Random matrices and chaos in nuclear physics: Nuclear structure

Evidence for the applicability of random-matrix theory to nuclear spectra is reviewed. In analogy to systems with few degrees of freedom, one speaks of chaos (more accurately, quantum chaos) in nuclei whenever random-matrix predictions are fulfilled. An introduction into the basic concepts of random-matrix theory is followed by a survey over the extant experimental information on spectral fluctuations, including a discussion of the violation of a symmetry or invariance property. Chaos in nuclear models is discussed for the spherical shell model, for the deformed shell model, and for the interacting boson model. Evidence for chaos also comes from random-matrix ensembles patterned after the shell model such as the embedded two-body ensemble, the two-body random ensemble, and the constrained ensembles. All this evidence points to the fact that chaos is a generic property of nuclear spectra, except for the ground-state regions of strongly deformed nuclei.

Figure 1

The total neutron cross section on ${}^{232}\mathrm{Th}$ vs neutron energy $E_n$ in eV. From 147, as reproduced in 41, Vol. 1, p. 178.

Figure 2

Bohr’s wooden toy model of the compound nucleus. From 146.

Figure 3

Six spectra with 50 levels each and the same mean level spacing. From right to left: The one-dimensional harmonic oscillator, a sequence of zeros of the Riemann zeta function, a sequence of eigenvalues of the Sinai billiard (see Sec. 2F), a sequence of resonances seen in neutron scattering on ${}^{166}\mathrm{Er}$, a sequence of prime numbers, and a set of eigenvalues obeying Poisson statistics (see Sec. 2F). From 32.

Figure 4

The nearest-neighbor-spacing (NNS) distribution of the GOE (solid line) vs $s$, the ratio of the actual level spacing and the mean level spacing. For comparison, we also show the NNS distributions for the GUE (dashed line) and the GSE (dotted line). The parameter $\beta$ is the Dyson index with $\beta =1$, 2, and 4 for GUE, GOE, and GSE, respectively.

Figure 5

The number variance vs the length $L$ of the interval ($L$ is in units of the mean level spacing) for the three canonical ensembles. Top curve, GOE; middle curve, GUE; bottom curve, GSE. The parameter $\beta$ is the Dyson index; see Fig. 4. From 91.

Figure 6

The ${\Delta }_3$ statistic for the Sinai billiard (open circles), the GOE prediction (solid line), and the Poisson result (dashed line). From 33.

Figure 7

The NNS distribution for the Sinai billiard (histogram), the GOE prediction (solid line), and the Poisson result (dashed line). While the theoretical literature commonly uses the label $s$ for the actual level spacing in units of the mean level spacing, that quantity is usually referred to as $x$ when RMT predictions are compared with data. We follow that usage here. Inset: The eigenvalues must be selected according to the symmetries of the eigenfunctions. From 33.

Figure 8

The yrast line, i.e., the excitation energy $E\left(J\right)$ of the lowest state with spin $J$ vs $J$ for an even-even nucleus with mass number around 160 (schematic). The shaded area indicates the domain where spectroscopic information is available; see 70. The letter $S$ denotes the particle threshold. Depending on $A$, $S$ typically lies between 5 and $8\mathrm{MeV}$.

Figure 9

The domains of excitation energy $E_x$ and mass number $A$ where spectral fluctuations have been investigated are shown for four classes of states (resonances, low-lying states, high-spin states, and states that belong to a complete spectrum). Complete spectra are known for three nuclei only, ${}^{26}\mathrm{Al}$, ${}^{30}\mathrm{P}$, and ${}^{116}\mathrm{Sn}$. The letter $S$ denotes the particle threshold.

Figure 10

The NNS distribution for the nuclear data ensemble vs $s$ as in Fig. 4 (histogram) and the GOE prediction (solid line). From 34. (By the time that paper was published, the nuclear data ensemble had grown to 1726 spacings.)

Figure 11

The ${\Delta }_3$ statistic for the nuclear data ensemble (data points) and the GOE and GUE predictions (solid lines). The dashed lines estimate the finite-range-of-data errors. From 99.

Figure 12

Level spacing distributions for six different regions. Upper panels: NNS distributions (histograms) are compared with the Wigner distribution (solid lines) and the Poisson distribution (dashed lines) for several ranges of mass numbers. Lower panels: Same for the cumulative distributions (number of spacings smaller than $x$) (cf. the caption of Fig. 7). The approximate number of levels in each region is as follows: $A=0–50$ $(N=150)$, $A=50–100$ $(N=50)$, $A=100–150$ $(N=270)$, $A=150–180$ $(N=450)$, $A=180–210$ $(N=60)$, $A>230$ $(N=190)$. From 172.

Figure 13

Comparison of the NNS distributions vs $x$, the level spacing in units of the mean level spacing, for $2^{+}$ and $4^{+}$ states in strongly deformed (left panel) and in spherical nuclei (right panel). Adapted from 172.

Figure 14

The distribution $\sqrt{y}P\left(y\right)$ for $y={\gamma }^2∕\overline{{\gamma }^2}$ for 1117 reduced widths. From 171.

Figure 15

The ${\Delta }_3$ statistic shown for the states in ${}^{26}\mathrm{Al}$ obtained by taking into account only the quantum numbers indicated at the top of each panel.

Figure 16

The spectral rigidity ${\Delta }_3$ vs $L$ for 75 levels with $T=0$ and 25 levels with $T=1$ measured in ${}^{26}\mathrm{Al}$ (dots with error bars). The excitation energies lie between 0 and $8\mathrm{MeV}$. The lower (upper) dashed line is the prediction for a single GOE (for two uncorrelated GOEs with fractional densities $3∕4$ and $1∕4$, respectively). The solid line is the result of a numerical simulation incorporating symmetry breaking. The strength of the symmetry-breaking interaction was fitted to the data. From 91.

Figure 17

Comparison between the Porter-Thomas distribution (written in terms of the variable $z={\log }_{10}y$) (solid lines) and the data (histograms) for several multipole transitions and isospin differences in ${}^{30}P$. From 170.

Figure 18

Level sequence in the nuclear shell model. From 135.

Figure 19

NNS distribution (histogram) and Wigner surmise (solid line) for the states with $m=12$ and $J=2$, $T=0$ in the $sd$ shell. From 198.

Figure 20

The ${\Delta }_3$ statistic (dots) and the GOE prediction (solid line) for states with $m=12$ and $J=2$, $T=0$ in the $sd$ shell. From 198.

Figure 21

The exponential of the information entropy (36) vs the energy of the state for the states with $m=12$, $J=2$, and $T=0$ in the $sd$ shell (dots). The GOE limit of 1578 is indicated by the horizontal line. From 198.

Figure 22

Two rotational bands in ${}^{174}\mathrm{Hf}$. Adapted from 41.

Figure 23

Mixing parameter $q$ for the ${\Delta }_3$ statistic (see text) (left-hand scale), average standard deviation ${\overline{\sigma }}_{E2}$ of the distribution of reduced $E2$ matrix elements (right-hand scale), and half-width at half maximum $\frac{1}{2}{\Gamma }_{\mathrm{rot}}$ (right-hand scale) for the same distribution, all vs $\Delta$, the strength of the two-body interaction which mixes the rotational bands. From 1.

Figure 24

Domains of regular or chaotic motion in the $\eta \text{−}\chi$ plane for two values of $\mathcal{l}$, the angular momentum per boson. The accessible parameter space has the shape of a triangle. Three domains in parameter space are separated by solid lines: Domains with nearly regular motion (fraction of chaotic phase-space $\text{volume}⩽0.3$; white areas); domains with nearly chaotic motion (fraction of chaotic phase-space $\text{volume}>0.7$; dotted areas); and the regions in between. From 10.

Figure 25

The deformation energy $V(\beta ,\gamma )$ for fixed total spin $J_0$ vs deformation (schematic). In some nuclei, $V(\beta ,\gamma )$ may display a second minimum at large deformation. The level shown in the second minimum, a member of the SD band with spin $J_0$, may decay either via gamma emission to the next lower level in the SD band (not shown) or, via tunneling, mix with the ND states with spin $J_0$ located in the first minimum. The latter decay dominantly by statistical $E1$ emission.

Figure 26

(Color online) Block structure of the matrix of the shell-model Hamiltonian for the $sd$ shell. Blocks in black, light gray, dark gray (online red, green, blue) indicate matrix elements that change the partition by zero, one, or two units, respectively, as explained in the text. White areas do not carry matrix elements. From 154.

Figure 27

(Color online) The square roots of the eigenvalues of the matrix $S_{\alpha \beta }\left(J\right)$ (see text) for the $j=19∕2$ shell with six identical nucleons vs total spin $J$. From 153.

Figure 28

(Color online) Six identical fermions in a $j=19∕2$ shell. Inset: Spectral radius $R_0$ vs spectral width ${\sigma }_0$ (data points) and the linear fit (line). Bottom: Scaling factor $r_J$ vs $J$. Top: Probability that the ground state has spin $J$ (points), probability that spin $J$ has the largest spectral width (solid line), and probability that the product $r_J{\sigma }_J$ is maximal (dashed line). From 153.