Journal of Nuclear and Radiochemical Sciences,Vol. 3, No. 1, pp. 195–200, 2002

195

Deformed Relativistic Mean-field Calculations on the Properties of Superheavy Nuclei
Zhongzhou Ren∗,a,b
a
Department of Physics, Nanjing University, Nanjing 210008, P. R. Chinab
Center of Theoretical Nuclear Physics, National Laboratory of Heavy-Ion Accelerator at Lanzhou,
Lanzhou 730000, P. R. China
Received: November 13, 2001; In Final Form: November 13, 2001
The structures of the nuclei with the proton number Z = 106–110 are systematically investigated using the deformed
relativistic mean-field (RMF) theory. The calculated binding energies are in good agreement with experimental ones.
The experimental data of α-decay energies are well reproduced by the calculations. Calculations show clearly thata prolate deformation is important for the ground state of these nuclei. The properties of the nuclei on the α-decay
chain of 269 Mt are predicted.
tons,25–35

1. Introduction
It was believed that there is still a long and difficult way toaccess the superheavy island around Z = 114. However the pioneering work of synthesizing elements Z = 110–112 by Hofmann et al. at GSI in Germany1–3 brings a hope to approach theisland in near future because the three elements were produced
within two years. This speeds up the researches on superheavy
nuclei both experimentally and theoretically.4–7 A breakthroughappeared at Dubna in Russia: the element Z = 114 was producedby Oganessian et al.5, 6 in 1999. A year later it was again reported that Z = 116 is synthesized at Dubna.8 At present many
big laboratories in the field of nuclear physics focus on the superheavy nuclei. New results are reported such as the synthesisof new nuclide 270 Hs in PSI (Ref. 9), 270 110 in GSI (Ref. 10),259
Db at Lanzhou.11
Theoretically there are some studies on superheavy nuclei
based on the self-consistent mean-field models or macrosocpicmicroscopic mass models.13–24 The Frankfurt group tested therelativistic mean-field (RMF) model for a known nucleus 264 Hs(Ref. 13). Ćwiok et al. investigated the ground state properties of the α-decay chain 289 114 within the framework of theSkyrme-Hartree-Fock-Bogoliubov model.14 Ren and Toki systematically calculated the properties of superheavy nuclei on the
decay chain of Z = 110–112 and Z = 114 in the RMF model.19
Shape coexistence is predicted in the ground state of superheavy
nuclei and deformation can be an important cause for the stability of superheavy nuclei based on a constraint RMF calculation.19
In this paper we study the properties of nuclei on the isotope
chain Z = 108, 109, and 110. Now these nuclei are possibly produced in many laboratories. 270 Hs and 270 110 have been synthesized in PSI and GSI, respectively.9, 10 After 259 Db is observedat Lanzhou in China,11 it is planned to investigated the properties of nuclei with Z = 108 and Z = 109 experimentally soon.12
The nuclei in this range also bridge the gap from the known actinide series to the unknown superheavy nuclei. Especially there
are some indications that deformations may be important in thisrange.
This paper is organized in the following way. Section 2 isthe formalism of the RMF model. The numerical results anddiscussions are given in sect. 3. Section 4 is a summary.
2. The Formalism of the Relativistic Mean-field Theory
In the RMF approach, we start from the local Lagrangiandensity for interacting nucleons, σ, ω, and ρ mesons, and pho∗E-mail: zren@nju.edu.cn; zren99@yahoo.com.

FAX: +86-25-3326028.

L = Ψ(iγ µ ∂µ − M)Ψ − gσ ΨσΨ − gω Ψγ µ ωµ Ψ

1
2

1
2

1
3

− gρ Ψγ µ ρaµ τa Ψ + ∂µ σ∂µ σ − m2σ σ2 − g2 σ3

1
1
1
1
4
4
4
2
1
1
1
− Raµν · Raµν + m2ρ ρaµ · ρaµ − F µν Fµν
4
2
4
µ µ1
3
− eΨγ A (1 − τ )Ψ2

− g3 σ4 + c3 (ωµ ωµ )2 − Ωµν Ωµν + m2ω ωµ ωµ

(1)

with
Ωµν = ∂µ ων − ∂ν ωµ ,Raµν = ∂µ ρaν − ∂ν ρaµ ,F µν = ∂µ Aν − ∂ν Aµ ,

(2)
(3)
(4)

where the meson fields are denoted by σ, ωµ , and ρaµ , and theirmasses are denoted by mσ , mω , and mρ , respectively. The nucleon field and rest mass are denoted by Ψ and M. Aµ is thephoton field which is responsible for the electromagnetic interaction, e2 /4π = 1/137. The effective strengths of the coupling
between the mesons and nucleons are, respectively, gσ , gω , andgρ . g2 and g3 are the nonlinear coupling strengths of the σ meson. c3 is the self-coupling term of the ω field. The isospin Paulimatrices are written as τa , τ3 being the third component of τa .
The equations of motion for the fields are easily obtained
from the variational principle.19, 25–29 In order to describe theground state properties of nuclei we need static solution of theabove Lagrangian. For this case the meson field and photon
fields are assumed to be classical fields and they are time independent (c-numbers). The nucleons move in classical fields as
independent particles (mean-field approximations). The Diracfield operator can be expanded in terms of single particle wave
functions Ψ = ∑i φi ai where ai is a particle creation operator25, 26
and φi is the single particle wave function. For actual calculations, we omit the contribution of the Fermi-sea under no-sea
approximations. The sum on single particles states runs on physical bound states, i.e. the occupied shell model states. Symmetries will simplify the calculations considerably. Time reversal
symmetry is used and therefore the spacial vector componentsfor ωµ , ρaµ , and Aµ are zero. Charge conservation guarantees that
only the third-component of the isovector field (ρ00 ) survives.25–28
We denote simply ρ00 as ρ0 . Finally we have the following Diracequations for nucleons and the Klein-Gordon equations for meson fields (for details: see References 25–28):[−i α  +βM ∗ (r) +V (r) ]φi (r) = εi φi (r),(5)

where the effective mass M ∗ (r) = M + gσ σ(r). The potential

c 2002 The Japan Society of Nuclear and Radiochemical Sciences
Published on Web 6/30/2002

196 J.Nucl.Radiochem.Sci.,Vol. 3, No. 1, 2002

Ren

TABLE 1: The ground state properties of even-even superheavy nuclei with Z = 108. The TMA force is inputted in the deformed RMF
calculation. The last two columns are the experimental data of the α-decay energy and the total binding energy.

Nuclei
256
Hs
258
Hs
260
Hs
262
Hs
264
Hs
266
Hs
268
Hs
270
Hs
272
Hs
274
Hs
276
Hs

Bthe. / MeV
1864.99
1882.16
1899.03
1915.26
1930.171944.46
1958.42
1971.80
1984.60
1996.62
2007.73

βn
0.26
0.25
0.25
0.25
0.24
0.24
0.22
0.22
0.21
0.20
0.19

βp
0.27
0.26
0.26
0.25
0.25
0.24
0.23
0.22
0.22
0.20
0.20

Rn
6.15
6.16
6.18
6.21
6.23
6.24
6.26
6.28
6.30
6.32
6.34

Rp
6.04
6.05
6.05
6.06
6.07
6.08
6.09
6.10
6.11
6.12
6.13

Qα (the.)
10.99
10.58
9.96
9.59
9.98
9.74
9.14
8.90
8.77
8.88
9.26

Qα (exp.)

Bexp. / MeV

10.54
10.18

1926.72

9.30

TABLE 2: The ground state properties of even-even superheavy nuclei with Z = 108. The NLZ2 force is inputted in the deformed RMF
calculation. The last two columns are the experimental data of the α-decay energy and the total binding energy.

Nuclei
256
Hs
258
Hs
260
Hs
262
Hs
264
Hs
266
Hs
268
Hs
270
Hs
272
Hs
274
Hs
276
Hs

Bthe. / MeV
1862.85
1879.66
1895.90
1911.48
1926.63
1941.35
1955.59
1969.22
1981.03
1992.08
2003.31

βn
0.30
0.30
0.29
0.28
0.28
0.28
0.27
0.27
0.26
0.25
0.20

βp
0.32
0.31
0.30
0.29
0.29
0.29
0.28
0.28
0.27
0.26
0.22

Rn
6.27
6.31
6.31
6.33
6.35
6.38
6.40
6.42
6.45
6.47
6.48

V (r) is a timelike component of a Lorentz vector,
V (r) = gω ω0 (r) + gρ τa ρa0 (r) + e((1 − τ3)/2)A0 (r),(6)(−  +m2σ )σ(r) = − gσ ρs (r) − g2 σ2 (r) − g3 σ3 (r), (7)(−  +m2ω )ω0 (r) = gω ρv (r) − c3ω30 (r),(8)
2(−  +mρ )ρ0 (r) = gρ ρ3 (r),(9)−  A0 (r) = eρ p (r),(10)
where ρs , ρv , and ρ p are, respectively, the densities of scalar,baryon, and proton. ρ3 is the difference between the neutron andproton densities. They will be c-numbers by taking expectation
values. Their expressions are as follows,Aρs (r) = ∑ φi (r)φi (r),

(11)

Rp
6.13
6.15
6.15
6.16
6.17
6.18
6.19
6.20
6.21
6.22
6.22

Qα (the.)
10.72
11.11
11.15
11.08
10.68
10.30
9.96
9.55
10.26
10.15
9.10

Qα (exp.)

Bexp. / MeV

10.54
10.18

1926.729.30

is chosen as N f = Nb = 20. This space is enough for the√calculations. The inputs of pairing gaps are ∆n = ∆ p = 11.2/ A MeV.
An axial deformation is assumed in calculations. For the detailsof calculations please see relevant publications.19, 27, 29, 31
At first let us focus on the global behaviour of the RMF
model. We calculate the average binding energy of nucleons(B/A) for Z = 98, 100, 102, 104, 106, and 108 isotopes and draw
the variation of average binding energy with nuclei in Figure 1.
In Figure 1, the X-axis is just used for the sequence of different
nuclei. We choose four even-even nuclei for every isotope where
the experimental data are available for many of them.36 The theoretical results are denoted by RMF1.dat and RMF0.dat where
they correspond to the results of TMA and NLZ2, respectively.
They are connected by solid lines. The experimental averagebinding energy36 is denoted by EXP.dat. It is seen from Figure 1

i=1
Aρv (r) = ∑ φ+i (r)φi (r),(12)

"RMF1.dat""EXP.dat""RMF0.dat"

7.5Aρ3 (r) = ∑ φ+i (r)τ3 φi (r),

(13)

i=1
A

ρ p (r) = ∑ φ+i (r)((1 − τ3)/2)φi (r).

(14)

i=1

Now we have a set of coupled equations for mesons and nucleons and they will be solved consistently by the iteration. After the final solution is obtained, we can calculate the binding
energies, root-mean-square radii of proton and neutron density
distributions, single particle levels, and quadrupole deformation. An axial deformation is assumed in our calculations forsuperheavy nuclei. The details of numerical calculations are described in References 19, 25–28.
3. Numerical Results and Discussions
We carry out RMF calculations with two typical sets of force
parameters, TMA19 and NLZ215, 20 where the method of the harmonic basis expansion19, 27, 29–31, 33 is used. The number of bases

Theoretical and Experimental B/A (MeV)

i=1

7.45

7.4

7.35

7.3

0

4

8

12
16
Z = 98, 100, 102, 104, 106, 108

20

24

Figure 1. The comparison of theoretical and experimental averagebinding energy of nucleons for nuclei with Z = 98, 100, 102, 104, 106,
and 108. For every isotope the results of four nuclei are shown. This
includes the present data of binding energy for even-even nuclei in theseisotopes. It is interesting to note that the experimental data lie between
two sets of the RMF results. The TMA force and NLZ2 force are usedin calculations.

Deformed Relativistic Mean-field Calculations

J.Nucl.Radiochem.Sci.,Vol. 3, No. 1, 2002

197

TABLE 3: The ground state properties of even-even superheavy nuclei with Z = 110. The TMA force is inputted in the deformed RMF
calculation. The last column is the experimental data of the α-decay energy.

Nuclei
268
110
270
110
272
110
274
110
276
110
278
110
280
110

Bthe. / MeV
1946.70
1961.39
1975.66
1989.43
2002.54
2014.562025.95

βn
0.23
0.22
0.21
0.20
0.19
0.18
0.17

βp
0.24
0.22
0.22
0.21
0.20
0.19
0.18

Rn
6.25
6.26
6.28
6.30
6.32
6.34
6.36

Rp
6.11
6.11
6.12
6.13
6.13
6.14
6.15

Qα (the.)
11.77
11.37
11.06
10.67
10.36
10.36
10.08

Tα (the.)4.37 µs33.6 µs
176 µs1.57 ms
9.71 ms
9.71 ms
54.2 ms

Qα (exp.)
10.97

TABLE 4: The ground state properties of even-even superheavy nuclei with Z = 110. The NLZ2 force is inputted in the deformed RMF
calculation. The last column is the experimental data of the α-decay energy.

Nuclei
268
110
270
110
272
110
274
110
276
110
278
110
280
110

Bthe. / MeV
1943.91
1958.87
1973.311986.07
1998.68
2010.862022.80

βn
0.26
0.26
0.25
0.24
0.21
0.20
0.18

βp
0.27
0.26
0.26
0.24
0.22
0.21
0.19

Rn
6.37
6.40
6.42
6.44
6.45
6.47
6.49

that the experimental points lie between two theoretical curves.
Therefore the RMF model can reliably limit the data in a very
narrow range. This can be useful for future experimental study.
Secondly we discuss in detail the theoretical results. The
variation of the total binding energy with nucleon number for
Z = 106 isotopes are given in Figure 2. The two sets of theoretical results with TMA and NLZ2 are represented by two curves(Z106RMF1.dat and Z106RMF0.dat). For this isotope chain,
only the binding energy of 260 Sg is measured and it is plotted bya point (Z106EXP.dat). The experimental datum36 is between
two curves. The difference between two sets of theoretical results is very small and this is a good indication for the stability
and reliability of the model in this mass range.
The theoretical results of Z = 108 isotopes with TMA and
NLZ2 are listed in Tables 1 and 2, respectively. In Tables 1
and 2, the first column is for nuclei. Bthe. is the theoretical binding energy, whereas R p and Rn are the root-mean-square radiifor the proton- and neutron-density distributions, respectively.
The symbols βn and β p in Tables 1 and 2 denote the quadrupole
deformations of neutrons and protons, respectively. Further, thesymbols Qα (the.) and Qα (exp.) are used for the calculated αdecay energies and the experimental ones. The experimental
binding energies are obtained from the nuclear mass table36 andthe experimental α-decay energies can be deduced accordingly.
They are listed in the last two columns for comparisons. The
binding energies of Z = 106 in Figure 2 are inputted for the calculation of α-decay energies of Z = 108.

Qα (the.)
11.02
10.78
10.58
11.45
10.65
9.52
8.81

Tα (the.)219 µs835 µs
2.63 ms
22.2 µs
1.76 ms
2.16 s361 sQα (exp.)
10.97

It is seen from Table 1 that the theoretical decay energies arevery close to the experimental data. The average difference between the theoretical binding energy and experimental one isapproximately 0.4 MeV. This shows that the RMF model cangive a reliable result for the decay energy. The maximum difference for the binding energy of this mass range is 3.45 MeV for264
Hs. It corresponds to a relative difference 0.2%. This is very
close to the predicting limit of the RMF model for the binding
energy of a nucleus.
Table 2 is the RMF results of Z = 108 with NLZ2. The theoretical decay energy is very close to the data. The binding energy agrees well with the experimental datum. If we compare
Tables 1 and 2 together, we find both results are very close. The
RMF model with TMA overestimates very slightly the data ofbinding energy and that with NLZ2 underestimate a little thedata. Both of them predict there are prolate deformation in these
nuclei. The parameter of quadrupole deformation (βn and β p )is approximately 0.2–0.3. A constraint RMF calculation with
NLZ2 is carried out for the nucleus 270 Hs. The energy surface of270
Hs with the variation of quadrupole deformation is plotted in
Figure 3. Here the number of the bases is chosen as N f = Nb = 18.
It is seen clearly that there exists a minimum at β2 = 0.27. This
corresponds to the ground state of 270 Hs in Table 2.
The variation of decay energy with nucleon number are drawn
in Figure 4. The experimental points are between two sets of-1950"Z108A270z.dat""Z106RMF1.dat""Z106EXP.dat""Z106RMF0.dat"

2020

-1955
2000
1980

Energy (MeV)

Theoretical and Experimental Binding Energy (MeV) Z=106

Rp
6.20
6.21
6.22
6.22
6.23
6.24
6.25

1960
1940

-1960

1920
-1965
1900
1880
-1970

1860
1840
250

-0.2

255

260

265

270
275
Nucleon Number (A)

280

285

290

Figure 2. The variation of the total binding energy of Z = 106 isotopes with nucleon number (A). The two sets of theoretical results areconnected by solid lines.

0

0.2
0.4
Quadrupole deformation parameter

0.6

0.8

Figure 3. The energy surface of 270 Hs. This is the variation of theground state energy with the quadrupole deformation. The points are
numerical results and they are connected by solid lines. A mimimum
appears at the deformation parameter 0.27 and it corresponds to thedeformation of the ground state.

198 J.Nucl.Radiochem.Sci.,Vol. 3, No. 1, 2002

Ren

TABLE 5: The binding energies, deformations,
nuclear radii, α-decay energies, and the lifetimes of nuclei on the α-decay chain 269 Mt. The√
input pairing gaps: ∆p = ∆n = 11.2/ A MeV. The TMA force is used.

Nuclei
269
Mt
265
Bh
261
Db
257
Lr
253
Md
249
Es
245
Bk

B / MeV
1960.17
1942.08
1923.191904.03
1883.85
1863.23
1841.84

βn
0.22
0.23
0.26
0.26
0.26
0.26
0.27

βp
0.23
0.24
0.27
0.27
0.27
0.27
0.28

Rn
6.26
6.24
6.23
6.20
6.17
6.14
6.12

Rp
6.10
6.07
6.05
6.01
5.98
5.94
5.91

Qα
10.21
9.41
9.14
8.12
7.68
6.91

Tα
68.8 ms
2.56 s3.33 s1.28 ×103 s
8.4 ×103 s
1.53 ×106 s

TABLE 6: The binding energies, deformations,
nuclear radii, α-decay energies, and the lifetimes of nuclei on the α-decay chain of 269 Mt.
√
The input pairing gaps: ∆p = ∆n = 11.2/ A MeV. The NLZ2 force is used.

Nuclei
269
Mt
265
Bh
261
Db
257
Lr
253
Md
249
Es
245
Bk

B / MeV
1957.41
1939.56
1920.89
1901.03
1880.24
1859.72
1839.00

βn
0.27
0.28
0.29
0.30
0.30
0.31
0.31

βp
0.28
0.30
0.31
0.31
0.31
0.31
0.32

theoretical results. The sudden increase of the decay energy on
one curve beyond A = 270 (NLZ2) is a signature of the deformed
shell closure for Z = 108 and N = 162. We get a similar result
from the single particle levels for NLZ2. This effect is not very
evident for TMA. We expect that there is a deformed shell closure for Z = 108 and N = 162 but its strength is between the twotheoretical results. We should notice that the shell gap in superheavy nuclei is not large as compared with that around 208 Pb.
Maybe the shell gap in superheavy nuclei can be only seen in
the decay energy or lifetime. This can be different from thataround 208 Pb where many quantities demonstrate the existenceof a large gap.
The numerical results for Z = 110 isotopes are listed in Tables 3 and 4. The α-decay energies of this isotope chain are plotted in Figure 5. In order to compare with future experiments, we
also list the theoretical lifetime of these nuclei in Tables 3 and 4.
The theoretical lifetime Tα (the.) is calculated according to theViola-Seaborg formula on α decays,21, 37
log(Tα ) = (aZ + b)(Qα )−1/2 + (cZ + d) + hlog ,(15)

where Tα is given in second and Qα in MeV, and Z is the proton number of the parent nucleus. This formula is usually used
to estimate the lifetime of α decays by the decay energies.21, 37
The constants in this expression have been determined as a =1.66175, b = −8.5166, c = −0.20228, d = −33.9069, hlog = 0.0

Rn
6.40
6.38
6.36
6.33
6.30
6.27
6.25

Rp
6.20
6.17
6.14
6.10
6.07
6.03
6.00

Qα
10.45
9.63
8.44
7.51
7.78
7.58

for even-even nuclei. These values are obtained by fitting theexperimental data of middle and heavy nuclei.21, 22, 24, 37
It is concluded from Tables 3 and 4 that there is prolate deformation for the ground state of these nuclei. The binding energies
for two sets of theoretical results are close. It is expected the experimental data of binding energy will be between two sets oftheoretical results. As a good estimate on the binding energy
and α-decay energy of unknown nuclei, we suggest to choose
the average value of two sets of theoretical results. The experimental data should be very close to this average value. On the
lifetime, there is a difficulty to predict it in a good precision. But
it is very important for experimental physicists to adjust the detectors to observe a new element or a new nuclide. We also suggest to take the average of two theoretical values as an estimatedvalue. But this average value on lifetime should not be chosen
as a sum of two values divided by two. It should be an average with an exponential weight. For example, if one theoreticalvalue is 10 ms (101 ) and another theoretical value is 1000 ms(103 ), people should choose 100 ms (102 ) as an estimate valueof measurements. It is stressed again that it is difficult to reproduce the experimental lifetime well by theory. The differencewith 100 times between theory and experiment should be considered very well.
It is seen from Figure 5 that the experimental datum of decay energy is between two theoretical values. There is a strong
deformed shell closure at N = 162 for NLZ2 (Z110RMF0.dat).
1212"Z110RMF1.dat""Z110EXP.dat""Z110RMF0.dat""Z108RMF1.dat""Z108EXP.dat""Z108RMF0.dat"
11.5

11.5

11

11

Alpha Decay Energy (MeV) Z=110

Alpha Decay Energy (MeV) Z=108Tα16.4 ms
595 ms568 s
2.4 ×105 s
3.58 ×103 s
3.16 ×103 s

10.5

10

9.5

9

10.5

10

9.5

9

8.5

8.5

8
255

260

265
Nucleon Number (A)

270

275

Figure 4. The variation of the α-decay energy of Z = 108 isotopes with
nucleon number (A). The two sets of theoretical results are connectedby solid lines. It is expected that experimental data lie between twotheoretical results.

8
266

268

270

272

274
276
Nucleon number (A)

278

280

282

Figure 5. The variation of the α-decay energy of Z = 110 isotopes with
nucleon number (A). The two sets of theoretical results are connectedby solid lines. It is expected that experimental data lie between twotheoretical results.

Deformed Relativistic Mean-field Calculations
It leads to a sudden increase of decay energy beyond N = 162.
This effect is not so evident for TMA (Z110RMF1.dat).
Finally let us make a prediction for planned experiments
at Lanzhou in China. After the nuclide 259 Db is identified atLanzhou in China,11 it is planned to synthesize new nuclide with
Z = 108 (Hs) or Z = 109 (Mt) soon.12 The properties of Z = 108
isotopes have been listed in Tables 1 and 2. Here we give thetheoretical prediction on the α-decay chain of 269 Mt in Tables 5
and 6. It is nice to see that the α-decay energy and the lifetimeof 269 Mt and 265 Bh from two sets of forces are very close. The
experimental data can be between them. Although there is adifference of decay energy and lifetime for low mass nuclei inTables 5 and 6, this is not important because the α-decay chain
is generally not so long. The fission can appear in the low mass
range.
4. Summary
We have investigated the structure of nuclei with proton number Z = 106–110 in the RMF model. The theoretical binding energy of the RMF model agrees well with the available data. The
calculations set an upper limit and a lower limit for the binding
energy based on the comparison with present data. This is useful for the good estimate of the binding energy, decay properties
on unknown nuclei. The RMF results show that there is prolate
deformation in these nuclei. There may exist a deformed shellclosure for Z = 108 and N = 162. The prediction for future experiments such as the properties of nuclei on the α-decay chainof 269 Mt is made.
Acknowledgments. Z. REN thanks Profs. T. Otsuka, H.
Toki, W. Q. Shen, H. Q. Zhang, Z. Qin, Z. G. Gan, J. S. Guo forkindly communicating him new progress related to superheavy
nuclei. This work is supported by the National Natural Science
Foundation of China (10125521), the 973 National Major State
Basic Research and Development of China (G2000077400), andby the CAS Knowlidge Innovation Project No. KJCX2-SWN02.
References(1) S. Hofmann, V. Ninov, F. P. Heßberger, P. Armbruster, H.
Folger, G. Münzenberg, H. J. Schött, A. G. Popeko, A. V.
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