- Email: [email protected]
Comparison of the logical structure of the liquid drop model and the interacting boson model
Comparison of the Logical Structure of the Liquid Drop Model and the Interacting Boson Model ROBERT GILMORE and DA HSUAN FENG Department of Physics and Atmospheric Science, Drexel University, Philadelphia, PA 19104, U.S.A.
The logical structures of the Liquid Drop Model and the Interacting Boson Model are compared and found to be remarkably similar. Minor modifications in these models are indicated which, i f implemented, will result in mathematically equivalent models. I.
INTRODUCTION
There are now two generally successful approaches to the description of collective nuclear motion. One is the Bohr Model of Collective Nuclear Motion, based on the Liquid Drop Model (LDM) (Bohr and Mottelson, 1953; 1975). The other is the Interacting Boson Model (IBM), (Arima and lachello, 1975; 1976; 1978; 1979). These two models are remarkably similar in their predictive capabilities, while at the same time exhibiting a number of differences. One way to compare the two models is to make detailed numerical computations on specific nuclei (such as 168Er) and to compare the results. This approach has been undertaken by others (Bohr and Mottelson, Ig80; Casten, 1982; Warner and Casten, Ig80). Based on the assumption that i t is more useful to compare the inputs, the computational algorithms, and the outputs of these models (a standard approach for comparing models in other disciplines, such as statistics, computer science, and numerical analysis), we undertake here a detailed comparison of the logical structure of these two models (Gilmore and Feng, 1982). This comparison is carried out in Section I f , after a brief description of the two models is presented. We see that the logical structures of the two models are isomorphic, with differences in the details. The 16 items we have identified in the logical structures of these models are listed in Table I I .
A direct consequence of this comparison is that, while similar, the models are not equlvalent (Gllmore and Feng, 1981) In view of the elegant geometric picture of collective nuclear motion provided by the Liquid Drop Model CBohr and Mottelson, 1953; 1975) and the slmple computational techniques afforded by the Interacting Boson Model ~Scholten, 1980), i t may be useful, even desirable, to determine what modifications in these models would be required to make them different representations ~analagous to wave mechanics and matrix mechanics) of the same underlylng physics. This study is carried out in Section I I I , and results are summarized in Table I I .
495
496
Robert Gilmore and Da Hsuan Feng
Throughout this work we make no attempt either to renormalize the history of the development of these models or to pass judgment on this renormalized history. II.
COMPARISONOF THE TWO COLLECTIVE NUCLEARMODELS
We begin with a brief description of these two models. a. Liquid Drop Model. The Bohr Model of Collective Nuclear Motion has its origins in the picture of an oscillating liquid drop (Bohr and Mottleson, 1953; 1975). In this picture, the surface oscillations of a rotating, vibrating liquid drop are represented by means of a classical expansion of the drop radius in terms of spherical harmonics. After a set of approximations which leave only the quadrupole degrees of freedom, the amplitudes of the Y~ are quantized after some fashion to become the basic operators of the Bohr collective model, or the 5-dimensional oscillator model (Bohr and Mottelson, 1953; 1975). In general, the parameters which appear in the collective hamiltonian are chosen to f i t observed nuclear properties (Eisenberg and Greiner, 1970), rather than input on the basis of their putative liquid drop values. While i t may be argued that the Bohr collective model is only a 5-dimensional oscillator model ( i . e . , no connection at all with the Liquid Drop Model), the standard interpretation of the classic parameters (B,y) as shape variables belies this view and makes evident the clear connection between the quadropole deformation of an osc i l l a t i n g fluid and i t s quantum counterpart, where each of the five modes Y~ is a quantum oscillator. b. Interacting Boson Model. This model has its origins in an attempt to extrapolate Hund's rules from valence electrons to valence nucleons in a logical way. I t was assumed that the strongest interaction among valence nucleons produced strong binding of nuclear pairs in J=O states, s l i g h t l y weaker binding in J=2 states, and either very weak binding or repulsion in higher angular momentum states. The s-boson and the five d-bosons then form the basis for a dynamical group SU(6). Residual interactions among nucleons are then assumed to be representable in terms of the generators of this dynamical group (Arima and lachello, 1975; 1976; 1978; 1979). The original formulation of the IBM was in terms of matrices (Arima and lachello, 1975; 1976; 1978; 1979). However, the subsequent introduction of coherent states for SU(6) has made i t clear that the IBM has a geometric formulation as well (Ginocchio and Kirson, 1980; Dieperink, Scholten and lachello, 1980; Feng, Gilmore and Deans, 1981). Many of the logical properties of the IBM will be presented in both their algebraic and geometric representations. The Liquid Drop Model has served as a nucleus about which a succession of more elaborate models has evolved to increase our understanding of collective nuclear motion: the two fluid model (Rowe, 1970), the nucleon coupled to surface oscillations (Bohr and Mottelson, 1953). The Interactin9 Boson Model has also served as a nucleus about which a succession of more elaborate models has developed: the Interacting Boson Approximation-2 (treating neutrons and protons separately) (Otsuka and coworkers, 1978), the Interacting Boson-Fermion Approximation (lachello and Scholten, 1979), the Supermultiplet Model (lachello, 1980). The veKy parallel evolution of these two models is summarized in Table I. The parallel developments obvious from this table already suggests a great deal of similarity between these two models. To make this similarity evident, we now undertake a detailed comparson of the logical structures of these two models. This comparison is undertaken only at the phenomenological level. No attempt is made to describe or compare whatever microscop! c foun~atlons exlst for these models.
Logical
Structure of the Liquid Drop Model and the IBM
497
TABLE I. The evolution of collective models has proceeded in a series of parallel developments. Liquid Drop Model
Interactin 9 Boson Model
Liquid Drop
IBA-I = IBM
Two Fluid Model
IBA-2
Nucleon coupled to surface oscillations
IBFA Supermultiplet model
I. Physical Basis. The LDM i~ based on the oscillations of a rotating-vibrating liquid drop around its equilibrium shape. The IBM is based on the Cooper pairing of nucleons into collective states with J=O, J=2. 2. Physical Bias. The starting point for the LDM is classical continuum mechanics. That for the IBM is the independent particle picture of the nucleus as expressed in the shell model. 3. Mathematical Bias. algebraic. 4.
The LDM is palpably geometric; the IBM is unabashedly
Shell Model Inputs. The LDM has none. The IBM has two: a) The nucleus has an inert closed core. b) Basis states are drawn from the valence orbitals of the next shell.
5. Starting Point. The radius of the liquid drop is expressed in terms of spherical harmonics: RCe,¢)
=
Z aL YM (0,@) > 0 L,M
In the IBM, nucleons are assumed to form Cooper pairs IJ> = Z C j j , l ( J J ' ) J > jj' Here nucleons with angular momenta j and j ' couple to form a boson with angular momentum J. 6. Approximations. In the LDM the following series of approximations reduce the complexity from that of continuum mechanics to that of the classical mechanics of a system with a small number of degrees of freedom: a) L=O term is constant (no monopole vibration) b) L=2 terms describe quadrupole oscillations, which are of primary importance. c) L=4 terms can be shown to be of much less importance than quadrupole deformation terms and are generally neglected (Bohr and Mottelson, 1953;1975). d) L=6,8,... terms are also neglected. e) L=I terms represent a displacement of the center of mass, and are neglected.
498
Robert
f)
Gilmore
and Da Hsuan Feng
L=3,5,7, . . . terms are neglected.
We remark here that onl£ i f this set of simplifying approxomations is made does the LDM evolve into the 5-dimensional oscillator. An analogous series of approximations exists for the IBM: a) J=O term is present. In fact, the S-boson is strongly bound. b) J=2 terms (D-bosons) are important, but not as strongly bound as the S-boson. c) J=4 states are either weakly bound or are slightly repulsive, and are neglected. d) J=6,8,... are not bound, and therefore neglected. e) J=l states represent a displacement of the center of mass, and are neglected.
f) J=3,5,7,-.. arise through the admixture of nucleons from adjacent opposite parity (3-dimensional) harmonic oscillator levels, and are therefore neglected as the IBM seeks to describe principally low-lying states. We remark here that inclusion of additional boson degrees of freedom (e.g., f or g) enlarges the dynamical group from SU(6) to SU(13) or SU(15). 7. Technical Conditions. The amplitudes a~ describing the amplitudes of quadrupole deformations obey a reality condition derived from the requirement that R(e,@) be real: 2* )M 2 aM = (a_M There are thus five real coordinates in this model. There is also a requirement that S a~* a~ (=B2) is "not too large", otherwise the liquid drop volume will become negative. The S- and D-boson operators obey the technical condition + Z
= N
where N is the total number of valence bosons present. valence nucleons for low excitation energies.
This is half the number of
2 8. Dynamical Variables. For the LDM, the five complex variables a M constltute the d¥namical variables. These may alternately be taken as the five re~l variables aZ=(-) M a2~ and the five canonically conjugate momenta ~ . For the IBM, the dynamiMcal variables are the S- and D-boson operators s,s t , d , d f .
9. Configuration Space. For the LDM, this is basis vectors a~ = (.)m a 2 ~ . In r e a l i t y , only origin is considered (iRS-÷IRS). For the IBM, group quotient (coset) space SU(6)/U(5) : slO,
the five dimensional spacelR5 with a sphere of radius < l around the the configuration sp-ace is the the IO-dimensional sphere.
lO. Phase Space. For the LDM, this is {5 = RIO spanne~ by the five complex parameters a~. This may alternatively be viewed aslR~®IR~ s anned bv aL = {_~m aZand the corresponding canonically conjugate momenta. F~r the IBM,Mphase'spac~ is the coset SU(6)/U(5).
11. Quantum-Classical Relation. Here the parallel development breaks down. Since the LDM has it s origins_in classical mechanics, i t must be quanttzed. This is done according to a~ ÷ b~, a~ ÷ b~, where [b u, b~t ] = S~v
Logical Structure of the Liquid Drop Model and the IBM
499
Conversely, since the IBM has quantum origins, we must look at its classical l i m i t . This requires = , = (-)~ 12. Dynamical Group. The LDM hamiltonian, (whatever i t is) w i l l not preserve the number of quadrupole oscillation quanta. The dynamical group then has spectrum generating algebra spanned by
I
(-2
+2)
These 36 operators span noncompact, nonsemisimple Lie algebra iu(5), the inhomogeneous unitary algebra in five dimensions (analog of the Poincare'group in its relation to the Lorentz group). The dynamical U(6) for the IBM has algebra u(6) spanned by
d~d, d~*s, s*%, s*s
(-2~,~<+2)
This algebra is compact and, removing the invariant operator sfs + Z.df d , is simple [su(5)]. 13. Underlying Hilbert Space. For the LDM, the Hilbert space is i n f i n i t e dimensional One model for this space is in terms of squ~re integrable functions defined over~5 with respect to an appropriate measure: C2C~ , v ~ . The metric is obtained from the kinetic energy term in the hamiltonian. Another model for this space is the algebraic model for a ladder representation of the 5-dimensional oscillator with basis states In_2,n l,no,nl,n2> , n~ = 0,I,2,..... For the IBM, the hilbert space carries the fully-symmetricrepresentation {N,O} of SU(6). This has basis states InR,n 2,n 1,no,nl,n2>, with ~ni = N, where i = s , - 2 , - l , O , l , 2 . An alternate model of this space'is the set of square-integrable functions defined on the coset SU(6)/U(5) w~th respect to the natural measure (Haar measure) for the representation {N,O} :
C ( S U ( 6 ) / U ( 5 ) ; ~ " ; {N,O}). 14. Dynamics. Hamiltonians for the LDM must be SO(3) scalars formed by coupling the spherical tensor (L=2) operators b~, b to angular momentum zero: e.g.: [[b * x bt] (2) x b] (0) These operators need not conserve excitation number. Hamiltonians for the IBM must be SO{3) scalars formed by coupling the spherical tensor operators s, st (L=O) and d , d~ (L=2) to angular momentum zero: e.g.: lid t x d*] (2) x [d x s](2)] (0) These operators must conserve boson number. 15. Transition Operators, Other Operators. Ml, E2, and other operators for the LDM and the IBM are mgdelled.principally by forming linear combinations of operator products (bt , b, or sT , s, d~, d) with the appropriate transformation properties, and adjusting the coefficients to reproduce experimental data. 16. Method of Solution. In principle, a Schr~dinger-like equation (Bohr's collective hamiltonian) must be solved to determine energy eigenvalues and eigenfunctions in the LDM (Bohr and Mottelson, 1953; 1975). Transition matrix elements are obtained after the usual prescription of wave mechanics. In practice, the discrete basis In-2,n-l,no, nl, n2> is used, the hilbert space is truncated to f i n i t e dimension by requiring Sni < N', all matrix elements of a model hamiltonian are computed in this basis within this truncated f i n i t e dimensional space, and the resulting matrix
Robert
500
Gilmore
and Da Hsuan Feng
diagonalized (Eisenberg and Greiner, 1970). Transition operators are computed by the usual prescription of matrix mechanics. In the IBM, the hilbert space is f i nite dimensional, all matrix elements are known both in principle (Arima and Iachello, 1975;1976;1978;1979) and in practice, the matrix diagonalization is easily carried out in a readily accessible code (Scholten, 1980) which can also be used to evaluate transition operator matrix elements. Ill.
MODELMODIFICATIONS
In this section we indicate what minimal modifications would be required in the assumptions underlying the Liquid Drop Model and the Interacting Boson Model, i f i t is desired to make the modified models different representations of the same underlying physics. The required modifications are almost self-evident from Table I I . The modifications themselves are summarized in the last column of this table. No modifications are required in the f i r s t three logical components. These represent preferences for distinct starting points in much the same way that Schr~dinger's Wave Mechanics (Schr~dinger, 1926a;1926b;1926c) and Heisenberg's Matrix Mechanics (see, for example, the discussions in Dirac, 1958) represent different starting points for the study of Quantum Mechanics (Schr~dinger, 1926d) or Feynman diagrams (Feynman, 1962) and Schwinger manifest covariance (Schwinger, 1970) represent distinct formulations of quantum electrodynamics. These representations were eventually shown to be equivalent by Schr~dinger (Schr~dinger, 1926d) and Dyson (Dyson 1949a;1951). We should not be surprised i f the Liquid Drop Model and the Interacting Boson Model, suitably modified, are also equivalent. In fact, this goal holds the promise of a deeper understanding of the geometric properties of bosons and the quantum nature of surface fluctuations. 4. Shell Model Inputs. The IBM has two shell model inputs, the LDM has none in its original phenomenological formulation. The presence of the inert closed core in the IBM suggests a classical counterpart for the LDM. This is the "tidal planet model" in which the liquid drop contains a solid undeformable spherical core of radius Rc (Krutov, 1968). We make no counterpart to the second shell model input to the I~M (Cooper pairing). Indeed, the liquid drop surface fluctuations will eventually be interpreted as S- and D-bosons. 5.
Starting Point.
We assume that the liquid drop radius is restricted by
R(8,@) ) Rc
6. Approximations. The approximations involved in the formulation of both models are remarkably similar. In fact, after the dust settles, both models have quadrupole degrees of freedom, and the IBM has the additional S-boson "degree of freedom". This is not really an extra degree of freedom because of the constraint that the total boson number must be conserved. The analogous constraint for the LDM-conservation of volume--is only enforced to lowest degree. For that reason, the L=O (monopole) mode is missing. We may ask: what happens i f we enforce volume conservation up to second order? The usual argument is that monopole excitations are volume changing and therefore cannot occur. This argument is only true i f only monopole fluctuations are present. I f other oscillations are present, then monopole oscillations can not only be present, but can be used to enforce the constant volume condition to higher order than in the usual LDM. We now show this by expressing the volume V in terms of the liquid drop radius R(O,@)
Logical
Structure of the Liquid Drop Model and the IBM
501
oR(B,@) ,qb)]3 V = f d~J r2dr = ~-fda[Z a~ yL (0
,. aM" ~.. L aM' = ~l Z aM
[~ M' ~. M"J ,.i [~ ~. L..] r~ ( T~, J~..~,'
where [ = 2L+l. We now assume that only the L=O and L=2 modes are present and set a° :v~T(R +s)
a2m = ~ d m then the expression for the volume is V=~
(R 3 + 3R2s + 3R s 2 + s 3) + 4Tr (Rd 2 + d 3)
where d2,d3 have their usual meanings. To second order in the amplitude s,d the volume is V=~
+ 4~R
[(;
+s
+Z
s ~< 0
The volume preserving condition (up to and including second order) is
(~÷#+ ~m, ~-(~)~
s.
This classical liquid-drop volume-preserving condition is directly analogous to the IBM boson number conserving condition:
(R)2 (I
R--~2)
dm 12
The identification is s* 1 +R-~
st ~
LDM
IBM dm* R/2
dmt v~-
The identification, based on the volume-preserving condition with monopole and quadrupole degrees of freedom, can be carried out to all orders using the method described in Gilmore, 1981.
Remark:
7. Technical Conditions. By introducing a classical monopole vibration and enforcing the volume-preserving condition to second order, we have created a classical LDM constraint of the form
502
Robert Gilmore and Da Hsuan Feng
s's* + Z dm'dm* = N s'
= v"ff _
1 + R--~ -
v~"
R/2 d
d'
The remaining LDM technical condition will be given an IBM counterpart in Part 9 below. 8. Dynamical Variables. For the LDM, the dynamical variables may be taken as a8 ' aM, ~ ao* 2* , subject ' to the constraint a8 = a~*, a~ =(-)M a2~. o , aM 9. Configuration Space. For the LDM, this is now the curved compact space consisting of the six real variables a° = 40* 42 _ (.)M a ~ , subject to the volumepreserving constraint. For the IBM~ we°o , °M - a 5-dimensional subspace of the must take lO-dimensional coset space SU(6)/U(5). The subspace is a lagrangian submanifold on which the following conditions are obeyed = , = (-)~ * lO.
For the LDM, this may be taken as the lO-dimensional space
Phase Space.
+ Z ]a212 = constant -a~ I> 0 For the IBM, i t is the coset space SU(6)/U(5) with coordinates Zi , i = s,-2,-l,O, +I,+2 subject to +2
Zs2+Z
IZll 2 =o
-2 Zs ;
0 ,
II.
2
Quantum-Classical Relation. The amplitude a~ : s , aM = dM' are coordinates fo r a 5-dimensional complex space. This space is curved and compact. I t is a Riemannian space which is symmetric and of rank I . There exists a complex structure on this space. This space is therefore uniquely SU(6)/U(5) (Gilmore, 1974). The amplitudes can be quantized by the Kostant-Sourian method (Kostant, 1970; Sou rian, 1970).The q-number counterparts of the c-numbers a~, a~:a~ bm,at , bmt are equivalent ("isomorphic") to the boson operators s, dm, st, dm .
12. Dynamical Group. The dynamical group IU(5) for the LDM and U(6) for the IBM are simply related by the group contraction /group expansion methods (Gilmore, 1974):
lu(5)
expand
- u(6)
"contract To contract U(6) to IU(5) we proceed as in the contraction of U(2) to IU(1) (Weyl group), well known in general and in quantum optics in particular (Gilmore, 1974; Arecchi and coworkers, 1972):
Logical Structure of the Liquid Drop Model and the IBM
503
d t d~ + d td d ts
+ c d~t s = dpt(cs)
std
÷ c std
: (cst ) d
sts
+ c2sts
= (cst)(cs)
This nonsingular transformation preserves the structure of the Lie algebra u(6) until the parameter c ÷ O. In this case, the transformation becomes singular, but the set of operators s t i l l closes under commutation. However, the structure of the Lie algebra changes from that of u(6) to that of iu(5). To proceed in the other direction, we begin with the operators b b t of the Lie algebra iu(5) and transform them as follows: ~' b tb
÷
b tb
bp t ÷
b t~
b ÷ at b I ÷ ~tq where [~,~t] = +I.
The resulting algebra is u(6).
13. Underlying Hilbert Space. The Hilbert spaces for both the LDM and the IBM have both an algebraic and a geometric realization. We compare f i r s t their algebraic realizations. For the LDM, the standard 5-dimensional oscillator basis is converted to a U(6) basis by e x p l i c i t l y including the degree of freedom associated with the ~ boson
In_2,n_l,nO,nl,n2>-~-
In Cfiu(1) > ~ In -2 ,n -I ,no,n 1 'n2>IU(5) = Incr; n_2,n.l,nO,nl,n2>u(6)
+2 no
+Z n =N -2 I;
Conversely, the IBM basis may be contracted to the 5-dimensional oscillator basis according to
Lim Ins;n_2,n_l ,no,n I ,n2> + In_2,n_l ,no,nI ,n2>
c+O
The l i m i t is taken as the parameter c + 0 such that cvIT: I. The Hilbert spaces may also be compared in their geometric realizations. The appropriate hilbert space for the IBM is
C2 (CLM; vr~T; {N,~})
504
Robert Gilmore and Da Hsuan Feng
where CLM is the lagrangian submanifold of SU(6)/U(5), g is the metric on i t induced from t h ~ of SU(6)/U(5), and {N,O} is the representation space for N bosons. For the LDM, the appropriate hilbert space is C2(CS; vTgT; ~) cs is the real configuration space of the modified LDM, g is the metric on this conpact space, and ~ is the r a t i o of f l u i d volume outside the inert core to the total liquid drop volume. 14. Dynamics. Before the dynamical details of the LDM and the IBM can be compared, a framework must be established for comparing them at the kinematical level. Having stated this caveat, we indicate that i t is an easy matter to make a correspondence between IBM and LDM hamiltonians. This is a many-I correspondence for the original LDM which becomes I - I when the LDM model is "compactified" or "truncated" by the addition of the ~ boson. For example, the term Q.Q in the IBM hamiltonian corresponds, under group contraction U(6) ÷ IU(5) to the term Q'.Q' in the LDM hamiltonian, where IBM:
Q = ( d t s + std) (2) + x(dfd) (2) + Q' : (bt + b) + x(btb) (2) : LDM
Conversely, the term [ ( b t x bt) (2) x b] (0) in the original LDM.bamiltonia~ has several IBM counterparts, the simplest of which is [ ( d r x d+)"C2) x (dxs)(2)](O). Other counterparts have additional number-preserving combinations sis. 15. Transition Operators. The group expansion-contractor method allows us to step back and forth between the LDM and the IBM-modified LDM. The general procedure is simple, and has already been indicated for r o t a t i o n a l l y invariant operator (the hamiltonlan). To obtain the LDM counterpart of an IBM operator, contract as above. To obtain the simplest IBM, or modified LDM counterpart of an LDM operator, multiply the LDM operator by the smallest number of ~ or ~+ operators required to preserve excitation number. I f operator order is not unique, ascertain using boundary values or asymptotic l i m i t s . 16. Method of Solution. The modified LDM appears as a matrix equation in a vector space which is automatically truncated at f i n i t e dimension by the introduction of the a boson. By identifying N, the total number of r o t a t i o n a l - v i b r a t i o n a l excitations present, with the number of S- and D-boson present (half the number of valence nucleons), the h i l b e r t spaces have the same dimension. By "expanding" the original LDM hamiltonian, a corresponding IBM hamiltonian is obtained. Numerical-algebraic solution of the modified LDM thus is equivalent to numerical solution of the IBM. Solution of the IBM or modified LDM can also be carried out in the geometric representation by solving of a SchrBdinger-like equation. Both the IBM and modified LDM must possess the same SchrBdinger equation with the same geometric (shape) parameters, since they obey the same equation in the algebraic representation. (Transformation between algebraic and geometric representations is affected by means of the SU(6)/U(5) coherent states (Gilmore and Feng, 1982; Gilmore and Feng, 1981; Gilmore, 1972). In view of the notorious d i f f i c u l t i e s in solving this SchrBdingerl i k e equation, a matrix diagonalization using PHINT w i l l eventually become standard procedure for solving the modified LDM. IV.
DISCUSSION
The analysis carried out in Section II has revealed the remarkable similarity in the logical structure of the two most important models of collective nuclear structure: the Liquid Drop Model and the Interacting Boson Model. Not only do the
Logical
Structure of the Liquid Drop Model and the IBM
505
models have a similar logical structure, but the details of each logical segment are also very similar. The models are not l o g i c a l l y equivalent, as is evident from their different dynamical groups: IU(5) for the LDM and U(6) for the IBM. A comparison of the logical foundations of the two models is presented in Table I I . A glance at this table is suf f icient to suggest the modification required in each model to bring the two into equivalence--in the sense that the two models are d i f ferent mathematical representations of the same underlying physical processes. Once these modifications have been made and the equivalence demonstrated, all features of either model (fermion pairing, surface fluctuations) become concepts available in the other. This allows the development of new insights for both models. For example, the conservation of S- and D-bosons becomes equivalent to the conservation of nuclear volume (in second order), a result of dramatic simplicity which ceases to be surprising in view of the s i m i l a r i t y between the models. ACKNOWLEDGEMENTS We thank Akito Arima, Rick Casten, Franco lachello, Michel Vallieres and John Wood for useful discussions. Work p a r t i a l l y supported by NSF grant #PHY810-2977. REFERENCES Arecchi, F.T., E. Courtens, R. Gilmore and H. Thomas (1972). Atomic Coherent States in Quantum Optics. Phys. Rev., A6, 2211. Arima A. and F. lachello (1975). Collective Nuclear States as Representations of a SU(6) Group. Phys. Rev. Lett., 35, I069. Arima A. and F. lachello (1976). Interaction Boson Model of Collective Nuclear States I. The Vibrational Limit. Ann. Phys. (NY), 99, 253. Arima A. and F. lachello (1978). Interaction Boson Model of Collective Nuclear States I I . The Rotational Limit. Ann. Phys. (NY), I l l . , 201. Arima A. and F. lachello (1979). Interacting Boson Model of Collective Nuclear States I I I . The 0(6) Limit. Ann. Phys. (NY), 123, 468. Bohr A. and B.R. Mottelson (1953). Collective and Individual-Particle Aspects of Nuclear Structure. Mat. Fys. Dan. Vid. Selsk.~ 27, 174. Bohr, A. and B.R. Mottelson (1975). Nuclear Structure, Vol. I I . Addison-Wesley, Reading, PA. Bohr A. and B.R. Mottelson (1980). Features of Nuclear-Deformations Produced by the Alignment of Individual Particles or Pairs. Phys. Scr., 22, 468 Casten, R.F. (1982). Lectures in these proceedings. Dieperink, A.E.L., O. Scholten and F. Iachello (1980). Classical Limit of the Interacting-Boson Model. Phys. Rev. Lett.~ 44, 1747. Dirac, P.A.M. (1958). The Principles of quan.tum Mechanics. Oxford University Press. Dyson, F. (1949). The Radiation Theories of Tomonaga, Schwinger and Feynman. Phys. Rev., 75, 486. Dyson, F. (1951). The SchrBdinger Equation in Quantum Electrodynamics. Phys. Rev., 83, 1207. Eisenberg, J.M. and W. Greiner (1970). Nuclear Models, Vol. I . North Holland, Amsterdam. Feng, D.H., R. Gilmore and S.R. Deans (1981). Phase Transitions and the Geometrical Properties of the Interacting Boson Model. Phys. Rev., C23, 1254. Feynman, R.P. (1962). quantum Electrodynamics., Benjamin, New York. Gilmore, R. (1972). Geometry of Symmetrized States. Ann. Phys.~ (NY) 74, 391. Gilmore, R. (1974). Lie Groups, Lie Algebras and Some of Their Applications. Wiley, New York. Gilmore, R. (1981). Catastrophe Theory for Scientists and Engineers. Wiley, New York.
506
Robert Gilmore and Da Hsuan Feng
Gilmore, R. and D.H. Feng (1981). Geometric and Dynamical Properties of the Interacting Boson Model. In F. lachello (Ed.), Interacting Bose-Fermi Systems in Nuclei_, Plenum Press, New York, pp. 149. Gilmore, R. and D.H. Feng (1982). The Interacting Boson Model in the Continuous Basis Representations. In D.H. Feng, M. Vallieres, M.W. Guidry and L.L. Riedinger (Ed.), Contemporarx Research in Nuclear Physics, Plenum Press, NY (in press). Ginocchio, J.N. and M.W. Kirson (1980). Relationship Between the Bohr Collective Hamiltonian and the Interacting-Boson Model. Phxs. Rev. Lett.~ 44, 1744 lachello, F. and O. Scholten (1979). Interacting Bose-Fermi States in Odd-A Nuclei. Phys. Rev. Lett.~ 43, 679. lachello, F. (1980). Dynamical Supersymmetries in Nuclei. Phxs. Rev. Lett., 44, 772. Kostant, B. (1970). Lecture Notes in Mathematics, Vol. 17O, Springer-Verla~, NY. Sourian, J.M. (1970). Structure des Sys.temes Dynamiques, Paris, Durod. Krutov, V.A. (1968). Rotational Motion of Deformed Nuclei. Annalen der Phxsi.k, 21, 263. Otsuka, T., A. Arima, F. lachello and I. Talmi (1978). Shell Model Description of the Interacting Boson Model. Phys. Letters~ 76B, 139. Rowe, D. (1970). Nuclear Collective Motion. Methuen, London. Scholten, O. (1980). Code Package PHINT (unpublished). SchrBdinger, E. (1926a). Ann. Phxsik IBerlin)p 79, 361. SchrBdinger, E. (1926b). Ann. Phxsik IBerlin)p 79, 489. SchrBdinger, E. (1926c). Ann. Ph~sik IBerlin)~ 80, 473. SchrBdinger, E. (1926d). Ann. Physik {Berlin)', 81, I09. Schwinger, J. (1970). quantum Kinematics and Dxnamics. Benjamin, NY. Warner, D.D., R.F. Casten and W.F. Davidson ('1'980). Detailed Test of the Interacting Boson Approximation in a Well-Deformed Nucleus--The Positive Parity States. Phys. Rev. Lett., 22, 1761
Logical Structure of the Liquid Drop Model and the IBM
507
TABLE II Logical Inputs to the Bohr-Mottelson Liquid Drop Model of Collective Nuclear Motion & the Arima-lachello Interacting Boson Model of Collective Nuclear States Bohr-Mottel son Liquid Drop Model of Nuclear Motion
Arima-lachell o Interacting Boson Model of Nuclear States (B)
I . Physical Basis
Liquid Drop Motion
Cooper Pairing
2. Physical Bias
Classical Continuu~ Mechanics
Independent Particle Motinn
3. Mathematical Bias
Geometry
Algebra
Assumptions
(A)
4. Shell Model
a. Inert closed core b. Valence orbital basis states
Inputs
5. Starting Point
Modifications and Remarks
(c)
LDM: Inert Closed Core of Radius R
c
R(B,qS) :
Z aLyL(~,*)>0
I J> : . j ~ j , ~ j j , I ( j j ' ) J >
LDM: R(8,@) ) Rc
LM 6. Approximations
L =0 L =2
Neglect. No monopol e vibrations.
S-Boson Strongly Bound
Quadrupole deforma- D-Boson is tion. P r i m a r y important importance
LDM: Conserve Volume to 2nd Order and l e t a00 : ~ ( R + s ) 2 aM
L = 4,6,8,-..
Much less important than L = 0,2 term.
Weakly bound or unbound
L =l
Represents nuclear displacement,
Requiresdifferent o s c i l l a t o r shells
1
J
=
~
then
J
dm
1+ s 12 ~Jdm 2 IBM: Boson number
conservation then L = 3,5,.-.
Not important
Requires different o s c i l l a t o r shells
Z
~-~>+m\ T / = s*
st
1 + R-7-f ~
LDM: d* R/-7-f *'~
1
,~:IBM
(T
m
~
All other L components neglect.
08
Robert
Gilmore
and Da Hsuan Feng
Table II (contd.) 7. Technical Conditions
8 •
Dynamical Variables
9. Configuration Space
2* , ,M 2 aM =t-) a_M 2* 2 Z aM aM not large M
2 2 aM,'~" M
'~5
Classical LDM constraint due to 2nd order volume preserving : s's'*
~ d ' dm' *m = N +m
d'=
R-T2-d
s,s t, d ,d~
0 2 O* 2* LDM: aO, aM, aO, aM
SU(B)IU(5)
LDM: Curved compact O, O*, space of ao~=a0 ) 2, , ,M 2"~ and aM£=t-) a_M) IBM: Choose a Lagrangian submanifold whic is 5-dimensional subspace of the IO-dim. coset space: SU(6)/ U(5) and obeying liB.
I0. Phase Space
C5~ IRlO
I 1. Quantum-
2 aM+ bM
Classical Relations
2* aM -~ b~
SU(6)IU(5) :
: (-~
LDM: Rank l Riemannia symmetric space. It is uniquely SU(6)/U(5 LDM: Quantization of classical amplitudes by Kostant-Sourian method.
[bM'bM'] = aMM' 12. Dynamical Group algebra spanned by 13.Underlying Hilbert Space
14. Dynamics
iu(5) iu(5) b"Fb ' bp, + bv, I
U(6) U(6)
d~du, dtps, stdv, sts
@5,v l-)
L2(SU(6)/U(5); ~/T~; {N,O})
S0(3) scalar Hamiltonian
S0(3) scalar Hamiltonian
expand lU(5)
~" U(6) ~contract
LDM: C2(CS;vr~; v) IBM: L2(CLM;V]~T;{N,O IBM< Man7 to 1 ) Orlglnal LDM Hamil tonian IBM 1 to 1 compactified "LDM Hamiltonian
Logical
Structure of the Liquid Drop Model and the IBM
509
Table I I (contd.) 15. Transition and Linear Combinations Linear Combinations Group expansionother operators of o~erator productsl of operator products contraction of 12c of bT,b. of s@, s,d t , d iallows step back and forth between LDM and IBM (=modified LDM). 16. Method of Sol ution
Solve Schr6dinger- Diagonalize like equation or Matrix diagonalize truncated 5-dimensional oscillator matrix hamiltonian
Construct transforma. tion theory between two representations.
The logical structures of the Liquid Drop Model and the Interacting Boson Model are compared and found to be remarkably similar. Minor modifications in these models are indicated which, i f implemented, will result in mathematically equivalent models. I.
INTRODUCTION
There are now two generally successful approaches to the description of collective nuclear motion. One is the Bohr Model of Collective Nuclear Motion, based on the Liquid Drop Model (LDM) (Bohr and Mottelson, 1953; 1975). The other is the Interacting Boson Model (IBM), (Arima and lachello, 1975; 1976; 1978; 1979). These two models are remarkably similar in their predictive capabilities, while at the same time exhibiting a number of differences. One way to compare the two models is to make detailed numerical computations on specific nuclei (such as 168Er) and to compare the results. This approach has been undertaken by others (Bohr and Mottelson, Ig80; Casten, 1982; Warner and Casten, Ig80). Based on the assumption that i t is more useful to compare the inputs, the computational algorithms, and the outputs of these models (a standard approach for comparing models in other disciplines, such as statistics, computer science, and numerical analysis), we undertake here a detailed comparison of the logical structure of these two models (Gilmore and Feng, 1982). This comparison is carried out in Section I f , after a brief description of the two models is presented. We see that the logical structures of the two models are isomorphic, with differences in the details. The 16 items we have identified in the logical structures of these models are listed in Table I I .
A direct consequence of this comparison is that, while similar, the models are not equlvalent (Gllmore and Feng, 1981) In view of the elegant geometric picture of collective nuclear motion provided by the Liquid Drop Model CBohr and Mottelson, 1953; 1975) and the slmple computational techniques afforded by the Interacting Boson Model ~Scholten, 1980), i t may be useful, even desirable, to determine what modifications in these models would be required to make them different representations ~analagous to wave mechanics and matrix mechanics) of the same underlylng physics. This study is carried out in Section I I I , and results are summarized in Table I I .
495
496
Robert Gilmore and Da Hsuan Feng
Throughout this work we make no attempt either to renormalize the history of the development of these models or to pass judgment on this renormalized history. II.
COMPARISONOF THE TWO COLLECTIVE NUCLEARMODELS
We begin with a brief description of these two models. a. Liquid Drop Model. The Bohr Model of Collective Nuclear Motion has its origins in the picture of an oscillating liquid drop (Bohr and Mottleson, 1953; 1975). In this picture, the surface oscillations of a rotating, vibrating liquid drop are represented by means of a classical expansion of the drop radius in terms of spherical harmonics. After a set of approximations which leave only the quadrupole degrees of freedom, the amplitudes of the Y~ are quantized after some fashion to become the basic operators of the Bohr collective model, or the 5-dimensional oscillator model (Bohr and Mottelson, 1953; 1975). In general, the parameters which appear in the collective hamiltonian are chosen to f i t observed nuclear properties (Eisenberg and Greiner, 1970), rather than input on the basis of their putative liquid drop values. While i t may be argued that the Bohr collective model is only a 5-dimensional oscillator model ( i . e . , no connection at all with the Liquid Drop Model), the standard interpretation of the classic parameters (B,y) as shape variables belies this view and makes evident the clear connection between the quadropole deformation of an osc i l l a t i n g fluid and i t s quantum counterpart, where each of the five modes Y~ is a quantum oscillator. b. Interacting Boson Model. This model has its origins in an attempt to extrapolate Hund's rules from valence electrons to valence nucleons in a logical way. I t was assumed that the strongest interaction among valence nucleons produced strong binding of nuclear pairs in J=O states, s l i g h t l y weaker binding in J=2 states, and either very weak binding or repulsion in higher angular momentum states. The s-boson and the five d-bosons then form the basis for a dynamical group SU(6). Residual interactions among nucleons are then assumed to be representable in terms of the generators of this dynamical group (Arima and lachello, 1975; 1976; 1978; 1979). The original formulation of the IBM was in terms of matrices (Arima and lachello, 1975; 1976; 1978; 1979). However, the subsequent introduction of coherent states for SU(6) has made i t clear that the IBM has a geometric formulation as well (Ginocchio and Kirson, 1980; Dieperink, Scholten and lachello, 1980; Feng, Gilmore and Deans, 1981). Many of the logical properties of the IBM will be presented in both their algebraic and geometric representations. The Liquid Drop Model has served as a nucleus about which a succession of more elaborate models has evolved to increase our understanding of collective nuclear motion: the two fluid model (Rowe, 1970), the nucleon coupled to surface oscillations (Bohr and Mottelson, 1953). The Interactin9 Boson Model has also served as a nucleus about which a succession of more elaborate models has developed: the Interacting Boson Approximation-2 (treating neutrons and protons separately) (Otsuka and coworkers, 1978), the Interacting Boson-Fermion Approximation (lachello and Scholten, 1979), the Supermultiplet Model (lachello, 1980). The veKy parallel evolution of these two models is summarized in Table I. The parallel developments obvious from this table already suggests a great deal of similarity between these two models. To make this similarity evident, we now undertake a detailed comparson of the logical structures of these two models. This comparison is undertaken only at the phenomenological level. No attempt is made to describe or compare whatever microscop! c foun~atlons exlst for these models.
Logical
Structure of the Liquid Drop Model and the IBM
497
TABLE I. The evolution of collective models has proceeded in a series of parallel developments. Liquid Drop Model
Interactin 9 Boson Model
Liquid Drop
IBA-I = IBM
Two Fluid Model
IBA-2
Nucleon coupled to surface oscillations
IBFA Supermultiplet model
I. Physical Basis. The LDM i~ based on the oscillations of a rotating-vibrating liquid drop around its equilibrium shape. The IBM is based on the Cooper pairing of nucleons into collective states with J=O, J=2. 2. Physical Bias. The starting point for the LDM is classical continuum mechanics. That for the IBM is the independent particle picture of the nucleus as expressed in the shell model. 3. Mathematical Bias. algebraic. 4.
The LDM is palpably geometric; the IBM is unabashedly
Shell Model Inputs. The LDM has none. The IBM has two: a) The nucleus has an inert closed core. b) Basis states are drawn from the valence orbitals of the next shell.
5. Starting Point. The radius of the liquid drop is expressed in terms of spherical harmonics: RCe,¢)
=
Z aL YM (0,@) > 0 L,M
In the IBM, nucleons are assumed to form Cooper pairs IJ> = Z C j j , l ( J J ' ) J > jj' Here nucleons with angular momenta j and j ' couple to form a boson with angular momentum J. 6. Approximations. In the LDM the following series of approximations reduce the complexity from that of continuum mechanics to that of the classical mechanics of a system with a small number of degrees of freedom: a) L=O term is constant (no monopole vibration) b) L=2 terms describe quadrupole oscillations, which are of primary importance. c) L=4 terms can be shown to be of much less importance than quadrupole deformation terms and are generally neglected (Bohr and Mottelson, 1953;1975). d) L=6,8,... terms are also neglected. e) L=I terms represent a displacement of the center of mass, and are neglected.
498
Robert
f)
Gilmore
and Da Hsuan Feng
L=3,5,7, . . . terms are neglected.
We remark here that onl£ i f this set of simplifying approxomations is made does the LDM evolve into the 5-dimensional oscillator. An analogous series of approximations exists for the IBM: a) J=O term is present. In fact, the S-boson is strongly bound. b) J=2 terms (D-bosons) are important, but not as strongly bound as the S-boson. c) J=4 states are either weakly bound or are slightly repulsive, and are neglected. d) J=6,8,... are not bound, and therefore neglected. e) J=l states represent a displacement of the center of mass, and are neglected.
f) J=3,5,7,-.. arise through the admixture of nucleons from adjacent opposite parity (3-dimensional) harmonic oscillator levels, and are therefore neglected as the IBM seeks to describe principally low-lying states. We remark here that inclusion of additional boson degrees of freedom (e.g., f or g) enlarges the dynamical group from SU(6) to SU(13) or SU(15). 7. Technical Conditions. The amplitudes a~ describing the amplitudes of quadrupole deformations obey a reality condition derived from the requirement that R(e,@) be real: 2* )M 2 aM = (a_M There are thus five real coordinates in this model. There is also a requirement that S a~* a~ (=B2) is "not too large", otherwise the liquid drop volume will become negative. The S- and D-boson operators obey the technical condition
= N
where N is the total number of valence bosons present. valence nucleons for low excitation energies.
This is half the number of
2 8. Dynamical Variables. For the LDM, the five complex variables a M constltute the d¥namical variables. These may alternately be taken as the five re~l variables aZ=(-) M a2~ and the five canonically conjugate momenta ~ . For the IBM, the dynamiMcal variables are the S- and D-boson operators s,s t , d , d f .
9. Configuration Space. For the LDM, this is basis vectors a~ = (.)m a 2 ~ . In r e a l i t y , only origin is considered (iRS-÷IRS). For the IBM, group quotient (coset) space SU(6)/U(5) : slO,
the five dimensional spacelR5 with a sphere of radius < l around the the configuration sp-ace is the the IO-dimensional sphere.
lO. Phase Space. For the LDM, this is {5 = RIO spanne~ by the five complex parameters a~. This may alternatively be viewed aslR~®IR~ s anned bv aL = {_~m aZand the corresponding canonically conjugate momenta. F~r the IBM,Mphase'spac~ is the coset SU(6)/U(5).
11. Quantum-Classical Relation. Here the parallel development breaks down. Since the LDM has it s origins_in classical mechanics, i t must be quanttzed. This is done according to a~ ÷ b~, a~ ÷ b~, where [b u, b~t ] = S~v
Logical Structure of the Liquid Drop Model and the IBM
499
Conversely, since the IBM has quantum origins, we must look at its classical l i m i t . This requires
I
(-2
+2)
These 36 operators span noncompact, nonsemisimple Lie algebra iu(5), the inhomogeneous unitary algebra in five dimensions (analog of the Poincare'group in its relation to the Lorentz group). The dynamical U(6) for the IBM has algebra u(6) spanned by
d~d, d~*s, s*%, s*s
(-2~,~<+2)
This algebra is compact and, removing the invariant operator sfs + Z.df d , is simple [su(5)]. 13. Underlying Hilbert Space. For the LDM, the Hilbert space is i n f i n i t e dimensional One model for this space is in terms of squ~re integrable functions defined over~5 with respect to an appropriate measure: C2C~ , v ~ . The metric is obtained from the kinetic energy term in the hamiltonian. Another model for this space is the algebraic model for a ladder representation of the 5-dimensional oscillator with basis states In_2,n l,no,nl,n2> , n~ = 0,I,2,..... For the IBM, the hilbert space carries the fully-symmetricrepresentation {N,O} of SU(6). This has basis states InR,n 2,n 1,no,nl,n2>, with ~ni = N, where i = s , - 2 , - l , O , l , 2 . An alternate model of this space'is the set of square-integrable functions defined on the coset SU(6)/U(5) w~th respect to the natural measure (Haar measure) for the representation {N,O} :
C ( S U ( 6 ) / U ( 5 ) ; ~ " ; {N,O}). 14. Dynamics. Hamiltonians for the LDM must be SO(3) scalars formed by coupling the spherical tensor (L=2) operators b~, b to angular momentum zero: e.g.: [[b * x bt] (2) x b] (0) These operators need not conserve excitation number. Hamiltonians for the IBM must be SO{3) scalars formed by coupling the spherical tensor operators s, st (L=O) and d , d~ (L=2) to angular momentum zero: e.g.: lid t x d*] (2) x [d x s](2)] (0) These operators must conserve boson number. 15. Transition Operators, Other Operators. Ml, E2, and other operators for the LDM and the IBM are mgdelled.principally by forming linear combinations of operator products (bt , b, or sT , s, d~, d) with the appropriate transformation properties, and adjusting the coefficients to reproduce experimental data. 16. Method of Solution. In principle, a Schr~dinger-like equation (Bohr's collective hamiltonian) must be solved to determine energy eigenvalues and eigenfunctions in the LDM (Bohr and Mottelson, 1953; 1975). Transition matrix elements are obtained after the usual prescription of wave mechanics. In practice, the discrete basis In-2,n-l,no, nl, n2> is used, the hilbert space is truncated to f i n i t e dimension by requiring Sni < N', all matrix elements of a model hamiltonian are computed in this basis within this truncated f i n i t e dimensional space, and the resulting matrix
Robert
500
Gilmore
and Da Hsuan Feng
diagonalized (Eisenberg and Greiner, 1970). Transition operators are computed by the usual prescription of matrix mechanics. In the IBM, the hilbert space is f i nite dimensional, all matrix elements are known both in principle (Arima and Iachello, 1975;1976;1978;1979) and in practice, the matrix diagonalization is easily carried out in a readily accessible code (Scholten, 1980) which can also be used to evaluate transition operator matrix elements. Ill.
MODELMODIFICATIONS
In this section we indicate what minimal modifications would be required in the assumptions underlying the Liquid Drop Model and the Interacting Boson Model, i f i t is desired to make the modified models different representations of the same underlying physics. The required modifications are almost self-evident from Table I I . The modifications themselves are summarized in the last column of this table. No modifications are required in the f i r s t three logical components. These represent preferences for distinct starting points in much the same way that Schr~dinger's Wave Mechanics (Schr~dinger, 1926a;1926b;1926c) and Heisenberg's Matrix Mechanics (see, for example, the discussions in Dirac, 1958) represent different starting points for the study of Quantum Mechanics (Schr~dinger, 1926d) or Feynman diagrams (Feynman, 1962) and Schwinger manifest covariance (Schwinger, 1970) represent distinct formulations of quantum electrodynamics. These representations were eventually shown to be equivalent by Schr~dinger (Schr~dinger, 1926d) and Dyson (Dyson 1949a;1951). We should not be surprised i f the Liquid Drop Model and the Interacting Boson Model, suitably modified, are also equivalent. In fact, this goal holds the promise of a deeper understanding of the geometric properties of bosons and the quantum nature of surface fluctuations. 4. Shell Model Inputs. The IBM has two shell model inputs, the LDM has none in its original phenomenological formulation. The presence of the inert closed core in the IBM suggests a classical counterpart for the LDM. This is the "tidal planet model" in which the liquid drop contains a solid undeformable spherical core of radius Rc (Krutov, 1968). We make no counterpart to the second shell model input to the I~M (Cooper pairing). Indeed, the liquid drop surface fluctuations will eventually be interpreted as S- and D-bosons. 5.
Starting Point.
We assume that the liquid drop radius is restricted by
R(8,@) ) Rc
6. Approximations. The approximations involved in the formulation of both models are remarkably similar. In fact, after the dust settles, both models have quadrupole degrees of freedom, and the IBM has the additional S-boson "degree of freedom". This is not really an extra degree of freedom because of the constraint that the total boson number must be conserved. The analogous constraint for the LDM-conservation of volume--is only enforced to lowest degree. For that reason, the L=O (monopole) mode is missing. We may ask: what happens i f we enforce volume conservation up to second order? The usual argument is that monopole excitations are volume changing and therefore cannot occur. This argument is only true i f only monopole fluctuations are present. I f other oscillations are present, then monopole oscillations can not only be present, but can be used to enforce the constant volume condition to higher order than in the usual LDM. We now show this by expressing the volume V in terms of the liquid drop radius R(O,@)
Logical
Structure of the Liquid Drop Model and the IBM
501
oR(B,@) ,qb)]3 V = f d~J r2dr = ~-fda[Z a~ yL (0
,. aM" ~.. L aM' = ~l Z aM
[~ M' ~. M"J ,.i [~ ~. L..] r~ ( T~, J~..~,'
where [ = 2L+l. We now assume that only the L=O and L=2 modes are present and set a° :v~T(R +s)
a2m = ~ d m then the expression for the volume is V=~
(R 3 + 3R2s + 3R s 2 + s 3) + 4Tr (Rd 2 + d 3)
where d2,d3 have their usual meanings. To second order in the amplitude s,d the volume is V=~
+ 4~R
[(;
+s
+Z
s ~< 0
The volume preserving condition (up to and including second order) is
(~÷#+ ~m, ~-(~)~
s.
This classical liquid-drop volume-preserving condition is directly analogous to the IBM boson number conserving condition:
(R)2 (I
R--~2)
dm 12
The identification is s* 1 +R-~
st ~
LDM
IBM dm* R/2
dmt v~-
The identification, based on the volume-preserving condition with monopole and quadrupole degrees of freedom, can be carried out to all orders using the method described in Gilmore, 1981.
Remark:
7. Technical Conditions. By introducing a classical monopole vibration and enforcing the volume-preserving condition to second order, we have created a classical LDM constraint of the form
502
Robert Gilmore and Da Hsuan Feng
s's* + Z dm'dm* = N s'
= v"ff _
1 + R--~ -
v~"
R/2 d
d'
The remaining LDM technical condition will be given an IBM counterpart in Part 9 below. 8. Dynamical Variables. For the LDM, the dynamical variables may be taken as a8 ' aM, ~ ao* 2* , subject ' to the constraint a8 = a~*, a~ =(-)M a2~. o , aM 9. Configuration Space. For the LDM, this is now the curved compact space consisting of the six real variables a° = 40* 42 _ (.)M a ~ , subject to the volumepreserving constraint. For the IBM~ we°o , °M - a 5-dimensional subspace of the must take lO-dimensional coset space SU(6)/U(5). The subspace is a lagrangian submanifold on which the following conditions are obeyed
For the LDM, this may be taken as the lO-dimensional space
Phase Space.
+ Z ]a212 = constant -a~ I> 0 For the IBM, i t is the coset space SU(6)/U(5) with coordinates Zi , i = s,-2,-l,O, +I,+2 subject to +2
Zs2+Z
IZll 2 =o
-2 Zs ;
0 ,
II.
2
Quantum-Classical Relation. The amplitude a~ : s , aM = dM' are coordinates fo r a 5-dimensional complex space. This space is curved and compact. I t is a Riemannian space which is symmetric and of rank I . There exists a complex structure on this space. This space is therefore uniquely SU(6)/U(5) (Gilmore, 1974). The amplitudes can be quantized by the Kostant-Sourian method (Kostant, 1970; Sou rian, 1970).The q-number counterparts of the c-numbers a~, a~:a~ bm,at , bmt are equivalent ("isomorphic") to the boson operators s, dm, st, dm .
12. Dynamical Group. The dynamical group IU(5) for the LDM and U(6) for the IBM are simply related by the group contraction /group expansion methods (Gilmore, 1974):
lu(5)
expand
- u(6)
"contract To contract U(6) to IU(5) we proceed as in the contraction of U(2) to IU(1) (Weyl group), well known in general and in quantum optics in particular (Gilmore, 1974; Arecchi and coworkers, 1972):
Logical Structure of the Liquid Drop Model and the IBM
503
d t d~ + d td d ts
+ c d~t s = dpt(cs)
std
÷ c std
: (cst ) d
sts
+ c2sts
= (cst)(cs)
This nonsingular transformation preserves the structure of the Lie algebra u(6) until the parameter c ÷ O. In this case, the transformation becomes singular, but the set of operators s t i l l closes under commutation. However, the structure of the Lie algebra changes from that of u(6) to that of iu(5). To proceed in the other direction, we begin with the operators b b t of the Lie algebra iu(5) and transform them as follows: ~' b tb
÷
b tb
bp t ÷
b t~
b ÷ at b I ÷ ~tq where [~,~t] = +I.
The resulting algebra is u(6).
13. Underlying Hilbert Space. The Hilbert spaces for both the LDM and the IBM have both an algebraic and a geometric realization. We compare f i r s t their algebraic realizations. For the LDM, the standard 5-dimensional oscillator basis is converted to a U(6) basis by e x p l i c i t l y including the degree of freedom associated with the ~ boson
In_2,n_l,nO,nl,n2>-~-
In Cfiu(1) > ~ In -2 ,n -I ,no,n 1 'n2>IU(5) = Incr; n_2,n.l,nO,nl,n2>u(6)
+2 no
+Z n =N -2 I;
Conversely, the IBM basis may be contracted to the 5-dimensional oscillator basis according to
Lim Ins;n_2,n_l ,no,n I ,n2> + In_2,n_l ,no,nI ,n2>
c+O
The l i m i t is taken as the parameter c + 0 such that cvIT: I. The Hilbert spaces may also be compared in their geometric realizations. The appropriate hilbert space for the IBM is
C2 (CLM; vr~T; {N,~})
504
Robert Gilmore and Da Hsuan Feng
where CLM is the lagrangian submanifold of SU(6)/U(5), g is the metric on i t induced from t h ~ of SU(6)/U(5), and {N,O} is the representation space for N bosons. For the LDM, the appropriate hilbert space is C2(CS; vTgT; ~) cs is the real configuration space of the modified LDM, g is the metric on this conpact space, and ~ is the r a t i o of f l u i d volume outside the inert core to the total liquid drop volume. 14. Dynamics. Before the dynamical details of the LDM and the IBM can be compared, a framework must be established for comparing them at the kinematical level. Having stated this caveat, we indicate that i t is an easy matter to make a correspondence between IBM and LDM hamiltonians. This is a many-I correspondence for the original LDM which becomes I - I when the LDM model is "compactified" or "truncated" by the addition of the ~ boson. For example, the term Q.Q in the IBM hamiltonian corresponds, under group contraction U(6) ÷ IU(5) to the term Q'.Q' in the LDM hamiltonian, where IBM:
Q = ( d t s + std) (2) + x(dfd) (2) + Q' : (bt + b) + x(btb) (2) : LDM
Conversely, the term [ ( b t x bt) (2) x b] (0) in the original LDM.bamiltonia~ has several IBM counterparts, the simplest of which is [ ( d r x d+)"C2) x (dxs)(2)](O). Other counterparts have additional number-preserving combinations sis. 15. Transition Operators. The group expansion-contractor method allows us to step back and forth between the LDM and the IBM-modified LDM. The general procedure is simple, and has already been indicated for r o t a t i o n a l l y invariant operator (the hamiltonlan). To obtain the LDM counterpart of an IBM operator, contract as above. To obtain the simplest IBM, or modified LDM counterpart of an LDM operator, multiply the LDM operator by the smallest number of ~ or ~+ operators required to preserve excitation number. I f operator order is not unique, ascertain using boundary values or asymptotic l i m i t s . 16. Method of Solution. The modified LDM appears as a matrix equation in a vector space which is automatically truncated at f i n i t e dimension by the introduction of the a boson. By identifying N, the total number of r o t a t i o n a l - v i b r a t i o n a l excitations present, with the number of S- and D-boson present (half the number of valence nucleons), the h i l b e r t spaces have the same dimension. By "expanding" the original LDM hamiltonian, a corresponding IBM hamiltonian is obtained. Numerical-algebraic solution of the modified LDM thus is equivalent to numerical solution of the IBM. Solution of the IBM or modified LDM can also be carried out in the geometric representation by solving of a SchrBdinger-like equation. Both the IBM and modified LDM must possess the same SchrBdinger equation with the same geometric (shape) parameters, since they obey the same equation in the algebraic representation. (Transformation between algebraic and geometric representations is affected by means of the SU(6)/U(5) coherent states (Gilmore and Feng, 1982; Gilmore and Feng, 1981; Gilmore, 1972). In view of the notorious d i f f i c u l t i e s in solving this SchrBdingerl i k e equation, a matrix diagonalization using PHINT w i l l eventually become standard procedure for solving the modified LDM. IV.
DISCUSSION
The analysis carried out in Section II has revealed the remarkable similarity in the logical structure of the two most important models of collective nuclear structure: the Liquid Drop Model and the Interacting Boson Model. Not only do the
Logical
Structure of the Liquid Drop Model and the IBM
505
models have a similar logical structure, but the details of each logical segment are also very similar. The models are not l o g i c a l l y equivalent, as is evident from their different dynamical groups: IU(5) for the LDM and U(6) for the IBM. A comparison of the logical foundations of the two models is presented in Table I I . A glance at this table is suf f icient to suggest the modification required in each model to bring the two into equivalence--in the sense that the two models are d i f ferent mathematical representations of the same underlying physical processes. Once these modifications have been made and the equivalence demonstrated, all features of either model (fermion pairing, surface fluctuations) become concepts available in the other. This allows the development of new insights for both models. For example, the conservation of S- and D-bosons becomes equivalent to the conservation of nuclear volume (in second order), a result of dramatic simplicity which ceases to be surprising in view of the s i m i l a r i t y between the models. ACKNOWLEDGEMENTS We thank Akito Arima, Rick Casten, Franco lachello, Michel Vallieres and John Wood for useful discussions. Work p a r t i a l l y supported by NSF grant #PHY810-2977. REFERENCES Arecchi, F.T., E. Courtens, R. Gilmore and H. Thomas (1972). Atomic Coherent States in Quantum Optics. Phys. Rev., A6, 2211. Arima A. and F. lachello (1975). Collective Nuclear States as Representations of a SU(6) Group. Phys. Rev. Lett., 35, I069. Arima A. and F. lachello (1976). Interaction Boson Model of Collective Nuclear States I. The Vibrational Limit. Ann. Phys. (NY), 99, 253. Arima A. and F. lachello (1978). Interaction Boson Model of Collective Nuclear States I I . The Rotational Limit. Ann. Phys. (NY), I l l . , 201. Arima A. and F. lachello (1979). Interacting Boson Model of Collective Nuclear States I I I . The 0(6) Limit. Ann. Phys. (NY), 123, 468. Bohr A. and B.R. Mottelson (1953). Collective and Individual-Particle Aspects of Nuclear Structure. Mat. Fys. Dan. Vid. Selsk.~ 27, 174. Bohr, A. and B.R. Mottelson (1975). Nuclear Structure, Vol. I I . Addison-Wesley, Reading, PA. Bohr A. and B.R. Mottelson (1980). Features of Nuclear-Deformations Produced by the Alignment of Individual Particles or Pairs. Phys. Scr., 22, 468 Casten, R.F. (1982). Lectures in these proceedings. Dieperink, A.E.L., O. Scholten and F. Iachello (1980). Classical Limit of the Interacting-Boson Model. Phys. Rev. Lett.~ 44, 1747. Dirac, P.A.M. (1958). The Principles of quan.tum Mechanics. Oxford University Press. Dyson, F. (1949). The Radiation Theories of Tomonaga, Schwinger and Feynman. Phys. Rev., 75, 486. Dyson, F. (1951). The SchrBdinger Equation in Quantum Electrodynamics. Phys. Rev., 83, 1207. Eisenberg, J.M. and W. Greiner (1970). Nuclear Models, Vol. I . North Holland, Amsterdam. Feng, D.H., R. Gilmore and S.R. Deans (1981). Phase Transitions and the Geometrical Properties of the Interacting Boson Model. Phys. Rev., C23, 1254. Feynman, R.P. (1962). quantum Electrodynamics., Benjamin, New York. Gilmore, R. (1972). Geometry of Symmetrized States. Ann. Phys.~ (NY) 74, 391. Gilmore, R. (1974). Lie Groups, Lie Algebras and Some of Their Applications. Wiley, New York. Gilmore, R. (1981). Catastrophe Theory for Scientists and Engineers. Wiley, New York.
506
Robert Gilmore and Da Hsuan Feng
Gilmore, R. and D.H. Feng (1981). Geometric and Dynamical Properties of the Interacting Boson Model. In F. lachello (Ed.), Interacting Bose-Fermi Systems in Nuclei_, Plenum Press, New York, pp. 149. Gilmore, R. and D.H. Feng (1982). The Interacting Boson Model in the Continuous Basis Representations. In D.H. Feng, M. Vallieres, M.W. Guidry and L.L. Riedinger (Ed.), Contemporarx Research in Nuclear Physics, Plenum Press, NY (in press). Ginocchio, J.N. and M.W. Kirson (1980). Relationship Between the Bohr Collective Hamiltonian and the Interacting-Boson Model. Phxs. Rev. Lett.~ 44, 1744 lachello, F. and O. Scholten (1979). Interacting Bose-Fermi States in Odd-A Nuclei. Phys. Rev. Lett.~ 43, 679. lachello, F. (1980). Dynamical Supersymmetries in Nuclei. Phxs. Rev. Lett., 44, 772. Kostant, B. (1970). Lecture Notes in Mathematics, Vol. 17O, Springer-Verla~, NY. Sourian, J.M. (1970). Structure des Sys.temes Dynamiques, Paris, Durod. Krutov, V.A. (1968). Rotational Motion of Deformed Nuclei. Annalen der Phxsi.k, 21, 263. Otsuka, T., A. Arima, F. lachello and I. Talmi (1978). Shell Model Description of the Interacting Boson Model. Phys. Letters~ 76B, 139. Rowe, D. (1970). Nuclear Collective Motion. Methuen, London. Scholten, O. (1980). Code Package PHINT (unpublished). SchrBdinger, E. (1926a). Ann. Phxsik IBerlin)p 79, 361. SchrBdinger, E. (1926b). Ann. Phxsik IBerlin)p 79, 489. SchrBdinger, E. (1926c). Ann. Ph~sik IBerlin)~ 80, 473. SchrBdinger, E. (1926d). Ann. Physik {Berlin)', 81, I09. Schwinger, J. (1970). quantum Kinematics and Dxnamics. Benjamin, NY. Warner, D.D., R.F. Casten and W.F. Davidson ('1'980). Detailed Test of the Interacting Boson Approximation in a Well-Deformed Nucleus--The Positive Parity States. Phys. Rev. Lett., 22, 1761
Logical Structure of the Liquid Drop Model and the IBM
507
TABLE II Logical Inputs to the Bohr-Mottelson Liquid Drop Model of Collective Nuclear Motion & the Arima-lachello Interacting Boson Model of Collective Nuclear States Bohr-Mottel son Liquid Drop Model of Nuclear Motion
Arima-lachell o Interacting Boson Model of Nuclear States (B)
I . Physical Basis
Liquid Drop Motion
Cooper Pairing
2. Physical Bias
Classical Continuu~ Mechanics
Independent Particle Motinn
3. Mathematical Bias
Geometry
Algebra
Assumptions
(A)
4. Shell Model
a. Inert closed core b. Valence orbital basis states
Inputs
5. Starting Point
Modifications and Remarks
(c)
LDM: Inert Closed Core of Radius R
c
R(B,qS) :
Z aLyL(~,*)>0
I J> : . j ~ j , ~ j j , I ( j j ' ) J >
LDM: R(8,@) ) Rc
LM 6. Approximations
L =0 L =2
Neglect. No monopol e vibrations.
S-Boson Strongly Bound
Quadrupole deforma- D-Boson is tion. P r i m a r y important importance
LDM: Conserve Volume to 2nd Order and l e t a00 : ~ ( R + s ) 2 aM
L = 4,6,8,-..
Much less important than L = 0,2 term.
Weakly bound or unbound
L =l
Represents nuclear displacement,
Requiresdifferent o s c i l l a t o r shells
1
J
=
~
then
J
dm
1+ s 12 ~Jdm 2 IBM: Boson number
conservation then L = 3,5,.-.
Not important
Requires different o s c i l l a t o r shells
Z
~-~>+m\ T / = s*
st
1 + R-7-f ~
LDM: d* R/-7-f *'~
1
,~:IBM
(T
m
~
All other L components neglect.
08
Robert
Gilmore
and Da Hsuan Feng
Table II (contd.) 7. Technical Conditions
8 •
Dynamical Variables
9. Configuration Space
2* , ,M 2 aM =t-) a_M 2* 2 Z aM aM not large M
2 2 aM,'~" M
'~5
Classical LDM constraint due to 2nd order volume preserving : s's'*
~ d ' dm' *m = N +m
d'=
R-T2-d
s,s t, d ,d~
0 2 O* 2* LDM: aO, aM, aO, aM
SU(B)IU(5)
LDM: Curved compact O, O*, space of ao~=a0 ) 2, , ,M 2"~ and aM£=t-) a_M) IBM: Choose a Lagrangian submanifold whic is 5-dimensional subspace of the IO-dim. coset space: SU(6)/ U(5) and obeying liB.
I0. Phase Space
C5~ IRlO
I 1. Quantum-
2 aM+ bM
Classical Relations
2* aM -~ b~
SU(6)IU(5)
LDM: Rank l Riemannia symmetric space. It is uniquely SU(6)/U(5 LDM: Quantization of classical amplitudes by Kostant-Sourian method.
[bM'bM'] = aMM' 12. Dynamical Group algebra spanned by 13.Underlying Hilbert Space
14. Dynamics
iu(5) iu(5) b"Fb ' bp, + bv, I
U(6) U(6)
d~du, dtps, stdv, sts
@5,v l-)
L2(SU(6)/U(5); ~/T~; {N,O})
S0(3) scalar Hamiltonian
S0(3) scalar Hamiltonian
expand lU(5)
~" U(6) ~contract
LDM: C2(CS;vr~; v) IBM: L2(CLM;V]~T;{N,O IBM< Man7 to 1 ) Orlglnal LDM Hamil tonian IBM 1 to 1 compactified "LDM Hamiltonian
Logical
Structure of the Liquid Drop Model and the IBM
509
Table I I (contd.) 15. Transition and Linear Combinations Linear Combinations Group expansionother operators of o~erator productsl of operator products contraction of 12c of bT,b. of s@, s,d t , d iallows step back and forth between LDM and IBM (=modified LDM). 16. Method of Sol ution
Solve Schr6dinger- Diagonalize like equation or Matrix diagonalize truncated 5-dimensional oscillator matrix hamiltonian
Construct transforma. tion theory between two representations.