619
Progress of Theoretical Physics, Vol. 27, No. 4, April 1962
The Excited Q+ State and Center-of-Mass Effects in Oxygen-16
Hiroyuki NAGAI
Department of Physics, Kyushu Institute of Technology, Tobata,
(Received September 4, 1961)
The first excited state of Ql6 is investigated by the use of the harmonic-oscillator shell
model with a nuclear interaction. A central potenti.al with a Yukawa shape and Rosenfeld
mixture is assumed as the residual interaction. The first excited state is assumed to be
some admixture. of possible configurations with an excitation energy of 21!w above the shell
model ground state which has a wave function corresponding to the double closed configuration (1s) 4 -(1p)l2, In regard to this excited state, there are two pieces· of data available~ the
excitation energy 6.06 Mev and the monopole transition probability to the ground state_ The
calculated matrix element for this transition (4.7XlQ-26cm~) is in good agreement with the
observed value (3.8X1Q-26cm2), while the calculated excitation energy is twice as ,large as
the experimental excitation energy. After considering the serious effects of the center-ofmass motion and removing the spurious states, the excitation energy is increased by 15% and
the monopole matrix element has increased to twice the experimental. value. It turns out
that the lowest exCited o+ state obtained by the usual shell model calculation corresponds
to the spurious state with the 2s center-of-mass motion.
§ I. Introduction
The first excited o+ state of 0 16 has been investigated by many authors/>
one has yet succeeded in explaining its properties, using either an inno
but
This state has two individual-particle shell-model or collective models_
teresting properties from a theoretical point of view. One of them is the low
value of 6.06 Mev for the excitation energy .and the other is the monopole transition to the ground state. As far as the excitation energy is concerned, all
calculations using an interaction consistent with low energy two-body data have
given values which are too large. On the other hand, the simple individualparticle model gives too small a value for the monopole transition probability,
while the collective models and the shell-model wave functions resembling them
have given values which are too large up to three to· five times as large as the
observed value.
Elliotti· •> has pointed out that if this o+ state is composed of the following
configurations,
= (c. s.)(1s) - 1 (2s),
(a)
(1s) 3 (1p) 12 (2s)
(b)
(1s) 4 (1p) 11 (2p),
(c)
(1s) 4 (1p) 10 (1d) 2 ,
(1)
H. Nagai
620-
(d)
(ls) 4 (lp) 10 (2s) 2 ,
(e)
(ls) 4 (lp) 10 (ld)(2s),
and if this state includes the configurations (a) and (b) to the extent of 50%,
the monopole matrix element theoretically predicted is in agreement with the
experimental value. Griffin1 ·g> has also supported Elliott's suggestion. This
mixing of several configurations may make it possible to effectively reduce the
energy.
The odd parity levels of 0 16 are well explained by Elliott and Flowers/>
who started from the jj-coupling shell model and introduced a residual interaction. This residual interaction causes many configurations to mix. They assumed a central potential with a Yukawa shape and Rosenfeld exchange character
as the residual interaction. This potential has been used also with 0 18 and P 9
by the same authors and they have obtained reasonable agreements with
experiment. s>
In this paper the harmonic oscillator shell model is · assumed. The twobody interaction is added to this simple model. The shape and exchange character
of this interaction is assumed to be just the same as in references 2) and 3) .
The spin-orbit interaction is neglected. On these assumptions, the configuration
mixing is calculated among the twelve states which belong to the configurations
of (1). For simplicity, only the states with T=S=L=O are taken into consideration where T, S, and L are the total isotopic spin, total spin, and total
orbital ~ngular momentum. However, these shell-model wave-functions include
the spurious states in which the center-of-mass motions are excited to lp and
2s states from the ground ls state. Here the ld state of the center-of-mass
motion is excluded from our wave functions, since this state must have L=2.
According to Elliott and Flowers, it is very important to remove these spurious
states to avoid any error caused by the center-of-mass motion in the lowest oddparity states of 0 16 •2>· 4 >· 5>· 6> It is expected that these spurious states must be
removed also in the even-parity states. In § 3, the wave functions in which the
center-of-mass motion is in the ls. state are constructed to calculate the configuration mixing. The excitation energy is evaluated in § 4 and the wave function obtained there is used to estimate the monopole transition matrix element
in § 5. The discussion is given in the last section.
§ 2. Model Hamiltonian and interaetion
The total Hamiltonian of an A-nucleon system is
H= _ _!!_ L:: Vl+
2m
i=I
L:: V(i,j)
'>'
(2)
where V,; 2 is the Laplacian operating on the coordinates of the i-th nucleon, m
is the mass of the nucleon, and V (i, j) is a potential which may depend upon
The Excited
o+
State and Center-of-Mass Effects in Oxygen-16
621
the spm and isotopic spin coordinates as well as space coordinates of nucleons
i and j. In order to solve
(3)
we assume that
lJf IS
a linear combination of q trial functions,
(4)
How should the set of trial functions be selected ? The qualitative success
of the harmonic oscillator shell model in light nuclei suggests as trial functions
the solutions lJfa of the Schrodinger equation
(5)
for sixteen nucleons, each of which moves in the same central harmonic oscillator field. Then,
(6)
where
U(ri) = m;,l r/=
2~
(hvri) 2 •
Since Eq. (5) is a sum of single particle equations, each of which expresses
a three-dimensional harmonic oscillator, lJfa is an anti-symmetrized linear combination of products of the harmonic oscillator wave functions ¢nlm ,
</Jnlm= Rnl(r) ylm(19, cp),
r
where
Rnl (r)
= Nnl exp ( -vr 2/2)
r 1 +1 v,. 1 (r 2) ,
Y 1m is a normalized spherical harmonic, Nn 1 a normalization factor, and vn 1 (r2)
an associated Laguerre polynomiaU 7J The parameter v which determines the
range of the harmonic oscillator wave functions is chosen so as to give a rootmean-square radius of 0 16 in the ground state. Assuming the radii of the 0 16,
0 15 , and N 15 to be equal, this v has been determined by Carlson and TalmFl
from the Coulomb energy difference between the 0 15 and N 15 ground states. The
experimental value of v is 0.349 X 1026 cm- 2 which gives hw = 14.5 Mev and
r 0 = 1.3 X 10-13cm.
Moreover, these trial functions lJfa are limited to those wave functions which
belong to the configurations (1). All the Wa are degenerate with the excitation
energy 2hw above the ground configuration, i.e. the double closed shells (ls) 4
(1p) 12 • A variational treatment with a trial function (4) is formally equivalent
to a perturbation theory for degenerate states. The unperturbed Hamiltonian IS
Eq. (6) and the perturbation H' is given by the equation
622
H. Nagai
H'= :L; V(i,j)- :L; U(r;),
i>J
(7)
i
where V (i, j) is taken as follows,
Here r 12 = [r1 -r 2 [, Vc=40 Mev, and a= 1.37 x I0-13cm. This charge-symmetric
interaction has the exchange character suggested by Rosenfeld. It has been
successfully used by Elliott and Flowers in the problems of the odd-parity states
of 0 16 2J and the low-lying states in 0 18 and P 9 •3J
Since V (i, j) is invariant under rotations in spin and isotopic spin spaces
as well as in ordinary space, IJl" can be characterized by the quantum numbers
T, S, and L. In the present paper we are interested in the lower excited o+
states of 0 16, so that T, S, and L may be assumed to be all equal t~ zero. As
will be shown in the next section, there exist twelve 118 states belonging to the
configurations (I). The wave function lJ! composed of such states is not a possible solution of the total Hamiltonian (2), since none of the lf!a are eigenfunctions of the center-of-mass motion. However, we can take a proper linear
combination of these wave functions corresponding to the Is center-of-mass
motion, so as to avoid an error induced by the excitation of higher ·center-ofmass motion.
§ 3 · I.
Shell-model wave funclions
Only the central force is taken into consideration and the spin-orbit interaction is neglected, so that the total isotopic spin T, total spin S, and total
orbital angular momentum L are good quantum numbers. In this calculation,
all of them are assumed to be zero.
The state belonging to a one-hole configuration,js unique, for example, (Is) 3
(T=I/2, S=I/2, L=O). The configurations (Ip)- 2 and (Id) 2 have several states
which are easily classified by T, S, and L. 8J, 9J We denote these states as
((Ip- 2 2 T+ 1 • 28 + 1L) and 8(Id 2 2 T+ 1' 28 +1L). The vector-coupled product is expressed
as [(, {}]rsL.
Antisymmetric 11 8 wave functions belonging to the configurations (I) can
be easily written down as follows :
(j) ( {
(Is 3, 2s) ([ 4] 11 8), (Ip12)
=
(j) (
j
{(I s4 )
=
j
3 ~~~!
([
~
([
444] 118)} [ 4444] 118)
( -I)P P[[~(ls3 ), ¢(2s)]oo0 , u(Ip12 )]ooo,
(9·a)
4] 11S), (Ip11 , 2p) ([ 444] 118)} [ 4444] 118)
4!11!"'(
p
[
4
·u
.<:...J -I) P lu(Is), ['1ClP ), ¢(2p)Jooo]ooo,
I6!
p
(9·b)
The Excited
o+
State and Center-of-Mass Effects in Oxygen-16
623
IP({(1s4 )([4]uS), (1p10)(TcScLc), (1d 2)(TcScLc)} 11S)
cc
.1
10'! 2' "'
4
=!IV 4' 16
7:' (-1) p P[(v(1s),
( 1p10), 82oC1 d2) J ooo Jooo,
(9 ·c)
IP({(1s4) ([4] 11 S, (1p10) (TdSdLd), (2s2) (TdSdLd)} 11 S)
I
=~
4! 10!! 2! "'
4
2 Jo00] 000 ,
7:' (-1) p P[(v(1s),
[( C1p10), 8o2(2s)
16
(9-d)
and
tP ( { (1 s4)
([
4] 11S), (1p10) (T. S. L.), (1d, 2s) (T.S. L.)} uS)
= / 4 ! ~~: 2 !
~
( -1)P P[(v(1s4 ) , [((1p10), 8u(1d, 2s)]ooo]ooo,
(9·e)
where the permutation P runs over all those permutations of sixteen nucleons
which preserve the natural ordering within the wave functions of the pure configurations w, (, r;, 8, etc., separately.
The
states of the configurations (1, c), (1, d), and (1, e) can be classified by the charge-spin multiplicity (e.g. 2Tc + 1, 2Sc + 1) and the resultant
orbital angular momentum of the (1p) 10 configuration. Explicitly they are written
as follows:
us
( 1 , c)
tss, ats, taD, stD, up, aap,
(1, d)
tas, ats,
(1, e)
taD, atD.
Thus twelve 11S states with the excitation energy 2ftw above the ground configuration are obtained.
Except for the two uS states, (9 · a) and (9 ·b) , these uS states do not belong
to any irreducible representation of the symmetric group S,(u=16). However,
since such a strong exchange interaction as the Rosenfeld type favours the most
symmetric wave functions in the orbital space, we shall take those
states
which are of the most symmetric type [ 4444] . According to Elliott, Hope and
Jahn,I 0l they can be easily wi'itten down as follows :
us
¢ 1= {(1s3, 2s)[4]S, (1p12)[444]}[4444] 11S
=IP({(1s3, 2s) ([4] 11S), (1p12) ([444] 11S)}[4444] 11S),
¢ 2 = {(1s4) , (1p11 , 2p)[444]S}[4444]uS
= tP( { (1 s4 )
([
4] 11S), (1p11 , 2p) ([444] 11S)} [ 4444]uS),
¢ 3= {(1s4) , (1p 10)[442]S, (1d 2)[2]S}[4444]uS
=
2-1;2[IP ( { (1 s4 )
([
4] 11 S), (1p10) ([ 442] 13S), (1d 2) ([2] 13S)} uS)
- IP( { (1 i) ([4] us)' (1p10)([442] 31S)' (1d 2) ([2] 31S)} us) J'
(101
H. Nagai
624
¢ 4 = {(1s4 ) , (lp10)[442]D, (ld 2)[2]D}[4444] 11S
= 2- 112 [tP( { (1 s4 )
([
4] 11S), (1p10)
([
442] 13D), (1d 2) ([2] 13D)} 11S)
-tP({(1s 4 ) ([4] 11S), (1p10) ([442] 31 D), (1d 2) ([2] 31D)} 11S)],
¢ 5 = {(1s4) , (1p10)[433]P, (1d 2)[ll]P}[4444] 11S
= (10)-1i 2 [tP({(1s4 ) ([4] 11S), (1p10) ([433] 11 P), (1d 2 ) ([11] 11 P)} 11S)
+ 3@( { (1 s4 )
([
4] 11S)' (1p10)
([
433] 33 P)' (1d 2) ([11] 33P)} 11S) J'
t./J6 = {(1s4 ) , (1p10)[442]S, (2s 2)[2]S}[4444] 11S
= 2- 112 [tP ( { (1 s4)
([
4] 11S)' (1p10)
([
442] 13S)' (2s2) ([2] 13S)} 11S)
-tP({(ls4 ) ([4] 11S), (1p 10) ([442] 318), (2s2) ([2] 31S)} 11S)],
¢ 7 = {(1i), (1p10)[442]D, (1d, 2;S)[2]D}[4444] 11S
=2- 112 [@( {(1s4 ) ([4] 11 8), (1p10) ([442] 13D), (1d, 2s) ([2] 13D)} 11S)
- tP( { (1 s4 )
§ 3 · 2.
([
4] 11S), (1p10 ) [ 442] 31 D), (1d, 2s) ([2] 31 D)} 11S)].
Center-of-mass motion and admissible wave functions
In the shell model, it is assumed that the central field U (ri), where individual nucleons move independently, is fixed at an artificial origin, so that the.
center-of-mass of a nucleus fluctuates around this point. Difficulties arise from
the appearance of spurious states in which only the center-of-mass motion is
excited. As pointed out by Elliott and Skyrme,6l the states considered in the
previous section are expected to include more spurious states than the states
belonging to the ground configuration and the odd parity configurations with the
excitation energy hw. Therefore, the center-of-mass motion has a worse effect
on the present case than on the later cases. 2l
The central field is assumed in this paper to be an oscillator potential which
is believed to be a good approximation for a lighter nucleus such as 0 16 • This
particular choice of the averaged potential guarantees that the states of the ground
configuration which have at most one unfilled shell can include only the 1s centerof-mass motion. However, any state belonging to excited configurations which
contain two or more unfilled shells includes an excited center-of-mass motion in
the oscillator potential. In order to avoid serious errors caused by the centerof-mass motion, it is necessary to take appropriate wave functions in which the
center-of-mass motion is in its ground state, i.e. 1s state. These wave functions
are linear combinations of several functions which are degenerate in the harmonic oscillator shell model. These linear combinations can be obtained by diagonalizing the energy matrix of the center-of-mass motion in the complete set
of the shell model wave functions ·of a given symmetry. The proper wave
functions for our purpose are eigenstates whose energies are (3/2) hw, corre-
The Excited
o+
625
State and Center-of-Mass Effects in Oxygen-16
sponding to the Is state of the center-of-mass motion.
valent to diagonalize the operator
This procedure is equi-
.
Here a single-body term L;r? which corresponds to a sum of potential energies
of individual 'nucleons is non-vanishing only in the diagonal elements which are
easily derived from the total sum of energies of nucleons moving in the oscillator
potential. The matrix of a simple two-body term I:; r i · r 1 can be also straight•>!
forwardly evaluated by means of fractional parentage coefficients,11 > Racah coefficients,11>'12> and Wigner's 9j- symbols. 13>
Thus, the matrix elements of the second term of R 2 between states in
(10) can be written as linear combinations of two-nucleon matrix elements
<nalanblbTSLirl ·r2lnclcn/dTSL). The properties of this element have been discussed by Unna and Talmi. 14> The 7 X 7 matrix of R 2 between the complete set
of [ 4444] 11S states of (10) is easily obtained and diagonalized. It is found
that there are five admissible states and two spurious states. The two nondegenerate spurious states of the center-of-mass motion can be immediately
determined as follows :
and
Wr=[V3cf1 + 1/16¢2+ 21/5¢s+ 1/35¢4-51/3¢5-2¢a+ 2 VI0¢7]/8 v3,
(12)
where lffa corresponds to the lp state of the center-of-mass motion and lJf7 corresponds to the 2s state. Five admissible states which are orthogonal to the
spurious states (12) are constructed from seven linear independent states
cfr. ¢2, cfa, ¢4, ¢5, lffa and Wr by Schmidt's method. As a result, they are
, /l =[53 1/3¢1-3 VI6¢2+ 2 v5 ¢s+ 1/35~~4-5 1/3¢5+22¢ -10 VI0¢7]/8 VI59,
6
/l.=[62 1/3¢2-6~vs-3 v7 ¢ 4+3 V15¢5+ 81/5¢6 -231/2¢ 7]/2 1/3286,
il.= [l9¢a-7 v7 ¢4+7 V15¢ -2 1/5¢e-2 1/2¢7]/1/7347,
5
~=[109 1/3¢4+21 1/35¢5-2 V105¢6 -2V 42¢7]/1/51666.
and
(13)
The percentage of the spurious states in the wave functions of the symmetry
type [ 4444] 11 S can be immediately found in (12). The results are given in
Table I.
H. Nagai
626
Table I.
The percentage of spurious states in the wave functions of [4444]
w-
--------------S-M
F I
spurious state~
ns states.
</>a
'
1p center-of-mass motion
15.6
3.1
4.2
7:3
33.3
1.6
7.8
10.4
18.2
15.6
39.1
20.8
2s center-of-mass motion
2.1
20.8
17.2
10.9
14.6
25.5
54.7
22.9
54.2
total percentage
Table I shows that the wave functions of the configurations arising from the twonucleon excitation include more spurious states than those coming from the
one-nucleon excitation, except for the wave functions where the two holes in the
lp-shell leads to the states of [ 442]S.
The spurious state with the 2s center-of-mass motion is calculated by operating R 2 on the ground state (Is\ lp12) 11S while the admissible [4444] 11 P state3J,al
of h(l) above the ground state is multiplied by R ii1 order to get the spurious
state of the lp center-of-mass motion.
§ 4.
E~ergy calculation
The energy matrix can be calculated by means of the fractional parentage
coefficients in a similar fashion as for the matrix elements of L;ri·r1 • However,
•<!
we follow the procedure taken by Condon and Shortley,I"l who considered the
configurations with almost closed shells.
According to them, the diagonal elements of V = L;V (i, j) can be separated into a sum of six terms :
i<t
(i) terms within double closed shells (c.s.); (ii) terms between an outer
nucleon and (c.s.) ; (iii) terms between a missingnucleon and (c.s.) ; (iv) terms
between outer nucleons; (v) terms be.tween missing nucleons; (vi) terms between outer nucleons and missing nucleons.
Now we are interest~d only in the energy differences between the ground
state (c.s.) and the excited o+ states, so that we need not calculate the interaction (i). The interactions (ii) and (iii) are constant within the same configuration. Thus, for example, in the (lp) - 2 eas) (ld) 2 C3S) 11S state, we have
(lp- 2 CaS) ld 2 CaS) 11 SIHI1p- 2 C3S) ld 2 CaS) 11S)- E (c. s.)
=2{e(Id) -e(Ip)} +2{CU(lp) -CU(ld)}
+ 2 {CV(ld; c. s.) - CV(Ip; c. s.)} + V(Ip- 2 CaS) ld 2 CaS)
11
S),
(14)
where E(c.s.) = ( (c.s.) IHI (c.s.)) , and e(nl), CU(nl) are the total and potential
energies of the harmonic oscillator in the nl state, respectively. Energy CV(nl; c,s.)
is the interaction between one nl nucleon and (c.s.), and V(lp- 2 (1aS) ld 2 CaS) 11S)
is a sum of (iv), (v) and (vi) in the hole-particle state (lp) - 2 C3S) (I d) 2 (1 3S\ 11S.
The Excited
o+ State and Center-of-Mass Effects in Oxygen-16
627
All the interaction energies except type (vi) are given in TAS and are:expressed
in terms of Slater integrals with an obvious modification caused by the exchange
character. The interaction energies of type (vi) for the configurations (1, a) and
(1, b) are easily given in tenns of Slater integrals and have been obtained for
the general case by Hope and Longdon/ 6J but those for the other configurations
require considerably more manipulations. The wave functions which are constructed by a vector coupling of the two-hole state and the two-particle state
can be transformed to the wave functions which are the vector coupling of twopairs of missing and outer nucleons. After this transformation, the matrix
elements of type (vi) can be easily calculated. The results are given in Table
II. In order to be sure of these results, the diagonal elements obtained above
have been compared with the results calculated by the use of fractional parentage coefficients.
Table II. Contributions to diagonal elements from terms. (vi) between missing and
outer nucleons of the central interactions (8).
State
State
12
--FD(1s 2s)
'
'
5
(1, a)
(1, c) SID
1~
{ -3FO (1p, 1d) + ;O F2 (1p, 1d)
7 c~
+ 10
2
-·
12 {F0(1p,
2p) } '
2p) +--sF2(1p,
---s
(1, b)
(1, c) asp
1~
-
c1p, 1d)
9
+~sc1p,
1d) } •
{-17FD(1p,1d)
~~ F2(1p 1d) +fGt(1p, 1d)
+ 365 GS (1p, 1d) } '
+~Gs (1p, 1d)} ,
(1, c) 18 8
_!_ {2G1 (1p, 1d)
75
(1, c) lSD
7~
{7Gl (1p,
(1, c) 11P
~:
{+G1(1p, 1d)
7
1~) + ~: Gs (1p, 1d)} ,
(1, d) 138
8
-G1(1p 2s)
'
'
15
(1, d)
1
2s) } ,
54 { -FD(1p, 2s) +-gG1(1p,
SIS
(1, e) tan
+tGs (1p, 1d)},
(1, c)
318
1~
{-3FD(1p, 1d) +
-~
Gl (1p, 1d)
+ ;5 GB (1p, ld) } '
(1, e) SID
a2
2
4
{5G1
(1p, 2s) } ,
(1p, 1d) +5Gt
+{
-FD(lp, 1d) -+F2(1p, 1d)
2
-FD(1p, 2s) +-gG1(1p,'1d)
++Gl (lp, 2s)} .
It is easily seen that only the mutual interactions :EV(i,j) in H have noni>f
vanishing contributions to the off-diagonal elements of the energy matrix. The
matrix elements of the mutual interactions between twelve states in (9) can be
628
H. Nagai
Table III.
Off-diagonal elements of the central interactions (8).
States
States
~i: ~?'ars
(1, a),
(1, b)
-+J 1~
{R1(1p2p, 1d1d)
+ : 4 R3(1p2p, 1d1d)} ,
~i: ~?'arn
-+J Jo
{R1(1p2p, 1d1d)
+ : 9 R3(1p2p, 1d1d)} ,
(1, b),
(1, c) aap
1 {R1
d
- viO
(1p2p, 1d1 )
-+
R3(1p2p, 1d1d)} ,
(1,b),
(1, d) ras
1 R
vlf 1(1p2p, 2s2s),
3
R
(1, a)'
(1, e) arn
3 Rl
d
)
(1s1 , 1p1p ,
5 vz
(1,b),
(1, d) ars
- 5v 6
~i: ~?'ras
J 125 {R1(1p2p, 1d1d)
(1, b),
(1, e) ran
- v13 .{Rl(lp2p, 1d2s)
+ : 4 RB(1p2p, 1d1d)},
a: ~?ian
J 3~
+R1(1p2p, 2s1d)} ,
(1, b),
(1, e) arn
{R 1 (1p2p, 1d1d)
5:
3
{R1(1p2p, 1d2s)
+ : 9 RB(1p2p, 1d1d)},
~i; ~? irp
- {0
+R1(1p2p, 2s1d)},
J ; {Rl (1p 2p, 1d1d)
-+R3(1p2p, 1d1d)},
1 (1p2p, 2s2s),
~i: ~? ~:~·
+
(1, e) ran,
(1, e) 31D
5 1 {7FD(1p, 2s) -
{F 0 (1p, 2s) -+G1 (1p. 2s)},
31
G1(1p, 2s) .}
7 {FD(1p, 1d) +5F2(1p,
1
+5
1d)
ca (1p
(1, c) ras,
(1, c) 13D
_16 v7
245
(1, c) ras,
(1, c) np
- 4 v5 3 { 115
'
1d)
1
+l5G1(1p, 1d)
'
!
cr (1p, 1d)
+ 2 5 G3(1p, 1d)} ,
- ; 5 GB(1p, 1d)},
(1, c) ras,
(1, c) ars
5
2 {
4
7FD(1p, 1d) +l5G1(1p, 1d)
+ 315
(1, c)
(1, c)
ras,
arn
- v5
(1, c) ars,
(1, d) ars
- 5 v5
1 G ( d
2 1 , 2s),
ca (1p, 1d)} ,
4 v'5 7 {_1_5 F2 (1p, 1d)
+ 419
(1, c) 138,
(1, d) ras
ca (1p, 1d)} .
3
G2(1d )
. '2s '
The Excited
o+ State and Center-of-Mass Effects in Oxygen-16
(Continued)
Table III.
States
(1, c) ISS,
(1, c) ssp
629
I, States I-------------------------
I
(1, c) ISS,
(1, e) ISD
_I6v3 {__!__GICip Id)
15
'
. 15
- 315 GS(lp, ld) }·'
(1, c) ISD,
(1, c) 11P
-v;r {1~
(1, c) 13S,
GI (lp, !d)
(1, e) SID
{ 7
2
5 v' 5 -r;R2(1p1d, 1p2s)
+ 2~5 GS (lp, !d) } '
(1, c) ISD,
(1, c) SIS
(1, c) ISD,
(1, c) SID
4 ~7f+F2(lp, !d)
(1, c) !3D,
(1, e) ISD
-J 57 RI(1p1d, 2slp).
-- 49
180
:9 GS(!p, !d)}'
+
+
++RI(1p1d, 2s1p)},
(1, c) ISD,
(1, e) SID
{14FO(!p, !d) -+F2(lp, !d)
-
+J; {
-+R2(1p1d,
11
+lSGI(Ip, ld)
_v2I {-~F2(Ip
35
15
Id)
'
+ 1~
+ ::5
(1, c) np,
(1, c) SIS
- 8 ~a
(1, c) np,
(1, e) ISD
-+JT
(1, c) np,
(1, e) SID
~ I 3 {__!__R2(lpld, Ip2s)
GS (lp, !d)} ,
{- 1~ Gl(lp, !d)
5 Ill 5
-~ { 335 F2(lp, ld)
(1, c) SIS,
(1, e) ISD
_ __!:§__ GS (lp !d) }
245
-
2~
'
5 ~ 5 { -+R2(1pld, lp2s)
++RI(1pld, 2slp)},
'
(1, c) SIS,
(1, e) SID
{56 FO (Ip, !d)
5 ~5 {+R2(1p1d, 1p2s).
+ 2: F2(1p, !d) ++G1 (1p, Id)
+
(1, c) SIS,
(1, c) SID
-+RI(1p1d, 2s1p)},
~: GS(lp, ld)},
(1, c) SID,
(1, e) ISD
- 8 ~ 7 { 315 F2(1p, ld)
+
(1, c) SIS,
(1, c) ssp
5
}
.
2
+gRI(lpld, 2slp) ,
2
-l5GI(Ip, !d)
(1, c) np,
(1, c) ssp
RI(1p!d, 2s1p),
GI (Ip, Id)
+ 115 GS (lp, !d) } '
(1, c) 11P,
(1, c) SID
lp~s)
++RI(1p1d, 2s1p)},
_ _E_GS(lp !d).}
'
'
245.
(1, c) ISD,
(1, c) ssp
f
-
++ / 1~ R2(1d2s, Idld),
- 4 Y3
15
___!_ I 7 {-_!_R2(1p1d, 1p2s)
5 Ill 5 . 5
4~ GS(Ip, !d)}'
(1, c) SID,
(1, e) SID
GI (Ip !d)
{-___!_
'
15
+ 315 GS (lp, !d)}
++RI(1p1d. 2s1p) }.
'
fJ ;
{+R2(1p1d, 1p2s)
-+RI(1p!d, 2s1p)}
+fJ
is
R2(1d2s, ldld),
630
H. Nagai
Table III.
States
(1, c) 31D,
(1, c) 33p
(Continued).
I
State
- /2-1 { 6
-Ts5 F 2(lp, ld)
1
-15
Gl (lp, !d)
!
- 2 5 G3 (lp, !d)} ,
(1, c) 33p·,
(1, e) 13D
-+J ! {
(l, d) 318'
(l,e)31D
1; R2(!pld, lp2s)
_i_
·{_!_ Rl (lp2s, !dip)
-
53
4
+JRl(lpld,
2slp) } , '
(1, c) 33P,
(1, e) 31D
-+R2(lp2s, lpld)},
(1. d) 138,
(1, e) 31D
_l_ I 3 {_li_R2(1pld, lp2s)
5 II/ 5
5
52
{
1
- 3 R1 (1p2s, ldlp)
+fR2(1p2s, lpld)},
+ + RI (lp ld, 2s lp)} ,
(1, d) 138, 8
(l, e) 13D 15R1 (Ip2s, ldlp),
written as linear combinations of two-nucleon matrix elements of the form
(nalan 6 l 6 TSL[V[nclcndld TSL). Finally, these elements are expressed in terms
of the Slater integrals F1c. The results are given in Table III.
The Slater integrals in Tables II and III are as follows :
"'"'
Fk(nala-, nblb)
=II
JJ R,.
2
1 (rl)
aa
R,.
2
1 (r2)f'"(rl,
bb
r2)
dr1dr2,
,
0 0
G"(nala nb lb) = .\' .\' R,.a1Jr1) Rn6 !b Cr2) R,.6t 6 (~1) Rna!a (r2) f'"(rl, r2) dr1 dr2,
0 0
and
"'"'
R"(nalanblb; nclcndld)
= .\' .\' R,.a!a(r1) Rnc!c(r1) Rnb!b(r2) R,.dt/r2)f"(r1,
r2)
dr1dr2.
0 0
Here f" is defined by
Vc exp (- r12/ a)/ Cr12/ a) =
"'L,f" Cr1, r2) Pk (cos lV),
/c
where P" (cos lV) is a Legendre polynomial of degree k for the angle (V between
the vectors r 1 and r 2 • For the convenience of calculation, the Slater integrals
are expressed in terms of the Talmi integrals / 1 .17) Some of the Slater integrals
which could not be found in reference 17) are given in the Appendix. Explicit expressions of the / 1 for the Yukawa potential have been calculated by
TalmP 7> and Longdon. 18>
It is interesting to investigate how far the excited o+ states are from the
The Excited
o+ State and Center-of-Mass Effects in Oxygen-16
631
ground o+ state. Instead 'of diagonalizing the 12 X 12 energy matrix the submatrix in each configuration is diagonalized and the eigenvectors for the lower
characteristic values of each configuration are calculated. The· results are given
in Table IV, where the reduction of energy denotes a sum of all terms except
for the first term in (14) and the n:umerical values in parentheses denote the
results of a somewhat ad hoc assumptiol). which is a semi-empirical "shell-model
assumption " that the harmonic oscillator central field is a very good approximation to the averaged central nuclear field. This assumption leads to the
simplification in which the second term LICU cancels the third term LIC(l in Eq.
(14). But actually this assumption does not hold in our interaction ~V1. 1 as is
i>J
shown in Table IV. The lower seven wave functions obtained in Table IV are
very similar to those of the symmetry [ 4444] in (10). This similarity willlbe
discussed in § 6.
Table IV. The excitation energies of excited o+ states on tl:e single configuration model
(in Mev).
Configuration
I Reduction
of .energy
I
Excitation energy
(1s) -1 (2s)
-
0.94 ( -11.06)
28.06(17.94)
-14.50
24.62
(1p)-1 (2p)
-11.06 ( -10.89)
17.94 (18.11)
-14.50
14.32
{ -13:38 (- 21.21)
-10.45(-18,28)
- 2.15(- 9.98)
r5.62( 7.79)
18.55 (10. 72)
26.85(19.02)
-14.50
22.33
6.43 ( -15.82)
22.57(13.18)
-14.50
23.89
- .6.03 ( -14.65)
22.97 (14.35)
-14.50
23.11
(lp)-2 (1d)2
(1p)-2 (2s)2
(1p)-2 (1d) (2s)
-
In order to understand the effects of configuration mixing, the 7 X 7 energy
matrix is constructed between the above seven states. The results are given in
the third rows of Tables V and VI. A comparison between these results and
the results in Table IV shows that the resonance energy amounts to about 1 Mev.
However, since the usual shell model wave functions used here include the
spurious states, these results 'might contain errors due to the center-of-mass
motion. To avoid these errors, only the admissible set of wave functions in (13)
are taken into consideration in calculating the energy. For the sake of comparison, these results are listed in Tables V and VI along with the results
obtained by Ferrell and Visscher. 1 ·~> Percentages of spurious states with 1p
and 2s center-of-mass motions [(P"w[~)[ 2 and [(P"w!Wr)l 2 are given in column 4
and 5, respectively. In contrast with the results obtained by Unna and Talmi14>
for nuclei of configurations in which a 1p112 nucleon is raised into the next 2s1;2
shell, the lowest shell model wave function contains a large amount of spurious
states. It may fairly be said that this wave function corresponds to the spurious
state with the 2s center-of-mass motion. The center-of-mass motion has a serious
effect not only on the calculated value of the excitation energy, but also on the
632
H. Nagai
matrix element for the electric monopole transition which will be calculated in
the next section. These tables show that the lowest eigenstate in the usual
shell-model calculation consists of the configurations with two-nucleon excitation.
These configuration have more spurious states than the one-nucleon excitation
configuration. The percentage of the configurations with two-nucleon excitation
is considerably reduced by removing the spurious states.
All numerical values in parentheses in Tables V, VI and the Table VII in
§ 5 correspond to the semi-empirical "shell-model assumption" which was intro- ·
duced in the beginning of this section.
Table V.
Reference
Admissible W-F
'
Usual S-M W-F
Ferrell & Visscher1,d)
Table VI.
Calculated excitation energy of the first excited
o+ state.
ExCitation energy
of the 1st excited
o+ state (Mev)
Percentage of
one-nucleon
excited
configuration
Percentage of spurious states
16.51 (12.87)
88.4 (12.1)
36.0( 3.8)
-
-
14.31 ( 7.00)
4.9(2.4)
81.7(73.9)
9.1
100.0
0.0
4.0
!p center,-of-mass 12s center_-of-mass
motwn
motwn
Amplitudes of the wave functions of tf.e ns states of (9) in the wave function
for the- first excited· o+ state given in Table. V.
(lp)-2 (ld)2
(Is) -1 (2s)
(lp)-1 (2p)
1ss
I
s1s
I
13D
I
s1n
Admissible W-F
0.18
(0.25)
0.92
(0.25)
0.16
(0.54)
-0.16
( -0.54)
0.08
(0.06)
-0.08
( -0.06)
Usual S-M W-F
0.44
(0.18)
0.41
(0.07)
0.29
(0.46)
-0.27
( -0.43)
0.23
(0.33)
-0.24
( -0.36)
(lp) -2 (!d) 2
np
I
(Ip)-2 (2s)2
33p
1as
I
(Ip) -2 (Id) (2s)
s1s
1sn
I
31D
Admissible W-F
-0.02
(0.07)
-0.06
(0.20)
-0.08
(0.20)
0.08
( -0.20)
0.15
(0.10)
-0.15
( -0.10)
Usual S-M W-F
-0.11
( -0.12)
-0.36
( -0.40)
-0.07
(0.02)
0.07
(-0.02)
0.32
(0.26)
-0.35
( -0.29)
§5.
The electric monopole transition
The transition rate for pair emission from the first excited o+ state to the
ground state has been measured by Devons et al_1 9> and the observed matrix
element is
11roton
P'ol ~
( 1
1'
1'p 2
11Jl'lst) = 3.8 X 10-26 cm2 ,
•
(15)
The Excited
o+ State and Center-of-Mass Effects in Oxygen-16
633
where the error is less than IO%. In (IS), lJf0 is the ground-state wave function
and lff1st is the wave function of the first excited o+ state.
Schif£1 • •l has considered two monopole transitions in 0 16 and C 12 on the
basis of collective and individual particle models. For 0 16 , he has shown that
both the liquid drop and the alpha-particle models give values which are about
five times as large as the experimental value. On the other hand, his calculation
for CI 2 based on the jj-coupling shell-model gives only one sixth of the observed
value. It is reasonable to believe that a similar calculation for 0 16 would also
give too small a value. Redmond1·bl has recently·~ calculated these matrix elements
for C 12 and 0 16 on the assumption that the 'first excited o+ states in both nuclei
are composed of the shell-model configuration in which one of the Is nucleons
in the " alpha " core is replaced by a 2s nucleon. The results are in excellent
agreement with the experimental data. However, he has given no reason why
the observed o+ states consist of the configuration with an excitation of a Is
nucleon rather than a lp nucleon or some mixture of (ls)- 1 (2s) and (Ip)- 1 (2p)
configurations. Elliote· cJ also has pointed out that if one assumes the excited
o+ state in 016 to be composed of configurations (I), i.e. (Is)- 1 (2s), (Ip)- 1 (2p),
(Ip)- 2 (Id) 2 , (Ip)- 2 (2s) 2 and (Ip)- 2 (Id) (2s), and if this configuration mixing
contains about 50 percent of configurations (Is) - 1 (2s) and (Ip) - 1 (2p), the matrix
element for the o+ ~o+ monopole transition will be in agreement with the experimental value. On the other hand, Ferrell and Visscher1'dJ have interpreted
the excited o+ state as a collective compressional-dilational mode, the wave
function of which is given to be some mi:::;:ture of the configurations of onenucleon excitation, (Is)- 1 (2s) and (Ip)- 1 (2p). They have obtained an excitation
energy of 9.I Mev, while the monopole matrix element is about twice as large
as the experimental value. As for the alpha-particle model, using the constants
determined by level-fitting and assumin~ that the o+ state at 6.06 Mev is 'a pure
breathing mode, Kamenyl,e) has obtained a value of I4.7 X 10- 26 cm2 for this matrix
element. Perring and Skyrme1 ' fJ also have obtained a value about three times
as large. Recently, Griffin1·gl has treated dilational and collective quadrupole
excitation of 0 16 by the method of generator coordinates. 20 l He concluded that
the first excited o+ state is not primarily a dilatation excitation and concurred
with Elliott's suggestion1 • cJ that some mixture of configurations (I) is required
to explain the experiment.
The wave functions obtained in the previous section are used to calculate
this matrix element for the o+ ~o+ monopole transition. The results are listed in Table VII together with various theoretical values1 l which were explained above.
No corrections due to the center-of-mass motion need to be made, provided
that only admissible wave functions are used to calculate any physical quantity
which is a function only of the relative coordinates and nuclear multipole
moments. However, in the above calculation the operator is not correct. The
H. Nagai
634
reason is beca~se our operator isc a function of the coordinates relative to the
fixed external origin as is usual in the shell model. The corrected ~onopole
operator is given ·by the equation
!2'= ~r~2 c= ~ (rp-R)2.
p
p
For the nuclei with N = Z = A/2 su_ch as 0 16 ,
Q' = ~ rp + A2 R2 -2(~ rp) ·R= ~ rp
2
p
'Table VII.
p
p
2-
A
2
R2 -.2(~ r/) ·R.
c
p.
Calculated values of the matrix element for the 0+-7o+ transition in 0 16 .
References
Model of o+ state
Schiffl• a)
Schif£1 • a)
Redmond!, b)
Elliott!; c)
0;00
0.6
Experimental value· 3.8§
Experimental value§
Pure two-nucleon excited configurations.
1st-order perturbation: mixture of twonucleon excited configurations by twonucleon forces.
(1s) -1 (2s) state.
-50% (1s)-1 (2s), (1p)-1(2p)
-50% (1p)-2 (1d)2, (1p)-2 (2s)2,
(1d) (2s).
Ferrell and Visscher!, dJ
9
Kamenyl,e)
14.7
Perring and Skyrmel, f)
11
Schiff1• a)
Schiff!, a)
Griffin!, gl
19
present paper
17
17-22
7.8(2.8)*l
(1p)-2
Linear combination of (1s).-1 (2s) and
(1p) -I(2p) corresponding to dilatational
mode.
a-particle model.
S-M W-F obtained from alpha-particle
dilatational mode state.
a-particle model.
Liquid drop model.
Generator coordinates.
Mixing of configurations (1).
4. 7 (1.2) **l
§ The wave function has been fitted to the experimental value of 3.8 X 10-26 cm 2 •
*l Calculated values with the admissible wave functions.
**l Calculated values with the usual shell-model wave functions.
By .means of the selection rule for the matrix element of R, the last term on the
right side vanishes between two states both of which belong to the same Is state
of the center-of-mass motion. The second term also vanishes· Qwing to the
orthogonal relation between initial and final " internal " wave functions. Therefore, in this case, the uncorrected operator may be used provided that· only the
admissible wave functions are used to ca.rry out the calculations.
§ 6.
Concluding remarks
In § 3, only seven 11S states of symmetry [ 4444] are used to remove the
spurious states. That is, we have diagonalize4 the R 2-matrix using the wave
The Excited
o+ State and Center-of-Mass Effects in Oxygen-16
635
functions (10) instead of those in · (9). This restriction is reasonable, since
the Rosenfeld force favours the states with the full symmetry [ 4444]. This is
easily confirmed by comparing the low-lying eigenstates in Table IV with the
wave functions in (10). For example, the (1p) - 2 (1d) 2 configuration has three
low·lying 11S states, the wave functions of which are written as follows:
1Jf = xf/J C3S)
+ yf/J (31S) + zf/J C3D)+ uf/J CS1D) + vf/J ( 11P) + wf/J (38P),
x=0.60, y= -0.56, z=0.35, u= -0.37, v= -0.07, w= -0.24 for the lowest,
x=0.27, y= -0.24, z= -0.12, u=0.12, v=0.30, w=0.87
for the second,
x=0.31, y= -0.26, z= -0.58, u=0.59, v= -0.13, w= -0.36 for the third,
where f/J(1 3S) is the abbreviation for f/J({(1s 4 )([4] 11S), (1p10)([442] 13S),
{1d2) (1 3S)} 11S). It should be noted that these wave functions are essentially
the same as those in (10), because
x~-y,
z~-u,
and
~w
V--
3
These conditions are satisfied by the wave functions (10).
The present calculations might conclude that for light nuclei, such as 0 16 ,
the center-of-mass motion in the shell-model calculation is more important for
the configurations of 2hcv-excitation, particularly those of two-nucleon excitation,
than for the ground and singly excited configurations.
Contrary to our expectation, we have improved only a little on the numerical
results for the monopole transition matrix element and the excitation energy of
the first excited o+ state. This is because of the smallness of the matrix
elements for configuration mixing and the center-of-mass effect. Since the energy
matrix elements are linear in the strength parameter Vc, a variation: V. gives
rise to a change of the excitation energy, but leaves the eigenfunction invariant,
hence the value of the monopole matrix. element. In order to fit our results
to the experimental value of 6.06 Mev for the excitation energy of the first excited state, we must take Vc=74 and 56 Mev, respectively, for calculations on
the pure harmonic ·oscillator shell-model and on the semi-empirical "shellmodel assumption".
On the other hand, a variation of the nuclear radius changes both excitation
energy and eigenfunction. A rough estimation with ro = 1.25 X 10-13cm and the
original Vc gives results . in which the excitation energy increases by about 3
Mev and the monopole matrix element decreases by about 10%. On the contrary, the energy decreases by about 5 Mev and the matrix element increases considerably when r 0 is increased to 1.4 X I0-13 cm. Therefore, as long as we take
the harmonic-oscillator individual-particle model, we could not find a set of values
of parameters Vc and r 0 for a reasonable range of values consistent with that
required for the deuteron and nuclei of mass 18 and 193> which accounts for the
636
H. Nagai
two pieces of data available, the excitation energy of the first excited o+ state
and the transition probability to the ground state. However, if the. semi-empirica:i assumption mentioned in § 4 is taken, such a set of values can be found.
In the usual shell model, one uses only one parameter which is always
equal to the ground state width parameter v. Recently, Sheline and Wildermuth21>
have proposed a change of the width parameter, i.e. they proposed a deviation
from the "oscillator assumption" (anharmonicity) in order to explain the low
excitation energies of light nuclei in their harmonic-oscillator cluster modeP 2>
Such a prescription might be efficient and appropriate for the excited o+ states
composed of 2h(v-excited configurations.
Unna and Talmi14> have reported that the energy of 6.90 Mev can be obtained for the (1p1 12) - 4(2s1i 2) 4 configuration of quadruple excitation. But this configuration assignment gives a vanishing matrix element for the monopole transition.
It is expected that the monopole matrix element is too small even if lower configurations are mixed by two-nucleon forces.
The author would like to thank Professor M. Kobayasi and Professor H.
Yukawa for their kind interest and encouragement. He is also very grateful to
Drs. A. Arima, T. Terazawa and H. Narumi for their kind help and advice.
Appendix
In the following expressions, the Slater integrals necessary in these calculations are given in terms of It.
F 0 (1p, 2p) = (1/96) (23I0 +24I1+42I2 -56I3 +63I4),
F 2 (1p, 2p) = (5/96) (23I0 -60I1 + 114I2-140Ia+63I4),
G 0 (1p, 2p) = (1/96) (23 Io- 32 I1 +58 I 2 -112 Ia + 63 I4),
G 2 (1p, 2p) = (5/96) (23I0 -116Id226I2-196I3 +63I4),
F 0 (1s, 2p) = (1/16) {7 (10 + Ia) + CI1 + I2)},
G 1(1s, 2p)
= (21/16) {(lo-Is) -3(I1-I2)},
. R 1(1p2s, 1d1p) = (5 V2 /16 V5) { -I0 +I1-7(I2 -I8 ) } ,
R2 (1p2s, 1p 1d) = (5 Y10/48) (- I 0 -5I1+ 13I2-7 Is),
R 1 (2s1p, 2p1s) = (Y15/16) (3I0 -13I1+ 17 I 2-7 Is),
R 0 (2s1p, 1s2p)
= (YS/16V3) (3Io+I1-11I2+7 Ia),
R 2(1d2s, 1d1d)
=
R 1(1p2p, 1d1d)
= (1/16 Y10) (7 I 0 -24I1 + 10I2 -56Ia+ 63I4 ) ,
(Y10/96) ( -21Io+ 1411 -28I2 +98Is-63I4),
R 3 (1p2p, 1d1d) = (49V2/96V5) (I0 -12I1+30I2-28Is+9I4),
R 1(1p2p, 2s2s)
= (YS/24) (11I0 -41I1+65I2-35I3) ,
The Excited
o+ State and Center-of-Mass Effects in Oxygen-16
637
R1 (1p2p; 1d2s) = (1/32) (11/0 -2611 -7612+154/3 -63/4) ,
R1 (1p2p, 2s1d) = (1/32) (11/0 -1811 -2812 +98/3 -63/4) ,
R 1 (1s2s, 1p1p) = ('t/3/4 V2) (10 -611 +6/2),
R 1 (1s1d, lp1p)
= (1/15/4) (10 -12).
References
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
16)
17)
18)
19)
20)
21)
22)
a) L. I. Schiff, Phys. Rev. 98 (195S), 1281.
b) R. J. Redmond, Phys. Rev. 101 (1956), 751.
c) J. P. Elliott, Phys. Rev. 101 (1956), 1212.
d) R. A. Ferrell and W. M. Visscher, Phys. Rev. 102 (1956), 450.
e) S. Kameny, Phys. Rev. 103 (1956), 358.
f) ]. K. Perring and T. H. R. Skyrme, Proc. Phys. Soc. (London) A 69 (1956), 600.
g) J. J. Griffin, Phys. Rev. 108 (1957), 328.
h) H. Nagai and Y. Yamaji, Bull. Kyushu Inst. Tech. 3 (1957). 15.
i) H. Nagai, Bull. Kyushu Inst. Tech. 4 (1958), 15.
J. P. Elliott and B. H. Flowers; Proc. Roy. Soc. (London) A 242 (1957), 57.
J. P. Elliott and B. H. Flowers, Proc. Roy. Soc. (London) A 229 (1955), 536.
J. P. Elliott and T. H. R. Skyrme, Proc. Roy. Soc. (London) A 232 (1955), 561.
J. P. Elliott and T. H. R. Skyrme, Nuovo Cimento 4 (1956), 164.
S. Gartenhaus and C. Schwarz, Phys. Rev. 108 (1957), 482.
B. C. Carlson and I. Talmi, Phys; Rev. 96 (1954), 436.
H. A. Jahn and H. van Wieringen, Proc. Roy. Soc. (London) A 209 (1951), 502.
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