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Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

Engineering 45. Chp 2 & 3 Problems. Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu. Prob 3.4. For HCP Xtal Show c/a =1.633. Lattice Constants a & c. Consider the TETRAHEDRON formed by the Atoms between Planes. c. The Tetrahedron height, J.

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Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

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  1. Engineering 45 Chp 2 & 3Problems Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

  2. Prob 3.4 • For HCP Xtal Show c/a =1.633 • Lattice Constants a & c • Consider the TETRAHEDRON formed by the Atoms between Planes c • The Tetrahedron height, J • Note that c = 2J • Thus need only find the height of a Tetrahedron with Edge Length, a J a

  3. a The HCP Tetrahedron • Consider Tetrahedron ABCE with SideLength a • Find height J • By Trig Length of CD = a/3 • CH = aCos(30º) = a3/2 • CD =(2/3)CH = a/3 • Then by Pythagorus CD =a/3 • Thus

  4. The HCP Tetrahedron cont • Thus c a • Then the c/a Ratio c J Q.E.D

  5. Prob 3.11 – HCP Titanium • a) Given = 4.51 g/cc, Find the Volume of the Unit Cell, VC • Use Eqn 3.5 for Theoretical Density • Recall From Lecture 3: • n = 6 at/cell for HCP • From Text inside Front Cover Find • ATi = 47.88 g/mol • Running the numbers find

  6. Prob 3.11 – HCP Titanium cont • b) if for Ti c/a = 1.58, then find a & c Find This Volume and Mult by 6

  7. Prob 3.11 – HCP Titanium cont • The Basal (or base) Triangle is equilateral • Thus the 1/6th Volume, and with c = 1.58a • Use VC from before • And c = 1.58a → c = 0.467 nm

  8. % Program E45_Prob_2_15.m: % Plot EA and ER vs r * Verify r0 numerically % Bruce Mayer, PE • ENGR455 • 24Jan10 % % Calc r0 numerically using anonymous fcn for En %% the eqn in text book is for r in nm % A = 1.436; B = 7.32e-6; En = @(r) B/r^8 - A/r % % find En,min at r = r0 us fminbnd command r0 = fminbnd(En, 0, 1); disp('InterAtomic spacing for Min E, r0 in nm =') disp (r0) % % Calc En,min = En(r0) En_min = En(r0); disp('Min E, En_min in eV =') disp (En_min)% % Set Plotting Vector as 300 points r_plt = linspace (0.1, .4, 300); % in nm % % The Energy Functions EA = -1.436./r_plt; ER = 7.32e-6./r_plt.^8; Etot = EA + ER; % % Plot on Same Graph plot(r_plt,EA, r_plt,ER, r_plt, Etot), xlabel('r (InterAtom Spacing)'),... ylabel('Energy'), title('ENGR45 Problem 2.14'), grid,... legend('EA', 'ER', 'Etot'), axis([.1 .4 -8 8]) P2.15 MATLAB m-File

  9. Prob 2.15 plot by MATLAB

  10. Prob 2.17 – Applied Calculus • Given Net Potential Energy by expression • Find Optimum (minimum) Energy, E0, at Optimum InterAtomic distance, r0, In terms of • D, , r0→ Need to Eliminate C • C, , r0 → Need to Eliminate D • Plan: Take 1st Derivative and Set to Zero to find r0 • Do On WhiteBoard

  11. WhiteBoard Work • Problem 3.47 • Given Three Plane-Views, Determine Xtal Structure Also:

  12. All Done for Today ½ Cubic BixbiteXtal • e.g. Indium Oxide

  13. All Done for Today

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