Microscopic approach to the interacting boson model

Nuclear

0

Physics

North-Holland

A419 (1984) 241-260 Publishing

Company

MICROSCOPIC APPROACH TO THE INTERACTING BOSON (II). Extension to neutron-proton systems and applications

MARTIN Max-Plan&-Insfitut

MODEL

R. ZIRNBAUER

,fiir Kernphysik. Heidelberg, k’RG

Received

31 October

1983

Abstract: Earlier work on the microscopic foundation of the interacting boson model is extended to neutron-proton systems. and a numerical application to the Sm isotopes is given. The calculated excitation energies turn out to be systematically larger than the experimental values. but good agreement is obtained for some qualitative features such as level orderings and the character of the spectrum. Renormalization effects due to the coupling with hexadecupole degrees of freedom (g-boson) are estimated and found to be rather small. The validity of the IBA truncation scheme is discussed in detail.

1. Introduction In recent years. the interacting boson model of Arima and Iachello has found wide application in the description of collective properties of even-even nuclei. Good fits to excitation energies and electromagnetic transition probabilities have been obtained for many regions of the periodic table ‘). Parallel to these phenomenological studies, much effort has gone into developing a microscopic theory that is capable of relating the parameters of the IBA hamiltonian to the nuclear single-particle energies as known from experiment, and an effective shell-model NN interaction. The connection with microscopic quantities is thought to be most direct for the so-called IBA-2 where one distinguishes between neutron and proton bosons. The neutron (proton) s-boson is put into correspondence with the collective neutron (proton) S-pair (the Cooper pair of BCS theory), while the neutron (proton) d-boson is associated with a neutron (proton) L = 2 excited mode (D-pair). A standard procedure for obtaining the boson image of the truncated shell-model hamiltonian is then to equate certain matrix elements in the shell-model space with the corresponding matrix elements in the boson Hilbert space. Such a method, now generally referred to as the OAI mapping, was first proposed by Otsuka, Arima and Iachello2) for the case of a single j-shell (or several degenerate shells). Using the formalism of the broken-pair approximation, various groups have extended the ideas of OAI to nondegenerate systems 3-5). By now, there exist several realistic calculations which show that the 241

242

M. R. Zirnbauer / Microscopic approach (II)

simple microscopic picture underlying the OAI approach can indeed account for certain systematic trends of the phenomenological parameters of the interacting boson model. Gambhir rt al. 3, investigated the dependence on neutron number (in the Sr isotopes) of the quadrupole interaction parameters xv and ti, and found that their dependence is similar to that seen in phenomenological fits. [However, no results for the single d-boson energies E: and E: were given in ref. “).I Similar, and more extensive, results of this kind can be found in a paper by Pittel, Duval and Barrett 5). We mention also that Van Egmond and Allaart4) have recently obtained IBA parameters which appear to be much larger than the empirical values. Although individual aspects of the interacting boson approximation have been explored in detail (and are, in fact, well understood), to our knowledge there exists as yet no published account of an IBA calculation which is complete, in the sense that it starts from a shell-model hamiltonian with realistic single-particle energies and an effective NN interaction, maps the truncated shell-model problem on a boson eigenvalue problem, and diagonalizes the boson hamiltonian to obtain excitation energies and electromagnetic transition rates. The present paper contains the first report of such a complete calculation for the Sm isotopes. Our work is based on the formalism of ref. ‘), which uses Dyson’s boson representation for mapping the (truncated) shell-model hamiltonian on a corresponding boson operator. Our formalism differs in several respects from the approach of OAI. The main advantage of using the Dyson mapping is that it terminates exactly in finite order, thereby yielding a boson hamiltonian of the IBA type without any need to neglect interaction terms of sixth and higher order in the boson Fock operators. This advantage is gained at the expense of having to work with a boson hamiltonian which is non-hermitian. The validity of the SD truncation scheme for strongly deformed nuclei has been called into question by Bts et al. ‘) and Maglione et al. *). These authors argue that the presence of G-pairs (L = 4) is essential for obtaining the correct value of such quantities as the quadrupole moment of the deformed intrinsic state. For deformed nuclei, pairing correlations become almost as strong as quadrupole correlations if the effects of G are neglected, whereas in the exact solution pairing plays a less significant role. Otsuka et al.9) rejected this criticism by showing that collective wave functions are dominated by SD components. However, we feel that “SD dominance” may not be sufficient for making the microscopic IBA (in its present formulation) successful in the calculation of observable quantities in rotational nuclei. We will therefore restrict our attention to vibrational and transitional nuclei. where the IBA truncation scheme is thought to be better justified. However, it will be seen that also for these nuclei, the coupling to other degrees of freedom cannot be neglected. The organization of this paper is as follows. In ref. 6, a method was described for constructing a non-hermitian IBA hamiltonian using Dyson’s finite boson

243

M. R. Zirnbauer / Microscopic approach (11)

representation.

In sect. 2, we extend

this method

to neutron-proton

mapping the E2 operator on a corresponding boson quadrupole validity of this mapping is discussed, and a prescription is given

systems,

by

operator. The for finding the

“optimal” D-pair when all degrees of freedom other than S and D are ignored. Results of numerical calculations for 14’Srn, 15’Srn and “‘Sm are presented in sect. 3. As one might expect, our results are worse for ls2Sm than for 148Sm and 15’Sm. due to the deformed character of the first nucleus. In sect. 4, renormalization effects due to G-pairs are considered and found to be rather small. Sect. 5 contains a detailed discussion of the difficulties encountered in the present work. Possible sources of renormalization other than the G-pair are suggested.

2. Dyson boson image of the quadrupole operator

In a previous paper6), we used Dyson’s finite boson representation to derive an IBA-like hamiltonian from the shell model. To keep the technical details of the derivation as transparent as possible, only systems with one kind of nucleon were discussed. However, the truncated sd-boson hamiltonian obtained in ref. 6, has a very limited physical meaning as long as it is restricted to single-closed shell nuclei. We know that the first excited 2+ state in such nuclei lies in the vicinity of many other states of two quasiparticle character. The fact that there is little separation indicates that the between the 2: state and other states with low seniority quadrupole degree of freedom plays no distinct role in semi-magic nuclei. On the other hand, nuclei with both neutrons and protons active display a rich variety of spectra whose main feature is an abundance of quadrupole collective states already below the pairing gap. Clearly, the complex E2 collectivity observed in nuclei is due to the interaction between neutrons and protons. The final step in our attempt to provide a link between the IBA and the shell model will be to understand how the strong quadrupole interaction between neutrons and protons can be mapped on a corresponding boson operator. The formal derivation given in ref. 6, was based on a combination of Dyson’s finite boson representation and the boson expansion of Marumori. In this way, it was possible to (approximately) remove the effects of spurious boson states, which are present for particle number n > 4. Although the Dyson representation is usually applied directly to the commutator algebra of ~+a+, au and a+~, the method of ref.“) is the correct procedure if only the collective part of the boson hamiltonian is retained, since it includes the (kinematic) renormalization effects due to bosons outside the collective subspace. However, for a one-body operator such as the E2 operator, Q,, both methods give the same result and we therefore use the simpler approach, which is to apply the Dyson representation directly. The boson image of fermion pair (creation and annihilation) operators has already been given

244 in

M. R. Zirnbauer refs,

IO. l1.h

1 Microscopic

approach

(II)

1.

4 + J’Gh,.

(2.lb)

[The notation used in the present paper is identical to that of ref. ’ ).] In addition to the boson image of pairs we now require the Dyson representation of a general particle-hole operator

It is given by ‘I.(‘) (2.3) where (2.4)

If the summation over CL[j etc. extends over a complete set of boson operators b,, then the boson mapping defined in eqs. (2.1) and (2.3) exactly preserves the commutator algebra of LI+O+. au and LI+U. In practice, however. an approximation has to be introduced by truncating the mapping in some way. Our present aim is to express the quadrupole operator Q, in terms of six collective boson operators s and d,,. The simplest approach, which we are going to follow here, is to retain only the sd part of the boson quadrupole operator in eq. (2.3) and neglect all other terms. In this way we obtain Q, -+

where the parameters

~o(d~.s+s+~~+~(d+d):2)) = k-,,QB+,

(2.5)

~~ and jl are given by

(2.6a)

x = -5(k_,s2)_’

c

-. .&!

jj,j,yJ~J~

121 {;,

;

:,)

‘:?

(2.6b)

and (2.7)

245

M. R. Zirnbauer / Microscopic approach (II)

It is interesting

to note that eqs. (2.6) can also be written ~0 =

as

&WV.

(2.8a)

i! = J;Ko (WQIIW,

(2.8b)

where IS) and ID) are states with one S- and one D-pair, respectively. In view of the central role in the IBA truncation scheme played by the boson quadrupole operator Q,, we now discuss the result (2.5) in more detail. First, let us state the conditions of validity of the approximations that lead to eq. (2.5). The truncation to s and d is exact if and only if the commutator [Q,, B+] lies in the SD subspace for any B+ which is a linear combination of S+ and 0:. Consider the case

B+ = S+.We can arrange for the corresponding

commutator

condition

[Q,, S+] E SD subspace to be satisfied

exactly.

by using our freedom

D; =

in the choice of

[Q,,S+].

(2.9

D to define (2.10

(This definition implies that D completely exhausts the E2 strength connecting S with all possible quadrupole pairs.) In the general case where D is determined from a different prescription (see below), several d-bosons contribute to the “senioritychanging” part of Q,. and then the term d,+s+s+li, in eq. (2.5) is only approximate. A second approximation we made in the derivation of eq. (2.5) was to neglect the coupling of D with pairs of higher angular momentum. This approximation is justified if pairing correlations are strong enough to make the formation of these pairs energetically unfavourable. Clearly, the effects of other pairs become more important as the pairing gap decreases (strongly deformed limit, high spins). A quantitative assessment of the effects of the G-pair in Sm will be made in sect. 4. An interesting feature of the Dyson image (2.5) of the quadrupole operator is that, in the absence of shell effects manifesting themselves as changes in the structure of S and D, the parameters tie and x remain constant. In contrast, the boson quadrupole operator obtained in the OAI approach explicitly depends on particle number ‘. 5). Further analysis shows that this difference is closely connected with the non-hermiticity of the Dyson mapping. In particular, the nonhermiticity serves to effectively weaken the parameter tie as the number of particles increases. We are now ready to address the problem which lies at the heart of any attempt to derive an IBA-like hamiltonian from the shell model: How should we choose

M. R. Zirnbauer 1 Microscopic approach (11)

246

the collective pairs S and D? To begin, it is clear that the need for a careful choice of collective pairs arises from the truncation of the shell-model space made in the derivation of the boson hamiltonian. If we had a fully renormalized theory at our disposal,

i.e. a theory

where effects of non-collective

states are property

taken

into

account, the choice of coflective boson would be largely immaterial because any error made in the definit.ion of S and D would be compensated by corresponding corrections due to renormalization. In practice we have to truncate in some way, and then an optimal choice of subspace is required in order to minimize the effects of states left out in the truncation. We need to determine S and D from a variational criterion which ensures that collective (SD) states lie lowest in the spectrum and, thus, are maximally separated from all other shell-model states. An interesting idea along these lines has been put forward by Klein and Vallieres I’). These authors propose to determine the collective bosons (pairs) by minimizing the trace of the hamiltonian in the corresponding subspace: &TrH)su

= 0.

(2.11)

We claim, however, that this approach meets with difficulties if one wants to stay within the framework of IBA-2, i.e. preserve the distinction between neutrons and protons. The problems one encounters are due to the off-diagonal character of the neutron-proton force. An attractive Q, Q, interaction strongly lowers states symmetric in neutrons and protons, but at the time pushes up states that are antisymmetric (or not fully symmetric). For this reason, application of the variational criterion (2.11) to the full subspace corresponding to IBA-2 will not select the “optimal” bosons. [We note that a similar criticism has recently been made by Pittel and Dukelsky “).] For the “minimal trace condition” (2.11) to be physicaIly meaningful, it is necessary to explicitly separate out the symmetric degree of freedom befbre carrying out the variation. In applications, such a separation will severely complicate the use of eq. (2.11). (Note, however, that the symmetric degree of freedom could be separated by going from IBA-2 to the less microscopic IBA-1 which makes no distinction between neutrons and protons.) We feel that a variational principle like (2.11) should be used together with some additiona input concerning the particular physical case in question. In vibrational nuclei a possible way of proceeding is the following. First, we determine the coefficients of S by minimizing the energy of the fully paired state [see eq. (6) of ref. 6, and also sect. 3 of the present paper], for neutrons and protons separately. This choice of S is on fairly safe ground because the importance of S derives mainly from the pairing interaction between like nucleons. Choosing the D-pair is more problematic, and no general consensus seems to exist as to how one should proceed. Two basic choices are available to us at this point: (a) Take D from a diagonalization of the identical nucleon interaction. (b) Maximize E2 transitions by putting Dlf= [Q,, S+].

M. R. Zirnbauer 1 Microscopic approach (II)

We note quadrupole

that

method

(b) emphasizes

the importance

force, while (a) gives the maximum

possible

347

of the neutron-proton weight

to the interaction

between like nucleons. However, we contend that neither of these two extreme that, in choices will be the optimal one in general +. Method (a) has a consequence large spaces, a considerable amount of E2 strength is lost in the truncation to Q, in eq. (2.5). Prescription (b), on the other hand, may lead to unreasonably large values for the single d-boson energies Ed_ In order to ensure that the spectrum calculated in the truncated space comes to lie as low as possible, we propose to minimize the energy of an approximate eigenstate of the shell-model hamiltonian. In the vibrational case the collective spectrum is built on the “one-phonon state”

12:) = (S,~)N~-‘D:(Sn+)N,IO)C,+(s,t )NG,‘)N~-‘D,ilO)cn.

(2.12)

The expectation value of the energy of this State depends quadratically on the coefficients. of I>,? and D:, and we can therefore determine these by diagonalizing the corresponding quadratic form. In order to derive this form we use the overlap approximation proposed in ref.6) rather than the Dyson hamiltonian. This is because we do not know how to apply the variational principle to a non-hermitian hamiltonian. The argument preceding eq. (2.12) requires further discussion. We have suggested that, for vibrational nuclei. minimization of the energy of the “onephonon state” (2.12) is the optimal way of resolving the ambiguity in the choice of collective D-pair. However, the very existence of this ambiguity points to a major difficulty encountered in the interacting boson approximation. In realistic cases where the “E2 boson”, defined by 1>: = [Q,. S+], differs substantially from the “Tamm-Dancoff boson”. defined by method (a). we cannot expect the truncation a single D,. (D,) pair, as implied by the simple interacting boson approximation, lead to a correct description of low-lying collective states. In such a situation

to to the

present prescription detines a certain compromise, but we should be aware that the IBA approach gets worse as the overlap decreases due to strong splittings in the single-particle energies. We consider this as one of the main reasons for the difficulties encountered in comparing the numerical results presented in sect. 3 with experiment.

+ We note that model interactions can be constructed where both methods yield the same result. Consider, for example, a surface-delta interaction between like m&eons (with degenerate single-particle energies) plus a Q. Q interaction between neutrons and protons. It is easy to show that the D-pairs obtained from methods (a) and (b) coincide in this case if we approximate the radial integrals (jl)r21j,) in the matrix elements of Q by a constant independent ofj, and j, [“surface quadrupole interaction” “)I.

M. R. Zimbauer 1 Microscopic approach (II)

248

3. Application In this section

to the Sm isotopes

we will use our formalism

to calculate

the excitation

spectra

of

some Sm isotopes (‘“%m, isoSm and iszSm). This choice of nuclei was motivated by the following considerations. In ref. ‘), it was shown that the Dyson mapping is exact for certain model hamiltonians with degenerate single-particle energies. An analysis of the projection approximation made in ref. 6 ). and also the discussion at the end of the preceding section. suggest that the method is best used in nuclei with a large effective single-particle degeneracy L&r‘. Furthermore, the method should only be used for particle number up to II 5 !& because for II > Qelf serious problems may arise, in general, due to coupling to spurious boson states. In Sm the neutron orbits hq and f; (g; and d+ for protons) are almost degenerate leading to a large effective space, which can easily accomodate l&l? (8-10) particles without a drastic change in the structure of the collective pairs S and D. Another advantage of applying our formalism to the Sm isotopes is that these nuclei have been studied in detail experimentally. Certain aspects of our numerical procedure have already been explained earlier but, for completeness, we now summarize the most essential points. In the first part of the calculation we select the collective SD subspace. The monopole pairs S, and S, are determined by minimizing the energy of the paired ground state, calculated in the number operator approximation (NOA) 14. ‘). This approximation has recently come into discredit 5. “), and it is therefore appropriate to discuss once again its limits of validity. In ref. 16) the NOA formula for the energy of the fully paired state IS”) was derived from a projection approximation. The conditions that guarantee the validity of this approximation were shown to be related to Talmi’s scheme of generalized seniority “). These conditions are much weaker than the conditions for the commutator approximation used in ref. 14) to be valid. The NOA scheme applied to the ground-state energy of a pairing-like hamiltonian is therefore a valid approximation even if the coefficients c(~ in S+ (S+ = x,i~iSf) depart noticeably from their SU(2) value crj = 1 or 0, for which the NOA is exact. However, no such general argument exists for other quantities like the average number of particles in a shell j ((nj)) and, for this particular example, the NOA may in fact violate the constraint (nj) < 2j+ 1. As was discussed earlier, the quadrupole pairs D,, and D, are obtained by diagonalization of the quadratic form corresponding to the energy of the onephonon state (2.12). Next, we map the truncated shell-model problem on a boson hamiltonian

H -+ H, = H,.,,+H,,+KQ;.Q;.

The neutron

(proton)

part

of this hamiltonian

was defined

(3.1)

in ref.6). The boson

249

M. R. Zirnbauer 1 Microscopic approach (II)

quadrupole

interaction

strength

r~ is given by

where GpQ is the strength of the shell-model quadrupole interaction. in the calculation is to solve the non-hermitian eigenvalue problem

in a complete

basis of sd-boson

states

The final step

14),

14)= IN,.,N,,n&n,".....LM).

(3.4)

1s calculated using a computer The matrix of the boson hamiltonian, (~IHsl#), program written by the author. (Without going into numerical details, we just mention that this code is somewhat similar to the Glasgow and Oxford shell-model codes in that it represents the boson wave functions in a basis of M-scheme states. The standard IBM codes available from KVI use c.f.p. techniques. and thus correspond more closely to the Oak-Ridge-Rochester shell-model code.) Due to the non-hermiticity of the boson hamiltonian H,, the eigenvectors defined by eq. (3.3) do not form an orthogonal set. The set of vectors dual to a+(R) is given by the left eigenvectors u,(L) of H, : xa;(L)afp(R) + The orthogonality

relation

(3.5) is invariant

= 6,,.

under

(3.5)

scale transformations

g&L) + sa)6(L), u;(R)+ s-'&,(R) and, therefore, determine the

solution correct

of the eigenvalue problem (3.3) does not suffice normalization factor s needed for the evaluation

(3.6) to of

electromagnetic transition probabilities. (Of course, this normalization factor is irrelevant for the calculation of excitation energies.) The correct normalization is difficult to obtain, in general, because this requires an explicit knowledge of the no results on E2 overlaps of SD shell-model states “- 16). For this reason, transition rates are given in the present paper. We now discuss the shell-model hamiltonian used in the present calculation. Our choice of single-particle space is to a large extent dictated by the approximations made in the derivation of the boson hamiltonian (3.1). In the mass region around A + 150, the neutron orbits h; and f; (Q and d+ for protons) are very close in

250

M. R. Zirnbauer / Microscopic approach (II)

energy, leading to a large effective single-particle degeneracy L&. Clearly, it wouid be desirable to include the vhy and rcgt hole orbits in the calculation. However, since these orbits are essentially closed they contribute only through particle-hole configurations like h&‘f, which cannot be incorporated into the present formalism in a simple way. On the other hand, inclusion of orbits higher up in energy does not affect the validity of our approximations, and we therefore took the complete 82-126 and 50-82 major shells for neutrons and protons, respectively. Since the single-particle energies for mass number A -+ 150 are not directly available from experiment, we decided to use those of Kumar’s calculation i8). For the interaction between like nucleons we took a surface-delta interaction (SDI). It is well-known that the J = 0 two-body matrix elements of an SD1 are identical to those of a pairing force. We could therefore have chosen the SD1 strength parameters as the pairing strengths used in ref. 18). However. these strengths were used together with two major shells for each kind of nucleon. As the present calculation is restricted to one major shell, we expect somewhat larger values for G,, and G, to be more appropriate. We decided to choose the SD1 strengths so as to reproduce the pairing gaps in ‘48Sm extracted from the experimental binding energies 19). The resulting values were G,, = 0.177 MeV and G, = 0.196 MeV. An independent check as to whether these interaction strengths are reasonable can be obtained by looking at the 2: excitation energy in ‘44Sm (IV = 82 closed shell). The empirical value is E(2:) = 1.66 MeV while our shell-model hamiltonian with G, = 0.196 MeV gives E(2:) = 1.71 MeV (obtained with the NOA). Finally, the interaction between neutrons and protons was taken to be a separable force of the quadrupole-quadrupole type. For reasons of consistency with the use of an SD1 between like nucleons, we approximated the radial integrals in the matrix elements of the quadrupole operator by a constant. This is a reasonable approximation since our single-particle space does not exceed one major shell. It also has the advantage of ensuring maxima1 coherence in the structure of the lowest excited 2’ state, lSNmlD), of the like nucleon system. (By this we mean that, for the schematic case of degenerate singie-particle energies, the state ISN-‘D) exhausts the entire E2 strength connecting the vibrational ground state lSN) with all other shell-model states.) The overall strength’of the neutron-proton interaction was determined by fitting the experimental ratio E(4~)/~(2~) in ““*Sm. This resulted in a value for G,, about 15% larger than the “self-consistent” value i9) Go, = 240A-‘i3 MeV. Again, such an increase is consistent with our choice of single-particle space (one major shell only). Originally we were planning to test our method by comparison with the pairing plus quadrupole model calculations of Kumar 18). However, this comparison failed quite dramatically, and not surprisingly so. Kumar modifies the kinetic energy terms in his collective hamiltonian in order to account for the moment of inertia of the “core”. In the early calculation for Pt and OS [ref. “)I, core contributions were added to the calculated mass parameters, while in Sm [ref. ‘“)I all inertial functions

M. R. Zirnbauer / Microscopic approach (II)

251

is quite large were multiplied with a constant factor “F,“. This renormalization (e.g. in “‘Srn, F, = 4.7) and serves to rescale the calculated spectrum in order to achieve agreement with experiment. [The resealing of inertial functions adopted by Kumar has been called into question by Pomorski et al. 20). These authors calculated core contributions to the kinetic energy using the cranking model and found that core effects are rather small. They concluded that the large renormalization of mass parameters needed to fit experimental data is due to interactions not included in the model.] It is not u priori clear how to separate the boson hamiltonian in eq. (3.1) into potential and kinetic energy, and we therefore find it hard to see how a similar modification could be made in the present approach. The above discussion already points to the major problem encountered in the present work, namely a systematic difference in scale between theoretical and experimental spectra. For the particular interaction strengths quoted earlier in the text, the calculated excitation energies came out to be too large by almost a factor of 2. We found that this discrepancy could be alleviated somewhat by lowering the choice of interaction strengths G,., G, and Go,. However, for no “reasonable” interaction parameters were we able to get close to the experimental values. This difference in scale is, in fact, a problem well known in boson expansion theories [see e.g. refs. ‘2,23)] +. Its origin lies, more likely than not, in the radical truncation of the shell-model space. Since the resolution of this problem will have to await a detailed, and consequently lengthy analysis of renormalization effects due to noncollective states, we did not attempt a fit of experimental spacings and concentrated on the comparison of other observable features of the spectrum such as ordering of levels, the ratio E(4:)/E(2:) etc. (A detailed discussion of the problem of too large spacings can be found in sect. 5.) Results for 148Sm, ’ 50Sm and “‘Srn are compared with experiment in figs. 1, 2 and 3. Consider the results for 148Sm first. Please do not be misled by the seemingly good agreement in scale between experimental and theoretical spectra! For the purpose of comparison, we have resealed the calculated excitation energies by a factor of 0.565 so as to bring the 2: state into the correct position. Note that the agreement in the relative position of the 4: state has no meaning either as it was fitted by adjusting the quadrupole interaction strength Go,. Let us therefore concentrate on the remaining parts of the spectrum. We observe that our calculation gives the correct ordering of collective levels, although there is a lack of splitting between the (two-phonon) triplet states Oz, 4: and 2:. (The experimental spectrum contains two additional O+ and 4+ states at 1.43 MeV and 1.73 MeV.

’ As discussed in the text, in Kumar’s calculation the problem is simply hidden by an ad hoc modification of the kinetic energy. Note, however, that the problem of too large spacings appears to have been solved by Kishimoto and Tamura 19).

M. R. Zimbauer 1 Microscopic approuch illi

252

‘48Sl-ll MeV 2.0

6+

+

3+ + ----•==(:+,.______---2+ p_

1.5

30+ 6+ 4+ _.

&________

_.

__\,.,,

_-

2+

1 .o

2+

2+

~______~

0.5

0.0 theory

experiment Fig. I. Excitation

spectrum

spectrum of ‘?3rn. The theoretical to give the 2: state in the correct

has been resealed position.

by a factor

MeV

1.5

1.0

2+ -

-------

5: ---._____ __---

2+

_.--

4+

0+

0.5 2+

2+

_______-

0.0 experiment Fig. 2. Excitation

spectrum

theory

of IsOSm. Same resealing

as in fig. 1.

of 0.565

~

M. R. Zirnhauer 1 Microscopic approach (II)

152S,

253

o+

MeV

3+ 4+

P

1.5

1.0 -

3+---+ ;+_

4+-

/ f*+ -

/

/

___

0.5 -

,’

&+-’

_--

2*

experiment spectrum

2+

/ /

o+-

Fig. 3. Excitation

6+

/_

of “?Sm.

/’

o+

,~

4+

2

__--

+

theory Same resealing

as in fig

These states are very likely of a non-collective nature.) Also, the position of the 0: comes out slightly too low. We found that these discrepancies could be removed by enhancing the quadrupole part of the interaction. However, such a modification invariably increased the ratio E(4:)/E(2:) and, thus, made the spectrum more transitional in character. A peculiar feature of the experimental spectrum is the low position of the 23” state. We see that the calculation reproduces (and in fact exaggerates) this feature. Another positive point is the good agreement for the threephonon states 4:) 3: and 6:. Experimentally, these states are nearly degenerate and lie at about 1.90 MeV. Comparison with 3E(2:) = 1.65 MeV shows that the spectrum is noticeably anharmonic, and this anharmonicity is reflected in the calculation. Consider next the results for lsoSm shown in fig. 2. It is important to note that these results were obtained with exactly the same shell-mode1 hamiltonian as for 148Sm. (We decided to keep the interaction constant because any changes in the parametrization would have made it more difficult to assess the results. In general, we feel of course justified in giving the interaction an appropriate A-dependence to improve the quality of the.fit.) In order to facilitate the comparison between theory and experiment, the calculated spectrum was again resealed by a factor of 0.565 (the same factor as for 14’Srn). It is then encouraging to find the 2: state in about the right position which means that we can reproduce the considerable drop in the excitation energy of this state when going from 14’Sm to 15’Srn. The experimental

M. R. Zimbauer / Mirroscopie approach (II)

254

values are E(2:) calculation E(4:)/E(2:) the

= 0.550 MeV (14%m) and E(2:)

= 0.334 MeV (15’Srn) while our

gives 0.974 MeV and 0.604 MeV, respectively. Also, the calculated ratio = 2.38 agrees quite well with the experimental value of 2.31 reflecting

transitional

character

of 15’Sm.

Turning

now

to higher

excited

states

we

observe that the ordering of collective levels is again correct and, in fact, has improved relative to 14%rn. (There exists an additional Of state at about 1.25 MeV which we suspect to be of the same nature as the corresponding “intruder” in 148Sm.) In particular, in going from 148Sm to “‘Srn the 6: state moves away from seen in the calculation. A considerable 3: and 4;, and this effect is distinctly problem which we have inherited from “‘Srn is the relatively low position of 0: and 2:. This difficulty becomes far more serious for “‘Sm as is evident from fig. 3. Clearly, due to the particular way in which the collective bosons are selected, the present method is most suitable for use in vibrational nuclei (14*Sm) and, to a lesser extent, in transitional nuclei ( 15’Sm). For well-deformed nuclei such as “‘Srn we cannot expect to obtain good results. Nevertheless, we thought it might be of interest to do the calculation and see what we get. Experimentally there is a very sharp transition from vibrational to rotational character which manifests itself as a strong decrease in the energy of the 2: state from 0.343 MeV in “‘Srn to 0.122 MeV in 152Sm. The calculated value for E(2:) is 0.341 MeV and, thus, only a factor of 2 (instead of the required factor of 3) lower than the corresponding excitation energy in “‘Sm. What the calculation does get correct are the relative positions of the states in the ground-state rotational band. For example, the empirical value for E(4:)/E(2:) is 3.01 while the calculation gives 2.99. The most conspicuous difference between theory and experiment concerns the b- and ybands. The corresponding band heads 0; and 2’ lie extremely low, indicating a lack of stability in the ,& and y-directions which may be associated with an insufficiently deep deformed minimum. It is conceivable that this problem can be solved by determining

the collective

4. Renormalization

pairs from a deformed

(Nilsson-like)

scheme.

effects due to the coupling with g-hosons

Our calculation for the Sm isotopes made very clear that, for a more quantitative understanding of the experimental excitation energies, it is necessary to go beyond the simple sd-boson (SD-pair) picture. An effect distinctly absent from our calculation is the lowering of the single d-boson energies (seen in phenomenological fits) as more and more particles are added to the closed core. It is commonly assumed that this lowering is mainly due to interactions with the socalled g-boson. As mentioned in the introduction, various schematic studies have been performed ’ - 9 ) in order to assess the role played by g. It was found that, even in the limit of strong deformation, the g-boson takes at most 15 “/, of the NilssonBCS wave function. However, these small admixtures enhance the probability of

M. R. Zirnbaurr / Microscopic approach (II)

finding

d in the intrinsic

deformed

state, and thus have a non-negligible

255

effect on

observables such as excitation energies, quadrupole moments etc. As formulated by Bes rt al. ‘). the presence of the g-boson serves to “further break the superfluidity and quadrupole-polarize the nucleus”. Our failure to reproduce the experimental excitation energies in Sm is another indication that the simple IBA truncation scheme to S and D may not be a good approximation. The fact that the discrepancy between theory and experiment is quite large already for 14%rn shows that effects of truncation are not restricted to the deformed region, but also occur for vibrational nuclei. Clearly, any coupling of s and d with other bosons must be due to the seniority violating neutron-proton quadrupole interaction. (The interaction between like nucleons conserves seniority and is, therefore, to a very good approximation diagonal in s and d.) Renormalization effects due to the coupling with the g-boson have been studied by Scholten 25), and we base the following discussion on his results. As shown by Scholten, the coupling with g, induced by the quadrupole interaction between neutrons and protons. leads to an effective quadrupole interaction between like bosons Veff = V,Q; where the boson quadrupole in eq. (2.5). In second-order V, is given by

Q;il; + V,,Q; . Q;fi;,

(4.1)

operator Q; (Q;) is formally identical to that defined perturbation theory, the effective interaction strength

5 V, = G;Q(~;ti;)2/(~~

+ E; - E; - E;),

(4.2)

and an analogous result holds for V,. The parameter K; is defined as the coefficient of (d,+&):2’+ (y:&,):2’ m the “extended” boson image of the proton part of the E2 operator Q,. For vibrational nuclei, we can approximate the operator Q;1. Q& by its expectation value in the fully paired ground state

Q;. Q; + (Q;.Q;) = 5N,, and then eq. (4.1) leads to an effective lowering As; = 5V,N,,

of the single d-boson

de,” = 51/,N,,.

(4.3) energies, (4.4)

In ref. 24), the parameter V,(V,) was determined by fitting the value of ad obtained in phenomenological IBA calculations. Here, we will use expression (4.2) to evaluate V, and V,, from the shell-model hamiltonian of sect. 3. The Dyson formalism is ill-suited for that purpose as it leads to a non-hermitian hamiltonian which cannot be treated by standard perturbative techniques. In order to estimate

256

M. R. Zirnbauer

/ Microscopic~ approach

(II)

the value of V, and V,. we therefore decided to use the framework Tamm-Dancoff approximation. This approach yields for xg:

where e+(BCS) Tamm-Dancoff interaction

is the phonon

strength

of the BCS-plus-

two-quasi-particle state maximally connected with the (d-boson) d’lBCS). Taking the same quadrupole

as in sect. 3, we obtain

- GQQti$i; = 0.305 MeV,

for L48Sm -GO&$;

= 0.197 MeV.

(4.6)

The effective coupling is smaller for protons because in “‘Srn this kind of nucleon is closer to the middle of the shell (where ~~ changes sign). Similarly, the g-boson energy 8%can be evaluated in the Tamm-Dancoff approximation, leading to E; = 2.95 MeV. Eqs. (4.6) and (4.7) yield the following A.$ = -0.190

MeV.

8;; = 2.98 MeV. corrections

(4.7)

for cd:

AE; = -0.026

MeV.

(4.8 1

The correction is much larger for neutrons since NJN,. = 3 (“‘Sm). We now recall that the theoretical value for the excitation energy of the 2: state in lssSm was E(2: ) = 0.97 MeV, while experiment gave E(2:) = 0.55 MeV. Using the corrections (4.8) to obtain an improved value for E(2:). we find that these corrections fall short of explaining the difference between theory and experiment. We therefore believe that the g-boson only partly accounts for the renormalization of the IBA hamiltonian seen in phenomenological calculations. Of course, our finding that the g-boson has a rather small effect in Sm does not contradict the results of Bes rt ul. ‘), obtained for a simple degenerate model. As is seen from eqs. (4.1) and (4.4), the influence of g increases as we add more bosons and thus approach the rotational regime. However, our analysis does show that in actual nuclei additional effects come into play. These effects are most likely connected with our choice of realistic (i.e. non-degenerate) single-particle energies, a point which is further elaborated in the following discussion.

5. Discussion We begin the analysis of our numerical results by discussing some advantages and disadvantages of the Dyson approach used in sect. 3. The most attractive “feature”

M. R. Zirnbauer 1 Microscopic approach (II)

of Dyson’s

boson

mapping

is its finiteness.

The mapping

breaks

257

off in fourth

order

and, therefore, leads to a boson hamiltonian of the IBA type without any further assumptions concerning the magnitude of higher-order interactions. In addition. schematic mode1 hamiltonians can be constructed [such as Ginocchio’s SO(S) and Sp(6) hamiltonians ‘“)I, for which the Dyson mapping is exact. In these schematic cases, spurious solutions appear only for n > Sz and decouple completely from the physical boson states. The attractive property of finiteness is achieved at the expense of allowing the boson hamiltonian to be non-hermitian. From the point of view of making actual calculations, the non-hermiticity of the Dyson hamiltonian is a disadvantage because of the increase in the amount of computer time required to diagonalize a non-symmetric matrix. As another disadvantage, we found that in the present formulation E2 transition rates are more difficult to calculate than excitation energies. Finally, in non-degenerate shell-model spaces the separation between physical and spurious boson states is no longer clear-cut, and spurious solutions can already appear for II 5 SZeff.The advantages that make the Dyson formalism attractive in certain model situations, inay therefore be lost in realistic nuclear structure calculations. A crucial step in microscopic IBA calculations is the determination of the collective subspace, as was discussed at some length in sect. 2. While the definition of S, and S, is very clear and unambiguous, choosing the collective D-pairs turned out to be more problematic. Two extreme choices were considered. The first choice consisted in defining the D-pair by diagonalizing the interaction between like nucleons. This can be done within the broken pair appoximation or, in cases where conservation of particle number is not essential, using the formalism of the BCSTamm-Dancoff approximation (D,,, “TD boson”). A second possibility is to take D as the quadrupole pair which exhausts the entire E2 strength connecting the fully paired state IS”) with all other shell-model states. This leads to eq. (2.10) defining D+ as the commutator of S+ with the E2 operator Q (D,,, “E2 boson”). However, in realistic applications neither of these prescriptions is the optimal one as long as the coupling to other degrees of freedom is neglected. In vibrational nuclei, it is best to determine D by minimizing the energy of the one-phonon state (2.12). This can (again) be done in the framework of the broken pair formalism although we followed a simpler approach, here, by calculating the energy of this state in the number operator approximation. Application of the resulting formalism to the Sm isotopes led to satisfactory agreement concerning some qualitative features such as level orderings and the transition from vibrational to transitional (rotational) character. However, the calculated excitation energies were systematically larger than the experimental values. We do not think that this systematic difference is in any way particular to the present approach. We strongly believe that an attempt to perform the same calculation within the OAI approach will lead to similar difficulties. In fact, these difficulties are only to be expected because none of the present theories (in their

258

M. R. Zirnbauer / Microscopic approach (II)

simple versions) contain any mechanism for reduction of the single d-boson energies with increasing particle number (hole number). This reduction is known to be essential for obtaining the right spacings of excited states. According to current thinking 3*24*27),the renormalization of sd is largely due to the residual coupling with hexadecupole degrees of freedom (G-pair or g-boson), induced by the strong quadrupole interaction between neutrons and protons. However, when we actually calculated in perturbation theory the effects of the g-boson in 14%Srn, the corrections to cd were found to be too small to explain the difference between theory and experiment. This leads us to reconsider the choice of collective quadrupole pairs, D. In sects. 2 and 3, these pairs were selected from a variational procedure which in some way interpolates between the two extreme choices, D-r,, and D,,. What we now require is a better understanding of how strongly the overlap between these two pairs decreases due to the non-degeneracy of the single-particle space. (By construction of the shell-model interaction, D,, and D EZ coincide in the limit of degenerate singleparticle energies.) Such a reduction is expected since ISN- ‘D,,) is mainly a superposition of quadrupole excitations near the Fermi surface, whereas (SN- ‘D,,) contains also components where particles from low-lying orbits are excited to energies high above the Fermi surface. As a first step towards a quantitative description of this effect. we computed the overlap ((SN-iDTDISN- ‘D,,)12 in the BCS-plus-Tamm-Dancoff approximation :

c2 -

((BCS&,6:,JBCS)~2

’ - (BCS(d,,&D~BCS)(BCS~~E2~;2~BCS)

(5.1)

Here, &,(BCS) and d,‘,lBCS) are the two quasi-particle states that, after projection on good particle number, correspond to the shell-model states lSN- rDTD) and ISN- ID,,). A numerical study of eq. (5.1) was performed for neutrons (protons) in the 82-126 (50-82) major shell. Of the closed shell orbits, vhr and ng% are expected to couple most strongly to the valence nucleon conligurations, and we therefore included these orbits in the calculation (“polarization of the core”). Our preliminary results are as follows. Using a surface-delta interaction with strengths as given in sect. 3, the overlap (5.1) is found to be roughly 0.70 for both kinds of nucleon at the beginning of the shell. With increasing particle number, ci increases to about 0.75 (for n = 8, lo), thereby confirming the expectation that the d-boson should be most collective in the middle of the shell. We have not yet explored in detail the consequences of these numerical results, but in the light of our calculation for the Sm isotopes the following qualitative statements can be made. (i) Excitation if the core cannot be neglected. The above result for the overlap l(SN-1D-rDlSN-1DE2)12 means that about 25 y, to 30”,t E2 strength is missing from the TD boson, D,,. This indicates that in realistic shell-model spaces truncation to

M. R. Zirnbauer 1 Microscopic approach (II)

a single L = 2 excited mode (D-pair)

259

may not give rise to enough

(E2) collectivity.

In the present example, the missing strength resides mainly in the configuration hq’f; for neutrons and gf ‘d+ for protons. (Configurations like h,‘h, and @‘g are less important because the corresponding Clebsch-Gordan coefficient is much smaller.) Admixtures of this kind are expected to increase the “moment of inertia” of the system and, thereby, effectively reduce its excitation energies. Such an effect cannot be taken into account by directly incorporating these configurations into the collective D-pair because this, while making the boson quadrupole interaction stronger, also increases the value of ed. (ii) The ,fbruwrd urld backward amplitudes of’ the collectire RPA solution do not have identical structure. The difficulties encountered within the simple IBA truncation scheme can also be discussed in the language of the (quasi-particle) random phase approximation (RPA). Consider the RPA matrix for the standard IBA-2 hamiltonian (eq. (3.1) with H,.,,(,,, replaced by E;(%~(~‘): E;

0

0

Ed”

1 0

0 0 -&; 0

0 -E;

+JW

+-

K

0

K

0

K

0

0

-K -K

0

-K

-K

.

(5.2)

0,

L

From eq. (5.2) we expect the forward RPA amplitude, structure of D,, (which minimizes Ed), while the backward more closely to the structure of D,, (which maximizes

X, to be similar to the amplitude, Y, corresponds the absolute value of K).

In the interacting boson approximation, Ed and K are calculated D-pair, which in the language of the RPA amounts to assuming are proportional, Y,, = const

X,,.

from the same that X and Y

(5.3 1

In vibrational nuclei for which this assumption is not justified, a microscopic IBA calculation (such as our calculation for i4*Sm) will give a higher 2: excitation energy than does the RPA. The difference in excitation energy is related to the deviation of ci from unity. In conclusion, there are many aspects of the present and previous work that support the belief that the basic ideas underlying the interacting boson approximation are correct. However, our calculation for Sm indicates the urgent need for a careful and detailed treatment of renormalization effects. Unfortunately, these effects are quite large (in 14’Sm, sd has dropped from 1.6 MeV to 0.7-O.yMeV!), and it is not at all clear whether second-order perturbation theory will be sufficient for their treatment. The future of the microscopic interacting boson model will very

M. R. Zirnbauer 1 Microscopic approach (11)

260

crucially depend on whether a simple renormalization can be found. I would

like to thank

and satisfactory

Dr. D. M. Brink

pushing me to complete the present by the Studienstiftung des deutschen

for many

manuscript. Volkes.

solution

helpful

to the problem

discussions

and

of

for

Part of this work was supported

References I) 2) 3) 4) 5) 6) 7) 8) 9) 10) I I) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27)

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