Crystal Defects Chapter 6

1. Crystal Defects Chapter 6

2. IDEAL vs. Reality

3. IDEAL Crystal An ideal crystal can be described in terms a three-dimensionally periodic arrangement of points called lattice and an atom or group of atoms associated with each lattice point called motif: Crystal = Lattice + Motif (basis)

4. Real Crystal Deviations from this ideality. These deviations are known as crystal defects.

5. Free surface: a 2D defect Is a lattice finite or infinite? Is a crystal finite or infinite?

6. Vacancy: A point defect

7. Defects Dimensionality Examples Point 0 Vacancy Line 1 Dislocation Surface 2 Free surface, Grain boundary Stacking Fault

8. Point Defects Vacancy

9. Point Defects: vacancy A Guess There may be some vacant sites in a crystal Surprising Fact There must be a certain fraction of vacant sites in a crystal in equilibrium.

10. Equilibrium? Equilibrium means Minimum Gibbs free energy G at constant T and P A crystal with vacancies has a lower free energy G than a perfect crystal What is the equilibrium concentration of vacancies?

11. Gibbs Free Energy G G = H – TS T Absolute temperature 1. Enthalpy H =E+PV E internal energyP pressure V volume 2. Entropy S =k ln W k Boltzmann constantW number of microstates

12. Vacancy increases H of the crystal due to energy required to break bonds D H = n D Hf

13. Vacancy increases S of the crystal due to configurational entropy

14. Configurational entropy due to vacancy Number of atoms: N Number of vacacies: n Total number of sites: N+n The number of microstates: Increase in entropy S due to vacancies:

15. 100!=933262154439441526816992388562667004907159682643816214685\100!=933262154439441526816992388562667004907159682643816214685\ 9296389521759999322991560894146397615651828625369792082\ 7223758251185210916864000000000000000000000000 Stirlings Approximation N ln N!N ln N- N 1 0 -1 10 15.10 13.03 100 363.74 360.51

17. DG DH G of a perfect crystal DG =DH- TDS neq n -TDS Change in G of a crystal due to vacancy Fig. 6.4

18. Equilibrium concentration of vacancy With neq<<N

19. Al: DHf= 0.70 ev/vacancyNi: DHf= 1.74 ev/vacancy

20. Contribution of vacancy to thermal expansion Increase in vacancy concentration increases the volume of a crystal A vacancy adds a volume equal to the volume associated with an atom to the volume of the crystal

21. Contribution of vacancy to thermal expansion Thus vacancy makes a small contribution to the thermal expansion of a crystal Thermal expansion = lattice parameter expansion + Increase in volume due to vacancy

22. Contribution of vacancy to thermal expansion V=volume of crystalv= volume associated with one atomN=no. of sites (atoms+vacancy) Total expansion Lattice parameter increase vacancy

23. Experimental determination of n/N Lattice parameter as a function of temperature XRD Problem 6.2 Linear thermal expansion coefficient

24. Interstitialimpurity vacancy Substitutionalimpurity Point Defects

25. Defects in ionic solids Frenkel defect Cation vacancy+cation interstitial Schottky defect Cation vacancy+anion vacancy

26. Line Defects Dislocations

27. Missing half plane A Defect

28. An extra half plane… …or a missing half plane

29. What kind of defect is this? A line defect? Or a planar defect?

30. No extra plane! Extra half plane

31. Missing plane No missing plane!!!

32. An extra half plane… EdgeDislocation …or a missing half plane

33. If a plane ends abruptly inside a crystal we have a defect. The whole of abruptly ending plane is not a defect Only the edge of the plane can be considered as a defect This is a line defect called an EDGE DISLOCATION

34. Callister FIGURE 4.3 The atom positions around an edge dislocation; extra half-plane of atoms shown in perspective. (Adapted from A. G. Guy, Essentials of Materials Science, McGraw-Hill Book Company, New York, 1976, p. 153.)

35. 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9

36. 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9

37. boundary 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 Burgers vector Slip plane b slip no slip = edge dislocation

38. t b Slip plane slip no slip Dislocation: slip/no slip boundary b: Burgers vectormagnitude and direction of the slip dislocation t: unit vector tangent to the dislocation line

39. Dislocation Line:A dislocation line is the boundary between slip and no slip regions of a crystal Burgers vector:The magnitude and the direction of the slip is represented by a vector b called the Burgers vector, Line vectorA unit vector t tangent to the dislocation line is called a tangent vector or the line vector.

40. 1 2 3 4 5 6 7 8 9 Two ways to describe an EDGE DISLOCATION 1. Bottom edge of an extra half plane 2. Boundary between slip and no-slip regions of a slip plane 1 2 3 4 5 6 7 8 9 What is the relationship between the directions of b and t? Burgers vector Line vector Slip plane b slip no slip t b  t

41. b t , b  t  Mixed dislocation In general, there can be any angle between the Burgers vector b (magnitude and the direction of slip) and the line vector t (unit vector tangent to the dislocation line) b  t  Edge dislocation b  t  Screw dislocation

42. t b 3 2 1 Screw Dislocation b || t Slip plane Screw Dislocation Line unslipped slipped

43. If b || t Then parallel planes  to the dislocation line lose their distinct identity and become one continuous spiral ramp Hence the name SCREW DISLOCATION

44. Positive Negative Extra half plane above the slip plane Extra half plane belowthe slip plane Edge Dislocation Left-handed spiral ramp Right-handed spiral ramp Screw Dislocation b parallel to t b antiparallel to t

45. Burger’s vector Burgers vector Johannes Martinus BURGERS Burgers vector

46. 1 2 3 4 5 6 7 8 9 10 11 12 14 15 16 13 1 9 2 8 3 7 4 6 5 5 6 4 7 3 8 2 9 1 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 S F A closed Burgers Circuit in an ideal crystal

47. b 13 14 16 1 4 5 6 7 8 9 10 11 12 2 3 15 1 9 2 Map the same Burgers circuit on a real crystal 8 3 7 4 6 5 5 6 4 7 3 8 2 9 13 12 11 10 9 6 3 2 1 16 15 14 8 7 5 4 1 The Burgers circuit fails to close !! F S  RHFS convention

48. A circuit which is closed in a perfect crystal fails to close in an imperfect crystal if its surface is pierced through a dislocation line Such a circuit is called a Burgers circuit The closure failure of the Burgers circuit is an indication of a presence of a dislocation piercing through the surface of the circuit and the Finish to Start vector is the Burgers vector of the dislocation line.

49. Those who can, do. Those who can’t, teach. G.B Shaw, Man and Superman Happy Teacher’s Day

50. b b is a lattice translation Surface defect If b is not a complete lattice translation then a surface defect will be created along with the line defect.