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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 well-established non-relativistic Skryme–Hartree–Fock 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.

Keywords: Relativistic Mean Field theory; Skryme–Hartree–Fock theory; the average

pairing gap; two-neutron 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 co-exist 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, single-particle 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 shell-closure for the superheavy region within the

relativistic and non-relativistic 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

cross-section 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 α-decay as well

as terminating spontaneous fission events. It is observed that Z = 115–117 nuclei

have long α-decay 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 (half-life 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 well-defined but distinct approaches (i) non-relativistic

Skryme–Hartree–Fock (SHF) with FITZ, SIII, SkMP and SLy4 interactions31,32

(ii) Relativistic Mean Field (RMF) formalism33–37with NL3, G1, G2 and NL-Z2

parameter sets. These models have been successfully applied in the description of

nuclear structure phenomena both in β-stable and β-unstable 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

NL-Z2

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 two-neutron (or two-proton) 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 single-particle 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 well-known 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

NL-Z2

NL3

G1

G2

Fig. 2.

of SHF and RMF theory.

The two-neutron 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 proton-rich to the neutron-rich 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, NL-Z2, 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 arc-like

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

NL-Z2

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

NL-Z2

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 two-neutron 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 two-neutron 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 single-particle and collective properties of the nuclear interaction, some sim-

ple phenomenological facts emerge from the bulk properties of the low-lying 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 two-neutron 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 liquid-drop 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 liquid-drop 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.

Single-particle 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 single-particle 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, two-neutron sepa-

ration energy S2n, shell correction energy Eshelland single-particle energy ǫp,nfor

the whole Z = 112–130 region covering the proton-rich to neutron-rich 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)/2012-EMR-I). 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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