A study of deformation in light nuclei

I 1.D.I

l

I

I

Nuclear Physics A126 (1969) 305--331; ~ ) North-Holland Publishing Co., Amsterdam N o t to be reproduced b y p h o t o p r i n t or microfilm without written permission f r o m the publisher

A STUDY

OF DEFORMATION

IN LIGHT

NUCLEI

(I). Application to the ground state band of 2°Ne and to the low-energy spectrum of 19F H. G. BENSON t and B. H. FLOWERS Schuster Laboratory, Manchester University, England Received 27 September 1968

Abstract: The properties of the ground state band of ~°Ne are shown to be well described by the set of wave functions projected from a single intrinsic state. Particular deformed states of nuclei around 1sO may be constructed by breaking up the leo core in such a manner as to produce two protons and two neutrons in the (2s, ld) shell. Following an observation by Christy and Fowler, the wave function for these four particles is described in terms of the ground state 20Ne band and deformed states are obtained by coupling lp hole states on to the members of this band. The odd-parity levels of I~F are described by coupling p~ and pk proton holes on to the ground state 2°Ne band and it is shown that the measured E1 transitions in 19F provide evidence for states of this structure. It is also shown that the low-lying even-parity levels of X°F are well described in terms of a single K = ½+ band, and these states are compared directly with those obtained from a full intermediate coupling calculation. Overlaps of over 99 % are obtained for the ground state (½+) and the ~+ members of the band. The structure of the K = ½+ band is interpreted on the basis of the nucleus having permanent P4 as well as P~ (Nilsson) type distortions. Electromagnetic transitions in 19F are discussed in detail and the results for the odd-parity states are compared with those of a similar calculation by Harvey in the SU3 scheme. A detailed study of the levels in XgF in the region of 4 to 5 MeV excitation is made and the extent to which core deformation is important in the even parity states is discussed on a semi-quantitative basis. The properties of the 3.91 MeV level (possibly a doublet) are crucial to the detailed interpretation of this nucleus. 1. Introduction A g r e a t d e a l o f e f f o r t has b e e n m a d e in the p a s t to d e s c r i b e t h e d e t a i l e d p r o p e r t i e s o f n u c l e i in the r e g i o n o f t h e d o u b l y m a g i c n u c l e u s 160. T h e c a l c u l a t i o n s o f E l l i o t t a n d F l o w e r s ~) o n t h e n u c l e i o f m a s s 18 a n d 19 a n d t h o s e o f R e d l i c h 2) on m a s s 19 i n i t i a t e d a t t e m p t s t o d e s c r i b e the l o w - l y i n g e v e n - p a r i t y levels o f t h e s e nuclei in t e r m s o f a c o m p l e t e set o f states d e r i v e d f r o m m i x i n g all p o s s i b l e c o n f i g u r a t i o n s w i t h i n t h e (2s, l d ) shell. T h e d e s c r i p t i o n o f 19 F a c h i e v e d by this m e t h o d was in e x c e l l e n t a g r e e m e n t w i t h the e x p e r i m e n t a l d a t a t h e n a v a i l a b l e . U n t i l v e r y r e c e n t l y little a d d i t i o n a l e x p e r i m e n t a l i n f o r m a t i o n a b o u t the e v e n - p a r i t y states h a d b e e n a c c r u e d . T h i s is in spite o f t h e f a c t t h a t m a n y levels p r e d i c t e d at a b o u t 4 M e V e x c i t a t i o n in t h e E l l i o t t a n d F l o w e r s c a l c u l a t i o n s a r e as yet u n i d e n t i f i e d . A l t e r n a t i v e t r e a t m e n t s o f the levels o f a 9F on the basis o f a r o t a t i o n a l m o d e l h a v e b e e n g i v e n by P a u l 3) a n d also by R a k a v y 4). T h e n u c l e u s 2 ONe is a n o t h e r e x a m p l e f r o m this m a s s r e g i o n w h i c h has a r o t a t i o n a l s p e c t r u m t y p i c a l o f a d o u b l y e v e n n u c l e u s a n d w h i c h has also b e e n well d e s c r i b e d b y full i n t e r m e d i a t e c o u p l i n g c a l c u l a t i o n s for t Present address: Nuclear Physics Laboratory, Oxford, England. 305

306

H . G . B E N S O N A N D B. H . F L O W E R S

configurations (2s, ld) 4 [refs. 5, 6)]. A relationship between these alternative descriptions of nuclear spectra has been demonstrated by Elliott 7) who showed that shell model wave functions, classified according to irreducible representations of the group SU3, have a structure similar to the states of a rotational band. In the present paper wavefunctions for the ground state band of 2 ONe and for a K = ½+ ground state band of 19 F are derived in a shell model formalism by projection from intrinsic wave functions constructed from a generalized Nilsson-like model. During the last few years a thorough experimental survey of the low-lying energy levels in the mass 18 nuclei has revealed the presence of states which cannot be described in terms of simple (2s, I d) 2 intermediate coupling calculations. The large electric quadrupole transitions observed in 180 and also the presence of additional levels suggested that core-excitation was important - as was also implied by the presence of low-lying even parity levels in 160. Engeland s) and Brown 9) proposed specific methods for constructing 4p-2h states in 1sO using the SU3 classification and introduced these additional states into the (2s, ld) 2 basis to produce a good description of the observed spectra and transitions. Alternative descriptions of 180 given by Federman and Talmi 1o) and Benson and Irvine 11) used the ground state K = 0 + band of 2 ONe on which to build these 4p-2h states. Following this procedure the odd-parity basis states of 19F a r e constructed by coupling a (lp) proton hole on to the ground state 2 ONe band. The results and implications of using a model of this type to describe the low-lying odd-parity states of 19F a r e presented here and compared with a similar description given by Harvey 12) in SU3. In similar calculations on 180 and laF [refs. 11, 13)] the deformed 4p-2h states are constructed by coupling a complete set of states of configuration (lp) -2 on to the ground state 2°Ne band and the basis for diagonalization is completed with the complete set of states from configurations

(2s, ld) 2. The construction of states in the manner outlined was first investigated by Christy and Fowler 14) in an empirical approach. They considered the excited states of 160 (0 + at 6.05 MeV), 170 (½- at 3.07 MeV), 17F (½- at 3.10 MeV), 19F (½- at 0.110 MeV) and 2°Ne (0 + ground state) as an addition of four (2s, ld) particles to the ground states of 12C, 13C, 13N, 15N and 160, respectively. They showed that there resulted a series of values for the binding energy of these four particles which increased by only 3.5 MeV from 12C to 160. This fact is well understood on the basis of the model adopted in this work on the assumption that in each case the four (2s, ld) particles be considered in the same K = 0 intrinsic state. The Pauli principle necessitates that deformed states obtained by breaking up the 160 core will lie at low excitation relative to the usual shell-model states only if there are up to two like nucleons in the (2s, ld) shell. This implies that for the positiveparity levels of 19F there is no need to invoke collective core effects (i.e. to include 5p-2h states) below an excitation of at least 4 MeV - a fact well substantiated by the success of calculations working within the space of (2s, ld) 3 configurations only. Calculations in which higher intrinsic states must be utilized to describe core defor-

DEFORMATION I N L I G H T NUCLEI (I)

307

mation have been performed by Federman 15) in 190. The deformed states in 190 are necessarily higher in energy relative to the usual shell-model states than is the case in asO. This is essentially an effect due to the Pauli principle and is well demonstrated in the recent calculations of Ellis and Engeland 16). No description of such nuclei is attempted in this work. The effects of c.m. motion can never be completely eliminated in these calculations [ref. 11)]. The admixture of spurious states can however be easily estimated by calculating <,5~R 2> in the eigenvectors obtained in the calculations, in the manner outlined by Palumbo 17). In all the states obtained in this work the spurious content is found to be very small ( < 0.5 ~ ) and the effects of centre of mass motion are therefore neglected throughout.

2. Application to 2°Ne 2.1. CONSTRUCTION OF G R O U N D STATE BAND OF ~°Ne

The Nilsson model 18) is taken as a guide to postulate the existence of a low-lying k = ½+ orbital in the (2s, ld) oscillator shell. This is expressed in the Nilsson approximation by expanding the creation operator for a particle in this orbital in terms of the (2s, ld) spherical creation operators with the aid of deformation parameters Cg a t = ~ C~af~.

(1)

J

Assuming there exists a plane of symmetry at right angles to the intrinsic z axis it follows that 5 -~ + = Z C j ( - 1)J-~a+_~, (2) J

and the normalization of the intrinsic function implies that Z C~ = 1.

(3)

J

Thus each single-particle orbital labelled by k is doubly degenerate and can accommodate two protons and two neutrons in accordance with the Pauli principle. This suggests a possible trial function for the ground state band of/ONe, namely [2°Ne J M T = 0 K = 0> = PM(~(n)c~-½(n)c~(P)~-~(P))[ s + + + + 16 O),

(4)

where the labels n a n d p denote neutron and proton respectively, [160> is the shell model closed-shell wave function, and p S is the angular momentum projection operator defined by Unna 19). Because of symmetry and because of the restriction on the sum in eq. (1), J can take only the values 0, 2, 4, 6 and 8. Using a shell-model Hamiltonian incorporating a reasonable interaction between the extra-core nucleons, the energy of each angular momentum state is then minimized separately with respect to the coefficients Ci as variational parameters.

308

H. G. BENSONAND B. H. FLOWERS

The wave function is the projected (but restricted) Hartree-Fock intrinsic wave function as used by Bassichis et al. 2o). The method described is different from their procedure because they minimized the energy of the intrinsic function with respect to the coefficients C i before projection. It will be pointed out later that their criterion is in fact a good one for picking out the important low lying intrinsic states in a deformed nucleus. This point has also been discussed by Dreizler et al. 55). Fig. 1 shows the energy spectrum obtained by minimizing the energy of each state [2°Ne J M T = 0 K = 0) separately with respect to the deformation parameters. Experiment

Theory

11-99

11.21

8+

8+

IMI2= IO.5

8.79 8-13

6+

6+

2 .t.7 IMI = 2 8 . e IMI~ 20-5

4*

4-25 3.68

44"

]

IMI~ 16.5 24-

1.62

O Binding

.1.5 IMI 2- I5.1-1.z

23.7 MeV

1.63

2÷

0

O* 23.5 M¢V

Fig. l. Energy levels and E2 transitions for the ground state band of 2°Ne (energy levels taken from refs. ~s,29), [M[~ in Weisskopf units defined in ref. ~a).)

The Hamiltonian used in this calculation and also in the remainder of the work is of the form Vijua~ aj atak. (5) i

ijkl

The e i are taken directly from the empirical level scheme of 170 (assumed single particle). The interaction Vused is the Kallio-Kolltveit interaction 21) and the MoszkowskiScott separation method 22) is used to deal with the hard core. The separation distances used are 0.925 fm for the triplet potential and 1.045 fm for the singlet, with an oscillator constant b -- 1.69 fm appropriate to the nuclear size in this region. The agreement of the calculated spectrum with the experimental one of Litherland et a[.

DEFORMATION IN LIGHT NUCLEI (I)

309

[ref. 24)] is good and the binding energy of the valence particles is accurately reproduced. When the calculation is repeated with the more realistic Yale potential using the matrix elements of Lynch and Kuo 25), a similar energy spectrum is obtained but with about one MeV less binding. In calculating the energy of the state (4) the fact is used that the projection method described by Unna 19) is of great flexibility. Thus for a product of n single-particle wave functions ~bjk

= ~

J1J2M1M2 Jl

(J1K~,J2K2IJK)(J1M,, JEM2IJM)

× PMI( lk

J2 " ' " ~jmkm)PM2((~j,.+ 1 k i n + , ' ' " ~j,,k.)'

(6)

wherel < m < n a n d

K=~k~,

K~=~k,,

i=l

i=1

K2= ~

k i,

(7)

i=m+l

and (J1M~, J2M2IJM), (JIK1, J2K2IJK) are Clebsch-Gordan coefficients. Thus the wave function projected from a n particle intrinsic state can be expressed in a variety of ways according to the manner in which the single-particle wave functions are partitioned before projection. The matrix element of the two-body interaction V between the states (4) can therefore be expressed quite simply as a linear combination of two-body matrix elements with overlaps of two-particle states without the complication of introducing recoupling coefficients (in this case 9-j symbols). This probably makes this method of projection easier to handle than the numerical evaluation of a Hill-Wheeler integral 26). 2.2. ELECTROMAGNETIC

TRANSITIONS

I N 2°Ne

Although it is quite satisfactory to calculate energies using harmonic oscillator wave functions and indisputably more convenient, it is inadequate to retain these wave functions when calculating expectation values of r n, which determine the electric transitions of multipolarity 2 n. In these calculations eigenvectors were obtained using harmonic oscillator wave functions in the diagonalization of the Hamiltonian (5). In calculating electric transitions however these same eigenvectors are used but with Woods-Saxon single-particle radial functions replacing those of the harmonic oscillator. The radial integrals are calculated numerically in the manner of Barton et al. 27). The calculated electric quadrupole transition rates for the 4 + --* 2 +, 2 + -~ 0 + transitions in 2 ONe are shown in fig. 2 as a function of the effective charge. Both these transitions are well reproduced with a small effective charge of about 0.1e, using WoodsSaxon wave functions in calculating the radial integrals. Using harmonic oscillator radial functions an effective charge of 0.25e is required to reproduce the transitions. The theoretical E2 transition strengths among all the members of the ground state 2 ONe band are shown in fig. 1. All these transitions are evaluated using an isoscalar effective charge of 0.08e.

I I

1

I I

I

O.le

O-2e

2+-'0*

o.2e

4+_.. 2 +

Effective

Charge

Effective Charge

/ / // / / /;

////y~///////,

o.le

y.~'~U////

Fig. 2. E2 transitions in ~°Ne as a function of effective charge. (WoodsSaxon wave functions are used in calculating the radial integrals.)

0

IO

20

IMI

0

I0

20

IMI

Fig. 3. Interaction energy (in MeV) of Y = 0 + ground state o f ~°Ne as a function o f deformation parameters.

T

c~

(ua

DEFORMATIONIN LIGHTNUCLEI (I)

311

2.3. OTHER BANDS IN ~0Ne Two extremely interesting and important features emerging from the calculation in subsection 2.2 are firstly that the energy minima are sharply defined (at least for the low angular m o m e n t u m states) and secondly that all the minima occur in the same region of Cj space. This first point is illustrated in fig. 3 which is a contour map of the energy of the lowest member of the K = 0 + ground state band of 2°Ne over the unit circle defined by taking C~ and C~ as axes and taking C~ > 0. It is seen that the topograph3 2

nn p p

Emin= -20.1 McV

Emin = -12.3 McV _ _ ~ ÷ 2

TL 0

Emin= -11.4 MeV

np

np v~

5+ 2

TL I

np XX

Emin =

- 13.8

McV

L+ Z

Fig. 4. [ntcractional energies o f some K = 0, T = 0 intrinsic states in 20Nc. ( T ' denotes the isotopic spin o f the n-p pair in each orbital).

of the surface is by no means simple and a subsidiary minimum occurs in the right upper quadrant. The fact that the minima for each member of the band occur in the same region in C s space means that the intrinsic state energy also has a minimum in this region. This validates the procedure of Bassichis et al. a0), which is to minimize in the intrinsic scheme before projection, and confirms the recent findings of Saltpathy and Nair [ref. ao)]. Following their approach a k = ___~z+ single particle orbital is created along with the k = +½+ orbital and the energies of the various K = 0, T = 0 intrinsic states which can be constructed are examined as a function of the deformation parameters (varied independently in both orbitals). These intrinsic states, their construction

312

H. G. BENSON AND B. H. FLOWERS

and ergy on minimization with respect to the deformation parameters, are exhibited in fig. 4. The energy gap of over 6 MeV between the original K = 0, T = 0 intrinsic state used to describe the ground state band and the other intrinsic states indicates that other K = 0 + bands in 2°Ne should start at about 6 MeV excitation above the ground state band. The next observed K = 0 ÷ band in 2 ONe starts gratifyingly enough at 6.75 MeV. The ground-state trial wave function for J = 0 prescribed in eq. (4) is essentially a function of two independent variational parameters. It would be interesting directly to compare this wave function with that obtained by diagonalizing the same Hamiltonian in the complete set of J = 0, T = 0 states of configurations (2s,ld) 4 of which there are 21. Such a calculation was performed and yielded an overlap of 98 ~ between the two states. This indicates that the construction of the wave function given in (4) is extremely good and also implies that band-mixing between this ground-state K = 0 band and bands derived from other intrinsic states must be small. The large interactional energy between the four particles in this K = 0 band is about the same as the binding energy of a free e-particle and this extreme stability leads to the use of this band as a base on which to construct the more complicated particle-hole deformed states encountered in is F and 180 [ref. 13)]. Using this concept of a low-lying single-particle k = ½+ orbital in the region above the 160 closed shell, together with the Pauli principle, deformed states are constructed by raising as many particles as are necessary to produce this four-particle K = 0 intrinsic state. In this way the energy expanded in raising the nucleons through one oscillator shell is supplied to a large extent by the large e-particle binding of the four (2s, 1d) nucleons. This prescription for the construction of deformed states is essentially the same as that postulated by Christy and Fowler 14) on the basis of empirical observations.

3. Application to 19F 3.1. E V E N - P A R I T Y

LEVELS

OF

19F

The first shell-model calculations describing states of normal parity in X9F were performed by Elliott and Flowers 1) and Redlich 2) by considering the 19F nucleus as three nucleons coupled to an inert 160 core. They obtained an excellent description of the observed low-lying even parity states, in particular the ground state ½+, the 5+ state at 198 keV and the ~+ state at 1.55 MeV were well described. This model was also successful in accounting for the observed ground state magnetic moment and later in also predicting the magnetic moment of the 198 keV level 51). This prediction was subsequently verified by experiment 31). Elliott and Flowers also predicted a ~+ state at about 2.5 MeV excitation and all subsequent models have predicted a 9 + state in this region of excitation. There is a state observed at 2.78 MeV for which the spins ~ and ~ remain possible 32). The spin ~z for the 2.78 MeV level is consistent with an angular correlation measurement of Huang et al. 33), although the possibility of J = ) was not considered in their analysis.

DEFORMATION IN LIGHT NUCLEI (I)

313

An alternative description of the even-parity levels of ~9F from the viewpoint of the rotational model has been given by Paul 3). Working with the Nilsson model as a guide for deriving reasonable values for the decoupling parameter, and also assuming reasonable values for the m o m e n t of inertia parameters, he mixed K = ½+ and K = ~÷ rotational bands. He required the final spectrum to reproduce the lower ~-÷ state at around 1.55 MeV and the remaining 3+ state at about 4 MeV excitation. Thomas et al. 32) have repeated this calculation with different parameters in order to produce the second ~+ 2 state at 4.56 MeV. This was in accord with their assignments of J = ~2 on the basis of angular correlation measurements to both the 3.91 MeV and 4.56 MeV levels and speculative assignments of negative parity to the 3.91 MeV and positive parity to the 4.56 MeV levels. The decay scheme of Thomas et aL 32) for the 3.91 MeV level indicates that they find the main decay branch is to the 110 keV (½-) level. This is in disagreement with the findings o f Allen et aL 35) who indicate that the main decay mode is to the ground state. Recent work by Allen and L a w s o n 36) and also by Wright 37) using germanium detectors indicates clearly a large decay branch to ground although a sizable branch to the 198 keV level is also seen. Although the angular correlation 32) should be repeated for the 3.91 MeV level on the basis of the new decay pattern the postulate of J = 3 is probably still valid since the major decay mode is still to a J = ½ level. As regards the 4.56 MeV level however the recent decay pattern of Allen and Lawson 36) is not consistent with that of Thomas et aL 32) and also suggests the existence of a doublet at this energy. For the purpose of this discussion it will be therefore assumed that the 3.91 MeV level has J = 3 but that the spin(s) of the 4.56 MeV level remain to be determined. The results of Paul 3) together with the experimental positive parity levels and the theoretical spectrum of a single K = ½+ band are shown in fig. 5. This K = ½+ band is constructed by placing three particles in deformed k = _½+ orbitals in the manner of Bassichis et al. 20). F r o m this K = ½+ intrinsic level states of good angular momentum are projected and the deformation parameters Cj are varied to minimize separately the energy of each angular m o m e n t u m state. Explicitly the states are ]19F T = ½ K ~ = ½ + J M ) = p~r .2, M~,o¢+~ ~}~P)~o¢+~ ~,n)~o¢+ -~t/n~,,160 ))l

(8)

and the Hamiltonian used is that described in the 2 ONe calculation. Minima are again found in the same region of Cj space as in the 2 ONe calculation - though not as sharply defined. The binding energy of the ground state and the energies of the first three excited states are well reproduced - assuming that the 9+ model state is identified with the 2.78 MeV level. Also in fig. 5 for comparison with the single K -- ½+ band is the energy spectrum obtained in a diagonalization of the Hamiltonian (5) in the complete basis of all the states derived from configurations (2s, ld) 3. The overlaps between many of the states of the single band and the corresponding states obtained in the full intermediate coupling calculation are extremely good - a fact which has been indicated recently

314

H.G.

BENSON A N D B. H. FLOWERS

by H a m a m o t o and Arima 38). In our calculations the overlaps are found to be over 99 ~o for the ½+, ~5 + , ~+, T1 3 + band members and 97.3 ~o for the ~+ band member. The full intermediate coupling calculation furnishes more than one state of j~ _- z+2 and also more than one state of J= = ~-~-+in the region of 5-7 MeV excitation. Thus it is not surprising to find that the lowest ~+ and ~ - + states in the full intermediate coupling calculation have very poor overlap with the 7 + and ~ - + members of the K = ½+ band. These calculations do suggest however that many of the low lying even parity states of 19 F can be interpreted as the members of a single K = ½+ band con-

,~+ '3"2÷ 5, 4.

S'Z+ "~/~÷

_{'~*)

4,.G44.-50

I~ ÷

4 3"ql

q~+

q, -1-

y

~+ Z

(,a)

(b)

1"55

~.+

0

-'3"2"~

Z

0 ENE~'C

~+

Cc)

*'2"

(d)

H EV Fig. 5. Even parity levels o f l g F . (a) R o t a t i o n a l m o d e l calculation o f Paul a), m i x e d K = ½% K = .~bands. (b) Single K = ½+ band. (c) Full intermediate coupling calculation. (d) Experimental level s c h e m e (refs. a2, 55,54)).

taining only small admixtures from other bands. This is contrary to the findings of Paul who describes these states in terms of quite strong mixtures of K = ½-+ and K = v3÷ bands. There is now experimental evidence a¢, 5 o) that the mixing is much weaker than is implied in Paul's calculation and that the situation is close to that of a single K = 1÷ band as clearly suggested by these calculations. 3.2. TRANSITIONS BETWEEN EVEN-PARITY LEVELS The transitions between the even-parity states of 19F obtained in the full intermediate coupling calculation are shown in table 1. It is to be stressed that this calcula-

315

DEFORMATION IN L I G H T N U C L E I ( I )

tion is exactly similar to that of Elliott and Flowers 1) except for the fact that different residual interactions are used in the two cases. The electric quadrupole transitions are calculated using an effective charge of 0.25e with Woods-Saxon wave functions. This value, larger than that used in the 2 ONe calculation, is required to reproduce the measured E2 transitions ~+ (198 keV) --~ ½+ (ground) and ~+ (1.55 MeV) ~ ½+ (ground). A more detailed discussion of the transitions and identification of these model states with known levels in 19 F will be left until sect. 5. TABLE 1 Transitions between even-parity states o f tgF obtained in a full intermediate coupling calculation Transition

IM] 2 M1

(~+, O) -+ (½+, O) (~+, O) --,'- (~+, O) (½+, 0) (~+, 0) ~ (~+, 0) (~+, 1) ~ (,~+, O) ({+, 0) (½+, O) (~+, O) --~ (~+, O) (~+, O) (~+, 0) ( ~ + , O) ~ (~+, O)

2.1 0.01 0.01 0.006 0.01 0.81 0.14

]M[~

Transition E2 8.0 2.3 7.2 7.7 0.32 0.13 0.14 0.77 1.9 0.14 5.2

( ~ + , O) ~ ( ~ + , O) -+ (]+, O) (½+, 1) --~ (~t+, 1) ~ (]+, 0) -,'- (~+, O) ~ (½+, 0) (~+, 1) -+ (~+, O) ~ (~+, 1) ~ (~+, O) (~+, 0) ~ (~+, 0) -+ (½+, O)

M1

E2

0.84 0.15 0.72 0.04

1.97 0.03 0.01 1.03 0.87

0.006 0.04 0.29 0.02 0.006

2.1 0.99 4.35 0.40 0.41 1.13

(in, m) refers to the mth excited state o f angular m o m e n t u m j and parity n.

3.3. D E S C R I P T I O N O F T H E O D D - P A R I T Y STATES O F 19F BY A S I N G L E K = ½- B A N D

Harvey i2) has discussed the low-lying negative parity states of 1 9 F within the framework of the Elliott SU3 model using a conventional interaction. Various investigators 39, 40) have tackled rotational-model calculations in which the low-lying negative-parity spectrum is obtained by coupling a p~ proton hole on to the ground state K = 0 ÷ rotational band of 2 ONe. These calculations naturally furnish a ~ - , 5 doublet at about 1.5 MeV excitation, corresponding to the 2 ÷ state at 1.63 MeV in 2°Ne, and a ~ - , 9 - doublet at about 4 MeV corresponding to the 4 ÷ (4.25 MeV) state of z ONe . Both these doublets are observed experimentally in the low-energy spectrum of 19F. In the spirit of the rotational model, but working in the formalism of the shell model, the possibility is now investigated of reproducing the observed properties of the low-lying negative parity states of 19 F by a single K = ½- band obtained by coupling a p~ proton hole on to the lowest K = 0 ÷ band of 2°Ne. An immediate consequence of using a wave function of this type, together with even-parity states which exclude

316

H . G . B E N S O N A N D B. H . F L O W E R S

core excitation, is that all E1 transitions must be extremely inhibited. For shell-model wave functions to be physically acceptable the centre of mass R of the nucleons must be in an s-state. Since in this description neutrons are not raised from the core it follows that R, (neutron mass-centre) must always be in an s-state; hence so must the proton centre of mass Rp. Since all negative parity states furnished by this model must contain protons in an s-state the E1 operator, which is essentially Rp, will not connect them with the positive-parity states (in which no core excitation is assumed). Thus in general the vanishing of E1 matrix elements is implied by any model which fur-

-I

9

o

-3

-s

7

Z.

3-

'2

~"

7

"~-

I-

s"

3.

7

~-

s.42 4.67 4o4

:

4.00

~'-K_

1.46 1.34

,~--

0

2

-9 -il Energy McV

I-

((l)

{b) (c)

3_ 2 55 -

~2(d)

Fig. 6. Odd parity levels o f Z~F. (a) single K = ½- band. (b) Additional K = ½-, K = ~- bands. (c) All bands mixed. (d) Experimental 63) levels with which these states are identified.

nishes physical wave functions described in terms of states in which only one type of nucleon is excited from the core. In the case of 1 9 F the vanishing of E1 matrix elements in the low energy region is in agreement with the experimental findings of Litherland et 71. 42). Their measurements by Coulomb excitation of the E1 transitions ~2-(1.46 M e V ) ~ ½+(ground) and 1-(110 keV)--* ½+(ground) yield values o f ]MI 2 ~ 10 - 3 W.u. These transitions are inhibited compared with the average for light nuclei 23) which have IMI 2 ,~ 0.04 W.u. The spectrum of the single K = ½- band after minimization with respect to the Cj is shown in the first column of fig. 6. An initial inspection would appear to yield reasonable qualitative agreement with the observed spectrum, but with the following reservations: (i) For the ~-, 7 - and ~-, ~:- doublets it appears that the calculated particle-hole terms give rise to a larger splitting from the 2 ONe band than is observed.

DEFORMATION IN L I G H T NUCLEI ( I )

317

(ii) This "one-band" model predicts a value of 18 W.u. for the E2 decay of the ~-(1.46 MeV) ~ ½-(0.110 MeV) transition, which is in agreement with the measured value of 18 + 8 W.u. However on this simple model all M1 transitions between levels obtained by coupling a Pqr hole on to different ]2°Ne K = 0 J M ) states are forbidden. This is not in accord with the measured value 43) of [MI 2 --- 0.12 W.u. for the ~-(1.46 MeV) ~ ½-(0.110 MeV) M1 transition. (iii) There are quite large admixtures ( ~ 4 ~ ) of spurious states in the wave functions obtained in this model. 3.4.

MIXING ADDITIONAL ODD-PARITY BANDS IN I~F

The shortcomings of the "one band" model together with the possibility of another ~ - level in 19F (possibly at 3,91 MeV [ref. 32)] or at 5.33 MeV) suggest that other odd parity bands might occur in the low energy region. Also another ~ - level has been recently observed in 19F at 5.42 MeV excitation 53). Bands obtained by coupling a p~ hole on to higher bands in 2 ONe are expected according to the work of subsect. 2.3 at an energy of about 6-7 MeV excitation above the ground state. The ½- level observed at 6.4.3 MeV might arise in this manner. Mixing of such bands with the ground state band is expected to be small and is therefore neglected in this work. Alternatively a K = ~ - band and a second K = ½- band may be formed by coupling a p~ proton hole on to the ground state z ONe band. Fig. 6 shows the spectrum of these additional states when mixed and minimized and also the resulting spectrum obtained when these states are mixed with those of the original K = ½- band. Thus in the final calculation the basis states are constructed by coupling a l p proton hole ( j = 1 or ~) 2 on to the states describing the ground state band of ZONe,

[JJ1 Z = ½ J M ) = ~ (jm, JiMllJM)(-1)~-maj_m(P)la°NeJ~M1).

(9)

mMl

The matrix elements of the Hamiltonian (5) are given by

(j'J'~JM T = ½ [ ~ [ j J 1 J M T = ½) = 5(j,,j,,)5(j,i,)((J1Ml[~lJ1M1)--ei)+ Vp_ h.

(10)

Here e] is the hole energy and Vp_ h is the particle-hole interaction given by

Vp-h = --4 2 (-1)J~+x+J[X][J~]~[Ji]÷int(X)Cj~Cj~nd(J3) JlJzJ3

"

J3 J2

× x-~(J~ ½,J3-½1s; o)u2 ½,J3 -½1A 0),

J~, J

(11)

where ¢4/" is given by (a°Ne Jl[2°NeJ~) × (Z°NeJ,1lZ°Ne J, 1) and int(X) = ½ ~ [ T ' ] ( j j ~ X T ' I V I j ~ i z X T ' + ( - 1 ) i 2 + J ' + x + r ~ i 2 j ' X T ' ) ,

(12)

T'

and nd(J3) is a three-particle overlap, being the norm of the state given in eq. (8) (IX] = 2X+ l, etc.).

318

H . G. B E N S O N A N D B. H. FLOWERS

The Hamiltonian (5) is then diagonalized in the complete basis given by (9) for many values of the deformation parameters Cj. In this way the energy of the lowest state of each J value is minimized separately with respect to the Cj as variational parameters. Fig. 6(c) gives the energy level results for the odd-parity states of 19F. For this case a p~-p, splitting of 4.25 MeV is used and a p~-d~ splitting of 7.1 MeV is required to reproduce the energy of the lowest ½- level at 110 keV. Both these values for the gap parameters are consistent with these values used in the 180 and lSF calculations 13). The p~-p~ gap is chosen in this calculation so that the energy of the (½-, 1) model state is close to the energy of the observed -~- level 53) at 5.42 MeV. The energy minima for each J value are located about the point C÷ = - 0 . 5 , C~ = 0.4 in deformation space. Eigenvectors for the lowest eight odd-parity levels are given in table 2 and it is apparent that mixing between states of different quantum n u m b e r j is quite small. These eigenvectors now contain very low admixtures of spurious states ( < 0.2 ~ ) . TABLE 2 Eigenvectors for low lying 19F odd-parity states (jn, m)

P½[O)

(½-, 0) (~-, 0) (~-, 0) (½% 0) (~-, 0) (~-, 1) (~-, 1)

0.956

p~lO)

p~t2)>

pkl2)

--0.291

--0.966 0.924

0.294 0.141 0.249 --0.378

0.779

0.078 --0.088

p4_14)

p.k[4)

p~[6)

--0.218 0.894 --0.970

0.622 --0.967

0.239 0.184

--0.158

--0.238

TABLE 3 Percentage analysis of wave functions for lowest odd-parity laF levels compared with those obtained by Harvey 13) in parentheses

~

J~> p~lO>

p~lO>

p4.12>

P~I ~'2-?

p~]4>

p~]4)

p~_16)

4.7(15.6) 5.7(9.5) 3.4(7.6)

2.5(9.1)

19F states ½-~~½-~-

91.4(73.3) 8.5(21.6)

85.3(62.7) 93.3(78.8)

8.6(26.7) 6.2(15.7) 2.0(5.6) 14.3(37.1) 80 (53.3) 94.1(83.3)

Harvey lZ) has analysed his SU3 eigenvectors for the odd parity states of 19 F into states of a similar construction to those used in this paper (i.e. in his case p~ and p~ proton holes coupled to the ground state band of 2 ONe in SU3). His quoted mixtures are in fair agreement with those obtained in this calculation as shown in table 3

DEFORMATION IN L I G H T NUCLEI (1)

319

except that in general H a r v e y ' s mixtures between states o f different j are appreciably stronger than those obtained here. 3.5. TRANSITIONS BETWEEN ODD-PARITY LEVELS IN 19F Fhe g a m m a decays o f the first six excited odd parity states are given in table 4 in which an effective charge of C.25e is used to describe E2 transitions and a m u c h larger effective charge o f 2e for the E3 transitions. The latter value would only be reduced by making specific allowances for octupole distortions. A detailed comparison with experiment is left until sect. 5. The transitions can be seen to follow certain rules which are basically derived f r o m the fact that the mixing of states o f different j is TABLE 4 Transitions between odd-parity eigenvectors in agF Transition (g-, 0) ~ (~-, 0) (~-, 0) ~ (½% 0) --+ (½% 0) (~-, 0) -+ (,',',}%0) (½-, 0) ~ (~-, 0) (~,-, 0)

IMI~I 0.17

rMl~2

26.8 27.5 ]MI~E3 = 10 32.6 33.5 0.22 4.15

Transition (~-, 1) -+ (.~-, 0) -+ (~}-,0) (~--, 0) (½-, 0) (.,]-, 1) --',-(~-, 1) --~ (~-, O) (½% 0) (23-, 0) -+ (~-, 0) -+ (½-, 0)

IMI~, 0.13 0.4 0.62 0.09 0.13 0.43 0.59

[Mf~ 0.1 0.21 1.2 0.89 48 0.1 0.59 0.1 1.4 0.2

rather weak. The lowest eigenvector of each angular m o m e n t u m is largely based on j = ½ while others are based largely on states h a v i n g j = ~. 2 Transitions o f the f o r m (J, 0) ~ ( J ' , 0) and (J, m) -~ (J', n) (where n and m are both ¢ 0) are characterized by large E2 strengths, which come essentially f r o m the large contribution o f the Z°Ne states. These transitions are also characterized by small M1 strengths which are zero in the limit o f j being a g o c d q u a n t u m number. Transitions o f the f o r m ( J ' , n) ~ (J, O) for n ¢ 0 are characterized by very small E2 strengths and quite large M 1 decays (of the order o f one Weisskopf unit) whenever the p r o t o n holes are fully available to participate in the transition. 4. Limitations of the model E1 selection rule Before a detailed discussion o f the decay schemes o f the 19 F levels is undertaken in the next section it might be advantageous to discuss the limitation o f the model E1 selection rule, described in subsect. 3.3, and the circumstances under which it can no longer reasonably be expected to operate. It has already been established experimentally 42) that this rule is applicable to the El decay of the ½- (110 keV) level and also to the (~--, 5 - , 2~+) triplet at about 1.4 MeV excitation. At higher energies o f excita-

320

H.G.

BENSON A N D B. H . F L O W E R S

tion the E1 rule can no longer be expected to hold by virtue of admixtures of states derived from configurations in which neutr6ns are raised from the 160 core. Measured E1 decays from such levels to low-lying states in 19 F should give a direct indication of such admixtures. Unfortunately there is no direct experimental information on absolute E1 strengths from levels in 19 F having greater than 2 MeV excitation. The breakdown of the E1 model selection rule for negative parity states can be expected to occur through admixtures of states obtained by coupling a l p neutron hole on to the low energy states of 2 OF. From the relative binding energies 44) of 2 oF and 2°Ne and of 170 and lVF such levels are expected at about 10 MeV excitation in 19F. Evidence from the 180(3He, d)19F stripping reaction suggests 47) the presence at about 6 MeV excitation in 19 F of states formed from configurations (2s, 1d) 2(2p, 1f)~. It is admixtures of this kind in to the low-lying negative-parity states of 19 F that can be expected to lead to a violation of the E1 model selection rule. The breakdown of the E1 model rule for the positive parity levels in 19 F is expected to occur through admixtures of states derived from 5p-2h configurations. The construction of deformed states from this manifold might reasonably be expected to be based on the low lying deformed 4-particle-2-hole state in l SF at 1.7 MeV [refs. 6, 13)]. Such a 5-particle2-hole wave function is of the form ]19Fj 2 = 1 J I J M T

= ½ ) = ~ (J2M 2,J1MllJM)[(p~)-zJ2 = 1 M 2 T = 0 ) M1M2 J~ + n + + + xPml(~½( )~_~(n)~r(p)~_~r(p ) (~+~( n ))] 1 6 O).

(13)

The approximate energy of such states above the ground state of 19 F is easily estimated. An estimate of the particle-particle term is obtained from the relative binding energies of ZlNe and Z°Ne and the particle-hole and hole.hole terms from an aSF calculation 13). A rough estimate on these lines gives the excitation of deformed bands at about 6 MeV above the ground state. There will also be other states of similar construction to eq. (13)based on the 180 deformed states 11). However in this case both the isospin of the particles (Tp)and the isospin of the holes (Th) differ from zero and Zamick 56) has shown that a large lowering of the particle-hole energy can be expected. Zamick's rule gives the energy of the 5p-2h state in which Tp = ½ and T h = 1 about 2 MeV lower than the energy of the state (13) Jn which Th = 0. 5. Detailed discussion of the low-energy states of

19F

5.1. THE ~+ STATE AT 198 keV The theoretical E2 mean lifetime for this level of 112 nsec (evaluated for an effective charge of 0.25e) is in good agreement with the experimental value 45) of 129.9+ 2.3 nsec. Coulomb effects have not been included in these calculations so the wave functions should give as good a description of the low-energy levels of 19Ne" The theoretical E2 mean lifetime for the ~-+ state in 19No at 0.241 MeV is 17.2 nsec for an effective charge of 0.25e, compared with the measured value 45) of 26.6+ 1.2 nsec. Both these experimental E2 mean lifetimes are better reproduced if the proton and

DEFORMATION IN LIGHT NUCLEI (I)

321

neutron effective charges are allowed to differ and in this instance take the values of 0.03e and 0.30e respectively. These observations are in agreement with the findings of Elliott and Wilsdon 52). In the remainder of the calculations however we have continued to use a single effective charge of 0.25e for both neutron and proton. 5.2. T H E (~-, ~-, ~+) T R I P L E T A T 1.4 M e V E X C I T A T I O N

For the decay of the ~+ state at 1.55 MeV, which is mainly by M1, the theoretical ratio F(~ + --* ½+)/F(~ + ~ ~+) is 0.95 ~ , to be compared with the measurement of 2.6_+ 0.6 ~ obtained by Olness and Wilkinson 46). As might be expected however this quantity is very sensitive to the basic parameters of the model. For example if the d~-d~ gap is increased from 5 MeV to 7 MeV the theoretical value of this ratio increases to 2.2 ~ . The theoretical value of [M[ 2 = 7 W.u. for the ~+ ~ ½+(ground)E2 strength is in good agreement with the experimental value of IMI 2 --- 9__+3 quoted in ref. 34). Taking the measured 46) E1 branch for the ~+ 2 ~ 1 - (110 keV) together with the theoretical lifetime obtained in this calculation this E1 transition has a predicted strength of [M] 2 = 0.004 W.u. This is still a strongly inhibited transition and therefore consistent with the model El rule. For the decay of the 23-- state at 1.46 MeV to the 110 keV (½-) state the theoretical values for both M1 and E2 transitions are in good agreement with the experimental values of [MI 2 = 0.12___0.1 W.u. for the M1 transition and IMI 2 = 18+8 W.u. for the E2 transition 43). It would be most useful if better experimental values could be obtained for these transitions. The theoretical E2 mean lifetime for the transition ~-(1.34 MeV) --* ½-(0.110 MeV) is 3.3 ps which is in good accord with a recent m e a s u r e m e n t 36) of 2.6_0. +2 8 ps. An effective charge of 2e is required to reproduce the observed ~2- ~ ½+ E3 transition even when Woods-Saxon wave functions are used. The E3 transition is better reproduced here than in Harvey's calculation in the sense that a smaller effective charge is required to reproduce the transition. The reason for this is that the mixtures obtained in this calculation are weaker than those of Harvey and the contributions to the E3 matrix element from the states p~12) and p~12) are of opposite sign. In either case both models agree that a large effective charge is necessary to explain the observed E3 transition. 5.3. T H E 2.78 MeV L E V E L

It seems natural to identify the (2~+, 0) model state, which is predicted by the independent-particle model (1PM) to lie at about 2.7 MeV excitation, with the observed level at 2.78 MeV. This assignment is consistent with the angular correlation measurement of Huang et al. 33) and also with the experimental decay scheme of Thomas et al. 32) in which this state is observed to decay entirely to the s ÷ level at 198 keV. The mean E2 lifetime is calculated to be 0.30 psec which is to be compared with a recent m e a s u r e m e n t 36) oic. ,--. ~ + 0 035 psec for the lifetime of the 2.78 MeV level. o.ZJ-olo3 Further support for the ~+ assignment comes from the decay of the 4.39 MeV level (subsection 5.6).

322

H . O . BENSON AND B. FI. FLOWERS

5.4. THE 3.91 MeV LEVEL

The experimental decay scheme 36) for this level is given in fig. 7 and the parity is as yet undetermined. The likely candidate if the level has negative parity is the (~-, 1) model state predicted to lie at about 3.8 MeV excitation. However, a comparison of the theoretical and experimental decay patterns in fig. 7 indicates a violation of the E1 model selection rule which forbids decay to the positive-parity levels. Thus on the basis of the predicted M1 width for the (3-, 1) --* (½-, 0) decay the (3-, 1) --* (1+, 0) E1 transition must have [M[ 2 ~ 0.07 W.u. in order to explain the observing branching ,o ,

~'~4.

-

l

v4,Go t'345

,7

30

"V

V V

I

o.1~ii ~r

O'llO

'~+

0

Experiment.

~-

Identified ~ I~l%,}modd state.

V

Identified with

(.~+)lml~lelstate.

Fig. 7. Decay properties of the 3.91 MeV level.

ratio of the 3.91 MeV level. This value is unusually large compared with the other observed E1 decays in 19F which have ]MI 2 ~ 0.001 W.u. If instead the 3.91 MeV level has positive parity the observed decay scheme becomes consistent with the operation of the E1 model selection rule. The ({+, 1) state furnished by the 1PM calculations is predicted to lie at about 5.5 MeV excitation. However, since the 3 + deformed (5-particle-2-hole) state of construction (13) is expected to lie at about 6 MeV, it seems reasonable that mixing between these two states could yield a state at an energy close enough to be identified with the 3.91 MeV level. The decay scheme of the ({+, 1) IPM state is given in fig. 7 and it is apparent that this identification gives a good description of the decay branching to the positive parity states. Thus the properties of the 3.91 MeV level itself are seen to be satisfactorily described in terms of an identification with the (3 +, 1) state but not with the (~-, 1) state. Further discussion concerning the 3.91 MeV level is left until sect. 7 when the implications of these identifications as regards model predictions for other levels in 19F are considered.

DEFORMATION IN L I G H T NUCLEI (I)

323

5.5. THE ]-, 3- DOUBLET AT 4 MeV IN laF The (~7- , 0) and (9-, 0) model states are identified with the observed doublet at 4.00 and 4.04 MeV respectively in accordance with the recent experimental findings of Aitken e t al. 5 a). A recent lifetime measurement of ,,--28g~+ 19 fsec has been obtained [ref. 36)] for the 4.04 MeV level which is observed to decay 100 ~ to the ~- state at 1.34 MeV. The theoretical E2 mean lifetime for the (9-, 0) state of 59 fsec is in excellent agreement with this value. A recent study of t9F using the J SN(g, ~)19F reaction 5 a) has yielded an assignment of 27-- for the 4.00 MeV level and established its decay pattern. The theoretical decay scheme of the (~-, 0) model state is in good accord with experiment, the major decay mode being through the ~- level at 1.34 MeV. Also the small observed branch ~ 10 ~o [ref. 53)] to the ~2-(1.46 MeV) level is exactly reproduced by this model. A theoretical mean lifetime of 4 fsec for the (~-, 0) state is an order of magnitude less than the experimental upper limit for the lifetime of the 4.00 MeV level 36). Allen e t al. 35) report a 22 % decay branch from the 4.00 MeV level to the ½- (i 10 keV) level which is hardly compatible with an assignment of ~-. However if the decay is actually to the ~+ state at 198 keV this would be consistent with a ½- assignment, implying a lm[ 2 ~ 0.002 W.u. for the (2~-, 0) ~ (~+, 0) E1 decay strength. Such an inhibited E1 transition would indeed be expected on the basis of the E1 model rule. Thus the assignments o f ~ - to the 4.00 MeV level and ~2 to the 4.04 MeV level are entirely consistent with both experiment and theory. 5.6. THE 4.39 MeV LEVEL Olness and Wilkinson 46) have shown that the identification of this level with the (zz+, 0) provided by the IPM is consistent with the observed fl-decay branch from the 190 ground state. They showed further that the theoretical gamma decay pattern of the (-~-, 0) state obtained in an IPM calculation 1) is in good agreement with their measured decay pattern. The experimental decay scheme of the 4.39 MeV level is shown in fig. 8 together with the theoretical decay pattern of the (27-+,0) state obtained in this calculation. The agreement is very good and reinforces the already strong support for a ~+ 2 assignment to the 2.78 MeV level. The theoretical mean lifetime is 0.23 fsec and the absence of any measurable branch to the ~- state at 1.34 MeV again implies an extremely inhibited E1 transition. This is well understood on the basis of the fact that a deformed ~+ state arising from the deformed band (13) is expected to occur at least 5 MeV above the (½+, 0) state arising from the IPM. 5.7. THE 4.56 MeV LEVEL A decay scheme obtained by Alien and Lawson 36) is shown in fig. 9. However they noticed that the branching pattern of this level appears to change with bombarding energy in the x9F(p, p,y)l 9F reaction. This suggests the presence of a doublet at this energy and since, if it exists, it is unresolved even with germanium detectors, the levels must be closer in separation than about 10 keV.

324

rI. G. BENSON"AND B. H. FLOWERS

The limit of < 22 fsec for the mean lifetime 36) of the 4.56 MeV level rules out the possibility of any of the observed decay branches being M2 radiation. If the level is a singlet the experimental decay pattern 36) suggests a unique assignment of J = ~ for

~-3q'

18~7

82¢7

24

7G

~

..

,~2+

L

Z.7q

o,Iq$

.q~4.

(,~)

[b)

Fig. 8. Decay properties of the 4.39 MeV level. (a) Experimental decay scheme (taken from ref. 4e)). (b) Theoretical decay scheme identified with the (~+, 0) IPM state.

4.556.131'5

6~

2515 3 9 1 8 2315

1.5S4

~+

1.46~

~-

,~'÷

98

(a)

½+

61

I0

29 I t.

17

30

53__:~÷

3

.+

44_ _ _ _ _ _ _ ~ ' - r--'

(b)

_~-I-

r_½-

~

(el

½÷

÷

-_½

"

-

cd)

Fig. 9. Decay properties of the 4.56 MeV level. (a) Experimental decay scheme (taken from ref. Be)). (b) Identified with (½+, 1) IPM state. (c) Identified with (~+, 1) IPM state. (c) Identified with ([+, 1) IPM state.

the spin. Positive parity is suggested by the E1 model selection rule, since the level is observed to decay primarily to positive-parity levels. On this basis the candidates are the (~+, 1), (2~÷, 1) and (½+, 1) model states furnished by the IPM in this region of excitation. If the 4.56 MeV level is in fact a doublet, as is suggested above, then any

DEFORMATION IN LIGHT NUCLEI (I)

325

two of these model states are possible candidates. The theoretical decay schemes for the (~+, 1), (~÷, 1) and (½+, 1) IPM states are shown in fig. 9. However the decays of these states to lower levels in 19F are predicted to be somewhat weak and consequently their relative strengths are quite sensitive to perturbations in the wave functions. 5.8. THE 4.67 MeV LEVEL The decay scheme of Thomas et al. 32) indicates that the major decay of this level proceeds through the{ - (1.35 MeV) and the ~- (1.46 MeV) states. This has recently been confirmed 36) and an upper limit of 0.15 fsec obtained for the mean lifetime. 5.42

6_½_

7

89s

4 3

'_{-

4.04 4.00

~,.3s

½-

Expt

a)

~r~O) --

Theory

Fig. 10. Decay properties of the 5.42 MeV resonance and ({% 1) model state, a) Ref. 6a).

Negative parity is strongly suggested by the model E1 selection rule and the candidate for this level, as indicated by its predicted energy of excitation, is the (~-, 1) model state. This state decays mainly through the ~ - and ~- levels at 1.46 MeV and 1.35 MeV respectively, in agreement with the observed decay pattern of the 4.67 MeV level. The theoretical branching ratio F(({ -, 1) ~ ({ -, O))/F((~z-, 1) ~ (~-, 0)) = 1.55 is comparable with the experimental value of .a. .~+4.s . 1.6 reported by Thomas et al. a2). They also report weak decays of the 4.67 MeV level through the 2.78 MeV and 198 keV levels. The latter transition when scaled with the theoretical width for the state implies an E1 transition of IMI 2 ,~ 0.001 W.u. which is reasonable on the basis of the other known E1 decays in 19F. The observed decay to the 2.78 MeV level is inexplicable on the basis of a { - ~ ~÷ 2 transition, since this would imply a M2 transition strength of IM[ 2 ~ 250 W.u. It could arise from a high spin state (~-+ or ~ + ) if one of these states had an energy sufficiently close to the 4.67 MeV level to be unresolved in the experiments of Thomas et al. a2). A state probably having J " = -TIa + has been located 54) at 4.64 MeV excitation in 19 F using the reaction J 9F(~, ~,y)l 9F.

326

H . G . BENSON AND B. H. FLOWERS

5.9. THE 5.42 MeV LEVEL This state is known to be { - from a study of the 15N(~, y)t9F reaction s3). The (~-, 1) model state is identified with this level and the experimental and theoretical decay schemes are compared in fig. 10. The general agreement is good and supports this model identification. 5.10. HIGH-SPIN STATES IN 19F The intermediate coupling calculations predict a 1~ + state in the region of 4-5 MeV excitation and an ~-+ state in the region of 5-6 MeV excitation. The rotational model calculation of Paul 3) predicts these levels at a much higher energy ( > 9 MeV excitation). The location of a state at 4.64 MeV in 19U probably with J~ = 1.~+ [ref. 54)] is in exact accordance with the IPM predictions. The experimental mean lifetime 54) of the order of one ps is well reproduced on identification with the ( ~ + , 0) IPM state. The mean lifetime of the (~-+, 0) IPM state is predicted to be at least 100 times smaller than that of the ~123+, 0) IPM state.

I 6. Physical significance of the deformation parameters In these calculations the deformation parameters Ci are determined through a variation principle. In the calculations on 2 ONe, the odd and even parity states of 19F, and also on the even parity states of the mass 18 nuclei 11, 13) roughly the same values for the coefficients Cj are obtained. The minima in these cases are not exactly coincident and may change a little in each angular momentum state but they lie in the same small region of Cj space. It is pertinent therefore to enquire what specific deformed axially symmetric potential can create such a distorted k = _ ½ intrinsic single particle orbital in the (2s, ld) shell and it transpires that the pure quadrupole distortion of the Nilsson potential is inadequate for this purpose. The most general axially symmetric potential under the approximation of the expansion (1) contains a P4 distortion in addition to the usual quadrupole term. In this case adapting the notation of Brihaye et al. 4s) the single-particle Hamiltonian is written oY" = T + V(r, O)+CI2 + D I " s,

(14)

where

V(r, O) = ½rnco20(f2)r2[1 -

8

6 , ±2 r e % 2t o ) r 2 Y,o(,O),

(15)

and I~2 is the ordinary Nilsson distortion parameter 18). A map is shown in fig. 11 indicating how the structure of the lowest k = ___½ orbital in the (2s, ld) shell (defined by parameters C~ and C~)varies as a function of the coefficients 52 and 64. The region of interest appropriate to the nuclei studied here is indicated by shading. It is noticed immediately that a Nilsson potential (64 = 0) cannot by itself produce the deformation required and a sizeable I"40 distortion is needed.

DEFORMATION IN LIGHT NUCLEI (I)

327

In particular for the ground state K = ½+ band in 19F the values 82 ~ 0.3 and 64 ~ 0.2 are required in order to reproduce the values for the deformation parameters obtained from the variational calculation. It is assumed that the nuclear shape can be written in the form g = go(1 + f12 Y2o(a~)+ f14 Y4o(O9))•

(16)

c~ o.s °yo.4~ /

o

/ y'"o.s--. 2 '2 "

-9.,

0.5

°05 0.4

-0.2

0.3

0.2

O.1

c~

O

Fig. 11. Amplitudes of Ida½+) and [s~½+) states in the lowest k = :~½ orbital in the (2s, ld) shell as a function of the distortion parameters 62 and 64.

The coefficients f12 and f14 are determined from 62 and 64 in the manner outlined in ref. 48). This furnishes in the case of 19 F theoretical values o f f l 2 = 0.39 and f14 = 0.17. A direct measurement of these shape parameters has been made for the 19 F nucleus by the inelastic scattering of protons 49). Reasonable values for the complex optical model potential are assumed and f12 and f14 are determined directly by fitting the angular distributions of protons scattered from the ½+, ~+ 2 and ]~ + members of the band in terms of the adiabatic-approximation rotational optical model of Barrett 30). The values o f f l 2 and f14 which fit the angular distributions are f12 = 0.43 and f14 = 0.14 with which the theoretical values are in remarkable agreement.

328

H . G . B E N S O N A N D B. H . F L O W E R S

7. Discussion and conclusion

It has been demonstrated that using a reasonable interaction the main features of the low energy spectra of 2°Ne and 19 F are satisfactorily explained on the basic assumption of a permanent nuclear distortion giving rise to a low lying k = ± 1 orbital in the (2s, ld) shell. The low-lying even-parity spectrum of 19F is well reproduced on the assumption that it can be derived from a pure K = ½+ band obtained by placing the three extra-core nucleons in this orbital. The physical states obtained by projection from the single intrinsic state are found to be very similar to those obtained in a full intermediate coupling calculation similar to that of Elliott and Flowers i). A simple description of higher even-parity levels in 19F in terms of other rotational bands has not been attempted. It appears on a preliminary inspection that a large number of excited intrinsic states giving rise to states in the region of 4-5 MeV excitation in 19F must be considered. These can reasonably be expected to mix and destroy the band structure. The full intermediate coupling calculation is therefore used in the discussion of the properties of the even parity states of 19F. In order to produce the required k = ±½ orbital it is found necessary to assume a large permanent deformation including ]140 as well as Y20 terms. This implies that the 160 core previously considered to be inert is itself somewhat deformed and th's is reflected in the necessity for the adoption of a small effective charge in order to reproduce the observed E2 transitions. The principal effect of introducing Woods-Saxon wave functions when calculating E2 transitions is to reduce the numerical value of the effective charge required to reproduce the experimental E2 decays. In order to reproduce the observed mean lifetimes of the ~+ level at 198 keV in 19F and the corresponding z~+ level in 19Ne at 241 keV it is necessary to adopt different values for the effective charges of neutron and proton. It is important that other E2 transitions in 19Ne should be measured in order to see whether this result applies to other corresponding transitions in 19Ne and 19F. An effective charge of 0.25e for both neutron and proton, required to reproduce the measured E2 mean lifetime of the ~+ (198 keV) level in 19F, is found to account satisfactorily for the other measured E2 decays in this nucleus. The identification of the (9+, 1) model state with the 2.78 MeV level is shown to result in a satisfactory account of its experimental mean lifetime. This identification is reinforced by the model assignment of ~+ to the 4.37 MeV level and its observed decay pattern. A rigorous experimental assignment of ~+ to the 2.78 MeV level has recently been made 54). For the odd parity states of 19Fit has been shown that the weak coupling "one-band" model, while presenting a delightfully simple account of the observed energy spectrum, is unable to explain the reported M1 transition 43) between the 1.46 MeV ~ state and the 110 keV½- state, and gives rise to states containing spurious centre-ofmass motion. There is also evidence from a lifetime measurement 36) for an M1 transition from the ½- state at 4.00 MeV to the ~z (1.34 MeV) state, which is also unexplained on the "one-band" theory. In the calculations reported in this paper the

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particle-hole terms (assumed to be small in the weak coupling theory) are calculated explicitly from the two-body interaction. They are found to be quite large and also found to vary appreciably depending on the angular momentum to which the hole is coupled to the particles. Agreement with the observed energy spectrum is restored by mixing in particular higher bands and this mixing is shown to give a natural account of the observed M 1 transitions - which are forbidden on the simple "one-band" model. Mixing between the ground state K = ½- band and these additional bands is found to be small so that the predicted M1 transitions are relatively insensitive to the p½-p~ gap chosen. This parameter essentially determines the excitation energies of the levels in these higher bands and is chosen so that the (~-, 1) model state is reproduced at the correct energy to be identified with the 5.24 MeV level (which is known to be ~-). The predicted decay pattern of this state is shown to be consistent with that observed for the 5.42 MeV level. Other implications of this identification are that the model now provides two additional negative parity states in the low-energy region. These are the (~-, 1) and (8-, 1) model states predicted to lie about 4.9 MeV and 3.8 MeV respectively. The former state is identified with the 4.67 MeV level and is shown to have a theoretical decay pattern closely similar to the experimental one af Thomas et al. a2). This is on the reasonable assumption that the state recently found 54) at an energy of 4.64 MeV, which is most probably - ~ + , is not resolved from the 4.67 MeV level in their analysis. The (8-, 1) model state is now naturally identified with the observed 8 level at 3.91 MeV. However the observed decay pattern of the 3.91 MeV level is not consistent with this assignment. The 3.91 MeV level is observed to decay mainly to positive-parity states which, assuming it has negative parity, implies a violation of the model E1 selection rule. In this instance the implication is that the E1 rule is seriously broken since the transition to the ½- 0.110 MeV level is predicted to be a fast M1 transition but is observed only as a 12 70 branch of the total decay. This implies a value of IMJ 2 ~ 0.07 W.u. for the 3.91 MeV (assumed 8 - ) --* ½+(ground) E1 transition. All other E1 decays from identified negative parity states in 19 F are shown to have JMJ 2 ~ 0.001 W.u. in accordance with the E1 rule. The breakdown of this rule is expected to occur mainly through admixtures of states derived from configurations (2s, Id)2(2p, lf) ~. It is possible that such states could admix selectively into the ( ~ - , 1) state thus causing a breakdown of the E1 rule. However, this seems unlikely since the ~ - level at the higher excitation of 5.42 MeV is observed to conform to the rule. If the 3.91 MeV level is instead assumed to have positive parity, then it has been shown that an identification with the (3+, 1) model state yields a reasonable account of its decay properties. In this event there is now no observed level with which the (8-, 1) model state may reasonably be identified. A possible solution of this dilemma would be that the 3.91 MeV level is a doublet consisting of a 8 - state together with a state of positive parity. Such an explanation is favoured for two reasons. Firstly it would eradicate the apparent decay anomaly of the 3.91 MeV level and preserve the E 1 model rule. Secondly there would now (as.suming the 4.57 MeV is a doublet as the experimental evidence suggests) be a one-to-one correspondence between the energy levels observed in the

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energy region below 4.7 MeV excitation and the states predicted by the model to lie in this region. The positive parity member of the postulated doublet must have spin J = ½, z2 or ~ from the observed decay pattern of the 3.91 MeV level. The most likely value from the model viewpoint is J = 3 since the (~+, 1) model state is predicted to decay to the positive parity states in a manner close to that observed for the 3.91 level. The properties of the 3.91 MeV level or levels in 19 F are therefore of great interest in terms of the application and prediction of this model. In particular the parities of the 3.91 MeV, 4.57 MeV and 4.67 MeV levels are of crucial importance in testing the validity of the model and its predictions. These properties in particular therefore demand experimental investigation with very high resolution. It is most probable s4) that the level found recently at 4.64 MeV in 19 F is 1.~+ which is gratifyingly close to the predicted energy of 4.5 MeV. On this basis the 1_~_+level, so far undetected, might be expected to occur around 5.6 MeV. A thorough experimental survey of states in the region of 4.5-6.5 MeV excitation in 19 F would also be most useful in determining the possible presence or influence of deformed (5p-2h) even parity states. This can most easily be done by examining the E1 transitions to the low-lying energy states in order to ascertain to what extent the E1 model rule is still applicable. Such information would be of great value before further calculations (in which core deformation is included explicitly) are attempted. It is a pleasure to acknowledge the help of Prof. K. W. Allen, Dr. P. G. Lawson, Dr. I. F. Wright and Dr. C. O, Lennon for communicating their experimental results prior to publication. Thanks are also due to Dr. J. W. Clark for his assistance in computing Woods-Saxon integrals and the service rendered by the ATLAS computing staff of Manchester University is gratefully acknowledged. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19)

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