The Reciprocal Lattice

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Appendix A THE RECIPROCAL LATTICE

When the translations of a primitive space lattice are denoted by a, b and c, the vector p to any lattice point is given p = ua + vb + we. The definition of the reciprocal lattice is that the translations a*, b* and c*, which define the reciprocal lattice fulfil the following relationships: (A. I) a*.b = b*.c = c*.a = a.b* = b.c* =c. a*= 0 (A.2) It can then be easily shown that: (1) Ic* I = 1/c-spacing of primitive lattice and similarly forb* and a*.

(2) a*= (b 1\ c)ja.(b 1\ c) b* = (c 1\ a)jb.(c 1\ a) c* =(a 1\ b)/c-(a 1\ b) These last three relations are often used as a definition of the reciprocal lattice. Two properties of the reciprocal lattice are particularly important: (a) The vector g* defined by g* = ha* + kb* + lc* (where h, k and I are integers to the point hkl in the reciprocal lattice) is normal to the plane of Miller indices (hkl) in the primary lattice. (b) The magnitude Ig* I of this vector is the reciprocal of the spacing of (hkl) in the primary lattice.

AI ALLOWED REFLECTIONS The kinematical structure factor for reflections is given by

Fhkl = I;J;exp[- 2ni(hu; + kv; + lw)] (A.3) where ui' v; and W; are the coordinates of the atoms and hkl the Miller indices of the reflection g. If there is only one atom at 0, 0, 0 in the unit cell then the structure factor will be independent of hkl since for all values of h, k and I we have Fhkl =f. Thus if b.c.c. and f.c.c. crystals are referred to their primitive cells then reflections from the simple metals such as niobium and copper, which have only one atom at each lattice point, will all have intensities given by f(0)2 • Since the three basic g vectors derived below, for the b.c.c. structure for 180 ELECTRON BEAM ANALYSIS OF MATERIALS

Table A.l Necessary conditions for allowed reflections in terms of the type of unit cells in crystals.

Possible Forbidden Unit cell reflections reflections

Primitive All values of h, k and I none Body centred (h + k + I) even (h + k +I) odd Face centred h, k, I all odd or even h, k and l mixed Base centred h and k both odd or even hand k mixed

example, are [11 OJ*, [0 11 ]* and [10 1]* it is clear that the allowed reflections are those derived by summing or subtracting these vectors, i.e. reflections such that (h + k + l) is always even. A similar conclusion is reached if the cubic structure cell, which contains atoms at 0, 0, 0 and ±. ±. ±. is used since

Fhkt = f { exp(- 2ni0) + exp[- 2ni( ~ + ~ + ~) ]}

= {f(l + 1) = 2f if (h + k +/)is even !(1-1)=0 if(h+k+/)isodd. The allowed reflections in the various possible unit cells are shown in Table A.1 If the number of atoms in a unit cell is large there is the possibility that some of the allowed reflections will have zero intensity, e.g. the 200 reflection in silicon. Nevertheless all the allowed reflections must conform with the above classifications. It is of course an important part of structure determination to recognize which reflections are absent (see Chapter 4). The absent reflections in metals such as zirconium (i.e. those for which (h + 2k) is a multiple of 3 and I is odd) show that the structure is h.c.p ..

A.2 RECIPROCAL LATTICE FOR F. C. C. AND B.C. C. CRYSTALS Referring the primitive translations to cubic axis for a primitive f.c.c. cell we have

where i, j and k are unit vectors along the cubic axes. Since

a*= b 1\ c a.(b 1\ c) APPENDIX A 181 then * _ !a[j + k] A !a[i + k] a - !a[i + jJ.!a[j + k] A !a[i + k] and

1 z[· · k] I I 4a 1 + J- = -[i + j- k] =-[III] ia3 [i+jJ.[i+j-k] a a Correspondingly,

b* =~[Ill] a with a b.c.c. structure and referring translations to cubic axes then a f] a _ a _ a=-[111 b=-[111] c=-[111] 2 2 2 so that * _ !a[- i + j + k] A !a[i-j + k] a -!a[i+j-kJ·!a[-i+j+k] A!a[i-j+k] which reduces to -!a 2 [i + j] I sa1 3['•+J- • k] . ['•+J '] = -[110]a

Similarly, I and c* = -[101] a Note that the basic vectors in the reciprocal lattice for these non-primitive cells are twice those of the crystal of the same system. Thus, the basic translations are aj2 (Ill) for a b.c.c. crystal and Ija (Ill), rather than I j2a ( Ill ) , for the reciprocal lattice for an f.c.c. crystal.

A.3 DOUBLE DIFFRACTION Because electrons are strongly scattered it is possible that rescattering of a diffracted beam can give rise to a strong diffracted beam where structure factor considerations suggest the beam should be of zero intensity. The conditions under which a forbidden diffraction spot may appear are most easily seen using the Ewald sphere construction (see Chapter 4). Thus, if the reciprocal lattice point corresponding to g 1 for which the structure factor is large, lies on the Ewald sphere then a strong diffracted beam will be produced in the direction A 1 , as shown on Fig. A.l. Similarly if g2 also 182 ELECTRON BEAM ANALYSIS OF MATERIALS

Fig. A.l Ewald sphere construction showing the conditions which must be fulfilled for double diffraction to occur. See text.

lies on the sphere a diffracted beam would be expected in the direction A2 . However, if the structure factor is zero for g2 the intensity of A2 would be zero but rediffraction of the beam A 1 by planes perpendicular to g2 - g 1

will give rise to a beam A3 which is parallel to and therefore indistinguishable from A2 . Diffraction maxima due to double diffraction can be distinguished by rotating the crystal about the direction which contains the spot in question. The intensity of this spot will be unchanged unless it is due to double diffrac•

tion when it will disappear when g 1 is no longer excited.

A.4 SHAPES OF DIFFRACTION MAXIMA Significant diffracted intensity is observed from thin samples even when the Bragg condition is not precisely satisfied ( [1 ], [2] ). It can be shown that only for an infinite crystal will the diffraction maxima be points and that as the crystal dimensions get smaller so the diffraction maxima get larger. For a parallelepiped crystal the diffracted amplitude is given by

F ¢ g = V fA IBIC exp [ - 2:rri(ux + vy + wz) J dxdydz (A.4) c 0 0 0 APPENDIX A 183

-3/A -2/A -1/A 0 1/A 2/A 3/A Fig. A.2 Predicted variation in intensity of a diffracted beam from a crystal of edge length A as a function of deviation from the Bragg condition where the deviation is measured in units of 1/A. See text. = ~ sin(nAu) sin(nBv) sin(nCw) (A.5) Vc nu nv nw where A, Band Care the edge lengths of the parallelepiped along x, y and z, and u, v, w are the values of s (the deviation from the Bragg condition) along x, y and z. Fig. A.2 shows how the intensity (obtained by multiplying ¢g by¢;, its complex conjugate), varies along u for v and w = 0. The central maximum has a width at half maximum height of 1/A and successive minima occur at intervals of 1/A. The intensity of the successive maxima decreases very rapidly, as shown in Fig. A.2 but because electron diffraction patterns have such a large dynamic range it is possible to detect maxima out to at least the fiftieth maximum for typical thickness samples. For very thin crystals, such as small precipitates, the spikes in reciprocal space give rise to visible intensity for all beam directions. The dimension of the central diffraction maximum along a given direction in reciprocal space is given by 1/d where d is the parallel dimension of the specimen. Thus a thin plate precipitate gives rise to a long spike in reciprocal space normal to the plate and a needle precipitate gives rise to a disc of intensity.

REFERENCES I. Hirsch, P.B., Howie, A., Nicholson, R.B., Pashley, D.W. and Whelan, M.J. (1965) Electron Microscopy of Thin Crystals, Butterworths, Sevenoaks. 2. James, R. W. ( 1958) Optical Principles of the Diffraction of X-rays, Bell, London. Appendix B INTERPLANAR DISTANCES AND ANGLES IN CRYSTALS. CELL VOLUMES. DIFFRACTION GROUP SYMMETRIES

B.! THE SEVEN SYSTEMS The axial lengths and angles and the symmetries exhibited m the seven crystal systems are shown in Table B.l.

Table B.l The axial relationships and symmetries of the seven crystal systems.

Axial length Crystal and angles Minimum symmetry elements

Cubic Three equal axes Four, threefold rotation axes at right angles a = b = c, IX = f3 = y = 90°

Hexagonal Two coplanar axes at One, sixfold rotation (or 120°, third axis at rotation-inversion) axis right angles a= b =/= C, IX= {3 = 90°, y = 120°

Trigonal (or Three equal axes One, threefold rotation rhombohedral) equally inclined (or rotation-inversion) axis a = b = c, IX = f3 = y =!= 90°

Tetragonal Three axes at right One, fourfold rotation (or angles rotation-inversion) axis a = b =!= c, IX = f3 = y = 90°

Orthorhombic Three orthogonal Three, perpendicular twofold unequal axes (or rotation-inversion) axis a =!= b =!= c, IX = f3 = y = 90°

Monoclinic Three unequal axes One, twofold rotation (or one pair not orthogonal rotation-inversion) axis a=!= b =f=c, IX= y = 90o =!= f3 APPENDIX B 185

Table 8.1 (Contd.)

Triclinic Three unequal axes None none at right angles a +b +c, a +fJ + y +90°

8.2 INTERPLANAR SPACING The value of d, the distance between adjacent planes in the set (hkl), may be found from the following equations. h2 + k2 + [2 Cubic:

Tetragonal:

2 2 Hexagonal: -1-- ~(h + hk + k ) + f_ (Miller indices) d 2 3 a2 c2 (h 2 + k 2 + l2 )sin2 a+ 2(hk + kl + hl)(cos2 a- cos a) Rhombohedral: d2 a 2 (l - 3 cos2 a + 2 cos3 a) 1 h2 k2 12 Orthorhombic: d2 = a2 + bz + c2

2 2 2 Monoclinic: _!__=_I_ ( h + k sin f3 + f_ _ 2hl cos /3) d 2 sin 2 f3 a 2 b2 c 2 ac

I 1 2 2 2 Triclinic: d 2 =v2 (S 11 h +S22 k +S33l +2S12hk+2S23kl

+ 2S13hl)

In the equation for triclinic crystals,

V = volume of unit cell (see below)

sll =b2 c2 sin2 r:t.

s22 = a 2 c2 sin2 f3

s33 = a 2 b 2 sin2 y

s12 = abc2 (cos r:t. cos f3- cosy)

s23 = a 2 bc(cos f3 cos y- cos r:t.)

s13 = ab2 c(cos y cos r:t.- cos /3) 186 ELECTRON BEAM ANALYSIS OF MATERIALS B.3 INTERPLANAR ANGLES

The angle¢ between the plane (h1k1 /1), of spacing d1, and the plane (h 2 k2 l2 ), of spacing d2 , may be found from the following equations (Vis the volume of the unit cell).

Cubic:

Tetragonal:

Hexagonal:

3a2 )]-112 x ( h~ + k~ + h2k2 + 4c2l~

a4did2[· 2 Rhombohedral: cos¢=~ sm rx(h 1h 2 +k1 k2 +l1 l2 ) + (cos2 rx- cos rx)

x(k1 l2 +k2l 1 +l1 h2 +l2h 1 +h1 k 2 +h2k 1 )]

Orthorhombic:

[(hi ki li)(h~ k~ /~)]-l/ 2 X 2 a + b2 + c 2 2 a + b2 + c 2

Monoclinic

Triclinic: cos¢= d~~2 [s II hl hz + s22kl kz + s33zlz2 + S23(kilz + k2ll) + Si3(lihz + l2hl) + Sdhik2 + h2kl)] 6) 8 fS)Q) t- + +

1 1. 0~. fS) fS) fS)~ + + + fSJ0 f:J0 6) 0 (:9 + 0 (]Y + 6) ~ 2 m. ~2mm (0 E9 0 Q)~ fS) fS) ISJQ) 0~ 4m.m. 4mm + + + fS) 2. 0 m Q)~.rnm. fSJ0 !&0 ~ ~ ~0 6) + (0 (9 + (]) 0 E9 ~ 5) + + + 0~ 0~ ~ 21. 0 m1. 0~rrm1. 4.mm. 4mm1.

eD ~ + + (?) e (!) e

3 31. el)(J)

+ + ~ ~ ~ ~ 3m. 3m ~'b

+ ~ ~ 3m1.

Fig. 8.1 Figures illustrating the symmetry properties of the 31 diffraction groups (see text). 188 ELECTRON BEAM ANALYSIS OF MATERIALS

8.4 CELL VOLUMES The following equations give the volume V of the unit cell. Cubic:

Tetragonal: V =a2 c

2 Hexagonal: V = acJ3 = 0.866a2 c 2

Rhombohedral: V = a 3 (1 - 3 cos2 r:x. + 2 cos3 r:x.) 112 Orthorhombic: V=abc Monoclinic: V =abc sin f3

Triclinic: V = abc(1 - cos2 r:x.- cos2 f3- cos2 y + 2 cos r:x. cos f3 cos y) 112

8.5 DIFFRACTION GROUP SYMMETRIES Schematic drawings illustrating the symmetries of the 31 diffraction groups, (taken from [1]) are shown in Fig. B.l. The number refers to the rotation symmetry, for example all the groups beginning with 3 exhibit some form of threefold rotation symmetry. The letter m denotes the presence of a vertical mirror plane, the group 3m1R for example has a mirror which bisects the individual pairs of dark field maxima in the figure. The subscript R is used to indicate that the operation which has a subscript associated with it requires a rotation of n about the individual point in the pattern, after the subscripted operation has been performed, in order to restore the initial symmetry. For example if the diffraction group 6 is compared with the diffraction group 6R it is clear that the original pattern is restored in diffraction group 6 simply by rotating the whole pattern in increments of 2n/6 but the individual symbols must be rotated by n after each whole pattern rotation of 2n/6 for the 6R group. The symbol 1R means that the original symmetry is retained if each individual symbol is rotated by n so that the diffraction group 6R therefore does not possess this property whereas the 61R does.

REFERENCES I. Buxton, B.F., Eades, J.A., Steeds, J.W. and Rackham, G.M. (1976) Phil. Trans. R. Soc. A, 281, 171. Appendix C KIKUCHI MAPS, STANDARD DIFFRACTION PATTERNS AND EXTINCTION DISTANCES

The schematic Kikuchi maps and diffraction patterns which are shown in this appendix (Figs. C.l-C.15) have been indexed so that the indices of poles correspond to the upward drawn directions. The indices of the Kikuchi • • • • • • • o2o • 2oo 220 • 111 • lil • • 000 • 202 • 000• 020• • • • • • • • • • • [1011 [0011

• • • • _! • • • 311 J11 111• ~11• • 000• 220• • • • • • • • 000 020 • • • • • • •

[ 1121 [1031

• • • • • ~31 240 202• 220• • • • . . . . 000 022 . 000 11f • • • • • • • [1111 [2131 Fig. C.l Schematic diffraction patterns for six selected electron beam directions for a f.c.c. crystal. The low order reflections are indexed and the upward drawn direction, i.e. the electron beam direction B is indicated under each diffraction pattern. 190 ELECTRON BEAM ANALYSIS OF MATERIALS

Fig. C.2 Schematic Kikuchi map for a f.c.c. crystal extending over two standard triangles.

Fig. C.3 Schematic Kikuchi map for a f.c.c. crystal centred on [001]. APPENDIX C 191

Fig. C.4 Schematic Kikuchi map for a f.c.c. crystal centred on (101].

Fig. C.5 Schematic Kikuchi map for a f. c. c. crystal centred on[ 111]. 192 ELECTRON BEAM ANALYSIS OF MATERIALS • • • • • • • • 200-· lOl 211-· • no• • 000• 110• • 000• 020• • • • • • • • • • • • •

I 111 I [001)

• • • • • • 121 211-• 200 21i -· • • • • 000• 011 • • 000• i10• • • • • • • • • • •

1011 I [113 I

• 301• • • • 121• i21• • • • • • • 000 011 • 000• 2oo• • • • • • • • •

[133] [012] Fig. C.6 Schematic diffraction patterns for six selected electron beam directions for a b.c.c. crystal. The low order reflections are indexed and the upward drawn direction, i.e. the electron beam direction B is indicated under each diffraction pattern.

Fig. C.7 Schematic Kikuchi map for a b.c.c. crystal extending over two standard triangles. APPENDIX C 193

Fig. C.8 Schematic Kikuchi map for a b.c.c. crystal centred on [001].

Fig. C.9 Schematic Kikuchi map for a b.c.c. crystal centred on [0 ll]. 194 ELECTRON BEAM ANALYSIS OF MATERIALS

Fig. C.lO Schematic Kikuchi map for a b.c.c. crystal centred on [111].

0001 0002 • • • • • ~~---·• ---..1011 . • 0002• • 0000~ ..___1010 X • • • • • 0000• 2Ho• • • • • • - • • X • • • 0002. • • • • • • • • •

[010) [1210) [1.20) [0110)

• • • • • • • • • • • X • nffi 1~10 • 0111• • • cU:o ~10. • • ~1~1o• • • • • X X • • • • • • 1ro1 • • • • • [001) 10001 I [011 J [1213 J

• • • • • • X • • 0~11 2021•- • X • • 2f1o lo1 • OCXJO 2/\o • 1011.. OCXXJ• • • • • • X • • • • • • • • X • • •

ff.22] I 01121 [1.21] [0111] Fig. C.ll Schematic electron diffraction patterns for six selected electron beam direc• tions for a h.c.p. crystal. The low order reflections are indicated and the Miller indices and Miller-Bravais indices of the upward drawn direction, i.e. the electron beam direction B is indicated below each pattern. The crosses indicate reflections which can occur due to double reflection. APPENDIX C 195

Fig. C.12 Schematic Kikuchi map for a h.c.p. crystal extending over two standard triangles.

Fig. C.13 Schematic Kikuchi map for a h.c.p. crystal centred on [0001]. 196 ELECTRON BEAM ANALYSIS OF MATERIALS

Fig. C.14 Schematic Kikuchi map for a h.c.p. crystal centred on [1210].

Fig. C.15 Schematic Kikuchi map for a h.c.p. crystal centred on [OTIO]. APPENDIX C 197 lines appear on the lines themselves with the sign of g marked to correspond to the indices of the diffraction maximum through which the line passes when the Bragg condition is satisfied. For example, considering the 020 reflection in the f.c.c. map for B = [00 1] the 020 Kikuchi line would pass through the 020 maximum when the 020 reflection is at the Bragg condition so that the line marked 010 would then pass through the origin of the re• ciprocal lattice, the direct beam. If the crystal is tilted appropriately the situation will be reversed and the 010 reflection will be at the Bragg condition and the 020 Kikuchi line will pass through the direct beam. It should be emphasized that the maps are not drawn to scale and should be regarded as topologically correct only. If detailed work is to be carried out on any particular crystal system it is a very valuable exercise to make up a composite experimental Kikuchi map. The usefulness of reflections in diffraction contrast is effectively measured by the extinction distance for the reflection (see Chapter 2) and a few selected values are given in Table C.l for 100 kV electrons. To obtain the extinction distances for other voltages these values should be scaled by the ratio of electron velocities.

Table C.l Typical extinction distances for 100 kV electrons (in nm).

Reflection AI Cu Ni Au Si Ge

Ill 56 24 24 16 60 43 200 67 28 28 18 220 106 42 41 25 76 45 311 130 51 50 29 135 76 222 138 54 53 31 400 167 65 65 36 127 66

Reflection Fe Nb Mo

110 27 26 23 200 39 37 32 211 50 46 41 310 71 62 58

Reflection Mg Co Zn Zr Cd

0002 81 25 26 32 24 ITO! 100 31 35 38 32 1120 141 43 50 49 44 !TOO 151 47 55 59 52 1122 171 52 58 59 68 2201 202 62 70 69 61 !102 231 70 76 84 68 Appendix D STEREOMICROSCOPY AND TRACE ANALYSIS

D.! STEREOMICROSCOPY Stereomicroscopy is used in both SEM and TEM to obtain qualitative and quantitative three-dimensional information from specimens. Stereo pairs are taken simply by tilting the specimen between micrographs. In the case of SEM stereopairs this is a very straightforward procedure since all that is required is to operate the appropriate graduated tilt control after first taking a micrograph and taking a second micrograph at this different perspec• tive. In the case of TEM using diffraction contrast imaging (Chapter 5) it is essential to change only B and to maintain g and s identical for the two micrographs. Clearly this is best done using Kikuchi ~aps and tilting along the appropriate Kikuchi line. The tilt angle can be calculated from the angle between the two beam directions. Qualitative three-dimensional information can be obtained by viewing the stereomicrographs in a stereoviewer with the tilt axis towards the viewer. Fracture surfaces in SEM and the defect arrangement within a thin foil in TEM and in particular HVEM are far more easily interpreted when viewed in stereo. Quantitative three-dimensional information can be obtained by using the height measuring attachments available on stereoviewers. The apparent height difference measured on the viewer (j is related to real height difference H through the tilt angle (J and the magnification M:

(j H =------c 2M sin (J

D.2 TRACE ANALYSIS When appropriate crystallographic information is available it is straight• forward to relate line directions of defects or directions in surfaces to the orientation of the crystal axes. For unambiguous analysis it is clear that information is required in more than one projection so that true directions can be extracted from projected directions. Thus in the case of the projected APPENDIX D 199

'.u. True A( '\.direction I ~a~ \ I \

Fig. D.l Schematic stereogram illustrating the technique of trace analysis to determine the true direction of a line which when viewed in 8 1 and 8 2 projects along [h1 k1 11]P and [h 2 k2 12 ]P respectively.

line direction [h 1 k1 l1]P of a dislocation viewed in a direction B 1 all that can be said is that the true direction lies in the plane defined by [h 1 k1 l1 ]P and B 1 . Micrographs taken in any other beam direction B2 result in a second pro• jected direction [h2 k2 l2 ]P and hence a plane defined by [h2 k2 l2 ]P and by B 2 . The zone axis for these two planes defines the true direction of the dislocation. This is illustrated in Fig. D.l. Appendix E TABLES OF X-RAY AND EELS ENERGIES

Tables of X-ray and EELS energies are shown on the following pages. Table E.l Energies (in ke V) of X-rays from the elements tabulated as a function of increasing energy.

Less intense Lesser intense Even less intense Principal minor line minor line minor line Atomic emission Energy Element number line Energy Line Intensity Energy Line Intensity Energy Line Intensity

0.851 Ni 28 L a 0.883 58 Ce M a 0.930 Cu 29 L " 0.972 Ba 56 M y 1.012 Zn 30 L. 1.041 Na 11 K " 1.081 Sm 62 M • 1.098 Ga 31 L. 1.185 Gd 64 M. 1.188 Ge 32 L, 1.253 Mg 12 K 1.282 As 33 L. " 1.379 Se 34 L• 1.480 Br 35 L. 1.486 AI 13 K. 1.521 Yb 70 M. 1.586 Kr 36 L 1.644 Hf 72 M". 1.694 Rb 37 L, 1.709 Ta 73 M 1.739 Si 14 K " > . '"C 1.774 w 74 M. m'"C 1.806 Sr 38 L z 1.842 Re 75 M". 0 1.914 Os 76 M • ->< 1.922 y 39 L m " 1.977 Ir 77 M •

N =..... = .... tT1

tT1 .... r ..., ;:>;! n z al

0 > tT1 > a: z -< > r C/l C/l 0 "11 a: - tT1 > ;:>;!

..., > r

- C/l

Intensity

intense

Line

line

less

Energy

Even minor

(6) (8) (5) (5) (5)

Intensity (20) (25) (17) (10) (10) (25) (17) (25) (20) (25) (17) (17)

(17)

KP KP

Lp, M Lp, MY L KP Lp, Lp, Lp, ~2 Lp, Lp, Lp, L Lp, Line

intense

line

3.171 3.465 2.137 2.163 2.465 3.528 3.904 3.347 3.563 3.001 3.369 3.713 4.935 4.507 2.815 4.100 4.72 4.301

Energy Lesser minor

(I (45) (20) (45) Intensity (50) (75) (50) (55) (50) (60) (40) (60) (15) (75) {50) (50) (45) (45) (15) (42) (75) (75) (50) (10) (40) (60) (75) (60) (40) (60) (15)

., .,

K.2 MP

MP KPI MP K MP MP LPI MP LPI ~I LPI ~I LPI MP ~I LPI KP LPI LPI LP L LPI ~I Line ~I

line

intense

2.028 2.257 3.150 3.336 3.843 4.42 2.127 2.362 2.524 2.990 3.145 3.487 4.619 2.204 2.683 3.239 3.589 2.322 2.631 3.190 3.316 3.662 4.460 2.124 2.282 2.442 2.834 4.012 4.029 2.394 4.220

Less

Energy minor

•

......

.I .I .I

.I .I .I .I .I .I .I "' .I .I . .

M L L K M M M M M. M K. L. K K L M K. L M L K. Principal L L L L L L L K L line

emission

15 16

19 18 54 55 17

52 53 21

78 80 42 79 41 81 83 Atomic 50 number 51 20 46 82 44 91 47 40 45 92 49 90 48

(Contd.)

Sc Cs

Hg Xe Pt Nb Mo I Pb p Bi Pd Pa K Sb Ca Au s Ar Te Ru Rh u Ag Zr Cl Sn Tl

Th In

E.l

3.937

4.286 3.769 2.013 2.166 3.443 4.088 2.042 2.195 3.604 4.109 2.048 2.267 2.419 3.690 2.121 2.293 2.558 3.312 2.307 2.621 2.984 2.342 2.696 2.838 3.286 2.957 3.077 3.133 2.991 3.165

Table Energy Element 4.465 Ba 56 La 4.827 (50) 5.193 Lp, (20) 4.508 Ti 22 4.931 ~I(20) Ka P1 4.650 La 57 La 5.041 LPI (50) 5.383 Lp, (20) 4.839 Ce 58 La 5.261 LPI (50) 5.612 Lp, (20) 4.949 v 23 Ka 5.426 KPI (20) 5.411 Cr 24 Ka 5.946 KPI (18) 5.635 Sm 62 La 6.204 LPI (50) 6.586 Lp, (20) 5.894 Mn 25 K a 6.489 KPI (20) 6.398 Fe 26 K a 7.057 KPI (20) 6.924 Co 27 K a 7.648 KPI (20) 7.471 Ni 28 K a 8.263 KPI (20) 72 9.021 (50) 9.346 (20) 10.514 L (10) 7.898 Hf La Lp, Y1 29 8.904 ~I(20) 8.040 Cu Ka PI 8.145 Ta 73 La 9.342 LPI (50) 9.650 Lp, (20) 10.893 L (10) 74 L 9.671 (50) 9.960 (20) 11.284 LYI (10) 8.396 w a Lp, Y1 30 9.570 ~I(20) 8.630 Zn Ka PI (20) 11.683 (10) 8.651 Re 75 La 10.008 LPI (50) 10.274 Lp, LYl 10.354 (50) (20) 12.093 L (10) 8.910 Os 76 La LPI 10.597 Lp, Yl 77 10.706 (50) (10) 9.174 Ir La L 10.919 Lp, (20) 12.510 LYl I(~I 9.241 Ga 31 K 10.262 p (21) . (10) 9.441 Pt 78 La 11.069 LPI (50) 11.249 Lp, (20) 12.940 L (20) LYI (10) 9.712 Au 79 La 11.440 (50) 11.583 Lp, 13.379 Yl ~I 9.874 Ge 32 K a 10.979 p (21) 11.922 (20) 13.828 L (10) 9.987 Hg 80 La 11.821 LPI (50) Lp, Yl (50) 12.270 (20) 14.289 (10) 10.267 T1 81 La 12.211 LPI Lp, LYl 10.542 As 33 K a 11.722 KP (22) (20) 14.762 (10) 10.550 Ph 82 La 12.612 LPI (50) 12.621 Lp, LYl (20) 15.245 (10) 10.837 Bi 83 La 13.021 (50) 12.978 Lp, LYl ~I 11.207 Se 34 K a 12.492 p (24) K 13.286 (24) 11.907 Br 35 a KP > 12.631 Kr 36 K a 14.107 KP (24) '"C 16.199 (50) 15.621 (20) 18.979 L (10) '"C 12.967 Th 90 Lal Lp, Y1 tTl ~I 13.373 Rb 37 K a 14.956 p (24) z 17.217 (50) 16.425 (20) 20.164 L (10) 0 13.612 u 92 Lal Lp, Yl ~I 14.140 Sr 38 K 15.830 (24) -><: a p y tTl 14.931 39 K a 16.734 KP (25) (27) 15.744 Zr 40 K a 17.663 KP (12) 16.581 Nb 41 K a 18.700 KP N Q .... t""' tTl

tTl ("l ..., i" z 2 0:1 0 ~ > tTl > z t""'

> -< en ..... en

'""l"j

0 ~ > ..., ..... t""' tTl > i"

Intensity

(5) (5) (5) (6) (6) (7) (5) (6) (7)

Line

Kp, Kp, Kp, Kp, Kp, Kp, Kp, Kp, Kp,

line

intense

Less Energy

minor 27.856 30.388 29.104 33.036 37.251 26.639 34.408 38.723 35.815

Intensity

(26) (24) (25) (26) (27) (29) (24) (28) (29) (30) (27) (29) (30) (32) (28)

3 3

3 3 3

, ,· ,' ,' ,· ,' '· ,' ,'

11 11 11 11 11 11 11

11 11 /1!,3 11 11 11

column.

Line KP KP K K K K K K K K K K K K ' K

line

intense

19.599

Less

Energy minor 23.807 21.646 24.921 28.467 32.271 36.317 22.712 26.080 29.705 33.598 37.771 27.252 34.962 39.232

listed

peaks

1%1.2

i%1,2

Principal K K"'·' K"'·' K"'·' line K K"'·' K"'·' K"'·' K"'·' K"'·' K"'·' K"'·' emission K"'·' K"'·' K"'·' K"'·'

unresolved

Atomic number

47 53 50 54 55 57 42 46 49 51 58 44 56 62 45 48

energies

Individual

Mo Element Rh Pd Ru Ag La Cd Sn In I Ba Sb Xe Sm Cs Ce 2

and

K.,

05) of

energies

energy

170,22.980)

714,34.273)

(Contd.)

Individual

(17.476,17.371) (2l.l74,2l.Ol7) (19.276,19.147) (22.159,21.987) (20.213,20.070) (23. (24.206,23.998) (28.607,28.312) (40.111,39.516) (25.267,25.040) (29.774,29.453) (26.355,26.1 (30.968,30.620) (34. (32.188,31.812) (33.436,33.028)

average

E.l

Weighted

17.441

19.233

Energy* Table 22.101 20.165 23.106 2l.l21 24.136 25.191 29.666 26.271 28.508 30.851 32.062 39.911 33.299 34.566

* tTl

N =

:><

tTl

>- z 0 "' ...... "'

(*)

Lp,

(45)

(*)

2.124

Lp,

. 1.922

1.098

1.282 1.379 1.480 1.586 1.694 1.806

1.012

1.188

2.042

0.851 0.930

0.341 0.395 0.452 0.511 0.573 0.637 0.705 0.733

elements.

the

(6)

(8)

(25) (27)

(21) (*) (20) (22) (20) (24) (20) (24) (24) (20) (24) (20) (10) (20) (20) (15) (20) (15) (15) (21)

(18) (24) (20)

X-rays

7.057 7.648

3.190 3.589 8.263 8.804

5.426 5.946 2.465 6.489 2.815

4.460 9.570 4.931

2.137

4.012

16.734 17.633

10.262 10.979 11.722 12.492 12.286 14.107

15.830

14.956 KP

characteristic

1.041 1.253

1.739

1.486

7.471

8.040 8.630

2.307 5.411 2.621 5.894 6.398 2.957 6.924

4.088 3.312 4.508 9.241 3.690 4.949 9.874

2.013

14.931

10.542 11.207

13.373 14.140

15.744

11.907 12.631

(in

14 15 16 17

11 13

18 29 19

30 31 32

25 34 35 27 36 28 37 38 number 20 21 22 23 24

Atomic

Energies

E.2

Element

Mg Kr Mn y

Co Na K Si Cu As p Cl Ga Ge AI s v Br Ni Rb Fe Sc Cr Se Zr Ar Zn

Ca Ti

Table

r r

trl trl

IJ:l IJ:l

trl trl

;:c ;:c

..., ..., (") (")

z z

tTl tTl

3:: 3::

3:: 3:: z z

0 0

r r

-< -<

r r

> > [JJ [JJ

[JJ [JJ

trl trl

.., ..,

= =

[JJ [JJ

N N

;:c ;:c

..., ...,

0 0

~ ~ -

1.081 1.081

1.131 1.131

1.240 1.240

1.185 1.185

1.293 1.293

0.461 0.461

0.331 0.331

M M 0.355 0.355

0.733 0.733

0.691 0.691

0.978 0.978 0.606 0.606

0.568 0.568

0.532 0.532

0.833 0.833 0.496 0.496

0.972 0.972

0.778 0.778

0.929 0.929

0.883 0.883

(*) (*)

y y

L L

(20) (20)

(20) (20) (25) (25)

(20) (20)

(17) (17)

(25) (25)

(20) (20) (25) (25) (17) (17)

(25) (25)

(20) (20)

(17) (17)

(20) (20)

(*} (*}

3.001 3.001

5.849 5.849

5.383 5.383 4.507 4.507

5.193 5.193

7.634 7.634

7.365 7.365

7.102 7.102

6.842 6.842

3.904 3.904

3.713 3.713

6.338 6.338

3.528 3.528

6.088 6.088

3.347 3.347

3.171 3.171

5.612 5.612

4.100 4.100

6.586 6.586

4.935 4.935

4.72 4.72

4.301 4.301

Lp, Lp,

(50) (50)

(40) (40)

(45) (45)

(50) (50)

(45) (45)

(50) (50) (45) (45)

(50) (50)

(50) (50) (75) (75)

(75) (75)

(50) (50)

(50) (50) (42) (42)

(50) (50) (40) (40)

(50) (50)

(75) (75)

(*) (*)

6.204 6.204

5.041 5.041

7.246 7.246

6.977 6.977

6.712 6.712

3.843 3.843

6.455 6.455

3.487 3.487

3.316 3.316 5.960 5.960

2.834 2.834

5.721 5.721

3.150 3.150

2.683 2.683 5.488 5.488

5.261 5.261

2.394 2.394

2.990 2.990

4.619 4.619

4.42 4.42

4.220 4.220

4.029 4.029

Lp, Lp,

a a

5.635 5.635

5.845 5.845

5.229 5.229

2.424 2.424 2.536

2.166 2.166 2.257

6.494 6.494

3.937 3.937

6.272 6.272

6.056 6.056

3.604 3.604

3.443 3.443 3.662

5.432 5.432

3.133 3.133

2.696 2.696

2.558 2.558 5.033 5.033

2.293 2.293

3.769 3.769

2.984 2.984 4.839 4.839

2.838 2.838 4.650 4.650

4.465 4.465 4.829

4.286 4.286

4.109 4.109

3.286 3.286

L L

(12) (12)

(31) (31) (24) (24)

(30) (30) (24) (24)

(30) (30) (25) (25)

(29) (29)

(28) (28)

(27) (27)

(26) (26)

(32) (32) (26) (26)

(29) (29)

(*) (*)

19.599 19.599

18.700 18.700

22.712 22.712

21.646 21.646 39.232 39.232

20.608 20.608 37.951 37.951

36.504 36.504

35.104 35.104

33.737 33.737

32.402 32.402

31.097 31.097

26.167 26.167

23.807 23.807

KP KP

29.805 29.805

28.564 28.564

24.921 24.921

a a

16.581 16.581

19.233 19.233

17.441 17.441 18.325

39.911 39.911

33.299 33.299

30.851 30.851

38.532 38.532

37.182 37.182 20.165 20.165

35.860 35.860

34.566 34.566

32.062 32.062 28.508 28.508

26.271 26.271

25.191 25.191

24.136 24.136 27.346

23.106 23.106

22.101 22.101

21.121 21.121

29.666 29.666

27.377 27.377

K K

(Contd.) (Contd.)

66 66

65 65

64 64

62 62

57 57

55 55

53 53

63 63

51 51

61 61

50 50

45 45

44 44

59 59

43 43 58 58

42 42

41 41

56 56

54 54

49 49

48 48

52 52

47 47

46 46

60 60

number number

Atomic Atomic

E.2 E.2

Nb Nb

Sm Sm

Dy Dy

Eu Eu

Rh Rh

Ru Ru

Tb Tb

Gd Gd Pm Pm

Pd Pd

I I

Ag Ag

In In

Cd Cd

Sb Sb

Ba Ba

Xe Xe Sn Sn

Mo Mo

Tc Tc

Cs Cs

Nd Nd

Element Element

Pr Pr

Table Table

La La

Ce Ce Te Te Ho 67 6.719 7.524 (50) 7.910 (20) 1.347 Er 68 6.947 7.809 (50) 8.188 (20) 1.405 Tm 69 7.179 8.100 (50) 8:467 (20) 9.424 (5) 1.462 Yb 70 7.414 8.400 (50) 8.757 (20) 9.778 (5) 1.521 Lu 71 7.654 8.708 (50) 9.038 (20) 10.142 (6) 1.581 Hf 72 7.898 9.021 (50) 9.346 (20) 10.514 (10) 1.644 Ta 73 8.145 9.342 (50) 9.650 (20) 10.893 (10) 1.709 w 74 8.396 9.671 (50) 9.960 (20) 11.284 (10) 1.774 Re 75 8.651 10.008 (50) 10.274 (20) 11.683 (10) 1.842 Os 76 8.910 10.354 (50) 10.597 (20) 12.093 (10) 1.914 Ir 77 9.174 10.706 (50) 10.919 (20) 12.510 (10) 1.977 Pt 78 9.441 11.069 (50) 11.249 (20) 12.940 (10) 2.074 Au 79 9.712 11.440 (50) 11.583 (20) 13.379 (10) 2.148 Hg 80 9.987 11.821 (50) 11.922 (20) 13.828 (10) 2.224 T1 81 10.267 12.211 (50) 12.270 (20) 14.289 (10) 2.301 Pb 82 10.550 12.612 (50) 12.621 (20) 14.762 (10) 2.380 Bi 83 10.837 13.021 (50) 12.978 (20) 15.245 (10) 2.458 Po 84 11.129 13.445 (50) 13.338 (20) 15.741 (10) At 85 11.425 13.574 (50) 14.065 (10) 16.249 (10) Rn 86 11.725 14.313 (50) 14.509 (10) 16.768 (10) Fr 87 12.029 14.768 (50) 14.448 (20) 17.300 (10) Ra 88 12.338 15.233 (50) 14.839 (20) 17.845 (10) Ac 89 12.650 15.710 (50) 15.929 (10) 18.405 (10) Th 90 12.967 16.199 (50) 15.621 (20) 18.979 (10) 3.058 Pa 91 13.288 16.699 (50) 16.022 (20) 19.565 (10) 3.148 u 92 13.612 17.217 (50) 16.425 (20) 20.164 (10) 3.239 Np 93 13.942 17.747 (50) 16.837 (20) 20.781 (10) >- "' "'trl (*) Approximate intensity relative to principal line of series. z 0 ...... :>< trl

1-J =-.J 208 ELECTRON BEAM ANALYSIS OF MATERIALS Table E.3 Energies (in eV) corresponding to edges in EELS data.

K L3 Lz L3 Lz Ms M4 Li 55 As 1323 1359 Be 111 Se 1435 1476 57 57 B 188 Br 1550 1596 69 70 c 284 Kr 1675 1727 89 89 N 402 Rb 1804 1864 110 112 0 532 Sr 1940 2007 133 135 F 685 y 2080 2156 157 160 Ne 867 Zr 2222 2307 150 152 Na 1072 Nb 205 207 Mg 1305 51 51 Mo 227 230 AI 1560 73 73 Tc 253 256 Si 1839 99 99 Ru 279 284 p 2146 132 132 Rh 307 312 s 165 165 Pd 335 340 Cl 200 202 Ag 367 373 Ar 245 247 Cd 404 411 K 294 296 In 443 451 Ca 346 350 Sn 485 493 Sc 402 407 Sb 528 537 Ti 456 462 Te 572 583 v 513 521 I 619 631 Cr 575 584 Xe 672 Mn 640 651 Cs 726 740 Fe 708 721 Ba 781 796 Co 779 794 La 832 846 Ni 855 872 Ce 883 901 Cu 931 951 Sm 1080 1106 Zn 1020 1043 Ga 1115 1142 Ge 1217 1248 INDEX

Aberration Deflection systems, 17 chromatic, 4, I 0 Detector focussing error, 43 energy dispersive, 56, 59 spherical, 2, 10 scintillation, 46 Allowed reflections, 179 wavelength dispersive, 56, 57 Antiphase boundary, 140 Deviation parameter, 75, 114, 128 Aperture Diffraction contrast, in TEM, 114 condenser aperture in convergent beam, 90 Diffraction from aperture, I 0 diffraction limit, II Diffraction group symmetry, 88, 90, 188 objective aperture, 40 Diffraction patterns, 40, 65 Auger electrons, generation of, 35 analysis of, 62, 65 Auger electron spectroscopy, 16, 53 errors in SAD, 45 interpretation of spectra in, 169 from twinned crystals, I 08 rotation with respect to image, 42 Backscattering of electrons, 30 Diffuse scattering, 103 images, 149 Dislocation images, 119 patterns, 112 Dispersion of spectrometer, 16 Black/white contrast, 132 Double diffraction, 181 Bragg's law, 68 Dynamical theory, 117 Bremsstrahlung, see X-rays Burgers vector, FS/RH definition, 124 Elastic scattering, of electrons, 19 determination of, 120 high angle, 23 loop analysis, 124 low angle, 22 Rutherford scattering, 20, 24 Channelling patterns, 109 Electron beam induced current signal, 37 Cluster formation, influence on diffraction Electron energy loss spectroscopy, 15, 163 patterns, 104 Electron lenses, II Contrast transfer function, 146 Electron microprobe analysis, 56 Convergence, 13 Electron sources, I in SEM, 46 Electron spectrometer Convergent beam diffraction, 52, 79 magnetic electron, 61 foil thickness determination, 80 Auger electron, 63 point group determination, 90 Ewald sphere, 68 space group determination, 95 Extinction distance, 22 unit cell determination, 95 effective, 23, 119 Coster-Kronig transition, 34 Coupling in energy loss, diffraction, 61, 62 Field depth, in SEM, 46 image, 61, 62 Fluorescent yield, 28, 32 Cross section for ionization, 28 partial, 167 High order Laue zones, 82 Crystal planes, angles between, 186 High resolution electron microscopy, 43, 144 spacing of, 70, 185 High voltage electron microscopy, 43 Crystal system, determination of, 66, 76 HT stability, 4

Debye-Waller factor, 23 Inelastic scattering, of electrons, 24 210 INDEX

plasmon, 25 Scanning transmission electron microscopy, single electron, 25 11, 13, 50, 143 thermal, 24 Short range order, 102 Source, electron, 3 Kikuchi lines, 72 brightness of, 2, 7 Kikuchi maps, 189 coherence, 4 Kinematic theory, 114 energy spread, 4 field emission, 6 Lattice, Bravais, 22 LaB6 ,2,6 Laue zones, indexing of, 82 size, 3, 4, 7 Lenses stability in Auger spectroscopy, 16 thermionic, 4 auxiliary, 12 Spatial resolution of analysis, 175 condenser, 12 Spectrometer, magnetic, 61 electrostatic, 7 Auger, 63 focal length, 10 Spinodal decomposition, 104 image forming, II Stacking fault contrast analysis, 134 magnetic, 7 in f.c.c. crystals, 137 probe forming, 11, 14 in h.c.p. crystals, 139 rotation, I 0 Stereomicroscopy, 198 Long range order, 66, 101 Stopping power, 31 Strain contrast, from precipitates, 138 Magnetic domains, in TEM, 142 Structure factor, 22, 181 in SEM, 151 Magnetic prism, 15 Thickness fringes, 116 Magnetic samples, 108 Trace analysis, 198 imaging of, 142 w-Transformation, 107 Miller indices, 70 Transmission electron microscopy, 39 Miller Bravais indices, 72 analysis of images in, 119 Moire patterns, 131 diffraction contrast in, 114 Monte Carlo calculations, 31 influence of electron optical conditions, 143 Omega transformation, 107 Tunnelling, electron, 2 Optics, electron, 7 Twinning, 108

Partial dislocation analysis, 141 Weak beam microscopy, 128 Planar defect, contrast from, 134 Wehnelt cap, 4, 6 Precipitation and diffuse scattering, I 03 Premartensitic phenomena, 105 Void contrast, 130 Pretransformation and diffuse scattering, 103, 105 X-rays Probe size, in SEM, 47 absorption and fluorescence of, !57 bremsstrahlung, 34 Reciprocal lattice, 68, 179 characteristic, 27 Resolution cross section for ionization, 28 of Auger spectroscopy, 55, 175 fluorescent yield, 28 of back scattered images, 50 from bulk samples, 162 of electron energy loss spectroscopy, 175 generation of, 26 of energy loss spectrometer, 16 influence of diffraction condition on of secondary images, 48 production, 162 of X-ray analysis, 50, 53, 175 interpretation of X-ray data from thin of X-ray spectrometer, 59, 61 foils, 153 Rocking curves, 115 X-ray spectrometer wavelength dispersive, 57 Scanning electron microscopy, 14, 44 energy dispersive, 59 interpretation of images, 147