Diapositiva 1

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Intrinsic Localized modes A mechanism for the
migration of defects
Jesús Cuevas Maraver Nonlinear Physics
Group Universidad de Sevilla
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Outline

• Nonlinear lattice dynamics Phonons vs Intrinsic
  Localized Modes (Discrete Breathers)
• A classical nonlinear model Frenkel-Kontorova
• Stationary and moving breathers
• Point defects Impurities, vacancies and
  interstitials
• Interaction between discrete breathers and
  vacancies
• Double vacancies and interstitials
• Conclusions
• References
• J. Phys. A 35 (2002) 10519
• Phys. Lett. A 315 (2003) 364

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Nonlinear lattices

• A great number of systems can be described by
  oscillator networks (crystals, biomolecules,
  Josephson-junctions arrays)
• The interactions between the oscillators is
  nonlinear, although most of the times they are
  approximated by linear functions
• An oscillator network is described by the
  following Hamiltonian

Interaction potential
On-site potential
Kinetic Energy
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Linear lattices
Linear vibrational modes phonons
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Nonlinear lattices
Phonons Intrinsic localized modes (breathers)
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Discrete breathers

• Exact periodic and localized solutions of the
  dynamical equations that exist due to
  nonlinearity and discreteness.
• They exist as long as two conditions are
  fulfilled (MacKay-Aubry theorem, 1994)
• The on-site potential is nonlinear
• The breather frequency does not resonate with
  phonons
• They have been generated in Josephson-junction
  arrays and observed in molecular crystals (PtCl).
• They are speculated to play an important role in
• DNA transcription and denaturation bubbles
• The appearance of dark lines of mica muscovite
• Reconstructive transformations in layered
  silicates

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The Frenkel-Kontorova model

• Introduced in 1938 to study the dynamics of
  dislocations.
• It consists of a on-site periodic potential
  (sine-Gordon)

• The particles are located at the bottom of the
  potential

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Breathers in Frenkel-Kontorova

• The Frenkel-Kontorova model supports discrete
  breathers due to the nonlinearity of the on-site
  potential

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Mobile breathers

• In some conditions, a static breather can be
  perturbed leading to a mobile state.
• These solutions are not exact can only be
  observed through numerical simulations.
• Contrary to static breather, they are not
  supported in every nonlinear lattices.
• One of the systems supporting moving breathers is
  the Frenkel-Kontorova model.

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Mobile breathers in Frenkel-Kontorova
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Point defects

• The Frenkel-Kontorova is the simplest way of
  modelling vacancies (an empty well) and
  interstitials (two particles in the same well).

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Moving breathers and vacancies

• We consider a modified Frenkel-Kontorova model
  with a nonlinear interaction potential

• W(x) is the Morse potential. b is the inverse
  width of the potential

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Moving breathers and vacancies

• We have studied the effect of varying the
  potential width in the interaction.
• For each value of the potential width, the
  interaction is studied in function of the kinetic
  energy of the moving breather.
• When the moving breather reaches the vacancy, the
  latter can move forwards, backwards or remain at
  rest. However, there is no correlation between
  the kinetic energy and the number of vacancy
  jumps

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Critical values

• Statistical analysis

• For bgtbf the vacancy does not move forwards.
• For bb1ltbltbb2 there exist a critical value of the
  kinetic energy for vacancy movement

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Vacancy moving backwards
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Vacancy moving backwards
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Vacancy moving forwards
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Vacancy moving forwards
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Double vacancy

• This configuration needs an narrow interaction
  potential.
• The double vacancy does not move forwards.
  Instead, it can be broken.
• The observed regimes are now rest, breaking and
  forwards movement (with breaking). The latter
  needs b to be small enough.
• The threshold kinetic energy is also observed.

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Double vacancy breaking
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Interstitials

• Preliminary results.
• A threshold kinetic energy is always observed.
• For bgtbc, the interstitial always moves forwards.
• For bltbc, the interstitial can move backwards,
  forwards or remain at rest.

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Interstitial moving forwards
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Conclusions

• We have described the interaction between moving
  breathers and vacancies when the interaction
  potential width is varied.
• Two critical values of the width exist
• Vacancy forwards movement
• Threshold kinetic energy
• These critical values can be determined through
  the existence and stability analysis of static
  breathers in the neighborhood of the vacancy.
• More information
• http//www.grupo.us.es/gfnl