Chaotic motion and collective nuclear rotation

Volume 167B, number 4

PHYSICS LETTERS

20 February 1986

CHAOTIC MOTION AND COLLECTIVE NUCLEAR ROTATION T.H. S E L I G M A N Insntuto de Ftstca, Autonomous University of Mextco, Ctudad Unwersttarta, Alvaro Obregbn, 04510 Mbxtco, DF, Mextco

J.J.M. V E R B A A R S C H O T 1 and H.A. W E I D E N M ( A L L E R Max-Planck-lnstttut fur Kernphystk, D-6900 Heidelberg, Fed Rep Germany

Received 30 October 1985

The regular and chaohc motion m the classtcal and quantal versions of a model hamlltonlan w~th two degrees of freedom are lnvesttgated Tins model contains a parameter which is identified w~th a conserved quantum number, the total spin In particular, transitions between states dlffenng m spin by one unit are stu&ed. The transmon is strongly collectwe for regular motion, and collectivity ~s destroyed with increasing stochastloty of the model

1. Introduction. Nuclear fluctuation properties like the distribution of level spacings of states of the same spin and parity, or the distribution of partial widths, display chaotic behaviour. There is evidence that this behaviour extends all the way down to the vicinity of the ground state [1 ]. On the other hand, many nuclear states show striking regularities of a single-particle, or of a collective, type. As an example for the latter, we recall the strong enhancement of B(E2) values characteristic of a rotational band. The interplay between regular and chaotic dynamics is one of the fascinating aspects of the nucleus, and is not fully explored yet. To shed further light on this interplay, we investigate in this letter the connection between regular rotational motion and chaotic behaviour. We do so in terms of a simple dynamical model involving two degrees of freedom. In contradistinction to standard models of classical and quantal chaotic motion [2], our model allows for the presence o f an additional parameter. This parameter appears in the model in a way in which a conserved quantum number would appear, and this causes us to identify the parameter with the total spin h J o f the system. Choosing the parameters of the model in such a way that for small J (J = 1 or

Present address Department of Physics, Umverslty of Illinois, Urbana, IL 61801, USA 0 3 7 0 - 2 6 9 3 / 8 6 / $ 0 3 . 5 0 © Elsevier Science Publishers B.V.

(North-Honand Physics Publishing Division)

so) the motion is chaotic, we study the influence of a change of J on the classical and quantal dynamics o f the system. In particular, we investigate the following questions. (a) Does a substantial increase in J make the system more regular? We expect this to be the case, at least near the ground state, since that part of the interaction which is responsible for chaotic motion becomes less important compared to the rotational terms as J increases. (b) Does an increase in regularity lead to an enhancement o f collectivity in the intraband transition strength? We expect this to be the case: Increasing regularity should funnel more of the transition strength into the collective intraband transition. (c) We expect, for states of fixed spin and parity, that chaoticity increases with excitation energy. Does this increase destroy the collectivity of the intraband transition strength? Does the transition strength become more and more spread out as the excitation energy grows? 2. The m o d e l hamiltonian. Scaling properties. To study these questions, we consider a system with two degrees of freedom and a hamiltonian H given by 1

2

2

t

H=~(pl+P2)+~(L

2 +~ 22 aikri rk . t
2

1 _ ~ ) r 1 2 + l ~ ( L22 _ 2~ ) r - 2

(1) 365

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Here, p/stands either for the classical momentum variable conjugate to r]/> 0, or for i~/Sr] withj = 1,2. The quantities r], Pl, L/, alk (both j and k take values 1 or 2) are dimensionless so that H is dimensionless, too. (The physical energy scale would be obtained by multiplying H by a suitable constant of dimension energy; this is not done in what follows.) We consider both classical motion governed by Hamilton's equations, and the solutions of the time-independent Schrodinger equation involving H. The quantities L1 and L2 are dimensionless parameters which we later identify with conserved quantum numbers. The appearance of the terms L 2 - ~ rather than L 2, a little awkward for the discussion of scaling properties, is caused by the desire to tackle the present problem by a minor modification of the computer programme used for the calculations in ref. [2]. ForL 2 - ~ = 0 =L22 - I, the hamiltonian (1) contains a potential which is a homogeneous polynomial m rl and r2. Potentials in this class were used in ref. [2] to study chaotic motion in classical and quantum physics. It was found that the coefficients az/can be chosen in such a way that the motion is either completely chaotic for all energies, or completely regular for all energies, or partly chaotic. Because of a scaling property, the fraction of the energy surface in phase space which is ffUed by chaotic trajectories, is independent of energy. While in ref. [2] the quantities r/ and pj (j = 1,2) were interpreted as the position and momentum variables, respectively, of two one-dimensional particles, we now read them as the radial coordinates and associated conjugate momenta, respectively, of two particles moving in three dimensions. This gives us the freedom to Introduce the centrifugal terms ~(L 2 - ~) rT 2, and to study the dependence of the degree of chaoticity on the angular momenta Li. The price we pay is that the polynomial interaction Y,i~k%kr2r 2 which has a plausible form when the r/. are position variables in one dimension, attains somewhat strange features. We have chosen the form (1) for H because the method of solution used in ref. [2] for the quantum problem applies without major modification also to the hamiltonian (1). We have no reason to doubt, however, that our results are generic and hold as well for other hamiltonians with a more "reasonable" interaction. For simplicity we put L1 = L2 = J and refer t o J as the angular momentum, or spin. 366

20 February 1986

The classical hamiltonian (1) possesses a scaling property which greatly simplifies the presentation and discussion of our results. For J ~> 1, we write j2 = 2,2 + ¼with 2, > 0 and consider for 3' > 0 the scaling transformation 2, ~ 3'32,, rI ~ 7r/, p~ ~ 72p/ (j = 1,2).

(2)

Under the transformation (2), the classical hamfltonian (1) scales as H ~ @H. Put differently, H2`-4/3 is invariant. Therefore, properties of H pertaining to different values of 2, are related by a scaling of the energy variable. For instance, let/2(E, )~) with 0 ~
wsc(e,x)=

f6,2 fdpl f dp20(E-HeI(h)).

Here, Hcl stands for the classical expression. (We put h = 1 .) Scaling yields Nsc(E, X') = (2`,/),)2 X Nsc(E(X/2`')4/3, 2`). The additional factor (2`'/X)2 absent in the scaling relation for the phase-space fraction/2(E, X) considered above - arises from the integration over phase space. A similar relation follows by differentiation for the average level spacing d. We note that as 2` grows (X' > X), the energy must be scaled up. This corresponds to the Yrast line in nuclei: The ground-state energy increases with increasing X. The scaling relation just obtained shows that for E fixed and J increasing, the angular momentum terms } j2r~-2 in eq. (1) tend to move more and more levels to the regular regime: The centrifugal forces stabilise the system against chaoticity. While we expect such a feature on general grounds, we see that for the simple model (1) it follows already from scaling. Our scaling -

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PHYSICS LETTERS

arguments are only based on the fact that the potential is homogeneous in r l , r2. Similar results would obviously follow for other homogeneous forms.

level spectrum obtained by diagonalisation [2] (the method used in this paper) with the semiclassical formula (3). For J = 10 we found excellent agreement for the first 200 levels (corresponding to E ~ 200). Tills suggests that these levels are reliable and can be used for a statistical analysis. A similar conclusion relating to the eigenvectors is reached with the help of a completeness relation as explained below. Varying J from zero to 30, and always using the first 200 eigenvalues (grouped into subgroups encompassing levels 1 - 5 0 or levels 5 0 - 1 5 0 or levels 1 0 0 - 2 0 0 , etc.), we have evaluated both the nearestneighbour spacing distribution and the A3 statistic for each subgroup. In fig. 1 we show the nearest-neighbour spacing distribution (upper part) and the A3 statistic (lower part) for J = 1, J = 10, and J = 30. Each plot is constructed from the eigenvalues labeUed 50 to 150. According to the behaviour o f # described above, this set of eigenvalues lies in the classically chaotic region for J = 1, in a transitional region for J = 10, and in a non-chaotic region ~ ~ 0.1) for J = 30. In spite o f this distinction, the difference in the A3 statistic in the three cases is not very striking. This is all the more amazing since plots o f the quantal step function N(E, dO show very different features in the three cases:

3. Results. Using the numerical methods described in ref. [2], we have calculated classical trajectories, and quantal energy spectra, for various choices o f the parameters ai/, and for a set o f integer values o f J. Here, we report on results obtained for Ctll = 0.4, or22 = 2.0 and a12 = - 1 . 6 . F o r A = 0, this choice o f parameters defines [2] a problem with completely > 0.99) chaotic classical motion. For X 4= 0, the ratio/a introduced above was calculated as follows. Using a Monte Carlo procedure, we determined the (positive) Lyapunov coefficient for a set of points in phase space chosen randomly according to the measure 6 ( E - H ) drl dr2 dpl @2. The cumulative histogram for the distribution o f the values o f this coefficient yielded #. To quote an example: F o r J = 10, the function/a(E, dO rises steeply with E from a value around 0.1 a t e = 50 to a value around 0.9 a t e = 70. F o r e 2 125, we have # > 0.99, and we expect complete quantum chaoticity there. To test for numerical reliability in the quantum case, we applied the tests described in ref. [2]. In particular, we compared the

10

1.5 p (x)

20

~x=s/d 10 20

30

20 February 1986

3=1

30

1.0

20

J=lO

3.0 J=30

10 0.5

, ~

8 6

i

i

J

,

,

i

r

~

i

J'=30'

J=l

A3(L] 4 2 0

0

' ; ' 2 ' '1 '

4

16 20

24

i

1'2 1; 2'o

I

/

I

4

8

12

I

I

16 20

I 21,

L

Fig. 1. Nearest-neighbour spacing distributions versus x = s/d (upper part) and Aa-statistics versus L (lower part) for the eigenvalues labelled 50 to 150 and for J = 1, J = 10, and J = 30, respecUvely. In the hamiltonlan (1), the parameters ui/were chosen as ¢'11 = 0.4, c,22 = 2.0 and ~12 = -1.6. The full curves correspond to the GOE prediction (complete chaoticity) and to the Poisson distribution of eigenvalues (complete regularity). The quantity s is the actual level spacing, d the mean. 367

Volume 167B, number 4

PHYSICS LETTERS

20 February 1986

N(E, 30) is nearly identical, at least for the first 80 levels or so, to the function N(E) evaluated for two harmonic oscillators with a frequency ratio of approximately three, and an anharmonic coupling. No such similarity is apparent f o r N ( E , 1) a n d N ( E , 10). This "oscillator anomaly" is obviously a consequence of the angular-momentum terms in eq. (1) which have the tendency to make the spectrum regular. We take the appearance of this anomaly, and the lack of evidence for it in A3, as a warning signal that A3 alone is not always a reliable measure for the attainment of chaoticlty. In the nearest-neighbour spacing distribution of fig. 1, the increasing regularity of the spectra with increasingJ is better visible than in the A3 statistic. The peak structure in the J = 30 distribution is indicative of the anomaly. It is clear, however, that no definitive conclusions can be drawn from either A 3 o r the nearestneighbour spacing distributions on the chaoticity of the spectra. We turn to the "collectivity" of transition matrix elements connecting states which differ by one unit in J. Specifically, we considered the matrix elements

MJv = (.1._ 1 #lrlr21J v) .

(3)

Here, p and v are running indices which label the levels of fixed angular momentum consecutively, starting with the ground state. If the transition considered in eq. (3) were strongly collective, we would expect one or a few matrix elements to be much bigger in magnitude than the others. As a quantitative test of collectivity, we have calculated the "participation ratios" (a term borrowed from the theory of Anderson localisation) defined as

~=(~lM~uvlj 2 ) 2/ ~ l M J v l 4.

(4)

With one matrix element much bigger than the rest, we have/~v "~ 1 while ~ >> 1 when the transition strength is spread over many final states p. As a test for the reliability o f the numerical evaluation o f PvJ, we have compared the numerical result for Y'u IMJv 12 with the numerical evaluation of the sum-rule expression ( J - 1 vlr2r21J - 1 v) and found satisfactory agreement. In fig. 2 we display the values of/~v versus v = 1 ..... 80 f o r J = 2 , J = 11 a n d J = 31. It is verystriking to see how the wide scatter of/~v contrasts with the more regular pattern for p~l and the very regular pattern for 368

Pv 15

,.1=2

10 5 I

i

i

i

i

i

i

i

,1=11

i

8

J

i

p

i

,

J=31

6

2 00

'2'o'

40''

6'0

80'

~ V

Fig. 2. Partacipataon ratios P J versus ~ for J = 2, J = 11, J = 31, and for v = 1, ..., 80. We note the reduced scale for the J= 2 distribution which optically diminishes the effect.

/o31. F o r P 31 and v ~< 80, not a single value exceeds 2.2! We conclude that a growing value of J has a very strong tendency to force the system towards increased collectivity. The lack of collectivity for small J is directly associated with chaotic motion. Indeed, we have also evaluated p2 for the hamiltonian (1) with a12 = 0 (a separable and therefore completely regular problem). In this case, we found p2 ~< 4 for v = 1, ..., 80. As a further test of disorder, we have checked whether the M2v for which P~v is bigger than ten have a gaussian distribution from the first three moments of 2 2, the spectral distribution of lM2v 12, i.e. 2~E~p Ig~vl p ~ 0,1,2. We have then scaled the matrix elements M~v accordingly. The distribution of the scaled matrix elements is shown as the histogram in fig. 3. The solid line is a gaussian calculated from the first two moments (p = 1,2) of the scaled M~v distribution. We see that the agreement is very good, which lends further support to the statement that large values o f p 2 are associated with stochastic behaviour.

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PHYSICS LETTERS

~3 o

'62 E 1 z -3

-2

-1 0 1 2 Matrix element

3

Fig. 3. Histogram of the distribution after scaling of the matrix elements Mmyfor which P2v/> 10. All cases with v ~<80 were sampled. For fixed v, we took into account only those cases for which the smooth curve reproducing the mean energy dependence of IM~vl was bigger than 5 percent of its maximum value. The sohd line is a gaussmn. Position and width were calculated from the moments of the scaled M2uvdistribution.

4. Conclusions. Before answering the questions raised in the introduction, we observe that as we change J by one unit, the relative change of the hamiltonian (1) is less when J is large than when it is small. We therefore expect trivially that the participation ratio decreases with increasing J. The only nontrivial change of this ratio is due to a change in the chaotic properties of the system. F o r this reason we first answer question (c). Although the evidence is not too strong, we beheve we can answer it with a qualified "yes". Looking at the distribution o f the participation ratios 11 Pv , we are tempted to identify the onset o f large fluctuations o f p 11 around v ~- 40 with the onset o f chaotic behaviour. This is consistent with the behaviour o f ~t(E) f o r J = 10: As v increases from around 5 to 35, /a(E) grows from ~10% to ~90%. A further increase o f v from 35 to 100 causes/a(E) to grow to ~99%. Since for a fLxed level number the chaoticity decreases with increasing J , the answer to question (a) is also

20 February 1986

yes. The participation ratios decrease drastically, and the transition become much more collective. The answer to question (b) is also affirmative. An increase of collectivity is found as the parameter a12 in eq. (1) is put equal to zero. It has to be remembered, that the system becomes much stiffer as a12 is turned off, and therefore the system changes less as J is increased by one unit. It would, of course, be interesting to use our results as the basis for speculations on nuclear coUectlve motion. It would appear that nuclear collective motion would, for fixed spin, have a tendency to be destroyed by increasing energy. Conversely, it would appear that for fixed excitation energy and increasing spin, collectivity is enhanced. Such qualitative conclusions are also suggested by an analysis of new experimental data reported in ref. [3]. We feel, however, that such speculations would be premature. Rather, we view the present paper as a first step in a direction of research which investigates the influence of chaotic motion on conserved quantum numbers different from the energy, and on the collectivity associated with them. We believe that both further model studies o f the type described above, and experimental investigations (of the "r-decay o f high-spin states and of the question whether rotational bands in nuclei exist an MeV or so above the Yrast line) are necessary for a deeper understanding o f these problems. The authors are grateful to O. Bohagas for a reading of the manuscript.

References [1 ] T.A. Brody, J. Flores, J.B. French, P.A. MeUo, A. Pandey and S.S.M. Wong, Rev. Mod. Phys. 53 (1981) 385. [2] T.H. Seligman, J.J.M. Verbaarschot and M.R. Ztrnbauer, J. Phys. A, to be published. [3] A. Abul-Magd and H.A. Weldenmiiller, Phys. Lett. 162B (1985) 223.

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