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Spectroscopic properties of 237,239Pu fission isomers from self-consistent calculations
Volume 95B, number 2
PHYSICS LETTERS
22 September 1980
SPECTROSCOPIC PROPERTIES OF 237,239pu FISSION ISOMERS FROM SELF-CONSISTENT CALCULATIONS J. LIBERT CSNSM, 91406 Orsay, France
M. MEYER 1PN Lyon (and IN2P3), 69622 Villeurbanne, France
and P. QUENTIN Institut Laue-Langevin, 156X, 38042 Grenoble-Cedex, France
Received 21 June 1980
We have studied 237,239puisomeric states: energy levels, M1 and E2 spectroscopic moments and reduced transition probabilities have been calculated within the rotor + quasi-particle model from self-consistently determined single-particle states. The electromagnetic properties of these states have been especially investigated. Without any ad hoc parameter adjustment, a very good reproduction of most of the known spectroscopic data is yielded which assessesthe predictive power of the whole approach.
Odd deformed nuclei are ideally suited for a precise determination of Hartree-Fock single-particle (or BCS quasi-particle) energies. Unlike the case of quasispherical systems corresponding to the vicinity of closed shells, the coupling between their individual and core vibration degrees of freedom is generally weak and the coupling with collective rotational motion is the only one being of some importance. This is why the experimental spectra of odd deformed nuclei have constituted the main ingredient of any fitting procedure of phenomenological single-particle potentials, as done in'ref. [1]. To the best of our knowledge, this procedure has not been carried out for the determination of most phenomenological effective forces (especially their spin-orbit component) in use in Hartree-Fock calculations. For reasons of computational simplicity, the parameter search has been done only for spherical solutions, see e.g. ref. [2]. This fact may lead in practice to rather large uncertainties in the spin-orbit force strength [3]. It turns out, however, that the Skyrme SIII effective
force [2] proves a posteriori to correspond to a reason. able choice of this strength. A hint at this may come from a detailed comparison between the Nilsson model and Hartree-Fock single-particle spectra in both the rare-earth and the actinide region [4]. Nevertheless this type of comparison was restricted so far to the "normal" range of nuclear deformation 032 0.2-0.3). Recent spectroscopic data for fission isomeric states in 239Pu [5,6] together with older data on 237pu [7] provide a challenging testing ground for any theoretical estimate of the mean nuclear field at larger deformations (/32 ~ 0.6). In the 237pu case, the available data consist of measurements of the g-factors for two fission isomers whose spins are essentially unknown, even though one has tried to draw conclusions [8] from measured fragment angular distributions. For the 239Pu states, a rotational band based on a 5/2 spin state has been identified, together with a 9/2 spin state lying 203 keV higher. From conversion electron measurements an experimental limit for the M1/E2 ratio for transitions 175
Volume 95B, number 2
PHYSICS LETTERS
within the ground band has been proposed [6]. The latter combined with the knowledge of a previously measured quadrupole moment [5] has led the authors of ref. [6] to suggest an antiparallel coupling of the orbital angular momentum and spin vectors in the single-particle 5/2 state. In this paper we report on calculations aiming at a description of spectroscopic properties of 237pu and 239pu isomeric states within the rotor + quasi-particle approach of Bohr and Mottelson [9]. Technical detafls of the method used in the present calculations can be found in ref. [10] for the diagonalization of the hamiltonian and in ref. [ 11 ] for the evaluation of multipole moments and transition probabilities. The underlying Hartree-Fock + BCS results (for 240Pu) have been published long ago [12]. It is worth recalling that such an approach does not imply any ad hoc parameter adjustment: (i) the effective force parameters are those discussed in ref. [2] and are valid for the whole chart of nuclides; (ii) the pairing gap parameters have been deduced [12] through a well-defined if not fully satisfactory prescription from odd-even (ground-state) binding energy differences; (iii) the rotor hamiltonian parameters (9(17) with the notation of ref. [10]) are determined by experimental results in 240pu [13]; (iv) there is neither any deformation parameter (/32,/34.... ) adjustment, nor any ambiguity due to the choice of a fission path (apart from an a priori choice of self-consistent symmetries) because we are considering a local extremum HartreeFock solution; (v) we have naturally included any possible "AN = 2" coupling; (4) we have taken into account the full rotor kinetic energy (thus including the so-called recoil one-body term/2) and we have not used any renormalized Coriolis coupling term (no attenuation factor). As is well known, a one-quasi-particle wave function does not correspond in general to a state with an integer average number of particles. That is why, starting from a BCS solution describing an even nucleus with A nucleons, the lowest one-quasi-particle states will describe an admixture ofA + 1 and A - 1 nuclear states. Consequently, in order to study for instance the 239pu nucleus, it will be suitable to compare theoretical results obtained in calculations corresponding to both 240pu and 238pu BCS solutions. To compute the latter we have assumed that the optimal basis parameters and pairing gaps were those of 240pu. 176
22 September 1980
This has been a posteriofi supported (at least for the basis deformation parameter) by the fact that the local equilibrium solutions of 238pu and 240pu correspond to about the same quadrupole moment (33.8 b in 238pu and 32.7 b in 240pu for the charge distribution). One observes in fig. 1 that the gross structure of the single-particle neutron levels is not very much affected when going from N = 144 to N = 146 which is consistent with the above-mentioned similarity of the fission isomer quadrupole moments. One may note, however, the usual occurrence of a bunching of the single-particle levels immediately below and above the Fermi surface, resulting particularly in an increased energy difference between the lowest 5/2 + and the 9/2- levels in 238pu (~500 keV instead of ~200 keV in 240pu). In ref. [12] we have already noted the striking similarity of our 24°pu spectrum with the one obtained by M611er, Nilsson and Nix (see fig. 12 of ref. [14]). For most of the single-particle states whose energies are given in fig. 1, we present in table 1 the main coefficients of the expansion in the axially sym-
~N
HF (MeV)
712+ 5/2 + 1/2 +
-5
912 -6 -
512 +
i
1112+~ -7
312-~-8
112 +
l t 2 - ~ 312312+ 238 240 Pu Pu
Fig. 1. Neutron Hartree-Fock spectra for the calculated fission isomeric states of 238pu and 24°pu. The Fermi level h resulting from our BCS calculations is also given.
Volume 95B, number 2
PHYSICS LETTERS
22 September 1980
Table 1 Characterization of the quasi-particle states near the Fermi level. The Hartree-Fock energies e are given together with the expanston coefficients in deformed harmonic oscillator states, in current notation, for the self-consistent 24°pu solution. The pairing occupation probabilities f/2 are also given for both the 238pu and the 24°pu solution. The expectation values of the one-body operator j 2 in the lowest 5/2 + and the 9/2- states are also given (for the 24°pu solution). Level
e (MeV) V~4o
V~as
5/2 + 7/2 + 9/2 + 5/2 + 11/2 + 1/2-
-4.71 -4.78 -5.82 -6.04 -6.92 -7.24
0.071 0.035 0.420 0.568 0.901 0.936
0.037 0.035 0.164 0.404 0.641 0.882
3/2-
-7.43
0.949
0.897
~/2) (~2) Main components in the harmonic oscillator basis
39 57
0.63[622] 0.78[624] 0.77[734] 0.55[862] 0.881615] 0.441510] 0.211310] 0.551512] 0.201312]
metric harmonic oscillator basis. They are only given for the 240pu solution since they are essentially identical in 238pu. In the same table we have also reported the pairing occupation probabilities for the two solutions (in both cases the neutron pairing gap was 0.732 MeV [12]). For the lowest 5/2 + and 9 / 2 - states we have listed the expectation value of the j2 operator showing that these states are indeed stemming from rather high-j spherical orbitals. As previously noted [4] the H a r t r e e - F o c k field mixes much more strongly than the modified harmonic oscillator basis states coming from different major shells (e.g. up to four shells significantly contribute to the 1 / 2 - and 3 / 2 - states). It is also noticeable that for all the states close to the Fermi surface, the mixing between spin-up and spin-down states is found to be very small (typically 80% of the wave function at least, corresponds to the same spin projection). The only spin-down states listed in table 1 are the 3 / 2 and 7/2 + states. The closest 5/2 spin-down state is a 5/2 + state lying rather deep in the Fermi sea (at a • H a r t r e e - F o c k energy of ~ - 9 . 9 MeV) and whose components are: 0.77 [633], 0.38 [853], - 0 . 2 5 [413]. As an illustration of our results, we have presented in fig. 2 the calculated 5/2 + and 9 / 2 - bands starting from the 238pu core in the three following cases: (i) no pairing and no RPC (Coriolis) terms, (ii) pairing and no RPC terms, (iii) full rotor + one quasi-particle hamiltonian (with pairing included). It has to be understood that, in all cases, the diagonal part of the rotor kinetic energy (namely (h2/2 9)[1(1+ 1 ) - K 2 {/2)] with current notation) is included. This term is
-0.40[822] 0.36[844] -0.37[954] 0.47[842] -0.371815] -0.43[750]
-0.311862] -0.25[824] -0.28[934] 0.3911082] 0.30[835] -0.40[970]
0.301422] 0.24[633] -0.271514] -0.23[642]
-0.48[752]
-0.341712]
-0.311972]
E (MeV) 06
-0.321710]
HF
HF +
+
--
BCS
BC.S
-HF
04
HF
HF
+
+
BCS
BCS COR
EXP
HF
--
C~)R
---~
- -
-
-
,, ',
-/ .
02
EXP
.
.
.
.
9/2-
5/2+ Fig. 2. Influence of the various parts of the rotor + quasiparticle hamiltonian on the ground 5/2 + and the 9/2- bands. In the calculations reported in this work, all quasi-particle states corresponding to Hartree-Fock energies lying within -+2 MeV around the Fermi level have been included. responsible for lowering the Hartree-Fock energy difference between the 9 [ 2 - and 5/2 + band heads to ~400 keV. Including further pairing correlation resuits for these unoccupied states in a sizeable compression of their quasi-particle energy difference (as compared to their particle energy difference). The inclusion of RPC terms yields only a small energy shift. Indeed, in this case, as in all results presented here, there is n o significant Coriolis band mixing (typically less than 1% of the final wave function). The net effect of all these terms is to shift the energy difference 177
Volume 95B, number 2
PHYSICS LETTERS E
238F~ + 2 4+0 p u
(MeV
lqp
1 qp
a % E,p 24o~, 0.5 - l q p
lqp
5i/21+
22 September 1980
_- -_
-- -- -
~=
~
2/,0 +1% lq. p
= __
--
9/2-
11/2+
__
~r
1//2-
3/2
7~+
5/2
Fig. 3. Calculated spectra for the 238pu(24°pu) + one-quasi-particlesystems. For the 5/2~ and 9/2- bands a comparison with available data is made. For the 7/2 + and 5/2~ bands, the results of 2aSpu core calculations are not reported smce the band heads are found at an excitation energy larger than 1 MeV. between the 9[2- and 5/2 + states from ~500 keV (as obtained in Hartree-Fock) to ~200 keV, which turns out to be exactly the experimentally proposed energy (see below however). A more extensive presentation of the calculated energies is given in fig. 3 where the results obtained with both the 238pu and the 240pu core are systematically compared (with the exception of'the 7/2 + and 5/2~ bands whose band heads are found at an energy above 1 MeV when evaluated from the 238pu core). It may be seen from this figure that the remarkable agreement between experimental 239pu energies and those calculated with the 238pu core, is lost with the 240pu core. Indeed, the pairing and the Coriolis terms, in the latter case, change the 5/2 + 9 / 2 - energy difference from ~.+200 keV for the pure Hartree-Fock spectrum to ~ - 1 0 0 keV. One may also notice the lowering of the 11/2 + band on the 238pu side as well as the anomalous ordering of the 1/2- band (3[2-, 1[2-, 7 / 2 - , 5 / 2 - , etc.) corresponding to a decoupling parameter of ~ - 1 . Experimental quadrupole moment measurements [5,6] lead for the lowest isomeric state of 239pu to Qmtrinsie = 36 -+ 4 b assuming K = 1 = 5/2. This result is consistent with a recent optical isomer shift measurement [ 15] for the spontaneous fission isomer of 240Am. Since our results show that there is almost no Coriolis band mixing and because of the wellknown dominance of the collective part of the quadrupole operator, it is no surprise that we obtain spectroscopic quadrupole moments for the relevant band heads corresponding to the even-even calculat178
ed intrinsic moment, namely ~32.7 b (~33.8 b) for the 240pu (238pu) core, which falls inside the experimental error bars. From the above reasons and due to the nice agreement yielded for Q, it follows that we should well reproduce B(E2) probabilities within the 5/2 + band (see our estimations in table 2). The reproduction ofB(Ml)probabilities would constitute a much more stringent test. From conversion electron spectra of ref. [6] two kinds of information may be extracted pertaining both to the determination of the mixing ratio 6 2 = T,r(E2)/T.r(M1) for the 9/2 + ~ 7/2 + transition, namely: (i) relative intensities of L I + LII to LII I conversion lines, (ii) ratio X of the total M1 + E2 (9/2 + ~ 7/2 +) intensity to the pure E2 (9/2 + ~ 5/2 +) intensity. It should first be noted that it is very difficult to extract reliable 8 2 values from such spectra. Apart from the statistical quality of the raw data, some problems may arise from the poor knowledge of conversion a-factors for such highly deformed nuclear charge distributions. Since the static effect of the nuclear charge structure is completely different for LI, LI! and LII I lines [16], errors on a-factors due to deformation effects might cast some doubts on the 6 2 factor evaluation from relative L intensites. Moreover experimental data of the above type (i) might not be fully consistent with those of the above type (ii). Indeed, to illustrate this, let us assume (which is highly tentative, in particular due to the nuclear deformation as stated previously) that the a-factors were 177 for the M1 (9/2 + ~ 7/2 +) transition, 3927 for the E2 (9/2 + ~ 7/2 +) transition and 251 for the E2 (9/2 + ~ 5/2 +) transition which
Volume 95B, number 2
PHYSICS LETTERS
22 September 1980
Table 2 Reduced E2 and M1 transition probabilities. Magnetic properties have been calculated for three "reasonable" choices of the gl and gs factors. The results correspond to the 2B8pu core calculations; they are found to be almost identical m the 24°pu core calculations. Transition
B(M1)
(g~q)
_(free) gs = 0.7 g~free) gs = g(free) gs = 0.7 gs gl = 0 gl = 0 gl = -0.06 9/2 + ~ 7/2 + 9/2 + ~ 5/2 + 7/2 + 4 5/2 +
0.385 0.266
0.264 0.182
are the total a-factors for spherical nuclei [17], as done [18] by the authors o f ref. [6]. We deduce then that 8 2 = 0.126/(X - 2.80) where X is the intensity ratio defined above. F r o m experimental intensities as given in fig. 3 o f ref. [6] we f'md X = 2.3 + 0.6 leading to a negative average 8 2 value with a lower limit on the positive side which is higher than the announced 0.7 value as deduced from relative L intensities. Should, however, this relatively high 8 2 value be confirmed, it would appear that our calculations are not able to reproduce it since we find it within the range 0 . 0 6 - 0 . 0 9 depending on whether we calculate it from one or the other core, and choice we make for the gl and gs factors from the possibilities considered in table 2 (where our B(M1) values are to be found). The interpretation of the value 8 2 = 0.7 in terms of a 5/2 + [633] band from the results of ref. [8] seems also to us (see a similar point in ref. [6]) a little premature, not only in view o f the rather ad hoc character of the phenomenological mean field parameterization in that work but mamly in consideration o f the strong coupling o f the [633] and [862] 5/2 + "asymptotic" states in the relevant deformation range. In 237pu, g-factors have been a-easured for two isomeric states [7] (g = 0.18 +- 0.02 for the 1.1/as isomer a n d g = 0.56 +- 0.06 for the 100 ns isomer). As seen in table 3 the K ~r = pr = 3 / 2 - state is a very good candidate for the 100 ns isomer. This assignment is consistent with the conclusions of ref. [8]. For the 1.1/Is isomer lying experimentally 300 -+ 150 keV [19] above the other isomeric state and having a higher spin [20] (ruling out the K ~r = pr = 1 / 2 - state), the K ~r = pr = 7/2 + might be a convenient candidate. It is found however a little too high with respect to the 3 / 2 - state (~1 MeV) in our calculation.
B(E2) (e2 fm4 )
0.343 X 106 0.113 × 106 0.405 X 106
0.300 0.207
Table 3 Calculated g-factors for three "reasonable" choices of the gl and gs factors. Levels
g s = g (free)
gs = 0.7gs(free)
gs = 0.7g
X"(ZO
gl = o
gl = o
gl = - 0 . 0 6
free)
5/2+(5/2) 9/2-(9/2) 3/2-(3/2) 1/2-(1/2) 3/2-(1/2) 11/2+(11/2) 7/2+(7/2)
-0.28 -0.23 0.84 1.77 -0.90 -0.23 0.39
-0.17 -0.14 0.64 1.39 -0.55 -0.14 0.30
-0,21 -0.19 0.59 1.38 -0.55 -0.19 0.25
In this paper we have shown that without any ad hoc parameter adjustment we have been able to give a very good account o f many known spectroscopic properties of 237pu and 239pu fission isomeric states: energy spectrum, quadrupole moments, confirmation of the spin assignments and parity predictions for the latter, magnetic moments at least for one o f the isomeric states o f the former. A remarkable result has been the finding that in spite of the rather high (]2) value for some o f the relevant quasi-particle states, no sizeable Coriolis band mixing has been found. The only remaining discrepancy might be the E2/M1 ratio for 5/2 + band transitions. However, the experimental difficulties o f its determination have been pointed out. As a general conclusion it does not yet appear clear to us, that any known spectroscopic data for such highly "abnormal" nuclear states crucially invalidate the parametrization (especially the s p i n - o r b i t force strength) o f the effective force used here. Obviously, the strong need for more data should be stressed. 179
Volume 95B, number 2
PHYSICS LETTERS
Interesting discussions with M. Asghar, R. B~raud, H. Faust, J. Genevey, F. G6nnenwein, D. Habs and J. Tr~herne are gratefully acknowledged.
References [1] C. Gustafson, I.-L. Lamm, B. Nilsson and S.G. Nilsson, Ark. Fys. 36 (1967) 613. [2] M. Beiner, H. Flocard, Nguyen Van Giai and P. Quentin, Nucl. Phys. A238 (1975) 29. [3] M. Brack, P. Quentin and D. Vautherin, in: Superheavy elements, ed. M.A.K. Lodhi (Pergamon, New York, 1978) p. 309. [4] P. M611erand P. Quentin (1975), unpublished. [5] D. Habs, V. Metag, H.J. Speeht and G. Ulfert, Phys. Rev. Lett. 38 (1977) 387. [6] H. Backe et al., Phys. Rev. Lett. 42 (1979) 490. [7] R. Kalish, B. Herskind, J. Pedersen, D. Schackleton and L. Strabo, Phys. Rev. Lett. 32 (1974) 1009; D. Schackleton, B. Herskind, R. Kalish, J. Pedersen and L. Strabo, in: Proc. Intern. Conf. on Hyperfine interactions studied in nuclear reactions and decay (Uppsala), Contributed papers, eds. E. Karlsson and R. W~'ppling (Upplands Grafiska, Uppsala, 1974) p. 132. [8] I. Hamamoto and W. Ogle, Nuel. Phys. A240 (1975) 54.
180
22 September 1980
[9] A. Bohr and B. Mottelson, Nuclear structure, VoL 2 (Benjamin, New York, 1975). [10] M. Meyer, J. Dani6re, J. Letessier and P. Quentin, Nucl. Phys. A316 (1979) 93. [11] J. Libert, M. Meyer and P. Quentin (1980), to be published; see also P. Quentin, M. Meyer, J. Letessier, J. Libert and M.-G. Desthuilliers-Porquet, in: Nuclear spectroscopy of fission products, ed. T. yon Egidy (The institute of Physics, Bristol, 1980) p. 280. [12] H. Flocard, P. Quentm, D. Vautherm, M. V~n6roni and A.K. Kerman, Nucl. Phys. A231 (1974) 176. [13] H.J. Speckt, J. Weber, E. Konecny and D. Heunemann, Phys. Lett. 41B (1972) 43. [14] R. Vandenbosch,in : Physics and chemistry of fission 1973, Vol. 1 (IAEA, Vienna, 1974) p. 251. [15] C.E. Bemis Jr., J.R. Beene, J.P. Young and S.D. Kramer, Phys. Rev. Lett. 43 (1979) 1854. [16] See e.g.M.E. Rose, in: Alpha-, beta-, gamma-ray spectroscopy, ed. K. Siegbahn, Vol. 2 (North-Holland, Amsterdam, 1968) p. 887. [17] F. R6sel, H.M. Fries, K. Alder and H.C. Pauli, At. Data Nucl. Data Tables 21 (1978) 91. [ 18 ] D. Habs, private communication. [19] R. Vandenbosch, P.A. Russo, G. Sletten and M. Mehta, Phys. Rev. C8 (1973) 1080. [20] P.A. Russo, R. Vandenbosch, M. Mehta, J.R. Tesmer and K.L. Wolf, Phys. Rev. C3 (1971) 1595.
PHYSICS LETTERS
22 September 1980
SPECTROSCOPIC PROPERTIES OF 237,239pu FISSION ISOMERS FROM SELF-CONSISTENT CALCULATIONS J. LIBERT CSNSM, 91406 Orsay, France
M. MEYER 1PN Lyon (and IN2P3), 69622 Villeurbanne, France
and P. QUENTIN Institut Laue-Langevin, 156X, 38042 Grenoble-Cedex, France
Received 21 June 1980
We have studied 237,239puisomeric states: energy levels, M1 and E2 spectroscopic moments and reduced transition probabilities have been calculated within the rotor + quasi-particle model from self-consistently determined single-particle states. The electromagnetic properties of these states have been especially investigated. Without any ad hoc parameter adjustment, a very good reproduction of most of the known spectroscopic data is yielded which assessesthe predictive power of the whole approach.
Odd deformed nuclei are ideally suited for a precise determination of Hartree-Fock single-particle (or BCS quasi-particle) energies. Unlike the case of quasispherical systems corresponding to the vicinity of closed shells, the coupling between their individual and core vibration degrees of freedom is generally weak and the coupling with collective rotational motion is the only one being of some importance. This is why the experimental spectra of odd deformed nuclei have constituted the main ingredient of any fitting procedure of phenomenological single-particle potentials, as done in'ref. [1]. To the best of our knowledge, this procedure has not been carried out for the determination of most phenomenological effective forces (especially their spin-orbit component) in use in Hartree-Fock calculations. For reasons of computational simplicity, the parameter search has been done only for spherical solutions, see e.g. ref. [2]. This fact may lead in practice to rather large uncertainties in the spin-orbit force strength [3]. It turns out, however, that the Skyrme SIII effective
force [2] proves a posteriori to correspond to a reason. able choice of this strength. A hint at this may come from a detailed comparison between the Nilsson model and Hartree-Fock single-particle spectra in both the rare-earth and the actinide region [4]. Nevertheless this type of comparison was restricted so far to the "normal" range of nuclear deformation 032 0.2-0.3). Recent spectroscopic data for fission isomeric states in 239Pu [5,6] together with older data on 237pu [7] provide a challenging testing ground for any theoretical estimate of the mean nuclear field at larger deformations (/32 ~ 0.6). In the 237pu case, the available data consist of measurements of the g-factors for two fission isomers whose spins are essentially unknown, even though one has tried to draw conclusions [8] from measured fragment angular distributions. For the 239Pu states, a rotational band based on a 5/2 spin state has been identified, together with a 9/2 spin state lying 203 keV higher. From conversion electron measurements an experimental limit for the M1/E2 ratio for transitions 175
Volume 95B, number 2
PHYSICS LETTERS
within the ground band has been proposed [6]. The latter combined with the knowledge of a previously measured quadrupole moment [5] has led the authors of ref. [6] to suggest an antiparallel coupling of the orbital angular momentum and spin vectors in the single-particle 5/2 state. In this paper we report on calculations aiming at a description of spectroscopic properties of 237pu and 239pu isomeric states within the rotor + quasi-particle approach of Bohr and Mottelson [9]. Technical detafls of the method used in the present calculations can be found in ref. [10] for the diagonalization of the hamiltonian and in ref. [ 11 ] for the evaluation of multipole moments and transition probabilities. The underlying Hartree-Fock + BCS results (for 240Pu) have been published long ago [12]. It is worth recalling that such an approach does not imply any ad hoc parameter adjustment: (i) the effective force parameters are those discussed in ref. [2] and are valid for the whole chart of nuclides; (ii) the pairing gap parameters have been deduced [12] through a well-defined if not fully satisfactory prescription from odd-even (ground-state) binding energy differences; (iii) the rotor hamiltonian parameters (9(17) with the notation of ref. [10]) are determined by experimental results in 240pu [13]; (iv) there is neither any deformation parameter (/32,/34.... ) adjustment, nor any ambiguity due to the choice of a fission path (apart from an a priori choice of self-consistent symmetries) because we are considering a local extremum HartreeFock solution; (v) we have naturally included any possible "AN = 2" coupling; (4) we have taken into account the full rotor kinetic energy (thus including the so-called recoil one-body term/2) and we have not used any renormalized Coriolis coupling term (no attenuation factor). As is well known, a one-quasi-particle wave function does not correspond in general to a state with an integer average number of particles. That is why, starting from a BCS solution describing an even nucleus with A nucleons, the lowest one-quasi-particle states will describe an admixture ofA + 1 and A - 1 nuclear states. Consequently, in order to study for instance the 239pu nucleus, it will be suitable to compare theoretical results obtained in calculations corresponding to both 240pu and 238pu BCS solutions. To compute the latter we have assumed that the optimal basis parameters and pairing gaps were those of 240pu. 176
22 September 1980
This has been a posteriofi supported (at least for the basis deformation parameter) by the fact that the local equilibrium solutions of 238pu and 240pu correspond to about the same quadrupole moment (33.8 b in 238pu and 32.7 b in 240pu for the charge distribution). One observes in fig. 1 that the gross structure of the single-particle neutron levels is not very much affected when going from N = 144 to N = 146 which is consistent with the above-mentioned similarity of the fission isomer quadrupole moments. One may note, however, the usual occurrence of a bunching of the single-particle levels immediately below and above the Fermi surface, resulting particularly in an increased energy difference between the lowest 5/2 + and the 9/2- levels in 238pu (~500 keV instead of ~200 keV in 240pu). In ref. [12] we have already noted the striking similarity of our 24°pu spectrum with the one obtained by M611er, Nilsson and Nix (see fig. 12 of ref. [14]). For most of the single-particle states whose energies are given in fig. 1, we present in table 1 the main coefficients of the expansion in the axially sym-
~N
HF (MeV)
712+ 5/2 + 1/2 +
-5
912 -6 -
512 +
i
1112+~ -7
312-~-8
112 +
l t 2 - ~ 312312+ 238 240 Pu Pu
Fig. 1. Neutron Hartree-Fock spectra for the calculated fission isomeric states of 238pu and 24°pu. The Fermi level h resulting from our BCS calculations is also given.
Volume 95B, number 2
PHYSICS LETTERS
22 September 1980
Table 1 Characterization of the quasi-particle states near the Fermi level. The Hartree-Fock energies e are given together with the expanston coefficients in deformed harmonic oscillator states, in current notation, for the self-consistent 24°pu solution. The pairing occupation probabilities f/2 are also given for both the 238pu and the 24°pu solution. The expectation values of the one-body operator j 2 in the lowest 5/2 + and the 9/2- states are also given (for the 24°pu solution). Level
e (MeV) V~4o
V~as
5/2 + 7/2 + 9/2 + 5/2 + 11/2 + 1/2-
-4.71 -4.78 -5.82 -6.04 -6.92 -7.24
0.071 0.035 0.420 0.568 0.901 0.936
0.037 0.035 0.164 0.404 0.641 0.882
3/2-
-7.43
0.949
0.897
~/2) (~2) Main components in the harmonic oscillator basis
39 57
0.63[622] 0.78[624] 0.77[734] 0.55[862] 0.881615] 0.441510] 0.211310] 0.551512] 0.201312]
metric harmonic oscillator basis. They are only given for the 240pu solution since they are essentially identical in 238pu. In the same table we have also reported the pairing occupation probabilities for the two solutions (in both cases the neutron pairing gap was 0.732 MeV [12]). For the lowest 5/2 + and 9 / 2 - states we have listed the expectation value of the j2 operator showing that these states are indeed stemming from rather high-j spherical orbitals. As previously noted [4] the H a r t r e e - F o c k field mixes much more strongly than the modified harmonic oscillator basis states coming from different major shells (e.g. up to four shells significantly contribute to the 1 / 2 - and 3 / 2 - states). It is also noticeable that for all the states close to the Fermi surface, the mixing between spin-up and spin-down states is found to be very small (typically 80% of the wave function at least, corresponds to the same spin projection). The only spin-down states listed in table 1 are the 3 / 2 and 7/2 + states. The closest 5/2 spin-down state is a 5/2 + state lying rather deep in the Fermi sea (at a • H a r t r e e - F o c k energy of ~ - 9 . 9 MeV) and whose components are: 0.77 [633], 0.38 [853], - 0 . 2 5 [413]. As an illustration of our results, we have presented in fig. 2 the calculated 5/2 + and 9 / 2 - bands starting from the 238pu core in the three following cases: (i) no pairing and no RPC (Coriolis) terms, (ii) pairing and no RPC terms, (iii) full rotor + one quasi-particle hamiltonian (with pairing included). It has to be understood that, in all cases, the diagonal part of the rotor kinetic energy (namely (h2/2 9)[1(1+ 1 ) - K 2 {/2)] with current notation) is included. This term is
-0.40[822] 0.36[844] -0.37[954] 0.47[842] -0.371815] -0.43[750]
-0.311862] -0.25[824] -0.28[934] 0.3911082] 0.30[835] -0.40[970]
0.301422] 0.24[633] -0.271514] -0.23[642]
-0.48[752]
-0.341712]
-0.311972]
E (MeV) 06
-0.321710]
HF
HF +
+
--
BCS
BC.S
-HF
04
HF
HF
+
+
BCS
BCS COR
EXP
HF
--
C~)R
---~
- -
-
-
,, ',
-/ .
02
EXP
.
.
.
.
9/2-
5/2+ Fig. 2. Influence of the various parts of the rotor + quasiparticle hamiltonian on the ground 5/2 + and the 9/2- bands. In the calculations reported in this work, all quasi-particle states corresponding to Hartree-Fock energies lying within -+2 MeV around the Fermi level have been included. responsible for lowering the Hartree-Fock energy difference between the 9 [ 2 - and 5/2 + band heads to ~400 keV. Including further pairing correlation resuits for these unoccupied states in a sizeable compression of their quasi-particle energy difference (as compared to their particle energy difference). The inclusion of RPC terms yields only a small energy shift. Indeed, in this case, as in all results presented here, there is n o significant Coriolis band mixing (typically less than 1% of the final wave function). The net effect of all these terms is to shift the energy difference 177
Volume 95B, number 2
PHYSICS LETTERS E
238F~ + 2 4+0 p u
(MeV
lqp
1 qp
a % E,p 24o~, 0.5 - l q p
lqp
5i/21+
22 September 1980
_- -_
-- -- -
~=
~
2/,0 +1% lq. p
= __
--
9/2-
11/2+
__
~r
1//2-
3/2
7~+
5/2
Fig. 3. Calculated spectra for the 238pu(24°pu) + one-quasi-particlesystems. For the 5/2~ and 9/2- bands a comparison with available data is made. For the 7/2 + and 5/2~ bands, the results of 2aSpu core calculations are not reported smce the band heads are found at an excitation energy larger than 1 MeV. between the 9[2- and 5/2 + states from ~500 keV (as obtained in Hartree-Fock) to ~200 keV, which turns out to be exactly the experimentally proposed energy (see below however). A more extensive presentation of the calculated energies is given in fig. 3 where the results obtained with both the 238pu and the 240pu core are systematically compared (with the exception of'the 7/2 + and 5/2~ bands whose band heads are found at an energy above 1 MeV when evaluated from the 238pu core). It may be seen from this figure that the remarkable agreement between experimental 239pu energies and those calculated with the 238pu core, is lost with the 240pu core. Indeed, the pairing and the Coriolis terms, in the latter case, change the 5/2 + 9 / 2 - energy difference from ~.+200 keV for the pure Hartree-Fock spectrum to ~ - 1 0 0 keV. One may also notice the lowering of the 11/2 + band on the 238pu side as well as the anomalous ordering of the 1/2- band (3[2-, 1[2-, 7 / 2 - , 5 / 2 - , etc.) corresponding to a decoupling parameter of ~ - 1 . Experimental quadrupole moment measurements [5,6] lead for the lowest isomeric state of 239pu to Qmtrinsie = 36 -+ 4 b assuming K = 1 = 5/2. This result is consistent with a recent optical isomer shift measurement [ 15] for the spontaneous fission isomer of 240Am. Since our results show that there is almost no Coriolis band mixing and because of the wellknown dominance of the collective part of the quadrupole operator, it is no surprise that we obtain spectroscopic quadrupole moments for the relevant band heads corresponding to the even-even calculat178
ed intrinsic moment, namely ~32.7 b (~33.8 b) for the 240pu (238pu) core, which falls inside the experimental error bars. From the above reasons and due to the nice agreement yielded for Q, it follows that we should well reproduce B(E2) probabilities within the 5/2 + band (see our estimations in table 2). The reproduction ofB(Ml)probabilities would constitute a much more stringent test. From conversion electron spectra of ref. [6] two kinds of information may be extracted pertaining both to the determination of the mixing ratio 6 2 = T,r(E2)/T.r(M1) for the 9/2 + ~ 7/2 + transition, namely: (i) relative intensities of L I + LII to LII I conversion lines, (ii) ratio X of the total M1 + E2 (9/2 + ~ 7/2 +) intensity to the pure E2 (9/2 + ~ 5/2 +) intensity. It should first be noted that it is very difficult to extract reliable 8 2 values from such spectra. Apart from the statistical quality of the raw data, some problems may arise from the poor knowledge of conversion a-factors for such highly deformed nuclear charge distributions. Since the static effect of the nuclear charge structure is completely different for LI, LI! and LII I lines [16], errors on a-factors due to deformation effects might cast some doubts on the 6 2 factor evaluation from relative L intensites. Moreover experimental data of the above type (i) might not be fully consistent with those of the above type (ii). Indeed, to illustrate this, let us assume (which is highly tentative, in particular due to the nuclear deformation as stated previously) that the a-factors were 177 for the M1 (9/2 + ~ 7/2 +) transition, 3927 for the E2 (9/2 + ~ 7/2 +) transition and 251 for the E2 (9/2 + ~ 5/2 +) transition which
Volume 95B, number 2
PHYSICS LETTERS
22 September 1980
Table 2 Reduced E2 and M1 transition probabilities. Magnetic properties have been calculated for three "reasonable" choices of the gl and gs factors. The results correspond to the 2B8pu core calculations; they are found to be almost identical m the 24°pu core calculations. Transition
B(M1)
(g~q)
_(free) gs = 0.7 g~free) gs = g(free) gs = 0.7 gs gl = 0 gl = 0 gl = -0.06 9/2 + ~ 7/2 + 9/2 + ~ 5/2 + 7/2 + 4 5/2 +
0.385 0.266
0.264 0.182
are the total a-factors for spherical nuclei [17], as done [18] by the authors o f ref. [6]. We deduce then that 8 2 = 0.126/(X - 2.80) where X is the intensity ratio defined above. F r o m experimental intensities as given in fig. 3 o f ref. [6] we f'md X = 2.3 + 0.6 leading to a negative average 8 2 value with a lower limit on the positive side which is higher than the announced 0.7 value as deduced from relative L intensities. Should, however, this relatively high 8 2 value be confirmed, it would appear that our calculations are not able to reproduce it since we find it within the range 0 . 0 6 - 0 . 0 9 depending on whether we calculate it from one or the other core, and choice we make for the gl and gs factors from the possibilities considered in table 2 (where our B(M1) values are to be found). The interpretation of the value 8 2 = 0.7 in terms of a 5/2 + [633] band from the results of ref. [8] seems also to us (see a similar point in ref. [6]) a little premature, not only in view o f the rather ad hoc character of the phenomenological mean field parameterization in that work but mamly in consideration o f the strong coupling o f the [633] and [862] 5/2 + "asymptotic" states in the relevant deformation range. In 237pu, g-factors have been a-easured for two isomeric states [7] (g = 0.18 +- 0.02 for the 1.1/as isomer a n d g = 0.56 +- 0.06 for the 100 ns isomer). As seen in table 3 the K ~r = pr = 3 / 2 - state is a very good candidate for the 100 ns isomer. This assignment is consistent with the conclusions of ref. [8]. For the 1.1/Is isomer lying experimentally 300 -+ 150 keV [19] above the other isomeric state and having a higher spin [20] (ruling out the K ~r = pr = 1 / 2 - state), the K ~r = pr = 7/2 + might be a convenient candidate. It is found however a little too high with respect to the 3 / 2 - state (~1 MeV) in our calculation.
B(E2) (e2 fm4 )
0.343 X 106 0.113 × 106 0.405 X 106
0.300 0.207
Table 3 Calculated g-factors for three "reasonable" choices of the gl and gs factors. Levels
g s = g (free)
gs = 0.7gs(free)
gs = 0.7g
X"(ZO
gl = o
gl = o
gl = - 0 . 0 6
free)
5/2+(5/2) 9/2-(9/2) 3/2-(3/2) 1/2-(1/2) 3/2-(1/2) 11/2+(11/2) 7/2+(7/2)
-0.28 -0.23 0.84 1.77 -0.90 -0.23 0.39
-0.17 -0.14 0.64 1.39 -0.55 -0.14 0.30
-0,21 -0.19 0.59 1.38 -0.55 -0.19 0.25
In this paper we have shown that without any ad hoc parameter adjustment we have been able to give a very good account o f many known spectroscopic properties of 237pu and 239pu fission isomeric states: energy spectrum, quadrupole moments, confirmation of the spin assignments and parity predictions for the latter, magnetic moments at least for one o f the isomeric states o f the former. A remarkable result has been the finding that in spite of the rather high (]2) value for some o f the relevant quasi-particle states, no sizeable Coriolis band mixing has been found. The only remaining discrepancy might be the E2/M1 ratio for 5/2 + band transitions. However, the experimental difficulties o f its determination have been pointed out. As a general conclusion it does not yet appear clear to us, that any known spectroscopic data for such highly "abnormal" nuclear states crucially invalidate the parametrization (especially the s p i n - o r b i t force strength) o f the effective force used here. Obviously, the strong need for more data should be stressed. 179
Volume 95B, number 2
PHYSICS LETTERS
Interesting discussions with M. Asghar, R. B~raud, H. Faust, J. Genevey, F. G6nnenwein, D. Habs and J. Tr~herne are gratefully acknowledged.
References [1] C. Gustafson, I.-L. Lamm, B. Nilsson and S.G. Nilsson, Ark. Fys. 36 (1967) 613. [2] M. Beiner, H. Flocard, Nguyen Van Giai and P. Quentin, Nucl. Phys. A238 (1975) 29. [3] M. Brack, P. Quentin and D. Vautherin, in: Superheavy elements, ed. M.A.K. Lodhi (Pergamon, New York, 1978) p. 309. [4] P. M611erand P. Quentin (1975), unpublished. [5] D. Habs, V. Metag, H.J. Speeht and G. Ulfert, Phys. Rev. Lett. 38 (1977) 387. [6] H. Backe et al., Phys. Rev. Lett. 42 (1979) 490. [7] R. Kalish, B. Herskind, J. Pedersen, D. Schackleton and L. Strabo, Phys. Rev. Lett. 32 (1974) 1009; D. Schackleton, B. Herskind, R. Kalish, J. Pedersen and L. Strabo, in: Proc. Intern. Conf. on Hyperfine interactions studied in nuclear reactions and decay (Uppsala), Contributed papers, eds. E. Karlsson and R. W~'ppling (Upplands Grafiska, Uppsala, 1974) p. 132. [8] I. Hamamoto and W. Ogle, Nuel. Phys. A240 (1975) 54.
180
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[9] A. Bohr and B. Mottelson, Nuclear structure, VoL 2 (Benjamin, New York, 1975). [10] M. Meyer, J. Dani6re, J. Letessier and P. Quentin, Nucl. Phys. A316 (1979) 93. [11] J. Libert, M. Meyer and P. Quentin (1980), to be published; see also P. Quentin, M. Meyer, J. Letessier, J. Libert and M.-G. Desthuilliers-Porquet, in: Nuclear spectroscopy of fission products, ed. T. yon Egidy (The institute of Physics, Bristol, 1980) p. 280. [12] H. Flocard, P. Quentm, D. Vautherm, M. V~n6roni and A.K. Kerman, Nucl. Phys. A231 (1974) 176. [13] H.J. Speckt, J. Weber, E. Konecny and D. Heunemann, Phys. Lett. 41B (1972) 43. [14] R. Vandenbosch,in : Physics and chemistry of fission 1973, Vol. 1 (IAEA, Vienna, 1974) p. 251. [15] C.E. Bemis Jr., J.R. Beene, J.P. Young and S.D. Kramer, Phys. Rev. Lett. 43 (1979) 1854. [16] See e.g.M.E. Rose, in: Alpha-, beta-, gamma-ray spectroscopy, ed. K. Siegbahn, Vol. 2 (North-Holland, Amsterdam, 1968) p. 887. [17] F. R6sel, H.M. Fries, K. Alder and H.C. Pauli, At. Data Nucl. Data Tables 21 (1978) 91. [ 18 ] D. Habs, private communication. [19] R. Vandenbosch, P.A. Russo, G. Sletten and M. Mehta, Phys. Rev. C8 (1973) 1080. [20] P.A. Russo, R. Vandenbosch, M. Mehta, J.R. Tesmer and K.L. Wolf, Phys. Rev. C3 (1971) 1595.