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Article

Simulation of the Dynamics of Supersonic N-Crowdions in fcc Lead and Nickel

by
Ayrat M. Bayazitov
1,†,
Alexander S. Semenov
2,† and
Sergey V. Dmitriev
1,2,*,†
1
Institute of Molecule and Crystal Physics, UFRC RAS, 450054 Ufa, Russia
2
Polytechnic Institute (Branch) in Mirny, North-Eastern Federal University, Tikhonova St. 5/1, 678170 Mirny, Russia
*
Author to whom correspondence should be addressed.
†
These authors contributed equally to this work.
Micro 2023, 3(3), 632-642; https://doi.org/10.3390/micro3030044
Submission received: 17 May 2023 / Revised: 7 June 2023 / Accepted: 19 June 2023 / Published: 4 July 2023
(This article belongs to the Section Microscale Materials Science)
Browse Figures
Review Reports Versions Notes

Abstract

:
In the case where an interstitial atom is located in a close-packed atomic row of the crystal lattice, it is called a crowdion. Crowdions play an important role in the processes of mass and energy transfer resulting from irradiation, severe plastic deformation, ion implantation, plasma and laser processing, etc. In this work, supersonic N-crowdions ( N = 1 ,   2 ) in fcc lattices of lead and nickel are studied by the method of molecular dynamics. Modeling shows that the propagation distance of a supersonic 2-crowdion in lead at a high initial velocity is less than that of a supersonic 1-crowdion. In other fcc metals studied, including nickel, supersonic 2-crowdions have a longer propagation distance than 1-crowdions. The relatively short propagation distance of supersonic 2-crowdions in lead is due to their instability and rapid transformation into supersonic 1-crowdions. This feature of the dynamics of supersonic N-crowdions in lead explains its high radiation-shielding properties.

1. Introduction

High-energy impacts on metals and alloys, such as laser treatment [1], plastic deformation [2], high-speed deformation [3,4], neutron irradiation [5,6,7,8,9,10], plasma surface treatment [11], etc., bring the crystal lattice into a non-equilibrium state and cause the formation of defects in the crystal structure [12]. These processes are accompanied by a strong deviation of atoms from equilibrium positions, which leads to an increase in the role of nonlinearity in interatomic interactions. In a nonequilibrium state, vacancies and interstitial atoms (Frenkel pairs) are intensively generated. An interstitial atom can occupy various positions in the crystal lattice; if it is located in a close-packed atomic row, it is called a crowdion [13,14,15].
Crowdions have a low migration barrier compared to vacancies [16,17]; therefore, they make a significant contribution to the processes of mass and energy transfer in metals. Due to their high migration ability, crowdions quickly annihilate, which complicates the experimental study of their dynamics [18]. This leads to an increase in the role of computer simulation in the study of the movement of crowdions, including molecular dynamics [19,20,21,22,23,24,25], the Monte Carlo method [26,27], ab initio calculations [28,29], and multiscale simulations [14,30,31,32].
The movement of the crowdion occurs as a result of a sequence of replacement collisions [33]. A supersonic crowdion can be excited by imparting a sufficiently large initial momentum to the atom in a close-packed direction; as a result, a vacancy is formed, and the interstitial atom performs a relay-race motion. If the initial velocity of an atom is insufficient for the formation of a crowdion, then a focuson appears [33]. Focuson relaxation leads to the restoration of an ideal crystal lattice. The elastic stress fields created by a static crowdion or a crowdion moving at subsonic velocity have been studied in many works [28,32,34,35], while crowdions moving at a supersonic velocity are much less studied. They were considered in two-dimensional triangular lattices [36,37], fcc lattices [29,38,39], Ni 3 Al alloy [40] and bcc lattices [41].
The next step in the study of supersonic crowdions was the introduction of the concept of supersonic N-crowdion, where instead of one atom, N neighboring atoms are provided with the same initial velocity along the close-packed atomic row [36,38,39]. Supersonic N-crowdions require less energy for excitation, while they travel a much longer distance compared to supersonic 1-crowdions, due to the more self-focusing nature of the motion of supersonic N-crowdions [38,42]. The bombardment of the crystal surface by biatomic molecules can excite a supersonic 2-crowdion ( N = 2 ) in a close-packed atomic row [39]. This study is important in such processes as, for example, ion implantation.
The present work is aimed at studying the dynamics of supersonic N-crowdions ( N = 1 , 2) in the crystal lattices of lead and nickel by molecular dynamics modeling using many-body interatomic potentials. Lead is of particular interest due to its ability to absorb X-ray and gamma radiation. In this regard, lead is used in X-ray installations and nuclear reactors.

2. Materials and Methods

Molecular dynamics simulations are based on a solution of Newtonian equations of motion which are integrated using the Verlet integration scheme with the time step of 0.5 fs. Simulations are performed using the Large-Scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). In this study, the interatomic potentials constructed following the embedded atom method (EAM) for fcc lead [43] and nickel [44,45,46] are employed. To visualize the results obtained, the OVITO program is used [47].
There are six close-packed directions in the fcc lattice, 〈 110 〉 , 〈 1 ¯ 10 〉 , 〈 101 〉 , 〈 1 ¯ 01 〉 , 〈 011 〉 , and 〈 0 1 ¯ 1 〉 . The Cartesian coordinate axes are chosen so that the x, y and z axes are directed along the crystallographic directions 〈 110 〉 , 〈 1 ¯ 10 〉 and 〈 001 〉 , respectively. With this choice, one of the close-packed directions in the computational cell is along the x axis. The equilibrium lattice constant at zero temperature found for the used potential is a = 4.9494 Å, and the interatomic distance is d = a/ 2 = 3.4997 Å. The elementary translational cell has the shape of a parallelepiped and contains 8 atoms. The number of unit cells along the x, y and z axes is set to be 50, 25 and 25, respectively. The computational cell contains 250,000 atoms, and its volume is 350 × 175 × 124 Å 3 .
Periodic boundary conditions are used for all three coordinate directions. The excitation of 1- and 2-crowdions is achieved by imparting at time t = 0 an initial momentum to one or two atoms, respectively, see Figure 1. The initial velocity component V x 0 varies from 40 to 150 Å/ps. Along the y and z axes, small components of the initial velocity V y 0 = V z 0 = 10 − 6 Å/ps are introduced to analyze the effect of small disturbances on the crowdion dynamics. In most cases, molecular dynamics modeling is carried out at a temperature of 0 K using the NVE thermodynamic ensemble (a constant number of atoms, the volume of the computational cell, and the total energy of the system). For some cases, the effect of temperature T = 297 K is also analyzed. In this case, the initial velocities selected from the Maxwell–Boltzmann distribution at the desired temperature are introduced, and the NVE ensemble is used. The NVT ensemble cannot be used in our simulations because the thermostat normalizes the velocities of the atoms in the crowdion core and affects its natural motion.
The initial energy of crowdion is in the form of kinetic energy and is equal to
E k = N m ( V x 0 ) 2 2 ,
where N = 1 or 2 for the 1- or 2-crowdion, m = 207.2 a.m.u. is the mass of the lead atom or m = 58.7 a.m.u. is the mass of the nickel atom, and the contribution of the transverse components of the initial velocity is not taken into account because of their smallness.
It is well known that the repulsive part of the potential is significant in modeling collision cascades. The potentials that we use in our study are taken from the LAMMPS repository without any modifications. To justify the use of these potentials, we calculate the energy of the colliding atoms. As stated above, the maximum initial velocity of the atoms in our simulations is 150 Å/ps, which gives a maximum collision energy of about 230 eV for lead and 70 eV for nickel. Cascades generated by ions with such energies are considered low-energy cascades [48], and the repulsive part of the potentials plays a less important role than for high-energy cascades.

3. Results and Discussion

We first compare the repulsive parts of the interatomic potentials for lead and nickel, and then study the dynamics of N-crowdions in these metals.

3.1. Comparison of Potentials for Pb and Ni

It is well known that the repulsive part of the interatomic potential is very important when discussing the dynamics of the N-crowdions since it affects the condition of the self-focusing collisions of atoms [49]. In Figure 2, the repulsive parts of the potentials are compared for Pb (blue line) and Ni (orange line) by plotting the potential energy of the computational cell as a function of the lattice parameter normalized to its equilibrium value. It can be seen that the repulsive part of the potential of Ni is stiffer than that of Pb. This fact is used to explain the difference in the dynamics of supersonic N-crowdions in lead and nickel.

3.2. Dynamics of Supersonic 1- and 2-Crowdions

Supersonic 1- and 2-crowdions are initiated in the crystal lattices of lead and nickel. Moving at supersonic speeds, the crowdion radiates energy and slows down, eventually transforming into a subsonic crowdion. The propagation distance of an interstitial atom in the supersonic regime is analyzed for 1- and 2-crowdions in Pb and Ni.
The propagation distance S of supersonic 1- and 2-crowdions depending on the initial velocity V x 0 is shown in Figure 3 for (a) lead and (b) nickel. The propagation distance is normalized to the interatomic distance d. Results for 1-crowdion (2-crowdion) are shown in blue (orange). In Figure 3a, the results are shown in green, taking into account the propagation distance of an interstitial atom after the transition of a supersonic 2-crowdion to a supersonic 1-crowdion.
The results presented in Figure 3 suggest that the dynamics of supersonic crowdions in Pb and Ni is different.
In Ni, see Figure 3b, 2-crowdions travel about five times longer distance than 1-crowdions launched with the same initial velocity. This is true for the initial velocities up to V x 0 = 130 Å/ps and for larger initial velocities the propagation distance of 2-crowdion starts to rapidly decrease. This happens due to the instability of supersonic 2-crowdion which, at large propagation speed, transforms into supersonic 1-crowdion.
In Pb, see Figure 3a, the 2-crowdion travels a longer distance than the 1-crowdion only for the initial velocities up to 50 Å/ps, and for greater V x 0 , the instability of the 2-crowdion results in its transformation to a 1-crowdion and the propagation distances of the 1- and 2-crowdions become equal. In Figure 3a, the distance traveled by the 2-crowdion is shown in orange color, and it is smaller than the distance traveled by the 1-crowdion launched with the same initial velocity. However, after the collapse of the supersonic 2-crowdion, the interstitial atom continues to travel as the supersonic 1-crowdion, and altogether the propagation distance (shown by green color) is the same as that of the 1-crowdion (shown by blue color).
Note that the 2-crowdion initiated with the velocity V x 0 has an initial energy two times greater than the 1-crowdion launched with the same velocity, see Equation (1). The 2-crowdion propagation distance in lead at V x 0 = 40 and 50 Å/ps is also approximately twice the 1-crowdion propagation distance, see Figure 3a. On the other hand, the propagation distance of a supersonic 2-crowdion in Ni is about five times greater than that of a supersonic 1-crowdion, see Figure 3b. This means that in lead, the energy radiation rate of the 2-crowdion is approximately equal to the radiation rate of the 1-crowdion, but in Ni, it is much lower. This difference in the rate of the energy radiation of supersonic crowdions in Ni and Pb can be caused by the difference in the repulsive parts of their potentials, see Figure 2.
Apparently, the instability of supersonic 2-crowdions in lead can be explained by the relatively soft repulsive part of the interatomic potential, see Figure 2.
The supersonic 2-crowdion is excited in lead with an initial velocity V x 0 = 60 Å/ps. The evolution of the crystal structure is shown in Figure 4 by two snapshots: (a) t = 0.84 ps and (b) t = 3.71 ps. Atoms with minimum energy are shown in blue, while atoms with maximum energy in red. In Figure 4a, one can see a vacancy at the cite of initially excited atoms and a moving supersonic 2-crowdion (their positions are indicated by arrows). After the collapse of the 2-crowdion, the supersonic 1-crowdion continues its motion along the x axis, see Figure 4b, while the atoms behind it move backward, filling the vacancy and forming an extended vacancy at some distance to the right of the site of excitation of the 2-crowdion. After complete relaxation, a vacancy forms at the site of the extended vacancy.
To demonstrate the instability of a supersonic 2-crowdion in lead, launched at a speed of more than V x 0 > 50 Å/ps, in Figure 5 we plot the time evolution of the kinetic energy of atoms belonging to the close-packed row where crowdions move. The results for Figure 5a 1-crowdion and Figure 5b 2-crowdion are presented. One curve per atom is plotted. The crowdions are excited by setting the initial velocity V x 0 = 60 Å/ps. The qualitative difference in the dynamics of supersonic 1- and 2-crowdions lies in that each curve in Figure 5a has one maximum, and in Figure 5b, for the 2-crowdion, each curve has two maximums as shown for the curve highlighted in red. After the 2-crowdion is destroyed, it transforms into a 1-crowdion with T n ( t ) curves having single maximum. The presented results show that for 0 < t < 2 ps, the kinetic energy of the 2-crowdion decreases more slowly than that of the 1-crowdion. For 2.0 < t < 2.5 ps, the kinetic energy of atoms in a 2-crowdion begins to decrease faster than in a 1-crowdion, and for t > 2.5 ps, the kinetic energies of atoms in both crowdions become approximately equal.
A distinctive feature of the 2-crowdion dynamics is that the T n ( t ) functions have two maxima since each atom moves to the nearest lattice position in two steps, see Figure 5b. For a 1-crowdion, the T n ( t ) function has only one maximum as seen in Figure 5a. The transformation of a supersonic 2-crowdion into a supersonic 1-crowdion can be seen in Figure 5b at approximately t = 1.5 ps. At this time, the curves T n ( t ) with two maxima are transformed into curves with one maximum.
Further insight into the dynamics of supersonic N-crowdions can be found in Figure 6, which shows the relative atomic displacements of atoms, Δ x n / d , in a close-packed atomic row where crowdions move. Panels (a) and (b) show the motion of 1- and 2-crowdions, respectively, excited with an initial velocity of V x 0 = 60 Å/ps. These are the same numerical runs as shown in Figure 5. In a 1-crowdion, see Figure 6a, the atoms quickly cross the potential barrier located at Δ x n / d = 0.5 and then slowly relax to the neighboring equilibrium position at Δ x n / d = 1. A propagating supersonic 1-crowdion constantly radiates its energy and at t = 3 ps, turns into a subsonic crowdion, which continues its movement along the x axis, transferring atoms from Δ x n / d = 0 to Δ x n / d = 1 . In a 2-crowdion, see Figure 6b, the atoms move very quickly in two steps to Δ x n / d ≈ 1.3 and then slowly relax back to the neighboring equilibrium at Δ x n / d = 1 . At t = 1.5 ps, the supersonic 2-crowdion transforms into a supersonic 1-crowdion, which at t = 3 ps transforms into a subsonic crowdion.
As it was said in Section 2, the small transverse components of the initial velocity V y 0 and V z 0 are introduced into the initial conditions to test the stability of N-crowdions with respect to small perturbations. Figure 7 shows the time variation of the V y velocity component for atoms in a close-packed atomic row where crowdions move. The results for lead are shown in Figure 7a,c, and for nickel, in Figure 7b,d. Panels (a) and (b) are for 1-crowdion, and (c) and (d) are for 2-crowdion. The initial velocity V x 0 = 60 Å/ps is used to excite the crowdions.
It can be seen from the vertical axes of Figure 7 that in both metals, the transverse component of atomic velocities for the 2-crowdion is greater than for the 1-crowdion. The second observation is that the transverse component of atomic velocities in lead is greater than in nickel. The latter fact explains the results shown in Figure 3 that the propagation of 2-crowdions in nickel is more stable than in lead.

3.3. The Effect of Temperature

It is known that thermal fluctuations reduce the propagation distance of supersonic crowdions [39]. To see the effect of temperature, a 2-crowdion is launched in lead at a temperature of T = 297 K along the x axis with an initial velocity of V x 0 = 60 Å/ps, see Figure 8a. After passing a distance of about a dozen interatomic distances, the supersonic 2-crowdion disappears, giving its energy to the newly excited supersonic 1-crowdions, denoted as C1 and C2, moving along the y axis, i.e., in the direction perpendicular to the propagation direction of the 2-crowdion. The results shown in Figure 8a are from a single MD run, but they are representative, as laterally moving supersonic 1-crowdions were observed in almost every run. In Figure 8b, the y-component of the velocity of atoms in the atomic row along which the crowdion C1 moves is presented as a function of time. These functions are typical for supersonic 1-crowdions. It is concluded that supersonic crowdions can be scattered by thermal fluctuations of the crystal lattice, forming 1-crowdions.

4. Conclusions

Supersonic 1- and 2-crowdions in the crystal lattices of lead and nickel are studied by the method of molecular dynamics using EAM interatomic potentials; the results can be compared with crowdion dynamics in other metals, for example, in aluminum [29], platinum [39], tungsten [41] and Ni 3 Al intermetallic compound [40]. In all metals studied in previous works, supersonic 2-crowdions showed much better transport properties since they propagate over longer distances compared to supersonic 1-crowdions. The reason is that the sequence of self-focusing collisions is realized only for atoms colliding with a speed below the threshold value. The very fast collisions of atoms cause defocusing, and an interstitial atom cannot propagate stably along a close-packed atomic row; its directed motion is destroyed, and energy is transferred to many atoms in the lattice [38]. In a supersonic 1-crowdion, only one atom moves at a high speed, while in a 2-crowdion, two atoms have a high speed, which means that the maximum energy of a 2-crowdion can be about twice that of a 1-crowdion. Having a higher energy, the 2-crowdion propagates over a greater distance at a velocity below the critical one.
It turns out that the propagation distance of supersonic 2-crowdions in lead is greater than that of supersonic 1-crowdions only at relatively low propagation velocities (see Figure 3a), while in all other materials studied, 2-crowdions propagate over longer distances in a wider range of propagation velocities (see Figure 3b for nickel). This difference is presumably related to the relatively soft repulsive part of the interatomic potential in lead compared to, for example, nickel, see Figure 2. The softness of the repulsive part of the potential leads to unstable propagation of the supersonic 2-crowdion and its rapid transformation into a supersonic 1-crowdion.
It is found that a supersonic 2-crowdion can scatter on thermal fluctuations, creating supersonic 1-crowdions moving in a direction perpendicular to the direction of movement of the 2-crowdion, see Figure 8.
The relatively short propagation distance of supersonic N-crowdions in lead explains its high radiation-shielding properties, which is important in practice. Indeed, due to the instability of the motion of supersonic 2-crowdions in lead, their ability to carry high-energy interstices over long distances is limited, and their energy is rapidly dissipated in the crystal lattice in the form of thermal vibrations.

Author Contributions

Conceptualization, A.S.S. and S.V.D.; methodology, A.M.B.; software, A.M.B.; data curation, A.M.B.; writing—original draft preparation, A.M.B., S.V.D. and A.S.S. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the Russian Science Foundation, grant number 22-22-00810.

Data Availability Statement

Data supporting the published results can be obtained from the authors upon reasonable request.

Conflicts of Interest

The authors declare no conflict of interest.

Abbreviations

The following abbreviations are used in this manuscript:
LAMMPSLarge-Scale Atomic/Molecular Massively Parallel Simulator
EAMEmbedded Atom Model
OVITOOpen Visualization Tool

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Figure 1. Atoms occupying points of the fcc lattice projected onto the ( x , y ) plane. Only part of the computational cell is shown. The axes of the Cartesian coordinate system x, y and z are directed along the crystallographic directions 〈 110 〉 , 〈 1 ¯ 10 〉 and 〈 001 〉 , respectively. One of the close-packed directions coincides with the x axis. The excitation of supersonic 1- and 2-crowdions is shown in (a,b), respectively. Atoms colored green at t = 0 are excited with initial velocity V x 0 . Other atoms have zero initial displacements and zero initial velocities.
Figure 1. Atoms occupying points of the fcc lattice projected onto the ( x , y ) plane. Only part of the computational cell is shown. The axes of the Cartesian coordinate system x, y and z are directed along the crystallographic directions 〈 110 〉 , 〈 1 ¯ 10 〉 and 〈 001 〉 , respectively. One of the close-packed directions coincides with the x axis. The excitation of supersonic 1- and 2-crowdions is shown in (a,b), respectively. Atoms colored green at t = 0 are excited with initial velocity V x 0 . Other atoms have zero initial displacements and zero initial velocities.
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Figure 2. Potential energy of the computational cell as a function of the lattice parameter normalized to its equilibrium value. The blue line corresponds to lead, the orange line to nickel.
Figure 2. Potential energy of the computational cell as a function of the lattice parameter normalized to its equilibrium value. The blue line corresponds to lead, the orange line to nickel.
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Figure 3. The propagation distance of supersonic 1- and 2-crowdions in (a) lead and (b) nickel depending on the initial velocity. The propagation distance is normalized to the interatomic distance. Results for 1-crowdion are shown in blue, and results for 2-crowdion are shown in orange. In (a), the propagation distance is shown in green, taking into account the distance traveled by an interstitial atom after the transformation of a supersonic 2-crowdion into a supersonic 1-crowdion.
Figure 3. The propagation distance of supersonic 1- and 2-crowdions in (a) lead and (b) nickel depending on the initial velocity. The propagation distance is normalized to the interatomic distance. Results for 1-crowdion are shown in blue, and results for 2-crowdion are shown in orange. In (a), the propagation distance is shown in green, taking into account the distance traveled by an interstitial atom after the transformation of a supersonic 2-crowdion into a supersonic 1-crowdion.
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Figure 4. Snapshots showing the development of instability of supersonic 2-crowdion and its transformation to a supersonic 1-crowdion in lead. Energy of atoms is color-coded according to the color gradient: Blue—minimal energy, red—maximal energy. The initial value of the velocity used for excitation of supersonic 2-crowdion is V x 0 = 60 Å/ps. Snapshots are taken at (a) t = 0.84 ps and (b) t = 3.71 ps.
Figure 4. Snapshots showing the development of instability of supersonic 2-crowdion and its transformation to a supersonic 1-crowdion in lead. Energy of atoms is color-coded according to the color gradient: Blue—minimal energy, red—maximal energy. The initial value of the velocity used for excitation of supersonic 2-crowdion is V x 0 = 60 Å/ps. Snapshots are taken at (a) t = 0.84 ps and (b) t = 3.71 ps.
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Figure 5. Time evolution of the kinetic energy of atoms T n ( t ) in a close-packed atomic row where crowdions move. A curve is plotted for each atom in the row. Panels (a,b) show the results for 1- and 2-crowdions in lead, respectively. The crowdions are excited with an initial velocity of V x 0 = 60 Å/ps. All curves in (a) have one maximum, and in (b), in the regime of motion of 2-crowdion, each curve has two maximums, as shown for the curve highlighted in red.
Figure 5. Time evolution of the kinetic energy of atoms T n ( t ) in a close-packed atomic row where crowdions move. A curve is plotted for each atom in the row. Panels (a,b) show the results for 1- and 2-crowdions in lead, respectively. The crowdions are excited with an initial velocity of V x 0 = 60 Å/ps. All curves in (a) have one maximum, and in (b), in the regime of motion of 2-crowdion, each curve has two maximums, as shown for the curve highlighted in red.
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Figure 6. The relative atomic displacements of atoms Δ x n / d in a close-packed atomic row, where crowdions move. Panels (a,b) show the results for 1- and 2-crowdions in lead, respectively. The crowdions are excited with an initial velocity of V x 0 = 60 Å/ps.
Figure 6. The relative atomic displacements of atoms Δ x n / d in a close-packed atomic row, where crowdions move. Panels (a,b) show the results for 1- and 2-crowdions in lead, respectively. The crowdions are excited with an initial velocity of V x 0 = 60 Å/ps.
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Figure 7. Time evolution of the velocity component V y for the atoms in a close-packed atomic row where crowdions move. Panels (a,c) show the results for 1- and 2-crowdions in lead, respectively. Panels (b,d) show the results for 1- and 2-crowdions in nickel, respectively. Crowdions are excited with the initial velocity V x 0 = 60 Å/ps.
Figure 7. Time evolution of the velocity component V y for the atoms in a close-packed atomic row where crowdions move. Panels (a,c) show the results for 1- and 2-crowdions in lead, respectively. Panels (b,d) show the results for 1- and 2-crowdions in nickel, respectively. Crowdions are excited with the initial velocity V x 0 = 60 Å/ps.
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Figure 8. (a) Scattering of supersonic 2-crowdion in lead on thermal fluctuation at T = 297 K and formation of two supersonic 1-crowdions C1 and C2 moving along the y axis. The 2-crowdion was excited along x axis with the initial velocity of V x 0 = 60 Å/ps. (b) Dependence of the velocity component V y for atoms in the row where the C1 crowdion moves on simulation time.
Figure 8. (a) Scattering of supersonic 2-crowdion in lead on thermal fluctuation at T = 297 K and formation of two supersonic 1-crowdions C1 and C2 moving along the y axis. The 2-crowdion was excited along x axis with the initial velocity of V x 0 = 60 Å/ps. (b) Dependence of the velocity component V y for atoms in the row where the C1 crowdion moves on simulation time.
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MDPI and ACS Style

Bayazitov, A.M.; Semenov, A.S.; Dmitriev, S.V. Simulation of the Dynamics of Supersonic N-Crowdions in fcc Lead and Nickel. Micro 2023, 3, 632-642. https://doi.org/10.3390/micro3030044

AMA Style

Bayazitov AM, Semenov AS, Dmitriev SV. Simulation of the Dynamics of Supersonic N-Crowdions in fcc Lead and Nickel. Micro. 2023; 3(3):632-642. https://doi.org/10.3390/micro3030044

Chicago/Turabian Style

Bayazitov, Ayrat M., Alexander S. Semenov, and Sergey V. Dmitriev. 2023. "Simulation of the Dynamics of Supersonic N-Crowdions in fcc Lead and Nickel" Micro 3, no. 3: 632-642. https://doi.org/10.3390/micro3030044

APA Style

Bayazitov, A. M., Semenov, A. S., & Dmitriev, S. V. (2023). Simulation of the Dynamics of Supersonic N-Crowdions in fcc Lead and Nickel. Micro, 3(3), 632-642. https://doi.org/10.3390/micro3030044

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