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Bremsstrahlung Radiation by a Tunneling Particle: a Time-Dependent Description

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Bremsstrahlung Radiation by a Tunneling Particle: a Time-Dependent Description RAPID COMMUNICATIONS PHYSICAL REVIEW C, VOLUME 60, 031602 Bremsstrahlung radiation by a tunneling particle: A time-dependent description C. A. Bertulani,1,* D. T. de Paula,1,† and V. G. Zelevinsky2,‡ 1Instituto de Fı´sica, Universidade Federal do Rio de Janeiro, 21945-970 Rio de Janeiro, Rio de Janeiro, Brazil 2Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, Michigan 48824-1321 ͑Received 21 December 1998; published 23 August 1999͒ We study the bremsstrahlung radiation of a tunneling charged particle in a time-dependent picture. In particular, we treat the case of bremsstrahlung during ␣ decay and show deviations of the numerical results from the semiclassical estimates. A standard assumption of a preformed ␣ particle inside the well leads to sharp high-frequency lines in the bremsstrahlung emission. These lines correspond to ‘‘quantum beats’’ of the internal part of the wave function during tunneling arising from the interference of the neighboring resonances in the open well. ͓S0556-2813͑99͒50709-7͔ PACS number͑s͒: 23.60.ϩe, 03.65.Sq, 27.80.ϩw, 41.60.Ϫm Recent experiments ͓1͔ have triggered great interest in the that a stable numerical solution of the Schro¨dinger equation, phenomena of bremsstrahlung during tunneling processes keeping track simultaneously of fast oscillations in the well which was discussed from different theoretical viewpoints in ͑‘‘escape attempts’’͒ and extremely slow tunneling, is virtu- Refs. ͓2–4͔. This can shed light on basic and still controver- ally impossible in this approach: for 210Po it would require sial quantum-mechanical problems such as tunneling times about 1030 time steps in the iteration process. Instead, we ͓5͔, tunneling in a complex and nonstationary environment study the bremsstrahlung in high energy ␣ decays, for which ͓6͔, preformation of a tunneling cluster in a many-body sys- the decay time is treatable numerically. This allows us to pay tem ͓7͔, and so on. It seems that ␣ decay offers a unique attention to qualitatively new aspects of the bremsstrahlung possibility to study these fundamental questions. The discus- in decay processes, and compare with stationary approaches. sion of the bremsstrahlung radiation in ␣ decay, initiated by Although one could envisage other ways of solving the time- a semiclassical calculation of Dyakonov and Gornyi ͓2͔ and dependent Schro¨dinger equation which could possibly continued by a quantum-mechanical calculation in perturba- handle the real situation, this is beyond the scope of the tion theory of Papenbrock and Bertsch ͓3͔ and a more de- present article, and we refer the reader to other methods, e.g., tailed comparison of quantum-mechanical calculations with those discussed in Ref. ͓10͔. classical and semiclassical results by Takigawa et al. ͓4͔, For an ␣ particle being accelerated from the turning point shows a complicated interference pattern arising from the to infinity, the classical bremsstrahlung can be calculated contributions to bremsstrahlung from inner, under the bar- analytically. Using well-known equations, see, for example rier, and outer, in the classically allowed region, parts of the ͓11,12͔, and integrating along the outward branch of the Ru- wave function. Currently the experimental data are not con- therford trajectory in a head-on collision we get for the en- clusive, and more exclusive experiments, with better statis- ergy emitted by bremsstrahlung per frequency interval d␻, tics, are needed to give a clearer understanding of the phe- in the long-wavelength approximation, nomena. ͓ ͔ The theoretical approaches of Refs. 3,4 assume a stan- 8␲␻2 dard stationary description of quantum tunneling which is dE͑␻͒ϭ Z2 e2͉p ͑␻͉͒2, ͑1͒ 2 3 eff r very successful for ␣-decay lifetime and probabilities. How- 3m c ever, a time-dependent picture of a decay process ͓8,9͔ may be essential in understanding physics of the bremsstrahlung where and similar processes, in particular, in obtaining the appro- priate bremsstrahlung spectrum. This can be shown, for ex- ma Ϫ␲␯/2 ample, by looking into a case where there is no Coulomb p ͑␻͒ϭ e KЈ␯͑␯͒ ͑2͒ r 2␲ i acceleration after the tunneling. In this Rapid Communica- tion we model the time evolution of a wave function during tunneling using again ␣ decay as an example. We do not is the Fourier transform of the particle momentum, m ϭ ϩ ϭ attempt to compare our results with experimental data. The mN 4A/(A 4) is the reduced mass, and Zeff (2A Ϫ • ϩ ␣ ϩ reason is simple. The ␣ decay time, e.g., the lifetime of 4Z)/(A 4) is the effective charge of the particle the 210Po, is many orders of magnitude larger than typical times daughter nucleus (A,Z) in the dipole approximation. In Eq. ͑ ͒ ϭ ˆ ϭ 2 ␯ϭ ប for an ␣-particle to traverse across the nucleus. This implies 1 , pr p r, a Ze /E␣ , E␣a/ v0 , E␣ is the energy of ␣ • the particle, and v0 its asymptotic velocity. The functions Ki␯(x) are the modified Bessel functions of imaginary order, Ј *Electronic address: [email protected] and Ki␯(x) are their derivatives with respect to the argument. † Electronic address: [email protected] Dividing this equation by the photon energy E␥ϭប␻,weget ‡ Electronic address: [email protected] the differential probability per unit energy for the brems- 0556-2813/99/60͑3͒/031602͑4͒/$15.0060 031602-1 ©1999 The American Physical Society RAPID COMMUNICATIONS C. A. BERTULANI, D. T. de PAULA, AND V. G. ZELEVINSKY PHYSICAL REVIEW C 60 031602 ͑ FIG. 1. Average momentum ͑in MeV/c) for ␣ energies of 23.1 FIG. 2. Classical bremsstrahlung emission probability dotted ͒ ͑ ͒ MeV ͑a͒ and 20 MeV ͑b͒.In͑a͒ the dotted line is the classical lines , quantum-mechanical spectrum dashed lines using only the momentum for an ␣ particle running away from the closest ap- part of the wave function outside the barrier, and using the full ͑ ͒ proach distance. The dashed line is the average momentum calcu- space wave function solid lines . The upper part of the graph is for ϭ ϭ lated according to Eq. ͑4͒, but using only the part of the wave E␣ 23.1 MeV, while the lower part is for E␣ 20 MeV. function outside the barrier, i.e., the integral in Eq. ͑4͒ extending ϭ 2 1͑b͔͒. The ␣-energies are changed by keeping the number of from Rcl 2Ze /E␣ to infinity. The solid line represents the mo- mentum calculated according to Eq. ͑4͒, but including the full space nodes constant ͑in this case, seven nodes͒, and varying the range of the wave function. depth of the potential from 14 to 18 MeV, respectively. The barrier height is 27.6 MeV. The dotted line is the classical strahlung, i.e., dP/dE␥ϭ(1/E␥)dE(␻)/dE␥ . At low photon momentum for an ␣ particle running away from the closest ␯2 Ϫ␲␯͓ Ј ␯ ͔2 approach distance. The dashed line is the average momentum energies, we use the relation e Ki␯( ) ˜1 to show that calculated according to Eq. ͑4͒, but using only the part of the wave function outside the barrier, i.e., the integral in Eq. ͑4͒ dP 2 1 v 2 ϭ 2 ϭ 2 ␣ͩ 0 ͪ ͑ ͒ extending from Rcl 2 Ze /E␣ to infinity. In this case, the Zeff . 3 ͑ ͒ dE␥ 3␲ E␥ c wave function entering 4 is normalized within this space range. We solve the time-dependent Schro¨dinger equation for The resulting ‘‘quantum-mechanical’’ momentum at the the ␣ particle in a potential well ϩ Coulomb barrier, and later stage is closer to the classical one. However, the calculate the radial momentum from quantum-mechanical momentum increases slower than the classical one, due to the extended nature of the particle’s ץ ប u͑r,t͒ wave function. As a consequence, one expects that the Fou- p ͑t͒ϭ ͵ dru*͑r,t͒ , ͑4͒ -r rier transform of the quantum momentum has its higher freץ r i quencies suppressed, as compared to the classical case. The where u(r,t) is the radial part of the total wave function. A solid line represents the momentum calculated according to similar numerical study of the time evolution for the problem Eq. ͑4͒, but including the full space range of the wave func- of the ␣ decay was performed by Serot et al. ͓9͔. tion. One observes a wiggling pattern associated with the The time-dependent Schro¨dinger equation is solved, start- interference of neighboring quasistationary states of the par- ing with an s-wave wave function for an ␣ particle confined ticle inside the nucleus. The leaking of the inner part of the ϭϪ Ͻ ϭϪ in a radial well V(r) V0 for r R0, and V(r) V0 wave function creates an effective oscillating dipole that ϩ 2 у ϭ 2Ze /R0 for r R0.WesetR0 8.75 fm, realistic for the emits radiation. The same leaking creates the perturbation ␣ϩ210Po case ͓3͔.Attϭ0 we switch the potential to a mixing different resonance states. The Fourier transform of Ϫ Ͻ spherical square well with depth V0 for r R0 and a Cou- the particle momentum should therefore contain appreciable 2 у lomb potential 2Ze /r for r R0 which triggers the tunnel- amplitudes associated with this motion, as observed in Fig. ing. At a given time t the wave function is found by using the 2, solid line. Asymptotically, the momenta calculated in all standard method of inversion of a tridiagonal matrix at each different ways coincide at large times. time step increment ⌬t ͑see, e.g., Ref. ͓15͔͒. The space is In Fig. 2 the dotted lines show the classical bremsstrah- discretized in steps of ⌬xϭ0.05 fm, and the time step used is lung emission probability, the dashed lines show the ⌬tϭmc(⌬x)2/ប.
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