85
Progress of Theoretical Physics, Vol. 45, No. 1, January 1971
Rotational Levels in a Simple Shell Model Configuration
Yasutoshi TAN AKA, Kengo 0GAW A and Akito ARIMA
Department of
Physics~
the University of Tokyo, Tokyo
(Received August 11, 1970)
Systems of (j12 (proton), h± 2 (neutron)) configurations are examined for the purpose of
studying the role of neutron-proton correlation. Residual interactions of o+ QQ type are
assumed. Protons and neutrons are strongly coupled by the attractive neutron-proton interaction (both short range and long range) which causes rotational features with the axial
symmetry in two-proton and two-neutron systems. However, because the coupling of protons
with neutron holes is weak, this leads the two-proton and two-neutron hole system to show
no rotational property with the axial symmetry. In the former system, the ground K=O+
band and also an excited K = 1 + band appear, provided that j 1 and h are large and the QQ
interaction is strong. Exact wave functions of the ground band members and K =1 + band
members are very similar to projected states from intrinsic wave functions given by the
aligned coupling scheme.
§ 1.
Introduction
Microscopic descriptions of rotational levels have been extensively studied
in recent years. The SU3 modeP) can successfully describe collective properties
in light nuclei; and in heavier ones, the aligned coupling scheme (A.C.S.y) and
the stretch scheme (S.S.Y) were introduced for the purpose of describing rotational levels in the spherical j-j shell model basis.
On the other hand, Mulhall and Sips 4) found that collective features persist
even in a system where four like nucleons occupy a single j shell, provided that
j is large and the QQ interaction is strong. Nomura and Arima (one of authors),
and recently Friedman and Kelson, 5) reexamined this model, and Nomura has
further shown that a K = 2 r-like band also appears when j is larger than 17/2.
In real nuclei, however, the correlation of protons and neutrons in open
shells may be very important for producing rotational features. In fact, rotational
features have not been observed in single closed-shell nuclei such as Ni and
Sn isotopes.
In order to study the role of the neutron-proton interaction in nuclear col. lective motion, we shall investigate here systems consisting of two protons in a
j 1-orbit and two neutrons or two neutron-holes in a j 2-orbit. The systems which
we shall examine in this paper are schematically shown in Fig. 1.
When j 1 and j 2 are large enough, the rotational level structures are found in
two-proton and two-neutron configurations (case (a)) under the strong QQ interaction. We compare the exact wave functions with those formed by projecting
the angular momentum from the intrinsic state in the A.C.S. When the intrinsic
86
Y. Tanaka, K. 0 gawa and A. Arima
states for both protons and neutrons are
assumed
to have the same oblate deformaj2
tion, the overlapping integrals are very
j,
large (0.995). For other shapes (for example, protons having an oblate deformation and neutrons a prolate one) the overcase (a)
case (b)
lapping integral is very small. We can
thus confirm that these two protons and
Fig. 1. Two types of configurations extwo neutrons interact coherently to make
amined in this paper. Case (a) is the
two-proton and two-neutron configurathe oblate deformed shape.
tion, and case (b) is the two-proton and
Two-proton and two-neutron-hole contwo-neutron-hole one. Particles are refigurations (case (b)), on the other hand,
presented by circles and holes are by
do not show the rotational structure with
cross signs.
axial symmetry. The spectrum resembles that of the asymmetric rotor.
The neutron-proton interaction increases the binding energy in both cases,
but the energy gain in the two-proton and two-neutron configurations are about
twice as large as those in the two-proton and two-neutron-hole configurations.
These facts suggest that couplings between protons and neutrons are stronger in
the former case than in the latter case.
Four different ( J1 J2) pairs are taken: (1d5/2• 2Pa;2), (Og112, Oh9;2), (Ohn12, Oh9/2)
and (Oh 1112 , Oi13; 2) . The proton orbits are located in the 50~82 major shell and
the neutron orbits are in the 82~ 126 major shell. The excitation energy is
calculated as a function of the strength of the QQ force in § 2. In § 3, electromagnetic properties of these states are studied. In § 4, wave functions of the
case (a) are compared with those of the A.C.S. and with those of the S.S.
Finally, conclusions are given in § 5.
(p)
(n)
(Pl
(n)
-e-e--
~
§ 2.
Energy levels
The Hamiltonian is composed of two parts, I.e. the spherical shell model
Hamiltonian Hs.M. and the residual interaction.
(2 ·1)
The delta plus QQ interaction is assumed for the residual interaction;
Vi 1 = (Yo+ V1 (rJi · rJ1 )) o(ri- r 1)
-
K'ri 2r/ P2 (cos
(J)i 1 ).
(2·2)
The energy matrix is constructed in the truncated space spanned by the following
vectors which are eigenstates. of Hs.M.:
for case (a)
(2·3)
for case (b).
(2 ·4)
and
Rotational Levels in a Simple Shell Model Configuration
87
The matrix elements are generally expressed as follows: In case (a)
<j1 2(JP)j2 2(Jn): JIHs.M. + .EVI j12(J/)j2 2(Jn'): J)
2
2
2
2
= 0JpJp'0JnJn' {2s1t + 2Sj 2 + <j1 Jpl VI j1 Jp) + <j2 Jnl VI j2 Jn)}
+ 4 :E (2A + 1) (2B + 1)
J (2Jp + 1) (2Jn + 1) (2Jp' + 1) (2Jn' + 1)
AB
(2·5)
and in case (b)
<j1 2 (Jp)j2- 2 (Jn): JIHs.M. + .EVI j12(JP')j2- 2(Jn'): J)
= 0JpJp'0JnJn' {2sh + (2j2 _:_ 1) CJ 2+ < j1 Jpl VI j12Jp) + (j2 2Jnl Vlj2 2Jn)
2
+
2 2
~ -3
2; 2 + 1
:E(2A+1)<j2 2AIVIj2 2A)}
A
-4 :E (2A + 1) (2B + 1) J (2Jp + 1) (2Jn + 1) (2J/ + 1) (2Jn' + 1)
AB
where A, B and C are intermediate angular momenta; <I VI) means a two body
matrix element between the like nucleons (T= 1); and (I VI) is a matrix element
between the proton and the neutron which contains equally T = 1 and T = 0 components. Only the direct integral part contributes to this matrix element of the
neutron-proton interaction in this paper, because the charge exchange forces are
neglected. All the Hartree energy terms and single particle energy terms can
be omitted in the course of diagonalization.
In order to evaluate the energy matrices mentioned above, it is necessary
to give general forms of the two-body matrix elements of the o and QQ interactions in nonantisymmetric states:
(j1j2JI (Vo+ V10"1·0"2)o(r1-r2) lj/j/J)
- ( -1)l2+L2'+J2-j2'J (2jl + 1) (2j2 + 1) (2j/ + 1) (2j2' + 1)
2(2J+1)
x [ { ( Vo + V1) - (1 + ( -1Yr+l2-J) 2 V1} (jri" j2- -!IJO) (ji'-! j / - tiJO)
+ ( -1Y 2+l2'+Ja-J 2'(Vo+ V1) (jl!j2-!IJ1) (j1'ij2'-!IJ1)]
(2·7)
Y. Tanaka, K. Ogawa and A. Arima
88
and
(j1j2Jir1 2r/·P2 (cos W12) I j/j2' J)
=
4
n' (n1l1ir
5
X
2
in/l/) (n2l2ir2in/ l/) ( -1)it+iz'+JW( jlj,j/j/: J2)
(2·8)
(jlll Y;ijj/) (j2ll Y;llj2'),
where
(2·9)
and
(2 ·10)
In these equations Rn~ are radial parts of the single particle wave functions.
Using Eqs. (2 · 7) and (2 · 8), one can easily calculate matrix elements in antisymmetric states.
Since our main objective _is to study the dependence of energy level structures
interaction
on the strength of. the QQ interaction, we fix the strength of the
6
as follows : )
o
(Vo-3V1)F 0 (nf2, n'ln) = - f(nl 2,n'l'2)CA- 112,
(2 ·11)
V 1 =l_
Vo 11
(2 ·12)
C=30MeV,
and
A=150.
A is the mass number, and f(nf2, n'l' 2) are dimensionless quantities which are the
ratios of F 0 (nl 2, n'ln) to F 0 (0s 2, Os 2) .
The only variable parameter is the strength of the QQ interaction. Instead
of K' we will use. K, which is defined as follows:
(2 ·13)
In this paper, the parameter K is varied from 0.000 to 0.020. A realistic value
of K should be about 0.0087) and therefore K = 0.020 corresponds to a very strong
interaction.
Constructions of energy matrices and their diagonalizations were carried out
by using the HITAC 5020E of the University of Tokyo. Results are illustrated
in Figs. 2 (a) and 2 (b). Here we show results only for (Ohi1;2, Oii312) and (Ohi112,
Oii3/2) as typical examples.
In thes_e figures, the excitation energy is given in MeV. It should be
noticed that this energy scale is rather arbitrary, because the only meaningful
parameter is the ratio of the strength of the o interaction to that of the QQ interaction. The figure on the left-hand side of each state indicates its spin. Parity
Rotational Levels in a· Simple Shell Model Configuration
K=O.OOO
K=0.004
K=0.008
K=O.OI2
K=0.016
89
K=0.020
E. E.
I I.
oo-7--
10.
6--
9.
I0--
7-5--
8.
6--
s-"1--
(a)
10-7--
7.
5-8 --
6--
4--
3--
·~-==--I
3--
6.
,6--=-
5.
6
--
5
--
s-4
3
---
6--
,t--==::-
~--=-
6--
3
1
-7
4.
3
3
2.
2
-10-6--
·~~
I
3.
--
~
'=
--
~~--
4-4--
6--
10-2
10--
4--
4--
8--
6--
6--
4--
4--
I.
2--
0.
0--
2--
2--
2--
2--
o--
o--
o--
o--
2--
o--
Fig. 2(a). Energy spectra plotted as functions of the QQ interaction strength: the (Ohw2 2, Oita;2 2)
configuration: Energy scale is the unit of MeV. A small figure on the left-hand side of each
level means the spin of its state.
signs are omitted, since all the states have positive parities. We also omit suffixes
which distinguish the states having the same spin. In order to avoid unnecessary
complication of the figures, only the lowest states with spins 1, 3, 5 and 7 are put
into Fig. 2 (a). Furthermore, all states with odd spins higher than 7 are omitted.
All these omitted states are located higher in energy than the second 6+ state.
In Fig. 2 (b), no state is omitted below the first s+ state.
Calculated levels for (1di12 , 2p~12 ) show no clear band structure even when the
QO force is strong. In energy levels for (Og; 12 , Oh~12 ) 0 1 +, 2 1 +, 4 1 +, 6 2 +, 8 2 + and
102 + states may belong to the ground band, and 1 1+, 2 2 +, 3 1 +, 4 2 +, 5 1 + and 6 3 + seem
to construct an excited band, because excitation energies obey approximately
Y. Tanaka, K. 0 gawa and A. A rima
90
E.E.
6.
K=0.004
K=O.OOO
K=0.008
K=0.012
1<=0.016
K=0.020
a--
5.
IO--
(b)
·-- ·-4--
10--
4.
a-IO--
11--
3.
10--
e
:s--
~--
.,-4--
s--
4--
8
4--
:s--
~--
:s--
8--
lo=
2.
4--
~=
1==
4-2--
4--
2c
I.
0.
o--
z--
2-.--
2--
z--
2--
o--
o--
0--
0--
o--
Fig. 2(b). Energy spectra plotted as functions of the QQ interaction strength: (Oh 11122, Oi1312 - 2) :
Energy scale is the unit of MeV. A small figure on the left-hand side of each level means the
spin of its state.
the I (I+ 1) rule. In the case of (Ohi112, Oh~ 12), 01+, 21 +, 41 +, 61 +, 8 2+ and lOs+
states can be assigned to the ground band, while 11+, 22+, 31+, 42+, sl+, 62+ and
71+ may belong to another band.
Figure 2a, for (Ohi 112, Oii312), shows that the 01 +, 21 +, 41 +, 61+, 81 + and 102+
states can construct the ground band, and the l1 +, 2 2+, 31 +, 4 2 +, 61 +, 62+ and 71 +
states may be members of an excited band. To observe these situations more
clearly, the excitation energy is plotted as a function of I(I + 1) in Figs. 3 (a) r-v
3 (d).
In the following, the states which seem to have the ground band structure
and an excited band structure will be specified by the suffixes "g" and "e ",
respectively.
In Fig. 2(b), for (Ohi112 , Oii3}2 ), the rotation-like spectrum does not appear even
when the QQ interaction becomes strong enough. The excitation energy of the
4 1+ state is about twice as high as that of the 21 + state, and the 2 2 + state is
nearly degenerate with the 41+ state. The 2 2+ state, however, should not be
considered as a two-phonon state, because of electromagnetic properties of this
state (e.g., the cross-over E2 transition from this 2 2+ state to the ground state
IS
not so small, etc.).
Detailed discussions will be given in the next section.
§ 3. Q-moment and B(E2)
In the previous section we have seen that a low-lying state can be regarded
as a member of the ground band or an excited band. In order to see more
Rotational Levels in a Simple Shell Model Configuration
91
E. E.
2.
Oi
2i
I (I+ I)
0."------------------------
12.
6.
4.
I (1+1)
Q~-------------------------------------
12.
Fig. 3. Band structures in each two-proton and two-neutron configuration, where the QQ interaction strength is equal to K=0.020. The ordinate indicates the excitation energy, and the abscissa,
l(I+!). (a) (lds;22, 2pa;22), (b) (Og712 2, Oh9;2 2), (c) (Oh1112 2, Oh9;22) and (d) (Oh1112 2, Oi1a;22).
clearly how far this argument can be justified, electromagnetic properties were
studied. Calculated quadrupole moments Q are shown in Tables I(a)"-/I(c),
together with the intrinsic quadrupole moments Q0 of the ground band members
and the excited band members assumed as K = o+ and K = 1 +, respectively.
These 0 0 are derived by assuming the rotational model, I.e.
Q
=
o
(J+ 1) (21+3) Q.
3K 2 -J(J+1)
(3 ·1)
Y. Tanaka, K. 0 gawa and A.· Arima
92
The reduced Ml and E2 transition probabilities are also shown in Tables II(a)
and II(b). In Table II( a), the figures in brackets are relative ratios of B (E2)
m each band. Examining these tables, we make the following observations:
1. Under the strong QQ interaction, intrinsic quadrupole moments indicate
as belonging to the ground band the 6 2 + state, not the 6 1 +, in the (Og; 12 , Oh~ 12 )
Table I(a). Calculated E2 moments of K=0.020 in the two-proton and two-neutron
configurations, which are given in unit of efm2. Q0 means an intrinsic :quadrupole
moment derived from Eq. (3·1). No effective charge is considered, i.e. ep= 1.0 and en
=0.0.
11+
(1d5;2 2, 2pa;2 2)
(Og7;2 2, Oh9;22)
(Oh11;2 2, Oh9;22)
Q
Qo
Q
Q
-34.80
-3.48
I
-3.48
I
Qo
Q
-5.56
-55.60
-5.30
-53.00
17.43
6.22
-60.99
-43.54
17.18
7.71
-60.14
-53.97
Qo
-34.80
(Ohu;2 2, Oi1312 2)
I
I
Qo
----
21+
22+
9.21
-4.70
-32.23
32.90
12.65
5.58
3t+
1.97
-7.88
7.22
-28.88
12.38
-49.52
12.32
-49.28
41+
42+
-11.24
7.99
30.90
-25.85
12.25
4.29
-33.69
-13.88
19.06
12.21
-52.41
-39.50
19.82
14.06
-54.51
-45.49
51+
-10.44
30.16
-1.18
3.41
12.68
-36.63
13.88
-40.10
61+
62+
6a+
-13.92
34.79
-5.65
26.30
15.37
5.19
-38.43
-13.97
18.64
11.81
-46.61
-31.80
...
...
-14.81
2.26
-9.77
-17.78
45.62
...
...
...
-29.24
-9.10
...
...
-4.28
-13.80
-44.27
-39.06
----
---~
71+
81+
82+
lOt+
102+
lOa+
...
...
...
...
...
...
...
...
...
--·--
...
.. .
21.61
...
31.75
...
...
...
...
...
-14.00
35.92
-39.56
4.54
...
9.96
-25.56
15.05
-35.74
-10.79
...
...
.. .
-37.13
5.83
-37.42
-11.17
-14.36
...
...
...
-13.41
...
...
33.03
Table I(b). Calculated M1 moments and E2 moments of K=0.020 in the two-proton and
two-neutron-hole configurations. Units are n.m. (nuclear magneton) and efm 2 respectively. No effective charge is considered.
(ld5;22, 2Pa;2 - 2 )
p.
I
Q
(Og112 2, Oh9;2 - 2)
p.
I
(Ohw2 2, Oh9;2 - 2)
Q
p.
I
Q
(Ohu;2 2, Oi1s12 - 2)
p.
I
Q
21+
22+
2.69
-1.05
9.21
-4.70
0.82
0.86
4.02
-4.46
1.90
1.64
10.89
-10.15
1.03
1.24
7.27
-6.98
31+
0.98
1.97
1.26
0.13
2.67
0.53
1.72
0.31
41+
6.79
-11.24
1.70
5.22
3.85
14.82
2.20
12.20
Rotational Levels in a Simple Shell Model Configuration
93
Table I(c). Calculated E2 moments for each QQ interaction strength in the (Oh 11122,0i1312 2) configuration.
K=O.OOO
Q
2t+
41+
6t+
8t+
102+
I Qo
K=0.004
Q
14.27 -49.95 16.33
14.25 -39.20 17.71
6.71 -16.77 13.72
-5.45 12.94
2.05
-25.04 57.60 -14.46
I Qo
K=0.008
Q
K=0.012
I Qo
Q
-57.17 16.82 -58.86 17.02
-48.71 18.87 -51.89 19.38
-34.30 16.70 -41.76 17.82
-4.87 10.47 -24~88 13.43
33.25 -9.85
22.66 -1.93
K=0.016
I Qo
Q
-59.55
-53.29
-44.56
-31.90
-4.43
17.12
19.65
18.35
14.53
3.28
I Qo
-59.92
-54.04
-45.87
-34.51
-7.55
K=0.020
Q
17.18
19.82
18.64
15.05
5.83
Qo
I
-60.14
-54.51
-46.61
-35.74
-13.41
Table II(a). Reduced E2 transition probabilities of K=0.020 in the two-proton and twoneutron configurations, measured in unit of e 2fm4.
'-c:l
2/~0g+
Ill
4g +~2g+
'-c:l
6g+~4g+
:::1
8g+~6g+
s::
,..a
s::
0
1-<
btl
lOu +~8" +
2e +~le +
'-c:l
s::
3e +~le +
3e +~2e +
4e +~2e +
Ill
4e+~3e+
'-c:l
56 +~3e +
,..a
Q)
·o.....!;<
Q)
5e +~4e +
6e +~4e +
6e +~5e +
7e +~5e +
7e +~6e +
16 +~2g +
26 +~o" +
2e +~2g +
26 +~4g +
3e +~2g +
3 6 +~4g+
'-c:l
4 6 +~2g+
Ill
4 6 +~4g+
s::
.-e
Q)
.....
.s
46 +~6g+
5e +~4g +
5e+~6g+
6 6 +~4g+
6e+~6g +
66 +~s" +
7 6 +~6g +
7e+~8/
57.50 (1)
41.55 (0.72)
6.40 (0.11)
.........
·········
37.05
4.79
67.92
37.03
1.82
35.34
18.73
(1)
(0.13)
(1.83)
(1.00)
(0.05)
(0.95)
(0.51)
.........
·········
.........
.........
7.67
3.67
1.63
0.34
7.96
6.08
6.14
8.28
47.11
0.19
7.97
·········
·········
.........
..........
.........
46.73
62.14
34.18
26.32
20.07
(1)
(1.33)
(0.73)
(0.56)
(0.43)
81.81
112.46
112.00
93.29
48.37
(1)
(1.37)
(1.37)
(1.14)
(0.59)
77.15
107.09
111.61
107.36
87.07
(1)
(1.39)
(1.45)
(1.39)
(1.13)
39.18
23.62
17.11
38.36
4.83
32.62
2.48
5.61
6.36
6.98
1.84
(1)
(0.60)
(0.44)
(0.98)
(0.12)
(0.83)
(0.06)
(0.14)
(0.16)
(0.18)
(0.05)
96.53
52.74
46.00
66.62
27.94
79.26
18.68
67.42
13.46
31.47
17.76
(1)
(0.55)
(0.48)
(0.69)
(0.29)
(0.82)
(0.19)
(0.70)
(0.14)
(0.33)
(0.18)
82.96
48.55
37.91
68.93
20.85
72.13
13.98
76.99
9.17
74.61
7.35
(1)
(0.59)
(0.46)
(0.83)
(0.25)
(0.87)
(0.17)
(0.93)
(0.11)
(0.90)
(0.09)
26.54
11.97
0.82
12.27
14.69
14.58
7.30
2.86
31.62
17.23
0.41
7.39
0.01
2.06
7.88
4.71
20.72
8.24
1.03
9.06
10.42
12.23
5.84
0.53
10.20
14.51
8.47
2.93
2.75
5.82
7.79
3.79
27.56
11.37
1.73
13.29
12.31
17.23
10.26
0.29
14.87
17.85
13.38
6.96
3.23
15.09
20.42
9.82
Y. Tanaka, K. 0 gawa and A. Arima
94
Table II(b). B(M1) and B(E2) of K=0.020 in the two-proton and two-neutron-hole configurations, in units of (n.m.) 2 and e 2fm 4 respectively.
(1d5;2 2 2pa;2 - 2)
(Of!7tz2 Oh9;2 - 2)
(Ohu;2 2 Oh912 - 2)
(Ohu;2 2 Oha12 - 2)
M1
E2
M1
E2
M1
E2
M1
57.50
3.67
...
...
41.04
72.92
...
21.62
...
...
0.26
22+--?o1+
...
...
21+--?o1+
22+--721+
2.39
41 +--721 +
41 +--722+
...
31 +--721+
0.11
0.21
0.42
31 +--722 +
31+ --741+
I
...
I
I
20.26
1.63
41.55
0.19
0.01
54.32
...
55.35
0.03
...
...
7.96
0.00
0.00
0.00
33.40
58.54
0.00
0.15
118.38
25.92
0.17
39.99
67.92
6.08
...
71.17
98.28
0.01
32.66
E2
I
64.03
28.37
94.20
...
0.66
...
...
o.oo
0.40
0.44
88.55
0.05
46.61
104.20
48.94
configuration; the 8 2 +, not the 81 +, in the (Ohi112, Oh~1 2) configuration; .and the 102 +,
not the 101 +, in the (Ohi112 , Oi{312 ) configuration. On the other hand, such a high
spin state in a configuration of small j 1 and j 2 orbits cannot retain the rotational
nature of the ground band.
2. Reduced transition probabilities also support this conclusion. According
to the pure rotational model, B (E 2: IK ~I' K) can be given by the following
formula:
B(E2: IK~I' K) =~e 2 Q 0 2 (1K20il' Kt
16n
(3·2)
Table III shows B(E2) ratios calculated by using this formula. B(E2) of transitions inside the ground band and the excited band assumed as K = 1 + are given
in units of B(E2: 2/~0/) and B(E2: 2e+~Ie+), respectively.
Table III. Relative ratios of B(E2) predicted by the rotational model.
is neglected for the K=1 +band.
K=1+ band
ground band
B(E2: 2g +--70g +)
B(E2: 4g +--72g +)
B(E2: 6g +--74g +)
B(E2: 8u+--76g+)
B(E2: 10g+--78g+)
The term (KIQ21-K)
1.000
1.429
B(E2: 2e +--71e +)
1.573
1.647
1.692
B (E2: 3e +--72e +)
B(E2: 3e +--71e +)
B (E2: 4e +--72e +)
B (E2: 4e + --73e +)
B(E2: 5e +--73e +)
B(E2: 5e +--74e +)
B(E2: 6e +--74e +)
B(E2: 6e +--75e +)
B(E2: 7e +--75e +)
B(E2: 7e +--76e +)
1.000
0.571
0.476
0.794
0.278
0.909
0.182
0.979
0.128
1.026
0.095
Relative B (E2) ratios inside the ground band and the excited band have fairly
95
Rotational Levels in a Simple Shell Model Configuration
good agreement with the prediction of the rotational model. This is particularly
true for (Ohi112, Oh~1 2) and (Ohi1 12, Oif312) configurations. In these. configurations,
B (E2) of the transitions between these two bands are also small: therefore, we
can say that the ground band and also the excited K = 1 + band appear, at least
in these configurations, provided that the QQ strength is twice or three times as
strong as the reasonable value.
3. The intrinsic quadrupole moments of the levels belonging to the ground
band become constant, as is assumed in the pure rotational model when the QQ
interaction becomes stronger. Such a tendency can also be seen in the relative
ratios of the B (E2) between these levels.
4. At first sight, the level structure of the two-proton and two-neutron-hole
system (case (b)) resembles the phonon levels. The 0 2 + state, however, is missing
and the large values of B(E2: 2 2 +~oi+), B(M1: 2 2 +~21+) and B(E2: 31+~21+)
indicate that this level structure is rather similar to that of asymmetric rotor. 8 )
This is also suggested by the fact that the following relation is nearly satisfied.
0 (21 +) _..._-- Q (22 +) ~0.
Calculated magnetic moments are also listed in Table I(b), where we can see that
the gyro magnetic ratio g = jJ.j J is nearly constant in each configuration. This is
consistent with the prediction of the asymmetric rotor model, though many other
models, including the phonon model and the symmetric rotor model, predict a
constant gyromagnetic ratio.
§ 4.
4.1
Comparison with the A.C.S. and with the S.S.
The aligned coupling scheme (A.C.S.)
This scheme was introduced by Mottelson and is known to be a fairly good
eigenstate of the QO interaction in a single j modeP) For the jn configuration,
two different K = o+ intrinsic states are possible;
/Xo) = u2Z {<11jj)<21 j - j)<31 jj-1)<41 j-j+ 1)· .. }
(4·1)
for the oblate deformation
and
/xp> =
ull{<11 jf)<21 j - !)<31 j!)<41 j - !) .. ·}
(4·2)
for the prolate deformation,
where <n / jm) means a single-particle wave function of the n-th particle with
the spin j and its z-component m.
We have no a priori reason to assume the same kind of deformation for
protons and neutrons. Therefore four different combinations of deformations are
possible: (1) protons and neutrons have an oblate deformation; (2) protons and
neutrons have a prolate deformation; (3) protons have an oblate. deformation,
96
Y. Tanaka, K. 0 gawa and A. Arima
while neutrons have a prolate deformations; and finally ( 4) protons have a prolate
deformation, while neutrons have an oblate deformation. Because the total angular
momentum J is not a good quantum number, we have to project out J from these
intrinsic states. Projection procedure is easily carried out by using the ClebschGordan. coefficients. For instance, the A.C.S. in which both protons and neutrons
have an oblate deformation produces a following wave function:
(4·3)
where N!o is a normalization constant and the suffix "oo" means that both protons
and neutrons have an oblate shape. The normalization constant is
(4·4)
For convenience, we will use following expressions:
(4·5)
We can use this scheme to see which type of deformation gives a better approximation to our exact solution.
Before calculating the overlapping integrals between the exact wave functions
calculated in § 2 and the A.C.S. wave functions, we have to examine the difference
between the projected states from the differe:q.t kinds of A.C.S. functions. Table
IV shows these results for the case of (j1 = 11/2, j 2 = 13/2).
Table IV. Overlapping integrals between· different types of the projected A.C.S. wave functions for the case of (h = 11/2, h= 13/2). IJ0 p), for example, is the abbreviation of
PJIXap>, in which the suffix "op" means that two protons have an oblate shape and
two neutrons have a prolate shape. Other notations are used in similar definitions.
J
o+
2+
4+
6+
g+
10+
0.0230
-0.0048
-0.0074
0.0141
-0.0080
-0.0069
-0.0180
0.0012
0.0088
-0.0089
-0.0035
0.0153
<J
-0.8771
0.5537
-0.4747
0.4429
-0.4341
0.4469
-0.9987
0.5605
-0.4978
0.4307
-0.3144
0.2039
-0.3709
0.2014
-0.1124
0.0449
-0.0161
..
~0.0217
0.3572
-0.1999
0.1222
-0.0559
0.0084
0.0234
Large overlappings of
= OoaiOvv) and <OoviOvo) indicate that pJ=o'Xoo is very
similar to pJ=o'Xvv; and the same thing is particularly true between pJ=o'Xov and
pJ= 0'Xvo·
Table V(a) shows overlapping integrals between exact solutions for each QQ
interaction strength obtained in § 2, and projected wave functions in the A.C.S.
with the oblate-oblate deformation for the (Oh~ 112 , Oi{312 ) configuration. For other
shapes, overlappings are very small except the IJ = Ovv>·
Rotational Levels zn a Simple Shell Model Configuration
97
Table V (a). Overlapping integrals between IJ 00 ) and our wave functions for each QQ interaction strength in the (Oh 11122, Oi13122) configuration.
o1+
21+
22+
41+
42+
61+
62+
81+
82+
101+
102+
j K=O.OOO
K=0.004
0.8649
0.9330
-0.2022
0.8618
-0.0787
0.6340
-0.1494
0.9525
0.9818
0.0540
0.9573
-0.0162
0.8772
-0.0975
0.6805
-0.3640
0.0698
0.3155
0.3576
-0.1435
0.1375
-0.0935
j
K=0.008
K=0.012
K=0.016
0.9772
0.9913
0.0208
0.9799
0.0005
0.9490
-0.0248
0.8827
-0.2841
0.0406
0.5471
0.9866
0.9948
0.0111
0.9881
0.0038
0.9717
-0.0040
0.9420
-0.1854
0.0287
0.7472
0.9911
0.9964
0.0070
0.9919
0.0040
0.9813
0.0007
0.9630
-0.1321
0.0224
0.8499
j
K=0.020
0.9936
0.9973
0.0047
0.9941
0.0035
0.9862
0.0013
0.9727
-0.1018
0.0185
0.8963
Very good overlappings show that the states belonging to the ground band
are very similar to the projected A.C.S. wave functions with the oblate~oblate
deformation. In other words, the A.C.S. wave function is the intrinsic state of
these ground band members. The other two configurations, except the (1df12 ,
2pi12), also exhibit similar results.
These results also provide us with other evidence that states up to 6 2+ in
the (Og~12, Oh~12), states up to 82 + in the (Ohi112, Oh~12) and states up to 102+ can
be included in the ground band.
On the other hand, the intrinsic state of an excited band is assumed to be
orthogonal with
(4·6)
A reason will be given as follows.
in terms of xk-= 1 and X~=~:
The intrinsic state can be generally expressed
where
Xk=l = c.Jl {<II jljl)<2J i1- i1 + 1)<31 i2i2>< 41 i2 -j2)},
X~=l =
c.Jl {<1J jlj1)<2J jl-j1)<3J i2j2)< 4J i2-i2 + 1)}
(4·8)
and the antisymmetrizer operates on the proton part and the neutron part, respectively.
In order to determine the mixing amplitude a and {3, we use an analogy
with the SU3 model. Operating the infinitesimal generator Q1 on the intrinsic
state Xo of K = 0 with the minimum quadrupole moment, we cannot obtain a K = 1
intrinsic state, because the irreducible representation (J., fl.) (which includes the
98
K
Y.
=
Tanaka~
K. Ogawa and A. Arima:
0 state) can have only the following K values: 1>
K
=
0,2,. · ·, min (A, f.J.).
We can further prove that the wavefunction PJQ 1Xo is the same as PJXo· The
operator 3Q1- Lr, which decreases the quadrupole moment, gives a vanishing
result when it operates on Xo, because Xo has the minimum quadrupole moment.
We thus obtain the following relation:
Q1Xo = tL1Xo .
The intrinsic state Xo can be expanded m terms of angular momentum
eigenstates x/; Xo = L;CJx/. Because the operator L 1 can change only the z component of x/, Q1Xo can be expanded in terms of the same XoJ·
This condition will be approximately satisfied in our case. The ratio a/ /3
IS given by
This value is practically one. On the other hand, L 1Xoo can also be expanded in
terms of xk=1 and Xk=t with the coefficients a' and /3'. The ratio a'/ /3' is to be
CJ1-}till J1-J1 + 1) / CJ2-j2lll J2-J2 + 1) = .J j 1/j2 which turns out to be approximately one in our cases. It is thus proved that Q 1Xoo is approximately equal to
L1Xoo· In other words the wavefunction PJ0 1Xoo can be very similar to the PJX00 ,
which is proved in our calculation. Thus the state which is orthogonal to this
01Xoo may be the intrinsic state of our K = 1 + band. This leads us to consider
the following intrinsic state:
~rc
1 +XK=1
2) .
XK=1=
..;z - XK=1
(4·9)
We have found that the 1 1 +, 2 2 +, 3 1+, 4 2 +, 5 1+ and 6 2 + states in the (Ohi112, Oh~;z)
and the 1 1+, 2 2 +, 3 1 +, 4 2 +, 5 1+, 6 2 + and 71 + states in the (Ohi112, Oi12s;z) can be derived from this intrinsic state ( 4 · 9). The calculated overlaps for the latter configuration are shown in Table V(b) for different QQ interaction strengths. In our
Table V(b). Overlapping integrals between PJjxK= 1(a= -1/ v2, /3=1/ v2)) and our wave functions for each QQ interaction strength in the (Oh 11122, Oi13122) configuration.
11+
22+
31+
42+
51+
62+
71+
K=O.OOO
K=0.004
K=0.008
K=0.012
K=0.016
K=0.020
0.6414
0.5126
0.8291
-0.7929
0.8524
-0.6609
0.6926
0.9457
-0.8623
0.9586
-0.9234
0.9520
-0.8062
-0.8487
0.9828
-0.9543
0.9824
-0.9623
-0.9754
-0.8965
-0.9095
0.9913
-0.9772
0.9900
-0.9767
-0.9840
-0.9360
-0.9359
0.9945
-0.9859
0.9933
-0.9836
-0.9880
-0.9541
-0.9495
0.9961
-0.9901
0.9951
-0.9873
-0.9903
-0.9639
-0.9574
Rotational Levels in a Simple Shell Model Configuration
99
calculation, PJXx= 1 has not been orthogonalized to PJXoo, but the orthogonalization
is approximately satisfied in our cases.
It becomes clear that the ground band members can be derived from one
intrinsic state, and the excited band members from another intrinsic state, provided
that large j1 and j2 orbits are taken and the QQ force is strong.
In case (b) (two-particle and two-hole configuration) 0 1 +, 2 1 + and 4 1 + states
may have different deformation of the proton part and the neutron part. Protons
may have the oblate shape and neutrons the prolate shape, or vice versa. Because
of the rather large overlapping between the IJo.v) and the IJ.vo), it is not definite
which combination of deformation these states have. Results for the case (b) of
the (Ohi112, Oii3}2 ) configuration are shown in Table V(c). It is interesting that one
intrinsic state with either the oblate-prolate deformation or the prolate-oblate
deformation can produce approximately our wave functions, particularly when
J=O.
Table V(c). Overlapping integrals between the all types of A.C.S. wave functions and our
wave functions of K = 0.020 in the (Oha122, Oi1312 - 2) configuration.
o1+
21+
22+
41+
4.2
0.1144
0.0056
0.0712
0.0129
0.9761
-0.8521
0.4353
0.8205
-0.9655
-0.8806
-0.3716
-0.8524
-0.4693
0.0233
0.5163
-0.0901
The stretch scheme (S.S.)
This coupling scheme was introduced by Danos and Gillet. 8) A typical configuration of j1 2N (proton) j 22N (neutron) consists of two chains. Each of these
has N protons and N neutrons and is coupled to the maximum angular momentum
C allowed by the Pauli principle. Then these two chains are coupled to a state
with total angular momentum J. These procedures
are illustrated in Fig. 4.
Usually the antisymmetrization between two
chains is omitted in calculating the energies. In
the course of calculation, however, the antisymrnetrized two-body matrix elements between two interacting particles from different chains were used. 3)
Consequently, an error arises from the lack of
antisyrnmetrization of two non-interacting particles
from two different chains.
Fig. 4. A schematic diagram of
The stretch wave function for a configurathe stretch scheme wave function with two protons and two neutrons is given by
tion.
100
Y. Tanaka, K. 0 gawci and A. A rima
(4·10)
and Jmax =jl + j2.
Here the Pauli principle Is omitted between two chains, while the state in
which the antisymmetri zation Is properly taken into account has the following
form:
( 4 ·11)
where the normalizatio n constant is given by
(4 ·12)
These are irrespective of J in so far as j 1 and j 2 are large because of the asymptotic behavior of the 9j symbol. The wave function ( 4 ·11) is the same as
that given by projecting out from the A.C.S. wave function with the oblate-oblate
3
deformation, because these two intrinsic wave functions are the same. )
We calculate the overlapping integrals between the A.C.S. wave functions
of ( 4 ·11) and those of ( 4 ·10). The calculation is easily carried out and the
results are
irrespective of J,
(4·13)
where IJs.s)N.A. means the S.S. wave function with the total angular momentum
J which is not antisymmetri zed. These values show that the overlapping integrals
between our wave functions and IJs.s)N.A. are not good, and that the antisymmetrization between particles in different chains is important.
Rotational Levels in a Simple Shell Model Configuration
101
A use of the antisymmetrized two-body matrix elements seems, however, to
diminish the error. In fact, this procedure works very effectively to decrease
the error. In other words, the energy spectra computed for the wave functions
(4·10) and (4·11) are hardly different from each other, except that the binding
energy is slightly larger in the former case than in the latter case. These facts
have already been noted by Raynal. 9)
§ 5. Conclusions
Because the configurations and the particle number are limited, we cannot
directly compare our results with real experimental data. We can, however,
make following assumptions :
1. the freedom of longitudinal flips (i.e., the particle jump between several
nearly degenerate orbits) can simulate the freedom of transverse flips (i.e., the
particle jump between different magnetic substates in a large j orbit).
2. the QQ interaction can be strong because of a renormalization effect
when a large number of nucleons are replaced by a few nucleons.
Under these assumptions, we can show that protons will be <:oupled with neutrons coherently when both nucleons start to fill up new shells or both are completing some shells. The proton-neutron interaction is not strong enough to make
both nucleons have the same kind of deformation when protons occupy the
beginning part of the shell and neutrons fill up the rest of the shell.
These features persist even when the QO interaction of protons is much
weakened, as compared with that of neutrons and that between protons and
neutrons, i.e.
OQ(P-P) =0.2 x QQ(N-N) =0.2 x QQ(P-N).
(5·1)
On the other hand when the 00 interaction between protons and neutrons 1s
switched off, i.e.
· OO(P-P) =QQ(N-N),
OQ(P-N) =0,
(5·2)
the rotational spectrum does not appear. Overlappings between the calculated
wave functions and the wave functions projected from the A.C.S. are still large
in the ground band as long as the neutron-proton part of the interaction exists.
If the o(P- N) is also switched off, overlappings become very small.
From this evidence, we can conclude that the QQ (P- N) is essential in
producing rotational spectra; the wave functions overlap very well with those
projected from the A.C.S.; and the (P- N) cooperates to make the overlappings
large.
wave functions of the lowest o+ and 2+ states also show interesting behavior.
In case (a), seniority 4 components as well as seniority 2 components are strongly
admixed even in the interaction limit (K = 0.000). This leads to large overlapping integrals between these states and those of the A.C.S. (see Table V(a)). Thi~
o
o
o
Y. Tanaka, K. Ogawa and A. Arima
102
situation is very different from that of four like nucleons, where the pure o interaction makes the seniority pure. In case (b), however, a main amplitude of
the lowest o+ state is concentrated in the seniority 0 component. For the lowest
2+ state, a main amplitude is distributed in the two seniority 2 components, even
when the QQ interaction is strong. In other words, the weak coupling model
of the simple shell model can be applicable to case (b).
The following wave functions for the (Ohi112, Oi1~f2 ) configurations is illustrative:
Case (a):
i)
K=O.OOO
101+)=0.79110 X 0)+0.40612 X 2)+ 0.31214 X 4)+ 0.24716 X 6)
+0.189l8x8)+0.119l10x10)+ ... ,
(5·3)
121+)=0.56310 X 2)+ 0.51712 X 0)-0.27812 X 2)+ 0.30912 X 4)
+ 0.25514 X 2)- 0.15614 X 4) + 0.22914 X 6) + 0.15616 X 4)
(5 ·4)
+ 0.17816 X 8) + 0.13818 X 10) + · ··.
ii)
K=0.020
101+)=0.60910 X 0)+ 0.70412 X 2)+ 0.34114 X 4)+ 0.12016 X 6)+ ···,
(5 ·5)
121+) =0.51610 X 2)+ 0.49612 X 0)-0.42512 X 2) + 0.36012 X 4)
+ 0.32814 X 2)-0.17914 X 4)+ 0.13714 X 6)+ 0.11616 X 4) + ···.
(5 ·6)
Case (b):
i)
K=O.OOO
I01+> =0.97810 X 0) -0.17112 X 2)+ ... ,
(5·7)
121+)=0.72810 X 2)-0.67412 X 0)+ ···.
(5 ·8)
ii) K=0.020
IOl+> = 0.83010 X 0)- 0.52112 X 2) + 0.19014 X 4) + .. ·,
(5 ·9)
121+)=0.65310 X 2)-0.61712 X 0)-0.29812 X 4)+ 0.29014 X 2)+ ···.
(5 ·10)
The small components less than 0.100 are omitted, and
(J;p=n), Oiir}~(Jn=m): J).
In X m) means IOhi1;2
There remains the question of why the noncollective states with high spins
which were omitted from the ground band members should appear so low in energy.
Such states appear first when the total angular momentum J is equal to 2] -1,
Rotational Levels in a Simple Shell Model Configuration
103
where J is the minimum of j 1 and j 2 • These states have relatively simple wave
functions. The 101+ state in the (Ohi112, Oif312) configuration, when K = 0.020, is
as follows:
1101+) = 0.591110 X 0) + 0.723110 X 2) + 0.339110 X 4)+
· ...
(5 ·11)
These states have always large amplitudes of IJx 0) or 10 X J) where J is the
highest spin of two nucleons in the j 1 or j 2 shell.
It must be noticed that, in a system of two identical particles in a single j
orbit, the state with highest spin I j 2J=2j-1) has a lower energy than the
I PJ = 2j- 3) under the strong QQ interaction.
This suggests that the appearance of such noncollective states in low energy
region may be caused by the OQ part of the effective interaction between identical
particles. This kind of low-lying high-spin states were already predicted by
Glendenning and Harada. 10 )
Acknowledgements
The authors would like to express their appreciation for the many useful
comments of Drs. M. Harvey, V. Gillet and J. Ginocchio; and also colleagues
in the theoretical nuclear structure group of the University of Tokyo. They are
also grateful to the Institute for Nuclear Study for financial support and to the
Computer Center of University of Tokyo for computational assistance. Thanks
are also due to Miss M. Ichimura for typing this manuscript.
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