6 Realistic and Model Applications of the Boson Expansion Theory Taro Tamura
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Department of Physics, University of Texas at Austin, Austin, TX 78712
The structure of the boson expansion theory is explained based on simple models. It gives a good idea on how this theory would work when it is applied to realistic cases.
Our group at the U n i v e r s i t y o f Texas has been engaged i n the boson expansion theory (BET), during the past decade or so. Our work s t a r t e d with two papers [1,2], which we s h a l l henceforth r e f e r to as K T l and KT2. The BET of KTl and KT2 are very s i m i l a r i n t h e i r forms, but d i f f e r c r i t i c a l l y i n what they d e s c r i b e . The BET of KT2 was used e x t e n s i v e l y for r e a l i s t i c c a l c u l a t i o n s , with notable success i n f i t t i n g a number of data of c o l l e c t i v e motions i n a v a r i e t y o f n u c l e i [ 3 ] . Subsequently, we renewed our formal study of BET as a whole, r e s u l t ing i n a paper [4], which we s h a l l c a l l KT3 henceforth. In both K T l and KT2, a method c a l l e d a commutator method was used, while the method used i n KT3 was (a g e n e r a l i z e d v e r s i o n of) that of MYT [ 5 ] . In p a r a l l e l with or f o l l o w i n g KT3, a few a d d i t i o n a l papers were published, c l a r i f y i n g f u r t h e r the general s t r u c t u r e of BET [ 6 , 7 ] . One aspect which c h a r a c t e r i s e s our r e a l i s t i c c a l c u l a t i o n s [2,3] i s the use of the BCS theory, making our theory be sometimes c a l l e d BCS+BET. As i s w e l l known, the BCS theory causes c e r t a i n types of e r r o r s . However, i t i s also w e l l known that one w i l l not commit too s e r i o u s e r r o r s , i f one uses BCS, as we d i d , w i t h i n the bound of i t s a p p l i c a b i l i t y . I t i s , n e r v e r t h e l e s s , h i g h l y d e s i r a b l e , t o construct a theory that maintains the s i m p l i c i t y of the BCS theory, yet removes the (major part of the) BCS e r r o r s . A theory, r e c e n t l y developed by L i [8], which uses the number-conserving q u a s i - p a r t i c l e (NCQP) method, i s such a theory. We are now working on r e p l a c i n g BCS+BET by NCQP+BET, and on resuming r e a l i s t i c c a l c u l a t i o n s . This time we s h a l l be able to go much beyond the bounds imposed by the BCS approximation. I t i s also valuable to i n v e s t i g a t e on how our new form o f BET would perform, when i t i s a p p l i e d to simple models. During the past year or so we indeed d i d t h i s , the models taken up being the s i n g l e - j s h e l l model (lj-SM) and the Ginocchio model [ 9 ] . 0097-6156/ 86/ 0324-0041 $06.00/0 © 1986 American Chemical Society
Meyer and Brenner; Nuclei Off the Line of Stability ACS Symposium Series; American Chemical Society: Washington, DC, 1986.
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In the f o l l o w i n g , we discuss mostly the use of our BET f o r these simple models. We do t h i s because i t w i l l help the reader to understand the essence of our BET, and a l s o because these models were studied by Otsuka e t . a l . [10,11], and by Arima et^. j d . [12]. (They d i d t h i s with a stated purpose of j u s t i f y i n g the i n t e r a c t i n g - b o s o n approximation (IBA) [13].) I t w i l l be seen below t h a t , i n s p i t e of the use of the same models, the conclusions we a r r i v e at d i f f e r s i g n i f i c a n t l y from those given by these authors. SINGLE-j SHELL MODEL The lj-SM i s c h a r a c t e r i s e d by the fact that i t i s very easy to construct i n i t a basis states i n the s e n i o r i t y scheme. Let a (basis) s t a t e |n,v;ot> contain η p a r t i c l e s , ν of which c o n t r i b u t i n g to make t h i s state have s e n i o r i t y v. ( a stands f o r a d d i t i o n a l quantum numbers.) This state then has S p a i r s (of Cooper type) of which number equals k=(n-v)/2. The states with n=v (and hence k=0) are c a l l e d highest s e n i o r i y (HS) s t a t e s . Once the states are thus constructed i n the s e n i o r i t y scheme, a step c a l l e d s e n i o r i t y r e d u c t i o n (SR) can follow. An example of i t i s given as μ
2 )
6· ν
6 . y
; v + 2
(la)
; v
ό· ν
; v
-
2
In ( l a ) , Β i s a p a i r c r e a t r i o n opeartor, C y i s a s c a t t e r i n g operator, and the U and V c o e f f i c i e n t s are defined as (Ω = j + 1/2) 2 μ
2
uU.jMUn-n-v-i-jMUiMv^j)] / 1
2
(lb) V(i,j)=[(n-v+i-j)/(2fi-2v-2j)]l/
2
As seen, the SR i s to express a matrix element of an operator, between states which are not ( u s u a l l y ) of HS nature, by a sum of matrix elements of r e l a t e d operators taken between HS s t a t e s . The SR formula of (1) i s w e l l known; see, e.g., Lawson [14]. We gave i t , however, i n a form which i s d i f f e r e n t from what i s u s u a l l y employed. Namely we expressed the c o e f f i c i e n t s on the rhs of ( l a ) i n terms of the U and V f a c t o r s . The reason we chose t h i s form i s that we want to emphasize the f a c t that (1) i s very s i m i l a r to what we get when we perform the usual Bogoliubov transformation, i . e . , use BCS. In f a c t , we obtain 0^=
2
2
^2 UV B^, + ( U - V ) C ^
U - [(2Ω- )/(2Ω)]1/2 . η
V
+ ^2 UV B j ^
= [η/(2Ω)]1/2
(2a) ( ) 2 b
The operators on the rhs of (2a) are q u a s i p a r t i c l e p a i r operators.
Meyer and Brenner; Nuclei Off the Line of Stability ACS Symposium Series; American Chemical Society: Washington, DC, 1986.
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Boson Expansion Theory
43
As seen, (1) and (2) are i n f a c t i n the same a l g e b r a i c forms. The U and V f a c t o r s i n (2b), however, lack the d e l i c a t e i , j and ν depndence, which the U and V f a c t o r s i n ( l b ) have. This r e s u l t s i n the number nonconservation problem of BCS. Note that the bra and ket states that appear on the rhs of ( l a ) contain S p a i r s , while the HS states on the rhs do not. This means that SR has eliminated completely the S p a i r s from our d e s c r i p t i o n , t h e i r i n i t i a l pre sence being, nevertheless, a c c u r a t e l y remembered by the emergence of the U and V f a c t o r s . The complete e l i m i n a t i o n of the S-pairs i s a l s o the case with BCS, the only d i f f e r e n c e being that U and V f a c t o r s now have somewhat poorer memory. The s i g f i f i c a n c e of the s i m i l a r i t y of (1) and (2) l i e s i n t h i s complete e l i m i n a t i o n of the Cooper p a i r s i n both SR and BCS. Let us now consider bosonizing [4] the p a i r operators, based on the reduced matrix elements that appear on the rhs of ( 1 ) . We then have an expan s i o n which contains no s^ bosons i n i t . This BET, which we may c a l l SR+BET, i s exact (assuming that the boson expansion i s c a r r i e d out to a desired o r d e r ) . An exact BET can thus be constructed without s_ bosons, even when the s e n i o r i t y scheme i s used, a f a c t which may s u r p r i s e those who know the OAI work [10]. I t appears that i t has been normaly b e l i v e d that OAI had s_ bosons, because i t used the s e n i o r i t y scheme, while, e.g., our BET d i d not, because BCS was used. What we showed above i s that such a b e l i e f i s unfounded. We may also note that, i n the lj-SM, the NCQP+BET, which replaces BCS+BET, i s exact and agrees e x a c t l y with SR+BET. OAI request that the fermion (S,D) space must always be mapped onto the boson (s,d) space; and obtain s-bosons. We showed, however, that an exact BET e x i s t without s-bosons. The s-bosons i n OAI may then be considered e s s e n t i a l l y as mathematical a r t i f a c t s . OAI claimed that they derived IBA, by c r e a t i n g s-bosons i n t h i s way. Then the s-bosons i n IBA may a l s o be mathema tical artifacts. OAI f u r t h e r claimed that the lowest order expansion was s u f f i c i e n t l y good. However, one sees that the OAI tables does not a c t u a l l y show that such i s the case. Higher order terms are d e f i n i t e l y needed [15], i n d i c a t i n g that a microscopic IBA i s not as simple as i s the phenemenological IBA. We also noted [15] that the OAI and OAIT [11] theories are c r u c i a l l y d i f f e r e n t . In OAI, the c o e f f i c i e n t s m u l t i p l y i n g s and d boson operators are constant; thus OAI can be an IBA. In OAIT, however, the c o e f f i c i e n t s depend on v, and thus OAIT i s not an IBA. (With OAIT, the SU(6) symmetry, which i s the key ingredient of the phenomenological IBA, may l a r g e l y be l o s t . ) In any case, the (somewhat b e t t e r looking) OAIT numerical r e s u l t s , rather than the (poorer) OAI r e s u l t s , were presented i n OAI t a b l e s , without mentioning at a l l that t h i s was done. Sometime ago, Arima [12] argued that the use of our BET i n lj-SM caused e r r o r s of about a f a c t o r of 3 i n some B(E2) values. We d i d the same c a l c u l a t i o n ourselves r e c e n t l y [16], and found that the e r r o r s were i n the range of 10% or so; never as large as Arima s t a t e d . We then noticed that Arima
Meyer and Brenner; Nuclei Off the Line of Stability ACS Symposium Series; American Chemical Society: Washington, DC, 1986.
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s t a r t e d with the BET of K T l , and then truncated to the d-boson component. However, t h i s should not have been done. When t r u c a t i o n i s done, the commutator equations of KTl must be reconstructed, and then r e s o l v e d . And t h i s i s e x a c t l y what was done i n KT2. I t appears that Arima f a i l e d to n o t i c e t h i s f a c t and presented very misleading r e s u l t s .
GINOCCHIO MODEL
Ginocchio model i s a degenerate 4-j SM, the j values ranging from £-3/2 to £+3/2, f o r a given £. In s p i t e of t h i s 4-j s h e l l nature, the s p e c i f i c combination of the 4 s h e l l s makes t h i s model very s i m i l a r to lj-SM. Thus, again the s e n i o r i t y scheme can be constructed e a s i l y . In a d d i t i o n to t h i s , the (S,D) space becomes a closed space. This makes the algerbra of the Ginocchio model even simpler than that of lj-SM. In F i g . l , we show the r e s u l t s of our recent c a l c u l a t i o n s [17]. This i s the case of the s o - c a l l e d S0(6) l i m i t [12], and the energy spectra predicted by various boson t h e o r i e s are compared with the exact fermion spectrum. I t i s seen f i r s t that the NCQP+BET r e s u l t agrees almost per f e c t l y with the exact spectrum. (The remaining discrepancy was caused by a f i n i t e boson expansion.) The OAIT also performs f a i r l y w e l l though not as good as does NCQP+BET. The performance of BCS+BET i s much poorer compared with these two, the energies g e t t i n g too high with increased s p i n s . The spectrum we denote i n F i g . l as OAI a l s o behaves r a t h e r poorly, t h i s time, however, underpredicting the h i g h - s p i n s t a t e energies. As we emphasized towards the end of Sec.II, OAI i s an IBA but OAIT i s not. Nevertheless, Arima e t . a l . presented [12] the r e l a t i v e l y b e t t e r OAIT r e s u l t as an evidence to j u s t i f y IBA. However, t h i s p r e s e n t a t i o n i s again very misleading. The OAI r e s u l t s should have been presented as IBA, as we have done so i n our F i g . l . In the above, we have repeatedly mentioned the the BCS theory. I t means more s p e c i f i c a l l y that the used only when the ( e f f e c t i v e ) space s i z e , and the both s u f f i c i e n t l y larger than i s the q u a s i - p a r t i c l e s e n i o p r i t y ν i n the Ginocchio model). Ω
bound of a p p l i c a b i l i t y of BCS theory i s to be p a r t i c l e number n, are number (which i s the
F i g . l . L e v e l spectra predicted by various boson theories i n the S0(6) l i m i t of the Ginocchio model. EXACT
SCQP •BET
BCS +BET
OA I (IDA)
OAIT
KXACT
Meyer and Brenner; Nuclei Off the Line of Stability ACS Symposium Series; American Chemical Society: Washington, DC, 1986.
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Boson Expansion Theory
Thus, we do not expect from the beginning that the BCS+BET w i l l f i t the high-spin states i n F i g . l . (Note that i n F i g . l , Ω 22 and n*16. For 1*16 state we thus have ν η.) However, we expect that the lower states are f i t s a t i s f a c t o r i l y , and F i g . l shows that t h i s i s indeed the case. Note that t h i s f a c t assures to a good extent that our r e a l i s t i c c a l c u l a t i o n s were done q u i t e meaningfully. We have considered [2,3] only lowlying s t a t e s , and thus have stayed w i t h i n the bound of a p p l i c a b i l i t y o f the BCS theory. β
β
DISCUSSIONS
In the present paper we have discussed mostly cases with simple models. In s p i t e of t h i s r e s t r i c t i o n , we seem to have a rather c l e a r view, on where we stand now with our BET, i n c l u d i n g i t s a p p l i c a t i o n s to réaliste cases. We have already shown [2,3] that our BCS+BET worked rather n i c e l y i n f i t t i n g a number of data. And S e c . I l l of the present paper showed that BCS+BET i s i n fact a dependable theory. We a l s o showed i n both Secs.II and I I I that NCQP+BET can remove the BCS e r r o r s . In any case, our future r e a l s i t i c c a l c u l a t i o n s w i l l be done i n terms of NCQP+BET. An important remark we want to make here i s that i t i s rather easy to construct NCQP+BET even for r e a l i s t i c cases with non-degenerate many-j s h e l l s . The same i s not the case, however, with OAI and OAIT (and SR+BET). In order to construct the many-j s h e l l versions of these t h e o r i e s , one must be able to handle the generalized s e n i o r i t y scheme. However, t h i s i s a very d i f f i c u l t task to achieve. In constructing NCQP+BET, we f i r s t construct BCS+BET. Since we use BCS f i r s t , we eliminate Cooper p a i r s completely, as we emphasized i n Sec.II. Thus NCQP+BET i s v o i d of s-bosons. Note that the switch from BCS+BET to NCQP+BET means to modify (improve) the U and V f a c t o r s . No change i s made of the b a s i c s t r u c t u r e of BET. As we remarked, the many-j s h e l l v e r s i o n of OAI (or OAIT), and hence the microscopic v e r s i o n of IBA (of many-j s h e l l nature), which have been coveted f o r , seem hard to come by. Note, however, that we showed i n Sec.II that SR+BET ( NCQP+BET), OAI and OAIT were e s s e n t i a l l y equivalent, i n s p i t e of the subtle d i f f e r e n c e s which were produced by d i f f e r e n t t a s t e s . Then, i t may make sense to consider that the many-j s h e l l NCQP+BET i s indeed the many-j s h e l l v e r s i o n of a l l the above t h e o r i e s . S
In other words, to construct the many-j s h e l l v e r s i o n NCQP+BET may be considered as an e x c e l l e n t way o f bypassing the d i f f i c u l t task o f handling the generalized s e n i o r i t y . However, an important remark to be repeated here i s that the NCQP+BET i s v o i d of s-bosons (and hence i s not of the IBA form). This i s not a problem f o r us. We have already shown [2,3], e.g., that we can produce v i b r a t i o n , gamma-unstable, r o t a t i o n and other t r a n s i s i o n a l s i t u a t i o n s , without having s-bosons. We do not need the SU(6) symmetry of IBA f o r such purposes.
Meyer and Brenner; Nuclei Off the Line of Stability ACS Symposium Series; American Chemical Society: Washington, DC, 1986.
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In c o n c l u s i o n , we b e l i e v e we are i n a r a t h e r good shape with our BET, p a r t i c u l a r l y since NCQP [8] has become a v a i l a b l e . We expect to s t a r t s h o r t l y to produce numerical r e s u l t s that could be compared with a v a r i e t y of experimental data. As we s t r e s s e d above, we can now go beyond the bounds of the BCS theory. A l s o , the r a t h e r general framework o f BET, constructed i n KT3, now allows us to take i n t o account the n o n - c o l l e c t i v e , as w e l l as c o l l e c t i v e components i n a r a t h e r systematic way. Further accumulation o f data, p e r t a i n i n g to both kinds o f l e v e l s , i s thus h i g h l y hoped f o r . ACKNOWLEDGMENTS The author i s very much indebted to V. G. Pedrocchi and C.-T. L i f o r t h e i r i n v a l u a b l e cooperations. This work was supporterd i n part by the U. S. Department o f Energy.
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Meyer and Brenner; Nuclei Off the Line of Stability ACS Symposium Series; American Chemical Society: Washington, DC, 1986.
