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IBA-1 as a model of IBA-2
Volume 160B, n u m b e r 1,2,3
PHYSICS LETTERS
3 October 1985
IBA-I AS A M O D E L OF IBA-2 A. NOVOSELSKY and I. TALMI Department of Nuclear Physzcs, The Wetzmann Insntute of Sczence, Rehovot, Israel Received 4 June 1985 Elgenstates of a reahstlc IBA-2 hamlltoman which gwes a very good description of the 17~Hf spectrum are rather far from having definite F-spins Still, tlns IBA-2 hamfltoman projected on the set of states with maximum symmetry, F = Fm,~ = 15/2, yields spacings between low-13qng levels m good agreement with the exact ones Shght modifications of the parameters m the projected harmltoman thus obtained yield a simple IBA-1 hamfltoman winch is a very good model of the more detailed IBA-2 hamiltoman
In the original version of the interacting boson model (IBA-1) states of even-even nuclei are obtained from a hamiltonian of s and d bosons whose total number N is fixed [1,2]. Proper choices of the parameters of the IBA-1 hamiltonian lead to various kinds of collective motion. Special values of these parameters give rise to simple limits of collective spectra. For example, at the SU(3) limit or near it, rotational levels (corresponding to an axially symmetric rotor) arise in a simple and transparent fashion from the structure of the hamiltonian. IBA-1 hamiltonians have been extensively and successfully used for quantitative descriptions of various nuclei [2]. In some strongly deformed nuclei energy levels and E2 transition probabilities were calculated and good agreement with experiment was obtained [3,4]. The search for a shell model basis of the boson model led to a correspondence between states obtained by couphng J = 0 and J = 2 pairs of ~dentical valence nucleons and boson states [5,6]. Hence a more detailed model, IBA-2, was introduced in which there are two kinds of bosons, s,,, d~, proton bosons and s,, d, neutron bosons [5,6]. If the IBA-2 hamiltonian is fully symmetric with respect to proton and neutron bosons, its eigenstates can be characterized by definite symmetry properties of definite values of F-spin [5]. The fully symmetric states have the highest
possible value Fm= = (N~N~)/2 = N / 2 . Such states uniquely correspond to states with one kind of bosons (e.g. for M F = + Fm~x). Thus, in case of such a hamiltonian IBA-1 elgenstates (and eigenvalues) form a subset of the eigenstates (and eigenvalues) of IBA-2. The situation for actual nuclei is drastically different. Empirical reformation about the effectwe interactions in the shell model indicates that the interaction between identical nucleons (i.e. the T = 1 part) leads to states with definite seniorities [7] (or generalized seniorities [8]). The proton-neutron (i.e. the T = 0 part), however, breaks seniority in a major way [9]. It can be approximated by a strong and attractive quadrupole-quadrupole interaction. Typical IBA-2 hamiltonians are written according to these features as ,,~d +. ~],, + ,~d~+. c]. + KQ.(x~) • Q.(X~) +
~
~k(d~xd,+)~k~.(a~×d~) ~)
k=l,3
+~2(d~+s~+ - s+d~+) • (c],~s,- s,,d~),
(1)
where the quadrupole operators are defined by a , ~ ( x ~ ) = s~+a, + d~+s~ + x , ( d ~+ × a~) ~2),
O , ( x , ) = s.+ d- . + d~+s. + x~(d + X a . ) ~2).
(2)
The last three terms in (1) determine the interactions in the antisymrnetric proton boson-neutron
0 3 7 0 - 2 6 9 3 / 8 5 / $ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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PHYSICS LETTERS
boson states. In the states (d~+ × d~+ )M[0), J with J = 1, 3, the interaction is given by - 41 and - 4 3 whereas in the antlsymmetric J = 2 state the interaction is equal to 242. The hamiltoman (1) is manifestly not symmetric in proton and neutron bosons and it can be expected that its eigenstates will not have definite F-spin values. The problem naturally arises whether it is at all possible to derive the simplicity and elegance of IBA-1 from the apparently complicated structure of realistic IBA-2 hamiltonians. For instance, it is not clear a priori how IBA-2 hamiltonians like (1) may lead to simple rotational spectra. Recently, the low-lying spectrum of 178Hf has been measured in detail [4]. Energy levels and E2 transitions were successfully reproduced by calculations using an IBA-1 hamiltonian [4]. Equally good agreement, and for some levels a better one, has been obtained by using an IBA-2 hamiltonian (1) [10]. It has been thus demonstrated, that, at least in this case, diagonalization of the rather large matrices obtained from an IBA-2 hamiltonian can give rise to a rotational spectrum. The same rotational spectrum emerges naturally from an IBA-1 hamiltonian. In view of this, 178Hf case seems to be an ideal one to investigate the connection between the IBA-2 hamiltonian of ref. [10] and IBA-1 hamiltonians which give comparable fits to the experimental data. A straightforward method of obtaining an IBA-1 hamiltonian from the IBA-2 one is to project the latter onto the subset of states with F = Fm~ = N / 2 . If F-spin is an exact quantum number, the eigenstates of the projected operator will be identical to some IBA-2 eigenstates with the same eigenvalues. Scholten [11] has carried out this projection but his choice of independent terms in the IBA-1 hamiltonian is not convenient for our purpose. In a recent paper, Harter, Gelberg and von Brentano [12] carry out this projection in a simpler way. They consider the F-spin tensor character of the various terms in the IBA-2 hamiltonian and use the Wigner-Eckart theorem to relate matrix elements between states with F = F m a x , M v = (N,, - N ~ ) / 2 to those with M F = Fma~. The latter can be identified with IBA-1 states. They also calculated the spectrum of an IBA-2 hamiltonian and compared it with the spectrum of the projected operator. They find that 14
3 October 1985
energy spacings between low-lying levels agree rather well with spacings between eigenvalues of the projected operator. The IBA-2 hamiltonian of ref. [12] has the following parameter values %=~v=0.5MeV,
•--0.18MeV,
X~ = X~ = 0.2.
(3)
In addition to these, they take 42 = - ~41 = - ~431 = ~, for which case the last three terms in (1) combine to form the Majorana operator (multiplied by 2) whose eigenvalues are ½N(½N + 1) F ( F + 1). Naturally, the agreement between IBA-2 spacings and those of the projected operator increases with )~. For h = 1 MeV the effects of states with F = Fm~x - 1 are strongly reduced, their unperturbed positions shifted by the Majorana operator by 6 MeV ( N = 6 in ref. [12]). The good agreement between energy spacings m a y indicate that the low-lying IBA-1 states of ref. [12] are rather pure F-spin states with F = FmaxWe calculated the weights of Fm~, states in the lowest eigenstates of (1) with the parameters (3). For X = 0.1 MeV, which is the value preferred by the authors of ref. [12], the percentage of Fmax = 3 states (and N~ = Nv = 3) in the lowest 0 + and 2 ÷ states 98.4% and 98.6%. For ~ = 0 these percentages are smaller but still considerable, 95.7% and 96.5% respectively (for X = 1 MeV the states are practically pure, 99.9% of Fm~x = 3 in both). There are, however, large deviations from pure F-spin states in a realistic case like 178Hf. The IBA-2 parameters used in that case were determined by a least squares fit to be [10] (c, x and 4, are in MeV) ~. = c v = 0.5259, X,, = - 1.1507,
x = -0.0872, X~ = -0.2212,
41 = - 0 . 5 0 5 6 , 42=-0.0228,
43=-0.3846.
(4)
In this set of parameters, unlike the set (3) of ref. [12], X~ and X~ have rather different values. This introduces additional asymmetry between proton bosons and neutron bosons. We calculated the F-spin composition of low-lying states and found large admixtures of states with F ~< F_,n~x - 1. The weight of the states with Fm~x = 1 5 / 2 in the 0 + ground state is 82.1% (16.4% for the F = Fm~x - 1
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PHYSICS LETTERS
state). The second 0 ÷ state (of the r - b a n d ) is only 63.6% in a state with Fr, a~ while the weight of states with F = Fm~x - 1 is 29.6%. Thus, F-spin is badly broken and the success of IBA-1 in 178Hf cannot be attributed to F-spin being a good q u a n t u m n u m b e r of IBA-2 states. A close look at the F-spin composition of 178Hf low-lying IBA-2 states reveals an interesting phenomenon. Admixtures of states with F = Fm~ 1 in the various states of the ground state band have rather constant amplitudes. In higher states, grouped into the r - b a n d and y-band, this behavior is less striking but is still evident. This pattern suggests that we may use the subset of states with Fm~x (e.g. IBA-1 states for M F = Fmax) as a model of the exact IBA-2 states. The first step in such a p r o g r a m is to consider matrix elements of the IBA-2 hamiltonian between states with Fm~,. In the following steps the contribution of non-diagonal elements connecting states of the model space ( F = Fmax) with other states (with F < F m a x ) should be added. Such contributions will certainly renormalize the parameters of the model hamiltonian (that of IBA-1) and may introduce more complicated terms. The first step of obtaining the submatrix of the hamiltonian (1) defined by F = Fm~x can be accomplished in two ways. We can add to (1) a very larger Majorana term which will decouple states with F = Fm~ from other states which will be pushed very high up m energy. Alternatively, we can project the IBA-2 hamiltonian (1) onto the F = Fm~, set of states according to ref. [12]. Either of these ways led in the case of the hamiltonian (1) with the parameters (4) to a ground state energy higher by 0.31 MeV than the lowest exact IBA-2 eigenvalue. Still, this shift, though rather large, turned out to be rather uniform for all low-lying states. Shifts in the spacings of these levels are considerably smaller. Positions of low-lying levels above the ground state are presented fig. 1. It is important to emphasize that this feature occurs only for the ground-state band and the lowest excited r - b a n d and ,/-band. The position of the second excited K = 0 band is changed from 1.346 MeV in the IBA-2 calculation to 2.048 MeV obtained with the project hamiltonian. Energy levels in that b a n d are well reproduced by the IBA-2 calculation as are measured branching
3 October 1985
MeV 20
m
6 - -
1,5
6 - 5 - -
_ _
4 - -
- -
- -
5 - -
- -
- -
2 - -
2 - -
~
- -
0 - -
_ _
- -
_ _
- -
~
_ _
4 - -
a
8--
I0
b
c
a
b
c
6--
05 4
-
~m o-a
m
m b
c
Fig 1 Spacings of energy levels m 17SHf Levels denoted by a, b and c were calculated from the IBA-2, the projected IBA-1 and a fitted IBA-1 hamdtoman, respectively
ratios of E2 transitions. The component of Fm~x in the lowest state of that band is only 37.7% while the components with Fro,x - 1 and Fm~x - 2 are considerable, 28.9% and 29.0% respectively. It is not surprising that the position of this band cannot be reproduced by IBA-1 calculations [4]. F r o m our calculation it appears that for spacings of low-lying levels the IBA-1 hamiltonian projected from a reahsnc IBA-2 hamiltonian is already a good model. We can see how its parameters should be modified or renormalized by trying to improve the agreement with the IBA-2 results. We m a y consider them as free parameters and determine their values which give the bestfit to the eigenvalues of the IBA-2 hamiltonian (which are very close to the experimental energies). The IBA-1 hamiltonian projected from the hamiltonian (1) with the parameters (4) is obtained as C + , ' ( d +. d) + r ' a ( x ) - Q(x) + x ' x n ( d + X 8) (2) "(d + X (~)(2).
(5)
In (5) the various parameters are given in terms of those in the IBA-2 hamiltonian by C -.~
,' Kt
5N,,N, ~ N-1 + g.N + (4 - X~X,) N N ( N - 1) ~' N,,N. N(N-
1) x,
x'=
x,).
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The ~, do not appear in (5) at all. In (1) they multiply operators wtuch measure the a m o u n t of coupling of two bosons in anttsymmetrw states. Hence, each of them vanishes when applied to fully symmetric states with F = Fra~x. In fact, each of them is a scalar with respect to F-spin. As such, their eigenvalues are independent of M F and each of them vanishes when applied to states with M v = + Fma~. The hamiltonian (5) is, for ~,, = ~ , identical with the one obtained in ref. [12]. There, however, the last term has been transformed and expressed as a hnear combination of other operators a n d the constant term dropped. Substituting into (6) the values of the parameters (4), with N~ = 5, N~ = 10, we obtain the following set of IBA-1 parameters c' = 0.4481 MeV, X = - 0.6859,
x' = - 0 . 0 2 0 8 MeV,
X' = - 0 . 4 6 4 7 -
X = -0.7033,
x' = - 0 . 0 2 0 7 MeV,
X' = - 0 . 4 8 9 9 .
(8)
T h e level spacings calculated with these values are also plotted in fig. 1 (denoted by c). As seen by c o m p a r i n g (7) and (8) a slight renormalization of the IBA-1 parameters gives a very good fit to level spacings calculated in IBA-2 which in turn agree very well with the experimental ones. Also E2 transition probabilities calculated from the IBA-1 hamiltonian (5) with the parameters (8) [or even with those in (7)] agree rather well with experimental b r a n c h i n g ratios. Better fits to the IBA-2 results can be obtained by adding two more independent operators to the IBA-1 hamiltonian (5). F o r the sake of the a r g u m e n t presented here, however, it is sufficient to consider IBA-1 hamiltonians given by (5). In summary, our results show how rotational features emerge from dlagonalization of very large
16
matrices (orders of thousands) constructed from a realistic IBA-2 hamiltonian. The overlap of low-lying states with m a x i m u m symmetry states ( F = Fm~x ) is not very high. Still, the latter form a g o o d model space of the IBA-2 hamiltonian. E n e r g y spacings of low-lying levels are well r e p r o d u c e d b y using an IBA-1 hamiltonian only slightly different from the one projected from IBA-2. The detailed IBA-2 hamiltonlan based on properties o f the effective interactions between nucleons can be well approximated for low-lying levels by a simple IBA-1 hamiltonian from which collective b a n d s arise naturally. The authors would like to express their thanks to Professor M.W. Kirson and Professor P. von B r e n t a n o and in particular to A. Leviatan for helpful discussions.
(7)
Using these parameters we obtain the level spacings gwen in fig. 1 denoted b y b. We looked for values of the parameters (7) that will give a better fit to level spacings calculated from the I B A - 2 hamiltonian. We found the following set which gives better agreement with the IBA-2 results e' = 0.4513 MeV,
3 October 1985
References [1] A Anma and F. Iachello, Phys Rev Lett 35 (1975) 1069; Ann Phys (NY) 99 (1976) 253; 111 (1978) 201; 123 (1979) 468. [2] A Anma and F Iachello, Annu Rev Nucl. Part Sci. 31 (1981) 75; Adv Nucl Phys. 13 (1984) 139 [3] D D Warner, R F Casten and W F Davadson,Phys Rev C24 (1981) 1713 [4] A.MI Haque et al, m: Proc 5th Intern Symp on Capture gamma-ray spectroscopy and related topics, AIP Conf Proc Vol 125, ed S Raman (AIP, New York, 1985), p 423 [5] A Arlma, T Otsuka, F Iachello and I Taln-a, Phys Lett 66B (1977) 205. [6] T Otsuka, A Anma, F Iachello and I Talrm, Phys Lett 76B (1978) 141 [7] I Talml, Rev Mod Phys. 34 (1962) 704 [8] I Talml, Nucl Phys A172 (1971)1, S Shlomo and I Talrm, Nucl Phys A198 (1972) 81, I Talml, Rav Nuovo Clmento 3 (1973) 85 [9] I Talml, m. Interacting bosons in nuclear physics, ed F Iachello (Plenum, New York 1979); in. From nuclei to particles, Proc Intern School of Physics "Enraco Fermf' (Varenna, 1980), ed A Molinan (North-Holland, Amsterdam, 1982), p 172 [10] A Novoselsky, Phys Lett 155B (1985) 299. [11] O Scholten, Ph D Thesis, (Gromngen, 1980), unpubhshed [12] H Harter, A Gelberg and P von Brentano, Phys Lett 157B (1985) 1
PHYSICS LETTERS
3 October 1985
IBA-I AS A M O D E L OF IBA-2 A. NOVOSELSKY and I. TALMI Department of Nuclear Physzcs, The Wetzmann Insntute of Sczence, Rehovot, Israel Received 4 June 1985 Elgenstates of a reahstlc IBA-2 hamlltoman which gwes a very good description of the 17~Hf spectrum are rather far from having definite F-spins Still, tlns IBA-2 hamfltoman projected on the set of states with maximum symmetry, F = Fm,~ = 15/2, yields spacings between low-13qng levels m good agreement with the exact ones Shght modifications of the parameters m the projected harmltoman thus obtained yield a simple IBA-1 hamfltoman winch is a very good model of the more detailed IBA-2 hamiltoman
In the original version of the interacting boson model (IBA-1) states of even-even nuclei are obtained from a hamiltonian of s and d bosons whose total number N is fixed [1,2]. Proper choices of the parameters of the IBA-1 hamiltonian lead to various kinds of collective motion. Special values of these parameters give rise to simple limits of collective spectra. For example, at the SU(3) limit or near it, rotational levels (corresponding to an axially symmetric rotor) arise in a simple and transparent fashion from the structure of the hamiltonian. IBA-1 hamiltonians have been extensively and successfully used for quantitative descriptions of various nuclei [2]. In some strongly deformed nuclei energy levels and E2 transition probabilities were calculated and good agreement with experiment was obtained [3,4]. The search for a shell model basis of the boson model led to a correspondence between states obtained by couphng J = 0 and J = 2 pairs of ~dentical valence nucleons and boson states [5,6]. Hence a more detailed model, IBA-2, was introduced in which there are two kinds of bosons, s,,, d~, proton bosons and s,, d, neutron bosons [5,6]. If the IBA-2 hamiltonian is fully symmetric with respect to proton and neutron bosons, its eigenstates can be characterized by definite symmetry properties of definite values of F-spin [5]. The fully symmetric states have the highest
possible value Fm= = (N~N~)/2 = N / 2 . Such states uniquely correspond to states with one kind of bosons (e.g. for M F = + Fm~x). Thus, in case of such a hamiltonian IBA-1 elgenstates (and eigenvalues) form a subset of the eigenstates (and eigenvalues) of IBA-2. The situation for actual nuclei is drastically different. Empirical reformation about the effectwe interactions in the shell model indicates that the interaction between identical nucleons (i.e. the T = 1 part) leads to states with definite seniorities [7] (or generalized seniorities [8]). The proton-neutron (i.e. the T = 0 part), however, breaks seniority in a major way [9]. It can be approximated by a strong and attractive quadrupole-quadrupole interaction. Typical IBA-2 hamiltonians are written according to these features as ,,~d +. ~],, + ,~d~+. c]. + KQ.(x~) • Q.(X~) +
~
~k(d~xd,+)~k~.(a~×d~) ~)
k=l,3
+~2(d~+s~+ - s+d~+) • (c],~s,- s,,d~),
(1)
where the quadrupole operators are defined by a , ~ ( x ~ ) = s~+a, + d~+s~ + x , ( d ~+ × a~) ~2),
O , ( x , ) = s.+ d- . + d~+s. + x~(d + X a . ) ~2).
(2)
The last three terms in (1) determine the interactions in the antisymrnetric proton boson-neutron
0 3 7 0 - 2 6 9 3 / 8 5 / $ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
13
Volume 160B, number 1,2,3
PHYSICS LETTERS
boson states. In the states (d~+ × d~+ )M[0), J with J = 1, 3, the interaction is given by - 41 and - 4 3 whereas in the antlsymmetric J = 2 state the interaction is equal to 242. The hamiltoman (1) is manifestly not symmetric in proton and neutron bosons and it can be expected that its eigenstates will not have definite F-spin values. The problem naturally arises whether it is at all possible to derive the simplicity and elegance of IBA-1 from the apparently complicated structure of realistic IBA-2 hamiltonians. For instance, it is not clear a priori how IBA-2 hamiltonians like (1) may lead to simple rotational spectra. Recently, the low-lying spectrum of 178Hf has been measured in detail [4]. Energy levels and E2 transitions were successfully reproduced by calculations using an IBA-1 hamiltonian [4]. Equally good agreement, and for some levels a better one, has been obtained by using an IBA-2 hamiltonian (1) [10]. It has been thus demonstrated, that, at least in this case, diagonalization of the rather large matrices obtained from an IBA-2 hamiltonian can give rise to a rotational spectrum. The same rotational spectrum emerges naturally from an IBA-1 hamiltonian. In view of this, 178Hf case seems to be an ideal one to investigate the connection between the IBA-2 hamiltonian of ref. [10] and IBA-1 hamiltonians which give comparable fits to the experimental data. A straightforward method of obtaining an IBA-1 hamiltonian from the IBA-2 one is to project the latter onto the subset of states with F = Fm~ = N / 2 . If F-spin is an exact quantum number, the eigenstates of the projected operator will be identical to some IBA-2 eigenstates with the same eigenvalues. Scholten [11] has carried out this projection but his choice of independent terms in the IBA-1 hamiltonian is not convenient for our purpose. In a recent paper, Harter, Gelberg and von Brentano [12] carry out this projection in a simpler way. They consider the F-spin tensor character of the various terms in the IBA-2 hamiltonian and use the Wigner-Eckart theorem to relate matrix elements between states with F = F m a x , M v = (N,, - N ~ ) / 2 to those with M F = Fma~. The latter can be identified with IBA-1 states. They also calculated the spectrum of an IBA-2 hamiltonian and compared it with the spectrum of the projected operator. They find that 14
3 October 1985
energy spacings between low-lying levels agree rather well with spacings between eigenvalues of the projected operator. The IBA-2 hamiltonian of ref. [12] has the following parameter values %=~v=0.5MeV,
•--0.18MeV,
X~ = X~ = 0.2.
(3)
In addition to these, they take 42 = - ~41 = - ~431 = ~, for which case the last three terms in (1) combine to form the Majorana operator (multiplied by 2) whose eigenvalues are ½N(½N + 1) F ( F + 1). Naturally, the agreement between IBA-2 spacings and those of the projected operator increases with )~. For h = 1 MeV the effects of states with F = Fm~x - 1 are strongly reduced, their unperturbed positions shifted by the Majorana operator by 6 MeV ( N = 6 in ref. [12]). The good agreement between energy spacings m a y indicate that the low-lying IBA-1 states of ref. [12] are rather pure F-spin states with F = FmaxWe calculated the weights of Fm~, states in the lowest eigenstates of (1) with the parameters (3). For X = 0.1 MeV, which is the value preferred by the authors of ref. [12], the percentage of Fmax = 3 states (and N~ = Nv = 3) in the lowest 0 + and 2 ÷ states 98.4% and 98.6%. For ~ = 0 these percentages are smaller but still considerable, 95.7% and 96.5% respectively (for X = 1 MeV the states are practically pure, 99.9% of Fm~x = 3 in both). There are, however, large deviations from pure F-spin states in a realistic case like 178Hf. The IBA-2 parameters used in that case were determined by a least squares fit to be [10] (c, x and 4, are in MeV) ~. = c v = 0.5259, X,, = - 1.1507,
x = -0.0872, X~ = -0.2212,
41 = - 0 . 5 0 5 6 , 42=-0.0228,
43=-0.3846.
(4)
In this set of parameters, unlike the set (3) of ref. [12], X~ and X~ have rather different values. This introduces additional asymmetry between proton bosons and neutron bosons. We calculated the F-spin composition of low-lying states and found large admixtures of states with F ~< F_,n~x - 1. The weight of the states with Fm~x = 1 5 / 2 in the 0 + ground state is 82.1% (16.4% for the F = Fm~x - 1
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state). The second 0 ÷ state (of the r - b a n d ) is only 63.6% in a state with Fr, a~ while the weight of states with F = Fm~x - 1 is 29.6%. Thus, F-spin is badly broken and the success of IBA-1 in 178Hf cannot be attributed to F-spin being a good q u a n t u m n u m b e r of IBA-2 states. A close look at the F-spin composition of 178Hf low-lying IBA-2 states reveals an interesting phenomenon. Admixtures of states with F = Fm~ 1 in the various states of the ground state band have rather constant amplitudes. In higher states, grouped into the r - b a n d and y-band, this behavior is less striking but is still evident. This pattern suggests that we may use the subset of states with Fm~x (e.g. IBA-1 states for M F = Fmax) as a model of the exact IBA-2 states. The first step in such a p r o g r a m is to consider matrix elements of the IBA-2 hamiltonian between states with Fm~,. In the following steps the contribution of non-diagonal elements connecting states of the model space ( F = Fmax) with other states (with F < F m a x ) should be added. Such contributions will certainly renormalize the parameters of the model hamiltonian (that of IBA-1) and may introduce more complicated terms. The first step of obtaining the submatrix of the hamiltonian (1) defined by F = Fm~x can be accomplished in two ways. We can add to (1) a very larger Majorana term which will decouple states with F = Fm~ from other states which will be pushed very high up m energy. Alternatively, we can project the IBA-2 hamiltonian (1) onto the F = Fm~, set of states according to ref. [12]. Either of these ways led in the case of the hamiltonian (1) with the parameters (4) to a ground state energy higher by 0.31 MeV than the lowest exact IBA-2 eigenvalue. Still, this shift, though rather large, turned out to be rather uniform for all low-lying states. Shifts in the spacings of these levels are considerably smaller. Positions of low-lying levels above the ground state are presented fig. 1. It is important to emphasize that this feature occurs only for the ground-state band and the lowest excited r - b a n d and ,/-band. The position of the second excited K = 0 band is changed from 1.346 MeV in the IBA-2 calculation to 2.048 MeV obtained with the project hamiltonian. Energy levels in that b a n d are well reproduced by the IBA-2 calculation as are measured branching
3 October 1985
MeV 20
m
6 - -
1,5
6 - 5 - -
_ _
4 - -
- -
- -
5 - -
- -
- -
2 - -
2 - -
~
- -
0 - -
_ _
- -
_ _
- -
~
_ _
4 - -
a
8--
I0
b
c
a
b
c
6--
05 4
-
~m o-a
m
m b
c
Fig 1 Spacings of energy levels m 17SHf Levels denoted by a, b and c were calculated from the IBA-2, the projected IBA-1 and a fitted IBA-1 hamdtoman, respectively
ratios of E2 transitions. The component of Fm~x in the lowest state of that band is only 37.7% while the components with Fro,x - 1 and Fm~x - 2 are considerable, 28.9% and 29.0% respectively. It is not surprising that the position of this band cannot be reproduced by IBA-1 calculations [4]. F r o m our calculation it appears that for spacings of low-lying levels the IBA-1 hamiltonian projected from a reahsnc IBA-2 hamiltonian is already a good model. We can see how its parameters should be modified or renormalized by trying to improve the agreement with the IBA-2 results. We m a y consider them as free parameters and determine their values which give the bestfit to the eigenvalues of the IBA-2 hamiltonian (which are very close to the experimental energies). The IBA-1 hamiltonian projected from the hamiltonian (1) with the parameters (4) is obtained as C + , ' ( d +. d) + r ' a ( x ) - Q(x) + x ' x n ( d + X 8) (2) "(d + X (~)(2).
(5)
In (5) the various parameters are given in terms of those in the IBA-2 hamiltonian by C -.~
,' Kt
5N,,N, ~ N-1 + g.N + (4 - X~X,) N N ( N - 1) ~' N,,N. N(N-
1) x,
x'=
x,).
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Volume 160B, number 1,2,3
PHYSICS LETTERS
The ~, do not appear in (5) at all. In (1) they multiply operators wtuch measure the a m o u n t of coupling of two bosons in anttsymmetrw states. Hence, each of them vanishes when applied to fully symmetric states with F = Fra~x. In fact, each of them is a scalar with respect to F-spin. As such, their eigenvalues are independent of M F and each of them vanishes when applied to states with M v = + Fma~. The hamiltonian (5) is, for ~,, = ~ , identical with the one obtained in ref. [12]. There, however, the last term has been transformed and expressed as a hnear combination of other operators a n d the constant term dropped. Substituting into (6) the values of the parameters (4), with N~ = 5, N~ = 10, we obtain the following set of IBA-1 parameters c' = 0.4481 MeV, X = - 0.6859,
x' = - 0 . 0 2 0 8 MeV,
X' = - 0 . 4 6 4 7 -
X = -0.7033,
x' = - 0 . 0 2 0 7 MeV,
X' = - 0 . 4 8 9 9 .
(8)
T h e level spacings calculated with these values are also plotted in fig. 1 (denoted by c). As seen by c o m p a r i n g (7) and (8) a slight renormalization of the IBA-1 parameters gives a very good fit to level spacings calculated in IBA-2 which in turn agree very well with the experimental ones. Also E2 transition probabilities calculated from the IBA-1 hamiltonian (5) with the parameters (8) [or even with those in (7)] agree rather well with experimental b r a n c h i n g ratios. Better fits to the IBA-2 results can be obtained by adding two more independent operators to the IBA-1 hamiltonian (5). F o r the sake of the a r g u m e n t presented here, however, it is sufficient to consider IBA-1 hamiltonians given by (5). In summary, our results show how rotational features emerge from dlagonalization of very large
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matrices (orders of thousands) constructed from a realistic IBA-2 hamiltonian. The overlap of low-lying states with m a x i m u m symmetry states ( F = Fm~x ) is not very high. Still, the latter form a g o o d model space of the IBA-2 hamiltonian. E n e r g y spacings of low-lying levels are well r e p r o d u c e d b y using an IBA-1 hamiltonian only slightly different from the one projected from IBA-2. The detailed IBA-2 hamiltonlan based on properties o f the effective interactions between nucleons can be well approximated for low-lying levels by a simple IBA-1 hamiltonian from which collective b a n d s arise naturally. The authors would like to express their thanks to Professor M.W. Kirson and Professor P. von B r e n t a n o and in particular to A. Leviatan for helpful discussions.
(7)
Using these parameters we obtain the level spacings gwen in fig. 1 denoted b y b. We looked for values of the parameters (7) that will give a better fit to level spacings calculated from the I B A - 2 hamiltonian. We found the following set which gives better agreement with the IBA-2 results e' = 0.4513 MeV,
3 October 1985
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