School of Materials Science & Engineering

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1 Problem Set 1 Solutions Reminder From Syllabus: Problem Set Format Checklist: At the top of the page, include your name, problem set number, course name & number, and date. Submitted problem sets must be legible, neat and decipherable. Show all work. Complete work means step-by-step. Multiple page problem sets must be stabled with problems kept in sequential order. Multiple page problem sets must be stabled except for ABET questions. Answers must be circled/boxed. If the answer has units, include the units as they are required. All graphs must be thoroughly labeled, have axis titles and units, figure captions, and detailed explanations. Include comments on trends you observe and what you want your audience to take away from the graph. Using legends, arrows, and text will facilitate trends you want to point out. Using color does not help if you print your problem set in black and white. Extra credit many times is given for exceptional graphs with thoughtful and thorough explanations. All software program code must be thoroughly documented/commented and follow the above criteria. Include units. If unit conversions are being performed, the manner in which they are performed (e.g., dimensional analysis) need to be complete. "ABET Problems" - These problems are to be handed in separately and follow the criteria above. ABET problems with multiple pages should not be stapled. ABET problems will be assessed more thoroughly than other problems relative to completeness, correctness, legibility, neatness and decipherability, so extra care should be taken when answering these questions. Grading The TA/grader will provide a cursory review of your solution and provide a grade based on being able to follow and understand your solution. The grade will be based on the final solution, the completeness and validity of the path to the solution, and the Problem Set Format Checklist. If the grader cannot decipher your solution, you will be graded accordingly. Ultimately, you are responsible for understanding the solution to each problem. Citing - Remember to cite your references using references that have been peer-reviewed (yes, need to do for this assignment as well). Note: Besides problem 1, the other problems are review problems for which answers can be found in your ENGR 245 Introduction to Materials Science & Engineering, first and second semesters of physics courses, and first semester chemistry course. 1. Ethics: [ABET question] Program Outcome H. The broad education necessary to understand the impact of engineering solutions in a global, economic, environmental, and societal context. Go to my Courses Website at: a. You will find a section called Scientific and Professional Misconduct. Read the articles about the physicist Jan Hendrik Schön and one of the stem cell researchers, either Woo Suk Hwang or Haruko Obokata, and the predicament they are in. Comment on what might have gone wrong and/or where they went wrong in their careers that led them to where they are today. An ethically responsible answer. b. Read through the sections Information on: Academic Honesty and Dismissal & Student Code of Conduct for Boise State. Comment on why Boise State takes this so seriously. Relate your comments to your comments in part a. An ethically responsible answer. c. Explain how might your reflection and discussion with respect to part a and b affect your undergraduate career and career beyond bachelorette degree? An ethically responsible answer. 1

2 2. Waves and Wave Motion a. Define the following for a wave in a sentence or two for each: Wavelength, Frequency, Velocity Wavelength: For a periodic wave, the distance between two successive parts of the wave that are in phase (that is, at identical points on the waveform) Frequency: Number of cycles per second (f = 1/period or v = 1/period), that is, the number of complete waveforms, or wavelengths that pass by a given point during each second. Velocity: Speed (i.e., magnitude of velocity) and direction of a wave. Velocity is a vector. b. Look up and write down the equations that relates i) wavelength, frequency, velocity and ii) energy and wavelength. Define all variables. i) velocity = λ f & hc E = where λ is wavelength, f is frequency, h is Planck s constant λ and c is the speed of light and E is energy. c. What is the difference between a transverse wave and a longitudinal wave? Use an illustrative (drawing) example that you have created to aid your explanation. You may use phonons (wave packet of crystalline lattice vibrations) in your illustration. That is, you may use examples of transverse phonons and longitudinal phonons. In a longitudinal wave, the displacement, u, (or motion) is parallel to the direction of the wave velocity or wave vector, K. This can be seen in figure 1.[1] In a transverse wave, the wave displacement, u, (or motion) is perpendicular to the direction of the wave velocity or wave vector, K, as shown in figure 2.[1] 2

3 References: [1] Charles Kittel, Introduction to Solid State Physics (8 th Edition, 2005, John Wiley & Sons, Inc.) Ch. 4, p. 90. d. Using a graph that you have created with a mathematical program (not Excel) to help you, explain the following: i. Constructive interference ii. Destructive interference Perfect constructive interference is when two waves with the same magnitude are exactly in phase. The crest of one wave coincides with the crest of the second wave. In the figure directly below, the sinusoidal wave in red shows partial constructive interference (sin[x] + cos[x]) where the initial two waves (cyan & green) have an amplitude of 1 and the final wave has an amplitude of ~1.5 with the same frequency. Perfect destructive interference is when two waves are completely out of phase (180 difference), that is a crest coincides with a trough. In the figure directly below, the sinusoidal wave in blue shows partial destructive interference (sin[x]*cos[ x]) where the initial two waves (cyan & green) have an amplitude of 1 and the final wave has half the original amplitude. A decrease in frequency for the sinusoidal wave in blue is also observed. 3

4 The figure above plots Sin and Cos and examples of partial constructive and destructive interference. 3. Electromagnetic spectrum: Calculate the range of frequencies or wavelengths (in meters) and energies (in ev) for parts a-g using mathematical expressions for each (which you should show). Show and define the mathematical expressions you use and the variables in the mathematical function you use in your calculations and cite the references you used using a peer reviewed source. Hint: use problem 2b to help you. In this problem, we look up the wavelength or frequency range, list them, and calculate the frequency or wavelength and energy (in ev) range for the following. Then, we can use the relationships between energy, E, frequency, f, and wavelength, λ, given by: hc hcf E = cf λ = 2π = to calculate the ranges of each. Note that h is Planck s constant, c is the speed of light. a. Radio and TV? Frequency Range: 1x10 6 to 1x10 9 Hz Wavelength: 1 to 600 m Energy: 4.1x10-9 to 4.1x10-6 ev b. Microwaves? Frequency: 1x10 9 to 1x10 11 Hz Wavelength: to 0.1 m Energy: 4.1x10-6 to 4.1x10-4 ev c. Infrared? Frequency: 1x10 11 to 4.3x10 14 Hz Wavelength: 1 x 10-3 to 7 x 10-7 m Energy: 4.1x10-4 to 1.77 ev d. Visible light? Frequency: 7.5x10 14 to 4.3x10 14 Hz Wavelength: 4 x 10-7 m to 7 x 10-7 m Energy: 3.1 to 1.77 ev e. Ultraviolet? Frequency: 7x10 14 to 1x10 17 Hz Wavelength: 1 x 10-7 to 1 x m Energy: 3 to 413 ev 4

5 f. X-Rays? Frequency: 1x10 17 to 1x10 19 Hz Wavelength: 1 x to 1 x m Energy: 413 to 4.1x10 4 ev g. Gamma Rays? Frequency: above 1x10 19 Hz Wavelength: below 1 x m Energy: >4.1x10 4 ev h. Using a mathematical program (not Excel), plot the electromagnetic (i.e. photon) energy (y-axis) from 0 ev to 10 ev as a function of wavelength (x-axis, units of nanometer) using the mathematical function you used in the first part of this problem. On your plot, label thoroughly and show the energy and wavelength ranges of the visible spectrum with lines that bound the region. Comment on whether or not you think that the energy range of the visible spectrum is easier to remember in units of ev or units of Joules Energy (ev) Visible regime within this closed area Wavelength (nm) In the figure, the photon energy in electron-volts is plotted as a function of photon wavelength in nm (solid blue line). The visible regime is the enclosed area by the intersection of the lines on the energy axis at 1.77 ev and 3.1eV and the lines on the wavelength axis at 400nm and 700 nm. Since the visible range varies from about 1.77 to 3.1 ev, the visible regime in electron-volts is much easier to recall for me. The plot was created in Mathematica. The energy range in ev is much easier to remember (1.77 to 3.1 ev) than in Joules (2.84x10-19 to 4.97x10-19 Joules). 5

6 4. Bonding: The Lennard-Jones potential is one way to mathematically describe the bonding interaction between two atoms. It provides the equilibrium bond length, r o, and the equilibrium bond energy, E o. a. Look up and write down the equation for the Lennard-Jones potential. Define all the variables. Cite the reference where you obtained the Lennard-Jones potential. b. Using a mathematical program (not Excel), plot the Leonard-Jones potential so that you obtain a graph somewhat (but not exactly) similar to figure 1.10 in your book using the units of ev for energy and nanometers for interatomic distance. Label your plot thoroughly including units, labeling r o, and labeling E o. Note: pay attention to the magnitude of energy and length in nanometers. If the values you obtain do not make physically sense, then you probably made a mistake (usually with unit conversion). c. For r > r o, why does your plot of energy never reach 0 or a positive value? d. At a given E=f(r), describe the mathematical process to step by step find the bond energy and the bond length of your plot. Mathematically, how could you ensure that r o was a minimum and not a maximum? = A B r + r where r is the interatomic distance and A and B are constants with a. E( r) (ev) 6 12 units that when divided by their respective denominators will provide the units of ev. S.O. Kasap, Principles of Electronic Materials & Devices, 3 rd Edition, (McGraw-Hill, 2006) page 23, example 1.4. b. We take the approach that is shown in Example 1.4 in Kasap to use the Lennard-Jones potential to find the van der Waals bond energy and length of solid Ar. We use the same parameters as in the example problem. Unit conversion is key in this problem, otherwise we would obtain physically unrealistic values for the bond length and energy. The problem is solved using the following Mathematica code: Clear En, r, M, B, m,, q, a, A initializing variables Assign Values A ; A term in the L J potential; J m 6 B B term in the L J potential; J m 12 ; Define Function En r : A r 10 9 nm m 6 B r 10 9 nm m ev J 12 ; Lennard Jones Potential or L J potential; p. 23 Kasap Note that 10 9 is converting meters to nm & is to convert J to ev Plotting E r in units of ev versus r in units of nm Plot En r, r,.2, 1, PlotRange.1,.1, Frame True, GridLines Automatic, PlotStyle RGBColor 1, 0, 0, PlotLabel "Solid Ar: Energy vs Interatomic Distance", FrameLabel "r nm ", "E r ev " Zooming in on the plot Plot En r, r,.3,.5, PlotRange 0,.1, Frame True, GridLines Automatic, PlotStyle RGBColor 1, 0, 0, PlotLabel "Energy vs Atomic Radius, Enlarged", FrameLabel "r nm ", "E r ev " 0.10 Solid Ar: Energy vs Interatomic Distance 0.05 E r ev E o 0.10 r o r nm You were not required to do this, but to find the equilibrium bond length and bond energy for solid Ar, we use the approach outlined in part d implementing the following Mathematica 6

7 code and obtain the following solutions. Note that only one of the r o solutions is real while the other solutions are imaginary. Using r o, we can plug that into E n(r o) = E o. Solving for thebondlength andbondenergy Using Calculus tofindthelocal minima in r ande which is roandeo rosolns NSolve r En r 0, r Solving for the bond length by taking the derivative of the potential energy and setting it equal to zero and solving for ro ro r. rosolns 6 ; The only real positive solution is the 6th solution, so this command extracts the 6th solution Printing out the solutions Print "Bond Length r o ", ro, " nm" Print "Bond energy E o ", En ro, " ev" r , r , r , r , r , r Bond Length r o nm Bond energy E o ev c. The reason that the plot of energy never reaches 0 or a positive value is because the Lennard-Jones potential always assumes an interaction between Ar atoms. That is, mathematically, the repulsion and attraction terms will never equal one another where r > r o. One can prove this by setting Repulsive Term = Attractive Term and solving for r. One will only obtain imaginary r values. d. The mathematical process to step-by-step find the bond energy and the bond length of the plot given the energy as a function of interatomic distance is given below: Step one: We know that E N versus- r has a minima at r = r o. Step 2: Thus, we can take the derivative of E and set this to zero. den dr = r ro = ; That is, the slope is equal to zero. 0 Step 3: Solve for r which is actually r o. Step 4: Substitute r o into E N and this gives E N(r o). In essence, you are finding at what r o is Forces = 0. That is, you are finding where molecular bond length is in equilibrium. To mathematically determine whether or not r o is a minimum and not a maximum, one would take the second derivative of the energy equation, and substitute in r o. If the number is negative, then it is concave down and is a maximum and if the number is positive, then it is concave up and is a minimum. 7

8 5. Crystallography:: Showing all steps in your work including drawing a cubic unit cell, determine the Miller indices for the following shown in the cubic unit cells. Show all work and label thoroughly (e.g., axes, origin, unit lengths). Show all steps. You may use a table to show your steps. a. b. Answers: a. 8

9 b. 6. Crystallography: In your crystallography notes in table 1-3, one of the 14 space lattices is incorrect. Which one is incorrect and why? The last one (14), FCC, since not all faces have a lattice point. 7. Crystallography: List the two components necessary to describe a crystal structure. crystal structure = bravais lattice + basis or crystal structure = space Lattice + motif 8. Defects: The formation energy of vacancies in Cu, Ag and Au is about 1 ev (p. 262 Kelly & Groves). Calculate the equilibrium concentration of vacancies at 300 K in Cu. You will need to know the density and AMU of the elements. What type of behavior does the plot of vacancy concentration versus 1/T exhibit? Need bulk concentration, n bulk, for Cu which we can calculate using the density and atomic mass of Cu with the help of dimensional analysis: nbulk n v 1 mole atoms 8.96 g 3 = = g 1 mole 1 cm = n bulk e ( ) E f = kt cm 22 3 = cm 6 3 e ( 1) ( ev / K )( 300K ) cm 22 3 The behavior exhibited by the plot of vacancy concentration versus 1/T is Arrhenius in nature and follows the same trend as the plot diffusivity 9

10 9. Defects: On the last page of the Bulk Defects in Materials Notes, cracks and pits are shown in GaN. What is the rotational symmetry of the pits? Explain why you think the symmetry exists. The pits are hexagonal in nature and thus have 6-fold or 3-fold rotational symmetry. The reason is most likely because the crystal structure of GaN is based on the hexagonal Bravais lattice and is Wurtzite. 10. Defects: Name and describe two of each type of defect listed in the following a. Point defect b. Line defect c. Planar defect Point defects: native point defects are defects that are intrinsic to the material such as self interstitials and vacancies. Point defects are atoms and thus zero dimensional defects. Line defects: dislocations are line defects described by the Burger s (displacement) and line (direction) vectors. An edge dislocation is a dislocation in which the Burger s (displacement) and line (direction) vectors are perpendicular while the Burger s (displacement) and line (direction) vectors are parallel for a screw dislocation. Planar defects: defects that have an interface that is planar such as stacking faults and grain boundaries are planar defects. Stacking faults are always bordered by partial dislocations. Grain boundaries are the border between grains that form polycrystalline materials and are highly defective containing point and line defects. 10