutron and proton occupation numbers. The agreement is interesting, however, because collective models use somewhat different spatial boundary conditions. It should be noted that our calculations were for natural-parity states, and there is no expectation that our Skyrme force would be good for unnatural-parity states. In general, our results are comparable with other calculations by Krewald and Speth as) and by Krewald et al. 39) where a RPA model with a density dependent interaction and single-particle wave function from a Woods-Saxon well are employed. Our results for the giant resonances are also consistent with other self-consistent RPA calculations by Bertsch and Tsai 40) using the Skyrme force I and II, except for the giant dipole states. We would like to thank Dominique Vautherin for the original suggestions motivating this work, and for use of the Hartree-Fock programs. Thanks are due to G. Bertsch and S. F. Tsai for kindly lending their computer programs and their stimulating discussions. One of the authors (K.F.L.) wishes to extend acknowledgements to G. Ripka and M. Rho for their encouragement and hospitality during his stay in Saclay where part of the work was carried out.
414
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415