“Independent-pair” property of condensed coherent fermion pairs and derivation of the IBM quadrupole operator

Volume 138B, number 1,2,3

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12 April 1984

"INDEPENDENT-PAIR" PROPERTY OF CONDENSED COHERENT FERMION PAIRS AND DERIVATION OF THE IBM QUADRUPOLE OPERATOR Takaharu OTSUKA Japan A tomie Energy Research Institute, Tokai, Ibaraki 319-11, Japan and Theoretical Division, Los Alamos National Laboratory, Los Alamo& NM 87545, USA Received 27 September 1983

It is shown that condensed coherent fermion pairs have a property that each pair can be treated as if it is independent of the other pairs. Utilizing this property, the S and D pairs in deformed nuclei are shown to carry the major fraction of the quadrupole moment. A new fermion-boson mapping method is proposed based on this property. The IBM quadrupole operator is derived microscopically.

The validity of the interacting boson model (IBM) [1] is based on dominant roles of monopole (S) and quadrupole (D) nucleon pairs in quadrupole collective states [ 2 - 9 ] . The S and D pairs are coherent nucleon pairs. The S pair condensed state is nothing but the BCS ground state in the spherical nucleus. It has been clarified [ 5 - 9 ] that quadrupole collective states of deformed nuclei are constructed from condensation of nucleon pairs each of which is dominated by S and D. However, it has been an open problem to what extent various pairs contribute to physical observables, since the S - D pairs have been found not to explain physical observables perfectly [ 3 - 5 , 7 ] . In this letter, I shall consider this problem, introducing the "independent-pair" property of condensed coherent fermion pairs which means that each condensed pair in a multipair system can be treated as if it is independent of the other pairs. Utilizing this property the quadrupole moment in deformed nuclei is analyzed. It turns out that the S - D pairs account for the major fraction of this quantity. Based on the "independent-pair" property, a new f e r m i o n - b o s o n mapping method will be proposed. This method will be useful for general problems with condensed coherent pairs. Boson quadrupole operators in the IBM for deformed nuclei are obtained microscopically by this mapping, and found to be in good agreement with phenomenologically determined ones. The applica0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

bility of the IBM to deformed nuclei has been questioned partly due to discrepancies between the phenomenological quadrupole operator and the microscopic one obtained by a previous method [10,11]. The present result provides the first microscopic justification for this IBM operator in deformed nuclei. The ground state band of the deformed nucleus is described by the intrinsic state [12], which is well approximated by a condensate state of the Cooper pair in the deformed single-particle orbits [ 13]. This Cooper pair is denoted hereafter as the A pair created by the A t operator. The amplitudes in A t are determined, for instance, by the BCS calculations in the deformed orbits. The intrinsic state ~b of an N-pair system is written as ~b cc (At)NI0> [6]. The A t operator is rewritten as A t = Z j x j A t (J), where x j denote amplitudes, and A t (J) is obtained by projecting from A t onto an angular momentum J (see ref. [6]). The operatorsA t(J) are denoted by S t, Dt and G t for J = 0, 2 and 4, respectively [6] ; A t = x0 St + x2D~ + x4G~ + ....

(1)

It has been shown [ 6 - 9 ] that the S - D probability, 2 2 x0 + x2, is more than 85% in deformed nuclei with the deformation parameter 6 ~ 0.30 and the pairing gap A ~ 1.0 MeV. For an N-pair system (2N = the number of valence nucleons), however, one has to consider the N t h power of A t :

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(A*Yv = (x0S* + x2D Yv + Nx4G~(xO St + x2D~)N-1 + ....

(2)

The pure S - D component in eq. (2) becomes less and less dominant for larger N and fixed xy's. This could be a serious problem for large N. It is then of great interest to see whether this fact is relevant to physical observables or not. As an example, I would like to consider the intrinsic quadrupole moment Qin (of the ground band), which is written as Qin = (AN[ Q01 AN> with IAN> being the normalized intrinsic state, and Qo being the m = 0 component of the one-body quadrupole operator (}m. I begin with considering a schematic example in order to make discussions as transparent as possible. This example is the purely aligned limit where the normal phase appears as the solution of the BCS calculation. In this limit, At is given by uniform v-factors below the Fermi level and vanishing v-factors above it [6]. In order to calculate Qin, we first evaluate an unnormalized matrix element, <01ANQ0(At)NI 0>, which is the overlap between state, (At)NI 0>, and state, O0(mt)NI0> =N(AI001A>(A? + 2?)(At)N-110>. (3) Here, the l~t operator is introduced as, [00, At] = (At + Zt), with orthogonality <01ANt 10> = 0. In the aligned limit, one can easily obtain = 0, and then, <01ANQ(A~)NI 0> = N<01NV(At)NI 0>. (4) In other words, 1~i"appears in eq. (3) as a consequence of the commutator [0., At], while it has no effect on Qin due to the complete cancellation among various terms in £t. Since the quantity in eq. (4) has to be normalized by the norm (01AN(At)NI 0>, one finally obtains a simple relation, ain = .

(5)

It should be pointed out that eq. (5) is exact in the aligned limit, and that all effects of the Pauli principle are included. The N-pair matrix element is given by the product of the number of pairs, N, and the onepair matrix element
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pairs. Eq. (5) implies that the fraction of the S - D contributions in is determined by the one-pair matrix elements . In the super-phase case, occupation probabilities are different among deformed single-particle orbits, and lower orbits are more occupied than higher orbits. Thus, lower orbit terms in Z t are more suppressed than higher orbit terms. Therefore, in spite of the orthogonality <01AZ t ]0) = 0, state £ t (At) N-1]0 ) is not orthogonal to (At)N] 0>. In fact, by examining the single-particle amplitudes of £ t , it can be seen that Zt (At)N-1 ]0 ) is comprised primarily of(At)NI 0)usually. I then introduce a quantity representing this non-orthogonality,

e = --/.

(6)

This parameter e has a positive value 0.3-0.4 for usual deformed nuclei, and is quite insensitive to 6 and A. This negative non-orthogonality therefore yields a blocking effect on the Q operator, reducing <0IANQ0(At)NI 0) from the rhs of eq. (4) by a factor ( 1 - e). Since Qin is an average of quadrupole matrix elements of low-lying ground-band members, e represents the mean non-orthogonality blocking effect in this region. Although there should be other blocking effects which are contained in the norm <0 IAN(At)NI 0>, this norm does not appear in the normalized matrix element. Similar to eq. (5), ain = is obtained. This expression is regarded as a generalization of eq. (5) and is also referred to as the "independent-pair" property, since e is very insensitive to 8 and A and 2~t(At)N-110> is primarily ~x(A?)N[ 0>. Using eq. (1), Qin is expanded as, Oin = N(1 - e) [2x0x 2 + x22+ ...] ,

(7)

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where , (D01 ()01 DO), ( D01001 GO> and ( G O1201 GO) are, respectively, 52%, 18%, 25%, and 4%. The S - D pairs account for 70% of Qin, while the total probability of the pure S - D components in IAN) is much less [see eq. (2)]. One thus finds dominant contributions from the S - D pairs to the N-pair matrix element. The same conclusion is obtained for other deformed nuclei. The "independent-pair" property which means essentially that the complicated fermion N-pair norm can be eliminated in the normalized N-pair matrix element, leads us to a new fermion-boson mapping method as described in the following. The nucleon A pair is mapped onto a boson, denoted X; At = x0S? + x2D ~ + x4G~ -+ ~tt = x0st + x2d ~ + x4g ~. Consequently, the nucleon intrinsic state is mapped as, IAN) -+ IxN). The boson image of the nucleon quadrupole operator is given as ~) ~ ()B = q l ( d t s + s]-~) + q2 [dtd] (2) + q3 [gt~ + dt~] (2) + ....

(8)

with coefficients ql = (1 - e)(S I I~)[ ID > / ~ , q2 = (1 e)(DI I ( ) 1 1 D ) / ~ , etc. In this mapping, the equality (ANI Qol AN) = (kArl O~l xN) holds [see eq. (7)]. I note that the "independent-pair" property -

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contains the bosonic structure characterized by the factorization (?,NI ~ I XN) = N(XIO0B I X). The validity of this mapping is examined by lookhag at other matrix elements. As an example, I consider matrix elements related to a state ID2AN-1), where D2 denotes the m = 2 component of Din. This state is considered to be one of the major components of the "/-band intrinsic state. Matrix elements are compared in table I with the corresponding boson predictions (d2 XN - 1 IQBI d2 XN - 1) and ( d 2 ) N - 11(~B I xN). Note that the diagonal one of the above nucleon matrix elements is related to the y-band intrinsic quadrupole moment which is the same order of magnitude as Qin- The off-diagonal one is related to the g r o u n d gamma E2 transition which is one order of magnitude less than Qin- Table 1 clearly demonstrates that these fermion matrix elements are reproduced very well by the one-body boson operator QB. The agreement exhibited in table 1 suggests that the non-orthogonality blocking effect for ID2 A N - 1) over deformed single-particle orbits is not very different from that for IAN). In fact, only one pair is different between the two states; it is either the D2 component of the D-pair or the A pair in which D O is one of the major components. I emphasize that the agreement in table 1 is not exceptional and a similar agreement can be seen generally in deformed nuclei. Further investigation on the validity of the present mapping will be reported in a forthcoming paper. Parameters of boson quadrupole operators are evaluated for 158Gd. Here, 6 = 0.25 was taken, and the pairing strength was chosen so that A ~ 0.9 MeV. The A pair was calculated from the Nilsson + BCS. Following the above mapping procedure, parameters of ()B in eq. (8) are calculated. In order to construct an s - d boson (or IBM) sys-

Table 1 Comparison between exact and boson quadrupole matrix elements (fm 2) related to the D2 pair for (a) the 16-neutron system in the N = 82-126 shell with 8 = 0.3 and & = 0.8 MeV and for (b) the 16-proton system in the Z = 50-82 shell with 8 = 0.25 and = 0.9 MeV. Qin = (=(hNIO-BIhN))is also shown. Case

(a) (b)

Qin

179 108

(D2AN- 11QolD2AN-1)

(D2AN-1 [Q2 jAN)

exact

boson

exact

boson

150 88

150 91

12 14

9 13 3

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tern, the g boson is eliminated in the next step, and its effects are taken into account by renormalization of s - d boson terms by the method of ref. [5]. The quadrupole field Q is responsible for the admixture of the g boson [5]. Since Q is mapped onto the one-body boson field OB, the renormalization can be done for each boson separately. In other words, when a d boson is coupled to a g boson by OB, the other bosons behave as spectators. This independence property simplifies appreciably the renormalization procedure [5]. Parameters of the s - d boson quadrupole operator are thus calculated microscopically with the renormalization. The result of this calculation should be compared to that obtained by phenomenological fitting IBM calculations. In table 2, these two results are shown in a convention that the boson quadrupole operator is defined as QB = d~sr + sz?~r + Xr [d~Lr] (2) with r = 7r (proton) or u (neutron) and Xr being a parameter, and that the boson p r o t o n - n e u t r o n QQ interaction is defined as - • 0 B. QB with the strength K. The strength of the nucleon QQ interaction is taken from ref. [14]. The boson E2 operator is written as ~(E2) = e~, B0.~B + euBO.u B with the boson effective charge e B. The nucleon effective charges are assumed to be 1.7e for protons and 0.7e for neutrons. In table 2, a reasonable agreement is seen between the microscopically calculated parameters and those obtained by fitting calculations. In table 2, unrenormalized values of the parameters are also indicated to show changes due to the renormalization. These parameters are calculated also by the method of ref. [10] (OAI), where the IBM parameters are determined by states ISN) and Is N - I D ) . This method is useful in and near spherical nuclei, while it yields,

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in deformed regions, too large K and too small IXI compared to results of phenomenological fitting. Large IXl is essential to obtain rotational spectra. In fact, IX[ = x/r772 in the SU(3) limit [15]. This discrepancy is now solved, by introducing the new mapping where the blocking from a coherent linear combination of S, D and G is included properly. It seems worthwhile to mention the general proper. ty of the mapping based on the "independent-pair" property. Since this property is general, the mapping should be useful for physical systems with condensed coherent pairs in a single particle potential which is not necessarily quadrupole. One can then construct a simple boson hamiltonian, which could include bosons other than s, d, etc. Since such a simple boson hamiltonian can be related to the classical or geometrical description [ 3 , 1 6 - 1 9 ] , the "independent-pair" property plays an important role for connecting a quantum mechanical multi-fermion system to a classical system. Returning to the IBM, I would like to remark on three points. First, the present mapping method is consistent with the OAI method in the limiting situation of A t = St + x D t with x -+ 0. Secondly, the mapping based on the "independent-pair" property can be applied to two-body interactions between like nucleons, for which N-pair matrix elements can be written in terms of N-, one- and two-pair matrix elements, and a parameter like e. This will be discussed in a future publication. Finally, the parameter e is treated as a state-independent constant so far. This is a good approximation in low excitation energy regions. The approximation will become, however, less reliable in going to highly excited states. In such case, many-boson

Table 2 Parameters in the boson QQ interaction and in the boson E2 operator for 158Gd" In the column microscopic, they are calculated microscopically by the present mapping method with the renormalization due to elimination of g-boson. In the column (unrenormalized), the unrenormalized result is shown in parentheses. Parameters obtained by a fitting calculation and those by the OAI method are also shown.

4

Parameter

Microscopic

(Urtrenormalized)

Fitting

OAI

(MeV) ×~r ×u e B (efm 2) eB (e fm 2)

0.094 -0.86 -1.18 12.0 10.0

(0.12) (-0.80) (-1.03) (10.6) (6.7)

0.08- 0.09 -0.8 - - 0 . 9 -1.1 - - 1 . 2 12 - 14 12 - 14

0.19 0.04 -0.55 14.3 8.4

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interactions or new kinds of bosons ~11 be required to take into account state-dependent e. In the method proposed in this paper, the one-body quadrupole boson operator is good for low-lying states, while one needs corrections for higher states. I would like to emphasize that this property is in good accordance with the basic idea of the IBM. I would like to thank Dr. N. Yoshinaga for fruitful discussions and Dr. J. Ginocchio for careful reading of the manuscript. I am grateful to Professor A. Arima for valuable discussions particularly on the analogy to the independent-pair model of the nuclear matter. I also thank Dr. K. Harada, Dr. A. Iwamoto, and Dr. N. Yoshida for useful comments on the manuscript.

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[4] T. Otsuka, Phys. Rev. Lett. 46 (1981) 710. [5] T. Otsuka, Nucl. Phys. A368 (1981) 244. [6] T. Otsuka, A. Arima and N. Yoshinaga, Phys. Rev. Lett. 48 (1982) 387. [7] D.R. Bes, R.A. Broglia, E. Maglione and A. Vitturi, Phys. Rev. Lett. 48 (1982) 1001. [8] J. Dukelsky, G.G. Dussel and H.M. Sofia, Nucl. Phys. A373 (1982) 267. [9] K. Sugawara-Tanabe and A. Arima, Phys. Lett. 100B (1982) 87. [10] T. Otsuka, A. Arima and F. Iachello, Nucl. Phys. A309 (1978) 1. [11] S. Pittel and J. Dukelsky, preprint. [12] A. Bohr and B.R. Mottelson, Nuclear structure, Vol. 2 (Benjamin, New York, 1975). [13] S.G. Nilsson and O. Prior, Mat. Fys. Medd. Dan. Vid. Selsk. 32, No. 16 (1961). [14] J.L. Egido and P. Ring, Nucl. Phys. A383 (1982) 189. [15] A. Arima and F. IacheUo, Ann. Phys. 111 (1978) 201. [16] J.N. Ginocchio and M.W. Kirson, Phys. Rev. Lett. 44 (1980) 1744; Nucl. Phys. A350 (1980) 31. [17] A.E.L. Dieperink, O. Scholten and F. IacheUo, Phys. Rev. Lett. 44 (1980) 1747. [18] A. Klein and M. VaUieres, Phys. Rev. Lett. 46 (1981) 586. [19] R. Hatch and S. Levit, Phys. Rev. C25 (1982) 614.