Collectivity Embedded in Complex Spectra:

Example of Nuclear Double-Charge Exchange Modes

S. Dro_zd_z1;2, S. Nishizaki3, J. Speth1 and M. W�ojcik1;21Institut f�ur Kernphysik, Forschungszentrum J�ulich,

D-52425 J�ulich, Germany2 Institute of Nuclear Physics, PL-31-342 Krak�ow, Poland

3 Faculty of Humanities and Social Sciences, Iwate University,

Ueda 3-18-34, Morioka 020, Japan

(March 20, 1997)

The mechanism of collectivity coexisting with chaos is in-

vestigated on the quantum level. The complex spectra are

represented in the basis of two-particle two-hole states de-

scribing the nuclear double-charge exchange modes in 48Ca.

An example of J� = 0� excitations shows that the residual

interaction, which generically implies chaotic behavior, under

certain speci�c and well identi�ed conditions may create tran-

sitions stronger than those corresponding to the pure mean-

�eld picture. Therefore, for this type of excitations such an

e�ect is not generic and in most cases the strength of tran-

sitions is likely to take much lower values, even close to the

Porter-Thomas distributed.

PACS numbers: 05.45.+b, 24.30.Cz, 24.60.Lz

Chaos is essentially a generic property of complexsystems such as atomic nuclei and this �nds evidencein a broad applicability of the random matrix theory(RMT) [1] to discribe level uctuations. Another char-acteristics connected with complexity, even more inter-esting and important from the practical point of view, iscollectivity. It means a cooperation, and thus the cou-pling, between the di�erent degrees of freedom in orderto generate a coherent signal in response to an exter-nal perturbation. Consequently, even though the realcollectivity implies a highly ordered behavior it involvese�ects beyond the mean �eld { the most regular part ofthe nuclear Hamiltonian [2]. At the same time the e�ectsbeyond the mean �eld are responsible for the uctuationproperties characteristic of the Gaussian orthogonal en-semble (GOE) of random matrices [3].Local level uctuations characteristic of GOE ap-

pear [4] to take place for the nuclear Hamiltonian actingalready in the space of two-particle { two-hole (2p2h)states and this is a crucial element for an appropriatedescription of the giant resonance decay properties [5].The ordinary giant resonances are, however, excited byone-body operators which directly probe the 1p1h com-ponents of the nuclear wave function. The 2p2h statesonly form the background which determines a decay-law.There exist, however, very interesting physical processes,represented by two-body external operators, which di-rectly couple the ground state to the space of 2p2h states.In view of the above mentioned local GOE uctuationsgiving evidence for a signi�cant amount of chaotic dy-namics already in the 2p2h space, the question of a pos-

sible coherent response under such conditions is a veryintriguing one and of interest not only for many branchesof physics but also for biological systems [6].Among various nuclear excitation modes which can be

considered in this context the double charge exchange(DCX) processes are of special interest. These modes,excited in (�+; ��) reactions [7], involve at least two nu-cleons within the nucleus and give rise to a sharp peakat around 50 MeV in the forward cross section. Theyare thus located in the energy region of the high den-sity of 2p2h states which points to the importance ofcoherence e�ects among those states. Consequently, thepresent investigation may also appear helpful in futurestudies of the mechanism of DCX reactions and in sep-arating the suggested [8] dibaryon contribution from theconventional e�ects [9].Diagonalizing the nuclear Hamiltonian in the subspace

of 2p2h states j2i � ayp1ayp2ah2ah1 j0i yields the eigenen-

ergies En and the corresponding eigenvectors jni =

�2cn2 j2i. In response to an external �eld F̂� a state

jF�i � F̂�j0i =P

nhnjF̂�j0ijni is excited. The two-

phonon operator F̂� can be represented as F̂� = ff̂�

f̂ g�, where f̂� and f̂ denote the single-phonon oper-ators whose quantum numbers � and are coupled toform �. The state jF�i determines the strength function

SF�(E) =X

n

SF�(n)�(E �En); (1)

where SF�(n) = jhnjF̂�j0ij2. In the basis of states j2i

SF�(n) =X

2

jcn2 j2jh2jF̂�j0ij

2 +X

26=20

cn�2 cn20h0jF̂ y�j2

0ih2jF̂�j0i:

(2)

This equality de�nes its diagonal (SdF�(n)) and o�-

diagonal (SodF�(n)) contributions. The second componentincludes many more terms and it is this component whichpotentially is able to induce collectivity, i.e. a strongtransition to energy En. Two elements are however re-quired: (i) a state jni must involve su�ciently many ex-pansion coe�cients cn2 over the unperturbed states j2i

which carry the strength (h2jF̂�j0i 6= 0) and this is equiv-alent to at least local mixing, but at the same time(ii) sign correlations among these expansion coe�cients

1

brought to you by COREView metadata, citation and similar papers at core.ac.uk

provided by CERN Document Server

should take place so that the di�erent terms do not cancelout.Optimal circumstances for the second condition to be

ful�lled read: cn2 � h0jF̂�j2i. This may occur if the in-teraction matrix elements can be represented by a sumof separable terms Q� of the multipole-multipole type

(Vij;kl =PM

�=1Q�ijQ

�kl with Q�

ij � hijf̂� jji). The successof the Brown-Bolsterli schematic model [10] in indicatingthe mechanism of collectivity on the 1p1h level points toan approximate validity of such a representation. Collec-tivity can then be viewed as an edge e�ect connected withappearance of a dominating component in the Hamilto-nian matrix and the rank M of this component is signi�-cantly lower (unity in case of the Brown-Bolsterli model)than the size of the matrix. This rank speci�es a num-ber of the prevailing states whose expansion coe�cientspredominantly are functions of Q� . Due to two-bodynature of the nuclear interaction which reduces its 2p2hmatrix elements to combinations of the ones representingthe particle-particle, hole-hole and particle-hole interac-tions [4] the separability may become e�ective also onthe 2p2h level.For quantitative discussion presented below we choose

the 48Ca nucleus, specify the mean �eld part of theHamiltonian in terms of a local Woods-Saxon potentialincluding the Coulomb interaction and adopt the density-dependent zero-range interaction of ref. [11] as a residualinteraction. Since we want to inspect the higher energyregion at least three mean �eld shells on both sides ofthe Fermi surface have to be used to generate the unper-turbed 2p2h states as a basis for diagonalization of thefull Hamiltonian. Typically, the number of such states isvery large and this kind of calculation can be kept underfull numerical control only for excitations of the lowestmultipolarity. For this reason we perform a systematicstudy of the DCX J� = 0� states. Our model spacethen develops N=2286 2p2h states. There are still sev-eral possibilities of exciting such a double-phonon moderepresented by the operator F̂� out of the two single-

phonons f̂� and f̂ of opposite parity. For de�nitness we

choose f̂� = rY1�� and f̂ = r2[Y2 �]1+

��. The �rstof these operators corresponds to the 1�h! dipole and thesecond to 2�h! spin-quadrupole excitation. The resultingtwo-phonon mode thus operates on a level of 3�h! excita-tions.The results of calculations are presented in Fig. 1. As

one can see, including the residual interaction (part (b))induces a spectacularly strong transition at 49.1 MeV.This transition is stronger by more than a factor of 2 thanany of the unperturbed (part (a)) transitions even thoughit is shifted to signi�cantly higher (� 10 MeV) energy.This is also a very collective transition. About 96% of thecorresponding strength originates from SodF�(n), as com-parison between parts (b) and (c) of Fig. 1 documents.The degree of mixing can be quanti�ed, for instance, interms of the information entropy [12] I(n) = ��ipi ln pi(pi = jcni j

2) of an eigenvector jni in the basis (part (d)).

Interestingly, the system �nds preferential conditions forcreating the most collective state in the energy regionof local minimum in I(n). Our following discussion issupposed to shead more light on this issue.

FIG. 1. (a) The unperturbed transition-strength distribu-

tion in 48Ca for the J� = 0� DCX excitation involving the

single-phonon dipole and 2�h! spin-quadrupole modes. (b)

The same as (a) but after including the residual interaction.

(c) Sd

F�components of the transition-strength as de�ned by

eq. (2) (notice a di�erent scale). (d) The information entropy

of the states jni in the unperturbed basis. The dashed line

indicates the GOE limit (ln(0:48N)).

As shown in Fig. 2(a) our Hamiltonian matrix displaysa band-like structure with spots of the signi�cant matrixelements inside. This together with the nonuniform en-ergy distribution �u(E) of the unperturbed 2p2h states,(Fig. 2(b)) which is a trace of the shell structure of thesingle particle states, sizably suppresses the range of mix-ing and locally supports conditions for the edge e�ect tooccur in the energy region of the minimum in �u(E).A comparison with Fig. 1(b) shows that the collectivestate is located at about this region. Moreover, the mini-mum survives diagonalization (�p(E) in Fig. 2(c)) and allthe above features are consistent with the e�ective bandrange [13] ((�Ei)

2 =P

j(Hii � Hjj)2H2

ij=P

i6=j H2ij)

shown in Fig. 2(d).Further quanti�cation of the character of mixing be-

tween the unperturbed states is documented in Fig. 3.The distribution of o�-diagonal matrix elements (a) isof the Porter-Thomas (P-T) { type which indicates thepresence of the dominating multipole-multipole compo-nents in the interaction [3,14].

2

FIG. 2. (a) The structure of the Hamiltonian matrix for

the J� = 0� DCX states. The states are here labeled by

energies, ordered in ascending order and the matrix elements

Hik � 0:1 are indicated by the dots. (b) The density of the

unperturbed 2p2h-states. (c) The density of states after the

diagonalization. (d) The energy range of interaction between

the unperturbed states.

An interesting novel feature is the asymmetry betweenthe positive and negative valued matrix elements (seeparameters in the caption to Fig. 3). The positive ma-trix elements are more abundant which expresses furthercorrelations among them and the fact that the interac-tion is predominantly repulsive for the mode considered.As a chaos related characteristics we take the spectralrigidity measured in terms of the �3 statistics [1]. We�nd this measure more appropriate for studying variouslocal subtleties of mixing than the nearest neighbor spac-ing (NNS) distribution because for a smaller number ofstates the latter sooner becomes contaminated by strong uctuations. Indeed, the spectral rigidity (Fig. 3(b)) de-tects di�erences in the level repulsion inside the string ofeigenvalues (i1) covering the �rst maximum in �p(u)(E)(35.0-41.75 MeV) and the one (i2) covering the minimumand thus including the collective state (41.75-50.1 MeV).Again, the deviation from GOE is more signi�cant in i2

which, similarly as the behavior of I(n), signals a moreregular dynamics in the vicinity of the collective state(n = 996).

FIG. 3. (a) The distribution of o�-diagonal matrix ele-

ments between the J� = 0� DCX states (histogram). The

solid lines indicate �t in terms of P (H) = ajHjb exp(�jHj=c)with the resulting parameters: a = 683, b = �1:22, c = 0:23

(left) and a = 538, b = �1:32, c = 0:32 (right). (b) The

spectral rigidity �3(L) for eigenvalues from the two inter-

vals: n = 351 � 700 (i1) and n = 701 � 1050 (i2). The

long-dashed line corresponds to Poisson level distribution and

the short-dashed line to GOE.

The conditions corresponding to the actual nuclearHamiltonian are not the most optimal ones from thepoint of view of the collectivity of our J� = 0� DCX ex-citation. By multiplying the o�-diagonal matrix elementsby a factor of 0.7 we obtain a picture as shown in Fig. 4(aand b). Now the transition located at 46 MeV is anotherfactor of 2 stronger than before. However, the range ofvalues of a multiplication factor which produces this kindof picture is rather narrow and this feature of collectivityresembles a classical phenomenon of the stochastic res-onance [15]. It is relatively easy to completely destroysuch a strong transition. By multiplying the o�-diagonalmatrix elements by a factor of 3 (which is equivalent toincreasing the density of states) the strength distributiondisplays a form as shown in Fig. 4(c).

3

FIG. 4. The transition-strength distribution and the in-

formation entropy for the same excitation as in Fig. 1 but

o�-diagonal matrix elements are now multiplied by a factor

of 0.7 ((a) and (b)) and by a factor of 3.0 ((c) and (d)).

Even though this strength remains largely localizedin energy (standard way of analysing experimental datamay even classify it as collective) the correspondingP (S(n)) is essentially P-T distributed (Gaussian dis-tributed amplitudes) which becomes evident from the en-tropy estimate. Normalizing the total strength to unity,identifying the so normalized S(n) with the probability�n to populate a state at energy En, and de�ning the cor-responding total entropy I = �

Pn �n ln �n one obtains

I = 6:82 while the P-T distribution for the same numberof states (N = 2286) gives I = 7:0. Further increase ofthe multiplication factor may again produce some tran-sitions which are more collective than those allowed byP-T. In particular, starting from values �5 some newstrong collective transitions appear at the upper edge ofthe whole spectrum but this is the e�ect of basis trunca-tion.In conclusion, a real collectivity, by which we mean

a transition stronger than those generated by the mean�eld, is a very subtle e�ect and is not a generic propertyof the complex spectra. Its appearance, as it happensfor one of the components of the J� = 0� DCX ex-citations considered here, involves several elements likecorrelations among the matrix elements, nonuniformitiesin the distribution of states and a proper matching ofthe interaction strength to an initial (unperturbed) lo-

cation of the transition strength relative to the scale ofnonuniformities in the distribution of states. If present,a collective state is then located in the region of moreregular dynamics. This aspect of collectivity parallels ananalogous property hypothesised for living organisms [6]and stating that collectivity is a phenomenon occuringat the border between chaos and regularity. Based onthe present study it is expected that majority of the two-phonon nuclear giant resonances is characterised by aspectrum of transitions which are signi�cantly smallerthan those corresponding to the mean �eld picture andmay even be compatible with the P-T distribution withthe largest transitions concentrated at similar energies.This latter e�ect may then lead to an illusion of collec-tivity.We thank Franz Osterfeld for helpful discussions. This

work was supported in part by Polish KBN Grant No. 2P03B 140 10.

[1] T.A. Brody et al.,, Rev. Mod. Phys. 53, 385(1981)

[2] V.G. Zelevinsky, Nucl. Phys. A555, 109(1993)

[3] V.G. Zelevinsky, B.A. Brown, N. Frazier and M. Horoi,

Phys. Rep. 276, 85(1996)

[4] S. Dro_zd_z, S. Nishizaki, J. Speth and J. Wambach, Phys.

Rev. C49, 867(1994)

[5] S. Dro_zd_z, S. Nishizaki and J. Wambach, Phys. Rev. Lett.

72, 2839(1994)

[6] S.A. Kau�man, in The Principles of Organisation in

Organisms, edited by J.E. Mittenthal and A.B. Baskin

(Addion-Wesley, New York, 1992), p. 303

[7] M.J. Leitch et al., Phys. Rev. C39, 2356(1989)

[8] R. Bilger, H.A. Clement and M.G. Schepkin, Phys. Rev.

Lett. 71, 42(1993)

[9] M.A. Kagarlis and M.B. Johnson, Phys. Rev. Lett. 73,

38(1994)

[10] G.E. Brown and M. Bolsterli, Phys. Rev. Lett. 3,

472(1959)

[11] B. Schwesinger and J. Wambach, Nucl. Phys. A426,

253(1984)

[12] F.M. Izrailev, Phys. Rep. 196, 299(1990)

[13] M. Feingold, D.M. Leitner and M. Wilkinson, Phys. Rev.

Lett. 66, 986(1991)

[14] V.V. Flambaum, A.A. Gribakina, G.F. Gribakin and

M.G. Kozlov, Phys. Rev. A50, 267(1994)

[15] K. Wiesenfeld and F. Moss, Nature 373, 33(1995);

A.R. Bulsara and L. Gammaitoni, Physics Today,

39(March 1996)

4