NUCLEAR PH’i‘SICS A
Nuclear Physics A570 (1994) lc-14c North-Holland, Amsterdam
Microscopic Akito
FolzndstioI1
of the
Illt,eractiIlg
Model
Boson
Arima”
a Institute
of Physical
and C’hemical
Research
(RIKEN),
Wako-shi,
Saita,ma
351-01,
Japan A microscopic foundation of the interact,ing boson model is described. The importa.nce of monopole and quadrupole pairs of nucleons is emphasized. Those pairs a.re mapped onto the s and d bosons. It is shown that, t.his mapping provides a good approximation in vibrational and transit,ional nuclei. In appendix; it is shown tha,t, the monopole pa,ir of elect,rons
plays possibly
an important
role in niet,al clusters.
1. INTRODUCTION Nuclea,r collective motions show a number of different, a.spect,s such as nuc1ea.r rotat,ional and vibrational motions and the giant, resonances. It has been recoguized that, nucleon pairs p1a.y a.11 important. role in uuc1ea.r structure. The pair of spin zero is especially important and known as the cooper pa.ir or t,he S pair. The pair of spin 2 which is often called t.he 1) pair has beeu shown to bc an important. building block of nuclear collective states in vibrational and rotational nuclei[l]. In the Interacting Boson Model, those S and D pairs are simulated by s and d hosons[2]. The model utilizes extensively the concept, of dynamical symmetries and is often called an algebraic model. The success of the Int,era.cting Boson Model has stimulated o&r algebraic models such as the Fermion Dyna.mical Symmetry Model[S] and the Rowe’s model based on the symplectic grolip[4]. The Interacting Boson Model has accomplished three ma,jor things. These a.re (i) the recognition of dynamical symmetries such as O(5); O(6)and $X1(3), (ii) the recognition of the importance of S-and D-pairs and (iii) predictability. Here the import,ance of O(6) was first pointed out in t,he fra.mc work of t,he Interacting Boson Model and proved experimentally by Casten and C!izewski[S]. Two examples of predictability are shown in Figure 1. In section 2, I will show how good the SD truncation is; where the SD truncation means that. the complet,e shell model space is trunca.ted into that construct,ed by using S and 1) pairs only[7]. In section 3, those pairs are mapped onto the s and cl bosons. It is then examined how good this mapping is. Here a method introduced by Otsuka, Arima and Ia,chello is used[8]. A method int.roduced by Li is also used in the case of non-degenerat,e single particle energies[Y, lo]. In t.h ese t,wo sections, we use a schematic model. In section 4; t,ransit,ional nuclei arc t,rcat,cd more realist,ically. Especially a new reskilt. obtained by Mizusaki and Otsuka is shomu[l I]. in t,heir calcula,tion the hoson Hamiltonian was derived microscopica.lly, and no frirthcr a.djustable paramet.ers are int.roduced. Their theoretical results are in good a.grcemcnts with observed data. 1994 - Elsevier Science B.V. SSDI 0375-9474(94)00073-V
A. Arima / Microscopic foundation of the IBM
2c
1
(X0)
r
(20.2)
(20.2)
-1,II
\ 11 5t 4_ 4'4-
6’-
6'+
5+_
IO'_
;f
:‘g
(16.4)
2*-3'= - 2' 0*
4Lk 2+_"Z 0+-2'
- *+_ o+-2+
6 ---
ls6Gd
Figure 1. a) An example of a spectrum with SU(3) symmetry: $$d92 and b) comparison between the experimental data a.ncl the predictions of the extended IBM-1 Hamiltonian. Both from the Ref. [6]
3c
A. Arima / Microscopic foundation of the IBM
An interesting result concerning the effective interaction between two electrons in Na clusters is discussed in the Appendix. The result suggests that the pairing correlation seems to play again an important role in the metal clusters[l2]. 2. PAIR
APPROXIMATIONS
Racah introducecl the seniority scheme in atomic physics[l3]. The enjoyed success in nuclear structure ra,ther than in atomic structure[lJ]. the zero angular momentum pair which is called the monopole pair essential building block of nuclear low lying states[l5]. If an effective attractive monopole pairing type is assumed, the ground state can be
concept, however: In t,his scheme or S-pair is a very interaction of the constructed
as
* = N(s+)“lo), where
of a nucleon in the orbit characterized by j and m. where a&, is the creation operator This pair is the analogue of the Cooper pair which was introduced by Cooper in order to explain the superconductivity. It, has been well known that even-even nuclei have very low 2 states. This fact, indicat,es thaL a quadrupole pair which is called the D-pair is a building block in addition to the S-pair. Namely the D-pair is a kind of generalization of the Cooper pair. A question here is ‘&how good is the S&D Pair Approximation?“.There have been many works in order to answer t,his question. In this section some results obtained by Yoshinaga are discussed[7]. H is calculations are based on a schematic Hamiltonian and model spa,ce. (i) Vibrational and transitional The model Hamiltonian is written
nuclei as
H = &PO + C&P2 + K&. Q. The first term is the monopole pairing s’+S and the second is the quaclrupole pairing which is roughly D+D. The last term is the usual quadrupole-quaclrupole interaction. One example of Yoshinagn’s calculations is shown in Figure 2. Here he put six particles in g7i2 and illjZ orbits. Since h: is weak, the case corresponds to a vibrational or tra.nsitiona,l nucleus. From this figure, one can 1ea.m that the SD truncation is good and the G(L = 4) pair is not necessary. (ii) Well deformed nuclei Yoshinaga took a step forward, keeping to simulate deformed nuclei. As a model six particles are put in. One example of that the SD truncation is not enough but deformed nuclei.
only D finite in his model Hamiltonian in order space, he assumes c&/2 a,nd gs/z orbits in which his results is shown in Figure 3. Now it is clear the G pairs must be taken into account in well
4C
Figure 2. lg7j2 and (SDG) in K = 0.08.
A. Arima I Microscopic foundation of the IBM
Energy levels of the pairing, quadrupole pairing and quadrupole interaction for OiII,2 orbits: (exact) in the full shell-model space; (SD) in the SD-subspace: the SDG-subspace; (sd) in the sd-boson space. Parameters are Go = 1.0 C:z = The number of phonons is indicated t.o t,he left of exact energy levels.
A. Arima I Microscopic foundation of the IBM
5C
-4 -6
-6 -_5 -30
-0 -2
-2
-
.
-3-2 --4
-4 --6 -_2 =r: --'3-O
-6
-6
-4 -3-o -4 -2
-4-2
--‘I
-2
-2
-4o-
-2
-0
-0
-0 exact
SDG
SD
Figure 3. Energy levels of the yuadrupole interaction in the fermion space for l&,/2 and Ogsjz orbits: (exact) in the full shell-model space; (SDG) in the SDG-subspace; (SD) in the SD-subspace.
A. Arim I Microscopic foundation of the IBM
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3. BOSON
MAPPING
The next question is how to map the fermion pairs onto bosons. In t,he Interacting Hoson moclel, one often uses a method introduced by Otsuka, Arima and Iachello (the OAI and succeeds very well in reproducmapping)[8]. Th is method gives a boson Hamiltonian ing numerical results based oh diagonalization of the corresponding fermion Hamiltonia,n. Some examples were already shown in Figure 2, where the levels indicated by sd are obtained with this procedure. One sees the agreement with the exa.ct result is excellent. This corresponds to the vibmtional spectra. Yoshinaga took a model which has six pa.rticle in the j = 23/2 orbit. A pure quadrupolequadrupole interaction is assumed. The numerical results shown in Figure 4 resemble the rotational levels. Here one need the G pair together with the S a,nd D pairs. The OAI method with s, d and g bosons simuIa,te very well the level structure obta,ined by the SDG truncation. In this figure, BKL stands for a mapping method introduced by Ronatsos, Klein and Li[16]. In deformed nuclei, the role of g bosons cannot. be ignored. They can be included cxplicitly in the Interacting Boson model or implicitly by a renorma.lization such a.s Rarrett[l7] and Otsuka and Ginocchiollti] did. This problem must be further studied. The number of single particle orbits is either one or two in the calculations discussed above. In realistic cases, more orbits compete with each other. Their energies are not clegenerated. The most powerful met,hod is the BCS to dea.1 with such many non-degenerate orbits. Using the Ginocchio model, however, we compared the Intera.cting Boson Model and the Boson Expansion Technique[lS]. We then found that the number conserva,tion which is violated by the BCS approximation is very crucial. Li, Pedrocchi and Tamura.[%O] reached the same conclusion. In this resea.rch, they use Li’s method in order to restore approximately the number conservation[9]. Nakada and I applied the method of Li to derive an Interacting Boson Hamiltonian[lO]. Figure 5 shows one example of such calculations. Here NCQP means ‘inumber-conserved quasi pa.rticle” in which the Li method is used. Comparing with results of calculations in which number conservation is not ta.ken into account, one sees that the boson mapping with the number conserva.tion produces very good agreement with the exact ca.lculation. However, In this calculation, only two single particle orbits are taken into consideration. the Li method
is found
4. REALISTIC Kzusaki
very promising.
CALCULATIONS
and Otsuka. started
Further
studies
are desired.
IN TRANSITIONAL
NUCLEI
from a shell model H~iitonia~~~llj;
fl=&r-t-H,+xwQ,+Q, where
HP = -G&S;S,
-
G;D,+D,+ ,yPQp . Qp
p = 7riT, u stands for protons or for neutrons. The parameters Co, GZ and ?: are determined by fitting to observed levels of the Z=50 (S n ) isotopes a,nd those of the N=82 isotones as shown in Figure 6.
A. Arima I Microscopic foundation of the IBM
7c
-6
-2o-
-4 -6 ZJ
-2~
SDG
s; sdg
-6
-3o----a
-35-
-2
liTLo
Figure 4. Energy levels of j = 2312 and 3 bosom quadrupole interaction for SDG-subspace.
(6 particles)
for the quadrupole-
A. Arima I Microscopic foundation of the IBM
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-2
$+L$
-0’
2&q;;
-f3+
O$+41+-
-2, ---
rp
,o+ 4+
-4
-
+a
to
2+
-2+ -2+
+
e -2+
-5/
j
I-++ -!L+ &2+2+ _q -2 -4+
21+-
I_$
4+
-
-oo+
ol+_
Exact
NCQP+BY
--+
eP+m
2+
1
-O+ 1 Qp+B=
Figure 5. Energy levels for fo;,,,.2 = ~~~~~~= O.OMeV. The valence particle number is fixed to be 6. The levels exhibited in the left most column in this figure are obtained as t,hose of quadrnpole collective states from t,he exa,ct diagonalization wit.hin the shell model. The way to specify quaclrupole collective states is described in the text. The levels with the symbol “NCQP+BM” show the results of the “number-conserved qua,si particle.” Those in the columns named “QP+BMl” and “QP+BM2” are obtained by the boson mapping to t.he sa.me order as “NCQP+BM,” but based on the quasi-particle approximation without number projection. In the column denoted as “QP+BMl,” the influence of the spurious component,s is removed, while in the column “QP+BM2” it, is not.
A. Arima I Microscopic foundation of the IBM
9c
Using the OAI mapping, they calculated the values of the paramet.ers xii and ,xy for (a) the Te isotopes, (b) the Xe isotopes and for (c) the Ba isotopes, as shown in Figure 7. Here xii and xU are the parameters appearing in the boson quadrupole moments of protons and neutrons;
Q; = s+& + d;s - ,yP[d+Ljl? Examples of their ca.lculations are shown in Figure 8. Here cally derived IBM-2 ca.lculations. One sees that, the IBM-2 with experiment,s. It should be not.iced that the parameters ically without further modifications. If small adjustments agreements will be obtained. Further calculations along this 5.
theory refers to microscopicalcula,tions agree very well are all obtained microscopare introduced, much better line are wanted.
CONCLUSIONS
The S and D pair approxima.tion is good in vibrational and transitional nuclei. However. G pairs are necessary in very well deformed nuclei. The OAI boson mapping is successful in vibrational and transitional nuclei. In well deformed nuclei, g-bosons must be taken into account, either explicitly or by renormalization. If one uses the BCS approxima,tion, the number projection is necessary. For this purpose, the Li method is promising. Finally I would like to comment why nuclear physics is still interesting. Nuclei provide good microscopical laboratories to study many body systems with finit,e number of constituents less than 300. We can a.pply the know-hows from nuclea,r physics to for example met.al clusters. Masses, life-t,imes and level schemes must, be predicted in order to apply nuclear physics to, for example, astrophysics, and fundamental physics such as double bet,a decay and the problem of quark degree of freedoms in nuclei.
APPENDIX:
THE PAIRING
CORRELATION
IN METAL
CLUSTERS
Metal clusters such as Na and K have attracted our interest very much. They show very similar LS coupling shell model patterns. I am interested in the effective interaction between two valence electrons. The most important int,eraction is definitely Coulomb interaction. However, our experience in nuc1ea.r physics suggests la.rge contributions from Brueckner-type corrections and renormalization corrections. I suggested Shimizu and Takayanagi[l2] to calculat,e t.hese corrections in a system with two electrons and a,n L,Yclosed core. An example is the cluster of ten Na atoms. We assumed an isotropic harmonic oscillator well as the average field for the electrons. The first part, of the calculation has been done for the Brueckner correlation. The second part concerns the core polarization and the 4p-2h excitation. The results are shown in Table 1, where the b parameter corresponding to hw = 0.95eV is assumed; and the two elect.rons a.re assumed t,o be in the Od orbit. The bare Coulomb potential favors of course the 3F state and pushes up the ‘S state very much as the Hund rule predicts. However, the present calculat,ions shows a surprising result. Namely, t.he ‘S state is brought down very much to compete with the 3F and “P
1oc
A. Arima I Microscopic foundation of the IBM
Sn isotopes 3.-x_. 6+ _ ..z.r. .I” 0““&:+A,
*
6’ ; s g’
_
4+/
l
.
_
_
*
“-;.._. -..- *;--_:&._.,;
_
~
_
,
I\ 2+ 0’
66
’ 66
I
I
I
I
I
I
70
72
74
76
76
60
Neutron Number N=82 isotone 3-
y-
6+\_._._&
*&s/zg
2+
0 50
I
I
I
55
60
65
Proton Number
Figure 6. a) Energy levels of 2:, 4: and 6: f or S n isotopes. Lines a.re calculations. Points are experiments; closed diamonds, open diamonds and triangles indicate, respectively, t’he 2:, 4t and 6: levels. b) Energy levels of 2:, 4: and 6: for N = 82 isotones.
A. Arima I Microscopic foundation of the IBM
*;::
-0.5
*
______ L e
_______........~~~~~~._._........~.~~..~.
0.0
-0.5
fcfBa
0.5
/f--l u
0.0
llc
xv
_..._.__._...._____..._.....“,...----.-..
t
I.
I 66
-
I.
1
72
76
Neutron
I
II 60
Number
Figure 7. parameters xn and xv for (a) Te, (b) Xe and for (c) Ra. isotopes.
12c
A. Arima I Microscopic foundation of the IBM
3-
I'34Xe
-
4+ o+
-3+ 9 22
3+4+2+
-4+
2+-
-2+
-2+
ii I-
0 IBM.2
exP
3-
4+ -3+ -o+ 4+
X2E d
3+-
l-
-4+
42+
2+
-2+
2+-
ev
theory
4+ -o+ 3+ 0+4+3+-
-4+ -2t
4+2+
ew
Figure 8. Experimental
theory
and IBM-2 energy levels of 130*132,*34Xe.
A. Arima I Microscopic foundation of the IBM
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Table 1 Interaction energies between two Od electrons. b.C. means bare Coulomb and Diagon. means diagonalizat,ion among two electrons. The two columns in “21iw + 4F& mean t.hc direct (screening type) a.nd excha.ngr second-order configuration mixings, respcctivaly. L
S
b.C.
Diagon.
0 1 2 3
0 1 0 1
3.30 2.57 2.55 2.28
2.35 2.16 2.14 2.06
2tLw + 4hw 0.45 -0.22 -0.14 -0.27
4
0
2.68
2.30
0.33
S uni -1.02 -0.19 -0.21 0
1.78 1.75 1.79 1.79
0
1.97
states. If the b parameter is increased slightly from it,s value used in the present, calculation, the ‘S state becomes even the ground st.ate; because the bare Coulomb matrix elements depend on l/b, while the second-order corrections are independent, of b. .4s t,he Od orbit is being filled up, the b paramet,er increase. It is then possible that the ‘S state then becomes the ground state in t,he middle of the O&shell. REFERENCES 1. 2.
T. Otsuka, A. Arima and F. Iachello, Phys. Lett. 76B (1975) A. Arima and F. Iachello, Phys. Rev. Lett. 35 (1975) 1069.
3. 4. 5. 6.
C.-L. Wu, D.H. Feng, X.G. Chen and M.W. Guidry, Phys. Lett. 168B (1986) 313. B. Rosensteel and D.J. Rowe. Phys. Rev. Lett. 38 (1977) 10. R.F. Casten and J. Cizewski, Phys. Lett,. 79B (1978) 5. D. Eionatsos, Interacting Boson Model of Nuclear Physics (Oxford Science Publicat.ions, 1988). N. Yoshinaga, Nucl. Phys. .4522 (1991) 99C. T. Otsuka, A. .4rima and F. lachello, Nucl. Phys. A309 (1978) 1. C.T. Li, Nucl. Phys. A417 (1984) 37. H. Nakada and A. Arima, Phys. Lett,. B209 (1988) 411. T. Mizusaki and T. Otsuka, to be published; T. Mizusaki, Ph. D. thesis, ITniversity of Tokyo (1992). A. Arima, K. Shimizu and K. Takayanagi, to be published elsewhere. G. Racah, Phys. Rev. 62 (1942) 138: 63 (1943) 367. I. Talmi, Simple Models of Complex Nuclei (Harvard Academic Publishers, 1993). I. Talmi, Nucl. Phys. Al72 (1971) I. D. Bonatsos, A. Klein and C. Li, Nucl. Phys. A425 (1984) 521. C. Druce, S. Pittel, B. Barrett, and P. Duval, Ann. Phys. 144 (1982) 168. T. Otsuka and J.N. Ginocchio, Phys. Rev. Lett. 55 (1985) 276. A. Arima, N. Yoshida and J.N. Ginocchio, Phys. Lett. 101s (1981) 209. C.T. Li, V.G. Pedro&i and T. ‘l~amura, Phys. Rev. C33 (1986) 1762.
7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
139.