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THE ANTISYMMETRIZED WEAK COUPLING SCHEME IN SHELL MODEL CALCULATIONS
S. Wong, A. Zuker
To cite this version:
S. Wong, A. Zuker. THE ANTISYMMETRIZED WEAK COUPLING SCHEME IN SHELL MODEL CALCULATIONS. Journal de Physique Colloques, 1971, 32 (C6), pp.C6-51-C6-55.
�10.1051/jphyscol:1971607�. �jpa-00214825�
JOURNAL DE PHYSIQUE Colloque C6, supplkment au no 11-12, Tome 32, Novembre-Dkcembre 1971, page C6-51
THE ANTISYMMETRIZED WEAK COUPLING SCHEME IN SHELL MODEL CALCULATIONS (*)
S. K. M. WONG and A. P. ZUKER Service de Physique ThCorique, CEN, Saclay, France
Rbume. - Nous presentons des calculs de modele en couches a partir d'une base antisymetrisee de couplage faible. Nous obtenons d'importantes reductions clans la dimension des matrices et une grande simplification dans I'interpretation des resultats. Le schema se prCte a une cornparaison presque directe avec les donnees exp6rimentales et nous en discutons quelques consequences.
Abstract. - Shell model calculations are carried out in an antisymmetrized weak coupling basis.
Dramatic reductions in dimensionality and great gains of physical insight are achieved, when compared with conventional results. The scheme is adapted to almost direct comparison with experiment and some implications are discussed.
Although the idea of weak coupling in nuclear physics is quite old it is only relatively recently that Arima, Horiuchi and Sebe [2] showed it was possible to interpret several low lying states in the oxygen region, as the result of coupling the ( ~ d ) ~ band in
"Ne to different p shell nuclei. This suggestion was borne out experimentally by a-transfer experiments (see Betghe's contribution to this Conference) and its theoretical soundness demonstrated by shell model calculations by one of us [3] (A. Z.).
The results of [3] contained a clear indication that the idea of weak coupling, though basically sound, was not as simple as one might have expected and that some care was necessary in the interpretation of what was meant by weak coupling.
Taking as an example the conventional 2 particle states in 18F we can illustrate the nature of the pro- blem : according to [3] the ground and other low lying, states of this nucleus are not simply <( 2 particle )) but contain strong 4p-2h admixtures. Fortunately, their presence can be reconciled with weak coupling, once we notice that 160 contains also strong 2p-2h components and that the 4p-2h terms are then a consequence of the schematic relation :
Weak coupling remains valid although the building blocks can no longer taken to be simple particle-hole excitations but involve true physical states and it becomes necessary to cope with the exclusion prin- ciple.
The question that arises naturally is whether we are
(*) Except for a few remarks and sonle more detailed dis- cussions the contents of this talk follow closely ref. [I], where the reader may find, on the other hand, some comments omitted here for the sake of brevity.
in the presence of a fortunate accident confined to the neighbourhood of closed shells, or whether it is a more general feature of nuclear wave functions independent of the existence of particle-hole configura- tions. The shell model results seem to indicate that the second alternative holds since weak coupling charac- teristics are preserved, in spite of the fact thatcouplings do not take place solely between states in inequivalent spaces. The reinterpretation of a <( 2 particle )) state made in the schematic equation above does not rely on having only hole orbitals in 1 6 0 .
We can then try to be more ambitious and decide to do away completely with particles and holes and simply try to couple states in equivalent spaces in an attempt to generate new ones. If there is a state in 160 that is the result of coupling 20Ne (a 4p state) to 12C (a 4h state) why not try to explain 24Mg (8 par- ticles) as 20Ne x 20Ne, or 20Ne itself as 18F x I8F ? Since the Rochester-Oak-Ridge shell model pro- gram [4] allows to construct eigenstates for most of the (sd) shell nuclei it seems natural to concentrate on this region to be able to check the validity of the schemes we-wish to explore and from now on our labels 18F, 20Ne,Iet~... will refer to ( ~ d ) ~ , ( ~ d ) ~ , etc ... ; the 1 6 0 core being tacitly assumed or ignored accord- ing to context.
Before proceeding to the specific calculations let us try to ask what would be a weak coupling theory and what advantages could be derived from it.
The only case'we know of rigorous weak coupling, in the sense we have adopted for this word, is the seniority scheme, which provides exact states by coupl- ing pairs of particles having J = 0, T = 1. More generally we can study sets of pairs (or of any other state 0perators;we-wish to try). Consider for example the set of eigenstates of 18F, denoted by { F ) , and suppose we restrict our attention to a small subset
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1971607
C6-52 S . K. M. WONG AND A. P. ZUKER
{ F ), c { F 1,. A weak coupling basis will be constructed by forming the (tensor) products of the form
For this basis to have any interest it must be closed under the operation of the Hamiltonian H :
H { F i x Fj x F k . . . ) , + { F , x F j x F, ...I,.
Ignoring for a moment the tensor couplings it is easy to see what this latter condition implies. Consider
and notice that the double commutator [[H, F i ] , Fj]
is again a state operator belonging to the set { F x F),.
Therefore we must require
[ [ H , FiI, F j ] E { F x F j, for Fi,j E { F ) , .
We do not really hope to have this condition satisfied for realistic cases. But we can look for cases where it holds approximately and this involves being able to define properly what is a minimal set s and under what conditions certain small operators can be neglected so that we are in the presence of some approximate (dynamical) symmetry group.
At the moment we are working on this problem encouraged by the numerical results of our first calculations which indicate such minimal sets and their associated group structures may exist.
Concerning the advantages of the weak coupling schemes, little remains to be said once we have shown they are equivalent to the existence of symmetries.
They will lead to drastic reduction in dimensionalities and to great simplicity in the description of wavefunc- tions as we will show now.
Our main theme is that it must be advantageous to couple eigenstates of smaller systems to generate a basis for larger ones.
To implement the program wc start with the two particle states of 18F calculated in the single particle space defined in "0 by experiment and using some good interaction [5].
We then restrict ourselves to the low lying ones (say the first of each spin and isospin) and recouple them to themselves to get the 4 particle basis for
"Ne. Once the secular problem for 20Ne in this basis is solved we can use the new eigenvectors as building blocks for another problem. Schematically : we are constructing a basis @(n, i) for n particles (i stands for JT and other quantum numbers) by coupling eigenvectors $(n,, i,) and $(n2, i2). Our solutions $(n, k) will be of the form
+(n, h ) = z: @(n, i) = 1 r [ +(n,, i , ) x $(n2, i2) ,
I i 1 i 7
n = n, + 1 1 , .
It is understootl that we deal with state operators in second quantization and that angular momentum
coupling is always assumed. We refer to [I] for further technical details and concentrate on the results shown in the tables which we will now explain.
The basic vectors defined above are not orthonormal and need not even be linearly independent. It means that the coefficients cr do not contain all the neces- sary information since the norm matrix
Therefore we have introduced the following quantities :
or in words : the partial overlaps (w) and contribu- tions to the total energy (h), from component @(n, i ) to the total wavefunction. The overlaps o with the exact wavefunction are also given under the columns we. It is clear that
Z
w i a, = 1 and 1 hi ori = EcOther quantities given are the following : total cal- culated energy EC, total exact energy E,, overlap between exact and calculated wavefunction 12, dimen- sionality for the weak coupling basis d, dimensionality in the shell model basis D. The extra quantum number x denotes the order in which eigenvalues for given JT appear.
A superficial look at the tables will reveal a number of interesting things : with small dimensionalities we can get arbitrarily close to the exact results. Further- more. the results are quite transparent since the overlaps w arc directly observable spectroscopic amplitudes.
More careful inspection leads to even more interest- ing discoveries : all the states studied seem t o be dominated by w r y few components and often we are quite close to what we would expect in rigorous weak coupling (only 1 component). Furthermore seniority seems to be quite a reasonable scheme (quite unexpect- ed for deformed nuclei) ; it is clear that J = 0 T = 1 pairs are everywhere important, followed by J = I T = 0. We seem to be not too far from the possibility of defining a minimal space and search for approximate dynamical symmetries as mentioned above.
As a consequence of this pairing behaviour we can see that strong ground state transitions should be expectcd in two proton transfer reactions ((3He, n) or (160, 14C)).
Probably the most interesting results obtain in 24Mg for which we have used the two approximations ''Ne x 2 0 ~ e and I8F x 22Na. From our point of view both possibilities are open and it is interesting to explore them. Howcver there is another good reason to tackle the former coupling, since Arima and Gillet [6] have proposed a (( roton )) scheme which recommends using e~genstates of the T = 0, 4 particle system as building blocks for calculations. Their
THE ANTISYMMETRIZED WEAK COUPLING SCHEME IN SHELL MODEL CALCULATIONS C6-53
reasons for this choice are more specific than our vague and general proposal of coupling eigenstates to eigenstates without further explanation.
The results of the calculations are rather surprising ; both approximations are equally good. In the strict sense of weak coupling (only one component)
"F x 22Na is slightly superior, specially if we notice that the entry 2 0 ~ e ( 0 0 1 ) x 20Ne(001) of the Mg table is not really what it says but rather the result of a prediagonalization involving the 4 lowest 0' 0 states of 20Ne. The outcome of this operation is an improve- ment of o, from 0.85 to 0.91 and reveals a softening of the original Ne(001) clusters. In any case there is not much to choose between the two schemes and the consequences of this equivalence may be quite interest- ing. If we imagine an experiment that could show the break up of the ( ~ d ) ~ configurations (not of the whole 24Mg ; here the difference between 160 x (sd)' and ( ~ d ) ~ becomes important) we would find that, within kinematics and other details coming from reaction mechanisms, the two channels
Notice the possible implications of such a state of affairs ; different asymptotic behaviour is not related to any particular form of clustering inside the nucleus.
Referring to the figure in A. Bromley's talk showing the break up (fission ?) of 24Mg in "Ne + a, 'Be + 160,
12C + 12C, etc ..., we could extrapolate from our
(( simple )) example and suggest that inside the nucleus there may be little or no difference between the possible forms of clustering and that the choice between 20Ne + a and I2C + 12C as an o ~ ~ t g o i n g channel is purely kinematic.
Responsible for such a situation is obviously the exclusion principle. It should be pointed out however that our calculations indicate that the goodness of the roton scheme is a direct consequence, in the case of 24Mg, of the goodness of the pair )) scheme as can be checked by the follo\ving sequence of equalities (or near equalities)
24Mg -+ "F + 2 2 ~ a = (sd)' -+ (sd) + (sd)
/
'"?vlg -+ " ~ e + ON^-
(sd)'-
(sd) 4 + (sdy ,should be equally favored. The physical realization of such a thought experiment would probably be
z0Ne(JTx) ~ ( J I T I X I , J z T2 XZ) [ I R ~ ( J , TI XI) X 18F(J2 T~x~)].IT ; the meaning of the symbols is given in the text
S. K . M. WONG A N D A . P. ZUKER
Comparing with the values given in the tables for a redundancy of 2 out 4 states for 20Ne x 20Ne.
and b it is verified that these coefficients are just what It could be thought that this is a consequence of the they should be to make the roton scheme work. low orbital degeneracies in our problem and that for The converse would not be necessarily true, since one heavier nuclei such situation would be corrected and would imagine rotons to be good approximations that rotons would start behaving as good <( bosons B.
while pairs were poor ones. Unfortunately this happy situation is not likely to A striking consequence of the Pauli principle is the evolve. The reason for our doubt is the remark that
T H E ANTISYMMETRIZED WEAK COUPLlNG SCHEME I N SHELL MODEL CALCULATIONS C6-55
the shell model basis for (sd)', J = 0 T = 0, has 325 states and it should be large enough to accomodate the 4 basic components of *'Ne x 20Ne. If they
(( choose )> to become non-linearly independent, there must be a deeper reason.
It is a remarkable fact that operators that d o not obey boson commutation rules and d o not stand any chance of obeying them should still share with bosons two striking properties :
1) the product of two such state operators is an
eigenvector (the tables show that this happens quite often, to a good approximation).
2) the energy of the composite system is the sum of the energies of the original blocks (true only for the roton case, and one of the starting points in the discussions of ref. [6]).
Although the numerical work involved is still quite hard and the theoretical ideas need much clarification we feel that AWC (antisymmetrized weak coupling) is a very promising approach which has produced some interesting results and is quite worth pursuing.
References
[I] WONG (S. K. M.) and ZUKER (A. P.), Phys. Letters, WONG (S. S. M.). in Advances in Nuclear Phy- 1971, 36B, 437. sics, vol. 3, Ed. by M. Baranger and E. Vogt [2] ARIMA (A.), HORIUCHI (H.) and SEBE (T.), Phys. Lef- (Plenum Press, New York, 1969).
fers, 1967, 24B, 129. [5] KAHANA (S.), LEE (H. C.) and SCOTT ( C . I.), Phys. Rev., [3] ZUKER (A. P.), Phys. Rev. Letters, 1969, 23, 983. 1969, 185, 1378.
[6] ARIMA (A.) and GILLET (V.), to be published in the [4] FRENCH (J. B.), HALBERT (E.), MC GRORY (J. B.) and De Shalit mcmorial volume in Annals of Physics.
