Analysis of the extended defects in 3C-SiC.pdf - Nelson Mandela ...

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38 which reduces to, 2d hkl sin for use in the electron microscope since the n is incorporated into the value of dhkl using the Miller index notation . It should be noted that the Bragg equation only describes the geometric representation of diffraction and does not give any information on the intensity of the diffracted wave. The equation is derived with respect to a specific plane and states that the incident and diffracted waves must travel in directions which lie on a conical surface with top in the diffracting plane and opening angle π/2-θ as illustrated in Fig. 4.1(b). 4.2.2 The Bragg Equation in Reciprocal Space The direct space Bragg equation however elegant and simple is not useful when the absolute direction of a diffracted wave needs to be determined. To do this the Bragg equation should be reformulated in terms of reciprocal space which in turn leads to the Ewald sphere construction. From the de Broglie relation it is known that a plane wave may be represented by its momentum vector p = hk where k has units of reciprocal length and that λ = 1/|k|. k fully characterises the direction and wavelength of the plane wave with respect to the crystal reference frame and thus a reciprocal space equivalent of the Bragg equation can be derived since the plane (hkl) may be represented by the vector ghkl in reciprocal space. Fig. 4.2. (a) The incident and diffracted wave vectors with reciprocal lattice vector g, (b) The redrawn vector construction (from De Graef (2003))
39 Consider Fig. 4.2 which illustrates the wave vectors of the incident and diffracted waves in Fig. 4.1 with the reciprocal lattice vector g. By redrawing the vector construction such that k' starts at point C Fig. 4.2 (b) is obtained which shows the relation, k' k g (4.3) This is equivalent to the direct space Bragg condition and is shown by projecting the above equation onto the vector g, k' g k g g g; k g sin k g sin g sin sin 1 d hkl 2 (4.4) from which the Bragg equation follows. The Bragg condition is thus satisfied whenever the point C falls on the perpendicular bisector plane of the vector g. The perpendicular bisector plane to the vector ghkl consists of all points that are at the same distance from the origin of reciprocal space and the reciprocal lattice point hkl. This leads to a simple geometric construction for the direction of a diffracted wave when the incident wave vector and crystal orientation are given. This is known as the Ewald sphere and is depicted in Fig 4.3. Fig. 4.3. The Ewald sphere construction (from De Graef (2003))
Page 1 and 2: Analysis of the Extended Defects in
Page 3 and 4: My sincere thanks to the following
Page 5 and 6: OPSOMMING Hierdie dissertasie hande
Page 7 and 8: 4.2.5. Two-Beam Scattering Matrix 4
Page 9 and 10: Dedicated to my wife and parents
Page 11 and 12: SUMMARY The dissertation focuses on
Page 13 and 14: CONTENTS Chapter 1: INTRODUCTION 1
Page 15 and 16: 1 CHAPTER ONE INTRODUCTION Silicon
Page 17 and 18: 2.1. Introduction 3 CHAPTER TWO OVE
Page 19 and 20: 2.2.2 The Zinc-Blende Structure 5 T
Page 21 and 22: 2.3 Point Defects, Defect Migration
Page 23 and 24: v m Em vm0 exp( ) (2.2) kT 9 wher
Page 25 and 26: 11 explained and the concept of par
Page 27 and 28: 13 Straight dislocations furthermor
Page 29 and 30: 2.5 Partial Dislocations and Slip 1
Page 31 and 32: 17 Fig. 2.13. The geometrical setup
Page 33 and 34: 19 extrinsic stacking faults are th
Page 35 and 36: 2.8 Twinning 21 Twinning is a proce
Page 37 and 38: 23 Fig. 2.20. The process of vacanc
Page 39 and 40: S n Td (3.2) 25 and is also referr
Page 41 and 42: 27 where a = a0(Z1 2/3 + Z2 2/3 ) -
Page 43 and 44: 29 Using this expression, curves of
Page 45 and 46: 31 As stated previously hard-sphere
Page 47 and 48: 33 where A is the atomic weight. Th
Page 49 and 50: 35 a vacancy dislocation loop if th
Page 51: 37 Fig. 4.1. (a) A wave incident on
Page 55 and 56: 41 V0 is the mean inner potential o
Page 57 and 58: 43 beam left-hand side g' = 0 g' =
Page 59 and 60: 45 In addition another set of indep
Page 61 and 62: 47 When this is incorporated into t
Page 63 and 64: S ( s , z ) e T i Se ( / 0 )
Page 65 and 66: z0 z2 . xn cos and (4.47) D 1 proj
Page 67 and 68: 5.1 Introduction 53 CHAPTER FIVE LI
Page 69 and 70: 55 different orientations. The inte
Page 71 and 72: 57 Furthermore Pirouz et al. (1987)
Page 73 and 74: 59 intersection of the fault, the f
Page 75 and 76: 61 each nuclei because of the low s
Page 77 and 78: 63 (2000) found that subsequent ann
Page 79 and 80: 6.3.1 Ion Range and Defect Producti
Page 81 and 82: 67 Fig. 6.3. Total number of displa
Page 83 and 84: 7.1 Introduction 69 CHAPTER SEVEN R
Page 85 and 86: Fig. 7.2. Simulated diffraction pat
Page 87 and 88: 73 Fig. 7.3. A HRTEM image of the S
Page 89 and 90: 75 Fig. 7.5. The stacking fault sel
Page 91 and 92: (Si/ni) 2 0.000018 0.000016 0.00001
Page 93 and 94: 79 may be concluded that the initia
Page 95 and 96: 81 Fig. 7.11. Bright field TEM micr
Page 97 and 98: 83 Fig. 7.12 which was obtained by
Page 99 and 100: 85 Fig. 7.13. Bright-field TEM micr
Page 101 and 102: 87 (a) (b) Fig. 7.15. The (a) exper
Page 103 and 104:
89 In the following paragraphs the
Page 105 and 106:
Fig. 7.18. CBED pattern taken of th
Page 107 and 108:
93 Table 7.5. The measured paramete
Page 109 and 110:
7.4.2 Results 95 Fig. 7.22. Bright-
Page 111 and 112:
97 Twin platelets on {111} planes
Page 113 and 114:
99 REFERENCES Blumenau, A.T., Jones
Page 115 and 116:
101 Hull, D., Bacon, D.J. : (1984)
Page 117:
103 Smith, D.J., Jepps, N.W. and Pa
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