Random matrices and chaos in nuclear physics: Nuclear reactions

The application of random-matrix theory (RMT) to compound-nucleus (CN) reactions is reviewed. An introduction into the basic concepts of nuclear scattering theory is followed by a survey of phenomenological approaches to CN scattering. The implementation of a random-matrix approach into scattering theory leads to a statistical theory of CN reactions. Since RMT applies generically to chaotic quantum systems, that theory is, at the same time, a generic theory of quantum chaotic scattering. It uses a minimum of input parameters (average $S$ matrix and mean level spacing of the CN). Predictions of the theory are derived with the help of field-theoretical methods adapted from condensed-matter physics and compared with those of phenomenological approaches. Thorough tests of the theory are reviewed, as are applications in nuclear physics, with special attention given to violation of symmetries (isospin and parity) and time-reversal invariance.

Figure 1

Level scheme of ${}^{18}\mathrm{O}$. The center part shows the low-lying levels, beginning with the ground state. On the left and right, various thresholds are indicated plus the energy dependence of some reaction cross sections. From 230.

Figure 2

Spectrum of neutrons (“evaporation spectrum”) emitted from the CN ${}^{104}\mathrm{Pd}$ in the $(p,n)$ reaction on ${}^{103}\mathrm{Rh}$ at an angle of 80° vs neutron energy $E_n$ (semilogarithmic plot). With all transmission coefficients set equal to unity (“black box” model for the CN), the cross section $\propto \exp (-E_n∕T)$ mirrors the level density in the residual nucleus and permits the determination of the nuclear temperature $T$. From 128.

Figure 3

Angular distribution of the compound-elastic cross section for the ${}^{30}\mathrm{Si}(p,p)$ reaction at a bombarding energy $E_p=9.8\mathrm{MeV}$. The data (dots), taken in the regime $\Gamma ⪢d$, show the symmetry of the CN cross section about 90° c.m. The solid lines (a) and (b) are predictions of the Hauser-Feshbach formula (9) without and with an elastic enhancement factor 2, respectively. From 143.

Figure 4

Elastic cross section for protons at $5\mathrm{MeV}$ scattered on ${}^{25}\mathrm{Mg}$ vs scattering angle. The data points with error bars are compared with predictions of the Hauser-Feshbach formula (NC) without (solid line) and with elastic enhancement factor (dashed line) and of the optical model (ID). From 102.

Figure 5

Distribution of the poles of the $S$ matrix in the complex energy plane. The distribution is shown for several values of the strength $\gamma$ of the average coupling to the channels. The abscissa spans the entire range of the spectrum. From 150.

Figure 6

(Color online) Flat microwave resonator (left-hand side) as a model for the compound nucleus (right-hand side). The height $d=0.84\mathrm{cm}$ of the flat resonator makes it a chaotic quantum billiard up to a frequency of $18.75\mathrm{GHz}$.

Figure 7

Part of the transmission spectrum of a superconducting microwave billiard. We note the extremely high resolution of the resonances. Two singlets (a) and (c) and a doublet (b) of resonances are magnified in the upper part. For all three cases $R$-matrix resonance formulas (16) based on expressions from 149 were fitted to the data. From 52.

Figure 8

The experimental distribution of the partial widths ${\Gamma }_{\mu 2}$ in units of their mean value $⟨{\Gamma }_{\mu 2}⟩$. The solid line corresponds to a Porter-Thomas distribution. From 8.

Figure 9

Autocorrelation function for isolated resonances. Upper part: The experimental autocorrelation function ${\left|C_2\left(\epsilon \right)\right|}^2$ (circles within the shaded band of errors), the prediction of the statistical theory (dashed line), and a Lorentzian (solid line). Lower part: Fourier coefficients of the autocorrelation function (dots) with errors indicated by the shaded band together with the prediction of the statistical theory (dashed line) and an exponential (Fourier transform of a Lorentzian). The nonexponential decay in time of the Fourier transform of the $S$-matrix autocorrelation functions is clearly visible. From 8.

Figure 10

Intensity measurements in microwave cavities. Transmitted (upper part) and reflected (lower part) intensity vs frequency between 9.0 and $9.5\mathrm{GHz}$. The resonances overlap and create a fluctuation pattern. The shape of the two-dimensional quantum billiard used in the experiment is shown in the inset. Points 1 and 2 indicate the positions of the antennas. From 61.

Figure 11

Autocorrelation function for weakly overlapping resonances. Upper part: The autocorrelation function ${\left|C_{ab}\left(\epsilon \right)\right|}^2$ for values of $a$ and $b$ as indicated. The values of ${\left|C_{ab}\left(\epsilon \right)\right|}^2$ were calculated from the measured $S$-matrix elements (points) and from the fit of Eq. (75) of the statistical theory to the data (solid line). Lower part: Fourier coefficients ${\tilde{C}}_{ab}\left(t\right)$ of the autocorrelation function (points) and the Fourier transform of the fit of $C_{ab}\left(\epsilon \right)$ [Eq. (75)] to the data (solid line). The nonexponential decay in time in the region of weakly overlapping resonances $(\Gamma ∕d\approx 0.2)$ is striking. From 61.

Figure 12

Distribution of scattering amplitudes. From left to right: Histograms for the scaled distributions of the real and imaginary parts of the reflection amplitude $S_{11}$ and the real part and the phase of the transmission amplitude $S_{12}$, respectively. The data were taken in the two frequency intervals $5–6\mathrm{GHz}$ (upper panels) and $9–10\mathrm{GHz}$ (lower panels). The scaling factors are given in each panel. The solid lines are best fits to Gaussian distributions. From 61.

Figure 13

Fluctuating excitation function of the reaction ${}^{28}\mathrm{Si}(n,p_{0+1}){}^{28}\mathrm{Al}$ in the regime of strongly overlapping resonances $(\Gamma ⪢d)$ for the CN ${}^{29}\mathrm{Si}$. The cross sections of proton exit channels $p_0$ and $p_1$ leading to the ground and first-excited states in ${}^{28}\mathrm{Al}$, respectively, were not resolved. From 15.

Figure 14

Ericson fluctuations in the giant dipole resonance. Upper part: Double differential cross section of the ${}^{40}\mathrm{Ca}(e,e^{\prime }{p}_0)$ reaction at an electron energy $E_0=183.5\mathrm{MeV}$. Middle part: Ratio of the measured cross section and the smoothed cross section. Lower part: Autocorrelation function. From 46.

Figure 15

Same as in Fig. 10 but in the Ericson regime of overlapping resonances. From 203.

Figure 16

Autocorrelation function in the Ericson regime. Autocorrelation functions (upper part) and Fourier coefficients (lower part) with a fit of Eqs. (75, 76, 77) of the statistical theory. Same as in Fig. 11 but for the Ericson regime. The exponential decay in time in the region of overlapping resonances $(\Gamma ∕D\approx 1.06)$ is striking. From 203.

Figure 17

Distribution of transmission amplitudes. Upper part: Histogram for the distribution of the real part of the matrix element $S_{12}$. Lower part: Histogram for the distribution of the phase of $S_{12}$. The data are taken in the Ericson regime $(\Gamma ∕D\approx 1.06)$. From 203.

Figure 18

Level scheme showing the parent nucleus, the isobaric analog state, and a state of lower isospin (schematic). Adapted from 20.

Figure 19

Partial widths of individual resonances (lower panel) and their integral (upper panel) vs proton bombarding energy $E_p$ in the elastic proton scattering on ${}^{92}\mathrm{Mo}$. From 21.

Figure 20

Same as in Fig. 19 but for the target nucleus ${}^{40}\mathrm{Ar}$. From 20. Adapted from 136.

Figure 21

Fine structure of the analog resonance with spin $3∕2$ in the CN ${}^{45}\mathrm{Sc}$. The partial width amplitudes $\gamma$ are indexed by the channel angular momentum and spin. The Robson asymmetry is clearly displayed. From 168.

Figure 22

Neutron transmission spectra for two helicity states near the $63\mathrm{eV}$ resonance in ${}^{238}\mathrm{U}$. The resonance appears as a dip in the transmission curve. The transmission at the resonance differs significantly for the two helicity states and parity violation is apparent by inspection. From 170.

Figure 23

Plot of the maximum-likelihood function $L\left(\nu \right)$ vs the root-mean-square matrix element $\nu$ for ${}^{238}\mathrm{U}$. The spins of all the $p$-wave resonances are known. From 170.

Figure 24

Normalized cross sections for the reaction ${}^{27}\mathrm{Al}(p,{\alpha }_0){}^{24}\mathrm{Mg}$ (solid line) and for the inverse reaction ${}^{24}\mathrm{Mg}(\alpha ,p_0){}^{27}\mathrm{Al}$ (open circles) near and at a deep minimum. The two cross sections are equal and detailed balance holds. From 22.

Figure 25

Magnitude and phase of $S_{12}$ (solid lines) and $S_{21}$ (dashed lines) measured in the frequency interval from $16\text{to}16.5\mathrm{GHz}$ for a fixed magnetization of the ferrite. From 59.

Figure 26

Cross-correlation coefficient. Upper panel: Values of $C_{\text{cross}}\left(0\right)$ calculated from the fluctuating matrix elements $S_{12}$ and $S_{21}$ of Fig. 25. Lower panel: The parameter $\xi$ vs frequency as deduced from the data. From 59.