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Velocity-dependent interactions in mass-14 nuclei
Nuclear Physics A l l l (1968) 63--80; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
VELOCITY-DEPENDENT INTERACTIONS IN MASS-14 NI~ICLEI N. F R E E D and P. O S T R A N D E R
Department of Physics, The Pennsylvania State University, University Park, Pennsylvania t Received 20 December 1967 Abstract: The non-local Tabakin interaction is applied to a shell-model calculation of 14C and a*N. In one approach, we consider the ground and low-lying states to arise from the configuration p-2; in another, we allow configuration mixing in the 2s-ld shell but restrict ourselves to a closed 12C core. In both calculations, second-order contributions to the t-matrix are taken into account in the plane-wave approximation. The low-lying spectra and 14C -+ ~*N beta-decay are studied. Calculations are carried out for leO and lSF and compared with results obtained from the hard-core H a m a d a - J o h n s t o n interaction.
1. Introduction
Because of their configurational simplicity, mass-14 nuclei have long served as objects of detailed experimental and theoretical study. Lying at the end of the Ip shell, these nuclei have been treated with varying degrees of success in versions of the shell model 1- 3), cluster model 4), Nilsson model 5) etc. Among the first thorough investigations were those of Inglis 3,6) and Kurath 2,7) who carried out extensive intermediate coupling calculations throughout the lp shell. These calculations, employing central two-body interactions in states of (1 s )4 (1 p)a - a, gave qualitatively good agreement for many low-lying energy levels. The need for inclusion of non-central interactions in mass-14 nuclei was first suggested by Inglis 3,6), who showed that a purely central interaction acting between the two holes in 14N could not produce the almost complete cancellation in the 14C ~ 14N beta-decay matrix element needed in order to explain the anomalously long lifetime (logfl ~ 9) for this transition. Subsequent calculations by Jancovici and Talmi s) and later by Visscher and Ferrell 9) showed that a tensor interaction if properly chosen could be made to yield the correct transition rate. The latter authors using a tensor force arising in lowest order of meson theory to mix states of p - 2 showed that it was possible to choose the interaction parameters in such a way that reasonably good agreement could be obtained not only for the beta-decay and low-lying levels in 14N but also for the spin-orbit splitting in A = 5, the 4He binding energy, the magnetic dipole and electric quadrupole moments of the 14N ground state, several ?-decay rates in ~4N and low energy cross sections for several charge exchange reactions. On the other hand, the results of such transfer reactions as 14N(d, p)l 5N and ~4C(d, t) ~3C indicated considerable ground-state configuration mixing for both ~4C and ~4N and were inconsistent with the Visschert Work supported in part by a grant f r o m the National Science Foundation. 63
64
N. FREED AND P. OSTRANDER
Ferrell predictions 10). An alternative approach based on the stripping reduced widths of the ground and first two excited states of 1aC was made by Baranger and Meshkov [ref. x1)] who attributed the beta-decay cancellation solely to the effects of configuration mixing without invoking any non-central two-body interaction. Since that time, there have been many other attempts at explaining the low-energy spectra, transition rates and reaction data of mass-14 nuclei 12-15). A serious deficiency in most of these calculations is the sensitive dependence of the results on the parametrization of the problem. It is usual in these attempts to assume the presence of some sort of average Hartree-Fock field and assign to phenomenological residual forces the task of reproducing the degree of configuration mixing needed to explain the spectra. The smooth local effective interactions traditionally chosen for shell-model calculations have forms dictated chiefly by considerations of ease of handling with parameters adjusted to reproduce a select body of data. This procedure is neither self-consistent nor unique, and its arbitrariness is heightened by the inclusion of non-central forces. The larger number of parameters, together with their sensitivity to size of the shell-model space, makes assignment of numerical values to these parameters strongly dependent on the single-particle levels included in firstorder, treatment of core configurations, etc. An entirely different approach, illustrated by the work of Unna and Talmi 16), assumes pure states with little configuration mixing and treats the interaction matrix elements themselves as parameters to be fit to the data. The effects of configuration mixing are then simulated by suitable adjustment of the parameters. The method is reasonable to apply only if little configuration mixing is allowed, and in addition considerable caution must be exercised in the choice of data which is used to initialize the matrix elements 13, 17). A third scheme which has become practicable only recently consists of calculating matrix elements of the potential or reaction matrix directly from an interaction which has been fit to the two-nucleon high-energy scattering data. The presence of strong short-range correlations between nucleon pairs has given rise to the use of two distinct types of interactions. The Yale is) and Hamada-Johnston 19) potentials are representative of the hard-core interactions in which the correlations are treated by means of an infinitely repulsive core, giving rise to wave functions that are strongly correlated at very small separation distances. Since straightforward use of such potentials would lead to infinite results, it is necessary to introduce the Brueckner reaction matrix in place of the potential and treat the leading terms in the t-matrix expansion in some approximate scheme 20). Although the evaluation of the resulting reaction matrix elements is quite involved, a number of shell-model 21,22) and Hartree-Fock 23) calculations with hard-core interactions have been carried out with good preliminary results. The second type of interaction is characterized by a smooth energy dependence giving rise to off-energy shell matrix elements which are usually small. Representative of this approach are the local soft-core potentials of Goldhammer 24) and Bressel
MASS-] 4 NUCLEI
65
et al. 25) and the velocity-dependent interactions of Razavy et al. 26), Green 27) and
Tabakin 2s). In particular, Tabakin has fit the S-, P- and D-state phase-shift parameters to 320 MeV by using a non-local separable interaction constructed in such a way as to make Hartree-Fock as well as shell-model calculations relatively easy to carry out. In addition to offering greater ease of computation, such a soft-core potential should lessen the need to consider mixing from highly excited intermediate states, an essential ingredient in building up the short-range repulsion of hard-core interactions. There are, however, two drawbacks involved in the use of the Tabakin force for structure calculations. One is that the fit to the scattering data is somewhat poorer than for either the Yale or HJ potentials. Secondly, the higher-order contributions to the reaction matrix can sometimes be substantial, especially for the coupled 3St + 3D t interaction in which the tensor force enters in first order. The first objection is mitigated to some extent by the observation that the HJ and Tabakin potentials have been found to yield very similar matrix elements in earlier calculations 29) while the second can be remedied in part by carrying out reliable calculations of second-order effects. It is our intent in this paper to use the Tabakin interaction with second-order tmatrix corrections taken into account in a shell-model calculation of the low-lying spectra of t 4C and t 4 N and to apply the calculated ground state wave functions to an analysis of the 14C ~ t 4N beta decay. We shall compare the results without configuration mixing with those in which we represent 12C as an inert spherical core and allow the two valence nucleons to mix in the s-d shell. We repeat our calculations for mass-18 nuclei and make a number of further comparisons. In sect. 2, we discuss briefly the experimental findings. Sect. 3 describes the Tabakin potential and our calculations, and sect. 4 contains the results and discussion.
2. Experimental data The experimental findings in t4C and 14N have been analysed and discussed in detail largely through the efforts of Warburton and colleagues ao) and will not be repeated here. Definite spin-parity assignments are now available for all bound levels in 14C (fig. 1) except for the state at 7.01 MeV which is believed to be the J~, T = 2 +, 1 analog of the largely core-excited 9.16 MeV level in 14N. The 0 + ground state and 6.58 MeV level are analogs of the 0 +, 1 states at 2.31 and 8.63 MeV in I4N which were assigned dominant configurations of p~ and s~, respectively, in earlier calculations t). The four odd-parity states arise largely from the p~ doublet configuration (table 1). The low-lying levels in t4N are also depicted in fig. 1. The ground and 2.31, 3.95 and 7.03 states are predominantly s4p I° and are the only states of this configuration bound against nucleon emission. The likelihood that these lowest-lying states contain significant amounts of core excitation (table 1) has been pointed out on several occasions i6, 3i) and will be discussed in detail later. The negative-parity states, which arise chiefly from p½s~ or p½d~, are similar to corresponding states in 160 which originate from the p~ ~(s, d) configuration.
J2-
" 2-
II--
[;I,} II0--
I+ 2-11} I-
2" _ 2# 0-~-= g,
9--
--24.11)
--O-ill ~-O+li) I-ill
I"
i ¢.
-3-1l)
2> 7-
2+
-
54,.
>uJ Z bJ
14. 35-
4-
I2tO-)
-
14
-
o*
ot'(i)
2I0
i÷
14N
14C
Fig. I. Energy levels o f 14C and a#N. All a#N states are T = 0 unless noted otherwise. The carbon ground state is drawn to match its isospin analog in nitrogen. Isospin analog states are connected by solid lines. Data from ref. s°L TABLE
I
Expected dominant configurations in 14C and 14N for levels to 8 MeV Nucleus 14C
"N
E (MeV)
J= (expt)
Dominant configuration
0
0+
(p½p
6.09 6.58 6.72
1-
(P½S½) (s~r)2 (P½d~r)
0+ 3-
6.89
O-
(P½S,!.)
7.01 7.34
(24 ) 2-
c.e. (p/td{)
0 2.31 3.95 4.91 5.10
1+ 040) 14 (0-) 2-
(p½p + c.e. (p½)2 (p½)2+c.e. (P½S½) (P½d~.)
5.69
1-
(P~S½)
5.83 6.23 6.44 7.03 7.97
31* 3+ 2+ 2-
(p~rd~r) (s½) 2 (s½d{) c.e. (P½d~)
D a t a are f r o m refs. 1, le, a0). A l l states in 14N are T = 0 e x c e p t as n o t e d , c.e. refers to e x c i t a t i o n o f t h e ~zC c o r e .
MASS-14 NUCLEI
67
3. Calculations We adopt two approaches in the following calculations. In one, we represent the lowest states of 14C and 14N as arising from the configuration p-2 and perform a modified intermediate coupling calculation, allowing the p-state hole splitting to play the role of an adjustable parameter in much the same way as the spin-orbit splitting is varied for each nucleus in the traditional intermediate coupling calculations. We do not vary the oscillator length parameter. In the second approach, we assume an inert spherical ~2C core and purejj-coupled wave functions with the two valence nucleons allowed to occupy the p½ state as well as the s-d shell. It is, of course, not expected that either scheme will produce results in detailed agreement with experiment. For example, the presence of low-lying negative-parity states in both t4C and ~4N argues against excluding excitations into the s-d shell, while at the same time the collective nature of the first excited 0 ÷ state in ~2C indicates the presence of considerable core excitation. Rather, our aim in these preliminary calculations is the more modest one of investigating the extent to which these commonly used schemes can provide a valid picture of the low-lying states of mass-14 nuclei when the appropriate configurations are admixed by a realistic soft-core interaction. It will be interesting, e.g. to see if such an interaction acting in states of p-2 is capable of explaining the 14C ~ ~4N beta decay or whether it will be necessary to include configuration mixing. Such a calculation was recently carried out for the HJ potential 32) and will serve as a comparison. 3.1. T A B A K I N P O T E N T I A L
In both approaches the particles or holes are assumed to interact via the Tabakin potential 28). This interaction is non-singular and separable and gives a fairly good fit to the high-energy scattering data. It reproduces the saturation effects of nuclear matter fairly well and has small higher-order corrections except in the coupled 3S 1+ 3D 1 state. The explicit form in momentum space is given by
V(k[k')-
i"-'[-#=,(k)#~,.(k')+h=,(k)h=,,(k')]~(k)~.'(k'),
2h2 ~
(1)
7~Frl ctMll"
where m is the nucleon mass, l and l' the relative orbital angular momenta, 0t the quantum numbers (JTS), the relative total angular momentum ( ~ = l +S), isospin and spin, respectively, M the z-projection of ~ and k, k' the relative wave numbers. The ~ ( k ) are normalized eigenstates of aCM and are given by
~21~(k) = ~. C~(MsMYIM,(k)XsMsPT,
(2)
where Y~u,(k) is a spherical harmonic, )~SM,the two-nucleon spin wave function and P r the isospin projection operator. Two separable potentials are assumed to act in each state with the g(k) denoting attraction and the h(k) repulsion. It is seen that total
68
N . F R E E D A N D P. O S T R A N D E R
relative m o m e n t u m , re', spin S a n d i s o s p i n T are c o n s e r v e d . A l t h o u g h p a r i t y is a l s o c o n s e r v e d , l is n o t a g o o d q u a n t u m n u m b e r for c o u p l e d states w h i c h are c o n n e c t e d by an effective t e n s o r force. H o w e v e r , since we a s s u m e the i n t e r a c t i o n to be effective o n l y for relative S-, P- a n d D - s t a t e s , it follows t h a t o n l y in the ( T = 0) aS 1 + 3D 1 case d o we find l :/: 1'. TABLE 2 Parameters of the Tabakin interaction I,"r
VB
1/a
ISo
115.9
235.6
0.834
0.801
3S I
164.7
10.3
0.763
0.990
1P1 3Po 3Pt ap, ID= aDI aD= aD a
44.3 267.7 107.6 103.7 297.1 189.3 389.7 143.5
1506.0 1067.0 53 ! .2 394.5 0.0 488.9 0.0 0.0
0.741 0.714 0.800 0.625 0.565 0.833 0.725 0.714
0.741 0.714 0.800 0.625
State
I/b
l/c
lid 0.694 0.590
0.909
2.00
The potential Vy = h=y=/ma and V# = h=fl2/mb. All potentials are in MeV and all lengths in fm. T h e explicit f o r m s for 9~a(k) a n d h~a(k ) are given b e l o w a n d the a s s o c i a t e d p a r a m eters are listed in t a b l e 2. I So(0r = 0, 1, 0) g=o(k) = v(k 2 + a 2 ) - ' ,
h=o(k ) = f l k 2 [ ( k - d ) 2 + b2] - ' [ ( k + d ) '
+ b2] - ',
3S1(~ = 1, 0, 1) #,o(k) = y(k 2 + a2) - ' ,
h~o(k ) = flk2[(k - d) 2 + b 2] - ' [ ( k + d) 2 + b 2] - ', 3po(~t = 0, 1, 1)
#=l(k) = ~k(k2 + a 2 ) -~, h , t ( k ) = flka(k 2 + b2) - ~ , 3Dl(Ct = 1, 0, 1) g,2(k) = ~k2[(k - c) 2 + a 23 - l [ ( k + c) 2 + a 2] - 1, h=2(k) _- flkZ(k 2 jr_ b2) - 2, 1D2(0t = 2, 1, 0) O=2(k) = 7k2(k 2 + a2) - 2, h=2(k) = O.
MASS-14 NUCLEI
69
The g,~(k) and h,l(k) for the IP1, 3P 1 and 3P 2 states are the same as those for 3P 0 except that the terms in the bracket o f e q . (1) are both taken as positive for 1P 1 and 3P 1 and negative for 3P 2. Similarly, the form of the 3D 2 and 3D 3 potentials is the same as that of ~D2. 3.2. C A L C U L A T I O N
OF t-MATRIX
The t-matrix elements to second-order are constructed in jj-coupling by transforming the two-particle oscillator states into an L S representation and applying the Moshinsky transformation in order to separate the relative and center-of-mass variables. The result may be written
(abJTltlcdJT) = ~ A
½ S
A
½ S'
x (nIN~q~Lln~ 1~n b Ib L)(n'I'N'~'IU[n¢ 1¢nd Id U ) x (nl, N . ~ ; LSJTItln'I', N ' ~ ' ; E S ' J T ) ,
(3)
where a - (n a, la,j~, r_.j), the A are t h e j j ~ L S coupling coefficients and L the total orbital angular momentum. The Moshinsky brackets a3) are elements of the transformation from the laboratory to the relative (nl) and c.m. (N.~) frame. The transformation is carried out subject to the conservation requirements
2na + la + 2nb + l b = 2n + l + 2N + ~CP, (--)'.+'~ = (-)'+~,
(4)
la+l b = 1+~,
representing conservation of energy, parity and angular momentum. The t-matrix may be written to second-order as
t =
v+
"-'?_,'l.)]l'l~n]l) ,.
- o + t2,
AE
(5)
where o is the two-nucleon potential, AE the energy difference between the initial bound states in the nucleus and the intermediate states in), and where the sum is over unoccupied states only. Although t 2 should be evaluated off the energy shell, it is a good approximation to replace the highly-excited intermediate states by plane waves [ref.22)], in which case the single-particle energies of the unoccupied states are replaced by kinetic energies.
70
N. FREED AND P. OSTRANDER
Since v is independent of the c.m. coordinates, we may easily calculate the firstorder term. We find in eq. (3)
(nl, N.~; LSJTlvln'l', N'.o~'; ES'JT)
= (_)l.+I;
× Y. U(lS.~J, JL)U(I'S~J; JtO
(6)
J
where the U-coefficients are defined by Edmonds a4). The reduced integral (nISJTlvln'l'SJT) for the Tabakin interaction is found to be 29)
(nlSJT[vln'l'SJT) = +_G(nIS,,gT)G(n'I'SJT) +_H(nlSJrT)H(nTSJT),
(7)
where the choice of signs was discussed earlier. The G and H are given by
G(nISJT) = ( 2h2] ~f~dkg,,(k)P.,(k)k z, \-~m /
H(nlS,/T)
Jo
= (2hZ)~f~dkh.,(k)P.,(k)k2. \l~m/
(8) (9)
do
The P,l(k) are the oscillator wave functions in momentum space and are given in a number of places [cf. e.g., ref. 29)]. The second-order contributions have been discussed recently by Kerman and Pal 3s). They use the plane wave approximation to represent the intermediate states, writing them after the Moshinsky transformation has been carried out as a product of c.m. and relative momentum components. The angle-averaged Pauli operator is then employed to take into account the requirement that the particles must scatter into unoccupied states above the Fermi sea. Although these approximations later lead to conservation of the c.m. angular momentum ~ , the quantum number N need not be conserved. In actual calculation, however, the matrix elements off-diagonal in N are much smaller than the diagonal terms and are omitted in the final results. To this approximation, the integrations can be carried out in relative and c.m. states, resulting in
(hi, N C.~a;LSJTlt2InT, N'~"; ES'JT) = 2ff-n-n(__)L+L" nh 2 x ~, U(IS~qPJ;JL)U(I'S.~J; JL')fNN, f~z" J
x I~°dkkZfNz(k, KF, A) Z ,tO
~
GU'(nlSJT)GCi)(n'l'Sa¢ "T)
i,j=l,2
(lo)
2
where the sum on i and j is understood to go over the contributions G(,flrT) and H ( J T ) and where the terms contributing to the 2 sum may be found from the listing
MASS-14 NUCLEI
71
of explicit forms for the potential components given earlier. The term fNse (k, K r, A) is defined as
fm~(k, KF, ,d)
f2rr K2Q(K, k, KF) = .~0-- dK[PNse(K)]2(h2/Em)(½K2+2k2) + A"
(11)
The quantities k, K and K~ are the relative, c.m. and Fermi momenta. An approximate value (d = 20 MeV) was used as the energy for all bound-state pairs. The angleaveraged Pauli operator is 22) Q ( r , k,
= o
k 2 +¼K 2 < K 2
=1
k-½K > K~
= (k ~ +¼K ~ - K~)/Kk
otherwise.
(12)
Since bound-state oscillator wave functions are sometimes used to represent the intermediate states a6), we carried out a comparison of the plane wave results with those for the lowest oscillator excitations in the special case of terms diagonal in the relative quantum numbers. To second-order, the oscillator reduced integral is given by
(nlSJTltlnlSJT)2 = (nlSafTlvlnlSJT) + ~, [(nlS°fTIvIn'I'SJT)[2, n'r *hi
(13)
Enl - - En, v
where, in order to conserve parity, N, and S¢, the sum is restricted to states an even number of shells higher in excitation. These matrix elements were evaluated for an oscillator parameter b = (h/mco)~ = (2.6) ~ fm with the sum including only states two shells up. The elements were then compared with similar results tabulated in ref. as). In most cases, the two differed by no more than 20 %. If we now collect results and include the exchange term, we find for the t-matrix elements to second order
=
[(1 +c5~b)(1 +tSca)]-~r
½ jji} (nlN.~Llnal.nblbL) x ~ A {1~ ½ ½ jj~){1~ A ½ S S' x (n'l'N.~L'[n¢ 1~nd la L')[-I - ( - )t +s + r-[
x (nl, N.~q';LSJT[v+t2[n'l', N.~q~;ESJT),
(14)
where 6~b -- 6..nb3,.,b~i.ib and where ([v+t2[) is given in eqs. (6)-(12). The matrices were constructed and diagonalized on the Penn State IBM 360-67 computer. Single-particle energies were taken from the spectrum of 1sC.
72
N. FREED
AND
P. OSTRANDER
4. Results and discussion
The results for the energy levels of ~+C and I+N in which the approximation has been made o f an inert spherical ~2C core are shown in figs. 2 and 3. The 14C ground - - 0 -
5- - 2 + - - 3 ÷ 2-
4-
2+ 0-
0+
o3;
2+
- - - 2 -
3-
:>
- - 2 4 .
,-
- - 0 -
3"
- - 1 4 .
2I0 +
O-
- - 3 - -
li+ 3-
I- - 2 -
- - 3 -
3- - 0 -
>(.9 O: bJ Z b.I
3+
3+
I"
2-
- - 0 -
I÷
ca
I-
>(.9 nw z hJ
O-
- 2 -
-
I0-)
_
_
14.
-I- - 0 + ( I I+
)
- - 2 - - 0 + ( i | - - i +
--2-
-3-
-I-
Oe(I]
0 4. _ _ 0
4.
-4-
-2-
-5-
-30+
i+
-6
-4 EXP
TI
EXP
T2
Fig. 2. Energy spectra for " C assuming a closed 12C core and configuration mixing in the s-d shell. First- and second-order results for the Tabakin potential are shown.
TI
T2
Fig. 3. Energy spectra for I+N. See caption to fig. 2.
state is placed at (binding-energy data from ref. 37)) E (t +C_g.s.) = _ B.E.(I #C) + B.E.(12C) +2[B.E.(taC)-B.E.(12C)]
= - 3 . 2 3 MeV,
(15)
and the 1+N ground state at E ( 1#N-g.s.) = - B.E.(t +N) + B.E.(I'C) + [B.E.( ~3 N ) - B.E.(12C)] +[B.E.(~aC)-B.E.(~2C)]
= - 5 . 6 1 MeV.
(16)
The oscillator spacing hco was taken to be 15.6 MeV. For comparison we have shown results with and without second-order corrections.
MASS-14
73
NUCLEI
I I
I I
0 o•
o o°
e-
E e-
c~c~ I
IIII
-i e-,
8 illl Z
oo ~o
~o o~
II
~ oo
III
II
O
II
O
-i
~.~
-~
I I
8 II
I +
I
+
I
r
+
II
l
I
I 0
I
,.s
il
il
74
N. FREED AND P. OSTRANDER
For both nuclei, the most obvious discrepancy lies in the insufficient binding of the ground' state. This result is due in part to the fact that these levels, being largely p-shell in origin, will mix strongly with the core states of our approximation. This point will be elaborated upon shortly. We also find, in contrast to some earlier predictions, that the inclusion of the d~ and s½ single-particle states plays a large role in lowering the ground-state binding energies. The carbon ground state was lowered by 1.5 MeV and the nitrogen ground state by 1.0 MeV by the inclusion of mixing in the s-d shell. Although the d~r state is important in reproducing certain electromagnetic transitions in this region 3o), we find that its inclusion has very little effect on the spectra. Sebe 38) has pictured the odd-parity levels as arising from the coupling of the singleparticle states in the s-d shell with the lowest p9 states in ~3C. One result of his calculation is the large departure of the non-normal-parity states from the configurations p~s~ or p~d~. The appearance of a core-excited state in ~3C at 3.7 MeV gives additional evidence that the odd-parity states will not be well reproduced in our calculation. The inadequate depression of the ground and low-lying excited states in nitrogen and ground state in carbon together with overbinding in the second 0 ÷ state in carbon and odd-parity states in carbon and nitrogen are strong indications that important modes of core excitation have been omitted. It has already been shown for mass-18 nuclei 22) that the inclusion of certain types of intermediate states 39) (three-particleone-hole in that case) not contained in the t-matrix causes a level spreading, the nature of which is a depression of ground and low-lying states, an elevation of high-lying states and an unalteration of the positions of states at intermediate energy. We return to this point below. The wave functions in first and second order are given in table 3. From them we calculate the ~4C(g.s.) ~ 14N(g.s.) beta transition which arises almost entirely from the allowed Gamow-Teller matrix element MGT, where in standard notation 39)
M~T = f o
2 = I(JrMrl
~ o°~IJ~M~)I
(17)
We find that these wave functions are totally unable to provide the almost complete cancellation of S- and P-state components needed to explain the transition. The calculations were then carried out for the 0 ÷, 1 states in t4N and 14C and 1+, 0 states in 14N under the assumption that they originate entirely from the configuration p-2. Once again we fix h¢o = 15.6 MeV and treat the s(.p~ 1)-s(p~ ~) = As splitting as an adjustable parameter. Although the centroid splitting is not known, the location of the first ½- state in ~SN at 6.3 MeV provides a rough orientation. In the calculations we varied the splitting until the beta-decay matrix element M2T vanished and based binding energy considerations on the energies resulting from that choice of As. We found complete cancellation to occur at As = 7.0 MeV in first order
M/~$-14 NUCLEI
75
and 6.8 MeV in second. The wave functions are listed in table 4 where they should be compared with results of Zamick's calculation 32) using the HJ potential. Although the wave functions are quite similar, it must be noted that in order to compensate for the very strong tensor force in the (state-independent) HJ calculation, it was necessary to use a rather large spin-orbit splitting to achieve cancellation. TABLE 4 Eigenvalues and eigenvectors o f 14C and 14N for states o f p-* Nucleus
J~
14C
0+
--1.920
0+ 14N
Energy (MeV) exp. calc.
p~-2
p~-Z
p ~ l p ~ 1 Ae(MeV)
--1.185 --1.572
0.949 0.949 0.975
0.316 0.316 0.220
7.0 6.8 9.2
4.680
12.642 10.521
0.316 0.316
--0.949 --0.949
7.0 6.8
1+
--4.830
--0.343 -- 1.405
1+
--0.880
0.654 --0.749
0.915 0.933 0.917 --0.394 --0.345
0.051 0.027 --0.014 0.331 0.348
0.401 0.358 0.399 0.857 0.872
7.0 6.8 9.2 7.0 6.8
The results are tabulated in the order (i) first-order Tabakin, (ii) second-order Tabakin and, where applicable, (iii) H a m a d a - J o h n s t o n s~).
With these wave functions we can calculate the ground-state magnetic dipole and electric quadrupole moments of ~4N. Leaving aside for the moment considerations of renormalization of electromagnetic operators arising from explicit velocity dependence of the two-nucleon interaction, we calculate that p(t4N-g.s.) = 0.33 nm and Q(~4N-g.s.) = +0.019 b, to be compared with the experimental values of 4~) 0.4036 n.m. and +0.01 b. With A~ = 6.8 MeV, we find that the binding energy of the 14C ground state is -1.57 MeV, and that of ~4N is -1.41 MeV. These should be compared with the "experimental" values of - 1.92 and -4.83 MeV, respectively, as calculated from the binding energy of two holes in ~60. The splitting between the two lowest 1 ÷, 0 states in nitrogen is 0.66 MeV versus the experimental 3.95. It is interesting to note that although the ~4N ground state is given about 3.5 MeV too high, the first 1 +, 0 excited state is correct to less than 0.15 MeV, and the 0 +, 1 first excited state is less than 1 MeV too high. For ~4C, the corresponding discrepancy for the ground state is 0.35 MeV. The second 0 ÷ state is poorly reproduced, in keeping with results of an earlier phenomenological calculation 12). The departures from experiment appearing in both approaches originate for the most part from the severe restrictions on the size of our model space. The great sensitivity of the lowest- and highest-lying levels in ~sO and ~SF to admixtures of 3p-lh
76
N . F R E E D A N D P. O S T R A N D E R
states (fig. 4b) has been mentioned earlier. In addition, a more recent calculation 42) with the HJ potential, in which the approximate state dependence of the t-matrix was taken into account, illustrated for lSF and 1sO the need for inclusion of 2p and
(o)
(b)
(c)
(d)
Fig. 4. Diagrams considered in the mass 18 calculation o f ref. 42). The first diagram (a) represents the t-interaction, while the other three represent 3p-lh, 2p and 2h correction terms, respectively.
4-
4÷
4+
3+
3+
0+
0+
4÷
3"
23• 0÷ Z+
>.
3+
3+
).-
oW Z bJ
~
4
÷
-I2+ 2+
-2-
- - 2 e
- - 0
_ _ 2 + - - 2
•
~
+
+
-3~
0
0+
-4-
EXP
TI
T2
HJI
+ 0
HJ2
Fig. 5. Energy spectra o f aaO. First- and second-order Tabakin results are compared to those o f K u o ,2). The notation HJ1 refers to the inclusion of diagrams (a) and (b) (fig. 4) whereas HJ2 refers to the inclusion o f (c) and (d) in addition.
2h diagrams (figs. 4c, 4d) which proceed mainly through the 2p-lf and lp shells, respectively. Especially important are the 4p-2h intermediate states of the 2h diagram, which are low-lying and highly deformed 4a) and contribute substantially to the lowering of the 1 +, 0 ground state of fluorine by about 2 MeV (figs. 5 and 6). Thus,
MASS-14
77
NUCLEI
it appears reasonable to attribute many of our discrepancies to the neglect of corresponding modes of core excitation not explicitly included in our t-matrix. In this regard, we felt that a direct comparison of the second-order Tabakin results for fluorine and oxygen with those of the HJ potential might prove interesting. Therefore, we repeated our calculations for mass-18 assuming an inert 160 core t. The results are shown in figs. 5 and 6 and summarized in tables 5 and 6. The spectra for the HJ results are from ref. 42) and the wave functions from ref. 22) [no wave functions were listed in ref. 42)]. The T2 results are quite similar to those for HJ2 (all
O3* - - i +
~ 1 -2-
>(.9 ne hj z hj
34.
~ 1
__21"
-I-
+ -
2÷ ÷ 2÷i I )
24"(I ]
2+
0+(I)
i+,Z -
,5 a+÷
~ 2 4 24. ( I )
~ 2 4 "
, 14.
• 2+ ( ! ) --24"(I) ~
2
+
--O+(l} -3-
I+
- -
i÷
~ o ~+_ -4-
~ - - 0 4 " ( i x-3+
i÷ 0+(11 54.
5+
--.3÷
~4.
~4. ~ 0 + ( I ) 3•
I - - i
+
-5"
.
,i ÷
-6
EXP
TI
T2
Hdt
Hd2
Fig. 6. Energy spectra of laF. See caption to fig. 5.
diagrams in fig. 4 included) with the exception of the energies of the 1 +, 0 ground state and 0 +, 1 excited state in lSF and the analog 0 + oxygen ground state. The 18F ground state is seen to be very sensitive to the inclusion of second-order effects in the Tabakin interaction and to the core diagrams (figs. 4c and d) in the HJ potential. In our calculations, a number of effects have been neglected in addition to those already mentioned. Some of these are minor (isospin impurities, Thomas-Ehrmann shifts and Coulomb force), but others should be included in any serious attempt to get detailed agreement with experiment. For example, center-of-mass corrections might be important for the 0 - , 1- and 2 - states 4s). We have ignored such corrections here because previous indications ~1) are that they are small. There also arises the possibility that higher-order terms in the t-matrix which have not been taken into account here are non-negligible. A recent work by Halbert et al. [ref. 46)], in which HJ calculations were carried out in the lp shell, indicated the t Similar first-order calculations have been reported by Lee and Baranger 44).
78
N. FREED AND P. OSTRANDER TABLE 5
Eigenvalues a n d eigenvectors o f asO J~
Energy (MeV) exp. calc.
d~ ~
0+
--3.90
--2.38 --2.64 --4.22
0.875 0.882 0.901
2+
--1.92
--1.59 -- 1.79 -- 1.89
0.757 0.754 0.782
4+
--0.35
--0.61 --0.75 --0.27
0.964 0.963 0.954
0+
--0.27
--0.05 --0.16 --0.06
--0.403 --0.399 --0.322
2+
0.02
-[-0.04 --0.11 0.23
--0.641 --0.647 --0.607
3+
1.47
0.66 0.52 I. 11
4+
3.22
3.48 3.37 4.26
d~s~.
d~d~
s~ 2
s~}d~
0.410 0.403 0.324 0.609 0.614 0.579
0.099 0.094 0.090
d~ ~
0.257 0.245 0.287 0.193 0.192 0.190
0.097 0.093 0.099
0.267 0.269 0.300 0.912 0.915 0.945 0.753 0.750 0.788
--0.023 --0.003 --0.008
0.008 0.003 0.025
1.000 1.000 1.000
--0.267 --0.269 --0.300
--0.082 --0.070 --0.055 0.145 0.137 0.105
0.007 0.011 --0.007
0.964 0.963 0.954
Results are tabulated in the order: (i) first-order T a b a k i n , (ii) second-order T a b a k i n a n d (iii) statei n d e p e n d e n t H a m a d a - J o h n s t o n ~2) (hto = 14 MeV). TABLE 6 Eigenvalues a n d eigenvectors o f ~SF J=
Energy (MeV) exp. calc.
dt2
1+
--5.00
--3.29 --4.33 --4.83
0.542 0.580 0.571
3+
--4.06
--2.98 --3.63 --4.04
0.550 0.552 0.527
5+
--3.88
--2.89 --3.43 --3.69
1.00 1.00 0.557 0.573 0.554
1+
--1.87
--0.54 --1.18 --1.23
2+
--1.65
--1.01 - - 1.86 --1.59
d~S~r
dtd~
--0.621 --0.612 --0.629 0.810 0.809 0.817
s~ 2
s~. d~}
d~}2
0.553 0.524 0.507
--0.103 --0.093 --0.143
--0.067 --0.072 --0.040
--0.201 --0.204 --0.234
--0.007 --0.012 +0.016
1.00
--0.186 --0.125 --0.163 0.764 0.768 0.753
--0.791 --0.797 --0.803
0.515 0.509 0.536
See table 5 for a r r a n g e m e n t o f table. H a m a d a - J o h n s t o n results f r o m ref. n ) .
--0.097 --0.070 0.110 0.388 0.389 0.382
--0.142 --0.127 --0.103
MASS-14
NUCLEI
79
possible importance of including such higher-correlation effects. In a recent calculation by Goldhammer et aL 47), the inclusion of a three-body vector force 4s) was found to make a contribution to the binding energy of mass-14 nuclei of ~ 2 MeV. Our results, then, are not inconsistent with those of earlier studies. In those cases where comparison with hard-core results is possible we obtain excellent agreement for the wave functions. We are able to explain the 14C ~ ~4N beta decay with a reasonable value of p-state spin-orbit splitting and at the same time predict the fast laNe ~ laF decay ( l o g f t = 2.89 calc. versus 3.05 exp.). The spectra are very sensitive to higher-order processes, some of which are included in the t-matrix, others of which are hot. Any attempt at fitting details of binding energies will have to incorporate such corrections in a reliable manner. It is a pleasure to thank Drs. T. T. S. Kuo and K. T. R. Davies for use of computer codes and Dr. F. Tabakin for a correspondence concerning the parameters of his potential. We are grateful to Dr. P. Goldhammer for interesting discussions and for information concerning his work in ref. 47). We also wish to acknowledge the computational assistance of The Pennsylvania State University Computation Centre.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20)
21) 22) 23) 24)
W. W. True, Phys. Rev. 130 (1963) 1530 D. Kurath, Phys. Rev. 101 (1956) 216 D. Inglis, Revs. Mod. Phys. 25 (1953) 390 R. K. Sheline and K. Wildermuth, Nucl. Phys. 21 (1960) 196 M. Soga, Phys. Rev. 163 (1967) 995 and preprint D. lnglis, Phys. Rev. 87 (1952) 915 D. Kurath, Phys. Rev. 106 (1957) 975 B. Jancovici and I. Talmi, Phys. Rev. 95 (1954) 289 W. M. Visscher and R. A. Ferrell, Phys. Rev. 107 (1957) 781 M . H . Macfarlane and J. B. French, Revs. Mod. Phys. 32 (1960) 567 E. Baranger and S. Meshkov, Phys. Rev. Lett. 1 (1958) 30 M. A. Nagarajan, Nucl. Phys. 42 (1963) 454 S. Cohen and D. Kurath, Nucl. Phys. 73 (1965) 1 A. B. Volkov, Nucl. Phys. 84 (1965) 33 R. K. Gupta and P. C. Sood, Phys. Rev. 152 (1966) 917 I. Unna and I. Talmi, Phys. Rev. 112 (1958) 452 D. Amit and A. Katz, Nucl. Phys. 58 (1964) 388 K. E. Lassila, M. H. Hull, Jr., H. M. Ruppel, F. A. McDonald and G. Breit, Phys. Rev. 126 (1962) 881 T. Hamada and I. D. Johnston, Nucl. Phys. 34 (1962) 382 J. F. Dawson, I. Talmi and J. D. Walecka, Ann. of Phys. 18 (1962) 339; H. A. Bethe, B. H. Brandow and A. G. Petschek, Phys. Rev. 129 (1963) 225; B. L. Scott and S. A. Moszkowski, Nucl. Phys. 29 (1962) 665 R. D. Lawson, M. H. Macfarlane and T. T. S. Kuo, Phys. Lett. 22 (1966) 209; R. D. Becker and A. D. McKeller, Phys. Lett. 21 (1966) 201 T. T. S. Kuo and G. E. Brown, Nucl. Phys. 85 (1966) 40 M. K. Pal and A. P. Stamp, Phys. Rev. 158 (1967) 924; C. M. Shakin, Y. T. Waghmare, M. Tomaselli and M. H. Hull, Jr., Phys. Rev. 161 (1967) 1015 P. Goldhammer, Phys. Rev. 116 (1959) 676
80 25) 26) 27) 28) 29) 30)
31) 32) 33) 34) 35) 36) 37) 38) 39) 40) 41) 42) 43) 44) 45) 46) 47) 48)
N . F R E E D A N D P. O S T R A N D E R
C. Bressel, A. Kerman and E. Lomon, Bull. Am. Phys. Soc. 10 (1965) 584 M. Razavy, G. Field and J. S. Levinger, Phys. Rev. 125 (1962) 269 A. M. Green, Nucl. Phys. 47 (1963) 671 F. Tabakin, Ann. of Phys. 30 (1964) 51 T. T. S. Kuo, E. Baranger and M. Baranger, Nucl. Phys. 81 (1966) 241 E. K. Warburton, H. J. Rose and E. N. Hatch, Phys. Rev. 114 (1959) 214; E. K. Warburton and W. T. Pinkston, Phys. Rev. 118 (1960) 733; D. E. Alburger and E. K. Warburton, Phys. Rev. 132 (1963) 790; E. K. Warburton, J. W. Olness, D. E. Alburger, D. J. Brodin and L. F. Chase, Jr., Phys. Rev. 134 (1964) B338; D. E. Alburger and E. K. Warburton, Phys. Rev. 148 (1966) 1050; J. W. Olness, A. R. Poletti and E. K. Warburton, Phys. Rev. 154 (1967) 971 D. E. Alburger and E. K. Warburton, Phys. Rev. 148 (1966) 1050 L. Zamick, Phys. Lett. 21 (1966) 194 M. Moshinsky, Nucl. Phys. 13 (1959) 104 A. R. Edmonds, Angular momentum in quantum mechanics (Princeton University Press, Princeton, 1957) A. K. Kerman and M. K. Pal, Phys. Rev. 162 (1967) 970 B. R. Barrett, Phys. Rev. 154 (1967) 955 L. A. K6nig, J. H. E. Mattauch and A. H. Wapstra, Nucl. Phys. 31 (1962) 18 T. Sebe, Prog. Theor. Phys. 30 (1963) 290 G. F. Bertsch, Nucl. Phys. 74 (1965) 234 E. J. Konopinsky and M. E. Rose, in Alpha-, beta- and gamma-ray spectroscopy, ed. by K. Sieghahn (North-Holland Publ. Co., Amsterdam, 1965) p. 1327 Nuclear Data Sheets, compiled by K. Way et al. (NAS-NRC, Washington, D.C.) T. T. S. Kuo, Nucl. Phys. A103 (1967) 71 G. E. Brown and A. M. Green, Nucl. Phys. 85 (1966) 87 C. W. Lee and E. Baranger, Nucl. Phys. 79 (1966) 385 N. Freed and P. Goldhammer, Phys. Rev. 136 (1964) B1260 E. C. Halbert, Y. E. Kim and T. T. S. Kuo, Phys. Lett. 20 (1966) 657 P. Goldhammer, J. R. Hill and J. Nachamkin, to be published A. M. Feingold, Phys. Rev. 101 (1956) 258, 105 (1957) 944, 114 (1959) 540
VELOCITY-DEPENDENT INTERACTIONS IN MASS-14 NI~ICLEI N. F R E E D and P. O S T R A N D E R
Department of Physics, The Pennsylvania State University, University Park, Pennsylvania t Received 20 December 1967 Abstract: The non-local Tabakin interaction is applied to a shell-model calculation of 14C and a*N. In one approach, we consider the ground and low-lying states to arise from the configuration p-2; in another, we allow configuration mixing in the 2s-ld shell but restrict ourselves to a closed 12C core. In both calculations, second-order contributions to the t-matrix are taken into account in the plane-wave approximation. The low-lying spectra and 14C -+ ~*N beta-decay are studied. Calculations are carried out for leO and lSF and compared with results obtained from the hard-core H a m a d a - J o h n s t o n interaction.
1. Introduction
Because of their configurational simplicity, mass-14 nuclei have long served as objects of detailed experimental and theoretical study. Lying at the end of the Ip shell, these nuclei have been treated with varying degrees of success in versions of the shell model 1- 3), cluster model 4), Nilsson model 5) etc. Among the first thorough investigations were those of Inglis 3,6) and Kurath 2,7) who carried out extensive intermediate coupling calculations throughout the lp shell. These calculations, employing central two-body interactions in states of (1 s )4 (1 p)a - a, gave qualitatively good agreement for many low-lying energy levels. The need for inclusion of non-central interactions in mass-14 nuclei was first suggested by Inglis 3,6), who showed that a purely central interaction acting between the two holes in 14N could not produce the almost complete cancellation in the 14C ~ 14N beta-decay matrix element needed in order to explain the anomalously long lifetime (logfl ~ 9) for this transition. Subsequent calculations by Jancovici and Talmi s) and later by Visscher and Ferrell 9) showed that a tensor interaction if properly chosen could be made to yield the correct transition rate. The latter authors using a tensor force arising in lowest order of meson theory to mix states of p - 2 showed that it was possible to choose the interaction parameters in such a way that reasonably good agreement could be obtained not only for the beta-decay and low-lying levels in 14N but also for the spin-orbit splitting in A = 5, the 4He binding energy, the magnetic dipole and electric quadrupole moments of the 14N ground state, several ?-decay rates in ~4N and low energy cross sections for several charge exchange reactions. On the other hand, the results of such transfer reactions as 14N(d, p)l 5N and ~4C(d, t) ~3C indicated considerable ground-state configuration mixing for both ~4C and ~4N and were inconsistent with the Visschert Work supported in part by a grant f r o m the National Science Foundation. 63
64
N. FREED AND P. OSTRANDER
Ferrell predictions 10). An alternative approach based on the stripping reduced widths of the ground and first two excited states of 1aC was made by Baranger and Meshkov [ref. x1)] who attributed the beta-decay cancellation solely to the effects of configuration mixing without invoking any non-central two-body interaction. Since that time, there have been many other attempts at explaining the low-energy spectra, transition rates and reaction data of mass-14 nuclei 12-15). A serious deficiency in most of these calculations is the sensitive dependence of the results on the parametrization of the problem. It is usual in these attempts to assume the presence of some sort of average Hartree-Fock field and assign to phenomenological residual forces the task of reproducing the degree of configuration mixing needed to explain the spectra. The smooth local effective interactions traditionally chosen for shell-model calculations have forms dictated chiefly by considerations of ease of handling with parameters adjusted to reproduce a select body of data. This procedure is neither self-consistent nor unique, and its arbitrariness is heightened by the inclusion of non-central forces. The larger number of parameters, together with their sensitivity to size of the shell-model space, makes assignment of numerical values to these parameters strongly dependent on the single-particle levels included in firstorder, treatment of core configurations, etc. An entirely different approach, illustrated by the work of Unna and Talmi 16), assumes pure states with little configuration mixing and treats the interaction matrix elements themselves as parameters to be fit to the data. The effects of configuration mixing are then simulated by suitable adjustment of the parameters. The method is reasonable to apply only if little configuration mixing is allowed, and in addition considerable caution must be exercised in the choice of data which is used to initialize the matrix elements 13, 17). A third scheme which has become practicable only recently consists of calculating matrix elements of the potential or reaction matrix directly from an interaction which has been fit to the two-nucleon high-energy scattering data. The presence of strong short-range correlations between nucleon pairs has given rise to the use of two distinct types of interactions. The Yale is) and Hamada-Johnston 19) potentials are representative of the hard-core interactions in which the correlations are treated by means of an infinitely repulsive core, giving rise to wave functions that are strongly correlated at very small separation distances. Since straightforward use of such potentials would lead to infinite results, it is necessary to introduce the Brueckner reaction matrix in place of the potential and treat the leading terms in the t-matrix expansion in some approximate scheme 20). Although the evaluation of the resulting reaction matrix elements is quite involved, a number of shell-model 21,22) and Hartree-Fock 23) calculations with hard-core interactions have been carried out with good preliminary results. The second type of interaction is characterized by a smooth energy dependence giving rise to off-energy shell matrix elements which are usually small. Representative of this approach are the local soft-core potentials of Goldhammer 24) and Bressel
MASS-] 4 NUCLEI
65
et al. 25) and the velocity-dependent interactions of Razavy et al. 26), Green 27) and
Tabakin 2s). In particular, Tabakin has fit the S-, P- and D-state phase-shift parameters to 320 MeV by using a non-local separable interaction constructed in such a way as to make Hartree-Fock as well as shell-model calculations relatively easy to carry out. In addition to offering greater ease of computation, such a soft-core potential should lessen the need to consider mixing from highly excited intermediate states, an essential ingredient in building up the short-range repulsion of hard-core interactions. There are, however, two drawbacks involved in the use of the Tabakin force for structure calculations. One is that the fit to the scattering data is somewhat poorer than for either the Yale or HJ potentials. Secondly, the higher-order contributions to the reaction matrix can sometimes be substantial, especially for the coupled 3St + 3D t interaction in which the tensor force enters in first order. The first objection is mitigated to some extent by the observation that the HJ and Tabakin potentials have been found to yield very similar matrix elements in earlier calculations 29) while the second can be remedied in part by carrying out reliable calculations of second-order effects. It is our intent in this paper to use the Tabakin interaction with second-order tmatrix corrections taken into account in a shell-model calculation of the low-lying spectra of t 4C and t 4 N and to apply the calculated ground state wave functions to an analysis of the 14C ~ t 4N beta decay. We shall compare the results without configuration mixing with those in which we represent 12C as an inert spherical core and allow the two valence nucleons to mix in the s-d shell. We repeat our calculations for mass-18 nuclei and make a number of further comparisons. In sect. 2, we discuss briefly the experimental findings. Sect. 3 describes the Tabakin potential and our calculations, and sect. 4 contains the results and discussion.
2. Experimental data The experimental findings in t4C and 14N have been analysed and discussed in detail largely through the efforts of Warburton and colleagues ao) and will not be repeated here. Definite spin-parity assignments are now available for all bound levels in 14C (fig. 1) except for the state at 7.01 MeV which is believed to be the J~, T = 2 +, 1 analog of the largely core-excited 9.16 MeV level in 14N. The 0 + ground state and 6.58 MeV level are analogs of the 0 +, 1 states at 2.31 and 8.63 MeV in I4N which were assigned dominant configurations of p~ and s~, respectively, in earlier calculations t). The four odd-parity states arise largely from the p~ doublet configuration (table 1). The low-lying levels in t4N are also depicted in fig. 1. The ground and 2.31, 3.95 and 7.03 states are predominantly s4p I° and are the only states of this configuration bound against nucleon emission. The likelihood that these lowest-lying states contain significant amounts of core excitation (table 1) has been pointed out on several occasions i6, 3i) and will be discussed in detail later. The negative-parity states, which arise chiefly from p½s~ or p½d~, are similar to corresponding states in 160 which originate from the p~ ~(s, d) configuration.
J2-
" 2-
II--
[;I,} II0--
I+ 2-11} I-
2" _ 2# 0-~-= g,
9--
--24.11)
--O-ill ~-O+li) I-ill
I"
i ¢.
-3-1l)
2> 7-
2+
-
54,.
>uJ Z bJ
14. 35-
4-
I2tO-)
-
14
-
o*
ot'(i)
2I0
i÷
14N
14C
Fig. I. Energy levels o f 14C and a#N. All a#N states are T = 0 unless noted otherwise. The carbon ground state is drawn to match its isospin analog in nitrogen. Isospin analog states are connected by solid lines. Data from ref. s°L TABLE
I
Expected dominant configurations in 14C and 14N for levels to 8 MeV Nucleus 14C
"N
E (MeV)
J= (expt)
Dominant configuration
0
0+
(p½p
6.09 6.58 6.72
1-
(P½S½) (s~r)2 (P½d~r)
0+ 3-
6.89
O-
(P½S,!.)
7.01 7.34
(24 ) 2-
c.e. (p/td{)
0 2.31 3.95 4.91 5.10
1+ 040) 14 (0-) 2-
(p½p + c.e. (p½)2 (p½)2+c.e. (P½S½) (P½d~.)
5.69
1-
(P~S½)
5.83 6.23 6.44 7.03 7.97
31* 3+ 2+ 2-
(p~rd~r) (s½) 2 (s½d{) c.e. (P½d~)
D a t a are f r o m refs. 1, le, a0). A l l states in 14N are T = 0 e x c e p t as n o t e d , c.e. refers to e x c i t a t i o n o f t h e ~zC c o r e .
MASS-14 NUCLEI
67
3. Calculations We adopt two approaches in the following calculations. In one, we represent the lowest states of 14C and 14N as arising from the configuration p-2 and perform a modified intermediate coupling calculation, allowing the p-state hole splitting to play the role of an adjustable parameter in much the same way as the spin-orbit splitting is varied for each nucleus in the traditional intermediate coupling calculations. We do not vary the oscillator length parameter. In the second approach, we assume an inert spherical ~2C core and purejj-coupled wave functions with the two valence nucleons allowed to occupy the p½ state as well as the s-d shell. It is, of course, not expected that either scheme will produce results in detailed agreement with experiment. For example, the presence of low-lying negative-parity states in both t4C and ~4N argues against excluding excitations into the s-d shell, while at the same time the collective nature of the first excited 0 ÷ state in ~2C indicates the presence of considerable core excitation. Rather, our aim in these preliminary calculations is the more modest one of investigating the extent to which these commonly used schemes can provide a valid picture of the low-lying states of mass-14 nuclei when the appropriate configurations are admixed by a realistic soft-core interaction. It will be interesting, e.g. to see if such an interaction acting in states of p-2 is capable of explaining the 14C ~ ~4N beta decay or whether it will be necessary to include configuration mixing. Such a calculation was recently carried out for the HJ potential 32) and will serve as a comparison. 3.1. T A B A K I N P O T E N T I A L
In both approaches the particles or holes are assumed to interact via the Tabakin potential 28). This interaction is non-singular and separable and gives a fairly good fit to the high-energy scattering data. It reproduces the saturation effects of nuclear matter fairly well and has small higher-order corrections except in the coupled 3S 1+ 3D 1 state. The explicit form in momentum space is given by
V(k[k')-
i"-'[-#=,(k)#~,.(k')+h=,(k)h=,,(k')]~(k)~.'(k'),
2h2 ~
(1)
7~Frl ctMll"
where m is the nucleon mass, l and l' the relative orbital angular momenta, 0t the quantum numbers (JTS), the relative total angular momentum ( ~ = l +S), isospin and spin, respectively, M the z-projection of ~ and k, k' the relative wave numbers. The ~ ( k ) are normalized eigenstates of aCM and are given by
~21~(k) = ~. C~(MsMYIM,(k)XsMsPT,
(2)
where Y~u,(k) is a spherical harmonic, )~SM,the two-nucleon spin wave function and P r the isospin projection operator. Two separable potentials are assumed to act in each state with the g(k) denoting attraction and the h(k) repulsion. It is seen that total
68
N . F R E E D A N D P. O S T R A N D E R
relative m o m e n t u m , re', spin S a n d i s o s p i n T are c o n s e r v e d . A l t h o u g h p a r i t y is a l s o c o n s e r v e d , l is n o t a g o o d q u a n t u m n u m b e r for c o u p l e d states w h i c h are c o n n e c t e d by an effective t e n s o r force. H o w e v e r , since we a s s u m e the i n t e r a c t i o n to be effective o n l y for relative S-, P- a n d D - s t a t e s , it follows t h a t o n l y in the ( T = 0) aS 1 + 3D 1 case d o we find l :/: 1'. TABLE 2 Parameters of the Tabakin interaction I,"r
VB
1/a
ISo
115.9
235.6
0.834
0.801
3S I
164.7
10.3
0.763
0.990
1P1 3Po 3Pt ap, ID= aDI aD= aD a
44.3 267.7 107.6 103.7 297.1 189.3 389.7 143.5
1506.0 1067.0 53 ! .2 394.5 0.0 488.9 0.0 0.0
0.741 0.714 0.800 0.625 0.565 0.833 0.725 0.714
0.741 0.714 0.800 0.625
State
I/b
l/c
lid 0.694 0.590
0.909
2.00
The potential Vy = h=y=/ma and V# = h=fl2/mb. All potentials are in MeV and all lengths in fm. T h e explicit f o r m s for 9~a(k) a n d h~a(k ) are given b e l o w a n d the a s s o c i a t e d p a r a m eters are listed in t a b l e 2. I So(0r = 0, 1, 0) g=o(k) = v(k 2 + a 2 ) - ' ,
h=o(k ) = f l k 2 [ ( k - d ) 2 + b2] - ' [ ( k + d ) '
+ b2] - ',
3S1(~ = 1, 0, 1) #,o(k) = y(k 2 + a2) - ' ,
h~o(k ) = flk2[(k - d) 2 + b 2] - ' [ ( k + d) 2 + b 2] - ', 3po(~t = 0, 1, 1)
#=l(k) = ~k(k2 + a 2 ) -~, h , t ( k ) = flka(k 2 + b2) - ~ , 3Dl(Ct = 1, 0, 1) g,2(k) = ~k2[(k - c) 2 + a 23 - l [ ( k + c) 2 + a 2] - 1, h=2(k) _- flkZ(k 2 jr_ b2) - 2, 1D2(0t = 2, 1, 0) O=2(k) = 7k2(k 2 + a2) - 2, h=2(k) = O.
MASS-14 NUCLEI
69
The g,~(k) and h,l(k) for the IP1, 3P 1 and 3P 2 states are the same as those for 3P 0 except that the terms in the bracket o f e q . (1) are both taken as positive for 1P 1 and 3P 1 and negative for 3P 2. Similarly, the form of the 3D 2 and 3D 3 potentials is the same as that of ~D2. 3.2. C A L C U L A T I O N
OF t-MATRIX
The t-matrix elements to second-order are constructed in jj-coupling by transforming the two-particle oscillator states into an L S representation and applying the Moshinsky transformation in order to separate the relative and center-of-mass variables. The result may be written
(abJTltlcdJT) = ~ A
½ S
A
½ S'
x (nIN~q~Lln~ 1~n b Ib L)(n'I'N'~'IU[n¢ 1¢nd Id U ) x (nl, N . ~ ; LSJTItln'I', N ' ~ ' ; E S ' J T ) ,
(3)
where a - (n a, la,j~, r_.j), the A are t h e j j ~ L S coupling coefficients and L the total orbital angular momentum. The Moshinsky brackets a3) are elements of the transformation from the laboratory to the relative (nl) and c.m. (N.~) frame. The transformation is carried out subject to the conservation requirements
2na + la + 2nb + l b = 2n + l + 2N + ~CP, (--)'.+'~ = (-)'+~,
(4)
la+l b = 1+~,
representing conservation of energy, parity and angular momentum. The t-matrix may be written to second-order as
t =
v+
"-'?_,'l.)]l'l~n]l) ,.
- o + t2,
AE
(5)
where o is the two-nucleon potential, AE the energy difference between the initial bound states in the nucleus and the intermediate states in), and where the sum is over unoccupied states only. Although t 2 should be evaluated off the energy shell, it is a good approximation to replace the highly-excited intermediate states by plane waves [ref.22)], in which case the single-particle energies of the unoccupied states are replaced by kinetic energies.
70
N. FREED AND P. OSTRANDER
Since v is independent of the c.m. coordinates, we may easily calculate the firstorder term. We find in eq. (3)
(nl, N.~; LSJTlvln'l', N'.o~'; ES'JT)
= (_)l.+I;
× Y. U(lS.~J, JL)U(I'S~J; JtO
(6)
J
where the U-coefficients are defined by Edmonds a4). The reduced integral (nISJTlvln'l'SJT) for the Tabakin interaction is found to be 29)
(nlSJT[vln'l'SJT) = +_G(nIS,,gT)G(n'I'SJT) +_H(nlSJrT)H(nTSJT),
(7)
where the choice of signs was discussed earlier. The G and H are given by
G(nISJT) = ( 2h2] ~f~dkg,,(k)P.,(k)k z, \-~m /
H(nlS,/T)
Jo
= (2hZ)~f~dkh.,(k)P.,(k)k2. \l~m/
(8) (9)
do
The P,l(k) are the oscillator wave functions in momentum space and are given in a number of places [cf. e.g., ref. 29)]. The second-order contributions have been discussed recently by Kerman and Pal 3s). They use the plane wave approximation to represent the intermediate states, writing them after the Moshinsky transformation has been carried out as a product of c.m. and relative momentum components. The angle-averaged Pauli operator is then employed to take into account the requirement that the particles must scatter into unoccupied states above the Fermi sea. Although these approximations later lead to conservation of the c.m. angular momentum ~ , the quantum number N need not be conserved. In actual calculation, however, the matrix elements off-diagonal in N are much smaller than the diagonal terms and are omitted in the final results. To this approximation, the integrations can be carried out in relative and c.m. states, resulting in
(hi, N C.~a;LSJTlt2InT, N'~"; ES'JT) = 2ff-n-n(__)L+L" nh 2 x ~, U(IS~qPJ;JL)U(I'S.~J; JL')fNN, f~z" J
x I~°dkkZfNz(k, KF, A) Z ,tO
~
GU'(nlSJT)GCi)(n'l'Sa¢ "T)
i,j=l,2
(lo)
2
where the sum on i and j is understood to go over the contributions G(,flrT) and H ( J T ) and where the terms contributing to the 2 sum may be found from the listing
MASS-14 NUCLEI
71
of explicit forms for the potential components given earlier. The term fNse (k, K r, A) is defined as
fm~(k, KF, ,d)
f2rr K2Q(K, k, KF) = .~0-- dK[PNse(K)]2(h2/Em)(½K2+2k2) + A"
(11)
The quantities k, K and K~ are the relative, c.m. and Fermi momenta. An approximate value (d = 20 MeV) was used as the energy for all bound-state pairs. The angleaveraged Pauli operator is 22) Q ( r , k,
= o
k 2 +¼K 2 < K 2
=1
k-½K > K~
= (k ~ +¼K ~ - K~)/Kk
otherwise.
(12)
Since bound-state oscillator wave functions are sometimes used to represent the intermediate states a6), we carried out a comparison of the plane wave results with those for the lowest oscillator excitations in the special case of terms diagonal in the relative quantum numbers. To second-order, the oscillator reduced integral is given by
(nlSJTltlnlSJT)2 = (nlSafTlvlnlSJT) + ~, [(nlS°fTIvIn'I'SJT)[2, n'r *hi
(13)
Enl - - En, v
where, in order to conserve parity, N, and S¢, the sum is restricted to states an even number of shells higher in excitation. These matrix elements were evaluated for an oscillator parameter b = (h/mco)~ = (2.6) ~ fm with the sum including only states two shells up. The elements were then compared with similar results tabulated in ref. as). In most cases, the two differed by no more than 20 %. If we now collect results and include the exchange term, we find for the t-matrix elements to second order
[(1 +c5~b)(1 +tSca)]-~r
½ jji} (nlN.~Llnal.nblbL) x ~ A {1~ ½ ½ jj~){1~ A ½ S S' x (n'l'N.~L'[n¢ 1~nd la L')[-I - ( - )t +s + r-[
x (nl, N.~q';LSJT[v+t2[n'l', N.~q~;ESJT),
(14)
where 6~b -- 6..nb3,.,b~i.ib and where ([v+t2[) is given in eqs. (6)-(12). The matrices were constructed and diagonalized on the Penn State IBM 360-67 computer. Single-particle energies were taken from the spectrum of 1sC.
72
N. FREED
AND
P. OSTRANDER
4. Results and discussion
The results for the energy levels of ~+C and I+N in which the approximation has been made o f an inert spherical ~2C core are shown in figs. 2 and 3. The 14C ground - - 0 -
5- - 2 + - - 3 ÷ 2-
4-
2+ 0-
0+
o3;
2+
- - - 2 -
3-
:>
- - 2 4 .
,-
- - 0 -
3"
- - 1 4 .
2I0 +
O-
- - 3 - -
li+ 3-
I- - 2 -
- - 3 -
3- - 0 -
>(.9 O: bJ Z b.I
3+
3+
I"
2-
- - 0 -
I÷
ca
I-
>(.9 nw z hJ
O-
- 2 -
-
I0-)
_
_
14.
-I- - 0 + ( I I+
)
- - 2 - - 0 + ( i | - - i +
--2-
-3-
-I-
Oe(I]
0 4. _ _ 0
4.
-4-
-2-
-5-
-30+
i+
-6
-4 EXP
TI
EXP
T2
Fig. 2. Energy spectra for " C assuming a closed 12C core and configuration mixing in the s-d shell. First- and second-order results for the Tabakin potential are shown.
TI
T2
Fig. 3. Energy spectra for I+N. See caption to fig. 2.
state is placed at (binding-energy data from ref. 37)) E (t +C_g.s.) = _ B.E.(I #C) + B.E.(12C) +2[B.E.(taC)-B.E.(12C)]
= - 3 . 2 3 MeV,
(15)
and the 1+N ground state at E ( 1#N-g.s.) = - B.E.(t +N) + B.E.(I'C) + [B.E.( ~3 N ) - B.E.(12C)] +[B.E.(~aC)-B.E.(~2C)]
= - 5 . 6 1 MeV.
(16)
The oscillator spacing hco was taken to be 15.6 MeV. For comparison we have shown results with and without second-order corrections.
MASS-14
73
NUCLEI
I I
I I
0 o•
o o°
e-
E e-
c~c~ I
IIII
-i e-,
8 illl Z
oo ~o
~o o~
II
~ oo
III
II
O
II
O
-i
~.~
-~
I I
8 II
I +
I
+
I
r
+
II
l
I
I 0
I
,.s
il
il
74
N. FREED AND P. OSTRANDER
For both nuclei, the most obvious discrepancy lies in the insufficient binding of the ground' state. This result is due in part to the fact that these levels, being largely p-shell in origin, will mix strongly with the core states of our approximation. This point will be elaborated upon shortly. We also find, in contrast to some earlier predictions, that the inclusion of the d~ and s½ single-particle states plays a large role in lowering the ground-state binding energies. The carbon ground state was lowered by 1.5 MeV and the nitrogen ground state by 1.0 MeV by the inclusion of mixing in the s-d shell. Although the d~r state is important in reproducing certain electromagnetic transitions in this region 3o), we find that its inclusion has very little effect on the spectra. Sebe 38) has pictured the odd-parity levels as arising from the coupling of the singleparticle states in the s-d shell with the lowest p9 states in ~3C. One result of his calculation is the large departure of the non-normal-parity states from the configurations p~s~ or p~d~. The appearance of a core-excited state in ~3C at 3.7 MeV gives additional evidence that the odd-parity states will not be well reproduced in our calculation. The inadequate depression of the ground and low-lying excited states in nitrogen and ground state in carbon together with overbinding in the second 0 ÷ state in carbon and odd-parity states in carbon and nitrogen are strong indications that important modes of core excitation have been omitted. It has already been shown for mass-18 nuclei 22) that the inclusion of certain types of intermediate states 39) (three-particleone-hole in that case) not contained in the t-matrix causes a level spreading, the nature of which is a depression of ground and low-lying states, an elevation of high-lying states and an unalteration of the positions of states at intermediate energy. We return to this point below. The wave functions in first and second order are given in table 3. From them we calculate the ~4C(g.s.) ~ 14N(g.s.) beta transition which arises almost entirely from the allowed Gamow-Teller matrix element MGT, where in standard notation 39)
M~T = f o
2 = I(JrMrl
~ o°~IJ~M~)I
(17)
We find that these wave functions are totally unable to provide the almost complete cancellation of S- and P-state components needed to explain the transition. The calculations were then carried out for the 0 ÷, 1 states in t4N and 14C and 1+, 0 states in 14N under the assumption that they originate entirely from the configuration p-2. Once again we fix h¢o = 15.6 MeV and treat the s(.p~ 1)-s(p~ ~) = As splitting as an adjustable parameter. Although the centroid splitting is not known, the location of the first ½- state in ~SN at 6.3 MeV provides a rough orientation. In the calculations we varied the splitting until the beta-decay matrix element M2T vanished and based binding energy considerations on the energies resulting from that choice of As. We found complete cancellation to occur at As = 7.0 MeV in first order
M/~$-14 NUCLEI
75
and 6.8 MeV in second. The wave functions are listed in table 4 where they should be compared with results of Zamick's calculation 32) using the HJ potential. Although the wave functions are quite similar, it must be noted that in order to compensate for the very strong tensor force in the (state-independent) HJ calculation, it was necessary to use a rather large spin-orbit splitting to achieve cancellation. TABLE 4 Eigenvalues and eigenvectors o f 14C and 14N for states o f p-* Nucleus
J~
14C
0+
--1.920
0+ 14N
Energy (MeV) exp. calc.
p~-2
p~-Z
p ~ l p ~ 1 Ae(MeV)
--1.185 --1.572
0.949 0.949 0.975
0.316 0.316 0.220
7.0 6.8 9.2
4.680
12.642 10.521
0.316 0.316
--0.949 --0.949
7.0 6.8
1+
--4.830
--0.343 -- 1.405
1+
--0.880
0.654 --0.749
0.915 0.933 0.917 --0.394 --0.345
0.051 0.027 --0.014 0.331 0.348
0.401 0.358 0.399 0.857 0.872
7.0 6.8 9.2 7.0 6.8
The results are tabulated in the order (i) first-order Tabakin, (ii) second-order Tabakin and, where applicable, (iii) H a m a d a - J o h n s t o n s~).
With these wave functions we can calculate the ground-state magnetic dipole and electric quadrupole moments of ~4N. Leaving aside for the moment considerations of renormalization of electromagnetic operators arising from explicit velocity dependence of the two-nucleon interaction, we calculate that p(t4N-g.s.) = 0.33 nm and Q(~4N-g.s.) = +0.019 b, to be compared with the experimental values of 4~) 0.4036 n.m. and +0.01 b. With A~ = 6.8 MeV, we find that the binding energy of the 14C ground state is -1.57 MeV, and that of ~4N is -1.41 MeV. These should be compared with the "experimental" values of - 1.92 and -4.83 MeV, respectively, as calculated from the binding energy of two holes in ~60. The splitting between the two lowest 1 ÷, 0 states in nitrogen is 0.66 MeV versus the experimental 3.95. It is interesting to note that although the ~4N ground state is given about 3.5 MeV too high, the first 1 +, 0 excited state is correct to less than 0.15 MeV, and the 0 +, 1 first excited state is less than 1 MeV too high. For ~4C, the corresponding discrepancy for the ground state is 0.35 MeV. The second 0 ÷ state is poorly reproduced, in keeping with results of an earlier phenomenological calculation 12). The departures from experiment appearing in both approaches originate for the most part from the severe restrictions on the size of our model space. The great sensitivity of the lowest- and highest-lying levels in ~sO and ~SF to admixtures of 3p-lh
76
N . F R E E D A N D P. O S T R A N D E R
states (fig. 4b) has been mentioned earlier. In addition, a more recent calculation 42) with the HJ potential, in which the approximate state dependence of the t-matrix was taken into account, illustrated for lSF and 1sO the need for inclusion of 2p and
(o)
(b)
(c)
(d)
Fig. 4. Diagrams considered in the mass 18 calculation o f ref. 42). The first diagram (a) represents the t-interaction, while the other three represent 3p-lh, 2p and 2h correction terms, respectively.
4-
4÷
4+
3+
3+
0+
0+
4÷
3"
23• 0÷ Z+
>.
3+
3+
).-
oW Z bJ
~
4
÷
-I2+ 2+
-2-
- - 2 e
- - 0
_ _ 2 + - - 2
•
~
+
+
-3~
0
0+
-4-
EXP
TI
T2
HJI
+ 0
HJ2
Fig. 5. Energy spectra o f aaO. First- and second-order Tabakin results are compared to those o f K u o ,2). The notation HJ1 refers to the inclusion of diagrams (a) and (b) (fig. 4) whereas HJ2 refers to the inclusion o f (c) and (d) in addition.
2h diagrams (figs. 4c, 4d) which proceed mainly through the 2p-lf and lp shells, respectively. Especially important are the 4p-2h intermediate states of the 2h diagram, which are low-lying and highly deformed 4a) and contribute substantially to the lowering of the 1 +, 0 ground state of fluorine by about 2 MeV (figs. 5 and 6). Thus,
MASS-14
77
NUCLEI
it appears reasonable to attribute many of our discrepancies to the neglect of corresponding modes of core excitation not explicitly included in our t-matrix. In this regard, we felt that a direct comparison of the second-order Tabakin results for fluorine and oxygen with those of the HJ potential might prove interesting. Therefore, we repeated our calculations for mass-18 assuming an inert 160 core t. The results are shown in figs. 5 and 6 and summarized in tables 5 and 6. The spectra for the HJ results are from ref. 42) and the wave functions from ref. 22) [no wave functions were listed in ref. 42)]. The T2 results are quite similar to those for HJ2 (all
O3* - - i +
~ 1 -2-
>(.9 ne hj z hj
34.
~ 1
__21"
-I-
+ -
2÷ ÷ 2÷i I )
24"(I ]
2+
0+(I)
i+,Z -
,5 a+÷
~ 2 4 24. ( I )
~ 2 4 "
, 14.
• 2+ ( ! ) --24"(I) ~
2
+
--O+(l} -3-
I+
- -
i÷
~ o ~+_ -4-
~ - - 0 4 " ( i x-3+
i÷ 0+(11 54.
5+
--.3÷
~4.
~4. ~ 0 + ( I ) 3•
I - - i
+
-5"
.
,i ÷
-6
EXP
TI
T2
Hdt
Hd2
Fig. 6. Energy spectra of laF. See caption to fig. 5.
diagrams in fig. 4 included) with the exception of the energies of the 1 +, 0 ground state and 0 +, 1 excited state in lSF and the analog 0 + oxygen ground state. The 18F ground state is seen to be very sensitive to the inclusion of second-order effects in the Tabakin interaction and to the core diagrams (figs. 4c and d) in the HJ potential. In our calculations, a number of effects have been neglected in addition to those already mentioned. Some of these are minor (isospin impurities, Thomas-Ehrmann shifts and Coulomb force), but others should be included in any serious attempt to get detailed agreement with experiment. For example, center-of-mass corrections might be important for the 0 - , 1- and 2 - states 4s). We have ignored such corrections here because previous indications ~1) are that they are small. There also arises the possibility that higher-order terms in the t-matrix which have not been taken into account here are non-negligible. A recent work by Halbert et al. [ref. 46)], in which HJ calculations were carried out in the lp shell, indicated the t Similar first-order calculations have been reported by Lee and Baranger 44).
78
N. FREED AND P. OSTRANDER TABLE 5
Eigenvalues a n d eigenvectors o f asO J~
Energy (MeV) exp. calc.
d~ ~
0+
--3.90
--2.38 --2.64 --4.22
0.875 0.882 0.901
2+
--1.92
--1.59 -- 1.79 -- 1.89
0.757 0.754 0.782
4+
--0.35
--0.61 --0.75 --0.27
0.964 0.963 0.954
0+
--0.27
--0.05 --0.16 --0.06
--0.403 --0.399 --0.322
2+
0.02
-[-0.04 --0.11 0.23
--0.641 --0.647 --0.607
3+
1.47
0.66 0.52 I. 11
4+
3.22
3.48 3.37 4.26
d~s~.
d~d~
s~ 2
s~}d~
0.410 0.403 0.324 0.609 0.614 0.579
0.099 0.094 0.090
d~ ~
0.257 0.245 0.287 0.193 0.192 0.190
0.097 0.093 0.099
0.267 0.269 0.300 0.912 0.915 0.945 0.753 0.750 0.788
--0.023 --0.003 --0.008
0.008 0.003 0.025
1.000 1.000 1.000
--0.267 --0.269 --0.300
--0.082 --0.070 --0.055 0.145 0.137 0.105
0.007 0.011 --0.007
0.964 0.963 0.954
Results are tabulated in the order: (i) first-order T a b a k i n , (ii) second-order T a b a k i n a n d (iii) statei n d e p e n d e n t H a m a d a - J o h n s t o n ~2) (hto = 14 MeV). TABLE 6 Eigenvalues a n d eigenvectors o f ~SF J=
Energy (MeV) exp. calc.
dt2
1+
--5.00
--3.29 --4.33 --4.83
0.542 0.580 0.571
3+
--4.06
--2.98 --3.63 --4.04
0.550 0.552 0.527
5+
--3.88
--2.89 --3.43 --3.69
1.00 1.00 0.557 0.573 0.554
1+
--1.87
--0.54 --1.18 --1.23
2+
--1.65
--1.01 - - 1.86 --1.59
d~S~r
dtd~
--0.621 --0.612 --0.629 0.810 0.809 0.817
s~ 2
s~. d~}
d~}2
0.553 0.524 0.507
--0.103 --0.093 --0.143
--0.067 --0.072 --0.040
--0.201 --0.204 --0.234
--0.007 --0.012 +0.016
1.00
--0.186 --0.125 --0.163 0.764 0.768 0.753
--0.791 --0.797 --0.803
0.515 0.509 0.536
See table 5 for a r r a n g e m e n t o f table. H a m a d a - J o h n s t o n results f r o m ref. n ) .
--0.097 --0.070 0.110 0.388 0.389 0.382
--0.142 --0.127 --0.103
MASS-14
NUCLEI
79
possible importance of including such higher-correlation effects. In a recent calculation by Goldhammer et aL 47), the inclusion of a three-body vector force 4s) was found to make a contribution to the binding energy of mass-14 nuclei of ~ 2 MeV. Our results, then, are not inconsistent with those of earlier studies. In those cases where comparison with hard-core results is possible we obtain excellent agreement for the wave functions. We are able to explain the 14C ~ ~4N beta decay with a reasonable value of p-state spin-orbit splitting and at the same time predict the fast laNe ~ laF decay ( l o g f t = 2.89 calc. versus 3.05 exp.). The spectra are very sensitive to higher-order processes, some of which are included in the t-matrix, others of which are hot. Any attempt at fitting details of binding energies will have to incorporate such corrections in a reliable manner. It is a pleasure to thank Drs. T. T. S. Kuo and K. T. R. Davies for use of computer codes and Dr. F. Tabakin for a correspondence concerning the parameters of his potential. We are grateful to Dr. P. Goldhammer for interesting discussions and for information concerning his work in ref. 47). We also wish to acknowledge the computational assistance of The Pennsylvania State University Computation Centre.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20)
21) 22) 23) 24)
W. W. True, Phys. Rev. 130 (1963) 1530 D. Kurath, Phys. Rev. 101 (1956) 216 D. Inglis, Revs. Mod. Phys. 25 (1953) 390 R. K. Sheline and K. Wildermuth, Nucl. Phys. 21 (1960) 196 M. Soga, Phys. Rev. 163 (1967) 995 and preprint D. lnglis, Phys. Rev. 87 (1952) 915 D. Kurath, Phys. Rev. 106 (1957) 975 B. Jancovici and I. Talmi, Phys. Rev. 95 (1954) 289 W. M. Visscher and R. A. Ferrell, Phys. Rev. 107 (1957) 781 M . H . Macfarlane and J. B. French, Revs. Mod. Phys. 32 (1960) 567 E. Baranger and S. Meshkov, Phys. Rev. Lett. 1 (1958) 30 M. A. Nagarajan, Nucl. Phys. 42 (1963) 454 S. Cohen and D. Kurath, Nucl. Phys. 73 (1965) 1 A. B. Volkov, Nucl. Phys. 84 (1965) 33 R. K. Gupta and P. C. Sood, Phys. Rev. 152 (1966) 917 I. Unna and I. Talmi, Phys. Rev. 112 (1958) 452 D. Amit and A. Katz, Nucl. Phys. 58 (1964) 388 K. E. Lassila, M. H. Hull, Jr., H. M. Ruppel, F. A. McDonald and G. Breit, Phys. Rev. 126 (1962) 881 T. Hamada and I. D. Johnston, Nucl. Phys. 34 (1962) 382 J. F. Dawson, I. Talmi and J. D. Walecka, Ann. of Phys. 18 (1962) 339; H. A. Bethe, B. H. Brandow and A. G. Petschek, Phys. Rev. 129 (1963) 225; B. L. Scott and S. A. Moszkowski, Nucl. Phys. 29 (1962) 665 R. D. Lawson, M. H. Macfarlane and T. T. S. Kuo, Phys. Lett. 22 (1966) 209; R. D. Becker and A. D. McKeller, Phys. Lett. 21 (1966) 201 T. T. S. Kuo and G. E. Brown, Nucl. Phys. 85 (1966) 40 M. K. Pal and A. P. Stamp, Phys. Rev. 158 (1967) 924; C. M. Shakin, Y. T. Waghmare, M. Tomaselli and M. H. Hull, Jr., Phys. Rev. 161 (1967) 1015 P. Goldhammer, Phys. Rev. 116 (1959) 676
80 25) 26) 27) 28) 29) 30)
31) 32) 33) 34) 35) 36) 37) 38) 39) 40) 41) 42) 43) 44) 45) 46) 47) 48)
N . F R E E D A N D P. O S T R A N D E R
C. Bressel, A. Kerman and E. Lomon, Bull. Am. Phys. Soc. 10 (1965) 584 M. Razavy, G. Field and J. S. Levinger, Phys. Rev. 125 (1962) 269 A. M. Green, Nucl. Phys. 47 (1963) 671 F. Tabakin, Ann. of Phys. 30 (1964) 51 T. T. S. Kuo, E. Baranger and M. Baranger, Nucl. Phys. 81 (1966) 241 E. K. Warburton, H. J. Rose and E. N. Hatch, Phys. Rev. 114 (1959) 214; E. K. Warburton and W. T. Pinkston, Phys. Rev. 118 (1960) 733; D. E. Alburger and E. K. Warburton, Phys. Rev. 132 (1963) 790; E. K. Warburton, J. W. Olness, D. E. Alburger, D. J. Brodin and L. F. Chase, Jr., Phys. Rev. 134 (1964) B338; D. E. Alburger and E. K. Warburton, Phys. Rev. 148 (1966) 1050; J. W. Olness, A. R. Poletti and E. K. Warburton, Phys. Rev. 154 (1967) 971 D. E. Alburger and E. K. Warburton, Phys. Rev. 148 (1966) 1050 L. Zamick, Phys. Lett. 21 (1966) 194 M. Moshinsky, Nucl. Phys. 13 (1959) 104 A. R. Edmonds, Angular momentum in quantum mechanics (Princeton University Press, Princeton, 1957) A. K. Kerman and M. K. Pal, Phys. Rev. 162 (1967) 970 B. R. Barrett, Phys. Rev. 154 (1967) 955 L. A. K6nig, J. H. E. Mattauch and A. H. Wapstra, Nucl. Phys. 31 (1962) 18 T. Sebe, Prog. Theor. Phys. 30 (1963) 290 G. F. Bertsch, Nucl. Phys. 74 (1965) 234 E. J. Konopinsky and M. E. Rose, in Alpha-, beta- and gamma-ray spectroscopy, ed. by K. Sieghahn (North-Holland Publ. Co., Amsterdam, 1965) p. 1327 Nuclear Data Sheets, compiled by K. Way et al. (NAS-NRC, Washington, D.C.) T. T. S. Kuo, Nucl. Phys. A103 (1967) 71 G. E. Brown and A. M. Green, Nucl. Phys. 85 (1966) 87 C. W. Lee and E. Baranger, Nucl. Phys. 79 (1966) 385 N. Freed and P. Goldhammer, Phys. Rev. 136 (1964) B1260 E. C. Halbert, Y. E. Kim and T. T. S. Kuo, Phys. Lett. 20 (1966) 657 P. Goldhammer, J. R. Hill and J. Nachamkin, to be published A. M. Feingold, Phys. Rev. 101 (1956) 258, 105 (1957) 944, 114 (1959) 540