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Static moments of 21+ states in light doubly even nuclei
1.E.3
I I
Nuclear Physics A154 (1970) 191--201; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
STATIC MOMENTS
OF 2+ STATES
IN LIGHT DOUBLY
EVEN NUCLEI
LARRY Z A M I C K t
Department of Nuclear Physics, Weizmann Institute of Science, Rehovot, Israel *t Received 14 January 1970 (Revised 21 April 1970) Abstract: Quite general arguments indicate that if the shell-model picture holds or even if one allows for effective charge corrections, then the quadrupole moment of the first excited state of 48Ti should be small, and the magnetic moment of the first excited state of 54Fe should be large, compared to the rotational value. These results are contrary to recent experiments. A comparison is made between perturbation theory and matrix diagonalization in the space two holes and three holes and one particle, for s 4Fe and s 4Co" One gets less magnetic quenching with matrix diagonalization in 54Fe (hence the deviation from experiment is more) than with perturbation theory. The possibility that the lowest 2 + state in S4Fe is mostly a 3p-lh state is ruled out. Corrections to the quasiparticle approximation i.e. the use of the magnetic moment of one hole (55Co) to calculate the magnetic moment two holes (S4Fe), are calculated, and again tends to reduce the quenching. The J = 7 J = 0 splitting of two holes - 54Co - is less than that of two particles - 42Sc - by 400 keV but in perturbation theory the admixture of 3h-lp in 54Co goes in the wrong direction by 600 keV. It is shown however that with matrix diagonalization the J = 0 state of 54Co gets pushed down much less than in perturbation theory, so that this 600 keV additional discrepancy is erased. A triplet of levels at 2.5 MeV in 54Fe is analyzed to consist of a (mostly) 2h J = 4 state, a 3h-lp particle J = 2 state, and 4h-2p J = 0 state. A bare realistic interaction puts the 4h-2p state much too high in energy but an interaction chosen to fit single-particle energy shifts gives excellent agreement.
1. I n t r o d u c t i o n M o s t m e a s u r e m e n t s o f electric q u a d r u p o l e m o m e n t s a n d magnetic m o m e n t s of e x c i t e d s t a t e s o f n u c l e i h a v e , a t t h e t i m e o f t h i s w r i t i n g , b e e n c a r r i e d o u t in h e a v y nuclei, b u t s o m e r e c e n t m e a s u r e m e n t s h a v e b e e n c a r r i e d o u t i n t h e lf~ shell r e g i o n . P a r t l y i n t h e h o p e o f s t i m u l a t i n g f u r t h e r w o r k i n t h i s r e g i o n , w e shall d i s c u s s t w o e x a m p l e s b o t h c f w h i c h a r e e v i d e n c e a g a i n s t a s i m p l e f~ p i c t u r e .
On leave from Rutgers University with the support of a Rutgers University Fellowship and a Weizmann Institute Senior Fellowship. tt Supported in part by the National Science Foundation. 191
192
L. ZAblICK 2. The quadrupole moments of the 2 + state of 4STi
It was reported by Hausser et al. 1), in the 1969 Montreal Conference contributions, that the quadrupole moment of the 2 + state of 4aTi had a value Q = -0.22__.0.08 b. They note that this value is large, in the sense that if one assumes a rotational formula then the value of the intrinsic quadrupole moment Qo is consistent with a value obtained from their measured value of the B(E2, 0 + ~ 2 + ) = 690___60 e 2- fm 4. Let us discuss the above quadrupole moment from the shell-model point of view. Allowing for effective charges, the quadrupole operator can be written as a sum of an isoscalar and isovector term O ° + O 1 0 0 = ½(ep+e.)
r2y 2,
~ all
particles
01 = --½(%--en) ~
r 2Y2~z,
all
particles
where ~z = + 1 for a neutron and - 1 for a proton. In the f~ model 48Ti consists of 2 protons and 2 neutrons holes in the f~ shell so the wave function can be written
~b' = Z C(LpL,)[Pf~2LfC ZLn]t, with C(Lp, Ln) the probability amplitude that the protons couple to Lp and the neutrons to L . . Assuming charge symmetry, and realizing that the interaction of two holes equals the interaction of two particles, it is easy to show that C(L,, Lp) -+__C(Lp, L,). For example, the following wave functions were obtained 2) for the 0 + and 2+ states of 48Ti in the f~ model in terms of the basis states [Lp L.] I J=
01
J = 21
~ = 0.91 [001 - 0 . 4 1 [22] - 0 . 0 2 [44] +0.01 [661, ~O = 0.69 [02] +0.17 [42]
- 0 . 6 9 [201 - 0 . 1 7 [24] - 0 . 0 1 [46] +0.01 [641.
Note that the J = 0~" state necessarily does not change sign under the interchange of protons and neutron holes. It turns out that the J = 2 + state does change sign under such an interchange, as seen above, but that the J = 2~- state does not. The J = 2+ state is said to have negative signature; states that do not change sign are said to have
positive signature. Using the fact that the quadrupole moment of a particle is minus that of a hole we can derive the following selection rule: The matrix element of the isoscalar part of the quadrupole operator 0 ° vanishes
STATIC MOMENTS
193
between states of the same signature, the matrix element of the isovector part O 2 vanishes between states of the opposite signature. Thus the E2 transition J = 0 + ~ J = 2~- will be proportional to ½(en + ep) whereas the quadrupole moment of the 2 + state (which can be regarded as a transition between two states of the same signature) will be proportional to ½(ep-e,). In the strict shell model ep = 1, e. = 0. To fit the transitions and quadrupole moments effective charges are required. It always turns out, both empirically and theoretically that the effective charges are such that ep > 1, en > 0 and in the f~ region en > ( e p - 1 ) i.e. the effective charge of the neutron is if anything, somewhat greater than the effective charge correction of the proton. Theoretically, the use of effective charge is in part equivalent to adding perturbative corrections to the shell-model wave functions. For example in the case of 4STi it is equivalent to improving the f~ wave functions via admixtures of the type f~ (PH), where P H the particle hole excitation consists of both An = 0 pairs of0f~ ~ 0f~, 0f~ 1 lp~ and An = 2 pairs such as 0f~- 1 0h,~, 0p~-1 0f~, 0d~ 1 0g~ etc. Both the An = 0 and An = 2 excitations are important. The point we wish to make is that such effective charge renormalizations as above will enhance the 0~ ~ 2,. transition, but will not collectively enhance the quadrupole moment of the 2 + state. This is clear from the fact that any collective enhancement of ep or e, will add coherently in the isoscalar part ½(ep + en), but will cancel in the isovector part ½(ep-e.). With the above f~ wave function of the 2 + state of 4STi we find Q/e = ( % - e ~ ) x 0.030 b which is not only smaller in magnitude but also the wrong sign. No reasonable manipulation of ep and en can get us to the Hausser et al. value. Note: I have been informed by Goode ~5) t that in perturbation theory there are effects beyond effective charge renormalization which can explain the discrepancy.
3. The magnetic moment of the 2+ state of S4Fe The magnetic moment of a proton or proton hole in the f~ shell is 5.79 n.m. This should be the value of the { - ground state of 55Co, but in actual fact is 4.2. This discrepancy may perhaps be explained by perturbative mixtures of the configuration f~ f~-2 into the ground state. Indeed a first order calculation was performed by Mavromatis et aL 10), with the result p(55Co) = 4.7. The correction was in the right direction but there was still a discrepancy. Whether this remaining discrepancy is significant or not is hard to say i.e. perhaps changing the two-body force or singleparticle energies might make a difference. It should be noted though that if a matrix diagonalization is performed in the space of lh and 2p-lh - this should be superior to perturbation theory - one obtains a smaller correction to the moment. Nevertheless the situation is somewhat marginal. t Evidently diagrams like fig. 16 which represent a breakdown of the effective charge idea are very important for quadrupole moments of 2~ states.
194
L. ZAMiCK
We now consider the 2~ state of 26Fe2a, 24 where we would like to point out that there should be a large difference between the shell-model and rotational-model prediction of the moment. The rotational formula It = 2 # R ,
gR = Z / A ,
yields a magnetic moment equal to 0.96 n.m. With the shell model the magnetic moment of the 2~- state of SaFe should equal 2 - - x 5.79 = 3.31/~,. 3.5 This is much larger than the rotational formula. I f instead of using the Schmidt value/~ = 5.79, we use the magnetic moment of the one proton hole nucleus s 5Co _ this is called the q u a s i p a r t i c l e a p p r o x i m a t i o n we get 2 #(2~-) = - - x4.2 = 2.4/2,. 3.5 This is still far away from the rotational value. Hence a measurement of this moment should distinguish between the rotational and shell model. A recent abstract by McDonald, Murnick and Lie a) gives a very small value for this moment - even less than the rotational value. Apparently, some other measurements of this moment are in progress, the results of which will certainly be of great importance in deciding between these two models. We have performed some calculations involving perturbation theory and matrix diagonalization in the space of two holes and three holes and one particle, which we shall now describe. 3.1. BREAKDOWN OF THE QUASIPARTICLE APPROXIMATION To what extent is one justified in using the experimental moment of 55 Co (one hole) in order to calculate the moment of 54Fe (two holes)? Even in first order perturbation theory, there is a term, shown in fig. lb, which involves a breakdown of the quasiparticle (or effective charge) idea. In fig. l a we have a process where one of the two holes polarizes the core (and this obtains a quenched magnetic moment) independent of the second nucleon. To the extent that this term dominates the quasiparticle idea holds. But in fig. lb we have a process in which both nucleons are involved in polarizing the core. A very simple expression can be obtained for the correction to the magnetic moment due to this second process 5 # = --
10---0--( g , - - g ~ ) ((f~)t=2V(f~tf~)'=2),
7x/3 where gt = 1, g~ = 5.588.
AE
195
STATIC MOMENTS
The two-body matrix element is antisymmetrized but not normalized, and ls rather than sl coupling of the individual nucleon orbit is assumed. The energy denominator is negative and about - 6 MeV. With the Kallio-Kolltveit interaction ~) (with mco/h --- 0.27 fm -1) the two-body matrix element is - 0 . 3 4 6 MeV. Thus we
o) THIS TERM IN54Fe IS INCLUDED WHEN
b) BREAKDOWN OF EFFECTIVE OPERATOR
ONE USES THE EXPERIMENTAL MOMENT
CONCEPT IN FIRST ORDER
FROM 55Co
Fig. 1. First order corrections to the magnetic moment of the 2~+ state of SaFe. find for the contribution due to two holes acting together c5# = 0.22 Pn. This is about 35 ~o of fig. la and so not negligible. However, note that the contribution goes in the direction of reducing the quenching, thus causing a larger divergence between the rotational- and shell-model values. That 8/z is positive should not be surprising. It is a Pauli effect. The presence of one hole partly prevents the second hole from polarizing the core. 3.2, MATRIX DIAGONALIZATION THEORY We shall now consider a matrix diagonalization in space of two f~ holes, and of three f~ holes and one particle in the p~, f~ and P~r orbits. One reason for doing this is that there is always the possibility that a 3h-lp "collective state" has an energy comparable to the 2h state, thus mixing with it very strongly. We again use the Kallio-Kolltveit interaction 4) with mco/h = 0.27 fro-2. Let us now quote the single-particle energies obtained from experiment for a S6Ni core although we shall not use them without some modification: /3f.I_ = 0,
gp~_ = 4.5 MeV,
ef~ = 5.1 MeV,
ep~r = 5.7 MeV.
The modification we refer to is for purposes of avoiding double counting, q-he "experimental" energy of the f~ hole consists mainly of the interaction of an f~ hole with sixteen f~ nucleons. On the other hand, there is a small contribution because the hole can polarize the f~6 core exciting an fff nucleon into the rest of the fp shell. This is shown in fig. 2b. But such a core polarization introduces a 2h-lp configuration of the type which we will be taking into account explicitly. We should therefore use a single-hole energy which is the experimental value minus the contribution of 2h-lp to the experimental value. We have performed a perturbation theory calculation and we find that this can be achieved by simply changing the f~ hole energy to ef~ = - 0 . 6 0 MeV.
196
L. ZAMICK
N o t e that this makes the s p t - s f t splitting bigger and therefore goes in the direction to reduce the configuration mixing, relative to a calculation in which experimental single-particle energies are used. It turns out that the changes in the wave functions are small.
0f7/2
a)
B A R E SINGLE PARTICLE ENERGY
b)
IP-Of5/2
PART OF "EXPERIMENTAL" SINGLE PARTICLE ENERGY THAT SHOULD BE REMOVED
Fig. 2. Modification of the experimental single-particle energies to avoid double counting.
Although we are interested in 54Fe, we can just as well do the calculation in S4Co - the T --- 1 states will be analogues of the states in S4Fe. We have p e r f o r m e d a diagonalization, not only for J = 2 T = 1 but also for J = 0 T = 1, J = 1 T = 0, J=7T=0. The results o f the matrix diagonalization are as follows: F o r all the spins considered J = 0 +, 1 +, 2 ÷, 7 + the lowest state still remained a basically 2h state. F o r example, the wave function o f the 0 + state was ~b = 0.95f~- 2 + ~/,.ut~** excitation-[- I/fprotonexcitation, with ~/.= = - 0 . 2 0 1 p ÷ [ 2 i - 1]+1° -0.091p~l47j-']~[ ° + 0 . 0 9 [ f t [ 2 i - ']~1 ° - 0.121f~14~- 1]~1°
ep,
+ 0.121f~E6V13~1°-0.071p~[4~-1]~1°, = 0.031p~1-2i- 1]~1° + 0.051f~E2V'3'1 °,
where I~Jp[Z~-l]~°lJ is a normalized state, with jp labelling the particle, L is the angular m o m e n t u m o f the two p r o t o n holes, Jo is the angular m o m e n t u m of the three holes and J to total angular m o m e n t u m of the state. Let us note here that although our basis functions do not have good isobaric spin, our total wave functions do because we are using a large enough space and a charge independent force. The amplitude of f2 z was 0.92 for J = 2, 0.93 for J = 7 and 0.92 for J --- 1. Thus, there is no "collective" 3 h - l p state which comes d o w n very close to a 2h state of the same spin. The energy level d i a g r a m is listed in fig. 3. The lowest four states J = 0, 7, 1 and 2 can obviously be associated with the four lowest calculated states, although one
STATIC MOMENTS
197
must admit that the J = 1 and J 7 states do not fit to well. Better results are obtained with other realistic interactions. =
5 ~
0
~
0
0
-! m
2
~
2
~ ~
7 0
-2
m
-5 TWO HOLES.
THREE HOLESI PARTICLE
0 t
COMPLETE 2H PLUS 3H-- IP.
EXPERIMENT
54Fe ond54Co SUPERIMPOSED
Fig. 3. The energy levels of 54Co and 54Fe in the space of 2h and 3h-lp. There is a triplet o f T = 1 levels found in S4Fe by Belote et aL 5) at about 2.5 MeV, two of the spins being J = 4 and J -- 0, but the third member has not been identified. From energy level systematics it would not be surprising if this were a J = 2 state. The J = 4 level fits in with the two-hole assignment but what about the J - 0 and J = 2 states? To answer this it is more reasonable to compare the difference between the unperturbed 3h-lp spectrum and the unperturbed two-hole spectrum. The reason for not using the "complete" spectrum is that in this spectrum the lowest states have been pushed down due to 3h-lp admixture, but the basically 3h-lp states have not correspondingly been pushed down by 4h-2p admixtures. From such a comparison we can definitely rule out a 3h-lp assignment to the 2.5 MeV 0 ÷ state; the calculated position is simply too high. However, an assignment 3h-lp J --- 2 to the alleged 2 + member of the triplet is possible, although in the calculation it comes 0.5 MeV too high. A quite possible state of affairs to the 2.5 MeV triplet in S4Fe is: J = 0
basic configuration
4h-2p,
J = 2
basic configuration
3h-lp,
J = 4
basic configuration
2h.
The 4h-2p assignment for J = 0 can be made plausible as follows: We picture this
198
L. ZAMICK
state as being formed by exciting two neutrons from the f~ into the p~ shell, thus having a state of the type 52Fe x SSNi. Using a monopole particle-hole interaction a + b t l • t2 of Bansal and French 6, 7), the excitation energy of the state will be E* = E ( S 2 F e ) - E ( 5 4 F e ) + E ( S S N i ) - E ( S 6 N i ) + 8 a = 1.6 M e V + 8 a . Now a is minus the center of gravity of the p~f~ interaction
E
(2J
+ 1)(2T + l)<['pl fl]~r V[p~ f.l.]~r>
,/T
E (2d + 1)(2T + 1) dT
Using the Kallio-Kolltveit 4) interaction we find a = 0.45. This would put the 4h-2p state much too high; E* = 5.2 MeV. This is a common feature of all realistic interactions. However, a smaller "value of a can be justified, by noting that this parameter enters in a calculation of the change in single-particle energies in going from 4~Ca to 56Ni. With 4°Ca as a core the p~f~ single-particle splitting as taken from experiment is 2.1 MeV; with 56Ni as a core the corresponding value is 4.5 MeV. This change can be calculated by considering the interaction of first a p~ nucleon and then an f÷ nucleon with sixteen f÷ nucleons in the core of S6Ni. The p~ interaction will be proportional to the parameter a. It was noted by Sartoris and Zamick s) that the realistic interactions did not increase the p~f~ splitting enough. This was noticed also by McGrory et al. 9) who modified the p~f~ interaction in T = 1 states from the realistic value. This change essentially ensured that the change in singleparticle energies in going from 4°Ca to 4SCa or to 56Ni would come out correctly. Thus, their T = 1 p~f~ matrix elements for J = 2, 3, 4 and 5 are - 0 . 5 6 , 0.25, 0.28 and 0.49 MeV. These should be compared with our Kallio-Kolltveit 4) matrix elements which are - 0 . 8 4 , 0, - 0 , 3 5 , 0. With the McGrory-Wildenthal modification the parameter a now becomes 0.115 and the energy of the 4h-lp J = 0 state in 54Fe becomes E* = 1.6 +0.92 = 2.5 MeV, in excellent agreement with experiment. In summary, if we are permitted to use a particle-hole interaction which determines our single-particle energies correctly, then we can understand why the many particlemany hole (deformed) states and in particular the J = 0 4h-2p state in 54Fe, come down so low. Note that our J = 0, J = 2, J = 4 configuration assignment is quite different from the most simple vibrational picture, which would identify all three states as two phonon excitations. It should be obvious from the large p ~ - f ~ splitting that the vibrational picture will be very bad in this region. We now turn to our original concern, the magnetic moment of the J = 2 + state. The calculated value of the moment is now 0, 3.31, - 0 . 3 4 , 2.97 nuclear magnetons, not very far from the Schmidt value. Let us now briefly summarize all the calculations
STATIC MOMENTS
199
of the magnetic moment: (i) (ii) (iii) (iv) (v)
Schmidt value quasiparticle approximation and experimental value for 55Co quasiparticle approximation and perturbation theory for 55Co perturbation theory for 54Co matrix diagonalization for 5~Co
3.31
Schmidt # 0
2.4
0.9
2.65 2.83 2.97
0.66 0.48 0.34
Note that in matrix diagonalization one gets even less quenching than in perturbation theory. This is universally true in such situations where one has nucleons or holes adjacent to closedjj but not L S closed shells e.g. 29Si, 55C0, 2°9Bi [ref. lo)]. In all these cases, first-order perturbation theory does not yield enough quenching, but higher-order corrections make matters worse. One reason for this is that the energy of the giant magnetic goes up several MeV relative to its unperturbed position, in a particle-hole calculation. At any rate we definitely predict a much larger magnetic moment than is given by the rotational model, and we feel that a measurement of this moment would be of great importance.
4. Other remarks concerning spherical or deformed pictures for this region A comparison o f the J = 7 - J = 0 splitting in 54C0 (2 holes) and 425c (2particles). M a t r i x diaoonalization versus perturbation theory. If the f~ model were perfect then
the energy spectrum of 42Sc and 54Co would be identical, since the interaction of two holes is equal to that of two particles. However, in 42Se the J = 0 level is 600 keV lower than the J = 7 whereas in 5 ,Co it is only 200 keV lower. One obvious mechanism which could cause this difference is the admixture in 54Co of 3h-lp, of the type that we had considered. We can now simply look at fig. 3 and compare the J = 0, J = 7 splitting in the two-hole spectrum and the complete spectrum. In order to explain the effect we would expect the J = 7 in the complete spectrum to be lower relative to J = 0, than it is for two holes (never mind the absolute spectra in the two cases for this particular point). However one notes that there is essentially no difference in the two splittings. This is disappointing, but there is a redeeming feature - things are much worse in perturbation theory. To show this we write down the sum of squares of the 2h-(3h-lp) matrix elements S = ~ah-1 p[(2h V 3p-lh)12 S s=° = 8.59 (MeV) 2, S s=7 = 5.40 (MeV) 2.
This means that in perturbation theory the J = 0 will get depressed more than the
200
L. ZAMICK
J = 7 due to 3h-lp admixture - the number being 1.55 and 0.98 MeV respectively, hence a net of 575 keV in the wrong direction. It is good that most of this 575 keV gets wiped out by matrix diagonalization. Note that the 3h-lp admixture consists in part of a core renormalization of the hole-hole interaction. The fact that for J = 0 this is large in perturbation theory but essentially zero in matrix diagonalization, is closely related to the Barrett-Kirson effect 11). These authors found that for J = 0 states of two particles in the 2s-ld shell, the large core polarization of the two-nucleon interaction, which results from a second (lowest possible) order perturbation theory gets wiped out in third order perturbation theory. The fact that in our example, the effect persists in matrix-diagonalization, which includes effects well beyond third order, indicates that the Barrett-Kirson 11) effect is a real effect, it will not be cancelled in higher orders. It is amusing to note that this effect must be buried in several shell-model calculations similar to the one carried out here, and could probably have been found earlier. Other calculations concerning 54Fe. A Hartree-Fock calculation was performed by Parikh and Svenne 12) through the entire 2p-lf shell. They obtained a deformed solution for 54Fe but they caution that the Hartree-Fock solution may be unstable to pairing effects. It was shown by Chandra 13), that whereas a Haltree-Fock calculation does give a deformed solution for SgFe, a Hartree-Fock Bogoliubov calculation which incorporates the pairing effects, restores the spherical symmetry. Clearly, the situation is too delicate to immediately accept or even to reject the above two possibilities. An operator way to check the Hartree-Fock solutions as was suggested by Parikh and Svenne 12), is to study the ground state occupation probabilities. Whereas for a spherical S4Fe ground state the f~ pickup strength for a neutron should be 8, of which ~ x 8 is to T = ½ states of S3Fe a n d ½ x 8 to the T = states, for the Hartree-Fock solution obtained by the above authors the strength is 5.1 for a prolate deformation and even less 4.6 for an oblate deformation. There should also be a large p~} pickup strength S = 1 for prolate and 1.7 for oblate. It should be further noted that in the extreme deformation limit for both 54Fe and 53Fe, the largest ground state to ground state spectroscopic factor is two, which is the number of neutrons in a single Nilsson orbit. The results of a neutron pickup experiment 54Fe(He3, ~)53Fe are reported in an MIT Physics Progress by Rapaport, Belote and Dorenbush 14). Their results definitely support a spherical ground state for 54Fe. They get a value C2S = 4.68 to the ground state of 54Fe and see strength to some ½- excited states with T = ½ so that the sum rule 5.33 is nearly exhausted. The ~ - (l = 1) strength is at most 0.16 according to their results. They also get the sum rule of 2.67 for the T = ½ isobaric analogue state. The spectroscopic factor to the ground state of SaFe obtained with MBZ 2) f~ model is 4.62, which is almost identical with the above experimental value. Thus, this experiment as well as our calculations are in favor of a spherical picture for the ground state of 54Fe. Our calculations also support a spherical description of the 2~- state and we eagerly await experiments which will prove or disprove this.
STATIC MOMENTS
201
I would like to thank P. G o o d e and M. Kirson for comments. I would like to thank I. Talmi and the Weizmann Institute for their hospitality. References I) O. Hausser, D. Pelte, T. K. Alexander and H. C. Evans, Montreal Conf. contributions, 4.7, 94 (1969), and preprint 2) J. D. McCullen, B. F. Bayman and L. Zamick, Phys. Rev. 134 (1964) B515 3) D. E. Murnick, J. R. MacDonald and H. P. Lie, Bull. Am. Phys. Soc., New York meeting (Jan., 1969); R. Kalish, private communication 4) A. Kallio and K. Kolltveit, Nucl. Phys. 53 (1969) 87 5) T. A. Belote, W. E. Dorenbush and O. Hansen, Nuclear-spin parity assignments, ed. N. B. Gove (Academic Press, New York, 1966) p. 350; D. J. Church, R. N. Horoshko and G. E. Mitchell, Phys. Rev. 160 (1967) 894 6) R. K. Bansal and J. B. French, Phys. Lett. 11 (1964) 145 7) L. Zamick, Phys. Lett. 19 (1965) 580 8) G. Sartoris and L. Zamick, Phys. Rev. 167 (1968) 1035 9) J. B. McGrory, B. H. Wildenthall and E. C. Halbert, Phys Rev. C2 (1970) 186 10) H. A. Mavromatis and L. Zamick, Nucl. Phys. A104 (1967) 17; H. A. Mavromatis, L. Zamick and G. E. Brown, Nucl. Phys. 80 (1966) 545 11) B. R. Barrett and M. W. Kirson, preprint 12) J. C. Parikh and J. P. Svenne, Phys. Rev. 174 (1969) 1343 13) H. Chandra, Phys. Rev. 185 (1969) 1320 t4) U. Rapaport, T. A. Belote and W. E. Dorenbush, Physics Progress rep., (Dec. 31, 1968) MIT 2098-540 15) P. Goode, to be published -
I I
Nuclear Physics A154 (1970) 191--201; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
STATIC MOMENTS
OF 2+ STATES
IN LIGHT DOUBLY
EVEN NUCLEI
LARRY Z A M I C K t
Department of Nuclear Physics, Weizmann Institute of Science, Rehovot, Israel *t Received 14 January 1970 (Revised 21 April 1970) Abstract: Quite general arguments indicate that if the shell-model picture holds or even if one allows for effective charge corrections, then the quadrupole moment of the first excited state of 48Ti should be small, and the magnetic moment of the first excited state of 54Fe should be large, compared to the rotational value. These results are contrary to recent experiments. A comparison is made between perturbation theory and matrix diagonalization in the space two holes and three holes and one particle, for s 4Fe and s 4Co" One gets less magnetic quenching with matrix diagonalization in 54Fe (hence the deviation from experiment is more) than with perturbation theory. The possibility that the lowest 2 + state in S4Fe is mostly a 3p-lh state is ruled out. Corrections to the quasiparticle approximation i.e. the use of the magnetic moment of one hole (55Co) to calculate the magnetic moment two holes (S4Fe), are calculated, and again tends to reduce the quenching. The J = 7 J = 0 splitting of two holes - 54Co - is less than that of two particles - 42Sc - by 400 keV but in perturbation theory the admixture of 3h-lp in 54Co goes in the wrong direction by 600 keV. It is shown however that with matrix diagonalization the J = 0 state of 54Co gets pushed down much less than in perturbation theory, so that this 600 keV additional discrepancy is erased. A triplet of levels at 2.5 MeV in 54Fe is analyzed to consist of a (mostly) 2h J = 4 state, a 3h-lp particle J = 2 state, and 4h-2p J = 0 state. A bare realistic interaction puts the 4h-2p state much too high in energy but an interaction chosen to fit single-particle energy shifts gives excellent agreement.
1. I n t r o d u c t i o n M o s t m e a s u r e m e n t s o f electric q u a d r u p o l e m o m e n t s a n d magnetic m o m e n t s of e x c i t e d s t a t e s o f n u c l e i h a v e , a t t h e t i m e o f t h i s w r i t i n g , b e e n c a r r i e d o u t in h e a v y nuclei, b u t s o m e r e c e n t m e a s u r e m e n t s h a v e b e e n c a r r i e d o u t i n t h e lf~ shell r e g i o n . P a r t l y i n t h e h o p e o f s t i m u l a t i n g f u r t h e r w o r k i n t h i s r e g i o n , w e shall d i s c u s s t w o e x a m p l e s b o t h c f w h i c h a r e e v i d e n c e a g a i n s t a s i m p l e f~ p i c t u r e .
On leave from Rutgers University with the support of a Rutgers University Fellowship and a Weizmann Institute Senior Fellowship. tt Supported in part by the National Science Foundation. 191
192
L. ZAblICK 2. The quadrupole moments of the 2 + state of 4STi
It was reported by Hausser et al. 1), in the 1969 Montreal Conference contributions, that the quadrupole moment of the 2 + state of 4aTi had a value Q = -0.22__.0.08 b. They note that this value is large, in the sense that if one assumes a rotational formula then the value of the intrinsic quadrupole moment Qo is consistent with a value obtained from their measured value of the B(E2, 0 + ~ 2 + ) = 690___60 e 2- fm 4. Let us discuss the above quadrupole moment from the shell-model point of view. Allowing for effective charges, the quadrupole operator can be written as a sum of an isoscalar and isovector term O ° + O 1 0 0 = ½(ep+e.)
r2y 2,
~ all
particles
01 = --½(%--en) ~
r 2Y2~z,
all
particles
where ~z = + 1 for a neutron and - 1 for a proton. In the f~ model 48Ti consists of 2 protons and 2 neutrons holes in the f~ shell so the wave function can be written
~b' = Z C(LpL,)[Pf~2LfC ZLn]t, with C(Lp, Ln) the probability amplitude that the protons couple to Lp and the neutrons to L . . Assuming charge symmetry, and realizing that the interaction of two holes equals the interaction of two particles, it is easy to show that C(L,, Lp) -+__C(Lp, L,). For example, the following wave functions were obtained 2) for the 0 + and 2+ states of 48Ti in the f~ model in terms of the basis states [Lp L.] I J=
01
J = 21
~ = 0.91 [001 - 0 . 4 1 [22] - 0 . 0 2 [44] +0.01 [661, ~O = 0.69 [02] +0.17 [42]
- 0 . 6 9 [201 - 0 . 1 7 [24] - 0 . 0 1 [46] +0.01 [641.
Note that the J = 0~" state necessarily does not change sign under the interchange of protons and neutron holes. It turns out that the J = 2 + state does change sign under such an interchange, as seen above, but that the J = 2~- state does not. The J = 2+ state is said to have negative signature; states that do not change sign are said to have
positive signature. Using the fact that the quadrupole moment of a particle is minus that of a hole we can derive the following selection rule: The matrix element of the isoscalar part of the quadrupole operator 0 ° vanishes
STATIC MOMENTS
193
between states of the same signature, the matrix element of the isovector part O 2 vanishes between states of the opposite signature. Thus the E2 transition J = 0 + ~ J = 2~- will be proportional to ½(en + ep) whereas the quadrupole moment of the 2 + state (which can be regarded as a transition between two states of the same signature) will be proportional to ½(ep-e,). In the strict shell model ep = 1, e. = 0. To fit the transitions and quadrupole moments effective charges are required. It always turns out, both empirically and theoretically that the effective charges are such that ep > 1, en > 0 and in the f~ region en > ( e p - 1 ) i.e. the effective charge of the neutron is if anything, somewhat greater than the effective charge correction of the proton. Theoretically, the use of effective charge is in part equivalent to adding perturbative corrections to the shell-model wave functions. For example in the case of 4STi it is equivalent to improving the f~ wave functions via admixtures of the type f~ (PH), where P H the particle hole excitation consists of both An = 0 pairs of0f~ ~ 0f~, 0f~ 1 lp~ and An = 2 pairs such as 0f~- 1 0h,~, 0p~-1 0f~, 0d~ 1 0g~ etc. Both the An = 0 and An = 2 excitations are important. The point we wish to make is that such effective charge renormalizations as above will enhance the 0~ ~ 2,. transition, but will not collectively enhance the quadrupole moment of the 2 + state. This is clear from the fact that any collective enhancement of ep or e, will add coherently in the isoscalar part ½(ep + en), but will cancel in the isovector part ½(ep-e.). With the above f~ wave function of the 2 + state of 4STi we find Q/e = ( % - e ~ ) x 0.030 b which is not only smaller in magnitude but also the wrong sign. No reasonable manipulation of ep and en can get us to the Hausser et al. value. Note: I have been informed by Goode ~5) t that in perturbation theory there are effects beyond effective charge renormalization which can explain the discrepancy.
3. The magnetic moment of the 2+ state of S4Fe The magnetic moment of a proton or proton hole in the f~ shell is 5.79 n.m. This should be the value of the { - ground state of 55Co, but in actual fact is 4.2. This discrepancy may perhaps be explained by perturbative mixtures of the configuration f~ f~-2 into the ground state. Indeed a first order calculation was performed by Mavromatis et aL 10), with the result p(55Co) = 4.7. The correction was in the right direction but there was still a discrepancy. Whether this remaining discrepancy is significant or not is hard to say i.e. perhaps changing the two-body force or singleparticle energies might make a difference. It should be noted though that if a matrix diagonalization is performed in the space of lh and 2p-lh - this should be superior to perturbation theory - one obtains a smaller correction to the moment. Nevertheless the situation is somewhat marginal. t Evidently diagrams like fig. 16 which represent a breakdown of the effective charge idea are very important for quadrupole moments of 2~ states.
194
L. ZAMiCK
We now consider the 2~ state of 26Fe2a, 24 where we would like to point out that there should be a large difference between the shell-model and rotational-model prediction of the moment. The rotational formula It = 2 # R ,
gR = Z / A ,
yields a magnetic moment equal to 0.96 n.m. With the shell model the magnetic moment of the 2~- state of SaFe should equal 2 - - x 5.79 = 3.31/~,. 3.5 This is much larger than the rotational formula. I f instead of using the Schmidt value/~ = 5.79, we use the magnetic moment of the one proton hole nucleus s 5Co _ this is called the q u a s i p a r t i c l e a p p r o x i m a t i o n we get 2 #(2~-) = - - x4.2 = 2.4/2,. 3.5 This is still far away from the rotational value. Hence a measurement of this moment should distinguish between the rotational and shell model. A recent abstract by McDonald, Murnick and Lie a) gives a very small value for this moment - even less than the rotational value. Apparently, some other measurements of this moment are in progress, the results of which will certainly be of great importance in deciding between these two models. We have performed some calculations involving perturbation theory and matrix diagonalization in the space of two holes and three holes and one particle, which we shall now describe. 3.1. BREAKDOWN OF THE QUASIPARTICLE APPROXIMATION To what extent is one justified in using the experimental moment of 55 Co (one hole) in order to calculate the moment of 54Fe (two holes)? Even in first order perturbation theory, there is a term, shown in fig. lb, which involves a breakdown of the quasiparticle (or effective charge) idea. In fig. l a we have a process where one of the two holes polarizes the core (and this obtains a quenched magnetic moment) independent of the second nucleon. To the extent that this term dominates the quasiparticle idea holds. But in fig. lb we have a process in which both nucleons are involved in polarizing the core. A very simple expression can be obtained for the correction to the magnetic moment due to this second process 5 # = --
10---0--( g , - - g ~ ) ((f~)t=2V(f~tf~)'=2),
7x/3 where gt = 1, g~ = 5.588.
AE
195
STATIC MOMENTS
The two-body matrix element is antisymmetrized but not normalized, and ls rather than sl coupling of the individual nucleon orbit is assumed. The energy denominator is negative and about - 6 MeV. With the Kallio-Kolltveit interaction ~) (with mco/h --- 0.27 fm -1) the two-body matrix element is - 0 . 3 4 6 MeV. Thus we
o) THIS TERM IN54Fe IS INCLUDED WHEN
b) BREAKDOWN OF EFFECTIVE OPERATOR
ONE USES THE EXPERIMENTAL MOMENT
CONCEPT IN FIRST ORDER
FROM 55Co
Fig. 1. First order corrections to the magnetic moment of the 2~+ state of SaFe. find for the contribution due to two holes acting together c5# = 0.22 Pn. This is about 35 ~o of fig. la and so not negligible. However, note that the contribution goes in the direction of reducing the quenching, thus causing a larger divergence between the rotational- and shell-model values. That 8/z is positive should not be surprising. It is a Pauli effect. The presence of one hole partly prevents the second hole from polarizing the core. 3.2, MATRIX DIAGONALIZATION THEORY We shall now consider a matrix diagonalization in space of two f~ holes, and of three f~ holes and one particle in the p~, f~ and P~r orbits. One reason for doing this is that there is always the possibility that a 3h-lp "collective state" has an energy comparable to the 2h state, thus mixing with it very strongly. We again use the Kallio-Kolltveit interaction 4) with mco/h = 0.27 fro-2. Let us now quote the single-particle energies obtained from experiment for a S6Ni core although we shall not use them without some modification: /3f.I_ = 0,
gp~_ = 4.5 MeV,
ef~ = 5.1 MeV,
ep~r = 5.7 MeV.
The modification we refer to is for purposes of avoiding double counting, q-he "experimental" energy of the f~ hole consists mainly of the interaction of an f~ hole with sixteen f~ nucleons. On the other hand, there is a small contribution because the hole can polarize the f~6 core exciting an fff nucleon into the rest of the fp shell. This is shown in fig. 2b. But such a core polarization introduces a 2h-lp configuration of the type which we will be taking into account explicitly. We should therefore use a single-hole energy which is the experimental value minus the contribution of 2h-lp to the experimental value. We have performed a perturbation theory calculation and we find that this can be achieved by simply changing the f~ hole energy to ef~ = - 0 . 6 0 MeV.
196
L. ZAMICK
N o t e that this makes the s p t - s f t splitting bigger and therefore goes in the direction to reduce the configuration mixing, relative to a calculation in which experimental single-particle energies are used. It turns out that the changes in the wave functions are small.
0f7/2
a)
B A R E SINGLE PARTICLE ENERGY
b)
IP-Of5/2
PART OF "EXPERIMENTAL" SINGLE PARTICLE ENERGY THAT SHOULD BE REMOVED
Fig. 2. Modification of the experimental single-particle energies to avoid double counting.
Although we are interested in 54Fe, we can just as well do the calculation in S4Co - the T --- 1 states will be analogues of the states in S4Fe. We have p e r f o r m e d a diagonalization, not only for J = 2 T = 1 but also for J = 0 T = 1, J = 1 T = 0, J=7T=0. The results o f the matrix diagonalization are as follows: F o r all the spins considered J = 0 +, 1 +, 2 ÷, 7 + the lowest state still remained a basically 2h state. F o r example, the wave function o f the 0 + state was ~b = 0.95f~- 2 + ~/,.ut~** excitation-[- I/fprotonexcitation, with ~/.= = - 0 . 2 0 1 p ÷ [ 2 i - 1]+1° -0.091p~l47j-']~[ ° + 0 . 0 9 [ f t [ 2 i - ']~1 ° - 0.121f~14~- 1]~1°
ep,
+ 0.121f~E6V13~1°-0.071p~[4~-1]~1°, = 0.031p~1-2i- 1]~1° + 0.051f~E2V'3'1 °,
where I~Jp[Z~-l]~°lJ is a normalized state, with jp labelling the particle, L is the angular m o m e n t u m o f the two p r o t o n holes, Jo is the angular m o m e n t u m of the three holes and J to total angular m o m e n t u m of the state. Let us note here that although our basis functions do not have good isobaric spin, our total wave functions do because we are using a large enough space and a charge independent force. The amplitude of f2 z was 0.92 for J = 2, 0.93 for J = 7 and 0.92 for J --- 1. Thus, there is no "collective" 3 h - l p state which comes d o w n very close to a 2h state of the same spin. The energy level d i a g r a m is listed in fig. 3. The lowest four states J = 0, 7, 1 and 2 can obviously be associated with the four lowest calculated states, although one
STATIC MOMENTS
197
must admit that the J = 1 and J 7 states do not fit to well. Better results are obtained with other realistic interactions. =
5 ~
0
~
0
0
-! m
2
~
2
~ ~
7 0
-2
m
-5 TWO HOLES.
THREE HOLESI PARTICLE
0 t
COMPLETE 2H PLUS 3H-- IP.
EXPERIMENT
54Fe ond54Co SUPERIMPOSED
Fig. 3. The energy levels of 54Co and 54Fe in the space of 2h and 3h-lp. There is a triplet o f T = 1 levels found in S4Fe by Belote et aL 5) at about 2.5 MeV, two of the spins being J = 4 and J -- 0, but the third member has not been identified. From energy level systematics it would not be surprising if this were a J = 2 state. The J = 4 level fits in with the two-hole assignment but what about the J - 0 and J = 2 states? To answer this it is more reasonable to compare the difference between the unperturbed 3h-lp spectrum and the unperturbed two-hole spectrum. The reason for not using the "complete" spectrum is that in this spectrum the lowest states have been pushed down due to 3h-lp admixture, but the basically 3h-lp states have not correspondingly been pushed down by 4h-2p admixtures. From such a comparison we can definitely rule out a 3h-lp assignment to the 2.5 MeV 0 ÷ state; the calculated position is simply too high. However, an assignment 3h-lp J --- 2 to the alleged 2 + member of the triplet is possible, although in the calculation it comes 0.5 MeV too high. A quite possible state of affairs to the 2.5 MeV triplet in S4Fe is: J = 0
basic configuration
4h-2p,
J = 2
basic configuration
3h-lp,
J = 4
basic configuration
2h.
The 4h-2p assignment for J = 0 can be made plausible as follows: We picture this
198
L. ZAMICK
state as being formed by exciting two neutrons from the f~ into the p~ shell, thus having a state of the type 52Fe x SSNi. Using a monopole particle-hole interaction a + b t l • t2 of Bansal and French 6, 7), the excitation energy of the state will be E* = E ( S 2 F e ) - E ( 5 4 F e ) + E ( S S N i ) - E ( S 6 N i ) + 8 a = 1.6 M e V + 8 a . Now a is minus the center of gravity of the p~f~ interaction
E
(2J
+ 1)(2T + l)<['pl fl]~r V[p~ f.l.]~r>
,/T
E (2d + 1)(2T + 1) dT
Using the Kallio-Kolltveit 4) interaction we find a = 0.45. This would put the 4h-2p state much too high; E* = 5.2 MeV. This is a common feature of all realistic interactions. However, a smaller "value of a can be justified, by noting that this parameter enters in a calculation of the change in single-particle energies in going from 4~Ca to 56Ni. With 4°Ca as a core the p~f~ single-particle splitting as taken from experiment is 2.1 MeV; with 56Ni as a core the corresponding value is 4.5 MeV. This change can be calculated by considering the interaction of first a p~ nucleon and then an f÷ nucleon with sixteen f÷ nucleons in the core of S6Ni. The p~ interaction will be proportional to the parameter a. It was noted by Sartoris and Zamick s) that the realistic interactions did not increase the p~f~ splitting enough. This was noticed also by McGrory et al. 9) who modified the p~f~ interaction in T = 1 states from the realistic value. This change essentially ensured that the change in singleparticle energies in going from 4°Ca to 4SCa or to 56Ni would come out correctly. Thus, their T = 1 p~f~ matrix elements for J = 2, 3, 4 and 5 are - 0 . 5 6 , 0.25, 0.28 and 0.49 MeV. These should be compared with our Kallio-Kolltveit 4) matrix elements which are - 0 . 8 4 , 0, - 0 , 3 5 , 0. With the McGrory-Wildenthal modification the parameter a now becomes 0.115 and the energy of the 4h-lp J = 0 state in 54Fe becomes E* = 1.6 +0.92 = 2.5 MeV, in excellent agreement with experiment. In summary, if we are permitted to use a particle-hole interaction which determines our single-particle energies correctly, then we can understand why the many particlemany hole (deformed) states and in particular the J = 0 4h-2p state in 54Fe, come down so low. Note that our J = 0, J = 2, J = 4 configuration assignment is quite different from the most simple vibrational picture, which would identify all three states as two phonon excitations. It should be obvious from the large p ~ - f ~ splitting that the vibrational picture will be very bad in this region. We now turn to our original concern, the magnetic moment of the J = 2 + state. The calculated value of the moment is now 0, 3.31, - 0 . 3 4 , 2.97 nuclear magnetons, not very far from the Schmidt value. Let us now briefly summarize all the calculations
STATIC MOMENTS
199
of the magnetic moment: (i) (ii) (iii) (iv) (v)
Schmidt value quasiparticle approximation and experimental value for 55Co quasiparticle approximation and perturbation theory for 55Co perturbation theory for 54Co matrix diagonalization for 5~Co
3.31
Schmidt # 0
2.4
0.9
2.65 2.83 2.97
0.66 0.48 0.34
Note that in matrix diagonalization one gets even less quenching than in perturbation theory. This is universally true in such situations where one has nucleons or holes adjacent to closedjj but not L S closed shells e.g. 29Si, 55C0, 2°9Bi [ref. lo)]. In all these cases, first-order perturbation theory does not yield enough quenching, but higher-order corrections make matters worse. One reason for this is that the energy of the giant magnetic goes up several MeV relative to its unperturbed position, in a particle-hole calculation. At any rate we definitely predict a much larger magnetic moment than is given by the rotational model, and we feel that a measurement of this moment would be of great importance.
4. Other remarks concerning spherical or deformed pictures for this region A comparison o f the J = 7 - J = 0 splitting in 54C0 (2 holes) and 425c (2particles). M a t r i x diaoonalization versus perturbation theory. If the f~ model were perfect then
the energy spectrum of 42Sc and 54Co would be identical, since the interaction of two holes is equal to that of two particles. However, in 42Se the J = 0 level is 600 keV lower than the J = 7 whereas in 5 ,Co it is only 200 keV lower. One obvious mechanism which could cause this difference is the admixture in 54Co of 3h-lp, of the type that we had considered. We can now simply look at fig. 3 and compare the J = 0, J = 7 splitting in the two-hole spectrum and the complete spectrum. In order to explain the effect we would expect the J = 7 in the complete spectrum to be lower relative to J = 0, than it is for two holes (never mind the absolute spectra in the two cases for this particular point). However one notes that there is essentially no difference in the two splittings. This is disappointing, but there is a redeeming feature - things are much worse in perturbation theory. To show this we write down the sum of squares of the 2h-(3h-lp) matrix elements S = ~ah-1 p[(2h V 3p-lh)12 S s=° = 8.59 (MeV) 2, S s=7 = 5.40 (MeV) 2.
This means that in perturbation theory the J = 0 will get depressed more than the
200
L. ZAMICK
J = 7 due to 3h-lp admixture - the number being 1.55 and 0.98 MeV respectively, hence a net of 575 keV in the wrong direction. It is good that most of this 575 keV gets wiped out by matrix diagonalization. Note that the 3h-lp admixture consists in part of a core renormalization of the hole-hole interaction. The fact that for J = 0 this is large in perturbation theory but essentially zero in matrix diagonalization, is closely related to the Barrett-Kirson effect 11). These authors found that for J = 0 states of two particles in the 2s-ld shell, the large core polarization of the two-nucleon interaction, which results from a second (lowest possible) order perturbation theory gets wiped out in third order perturbation theory. The fact that in our example, the effect persists in matrix-diagonalization, which includes effects well beyond third order, indicates that the Barrett-Kirson 11) effect is a real effect, it will not be cancelled in higher orders. It is amusing to note that this effect must be buried in several shell-model calculations similar to the one carried out here, and could probably have been found earlier. Other calculations concerning 54Fe. A Hartree-Fock calculation was performed by Parikh and Svenne 12) through the entire 2p-lf shell. They obtained a deformed solution for 54Fe but they caution that the Hartree-Fock solution may be unstable to pairing effects. It was shown by Chandra 13), that whereas a Haltree-Fock calculation does give a deformed solution for SgFe, a Hartree-Fock Bogoliubov calculation which incorporates the pairing effects, restores the spherical symmetry. Clearly, the situation is too delicate to immediately accept or even to reject the above two possibilities. An operator way to check the Hartree-Fock solutions as was suggested by Parikh and Svenne 12), is to study the ground state occupation probabilities. Whereas for a spherical S4Fe ground state the f~ pickup strength for a neutron should be 8, of which ~ x 8 is to T = ½ states of S3Fe a n d ½ x 8 to the T = states, for the Hartree-Fock solution obtained by the above authors the strength is 5.1 for a prolate deformation and even less 4.6 for an oblate deformation. There should also be a large p~} pickup strength S = 1 for prolate and 1.7 for oblate. It should be further noted that in the extreme deformation limit for both 54Fe and 53Fe, the largest ground state to ground state spectroscopic factor is two, which is the number of neutrons in a single Nilsson orbit. The results of a neutron pickup experiment 54Fe(He3, ~)53Fe are reported in an MIT Physics Progress by Rapaport, Belote and Dorenbush 14). Their results definitely support a spherical ground state for 54Fe. They get a value C2S = 4.68 to the ground state of 54Fe and see strength to some ½- excited states with T = ½ so that the sum rule 5.33 is nearly exhausted. The ~ - (l = 1) strength is at most 0.16 according to their results. They also get the sum rule of 2.67 for the T = ½ isobaric analogue state. The spectroscopic factor to the ground state of SaFe obtained with MBZ 2) f~ model is 4.62, which is almost identical with the above experimental value. Thus, this experiment as well as our calculations are in favor of a spherical picture for the ground state of 54Fe. Our calculations also support a spherical description of the 2~- state and we eagerly await experiments which will prove or disprove this.
STATIC MOMENTS
201
I would like to thank P. G o o d e and M. Kirson for comments. I would like to thank I. Talmi and the Weizmann Institute for their hospitality. References I) O. Hausser, D. Pelte, T. K. Alexander and H. C. Evans, Montreal Conf. contributions, 4.7, 94 (1969), and preprint 2) J. D. McCullen, B. F. Bayman and L. Zamick, Phys. Rev. 134 (1964) B515 3) D. E. Murnick, J. R. MacDonald and H. P. Lie, Bull. Am. Phys. Soc., New York meeting (Jan., 1969); R. Kalish, private communication 4) A. Kallio and K. Kolltveit, Nucl. Phys. 53 (1969) 87 5) T. A. Belote, W. E. Dorenbush and O. Hansen, Nuclear-spin parity assignments, ed. N. B. Gove (Academic Press, New York, 1966) p. 350; D. J. Church, R. N. Horoshko and G. E. Mitchell, Phys. Rev. 160 (1967) 894 6) R. K. Bansal and J. B. French, Phys. Lett. 11 (1964) 145 7) L. Zamick, Phys. Lett. 19 (1965) 580 8) G. Sartoris and L. Zamick, Phys. Rev. 167 (1968) 1035 9) J. B. McGrory, B. H. Wildenthall and E. C. Halbert, Phys Rev. C2 (1970) 186 10) H. A. Mavromatis and L. Zamick, Nucl. Phys. A104 (1967) 17; H. A. Mavromatis, L. Zamick and G. E. Brown, Nucl. Phys. 80 (1966) 545 11) B. R. Barrett and M. W. Kirson, preprint 12) J. C. Parikh and J. P. Svenne, Phys. Rev. 174 (1969) 1343 13) H. Chandra, Phys. Rev. 185 (1969) 1320 t4) U. Rapaport, T. A. Belote and W. E. Dorenbush, Physics Progress rep., (Dec. 31, 1968) MIT 2098-540 15) P. Goode, to be published -