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Tunneling and Collision times of Unstable Particles and Nuclei N. G. Kelkar Universidad de los Andes, Bogota, Colombia I Collision times (i) Wigner´s time delay (ii) Smith´s generalization (iii) Applications and Modifications II Tunneling times (i) Dwell times and Phase times (ii) Where has the alpha been? Talk given at the Third Workshop on Nuclei and Mesoscopic Physics Michigan State University, March 2011 I COLLISION TIMES (i) Wigner and Eisenbud’s time delay of a radial wave packet The first term has a peak at and the second one at The interaction has delayed the radial wave packet by E.P. Wigner, Phys. Rev. 98, 145 (1955) A more general proof of Wigner’s radial time delay Here Now replace by For scattering with multiple channels The complex multichannel S-matrix can in general be parameterized as This is indeed Smith’s single channel time delay for a particle injected into the ith channel and going out into the jth channel (ii) SMITH´S GENERALIZATION: THE LIFETIME MATRIX Starting with Smith’s single channel time delay one can construct an average time delay (weighted by the probabilities) which are diagonal elements of the lifetime matrix Q The Q matrix has some elegant properties: F. T. Smith, Phys. Rev. 118, 349 (1960) (iii) APPLICATIONS An intuitive picture (delay due to the creation and propagation of a resonance): Time delay in a resonant (R) scattering process is much larger than in a non-resonant (NR) scattering process A+BA+B A Pedagogic example of time delay – the deuteron The S-matrix for a neutron-proton system constructed from a square well potential which reproduces the correct binding energy of the deuteron is given as a function of l as, We use the above with What do we expect? Bound state never decays, time delay must be infinite at the real negative energy where the bound state occurs Virtual state also occurs at a real negative energy. However the wave function is not normalizable, state is unphysical, hence, infinite negative time delay Quasibound (or unstable bound) state – complex energy Bound real part of energy is negative, finite imaginary part gives width … positive finite time delay at real negative energy n-p system has one bound state at -2.22 MeV – the deuteron n-p system has one virtual state at -0.1 MeV We put a small imaginary potential and get a fictitious quasibound state too. A more realistic example from the unstable states of the eta mesons and nuclei S-matrix Pole in the complex momentum plane N. G. Kelkar, K. P. Khemchandani, B. K. Jain, J. Phys G 32, 1157 (2006) Time delay peak Applications of the Time Delay Method to Characterize Resonances 1. UNFLAVOURED BARYON RESONANCES N. G. Kelkar, M. Nowakowski, K. P. Khemchandani and S. R. Jain, Nucl. Phys. A 730, 121 (2004). 2. MESON RESONANCES N. G. Kelkar, M. Nowakowski, K. P. Khemchandani, Nucl. Phys A 724, 357 (2003) 3. EXOTIC – PENTAQUARK STATES N. G. Kelkar, M. Nowakowski, K. P. Khemchandani, Mod. Phys. Lett. A 19, 2001 (2004) 4. EXOTIC QUASIBOUND STATES OF ETA-MESIC NUCLEI N. G. Kelkar, K. P. Khemchandani and B. K. Jain, J. Phys. G 32, 1157 (2006); J. Phys. G 32, L19 (2006); N. G. Kelkar, Phys. Rev. Lett. 99, 210403 (2007). Time delay in π N π N elastic scattering P33 Δ (1232) π N (100%) D13 N(1520) π N (50 – 60 %) 8Be levels from time delay in alpha alpha elastic scattering α + α 8Be α + α B = S = +1 baryonic state Ѳ (1540) Exotic states which historically appeared (1960´s), disappeared for 20 yrs, reappeared (2003) … remain controversial There exist several ways to characterize a resonance: 1. S-matrix pole (most standard) 2. Argand diagrams 3. Cross section bumps 4. Speed Plots Time delay seems to be a useful complementary tool to characterize resonances at all energies and in all partial waves except the near threshold resonances in s-waves (l = 0) With the scattering phase shift, we get, and hence For l = 0, Wigner’s time delay blows up near threshold. One more application … determination of the non-exponential decay law Spontaneous decay is inherently a quantum mechanical process The exponential nature of the decay law which appears in many branches of physics and follows from classical physics is an approximation Quantum mechanical survival probability (apart from very short times) displays three regions: 1. A dominant exponential (E) law – exp(-Г t) 2. Transition region 3. Non-exponential (NE) power law at large times Critical time tC for transition from E to NE depends on the width of the state Narrow states tC is large exponentially decaying sample diminishes to an unmeasurable amount Broad states tC is small but Г large and hence exp(-Г t) reduces sample rapidly Nature conspires to hide the non-exponential decay law at large times Quantum mechanically the survival probability is: Since the unstable state cannot be an eigenstate of the (hermitian) Hamiltonian we expand it (assuming a continuous energy spectrum) in terms of the energy eigenstates of the decay products Thus in the Fock-Krylov method Is the probability density of states in a resonance Typically this probability density is taken to be of a Breit-Wigner form in the vicinity of the resonance: Interesting to note is the fact that if one starts with a Breit-Wigner amplitude or phase shift, one can derive: In calculating the second virial coefficients for the equation of state in an ideal gas Beth and Uhlenbeck found that the difference in the density of states with and without interaction is given by the energy derivative of the scattering phase shift Since time delay is the time spent with interaction – time spent without interaction Once the scattering phase shift is known from experiment the survival probability can be found and the non-exponential decay law determined Starting from the scattering phase shift data gives the experimental spectral density The survival amplitude and hence the survival probability is thus determined from experimental data The survival probability of 8Be α + α decay Non-exponential decay law at large times determined from an analysis of scattering phase shift data N. G. Kelkar, M. Nowakowski and K. P. Khemchandani, Phys. Rev. C 70, 024601 (2004) A peculiar case: the σ meson Using a parametrization of the scattering amplitude which fits data on π π elastic scattering, the spectral density is given by: Parametrization by D. V. Bugg, Phys. Lett. B 572, 1 (2003) π+πσπ+π Mass and width of the σ ER=542 MeV ГR = 498 MeV The only realistic case of a hadron with a non-exponential decay law at all times! N. G. Kelkar and M. Nowakowski, J. Phys. A 43, 385308 (2010) II TUNNELING TIMES In an attempt to the answer the question … How much time does a particle need to tunnel through a barrier? Physicists invented several time concepts such as: Dwell time (sojourn or residence time) Phase time (extrapolated phase time) Larmor time Group delay time Traversal time, Buettiker Landauer time and even complex times The physical significance of these times and the subject in general remains controversial A RATHER SIMPLE DEFINITION Dwell Time = Probability of finding the particle in a given region of space _________________________________________________ Flux through the surface Starting classically: with Among most definitions of dwell time, the current appears outside the integral Kapur and Peierls - 1938 Proc. of the Royal Society of London A 166, 277 (1938). Dwell time – a by product of the formalism for cross sections with resonances in nuclear reactions. With typical pole of an unstable state Kapur and Peierls did not identify this expression as a dwell time, but simply as the width of a resonant level. A similar derivation by Smith followed much later in 1960. Phase time and dwell time Reflection amplitude: Transmission amplitude: Following the peak of a “sharp” wave packet (transmitted) Transmission phase time : Similarly, the reflection phase time: H. G. Winful, Phys. Rev. Lett. 91, 260401 (2003) In 3-dim collisions: N. G. Kelkar, Phys. Rev. Lett. 99, 210403 (2007). Time delay of eta-mesic states (from a theoretical 3-body calculation) Dwell time rather than the phase time delay displays the correct threshold behaviour N. G. Kelkar, Phys. Rev. Lett. 99, 210403 (2007). Where has the alpha been? A tunneling time approach to evaluate alpha decay half-lives Radioactive nucleus cluster of its daughter nucleus + alpha particle (4He) Alpha Decay tunneling of alpha through the Coulomb barrier of the alpha-daughter system - Nuclear interaction - Coulomb - Langer term which ensures the correct behaviour of the WKB radial wave function near the origin Different methods to evaluate the widths A. The two potential approach: S. A. Gurvitz and G. Kalbermann, Phys. Rev. 59, 262 (1987); S. A. Gurvitz, Phys. Rev. A 38, 1747 (1988). B. Double Humped Barrier H. K. Shepard, Phys. Rev. D 27, 1288 (1983) C. Path Integral Method G. Drukarev, N. Froeman and P. O. Froeman, J. Phys. A 12, 171 (1979). A critical view of WKB widths can be found in N. G. Kelkar and H. M. Castañeda, Phys. Rev. 76, 064605 (2007) The Alpha-Nucleus Potential A double folding model with realistic nucleon-nucleon interactions , - Densities of the alpha and daughter nucleus - nucleon-nucleon interaction potential distance between a nucleon in the alpha and a nucleon in the daughter The M3Y NN interaction: G. R. Satchler and W. G. Love, Phys. Rep. 55, 183 (1979); A. M. Kobos, G. R. Satchler and A. Budzanowski, 384, 65 (1982) The alpha particle density is given using a standard Gaussian form: The daughter nucleus density is taken to be: Is obtained by normalizing the nuclear density to the number of nucleons The Coulomb potential is obtained using a similar double folding procedure In what follows we look at the half-lives of heavy, super heavy nuclei and the light nucleus 8Be. Dwell times in alpha tunneling Recall the standard definition of (average) dwell time: One can also define the transmission or reflection dwell time: M. Goto et al., J. Phys. A 37, 3599 (2004) Writing the wave functions in region I and II Using the Wentzel-Kramers-Brillouin (WKB) semiclassical approximation, the dwell times in region I and II are: RECALL: - width - Transmission coefficient - Dwell time in region I The decay width is given by the inverse of the transmission dwell time in front of the barrier N.G. Kelkar, H.M. Castañeda and M. Nowakowski, Europhys. Lett. 85, 20006 (2009). The half-life of alpha decay which is evaluated at E=Q (amount of energy released in the decay) is then given by Arises from the normalization of the bound wave within the WKB approximation This factor is related to the assault frequency of the tunneling particle The period defined this way turns out to be the definition of traversal time as given by Buettiker and Landauer The assault frequency we already saw: with The number of assaults that the particle makes before tunneling is: Half-lives of medium heavy nuclei Half-lives of super heavy nuclei Number of assaults Transmission coefficient tiny reflection dwell times << transmission dwell times The decay of 8Be α + α Transmission coefficient Reflection and transmission dwell times are comparable Alpha decay half-lives The (transmission) dwell time calculations of alpha decay indicate that the alpha particle spends a major part of its lifetime in front of the barrier (in case of alpha decay of heavy and super heavy nuclei) The light nuclear case is different: for 8Be α + α the alpha spends the same order of time in front of the barrier and within it. The entire lifetime – should be given by the sum of times in front of the barrier as well as within the barrier SUMMARY 1. Wigner´s phase time delay finds applications in the characterization of nuclear and particle unstable states (with the exception of s-wave states near threshold) 2. Dwell time delay - closely related to phase time delay and valid for all energies as well as all partial waves 3. A new look at the tunneling problem of nuclear decay is provided through the dwell time calculations in alpha decay 4. The connection between density of states and time concepts allows the extraction of the non exponential decay from an analysis of experimental data on scattering of the decay products ¡GRACIAS!