Download Diapositiva 1 - Indico - Universidad de los Andes

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+BA+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+πσπ+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!