A Short Note on Even-Parity States of
Oxygen-16
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Nagai Hiroyuki, Itaya Mitsuo
九州工業大学研究報告. 自然科学
17
47-54
1970-03-30
http://hdl.handle.net/10228/3093
Bull. Kyushu Inst. Tech.
(M. & N. S.) No. 17, 1970
A SHORT NOTE ON EVEN-PARITY STATES OF OXYGEN-16
Hiroyuki NAGAi
Department of Physics, Kyushu Institute of Technology, Tobata, Kitakyushu
and
Mitsuo ITAyA
Department of Physics, Kyushu Industrial University, Kashii, Fukuoka
(Received Dec, 22, 1969)
In double closed shell nuclei, such as Oxygen-16 and Calcium-40 all nuclear
states except for the ground state are arizen from core excitation. Most striking facts in both nuclei are that even-parity excited states, particularly the
first exeited O" states are very low, and that the monopole transition from this
state to the ground state is very large.
Several attempts to explain these properties have been performed by
many authors. These attempts on the individual particle models are roughly
divided into two classes. One is based on spherical scheme')'-iO) (conventional
shell model), and the other is based on deformed seheme, including SU3Modeli')-'`) and Hartree-Fock-modeli5)-26). The latter has been stimulated by
the experimental evidence25) that certain of these excited states form rotational
bands. In the early works the Hilbert space was truncated to only particlehole configurations of 2fico (one particle-one hole (lp-lh) and 2pT2h) excitation.
This corresponds to the lowest order of the Tamm-Dancoff approximation.
Next, the admixture of the closed shell configuration (or mixing of the 2hto
particle-hole configurations in the ground state) was taken into account. If it
is included, then to be consistent the 4ntu configurations (4p-4h states) should
be included as well. Brown and Green,i8) and Gerace and/Green'9) described
the lowest three O'(T=O) states in '60 and `OCa, respectively, as linear combinations of a spherical Op-Oh state and deformed 2p-2h and 4p-4h states.
The 6.06 Mev level turns out to be dominantly the 4p-4h state rather than the
2p-2h state and the first level of a rotational band of which the E2 transition
probabilities are in fair agreement with the experimental results. However,
since we have had no information about the values of the unpertubed configuration energies in these nuclei with core excitation, Brown and Greeni8)
have considered them as free parameters adjusted to give the observed spectra.
Although we have not any information about values of the single particle
and hole energies in double-.closed-shell nuclei with core excitation, as in the
usual shell-model calculation these values have customarily been taken from
48 H. NAGAi and M. ITAyA
the spectra of the neighboring (closed shellÅ}one nucleon) nuclei. The values
of the single particle and hole energies in '60 are taken from the spectra of
'5N and i7F (proton levels), and of i50 and '70 (neutron levels). The various
authors have estimated the particle-hole excitation energies (hto in the harmonic-
oscillator shell-model) to be about 13--17 MeV from these values. Therefore,
the unperturbed (zero-th Order) excitation energies of the even-parity states
are to be about 30-v50 MeV. Even if these high excitation energies could be
redueed to the very low experimental values by large configuration interaction, then could configurations maintain their physical significance ?
The spectra of the extra single particle and hole result from the motion in
the average central field given rise to by the inert closed-shell core. If so, in
the case of the excited states of double-closed-shell nuclei with the eore exeitation which represents a break-up of this core, the values of the single particle
and hole energies may be considerably different from the above-mentioned esti-
mations, and rather appreciably reduced. From investigations of many results
given by several authors we might infer that we should attribute the large
reduction of the excitation energies not to the configuration interaction (or
off-diagonal elements of the energy matrix), but to the unperturbed configuration energies (or diagonal matrix elements).
In i60, 10 .ooi -v30 .oo! of the core nucleons are promoted from the states in
the spherical core to the higher states without the core. Therefore, it is
natural to consider that the nucleons in the excited i60 nucleus move in the
different, average self-consistent central field which gives rise to the more
loosely bound orbital motion than in the ground state.
For simplicity, we adopt the harmonic-oscillator shell model in this short
note. Thus, we assume that different configurations with the different number
of excitation quanta correspond to the average harmonic oscillator potential
with the different oscillator constant, v=(mto)/h. We have considered only the
particle-hole configurations of 2hto excitation.
The spherical ground state have the double-closed-shell configuration (ls)`
(lp)i2 and the oscillator constant y which has been determined by Carlson and
Talmi28) from the Coulomb energy difference between the i50 and i5N ground
states and this value of v is O.349Å~1026cm-2(Z==O.571) which gives htu=14.5
MeV. Next, it comes into question how the oscillator constant, y'=(mco')/h, of
the 2hal configurations is determined. In this note we have determined the
value of v' according to the following assumption : "FollJwing the way of thinking on the kinetic theory of gases, the nucleons begin to move more rapidly in
the excited state than in the ground-state nucleus. Beeause the mean kinetic
. the nuclear temperature rises.
energy of the nucleon in the nucleus increases,
Consequently, if the nuclear volume remains invariant, the pressure rises, and
AShort Note on Even-Parity States of Oxygen-16 49
then the nuclear spherical wall is more strongly acted on by the inner nucleons.
The excited nucleus goes on to expand adiabatically until the temperature, or
the mean kinetic energy of the nucleons, is equal to the value in the ground
state nucleus."
4•
(-g- ha)+12e( : ha)
==4•
( : htu')+12•( 52 nco')+2htoi, . ........ (1)
Consequently,
i 18 r 18
(D =u rt" tu, v = lg Y)
ha!==13.7 MeV,
yi..O.331Å~1026cm-2. ....................................... (IX)
(Z! -= O.557)
We employ standard shell model techniques to calculate the energies of
the excited Oi states. Terminology used here is just the same as in the previous works3)•`). Here we give only a brief description of states and residual
interactions adopted by us.
The eonfigurations of 2htu excitation considered here are as follows: (a)
(ls)-'(2s), (b) (lp)`i(2p), (c) (lp)'2(ld)2, (a) (lp)-2(2s)2, (e) (lp)-2(ld)(2s).
All calculations are carried out in the LS coupling scheme and only the states
with T=S=L=O are taken into account. The states belonging to the ground
and lp-lh configurations are unique, respectively, while the O' states of the
2p-2h configurations (c), (a) and (e) can be classified by the charge-spin multiplicity and the resultant orbital angular momentum of the 2h configuration
(lp)-2. Consequently, the ground state and twelve 'iS states belonging to the
2ha-excited configurations can be written as follows :
Åë (as)4(lp) i2) =- Åë ((c. s.)) -- gy',,
Åë ((ls)mi(2s)) i -- O ((a)), Åë ((lp)-'(2p)) =-TÅë ((b)),
di({(lp)-2 i3,3iS, (ld)2 i3•3iS}iiS)...mÅë((.)i3,3iS),
Åë({(lp)-2 i3•3iD, (ld)2 '3•3iD}i'S) :.ii.i. di ((c)'3•3iD),
Åë({(lp)m2 ii•33P, (ld)2 ii•33P}iiS)iiir..Åë((c)i'•33P),
50 H. NAGAi and M. ITAyA
Åë({(lp)-2 i3,3is, (2s)2 i3,3is}iis)=.Åë((a)i3,3is),
Åë({(lp)-2 i3•3iD, (ld)(2s)i3•3iD}iiS)=.di((e)i3•3'D). ......... (2)
The effective two-body interaction is chosen as the Yukawa potential with
the Rosenfeld mixture, used by Elliott and Flowers29). Namely
V(1, 2) = -g---- V, (T,•T2) {O.3 + O.7 (6i •62)} eXP (- ri 2/a)/(ri 2/a), - - • T - (3)
with a=1.37Å~10Lmi3cm. The interaetion strength V. is fixed at 40 MeV. This
value was used in the previous works3)•4).
Due to the assumption (1), the energy differences between the states result
from only the effective two-body interactions (3) among all of the 16 nucleons.
16state, Åq(c.s.) 1Z J7(i,i)l(c.s.)År, is calThis expeetation value for the ground
iÅrj' --1
culated with the value of the oscillator constant, v =O.349Å~10L'6cm-2, and the
16 elements of Z V(i, 7•) are calculated by
result is -204.13 MeV. Those matrix
iÅr1' =1
taking as a basis states (2) of the 2hto-excited configurations, whose numerical
values are evaluated with the excited value of the oscillator constant, i.e. yt=
O.331Å~1026cm'2. Then, the diagonalization of this 12Å~12 interaction energy
matrix is carried out on the OKITAC-5090 computer to give eigenvalues and
eigenvectors for excited O' states. The results thus obtained are listed in
Table 1.
Our shell-model wave functions for the 2ha-excited configurations include
the spurious states30) in which the center-of-mass motions are excited to 2s and
lp states from the ground state3)•4). However, such spurious states are completely separated from the admissible states by a diagonalization of the matrix
16 interaction energy Z V(i, i) deof the twe-body operator, such as the mutual
iÅri-;:1
pending on only relative coordinates among the nucleons.
The lowest eigenvalue, listed in Table 1., is completely equal to the value
which is obtained for the ground state, i.e. (c.s.) with the excited value v/.
Therefore, the state which corresponds to this lowest value, is probable to be
the spurious state with the 2s center-of-mass motion. In faet, we estimate the
overlap of the eigenfunction belonging to the lowest eigenvalue with the 2sspurious-state wave function obtained by us in reference 3) to be almost perfect, i.e. O.992. Therefore the actual lowest eigenvalue is -192.90 MeV and
then the first excited O' state is at 11.23 MeV above the ground state. This
value is about twice the experimental one, but since this quantity in our model
is the difference between the large quantities, excitation energies ma: appreciably depend on a choice of the effective two-body interaction.
AShort Note on Even-Parity States of Oxygen-16 51
Table 1. Eigenvalues and eigenvectors of the matrix of the eflETective two-body interactions
.Z16
V(i, i) for the state (2) of the 2hto-excited configurations.
pÅrz=1
(v'=O.331 Å~ 1026cmF2, 2' == O.557)
-177. 5517
- 194. 5252
O. Ol61
O. 2041
-O. OO15
-O. OO03
O. 3354
d) ((c)i3s)
O. 5323
di ((c)3iS)
- 190. 6646
- 161. 8030
- 179. 0625
O. 0207
O. O085
O. 8170
O. 3215
-O. Ol33
-O. OO08
-O. 0848
O. 1917
O. 6332
-O. 1414
-O. 0618
O. O052
O. 6698
-O. 1969
-O. 5354
-O. 0572
-O. 0821
-O. 0239
di ((c)i3D)
-O. 2405
O. 2788
-O. 1209
O. 6167
-O. 21 18
O. 0319
di ((c)3`D)
-O. 4194
-O. 2937
O. 0495
-O. 5813
-O. 1603
-O. 0261
a) ((c)iiP)
-O. 0217
-O. 2170
O. 0898
O. 2013
O. 8919
-O. 0314
di ((c)33P)
O. 0542
-O. 6109
0, 3113
O. 4656
-O. 3342
-O. 0891
di ((d)itgS)
O. 1080
-O. 0732
O. 2038
O, OO05
O. 0200
-O. 0447
di.((a)3iS)
-O. 1807
-O. 0242
O. Ol83
O. 0278
-O. 0787
O. O024
O. 0813
-O. 2933
O. 0824
-O. O090
- O. 0660
O. 4746
- l87. 7479
-173. 2211
- 167. 3653
- 192. 8983
-O. 0935
O. !857
-O. Ol11
O. 7613
- O. 0556
O. 1240
-O. 2360
Åë ((a))
d) ((b))
O. 1358
O. 0770
di ((e)13D)
-O. 0217
O. 2944
di ((e)3iD)
-O. O091
-O. 2969
,
- 182. 2404
--
1
LXxx eig. en-value
---- -........ -170. 7538
- 187. 2329
x. X--sx
basis
1
Åë ((a))
- O. I729-O.
0398
O. 0298
O. 4631
-O. O028
-O. 1465
di ((c)ltgs)
O. 3361
-O. 0994
-O. 241I
di ((c)3iS)
O. 3096
O. 1028
O. 2225
-O. 0635
O. I060
O. 2105
Åë ((c)i3D)
O. 6013
-O. 1155
O. 0999
O. 1l74
O. 0845
-O. 1336
op ((c)3iD)
O. 5548
O. 1213
-O. 0666
O, 1192
O. 07488
O. 1561
di ((c)iiP)
O. 2399
O. 0746
-O. 0705
O. 0463
O. I556
O. 1141
cZ) ((c)3np)
-O. 0721
O. 2005
-O. 2311
-O. Ol13
-O. 0353
O. 3008
di.((a)i,as)
-O, 0245
O. 4078
O. 5000
O. 6230
-O. 2475
O. 2700
di ((a)3is)
-O. 0221
-O. 3525
-O. 4939
O. 6812
-O. 2323
O. 2336
di.((e)i3D)
-O. 1786
O. 4443
-O, 221O
O. 2346
O. 6884
O. 0832
op ((e)3iD)
-O. 1589
-O. 4264
O. 2674
O. 2317
O. 5807
-O. 0767
di ((b))
O. 41 89 i O. O134,
The transition rate for pair emission from the first exeited O" state to the
ground state has been measured by Devons et al3i) and the observed monopole
52 H. NAGAi and M. ITAyA
matrix element is
1 ÅqTG 116
E r?- (1 - Ti.)/2 1 gV'i,tÅr l =3.s ÅrNÅq o-26cm2.
i-1
gFG is the ground state wave function and Ti,t is the eigenfunction which has
been obtained by us and corresponds to the eigenvalue -192.90 Mev. Here, we
must call our attension to the fact that the value of the oscillator constant in
the ground state (the closed-shell configuration) is different from that in the
2hc,)-excited configuration in our model.
Therefore, in order to calculate the monopole matrix element between the
ground and excited states, we must beforehand expand the single-particle
harmonic-oscillator functions, say in the ground-state wave-function in terms
of oscillator functions with the different oscillator constant, i.e. the excited
value v', according to the same way as the method given by Bailey32). But,
since the value of v is almost equal to that of v' in our case, the radial harmonic oscillator wave functions R.ti(r; v) are expanded at the point v==v' in
terms of the Taylor's series, vLThere ni=n-1 for the harmonic orbital state
(nl). We take into account the terms of these series up to the second order of
(--1). Namely,
vV,
1 (v-y')+-2-R6',(rs 7, i) (v-yi)2+•••
Roi(r; y)=Roi(r; y')+Rtoi(r; y/)
=Roi (r ; v!) +(-il,,- -1)(212t-III-)if2R,,(r ; v')
+ -21-(-,"7 ' 1)2[( (21 + 3SE21+ 5) l'i2R,,(r ; v')
- (--2+l2:tT. T3+ )ii2R ,, (r ; v') -( 212Fl-;-1-III-) R oi (r ; y/) ]/ + ' ' '• • • • • • • (4)
The results obtained above are used to calculate the matrix element of the
monopole operator"r \O"r2 up to the second order with respect to(-IY,='-1). The
calculated value is 2.6Å~10-"26cm2 which is in fair agreement with the experimental value.
Since pointing out the existence of rotational bands in the energy spectra
of light nuclei (such as lp-shell and begining of ld, 2s-shell), by taking as a
intrinsic wave function a superposition of single-nucleon Nilsson wave functions33), fiIIing the deformed-field orbits in order, several authors3`)•35) have
shown that vv'ave functions obtained through the projection technique agree
AShort Note on Even-Parity States of Oxygen-16 53
remarkably well in the case of these light nuclei with the wave functions
obtained from shell-model calculations36)'29) with effective interaetions, in the
sense t•hat overlap integrals of both model wave functions are of the order of
90.o/..
Then, we presume that :
"For those states of the 2ha-excited configuration which eorrespond to the
other total angular momenta we could construct the vector-eoupled wave func-
tions in the same way as the wave functions (2) for the O" states have been
constructed. For each total angular momentum the energy matrix of the
effective two-body interactions would be constructed with these wave functions
and be diagonalized to obtain eigenvalues and eigenfunctions. A set of lowest
states for the respective even total-angular momenta would form the rotational
band. If we should carry out the similar calculation for the 4htu-excited configuration, we would have the other rotational band."
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