^{1,a)}, Giovanni Ciccotti

^{2,b)}and Simone Meloni

^{2,c)}

### Abstract

To locate the position and characterize the dynamics of a vacancy in a crystal, we propose to represent it as the ground state density of a pseudo–quantum probe particle associated to the Hamiltonian which has, for the potential energy, the field generated by the atoms in the sample. In this description, the coefficient of the kinetic energy term is a tunable parameter controlling the density localization in the regions of the relevant minima of the potential energy field. Based on this description, we derive a set of collective variables that we use in rare event simulations to identify some of the vacancy diffusion paths in a 2D crystal. Our simulations reveal that the vacancy migrates according to local and non-local mechanisms, the second involving several lattice sites and producing a long range migration. We also observed a vacancy induced crystal reorientation process.

The research proposed here originated from discussions with Eric Vanden-Eijnden at a summer school in Italy in 2008; we would like to thank Eric for the fruitful discussions and suggestions. We would like to thank Sara Bonella for the careful reading of the manuscript and the many stimulating discussions and suggestions. S.M. acknowledges financial support from the European Community under the Marie Curie Intra-European Fellowship for Career Development Grant No. 255406. G.C. and S.M. acknowledge financial support from SFI Grant No. 08- IN.1-I1869 and the Istituto Italiano di Tecnologia under the SEED project Grant No. 259 SIMBEDD - Advanced Computational Methods for Biophysics, Drug Design and Energy Research. S.M. acknowledges financial support from the MIUR-FIRB Grant No. RBFR10ZUUK.

I. INTRODUCTION

II. THE PSEUDO-QUANTUM PROBE PARTICLE REPRESENTATION TO IDENTIFY A VACANCY IN A SYSTEM

III. COLLECTIVE VARIABLES TO STUDY VACANCY MIGRATION BY RARE EVENT SIMULATIONS

IV. RESULTS AND DISCUSSION

A. Use of the pseudo-quantum probe particle to identify the vacancy in a 2D crystal

B. Vacancydiffusion path in a 2D WCA crystal by Temperature Accelerated Monte Carlo

V. CONCLUSIONS

### Key Topics

- Vacancies
- 113.0
- Diffusion
- 12.0
- Wave functions
- 11.0
- Free energy
- 10.0
- Ground states
- 10.0

## Figures

Series of atomic configurations along the local migration path in a 2D trigonal lattice of WCA particles superimposed to the corresponding *V*(**x**; **R**) potential (see Sec. IV A for all the details). *V*(**x**; **R**) is reported in Lennard-Jones units (these units are used throughout the text) and represented according to the colormap reported in the bar at the bottom of the figure. The simulation box is also reported in the figure (periodic boundary conditions are used in all calculations discussed in the text). The (ideal) local migration path is the (linear) path followed by a vacancy migrating from a lattice site to one of its nearest neighbor sites (indicated by the white arrow in the top panel). Panel (a) corresponds to the configuration in which the vacancy is located on a lattice site. In the next two panels, (b) and (c), the atomic configuration and the potential *V*(**x**; **R**) at intermediate states between the initial and the mid-way states are shown. In these states, the potential *V*(**x**; **R**) already presents the characteristic “double-well” shape. In panel C is reported the mid-way configuration and the corresponding potential *V*(**x**; **R**). In this state, the two wells of *V*(**x**; **R**) are symmetric.

Series of atomic configurations along the local migration path in a 2D trigonal lattice of WCA particles superimposed to the corresponding *V*(**x**; **R**) potential (see Sec. IV A for all the details). *V*(**x**; **R**) is reported in Lennard-Jones units (these units are used throughout the text) and represented according to the colormap reported in the bar at the bottom of the figure. The simulation box is also reported in the figure (periodic boundary conditions are used in all calculations discussed in the text). The (ideal) local migration path is the (linear) path followed by a vacancy migrating from a lattice site to one of its nearest neighbor sites (indicated by the white arrow in the top panel). Panel (a) corresponds to the configuration in which the vacancy is located on a lattice site. In the next two panels, (b) and (c), the atomic configuration and the potential *V*(**x**; **R**) at intermediate states between the initial and the mid-way states are shown. In these states, the potential *V*(**x**; **R**) already presents the characteristic “double-well” shape. In panel C is reported the mid-way configuration and the corresponding potential *V*(**x**; **R**). In this state, the two wells of *V*(**x**; **R**) are symmetric.

*V*(**x**; **R**) (left) and ρ(**x**|**R**) (right) for four atomic configurations along the ideal local vacancy migration path in a system of WCA particles.

*V*(**x**; **R**) (left) and ρ(**x**|**R**) (right) for four atomic configurations along the ideal local vacancy migration path in a system of WCA particles.

*V*(**x**; **R**) (left) and ρ(**x**|**R**) (right) for a system of hard disks in the equilibrium and mid-way configurations along the ideal local migration path. The blue color in the left-hand panel denotes *V*(**x**; **R**) = 0; the red *V*(**x**; **R**) = ∞.

*V*(**x**; **R**) (left) and ρ(**x**|**R**) (right) for a system of hard disks in the equilibrium and mid-way configurations along the ideal local migration path. The blue color in the left-hand panel denotes *V*(**x**; **R**) = 0; the red *V*(**x**; **R**) = ∞.

and as a function of *a* at the configurations of panels (a) (●), (b) (■), (c) (◆), and (d) (**x**) of Fig. 1 . *a* is reported on a logarithmic scale. Vertical lines are inserted to facilitate the comparison of values of and at a given value of *a*. Zone 1, 2, and 3 denote the three regions of *a* in which and have different behaviors (see text).

and as a function of *a* at the configurations of panels (a) (●), (b) (■), (c) (◆), and (d) (**x**) of Fig. 1 . *a* is reported on a logarithmic scale. Vertical lines are inserted to facilitate the comparison of values of and at a given value of *a*. Zone 1, 2, and 3 denote the three regions of *a* in which and have different behaviors (see text).

*V*(**x**; **R**) (left) and ρ(**x**|**R**) (right) at few atomic configurations selected along the migration event observed during an unbiased simulation at *T* = 2.5. The arrows in the left column point to representative narrow minima of the potential *V*(**x**; **R**) not associated to the vacancy (see text).

*V*(**x**; **R**) (left) and ρ(**x**|**R**) (right) at few atomic configurations selected along the migration event observed during an unbiased simulation at *T* = 2.5. The arrows in the left column point to representative narrow minima of the potential *V*(**x**; **R**) not associated to the vacancy (see text).

The two components, and , of the first moment (top) and the trace of the second moment (bottom) of the density ρ(**x**|**R**) as measured along a high temperature (*T* = 2.5) simulation. The arrow in the bottom panel indicates a failed migration event.

The two components, and , of the first moment (top) and the trace of the second moment (bottom) of the density ρ(**x**|**R**) as measured along a high temperature (*T* = 2.5) simulation. The arrow in the bottom panel indicates a failed migration event.

Potential (left) and density ρ(**x**|**R**) (right) computed at configurations taken along the avoided migration event around timestep 5600 of the high temperature MD simulation. The white cross indicates the position of . When the vacancy is located at a crystal site (top and bottom panels), is approximately at the maximum of ρ(**x**|**R**) and the vacancy minimum of the potential . During the attempted migration event (central panels), the main peak of the density is shifted towards the bottom-left with respect to , while a long tail is formed at its top-right.

Potential (left) and density ρ(**x**|**R**) (right) computed at configurations taken along the avoided migration event around timestep 5600 of the high temperature MD simulation. The white cross indicates the position of . When the vacancy is located at a crystal site (top and bottom panels), is approximately at the maximum of ρ(**x**|**R**) and the vacancy minimum of the potential . During the attempted migration event (central panels), the main peak of the density is shifted towards the bottom-left with respect to , while a long tail is formed at its top-right.

Comparison of the first moment and the trace of the second moment of the density ρ(**x**|**R**) as obtained from (unbiased) MC at *T* = 0.5 (left) and TAMC at the same *T* and at *T** = 8.5 (right). The first component of ( ) is denoted by a continuous black line, while the second component ( ) by a dotted red line. The dashed ellipses indicate events that are discussed in the text. The inset shows a zoom of the along the TAMC simulation in the interval in which the system was in a reoriented crystalline state (see text). The red line in the inset corresponds to the bottom of the curve in the simulation time complementary to the inset interval.

Comparison of the first moment and the trace of the second moment of the density ρ(**x**|**R**) as obtained from (unbiased) MC at *T* = 0.5 (left) and TAMC at the same *T* and at *T** = 8.5 (right). The first component of ( ) is denoted by a continuous black line, while the second component ( ) by a dotted red line. The dashed ellipses indicate events that are discussed in the text. The inset shows a zoom of the along the TAMC simulation in the interval in which the system was in a reoriented crystalline state (see text). The red line in the inset corresponds to the bottom of the curve in the simulation time complementary to the inset interval.

Trajectories along a local (a) and non-local (b) migration event. The particles are colored in red at the beginning of the trajectory and, passing by green, become blue at the end. The boundary of the simulation box is also reported.

Trajectories along a local (a) and non-local (b) migration event. The particles are colored in red at the beginning of the trajectory and, passing by green, become blue at the end. The boundary of the simulation box is also reported.

Atomic configurations corresponding to the misoriented crystal. The stress due to the wrong orientation is accommodated by the formation of a wide vacancy defect. During the interval ∼(2.3–5) × 10^{5} TAMC steps the misoriented crystal maintained the same geometry (trigonal) but its orientation and the position of the vacancy changed several times. The misorientation of the crystal is emphasized by reporting on the plot the original lattice vectors *a* and *b*, and the lattice vectors of the reoriented crystal, *a*′ and *b*′, and *a*″ and *b*″.

Atomic configurations corresponding to the misoriented crystal. The stress due to the wrong orientation is accommodated by the formation of a wide vacancy defect. During the interval ∼(2.3–5) × 10^{5} TAMC steps the misoriented crystal maintained the same geometry (trigonal) but its orientation and the position of the vacancy changed several times. The misorientation of the crystal is emphasized by reporting on the plot the original lattice vectors *a* and *b*, and the lattice vectors of the reoriented crystal, *a*′ and *b*′, and *a*″ and *b*″.

3D representation of the classical (a) and quantum (b) densities in the finite temperature system at the same configuration of the third panel of Fig. 5 . The classical density has been computed at = 0.1, the lowest with associated negligible density values at crystal minima. The scales of the panels (a) and (b) are set such that the densities’ maxima have a comparable magnitude.

3D representation of the classical (a) and quantum (b) densities in the finite temperature system at the same configuration of the third panel of Fig. 5 . The classical density has been computed at = 0.1, the lowest with associated negligible density values at crystal minima. The scales of the panels (a) and (b) are set such that the densities’ maxima have a comparable magnitude.

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