Title: The properties of the infinite nuclei in the relativistic mean field theory
1The properties of the infinite nuclei in the
relativistic mean field theory
- Zhang Hongfei
- Institute of Modern Physics ,Lanzhou China
2Collaborators
- Wei Zuo¹, Junqing Li¹, Soojae Im¹, Zhong-yu
Ma², Bao-qiu CHen². - ¹Institute of Modern Physics, CAS,
Lanzhou, P.R.China - ²China Institute of Atomic Energy,
Beijing, P.R.China
3Contents
- 1. Introduction
- 2. The formation of RMF
- 3. Properties of the superheavy element
- 287115 and its a-decay time
- 4. The BCS-type pairing with DDDI in the
- relativistic mean field theory
- 5. Summary
-
41.IntrodcutionThe investigation of the
properties of superheavy nuclei are extremely
intriguing for exploring some new, unexpected
features of the nuclear structure. It is unlikely
that SHN exists in nature, and the extreme
difficulties to synthesize SHN restrict the
experimental studies on it. Therefore,
theoretical studies are very important.
5- It is well known that the relativistic mean
field calculation gives a good description
of the structure of nuclei throughout the
periodic table. - In the first part of this report,
properties of the superheavy element 287115 and
its a-decay time are studied.
6In the second part of this report, a
density-dependent delta interaction (DDDI) is
proposed for the BCS-type pairing, in order to
pick up those states whose wave functions are
concentrated in the nuclear region and employ
the standard BCS approximation for the
single-particle states obtained from the RMF
theory with deformation.The RMFBCS(DDDI)
calculations for the isotopic chain 74-136Sr
are presented for demonstration purposes.
72. The formation of RMF
- We have used the effective Lagrangian in the
following form,
8The pairing correlation is treated by the
BCS theory. We have introduced the isospin
dependent strength of the pairing forces in the
following forms (Zhong-yu Ma, CPL,2000)for
neutrons and protons, respectively
9A blocking method is used to deal with the
unpaired nucleons, the states of
whichare chosen in such a way that the total
sing particle energy is the minimum. If the
state of the last odd nucleon is k with
energy , the total single-particle energy
can be written as
103. Properties of the superheavy element
287115 and its a-decay time
11TABLE I. Calculated binding energies, quadrupole
deformations, and ground state spin-parities of
nuclei belonging to the a-decay chainof 287115
by RMFBCS(NL-Z2)(Hongfei Zhang et, al, PRC 71,
054312 (2005)), compared with those from FRDM and
Ref.19(Sankha Das and G. Gangopadhyay, J. Phys.
G 30, 957(2004).
Result our calculated binding energies from the
RMF BCS are more consistent with the values of
FRDM than those by Sankha Das et al.
12 The stability of the newly discovered
odd-Z nucleus 287115 in the element 115 isotope
chain is interesting. There are many
physical quantities to judge the stability of a
nucleus. Among them, for a given A the nucleus
with the biggest absolute mean binding energy is
the most stable against ß decay, which is related
to the minimum Q value of ß decay (the ß
stability line) and the maximum absolute binding
energy per nucleon for a given Z connects the
fission stability line, which is related to the
minimum Q value of fission S. K. Patra et al.,
NPA651, 117 (1999). For SHN, the fission
stability is prior to the ß stability Hongfei
Zhang,et. al, Commun. Theor. Phys. 42, 871 (2004)
.
13(Hongfei Zhang et, al, PRC 71, 054312 (2005)),
From the results of both theories, the nucleus
287115 is one of the most stable in the Z 115
isotope chain.
14(Hongfei Zhang et, al, PRC 71, 054312 (2005)),
Results 1. The gaps appear at N 174, (172),
(168), (162), (160), and (186), among which N
174 is the most distinct with an energy gap of
2.64 MeV. N 172 appears to be a subshell
closure with an energy gap of 0.56 MeV. In the
SHN region, it seems that each nucleus has its
different sequence of magic numbers. 2. The
magnitudes of the shell gaps in SHN are much
smaller than those of nuclei before the actinium
region, and the Fermi surfaces are close to the
continuum. Thus, the SHN are usually not stable.
15Results 1. A distinct shell gap of about 1.61
MeV exists at Z 116. There exist some
relatively smaller gaps, which we hereafter put
in parenthesis. With this parameter set, the
shell gap at Z 120 is absent, and some smaller
shell gaps appear at Z (110), (106) and
(122), (126), (128) in addition to Z 116,
(114). 2. there is some probability that protons
will be distributed at positive levels, which are
resonant continuums. It is a common phenomenon
that the resonant continuums exist in the SHN in
the same way as they exist in very neutron- or
proton-rich nuclei. It is desirable to correctly
deal with the resonant continuum in further study
of the SHN.
16TABLE II. a-decay energies (in MeV) of 287115
chain from the RMF BCS, FRDM, and other
calculations and from the experimentaldata.
(Hongfei Zhang et, al, PRC 71, 054312 (2005)),
From Table II, we can find that the deviation
between Qth And Qexp is very small, which is not
more than 0.4MeV, indicating that the current RMF
BCS theory works well for the study of a-decay
properties in the SHN region. Ref. 9 Sankha
Das and G. Gangopadhyay, J. Phys. G 30, 957
(2004). Ref.37 L. S. Geng, H. Toki, and J.
Meng, Phys. Rev. C 68, 061303(R) (2003).
17- For the SHN, the main and often the only observed
decay is the a emission. The lifetime of the
a-decay chain of the SHN in contrast with
measured data will shed light on the nuclear
structure. - We use two different formulas to calculate the
lifetimes of this a-decay chain and compare the
results with the experimental data and the values
by Basu D. N. Basu, J. Phys. G 30, B35 (2004)
18- A fitting formula (FF) to evaluate the lifetimes
of the odd-even 287115 a-decay chain in the
generalized liquid dropmodel is (G. Royer, J.
Phys. G, Nucl. Part. Phys. 26, 11491170 (2000))
19The lifetime could also be calculated by the
Viola-Seaborg formula (VSF) V. E. Viola and G.
T. Seaborg, J. Inorg. Nucl. Chem. 28, 741(1966)
as
20(Hongfei Zhang et, al, PRC 71, 054312 (2005)),
- From Table III, one can find that the lifetimes
estimated - with FF and WKBDDM3Y agree with the
experimental - values better than those obtained by the VSF.
214. The BCS-type pairing with DDDI in the
relativistic mean field theory and its effects
for the neutron-rich nucleus
- (1) Pairing with the density-dependent delta
function interaction. - Based on the single-particle spectrum
calculated with the RMF method described above,
we carry out a state-dependent BCS calculation.
In order to improve upon the seniority pairing
calculation, we introduce the DDDI
22For either neutrons or protons, the pairing
matrix elements may be written as
23It is necessary to prevent the unrealistic
pairing of highly excited states, and to confine
the region of influence of the pairing potential
to vicinity of the Fermi surface. As discussed in
ref.S. J. Krieger, et. al,NPA 517,275(1990),
this is accomplished by defining the pairing
contribution Epair to the total energy as
24- With this definition of the pairing energy, the
state dependent energy gap are the solution of
the equations
25The pairing energy and occupation probabilities
may be written as
26(2) We apply RMF theory with the improved
BCS-type pairing to the ground properties of even
Sr isotopes including the nuclei far away from
the stability line..
- The calculations have been carried out using the
Lagrangian parameter set NL-SH, which has been
fitted to reproduce the experimental neutron
radius, including deformation and - highly exotic nuclei provides on both sides of
the stability line.
27Binding energyFig. 1 shows the binding energy
per nuclei (E/A) for Sr isotopes from the
improved RMFBCS(DDDI) and RMF. The known
empirical values (expt) are also shown. The
figure also includes the prediction of the FRDM
for comparison.
Results Our calculation results in the two
cases are in agreement with those from FRDM and
experiment. All shown parabolic shapes with a
minimum binding energy per nucleon located at A
88, where the neutron number N 50 is a magic
number.
28b. Quadrupole deformationWe show in Fig. 2 the
quadrupole deformation for the shape
corresponding to the lowest energy. The
predictions of FRDM and known experimental values
are also shown for comparison.
Results The RMFBCS(DDDI) and the RMF give
nearly the same quadrupole deformation almost for
all Sr isotopes and the quadrupole deformation
values are consistent with the results of the
FRDM and known experiment.
29C. Shap coexistence In addition to the
lowest minimum, several isotopes exhibit a second
minimum, thus implying a shape coexistence, i.e.,
the prolate and the oblate shapes differ in the
total binding energy only by several hundreds
keV.
30d. Isotope shift
With increasing mass number A, the
isotope shift change only slightly until it
reaches the magic number N50. Beyond the magic
number it increases rapidly with mass number A.
Such an anomalous behavior is a generic feature
of deformed nuclei.
31e. Two neutron separation energy
- Results
- The neutron shell closures at N50 and N82 are
found in Figs.5. - 2. Two-neutron separation energy for nucleus
A120 is positive, while that for the nucleus
A122 negative. Therefor the figure show 120Sr is
the last stable nucleus against the neutron
emission, i.e., the neutron drip-line nucleus.
32f. Single-particle states and their occupation
probabilities of rich-neutron nucleus 122Sr
- In this context, it is very interesting to study
the amount by which the contributions from the
continuum differ for the calculations with a
constant pairing interaction and the calculations
with the DDDI. - In Fig. 6, we plot the occupation probabilities
of 122Sr for the neutron levels and present the
occupation probabilities of neutron
single-particle states for two cases, the DDDI
and the constant pairing interaction usually used
in the BCS framework. - In the constant pairing calculation, we used Gn
15/A and Gp 20/A, which can give the same
pairing gaps as used DDDI pairing strength.
33- 1. There are so many degenerated states for the
deformation is nearly zero. - 2. The single particle levels under the Fermi
level from the two cases are same and all magic
neutron number below N126 are reproduced
implying the pairing effect is mostly the surface
effect. - 3. The BCS calculation with DDDI can include more
of the continuum states that are more localized
inside the nuclear region, which correspond to
resonance states.
345. Summarys
- (1). In the first part of this report, we use the
RMFBCS with the isospin dependent pairing
strength and block effect to study the properties
of the SHN 287115 and its alpha decay chain. - Due to the neutron-deficient character of the
SHN, there is some probability that protons will
be distributed at the positive levels, which form
the resonance continuum the proper treatment of
that continuum is to take account of not only the
energy of the resonant state, but also its width.
The pairing gaps, Fermi energies, pairing
correction energies, and binding energies may be
affected with the - proper consideration of the width of
resonant states, thus it is worth further
investigation. - In the SHN region, it seems that each nucleus
has its different sequence of magic numbers. The
magnitudes of the shell gaps in SHN are much
smaller than those of nuclei before the actinium
region, and the Fermi surfaces are close to the
continuum. Thus, the SHN are usually not stable.
When the number of nucleons changes, the level
diagram alters considerably the properties are
quite different for each superheavy nucleus. -
35(2) In the second part of this report, we adopt
the DDDI for the BCS-type pairing to improve
the pairing correlation. The RMFBCS(DDDI)
calculations for the isotopic chain 74-136Sr
are presented for demonstration purposes.
- We calculated the binding energies, deformation
parameters - of these nuclei. We also calculated the
binding energy per nucleon and the two-neutron
separation energy. We found that the agreement
with experimental values is very satisfactory. - The occupation probabilities in the continuum are
important for the purpose of determining if the
present pairing method is effective in the
treatment of the pairing in the continuum. While
the occupation probabilities decrease
monotonically as the states deviate from the
Fermi energy in the case of a constant pairing
interaction, they exhibit characteristic behavior
for the case of a DDDI.
36?? ??!Thank you!