c Pleiades Publishing, Ltd., 2011.
ISSN 1063-7788, Physics of Atomic Nuclei, 2011, Vol. 74, No. 11, pp. 1521–1536. c O.V. Bespalova, I.N. Boboshin, V.V. Varlamov, T.A. Ermakova, B.S. Ishkhanov, A.A. Klimochkina, S.Yu. Komarov, H. Koura, E.A. Romanovsky,
Original Russian Text T.I. Spasskaya, 2011, published in Yadernaya Fizika, 2011, Vol. 74, No. 11, pp. 1555–1569.
NUCLEI
Experiment
Shell Structure of Even–Even Nickel Isotopes Containing Twenty
to Forty Neutrons
O. V. Bespalova1)* , I. N. Boboshin1), V. V. Varlamov1), T. A. Ermakova1),
B. S. Ishkhanov1), A. A. Klimochkina1), S. Yu. Komarov1),
H. Koura2), E. A. Romanovsky1), and T. I. Spasskaya1)
Received December 21, 2010
Abstract—Shell parameters of even–even nickel isotopes involving twenty to forty neutrons are analyzed,
and the results of this analysis are presented. A detailed comparison of the results obtained by calculating, on the basis of the mean-field model with the Koura–Yamada potential and the dispersive optical
potential, single-particle energies of proton and neutron subshells with experimental data on the isotopes
56,58,60,62,64,68
Ni and with evaluated data on the neutron-deficient isotopes 48,50,52,54 Ni is performed.
DOI: 10.1134/S1063778811110056
1. INTRODUCTION
Investigation of neutron-deficient and neutronrich nuclei is one of the main lines of research in modern nuclear physics. A number of microscopic and
phenomenological nuclear-structure models were
developed in order to calculate the properties of such
nuclei. In order to test the predictive power of such
models, one needs, first of all, experimental data—
in particular, experimental data on single-particle
energies of nuclei.
As a rule, reliable information on single-particle
energies Enlj in nuclei is available only for a very
small number of nuclei in the vicinity of the Fermi
energy EF , but, in order to test the applicability of
one nuclear-structure model or another, information
about energies for a substantially greater number of
nuclei is necessary. If experimental information about
neutron energies is available for some isotopes of the
element being studied, it is sometimes possible to find
eval for some other isotopes
the evaluated energies Enlj
of this element by using regularities in the changes
in Enlj as the number of neutrons N changes. In
this case, a comparison of the calculated values of
Enlj with experimental and evaluated data becomes
possible for a larger number of isotopes.
Among even–even nickel isotopes, there are four
stable isotopes 58,60,62,64 Ni for which the application
1)
Skobeltsyn Institute of Nuclear Physics, Moscow State
University, Moscow, 119991 Russia.
2)
Advanced Science Research Center, Japan Atomic Energy
Agency, Tokai, Ibaraki, 319-1195, Japan.
*
E-mail: [email protected]
of the method of a joint evaluation of data from
nucleon-stripping and nucleon-pickup reactions on
the same nucleus (for the sake of brevity, the jointevaluation method in the following) in [1,2] made
it possible to obtain the most reliable and the most
comprehensive experimental information about the
energies Enlj of single-particle states and about
their occupation numbers Nnlj . The joint-evaluation
method permits deducing Nnlj and Enlj values from
self-consistent spectroscopic strengths obtained in
stripping and pickup reactions. The matching of
data from stripping and pickup reactions is achieved
on the basis of the application of sum rules and the
renormalization of spectroscopic strengths, as well
as on the basis of taking into account experimental
data on spins and parities of nuclear levels. For the
doubly magic long-lived unstable nucleus 56
28 Ni28 ,
there is information in the literature on Enlj from an
analysis of the schemes of decay of nuclei neighboring
it in N and Z. Information about the nucleus of
the isotope 68 Ni, which is a candidate for a doubly
magic nucleus, was obtained in just the same way.
Since information about single-particle energies in
neutron-deficient nickel isotopes has not yet been
obtained from experiments with radioactive beams,
eval were evaluated in [3] in analyzing
the energies Enlj
experimental data on Enlj for mirror nuclei, which are
stable and magic, involving N = 28 nucleons. For
eval were found for 48 Ni
example, the energies Enlj
28 20 by
48
50
rescaling data on 20 Ca28 , for 28 Ni22 on the basis of
52
rescaling data on 50
22 Ti28 , for 28 Ni24 on the basis of
1521
1522
BESPALOVA et al.
π,expt
Table 1. Single-particle energies Enlj
of proton subshells in the isotopes 58,60,62,64 Ni and their occupation probabilities
π,expt
Nnlj
58
nlj
π,expt
Nnlj
1
2
π,expt
−Enlj
,
MeV
3
60
Ni
π,expt
Nnlj
4
π,expt
−Enlj
,
MeV
5
π,expt
Nnlj
6
62
Ni
π,expt
−Enlj
,
MeV
7
1g9/2
π,expt
Nnlj
8
64
Ni
π,expt
−Enlj
,
MeV
9
π,expt
Nnlj
Ni
π,expt
−Enlj
,
MeV
10
0.04(4) –0.32(52) 0.02(2)
11
–1.11(33)
1f5/2 0.04(4)
1.44(52) 0.08(8)
1.12(124) 0.12(7)
2.35(103) 0.07(7)
2.97(85) 0.09(9)
3.33(122)
2p1/2 0.01(1)
0.98(3)
0.10(2)
1.80(39)
0.04(4)
0.90(65)
0.06(6)
3.37(67) 0.02(2)
3.74(21)
2p3/2 0.16(2)
3.63(17) 0.12(5)
2.30(56)
0.09(4)
2.55(70)
0.23(8)
5.72(62) 0.10(1)
5.48(51)
1f7/2 0.92(1)
8.06(20) 0.86(7)
7.47(87)
0.87(13)
9.20(149) 0.89(5) 10.96(44) 0.91(7)
12.68(74)
1d3/2 1.00(0) 12.62(14) 0.97(2) 12.25(28)
0.97(2)
54
rescaling data on 52
24 Cr28 , and for 28 Ni26 on the basis
54
of rescaling data on 26 Fe28 .
In the present study, we explore regularities in
the experimental and evaluated neutron and proton
expt(eval)
for even–even nickel isotopes in
energies Enlj
which 20 N 40. These data are analyzed on
the basis of the mean-field model with a dispersive
optical potential (see [4] and references therein) and
the Koura–Yamada potential [5], and the predictive
power of the respective approaches is tested.
2. PROTON SINGLE-PARTICLE STRUCTURE
OF EVEN–EVEN NICKEL ISOTOPES
π,expt
π,expt
The experimental values Enlj and Nnlj for the
proton states of the isotopes 58,60,62,64 Ni were obtained in [1, 2] by the joint-evaluation method [6] and
are presented in Table 1. The quoted errors only take
into account the uncertainties in the respective spins,
as well as the uncertainties that arise in generalizing
results based on data from studying different stripping
and pickup reactions.
The single-particle properties of proton states in
the 58 Ni nucleus were determined on the basis of
data on spectroscopic strengths, spins, and parities
of states in 57 Со (pickup) and 59 Со (stripping) nuclei.
The results of the experiments that studied the reaction 58 Ni(d, 3 He) induced by polarized deuterons [7,
8] were used for data from pickup reactions, while
the results of the experiments that studied the reaction 58 Ni(d, n) [9, 10] and the reaction 58 Ni(3 He,
d) [11, 12] were taken from data on stripping reactions
(columns 2 and 3 in Table 1 for the first stripping
process and columns 4 and 5 in this table for the
second one). The results obtained on the basis of data
12.97(58)
0.99(1) 14.49(38) 0.92(4)
15.58(47)
both on the reaction 58 Ni(d, n) and on the reaction
d) can be taken for the ultimate values of
the parameters of proton subshells in the 58 Ni nucleus since neither group of results is preferable. The
two versions of ultimate results are equivalent from
the point of view of their compliance with available
experimental data, since they are characterized by
the identical degree of consistency of experimental
data on stripping and pickup reactions. At the same
time, the data obtained for occupation probabilities
and single-particle energies show a significant scatter
for some states (see Table 1).
In addition to the shell-structure parameters in
π,expt
π,expt
Table 1, the values of N2s1/2 = 0.99(1) and E2s1/2 =
58 Ni(3 He,
π,expt
−13.49(13) MeV for 62 Ni and the values of N1d5/2 =
π,expt
0.98(2) and E1d5/2 = −18.81(146) MeV for 60 Ni
were found in [1].
According to the single-particle shell model where
the subshells are filled consecutively, the 1d5/2 , 1d3/2 ,
and 2s1/2 subshells and the 1f7/2 shell in nickel
nuclei are filled, while the 2p3/2 , 1f5/2 , and 2p1/2
subshells are empty. The experimental data reported
in [1] indicate that the occupation probabilities for the
2s1/2 and 1d3/2 subshells are indeed close to unity.
At the same time, the occupation probability for the
1f7/2 shell in various nickel isotopes ranges from 87%
(60 Ni) to 92% (58 Ni), while the occupation probability
for the 2p3/2 subshell changes from 8% (58 Ni) to 23%
(60 Ni). The shell-model gap between the 1f7/2 and
2p3/2 states that corresponds to the magic number of
Z = 28 is 4.5 to 7.2 MeV.
π of proton states
The single-particle energies Enlj
are related to the single-particle energies of neutron
PHYSICS OF ATOMIC NUCLEI
Vol. 74 No. 11
2011
SHELL STRUCTURE OF EVEN–EVEN NICKEL ISOTOPES
1523
π,eval
ν,m.n.
of proton states in 48,50,52,54 Ni nuclei, single-particle energies Enlj
of neutron
Table 2. Single-particle energies Enlj
ν,m.n.
C
states in mirror nuclei and their occupation probabilities Nnlj , and Coulomb shift energies Δnlj
nlj
ν,m.n.
−Enlj
,
48
MeV 20 Ca28
ν,m.n.
Nnlj
48
20 Ca28
ΔС
nlj ,
MeV
π,eval
−Enlj
,
Ni
MeV 48
20
28
ν,m.n.
−Enlj
,
50
MeV 22 Ti28
ν,m.n.
Nnlj
50
22 Ti28
ΔС
nlj ,
MeV
1f5/2
1.57(37)
0.03(3)
9.16
−7.6
4.14(67)
0.13(4)
9.32
−5.2
2p1/2
2.87(1)
0.00(0)
9.14
−6.3
4.60(24)
0.03(3)
9.26
−4.7
2p3/2
4.68(1)
0.01(1)
9.46
−4.8
6.37(7)
0.09(1)
9.47
−3.1
1f7/2
10.10(10)
1.00(0)
9.54
0.6
10.89(58)
0.89(4)
9.49
1.4
1d3/2
15.22(94)
0.99(1)
10.11
5.1
14.57(27)
0.87(14)
9.98
4.6
2s1/2
15.07(27)
1.00(0)
10.09
5.0
16 .0
10.02
6.0
nlj
ν,m.n.
−Enlj
,
Cr
MeV 52
28
24
ν,m.n.
Nnlj
52
24 Cr28
ΔС
nlj ,
MeV
π,eval
−Enlj
,
MeV 52
Ni
24
28
ν,m.n.
−Enlj
,
Fe
MeV 54
28
26
ν,m.n.
Nnlj
54
26 Fe28
ΔС
nlj ,
MeV
π,eval
−Enlj
,
Ni
MeV 50
22
28
π,eval
−Enlj
,
MeV 54
Ni
26
28
9.41
−3.9
7.5
9.52
−2.0
0.04(9)
9.29
−3.1
7.6
9.30
−1.7
7.25(40)
0.05(3)
9.49
−2.2
8.38(42)
0.07(1)
9.47
−1.1
1f7/2
12.78(63)
0.86(7)
9.43
3.4
14.97(38)
0.95(0)
9.36
5.6
1d3/2
16.03(135)
0.89(10)
9.87
6.2
17.22(96)
0.88(1)
9.79
7.4
2s1/2
16 .9
9.93
7.0
17.82(44)
1.00(0)
9.83
8.0
1f5/2
5.5
2p1/2
6.20(99)
2p3/2
Note: Energies evaluated for mirror nuclei on the basis of interpolated data for neighboring nuclei are italicized.
ν,m.n.
states in the mirror nucleus (m.n.), Enlj
, by the
equation
ν,m.n.
π
= Enlj
+ ΔC
Enlj
nlj ,
(1)
ΔC
nlj
is the Coulomb shift energy. Similarly,
where
ν of neutron states are
the single-particle energies Enlj
related to the single-particle energies of proton states
π,m.n.
, by the equation
in the mirror nucleus, Enlj
π,m.n.
ν
= Enlj
− ΔC
Enlj
nlj .
(2)
In order to find the evaluated single-particle energies of proton states in the neutron-deficient isotopes
48,50,52,54 Ni, use was made of experimental information about single-particle energies of neutron states
in the 48 Са [13], 50 Ti, 52 Cr, and 54 Fe [14] mirror nuclei
that was obtained by the global-evaluation method.
The shift energies ΔC
nlj were determined from data on
single-particle energies calculated with the Koura–
Yamada potential [5] for the corresponding mirror
nuclei.
π,eval
evaluated for the neutronThe energies Enlj
deficient nickel isotopes on the basis of expression (1),
PHYSICS OF ATOMIC NUCLEI
Vol. 74 No. 11
2011
ν,m.n.
as well as the energies ΔC
used in evalnlj and Enlj
uations, are given in Table 2 along with the correν,m.n.
. If we disregard inaccusponding values of Nnlj
racies in calculating ΔC
nlj , then the error in determinπ
ing the values Enlj can be taken to be close to the
ν,m.n.
for the mirror
error in determining the energy Enlj
nucleus. In the single-particle model, it is postulated
that the spectroscopic factors of neutron and proton
states in mirror nuclei are equal to each other. In
accordance with this, the occupation probabilities for
proton subshells in nickel isotopes can be set to the
occupation probabilities for the neutron subshells in
mirror nuclei. Therefore, it is natural to expect that
π
is equal to unity
the occupation probability N1f
7/2
in the 48 Ni nucleus and presumably changes from
0.9 to 0.95 in the 50 Ni–54 Ni chain. On the basis of
ν,v.n
that are presented
an analysis of the values of Nnlj
in Table 2, one can find that, in the doubly magic
nucleus 48
28 Ni20 , the 1d3/2 , 2s1/2 , and 1f7/2 states are
completely filled, while the 2p and 1f5/2 states are
empty. For the isotopes 50,52,54 Ni, the populations of
subshells are similar to the populations of subshells in
the stable isotopes 58,60,62,64 Ni.
1524
BESPALOVA et al.
The single-particle structure of a nucleus is also
characterized by the second energy moment of the
spectroscopic strength of the ith subshell or the fragmentation width specified by the formulas
−(+) −(+)
−(+) 2
S
−
Ē
E
ij
i
j ij
−(+)
(M2 )i
=
(3)
−(+)
j Sij
for the pickup (–) and stripping (+) reactions, re −(+)
is the sum of all pickup
spectively. In (3), j Sij
(M2 )nlj =
Table 3 gives the values of
(stripping) spectroscopic strengths for the ith subshell, Eij stands for the energies of excited states, and
−(+)
Ēi
are the energy centroids for the corresponding
shell. The total fragmentation width of the distribution of spectroscopic strengths for the ith subshell
with allowance for the distribution of the stripping
and pickup spectroscopic strengths was determined
by the formula
+
Nnlj (2j + 1) (M2 )−
nlj + (1 − Nnlj ) (2j + 1) (M2 )nlj .
(M2 )nlj for four stable
nickel isotopes. For the 58 Ni nucleus, we present two
sets of values for the fragmentation width. Of these,
one (column 2) corresponds to the shell parameters
given in column 3 of Table 1, while the other (column 3) corresponds to the shell parameters given in
column 5 of Table 1.
(4)
ν
ν
−19.83(6) MeV, N1f
= 0.89(2), and E1f
=
7/2
7/2
ν
=
−15.21(17) MeV for 58 Ni and the values of N1f
7/2
ν
= −15.16(22) MeV for 60 Ni. The
0.92(2) and E1f
7/2
quoted errors correspond to the uncertainties in the
spins of final states. From Table 4, it can be seen
that, in the isotopes 58,60,62,64 Ni, the 2p3/2 , 1f5/2 , and
2p1/2 neutron subshells are predominantly filled as
the number of neutrons grows. The resulting values
3. NEUTRON SINGLE-PARTICLE
of the occupation probabilities differ sizably from 0
STRUCTURE OF EVEN–EVEN NICKEL
and 1. As was indicated above, the 1f7/2 state in the
ISOTOPES
isotopes 58,60 Ni is filled almost completely. One can
ν,expt
ν,expt
for the isotopes assume that a similar situation takes place for the
The values Enlj and Nnlj
58,60,62,64 Ni were obtained in [2] by the joint-evaluation
isotopes 62,64 Ni as well. Thus, the boundary defined
method from data on stripping and pickup reactions by convention between occupied and empty subshells
and spin–parity data; they are given in Table 4. In proves to be strongly smeared. From the data in
addition to values presented in this table, we have Table 4, it follows that each added pair of neutrons is
ν
ν
= 0.99(1), E1d
= distributed almost uniformly among the 2p3/2 , 1f5/2 ,
also found the values of N1d
3/2
3/2
and 2p1/2 subshells. In the isotopes 62,64 Ni, the filling
of the 1g9/2 and 2d5/2 subshells, which lie higher, also
Table 3. Total fragmentation widths for proton subshells in
begins. The proximity of the occupation probabilities
58,60,62,64
Ni
the isotopes
for the 2p3/2 , 1f5/2 , and 2p1/2 neutron subshells in
the isotopes 58,60,62,64 Ni is accompanied by close
(M2 )nlj , MeV
ν,expt
nlj
values of Enlj (see Table 4), which lie in a corridor
58
58
60
62
64
Ni
Ni
Ni
Ni
Ni
of width 1 to 2 MeV, so that the 2p3/2 , 2p1/2 , and
1f5/2 states are degenerate to a considerable extent.
1
2
3
4
5
6
A significant degeneracy of this subshells in energy in
1f5/2
4.34
5.23
6.19
8.51
11.15
the isotopes 56−66 Ni was predicted in [15] on the basis
of calculations performed within the shell models.
2.85
3.03
4.22
4.86
6.16
2p1/2
2p3/2
3.54
4.81
5.69
6.37
7.89
1f7/2
3.38
3.47
4.72
3.14
3.52
1d3/2
2.25
2.21
1.28
1.81
1.88
According to the single-particle shell model that
does not involve any mixing of configurations, the
subshells in question are filled consecutively. The
spin–parities of the ground states of the isotopes
57,59,61,63 Ni are 3/2− , 3/2− , 3/2− , and 1/2− , respecPHYSICS OF ATOMIC NUCLEI
Vol. 74 No. 11
2011
SHELL STRUCTURE OF EVEN–EVEN NICKEL ISOTOPES
1525
ν,expt
Table 4. Single-particle energies Enlj
of neutron subshells in the isotopes 58,60,62,64 Ni and their occupation probabilν,expt
ities Nnlj
58
nlj
ν,expt
Nnlj
60
Ni
ν,expt
−Enlj ,
MeV
ν,expt
Nnlj
62
Ni
ν,expt
−Enlj ,
MeV
ν,expt
Nnlj
64
Ni
ν,expt
−Enlj ,
MeV
2d5/2
1g9/2
ν,expt
Nnlj
Ni
ν,expt
−Enlj , MeV
0.04(1)
2.65(3)
0.04(4)
5.42(26)
0.09(2)
5.56(13)
2p1/2
0.15(4)
8.63(72)
0.25(4)
8.17(19)
0.32(11)
7.62(63)
0.42(1)
7.62(6)
2p3/2
0.28(1)
9.81(8)
0.48(6)
8.80(35)
0.60(4)
8.85(40)
0.78(1)
9.10(4)
1f5/2
0.32(2)
10.55(15)
0.39(5)
9.20(24)
0.47(12)
8.47(70)
0.64(7)
8.72(60)
tively. These values do not comply with the consecutive character of filling of the neutron subshells,
confirming the parallel character of filling of the 2p3/2 ,
1f5/2 , and 2p1/2 subshells in the isotopes 58,60,62,64 Ni.
This special feature of the filling of neutron subshells
in nickel isotopes was highlighted in [16]. In the
present study, we confirm its presence and refine
quantitative properties of the respective degeneracy.
In order to find evaluated single-particle energies
of neutron states in the neutron-deficient isotopes
48,50,52,54 Ni, we employed experimental information
about single-particle energies of proton states in the
mirror nuclei 48 Са, 50 Ti, 52 Cr, and 54 Fe [17] that was
obtained by the joint-evaluation method. The evaluν,eval
were determined by formula (2).
ated energies Enlj
The Coulomb shift energies ΔC
nlj were found in just
the same way as was described in Section 1. The
ν,eval
, as well as the energies ΔC
evaluated energies Enlj
nlj
π,m.n.
and Enlj
used in evaluations, are given in Table 5
π,m.n.
. It should be
along with the respective values Nnlj
noted that the evaluated data are presented in [17] as
π for all subshells of 50 Ti and for some
the values of Enlj
subshells of 52 Cr and 54 Fe.
On the basis of the hypothesized equality of the
neutron and proton spectroscopic factors in mirror
nuclei, it can be found that the occupation probabilities for neutron subshells in neutron-deficient nickel
isotopes are in accord with the corresponding occupation probabilities for proton subshells in mirror
nuclei. According to evaluations (see Table 5), the
48 Ni nucleus has the totally occupied neutron sub28 20
shell 1d3/2 , which corresponds to the magic number
of N = 20. In the isotopes 50,52,54 Ni, the 1f7/2 shell
is filled consecutively.
For the stable isotopes 58,60,62,64 Ni, we have determined the values of the total fragmentation width
PHYSICS OF ATOMIC NUCLEI
Vol. 74 No. 11
2011
(M2 )nlj and presented them in Table 6. From a
comparison of the width (M2 )nlj for proton (Table 3) and neutron states, we can conclude that their
fragmentation in the 58 Ni nucleus is approximately
identical, but that, for all of the remaining isotopes,
π
E nlj, MeV
10
5
0
–5
1
–10
2
3
4
–15
5
6
–20
48
52
56
60
64
68
A
Fig. 1. Single-particle energies of proton subshells in the
48−68
Ni nuclei according to (points) experimental (with
error bars) and evaluated data and (curves 1–6) theoretical calculations with the dispersive optical potential for
the following subshells: (closed boxes, 1) 1f5/2 , (closed
circles, 2) 2p1/2 , (closed triangles, 3) 2p3/2 , (closed diamonds, 4) 1f7/2 , (stars, 5) 2s1/2 , and (stars, 6) 1d3/2 .
1526
BESPALOVA et al.
ν,eval
of neutron states in the 48,50,52,54 Ni nuclei, single-particle energies
Table 5. Evaluated single-particle energies Enlj
π,m.n.
π,m.n.
Enlj
of proton states in mirror nuclei and their occupation probabilities Nnlj
, and Coulomb shift energies ΔC
nlj
π,m.n.
−Enlj
,
MeV
48
20 Ca28
π,m.n.
Nnlj
48
20 Ca28
ΔC
nlj ,
MeV
ν,eval
−Enlj
,
MeV
48
28 Ni20
π,m.n.
−Enlj
,
MeV
50
22 Ti28
ΔC
nlj ,
MeV
ν,eval
−Enlj
,
MeV
50
28 Ni22
1f5/2
4.17(50)
0.00(1)
7.08
11.3
3.43
7.68
11.1
2p1/2
2.41(70)
0.01(1)
6.91
9.3
2.60
7.54
10.1
2p3/2
4.06(50)
0.01(1)
7.00
11.1
3.70
7.62
11.3
1f7/2
8.76(65)
0.02(1)
7.28
16.0
8.30
7.92
16.2
1d3/2
15.96(60)
0.95(4)
6.99
25.2
14.64
7.66
25.5
2s1/2
14.84(108)
0.86(10)
7.00
23.4
nlj
nlj
π,m.n.
−Enlj
,
MeV
52
24 Cr28
π,m.n.
Nnlj
52
24 Cr28
ΔC
nlj ,
MeV
ν,eval
−Enlj
,
MeV
52
28 Ni24
π,m.n.
−Enlj
,
MeV
54
26 Fe28
π,m.n.
Nnlj
54
26 Fe28
ΔC
nlj ,
MeV
ν,eval
−Enlj
,
MeV
54
28 Ni26
1f5/2
2.19
0.0
8.28
10.5
0.95
0.01
8.88
9.8
2p1/2
2.30
0.0
8.14
10.4
1.00
0.05
8.75
9.8
2p3/2
3.47(23)
0.17(3)
8.26
11.7
2.4
0.09(6)
8.87
11.2
1f7/2
7.51(88)
0.50(9)
8.52
16.0
7.88(22)
0.78(8)
9.12
17.0
1d3/2
13.16(32)
0.96(3)
8.33
21.5
12.869(144)
0.95(5)
8.99
21.9
12.11(10)
1.00(3)
9.07
21.2
2s1/2
Table 6. Total fragmentation widths of neutron subshells
in the isotopes 58,60,62,64 Ni
(M2 )nlj , MeV
nlj
58
Ni
60
Ni
62
Ni
64
Ni
2d5/2
2.9
2.8
1g9/2
1.8
1.1
1f5/2
4.5
1.4
2.2
2.5
2p1/2
2.5
1.6
1.2
0.7
2p3/2
3.1
2.9
1.3
1.1
the fragmentation of proton states is substantially
higher than the fragmentation of neutron states.
4. CALCULATION OF SINGLE-PARTICLE
ENERGIES ON THE BASIS OF THE SHELL
MODEL WITH THE KOURA–YAMADA
POTENTIAL
The real central part of the Koura–Yamada potential is expressed in terms of the Woods–Saxon function modified in the surface region. The parameters of
the Koura–Yamada potential are independent of energy and make it possible to describe single-particle
subshells of nuclei in the region Enlj −20 MeV.
KY
calculated in
For deeper subshells, the energies Enlj
this way are inconsistent with experimental data. The
global parameters of the Koura–Yamada potential,
which depend smoothly on Z and N , were found by
KY
to the experimental values of
fitting the energies Enlj
the single-particle energy in the vicinity of the Fermi
energy EF for the 4,8 He, 12,14 C, 16 O, 36 S, 40,48 Ca,
56,66,68 Ni, 88 Sr, 90 Zr, 132 Sn, and 208 Pb nuclei. Here,
a state characterized by the maximum value of the
spectroscopic factor was chosen for a single-particle
state. This definition of the energy of a single-particle
state corresponds to the assumption that there is no
effect of its fragmentation.
π,KY
ν,KY
and Enlj
calculated with the
The energies Enlj
Koura–Yamada potential for the 20 N 40 even–
even nickel isotopes are presented in Tables 7 and 8,
respectively, along with the experimental and evaluated data from Tables 1 and 2 and Tables 4 and 5.
π,expt
In addition, Tables 7 and 8 give the values of Enlj
ν,expt
and Enlj
for
56 Ni
from [18] and for
expt(eval)
19]. The mass dependences of Enlj
PHYSICS OF ATOMIC NUCLEI
68 Ni
from [5,
(A) for proton
Vol. 74 No. 11
2011
4.98
2s1/2
8.93
12.68
12.31
2.30(56)
7.47(87)
2p3/2
1f7/2
1d3/2 12.22(28)
2s1/2
3.56
1.79
1.80(39)
2p1/2
1.63
KY
1.12(124)
expt.
58
28 Ni28
3.42
1f5/2
nlj
5.10
1d3/2
3.37
0.58
−0.20
1f7/2
2011
0.56
−4.78
−4.03
2p3/2 −4.78
Vol. 74 No. 11
12.32
12.72
7.31
2.44
1.31
1.99
DOP
4.74
4.07
−6.36
−6.80
DOP
−5.60
KY
2p1/2 −6.27
evaluation
48
28 Ni20
−7.20
PHYSICS OF ATOMIC NUCLEI
1f5/2 −7.59
nlj
12.97(58)
9.20(149)
2.55(70)
0.90(65)
5.15
5.11
1.45
−2.66
−4.37
−5.70
KY
13.69
14.15
10.30
4.80
3.03
3.07
KY
60
28 Ni32
2.35(103)
expt.
6.00
4.59
1.40
−3.11
−4.66
−5.18
evaluation
50
28 Ni22
13.09
12.62
8.97
2.77
1.18
1.43
DOP
5.95
4.90
1.55
−3.20
−4.46
−5.36
DOP
6.78
6.79
3.03
−1.32
−3.06
−4.20
KY
15.02
15.59
11.64
6.00
4.24
4.48
KY
62
28 Ni34
13.49(131)
14.48(38)
10.95(44)
5.72(62)
3.37(67)
2.97(85)
expt.
6.97
6.16
3.35
−2.24
−3.09
−3.91
evaluation
52
28 Ni24
π
–Enlj
, MeV
15.28
14.65
11.45
5.31
3.69
4.03
DOP
7.59
6.68
3.22
−2.11
−3.47
−4.10
DOP
15.58(47)
12.68(74)
5.48(51)
3.74(21)
8.36
8.39
4.55
0.00
−1.76
−2.77
KY
16.29
16.99
12.93
7.16
5.41
5.85
KY
64
28 Ni36
3.33(122)
expt.
7.99
7.43
5.61
−1.09
−1.70
−2.02
evaluation
54
28 Ni26
16.60
16.10
12.65
5.54
3.67
4.01
DOP
8.90
7.88
4.78
−0.43
−1.80
−1.94
DOP
Table 7. Calculated single-particle proton energies along with respective experimental and evaluated data for the even–even isotopes 48−68 Ni
15.69
9.56
8.26
8.66
expt.
10.72
10.08
7.16
0.74
−0.37
−0.29
expt.
18.70
19.65
15.40
9.36
7.64
8.47
KY
68
28 Ni40
10.88
11.16
7.52
1.29
−0.48
−1.24
KY
56
28 Ni28
20.21
20.48
15.95
9.36
7.78
8.76
DOP
11.14
10.48
6.91
0.97
−0.58
−0.59
DOP
SHELL STRUCTURE OF EVEN–EVEN NICKEL ISOTOPES
1527
1528
BESPALOVA et al.
ν
odds with experimental data from [20], which indicate
that the 48 Ni nucleus is stable against one-proton
decay. In the neutron-deficient isotopes 48,50,52,54 Ni,
ν,eval
ν,KY
and Enlj
of
compliance between the energies Enlj
the 2s1/2 , 1d3/2 , 1f5/2 , and 2p1/2 subshells (see Table 8) holds within the errors in the evaluating energies, this probably being due to a small fragmentation
width of these states. For the 1f7/2 and 2p3/2 subshells, the fragmentation effect leads to a decrease in
the subshell binding energy, which corresponds to the
centroid of the distribution of fragments, in relation to
ν,KY
ν,KY
and E2p
.
the calculated values of E1f
3/2
7/2
E nlj, MeV
2
1
3
–10
4
–15
ν,KY
(A) for the
The calculated dependences Enlj
2s1/2 , 1d3/2 , and 1f7/2 subshells are characterized by
the presence of jumps upon going over from A = 56
to A = 58 (the binding energies decrease). For the
2p3/2 , 2p1/2 , and 1f5/2 subshells, similar jumps occur
upon going over from A = 54 to A = 56.
For the 1f7/2 subshells in 56 Ni, 58 Ni, and 60 Ni,
5
–20
6
48
52
56
60
64
68
A
ν,expt
Enlj
ν,KY
and Enlj
agree within the experimental erν,expt
Fig. 2. As in Fig. 1, but for neutron subshells of the
48−68
Ni nuclei.
and neutron states are on display in Figs. 1 and 2,
respectively.
π,KY
From Table 7, one can see that the values of Enlj
for the 2s1/2 , 1d3/2 , and 1f7/2 proton states increase
in absolute value upon going over from 54 Ni to the
doubly magic nucleus 56 Ni and that, for 2p3/2 , 1f5/2 ,
and 2p1/2 states, the respective values increase upon
going over from 56 Ni to 58 Ni. The total fragmentation widths (Table 3) of the 1d3/2 and 1f7/2 states
are smaller than the total fragmentation widths of
π,KY
do
the 1f5/2 and 2p states. The energies Enlj
not differ very strongly from the evaluated energies
π,expt(eval)
of the 1d3/2 and 1f7/2 states of the isoEnlj
58−64
Ni. For the 1f5/2 and 2p states, the deviatopes
π,expt(eval)
π,KY
from Enlj
is qualitatively consistion of Enlj
tent with the values of the total fragmentation width.
π,KY
from
It is also noteworthy that the deviation of Enlj
π,expt(eval)
is smaller in the case of neutron-deficient
Enlj
isotopes for all states than in the case of stable isotopes. Possibly, this is due to the smallness of the
fragmentation widths of neutron states in the respective mirror nuclei. Here, it is worth noting, however,
π,KY
= +0.20 MeV for 48 Ni is at
that the value of E1f
7/2
rors in determining Enlj , and this gives sufficient
ν,KY
for the isogrounds to employ the values of E1f
7/2
topes 62,64,68 Ni as evaluated data. For the 2p1/2
subshells, this agreement takes place for the isotopes
56,58,60,62,64,68 Ni. For the 2p
3/2 and 1f5/2 subshells
of the isotopes
58,60,62,64 Ni,
ν,KY
the values of Enlj
are
ν,expt
beyond the corridor of experimental errors in Enlj
and do not comply with the aforementioned effect of
degeneracy of the 2p and 1f5/2 subshells in these
isotopes.
For the 2s1/2 and 1d3/2 subshells, the evaluated
and experimental energies are available only for
the isotopes 48,50,52,54,56,58 Ni. Since, for the subshells in question, there is good agreement between
ν,expt(eval)
ν,KY
and Enlj
, one can take, with a precision
Enlj
ν,KY
for the evaluated enof 10 to 15%, the values of Enlj
ergies of these subshells in the isotopes 60,62,64,68 Ni.
The energies of the 1d5/2 , 1p1/2 , 1p3/2 , and 1s1/2
subshells, which lie deeper, cannot be evaluated on
the basis of the Koura–Yamada potential.
Graphs representing experimental A dependences
of single-particle energies of deep-lying proton and
neutron states of some nuclei are given in [21] according to data on the respective (p, pn) and (p,
2p) reactions induced by projectile protons of energy
about 1 GeV. By way of example, we indicate that,
according to the data in Fig. 4 from [21], the energy
π,expt
E1s1/2 for 50 A 70 nuclei falls within the range
PHYSICS OF ATOMIC NUCLEI
Vol. 74 No. 11
2011
SHELL STRUCTURE OF EVEN–EVEN NICKEL ISOTOPES
1529
Table 8. Calculated single-particle neutron energies along with the experimental and evaluated data for the even–even
isotopes 48−68 Ni
ν
−Enlj
, MeV
nlj
48
28 Ni20
50
28 Ni22
52
28 Ni24
54
28 Ni26
evaluation KY DOP evaluation KY DOP evaluation KY DOP evaluation KY DOP expt.
56
28 Ni28
KY DOP
1f5/2 11.25
10.32 11.24 11.1
10.14 10.80 10.47
9.95 9.98 9.83
9.78 10.28 9.14
8.24 8.81
2p1/2 9.32
11.05 9.61 10.0
10.77 9.33 10.44
10.46 8.77 9.75
10.16 9.22 9.48
8.82 8.87
2p3/2 11.06
13.05 11.05 11.4
12.73 10.93 11.73
12.41 10.49 11.24
12.08 11.11 10.25
10.74 10.60
1f7/2 16.04
18.08 15.92 16.2
17.77 16.10 16.03
17.42 15.85 17.00
17.09 16.79 16.65
16.79 16.26
1d3/2 22.95
22.62 22.83 22.3
22.10 22.51 21.49
21.61 21.68 21.85
21.18 22.14 19.84
20.80 19.77
2s1/2 22.81
22.65 22.24
22.13 22.15
21.60 21.58 21.18
21.10 22.24 20.40
20.62 21.15
56
28 Ni30
nlj
expt.
KY DOP
60
28 Ni32
expt.
KY DOP
62
28 Ni34
expt.
KY DOP
64
28 Ni36
expt.
KY DOP expt.
68
28 Ni40
KY DOP
1f5/2 10.55(15) 8.22 10.08 9.20(24) 8.20 9.17 8.47(70) 8.21 8.79 8.72(60) 8.23 8.88 8.3(9) 8.30 8.73
2p1/2 8.63(72) 8.71 8.78 8.17(19) 8.37 8.30 7.62(63) 8.16 8.00 7.62(6)
7.96 8.04 7.8(8) 7.67 7.81
2p3/2 9.81(8) 10.62 9.94 8.80(35) 10.26 9.69 8.85(40) 10.03 9.40 9.10(4)
9.81 9.38 9.0(9) 9.49 9.11
1f7/2 15.21(17) 15.13 14.81 15.16(22) 14.96 15.12
14.81 14.88
14.69 14.41
14.48 14.53
1d3/2 19.83(6) 19.42 20.20
19.22 20.84
19.05 20.25
18.92 19.36
18.72 20.05
2s1/2
18.85 20.59
18.52 20.06
18.21 19.37
17.63 19.70
19.22 19.92
π,KY
between –55 and –60 MeV. The energies E1s
1/2
are between –23 and –37 MeV, while the energies
π,RMFM
calculated on the basis of the relativistic
E1s
1/2
mean-field model (RMFM) [22] lie in the interval from
–47 to –57 MeV, this being within the experimental
errors [21]. A similar situation prevails for the 1s1/2
neutron states.
π,KY
The energies E1s
fall between
1/2
π,RMFM
lie
−36 and –45 MeV, while the energies E1s
1/2
between –61 and –63 MeV, the latter being close to
π,expt
the value of E1s1/2 from [21]. In the next section,
the results of the RMFM calculations from [22] are
DOP as the evaluated energy of
used to determine Enlj
the 1s1/2 level.
5. ANALYSIS ON THE BASIS
OF THE MEAN-FIELD MODEL
WITH A DISPERSIVE OPTICAL POTENTIAL
The dispersion approach to determining the mean
nuclear field unified for positive and negative energies [23] was successfully used to analyze data
on nucleon–nucleus scattering and single-particle
properties of nuclei. Within the dispersion approach,
PHYSICS OF ATOMIC NUCLEI
Vol. 74 No. 11
2011
the mean field is complex-valued. Its real and imaginary parts are related by a dispersion equation, so
that the real part of the dispersive optical potential
is the sum of a component featuring a weak energy dependence and belonging to the Hartree–Fock
type, VHF , and a dispersive component, ΔVs,d, which
sharply depends on energy in the vicinity of the Fermi
energy EF . The dispersive component is calculated
on the basis of data on the imaginary part of the
dispersive optical potential and, in just the same way
as this imaginary part, can be broken down into the
volume and surface components, the subscripts s and
d, respectively, being used to label them.
Methods developed earlier for constructing a dispersive optical potential require experimental data
on nucleon scattering over a broad energy interval.
However, the range of nuclei for which there are
experimental data on nucleon–nucleus scattering is
not wide. In particular, it does not contain unstable
nuclei. A new method for constructing a dispersive
optical potential was developed in [4]. This method
makes it possible to calculate single-particle properties for a broad range of magic and near-magic nuclei,
which includes unstable neutron-rich and neutrondeficient nuclei.
The method proposed in [4] does not require the
presence of experimental data on the scattering of
1530
BESPALOVA et al.
nucleons by the target nucleus being studied, since
one takes the required data on the imaginary part from
existing systematics of global potential parameters in
the traditional (nondispersive) optical model. In the
present study, we fix the parameter of radius of the
dispersive optical potential and its diffuseness parameter, rs,d and as,d , and the respective parameters of
the spin–orbit potential, rso and aso (hereafter, we
follow the notation adopted in [4]), in accordance with
the systematics presented in [24]. Two additional
parameters of the energy dependence of the imaginary
part of the dispersive optical potential were also found
with the aid of results reported in [24]. Namely, the
parameter αI , which determines the height of the
plateau of the volume integral JI (E), was found by
formula (5) from [4], while the parameter βs , which
determines the steepness of the slope of the volume
integral Js (E), was found by formula (7) from [4]. In
doing this, the dependences JI (E) and Js (E) were
approximated by expression (4) from [4] at n = 4 and
for E0 = EF .
The parameter γ, which characterizes the energy
dependence of the Hartree–Fock component of the
dispersive optical potential, was determined by formula (11) from [4]. For this, the evaluated energies
eval were obtained with the aid of the experimental
E1s
1/2
expt
dependences E1s1/2 (A) [21] and the results of the
RMFM calculations from [22].
In the course of searches for optimum values of
the radius and diffuseness parameter (rHF and aHF ,
respectively) of the Hartree–Fock component of the
dispersive optical potential, it turned out that they can
be set to the values of the parameters rd and ad , respectively, from the systematics in [24]. Probably, this
is because the search for optimum parameter values
was performed on the basis of fits to the experimental
energies of states lying in the vicinity of the nuclear
surface. As a rule, the sensitivity of the parameters
rHF and aHF determined from fits to energies of deeplying states is not very high.
As a result, only three parameters were determined
on the basis of mesh-based searches. These were
βI , E0 , and Vso . The search was terminated upon
reaching a minimum of the χ2 functional
DOP
2
1 (Enlj − Enlj )
,
(5)
n
Δ2
where n is the number of the subshells in the vicinity
of the Fermi energy EF and Δ is the error in deterexpt
mining Enlj .
expt
χ2 =
The dispersion component of the real part of the
dispersive optical potential is calculated on the basis
of data on its imaginary part, which is assumed to
be symmetric with respect to the energy identified
with the Fermi energy EF . A determination of this
parameter is the first and very important step in analyzing data within the dispersion approach. Since the
dispersive optical potential describes states of system
A + 1 (particle state) and system A – 1 (hole state),
the energy EF for systems n, p + A can be determined
as half-sums
E+ + E−
,
(6)
EF =
2
where E+ is the energy of first particle state (in the
most strongly bound predominantly unfilled orbit)
and E− is the energy of the last hole state (in the most
loosely bound predominantly filled orbit).
Within the single-particle shell model without
mixing of configurations, the role of the energies
E+ and E− is played by, respectively, the negative
nucleon-separation energy S(A + 1) from nucleus
A + 1 and the negative nucleon-separation energy
S(A) from nucleus A. According to this model, we
have
S(A) + S(A + 1)
.
(7)
EFs.e. =
2
In the case where it is difficult to single out states
that can be thought to be the first particle and last hole
states, the respective formula of Bardeen–Cooper–
Schrieffer (BCS) theory can be used to determine the
Fermi energy EF , since this formula makes it possible
to describe the energy dependence of the occupation
probabilities for single-particle orbits; that is,
⎛
=
1⎝
1 − 2
Nnlj
⎞
(8)
Enlj − EFBCS
⎠,
2
BCS
2
+ ΔBCS
Enlj − EF
where ΔBCS is the gap parameter. By using the
expt
expt
values of Enlj and Nnlj from Tables 1 and 4, we
have calculated by formula (8) the values of EFBCS and
ΔBCS for proton and neutron shells in the isotopes
58,60,62,64 Ni.
The energy EF plays the role of a model parameter,
on one hand, and one of the shell characteristics of
nuclei, on the other hand. The self-consistent dispersive optical potential must reproduce the Fermi
energy EF (see [23]). We ensure this here by determining EF at the initial step on the basis of data on
expt
the energies Enlj that are used in searches for the
best agreement at the final stage of determining the
parameters of the dispersive optical potential. Table 9
gives the values that we found for EFBCS , EFs.e. , and
expt
EF
for proton and neutron subshells of stable nickel
PHYSICS OF ATOMIC NUCLEI
Vol. 74 No. 11
2011
SHELL STRUCTURE OF EVEN–EVEN NICKEL ISOTOPES
1531
Table 9. Fermi energies EF and gap parameter ΔBCS for proton and neutron states of the isotopes 58,60,62,64 Ni (all of the
values here are given in MeV units)
Proton shells
Nucleus
58
Ni
Neutron shells
expt
−EF
−EFs.e.
−EFBCS
ΔBCS
−EF
expt
−EFs.e.
1.79
5.84(26)
5.79
11.34
3.11
12.9(2)
10.60
5.04
2.37
4.88(103)
−EFBCS
ΔBCS
5.29
60
Ni
6.31
2.62
5.87(164)
7.17
9.87
2.92
12.1(6)
9.60
62
Ni
7.50
2.77
8.34(76)
8.63
8.63
2.62
11.8
8.72
64
Ni
8.95
2.61
9.08(90)
10.00
7.90
1.72
11.9
7.88
Table 10. Parameters of the proton dispersive optical potential for the 48−68 Ni nuclei
Parameter of the dispersive
optical potential
52
28 Ni24
54
28 Ni26
56
28 Ni28
58
28 Ni30
60
28 Ni32
62
28 Ni34
64
28 Ni36
−0.8
0.5
2.2
3.9
4.9
5.9
8.3
9.1
12.6
100.0
96.2
100.0
100.0
100.0
101.0
101.7
102.5
103.0
104.3
βI [MeV]
12.0
10.0
11.0
8.0
10.0
7.0
10.0
8.0
12.0
10.0
βs [MeV]
50.0
52.0
51.0
52.0
57.4
59.0
61.0
63.5
64.0
69.0
VHF (EF ) [MeV]
44.81
46.33
47.52
49.08
50.61
51.68
51.57
53.96
54.10
57.56
−EF [MeV]
3
αI [MeV fm ]
48
28 Ni20
50
28 Ni22
−2.1
γ
0.480
rHF [fm]
1.285
1.284
1.283
1.282
1.282
1.281
1.280
aHF [fm]
0.45
0.545
0.546
0.547
0.548
0.549
0.550
rd [fm]
1.285
1.284
1.283
1.282
1.282
1.281
1.280
ad [fm]
0.544
0.545
0.546
0.547
0.548
0.549
rs [fm]
1.192
1.194
1.195
1.197
1.198
0.670
0.670
0.670
0.670
Vso [MeV fm ]
7.5
7.5
7.5
rso [fm]
1.007
1.010
aso [fm]
0.59
rC [fm]
1.271
as [fm]
2
−E1s1/2 [MeV]
50.6
0.488
0.478
0.456
1.280
1.279
1.278
0.551
0.552
0.554
1.280
1.279
1.278
0.550
0.551
0.552
0.554
1.200
1.200
1.201
1.202
1.205
0.669
0.669
0.669
0.669
0.668
0.668
7.5
7.5
6.0
7.5
8.0
8.0
7.0
1.012
1.014
1.016
1.020
1.020
1.020
1.024
1.027
0.59
0.59
0.59
0.59
0.59
0.59
0.59
0.59
0.59
1.269
1.266
1.264
1.261
1.259
1.258
1.256
1.254
1.251
52.0
0.490
51.7
expt
isotopes. We have calculated the values of EF by
formula (6), taking the experimental and evaluated
energies of the 1f7/2 shell for the energies E− ; as for
the energies E+ , they were set to the energies of the
2p3/2 subshells for the proton structure and the energies of the 1f5/2 and 2p3/2 subshells for the neutron
structure of, respectively, the 58,60 Ni and 62,64 Ni nuclei. The resulting sets of parameters of the dispersive
optical potential for proton and neutron subshells of
the even–even isotopes 48−68 Ni are given in Tables 10
and 11, respectively. In those tables, we also present
our evaluated energies of the 1s1/2 level, which were
PHYSICS OF ATOMIC NUCLEI
Vol. 74 No. 11
2011
0.492
54.0
0.479
52.0
0.473
52.5
0.482
53.2
0.472
68
28 Ni40
53.9
54.6
56.0
used to determine the slope parameter γ for the energy
dependence of the Hartree–Fock component of the
dispersive optical potential. The calculated energies
DOP are given in Tables 7 and 8 and in Figs. 1 and 2
Enlj
for, respectively, proton and neutron subshells of the
nuclei under study.
5.1. Proton Subshells
For the 58 Ni nucleus, we determined two values of
EFπ,BCS (see Table 9) that correspond to two versions
of the experimental parameters of the shell structure,
1532
BESPALOVA et al.
Table 11. Parameter of the neutron dispersive optical potential for the 48−68 Ni nuclei
Parameter of the dispersive
optical potential
48
28 Ni20
50
28 Ni22
52
28 Ni24
54
28 Ni26
56
28 Ni28
58
28 Ni30
60
28 Ni32
62
28 Ni34
64
28 Ni36
68
28 Ni40
−EF [MeV]
19.5
19.3
18.8
19.4
13.4
11.3
12.1
11.8
11.9
11.8
−E0 [MeV]
19.5
19.3
18.8
19.4
13.4
3.0
3.0
3.0
3.0
3.0
107.0
99.3
96.5
93.8
91.4
89.0
87.3
85.0
83.0
80.0
βI [MeV]
10.0
10.0
10.0
10.0
7.5
0.7
0.7
0.7
0.7
0.7
βs [MeV]
61.0
76.0
76.0
74.7
60.0
67.0
62.4
59.0
53.5
62.0
VHF (EF ) [MeV]
55.18
54.19
52.68
52.75
50.42
49.50
48.40
47.2
46.6
44.9
αI [MeV fm3 ]
γ
0.475
0.461
0.474
0.476
0.463
0.437
0.452
0.459
0.462
0.463
rHF [fm]
1.285
1.284
1.283
1.282
1.282
1.281
1.280
1.280
1.280
1.278
aHF [fm]
0.536
0.536
0.536
0.536
0.536
0.536
0.535
0.534
0.534
0.533
rd [fm]
1.285
1.284
1.283
1.282
1.282
1.281
1.280
1.280
1.280
1.278
ad [fm]
0.536
0.536
0.536
0.536
0.536
0.536
0.535
0.534
0.534
0.533
rs [fm]
1.192
1.194
1.195
1.197
1.198
1.198
1.200
1.201
1.203
1.205
as [fm]
0.671
0.670
0.670
0.670
0.670
0.670
0.669
0.669
0.668
0.668
Vso [MeV fm2 ]
4.5
5.0
5.5
6.0
8.0
4.5
5.5
5.7
5.4
5.5
rso [fm]
1.01
1.01
1.01
1.01
1.016
1.018
1.020
1.022
1.022
1.027
aso [fm]
0.59
0.59
0.59
0.59
0.59
0.59
0.59
0.59
0.59
0.59
−E1s1/2 [MeV]
π,expt
62.6
63.2
63.2
π,expt
Enlj and Nnlj , from Table 1. It turns out that the
best agreement between the calculated and experimental energies is attained in the case of choosing the
value of EFπ,BCS = −5.04 MeV and the corresponding
π,expt
experimental energies Enlj from Table 1 (columns 4
and 5). That set of parameters of the dispersive
optical potential for proton states of the 58 Ni nucleus
π,DOP
which ensured the best agreement between Enlj
π,expt
and Enlj is given in Table 10, and the energies calculated by using this set of parameters are presented
in Table 7. It should be noted that, for all four stable
π,DOP
and
nickel isotopes, good agreement between Enlj
π,expt
Enlj
was obtained upon employing, for the Fermi
expt
energy, the values of EF , which differ from EFs.e.
by about 1 MeV. This difference correlates with large
values of the total fragmentation width of the 1f7/2
and 2p3/2 subshells in the isotopes 58,60,62,64 Ni (see
Table 3).
63.4
63.9
63.0
62.6
62.0
61.5
61.0
For the neutron-deficient isotopes 48,50,52,54 Ni,
there is no possibility for determining EFBCS at the
present time. Relying on the correspondence between
π,expt
that was found for
the energies EFπ,BCS and EF
stable nickel isotopes, we have set the Fermi energy
for the aforementioned unstable nickel isotopes to
π,expt
as determined by using the evaluated
values of EF
π,eval
energies Enlj from Table 7. It is noteworthy that the
π,expt
for 48,50 Ni are positive (see Table 10).
values of EF
Nevertheless, these values correspond to the 1f7/2
bound state (see Table 7), which is the last occupied
proton state in nickel isotopes. It follows that
π,expt
for the 48,50 Ni nuclei do not
positive values of EF
contradict the experimental fact of stability of these
nuclei against proton emission, and we consider them
as model-parameter values. For the 52,54 Ni nuclei,
π,expt
are negative and are equal to
the values of EF
–0.55 and –2.25 MeV, respectively. The energies
π,DOP
calculated for the 48,50,52,54 Ni nuclei with
Enlj
PHYSICS OF ATOMIC NUCLEI
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2011
SHELL STRUCTURE OF EVEN–EVEN NICKEL ISOTOPES
the parameter values from Table 10 are presented
in Table 7 and in Fig. 1. They comply well with the
π,eval
evaluated energies Enlj
.
We have also attained good agreement between
π,expt
π,DOP
and Enlj
for the isotopes
the energies Enlj
π,expt
1533
(see Table 11) and to match the calculated energies
ν,expt
ν,DOP
Enlj
with Enlj . A comparison of these energies
for the 1d3/2 , 1f , and 2p neutron states is illustrated
in Table 8. In addition, we note that the values of
ν,DOP
ν,DOP
= −5.51 MeV and E2d
= −2.61 MeV are
E1g
9/2
5/2
ν,KY
In determining EF
= EFπ,s.e. = −3.95 MeV close to the energies of E1g
= −5.69 MeV and
9/2
ν,KY
Ni
,
we
employed
for the doubly magic nucleus 56
58
28 28
E2d5/2 = −3.12 MeV for the Ni nucleus and to the
π,expt
data from [18] for the energies Enlj . We note that experimental energies of the 1g and 2d subshells
9/2
5/2
RMFM
calculated within in the 62,64 Ni nuclei.
agreement of the energies Enlj
RMFM approach [22, 25] with experimental data is
The choice of EFν,BCS ≈ EFν,s.e. for the isotopes
reached only for the 1f7/2 state. For the 1d3/2 and 60,62,64
Ni as the Fermi energy gave no way to match
π,expt
π,RMFM
ν,expt
ν,DOP
differ from Enlj
2s1/2 states, the energies Enlj
the energies Enlj
with Enlj for the 1d3/2 –2d5/2
by, respectively, six to seven MeV units and two to
states to the same degree of precision as for 58 Ni. For
three MeV units. As for the 2p3/2 state, it is not
this reason, the search for optimum parameters of the
bound according to [22].
neutron dispersive optical potential for the 60,62,64 Ni
A comparative analysis of the parameters of the
ν,expt
.
proton dispersive optical potential for nickel nuclei nuclei was performed by using the values of EF
In our attempts at finding parameters of the neuwhose mass number A ranges between 48 and 68
π,DOP
tron
dispersive optical potential for the 60,62,64 Ni nuand
reveals that good agreement between Enlj
clei, we run into the problem of insufficiency of exπ,expt(eval)
is attained at close values of the majorEnlj
ν,expt
perimental information about the values of Enlj
ity of parameters for all isotopes under study. For
(see Table 8). By using this limited experimental
example, the parameters rHF and aHF change in the
information, one can find several sets of parameters
intervals 1.278 rHF 1.285 fm and 0.545 aHF of the dispersive optical potential such that they lead
0.554 fm, respectively, for all isotopes, with the exν,expt
ν,DOP
48
and Enlj .
ception of 28 Ni20 , for which the optimum value of to very good agreement between Enlj
the parameter aHF is aHF = 0.45 fm. The strength One of such sets was determined in [2]. However, an
parameter of the spin–orbit potential is identical for additional analysis revealed that, for a less ambiguous
all isotopes, Vso = 7.0 ± 1.0 MeV fm2 . The values search for parameters of the dispersive optical potenof the parameters αI and βs fall within the ranges tial, it is necessary to have data on the energies of the
100 αI 104 MeV fm3 and –50 βs 70 MeV. 1f7/2 , 2s1/2 , and 1d3/2 neutron states. Moreover, we
As matter of fact, the parameters βI and EF proved do not have at our disposal experimental data on the
to be individual parameters that characterize special energies of the 1g9/2 and 2d5/2 states for the isotopes
features of the proton structure. This is because 58,60 Ni or experimental data on the energy of the 2d5/2
all nickel isotopes proved to be magic nuclei whose state for the isotope 62 Ni. In view of this, the energies
charge number is Z = 28.
calculated with the Koura–Yamada potential were
included here in our analysis as evaluated energies
in the case where there were no experimental data.
5.2. Neutron Subshells
In particular, we took into account the calculated
From a comparison of the values of EFν,BCS , energies E ν,KY for the 1d , 2s , 1f , 1g , and
3/2
1/2
7/2
9/2
nlj
ν,expt
, and EFν,s.e. (see Table 9) for the stable isotopes 1d5/2 states. This was motivated by the proximity
EF
58,60,62,64 Ni, one can see that, for all of the respective
ν,KY
in
to the experimental data on the energies E2s
1/2
ν,BCS
and
nuclei, there is no agreement between EF
ν,KY
the 56 Ni nucleus, E1f7/2 ,1d3/2 in the 58 Ni nucleus,
ν,expt
, but there is good agreement between EFν,BCS
EF
ν,KY
62,64 Ni nuclei, and E ν,KY in the 64 Ni
and EFν,s.e. . This is likely to be due to special features E1g9/2 in the
2d5/2
in the filling of those neutron subshells in stable nickel nucleus (see Table 8). The parameters found for the
isotopes that are not magic in intranuclear neutrons.
neutron dispersive optical potential in the 60,62,64 Ni
of the above estimates are given
The choice of EFν,BCS for the Fermi energy of 58 Ni nuclei with the aidν,DOP
values calculated with these
made it possible to determine the set of optimum in Table 11. The Enlj
parameters of the neutron dispersive optical potential parameter values (see Table 8 and Fig. 2) agree well
56,68 Ni.
PHYSICS OF ATOMIC NUCLEI
Vol. 74 No. 11
2011
1534
BESPALOVA et al.
ν
N nlj
while, in Table 8, we present the corresponding values
ν,DOP
of Enlj
.
2p3/2
1.0
ν,expt
0.8
1f5/2
0.6
2p1/2
0.4
1g9/2
0.2
0
58
60
62
64
66
68
A
Fig. 3. Occupation probabilities for neutron subshells in
the 58−68 Ni nuclei. The displayed points are experimental
data for the (closed boxes) 1f5/2 , (closed circles) 2p1/2 ,
(closed triangles) 2p3/2 , and (closed diamonds) 1g9/2
subshells. The lines represent extrapolations to the region
of 68 Ni.
with experimental and evaluated data for all states
from the 1d3/2 to the 2d5/2 ones in the 60,62,64 Ni
nuclei.
It is noteworthy that the 1f5/2 neutron state is
somewhat more strongly bound in the 58 Ni and 60 Ni
nuclei than in the doubly magic nucleus 56
28 Ni28 (see
Table 8 and Fig. 2). This follows both from the
results obtained by the joint-evaluation method and
the results of the calculation with the dispersive optical potential and may be due to the j> –j< tensor interaction between the 1f5/2 neutron state and
the 1f7/2 proton state, which have close energies.
ν,expt
For example, E1f5/2 = −10.55 and –9.20 MeV and
π,expt
E1f7/2 = −8.09 and –9.20 MeV in, respectively, the
58 Ni
and the 60 Ni nucleus. In that case, it can be
concluded that the effect of the j> –j< interaction
on single-particle spectra of nickel isotopes is taken
into account via choosing optimum parameters of the
dispersive optical potential.
ν,expt
For the 48,50,52,54 Ni nuclei, the values of EF
and EFν,s.e do not differ strongly from each other,
in just the same way as for the 58 Ni nucleus. We
have determined optimum parameters of the dispersive optical potential for these isotopes, employing
ν,expt
and EFν,s.e. . In either case, we reached
both EF
ν,DOP
ν,eval
and Enlj
for all
good agreement between Enlj
of these isotopes and for the states being considered.
In Table 11, we give the set of parameters of the disν,expt
,
persive optical potential for the case of EF = EF
The Fermi energy EF = EF
= EFν,s.e. for the
doubly magic nucleus 56
28 Ni28 was determined on the
basis of data from [18]. In just the same way as for
other neutron-deficient nickel isotopes, good agreeν,DOP
ment between the calculated energies Enlj
(see
Table 8) and their experimental counterparts was obtained for this nucleus.
In [26], our group found a local set of parameters
of the dispersive optical potential for the case of EF =
EFν,s.e. = −6.19 MeV and attained good agreement
ν,DOP
ν,eval
and the energies Enlj
between the energies Enlj
evaluated for the 1f , 2p, 1g9/2 , and 2d5/2 neutron
states in the 68 Ni nucleus (see Table 1 in [26]). In
that study, the energies of the 1d3/2 and 2s1/2 states
were not analyzed because of the absence of experν,DOP
and
imental data for them. The energies E1d
3/2
ν,DOP
calculated with the dispersive optical potenE2s
1/2
tial in [26] proved to lie 6 to 7 MeV deeper than
KY
KY
= −18.7 MeV and E2s
= −17.6 MeV.
E1d
1/2
3/2
In [26], it was assumed that the 1f and 2p subshells are fully occupied, but that the 1g9/2 subshell
ν,eval
= −4.6(5) MeV is empty, which
of energy E1g
9/2
is in accord with the idea that the 68 Ni nucleus is
magic in neutrons. In the present study, we have
ν,DOP
addressed the question of how the energies Enlj
change in response to changes in EF . Figure 3 shows
the mass dependences of the experimental occupation
probabilities for the 2p, 1f5/2 , and 1g9/2 subshells in
ν,expt
the stable nickel isotopes from Table 4, Nnlj (A).
From Fig. 3, one can see that these dependences
are close to linear ones. Linear extrapolations make
it possible to obtain evaluated occupation probabilν
for the 68 Ni nucleus. In accorities N2p,1f
5/2 ,1g9/2
dance with these estimates, the 2p3/2 neutron subshell in the 68 Ni nucleus is completely occupied—
ν,eval
= 1.0—while the 1f5/2 and 2p1/2 subshells
N2p
3/2
ν,eval ∼
are filled only partly—N ν,eval ∼
= 0.85 and N
=
2p1/2
1f5/2
0.6, respectively; as for the 1g9/2 subshell, it is partly
filled rather than free, its evaluated occupation probν,eval ∼
ability being N1g
= 0.2. It should be borne in
9/2
ν (A) obmind, however, that the dependences Nnlj
tained from experimental data are not always linear, sometimes showing jumps as the mass number
approaches that of a magic nucleus. For example,
PHYSICS OF ATOMIC NUCLEI
Vol. 74 No. 11
2011
SHELL STRUCTURE OF EVEN–EVEN NICKEL ISOTOPES
the occupation probability for the first 2p3/2 neutron
subshell, which is predominantly free, is lower for the
48 Ca than for the 46 Ca nucleus [13]. Similarly, the
occupation probability for the 3s1/2 subshell is lower
for the 96 Zr than for the 94 Zr nucleus [27]. Therefore,
only with this reservation are linear extrapolations
ν
for the 68 Ni nujustified in evaluating N2p,1f
5/2 ,1g9/2
cleus.
In accordance with the presumed filling of neutron
subshells in the 68 Ni nucleus, we determined three
values of the Fermi energy. Under the assumption
that the 2p3/2 subshell is filled to an extent less than
that which is predicted by the linear extrapolation (see
Fig. 3), the Fermi energy EF can be defined as the
half-sum of the energies of the 1f7/2 and 2p3/2 subν,eval
< 1.0, the halfshells (EF = −11.8 MeV) if N2p
3/2
sum of the energies of the 2p3/2 and 1f5/2 subshells
∼ 1.0, and the half-sum
(EF = −8.7 MeV) if N ν,eval =
2p3/2
of the energies of the 2p1/2 and 1g9/2 subshells (EF =
ν,eval
ν,eval
ν,eval
= N1f
= N2p
= 1. We
−6.2 MeV [26]) if N2p
3/2
1/2
5/2
ν,DOP
at three values
have calculated the energies Enlj
of the Fermi energy EF and the values from Table 11
for the parameters of the dispersive optical potential. It turned out that (i) the energies of the 1g9/2
and 2d5/2 subshells are virtually independent of the
Fermi energy EF ; (ii) the energies of the 2p and 1f
subshells change within an interval of width smaller
than 1 MeV; and (iii) the energies of the 2s1/2 and
1d3/2 subshells change within an interval of width 4
to 4.5 MeV, the 1s1/2 level remaining in the vicinity of –60.8 MeV. The value of EF = −11.8 MeV
presented in Table 11 corresponds to a minimum of
the χ2 (5) functional calculated with allowance for
KY
as evaluated energies. We
the values of E1d
3/2 ,2s1/2
note that the values of the Fermi energy EF from
Table 11 exhibit a jump upon going over from the 54 Ni
to the 56 Ni nucleus, but that there is no such jump
upon going over to the 68 Ni nucleus. Nevertheless,
the energy gap between the 2p1/2 and 2g9/2 neutron
states that corresponds to calculations with either
of the Fermi energy values EF = −11.8 MeV and
EF = −6.19 MeV [26] is larger in the 68 Ni nucleus
than in the neighboring N = 40 nuclei, this being
characteristic of a magic nucleus. The result in question suggests an intricate character of the evolution
of the shell structure of nuclei in the vicinity of 68 Ni.
However, it is necessary to consider that this may be
attributed to a specific definition of the parameter EF
in the dispersive optical potential. In order to find
the dispersive optical potential more precisely, one
PHYSICS OF ATOMIC NUCLEI
Vol. 74 No. 11
2011
1535
needs experimental data on single-particle energies
with respect to the deep-lying 1d3/2 and 2s1/2 states.
6. CONCLUSIONS
expt
The occupation probabilities Nnlj for proton and
neutron subshells of the isotopes 58,60,62,64 N and their
expt
single-particle energies Enlj were determined previously by applying the joint-evaluation method to
data on stripping and pickup reactions. These data take into account the effect of fragmentation of
states because of the role of residual interactions in
nuclei. The same method was used to determine the
expt
expt
values of Nnlj and Enlj for the proton and neu50
52
tron subshells in the isotopes 48
20 Ca28 , 22 Ti28 , 24 Cr28 ,
and 54
26 Fe28 . Since the nuclei listed above are mirror
with respect to the neutron-deficient nickel isotopes
48,50,52,54 Ni, the energies of the respective subshells
and the occupation probabilities could be estimated
for the latter. Thus, we were able to obtain here the
most comprehensive experimental information about
the energies of proton and neutron subshells in nickel
isotopes containing 20 to 40 neutrons, invoking data
from the literature on the single-particle energies of
68
the isotopes 56
28 Ni28 and 28 Ni40 .
We have found special features in the dependence
expt
of the energies Enlj for proton and neutron subshells
on the number of neutrons in respective isotopes. For
the energies of the 2p3/2 and 2p1/2 proton subshells
in the isotopes 58,60,62,64 Ni, there are special features
in the dependences of proton single-particle energies
π,expt
Enlj (N ) in the vicinity of N = 28 (56
28 Ni28 nucleus).
Special features in the dependences of the neutron
ν,expt
single-particle energies Enlj (N ) are also observed
in the vicinity of the doubly magic nucleus 56
28 Ni28 and
ν,expt
in the dependence E1f5/2 (N ) in the region of N = 30,
32. The degeneracy of the 2p3/2 , 2p1/2 , and 1f5/2
neutron subshells in the nickel isotopes being studied
that was observed earlier in [16] has been confirmed
in the present study.
A detailed comparison of a number of theoretical
calculations purporting to provide a unified description for a large number of nuclei with the experimental
expt
dependences Enlj (N ) for nickel isotopes from 48 Ni
to 68 Ni has been performed. In particular, this was
done for the calculations within the relativistic meanfield model and the mean-field model with the Koura–
Yamada potential and with the dispersion optical potential.
Although the energies calculated with the Koura–
Yamada potential disregard the effect of subshell
1536
BESPALOVA et al.
fragmentation, their deviation from experimental data
does not exceed 1 MeV, on average. The largest
distinctions are observed for the 1f5/2 state. For
π,KY
in the 64 Ni nucleus differs
example, the value of E1f
5/2
from experimental data by 2.5 MeV, while the value of
ν,KY
in the 58 Ni nucleus differs from experimental
E1f
5/2
data by 2.3 MeV. Nevertheless, the calculations with
the Koura–Yamada potential predict correctly special
π,ν
(N ) and are useful
features in the dependences Enlj
in evaluating subshell energies in unstable nuclei.
We have shown that the method used in [4] to construct a dispersive optical potential makes it possible
to match, within the experimental errors, the calcuDOP of proton and neutron subshells
lated energies Enlj
in the nuclei from the range between 48 Ni and 68 Ni;
to describe the observed special features in the deexpt
pendences Enlj (N ); and to make well-substantiated
predictions for single-particle energies in unstable
nuclei, which are members of isotopic chains.
ACKNOWLEDGMENTS
We are grateful to S.A. Goncharov for stimulating
discussions.
This work was supported by the Federal Agency
for
Science
and
Innovation
(Contract
no. 02.740.11.0242 within Part 1.1. “Implementation
of Scientific Investigations by the Staff of Research
and Educational Centers”).
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Translated by A. Isaakyan
PHYSICS OF ATOMIC NUCLEI
Vol. 74 No. 11
2011