MSE-561/MEAM-553 Homework 1 (due 09/23/21) Notes: I will introduce the Lennard-Jones (L-J) pair potential for solid Argon in the next class. You need to know this potential and the smooth cut-off to zero that is needed in modeling. Both the potential and the cut-off are described in Prof. Vitek’s notes in a separate file titled the L-J potential. Please read this before attempting the homework or wait until Tuesday’s class to know more. In the meantime, you may look up or convince yourself what close-packed crystal structures are in one, two and three dimensional space. You may search if you are not familiar with this concept. If anybody needs a pointer, please email me and I can suggest some good websites. Once you know what close-packed structure you should consider in two dimensions, please draw a rough sketch by hand or make a plot using a code and identify how many neighbors an atom has and at what distances given the lattice parameter. The pair potential only depends on the distance between the atoms and not their orientation. Such potentials are called “central” potentials in physics. With this background, you may proceed to do the homework. Please prepare your answers in the form of a report where you explain your thinking and procedure in words briefly before solving the problem. The numerical code or relaxation code in subsequent homework should be attached here and also uploaded separately. Consider a material in which atoms interact via the Lennard-Jones potential that is smoothly cut-off at = 7.5Å, as explained in the description of the Lennard-Jones potential (see Vitek’s notes). Do not forget that the potential has a different form for r < and for . 1. Determine analytically the equilibrium lattice parameter for the two-dimensional periodic close-packed structure with no surfaces. The L-J potential parameters given in the notes are for three-dimensional face-centered cubic close-packed structure. In this HW 1, you are asked to determine the equilibrium lattice parameter for a close- packed lattice in two dimensions with the same L-J potential. Use the following method: Decide first geometrically the close-packed two-dimensional structure. Make a sketch showing several tens of atoms. Determine how many first, second, third etc. interacting neighbors are within the cut-off radius. First, choose just the neighbors which are the closest to the central atom which is chosen as the origin. Analytically, minimize the potential energy with respect to the lattice parameter and obtain its value. Now check, if the second neighbor shell atoms lie within the cut-off value of the potential. Repeat the rcut ro ro < r < rcut 2 minimization analytically by including the first and second neighbors. Obtain the equilibrium lattice parameter. Now, check if the third neighbors lie within the cut-off of the potential. If they do, repeat the procedure and find the two dimensional equilibrium lattice parameter for your chosen close-packed lattice. Include more neighboring shells if necessary and repeat. 2. In order to get the first training in coding solve this problem numerically. Proceed as in the analytical case: build your 2D close-packed lattice and calculate the radius of first, second, third and fourth neighbor shells. Determine the number of neighbors in each shell. Evaluate the energy of the system as a function of the lattice parameter and taking into account only the first shell, the first and second shell, etc., up the fourth shell of neighbors. Plot this dependence of the potential energy as a function of the lattice parameter. The minimum of the potential energy determines the equilibrium lattice parameter. In the computer code you have to make logical decisions whether two atoms interact with each other or not, what is their separation and what is the contribution of this interaction to the potential energy. Comment on the agreement/disagreement between your analytical and numerical answers to the first and second part of this homework respectively. Please upload your homework with the numerical code, details of analytical method and plot of the variation of energy with lattice parameter. ______ 学霸联盟