Manifestations of Fermion Dynamical Symmetries in Collective

4 Manifestations of Fermion Dynamical Symmetries in Collective Nuclear Structures Da Hsuan Feng

Downloaded by UNIV OF NEW HAMPSHIRE on June 17, 2014 | http://pubs.acs.org Publication Date: November 14, 1986 | doi: 10.1021/bk-1986-0324.ch004

Department of Physics and Atmospheric Science, Drexel University, Philadelphia, PA 19104

A Fermion dynamical symmetry model which can account for both the low as well as high spin nuclear collective phenomena is presented.

The phenomenological Interacting Boson Model (IBM), introduced about a decade ago by Arima and Iachello , has linked the collective phenomena in the low energy region 1

for nuclei (E < 2 MeV and J < 10, say) with the concept of dynamical symmetries. The profound program of the IBM is remarkably simple and can be understood as follows. By simulating the L=0 and 2 valence coherent nucléon pairs as s and d bosons, the IBM has the highest symmetry U ( 6 ) which, together with the lowest symmetry 0 ( 3 ) B

B

physically demanded by rotational invariance, resulted in three limiting dynamical symmetries: U ( 5 ) , S U ( 3 ) and 0 ( 6 ) . B

B

multi-chain

Using the generalized coherent

B

states of Perelomov and G i l m o r e ( See also the book recently edited by Klauder and 2

3

S k a g e r s t a m ) , one can show that each of these chains depicts a particular type of 4

geometrical motion (vibrational: U ( 5 ) , rotational: S U ( 3 ) and y-soft: 0 ( 6 ) ) . It should B

B

B

5

be mentioned that everyone of these dynamical symmetries of the IBM is realized in nuclear structure. This point is well discussed by the many talks in this symposium, especially the overview talk of Iachello . Hence, the IBM treats all the collective 1

motions on equal footing:

each has its own characteristics and its own set of

eigenstates and more importantly, its own geometrical interpretation. For example, a special characteristic of the 0 ( 6 ) limit is to predict the staggering of the states in the B

γ-band while the S U ( 3 ) limit does not. Clearly, the multi-chain concept of the IBM, B

which is perhaps the most important

lesson

one learns from the model, is a clear

0097-6156/ 86/ 0324-0027506.00/ 0 © 1986 American Chemical Society

In Nuclei Off the Line of Stability; Meyer, R., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

N U C L E I O F F T H EL I N E O F STABILITY

28

departure from the conventional usage of dynamical symmetry of Elliott and its many subsequent

d e v e l o p m e n t s where there is only one 6

dynamical symmetry chain

( S U ( 3 ) or pseudo-SU (3)). F

F

Although the multi-chain dynamical symmetries concept of the IBM is successful in anchoring the various types of collective structures, the full microscopic justification for each chain is still not entirely transparent. Thus, on a purely theoretical level, we deem it important to know whether the multi-chain dynamical symmetries of the IBM are:


(a) merely a fortuituous consequence of the boson assumption for the coherent valence fermion pairs; or (b) inherent in the "raw" fermionic shell structure, i.e. the most general shell model hamiltonian with one and two body interactions, simplified only by the same "physics input" of the IBM. Also, there are experimental reasons as to why these questions require serious consideration. For example, it was recently noted by Casten and von Brentano that nuclei with A«130 (a large number of the Xe and Ba isotopes), just as the previously studied A« 196 system (Pt isotopes) , are very well described by the IBM's 0 ( 6 ) limit. Of course, the boson assumption of the IBM prevents an obvious way to explore the reason (or reasons) why these two mass regions should possess the same dynamical symmetry even though they manifestly have different underlying shell structures. Thus, such experimental observations clearly hasten the necessity to seek answers to the above raised questions. 7

8

B

There is another equally important reason as to why it is necessary to seek the dynamical symmetries from the fermionic point of view. One knows that the physics of nuclei in states of large angular momentum constitutes an important branch of nuclear structure physics as well. Experiments are now routinely done yielding detailed spectroscopy of states in the range of J = 35 ~ 45 (typically Ε is about 10 MeV or so) No comprehensive theory of nuclear structure can ignore these facts - such a theory must adequately address both the low as well as the high spin phenomena. Therefore, to propose a fermionic dynamical symmetry model of collective nuclear structure which is only applicable in the low energy, low spin region must be inadequate by definition. 9

Motivated by these considerations, we have recently proposed a multi-chain fermionic dynamical symmetry model (FDSM) which was developed to specifically address the above raised questions. Our starting point is the Ginocchio SO(8) model ( s i n c e from now on only fermion groups will be mentioned, we shall drop the use of the F superscript to denote them). In our opinion, Ginocchio was the first person to seriously pursue the concept of multi-chain dynamical symmetries from a fermionic viewpoint. The main ingredients of the Ginocchio model can be summarized as follows. If one were to take the fermion pair ( i.e. a a type of operators) with X=0(S) and 2(D) and certain multipole operators (i.e. a a type of operators), both types are constructed from 10

+

+

+


4.

29

Fermion Dynamical Symmetries

FENG

recoupling the single particle j's into pseudo-spin (i ) and pseudo-orbit (k), then we will obtain either the Sp(6) or SO(8) algebras. The SO(8) and Sp(6) algebras have the following group chains : SO(8) 3 SO(5)xSU(2) 3 SO(5) z>SO(3)

(1 a)

DSO(6)

DSO(5)DSO(3)

(1b)

=>SO(7)

z>SO(5) 3 S 0 ( 3 )

(1c)


and Sp(6) => SU(3) 3 SU(2)xSO(3)

3S0(3)

(2a)

=>SO(3)

(2b)

The eigenstates of (1a) and (1b) are identical with the U(5) vibrational and 0 ( 6 ) γ-soft limits of the IBM respectively, while (1c) has no IBM counterpart. The eigensates of (2a) and (2b) are identical with the SU(3) rotational and U(5) vibrational limits of the IBM respectively. There are many interesting features of the Ginocchio model which we do not have the space to cover here. The two most notable ones are: (a) the S 0 ( 8 ) => S 0 ( 6 ) has similar characteristics of the 0 ( 6 ) limit of the IBM. This is the first fermionic (dynamical symmetry) description of the γ-soft collective mode in nuclei. This mode has so far not been understood microscopically, (b) Although there exists a "rotational" chain Sp(6) => SU(3), it was ruled out on the grounds that the most important representation (2N,0) (where Ν is the pair number) is disallowed due to the Pauli principle (more about this later when we get into the description of our model). What is perhaps most lacking in the Ginocchio model is how one might link such algebraic structures to the shell structures, without which the model cannot be used for the study of real nuclei(i.e. toy model) The model which we have developed is called the Fermion Dynamical Symmetry Model ( F D S M ) which is the subject matter of two recent preprints. The FDSM begins with a shell model Hamiltonian in one major valence shell. 1 1

H = Xjejiij + Vp + V

(3)

q

where V ( q j is the pairing (multiple) part of the residual interaction. Our model makes p

three assumptions, (a) The dominant parts of the pairing interaction V and quadrupole. (b) The two body matrix elements of V ( ) p

q

p

are monopole

are proportional to the

degeneracy of each major valence shell, (c) Terms involving the single particle energies and the multipole interactions are approximated so that the Hamiltonian is a


N U C L E I O F F T H E LINE O F STABILITY

30

function of the generators of a tractable Lie algebra. Clearly, the first assumption is a result of the successes of the IBM which we have used as input physics. The second assumption is the simple generalization of the usual pairing assumption in nuclear physics and the third is derived from the belief that a dynamical symmetry in the Hamiltonian corresponds to a certain collective mode of the system, which is perhaps the most important lesson one learns from the IBM. As a result of our assumption (a), we can reclassify all the single particle levels in the major valence shell (see F i g . 1) in terms of the pseudo orbit (k) and pseudo spin (i), thus allowing us to make contact with the Ginocchio model for symmetry classifications. The result is that for the normal parity levels, we will have either the Sp(6) or SO(8)


symmetry and for the "intruder" level, it is an SU(2) algebra, meaning that for the intruder, only monopole pair is allowed in the lowest energy levels.

Thus we have

either Sp(6)xSU(2) or SO(8)xSU(2), depending on which major shell we are referring to.

For example, for the major shell with normal single particle levels s

g

(k=2, i=3/2) and intruder single particle h

7 / 2

(uniquely) SO(8)xSU(2).

1 1 / 2

(

k = 0

- 3 / 2 » 5/2» D

1 / 2

D

» i=11/2) level, the symmetry is

Let me mention in passing that this is fully consistent with

the experimental observation of a large number of stable 0 ( 6 ) nuclei for this mass region . 7

As we have mentioned earlier, although one could construct an Sp(6) 3

SU(3) chain in the Ginocchio model, it was nevertheless discarded because the (2N.0) representation is ruled out for most deformed nuclei due to the Pauli principle ( Ν < Ω/3 where Ω , the full degeneracy of a major physical shell, was nebulously specified). This so-called "major flaw" of the model was also reiterated by H e c h t . 6

However, in our

model, the Pauli restriction applies only for the normal parity levels, i.e. where

and

< Ω^3

are the pair number and the total degeneracy of the normal levels,

respectively. There is no requirement in our model that the total number of pairs in one major physical shell, defined as Ν = NQ + N j must be < Ω/3.

Hence Ν can be

approximately Ω/2 (a condition for most deformed nuclei) while N j £ Ω^β. this means that the Pauli principle squeezes out the remaining N

0

Physically,

pairs to the unique

parity levels. In a forthcoming paper, we will discuss in detail an analysis based on the Nilsson scheme. The result indicates that for the strongly deformed nuclei, N-j is generally < û j / 3 thus eliminating the reason for rejecting the Sp(6) chain. Thus, we now have, within the context of the FDSM, a handle on how the multi-chain dynamical symmetries, all treated on equal footing just as ther IBM, can manifest themselves. As we have emphasized earlier, any comprehensive theory of nuclear structure must adequately address the high spin phenomena. By simulating the coherent fermion pairs as bosons, the IBM essentially ruled out the possibility of discussing high spin p h e n o m e n a without additional ingredients. In our m o d e l , however, this is accomplished in a rather simple fashion. Note that the S 0 ( 8 ) [Sp(6)] and the SU(2)


4.

31


FENG

possess the quantum numbes u and υ

0

where they denote the generalized seniority

(for S and D pairs in the normal levels) and seniority (for the S pairs in the intruder level) respectively. For u = υ = 0, one can find a one to one association with the IBM 0

dynamical symmetries for the FDSM. For example, in the SU(3) limit, for the ground band, one can obtain the usual rigid rotation situation. On the other hand, when either u * 0 or υ φ 0 or both, we have the possibility of broken (either generalized or S) 0


pair(s) which corresponds to the well known phenomena in high spin physics, i.e. rotational alignment, Coriolis antipairing, multiple band crossing and the associated backbendings. The ability to incorporate such features is a consequence of this being a fermion (not boson) model! We present, as an example of this work, the results for the nucleus T h as predicted by the FDSM. It is of course known that for this nucleus, the neutrons occupy the 8th major shell while the protons occupy the 7th major shell. Since our results are based on dynamical symmetries, no interband mixing is introduced. In Fig. 2, the results of the yrast states plotted as a function of J(J+1) (although only the J value is indicated) are given. It is seen very clearly that roughly between J=10 to 12, the band switches from the ground band (with u = υ = 0) to the i = 7/2 pair band ("broken" proton normal 2 3 2

0

pairs with u=2 and υ = 0). Similarly, another band crossing occurs between J=18 to 20. The new band is the i=9/2 pair band ("broken" neutron normal pairs with u=2 and υ = 0). Finally, for a much higher J (24 or so), the new band is associated with the "breaking" of the proton S pair in the j=i=13/2 orbit. These band crossing phenomena is even more transparent in the B(E2)s\ plotted as a function of J , as can be seen from Fig. 3. (The data is joined by the dash line to guide the eye). For the j=13/2 band crossing, there is a sharp drop of the B(E2) values. The physical reason for this to occur is quite clear, whenever there is a band crossing (i.e., the breaking of a pair of coherent fermions), the "core" of the rotor will slow down because of angular momentum conservation. This slowing down of the core (the core is the primary contributor of the B(E2)s) is reflected in the data. For the FDSM, the same physics applies also for the low J band crossing phenomena. Our B(E2) calculations, which ignore band mixing, show very clearly this predicted behavior, i.e., whenever band crossing occurs, there is a reduction of the B(E2) (The dotted line in Fig. 3). The reduction is in fact zero when there is no band mixing and with just a small amount of band mixing, the B(E2) can be fitted rather well. Also, whether these primordial structures can remain depends very much on the strength of band mixing. For the actinides ( T h , U isotopes, for example), such structures prevail. Hence, since the FDSM incorporates fermion degrees of freedom, it gives a better description of high-spin B(E2) values than the algebraic boson model. 0

0

1 2

2 3 2

So far, we have concentrated only on the even-even nucleus. Of course, our model encompass the even-odd as well as odd-odd nuclei as well (by the study of other seniority states).


N U C L E I O F FT H E LINE O F STABILITY


Th

2 3 2

E(MeV)

F i g u r e 1. Dynamical s y m m e t r i e s o f t h e s h e l l m o d e l . be f o u n d i n R e f s . 10-11.

F i g u r e 2. 12.

Yrast states of

Th.

D e t a i l s may

The d a t a a r e t a k e n f r o m R e f .



No

1

2

3

4

η

0

1

2

3

5 4

4

1

0

1

0

i

1/2

1/2

3/2

7/2

3/2

9/2

3/2 11/2

1/2

Pi/2

S

9/2

s h 1/2 11/2

P3/2

d

3/2

d

5/2

f

CONFIGURATION

1

1/2

f

7/2

P

1/2

9

b

r,

P3/2

d

3/2

5/2

d

5/2

5

1

0

0

S

5

5

2

8

7

6

3

k

1

1/2

P

1/2

6

6

6

7

0

1

1

0

7/2 13/2

f

5/2 'l3/2

P3/2 7/2 f

h

9/2

3/2

9/2 15/2

1/2

9

d

3/2

9

d

5/2

S

7/2 15/2 J

9/2 'l1/2

9?/2

SYM


FENG

G

6

G

8

G

3

G

6

G

8

G

3

G

8

G

6

G

6

0

0

0

0

5

6

7

8

Qi

1

3

6

4

6

10

15

21

Ω

1

3

6

4

11

16

22

29

126

184

il

2

G = 6

G

3

8

20

50

28

(SpgXS0 ) χ ( S & 2 3

X

82

S6> ) 3

= (SU3X SOg) x ( S ^ x S6> ) 2

3

G = (SOgX SO3) x ( S ^ x S6>) 8

2

3

F i g u r e 3. The t h e o r e t i c a l are taken from Ref. 12.

and e x p e r i m e n t a l

B(E2) v a l u e s .

The d a t a


NUCLEI O F F T H E LINE O F STABILITY

34

In conclusion, just as the IBM, the FDSM contains, for each low energy collective mode, a dynamical symmetry. For no broken pairs, some of the FDSM symmetries correspond to those experimentally known and studied previouly by the IBM. Thus all the IBM dynamical symmetries are recovered. In addition, as a natural consequence of the Hamiltonian, the model describes also the coupling of unpaired particles to such modes. Furthermore, since the model is fully microscopic, its parameters are calculable from effective nucleon-nucleon interactions. The uncanny resemblance of these preliminary results to well-established phenomenology leads us to speculate that fermion dynamical symmetries in nuclear structure may be far more pervasive than has commonly been supposed.


Acknowledgments

This work is supported by the National Science Foundation. It is indeed a pleasure for me to thank my colleagues and collaborators in China and the US in this work. They are C h e n g - L i W u of Jilin University, X u a n - G e n C h e n of the Nanjing Military College, J i n - Q u a n C h e n of Nanjing University and M i c h a e l W . G u i d r y of the University of Tennessee, all of whom I have known and learned from for a number of years. It must also be mentioned that during the development of this work, J o e G i n o c c h i o of the Los Alamos Scientific Laboratory has played the role of a "knowledge provider" and critical evaluator of our ideas. Finally, I must thank the two organizers of this symposium, Dr. R. A. Meyer and Prof. Daeg Brenner, for inviting me to participate in my first (and hopefully not last) ACS meeting.

References 1. A. Arima and F. Iachello, see Professor Iachello's contribution in this volume. 2. A. M. Perelomov, Commun. Math. Phys. 26, 222 (1972). 3. F. T. Arecchi, E. Courtens, R. Gilmore and H. Thomas, Phys. Rev. A6, 2211(1972); R. Gilmore, Rev. Mex. de Fisica 23, 143 (1974). 4. J. N. Ginocchio and M. W. Kirson, Phys. Rev. Lett. 44, 1744(1980); A. E. L Dieperink, O. Scholten and F. Iachello, Phys. Rev. Lett 44, 1747(1980); D. H. Feng, R. Gilmore and S. R. Deans, Phys. Rev. C23, 1254(1981). 5. COHERENT STATES, ed. by J. R. Klauder and B.-S. Skagerstam, World Scientific, Singapore, 1985. 6. J. P. Elliott, Proc. Roy. Soc. A245, 128(1958); A245, 562(1958); Κ. T. Hecht, Lectures delivered at the VIIIth symposium on Nuclear Physics, Oaxtepec, Mexico, Jan. 1985 and Phys. Rev. C (to be published). J. P. Draayer, Lectures delivered at the VIIIth symposium on Nuclear Physics, Oaxtepec, Mexico, Jan. 1985. 7. R. F. Casten and P. von Brentano, Phys. Lett. B152 , 22(1985). 8. J. A. Cizewski et al., Phys. Rev. Lett. 40, 167(1980). 9. For a comprehensive discussion, see R. Bengtsson and J. D. Garrett in COLLECTIVE PHENOMENA IN ATOMIC NUCLEI, edited by T. Engeland, J. Rekstad and J. S. Vaagen (World Scientific, Singapore, 1985). 10.J. N. Ginocchio, Ann. of Phys. 126, 234(1980). 11.C.-L. Wu, D. H Feng, X.-G. Chen, J.-Q. Chen and M. W. Guidry, Preprint, 1985. Submitted to Phys. Lett. and Ann. of Phys. 12. This is the so-called "GSI catastrophe" which shows that as a function of J, the IBM predictions of the B(E2) values will eventually drop off. See H. Ower, Ph.D. thesis, University of Frankfurt, 1980. RECEIVED May 2, 1986


Manifestations of Fermion Dynamical Symmetries in Collective

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