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IBA and the bohr-mottelson model
Nuclear Physics A396( 1983)291 c-306c. © North-HoUand Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher.
291c
IBA AND THE BOHR-MOTTELSONMODEL* Amand FAESSLER I n s t i t u t fur Theoretische Physik, Universit~t TUbingen D-7400 TUbingen, W.-Germany Abstract: The Bohr-Mottelson Model (BMM) and the Interacting-Boson Approximation (IBA) of Arima and l a c h e l l o are compared with each other on the phenomenological and the microscopic level and on the level of selected data. The aim is here not to demonstrate the great successes of both models. But on the phenomenological level we discuss how f a r the models are id e n t ic al and where they are d i f f e r e n t . For the microscopic foundation we mainly investigate how well the S-D subspace represents strongly deformed nuclei. F i n a l l y we look into data to search f o r i n d i c a ti o n s of the boson c u t - o f f and of t r i a x i a l i t y . The f i r s t is a property of IBA, the l a t t e r can not be described by i t . 1. Introduction The aim of this talk is not to show the d e f i n i t e l y large successes of the Bohr-Mottelson model (BMM)I) and of the interacting boson approximation of Arima and lachello2), but the intention is to compare the BMM and the IBA on three levels: ( i ) On a phenomenologicallevel we want to see how far the two models are identical and where they d i f f e r . We shall see that the f i n i t e and constant boson number N plays the central role to distinguish the IBA from the BMM. ( i i ) To have a conserved boson number the IBA boson has to be b u i l t as part i c l e - p a r t i c l e or hole-hole bosons. The model allows that they are only coupled to angular momentum 0 or 2. The question we want mainly to investigate is i f the S-D subspace of the many Fermion-Hilbert space is good enough to describe the lowlying collective states. Especially in the deformed nuclei we shall investigate the question i f one needs also higher angular momentum bosons to describe correctly the nuclear properties. ( i i i ) In a third part we w i l l look to selected data to see how BMM and IBA can describe them. We shall especially search for the boson cut-off of the IBA. Bvt we also w i l l look to indications of t r i a x i a l i t y which is excluded in the IBA at least in its usual form. 2. Phenomenology The Bohr-Mottelson model (BF~I) can be introduced in different ways. The geometrical method is only the best known one. In i t one introduces the collective quadrupole coordinates by defining the surface of the nucleus. R(~,~) = Ro[1 +
2 ~
The c o l l e c t i v e quadrupole coordinates ~2~(t) allow to wr i t e down a classlcal Hamlltonlan.
*Supported by BMFT.
~2~(t) Y2~(~,~)] which depend c l a s s i c a l l y on time,
(I)
292c
A. FAESSLER 1
-,
HBMM = ~ ~ B~v(~Z)a~&2v + V(~2) (2)
1B~.m. i ~ ,, • =~ ~2 ~2~ + ~ C ~2 e2~ .... In the second form of the Hamiltonian in eq. (2) we expanded the mass tensor Buv(~2) and the p o t e n t i a l energy surface V(~2) i n t o a power series of r o t a t i o n a l iihvar~ant expressions of the ~2u" The f i r s t ~wo terms describe a f i v e dimensional harmonic o s c i l l a t o r with mass parameter B and r e s t o r i n g force constant C. This Hamiltonian can be e i t h e r quantized in the l a b o r a t o r y system or in the i n t r i n s i c system. In the l a b o r a t o r y s~stem one w r i t e s the Hamiltonian in the creation b~ and the a n n i h i l a t i o n operators B =(-)~b_ u of the f i v e dimensional harmonic o s c i l . ~ tions.
(3)
The BMM Hamiltonian is in the quadrupole-boson operators a power series which has to be broken o f f at some order. : cn
Z
+ Y
,
°
~=0,2,4
(4) + C21 {[[b+b+]2b]o
+ h.c.} + ....
This Hamiltonian can be diagonalized in the basis of the f i v e dimensional harmonic o s c i l l a t o r . I f we neglect at the moment the coupling to the correct t o t a l angular momentum (which is not a t r i v i a l matter) the basis w r i t e s :
(b+)n.
1 .... . ~-
2
Inb> = ]I
i0>
(5)
To quantize the Hamiltonian (2) in the i n t r i n s i c system one transforms the f i v e quadrupole v a r i a b l e s ~2, i n t o the i n t r i n s i c system and the f i v e v a r i a b l e s are replaced by the three Eul~r angles ~,o,~ and the two deformation parameters ao, a2 or B and y. c~2p = D~0(C,,e,~,)a 0 + (D~2+D2_2)a 2
(6)
1 ; a2 = - - 6sinx
a 0 = Bcosx
/E
A f t e r t h i s transformation the Bohr-Mottelson Hamiltonian can be quantized in the f o l l o w i n g form: ^
~T2
1
~
84
+
-h-2
2BB2 +
v(s,y)
c(s%)
IBA AND THE BOHR-MOTTELSON MODEL
C(SO5) =
12 X k k=1,2,3 2sin2(x_ ~
- ~
i
293c
k)
a T~ sin3y 3-~
(7)
The f i r s t term of the Hamiltonian is the k i n e t i c energy of the B-vibrations. The second term contains the Casimir operator C(S05). I t describes the r o t a t i o n a l energy and the k i n e t i c energy of the y - v i b r a t i o n s . The l a s t term is the potential energy as a function of the deformation parameters B,y. I t is easy to give solutions of the BMM-Hamiltonian in the v i b r a t i o n a l (see the book of Eisenberg and Greiner 3) on page 43 and f o l l o w i n g ) , in the r o t a t i o n a l (book of Eisenberg and Greiner on page 127 and f o l l o w i n g 3 , 4 ) ) and f or the y-unstable l i m i t S ) . This is i n t e r e s t i n g in the connection that also f o r the IBA one can find in the samel i m i t s exact solutions. Concluding t h i s short discussion of the BMM I should mention that the BMM has been extended by Faessler6) to include quadrupole e x c i t a t i o n s of the protons and the neutrons separately. Recently t h i s model has been revived by the Frankfurt group7). This extension corresponds in IBA to IBA2, But I do not want to discuss them here in d e t a i l since I also w i l l not discuss IBA2. A model f o r the quadrupole degrees of freedom which is mathematically complet e l y equivalent to the IBA of Arima and l a c h e l l o 2) has been f i r s t introduced by Jansen, Jolos and D~nau8) in 1974 as the truncated quadrupole model (TQM). An SU~ space is spanned up by the f i v e quadrupole-boson creation operators b~ and the square root [N-~h]l/2. This is the Holstein-Primakoff representation9~ of SU6, Arima and lachelTo2) used the f i v e quadrupole-boson creation operators d~ and an s-boson s+. This is the Schwinger representation 1°) of SU6. The equivalence of the IBA model of Arima and l a c h e l l o and of the TQM model of Jansen, Jolos and D~nau has been shown in sever al publications 2,11,12,13~14 ). Already Jansen et a l. 8) showed that the square root can be replaced by a scalar boson. In a nonrigorous way this equivalence can be made plausible by using conservation of the t o t a l number of bosons. ^ + N = nd + ns = X d ~ d + S S
(8) s + ~ NV~-6b
Here we put the boson number operator fo r the d-bosons of IBA ident ic a l to the boson number operator of the b-bosons of TQM. The Hamiltonian f o r TQM and IBA can now be constructed as a l i n e a r and b i l i n e a r expression of the 36 generators of SU6. For IBA they have the form d~dv, d~s, s + d , s+s
(9)
The number of parameters in the Hamiltonian is r e s t r i c t e d by the requirement that i t has to be r o t a t i o n a l i n v a r i a n t . The boson number operator (8) is the Casimir operator of SU6 and is a good quantum number f o r each representation i f H is only b u i l t by the generators (9). HIBA = Linear + B i l i n e a r Form in Generators (9)
(i0)
I f one has a computer code f o r the BMM in the v i b r a t i o n a l basis Is) i t is a simple step in the Holstein-Primakoff representation modifying i t to a IBA code by using the square root instead of the s-bosons, Details of the IBA including the extension to IBA2 where one treats the
294c
A. FAESSLER
proton and the neutron degrees of freedom s e p a r a t e l y have been discussed in previous t a l k s at t h i s conference. Here I only want t o discuss in as f a r IBA is e q u i v a l e n t to the BMM and where i t i s d i f f e r e n t . To see t h i s I w i l l give a short summary of the work of Ginocchio and Kirson16). S i m i l a r c o n s i d e r a t i o n s can also be found in other Publications]7'18). The work of Ginocchio and Kirson16) has been c r i t i c i z e d f o r n o t u s i n g complex collective variables ~2~" Welguny19) extended their considerations to complex ~2-, so that one finds automatically a hermitian Hamilton operator in the lntrlns]c system. Here I am presentlng the conclus10ns of Ginocchio and KirsonIG) modified by Weigunyl9). Ginocchio and KirsonIG) introduce deformed bosons in the intrinsic system. •
+ + Bsiny (d~+d+2) B+ = s+ + ~cosy do ~
IN;S,~>
=
(ii)
C(B+)NIo >
The i n t r i n s i c N boson s t a t e depends on the deformation v a r i a b l e s 8 and y (6). I t can be shown t h a t from t h i s i n t r i n s i c s t a t e ( i i ) one can o b t a i n a l l the IBA s t a t e s in the l a b o r a t o r y system by angular momentum p r o j e c t i o n PIK in connection w i t h using 8 and y as generator c o o r d i n a t e s .
JN~ '">LAB"
60 ° =o ~ 84dB fo sin3ydy ~KfN~;K(B'Y)PIKIN;B'Y>
(12)
c h a r a c t e r i z e s a l l the quantum numbers of the IBA s t a t e a p a r t of the t o t a l boson number N. K is the p r o j e c t i o n of the t o t a l angular momentum I to the i n t r i n s i c za x i s . The deformation energy surface in the i n t r i n s i c system f o r the IBA can now be defined by t a k i n g the e x p e c t a t i o n value o f the IBA Hamiltonian w i t h the i n t r i n s i c wave f u n c t i o n (11). ^
VN(~,y) = #2 = N~s + . N(Ed-Es) 1+82
(13)
+ N(N+I) [C1+C282 + C3B3cos3y+C4B4] (1+~2) 2 One can now show t h a t one obtains f o r the overlap o f the r o t a t e d i n t r i n s i c wave f u n c t i o n IN;~,y;@,~,~> w i t h the IBA s t a t e IN,~> a second o r d e r d i f f e r e n t i a l equat i o n i f one replaces the c r e a t i o n and a n n i h i l a t i o n operators o f the bosons by d i f f e r e n t i a l o p e r a t o r s , an equivalence which i s w e l l known f o r the l i n e a r harmonic oscillator: b+
/%
(x
-
~) OA
(14)
b =_~_i (x + ~ x ) In our case the relation is a l i t t l e bit more complicated, but in principle i t is the same. ~eiguny19) shows that the d i f f e r e n t i a l equation obtained in this way goes over into equation (7) for the BMMif one takes the leading terms for N ÷ ~. Thus IBA goes over into BK~I i f the boson number N goes to i n f i n i t y . Therefore the deformation energy surface of the equivalent Bohr-Mottelson model i f we take the leading term for N to i n f i n i t y which is the factor of N(N+I) is:
IBA AND THE BOHR-MOTTELSON MODEL V(~,y)
= [C~+C2B2+C~3 cos3¥ + C ~ 4 ] / ( I + ~ 2 ) 2
295c (15)
One sees t h a t we get a gamma unstable p o t e n t i a l in (13) and (15) i f we put C3 or l C~ equal to zero. By choosing C3 p o s i t i v e one obtains an o b l a t e and f o r a negative cnoice a p r o l a t e shape. IBA is not able to describe a t r i a x i a l deformed nucleus. To obtain a t r i a x i a l i t y we would need at least a term with tcos(3y)] 2. To obtain such a term one needs t h i r d order terms in the generators (9). A term which would lead to t r i a x i a l i t y would f o r example be:
[[d+d+]2 d+]
O/3
[[d~]2d]o/3
(16)
We can therefore summarize our comparison between BMM and IBA (or TQM) in the following way: The IBA is distinguished from the BI~ by the f i n i t e boson number N. IBA goes over into BMM i f tile boson number goes to i n f i n i t y . In this sense IBA is more general than BMM. On the other side the Hamiltonian of IBA is more restricted than the BMM Hamiltonian. IBA can for example not describe t r i a x i a l shapes in i t s present form. 3. Microscopic foundation We have seen that the major d i f f e r e n c e between the i n t e r a c t i n g boson model (IBA) or the truncated quadrupole model (TQM) and the Bohr-Mottelson model (BMM) is the f i n i t e and constant number o f bosons so t h a t we a u t o m a t i c a l l y obtain the BMM i f the number of bosons N goes to i n f i n i t y . I f we use the Tamm-Dancoffapproximation (TDA) f o r the d e f i n i t i o n of the c o l l e c t i v e states and i f we are mapping the most c o l l e c t i v e TDA state in lowest order onto the boson we can use in p r i n c i p l e an expansion i n t o p a r t i c l e - h o l e or p a r t i c l e - p a r t i c l e ( h o l e - h o l e ) or two quasip a r t i c l e states. +
+ = Z . C~" B~ ml ml
amai
f o r a ph-basis
ama + +i + + ~m~i
f o r a pp(hh)-basis
l
(17)
f o r a 2q.p. basis
To get a good boson numbe~ we have to request f o r IBA a p a r t i c l e - p a r t i c l e or a h o l e - h o l e basis. Since IBA r e s t r i c t s the many-Fermion-Hilbert-space to the most c o l l e c t i v e angular momentum zero and angular momentum 2 s o l u t i o n s one has to inv e s t i g a t e the question how well does a S-D subspace represent the low l y i n g states. One expects t h a t t h i s S-D subspace is describing b e t t e r the low l y i n g states in spherical than in s t r o n g l y deformed n u c l e i . Thus we w i l l study here mainly t h i s question in s t r o n g l y deformed nuclei by looking i n t o d i f f e r e n t i n v e s t i g a t i o n s which have been published r e c e n t l y 2 ° , 2 1 , 2 2 , 2 3 ) . A comparison of the r e s u l t s in a S-D subspace with exact shell model c a l c u l a t i o n s in a f i n i t e space in the laborat o r y system has been performed by OtsukaZ°). He defines a degenerate s i n g l e - p a r t i c l e
space of Og712, Ids/~, ld~/~ and 2sl/2 levels for protons and neutrons, lhe protons and neutrons lh~erac~ between themselves by the surface delta interaction, while the proton-neutron interaction is represented by a quadrupole-quadrupole force. HFermion = H~ + H~ - f ~ H = - 4x g
~ Y2(x)Y2(v) ~,v
~ Yk(i) Yk(j) i,j;k
f=1.5 MeV
(18) ; g:O.5 MeV
296c
A. FAESSLER
For 6 protons and 6 neutrons in the above degenerate s i n g l e - p a r t i c l e levels one obtains from the Hamiltonian (18) a good approximation to a rotor spectrum including also the corresponding quadrupole t r a n s i t i o n p r o b a b i l i t i e s . The S-D subspace is constructed with the help of the generalized s e n i o r i t y formalism as described in ref. 24. Fig. I shows the overlap II with the shell model states. One sees that for angular momentum zero the S-D subspace contains 76% of the angular momentum zero shell model state. The angular momentum 12 state is only contained in the subspace by 7.6%. This percentage w i l l be increased 22) for heav i e r nuclei in a larger s i n g l e - p a r t i c l e space but the q u a l i t a t i v e feature t h a t w i t h increasing angular momentum one needs angular momentum 4 and higher nucleon ~airs to describe the r o t a t i o n a l states w i l l be correct. Sugawara-Tanabe and Arima 2) find for the 10+ in l~4Er a value around 40%.
Prob, (%) [
i
[
i
]
i
i
fO0 80
1.5 MeV
/
60
/
4O 20 0
I
~
I
I
I
I
I
0
2
4
6
8
10
t2
Fig. 1: Overlap of the shell model solution f o r the r o t a t i o n a l states of the ground state. band in a Og~,o, Id. ~.L . .z . ld . .L .~ . 2sI,I 2 s i n g l e - p a r t i c l e space with the • . ll L Fermlon Hamlltonlan (18) accordlng to O~suka2°l. With increasing angular momentum the overlap with the S-D subspace is decreasing and one needs nucleon pair states coupled to angular momentum 4 and higher to describe the many-body Fermion states. Most of the investigations how much the S-D subspace is exhausting of the many-Fermion states have been performed in the i n t r i n s i c system21,22,23). Looking to these studies one should not forget that the typical average angular momentum in an i n t r i n s i c state is about 2 f o r r e a l i s t i c nuclei. Thus these studies give only information about the very low angular momentum states and say nothing about the q u a l i t y of the high spin states in IBA. These investigations s ta r t from a Nilsson Hamiltonian and a pairing force. H = hNilsso n (~)
-
G ~ aia + ~ + a~ak ik
(19)
The solution of this Hamiltonian can be found in the BCS approximation. In t h i s approach the many-Fermion wave function has not a good nucleon number, but one can project with the projection o~erator P2N onto the nucleon number 2N.
IBA AND THE BOHR-MOTTELSON MODEL +
297c
+
P2N ]BCS> = P2N 11 (u k + v k akaE)IO> k>O =
vk
oF[X
k>OUkk
++ N aka ~] I0 > (2O)
= Or [A+]NIO> +
with:
A+ = [
x I AIO
I
x I = C v~2C(jjllm-mO) k~>OV~kuk The nucleon p a i r operator A+ is expanded i n t o nucleon p a i r s w i t h good angular momentum I, The expansion c o e f f i c i e n t s x I are given f o r a s i n g l e j - s h e l l . To get a shell of s i m i l a r size as the neutron shell between the magic numbers 82 and 126 one uses a j=41/2 s i n g l e - p a r t i c l e s h e l l . This n a t u r a l l y simulates only the neu-trons and not the protons and by looking to the overlaps one should take t h i s i n t o account. The t o t a l overlap is reduce~ by i n c l u d i n g also the proton part of the wavefunction. The p r o b a b i l i t y PI = Xl is p l o t t e d 22) in f i g . 2. This procedure f o r c a l c u l a t i n g the wave f u n c t i o n corresponds to p r o j e c t i o n onto the S-D subspace a f t e r v a r i a t i o n . This method does not y i e l d the best wave f u n c t i o n in the model space.
I
'
'
'
'
1
'
'
'
'
I
0+-2+-4+ 100 . . . . . . . . . -8.0
5o
i
-4.0
13_
0.0
0.5 N I~
1.0
cg
0.0
Fig. 2: P r o b a b i l i t i e s to f i n d 0+, 2+ , 4+ and 6+ pairs in the Cooper p a i r A+ o b t a i ned in the Nilsson model. The s i n g l e j=41/2 s i n g l e - p a r t i c l e shell is taken w i t h the deformed s i n g l e - p a r t i c l e f i e l d of B=-0.32 and the p a i r i n g force of G=O.2 MeV. The p r o b a b i l i t i e s Pl=X~ and the sum of such q u a n t i t i e s are p l o t t e d as a f u n c t i o n o f the number of nucleon p a i r states N over the t o t a l number of p a i r states ~ = j + I / 2 . The dashed l i n e gives the i n t r i n s i c quadrupole moment of which the scale is p l o t t e d on the r i g h t hand side. In a boson approximation only r e s u l t s f o r N/~
298c
A. FAESSLER
The deformation is negative (~=-0.32) and the p a i r i n g s t r e n g t h G=O.2 MeV. The prob a b i l i t y o f f i n d i n g angular momentum 0+, 2+ or 4 + in the p a i r s t a t e A+ i s p l o t t e d as a f u n c t i o n o f the number of nucleon p a i r s N over the t o t a l number o f p a i r states = j + i / 2 . At the value o f the maximum quadrupole moment (N/~ = 0.38) the overlap o f the two nucleon states A+ w i t h the S-D subspace is 92%. This overlap is l a r g e and is increasing i f one goes to the h a l f - f i l l e d s h e l l (N/~ = 0 . 5 ) . Values l a r g e r than N/A = 0.5 should not be considered since there one would look i n t o two hole p a i r s t a t e s . I f one performs a lowest order mapping of A+ onto bosons +
A+(boson) = Xos+ + x2d 0 + . . .
(21)
one can e a s i l y c a l c u l a t e the overlap of the mapped many Fermion s t a t e w i t h the s-d boson space. For the maximum quadrupole moment one has N=8 bosons and the overlap i s 73%. Everyone can decide by h i m s e l f i f he wants to c a l l t h i s dominance of monopole and quadrupole p a i r s in the Nilsson model or i f he wants to conclude t h a t 73% i s a too small oercentage to give a good d e s c r i p t i o n . Otsuka et a l . 22) give also the r e s u l t s f o r many non-degenerate s i n g l e - p a r t i c l e s h e l l s Ohg/2, I f T / 2 , I f ~ / ~ , 2p~/~, 2P1/2 and O i ~ / ~ f o r the p o s i t i v e deformation 6=0.26 and the p a i r i ~ g - f o r c ~ - ~ = O . 2 MeV. The ~ # h ~ l e - p a r t i c l e energies are taken from r e f . 25. The p r o b a b i l i t y PQ-2 f o r f i n d i n g angular momentum 0+ and 2+ in the Cooper p a i r A+ is now i n c r e a s e d - t o 95.5%.The number of p a i r s t a t e s a t the maximum quadrupole moment is N=II. I f one performs the lowest order mapping onto bosons t h i s corresponds t o an overlap o f the t o t a l wavefunction w i t h the s-d boson space of 77%.
[
.
.
.
.
I
'
.
.
.
.
0+-2+_4+
I00
400.~E •o"-; 50
1 200
13_
o.o
b
o.5 N/a ,b 20
o
N
Fig. 3: P r o b a b i l i t i e s to f i n d 0+, 2+ , 4 + and 6+ p a i r s i n the Cooper p a i r obtained in the Nilsson model • Non-degenerate s i n g l e j - s h e l l s Ohq/~, i f T / ~_ , I f. 5 2/ , 2pB/~, . _ _ . _ 2Pi/2 and 0i13/2 have been used w l t h a deformation B=0.25 and the palr~ng f o r c e s t r e n g t h G=O.2"MeV. S i n g l e - p a r t i c l e energies are taken from r e f . 25. The overlap w i t h the S-D subspace f o r the maximum value of the i n t r i n s i c quadrupole moment QO which is i n d i c a t e d as a dashed l i n e and has i t s scale on the r i g h t hand side i s 95.5% f o r the Cooper p a i r A+ obtained in the Nilsson model. In lowest order mapping onto bosons the overlap of t h i s wavefunction at the maximum quadrupole moment w i t h the t o t a l wavefunction is 77%.
IBA A N D T H E B O H R - M O T T E L S O N M O D E L
299c
Beset al. 23) use practically the samemodel for the single j=41/2 shell. But they minimize the energy after projection onto the S-D subspace. Their results are given in f i g . 4. In part a and b one sees that the model with only S-D pairs underestimates the intrinsic quadrupole moment and overestimates the pairing gap.
o.......... 0.3
o_ ({:) ......
(a) N - B C S , ~•
oar,r
n/O
/
= 0.38
° !ii°,"°I_ I~
.--"
oc
o,
o,
o
i "N-BCS o,.
o,
nlQ 0,~
,"
,
b)
,
,
,
,
..._S_O_ ....
(d)
~, },
OI 0,3
~ az N+BCS
SO
SDG SDGI
0
081
0.60
051
2
0.59
026
025
0.25
0.40
Z, 6
;
/ 0.2
i
..o
mlP
i o J,
0.09
i 0.6
n/O
Fig. 4: Properties of the Nilsson plus pairing wavefunction of the Hamiltonian (19) calculated in a single j=41/2 ~hell according to (20). (a) The intrinsic static quadrupole moments in units of~ as a function of the f i l l i n g of the shell N/~(n/~ in the figure). The solid curve corresponds to the f u l l solution of the Nilsson plus BCS model. The dashed-dotted and the dashed lines give solutions where the expectation value of the Hamiltonian is minimized after truncation onto the S-D and the S-D-G subspace opposite to figs. 2 and 3 where the solution in the subspace is only projected from the Nilsson-BCS solution. In part (b) the pairing gap and in (c) the occupation probability as a function of the magnetic substates m/~ is plotted. Part (d) shows the amplitude x I for the different angul a r momenta of the nucleon pairs with angular momentum I in the Cooper pair. These amplitudes are called in the f i g . ~x" They are given for minimization in the indicated subspaces up to angular momentum 6~. In the Nilsson plus BCS intrinsic state the amplitudes for angular momentum zero and two are: Xo~0 = 0.57 and x2~2=0.77. From f i g . 4 one reads o f f that the pairing gap obtained by minimization of the expectation value of the many-body Hamiltonian in the S-D subspace is 1.6 times larger than for N-BCS and the i n t r i n s i c quadrupole moment by a factor 0.75 smallen Beset al. 23 ) used also the Otsuka20 ) many Fermion Hamiltonian I"n the same single-particle space to calculate the intrinsic quadrupole moment and the pairing gap for the Nilsson plus BCS model and for the S-D subspace. Only the surface delta interaction between nucleons of the same charge is replaced by the monopole and the quadrupole pairing force. The results are given in f i g . 5. Although the Cooper pair has a large overlap with the S-D subspace one finds in figs. 4 and 5 appreciable changes of different properties of the nuclei i f the space is restricted to S-D nucleon pairs. Bohr and Mottelson21) pointed out in a recent preprint that this is connected with the fact that the properties of the nuclei are determined by a few valence nucleons. The many low lying nucleons in-
300c
A. FAESSLER
crease the overlap with the S-D subspace but they do not determine the properties of the nuclei near the Fermi surface which we measure in the low lying states.
A 20
(a )
I
]o'
I I
&
(c)
SD
~o6 1.0
15
20
I
10
~5
-f
~
>
~ °'5 FsD<¢ff]l *(f])N~BC;'" L 1.0
I
"'.\
1
'i
"'" @I
15
20
-f
20
_ " ~
J
Fig. 5: I n t r i n s i c quadrupole moment (a) and overlap of the many Femlion wavefunction (b) f o r f i n i t e strength f of the quadrupole-quadrupole force with the wavefunction f o r i n f i n i t e strength f=~ as a function of the quadurpole-quadruDoleforc~ strength f (18). Part (c) and (d) show the pairing gap and the occupation probabil i t y as a function of f and the deformed single p a r t i c l e energies c, respectively. The dashed-dotted l i n e is the r e s u l t of Otsuka2°), the solid l i n e the result of the N plus BCS model, the dashed l i n e corresponds to the results of a minimization of the expectation value of the Fermion Hamiltonian in the S-D subspace. The wavefunctions are calculated fo r 6 protons and 6 neutrons. 4. Comparison with data As I stressed already above the purpose of this t a l k and t h i s chapter is not to show the undoubtedly large successes of the IBA to explain data. We want to investigate here the l i m i t s of the v a l i d i t y of the IBA and suggest generalizations. In chapter 2 we saw that i f the boson number N goes to i n f i n i t y IBA is ident i c a l with BMM. On the other side the deformation energy surface which one obtains in t h i s way from IBA is more r e s t r i c t e d than the deformation energy surface allowed in the BMM. The usual IBA deformation energy surface is not able to describe t r i a x i a l i t y . We therefore want to study here the experimental evidence f or the boson cuto f f which is connected with the f i n i t e number of bosons N. Furthermore we want to look into indications of t r i a x i a l i t y which cannot be described in IBAI or IBA2 (without a quadrupole force between protons and between neutrons). 158Er with 68 protons and 90 neutrons has a t o t a l number of bosons N=13. This corresponds to a maximum angular momentum of 26. In the crystal ball in Heidelberg 26) a maximum angular momentum of Imax(exp)=44+ has been found. Although there have been t r i a l s to explain the reduction of the E2 t r a n s i t i o n p r o b a b i l i t i e s near
IBA AND THE BOHR-MOTTELSON MODEL
301c
the f i r s t backbending and above by the boson c u t - o f f , I would expect that todaynoone is surprised that IBA is not describing the maximum angular momentum c o r r e c t l ~ Due to alignment of a i1~,~ neutron pair at f i r s t backbending, alignment of a h proton palr at the secon~ anomaly of the moment of i n e r t i a and possibly a t h l r ~ I/2 alignment of a h or h . . . . neutron p a i r the maximum angular momentum can be much larger than a l l o ~ by t ~ / ~ o s o n c u t - o f f . The standard example f o r the boson c u t - o f f 27) is 78Kr with 36 protons and 42 neutrons and a boson number N=8. Fig. 6 shows27) that from d i f f e r e n t models only IBA is able to reproduce the reduction of the E2-transition p r o b a b i l i t i e s at high angular momenta in the yrast states. But in Fig. 7 one sees that between angular momentum 6 and 12 one has an alignment of probably two g9/2 protons. •
/ Z
78 3sKr~2
N=8 °
r'n
i
i LIJ
I
I
I
I
I
I
I
I
I
I
2 6 10 1L 18 Anguler Momentum [lh]
Fig. 6:E2 transition probabilities between angular momentum I and I-2 in units of the reduced transition probability between angular momentum 2+ and 0+. The f i g . is taken from ref. 27. The data are measured by Hellmeister et ai.27). The theoret i c a l results for IBAI, a rigid rotor and the asymmetric rotor (AR) are indicated. Reduction of the reduced transition probability with increasing angular momentum has been connected with the boson cut-off27). But f i g . 7 indicates an alignment of a g9/2 proton pair which d e f i n i t e l y cannot be described by IBA. The total gain of angular momentum is roughly 4. Such an alignment of 4 units of angular momentum can not be described in the framework of IBA. At least part of the reduction of the E2 transition probabilities found in 78Kr at high angular momentum states (see f i g . 6) is due to this alignment. Even careful discussions of the situation in 79Rb cannot remove the fact that IBA cannot be applied in 78Kr. I would therefore expect that this standard example of the boson cut-off is not a convincing one. One could even have general doubts i f one can see indications of the boson cut-off in the way of a termination of a rotational band ( i f one wants not to look iinto very l i g h t nuclei as 2°Ne where the boson cut-off is identical with the shell model cut-off). Normally the boson cut-off w i l l not exist since for high spin states in rotational nuclei g and i-pairs w i l l be important and the Coriolis force
302c
A. FAESSLER
w i l l a l i g n nucleon p a i r s w i t h a l a r g e s i n g l e - p a r t i c l e angular momentum j which y i e l d s a d d i t i o n a l c o n t r i b u t i o n s to the t o t a l angular momentum.
15
E E
_2 .....
/[___
E
alignment of 1~g9/212
.<
o o
I
I
l
I
0.2 O.L. hw [MeVl
I
I
08
t
I
0.8
I
I
I
1.0
Fig. 7: Angular momentum as a f u n c t i o n o f the r o t a t i o n a l frequency ~. The curve • owsbetween angular momentum 7 and 12 the alignment of a nucleon p a i r 27) This is probably a g9/2 proton p a i r which y i e l d s an aligned s i n g l e - p a r t i c l e angular momenGum of i=4~. Such an alignment is not contained in IBA. In eqs. (13) and (15) we saw t h a t IBA cannot describe t r i a x i a l n u c l e i . Are t h e r e experimental i n d i c a t i o n s t h a t such t r i a x i a l l y deformed nuclei e x i s t ? This question is a very o l d one and i s not easy to answer. Recent data o f Stachel et a l . 2e) might i n d i c a t e such a t r i a x i a l deformation. By Coulomb e x c i t a t i o n they measured the B(E2)-values up t o the 10 + to 8+ t r a n s i t i o n . They furthermore obtained also the diagonal quadrupole moment f o r the ground s t a t e r o t a t i o n a l band up to the 8 + . The r o t o r model, IBAI and IBA2 are not able t o reproduce the t r a n s i t i o n prob a b i l i t i e s . One obtains a good f i t to the data i f one extends IBA w i t h a g-boson or i f one uses the asymmetric r o t o r . Furthermore Tamura 29) o b t a i n s also good res u l t s w i t h a boson expansion i n t o p a r t i c l e - h o l e TDA c o l l e c t i v e s t a t e s . But i f one looks to the diagonal quadrupole m a t r i x elements only the asymmetric r o t o r is able t o describe the data.(Tamura 29) does not give the diagonal quadrupole m a t r i x elements but he i n d i c a t e s t h a t t h e i r values are decreasing and not i n c r e a s i n g l i k e the d a t a . ) The d i f f e r e n t t h e o r e t i c a l d e s c r i p t i o n s and the data are given in f i g . 8a and 8b. The IBA can n a t u r a l l y be extended to include asymmetric deformations. For t h a t one needs only to add t o the t o t a l Hamiltonian three body boson i n t e r a c t i o n terms o f the type given in eq. (16). I t would be i n t e r e s t i n g t o t e s t i f w i t h the i n c l u s i o n of such terms 1°~Ru can be described in agreement w i t h the data. Another extension o f IBA2 to t r i a x i a l i t y has r e c e n t l y been given by Dieperink and B i j k e r by a l l o w i n g f o r quadrupole forces among the protons and among the neutrons.
IBA AND THE BOHR-MOTTELSON MODEL
0.6
I
i I I 10LRu
i
I
l
I
I
I T
B(E2)
ra..~I-.-IBA+g
..;'.!-';---~.:---T--~ot o,-
l
I
~O~Ru
Lk1 /
-0.5 Tamu
I
303c
-~ -1.0
-o.2 f
IBA 2
UJ
\
-~.s
\,
W rn
IBA2 0
I
I
I
I
I
10
I
I
2 6 Angulor Nomentum [hl
I
J
I
2 6 10 Angulor Momentum[hi
(a)
(b)
Fig. 8: Reduced t r a n s i t i o n p r o b a b i l i t i e s (a) and diagonal quadrupole moments (b) f o r the y r a s t band in l°4Ru according to Ref. 28. Only the asjanmetric r o t o r w i t h ~=25 ° is able t o d e s c r i b e these data. Results f o r IBA2, the r i g i d r o t o r , IBA and a g-boson, the asymmetric r o t o r and the r e s u l t s o f a boson expansion by Tamura 28) are given. 5. Summary We t r i e d t o compare the Bohr-Mottelson model (BMM) and the i n t e r a c t i n g boson a p p r o x i m a t i o n (IBA). The l a s t model is e q u i v a l e n t to the t r u n c a t e d quadrupole model (TQM) o f Jansen, Jolos and D~naue)). We t r i e d to see how f a r BMM and IBA are i d e n t i c a l and where they are d i f f e r e n t . On the phenomenological l e v e l we saw t h a t the IBA i s i d e n t i c a l w i t h the BMM i f the boson number N goes t o i n f i n i t y . For small boson numbers those models give d i f f e r e n t r e s u l t s although they are q u a l i t a t i v e l y very s i m i l a r . On the o t h e r side the Hamiltonian of IBA i s more r e s t r i c t e d than the BMM Hamiltonian since one r e quests boson number c o n s e r v a t i o n and a l l o w s o n l y two body boson i n t e r a c t i o n terms. Thus the IBA Hamiltonian can f o r example not describe t r i a x i a l deformed n u c l e i . For both models BMM and IBA one can f i n d the exact s o l u t i o n in t h r e e l i m i t s : The f i v e dimensional harmonic quadrupole v i b r a t o r , the r o t o r and the v - u n s t a b l e case. From the l e v e l of the m i c r o s c o p i c f o u n d a t i o n s of the two models we saw t h a t
304c
A. FAESSLER
we have to build the bosons fo r IBA by two nucleon p a r t i c l e s (pp) or by nucleon holes (hh). This y i e l d s automatically boson number conservation which is the essential difference between IBA and BMM. The second question is i f low lying solutions of the many-body Fermion Hamiltonian can be represented in a subspace where a l l the nucleons are coupled pairwise to angular momentum zero and angular momentum 2 and where fo r each angular momentum only one such c o l l e c t i v e pair is taken into account. This has been contested by the Copenhagen School. We discussed in detail the l i t e r a t u r e on this question. We saw that the Cooper pairs even in strongly deformed nuclei are b u i l t mainly by angular momentum 0 and 2 pairs with p r o b a b i l i t i e s larger than 90% f o r the largest deformation in a s h e l l . Nevertheless observables l i k e the pairing gap and the quadrupole moment are quite d i f f e rent with and without inclusion of higher angular momentum pairs. In a j=41/2 shell one finds for the maximum value of the quadrupole moment a pairing gap which is f o r only S-D pairs by a factor 1.6 larger than in the Nilsson BCS model. The quadrupole moment on the other side is by a factor 0.75 smaller. Bohr and Mottelson2]) recently pointed out that the large overlap found f or the Cooper pairs in the Nilsson model is due to the almost perfect overlap of the nucleon pairs in the lower shells or for the non-valence pairs in a large s i n g l e - p a r t i c l e shell with the S-D space. The valence nucleons which determine the physics of the low lying c o l l e c t i v e states have an appreciable admixture of G and l - p a i r s . This might y i e l d the differences of the results for pairing gaps and quadrupole moments for the f u l l and the S-D r e s t r i c t e d space. F i n a l l y we looked into the data to search f o r indications of the boson cuto f f . We found that this c u t - o f f is hard to see and that the standard example for the boson c u t - o f f , the reduction of the B(E2) values in 78Kr might be at least p a r t i a l l y connected with the alignment of two g9/2 protons. With 1°4Ru we found a nucleus which seems to request a t r i a x i a l shape which can not be described by IBA i f one r e s t r i c t s the Hamiltonian to at most two-body boson interactions. An extension which includes three-body terms or quadrupole forces among the protons and among the neutrons might be possible to describe also in IBA this nucleus.
References 1. 2. 3. 4. 5. 6. 7. 8.
i0. 11. 12. 13.
A. Bohr, Kgl. Danske Vid. Selsk. Mat. Fys. Medd. 27 (1952) 14. A. Bohr, B. Mottelson, Kgl. Danske Vid. Selsk. Mat-- Fys. Medd. 27 (1953) 16. A. Arima, F. l a c h e l l o , Phys. Rev. Lett. 35 (1975) 1069; Ann~ Phy-s. (N.Y.) 99 (1976) 253; 111 (1978) 201; 123 (1979) 4~-~. J.M. E i s e n b e ~ W . Greiner, ~ l e a r Theory, North Holland Publ. Comp., Amsterdam 1975. A. Faessler, W. Greiner Z. Phys. 168 (1962) 1965 and A. Faessler, W. Greiner, R.K. Sheline, Nucl. Phys. 70 (196~-~--33. M. Jean, L. Wilets, Comptes--Rendus 241 (]955) 1108 and Phys. Rev. 102 (1956) 788. A. Faessler, Nucl. Phys. 85 (1966) 653. P.O. Hess, J.A. Maruhn, .~FT--Greiner, in "Future Directions in Studies of Nucl e i f a r from S t a b i l i t y " , eds. J.H. Hamilton et a l . ; North Holland, 1980, Amsterdam. D. Jansen, R.V. Jolos, F. D~nau, Nucl. Phys. A224 {1974) 93. R.V. Jolos, F. D~nau, D. Jansen, Theor. Mat. Phys. 20 (1974) 112. G. Holzwarth, D. Jansen, R.V. Jolos, Nucl. Phys. A2-G-I-(1976) i . T. Holstein, H. Primakoff, Phys. Rev. 58 (1940) I ~ J. Schwinger, in "Quantum Thoery of A n ~ l a r Momentum", eds. L. Biedenharn, H. van Dam, Academic Press, N.Y. (1965) p. 229. J.P. B l a i z o t , E.R. Marshalek, Nucl. Phys. A309 (1978) 422. G. Kyrchev, Nucl. Phys. A349 (1980) 416. A. Klein, M. V a l l i e r e s , ~ . Rev. Lett. 46 (1981) 586.
IBA AND THE BOHR-MOTTELSON MODEL
305c
14. M. Moshinsky, Nucl. Phys. A338 (1980) 156 and A354 (1981) 257c. 15. W. Gneuss, W. Greiner, NucTT-l~hys. 171 (1971) -4-4-9-7. P.O. Hess, J. Maruhn, W. Greiner, J.----P~qys. G7 (1981) 737. 16. J.N. Ginocchio, M.W. Kirson, Nucl. Phys. 35-0--(1980) 31. 17. A. Bohr, B.R. Mottelson, Phys. Scripta 22-~/980) 468. 18. A.E.L. Dieperink, O. Scholten, F. lacheTTo, Phys. Rev. Lett. 44 (1980) 1747. 19. A. Weiguny, Z. Phys. A301 (1981) 335. 20. T. Otsuka, Nucl. P h y s - - 6 8 (1981) 244. 21. A. Bohr, B.R. Mottelson,-~Pl~ys. Scripta 2_22(1980) 468 and to be published in Phys. Scripta 1982. 22. T. Otsuka, A. Arima, N. Yoshinaga, Phys. Rev. Lett. 48 (1982) 387. K. Sugawara-Tanabe, A. Arima, Phys. Lett. IIOB (1982787. 23. D.R. 8es, R.A. Broglia, E. Maglione, A. V i t t u r i , Phys. Rev. Lett. 48 (1982) (1982) 1001. 24. T. Otsuka, A. Arima, F. lachello, Nucl. Phys. A309 (1978) I . 25. C. Gustafson et a l . , ARk. Fys. 36 (1967) 613. - 26. D. Habs, V. Metag, R.S. Simon et a l . , private communications, 1982. 27. H.P. Hellmeister, K. Lieb, J. Panqueva, in "Interacting Bose-Fermi Systems in Nuclei", ed. F. lachello, Plenum Pub1. Corp., 1981. 28. J. Stachel, N. K a f f r e l l , E. Grosse, H. Emling, H. Folger, R. Kulessa, D. Schwalm, preprint 1982. 29. T. Tamura, Proceedings of the INS International Conference on Dynamics of Nuclear Collective Motions, July 6-10, 1982, Mt. Fuji, Japan.
291c
IBA AND THE BOHR-MOTTELSONMODEL* Amand FAESSLER I n s t i t u t fur Theoretische Physik, Universit~t TUbingen D-7400 TUbingen, W.-Germany Abstract: The Bohr-Mottelson Model (BMM) and the Interacting-Boson Approximation (IBA) of Arima and l a c h e l l o are compared with each other on the phenomenological and the microscopic level and on the level of selected data. The aim is here not to demonstrate the great successes of both models. But on the phenomenological level we discuss how f a r the models are id e n t ic al and where they are d i f f e r e n t . For the microscopic foundation we mainly investigate how well the S-D subspace represents strongly deformed nuclei. F i n a l l y we look into data to search f o r i n d i c a ti o n s of the boson c u t - o f f and of t r i a x i a l i t y . The f i r s t is a property of IBA, the l a t t e r can not be described by i t . 1. Introduction The aim of this talk is not to show the d e f i n i t e l y large successes of the Bohr-Mottelson model (BMM)I) and of the interacting boson approximation of Arima and lachello2), but the intention is to compare the BMM and the IBA on three levels: ( i ) On a phenomenologicallevel we want to see how far the two models are identical and where they d i f f e r . We shall see that the f i n i t e and constant boson number N plays the central role to distinguish the IBA from the BMM. ( i i ) To have a conserved boson number the IBA boson has to be b u i l t as part i c l e - p a r t i c l e or hole-hole bosons. The model allows that they are only coupled to angular momentum 0 or 2. The question we want mainly to investigate is i f the S-D subspace of the many Fermion-Hilbert space is good enough to describe the lowlying collective states. Especially in the deformed nuclei we shall investigate the question i f one needs also higher angular momentum bosons to describe correctly the nuclear properties. ( i i i ) In a third part we w i l l look to selected data to see how BMM and IBA can describe them. We shall especially search for the boson cut-off of the IBA. Bvt we also w i l l look to indications of t r i a x i a l i t y which is excluded in the IBA at least in its usual form. 2. Phenomenology The Bohr-Mottelson model (BF~I) can be introduced in different ways. The geometrical method is only the best known one. In i t one introduces the collective quadrupole coordinates by defining the surface of the nucleus. R(~,~) = Ro[1 +
2 ~
The c o l l e c t i v e quadrupole coordinates ~2~(t) allow to wr i t e down a classlcal Hamlltonlan.
*Supported by BMFT.
~2~(t) Y2~(~,~)] which depend c l a s s i c a l l y on time,
(I)
292c
A. FAESSLER 1
-,
HBMM = ~ ~ B~v(~Z)a~&2v + V(~2) (2)
1B~.m. i ~ ,, • =~ ~2 ~2~ + ~ C ~2 e2~ .... In the second form of the Hamiltonian in eq. (2) we expanded the mass tensor Buv(~2) and the p o t e n t i a l energy surface V(~2) i n t o a power series of r o t a t i o n a l iihvar~ant expressions of the ~2u" The f i r s t ~wo terms describe a f i v e dimensional harmonic o s c i l l a t o r with mass parameter B and r e s t o r i n g force constant C. This Hamiltonian can be e i t h e r quantized in the l a b o r a t o r y system or in the i n t r i n s i c system. In the l a b o r a t o r y s~stem one w r i t e s the Hamiltonian in the creation b~ and the a n n i h i l a t i o n operators B =(-)~b_ u of the f i v e dimensional harmonic o s c i l . ~ tions.
(3)
The BMM Hamiltonian is in the quadrupole-boson operators a power series which has to be broken o f f at some order. : cn
Z
+ Y
,
°
~=0,2,4
(4) + C21 {[[b+b+]2b]o
+ h.c.} + ....
This Hamiltonian can be diagonalized in the basis of the f i v e dimensional harmonic o s c i l l a t o r . I f we neglect at the moment the coupling to the correct t o t a l angular momentum (which is not a t r i v i a l matter) the basis w r i t e s :
(b+)n.
1 .... . ~-
2
Inb> = ]I
i0>
(5)
To quantize the Hamiltonian (2) in the i n t r i n s i c system one transforms the f i v e quadrupole v a r i a b l e s ~2, i n t o the i n t r i n s i c system and the f i v e v a r i a b l e s are replaced by the three Eul~r angles ~,o,~ and the two deformation parameters ao, a2 or B and y. c~2p = D~0(C,,e,~,)a 0 + (D~2+D2_2)a 2
(6)
1 ; a2 = - - 6sinx
a 0 = Bcosx
/E
A f t e r t h i s transformation the Bohr-Mottelson Hamiltonian can be quantized in the f o l l o w i n g form: ^
~T2
1
~
84
+
-h-2
2BB2 +
v(s,y)
c(s%)
IBA AND THE BOHR-MOTTELSON MODEL
C(SO5) =
12 X k k=1,2,3 2sin2(x_ ~
- ~
i
293c
k)
a T~ sin3y 3-~
(7)
The f i r s t term of the Hamiltonian is the k i n e t i c energy of the B-vibrations. The second term contains the Casimir operator C(S05). I t describes the r o t a t i o n a l energy and the k i n e t i c energy of the y - v i b r a t i o n s . The l a s t term is the potential energy as a function of the deformation parameters B,y. I t is easy to give solutions of the BMM-Hamiltonian in the v i b r a t i o n a l (see the book of Eisenberg and Greiner 3) on page 43 and f o l l o w i n g ) , in the r o t a t i o n a l (book of Eisenberg and Greiner on page 127 and f o l l o w i n g 3 , 4 ) ) and f or the y-unstable l i m i t S ) . This is i n t e r e s t i n g in the connection that also f o r the IBA one can find in the samel i m i t s exact solutions. Concluding t h i s short discussion of the BMM I should mention that the BMM has been extended by Faessler6) to include quadrupole e x c i t a t i o n s of the protons and the neutrons separately. Recently t h i s model has been revived by the Frankfurt group7). This extension corresponds in IBA to IBA2, But I do not want to discuss them here in d e t a i l since I also w i l l not discuss IBA2. A model f o r the quadrupole degrees of freedom which is mathematically complet e l y equivalent to the IBA of Arima and l a c h e l l o 2) has been f i r s t introduced by Jansen, Jolos and D~nau8) in 1974 as the truncated quadrupole model (TQM). An SU~ space is spanned up by the f i v e quadrupole-boson creation operators b~ and the square root [N-~h]l/2. This is the Holstein-Primakoff representation9~ of SU6, Arima and lachelTo2) used the f i v e quadrupole-boson creation operators d~ and an s-boson s+. This is the Schwinger representation 1°) of SU6. The equivalence of the IBA model of Arima and l a c h e l l o and of the TQM model of Jansen, Jolos and D~nau has been shown in sever al publications 2,11,12,13~14 ). Already Jansen et a l. 8) showed that the square root can be replaced by a scalar boson. In a nonrigorous way this equivalence can be made plausible by using conservation of the t o t a l number of bosons. ^ + N = nd + ns = X d ~ d + S S
(8) s + ~ NV~-6b
Here we put the boson number operator fo r the d-bosons of IBA ident ic a l to the boson number operator of the b-bosons of TQM. The Hamiltonian f o r TQM and IBA can now be constructed as a l i n e a r and b i l i n e a r expression of the 36 generators of SU6. For IBA they have the form d~dv, d~s, s + d , s+s
(9)
The number of parameters in the Hamiltonian is r e s t r i c t e d by the requirement that i t has to be r o t a t i o n a l i n v a r i a n t . The boson number operator (8) is the Casimir operator of SU6 and is a good quantum number f o r each representation i f H is only b u i l t by the generators (9). HIBA = Linear + B i l i n e a r Form in Generators (9)
(i0)
I f one has a computer code f o r the BMM in the v i b r a t i o n a l basis Is) i t is a simple step in the Holstein-Primakoff representation modifying i t to a IBA code by using the square root instead of the s-bosons, Details of the IBA including the extension to IBA2 where one treats the
294c
A. FAESSLER
proton and the neutron degrees of freedom s e p a r a t e l y have been discussed in previous t a l k s at t h i s conference. Here I only want t o discuss in as f a r IBA is e q u i v a l e n t to the BMM and where i t i s d i f f e r e n t . To see t h i s I w i l l give a short summary of the work of Ginocchio and Kirson16). S i m i l a r c o n s i d e r a t i o n s can also be found in other Publications]7'18). The work of Ginocchio and Kirson16) has been c r i t i c i z e d f o r n o t u s i n g complex collective variables ~2~" Welguny19) extended their considerations to complex ~2-, so that one finds automatically a hermitian Hamilton operator in the lntrlns]c system. Here I am presentlng the conclus10ns of Ginocchio and KirsonIG) modified by Weigunyl9). Ginocchio and KirsonIG) introduce deformed bosons in the intrinsic system. •
+ + Bsiny (d~+d+2) B+ = s+ + ~cosy do ~
IN;S,~>
=
(ii)
C(B+)NIo >
The i n t r i n s i c N boson s t a t e depends on the deformation v a r i a b l e s 8 and y (6). I t can be shown t h a t from t h i s i n t r i n s i c s t a t e ( i i ) one can o b t a i n a l l the IBA s t a t e s in the l a b o r a t o r y system by angular momentum p r o j e c t i o n PIK in connection w i t h using 8 and y as generator c o o r d i n a t e s .
JN~ '">LAB"
60 ° =o ~ 84dB fo sin3ydy ~KfN~;K(B'Y)PIKIN;B'Y>
(12)
c h a r a c t e r i z e s a l l the quantum numbers of the IBA s t a t e a p a r t of the t o t a l boson number N. K is the p r o j e c t i o n of the t o t a l angular momentum I to the i n t r i n s i c za x i s . The deformation energy surface in the i n t r i n s i c system f o r the IBA can now be defined by t a k i n g the e x p e c t a t i o n value o f the IBA Hamiltonian w i t h the i n t r i n s i c wave f u n c t i o n (11). ^
VN(~,y) =
(13)
+ N(N+I) [C1+C282 + C3B3cos3y+C4B4] (1+~2) 2 One can now show t h a t one obtains f o r the overlap o f the r o t a t e d i n t r i n s i c wave f u n c t i o n IN;~,y;@,~,~> w i t h the IBA s t a t e IN,~> a second o r d e r d i f f e r e n t i a l equat i o n i f one replaces the c r e a t i o n and a n n i h i l a t i o n operators o f the bosons by d i f f e r e n t i a l o p e r a t o r s , an equivalence which i s w e l l known f o r the l i n e a r harmonic oscillator: b+
/%
(x
-
~) OA
(14)
b =_~_i (x + ~ x ) In our case the relation is a l i t t l e bit more complicated, but in principle i t is the same. ~eiguny19) shows that the d i f f e r e n t i a l equation obtained in this way goes over into equation (7) for the BMMif one takes the leading terms for N ÷ ~. Thus IBA goes over into BK~I i f the boson number N goes to i n f i n i t y . Therefore the deformation energy surface of the equivalent Bohr-Mottelson model i f we take the leading term for N to i n f i n i t y which is the factor of N(N+I) is:
IBA AND THE BOHR-MOTTELSON MODEL V(~,y)
= [C~+C2B2+C~3 cos3¥ + C ~ 4 ] / ( I + ~ 2 ) 2
295c (15)
One sees t h a t we get a gamma unstable p o t e n t i a l in (13) and (15) i f we put C3 or l C~ equal to zero. By choosing C3 p o s i t i v e one obtains an o b l a t e and f o r a negative cnoice a p r o l a t e shape. IBA is not able to describe a t r i a x i a l deformed nucleus. To obtain a t r i a x i a l i t y we would need at least a term with tcos(3y)] 2. To obtain such a term one needs t h i r d order terms in the generators (9). A term which would lead to t r i a x i a l i t y would f o r example be:
[[d+d+]2 d+]
O/3
[[d~]2d]o/3
(16)
We can therefore summarize our comparison between BMM and IBA (or TQM) in the following way: The IBA is distinguished from the BI~ by the f i n i t e boson number N. IBA goes over into BMM i f tile boson number goes to i n f i n i t y . In this sense IBA is more general than BMM. On the other side the Hamiltonian of IBA is more restricted than the BMM Hamiltonian. IBA can for example not describe t r i a x i a l shapes in i t s present form. 3. Microscopic foundation We have seen that the major d i f f e r e n c e between the i n t e r a c t i n g boson model (IBA) or the truncated quadrupole model (TQM) and the Bohr-Mottelson model (BMM) is the f i n i t e and constant number o f bosons so t h a t we a u t o m a t i c a l l y obtain the BMM i f the number of bosons N goes to i n f i n i t y . I f we use the Tamm-Dancoffapproximation (TDA) f o r the d e f i n i t i o n of the c o l l e c t i v e states and i f we are mapping the most c o l l e c t i v e TDA state in lowest order onto the boson we can use in p r i n c i p l e an expansion i n t o p a r t i c l e - h o l e or p a r t i c l e - p a r t i c l e ( h o l e - h o l e ) or two quasip a r t i c l e states. +
+ = Z . C~" B~ ml ml
amai
f o r a ph-basis
ama + +i + + ~m~i
f o r a pp(hh)-basis
l
(17)
f o r a 2q.p. basis
To get a good boson numbe~ we have to request f o r IBA a p a r t i c l e - p a r t i c l e or a h o l e - h o l e basis. Since IBA r e s t r i c t s the many-Fermion-Hilbert-space to the most c o l l e c t i v e angular momentum zero and angular momentum 2 s o l u t i o n s one has to inv e s t i g a t e the question how well does a S-D subspace represent the low l y i n g states. One expects t h a t t h i s S-D subspace is describing b e t t e r the low l y i n g states in spherical than in s t r o n g l y deformed n u c l e i . Thus we w i l l study here mainly t h i s question in s t r o n g l y deformed nuclei by looking i n t o d i f f e r e n t i n v e s t i g a t i o n s which have been published r e c e n t l y 2 ° , 2 1 , 2 2 , 2 3 ) . A comparison of the r e s u l t s in a S-D subspace with exact shell model c a l c u l a t i o n s in a f i n i t e space in the laborat o r y system has been performed by OtsukaZ°). He defines a degenerate s i n g l e - p a r t i c l e
space of Og712, Ids/~, ld~/~ and 2sl/2 levels for protons and neutrons, lhe protons and neutrons lh~erac~ between themselves by the surface delta interaction, while the proton-neutron interaction is represented by a quadrupole-quadrupole force. HFermion = H~ + H~ - f ~ H = - 4x g
~ Y2(x)Y2(v) ~,v
~ Yk(i) Yk(j) i,j;k
f=1.5 MeV
(18) ; g:O.5 MeV
296c
A. FAESSLER
For 6 protons and 6 neutrons in the above degenerate s i n g l e - p a r t i c l e levels one obtains from the Hamiltonian (18) a good approximation to a rotor spectrum including also the corresponding quadrupole t r a n s i t i o n p r o b a b i l i t i e s . The S-D subspace is constructed with the help of the generalized s e n i o r i t y formalism as described in ref. 24. Fig. I shows the overlap I
Prob, (%) [
i
[
i
]
i
i
fO0 80
1.5 MeV
/
60
/
4O 20 0
I
~
I
I
I
I
I
0
2
4
6
8
10
t2
Fig. 1: Overlap of the shell model solution f o r the r o t a t i o n a l states of the ground state. band in a Og~,o, Id. ~.L . .z . ld . .L .~ . 2sI,I 2 s i n g l e - p a r t i c l e space with the • . ll L Fermlon Hamlltonlan (18) accordlng to O~suka2°l. With increasing angular momentum the overlap with the S-D subspace is decreasing and one needs nucleon pair states coupled to angular momentum 4 and higher to describe the many-body Fermion states. Most of the investigations how much the S-D subspace is exhausting of the many-Fermion states have been performed in the i n t r i n s i c system21,22,23). Looking to these studies one should not forget that the typical average angular momentum in an i n t r i n s i c state is about 2 f o r r e a l i s t i c nuclei. Thus these studies give only information about the very low angular momentum states and say nothing about the q u a l i t y of the high spin states in IBA. These investigations s ta r t from a Nilsson Hamiltonian and a pairing force. H = hNilsso n (~)
-
G ~ aia + ~ + a~ak ik
(19)
The solution of this Hamiltonian can be found in the BCS approximation. In t h i s approach the many-Fermion wave function has not a good nucleon number, but one can project with the projection o~erator P2N onto the nucleon number 2N.
IBA AND THE BOHR-MOTTELSON MODEL +
297c
+
P2N ]BCS> = P2N 11 (u k + v k akaE)IO> k>O =
vk
oF[X
k>OUkk
++ N aka ~] I0 > (2O)
= Or [A+]NIO> +
with:
A+ = [
x I AIO
I
x I = C v~2C(jjllm-mO) k~>OV~kuk The nucleon p a i r operator A+ is expanded i n t o nucleon p a i r s w i t h good angular momentum I, The expansion c o e f f i c i e n t s x I are given f o r a s i n g l e j - s h e l l . To get a shell of s i m i l a r size as the neutron shell between the magic numbers 82 and 126 one uses a j=41/2 s i n g l e - p a r t i c l e s h e l l . This n a t u r a l l y simulates only the neu-trons and not the protons and by looking to the overlaps one should take t h i s i n t o account. The t o t a l overlap is reduce~ by i n c l u d i n g also the proton part of the wavefunction. The p r o b a b i l i t y PI = Xl is p l o t t e d 22) in f i g . 2. This procedure f o r c a l c u l a t i n g the wave f u n c t i o n corresponds to p r o j e c t i o n onto the S-D subspace a f t e r v a r i a t i o n . This method does not y i e l d the best wave f u n c t i o n in the model space.
I
'
'
'
'
1
'
'
'
'
I
0+-2+-4+ 100 . . . . . . . . . -8.0
5o
i
-4.0
13_
0.0
0.5 N I~
1.0
cg
0.0
Fig. 2: P r o b a b i l i t i e s to f i n d 0+, 2+ , 4+ and 6+ pairs in the Cooper p a i r A+ o b t a i ned in the Nilsson model. The s i n g l e j=41/2 s i n g l e - p a r t i c l e shell is taken w i t h the deformed s i n g l e - p a r t i c l e f i e l d of B=-0.32 and the p a i r i n g force of G=O.2 MeV. The p r o b a b i l i t i e s Pl=X~ and the sum of such q u a n t i t i e s are p l o t t e d as a f u n c t i o n o f the number of nucleon p a i r states N over the t o t a l number of p a i r states ~ = j + I / 2 . The dashed l i n e gives the i n t r i n s i c quadrupole moment of which the scale is p l o t t e d on the r i g h t hand side. In a boson approximation only r e s u l t s f o r N/~
298c
A. FAESSLER
The deformation is negative (~=-0.32) and the p a i r i n g s t r e n g t h G=O.2 MeV. The prob a b i l i t y o f f i n d i n g angular momentum 0+, 2+ or 4 + in the p a i r s t a t e A+ i s p l o t t e d as a f u n c t i o n o f the number of nucleon p a i r s N over the t o t a l number o f p a i r states = j + i / 2 . At the value o f the maximum quadrupole moment (N/~ = 0.38) the overlap o f the two nucleon states A+ w i t h the S-D subspace is 92%. This overlap is l a r g e and is increasing i f one goes to the h a l f - f i l l e d s h e l l (N/~ = 0 . 5 ) . Values l a r g e r than N/A = 0.5 should not be considered since there one would look i n t o two hole p a i r s t a t e s . I f one performs a lowest order mapping of A+ onto bosons +
A+(boson) = Xos+ + x2d 0 + . . .
(21)
one can e a s i l y c a l c u l a t e the overlap of the mapped many Fermion s t a t e w i t h the s-d boson space. For the maximum quadrupole moment one has N=8 bosons and the overlap i s 73%. Everyone can decide by h i m s e l f i f he wants to c a l l t h i s dominance of monopole and quadrupole p a i r s in the Nilsson model or i f he wants to conclude t h a t 73% i s a too small oercentage to give a good d e s c r i p t i o n . Otsuka et a l . 22) give also the r e s u l t s f o r many non-degenerate s i n g l e - p a r t i c l e s h e l l s Ohg/2, I f T / 2 , I f ~ / ~ , 2p~/~, 2P1/2 and O i ~ / ~ f o r the p o s i t i v e deformation 6=0.26 and the p a i r i ~ g - f o r c ~ - ~ = O . 2 MeV. The ~ # h ~ l e - p a r t i c l e energies are taken from r e f . 25. The p r o b a b i l i t y PQ-2 f o r f i n d i n g angular momentum 0+ and 2+ in the Cooper p a i r A+ is now i n c r e a s e d - t o 95.5%.The number of p a i r s t a t e s a t the maximum quadrupole moment is N=II. I f one performs the lowest order mapping onto bosons t h i s corresponds t o an overlap o f the t o t a l wavefunction w i t h the s-d boson space of 77%.
[
.
.
.
.
I
'
.
.
.
.
0+-2+_4+
I00
400.~E •o"-; 50
1 200
13_
o.o
b
o.5 N/a ,b 20
o
N
Fig. 3: P r o b a b i l i t i e s to f i n d 0+, 2+ , 4 + and 6+ p a i r s i n the Cooper p a i r obtained in the Nilsson model • Non-degenerate s i n g l e j - s h e l l s Ohq/~, i f T / ~_ , I f. 5 2/ , 2pB/~, . _ _ . _ 2Pi/2 and 0i13/2 have been used w l t h a deformation B=0.25 and the palr~ng f o r c e s t r e n g t h G=O.2"MeV. S i n g l e - p a r t i c l e energies are taken from r e f . 25. The overlap w i t h the S-D subspace f o r the maximum value of the i n t r i n s i c quadrupole moment QO which is i n d i c a t e d as a dashed l i n e and has i t s scale on the r i g h t hand side i s 95.5% f o r the Cooper p a i r A+ obtained in the Nilsson model. In lowest order mapping onto bosons the overlap of t h i s wavefunction at the maximum quadrupole moment w i t h the t o t a l wavefunction is 77%.
IBA A N D T H E B O H R - M O T T E L S O N M O D E L
299c
Beset al. 23) use practically the samemodel for the single j=41/2 shell. But they minimize the energy after projection onto the S-D subspace. Their results are given in f i g . 4. In part a and b one sees that the model with only S-D pairs underestimates the intrinsic quadrupole moment and overestimates the pairing gap.
o.......... 0.3
o_ ({:) ......
(a) N - B C S , ~•
oar,r
n/O
/
= 0.38
° !ii°,"°I_ I~
.--"
oc
o,
o,
o
i "N-BCS o,.
o,
nlQ 0,~
,"
,
b)
,
,
,
,
..._S_O_ ....
(d)
~, },
OI 0,3
~ az N+BCS
SO
SDG SDGI
0
081
0.60
051
2
0.59
026
025
0.25
0.40
Z, 6
;
/ 0.2
i
..o
mlP
i o J,
0.09
i 0.6
n/O
Fig. 4: Properties of the Nilsson plus pairing wavefunction of the Hamiltonian (19) calculated in a single j=41/2 ~hell according to (20). (a) The intrinsic static quadrupole moments in units of
300c
A. FAESSLER
crease the overlap with the S-D subspace but they do not determine the properties of the nuclei near the Fermi surface which we measure in the low lying states.
A 20
(a )
I
]o'
I I
&
(c)
SD
~o6 1.0
15
20
I
10
~5
-f
~
>
~ °'5 FsD<¢ff]l *(f])N~BC;'" L 1.0
I
"'.\
1
'i
"'" @I
15
20
-f
20
_ " ~
J
Fig. 5: I n t r i n s i c quadrupole moment (a) and overlap of the many Femlion wavefunction (b) f o r f i n i t e strength f of the quadrupole-quadrupole force with the wavefunction f o r i n f i n i t e strength f=~ as a function of the quadurpole-quadruDoleforc~ strength f (18). Part (c) and (d) show the pairing gap and the occupation probabil i t y as a function of f and the deformed single p a r t i c l e energies c, respectively. The dashed-dotted l i n e is the r e s u l t of Otsuka2°), the solid l i n e the result of the N plus BCS model, the dashed l i n e corresponds to the results of a minimization of the expectation value of the Fermion Hamiltonian in the S-D subspace. The wavefunctions are calculated fo r 6 protons and 6 neutrons. 4. Comparison with data As I stressed already above the purpose of this t a l k and t h i s chapter is not to show the undoubtedly large successes of the IBA to explain data. We want to investigate here the l i m i t s of the v a l i d i t y of the IBA and suggest generalizations. In chapter 2 we saw that i f the boson number N goes to i n f i n i t y IBA is ident i c a l with BMM. On the other side the deformation energy surface which one obtains in t h i s way from IBA is more r e s t r i c t e d than the deformation energy surface allowed in the BMM. The usual IBA deformation energy surface is not able to describe t r i a x i a l i t y . We therefore want to study here the experimental evidence f or the boson cuto f f which is connected with the f i n i t e number of bosons N. Furthermore we want to look into indications of t r i a x i a l i t y which cannot be described in IBAI or IBA2 (without a quadrupole force between protons and between neutrons). 158Er with 68 protons and 90 neutrons has a t o t a l number of bosons N=13. This corresponds to a maximum angular momentum of 26. In the crystal ball in Heidelberg 26) a maximum angular momentum of Imax(exp)=44+ has been found. Although there have been t r i a l s to explain the reduction of the E2 t r a n s i t i o n p r o b a b i l i t i e s near
IBA AND THE BOHR-MOTTELSON MODEL
301c
the f i r s t backbending and above by the boson c u t - o f f , I would expect that todaynoone is surprised that IBA is not describing the maximum angular momentum c o r r e c t l ~ Due to alignment of a i1~,~ neutron pair at f i r s t backbending, alignment of a h proton palr at the secon~ anomaly of the moment of i n e r t i a and possibly a t h l r ~ I/2 alignment of a h or h . . . . neutron p a i r the maximum angular momentum can be much larger than a l l o ~ by t ~ / ~ o s o n c u t - o f f . The standard example f o r the boson c u t - o f f 27) is 78Kr with 36 protons and 42 neutrons and a boson number N=8. Fig. 6 shows27) that from d i f f e r e n t models only IBA is able to reproduce the reduction of the E2-transition p r o b a b i l i t i e s at high angular momenta in the yrast states. But in Fig. 7 one sees that between angular momentum 6 and 12 one has an alignment of probably two g9/2 protons. •
/ Z
78 3sKr~2
N=8 °
r'n
i
i LIJ
I
I
I
I
I
I
I
I
I
I
2 6 10 1L 18 Anguler Momentum [lh]
Fig. 6:E2 transition probabilities between angular momentum I and I-2 in units of the reduced transition probability between angular momentum 2+ and 0+. The f i g . is taken from ref. 27. The data are measured by Hellmeister et ai.27). The theoret i c a l results for IBAI, a rigid rotor and the asymmetric rotor (AR) are indicated. Reduction of the reduced transition probability with increasing angular momentum has been connected with the boson cut-off27). But f i g . 7 indicates an alignment of a g9/2 proton pair which d e f i n i t e l y cannot be described by IBA. The total gain of angular momentum is roughly 4. Such an alignment of 4 units of angular momentum can not be described in the framework of IBA. At least part of the reduction of the E2 transition probabilities found in 78Kr at high angular momentum states (see f i g . 6) is due to this alignment. Even careful discussions of the situation in 79Rb cannot remove the fact that IBA cannot be applied in 78Kr. I would therefore expect that this standard example of the boson cut-off is not a convincing one. One could even have general doubts i f one can see indications of the boson cut-off in the way of a termination of a rotational band ( i f one wants not to look iinto very l i g h t nuclei as 2°Ne where the boson cut-off is identical with the shell model cut-off). Normally the boson cut-off w i l l not exist since for high spin states in rotational nuclei g and i-pairs w i l l be important and the Coriolis force
302c
A. FAESSLER
w i l l a l i g n nucleon p a i r s w i t h a l a r g e s i n g l e - p a r t i c l e angular momentum j which y i e l d s a d d i t i o n a l c o n t r i b u t i o n s to the t o t a l angular momentum.
15
E E
_2 .....
/[___
E
alignment of 1~g9/212
.<
o o
I
I
l
I
0.2 O.L. hw [MeVl
I
I
08
t
I
0.8
I
I
I
1.0
Fig. 7: Angular momentum as a f u n c t i o n o f the r o t a t i o n a l frequency ~. The curve • owsbetween angular momentum 7 and 12 the alignment of a nucleon p a i r 27) This is probably a g9/2 proton p a i r which y i e l d s an aligned s i n g l e - p a r t i c l e angular momenGum of i=4~. Such an alignment is not contained in IBA. In eqs. (13) and (15) we saw t h a t IBA cannot describe t r i a x i a l n u c l e i . Are t h e r e experimental i n d i c a t i o n s t h a t such t r i a x i a l l y deformed nuclei e x i s t ? This question is a very o l d one and i s not easy to answer. Recent data o f Stachel et a l . 2e) might i n d i c a t e such a t r i a x i a l deformation. By Coulomb e x c i t a t i o n they measured the B(E2)-values up t o the 10 + to 8+ t r a n s i t i o n . They furthermore obtained also the diagonal quadrupole moment f o r the ground s t a t e r o t a t i o n a l band up to the 8 + . The r o t o r model, IBAI and IBA2 are not able t o reproduce the t r a n s i t i o n prob a b i l i t i e s . One obtains a good f i t to the data i f one extends IBA w i t h a g-boson or i f one uses the asymmetric r o t o r . Furthermore Tamura 29) o b t a i n s also good res u l t s w i t h a boson expansion i n t o p a r t i c l e - h o l e TDA c o l l e c t i v e s t a t e s . But i f one looks to the diagonal quadrupole m a t r i x elements only the asymmetric r o t o r is able t o describe the data.(Tamura 29) does not give the diagonal quadrupole m a t r i x elements but he i n d i c a t e s t h a t t h e i r values are decreasing and not i n c r e a s i n g l i k e the d a t a . ) The d i f f e r e n t t h e o r e t i c a l d e s c r i p t i o n s and the data are given in f i g . 8a and 8b. The IBA can n a t u r a l l y be extended to include asymmetric deformations. For t h a t one needs only to add t o the t o t a l Hamiltonian three body boson i n t e r a c t i o n terms o f the type given in eq. (16). I t would be i n t e r e s t i n g t o t e s t i f w i t h the i n c l u s i o n of such terms 1°~Ru can be described in agreement w i t h the data. Another extension o f IBA2 to t r i a x i a l i t y has r e c e n t l y been given by Dieperink and B i j k e r by a l l o w i n g f o r quadrupole forces among the protons and among the neutrons.
IBA AND THE BOHR-MOTTELSON MODEL
0.6
I
i I I 10LRu
i
I
l
I
I
I T
B(E2)
ra..~I-.-IBA+g
..;'.!-';---~.:---T--~ot o,-
l
I
~O~Ru
Lk1 /
-0.5 Tamu
I
303c
-~ -1.0
-o.2 f
IBA 2
UJ
\
-~.s
\,
W rn
IBA2 0
I
I
I
I
I
10
I
I
2 6 Angulor Nomentum [hl
I
J
I
2 6 10 Angulor Momentum[hi
(a)
(b)
Fig. 8: Reduced t r a n s i t i o n p r o b a b i l i t i e s (a) and diagonal quadrupole moments (b) f o r the y r a s t band in l°4Ru according to Ref. 28. Only the asjanmetric r o t o r w i t h ~=25 ° is able t o d e s c r i b e these data. Results f o r IBA2, the r i g i d r o t o r , IBA and a g-boson, the asymmetric r o t o r and the r e s u l t s o f a boson expansion by Tamura 28) are given. 5. Summary We t r i e d t o compare the Bohr-Mottelson model (BMM) and the i n t e r a c t i n g boson a p p r o x i m a t i o n (IBA). The l a s t model is e q u i v a l e n t to the t r u n c a t e d quadrupole model (TQM) o f Jansen, Jolos and D~naue)). We t r i e d to see how f a r BMM and IBA are i d e n t i c a l and where they are d i f f e r e n t . On the phenomenological l e v e l we saw t h a t the IBA i s i d e n t i c a l w i t h the BMM i f the boson number N goes t o i n f i n i t y . For small boson numbers those models give d i f f e r e n t r e s u l t s although they are q u a l i t a t i v e l y very s i m i l a r . On the o t h e r side the Hamiltonian of IBA i s more r e s t r i c t e d than the BMM Hamiltonian since one r e quests boson number c o n s e r v a t i o n and a l l o w s o n l y two body boson i n t e r a c t i o n terms. Thus the IBA Hamiltonian can f o r example not describe t r i a x i a l deformed n u c l e i . For both models BMM and IBA one can f i n d the exact s o l u t i o n in t h r e e l i m i t s : The f i v e dimensional harmonic quadrupole v i b r a t o r , the r o t o r and the v - u n s t a b l e case. From the l e v e l of the m i c r o s c o p i c f o u n d a t i o n s of the two models we saw t h a t
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we have to build the bosons fo r IBA by two nucleon p a r t i c l e s (pp) or by nucleon holes (hh). This y i e l d s automatically boson number conservation which is the essential difference between IBA and BMM. The second question is i f low lying solutions of the many-body Fermion Hamiltonian can be represented in a subspace where a l l the nucleons are coupled pairwise to angular momentum zero and angular momentum 2 and where fo r each angular momentum only one such c o l l e c t i v e pair is taken into account. This has been contested by the Copenhagen School. We discussed in detail the l i t e r a t u r e on this question. We saw that the Cooper pairs even in strongly deformed nuclei are b u i l t mainly by angular momentum 0 and 2 pairs with p r o b a b i l i t i e s larger than 90% f o r the largest deformation in a s h e l l . Nevertheless observables l i k e the pairing gap and the quadrupole moment are quite d i f f e rent with and without inclusion of higher angular momentum pairs. In a j=41/2 shell one finds for the maximum value of the quadrupole moment a pairing gap which is f o r only S-D pairs by a factor 1.6 larger than in the Nilsson BCS model. The quadrupole moment on the other side is by a factor 0.75 smaller. Bohr and Mottelson2]) recently pointed out that the large overlap found f or the Cooper pairs in the Nilsson model is due to the almost perfect overlap of the nucleon pairs in the lower shells or for the non-valence pairs in a large s i n g l e - p a r t i c l e shell with the S-D space. The valence nucleons which determine the physics of the low lying c o l l e c t i v e states have an appreciable admixture of G and l - p a i r s . This might y i e l d the differences of the results for pairing gaps and quadrupole moments for the f u l l and the S-D r e s t r i c t e d space. F i n a l l y we looked into the data to search f o r indications of the boson cuto f f . We found that this c u t - o f f is hard to see and that the standard example for the boson c u t - o f f , the reduction of the B(E2) values in 78Kr might be at least p a r t i a l l y connected with the alignment of two g9/2 protons. With 1°4Ru we found a nucleus which seems to request a t r i a x i a l shape which can not be described by IBA i f one r e s t r i c t s the Hamiltonian to at most two-body boson interactions. An extension which includes three-body terms or quadrupole forces among the protons and among the neutrons might be possible to describe also in IBA this nucleus.
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