Jochen Aufrecht

Wolfgang Baumann

Prof. Dr Ir Andreas E J Mittemeijer ; Leineweber

Viola Duppel

Eric Jan Mittemeijer

-Cr Laves phases and their relation with polytypic phase transformations

Keywords: intermetallics, phase transformations, phase transitions, TEM, X-ray diffraction, crystal structure, crystal defects, HRTEM line-broadening analysis

published or not. The documents may come

Layer-stacking irregularities in C36-type Nb-Cr and Ti-Cr Laves phases and their relation with polytypic phase transformations

Introduction 1.1 Laves phases; layer-stacking rules

In the last ten years the interest in Laves phases (general formula AB 2 ; A, B: metal atoms) has grown pronouncedly. From a practical point of view, Laves phases have been proposed as candidates for structural materials, but also for functional materials, like magnetic, superconducting materials and as hydrogen-storage materials [START_REF] Sauthoff | Intermetallics[END_REF]. From a scientific point of view, the Laves phases constitute the largest group of intermetallic phases occurring in a very large number of binary and ternary systems [START_REF] Stein | [END_REF]3], which makes it possible to study specific properties as function of the elemental constituents of Laves phases are characterised by their specific crystal structures, which are tetrahedrally close-packed structures and thus belong to the so-called Frank-Kasper phases [4,5]. The different Laves-phase crystal-structure types can be conceived as polytypes generated by stacking several types of atomic layers subject to some particular stacking principles (Fig. 1). These crystal structures imply that the atomic-radius ratio of the A and B atoms, r A /r B , equals (3/2)1/2 ≈ 1.225, regarding the atoms as hard spheres.

Different descriptions of stacking rules leading to the Laves phases are available in the literature. For the current purposes it suffices to focus on the stacking of layersandwich units, each composed of a specific stack of three atomic layers, AB 3 A (this notation refers to the composition of this stacking unit; called layer triplets in Ref. [6]; not identical with the triple layers considered in some other works [7,8]). Between these layer-sandwich units additional single-layer units, B are located (see Fig. 1). With respect to a given two-dimensional hexagonal unit mesh defined by the lattice-structure basis vectors a and b perpendicular to the stacking direction, the A atoms of the layer-sandwich units and the B atoms of the single-layer units can each assume three different relative positions, i.e. at fractional coordinates either (0, 0), or (1/3, 2/3) or (2/3, 1/3). 1 Representing the position of the layer-sandwich units by X, Y, and Z, and of the singlelayer units by x, y, and z, depending on which of the former three positions is assumed by the layer concerned (i.e. (0, 0): X, x; (1/3, 2/3): Y, y; and (2/3, 1/3): Z, z), the layerstacking sequence of a Laves phase can be represented by an alternate sequence of these upper-and lower-case letters. The following rules apply additionally:

(i) Taking first only into account the upper-case letters, a given letter, say X, may only be followed by a different other upper-case letter, in this case Y or Z, i.e. an X cannot be followed by an X (Y not by Y; Z not by Z).

(ii) A single-layer unit located between two layer-sandwich units is represented by a lower-case letter different from the upper-case letters representing the adjacent layer- Since the positions of the single layer units are unambiguously prescribed by the positions of the adjacent layer-sandwich units (see above), it is sufficient for most purposes to consider only the upper-case letters to indicate the stacking sequence, as done at most places in the present paper. In terms of only the X, Y, and Z symbols describing the stacking sequences following the rules given above, the possible polytypes of Laves phases constitute one-to-one analogues of the polytypes possible for closedpacked structures of pure elements, and also of the polytypes possible for SiC, CdI 2 etc.

Against this background, see also Table 1 summarizing the most important Laves-phase polytypes. a The Jagodzinkski symbol indicates the stacking transition associated with a certain layer-sandwich unit: "h", if the layer-sandwich units above and below the respective layer are both at the same position and "c" if they are at different positions. The respective crystal structures are built up by infinite repetition of the stacking transitions given by the Jagodzinski symbols. To obtain one single, complete unit cell in stacking direction of the respective crystal structure, the number of layers given by the Ramsdell symbol is necessary; i.e. …ccc… for 3C, …hh… for 2H, …chch… for 4H and …cchcch… for 6H. b This 6H-type structure is often referred to as 6H 1 . The 6H 2 structure with the Jagodzinski symbol hchhhc is not considered here. The phase diagrams of many binary and ternary systems comprise several Lavesphase polytypes. The stability of a particular polytype can vary as function of composition or temperature (at constant pressure). A transformation from one to another polytype is in principle possible without need of long-range diffusion. A number of works have been dedicated in the past to the problem how such a polymorphic transformation can occur [START_REF] Allen | Proceedings of the ICOMAT Boston[END_REF][START_REF] Liao | Proceedings of the International Conference of Solid-Solid Phase Transformations[END_REF][START_REF] Allen | [END_REF]7,8]. One main idea is that the layer-stacking sequences, considering only the layer-sandwich units, are changed by glide of partial dislocations, with six different Burgers vectors ±b 1 = ± 1 3 10 10

    , ±b 2 =± 1 3 1100   
 , and

±b 3 = ± 1 3 0 110   
 (Fig. 2; employing Miller-Bravais indices for hexagonal crystal structure lattices, where the basis vectors a and b correspond to those used for the unit mesh mentioned in section 1.1; cf. Fig. 1), along the basal (0001) plane between two (assumedly rigid) layer-sandwich units. Thereby the relative position of the layersandwich unit, say "above" the (0001) glide plane becomes changed with respect to that of the layer-sandwich unit "below" the (0001) glide plane (Fig. 3). Note that, as consequence of stacking rule (ii) indicated above, this glide process requires that the single-layer unit between the two layer-sandwich units has to move synchronously in another direction, but also within the glide plane. Thus the shear process associated with one of the above-mentioned Burgers vectors is actually a synchroshear process, which has been discussed in more detail elsewhere [7,8,15]. The corresponding partial dislocations are, therefore, called synchro-Shockley partial dislocations [15].

It has been proposed [START_REF] Allen | [END_REF]15], supported by some experimental evidence (see below) that, transformation proceeds, revealed C36 domains growing parallel to the (0001) plane into the C14 matrix [START_REF] Liao | Proceedings of the International Conference of Solid-Solid Phase Transformations[END_REF]. This observation is compatible with the glide of synchro-Shockley partial dislocation dipoles as transformation mechanism. In the same work [START_REF] Liao | Proceedings of the International Conference of Solid-Solid Phase Transformations[END_REF] it was indicated that the apparent sluggishness of formation of 3C (C15)-TiCr 2 from 2H (C14)-TiCr 2 (in contrast to that of formation of 4H and also 6H) might be ascribed to the impossibility to accomplish the 2H → 3C transformation by a partial dislocation dipole mechanism as discussed above; the 2H → 3C transformation has to involve macroscopic shear for the crystal considered, which can be difficult to realize in a polycrystalline specimen.

in
In the present publication layer faulting in three C36-type Laves phases (NbCr 2 , TiCr 2 , NbCo 2 ; see the phase diagrams provided by Fig. 6) have been studied by transmission electron microscopy and X-ray powder diffraction. It was found that the investigated C36-NbCr 2 and C36-TiCr 2 Laves phases contain specific stacking irregularities. These are compatible with the occurrence of specific irregularities in the periodic passage of the synchro-Shockley partial dislocation dipoles in the course of the 2H (C14) → 4H (C36) transformation, experienced by the NbCr 2 and TiCr 2 specimens.

In contrast with these observations, the C36-NbCo 2 Laves phase does not contain such layer-stacking faults: a 2H (C14) → 4H (C36) transformation does not occur during formation of C36-NbCo 2 (cf. Fig. 6c).

Diffraction from layered hexagonal structures containing stacking irregularities

The effect of a stacking irregularity like a single stacking fault, and also of a combination of stacking faults, in a layered structure on its diffraction pattern can be understood as follows: a single stacking irregularity separates two halves of a crystal (Fig. 7). Then the unit cells of the one half of the crystal with respect to the other half may be shifted by a The actual line broadening in case of a polycrystal or powder containing many stacking faults will depend on the probability distribution of the different types of faults.

Corresponding calculations have been performed for many types of layered structures, and computer programs are available to predict for general layered structures the layerfault induced line broadening [16]. The line broadening (and the line shift) depends on the relative phases of the diffracted waves emanating from the two crystal parts separated by a stacking irregularity. Thus, for the Laves-phase crystal structures considered here, it suffices to refer to previously worked out predictions on line broadening in similar layered crystal structures in which same types of displacement vectors occur upon varying the layer-stacking sequence. For 4H-type structures, as the C36-type Laves phase with its XYXZ stacking sequence, such results have been presented in Refs. [17,18]. The different types of stacking faults considered in these works can be ascribed to only five types of displacement vector ∆r, which are indicated in the following by its coefficients reflections have an integrated intensity of exactly or close to 0 (these classes were not considered explicitly in Refs. [17,18]). Three further classes show line broadening characteristic for the corresponding type of ∆r, see Table 2. Note that the appearance of a "0" in a cell in Table 2 means that the reflections of that reflection class remain unbroadened for the type of displacement vector considered, which implies that g⋅∆r = u'h + v'k + w'l is an integer.

±(0,0, 1/4) (0, 0, 1/2) ±(1/3, 2/3, 0) ±(1/3, 2/3, ±1/4) ±(1/3, 2/3, ±1/2) h -k = 3N l = 4M 0 0 0 0 0 F o r P e e r R e v i e w O n l y 8 h -k ≠ 3N l = 4M 0 0 3 3 3 h -k ≠ 3N l = 4M ± 1 2 4 3 2 1 h -k ≠ 3N l = 4M + 2 4 0 3 1 3
a) The two additional reflection classes indicated by hk = 3N and l = 4M ± 1, and hk = 3N and l = 4M + 2 are irrelevant due to the zero or negligible integrated intensity of the corresponding hkl reflections.

Experimental

Preparation of alloys

The NbCr 2 (target composition 34.5 at.% Nb) and the TiCr 2 (target composition 35.8

at.% Ti) alloys were prepared by arc melting high-purity metals (niobium 99.99 wt%, titanium 99.999 wt%; chromium 99.999 wt%) under a titanium-gettered argon atmosphere. To ensure chemical homogeneity, the samples were flipped, remelted and subsequently cooled down on a water-cooled copper hearth for several times. Although samples of various compositions were prepared, the observations presented here were found to be composition-independent and can be discussed by considering, exemplarily, specimens of the two compositions indicated above.

The NbCr 2 ingot was used as obtained after arc-melting because C36 material can only be obtained in as-cast conditions [19]. The TiCr 2 ingot was subjected to further heat treatments. First, the sample was annealed for 50 h at 1395 °C (in the bcc-solid solution region; cf. Fig. 6b) to ensure chemical homogeneity, i.e. to eliminate the chemical inhomogeneity inherent to the dendritic as-cast structure. Thereafter, the specimen was annealed for 50 h at 1200 °C (in the C36 Laves-phase region; cf. Fig. 6b). These heat treatments were carried out in an induction furnace under a high-purity argon atmosphere. To this end, the sample was placed into an Y 2 O 3 crucible which was covered with three Ti-getter sheets. Before starting the annealing the furnace was evacuated and back-filled with argon for several times. After the annealing, the specimens were rapidly cooled down by switching off the furnace. Preparation of the C36-type Nb-Co Laves phase (25.0 at.% Nb) powder, has been described in Ref. [20]. Note that this material has been considered here on the sole basis of its X-ray powder diffraction pattern.

Both the as-cast NbCr 2 alloy and the as-cast and heat treated TiCr 2 alloy were characterised by optical microscopy, scanning electron microscopy (SEM), and energy/wavelength dispersive X-ray analysis (EDX/EPMA-WDX electron microprobe analysis), X-ray powder diffractometry (XRPD) and high resolution transmissionelectron microscopy (HRTEM).

For the TEM and XRPD measurements powders of the NbCr 2 and TiCr 2 alloys were produced within a mortar. In case of NbCr 2 the material was taken from the top of the ingots containing predominantly C36 phase [19]. In case of TiCr 2 , for the XRPD both a fine and a coarse powder batch were produced. For the SEM and EDX/EPMA-WDX analyses metallographic cross sections were prepared which were manually ground and polished. A Cameca SX100 microprobe was employed for the EPMA-WDX measurements, applying an acceleration voltage of 15 kV and a current of 40 nA.

Elemental standards (Nb, Ti, Cr) were employed for quantification according to the Φ(ρz)

approach [21]. The electron backscattering diffraction (EBSD) measurements were conducted using a Zeiss LEO 438 VP scanning electron microscope equipped with an EBSD system (TSL, EDAX, Inc.) and the software OIM 4.5.

Chemical analysis by carrier-gas hot extraction (ELTRA ONH 2000), combustion

technology (ELTRA CS-800) and Inductively Coupled Plasma Optical Emission (Spectro -CIROS CCD) showed that no uptake of metallic or non-metallic impurities had occurred during sample production and treatment.

High resolution transmission electron microscopy (HRTEM)

Due to the brittle nature of the investigated coarsely ground NbCr 2 and TiCr 2 powders, the edges of the powder particles were thin enough for electron transmission. The powder was suspended in n-butanole. Afterwards, a holey-carbon covered copper net was covered with that suspension leaving behind, after evaporation of the liquid, wedgeshaped crystallites in random orientation transparent for electrons. HRTEM was performed using a Philips CM 30 with an accelerating voltage of 300 kV.

X-ray powder diffraction (XRPD)

XRPD was performed using a "Philips X`Pert MPD" diffractometer equipped with a germanium-monochromator in the incident beam, selecting Cu-Kα 1 radiation.

In case of the TiCr 2 powder specimen, for correction of instrumental broadening, the instrumental line-profile contributions were determined using a LaB 6 powder sample (NIST standard reference material, SRM660a) using TOPAS [START_REF]Topas, General Profile and Structure Analysis Software for Powder Diffraction Data[END_REF]. These instrumental contributions were then convoluted with pseudo-Voigt functions which were fitted to the reflections of the experimental TiCr 2 diffraction patterns by least squares minimization.

The fitting parameters were the reflection-maximum position, 2θ max and the full width at half maximum (FHWM) of the only structurally broadened line profiles; the line-shape parameter η was fixed to a value of 1.2 for all reflections 2 . Similar fits using pseudo-Voigt functions were conducted for NbCr 2 , however, without separate consideration of the instrumental contribution (because of the considerably larger structural line broadening in case of NbCr 2 ) and with η as an additional fit parameter.

Diffraction patterns of the Nb-Co alloy powder specimen were recorded using a Huber G670 Guinier camera equipped with an incident-beam germanium monochromator, selecting Co-Kα 1 radiation. The powder was put as a thin layer on a Kapton foil. Like for NbCr 2 , because of the relative large structural line broadening, without making separate consideration of the instrumental contribution, the reflection positions and widths were determined by fitting Lorentzian functions (corresponding to pseudo-Voigt functions with fixed η = 1; see also footnote 2) to each reflection.

Results

Nb-Cr

Phase analysis and composition

For the as-cast NbCr 2 specimens, a majority of crystallites of the hexagonal C36 modification occur at the top of the arc-melted ingot, together with a minor amount of 2 Fixing the peak-shape parameter η to a certain, reasonable value, removes the possible correlation of the FWHM and η in a fitting procedure and thereby the hkl dependency of the FWHM is more clearly exhibited. This procedure was necessary for the TiCr 2 powder specimen and for also the Nb-Co alloy powder specimen. (retained; cf. Fig. 6a) hexagonal C14-type crystallites (see Fig. 10a) and cubic C15 type crystallites [19]. The bottom part of the ingot mainly consists of the C15 modification.

Recent research performed in the present project has demonstrated that the C14 and C36

modifications are metastable at all temperatures and form only upon solidification from the melt at the top of the initially arc-melted ingots, with the C36 phase forming by solid state transformation from the initially crystallized C14 (see Refs. [19,23]). The powder used for the present study was taken from the top of the arc-melted ingot (cf. section 2.1).

Because, in order to maintain the C36 modification, the sample could not be homogenized rigorously (this would have led to transformation of the C14/C36 crystallites to C15; cf. Fig. 6a) compositional inhomogeneities as a result of the solidification process are possible. The average niobium content of the sample was determined by WDX-EPMA to be equal to 34.9 ± 0.2 at.%. A niobium content somewhat higher than the target composition (34.5 at.%) can be ascribed to chromium evaporation during arc-melting.

XRPD

Although the powder taken from the top part of the as-cast specimen contains C14and C15-type crystallites additional to the C36 majority phase crystallites (cf. section 3.1.1), separate reflections of these minority phases are not visible in the X-ray diffraction pattern (Fig. 8a) because they are superimposed on the C36 reflections occurring at the same values of the diffraction vector Q. In view of the tiny amounts of C14 and C15 phase, for the current purpose the diffraction patterns can yet be conceived as characteristic for the C36 phase.

The recorded diffraction pattern exhibits a distinct hkl-dependent broadening of reflections: The broadening occurs for (and is restricted to; see later) reflections with h-k ≠ 3N and l = 4M ± 1 (see Figs. 8a and9a), as it is expected for layer-faulting induced line broadening for the C36 modification (cf. Table 2). The extent of the line broadening for these reflections increases with l (Fig. 9a), leading to almost complete undetectability of reflections of high l values (see e.g. the 105 and 107 reflections in Fig. 8a). All other reflections exhibit no appreciable broadening. Hence, (0,0,1/2) is the predominant displacement vector (cf. Table 2). This conclusion is supported by the observation that in narrow, whereas it would have broadened by the occurrence of all types of displacement vectors, in particular as promoted by its high l value, except for the occurrence of (0,0,1/2) as displacement vector (Table 2).

HRTEM

A total of 15 crystallites from the NbCr 2 powder obtained from the top part of the as-cast ingot was examined by HRTEM. The C36 structure was found in 13 crystallites; in two cases together with some regions exhibiting the C14 modification and in two other cases together with some regions exhibiting the C15 modification. Only one crystallite was fully of the C14 modification, and one other crystallite was fully of the C15 modification.

In many crystallites, a high density of layer-stacking irregularities was found, in association with streaks along the (hexagonal) 〈0001〉 direction in the selected area diffraction (SAD) patterns (Fig. 10).

Single faults (i.e. only one isolated "incorrect" transition between two layers (actually, two layer-sandwich units) with respect to the C36 layer-stacking sequence; cf. Fig. 3)) were generally not found in the hexagonal C36 crystallites: instead, combinations of successive faults were observed which led to the emergence of intergrowth-type irregularities within the dominant C36 modification (Figs. 10 a-c). Similarly, intergrowth-type irregularities were observed within the minor retained C14 modification as well (Fig. 10d).

With reference to Table 1, in case of the C36 modification (stacking sequence …chch…), one or multiple 2H-type (C14-type) sequences (with stacking sequence -hh-) are observed (Fig. 10a) and one or multiple 6H-type sequences (with stacking sequencecchcch-; Figs. 10b,c).

In case of the C14 modification (stacking sequence …hh…), frequently one or multiple 4H-type (C36-type) sequences (with stacking sequence -chch-) are found (Fig. 10d). For distinction between the periodic Laves-phase crystal structure matrices and the local stacking irregularities, the Ramsdell notation is used for the stacking irregularities and the Strukturbericht designation for the matrices. Further, the periodic stacking sequences of the crystal structures will be enframed by the symbol "…", whereas the local stacking irregularities will be enframed by the symbol "-" to distinguish between the periodic stacking sequences of the matrices and the local layer configuration at the layer-stacking irregularities.

TiCr 2

Phase analysis and composition

The as-cast and heat treated (cf. section 2.1) specimen consisted fully of C36-Laves phase (as determined by X-ray diffraction). According to wavelength dispersive X-ray analysis (WDX-EPMA) the composition of the Laves phase obeys (35.6 ± 0.3) at.% Ti and (64.4 ± 0.3) at.% Cr in agreement with the target composition (section 2.1).

By optical microscopy and SEM the presence of a minority phase was detected (volume fraction approx. 1 % as determined by area analysis). The phase was identified by EBSD as a bcc phase. Standardless EDX analysis revealed that the minority phase is a Ti-rich phase. In view of the Ti-Cr phase diagram this suggests that this phase is the β-Ti solid solution.

XRPD

The diffraction pattern of the coarse powder (cf. section 2.1) of the C36 phase is shown in Fig. 8b. The FWHM of the reflections (shown in Fig. 8b), after correction for instrumental broadening (see section 2.3), has been plotted as function of the diffraction vector Q in Fig. 9b.

The 110 reflection belongs to the class of reflections with hk = 3N and l = 4M.

These reflections show no peak broadening due to layer-stacking irregularities (see Table 2), so that only broadening due to finite crystallite size and due to microstrains, caused by defects or composition variations, is possibly exhibited by these reflections. Assuming that the whole broadening of the 110 reflection is due to microstrain-like broadening according to the method given in Ref. [24]. This B ξ is the FWHM of a super-Lorentzian (η = 1.2, see section 2.3) probability density function of the composition on the at.%scale. Thus the maximum compositional variation was determined to be ±0.28 at.%. This value is within the standard deviation of the WDX-EPMA analysis (cf section 3.2.1):

such compositional variation cannot be exposed by WDX-EPMA analysis.

The peak broadening is affected by the grinding procedure: the peaks of the fine powder (more severely ground) are somewhat broader than the peaks of the coarse powder (less severely ground; see Fig. 9b). This includes the reflection class with hk = 3N and l = 4M, as well. Since the hk ≠ 3N reflections (which should show peak broadening due to (also) the layer-stacking irregularities, in addition to microstrains, as discussed above (cf. Table 2)) of the fine powder, as compared to the coarse powder, reveal extra broadening to about the same extent as reflections of the reflection class with hk = 3N and l = 4M, it can be concluded that the grinding procedure does not influence the layer-stacking irregularity density.

The powder-diffraction pattern of the C36 phase exhibits a distinctly hkldependent broadening of reflections, which is of the same type as observed for C36-NbCr 2 (see section 3.1.2): The reflections with hk ≠ 3N and l = 4M ± 1 show a considerable peak broadening. This means that in the case of TiCr 2 also (0,0,1/2) is the predominant displacement vector. However, in contrast to C36-NbCr 2 , the reflections with hk ≠ 3N and l = 4M (e.g. 204) and those with hk ≠ 3N and l = 4M + 2 (e.g. 106)

show a slightly larger peak broadening than the reflections with hk = 3N and l = 4M.

This indicates that in case of TiCr 2 also layer-stacking irregularities with displacement vectors different from (0,0,1/2) occur.

HRTEM

A total of eight crystallites were investigated by HRTEM. All crystallites are dominantly composed of the C36 modification. In one crystallite within the C36 phase a 2H-type (C14-type) stacking irregularity was found and in another a 6H-type stacking irregularity was found. Furthermore in one crystallite both an accumulation of three consecutive 2H- 15 type (-hh-) and a 6H-type (-cchcch-) stacking irregularity was found (see Fig. 11a). In addition one crystallite contained a "split" 6H-type stacking sequence (i.e. two -cchstacking sequences separated by a larger C36 block; see Fig. 11b). In general, in the investigated C36-TiCr 2 specimens much fewer stacking irregularities occur than in the investigated C36-NbCr 2 specimens.

NbCo 2

The line broadening visible in the XRPD data of C36-NbCo 2 differs distinctly from that observed for the NbCr 2 and TiCr 2 analogues (Figs. 8a,b vs. Fig. 8c). The reflections pertaining to all reflection classes with hk ≠ 3N experience considerable line broadening (Fig. 9c), in particular the prominent 106 reflection (belonging to the hk ≠ 3N and l = 4M + 2 class of reflections) which is only marginally broadened in the case of the C36-TiCr 2 specimens (Fig. 9b) and not broadened at all for the C36-NbCr 2 specimens (Fig. 9a). Moreover, distinctly asymmetric diffraction-line broadening occurs for the hk ≠ 3N and l = 4M + 1 class of reflections, which has been reported in Ref. [18] as representative for stacking-layer faulting not characterized by the (0,0,1/2) displacement vector.

Discussion

Stacking irregularities and associated displacement vectors

Two predominant types of stacking irregularities are found in the equilibrated C36-TiCr 2 specimen and in the as-cast C36-NbCr 2 specimen by HRTEM: single and/or multiple 2Htype and 6H-type (intergrowth-type) stacking irregularities. Insertion of a 2H-type irregularity in a perfect C36 (4H) crystal leads to a displacement vector of (0,0,1/2) (cf. section 1.2); insertion of a 6H type irregularity requires a displacement vector of (0,0,3/2), which is equivalent to the displacement vector (0,0,1/2) due to the translational symmetry of the C36 (4H) crystal structure, comprising four layer-sandwich units (cf.

Table 1). Further, because of this translational symmetry, accumulations of multiple 2Htype or 6H-type stacking irregularities, as frequently observed in the HRTEM images (e.g. Figs. 10c, 11a), can only add up to a displacement vector equivalent to (0,0,0), which causes no peak broadening, or (0,0, 1/2) which leads to broadening of reflections 3N and l = 4M + 2 remain narrow, which would be not the case for (all) other possible displacement vectors (cf. Table 2).

Exactly these above-indicated broadening characteristics are observed in the XRPD patterns of C36-NbCr 2 and in those of C36-TiCr 2 (where the broadening was less pronounced).

A special case is the "split" 6H-type irregularity observed in TiCr 2 (cf. Fig. 11b).

The displacement between the middle 4H (C36) part, which is bounded by the two -cchstacking irregularities, which can be regarded as a split-up 6H (-cchcch-) stacking sequence, and the two adjacent parts of the 4H crystal is of type ±(1/3, 2/3, +1/4); the displacement vector between the two adjacent parts of the 4H crystal separated by the whole "split" 6H-type irregularity is again of type (0,0,1/2).

HRTEM and XRPD line-profile analysis give complementary results: HRTEM reveals the detailed structure of individual stacking irregularities, whereas XRPD provides data pertaining to the average microstructure. It thus follows that the 2H-type and 6H-type stacking irregularities are the dominating layer-stacking defects in C36-type NbCr 2 and TiCr 2 Laves phases.

Relation with phase-transformation mechanism

General remarks

The similarity of the stacking irregularities occurring in C36-TiCr 2 and C36-NbCr 2 (i.e.

the occurrence of the same type of displacement vector; cf. section 4.1) suggests that the origin of these stacking irregularities is the same for both Laves phases. A very remarkable feature of the stacking irregularities with a displacement vector of (0,0,1/2), is that no shear displacement of the part of the crystal above the irregularity occurs with respect to the part of the crystal below the irregularity: i.e. u' = v' = 0 (see Fig. 7). It can thus be concluded that these stacking irregularities are not induced by plastic deformation processes possibly occurring during sample preparation, e.g. by thermal stresses. Indeed, as demonstrated in section 3.1.3, the grinding procedure has no influence on the density of the stacking irregularities. from a C14-type structure, which evolves in case of NbCr 2 as a metastable state (cf. section 3.1.1) during solidification at certain locations in the ingot [19,23] and in case of TiCr 2 as a stable state during the decrease of annealing temperature from 1395°C (bccsolid solution) to 1200°C (cf. phase diagram, Fig. 6b); in this temperature range, the C14 phase field of TiCr 2 is passed. In section 1.2 it has been discussed how a C14 crystal can transform into C36 by passage of a periodic series of synchro-Shockley partial dislocation dipoles ("ordered glide", Figs. 4,5). The passage of one synchro-Shockley partial dislocation dipole within a C14 structure (as shown in Fig. 4), generates a layerstacking irregularity of 4H-type (C36-type) -YZYX-(-chch-) into the …XYXY… (…hh…) stacking sequence of C14. Due to the opposite sign of the Burgers vectors of the partial dislocations forming the dipole, the overall Burgers vector associated with this defect is zero, and thus no macroscopic displacement will be induced between the parts of the C14 crystal adjacent to such a defect (see Fig. 4). Such 4H-type layer stacking irregularities are experimentally observed in the retained NbCr 2 -C14 crystallites (Fig. 10d). Only "ordered glide", i.e. glide of a synchro-Shockley partial dislocation dipole on (i.e. above and below) every fourth layer-sandwich unit over a certain range of the C14 crystal, generates the C36 modification. This "ordered glide" requires coordination of the gliding partial-dislocation dipoles in order to produce C36. If this order is kept perfectly over the whole crystal, a defect-free, perfect C36 crystal would result, and hence no layer-stacking irregularities (with respect to the C36 layer stacking) and no corresponding reflection broadening would occur.

Irregularities in the passage of the synchro-Shockley partial dislocation dipoles (i.e. deviations from "ordered glide") can be the origin of the layer-stacking irregularities in the C36-Laves phases as observed by HRTEM and XRPD in this work. Two mechanisms, which can occur simultaneously, are proposed in the following to explain the observed deviations. The first mechanism (section 4.2.2) derives from the recognition that four different but energetically equivalent modes for formation of C36 are possible, depending on which of the four possible layers within two consecutive C14 unit cells (i.e.

XYXY) is shifted to produce C36 (cf. Fig. 4). If more than one of these modes is operative within one parent C14 crystal, initiating from different locations within the impingement of these domains (Fig. 12). Yet, this first mechanism involves perfect, "ordered glide" within each domain. The second mechanism (section 4.2.3) involves the occurrence of local deviation from "ordered glide", i.e. within a domain, leading to breaking up of the domain into regimes.

Domain formation

The requirement that every fourth layer-sandwich unit has to be displaced for the formation of C36 from C14 necessitates to distinguish between every two adjacent layersandwich units in C14, which is done here by introducing Roman numerals as subscript (see Fig. 13); the stacking sequence of a C14 crystallite would thus be …X I Y I X II Y II …. A C36 stacking can then be formed by shifting either all X I to Z or all X II to Z or all Y I to Z or all Y II to Z (cf. also Fig. 13). Taking the initial C14 structure as reference, four different C36 product domains can thus be formed (similar to the domains found e.g. in ordered substitutional intermetallics/phases). In the first two cases, the shift is realized by a dipole consisting of the synchro-Shockley partial-dislocation pair b i /-b i , while in the last two cases, the pair of synchro-Shockley partial dislocations is -b i /b i (cf. Fig. 2) Note that the first Burgers vector of the pair indicates the shift of the layer-sandwich unit with respect to the, say, bottom part of the crystal concerned, while the second Burgers vector is necessary to shift (reversely) the, say, top part of the crystal back to its original position, thereby avoiding macroscopic shear of the crystal (see Fig. 4). The four possible C36 structures formed by the four layer-shift options are not identical, but displaced with respect to each other by the displacement vectors indicated in Fig. 13 and provided by shift options is adopted throughout the crystal. In this case, for transitional stages as well as for the end stage of the transformation, the displacement vectors between the various C36 domains would always be (0,0,0), and thus no phase shift of diffracted waves and hence no reflection broadening of reflections originating from the C36 domains in the XRPD pattern can occur. In this case, after impingement of the growing domains, a perfect C36 stacking sequence over the whole initial C14 crystallite results.

If the four types of domains shown in Fig. 13 would occur randomly within a parent C14 crystal, the impingement (of the domains growing in the c-direction) between different domains, will lead to "anti-phase boundaries" (see footnote 3) because of the displacement between two impinging C36 structures: stacking irregularities occur associated with corresponding broadening of XRPD reflections according to the theoretical considerations in section 1.3. Two types of displacement vectors can be indicated: (1/3,2/3,1/4) and (0,0,1/2), with the first ones occurring statistically twice as frequently as the latter one if all four types of domains occur randomly (see Fig. 13). This last phenomenon is not compatible with the present observations for the C36-NbCr 2 and C36-TiCr 2 specimens, revealing a predominant broadening of the reflections of the class with hk ≠ 3N and l = 4M ± 1, which implies that the (0,0,1/2) displacement vector is dominant, i.e. only X or only Y layer-sandwich units have been displaced within a parent C14 crystal (cf. Fig. 13). In the case of TiCr 2 , also hk ≠ 3N reflections not pertaining to l = 4M ± 1 are slightly broadened suggesting a minor mixed occurrence (i.e. both X and Y) of layer-sandwich unit displacements within a parent C14 crystal. The "split" 6H-type irregularity (Fig. 11b; see section 3.2.3) can thus be conceived as the outcome of impingement of three C36 domains. The upper and lower one are formed by e.g. shifts of X layers (e.g. X I in the lower and X II in the upper domain), whereas the middle domain is formed by e.g. shifting Y II layers. the parent C14 structure to Z position (see Fig. 13).

X I to Z X II to Z Y I to Z Y II to Z X I to Z (0,0,0) (0,0,1/2) (1/3,2/3,1/4) (1/3,2/3,1/4) X II to Z (0,0,1/2) (0,0,0) (1/3,2/3,1/4) (1/3,2/3,1/4) Y I to Z (1/3,2/3,1/4) (1/3,2/3,1/4) (0,0,0) (0,0,1/2) Y II to Z (1/3,2/3,1/4) (1/3,2/3,1/4) (0,0,1/2) (0,0,0)

Transformation errors

The "ordered glide" of synchro-Shockley partial dislocation dipoles is induced and controlled by the associated decrease in Gibbs energy of the transforming Laves phase (cf. section 4.2.1). The "ordered glide" and thus the stacking sequence of the resulting C36 crystal will be practically perfect, if the energy "penalty" for deviations from this perfectly ordered glide is high compared to the decrease of Gibbs energy achieved by the transformation from C14 to C36. In case of Laves phases, the Gibbs energies of the different layer-sandwich unit stacking modifications differ only slightly (e.g. [25]) and hence the stacking fault energy is also low. Thus, deviations from perfect "ordered glide" are likely.

Irrespective of the precise origin of deviations from perfect "ordered glide", the most likely deviations are those which correspond most closely to the state of perfect order. Such most likely deviations are those which, nevertheless, satisfy the following characteristics of the perfect phase transformation:

• Glide of only synchro-Shockley partial dislocation dipoles is required to realize the irregularity to Y II for the shifting layer-sandwich unit (Fig. 14). Such changes can occur if either of the following scenarios takes place:

•
(i) One (say, either X I or Y I ) layer, which should have shifted according to perfect "ordered glide", remains immobile and "ordered glide", supposed to proceed in the c-direction, is continued by shifting the other representatives of the layers of the same stacking position (i.e. either X II or Y II , Fig. 14a) (ii) A change of shifting layer (from either X I to X II or Y I to Y II ) occurs without such preceding immobilisation, i.e. the distance between two shifted layers is two instead of four layers (Fig. 14b).

Both such local deviations from perfect "ordered glide" are accompanied by the introduction of 2H-type layer-stacking irregularities; the displacement vector between the different 4H (C36) regimes is (0,0,1/2).

Another possibility to introduce stacking irregularities in the product C36 crystal is that some of the synchro-Shockley partial dislocation dipoles are not separated by only one (cf. Fig. 4) but by three layer-sandwich units (see Fig. 15a). Such deviations also bring about a change of shifted layer type from either X I to X II or Y I to Y II , or vice-versa.

The local stacking irregularity formed by such an "expanded" dipole is of 6H type, the corresponding displacement vector between the adjacent 4H (C36) regimes is (0,0,1/2).

Formation of 6H-type layer-stacking irregularities (Fig. 15) may be a way to lower the energy increase associated with the introduction of the 2H-type layer-stacking irregularities described above (Fig. 14). This could occur upon further cooling (i.e. after completion of the C14 → C36 transformation, assuming that (for some temperatures range) below the C14 → C36 transformation temperature, a 6H stacking sequence has a lower Gibbs energy than a 2H stacking sequence. This process is shown in Fig. 15b. It requires (with respect to Burgers vector pair) passage of the type of synchro-Shockley partial dislocation dipole with Burgers vectors of opposite signs as compared to those operating upon formation of the C36 structure from the parent C14 structure.

Overall discussion

In the two preceding sections, two mechanisms have been presented which induce stacking irregularities into a C36 product structure formed from a C14 parent structure by According to the first mechanism (section 4.2.1) four different types of domains can be formed, which are not "in phase" with respect to each other, causing layer-stacking irregularities upon impingement of the domains. Within each domain perfect "ordered glide" of synchro-Shockley partial dislocation dipoles takes place and thus no layerstacking irregularities occur within the domains. Two different displacement vectors occur between the different domains in this model: (1/3,2/3,1/4) and (0,0,1/2). In case of random occurrence of all four domain types the first displacement vector (1/3,2/3,1/4) occurs statistically twice as often as the displacement vector (0,0,1/2). The experimentally observed dominance of the (0,0,1/2) displacement vector thus is ascribed to the presence of 2H-and 6H-type irregularities, induced by the second mechanism to generate stacking irregularities, i.e. by deviations from perfect "ordered glide" of synchro-Shockley partial dislocation dipoles during the transformation in a single domain (section 4.2.2), leading to regime formation within the domains. It is shown that the smallest possible and simplest deviations lead to the 2H-and 6H-type layer-stacking irregularities as observed by HRTEM, associated with a displacement vector between the adjacent C36 crystal regimes of (0,0,1/2), as deduced from XRPD-line profile analysis.

The two mechanisms for generating stacking irregularities can operate jointly.

Domain formation and impingement can occur together with flawed "ordered glide" during the transformation within the forming domains (transformation errors). The case that regime boundaries occur more frequently than domain boundaries has been schematically illustrated in Fig. 16. In this case, using HRTEM, the probability of finding a 2H-type or 6H-type layer-stacking irregularity would be much higher than finding a domain boundary. Also the line broadening in XRPD patterns originating from such C36 structures would be dominated by the displacement vector (0,0,1/2), inducing pronounced broadening of reflections of the class with h-k ≠ 3N and l = 4M ± 1. These predictions regarding the HRTEM and XRPD data agree fully with the corresponding observations made on C36-NbCr 2 . Domain boundaries associated with a displacement vector of The C36-NbCo 2 structure does not form by a C14 → C36 transformation. Although a C14-NbCo 2 Laves phase exists at high Nb content, it is separated from the C36-phase field in the Nb-Co phase diagram by a C15-NbCo 2 phase [16] (see Fig. 6c). Thus, the preferential formation of stacking irregularities with a (0,0,1/2) displacement vector, based on formation of C36 from (fault-free) C14, cannot pertain to C36-NbCo 2 . The discussion of possible origins for the stacking-layer irregularities in C36-NbCo 2

(associated with only a minor occurrence of the displacement vector (0,0,1/2), but with major occurrence of other displacement vectors as listed in Table 2; see results reported in section 3.1) is beyond the scope of the present paper. ii. Two mechanisms bringing about the (layer-sandwich unit) stacking irregularities have been identified:

• Domain formation: Four different types of perfect C36 domains, which are displaced with respect to each other, occur simultaneously within one parent C14 crystal, leading to layer-stacking irregularities upon impingement of the growing domains.

• Transformation errors: Deviations occur from perfect "ordered glide" during the transformation within a domain.

iii. The predominant layer-stacking irregularities are 2H-and 6H-type stacking sequences within a domain, as evidenced by HRTEM and XRPD (selective broadening of the class of XRPD reflections with hk ≠ 3N and l = 4M ± 1 implying that stacking faults with a displacement vector of (0,0,1/2) are phases. The contribution of domain boundaries is negligible compared to that of the regime boundaries in case of C36-NbCr 2 ,but less so in case of C36-TiCr 2 .

iv. C36-NbCo 2 , which is not formed from a C14 precursor, (indeed) shows faulting distinctly different from C36-TiCr 2 and C36-NbCr 2 .

Acknowledgement

2 the

 2 Laves phase. Such information allows tailoring the properties for specific applications.

  Consequently, a z-type single layer is located between X and Y, i.e. the corresponding part of the stacking sequence reads ...XzY.. or ..YzX...

4 1. 2

 42 Partial-dislocation dipole mechanism for polymorphic transformations in Laves phases

  case of a C14 (2H) → C36 (4H) transformation, such synchro-Shockley partial dislocations as described above do not glide through the crystal independently. Instead they glide as dislocation dipoles, where the two dislocations comprising the dipole have Burgers vectors of opposite sign (e.g. b 1 and -b 1 ) and with one dislocation travelling above and one dislocation travelling below a given layer-sandwich unit. As net effect, only the middle layer-sandwich unit and the two adjacent single layer units are shifted with respect to the initial stacking sequence, whereas the upper layer-sandwich unit original position. Hence, glide of such dislocation dipoles does not lead to a macroscopic shear of the crystal (see Fig.4) and thus can occur under the constraint of conservation of shape of the crystal. Thus, simple, ordered (i.e. periodic) sequences of such dislocation dipoles can bring about the 2H (C14) → 4H (C36) (Fig.5) and the 2H (C14) → 6H transformation, which have been frequently observed in real systems. Early high-resolution electron microscopy (HRTEM) studies on TiCr 2 , in which a C14 → C36

  in general, real-valued u', v', and w', and a, b, c as the basis vectors of the translation lattice. Non-integer u', v' and w' imply that for certain (some, all or most) diffraction vectors g = ha* + kb* +lc*, with h, k and l as integer-valued Miller (actually Laue) indices and a*, b* and c* as the reciprocal basis vectors of the crystal, a phase shift occurs between the diffraction waves emanating from the two halves of the crystal. This phase shift can be shown to amount to 2πg⋅∆r = 2π(u'h+ v'k + w'l).No phase shift will occur for certain hkl if g⋅∆r = u'h + v'k + w'l assumes an integer value, and thus the reflections pertaining to these hkl do not broaden and do not shift (in reciprocal space). For a start, the absence of other line-broadening contributions is assumed. If g⋅∆r assumes an arbitrary, non-integer value, the otherwise delta-function shaped diffraction peak gets broadened and possibly shifted in reciprocal space into a direction perpendicular to the fault plane. For predicting how such broadening in reciprocal space will show up in an (X-ray) powder-diffraction pattern, it has to be considered that the line broadening in reciprocal space then gets projected onto the diffraction vector. For example, considering one crystal, if the stacking direction is parallel to c* and thus the fault plane is perpendicular to c*, the line broadening in reciprocal space leads to a large line broadening in the X-ray diffractogram if g || c*. If g ⊥ c*, no line broadening in the X-ray diffractogram results.

Fig. 7 .

 7 Fig. 7. Because of the translational symmetry of the crystals, all components u', v', w' of ∆r are equivalent to u' mod 1, v' mod 1 and w' mod 1, respectively. The occurrence of one of these displacement vectors leads to a specific characteristic hkl dependence of the line broadening (and the line shift) in reciprocal space. The hkl dependence of the line broadening can be systematised by considering six classes of hkl reflections, depending on the values of hk and of l. Of these classes, one with hk = 3N and l = 4M (with integer N, M) shows no line broadening for any of the five ∆r. Two further classes, with h k = 3N and l = 4M ± 1, and with hk = 3N and l = 4M + 2, are irrelevant because their

  reflection (belonging to the h-k ≠ 3N and l = 4M ± 2 reflection class) is

  variations, an upper boundary for the width of the composition distribution equal to B ξ = 0.56 at.% was calculated from the FWHM of the 110 reflection

  with hk ≠ 3N and l = 4M ± 1. Then also reflections of the class with hk ≠

  that both C36 Laves phases, NbCr 2 and TiCr 2 , have been formed

  C36 domains form, and layer-stacking irregularities occur upon

  Within one transforming domain, either only X or only Y layers are shifted to Z. The last point involves that glide is realized by either only dipoles with the Burgers vectors b i /-b i or only dipoles with the Burgers vectors -b i /b i . Under these constraints, disorder in the passage of synchro-Shockley partial dislocation dipoles during the transformation occurs by a change of either X I to X II or Y I

  ordered glide" of synchro-Shockley partial dislocation dipoles.

( 1 / 3 , 2 / 3 , 1 / 4 )

 132314 would also induce some minor broadening of reflections of the classes h-k ≠ 3N with l = 4M and h-k ≠ 3N with l = 4M + 2, in addition to broadening of reflections of h-k ≠ 3N and l = 4M ± 1. This is in agreement with the experimental observations made on C36-TiCr 2 .

  in the layer-sandwich unit stacking sequences of C36 Laves phases (TiCr 2 and NbCr 2 ), which form from C14-type precursors, have been exhibited by HRTEM and XRPD-line profile analysis. The occurrence of such faulting provides validation of the proposed mechanism for the C14 → C36 phase transformation by glide of a series of ordered synchro-Shockley partial dislocation dipoles ("ordered glide").

  The transformation errors are the dominating layer-sandwich unit stacking irregularities for the investigated C36-NbCr 2 and C36-TiCr 2 Laves

Fig. 1 :

 1 Figure captionsFig. 1: Schematic crystal structure of (a) C14 type Laves phase and (b) C36 type Laves phase, built up by alternate stacking of AB 3 A layer-sandwich units and B single-layer units. In the upper part of the figure, the layer-stacking sequences have been displayed in a 1120     projection, i.e. the 1120     direction is perpendicular to the plane of drawing. In the lower part of the figure, the stacking positions have been indicated in [ ] 0001 projection. The basis vectors a and b, perpendicular to the stacking [0001] direction, span a two-dimensional hexagonal unit mesh.

Fig. 2 :Fig. 3 :Fig. 4 : 3 .

 2343 Fig. 2: Stacking positions and Burgers vectors of partial dislocations in a (0001) plane of a Laves phase.

28 Fig. 5 :

 285 Fig. 5: Formation of the 4H stacking sequence of C36 from the 2H stacking sequence of C14 by passage by glide of an ordered sequence of synchro-Shockley partial dislocation dipoles (the symbols ╤ and ╧ denote single synchro-Shockley partial dislocations with Burgers vectors of opposite sign).

Fig. 6 :

 6 Fig.6: Laves-phase containing composition ranges of the phase diagrams of the systems (a) Nb-Cr[26], (b) Ti-Cr[27,28] and (c) Nb-Co[29]. Note that in case of NbCr 2 the C14-NbCr 2 Laves phase (and also the C36-NbCr 2 Laves phase) is no equilibrium phase, as shown recently in Ref.[23].

Fig. 7 :

 7 Fig. 7: Graphical illustration of the five different stacking-fault displacement vectorsbetween two halves of a C36 crystal separated by a stacking irregularity. The actual, here arbitrarily chosen atomistic structure of the irregularity itself has no or negligible influence on the line broadening in reciprocal space. Because of the periodicity of the crystal structure, the two displacement vectors +(0, 0, 1/2) and -(0, 0, 1/2) are identical, whereas displacements as +(0, 0, 1/4) and -(0, 0, 1/4) are not identical but equivalent with respect to the resulting line broadening.

Fig. 8 :

 8 Fig. 8: X-ray powder diffraction patterns: (a) C36-NbCr 2 , powder as obtained from top of the arc-melted, solidified ingot (Cu-Kα 1 radiation); (b) C36-TiCr 2 produced by solid-state annealing for 50 h at 1200°C (coarse powder; Cu-Kα 1 radiation); (c) C36-NbCo 2 (Co-Kα 1 radiation). The Laue indices have been indicated.

Fig. 9 :

 9 Fig. 9: FHWM values of the reflections in the XRPD patterns of (a) C36-NbCr 2 as obtained from top of the arc-melted, solidified ingot (Cu-Kα 1 radiation), (b) C36-TiCr 2 produced by solid-state annealing for 50 h at 1200°C (coarse powder and fine powder, Cu-Kα 1 radiation) and (c) C36-NbCo 2 (Co-Kα 1 radiation).

Fig. 10 :

 10 Fig. 10: HRTEM images (C36 zone axis: 1120 ) of material taken from the top of an as-

Fig. 11 :Fig. 12 :Fig. 13 :

 111213 Fig. 11: HRTEM images (C36 zone axis: 1120 ) of annealed C36-TiCr 2 specimens containing stacking irregularities. (a) A 2H-type (C14-type) and a 6H-type stacking irregularity, (b) a "split" 6H-type stacking irregularity. The arrows indicate the 〈0001〉 direction.

Fig. 14 :Fig. 15 :

 1415 Fig.14:Possible deviations from perfect "ordered glide" of synchro-Shockley partial dislocation dipoles during a C14 → C36 transformation, subject to the constraint that only one type of synchro-Shockley partial dislocation dipoles is involved in the phase transformations, i.e. either only X or only Y layers are shifted to Z positions (cf. Fig.13).For both cases shown in the figure, the layer shift switches from X I to X II . Contrarily to the occurrence of perfect "ordered glide" in the parent C14 crystal, where every second X

Fig. 16 :

 16 Fig. 16: Hierarchy of stacking irregularities in a C36 crystal, as deduced from the results of the present XRPD and HRTEM investigations. The C36 crystallite has formed from a C14 crystal by glide of synchro-Shockley partial dislocation dipoles. The displacement vectors between the crystal parts adjacent to the various irregularities have been indicated, including the probabilities for the displacement vectors associated with the domain boundaries for the case that the four layer-sandwich unit shift possibilities (X I , X II , Y I , Y II ) contribute equally.

Table 1 :

 1 The most important Laves-phase stacking types described as polytypes (see text)

	stacking	stacking	Jagodzinski	Ramsdell	Strukturbericht	prototype	space
	sequence	sequence,	symbol a	symbol	symbol		group
		"reduced"	[9]	[10]	[11]
	XzYxZy	XYZ	c	3C	C15	MgCu 2	Fd 3 m
	XzYz	XY	h	2H	C14	MgZn 2 P6 3 /mmc
	XzYxZxYz	XYZY	ch	4H	C36	MgNi 2 P6 3 /mmc
	XzYxZyXyZxYz XYZXZY	cch	6H b	--	--	P6 3 /mmc

Table 2

 2

	. Extent of line broadening for a 4H-type crystal (as C36) along c* in reciprocal
	space as function of the displacement vector for four of the six reflection classes a . The
	width in reciprocal space units (here not specified) is proportional to the indicated
	numbers, as well as to the specimen-dependent probability for the occurrence of the
	corresponding displacement vectors. N and M are integers.
	reflection-	displacement vector, ∆r = (u', v', w')
	class;
	conditions
	for hkl

Table 3

 3

	3 . The C36 domains thus formed either grow in c (stacking) direction by "ordered
	glide" of more synchro-Shockley partial-dislocation dipoles shifting the same type of
	layer-sandwich unit, or new domains form between the existing ones, which is shown
	schematically in Fig. 12. Growth of existing domains and formation of new domains may
	also occur simultaneously. The crystal can be regarded as completely transformed from
	C14 to C36 after impingement of these domains.

Table 3 :

 3 Displacement vectors between C36 domains, produced in a parent C14 crystal by the synchro-Shockley partial dislocation dipole mechanism, by shifting for each the four different layer-sandwich units of the …X I Y I X II Y II … sequence in

	domain one of
	F o
	r
	P
	e e r
	R e v i e w
	O n
	l y
	20

Use of the symbols A, B, and C to indicate the relative layer positions is prevented to avoid confusion with the symbols A and B used for the two different metals constituting the Laves phase AB

.

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The layer-stacking irregularity occurring between a pair of such domains can be conceived as an antiphase boundary.

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The authors are grateful to Mr. Thomas Meisner (Max Planck Institute for Metals Research) for indispensible collaboration during alloy preparation. The diffraction pattern of NbCo 2 was kindly provided by Dr. Daniel Grüner and Dr. Guido Kreiner (Max Planck Institute for Chemical Physics of Solids, Dresden, Germany). The authors also thank Prof. Dr. Arndt Simon (Max Planck Institute for Metals Research, Stuttgart) for use of his TEM facilities. This work has been performed within the framework of the Inter-Institutional Research Initiative "The Nature of Laves Phases" funded by the Max Planck Society.