The Music of Nuclear Energy Level Spacings

High-resolution experiments have recently led to a complete identification (energy, spin, and parity) of 151 nuclear levels up to an excitation energy of Ex = 6.20 MeV in 208Pb [1], The number of levels and their spacings are consistent with predictions made by Alex Brown [2]. The energy spacings of the experimental and theoretical levels for each spin and parity have recently been studied [3]. The figure shows three types of energy spacings for 100 levels. The completely ordered equal-spacing distribution is shown by (a). The Poisson distribution based upon random numbers for the energies is shown by (b). The right-hand panel (c) is based on a distribution first proposed by Wigner [4] that emerges from the random matrix distribution of the Gaussian Orthogonal Ensemble (GOE). The GOE is an approximation to the matrix obtained from large-basis configuration interaction calculations [5]. The experimental results for 208Pb were analyzed in terms of the combination f dGOE + (1-f) dPoisson . The experimental result of f = 0.9 is so far the closest agreement with a GOE observed in the spectra of bound states in a nucleus. The theoretical result depends upon the type residual interaction. The realistic M3Y (Michigan sum of three Yukawa) interaction gives f=0.73 (close to experiment) whereas a simple delta-function interaction gives f=0.18 that is much smaller that experiment.

A computer program by Alex Brown can turn these distributions into musical notes. Starting from the bottom, the time interval between notes is determined by energy interval to the next level. This energy interval is also used to change the relative pitch, alternating up and down relative to the last note. Here are the results:

(a) equal spacing mp3, (b) Poisson mp3, and (c) GOE mp3. And here thay are all combined [trio mp3].


Read more here:

B. Dietz, A. Heusler, K. H. Maier, A. Richter, and B. A. Brown Phys. Rev. Lett. 118, 012501 (2017)

[1] Heusler et al., Phys. Rev. C 93, 054321 (2016)].

[2] Double Octupole States in 208Pb, B. A. Brown, Phys. Rev. Lett. 85, 5300 (2000). [link to paper].

[3] Chaos and regularity in the doubly magic nucleus 208Pb, B. Dietz, A. Heusler, K. H. Maier, A. Richter and B. A. Brown, Phys. Rev. Lett. 118, 012501 (2017). [link to paper].

[4] E. P. Wigner, Ann. Math. 62, 548 (1955).

[5] The Nuclear Shell Model as a Testing Ground for Many-Body Quantum Chaos, V. Zelevinsky, B. A. Brown, N. Frazier and M. Horoi, Physics Reports 276, 85 (1996). [link to paper].