Dipole and octupole pairs in deformed actinide nuclei

Volume 182, number 3,4

PHYSICS LETTERS B

25 December 1986

DIPOLE AND OCTUPOLE PAIRS IN D E F O R M E D ACTINIDE NUCLEI Takaharu OTSUKA Departmentof Physics, JapanAtomic EngergyResearchInstitute, Tokai, Ibaraki, 319-11, Japan Received 26 July 1986

The structure of deformed actinide nuclei is studied in terms of an intrinsic hamiltonian containing an octupole mean field in addition to a quadrupole field (Nilsson potential). It is shown that there is a significant amount of the collective dipole ( 1 - ) nucleon pair in the wave function of low-lying states, and that the dipole mode, for instance the dipole boson, has to be included in the description of such nuclei as well as the octupole one.

Low-lying negative-parity collective states are observed in even-even deformed actinide nuclei [ 1]. The lowest negative-parity state in these nuclei is 1in most cases, and its excitation energy comes down to 200-300 keY for A ~ 222-228 [ 1 ]. Two algebraic approaches to describe these actinide nuclei have been proposed recently. One approach [2-4] includes the dipole (1 - ) boson as a negative-parity degree of freedom, while the other [ 5 ] employs the dipole and octupole ( 3 - ) bosons. The dipole mode is introduced in both approaches as an essential ingredient. On the other hand, it has been suggested [6-11 ] that strong and static octupole deformation occurs in the same region of the periodic table to the extent that the RPA treatment of the octupole vibration breaks down. Here I shall study the structure of these nuclei in terms of an intrinsic hamiltonian containing an octupole mean field in addition to a quadrupole one. This parity-violating intrinsic hamiltonian seems to be reasonable for nuclei where positive- and negative-parity states are close to each other. It will be shown that the intrinsic wave function has a dipole ( 1 ) pair correlation which is comparable to an octupole one. This rather strong dipole correlation has never been reported. The result here suggests, under strong quadrupole-octupole deformation, the need for the dipole mode in the description of actinide nuclei which is one of the current intriguing subjects in the nuclear structure study. The intrinsic hamiltonian is assumed to be h=ho +h2 + h 3 , 256

(1)

where ho stands for the spherical part, and h2 and h 3 imply the quadrupole and octupole mean fields. The axially symmetric deformation is considered here. The first two terms on the right-hand side of eq. (1) are included in the usual Nilsson model [ 12,13 ]. The strength of h2 is proportional to the deformation parameter, 62; h 2 = -- 262MO92r2C~2)(O), where M and o9 denote the nucleon mass and the oscillator frequency for the spherical potential (ho9 = 41 A - ~/3 MeV), and r and 0 are coordinates in the intrinsic system. Utilizing the formula ( r 2) = (N+~)h/Mog, and evaluating the harmonic-oscillator quanta N by N + 2 ~ (3A/2) ~/3 (ref. [ 141 ] h2 is expressed as hz~ -3162C~2)(0)

•

(2)

This approximation should suffice, since the basic property is discussed here. The octupole mean field is parametrized as h3 ~, - f t / 3 C63) ( 0 ) ,

(3)

where t/3 is a dimensionless parameter similar to 62, and fdenotes a scaling parameter of the dimension of energy. The same scaling as h2 is taken for simplicity; f = 31 (MeV). The origin of the octupole mean field should be the octupole-octupole term in the nucleon-nucleon interaction [ 7,10,11,15 ] of which the r-dependence is not known precisely and may not be simple. The average effect of such an r-dependence should be included i n f a n d r/3. The evaluation of ~/3is a difficult task at present, and thereby various values are used. Note that the major conclusion is quite insensitive to r/3, as will be shown.

Volume 182, number 3,4

PHYSICS LETTERSB

The spherical part h0 in eq. (1) is comprised of the spherical single-particle energy and the monopole pairing interaction. The single-particle energy is calculated by the density-dependent Hartree-Fock calculation [16,17] with some correction so that the energy difference between two major shells is ~ 1hog. The strength of the monopole pairing interaction is taken so that the pairing gap is ~ 1 MeV. Again, the precise values of the spherical single-particle energy and the pairing interaction strength are not relevant to the final conclusion. I shall begin with an actinide nucleus 226Th as an example. This nucleus has the ground-state rotational band and a low-lying negative-parity rotational band starting from 1 at Ex = 230 keV which is only 4 keV above 4 +. The doubly closed core 2°8pb is assumed. Two major shells on the top of this core are taken both for protons and neutrons. The centerof-mass (CM) motion of the total system is treated as follows. The CM motion of valence nucleons can be cancelled by the core CM motion to the opposite direction. The core CM motion is nothing but a particle-hole excitation. The valence-nucleon states considered in this work are assumed to carry implicitly this core CM excitation so that the total CM motion vanishes. The mixing amplitude of the core CM excitation is indeed small for actinide nuclei since it is an effect of O(1/A). It is then assumed that the correct treatment of the CM motion does not change significantly the following discussions and final conclusion. A more detailed account of this point will be presented elsewhere. The quadrupole deformation parameter is assumed to be 62=0.20, while the 62-dependence will be discussed later. The deformed single-particle energies are calculated from ho, h2 and h3, resulting in a deformed and parity-mixed single-particle basis. The hamiltonian in eq. ( l ) is solved by the BCS approximation because of the pairing interaction. The BCS ground state is analyzed in a similar way to ref. [ 18]. The only difference between this work and ref. [ 18] is that parity is not conserved here. The BCS ground state is projected onto an N-pair condensate I ~r/) oc (At) N] ) with

A* oc Z (vu/uu)a*ua~" /t>0

(4)

where A * is the normalized pair creation operator,/2

25 December 1986

stands for a single-particle state in the quadrupoleplus-octupole potential,/Z denotes its time reversal state, v, and u~ are obtained by the BCS calculation, a~ creates a nucleon in the state ~t, and I ) implies the closed shell. Eq. (4) is obtained for protons and neutrons separately. I shall consider the neutron system here only, since the proton system can be analyzed in the same way. Hence, the number of pairs refers to the number of neutron pairs, denoted by N, here. Note that N~ = 5 for 226Th. By decomposing the deformed and parity-mixed state/z in terms of the original spherical basis, the A pair can be expanded into a linear combination of nucleon pairs of definite angular momentum and parity;

A t =z+A*+ +z_A*_ , At+ =Xo St +xzD t + X 4 G't ....

A~- =Yl p* +y3F* +ysH ~ +...,

(5)

where At+ (A*_) creates the positive (negative) parity component of the A pair with z+ ( z ) being its amplitude ( z2+ + z 2 = 1 ), S t, Dr, G ~, P*, F*, and H t denote the normalized creation operators of the collective nucleon pairs o f J ~ = 0 +, 2 ÷, 4 ÷, 1 , 3-, and 5 - with x's and y's being their amplitudes. Note that Yx~ = ~y~_ = 1. Only natural-parity pairs appear in eq. (5). The magnetic quantum number is zero for all pairs in eq. (5) because of the axial symmetry. Fig. 1 shows z2_, y~, y2 andy] as function oft/3 in eq. (3). The probability z 2_ becomes larger for r/3 larger. The probabilities y2 are, on the other hand, nearly unchanged. In other words, the total probability of the negative-parity component is approximately proportional to q3 in the range of fig.l, whereas the relative probabilities among pairs within the A pair are almost constant. In an improved approximation, the intrinsic state is obtained by the variation after parity projection [ 7 ]. The major effect of parity projection is that the strength of the octupole mean field for the positive-parity state is different from that for the negative-parity state [ 7 ]. Fig. 1 indicates that YL is almost independent of the parity projection, while z 2_ may be sensitive. Fig. lb shows the probabilities y~, y32 and y2. It should be noticed that the P-pair probability is finite, 257

Volume 182,number 3,4 50

F

PHYSICSLETTERSB .

,

,

,

i

,

,

(a)

O~

(b)

5O P 1

~

l

~

q

l

q

q

b

o 50I (c) caH 0

011

OL.O . . . . %

Fig. 1. Structure of the A pair (Cooper pair) obtained by the intrinsic hamiltonian with the quadrupole-octupole single-particle potential (dashed lines), and by the same hamiltonian plus a weak dipole potential (solid lines). The abscissa shows the strenghth of the octupole potential. (a) Total probability of the negative-parity componentsin the A pair. (b) and (c) Probabilities ofJ ~= 1- (P), 3- (F) and 5- (H) pairs in the A pair devided by the probability in (a). and is even of the same order of magnitude as the Fpair probability y]. This is of great interest because there is no dipole field implemented in h in eq. (1). The P pair is produced when the quadrupole field acts on the F pair or the octupole field acts on the D pair. The distribution of nucleons over deformed single-particle orbits is more spread out in the P pair than in the F pair. In other words, the F pair has more complicated inner structure than the P pair, mainly because the Clebsch-Gordan coefficient for J = 3 is not as simple as that for J = 1. Hence, the P pair is favoured by the Pauli principle and the pairing interaction. The same mechanism was found for the S - D - G pair dominance of the Nilsson wave function [ 18]. The P pair thus appears as a consequence of the quadrupole deformation and many-body correlation effects. There is also the H pair. Although y2 is larger than y2, the relative importance of the P pair over the H pair is enhanced in low-spin states, because the probability of the P pair in low-spin states after the angular momentum projection is larger than that in the intrinsic state, whereas the probability of the H pair is less increased or probably decreased. This is an important point and is being studied in terms of the parity and angular-momentum projection. It is expected that H-pair effects can be included effectively by an appropriate renormalization simi258

25 December 1986

larly to G pair effects [ 19 ]. There is another reason for the importance of the P pair over the H pair, as will be shown later. The expectation value of C~ K) (0) with respect to the intrinsic state is calculated as ( C ( K ) ) = 2 . 6 9 , 4.88, 2.40, 1.67, and 1.00 for K = 1-5, respectively. One finds that ( C (t)) is larger than (C(3)). This means that the dipole-dipole term in the nucleon-nucleon interaction cannot be neglected for the intrinsic state. To see this point more precisely, the surface delta interaction (SDI) [20,21 ] is used. The SDI can be written as Voc E • (2K+ 1 )(C (K) (Oi)" C (K) (0j)).

(6)

i,j K

where i a n d j refer to nucleons, and ( ) means a scalar product. The mean field produced by eq. (6) is written as Foc ~] (2K+ 1 ) C(oK) ( C (K)) ,

(7)

K

where the exchange process is neglected as usual for simplicity. Since ( C ° ) ) is larger than (C(3)), the strength of the dipole mean field given by eq. (7) is larger than ~ of that of the octupole mean field. The relative ratio of the octupole-octupole interaction over other multipole-multipole interactions may be larger in the actinide region than the ratio given by eq. (6), due to the strong octupole core-polarization. I then introduce a reduction facter ½ for the dipole mean field. This reduction factor is considered to be a "safety factor" against overestimation. Combining these factors with actual values of [C(1'3)), the following weak dipole field is added to h in eq. (1): ht = - f q l C(t)(O) ,

(8)

where q~ is a parameter with q~ =~q3. Although the same argument applies to other higher multipoles, the sensitivity of < c¢K)) to inclusion of the Kth multipole mean field rapidly decreases as K increases. I hence consider only the dipole mean field beside the quadrupole and octupole fields. A more comprehensive self-consistent calculation is desired. The intrinsic hamiltonian then becomes

h=ho+hl

+h2+h3

.

(9)

Fig. 1c shows y~ (L = 1, 3, 5) calculated by h in eq.

Volume 182,number 3,4

PHYSICS LETTERSB

(9). The essential feature does not change from fig. lb, while the P pair is more favoured and becomes the second largest among the negative-parity pairs. I now survey the N~ dependence ofyL and ( C (K)). Since the quadrupole deformation increases as N~ in the actinide region, 62 is assumed as 6 2 = 0 . 2 0 + 0 . 0 3 × ( N ~ - 5 ) ( 2 ~ < N ~ 11 in this case). The other parameters are fixed at the values for the above calculation with 713 0.10. Fig. 2a shows enormous enhancement of ( C t2) ) as N, increases. This enhancement occurs even without the increase of 62, but is accelerated by enlarging 62. Fig. 2a shows also ( C (1'3)). These quantities are saturated around N~=6-8, and begin to decrease around N~= 8-10. Thus, fig. 2a demonstrates that, as function ofN~, ( C °'3) ) and ( C (2)) are comparable for N,~< 6, whereas ( C (2)) dominates for N,>_.8. This seems to explain why the K = 0 - band is so low at N~~ 5 (for instance 226Th), while it goes up rapidly at N~>~8 (for instance 2 3 6 U ) . Fig. 2b shows z 2_ in eq. (5). This probability is approximately constant as function of N,, implying that the different behaviours of ( C ( 2 ) > and ( C (1'3)) are not due to z 2_. The most striking result seen in fig. 2 is y2. The Ppair probability y~ increases impressively as N~, and =

10 (o)

'C{21J

]

(_> ~/

C(1) C13)] 0

>.

P

I

,

,

,

,

P

~-- 50~I (c) ~

p

F

n

H

0

i

0

i

i

i

i

i

10

i

Nu Fig. 2. (a) Expectation values of the single-particle operator C~x) (0) with respectto the intrinsicstate. The numberof valenceneutron pairs, N~, is varied. (b) Sameas fig. la as functionof N,. (c) Sameas fig. lc as functionof N~.

25 December 1986

becomes almost equal to the F-pair one y32 around N~ = 10. This P-pair enhancement is, as emphasized already, due to the many-body effect and the growing quadrupole deformation. Note that the octupole and dipole fields are not changed in fig. 2. This P-pair enhancement in heavier actinide nuclei may be related to the fission mechanism, because it is basically the enhancement of the linear motion of the valence nucleons and the core in the opposite directions. The H pair decreases as N~ increases in fig. 2c. The importance of the P pair over H is thus evident. If r/t = 0, the P pair is less favoured, but its probability is still larger for N~>~7 than that of the H pair. I add a comment on the CM motion. The intrinsic state considered here contains the CM spurious motion of the total system, although its amplitude should be small as already mentioned. This CM motion is carried not only by the P pair but also by all other pairs, if there is some quadrupole deformation. The accurate treatment of the CM motion therefore does not necessarily reduce only the amplitude of the P pair, and probably has effects on all pairs. The probability of the P pair becomes smaller as the quadrupole deformation decreases. The F pair is the dominant negative-parity in lowest negative-parity states of spherical nuclei. The conventional octupole phonon or boson picture is valid there. On the other hand, there are negative-parity states in deformed rare-earth nuclei. The P pair should be needed to describe these states [ 22 ]. The A pair in eq. (5) can be mapped to an intrinsic boson by the method of the author and Yoshinaga [23]. As already mentioned, the H pair will be removed as well as the G pair by some appropriate renormalization [ 19]. The present work thus suggests, from the microscopic point of view, a boson model which includes s(0+), d(2+), p ( 1 - ) and f ( 3 - ) bosons. This suggestion coincides with an algebraic model proposed recently by Engel and Iachello [ 5 ]. In summary, the dipole pair has to be included in the description of deformed actinide nuclei as well as the octupole pair, for instance, by introducing the dipole boson in addition to the octupole boson. The author acknowledges Professor A. Arima, Dr. N. Yoshida, Dr. N. Yoshinaga and Dr. M. Sugita for valuable discussions. 259

Volume 182, number 3,4

PHYSICS LETTERS B

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[ 13 ] S.G. Nilsson, Mat. Fys. Medd. Dan. Vid. Selsk. 29, No. 16 (1955). [ 14] D,R. Bes and R.A. Sorensen, in:Advances in nuclear physics, eds. M. Baranger and E. Vogt, Vol. 2 (Plenum, New York, 1969). [ 15 ] L.M. Robledo, J.L. Egido and P. Ring, preprint LBL- 19520 (1985). [ 16] D. Vautherin and D.M. Brink, Phys. Rev. C 5 (1972) 626. [ 17 ] N. Van Giai and H. Sagawa, Phys. Lett. B 106 (1981 ) 379. [ 18] T. Otsuka, A. Arima and N. Yoshinaga, Phys. Rev. Lett. 48 (1982) 387; D.R. Bes, R.A. Broglia, E. Maglione and A. Vitturi, Phys. Rev. Lett. 48 (1982) 1001; K. Sugawara-Tanabe and A. Arima, Phys. Lett. B 110 (1982) 87. [19] T. Otsuka and J.N. Ginocchio, Phys. Rev. Lett. 55 (1985) 276. [20] P.J. Brussaard and P.W.M. Glaudemans, Shell-model applications in nuclear spectroscopy (North-Holland, Amsterdam, 1977). [ 21 ] I. Green and S.A. Moszkowski, Phys. Rev. 139 (1965 ) 790. [22] H.J. Daley and M.A. Nagarajan, Phys. Lett. B 166 (1986) 379. [23] T. Otsuka, Phys. Lett B 138 (1984) 1; T. Otsuka and N. Yoshinaga, Phys. Lett. B 168 (1986) 1.