Individual and collective properties of fermions in
nuclear and atomic cluster systems
D N Poenaru, R A Gherghescu, Walter Greiner
To cite this version:
D N Poenaru, R A Gherghescu, Walter Greiner. Individual and collective properties of fermions
in nuclear and atomic cluster systems. Journal of Physics G: Nuclear and Particle Physics, IOP
Publishing, 2010, 37 (8), pp.85101. .
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Individual and collective properties of fermions in
nuclear and atomic cluster systems
D N Poenaru1,2 , R A Gherghescu2,1 and Walter Greiner1
1
Frankfurt Institute for Advanced Studies, Ruth-Moufang-Str. 1
60438 Frankfurt am Main, Germany
2
Horia Hulubei National Institute for Physics and Nuclear Engineering
PO Box MG-6, 077125 Bucharest-Magurele, Romania
E-mail: [email protected]
Abstract.
Usually in Nuclear Physics the minimum of the liquid drop model (LDM) energy
for synthesis of superheavy nuclei or cluster decay modes occurs at a mass asymmetry
which is different from the minimum of shell correction. Three examples are given:
potential energy surfaces of 300 120; the table showing the magic numbers of target
nuclei used to produce superheavies at GSI and RIKEN, and the table of magic
numbers of daughter for identified cluster emitters. On the other hand, charged
metallic clusters are ideal emitters of singly ionized trimers because both LDM and
shell correction are reaching a minimum for the same mass asymmetry corresponding
to the emission of a charged particle with two delocalized electrons. Example: fission
of alkali (Cs, K, Na, and Li) clusters with 146 atoms and an excess charge z = 6.
Calculations of Q2 -values for Cs, Na, Au, and Cu atomic clusters multiply ionized
(z = 2, 4, 6, 8, 10) and spheroidally deformed proove that large dissociation energy is
obtained for metallic clusters with high surface tension and low Wigner-Seitz radius
(transition metals).
PACS numbers: 25.60.Pj, 23.70.+j, 34.80.Ht, 36.40.Wa
Submitted to: J. Phys. G: Nucl. Phys.
Individual and collective properties of fermions
2
1. Introduction
The electronic shell structure in alkali metal clusters [1] has shown a strong analogy
with the single-particle states of atomic nuclei since delocalized electrons of a metallic
cluster are fermions in a confined space like the nucleons (protons and neutrons).
This analogy was practically exploited to adapt the nuclear macroscopic-microscopic
method (MMM) [2] to atomic cluster physics [3-8], (see also the review papers [9-11]
and the references therein) as an alternative to the density functional theory or quantum
molecular dynamics used with great success to study the ground state properties and
the fission process of atomic clusters. The detailed calculations for triaxial shapes [4]
explained very well the experimental ground state properties of metallic (potassium,
copper, and sodium) clusters, as well as the fission energetics of doubly ionized clusters.
The electronic-entropy effects in the size-evolutionary patterns of small clusters have
been properly investigated [6].
Within MMM the collective properties of fermions are well accounted for by the
liquid drop model (LDM). The shell correction adds the contribution of the individual
properties based on a quantum single-particle shell model. We contributed to this field
by investigating within MMM the nanophysics of hemispheroidal clusters deposited on
planar surfaces [12-16].
We would like to emphasize the importance of microscopical corrections and of
the phenomenological LDM deformation energies for the synthesis of superheavy nuclei,
cluster radioactivity, as well as for the singly charged trimer emission from a multiply
ionized metallic cluster. By comparing these processes one can see the uniqueness of
the latter one, for which both the LDM deformation energy and the shell correction are
reaching the minima simultaneously at the same mass asymmetry. We hope that this
remarkable property could be exploited in applications using the released energy in a
similar way to the rich variety of applications of nuclear α-decay.
2. Superheavy nuclei
This year on February 19, the Nicolaus Copernicus birth day, the International Union
of Pure and Applied Chemistry (IUPAC) officially recognized the sixth superheavy
element, Copernicium (symbol Cn, Z = 112) discovered at GSI by an international
team of scientists during the past 30 years. As shown in table 1 the “cold fusion”
method of producing superheavy elements, leading to low excitation of the compound
nucleus followed by one evaporated neutron at GSI Darmstadt [17, 18] and RIKEN [19],
is based on closed shell nuclear targets 208 Pb and 209 Bi (the cold valley idea [20, 21]).
The excited compund nucleus, A Z, is produced by bombarding the target, At Zt , with
the projectile, Ap Zp :
At
Zt +
Ap
Zp →
A
Z
(1)
where the hadronic numbers are conserved: A = Ap + At and Z = Zp + Zt . The neutron
number N = A − Z. The hot fusion method based on 48 Ca induced reactions, promoted
Individual and collective properties of fermions
3
at JINR Dubna [22, 23], allows to produce less neutron-defficient superheavies closer to
the valley of beta-stability.
The magic numbers of nucleons are marked with bold face fonts in table 1. For
all seven superheavy elements the target neutron number is magic, Nt = 126, for 3 of
them (even atomic number Z) the target atomic number is magic, Zt = 82, and for Ds
and Rg also the projectile atomic number, Zp = 28, is magic.
We illustrate the idea of cold valley for production of superheavy nuclei on potential
energy surfaces (PES) versus the relative distance between centers ξ = (R−Ri )/(Rt −Ri )
and the mass asymmetry η = (At − Ap )/(A) shown in figure 1 for 300 120. Here R is
the separation distance of the fragments with an initial value Ri (equal to R0 − Rp
for spherical shapes) and the touching point value Rt (equal to Rp + Rt for spherical
fragments). R0 , Rt , Rp are the radii of compound nucleus, of the target, and of the
projectile. The phenomenological (LDM-like) Yukawa-plus-exponential (Y+E) model
deformation energy, EY +E , is shown at the bottom of this figure, where one can see an
almost zero fission barrier, and the Businaro-Gallone mountains. The calculations using
an advanced two-center shell model [24, 25] are outlined elsewhere [26]. At the center is
plotted the shell plus pairing corrections, δEsh+p , producing a finite fission barrier which
makes possible the existence of this superheavy nucleus. Different valleys at various mass
asymmetry are obtained whenever the nucleons of one or the two fragments are reaching
a magic value during the deformation process. By adding EY +E + δEsh+p we get the
MMM deformation energy Edef plotted at the top.
Figure 2 is a plot of cuttings through the three PES at the touching point
configuration, R = Rt , ξ = 1. The most important cold valley for synthesis of
a superheavy nucleus with Z = 120 and A = 300 corresponds to the reaction
208
92
→ 300 120, for which η ' 0.4. This is indeed the deepest valley
82 Pb126 +38 Sr54
of the shell and pairing corrections (the dotted curve in figure 2) because the doubly
magic 208 Pb has a very strong shell effect. Nevertheless, after adding the Y+E model
deformation energy, having a steep variation toward the Businaro-Gallone peak, this
valley becomes shallower in the final result Edef (the solid curve). The deepest valley of
the total deformation energy remains that produced by the doubly magic light fragment
132
50 Sn82 at η ' 0.12, which is also responsible, as a heavy fragment, for the asymmetric
mass distribution in the region of heavy nuclei lighter than Fm.
3. Cold fission and cluster radioactivity
Cluster radioactivity was predicted in 1980 [27] and the first experimental confirmation
was reported in 1984 [28]. A typical PES for heavy nuclei decaying by fission or cluster
radioactivity
A
Z →
Ad
Zd +
Ae
Ze
(2)
within LDM or Y+E model will always have a minimum at mass symmetry η =
(Ad − Ae )/A = 0, so that collective and single-particle properties in nuclei are
Individual and collective properties of fermions
4
driving the system toward different mass asymmetry. One exception is 264 Fm which
fissions symmetrically in two identical doubly magic fragments 132 Sn [29]. In this case
both phenomenological deformation energy and the shell corrections are exhibiting a
minimum value at η = 0. Other exceptions of symmetrical mass distribution of fission
fragments were experimentally observed in fission of few Fm, Md, No, and Rf isotopes
[30, 31].
In the region of nuclei with Z = 87 − 96, where cluster radioactivity was confirmed
[32] (experiments performed in Oxford, Moscow, Orsay, Argonne, Berkeley, Dubna,
Livermore, Geneva, Milano, Vienna, Beijing), the 208 Pb valley proved again to be of
practical importance, as can be seen from the table 2.
Even for alpha decay it is possible to see such a valley if the emitter is 212 Po or 106 Te
[33]. In the latter case the heavy fragment 102 Sn with proton magic number Z = 50
plays the important role.
As can be seen from the table 2, in all successful measurements of cluster decay
modes performed until now the daughter has at least one magic number of nucleons,
either Zd = 82, Nd = 126, or both (daughter 208 Pb). There is one exception: the
28
Mg radioactivity of 236 U where the daughter has an atomic number 80 and a neutron
number 128, not very far from the magicity. Also two of the emitted clusters 14 C, and
34
Si possess a magic neutron number (Ne = 8 and Ne = 20, respectively), and there is
one emitted cluster, 20 O, with magic number of protons Ze = 8.
4. Fission of multiply charged metallic clusters
Now let us turn to the metallic cluster fission process [5,11,34-41] :
MNz+ → MNz11+ + MNz22+
(3)
where the excess charge z and the number of atoms N are conserved: z = z1 + z2 and
N = N1 + N2 . Also the corresponding number of delocalized electrons is conserved:
ne = ne1 + ne2 , where ne = N − z, ne1 = N1 − z1 and ne2 = N2 − z2 . The experiments
on charged metallic cluster fission are showing a clear high yield for a singly charged
trimer light fragment, z2 = 1, N2 = 3, ne2 = 2 in eq. 3, the analog of the α-decay, having
a magic number of 2 delocalized electrons. Since the minima of the two valleys coincide
we may say that in atomic cluster physics the “ideal” conditions of a superasymmetric
fission (alpha-like emission) are frequently fulfilled.
We present in figure 3 the large asymmetry part of our calculations for fission
6+
5+
+ M31+ of alkali metal clusters Cs, K, Na, Li (left-hand side) and transition
M146 → M143
metal clusters Al, Ag, Au, Cu (right-hand side) with a number of atoms 146, and a
positive charge excess z = 6. The particular split leading to a singly charged trimer
emission M31+ illustrates, particularly for the alkali metals, how the minmum of the
LDM energy takes place at the same mass asymmetry, corresponding to p = ne1 = 2
or to ne2 = n − p = 138, where n = ne . Both numbers of fission fragments delocalized
electrons 143 − 5 = 138 and 3 − 1 = 2 are magic numbers.
Individual and collective properties of fermions
5
A simple explanation of the existence of a minimum of LDM deformation energy at
the large mass asymmetry corresponding to p = 2 can be obtained by plotting in figure 4
the derivative dQLD /dp of the released (dissociation) energy calculated within LDM for
spherical shapes [36] as a sum of three contributions coming from the Coulomb, surface,
and ionization energies: QLD = QC + Qs + QIP ,
2
dQLD
e2
1
(z − z1 )2
z1
2as
=
−
−
+
4/3
4/3
dp
6rs p
(n − p)
3 (n − p)1/3
1
z1
e2
z − z1
−
+
(4)
p1/3
24rs (n − p)4/3 p4/3
where as = 4πrs2 σ is the surface energy constant proportional to the surface tension
σ, e2 /2 = 7.1998259 eV·Å, e is the electron charge and rs is the Wigner-Seitz radius.
Due to the smooth variation with p of the interaction energy between the separated
fragments, the fission barrier is minimum when the Q-value is maximum, dQLD /dp = 0,
which happens very close to p = 2 for alkali clusters. Transition metals (right-hand side
of figure 4) may not have a minimum fission barrier at p = 2, but anyhow the slope of
deformation energy variation with p is very small in this region.
We can see in figure 5 the influence of deformation and shell effects (points) and
the spherical LDM values (lines), increasing the Q2 -values particularly for any parent
number of delocalized electrons leading to a magic daughter. Spherical magic numbers:
2, 8, 40, 58, 92, 138, 198, 264, 344, 442, 554, 680, ... The points were obtained by
making the calculations after minimization within MMM [16] of the parent and daughter
deformation energies Edef = ELDM + δEsh+p vs. spheroidal deformation.
Possible applications in nanotechnology may be envisaged in which the kinetic
energy of the singly charged trimer can be used in analogy with the wide spread
applications of nuclear α-decay. The Q-value for metallic cluster fission increases with
the charge z (see figure 5). It is large when the surface tension, σ, is large and the
Wigner-Seitz radius, rs , is small (larger for transition metallic clusters and smaller for
alkali metal clusters). At a given value of z the larger Q-values are obtained for smaller
ne when the fissility parameter X = EC0 /(2Es0 ) = (ez)2 /(16πrs3 σne ) approaches unity.
In conclusion the valleys produced by shell effects on PES of superheavy nuclei
and cluster radioactive heavy nuclei are usually shallower because the minima of shell
corrections occur at a mass asymetry η 6= 0, while the minimum of LDM energy is
placed at η = 0. Charged clusters are ideally “alpha” emitters because both LDM and
shell corrections are reaching a minimum for the corresponding mass asymmetry. When
the number of delocalized electrons of both fragments are magic, the Q-value exhibits
local maxima. In order to have large absolut values one has to choose metallic clusters
with high σ and low rs (transition metals).
Acknowledgments
This work is partially supported by Deutsche Forschungsgemeinschaft Bonn, partially
within IDEI Programme under contracts 123/01.10.2007 and 124/01.10.2007 with
Individual and collective properties of fermions
6
UEFISCSU, and partially within PN09370102 of the Nucleu programme of Ministry
of Education and Research, Bucharest.
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[36]
[37]
[38]
[39]
Individual and collective properties of fermions
8
Table 1. Magic numbers of nucleons in the target and projectile producing cold valleys
for synthesis of superheavy nuclei at GSI Darmstadt and RIKEN
Element
Symbol
Name
Z
107
108
109
110
111
112
Bh
Hs
Mt
Ds
Rg
Cn
Projectile
Np Zp
Bohrium
Hassium
Meitnerium
Darmstadtium
Roentgenium
Copernicium
113
54
Cr
Fe
58
Fe
62
Ni
64
Ni
70
Zn
30
32
32
34
36
40
24
26
26
28
28
30
70
40
30
58
Zn
Target
Nt
Zt
Bi
Pb
209
Bi
208
Pb
209
Bi
208
Pb
126
126
126
126
126
126
83
82
83
82
83
82
209
126
83
209
208
Bi
Table 2. Magic numbers of nucleons in the daughter and emitted cluster of the
experimentally confirmed cluster radioactivities.
Cluster
Ze Ne
14
C
6
8
Parent, Daughter
Zd Nd
221
Fr
Ra
224
Ra
223
Ac
228
Th
230
U
232
U
234
U
233
U
234
U
236
U
238
Pu
238
Pu
242
Cm
222
20
O
Ne
24
Ne
8
10
10
12
12
14
25
10
10
12
15
16
16
22
Ne
Ne
28
Mg
26
30
Mg
34
Si
12
14
18
20
81
82
82
83
82
82
82
82
82
82
80
82
82
82
126
126
128
126
126
126
126
128
128
126
128
128
126
126
Cluster
Ze Ne
14
C
6
8
Parent, Daughter
Zd Nd
221
Ra
Ra
226
Ra
225
Ac
231
Pa
231
Pa
233
U
235
U
235
U
234
U
236
Pu
236
U
238
Pu
223
23
F
Ne
9
10
14
14
Ne
Mg
10
12
15
16
Mg
Si
12
14
18
18
24
25
28
30
32
82
82
82
83
82
81
82
82
82
80
82
80
80
125
127
130
128
126
126
127
129
128
126
126
126
126
9
20
0
-20
0
ξ 0.5
δEsh+p (MeV)
Edef (MeV)
Individual and collective properties of fermions
10
5
0
0
ξ
-0.5 0
1
0.5
EY+E (MeV)
1
0
-0.5
0.5 η
0.5
η
20
0
-20
0
0.5
1
ξ
-0.5
0
0.5
η
Figure 1. Y+E model deformation energy (bottom), shell plus pairing corrections
(center), and the total deformation energy (top) PES of 300 120 vs ξ = (R−Ri )/(Rt −Ri )
and η = (At − Ap )/(A).
EY+E,
Esh+p, Edef (MeV)
Individual and collective properties of fermions
10
Edef
Esh+p
EY+E
25
20
15
10
5
0
-5
-10
-0.8
-0.4
0.0
0.4
0.8
Figure 2. Touching point (R = Rt , ξ = 1) deformation energies of
asymmetry η.
300
120 vs. mass
Individual and collective properties of fermions
2
4
6
8
2
4
6
2
4
6
2
4
6
8
5
E (eV)
1.5
E (eV)
11
1.0
0.5
0.0
2
4
6
Li
4
3
2
1
8
1.0
0
Cu
8
5
4
3
0.5
2
0.0
2
4
6
Na
1
8
0
Au
8
1.0
3
0.5
2
1
K
0.0
Ag
0
1.0
3
0.5
2
Cs
ELDM
ELDM + E
0.0
0
2
4
p=ne1
6
8
Al
1
ELDM
ELDM + E
0
0
2
4
p=ne1
6
8
Figure 3. The large asymmetry part of the scission point deformation energies ELDM
6+
5+
(dotted lines) and ELDM +δE (full lines) for the fission of cations M146
→ M143
+M31+
of alkali clusters (left-hand side) and transition metal clusters (right-hand side).
Individual and collective properties of fermions
12
dQs/dp
dQC/dp
dQIP/dp
dQLD/dp
dQLD/dp (eV)
0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
0
1
2
3
4
p=ne1
5
6
7
Figure 4. The derivatives of LDM Q-values vs the number of delocalized electrons of
the light fragment. dQLD /dp ' 0 for p = 2. Fission of Cs6+
100 with singly charged light
fragments.
Individual and collective properties of fermions
0
100
200
300
400
500
0
100
200
300
400
500
Cu
12
Q2 (eV)
13
Au
12
8
8
4
4
0
0
z=10
6
6
8
4
4
6
4
2
Na
2
0
100
200
300
400
500
Cs
0
100
200
300
ne
400
500
Figure 5. Q2 values vs the number of delocalized electrons of the parent for singly
charged trimer emission from metallic clusters multiply ionized Cuz+ , Auz+ , Naz+ ,
and Csz+ , z = 2, 4, 6, 8, 10, with spherical (lines) and spheroidal (points) shapes.
2