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MSE 561/MEAM 553 - Homework  

Notes: As mentioned in class lectures, HW 1 results give the equilibrium lattice parameter where the L-J potential energy of a 2D close packed lattice is a minimum. In HW 1, you should have obtained good agreement with your analytical results and numerically obtained result. The equilibrium lattice parameter for the 2D hexagonal close packed lattice is 3.80 Angstroms. In HW 2, you will build a square lattice. You may keep the lattice parameter for this 2D square lattice to be the same as the original value for 3D L-J potential for Argon (refer to L-J notes). The square lattice is not relaxed in this homework. You are merely calculating neighbors of each atom in the array, atomic stress components and the RDF. If you wish to minimize the L-J potential for the 2D square lattice as in HW 1 (either analytically or numerically or by both), bonus points (5 marks) will be awarded. Please upload your results, plots and code as a pdf report and the code separately. (1) Construct a two-dimensional block containing 400 atoms, with the atoms occupying the sites of a square lattice. Let the lattice parameter be equal to the rmin of the Lennard-Jones potential for argon (3D). Display the block such that each atom is shown as a circle. Label all atoms with numbers clearly. (2) Decide the boundary conditions you will use for your block and state them. You may adapt your code for the square array accordingly for the calculations below. (3) Assume that the interaction between the atoms is described by the Lennard- Jones potential for argon which is smoothly cut-off at rcut = 7.5Å as explained in the description of the Lennard-Jones potential. Write the code that produces the map of neighbors that will be used in later calculations of forces, energies etc. Tabulate the number of neighbors and their distance for three types of atoms (give the atom number corresponding to the figure in part 1) in the block: central atom, an atom on an edge and a corner atom. 2 (4) Determine the radial distribution function for this block. In the RDF use the maximum value of r equal to 12Å. Plot this function up to the maximum value. (5) Determine the atomic level stresses in the block. Calculate these on every atom and identify the symmetry exhibited by the stress components by examining the stress components for a number of atoms. Use appropriate units for the stress. Include all formulae in your report. Display the entire block of atoms using circles and use color variation to depict the variation of atomic level stress around each atom in the block. Provide a color index label for the stresses you depict. (6) Display the atomic stress in matrix form for the three types of atoms you selected in part (3). Specify the boundary condition you used for the block.