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Scaling- and antiscaling-type oscillations in isoscalar and isovector nuclear monopole vibrations
Nuclear Physics A485 (1988) 233-257 North-Holland, Amsterdam
SCALINGAND
AND ANTISCALING-TYPE ISOVECTOR
NUCLEAR
S.S. DIMITROVA,
I.Zh.
OSCILLATIONS MONOPOLE
PETKOV
and
M.V. STOITSOV
Instiiute for Nuclear Research and Nuclear Energy, Bulgarian Academy Received (Revised
IN ISOSCALAR
VIBRATIONS*
ofSciences,Sofia,
Bulgaria
29 January 1987 16 March 1988)
Abstract: Isoscalar (T = 0) and isovector (T = 1) giant monopole resonances are studied using a localscale version of the ATDHF theory developed on the basis of a rigorous energy-density functional approach. Due to the strong coupling between the bulk and surface density vibrations, the monopole collective motion is split into four normal modes. Two of them, lower in energy, correspond to scaling-type density vibrations. The other two are of antiscaling-type in which the nuclear surface oscillates opposite in phase to the scaling-type vibrations. Excitation energies, transition densities, T = 0 and T = 1 energy weighted sum rules and other properties of breathing even-even nuclei are calculated using different Skyrme-type effective forces. The strong sensitivity of the antiscaling-type vibrations to the particular form of the approximate energy-density functionals is demonstrated.
1. Introduction In recent years, much effort has been devoted
to the experimental
and theoretical
study of nuclear giant monopole resonances (GMR) I). Presently, the breathing mode has been established experimentally for about fifty nuclei over the whole mass table l,‘). Microscopic approaches such as the random phase approximation (RPA) 3-7), the generator coordinate method (GCM) ‘-‘O), and the adiabatic timedependent Hartree-Fock (ATDHF) method 1’-‘3) have been applied to the theoretical study of giant monopole resonances (GMR). General features of GMR have been successfully described by fluid-dynamical approaches 14-i7) with a few parameters related to bulk and surface nuclear characteristics. The experimental data of isoscalar giant monopole resonances are usually described
within the distorted-wave
Born approximation
(DWBA)
i8) using a simple
scaling model of the breathing vibrations ‘*19). Similar transition densities follow from the constrained Hartree-Fock (CHF) method 20) where the nucleus is constrained to have a given mean squared radius. The analysis of the experimental data from both (Y- and d-inelastic scattering with excitation of GMR in 208Pb shows, however, that the appropriate transition density is of the form 21*22) pTr3(r)
’ Work partially
supported
=
6bap(r;
by Bulgarian
R, b)/ab+GRdp(r; Science
03759474/88/$03.50 @ Elsevier Science Publishers (North-Holland Physics Publishing Division)
Committee, B.V.
R, b)/aR Contract
No. 325.
(1.1)
234
S.S. Dimitrooa et al. / Scaling and antiscaling oscillations
where p(r; R, b) is the ground-state density distribution. Such a transition density pTr3 is quite different from that of the scaling model since in general (6R/ 6b) # (R/b). A surface mixed
monopole
to some extent
mode
(diffuseness
(0 = arctan(aR/?%))
oscillations)
is included
with a bulk mode
in eq. (l.l), (half-radius
being oscilla-
tions). In the present paper, attention is undertaken to relate quantitatively the GMR, as dynamically coupled bulk and surface vibrations of the breathing nucleus, back to the effective nucleon-nucleon interaction in nuclei. Some stimulation for such a theoretical interpretation of GMR is provided by the recently reported experimental data 23). They allow an antiscaling-type mode associated with O-arctan( 6R/6b) < 0 to be investigated. There are three papers 17*24,25)in which the interplay between surface and bulk vibrations in nuclei has been considered with application to GMR. In ref. 24), the spin-orbit interaction and the corrections to the Thomas-Fermi (TF) kinetic energy have been omitted. Therefore, as mentioned in ref. “), a quantitative agreement with experimental data could not be obtained. In refs. 17,25)only the isoscalar GMR have been considered. The aim of the present work is to investigate in a unified way the main features of the scaling- and antiscaling-type vibrations in both isoscalar and isovector giant monopole resonances. We start from a rigorous energy-density functional (EDF) approach for constructing approximate energy-density functionals E[p] within a given trial orbit 0 of the Hilbert space X with respect to a local-scale point transformation group (sect. 2). In sect. 3, we present a local-scale version of the ATDHF approach ‘i-13). Assuming the constrained densities pn(r, t) and pp(r, t) to be of the symmetrized Fermi (SF) density form with a time-dependent set of parameters u(f) = (R,(t), RJ t), b,(t), bp( t)) we have a natural choice of collective variables for describing small density oscillations around the HF ground state. Numerical results for a number of even-even nuclei obtained with Skyrme forces are given in sect. 4. Summary and conclusions are made in sect. 5. 2. Ground-state 2.1. A RIGOROUS
ENERGY
energy density functional
DENSITY
FUNCTIONAL
description
APPROACH
In previous papers 26-28), the local-scale point transformation (LST), r + r’ =f( r) = (r/r)f(r), has been used. It depends on an unknown scalar LST function f(r). A set of invertable transformations of this type forms a LST group @. Its isomorphic group Ou, of unitary operators s, E %% produces local-scale wave function transformations %(r,
. . . r,4)+(rl
. *. 4
=
(2.1)
S.S. Dimitrova
et al. / Scaling and antiscaling oscillations
r,J belongs where +=+(r,,..., system with a hamiltonian fi. For any given function
to the Hilbert
space
&E X (the so-called
WI = 02l,*
= %(h,.
235
X of the A-particle
model
function
. . , rd+(rl,.
. . , rd
nuclear
‘“)) the functions (2.2)
form an orbit 6’ = {%f$,f~ 9”) of X’ with respect to 9. An one-to-one relationship between LST functions f(r) and local densities associated with $[f] E Q can be achieved in a given orbit 0 c X -_ =f2(r) ““)
p(r)
Here, the model
density
r*
distribution
p(r)=A
corresponds
p(r)
giving
that density.
the wave function
Thus,
. . , rA;
[PI)
=
= wNm4
J%Jl
I$E 0 (2.4)
been analyzed function ).
value
an energy
to the model function
equation with respect to f(r) which r ) is obviously one-to-one related to
%[p](rl
the expectation
(2.5) becomes
(2.3)
rA)12dr2*-.dr,.
In fact, eq. (2.3) is a first-order differential we solve numerically. The solution f[~] =fi,,,( the local density p(r). Its particular form has given density p(r) there is an A-particle wave +[P] = (cl(rl).
.
i3r
I$(r,r, ,...,
I
p(f(r))
p(r)
in refs. 26,27). For any
. . , rA)&(rl
).
. . , *A),
of the nuclear
density
hamiltonian
(2.5)
fi on
functional
(2.6)
= UPI + VP1 .
It is important that the wave function (2.5) allows the kinetic T[p] and potential V[p] energy in (2.6) to be obtained in an unified way as local density functionals in O= ZY,i.e.,
7bls
(~[~llfl$bl)
= (h2/2m)
UPI = (+bll ~I$bl> = In the hamiltonian fi = ?+ operators, respectively.
2.2. HARTREE-FOCK
Let us consider Slater-determinant
ENERGY
t
J
f are kinetic
DENSITY
FUNCTIONAL
energy-density 6 = C$sL which
hl
dr,
(2.7)
(2.8)
E~&I dr.
? and
an approximate model function
I
and two-body
potential
energy
functional E[p] obtained by a involves given model s.p. wave
236
S.S. Dimitrova et ai. / Scaling and antiscaling oscilIa?ions
functions (Q,(r), i = 1,2,. . . , A). The density-dependent 0 SL={021,.~sL,f~ S}C X has the form2’) llrEP]=9(rI,...,r,4;
[PI) = (A!)-“* dct \rpAq; Cp])I,
with local-scale transformed
wave function
(2.5) in
(&j = I,. . . , A)
(2.9)
s.p. wave functions (2.10)
They depend on the LST function &,(r) which is the solution of eq. (2.3). The model density distribution (2.4) is expressed in 6&c X as (2.11) The kinetic-energy only two terms
density in (2.7) depends on the orbit Os,c Z and contains
Q3 = G.&PI+ ~&Cl *
(2.12)
The first term T&P] is the original Weizsaecker term *9) while the second is of the form 30) (2.13) For the most general Skyrme-type forces and time-reversal invariant systems the potential-energy density E&‘P] in (2.8) is defined in fys,c X as +Xlpl-4~0[(~
+&o)P2 -(~o+4)(p~+p2,)l+ht~p*[(l+~X3)p2-(X~+t)(p2nCp2p)l
+~[tl(l+~x,)+t,(l+~x,)]pl-[p]+$~fZ(X~+~)-t,(X,+f)~ x (Pn7n[P1+Pp7pEPI)+~E3t,(l+tx*>-- f2(1+$~2)1ow’ --&i’3f,(~,+1, + G%+3)1~m”)2+
o%,Yl
+~W~(J.OP+J,.V~,+J,*V~,)+EC~~,[PI
(2.14)
where to, t,, fZ, f,, x0, x,, xz, x3, a and W. are parameters of the Skyrme forces (see the notation of ref. “I)). The spin density is also a local density functional
Jbl E J(r; [PII= ? dr; bYI)!*wf(r; [PI).
(2.15)
i=l
The index n (p) in (2.14) refers to neutron ~proton~ densities and restricts the summations to neutron (proton) occupied states in formulae (2.11), (2.13) and (2.15). The Coulomb energy density E~~~,[J-I]in (2.14) is the sum of the direct term and the exchange term, the latter taken in the weli-known Slater approximation 32).
S.S. Dim&ova
In what follows we consider 0g;O.c 2, functions. teristics
et al. / Scaling and antiscaling oscillations
a harmonic-oscillator
237
single Slater determinant
orbit
{g,(r)} to be harmonic-oscillator shell-model s.p. wave i.e. assuming In this case the self-consistent HF results 33) for average nuclear charac-
(binding
tics sensitive reproduced
energies,
local densities,
with a quite satisfactory
2.3. DENSITY
r.m.s. radii, etc.) as well as the characteris-
to shell effects (s.p. spectra,
VARIATIONAL
shell oscillations,
etc.) are
27).
CALCULATIONS
For spherical nuclei we have chosen p,(r) densities to be of a symmetrized p(r)‘=p’(r;
local-density
accuracy
in refs. 30,34,35)the neutron p,(r) Fermi-type (SF) radial form
R,, b,) =~04 sh (R,lb,)l[ch
(R,lb,)+ch
where the index q stands for neutrons (q = n) and protons condition is satisfied through the relation
~04 = W,/4&/[1
(q =
and proton
(r/b,)1
,
p). The normalization
(2.17)
+WqlRq)21,
where the number of neutrons (protons) is denoted N+Z. Eq. (2.17) and the r.m.s. radius expression
as A,=
(2.16)
N (A,=Z)
and A=
%~=(r2p4)=~A,R~[l+~(rrb,/R,)2]
(2.18)
are exact relationships when py Coulomb energy density .zCou,[p]
is represented by eq. (2.16). Furthermore, the in (2.14) can also be represented in a closed
analytical form 36). Inserting (2.16) into (2.6)-(2.8)
and (2.12)-(2.15)
the functional
E[p]
an algebraic function E (R, , R,, b,, bJ of the four density variational II = (R, , R,, b,, bJ. We minimize E(u) = E (R,, R,, b, , b,) by solving the following system of non-linear algebraic equations aE(u)/dq using the autoregularized in ref. 38).
= 0,
Gauss-Newton
i = 1,2,3,4 iterative
becomes
parameters numerically
(2.19)
process “) and the code described
As already reported in ref. 30) the obtained results are in satisfactory agreement with respect to the self-consistent HF ground-state results 39). Therefore, on one hand, the wave function $[psF] (eqs. (2.9) and (2.16)), allows us to reproduce the HF ground-state results quite well. On the other hand, t+b[psF] directly depends on the density parameters II = (R,, R,, b,, bJ as natural collective coordinates and can be used within a number of collective methods as, for example, the GCM or the ATDHF method.
SS. Dimirroua cf al. / Scaling and antiscaling oscillations
238 3.
3.1. LO&AL&ZALE
Locat-scale
VERSfON
dem-iption of monopole vjbr~~ia~s
ATDHF
OF THE ATDHF
AP~RUA~H
F~~~~wi~~ closely the notations of refs. ‘iX1’ ), the classicat harnj~t~~~a~ of the nuclear system under c~~~ider~ti~n can be expressed in the ATIXIF ~~proxi~a~~~~ l3S
(3.2) and (3.3)
2’=&W)lfilW)>
are the collective kinetic and potent&I energies, respectively, An approximate approach to avoid the solution of the ATDHF ~~~ati~n “) is supplied by using s.p. functions {pipi(t; t)f which satisfy the gen~raIi~ed scaling property I’), This could in principle be done with the help of additional timedependent external fields within the CHF method. It appears, however, more appealing and at the same time easier to use the present e~ergy~d~us~t~ functiona‘t ~~~~~ac~. ‘l’?~ebasic. idea is to start fram a tile-de~ende~t density dist~b~tia~s p(~, f) = P,(P) ~~~stra~~ed EO depend on the tiime I Ety means of a few ~~l~-~h~se~ density ~ha~~ter~sti~s ti = n(t) = (q(f), rz252(f), . . _>, e.g. density rn~rne~ts~ deformation parameters and so an, For any given values of u(t), the density p,,(r), must be then obtained by rni~~i~li~~~gE[p,(r)] with respect to p,(r) within a density variational space in which all densities give the same values of u(t). Such a minimization of the expectation value I?[p, ( P)] of 2 in ossLc 2? corresponds to a specific density CHF problem in which the densities are constrained to have given nuclear characteristics u(t) at each moment f. Inserting pU(t) into eqs. (2.9) and (2.10) we obtain the time-dependent wave function 44 t) f 44&J = (A Yr”
det iFit%, t)f ,
fi,j=l,.,.,A)
(3s)
which involves the s.p. wave functions
of eq. CZ.3) with the ~~~sit~ p,(r). For even-even f3S) is time-even and a family of tirne~e~~~ one-body
with flewI(r) being the solution
nuclei the Slater dete~i~ant
S.S. Dimitrova et al. j Scaling and antiscaling oscillations
239
density matrices pO(t)
can be applied to simplify the ATDHF approach. Further, it is important that the time-dependent the generalized scaling property y5;9(r,t) = --uq * Vp’(r,
s.p. wave functions (3.6) satisfy
t) -#(r,
(3.8a)
t)V ’ t$
with real vector functions (3.8b)
r+ = uy(l; t) = -~E$~:(r)/f~:(r)l,
where jPf(r) =dj&r)/dt and fi$r) = r. * Vf,;;(r). As well known 14), under the conditions (3.8), all nucleons are forced to move in a common (for each kind of nucleons, 9 = n, p) velocity field v, and the collective kinetic energy (3.2) is related to the current density in ATDHF approximation I*) as
Inserting (3.8) into (3.2) and using eq. (3.9) we obtain after a partial integration and some rearrangements the expression (3.10)
As shown in “), the ATDHF equation of motion leads to the continuity equation ~~(r)+(~/~)V
*J,(r, t>=O
(3.11)
with a generalized current density Jg = J,(r, t) =.Qr, r>+P~iC&(r,
r) -iC_Lj(r,
f)l
(3.12)
where 4 = (p, n) when 9 = (n, p) and the non-locality parameter @= (m/2~~}~~~i- t2) comes from the exchange Skyrme forces. On the other hand, the time-derivative Pz(r) entering eq. (3,1X) can be obtained by the equation PZ(rI+V
* Cd(rb,l
=O,
(3.13)
which follows after taking into account eqs. (3.7) and (3.8). Comparing (3.13) with (3.11) we obtain the relation V *Id(rh,l=
(~/~)V %,+BtX_iJr, ~~-~tj,.&(~, f)lI
(3.14)
which allows the current density &(r, t> to be expressed by means of the local densities pi(r) and the velocity fields ug, q = (n, p).
240
S.S. Dimitrova
In the case of breathing and the relation
et al. / Scaling and antiscaling oscillations
vibrations
nuclei both o, and& are irrotational
m + pt.
ql+PP:
pip,”
hj,=p
where pu = p; i.e. V~= r,,v,(r,
of spherical
(3.14) gives
l+ppu
uq+pU l+ppu
Due to the spherical
symmetry,
(3.15)
uq
U, has a radial component
only,
t), and eqs. (2.3) and (3.8) yield
r (ap:/&)rr2 vq(r, t)=
I
O
dr’
pzr2
Inserting (3.15) and (3.16) into (3.10), the collective form of a local density functional ZJ;+p---- p”upt: l+PP”
(3.16)
’
zl,u,+
kinetic
l+PP: ___ 1+PPU
(pfy
The same holds reasonably for the collective potential be identified with the energy-density functional
energy
X takes the
dr.
ZI’,
(3.17)
I
energy (3.3), (3.5) which can
"Irr_Pul= HPUI = (~~Pull~l~cI[Pul)~
(3.18)
A number of approaches exist which assume the internal energy in the dynamical case to be static TF-like approximate energy-density functionals of the time-depensince different approximadent density r5-“). Such a procedure may be questioned tions are used in determination of both X[p] and V[p]. In this respect, the local-scale version of the ATDHF approach is more rigorous than the irrotational dynamics, although we also assume irrotational flow. In fact, the local-scaling both the kinetic and the potential collective energies to be obtained and hence the inconsistency mentioned above to be avoided.
3.2. MONOPOLE
Let us assume
VIBRATIONS
AS NORMAL
that the local density
ONE-PHONON
p(r)
applied
EXCITED
to breathing
fluid allows
simultaneously
STATES
spherical
nuclei
is
represented by the SF density distribution (2.16) with a time-dependent set of parameters u = u(t) = (R,(t), R,(t), b,(t), bp( t)). Inserting eq. (2.16) into eqs. (3.17) and (3.18) we obtain the ATDHF classical hamiltonian
X=5
,$ m,(u)ti,ti,+clr(u),
(3.19)
‘.I 1
where ri = (&,, &,, d,, 6,) is the time derivative of u = (R,, R,, b,, b,). The matrix elements mq(u) of the inertial tensor M(u) in (3.19) are given as PP~P4,,(1-~,9,)+P~(1+PP~)s,,,, (1 +PPu)
1viv, dr,
(3.20)
S.S. Dimitrova
et al. / Scaling and antiscaling osciliati~ns
241
where the isospin index 4 = n (p) for i = 1, 3 (2,4) is ordered to correspond to the index i = 1,2,3,4 labeling the collective vector u = (u, , u2, u3, u,) = (R,, R,, b,, bJ. Taking into account eqs. (2.16) and (3.16) we express the velocity geld U, in (3.20) (measured in units “ii= dUi( t)/dt) as
(a~~/a~~)r’~dr’ Uj(P,t>=
p%r’
(3.21)
*
In the case of SF densities (2.16), the collective potentia1 energy (3.18) is represented as an algebraic function of collective variables (3.22) whose explicit expression is well defined in Ok;O.c X by the energy-density functional (2.6)-(2.8) and (2.12)-(2.15). Furthermore, the ATDHF hamiltonian (3.19) allows one to apply the standard method of quantization ‘I). We solve the latter problem in the same context as in RPA looking for small oscillations around equilibrium of spherical nuclei. In harmonic approximation, the classical hamiltouian (3.19) reads fi =~mziMti*+$uKu*+ =
$m ii,
E,,
m&i, t-f i r,j=
I
KijUiUj4 Eb
(3.23)
where the matrix elements m, = mii(uO) are obtained by eqs. (3.20) and (3.21) at the equiIibrium values u0 = (Rz, RE, bz, b:). The matrix K = K(u”) in (3.23) contains the matrix elements K, = K,( u”) =
a’E(R,,
R,, h,, &,I U=Uo ’ auj auj
i,j=1,2,3,4
(3.24)
which follow after expanding the collective potential energy (3.22) around its equiIibrium value E, = E ( II’) up to second order in the deviations (uf r) - u”) denoted here by u = n(t), again. The potential energy Y(H) appears to be reasonably we11 approximated by its harmonic expansion around the HF ground-state point u =0 (fig. 1). It is appropriate to mention here that due to the harmonic approximation our The main difference considerations resemble the approaches given in refs. 17*24*25). lies in the energy-density between the present work and the previous ones ‘7*24*2s) functional used. As demonstrated later on, the particular form of the kinetic energy as a functional of p(r) has an impact on the properties of antiscaling-type modes of excitation.
242
S.S. Dimitrova
et al. / Scaling and antiscaling
“‘pb
oscillations
16(
S;M
),M 0 8OC
1000
1200
I I
1LOO
1600 -2
2-2
0
(R-d),fm
0
2
(b-b'Lfm
Fig. 1. The collective potentials for pure bulk and pure surface isoscalar monopole vibrations in I60 and “sPb calculated with SkM* forces within the present local-scale version of ATDGF approach. The first excited isoscalar monopole states are given. The dashed curves indicate the potentials obtained by the harmonic approximation.
We diagonalize
the hamiltonian
(3.23) by solving (K-w2,M)Sa
the matrix
equation
=o
(3.25)
where o, ( CY= 1,2,3,4) are the eigenvalues which satisfy the characteristic equation det\K - w2A41 = 0. The eigenvectors S” associated with o, of eq. (3.25) relate the original
collective
variables
u = (R,, R,, b,, bp) to a new set of normal
coordinates
Q = (Q, , Qz, Q3, QJ. In this way, we can consider the quantized normal vibrations as elementary excitations (monopole phonons ““)) with excitation energies hw,. The phonon vacuum IO) (a, IO) = 0) defines the ground state of the breathing nucleus Eo+5C4,=, 192,394).
3.3. PHYSICAL
fi%, while the one-phonon
CHARACTERISTICS
excited
OF BREATHING
states are given as ICX)= aLlO) (cy =
NUCLEI
In this section, we obtain the matrix elements of some physical operators defined in the collective space by means of the phonon states, given above. Using eq. (2.16), the local density operator is represented in the form p^(r)=p&(r)+
i
a=,
Dt(r)(aL+a,)+i
i
a,p=1
~‘,,(r)(a~a~+a~a~+a,a~+a,a~).
(3.26a)
S.S. Dimiirova
et al. / Scaling and antiscaling osciliations
243
Thereby, (3.26b) and
@dr)=
[&-]“2[&y2i,i, [*]“=“os:sf > (326c)
where the density P&r) and its first and second derivatives are taken at the equilibrium value u”. The representation (3.26) leads to the following ground-state density distribution of the vibrating nucleus 4
POO(~)
=(OI~IO>=PW(~)+~ C Apa(r) a=1
(3.27)
with Ap,(r) = I?:&)
=
SST.
(3.28)
All four norma vibrations take part in forming of poo(r) by means of the dynamical contributions Ap, (r) (eq. (3.28)). The latter contributions, added to poa(r), determine the locai density in the first excited a-type one-phonon state P,*(r)=(O/a,~a~lO)=Poo(t)+Ap,(r).
(3.29)
Similarly, isoscatar transition densities between the ground 10)and the one-phonon excited state a’,lO), are expressed as
(3.30) Eq. (3.30) reproduces the phenomenological transition density (1.1) used in the DWBA analysis of ISMR data 22,23).For example, in the propo~ional SF density approximation (SFP) for equilibrium density dist~butions, i.e. assuming that R, = R, = R and b, = b, = b, we obtain
pbS;=“‘( r) = (t,, + tzm)
244
S.S. Dimitrova et al. / Scaling and antiscaling oscillations
which has to be compared with eq. (1.1). The amplitude coefficients ti, in (3.30) and (3.31), however, do not contain any ajustable parameters and they depend only on the Skyrme
forces parameters
h I/2 1
by means t,, =
of w, and ST
[ 2mw,
sp .
(3.32)
The isovector transition density p$z=” (r) “) also follows from eq. (3.30) by substituting tie + -tie for LY= 2, 4 and i = 1, 2, 3, 4. In a similar manner, the r.m.s. radii associated with p,,,,(r), paa (r) and pbz=” (r) can be obtained in an explicit form by using the r.m.s. radius operator 9%:’and the definition (2.18). Monopole resonances are related to the strength distribution one-body operators of either isoscalar or isovector type, 6”=irf
or
6’=
i=l
where
T = 1 for neutrons
the transition
strengths
corresponding
2 rf7,,
(3.33)
i=l
and T = -1 for protons.
If, for a given nuclear
state ]a),
‘) S,(r)
= l((“l@10)12
for T=O,
1
(3.34)
are such that S,(O) > S,(l), (S,( 1) * S,(O)), then this state is predominantly (T = l), mode. As is also known 9), EWSR must be saturated EWSR(T)=(O]$[$-,[Ei,~T]]]O)= In order quantities
to
to study both
strengths
i (&-E,)&(T). a=,
(3.34) and the EWSR
EMSJ~a-~o)S,(o)x EWSR(0)
a T = 0,
1oo=
(3.35)
(3.35), we consider
c%m)2x100
fi%
4(h2/2m)
LZ!~,
the
(3.36)
and
EM”=(E,-Eo)s~(l)xlOO= EWSR( 1)
4(h2/2m)(
fw>”--&t012 5?;,+4@
I pgopior2 dr)
(3.37) ’ loo ’
where pOgoand %ioC, are the neutron (q = n) or proton (q = p) local densities and r.m.s. radii, respectively, while CT?,;,= 9,iom+ 9?ion and poo = pie+ pEo. In order to obtain a more transparent physical meaning of the monopole vibrations under consideration we introduce a new set of isoscalar and isovector coordinates
$(R, - RJ
Ro= %R”+ &I,
R, =
bo=5(b”+b,L
b, = ;( b” - bJ
(3.38)
S.S. ~~~it~oua et al. / Scaling and aatjscaling oseii~atio~s
245
instead of the original density variables (R,, R,, b,, bp). In the new collective variables ii = (R,, R,, b,, b,), the hamiltonian (3.23) is transformed into a new quadratic form with matrices A? = 16&land I? = igijl, the latter being uniquely related to the matrices M and K in the original variables u = (R,, R,, b,, bp)_ One can obtain the following representation of the matrix A?
(N-Z)
M(T=O'
~'7==O'
RR
Rh
A
(N-Z) ___
j$,fkTR’1)
A
n;i=
M(p,j Rb
&p’r=t) Rh 9
(N-Z) 7
(M$$,=‘) (N-2) A
&pO)
M(T=O) Rb
ML;=])
Rb
lN - ‘1 ~ A
&p;=w (N-Z) A
M(T=o) bh
(3.39)
M(T=O;
bb
Mi;=‘)
where we have assumed the propo~ional approximation (SFP) for the equilibrium densities p(u”,, r): R = R, = R, and b = b, = b,. In eq. (3.39)
(3.41) where cr and CT’stand for R or b, and the velocity field o,, follows from eq. (3.21) in SFP. By analogy, a similar structure can be easily found for the matrix g.
4. Results and discussion We have performed calculations for a number of even-even nuclei and Skyrme effective forces. The equilibrium set u” = (Rf,, RE, bz, b;) has been derived by solving numerically eqs. (2.19) for each nucleus and each force, (see e.g. ref. “‘)). Having determined u’, we then diagonalize eq. (3.25) and calculate the physical characteristics of interest.
4.1. NATURE
OF THE
FOUR
NORMAL
MODE
VIBRATIONS
We first consider the results obtained in the particular case when all off-diagonal matrix elements of the matrices G and R are equal to zero. The corresponding excitation energies ZIw,, EWSR and transition density coeficients rim (eq. (3.32)) calculated with SkM* forces, are given in table 1.
246
S.S. Dimitrova
et al. / Scaling and antiscaiing
oscillations
TABLE 1 Characteristics
Nudeus
of the four normal modes calculated with SkM* forces in the particular case of independent pure bulk, surface, isoscalar and isovector monopole density vibrations (Y
&
EMS
EMV
11,
I60
4 3 2 1
42.40 31.12 30.53 23.55
0.00 0.00 88.39 84.11
93.07 78.82 0.00 0.00
0.309 0.000 0.328 0.000
-0.309 0.000 0.328 0.000
0.000 0.082 0.000 0.090
0.000 -0.082 0.000 0.090
‘Wa
4 3 2 1
38.21 29.94 23.82 22.12
0.00 0.00 94.41 82.69
91.54 74.45 0.00 0.00
0.196 0.000 0.220 0.000
-0.196 0.000 0.220 0.000
0.000 0.063 0.000 0.069
0.000 -0.063 0.000 0.069
“Zr
4 3 2 1
34.63 29.94 22.01 18.79
1.57 1.06 76.85 97.26
98.81 66.78 0.74 0.93
0.132 0.000 0.000 0.157
-0.132 0.000 0.000 0.157
0.000 0.050 0.055 0.000
0.000 -0.050 0.055 0.000
“‘Pb
4 3 2 1
29.67 28.40 20.66 13.87
5.80 3.46 69.37 98.67
99.04 59.13 2.37 3.38
0.091 0.000 0.000 0.116
-0.091 0.000 0.000 0.116
0.000 0.041 0.045 0.000
0.000 -0.041 0.045 0.000
Excitation energies E, (in MeV), isoscalar (in %), eqs. (3.34) and (3.373, and transition within the SFP approximation.
(EMS) density
r2,
*3n
*4a
and isovector (EMV) energy-weighted sum rules coefficients t,, (in fm), eq. (3.32), are calculated
In this case all four normal modes correspond to independent bulk, surface, isoscalar and isovector density vibrations, as can be easily seen form eq. (3.39) and table 1. For example, in the nucleus “*Pb, the first normal mode LY= 1 includes pure isoscalar (EMS Z+EMV) bulk vibrations of the density in which R, and R, vibrate in phase with equal amplitudes (see columns 6 and 7 in table l), while the amplitude of b-type vibrations are equal to zero (see columns 8 and 9 in table 1). The second
normal
mode
(Y= 2 is a pure
isoscalar
surface
mode,
while
normal
modes LY= 3 and LY= 4 are pure isovector surface and bulk modes, respectively. From table 1, it is also evident that isovector modes (bulk and surface) have appreciably higher energies than the isoscalar modes. Systematically, the isovector R-modes CY= 4 are with higher energies than the isovector b-modes cy = 3. At the same time, the isoscalar R- and b-modes have similar positions in light nuclei (up to 4oCa) while for heavier nuclei the isoscalar R-mode has lower energies than the isoscalar b-type vibrations. Considering EWSR (table l), we conclude that in heavy nuclei, the surface mode (isoscalar and isovector) is not so collective as the bulk vibrations. For light nuclei the degree of collectivity is nearly the same for both pure bulk and pure surface (isoscalar and isovector) vibrations. Regarding the strengths, it appears that the sum rules are grossly violated in table 1 (see tables 2,3 and 4). It should be remembered,
S.S. Dim&ova
however,
that
decoupled.
we consider
pure
a particular
need be performed
In order to investigate both
here
Thus, the EWSR concerns
no summation bulk
with respect
surface
between
vibrations,
diagonal (2 x 2) block matrices in G shown in table 2. In this case we have However, neutron and proton R- and and, therefore, an isoscalar (isovector) type vibrations.
collective
all four
to the separate isoscalar
are
vibrations.
and isovector
we consider
modes
mode only. Therefore, vibrations
at
the case in which
all
and i are included. The results obtained are pure bulk and pure surface oscillations, again. b-variables oscillate with non-equal amplitudes contribution exists in the isovector- (isoscalar-)
TABLE Same as table 1 in the case of coupled
case in which
the associated
the interplay
and pure
247
et al. / Scaling and antiscaling oscillations
isoscalar
2
and isovector
density vibrations
of pure bulk and
surface type Nucleus
Comparing
a
K
EMS
EMV
t,a
4
42.40
0.00
94.06
0.310
3
31.12
0.03
19.96
0.000
2
30.53
77.40
0.00
1
23.55
74.12
0.02
t2o -0.309
ha 0.000
t‘le 0.000
0.000
0.084
0.328
0.329
0.000
0.000
0.000
0.000
0.089
0.092
4
38.22
0.02
98.26
0.193
3
29.94
0.02
75.16
0.000
0.000
0.064
2
23.82
87.47
0.01
0.223
0.217
0.000
1
22.12
76.80
0.02
0.000
0.000
0.068
4
34.78
0.02
94.87
0.119
-0.198
0.000
-0.081
0.000 -0.062 0.000 0.07 1 0.000
-0.146
3
30.11
0.10
65.48
0.000
0.000
0.043
2
22.00
76.16
1.31
0.000
0.000
0.057
0.053
1
18.78
93.03
0.65
0.154
0.160
0.000
0.000
4
30.22
0.08
95.11
0.075
3
28.74
0.00
56.85
0.000
0.000
2
20.66
69.37
2.3 1
0.000
0.000
0.044
0.045
1
13.85
96.14
2.39
0.113
0.121
0.000
0.000
the results
shown
in tables
-0.110
1 and 2 we conclude
0.000 0.032
-0.058
0.000 -0.049
that the mixing
of
the T = 0 and T = 1 modes has a very small effect on the level energies and other main characteristics. Consequently, the classification of the modes into T = 0 and T = 1 types is still meaningful for the pure bulk and surface vibrations. A strong effect on the main physical characteristics is observed considering the interplay between bulk and surface vibrations of the nuclear densities. In table 3, we show the results obtained in the case when only those matrix elements of IL? and k are neglected which are proportional to the difference (N -Z)/A (see e.g. eq. (3.39)). This is the case of pure isoscalar or isovector vibrations in which the R-type and b-type vibrations are naturally coupled. In fact, this is the case considered in the paper “) on the basis of an ETF energy-density functional.
248
S.S. Dimiirova
et al. / Scaling and antiscaling oscillations TABLE 3
Same as table 1 in the case of scaling and antiscaling
density vibrations
of pure isoscalar
and isovector
type
Nucleus IhO
Ta
90Zr
“‘Pb
E,
EMS
EMV
t,,
4
58.69
0.00
23.28
0.529
3
50.84
4.45
0.00
0.509
2
30.64
0.00
71.28
1
23.33
81.08
0.00
0.053
4
51.90
0.00
26.50
0.318
a
-0.080
3
40.52
2.73
0.00
2
29.90
0.00
71.30
0.309
1
20.77
90.66
0.00
0.097
4
47.17
0.36
22.93
0.196
3
36.17
0.42
0.00
0.179
2
29.71
1.20
75.70
0.025
1
17.81
95.80
0.93
0.114
4
41.02
0.95
16.21
0.120
3
31.35
0.00
0.00
0.103
2
27.18
4.83
82.65
0.041
1
13.64
98.02
3.37
0.103
-0.016
‘2, -0.529 0.509
t3, -0.098 -0.111
0.080
0.098
0.053
0.079
-0.318 0.309
-0.075 -0.091
0.016
0.067
0.097
0.045
-0.196 0.179 -0.025 0.114 -0.120 0.103 -0.041 0.103
-0.061 -0.073 0.043 0.020 -0.050 -0.058 0.026 0.008
t‘l, 0.098 -0.111 -0.098 0.079 0.075 -0.091 -0.067 0.045 0.06 1 -0.073 -0.043 0.020 0.050 -0.058 -0.026 0.008
As evident from table 3, bulk and surface vibrations (of pure T = 0 or T = 1 type) are mixed and the normal modes can be considered as a scaling- or antiscaling-type vibrations of the nuclear densities. For example, in the nucleus *“Pb, the normal mode (Y= 1 contains isoscalar bulk oscillations with equal amplitudes as well as isoscalar surface vibrations with equal amplitudes, too. The variable R,(R,) oscillates in phase to the variable b,, (b,) and, consequently, a scaling-type isoscalar normal mode arises. The normal mode (Y= 3 is pure isoscalar mode too, but R,(R,) oscillates out of phase to b,(b,) and therefore we observe an antiscaling-type isoscalar mode. In the same way we conclude that the normal modes (Y= 2 and (Y= 4 are isovector scaling- and antiscaling-type vibrations, respectively. From table 3 also follows that the isoscalar scaling-type mode is the lowest normal mode in all nuclei
and exhausts
almost
completely
the isoscalar
EWSR. Therefore,
this mode could be identified with the isoscalar giant monopole resonance (ISMR). The second normal mode LY= 2, i.e. the isovector scaling-type mode, exhausts to a large extent the isovector EWSR and it could be identified with the isove.ctor giant monopole resonance (IVMR). The antiscaling-type modes have no collective character with respect to EWSR. However, we should mention that the isovector antiscalingtype oscillation exhausts within 16-22% the EWSR (T = 1) in heavy nuclei. In fact, in the case of light nuclei, R, (R,) oscillates out of phase to b, (bJ in the two T = 1 normal modes. Nevertheless, the (Y= 2 (a = 4) normal vibrations are of an isovector scaling) (antiscaling-) type with respect to both the EWSR (see table 3) and the isovector transition density shapes (see the next section).
249
S.S. Dimitroca et al. / Scaling and anrisraling oscillations TABLE
4
Same as table 1 in the general case when all four density collective variables are coupled Nucleus
a
EC,
EMS
EMV
IhO
4
58.99 50.53 30.59 23.31
0.07 4.34 0.08 79.35
20.13 1.59 65.01
51.98 40.46 29.85 20.77
0.00 2.80 0.00 91.40
2 @Ca
4 2
t,a 0.609 0.393 -0.05 I 0.067
-0.427 0.608 0.113 0.037
-0.117 -0.092 0.094 0.075
0.076 -0.126 -0.104 0.085
25.52 0.94 69.14 0.01
0.339 0.277 0.005 0.102
-0.293 0.337 0.040 0.091
-0.082 -0.086 0.060 0.044
0.067 -0.094 -0.074 0.046
0.01
“Zr
4 3 2
47.57 36.10 29.83 17.79
0.02 0.46 0.00 97.29
22.46 0.74 74.37 0.79
0.164 0.195 0.013 0.109
-0.229 0.158 -0.038 0.121
-0.049 -0.076 0.042 0.022
0.073 -0.069 -0.044 0.018
“‘Pb
4
42.21 31.22 27.50 13.59
0.00 0.00 0.21 98.93
16.23 1.23 79.79 2.08
0.086 0.117 0.027 0.102
-0.158 0.081 -0.053 0.106
-0.034 -0.061 0.026 0.006
0.066 -0.052 -0.028 0.010
2
Finally, in table 4, we show the results obtained in the general case when all four collective coordinates (R,, R,, b,, bp) or (R,, R,, b,, b,) are coupled. In this case, the complete structure of the matrices A% and 2 lead to an isoscalar-isovector coupiing of the scaling- (antiscaling-) type vibrations. The comparison of the results in tables 3 and 4 shows that the mixing of the T = 0 and T = 1 modes has very little effect on the main normal-mode characteristics. Therefore, we can identify the ISMR and IVMR vibrations with 0: = 1 and CY= 2 normal density vibrations, respectively. In this way, we obtain a quite natural interpretation of ISMR (IVMR) as dynamically coupled scaling-type bulk and surface density isoscalar (isovector) vibrations in which a small contribution of isovector (isoscalar) scaling-type oscillations appears.
4.2. COMPARISON
WITH
EXPERIMENTAL
DATA
AND
OTHER
THEORETICAL
ESTIMATES
The ISMR excitation
energies
(normal
mode (Y= l), calculated
with SkM* forces
and interpolated by a smooth curve, are compared in fig. 2 with some experimental data ‘). A remarkable agreement is observed between calculated values and the experimental peak energies within their error bars for the region of the heavy nuclei. For the light nuclei (A < 70), the present results are in better agreement with the experimental values ‘) than with the classical average trend (80.6A-“3 MeV ‘*)), which deviates significantly from the experimental data in this region. Due to the center-of-mass motion correction, the calculated ISMR energies decrease with about
250
S.S. Wimitrova
et al. / Scaling and antiscaling
EISMR ’ MeV
c I
:
23
\ ‘\
S*kM
-113 80.6 A *
‘,~~,T~~~T
osciilations
CENTER-OF-MASS
10
, 20
60
100
110
180
P
: 0
Fig. 2. Energies of ISMR of spherical nuclei obtained in our calculations using the SkM* forces (solid line). The dashed curve corresponds to the estimate in ref. ‘). The short-dashed curve presents our results obtained without centre-of-mass correction. Some experimental ISMR peak energies I,*) are also given.
3.5 MeV in 4He, 0.8 MeV in I60 and less than 0.1 MeV in heavy nuclei (see the short-dashed line in fig. 2). The value of the nuclear-matter incompressibility I&,,. apparently determines the variations in ISMR energies calculated in the framework of the scaling model (see e.g. ref. ‘“)). In order to verify the latter property in our approach we have calculated the 1SMR energies using a set of Skyrme forces SkM* 39), SkA 4’), SkIII 42) and SkV4*) with coefficient K,.,. ranging from 200 to 365 MeV. The results are shown in fig. 3. In spite of the complex nature of the ISMR the calculated excitation energy for each nucleus is a smooth almost linear function of K,.,. . Consequently, bearing in mind the agreement between ISMR energies calculated with SkM* forces and experimental data, we can conclude that the appropriate value for K,.,, should be near to that obtained with SkM* forces, i.e. K,.,.-215 MeV. In table 5 the present excitation energies calculated in the general case with Sk111 forces are compared with the results obtained within the RPA sum rules approach 3*4). The RPA values correspond to the mean energy (m,/m_,)“*, where mk is the kth energy-weighted moment of the strength 43). It is obvious that a good agreement does exist among the present results and the corresponding RPA energies for ISMR ‘) and IVMR “). It should be mentioned that our results also agree with those following from the genuine RPA calculations 44). In table 5 we also list the nuclear fluiddynamical (NDF) results 16) based on the familiar scaling assumption. As pointed out in ref. 16) the NFD results are in agreement with the RPA values (~~/~,)“2. In figs. 4 and 5, the isoscalar T = 0 and isovector T = 1 transition densities in *“Pb calculated with Sk111 forces are compared with available RPA transition densities ‘> obtained with the same Sk111 forces for T =0 and T = 1 monopole
S.S. Dimirrova
et al. / Scaling and anriscafing
SkV
Ska
SCM IO 200
250
300
oscillations
251
Sk Ill $++eL 350
Fig. 3. Calculated ISMR excitation energies versus incompressibility K,,, . The Skyrme forces used are Skill, SkV, SkA and SkM*, respectively.
Isoscalar and isovector giant monopole excitation energies obtained with Sk111
s=o
s=o
T=O
T=l
Nucleus
lb0
‘%a %Zr 10SPb
PW
RPA
NFD
PW
RPA
NFD
28.5 25.7 22.5 17.7
28.2 25.6 22.1 16.9
32.8 27.5 22.2 17.8
34.7 33.5 32.9 30.1
32.1 31.8 32.1 26.8
43.9 39.8 35.3 30.5
Columns labelled PW correspond to results obtained in the present work. RPA energies are calculated in ref. ‘) for T = 0 and in ref. 4, for T = 1. NFD results are obtained in ref. la).
vibrations. From these figures it is obvious that ISMR and IVMR can be actually identified with the isoscalar and isovector scaling-type normal modes CY= 1 and LY= 2, respectively. Our transition densities have a node at about 6.3 fm for (Y= I and 6.7 fm for a = 2 which is very close to the values obtained for the RPA transition densities 5,6). In contrast, the antiscaling-type transition densities (normal modes LY= 3 and (Y= 4) have two nodes for all considered nuclei, independent of the mass region. Furthermore, the quantitative agreement with the RPA results shows the
252
S.S. Dimitrouu
0.8
et al. / Scaling and antiscaling
oscillations
I
ANTlSCALING
-TYPE
Fig. 4. Isoscalar transition densities for scaling-type LY= 1 and antiscaling-type (Y= 3 monopole vibrations (solid curves) compared with the corresponding RPA transition densities (dashed lines).
Sk lit ,20apb (d=2) SCALING -TYPE
i
Fig. 5. Same as in fig. 4 for isovector transition densities calculated with SkIII.
usefulness of the present transition densities in a more elaborate analysis of the experimental data. They have a simple analytical form (eqs. (3.30) and (3.31)) depending only on the Skyrme forces parameters.
4.3. ANTISCALING-TYPE
VIBRATIONS
As already demonstrated, due to the existence of rather strong coupled bulk and surface density vibrations, the monopole oscillation is split into four normal modes. The two modes lower in energy, (Y= 1 and cy= 2, have been identified as ISMR and
S.S. Din&ova
IVMR,
respectively.
classified nuclear table
The other
two normal
(see sect. 4.1) as isoscalar densities.
The associated
modes,
and isovector excitation
4) as well as the unperturbed
shown
253
et al. / Scaling and antiscaling oscillations
(Y= 3 and
antiscaling-type
vibrations
of the
E,=, , Em=*, Eac3, Eac4, (see
energies E,,
energies
(Y= 4, have been
ER,, Ebo, Eb,, (see table
1) are
in fig. 6.
The considerable strength of the coupling can be seen from the fact that unperturbed T =0 (T = 1) splitting E, - ER,( E, - E,,) is increased several times by the coupling (see fig. 6). In particular, the differences between ISMR energies En=, and the antiscaling-type T = 0 energies E, =3 are much larger than the experimental widths of the ISMR (r = 2-4 MeV, ref. ‘)) and should be experimentally observed. The present calculations reveal a strong sensitivity of the antiscaling-type monopole vibrations to the particular form of the energy-density functional under consideration. In order to illustrate this conclusion we have performed calculations using the TF-like energy-density functional i.e. assuming the kinetic energy density ~[p] to be approximated by the familiar TF kinetic-energy density expression r[p] - p5”( r) 32). The results obtained with SkM” forces are presented with dashed lines in fig. 6. The comparison made in this figure between the results obtained using both the TF approximation and the present energy-density functional shows that the TF approximation do not change ISMR and IVMR energies (scaling-type modes) very much. However, the antiscaling-type mode energies are strongly affected. For example, the resulting isoscalar scaling-antiscaling splitting obtained in TF approximation is remarkably constant, about 10 MeV for all nuclei heavier
SKIM, S,=J,
‘60
,50
4oca
.-Ed.‘
LO
----
ET+31
/,/-Ed’
E d.‘
3
_I_ f---_
E d.3
ERO
’’ A’--+___
.-~.L(T=l) I
i ,’
ERI E, -
Pb Ed;4
/
i ER, -
Ebl 4.----Ed;>
/_
EJa2 E b, -_
-J-=-
Ed.3
i----
Edz2
/‘,,__-_ 20
--.-._
71
208
-
Et71--‘I
E b0
c
Zr
i 30
Etsl, o,,
90
I 50
HO
-
4;,
Em2-N Et0
.
Ebl
Ed_, Emi.’ Em---
ERl-,/
i //;---
E,:$ T=o)
/-.-
E&2( T=l)
EM i.-.-
----
EL,
hl --.____
E&T=O)
10 -
Fig. 6. Comparison between results for monopole excitation energies obtained using both the TF approximation (dashed lines) and the present energy-density functional (solid lines) for some spherical nuclei with SkM* forces. The monopole vibration is split into four normal modes with excitation energies E,, (CY= 1,2,3,4). The unperturbed bulk isoscalar (isovector) excitation energies E,, (ER,) are also given together with surface ones E,, (Eh,).
S.S. Dimirrova
254
than the nucleus ETF within
40Ca. Similar
energy-density the present
160. Obviously,
functional functional
et al. / Scaling and antiscaling
results have been found in ref. I’) by using the TF-like 32). However, E[p]
([email protected]
there is a strong sensitivity
tions to the particular
oscillations
the
corresponding
splitting
varies
X) from 18 MeV in *“Pb to 28 MeV in of the antiscaling-type
choice of the energy-density
functionals.
monopole
vibra-
Thus we can conclude
that an experimental observation of the antiscaling-type excitations would be an effective way to estimate the goodness of the different approximate energy-density functionals for the theoretical description of the nuclear dynamics. Regarding the prospect to experimentally observing the antiscaling modes given in ref. 23) we should mention that the present results show minor possibilities in this direction. For isoscalar modes the strength in vanishingly small, while for isovector modes they are rather high energies and not very collective. Nevertheless, we can compare our results with those given in ref. 23). Considering the experimental observation of the isoscalar antiscaling monopole vibrations, Morsch et al 22) have assumed transition densities of the form of eq. (1.1). The dependence of the a-scattering cross section on surface vibrations is shown in fig. 7, where 8 is related to the surface (6b) and bulk (SR) amplitudes in the form 0 = arctan(Gb/SR). In fig. 7 the experimental limits on the monopole cross section are also given by the hatched area 29). As evident from this figure, the experimental data 23) rule out the antiscaling-type vibrations corresponding a-scattering TF-like calculations giving 8 = -10” to -20”. The calculated
I
z -[mb/sr]
ANTISCALING
-L-
to the cross
SCALING
a-
Fig. 7. The calculated in 23) (a, o’) cross section for the ISMR excitation in *‘aPb as a function of the relative amount of surface contribution. The solid lines indicate contributions of scaling-type (0 > 0) and the antiscaling-type (0 < 0) ISMR transition densities (see the text) calculated with SkIII, SkV, and SkM*, respectively. The hatched area represents the experimental L = 0 cross section determined in ref. 23).
SS. Dimitrooa
sections
in this ®ion
bulk vibration
et al. / Scaling and antiscaling
are a factor of five smaller
(8 = 0”) which can be considered however,
we have obtained
which
of
form
actually
the
than those obtained
from a pure
as the lower limit of the experimental
monopole cross section. As mentioned above, are
255
oscillations
of
eq.
the transition (1.1)
with
densities a
quantity
(3.31) 8=
arctan((S7 + ST)/(ST + ST)) being related to the Skyrme force parameters and the normal mode vibrations according to eq. (3.32). Thus, the calculated isoscalar scaling- and antiscaling-type transition densities (3.3 1) can be compared with experimental data 23). In fig. 7 we show the values of 0 for both ISMR and antiscaling-type T = 0 modes calculated with SkM*, SkV and Sk111 forces. It is obvious that the obtained scaling (19= 4” to 6”) and antiscaling values (8 = -29” to -31”) cannot be distinguished in the corresponding experimental a-scattering cross section data. Consequently, the presently obtained antiscaling-type vibrations cannot be ruled out by these experimental data 23).
5. Summary and conclusions In the present paper, we have reported a rigorous EDF approach using the local-scale point transformation method 26). The energy-density functional E[p] has been generalized to time-dependent density distributions which follow as a solution of the proposed local-density CHF problem. Assuming the constrained densities to be of the SF form with time-dependent density parameters we have oscillations around equilibrium ground-state values. From the results obtained we draw the following conclusions. modes of monopole vibrations are separated into two scaling-type in energy) and other two antiscaling-type density oscillations. modes are identified with the ISMR and IVMR. Small contributions scaling-type
vibrations
have been seen to appear
considered
small
The four normal vibrations (lower The scaling-type of T = 1 (T = 0)
in the ISMR (IVMR).
Nevertheless,
in all nuclei both ISMR and IVMR exhaust almost completely the corresponding T = 0 and T = 1 EWSR. The ISMR excitation energies are smooth, almost linear, functions of the nuclear matter incompressibility K,.,. . Calculated with SkM* forces (I&,,.-
215 MeV), they show a remarkable
agreement
with the experimental
peak
energies. The ISMR and IVMR excitation energies and transition densities, calculated with Sk111 forces, are in excellent agreement with the RPA results 3-5). The two antiscaling-type modes also have isoscalar and isovector nature, respectively, slightly mixing. They carry relatively little strength and therefore do not show collective structure. The corresponding T = 0 and T = 1 antiscaling-type mode excitation energies are pushed up strongly by the coupling between unperturbed pure bulk and surface vibrations. The resulting splitting between scaling- and antiscalingtype isoscalar mode energies are much larger than the experimental widths ‘) and should therefore be experimentally observed. Furthermore, the calculated scalingand antiscaling-type transition densities and the corresponding a-scattering cross
256
S.S. Dimitroua
et al. / Scaling and antiscaling oscillations
section values cannot be separated by the experimental data 23) and consequently it is not possible to rule out the antiscaling-type vibrations by these experiments. The experimental observation of the antiscahng-type monopole modes would be an appropriate way to check the approximate energy-density functionals with respect to their applicability to dynamical considerations, because it has been demonstrated that these modes are strongly sensitive to the particular choice of the energy-density functionals.
References 1) F.E. Bertrand, Nucl. Phys. A354 (1981) 129 A. van der Woude, Kernfysisch Versneller Institut preprint KVI-583, Groningen, 1986 2) M. Buenerd, J. de Phys. 45 (1984) C4-115 3) 0. Bohigas, A.M. Lane and J. Martorell, Phys. Reports 51 (1979) 267 4) K. Goeke, B. Caste1 and P.-G. Reinhard, Nucl. Phys. A339 (1980) 337 5) J.P. Blaizot, Phys. Reports 65 (1980) 171 6) N. Auerbach and A. Klein, Phys. Rev. CZ8 (1983) 2075 7) N. Auerbach, A. Klein and E.R. Siciliano, Phys. Rev. 61 (1985) 682 8) Y. Abgrall and E. Caurier, Phys. Lett. BS6 (1975) 229 9) H. Flocard and D. Vautherin, Nucl. Phys. A264 (1976) 197 10) IZh. Petkov and M.V. Stoitsov, Joint Institute of Nuclear Research report JINR-P4-75-785, Dubna, USSR, 1985 11) M. Baranger and M. Veneroni, Ann. of Phys. 114 (1978) 123 12) Y. Engel, D.M. Brink, K. Goeke, S.J. Krieger and D. Veneroni, Nucl. Phys. A249 (1975) 215 13) I.Zh. Petkov and M.V. Stoitsov, International Centre for Theoretical Physics preprint ICTP-IC/85/45, Trieste, Italy, 1985 14) G. Bertsch, Nucl. Phys. A249 (1975) 253 15) G. Eckhart, G. Holzwarth and J.P. da Providencia, Nucl. Phys. A364 (1981) 1 16) H. Krivine, J. Treiner and 0. Bohigas, Nucl. Phys. A336 (1980) 115 17) M. Brack and W. Stocker, Nucl. Phys. A406 (1983) 413 18) f. Speth and A. van der Woude, Rep. Progr. Phys. 44 (1981) 719 19) L.J. Tassie, Austral. J. Phys. 9 (1956) 407 20) S.K.M. Wong, G. Saunier and B. Rouben, Nucl. Phys. Al69 (1971) 294 21) H.P. Morsch, P. Decowski, Phys. Lett. B82 (1979) 1 22) H.P. Morsch, C. Sukosd, M. Rogge, P. Turek and C. Mayer-Boericke, Phys. Rev. C22 (1980) 489 23) H.P. Morsch, P. Decowski, M. Rogge and P. Turek, Phys. Rev. C28 (1983) 1947 24) A.S. Jensen and SM. Larsen, Phys. Scripta 24 (1981) 534 25) S. Shlomo and P.J. Siemens, Phys. Rev. C31 (198.5) 2291 26) I.Zh. Petkov and M.V. Stoitsov, Theor. and Math. Phys. 55 (1983) 584; LZh. Petkov, M.V. Stoitsov, ES. Kryachko, Int. J. Quantum Chem. 25 (1986) 149 27) I.Zh. Petkov and M.V. Stoitsov, Sov. J. Nucl. Phys. 37 (1983) 692 28) I.Zh. Petkov and M.V. Stoitsov, Comp. Rend. Bulg. Acad. Sci. 33 (12) (1980) 1623; 34 (12) (1981) 1651 29) C.F. von Weizsaecker, Z. Phys. 96 (1935) 431 30) S.S. Dimitrova, LZh. Petkov and M.V. Stoitsov, 2. Phys. A325 (1986) 15 31) M. Brack, C. Guet and H.-B. Hakansson, Phys. Reports 123 (1985) 275 32) C. Titin-Schnaider and P. Quentin, Phys. Lett. B49 (1974) 397 33) D, Vautherin and D. Brink, Phys. Rev. C5 (1972) 626 34) M.V. Stoitsov, I.Zh. Petkov and S.S. Dimitrova, Izv. AN USSR (ser. fiz.) 50 (1986) 2071 35) S.S. Dimitrova, LZh. Petkov and M.V. Stoitsov, Comm. JINR E4-86-658, Dubna, 1986 36) J. Jaenecke, Nucl. Phys. Al81 (1972) 49
S.S. Dimitrova 37) L. Aleksandrov,
38) 39) 40) 41) 42) 43) 44)
et al. / Scaling and antiscaling
oscillations
251
Zh. Vychisl. Math. Fiz. 11 (1971) 36; Joint Institute of Nuclear Research preprint P5-1259, Dubna, USSR, 1973 L. Aleksandrov, M. Drenska and D. Karadjov, Program System REGN for solution of nonlinear system of equations, PSR-RSIC-31, Oak Ridge, USA J. Bartel, P. Quentin, M. Brack, C. Guet and H.-B. Hakansson, Nucl. Phys. A386 (1982) 72 L.D. Landau and E.M. Lifshitz, Mechanics - A course of theoretical physics, vol. 1 (Pergamon, NY, 19S9) sect. 23 H.S. Koehler, Nucl. Phys. A258 (1976) 301 M. Beiner, H. Flocard, N. Van Giai and P. Quentin, Nucl. Phys. A238 (1975) 29 K. Goeke, J. Speth, Ann. Rev. Nucl. Part. Sci. 32 (1982) 65 V.E. Starodubsky, Yad. Fiz. 42 (1985) 78
SCALINGAND
AND ANTISCALING-TYPE ISOVECTOR
NUCLEAR
S.S. DIMITROVA,
I.Zh.
OSCILLATIONS MONOPOLE
PETKOV
and
M.V. STOITSOV
Instiiute for Nuclear Research and Nuclear Energy, Bulgarian Academy Received (Revised
IN ISOSCALAR
VIBRATIONS*
ofSciences,Sofia,
Bulgaria
29 January 1987 16 March 1988)
Abstract: Isoscalar (T = 0) and isovector (T = 1) giant monopole resonances are studied using a localscale version of the ATDHF theory developed on the basis of a rigorous energy-density functional approach. Due to the strong coupling between the bulk and surface density vibrations, the monopole collective motion is split into four normal modes. Two of them, lower in energy, correspond to scaling-type density vibrations. The other two are of antiscaling-type in which the nuclear surface oscillates opposite in phase to the scaling-type vibrations. Excitation energies, transition densities, T = 0 and T = 1 energy weighted sum rules and other properties of breathing even-even nuclei are calculated using different Skyrme-type effective forces. The strong sensitivity of the antiscaling-type vibrations to the particular form of the approximate energy-density functionals is demonstrated.
1. Introduction In recent years, much effort has been devoted
to the experimental
and theoretical
study of nuclear giant monopole resonances (GMR) I). Presently, the breathing mode has been established experimentally for about fifty nuclei over the whole mass table l,‘). Microscopic approaches such as the random phase approximation (RPA) 3-7), the generator coordinate method (GCM) ‘-‘O), and the adiabatic timedependent Hartree-Fock (ATDHF) method 1’-‘3) have been applied to the theoretical study of giant monopole resonances (GMR). General features of GMR have been successfully described by fluid-dynamical approaches 14-i7) with a few parameters related to bulk and surface nuclear characteristics. The experimental data of isoscalar giant monopole resonances are usually described
within the distorted-wave
Born approximation
(DWBA)
i8) using a simple
scaling model of the breathing vibrations ‘*19). Similar transition densities follow from the constrained Hartree-Fock (CHF) method 20) where the nucleus is constrained to have a given mean squared radius. The analysis of the experimental data from both (Y- and d-inelastic scattering with excitation of GMR in 208Pb shows, however, that the appropriate transition density is of the form 21*22) pTr3(r)
’ Work partially
supported
=
6bap(r;
by Bulgarian
R, b)/ab+GRdp(r; Science
03759474/88/$03.50 @ Elsevier Science Publishers (North-Holland Physics Publishing Division)
Committee, B.V.
R, b)/aR Contract
No. 325.
(1.1)
234
S.S. Dimitrooa et al. / Scaling and antiscaling oscillations
where p(r; R, b) is the ground-state density distribution. Such a transition density pTr3 is quite different from that of the scaling model since in general (6R/ 6b) # (R/b). A surface mixed
monopole
to some extent
mode
(diffuseness
(0 = arctan(aR/?%))
oscillations)
is included
with a bulk mode
in eq. (l.l), (half-radius
being oscilla-
tions). In the present paper, attention is undertaken to relate quantitatively the GMR, as dynamically coupled bulk and surface vibrations of the breathing nucleus, back to the effective nucleon-nucleon interaction in nuclei. Some stimulation for such a theoretical interpretation of GMR is provided by the recently reported experimental data 23). They allow an antiscaling-type mode associated with O-arctan( 6R/6b) < 0 to be investigated. There are three papers 17*24,25)in which the interplay between surface and bulk vibrations in nuclei has been considered with application to GMR. In ref. 24), the spin-orbit interaction and the corrections to the Thomas-Fermi (TF) kinetic energy have been omitted. Therefore, as mentioned in ref. “), a quantitative agreement with experimental data could not be obtained. In refs. 17,25)only the isoscalar GMR have been considered. The aim of the present work is to investigate in a unified way the main features of the scaling- and antiscaling-type vibrations in both isoscalar and isovector giant monopole resonances. We start from a rigorous energy-density functional (EDF) approach for constructing approximate energy-density functionals E[p] within a given trial orbit 0 of the Hilbert space X with respect to a local-scale point transformation group (sect. 2). In sect. 3, we present a local-scale version of the ATDHF approach ‘i-13). Assuming the constrained densities pn(r, t) and pp(r, t) to be of the symmetrized Fermi (SF) density form with a time-dependent set of parameters u(f) = (R,(t), RJ t), b,(t), bp( t)) we have a natural choice of collective variables for describing small density oscillations around the HF ground state. Numerical results for a number of even-even nuclei obtained with Skyrme forces are given in sect. 4. Summary and conclusions are made in sect. 5. 2. Ground-state 2.1. A RIGOROUS
ENERGY
energy density functional
DENSITY
FUNCTIONAL
description
APPROACH
In previous papers 26-28), the local-scale point transformation (LST), r + r’ =f( r) = (r/r)f(r), has been used. It depends on an unknown scalar LST function f(r). A set of invertable transformations of this type forms a LST group @. Its isomorphic group Ou, of unitary operators s, E %% produces local-scale wave function transformations %(r,
. . . r,4)+(rl
. *. 4
=
(2.1)
S.S. Dimitrova
et al. / Scaling and antiscaling oscillations
r,J belongs where +=+(r,,..., system with a hamiltonian fi. For any given function
to the Hilbert
space
&E X (the so-called
WI = 02l,*
= %(h,.
235
X of the A-particle
model
function
. . , rd+(rl,.
. . , rd
nuclear
‘“)) the functions (2.2)
form an orbit 6’ = {%f$,f~ 9”) of X’ with respect to 9. An one-to-one relationship between LST functions f(r) and local densities associated with $[f] E Q can be achieved in a given orbit 0 c X -_ =f2(r) ““)
p(r)
Here, the model
density
r*
distribution
p(r)=A
corresponds
p(r)
giving
that density.
the wave function
Thus,
. . , rA;
[PI)
=
= wNm4
J%Jl
I$E 0 (2.4)
been analyzed function ).
value
an energy
to the model function
equation with respect to f(r) which r ) is obviously one-to-one related to
%[p](rl
the expectation
(2.5) becomes
(2.3)
rA)12dr2*-.dr,.
In fact, eq. (2.3) is a first-order differential we solve numerically. The solution f[~] =fi,,,( the local density p(r). Its particular form has given density p(r) there is an A-particle wave +[P] = (cl(rl).
.
i3r
I$(r,r, ,...,
I
p(f(r))
p(r)
in refs. 26,27). For any
. . , rA)&(rl
).
. . , *A),
of the nuclear
density
hamiltonian
(2.5)
fi on
functional
(2.6)
= UPI + VP1 .
It is important that the wave function (2.5) allows the kinetic T[p] and potential V[p] energy in (2.6) to be obtained in an unified way as local density functionals in O= ZY,i.e.,
7bls
(~[~llfl$bl)
= (h2/2m)
UPI = (+bll ~I$bl> = In the hamiltonian fi = ?+ operators, respectively.
2.2. HARTREE-FOCK
Let us consider Slater-determinant
ENERGY
t
J
f are kinetic
DENSITY
FUNCTIONAL
energy-density 6 = C$sL which
hl
dr,
(2.7)
(2.8)
E~&I dr.
? and
an approximate model function
I
and two-body
potential
energy
functional E[p] obtained by a involves given model s.p. wave
236
S.S. Dimitrova et ai. / Scaling and antiscaling oscilIa?ions
functions (Q,(r), i = 1,2,. . . , A). The density-dependent 0 SL={021,.~sL,f~ S}C X has the form2’) llrEP]=9(rI,...,r,4;
[PI) = (A!)-“* dct \rpAq; Cp])I,
with local-scale transformed
wave function
(2.5) in
(&j = I,. . . , A)
(2.9)
s.p. wave functions (2.10)
They depend on the LST function &,(r) which is the solution of eq. (2.3). The model density distribution (2.4) is expressed in 6&c X as (2.11) The kinetic-energy only two terms
density in (2.7) depends on the orbit Os,c Z and contains
Q3 = G.&PI+ ~&Cl *
(2.12)
The first term T&P] is the original Weizsaecker term *9) while the second is of the form 30) (2.13) For the most general Skyrme-type forces and time-reversal invariant systems the potential-energy density E&‘P] in (2.8) is defined in fys,c X as +Xlpl-4~0[(~
+&o)P2 -(~o+4)(p~+p2,)l+ht~p*[(l+~X3)p2-(X~+t)(p2nCp2p)l
+~[tl(l+~x,)+t,(l+~x,)]pl-[p]+$~fZ(X~+~)-t,(X,+f)~ x (Pn7n[P1+Pp7pEPI)+~E3t,(l+tx*>-- f2(1+$~2)1ow’ --&i’3f,(~,+1, + G%+3)1~m”)2+
o%,Yl
+~W~(J.OP+J,.V~,+J,*V~,)+EC~~,[PI
(2.14)
where to, t,, fZ, f,, x0, x,, xz, x3, a and W. are parameters of the Skyrme forces (see the notation of ref. “I)). The spin density is also a local density functional
Jbl E J(r; [PII= ? dr; bYI)!*wf(r; [PI).
(2.15)
i=l
The index n (p) in (2.14) refers to neutron ~proton~ densities and restricts the summations to neutron (proton) occupied states in formulae (2.11), (2.13) and (2.15). The Coulomb energy density E~~~,[J-I]in (2.14) is the sum of the direct term and the exchange term, the latter taken in the weli-known Slater approximation 32).
S.S. Dim&ova
In what follows we consider 0g;O.c 2, functions. teristics
et al. / Scaling and antiscaling oscillations
a harmonic-oscillator
237
single Slater determinant
orbit
{g,(r)} to be harmonic-oscillator shell-model s.p. wave i.e. assuming In this case the self-consistent HF results 33) for average nuclear charac-
(binding
tics sensitive reproduced
energies,
local densities,
with a quite satisfactory
2.3. DENSITY
r.m.s. radii, etc.) as well as the characteris-
to shell effects (s.p. spectra,
VARIATIONAL
shell oscillations,
etc.) are
27).
CALCULATIONS
For spherical nuclei we have chosen p,(r) densities to be of a symmetrized p(r)‘=p’(r;
local-density
accuracy
in refs. 30,34,35)the neutron p,(r) Fermi-type (SF) radial form
R,, b,) =~04 sh (R,lb,)l[ch
(R,lb,)+ch
where the index q stands for neutrons (q = n) and protons condition is satisfied through the relation
~04 = W,/4&/[1
(q =
and proton
(r/b,)1
,
p). The normalization
(2.17)
+WqlRq)21,
where the number of neutrons (protons) is denoted N+Z. Eq. (2.17) and the r.m.s. radius expression
as A,=
(2.16)
N (A,=Z)
and A=
%~=(r2p4)=~A,R~[l+~(rrb,/R,)2]
(2.18)
are exact relationships when py Coulomb energy density .zCou,[p]
is represented by eq. (2.16). Furthermore, the in (2.14) can also be represented in a closed
analytical form 36). Inserting (2.16) into (2.6)-(2.8)
and (2.12)-(2.15)
the functional
E[p]
an algebraic function E (R, , R,, b,, bJ of the four density variational II = (R, , R,, b,, bJ. We minimize E(u) = E (R,, R,, b, , b,) by solving the following system of non-linear algebraic equations aE(u)/dq using the autoregularized in ref. 38).
= 0,
Gauss-Newton
i = 1,2,3,4 iterative
becomes
parameters numerically
(2.19)
process “) and the code described
As already reported in ref. 30) the obtained results are in satisfactory agreement with respect to the self-consistent HF ground-state results 39). Therefore, on one hand, the wave function $[psF] (eqs. (2.9) and (2.16)), allows us to reproduce the HF ground-state results quite well. On the other hand, t+b[psF] directly depends on the density parameters II = (R,, R,, b,, bJ as natural collective coordinates and can be used within a number of collective methods as, for example, the GCM or the ATDHF method.
SS. Dimirroua cf al. / Scaling and antiscaling oscillations
238 3.
3.1. LO&AL&ZALE
Locat-scale
VERSfON
dem-iption of monopole vjbr~~ia~s
ATDHF
OF THE ATDHF
AP~RUA~H
F~~~~wi~~ closely the notations of refs. ‘iX1’ ), the classicat harnj~t~~~a~ of the nuclear system under c~~~ider~ti~n can be expressed in the ATIXIF ~~proxi~a~~~~ l3S
(3.2) and (3.3)
2’=&W)lfilW)>
are the collective kinetic and potent&I energies, respectively, An approximate approach to avoid the solution of the ATDHF ~~~ati~n “) is supplied by using s.p. functions {pipi(t; t)f which satisfy the gen~raIi~ed scaling property I’), This could in principle be done with the help of additional timedependent external fields within the CHF method. It appears, however, more appealing and at the same time easier to use the present e~ergy~d~us~t~ functiona‘t ~~~~~ac~. ‘l’?~ebasic. idea is to start fram a tile-de~ende~t density dist~b~tia~s p(~, f) = P,(P) ~~~stra~~ed EO depend on the tiime I Ety means of a few ~~l~-~h~se~ density ~ha~~ter~sti~s ti = n(t) = (q(f), rz252(f), . . _>, e.g. density rn~rne~ts~ deformation parameters and so an, For any given values of u(t), the density p,,(r), must be then obtained by rni~~i~li~~~gE[p,(r)] with respect to p,(r) within a density variational space in which all densities give the same values of u(t). Such a minimization of the expectation value I?[p, ( P)] of 2 in ossLc 2? corresponds to a specific density CHF problem in which the densities are constrained to have given nuclear characteristics u(t) at each moment f. Inserting pU(t) into eqs. (2.9) and (2.10) we obtain the time-dependent wave function 44 t) f 44&J = (A Yr”
det iFit%, t)f ,
fi,j=l,.,.,A)
(3s)
which involves the s.p. wave functions
of eq. CZ.3) with the ~~~sit~ p,(r). For even-even f3S) is time-even and a family of tirne~e~~~ one-body
with flewI(r) being the solution
nuclei the Slater dete~i~ant
S.S. Dimitrova et al. j Scaling and antiscaling oscillations
239
density matrices pO(t)
can be applied to simplify the ATDHF approach. Further, it is important that the time-dependent the generalized scaling property y5;9(r,t) = --uq * Vp’(r,
s.p. wave functions (3.6) satisfy
t) -#(r,
(3.8a)
t)V ’ t$
with real vector functions (3.8b)
r+ = uy(l; t) = -~E$~:(r)/f~:(r)l,
where jPf(r) =dj&r)/dt and fi$r) = r. * Vf,;;(r). As well known 14), under the conditions (3.8), all nucleons are forced to move in a common (for each kind of nucleons, 9 = n, p) velocity field v, and the collective kinetic energy (3.2) is related to the current density in ATDHF approximation I*) as
Inserting (3.8) into (3.2) and using eq. (3.9) we obtain after a partial integration and some rearrangements the expression (3.10)
As shown in “), the ATDHF equation of motion leads to the continuity equation ~~(r)+(~/~)V
*J,(r, t>=O
(3.11)
with a generalized current density Jg = J,(r, t) =.Qr, r>+P~iC&(r,
r) -iC_Lj(r,
f)l
(3.12)
where 4 = (p, n) when 9 = (n, p) and the non-locality parameter @= (m/2~~}~~~i- t2) comes from the exchange Skyrme forces. On the other hand, the time-derivative Pz(r) entering eq. (3,1X) can be obtained by the equation PZ(rI+V
* Cd(rb,l
=O,
(3.13)
which follows after taking into account eqs. (3.7) and (3.8). Comparing (3.13) with (3.11) we obtain the relation V *Id(rh,l=
(~/~)V %,+BtX_iJr, ~~-~tj,.&(~, f)lI
(3.14)
which allows the current density &(r, t> to be expressed by means of the local densities pi(r) and the velocity fields ug, q = (n, p).
240
S.S. Dimitrova
In the case of breathing and the relation
et al. / Scaling and antiscaling oscillations
vibrations
nuclei both o, and& are irrotational
m + pt.
ql+PP:
pip,”
hj,=p
where pu = p; i.e. V~= r,,v,(r,
of spherical
(3.14) gives
l+ppu
uq+pU l+ppu
Due to the spherical
symmetry,
(3.15)
uq
U, has a radial component
only,
t), and eqs. (2.3) and (3.8) yield
r (ap:/&)rr2 vq(r, t)=
I
O
dr’
pzr2
Inserting (3.15) and (3.16) into (3.10), the collective form of a local density functional ZJ;+p---- p”upt: l+PP”
(3.16)
’
zl,u,+
kinetic
l+PP: ___ 1+PPU
(pfy
The same holds reasonably for the collective potential be identified with the energy-density functional
energy
X takes the
dr.
ZI’,
(3.17)
I
energy (3.3), (3.5) which can
"Irr_Pul= HPUI = (~~Pull~l~cI[Pul)~
(3.18)
A number of approaches exist which assume the internal energy in the dynamical case to be static TF-like approximate energy-density functionals of the time-depensince different approximadent density r5-“). Such a procedure may be questioned tions are used in determination of both X[p] and V[p]. In this respect, the local-scale version of the ATDHF approach is more rigorous than the irrotational dynamics, although we also assume irrotational flow. In fact, the local-scaling both the kinetic and the potential collective energies to be obtained and hence the inconsistency mentioned above to be avoided.
3.2. MONOPOLE
Let us assume
VIBRATIONS
AS NORMAL
that the local density
ONE-PHONON
p(r)
applied
EXCITED
to breathing
fluid allows
simultaneously
STATES
spherical
nuclei
is
represented by the SF density distribution (2.16) with a time-dependent set of parameters u = u(t) = (R,(t), R,(t), b,(t), bp( t)). Inserting eq. (2.16) into eqs. (3.17) and (3.18) we obtain the ATDHF classical hamiltonian
X=5
,$ m,(u)ti,ti,+clr(u),
(3.19)
‘.I 1
where ri = (&,, &,, d,, 6,) is the time derivative of u = (R,, R,, b,, b,). The matrix elements mq(u) of the inertial tensor M(u) in (3.19) are given as PP~P4,,(1-~,9,)+P~(1+PP~)s,,,, (1 +PPu)
1viv, dr,
(3.20)
S.S. Dimitrova
et al. / Scaling and antiscaling osciliati~ns
241
where the isospin index 4 = n (p) for i = 1, 3 (2,4) is ordered to correspond to the index i = 1,2,3,4 labeling the collective vector u = (u, , u2, u3, u,) = (R,, R,, b,, bJ. Taking into account eqs. (2.16) and (3.16) we express the velocity geld U, in (3.20) (measured in units “ii= dUi( t)/dt) as
(a~~/a~~)r’~dr’ Uj(P,t>=
p%r’
(3.21)
*
In the case of SF densities (2.16), the collective potentia1 energy (3.18) is represented as an algebraic function of collective variables (3.22) whose explicit expression is well defined in Ok;O.c X by the energy-density functional (2.6)-(2.8) and (2.12)-(2.15). Furthermore, the ATDHF hamiltonian (3.19) allows one to apply the standard method of quantization ‘I). We solve the latter problem in the same context as in RPA looking for small oscillations around equilibrium of spherical nuclei. In harmonic approximation, the classical hamiltouian (3.19) reads fi =~mziMti*+$uKu*+ =
$m ii,
E,,
m&i, t-f i r,j=
I
KijUiUj4 Eb
(3.23)
where the matrix elements m, = mii(uO) are obtained by eqs. (3.20) and (3.21) at the equiIibrium values u0 = (Rz, RE, bz, b:). The matrix K = K(u”) in (3.23) contains the matrix elements K, = K,( u”) =
a’E(R,,
R,, h,, &,I U=Uo ’ auj auj
i,j=1,2,3,4
(3.24)
which follow after expanding the collective potential energy (3.22) around its equiIibrium value E, = E ( II’) up to second order in the deviations (uf r) - u”) denoted here by u = n(t), again. The potential energy Y(H) appears to be reasonably we11 approximated by its harmonic expansion around the HF ground-state point u =0 (fig. 1). It is appropriate to mention here that due to the harmonic approximation our The main difference considerations resemble the approaches given in refs. 17*24*25). lies in the energy-density between the present work and the previous ones ‘7*24*2s) functional used. As demonstrated later on, the particular form of the kinetic energy as a functional of p(r) has an impact on the properties of antiscaling-type modes of excitation.
242
S.S. Dimitrova
et al. / Scaling and antiscaling
“‘pb
oscillations
16(
S;M
),M 0 8OC
1000
1200
I I
1LOO
1600 -2
2-2
0
(R-d),fm
0
2
(b-b'Lfm
Fig. 1. The collective potentials for pure bulk and pure surface isoscalar monopole vibrations in I60 and “sPb calculated with SkM* forces within the present local-scale version of ATDGF approach. The first excited isoscalar monopole states are given. The dashed curves indicate the potentials obtained by the harmonic approximation.
We diagonalize
the hamiltonian
(3.23) by solving (K-w2,M)Sa
the matrix
equation
=o
(3.25)
where o, ( CY= 1,2,3,4) are the eigenvalues which satisfy the characteristic equation det\K - w2A41 = 0. The eigenvectors S” associated with o, of eq. (3.25) relate the original
collective
variables
u = (R,, R,, b,, bp) to a new set of normal
coordinates
Q = (Q, , Qz, Q3, QJ. In this way, we can consider the quantized normal vibrations as elementary excitations (monopole phonons ““)) with excitation energies hw,. The phonon vacuum IO) (a, IO) = 0) defines the ground state of the breathing nucleus Eo+5C4,=, 192,394).
3.3. PHYSICAL
fi%, while the one-phonon
CHARACTERISTICS
excited
OF BREATHING
states are given as ICX)= aLlO) (cy =
NUCLEI
In this section, we obtain the matrix elements of some physical operators defined in the collective space by means of the phonon states, given above. Using eq. (2.16), the local density operator is represented in the form p^(r)=p&(r)+
i
a=,
Dt(r)(aL+a,)+i
i
a,p=1
~‘,,(r)(a~a~+a~a~+a,a~+a,a~).
(3.26a)
S.S. Dimiirova
et al. / Scaling and antiscaling osciliations
243
Thereby, (3.26b) and
@dr)=
[&-]“2[&y2i,i, [*]“=“os:sf > (326c)
where the density P&r) and its first and second derivatives are taken at the equilibrium value u”. The representation (3.26) leads to the following ground-state density distribution of the vibrating nucleus 4
POO(~)
=(OI~IO>=PW(~)+~ C Apa(r) a=1
(3.27)
with Ap,(r) = I?:&)
=
SST.
(3.28)
All four norma vibrations take part in forming of poo(r) by means of the dynamical contributions Ap, (r) (eq. (3.28)). The latter contributions, added to poa(r), determine the locai density in the first excited a-type one-phonon state P,*(r)=(O/a,~a~lO)=Poo(t)+Ap,(r).
(3.29)
Similarly, isoscatar transition densities between the ground 10)and the one-phonon excited state a’,lO), are expressed as
(3.30) Eq. (3.30) reproduces the phenomenological transition density (1.1) used in the DWBA analysis of ISMR data 22,23).For example, in the propo~ional SF density approximation (SFP) for equilibrium density dist~butions, i.e. assuming that R, = R, = R and b, = b, = b, we obtain
pbS;=“‘( r) = (t,, + tzm)
244
S.S. Dimitrova et al. / Scaling and antiscaling oscillations
which has to be compared with eq. (1.1). The amplitude coefficients ti, in (3.30) and (3.31), however, do not contain any ajustable parameters and they depend only on the Skyrme
forces parameters
h I/2 1
by means t,, =
of w, and ST
[ 2mw,
sp .
(3.32)
The isovector transition density p$z=” (r) “) also follows from eq. (3.30) by substituting tie + -tie for LY= 2, 4 and i = 1, 2, 3, 4. In a similar manner, the r.m.s. radii associated with p,,,,(r), paa (r) and pbz=” (r) can be obtained in an explicit form by using the r.m.s. radius operator 9%:’and the definition (2.18). Monopole resonances are related to the strength distribution one-body operators of either isoscalar or isovector type, 6”=irf
or
6’=
i=l
where
T = 1 for neutrons
the transition
strengths
corresponding
2 rf7,,
(3.33)
i=l
and T = -1 for protons.
If, for a given nuclear
state ]a),
‘) S,(r)
= l((“l@10)12
for T=O,
1
(3.34)
are such that S,(O) > S,(l), (S,( 1) * S,(O)), then this state is predominantly (T = l), mode. As is also known 9), EWSR must be saturated EWSR(T)=(O]$[$-,[Ei,~T]]]O)= In order quantities
to
to study both
strengths
i (&-E,)&(T). a=,
(3.34) and the EWSR
EMSJ~a-~o)S,(o)x EWSR(0)
a T = 0,
1oo=
(3.35)
(3.35), we consider
c%m)2x100
fi%
4(h2/2m)
LZ!~,
the
(3.36)
and
EM”=(E,-Eo)s~(l)xlOO= EWSR( 1)
4(h2/2m)(
fw>”--&t012 5?;,+4@
I pgopior2 dr)
(3.37) ’ loo ’
where pOgoand %ioC, are the neutron (q = n) or proton (q = p) local densities and r.m.s. radii, respectively, while CT?,;,= 9,iom+ 9?ion and poo = pie+ pEo. In order to obtain a more transparent physical meaning of the monopole vibrations under consideration we introduce a new set of isoscalar and isovector coordinates
$(R, - RJ
Ro= %R”+ &I,
R, =
bo=5(b”+b,L
b, = ;( b” - bJ
(3.38)
S.S. ~~~it~oua et al. / Scaling and aatjscaling oseii~atio~s
245
instead of the original density variables (R,, R,, b,, bp). In the new collective variables ii = (R,, R,, b,, b,), the hamiltonian (3.23) is transformed into a new quadratic form with matrices A? = 16&land I? = igijl, the latter being uniquely related to the matrices M and K in the original variables u = (R,, R,, b,, bp)_ One can obtain the following representation of the matrix A?
(N-Z)
M(T=O'
~'7==O'
RR
Rh
A
(N-Z) ___
j$,fkTR’1)
A
n;i=
M(p,j Rb
&p’r=t) Rh 9
(N-Z) 7
(M$$,=‘) (N-2) A
&pO)
M(T=O) Rb
ML;=])
Rb
lN - ‘1 ~ A
&p;=w (N-Z) A
M(T=o) bh
(3.39)
M(T=O;
bb
Mi;=‘)
where we have assumed the propo~ional approximation (SFP) for the equilibrium densities p(u”,, r): R = R, = R, and b = b, = b,. In eq. (3.39)
(3.41) where cr and CT’stand for R or b, and the velocity field o,, follows from eq. (3.21) in SFP. By analogy, a similar structure can be easily found for the matrix g.
4. Results and discussion We have performed calculations for a number of even-even nuclei and Skyrme effective forces. The equilibrium set u” = (Rf,, RE, bz, b;) has been derived by solving numerically eqs. (2.19) for each nucleus and each force, (see e.g. ref. “‘)). Having determined u’, we then diagonalize eq. (3.25) and calculate the physical characteristics of interest.
4.1. NATURE
OF THE
FOUR
NORMAL
MODE
VIBRATIONS
We first consider the results obtained in the particular case when all off-diagonal matrix elements of the matrices G and R are equal to zero. The corresponding excitation energies ZIw,, EWSR and transition density coeficients rim (eq. (3.32)) calculated with SkM* forces, are given in table 1.
246
S.S. Dimitrova
et al. / Scaling and antiscaiing
oscillations
TABLE 1 Characteristics
Nudeus
of the four normal modes calculated with SkM* forces in the particular case of independent pure bulk, surface, isoscalar and isovector monopole density vibrations (Y
&
EMS
EMV
11,
I60
4 3 2 1
42.40 31.12 30.53 23.55
0.00 0.00 88.39 84.11
93.07 78.82 0.00 0.00
0.309 0.000 0.328 0.000
-0.309 0.000 0.328 0.000
0.000 0.082 0.000 0.090
0.000 -0.082 0.000 0.090
‘Wa
4 3 2 1
38.21 29.94 23.82 22.12
0.00 0.00 94.41 82.69
91.54 74.45 0.00 0.00
0.196 0.000 0.220 0.000
-0.196 0.000 0.220 0.000
0.000 0.063 0.000 0.069
0.000 -0.063 0.000 0.069
“Zr
4 3 2 1
34.63 29.94 22.01 18.79
1.57 1.06 76.85 97.26
98.81 66.78 0.74 0.93
0.132 0.000 0.000 0.157
-0.132 0.000 0.000 0.157
0.000 0.050 0.055 0.000
0.000 -0.050 0.055 0.000
“‘Pb
4 3 2 1
29.67 28.40 20.66 13.87
5.80 3.46 69.37 98.67
99.04 59.13 2.37 3.38
0.091 0.000 0.000 0.116
-0.091 0.000 0.000 0.116
0.000 0.041 0.045 0.000
0.000 -0.041 0.045 0.000
Excitation energies E, (in MeV), isoscalar (in %), eqs. (3.34) and (3.373, and transition within the SFP approximation.
(EMS) density
r2,
*3n
*4a
and isovector (EMV) energy-weighted sum rules coefficients t,, (in fm), eq. (3.32), are calculated
In this case all four normal modes correspond to independent bulk, surface, isoscalar and isovector density vibrations, as can be easily seen form eq. (3.39) and table 1. For example, in the nucleus “*Pb, the first normal mode LY= 1 includes pure isoscalar (EMS Z+EMV) bulk vibrations of the density in which R, and R, vibrate in phase with equal amplitudes (see columns 6 and 7 in table l), while the amplitude of b-type vibrations are equal to zero (see columns 8 and 9 in table 1). The second
normal
mode
(Y= 2 is a pure
isoscalar
surface
mode,
while
normal
modes LY= 3 and LY= 4 are pure isovector surface and bulk modes, respectively. From table 1, it is also evident that isovector modes (bulk and surface) have appreciably higher energies than the isoscalar modes. Systematically, the isovector R-modes CY= 4 are with higher energies than the isovector b-modes cy = 3. At the same time, the isoscalar R- and b-modes have similar positions in light nuclei (up to 4oCa) while for heavier nuclei the isoscalar R-mode has lower energies than the isoscalar b-type vibrations. Considering EWSR (table l), we conclude that in heavy nuclei, the surface mode (isoscalar and isovector) is not so collective as the bulk vibrations. For light nuclei the degree of collectivity is nearly the same for both pure bulk and pure surface (isoscalar and isovector) vibrations. Regarding the strengths, it appears that the sum rules are grossly violated in table 1 (see tables 2,3 and 4). It should be remembered,
S.S. Dim&ova
however,
that
decoupled.
we consider
pure
a particular
need be performed
In order to investigate both
here
Thus, the EWSR concerns
no summation bulk
with respect
surface
between
vibrations,
diagonal (2 x 2) block matrices in G shown in table 2. In this case we have However, neutron and proton R- and and, therefore, an isoscalar (isovector) type vibrations.
collective
all four
to the separate isoscalar
are
vibrations.
and isovector
we consider
modes
mode only. Therefore, vibrations
at
the case in which
all
and i are included. The results obtained are pure bulk and pure surface oscillations, again. b-variables oscillate with non-equal amplitudes contribution exists in the isovector- (isoscalar-)
TABLE Same as table 1 in the case of coupled
case in which
the associated
the interplay
and pure
247
et al. / Scaling and antiscaling oscillations
isoscalar
2
and isovector
density vibrations
of pure bulk and
surface type Nucleus
Comparing
a
K
EMS
EMV
t,a
4
42.40
0.00
94.06
0.310
3
31.12
0.03
19.96
0.000
2
30.53
77.40
0.00
1
23.55
74.12
0.02
t2o -0.309
ha 0.000
t‘le 0.000
0.000
0.084
0.328
0.329
0.000
0.000
0.000
0.000
0.089
0.092
4
38.22
0.02
98.26
0.193
3
29.94
0.02
75.16
0.000
0.000
0.064
2
23.82
87.47
0.01
0.223
0.217
0.000
1
22.12
76.80
0.02
0.000
0.000
0.068
4
34.78
0.02
94.87
0.119
-0.198
0.000
-0.081
0.000 -0.062 0.000 0.07 1 0.000
-0.146
3
30.11
0.10
65.48
0.000
0.000
0.043
2
22.00
76.16
1.31
0.000
0.000
0.057
0.053
1
18.78
93.03
0.65
0.154
0.160
0.000
0.000
4
30.22
0.08
95.11
0.075
3
28.74
0.00
56.85
0.000
0.000
2
20.66
69.37
2.3 1
0.000
0.000
0.044
0.045
1
13.85
96.14
2.39
0.113
0.121
0.000
0.000
the results
shown
in tables
-0.110
1 and 2 we conclude
0.000 0.032
-0.058
0.000 -0.049
that the mixing
of
the T = 0 and T = 1 modes has a very small effect on the level energies and other main characteristics. Consequently, the classification of the modes into T = 0 and T = 1 types is still meaningful for the pure bulk and surface vibrations. A strong effect on the main physical characteristics is observed considering the interplay between bulk and surface vibrations of the nuclear densities. In table 3, we show the results obtained in the case when only those matrix elements of IL? and k are neglected which are proportional to the difference (N -Z)/A (see e.g. eq. (3.39)). This is the case of pure isoscalar or isovector vibrations in which the R-type and b-type vibrations are naturally coupled. In fact, this is the case considered in the paper “) on the basis of an ETF energy-density functional.
248
S.S. Dimiirova
et al. / Scaling and antiscaling oscillations TABLE 3
Same as table 1 in the case of scaling and antiscaling
density vibrations
of pure isoscalar
and isovector
type
Nucleus IhO
Ta
90Zr
“‘Pb
E,
EMS
EMV
t,,
4
58.69
0.00
23.28
0.529
3
50.84
4.45
0.00
0.509
2
30.64
0.00
71.28
1
23.33
81.08
0.00
0.053
4
51.90
0.00
26.50
0.318
a
-0.080
3
40.52
2.73
0.00
2
29.90
0.00
71.30
0.309
1
20.77
90.66
0.00
0.097
4
47.17
0.36
22.93
0.196
3
36.17
0.42
0.00
0.179
2
29.71
1.20
75.70
0.025
1
17.81
95.80
0.93
0.114
4
41.02
0.95
16.21
0.120
3
31.35
0.00
0.00
0.103
2
27.18
4.83
82.65
0.041
1
13.64
98.02
3.37
0.103
-0.016
‘2, -0.529 0.509
t3, -0.098 -0.111
0.080
0.098
0.053
0.079
-0.318 0.309
-0.075 -0.091
0.016
0.067
0.097
0.045
-0.196 0.179 -0.025 0.114 -0.120 0.103 -0.041 0.103
-0.061 -0.073 0.043 0.020 -0.050 -0.058 0.026 0.008
t‘l, 0.098 -0.111 -0.098 0.079 0.075 -0.091 -0.067 0.045 0.06 1 -0.073 -0.043 0.020 0.050 -0.058 -0.026 0.008
As evident from table 3, bulk and surface vibrations (of pure T = 0 or T = 1 type) are mixed and the normal modes can be considered as a scaling- or antiscaling-type vibrations of the nuclear densities. For example, in the nucleus *“Pb, the normal mode (Y= 1 contains isoscalar bulk oscillations with equal amplitudes as well as isoscalar surface vibrations with equal amplitudes, too. The variable R,(R,) oscillates in phase to the variable b,, (b,) and, consequently, a scaling-type isoscalar normal mode arises. The normal mode (Y= 3 is pure isoscalar mode too, but R,(R,) oscillates out of phase to b,(b,) and therefore we observe an antiscaling-type isoscalar mode. In the same way we conclude that the normal modes (Y= 2 and (Y= 4 are isovector scaling- and antiscaling-type vibrations, respectively. From table 3 also follows that the isoscalar scaling-type mode is the lowest normal mode in all nuclei
and exhausts
almost
completely
the isoscalar
EWSR. Therefore,
this mode could be identified with the isoscalar giant monopole resonance (ISMR). The second normal mode LY= 2, i.e. the isovector scaling-type mode, exhausts to a large extent the isovector EWSR and it could be identified with the isove.ctor giant monopole resonance (IVMR). The antiscaling-type modes have no collective character with respect to EWSR. However, we should mention that the isovector antiscalingtype oscillation exhausts within 16-22% the EWSR (T = 1) in heavy nuclei. In fact, in the case of light nuclei, R, (R,) oscillates out of phase to b, (bJ in the two T = 1 normal modes. Nevertheless, the (Y= 2 (a = 4) normal vibrations are of an isovector scaling) (antiscaling-) type with respect to both the EWSR (see table 3) and the isovector transition density shapes (see the next section).
249
S.S. Dimitroca et al. / Scaling and anrisraling oscillations TABLE
4
Same as table 1 in the general case when all four density collective variables are coupled Nucleus
a
EC,
EMS
EMV
IhO
4
58.99 50.53 30.59 23.31
0.07 4.34 0.08 79.35
20.13 1.59 65.01
51.98 40.46 29.85 20.77
0.00 2.80 0.00 91.40
2 @Ca
4 2
t,a 0.609 0.393 -0.05 I 0.067
-0.427 0.608 0.113 0.037
-0.117 -0.092 0.094 0.075
0.076 -0.126 -0.104 0.085
25.52 0.94 69.14 0.01
0.339 0.277 0.005 0.102
-0.293 0.337 0.040 0.091
-0.082 -0.086 0.060 0.044
0.067 -0.094 -0.074 0.046
0.01
“Zr
4 3 2
47.57 36.10 29.83 17.79
0.02 0.46 0.00 97.29
22.46 0.74 74.37 0.79
0.164 0.195 0.013 0.109
-0.229 0.158 -0.038 0.121
-0.049 -0.076 0.042 0.022
0.073 -0.069 -0.044 0.018
“‘Pb
4
42.21 31.22 27.50 13.59
0.00 0.00 0.21 98.93
16.23 1.23 79.79 2.08
0.086 0.117 0.027 0.102
-0.158 0.081 -0.053 0.106
-0.034 -0.061 0.026 0.006
0.066 -0.052 -0.028 0.010
2
Finally, in table 4, we show the results obtained in the general case when all four collective coordinates (R,, R,, b,, bp) or (R,, R,, b,, b,) are coupled. In this case, the complete structure of the matrices A% and 2 lead to an isoscalar-isovector coupiing of the scaling- (antiscaling-) type vibrations. The comparison of the results in tables 3 and 4 shows that the mixing of the T = 0 and T = 1 modes has very little effect on the main normal-mode characteristics. Therefore, we can identify the ISMR and IVMR vibrations with 0: = 1 and CY= 2 normal density vibrations, respectively. In this way, we obtain a quite natural interpretation of ISMR (IVMR) as dynamically coupled scaling-type bulk and surface density isoscalar (isovector) vibrations in which a small contribution of isovector (isoscalar) scaling-type oscillations appears.
4.2. COMPARISON
WITH
EXPERIMENTAL
DATA
AND
OTHER
THEORETICAL
ESTIMATES
The ISMR excitation
energies
(normal
mode (Y= l), calculated
with SkM* forces
and interpolated by a smooth curve, are compared in fig. 2 with some experimental data ‘). A remarkable agreement is observed between calculated values and the experimental peak energies within their error bars for the region of the heavy nuclei. For the light nuclei (A < 70), the present results are in better agreement with the experimental values ‘) than with the classical average trend (80.6A-“3 MeV ‘*)), which deviates significantly from the experimental data in this region. Due to the center-of-mass motion correction, the calculated ISMR energies decrease with about
250
S.S. Wimitrova
et al. / Scaling and antiscaling
EISMR ’ MeV
c I
:
23
\ ‘\
S*kM
-113 80.6 A *
‘,~~,T~~~T
osciilations
CENTER-OF-MASS
10
, 20
60
100
110
180
P
: 0
Fig. 2. Energies of ISMR of spherical nuclei obtained in our calculations using the SkM* forces (solid line). The dashed curve corresponds to the estimate in ref. ‘). The short-dashed curve presents our results obtained without centre-of-mass correction. Some experimental ISMR peak energies I,*) are also given.
3.5 MeV in 4He, 0.8 MeV in I60 and less than 0.1 MeV in heavy nuclei (see the short-dashed line in fig. 2). The value of the nuclear-matter incompressibility I&,,. apparently determines the variations in ISMR energies calculated in the framework of the scaling model (see e.g. ref. ‘“)). In order to verify the latter property in our approach we have calculated the 1SMR energies using a set of Skyrme forces SkM* 39), SkA 4’), SkIII 42) and SkV4*) with coefficient K,.,. ranging from 200 to 365 MeV. The results are shown in fig. 3. In spite of the complex nature of the ISMR the calculated excitation energy for each nucleus is a smooth almost linear function of K,.,. . Consequently, bearing in mind the agreement between ISMR energies calculated with SkM* forces and experimental data, we can conclude that the appropriate value for K,.,, should be near to that obtained with SkM* forces, i.e. K,.,.-215 MeV. In table 5 the present excitation energies calculated in the general case with Sk111 forces are compared with the results obtained within the RPA sum rules approach 3*4). The RPA values correspond to the mean energy (m,/m_,)“*, where mk is the kth energy-weighted moment of the strength 43). It is obvious that a good agreement does exist among the present results and the corresponding RPA energies for ISMR ‘) and IVMR “). It should be mentioned that our results also agree with those following from the genuine RPA calculations 44). In table 5 we also list the nuclear fluiddynamical (NDF) results 16) based on the familiar scaling assumption. As pointed out in ref. 16) the NFD results are in agreement with the RPA values (~~/~,)“2. In figs. 4 and 5, the isoscalar T = 0 and isovector T = 1 transition densities in *“Pb calculated with Sk111 forces are compared with available RPA transition densities ‘> obtained with the same Sk111 forces for T =0 and T = 1 monopole
S.S. Dimirrova
et al. / Scaling and anriscafing
SkV
Ska
SCM IO 200
250
300
oscillations
251
Sk Ill $++eL 350
Fig. 3. Calculated ISMR excitation energies versus incompressibility K,,, . The Skyrme forces used are Skill, SkV, SkA and SkM*, respectively.
Isoscalar and isovector giant monopole excitation energies obtained with Sk111
s=o
s=o
T=O
T=l
Nucleus
lb0
‘%a %Zr 10SPb
PW
RPA
NFD
PW
RPA
NFD
28.5 25.7 22.5 17.7
28.2 25.6 22.1 16.9
32.8 27.5 22.2 17.8
34.7 33.5 32.9 30.1
32.1 31.8 32.1 26.8
43.9 39.8 35.3 30.5
Columns labelled PW correspond to results obtained in the present work. RPA energies are calculated in ref. ‘) for T = 0 and in ref. 4, for T = 1. NFD results are obtained in ref. la).
vibrations. From these figures it is obvious that ISMR and IVMR can be actually identified with the isoscalar and isovector scaling-type normal modes CY= 1 and LY= 2, respectively. Our transition densities have a node at about 6.3 fm for (Y= I and 6.7 fm for a = 2 which is very close to the values obtained for the RPA transition densities 5,6). In contrast, the antiscaling-type transition densities (normal modes LY= 3 and (Y= 4) have two nodes for all considered nuclei, independent of the mass region. Furthermore, the quantitative agreement with the RPA results shows the
252
S.S. Dimitrouu
0.8
et al. / Scaling and antiscaling
oscillations
I
ANTlSCALING
-TYPE
Fig. 4. Isoscalar transition densities for scaling-type LY= 1 and antiscaling-type (Y= 3 monopole vibrations (solid curves) compared with the corresponding RPA transition densities (dashed lines).
Sk lit ,20apb (d=2) SCALING -TYPE
i
Fig. 5. Same as in fig. 4 for isovector transition densities calculated with SkIII.
usefulness of the present transition densities in a more elaborate analysis of the experimental data. They have a simple analytical form (eqs. (3.30) and (3.31)) depending only on the Skyrme forces parameters.
4.3. ANTISCALING-TYPE
VIBRATIONS
As already demonstrated, due to the existence of rather strong coupled bulk and surface density vibrations, the monopole oscillation is split into four normal modes. The two modes lower in energy, (Y= 1 and cy= 2, have been identified as ISMR and
S.S. Din&ova
IVMR,
respectively.
classified nuclear table
The other
two normal
(see sect. 4.1) as isoscalar densities.
The associated
modes,
and isovector excitation
4) as well as the unperturbed
shown
253
et al. / Scaling and antiscaling oscillations
(Y= 3 and
antiscaling-type
vibrations
of the
E,=, , Em=*, Eac3, Eac4, (see
energies E,,
energies
(Y= 4, have been
ER,, Ebo, Eb,, (see table
1) are
in fig. 6.
The considerable strength of the coupling can be seen from the fact that unperturbed T =0 (T = 1) splitting E, - ER,( E, - E,,) is increased several times by the coupling (see fig. 6). In particular, the differences between ISMR energies En=, and the antiscaling-type T = 0 energies E, =3 are much larger than the experimental widths of the ISMR (r = 2-4 MeV, ref. ‘)) and should be experimentally observed. The present calculations reveal a strong sensitivity of the antiscaling-type monopole vibrations to the particular form of the energy-density functional under consideration. In order to illustrate this conclusion we have performed calculations using the TF-like energy-density functional i.e. assuming the kinetic energy density ~[p] to be approximated by the familiar TF kinetic-energy density expression r[p] - p5”( r) 32). The results obtained with SkM” forces are presented with dashed lines in fig. 6. The comparison made in this figure between the results obtained using both the TF approximation and the present energy-density functional shows that the TF approximation do not change ISMR and IVMR energies (scaling-type modes) very much. However, the antiscaling-type mode energies are strongly affected. For example, the resulting isoscalar scaling-antiscaling splitting obtained in TF approximation is remarkably constant, about 10 MeV for all nuclei heavier
SKIM, S,=J,
‘60
,50
4oca
.-Ed.‘
LO
----
ET+31
/,/-Ed’
E d.‘
3
_I_ f---_
E d.3
ERO
’’ A’--+___
.-~.L(T=l) I
i ,’
ERI E, -
Pb Ed;4
/
i ER, -
Ebl 4.----Ed;>
/_
EJa2 E b, -_
-J-=-
Ed.3
i----
Edz2
/‘,,__-_ 20
--.-._
71
208
-
Et71--‘I
E b0
c
Zr
i 30
Etsl, o,,
90
I 50
HO
-
4;,
Em2-N Et0
.
Ebl
Ed_, Emi.’ Em---
ERl-,/
i //;---
E,:$ T=o)
/-.-
E&2( T=l)
EM i.-.-
----
EL,
hl --.____
E&T=O)
10 -
Fig. 6. Comparison between results for monopole excitation energies obtained using both the TF approximation (dashed lines) and the present energy-density functional (solid lines) for some spherical nuclei with SkM* forces. The monopole vibration is split into four normal modes with excitation energies E,, (CY= 1,2,3,4). The unperturbed bulk isoscalar (isovector) excitation energies E,, (ER,) are also given together with surface ones E,, (Eh,).
S.S. Dimirrova
254
than the nucleus ETF within
40Ca. Similar
energy-density the present
160. Obviously,
functional functional
et al. / Scaling and antiscaling
results have been found in ref. I’) by using the TF-like 32). However, E[p]
([email protected]
there is a strong sensitivity
tions to the particular
oscillations
the
corresponding
splitting
varies
X) from 18 MeV in *“Pb to 28 MeV in of the antiscaling-type
choice of the energy-density
functionals.
monopole
vibra-
Thus we can conclude
that an experimental observation of the antiscaling-type excitations would be an effective way to estimate the goodness of the different approximate energy-density functionals for the theoretical description of the nuclear dynamics. Regarding the prospect to experimentally observing the antiscaling modes given in ref. 23) we should mention that the present results show minor possibilities in this direction. For isoscalar modes the strength in vanishingly small, while for isovector modes they are rather high energies and not very collective. Nevertheless, we can compare our results with those given in ref. 23). Considering the experimental observation of the isoscalar antiscaling monopole vibrations, Morsch et al 22) have assumed transition densities of the form of eq. (1.1). The dependence of the a-scattering cross section on surface vibrations is shown in fig. 7, where 8 is related to the surface (6b) and bulk (SR) amplitudes in the form 0 = arctan(Gb/SR). In fig. 7 the experimental limits on the monopole cross section are also given by the hatched area 29). As evident from this figure, the experimental data 23) rule out the antiscaling-type vibrations corresponding a-scattering TF-like calculations giving 8 = -10” to -20”. The calculated
I
z -[mb/sr]
ANTISCALING
-L-
to the cross
SCALING
a-
Fig. 7. The calculated in 23) (a, o’) cross section for the ISMR excitation in *‘aPb as a function of the relative amount of surface contribution. The solid lines indicate contributions of scaling-type (0 > 0) and the antiscaling-type (0 < 0) ISMR transition densities (see the text) calculated with SkIII, SkV, and SkM*, respectively. The hatched area represents the experimental L = 0 cross section determined in ref. 23).
SS. Dimitrooa
sections
in this ®ion
bulk vibration
et al. / Scaling and antiscaling
are a factor of five smaller
(8 = 0”) which can be considered however,
we have obtained
which
of
form
actually
the
than those obtained
from a pure
as the lower limit of the experimental
monopole cross section. As mentioned above, are
255
oscillations
of
eq.
the transition (1.1)
with
densities a
quantity
(3.31) 8=
arctan((S7 + ST)/(ST + ST)) being related to the Skyrme force parameters and the normal mode vibrations according to eq. (3.32). Thus, the calculated isoscalar scaling- and antiscaling-type transition densities (3.3 1) can be compared with experimental data 23). In fig. 7 we show the values of 0 for both ISMR and antiscaling-type T = 0 modes calculated with SkM*, SkV and Sk111 forces. It is obvious that the obtained scaling (19= 4” to 6”) and antiscaling values (8 = -29” to -31”) cannot be distinguished in the corresponding experimental a-scattering cross section data. Consequently, the presently obtained antiscaling-type vibrations cannot be ruled out by these experimental data 23).
5. Summary and conclusions In the present paper, we have reported a rigorous EDF approach using the local-scale point transformation method 26). The energy-density functional E[p] has been generalized to time-dependent density distributions which follow as a solution of the proposed local-density CHF problem. Assuming the constrained densities to be of the SF form with time-dependent density parameters we have oscillations around equilibrium ground-state values. From the results obtained we draw the following conclusions. modes of monopole vibrations are separated into two scaling-type in energy) and other two antiscaling-type density oscillations. modes are identified with the ISMR and IVMR. Small contributions scaling-type
vibrations
have been seen to appear
considered
small
The four normal vibrations (lower The scaling-type of T = 1 (T = 0)
in the ISMR (IVMR).
Nevertheless,
in all nuclei both ISMR and IVMR exhaust almost completely the corresponding T = 0 and T = 1 EWSR. The ISMR excitation energies are smooth, almost linear, functions of the nuclear matter incompressibility K,.,. . Calculated with SkM* forces (I&,,.-
215 MeV), they show a remarkable
agreement
with the experimental
peak
energies. The ISMR and IVMR excitation energies and transition densities, calculated with Sk111 forces, are in excellent agreement with the RPA results 3-5). The two antiscaling-type modes also have isoscalar and isovector nature, respectively, slightly mixing. They carry relatively little strength and therefore do not show collective structure. The corresponding T = 0 and T = 1 antiscaling-type mode excitation energies are pushed up strongly by the coupling between unperturbed pure bulk and surface vibrations. The resulting splitting between scaling- and antiscalingtype isoscalar mode energies are much larger than the experimental widths ‘) and should therefore be experimentally observed. Furthermore, the calculated scalingand antiscaling-type transition densities and the corresponding a-scattering cross
256
S.S. Dimitroua
et al. / Scaling and antiscaling oscillations
section values cannot be separated by the experimental data 23) and consequently it is not possible to rule out the antiscaling-type vibrations by these experiments. The experimental observation of the antiscahng-type monopole modes would be an appropriate way to check the approximate energy-density functionals with respect to their applicability to dynamical considerations, because it has been demonstrated that these modes are strongly sensitive to the particular choice of the energy-density functionals.
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