MAGIC NUCLEI IN SUPERHEAVY VALLEY
ABSTRACT An extensive theoretical search for the proton magic number in the superheavy
valley beyond $Z=$82 and corresponding neutron magic number after $N=$126 is
carried out. For this we scanned a wide range of elements $Z=112130$ and their
isotopes. The well established nonrelativistic SkrymeHartreeFock and
Relativistic Mean Field formalisms with various force parameters are used.
Based on the calculated systematics of pairing gap, two neutron separation
energy and the shell correction energy for these nuclei, we find $Z=$120 as the
next proton magic and N=172, 182/184, 208 and 258 the subsequent neutron magic
numbers.
 [Show abstract] [Hide abstract]
ABSTRACT: We calculate the ground state properties of recently synthesized superheavy nuclei starting from $Z$=105120. The nonrelativistic and relativistic mean field formalisms is used to evaluate the binding energy, charge radius, quadrupole deformation parameter and the density distribution of nucleons. We analyzed the stability of the nuclei based on the binding energy and neutron to proton ratio. We also studied the bubble structure of the nucleus which reveals about the special features of the superheavy nucleus.International Journal of Modern Physics E 07/2012; 22(1). · 0.63 Impact Factor
Page 1
Modern Physics Letters A
Vol. 27, No. 30 (2012) 1250173 (9 pages)
c ? World Scientific Publishing Company
DOI: 10.1142/S0217732312501738
MAGIC NUCLEI IN SUPERHEAVY VALLEY
M. BHUYAN
School of Physics, Sambalpur University, Jyotivihar, Burla 768019, India
bunuphy@iopb.res.in
S. K. PATRA
Institute of Physics, Sachivalaya Marg, Bhubaneswar 751005, India
patra@iopb.res.in
Received 7 January 2012
Revised 10 August 2012
Accepted 28 August 2012
Published 18 September 2012
An extensive theoretical search for the proton magic number in the superheavy valley
beyond Z = 82 and the corresponding neutron magic number after N = 126 is carried
out. For this we scanned a wide range of elements Z = 112–130 and their isotopes.
The wellestablished nonrelativistic Skryme–Hartree–Fock and Relativistic Mean Field
formalisms with various force parameters are used. Based on the calculated systematics
of pairing gap, twoneutron separation energy and the shell correction energy for these
nuclei, we find Z = 120 as the next proton magic and N = 172, 182/184, 208 and 258
the subsequent neutron magic numbers.
Keywords: Relativistic Mean Field theory; Skryme–Hartree–Fock theory; the average
pairing gap; twoneutron separation energy; shell correction energy.
PACS Nos.: 21.10.Dr., 21.60.n., 23.60.+e., 24.10.Jv
After the discovery of artificial transmutation of elements by Sir Ernest Rutherford
in 1919,1the search for new elements is an important issue in nuclear science. The
existence of elements beyond the last heaviest naturally occurring238U, i.e. the
discoveryof Neptunium, Plutonium and other 14 elements (transuranium elements),
which make a separate block in Mendeleev’s periodic table was a revolution in
nuclear chemistry. This enhancement in the periodic table raises a few questions:
• Whether there is a limited number of elements that can coexist either in nature
or can be produced from artificial synthesis by using modern technique?
• What is the maximum number of protons and neutrons that of a nucleus?
• What is the next double shell closure nucleus beyond208Pb?
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M. Bhuyan & S. K. Patra
To answer these questions, first we have to understand the agent which is re
sponsible to rescue the nucleus against Coulomb repulsion. The obvious reply is the
shell energy, which stabilizes the nucleus against Coulomb disintegration.2Many
theoretical models, like the macroscopic–microscopic (MM) calculations involves
to explain some prior knowledge of densities, singleparticle potentials and other
bulk properties which may accumulate serious error in the largely extrapolated
mass region of interest. They predict the magic shells3–7at Z = 114 and N = 184
which could have surprisingly long lifetime even of the order of a million years.6,8–11
Some other such predictions of shellclosure for the superheavy region within the
relativistic and nonrelativistic theories depend mostly on the force parameters.12,13
Experimentally, till now, the quest for superheavy nuclei has been dramatically
rejuvenated in recent years owing to the emergence of hot and cold fusion reactions.
In cold fusion reactions involving a doubly magic spherical target and a deformed
projectile were used at GSI14–19to produce heavy elements upto Z = 110–112. In
hot fusion evaporation reactions with a deformed transuranium target and a doubly
magic spherical projectile were used in the synthesis of superheavy nuclei Z =
112–118 at Dubna.20–26At the production time of Z = 112 nucleus at GSI the fusion
crosssection was extremely small (1 pb), which led to the conclusion that reaching
still heavier elements will be very difficult. At this time, the emergence of hot fusion
reactions using48Ca projectiles at Dubna has drastically changed the situation and
nuclei with Z = 114–118 were synthesized and also observed their αdecay as well
as terminating spontaneous fission events. It is observed that Z = 115–117 nuclei
have long αdecay chains contrast to the short chains of Z = 114–118. Moreover,
the lifetimes of the superheavy nuclei with Z = 110–112 are in milliseconds and
microseconds whereas the lifetime of Z = 114–118 up to 30 s. This pronounced
increase in lifetimes for these heavier nuclei has provided great encouragement to
search the magic number somewhere beyond Z = 114. Moreover, it is also an
interesting and important question for the recent experimental discovery27–30say
chemical method of Z = 122 from the natural211,213,217,218Th which have long lived
superdeformed (SD) and/or hyperdeformed (HD) isomeric states 16 to 22 orders
of magnitude longer than their corresponding ground state (halflife of292122 is
t1/2≥ 108years).
Here, our aim is to look for the next double closed shell nucleus beyond
208Pb which may be a possible candidate for the experimentalists to look for.
For this, we have used two welldefined but distinct approaches (i) nonrelativistic
Skryme–Hartree–Fock (SHF) with FITZ, SIII, SkMP and SLy4 interactions31,32
(ii) Relativistic Mean Field (RMF) formalism33–37with NL3, G1, G2 and NLZ2
parameter sets. These models have been successfully applied in the description of
nuclear structure phenomena both in βstable and βunstable regions throughout
the periodic chart. The constant strength scheme is adopted to take care of pairing
correlation38,39and evaluated the pairing gaps ∆nand ∆pfor neutron and proton
respectively from the celebrity BCS equations.40This type of prescription for pair
ing effects, in both RMF and SHF has been used by us and many others.37–39,41
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Magic Nuclei in Superheavy Valley
0.01
0
0.01
0
0.5
1
1.5
0.1
0.5
0.2
0.3
0.4
0
0.5
1
1.5
120
126
132138
0.3
0.4
FITZ
SIII
SkMP
SLy4
NLZ2
NL3
G1
G2
120
126
132138
0
0.5
1
1.5
Z=78
Z=80
Z=82
∆n (MeV)
∆p (MeV)
N
N
Z=82
Z=80
Z=78
Fig. 1. The proton (neutron) average pairing gap ∆p (∆n) for Z = 78–82 with N = 120–140.
Within this pairing approach, it is shown that the results for binding energy are
almost identical with the predictions of the Relativistic Hartree–Bogoliubov (RHB)
approach37,42
It is well understood and settled that the properties of a magic number for a
nuclear system has the following characteristics:
• The average pairing gap for proton ∆pand neutron ∆nat the magic number is
minimum.
• The binding energy per particle is maximum compared to the neighboring one,
i.e. there must be a sudden decrease (jump) in twoneutron (or twoproton) sep
aration energy S2njust after the magic number in an isotopic or isotonic chain.
• At the magic number, the shell correction energy Eshellis minimum negative. In
other words, a pronounced energy gap in the singleparticle levels ǫn,p appears
at the magic number.
We focus on the shell closure properties in the superheavy valley based on the
above three important observables and identified the magic proton and neutron
numbers. Before going to that region, first of all, we have tested these observ
ables for a wellknown and experimentally verified double closed Pb isotopes. For
this representative case, we have taken the region Z = 78–82 with N = 120–140
and calculated the average pairing gap (for proton ∆pand neutron ∆n), and two
neutron separation energy S2n. These are shown in Figs. 1 and 2 respectively. From
Fig. 1, it is clear that the ∆pgives the minimum value (almost zero) for Z = 82 and
∆nis minimum at N = 126 in the isotopic chain of all these atomic nuclei. Further
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M. Bhuyan & S. K. Patra
6
12
18
Z=78
Z=80
Z=82
Z=80 (Expt.)
Z=82 (Expt.)
6
12
18
6
12
18
6
12
18
6
12
18
6
12
18
120
126
132138
6
12
18
120
126
132
N
138
6
12
18
FITZ
N
S2n (MeV)
S2n (MeV)
SIII
SkMP
SLy4
NLZ2
NL3
G1
G2
Fig. 2.
of SHF and RMF theory.
The twoneutron separation energy S2nfor Z = 78–82 and N = 120–140 in the framework
the values of S2nenergy, repeat the prediction as achieved by the average pairing
gap calculation. It is worthy to mention that the above defined magic properties
are clearly observed from figures (Figs. 1 and 2) for Z = 82 with N = 126.
Our aim is to find the next magic nuclei beyond208Pb. We scanned a wide range
of nuclei starting from the protonrich to the neutronrich region in the superheavy
valley (Z = 112 to Z = 130). The average pairing gap for proton ∆pand for neutron
∆nare the representative of strength of the pairing correlations. The curves for ∆p
are displayed in Fig. 3 obtained by SHF and RMF with FITZ, SIII, SLy4, SkMP and
NL3, NLZ2, G1, G2 force parametrizations. If we investigate the figure carefully,
it is clear that the value of ∆palmost zero for the whole Z = 120 isotopic chain in
both theoretical approaches. A similar ∆pis observed for a few cases of Z = 124
and Z = 114 isotopes.
To predict the corresponding neutron shell closure of the magic Z = 120, we
have estimated the neutron pairing gap ∆nfor all elements Z = 112–130 with their
corresponding isotopic chain. As a result of this, the calculated ∆nfor the whole
atomic nuclei in the isotopic chains are displayed in Fig. 4. We obtained an arclike
structure with vanishing ∆nat N = 182, 208 and N = 172, 184, 258 respectively
for SHF and RMF of the considered parameter sets. Further, the neutron pairing
gap is found to be minimum among the isotopic chains pointing towards the magic
nature of Z = 120. Therefore, all of these force parameters are directing Z = 120
as the next magic number after Z = 82.
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Magic Nuclei in Superheavy Valley
0
0.4
0.8
0
0.4
0.8
112
114
116
118
120
122
124
126
128
130
0
0.4
0.8
0
0.4
0.8
0
0.4
0.8
0
0.4
0.8
168
182
196
210
0
0.4
0.8
162 180 198 216 234 252
N
0
0.4
0.8
RMF
SHF
FITZ
SIII
SkMP
SLy4
NLZ2
NL3
G1
G2
∆p (MeV)
N
∆p (MeV)
Fig. 3.
with N = 162–260.
The proton average pairing gap ∆pfor Z = 112–126 with N = 162–220 and Z = 112–130
0
0.5
1
0
2
1
2
112
114
116
118
120
122
124
126
128
130
0
0.5
1
0
2
1
0
0.5
1
0
2
1
168
180192
N
204
216
0
0.5
1
162 180 198 216 234 252
N
0
1
RMF
SHF
FITZ
SIII
SkMP
SLy4
NLZ2
NL3
G1
G2
∆n (MeV)
∆n (MeV)
Fig. 4.Same as Fig. 3 but for neutron average pairing gap ∆n.
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M. Bhuyan & S. K. Patra
160176192208 224
0
8
16
24
112
114
116
118
120
122
124
126
75
0
75
0
8
16
75
300
0
75
0
8
16
525
450
375
160176
192
N
208224
0
8
16
160176
192
N
208224
450
375
300
225
Two Neutron Separation Energy
Shell Correction Energy
FITZ
S2n (MeV)
Eshell (MeV)
SIII
SkMP
SLy4
FITZ
SIII
SkMP
SLy4
Fig. 5.
112–126 and N = 162–220 in the framework of SHF theory.
The twoneutron separation energy S2n and the shell correction energy Eshellfor Z =
As mentioned earlier, the binding energy per particle (BE/A) is maximum for
double closed nucleus compared to the neighboring one. For example, the BE/A
with SHF (FITZ set) for300,302,304120 are 7.046, 7.048 and 7.044 MeV correspond
ing to N = 180, 182 and 184 respectively. Similarly with SLy4 these values are
6.950, 6.952 and 6.933 MeV. This is reflected in the sudden jump of S2n from a
higher value to a lower one at the magic number in an isotopic chain. This lower
ing in twoneutron separation energy is an acid test for shell closure investigation.
Figure 5(left) shows the S2n as a function of neutron number for all the isotopic
chain of the considered elements for SHF formalisms. In spite of the complexity
about singleparticle and collective properties of the nuclear interaction, some sim
ple phenomenological facts emerge from the bulk properties of the lowlying states
in the even–even atomic nuclei. The S2nenergy is sensitive to this collective/single
particle interplay and provides sufficient information about the nuclear structure
effects. From Fig. 5, we notice such effect, i.e. jump in twoneutron separation en
ergy at N = 182 and 208 with SHF. The shell correction energy Eshell is a key
quantity to determine the shell closure of nucleon. This concept was introduced by
Strutinski43,44in liquiddrop model to take care of the shell effects. As a result,
the whole scenario of liquid properties converted to shell structure which could ex
plain the magic shell even in the framework of liquiddrop model. The magnitude
of total (proton plus neutron) Eshellenergy is dictated by the level density around
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Magic Nuclei in Superheavy Valley
Table 1.
in SHF(SLy4 and FITZ) and304120 in RMF (NL3 and G2). The energy are in MeV.
Singleparticle levels for neutron ǫn and proton ǫp nearer to the Fermi level for302120
Single particle energy level for neutron (ǫn)Single particle energy level for proton (ǫp)
OrbitSLy4FITZNL3G2OrbitSLy4FITZ NL3G2
3s1/2
1h9/2
2f7/2
1i13/2
2f5/2
3p3/2
3p1/2
1i11/2
2g9/2
1j15/2
2g7/2
3d5/2
3d3/2
4s1/2
1j13/2
−25.1
−24.1
−24.8
−23.32p3/2
2p1/2
1g9/2
1g7/2
2d5/2
1h11/2
2d3/2
3s1/2
1h9/2
2f7/2
1i13/2
2f5/2
3p3/2
−19.1
−16.5
−19.8
−19.0
−22.7
−19.8
−22.2
−19.2
−23.8
−20.5
−21.8
−18.7
−18.9
−17.9
−16.3
−15.3
−19.7
−19.3
−18.7
−18.1
−18.5
−18.1
−19.6
−18.1
−15.3
−13.4
−17.0
−15.4
−17.7
−16.5
−17.5
−15.9
−19.4
−16.9
−16.9
−14.9
−11.9
−11.1
−9.5
−8.8
−12.9
−12.5
−11.6
−11.3
−16.2
−15.7
−16.7
−14.4
−10.9
−8.7
−12.3
−10.7
−13.3
−11.7
−14.1
−11.9
−15.6
−13.2
−13.1
−11.0
−9.8
−7.3
−7.2
−6.0
−10.2
−9.3
−9.3
−7.5
−10.3
−8.8
−10.9
−9.6
−12.1
−11.5
−10.5
−8.6
−4.5
−4.0
−2.4
−2.0
−5.8
−5.5
−4.1
−4.1
−8.0
−8.5
−9.5
−7.2
−2.6
−0.9
−4.7
−2.4
−7.0
−3.6
−7.7
−5.7
−9.2
−8.2
−6.6
−6.0
−1.40.4
−2.6
−0.8
−7.3
−4.3
the Fermi level. A positive Eshellreduces the binding energy and a negative shell
correction energy increases the stability of the nucleus. As a representative case, we
have depicted our SHF result of Eshellin Fig. 5(right). It is clear from the figure the
extra stability of302,328120. However, we find similar results S2nand Eshellenergies
for RMF calculation at neutron numbers 172, 184, 258. Such calculations for a few
cases are reported in Ref. 45.
It is well known that, the double magic nuclei,4He,16O,40Ca,48Ca,56Ni,90Zr,
132Sn and208Pb are spherical in shape. Here we are enthusiastic to know the shape
of these predicted magic nuclei292,304120 in their ground state. For this we have
calculated the quadupole deformation parameter β2, using axially deformed RMF
and SHF formalisms for these force parameters. The results obtained from these
calculations are interesting because the ground state solution appears at β2∼ 0.0,
which is also spherical in nature.
As a further confirmatory test, the singleparticle energy levels for neutrons and
protons ǫn,pare analyzed. The calculated ǫn,pnearer to the Fermi levels are listed
in Table 1 for302120 SHF(SLy4 and FITZ) and for304120 RMF(NL3 and G2) as
representative cases. From Table 1, one can estimate the energy gaps ∆ǫn,pfor neu
tron and proton orbits. For example, in302120 (FITZ), the gap ∆ǫn= ǫn(3d3/2) −
ǫn(4s1/2) at N = 182 is 1.977 MeV and ∆ǫp= ǫp(2f5/2)−ǫp(3p3/2) = 1.340 MeV at
Z = 120, which is a considerably large value. Almost identical behavior is noticed
with RMF (at N = 184) calculations, irrespective of parameter used, confirming
Z = 120 as a clear magic number. It is well accepted that the sequence of the magic
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M. Bhuyan & S. K. Patra
number for exotic system is much different from that of the normal nuclei.46–50This
phenomenon is quite normal in superheavy region.
In summary, we have analyzed the pairing gap ∆pand ∆n, twoneutron sepa
ration energy S2n, shell correction energy Eshelland singleparticle energy ǫp,nfor
the whole Z = 112–130 region covering the protonrich to neutronrich isotopes. To
our knowledge, this is one of the first such extensive and rigorous calculations in
both SHF and RMF models using a large number of parameter sets. The recently
developed effective field theory motivated relativistic mean field forces G1 and G2
are also involved. Although the results depend slightly on the forces used, the gen
eral set of magic numbers beyond208Pb are Z = 120 and N = 172, (182 or 184)
and 258. The highly discussed proton magic number Z = 114 in the past (last four
decades) is found to be feebly magic in nature.
Acknowledgments
This work is supported in part by Council of Scientific & Industrial Research (File
No. 09/153 (0070)/2012EMRI). We thank Profs. L. Satpathy, C. R. Praharaj and
K. Kundu for discussions and a careful reading of the manuscript. M.B. thanks the
Institute of Physics for hospitality.
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