AECL-5556

ATOMIC ENERGY &S& L ENERGIE ATOMIQUEOF CANADA LIMITED V j & j r DU CANADA LIMITEE

COMPUTER ANALYSIS OF ELECTRON

DIFFRACTION PATTERNS

by

R.A. PLOC

Chalk River Nuclear Laboratories

Chalk River, Ontario

July 1976

COMPUTER ANALYSIS OF ELECTRON DIFFRACTION PATTERNS

bv

R.A. PlocMaterials Science Branch

Atomic Energy of Canada LimitedChalk River Nuclear Laboratories

Chalk River, Ontario KOJ 1J0July, 19 76

AECL-5556

Analyses par ordinateur

<3es modeles de diffraction electronique

par

R.A. Ploc

Resume

On passe en revue les techniques disponibles d'indexationpar ordinateur des modeles Kossel, canalisation zonale selective,Kikuchi, point et anneau SAD. On comments egalement la determinationdes orientations et des parametres de reseaux.

L'Energie Atomique du Canada, LimiteeLaboratoires Nuclaaires de Chalk River

Chalk River, OntarioKOJ 1J0

Juillet, 1976

AECL-5556

COMPUTER ANALYSIS OF ELECTRON DIFFRACTION PATTERNS

b y

R. A. PlocMaterials Science Branch

ABSTRACT

A review is given of the computer techniques available forindexing Kossel, Selected Area Channelling, Kikuchi and SAD spot andring patterns. Also discussed are orientation and lattice parameterdeterminations.

Atomic Energy of Canada LimitedChalk River Nuclear Laboratories

Chalk River, Ontario KOJ 1 JOJuly, 1976

AECL-5556

TABLE OF CONTENTS

Page

INTRODUCTION

II. KOS3EL PATTERNS • 2

(i) Introduction 2(ii) Orientation Determination 3(iii) Lattice Parameter Determination 5

(iv) Preferences 6(v) Figures , 7

III. SELICCTED AREA CHANNELLING PATTERNS (SACP) 18

(i) Introduction 18(ii) Indexing by Comparison 18(iii) Analytical Indexing 18(iv) References 19(v) Figures 19

IV. KIKUCHI PATTERNS 22

(i) Introduction 22(ii) Indexing 22(i.ii) Orientation 23(iv) Lattice Parameter and Wavelength

Determination 2 4(v) Convergent Beam Diffraction 25(vi) References 25(vii) Figures 27

V. SELECTED AREA ELECTRON DIFFRACTION (SAD) 36

(i) Introduction 36(ii) SAD-Ring Patterns 36(iii) SAD-Spot Pat terns „ 37(iv) Structure Factors and Extinction Distances , 38(v) References 39(vi) Figures 40

VI. DIFFRACTION PROGRAMS WRITTEN FOR A HEWLETT-PACKARD 9810A DESK TOP CALCULATOR 53

(i) Introduction 53(ii) SAD-SP 53(iii) Triclinic d-Spacings 54(iv) Interplanar Angles 54

Page

(v) Interdirecfcion Angles 35(vi) Angle Between a Crystal Plane and Direction. 55'vii) Bragg Angles 55(viii) Kikuchi Pat terns - STEREO Data 55

VII. ACKNOWLEDGEMENTS ' 56

VIII. APPENDIX A 57

COMPUTER ANALYSIS OF ELECTRON DIFFRACTION PATTERNS

R. A. PLoc

Materials Science BranchAtomic Energy of Canada LimitedChalk River Nuclear Laboratories

Chalk River, Ontario KOJ 1J0

I. INTRODUCTION

The focus of this presentation is on the computer analysis ofelectron diffraction patterns from crystalline, inorganic materials. Thesubjects, in order of discussion, will be from long to short wavelengths(X), i.e. , from approximately 10"1 to 10"3 nm or from electronaccelerating voltages of 102 to 107 eV. Low energy electron diffractionwill not be discussed.

Kossel patterns should not be part of this report; however,divergent X-ray diffraction does occur in the Scanning Electron Microscope(SEM) and scanning electron Microprobe Analyzer (MPA). Since analysisof diffraction patterns is primarily geometrical, there is effectively nodifference between Kossel and Kikuchi patterns except one of wavelengthand hence, Bragg angle. Our treatment will progress from Kossel throughSelected Area Channelling (SACP) to Kikuchi patterns. Finally, diffractionfrom non-divergent sources will be discussed; i. e. , Selected Area spot andring diffraction patterns (SAD).

There are two reasons for using computers or programabledesk-top calculators to analyze diffraction patterns. First, for non-cubiccrystal systems the difficulty of indexing and orientation determination canbe considerable. Secondly, the computer is unbiased. It is a falsepremise to assume that computer analysis relieves the researcher of thenecessity of having an understanding of the basic principles of diffraction.The greatest attribute in computerized diffraction analysis is an intelligentand knowledgeable use of the tools available.

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A literature survey reveals that few investigators actively seekout published programs and adapt them to their own systems. Rather,programs are generally written for a specific topic and this is often reflectedin the lack of generality of the program as well as the input and outputparameters . The approach adopted here is to locate as general a programas possible so that the effort necessary to make foreign programscompatible with local systems will be well spent.

Numerous programs for desk-top calculators have been writtenbut are unpublished because of their smallness. Several sucn programswritten for a Hewlett-Packard, Model 9^10A calculator with options 001(111 storage regis ters ) , 003 (2036 programming steps) and 004 (printer)with (he Mathematics and Pr inter Alpha ROM are included in this report.The use of these programs will be illustrated since often they easilyduplicate many of the published analysis techniques.

Each section of the report is self-contained. For instance,sections on Kossel pat terns, etc. will contain pertinent references andphotographs. Not only will references to specific computer programs begiven but also to related techniques. For instance, in the "SAD-spot"section, reference to computer programs is made and also to reports whichlist typical diffraction patterns for analysis by comparison.

II. KOSSEL PATTERNS

(i) Introduction

Use of Kossel patterns has not been extensive in the area ofscanning electron microscopy or microprobe analyses. The reasons forthis are three-fold:

1) Lack of knowledge of the capabilities of the technique.

2) Limited application to mater ials for which the X-ray wavelength \is less than 2d where d is the interplanar d-spacing. The pseudo-Kossel pattern, produced by placing a thin foil of suitable mater ia labove the sample to be analyzed is an even less widely used technique.

3) Modification to the SEM specimen chamber is necessary to incorporatea Kossel camera (See references 1, 9-11).

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Computer analysis of Kossel patterns falls naturally into twosections; orientation and lattice parameter determinations. An excellentreview of Ltie subject can be found in reference (2.).

! ii) Orientation Determination

To determine the crystal orientation from a Kossel pattern thelines (henceforth meaning the trace of the Kossel cone on the photographicplate) must be indexed. No program has been published which solvesthis problem though, in theory, it could be written. Rather, numerousprograms exist to plot simulated patterns (see references 2-6) andindexing is made by comparing experimental and calculated patterns.

a) Indexing lines

A program was developed in our laboratory to constructgeneral stereographic projections (6). This computer program possessesthe capability of producing simulated Kossel patterns for any crystal systemin any orientation. Figures 1 and 2 illustrate plotted output from STEREO.The quantities PA, PB and PC refer to crystal poles pointing out of theplane of the paper at the center of the stereogram, in the piane of the paperbut from the center to the bottom of the plot and PC at right angles to bothPA and PB respectively. Atomic scattering data and atom positions mustbe "read in" as well as a cutoff value for determining which Kossel linesare to be plotted (structure factor considerations). Figure 3 shows theinput parameters used; a fuller description of input and output parameterscan be obtained by request. The program will also produce standardstereographic plots for poles and/or directions of the same orientationwhich can also be plotted on the Kossel pattern if so desired.

Two difficulties exist with STEREO,

1) The Kossel pattern simulation is a stereographic rather thana gnotnonic projection, and

2) Kossel line indexing can overlap.

Item (1) is not a limitation of any significance. For example, comparethe experimental Kossel pattern of F ;gure 4 with that computed andreproduced in Figure lb. Indeed, the majority of the publishedcomputer programs produce stereographic projections. Item (2) ismore difficult to rectify though not impossible.

The programs of both Ploc and Barnett (6) and Frazer and Arrhenius

(4) can expand central regions of the Kossel pattern (see Fig. 5). The use of

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the programs of Ploc and Barnett (6), Tixicr, et al. (2) or Frazer andArrhenius (4) are adequate to index Kossel lines.

b) Crystal orientation from indexed patterns

Tixier et al. (2) give a review of the several techniquesavailable for crystal orientation determination though not all of these arecomputerized, Two programs are to be noted in this section: those ofMorris (5) and Halbig ct al. (1), the latter being more general.

To utilize the program of Morris the center position of the Kosselpattern and the crystal to film distance are required. Three points oneach conic section are measured and from these are calculated theassociated d-spacings and interplanar angles which are compared manuallyto computed tables. The program is therefore, limited to relativelysimple crystal systems due to the increasing number of d-spacing and inter-planar calculations and. comparisons which accompanies low symmetrysystems. In principle, the program could be extended to cope with thislimitittion.

The program of Halbig, et al. (1), does not require a knowledgeof specimen, to film distance or the location of the pattern center. Further,the use of vector analysis makes the program attractive because of itsgenerality. Fialbig et al. (1) give an example in the cubic system but thecoding appears to be sufficiently general to handle triclinic systems.

Figure 6a illustrates the analysis technique of Halbig et al. (1).The points P | to P4 represent the intersection points of Kossel conies onthe photographic plate and are illustrated in Figure 6b. The point Qrepresents the exit surface of the crystal and the vectors QPn (n = 1-4),directions lying along the intersection of Kossel cones. The lengths dj tod.|, the Miller indices of the Kissel cones forming the intersections and theangle between an arbitrary (but fixed within the camera) x-axis and thevector joining Pj to P^ are all part of th^ input data. Output lists the anglesbetween the x, y and z (perpendicular to the film) axes and the crystaldirections [ 100] , [010] and [ 001] . AI30 given are the angles between thex/y plane =>r..d the planes which contain the x-axis and the [ 100], the y-axisand the [ 010] and the z-axis and the [ 001] . This information is used toplot the orientation of the crystal on a standard stereographic projection.

Halbig et al. (I1, calculate the vector which is normal to the filmand the crystal exit surface (in their notation r^4 x r ^ ) . This informationalong with any of the vectors lying between the points P n (n -.-. 1-4) can beused as input data in the program of Ploc and Barnett (6) to produce astereographic projection with the crystal normal at the center of the

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s te reogram and r , 7 at the cor rec t angle with respect to the x-ax is . Asa double check on interpretat ion, a Kossel pattern can be produced in thesame orientation as the crys ta l . For example compare Figures 6b and 7b;Figure 6a is a s tereographic projection of the crystal in the same orienta-tion.

(iii] Lattice Pa r ame te r Determination

Tixier and Wache" (2) have reviewed the methods used to determinelattice pa r ame te r s from Kossel pa t te rns . The three most usefultechniques involve:

1) Lens shaped intersect ions ^f Kossel conies.

2) Exact tr iple intersect ions (Lonsdale (12)).

3) Near triple in tersect ions .

The method of Phi l iber t (3) ut i l izes lens shaped intersect ions(see Figure 8a) to calculate latt ice p a r a m e t e r s . Kossel lines must beindexed and the angular separa t ions , AS, between the intersect ion pointsmeasured . Morr i s (5) points out that the major e r r o r s in the technique areones of measur ing the AS and the crys ta l to film dis tance. P a r a m e t e raccuracy is of the order of ± 10"5 nm. The key to the analysis is todetermine the sensitivity of the AS to the latt ice parameter ( s ) involved.Bevis and Swindells (11) and Morr i s (computer p rogram for general c rys ta lsystem (5)) have published the necessa ry equations. Knowing the c rys ta ldirect ions from Halbig et al. (1) a simple calculation for interdirect ionangles would seem sufficient to es tabl ish AS values.

The technique of Phi l ibert et al . (3), for a typical calculation inthe cubic system requ i res six intersect ions; four with low displacementvalues (not sensitive to variat ions in the lat t ice paramete r ) and two with highvalues (sensitive to variation). The displacement value is the measu re ofshift of the intersect ion points for two slightly different latt ice p a r a m e t e r s .The four insensitive points are used to determine the c rys ta l orientat ionusing the technique of Halbig et al . (1) (described in section II ii b). Thetwo sensitive points are used in calculating their theoret ical separat ion,dn (n •_- 0, 1, . . , ) , for a given latt ice pa ramete r a o . The calculation isrepeated for a second pa rame te r , a, where a = 1.001 aQ. The values of da re plotted against their respect ive a values from which the pa rame te r of themate r i a l can be approximated Knowing d , the measured distance betweenthe points. The i teration is repeated until the difference between the twovalues is l ess than the accuracy of the film measurement ; see Figure 8b.

- 6 -

For crys ta ls of low symmetry the displacement values need to becalculated as a function of each paramete r separately.

(iv) References

a) C "niter programs

!) II. il'albig, P. L. Ryder and W. Pitsch; Technique for OrientationDeterminations by Means of Kossel Diffraction in the Electron Microprobe,5th International Congress on X-Ray Optics and Microanalys is , Tubingen,Germany, Tubingen Universi ty, Inst. Appl. Phys. , Sprin r e r -Ver l ag , N. Y. ,edited by G. Mollenstedt and K. H. Saukler, Sept. (1968) 388-395.

2) R. Tixier and C. Wache*; Kossel Pa t t e rn s , J. Appl. Crys t . 3 (1970) 466-485.

3) J. Pbi l ibert , R. Tixier and C. Wache"; Lattice P a r a m e t e r Computation onKossel Pa t t e rns and Related Crys tallographic Computations, P roc . 25thAnniversary Meeting of EMAG, Inst. Physics (1971) 202-205.

4) J. F r a z e r and G. Arrhenius ; Geometry of Divergent Beam Diffraction,Quatrieme Congres International sur L'Optique des Rayons-X et laMlcroanalyse , Orsay , F rance (1965) 516-533.

5) W. G. M o r r i s ; Crysta l Orientation and Latt ice P a r a m e t e r s from KosselLines , J. Appl. Phys . =S31, ^9 (1968) 1813-1823.

6) R. A. Ploc and P. C. Barnett; Computer-Generated StereographicProjec t ions , J. Appl. Cryst . J3(1972) 135-136.

7) J. L. Bomback and L. E. Thomas; Generation and Application of ComputerDrawn Kikuchi Maps, J. Appl. Cryst . 4 (1971) 356.

8) R. Seguin and F . Maurice; Utilisation des Diagrammes de Kossel pourla Determination P r e c i s e des P a r a m e t r e s Cris ta l l ins d'Alliages Diluesd1 Aluminum, J. Appl. Crys . 8 (1975) 266-274.

b) Related publications

9) F . Maur ice , J. Phi l iber t , R. Seguin and R. Tixier; A Kossel CameraDesigned for the CAMECA Electron Probe Microanalyser , J. Appl.Cryst. 8 (1975) 287-291.

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10) !.•;. T. Peters and R. E. Ogilvie; X-Ray Orientation and DiffractionStudies by Kossel Lines, Trans. Met. Soc. of AIME 233 (1965) 89-95.

1 1) M. Bevis and N. Swindells; The Determination of the Orientation ofMicro-Crystals Using a Back-Reflection Kossel Technique and anHcdro i i Probe Microanalyser, Phys. Stat. Sol. _2_0_( 1967) 197-212.

12) K. Lonsdale; Divergent Beam X-Ray Photography of Crystals , Phil.Trans. Roy. Soc., London A24Q (1947) 2 19-250.

13) P. G. Gielen, H. Yakowitz, D. Ganow and R. Ogilvie; Evaluation ofKossel M.ic rodiffraction Procedures: The Cubic Case, J. Appl. Phys.36 (1965) 773-782.

14) K. J. H. MacKay; The Use of Miller Indices and the Reciprocal LatticeConcept in the Interpretation of Divergent-Beam X-Ray DiffractionPat terns, Quatrieme Congres International sur l'Optique des Rayons-Xet la Microanalyse, Orsay, France, (1965) 544-554.

15) P. Rowlands and M. Bevis; A Rapid Method for Orientating Crystalsfrom Kossel Pat terns , Phys. Stat. Sol. 26 (1968) K25-K28.

16) D. J. Dingley; Use of Kossel Line Techniques in Conjunction with TensileTesting in the Scanning Electron Microscope, Proc . 25th AnniversaryMeeting of EMAG, Inst. Physics (1971) 206-209.

(v) Figures

POLL

- 8 -

•K'

V- -/;//

FT:Q-2 Ht,r,1 3 -

-L-CNCTH-

..e.:,r-.

.IS3G

. . ' . * • • .

Q00303

. - . , •

SQNOnc

<u- 90

-FRS

ys. jrrq °U I . ^ . . .PC 5. i . S.

Figure la Stereographic projection of Kossel circles for Fe with \-- 0, 1936 nm, a . 0. 28664 nm, v 100) pole at center(PA . 100); (hkl) limit, 5.

3i.FI INDICES

Q-Z.9664 B-Z.8664 C-Z.B6&q O_PHP- 90.000 BETP- 90.000 GPTTC- 30.000

.1936000000 NPMQnETERS

PB 1 . 1 . 1 .PC-1 . 0. 1 .

F i g u r e Jb - A s figure la e x c e p t P A = 111; (hkl) l i m i t , 3 .

- 10 -

POLL INDICES

L-J

NIQ - 3 . S Z 3 0 Q - 3 . S Z 9 0 C - 3 . S Z 9 0 Q L P H P - 9 0 . 0 0 0 B C T P - 9 0 . 0 0 0 C d n n Q - 3 0 . 0 0 0

U O U U X N G T H - . 1 G 5 7 B 0 0 0 0 0

pp i . o . o .P C o . i . o .

Figure 2a - As figure la but for Ni with X = 0.16578 nm, a = 0. 3529 nmPA = 100; (hkl) limit, 3.

- n -

POLE INDICES

Ui

Q - T . S 2 3 0 B""3.S290 O 3 . S 2 9 0 CLPHQ- 9 0 . 0 0 0 3 E ' Q - 9 0 . 0 0 0 G O T O - 9 0 . 0 0 0

J Q - C L E N C T H . . ! B S ? 8 0 0 0 0 0 NQMCMETrR1?

PQ I . • . I .P C - ! . D . I .

Figure 2b - As figure 2a except PAr 111; (hkl) limit, 3.

- 12 -

ATOMIC POSITIONS IN UNIT CELL

s .SCATTERING PARAMETERS

z <o St a

TITIE CARD

LATTICE PARAMETERS

1 HI 111 II H 1 II 1

j i i ? } : ! ' ! ! : • i : J : i

Figure 3 - Example of input cards to produce Figure 2b except PA110 rather than 111.

Figure 4 Kossel pattern taken with Ka X-rays corresponding toFigure lb (taken from reference (2),

- 13 -

POLE INDICES

U35IU-4.71B2 0-4.7182 C-4.71SZ HJWJ- 90.000 OTTO- 30.000 BJTB- 90.000

0000

PQ 0. 0. 1 .PC 0. I . 0.

Figure 5 - Central region of the Stereographic Projection of Kosselcircles for V3Si using Cr Ka radiation; limit, 3, after thework of Frazer and Arrhenius (4). Kossel lines not drawnfor planes with a structure factor s 20.

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Figure 6a - Measurements required to determine orientation of Kosselpattern. Each vector QPn where n = 1 to 4 represents theline of intersection of two Kossel cones. The points P n aremultiple intersection points (of Kossel conies) on the photo-graphic film (taken from reference (1).

Figure 6b - Kossel pattern from austenite phase of an Fe-31% Ni-0. 66% Palloy with a = 0. 3584 nm, \ = 0. 1937 nm. Dashed lines passthrough intersection points and correspond to the trace of lowindex planes (taken from reference (1).

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Q-3.S810 D-3.S810 C-3.5810 a_PHa-90.DOO DCTR- 90.000 GOrrn- 90.000 DU-0. 1 . D. PHI1ZQ- 90.D0O PHI I B - 95.ODDD l 0 . 0 . - 1 .

Figure 7a - Stereographic projection of the crystal depicted in Figure 6b.

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Figure 7b Kossel pattern in same crystal orientation as given inFigure 7a, compare with Figure 6b.

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yr (hkt). POLE

= (90 - S B |)

KOSSEL CONIC (hkj)

LENS INTERSECTION

KOSSEL CONIC (tiki!).

Figure 8a - Lens shape intersection of two Kossel conies showing theangular separation d .

Figure 8b - Curve of a vs d where d,, is the displacement value, a ann _ n n r * n

assumed lattice parameter; diri is the measured value and a r

the resultant lattice parameter. Taken from the work ofPhilibert et al (3).

- l i

III. FELECTKl) AREA CHANNELLING PATTERNS (SACP)

(i) Introduction

Computer programs written to produce simulated Kosselpatterns are normally capable of producing SACP by changing the magnitudeof the wavelength; typically, 0. 1 nm for X-rays and 0. 01 nni for SACP(i.e. electrons are accelerated through 20 keV). Again, indexing ofSACP is accomplished by comparison of the experimental and computedpatterns. As thu wavelength decreases, however, bands of intensityvariation often appear between the two separating K-lines making com-parisons more difficult than in the Kossel case, see for example, Fig. 1which is a 25 keV SACP from Si. For this reason, experimentally recordedSACP maps over large angular ranges have been assembled andoccasionally published. Further, the two band edgca nwy not be presentin the same micrograph. Band widths are often determined by changingthe electron accelerating voltage aid measuring the line shift (5).

(ii) Indexing by Comparison

Figure 2 was produced by the program of Ploc and Barnett (1)and many other examples exist in the literature. In practice, channellinglines (lines on the SACP) are relatively straight. Pirouz and Boswarva(2, 3) have developed a program for triclinic crystals with the correctgeometrical simulation. Here the lines are straight and more similarto the SACP. Input consists of crystal parameters (a, b, c, a, (3, y), atompositions in the unit cell, Miller indices of the surface being investigated,magnification, voltage, screen size and the maximum (hs + k2 + I3)allowed. It is also possible to interchange the Miller indices of thesurface plane with the direction indices of the incident electron beam.Unlike the program of Ploc and Barnett (1) this program would appear to belimited to rational surface planes.

(iii) Analytical Indexing

Newbury and Joy (4) developed an analytical technique forindexing SACP. Their program does not appear to be available and hasbeen applied only to the cubic system.

Newbury and Joy (4) take channelling patterns at two differentvoltages and measure the shift of three lines which form a triangle in the

SACP. By using the d.fferentiated Bragg lav/ and calibrating the SEM,CRT (to measure in radians or degrees) , values of (h2 + k2 + I2) wereassigned to the lines. Using these data in conjunction v/ith SACP bandseparations (interplanar separations) the lines were indexed. Crystalorientation was established by measuring the angular distance between theintersection point of the optic axis with the crystal surface and the threelines of an SACP triangle. The co-ordinates of this point in a s tereogramare given.

Once the channelling pattern is indexed it should be possible touse the generalized input feature of Ploc and Barnett 's (1) program tocalculate PA, the normal to the crystal surface. For instance, theangular deviation of a plane from the symmetrical position can be measuredand used as input data. This aspect will receive further consideration insection IV iii.

References

a) Computer programs

1) R. A. Ploc and P. C. Barnett; Computer-Generated StereographicProjections, J. Appl. Cryst. 5 (1972) 135-136.

2) P. Pirouz and I. M Boswarva; Computer Generation of Pseudo-KikuchiPa t te rns , Scanning Electron Microscopy: Systems and Applications,The Institute of Physics , London and Bristol (1973) 324-327.

3) P. Pirouz and I. M. Boswarva; Computer Generation of Pseudo-Kikuchi

Pat terns for Triclinic Crys ta ls , Phys. Stat. Sol. (a) 26 (1974) 407-415.

4) D. E. Newbury and D. C, Joy; A Computer Technique for the Analysisof Electron Channelling P? t te rns , P roc . 25th Anniversary Meeting ofEMAG, Inst. Physics (1971) 306-30^.

b) Related publications

5) E. M. Schulson; Some Considerations of Selected Area Channelling inthe Scanning Electron Microscope, J. Sci. Instr . Z (1969) 361-364.

(v) Figures

- 20 -

Figure 1 - Typical SACP from a Si single crystal taken with 25 keVprimary electrons. Note that full band widths are notpresent on the micrograph.

1 -

POi E I INDICES

H-Z.SG64 B-Z.B664 C-Z.B664 PLPH3- 90.000 SETO- 90.0D0 O * T P - 90.D0ORCCO_CRPTING UOLTHGC- 20COO.D CoIHJQ-OCTH- .00Hbafel63 NCWCTETERS

PO I . D. 0 .PC 0 . i . 0 .

Figure 2 - Stereographic SACP for Fe (a - 0.28664 nm) with PA = 100;limit, 3 and for 20 keV electrons.

IV. KIKUCHI PATTERNS

(i) Introduction

Kossel, SACP and Kikuchi patterns arc geometricallysimilar, the significant difference being wavelength which is typically anorder of magnitude less in each instance. This difference causes the K-cone semi apex-angle to be larger (90° -Bragg angle) and the radius ofcurvature of a K-line to be smaller (on the pattern). For Kikuchi patternsthe area between the (hki) and (hkl) lines, referred to as a band, canassume significant contrast effects, see for example the work of Thomas{(>). For this latter reason several investigators have published Kikuchi-maps which are composites of actual patterns covering large angular areasof standard stereographic projections; see, for example, references (7,8).

(ii) Indexing

Most computer programs produce stereographic Kikuchi projectionsfor the indexing of experimental patterns by comparison (see 1 to 3). Insome instances, lines for all possible h .k . l ' s are drawn (Bomback andThomas (1)) and in other instances they are eliminated (Young and Lytton(2), Ploc and Barnett (3)) based on structure-factor considerations. Youngand Lytton (2) have attempted to overcome the problem of overlappingline indices by producing tables locating line positions.

Both the programs of Young and Lytton (2) and Ploc and Barnett(3) can produce superimposed or separate pole projections. The lattercan also produce projections of crystal directions (see, Figs, lb, 2a and2b). The computed figures in this section were produced by the programof Ploc and Barnett (3). With this program it is possible to use any formof input data; two poles, two directions, a mixture of one pole and onedirection, anywhere within the stereogram and for any crystal system.The projections in Figure 1 are simply standards often repeated in theliterature. The electron accelerating voltage can be infinitely variedfrom the lower diffraction limit of A. <, 2d to any upper value. Forinstance see Figures 3a to 3d.

(iii) Orientation

No programs have been published which directly yield crystalorientation though several techniques are available, three examples being

- 23 -

the work of von Heimendahl (9), Faivre (10), and Hartley and Keown (11).In principle analysis should not be difficult if the Kikuchi pattern is treatedas a SAD spot pattern and rotations measured. For instance, theseparation between the (hkl) and (hkl) lines is proportional, to a good approx-imation, to the reciprocal la.ttice spacing for the plane giving rise to theKikuchi cone. By measuring two independent band widths and the anglebetween them, computer programs developed for SAD spot patterns cananalyze the data. The rotations from the zone axis can be measured andfed into the program of Ploc and Barnett (3) to yield a computed Kikuchipattern. The pole perpendicular to the foil surface will also be given.Following is an example,

Figure 4a is a 100 keV Kikuchi pattern from Zr3Al (chosen forconvenience rather than the simplicity of crystal symmetry). The bandsmarked with arrows were measured for linear and interangular separationsand these data used in the program SADSP (to be described in section V iii)

( Camera Constant = 0. 15302 nm cmRl = 1. 565 cm

Input R2 - 1. 160 cmSeparation = 19. 25°Zr3Al; a r b r: c = 0. 4374 nm; a = (3 = y = 90°.

(hkl) 1 -_• 024; % error in d-spacing 0.03%(hkl) 2 = 113; % error in d-spacing -0.03%% error in angular separation 0. 24%Zone axis - [ 121]

Knowing the line indices the crystal tilt was determined bymeasuring the deviation of the mid-plane of the band from the central pointof the pattern in a direction perpendicular to the band. This deviation, d ,can be converted to an angular deviation, Q (to a good approximation), fromthe relationship

tan a = d \ /rdm

where X is the wavelength, r is the Kikuchi band separation and d, the inter-planar spacing corresponding to the planes giving rise to the Kikuchi band.The angular deviations were 0. 9455° in the Sn^d direction (g being areciprocal lattice vector) and 0.3050° in the g..5 direction; i .e. the crystalwas tilted so that the normal to the plane gnk-l rose up out of the paper in animaginary plane, at right angles to the photograph, whose intersection withthe paper was parallel to g, , , . Knowing that the vector g, , . is at right anglesto the plane (hkl) the location of the (024) and (113) were easily positioned on

- 2 4 -

the stereogram (i .e. sets the variables cp^ and (£>-,, in STEREO). Thesepoles do not lie along the edge of the stereogram since the crystal wastilted. The tilt however does not affect (pnb (n _ 1,2) but only <Pnza

(positioning variable in STEREO) which can be calculated from

(0 -• c o s " 1 f c o s <p / s i n to , ]nza na ^nb

where <r> is equal to (90°-a ). The data so obtained and used to or ientthe Kikucni and stereographuc project ions were :

PI (pole #1) . 024, 0 = 88 .83° , o = 54°1 za 1 b

P2 (pole 42) = 113, o r 89 .66% o = 106.75°L. Zi 3, £ D

The edge of the photographic plate was used as the x-axis. In this casethe printed output of STEREO indicates that the pole indices or direction(both are given and are not necessarily equal in non-cubic crystal systems)normal to the foil surface was (-0. 0799. -0. 1933, -0. 0924). Figure 4b isthe computed Kikuchi pattern. This technique can be easily applied to thecalculation of misorientations between matrix and second phase particles ormisorientations between subgrains, etc. STEREO produces Kikuchipatterns as well as pole or direction stereographic projections.

(iv) Lattice Parameter and Wavelength Determination

In 1969 H<iier (12) published a technique to determine latticeparameters from Kikuchi patterns, analogous to that for Kossel patterns. Thistechnique was extended to triclinic crystals by Olsen (4) and his program isavailable upon request.

Exact triple intersections of Kikuchi lines are relatively rare andso the technique was developed for near intersections as schematicallydepicted in Figure 5. Though not physically possible, it was assumed thatan exact intersection was brought about by changing the wavelength for H,alone. The corresponding AXo was approximated from the ratio of &R3(the distance H3 was below the apex of the triangle) to R3 (band separation).If the number of near intersections was greater than the number of unknownlattice parameters, the lattice parameters were determined by a least-squares method (4). The influence of measurement errors in AR3 wereminimized by ensuring that H3 was chosen such that there was a rapid changein &R.3 with / \ \ . This occurred when all the Hn (n = 1,2,3) possessed high

- 25 -

indices. Also, care was exercised to avoid intersections where thelines were displaced due to dynamical electron interactions.

The techniques of Hciier (12) and Olsen (4) determine the ratio ofthe lattice parameter to the wavelength. To isolate one of these variables(e. g. lattice parameter) it is necessary to do two experiments. The firstwith a crystal of known lattice parameter such as Si (a = 0. 543095 ± 0. 000005nm) to determine X then secondly, with the unknown crysta l to establish thelattice parameter .

(v) Convergent Beam Diffraction

Some conventional TEM's have been modified to produce a highlyexcited (magnetically) objective lens; the so called, condenser-objective.Because of '•he large electron beam divergence, K-line patterns analogousto Kikuchi lines appear in the diffraction pattern. The indexing and use ofthese patterns follows the same procedures as for Kikuchi pat terns .

(vi) References

a) Computer programs

1) J. L. Bomback and L. E. Thomas; Generation and Application of Com-puter Drawn Kikuchi Maps, J. Appl. Cryst . 4 (1971) 356.

2) C. T. Young and J. L. Lytton; Computer Generation and Identificationof Kikuchi Projections, J. Appl. Phys. 43 #4 (1972) 1408-1417.

3) R. A. Ploc and P. C. Barnett; Computer-Generated StereographicProjections, J. Appl. Cryst. 5 (1972) 135-136.

4) A. Olsen; Lattice Paramete r Determination using Kikuchi-LineIntersections: Application to Olivine and Feldspar , J. Appl. Cryst .9 (1976) 9-13.

5) P . P i rouz and I. M. Boswarva; Computer Generat ion of Pseudo-Kikuchi P a t t e r n s for Tr ic l in ic C r y s t a l s , P h y s . Stat. Sol. (a) 2b, (1974)407-415.

- 26 -

b) Related publications

6) L. E. Thomas; Kikuchi Patterns in High Voltage Electron Microscopy,Phil. Mag., 46, .26(1972) 1447-1465.

7) E. Levine, W. L. Bell and G. Thomas; Further Applications of KikuchiDiffraction Patterns; Kikuchi Maps, J. Appl. Phys. _3_7 £ 5 (1966) 2141-2148.

8) D. S. Gelles; The Hexagonal Close-Packed Kikuchi Map, Acta Cryst.A28 (1972) 471.

9) M. von Heimendahl; Precise Orientation Determination by ElectronDiffraction Single-Pole Kikuchi Pat terns , Phys. Stat. Sol. (a) _5 (1971)137-146.

10) P. Faivre; A Method for Calculating Misorientations Between Subgrainsfrom Kikuchi Line Pat terns , J. Appl. Cryst. 8 (1975) 356-360.

11) A. J. Hartley and S. R. Keown; The Determination of MisorientationsAcross Subgrain Boundaries from 'Split1 Kikuchi Lines, Micron, 3(1972) 374-382.

12) R. H0ier; A Method to Determine the Ratio Between Lattice Parameterand ELectron Wavelength from Kikuchi Line Intersections, Acta Cryst.A25 (1969) 516-518.

13) M. Lenc; The Equation of Kikuchi Lines onthe Basis of the KikuchiGeometrical Model, Acta Cryst. 23 (1967) 710-712.

14) G. J. C. Carpenter and J. F . Watters; Electron Diffraction Patterns forHCP a-Zirconium, Atomic Energy of Canada Limited report AECL-4355(1972).

15) W. Wu, L . - J . Chen, J. Washburn and G. Thomas; Indexing ofDiffraction Planes Using the Kikuchi Pattern, J. Appl. Phys. 45, #2(1974) 563-566. '

16) R. Bonnet and F, Durand; Precise Determination of the RelativeOrientation of Two Crystals from the Analysis of Two Kikuchi Pat terns ,Phys. Stat. Sol. (a) 27 (1975) 543-549.

- 27 -

(viU Figures

Figure la - Kikuchi stereographic projection for iron with a [ 001] zoneaxis (PA), accelerating voltage - 100,000 volts, limit ~ 3.

3-4.S93? 0-1.S93? C-2.958) O.PHQ- 9Q.00DnCCCLCRO'I-G UOLTPSC- 100000.0 PJ^CUELE^GTM- .0037D14338

90.000 GOrrC- 90.000 ran. o. i .<=C 0 . 1 . 0 .

F i g u r e 1b - As F i g u r e l a but for TiO,, with a [001] zone a x i s , l imi t _ 5.

^ J D . L. \.PD ! . D. C.

~igure lc - As Figure lb but for Si; limit = 5.

- 30 -

T] 02P-1.5937 t-«.S337 O7.95S1 90.000 SCTP- 90.000 GOTO- S0.0D0 PO 0. 17. 1.

PC D. 1. 0.

Figure 2a - Stereographic projection of poles for TiO3 in the orientationgiven in Figure 1b.

.. 31 -

LJlRrCl JOIN INDITES

TI 020-4.S33.-' (3-1.S337 C-2.9581 PLPK- 90.000 BETP- 90.000 GOmR- 90.000 PB 0 . 0 . 1 .

PC 0 . 1 . 0 .

Figure 2b - As Figure 2a but for directions.

- 32 -

Figure 3a - Stereographic Kikuchi, plot for Zr for an electron acceleratingvoltage of 10 eV, note the exclusion of lines due to diffractionlimitation; limit = 5.

Figure 3b - As Figure 3a but for 10,000 eV.

- 33 -

Figure 3c - As Figure 3a but for 100,000 eV.

Figure 3d - As Figure 3a but for 10,000. 000 eV.decreasing Bragg angle.

Note the effect of

- 34 -

Figure 4a - Kikuchi pattern from Zr3Al (cubic, a = 0. 4374 nm),accelerating voltage = 100 keV, Cracks in micrograph arefrom damage to the plate during final printing. The darkarea in the upper right is from dirt in the TEM. The patterncenter is marked with a dot.

Figure 4b - Calculated Kikuchi pattern corresponding to 4a, Some of themore prominent lines are dotted. The pattern center ismarked with a dot.

- 35 -

Figure 5 - Schematic drawing of a near triple Kikuchi line intersection.The heavy lines refer to the position of the Kikuchi lines at awavelength of \ and the thin lines to the position when \ = \ +A\ where AX > 0. Taken from the work of Htfier (12),

- 36 -

SELECTED AREA ELECTRON DIFFRACTION (SAD)

(i) Introduction

The computer analysis of SAD spot-patterns has received by farmore attention in the literature than the previously discussed K-patterns.Thij is an immense field and cannot be discussed fully in this presentation.The use of programs to solve identity relationships between matrix andsecond phase particles, epitaxy, domain structures, twinning, etc. , mustbe left to the individual investigator. Generally, the published computerprograms have been dedicated to the geometrical analysis of spot arrays w;thlittle, or no attention being given to double diffraction, superlatticereflections, etc.

(ii) SAP-Ring Patterns

A computer program entitled RINGS has been developed at thislaboratory (1) to index diffraction rings obtained from polycrystallinematerials. For a given camera constant and ring radii, RINGS attemptsto select the best solution set of Miller indices for a particular crystalsystem (triclinic permissible), i.e. the set of crystal planes with the mostconsistent relative errors in the d-spacing. A detailed description of theanalysis can be obtained from the author.

Figure la is an example of the output. The program was runfor the nine ring radii given. All the possible d-spacing solutions arelisted and the BEST SOLUTION is isolated. Next the camera constant isallowed to vary and a new set of rings selected (in this instance, the same).The same input data as in Figure lb were used for aZrO2 (monoclinic,a = 0. 5145, b = 0. 52075, c = 0. 53107 nm, (3 = 99.2333°) and the finalanalysis shown in Figure lb. The results of Figure la suggested cubicZrO3 were it not for ring 4 (no solution). Reference to Figure 1bdemonstrated that many of the diffraction rings could have originated fromthe cubic and/or monoclinic phase and that, using the same camera constant(XL =•= 48), ring 4 is the (111) aZrO2. Clearly, the oxide consisted of amixture of cubic and monoclinic ZrO3.

RINGS can be used to process one set of ring radii for manymaterials or alternatively, one material for many sets of ring radii. Inthe former case, analysis of unknown materials becomes a possibility.

- 37 -

(in) SAP-Spot Patterns

a) Comparison

Spot patterns (SAD-SP) have been indexed by comparison withhand-drawn simulated patterns (15-17) or with computer simulatedreciprocal lattice (RL) sections (2-7). The projection can bo either planarRL sections or take into account the geometrical relationships between theRL and Ewald sphere (8,9). Normally, the simulated sections give theratio of two radii as well as the angle between the two vectors from the patternorigin to each of the reflections.

The program PATTERN (8) allows simulation of any diffractionpattern geometry for the triclinic crystal system. Also, the program cansuperimpose patterns to facilitate the study of epitaxy, twinning, etc. aswell as the ability to tilt pattern composites. The tilting of the electronbeam or crystal eliminates ambiguity (18). The reciprocal lattice spike canalso be tilted or extended. This generalized program makes possible thesophisticated analysis of any diffraction pattern(s).

Figures 2a to 2d show how complicated RL spike shapes can beinvestigated. In this instance, it was assumed that the spike had threecomponents - 0, ± 45° to the vertical and in a plane perpendicular to thepaper but containing a trace parallel to the bottom edge of the page. If thespike length was allowed to increase, second-order Laue reflections wouldshow in their geometrically correct positions. Figure 2c is a partialprinted output corresponding to Figure 2a. The effect of the Ewald spherecurvature can be seen from Figure 3d where the crystal depicted in Figure2b has been ti'.ted 41" along the x-axis (parallel to bottom edge of the paper).

Figures 3a and 3b are further examples of how PATTERN can beused. Showu in Figure 3a is the basic growth model of thermally formedaZrO2 on Zr. Here many individual superimposed oxide crystallitepatterns are collectively tilted to produce Figure 3b. If the experimentalresult matches the predicted result the model can be assumed correct.If there is partial disagreement, then each fundamental crystalliteorientation can be treated separately to test its validity.

b) Analytic

Two programs exist for the analytic indexing of SAD-SP,XIDENT (10) and SADSP (1). The latter program can be applied to patternsfrom any crystal system for any number of combinations of materials andsets of \L, Rl, R2, <pl2 where Rl and R2 are the distances from the pattern

- 38 -

origin to the reflections 1 and 2, (̂ 12 is the interangular separation. Theprogram has proved useful in separating multiple superimposed spot patternsand in determining lattice parameters (i.e. , using an accurate XL and severalsets of lattice parameters). Also, using one set of lattice parameters manypatterns can be consecutively analyzed or several pairs of spot^ considered.Neither XIDENT nor SADSP will handle complicated double diffractioneffects. The investigator's basic knowledge of diffraction must be used.

Figures 4a and 4b are examples from SADSP. The material isa proposed phase of zirconium hydride with orthorhombic lattice parameters.The camera constant is known to be 2. 628 (Au deposition). In Figure 4athe diffracting material has a [ 141] type zone axis as opposed to a [ 010] .This is evident because when the camera constant is allowed to vary to yielda minimum error solution, the change is less than 0. 05%, well within anymeasurement error. Alternatively, if the [010] zone axis was selectedthe 4. 7% error in D2 (interplanar spacing for the second reflection) must becontended with. In Figure 4b, the proposed camera constant is sufficientlyclose to the input value that the solution of the pattern must be based onother factors. For instance, the presence of the (300) reflection necessi-tates a (100) and/or (200). If these reflections are not present then thematerial probably has its [ 011] zone axis parallel to the electron beam. Ifneither solution is acceptable then the proposed structure for ZrH_ is likelyin error.

Though such an analytic analysis of SAD-SP yields rapid and usefulsolutions further checks on the data are possible. For example, using thedata from SADSP, a simulated SAD pattern can be produced; this is thentilted and the predicted pattern compared to the experimental result.Ambiguity problems (18) in indexing can be solved in this manner.

Structure Factors and Extinction Distances

The calculation of structure factors, extinction distances (Debye-Vv aller corrected) and d-spacings are almost trivial. A program labelledSFAC is available from this laboratory, an example of the output is shownin Figure 5a, A second program, DSORT, calculates and tabulatesd-spacings in descending order of d and ascending h,k,l (i.e. (666) ^ (hkl)< (666)), see Figure 5b.

- 39 -

(v) References

a) Computer programs

1) R. A. Ploc; Diffraction Analysis by Computer, Proc . Thirtieth AnnualEMSA Meeting, Los Angeles (1972) 632-633.

2) M. Booth, M. Gittos and P. Wilkes; A General P rogram for InterpretingElectron Diffraction Pa t te rns , Metallurgical Transactions 5 (1974)775-776.

3) P . Rao and R. P. Goehner; Phase Identification in the Cobalt-SamariumSystem with Computer-Simulated Electron Diffraction Pa t te rns , J. Appl.Cryst. _7 (1974) 482-488.

4) J. D. Meakin; Computer Generation and Automatic Plotting of ElectronDiffraction Pa t te rns , Trans . Met. Soc. AIME 2 45 (1969) 170-171.

5) C. M. Maucione, D. L. Formenti and L,. A. Heldt; Computer Generationand Automatic Plotting of Perfect and Twinned Electron DiffractionPat terns for Cubic Crystal Structures , Metallurgical T r a n s a c t i o n s ^(1971) 2289.

6) A. Messerschmidt; Computer Simulation of the Geometry of TED andQuasi-Laue RHEED Pat te rns , Phys. Stat. Sol. (a) 43 (1974) 43-49.

7) G. H. Olsen and W. A. Jesse r ; Computer-Simulated Electron DiffractionPa t te rns , Mater. Sci. Eng. 5 (1969/70) 135-141.

8) R. A. Ploc and G. H. Keech; Transmission Electron Diffraction Analysisby Computer, Proc . Twenty-Eighth Annual EMSA Meeting, Houston (1970)40-41.

9) R. A. Ploc and G. H. Keech; Computer Assisted Analysis of TransmissionElectron Diffraction, Res. des Communications, Septieme CongresInternational, Grenoble II (1970) 203-204.

10) B .L . Rhoades; Indexing of Electron Diffraction Spot Pa t te rns byComputer, Micron _6 (1 975) 123-127.

11) R. E. Villagrana and P . H. White; Computer Generated ElectronDiffraction Pa t te rns , Septieme Congres International de Mi.croscopieElectronique, Grenoble _2_ (1970) 205-206.

12) R. A. Ploc and G. H. Keech; Computer-Generated Transmission ElectronDiffraction Pa t te rns , J. Appl. Cryst . 5 (1972) 244-247.

- 40 -

13) P. Wilkes; Complete Indexing of Electron Diffraction Patterns by-Computer, J. Mat. Science _9 (1974) 517-518,

14) P. Rao and R. P. Goehner; Computer-Aided Indexing and Simulation ofTransmission Electron Diffraction Patterns, Proc. Thirty-ThirdAnnual EMSA Meeting, Las Vegas (1975) 222-223.

b) Related publications

15) G. J. C. Carpenter and J. F. Watters; Electron Diffraction Patternsfor HCP a-Zirconium, Atomic Energy of Canada Limited reportAECL-4355 (1972).

16) A, J. Bedford; On Indexing Electron Diffraction Patterns for HexagonalZirconium Alloys, J. Less Common Metals 22 (1970) 141-148.

17) K. W. Andrews, D. J. Dyson and S. R. Keown; Interpretation of ElectronDiffraction Patterns, Adam Hiiger Ltd. , London, (1967).

18) W. Griem and P. Schwaab; Ambiguity Problems in the Indexing ofElectron Diffraction Patterns, Practical Metallography 11 (1974) 336-348.

19) S. R. Keown; The Simulation of Electron Diffraction Patterns ofMetallurgical Structures by Optical Transforms, Proc. Twenty-FifthAnniversary Meeting of EMAG, Inst. Physics (1971) 234-237.

20) D. F. Lupton and D. H. Warrington; Computer-Aided Determination ofOrientation Relationships from Pearlite Electron Diffraction Patterns,Metallography^ (1972) 325-331.

21) M. A. Meyers and R. N. Orava; A Geometrical Method for theDetermination and Indexing of Electron Diffraction Patterns, Metallo-graphy 7 (1974) 231-240.

(vi) Figures

- 41 -

ZRO?{CUBIC)? .o7o

9 0 , 0 0 0 9 0 , 0 0 05.070GAMMAS 90,000

•—CAMERA CONSTAMTs/Jft.660 PLATE. NUMBfcRgi-5/ARING R A D I I

1 9 , 5 0 0 0 0

3 1 6 ^ 5 5 0 0 01 7 , 2 0 0 0 0

3FACIN(,S

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-3,786

1.U6J5H

1,195011,229661,26750

-5,0323,385

L<4 1 0« 0 0

2,053,994

Figure la - Analysis by RINGS of nine diffraction rings for cubic ZrO2.

- 42 -

***** PLANE 1 1 1 FOR RING a DISCARDED ••• ANALYSIS 1 *****

PLATE NUMHER»3»373BEST SOLUTION SET FO* T H E GIVEN

1 1 0 0 5,070002 1 1 0 3,38503

"3 m TTfTTTT4 NO SOLUTIONS5 2 0 0 2,535006 2 2 0 ~7 3 1 18 2 2 2

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1,79757"1,528661,46358

.345

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***** PtANt 1 1 1 FOR RING U DISCARDEO --• ANALYSIS 2 *****

PLATE NUMBERa'3-373BEST SOLUTION SET FOR A C A M E R A CONSTANT = <I8.O«7RING H K L 0 SPACING PERCENT ERROR

23a5b

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3.585572,90312

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7

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1,478361,25612

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PERCENT DIFFERENCE IN CAMERA CONSTANT* ,097

Figure la - continued

- 43 -

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P E R C E N T OlFf-EkFMCE IN CONSTANTS 1,156

Figure lb - Same analysis as in Figure la but for aZrO.

- 44 -

03& D « osr.

323 D3J 041 053

Figure 2a - Simulated [ 1001 zone axis pattern in aZrOa; no tilt, RLspike perpendicular to plane of the paper.

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Z fi 0 2 L*TTICE C O N S T A N T S ii «; t 1 « s 0 0 P = S , ? 0 7 S 0 C = 5,31070 *L P H*sgo fil0O0O

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b 0b 1 «h 2b 3 •

-6 0

-6 6- 5 . 6 •- 5 - 5- 5 - ( I •- 5 - 3- 5 - 2 •- 5 - 1- 5 0- 5 1• 5 2 •• 5 1-

-

••

-

) 0 *, S

- 6. 5- a «

a - 3 •l - 2

- 1 •4 0 •

1 •

Figure 2c - Pa r t of the printed output corresponding to Figure 2a. The quantities FG2, XI, RX(N),R.Y(N) are the squared structure factors, the extinction distance and the x/y co-ordinatesof the reflection (HN KN LN) in the plane of the paper. A fuller description of the input/output variables can be found in reference (12).

- 46 -

ZROZ H'OS'TS M=LOT- 1

QXIS 1 0 D HX- 0 . 0 0 0 KX- S . 2 0 8 LX- 0 . 0 0 0 ROC LENGTH- . 0 7 3 9

B E T q p - 0 J ) G O m G P - 0 . 0 I - 0 . 0 0 0 P H I - 0 . 0 0 0 PH1X- 0 . 0 0 0 OELTQ- - 4 1 . 0 0 0

•464

3G3

262

161

060X X X

16!

262

363

464

6S 4 31 I'l'l • 1—•

4S4

3S3

2S2a K •

1S1

050

151

252

353

21

545

343

242MM

141

040MK

141ran

242

343

444

545*

535

434

333

232

131

030

131

232

333

434

S3S

525

424

323XXK

222

w

121

020

a

121

m

222

323

424

S2SK

1

sis

414

3l3

212

1 1 1m

0I0

Ml

212

313

414

sis•

SOS

404

303

202

m

101•

000

101K

202

303

404

SOS

SIS

414

313

212

m

111•

010

111•

212

313

414

SIS

S2S

424

323

222

121•

020

B

121

222

323

424

S2S

S3S

434

333

232

131

030

131

232

333

434

S3S*

S4S

444

343

2-i2UK

141

040

141

242

343

444

S4S

*

454

3S3

252• X •

151

osoXXX

ISlXXX

2S2

353

4=4

464

363

262x x x

161

050

is:

262

363

464

Figure 2d - The crystal depicted in Figure 2b ([ 100] zone axis) has beentilted 41° about an axis passing from the (000) to (060) points.Note the curvature in the rows due to the relative orientationof the R.L spikes and Ewald sphere.

- 47 -

Z ZlS0Z'7l HPLOT- 1

fl -1 1Z EFITPXIPL GROUTH nOOEL OT PLFMP ZKOi DM iRCDODI). THIS DirrRPCTIDM

PSTTEKIN SHOULD RESULT U*HCM THC inciDCNT ELECTRON OERM IS PBROLI.EL TO THE -nz

REFLECTIDM5 UITM SaUPRTJ STRUCTURE TPCTDRS <«O HPME DEEM SCT EOUPL TO iERO.

Figure 3a - Simulated SAD-SP from a aZrOg growth model containingmany individual oxide orientations.

- 48 -

«_PHP ZROZ Z3.'OZ'71 MPLOT- Z

THIS DIFfRPCTIOM PPTTERN SHOULD RESULT IF THE PREVJIOUSLV DEPICTED DXIOE

nui. coMToiNiric npnr DRIENTPTIDNS, RECEIUES a RICH H(*OED ROTOTIOM or 10

PODUT P g E R T I C P L Q X I S f I N TMC P L P T C OF THE O R I C I ^ W L - I 1 Z P P T T E R T " ] .

• • „

Figure 3b - Same pattern as in Figure 3a but tilted 10° about an axis inthe plane of the paper and parallel to the vertical.

? R H ; SAUII

A= u,9700

=iiiBb Cb

ALPHAS 90,0000RF.TA = 9 1 , 0 0 0 0

(JAMMAe 9 0 , 0 0 0 0O i l 1 . 6 8 7 30 ? 3 1 . 5 8 3 8

L i t 2,b2l<0

THET12» 6 1 , 0 0 0 0R l = 1 , 5 5 7 5R 2 » 1 , 9 0 0 0

H K L RANGE • 6T0 6

H1ML10002

2

22D0ft

222iaa

-i

i-i-i-l

-i"inI

L2333333

11

111

OC1.6576

,657b,6576.6780

,6780,67«0

OC2.3896

, 3196

• 38^6,U506,3896

616061605960

THE T A,9080

,6300,9080

. PSC1, Muug

,8580

nI11

/D-^ !. 79SU, 79SU, 795u,^)5S0.5550.5550

0/"-D2-,UbOu

-il.U 30 1-jubOu• , u b O a

•a,6509»,ybuy

U/C-T H£ f fl

- 1,ub«^,bO66

- 1 ,Mft«SI i? i^7

8»3£0 ON «

0 2

«(31Irrvtr 48S0LUT6 PMJKJL2 OCI

-1 »\ 3 1,6760

CE'-T*( ' f t » » n H , Tht REST S O I I IDCS T H t T t O/r)»Dl

l . l « 9 6 6 0 , 8 5 8 0 , 5 5 5 0

O'I IS AS0 / O - L i n /O-TMtT»..UftOo ,23<>7

" ' 7;iii

u = c- = ?A S ?a : 4

A = ZA S V

Z J = 4 1 - u - l V

A L T E R N A T I V E 1 . V , I G V O B J N G T H f c M A G N l T U O f n F T H f S N G U L A H t c ^ n o , T M t H t S T S O L U T I O N F C L I O - S I F T H F C IH ) K 1 L 1 H J K 2 L J D C 1 n C ? T M f T A O/O-ni (l ' ll"0a i i / 5 - I t ' f . l a (•!

2 0 2 - 1 -1 J 1.67H0 l ,3f l»6 60,1S8il - , "j 101 ,50u9 1 , 01*50 : I \ - u - l \

T H E > I ; « I M I I M T O T A L F 9 » 0 9 I S G W F N » » T U F f n i L O x t ^ t i I F T H E C » M E O » C O N S T A N T = 2 , h ^ b BH I X I L I rtj«ju2 oci ncj TKETA D/n.rn n/o-1)£ o /o - r« t i«

2 0 2 - I • ! 5 1 . 6 7 H 0 1 ,3«<>6 6 0 , 1 S 0 0 - , 5 I " I , 5 0 " < ( , ? 3 J 7

AHS t ^ ^ ' ^ f l / l )

F i g u r e 4a - SADSP output for a p roposed o r t h o r h o m b i c Z rH g s t r u c t u r e for the given inputcondi t ions; i . e . Rl = 1.5575, R2 r. 1.9, o*2 - 61 and XL r, 2 . 6 2 8 .

Z3H? SAUU P L » T t

As 11,9700R2 u , 8 u 0 0r « U.5S00

9 0 , 0 0 0 0RETAJ 9(1 ,0000

GA««Ai 9 0 . " 0 0 00 | i | . 0 8 1 502s 2.«561

11 = 2,6280

T(^s 90.0000B| = s.Sb100 2 = 1,0700

6TC 6

00000

2?

3

1«

f??

3

(<0Q

1L1

a0

20Q

?-22-22

00

o

»2H2L20 0a oo o0 00 0

.-> 02 0

-J 0

DC!.6576

, h5761,6133.6133

.6760

.67*11, &5(.7,6567

0C2

2.aflbC2.UA50

2,ufl502.U850

2,»2002.y?002.U2. or

90

•JO90

90

90

90909090

T«f TA.0000,C0OO.COCO."000,0000

.0000,0000,0000,0000

1!11

"

11

, 567P,567?.3511,3MU

'3295.3295,6219.6219

• 1-1- y-1-1

,111

/r-r,2,16"0, 16UC.11. Ill1 1 6«J0

• 16J0

!«907,U9O7,U9O7,U9O7

°'>'p0Co"

-,00 000,000c-, ̂ D c r

, 0 0 0 C-,CC!uO

0,0000-,oocr

/» = i :-:-!?* = .• :-) iIA - t r 1 - J

/» =

/• :

1A -

H -l * =/ * -

--' 1

1 t-1

-1 '. 1? '-11 ' 1

H1K1L15 0 - i

Afinirlv£ iwson.Tf IS »SOC 1

1 . 6 7 8 0P C ? T H F T A

2 . U 2 O O 9 0 , 0 0 0 0 , 5 2 9 5I V ! •-11Jl , u 9 0 7

O/S-'»fl«0,"00r

ALTERNtTlvELY. IGNflHIHG T H E M A G M I T U P E OF Th^ A\GUL*B tBHQP, T>-t Bf ST SOLuTlus

H|K1L1J 0 0

MJKJL20 • ? 0

OC 11 . 6 5 6 7

nC22.U200

Tnf T A90,0000

0/0-0!

-,0606

0/0-1)2, 06U6

P/L-Tt-f- TA

D , 0 0 0 (!

T M E M I K I M U M T O T A I E R B O B I S G 1 V E > . » v t i f F 0 L L O 1 N S I F T H J C A " t » A C O N S T A N T » 2.sH 1 K 1 L 1

3 0 0D C 1

] . 6 5 1 - 7O C ?

2,1200T»ETA

90,0000o/n-ci- , 0 6 U 6

0 / 0 - T " l T A0 , 0 0 0 0

Figure 4b - As Figure 4a but for a different set of input parameters.

- 51 -

»a 5 . 1 U 5 0 B « 5 . P 0 7 5 C s 5 . 1 1 0 7» L P H * z 9 0 . 0 0 0 0 R F T 1 = 0 9 , 2 3 3 3 G A M M A : 9

A T O M I C S C A T T E R I N G F A C T O R S i » t E S T I M A T E D ( P A H A B C L t C ) FOff « E f L t C T I O V S « I T « * O - S OF L F S S T x i M

VOLTAGE» lOOOOfl.EVDEB*F. H i i LE» F

I j 7 ï » 0 »151'6O

z.T91I0. P ' l l O.70890

AT0M»O NX.06900.93100.911(10,06900,US!O0.5U900.5U900.açioo

"FFLFCTlnN

UMBER OF »Tn"S= 8Y.3U200.fcSBOO,((u?(10.15800.75*00.JU200.25800.7UJ00

D-SPACTWG STUUCTOBF

Z,3«500.(.S500,15500.OU500,u7900.S2100.nilOO,97900

FACTO DÎSTAWCt (C) STRUCTuBf0-» COBPtCTfO

O I S T « M C f t r - )

-u-<t

-u• <!-a 3

a

,«9 ' i ia9,nu7tflu.T«0HO

,8926372,61690»

.00U955

?7U8,?6OlSSuT.'esU55J.965

?Htu828,5lO

a,O2Ji7O.8252U7

Î .J986651.2J58,309

2668770.51»

•1 «5 1 1/11IH 7,817663- t -Ï ? ,999?90 7,866357• " «5 3 -9SKS77 5.36801S'1 -3 " .863320 3.Ï76063

152U.5261515,073?2?0,157}529,97u

7.371J9J7,3891015,0003613.10U503

1616.8221612,9502S83,39S3838,750

-« -2 1 1,15PO«6 2.21910a»« -2 2 1,1O»,312 «.118U7»'" 'Z 3 t ^ j s s 1 ; ? ^,o^53^l-« -2 " .9295119 3,25137î

5370,9102893,9055913,«033666,591

2.1166793,91Juu61,8990503,023566

5630,80630U5.5216275,87939U1.A53

Figure 5a - Example of part of the output from SFAC.

- 52 -

ALPHA ZR02

4 = 5 . 1 V 5 : 8 = 5 . 2 0 7 5 C= 5 . 3 1 0 7 ALPHA=

D

5 .2V195 . 2 0 7 5CO Z.8JL

~3.9~8 053.69VV

H K L

0 00 11 0 01 0 - 1O i l

1/0

. 1933

. 1920

. 1969

. 2512

. 2707

H <

-6 1- 6 1- 6 1- 6 1-6 1

BETA = 9 9 . 2 3 3 3

0

. 8 V 61. 8 3 5 2. 80V9, 7 6 1 2. 71Q5

1/0

1.18191.197V1.2V271.3137l.VO 75

= 9 0 . 1 0 0 0

3.69VV3.63573 ,63573.38fcO3.162V

0 1-11 1 01-1 I1 C 11 1-1

. 2707

.2750

.2753

. 2953

. 3162

-6 1 6l6_2__l_-6 2 2•6 2 3-6 2 V

. 6581

.80V7

. 7 776

. 7379

1.51=51.22781.2V271.28611.3551

3 . 1 6 2 V - i l l 31622 . 8 3 8 72 . 8 3 8 72 . 6 2 0 92 . 6 0 3 7

1 1 11-1 10 0 20 2 0

, 3 5 2 3, 3 5 2 3, 3815,38VL

-62 5_- 6~ Y ~6-6 3 1-6 3 2-6 3 3

.691V. 6 V 29. 7688. 7606. 7375

1.VV62Y. 55 5 51.30071.31V81,3559

2 . 5 3 9 22.V9812 .VW22.3V112.3V11

2 c :1 0 - 22 u - 10 1 21 1-2

, 3938, V0D3

, V271, V271

-6•6

•6 3

3 V3 5

6-6 V•h V

. 70 35

. 6629

. 6 1 9 7

. 7161,709V

1.V2151.50861.61371.39«V1.V096

2.33192.3319

2.317J2.2823

a 2 I0 2 - 11 2 01-2 C2 1 0

. V288

. V288

. V316

. V315

. V382

•6 v- 6 V- 6 <f

-6 v•6 5

.69G6

.662V

.6282

. 5 9 1 1

.6620

1.VV791.50961.59191.69181.51C6

2 . 2 « 2 32.252V2.252V2 .2126

2-1 0- 1 1 21 i - 22 1-1

2 . 2 1 2 6 -2 1 1

. V332

. VVVO

. VVVJ

.V520

.V520

-6 5- 6 •>

-6 5•6 5-6 5 6

. 6567• 6V17.6189

.5595

1.5227

1.6158__1.69291.78 72

2.19022.179U2.1790

1 0 21 2 - 1

- 1 2 12 . 1 5 3 62 . 0 6 V J

2 0 11 2 1

, V556. V589. V589. V6V3. V8V5

- 6 6 1- 6 o 2- 6 6 3-6 b-6 6

, 6 i c e.60 59!_L9J«JL,5758.5529

1.63SV1.65061.663V1.7368

Figure 5b - Example of part of the output from DSORT.

- 53 -

VI. DIFFRACTION PROGRAMS WRITTEN FOR A HEWLETT-PACKARD,9810A DESK TOP CALCULATOR

(i) Introduction

Often a great deal of pattern analysis can be accomplished withthe use of simple desk-top, programmable calculators. Several suchprograms written for a Hewlett-Packard, desk top calculator (Model 98! OA,111 storage locations, 2036 program steps, printer, Mathematics andPrinter Alpha ROMS) are listed in the following Appendix. The first pageof each listing is an example of the program print-out. The programswere not written to minimize the number of programming steps as the timerequired to optimize operation was not considered justifiable. Allprograms listed have been proven many times over and have been writtenfor the triclinic crystal system.

(ii) SAD-SP

This program searches the (hkl) range for d-spacings corres-ponding to the first reflection. When a possibility is located the range isagain scanned for (hkl) 2, the theoretical interanguiar reparation iscalculated and compared to input. If acceptable (±2°), the results areprinted and the search continues for further solutions. No attempt is madeto eliminate symmetrically equivalent solutions.

Page 57 of Appendix A shows the printer output from the program.Entered are the lattice parameters, the (hkl) limit, the camera constant(consistent with the units of the reflection distances) and the interanguiarseparation (in degrees). The calculations proceed and solutions withconcomitant errors (between input and crystal formula) are printed. Thesecond column on page 57 is a second solution (symmetrically equivalent).When all possible (hkl) l/(hkl)2 pairs are examined the program printsRl = ? , ready for a new set of input data.

This program is not as thorough or as rapid as SADSP but isnormally sufficient for indexing spot patterns. For the input data on page57 typical running times are shown in Table 1.

54 -

TABLE 1

Time in minutes to locate and print solution *n (n = 1 to 4) for the(hkl) range h (h _-. 2 8).

(hkl)Range

2345678

Solution

# 1

. 2 7

. 741. 552.854. 777.40

10.85

Solution*2

1. 102.464.818. 38

13. 5720. 4329. 31

Solution#3

1.793.967.70

13. 4521.6232,6946. 93

Solution# 4

2.275, 049. 85

17. 3228. 0642.6561,60

TotalTime

2.926. 58

13.0223. 0537.43

56.9782. 34

(iii) Triclinic d-Spacings

Shown on page 65 of Appendix A is the printed output for theprogram which calculates d-spacings for the triclinic crystal-system.Firstly, the calculator requests the lattice parameters then cycles aboutthe request for h, k, 1. Pages 66 to 68 gives the program listing.

(iv) Interplanar Angles

On page 69 is an example of the printed output from the programused to calculate interplanar angular spearations. As in the previouscalculator programs the initial request is for lattice parameters thencycling occurs about the request for (hkl) 1 and (hkl) 2 values. Pages70 to 72 are the program listing.

- 55 -

(v) Interdirection Angles

The output shown on page 73 is an example from the calculationof the angle between two crystal directions. This calculation is oftenuseful in determining the angle between zone axes. Pages 74 to 76are the progi i listing.

(vi) Angle Between a Crystal Plane and Direction

This program, output shown on page 77 , does not calculate theangle between a crystal plane and direction but the angle between a pole,which is 90° to the plane, and a direction. Pages 78 to 81 are theprogram listing.

(vii) Bragg Angles

Shown on page 82 is the printed output from the program usedto calculate Bragg angles for any electron accelerating voltage. The programcycles about requests for h,k, l's. Also listed are the d-spacings andrelativistic wavelengths. Pages 83 to 86 are the program listing.

(viii) Kikuchi Patterns - STEREO Data

In section IV, iii an example was given of indexing Kikuchipatterns using SADSP. It was noted that the crystal tilt was calculated andsupplied as input data for STEREO. The present program (page 87) requiresthe x-co-ordinates (an x-axis being perpendicular to the Kikuchi band) of thespot (if present) and the line closest and furthest from the pattern origin,(000), respectively. The shift, d , (see section IV, iii) of the plane fromthe symmetrical position is calcula^d (D(MEASURED)). When the d-spacing of the diffracting plane is entered, the crystal tilt (angular move-ment of the pole vector in a perpendicular plane containing the RL vectorand x-axis) is calculated. The quantity PHI(A)-STEREO is 90°-6 where6 is the crystal tilt, PHI(B)-STEREO is the angle the RL vector makes withthe bottom edge of the photographic plate, PHI(ZA)-STEREO is calculated.The two last variables are required STEREO input.

Page 88 gives the program listing for calculating the samevariables as shown on page 87 except, in this instance, the x-parameters

- 56 -

entered are x,, x and x (origin) rather than x which might not be visible.The order of entry of xc and X£ are not important.

VII. ACKNOWLEDGEMENTS

Thanks are given to the various authors for permission toreproduce their work. Also, I wish to acknowledge the large contributionof M. A. Miller in modifying, "debugging" and running the programs atthis laboratory.

VIII. APPENDIX A

i 4 4 3 b

• H F T H •••-

• O1. 6 i 2 4

- 57 -

SAD SPOT PATTERNS

i • J 4

1 • . •-• J c\ 4

- 58 -

- 59 -

1 . 1 , I ' I C- I'Jr.

1 '.-:• ii •

. t . - '.- r !

4 0 i.'1- ' . ' > • • '

i - iO

i - 1 • •;• r ;

1 i - i i-.'

i i - '"'

i • ' i

U 'i

114 .'i . i • ; ; •

- g i.j

- U C1

- -3 5K.< ••''

-1 o

11'",

111.1

i V .

r .1 ,

- 60 -

u'".'

> i i

Mil

111 ' ,1

I I I - '+ ; 11

i - . l ' , . ;

t~1 I ' -

I'i'j

j i •• •! • :

- i 1 J

- I t

l i 1 ':M''

I ' I :V"! i

i i:;. i;

i • i r ; - i ' ,

uMh

i i

•I

J

I n . •

h i

: h . ' .

i t . • i

f (!-.

• !J K' 11 .

- 61 -

I H II

• ... u

i l l I

'••. 11 I

• i i 5 i •'

; 1 M : i

I i - - ,

11 JU.'1

U1.-V- •i r -: . n -

Mi.;

!.1 I

Ij-i

I I !

- 63 -

i

- 64 -

MM

•1L"i

0 4- 0 ;.•

• f - K fa

- 0 4

"! f 1

1.14

- 65 -

TRICLINIC d-SPACINGS

- 66 -

- 67 -

, : i , i; , .

i l l . I : .

• • , ! • • I

| I I

I I > r

i l l : M.

I '• , '

I1 L

• I ' ' ' I

• i l l I ;

H l ' l

i.l •'

• I , . 1

- 68 -

i i

.1 U

4 ('.

b 1

• 4 ,":

4 P.• 1 c ; i

I . J • I i

- 69 -

INTER-PLANAR ANGLES

1 1 1 1

Ml

- 70 -

) ) •: ; i

4

+ U'

• ^ L . : . i I

1 > . ,

'1 1• O S

: -i

i t , '

I.I I :

11 •'-. 11:

II n 1

I I . - ' i i

i i.' ]

- 71 -

•; H

• Ml i l

i ' 1 .

MM

F • i , ;

i j I• 1 " .

11 :

i-1 •

n :i •

I i-

I : ,

M l I i

I ' l l

l i M

H , '

1 i : ; .

M J ' i.:

I'I. ' i L

iV : i

4 1i '.3

Mi'i 1 4

1 1 ••! : 1

I ' . I • ; • > ' ; ;

1 " I . , :..i

11.' i l .

I j -i-i i '11,.. i 1.

M , ' • . . , •

11 • i ; . !

II ,:.., ;i i • : i ,

" . ; ' : • • • •

i i ; ;1,!;

M •?•'• < > • '

I . I ; '-, ' • - •

M . ; S ; > -

i

1I'

j-

I

! iF I'i

t r1 ' , - •

l •• i

•>•

i\[ i

• • 4

• 4

• • • 4

• - • . . . i .

• 1

11'..;

m'i

•: i M i,

- 72 -

; I ,

I . i •

I I I .

l .n .

I I , ; • ' - ,

• 1 ! . , i • ' • i - :

I I1

. , ' ! : • • \

*,..' i i i . . . ' : - ;

4 i Ml• 4 •-• H I .

OI.I in.

iiii

• S 1

Ml.;, :.+

i l l ' >'"

lit ' K

i.l J . •

-.1 ,.

- ,if I.!' ; . .

I- • .

11 j

I I I : I'

i l l , V I

l i t . ••;

0 4

U f : - 4 '

1 l l ' i - i .

1.

1 1 • l-.l

13 c: , j • .

M,::.'

0;:

i t ;

•~-/i'

114

l.ll;- .',

i : l t '• •'

M i - ; . r i l . .

Mi ,:,; ;.

M i ; . • - :

Mt. 5 ' :I ' l l ' r-~lfr

U L. j "'

H ij,r.:;

f1hf,h

• I-. - • - h ,

Uli'

•- O i l '

- 73 -

INTER-DIRECTION ANGLES

- 74 -

11..: iin : i m

4,"'

> 1 ...

1.11 u-.1:1, .

I II I '

I ! I

I I ; ,I , . . . ,

I'.1.1 'l

1 ', \ 1

111.1 II

'.1 0 I I

'.1 -. I i

4 , ' i '4 : • : i i

I I ; 1

: ! M '1 i l l

. 1 ' .

M. ; :

1 1 > • ' ; : .

I ii i

I ! i. 1

I.I I I

I ii i

I ' IJ

111-

> • • L

: I!I..::i n •, • ;

i -I

:, IPKit-'

n i l ' .i;-::

tJ.

M

I !

1 I

] 1

1 1M ;

( I

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