AUSTRALIAN ATOMIC ENERGY COMMISSION · AUSTRALIAN ATOMIC ENERGY COMMISSION RESEARCH ESTABLISHMENT...

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AUSTRALIAN ATOMIC ENERGY COMMISSIONRESEARCH ESTABLISHMENT

LUCAS HEIGHTS

CRYSTALLOGRAPHIC TECHNIQUES AND DATA FOR TRANSMISSION

ELECTRON MICROSCOPY OF ZIRCONIUM

by

A. JOSTSONS

J.G. NAPIER

February 1970

AUSTRALIAN ATOMIC ENERGY COMMISSION

RESEARCH ESTABLISHMENT

LUCAS HEIGHTS

CRYSTALLOGRAPHIC TECHNIQUES AND DATA FOR TRANSMISSION

ELECTRON MICROSCOPY OF ZIRCONIUM

by

A, JOSTSONS

J. G.NAPIER

ABSTRACT

The crystallography of hexagonal close packed metals is discussed briefly in termsof the four-axis hexagonal reference basis and the Miller-Bravais notation which are usedthroughout the report. Electron diffraction problems are treated with reference to the four-axis hexagonal reciprocal lattice rather than the more usual three-axis hexagonal system.Using these concepts, analysis of el?,:trou diffraction spot andKikuchi patterns is illustratedand applied to orientation and dislocation Burgers vector determinations. Computed values ofinterplanar spacings, interplanar angles, angles between directions, and extinction distancesfor zirconium are listed.

CONTENTS

Page

1. INTRODUCTION 1

2. CRYSTALLOGRAPHY OF h.c.p. METALS 1

2.1 Crystal Structure 12.2 Crystallographic Indices 22.3 Stereographic Projections 42.4 Dislocations in h.c.p. Metals 4

3. ELECTRON MICROSCOPY OF h.c.p. METALS 5

3.1 The Reciprocal Lattice 53.2 Electron Diffraction Patterns 73.3 Kikuchi Patterns 93.4 Determination of Dislocation Burgers Vectors 12

4. REFERENCES 14

APPENDIX 1 Interplanar Spacings in a-Zirconium

APPENDIX 2 Angles Between Crystallographic Planes in Alpha Zirconium, c/a = 1.593

APPENDIX 3 Angles Between Directions in Alpha Zirconium, c/a- =1.5927

APPENDIX 4 Standard (0001) Projection for h.c.p. Zirconium

APPENDIX 5 Crystallographic Formulae for h.c.p. Metals

APPENDIX 6 Electron Diffraction Patterns Frequently Obtained from h.c.p, Metals

APPENDIX 7 Kikuchi Patterns

APPENDIX 8 Values of g.b. for Eight Reflections and^ <1120> -, ^ <1123> -, and

<0001> - Type Burgers Vectors

APPENDIX 9 Extinction Distances for Various Reflections for 100 kV Electrons in a-Zirconium

Figure 1 Model of hexagonal close-packed structure with unit cell shown in heavy outline.

Figure 2 Interstitial voids in the h.c.p. structure with ideal axial ratio c/a = 1 8/3.(a) Octahedral voids; (b) Tetrahedral voids. (Barrett and Massalski 1966).

Figure 3 Model of hexagonal close-packing of spheres with octahedral and tetrahedralinterstitial sites, looking along the c-axis. (Gehman I960).

Figure 4 Indices of planes in hexagonal crystals. (Barrett and Massalski 1966).

Figure 5 Indices of directions in the hexagonal system with both three - and four - digit indices.The c-axis is normal to the plane of the drawing; a i ? a^ and as in h.c.p.- crystals are alongthe close-packed rows of atoms. (Barrett arid Massalski 1966).

(continued)

CONTENTS (continued) 1. INTRODUCTION

Figure 6 Burgers vectors in the hexagonal close-packed lattice. (Berghezan, Fourdeux andAmelinckx, 1961).

Figure 7 Ewald sphere construction in reciprocal lattice.

Figure 3 -Indexing an unknown diffraction pattern.

Figure 9 Alternative ways of indexing a diffraction pattern from a thin foil with the upward drawnnormal [ 12l3] parallel to the electron beam.

Figure 10 Stereo graphic projections of plane normals; (a) and (b) correspond to diffraction patternsin Figures 9(a) and 9(b) respectively. The broken lines show reflections which wouldappear on the diffraction pattern after tilting about [ 1010] in the sense given on theprojections.

Figure A7.1 The [0001] Kikuchi map for h.c.p. titanium (c/a = 1.588). All poles are indexed interms of directional indices.

Figure A7.2 (a) A Kikuchi pattern from zirconium, (b) Schematic drawing of centre lines of indexedKikuchi pairs in (a).

Figure A7.3 An enlarged section of the [0001] standard projection for directions for h.c.p. zirconium.The foil orientation in Figure A7.1 is shown at B.

Zirconium and most of its alloys of interest in nuclear power technology have hexagonalclose-packed (h.c.p.) structure. Many aspects of physical metallurgical studies of these alloysrequire a knowledge of their crystallography whichin contrast to that of cubic metals, is notconsidered in any detail in the many available texts on physical metallurgy. Most of the scatteredinformation is in original papers in the literature and unfortunately some of them contain errorsbecause of confusion over different crystallographic systems and notation used to describe h.c.p.structures. An exception is the review prepared by Partridge (1967) but he does not list specificcrystallographic information such as values of interplanar angles and angles between directions forzirconium.

The aim of this report is to describe briefly the fundamentals of crystallography of h,c.p.metals and their application to problems in transmission electron microscopy of zirconium. Thespecific crystallographic formulae and data for zirconium required for quantitative electron microscopyhave been calculated and are tabulated in the Appendices. Some of these data have more generalapplication in X-ray diffraction studies and analyses of twinning and deformation modes in bulk mat-erials.

2. CRYSTALLOGRAPHY OF h.c.p. METALS

2.1 Crystal Structure

The arrangement of atoms in a h.c.p. metal can be shown in terms of a hexagonal prism,Figure 1, where the filled circles represent atom centres. The primitive unit cell, (heavy lines)

which does not immediately reveal the hexagonalsymmetry, has axes §! = a2 ^ c with the anglebetween aa and a2 equal to 120° and c perpendic-ular to both §1 and §2 . The unit cell contains twoatoms with positions given by the coordinates 000and 2- .L i. Sirice the surroundings of the interior

atom differ from those at the cell corners the atompositions in the h.c.p. structure do not constitute aspace lattice. The actual space lattice remains prim-itive with points at cell corners only if two atoms areconsidered to be associated with each lattice point.

If the atoms are considered as hard spheres,the plane containing both a* and a2 is a close-packedplane. The hexagonal close-packed structure is char-acterised by an ABABABA. ... stacking sequence ofclose-packed planes. In an ideal close-packed struc-ture the axial ratio c/a is equal to

Figure I. Model of hexagonal close-packed structure withunit ceii bhcwn in heavy outline. 1.633

Zirconium and its alloys have a strong affinity for hydrogen, oxygen, nitrogen and carbon whichin solid solution occupy : the interstitial holes. The locations of the octahedral and tetrahedral inter-stices in a h.c.p. structure are shown in Figure 2. There are 2 octahedral and 4 tetrahedral interstices

2. 1 7 ? 00 3 ^ 005per unit cell centred on coordinates ~ -2. ! , 1 2 land 2 i I

respectively. In an ideal h.c.p. structure of rigid spheres of radius r, the maximum radius of a spherethat can be accommodated in an octahedral or tetrahedral interstice is 0.41 r and 0.22 r respectively.The geometry of interstitial sites in a h.c.p. structure is revealed more clearly by the model shown inFigure 3.

-2- -3-

V5/2V?

Metal atoms

O Octahedral interstices

Metal atoms

O Tetrahedral interstices

(a) Octahedral voids (b) Tetrahedral voids

Figure 2. Interstitial voids in the h.c.p. structure with ideal axial ratio c/ft = N 8/3(Barrett and Massalski 1966)

TETRAHEORALINTERSTITIALS

OCTAHEDRALIMTERSTITIALS

Figure 3. Model of hexagonal close-packing of spheres with octahedraland tetrahedral interstitial sites, looking along the c-axis.

(Gehman i960)

2.2 Crystallographic Indices

Crystallographic indices constitute a convenient system of notation of crystal planes anddirections. The different axial systems commonly used to define the indices of directions andplanes in hexagonal crystals are three-axis hexagonal (Miller indices), four-axis hexagonal (Miller-Bravais indices) and orthohexagonal. Only the four-axis hexagonal system gives rise to similarindices for crystallographically equivalent directions and planes. This feature of the Miiier-Bravaissystem is an asset when dealing with typical problems in physical metallurgy, for example, descrip-tion of slip modes and Burgers vectors of dislocations, and accounts for its general acceptance. Themain arguments against universal use of the Miller-Bra vais notation are the alleged greater complexityof Crystallographic formulae and claims that the four-axis hexagonal reciprocal lattice is physicallymeaningless (Partridge and Gardiner 1967 a). The simplifications introduced by Nicholas (1966)

into Miller-Bravais formulae, however, lead to expressions of equal simplicity to those based on thealternative systems while still retaining the virtue of symmetry. Frank (1965), and more recently,Okamoto and Thomas (1967, 1968) have demonstrated that the four-axis hexagonal reciprocal lattice,though not strictly a reciprocal lattice according to the usual mathematical definitions, permits theanalysis of the necessary Crystallographic relationships directly in terms of Miller-Bravais indicesusing elementary vector analysis. Consequently, the four-axis hexagonal system with Miller-Bravaisindices does not introduce any additional limitations for normal electron microscopy, compared withalternative systems of indexing, -and is used throughout this report.

The nomenclature of planes in the Miller-Bravais system is shown in Figure 4 in which theprimitive hexagonal unit cell with the three axes,ai, a2 and a3 at 120° to each other in the basalplane, is shown within the hexagonal prism. Thus, •a three dimensional crystal is represented in fourdimensional space. From the symmetry of the axesin the basal plane it is noted that(1IOO)i I f~1Y"f (11*0)

I i a i i i i , ,. ,,.aa = -(§! + a.2) . (1)

Consequently, when reciprocal intercepts of aplane on all four axes are determined and reduced tothe smallest integers, the indices of a plane will beof the type (hki£), with the restriction that

(lOlO)

-o, h 4 k + i = 0 . (2)Figure 4. Indices of planes in hexagonal

crystals.(Barrett and Massalski 1966)

with the restriction that

u + v + t = 0

Similarly, an arbitrary crystal vector £ can bewritten in the form

= v§2 H- ta3 we. , (3)

(4)

•fa

The indexing of directions in the Miller-Bravais notation is illustrated in Figure 5 whera thetranslations leading from the origin to several lattice points are indicated. The comparatively unusual

translations used in deriving the indices of adirection are necessitated by the restrictionsimposed by Equation 4. The interplanar spaoings in h.c.p. metals are given by

«*_.!_

(5)

,2 -[010)

-dooi[lOlOj-(2101

[1120]-(HOI

Figure 5. indices of directions in the hexagonal system "with boththree—'and four—digit indices. The c-axis is normal to the planeof the drawing; a l f a2 and aa in h.c.p. crystals are along theclose-packed rows of atoms.

(Barrett and Massalski 1966)

where A? =~«j

The symbol A has been retained in the cry stalk-graphic formulae to conform with the literaturebut it is used later to denote the wavelength ofelectrons also, The meaning of these symbolswill be clear from the context.

Values of interplanar spacings of zircon-ium are tabulated in Appendix 1,

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In the h.c.p. system, in contrast to the cubic system, a direction is not normal to a plane withthe same Miller-Bravais indices, except for directions of the form <000£> and <hkiO > . The indicesof the normal to a plane (hki2) are [ hki (AT 2 £)] .

Interplanar angles and angles between directions in h.c.p. metals depend on the c/a ratio andare tabulated for zirconium in Appendices 2 and 3 respectively.

2.3 Stereographic Projections

Although electron micrographs can be readily interpreted analytically, it is often more convenientto use Stereographic methods. Since directions in h.c.p. metals are not necessarily normal to planes ofthe same indices, -two Stereographic projections are required; one for plane normals (poles) and one forcrystal directions. Packer and Miller (1967) have demonstrated the application of two standard projec-tions printed on transparent film; one for poles and the other for directions; which when superimposedcorrectly permit the necessary operations. However Rarey et al. (1966) advocate the use of a doublestereogram' on which are plotted both the poles and directions with rational indices.

A stereogram in the standard (0001) projection is shown in Appendix 4.

2,,4 Dislocations in h.c.p. Metals

Since much of transmission electron microscopy is concerned with the study of dislocations,their geometry in h.c.p. metals will be considered. A notation based on the bipyramid shown in Figuie 6,devised by Berghezan et al. (1961), is often used to describe dislocations' in h.c.p. metals. Becausethe h.c.p. structure is a double lattice structure^ not all nearest neighbour atomic translations are poss-

ible Burgers vectors of perfect dislocations. Thetypes of dislocations which are likely to be stablehave been discussed by Frank and Nicholas (1953)and Berghezan et al. (1961) and are summarised inTable 1.

A comprehensive review of deformationmodes in h.c.p. metals has been made by Dornand Mitchell (1965). In zirconium the primaryslip mode is {1010} <1210> . There is verylittle unambiguous evidence for other slip systemsin zirconium, although basal plane slip of disloca-tions with ^-<^1120)> Burgers vectors has been

(a) «Jproposed to explain the formation of kink bands inzirconium deformed above 500°C (Reed Hill 1964).Martin and Reed-Hill (1964) have reported slipmarkings which could not be explained in terms ofthe above slip systems in grains of deformed poly-crystalline zirconium. Howe et al. (1962) suggestthe operation of slip on {1013} and {111!} planesfrom observation of slip traces in thin foils but theBurgers vectors of these dislocations were notidentified. There is no unambiguously proven casein zirconium of slip systems that will produce strainin a direction not contained in the basal plane, thatis, -dislocations with a Burgers vector with a non-

zero fourth index. However dislocations with a Burgers vector 4 <,1123> have been proposed by Rosen-

baum (1964) to explain atom movements associated with {1122} deformation twinning. The possibility ofslip iri<(ll23>directions is also suggested by analyses of deformation systems required to produce theobserved rolling textures in zirconium (Picklesimer 1966)0

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TABLE 1

CD)

Figure 6« Burgers vectors in the hexagonalclose-packed lattice.

(Berghezan, Fourdeux and Amelinckx, 1961)

BURGERS VECTORS OF DISLOCATIONS IN h.c.p. METALS

Type of Dislocations(Figure 6)

Perfect dislocations

(1) AB, BC, CA, BAS CB, AC

(2) ST, TS

(3) 1ST ±AB, ±ST +BC, ±ST ±CA

Partial dislocations

(4) ±Ac7, ±Ba, ±Ccr

(5) ±crS, ±crT

(6) ±AS, ±BS, ±CS, ±AT, ±BT, ±CT

Total No. ofDislocations

6

2

12

6

4

12

Vector

J- 0.•^£1

±c

±£ + §i

±(3>-i + 3aJ i

± l c

(4) + (5)above

DirectionIndices of

Vector

•J-<H2D>

<0001>

A ./l-lTToV^ XAAA^/

§ <1010>

i<0001>

•~<2023>

Magnitudeof Vectorin Terms ofLatticeParameters

a

l e i

(a2 + c2)*

awI c l2

te' + c?)7

1-3 4 /

RelativeEnergies of)i sloe at ions

for c/a =1.633

a2

c' = f a'11 .T a'

i a2

3 a

-2- a'3 a

a2

Generally, the stability of various dislocations varies as the square of the Burgers vector as hasbeen assumed in values given in Table 1. This, however, is only approximately true in an isotropic med-ium. Theoretical analyses of dislocation stability after including effects of elastic anisotropy in zircon-ium have been made by Roy (1967). Fisher and Alfred (1968) and Yoo (1968). Apparently, basal and pris-matic slip of A <[1120)> dislocations is energetically equally probable and Roy (1967) suggests that theopredominant prismatic slip in zirconium may be due to the presence of interstitial impurities. Fisher andAlfred's calculations suggest that with increasing temperature, effects of elastic anisotropy render edgedislocations with b = -i- <ll20> energetically unstable in the basal plane.«5

The stacking fault energy of zirconium is high and it is expected that partial dislocations wouldnot be resolvable in the electron microscope.

3. ELECTRON MICROSCOPY OF h.c.p. METALS

3.1 The Reciprocal Lattice

For the interpretation of electron diffraction patterns the reciprocal lattice concept is the mostconvenient but some confusion in the literature is due to there being two alternative ways of defining thereciprocal lattice and the four index labelling of points (Frank 1965).

Mathematically the reciprocal lattice is defined by the following relationship between the recip-rocal lattice axes aj, a*, -ja* and the direct lattice axes a ± , a2 , a3:

-6- -7-

a?

a?

Y ( a 2 x a 3 )

§2)

(6)

where V is the volume of the unit cell in the direct lattice, that is, a t . a_2 x a3. This leads to their with 8:: = 1 when i = j and otherwise zero.

*(7)general relation a^ . a.j •

The reciprocal lattice is built up by repeated translations of the unit cell by the vectors £i,a2 and as.

Frank (1965), Otte and Crocker (1965). Partridge and Gardiner (1967) and Okamoto and Thomas(1967, 1968) have shown that the reciprocal lattice defined by Equation 6 leads to difficulties in thefour-axis hexagonal system using Miller-Bravais indices. The normal reciprocal lattice in hexagonalcrystals has ?xes at, a*, and c* which are related by Equations 6 and 7 to the axes a1? a2 and cin the three-axis hexagonal system with Miller indices. The angle between a? and a2 in this recip-rocal lattice is 60 ° and thus, because^they do not define a conventional hexagonal unit cell, 'theintroduction of a fourth axis a? = -(a* + a*) will not have the same geometrical significance as itsanalogue in the direct lattice. It is for this reason that the use of the three-axis hexagonal systemwith the Miller notation has been advocated for diffraction studies of hexagonal metals (Partridgeand Gardiner 1967). This procedure is unattractive because the symmetry of the hexagonal systemis not fully indicated and frequent transformation of indices is required if the end product of the cal-culation is to be Miller-Bravais indices.

The alternative approach, suggested by .Frank (1965). Otte and Crocker (1965) and developedin greater detail by Okamoto and Thomas (1967,1968) is to use a four-axis reciprocal lattice in whichthe introduction of the fourth axis has the same significance as in the direct lattice. The four-axishexagonal reciprocal lattice is constructed according to the alternative definition of a reciprocallattice, that is, a reciprocal lattice point hki£ lies at a distance from the origin which is the inverseof the spacing between (hki£) planes in the direct lattice, in a direction normal to^these planes. Thisdefinition of the reciprocal lattice is a corollary of the usual mathematical definition of a reciprocallattice. The reciprocal axes aj, a£, a*, c* are not related to the .direct lattice axes in the customaryway as defined by Equation 6 and do not satisfy the usual conditions of Equation 7. The reciprocallattice axes are parallel to the respective axes in the direct lattice. The mathematical relationshipsbetween the basis vectors in the four-axis hexagonal reciprocal lattice and the axes in the directlattice are given by:

a?

n*a.2

at

- U a> §,1

(8)

It is obvious, that in the four-axis hexagonal reciprocal lattice the following restrictions apply:

.*al)and

h + k + i = 0

(9)

Okamoto and Thomas (1967, 1968) have shown that only one out of every three reciprocal latticepoints in the four-axis reciprocal lattice coincides with a conventional (three-axis hexagonal) recip-rocal lattice point. However the additional points in the four-aids hexagonal reciprocal lattice havenon-integral indices and cannot represent real crystal planes. They can be considered to representreciprocal lattice points for which the structure factor is zero and hence will not be present in actualdiffraction patterns.

Although the four-axis hexagonal reciprocal lattice is not a true reciprocal lattice in the usualmathematical sense, the concept is most useful because it permits the calculation of necessary crystal-lographic relationships directly in terms of Miller-Bravais indices using elementary vector analysis. Forexample, the scalar product of a reciprocal lattice vector |hki£ and a direct lattice vector juvtw is givenby

= hu + kv + it + E w (10)

The direct lattice vector r_ is normal to the reciprocal lattice vector g when g.r = 0. This permitsthe determination of zonal relationships and invisibility criteria for dislocatiolis~directly in the Miller-Bravais notation.

Crystallographic relationships between crystal vectors and reciprocal lattice vectors in theMiller-Bravais notation have been derived by Otte and Crocker (1965, 1966), Nicholas (1966), Okamotoand Thomas (1968), Neumann (1966) and Schwartzkopff (1968). A summary of relationships useful intransmission electron microscopy is given in Appendix 5.

3.2 Electron Diffraction Patterns

Electron diffraction is interpreted most readily in terms of the reciprocal lattice and the Ewaldsphere construction (Kitsch et al. 1965). In Figure 7, consider the sphere of radius I/A. (the reciprocal

of the electron wavelength) that intersects theorigin of the reciprocal lattice at 0, wherecrystal planes spaced d^kig are inclined atan angle 6 to the incident electron beam alongEG. The reciprocal lattice vector ghki£ equalto OP will be on the sphere if ~~

.'Incident electronbeam

Sphere ofreflection

sin 9 - -QB- 1/dhkif _sin e ~ OE - 2A " (ID

Figure 7. Ewald sphere construction in reciprocal lattice

This relation is equivalent to Bragg' slaw. Thus, depending on the structure factor,a diffracted beam FP will arise whenever thereflecting sphere intersects a reciprocal latticepoint.

For electrons accelerated by 100 kVin the electron microscope, A.-* 0.037 A , theradius of the reflecting sphere is ~ 27 A" 'which is large compared with the lattice spac-ing of metals. The Bragg angles are also verysmall (about 1° ) and thus the reflecting spherecan be approximated to a plane over the visiblearea of the diffraction pattern. Consequently,if a plane in the reciprocal lattice is tangentialto the reflecting sphere, a simple ctoss-graiiagspot pattern will be formed. The normal to the

reciprocal lattice plane is a direct lattice vector t_ = [uvtw], the crystal zone axis, -which is parallel tothe direction of the incident electron beam. The indices of r can be determicollihear spots £1 =

determined from any two non-and gz = [h2k2i222] on the diffraction pattern since gi.r = g2.£ = 0.

-8-

Okamoto and Thomas (1968) have shown that in the four-axis hexagonal notation this leads to threesimultaneous equations:

gi .£ = hi.u

g 2 . £ = h 2 u

+ i i t + £ ± w = 0

+ i2 t + £ 2 w = 0

u + v + t = 0

(12)

which can be rearranged to give:

h ll2 —

W

w

1 = -£;w

\__w

w(13)

Jt_w

and using Cramer's rule the ratio u :v: t: w: can be solved to give:

r = [uvtw] = F2 k2 i2

o i i

hi £1 ii

h2 £2 i2

1 0 1 >

ht ki £±

h2 k2 £2

1 1 0 >

hi ki ii

h2 k 2 iz

1 1 1

(14)

Equation 14 contains a built-in symmetry check since the sum of the first three determinants must vanish.The indices of jr. represent the upward drawn normal to the diffraction pattern if £1 and £2 are chosen sothat an anticlockwise rotation of £1 through an angle less than 180 ° would bring the direction of gi intothe direction of £2«

Consider the stek s in indexing the electron diffraction pattern shown in Figure 8.

(i) Determine the interplanar spacings (d) of planesgiving rise to the spots Pi, P*, P3 on the diffractionpattern using the relationship

Jhki£ R (15)

Figure 8. Indexing an unknown diffraction pattern

where R is the distance of the spots from the originand XL is the camera constant obtained by calibra-tion of the electron microscope with a material ofknown lattice spacing.

(ii) Usually a set of indices hikiiiEi giving thecorrect value of d, can be assigned to gi providedthat not more than one crystallographically distinctset of such indices exists.

(iii) After assigning s specific set of indices to jji >there will still be many equivalent ways of indexing£3, corresponding to the multiplicity factor for theset of planeshjj-ksigEsk. The multiplicity factorsfor olanes in the h.c.o. system are listed below:

{hki£} = 24 , {hhii} = 12, {oki£| = 12, Ihkio} = 12,

{hhio} = 6, tokiol = 6, {ooo£} = 2.

Many of the permutations of indices o f h 3 k 3 i 3 £ 3 may lead to results which are not distinguishablecrystallographically and a correct set can be chosen by measuring the angle between gi and g3andcomparing the results with tables of interplanar angles in conjunction with a standard~projection ofplane normals.

(iv) Once two points on the diffraction pattern have been indexed consistently, all other pointscan be indexed by simple vector addition, for example,

[ h 2 k 2 i 2 £ 2 ] [ h s k s i 3 £ 3 ] .

(v) When the indices of two reciprocal lattice vectors gi and £2 are known, the normal to thecross grating pattern, that is, the zone axis of the crystal parallel tolhe directions of the incidentelectron beam, can be obtained using Equations 12 and 14. The sense of the direction of the normalto the diffraction pattern in the usual right-handed system of axes can be obtained using the conventionstated previously or by inspection using a double stereogram.

(vi) It must be noted that even if the sense of the normal to the diffraction pattern is known,there is still an ambiguity in indexing corresponding to a rotation of 180 ° about the normal to the foil.This ambiguity arises from the arbitrary assignment of the sign of the diffraction vector i in step (ii)above and can lead to erroneous results in some studies. This ambiguity can be removed by analysingthe Kikuchi pattern (to be considered in the next section) or by noting the effect of an applied tilt asillustrated in the following example. Consider a diffraction pattern with the upward drawn normal givenby [1213] . Such a pattern can be indexed in two ways as shown in Figures 9(a) and 9(b). The cor-responding stereographic projections of plane normals giving the crystal orientation are shown in Fig-ures 10(a) and 10(b) respectively. The correct indexing can be obtained by distinguishing the effectsof tilting the crystal about [lOlO] in the sense indicated on Figures 10(a) and.lO(b). A tilt of 19°would bring the crystal into an orientation such that spots corresponding to Oll2, 1102 and 1214 wouldappear on the diffraction pattern if the crystal orientation assumed in Figure 10(a) is correct. Other-wise, a crystal tilt of 14 ° would have caused reflections 0221, 2201 and f 2lf to appear as implied byFigure 10(b) which corresponds to the indexing assumed in Figure 9(b). It is important to realise thatthe alternative methods of indexing can lead to a deduction of opposite sense of inclination of planesin the [0001] zone for example.

It is often convenient to have available examples of indexed single crystal electron diffractionpatterns to aid the identification of unknown patterns on the microscope screen by simple inspection.A number of the most commonly bccurring diffraction patterns from h.c.p. crystals, indexed systematic-ally in terms of Miller-Bravais indices, have been published by Partridge (1967) and Partridge andGardiner (1967 b) and are reproduced with some variations in Appendix 6.

In h.c.p. crystals, the structure factor becomes zero and there is no diffracted beam when(h + 2k) is a multiple of 3 and £ is odd, for example 1121, 3033 etc. These conditions, however, arerelaxed for electron diffraction where extra spots may arise as a result of double difiraction, forexample "forbidden" spots 0001 may arise from double diffraction involving reflections lOll and1010. Spots due to double diffraction have been differentiated from the normal spots with non-zerostructure factors in the diffraction patterns shown in Appendix 6.

3.3 Kikuchi Patterns

A most useful method of determining the crystal orientation involves the analysis of Kikuchipatterns associated with the spots in electron diffraction patterns. A Kikuchi pattern is formed as aresult of Bragg diffraction of electrons which have suffered inelastic collisions involving only a smallenergy loss. The inelastic scattering provides the main component of the background intensity of thediffraction pattern. The Bragg scattered inelastic electrons, which form cones with the normal to thereflecting planes as axis, intersect the photographic plate in hyperbolae which under the geometrical

-II-

-10-

conditions in the electron microscope can be considered as nearly straight lines. Each set of planesgives rise to two parallel lines in a direction perpendicular to the projection of j^; the line separationR is given by ~~

IghkiER

(16)

1512

* • • Xiloi oin

tola 00*00 Toiott • • x

oifi TioT• • •

1215

(a)

1212K • • X

JlOl Olfl

50*0 0000 1010

01119 •

>212

(b)

Figure 9. Alternative ways of indexing z d i f fract ion pattern from a thin foilwith the upward drawn normal 11213] parallel to the electron beam

(OtH)

Figure 10. Stenographic projections of plane normals; (a) and (b) correspond to diffractionpatterns in Figures 9(a) and 9(b) respectively. The broken lines show reflectionswhich would appear on the diffraction pattern after tilting about [lOlO] in the sensegiven on the projections

where XL is the camera constant. The angular separation of the lines is thus 2 6 and the lines aresituated symmetrically on either side of the trace of the reflecting plane. This property leads to animportant difference in behaviour between the Kikuchi lines and the normal spot pattern when thespecimen is tilted. The spot pattern does not move on tilting over a small angular range, but theindividual spots change in intensity; some spots disappear and others appear in different positions.On the other hand, the Kikuchi lines move as though rigidly fixed to the crystal, so that their direc-tion and magnitude of movement reveals the orientation change with high accuracy. The determinationof accurate orientations from Kikuchi patterns will be considered in this section.

Let us consider first, however, the limitations inherent in electron diffraction spot patternsfor accurate orientation determinations.

(i) Because of the extension of reciprocal lattice points into iclrods, reflections canremain unchanged over a large angular range of tilt (±5°) leading to errors of atleast this order of magnitude in the determined orientations. In very thin foi!s>spot patterns become complicated because second-Laue-layer relrods intersectthe reflecting sphere.

(ii) A symmetrical diffraction pattern may be obtained from foils in non-symmetricalorientations as a result of foil buckling.

(iii) Perhaps the most important limitation in diffraction studies is the need to usetwo-beam conditions. It is not possible, of course, to determine an accurateorientation from one diffracted beam.

The indexing and application of Kikuchi patterns in orientation determinations is well coveredin the text by Hirsch et al. (1965) and the papers by von Heimendahl et al. (1964) and Otte et al. (1964).The application of Kikuchi methods to h.c,p, metals has been discussed by Okamoto et al. (1967) andOkamoto and Thomas (1968). The essential features are discussed here.

The initial step in orientation determination is the identification of Kikuchi lines from measure-ments of the pair separation, whereby dhki£ derived using Equation 16 is compared with values of inter-planar spacings for the material studied. Ambiguities in indexing Kikuchi pairs with very similar sep-aration can be avoided by comparison of measured angles between, different sets of Kikuchi pairs withinterplanar angles between the sets of planes responsible for the Kikuchi lines.

The intersection of the centre lines of different pairs of Kikuchi lines, the Kikuchi pole,corresponds to a crystal zone axis containing the planes giving rise to the Kikuchi lines. The indicesof the zone axis can be obtained by substitution in Equation 12 and 14 of indices of the intersecting setof Kikuchi pairs. Thus, when a Kikuchi pole is located at the origin of the diffraction pattern theorientation is determined accurately.

The most common situation, however, is one where the Kikuchi poles are not located on theorigin of the diffraction pattern. Then the orientation is determined by indexing three non-parallelKikuchi pairs and the Kikuchi poles of their intersections (von Heimendahl et al. 1964). The orienta-tion is then obtained from angles subtended between these poles and the origin of the diffraction pattern,either by calculation or stereographic projection. Since Kikuchi poles are zone axes in the crystal,that is, directions, a standard stereographic projection of directions must be employed for hexagonalmetals. The method of orientation determination using two Kikuchi poles has been discussed by Otteet al. (1964).

-12-

Okamoto et al. (1967) and Okaraoto and Thomas (1968) have constructed composite Kikuchi mapswhich cover a significant region of reciprocal space. These maps enable unknown Kikuchi patterns tobe solved quickly even when they do not contain two Kikuchi poles. The detailed geometry of thesemaps for h.c.p. metals depends on the c/a ratio. Okamoto and Thomas (1968) have constructed a Kikuchimap for titanium (c/a = 1.588) which is reproduced in Appendix 7 and should be applicable to zirconiumwhere c/a = 1.593.

Although Kikuchi patterns can be constructed for various orientations, the curvature of Kikuchilines gives rise to difficulties if large angular ranges are required. The most satisfactory solution,according to Hirsch et al. (1965), is to present a large angle plot as a stenographic projection of plasetraces, that is, the centre lines of Kikuchi pairs.

In conclusion, it is emphasised that because a Kikuchi pattern is uniquely representative ofcrystal symmetry the 180° ambiguity in orientation determination, which arises from the arbitrary assign-ment of the sign of the diffraction vector £ in electron diffraction patterns, is not present in orientationsdetermined from Kikuchi patterns.

3.4 Determination of Dislocation Burgers Vectors

Contrast in transmission electron microscope images of crystalline materials arises mainly fromBragg scattering cf the incident electron beam. In bright field microscopy a small aperture is insertedin the objective lens so that diffracted electrons are blocked out and only the undeviated electrons arepermitted to form the image. Thus, dark regions in the image correspond to regions of the thin foil whichare favourably oriented to give strong diffracted beams. The interpretation of bright field contrast is fac-ilitated if only one diffracted beam is operating, that is, the 'two-beam' case of one undeviated beam anda diffracted beam. The two-beam theory of dislocation image contrast is presented in detail in the textby Hirsch et al. (1955).

The usual method of Burgers vector determination makes use of the invisibility criteria. Thusa perfect screw dislocation, which produces atomic displacements in the direction of the Burgers vector,will be invisible, in an elastically isotropic material, if the condition £ . b * 0 is satisfied, where g isthe diffracting vector and b is the Burgers vector of the dislocation. This, of course, is a general prin-ciple stating that displacements or lattice distortions parallel to the reflecting plane produce no contrast.Similarly, an edge dislocation will be nearly invisible if £. b^= 0 although some residual contrast may beobserved because a pure edge dislocation produces minor atomic displacements perpendicular to t) . Thecondition for exact invisibility for a perfect edge dislocationis given by £. b = 0 together with g.bx u ^0,where £ is a unit vector along the dislocation line. For mixed dislocations some contrast is expectedunder all two-beam conditions, but it is found (Hirsch et al- 1965) that dislocations are effectively invisibleif £ . b^ = 0, and if £ . _b x u < 0.64 . At higher values of ^ - b.x u some contrast is expected.

The above account of the invisibility criteria is adequate for materials which are elastically iso-tropic. Head et al. (1967) and Humble (1967) have demonstrated that as the degree of elastic anisotropyis increased, the residual contrast associated with the condition g. b = 0 and g, . b x u_ = 0 generally,tends to increase. In practice, thus, ''near invisibility' is generally accepted as corresponding to thecondition = 0.

The degree of elastic anisotropy in h.c.p. crystals cannot be described by a single parameter asin cubic crystals. For a h.c.p. metal to be elastically isotropic, 'the following three ratios must be sim-ultaneously equal to unity:

A = 2C44/(Cll-Ci2) = C44/C66

B = Cas/Cn

•13

Fisher and Alfred (1968) showed that for zirconium at 300 °K, A = 0.907, B = 1.149, C = 1.115 whereasat 1133 °K the ratios are 2.00, 1.292 and 1.281 respectively. Thus, at room temperature elastic isotropyappears to be a reasonable approximation for zirconium and should permit the determination of the con-dition . b> = 0 from observations of 'near-invisibility'.

-13-

The unambiguous determination of J> requires two values of g which correspond to the conditiong . b = 0. The Burgers vector is then parallel to the direction which'is common to these two reflectingplanes, that is, £1 x g^ . The indices of b^ in the Miller-Bravais notation are obtained by solving Equa-tions 12 and 14. ~~ "~

Values of j* . b^ for perfect dislocations in h.c.p. metals with various low Border reflections havebeen tabulated by Partridge (1967) and are reproduced, with the addition of the 1124 reflection, inAppendix 8. Using these values of g . b,. the fraction of perfect dislocations visible for any of the low-order reflections included in Appendix 8 can be deduced and are shown in Table 2. The values shownin Table 2 are based on the assumptions:

(a) all Burgers vectors of similar form are equally represented.

(b) dislocations are visible only when g . t) / 0.

TABLE 2

THE FRACTION OF PERFECT DISLOCATIONS VISIBLE IN H.C.P. METALS

(ASSUMING ALL BURGERS VECTORS OF A FORM ARE EQUALLY REPRESENTED)

Reflection

10TO

0002

lOll

1012

1013

1120

1122

1124

Fraction of dislocation visible

J <1120>

2/3

0

2/3

2/3

2/3

1

1

1

J <1123>

2/3 '

1

2/3

1

1

1i

5/6

1

< 0001>

0

1

1

1

1

0

11

In addition to uncertainties introduced in bright-field image analysis of Burgers vectors inmaterials exhibiting elastic anisotropy, France and Loretto (1968) have demonstrated that undercertain two-beam conditions dislocations are predicted to be invisible when jg . b / 0. These calcula-tions have been supported by observation. This behaviour is connected with large values of w, thedeviation parameter in the dynamical theory (Hirsch et al. 1965) which are usually associated withlarge diffr? tion vectors.

To avoid errors in the determination of Burgers vectors of dislocations, the observed brightfield images should be compared with calculated images which take into full account the foil, dis-location and diffraction geometry under which the observations were made. Head (1967) has developeda method of computer calculation of theoretical images as pictures taking into account elastic aniso-tropy. This technique has been generalised by Humble (1968) for the case of a dislocation in a tiltedfoil. An important parameter in these calculations is the extinction distance corresponding to thevarious diffraction vectors used. Extinction distances for the common reflections in zirconium havebeen computed and are listed in Appendix 9.

-14-

4. REFERENCES

Barret, C.S. and Massalski. T. B. (1966). - Structure of Matals, 3rd ed., McGraw-Hill, New York.

Berghezan, A., Fourdeux, A., and Amelinckx. S. (1961). - Acta Met. _9: 464.

Dora. J.E. and Mitchell, J.B. (1965). - High Strength Materials, p.510 (V.F. Zackay editor)

J. Wiley, New York.

Fisher, E,S. and Alfred, L.C.R. (1968). - Trans. Met. Soc. AIME 242: 1575.

France, L.K. and Loretto, M.H. (1968). - Proc. Roy. Soc. A307: 83.

Frank. F.C. (1965). - Acta Cryst 18: 862.

Frank, F.C. and Nicholas, J.F. (1953). - Phil. Mag.^4: 1213.

Gehman, W.G. (1960). - NAA - SR - 6003.

Head, A.K. (1967). - Aust. J. Phys. 20: 557.

Head, :A.K., Loretto., M.H. and Humble, P. (1967). - Phys. Stat Sol. 20 : 505.

Hirsch, P.B., Howie, A.. Nicholson, R.B., Pashley, D.W. and Whelan, M.J. (1965). - ElectronMicroscopy of Thin Crystals, Butterworths, London.

Howe, L.M., Whitton, J.L. and McGurn, J.F. (1962). - Acta Met. 10: 773.

Humble, P. (1967). - Phys. Stat. Sol. 21: 733.

Humble, P. (1968). - Aust. J. Phys. 21: 325.

Martin, J.L. and Reed-Hill, R.E. (1964). - Trans. Met. Soc. AIME 230: 780.

Neumann, P. (1966). - Phys. Stat. Sol. 17: K71.

Nicholas, J.F. (1966). - Acta Cryst. 21: 880.

Okamoto, P.R. and Thomas, G. (1967). - Scripta Met. I: 25.

Okamoto, P.R. and Thomas, G. (1968). - Phys. Stat. Sol. 25: 81.

Okamcto, P.R., Levine. E. and Thomas, G. (1967). - J. Appl. Phys. 38: 289.

Otte, H.M. and Crocker. A,G. (1965). - PhySc Stat. Sol. 16: K25.

Otte, H.M. and Crocker, A.G. (1966). - Phys. Stat. Sol. 16 : K25.

Otte, H.M., Dash, J, and Schaake, H.F. (1964). - Phys. Stat. Sol._5: 527.

Packer, M.E. and Miller, D.R. (1967). - J. Australian Inst. Met. 12: 299

Partridge, P.G. (?u967). - Met. Reviews J18: 169.

Partridge, P.G. and Gardiner, R.W. (1967a). - Acta Met. 15: 387.

Partridge, P.G. and Gardiner, R.W. (1967 b). - Scripta Met. 1: 139.

-15-

Picklesimer; M.L-. (1966). - Electrochem Tech. 4j 289.

Earey, C.RV Stringer, J. and Edington, J.W. (1966). - Trans. Met. Soc. AIME 236: 811.

Reed-Hill., R.E. (1964). - Deformation Twinning p. 295 (Reed-HilL R.E., Hirth, J.P. and Rogers, H.C.editors), Gordon and Breach. New York.

Rosenbaum, H.S. (1964). - Deformation Twinning p,s3 (Reed-Hill, R.E., Hirth, J.P. ana Rogers. H.C.editors). Gordon and Breach; New York.

Roy; R.B. (1967), - Phil. Mag. 15: 477.

Schwartzkopff. K. (1968). - Scripta Met. 2_: 227.

Smith, G.H. and Burge, R.E. (1962). - Acta Cryst. IS: 182.

von Heimendahl, M., Bell, W. and Thomas, G. (1964). - J. Appl. Phys. 35: 3614.

Yoo? M.H. (1968). ~ Scripta Met. ,2: 537.

APPENDIX 1

INTERPLANAR SPACINGS IN a-ZIRCONIUM

Interplanar spacings for a-zirconium, shown in the table below, were calculated from latti(parameters determined with powder X-ray methods on zone refined zirconium supplied by MaterialsResearch Corporation. The lattice parameters at 21 °C are:

a - 3.2330

c 5.1497

c/a = 1.5927

{hklEl

lOfO

0002

1011

10l2

1120

1013

2020

1122

2021

•000,4

2022

2023

1230

1231

d in A

2.7999

2.5749

2.4598

1.8953

1.6165

1..4634

1.3999

1,3691

1.3509

1.2874

1.2299

1.0849

1.0583

1.0366

{hkift

1124

1232

2024

3030

1233

3032

1234

2240

..

d in A

1.0071

0.9788

0.9476

0.9333

0.9008

0.8774

0.8175

0.8083

APPENDIX 2 PAGE 1ANGLES BETWEEN CRYSTALLOGRAPHIC PLANES IN ALPHA ZIRCCMUM

C/A = 1.5927

H K I L H K I L

0 0 0 1

I 0 -I

1 0 - 1 1

ANGLES IN DEGREES

c111II22331111.11111

"t.1,

1.II

22

311I111I11

111122331111111

CCCcccrCCc111112222

cCccrCCcc1111I2222

CCCCCCCC111112#*•

C-1

•*.

-2

_3— 2— 2-2-2•"" £iri -3

.3

**** A

-i— i

-"• 2

— 3-2

— 2-2_ ^.-3-2-2— 2

-i-1— •}-1-2

•5**•£

- 3-3-2-2-2•• 3

-?«3-3

r

2

4

312G1234C1

3

c1234

3I2C1234C123

12341312C12340i

C.9C.61.42.31.

74.5C.79.7Co9C.72.57.46.3B.9C.76.67.58.

0.28.47.56.65.15.39..V C. a

19.3C.34.42.5C.57.19.22.29.36.

0.18.29.36*13.1C.16.8.

40.29.26.28.32.33.24.

0oc466051

8C7307OC5788725300396634

053404931212G2793002882923511250846

08696773367266146660619048959

60.63.7C.74*77.61.67*60.61.9C.9C.9C.90.90.4C.42.45.49.

52.49.5C.51.43.5C.38.48.90.54.67.7C.68*48.41.

009422859415205396CO0000000089236495

1154448774G3814600350288C6397C

79.79.79.80.

57.75.79.75.56.67.58.54.

81.75.76.84.80.44*

11339374

O /94774969748485

772892244420

80.86.87.86.72.87.69.75.

56.

9389031663806951

33 75.00 86*19

APPENDIX 2 PAGE 2ANGLES BETWEEN CRYSTALLOGRAPHIC PLANES IN ALPHA ZIRCCNIUH

C/A = 1.5<527

H K I L

1 0 - 1 1

1 0 - 1 2

1 0 - 1 3

I 0

2 0 - 2 1

H K I ANGLES IN DEGREES

1 21 2

1 C1 01 C2 02 C3 C3 C1 11 11 11 11 1I 21 21 21 2

1 C1 C2 C2 C3 C3 01 11 11 11 11 11 21 21 21 2

1 C2 G2 C3 03 01 11 11 11 11 11 21 21 21 2

2 02 03 0

-3«3

-1-1-1— n

-2-3-3-2-2-2

*»

-2«. -3—3-3

-1-1-2-2-3-3-2-2-2-2-2-3-3-3— 3

-1•=» 2-2-3

**

*•* 2-2-2-2-2-.-a•«*

-3-3-3

-2-2.-3

23

2341312C12340123

34131201234C123

4i312012340123

131

18.16.

0.11.17.32.

6.37.27.54.38.27.21.19.5C.39.29.21.

0.6.

43.19.48.38.63.46.33.23.18.60.49.38.29.

0.50.26.55.45.68.51.37.27.20.66.55.44.35.

Ow23.4.

3080

0099119201347117639357824223845

08228292256095921934041056680

010110438798886595975287263

09994

37.35.

39.36.35.58.43.57.55.90.7C.66.59.54.59.49.41.34.

30.27.61.42.65.57.90.75.63.54.48.66,56.46.38.

24.63.42.68.59.90.74.61.51.44.71.60.50.41.

30.54.25.

2631

56448969326732002096698422511473

309059158557002104231773044140

1190604360GO2111477159514079

424148

54.62.

71.63.58.62.78.62.67.

77.83.85.77.82.61.66.6C.

53.48.73.70.68.78.

79.85.75.67.84.71.65.57.

42 «80.65.75.83.

85.79.68.60.85.74.65.57.

57.

1188

77211761293433

2697538465425432

822870357642

8399233633814589

4252645850

8182939647922604

7057.3458.56

64.66.

85.74.67.82.86.78.86.

69.71.73.

63.56.88.82.83.87.

74.76.68.

49.87.75o87.85.

78.74.65.

66.

3792

20112933603614

338489

022037319743

435469

3889495123

241982

62

70.42 88.3971.66 83.72

—

74.13 88.6978.85 80.7080.90 87.18

77.56 85.7182.38 87.6383.62 88.45

82.74 83.9486.94 88.8777.99 81.90

77.9964.66

APPENDIX 2 PAGE 3ANGLES BETWEEN CRYSTALLOGRAPHIC PLANES

C/A = 1.5927,<LPHA ZIRCONIUM

H K I L

2 0 - 2 1

2 0 - 2 3

3 0 - 3 1

3 0 - 3 2

1 1 -2 0

H K I ANGLES IK Dcb^EES

3 G1 11 11 11 11 11 21 21 21 2

2 03 03 01 11 11 11 11 11 21 2I 21 2

3 03 01 11 11 11 11 11 21 21 21 2

3 C1 1111 11 11 11 21 21 21 2

111111111 11 .21 2

— 3-2-2-2-2-2-3-3-3-3

-2-3-3-2-2-2-2-2-3-3-3-3

2012340123

3120123401?.i

4332832374324181923

02819473325222442322317

.72

.31

.85

.09

.98

.46

.24

.92

.42

.96

.0

.93

.28

.85

.94

.25

.80

.15

.93

.38

.43

.25

~" 1 0.0-3-2-2-2-2-2-3-3-3-3

-3-— 2>2-2-2-2-3-3-3—3

-2>2--2>2-2-3-3

2G12340123

2012340I23

0i2340 3

i

9312935424721182127

0352829343927201720

017324351j i<315

.65

.55

.95

.26

.05

.91

.60

.80

.91

.75

.0

.49«49.45.27.30.33.15.97.77

*0•43.12.28.47.89.87

35.1490.0044.0555.3764.6371.6243.1632.8239.2540.66

45.5949.4754.5790.0063.1970.3664.326C.3754.1445.4938.5533.87

20.5530.2090.0040.5751.1860.1366.9641.9428.9837.6143.39

^9.8590.0047.5759.4568.9674.5444.7136.7238.1138*39

60. UO£1.5164.9568.6571.85

• * 49i'ii50.11

57

85ei7978793941ec786059

7976868581536462

5858

8684828179404045

56

8479767679404654

t 70j <71

.11

.49

.98

.64,16.49.89.96.32

.40

.40

.13

.09

.57

.83

.62

.58

.84

.09

.85

.94

.45

.94

.56

.98

.98

.28

.14

.91

.75

.08

.14

.56;«49.08.77.12.20.74

.89;so •

68.

48.54.61.

OT»

74.81.

63.67.73.

63.66.

46.51.57.

70.

51.58.64.

64

579010

014244

4393-05

1131

166735

99

131578

76.6274.4372.97

74.2972.4578.05

77.4176.1375.41

75.9672.9170*73

82.7786.0586.99

89.0783.9880.39

81.5984.0286.29

-

83.95£8.0183.42

1

APPENDIX 2 PAGE 4ANGLES BETWEEN CRYSTALLOGRAPHIC PLANES ifc ALPHA ZIRCONIUM

C/A = 1.5927

H K I L

1 1 - 2 0

1 1 - 2 1

1 1 - 2 2

i 1 -2 3

1 1 - 2 4

1 2 - 3 0

1 2 - 3 1

1 2 -3 2

1 2 -3 3

H K I L ANGLES IN DEGREES

1 2I 2

I II II II II 2I 2I 2I 2

I II II II 2I 2I 2I 2

I II II 21 2I 2I 2

I II 2I 2I 2I 2

1 21 2I 21 2

1 2

1 2

1 2

1 2

1 2

1 2

— 3

-2-2-2-2-3-3-3-3

-2-2-2— 3-3-3-3

-2-2-3-3-3-3

-2-3-3— 3-3_3

«3M "2

-3

-3

*™ .j

— 3

-3

-3

-3

23

123

C123

23AC123

34C123

40123

C123

1

2

3

2

3

3

2433

C1425342C121117

01119332213

9

08

44332214

052

.74

.29

.0

.69

.85

.04

.46

.04

.36

.31

.0

.16

.34

.73

.83

.75

.26

.0

.18

.37

.05

.80

.46

.0

.2940*833021

0112231

C581C582058

058

965

067

.32

.37

.0

.61

.34

.66

.0

.65

.73

.03

.04

.50

.0

.15

.31

.22

.0

.73

52.56.

34.

56.57.51.30.41.46.

50.47.47.56.44.44.41.

42.40.61.52.45.40.

36.65.56.47.4C.

21.24.30.37.

21.63.23.67.28.71.

20.73.

7413

8655458935971846

11711833963632

692654784605

3093165874

79568178

349539893787

1355.

21.4771.

18.72.

82

51'91

72*3873.82

56.9955.726C.7168.9071. 8G47.7746.2550.19

64.2575.4161.2573.9149.4555.4659.00

78.1771.9776.2155.7961.2555.76

65.3078.2363.7960.9254.26

38.2139.6843.3948.02

23.2379.7733.9678.114C.4777.01

35.2574.5335.1078.81

32.3680.10

68.75.81.86.

56.62.64.

35.86.83.

64*62.64.

86.85.

. 68.66.75.

77.

69.74.76.

60.60.62.64.

31.84*37.86.43.89.

44.88.53.85.

50.85.

57838440

533797

657759

1571kl

5725

•203960

06

074329

00674581

744697972723

69726201

3802

68.5266.2568.01

67.7571.9078.88

70.8079.6280.97

76.0083.7583.68

81.7981.9682.4183.02

37,40

40.12

48.04

49.47

54.00

63.31

-

75.7879.9283.76

80.5386*8987.53

84.5387.6987.37

87.5885,4286.36

44.49

50.56

56.67

55.09

57.89

66.59

U V T a

0 0 0 1

APPENDIX 3 PAGE 1A N G L E S BETWEEN DIRECTIONS Ih ALPHA ZIRCC.NIUH

C/A * 1.5927

U V T .

0 0 C 11 0 - 1 01 0 - 1 1I 0 1-1 210-1 31 C -I 41-0-1 51 C -1 62 0 - 2 12 C -2 32 0 - 2 53 0 - 3 13 0 - 3 23 0 - 3 43 0 -3 54 0 - 4 14 C -4 31 1 - 2 01 1 - 2 1I 1-2 21 1 - 2 31 1-2 41 1 - 2 61 1 -2 92 2 - 4 12 2 *4 32 2 - 4 52 2 -4 91.2-3 01 2 -3 11 2 r3 21 2 - 3 31 2 -3 41 2 -3 51 2 -3 613 -4 01 3-4 113: --4 21 3 -4 31 :4 -5 0I :4 -5 11 4 ^5 214 : -5 32 3-5 0* 3 -5 12 3 -5 2t 3-5 3

0.090.00047.40028.53519.92515.21012.27110.27365.30835.94223.50972.95958.49039.20133.12477.05355.40890.00062.03643.28332.12325.21617.42911.82175.13451.46836.99622.71390.00070.83555.19643.80335.72829.91825.61990wOOO75.69262.97552.58090.00078.65468.13356.95390.00078.08867*12457.671

I 0 «l;o 1-0,-10

Wl 3

>-i ..5

ANGLES IN DEGREES

<JiO : 60.00042.600 68,405

80¥18974*790 82*462

83«SCO

U V T H

1 0 - 1 0

1 0 - 1 1

APPENDIX 3 PAGE 2ANGLES BETWEEN DIRECTIONS Ifc ALPHA ZIRCONIUM

C/A = 1.5927

U V AnGtES IN DEGREES U V T

10-1 62 0 - 2 12 0 - 2 32 6 - 2 53 0 - 3 13 6 - 3 23 0 - 3 43 0 -?3 54 0 - 4 14 0 - 4 3I 1 -2 011 -2 11 1 - 2 21 1-2 31 1 - 2 41 1 - 2 611 ~2 92 2 -4 I'I 2 - 4 32 2 - 4 52 2 -4 91 2 -3 012-3 I1 2-3 21 2 - 3 31 2 - 3 41 2 - 3 51 2 -3 61 3 - 4 01 3 - 4 I1 3-4 21 3-4 31 4 - 5 01 4 -5 11 4 -5 21 4 - 5 32 3 -5 02-31-5 i -2 3 -5 22 3-5 3_..10-1 11 0 -1 210-1 31 0 "•! 41 0-1 51:0-1 62 < 0" -2 1 :2 0 >2 32 0 - 2 53 0 - 3 13<0. -3 23 0 -3 43' a -3 5

79.7S724.69254.05866.49117.04131.51050.79956.87612.94734.59230.00040.10153.57662.58068.34974.96679.78133.1^7147.35558.59170.46519.10726.80639.U549.15256,51261.88265.88513.89819.84536*14839.56010.89315.67924.31032.72223.41326*11732.27639.157' •-• . . .0.0

18.86527*47532.19035*12937*12717.903

• ll*45823.89125*55911.0908.198

14*276

84.88462*98172.93378.49561*44264.76871.57774.14360.83765.69490.00090.00090.00090.00090.00090*00090.00090.00090.00090.00090.00040.89344.43651 -.63 258.45063.60667.85070.92246.10247-78851.85456.58649*107t50.06952.58755,88436*58738.21842.28547*274*• *43.29139.60440.37841.43442.30042.97451*892*0*ltf 9

• i

56.616' 48.12240*78639.826

79.10779.71781.07382.48483.66484*59185.31373.89874.41075.6967T.27570.89371.28172.31573.71483.41383.55683*933t

84.438; r ~ . *. .u .:

, . 79.20865.23959.27556.17854*32553.10367.292

, 7G,'01;1L';./•-f61.'i7-13 -

59.64174.10973*02868.546 -t

85.20075.93567.32562.61059.67157.67387.040$3.34270.90981*16987.70986.60180.524

1 0 - 1 1

1 0 - 1 2

U V V5,

** m %v - / A

i I irs

4 C4 C1 11 I1 11 11 11 I1 12 22 22 22 21 21 2I 21 21 21 21 21 31 31 31 31 41 41 41 42 32 32 32 3

1 G1 C1 01 01 C2 C2 02 03 03 C3 03 04 G4 01 11 i111 11 111X 1

-4«*.

-2-2-2-2-2-2-4

-4

-3— 3-2«3-3-3

—3-4-4-4-4-5-.5-5-5-5

— 5— 5-5

-1-1-1-1-1-2-2-2— 3-3-3~3«4-4-2-••2-2-2-2-2-2

3012346913590123456G12301230I23

234561351245130123469

S . G G f i

28.30421.59524.18527.87633»20337.52337.83323.02522.45029.48C45.92928.16.14.17.2C.24.44.30.

453772079129999345394722

19.25111.43.32.22.14.47.36.27.21.

0*8.

13.16.18.36.7.5.

44.29.1C.4.

46«26.65.38.22.15.13.15.19»

818711661642447508807689131

028609325264262773407026^424955666589518872563982605419819824154

5* <t •' *. £A A * • ^*.»*•.• • •>

46. 3d 6

71.4946C.47955.0235,1. 23 P

63.7 M?65.(V-f57.27^51.36356. ISC4! ,59^32.51729.11229.0253C.20331 .66759.31048-56C4C.34-C35.24861.1 9152.72545.62840.36453.76844.10736.12930.524

27.63724.86124.43724.60324.90154.26831.62925.72660.93448.48733.72029.97164.55845.94190.00065.67250.24241.92437.36333.05030.695

• » O - -• i ••« 'T •".• • • • • > » . >

77. i-,£ *:>

70.07S

57.36'V79.9-^935.583GO. 96?67.77^82.004> 45;?5<504Q47785860557654615685586864

4841383634815544787558527472

876958514439

.228

.^'63

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.897

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.220

.32'r

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.253

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.874

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.271

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.730

.374

.175

.738

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.195

.006

.754

.412

.413• •

.331

.151

.396

.789*416.181

6-57<f66624..i *j:J6

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74

637073

5748434038866452838767618783

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« 2 '1 2. .-. 3 '..».307.079• -S •'- •"?. 6 ii .1

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.794

.572

.321

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.104

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.460

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.806

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.157

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.044

.333»025.736.659.940.943

72. .339 84V9.351 85>-"V.06a 8977.023 Oi7 ' J . V » 9 2 76vr t . bOl 71

7'} .(S^6 688: .541 82-« i . 01 4 89

n*366 847->.7C3 6881 .793 B6

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77.154 8673.654 7977.947 82

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.75^,120.966

.249

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.OB 4

.367

.344

.727

.320

.099

APPENDIX 3 PAGE 4ANGLES BETWEEN DIRECTIONS IK ALPHA ZIRCONIUM

C/A = 1.5927

U V T W

1 0 - 1 2

1 0 - 1 3

U V T W ANGLES IN DEGREES

2 2 - 4 12 2 - 4 32 2 - 4 52 2 - 4 91 2 - 3 01 2 -~ 1A fc. — *1 2 - 3 24» 9m -^ *•*

1 2 - 3 3-% «• *» «^

1 2 - 3 4A **• ** *

1 2 -* 5A C . ».* -^

1 2 - 3 6\ a -4 0tV *•* T V .

31 * -4 1Jt MT T A

1 3 - 4 2«» «^ • **•

1 3 - 4 31 4 -5 C•V • -* '-'

1 4 - 5 1•4* * ^ ^*

1 4 - 5 2«V • ^ *»

1 4 - 5 32 3 - 5 0v» «^ -* *

2 3 - 5 1w • ** *

? 3 - 5 2•^ ^ «r AM

2 3 - 5 3

1 C -1 3<c» ** ^ *

1 0 - 1 41 C -1 5\ 0 - 1 62 C -2 12 0 - 2 32 0 - 2 53 0 - 3 13 C -3 23 0 - 3 43 0 - 3 54 C -4 14 C -4 31 1 - 2 01 1 - 2 11 1 - 2 21 1 - 2 31 1 - 2 41 1 - 2 61 1 - 2 92 2 - 4 12 2 - 4 32 2 - 4 52 2 - 4 91 2 - 3 0^^ • %• «*• v

1 2 - 3 11 2 - 3 21 2 - 3 31 2 - 3 41 2 - 3 5

51.300 76.974 79.95329.437 56.819 T7.07618.083 45.44C 63.08214.039 35.865 49.40963.168 68.832 84.82044.375 5C.986 68.057 78.279 82.071 86*98029.300 37.066 54.861 64.703 78.175 82.48418.829 27.871 45.850 55.141 67.412 71.24012.376 22.475 40.013 48.664 59.846 63.2819.396 19.685 36.247 44.240 54.455 57.5669.131 18.506 33.779 41.139 50.505 53.346

62.373 7C.657 82.38748.207 57.449 69.789 76.572 84.039 84.91035.683 46.033 58.855 73.671 84.023 89.20425,562 37.165 50.278 64.620 74.289 80.47062.025 71*776 81.00450.746 61.35C 70.965 73.317 32.312 88.38140.311 51.871 61.815 79.508 93.792 87.88131.242 43,861 54*051 71.390 79.329 67.06564.000 67.445 86.85952.391 56.176 75.665 76.411 78.816 82.66241.807 45.98C 66.921 73.081 86,424 89.31932.836 37.446 58.928 64.943 81.622 84.295

0.0 19.622 34.332 39.8514.716 17.841 30.401 35.1357.655 17.278 28,061 32.1969.652 17.166 26.533 30.199

45.383 56,801 76.237 85.23416.016 30.552 48.614 55.867

3*583 21.554 37,427 43.43453.033 63.996 83.535 87.11638.565 50*459 69.752 78.41619.276 33.255 51.623 39,12713.199 28.301 46.034 53.05057.128 67.871 83.02i 87.44635.482 47.622 66.829 75.33372.834 90.00045.451 63.842 79.62127.533 46*812 61.18117.608 37.231 50.26312.502 31.728 43.547

9.811 26.237 36.04311.287 23.045 30.71358*233 76.042 87.47535.260 54.150 69.22021.302 41.334 55.02211,13.1 29.862 41.12571.215 75.071 86.30752.207 56.497 68.317 75.652 86.256 89.74436.773 41.571 53.880 61,072 71.031 74.20625.655 31.037 43.689 50.659 59.987 62.89517.965 23.987 36.792 43,482 52.210 54.88912.718 19. ,379 32.116 38.489 46.657 49.138

U V

1 0

T W

-1 3

1 0 - 1 4

APPENDIX 3 PAGE 5ANGLES 8ETKSEN DIRECTIONS IN ALPHA ZIRCONIUM

C/A = 1.5927

U V T ANGLES IK DEGREES

1111111112222

11122"5 -

33334411111112222I1111111111111122

2333344443333

0e000o.000000I.111i

•1.i22222222222

.333

;34

;4.<:4.:43,3

-3-4-4-4-4-5-5-5_ c

^

-5-5-5— c««

-i-1-1-2-2-2-3-3-3-3-4-4-2-2-2-2-2-2-2-4-4-4-4-3«3-3-3-3-3-3-4-4-4-4-5-5-5-5-5-5

6C123C1230123

4561351245130123469135901234560123012301

9.26770.68156.43443.79033.48770.44859.13146.6443$. 50471.77659.99649.18639.913

0.02.9394.936

50*09920.732

8.29957.74943.28123.99217.91561.84440.19876.86849.21930.88020.27314.1138.6137.6 1

62.14938.84324.84812.07775.64756.56841.03429.76921.85616.25612.23475.24660.97248.29337.94375*87163.74153.23844.07876.06964.231

16.44376.33262.5295C.38140.62977.10966.19056.13747.46774.11962.49651.86042*778

15.07513.87913.37056.51430.87920.41365.90851.96333.83028.37869.87749.01790,00063.09645<»37435..19029..18822*97619«17775-66653*04939.58327.10978.56159.72744.47033.51825.95120.72417.08879.51965.49753.09443;. 04380.111069.00858.74949.84877.839

, 66.090

28.88984.57671.10059.24449,73383.59572.88263.02054.51887.76076.56766.32957.602

26.26523.78722.16473.50645.23033*63880.94566.89048.32742.56884.93163.905

. . . .75.41556.85945.85939.08431.50226.10888.39764*95150.65536.63987.15868.67753.73843.050,35.67730.55926.93385.82772.00259.78349.89885.07474.09563*95455.16288.27576.783

34.929

81.90969.94260.251

3175

o77C.722

67.089

817061

302725805138S37354488770

745948413531

806858

837364

.036

.765

.983

.419

.480

.483

.518

.152

.718

.168

.700

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.338

.332

.538

.026

.742

.933

.331

.059

.090•

»935.757.906

80.227

42.582

847767

858172

838374

826756484238

867363

867869

87

.938

.486

.443

.665

.773

.919

.573

.492

.244

.561

.174

.002

.121

.484

.339

.434

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.915

44.890

89.80782.38772.029

88.06487c73278.571

S5.77485.56576.202

85.26069.68356^34250.31244.54240.278

89.52477 .78467.412

f

88.77163.09073.9*1

89.603

A P P E N D I X 3 PAGE 6ANGLES BETV€EN DIRECTIONS IK ALPHA ZIRCONIUM

U V T W

1 0 - 1 4

1 0 - 1 5

1 0 - 1 6

C/A = 1.5927

U V

2 0 - 2 1 :

AhSLES I N DEGREES

22

11222333311-1111111

12223333111111111

222333344

: 1

11i. i?

33

CCccc000c111112

.222

C0000

•000111112222

0, Q

000

• 000011

,111

-5— 5

-1-1— 2-2-2— 3-3-3— 3«•» 7

-2-2-2.-2-3-3-3-3

-1-2-2-2-3-3— 3— 3-2-2-2-2-2-3-3-3-3

-2«2-2-3-3

; -.3,

~3. -4-4>2-2-2-2

--2-

23

5

1351245C12340123

61351245C1234C123

13 •5124§1301234

53.35043.989

C.O1.997

53.03823,67111.23860,68846,22026*93120.8547^.3*5451.62833.11922.26715.75078.41559.30443.72932.410

0.055.03525.66913.23662.68648.21728.92822,85181.11553.28734*69523.72917.05880.29861.17045.57434.229

0.0 i29.36641*799

7,6516.818

26.10732.18411.745

9.90138.10827*01932.48239.45044.509

55,30546.049

12,20011,33159.68631,40720.20967.17253.03734,47328.78690,00062.72944,65634.15127,86680.75561.80046,39835*270

10,23260*52431.90020,30468*06153.82J335.02829.21690,00062.52344,25033.55827.10182.25263.23247.75436.538

49.38352.78255*64841«T73352*72652.34953.26637.63852.32890.00060.06072.29669.28167.794

66.24457.216

21.21219,51871,84243.21931.38879,35365,15946.36540,511

72.80454,18643.14036,32987.69868,97853.79342.859

17.77070.73041 .90229.91978.28164*00845.07839.165

71.03652.38141.30534.47288.06869.215F/3.89342.821

54.03878.75078.35556.16856.20175.49081.56757*56659.284

78.70476.38786.29587.552

69.672 79.57060*619 70.242

24.54122.54477.57948.21335.78085.23070.76151.47245.395

73.569 80,26858.349 64.79847*357 53.557

20*54775.58246.21533.78283.23268.76449.47543.398

73.071 78.71557.728 63.19746.621 51.915

76.21785.89788*81771.825

: 80.27287.90384.16969.56582.137

r

81.18271.783

82.46766.86955,513

80*57064.96153*594

U V T W

2 0 - 2 1

2 0 - 2 3

A P P E N D I X 3 PAGE 7ANGLES BETWEEN DIRECTIONS I K ALPHA Z I R C O N I U M

C/A = 1.5927

U V T H' ANGLES IN DEGREES

1 11 12 22 22 22 21 21 21 21 21 21 21 21 31 31 31 31 41 41 41 42 32 32 32 3

2 02 03 C3 03 C3 C4 04 C1 11 11 11 11 11 11 12 22 22 22 21 21 21 21 21 21 21 21 313

-2-2-4-4-4-4-3-3-3-3-3-3-3-4-4-4-4-5-5-5-5-5-5-5— 5

_ p-2-3— 3-3-3-4-4-2-2-2-2-2-2-2-4-A— 4-4-3-3-3-3-3-3~3-4-4

6913590123456012301230123

351245130123469135901234560 f "I

50.63755.24529,81028.86736.18446.43730.84918.54519.37526.39632.81437.79041.59328.11916.69512.71517.38826*84916.87410.39411*52833.51325.57421.47621.888

C.O12.43337.01722.548

3*2592.818

41.11119.46659.44734.04320.30017.06618.37322.37726.28945.65125*57917.72919.42156.31537*84423.43514.48311.1541U94414*10955.26541.220

66.51265.86649.20769.1897C.51067.33546.62238.19736.39.42.45.47,50.41.41.41.53*37.44.43.43.36.33.32.

34.30.58.47.35.33.62.45.96.67.53.46.42.39.37.73.59.49.41.63.46,34.26.23.22*22.65.53.

642023254201651951290323047502573961528152865431876

133742811674621027134473000689887712907426586525712713684659772266945604628799984527

80.62675.65983.84774.91681.96185.32480.11347.63962.19265.14463.93663.40363.19875.40544.46553.42663.00172.69946.29047.74656.70684.01843.14952.71461.241

61.10551.29371.09980.04263.77558.82467.00577.406

-86*02476*06265.45858.94251.67046.50378.01183.87570.07856.59583.63168.24956.41448.61743.77940.79538.92480.63169.033

59.67.72.76.73.72.

59.65.63.

59.64.61.

51.59.66.

71.59.87.85.75.69.63,88.

75.68.59.53.49.46.

229702976181953397

522526472

935432950

127351855

884451516568143066994650

043222500668730996

69(»388

727179808685

696875

686672

797571

808473666157

78

«383.,003.477.665.208.429

.671

.199

.730

.052

.645

.894

.152

.024

.853

.734

.389

.908

.538

.287

.441

.797

88.56584.40879.98486.44988.8d289.688

81.89888.01386.921

77.91383.08487.745

89.09886.19582.222

81.18009.61778,4^970,52664.34360.646

87.574

U V T W

2 0 - 2 3

2 0 - 2 5

3 0 - 3 1

APPENDIX 3 PAGE 8ANGLES BETWEEN DIRECTIONS IN ALPHA ZIRCONIUM

C/A = 1.5927

U V T A:\6LES IN DEGREES

111I112222

23333111111111

3333441111111222211111111111

3344443333

C0C00111112222

C000CC111111122222222

12:.2.;-2-;33

,3;3

-4-4— c

— 5-5— .>-5-5-5-5

-2-3-3-3-3-2-2-2-2-2-3-3-3-3

-3-3«3-3-4-4— 2-2-2-2-2-2-2-4-4-4-4-3-3-3-3-3-3-3-4-4-4-4

23C123G123

51245012340123

1245130123469135901234560123

28.90319.18154.80343.58033.23224.30557.40946.04035.80727.364

0.049.45034.98115.6929.615

69.79142.68325.25816.19812.36967.85748.92333.60422.674

0.014.46833*75739.835

4*09417.55134,10629.68738.64646.49051,83858.15662.84128.89933.87342.91953.84025.38818.27524.62133.19240.05345.203

43 * 07935.39467.40257.48.41.61.51.41.33.

23,62.49.33.28.90.64.48.39.33.72.54.39.29.

34.48.58*59.29.51.90.53.69.75«74.73.73.43.62.74.74.43.38.40.44.46.

• - 5 2 .

586841689881077499777

OC9650511210722000533122049941450136558479

082551045558988633000565247629625763331523280681316719841472620705086

49*089 34.77?21.85913.65316.27623.768

48.34.43.

476219639

45.204

59.14451.59278.92366.05661.32554.39786.13968.90567.86760.669

40.41983.53271.98854.22948.777

82.82664.48753.63546*96585.67768.14654.17744.439

57.11655.89467*84073.91757.89055*950

82.10177.68278.92584.97388.189

, 83.26885.68879.48176.46487.17079*59040*78754.93265.55469.91569.87269c,97974*62344*37446.08456.011

77.12368.736

69.65676.50475.348

72.90075,36367.910

47.01885.52481.99962.71056.633

76.71262.51752.437

68.21275.26285.69689.09566.40776.870

54.09364.77470.33273.16478.66279.274

55.25362*78167.128

81.97080

778284

767988

867363

747173798082

706869

.286

s440.921.932

.526

.540

.011

.848

.990

.057

.521

.601

.216

.390

.567

.739

.769

.323

.609

89.69587.740

88.33386.84385.626

84.19083.13688.729

89.07077.64966.364

85.73088. 91785.04182.39283.88887.234

79 .36684*08588.136

U V T W

3 0 - 3 1

3 0 - 3 2

3 0 ^ - 3 4


C/A = 1.5927

U V ANGLES IN DEGREES

11112222

3334411111112222111i11111111111222.2

331III1II

44443333

CCGCC111111122222222222333'344443333

C01111Ia2

-5— c_ c

-5— 5-5— 5-5

-3-3^ Q

-4-4-2-2-2-2-2-2-2-4-4-4-4-3-3-3— 3-3-3-3-4-4— 4-4-5<

— 5-5

— 5-5

— 5-5«i- *

-3-3^•2-2**2-2'-*2-3T'3-

Ci23C123

24513012346<313590123456012301230123

45,G •: •1234.01

2G.13811.99311.34317.15528.67323.22422.73726.091

0.019.28925.36618.563

3.08342.41026.20727.54233.36038.05843.96348.48632.03725.42530.49039.89836.33321.16916.30420.80226.53131.26334.95634.14721.39612.91412.87433.15522.53413.684

9.31838,52329,14522,48319.850

,.• c.o' .' ;'

6*077 ,56.813 •32.43120*040 j.18.675 :20.88$53.3,2835.062

51.25130*3614C.34446.57639.85235.74234.79136.349

5C.46347.59047.88344.45649.6369C.OOO65.98067.63763,72861.78160.08959.23254.5837C.99965.32861.17749.87438.70934.15834.58436.81239.24741.41653.76245.41540.17638.08856.07444.11044.55141.60746.79938.94933.50030,913

36,84934,74$9CU00QV

-6 9. 692p5.,£!r8/, ft fk Q O

~^« \3 ^r O jfi.

45,49661.4-59«5,U3

71.76347.83046.36849.21583.70437.06246.01754.221

63.01982.20878.18457.82366.102

75.81482.77787.13380.89273.88968.88182.29475.40588.46078.63780.72853.88864.49460.74358.77857.76857.25376.32347.71960.01659.64473.79649.45554.38759.42784,38848,88158.82567,568

^6*373 •'6 J i f f $6 -'V

'• • . i i

83.10379..1314:

68,578 '62*10283 . 14068,455,

56.21961.85365.212

68.62366.44867.350

75.57179.55583.282

46.31653.60160.529

84.82182.30888.33572,64286.893

•64.07668.70574.59370.71968.12266.322

63.66363.37770.142

63.61563,02263,403

55,81064*68868.769

V

78,403S72.32$•' ' -;' '"'

' . ' •

', ,

'' "

71.957

80 .34077.58675.558

71.10976.65179.60687.25182.44078.896

69.00573.19581.263

,

67.88771.13677.958

78.25472 ,94672,605

78,644

87.31689.26286.328

88.80.86.87.87*82.

84.88.82.

-80.86.88.

89.83.78,

81.

889444050356065939

263462543

166312254

301508647

861

U V T W

3 0 -3 A

3 0 - 3 5

4 0 - 4 1

4 0 - 4

I I -2 0

APPENDIX 3 FACE 10ANGLES BETWEEN OIRECTICNS IN ALPHA ZIRCONIUM

C/A * 1,5927

U V T Vi ANGLES IN DEGREES i« V T W

1 2 - 3 21 2 - 3 3

3 C -3 51 1 - 2 01 1 - 2 11 1 - 2 21 1 - 2 31 1 - 2 41 2 -3 C1 2 - 3 11 2 - 3 21 2 - 3 3

1 1 - 2 01 1 - 2 11 1 - 2 21 1 - 2 31 1 - 2 41 2 -3 C1 2 - 3 11 2 - 3 21 2 - 3 3

1 1 - 2 01 1 - 2 11 1 - 2 21 1 - 2 31 1 - 2 41 2 - 3 01 2 - 3 11 2 - 3 21 2 - 3 3

1 1 -2 C1 1 - 2 1I I -2 21 1 - 2 31 1 - 2 41 1 - 2 61 1 - 2 92 2 - 4 12 2 - 4 32 2 - 4 52 2 - 4 91 2 - 3 01 2 - 3 11 2 - 3 21 2 - 3 31 2 -3 41 2 - 3 51 - 2 - 3 61 3 -4 01 3 -4 1

21.14113.425

C.O61.75535.8362C.91216.07116.40358.9124C.29925*57915.879

32.43431.73142.11950.31755.78722.94319.36927.86736.953

44.52726.37525*56730.69435.17938.93522.90815.68618.528

0.027.96446.71757.87764.78472.57178.17914.86638.53253.00467.28710.89321.94336.26247.17955.01160*67464.87416.10221.413

33.42427.127

31.7139C.OOO66.87752.43444.8244C.74C65.60148.30835*19527.081

9C.OOO50.17865.44574.98678.30542.54837.22242.87847.8G6

9C.OOC66.68765.58361.26159.09451.51639.26933*38332.809

6C.OOC63.79369.95274.58177.7C181*38784.12161.10166.97572.49078.86949.10751.80357,48363.05567.52570.94273.55743.89845.715

57.290 69.86650.062 61.536

56.490 66.24^

88.55173.42962.76656.21684.07268.131 77.71455.750 66*84647.475 57.795

83.97080.61379*06280.97479.38739.697 51.51451.077 61.50761.593 69.620

-

74.56185.67184.16177.88381.05056.745 66.33363.142 71.65958.840 72.419

70.89371.99074.40976.90578.98180.60481.86376.10276.541

87.135 87.24476.787 81.611

79.782 83.38232.020 86.90371.428 75.699

75.669 84.23773.793 88.66273.191 88.039

70.529 87.74078.679 79.22682.607 88.797

1 1 - 2 0

1 1 - 2 1

1 1 -2 2

APPENDIX 3 PAGE 11ANGLES BETWEEN OIRECTICNS IN ALPHA ZIRCONIUM

C/A = 1.5927


1 31 31 41 41 41 42 32 32 32 3

1 11 11 11 11 I1 12 22 22 22 21 21 21 21 21 21 21 21 31 31 31 31 41 41 41 42 32 32 32 3

1 11 11 111112 22 22 22 21 21 21 21 2

-4-4-5-5-5-5— 5-5— 5-5

-2-2-2-2

-2-4-4-4-4-3— •a.-3-3-3-3-3-4-4-4-4-5-5-5-5-5-5-5

~!-2-2-2-2-2-4-4-4-4-3-3-3-3

2301230123

12346913590123456012301230123

2346913590123

31.40.19.22.28.35.6.

13.23.32.

0.18.29.36.44.50.13.1C.25.39.29.13.11.20.27.32.37.31.20.14.16.33.24.18.16.28.17.

145266107113726948587589756921

0753913820607215097568040323850295532153485964076941256304485427421343890668194

7*8257.

0.11.18.25.31.31.

8.6.

20.47.28.14.t.

173

0160067854462850185287570682964461516

50.06755.09040.89342.17045.44949.63653.41354.32456.69159.756

52.41549.89950.80552.23954.57356.67242.8305C.38950.18952.91054.67545.56442.06242.38344.08545.98747.71450.47342.89638.73837.77748.11341.68037.41335.52556.23440.38348.13945.924

40.09536.98436o42437.13438.48658-79543.80938.02936.49363.3B148.47533.36733.273

77.64579.00379.10779.32279.89980.68266.58767.12068.52470.381

55.92874.68180*66076.34471.63268.37756.82966.49680.96874.80973.19548.29359.66757.41656.66856.59056.78277^75245.05857.12163.03680.39143.44653.13861.75969.45452.27351.23556.816

72.84764«26359.16553.71050.03561.58379.31967.96757*37477.03163.19453.14647.099

80.87.85.87.79.73.72.86.83.

. 84.

64.63.74.77.74.72.

59.66.67.

55.63.67.

63.59.60.

86.75.68.60.55.81.85.80.65.\

66.76.68.

202789840252465857150942756749

721606815770798674

945294i O6

780577366

883626617

566406499712104688249279996

612630281

667882828380

716977

757070

657278

798277

.907

.045

.050

.773

.221

.050

.249

.275.260

.173

.758

.720

•271;.376.809

.341•1L08.601

83.15788.26786.45687.534as. 50187.261

84.85488.62083.313

85.90588.86984.328

75.72381.89087.375

- • ••

88.45287.30886.596

U V T W

1 1 - 2 2

1 1 - 2 3

1 1 - 2 4

APPENDIX 3 PAGE 12ANGLES BETWEEN DJRfcCTlCNS IN ALPHA ZIRCCMUH

C/A = 1.5927


111111111112222

111122221111111111111112222

1112222111

222333344443•3

33

111122222222222333344443333

1112222222

-3_3— •a-•*— • 4

-4-4-4_ c

•*

• c_ c

**

— K

-5— 5_ c

«. t

-2-2-2-2-4-4-4-4

— 3-3-3-3-3-3-3-4-4-4-4-5-5-5— 5-5-5-5— 5

-2-2-2-4-4-4-4-3— 3-3

456C123C1230I23

346913590»

23456012301230123

4691359012

1C. 22914.81518.64648.7S935.0S723.42815,09349.62138.8662<=.26621.49147.07235.24724.42515.245

C.O6,907

14.6942C.30243,01019.345

4.8729.410

58,52339.53324.19713.428

7*0566.0238,341

59.27745.22532,86923.02059,83848.75738.58129.85958,11446,23735,32325.939

0.07.787

13.39549,91826.25211,7802,502

65.27046,19830.696

31.45431,26031.73660.39446.8063S.56933.41456.78449.35641.22634.96865.88C56.62546.73942.726

30.83728,42127.42027*94461.68942e64533.23827.88565.62852.63839.71631.55027.02724.81523.91367.47C54.50943.43635.02266.29955.89446.49138.63371.52260.99251.59443.886

24.59822.03221.69664.02443.09531.70623.53873,80555. 91341.808

43*77942.02141.11180.52262.71S61.48055.03082.55560,53166.96060.89274.19158.926?-7.724!>1.720

54.33949.23743.24839.24272.74371.36158.90847.26879.97663.74051.22242.97037.89334.82932.97682.66270.54660.09351.97134.23370.97365.87858.47477.80267.57058.40650*843

43.30236.93032.66079.65166.61253.53141.21081,98464.61350.8S5

62.61858.73355.988

70.15575.16771.846

68.59770.72674.666

65.36569.84779.269

64.24757.33949.55243.94487.72683.59169.11954.836

77.57370.09560.61054.13549.66746.493

73.39774.27465.595

74.62477.16469.470

69.99980.94074.076

50.43142.64437.036

. 88.49776.684

; 62*21147.9I>9

- •80*43666,305

70.79965.97762.462

72.61479.39385.366

74.3207779

758480

837868615552

808577

768184

828279

.887

.655

.547

.634

.845

,,476,,606.268.037.915.184

.678

.970

.859

.851

.323

.698

.224

.268

.333

84.36773.299

78.59072.83268.574

88.83583.72487.134

89.07281,.32286*066

83,8887,

878675676157

858783

868689

888789

6880

.321

.167

.478

.089

.868

.535

.505

.731

.462

.093,507.749

.095,697•638

.170

.863

.625

«G76«046

U V T

1 1 - 2 6

1 1 - 2 9

2 2 - 4 1

2 2 - 4 3

2 2 - 4 5

2 2 ~4 9

1 2 - 3 0


C/A = 1.5927

U V AM51ES IN DEGREES

1 21 21 21 21 31 31 31 31 41 41 41 42 32 32 32 3

1 21 2I 21 2

1 21 21 21 2

1 21 21 21 2

1 21 21 2I 2

1 21 21 21 2

1 ;21 21 21 2

1 21 21 2121 21 21 2

-3-3-3-3-4-4-4-4-5-5-5

— c— 5-5_ c

— 5

-3-3-3— 3

-3-3-3-3

-3-3-3-3

-3-3-3-3

-3-3-3-3

-3-3-3-3

-3— 3-3-3~3

—3-3

345601230 '1230123

0123

0I23

0123

0123

0123

0123

0123456

1S.52C11.8376*8774.68765.83651*67039.13128.97566.26255.06344*72035.75864.96253.06942.13032.708

72.89553.76938.18026*852

78.39559.24743,62832.258

18.35711*26422.21632.634

39.81121.5509.49211.095

53.77834.87919.7699.768

67-71848.62533.08821*844

0.019.16534,80446,19754.27260.08264.380

32,22126.19422.5942C*55172-12256.59846.80037«48171.2136C.41350.51842.05475.29064.20254.15145.702

78.6S26C.10345.12534.481

82.29363.38448.02736.942

5C.7473S.68148.24251.454

59.19546.50939.09936.479

66.80050.63138.80131.886

75.35857.20842.77232.784

21*78726.70540.31850.00357.16562.41066.327

41.45535.32631.38428* \ 4 '

' 3 C, ""*

56494440

.191-231.390*934

62554945

.620

.139

.831

.959

68.69360.64954.36650.590

34*12771.17259.85250.87585.75.65*57.80.69 s59.51.

84.66.51.40.

86.67.52.41.

71.47.50.61.

75.58.55.51.

78.63.51.44.

82.65,50,41.

38.42.49,57.62.66,70.

382112690617254367492181

374057313848

155377141164

556008736849

164595542183

640368859540

739025929129

2130848220536SO928139

807162

777466

767870

776251

755948

596666

637474

727263

796454

606165

.035

.342,064

.485

.797

.570

.859

.713

.076

.255

.351

.631

.049

.761

.700

.106

.033• 162

.489

.027

.195

.785

.395

.891

.429

.993

.671

.000

.817

.76069.7517375.024.559

77.514

828272

828779

868374

826756

786352

676875

738184

838272

367160

81828384858586

.853

.090

.185

.082

.817

.007

.462

.239

.3nO

,646.446.437

.775

.350

.151

.476

.092

.356

.787

.645

.542

.690

.392

.317

.326

.401

.608

.787

.245«264.325.215.914.459

85.76387.33777.016

84,31587.90083,009

88,79187.8098?,TS«

87.97172.35960,988

82,45466,82855,447

77,60?83,49288.062

87.85986.27785.290

85.63388.31880.363

86,82977.57466.215


C/A = 1.5927


C/A = 1.5927

U V T W

1 2 - 3 0

U V T W ANGLES IN DEGREES

1 3 - 4 01 3 - 4 11 3 - 4 21 3 - 4 31 4 - 3 01 4 - 5 11 4 - 5 21 4 - 5 3A ^ ^ --»

2w22m,

2I

1

1

1

1

1

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5•i.

1

1

1•V

1

1

1

2•*»

2**•

2t*

j-i«

1

1

1

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3 -5«*^ ^

3 -5• •*»

3 -53 -52 -3

2 -2

2 ~3

2 -3

2 -3

2 -3

3 -43 -4

3 -4

3-4

4 -54 -5

4 ~5

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3 ..53 -5««* v

3 — 5

3 -5

2 -2

2 -3

2 «3

mr

o1231

2

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4

5

6

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2

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2

3

2

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5.2C815*20727.48537.728

8.21313.97523*28732*010

4*3C'5

12.65623.25632.583

C*056*36615*63865.03727*03170.71135.1C775.7794C«91779.24745*21583.54619,835

6.96152*494

9*20256.18718*81362.03320*79011.12665.9018.151

65.16614*0057C.65319.625

8.35453.9365.470

59,08113.71956*560

C.O72*52511.39381*00019*46977.079

26.9953G.29937.46344.9543C.COC31.88636.51342,10117.48021.04S28.50236*29620.5667C.22424*82772.65232.42973.93238.86C79.18143.77981.20447.52584.73332.68426.27562.29525.9S359.86829.78064*55535.11229.93173.79126.18378.28229.52480.25225.71618.31962*06516.72159.6732C.46865. 6?. 4

17*85476.735i\>*99982.66024*69185.031

33,004-35.64341.66448.23851*78752.66354.96457*99642.52043.84947.22951.47936*02076.39437*17678*43841.35381.74845.58280.39449*10285*08851.90885,22537.61331.92963*47531,26070.36934.04376,18454*24631.56786.29741.81082.97747.94082.47045.87831.36663.05439.73168.72240.1966<5«5<?<}

31.18278.23030.85183.34832.85288.962

54.79156.03659.09662.74868.21368.66069.85271.45955.69356.53158,71561.55938.33G88.87253.96935.60855*67864.33857.16089.46858.66187.19259*98588.57457.00333.B6372.38346,46578.24950.80176*52469.47742.465

48.353

50*829

57.83335.53270.55342.25171*562•51.? 877 4-. 080

48.48089*35045.88438.01545.32589.076

65.20966.02768.06770.54890.00090 .00090 .00090.00064.307€4.89966*456bfc.50943.38??

54.888

65*362

69.800

69.400

69.246

66.66742.74282.58250*63270 .tt'i'r.bo .7 9780 »78:i90.00050*396

50.439

57 *892

65.82541.49974 .44845.37475*54055 -659r.o.i&'-v

65.033

60*454

57.870

86*99587.08987*32487.614

77.48077.75473.47979.44532*£!1

57.803

68.273

73.437

77.453

76.251

87.3.6246.6503 8.? 00S i . O B H^ 3.96 V5 9.0 60• •'•'2. «~' '-• •.

59.449

63*403

61.987

7a*18452.16282.38151.82086.5005 4 « 1 4 !589 .Sv'Y

69.607

70.686

66.743

U V T W

1 2 - 3 2

1 2 - 3 3

U V

1 2

T

^t•**

K

5 25.27873.146

1 2

I 31 3

1 3

1 3

1 41 4

1 4

1 4

2 32 3

2 3

2 3

1 2

1 2

1 2

1 2

1 31 3

1 3

1 3

1 41 4

1 4

1 4

2 32 3

2 3

-3

-4-4

-4

-4

— c-5

«». c;

-5

-5-5

.- 5

— 5

— 3

— 3

- 2

— -a-***

-4-4

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~4

— c

-5

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-5

-5«c.*f

— 5

6

Ci

2

3

C1

2

3

01

2

3

3

4

5

6

C1

2

3

01

2

3

0V

2

29.70.35.21*58.e.

66.4.

71.35.24.67.14.75.7.

72.35.23.61.12.60.4.

67*C.

73.8*

67*33*63.18*59*46.32*65.19.72.9.

65.46.35.73e

25.74.16.81.46*34.63*23.

577298142027251967895•55473164061931280S0018478800362236835060183582540661075431885069184926423188211611806592an758542870231407430250352478729580

69.1832 3 -5 3 14.257

74.979

28.80.32.76.42.31.61.24.6<;«2U72.44.35.79*29.77.25.81.38.27.62.19.67.14.69.15.61.16.74.18.69.21.65,,51.39.67.28.73.21.77.53.43.81.34.82.27.81.48.37,

ANGLES IK DEGREES

956 35.265 45.61504239936897578866234226389038567595325124772636490644685421635730275411103369601043689425077C443920105410675374814992169137837399613650874683347

69.50027.70.19.78.

271868293622

83.43037.47579.34146.48036.07071.47729.21972.68226.65576.451\59.47446.30283.55246.84585.97143.15288.09352.75744*76770.42638.78370.38335.25278.10326.19185.64225.38277.84426.26072.24427.63468.10^54.51542.19777.67232.32477.20325.89783.01664.64855c82083 . 6 9148.22788.25942.34889.69259.32549.59772.79541.35779.07735.21080.100

85*11546.19880.3166i.?4«4-49.36078.89747.07377.23243.71278.45972.25652.389

57.205

56.257

62.43246.89473.0234-9*58878.19845.87081.09340.49737.60738.00279.53137.04573.72236.83169.42366.47955.59377.97446.88779.09640.93083.74775.11358.014

59.521;

53.678

67.03757.94776.54150.20881.81944.30682.393

56.412

55

695379558051859054

60

66

6949765683528553

49

47

46

736081548646879063

68

68

72588156855086

.572

.861

.145

.470

.525

.65i>

.664

.792

.000

.203

.303

.303

.145

.570

.672

.642

.908

.696

.076

.891

.937

.557

.091

.127

.686

.767

.092

.022

.005

.060

.000

.529

.470,

.l<7

.536

.238•682.148.475.242.773

64.138

62.359

87.53355.39684.29862.02587*28567.60488.329

65.726

63.425

71.647

79.74755.20586.77057.82486.68262*89489.74763.100

58.119

54.815

52.554

87.92162.63984.09568.88488.42062.13688.712

66.804

73.290

77.601- -

81.37060,19689.87765.24189.76759.15387.413

APPENDIX 5

APPENDIX 4

STANDARD (0001) PROJECTION FOR H.C.P. ZIRCONIUM

1010

iiso

1150Sttt

1100

olio

1370

S9IO

QUO

1100

iSo

silonio

IliO

CRYSTALLOGRAPHIC FORMULAE FOR H.C.P. METALS

Errors exist in ctystallographic formulae listed in the literature for h.c.p. metals due toconfusion between Miller and Miller-Bravais indices. Correct formulae in the Miller- Bravais notationhave been listed by Otte and Crocker (1966), Nicholas (1966) and Okamoto and Thomas (1968) andare reproduced below. These formulae are simplified by using a parameter X defined as

1.

2.

3.

7.

x1 - 2 / c \2A. ~ "5" 1— I3 \ a j

= 1.6914 for ct-Zr.

The direction [uvtw] normal to the plane (hki£) is given by

[uvtw] = [hki£X:2] .

Conversely the plane (hki£) normal to the direction [uvtw] is given by

(hki£) - (uvtwX2) • '

The interplanar spacing of planes (hki£) is given by

(A5.1)

(A5.2)

(A5.3)

4. The identity distance d° along a direction [uvtw] is given by

(AS. 4)

5. The angle between two directions

cos 6

and [ u 2 V 2 t 2 w 2 ] is given by

ti t2 W i W 2

(ul v} ti + X2 W i ) y j ( u 2 X2 (A5.5)

6. The angle between two planes (h ik i i i£ i ) and (h 2k 2 i 2£ 2 ) is given by

cos 6 =h ih X"a

/< 2 , , 2 , .2 . —2 - 2 vl/i /« 2 , 1 2 , • 2 . \~'? / > 1 \ I A(hi + ki + u + X £1 )%(h2 + k2 + i2 + X £2)^(A5.6)

Direction [uvtw] common to planes ( h i k i i j £ i ) and (h 2 k 2 i 2 £ 2 ) is given by the solutionof the three simultaneous equations

tuu + k iu + i i t + £ i W = 0

h 2 u + k 2 v + i2 t -i- £ 2 w = 0

u + v + t = 0

which gives

[uvtw]

Iikiii

£ 2 k 2 i 2

O i l »

hifiu

h2F2i2

1 0 1 1

•w

h2k2I2

1 1 0 } .

h.ki i i

h,k.i,

111

(A5.7)

-=-_

APPENDIX 5 (continued)

8. Plane (hki£) containing directions [ u a V a t i W i ] and [u 2 v 2 t 2 w 2 ] is given by the solution ofthe three simultaneous equations

Uih + Vik + tii + wi£ = 0

u 2 h + Vsk + t2i + w2£ = 0

h + k + i = 0

which gives

(hki£)

W i V i t i

" W 2 V 2 t 2

0 1 1 >

U i W i t i

U 2 W 2 t 2

1 0 1 >

Ui Vi Wi

U 2 V 2 W 2

1 1 0 > '

U i V i ti

U 2 V 2 t 2

11 1

9. Direction [uvtw] perpendicular to direction [ui Vthe solution of the following simultaneous equations

(AS. 8)

and lying in plane (hki£) is given by

APPENDIX 6

ELECTRON DIFFRACTION PATTERNS FREQUENTLY OBTAINEDFROM H.C.P. METALS

(Zone axes are given in terms of Miller-Bravais indices.The structure factor is zero for reflections marked x ).

X

»

X

oX

0

V

0

x

•X

00

00

laiz

011

Toio_0toil4

Toil

0

•

X

0

X

0000}

X0031

0000X

oo oT90002

X

0

X

0

00

^1012•

1011

10100_

101?0

IC120

0

0

X

0

X

0

X

0

X

0

X

0

X

0 012U

• 0

1212

0 012*10

• 0

1212

0 0

l2iZ

00004

t0002

00000

0

OOC2

0 _

OOOA

_•12U

0

1212

0

1210

01215

0 _

12U

[1210] [ioTo]

hu kv it + £w = 0

u

which gives

uvtw =

tit

= 0

f k i

X2 f

0 I 1 i •

h r iX2 w t

1 0 1 »

h k £

Ul Vl~A.2 wi

1 1 0 5

Ui Vi ti

i l l(A5.9)

• • • 0 0 *

o • • „ • « • •

O • 0 • •

M2D 1010 1100 ItfO0 0 « 0 0 •

0110 0006 OllO-• ,° t t •

1210 llOO itfO 11200 0 0 »

2110

• -•121

0 a 0

« 0 «t » 0 0

[0001]

X » •

0 0 0 0 *

0 M 0 0 X21U ItOI

0 • _ • • « «

1010 0000 tQIO

0 M « 0 X

ilzt out liol zTR* 0 0 0 • 0

0 X O » K

[313]

• * • * 0 0 0

. _ _ _2112 1102 0112 1122

0 » _0 «- • * * .Toio oooo 10100 0

_ • « • • ' • *1122 0112 1102 2112

9 0 0 0 • *

[2423]

o 0 0 • • x «23fl iJlO Ollt 101?, 2113

M 0 0'-' 0 //• « K

2^02 1101 0000 ;1l01 5202

• x * • * *2113 1012 OlTJ 1210

[JOTl]

-2-

APPENDIX 7

KIKUCHI PATTERNS

The Kikuchi map of titanium, constructed by O'lamoto and Thomas (1968) is reproduced in-ireA71

o X

TlV

022!x_

iJiT•

220T

32TT

•

2020

Toio•

0000

•loTo

202*0

X •

•3211

• ow

2201X •

_ ^4 •0221• •Illl

•

•

•

•1216•

•

•

X1123•

on 5A

110);rfX

2113•

•

*2020•

1010•

0000•

ioTo

20^0

•

X

*113

*1103•

Oil 3

11?3•

•

•

•

•l2i«•

9

[1216]

1232

02*21

21T5

io°T5

o*n

3033

2022

Ion

§212

2201

[1211]

0333

li2tt

oTn Toil

0000 12TO

22oT icTT 022T

3515 20?? 1232x x x

3033

[1012]

X3S03

2?02

0

• - *"02

,o

0000

*fT oil? T

*

X

ojJ5

0333[5U3]

0000

oTu

Ton

2112

• - §2115205X

9301

[2S?3]

Figure A7.1 The[000l] Kikuchi map for h.c.p, titanium (c/a = 1,588).All poles are indexed in terms of directional indices.

Determination of the Exact Foil Orientation from Kikuchi Patterns

The unknown Kikuchi pattern is shown in Figure A7.2(a) together with a tracing of some ofthe centre lines of the Kikuchi pairs (trace?:of crystal planes) in Figure A7.2(b). The Kikuchi polesB and C are obtained by extrapolation of the relevant traces. The traces of planes giving rise to theKiktichi pairs were indexed using Equation 16 and the Kikuchi map in Figure A7.1. Thus the polesA, 3, C and D can be shown to be [0114], -[0112] , -[11263 and [0113] respectively. From thescale factor of the pattern it can be shown that the origin of the pattern, and hence the direction ofthe electron beam, is 4.3° from [0114], 9.7° from [0112] and 8.2° from [1126]. The beam direc-tion is shown on a section of the standard projection in Figure A7.3.

The beam orientation can also be given analytically because it is obvious that the origin ofthe Kikuchi pattern lies almost on the intersection of the (1121) trace and the traro of a plane con-taining poles A and H.

Pole H lies on the iritei'section of the (2422) trace and the trace of the plane containing polesB and G. Pole G, which lies on the intersection of traces (23ll) and (2232). has indices [I 4 5 15](use Equation A5.7). Hence tifie trace containing poles B and G is given by (4131) (use EquationA5.8). Thus pole H, lying at the intersection of traces (2422) and 04131), is given by. [1 7 8 21].

-2-

The plane containing [0114] and [1 7 8 21] is (6511). Therefore, the beam orientation, givenby the intersection of planes (2242) and (6511), is [110 11 331 -

Figure A7.2. (a) A Kikuchi pattern fiom zirconium.

>[,T02] ,[,5,3]

O [0111]

O[,T03J

O[,T04]

OJ2429J

o[oT«]

o[om]

O [0051J O[mi)

Figure A7.2 (b) Schematic drawing ofcentre lines of indexed Kikuchi pairs

Figure A7.3 An enlarged section of the [OOOl] standardprojection for directions for h.c.p. zirconium. The foilorientation in Freure A7.1 is shown at R.

APPENDIX 8

VALUES OFg.b FOR EIGHT REFLECTIONS AND^ <ll20> -,j <ll23>-. AND <0001> -

TYPE BURGERS VECTORS

Reflei

1

2

3

4

5

6

7

8

:tion

lOlO

oiToHop0002

lOil

lOlf

Olll

OlII

1101

Hoi10f2

1012

Oll2

Oll2

1102

1102

1120

1210

2110

1013

10l3

0113

0113

1103

I1C3"

1122

1122

I2T2

1212

2112

3112

1124

1124

1214

1214

2114

2114

±[1120]

±1

±1

0

0

±1

±1

±1

±1

0

0

±1±1±1±1

0

0

± 2

±1

Tl

±1

±1

±1

±1

0

0

± 2

±2

±1

. ±1

+ 1

Tl

± 2

±2

±1

±1

±1

±1

Burgers Vectors of Perfect Dislocations (xi.)

±[1210]

0

±1 .

±1

0

0

0

±1

±1

±1

±1

0

0

±1

±1

±1

±1

:>1

± 2

±1

0

0

±1

±1

±1

±1

±1

Tl

± 2

±2

±1

±1

±1

±1

±2

± 2

±1

±1

±[2110] :

+ 1

0

±10

+ 1+ 1

0

0

±1

±1

±1

±1

0

0

±1±1+1±1± 2

Tl

Tl

0

0

±1

±1

Tl

Tl

±1

±1

±2

± 2

±1

±1

±1

. ±1

± 2 .

±2 .

t[1123]

±1

±1

0

±2

± 2

0

± 2

0

±1

Tl

±3

Tl

±3Tl

±2

T2

. ±2

±1

Tl

± 4

T2± 4

T2

±3

T3

± 4

0

±.3

Tl

±1

T3

±6

T2

±5

T3

±3

T.5

± [1213]

0

±1

±1

±2

±1

Tl

± 2

0

± 2

0

± 2

T2

±3Tl

±3

Tl

±1

±2

±1

±3

T3±4

T2

± 4

T2

±3Tl

± 4

0±3Tl

±5T3

±6T2

±5

T3

±[2113]

+ 1

0

±1

± 2

0

T2

±1

Tl

± 2

0

±1

T3

± 2

T2

±3

Tl

Tl

±1

± 2

± 2

T4

±3

T3

± 4

T2

±1

T3

±3

Tl

± 4

0

±.3

T5

±5

T3

±6

T2

± 1123]

±1

±1

0

T2

0

± 2

0

±2

Tl

±1

Tl

±3Tl

. ± 3

T2

± 2

± 2

±1

Tl

T2

± 4

+ 2

± 4

+ 3

±3

0

±4

Tl

±3

T3

±1

T2

±6

T3

±5

T5

±3

±[1213]

0

±1

±1

t2

Tl

±1

0

± 2

d± 2

T2

± 2

Tl

±3

Tl

±3

±1

± 2

±1

T3

T3

± 4

T2

± 4

T2

Tl

±3

0

0

±3

±1

T3

±5

T2

±6

T3

±5

t[2ll3]

Tl

0

±1

T2

T2

0Tl

±1

0

± 2

T3

±1

T2

± 2

Tl

±3

Tl

±1

± 2

T4

±2T3

±3

T2 .

± 4

T3

±1

Tl

±3

0

± 4

T5

±3

T3

±5

T2

±6

±[0003]

0

0

0

±2

±1

Tl

±1

Tl

±1

Tl

± 2

T2

± 2

T2

±2

T2

0

0

0

±3

T3±3

T3

± 3

T3

±2

T2

± 2

T2

±2 '

T2

± 4

T4

±4

T4

±4

T4

APPENDIX 9

EXTINCTION DISTANCES FOR VARIOUS REFLECTIONS FOR

100 kV ELECTRONS IN a-ZIRCONIUM

The extinction distance dfg is an important parameter in the dynamical theory of contrast andis given by the relationship

77 V COS 6

B

where V is the volume of the unit cell and Fg is the structure factor for reflection £. This relation-ship is only valid when s = 0, that is, when the crystal is at the exact Bragg position and when onlyone reflection is operating. For 100 kV electrons it is necessary to use relativistically correctedvalues of Fg and X.

In the dynamical theory of contrast, the deviation from the exact Bragg reflection position isdenoted by a dimensionless parameter & = sgg. The effective extinction distance g g when thecrystal is tilted from the exact Bragg position is given by

Thus, when using observation of thickness fringes to detennine the thickness of the foil it is necessaryto note the deviation from the Bragg position, co, of the operating reflection and to use values of g g.

The values of £g for a-zirconium for 100 kV electrons, given in Table A9.1 have been calcu-lated using the analytic representation of atomic scattering amplitude given by Smith and Burge (1962).Relativistic effects have been taken info account using correction factors listed in Appendix' 4 of thetext by Hirsch et al. (1965).

TABLE A9.1

EXTINCTION DISTANCES IN a-ZIRCONIUM FOR

100 kV ELECTRONS

hkig

0002

0004

1010

lOll

10f2

10L3

10l4

2020

2021

2022

2023

£g

315

622

592

378

830

625

1387

11.36

681

1310

877

hki£

3030

3032

1120

1122

1124

1230

1231

1232

1233

1234

2240

£g

927

1012

488

582

83c

1567

930

1736

1125

2244

1982

AUSTRALIAN ATOMIC ENERGY COMMISSION · AUSTRALIAN ATOMIC ENERGY COMMISSION RESEARCH ESTABLISHMENT...

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