| Acknowledgments |
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xiii | |
| Preface |
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xv | |
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1 Computer Modeling of Physical Phenomena and Processes |
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1 | (34) |
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1.1 Application of Computers in Physics |
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1 | (7) |
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1.1.1 Role of Models in Theoretical Study |
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1 | (1) |
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1.1.2 Methods of Computer Modeling of Physical Processes |
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2 | (1) |
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1.1.3 Influence of Computers on Methods of Physical Researches |
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3 | (2) |
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1.1.4 The Basic Aspects of Computer Application in Physics |
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5 | (2) |
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1.1.5 Computational Experiments and Their Role in Modern Physics |
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7 | (1) |
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1.2 Determination of Statistical Characteristics of Systems by the MC Method |
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8 | (16) |
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1.2.1 Determination of Average Values of Physical Quantities |
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8 | (4) |
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1.2.2 Application of the MC Method to Physical Problems |
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12 | (2) |
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1.2.3 The Metropolis Algorithm and the Thermostat Algorithm |
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14 | (3) |
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1.2.4 Boundary Conditions |
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17 | (3) |
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1.2.5 The Classical Atomic Interaction Potential Functions |
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20 | (4) |
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1.2.6 Typical Errors in the MC Method |
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24 | (1) |
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1.3 The MD Method and Its Application |
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24 | (11) |
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1.3.1 Algorithms for Numerical Solution of the Equation of Motion |
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25 | (5) |
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1.3.2 Near-Neighbor Calculations |
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30 | (1) |
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1.3.3 Typical Elements of the Program for MD Modeling |
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31 | (1) |
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32 | (3) |
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2 Basic Concepts of Theory of Phase Transformations |
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35 | (36) |
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2.1 The Method of Thermodynamic Functions |
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35 | (5) |
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36 | (1) |
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2.1.2 The Helmholtz Free Energy |
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37 | (2) |
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2.1.3 The Gibbs Free Energy |
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39 | (1) |
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2.2 Thermodynamic Functions of One-Component Systems |
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40 | (1) |
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2.3 Conditions of Equilibrium in the Thermodynamic System |
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41 | (1) |
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2.4 Equilibrium Conditions for Multiphase Systems |
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42 | (1) |
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2.5 Different Types of Phase Transformations |
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43 | (10) |
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2.5.1 Equilibrium Conditions for the First-Order Phase Transitions |
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46 | (3) |
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2.5.2 The Ehrenfest Equations |
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49 | (1) |
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2.5.3 The Gibbs Phase Rule |
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50 | (3) |
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2.6 Influence of the Interfacial Tension on Crystallization of Liquids |
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53 | (4) |
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2.7 Phenomena Connected with Formation of Solutions |
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57 | (14) |
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2.7.1 Heat Effects at the Solution Formation |
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57 | (1) |
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2.7.2 The Raoult's and Henry's Laws |
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57 | (4) |
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2.7.3 Partial Thermodynamic Functions |
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61 | (1) |
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2.7.4 Ideal Solutions; the van't Hoff Equation; the Distribution Coefficient |
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62 | (2) |
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2.7.5 Real Solutions and Regular Solutions |
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64 | (2) |
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2.7.6 The Basic Positions of the Quasi-Chemical Theory of Solutions |
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66 | (2) |
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2.7.7 Calculation of Interatomic Binding Energies |
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68 | (1) |
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69 | (2) |
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3 Diffusion Problems of Crystal Growth: Methods of Numerical Solutions |
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71 | (34) |
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3.1 Differential Equations for the Heat and Mass Transport Processes |
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71 | (9) |
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71 | (4) |
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3.1.2 Thermal Conductivity and Heat Emission |
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75 | (2) |
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3.1.3 Differential Equations of Convective Heat Transfer |
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77 | (2) |
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3.1.4 Euler's Algorithm for the Numerical Solution of Differential Equations |
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79 | (1) |
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3.2 Boundary Value Problems |
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80 | (2) |
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3.2.1 Boundary Conditions |
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80 | (1) |
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3.2.2 The Boundary Value Problem in the Dimensionless Variables |
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81 | (1) |
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3.3 Analytical Solutions of Heat and Mass Transport Problems for Crystal Growth |
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82 | (10) |
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82 | (1) |
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3.3.2 Stefan's Problem in the Initial Statement |
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83 | (3) |
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3.3.3 Boundary Conditions for the Diffusion Problem of Crystal Growth |
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86 | (1) |
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3.3.4 Growth of a Cylinder and a Sphere from Solution at Constant Surface Concentration |
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87 | (2) |
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3.3.5 On the Heat and Mass Transport During Growth of Single Crystals |
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89 | (3) |
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3.4 Numerical Solutions for the Heat and Mass Transport Problems |
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92 | (13) |
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3.4.1 The Finite Difference Schemes for Solution of the Heat and Mass Transport Problems |
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92 | (3) |
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3.4.2 Boundary Conditions at Interfaces during Crystal Growth |
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95 | (1) |
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3.4.3 The First Numerical Solutions of Diffusion Problems of Crystal Growth |
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96 | (2) |
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3.4.4 A Technique for the Numerical Analysis of Growth or Dissolution of Spherical or Cylindrical Crystals |
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98 | (3) |
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3.4.5 Study of the Transport Phenomena in the Framework of the Lattice Boltzmann Method |
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101 | (2) |
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103 | (2) |
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4 Structure of the Boundary Surfaces |
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105 | (22) |
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105 | (1) |
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4.2 The Major Discoveries Contributing to the Development of Surface Science |
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106 | (1) |
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4.3 On the Experimental Research Techniques of Surfaces |
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107 | (4) |
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4.4 Features of the Surface Phase Transitions |
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111 | (2) |
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113 | (5) |
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4.6 Transition from an Atomically Smooth to an Atomically Rough Surface Structure |
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118 | (2) |
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120 | (7) |
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124 | (3) |
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5 Adsorption. The Gibbs Adsorption Equation |
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127 | (24) |
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5.1 Adsorption on Solid Surfaces |
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127 | (8) |
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5.1.1 Physical and Chemical Adsorption. Different Types of Adsorption Isotherms |
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127 | (2) |
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5.1.2 Langmuir's Equation |
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129 | (1) |
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5.1.3 Model for the Computer Analysis of the Adsorption |
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130 | (2) |
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5.1.4 The BET Isotherm of the Multimolecular Adsorption |
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132 | (3) |
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5.2 The Gibbs Adsorption Equation |
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135 | (16) |
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5.2.1 The Physical Phase Boundary |
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135 | (2) |
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5.2.2 The Elementary Strain Energy: Interfacial Tension |
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137 | (1) |
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5.2.3 The Gibbs Method in Thermodynamics of Surface Phenomena |
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138 | (5) |
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5.2.4 Different Ways of Choice of the Separating Surface |
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143 | (2) |
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5.2.5 Adsorption Equilibrium in Multi-Component Systems |
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145 | (4) |
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149 | (2) |
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6 Simulation Techniques for Atomic Systems |
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151 | (36) |
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6.1 Nonclassical Potentials of Atomic Interaction |
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151 | (8) |
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6.1.1 The Empirical Pseudopotential Method |
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151 | (1) |
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6.1.2 DFT and Ab Initio Calculations |
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152 | (2) |
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6.1.3 Embedded Atom Method and Modified Embedded Atom Method |
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154 | (2) |
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6.1.4 Definition of Potentials of Atomic Interaction for Mixed Systems |
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156 | (2) |
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6.1.5 The Problem of Choice of the Pair Potential Function |
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158 | (1) |
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6.2 Finding the Equilibrium Structures by the MC Method and Their Analysis |
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159 | (9) |
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6.2.1 Searching for Equilibrium Structures |
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159 | (1) |
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6.2.2 Evaluation of Structural Properties |
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160 | (1) |
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6.2.3 The Radial Pair Distribution Function |
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161 | (3) |
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6.2.4 The Topological Analysis of the Simulated Atomic Configurations by the Voronoi -- Delone Method |
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164 | (3) |
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6.2.5 Evaluation of Pressure and Definition of the State Equation |
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167 | (1) |
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168 | (7) |
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6.3.1 The Basic Relations for the Transition Probabilities |
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168 | (4) |
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6.3.2 Developing More Realistic Models for Study of the Surface Processes |
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172 | (3) |
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6.4 Particularities in Application of the Molecular Dynamics Method in the Case of Phase Transitions |
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175 | (12) |
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6.4.1 Application of the Molecular Dynamics Method in Different Ensembles |
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175 | (3) |
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6.4.2 Reaching the Equilibrium State and Measuring Macroscopic Parameters |
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178 | (2) |
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180 | (3) |
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183 | (4) |
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7 The Surface Processes During Crystallization |
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187 | (58) |
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7.1 Surface Energy and Equilibrium Forms of Crystals |
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187 | (4) |
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7.1.1 Surface Energy in the First Approximation and its Anisotropy |
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187 | (2) |
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7.1.2 Equilibrium Forms of Crystals |
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189 | (1) |
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7.1.3 The Curie--Wulff Principle |
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190 | (1) |
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7.2 Atomic Structure of Crystal Surfaces |
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191 | (8) |
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7.2.1 Lifetime and Diffusion of Adsorbed Atoms |
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191 | (2) |
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7.2.2 Structure of Steps on the Crystal Surface |
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193 | (2) |
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7.2.3 Roughness of the Crystal Faces |
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195 | (3) |
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7.2.4 Simulation of Crystal Growth Within the "Solid-on-Solid" Model |
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198 | (1) |
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199 | (7) |
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199 | (1) |
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7.3.2 The Dislocation Mechanism of Growth |
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200 | (1) |
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7.3.3 Two-Dimensional Nucleation Growth Mechanism |
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201 | (1) |
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7.3.4 Growth Rate by the Normal Growth Mechanism |
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202 | (1) |
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7.3.5 Role of Bulk Transport Processes During Crystal Growth |
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203 | (1) |
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7.3.6 Application of MC Simulation Technique to Study Growth of Small Crystals |
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203 | (3) |
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7.4 Formation of Thin Films |
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206 | (12) |
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7.4.1 Atomic Mechanisms of the Film Formation |
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206 | (4) |
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7.4.2 Kinetics of Epitaxial Growth of Thin Films |
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210 | (2) |
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7.4.3 Formation of Films Through the Liquid Phase at Deposition |
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212 | (3) |
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7.4.4 Kinetic Modeling of Film Deposition from a Gas Phase |
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215 | (3) |
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7.5 Shapes of Crystal Growth and Their Stability |
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218 | (11) |
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7.5.1 Shapes of the Free Crystal Growth |
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218 | (1) |
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7.5.2 Stability of Spherical Crystals |
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218 | (3) |
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7.5.3 Stability of Polyhedrons |
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221 | (3) |
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7.5.4 Numerical Calculations of Evolution of the Crystal Shapes |
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224 | (5) |
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7.6 Development of Cellular Structure During Directional Solidification |
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229 | (16) |
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7.6.1 Concentration (Diffusion) Supercooling |
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229 | (2) |
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7.6.2 The Basic Results of the Theory of Small Perturbations |
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231 | (4) |
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7.6.3 Modeling Directional Solidification Using Finite Difference Schemes |
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235 | (4) |
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7.6.4 Kinetic Modeling of Directional Solidification by the MC Method |
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239 | (1) |
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240 | (5) |
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8 Modern Simulations by the Molecular Dynamics Method |
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245 | (56) |
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8.1 Cluster Structure of Supercooled Liquids and Glasses |
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245 | (15) |
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8.1.1 Amorphous and Nanocrystalline Materials |
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245 | (1) |
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8.1.2 Techniques for Local Structure Analysis of Simulated Models |
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246 | (4) |
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8.1.3 Cluster Structure of Supercooled Liquids and Glasses |
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250 | (10) |
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260 | (9) |
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8.2.1 The Main Classical Equation for the Nucleation Kinetics |
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260 | (4) |
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8.2.2 The Dependences of the Surface Tension on the Temperature and Radius of Nuclei |
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264 | (2) |
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8.2.3 Critical Radii and Waiting Times: Results of Simulations for Pure Elements |
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266 | (3) |
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8.3 Imperfect Structures of Small Crystallization Centers |
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269 | (4) |
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8.3.1 Local Distribution Functions for Crystals of Different Size |
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269 | (1) |
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8.3.2 Calculations of the Macroscopic Thermodynamic Driving Force |
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269 | (3) |
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8.3.3 The Size-Dependent Thermodynamic Driving Force |
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272 | (1) |
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8.3.4 Sizes of Critical Nuclei at Large Supercoolings |
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272 | (1) |
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8.4 Crystal Growth Kinetics in MD Models |
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273 | (6) |
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8.4.1 On Mechanism and Kinetics of Growth of Metal Crystals |
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273 | (2) |
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8.4.2 The Simulated Growth Velocities of Single Crystals |
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275 | (3) |
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8.4.3 The Size Effect in Growth Velocity |
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278 | (1) |
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8.5 Recent MD Results on Crystallization from Alloy Melts |
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279 | (22) |
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8.5.1 Growth of Disordered Solid Solutions from Alloy Melts. Solute Trapping and Solute Drag Effects |
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279 | (5) |
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8.5.2 Crystallization of the Intermetallic Compound: Kinetics and Disorder Trapping |
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284 | (11) |
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295 | (6) |
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9 Computational Experiments in Materials Science |
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301 | |
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301 | (6) |
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9.1.1 Model for Algorithm Construction |
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302 | (4) |
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9.1.2 Recommended Experiments |
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306 | (1) |
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9.2 Stefan's Problem of Ice Growth |
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307 | (1) |
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9.2.1 Recommended Experiments |
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308 | (1) |
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9.3 Growth of a Spherical Crystals from a Binary Melt |
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308 | (4) |
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9.3.1 Recommended Experiments |
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312 | (1) |
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9.4 Crystallization After Laser Processing of a Metal Surface |
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312 | (4) |
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9.4.1 Recommended Experiments |
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316 | (1) |
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9.5 Directional Solidification |
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316 | (5) |
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9.5.1 Recommended Experiments |
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321 | (1) |
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321 | (3) |
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9.6.1 Recommended Experiments |
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324 | (1) |
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324 | (3) |
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9.7.1 Recommended Experiments |
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327 | (1) |
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9.8 Determination of the Equilibrium Structure by the Monte Carlo Method |
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327 | (4) |
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9.8.1 Recommended Experiments |
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330 | (1) |
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9.9 Modeling of Crystal Growth by the Monte Carlo Method |
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331 | (3) |
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9.9.1 The Crystal Growth Forms |
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331 | (1) |
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9.9.2 Probabilities of Transitions |
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331 | (1) |
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9.9.3 Modeling of Growth of Kossel's Crystal |
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331 | (2) |
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9.9.4 Recommended Experiments |
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333 | (1) |
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9.10 The Method of Molecular Dynamics |
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334 | (12) |
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9.10.1 Potentials and Forces |
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335 | (2) |
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9.10.2 Algorithms for Calculating Velocities and Coordinates |
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337 | (2) |
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9.10.3 Designations of Principal Constants and Variables in the Program |
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339 | (1) |
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9.10.4 Procedures for Calculations of the System Characteristics |
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339 | (3) |
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342 | (3) |
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9.10.6 Recommended Experiments |
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345 | (1) |
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9.11 Fractal Dimension and Renormalization |
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346 | (11) |
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9.11.1 Definition of Fractal Dimension |
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346 | (2) |
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9.11.2 Fractal Dimensionality of Isolated Clusters |
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348 | (2) |
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9.11.3 Groups of Renormalization |
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350 | (3) |
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9.11.4 The Renorm-Group Calculation of the Fractal Dimensionality |
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353 | (3) |
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9.11.5 Recommended Experiments |
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356 | (1) |
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9.12 Complex Analysis of Microstructures |
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357 | (3) |
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9.12.1 Recommended Experiments |
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360 | (1) |
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9.13 How to Prepare Directives for Simulations with LAMMPS |
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360 | |
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9.13.1 From Official LAMMPS Information |
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361 | (1) |
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9.13.2 Some Package Command and Building of the Executable File with the GPU Package |
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362 | (3) |
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9.13.3 The LAMMPS Input Script |
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365 | (5) |
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370 | |
Lisainfo e-raamatute kohta