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A semi-microscopic study of negative-parity α-cluster states in 19F
Nuclear Physics A417 (1984) 109-127 @ North-Holland Publishing Company
A SEMI-MICROSCOPIC STUDY OF NEGATIVE-PARITY a-CLUSTER STATES IN ‘9 A.C. MERCHANT SERC Daresbury Laboratory, Warrington WA4 4AD, UK Received 23 September 1983 Abstract: The bands of negative-parity a-cluster states in 19F which may be formed by coupling an a-particle to a “N core in its ground (f) or lowest negative-parity excited @) state are studied. The central part of the effective cluster-core potential is obtained from a simple folding procedure, and the non-central terms are microscopically derived from an aN scattering potential. The spectrum of o-cluster states is calculated, taking into account mixing of states in the two bands by the tensor forces, and found to give generally good agreement with the experimentally determined spectrum up to excitation energies of 10 MeV, although mixing with 3H-chrster states may sometimes be important.
1. Introduction Alpha-particle transfer reactions induced by heavy ions are known to strongly and selectively excite a small subset of states of the final nucleus. These states appear to have a cluster-core structure, in that their spectra display rotational energy spacings, they are connected by strong electromagnetic E2 transitions and those above the a-threshold have large n-decay widths. Buck, Dover and Vary ‘) have introduced a simple cluster model to account for the properties of these states. They obtained a local, effective cluster-core potential by folding the cluster and core densities with an effective NN interaction, and were able to give a good description of the spectra, a-decay widths and electromagnetic transitions between the a-cluster states in I60 and 20Ne. Their model was subsequently applied with equal success to the study of a-cluster states in 180, 18F [ref. 2)] and 19F [ref. 3)]. In the latter two cases the 14Nand “N cores carry spins of 1 and i respectively, so that non-central forces may be present in the cluster-core interactions. Initially, the effects of these forces were parametrized phenomenologically, but later they were calculated microscopically4), so that the model as applied to a-cluster states in nuclei in the mass range A = 16-20 is now well established. All the calculations mentioned above have considered the cluster and core to be in their ground states. Since the free a-particle has its first excited state at an energy in excess of 20 MeV, this seems to be a good approximation for the cluster. However, the possibility of the core being in an excited state needs to be borne in mind. A study of the 14N-a system in which the 14N core was allowed to occupy its ground and first two excited states “) showed that the matrix elements connecting a-cluster 109
110
A.C. Merchant / a-cluster
states
states in r8F with equal total angular momentum, J, but different core states, were quite large, but that these states were generally so far apart in energy that mixing between them was not very great. In one particular case, however, two states with J” = 3+ (formed by coupling the a-cluster to a ground state 14N core with relative orbital angular momentum L = 4, and to a i4N core in its first excited (l+) state with L = 2) were found to be nearly degenerate, and hence were strongly mixed. The reason that, up to now, interest has centred on the case of cluster and core in their ground states is that a-transfer experiments are only likely to populate heavily those cluster states which contain a large ground-state core component, whereas to populate states based on an excited core, a two-step process of core excitation and a-transfer is necessary. An alternative method of observing ru-cluster states is to inelastically excite the final nucleus and watch it decay into its cluster and core components. This is currently being done at Daresbury and Oxford “f by inelastically scattering a 19F beam from a “C foil and simultaneously observing the a-particle and 15N fragment emitted in the subsequent break-up. The angles of emission of the two components are measured and the excitation energy of the 15N fragment plus the relative orbital angular momentum of the a-particle and i5N can be determined. Thus, candidates for a-cluster structure in 19F lying above the a-particle emission threshold, based on an excited “N core, together with their spin-parity values, can be located. It is therefore the purpose of this paper to predict the spectra of the two lowest-lying bands of negative-parity a-cluster states in 19F formed by coupling an a-particle to a i5N core in its ground (f) or first excited negative-parity (r) state, within the framework of the Buck, Dover and Vary folded potential cluster model, in anticipation of these new experimental data. 2. The cluster-core potential The central part of the cluster-core potential for the i5N-a! system has already been given by Buck and Pilt 3), and in view of its success in calculations of the properties of a-cluster states based on a ground-state “N core the same form will be used here. It is written as V(R) = V,(R) f V,,,,(R)
.
(2.1)
V,,,,(R) is the Coulomb potential appropriate to a uniform spherical charge distribution of radius 3.1 fm. V,(R)is the local potential obtained by folding the cluster and core densities with a b-function form of the effective nucleon-nucleon interaction, and was parametrized as
Vo(1+cosh (Rolao)) cash (R/a,)+cosh
(&/a,)
where V,= -172 MeV, R. = 2.25 fm and a0 = 0.8 fm.
’
(2.2)
A.C. Merchant / a-cluster states
111
cluster is treated as a single particle in orbit around the core, characterized by the principal quantum number N (the number of nodes in the radial wave function) and L (the orbital angular momentum).,The major requirements of the Pauli exclusion principle are satisfied by restricting the cluster nucleon quantum numbers to values appropriate to shell-model orbitals outside the core (sd or higher shells in the present case) so that The
2N+La8.
(2.3)
With this restriction, the single-particle Schriidinger equation can be solved with the potential of (2.1) to give energies and wave functions for the relative motion of cluster and core. For higher-lying quasi-bound states, Jackson’s procedure ‘) of finding the position, RB, and height, E u, of the potential barrier and setting V(R) = EB for R 2 RB is followed. All levels are thus treated as bound states, with minimal errors in their energies. When this is done, a series of almost perfect rotationally spaced levels is produced, whose energies may be written E(L) = C+AL(L+l)
(2.4)
with C the bandhead and A the rotational parameter, approximately 0.2 MeV for the 15N-(u system. Since the core carries a spin of i or $ there can be non-central terms in the cluster-core potential which may split the rotationally spaced centroids of (2.4). In general, these terms may be written as scalar products of rank-K spherical tensor operators formed from the spin of the core and equal-rank tensor operators formed from the variables of relative motion of cluster about core (namely, R, the relative position, P, the relative momentum and L, the relative orbital angular momentum). If the interaction is required to be invariant with respect to space exchange and time reversal, the only possible vector-vector contraction is the normal spin-orbit force L * S. When the core spin is 1, this is in fact the only non-central force possible between cluster and core. However, when the core spin is 1, scalar products of tensors of ranks 2 and 3 are also possible. The rank-2 contractions may be readily derived from the relation “) S,~R2(a,b)=(S*a)(S~b)-~iS+z~b)-$*(a~b),
(2.5)
where S2 is the rank-2 tensor formed from the core spin, SF =c (l/_&t4-~~2M)S~S”-~, Ir
(2.6)
and R2(a, b) is the rank-2 tensor formed from the vectors u and b, R2M(u,b)=C(l~~lM-~~12M)a~‘b~-~. I”
(2.7)
The vectors R, P and L may not be mixed in (2.7) if the interaction is to remain invariant under space inversion and time reversal, so the only possible contractions
112
A.C. Merchant / a-cluster states
are
s, *R,(d, Ii)
=[(S*
ii)*-fs*1,
(2.8)
s**R,(P,P)=[(S* P)*-~s*P*]) Sz * R,(L,L) =[(S*
(2.9)
L)*+$(S- L)-fS2~2],
(2.10)
where the circumflex accent denotes a unit vector, and all the contractions may be multiplied by an arbitrary radial factor, F(R). The possible scalar products of rank-3 tensor operators which may be present in the cluster-core potential are a little more complicated, but have recently been studied in connection with ‘Li scattering ‘-ll). The terms which are invariant under parity- and time-reversal operations consist of the contraction of a rank-3 tensor operator formed from the core spin, S,(S,, S), and a rank-3 tensor operator, formed from L and any of the rank-2 tensors, &(a, a), occurring in (2.8)-(2.10), and written ?‘,(&(a, a), L). The scalar product S3 - T3 may be simplified by using the relation ‘*) Z-“(k,, k,).
7”(k3, kq)=(-l)kl+k4(2K+1)
k3) * T&k,,
x C W(k~k~k~k~; KK’)T’~(k,, K’
kd,
(2.11)
which allows the scalar product of tensor operators of rank-K, formed from ranks k,, k2 and k3, k4, to be written as a linear combination of scalar products of tensor operators of all ranks K’ (with the structures indicated) allowed by triangulation, provided the tensor operators of ranks k2 and k3 commute. Therefore, (s,
’ R,)(S*
(s*
h)(s2
L)
=
* R2)
=
Kil
(-1)K’TK’(S2,
i
s)
(-l)K’TK’(s,
s2)
’ TK@,,
* TK4I.3
L,
,
(2.12) (2.13)
R2),
K'=l
where TK,(Sz, S) = T’$(S, S,) when K’ = 1 or 3 but T,(S,, S) # T2(S, S,) and similarly for TKp(R2, L). In fact, T,(S,, S) = Tl(S, S,) =(-&s*
s+@s,
%G(S,, S) + 7-0, S,)l= -GSz
(2.14) (2.15)
*
The tensors TK,(R;?(L, A), L) have an analogous form to the TK(S2, S) whereas TK*(Rz(k, k), L) and TKc(R2(P, P), L) obey the relations T,(R*(%
0-2(&b,
The results of (2.12)-(2.17)
a),
L)
=
T,W,
&(a,
a))
a),
L)
+
T2(&
R,(a,
a))3
=&am
=
--&Ma,
a)&
(2.16)
7
a)
.
(2.17)
show that there may be three scalar products of rank-3
113
A.C. MerchantI a-cluster states
tensor operators in the cluster-core
potential, and they may be written as
k>,L) =iW US, - R,& &+(Sz *R,U@, *MS*VI
S,(S,, S) - MMk
+;S,
%(SZ, S)
*~&W-‘, PI, U =iW
*R@, I@+[#.
S)-&](S- L) ,
(2.18)
US, - &VT Z-7)+(& - &VT P)W* L)l
+$S, - R,(P, P)+[&(S
S)-&,](P.
P)(S* L) ,
(2.19) S,(S,, S)
*T,(R,(L 0, U =&S* WS,(R,W, Q)+(S, *ML, L)W- 01 +$S2. R,(L, L)-&(S. +$s.
S)+(L*
A microscopic derivation of these non-central considered in the next section. 3. Microscopic calculation of the non-central
S)(L’ L)(S. L)
L)--:](s*
L).
(2.20)
forces in the “N-CY system will be
terms in the duster-core
potential
In this section the non-central terms present in the potential between a 15N core in its ground or first excited negative-parity state and an a-particle will be derived, using the microscopic model introduced in ref. “). The hamiltonian, H, for a 19nucleon system interacting through two-body forces may be written &P:+C i=r 2WI
v.. i
(3.1)
"
where m is the nucleon mass, pi is the momentum of the ith nucleon and Vij is the interaction between nucleons i and j. To describe states with the structure of an a-particle coupled to a 15N core this may be approximately decomposed as H=H,+H,+H,,,
,
(3.2)
where H, and Ha are the internal hamiltonians of the core and cluster respectively and Hreldescribes their relative motion (depending only on their relative coordinates). The eigenfunctions of this hamiltonian may be written as a sum of product states: YJM
=
da
JZ, (SM’LM-M’IJM)~,(15N)~~~-~,(R)y
(3.3)
where &&15N) is a fully antisymmetrized 15N wave function (and r5N has an intrinsic angular momentum of S), +(a) denotes the ground state of the a-particle, and $LM_M,(R) is a wave function for their relative motion. The microscopic model now seeks to build up an effective cluster-core potential, V,, in Hrel, by summing an (YN potential over all 15 core nucleons.
114
A.C. Merchant / a-cluster states
The 15N core in its ground or first excited negative-parity state is treated as a pi/2 or p3/2 hole in the 160 closed shell, IO),so the cluster-core potential can actually be derived by considering the interaction of the a-particle with this hole. Applying the symmetry arguments invoked in the last section to the possible form of the CXN potential, its most general form is seen to contain only central and spin-orbit terms: u,, = UJ(t) + G(r)L
(3.4)
- s,
where r is the separation of the nucleon and the centre of mass of the a-particle, Ire1is their relative orbital angular momentum and s is the spin of the nucleon. Writing out Ire, explicitly gives Ire,. s=$[I.
s-R*p.
s]+$L.
S-XhP.
S],
(3.5)
where M = m + m, is the sum of nucleon and alpha masses, R, P and L are the position, momentum and angular momentum of the a-particle with respect to a given origin (to be identified as the centre of the core) and x, p and 1 are analogous quantities for the nucleon. Since this expression only contains L and P to first order, the scalar products (2.9), (2.10), (2.19) and (2.20) will not be present in the cluster-core potential derived from this model. Tensor decompositions of the radial functions U,,(r) and ULS(r) may be made as follows: U,(r) = IJ,(lx--Rl)
=z (2y+I)(IJ,(x,
R))&&)
*
G@>,
(3.6)
where the circumflex accents again denote unit vectors, (3.7) with Yg($
the spherical harmonics, and (U&,R))z=S
I’
U&-
RI)Pz(cos 4) d(cos 4) ,
(3.8)
-1
where cos (8) = f * 8. Eq. (3.6) immediately gives the central potential, U,(r), as a sum of scalar products of rank-K spherical tensor operators, AK (say) in the cY-coordinates and BK (say) in the nucleon coordinates. It is possible to write the spin-orbit potential, U,,(r) lrel * s in the same way, so as to produce a potential between the ith nucleon and the a-particle of u’,iL =&jL(XitR)AK(a). This interaction will give a contribution
({#s(“N)$L(R))J
&(xt).
(3.9)
to the energy of
Ii!l@,,I
Md5NMR)h0
(3.10)
A.C.
and integration
Merchant
/ a-cluster
over the nucleon coordinates
(k(R)(SL)J
115
states
yields
; AK(~) - NK
(3.11)
with NKQ=(4s(15N)SM+Q
f ,fK(Xi,R)Bz(Xi)
+s("N)SM).
(3.12)
i=l
The rank-K tensor operator, NK, may be expressed in terms of the rank-K tensor operator formed from the core spin, SK, as @
=
(4d15N)ll C;:,f~(xi,R)B~(xi)llh(‘~N)) s,“,
ww)
(3.13)
so that the interaction has been written in terms of induced operators involving the core spin, and the cluster-core potential may be written v,,=C&(R)&(~). K
(3.14)
SK.
To calculate the form factors FK(R), the reduced matrix elements of the nucleon operators &(&) must be evaluated. In general, &(&) is formed by combining a nucleon spatial tensor operator of rank kI, C,, (say), with a nucleon spin tensor operator of rank kZ, Dkz (say). Evaluation of the reduced matrix elements of BK (xi) between wave functions of 15N gives (15N; SllBKll’5N S>= ((W’Il&lI(S)-‘)
(3.15)
with (S)-’ a hole in the 160 closed shell. Using the same arguments concerning particle-hole conjugation and the evenness of K + k, + k2 as in ref. 4), one finds (‘5N~lB~ll’5N
=
(OIIBKIIWK,O
(~~lC,,lll>(sllD,,lls),
(3.16)
where 1= 1 for a p-shell-hole description of the core, s = 4 and s^= (2s + l)r” etc. The symmetry properties of the 9-i symbol in (3.16) place the following restrictions on the combinations of (Kklk2): _ scalar:
(0,070)
(0,17 1)
vector:
(1,&l)
(L2,l)
rank-2 tensor:
(2,290)
(2,171)
rank-3 tensor:
(3,271)
(1,LO)
116
A.C. Merchant j a-cluster states
The terms in the tensor decompositions of U,(r) and ULs( r) irei 1 s corresponding to the above combinations can now be found, and the forces in the cluster-core potential evaluated in the form V,,(S=$)=V,(R)+V,(R;S=$)LS,
(3.17)
V,,(S=;)=V,(R)+V,(R;S=~)LS+V,(R)S,~R,(~,~) + V,(R)% - NW%
&I,
L) .
(3.18)
As explained in sect. 2 and discussed in ref. “) the cash potential (2.2) will be used for V,,(R). The spin-orbit and rank-2 tensor form. factors, V,(R; S=$), V,(R; S =$) and V,(R), may be obtained immediately from ref. 4), where an effective 14N-~ potential was derived treating the r4N core as two p-shell holes in the 160 closed shell (with their spins coupled to a value of 1). There, the two-hole matrix elements analogous to (3.15) were expressed in terms of single-hole matrix elements and some factors depending on the amplitudes for finding both holes in p1,,2states, ~312 states, or one in each state, (C,, C, and C,, res~ctively). By setting = 0 for S = 1 the 14N-a form factors C,=1,C2=C3=OforS=$andC,=1,CI=C2 “N-a! form factors which are required here. can be simply related to the equivalent Combining eq. (4.23) of ref. “) with (3.16) above yields Vl(14N-a; Cl = 1)
for S=$:
(3.19)
VI(‘5N_a; S ~1) = 1 ’
and for S = 3:
V,(‘4N-cx; C, = 1)
Vr(15N_(U;S+)
=’ ’
V2(‘4N-~; C, = 1)
V,(‘5N_(r)
=-+*
(3.20)
(3.21)
Defining the integrals of the p-shell radial wave functions, 8 (x), with the components (Ua(x, R)), of (3.8) as
and, since similar integrals multiplied by (x/R) defining
or (R/x)
and using eqs. (4.36) of ref. “) and (3.19)-(3.21)
above allows the spin-orbit and
also occur frequently,
117
A.C. Merchant / a-cluster states
rank-2 tensor form factors for the 15N-~ system to be written as
V,(R; s=t> =~[~(FLs(R))o-~(FIS(R))L+~(FLS(R))~I
(3.25)
9
V,(R) = (F,(R)),+~[(F,s(R)),-(Ft(R)),l .
(3.26)
It only remains to evaluate V,(R) to complete the derivation of the cluster-core potential of (3.18). There are two contributions to this form factor which arise as follows: (i) When 8= 2 in the tensor decomposition of the term (m/M) U&r)L * s one obtains the operator 543;
(&(x,
R))zTJG(i),
s) - GUMk
h,
U
which, when inserted in (3.16) and using (3.13), gives a contribution
to
V,(R)
of
-(WM(&(R)),.
(ii) When LZ= 3 in the tensor decomposition of the term -(m/M) one obtains, after some recoupling, the operator -dE
;
which gives a contribution
(ULS(X,
R))3T3(C2(-5),
s>
to V,(R) of (m/M)(F,,
V,(R) =~[(G
UUL-VAR)M
* 7-3(R,(k,
d’),
U,,(r)x
AP- s
L)
(R))3. Therefore,
.
(3.27)
It should be noted that the spin-orbit and rank-3 tensor form factors all contain the mass factor (m/M) = 0.2 and are derived entirely from the spin-orbit part of the cuN potential, whereas the rank-2 tensor form factor contains a (much larger) contribution from the central part of the potential.
4. Calculation
of the cluster state spectrum
The form factors (3.24)-(3.27) may be evaluated analytically, as in ref. 4), by the following choices of p-shell radial wave function and cuN potential. A harmonicoscillator radial wave function of the form -x2/26=
(4.1)
118
A.C. Merchant / a-cluster states
is used for the hole, where b = 1.77 fm is consistent with the measured charge radius of 160 [ref. 13)]. A gaussian shape for the (YN potential is used 4T14P15), U,, = ( U, + U&i
f s) e-r2’n2
(4.2)
(where U,, = -43.0 MeV, U,, = -7.0 MeV and l/a = 0.526 fm-i), which enables the tensor decomposition (3.6) and the radial integrals (3.22) and (3.23) to be evaluated in closed form. The precise value of ULs is not too important, since it has only a small effect on the cluster state spectrum through its linear appearance in V,(R; S =&, V,(R; S =s) and V,(R), and a very minor influence on V,(R). The form factors, as calculated with this prescription, are shown in figs. 1 and 2. The main point to note is that the second-rank tensor form factor, V,(R), is much more important than any of the others. Its analytic form is proportional to R2 exp (-R’/(a*+ b’)) and it has a minimum of about -2 MeV around R = 2.5 fm. This is more than fifty times greater in magnitude than the maximum of the similarly shaped third-rank tensor form factor V-,(R). This difference in scale is almost entirely accounted for by the mass factor m/M = f and the ratio of (l&J U,) = &. The spin-orbit form factors are non-zero at the origin, but at cluster-core separations in excess of around 2 fm they are smaller in magnitude than V,(R). It is therefore already apparent that the a-cluster states based on a spin-4 i5N core will consist of almost degenerate doublets, whilst those based on a spin-g core will form quadruplets with energy splittings characteristic of the tensor force (Se s)‘-fS2.
-2
Fig. 1. The second-rank tensor form factor, V,(R), for the l’N*-a separation.
system as a function of the cluster-core
119
A.C. Merchant / a-cluster states
Fig. 2. The spin-orbit and third-rank the “N-a and “N*-a
tensor form factors, V,(R; S=$), V,(R; S=$) and systems as functions of the cluster-core separation.
V,(R), for
The precise form of the splitting of the various doublets and quadruplets from the nearly rotational centroids of (2.4) can be found by evaluating matrix elements of the form factors (3.24)-(3.27), and their associated operators, between states of total angular momentum, J, obtained by coupling the core spin, S, with the relative orbital angular momentum of cluster about core, L.For matrix elements connecting states based on the same core state, it is readily found that 10*12*16)
((LS)J((V,(R; S)L*s~~(L’s)J>=s,&W+1)-L(L+1)-s(s+1)1 X
XL*(R) V,(R;
S)XL,@)
dR
(4.3)
120
A.C. Merchant / a-cluster states
with xL(R) the radial wave function for the relative motion of cluster about core. The second- and third-rank tensor forces are only applicable to states based on a S = 1 core and have matrix elements
WWtt V,(R)& * R,Otd.)[[(L’S)J) co
=
I
A,($ L, L’)
KLS)J((V,(R&
o
x:(R) VAR)xtW dR,
- URz(&
&
(4.4)
L)ll(L’S)J)
m
=
A#,
L, L’)
I
0
xT.W)V~R)XLW dR
(4.5)
with A&, L, L’) and A,(& L, L’) listed in table 1 under the heading RME. [Note that the values of A,($ L, I.‘) listed in ref. lo) are up to a factor of 10 too large.] So far, the cluster-core interaction has been written in terms of scalar products of tensor operators in the ru-cluster coordinates with others formed from the core spin, S, which have the same matrix elements as the actual operators involving the nucleon coordinates. This works very well as long as the core spin is a good quantum number. However, the second- and third-rank tensor contractions in the cluster-core potential actually involve quantities like 5( U,(X, R))&&) - C,(k). These have non-zero matrix elements between states of equal total angular momentum, J, formed by coupling different states of the core to L [ref. 5>],since the matrix elements of C#) must themselves be taken between coupled basis states, whose total angular
TABLE
1
Reduced matrix elements of tensor forces RME
J=L+$
((L$)Jb . Rzli(@J>
-2L+3
((mw,
. ~3llIGm RME
W~)JllS,. &ll(L+233J) G$)JllS,
. T&L+- mo
L+3
L-2
-(L+l)
2L+3
2L-1
2L-1
-L
-3L(L-
1)
9(L+2)(L-
1)
-9(L+2)(L-1)
J=L+$
-m 2L+3
2L+3
(L+3)43L(L+2)
l)(L+3)
2(2L+3)
2(2L.+3)
Angular dependence of the reduced matrix elements of S, . R,(& coupled cluster-core basis states having S =$.
10(2L-1)
J=L+$
-J3(L+l)(L+3)
-LJ3(L+
3(L+lf(L+2)
lO(ZL- 1)
lO(2Lf3)
10(2L+3)
JtL-2
J=L-f
J=L+$
i)
and S, * T,(R&,
k), L) between
121
A.C. Merchant / a-cluster states
momentum of $ or 3 is formed by combining the orbital angular momentum of 1 associated with a p-shell hole with the nucleon spin of s = 4. The same combinations of radial factors occur in these matrix elements between states based on cores with different spins as for those based on cores having equal spins, so that they again involve radial integrals of V,(R) and V,(R) with xf (R) and xL@). The angular parts of the integrals are given in table 2. TABLE
Reduced
matrix
2
elements
of tensor
forces JCL-~
J=L+f
RME
((L# JIP% R,II(L$l J) ((L#JII& . T,II(L%J)
-$JL(2L+3)
-$J(L+1)(2L-1)
Wf)JIP, . R,Il(L- ‘%J) ((Lt) JIIB, . 7.&L- 2%J)
Uf)JlP,
. R,lb+2%J)
((Lt)JII& . &(lW+2$)J)
$J3(L-1)(2L-1) $43(L+2)(2L+3)
The angular part of the reduced matrix elements connecting states different spins of S = f or $. B, is a nucleon operator of rank-k.
based
on 15N cores
having
Table 3 shows the values of the radial integrals obtained using the wave functions of relative motion found by solving the Schrodinger equation with the potential of (2.1) subject to the restriction 2N+L = 8. It confirms the deduction made from the form factors displayed in fig. 2 that the spin-orbit and third-rank tensor forces are very small and produce hardly any energy splitting from the rotational centroids. The second-rank tensor force, on the other hand, is much bigger, and capable of producing a sizeable splitting of the quadruplets of a-cluster states based on a spin-$ “N core. The diagonal radial integrals of V,(R) show a similar increase with L and are about half the magnitude of those calculated for the 14N-~ system 4), while the off -diagonal radial integrals are again of comparable magnitude to the diagonal ones. The results given in tables l-3 can now be combined with the centroid energies to find the spectrum of negative-parity a-cluster states in 19F formed by coupling an a-particle to a “N core in its ground or first excited negative-parity state. The bandhead of the higher-lying band is taken to be at 4.7 MeV [refs. 17*18)].It is then necessary, in general, to diagonalize a 3 X 3 matrix, since coupling L (= 0, 2, 4, 6, 8) with S = $ produces all the possible half-odd-integral angular momenta from 4 to y once each, whilst coupling L with S = s produces $, y and y once each and all the possible half-odd-integral angular momenta from 9 to 9 twice each. The
122
A.C. Merchant / a-cluster states TABLE 3 Radial integrals of form factors L=O
L=2
L=4
L=6
L=8
dRlx,W)lzV,(R; S=t)
-0.0120
-0.0111
-0.0092
-0.0066
-0.0040
dR(xrW)l*V,W;
-0.0101
-0.0104
-0.0111
-0.0124
-0.0147
dRIx,(R)i*V,(R)
-0.9368
-0.9746
-1.0648
-1.2219
-1.4825
dRx2CR) V,(R)xr+M)
-0.9311
-0.8857
-0.8083
-0.6769
Integral m
I
0 co
I0 co
S=$
I0 W I0 oz I0
dRIxtWizV,W)
0.0184
0.0192
0.0210
0.0241
dRxEWV,(R)x,+,W
0.0183
0.0174
0.0159
0.0133
0.0292
Radial integrals of the form factors V,(R; S=&, V,(R; S =$), V,(R) and V,(R) with radial wave functions obtained by solving the Schriidinger equation with the potential of (2.1) subject to the restriction 2N+L=8.
TABLE
4
Cluster states based on a “N ground-state core Excitation energy (MeV) .I”
I-
5 3-
T
s:-
2
9-
&%+l-l-
-T
theory
experiment
0.14 1.32 1.34 4.19 4.25 8.73, 9.53 9.14 16.63 16.60
0.11 1.46 1.35 4.00 4.03 8.95, 9.87 8.29 (12.67) (12.26)
The theoretical spectrum of a-cluster states in i9F based on a “N core, predominantly in its ground state, compared with experimental states which have been found to be heavily populated in a-transfer reactions 19-22). The y- states are predicted to be strongly mixed.
123
A.C. Merchanr / a-cluster states TABLE 5 Cluster states based on a 15N core in an excited state Excitation energy (MeV) J”
theory experiment 15N+a
l-
5 ;f5-
I I2
Z2 7-
5 9-
5 9-
1:;:3T sAT 1% -T u+-
‘60+3H
7.02 4.56 6.16 5.22 9.76 6.17 8.65 8.17 14.70 8.73, 9.53 13.35 13.04 22.34 14.01 20.66 20.42 21.59
7.47 6.34 9.98 7.42 13.92 10.13
18.96 14.77
6.99, 7.26 4.56 6.09 5.62 9.82 6.16 8.63, 6.93 (7.93) 8.95, 9.87
14.12
The theoretical spectrum of a-cluster states in 19F based on a 15N* core, predominantly in its first negative-parity excited state, compared with suggested experimental counterparts 6S23).The predicted energies of some neighbouring I60 + 3H states are also shown ‘).
energies obtained from this diagonalization are shown in tables 4 and 5, and those levels below 10 MeV excitation in 19F are displayed in fig. 3. 5. Discussion The calculations of the energies and wave functions for the two lowest-lying bands of negative-parity a-cluster states in ‘9 show that there is, in general, very little mixing between states based on a ground-state 15N core and an excited-state “N* core. The wave functions of the states based on the ground-state core are usually found to contain less than 5% admixtures of other configurations. Even so, the energy shifts in the ground-state core band due to these small mixtures (caused essentially by the second-rank tensor force) are greater in magnitude than those due to the spin-orbit force. This is clearly shown by the fact that the calculation does not place the J = L + f member of a doublet of the ground-state core band below the corresponding J = L-i member. Instead, one finds a series of nearly degenerate doublets with no regular pattern noticeable in the small splittings which
A.C. Merchant / a-cluster states
124
19
F
THEORY
L E 5 8
P 5 ;; :
+
+
--273
f32-r 5T
I 2
57-z_T 2
O
EXPERIMENT
I1
3T
5 7-977-===--
3
J
Fig. 3. The calculated spectrum of a-cluster states lying below 10 MeV excitation energy in “F, based on a 15N core predominantly in either its ground (S = f) or first negative-parity excited state (S = s), is compared with some experimental levels. Those experimental levels in the column labelled S = $ are heavily populated in a-transfer reactions lgez2), and those in the column labelled S = $ are seen as resonances in the “N(a, y)r9F reaction 23) and/or in the break-up of “F into “N and a-components “).
do occur. Conversely, the members of the excited core band form quadruplets with large energy splittings, more or less conforming to the pattern predicted by the first row of table 1, with some small modifications due to tensor force mixing of states having equal .I, but values of L differing by two units. However, there is one spectacular exception to these generalizations, involving the two ‘jl?-states formed
A.C. Merchant / a-cluster states
125
by coupling the a-particle to a ground-state core with L = 6 (0 S = 1) and to an excited core with L = 4 (OS =f). These states are nearly degenerate if mixing between bands is ignored, so that when the mixing. is “switched on” the final states are so thoroughly mixed that they essentially contain 50% of each cluster-core configuration. For this reason the two y- states are included in both tables 4 and 5, and their appearance in the various columns of fig. 3 is rather arbitrary and mainly for ease of presentation. The experimental implications of this lack of mixing are that the members of the ground-state core band (including both the strongly mixed y- states) should be heavily populated in a-transfer reactions onto 15N targets, whereas the members of the band base on an excited core are much less likely to be heavily populated by this method. The best chance of observing them seems to be through inelastic excitation of 19F followed by break-up into “N-a components or possibly as resonances in the 15N(a, r)‘v reaction. Enough a-transfer reactions onto 15N targets have now been performed for most of the negative-parity a-cluster states in ‘v, based on a ground-state core, to be confidently identified. A summary of the candidates is given in table 4. The earliest of these a-transfer experiments 19) was performed with a 20 MeV ‘Li beam. However, at this energy compound nucleus effects may have been appreciable. Subsequent investigations using a 105 MeV 13Cbeam 2oY21), a 115 MeV ’ 'B beam ‘l) and a 40 MeV ‘Li beam 22) have led to the sure assignment, on the basis of their strong, selective excitation, of the following states to the cr-cluster spectrum, $-(O.llO), s-(1.346), z-(1.459), s-(3.999), z-(4.033) and y-(8.288). The bracketed numbers following the spin-parity values are the excitation energies of the states in MeV. In addition the q-(8.953) state is quite heavily populated and appears to carry some of the cluster strength, although it has also been seen strongly in 3H transfer onto 160 [ref. ‘*)I. Two other strongly excited states at 12.26 MeV and 12.56 MeV were proposed by Pilt ef al. *‘) as the remaining y- and y- members of the band. The agreement in energy between the experimentally located states and the calculation presented in the last section is fairly good. The rather erratic splittings of the closely spaced doublets can not be reproduced in detail, but since these are of the order of 100 keV or less, that is hardly surprising, and the important point is to note that a microscopic explanation of their smallness and their failure to follow a classic spin-orbit pattern has been given. A large mixing of the y- states is predicted, although some additional mixing with triton cluster states may also be present and complicating the picture. The rotational parameter of the band seems to be a little overestimated with respect to its seven lowest-lying members and the final ye-$? doublet is predicted to lie much higher than the experimental candidates (around 16 MeV instead of 12 MeV). This is analogous to the situation in “Ne [ref. ‘)I, where a similar overestimate of the energy of the 8+ member of the ground-state band is made. However, a little caution should still be shown concerning the last two assignments in 19F because the spin-parity values of the excited states seen at 12.26 MeV and 12.67 MeV have not yet been conclusively measured.
126
A.C. Merchant / a-cluster
states
Identification of the members of the band based on an excited “N* core is, at present, rather more speculative. However, the energies and spin-parity values of many states in ‘? below 10 MeV have been identified 23) and merely comparing these quantities with the results of the calculation of the last section shows a remarkable correspondence. All the cluster states predicted to lie below 10 MeV are found to have a possible experimental counterpart and these are listed in table 5 and displayed in fig. 3. The i-(4.56) is fitted by choosing the head of the “N*-cy band to lie at 4.7 MeV 17,18) (mixing with the other two $- cluster states then depresses this slightly) and all other members of the band are calculated without any further free parameters. The predicted s-(6.16), g-(6.17), i-(7.02), z-(8.65) and g-(9.76) all lie within 0.07 MeV of possible experimental counterparts having well-identified spin-parity values 23), while there is a candidate for the predicted G-(5.22) at 5.62 MeV which also has J” = $-. The two remaining states predicted to lie below 10 MeV are the q-(8.17) and q-(9.53). There is an experimentally known (p, 5) state at 7.93 MeV which may be the z- and states at 9.71, 9.83 and 9.87 MeV, one of which may be the $-. This last assignment is complicated by the possibility of mixing with a triton cluster state, predicted to lie at 10.13 MeV [ref. ‘)I. In addition to this remarkable correspondence in energy it should be noted that the experimental candidates g-(5.62), s-(6.09), g-(6.16), (I’, $(7.93), g-(8.63), g-(9.82) and possible y-(9.83,9.87) are all seen as resonances in the 15N((u,y)19F reaction23). The y-(8.29) and y-(8.95) of the 15N-(u band are also seen in this reaction. Preliminary data from the inelastic excitation of 19F followed by break-up into 15N-a components from Oxford and Daresbury “) show several states of 19F between 5.4 MeV and 11.7 MeV which are presumed to have a predominantly 15N-~ structure. Among these are prominent peaks at 5.62, 6.12, 6.92, 8.01, 8.60 MeV which are in the right energy region to correspond to some of the predicted “N*-cy states. Strong excitations at 8.28 and 8.97 MeV (the energies of the ‘jl?-and y- members of the ground-state core 15N-~ band) are also seen. As yet no spin-parity assignments have been made. The experimental identification of any higher lying states, which might be associated with the remaining two quadruplets of “N*-cy states, is currently out of the question, although several states with a 15N-a structure above 12 MeV have been seen 21,22), One further difficulty in trying to compare the predicted “N*-cu negative-parity states with experiment is that the calculation presented here is not able to include mixing with 160-3H (triton) cluster states. In actual fact, a band of triton cluster states having 2N + L = 7 (and hence negative parity) is predicted to fall in the same energy region as the band of “N*-a cluster states and the predicted energies of its eight members are listed in table 5, together with a few suggested experimental counterparts 3). The overlap between the alpha and triton cluster states is not expected to be very large (analogies with the SU(3) model suggest typical values considerably less than 30%). It has also been shown above that mixing between states with the more closely similar structure of “N-(u and “N*-a is usually very
A.C. Merchant / wcluster states
127
small. However, since a “N*-a g- state is predicted to lie at 6.16 MeV compared with a 160-3H $- state at 6.34 MeV and a “N*-CY I- state at 9.76 MeV compared with a 160-3H state at 9.98 MeV, mixing of alpha and triton cluster structure in these cases may be significant. A coupled-channel orthogonality condition model calculation suggests that this is indeed the case r7,18), but in view of the apparently excellent agreement between the predicted “N*-cu spectrum and some highly plausible experimental candidates, the effects of such mixing on the energies of the states in question do not seem, in general, to be very great. It is hoped that this question of the purity of the a-cluster structure of the proposed members of the “N*-a band will be clarified by the forthcoming results of some new experimental investigations 6). The author would like to thank Dr. B.R. Fulton and Dr. W.D.M. Rae for permission to refer to their data prior to publication, and Drs M.A. Nagarajan and N. Rowley for useful discussions. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23)
B. Buck, C.B. Dover and J.P. Vary, Phys. Rev. Cl1 (1975) 1803 B. Buck, H. Friedrich and A.A. Pilt, Nucl. Phys. A290 (1977) 20.5 B. Buck and A.A. Pilt, Nucl. Phys. A280 (1977) 133 B. Buck, A.C. Merchant and N. Rowley, Nucl. Phys. A327 (1979) 29 N. Rowley and A.C. Merchant, Phys. Lett. 97B (1980) 341 B.R. Fulton and W.D.M. Rae, private communication D.F. Jackson and M. Rhoades-Brown, Nucl. Phys. A286 (1977) 354 G.R. Satchler, Nucl. Phys. 21 (1960) 116 D. Mukhopadhyay and G. Grawert, Nucl. Phys. A385 (1982) 133 J. Cook and R.J. Philpott, Nucl. Phys. A385 (1982) 157 R.C. Johnson, Invited talk at the RCNP-Kikuchi Summer School, Kyoto, Japan (1983), to be published D.M. Brink and G.R. Satchler, in Angular momentum, 2nd ed. (Clarendon, Oxford, 1968) p. 148 et seq. T.W. Donnelly and G.E. Walker, Phys. Rev. Lett. 22 (1969) 1121 C.J. Batty, E. Friedman and D.F. Jackson, Nucl. Phys. Al75 (1971) 1 C.J. Batty and E. Friedman, Phys. Lett. 34B (1971) 7 L.C. Biedenham, J.M. Blatt and M.E. Rose, Rev. Mod. Phys. 24 (1952) 249 T. Sakuda and F. Nemoto, Prog. Theor. Phys. 62 (1979) 1274, 1606 H. Furutani, H. Kanada, T. Kaneko, S. Nagata, H. Nishioka, S. Okabe, S. Saito, T. Sakuda and M. Seya, Prog. Theor. Phys. Suppl. 68 (1980) 193 R. Middleton, in Proc. Int. Conf. on nuclear reactions induced by heavy ions, Heidelberg, 1969, ed. R. Bock and W.R. Hering (North-Holland, Amsterdam, 1970) p. 263 A.A. Pilt, D.J. Millener, H. Bradlow, 0. Dietzsch, P.S. Fisher, W.J. Naude, W.D.M. Rae and D. Sinclair, Nucl. Phys. A273 (1976) 189 H.S. Bradlow, W.D.M. Rae, P.S. Fisher, N.S. Godwin, G. Proudfoot and D. Sinclair, Nucl. Phys. A314 (1979) 207 L.M. Martz, S.J. Sanders, P.D. Parker and C.B. Dover, Phys. Rev. C20 (1979) 1340 F. Ajzenberg-Selove, Nucl. Phys. A392 (1983) 1
A SEMI-MICROSCOPIC STUDY OF NEGATIVE-PARITY a-CLUSTER STATES IN ‘9 A.C. MERCHANT SERC Daresbury Laboratory, Warrington WA4 4AD, UK Received 23 September 1983 Abstract: The bands of negative-parity a-cluster states in 19F which may be formed by coupling an a-particle to a “N core in its ground (f) or lowest negative-parity excited @) state are studied. The central part of the effective cluster-core potential is obtained from a simple folding procedure, and the non-central terms are microscopically derived from an aN scattering potential. The spectrum of o-cluster states is calculated, taking into account mixing of states in the two bands by the tensor forces, and found to give generally good agreement with the experimentally determined spectrum up to excitation energies of 10 MeV, although mixing with 3H-chrster states may sometimes be important.
1. Introduction Alpha-particle transfer reactions induced by heavy ions are known to strongly and selectively excite a small subset of states of the final nucleus. These states appear to have a cluster-core structure, in that their spectra display rotational energy spacings, they are connected by strong electromagnetic E2 transitions and those above the a-threshold have large n-decay widths. Buck, Dover and Vary ‘) have introduced a simple cluster model to account for the properties of these states. They obtained a local, effective cluster-core potential by folding the cluster and core densities with an effective NN interaction, and were able to give a good description of the spectra, a-decay widths and electromagnetic transitions between the a-cluster states in I60 and 20Ne. Their model was subsequently applied with equal success to the study of a-cluster states in 180, 18F [ref. 2)] and 19F [ref. 3)]. In the latter two cases the 14Nand “N cores carry spins of 1 and i respectively, so that non-central forces may be present in the cluster-core interactions. Initially, the effects of these forces were parametrized phenomenologically, but later they were calculated microscopically4), so that the model as applied to a-cluster states in nuclei in the mass range A = 16-20 is now well established. All the calculations mentioned above have considered the cluster and core to be in their ground states. Since the free a-particle has its first excited state at an energy in excess of 20 MeV, this seems to be a good approximation for the cluster. However, the possibility of the core being in an excited state needs to be borne in mind. A study of the 14N-a system in which the 14N core was allowed to occupy its ground and first two excited states “) showed that the matrix elements connecting a-cluster 109
110
A.C. Merchant / a-cluster
states
states in r8F with equal total angular momentum, J, but different core states, were quite large, but that these states were generally so far apart in energy that mixing between them was not very great. In one particular case, however, two states with J” = 3+ (formed by coupling the a-cluster to a ground state 14N core with relative orbital angular momentum L = 4, and to a i4N core in its first excited (l+) state with L = 2) were found to be nearly degenerate, and hence were strongly mixed. The reason that, up to now, interest has centred on the case of cluster and core in their ground states is that a-transfer experiments are only likely to populate heavily those cluster states which contain a large ground-state core component, whereas to populate states based on an excited core, a two-step process of core excitation and a-transfer is necessary. An alternative method of observing ru-cluster states is to inelastically excite the final nucleus and watch it decay into its cluster and core components. This is currently being done at Daresbury and Oxford “f by inelastically scattering a 19F beam from a “C foil and simultaneously observing the a-particle and 15N fragment emitted in the subsequent break-up. The angles of emission of the two components are measured and the excitation energy of the 15N fragment plus the relative orbital angular momentum of the a-particle and i5N can be determined. Thus, candidates for a-cluster structure in 19F lying above the a-particle emission threshold, based on an excited “N core, together with their spin-parity values, can be located. It is therefore the purpose of this paper to predict the spectra of the two lowest-lying bands of negative-parity a-cluster states in 19F formed by coupling an a-particle to a i5N core in its ground (f) or first excited negative-parity (r) state, within the framework of the Buck, Dover and Vary folded potential cluster model, in anticipation of these new experimental data. 2. The cluster-core potential The central part of the cluster-core potential for the i5N-a! system has already been given by Buck and Pilt 3), and in view of its success in calculations of the properties of a-cluster states based on a ground-state “N core the same form will be used here. It is written as V(R) = V,(R) f V,,,,(R)
.
(2.1)
V,,,,(R) is the Coulomb potential appropriate to a uniform spherical charge distribution of radius 3.1 fm. V,(R)is the local potential obtained by folding the cluster and core densities with a b-function form of the effective nucleon-nucleon interaction, and was parametrized as
Vo(1+cosh (Rolao)) cash (R/a,)+cosh
(&/a,)
where V,= -172 MeV, R. = 2.25 fm and a0 = 0.8 fm.
’
(2.2)
A.C. Merchant / a-cluster states
111
cluster is treated as a single particle in orbit around the core, characterized by the principal quantum number N (the number of nodes in the radial wave function) and L (the orbital angular momentum).,The major requirements of the Pauli exclusion principle are satisfied by restricting the cluster nucleon quantum numbers to values appropriate to shell-model orbitals outside the core (sd or higher shells in the present case) so that The
2N+La8.
(2.3)
With this restriction, the single-particle Schriidinger equation can be solved with the potential of (2.1) to give energies and wave functions for the relative motion of cluster and core. For higher-lying quasi-bound states, Jackson’s procedure ‘) of finding the position, RB, and height, E u, of the potential barrier and setting V(R) = EB for R 2 RB is followed. All levels are thus treated as bound states, with minimal errors in their energies. When this is done, a series of almost perfect rotationally spaced levels is produced, whose energies may be written E(L) = C+AL(L+l)
(2.4)
with C the bandhead and A the rotational parameter, approximately 0.2 MeV for the 15N-(u system. Since the core carries a spin of i or $ there can be non-central terms in the cluster-core potential which may split the rotationally spaced centroids of (2.4). In general, these terms may be written as scalar products of rank-K spherical tensor operators formed from the spin of the core and equal-rank tensor operators formed from the variables of relative motion of cluster about core (namely, R, the relative position, P, the relative momentum and L, the relative orbital angular momentum). If the interaction is required to be invariant with respect to space exchange and time reversal, the only possible vector-vector contraction is the normal spin-orbit force L * S. When the core spin is 1, this is in fact the only non-central force possible between cluster and core. However, when the core spin is 1, scalar products of tensors of ranks 2 and 3 are also possible. The rank-2 contractions may be readily derived from the relation “) S,~R2(a,b)=(S*a)(S~b)-~iS+z~b)-$*(a~b),
(2.5)
where S2 is the rank-2 tensor formed from the core spin, SF =c (l/_&t4-~~2M)S~S”-~, Ir
(2.6)
and R2(a, b) is the rank-2 tensor formed from the vectors u and b, R2M(u,b)=C(l~~lM-~~12M)a~‘b~-~. I”
(2.7)
The vectors R, P and L may not be mixed in (2.7) if the interaction is to remain invariant under space inversion and time reversal, so the only possible contractions
112
A.C. Merchant / a-cluster states
are
s, *R,(d, Ii)
=[(S*
ii)*-fs*1,
(2.8)
s**R,(P,P)=[(S* P)*-~s*P*]) Sz * R,(L,L) =[(S*
(2.9)
L)*+$(S- L)-fS2~2],
(2.10)
where the circumflex accent denotes a unit vector, and all the contractions may be multiplied by an arbitrary radial factor, F(R). The possible scalar products of rank-3 tensor operators which may be present in the cluster-core potential are a little more complicated, but have recently been studied in connection with ‘Li scattering ‘-ll). The terms which are invariant under parity- and time-reversal operations consist of the contraction of a rank-3 tensor operator formed from the core spin, S,(S,, S), and a rank-3 tensor operator, formed from L and any of the rank-2 tensors, &(a, a), occurring in (2.8)-(2.10), and written ?‘,(&(a, a), L). The scalar product S3 - T3 may be simplified by using the relation ‘*) Z-“(k,, k,).
7”(k3, kq)=(-l)kl+k4(2K+1)
k3) * T&k,,
x C W(k~k~k~k~; KK’)T’~(k,, K’
kd,
(2.11)
which allows the scalar product of tensor operators of rank-K, formed from ranks k,, k2 and k3, k4, to be written as a linear combination of scalar products of tensor operators of all ranks K’ (with the structures indicated) allowed by triangulation, provided the tensor operators of ranks k2 and k3 commute. Therefore, (s,
’ R,)(S*
(s*
h)(s2
L)
=
* R2)
=
Kil
(-1)K’TK’(S2,
i
s)
(-l)K’TK’(s,
s2)
’ TK@,,
* TK4I.3
L,
,
(2.12) (2.13)
R2),
K'=l
where TK,(Sz, S) = T’$(S, S,) when K’ = 1 or 3 but T,(S,, S) # T2(S, S,) and similarly for TKp(R2, L). In fact, T,(S,, S) = Tl(S, S,) =(-&s*
s+@s,
%G(S,, S) + 7-0, S,)l= -GSz
(2.14) (2.15)
*
The tensors TK,(R;?(L, A), L) have an analogous form to the TK(S2, S) whereas TK*(Rz(k, k), L) and TKc(R2(P, P), L) obey the relations T,(R*(%
0-2(&b,
The results of (2.12)-(2.17)
a),
L)
=
T,W,
&(a,
a))
a),
L)
+
T2(&
R,(a,
a))3
=&am
=
--&Ma,
a)&
(2.16)
7
a)
.
(2.17)
show that there may be three scalar products of rank-3
113
A.C. MerchantI a-cluster states
tensor operators in the cluster-core
potential, and they may be written as
k>,L) =iW US, - R,& &+(Sz *R,U@, *MS*VI
S,(S,, S) - MMk
+;S,
%(SZ, S)
*~&W-‘, PI, U =iW
*R@, I@+[#.
S)-&](S- L) ,
(2.18)
US, - &VT Z-7)+(& - &VT P)W* L)l
+$S, - R,(P, P)+[&(S
S)-&,](P.
P)(S* L) ,
(2.19) S,(S,, S)
*T,(R,(L 0, U =&S* WS,(R,W, Q)+(S, *ML, L)W- 01 +$S2. R,(L, L)-&(S. +$s.
S)+(L*
A microscopic derivation of these non-central considered in the next section. 3. Microscopic calculation of the non-central
S)(L’ L)(S. L)
L)--:](s*
L).
(2.20)
forces in the “N-CY system will be
terms in the duster-core
potential
In this section the non-central terms present in the potential between a 15N core in its ground or first excited negative-parity state and an a-particle will be derived, using the microscopic model introduced in ref. “). The hamiltonian, H, for a 19nucleon system interacting through two-body forces may be written &P:+C i=r 2WI
v.. i
(3.1)
"
where m is the nucleon mass, pi is the momentum of the ith nucleon and Vij is the interaction between nucleons i and j. To describe states with the structure of an a-particle coupled to a 15N core this may be approximately decomposed as H=H,+H,+H,,,
,
(3.2)
where H, and Ha are the internal hamiltonians of the core and cluster respectively and Hreldescribes their relative motion (depending only on their relative coordinates). The eigenfunctions of this hamiltonian may be written as a sum of product states: YJM
=
da
JZ, (SM’LM-M’IJM)~,(15N)~~~-~,(R)y
(3.3)
where &&15N) is a fully antisymmetrized 15N wave function (and r5N has an intrinsic angular momentum of S), +(a) denotes the ground state of the a-particle, and $LM_M,(R) is a wave function for their relative motion. The microscopic model now seeks to build up an effective cluster-core potential, V,, in Hrel, by summing an (YN potential over all 15 core nucleons.
114
A.C. Merchant / a-cluster states
The 15N core in its ground or first excited negative-parity state is treated as a pi/2 or p3/2 hole in the 160 closed shell, IO),so the cluster-core potential can actually be derived by considering the interaction of the a-particle with this hole. Applying the symmetry arguments invoked in the last section to the possible form of the CXN potential, its most general form is seen to contain only central and spin-orbit terms: u,, = UJ(t) + G(r)L
(3.4)
- s,
where r is the separation of the nucleon and the centre of mass of the a-particle, Ire1is their relative orbital angular momentum and s is the spin of the nucleon. Writing out Ire, explicitly gives Ire,. s=$[I.
s-R*p.
s]+$L.
S-XhP.
S],
(3.5)
where M = m + m, is the sum of nucleon and alpha masses, R, P and L are the position, momentum and angular momentum of the a-particle with respect to a given origin (to be identified as the centre of the core) and x, p and 1 are analogous quantities for the nucleon. Since this expression only contains L and P to first order, the scalar products (2.9), (2.10), (2.19) and (2.20) will not be present in the cluster-core potential derived from this model. Tensor decompositions of the radial functions U,,(r) and ULS(r) may be made as follows: U,(r) = IJ,(lx--Rl)
=z (2y+I)(IJ,(x,
R))&&)
*
G@>,
(3.6)
where the circumflex accents again denote unit vectors, (3.7) with Yg($
the spherical harmonics, and (U&,R))z=S
I’
U&-
RI)Pz(cos 4) d(cos 4) ,
(3.8)
-1
where cos (8) = f * 8. Eq. (3.6) immediately gives the central potential, U,(r), as a sum of scalar products of rank-K spherical tensor operators, AK (say) in the cY-coordinates and BK (say) in the nucleon coordinates. It is possible to write the spin-orbit potential, U,,(r) lrel * s in the same way, so as to produce a potential between the ith nucleon and the a-particle of u’,iL =&jL(XitR)AK(a). This interaction will give a contribution
({#s(“N)$L(R))J
&(xt).
(3.9)
to the energy of
Ii!l@,,I
Md5NMR)h0
(3.10)
A.C.
and integration
Merchant
/ a-cluster
over the nucleon coordinates
(k(R)(SL)J
115
states
yields
; AK(~) - NK
(3.11)
with NKQ=(4s(15N)SM+Q
f ,fK(Xi,R)Bz(Xi)
+s("N)SM).
(3.12)
i=l
The rank-K tensor operator, NK, may be expressed in terms of the rank-K tensor operator formed from the core spin, SK, as @
=
(4d15N)ll C;:,f~(xi,R)B~(xi)llh(‘~N)) s,“,
ww)
(3.13)
so that the interaction has been written in terms of induced operators involving the core spin, and the cluster-core potential may be written v,,=C&(R)&(~). K
(3.14)
SK.
To calculate the form factors FK(R), the reduced matrix elements of the nucleon operators &(&) must be evaluated. In general, &(&) is formed by combining a nucleon spatial tensor operator of rank kI, C,, (say), with a nucleon spin tensor operator of rank kZ, Dkz (say). Evaluation of the reduced matrix elements of BK (xi) between wave functions of 15N gives (15N; SllBKll’5N S>= ((W’Il&lI(S)-‘)
(3.15)
with (S)-’ a hole in the 160 closed shell. Using the same arguments concerning particle-hole conjugation and the evenness of K + k, + k2 as in ref. 4), one finds (‘5N~lB~ll’5N
=
(OIIBKIIWK,O
(~~lC,,lll>(sllD,,lls),
(3.16)
where 1= 1 for a p-shell-hole description of the core, s = 4 and s^= (2s + l)r” etc. The symmetry properties of the 9-i symbol in (3.16) place the following restrictions on the combinations of (Kklk2): _ scalar:
(0,070)
(0,17 1)
vector:
(1,&l)
(L2,l)
rank-2 tensor:
(2,290)
(2,171)
rank-3 tensor:
(3,271)
(1,LO)
116
A.C. Merchant j a-cluster states
The terms in the tensor decompositions of U,(r) and ULs( r) irei 1 s corresponding to the above combinations can now be found, and the forces in the cluster-core potential evaluated in the form V,,(S=$)=V,(R)+V,(R;S=$)LS,
(3.17)
V,,(S=;)=V,(R)+V,(R;S=~)LS+V,(R)S,~R,(~,~) + V,(R)% - NW%
&I,
L) .
(3.18)
As explained in sect. 2 and discussed in ref. “) the cash potential (2.2) will be used for V,,(R). The spin-orbit and rank-2 tensor form. factors, V,(R; S=$), V,(R; S =$) and V,(R), may be obtained immediately from ref. 4), where an effective 14N-~ potential was derived treating the r4N core as two p-shell holes in the 160 closed shell (with their spins coupled to a value of 1). There, the two-hole matrix elements analogous to (3.15) were expressed in terms of single-hole matrix elements and some factors depending on the amplitudes for finding both holes in p1,,2states, ~312 states, or one in each state, (C,, C, and C,, res~ctively). By setting = 0 for S = 1 the 14N-a form factors C,=1,C2=C3=OforS=$andC,=1,CI=C2 “N-a! form factors which are required here. can be simply related to the equivalent Combining eq. (4.23) of ref. “) with (3.16) above yields Vl(14N-a; Cl = 1)
for S=$:
(3.19)
VI(‘5N_a; S ~1) = 1 ’
and for S = 3:
V,(‘4N-cx; C, = 1)
Vr(15N_(U;S+)
=’ ’
V2(‘4N-~; C, = 1)
V,(‘5N_(r)
=-+*
(3.20)
(3.21)
Defining the integrals of the p-shell radial wave functions, 8 (x), with the components (Ua(x, R)), of (3.8) as
and, since similar integrals multiplied by (x/R) defining
or (R/x)
and using eqs. (4.36) of ref. “) and (3.19)-(3.21)
above allows the spin-orbit and
also occur frequently,
117
A.C. Merchant / a-cluster states
rank-2 tensor form factors for the 15N-~ system to be written as
V,(R; s=t> =~[~(FLs(R))o-~(FIS(R))L+~(FLS(R))~I
(3.25)
9
V,(R) = (F,(R)),+~[(F,s(R)),-(Ft(R)),l .
(3.26)
It only remains to evaluate V,(R) to complete the derivation of the cluster-core potential of (3.18). There are two contributions to this form factor which arise as follows: (i) When 8= 2 in the tensor decomposition of the term (m/M) U&r)L * s one obtains the operator 543;
(&(x,
R))zTJG(i),
s) - GUMk
h,
U
which, when inserted in (3.16) and using (3.13), gives a contribution
to
V,(R)
of
-(WM(&(R)),.
(ii) When LZ= 3 in the tensor decomposition of the term -(m/M) one obtains, after some recoupling, the operator -dE
;
which gives a contribution
(ULS(X,
R))3T3(C2(-5),
s>
to V,(R) of (m/M)(F,,
V,(R) =~[(G
UUL-VAR)M
* 7-3(R,(k,
d’),
U,,(r)x
AP- s
L)
(R))3. Therefore,
.
(3.27)
It should be noted that the spin-orbit and rank-3 tensor form factors all contain the mass factor (m/M) = 0.2 and are derived entirely from the spin-orbit part of the cuN potential, whereas the rank-2 tensor form factor contains a (much larger) contribution from the central part of the potential.
4. Calculation
of the cluster state spectrum
The form factors (3.24)-(3.27) may be evaluated analytically, as in ref. 4), by the following choices of p-shell radial wave function and cuN potential. A harmonicoscillator radial wave function of the form -x2/26=
(4.1)
118
A.C. Merchant / a-cluster states
is used for the hole, where b = 1.77 fm is consistent with the measured charge radius of 160 [ref. 13)]. A gaussian shape for the (YN potential is used 4T14P15), U,, = ( U, + U&i
f s) e-r2’n2
(4.2)
(where U,, = -43.0 MeV, U,, = -7.0 MeV and l/a = 0.526 fm-i), which enables the tensor decomposition (3.6) and the radial integrals (3.22) and (3.23) to be evaluated in closed form. The precise value of ULs is not too important, since it has only a small effect on the cluster state spectrum through its linear appearance in V,(R; S =&, V,(R; S =s) and V,(R), and a very minor influence on V,(R). The form factors, as calculated with this prescription, are shown in figs. 1 and 2. The main point to note is that the second-rank tensor form factor, V,(R), is much more important than any of the others. Its analytic form is proportional to R2 exp (-R’/(a*+ b’)) and it has a minimum of about -2 MeV around R = 2.5 fm. This is more than fifty times greater in magnitude than the maximum of the similarly shaped third-rank tensor form factor V-,(R). This difference in scale is almost entirely accounted for by the mass factor m/M = f and the ratio of (l&J U,) = &. The spin-orbit form factors are non-zero at the origin, but at cluster-core separations in excess of around 2 fm they are smaller in magnitude than V,(R). It is therefore already apparent that the a-cluster states based on a spin-4 i5N core will consist of almost degenerate doublets, whilst those based on a spin-g core will form quadruplets with energy splittings characteristic of the tensor force (Se s)‘-fS2.
-2
Fig. 1. The second-rank tensor form factor, V,(R), for the l’N*-a separation.
system as a function of the cluster-core
119
A.C. Merchant / a-cluster states
Fig. 2. The spin-orbit and third-rank the “N-a and “N*-a
tensor form factors, V,(R; S=$), V,(R; S=$) and systems as functions of the cluster-core separation.
V,(R), for
The precise form of the splitting of the various doublets and quadruplets from the nearly rotational centroids of (2.4) can be found by evaluating matrix elements of the form factors (3.24)-(3.27), and their associated operators, between states of total angular momentum, J, obtained by coupling the core spin, S, with the relative orbital angular momentum of cluster about core, L.For matrix elements connecting states based on the same core state, it is readily found that 10*12*16)
((LS)J((V,(R; S)L*s~~(L’s)J>=s,&W+1)-L(L+1)-s(s+1)1 X
XL*(R) V,(R;
S)XL,@)
dR
(4.3)
120
A.C. Merchant / a-cluster states
with xL(R) the radial wave function for the relative motion of cluster about core. The second- and third-rank tensor forces are only applicable to states based on a S = 1 core and have matrix elements
WWtt V,(R)& * R,Otd.)[[(L’S)J) co
=
I
A,($ L, L’)
KLS)J((V,(R&
o
x:(R) VAR)xtW dR,
- URz(&
&
(4.4)
L)ll(L’S)J)
m
=
A#,
L, L’)
I
0
xT.W)V~R)XLW dR
(4.5)
with A&, L, L’) and A,(& L, L’) listed in table 1 under the heading RME. [Note that the values of A,($ L, I.‘) listed in ref. lo) are up to a factor of 10 too large.] So far, the cluster-core interaction has been written in terms of scalar products of tensor operators in the ru-cluster coordinates with others formed from the core spin, S, which have the same matrix elements as the actual operators involving the nucleon coordinates. This works very well as long as the core spin is a good quantum number. However, the second- and third-rank tensor contractions in the cluster-core potential actually involve quantities like 5( U,(X, R))&&) - C,(k). These have non-zero matrix elements between states of equal total angular momentum, J, formed by coupling different states of the core to L [ref. 5>],since the matrix elements of C#) must themselves be taken between coupled basis states, whose total angular
TABLE
1
Reduced matrix elements of tensor forces RME
J=L+$
((L$)Jb . Rzli(@J>
-2L+3
((mw,
. ~3llIGm RME
W~)JllS,. &ll(L+233J) G$)JllS,
. T&L+- mo
L+3
L-2
-(L+l)
2L+3
2L-1
2L-1
-L
-3L(L-
1)
9(L+2)(L-
1)
-9(L+2)(L-1)
J=L+$
-m 2L+3
2L+3
(L+3)43L(L+2)
l)(L+3)
2(2L+3)
2(2L.+3)
Angular dependence of the reduced matrix elements of S, . R,(& coupled cluster-core basis states having S =$.
10(2L-1)
J=L+$
-J3(L+l)(L+3)
-LJ3(L+
3(L+lf(L+2)
lO(ZL- 1)
lO(2Lf3)
10(2L+3)
JtL-2
J=L-f
J=L+$
i)
and S, * T,(R&,
k), L) between
121
A.C. Merchant / a-cluster states
momentum of $ or 3 is formed by combining the orbital angular momentum of 1 associated with a p-shell hole with the nucleon spin of s = 4. The same combinations of radial factors occur in these matrix elements between states based on cores with different spins as for those based on cores having equal spins, so that they again involve radial integrals of V,(R) and V,(R) with xf (R) and xL@). The angular parts of the integrals are given in table 2. TABLE
Reduced
matrix
2
elements
of tensor
forces JCL-~
J=L+f
RME
((L# JIP% R,II(L$l J) ((L#JII& . T,II(L%J)
-$JL(2L+3)
-$J(L+1)(2L-1)
Wf)JIP, . R,Il(L- ‘%J) ((Lt) JIIB, . 7.&L- 2%J)
Uf)JlP,
. R,lb+2%J)
((Lt)JII& . &(lW+2$)J)
$J3(L-1)(2L-1) $43(L+2)(2L+3)
The angular part of the reduced matrix elements connecting states different spins of S = f or $. B, is a nucleon operator of rank-k.
based
on 15N cores
having
Table 3 shows the values of the radial integrals obtained using the wave functions of relative motion found by solving the Schrodinger equation with the potential of (2.1) subject to the restriction 2N+L = 8. It confirms the deduction made from the form factors displayed in fig. 2 that the spin-orbit and third-rank tensor forces are very small and produce hardly any energy splitting from the rotational centroids. The second-rank tensor force, on the other hand, is much bigger, and capable of producing a sizeable splitting of the quadruplets of a-cluster states based on a spin-$ “N core. The diagonal radial integrals of V,(R) show a similar increase with L and are about half the magnitude of those calculated for the 14N-~ system 4), while the off -diagonal radial integrals are again of comparable magnitude to the diagonal ones. The results given in tables l-3 can now be combined with the centroid energies to find the spectrum of negative-parity a-cluster states in 19F formed by coupling an a-particle to a “N core in its ground or first excited negative-parity state. The bandhead of the higher-lying band is taken to be at 4.7 MeV [refs. 17*18)].It is then necessary, in general, to diagonalize a 3 X 3 matrix, since coupling L (= 0, 2, 4, 6, 8) with S = $ produces all the possible half-odd-integral angular momenta from 4 to y once each, whilst coupling L with S = s produces $, y and y once each and all the possible half-odd-integral angular momenta from 9 to 9 twice each. The
122
A.C. Merchant / a-cluster states TABLE 3 Radial integrals of form factors L=O
L=2
L=4
L=6
L=8
dRlx,W)lzV,(R; S=t)
-0.0120
-0.0111
-0.0092
-0.0066
-0.0040
dR(xrW)l*V,W;
-0.0101
-0.0104
-0.0111
-0.0124
-0.0147
dRIx,(R)i*V,(R)
-0.9368
-0.9746
-1.0648
-1.2219
-1.4825
dRx2CR) V,(R)xr+M)
-0.9311
-0.8857
-0.8083
-0.6769
Integral m
I
0 co
I0 co
S=$
I0 W I0 oz I0
dRIxtWizV,W)
0.0184
0.0192
0.0210
0.0241
dRxEWV,(R)x,+,W
0.0183
0.0174
0.0159
0.0133
0.0292
Radial integrals of the form factors V,(R; S=&, V,(R; S =$), V,(R) and V,(R) with radial wave functions obtained by solving the Schriidinger equation with the potential of (2.1) subject to the restriction 2N+L=8.
TABLE
4
Cluster states based on a “N ground-state core Excitation energy (MeV) .I”
I-
5 3-
T
s:-
2
9-
&%+l-l-
-T
theory
experiment
0.14 1.32 1.34 4.19 4.25 8.73, 9.53 9.14 16.63 16.60
0.11 1.46 1.35 4.00 4.03 8.95, 9.87 8.29 (12.67) (12.26)
The theoretical spectrum of a-cluster states in i9F based on a “N core, predominantly in its ground state, compared with experimental states which have been found to be heavily populated in a-transfer reactions 19-22). The y- states are predicted to be strongly mixed.
123
A.C. Merchanr / a-cluster states TABLE 5 Cluster states based on a 15N core in an excited state Excitation energy (MeV) J”
theory experiment 15N+a
l-
5 ;f5-
I I2
Z2 7-
5 9-
5 9-
1:;:3T sAT 1% -T u+-
‘60+3H
7.02 4.56 6.16 5.22 9.76 6.17 8.65 8.17 14.70 8.73, 9.53 13.35 13.04 22.34 14.01 20.66 20.42 21.59
7.47 6.34 9.98 7.42 13.92 10.13
18.96 14.77
6.99, 7.26 4.56 6.09 5.62 9.82 6.16 8.63, 6.93 (7.93) 8.95, 9.87
14.12
The theoretical spectrum of a-cluster states in 19F based on a 15N* core, predominantly in its first negative-parity excited state, compared with suggested experimental counterparts 6S23).The predicted energies of some neighbouring I60 + 3H states are also shown ‘).
energies obtained from this diagonalization are shown in tables 4 and 5, and those levels below 10 MeV excitation in 19F are displayed in fig. 3. 5. Discussion The calculations of the energies and wave functions for the two lowest-lying bands of negative-parity a-cluster states in ‘9 show that there is, in general, very little mixing between states based on a ground-state 15N core and an excited-state “N* core. The wave functions of the states based on the ground-state core are usually found to contain less than 5% admixtures of other configurations. Even so, the energy shifts in the ground-state core band due to these small mixtures (caused essentially by the second-rank tensor force) are greater in magnitude than those due to the spin-orbit force. This is clearly shown by the fact that the calculation does not place the J = L + f member of a doublet of the ground-state core band below the corresponding J = L-i member. Instead, one finds a series of nearly degenerate doublets with no regular pattern noticeable in the small splittings which
A.C. Merchant / a-cluster states
124
19
F
THEORY
L E 5 8
P 5 ;; :
+
+
--273
f32-r 5T
I 2
57-z_T 2
O
EXPERIMENT
I1
3T
5 7-977-===--
3
J
Fig. 3. The calculated spectrum of a-cluster states lying below 10 MeV excitation energy in “F, based on a 15N core predominantly in either its ground (S = f) or first negative-parity excited state (S = s), is compared with some experimental levels. Those experimental levels in the column labelled S = $ are heavily populated in a-transfer reactions lgez2), and those in the column labelled S = $ are seen as resonances in the “N(a, y)r9F reaction 23) and/or in the break-up of “F into “N and a-components “).
do occur. Conversely, the members of the excited core band form quadruplets with large energy splittings, more or less conforming to the pattern predicted by the first row of table 1, with some small modifications due to tensor force mixing of states having equal .I, but values of L differing by two units. However, there is one spectacular exception to these generalizations, involving the two ‘jl?-states formed
A.C. Merchant / a-cluster states
125
by coupling the a-particle to a ground-state core with L = 6 (0 S = 1) and to an excited core with L = 4 (OS =f). These states are nearly degenerate if mixing between bands is ignored, so that when the mixing. is “switched on” the final states are so thoroughly mixed that they essentially contain 50% of each cluster-core configuration. For this reason the two y- states are included in both tables 4 and 5, and their appearance in the various columns of fig. 3 is rather arbitrary and mainly for ease of presentation. The experimental implications of this lack of mixing are that the members of the ground-state core band (including both the strongly mixed y- states) should be heavily populated in a-transfer reactions onto 15N targets, whereas the members of the band base on an excited core are much less likely to be heavily populated by this method. The best chance of observing them seems to be through inelastic excitation of 19F followed by break-up into “N-a components or possibly as resonances in the 15N(a, r)‘v reaction. Enough a-transfer reactions onto 15N targets have now been performed for most of the negative-parity a-cluster states in ‘v, based on a ground-state core, to be confidently identified. A summary of the candidates is given in table 4. The earliest of these a-transfer experiments 19) was performed with a 20 MeV ‘Li beam. However, at this energy compound nucleus effects may have been appreciable. Subsequent investigations using a 105 MeV 13Cbeam 2oY21), a 115 MeV ’ 'B beam ‘l) and a 40 MeV ‘Li beam 22) have led to the sure assignment, on the basis of their strong, selective excitation, of the following states to the cr-cluster spectrum, $-(O.llO), s-(1.346), z-(1.459), s-(3.999), z-(4.033) and y-(8.288). The bracketed numbers following the spin-parity values are the excitation energies of the states in MeV. In addition the q-(8.953) state is quite heavily populated and appears to carry some of the cluster strength, although it has also been seen strongly in 3H transfer onto 160 [ref. ‘*)I. Two other strongly excited states at 12.26 MeV and 12.56 MeV were proposed by Pilt ef al. *‘) as the remaining y- and y- members of the band. The agreement in energy between the experimentally located states and the calculation presented in the last section is fairly good. The rather erratic splittings of the closely spaced doublets can not be reproduced in detail, but since these are of the order of 100 keV or less, that is hardly surprising, and the important point is to note that a microscopic explanation of their smallness and their failure to follow a classic spin-orbit pattern has been given. A large mixing of the y- states is predicted, although some additional mixing with triton cluster states may also be present and complicating the picture. The rotational parameter of the band seems to be a little overestimated with respect to its seven lowest-lying members and the final ye-$? doublet is predicted to lie much higher than the experimental candidates (around 16 MeV instead of 12 MeV). This is analogous to the situation in “Ne [ref. ‘)I, where a similar overestimate of the energy of the 8+ member of the ground-state band is made. However, a little caution should still be shown concerning the last two assignments in 19F because the spin-parity values of the excited states seen at 12.26 MeV and 12.67 MeV have not yet been conclusively measured.
126
A.C. Merchant / a-cluster
states
Identification of the members of the band based on an excited “N* core is, at present, rather more speculative. However, the energies and spin-parity values of many states in ‘? below 10 MeV have been identified 23) and merely comparing these quantities with the results of the calculation of the last section shows a remarkable correspondence. All the cluster states predicted to lie below 10 MeV are found to have a possible experimental counterpart and these are listed in table 5 and displayed in fig. 3. The i-(4.56) is fitted by choosing the head of the “N*-cy band to lie at 4.7 MeV 17,18) (mixing with the other two $- cluster states then depresses this slightly) and all other members of the band are calculated without any further free parameters. The predicted s-(6.16), g-(6.17), i-(7.02), z-(8.65) and g-(9.76) all lie within 0.07 MeV of possible experimental counterparts having well-identified spin-parity values 23), while there is a candidate for the predicted G-(5.22) at 5.62 MeV which also has J” = $-. The two remaining states predicted to lie below 10 MeV are the q-(8.17) and q-(9.53). There is an experimentally known (p, 5) state at 7.93 MeV which may be the z- and states at 9.71, 9.83 and 9.87 MeV, one of which may be the $-. This last assignment is complicated by the possibility of mixing with a triton cluster state, predicted to lie at 10.13 MeV [ref. ‘)I. In addition to this remarkable correspondence in energy it should be noted that the experimental candidates g-(5.62), s-(6.09), g-(6.16), (I’, $(7.93), g-(8.63), g-(9.82) and possible y-(9.83,9.87) are all seen as resonances in the 15N((u,y)19F reaction23). The y-(8.29) and y-(8.95) of the 15N-(u band are also seen in this reaction. Preliminary data from the inelastic excitation of 19F followed by break-up into 15N-a components from Oxford and Daresbury “) show several states of 19F between 5.4 MeV and 11.7 MeV which are presumed to have a predominantly 15N-~ structure. Among these are prominent peaks at 5.62, 6.12, 6.92, 8.01, 8.60 MeV which are in the right energy region to correspond to some of the predicted “N*-cy states. Strong excitations at 8.28 and 8.97 MeV (the energies of the ‘jl?-and y- members of the ground-state core 15N-~ band) are also seen. As yet no spin-parity assignments have been made. The experimental identification of any higher lying states, which might be associated with the remaining two quadruplets of “N*-cy states, is currently out of the question, although several states with a 15N-a structure above 12 MeV have been seen 21,22), One further difficulty in trying to compare the predicted “N*-cu negative-parity states with experiment is that the calculation presented here is not able to include mixing with 160-3H (triton) cluster states. In actual fact, a band of triton cluster states having 2N + L = 7 (and hence negative parity) is predicted to fall in the same energy region as the band of “N*-a cluster states and the predicted energies of its eight members are listed in table 5, together with a few suggested experimental counterparts 3). The overlap between the alpha and triton cluster states is not expected to be very large (analogies with the SU(3) model suggest typical values considerably less than 30%). It has also been shown above that mixing between states with the more closely similar structure of “N-(u and “N*-a is usually very
A.C. Merchant / wcluster states
127
small. However, since a “N*-a g- state is predicted to lie at 6.16 MeV compared with a 160-3H $- state at 6.34 MeV and a “N*-CY I- state at 9.76 MeV compared with a 160-3H state at 9.98 MeV, mixing of alpha and triton cluster structure in these cases may be significant. A coupled-channel orthogonality condition model calculation suggests that this is indeed the case r7,18), but in view of the apparently excellent agreement between the predicted “N*-cu spectrum and some highly plausible experimental candidates, the effects of such mixing on the energies of the states in question do not seem, in general, to be very great. It is hoped that this question of the purity of the a-cluster structure of the proposed members of the “N*-a band will be clarified by the forthcoming results of some new experimental investigations 6). The author would like to thank Dr. B.R. Fulton and Dr. W.D.M. Rae for permission to refer to their data prior to publication, and Drs M.A. Nagarajan and N. Rowley for useful discussions. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23)
B. Buck, C.B. Dover and J.P. Vary, Phys. Rev. Cl1 (1975) 1803 B. Buck, H. Friedrich and A.A. Pilt, Nucl. Phys. A290 (1977) 20.5 B. Buck and A.A. Pilt, Nucl. Phys. A280 (1977) 133 B. Buck, A.C. Merchant and N. Rowley, Nucl. Phys. A327 (1979) 29 N. Rowley and A.C. Merchant, Phys. Lett. 97B (1980) 341 B.R. Fulton and W.D.M. Rae, private communication D.F. Jackson and M. Rhoades-Brown, Nucl. Phys. A286 (1977) 354 G.R. Satchler, Nucl. Phys. 21 (1960) 116 D. Mukhopadhyay and G. Grawert, Nucl. Phys. A385 (1982) 133 J. Cook and R.J. Philpott, Nucl. Phys. A385 (1982) 157 R.C. Johnson, Invited talk at the RCNP-Kikuchi Summer School, Kyoto, Japan (1983), to be published D.M. Brink and G.R. Satchler, in Angular momentum, 2nd ed. (Clarendon, Oxford, 1968) p. 148 et seq. T.W. Donnelly and G.E. Walker, Phys. Rev. Lett. 22 (1969) 1121 C.J. Batty, E. Friedman and D.F. Jackson, Nucl. Phys. Al75 (1971) 1 C.J. Batty and E. Friedman, Phys. Lett. 34B (1971) 7 L.C. Biedenham, J.M. Blatt and M.E. Rose, Rev. Mod. Phys. 24 (1952) 249 T. Sakuda and F. Nemoto, Prog. Theor. Phys. 62 (1979) 1274, 1606 H. Furutani, H. Kanada, T. Kaneko, S. Nagata, H. Nishioka, S. Okabe, S. Saito, T. Sakuda and M. Seya, Prog. Theor. Phys. Suppl. 68 (1980) 193 R. Middleton, in Proc. Int. Conf. on nuclear reactions induced by heavy ions, Heidelberg, 1969, ed. R. Bock and W.R. Hering (North-Holland, Amsterdam, 1970) p. 263 A.A. Pilt, D.J. Millener, H. Bradlow, 0. Dietzsch, P.S. Fisher, W.J. Naude, W.D.M. Rae and D. Sinclair, Nucl. Phys. A273 (1976) 189 H.S. Bradlow, W.D.M. Rae, P.S. Fisher, N.S. Godwin, G. Proudfoot and D. Sinclair, Nucl. Phys. A314 (1979) 207 L.M. Martz, S.J. Sanders, P.D. Parker and C.B. Dover, Phys. Rev. C20 (1979) 1340 F. Ajzenberg-Selove, Nucl. Phys. A392 (1983) 1