APPENDIX 1. BASIC CRYSTALLOGRAPHY Al.l Introduction external symmetry of the structure. The trans­ lational symmetry leads to the concept of a lattice In order to perform electron microscope studies it (an array of atoms at points in space with identical is necessary to understand the basic principles of surroundings). Arising from considerations of this crystallography set out below. type we arrive at fourteen distinct Bravais lattices, It is the regular arrangement of atoms in space figure Al.l. These fall into the seven fundamental which constitutes the distinguishing feature of the crystal classes listed in table Al.l. Most metals are crystalline state. These regular arrays, called either cubic, hexagonal or tetragonal in structure, crystal structures, which are characterised in see table Al.3. However, many non-metallic various different ways, give rise to the internal and materials have more complex structures. stmple body- centred face -centred cubtc (P) cubic (I) cubic(F) stmple body-centred stmple body-centred tetragana I tetragonal orthorhombic orthorhombiC (P) (I) (P) (I) base -centred face-centred rhombohedral hexagonal orthorhombic orthorhombic (R) (P) (C) (F) c stmple bose-centred tnCllniC monocltnic (P) monoclimc(C) (P) Figure Al.l The fourteen Bravais lattices 80 Practical Electron Microscopy Table Al.1 Crystallographic formulae for interplanar spacings, Crystal system Interplanar spacing of the (hkl) plane cubic a=b=c a.={J=y=90° tetragonal a=bioc a.={J=y=90° orthorhombic a # b # c a.={J=y=90° hexagonal a=bioc 1 4(h2 hk k2 12 = 3a2 l a. = {J = 90°; y = 120° d2 + + ) + C2 2 rhombohedral a = b = c 1 (1 + cos a.){(h2 + k2 + 12) - (1 - tan !aXhk + kl + lh)} a. = {J = y < 120° # 90° d2 = Q2 1 + cos a. - 2 cos2 a. monoclinic aiobioc a.=y=90°#{J triclinic 2 2 : 2 = ~2 (s11h + s22kl + s331 + 2s 12hk + 2s23kl + 2s33lh) where V 2 = a2b2c2(1 - cos2 a. - cos2 {J - cos2 y + 2 cos a. cos {J cos y) and s 11 = b2c2 sin2 a. s22 = a2c2 sin2 {J s33 = a2b2 sin2 y s12 = abc2(cos a. cos {J - cosy) s23 = a2bc(cos {J cos y - cos a.) s31 = ab2c(cos y cos a. - cos {J) Al.l Indexing Planes in the plane are afh, bfk and cfl. A negative intercept results in a negative component ofthe Miller index Planes in any one of the fourteen Bravais lattices written as h. caD. be indexed in the same way. Axes are chosen Formulae for the angles between different (hkl) defining a unit cell x, y and z at angles a, fJ and y planes in the fourteen Bravais lattices are given in with the unit translation distances a, b and c table Al.l. (figure Al.2(a)). A plane is defined in terms of its intercepts on these axes. For example, in figure A1.2(b) the Al.3 Indexing Lattice Diredions plane cuts the axes at a/h, bjk and cfl where h, k and l are the Miller indices of the plane. For any Bravais lattice, such as that shown in Thus, in figure A1.2(c), the plane cuts the axes figure A1.3, the direction OA has the indices [121], at unit translation distances in the x and y axes that is the path from 0 to A involves moving 1 unit and two translation distances in the z axis. The translation parallel to x, 2 parallel to y and 1 index of the plane is then OX/OX, OY/OY and parallel to z. The other directions [Oll] and [ITO] OZ/OW, that is llf which is rationalised to 221, are self-evident. The symbol I means one trans­ the hkl indices for the plane, that is the intercepts lation in the negative direction. The general symbol Electron Diffraction in the Electron Microscope 81 angles, and angles between directions for the seven crystal systems [After Andrews et al. (1971)] cos p = {(u~ + v~ + w~)(u~ + v~ + w~)) 112 cost/>= [{ I 2 a2 1212}{ c 2 12}] 112 ? (hi + ktl + ? II ? (h2 + k2) + ? 12 convert to corresponding hexagonal indices (see appendix 2) and use the above two formulae 2 2 a2u1u2 + b v1v2 + c w 1w 2 + ac(w 1u2 + u1w 2)cosp cosp= 2 22 22 p) {(a 2 u1 + b v1 + c w 1 + 2acu 1w1 cos 112 x (a 2 u~ + b 2 v~ + c 2 w~ + 2acu 2 w2 cos P)} L cos p = -=---=-­ IUtVtWt IU2V2W2 2 2 2 2 2 where F = h1 h2 b c2 sin a + k 1 k2a c2 sin p + l112a2b2 sin y where L = a2 u1u2 + b2 v1v2 + c2 w 1w 2 2 + abc (cos a cos P - cos y)(k 1h2 + h1k2) + bc(v 1 w2 + w1 v2)cosa 2 + ab c(cos y cos a -cos P)(h 112 + 11h2) + ac(w1u2 + u1 w 2)cosP 2 + a bc(cos P cosy - cos a)(k 112 + 11k2) + ab(u 1 v2 + v1 u2) cosy and and Ahkl = {h2b2c2 sin2 a + k2a2c2 sin2 p + 12a2b2 sin2 y I uvw = ( a2u2 + b2v2 + c2w2 + 2hkabc2(cos a cos p - cosy) + 2bcvw cos a + 2hlab2c(cos y cos a - cos p) + 2cawu cos p 1 2 + 2kla2bc(cos p cosy - cos a)} ' + 2abuv cos y) 1' 2 for a direction is [uvw] where r = OA, figure A1.5 Zones and the Zone Law Al.3, = ua + vb + we. Consequently, r is a vector with components Any two lattice planes intersect in a line which can u, v and w along the axes. be defined by the directional indices [ uvw]. This Formulae for the angles between different [ uvw] is the axis for a prism of planes with this common directions in the fourteen Bravais lattices are given direction. The planes are known as the zone of in table Al.l. planes and the long axis is the zone axis, z, given the symbol [UVW]. The zone indices for any pair of planes, that is (h'k'l') and (hkl) can be obtained A1.4 Plane Normals in the following way: In the cubic crystal system only the direction h k h k normal to the plane (hkl) has indices [hkl], for example [111] is the (111) plane normal. In all X X X h' k' l' h' k' l' other crystal systems this is not true and table A1.2 gives the formulae for determining the indices of that is U = kl' - k'l, V = lh' - l'h, W = hk' - h'k. the directions [ uvw] normal to the plane (hkl) and This is the cross product between plane indices or vice versa. the directions of the plane normal. 82 Practical Electron Microscopy z c y X Figure A1.3 Crystal directions, OA is [121] If it is necessary to find out if a plane (hkl) lies in a zone [UVW] then the condition is X (a) hU + kV + IW = 0 and this is called the Weiss zone law. This is z essentially the condition that the normal to the plane (hkl) is perpendicular to [UVW] direction. Notation is as follows: (111) means single set of parallel planes {111} means equivalent planes of the type, that is (111), (ll1), etc. [111] means a single zone axis or direction ( 111) means directions of equivalent type A1.6 Stereograpbic Projection Although other methods of projection of the three­ dimensional crystal into two dimensions exist, the stereographic projection is the most common way of describing crystals. This is because the projection X preserves angular truth. The advantage of working (b) with the stereographic projection lies in the ease and rapidity of performing those crystallographic z analyses necessary in the electron microscope. Imagine a crystal at the centre of a sphere, see figure A1.4, with plane normals drawn from the centre of the sphere to its surface. In figure Al.5, one of these plane normals P is shown projected into the equatorial plane about the south pole of the sphere. Normally only those planes above the equator are projected. For planes underneath, the north pole is used as the projection point, indicated by open circles in the projection. The stereographic projection of the cubic crystal in figure A1.4 with [001] parallel to the south-north direction SN and [010] parallel to OD, is shown in figure A1.6, .each point being indexed as the normal to a particular plane. Standard projections for cubic crystal structures are shown in figure Al.12 with different planes X (c) in the centre of the stereogram. It is necessary to be able to measure the angles Figure A1.2 (a) The axes x, y, z defining the unit cell (dashed lines). (b) A general plane intersecting between planes using the stereographic projection. the axes. (c) A (221) plane The normals to any two crystal planes, for example Electron Diffraction in the Electron Microscope 83 Table A1.2 Formulae defining the indices of the direction [uvw] perpendicular to plane (hkl) for the seven crystal systems [After Andrews et a/. ( 1971)] Crystal system Equations for finding [ uvw] given (hkl) Equations for finding (hkl) given [ uvw] u w h k cubic h=k=l u w u w (c)2 h k tetragonal h- k -~ (a)2 u - = w(cfa)2 h k I orthorhombic ~ a2 = !!_ b2 = ~ c2 h k I ua2 u v 2w (c) 2 ,, k hexagonal 2k + h - h + 2k - 3T (a) 2 2u - v = 2v - u = 2w(cfa)2 u h k rhombohedral h sin2 tx + (k + 1Xcos2 tx - cos tx) k sin2 tx + (I + hXcos 2 tx - cos tx) u + (v + w) cos tx v + (w + u) cos tx w + (u + v) cos tx w I sin2 tx + (h + kXcos 2 tx - cos tx) u v w h k monoclinic hb2 c2 - lab2 c cos (J kc2 a2 sin2 (J la2 b2 - hab2 c cos (J ua2 + wca cos (J = vb2 = uca cos (J + wc 2 u w h k triclinic hs 11 + ks12 + ls13 hs12 + ks22 + ls23 hs 13 + ks23 + ls33 ua2 + vab cos y + wca cos (J uab cos y + vb2 + wbc cos tx (s 11 = b2 c2 sin2 tx, etc; s12 = s21 = abc2 (cos tx cos (J -cosy), etc.) uca cos.(J + vbc cos tx + wc2 001 ooT Figure A1.4 A crystal with cubic crystal structure situated at the centre of a sphere 84 Practical Electron Microscopy N Too projection on equatonal --- plane ---- -------- 100 5 Figure Al.6 The stereo graphic projection for a cube crystal with [001] parallel to the south north Figure Al.S The projection of a plane normal OP direction in the sphere of figure Al.4.