- Email: [email protected]
Chaotic behavior of nuclear spectra
446
Nuclear
Instruments
and Methods
in Physics
Research
B56/57
(1991) 446-450 North-Holland
Section IV. Nuclear physics and related phenomena
Chaotic behavior of nuclear spectra * G .E. Mitchell North Carolina State University, Raleigh, NC, USA, and Triangle Universities Nuclear Laboratory, Durham, NC, USA
E.G. Bilpuch Duke University and Triangle Universities Nuclear Laboratory,
Durham,
NC, USA
P.M. Endt Rijksuniversiteit
Utrecht, Utrecht, The Netherlands
J.F. Shriner, Jr. Tennessee
Technological
University,
Cookeville,
TN, USA
T. von Egidy Technische
Universitiit Miinchen,
Garching, Germany
Bohigas, Giannoni and Schmit conjectured that quantum analogs of classically chaotic systems have level statistics which are consistent with the Gaussian orthogonal ensemble (GOE) of random matrix theory. This suggests that the fluctuation properties of nuclear levels may be used as a signature for chaos or regularity. Resonance data are known to display the long and short range order predicted by GOE. In one nucleus, 26A1, all levels are known from the ground state to the resonance region. This provides the first experimental test of the behavior of fluctuation properties as a function of symmetry breaking - in this case, isospin. In addition there are many nuclides where level schemes are complete over a more limited range of energies and spins. Our analysis of a large data set ( - 160 sequences in 60 nuclides) reveals a strong dependence on the mass number A.
1. Introduction Searches
for quantum
systems
which
display
chaotic
in statistical nuclear spectroscopy. Bohigas et al. [l] conjectured that quantum analogs of classically chaotic systems display fluctuation properties which coincide with the Gaussian orthogonal ensemble (GOE) of random matrix theory [2]. It is known that the level statistics of high quality nuclear resonance data display the long and short range order predicted by GOE [3,4]. Much attention has been paid recently to energy level fluctuations of quantum systems and their possible role in understanding quantum chaos [5,6]. A review of applications to nuclear behavior
have
renewed
interest
* Supported in part by the U.S. Department of Energy, Office of High Energy and Nuclear Physics, under Contract Nos. DE-FGW87ER40353, DE-AC05-76ER01067 and DEFG05-88ER40441, and the Bundesministerium fur Forschung und Technologie, Bonn, Germany. 0168-583X/91/$03.50
0 1991 - Elsevier Science Publishers
physics is given by Bohigas and Weidenmtiller [7]. The literature is extensive on the generalization (that the analogs of classically chaotic systems show GOE behavior and analogs of classically regular systems show Poisson behavior), although there are exceptions to the generalization. Here we assume that the conjecture is valid and that fluctuation properties of nuclear states may be used as signatures of chaos or regularity. However, until very recently the only suitable data were for the resonance region. We felt that it was important to establish the empirical fluctuation properties in other regions of excitation energy and throughout the periodic table. We have performed an analysis [8,9] of levels from the ground state to the resonance region for one nucleus, 26Al, and have analyzed an extensive set of data on low-lying levels in many nuclei [lo]. Section 2 provides a brief review of the GOE and its success in describing the fluctuation properties in the resonance region. In section 3 the results for ‘“Al are briefly summarized. Although ‘“Al is unique in that a complete
B.V. (North-Holland)
441
G.E. Mitchell et al. / Chaotic behavior of nuclear spectra
set of levels is known from the ground state to the resonance region, there are many nuclides whose level schemes are complete over a more limited range of energies. The analysis of a large set of data (160 sequences in 60 nuclides) is described in section 4. A brief summary and conclusions are given in section 5.
2. Fluctuation properties of nuclear resonances The fluctuation properties of high quality neutron and proton resonance data have been tested in detail [3,4] and show the long and short range order predicted by the GOE. Random matrix theory might be called a minimalist theory, since one relies only on the existence of a few general symmetries and on statistical assumptions. [The fluctuation properties with (without) time reversal invariance obey the GOE (GUE). Here we consider only the GOE.] Since the theory makes definite (and striking) predictions for the fluctuation properties and has no adjustable parameters, there are hints of universality. An attractive current view is that the average properties of nuclei are determined by the nuclear interactions, but that the fluctuation properties are generic rather than specific to nuclei. We focus on the determination of the fluctuation properties, first in the resonance region and then at other excitation energies in nuclei. The fluctuation properties are most simply examined with the nearest neighbor spacing (NNS) distribution and the Dyson-Mehta A, statistics. We restrict the discussion to these two statistic. The nearest neighbor spacings S, are the energy differences between adjacent levels of the same symmetry (usually the same J and T). It is convenient to consider the dimensionless spacing parameter x = S/D, where D is the average spacing. For the GOE the NNS distribution is very close to the Wigner distribution P(x)
= G_x ee*X2/4.
If the spacings P(x)
are Poisson
distributed,
then
= eex.
The Dyson-Mehta A, statistic for a sequence levels with energy limits E,, and Em, is
of
The key predictions of the GOE are that levels of the same symmetry repel linearly and that for levels of the same symmetry there is a striking long range order (arising from the logarithmic dependence of A, on L). This so-called spectral rigidity or quasi-crystalline order has been verified by analysis of resonance data [3,4]. Thus the GOE has achieved a major success. However, it should be emphasized that this success was obtained only with very high quality data and that these data are rather restricted in scope - neutron resonances in heavy nuclei and proton resonances in the f-p shell. The level repulsion is rapidly destroyed when states of a different symmetry are admixed. The A, parameter is exceptionally sensitive both to missed levels and to misassigned levels. Therefore a major problem in determining the fluctuation properties for other regions of nuclear excitation energy is the rather stringent requirement that the level sequence be both pure (no levels with different symmetry) and complete (no missing levels). This proves a very severe limitation, as we discuss in sections 3 and 4.
3. Fluctuation properties in 26Al For a study of fluctuation properties from the ground state to the resonance region, the ideal nucleus would have a pure and complete (and large) set of states. At present 26A1 is unique in that a complete level scheme exists from the ground state far into the resonance region. Some 160 levels have been identified from 0 to 8 MeV in 26A1H-131. The various statistics (GOE or Poisson) assume a constant level density. Clearly this is not true for such a large energy range. A detailed description of the unfolding procedure utilized is given by Shriner et al. [9]. Here we simply assume that the correction for variable level density has been performed. For the NNS distribution x is determined for each level spacing in each sequence (each set of states with the same spin and parity). Since the spirit of the analysis is to determine empirically the actual distribution, it is convenient to define an interpolation formula [14] P(x,
‘3(L)
= F;
E,,,!E
with
x/:ma~[iV(E;‘:ai-U,2
dE, (Y=
In,” where L is the number of val [ Eti,,, Em,,] and N(E) energies less than or equal the expected value is L/15, expected value for large L Aa,,,(
L) = $
cd) = a(~ + 1)~~ e--axe+‘,
spacings in the energy interis the number of levels with to E. For Poisson statistics while for GOE statistics the is approximately
[In L - 0.0681.
[r[y1”‘.
In this purely empirical formula w = l(0) corresponds to GOE (Poisson) statistics. For each sequence the distribution is calculated and errors determined using the bootstrap method [15]. The distribution of each sequence (or average of several sequences) is characterized by the value of w which best fits the distribution. IV. NUCLEAR
PHYSICS
448
G.E. Mitchell et al. / Chaotic behavior of nuclear spectra
An interpolation formula is also defined for
A,(L) =Asx,&4
Unfortunately the available data set is too limited; the present data are consistent with the theoretical predictions but do not conclusively establish them. Additional data are needed; we plan experiments on neighbouring N = Z odd-odd nuclei to provide more experimental data.
A, [16]
PIL).
+A,p_W-
Then p = 0 yields a Poisson A, and p = 1 corresponds to a GOE description. The results for ‘6A1 [8,9] are that both the NNS distribution and A, are intermediate between the limiting Poisson and GOE values. As far as can be determined with the present statistics, the NNS distributions do not change with excitation energy. The isospin values for the states in 26A1are considered to be well known, and isospin is thought to be a rather good (but approximate) symmetry. It is therefore surprising that the fluctuation properties appear to be independent of isospin. This result can be understood from the theory of Dyson [17] and Pandey [18] on the effects of symmetry breaking on fluctuations. Even a small breaking of a symmetry yields fluctuation patterns which would be found in the absence of the symmetry. Guhr and Weidenmiiller [19] studied the effects of isospin breaking on fluctuations and confirm this prediction. This result seems very significant for symmetry breaking tests using fluctuations. The result also suggests that the observation of Poisson behavior would be especially significant, since weak perturbations apparently drive the system towards GOE behavior.
4. Fluctuation properties of low-lying states Although there is no other data set of sufficient quality to permit a complete analysis in a single nuclide, there are many nuclides whose level schemes are complete over a more limited range of energies [20,21]. We have analyzed the data compiled by von Egidy in order to assess empirically the fluctuation properties of lowlying states. (An analysis [22] of an earlier, smaller data set was performed by Abul-Magd and Weidenmtiller.) Since the sample sizes are extremely limited, it was necessary to combine data from different nuclei. Thus our results reflect averaged fluctuation properties. Since the sample sizes are so small, we performed extensive modelling studies to investigate the possibilities of bias. We concluded that for small samples the NNS distribution was the most reliable of the statistical parameters. Our results then consist of values of w determined for some 160 sequences and approximately 50
O
lOO
100
A::
0.8 1.0
0.6
3
0.6
z
z
0.4 0.2 0.0
E
1.0 5 P-l 5
0.6 0.6 0.4 0.2 0.0
2 z 5 0
1 x
2
0.6 0.4 0.2 0.0
?? ;r:
1.0 0.6 0.6 0.4 0.2 0.0
E a 5 0
3
S/D
=
150
2 x
4
0
2 =
4
2
%- 0:6
% 0.0
z
1.0 0.6 0.6 0.4 0.2 0.0
E a 5 0
2 x
neighbor
4
230
0.6
S/D
Fig. 1. Nearest
2
x = S/D
180
180
A 3 a \
x
1.0 0.6 0.6 0.4 0.2 0.0
S/D
=
1.0 0.6
0
0.6 0.4 0.2 0.0
spacing
=
distribution
4
S/D for six different
0.4 0.2 0.0 1.0 0.6 0.6 0.4 0.2 0.0 0
2 x
mass regions.
=
S/D
4
449
GE. Mitchell et al. / Chaotic behauior of nuclear spectra
1000 levels.The question is how to group these levels in order to examine them. In fig. 1 levels are grouped by mass number A (A I 50, 50 < A 2 100, 100 < A d 150, 150 c A I 180, 180 c A I 210 and 230
5. Summary
and conclusions
The suggestion that the level statistics of nuclear states can be used as signatures of the degree of order or chaos has stimulated the present efforts. The goal was to determine empirically the fluctuation properties of nuclear states at lower excitation energies, in order to supplement the information obtained from the resonance data. Our major conclusions are: (1) For only one nucleus, ?Al, can the fluctuation properties be examined from the ground state to the resonance region, with even marginally sufficient statistics. The fluctuation properties seem independent of energy and isospin. The coexistence of T = 0 and T = 1 states complicated the analysis but it provides the first direct evidence concerning the effects of symmetry breaking on fluctuation properties. The data are consistent with, but do not conclusively establish, the predictions of Dyson, Pandey, and Guhr and Weidenmtiler. Additional data on similar (odd-odd N = 2) nuclei such as 22Na or 30P would be extremely valuable. Such experiments are being planned. (2) Results on the fluctuation properties of low-lying states throughout the periodic table provided averaged
information. The most striking overall result is a strong mass dependence: light nuclei show GOE-like behavior, while heavy nuclei show Poisson-like behavior. It is not clear whether the Poisson distributions for heavy nuclei reflect regularity or the neglect of additional symmetries. However, in any case these data provide the only known example in which a pure and complete set of states of the same spin and parity show a NNS distribution which is Poisson rather than Wigner. (States of the same spin and parity do not always repel!) More extensive data sets in a given nuclide are clearly needed. However, the requirement of purity and completeness makes these measurements extremely difficult. Such experiments are now under consideration. (3) Since the experiments to obtain the required data are both tedious and complicated, it would be extremely valuable to have methods of analysis which provide information on level repulsion, long-range correlations, etc., from impure and incomplete data.
Acknowledgement The authors would like to thank H.A. for valuable discussions.
Weidenmiiller
References and C. S&nit, Phys. Rev. [II 0. Bohigas, M.J. Giannoni Lett. 52 (1984) 1. PI T.A. Brody, J. Flores, J.B. French, P.A. Mello, A. Pandey and S.S.M. Wong, Rev. Mod. Phys. 53 (1981) 381. i31 RU. Haq, A. Pandey and 0. Bobigas, Phys. Rev. Lett. 48 (1982) 1086. 141 0. Bohigas, R.U. Haq and A. Pandey, Phys. Rev. Lett. 54 (1985) 1645. Systems: [51 G. Casati (ed.), Chaotic Behavior in Quantum Theory and App~cations (Plenum, New York, 1985). Chaos 161 T.H. Seligman and H. Nishioka (eds.), Quantum and Statistical Nuclear Physics (Springer, Berlin, 1986). Ann. Rev. Nucl. Part. 171 0. Bohigas and H. Weidenmiiller, Sci. 38 (1988) 421. PI GE. Mitchell, E.G. Bilpuch, P.M. Endt and J.F. Shriner, Jr., Phys. Rev. L&t. 61 (1988) 1473. R J.F. Shriner, Jr., E.G. BiIpuch, P.M. Endt and GE. w [Ill P21 P31 u41 [I51
Mitchell, Z. Phys. A335 (1990) 393. J.F. Shriner, Jr., GE. Mitchell and T. von Egidy.
to be published in Z. Phys. A. P.M. Endt, P. de Wit and C. Alderliesten, Nucl. Phys. A459 (1986) 61. P.M. Endt, P. de Wit and C. Alderliesten, Nncl. Phys. A476 (1988) 333. P.M. Endt, P. de Wit, C. Alderliesten and B.H. Wildenthal, Nucl. Phys. A487 (1988) 221. T.A. Brody, Lett. Nuovo Cimento 7 (1973) 482. B. Efron, SIAM Review 21 (1979) 460. IV. NUCLEAR
PHYSICS
450
G.E. Mitchell et al. / Chaotic behavior
[16] T.H. Seligman, J.J.M. Verbaarschot and M.R. Zirnbauer, J. Phys. A18 (1985) 2227. [17] F.J. Dyson, J. Math. Phys. 3 (1962) 1191. [18] A. Pandey. Ann. Phys. (NY) 130 (1981) 110. [19] T. Guhr and H.A. Weidenmiiller, Ann. Phys. (NY) 199 (1990) 412.
of nuclear spectra
[20] T. van Egidy, A.N. Behkami and H.H. Schmidt. Nucl. Phys. A454 (1986) 109. [21] T. van Egidy, H.H. Schmidt and A.N. Behkami. Nucl. Phys. A481 (1988) 189. [22] A.Y. Abul-Magd and H.A. Weidenmiiller, Phys. Lett. 162B (1985) 223.
Nuclear
Instruments
and Methods
in Physics
Research
B56/57
(1991) 446-450 North-Holland
Section IV. Nuclear physics and related phenomena
Chaotic behavior of nuclear spectra * G .E. Mitchell North Carolina State University, Raleigh, NC, USA, and Triangle Universities Nuclear Laboratory, Durham, NC, USA
E.G. Bilpuch Duke University and Triangle Universities Nuclear Laboratory,
Durham,
NC, USA
P.M. Endt Rijksuniversiteit
Utrecht, Utrecht, The Netherlands
J.F. Shriner, Jr. Tennessee
Technological
University,
Cookeville,
TN, USA
T. von Egidy Technische
Universitiit Miinchen,
Garching, Germany
Bohigas, Giannoni and Schmit conjectured that quantum analogs of classically chaotic systems have level statistics which are consistent with the Gaussian orthogonal ensemble (GOE) of random matrix theory. This suggests that the fluctuation properties of nuclear levels may be used as a signature for chaos or regularity. Resonance data are known to display the long and short range order predicted by GOE. In one nucleus, 26A1, all levels are known from the ground state to the resonance region. This provides the first experimental test of the behavior of fluctuation properties as a function of symmetry breaking - in this case, isospin. In addition there are many nuclides where level schemes are complete over a more limited range of energies and spins. Our analysis of a large data set ( - 160 sequences in 60 nuclides) reveals a strong dependence on the mass number A.
1. Introduction Searches
for quantum
systems
which
display
chaotic
in statistical nuclear spectroscopy. Bohigas et al. [l] conjectured that quantum analogs of classically chaotic systems display fluctuation properties which coincide with the Gaussian orthogonal ensemble (GOE) of random matrix theory [2]. It is known that the level statistics of high quality nuclear resonance data display the long and short range order predicted by GOE [3,4]. Much attention has been paid recently to energy level fluctuations of quantum systems and their possible role in understanding quantum chaos [5,6]. A review of applications to nuclear behavior
have
renewed
interest
* Supported in part by the U.S. Department of Energy, Office of High Energy and Nuclear Physics, under Contract Nos. DE-FGW87ER40353, DE-AC05-76ER01067 and DEFG05-88ER40441, and the Bundesministerium fur Forschung und Technologie, Bonn, Germany. 0168-583X/91/$03.50
0 1991 - Elsevier Science Publishers
physics is given by Bohigas and Weidenmtiller [7]. The literature is extensive on the generalization (that the analogs of classically chaotic systems show GOE behavior and analogs of classically regular systems show Poisson behavior), although there are exceptions to the generalization. Here we assume that the conjecture is valid and that fluctuation properties of nuclear states may be used as signatures of chaos or regularity. However, until very recently the only suitable data were for the resonance region. We felt that it was important to establish the empirical fluctuation properties in other regions of excitation energy and throughout the periodic table. We have performed an analysis [8,9] of levels from the ground state to the resonance region for one nucleus, 26Al, and have analyzed an extensive set of data on low-lying levels in many nuclei [lo]. Section 2 provides a brief review of the GOE and its success in describing the fluctuation properties in the resonance region. In section 3 the results for ‘“Al are briefly summarized. Although ‘“Al is unique in that a complete
B.V. (North-Holland)
441
G.E. Mitchell et al. / Chaotic behavior of nuclear spectra
set of levels is known from the ground state to the resonance region, there are many nuclides whose level schemes are complete over a more limited range of energies. The analysis of a large set of data (160 sequences in 60 nuclides) is described in section 4. A brief summary and conclusions are given in section 5.
2. Fluctuation properties of nuclear resonances The fluctuation properties of high quality neutron and proton resonance data have been tested in detail [3,4] and show the long and short range order predicted by the GOE. Random matrix theory might be called a minimalist theory, since one relies only on the existence of a few general symmetries and on statistical assumptions. [The fluctuation properties with (without) time reversal invariance obey the GOE (GUE). Here we consider only the GOE.] Since the theory makes definite (and striking) predictions for the fluctuation properties and has no adjustable parameters, there are hints of universality. An attractive current view is that the average properties of nuclei are determined by the nuclear interactions, but that the fluctuation properties are generic rather than specific to nuclei. We focus on the determination of the fluctuation properties, first in the resonance region and then at other excitation energies in nuclei. The fluctuation properties are most simply examined with the nearest neighbor spacing (NNS) distribution and the Dyson-Mehta A, statistics. We restrict the discussion to these two statistic. The nearest neighbor spacings S, are the energy differences between adjacent levels of the same symmetry (usually the same J and T). It is convenient to consider the dimensionless spacing parameter x = S/D, where D is the average spacing. For the GOE the NNS distribution is very close to the Wigner distribution P(x)
= G_x ee*X2/4.
If the spacings P(x)
are Poisson
distributed,
then
= eex.
The Dyson-Mehta A, statistic for a sequence levels with energy limits E,, and Em, is
of
The key predictions of the GOE are that levels of the same symmetry repel linearly and that for levels of the same symmetry there is a striking long range order (arising from the logarithmic dependence of A, on L). This so-called spectral rigidity or quasi-crystalline order has been verified by analysis of resonance data [3,4]. Thus the GOE has achieved a major success. However, it should be emphasized that this success was obtained only with very high quality data and that these data are rather restricted in scope - neutron resonances in heavy nuclei and proton resonances in the f-p shell. The level repulsion is rapidly destroyed when states of a different symmetry are admixed. The A, parameter is exceptionally sensitive both to missed levels and to misassigned levels. Therefore a major problem in determining the fluctuation properties for other regions of nuclear excitation energy is the rather stringent requirement that the level sequence be both pure (no levels with different symmetry) and complete (no missing levels). This proves a very severe limitation, as we discuss in sections 3 and 4.
3. Fluctuation properties in 26Al For a study of fluctuation properties from the ground state to the resonance region, the ideal nucleus would have a pure and complete (and large) set of states. At present 26A1 is unique in that a complete level scheme exists from the ground state far into the resonance region. Some 160 levels have been identified from 0 to 8 MeV in 26A1H-131. The various statistics (GOE or Poisson) assume a constant level density. Clearly this is not true for such a large energy range. A detailed description of the unfolding procedure utilized is given by Shriner et al. [9]. Here we simply assume that the correction for variable level density has been performed. For the NNS distribution x is determined for each level spacing in each sequence (each set of states with the same spin and parity). Since the spirit of the analysis is to determine empirically the actual distribution, it is convenient to define an interpolation formula [14] P(x,
‘3(L)
= F;
E,,,!E
with
x/:ma~[iV(E;‘:ai-U,2
dE, (Y=
In,” where L is the number of val [ Eti,,, Em,,] and N(E) energies less than or equal the expected value is L/15, expected value for large L Aa,,,(
L) = $
cd) = a(~ + 1)~~ e--axe+‘,
spacings in the energy interis the number of levels with to E. For Poisson statistics while for GOE statistics the is approximately
[In L - 0.0681.
[r[y1”‘.
In this purely empirical formula w = l(0) corresponds to GOE (Poisson) statistics. For each sequence the distribution is calculated and errors determined using the bootstrap method [15]. The distribution of each sequence (or average of several sequences) is characterized by the value of w which best fits the distribution. IV. NUCLEAR
PHYSICS
448
G.E. Mitchell et al. / Chaotic behavior of nuclear spectra
An interpolation formula is also defined for
A,(L) =Asx,&4
Unfortunately the available data set is too limited; the present data are consistent with the theoretical predictions but do not conclusively establish them. Additional data are needed; we plan experiments on neighbouring N = Z odd-odd nuclei to provide more experimental data.
A, [16]
PIL).
+A,p_W-
Then p = 0 yields a Poisson A, and p = 1 corresponds to a GOE description. The results for ‘6A1 [8,9] are that both the NNS distribution and A, are intermediate between the limiting Poisson and GOE values. As far as can be determined with the present statistics, the NNS distributions do not change with excitation energy. The isospin values for the states in 26A1are considered to be well known, and isospin is thought to be a rather good (but approximate) symmetry. It is therefore surprising that the fluctuation properties appear to be independent of isospin. This result can be understood from the theory of Dyson [17] and Pandey [18] on the effects of symmetry breaking on fluctuations. Even a small breaking of a symmetry yields fluctuation patterns which would be found in the absence of the symmetry. Guhr and Weidenmiiller [19] studied the effects of isospin breaking on fluctuations and confirm this prediction. This result seems very significant for symmetry breaking tests using fluctuations. The result also suggests that the observation of Poisson behavior would be especially significant, since weak perturbations apparently drive the system towards GOE behavior.
4. Fluctuation properties of low-lying states Although there is no other data set of sufficient quality to permit a complete analysis in a single nuclide, there are many nuclides whose level schemes are complete over a more limited range of energies [20,21]. We have analyzed the data compiled by von Egidy in order to assess empirically the fluctuation properties of lowlying states. (An analysis [22] of an earlier, smaller data set was performed by Abul-Magd and Weidenmtiller.) Since the sample sizes are extremely limited, it was necessary to combine data from different nuclei. Thus our results reflect averaged fluctuation properties. Since the sample sizes are so small, we performed extensive modelling studies to investigate the possibilities of bias. We concluded that for small samples the NNS distribution was the most reliable of the statistical parameters. Our results then consist of values of w determined for some 160 sequences and approximately 50
O
lOO
100
A::
0.8 1.0
0.6
3
0.6
z
z
0.4 0.2 0.0
E
1.0 5 P-l 5
0.6 0.6 0.4 0.2 0.0
2 z 5 0
1 x
2
0.6 0.4 0.2 0.0
?? ;r:
1.0 0.6 0.6 0.4 0.2 0.0
E a 5 0
3
S/D
=
150
2 x
4
0
2 =
4
2
%- 0:6
% 0.0
z
1.0 0.6 0.6 0.4 0.2 0.0
E a 5 0
2 x
neighbor
4
230
0.6
S/D
Fig. 1. Nearest
2
x = S/D
180
180
A 3 a \
x
1.0 0.6 0.6 0.4 0.2 0.0
S/D
=
1.0 0.6
0
0.6 0.4 0.2 0.0
spacing
=
distribution
4
S/D for six different
0.4 0.2 0.0 1.0 0.6 0.6 0.4 0.2 0.0 0
2 x
mass regions.
=
S/D
4
449
GE. Mitchell et al. / Chaotic behauior of nuclear spectra
1000 levels.The question is how to group these levels in order to examine them. In fig. 1 levels are grouped by mass number A (A I 50, 50 < A 2 100, 100 < A d 150, 150 c A I 180, 180 c A I 210 and 230
5. Summary
and conclusions
The suggestion that the level statistics of nuclear states can be used as signatures of the degree of order or chaos has stimulated the present efforts. The goal was to determine empirically the fluctuation properties of nuclear states at lower excitation energies, in order to supplement the information obtained from the resonance data. Our major conclusions are: (1) For only one nucleus, ?Al, can the fluctuation properties be examined from the ground state to the resonance region, with even marginally sufficient statistics. The fluctuation properties seem independent of energy and isospin. The coexistence of T = 0 and T = 1 states complicated the analysis but it provides the first direct evidence concerning the effects of symmetry breaking on fluctuation properties. The data are consistent with, but do not conclusively establish, the predictions of Dyson, Pandey, and Guhr and Weidenmtiler. Additional data on similar (odd-odd N = 2) nuclei such as 22Na or 30P would be extremely valuable. Such experiments are being planned. (2) Results on the fluctuation properties of low-lying states throughout the periodic table provided averaged
information. The most striking overall result is a strong mass dependence: light nuclei show GOE-like behavior, while heavy nuclei show Poisson-like behavior. It is not clear whether the Poisson distributions for heavy nuclei reflect regularity or the neglect of additional symmetries. However, in any case these data provide the only known example in which a pure and complete set of states of the same spin and parity show a NNS distribution which is Poisson rather than Wigner. (States of the same spin and parity do not always repel!) More extensive data sets in a given nuclide are clearly needed. However, the requirement of purity and completeness makes these measurements extremely difficult. Such experiments are now under consideration. (3) Since the experiments to obtain the required data are both tedious and complicated, it would be extremely valuable to have methods of analysis which provide information on level repulsion, long-range correlations, etc., from impure and incomplete data.
Acknowledgement The authors would like to thank H.A. for valuable discussions.
Weidenmiiller
References and C. S&nit, Phys. Rev. [II 0. Bohigas, M.J. Giannoni Lett. 52 (1984) 1. PI T.A. Brody, J. Flores, J.B. French, P.A. Mello, A. Pandey and S.S.M. Wong, Rev. Mod. Phys. 53 (1981) 381. i31 RU. Haq, A. Pandey and 0. Bobigas, Phys. Rev. Lett. 48 (1982) 1086. 141 0. Bohigas, R.U. Haq and A. Pandey, Phys. Rev. Lett. 54 (1985) 1645. Systems: [51 G. Casati (ed.), Chaotic Behavior in Quantum Theory and App~cations (Plenum, New York, 1985). Chaos 161 T.H. Seligman and H. Nishioka (eds.), Quantum and Statistical Nuclear Physics (Springer, Berlin, 1986). Ann. Rev. Nucl. Part. 171 0. Bohigas and H. Weidenmiiller, Sci. 38 (1988) 421. PI GE. Mitchell, E.G. Bilpuch, P.M. Endt and J.F. Shriner, Jr., Phys. Rev. L&t. 61 (1988) 1473. R J.F. Shriner, Jr., E.G. BiIpuch, P.M. Endt and GE. w [Ill P21 P31 u41 [I51
Mitchell, Z. Phys. A335 (1990) 393. J.F. Shriner, Jr., GE. Mitchell and T. von Egidy.
to be published in Z. Phys. A. P.M. Endt, P. de Wit and C. Alderliesten, Nucl. Phys. A459 (1986) 61. P.M. Endt, P. de Wit and C. Alderliesten, Nncl. Phys. A476 (1988) 333. P.M. Endt, P. de Wit, C. Alderliesten and B.H. Wildenthal, Nucl. Phys. A487 (1988) 221. T.A. Brody, Lett. Nuovo Cimento 7 (1973) 482. B. Efron, SIAM Review 21 (1979) 460. IV. NUCLEAR
PHYSICS
450
G.E. Mitchell et al. / Chaotic behavior
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