Shell-model and deformed states in oxygen isotopes

Nuclear Physics A95 (1967) 443--472; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

S H E L L - M O D E L AND D E F O R M E D STATES I N OXYGEN I S O T O P E S P. FEDERMANt AERE Harwell, England Received 15 November 1966 Abstract: The energies and some other properties of low-lyinglevels in the oxygen isotopes are studied in the frame of a model including shell model and deformed states, as well as the interplay among them. The calculations are performed using the effectiveinteraction method, and very good agreement with experiment is obtained. The results are compared with previous effective interaction calculations, and also with the predictions of phenomenological forces. Particular attention is given to the determination of the parameters and unperturbed positions of the deformed states. It is concluded that the influence of the deformed states is important in 180, but their influence in 190 and 3°0 is small.

1. Introduction 1.1. SHELL-MODEL CALCULATIONS IN THE OXYGEN ISOTOPES The low-lying energy levels of the oxygen isotopes have been dealt with in the past by several authors 1-4). Shell-model calculations were performed using phenomenological forces as well as effective interactions. In particular, good agreement was obtained in the later cases. The spectrum of 170 (see fig. 1) shows a J = ~+ ground state and a J = ½+ level lying 0.87 MeV above it. A J = 3 ÷ leyel is observed at an excitation energy of 5.08 MeV. These levels are interpreted as the single-particle levels corresponding to the ld~, 2s~r and ld~ orbitals of a neutron outside the closed shells of 160. The ld~-ld~ splitting due to spin-orbit interaction is rather large. This justifies the usual approximation of neglecting the influence of the ld~ orbit when dealing with low-lying levels. Other indications for the small influence of the ld~ orbital are furnished by the l = 2(d, p) reaction on 180 going to the first J = 3 + state in 190 and the M1 transition from the same level to the ground state. Both transitions have to proceed only through the ld~ part of the J -- ~+ level. Experimentally both are weak, imposing a rough upper limit of 3 ~ for the ld~ contribution 4). Talmi and Unna 3) discussed the oxygen isotopes up to 2 ° 0 in terms of pure configurations, using effective interactions. First, levels belonging presumably to ld~ configurations were considered. It turned out that these configuration assignments gave a very good description of the ground and some of the excited states in the oxygen isotopes. Predicted energies are also given for some levels. Later on these levels were found experimentally at energies in fair agreement with the calculated ones. In 190 t Present address: The Niels Bohr Institute, University of Copenhagen, Denmark. 443

444

P. FEDERMAN

a J = ~+ state was calculated to lie between 2.6 MeV and 3.1 MeV, and later observed at 2.77 MeV. In 2°0 a J = 4 + level was predicted around 3.9 MeV and later found at 3.57 MeV. Secondly, states arising from ld~ 2s~ configurations were considered. In this case the number of unknown parameters was just equal to the number of known levels. Thus, it was not possible in this case to check the consistency of the values obtained for the theoretical parameters. Still, the energies tentatively calculated for levels belonging to the ld~ 2s~ configurations were later found to fairly agree with the experimental values. More recently, Cohen et al. 4) repeated the analysis of the oxygen isotopes. This time more experimental information was available, and it was possible to include in the calculations not only the configurations considered by Talmi and Unna, but also

5.08

3/2

O .871

I12

9.s.

5/2

Fig. 1. Low-lying positive parity levels in 1~O.

the interactions among them. The results are substantially in agreement with those obtained by Talmi and Unna. Thus, the shell model using effective interactions within configurations arising from the ld~ and ls~ orbitals seemed to give a reasonable description of the oxygen isotopes. 1.2. THE NEW "PUZZLING" LEVELS Recently s,6), two new experimental levels were found in 180 that do not fit into such a simple shell model picture. They are the rather low-lying J = 0 + (5.33 MeV) and J = 2 + (5.25 MeV) levels. These states cannot be understood in terms of the simple shell-model configurations of two neutrons outside the closed shells of 160. The two lower J = 0 + and J = 2 + levels are interpreted as arising respectively from the ld~ ( J = 0), 2s~ ( J = 0) and ld~ ( J = 2), ld~t 2s~r ( J = 2) shell-model states. The next J = 0 + shell-model state would arise from the ld~ configuration. But as mentioned before, such a state including the ld~ orbital, cannot give rise to a level at the rather low required energy. This can be seen easily as follows. The single-particle energy of the ld~ configuration lies 10.16 MeV above the single-particle energy of the ld~ one. On the other hand, the interaction between the two neutrons in J = 0 + states belonging to those configurations can be expressed for central forces, going to

STATES IN 0 ISOTOPES

445

L S coupling, as:

V(ld~S = O) = 3V(d 2 ' S ) + ~-V(d2 3p), V(ld~J -- O) = CV(d 2 ' S ) + 3 V ( d 2 3p). For any reasonable short range central force the interaction in the S-state is strongly attractive. For higher even L-values it becomes weaker, and for odd L-values small attractive or even repulsive. Thus, the expressions written above dearly show that the interaction of the two neutrons will be in general stronger in the ld~ ( J --- 0) state than in the ld~ ( J = 0) one. Therefore, the J = 0 + state arising from the ld~ configuration will lie at least 10 MeV above the one 7, 8) arising from ld~. A completely similar argument shows that a ld~ ( J = 2) state would not appear below around 12 MeV. But for J = 2 + it is possible to have a state including only one ld~ neutron, therefore with an initial single-particle loss of only 5.08 MeV. This is the ld~ ld~ ( J = 2) state. The wave function of such a state does not contain any S-component (and the D-component is small). Hence, states containing only one ld~ neutron are not likely to appear below 7 MeV. Therefore we see that the simple shell-model configurations of two neutrons outside the closed shells of 160 cannot yield an explanation for the appearance of three J = 0 + and three J = 2 + states below 5.5 MeV in 180. Up to 7 MeV, such a model would predict only two J = 0 + and two J = 2 + levels, belonging to the ld~, ld~ 2s~r and 2s~ configurations. I f the new levels are to be understood, new degrees of freedom have to be introduced. 1.3. A P O S S I B L E S O L U T I O N

This peculiar situation is not competely new. A similar puzzle is already known in ~60. Its spectrum (see fig. 2) presents a first excited J = 0 + state at 6.06 MeV. An IO . 3 6

4

6.g2 6.05

2 0

q.s.

0

Fig. 2. Low-lying positive parity levels in 160.

even number o f nucleons has to be raised into the (2s, ld) shell in order to obtain a positive parity state. Raising one lp~ particle into the (2s, ld) shell means loosing a big amount of single-partide energy (roughly around 10 MeV). One would therefore

446

P. FEDERMAN

be tempted to try raising the minimum number of nucleons, namely two. By raising two nucleons from the lp~ orbital into the (2s, ld) shell a state at around 14 MeV is obtained. But if one calculates the position of a state obtained by raising the four lp~ nucleons into the (2s, ld) shell, a much lower state is obtained 9) (at around 6.9 MeV). The extra loss in single-particle energy is more than compensated by the very strong interactions of neutrons and protons in the same orbit. This is a very general feature that can be already seen simply by comparing the binding energies of nuclei like 19F and 190, or 2°Ne and 2°0. In the latter case, 2°Ne is 9.2 MeV more bound in spite of the loss due to Coulomb repulsion (around 8 MeV). We therefore conclude that the 6.06 MeV J = 0 ÷ state in 160 contains as a main component a state obtained by raising four nucleons into the (2s, ld) shell. The spectrum of 160 presents another peculiar characteristic. A rotational band can be seen on the 6.06 MeV J = 0 ÷ level. This naturally leads to conclude that the four l p t nucleons are raised into an intrinsic deformed state in the (2s, ld) shell.

4:25

4

1-63

2

q,$.

O

Fig. 3. Low-lying positive parity levels in 2°Ne.

Such a situation resembles very much the case of 2°Ne. In this nucleus there are already in the ground state two neutrons and two protons in the (2s, Id) shell, outside the 160 core. And also in this case, a rotational band appears (see fig. 3). Actual calculations 1o) support such a description, showing that the low-lying spectrum of 2°Ne can be understood in terms of an intrinsic deformed state built in the (2s, ld) shell, and giving rise to a rotational band. Moreover, the energy of such an intrinsic deformed state is much lower than the spherical state 1o). Thus, the four lp~ nucleons in 160 are raised into an intrinsic deformed state in the (2s, ld) shell which is a kind of analogue of the 2°Ne ground state. Such a model leads to actual calculations that support this conclusion 11,12,13). Some admixture between this 4p-4h state and the spherical state is necessary to understand also the E0 transition between the two J = 0 + states. This requires the introduction of some other states (which can lie much higher) through which the deformed 4p-4h and the spherical states could admix. Those states are likely to be 2p-2h states 1a). A possible solution for the 180 puzzle is now quite naturally suggested from the introduction of similar deformed states in this nucleus. This was proposed by Brown and Engeland 14,1s).

447

STATES IN O ISOTOPES

In 1So there are already two neutrons in the (2s, ld) shell outside the 160 core. Obviously to put these two neutrons in a deformed orbital built in the (2s, Id) shell does not lead to anything new. Such a state is not even orthogonal to the shell-model states. In 160 the lowest state was obtained by raising four nucleons. This will not be the case in 1sO. As there are already two neutrons in the (2s, ld) shell, the lowest state will be obtained by raising only two nucleons. Moreover, in order to gain the strong attraction among neutrons and protons, the two nucleons lifted should be protons. The deformed state in 1SO is therefore built by raising the two l p , protons, which are put together with the two valence neutrons in a deformed orbital in the (2s, ld) shell. It was pointed out before that the Idg orbital can be ignored when considering shellmodel neutron configurations in the oxygen isotopes. But this is not the case when neutrons and protons are present. The importance of the I d t orbital when both kinds 3.06

2 2.07

1.70 I. 1 3

I

2

I.O6

I

0.42

3 o 5

~ 0.23: q.$.

1.05

0.94

9 .~-

I

F 18

A~ 26

Fig. 4. C o m p a r i s o n o f the spectra of 18F and a6A1.

of nucleons are present can be seen already from the spectrum (refs. 7,8)) of lSF. The ld~ configuration of lSF should lead to a spectrum identical with the one of the ld~ 2 holes configuration of 26A1 (see fig. 4). Due to their high spin, the J = 5 + levels in both nuclei are probably quite pure ld~ and ld~ 2 states. They are therefore matched in fig. 4. We see that the T - - 1, J = 0 ÷ and J = 2 + levels appear about the same position in both nuclei. But the discrepancies for the T = 0 states are very strong. The ground J = 1 ÷ level in lSF is clearly pushed down, and a second J = 1 ÷ level appears below the first J = 1 ÷ level in 26A1. This clearly indicates that there must be a third J = 1 + state in lSF that pushes them down. A reasonable configuration, that could provide such a state is the ld I ld~. If one expands the ld~ l d t ( J = 1) state for T --- 0 in L S coupling, it can be seen that in this case large S and D components are obtained. This leads to a strong and attractive diagonal matrix element, even larger than the ld~ ( J -- 1) one, partially compensating the 5.08 MeV initial loss in singleparticle energy. A rough estimate 7) gives for the unperturbed splitting of both levels about 2.5 MeV, and around 2 MeV for the nondiagonal matrix element between them (it also contains a S component). Thus, it seems quite possible that the ld~ ld~ configuration plays an important role in this case. Its admixtures with the other states would explain the differences in the lSF and 26A1 T = 0 spectra. Hence, while it may

448

e. FEDERMAN

be a reasonable approximation to neglect the ld t orbital when dealing with identical nucleons, it has to be taken into account when both protons and neutrons are present. We therefore include the ld~ orbital in the deformed orbital constructed in the (2s, ld) shell. In a preliminary calculation 16), it was shown that the inclusion of deformed states makes indeed possible a description of the low-lying spectrum of 1So. The question of similar states affecting the spectra of 190 and 200 then naturally arises. In the present work iSo, 190 and 2oO are considered together, and the influence of deformed states is systematically included in the three isotopes. The intrinsic deformed states are built by raising the two lp~ protons into Nilssonlike orbitals in the (2s, ld) shell. Actual states are obtained from them by projecting out the part with a given value of the total angular momentum J. The formalism used to deal with the deformed states is described in sect. 2. The calculations are performed in the scheme of effective interactions. A consistent picture accounting for the energies and some other properties of the low-lying levels in the oxygen isotopes is obtained. The agreement between experimental and calculated energies is very good. It turns out that the influence of the deformed states is significant for i So, but small for 190 and 2°0. This is due mainly to the higher unperturbed positions of the deformed states in the heavier oxygen isotopes. The calculations were also performed for some phenomenological forces: the one used by Elliott and Flowers 2)(EF), which consists of a Yukawa potential with a Rosenfeld admixture and the one used by KaUio and Kolltveit 17). For the second one, two cases are considered: with (KKB) and without (KK) corrections due to core polarization as calculated by Bertsch is). A short note including part of the results presented here was published earlier 19). After the completion of the present work, a similar calculation but only for the 180 case was performed by Benson and Irvine 2s). They used the (KK) force and allowed for changes in the pi-d~ single-particle splitting. This effectively amounts to vary the unperturbed position of the deformed state. In this way, reasonable agreement is obtained for 1sO. It is perhaps worth mentioning that if the same two particle interaction matrix elements were used to calculate the spectra of 190 and 2°0, the results would be rather poor. 2. Wave functions of the deformed states

2.1. WAVE FUNCTIONS IN A DEFORMED FIELD The wave function of a single-particle moving in an axially symmetric deformed field can be expanded in aj-j coupling basis as follows =

(1)

J Here the ~kj,~, are eigenfunctions of the total single-particle angular m o m e n t u m j and

STATES IN O ISOTOPES

449

its projection m, the X~,k depend on the form of the field and are called deformation parameters, and k is the projection o f j on the symmetry axis. Levels with + k and - k are degenerate, and for cigar-shaped potentials the lowest state has k = _+½. Many-particle wave functions are built by products of deformed single-particle orbitals, properly antisymmetrized among identical nucleons. Thus, the wave function of n protons and rn neutrons will be 20, a)

I kl'k2'''''k~

>

kin+l, k.+2 . . . . . kn+m

(2)

= ~¢[~bk,(1)... ~bk.(n)]dl'~bk.+t(n+ 1 ) . . . ~bk.+,~(n+m)], where d is the normalized antisymmetrization operator. The projection of the total angular momentum on the symmetry axis is given by:

K=Xk,, f where the sum extends over all the particles. 2.2. P R O J E C T I O N O F W A V E F U N C T I O N S W I T H A D E F I N I T E Jr

The wave function (2) is not an eigenfunction of the total angular momentum or. If it is to represent a nuclear wave function one has to obtain from it an eigenfunction of the total angular momentum. This is usually done by rotating the intrinsic wave function to the laboratory fixed frame of reference, using the D~K(R) functions of the rotation R. Linear combinations of intrinsic wave functions are then obtained, including all R values with weight D~K(R) which are eigenfunctions of the total angular momentum. Let ri and r; represent the coordinates of t h e / t h particle in the laboratory and intrinsic frames of reference, respectively. The rotation operator J~ transforms r~ into r; :

r; =/~r~. If the wave function in the intrinsic frame is ~k ~

~

(r 'lj''', rn,),

the wave function in the laboratory frame is given by:

~ , r(r i ..... r~) = f D~,(R)~r(r'~ . . . . . . . .

, r')dR

,/~r~)dR,

where the integration is performed over the parameters specifying R, that is the Euler angles. This expresses the laboratory wave function in terms of Hill-Wheeler integrals over the intrinsic wave functions.

450

P. FEDERMAN

The same result can be achieved by defining a kind of "projection" operator 21,2 0, S). This operator, when applied on an intrinsic wave function ~ r projects out the part with a value J for the total angular momentum and then changes K into M, where the resulting function is to be taken in the laboratory fixed frame. For one particle from (1) it follows that

e~

= z~,~Oj,.,

(3)

while for a neutron and a proton in states Ikl> and Ik2> it follows, using (1) and (2)

Ikl~ Ik2klJM > = JlJzEZJ,,ktXJ2,k2PMEOj~,ktOJ2,k2]" '

/~M k 2 / = But

where by

, ~rJ [~,~'1

(4)

we have denoted the usual Clebsch-Gordan coefficient

(Jl k 1J2k2[JlJ2J' K) and by O~j~1' the wave function obtained by coupling Jl and J2 to J', K. Both notations will be used in this paper. Using the definition of P~ it follows from (5):

p~[~yj,,kl~ljz, k2] _~ [Jl J2 JK] lp~lj2,l kl

(6)

k2

Hence, from (4) and (6):

k2klJ M ~ = Jlj2 ~ Z1,,kt~j2,k2 kl k2 In the case of two identical particles, it follows from (1):

Ikt k2)

1

= . ~ [~bkx(l)q~k~(2)--q~kl(2)(kk2(1)].

Using (6) one obtains:

e~Mlklk2) = Ik~ k2JM) = ~ xi,. kl xj~. ~ [Jl . J2

hi2

L_kl k2

1 [*~}'~J(1, 2) - O~ j2, (2,1)]. JK] ~-~

(8)

Let us define if J1 = J2, if Jl # J2-

(9)

Replacing the definition (9) in (8) it follows:

Ikl k2JM) = ~ Xj,,klXj2,k~ kl jlJ2 where ~ 2 ~ is antisymmetric.

J2 J ] ~J,J2Jr k2

K J WM t-'12,

(10)

STATESIN O ISOTOPES A simple relation exists

22) between l j , - k

451

and XJ,k:

X], k = ( - - 1)J-~RC~Zj, --k

where lr÷ = ( - 1 ) t is the parity of ek. In the case of even l (e.g. the (2s, ld) shell), it follows that:

Zj, k = (-- 1)J-'i')cj,-k"

(11)

TO project out the part with good J i n the case of many-particle wave functions (2), use is made of Araki's theorem 20) to obtain: I kl, . . . , k,n

kn+l .... kn+mJ M ) = JtJz Z

I J1 J2 JK] t~M [J1 J2 JM]p~,]lcl, [On> gl K2 "'" M , 2 M1 M2 (12)

x P ) ~ l k , + l , . . . . kn+,.>,

where

K1 =

/=1

kt,

n+m

K2 = ~ ki. i=n+l

2.3. THE CASE OF 1sO As we mentioned in the introduction, the single-particle deformed orbit is built in the (2s, ld) shell. Hence it includes the ld~, 2s~ and ld~ orbits. We shall assume the lowest deformed single-particle orbit to have k = + ½. From (12), the wave function of four nucleons in a k = +½ orbit is given by:

[deflSjM>= I~2~JM ) = Z IO1 J2 dlJ2

J1 Z

0

M1M2

IJ1J2JMI J1PMI[½-½>P'1M2,2__2>. ~21 M1 M2

(13)

Using (10) it follows: PMI½--½> "~- Z Zjl,~gZj2,--'~ jlJ2 L-~

--½

Let us define: Zj. ~r =

Zt.

Then, using (11), we obtain: p~t]½_½> = EX1Z2(_I)j2-~-[Jl m~

½

J2

-½

J l C12~J2J

"

(14)

452

P. FEDERMAN

Using (14) in (13), it follows:

,~o'"'>-- :,:2 ~

0 'I ['o''~

~" ZIZ2ZaZ4(-- 1)J2+J'- 1

Jlj2j3j4

X [~1~1 -½ J2 J1][13

j4, Jo21C12C341~[(jIj2)aJIMl(j3J4)aJ2M2JM], (15) -½

where the subindex a indicates antisymmetrized wave functions. The norm N of this wave function is given by:

N2(J)= E ['~1 JIJ2

0

× Let

Ha be an

da] z

J2

½

[ZlZ2Z~X,]2C212Ch

JlJ2Jaj4

u j

-½

Y

oJ L½

-½

(16)

"

one-particle operator such that

Ho¢~;;

on.~.,... oj Vjm,

=

where the superscript n, p denotes neutron or proton, respectively. It follows then for the four-nucleon wave function:

<~of~+,~o,~of.+,~> +~.[,~ ,~0 ,l ~.llJ2ja.i4 ~" [ZlZ2Z3Z']z ½

oJ L½ -½

-½

(~;,+ej2+8;3+~;,).

(17)

We shall also need the matrix dement of a two-body interaction

v

E~k,

=

i
which is given by s):
JMlVIdefts JM>

= 4E

J,:2

~,, J2 ,l~N,[2+(_l):+]a•, 0

0

+4~ E'~i J21J]2N2b,,-8~ [J0'J2 ~]E'~,J2 J] Nac h (18) JIJ2

JlJ2

--

0

0

for J even. The N~ in the previous expression are the two-particle normalization factors:

=/

½j

STATESIN O ISOTOPES

453

From (7) the norms (19) can easily be expressed as functions of the deformation parameters. The / 3j J1/~, =

(20) c~=

{~0

-

~J~JV]½J~ ½

for J, odd, for

are two-particle matrix elements. They can be expressed

I

JlJ2

J1 even,

in j-j coupling using (7):

1g] ~jlj2J

(21) where

~bj'j2j

refers to the symmetric combination;

Jlj2

--3

--3

Jl~_J2

C12 "

Here the antisymmetric (symmetric) combination is to be taken for ~kJ'j2J for even (odd) jr. From the wave functions (21) we obtain:

X~ZzXaX+(-1)i2+j'-1 2 Jl~_J2 C12 C34 Ja --~J4

aj = E

where again the antisymmetric (symmetric) combination is to be taken for and IJ3J+J) for even (odd) J

j,_~j~ 0 IJ,~-J2 J3:~]4

L

½ ½

½ ½

IJxAJ)

Jl - -2 C~2 Ca+

for J even

3 -3

(22) ½

C12 C3+

x

for J odd.

454

P. FEDERMAN

In the last two expressions the symmetric combinations have to be taken for ]JlJ2J) and ]j3j4Y). The non-diagonal element with a shell model state [lp~(0)jtj2 ( J ) ) of two protons in the lp~ orbit and two neutrons in t h e j l , j 2 ones is obtained withthe help of (15) as: (def18 JlVllp~(O)jtj2(j)j ) = Z1z2Ct 2 [~t

--½J2

J] (_l)j2-~

Z~

x42~42j3+l

(J~J1 = 0[Vllp~Jt = 0).

Let us define:

Z-----~23(j~S~ -- 0IVIlp~J1 - 0).

(23)

(--1)J2-~N/2S.

(24)

S = ~ x/'2j3+l Then: (def is dlV[lp~(O)jxj2(J)J ) = ~1x2C12

½ _½

The same parameter S enters into the expression of all nondiagonal elements between shell-model and deformed states for 1sO, 190 and 2°0. 3. Isospin of the deformed states

Provided the Pauli principle is properly taken care of, it is not necessary in general to use the isospin formalism. Moreover, when the Coulomb energy becomes important it introduces a considerable difference between the states of neutrons and protons, and there is no apriori reason to assume that isospin will be a good quantum number. But in the case of charge independent Hamiltonians, isospin is a constant of motion. Hence, the use of isospin eigenfunctions right from the beginning allows a reduction of the energy matrix. This simplifies the calculations and insures that the wave functions have the correct properties with respect to isospin. Let us consider the case when neutrons and protons occupy different unfilled orbits. In this case, there is no need to use the isospin formalism. This originates in the fact that if a charge coordinate is introduced and the complete antisymmetrization is carried out, only one value of the total isospin T can be obtained. Thus, the use of the isospin formalism in this case is superfluous and has no advantage. That only one value of T can be obtained in the case mentioned above can be seen from the following argument2a). Let the Jx orbit be fully occupied for neutrons and partially filled with n protons, and the j2 orbit partially filled with m neutrons and empty of protons. The minimum value for the total isospin T will be:

T > ½(2jl+l-n+m ).

(25)

On the other hand, the maximum value for the isospin of all the nucleons in the Jl

STATES IN 0 ISOTOPES

455

orbit is: T~, = ½ ( 2 j l + l - n ) , because if Tj, were bigger than ½(2jl + 1 - n ) , the completely failed orbital would have T # 0. Moreover, the total isospin for the m neutrons in the j2 orbital is: TJ2 ~ ½Fno

Thus, the maximum value of the total isospin is:

(26)

T <=Tj,+T~, = ½(2jx+l-n+rn ). From (25) and (26), the only value the total isospin can take is:

(27)

T = ½(2jl + l - n + r n ) .

In the case of deformed orbits, the single-particle state k may contain not more than two equal particles, with projections k and - k . In complete analogy to the previous case, only when a k orbit is partially filled with neutrons and protons is the T content not unique. This corresponds only to the case when only one neutron and one proton occupy the same deformed orbit (because the orbit is filled with two identical particles). Thus, states of the type: I kl - k l \

kl-kl/ give rise to only T = 0. Wave functions like:

Ikl>,

[kl

kl

-kt

k2~/

contain only T = ½, and states like:

Ikl

~ kz ~ ~

I kl

-kl

I tel --kl

k2 - k z ~

/

give rise only to T = 1 wave functions. When both kinds of nucleons occupy the same single particle state, like:

Ikl\ kl/'

k,\ I -kt/

two values of the isospin, namely T = 0 and T = 1, are contained. The cases dealt with here when expressing the deformed wave functions for the oxygen isotopes do not include this last type. Hence, in spite of not using the isospin formalism, they contain only one value of the isospin.

456

P. FEDERMAN

4. Shell-model configurations and deformed states 4.1. O X Y G E N

18

As mentioned earlier, it is necessary to raise four nucleons into Nilsson-like orbitals in the (2s, ld) shell in order to obtain the 6.06 MeV J = 0 + level in xeO. Additional evidence in this direction is furnished by a recent calculation including 4p-4h states as well as 2p-2h ones a4). A similar state is obtained in 1SO by raising only the two lp~ protons into the (2s, ld) shell, due to the presence of the two valence neutrons. Our description of 1SO is then based on the deformed states mentioned above and the shell model states arising from the configurations of the two extra neutrons outside the 160 core. The states for different values of the total angular momentum J are listed below. S = 0: Idef Is 0>,

Ild~>,

12s~>,

J = 2: Idef Is 2>,

Ild~>,

Ildt2s~>,

J = 3: Ild~2s½>, J = 4: [def is 4), 4.2. O X Y G E N

Ild~).

19

In 190 there are three valence neutrons outside 160. The k = +_½ orbital is filled in 1SO. Hence, the third neutron in 190 has to occupy the next deformed orbital with k = ---3. The deformed states are thus given by: [ d e f t ° J ) = P ~ 1½-½ I ½-½+3/" The minimum total angular momentum for such a state is J = 3. Including the shell model states arising from the configurations of the three extra neutrons, the possible states are: S = ½: Ild~(0)2s,), S = ½: ldef t9 ½>,

lld~>,

lld~(2)2s~r>,

S = ~r: Idef ~9 ~r>,

lld~>,

]Id~(2)2s~r>,

J = ~: ]def 19 ~>,

]ld~(4)2s~>,

J = ~: Idef 19 ~),

Ild~),

4.3. O X Y G E N

Ild~2s~(0)>,

Ild~(4)2st).

20

The four valence neutrons together with the protons raised from the lp~ orbit give rise to the deformed states [def z° J ) = PM[~-~ ~-~)-

457

STATES I N O ISOTOPES

In this case the states included are: J = 0: Idef 2° 0>,

Ild~>,

Ild~(0)2s~r(0)>,

J = 2: Idef 2° 2>,

Ild~>,

Ild~(½)2si>,

Ild~(~)2s~>,

Ild~>,

Ild~(~)2s~>,

Ild~(4)2s~(0)>.

Ild~(2)2s~(0)>, J -- 4: Idef ~° 4>,

The wave functions listed in the three last subsections are the basis in which the energies and wave functions for the low-lying levels of 1s O, 190 and 2oO are calculated. The omission of shell-model configurations including ldg orbitals was justified previously. In the previous section, the formalism to deal with the deformed states in XSO was presented in detail. The deformed states in 190 and 2°0 are dealt with in a similar fashion. The procedure is much more involved algebraically and an outline for the case of 200 is presented in the appendix. 5. Determination of the deformation parameters and unperturbed positions of the deformed states After raising the two lp~r protons in 1so into the deformed orbitals, we are left with a 14C core and the four nucleons in the deformed state. Hence, the unperturbed energy of the deformed states is given by" g(def 18 J) = B.E.(14C)+ (Ho> +
(28)

where by B.E.Q4C)we denote the binding energy of the 14C core. H o and V are the single-particle and two-body interaction operators, respectively, as defined in sect. 2. The expectation values are taken in the deformed states, and the corresponding expressions are given in (17) and (18). The ej entering in Ho are the single particle energies in the 14C core. Hence, through (17) and (18) equation (28) gives the unperturbed energies of the deformed states in 180 as a function of the deformation parameters, the single-particle energies in the ~4C core and the two particle matrix elements of the interaction V. A rough estimate of E(def is J = 0) can be obtained as follows. Assume the deformation parameters are about the same in 180 and 2°Ne. In Z°Ne the J = 0 deformed state is the ground state. Thus, B.E.(2°Ne) = B.E.(160) + < H i ) +
(29)

where we wrote the same interaction < V> as in the 1sO case, assuming the equality of the deformation parameters. From experiment we take B.E.(2 ONe) and B.E.(160) . The spectra of 170 and 17F furnish the single-particle energies in the 160 core (see next sect.). Thus, taking the deformations obtained in other works 1o, a) for 2ONe we can calculate . Using the single-particle energies in the 14C core (see below), we cal-

458

I,. FEDERMAN

culate (Ho> and with this and the previously obtained value of (V> we obtain E(defla J = 0) from (28). In this way a value around 4.5 MeV above the actual ground state of xaO is obtained for E(def is J = 0). Thus, the unperturbed position of the deformed state would lie higher than the two J = 0 + shell-model states. A more accurate procedure would be the following. As we mentioned before, E(def is J ) is given by (28) as a function of the single-particle energies sj in the 14C core, the two-particle matrix elements of V and the deformation parameters, as seen in sect. 2. The single-particle energies s~ can be obtained from the spectrum of 15C, in which the neutron moves in the (2s, ld) shell outside 14C: an = B.E.(15C)_ B.E.(I•C), n e,~ = ~,~r--0.66 MeV.

(30)

For the ld~ single-particle energies, the same l. s splitting as in the 160 core is assumed. The situation is somewhat complicated in the case of the proton single-particle energies. The spectrum of 15N cannot be interpreted as a single-particle spectrum 2 3 ) . Therefore in this case we start from the 12C core and obtain the single-particle energy values for the 14C core from: 2 ~ < lp½ JJI Vnv[lp~ jJ>(EJ + 1) .V(I"c) = 4 ( 1 2 c ) + • , (31)

E(2J+1) J

where the 8~(12C) are taken from the 13N spectrum. The second term adds to the single-particle energy in the ~2C core the effect of the interaction of the j-proton with the two extra neutrons of 14C. The matrix elements of the neutron-proton interaction Vpn are taken from effective interaction calculations 23):

(lp½1d~J = 2[I'npllp~ld~J = 2> = 1.75 MeV, (lp~ld~cJ = 3[Vnpllp~rld~J = 3) = 1.60 MeV, (lp½2s~rJ = 011/npllp~r2s,J = 0> = 1.12 MeV, (lp½2s~J = llVnpllp½2s½J = 1) = 0.97 MeV.

(32)

The following values are then obtained for the single-particle energies in the 14C core: s~a~ = 0.47 MeV, e~d~ = 1.71 MeV, 8~s4 = 1.22 MeV,

e~s, = 1.59 MeV,

s~a~ = -4.61 MeV,

~dg_ = --3.35 MeV.

(33)

Inserting the values (33) in (17), ( H o ) is obtained as a function of the deformation parameters only. If the two-particle matrix elements of the interaction Vin the (2s, 1d) shell were known, inserting them in (18) through (22), also (V> would be expressed as a function of the deformation parameters only. Thus, taking B.E.(t4C) from tables we would have an expression of E(def la J) as a function of only the deformations.

459

STATES I N O ISOTOPES

We could then determine the deformation parameters by looking for the minimum of E(def is J ) with respect to them. But this is not the case. The two particle matrix elements of V are not known. Obviously an alternative is to calculate those matrix elements whith some phenomenological force. But such forces do not in general account for the detailed properties of nuclear spectra. Thus, the values obtained by using such a force should be considered only as approximate estimates. Taking for Vthe interaction used by KaUio and Kolltveit 17) we calculated all the two-particle matrix elements in the (2s, ld) shell. Harmonic oscillator wave functions were considered, and the method of relative coordinates was used 25). The transformation coefficients were taken from the table given by Balashov and Eltekov 26), and the oscillator constant from Dawson et al. 27). Inserting those matrix elements in (28), we thus have an expression of E(def is J ) depending only on the deformation parameters. We look then for the minimum of E(def is j = 0) with respect to these deformation parameters, and obtain for them the following values: Z~a~ = 0.48,

Z2a~ = 0.15,

Z2s~ = 0.37.

(34)

It was shown by Kelson 11) that the values of the deformation parameters corresponding to the minimum energy do not depend strongly on J in the 1~O case. Benson and Irvine 28) showed that the same is true for 180, in a calculation using the K K force. We use the values obtained above for all the J states obtained from the same intrinsic state by the projection method. The unperturbed position of the J = 0 deformed state corresponding to the deformations (34) is around 4 MeV above the actual ground state of 180. In order to calculate the unperturbed positions of the deformed states in 190 and 2°0 the deformation parameters corresponding to the k = + 3 orbital are needed. The k = ½ orbital in the (2s, ld) shell includes only the Id a and ld~ orbitals. Due to the much higher position of the ld~ single-particle state, it is expected that the parameter corrsponding to it will be smaller than the one corresponding to the ld~ orbital. We have verified that the results are quite insensitive to the value of the parameter corresponding to ld a, by varying the square of it between 0.1 and 0.4. The actual calculation of the matrix elements of the two-body interaction between the deformed states in 190 and 2°0 is very complicated, and the values to be obtained can be considered only as approximate. We therefore estimate them by assuming the deformed states in 190 and 2 ° 0 to be the analogs of the 21Ne and 22Ne ground states, respectively. The single-particle contributions to the total energies are calculated as done for 180 in (17), but with the single-particle energies in the 1~O core (the procedure is outlined in the appendix for 2°O) and the two-body interaction contributions extracted from the binding energies of 21Ne and 22Ne. The estimated values for the unperturbed positions of the deformed states above the actual ground states were 6.0 MeV for 190 and 7.5 MeV for 2°0. As we see, they lie considerably higher than in 180, thus indicating that their influence will most probably not be as important.

460

P. FEDERMAN

The deformation parameters obtained by the same procedure used for 1sO in the case of 2°Ne are: ~2d~ ~---0.61, ~2d.,} = 0.17, Z2,~ = 0.22. (35) In this case the single-particle energies in the 160 core are inserted in the expression of ( H o ) given by (17). As seen from 170 and X7F, now the ld~ orbital lies lower than the 2s, one. This is reflected in the change of the parameters (35) with respect to the values (34). If one wants to use the EF force to obtain the deformation parameters, the strength Vo has to be fixed. We did this by looking for the minimum of the total energy of the deformed state in 2ONe (in this case the ground state) with respect to the deformation parameters, and fixed the value of Vo to reproduce the binding energy. The values obtained were: Vo = 48 MeV,

Z12dt. = 0.51,

Z2d~ = 0.18,

Z2s½ = 0.31,

(36)

which agree with others obtained previously 1o). In this case the single-particle energies in the 160 core are used. If the same value of V0 is now used for the 1sO case, the minimum is obtained for:

Z2d~ ~-- 0.36,

Z2d~. ---- 0.21,

X2s~ = 0.43.

(37)

The very big increase of 122s½in detriment of X2d~ reflects the fact that the EF force favours the 2s~ orbital, probably enhancing its influence. This can be seen for instance from the (2s~ J = 01Vl2s~J--- 0) matrix element being bigger than the (ld~ J = = 0[ VI ld~ J = 0) one. For the values of the deformation parameters given in (37) the unperturbed position of the deformed state in 1sO is about 5 MeV above the actual ground state. 6. Calculations

Using standard shell model techniques all matrix elements of the two-body interaction between shell-model states can be reduced to linear combinations of twoparticle matrix elements. We list below the formulae used in the present calculation. For the matrix elements in the d~ configuration the expression:

(j~uJIVlfl£J) - n ( n - 1 ) ~ [jn~j{lff_2(O~2j2)j2(j,)j ] 2

~252J"

x [jn-2(~2J2)j2(j')Jl}jnct'J-I(j2J'lVlj2j' )

(38)

is used 29). Expression (38), as well as (39), (40) and (41) are given here only for the sake of completeness. They are derived in ref. 29) in the general form listed here. In the actual case of the present calculation, some simplifications occur. For instance the u denoting quantum numbers other than J necessary to specify the state is not needed, and some 9-j symbols reduce to 6-j due to the unique coupling of ls~ to J=0.

STATESIN O ISOTOPES

461

The [j"-e(otzJz)je(j')Jl}j"J] is the n ~ n - 2 c.f.p. (coefficient of fractional parentage) obtained from the n ~ n - 1 c.f.p, ones 29). To calculate the matrix elements in the d~ s~-m configurations we use: I v "n2 "n2 v ! = (Jx•hi ~lJdVIJx-nI~iJx>(5:a:,~+(J2 ~2Jzl Vl:z ~2 J2>¢~/~z'~ .-I-n, nz E [j~'~,J,{lj~'-'(~,,J,,)j,J,'] • 11J11~22J22

× [ j ~ % s ~ { I j ~ ~- ~(~ese~)j~s~]

× ['J;' - '(~H J1 ~)J, J; I}J;'~; J;] [j]~- ~(~2 Jzz)J2 J'2l}j]'~'2 J'~] x [(2J, -I- 1)(2Ji -t- 1)(2J2 -1-1)(2J[ -I- 1)] ~r~ (2J' q- 1) j,

(JlI × E (2J12 "Jr-1) I J 2 2

s,,

Jl J2

I'J~2 J'

J1)(J11 J2[/J22

Jl J2

J1] J2~o

(39)

J J~,J~z J' J )

Here ']22

J2

J12

J'

denotes the usual 9-j coefficient. The first two terms in (39) are calculated using (38). The interaction between the configurations d] and d]-x s~ is given by:

(j%tJ[ VIi"- 1(~l J1)Y'J> _ ( n - 1)x/n ~. [j._l~ 1j~{ij._2(~ej2)jj1][fl_2(a2j2)fl(j,)ji}j%0. ]

2

Ct2J2J"

The interaction of the d] and d~-2 s~ configurations is obtained from: (J~JIVIj~-2(~IJ~)J~(J2)J> .n ..-2 -- 4 ~ - n ( n -- 1)[:~J{Ijl

.2 .2 .2 (~lJ~)jl(J2)J](j~J2lVIj2J2).

(41)

Finally, the interaction between d~-~ s~ and d] -2 s~ configurations is given by: t t .2 (Ja•hi (~iJx)j,• JIVIjx""1--1 (o~lJ1)J2(J2)J>

= ~/'~[j~'- ~(o~'1Ji)JlJll} J~'~l J , ] ( - 1)J - ' ' +J' +JN/(2J~ + 1)(2J z + 1) X {Jl

JIJ2

Jj~},fflJ2J2[Vlj22J2~ ¢glIJl 1

J12J"

462

P. FEDERMAN x [j~'-2(all J l , ) J l J i I}J'~~- xctl J i ] ( - 1)J' +h~ +J2(2j1 a + 1)

x[(2j1.4.1)(2jtl..l_1)(2J2.4.1)(2j,.{.1)]~} {J~l J} J12 / Jl!

X t~ J2J2



(42)

for Jl = dl, J2 = s& and therefore J2 = 0. The two-body interaction matrix elements between shell model states are built for 1so, 190 and 2°0 using expressions (38) to (42), and the single-particle energies are taken from the 170 spectrum:

e'~ag = B . E . ( 1 7 0 ) - B . E . ( ' 6 0 ) = 4.14 MeV, e~s½ = e~dg--0.87 MeV = 3.27 MeV,

e~at = e~ag- 5.08 MeV = - 0.94 MeV.

(43)

The energies given above and the corresponding to protons taken from the 17F spectrum are the ones used to calculate the unperturbed position of the deformed state in 2 ONe" For the unperturbed positions of the deformed states we assume a rotational structure. For simplicity we take the same moment of inertia for the three isotopes. Clearly this does not affect much the calculations. The moment of inertia will be determined mainly by its value in iSo. Due to the higher unperturbed positions of the deformed states in 190 and 2°0, a modification of the rotational constant R = h 2 / 2 ~ ¢" would not affect the low lying states. An estimation of R can be obtained from the spectrum of 160, where R = 0.15 MeV. All matrix elements between shell model and deformed states are proportional to the single parameter S. The two-particle matrix elements entering in the expressions of shell-model matrix dements are:

,

J = O, 2, 4,


J = 2, 3,

<2s~ JI Vl2s~ J>,

J = O,

<1d~JIVI1dg2%J),

J = 2,

,

J = O.

Their total number is eight. Including R and S it is thus possible to build all the energy matrices as functions of ten parameters. The calculations are performed in the effective interactions scheme. The values for the theoretical parameters are determined by diagonalizing the energy matrices and

STATES IN 0 ISOTOPES

463

r e q u i r i n g the eigenvalues to be as close as possible to the e x p e r i m e n t a l values. W i t h the values o b t a i n e d , the energies a n d wave functions are calculated. T h e n u m b e r o f e x p e r i m e n t a l energies included is nineteen. S o m e o f t h e m were recently o b t a i n e d in 180 (refs. 2,6, 30)), 190 (refs. 31,32,33)) a n d 2 0 0 (refs. 5,32)). A n i t e r a t i o n p r o c e d u r e is used to o b t a i n the best values o f the ten theoretical p a r a meters giving the least square fit to the data.

7. Results U s i n g the p r o c e d u r e described in the previous section the energies listed as case 1 in table 1 are o b t a i n e d for the d e f o r m a t i o n s (34). The a g r e e m e n t between calculated TABLE 1

Calculated and experimental energies of the low-lying positive parity levels in the oxygen isotopes (in MeV). The binding energy of 1~O has been substracted. Also the main components of the calculated wave functions for 1~O and 2°0 are listed Nucleus

J

Case l

Case 2

Experimental

180

0 0 0 2 2 2 3 4 4 ½

12.25 8.53 6.86 10.20 8.17 6.93 6.86 8.36 5.08 14.64 16.09 14.15 10.74 16.00 12.88 11.74 10.61 12.47 13.38 10.61 23.75 19.33 16.75 22.02 19.75 18.99 17.41 20.30 17.79

12.13 8.62 6.83 10.05 8.15 6.93 6.81 8.28 5.08 14.56 16.07 13.70 10.84 16.10 12.97 11.72 10.40 12.69 13.34 10.62 23.85 19.28 16.49 22.00 19.21 19.11 17.26 20.25 17.96

12.19 8.56 6.86 10.21 8.27 6.94 6.82 8.64 5.07 14.68 16.04

x~o

{t {r { 200

0 0 0 2 2 2 2 4 4

16.14 12.98

13.37 23.75 19.30 22.08 19.68 20.18

Main configuration

d~~(0)st_ d~a d~t2(2)s~: def d¢a d~Z(2)s{ s.~2(0)d{. def d~(4)s~r d{ s d~2(4)s~r d{_~ d{t~(0)s~.2(0) def d{r~ d~_a({)s½ d{ a({)s td{_2(2)s4r2(0) d~r4 dg_~(~)s~

a n d e x p e r i m e n t a l energies (also listed in table 1) is very good, as can be seen f r o m the small deviations. T h e values listed c o r r e s p o n d to the e s t i m a t e d u n p e r t u r b e d energies

464

P. FEDERMAN

TABLE 2 Values of the theoretical parameters obtained here and in other calculations in MeV Matrix element

Case 1

( l d ~ 0 ] Vlld+20> (ld~22] VIld~Z2) ( 1d~4l v[1d½M) (2sjrZ0[ Vl2s~r~0> ( 1d~2s~r~[ V [1d~.2s~.~> (ld~2s,~3j[Vlld~r2s½3 > (1 d½20[V[s~20) (ld~r22[ Vlld~2s~? ) S R

Case 2 Talmi and Unna

3.15±0.20 1.62 :t:0.17 -- 0.05 :t:0.08 1.47 4:0.33 0.69±0.17 --0.55-4-0.15 1.00±0.29

3.24 1.59 --0.03 1.97 0.76 --0.72 0.77

0.42=}=0.31 1.82±0.24 0.15±0.01

3.37 1.50 --0.15 1.70 0.79 --1.16

0.48 1.40 0.11

Cohen e t al.

3.33 1.38 -- 0.02 2.07 0.87 --1.00 0.93

EF force

Dawson

2.31 0.47 --0.02 2.71

0.69

e t al.

KK force

KKB force

1.96 1.60 0.77 3.02

2.67 0.87 0.48 2.40

1.08

1.69

1.35

--0.75 0.82 0.52

0.65 0.95 0.78

0 0.85 0.64

3.64 0.93 --0.21 3.07 1.35 0 1.13 0.64

1.15

1.54

1.17

1.17

TABLE 3 Wave functions obtained here for the J = 0+ and 3. = 2 + states in 180 Y ----0

def.

d½2

s~rS

Case 1

0.35 0.64 0.69 0.22 0.67 0.70

0.88 --0.47 --0.02 0.88 --0.44 --0.15

0.31 0.61 --0.73 0.42 0.59 --0.69

3" = 2

def.

d~ ~

Case 1

0.26 0.37 0.90 0.14 0.25 0.96

--0.93 0.37 0.14 --0.94 0.33 0.05

Case 2

Case 2

Energy 0.06 3.66 5.33 --0.06 3.57 5.36

d~ s~.

Energy

--0.26 --0.88 0.40 --0.31 --0.91 0.28

1.99 4.02 5.26 2.14 4.04 5.26

TABLE 4 Matrix elements between shell model and deformed states as obtained here for leo Matrix element

( d e f y ~ 0] V]dt.t Y = 0) ( d e f y ~ 0IV[s~ Y ----0) ( d e f y ----2Irides# y = 2>

Case 1

Case 2

1.18 1.24 0.66 0.59

0.59 1.20 0.30 0.41

HamadaJohnston force 1.31 0.96 0.75 1.12

Dawson, Talmi and Walecka 0.99 1.04 0.55 0.49

EF force

KK force

0.74 0.78 0.41 0.37

0.75 0.79 0.42

0.37

465

STATES IN O ISOTOPES

o f the d e f o r m e d states m e n t i o n e d in sect. 5 (4.0 MeV, 6.0 M e V a n d 7.5 MeV, respectively, a b o v e the a c t u a l g r o u n d states o f 1 s o , 190 a n d 2 ° 0 ) . I t was verified t h a t slight shifts in these values d o n o t give better agreement. T h e c o r r e s p o n d i n g values for the t h e o r e t i c a l p a r a m e t e r s are listed as case 1 in table 2. The value o b t a i n e d for R coincides w i t h the one in 160. W a v e functions for 180 are i n c l u d e d in case 1 o f table 3, a n d m a i n c o m p o n e n t s o f the wave functions for 190 a n d 2 0 0 are listed in table 1. M a t r i x elements between shell m o d e l a n d d e f o r m e d states are given in table 4 for the case o f 1sO, c o m p a r e d w i t h the values o b t a i n e d for different p h e n o m e n o l o g i c a l forces. T h e s p e c t r u m o f 1So is nicely r e p r o d u c e d . F u r t h e r m o r e , the wave functions are fairly consistent w i t h the single a n d d o u b l e s t r i p p i n g d a t a 5,6, a2). W e o b t a i n for the g r o u n d state wave f u n c t i o n m a i n l y a Id~ c o n t r i b u t i o n . This agrees with the values e x t r a c t e d f r o m t h e e x p e r i m e n t a l d a t a o f the 1 s O ( s H e , d) r e a c t i o n 3¢) a n d 170 (d, p ) r e a c t i o n a5,6) t. T h e generally g o o d a g r e e m e n t between the calculated a n d extracted (ref. 35)) wave functions c a n be seen in t a b l e 5 t t TABLE 5 Comparison between the calculated amplitudes for the wave functions of 180 and the ones extracted from (d, p) experiments State

Configurations

Case 1

Experimental

J = 0+ g.s.

d½~ s½s def

0.71 0.14 0.15

0.81 0.15 0.04

d" = 2+ 1.98

d~2 d~s~ clef

0.89 0.09 0.02

0.56 0.23 0.21

dt2 s.~~ def

0.25 0.66 0.09

<0.18

d = 0+ 3.63

d = 2+ 5.25

d~r* d~rs~r def

0.08 0.31 0.61

<0.08 0.32 >0.6

d~ ~ s~2 def

0.04 0.20 0.66

<0.12

d = 0 + 5.33

t It is worth noting that, in spite of the differences in the amplitudes listed in ref. ss) and ref. 0, the experimental information obtained in both works is quite the same. These differences arise from the different ways of extracting the amplitudes, mainly the different single-particle reduced widths considered ae). In ref. e) a constant value is used, while in ref. a6) a variation with the Q-value is included. We are indebted to Dr. R. Moreh for pointing out this to us. tt Still, it should be mentioned that the amplitudes listed here do not include any effects from the deformed components, The (d, p) reaction was assumed to proceed only through the shell-model components. The explicit inclusion of the deformed components in the reaction process may be of importance if the deformed component in the ground state of tTO is appreciable.

466

P. FEDERMAN

The ~60(t, p) reaction shows a small L = 0 double stripping to the second J = 0 + state in 1so (ref. s)). As can be seen in table 3 we obtain for this state comparable amplitudes with opposite phases for the ld~ and 2s~ components. The L = 0 double stripping to the ground and 5.33 MeV J -- 0 + states is much stronger, showing that the two-particle components in those states are rather large. The wave functions obtained here show such a feature. Also the spectra of 190 and 200 are reproduced nicely. Some levels are predicted in these nuclei. In 190 levels with spins 5 +, ~+ and 5 + are calculated respectively around 2.0 MeV, 3.8 MeV and 4.3 MeV. There are observed levels near those energies, but no definite assignments were yet made to them. Also a J = 5 + level is calculated around 5.4 MeV, and there is a such level at 5.45 MeV. Still, the inclusion of configurations including the ld~ orbital may affect the result for this level. The lowest of the extra levels predicted in 200 lies around 4.8 MeV. This is in agreement with the experimental data, which do not show any additional level below 4.84 MeV. For comparison the calculations were also performed for the deformation parameters (37). The general agreement is in this case somewhat poorer, probably due to the fact that the EF force overestimates the value of X2s½. The corresponding matrix elements and energies are listed in case 2 in the corresponding tables. In table 2 we include also for comparison the two-particle interaction matrix elements obtained in previous shell-model calculations. These include effective interaction calculations by Talmi and Unna 3) and Cohen et al. 4); the EF matrix elements (for Iio = 48 MeV); calculations using the free nucleon interaction obtained by the approximate solution of the Bethe-Goldstone equation by Dawson et al. 27), and the K K interaction acting for T = 1 only in singlet even states 17). The last matrix elements including corrections due to core polarization as calculated by Bertsch 18) for the diagonal ones are also listed. In table 4 the matrix elements between shell model and deformed states calculated using the different forces with the deformations (34) are also listed. Recently Brown and Green 37) calculated the matrix elements starting from the Hamada-Johnston nucleon-nucleon interaction and adding the effects of higher configurations. The values obtained by them are also given in table 4. It is noticeable that the results of this last calculation as well as the values obtained from the interaction used by Dawson et al. 27) are in fair agreement with the ones obtained here. In order to compare with our results we also carried out the calculations for the EF, K K and KKB phenomenological forces. The resulting energy levels are presented in figs. 5, 6 and 7, respectively, for ~SO, 190 and 2°O. From left to right in each figure the spectra respectively correspond to: experiment, K K force, K K B force and EF force. It is clear that in general the agreement is poor. The detailed structure of the low-lying spectra of the oxygen isotopes is not reproduced by phenomenological forces.

STATES IN 0 ISOTOPES

467

McV 4

4

O

4

0 "=='===3 2

~

~

- - 3

2 0 3 4

2

2 0 4

0

~

~ 2 - - 0

2

2

- - 2

0 ~ 4 2

~

O ~

O

O KK

EXPERIMENTAL

O

KKB

EF

Fig. 5. Low-lying positive parity levels of zeO calculated using phenomenologJca] forces. The experimental spectrum is included for comparison. MeV 6

....

%

9 / 2 S/2 - -

- -

~3/2s/2

3/2 st2

~2

EXPERIMENTAL

3/2

- -

%

- -

%2 st2

- -

%

s/2

--9/2 I/2 - - s / 2

o

s/2

- -

KK

- -

s/2 t/2

KKB

312 EF

Fig. 6. Low-lying positive parity levels o f z°O calculated using phenomenological forces. The exper£mental spectrum is included for comparison. MeV

- - 0 - - 0

2 ~

0

4

4 4 2 = = = ~ . 42

~ 2

~ 2

- - 0

- - 0

EXPERIMENTAL

KK

- - 2

4

- - 2 0 2

- - 4

4 4 2 2

0 KKR

2

0 EP

Fig. 7. Low-lying positive parity levels of ~°O calculated using phenomenological forces. The experimental spectrum is included for comparison.

468

e. rEDEmC~N

8. Discussion In general the agreement between calculated and experimental energies is very good. The influence of deformed states is found to be important in 1So, but not in 190 and 2 oo. In the later isotopes, the admixture with the shell-model states is small. We would like to make some comments about some of our assumptions. The influence of a J = 4 + state due to the ld~ ld~ configuration in 1SO was not included. Such a level is expected around 8 MeV above the ld~ ( J --- 0) state. Thus, the J -- 4 + level at 7.12 MeV seems to lie too low as to belong to the ld~ ld~ configuration. Due to the smaller value of Xa, the matrix element to the ld~ ld~ state with the deformed state will be relatively small. It is thus possible that its inclusion would then only push down the first J = 4 + state, bringing it closer to the experimental value. Another point is the influence of the ld~ ( J = 0) state. Assuming its unperturbed position to be 10 MeV above the ld~ ( J = 0) state, it would not affect the present results by more than around 0.1 MeV. The deformed states included in the present calculations are not the only ones possible to construct, but they are the lowest. For instance, instead of raising the two lp~ protons into the (2s, ld) shell, it is possible to raise the two lp~r neutrons. But in this case, they have to occupy the higher k -- + ½ orbital, the k = _ ½ being already filled with the two valence neutrons. Also the symmetry energy due to the strong interaction of protons and neutrons in the same orbit is lost. The result is that such a state is less much favoured in energy. An estimate of its position gives it at around 10 MeV above the ground state as). It is worth mentioning that a possible overlap factor arising from the deformation o f the 14C core 13) does not appear explicitly in the present calculation. The parameter S determines the effective non-diagonal matrix elements between shell model and deformed states. If similar states to those considered here obtained from the excitation of particles in the core would affect differently the ground states of 160 and 170, the experimental single-particle energies extracted from the spectrum of 170 should be corrected. Brown and Green 1a, a7) obtain the same shifts for the ground states of 160 and 170. Moreover, we repeated the calculations allowing the single-particle energies to vary, and obtained no significant change. Thus, it seems that if the ground states of 160 and 170 are depressed, it is by the same amount. In 190 we assumed the first excited J = ~+ level to be at 3.16 MeV. The level at 2.35 MeV was assigned one of the spins ½+, ~+ or ~+. We therefore repeated the present calculations without this hypothesis, and did not obtain any significant modification. The next region where similar "puzzles" were observed is the calcium region. Analogously to 160, also 4°Ca is a double-closed nucleus, showing a very low second J = 0 + state. Moreover, 42Ca being the analogue of 1sO, also shows a similar behaviour.

STATES I N 0 ISOTOPI~

469

Recent calculations ag, +o) show that the spectra of the calcium isotopes can be understood in the frame of the present model. In conclusion, a consistent picture describing the low-lying levels of the oxygen isotopes is obtained in the scheme of effective interactions, including the interaction of shell model and deformed states. The same model also accounts for the low-lying levels of nuclei in a region of similar behaviour. The identification of more states would be of great interest, furnishing a more severe test for the validity of the present picture. I wish to thank Professor 1. Talmi for suggesting the subject of this work and for his constant guidance and constructive help. I would like also to thank the authorities and staff of The Weizmann Institute of Science, in particular Professors A. de-Shalit and P. Hillman, for the warm hospitality enjoyed during my stay at Rehovoth, where this work was performed. The partial financial support of the Consejo Nacional de Investigaciones Cientificas y Tecnicas, Buenos Aires, and The Weizmann Institute of Science, Rehovoth is gratefully acknowledged. Thanks are also due to Dr. N. Auerbach for helpful discussions and to Dr. A. M. Lane for critical reading of the manuscript.

Appendix 1 WAVE FUNCTION

O F T H E 300 D E F O R M E D

STATE

In 2°0 two neutrons occupy the k = -+3 deformed orbit. Therefore the wave functions of the deformed states are given by:

I s~s2

0

{P~,I½ -½ ½ -½>~21½ _½>}SM.

(A.1)

We already obtained a j-j coupling expression for the two particles component (see eq. (10)). We shall proceed here to obtain a j-j coupling expression for the four particles component. Using (1 1) it follows: =

l ~ J 2 + j 4 - IZ

'

.

JlJ2J3J4

where as before q~k denotes the deformed orbital [k> and ~j, m the spherical orbital ]j, m). The notation convention gj, k=+ -- Yj is used. If by ~ we denote the antisymmetrization operator it follows from (A.2): P~,d[~+tb_+q~+~_+] =

~

(-1)s2+s*-lZ~g2yay,,

JlJ2Jaj4 J,,,~. -½ -3 × d[{tks ' ~ky~}J,~{~bs+~s+}S~,]s,, M,.

0

(A.3)

470

P. FEDERMAN

The reduction of the different components in eq. (A.3) to standard shell-model wave functions depends on how many of the j~ are equal among them. Each different case is separated, and using coupling and recoupling algebra all components are reduced to one of the standard forms $(j4),$(j~,j2 ), $(j2,j2) or $(j2,j2j3 ). For instance the contribution to (A.3) of the component with all thej~ equal is:

J34. i

x 4N(Jt2 , S3*,

J1)$(J*, Sl),

(A.4)

where $(j*, Jr) is completely antisymmetric and

E

N 2 ( J 1 2 , J 3 4 , J , ) = 6 1+c5~,2j3,-4

{!

J

J3,t~ ( 2 J 1 2 + l ) ( 2 J 3 , + l )

J 2 J34- J1 )

1

•

Once the different components are all expressed in one of the standard forms, the matrix elements of the single-particle energy operator are calculated using standard shell-model techniques.

Appendix 2

NON-DIAGONAL MATRIX ELEMENTS BETWEEN SHELL MODEL AND DEFORMED STATES IN I°O After raising the two l p , protons into the deformed orbitals, the wave function thus obtained consists of two neutrons in the lp½ orbit (coupled to zero angular momentum) and the deformed state, which includes four neutrons and two protons filling deformed orbitals in the (2s, ld) shell. Therefore, the only contribution of a two-body interaction V = ~i~Vi~ to a nondiagonal matrix element between a shell-model state (containing two IP½ protons) and a deformed state (containing no lp¢ protons) will come from the term in which both i a n d j refer to the two protons occupying different orbits in both states. Hence, given a shell-model state ~bj it follows (using (A.1)): (~ksl Vldef 2° J ) ="

- 20 ~ x/N(def j) s~s2

0

<0:1½-½ ½ - t J1)

x 1 = ~/)V(def2O J) <$JI½-½ ~ - ½ J),

(A.6)

STATESIN 0 ISOTOPES

471

where N(def 2° J ) denotes the norm of [def 2° J ) . From eq. (14), = E ~j2..(__ 1)j--.~- I..2 Ij

j

½

00] 4 2 < P ~ r J 2 = OI ~zrij2J2 ~" O> ~---4 2 ~ ,

where S was defined in (23). On the other hand, the only contribution of [ ½ - ½ ½ - ½ J ) to <~kJI½-½½-½ J ) will come from the term of the same form as ~j. For example, if ~kj = ~k(j~ J), it follows from (A.4) that

< ,(JfJ)1½- ½

22

= ZlYl

J> E

4

J12 o,en J34J

[J' Jt J~2][Jl J~ ~3';][J~2 J34. J] N(Jx2,J34,J), ½ --½

0

where N(J12 , J34, J) is defined in (A.5).

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) 25) 26) 27) 28)

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