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The effect of deformed states in the Ca isotopes
~
Nuclear Physics A93 (1967) 110---132; ( ~ North-Holland Publishing Co., Amsterdam
I
Not to be reproduced by photoprint or microfilm without written permission from the publisher
THE EFFECT OF D E F O R M E D STATES IN THE Ca I S O T O P E S W. J. GERACE
Palmer Physical Laboratory, Princeton University, Princeton, New Jersey and A. M. GREEN
The Institute for Advanced Study, Princeton, New Jersey t Received 14 October 1966 Abstract: An attempt is made to explain the low-lying levels of 4°Ca, 41Ca, 42Ca and 41Sc as mixtures of shell-model and deformed states. The deformed states are constructed using the isospin coupling scheme of Zamick, and the matrix elements between spherical and deformed states are calculated using the Hamada-Johnston nucleon-nucleon force. Wave functions are obtained for these states and predicted values for some transition rates and single-particle strengths are compared with experiment.
1. Introduction
For several years it has been known that the ordinary shell model is inadequate, even near double dosed shells, which is the region in which it has had its greatest success. The old model cannot explain either the occurrence of large electric quadrupole transitions or the number of low-lying energy levels, many of which can be fitted into rotational bands. A group of nuclei having these complications is the oxygen isotopes. These have been studied in some detail by Engeland 1) and Brown 2) and by Brown and Green a, 4), where a remedy has been proposed which is capable of explaining several of the new experimental features. The method consists of introducing low-lying deformed states in addition to the usual shell-model states. The mixing of these states can account for the large transition rates and the rotational character of the spectra. In this paper these techniques are applied to the calcium region with detailed studies of 4°Ca, 41Ca, 42Ca and 41Sc. The procedures employed here are an extension of the methods used by Bertsch 6) in which only two-hole states were introduced and in which the matrix elements did not contain core polarization effects. t This work was supported by the U.S. Atomic Energy Commission and the Higgins Scientific Trust Fund. This report made use of the Princeton Computer Facilities supported in part by the National Science Foundation Grant NSF-GP 579. 110
CaDEFORMED STATES
111
2. Even-parity states of 4°Ca In this section we interpret the even-parity states of 40Ca in a way that is completely analogous to the description of the lowest, even-parity states of 160 given in ref. 4). Namely, we consider these states as mixtures of the double closed 2s-ld shell-model state with two intrinsic deformed states formed by raising two and four particles from the 2s-ld shell into the 2 p - l f shell. These two deformed states can be illustrated by means of the Nilsson diagrams shown on figs. la and b. The intrinsic states contain components with different angular momenta, whereas the double closed shell contains only J = 0. Once the matrix elements between the states and the unperturbed energy of each state is known, wave functions can be derived. These wave functions can then be used to estimate the electromagnetic transition rates between levels. 4.5
- ~
4.5
. z
~P312
7/2-
I7
'
I
'
I/2
4.0
i
o
~
X~ :I
3.5
I
-0.2
I
-O.I
0
_
--.-''" 3/2+
I
I
0.1
0.2
3.5
I
-0.2
I
-0.1
0
I
I
0.I
0.2
Fig. 1. Relevant section of Nilsson diagram showing a) two-particle-two-hole state and b) fourparticle-four-hole state, ×-particle, e-hole. 2.1. MATRIX ELEMENTS IN 4°Ca The basic matrix element involved in this description is J=O
__J=O
M1 = (0p-0hr=°lVl2p-2hr=°),
(1)
and this we shall calculate in two ways. From the work of Zamick s), which involves only qualitative arguments, we find in 4°Ca that the 2p-2h state with [2p Tp= 1-2h T" = 1IT= 0 is about 5 MeV lower in energy than the configuration with Tp = Th = 0; and so in general, where these two configurations have been mixed, we expect the lowest 2p-2h state to be mainly [2prp = L2hTh = 1]T = o
112
W. J. GERACE AND A. M. GREEN
For simplicity, we assume that the 2p-2h state which we mix with the 0p-0h and 4p-4h state is only this configuration. This prescription of coupling the particles and holes separately to a given isospin ensures that we are antisymmetrizing correctly when we relate matrix elements in other calcium isotopes to the M 1 matrix element of 4°Ca. Taking Nilsson's no. 14 and no. 8 orbitals at fl = 0.2 and projecting out a 2p-2h wave function with zero angular momentum, we reduce the matrix element M1 to the form '
I a 2
M~ = ~/N
J=O
J=o
[~
((lf~)gr=l [Vl(ld,)2r=l 0
)
0
2 21 ld 0 2 1) +½b E((p,)IVl( J=2
Jr2
5 a2((lf~)r=l[Vl(ld~)r=,) 2x/42 2
2
- ½b2((2p~)*lVl(ld~)l )
3ab
2
2}
414 ((lf~2p~)llVl(ld~)x> /
x overlap.
(2)
In this we have used the simplified orbitals ffno.14 =
allf~,~)+bl2p~,~),
~k,o.S = Ild~.,). The factor N normalizes the 2p-2h a = o wave function and has the value 0.363 when a = 0.880, b = -0.476, remembering that the 2p wave function used here is of the form ~ - (at) 2, i.e. opposite in sign to that used by Nilsson. The overall factor of x/3 arises from the isospin, when we convert particle-hole matrix elements into particleparticle matrix elements. The overlap factor is introduced to take care of the fact that the 38 inert particles on each side of the two-body interaction are in different onebody potentials; the 0p-0h state is in a spherically shaped well and the 2p-2h state is in a deformed well. This lack of overlap has been estimated by Bertsch 6) in two ways. First, he calculated the equilibrium deformation of various particle-hole configurations by summing the energies of the single-particle orbitals in a deformed well. In 4°Ca, using wave functions derived by Rost for a deformed Saxon-Woods potential, he obtained a deformation of about fl = 0.15 for the 2p-2h configuration and fl = 0.20 for the 4p-4h configuration. However, this is expected to be an underestimation for fl since the Rost wave functions, just like the Nilsson wave functions, introduce only quadrupole correlations, and so the additional correlations with non-zero angular momentum tend to increase these estimates of ft. In his second calculation, he evaluated the overlap factor directly by perturbation theory and got the value 0.87.
Ca DEFORMED STATES
113
Throughout the rest of this paper we shall use an overlap factor of 0.8 which is consistent with the value calculated by Bertsch. This value for the overlap corresponds to a fl of about 0.24. The strong dependence of this factor on fl is shown in table 1. TABLE
1
Factors from the overlap of spherical and deformed cores fl
0.15
0.23
0.31
4°Ca overlap factor
0.92
0.84
0.74
It has been calculated assuming that the lowest 38 orbitals are filled in a deformed oscillator potential without spin-orbit splitting. In eq. (2), the two-body matrix elements taken are from the work of Kuo 7), who has an extensive program for deriving effective two-body interactions in nuclei from realistic, free nucleon-nucleon interactions (in this case the Hamada-Johnston potential). His results are given in column 2 of table 2 for ho~ = 10.5 MeV. TABLE 2
Two-body matrix elements Kuo 7) J=O
Effect of core
Total effective
polarization
interaction
1.92
1.13
3.05
0.36
0.03
0.39
0.22
0.63
0.85
0.52
1.32
1.84
0.02
0.32
0.34
J=O
((lf~)T=l[ Vl(ld~) T=t) 2
2
((1 f~) t ]V[(1 d~) 1) 2
2
o
((2p~)a] V] (ld~r)a > 2
2
((2p~) 11V[(ld~)X>
Inserting these numbers into eq. (2) results in a value of M1 = - 1.49 MeV. However the recent work of Bertsch 6) has shown that in 180 and 42Ca there is an important correction to the type of effective two-body matrix elements used in the previous paragraph. He finds that the term in which the two-valence particles interact with each other by first exciting the double closed shell core is by no means insignificant. The analogous correction to the (0p-0h°lVI2p-2h °) matrix element is best illustrated by fig. 2b, the first-order term being illustrated by fig. 2a.
(a) Fig. 2. a) First-order graph.
(b) b) Second-order graph.
114
W . J. GERACE A N D A. M. G R E E N
These higher-order terms have also been evaluated by Kuo with the HamadaJohnston potential. The energy denominator that enters is taken to be the difference in energy between the external particle-hole lines and the internal p-h bubble and ranges from 2.5 MeV to 13 MeV. These corrections are given in column 3 of table 2. Using the modified matrix elements of column 4 in eq. (2), we then get M1 = - 2 . 2 4 MeV, i.e. the effect of core polarization results in a 50 % increase over the first-order term of fig. 2a. In 160 the corresponding increase was 35 %. One complication that may be thought to arise in the evaluation of fig. 2b is the effect of spurious states in the excited bubble. However for the leading term, the vertices involved - namely lf~ld~ 1 _ do not interact with the J = 1 multipole of the twobody interaction, and a J = 1 p-h bubble cannot be excited. In other less important terms where the p-h bubble can have spurious components, their elimination has a negligible effect on M1. In general, it seems that spurious components do not affect results when they enter into the calculation of specific terms in perturbation theory. It is only when we diagonalize a matrix containing these spurious components that trouble can arise. In that case we are effectively summing an infinite series of p-h bubbles, and so the small spurious components in each term can build up coherently into a collective effect. In the above estimate of M1 we have leaned heavily on the Nilsson wave functions. This is because it is well known that for a few particles outside a closed shell an approximate ground state wave function can be obtained by projecting out the appropriate angular momentum from a product of Nilsson wave functions; the reason being that the Nilsson wave functions are essentially the result of diagonalizing a quadrupole-quadrupole interparticle interaction. However we have no right to expect good particle-hole wave functions when we couple Nilsson orbitals for particles and holes, since the Nilsson model does not allow for particle-hole interactions and, as Zamick 5) has shown, these can be very important. To overcome this objection we use Kuo's particle-particle and particle-hole matrix elements and diagonalize a 3 x 3 matrix formed from the leading 2p-2h states. This results in a vector J=0
J=0
0
+ 0.9671 {[lf~] r = 111d~-2] r = 1} 05 0
0 0
+ 0.1371{[2p~]* [ld~211}°5 2
2 0
- 0.2131{[lf2]~ [ld~ 211}° 5 for the lowest 2p-2h state and so the basic matrix element M , becomes - 2 . 3 3 MeV. The effects of higher configurations are added by perturbation theory as before and increase the value of M~ to - 2 . 6 MeV; this value is adopted in the rest of this paper. Before we can interpret the spectrum of 4°Ca, we also need the matrix elements
M~ = <2pZ~-2hJIVI4-~-4hJS.
Ca DEFORMED STATES
115
A complete calculation of this would require Hill-Wheeler integrals, and therefore we only make estimates in the limits o f extreme deformation and SU3 symmetry. The approximations involved here follow closely those made in ref. 4 ) , for 160 and result in a value o f M~ ranging f r o m - 0 . 9 MeV to - 1 . 1 MeV for J = 0 and 2 and a value o f M24 o f - 0 . 8 MeV. Consequently, in the following calculation, we take J=0
J=O
M J= o = (0p_0h T ~ o I V l~p.Eh T = o) = _ 2.6 MeV, _J=O
_ _ J = O
M~ =° = M~ =2 -- (2p-2hT=° I Vl4p-4hr=°> = - 1.0 MeV, J=4
- - J = 4
M J2=4 = ( 2 p - 2 h r = ° [ V I 4 p - 4 h r = ° > = - 0 . 8 M e g .
Quantities such as transition probabilities and single-particle strengths to be studied in later sections are found to be quantitatively the same over the range o f M2 quoted here. 2.2. ENERGIES OF UNPERTURBED CONFIGURATIONS In 016 the energies of the multi-particle-multi-hole configurations have been calculated by several people in different ways. The simplest estimates were made by Z a m i c k 5) using a particle-hole interaction that had only a m o n o p o l e c o m p o n e n t but with a strong isospin dependence. He f o u n d that the 4p-4h state, where the holes are in the l p shell and the particles in the 2s-ld shell, lay at about the same energy as the corresponding 2p-2h state. This rather surprising result had been f o u n d earlier in the more refined calculations o f Ripka and Bassichis 8) and H a y w a r d 9) by means o f Hartree-Fock and variational techniques. In fact these calculations predict the 4p-4h state to be below the 2p-2h state. Moreover this ordering of unperturbed levels was needed in our interpretation o f 160, since the calculated matrix element between the double closed shell and the 2p-2h state was too large to allow a fit to the observed energy levels if we permitted the 2p-2h state to be below the 4p-4h state. In 4°Ca, a similar situation exists. The only calculation o f the ordering o f the 2p-2h and 4p-4h is that o f Zamick 5), which is indecisive; his only conclusion being that these two states are more or less degenerate. However as in 160, the value o f M t exceeds one half the splitting o f the first two 0 ÷ levels, making it impossible to fit the observed levels with the 2p-2h level lower than the 4p-4h. ? In the previous work 4) on 1~O, in the expression corresponding to eq. (1) of this paper the isospin factor was ,/6 and not ,/3 as we have used here. This is because in ref. 4) the 2p-2h wave function is taken to be equal mixtures of [2pTp=°-2hTh=°]T~° and[2pTr, sa-2hTh=l]T=°. However the Zamick 5) argument that the latter state lies lower in energy also applies in 1~O, and therefore a factor of,/3 would have been more realistic. Fortunately in 160 the decrease in the matrix dement due to this correction is more or less cancelled by the core polarization which was also omitted. The effect of both corrections is to raise the 160 matrix element by a factor of 1.34, placing it within 0.2 MeV of the matrix elements used in ref. 4).
116
W. J. GERACE AND A. M. GREEN
We therefore place the 4p-4h lower than the 2p-2h at about 3.8 MeV. A more sophisticated calculation recently made by Arima 27) supports the case for a lowlying 4p-4h state. By considering the excitation of the low-lying 1 + state in ~aSc, he estimates the unperturbed position of the 4p-4h state in 4°Ca to be at 4.1 MeV above the final ground state of 4°Ca. With this ordering of states we adjust the unperturbed energies of the 2 + levels to ensure a good fit to the known levels. I f we are consistent we would expect the energies of the 0 +, 2 + and 4 + levels in each of the two unperturbed deformed bands to bear a (hZ/2J)J(J+ l) relationship with some reasonable hz/2J. It is also required that the perturbed energies correspond to observed levels. These two conditions are sufficient to set the values for the unperturbed energies of the J = 0 and J = 4 levels. 2.3. C A L C U L A T I O N
O F T H E 4°Ca W A V E F U N C T I O N S
We now have matrix elements and a prescription for determining the positions of the unperturbed levels. Before we can carry out this type of calculation we have to decide for which observed levels we are trying to account. The spectrum of interest is shown in fig. 3. The two J = 2 levels arising from the 2p-2h and 4p-4h mixture are immediately fixed to be those two 2 + levels at 3.90 MeV and 6.94 MeV strongly excited in inelastic electron 1o) and proton 11) scattering. However, this creates a problem when we have to decide which three J = 0 levels to fit as 0p-0h, 2p-2h and 4p-4h mixtures. Clearly the ground state and the 3.35 MeV 0 + state are two of the three levels, but the only other J -- 0 level known is one recently seen by Grace and Poletti 12) at 5.20 MeV. Unfortunately, if we consider this to be our third level we do not satisfy our consistency criterion of having a reasonable J = 0, J = 2 relation2+
6.94 6.75 6.54 6.50
2+ 4 2 4 (I)0
5.62 5.61 5.27 5.24 5.20
n+ a
3.90
0+
3.35
0+
0
Go
40
Fig. 3. E n e r g y levels o f 4°Ca w i t h even o r u n k n o w n spin.
Ca DEFORMED STATES
117
ship, since this requires the two unperturbed levels in a given b a n d to be about ½ M e V apart. We prefer to think that the third 0 ÷ we should fit is a level about 1 to 2 M e V higher that has yet to be identified. Actually several levels o f u n k n o w n spin and parity are seen in this region. We would then say that the 0 ~+) seen at 5.2 MeV has the 2 + at 5.24 or 5.6 MeV associated with it. With our value o f M 2 = - 1 . 0 MeV, it is necessary to take E(2p-2h, J = 2) = 6.52 MeV, E(4p-4h, J = 2) = 4.28 MeV, in order to obtain the eigenvalues and eigenvectors given in table 3. TABLE 3 W a v e functions of 2 + states in 4°Ca State
Energy
21+ 2z +
3.90 6.90
2p-2hJ = o
4p-4hJ= o
0.355 0.935
0.935 --0.355
In order to predict the unperturbed positions of the 0 ÷ levels f r o m unperturbed positions of the 2 ÷ levels, a value o f h2/2~¢must be given. We use a value o f hZ/2S o f about 0.1 which is indicated by the 0 ÷, 2 ÷ and 4 + states at 3.35, 3.90 and 5.2 MeV, respectively. These states already f o r m an almost perfect rotational b a n d even t h o u g h they are mixtures of deformed states. F r o m this we obtain for the J = 0 levels, E(0p-0h, J = 0) = 1.20 MeV, E(2p-2h, J = 0) = 5.95 MeV, E(4p-4h, J = 0) = 3.75 MeV. The final energies and wave functions are given in table 4. TABLE 4 W a v e functions o f 0 + states in 4°Ca State 01+ 02+ 0a+
Energy
0p.0hJ= o
2p.2hJ=O
4p.4hJ= o
0 3.55 7.33
0.904 0.202 0.378
0.414 --0.183 --0.892
0.111 --0.962 0.249
We also use E(2p-2h, J = 4) =
7.8 MeV,
E(4p-4h, J = 4) =
5.5 MeV,
118
w.J.
GERACE AND A. M. GREEN
to obtain the eigenvectors appearing in table 5. TABLE 5
Wave functions of 4 + states in 4°Ca State
Energy
2p-2hJ = 4
4p-4hJ = 4
41+
5.25
0.31
0.95
42+
8.00
0.95
--0.31
There have been several experiments carried out to study the structure o f the ground state of 4°Ca. In each case they seem to require larger deformed components than the 20 ~ we predict in table 4. For example Bock et al. 13) with the reaction 4°Ca(3He, c~) 39Ca and assuming the simplest possible structure for the residual 39Ca states f o u n d that about 45 ~ o f the 4°Ca ground state was deformed. Belote et al. 1~) with the reaction 4°Ca(d, p)41Ca also obtained large percentages in the region o f 40 ~ deformed. However, it has been recently shown by Rost 15) that the usual way o f analysing the above type o f experiment can be very misleading when deformed states are involved, and considering the reliability o f this calculation we do not find this discrepancy disturbing. 2.4. ELECTROMAGNETIC TRANSITIONS IN ~°Ca The electromagnetic transitions are calculated in precisely the same way as in ref. 4) for 160. Namely, the B(E2) between levels o f a given rotational band are put in terms o f the intrinsic quadrupole m o m e n t o f that band, which is then evaluated in the SUa limit. For hco = 10.5 MeV, this gives for in-band transitions B(E2, 2p-2h, fl = 0.17, 2 + --> 0 ÷) =
68 e2fm 4,
B(E2, 4p-4h, fl = 0.17, 2 + ~ 0 +) = 170 e2fm 4, B(E2, 4p-4h, fl = 0.23, 2 + ~ 0 ÷) = 230 eZfm *. Using these values and the 0 +, 2 + and 4 + wave functions given in tables 3-5, we obtain the B(E2) values given in table 6. TABLE 6
Electromagnetic transitions in
4°Ca
Transition
fl4p-4h : 0.17
fl4p-4h= 0.23
Experiment 10)
21+(3.90) --> 01+(0) 23+(6.94) --> 01+(0)
6.6 e~fm4 7.2
7.8 6.7
29±9 e~fm4 32±4
21+(3.90) --->02+(3.35) 41+(5.27) --->21+(3.90) 22+(6.94) --->03+(3.35)
152 220 9.2
203 300 14.3
Ca DEFORMED STATES
119
From this we can see that in the two cases where a comparison can be made with experiment the agreement is not good. A measurement of the 2 + ~ 0 + or 4 + ~ 2+ in-band transitions would help greatly in diagnosing the trouble. If our predictions for these in-band transitions are satisfactory then the problem is that our ground state wave function does not contain enough deformed component.
3. Odd-parity states of 41Ca and
41Se
Both 4~Ca and 41Sc would be described by the simple shell model as having a single f~ nucleon outside a double closed shell. However, instead of finding a single lowlying 3- excited state, which we would interpret as the valence nucleon excited to the 2p~ level, we actually observe two -~- levels which are 0.52 MeV apart in 4~Ca and 0.7 MeV apart in 41Sc. Experimentally each of these levels is strongly excited in singlenucleon transfer processes such as (d, p) (ref. t6)) and (3He, d)(ref. 13)) indicating that the single-particle strength is strongly mixed between them. The state being mixed with the single 2p~ nucleon level we interpret as a 3p-2h deformed state, where the holes are in the no. 8 Nilsson orbit and the particles are in the no. 14 orbit. To show that a 3p-2h state can lie sufficiently low in energy to mix strongly with the 2p~ level we again rely on the work of Zamick, who predicts the [3pTp=~-2hTh= 1]T=-,t- s t a t e to be about 3.5 MeV above ground in 4aCa and about 4.5 MeV above ground in 4~Sc. As we shall see later it is also possible to obtain a solution including the 5p-4h deformed state. Unfortunately in order to do this several more assumptions are necessary. Therefore, for the purpose of calculating transition rates and single-particle strengths we include only the 3p-2h state. 3.1. M A T R I X E L E M E N T S I N 4XCa
The basic matrix element involved is _ _ d
M~ = (lp-0hT=½1VI3pTp=~-2hT"=I
J
r=½),
(3)
where the isospin coupling of the deformed state is chosen in accordance with the prescription of Zamick. This then enables us to factor the matrix element and thereby relate it to the M1 in eq. (1) for 4°Ca M~ = ~22 Ml(4°Ca)"
(4)
The factor F takes into account the angular momentum recoupling and wave function renormalization and has the value 0.42 for J = ~ and 0.60 for J = ~. The ,,/2 is a correction to the isospin factor and represents the blocking of Nilsson's no. 14 orbital against double neutron excitation. Inserting these numbers and using the value
120
w . J . GERACE AND A. M. GREEN
Ml(4°Ca) = - 2 . 6 MeV calculated earlier, we get: M s=~ = - 0 . 7 8 MeV, MS3=~ = - 1.13 MeV,
(5)
when we use Nilsson's phase convention for the 2p wave function. If we are to conside1 the two ~2 levels as mixtures of only a lp-0h and a 3p-2h state, then the matrix element between the two should have a magnitude which is less than or equal to onehalf the separation - namely 0.26 MeV in 41Ca. Unfortunately our value exceeds this. Of course our method of calculation or choice of deformed wave function could be in error, but there is an alternative possibility. It is conceivable that other ~ - deformed levels be low enough to affect the positions of these states. In fact, the Zamick model predicts a 5p-4h state at about 5 MeV above the ground state. We shall consider this possibility in greater detail later, but until then we shall assume that M3J =~ = - 0 . 2 6 MeV and that this is simply a two state problem. 3.2. E N E R G Y
OF UNPERTURBED
3p-2h C O N F I G U R A T I O N
The assumption that M~ =~ = - 0 . 2 6 MeV, forces us to take the energy of the 3p-2h J =~ state to be degenerate with the 1 ~ J =~ level at 2.20 MeV, if we are to fit exactly the observed levels at 1.95 MeV and 2.47 MeV. As mentioned earlier, the Zamick model places the 3p-2h state at about 3.5 MeV above the ground state. However this is the centre of gravity of all the possible states of different angular momenta that can be projected out of this intrinsic state, and therefore the two energies are not inconsistent. The 3p-2h state we are dealing with here has K = 1, and thus this has the additional complication of the decoupling effect for an odd particle in the no. 14 Nilsson orbit. In fact the decoupling factor for such a particle is about -3.35, using Nilsson's wave function at fl = 0.2. Therefore, if we consider the J = ~ level to be at 2.20 MeV and assume a reasonable inertial parameter of h 2 / 2 J = 0.095 as indicated in 4°Ca, then we find that the K = ½ band is as follows: E ( J = ½) = 2.91 MeV, E ( J = -~) = 2.20 MeV, E ( J = ~ ) = 4.26 MeV and E ( J = ~ ) = 2.71 MeV. Experimentally a weak J = ½ level is seen at 2.97 MeV; but no J = ~ has been identified although several unknown levels do exist in this region. This decoupling effect has the desirable feature of moving down the J = ~ level towards the single-particle shell model level usually thought to be the A = 41 ground state. This results in a non-negligible amount of 3p-2h component in the A = 41 ground state, as is needed to account for the large electromagnetic transition observed. The deformed component in our ground state, although less than the amount found by Bock et al. ~ 3), is still consistent with their results. The assumptions made above as well as the difficulty in analysing singleparticle pick-up and stripping data when deformed states are present, as explained by Rost 15), could more than account for the differences. In fact the wave functions of Bock et al. 13) give transition rates not too different from the experimental values.
Ca DEFORMED STATES
121
3.3. WAVE FUNCTIONS OF 41Ca and 4xSc The assumption, in 4~Ca, that the (lp-Oh) s =~ and (3p-2h) J =~ levels are degenerate at 2.20 MeV with a matrix element between them of - 0 . 2 6 MeV, gives the wave functions in table 7. TABLE 7
Wave functions in 4xCa State
Energy (MeV)
(lp-0h)~
{1{2-
1.95 2.47
1/~/2 1/~/2
(3p-2h)~1/~/2 -- 1/~/2
In 41Sc we assume we have the same lp-0h to 3p-2h matrix element. Therefore, in order to get an exact fit to the observed levels at 1.71 MeV and 2.42 MeV we require the unperturbed lp-0h and 3p-2h states to be at 1.82 MeV and 2.31 MeV respectively, with the final wave functions given in table 8. TABLE 8
Wave functions in 41Sc State
Energy (MeV)
(lp-0h)k
(3p-2h)~
{a{2-
1.71 2.42
0.916 0.400
0.400 --0.916
This position of the unperturbed (lp-0h) ~ level is consistent with the calculation of Jones et al. 28) that the (lp-0h) ~ state in 41Sc is depressed by about 0.3 MeV below that in 41Ca. However we get that the deformed state in 4XSc is raised by only 0.1 MeV above the one in 4~Ca, whereas Zamick's argument gives the centre of gravity of the deformed 4~Sc state to be about 1 MeV above the centre of gravity of the corresponding 41Ca state. Probably Zamick's method is not sufficiently accurate to make such a difference significant. The ~- wave functions are found assuming that the deformed 5 - state has an unperturbed energy of 2.71 MeV, the value calculated earlier assuming a rotational band built on a Nilsson no. 14 orbital. To obtain a 5 - ground state, we must then take 0.48 MeV as the unperturbed energy of the ]lp-0h) ~ state. Since the deformed state in 4~Sc is only 0.1 MeV above the one in 4~Ca, we use the ~- wave functions given in table 9 for both 41Ca and 41Sc. TABLE 9
Wave functions of 5- states in ~lCa and 41Sc State
Energy
[1p-0h)-~
]3p-2h)÷
5152-
0 3.22
0.924 0.388
0.388 -- 0.924
122
w.J.
G E R A C E A N D A. M. G R E E N
3.4. ELECTROMAGNETIC TRANSITIONS IN *1Ca and 41Sc At first sight the electromagnetic transitions from the two ~ - levels in 41Ca seem rather peculiar. The lower level decays strongly to the ground state with a B(E2) of 7 W.U., whereas the upper level only has a B(E2) of less than 0.1 W.U. to the ground state. In fact the higher ~ - prefers to decay to the lower ) - by an M1 transition. This is so even though our model says that both ~ - levels have a similar structure except for a phase change between the two components (see table 7). Of course, it is just this phase difference which causes the B(E2, 32 ~ g.s.) to be inhibited, because now the contributions from the lp-0h and 3p-2h components tend to cancel each other. Of course, before we can get any contribution from the lp-0h term in 41Ca we need to introduce an effective charge for the valence neutron. At present, no complete calculation of effective charges has been made with realistic forces. Early estimates by Horie and Arima 19) with a 6-function force gave the effective charge of a neutron e~ff at the end of the s-d shell to be somewhat greater than e and that of the proton e~ff to be between ½e and e. Zamick 28) using more realistic forces came to the result n ~ n that e,ff = 0.6e and e~ff 0.2e. Therefore, it appears that eCff can be as large as e, whereas e~ff p seems to be more in the region of ½e. The reason for the larger value for e~ff is that the effective charge is essentially a measure of the number of protons excited from the core. When a proton is the valence nucleon, the protons in the core can only be excited via the T = 1 component of the internucleon interaction, whereas when a neutron is the valence nucleon the T = 0 component is also involved, and this is somewhat stronger than the T = 1 component. Another uncertainty in the electromagnetic transition of the lp-0h component is the single-particle wave function used. Using simple harmonic oscillator wave functions we get Bs.p.(2p~ ~ lf~, E2) = 35.2e2fm 4. However in a recent calculation using Woods-Saxon wave functions which fitted the spectrum of 4aCa and 4~Sc, Schiffer 29) found the result Bs.p.(41Ca, e~"ff = 1, 2p~ ~ lf~) = 36 e2fm4,
Bs.p.(41Sc, eePff = 0, 2pt --, lfi) = 47 e2fm 4. In other words there is no great change when the better wave functions are used. This is different from the effect in 170 where the more realistic wave functions increased the B~.p. (2s~ --, ld~, E2) by a factor of 4. In that case the absence of the centrifugal barrier is mainly responsible for the large change. The basic B(E2) values for the 3p-2h states for hco = 10.5 MeV appear in table 10. TABLE
10
B(E2) values of intrinsic states in e~fm* fl = 0.17 /3 = 0.23 B(E2, 3p-2h, a~Ca, ~2- ~ ~-) 340 460 B(E2, 3p-2h, 4XSc,3- ~ 7~-) 200 303
Ca DEFORMED STATES
123
Using these in-band transitions and the single-particle transitions o f Schiffer together with the 4aCa 3 - wave functions appearing in table 7 and the g r o u n d state wave function o f table 9, we obtain the results in table 11. TABLE 11 Electromagnetic transitions in 4~Ca (units e2fm4) Exp. a) B(E2, 41Ca, ~l- --~ g.s.) B(E2, 41Ca, ~2- --~ g.s.) B(M1, 41Ca, ~2- -~ ~1-)
59zL13
fl, = 0.17 80 1.2
f12 = 0.23 97 3.7 0.008
a) Refs. s0,21). Here we have chosen enff ---- 1. This is in the region indicated by the direct calculations o f Horie and A r i m a 19) and Zamick. Using the same g r o u n d state wave function and the 3 - wave functions o f 41Sc given in table 8, we get the results shown in table 12. TABLE 12 Electromagnetic transitions in 41Sc (units e2fm4)
Exp. a) B(E2, 41Sc, ~tl- -+ g.s.) B(E2, 41Sc, ~-2--+ g.s.) B(M1, 41Sc, ~2- ~ ~1-)
101 :t:34
fl = 0.17 110 1.3
fl -----0.23 120 5.4 0.005 e~fm~
s) Ref. 33).
I n 41Sc we simply chose e~Pff = ½ ee~f as this seems to be the trend in calculations o f these quantities. The result is in g o o d agreement with the one observed transition. However this transition is dominated by the lp-0h c o m p o n e n t and is therefore not very sensitive to the a m o u n t o f deformed c o m p o n e n t in the ground state. 3.5. SINGLE-PARTICLE STRENGTHS One o f the first indications that 41 Ca deviated f r o m the simple shell-model description was the excitation o f two strong 3 - levels in the 4°Ca(d, p)4aCa reaction 16,17). I n this experiment it was f o u n d that the single-particle strength, instead o f being concentrated in one o f these two levels, was in fact distributed in about a two to one ratio between the lower and the upper levels. I n this subsection we make r o u g h estimates o f these strengths using the wave functions o f tables 7 and 8. I f we consider the 4°Ca ground state has a wave function o f the f o r m ~k(4°Ca, g.s.) = a(0p-0h) J~ o +b(2p_2h)O + c(4p_4h)O,
(6)
and that the states o f 41Ca have the f o r m ~bs(41Ca) -- d(lp-0h) s + e(3p-Eh) J +f(Sp-4h) J,
(7)
124
w.J.
G E R A C E A N D A. M. G R E E N
then a simple estimate of the ratio of single-particle strengths for a fixed J is given by SPS ~:
lad+AJbe+BJfcl 2,
(8)
where Z J = R~ C~ = R~([J2p-2h°]Jl3p-2hJ ), B J = R~ C~ = R~([J4p-ala°]Jl5p-ahJ). The factors C1 and Ca represent the angular momentum structure of the 3p-2hJ and 5p~4h ~ states and have the values C~ = 0.435, C~ = 0.614 and C2~ = 0.400. We do not consider the effect of C2~. In fact we do not introduce even C2~ until the next section, where the effects of 5p-4h states are introduced. The R J factor which appears in the second and third terms of eq. (8) is there because in that case the transferred nucleon is in a deformed well. This effect has recently been studied by Rost 15) who shows that deformation of the well modifies the radial wave function in the surface region. We can approximate this effect 30) by using R~ = R2 = 1.3. Using these factors together with the values of the coefficients a, b, c, d and e given in tables 4 and 7, we get the single-particle strengths in the two strong ~- levels as shown in table 13. TABLE 13 Single-particle strengths in 4xCa and 4xSc Exp. 41Ca SPS ~x-(1.95 MeV)
2.9
2.3 ~)
SPS~,~-(2.47 MeV) 41Ca-4xSc SPS {2-
2.5 × 10-a
SPS {1-(g.s.) 41Sc SPS ~1-(1.71 MeV)
38
10 b)
SPS 22-(2.42 MeV) a) Ref. 17).
b) Ref. 13).
Also shown in table 13 are the results for 41Sc using the wave functions of tables 4 and 8. We see reasonable agreement with the observed ratios of single-particle strengths. As we go from 41Ca to 41Sc this increase in the ratios as the two mixed levels get further apart is a good indication of the basic validity of our model. In the next subsection where we introduce the effects of the 5p-4h state, we show that results comparable to those above can still be obtained. Another important point is the amount of SPS in the ½- state which we expect near 3.2 MeV. This must not be too large or the state would probably have been already identified. Thevalue of the ratio of the SPS of this state to that of the ground state is shown in table 13 to be quite small.
Ca DEFORMED STATES
125
3.6. E F F E C T O F 5p-4h S T A T E S
Thus far in 41Ca we have been using the value of - 0 . 2 6 MeV for the matrix element M~ =~ = (lp--0h~l V I ~ ~) even though we calculate it to be nearer - 0 . 7 8 MeV. This was done so that in subsect. 3.3 we could obtain the exact splitting using only two states. The purpose of this section is to show that it is possible to obtain a solution using the calculated value of M~ provided a third state, the 5p-4h state, is introduced. If we consider the 5 p - ~ r=½ state as being formed by adding an extra neutron in the no. 17 orbital of fig. lb, then its matrix element to the 3p-2h T=~ state is 0.37 MeV. Because of the large value of Ma~ it is necessary now to have the 5p-4h state lower than the 3p-2h state. The problem is whether or not a solution can be obtained without disturbing the ratio of the single-particle strength of the lower two states and without having the third state absorb an observable amount of single-particle strength. To obtain such a solution it is necessary to have the lp-0h and 5p-4h states degenerate at 2.45 MeV. Locating the 3p-2h at 3.35 MeV results in the wave functions appearing in table 14. TABLE 14 W a v e functions in 41Ca State
Etina~
Ilp-Oh)S=k
]3p-2h)~r ="l"
[5p-4h)1=k
~1-
1.95
0.77
0.52
--0.37
j}~-
2.47
0.44
--0.07
0.90
~3-
3.90
0.47
--0.86
--0.22
In 4XSc it is necessary to take for the unperturbed energies of the lp-0h 4, 3p--p-~2h~ and 5p~4h ~ states the values 2.12, 3.37 and 2.47, respectively. Here we have shifted the unperturbed position of the (lp-0h) ~ state by a value of 0.35 MeV relative to 41Ca. This shift is still consistent with the calculations of Jones et al. ~8). The resulting wave functions are given in table 15. TABLE 15 W a v e functions in 41Sc State
Etlna 1
11p - 0 h ) J = k
[3p-2h)J = ~-
~1-
1.71
0.86
0.45
-- 0.22
~z-
2.42
0.30
--0.12
0.94
~3-
3.93
0.40
--0.88
--0.24
These wave functions satisfy 4 ~ C a - SPS(~-)/SPS(~2) = 3.3, 4 1 C a - SPS(~7)/SPS(~-3-) = 17.2, 4 1 S c - SPS(~I)/SPS(~z~-) = 9.1, 41Sc- SPS(kT)/SPS(kg) = 38.0.
15p-4h)3= k
126
W.
J. GERACE
AND A. M. GREEN
In our model M& the matrix element between deformed states is approximately independent of J in the limit of large deformation, and therefore we take M$ = M$ = +0.37 MeV. Also since we have tixed the unperturbed positions of the 3p-2h3 and 5p-4h* states, we are able to estimate the unperturbed position of the corresponding J = 3 states. This is done using a value of h2/23 of 0.095 as before, and the decoupling factors for an odd particle in the no. 14 and no. 17 Nilsson orbitals. This gives E(3p-2h$) = 3.86, E(5p-4h3) = 3.45, and a ground state wave function I++,.,.= 0.9611p-Oh)+0.28~3p-2h)-0.05~5p-4h). This wave function is very similar in structure to that obtained using only two states, and consequently it is still true that almost all of the 3- strength is in the ground state. The other two weak f- states occur at about 3.3 and 4.4 MeV. As before we use this ground state for both 41Sc and 41Ca. The only remaining question is whether these wave functions will still give reasonable reduced transition rates. In addition to the transition rates in table 10, we need B(E2, 5p-4h 41Ca, $- --f $-) = 581 e2fm4 for ps = 0.23, B(E2, 5p-4h, 41Ca, $- + 3-) = 745 ezfm4 for ps = 0.30, B(E2, 5p-4h, 41Sc, $- + $-) = 770 e2fm4 for /3s = 0.23, B(E2, 5p-4h, 41Sc, 3- + 3-) = 970 e2fm4 for p5 = 0.30. The final reduced transition rates for 41Ca and 41Sc are given in tables 16 and 17, respectively. Here we have taken as before e& = e, e& = 0.5e. TABLE
16
Transition rates in %a Exp. “)
& = 0.17 b6 = 0.23
,!$ = 0.23 /?&= 0.30
B(E2, &- + p.s.)
59f13
57.2
64.6
B(E2, &- -+ g.s.)
< 0.4
1.2
0.8
2.1
4.7
B(E2, is- --f g.s.) B(Ml,
a) Refs. 20,21).
in e2fm4
Qz- + $1~)
5 0.03
0.0035 e2fm2
127
Ca DEFORMED STATES
We see therefore that the three level model with calculated matrix elements gives an equally g o o d interpretation o f the experimental data as the two level model. This suggests that the states in the two level model are actually the Il p - 0 h ) ~ and 15p-4h) ~ states b o t h renormalized by ]3p-2h) ~ state. The interaction between these two renormalized states then arises only t h r o u g h the 13p-2h) ~ c o m p o n e n t which justifies our use o f an effective matrix element which is small c o m p a r e d to the calculated value o f M~. TABLE 17 Transition rates in 41Scin e2fm4
B(E2, ~x- -+ g.s.) B(E2, ~z- ~ g.s.)
Exp. a)
f13 = 0.17 f15 = 0.23
f13 = 0.23 f15 = 0.30
1014-34
112
122
1.4 0.64
B(E2, ~3- ~ g.s.) B(M1, ~ - --> ~1-)
0.8 0.01 0.0026 e~fm2
a) Ref. 22).
4. Even-parity states of 42Ca According to the simple shell model, 42Ca should be described in terms o f configurations o f two valence particles in the lf-2p shell, the core providing the shellmodel potential. However, there exist states, such as the 0 + at 1.84 MeV and the 2~at 2.42 M e V shown in fig. 4, which cannot be explained by so simple a model. A further indication o f the inadequacy o f the model is the absence o f corresponding levels in the spectrum o f 50Ti ' while at the same time the excitation energies o f the 2 +, 4 + 6+
3.19
4+
2.75
2+
2.42
0+
-
2 +"
- 1.52
0 +"
1.84
O.
Ca
42
Fig. 4. Low-lying levels of 4~Ca having even spin and parity.
128
W, J. GERACEAND A. M. GREEN
and 6+ levels are the same to within 0.05 MeV in both spectra 23). The occurrence of a second 0 ÷ and 2 + in 4ZCa but not in 50Ti also indicates the nature of these states. These two levels must be formed by the excitation of protons from the 2s-ld to the lower lf-2p orbitals. In s 0Ti the corresponding levels would be formed by raising two neutrons, but these are blocked by the completely filled lf÷ shell of neutrons. The Zamick model cannot be used to predict the unperturbed position of the 4p-2h states because the 0~- at 1.84 MeV was used to determine the force constants in the calcium region assuming it to be the [4p rp = °-2hrh = 1]r = 1 state. 4.1. M A T R I X E L E M E N T S I N ~2Ca
Again we write the basic matrix elements as __J
I
M~(2p) = ( 2 p - 0 h r = 1] V]4pTp = 0_2hTh =1 T = 1 ) ,
(9)
Zamick's prescription of coupling particles and holes separately to definite isospins again enables us to factorize the 4p-2h wave functions and thereby relate M4s to M 1, the matrix element for 4°Ca, giving M4S(2p)- F(2P)M,(4°Ca).
(10)
43
Wave function normalization and angular momentum recoupling factors are collected into the factor £'(2p). Table 18 summarizes the value taken by _P(2p) for certain 2p configurations appearing on the left-hand side of the matrix element in eq. (9). TABLE 18 Factors relating matrix elements in 42Ca and 4°Ca [lf~]J=o
[2p~2]j=o
0.505
0.208
[lf~2]J=2 --0.272
[1 fk-2p~]J= 2
[2p$2]J =2
--0.306
--0.I01
The x/3 in the denominator of eq. (10) reflects the complete blocking of neutron excitations to the no. 14 orbital since it already contains two neutrons. Using these values together with M 1 = - 2 . 6 MeV, we obtain M°(lf~) = - 0 . 7 6 MeV
MZ(2p~) = 0.15 MeV
M°(2p~) = - 0 . 3 1 MeV 2 2 M4(lf~) = 0.41 MeV
M2(lf~2p~) = 0.46 MeV. 4.2. W A V E F U N C T I O N S O F 42Ca
Here we use Kuo's 7) wave functions for the 0~- and 2+ states and adjust their energies so that we get exactly the observed eigenvalues for the lowest two 0 ÷ and 2 + levels
129
Ca DEFORMED STATES
of fig. 4, when we insert the 4p-2h states appropriately. 0 [ (Kuo) = 0.927(f~) ° + 0.203(p~) ° + 0.203(f~) ° + 0.109(p~) °, 2 ~ ( K u o ) = 0.935(f~) 2 + 0.278(f~p~) 2. The resulting wave functions are given in tables 19 and 20. TABLE 19 Wave functions of 0+ states in 42Ca State
EflnaI
Einitia 1
(lf~2)°
(2p~2)°
(lf~2)°
(2p,2) °
01+
0
0.40
0.820
0.180
0.180
0.096
02+
1.84
1.44
0.432
0.094
0.094
0.051
Def. J=o 0.465 --0.884
TABLE 20
Wave functions of 2+ states in 42Ca
State
Eflnal
•tatttal
(lf~2)~
(lf~2P~) 2
Def. d=2
2x+
1.52
1.97
0.661
0.197
--0.707
22+
2.43
1.97
0.661
0.197
0.707
Again we see that our consistency criterion is satisfied in that the unperturbed positions o f the deformed 2 + and 0 + differ by a b o u t 0.5 MeV. The type o f matrix elements defined by eq. (9) correspond, in infinite nuclear matter, to the inclusion o f unlinked diagrams, which merely add a constant energy shift to all levels. This effect seems to be holding true even in the finite nucleus case we have here, since we see that the two lowest states 0~- and 2 + were b o t h depressed by 0.45 M e V f r o m their unperturbed positions. F o r the 4 + states we obtain the wave functions and energies shown in table 21. TABLE 21 Wave functions o f 4 + states in 4zCa
State
Ennai
Einlttal
4x+ 42+
2.75 3.56
3.10 3.21
~04+(Kuo) 0.75 0.66
~p4÷ (def.) 0.66 --0.75
4.3. ELECTROMAGNETIC TRANSITIONS IN *"Ca W i t h the wave functions given in the previous section and the following basic B(E2) values --J=2
B½(E2, eenff = 1.0, f~
__J=O
-~ f2~
) = - 4 . 3 5 elm 2,
130
W. J. GERACE AND A. M. GREEN -. 2
__0
B~(E2, ee"ff = 1.0, f~-Pl- -" f~ ) = - 2 . 6 4 efm 2, __2
B~(E2,
eenff
= 1.0, f~p~
~
__0 p 2~ )
= -3.74
e f m 2,
B~(E2.4p-2h, 2 + -~ 0+fl = 0.17) = 13.0 elm 2, B~r(E2, 4p-2h, 2 + ~ 0+fl = 0.23) = 15.0 elm 2, we get the transition rates given in table 22. TABLE 22 E l e c t r o m a g n e t i c transitions in 42Ca (units e~fm 4) eeff n
Transition
Expt. a)
B(E2, 21 + --* 01 +)
742L17
B(E2, 02 + ~ 21 +)
~ 500
B(E2, 22+ --* 0x +)
~
B(E2, 2~+ ~ 02 +)
<150
10
fl = 0.17
~ 0.5e fl = 0.24
eetf n ~ e fl = 0.17
fl = 0.24
33
42
52
62
274
375
222
315
8.0 80
12.3 104
2.0 95
4.4 120
a) Ref. 24).
In the above, we quote the square root of the basic B(E2) values to bring out the sign difference that enters between the single particle and deformed components. This now ensures that the B(E2, 2 + --, 0 +) is the coherent combination of the two components. It is seen that we again underestimate the two known B(E2) values by about a factor of 2 or 3 for the case of e~fr = e. Even so we consider these results as encouraging since the B(E2) are large. 4.4. EFFECT OF 6p-4h STATES In the last section our estimates of B(E2) fell below the observed values indicating that we were not sufficiently mixing the deformed components into the shell-model states. One possibility is that the configuration [6prp=l-4hrh=°] r = l is important. Such a state, by the arguments of Zamick, lies between 3.5 and 4.5 MeV above the ground state and has a matrix element of ~ 0.6 MeV with the 4p-2h state. The effect of this on the final transition rates can be approximately allowed for by an increase of less than 5 ~ in the basic 4p-2h B(E2) values and thus the results of the previous section are changed by about this same amount. 5. Conclusion In this paper we have applied to the calcium isotopes a model that was first intro-
Ca DEFORMED STATES
131
duced in the oxygen region where it proved successful. Within this model the electromagnetic transition probabilities and single-particle strengths, so far calculated, have all been in reasonable agreement (to within a factor of ~ 2) with the observed values. Further justification of these ideas depends on the measurement of additional B(E2) values such as 2 + ~ 0~- or 4 + ~ 2 + in 4°Ca. Also we predict states thus far unseen, such as another 0 + in 4°Ca near 7.3 MeV and weak ~- and ½- levels in 41Ca in the vicinity of 3.2 MeV. Identification of these levels would put our model on a much firmer foundation. Until now, no mention has been made about calcium isotopes with mass greater than 42Ca. However in these nuclei we expect the effect of deformed states on the lowest shell-model levels to decrease with increasing neutron excess. The reasons for this are twofold. Firstly, the matrix element between the deformed and spherical states decreases through the normalization of the deformed wave function 25). This is because the more neutrons there are, the greater number of components in the deformed wave function and thus the amplitude of that portion which connects with the spherical state decreases. Secondly, the unperturbed energies of these deformed states appear to increase with the number of neutrons. This is indicated by the rise in energy of the second 0 ÷ levels which we say contain mainly the deformed component. For example in 42Ca, 44Ca, 46Ca and 4SCa these second 0 + levels occur at 1.8, 1.8, 2.42 and 4.3 MeV, respectively. Also, estimates with the Zamick model put the deformed states in these nuclei at 1.8, 2.5, 3.2 and 5.1 MeV, respectively. This decrease in the effect of the deformed state on the ground state is also seen in (d, p) reactions. From these we can get a measure of the number of holes in the 2s-ld shell and thus estimate the amplitude of the deformed component in the ground state. Belote et aI. 26) show that this amplitude decreases by about a factor of three in going from 4°Ca to 46Ca. Finally, when our results are compared with those of Bertsch 6) we see that the inclusion of four-hole components and core renormalized matrix elements gives a significant improvement. The authors wish to thank Professor G. E. Brown for his guidance and continued interest, Dr. T. Kuo for his two-body matrix elements and Dr. G. Bertsch for a critical reading of the manuscript. We also acknowledge an enlightening conversation with Dr. J. Raz. One of us (A.M.G.) also thanks Professor J. R. Oppenheimer for his warm hospitality at the Institute for Advanced Study. References 1) 2) 3) 4) 5) 6)
T. Engeland, Nuclear Physics 72 (1966) 68 G. E. Brown, Compt. Rend. Int. de Phys. Nuc16aire, Vol. I (1964) p. 129 G. E. Brown and A. M. Green, Phys. Lett. 15 (1965) 168 G. E. Brown and A. M. Green, Nuclear Physics 75 (1966) 401 L. Zamick, Phys. Lett. 19 (1965) 580 G. Bertsch, Nuclear Physics 79 (1966) 209 and to be published
132 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30)
W. J. GERACE AND A. M. GREEN T. T. S. Kuo, private communication W. H. Bassichis and G. Ripka, Phys. Lett. 15 (1965) 320 J. Hayward, Nuclear Physics 81 (1966) 193 D. Blum, P. Barreau and J. Bellicard, Phys. Lett. 4 (1963) 109 W. S. Gray, R. A. Kenefick and J. J. Kraushaar, Nuclear Physics 67 (1965) 542 M. A. Grace and A. R. Poletti, Nuclear Physics 78 (1966) 273 R. Bock, H. H. Duhm and R. Stock, Phys. Lett. 18 (1965) 61 T. A. Belote, A. Sperduto and W. W. Buechner, Phys. Rev. 139 (1965) B80 E. Rost, Phys. Lett. 21 (1966) 87 L. L. Lee, Jr. and J. P. Schiffer, Phys. Rev. Lett. 12 (1964) 108 L. L. Lee, Jr., et al., Phys. Rev. 136 (1964) B971 K. W. Jones et al., Phys. Rev. 145 (1966) 894 H. Horie and A. Arima, Phys. Rev. 99 (1955) 778 S. I. Baker et al., Bull. Am. Phys. Soc. 11 (1966) 477 D. S. Gemmell, L. L. Lee, Jr. and J. P. Schiffer, private communication D. H. Youngblood, J. P. Aldrich and C. M. Class, Phys. Lett. 18 (1965) 291 P. Federman, Phys. Lett. 20 (1966) 174 F. R. Metzger and G. K. Tandon, Phys. Rev. 148 (1966) 1133; A. Bernstein, private communication G. E. Brown and A. M. Green, Nuclear Physics 85 (1966) 87 T. A. Belote, H. Y. Chen, O. Hansen and J. Rapaport, Phys. Rev. 142 (1966) 624 A. Arima, private communication L. Zamick, private communication J. P. Schiffer, private communication E. Rost, private communication
Nuclear Physics A93 (1967) 110---132; ( ~ North-Holland Publishing Co., Amsterdam
I
Not to be reproduced by photoprint or microfilm without written permission from the publisher
THE EFFECT OF D E F O R M E D STATES IN THE Ca I S O T O P E S W. J. GERACE
Palmer Physical Laboratory, Princeton University, Princeton, New Jersey and A. M. GREEN
The Institute for Advanced Study, Princeton, New Jersey t Received 14 October 1966 Abstract: An attempt is made to explain the low-lying levels of 4°Ca, 41Ca, 42Ca and 41Sc as mixtures of shell-model and deformed states. The deformed states are constructed using the isospin coupling scheme of Zamick, and the matrix elements between spherical and deformed states are calculated using the Hamada-Johnston nucleon-nucleon force. Wave functions are obtained for these states and predicted values for some transition rates and single-particle strengths are compared with experiment.
1. Introduction
For several years it has been known that the ordinary shell model is inadequate, even near double dosed shells, which is the region in which it has had its greatest success. The old model cannot explain either the occurrence of large electric quadrupole transitions or the number of low-lying energy levels, many of which can be fitted into rotational bands. A group of nuclei having these complications is the oxygen isotopes. These have been studied in some detail by Engeland 1) and Brown 2) and by Brown and Green a, 4), where a remedy has been proposed which is capable of explaining several of the new experimental features. The method consists of introducing low-lying deformed states in addition to the usual shell-model states. The mixing of these states can account for the large transition rates and the rotational character of the spectra. In this paper these techniques are applied to the calcium region with detailed studies of 4°Ca, 41Ca, 42Ca and 41Sc. The procedures employed here are an extension of the methods used by Bertsch 6) in which only two-hole states were introduced and in which the matrix elements did not contain core polarization effects. t This work was supported by the U.S. Atomic Energy Commission and the Higgins Scientific Trust Fund. This report made use of the Princeton Computer Facilities supported in part by the National Science Foundation Grant NSF-GP 579. 110
CaDEFORMED STATES
111
2. Even-parity states of 4°Ca In this section we interpret the even-parity states of 40Ca in a way that is completely analogous to the description of the lowest, even-parity states of 160 given in ref. 4). Namely, we consider these states as mixtures of the double closed 2s-ld shell-model state with two intrinsic deformed states formed by raising two and four particles from the 2s-ld shell into the 2 p - l f shell. These two deformed states can be illustrated by means of the Nilsson diagrams shown on figs. la and b. The intrinsic states contain components with different angular momenta, whereas the double closed shell contains only J = 0. Once the matrix elements between the states and the unperturbed energy of each state is known, wave functions can be derived. These wave functions can then be used to estimate the electromagnetic transition rates between levels. 4.5
- ~
4.5
. z
~P312
7/2-
I7
'
I
'
I/2
4.0
i
o
~
X~ :I
3.5
I
-0.2
I
-O.I
0
_
--.-''" 3/2+
I
I
0.1
0.2
3.5
I
-0.2
I
-0.1
0
I
I
0.I
0.2
Fig. 1. Relevant section of Nilsson diagram showing a) two-particle-two-hole state and b) fourparticle-four-hole state, ×-particle, e-hole. 2.1. MATRIX ELEMENTS IN 4°Ca The basic matrix element involved in this description is J=O
__J=O
M1 = (0p-0hr=°lVl2p-2hr=°),
(1)
and this we shall calculate in two ways. From the work of Zamick s), which involves only qualitative arguments, we find in 4°Ca that the 2p-2h state with [2p Tp= 1-2h T" = 1IT= 0 is about 5 MeV lower in energy than the configuration with Tp = Th = 0; and so in general, where these two configurations have been mixed, we expect the lowest 2p-2h state to be mainly [2prp = L2hTh = 1]T = o
112
W. J. GERACE AND A. M. GREEN
For simplicity, we assume that the 2p-2h state which we mix with the 0p-0h and 4p-4h state is only this configuration. This prescription of coupling the particles and holes separately to a given isospin ensures that we are antisymmetrizing correctly when we relate matrix elements in other calcium isotopes to the M 1 matrix element of 4°Ca. Taking Nilsson's no. 14 and no. 8 orbitals at fl = 0.2 and projecting out a 2p-2h wave function with zero angular momentum, we reduce the matrix element M1 to the form '
I a 2
M~ = ~/N
J=O
J=o
[~
((lf~)gr=l [Vl(ld,)2r=l 0
)
0
2 21 ld 0 2 1) +½b E((p,)IVl( J=2
Jr2
5 a2((lf~)r=l[Vl(ld~)r=,) 2x/42 2
2
- ½b2((2p~)*lVl(ld~)l )
3ab
2
2}
414 ((lf~2p~)llVl(ld~)x> /
x overlap.
(2)
In this we have used the simplified orbitals ffno.14 =
allf~,~)+bl2p~,~),
~k,o.S = Ild~.,). The factor N normalizes the 2p-2h a = o wave function and has the value 0.363 when a = 0.880, b = -0.476, remembering that the 2p wave function used here is of the form ~ - (at) 2, i.e. opposite in sign to that used by Nilsson. The overall factor of x/3 arises from the isospin, when we convert particle-hole matrix elements into particleparticle matrix elements. The overlap factor is introduced to take care of the fact that the 38 inert particles on each side of the two-body interaction are in different onebody potentials; the 0p-0h state is in a spherically shaped well and the 2p-2h state is in a deformed well. This lack of overlap has been estimated by Bertsch 6) in two ways. First, he calculated the equilibrium deformation of various particle-hole configurations by summing the energies of the single-particle orbitals in a deformed well. In 4°Ca, using wave functions derived by Rost for a deformed Saxon-Woods potential, he obtained a deformation of about fl = 0.15 for the 2p-2h configuration and fl = 0.20 for the 4p-4h configuration. However, this is expected to be an underestimation for fl since the Rost wave functions, just like the Nilsson wave functions, introduce only quadrupole correlations, and so the additional correlations with non-zero angular momentum tend to increase these estimates of ft. In his second calculation, he evaluated the overlap factor directly by perturbation theory and got the value 0.87.
Ca DEFORMED STATES
113
Throughout the rest of this paper we shall use an overlap factor of 0.8 which is consistent with the value calculated by Bertsch. This value for the overlap corresponds to a fl of about 0.24. The strong dependence of this factor on fl is shown in table 1. TABLE
1
Factors from the overlap of spherical and deformed cores fl
0.15
0.23
0.31
4°Ca overlap factor
0.92
0.84
0.74
It has been calculated assuming that the lowest 38 orbitals are filled in a deformed oscillator potential without spin-orbit splitting. In eq. (2), the two-body matrix elements taken are from the work of Kuo 7), who has an extensive program for deriving effective two-body interactions in nuclei from realistic, free nucleon-nucleon interactions (in this case the Hamada-Johnston potential). His results are given in column 2 of table 2 for ho~ = 10.5 MeV. TABLE 2
Two-body matrix elements Kuo 7) J=O
Effect of core
Total effective
polarization
interaction
1.92
1.13
3.05
0.36
0.03
0.39
0.22
0.63
0.85
0.52
1.32
1.84
0.02
0.32
0.34
J=O
((lf~)T=l[ Vl(ld~) T=t) 2
2
((1 f~) t ]V[(1 d~) 1) 2
2
o
((2p~)a] V] (ld~r)a > 2
2
((2p~) 11V[(ld~)X>
Inserting these numbers into eq. (2) results in a value of M1 = - 1.49 MeV. However the recent work of Bertsch 6) has shown that in 180 and 42Ca there is an important correction to the type of effective two-body matrix elements used in the previous paragraph. He finds that the term in which the two-valence particles interact with each other by first exciting the double closed shell core is by no means insignificant. The analogous correction to the (0p-0h°lVI2p-2h °) matrix element is best illustrated by fig. 2b, the first-order term being illustrated by fig. 2a.
(a) Fig. 2. a) First-order graph.
(b) b) Second-order graph.
114
W . J. GERACE A N D A. M. G R E E N
These higher-order terms have also been evaluated by Kuo with the HamadaJohnston potential. The energy denominator that enters is taken to be the difference in energy between the external particle-hole lines and the internal p-h bubble and ranges from 2.5 MeV to 13 MeV. These corrections are given in column 3 of table 2. Using the modified matrix elements of column 4 in eq. (2), we then get M1 = - 2 . 2 4 MeV, i.e. the effect of core polarization results in a 50 % increase over the first-order term of fig. 2a. In 160 the corresponding increase was 35 %. One complication that may be thought to arise in the evaluation of fig. 2b is the effect of spurious states in the excited bubble. However for the leading term, the vertices involved - namely lf~ld~ 1 _ do not interact with the J = 1 multipole of the twobody interaction, and a J = 1 p-h bubble cannot be excited. In other less important terms where the p-h bubble can have spurious components, their elimination has a negligible effect on M1. In general, it seems that spurious components do not affect results when they enter into the calculation of specific terms in perturbation theory. It is only when we diagonalize a matrix containing these spurious components that trouble can arise. In that case we are effectively summing an infinite series of p-h bubbles, and so the small spurious components in each term can build up coherently into a collective effect. In the above estimate of M1 we have leaned heavily on the Nilsson wave functions. This is because it is well known that for a few particles outside a closed shell an approximate ground state wave function can be obtained by projecting out the appropriate angular momentum from a product of Nilsson wave functions; the reason being that the Nilsson wave functions are essentially the result of diagonalizing a quadrupole-quadrupole interparticle interaction. However we have no right to expect good particle-hole wave functions when we couple Nilsson orbitals for particles and holes, since the Nilsson model does not allow for particle-hole interactions and, as Zamick 5) has shown, these can be very important. To overcome this objection we use Kuo's particle-particle and particle-hole matrix elements and diagonalize a 3 x 3 matrix formed from the leading 2p-2h states. This results in a vector J=0
J=0
0
+ 0.9671 {[lf~] r = 111d~-2] r = 1} 05 0
0 0
+ 0.1371{[2p~]* [ld~211}°5 2
2 0
- 0.2131{[lf2]~ [ld~ 211}° 5 for the lowest 2p-2h state and so the basic matrix element M , becomes - 2 . 3 3 MeV. The effects of higher configurations are added by perturbation theory as before and increase the value of M~ to - 2 . 6 MeV; this value is adopted in the rest of this paper. Before we can interpret the spectrum of 4°Ca, we also need the matrix elements
M~ = <2pZ~-2hJIVI4-~-4hJS.
Ca DEFORMED STATES
115
A complete calculation of this would require Hill-Wheeler integrals, and therefore we only make estimates in the limits o f extreme deformation and SU3 symmetry. The approximations involved here follow closely those made in ref. 4 ) , for 160 and result in a value o f M~ ranging f r o m - 0 . 9 MeV to - 1 . 1 MeV for J = 0 and 2 and a value o f M24 o f - 0 . 8 MeV. Consequently, in the following calculation, we take J=0
J=O
M J= o = (0p_0h T ~ o I V l~p.Eh T = o) = _ 2.6 MeV, _J=O
_ _ J = O
M~ =° = M~ =2 -- (2p-2hT=° I Vl4p-4hr=°> = - 1.0 MeV, J=4
- - J = 4
M J2=4 = ( 2 p - 2 h r = ° [ V I 4 p - 4 h r = ° > = - 0 . 8 M e g .
Quantities such as transition probabilities and single-particle strengths to be studied in later sections are found to be quantitatively the same over the range o f M2 quoted here. 2.2. ENERGIES OF UNPERTURBED CONFIGURATIONS In 016 the energies of the multi-particle-multi-hole configurations have been calculated by several people in different ways. The simplest estimates were made by Z a m i c k 5) using a particle-hole interaction that had only a m o n o p o l e c o m p o n e n t but with a strong isospin dependence. He f o u n d that the 4p-4h state, where the holes are in the l p shell and the particles in the 2s-ld shell, lay at about the same energy as the corresponding 2p-2h state. This rather surprising result had been f o u n d earlier in the more refined calculations o f Ripka and Bassichis 8) and H a y w a r d 9) by means o f Hartree-Fock and variational techniques. In fact these calculations predict the 4p-4h state to be below the 2p-2h state. Moreover this ordering of unperturbed levels was needed in our interpretation o f 160, since the calculated matrix element between the double closed shell and the 2p-2h state was too large to allow a fit to the observed energy levels if we permitted the 2p-2h state to be below the 4p-4h state. In 4°Ca, a similar situation exists. The only calculation o f the ordering o f the 2p-2h and 4p-4h is that o f Zamick 5), which is indecisive; his only conclusion being that these two states are more or less degenerate. However as in 160, the value o f M t exceeds one half the splitting o f the first two 0 ÷ levels, making it impossible to fit the observed levels with the 2p-2h level lower than the 4p-4h. ? In the previous work 4) on 1~O, in the expression corresponding to eq. (1) of this paper the isospin factor was ,/6 and not ,/3 as we have used here. This is because in ref. 4) the 2p-2h wave function is taken to be equal mixtures of [2pTp=°-2hTh=°]T~° and[2pTr, sa-2hTh=l]T=°. However the Zamick 5) argument that the latter state lies lower in energy also applies in 1~O, and therefore a factor of,/3 would have been more realistic. Fortunately in 160 the decrease in the matrix dement due to this correction is more or less cancelled by the core polarization which was also omitted. The effect of both corrections is to raise the 160 matrix element by a factor of 1.34, placing it within 0.2 MeV of the matrix elements used in ref. 4).
116
W. J. GERACE AND A. M. GREEN
We therefore place the 4p-4h lower than the 2p-2h at about 3.8 MeV. A more sophisticated calculation recently made by Arima 27) supports the case for a lowlying 4p-4h state. By considering the excitation of the low-lying 1 + state in ~aSc, he estimates the unperturbed position of the 4p-4h state in 4°Ca to be at 4.1 MeV above the final ground state of 4°Ca. With this ordering of states we adjust the unperturbed energies of the 2 + levels to ensure a good fit to the known levels. I f we are consistent we would expect the energies of the 0 +, 2 + and 4 + levels in each of the two unperturbed deformed bands to bear a (hZ/2J)J(J+ l) relationship with some reasonable hz/2J. It is also required that the perturbed energies correspond to observed levels. These two conditions are sufficient to set the values for the unperturbed energies of the J = 0 and J = 4 levels. 2.3. C A L C U L A T I O N
O F T H E 4°Ca W A V E F U N C T I O N S
We now have matrix elements and a prescription for determining the positions of the unperturbed levels. Before we can carry out this type of calculation we have to decide for which observed levels we are trying to account. The spectrum of interest is shown in fig. 3. The two J = 2 levels arising from the 2p-2h and 4p-4h mixture are immediately fixed to be those two 2 + levels at 3.90 MeV and 6.94 MeV strongly excited in inelastic electron 1o) and proton 11) scattering. However, this creates a problem when we have to decide which three J = 0 levels to fit as 0p-0h, 2p-2h and 4p-4h mixtures. Clearly the ground state and the 3.35 MeV 0 + state are two of the three levels, but the only other J -- 0 level known is one recently seen by Grace and Poletti 12) at 5.20 MeV. Unfortunately, if we consider this to be our third level we do not satisfy our consistency criterion of having a reasonable J = 0, J = 2 relation2+
6.94 6.75 6.54 6.50
2+ 4 2 4 (I)0
5.62 5.61 5.27 5.24 5.20
n+ a
3.90
0+
3.35
0+
0
Go
40
Fig. 3. E n e r g y levels o f 4°Ca w i t h even o r u n k n o w n spin.
Ca DEFORMED STATES
117
ship, since this requires the two unperturbed levels in a given b a n d to be about ½ M e V apart. We prefer to think that the third 0 ÷ we should fit is a level about 1 to 2 M e V higher that has yet to be identified. Actually several levels o f u n k n o w n spin and parity are seen in this region. We would then say that the 0 ~+) seen at 5.2 MeV has the 2 + at 5.24 or 5.6 MeV associated with it. With our value o f M 2 = - 1 . 0 MeV, it is necessary to take E(2p-2h, J = 2) = 6.52 MeV, E(4p-4h, J = 2) = 4.28 MeV, in order to obtain the eigenvalues and eigenvectors given in table 3. TABLE 3 W a v e functions of 2 + states in 4°Ca State
Energy
21+ 2z +
3.90 6.90
2p-2hJ = o
4p-4hJ= o
0.355 0.935
0.935 --0.355
In order to predict the unperturbed positions of the 0 ÷ levels f r o m unperturbed positions of the 2 ÷ levels, a value o f h2/2~¢must be given. We use a value o f hZ/2S o f about 0.1 which is indicated by the 0 ÷, 2 ÷ and 4 + states at 3.35, 3.90 and 5.2 MeV, respectively. These states already f o r m an almost perfect rotational b a n d even t h o u g h they are mixtures of deformed states. F r o m this we obtain for the J = 0 levels, E(0p-0h, J = 0) = 1.20 MeV, E(2p-2h, J = 0) = 5.95 MeV, E(4p-4h, J = 0) = 3.75 MeV. The final energies and wave functions are given in table 4. TABLE 4 W a v e functions o f 0 + states in 4°Ca State 01+ 02+ 0a+
Energy
0p.0hJ= o
2p.2hJ=O
4p.4hJ= o
0 3.55 7.33
0.904 0.202 0.378
0.414 --0.183 --0.892
0.111 --0.962 0.249
We also use E(2p-2h, J = 4) =
7.8 MeV,
E(4p-4h, J = 4) =
5.5 MeV,
118
w.J.
GERACE AND A. M. GREEN
to obtain the eigenvectors appearing in table 5. TABLE 5
Wave functions of 4 + states in 4°Ca State
Energy
2p-2hJ = 4
4p-4hJ = 4
41+
5.25
0.31
0.95
42+
8.00
0.95
--0.31
There have been several experiments carried out to study the structure o f the ground state of 4°Ca. In each case they seem to require larger deformed components than the 20 ~ we predict in table 4. For example Bock et al. 13) with the reaction 4°Ca(3He, c~) 39Ca and assuming the simplest possible structure for the residual 39Ca states f o u n d that about 45 ~ o f the 4°Ca ground state was deformed. Belote et al. 1~) with the reaction 4°Ca(d, p)41Ca also obtained large percentages in the region o f 40 ~ deformed. However, it has been recently shown by Rost 15) that the usual way o f analysing the above type o f experiment can be very misleading when deformed states are involved, and considering the reliability o f this calculation we do not find this discrepancy disturbing. 2.4. ELECTROMAGNETIC TRANSITIONS IN ~°Ca The electromagnetic transitions are calculated in precisely the same way as in ref. 4) for 160. Namely, the B(E2) between levels o f a given rotational band are put in terms o f the intrinsic quadrupole m o m e n t o f that band, which is then evaluated in the SUa limit. For hco = 10.5 MeV, this gives for in-band transitions B(E2, 2p-2h, fl = 0.17, 2 + --> 0 ÷) =
68 e2fm 4,
B(E2, 4p-4h, fl = 0.17, 2 + ~ 0 +) = 170 e2fm 4, B(E2, 4p-4h, fl = 0.23, 2 + ~ 0 ÷) = 230 eZfm *. Using these values and the 0 +, 2 + and 4 + wave functions given in tables 3-5, we obtain the B(E2) values given in table 6. TABLE 6
Electromagnetic transitions in
4°Ca
Transition
fl4p-4h : 0.17
fl4p-4h= 0.23
Experiment 10)
21+(3.90) --> 01+(0) 23+(6.94) --> 01+(0)
6.6 e~fm4 7.2
7.8 6.7
29±9 e~fm4 32±4
21+(3.90) --->02+(3.35) 41+(5.27) --->21+(3.90) 22+(6.94) --->03+(3.35)
152 220 9.2
203 300 14.3
Ca DEFORMED STATES
119
From this we can see that in the two cases where a comparison can be made with experiment the agreement is not good. A measurement of the 2 + ~ 0 + or 4 + ~ 2+ in-band transitions would help greatly in diagnosing the trouble. If our predictions for these in-band transitions are satisfactory then the problem is that our ground state wave function does not contain enough deformed component.
3. Odd-parity states of 41Ca and
41Se
Both 4~Ca and 41Sc would be described by the simple shell model as having a single f~ nucleon outside a double closed shell. However, instead of finding a single lowlying 3- excited state, which we would interpret as the valence nucleon excited to the 2p~ level, we actually observe two -~- levels which are 0.52 MeV apart in 4~Ca and 0.7 MeV apart in 41Sc. Experimentally each of these levels is strongly excited in singlenucleon transfer processes such as (d, p) (ref. t6)) and (3He, d)(ref. 13)) indicating that the single-particle strength is strongly mixed between them. The state being mixed with the single 2p~ nucleon level we interpret as a 3p-2h deformed state, where the holes are in the no. 8 Nilsson orbit and the particles are in the no. 14 orbit. To show that a 3p-2h state can lie sufficiently low in energy to mix strongly with the 2p~ level we again rely on the work of Zamick, who predicts the [3pTp=~-2hTh= 1]T=-,t- s t a t e to be about 3.5 MeV above ground in 4aCa and about 4.5 MeV above ground in 4~Sc. As we shall see later it is also possible to obtain a solution including the 5p-4h deformed state. Unfortunately in order to do this several more assumptions are necessary. Therefore, for the purpose of calculating transition rates and single-particle strengths we include only the 3p-2h state. 3.1. M A T R I X E L E M E N T S I N 4XCa
The basic matrix element involved is _ _ d
M~ = (lp-0hT=½1VI3pTp=~-2hT"=I
J
r=½),
(3)
where the isospin coupling of the deformed state is chosen in accordance with the prescription of Zamick. This then enables us to factor the matrix element and thereby relate it to the M1 in eq. (1) for 4°Ca M~ = ~22 Ml(4°Ca)"
(4)
The factor F takes into account the angular momentum recoupling and wave function renormalization and has the value 0.42 for J = ~ and 0.60 for J = ~. The ,,/2 is a correction to the isospin factor and represents the blocking of Nilsson's no. 14 orbital against double neutron excitation. Inserting these numbers and using the value
120
w . J . GERACE AND A. M. GREEN
Ml(4°Ca) = - 2 . 6 MeV calculated earlier, we get: M s=~ = - 0 . 7 8 MeV, MS3=~ = - 1.13 MeV,
(5)
when we use Nilsson's phase convention for the 2p wave function. If we are to conside1 the two ~2 levels as mixtures of only a lp-0h and a 3p-2h state, then the matrix element between the two should have a magnitude which is less than or equal to onehalf the separation - namely 0.26 MeV in 41Ca. Unfortunately our value exceeds this. Of course our method of calculation or choice of deformed wave function could be in error, but there is an alternative possibility. It is conceivable that other ~ - deformed levels be low enough to affect the positions of these states. In fact, the Zamick model predicts a 5p-4h state at about 5 MeV above the ground state. We shall consider this possibility in greater detail later, but until then we shall assume that M3J =~ = - 0 . 2 6 MeV and that this is simply a two state problem. 3.2. E N E R G Y
OF UNPERTURBED
3p-2h C O N F I G U R A T I O N
The assumption that M~ =~ = - 0 . 2 6 MeV, forces us to take the energy of the 3p-2h J =~ state to be degenerate with the 1 ~ J =~ level at 2.20 MeV, if we are to fit exactly the observed levels at 1.95 MeV and 2.47 MeV. As mentioned earlier, the Zamick model places the 3p-2h state at about 3.5 MeV above the ground state. However this is the centre of gravity of all the possible states of different angular momenta that can be projected out of this intrinsic state, and therefore the two energies are not inconsistent. The 3p-2h state we are dealing with here has K = 1, and thus this has the additional complication of the decoupling effect for an odd particle in the no. 14 Nilsson orbit. In fact the decoupling factor for such a particle is about -3.35, using Nilsson's wave function at fl = 0.2. Therefore, if we consider the J = ~ level to be at 2.20 MeV and assume a reasonable inertial parameter of h 2 / 2 J = 0.095 as indicated in 4°Ca, then we find that the K = ½ band is as follows: E ( J = ½) = 2.91 MeV, E ( J = -~) = 2.20 MeV, E ( J = ~ ) = 4.26 MeV and E ( J = ~ ) = 2.71 MeV. Experimentally a weak J = ½ level is seen at 2.97 MeV; but no J = ~ has been identified although several unknown levels do exist in this region. This decoupling effect has the desirable feature of moving down the J = ~ level towards the single-particle shell model level usually thought to be the A = 41 ground state. This results in a non-negligible amount of 3p-2h component in the A = 41 ground state, as is needed to account for the large electromagnetic transition observed. The deformed component in our ground state, although less than the amount found by Bock et al. ~ 3), is still consistent with their results. The assumptions made above as well as the difficulty in analysing singleparticle pick-up and stripping data when deformed states are present, as explained by Rost 15), could more than account for the differences. In fact the wave functions of Bock et al. 13) give transition rates not too different from the experimental values.
Ca DEFORMED STATES
121
3.3. WAVE FUNCTIONS OF 41Ca and 4xSc The assumption, in 4~Ca, that the (lp-Oh) s =~ and (3p-2h) J =~ levels are degenerate at 2.20 MeV with a matrix element between them of - 0 . 2 6 MeV, gives the wave functions in table 7. TABLE 7
Wave functions in 4xCa State
Energy (MeV)
(lp-0h)~
{1{2-
1.95 2.47
1/~/2 1/~/2
(3p-2h)~1/~/2 -- 1/~/2
In 41Sc we assume we have the same lp-0h to 3p-2h matrix element. Therefore, in order to get an exact fit to the observed levels at 1.71 MeV and 2.42 MeV we require the unperturbed lp-0h and 3p-2h states to be at 1.82 MeV and 2.31 MeV respectively, with the final wave functions given in table 8. TABLE 8
Wave functions in 41Sc State
Energy (MeV)
(lp-0h)k
(3p-2h)~
{a{2-
1.71 2.42
0.916 0.400
0.400 --0.916
This position of the unperturbed (lp-0h) ~ level is consistent with the calculation of Jones et al. 28) that the (lp-0h) ~ state in 41Sc is depressed by about 0.3 MeV below that in 41Ca. However we get that the deformed state in 4XSc is raised by only 0.1 MeV above the one in 4~Ca, whereas Zamick's argument gives the centre of gravity of the deformed 4~Sc state to be about 1 MeV above the centre of gravity of the corresponding 41Ca state. Probably Zamick's method is not sufficiently accurate to make such a difference significant. The ~- wave functions are found assuming that the deformed 5 - state has an unperturbed energy of 2.71 MeV, the value calculated earlier assuming a rotational band built on a Nilsson no. 14 orbital. To obtain a 5 - ground state, we must then take 0.48 MeV as the unperturbed energy of the ]lp-0h) ~ state. Since the deformed state in 4~Sc is only 0.1 MeV above the one in 4~Ca, we use the ~- wave functions given in table 9 for both 41Ca and 41Sc. TABLE 9
Wave functions of 5- states in ~lCa and 41Sc State
Energy
[1p-0h)-~
]3p-2h)÷
5152-
0 3.22
0.924 0.388
0.388 -- 0.924
122
w.J.
G E R A C E A N D A. M. G R E E N
3.4. ELECTROMAGNETIC TRANSITIONS IN *1Ca and 41Sc At first sight the electromagnetic transitions from the two ~ - levels in 41Ca seem rather peculiar. The lower level decays strongly to the ground state with a B(E2) of 7 W.U., whereas the upper level only has a B(E2) of less than 0.1 W.U. to the ground state. In fact the higher ~ - prefers to decay to the lower ) - by an M1 transition. This is so even though our model says that both ~ - levels have a similar structure except for a phase change between the two components (see table 7). Of course, it is just this phase difference which causes the B(E2, 32 ~ g.s.) to be inhibited, because now the contributions from the lp-0h and 3p-2h components tend to cancel each other. Of course, before we can get any contribution from the lp-0h term in 41Ca we need to introduce an effective charge for the valence neutron. At present, no complete calculation of effective charges has been made with realistic forces. Early estimates by Horie and Arima 19) with a 6-function force gave the effective charge of a neutron e~ff at the end of the s-d shell to be somewhat greater than e and that of the proton e~ff to be between ½e and e. Zamick 28) using more realistic forces came to the result n ~ n that e,ff = 0.6e and e~ff 0.2e. Therefore, it appears that eCff can be as large as e, whereas e~ff p seems to be more in the region of ½e. The reason for the larger value for e~ff is that the effective charge is essentially a measure of the number of protons excited from the core. When a proton is the valence nucleon, the protons in the core can only be excited via the T = 1 component of the internucleon interaction, whereas when a neutron is the valence nucleon the T = 0 component is also involved, and this is somewhat stronger than the T = 1 component. Another uncertainty in the electromagnetic transition of the lp-0h component is the single-particle wave function used. Using simple harmonic oscillator wave functions we get Bs.p.(2p~ ~ lf~, E2) = 35.2e2fm 4. However in a recent calculation using Woods-Saxon wave functions which fitted the spectrum of 4aCa and 4~Sc, Schiffer 29) found the result Bs.p.(41Ca, e~"ff = 1, 2p~ ~ lf~) = 36 e2fm4,
Bs.p.(41Sc, eePff = 0, 2pt --, lfi) = 47 e2fm 4. In other words there is no great change when the better wave functions are used. This is different from the effect in 170 where the more realistic wave functions increased the B~.p. (2s~ --, ld~, E2) by a factor of 4. In that case the absence of the centrifugal barrier is mainly responsible for the large change. The basic B(E2) values for the 3p-2h states for hco = 10.5 MeV appear in table 10. TABLE
10
B(E2) values of intrinsic states in e~fm* fl = 0.17 /3 = 0.23 B(E2, 3p-2h, a~Ca, ~2- ~ ~-) 340 460 B(E2, 3p-2h, 4XSc,3- ~ 7~-) 200 303
Ca DEFORMED STATES
123
Using these in-band transitions and the single-particle transitions o f Schiffer together with the 4aCa 3 - wave functions appearing in table 7 and the g r o u n d state wave function o f table 9, we obtain the results in table 11. TABLE 11 Electromagnetic transitions in 4~Ca (units e2fm4) Exp. a) B(E2, 41Ca, ~l- --~ g.s.) B(E2, 41Ca, ~2- --~ g.s.) B(M1, 41Ca, ~2- -~ ~1-)
59zL13
fl, = 0.17 80 1.2
f12 = 0.23 97 3.7 0.008
a) Refs. s0,21). Here we have chosen enff ---- 1. This is in the region indicated by the direct calculations o f Horie and A r i m a 19) and Zamick. Using the same g r o u n d state wave function and the 3 - wave functions o f 41Sc given in table 8, we get the results shown in table 12. TABLE 12 Electromagnetic transitions in 41Sc (units e2fm4)
Exp. a) B(E2, 41Sc, ~tl- -+ g.s.) B(E2, 41Sc, ~-2--+ g.s.) B(M1, 41Sc, ~2- ~ ~1-)
101 :t:34
fl = 0.17 110 1.3
fl -----0.23 120 5.4 0.005 e~fm~
s) Ref. 33).
I n 41Sc we simply chose e~Pff = ½ ee~f as this seems to be the trend in calculations o f these quantities. The result is in g o o d agreement with the one observed transition. However this transition is dominated by the lp-0h c o m p o n e n t and is therefore not very sensitive to the a m o u n t o f deformed c o m p o n e n t in the ground state. 3.5. SINGLE-PARTICLE STRENGTHS One o f the first indications that 41 Ca deviated f r o m the simple shell-model description was the excitation o f two strong 3 - levels in the 4°Ca(d, p)4aCa reaction 16,17). I n this experiment it was f o u n d that the single-particle strength, instead o f being concentrated in one o f these two levels, was in fact distributed in about a two to one ratio between the lower and the upper levels. I n this subsection we make r o u g h estimates o f these strengths using the wave functions o f tables 7 and 8. I f we consider the 4°Ca ground state has a wave function o f the f o r m ~k(4°Ca, g.s.) = a(0p-0h) J~ o +b(2p_2h)O + c(4p_4h)O,
(6)
and that the states o f 41Ca have the f o r m ~bs(41Ca) -- d(lp-0h) s + e(3p-Eh) J +f(Sp-4h) J,
(7)
124
w.J.
G E R A C E A N D A. M. G R E E N
then a simple estimate of the ratio of single-particle strengths for a fixed J is given by SPS ~:
lad+AJbe+BJfcl 2,
(8)
where Z J = R~ C~ = R~([J2p-2h°]Jl3p-2hJ ), B J = R~ C~ = R~([J4p-ala°]Jl5p-ahJ). The factors C1 and Ca represent the angular momentum structure of the 3p-2hJ and 5p~4h ~ states and have the values C~ = 0.435, C~ = 0.614 and C2~ = 0.400. We do not consider the effect of C2~. In fact we do not introduce even C2~ until the next section, where the effects of 5p-4h states are introduced. The R J factor which appears in the second and third terms of eq. (8) is there because in that case the transferred nucleon is in a deformed well. This effect has recently been studied by Rost 15) who shows that deformation of the well modifies the radial wave function in the surface region. We can approximate this effect 30) by using R~ = R2 = 1.3. Using these factors together with the values of the coefficients a, b, c, d and e given in tables 4 and 7, we get the single-particle strengths in the two strong ~- levels as shown in table 13. TABLE 13 Single-particle strengths in 4xCa and 4xSc Exp. 41Ca SPS ~x-(1.95 MeV)
2.9
2.3 ~)
SPS~,~-(2.47 MeV) 41Ca-4xSc SPS {2-
2.5 × 10-a
SPS {1-(g.s.) 41Sc SPS ~1-(1.71 MeV)
38
10 b)
SPS 22-(2.42 MeV) a) Ref. 17).
b) Ref. 13).
Also shown in table 13 are the results for 41Sc using the wave functions of tables 4 and 8. We see reasonable agreement with the observed ratios of single-particle strengths. As we go from 41Ca to 41Sc this increase in the ratios as the two mixed levels get further apart is a good indication of the basic validity of our model. In the next subsection where we introduce the effects of the 5p-4h state, we show that results comparable to those above can still be obtained. Another important point is the amount of SPS in the ½- state which we expect near 3.2 MeV. This must not be too large or the state would probably have been already identified. Thevalue of the ratio of the SPS of this state to that of the ground state is shown in table 13 to be quite small.
Ca DEFORMED STATES
125
3.6. E F F E C T O F 5p-4h S T A T E S
Thus far in 41Ca we have been using the value of - 0 . 2 6 MeV for the matrix element M~ =~ = (lp--0h~l V I ~ ~) even though we calculate it to be nearer - 0 . 7 8 MeV. This was done so that in subsect. 3.3 we could obtain the exact splitting using only two states. The purpose of this section is to show that it is possible to obtain a solution using the calculated value of M~ provided a third state, the 5p-4h state, is introduced. If we consider the 5 p - ~ r=½ state as being formed by adding an extra neutron in the no. 17 orbital of fig. lb, then its matrix element to the 3p-2h T=~ state is 0.37 MeV. Because of the large value of Ma~ it is necessary now to have the 5p-4h state lower than the 3p-2h state. The problem is whether or not a solution can be obtained without disturbing the ratio of the single-particle strength of the lower two states and without having the third state absorb an observable amount of single-particle strength. To obtain such a solution it is necessary to have the lp-0h and 5p-4h states degenerate at 2.45 MeV. Locating the 3p-2h at 3.35 MeV results in the wave functions appearing in table 14. TABLE 14 W a v e functions in 41Ca State
Etina~
Ilp-Oh)S=k
]3p-2h)~r ="l"
[5p-4h)1=k
~1-
1.95
0.77
0.52
--0.37
j}~-
2.47
0.44
--0.07
0.90
~3-
3.90
0.47
--0.86
--0.22
In 4XSc it is necessary to take for the unperturbed energies of the lp-0h 4, 3p--p-~2h~ and 5p~4h ~ states the values 2.12, 3.37 and 2.47, respectively. Here we have shifted the unperturbed position of the (lp-0h) ~ state by a value of 0.35 MeV relative to 41Ca. This shift is still consistent with the calculations of Jones et al. ~8). The resulting wave functions are given in table 15. TABLE 15 W a v e functions in 41Sc State
Etlna 1
11p - 0 h ) J = k
[3p-2h)J = ~-
~1-
1.71
0.86
0.45
-- 0.22
~z-
2.42
0.30
--0.12
0.94
~3-
3.93
0.40
--0.88
--0.24
These wave functions satisfy 4 ~ C a - SPS(~-)/SPS(~2) = 3.3, 4 1 C a - SPS(~7)/SPS(~-3-) = 17.2, 4 1 S c - SPS(~I)/SPS(~z~-) = 9.1, 41Sc- SPS(kT)/SPS(kg) = 38.0.
15p-4h)3= k
126
W.
J. GERACE
AND A. M. GREEN
In our model M& the matrix element between deformed states is approximately independent of J in the limit of large deformation, and therefore we take M$ = M$ = +0.37 MeV. Also since we have tixed the unperturbed positions of the 3p-2h3 and 5p-4h* states, we are able to estimate the unperturbed position of the corresponding J = 3 states. This is done using a value of h2/23 of 0.095 as before, and the decoupling factors for an odd particle in the no. 14 and no. 17 Nilsson orbitals. This gives E(3p-2h$) = 3.86, E(5p-4h3) = 3.45, and a ground state wave function I++,.,.= 0.9611p-Oh)+0.28~3p-2h)-0.05~5p-4h). This wave function is very similar in structure to that obtained using only two states, and consequently it is still true that almost all of the 3- strength is in the ground state. The other two weak f- states occur at about 3.3 and 4.4 MeV. As before we use this ground state for both 41Sc and 41Ca. The only remaining question is whether these wave functions will still give reasonable reduced transition rates. In addition to the transition rates in table 10, we need B(E2, 5p-4h 41Ca, $- --f $-) = 581 e2fm4 for ps = 0.23, B(E2, 5p-4h, 41Ca, $- + 3-) = 745 ezfm4 for ps = 0.30, B(E2, 5p-4h, 41Sc, $- + $-) = 770 e2fm4 for /3s = 0.23, B(E2, 5p-4h, 41Sc, 3- + 3-) = 970 e2fm4 for p5 = 0.30. The final reduced transition rates for 41Ca and 41Sc are given in tables 16 and 17, respectively. Here we have taken as before e& = e, e& = 0.5e. TABLE
16
Transition rates in %a Exp. “)
& = 0.17 b6 = 0.23
,!$ = 0.23 /?&= 0.30
B(E2, &- + p.s.)
59f13
57.2
64.6
B(E2, &- -+ g.s.)
< 0.4
1.2
0.8
2.1
4.7
B(E2, is- --f g.s.) B(Ml,
a) Refs. 20,21).
in e2fm4
Qz- + $1~)
5 0.03
0.0035 e2fm2
127
Ca DEFORMED STATES
We see therefore that the three level model with calculated matrix elements gives an equally g o o d interpretation o f the experimental data as the two level model. This suggests that the states in the two level model are actually the Il p - 0 h ) ~ and 15p-4h) ~ states b o t h renormalized by ]3p-2h) ~ state. The interaction between these two renormalized states then arises only t h r o u g h the 13p-2h) ~ c o m p o n e n t which justifies our use o f an effective matrix element which is small c o m p a r e d to the calculated value o f M~. TABLE 17 Transition rates in 41Scin e2fm4
B(E2, ~x- -+ g.s.) B(E2, ~z- ~ g.s.)
Exp. a)
f13 = 0.17 f15 = 0.23
f13 = 0.23 f15 = 0.30
1014-34
112
122
1.4 0.64
B(E2, ~3- ~ g.s.) B(M1, ~ - --> ~1-)
0.8 0.01 0.0026 e~fm2
a) Ref. 22).
4. Even-parity states of 42Ca According to the simple shell model, 42Ca should be described in terms o f configurations o f two valence particles in the lf-2p shell, the core providing the shellmodel potential. However, there exist states, such as the 0 + at 1.84 MeV and the 2~at 2.42 M e V shown in fig. 4, which cannot be explained by so simple a model. A further indication o f the inadequacy o f the model is the absence o f corresponding levels in the spectrum o f 50Ti ' while at the same time the excitation energies o f the 2 +, 4 + 6+
3.19
4+
2.75
2+
2.42
0+
-
2 +"
- 1.52
0 +"
1.84
O.
Ca
42
Fig. 4. Low-lying levels of 4~Ca having even spin and parity.
128
W, J. GERACEAND A. M. GREEN
and 6+ levels are the same to within 0.05 MeV in both spectra 23). The occurrence of a second 0 ÷ and 2 + in 4ZCa but not in 50Ti also indicates the nature of these states. These two levels must be formed by the excitation of protons from the 2s-ld to the lower lf-2p orbitals. In s 0Ti the corresponding levels would be formed by raising two neutrons, but these are blocked by the completely filled lf÷ shell of neutrons. The Zamick model cannot be used to predict the unperturbed position of the 4p-2h states because the 0~- at 1.84 MeV was used to determine the force constants in the calcium region assuming it to be the [4p rp = °-2hrh = 1]r = 1 state. 4.1. M A T R I X E L E M E N T S I N ~2Ca
Again we write the basic matrix elements as __J
I
M~(2p) = ( 2 p - 0 h r = 1] V]4pTp = 0_2hTh =1 T = 1 ) ,
(9)
Zamick's prescription of coupling particles and holes separately to definite isospins again enables us to factorize the 4p-2h wave functions and thereby relate M4s to M 1, the matrix element for 4°Ca, giving M4S(2p)- F(2P)M,(4°Ca).
(10)
43
Wave function normalization and angular momentum recoupling factors are collected into the factor £'(2p). Table 18 summarizes the value taken by _P(2p) for certain 2p configurations appearing on the left-hand side of the matrix element in eq. (9). TABLE 18 Factors relating matrix elements in 42Ca and 4°Ca [lf~]J=o
[2p~2]j=o
0.505
0.208
[lf~2]J=2 --0.272
[1 fk-2p~]J= 2
[2p$2]J =2
--0.306
--0.I01
The x/3 in the denominator of eq. (10) reflects the complete blocking of neutron excitations to the no. 14 orbital since it already contains two neutrons. Using these values together with M 1 = - 2 . 6 MeV, we obtain M°(lf~) = - 0 . 7 6 MeV
MZ(2p~) = 0.15 MeV
M°(2p~) = - 0 . 3 1 MeV 2 2 M4(lf~) = 0.41 MeV
M2(lf~2p~) = 0.46 MeV. 4.2. W A V E F U N C T I O N S O F 42Ca
Here we use Kuo's 7) wave functions for the 0~- and 2+ states and adjust their energies so that we get exactly the observed eigenvalues for the lowest two 0 ÷ and 2 + levels
129
Ca DEFORMED STATES
of fig. 4, when we insert the 4p-2h states appropriately. 0 [ (Kuo) = 0.927(f~) ° + 0.203(p~) ° + 0.203(f~) ° + 0.109(p~) °, 2 ~ ( K u o ) = 0.935(f~) 2 + 0.278(f~p~) 2. The resulting wave functions are given in tables 19 and 20. TABLE 19 Wave functions of 0+ states in 42Ca State
EflnaI
Einitia 1
(lf~2)°
(2p~2)°
(lf~2)°
(2p,2) °
01+
0
0.40
0.820
0.180
0.180
0.096
02+
1.84
1.44
0.432
0.094
0.094
0.051
Def. J=o 0.465 --0.884
TABLE 20
Wave functions of 2+ states in 42Ca
State
Eflnal
•tatttal
(lf~2)~
(lf~2P~) 2
Def. d=2
2x+
1.52
1.97
0.661
0.197
--0.707
22+
2.43
1.97
0.661
0.197
0.707
Again we see that our consistency criterion is satisfied in that the unperturbed positions o f the deformed 2 + and 0 + differ by a b o u t 0.5 MeV. The type o f matrix elements defined by eq. (9) correspond, in infinite nuclear matter, to the inclusion o f unlinked diagrams, which merely add a constant energy shift to all levels. This effect seems to be holding true even in the finite nucleus case we have here, since we see that the two lowest states 0~- and 2 + were b o t h depressed by 0.45 M e V f r o m their unperturbed positions. F o r the 4 + states we obtain the wave functions and energies shown in table 21. TABLE 21 Wave functions o f 4 + states in 4zCa
State
Ennai
Einlttal
4x+ 42+
2.75 3.56
3.10 3.21
~04+(Kuo) 0.75 0.66
~p4÷ (def.) 0.66 --0.75
4.3. ELECTROMAGNETIC TRANSITIONS IN *"Ca W i t h the wave functions given in the previous section and the following basic B(E2) values --J=2
B½(E2, eenff = 1.0, f~
__J=O
-~ f2~
) = - 4 . 3 5 elm 2,
130
W. J. GERACE AND A. M. GREEN -. 2
__0
B~(E2, ee"ff = 1.0, f~-Pl- -" f~ ) = - 2 . 6 4 efm 2, __2
B~(E2,
eenff
= 1.0, f~p~
~
__0 p 2~ )
= -3.74
e f m 2,
B~(E2.4p-2h, 2 + -~ 0+fl = 0.17) = 13.0 elm 2, B~r(E2, 4p-2h, 2 + ~ 0+fl = 0.23) = 15.0 elm 2, we get the transition rates given in table 22. TABLE 22 E l e c t r o m a g n e t i c transitions in 42Ca (units e~fm 4) eeff n
Transition
Expt. a)
B(E2, 21 + --* 01 +)
742L17
B(E2, 02 + ~ 21 +)
~ 500
B(E2, 22+ --* 0x +)
~
B(E2, 2~+ ~ 02 +)
<150
10
fl = 0.17
~ 0.5e fl = 0.24
eetf n ~ e fl = 0.17
fl = 0.24
33
42
52
62
274
375
222
315
8.0 80
12.3 104
2.0 95
4.4 120
a) Ref. 24).
In the above, we quote the square root of the basic B(E2) values to bring out the sign difference that enters between the single particle and deformed components. This now ensures that the B(E2, 2 + --, 0 +) is the coherent combination of the two components. It is seen that we again underestimate the two known B(E2) values by about a factor of 2 or 3 for the case of e~fr = e. Even so we consider these results as encouraging since the B(E2) are large. 4.4. EFFECT OF 6p-4h STATES In the last section our estimates of B(E2) fell below the observed values indicating that we were not sufficiently mixing the deformed components into the shell-model states. One possibility is that the configuration [6prp=l-4hrh=°] r = l is important. Such a state, by the arguments of Zamick, lies between 3.5 and 4.5 MeV above the ground state and has a matrix element of ~ 0.6 MeV with the 4p-2h state. The effect of this on the final transition rates can be approximately allowed for by an increase of less than 5 ~ in the basic 4p-2h B(E2) values and thus the results of the previous section are changed by about this same amount. 5. Conclusion In this paper we have applied to the calcium isotopes a model that was first intro-
Ca DEFORMED STATES
131
duced in the oxygen region where it proved successful. Within this model the electromagnetic transition probabilities and single-particle strengths, so far calculated, have all been in reasonable agreement (to within a factor of ~ 2) with the observed values. Further justification of these ideas depends on the measurement of additional B(E2) values such as 2 + ~ 0~- or 4 + ~ 2 + in 4°Ca. Also we predict states thus far unseen, such as another 0 + in 4°Ca near 7.3 MeV and weak ~- and ½- levels in 41Ca in the vicinity of 3.2 MeV. Identification of these levels would put our model on a much firmer foundation. Until now, no mention has been made about calcium isotopes with mass greater than 42Ca. However in these nuclei we expect the effect of deformed states on the lowest shell-model levels to decrease with increasing neutron excess. The reasons for this are twofold. Firstly, the matrix element between the deformed and spherical states decreases through the normalization of the deformed wave function 25). This is because the more neutrons there are, the greater number of components in the deformed wave function and thus the amplitude of that portion which connects with the spherical state decreases. Secondly, the unperturbed energies of these deformed states appear to increase with the number of neutrons. This is indicated by the rise in energy of the second 0 ÷ levels which we say contain mainly the deformed component. For example in 42Ca, 44Ca, 46Ca and 4SCa these second 0 + levels occur at 1.8, 1.8, 2.42 and 4.3 MeV, respectively. Also, estimates with the Zamick model put the deformed states in these nuclei at 1.8, 2.5, 3.2 and 5.1 MeV, respectively. This decrease in the effect of the deformed state on the ground state is also seen in (d, p) reactions. From these we can get a measure of the number of holes in the 2s-ld shell and thus estimate the amplitude of the deformed component in the ground state. Belote et aI. 26) show that this amplitude decreases by about a factor of three in going from 4°Ca to 46Ca. Finally, when our results are compared with those of Bertsch 6) we see that the inclusion of four-hole components and core renormalized matrix elements gives a significant improvement. The authors wish to thank Professor G. E. Brown for his guidance and continued interest, Dr. T. Kuo for his two-body matrix elements and Dr. G. Bertsch for a critical reading of the manuscript. We also acknowledge an enlightening conversation with Dr. J. Raz. One of us (A.M.G.) also thanks Professor J. R. Oppenheimer for his warm hospitality at the Institute for Advanced Study. References 1) 2) 3) 4) 5) 6)
T. Engeland, Nuclear Physics 72 (1966) 68 G. E. Brown, Compt. Rend. Int. de Phys. Nuc16aire, Vol. I (1964) p. 129 G. E. Brown and A. M. Green, Phys. Lett. 15 (1965) 168 G. E. Brown and A. M. Green, Nuclear Physics 75 (1966) 401 L. Zamick, Phys. Lett. 19 (1965) 580 G. Bertsch, Nuclear Physics 79 (1966) 209 and to be published
132 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30)
W. J. GERACE AND A. M. GREEN T. T. S. Kuo, private communication W. H. Bassichis and G. Ripka, Phys. Lett. 15 (1965) 320 J. Hayward, Nuclear Physics 81 (1966) 193 D. Blum, P. Barreau and J. Bellicard, Phys. Lett. 4 (1963) 109 W. S. Gray, R. A. Kenefick and J. J. Kraushaar, Nuclear Physics 67 (1965) 542 M. A. Grace and A. R. Poletti, Nuclear Physics 78 (1966) 273 R. Bock, H. H. Duhm and R. Stock, Phys. Lett. 18 (1965) 61 T. A. Belote, A. Sperduto and W. W. Buechner, Phys. Rev. 139 (1965) B80 E. Rost, Phys. Lett. 21 (1966) 87 L. L. Lee, Jr. and J. P. Schiffer, Phys. Rev. Lett. 12 (1964) 108 L. L. Lee, Jr., et al., Phys. Rev. 136 (1964) B971 K. W. Jones et al., Phys. Rev. 145 (1966) 894 H. Horie and A. Arima, Phys. Rev. 99 (1955) 778 S. I. Baker et al., Bull. Am. Phys. Soc. 11 (1966) 477 D. S. Gemmell, L. L. Lee, Jr. and J. P. Schiffer, private communication D. H. Youngblood, J. P. Aldrich and C. M. Class, Phys. Lett. 18 (1965) 291 P. Federman, Phys. Lett. 20 (1966) 174 F. R. Metzger and G. K. Tandon, Phys. Rev. 148 (1966) 1133; A. Bernstein, private communication G. E. Brown and A. M. Green, Nuclear Physics 85 (1966) 87 T. A. Belote, H. Y. Chen, O. Hansen and J. Rapaport, Phys. Rev. 142 (1966) 624 A. Arima, private communication L. Zamick, private communication J. P. Schiffer, private communication E. Rost, private communication