We provide for the first time accurate assessments of the consequences of
violations of self-consistency in the Hartree-Fock based random phase
approximation (RPA) as commonly used to calculate the energy $E_c$ of the
nuclear breathing mode. Using several Skyrme interactions we find that the
self-consistency violated by ignoring the spin-orbit interaction in the RPA
calculation causes a spurious enhancement of the breathing mode energy for spin
unsaturated systems. Contrarily, neglecting the Coulomb interaction in the RPA
or performing the RPA calculations in the TJ scheme underestimates the
breathing mode energy. Surprisingly, our results for the $^{90}$Zr and
$^{208}$Pb nuclei for several Skyrme type effective nucleon-nucleon
interactions having a wide range of nuclear matter incompressibility ($K_{nm}
\sim 215 - 275$ MeV) and symmetry energy ($J \sim 27 - 37$ MeV) indicate that
the net uncertainty ($\delta E_c \sim 0.3$ MeV) is comparable to the
experimental one.Comment: Revtex file (11 pages), Accepted for the publication in Phys. Rev. 

A. Bohr

B. K. Agrawal

P. G. Reinhard

S. Shlomo

English

arXiv

Agrawal, B. K.

Shlomo, S.

Texas A&M University

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 Ju
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04
Consequences of self-consistency violations in Hartree-Fock
random-phase approximation calculations of the nuclear
breathing mode energy
B. K. Agrawal and S. Shlomo
Cyclotron Institute, Texas A&M University, College Station, TX 77843, USA
Abstract
We provide for the first time accurate assessments of the consequences of violations of self-
consistency in the Hartree-Fock based random phase approximation (RPA) as commonly used to
calculate the energy Ec of the nuclear breathing mode. Using several Skyrme interactions we find
that the self-consistency violated by ignoring the spin-orbit interaction in the RPA calculation
causes a spurious enhancement of the breathing mode energy for spin unsaturated systems. Con-
trarily, neglecting the Coulomb interaction in the RPA or performing the RPA calculations in the
TJ scheme underestimates the breathing mode energy. Surprisingly, our results for the 90Zr and
208Pb nuclei for several Skyrme type effective nucleon-nucleon interactions having a wide range
of nuclear matter incompressibility (Knm ∼ 215 − 275 MeV) and symmetry energy (J ∼ 27 − 37
MeV) indicate that the net uncertainty (δEc ∼ 0.3 MeV) is comparable to the experimental one.
PACS numbers: 21.60.Jz, 24.30.Cz, 21.65.+f
1
The Hartree-Fock (HF) based random phase approximation (RPA) provides a microscopic
description of the nuclear compressional modes. The most special of these compressional
modes is the isoscalar giant monopole resonance (ISGMR) also referred to as the breathing
mode. The centroid energy Ec of the ISGMR enables one to determine the value of nuclear
matter incompressibility coefficient Knm which plays an important role in understanding a
wide variety of phenomena ranging from heavy-ion collision to supernova explosions. Recent
experimental data [1] for the Ec in heavy nuclei have rather small uncertainties (∼ 0.1 −
0.3 MeV). Since, the uncertainty δEc associated with Ec is approximately related to the
uncertainty δKnm in Knm by,
δKnm
Knm
= 2
δEc
Ec
, (1)
the value of δKnm is only about 10 MeV, for Knm = 250 MeV and Ec = 14.17 ± 0.28
MeV for the 208Pb nucleus. On the theoretical side, most of the results obtained within
the HF-RPA theory, as employed for the determination of Ec, are plagued by the lack of
self-consistency, see however, Ref. [2]. Self-consistency is violated due to the neglect of the
spin-orbit and Coulomb terms in the particle-hole interaction used in the RPA calculations.
Furthermore, some of the RPA calculations are performed in the TJ (isospin) scheme. Un-
certainties in the value of Ec calculated in the HF-RPA approach arise also due to various
numerical approximations such as the use of very small particle-hole excitation energies and
the introduction of the smearing parameter Γ to smoothen the strength function. In Ref.
[3], we have investigated in detail the effects of these numerical approximations and empha-
sized the need to perform a highly accurate calculation of the strength function. So, it is
of utmost importance that in order to determine an accurate value of Knm one must have
an accurate knowledge of the effects of violations of self-consistency on the centroid energy
of the ISGMR. We may also point out that modifying the particle-hole interaction in an ad
hoc manner in such a way that the spurious state associated with the center of mass motion
appears at zero energy [4] does not restore the self-consistency (see also Ref. [3]).
In this work we provide for the first time accurate assessments of the effects of violations
of self-consistency on the constrained and the scaling energies,
Econ =
√
m1
m−1
, and Es =
√
m3
m1
, (2)
of the ISGMR. Here
mk =
∫
∞
0
ωkS(ω)dω (3)
2
is the energy moments of the strength function
S(ω) =
∑
n
|〈n | f | 0〉|2 δ(ω − ωn), (4)
for the monopole operator f(r) =
∑A
i=1 r
2
i .
In the following the fully self-consistent and highly accurate values of the constraint
energy, Econ, and the scaling energy, Es, are obtained by calculating the energy moments
mk for k = −1, 1 and 3 using the constrained HF (CHF), the double commutator and the
generalized scaling approaches, respectively [5]. The self-consistent values of Econ and Es are
then compared with those obtained within the HF-RPA approach [3, 4] in order to assess the
effects of each type of violation of self-consistency. It may be pointed out that very recently
in Ref. [6] some preliminary attempts are made to estimate the effects of self-consistency
violation on the value of Econ. However, Es was not considered and the different components
contributing to the self-consistency violations are not dealt separately. We show here that
the appropriate knowledge of the effects of each of these components on the ISGMR energies
are instrumental in validating the HF-RPA calculations.
We consider here the HF solution for the Hamiltonian,
H =
A∑
i=1
Ti +
1
2
A∑
i,j=1
Vij +
1
2
Z∑
i,j=1
V Cij , (5)
where T is the kinetic energy operator, V C is the Coulomb interaction and V is the effective
two-body interaction of the Skyrme type [7],
V12 = t0 (1 + x0P
σ
12) δ(r1 − r2)
+
1
2
t1 (1 + x1P
σ
12)×
[←−
k 212δ(r1 − r2) + δ(r1 − r2)
−→
k 212
]
+t2 (1 + x2P
σ
12)
←−
k 12δ(r1 − r2)
−→
k 12
+
1
6
t3 (1 + x3P
σ
12) ρ
α
(
r1 + r2
2
)
δ(r1 − r2)
+iW0
←−
k 12δ(r1 − r2)(
−→σ1 +
−→σ2)×
−→
k 12, (6)
where P σ12 is the spin exchange operator,
−→σ i is the Pauli spin operator,
−→
k 12 = −i(
−→
∇1−
−→
∇2)/2
and
←−
k 12 = −i(
←−
∇1 −
←−
∇2)/2 . Here, the right and left arrows indicate that the momentum
operators act on the right and on the left, respectively. The parameters of the Skyrme force
ti, xi, α and W0 are obtained by fitting the HF results to the experimental data for the bulk
properties of finite nuclei.
3
Once the HF equations are solved, the m1 and m3 can be expressed in terms of the
ground state density ρ, kinetic energy density τ and spin density J. The value of m1 can be
obtained from the corresponding double commutator [8] and is given as,
m1 = 2
~
2
m
〈r2〉, (7)
where,
〈r2〉 =
∫
r2ρ(r)dr. (8)
The cubic moment m3 can be evaluated in the generalized scaling approach and it is given
by[5, 9],
m3 =
1
2
(
2~2
m
)2
[2T + 6Eδ + 20(Efin + Eso)+
(3α + 2)(3α+ 3)Eρ] , (9)
where,
T =
∫
~
2m
τdr, (10)
Eδ =
∫ [
3
8
t0ρ
2 −
1
4
t0(x0 +
1
2
)ρ21
]
dr (11)
Efin =
∫ [
a0ρτ + a1ρ1τ1 + c(∇ρ)
2 + d(∇ρ1)
2
]
dr (12)
Eρ =
∫ [
3
48
t3ρ
α+2 −
1
24
t3(x3 +
1
2
)ραρ21
]
dr (13)
Eso = −
1
2
W0
∫
[ρ∇ · J+ ρp∇ · Jp + ρn∇ · Jn] dr (14)
with ρ1 = ρn − ρp, τ1 = τn − τp, a0 =
1
16
(3t1 + (5 + 4x2)t2), a1 =
1
8
t2(x2 +
1
2
) − t1(x1 +
1
2
),
c = 1
64
(9t1− (5 + 4x2) + t2) and d = −
1
32
(3t1(x1 +
1
2
) + t2(x2 +
1
2
)). As described in detail in
Ref. [5], m−1 can be evaluated via the CHF approach and is given as,
m−1 =
1
2
d
dλ
〈r2λ〉
∣∣∣∣
λ=0
=
1
2
d2
dλ2
〈Hλ〉
∣∣∣∣
λ=0
(15)
where, 〈Oλ〉 = 〈Φλ |O|Φλ〉, with Φλ being the HF solution for the constrained Hamiltonian,
Hλ = H − λf.
The corresponding HF-RPA values of mk are obtained from Eq. (3) using,
S(ω) =
1
pi
Im [Tr(fG(ω)f)] (16)
4
with G(ω) being the RPA Green’s function [3, 4]. The differences between the values of mk
and consequently the difference
δE = E(RPA)− E(SC), (17)
between the energies obtained within the HF-RPA and the self-consistent (SC) values ob-
tained within the HF approach are due to the violation of self-consistency in the HF-RPA.
Since we are unable to obtain a fully self-consistent value for m0, we only provide the value
of the centroid energy
Ec =
m1
m0
, (18)
obtained within the HF-RPA. Thus we use δEcon and δEs as measures of δEc. We point out
that we find Ec ≈ Econ.
Before we present the main results, some remarks about the numerical accuracies are in
order. We carry out the HF calculations using a box of 18 fm and a mesh size of δr = 0.02
fm. To determine m−1 given by Eq. (15) with an accuracy of 0.1% we calculate the first
derivative of 〈r2λ〉 using a five-point formula with the increment δλ = 0.02. In order to
limit the spurious enhancement in mk ( particularly m3), due to the Lorentzian smearing
of the RPA Green’s function, to below 0.2%, we carry out the continuum RPA (CRPA)
and the discretized RPA (DRPA) calculations using the very small smearing parameter of
Γ/2 = 5× 10−2 MeV and the maximum energy of 4Ec in the integral of Eq. (3). In Ref. [3]
we show that in order to calculate the centroid energy accurate to within 0.1 MeV, one must
set the cut-off for the p-h excitations Emaxph > 400 MeV. In what follows, we shall present
our DRPA results obtained with Emaxph = 950 MeV.
In Table I we demonstrate the accuracy of our calculations by employing a simplified
Skyrme interaction [3] with t0 = −1800 MeVfm
3, t3 = 12871 MeVfm
4 and α = 1/3. We
consider the 80Zr nucleus and keep the spin-orbit and the Coulomb interactions switched off
in the HF as well as in RPA calculations, so that the HF-RPA calculations are fully self-
consistent. Self-consistent (SC) values of mk and the energies Econ and Es shown in the first
row of this table are obtained using Eqs. (2), (7), (9) and (15). These values are accurate
to within 0.1%. We compare these highly accurate results with the corresponding ones
obtained within the HF-RPA approach. The CRPA and DRPA results presented in Table
I are calculated by integrating the RPA strength function appearing in Eq. (3) up to the
different values of the maximum energy ωmax. One can easily verify that the CRPA as well as
5
DRPA results are quite close to the highly accurate SC results (first row). For instance, the
maximum uncertainty is associated with the Es and it is about 0.3% for ωmax = 100 MeV.
Since, the RPA energies Econ, Ec and Es obtained by integrating Eq. (3) up to ωmax = 100
and 120 MeV are very close, we shall adopt in the following the value of ωmax ≈ 4Ec for the
RPA calculations.
Having demonstrated in Table I the high accuracy of our numerical calculations, we now
consider the effects of the violations of self-consistency. To investigate effects of each of the
components contributing to the self-consistency violations, we carry out the calculations for
some specific nuclei using several interactions [10, 11, 12, 13]. We denote by θls(θc) = 0 or
1 that the spin-orbit (Coulomb) interaction is turned off or on in the HF calculations. In
the RPA calculations the p-h spin-orbit and Coulomb interactions are always absent. Thus,
the HF-RPA calculations for symmetric nuclei (N = Z) with θls = θc = 0 are fully self-
consistent. The effects of ignoring the p-h spin-orbit interaction on the ISGMR energies can
be singled out by carrying out the calculations for symmetric nuclei with θls = 1 and θc = 0.
Similarly the effects due to the neglect of p-h Coulomb interaction can be estimated by
performing the calculations for the cases with θls = 0 and θc = 1. The HF-RPA calculations
for asymmetric nuclei with θls = θc = 0 yields the effect of working in the TJ scheme. In
Table II we present the results obtained using the SGII interaction [10] for several nuclei for
different combinations of θls and θc. Results for other Skyrme interactions exhibit similar
features. The 40,60Ca and 80,110Zr nuclei considered here are spin saturated systems. But,
56Ni and 100Sn are spin unsaturated nuclei having 1f7/2 and 1g9/2 orbits as the last occupied
ones, respectively. As seen in Table I, it is once again evident that the fully self-consistent
HF-RPA calculations for the 40Ca and 80Zr nuclei (θls = θc = 0 case) yield small values
for δEcon and δEs (< 0.1 MeV). Considering the results for the cases with θls = 1, we find
that for the spin saturated nuclei the ISGMR energy do not get altered much even if the
p-h spin-orbit interaction is not included. However, in case of the spin unsaturated nuclei,
the effect of the neglect of the p-h spin-orbit interaction in the RPA leads to a significant
spurious enhancement of the ISGMR energy. Following Eq. (1), one can easily verify that
such enhancements in the ISGMR energies are equivalent to overestimating the value of
Knm by about 30-40 MeV, which is unreasonably large in view of the current accuracy of
the experimental data. Let us now focus on the results obtained for 60Ca and 110Zr nuclei
with θls = θc = 0 and
40Ca and 80Zr nuclei with θls = 0 but θc = 1. Compilation of these
6
results enables us to investigate the effects of working in the TJ scheme as well as the neglect
of the Coulomb term in the p-h interaction used in the RPA calculations. One can conclude
from Table II that both of these factors, contributing to the violation of self-consistency,
tend to underestimate the ISGMR energy compared with its SC value.
So far we have seen in a systematic manner how the inconsistency in implementing the
HF-RPA approach introduces uncertainty in the ISGMR energy. Considering the results
of Tables II, one may expect that the total effects of the self-consistency violation on the
ISGMR energy may be smaller. Because, the absence of p-h spin-orbit interaction tends to
increase the ISGMR energy while working in TJ scheme or ignoring the Coulomb term in p-h
interaction used in the RPA tends to underestimate the ISGMR energy. As an illustration,
we present in Table III the results for 90Zr and 208Pb nuclei which are widely used to extract
the value of Knm from the ISGMR energies. These calculations are carried out for several
Skyrme interactions such as SGII, SkM∗, SLy4, SK255 and SK272 [10, 11, 12, 13]. The choice
of these interactions cover a wide range of incompressibility coefficientKnm = 214−272 MeV,
symmetry energy coefficient J = 27 − 37 MeV and the spin-orbit strength W0 = 95 − 130
MeVfm5. We see that in the case of 90Zr, the ISGMR energy Econ obtained in DRPA agrees
with the corresponding self-consistent values to within 0.3 MeV. For the 208Pb nucleus, we
have | δEcon |≤ 0.4 MeV. For both nuclei, the behaviour of δEs is similar to that of δEcon
except for the fact that former one is larger by about 0.2 MeV. Thus, the net uncertainties
in the ISGMR energies due to the self-consistency violations are reasonably small for the
90Zr and 208Pb nuclei and are comparable to that attained in recent experiments. Of course,
this may not be the case for spin saturated nuclei near the drip line; the inconsistency in the
HF-RPA calculations introduced by the Coulomb interaction and the asymmetry in these
nuclei may overshadow the effect of neglecting the p-h spin-orbit interaction.
In summary, we have investigated in detail the effects of the violations of the self-
consistency in the HF-RPA calculation of the ISGMR energies Econ and Es. In particular,
we considered the self-consistency violations caused by ignoring the spin-orbit and Coulomb
terms in the p-h interaction and by carrying out the RPA calculations in the TJ scheme. We
performed the HF-RPA calculations for the ISGMR energies for several nuclei with the SGII
Skyrme interaction and demonstrated that ignoring the spin-orbit term in the p-h interac-
tion gives rise to a spurious enhancement in the values of Econ and Es for spin unsaturated
nuclei. On the contrary, neglect of the Coulomb term in the p-h interaction and performing
7
the RPA calculations in the TJ scheme underestimates the ISGMR energies. Finally, we
calculated the ISGMR energies for the 90Zr and 208Pb nuclei for the five different Skyrme
interactions SGII, SkM∗, SLy4, SK255 and SK272 and show that in these nuclei, widely
used to extract the value of Knm, the various elements contributing to the self-consistency
violations tend to counterbalance their effects leading to an uncertainty of about 0.1 − 0.4
MeV in the values of ISGMR energies, which is quite acceptable in view of the accuracy of
the experimental data currently available.
This work was supported in part by the US Department of Energy under grant # DOE-
FG03-93ER40773 and the National Science Foundation under grant #PHY-0355200.
[1] D. H. Youngblood, H. L. Clark and Y. W. Lui, Phys. Rev. Lett. 82, 691 (1999).
[2] P. G. Reinhard, Ann. Phys. (Leipzig) 1, 632 (1992) and private communication.
[3] B. K. Agrawal, S. Shlomo and A. I. Sanzhur, Phys. Rev. C 67, 034114 (2003).
[4] S. Shlomo and G. F. Bertsch, Nucl. Phys. A243, 507 (1975).
[5] O. Bohigas, A. M. Lane and J. Martorell, Phys. Rep. 51, 267 (1979).
[6] G. Colo and Nguyen Van Giai, Nucl. Phys. A731, 15c (2004).
[7] D. Vautherin and D. M. Brink, Phys. Rev. C 5, 626,(1972).
[8] A. Bohr and B. Mottelson, Nuclear Structure (Benjamin, London, 1975), Vol. II, Chap. 6.
[9] E. Lipparini and S. Stringari, Phys. Rep 175, 103 (1989).
[10] Nguyen Van Giai and H. Sagawa, Phys. Lett. B 106, 379 (1981).
[11] J. Bartel, P. Quentin, M. Brack, C. Guet and H. -B. Hakansson, Nucl. Phys. A386, 79 (1982).
[12] E. Chabanat, P. Bonche, P. Haensel, J. Meyer and R. Schaeffer, Nucl. Phys. A635, 231 (1998).
[13] B. Agrawal, S. Shlomo and V. Kim Au, Phys. Rev. C68, 031304(R) (2003).
8
TABLE I: Comparison of self-consistent (SC) values of the moments mk (in MeV
kfm4) and the
ISGMR energies (in MeV) for the 80Zr nucleus calculated in the HF approach with the ones
obtained within the continuum and discretized RPA using the smearing parameter Γ = 0.01 MeV.
The RPA values are obtained by integrating Eq. (3) over the energy range 0 − ωmax (in MeV).
The calculations are performed using a simplified Skyrme interaction with t0 = −1800 MeVfm
3,
t3 = 12871 MeVfm
4 and α = 1/3.
ωmax m−1 m0 m1 m3 Econ Ec Es
SC 185.1 96346 54490227 22.81 23.78
CRPA 80 185.0 4189 96229 53549803 22.80 22.97 23.59
CRPA 100 185.1 4190 96289 54024448 22.81 22.98 23.69
CRPA 120 185.1 4190 96303 54182661 22.81 22.98 23.72
DRPA 80 185.4 4194 96359 53721598 22.80 22.97 23.61
DRPA 100 185.4 4196 96426 54292408 22.80 22.98 23.73
DRPA 120 185.4 4196 96449 54544736 22.81 22.99 23.78
9
TABLE II: Comparison between the SC and DRPA results for the ISGMR energies (in MeV)
for several nuclei obtained for the SGII interaction [10]. The notation θls(θc) = 1 or 0 indicates
that the HF calculations are performed with or without the inclusion of the spin-orbit (Coulomb)
interaction. Note that the p-h interaction used in the RPA calculations does not include the
spin-orbit and Coulomb terms.
DRPA SC
Nucleus θls θc Econ Ec Es Econ Es δEcon δEs
40Ca 0 0 22.53 22.76 23.85 22.60 23.92 -0.07 -0.07
1 0 22.31 22.63 23.96 22.58 23.93 -0.27 0.03
0 1 20.97 21.24 22.44 21.38 22.69 -0.41 -0.25
80Zr 0 0 19.57 19.69 20.26 19.60 20.32 -0.03 -0.06
1 0 19.49 19.65 20.36 19.58 20.33 -0.09 0.03
0 1 17.38 17.54 18.20 17.81 18.63 -0.43 -0.43
56Ni 1 0 23.14 23.35 24.44 21.49 22.41 1.65 2.03
100Sn 1 0 20.16 20.27 20.95 19.04 19.57 1.12 1.38
60Ca 0 0 16.08 16.81 19.13 17.32 19.57 -1.24 -0.44
110Zr 0 0 15.64 16.04 17.35 16.33 17.52 -0.69 -0.17
10
TABLE III: Comparison between the SC and the DRPA results for the ISGMR energies (in MeV)
for the 90Zr and 208Pb nuclei. The DRPA results are obtained using the smearing parameter
Γ = 0.01 MeV and the values of ωmax = 70 and 50 MeV for the
90Zr and 208Pb nuclei, respectively.
Note that the values of δEcon are comparable to the experimental uncertainties.
DRPA SC
Nucleus Int. Econ Ec Es Econ Es δEcon δEs
90Zr SGII 17.93 18.05 18.70 17.70 18.30 0.23 0.40
SkM∗ 17.72 17.85 18.55 17.43 18.00 0.29 0.55
SLy4 18.19 18.33 19.07 17.86 18.46 0.33 0.61
SK255 18.83 18.97 19.64 18.87 19.57 - 0.04 0.07
SK272 19.55 19.70 20.45 19.49 20.28 0.06 0.17
208Pb SGII 13.53 13.64 14.22 13.42 13.94 0.11 0.28
SkM∗ 13.46 13.61 14.31 13.36 13.88 0.10 0.43
SLy4 14.04 14.16 14.80 13.68 14.22 0.36 0.58
SK255 13.83 13.98 14.65 14.26 14.88 - 0.43 -0.23
SK272 14.42 14.58 15.28 14.73 15.42 - 0.31 -0.14
11


Physical Review C

Consequences of self-consistency violations in Hartree-Fock random-phase approximation calculations of the nuclear breathing mode energy

Crossref

Texas A&amp;M Repository

arXiv:nucl-th/0406013v1  7 Jun 2004Consequences of self-consistency violations in Hartree-Fockrandom-phase approximation calculations of the nuclearbreathing mode energyB. K. Agrawal and S. ShlomoCyclotron Institute, Texas A&M University, College Station, TX 77843, USAAbstractWe provide for the first time accurate assessments of the consequences of violations of self-consistency in the Hartree-Fock based random phase approximation (RPA) as commonly used tocalculate the energy Ec of the nuclear breathing mode. Using several Skyrme interactions we findthat the self-consistency violated by ignoring the spin-orbit interaction in the RPA calculationcauses a spurious enhancement of the breathing mode energy for spin unsaturated systems. Con-trarily, neglecting the Coulomb interaction in the RPA or performing the RPA calculations in theTJ scheme underestimates the breathing mode energy. Surprisingly, our results for the 90Zr and208Pb nuclei for several Skyrme type effective nucleon-nucleon interactions having a wide rangeof nuclear matter incompressibility (Knm ∼ 215 − 275 MeV) and symmetry energy (J ∼ 27 − 37MeV) indicate that the net uncertainty (δEc ∼ 0.3 MeV) is comparable to the experimental one.PACS numbers: 21.60.Jz, 24.30.Cz, 21.65.+f1The Hartree-Fock (HF) based random phase approximation (RPA) provides a microscopicdescription of the nuclear compressional modes. The most special of these compressionalmodes is the isoscalar giant monopole resonance (ISGMR) also referred to as the breathingmode. The centroid energy Ec of the ISGMR enables one to determine the value of nuclearmatter incompressibility coefficient Knm which plays an important role in understanding awide variety of phenomena ranging from heavy-ion collision to supernova explosions. Recentexperimental data [1] for the Ec in heavy nuclei have rather small uncertainties (∼ 0.1 −0.3 MeV). Since, the uncertainty δEc associated with Ec is approximately related to theuncertainty δKnm in Knm by,δKnmKnm= 2δEcEc, (1)the value of δKnm is only about 10 MeV, for Knm = 250 MeV and Ec = 14.17 ± 0.28MeV for the 208Pb nucleus. On the theoretical side, most of the results obtained withinthe HF-RPA theory, as employed for the determination of Ec, are plagued by the lack ofself-consistency, see however, Ref. [2]. Self-consistency is violated due to the neglect of thespin-orbit and Coulomb terms in the particle-hole interaction used in the RPA calculations.Furthermore, some of the RPA calculations are performed in the TJ (isospin) scheme. Un-certainties in the value of Ec calculated in the HF-RPA approach arise also due to variousnumerical approximations such as the use of very small particle-hole excitation energies andthe introduction of the smearing parameter Γ to smoothen the strength function. In Ref.[3], we have investigated in detail the effects of these numerical approximations and empha-sized the need to perform a highly accurate calculation of the strength function. So, it isof utmost importance that in order to determine an accurate value of Knm one must havean accurate knowledge of the effects of violations of self-consistency on the centroid energyof the ISGMR. We may also point out that modifying the particle-hole interaction in an adhoc manner in such a way that the spurious state associated with the center of mass motionappears at zero energy [4] does not restore the self-consistency (see also Ref. [3]).In this work we provide for the first time accurate assessments of the effects of violationsof self-consistency on the constrained and the scaling energies,Econ =√m1m−1, and Es =√m3m1, (2)of the ISGMR. Heremk =∫∞0ωkS(ω)dω (3)2is the energy moments of the strength functionS(ω) =∑n|〈n | f | 0〉|2 δ(ω − ωn), (4)for the monopole operator f(r) =∑Ai=1 r2i .In the following the fully self-consistent and highly accurate values of the constraintenergy, Econ, and the scaling energy, Es, are obtained by calculating the energy momentsmk for k = −1, 1 and 3 using the constrained HF (CHF), the double commutator and thegeneralized scaling approaches, respectively [5]. The self-consistent values of Econ and Es arethen compared with those obtained within the HF-RPA approach [3, 4] in order to assess theeffects of each type of violation of self-consistency. It may be pointed out that very recentlyin Ref. [6] some preliminary attempts are made to estimate the effects of self-consistencyviolation on the value of Econ. However, Es was not considered and the different componentscontributing to the self-consistency violations are not dealt separately. We show here thatthe appropriate knowledge of the effects of each of these components on the ISGMR energiesare instrumental in validating the HF-RPA calculations.We consider here the HF solution for the Hamiltonian,H =A∑i=1Ti +12A∑i,j=1Vij +12Z∑i,j=1V Cij , (5)where T is the kinetic energy operator, V C is the Coulomb interaction and V is the effectivetwo-body interaction of the Skyrme type [7],V12 = t0 (1 + x0Pσ12) δ(r1 − r2)+12t1 (1 + x1Pσ12)×[←−k 212δ(r1 − r2) + δ(r1 − r2)−→k 212]+t2 (1 + x2Pσ12)←−k 12δ(r1 − r2)−→k 12+16t3 (1 + x3Pσ12) ρα(r1 + r22)δ(r1 − r2)+iW0←−k 12δ(r1 − r2)(−→σ1 +−→σ2)×−→k 12, (6)where P σ12 is the spin exchange operator,−→σ i is the Pauli spin operator,−→k 12 = −i(−→∇1−−→∇2)/2and←−k 12 = −i(←−∇1 −←−∇2)/2 . Here, the right and left arrows indicate that the momentumoperators act on the right and on the left, respectively. The parameters of the Skyrme forceti, xi, α and W0 are obtained by fitting the HF results to the experimental data for the bulkproperties of finite nuclei.3Once the HF equations are solved, the m1 and m3 can be expressed in terms of theground state density ρ, kinetic energy density τ and spin density J. The value of m1 can beobtained from the corresponding double commutator [8] and is given as,m1 = 2~2m〈r2〉, (7)where,〈r2〉 =∫r2ρ(r)dr. (8)The cubic moment m3 can be evaluated in the generalized scaling approach and it is givenby[5, 9],m3 =12(2~2m)2[2T + 6Eδ + 20(Efin + Eso)+(3α + 2)(3α+ 3)Eρ] , (9)where,T =∫~2mτdr, (10)Eδ =∫ [38t0ρ2 −14t0(x0 +12)ρ21]dr (11)Efin =∫ [a0ρτ + a1ρ1τ1 + c(∇ρ)2 + d(∇ρ1)2]dr (12)Eρ =∫ [348t3ρα+2 −124t3(x3 +12)ραρ21]dr (13)Eso = −12W0∫[ρ∇ · J+ ρp∇ · Jp + ρn∇ · Jn] dr (14)with ρ1 = ρn − ρp, τ1 = τn − τp, a0 =116(3t1 + (5 + 4x2)t2), a1 =18t2(x2 +12) − t1(x1 +12),c = 164(9t1− (5 + 4x2) + t2) and d = −132(3t1(x1 +12) + t2(x2 +12)). As described in detail inRef. [5], m−1 can be evaluated via the CHF approach and is given as,m−1 =12ddλ〈r2λ〉∣∣∣∣λ=0=12d2dλ2〈Hλ〉∣∣∣∣λ=0(15)where, 〈Oλ〉 = 〈Φλ |O|Φλ〉, with Φλ being the HF solution for the constrained Hamiltonian,Hλ = H − λf.The corresponding HF-RPA values of mk are obtained from Eq. (3) using,S(ω) =1piIm [Tr(fG(ω)f)] (16)4with G(ω) being the RPA Green’s function [3, 4]. The differences between the values of mkand consequently the differenceδE = E(RPA)− E(SC), (17)between the energies obtained within the HF-RPA and the self-consistent (SC) values ob-tained within the HF approach are due to the violation of self-consistency in the HF-RPA.Since we are unable to obtain a fully self-consistent value for m0, we only provide the valueof the centroid energyEc =m1m0, (18)obtained within the HF-RPA. Thus we use δEcon and δEs as measures of δEc. We point outthat we find Ec ≈ Econ.Before we present the main results, some remarks about the numerical accuracies are inorder. We carry out the HF calculations using a box of 18 fm and a mesh size of δr = 0.02fm. To determine m−1 given by Eq. (15) with an accuracy of 0.1% we calculate the firstderivative of 〈r2λ〉 using a five-point formula with the increment δλ = 0.02. In order tolimit the spurious enhancement in mk ( particularly m3), due to the Lorentzian smearingof the RPA Green’s function, to below 0.2%, we carry out the continuum RPA (CRPA)and the discretized RPA (DRPA) calculations using the very small smearing parameter ofΓ/2 = 5× 10−2 MeV and the maximum energy of 4Ec in the integral of Eq. (3). In Ref. [3]we show that in order to calculate the centroid energy accurate to within 0.1 MeV, one mustset the cut-off for the p-h excitations Emaxph > 400 MeV. In what follows, we shall presentour DRPA results obtained with Emaxph = 950 MeV.In Table I we demonstrate the accuracy of our calculations by employing a simplifiedSkyrme interaction [3] with t0 = −1800 MeVfm3, t3 = 12871 MeVfm4 and α = 1/3. Weconsider the 80Zr nucleus and keep the spin-orbit and the Coulomb interactions switched offin the HF as well as in RPA calculations, so that the HF-RPA calculations are fully self-consistent. Self-consistent (SC) values of mk and the energies Econ and Es shown in the firstrow of this table are obtained using Eqs. (2), (7), (9) and (15). These values are accurateto within 0.1%. We compare these highly accurate results with the corresponding onesobtained within the HF-RPA approach. The CRPA and DRPA results presented in TableI are calculated by integrating the RPA strength function appearing in Eq. (3) up to thedifferent values of the maximum energy ωmax. One can easily verify that the CRPA as well as5DRPA results are quite close to the highly accurate SC results (first row). For instance, themaximum uncertainty is associated with the Es and it is about 0.3% for ωmax = 100 MeV.Since, the RPA energies Econ, Ec and Es obtained by integrating Eq. (3) up to ωmax = 100and 120 MeV are very close, we shall adopt in the following the value of ωmax ≈ 4Ec for theRPA calculations.Having demonstrated in Table I the high accuracy of our numerical calculations, we nowconsider the effects of the violations of self-consistency. To investigate effects of each of thecomponents contributing to the self-consistency violations, we carry out the calculations forsome specific nuclei using several interactions [10, 11, 12, 13]. We denote by θls(θc) = 0 or1 that the spin-orbit (Coulomb) interaction is turned off or on in the HF calculations. Inthe RPA calculations the p-h spin-orbit and Coulomb interactions are always absent. Thus,the HF-RPA calculations for symmetric nuclei (N = Z) with θls = θc = 0 are fully self-consistent. The effects of ignoring the p-h spin-orbit interaction on the ISGMR energies canbe singled out by carrying out the calculations for symmetric nuclei with θls = 1 and θc = 0.Similarly the effects due to the neglect of p-h Coulomb interaction can be estimated byperforming the calculations for the cases with θls = 0 and θc = 1. The HF-RPA calculationsfor asymmetric nuclei with θls = θc = 0 yields the effect of working in the TJ scheme. InTable II we present the results obtained using the SGII interaction [10] for several nuclei fordifferent combinations of θls and θc. Results for other Skyrme interactions exhibit similarfeatures. The 40,60Ca and 80,110Zr nuclei considered here are spin saturated systems. But,56Ni and 100Sn are spin unsaturated nuclei having 1f7/2 and 1g9/2 orbits as the last occupiedones, respectively. As seen in Table I, it is once again evident that the fully self-consistentHF-RPA calculations for the 40Ca and 80Zr nuclei (θls = θc = 0 case) yield small valuesfor δEcon and δEs (< 0.1 MeV). Considering the results for the cases with θls = 1, we findthat for the spin saturated nuclei the ISGMR energy do not get altered much even if thep-h spin-orbit interaction is not included. However, in case of the spin unsaturated nuclei,the effect of the neglect of the p-h spin-orbit interaction in the RPA leads to a significantspurious enhancement of the ISGMR energy. Following Eq. (1), one can easily verify thatsuch enhancements in the ISGMR energies are equivalent to overestimating the value ofKnm by about 30-40 MeV, which is unreasonably large in view of the current accuracy ofthe experimental data. Let us now focus on the results obtained for 60Ca and 110Zr nucleiwith θls = θc = 0 and40Ca and 80Zr nuclei with θls = 0 but θc = 1. Compilation of these6results enables us to investigate the effects of working in the TJ scheme as well as the neglectof the Coulomb term in the p-h interaction used in the RPA calculations. One can concludefrom Table II that both of these factors, contributing to the violation of self-consistency,tend to underestimate the ISGMR energy compared with its SC value.So far we have seen in a systematic manner how the inconsistency in implementing theHF-RPA approach introduces uncertainty in the ISGMR energy. Considering the resultsof Tables II, one may expect that the total effects of the self-consistency violation on theISGMR energy may be smaller. Because, the absence of p-h spin-orbit interaction tends toincrease the ISGMR energy while working in TJ scheme or ignoring the Coulomb term in p-hinteraction used in the RPA tends to underestimate the ISGMR energy. As an illustration,we present in Table III the results for 90Zr and 208Pb nuclei which are widely used to extractthe value of Knm from the ISGMR energies. These calculations are carried out for severalSkyrme interactions such as SGII, SkM∗, SLy4, SK255 and SK272 [10, 11, 12, 13]. The choiceof these interactions cover a wide range of incompressibility coefficientKnm = 214−272 MeV,symmetry energy coefficient J = 27 − 37 MeV and the spin-orbit strength W0 = 95 − 130MeVfm5. We see that in the case of 90Zr, the ISGMR energy Econ obtained in DRPA agreeswith the corresponding self-consistent values to within 0.3 MeV. For the 208Pb nucleus, wehave | δEcon |≤ 0.4 MeV. For both nuclei, the behaviour of δEs is similar to that of δEconexcept for the fact that former one is larger by about 0.2 MeV. Thus, the net uncertaintiesin the ISGMR energies due to the self-consistency violations are reasonably small for the90Zr and 208Pb nuclei and are comparable to that attained in recent experiments. Of course,this may not be the case for spin saturated nuclei near the drip line; the inconsistency in theHF-RPA calculations introduced by the Coulomb interaction and the asymmetry in thesenuclei may overshadow the effect of neglecting the p-h spin-orbit interaction.In summary, we have investigated in detail the effects of the violations of the self-consistency in the HF-RPA calculation of the ISGMR energies Econ and Es. In particular,we considered the self-consistency violations caused by ignoring the spin-orbit and Coulombterms in the p-h interaction and by carrying out the RPA calculations in the TJ scheme. Weperformed the HF-RPA calculations for the ISGMR energies for several nuclei with the SGIISkyrme interaction and demonstrated that ignoring the spin-orbit term in the p-h interac-tion gives rise to a spurious enhancement in the values of Econ and Es for spin unsaturatednuclei. On the contrary, neglect of the Coulomb term in the p-h interaction and performing7the RPA calculations in the TJ scheme underestimates the ISGMR energies. Finally, wecalculated the ISGMR energies for the 90Zr and 208Pb nuclei for the five different Skyrmeinteractions SGII, SkM∗, SLy4, SK255 and SK272 and show that in these nuclei, widelyused to extract the value of Knm, the various elements contributing to the self-consistencyviolations tend to counterbalance their effects leading to an uncertainty of about 0.1 − 0.4MeV in the values of ISGMR energies, which is quite acceptable in view of the accuracy ofthe experimental data currently available.This work was supported in part by the US Department of Energy under grant # DOE-FG03-93ER40773 and the National Science Foundation under grant #PHY-0355200.[1] D. H. Youngblood, H. L. Clark and Y. W. Lui, Phys. Rev. Lett. 82, 691 (1999).[2] P. G. Reinhard, Ann. Phys. (Leipzig) 1, 632 (1992) and private communication.[3] B. K. Agrawal, S. Shlomo and A. I. Sanzhur, Phys. Rev. C 67, 034114 (2003).[4] S. Shlomo and G. F. Bertsch, Nucl. Phys. A243, 507 (1975).[5] O. Bohigas, A. M. Lane and J. Martorell, Phys. Rep. 51, 267 (1979).[6] G. Colo and Nguyen Van Giai, Nucl. Phys. A731, 15c (2004).[7] D. Vautherin and D. M. Brink, Phys. Rev. C 5, 626,(1972).[8] A. Bohr and B. Mottelson, Nuclear Structure (Benjamin, London, 1975), Vol. II, Chap. 6.[9] E. Lipparini and S. Stringari, Phys. Rep 175, 103 (1989).[10] Nguyen Van Giai and H. Sagawa, Phys. Lett. B 106, 379 (1981).[11] J. Bartel, P. Quentin, M. Brack, C. Guet and H. -B. Hakansson, Nucl. Phys. A386, 79 (1982).[12] E. Chabanat, P. Bonche, P. Haensel, J. Meyer and R. Schaeffer, Nucl. Phys. A635, 231 (1998).[13] B. Agrawal, S. Shlomo and V. Kim Au, Phys. Rev. C68, 031304(R) (2003).8TABLE I: Comparison of self-consistent (SC) values of the moments mk (in MeVkfm4) and theISGMR energies (in MeV) for the 80Zr nucleus calculated in the HF approach with the onesobtained within the continuum and discretized RPA using the smearing parameter Γ = 0.01 MeV.The RPA values are obtained by integrating Eq. (3) over the energy range 0 − ωmax (in MeV).The calculations are performed using a simplified Skyrme interaction with t0 = −1800 MeVfm3,t3 = 12871 MeVfm4 and α = 1/3.ωmax m−1 m0 m1 m3 Econ Ec EsSC 185.1 96346 54490227 22.81 23.78CRPA 80 185.0 4189 96229 53549803 22.80 22.97 23.59CRPA 100 185.1 4190 96289 54024448 22.81 22.98 23.69CRPA 120 185.1 4190 96303 54182661 22.81 22.98 23.72DRPA 80 185.4 4194 96359 53721598 22.80 22.97 23.61DRPA 100 185.4 4196 96426 54292408 22.80 22.98 23.73DRPA 120 185.4 4196 96449 54544736 22.81 22.99 23.789TABLE II: Comparison between the SC and DRPA results for the ISGMR energies (in MeV)for several nuclei obtained for the SGII interaction [10]. The notation θls(θc) = 1 or 0 indicatesthat the HF calculations are performed with or without the inclusion of the spin-orbit (Coulomb)interaction. Note that the p-h interaction used in the RPA calculations does not include thespin-orbit and Coulomb terms.DRPA SCNucleus θls θc Econ Ec Es Econ Es δEcon δEs40Ca 0 0 22.53 22.76 23.85 22.60 23.92 -0.07 -0.071 0 22.31 22.63 23.96 22.58 23.93 -0.27 0.030 1 20.97 21.24 22.44 21.38 22.69 -0.41 -0.2580Zr 0 0 19.57 19.69 20.26 19.60 20.32 -0.03 -0.061 0 19.49 19.65 20.36 19.58 20.33 -0.09 0.030 1 17.38 17.54 18.20 17.81 18.63 -0.43 -0.4356Ni 1 0 23.14 23.35 24.44 21.49 22.41 1.65 2.03100Sn 1 0 20.16 20.27 20.95 19.04 19.57 1.12 1.3860Ca 0 0 16.08 16.81 19.13 17.32 19.57 -1.24 -0.44110Zr 0 0 15.64 16.04 17.35 16.33 17.52 -0.69 -0.1710TABLE III: Comparison between the SC and the DRPA results for the ISGMR energies (in MeV)for the 90Zr and 208Pb nuclei. The DRPA results are obtained using the smearing parameterΓ = 0.01 MeV and the values of ωmax = 70 and 50 MeV for the90Zr and 208Pb nuclei, respectively.Note that the values of δEcon are comparable to the experimental uncertainties.DRPA SCNucleus Int. Econ Ec Es Econ Es δEcon δEs90Zr SGII 17.93 18.05 18.70 17.70 18.30 0.23 0.40SkM∗ 17.72 17.85 18.55 17.43 18.00 0.29 0.55SLy4 18.19 18.33 19.07 17.86 18.46 0.33 0.61SK255 18.83 18.97 19.64 18.87 19.57 - 0.04 0.07SK272 19.55 19.70 20.45 19.49 20.28 0.06 0.17208Pb SGII 13.53 13.64 14.22 13.42 13.94 0.11 0.28SkM∗ 13.46 13.61 14.31 13.36 13.88 0.10 0.43SLy4 14.04 14.16 14.80 13.68 14.22 0.36 0.58SK255 13.83 13.98 14.65 14.26 14.88 - 0.43 -0.23SK272 14.42 14.58 15.28 14.73 15.42 - 0.31 -0.1411

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Consequences of self-consistency violations in Hartree-Fock random-phase approximation calculations of the nuclear breathing mode energy

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Consequences of self-consistency violations in Hartree-Fock random-phase approximation calculations of the nuclear breathing mode energy

Abstract

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Texas A&M University

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