representation of misorientations in rodrigues–frank space

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Representation of misorientations in RodriguesFrank space:application to the Bain, KurdjumovSachs, NishiyamaWassermann

and Pitsch orientation relationships in the Gibeon meteorite

Youliang He * , Ste phane Godet, John J. JonasDepartment of Metals and Materials Engineering, McGill University, Wong Building, 3610 University Street, Montreal, Que., Canada H3A 2B2

Received 7 July 2004; received in revised form 14 September 2004; accepted 5 November 2004Available online 24 December 2004

Abstract

The three classical orientation relationships describing the c -to- a transformation, namely the Bain, KurdjumovSachs (KS) andNishiyamaWassermann (NW), are represented in RodriguesFrank (RF) space. Two alternative reference systems are used tohighlight the differences between the three types of misorientation. Some observations obtained on the Gibeon meteorite are ana-lyzed using the two classes of reference system to reveal features of the transformation under conditions of very slow cooling. It isshown that the Bain correspondence relations are never satised, while the measurements fall in the full range of direction parallelconditions extending from the KS to the NW. The crystallographic features of the Pitsch orientation relation are presented in RFspace in Appendix A. The experimental observations conform to this type of transformation to a considerably lesser extent than tothe classical KS and NW relations. 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Metals; Phase transformation; Misorientation; EBSD; RodriguesFrank space

1. Introduction

Misorientation, the crystallographic orientation dif-ference between two individual crystallites, is an impor-tant parameter used to describe the microtexture of materials. For instance, the grain boundary texture iscommonly specied in terms of a rotation about an axis

common to both crystallites that brings the coordinatesystem of the rst into coincidence with that of theother. This is the so-called angleaxis pair descriptionand it provides signicant information about the grainboundary geometry. An important example is that of

coincident site lattice (CSL) boundaries, which are un-iquely described by the axis and angle of misorientationbetween the two neighboring grains [1].

In other cases, such as phase transformations, wherethe misorientation between the initial phase and itstransformed products is the major concern, it is moreconvenient to represent the misorientation between the

two phases using the RodriguesFrank (RF) vector,since the latter takes the lowest angle solution and inte-grates the four parameters (i.e. the rotation angle andthe three components of the rotation axis) into athree-component vector that can be readily displayedin a three-dimensional Cartesian space (RF space) [2].One of the advantages of RF parameterization is thateither the specimen or the crystal axes can be chosenfor reference; according to this system of representation,the rotation angle and axis are directly related to a

1359-6454/$30.00 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.doi:10.1016/j.actamat.2004.11.021

* Corresponding author. Tel.: +1 514 3984755x09501; fax: +1 5143984492.

E-mail address: [email protected] (Y. He).

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vector drawn from the center of the space to the pointrepresenting the rotation axis.

The use of RF space to represent orientations andmisorientations is relatively new compared to that of using rotations about three successive axes, i.e. that of Euler space, and its advantages over the employment

of such other spaces have been addressed by severalresearchers [35]. However, most of these investigationswere focused on the representation of orientations inRF space; only a few were concentrated on the repre-sentation of misorientations [2,6,7]. In this study, themisorientation between two crystallites, which is calcu-lated from orientations measured by electron backscat-ter diffraction (EBSD) techniques, is represented as anRF vector, taking one of the crystallites as the refer-ence system. Specically, the three classical correspon-dence relationships that describe the FCC to BCCtransformation, namely the Bain [8], KurdjumovSachs[9] and NishiyamaWassermann [10,11], are representedin this space. A second reference system is then intro-duced, which has certain advantages in the present case.

The variants of the three orientation relationships arederived directly from the parallelism conditions apply-ing to the crystallographic planes and directions that de-ne these relationships. Some recent results concerningthe transformation of taenite (FCC austenite) in theGibeon meteorite are then presented to illustrate theadvantages of the use of this space. In Appendix A,the variants of the Pitsch [12] transformation relation-ship are also derived from the parallelism conditions.These are then represented in RF space using the two

frames of reference (austenite and Bain). Their positionson a {001} pole gure are compared with those of KSand NW. Some meteorite observations are then plottedin these two forms of representation (pole gure andRF space) and the extent to which the Pitsch relationapplies is evaluated.

2. Misorientation and RodriguesFrank space

Here, the orientation matrices for two crystallites Aand B are M A and M B , respectively. Then the misorien-tation matrix M AB relating these crystallites, arbitrarily

taking crystallite A as the reference system, can be writ-ten as

M AB M BM 1A : 1This matrix denes a rotation that transforms thecoordinate system of the reference crystallite into coinci-dence with that of the other crystallite. 1 The angleaxis

form associated with this misorientation matrix canthen be calculated as: h arccos 12Tr M AB 1 and[u, v, w] = [m23 m32 , m31 m13 , m12 m21 ] [13], whereTr( M AB ) is the trace of matrix M AB and mij (i , j = 1,2,3) are the elements of M AB .

The four parameters can be further reduced to three

using the Rodrigues formula: R tanh2u

; v ; w [14],which denes the three components ( R 1 , R 2 , R 3 ) of theRF vector. Each misorientation is now represented asan RF vector or more specically as the endpoint of the vector in RF space. To avoid the singularity asso-ciated with the RF vector approaching innity whenthe rotation angle h reaches its upper limit p , the spaceis reduced to a nite subspace called the fundamental zone by utilizing the minimum angleaxis pair represen-tation or disorientation . The latter is obtained by takingthe crystal symmetry into account, i.e. employing the 24symmetry operations for cubic crystals [4,7].

The fundamental zone of RF space for cubic sym-metry is reproduced here in Fig. 1 since most of the dis-cussion that follows about the four transformationrelationships will be presented in this subspace. Someauthors have reduced the fundamental zone even furtherby considering only 1/48 of this space [7]. However, thisapproach is not satisfactory for the present study since,as will be evident in what follows, both the signs and or-ders of the components of the rotation axis are of importance.

The three points A, B and C illustrated in the dia-gram typify the centers of the octahedral (A) and trian-gular (B) faces of the fundamental zone, while the

vertices of the triangles are characterized by C. Thereare 6, 8 and 24, respectively, such points in the funda-mental zone. It should be emphasized that it is a prop-

1 It should be noted that the matrices used here are associated withthe coordinate frame transformations that are often cited in materialscience rather than the body rotations usually employed in other elds.Moreover, the transformation is always expressed as a conversion of the reference coordinate system into that of the product.

Fig. 1. Fundamental zone of RF space for cubic symmetry. The threeillustrated points correspond to the following angleaxis pairs: A: 45 [100], B: 60 [111] and C: 62.8 1 1

ffiffiffi 2p 1.

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erty of RF space that points on the boundary faceshave their equivalents on the opposite faces [7]. For in-stance, points on the octahedra (the {10 0} bounds) havetheir equivalents on the opposite faces but these are off-set by rotations through p /4 about the corresponding1 0 0 axis. Similarly, points on the triangles (the

{111} bounds) have their equivalents on the oppositefaces but offset by rotations through p /3 about the1 1 1axis [15].

3. Bain, KS and NW relationships in RF space

The parallelism conditions for the four orientationrelationships considered here, namely the Bain, KS,NW and Pitsch, are summarized in Table 1 [16]. Alsoincluded are the minimum angleaxis pairs and the cor-responding RF vectors derived from the parallelism

relationships. The pole gure representation of the threeclassical c -to- a transformation relationships is shown inFig. 2 , where the {100} poles of the product phase areprojected onto the parent (111) plane. The KS variantsare numbered in terms of the associated c slip systemsdenoted using the nomenclature of Bishop andHill [17,18]. The Pitsch representation is provided inAppendix A.

It is clear from Fig. 2 that the KS and NW variantsare clustered around the Bain variants and that there areeight of the KS and four of the NW variants aroundeach Bain. Each NW variant is located between a pairof KS variants, e.g. NW variant 1 is located betweenKS variants 7 and 20. These three form a set of copla-nar variants that are characterized by having their crys-tallographic {110} a planes parallel to the same {111} cplane. This can be seen from Table 2 , where the variantsof the Bain, KS and NW relationships are listed to-gether with the detailed parallelism conditions that ap-ply to the crystallographic planes and directions of thetwo phases.

It is important to note that the direction parallelismspecied by the KS relationship, which is that of close-packed directions in the two phases, calls for the

pair of coplanar KS variants to have their crystallo-graphic 1 1 0c directions parallel to the same 11 1adirection. These are 10.53 away from each other inRF space. Each KS variant also has a closely spacednon-coplanar variant, which is again 10.53 away inRF space. For example, KS variants 1(aI) and 7(cI)are closely spaced non-coplanar variants. These twoproducts are cross-slip related in terms of the slip sys-tems, which means that the two variants are associatedwith dislocations on intersecting slip planes [19].

If the coordinate system of the parent crystal is takenas the reference system, which corresponds to matrixM A in Eq. (1), the variants of the three correspondencerelationships can be illustrated in RF space as shown inFig. 3 . From this gure and Table 1 , it can be seen thatthe numbers of minimum angleaxis variants, and thusthe numbers of RF vectors, for the Bain, NW andKS relationships are 6, 24 and 24, respectively. These

Table 1Orientation relationships between c and a

Orientation relationship Parallelism Number of variants Minimum angleaxis pairs RF vectors

Bain {0 0 1} c //{001} a 3 45 1 0 0 0.414 0 01 0 0c //1 1 0a (6 points)

KurdjumovSachs (KS) {1 1 1} c //{110} a 24 42.85 0.9680.1780.178 0.380 0.07 0.07 1 1 0c //1 1 1a (24 points)

NishiyamaWassermann (NW) {111} c //{110} a 12 45.98 0.9760.0830.201 0.414 0.035 0.085 1 1 2c //1 1 0a (24 points)

Pitsch (P) {1 0 0} c //{110} a 12 45.98 0.0830.2010.976 0.035 0.085 0.414 0 1 1c //1 1 1a (24 points)

Fig. 2. Stereographic plot of {10 0} a poles for the Bain, KS and NWvariants projected onto the (11 1) c plane. Circled and underlinednumbers represent the Bain and NW variants, respectively.

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should be compared with the numbers of variants thatare evident on the pole gure (3, 12 and 24, respectively).Note that there are twice as many variants for the Bainand NW relationships in RF space as in the pole g-ure because the endpoints of these variants in RF spaceare located on the external surfaces of the fundamentalzone; as a result, each point has its equivalent onthe opposite surface. By contrast, the endpoints of theKS variants are situated on the six planes located ata normalized distance of 0.918 from the origin, with fourpoints on each plane (i.e. these planes are inside the fun-damental zone).

As indicated above, the endpoints of the Bain andNW variants reside on the six octahedra of the fun-damental zone. Here, one Bain variant is located atthe center of each face and four NW variants residearound it. These relationships are claried in Fig. 3 (b)and (d), where all the points contained in the RF

space of Fig. 3 (b) are projected onto the bottom facein Fig. 3 (d). The two Bain variants on the top andbottom faces are collapsed into one point in thecenter.

In a similar manner, the eight KS variants onopposite planes in Fig. 3 (a) are represented by fouroverlapping points in Fig. 3 (c) [1(14), 23(10), 20(7)and 4(17)]. Note that, in contrast to the KS caseof Fig. 3 (c), the four NW points on each of theopposite faces of Fig. 3 (b) are not superimposed. Asa result, eight NW vectors and four KS vectorscan be seen around each Bain axis in the two projected views, which is just the opposite of the pole g-ure view, where four NW variants and eight KSvariants can be seen around each Bain variant. Thesupplementary NW variants are numbered13,14, . . ., 24 here; these correspond to the originalset of 1,2, . . ., 12, respectively. The reason for this

Table 2The Bain, KS and NW variants

Bain variant Coplanar NW and KSvariants

{111} c //{110} a 1 1 0c or 1 1 2c //1 1 1a or1 1 0a

1 NW 5 a (1 1 1) (0 1 1 ) [2 1 1 ] [0 1 1 ]KS 2 (aII) ( 1 0 1) [1 0 1] [111]

15 (

aIII) (

1 1 0) [

110]

NW 8 b (11 1) (0 1 1 ) [2 1 1 ] [0 1 1 ](100) c //(100) a KS 5 (bII) (1 0 1) [1 0 1] [1 1 1]18 (bIII) (1 1 0) [11 0]NW 11 c (1 1 1) (0 1 1 ) [2 1 1 ] [0 1 1 ]

[010]c //[011]a KS 8 (cII) (1 0 1) [1 0 1] [1 11]21 (cIII) (1 1 0) [1 1 0]NW 2 d (1 1 1) (0 1 1 ) [2 1 1 ] [0 1 1 ]

KS 11 (dII) ( 1 0 1) [1 0 1] [1 11]24 (dIII) ( 11 0) [11 0]

2 NW 6 a (1 1 1) (1 0 1 ) [1 2 1 ] [1 0 1 ]KS 3 (aIII) (1 1 0) [11 0] [111]

13 (aI) (0 1 1) [01 1]NW 3 b (11 1) (1 0 1 ) [1 2 1 ] [1 0 1 ](010) c //(010) a KS 6 (bIII) ( 1 1 0) [1 1 0] [1 11]

16 (bI) (0 1 1) [0 1 1]NW 9 c (1 1 1) (1 0 1 ) [1 2 1 ] [1 0 1 ][001]c //[101]a KS 9 (cIII) ( 11 0) [11 0] [111]

19 (cI) (0 1 1) [01 1]NW 12 d (1 1 1) (1 0 1 ) [1 2 1 ] [1 0 1 ]KS 12 (dIII) (1 1 0) [1 1 0] [ 1 11]

22 (dI) (0 1 1) [0 1 1]3 NW 4 a (1 1 1) (1 1 0 ) [1 1 2 ] [1 1 0 ]

KS 1 (aI) (0 1 1) [0 11] [111]14 (aII) (1 0 1) [1 01]NW 10 b (11 1) (1 1 0 ) [1 1 2 ] [1 1 0 ]

(001) c //(001) a KS 4 (bI) (0 11) [011] [1 11]17 (bII) ( 1 0 1) [1 0 1]NW 1 c (1 1 1) (1 1 0 ) [1 1 2 ] [1 1 0 ]

[100]c //[110]a KS 7 (cI) (0 1

1) [0 1

1] [1

1

1]

20 (cII) (1 0 1) [1 0 1]NW 7 d (1 1 1) (1 1 0 ) [1 1 2 ] [1 1 0 ]KS 10 (dI) (0 11) [011] [1 11]

23 (dII) (1 0 1) [1 01]The c {111} planes and KS variants are identied using the notation of Bishop and Hill.



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apparent redundancy will be taken up below after asecond frame of reference is introduced.

This situation is changed if one chooses one of theBain variants, rather than the parent c grain, as the ref-erence system. As illustrated in Fig. 4 , in this case 24KS and 12 NW variants can be seen, just as in thepole gure representation. The relation between theKS and NW variants around the reference Bain vari-ant (in the centers of Fig. 4 (c) and (d)) also resemblesthat of the pole gure, with one NW variant betweenpairs of KS variants. Under such conditions, the othertwo Bain variants are located at the vertices of the fun-damental zone, which are the triple junctions of threeboundary faces (two {10 0} bounds and one {11 1}bound). As noted above, each point on a boundary facehas an equivalent on the opposite face. Consequently,there are four equivalent points for each of the tworemaining Bain variants, one on each of the four pairsof opposing bounds. Therefore, in Fig. 4 , there are a to-

tal of eight Bain points on the triangular faces in addi-tion to the one located at the center of the gure (thereference Bain variant).

The geometry of RF space as employed here can besummarized as follows. A point situated inside the fun-damental zone is unique . This is the case for the 24 KSvariants or the 12 NW variants when a Bain variantis taken as the frame of reference. A point that is situ-ated on a face of the fundamental zone has one equiva-lent on the opposite face, for a total of two points(multiplicity of 2). These points are rotated by 45 10 0 about the center of the opposite face if they lieon an octahedron, or by 60 1 1 1 if they lie on atriangle.

This is the case for the RF vectors that correspondto the Bain and NW relationships when the austeniteis chosen as the reference frame. Similarly, a point thatlies on an edge of the fundamental zone is shared by twofaces and, therefore, has one equivalent point on each of

Fig. 3. The Bain, KS and NW orientation relationships represented in the fundamental zone of RF space: (a) the Bain and KS relationships,(b) the Bain and NW relationships, (c) projection of (a) onto the bottom face, (d) projection of (b) onto the bottom face. The KS variants arenumbered according to their associated BishopHill slip systems, numbers in brackets being variants on the bottom plane. Underlined numbersrepresent the equivalent NW variants induced by symmetry rotations.



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the two opposite faces. This corresponds to two equiva-lent points (in addition to the rst) for a total of three(multiplicity of 3). Finally, a point that resides on a ver-tex , and is thus shared by three faces, has three equiva-lents (multiplicity of 4) on the corresponding oppositefaces. This is the case for the remaining Bain variantswhen one Bain variant is taken as the reference, aspointed out above.

The misorientations between the Bain, KS and NWvariants can be readily calculated once all the variantshave been identied. Between the Bain variants, theseare 62 : 8 h1 1

ffiffiffi 2p 1i, represented as angleaxis pairs.Thus, the misorientation between two Bain variants

takes the maximum allowable value of disorientationangle between two cubic crystallites [20]. From Fig.

4(b), it is apparent that each KS variant has two closeneighbors, each with an RF misorientation angle of 10.53 . For example, the two closest neighbors of vari-ant 1 are 14 (coplanar variant) and 7 (cross-slip relatedvariant), with misorientations of 10.53 [0 11]a and10.53 [111]a , respectively. By contrast, the misorienta-tion between two adjoining NW variants is13.76 ; e.g. the two closest neighbors of NW variant1 are 4 and 10, with misorientations of 13.76 [0.706 0.06 0.706] a and 13.76 [0.706 0.06 0.706] a ,respectively. The misorientation between one NW var-iant and its two nearest KS neighbors is 5.26 1 1 0a .

In Fig. 4 , the 24 KS and 12 NW variants are dis-tributed on the surfaces of four spheres (two inner andtwo outer) centered on the origin. The radii of the inner

Fig. 4. The Bain, KS and NW relationships in RF space with one of the Bain variants taken as the reference system: (a) Bain variant 1 asreference, (b) Bain variant 3 as reference, (c) projection of (a) onto the bottom face, (d) projection of (b) onto the bottom face. Here, only 12 NWpoints can be seen. Circled and underlined numbers represent the Bain and NW variants, respectively.



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spheres for KS and NW are 0.097 and 0.085, respec-tively. These distances are calculated from misorienta-tions of 11.07 0.879 0.364 0.307 a and 9.74 10 0a ,respectively. There are eight KS variants and four N W variants on the surfaces of these inner spheres. Theradii of the outer spheres for KS and NW are 0.526

and 0.538, respectively. These correspond to misorienta-tions of 55.53 0.419 0.648 0.636 a , 55.530.162 0.754 0.636 a for the KS variants and 56.60 0.717 0.371 0.590 a for the NW variants. There are16 KS variants and eight NW variants on the surfacesof these outer spheres.

4. Application to misorientations in the Gibeon meteorite

Iron meteorites have long been of interest to metal-lurgists because of their special microstructural andcrystallographic characteristics. These materials passthrough the c -to- a phase transformation at coolingrates of 0.1500 C/million years [21], which cannotof course be reached under laboratory conditions.Iron meteorites usually consist of two major phases,kamacite and taenite, as well as their eutectoid mix-ture, plessite [21]. Kamacite is the BCC a phaseknown to metallurgists as ferrite, while taenite is theFCC c phase known as austenite [22]. Plessite is anintimate mixture of kamacite and taenite consistingof tiny a and c grains. The most interesting featureof iron meteorites is the very coarse Widmanstatten

structure that forms by diffusion-controlled nucleationand growth at these extremely slow cooling rates. TheWidmansta tten pattern is developed when the individ-ual lamellae of kamacite thicken by solid-state diffu-sion and eventually contact one another. Sometraces of the precursor phase almost always remainas rims or blebs between or within the kamacitelamellae. Their presence makes it possible to deter-mine the orientation of the prior austenite phase.

Some typical microstructures of the Gibeon meteorite(belonging to group IVA, ne octahedrite, discovered inGibeon, Namibia) are illustrated in Fig. 5 . In this gure,the colors of the kamacite lamellae are associated withcrystal orientation differences. The white rims betweenand within the kamacite grains consist of taenite.Numerous Neumann bands (mechanical twins) can alsobe seen across and within the individual kamacite grains.Plessite regions are separated from kamacite lamellae bytaenite rims.

The Widmansta tten patterns in iron meteorites notonly display special microstructural features, but alsoprovide evidence of specic orientation relationships be-tween the kamacite lamellae and the retained taeniterims.These relationships have been investigated bymeans of X-ray diffraction [23], TEM [24] and neutron

diffraction [25]. The results obtained have produced evi-dence for both the KS and NW relationships, as wellas for intermediate orientations. More recently, Bungeet al. [26] measured the orientation distribution of theWidmansta tten plates in a sample of the Gibeon meteor-ite using high-energy synchrotron radiation. Their mea-

surements revealed a continuous range of orientationsstretching out from the NW orientation to both of the adjoining KS orientations. However, the orienta-tion relationship between the taenite and kamacite inthe plessite regions was not studied intensively. Never-theless, an investigation by Hasan and Axon [24] usingTEM indicated that there was a near- KS relationshipbetween the two phases.

Some typical EBSD maps obtained from the Gibeonmeteorite are illustrated in Fig. 6 . The large plate-likegrains in the upper part are kamacite (BCC) lamellae.At the bottom of these maps, there is a plessite regionthat consists of tiny kamacite plates (about 24 l mwide) and taenite particles (1 l m or less in diameter).In the 844 l m 1068 l m scanned area, the taenite(FCC) phase (including the tiny grains in the plessite re-gion) shares essentially the same orientation, as shownin Fig. 7 (a). The mean orientation of the measured tae-nite points is ( u 1 = 101.8 , U = 51.0 , u 2 = 28.7 ), withan average deviation of 4.4 . If the taenite orientationand its immediately neighboring kamacite orientationsare compared, orientation relationships close to bothNW and KS are observed, usually a few degrees awayfrom the exact NW or KS variants.

In the plessite region, the tiny kamacite (BCC) grains

are also related to the parent taenite phase by relation-ships that are close to KS or NW. This is shown inFig. 7 (b), where the variants predicted by the Bain, KSand NW relationships based on the mean orientationof the taenite phase are superimposed on the measuredkamacite orientations. It can be seen that the observedorientations are distributed continuously along the lineof the three coplanar NW and KS variants. A contin-uous spread of orientations is also observed between thetwo non-coplanar but cross-slip-related KS variants,though with lower density. This corresponds to theneighborhood of the Pitsch orientation, as consideredbelow in more detail, and resembles the orientation rela-tionships between the kamacite lamellae and the taenitereported by Bunge et al. [26]. It should be noted, how-ever, that the Bain relationship is not observed.

These relationships are seen more clearly in RFspace, as shown in Fig. 8 . The measured kamacite orien-tations are compared against the predictions obtainedfrom the KS ( Fig. 8 (a)) and NW ( Fig. 8 (b)) relation-ships. In these cases, the axes of the average taenite ori-entation were chosen for reference. From this gure, it isevident that the measured kamacite orientations arespread over both the KS and NW locations, whichmeans that both of these relationships were obeyed. It



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should be added that for the NW relationship, all 24RF vectors (including the 12 equivalents induced bythe symmetry rotations) should be employed to evaluate

the experimental data since data points that are not ex-actly on the surface will have only one reection andshould be considered as either near one NW variant

Fig. 6. EBSD maps for the Gibeon meteorite: (a) image quality map for the kamacite and taenite, (b) inverse pole gure map of the kamacite,(c) orientation map of the taenite.

Fig. 7. {00 1} pole gures of the taenite and kamacite phases in the plessite region: (a) taenite (FCC), (b) kamacite (BCC), both measured andpredicted.

Fig. 5. Typical microstructures of the Gibeon meteorite: (a) kamacite lamellae (K), (b) taenite (T) and plessite (P). Etched using an aqueous solutioncontaining 10% sodium thiosulfate and 3% potassium metabisulte.



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or near its equivalent on the opposite face. Moreover, all24 variants predicted by both the KS and NW formsof representation appear to be present and evenly dis-tributed in RF space. This indicates that no variantselection occurred under the equilibrium cooling con-ditions that applied in the present case.

In Appendix A, some further kamacite observationsare presented in RF space, but compared with the pre-dictions of the Pitsch correspondence relations.

Fig. 9 shows the measured taenite and kamacite ori-entation relationships in the plessite region as referredto one of the Bain variants. Here, the observations areagain compared with the predictions. In this case, theyindicate in more detail that both the KS and NW rela-

tionships are satised, with almost all the measuredpoints in the vicinities of the variants predicted by thetwo sets of correspondence relations. It is of particularnote that the Bain relationship is never observed.

5. Conclusions

1. Three of the correspondence relationships (the Bain,KS and NW) that describe the c -to- a transforma-tion have been represented in RF space with the ccrystal chosen as the reference system. The Bainand NW variants are seen to be symmetrically dis-tributed on the boundary surfaces of the fundamental

Fig. 8. Measured kamacite orientations in the plessite region as represented in RF space: (a) compared with the predicted KS variants,(b) compared with the predicted NW variants. The average taenite orientation is taken as the reference.

Fig. 9. Orientation relationships in RF space referred to one of the Bain variants: (a) 3-D view of the predicted and measured variants,(b) projection onto the bottom face.

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zone. By contrast, the KS variants are located on thefaces of an inner cube located within the fundamentalzone. The Bain, KS and NW relationships are asso-ciated with 6, 24 and 24 endpoints, respectively, in thefundamental zone.

2. When one of the Bain variants is chosen as the ref-

erence system, the neighboring KS and NW vari-ants are located on the surfaces of two innerspheres; their radii are 0.085 and 0.097 for theNW and KS variants, respectively. The tworemaining Bain variants are situated at selected ver-tices of the fundamental zone, with each variantbeing represented by four equivalent points on theboundary faces. In this case, the relationshipbetween the Bain, KS and NW variants resem-bles that of the pole gure representation, particu-larly when an appropriate 2D projection of RFspace is employed.

3. Observations regarding the transformation of c to ain the plessite region of the Gibeon meteorite werecarried out by means of an automatic EBSD tech-nique. Analysis reveals that both the KS and NWrelationships are obeyed, together with all thosecalled for by the intermediate direction parallelismconditions. By contrast, the Bain relationship is neverobserved.

4. Most of the measured orientations are distributedaround the three coplanar KS and NW variants;a smaller number are located adjacent to the twocross-slip-related KS variants. Essentially all theKS and NW variants are observed without any

variant selection.

Appendix A. Crystallographic features of the Pitschrelationship and applicability to the present results

The Pitsch relationship was originally proposed in1962 [27] and can be described as follows: {10 0} c //{110} a 01 1c //11 1a . It applies principally toprecipitation in cubic systems, such as CuCr, andhas been observed and reported by numerousresearchers [2832]. Here, it is of interest to describethe features of this transformation relation in RFspace, to compare these to those of KS and NW,and then to assess the present results in terms of thedegree to which the present observations satisfy theabove parallelism conditions.

The conditions themselves lead to 24 possible rota-tion axes. However, as was the case for the NW rela-tionship, 12 of these are redundant, so that there areonly 12 (and not 24) physically distinct variants, seeTable A1 . The locations of the P rotation axes inRF space are compared with those of the KS andNW axes, respectively, in Fig. A1 (a) and (b). These

Table A1The Bain and Pitsch variants

Bain variant Pitsch variant {001} c //{110} a 1 1 0c //1 1 1a

1 2 (0 0 1)//(1 1 0) [11 0]//[11 1]5 (0 1 0)//(1 01) [101]//[11 1]

(100) c //(100) a 8 (0 0 1)//(11 0) [110]//[1 11][010]c //[011]a 11 (0 1 0)//(1 0

1) [1 0 1]//[1

1 1]

2 3 (1 0 0)//(0 1 1) [0 11]//[111]6 (0 0 1)//(11 0) [110]//[111]

(010) c //(010) a 9 (1 0 0)//(0 11) [011]//[11 1][001]c //[101]a 12 (0 0 1)//(11 0) [11 0]//[11 1]3 1 (0 1 0)//(1 0 1) [1 0 1]//[11 1]

4 (1 0 0)//(0 11) [0 11]//[1 11](001) c //(001) a 7 (0 1 0)//(1 01) [1 01]//[111][100]c //[110]a 10 (1 0 0)//(0 11) [0 11]//[111]

Fig. A1. (a) The Bain, KS and Pitsch relationships in RF space.(b) The Bain, NW and Pitsch relationships in RF space. The NW andPitsch variants are numbered according to Tables 2 and A1 ,respectively.



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axes are distributed around the Bain positions inmuch the same way as the NW axes. (Note thatthe rotation angle - is identical and that the compo-nents of the RF vector are also identical, but aresimply permuted in a different manner.) As a result,the sets of P and NW axes are simply related by a

rotation of 45 about the Bain axis.This is seen to better effect in Fig. A2 (a), where thepositions of the KS, NW and P rotation axes are projected onto the bottom face of the RF cube using the

parent austenite as the reference frame. Although theNW and P axes appear to be superimposed they are sit-uated on opposite faces (upper and lower) of the cube. Itis worthy of note that the P axes are located on the surface of the cube, as are the NW axes, while the KSaxes are slightly displaced towards the center. A similar

plot, but employing the Bain reference frame in thiscase, is presented in Fig. A2 (b). Here, the P axes displaya multiplicity of two around the periphery because theyare situated on the faces (octahedral faces or truncatedtriangular corners).

The relative positions of the KS, NW and P reec-tions are illustrated on a {001} pole gure in Fig. A3 .As indicated above, the NW reections are located be-tween two co-planar KS reections, while the Preections are situated midway between two KS reec-tions that are cross-slip related (in terms of the Bishopand Hill notation).

A {001} pole gure displaying the measured orienta-tions of the kamacite plates within a single taenite grainis displayed in Fig. A4 (a). Here, it can be seen that theobservations are clustered around the KS and NWpositions, while the intensities in the neighbourhood of the P axes are relatively low. There are no reectionsin the Bain position. The same measurements are illus-trated on the RF cube in Fig. A4 (b), where once againthe lack of correspondence with the rotation axes thatcorrespond to the P parallelism conditions is readilyevident.

Fig. A3. Stereographic plot of {1 00} a poles for the Bain, KS, NWand Pitsch variants projected onto the (111) c plane. The NW variants(underlined) and Pitsch variants are numbered according to Tables 2and A1 , respectively.

Fig. A2. (a) The Bain, KS, NW and Pitsch relationships in RFspace. Projection onto the bottom face, with the parent austeniteorientation as the reference frame. (b) The Bain, KS, NW and Pitschrelationships in RF space. Projection onto the bottom face, with Bainvariant 3 taken as the reference frame.



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Conclusions

1. The Pitsch relations can be described in terms of thesame rotation angle as the one that applies to the NWrelationship and the magnitudes of the RF vectorcomponents are identical, except that they are per-muted in a different order. This leads to considerablesimilarity between the distributions of the P and NWreections around the Bain position in a pole gure(there is a rotation angle of 45 between the two sets).

2. The present observations indicate that the Bain trans-formation is never observed, that both the KS andNW correspondence relations apply to the bcc phasein the Gibeon meteorite, but that few reections arelocated in the vicinities of the Pitsch rotation axes.

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Fig. A4. (a) {001} pole gure for the kamacite. (b) The kamaciteorientations in RF space; the Bain and Pitsch orientation relation-ships are also displayed.


representation of misorientations in rodrigues–frank space

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