Massachusetts Institute of TechnologyMathematical Methodsfor Materials Scientists and Engineers3.016 Fall 2005W. Craig CarterDepartment of Materials Science and EngineeringMassachusetts Institute of Technology77 Massachusetts Ave.Cambridge, MA 02139Problem Set 1: Due Friday Sept. 16, Before 5PM: email to [email protected] will be no group assignments for this first problem set. You should submit your homeworkby attaching a Mathematica notebook as an attachment of an email to [email protected]. Toensure that you receive credit, you should name your notebook HW01Lastname.nb before youattach it.Individual Exercise I1-1Design a random walk on a finite o ne-dimensional lattice simulator. Suppose a particle begins atposition 0 at iteration 0. At each iteration, the particle will either jump to the right (occupyingposition 1 at iteration 1) with pro bability12or to the left (occupying position −1 at iteration 1 )with probability12.Suppose the lattice occupies positions −100 to +100 and simulate how many iterations arerequired for the particle to exit the lattice by reaching the ends of the lattice.Each simulation is called “a trial.” Plot the number of steps for each trial versus the trials for100 trials.Individual Exercise I1-2In many simple models, the potential between two atoms is taken to b e the Lennard-JonespotentialLJ(r) =ar12−br61. Calculate the distance rminat which the Lennard-Jones potential is a minimum in terms ofa and b2. Calculate the minimum energy Emin= LJ(rmin) in terms of a and b3. The parameters a and b are not very “physical.” Re-express the Lennard-Jones potentialin terms of the minimum energy (Emin) and the equilibrium two-atom separation (rmin)—inother words what is LJ(r) written with parameters Eminand rmininstead of a and b.14. Calculate the force, F , between two atoms as a function of their separation r.5. It is good practice to create “normalized” or “dimensionless” representations of physicalva r ia bles. Explain whyF ≡ F rmin/Eminand r ≡ r/rminare normalized variables.6. Plot LJ(r)/Eminand F (r) together.7. A mass m in a simple linear spring system has potential energy given byU(x) = A + Bx +k2x2and kinetic energy given byK( ˙x) =m2˙x2and will have a vibrational frequency near its equilibrium position given byν =12πrkmRewrite the potential energy of a spring system with a more physical parameterization.8. The kinetic energy of a single atom has the same f orm as that of a simple massK( ˙x) =m2˙x2Expand the Lennard-Jones potential about the equilibrium separation, rmin, to second-orderto find the effective spring constant for small forces.9. Calculate the vibrational frequency for a atom near its equilibrium
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