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 3: Due Thu. O ct. 12, Before 5PM: email to [email protected] following are t his week’s randomly assigned homework groups. The first member of thegroup is the “Homework Jefe” who will be in charge of setting up work meetings and have respon-sibility for turning in the group’s homework noteb ook. If some some reason, the first member inthe list is incapacited, recalcitrant, o r otherwise unavailable, then the second member should takethat position. Attention slackers: The Jefe should include a line at the top of your notebook listingthe group members that participated in the notebook’s production. Group names are boldfacedtext.Austen: Maricela Delgadillo (marice l a), John Pavlish (jpavlish), Kimberly Kam (kimkam), GrantHofmeister (ghof meis), Jina Kim (jinakim)Bronte: Vladimir Tarasov (vtarasov), Omar Fabian (ofabian), Allison Kunz (akunz), KatrineSivertsen (katsiv), Rene Chen (rrchen), Ke l s e Vandermeulen (kvan der)Eliot: Lauren Oldja (oldja), Samuel Seong (sseong), Michele Dufalla (mdufalla), Cha d Iverson(civerson), Kyle Yazzie (keyazzie), John Rogosic (jrogosic)Lively: Emily Gullotti (emgull), Charles Cantrell (cantrell), Richard Ramsaran (rickyr21), LeanneVeldhuis (lve l dhui), Annika Larsson ( alarsson), Eugene Settoon (geneset)Potter: Bryan Gortikov (bryho), Sophia Harrison (sophiah), Lisa Witmer (witmer), EunRae Oh(eunraeoh), Saahil Mehra (smehra)Woolf: Jill Rowehl (jillar), Jonathon Teja da (tejada), JinSuk Kim (jk i m123), Ka therine Hart-man (khartman), Talia Gershon (tgershon)1Individual Exercise I3-1Kreyszig MathematicarComputer Guide: problem 6.4, page 77Individual Exercise I3-2Kreyszig MathematicarComputer Guide: problem 7.10, page 87Individual Exercise I3-3Kreyszig MathematicarComputer Guide: problem 7.12, page 87Group Exercise G3-1An edge dislocation generates a stress field around it. A straight edge dislocation lying along thez-axis will not generate forces in the z-direction and therefore its stress state can be representedin two dimensions.For an infinitely long edge dislocation with its extra lattice plane inserted the y > 0— z half-plane, the stress state is given byσxxσxyσy xσy y=Gb2π(1 − ν) −y(3x2+y2)(x2+y2)2x(x2−y2)(x2+y2)2x(x2−y2)(x2+y2)2y (x2−y2)(x2+y2)2!where b is the Burgers vector magnitude, and G and ν are the the shear modulus and Poisson’sratio for an isotropic elastic material.1. Convert these stresses to a representation in terms of polar coordinates r, θ.2. Calculate the hydrostatic pressure due to an edge as a function of position in the x, y-plane.Plot some of the isobars (contours o f constant pressure).3. Calculate the rotation of the principal a xis as a function of position and plot it.4. The maximum shear stress is the σxycomponent rotated by π/4 from the principal axissystem—and the Mohr’s circle construction provides a simple way to calculate the maximumshear stress in terms of the eigenvalues. Plot the maximum shear stress as a function ofposition.Hint: The principal coordinate system is a special coordinate system where the stress matrix isa diagonal matrix. It is related by a rotation from the laboratory coordinate system in which theproblem is posed. The trace of a matrix does not change when the coordinate s ystem is rotated.2Group Exercise G3-2The purpose of this problem is to calculate the entropy of a very small simple system exactly.By solving this problem, I hope you will understand Boltzmann’s formula for entropy S(E) =k log Ω(E) a little better and also understand why approximations are needed to calculate entropyin larger or more complex systems. In Boltzmann’s fo r mula, S is the entropy of the total system;E is the energy of the tota l system; k is Boltzmann’s constant; Ω(E) is the number of states ofthe system that have energy E.Consider a system of three isolated hydrogen atoms.Let the “zero of energy” be the ground state of the hydrogen atom, so that the energy of asingle hydrogen atom with its electron in state n is:E(n) = Eo(1 −1n2)As yo u know, counting the quantum numbers for each energy state (electron spin s, angularmomentum l, etc.), there are two states for n = 1; eight states for n = 2; and 18 states for n = 3,etc.So that the problem can be done with a reasonably small amount of RAM, supp ose that allthree electrons are either in n = 1 or n = 2 and no other states.1. Calculate and illustrate the total entropy of this simple system of three non-interactinghydrogen atoms;2. Calculate S in multiples of k (e.g., plot S(E)/k vs. E).3. Can you calculate S(E) for a system of three atoms if the quantum states are restricted ton = 1, 2, 3? Five atoms?4. Extra Credit: The assumption of a limited number of high-energy orbitals results in an anunphysical result. Can you identify what is unphysical abo ut your results?Hint: dU = T dS − P dV for this simple system and the vo lume can be fixed.Hint: One strategy is to enumerate all of the possible energies you can obtain by adding the allthe e nergies from two systems EAand EB; call this EAB. For each energy in EAB, count numberof different ways each energy can be added up. Once you have established an algorithm for addingtwo systems, you can add any number of systems by adding one at a time; i.e., combine EABandECto ge t EABC.Group Exercise G3-3Recall from 3.012 that the time-independent Schr¨odinger equation isˆHφ = EφφwhereˆH is the Hamiltonian operator on the wavefunction φ and Eφis t he (scalar) eigenvalue forthe particular wavefunction φ. This is also an eigenvalue equation which is similar to the matrix–eigenvector–eigenvalue systems we have been discussing, except thatˆH operates on a functionand φ is an eigenfunction.3For a one-dimensional problem, the eigenvalues can be calculated fromEφ=R¯φ(x)ˆHφ(x)dxR¯φ(x)φ(x)dxSuppose that we do not know the eigenfunctions φ0(x), φ1(x), . . . , φn(x), we may still wish tofind an approximate way to calculate the observable energies, E1, E2, . . . , EN.One method is to approximate the φ with a series of functions that match the boundaryconditions. For an electron in a one-dimensional box of length L, we could approximate φ withφ = c1x(L − x) + c2[x(L − x)]2+ . . .