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

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40 The following procedure is followed. First draw the reciprocal lattice with origin O. Then draw the incident wave vector k such that its end point coincides with O. The point C is then taken as the center of a sphere with radius |k|. Whenever a reciprocal lattice point falls on this sphere, the Bragg condition is satisfied and a diffracted vector k+g may occur. Note also that more than one reciprocal lattice point may lie on the Ewald sphere which indicates that multiple beam diffraction may occur. This is due to a relaxation of the Bragg condition due to the sample geometry causing the reciprocal points to extend into rods (see Hirsch et al. (1965)). 4.2.3 Darwin-Howie-Whelan Equations Bragg’s law shows that electrons are diffracted in directions k' given by k+g for all reciprocal lattice points g lying on the Ewald sphere. Hence it may be anticipated that the total wave function at the exit surface of the crystal will be a superposition of plane waves, one in each of the directions predicted by the Bragg equation. All that remains is the computation of the complex amplitudes of each of the diffracted waves which are given by the Darwin-Howie-Whelan (DHW) equations: d dz g 2is g g i i g g ' e ' g q ' g g ' g (4.5) These DHW equations are derived from the Schrödinger equation for dynamical electron scattering, 2 2 2 2 8 me 4 K 0 4 U ( r) V ( r) 2 c (4.6) h where Vc is a complex electrostatic lattice potential Vc=V + iW. The Fourier transform of which is given by, V ( r) V V '( r) iW ( r) V c 0 0 2igr Vg e i g0 g W g e 2igr (4.7)
41 V0 is the mean inner potential of the crystal and is positive and causes acceleration of the electrons and hence the relativistic acceleration of the electron increases to, ^ ^ C V (4.8) 0 from which the wave number k0 is given by, k 0 ^ 2me (4.9) h which in turn is used to simplify the Schrödinger equation to, 2 2 2 4 k 4 [ U iU ] (4.10) with, 0 2me U ( r) 2 h g 0 V 2 igr ge ; (4.11) igr g e W 2me 2 U ' ( r) ; (4.12) 2 h g To solve equation 4.10 an empty crystal approximation (i.e. U+iU’= 0) is used to obtain a general solution of the form, r ( ) g (4.13) i k g r g e 2 ( 0 ) which upon substitution into equation 4.10 and using a high energy approximation the DHW equations are obtained. To follow the complete derivation the reader is referred to de Graef (2003). The symbol θg in equation 4.5 is known as the phase factor, sg is the deviation from the Bragg condition and, 1 1 e i q g g ' g i g ' g (4.14)
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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 and 52: 37 Fig. 4.1. (a) A wave incident on
Page 53: 39 Consider Fig. 4.2 which illustra
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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