fulltext

Study of

Mo nucleus in microscopic frameworks

R. DEVIand S. K. KHOSA

Department of Physics, University of Jammu - Jammu 180004, India (ricevuto il 27 Febbraio 1997; approvato il 19 Dicembre 1997)

Summary. — The yrast spectra, reduced E2 transition matrix elements and QJ1

values are calculated for 108_{Mo isotope by carrying out variation-after-projection}

(VAP) calculations. The experimentally available spectra upto 101 _{are reasonably}

well reproduced. The yrast spectra upto 121 _{and root-mean-square radii are also}

obtained in the framework of Cranked-Hartree-Fock-Bogoliubov (CHFB) Method. These calculations have been performed by employing a pairing-plus-quadrupole-quadrupole effective interaction operating in a reasonably large valence space outside the 76_{Sr core. Our results indicate that the axial VAP framework reproduced the}

observed spectra upto 101_{much better than the non-axial CHFB framework. The}

cal-culations seem to predict an axial symmetry for108_{Mo to be confirmed experimentally.}

PACS 21.20 – Properties of nuclei; nuclear energy levels. PACS 27.60 – 90 GAG149.

Sometimes ago Hotchkis et al. [1] has studied the rotational bands upto spins of B 10 ˇ in even-even and odd-A nuclei in the mass 100 region, by observing prompt g-rays from the spontaneous fission of248Cm. Transitions in108Mo have been observed for the first time. Guessous et al. [2] have very recently predicted that the experimental results could be reproduced satisfactorily within a description of 108Mo as an axially symmetric nucleus. For this reason we are motivated to carry out a microscopic study of the properties of 108_{Mo involving axial and non-axial frameworks. To this end, we} have employed the usual pairing-plus-quadrupole-quadrupole effective interaction operating in a valence space spanned by the 3 s_{1 O2}, 2 p_{1 O2}, 2 d_{3 O2}, 2 d_{5 O2}, 1 g_{7 O2}, 1 g_{9 O2}and 1 h_{11 O2}orbits for protons as well as neutrons. The nucleus76Sr has been considered as an inert core.

From the results of our calculations it turns out that the available experimental data on 108_{Mo can be explained reasonably well in the axially-symmetric VAP framework,} confirming thereby the axial character of108_Mo.

The spherical single-particle energies (SPEs) that we have employed are (in MeV): ( 2 p_{1 O2) 420.8, (1g9 O2) 40.0, (2d5 O2) 45.4, (3s1 O2) 46.4, (2d3 O2) 47.9, (1g7 O2) 48.4} and ( 1 h_{11 O2) 49.0. This set of input SPEs is exactly the same as that employed in a} number of successful shell model calculations in A B100 nuclei by Vergados and

Kuo [3] as well as by Federman and Pittel [4] except for a slight lowering of the 1 g_{9 O2} energy by about 0.4 MeV.

The two-body effective interaction that we have employed is of pairing-plus quadrupole-quadrupole (q-q) type [5]. The strengths for the like particle (n-n) as well as neutron-proton (n-p) components of the q-q interactions are the same as the ones employed in an earlier study [6] of the deformation systematics in the A B100 region.

xnn(4 xpp) 420.0105 MeV b24, xnp4 2 0 .0231 MeV b24. Here, b(4kˇOmv) is the oscillator parameter. The strength for the pairing interaction was fixed through the approximate relation G 418221OA at G420.18 MeV.

The calculation of the energies of the yrast levels has been carried out by two methods.

Firstly, by employing the variation-after-projection (VAP) [7] formalism in conjunction with Hartree-Fock-Bogoliubov (HFB) [8] ansatz for the axially-symmetric intrinsic wavefunctions. The calculation of the energies of the yrast levels in this case has been carried out as follows.

We have first generated the self-consistent axially-symmetric HFB solutions f_k40(b) resulting from the Hamiltonian (H 2bQ2

0), where b is a parameter. The optimum intrinsic state for each J, fopt(bJ) has been selected by determining the minimum of

the projected energy,

EJ(b) 4 af(b)NHP00JNf(b)b Oaf(b) NP00JNf(b)b

as a function of b . In order words, the intrinsic state for each J satisfies the following condition:

¯O¯b[af(b) NHP00JNf(b)b Oaf(b) NP00JNf(b)b] 4 0 .

Here the operator PJ _{projects out the eigenstates of J}2 _{from the intrinsic states f(b).} Secondly, we have carried out a calculation of the yrast states in the Cranked-Hartree-Fock-Bogoliubov (CHFB) framework for which the wavefunction has been kept axially asymmetric. CHFB calculation amounts to carrying out an HFB type of calculation with the Hamiltonian (H 2vJx) with the subsidiary condition that

aJxb 4

k

1. – VAP calculations

In table I, we present results for the occupation probabilities of the various proton and neutron subshells obtained for ground states. From the occupation probabilities for protons, it is found that the down sloping K 41O2 and 3O2 component of ( g9 O2)p

orbit are occupied. Besides this, the ( g_{7 O2})nand (h11 O2)norbits are nearly half-filled. This

type of configuration of nucleons in the underlying valence orbits could be thought to be responsible for the deformed nature of this nucleus as it can generate a large value of intrinsic quadrupole moment for this nucleus which could produce large quadrupole deformation. It may be noted that E411OE211 value for

108

Mo is B 3.0, which is near the value of 3.3 for a perfect rotor. In fig. 1, the yrast spectra of108Mo is displayed. In this figure one observes that the present calculation (VAP) yields satisfactory overall agreement with the experiments; for example the 81_{state is theretically found to be at} 1.94 MeV when the experimentally observed value is 1.75 MeV. The agreement

TABLEI. – The calculated values of the occupation numbers of various orbits for a) protons and b) neutrons in the ground state of108_Mo.

3 s_{1 O2} 2 p_{1 O2} 2 d_{3 O2} 2 d_{5 O2} 1 g_{7 O2} 1 g_{9 O2} 1 h_{11 O2}

a) 0.11 0.00 0.06 0.85 0.05 2.93 0.0

b) 1.06 1.99 1.66 4.19 3.56 9.89 5.65

between the theoretical and observed yrast spectra can be thought to be reasonably good.

In table III, we present the results of the calculation of interstate B(E2 ) transition probabilities and quadrupole moments QJ1 calculated by using rigorous projection

technique. The calculated values are expressed in parametric form in terms of proton (ep) and neutron (en) effective charges. These two parameters have been chosen so that ep4 1 1 eeff and en4 eeff, therefore only one parameter is introduced in the calculation. The B(E2 ; Ji1K Jf1) values have been calculated in units of e2Q b2. The results indicate that by choosing eeff4 0 .49 , a good agreement with the observed value [10] for B(E2 ; 011K 211) transition probability is obtained for 108Mo isotope. We have calculated the QJ1 values for the same value of effective charge as the one taken for B(E2 ) values.

Due to non-availability of the experimental QJ1 values, it is not possible to make any

comment regarding the degree of agreement with the observed values. In this regard,

Fig. 1. – Comparison of the experimental and calculated yrast spectra by employing VAP and CHFB frameworks in108_{Mo. Here Th. 1 and Th. 2 correspond to VAP and CHFB yrast spectra,}

TABLEII. – The intrinsic quadrupole moments aQ02b and aQ22b for each CHFB state and root mean

square radii in108_{Mo isotope. Here rp(rn) give root-mean square radii for protons (neutrons) in}

fermis. The quadrupole moments have been computed in units of b2_{, where b 4k}

ˇOmv is the oscillator parameter. The radius for108_{Mo calculated from the relation R 4r}

0A1 O3is 5 Q 10 fm . Jp aQ2 0bCHFB aQ22bCHFB rp rn 01 21 41 61 81 101 121 63.64 64.51 65.30 65.73 65.15 64.40 63.48 0.73 1.33 7.18 7.40 7.89 8.44 9.06 5.17 5.18 5.19 5.20 5.20 5.21 5.22 5.24 5.24 5.25 5.25 5.26 5.26 5.26

TABLEIII. – The reduced E2 transition matrix elements and static quadrupole moments for the yrast levels in the nucleus108_{Mo. Here e}

p(en) denote the effective charge for protons (neutrons).

These two parameters have been chosen such that ep4 1 1 eeff and en4 eeff, so that only one

parameter is introduced in the calculation. The entries presented in the third and sixth column correspond to the calculated values of B(E2 ) s and QJ1s for e_eff4 0 .49 . The E2 matrix elements

are in units of e2_{Q b}2_{, and static moments have been given in units of e Q b .}

Transition (J1 i K Jf1) [B(E); J1 i K Jf1]1 O2 B(E2 ; Ji1K Jf1)1 O2 [QJ_f1] [QJ_f1]_{Th .} Th. Exp. [10] 01_{K 2}1 21_{K 4}1 41_{K 6}1 61_{K 8}1 81_{K 10}1 0.46ep1 0 .96 en 0.55ep1 1 .15 en 0.58ep1 1 .20 en 0.58ep1 1 .21 en 0.58ep1 1 .23 en 1.34 1.91 2.10 2.12 2.15 1.34(31) — — — — 20 .29 ep2 0 .61 en 20 .38 ep2 0 .78 en 20 .42 ep2 0 .88 en 20 .44 ep2 0 .90 en 20 .46 ep2 0 .96 en 20.73 20.95 21.04 21.09 21.15

the measurements of QJ1 values for this nucleus would provide a good test of the

theory and should therefore be made. This observation has also been made by Hotchkis et al. [1].

2. – Results of CHFB calculations

In fig. 1, the observed spectra is compared with the theoretical spectra (Th. 2) obtained in the CHFB framework. The theoretical spectra has been obtained upto 121 by using the same parameters of two-body effective interactions as in the case of VAP calculations. The calculated spectra is found to be more compressed than the observed spectra, for example, the experimental 81 _{state is at energy 1.75 MeV, whereas the} calculated value is 1.45 MeV. The root mean square radii for protons and neutrons rp

and rn presented in table II are observed to be reasonably constant with spin. These

values are seen to compare reasonably well with the radius of 5 Q 10 fm obtained from the empirical relation R 4r0A1 O3for108Mo. It is noteworthy that rnvalues all along the

yrast states are higher than rpvalues. Thus, there is a greater probability for neutrons

Fig. 2. – Moment of inertia (I) vs. the square of the angular velocity for the yrast levels in

108_Mo.

In fig. 2, we present the usual I vs. v2 _{curves for} 108_{Mo, calculated in VAP and} CHFB frameworks. The following expressions have been used to compute the moment of inertia (I) and the squared angular frequency (v2_{) in terms of the yrast states:}

2 I ˇ2 4 ( 4 J 22) EJ2 EJ 22 and (ˇv)2 4 [ (J 2 2 J 1 1 )(EJ2 EJ 22)2] ( 2 J 21)2 .

The results presented in fig. 2 indicate an overall qualitative agreement between the calculated variation of I as a function of v2 in the VAP framework with the experimentally observed one. The CHFB results are seen to predict a back-bend around 41_{which is not experimentally observed. It thus appears from our calculations} that the observed properties of 108_{Mo can be explained reasonably well in the axially} symmetric framework rather than non-axial framework.

Summarising the observed deformed character of 108Mo can be understood in the axially symmetric HFB framework in terms of the nearly half-filled subshells of g_{7 O2} and h_{11 O2} neutron orbits. It turns out that the axial VAP framework reproduces the observed yrast spectra upto 101_{much better than the non-axial CHFB framework. The}

calculations seem to predict an axial symmetry for 108_{Mo to be confirmed} experimentally. For this data on interstate B(E2 ) transition probabilities and QJ1

should be obtained experimentally. The calculations in the VAP framework have made a prediction of these values for comparison with experiments in the future.

* * *

One of the authors (RD) is thankful to CSIR New Delhi for the award of Senior Research Fellowship.

R E F E R E N C E S

[1] HOTCHKISM. A. C., DURELLJ. L., FITZGERALDJ. B., MOWBRAYA. S., PHILLIPSW. R., AHMAD I., CARPENTERM. P., JANSSENSR. V. F., KHOOT. L., MOOREE. F., MORSSL. R., BENETPH. and YED., Nucl. Phys. A, 530 (1991) 111.

[2] GUESSOUSA., SCHULZN., BENTALEBM., LUBKIEWICZE., DURELLJ. L. et al., Phys. Rev. C, 53 (1996) 1191.

[3] VERGADOSJ. D. and KUOT. T. S., Phys. Lett. B, 35 (1971) 93. [4] FEDERMANP., PITTELS. and COMPOSR., Phys. Lett. B, 82 (1979) 9. [5] BARANGERM. and KUMARK., Nucl. Phys. A, 110 (1968) 490.

[6] KHOSAS. K., TRIPATHIP. N. and SHARMAS. K., Phys. Lett. B, 119 (1982) 257. [7] ONSHIN. and YOSHIDAS., Nucl. Phys., 80 (1966) 367.

[8] BARANGERM., Phys. Rev., 130 (1963) 1244. [9] GOODMANA. L., Nucl. Phys. A, 230 (1974) 466.

[10] RAMANS., MALARKEYC. H., MILNERW. T., NESTERC. W. and STELSONP. H., At. Data Nucl. Data Tables, 36 (1987) 21.