774
Progress of Theoretical Physics, Vol. 59, No. 3, March 1978
Alpha- and 8Be-Spec troscopic Factors of 20Ne and 24Mg
Kiyoshi KA TO and Hiroharu BANDa*
Departmen t of Physics, rlokkaido University, SapjJoro 060
*Division of Jl.1athemati cal Physics, Fukui University, Ful::ui 9.10
(Received October 26, 1977)
The eigenvalue problem of the norm kernel for the a+a+ 12 C system is solved with a
method of which the point is the pre-diagonaliza tion procedure on the basis of the generating function technique. By using the results, a- and 'Be-spectrosco pic factors are calculated
and discussed for various parent ( 20 Ne) and daughter ( 16 0, 12 C, 'Be) states. Results for
24 Mg are
also presented and discus;;ecl.
§ I.
Introductio n
Recent experimenta l and theoretical studies on light nuclei seem to be revealmg ,-arieties of multi-cluste r structures in the excited states. 11 New progress has
been made in the microscopic cluster theory. A ren:tarkable recognition is that
microscopic cluster wave functions, in particular, of multicluster s can express not
only typical cluster structures but also most of the important shell-model configurations.21~61
Therefore one can treat the coexistence and interplay of the shell
and cluster structures in a unified way. 31 ·' 1• 61
The microscopic cluster theory needs the knowledge of the norm kernel associated vvith the multi-cluste r system. Non-redund ant eigenstates and eigen\·alues of
the norm kernel are necessary to proYide the basis functions for the microscopic
cluster model calculation and also to eyaluate the spectroscopi c amplitudes for the
reley;:mt various cluster channels. Theoretical studies on spectroscopi c amplitudes
are very important to identify the structure of the noted state by using experimenta l
spectroscopi c information. Since the multi-cluste r problem is so complicated, it
is still in a stage of beginning and requires more studies.
In a preyious paper 41 we proposed a method for sohing the eigenvalue problem
of multi-cluste r norm kernel, in which the "pre-diagona lization procedure" is used
on the basis of the generating function technique. 21 ' n. 81 This method has been
given a mathematica l transparency by Horiuchi in his elegant complex generatorcoordinate theory. 9 )
Although we applied our method to the 3a and 2a- 16 0
systems, the method can be naturally generalized for unclosed-she ll cluster systems.
In this paper we solve the eigen-value problem for the [ 12 C+a+a] norm kernel
of 20 Ne as an illustration. By using the results of this and the previous paper
we calculate the alpha- and 8 Be-spectrosc opic factors of 20 Ne and 2 'Mg. The spectroscopic factors of light nuclei have also been investigated by using shell-model
Alpha- and 8 Be-Spectroscopic Factors of
20
Ne and
24
i'vfg
775
techniques 101 and recently by a general construction procedure of the allowed
states.w
The [ 1'C +a+ a J cluster configuration is particularly interesting, because it
will be possible to describe all bands starting below 8.5 MeV excitation which
have individually different characters; the K" = 01 +(ground) band with a dominant
()., p) = (8, 0) component, the K" = 02 + (6.72 MeV) band with a dominant (A, p)
= ( 4, 2) component, the K" = 0 3 ;- (7.20 MeV) band which has been assigned as
having "8p-4h" configuration, the K" = o,+ (8.5 MeV) band which is the so-called
16 0 +a higher nodal state, the K" = 2- ( 4.97 MeV) band assigned as
(A, 11) = (8, 2)
and the K" = o- (5.80 MeV) band which is the inversion doublet partner of the
ground band. The theoretical spectroscopic factors for the a+ 16 0* and 8 Be* + 12 C*
channels will be useful for the analysis of transfer and pick-up reaction data.
In § 2, the method for solving the eigenvalue problem of the norm kernel
is explained for the case of the [ 12 C +a+ a] system. And the non-redundant
eigenstates and eigenvalues are calculated and listed. In § 3, a- and 8Be-spectroscopic factors are calculated and listed for 20 Ne and 24 Mg followed by brief discussion m comparison -with experimental transfer and pick-up reaction data.
§ 2.
Eigenvalue problem of norm kernel
- - 12
C +a+ a - -
The method proposed and applied to the 3a and 16 0 +a+ a systems]) is genAs an illustration, the [ 12 C +a+ a Jerali~ed to unclosed-shell cluster systems.
norm kernel of ' 0Ne is treated explicitly here.
Matrix elements of the norm kernel X with respect to harmonic oscillator
wave functions in Cartesian coordinates are defined by
A (k, n, N; k', n', N')
where ell' Is the antisymmetri~er and (j)k represents the internal wave function of
the clusters;
(2. 2)
We take the (Os) -closed h.o.w.f. for ¢(a) and the (}., p) = (0, 4) w.f. for ¢k C'C)
with a common size parameter: Therefore the latter is expressed as
(2· 3)
where the operator ai annihilates a h.o. quantum in the i-th Cartesian coordinate.
776
K. Kato and H. Banda
R
(a)
(b)
Fig.l.
The functions u(n; r~r) and u(N: r 2 R) represent h.o.w.f. for appropriate relative
coordinates r and R with h.o. quanta n and N, respectively. If we adopt the
coordinates in Fig. 1, the reduced h.o. constants are
r1 = v'2, r2= v'24/5
for (a),
r1=v'7;, r2=v'16/5
for (b).
(2·4)
In Eq. (2 ·1), conservation of h.o. quanta leads to the relation
n+N-k=n' +N' -k'.
(2·5)
In order to evaluate A (k, n, N; k', n', N'), we make use of the h.o. generating
function technique and the "pre-diagonalization procedure" for the small-norm kernel X, exactly parallel to the previous paper. 4> The final expression in the present
case is written as
. ' '
' - 1. --,-,--_,,,,(2)(N+N')/2jk fjV!
A(k,n,N,k,n,N)_4J 41_
2 !/n.N.n.N. ri
X
4
I:; :E
I:;
d(6 1 ; i/,
m/, M/)
I:;
I:;
I:;
I:;
ITA(6 1 )d(6 1 ;ihmhM1 )
6t•··-4, i1......,i, m1......,m, 1Jf1 .......M, i1' ...... i,' nt1'""1n,' .Jltl'......,lli"' j=l
X
(2·6) *>
with the conditions; 2.:::~= 1 i1 = k, 2.:::~= 1 m 1 = n, 2.:::~= 1 ~ = N and similarly for the
vectors with prime. The vector i denotes one of the three unit vectors, (1, 0, 0), (0,
1, 0) and (0, 0, 1), corresponding to a hole in the P-orbit of 16 0. In Eq. (2 · 6), A(6)
and d(6; i, m, M) are the 6-th eigenvalue and eigenvector of the small norm
kernel X: The matrix element of X is given explicitly in the Appendix for the
present [ 12 C +a+ a] case. Due to this pre-diagonalization procedure, we have
only to sum over non-redundant 6 (non-zero A(6)).
If we use the following concise notations:
I={k,n,N},
*> Abbreviated notations are used; n! =n,! n,! n, !,
qn=q,"Jq,"aq,•a, etc.
Alpha- and 8 Be-Spectroscopic Factors of 20 Ne and
24
Mg
777
4
Dr (s) = Jn! N! k!/4! (2/r2 2)N12 II d (rJ i; ih mh MJ),
(2·7)
j~l
1
2!
fJ.(s)=-
4
II .l.(rJ1 )w(s)
J~l
and
w (s) =multiplicity factor of the set s,
Eq. (2 · 6) shows up as simple as
A (I,
I')=~
s
D 1 (s) fl. (s) Dl' (s).
(2·8)
This expression suggests us that diagonalization of the norm kernel A 1s
in the s-basis rather than in the original 1-basis. In fact, dimensionality
s-basis is much less than that of the 1-basis, if total h.o. quanta are not too
Note that the s-bases are non-orthogonal. Actually we diagonalize the
defined by
eas1er
of the
large.
matrix
(2·9)
For example, dimensionality is 10 in the s-basis and 348 in the !-basis for the
case of n + N- k = (0, 0, 10). This is because many redundant components are
already dropped by the pre-diagonalization procedure. By performing the above
procedure for each value of total h.o. quanta Q=n+ N-k, we obtain the eigenvalues and eigenvectors of the norm kernel A; some non-redundant and the others
redundant (7 non-redundant and 3 redundant solutions in case of the above example).
It has been recognized that the eigenstates of the norm kernel should be
labeled with SU(3) quantum number (A, fl.) .21 'n In order to identify the eigenstates
with CJ-, 1-1) -labels, we use the shift operator of h.o. quanta. We start with the
"lowest distribution" of the Q h.o. quanta (Q=~~~ 1 Qi, Q=n+N-k) in which
the quanta are maximally distributed to i = 1 axis and then to i = 2 axis, as far as
the condition Q 1 <Q2 <Q 8 is satisfied: (Q~o Q 2 , Q 3) for the "lowest distribution"
is denoted by (Q1 , Q 2 , Qs) . In the present 12 C +a+ a case, Q 1 is allowed to be
chosen as 0 without loss of generality and hence -4<Q1 <0. We repeat the
operation of the shift operators SH2 and s.~s on a non-redundant eigenstate of the
lowest distribution until both operations produce a null state; then the last nonvanishing configuration denoted here by (Q~o Q2 , Q 3) should correspond to the maximum weight (or highest distribution) of a SU(3) (A, fl.) state with A=Q 3 -Q 2 and
11 = Q2 - Q1 • Thus we can identify each non-redundant state with a (A, fl.) label
and at the same time obtain its maximum weight function. The shifting operation
is so simple, because the shift operator S is expressed as a simple sum of those
for the holes (k) and two relative coordinates (n, N). Diagonalization of the
norm kernel is needed only for the lowest distribution where all non-redundant
778
K. Kato and H. Banda
eigenstates are generated.
Needless to say, our approach is free from the spurious center-of-ma ss contamination which is often a nasty problem in the shell-model approach.
We denote the eigenvalue of the state (A, 11) [, by A ( (}c,u) [,), l, being an
additional quantum number. The normalized wave function is written as
Iff Q (
().,
p) [,) = -j1~j 4 ! 4! !?!
·2!
20!
- 7=c-1~~~Jl' 2...;
vll((l.,/J.)()
D ((A, !t) l,; k, n, N)
k.n.x
(2 ·10)
In Table I we list the non-redunda nt eigenstates and eigenvalues of the 12 C
-1- a+ a norm kernel for Q = 8, 9 andlO.*l While the 16 0 +a configuratio n represents only (A, p) = (Q, 0) component, the present 12 C +a+ a configuratio n can describe much more components of 20 Ne. For Q=8, we have three out of four
possible [ 4] -symmetry states; only (A, p) = (2, 0) is missing. The Q = 9 space includes the (A,/.!.) = (8, 2) component. These components are essential to reproducing the K"=0 2 + and K"=2 1 - bands. Of course, enough components are included
to describe the K"=0 1 +, 0 4 ' , K"=0 1 - C60+a like) and K"=0 3 +C 2 C+ct+n: like)
bands. It is noted that multiplicity of a (}c, p) state increases quickly with number
of quanta Q, as illustrated by three (10, 0) states already in the Q = 10 quanta space.
Table I. Eigenvalues of the "C+a+a norm kernel.
SU (3) label characterizing the eigenstate.
Q is the total quanta and (A, /1) is the
Q
- · - - -
8
(8, 0)
1. 9629
9
(9, 0)
2.0457
(8, 2)
0.8642
(2, 5)
0.2981
(1,4)
0.2224
(4, 2)
0.6293
---------
- - - - - - - -
(0, 4)
0.4409
----------
(7, 1)
0.7749
- -
(6, 3)
0.4759
(5, 2)
0.2819
0.5640
-
(4, 4)
0.3745
(3, 3)
0.1515
0.3919
-------------------
10
(11, 1)
0.6227
(5, 4)
0.2912
0.2519
0.1372
---
--
(10, 0)
2.6582
1. 0619
0.4587
(4, 6)
0.2783
0.1978
(9, 2)
0. 7009
(4, 3)
0.5488
0.2259
0.1514
(8, 4)
0.5014
(8, 1)
0.9763
0.5664
0.2812
(3, 5)
0.2988
0.1432
(2, 4)
0.4145
0.1614
-
*J Those for 3a and
16
0-l-a-t-a arc found in Ref.
4).
(7, 3)
0.5263
0.3986
0.0659
(6, 5)
0.3537
(1, 6)
0.2301
(6, 2)
0.8033
0.5627
0.3812
0.2383
0.1523
(0, 8)
0.1505
Alpha- and BEe-Spectroscopic Factors of 20 Ne and 24 Mg
§ 3.
779
Spectroscopic factors
We calculate the spectroscopic factors (S-factors) into a- and BEe-channels for
the eigenstates of the norm kernel. The expressions are illustrated for the case
of 20 N e. The angular momentum projected wave function of the (A, fl.) ( state is
obtained from the maximum weight w.f. 7JfQ,Q,Q, ((A, fl.)() by the projection procedure;
7Jf xnr( (A, fl.)()= ZKJ( (A, fl.)() - 112FKJM7Jf Q,Q,Q, ((A, fl.) (),
(3 ·1)
where PKJ,,1 is the angular momentum projection operator and ZxJ( (A, fl.)() is the
normalization factor. The S-factor of the [(A, fl.)(, KJ] state into the [l C2 C)
(g;N1L1 (BBe) ]L,Q9L 2 (relative) channel is given by definition as
Sc•. 11 J~;xJ([ZC 2 C), NlLl(BBe)]L, L2)
°)
= (28 I<C [¢z C2C) 0¢N,L, (BEe) ]L,&JuN,L, Cr2, R) ]J[IfKJ ((A, fl.)()) [2,
(3 · 2)
where
The explicit expressions of ¢, C2 C) and rPN,L, (BBe) are given by
(3· 3)
and
¢N,L,M, (BBe) = ) 2 !
j 4 ~tzN, (BBe)- 112 Jl' {¢ (a1) ¢ (a2) uN,L,M, Cr1, r)}
ZN,(BBe) =1-2 2-N'+3oN,.o.
(3·4)
In Eq. (3 · 3) we introduce the transformation coefficient of h.o.w.f. c (N, l, m; n~> n 2 ,
n 3) between the Cartesian and spherical coordinate representations. By using the
transformation coefficient extensively, Eq. (3 · 2) is reduced to
Sc•• 11 JCKJ([l C2C), N1L1 (8Be) ]L,, L2)
=
ZN, (8Be) -JA ((A, fl.)() ACA. 11 1~;KJ([l C2 C), N1L 1(8Be) ]L,, L2) 2 ,
·
(3 · 5)
Ao. 11 Ji;KJ([l C2C), N1L1 (BBe) ]L,, L2)
=ZKJ( (A, fl.)() - 112 ~ C(4l, N 1L 1(Ls), N2L2(JK); k, n, N)
k,n.N
XD((A,fl.)(; k, n, N),
(3·6)
where
C(4l, N 1L 1(Ls), N 2L 2(JK); k, n, N)
=
~ (lmLlMliLsMa) (LsMsL2M2 [JK)c(4lm;k)*c(N1 L 1 ~M1 ;n)
m.,Ma
(3·7)
The factor ZxJ( (A, fl.)() normalizes the amplitude A of Eq. (3 · 6).
Similarly, the 16 0 +a S-factor into the [l C2C) (g;Nl.N;] L, C60) (g;L2 (relative)
780
K. Kati5 and H. Banda
channel 1s expressed as
s(l.t<)~KJ ([l C2C)' NlLl]L,
= (
C60)' L2)
24°) I<[¢[!(12C)@N,L,h, C 0) ®uN,L, Cr2, R) ]J¢ (a) I?F U, ,u) C)> I2
6
KJ
=2·ZclC"CJ®N,L,h,C6 0)- 1A((,l., ,u)()Acx.~l!.'KJ([LC 2 C), N1Ll]L,C 60), L2Y. (3·8)
A<l.t~),KJ([l C2 C)'
NlLl]L, C6 0)' L2)
=Z ( (J.., ,u) (KJ) - 112 :E C(4l, N;L1 (L,), N2L2 (JK); k, n, N)
k,n,N
X
D ( (J.., ,u) (; k, n, N),
(3·9)
where all quantities are associated with the coordinates in Fig. 1 (b). The internal
wave function of 16 0 specified by [l C2C) Q9N;L 1 ]L, is defined here by
ql[l("C)®N 1 L1 h a C60)
= j 1 ~~Fz[l(12C)@N,L,h, C 0)- 112Jl' { [¢! C2 C) ®uN,L, Cr1, r) ]L,¢ (a)}.
6
(3 ·10)
The normalization factor Z for
16
0 in Eq. (3 ·10) was already calculated!),J 2)
_2oNe-
Tables II and III list the
12
C
+ Be
8
and
16
0
+a
S-factors, respectively, of the
Table II.
The "C+'Be S-factors SBe of the (8, Oh-J-o•, (4, 2h-J-o•, (0, 4h-J-o•, (8, 2)x_J_,.,
(9, Oh-o·,J-•· and (8, 8)x-J-o• states of "'Ne. The "C and 'Be wave functions are taken
as (J., p.) = (0, 4) •-•·•·• and (4, 0)L 1-o,,,,, respectively. The first column specifies l, L 1
and their coupled angular momentum La. The number in each entry is a sum of SBe
over possible "C-'Be relative angular momenta. The channels for which all S-factors in
the corresponding rows are smaller than 0.03 are not listed. The parenthesized numbers in
the first row are experimental"C("C, a) "Ne cross sections (da/d!J) (mb/sr) at f:hab=5".")
[ 12 CQ9'Be]
[l L,] La
0
0
2
2
2
2
2
4
4
4
4
4
4
0
2
0
2
2
4
4
0
2
2
4
4
4
0
2
2
2
4
4
6
4
4
6
4
6
8
i
(8, Oh-J-o•
(4, 2)x-J-o+
(0. 56)
0.070
0.011
0.063
0.084
0.063
0.001
0.179
0.002
0.064
0.317
0.049
0.192
0.226
(0. 21)
0.025
0.016
0.073
0.022
0.011
0.003
0.007
0.027
0.033
0.012
0.002
0.003
0.318
I
I
(0, 4) K-J-o•
(8, 2h-J -·· ' (9, Oh-o-,J-1·
(0. 24)
0.015
0.014
0.058
0.012
0.046
0.005
0.034
0.050
0.009
0.061
0.001
0.006
0.097
0.039
0.039
0.004
0.064
0.020
0.094
0.020
0.016
0.026
o.ooo
0.016
0.124
(0. 74)
0.013
0.014
0.014
0.000
0.052
0.007
0.006
0.028
0.005
0.161
0.003
0.006
0.093
(8, 8h-J-o+
(1.00)
0.040
0.036
0.023
0.095
0.003
0.033
0.000
0.003
0.011
0.000
0.005
0.001
0.000
Alpha- and BEe-Spectroscopic Factors of ' 0 Ne and
24
Mg
781
The "O+a S-factors S. of the (S, O)x~J~o•, (4, 2)x~J~o•, (0, 4)x~J~o•, (S, 2)x~J~•-,
20 Ne.
Some interesting "0 states are chosen and
listed. The first column specifies the "0 states in terms of the quantum numbers associated
with the 12CQ9a structure. (See Eq. (3 ·10) in the text.) Jn" in parentheses indicate the
observed levels of 160 which are considered to be most closely related with the listed
states. The number in each entry is a sum of S. over possible 16 0-a relative angular
momenta.
Table III.
(9,0h~o-,J~•- and (S,S)x~J~o• states of
"0
[l N,L,]L,
j
o 40 o+ co.+) •
2 51 1- (1,-)
2- cz.-)
3- (3,-)
0 so
s2
S4
S6
2 so
s2
o+
z+
4+
6+
2+
o+
1+
2+
3+
4+
0 91 193 395 s-
co,+)
(2, +)
(4,+)
(6,+)
(2, +)
co,+)
(1,+)
(2, +)
(3, +)
(4,+)
(1, -)
(3,-)
cs.-)
0.229
O.lSl
0.194
0.395
0.033
0.059
0.006
O.OS9
0.007
0.004
0.000
0.001
0.000
0.000
O.lSl
0.040
0.014
0.002
0.000
0.001
0.101
0.001
0.000
0.234
0.002
0.019
0.013
0.000
0.001
o.ooo
0.005
0.073
0.127
,
.
.
!
:
i
!
I
0.015
0.045
0.021
0.015
0.001
0.010
0.022
o.06S
0.01S
0.004
0.001
(9, Oh-o-,
0.344
0.025
0.001
0.037
0.003
0.031
0.026
o.ooo
0.042
0.01S
0.010
0.020
0.045
0.056
0.004
0.044
0.001
J~•-1
0.024
0.003
0.021
0.052
0.014
0.009
0.030
0.000
0.022
0.050
0.005
band heads of the (8, 0), ( 4, 2), (0, 4), (8, 2), (9, 0) and (8, 8) states of 20 Ne.
For the BEe wave functions, we take simply (A, /.1) = (4, 0), i.e., M =4, L 1 =0, 2, 4,
in Eq. (3 · 4). In Table III, the results only for some interesting 16 0 states are
listed. Although these tables are for some pure SU(3) states, they would be
useful as a sort of standard for more realistic :wave functions. As mentioned in
§ 1, the (8, 0) state is dominating in the ground band 13J and the (4, 2) state is
a dominant component of the second Kn = o+ band, although mixing of higher
(Q, 0) components is also important to reproduce the fairly large a-reduced
width. BJ, 14 ) The (8, 2) state is considered to correspond to the Kn = 2- band. The
(9, 0) state is a starting component of the Kn = o- band which has well-developed
a-clustering and therefore includes large amount of higher (Q, 0) components.
The (8, 8) state could be assigned for the third Kn = o+ band called a "8P-4h"
state at least as an important component.
Referring to Tables II and III, we discuss some interesting points. The direct
BEe-transfer on 12 C corresponds to the second row in Table II if the BEe is assumed
to be transferred in its ground state. For reference, experimental 12 CC 2 C, a) 20 Ne
reaction cross sections 15J leading to the states of 20Ne which are assigned as the
782
K. Kato and H. Banda
(A, ,!i) states 111 Table II are tabulated in the first row with parentheses. The
(8, 0) state has the largest SBe (0.07), followed by the (8, 8) state (0.04). It
is unexpected that SBe for the (8, 8) state are generally small, as evident from
the last column in Table II. Observed large 8 Ee-transfer strength 151 ' 151 for the
.K" = 0 3 + band may suggest the necessity of coherent mixing of higher quantum
states. Experimental 8Ee-transfer strength for the other states is not small in
comparison with the K" = 0 3 + state. It is remarkable that this obsen-ecl feature
can be explained by theoretical SEe values in the second row of Table II. The
cross section for the unnatural parity state, 2 1 -, may be due to the large amplitude
of the excited 8Be (2+) transfer. The (8, 0) column in Table II corresponds to
8 Be-pick-up processes
from the ground state of 20 Ne going to excited 12 C states.
The SBe values tend to increase with the spin of 12 C. It is curious that in the last
row of Table II in which the spins of 12 C and 8Be are stretched in parallel, some
of the SBe values are exceptionally large.
The Sa associated with the direct a-transfer on 16 0 (the first row in Table
III) going to the ground state of 20 Ne (0.23) is about three times as large as
SBe (0.07). By the a-pick-up reaction from the ground state of 20 Ne, the residual
16 0 states can be excited with
the Sa-factors shown in the (8, 0) column in Table
III. In addition, we calculate the Sa-factors by using more realistic 16 0 wave
functions obtained by Suzuki. 51 The results are shown in the s"mix column of
Table IV together with the experimental 20 Ne (d, 6 Li) 16 0 reaction cross section 171
and the duplication of the (8, 0) column of Table III listed under s"vure. The
use of realistic 16 0 wave functions leads to enhancement or reduction with respect
to Sa pure depending on the 16 0 state. The surprisingly large Sapure for the 3 1 - and
1 1 - states are preserved with some reductions in the more realistic S"mix, in contrast
to the experimentally weak cross sections. This reduction is favorable for the weak
cross section of the 1 1 - state. However, it is surprising that the Sa value of the
Table IV. Cross sections observed for 20 Ne(d, 'Li) 16 0, and the calculated S-factors associated
with a-pick-up reaction from 20 Ne leading to the excited states of 1'0. The numbers in
S"mix and S"pure columns are for the use of realistic 16 0 wave functions obtained by
Suzuki'1 and the duplications of the (8, 0) column of Table III, respectively.
Excitation energy
in 16 0(MeV)
0.00
6.06
6.13
6.92
7.12
8.88
9.59
9.85
10.36
d!J I dJJ (15")
(pb/sr) 171
J"
o1+
o,+
3121+
11')-
~1
1,2,+
41+
80
}
45
20
weak
Sa: mix
Sapure
0.208
0.050
0.298
o. 018
0.093
0.229
0.033
0.395
0.059
0.181
0.019
0.039
0.017
0.002
0.181
0.006
4.0
weak
21
3.5
Alpha- and 8 Be-Spectroscopic Factors of 20Ne and
24
Mg
783
3 1- state is still larger than the 0 2+ state which has clearly large 12C plus alpha
cluster component. 51 In Suzuki's calculation, the 31- state has dominantly 1P-1h
configuration. It may be dangerous to consider that the a-cluster-like state is
always strongly enhanced in the a-pick-up reaction. Unfortunately, the observed
cross sectionm has not been separated for the 02 + and 31- states. If a separated
data is obtained in the a-pick-up reaction, it will be a very important test for the
microscopic study of the a-pick-up reaction mechanism. On the other hand, for
the 12- and 32- states we obtain about ten times enhancement of mix with respect
to Sapure. In Suzuki's wave function for the 1 2 - state of 16 0, 85% of the components
are distributed over h.o. quanta N 1>9 with N 1=5 and 7 being 10% and 5%
respectively, while the Sa-factor to this 1 2 - state, 0.019, arises predominantly from
the 10% N 1=5 component. This implies that the a-pick-up reaction strengths
from 20 Ne going to 1 2 - and 32 - of 16 0 are determined by the coupling with the
N 1 =5 component. The Samix_factors for the 2/ and 2/ states are reduced by
a factor of 3r'-/4 with respect to the corresponding Sapure, and still Samix (22+) is
sa
sa
mix (21 +) in comparison with the same magnitude of the experitwice as large as
mental cross section. Note that the 21 + transition is forbidden in the simple weak
coupling picture. Now let us see 20 Ne*---'> 16 0*+a decays. The (4, 2) state has
relatively large Sa for the 1p-1h states (11 -) and (3 1-). The (0, 4) state has
essentially no probabilities for listed 16 0 states. The (8, 2) state is found to go
to the (1 1-), (21-) and (31-) states with clear selectivity. It is understandable
that the (9, 0) state has generally smaller Sa for excited 16 0 states than the (8, 0)
state, since a (Q, 0) configuration tends to approach the a+ 16 0 (01+) structure as
Q increases. We note here that Sa from (8, 0) to an 16 0* state ([l, N;L1] L 3
= [ 4, 82] 4 +) and from (9, 0) to [ 4, 80] 4 + are specifically large, 0.43 and 0.23,
respectively, although they are not listed in Table III. The Sa of the (8, 8) state
are generally small with some selectivities among the 16 0* states under consideration.
_24MgTables V and VI list the
8
Be + 16 0
and
20
Ne +a S-factors, respectively, of
(8, 4h•~o•, (8, 4h·~2•, (4, 6h·~o•, (0, 8h•~o• and (9, 4h•~o- band members of
the
24Mg.
The (4, Oh~o·~4 • and (0, Oh~o• state wave functions are taken for 8Be and
16 0,
respectively, and the (8, 0) L~o·~a• and (9, 0) £~ 1 -~9 - for 20 Ne. The present
(8, 4) x•~ 2 ·.J is orthogonalized to (8, 4) x•~o•,J· The (8, 4) x•~o• and (8, 4) x•~ 2 • should
be dominant components of the ground K" = o+ and the first K" = 2+ bands. 1a1 The
second K" = o+ band (6.44 MeV) is considered to be a mixture of the ( 4, 6) and
(0, 8) states. The first K" = o- band (7.55 MeV) can be regarded as an inversion
doublet partner of the ground band and consists mainly of the (9, 4) state. 191 Calculations based on the a+ a+ 16 0 model for 24 Mg will be reported in a separate paper.
Concerning SBe in Table V, note that the (4, 6) and (0, 8) states have no
probabilities for the channels under consideration. 41 Direct aBe-transfer reactions
on 16 0 leading to the (A, f.!.) K,J state of 24 Mg are related to the S 8 e in the L (aBe) = 0
784
K. Kato and
~I.
Banda
Table V. The 'Be+"O S-factors SBe of the (8,4h~o•, (8,4)K~2• and (9,4h~o- states of 24 Mg,
where we take the ground state for 16 0 and the (A, ~1) = (4, 0) L~o,,,, states for 'Be. The first
column specifies the angular momenta of 'Be (L ('Be)) and of the 16 0- 'Be relative motion
(L(rel.)). J is the angular momentum of "Mg. Note that the (4,6) and (0,8) states
have no probabilities for the 'Be+ 16 0 channels under consideration.'l
(l., tt) = (8, 4), K=O+
L('Bc)
.f(rel.)
4'
J
0
2
0.037
0.013
0.002
0.003
0.010
2+
4+
(l.,,Lt)=(8,4), K=2+
6+
8-
0.000
0.021
0.038
0.003
0.000
0.008
0.041
0.009
0.001
0.001
0.002
0.000
0.004
0.033
0.012
0.000
0.001
0.005
0.006
0.021
0.000
0.001
0.019
0.013
0.000
0.001
0.010
0.001
0.038
7+
9+
0.014
0.033
0.000
0.005
0.002
0.028
0.007
0.029
0.000
0.007
J
J+2
J+2
L('Be) !
_ __,_I_L--'(rel.2 _
0
2
J
J
J-2
J
J+2
J-4
J-2
4
J
0.003
0.013
0.003
J~-·)
J+4
L('Be)
L (rel.)
2
J-1
4
J+1
J-3
0.036
0.025
J-1
i
L ('Be)
I
L(rel.)
O.Oll
0.010
J
0.003
0.003
0.015
J+1
J+3
J
0.031
0.002
0.020
0.000
0.000
0.001
0.004
0.012
0.033
J+2
J-4
J-2
4
8+
I
0.025
0.001
0.017
0. 016
0.000
0.000
0.002
0.007
0.014
J-2
J
0.035
0.003
0.008
0.021
0.023
0.032
0.000
0.004
0.000
0.022
0. Oll
(A,,Lt)=(9,4), K=o5-
1-
0.019
0.000
0.027
0.009
0.000
0.000
0.004
0.012
0.011
0.013
0.000
0.043
0.000
0.000
0.001
0.023
0.003
O.Oll
0.027
7-
9-
0.035
0.001
0.0:27
0.021
0.000
0.000
0.003
0.027
.~~--
0
2
J
J
4
0.052
J-2
J+2
J-4
J-2
0.015
0.032
J
J+2
J+4
0.005
0.013
0.049
0.005
0.013
0.029
0.001
0.002
0.004
0.015
0.043
0.003
0.018
0.027
0.000
0.000
0.002
0.006
0.018
0. Oll
0. 019
o.ooo
0.042
0. Oll
0.000
0.000
0.007
0.017
o. 014
Alpha- and 8 Be-Spectroscopic Factors of 20 Ne and
785
24 ]).1g
rows, if the 8Be is assumed to be transferred in the ground state. In the (8, 4) x•~o•
band, SBe decreases gradually from 0.037 for o+ to 0.019 for 8+, and is 1.4 times
tts small as that for the (9, 4) x•~o• band. For the (8, 4) x•~ 2 • band, SBe are essenTable VI. The a+ "Ne S-factors S. of the (8, 4h~o•, (8, 4h~•·· (4, 6h~o•, (0, 8h~o• and
(9, 4) K-o- states of "Mg, where the (J., JL) = (8, 0) L~o•-•• and (9, 0) L~•--•- states. are taken
for "Ne. J is the angular momentum of "Mg. To make the Table not too large, each
entry is a sum of Sa over possible a-"Ne relative angular momenta. Note that the (0, 8)
states are exhausted by the channels in which 20 Ne is in the (8, 0) states.
(J., JL) = (8, 4), K =0+
J
-----·~---
L("Ne)
o+
0.081
0.010
0.065
0.148
0.082
0.031
0.070
0.007
0.107
0.030
0.074
0.073
0.113
0.096
0.045
0.026
0.064
0.059
0.020
0.000
0.100
0.098
0.076
0.112
0.019
0.066
0.052
0.036
0.043
0.012
0.061
0.101
0.099
0.113
0.012
0.053
0.051
0.047
0.053
0.000
0.005
0.130
0.127
0.124
0.000
0.002
0.085
0.068
0.059
(J., JL) = (8, 4), K =2+
L("Ne)
J
8+
0.004
0.094
0.051
0.160
0.078
0.017
0.068
0.037
0.087
0.005
0.037
0.026
0.141
0.073
0.110
0.022
0.068
0.040
0.049
0.036
0.057
0.058
0.056
0.108
0.107
0.059
0.018
0.030
0.055
0.053
0.004
0.103
0.088
0.110
0.081
0.002
0.044
0.044
0.083
0.042
9+
J :
---------------- -------:
L("Ne)
0.082
0.098
0.111
0.095
0.034
0.027
0.083
0.047
0.024
0.073
0.132
0.062
0.119
0.022
0.045
0.080
0.013
0.055
0.089
0.064
0.122
0.111
0.002
0.127
0.150
0.107
0.000
0.070
0.041
0.035
0.068
0.001
0.074
0.073
0.067
0.001
0.026
0.125
0.233
o.ooo
0.012
0.096
0.107
786
K. Kato and H. Banda
Table VI continued
L("Ne)
(X, ,u) = (4, 6), K=o•
J
.
o•
I
2•
6+
-------~--~~~---------
o+
L("Ne)
1
J
o.069
0.055
0.009
0.016
0.185
0.002
0.011
0.021
0.022
o.o15
0.051
0.071
0.095
0.103
0.014
o. 011
0.009
0.009
0.013
-------- - -
o.oo4
0.088
0.078
0.078
0.086
o.ooo
0.018
0.013
0.012
0.012
---
-~-
0.000
0.007
0.119
0.098
0.110
o.ooo
o.ooo
0.024
0.016
0.015
(A,,u)=(0,8), K=o•
o•
6+
----------~--~------
o•
0.049
0.061
0.062
0.062
0.062
2•
4+
6+
8•
0.012
0.086
0.068
0.066
0.065
0.007
0.038
0.104
0.077
0.072
0.005
0.025
0.053
0.125
0.090
0.004
0.019
0.038
0.068
0.168
1-~9-
L("Ne)
(X,,u)=(9,4), K=o-
J
------------
9-
0.019
0.009
0.024
0.036
0.035
0.059
0.074
0.033
0.148
0.027
0.004
0.029
0.023
0.030
0.037
0.060
0.043
0.106
0.073
0.059
0.001
0.028
0.030
0.027
0.038
0.016
0.110
0.075
0.058
0.082
0.003
0.015
0.035
0.033
0.037
0.018
0.065
0.087
0.081
0.090
o.ooo
0.002
0.038
0.042
0.042
0.000
0.004
0.123
0.112
0.102
tially zero except for the 10+ state, entirely due to orthogonality to the K" = o+ band.
With reference to Table VI, the Sa values in the LC 0Ne) =0+ rows are related
to direct transfer reactions on 20 Ne. They are fluctuating with J in the (8, 4) x•~o•
and (8, 4) x•~ 2 • bands, while decreasing monotonically with J in the other bands.
The largest Sa is for the transfer to the (8, 4) K"~J~o· state. Note that this value
Sa= 0.08 is twice as large as Sae = 0.04 for the direct 8Be-transfer. The a-pick-up
reaction data from the ground state of 24 Mg leading to the ground and excited
Alpha- and 8Be-Spectroscopic Factors of 20 Ne and
24
Mg
787
states of 20 Ne provide Sa corresponding to the J = o+ column for ()., fJ.) = (8, 4) K"~o·
in Table VI. As seen here the theoretical Sa values show strong dependence
on states of 20 Ne which is expected to be checked by experiments, although presently available datam' 20 > are not consistent with each other.
§4.
Summary
We have calculated the eigenvalues and eigenstates of the norm kernel for the
a+ a+ 12 C system by applying the method which was used previously for closedshell-cluster systems such as 3a and a+ a+ 16 0. 4 > The method employs the prediagonalization procedure for the small norm kernel on the basis of the generating
function technique. This method is testified to be useful even for such complicated
systems. Allowed eigenstates are found to include many important SU (3) states
which are expected to be enough to describe all the bands of 20 Ne starting below
8.5 MeV excitation.
By using the present results for 20 Ne and the previous results for 24 Mg,4l we
have evaluated a- and 8Be-spectroscopic factors of some interesting states of 20Ne
and 24 Mg. Some selectivities are found in those states and very interesting results
are discussed in comparison with the weak coupling model of 16 0 and also with
a-pick-up reaction data. The results can be useful as a guide for the analysis
of experimental reaction data and for works using more realistic wave functions.
Acknowledgemen ts
The authors would like to thank Professor H. Horiuchi for helpful comments
and discussions. They are also thankful to Professor K. Ikeda, Professor R. Tamagaki and Professor H. Tanaka for continual encouragement and to Dr. Y. Suzuki
for providing them with his 16 0 wave functions.
This work was performed as a part of the annual research projects on the
"Alpha-Like Four-Body Correlations and Molecular Aspects in Nuclei" in 1975
and 1976 organized by the Research Institute for Fundamental Physics, Kyoto
University.
They are indebted to a "Grant-in-aid for Scientific Research of Ministry of
Education, Science and Culture under the Special Research Project on Heavy Ion
Science."
The numerical computations are carried out on the Computer Systems at the
Data Processing Center of Kyoto University and Computer Center of the University
of Tokyo.
Appendix
The matrix elements ). (i', m', M'; i, m, M) of the small norm kernel for
12 C-a-a is given by ). (i', m', M'; i, m, M) = {1- (- )m'} {1- (- )""} ~~~ 1 ).i (i', m',
M'; i, m, M) where i and i' indicate hole quanta in the i- and i'-th direction
788
K. Kati5 and H. Banda
respectively and
)q=
-~(-2)JI_l_
32
M!
5
or
X [ 0mi'· oO mi, oO}Ii',}f/i
- ;_7\;f
jo m'
J•
(lvfzO m/,10 m 1, rO mz', oO ml, oOJI1',JI10fllt'.JI z
10 mj. oO mt'· oO mt.lolffJ' +1,1Wjo Mz',flfz+l
+ 111 /:Jmj',oOmi, oOm,',lOm,,l!J M;'.Ml J.f ',J,f
1
1
- .1'vf!om/.oOmj,tOm,',/Jm,,ooM/.N.;+tO,w,'+l,Mz+t)]
(i, J, l cyclic),
for i' = i
for
•I
L
•
=t
for i' ~i.
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1)
2)
3)
4)
5)
6)
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789
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