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Dynamics of phason diffusion in icosahedral Al–Pd–Mn quasicrystals
Acta Materialia 54 (2006) 3233–3240 www.actamat-journals.com
Dynamics of phason diffusion in icosahedral Al–Pd–Mn quasicrystals M. Feuerbacher a
a,*
, D. Caillard
b
Institut fu¨r Festko¨rperforschung, Forschungszentrum Ju¨lich GmbH, D-52425 Ju¨lich, Germany b CEMES-CNRS, 29 rue Jeanne Marvig, BP4347, F-31055 Toulouse Cedex 4, France
Received 14 December 2005; received in revised form 8 March 2006; accepted 9 March 2006 Available online 22 May 2006
Abstract We present a direct experimental investigation of the dynamics of phason defects in an icosahedral quasicrystal. The activation 21 1 parameters of the diffusion of phasons were determined as DH = 4.28 eV and t1 s . d0 ¼ 4 10 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Transmission electron microscopy; Quasicrystals; Defects; Phasons; Diffusion
1. Introduction Quasicrystals are materials possessing long-range order and a structural symmetry incompatible with translational symmetry. Diffraction patterns of quasicrystals exhibit sharp Bragg peaks, which, for instance in the case of icosahedral phases, show fivefold rotational symmetry. The structure of quasicrystals can be described in terms of a higher-dimensional hyperspace, which introduces additional degrees of freedom for the formation of defects. Phasons are specific defects of a quasicrystalline structure, which in a tiling description can be visualized as configuration flips leading to local matching-rule violations [1]. Physically, these flips are realized by atomic displacements, which may involve numerous individual local atom jumps. Levine et al. [2], in a very general treatment, considered the hydrodynamic variables of icosahedral and pentagonal quasicrystals and concluded that the additional degrees of freedom associated with the hyperspace description imply the presence of additional diffusive modes. Later on, Kalugin and Katz [3] explicitly elaborated a scenario for long-range transport of matter by a phason-mediated mechanism in quasicrystals. Since then, numerous studies have been devoted to the search for experimental evidence *
Corresponding author. E-mail address: [email protected] (M. Feuerbacher).
for such a process. Experimental work has been carried out by different means such as conventional tracer diffusion [4,5], nuclear magnetic resonance studies [6], internal friction studies [7], magnetic susceptibility measurements [8] and coherent X-ray scattering [9]. However, no direct observation of phason diffusion processes has been reported, and the interpretation of these experimental results remains a matter of debate. Therefore, the dynamics of phason defects is one of the open issues in quasicrystal physics today. In this paper we present for the first time a direct and quantitative investigation of phason diffusion performed by means of in situ heating studies using transmission electron microscopy (TEM). 2. Experimental The experiments were carried out on icosahedral Al70.5Pd21.0Mn8.5 single quasicrystals produced by the Czochralski technique [10]. During cooling after growth, these crystals are subjected to process-related thermal stresses, which induce some amount of dislocation motion. By dislocation movements in the latest stages of the cooling process planar defects are created, which show stackingfault-like fringe contrast in TEM images. Contrast extinction studies and in situ heating experiments were carried out on these defects using a JEOL 2010 TEM instrument equipped with a Gatan double-tilt
1359-6454/$30.00 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2006.03.020
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high-temperature specimen holder. The temperature was measured with a calibrated thermocouple attached to the holder close to the specimen. Particular care was taken to provide a good thermal contact between the heating elements and the specimen. After each experiment the sample was melted in the microscope in order to check the precision of the temperature measurement. With this procedure we achieved an accuracy of ±5 C. The progress of the heat treatment was recorded by means of a video camera. Additionally, at temperatures below about 600 C the processes observed were sufficiently slow to allow for the recording of still pictures on photographic plates during different stages of the heating process. Experiments were performed in the temperature range from 560 to 750 C, which, according to the kinetics of the process investigated, corresponds to the span of experimentally accessible timescales of some tenths of seconds to several hours.
3. Results Fig. 1(a) shows a bright-field Bragg contrast image of a planar defect imaged with the diffraction vector ~ g k ¼ ð 1=0 0=1 0=0Þ (inset). Here and in the following, the indexing system of Cahn et al. [11] is used. A clear fringe contrast, resembling a stacking fault contrast, trailed by a dislocation (black arrow) is visible. By means of spec-
imen tilting, the plane normal of the fault was determined to be parallel to the twofold ½0=1 1=0 1=1 direction. Fig. 1(b) shows the same specimen area using ~ gk ¼ ð1=1 0=1 1=0Þ for imaging. The white arrows indicate a contrast feature present in both images. Under these imaging conditions and with ~ gk ¼ ð1=0 0=1 0=0Þ, the fault contrast is extinct. Using the invisibility criterion ~ gk~ rk ¼ 0 [12] we can therefore conclude that the displacement vector of the planar fault, ~ rk , is parallel to the twofold ½0=1 1=0 1=1 direction. The contrast of the dislocation terminating the planar fault is extinct under the same conditions. With the invisibility criterion ~ gk~ bk ¼ 0 (strong extinction condition [13]) we find that the physical space component of the dislocation Burgers vector, ~ bk , is parallel to ½0=1 1=0 1=1 as well. Fig. 1(c) shows the same specimen area after heating the specimen to 580 C imaged with ~ gk ¼ ð1=0 0=1 0=0Þ. The surface of the specimen shows a cloudy contrast due to degradation of the sample upon heating. Nevertheless, the fringe contrast of the fault as well as the leading dislocation (black arrow) are clearly visible. During the course of the heat treatment we observe a gradual weakening of the fringe contrast eventually leading to its total disappearance. This process does not involve any dislocation motion or size changes of the contrast. Fig. 1(d) shows the same specimen area imaged under the same conditions and with the same diffraction vector (inset) after 100 min at 580 C.
Fig. 1. Bright-field Bragg contrast images of a planar defect. (a) In contrast using ~ gk ¼ ð1=0 0=1 0=0Þ. (b) Contrast extinction using ~ gk ¼ ð1=1 0=1 1=0Þ. (c) After heating to 580 C imaged with ~ gk ¼ ð1=0 0=1 0=0Þ. (d) After 100 min at 580 C, same imaging conditions.
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The fringe contrast has entirely vanished. Note that the contrast of the dislocations remains unchanged. Fig. 2 shows an example of a heating experiment carried out at 600 C. Fig. 2(a) shows a set of planar fault fringe contrasts imaged using ~ gk ¼ ð1=1 0= 1 1=0Þ. The plane normal as well as the displacement-field vector were determined to be parallel to the [1/0 1/1 1/0] fault plane normal. Fig. 2(b) shows the same specimen area imaged with ~ gk ¼ ð0=1 0=0 1=0Þ. The faults visible in Fig. 2(a) are out of contrast under these imaging conditions, and we see a second set of faults in contrast. Fig. 2(c) and (d) show the specimen area after heat treatment at 600 C for 8 and 25 min, respectively. The imaging conditions are identical to those of Fig. 2(a) (inset). It is clearly seen that the fringe contrast of the planar faults decreases in Fig. 2(c) while it has almost entirely vanished in Fig. 2(d). Additional defects showing ‘‘coffee bean’’ shaped contrast are visible in Fig. 2(c) and (d). These are most probably due to evaporation processes taking place in during the heat treatment. We carried out these investigations for five samples in the temperature range 560–640 C. In all cases we found a displacement vector and a physical space component of the terminating dislocations’ Burgers vector perpendicular to the twofold fault plane normal, and a gradual vanishing behavior of the planar faults. Fig. 3 shows the time dependence of the contrast of a planar fault fringe system at 570 C. The ‘‘normalized con-
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Fig. 3. Exponential decay of the normalized contrast of a planar fault fringe system at 570 C.
trast’’ of the planar fault fringe system as plotted along the ordinate of the plot was determined as the average grayscale contrast of the fringe system normalized by that of a dislocation line in the same photographic plate. This normalization is necessary, since, according to the chosen exposure times, imaging conditions, and specimen degradation, the overall contrast may vary from plate to plate. The gradual vanishing of the normalized fault contrast is clearly evident. The line through the experimental data points represents a fit to the exponential decay function for the normalized contrast C as a function of the experiment time t:
Fig. 2. Bright-field Bragg contrast images of a planar defect. (a) In contrast using ~ gk ¼ ð1=1 0=1 1=0Þ. (b) Contrast extinction using ~ gk ¼ ð0=1 0=0 1=0Þ. After heat treatment at 600 C for (c) 8 min and (d) 25 min.
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CðtÞ C 1 ¼ ½C 0 C 1 expðt=sÞ; ð1Þ where C0 and C1 are the contrast values for t = 0 and t ! 1, respectively. In the exponential term we find the constant s, representing the timescale on which the disappearance of the fault contrast takes place. While the constants C0 and C1 are of no particular interest, the timescale parameter s is an important experimental parameter. For a temperature of 570 C we find s = 1081 s. Fig. 4 shows a series of video frames taken of a moving dislocation at 750 C. One can clearly see the motion of the
dislocation line (marked by an arrow in the first frame) towards the upper left direction of the frame. The velocity amounts to about 0.6 · 106 m s1. The plane of dislocation movement was determined to be perpendicular to a twofold quasilattice direction. The dislocation trails two types of contrast in its wake. Firstly, we see a long trace perpendicular to the direction of motion, showing dark outer contrast and an interior contrast approximately equal to the background. This contrast remains stable in the sample over the complete duration of the experiment.
Fig. 4. Series of video frames of a moving dislocation at 750 C. The rightmost image shows a contrast-enhanced enlargement of the boxed area.
Fig. 5. Series of video frames of a moving dislocation at 720 C. The last image shows a contrast-enhanced enlargement of the boxed area.
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The occurrence of such a persisting slip-trace contrast in the wake of moving dislocations is well known for all kinds of metals (see e.g. Ref. [14]) and is due to the lattice displacement at the surface of the specimen foil. The second type of contrast in the wake of the dislocation seen in Fig. 4 has a completely different appearance. It shows a fringe contrast with three bright internal fringes, and is visible only within a range of about 200 nm behind the dislocation line as marked by the arrowheads in the enlarged and contrast-enhanced view of the boxed area. A direct measurement of the time before the contrast has disappeared for a given specimen position crossed by the dislocation yields a value of 0.3 s for the temperature of 750 C. Fig. 5 shows an observation made at 720 C. A dislocation moves rapidly at about 2.5 · 106 m s1 from the lower left to the upper right corner of the images crossing a dark persistent dislocation trace. No contrast corresponding to the dislocation line can be seen due to its rapid motion – the time resolution of the video system is too low. The fringe contrast in the wake of the moving dislocation can clearly be seen. A disappearance time of 1.8 s was determined. The vanishing behavior of the contrast can be characterized by analyzing the grayscale in the wake of the moving dislocation on still images of the video sequence. We find a behavior that can be described by Eq. (1) with time constants of the order of 0.1–1 s, i.e. much smaller than those found for the lower temperatures. For all temperatures investigated, the time until the fringe contrast has completely disappeared, td, was determined. Fig. 6 shows td plotted logarithmically against the inverse absolute temperature. We conducted in situ heat treatments in the temperature range 560–750 C. The leftmost data points for the high-temperature range, 750 and 720 C, were obtained by direct observation of vanishing fringe contrast in the wake of moving dislocations as shown in Figs. 4 and 5. The other data points, for the low-temperature range 560–640 C, were obtained by heat
Fig. 6. Arrhenius representation of the temperature dependence of the time until the fringe contrast completely disappeared, td.
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treatment of planar fault fringe contrast present in the as-grown material as shown in Figs. 1 and 2. The results of all experiments conducted follow perfectly a linear behavior in the Arrhenius-type representation of Fig. 6. 4. Discussion 4.1. General discussion We have investigated the contrast dynamics of planar faults in icosahedral Al–Pd–Mn quasicrystals. In the lowtemperature range (560–640 C) we studied planar faults in as-grown material, and, at more elevated temperatures of 720–750 C, we studied faults, the creation of which by moving dislocations was observed in situ. The visual appearance of the faults as well as the time and temperature dependence of their contrast are compatible in the low- and high-temperature ranges. This fact provides strong evidence that all faults investigated are of the same physical origin, that is, they are all created by moving dislocations. The plane normals of the planar faults investigated were in all cases found to be parallel to twofold quasilattice directions. For the faults in the as-grown material, we have determined the displacement vectors and the Burgers vectors of the terminating dislocations to be parallel to the plane normals. Hence, the corresponding dislocation motion takes place by pure climb. Corresponding observations indicating that pure climb is a prominent mode of dislocation movement in icosahedral Al–Pd–Mn were previously made by other authors [15,16]. The fact that dislocation movement in the icosahedral structure leads to the introduction of a planar fault can be straightforwardly related to the quasiperiodicity of the material. In any ordered non-periodic structure, the motion of a dislocation for strictly geometrical reasons must alter the structure in its wake: due to the lack of translational symmetry, the Burgers vector of a dislocation cannot be translationally invariant and hence, upon its movement, a planar defect must be created. The nature of the defect created for the case of quasicrystals was first described in a theoretical treatment by Socolar et al. [1]. Those authors demonstrated that an agglomeration of local matching-rule violations is introduced in the plane swept over by a dislocation. In other words, a moving dislocation in a quasicrystal creates a plane of phason defects. This finding was confirmed by molecular dynamics simulations [17] and corresponds to dislocation simulations for icosahedral Al–Pd– Mn [18]. Such a fault plane, consisting of a dense agglomeration of individual phason defects, is in the following referred to as a ‘‘phason wall’’, a nomenclature introduced by Mikulla et al. [19] and also used in Ref. [15]. The faults characterized in the present paper can be clearly distinguished from other types of planar fault in icosahedral quasicrystals, i.e. stacking faults [20], antiphase boundaries [21], or decagonal precipitates [22]. The latter can directly be ruled out, as they necessarily possess fivefold habit planes. Antiphase boundaries can be ruled out
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as they obey very specific imaging conditions [21], i.e. they are only visible if superstructure reflections are used for imaging, which is not the case for the faults in this study. Imaged under Bragg contrast conditions, the present faults show a characteristic fringe contrast, a characteristic behavior under variation of imaging conditions, and possess twofold habit planes. Previous experimental characterizations of these faults have revealed their phason-type nature. Wang et al. [12] performed contrast extinction analyses as well as high-resolution TEM studies of the faults in edge-on orientation in material subjected to different degrees of plastic deformation. Among other things, they directly imaged the matching-rule violations along lattice planes perpendicular to the plane of dislocation motion. The authors concluded that the fringe contrast is due to planes of phason defects introduced into the structure by moving dislocations. Caillard et al. [15] carried out fringe contrast analysis experiments on planar fault defects in the same material. By means of a dedicated evaluation procedure these authors could unambiguously demonstrate that the faults were phason walls. Upon annealing, the contrast of the planar faults vanishes gradually without involving any movement of the bounding dislocations. In agreement with our previous conclusions, this rules out ordinary stacking faults, antiphase boundaries, or vacancies condensed in planes as an origin of the contrast observed. The removal of these defects would necessarily require movement of the bounding dislocations. The gradual vanishing shows that the faults consist of an agglomeration of individual defects that can be dissolved by essentially independent movement of the latter. The exponential disappearance of the contrast (Fig. 3) shows that the underlying mechanism is a firstorder reaction, which indicates a diffusive dissolution of the faults. We can therefore draw the following conclusions: (1) The planar faults investigated are created by moving dislocations. (2) The faults are phason walls, i.e. planar agglomerations of individual phason defects. (3) Upon annealing the faults undergo dissolution by diffusion of the individual phasons. It should be noted that (3) verifies the anticipation of a diffusive phason-relaxation mechanism made on the basis of hydrodynamic considerations in a continuum elastic picture [1]. Further insight into the origin of the fringe contrast and the mechanism of fading out of the contrast is expected from simulations being carried out by Gratias and Beauchesne [23]. 4.2. Activation parameters For all temperatures investigated, the time until the fringe contrast completely disappeared, td, was determined.
Fig. 6 shows td plotted logarithmically against the inverse absolute temperature. The behavior can, as for ordinary diffusion processes, be described by an Arrhenius-type equation: 1 t1 d ¼ t d0 expðDH =k B T Þ;
ð2Þ
where DH is the activation enthalpy, t1 d0 is the frequency constant of the process, and kB is the Boltzmann constant. The disappearance of the fault contrast in the TEM images is thus based on a thermally activated diffusion process. A linear fit (solid line) to the data points yields the activation parameters: DH ¼ ð4:28 0:17Þ eV; t1 d0
¼ 4 10
211
1
s :
ð3aÞ ð3bÞ
The creation of phason walls by moving dislocations implies the introduction of strong phason density gradients into the structure – a high density is narrowly located in the plane of dislocation movement while the average phason density in the bulk is much lower. Since an increase in the concentration of phason defects raises the free energy, this gradient leads to a driving force for a net flux of phason defects into the bulk. Hence, if the temperature is high enough, diffusion of the phason defects away from the plane of dislocation movement will set in. In TEM the fringe contrast of the initial contrast observed under twobeam conditions occurs due to a phase shift imposed on the electrons crossing the defect relative to those crossing the specimen in undistorted regions. Therefore, a broadening of the defect region leads to weakening of the contrast, which eventually, when diffusion has proceeded such that no considerable defect gradient is left in the material, entirely vanishes. The parameters determined (Eq. (3)) can therefore be understood as the activation parameters for the diffusion of phason defects in icosahedral Al–Pd–Mn. We should point out that the activation parameters found for the diffusion of phason defects are high in comparison with diffusion parameters found for ordinary metals. For the latter, diffusion or self-diffusion processes take place by vacancy or interstitial mechanisms and activation enthalpies of the order of 1–2 eV and pre-exponential frequency factors close to the Debye frequency (e.g. of the order 1013 s1) are found [24]. The parameters are, however, compatible with processes involving collective jumps of numerous atoms. It is well known that such diffusion mechanisms lead to high activation enthalpies and preexponential factors. This is the case, for example, for selfdiffusion in Si and Ge, for which activation enthalpies of about 5 and 3 eV, respectively, and significantly larger pre-exponential factors than for metals were observed [25]. Also, diffusion processes in metallic glasses are known to show high activation enthalpies approaching values up to about 4 eV and pre-exponential terms showing large variations over 30 orders of magnitude (see e.g. Refs. [26,27]). In these materials, the activation parameters are also generally interpreted in terms of collective atomic
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jumps. Diffusion mechanisms involving collective atom jumps were indeed also discussed for quasicrystals by Coddens et al. [28], who found evidence for correlated simultaneous phason jumps in quasielastic neutron scattering studies of icosahedral Al–Pd–Mn. Furthermore, in structure models based on embedding in a higher-dimensional hyperspace, a common and approved approach for quasicrystal structure description [29,30], collective atomic jumps occur as a direct consequence of the specific decoration of the atomic surfaces in many structural models [31,32]. There are two previous studies in the literature in which the phason diffusion parameters for icosahedral Al–Pd–Mn were estimated. Feuerbacher et al. [7] performed internal friction studies on Al–Pd–Mn single quasicrystals. In that study, two major internal friction peaks were observed, and were attributed to the occurrence two different phason relaxation mechanisms. The high-temperature peak, as in the present study, was interpreted in terms of collective phason jumps. The activation enthalpy for the underlying process was determined as DH = 4.0 eV and the authors found a frequency constant of 3.0 · 1024 s1. These parameters compare well to those obtained in the present study. More recent experiments on the dynamics of phason fluctuations by means of coherent X-ray scattering were performed by Francoual et al. [9]. The activation energy of the long-wavelength phason fluctuations was estimated from temperature-dependent measurements of the time evolution of diffuse scattering patterns. Those authors obtained a value of 3 eV, which, as for the values obtained in the present study, is a rather high value compared to diffusion parameters of ordinary metals. The time evolution found by Francoual et al. is dominated by an exponential decay. For 650 C they find a characteristic time constant of about 60 s. These results also fit very well to those obtained in the present study. The diffusion coefficient D for the phason diffusion propffiffiffiffiffiffiffi ffi cess can be estimated using w ¼ 2Dt [33], where w, the width of a Gaussian distribution of the diffusing species after a diffusion time t, is a measure of the diffusion distance. The contrast of the planar faults in TEM is assumed to vanish as soon as the phasons have diffused a certain distance into the bulk, such that the distribution of phasons is too wide to introduce a defined phase shift to the electron beam. If, say, the diffusion distances are of the order of half the specimen thickness, the contrast should disappear. Usual specimen thicknesses d were determined to amount to about 200 nm [34]. Setting w = d/2 and entering our experimental value for td0, we obtain a rough estimate of the diffusion constant, D0, of 2 · 107 m2 s1. This value is much higher than those obtained for diffusion processes in crystalline metals, which show diffusion constants of the order of 105 m2 s1 with little spread. However, for metallic glasses values of this order of magnitude are frequently observed [27,35,36]. Note that our results differ considerably from those of tracer diffusion studies of icosahedral Al–Pd–Mn [4],
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revealing DH and D0 values of the order of 2 eV and 103 m2 s1, respectively. In these experiments the longrange transport of tracer atoms is measured, which may be a fundamentally different physical process than the diffusion of phason defects considered in the present study. The diffusion parameters obtained in the tracer diffusion studies are, however, comparable to the diffusion parameters found for ordinary metals, and, indeed, the tracer diffusion studies in icosahedral Al–Pd–Mn were interpreted in terms of a conventional vacancy mechanism. 4.3. Consequences on plasticity and dislocation mobility The introduction of phason walls by moving dislocations is a geometrical necessity, which costs energy and therefore leads to a corresponding flow stress contribution. For icosahedral Al–Pd–Mn, values for the latter were estimated to be of the order of 100 MPa [37,38], which is a considerable portion of the experimentally observed flow stress values [39]. Even though at high temperatures the diffusion of the phason defects into the bulk takes place very rapidly, the energy for the initial creation of the planar fault nevertheless has to be expended. Our experiments show that the diffusive dissolution of the phason wall takes place in an independent subsequent step, decoupled from the dislocation propagation process. This is clearly seen in the case of the low-temperature experiments of the present study, where the creation of the faults took place right after the growth of the quasicrystals, whereas their diffusive dissolution happened during our heat treatment. Note furthermore that the timescales of td and the dislocation velocities are uncoupled. The comparison of the sequences shown in Figs. 4 and 5 reveal that in the latter the dislocation velocity is much larger while td is shorter. The diffusive dissolution process and therefore td depend only on temperature, whereas the dislocation velocity is a function of temperature and stress. As a consequence, the flow stress contribution due to the creation of a phason wall is not only present at low temperatures but also at high temperatures. While the introduction of phason defects is an athermal process, the diffusional step is not. At very low temperatures the kinetics of the diffusion process is low, so that the phason defects introduced essentially remain in the planes of dislocation motion, which may contribute to the low-temperature brittleness of the material. At high temperatures, in contrast, the phason diffusion kinetics becomes so fast that no permanent defect planes are created. The defects immediately diffuse away from the dislocation, which is then surrounded by a ‘‘cloud’’ of phason defects. Indeed, in post-mortem studies of icosahedral Al–Pd–Mn plastically deformed in the ductile high-temperature range, residual fault planes have never been observed [39]. Nevertheless, phason defects are continuously produced during dislocation motion and hence plastic deformation leads to their accumulation in the structure. This was confirmed experimentally by the electron diffraction
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studies of Franz et al. [40] and has considerable effects on the macroscopic plastic behavior of icosahedral quasicrystals. The continuous accumulation of phason defects leads to a decreasing resistance against dislocation mobility, which, in combination with fast dislocation annihilation [41], is a major ingredient of the characteristic strain-softening phenomenon of icosahedral quasicrystals [39]. In conclusion, we have presented the first direct experimental investigation of the dynamics of phason defects in icosahedral quasicrystals and determined the thermodynamic activation parameters of phason diffusion. We have discussed the consequences of our findings of phason dynamics on dislocation mobility and plasticity of icosahedral quasicrystals. Acknowledgements The authors thank C. Thomas for the growth of the single quasicrystals, Dr. F. Mompiou for fruitful discussions, and Prof. H. Mehrer for critical reading of the manuscript. The experiments were carried out during the stay of one of the authors (M.F.) at the CEMES-CNRS in Toulouse, France, the support and hospitality of which is kindly acknowledged. The work was supported by the Deutsche Forschungsgemeinschaft. References [1] Socolar JES, Lubensky TC, Steinhardt PJ. Phys Rev B 1986;34:3345. [2] Levine D, Lubensky TC, Ostlund S, Ramaswamy S, Steinhardt PJ. Phys Rev Lett 1985;54:1520. [3] Kalugin PA, Katz A. Europhys Lett 1993;21:921. [4] Zumkley Th, Mehrer H, Freitag K, Wollgarten M, Tamura N, Urban K. Phys Rev B 1996;54:R6815. [5] Blu¨her R, Scharwa¨chter P, Frank W, Kronmu¨ller H. Phys Rev Lett 1998;80:1014. [6] Gavilano JL, Mushkolaj S, Ott HR, Apih T, Dolinsek J, Dubois JM, et al. Physica B 2000;284:1167. [7] Feuerbacher M, Weller M, Diehl J, Urban K. Philos Mag Lett 1996;74:81. [8] Matsuo S, Takano H, Ishimasa T. Solid State Commun 1997;102:575. [9] Francoual S, Livet F, De Boissieu M, Yakhou F, Bley F, Le´toublon A, et al. J Phys Rev Lett 2003;91:225501.
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Dynamics of phason diffusion in icosahedral Al–Pd–Mn quasicrystals M. Feuerbacher a
a,*
, D. Caillard
b
Institut fu¨r Festko¨rperforschung, Forschungszentrum Ju¨lich GmbH, D-52425 Ju¨lich, Germany b CEMES-CNRS, 29 rue Jeanne Marvig, BP4347, F-31055 Toulouse Cedex 4, France
Received 14 December 2005; received in revised form 8 March 2006; accepted 9 March 2006 Available online 22 May 2006
Abstract We present a direct experimental investigation of the dynamics of phason defects in an icosahedral quasicrystal. The activation 21 1 parameters of the diffusion of phasons were determined as DH = 4.28 eV and t1 s . d0 ¼ 4 10 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Transmission electron microscopy; Quasicrystals; Defects; Phasons; Diffusion
1. Introduction Quasicrystals are materials possessing long-range order and a structural symmetry incompatible with translational symmetry. Diffraction patterns of quasicrystals exhibit sharp Bragg peaks, which, for instance in the case of icosahedral phases, show fivefold rotational symmetry. The structure of quasicrystals can be described in terms of a higher-dimensional hyperspace, which introduces additional degrees of freedom for the formation of defects. Phasons are specific defects of a quasicrystalline structure, which in a tiling description can be visualized as configuration flips leading to local matching-rule violations [1]. Physically, these flips are realized by atomic displacements, which may involve numerous individual local atom jumps. Levine et al. [2], in a very general treatment, considered the hydrodynamic variables of icosahedral and pentagonal quasicrystals and concluded that the additional degrees of freedom associated with the hyperspace description imply the presence of additional diffusive modes. Later on, Kalugin and Katz [3] explicitly elaborated a scenario for long-range transport of matter by a phason-mediated mechanism in quasicrystals. Since then, numerous studies have been devoted to the search for experimental evidence *
Corresponding author. E-mail address: [email protected] (M. Feuerbacher).
for such a process. Experimental work has been carried out by different means such as conventional tracer diffusion [4,5], nuclear magnetic resonance studies [6], internal friction studies [7], magnetic susceptibility measurements [8] and coherent X-ray scattering [9]. However, no direct observation of phason diffusion processes has been reported, and the interpretation of these experimental results remains a matter of debate. Therefore, the dynamics of phason defects is one of the open issues in quasicrystal physics today. In this paper we present for the first time a direct and quantitative investigation of phason diffusion performed by means of in situ heating studies using transmission electron microscopy (TEM). 2. Experimental The experiments were carried out on icosahedral Al70.5Pd21.0Mn8.5 single quasicrystals produced by the Czochralski technique [10]. During cooling after growth, these crystals are subjected to process-related thermal stresses, which induce some amount of dislocation motion. By dislocation movements in the latest stages of the cooling process planar defects are created, which show stackingfault-like fringe contrast in TEM images. Contrast extinction studies and in situ heating experiments were carried out on these defects using a JEOL 2010 TEM instrument equipped with a Gatan double-tilt
1359-6454/$30.00 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2006.03.020
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high-temperature specimen holder. The temperature was measured with a calibrated thermocouple attached to the holder close to the specimen. Particular care was taken to provide a good thermal contact between the heating elements and the specimen. After each experiment the sample was melted in the microscope in order to check the precision of the temperature measurement. With this procedure we achieved an accuracy of ±5 C. The progress of the heat treatment was recorded by means of a video camera. Additionally, at temperatures below about 600 C the processes observed were sufficiently slow to allow for the recording of still pictures on photographic plates during different stages of the heating process. Experiments were performed in the temperature range from 560 to 750 C, which, according to the kinetics of the process investigated, corresponds to the span of experimentally accessible timescales of some tenths of seconds to several hours.
3. Results Fig. 1(a) shows a bright-field Bragg contrast image of a planar defect imaged with the diffraction vector ~ g k ¼ ð 1=0 0=1 0=0Þ (inset). Here and in the following, the indexing system of Cahn et al. [11] is used. A clear fringe contrast, resembling a stacking fault contrast, trailed by a dislocation (black arrow) is visible. By means of spec-
imen tilting, the plane normal of the fault was determined to be parallel to the twofold ½0=1 1=0 1=1 direction. Fig. 1(b) shows the same specimen area using ~ gk ¼ ð1=1 0=1 1=0Þ for imaging. The white arrows indicate a contrast feature present in both images. Under these imaging conditions and with ~ gk ¼ ð1=0 0=1 0=0Þ, the fault contrast is extinct. Using the invisibility criterion ~ gk~ rk ¼ 0 [12] we can therefore conclude that the displacement vector of the planar fault, ~ rk , is parallel to the twofold ½0=1 1=0 1=1 direction. The contrast of the dislocation terminating the planar fault is extinct under the same conditions. With the invisibility criterion ~ gk~ bk ¼ 0 (strong extinction condition [13]) we find that the physical space component of the dislocation Burgers vector, ~ bk , is parallel to ½0=1 1=0 1=1 as well. Fig. 1(c) shows the same specimen area after heating the specimen to 580 C imaged with ~ gk ¼ ð1=0 0=1 0=0Þ. The surface of the specimen shows a cloudy contrast due to degradation of the sample upon heating. Nevertheless, the fringe contrast of the fault as well as the leading dislocation (black arrow) are clearly visible. During the course of the heat treatment we observe a gradual weakening of the fringe contrast eventually leading to its total disappearance. This process does not involve any dislocation motion or size changes of the contrast. Fig. 1(d) shows the same specimen area imaged under the same conditions and with the same diffraction vector (inset) after 100 min at 580 C.
Fig. 1. Bright-field Bragg contrast images of a planar defect. (a) In contrast using ~ gk ¼ ð1=0 0=1 0=0Þ. (b) Contrast extinction using ~ gk ¼ ð1=1 0=1 1=0Þ. (c) After heating to 580 C imaged with ~ gk ¼ ð1=0 0=1 0=0Þ. (d) After 100 min at 580 C, same imaging conditions.
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The fringe contrast has entirely vanished. Note that the contrast of the dislocations remains unchanged. Fig. 2 shows an example of a heating experiment carried out at 600 C. Fig. 2(a) shows a set of planar fault fringe contrasts imaged using ~ gk ¼ ð1=1 0= 1 1=0Þ. The plane normal as well as the displacement-field vector were determined to be parallel to the [1/0 1/1 1/0] fault plane normal. Fig. 2(b) shows the same specimen area imaged with ~ gk ¼ ð0=1 0=0 1=0Þ. The faults visible in Fig. 2(a) are out of contrast under these imaging conditions, and we see a second set of faults in contrast. Fig. 2(c) and (d) show the specimen area after heat treatment at 600 C for 8 and 25 min, respectively. The imaging conditions are identical to those of Fig. 2(a) (inset). It is clearly seen that the fringe contrast of the planar faults decreases in Fig. 2(c) while it has almost entirely vanished in Fig. 2(d). Additional defects showing ‘‘coffee bean’’ shaped contrast are visible in Fig. 2(c) and (d). These are most probably due to evaporation processes taking place in during the heat treatment. We carried out these investigations for five samples in the temperature range 560–640 C. In all cases we found a displacement vector and a physical space component of the terminating dislocations’ Burgers vector perpendicular to the twofold fault plane normal, and a gradual vanishing behavior of the planar faults. Fig. 3 shows the time dependence of the contrast of a planar fault fringe system at 570 C. The ‘‘normalized con-
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Fig. 3. Exponential decay of the normalized contrast of a planar fault fringe system at 570 C.
trast’’ of the planar fault fringe system as plotted along the ordinate of the plot was determined as the average grayscale contrast of the fringe system normalized by that of a dislocation line in the same photographic plate. This normalization is necessary, since, according to the chosen exposure times, imaging conditions, and specimen degradation, the overall contrast may vary from plate to plate. The gradual vanishing of the normalized fault contrast is clearly evident. The line through the experimental data points represents a fit to the exponential decay function for the normalized contrast C as a function of the experiment time t:
Fig. 2. Bright-field Bragg contrast images of a planar defect. (a) In contrast using ~ gk ¼ ð1=1 0=1 1=0Þ. (b) Contrast extinction using ~ gk ¼ ð0=1 0=0 1=0Þ. After heat treatment at 600 C for (c) 8 min and (d) 25 min.
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CðtÞ C 1 ¼ ½C 0 C 1 expðt=sÞ; ð1Þ where C0 and C1 are the contrast values for t = 0 and t ! 1, respectively. In the exponential term we find the constant s, representing the timescale on which the disappearance of the fault contrast takes place. While the constants C0 and C1 are of no particular interest, the timescale parameter s is an important experimental parameter. For a temperature of 570 C we find s = 1081 s. Fig. 4 shows a series of video frames taken of a moving dislocation at 750 C. One can clearly see the motion of the
dislocation line (marked by an arrow in the first frame) towards the upper left direction of the frame. The velocity amounts to about 0.6 · 106 m s1. The plane of dislocation movement was determined to be perpendicular to a twofold quasilattice direction. The dislocation trails two types of contrast in its wake. Firstly, we see a long trace perpendicular to the direction of motion, showing dark outer contrast and an interior contrast approximately equal to the background. This contrast remains stable in the sample over the complete duration of the experiment.
Fig. 4. Series of video frames of a moving dislocation at 750 C. The rightmost image shows a contrast-enhanced enlargement of the boxed area.
Fig. 5. Series of video frames of a moving dislocation at 720 C. The last image shows a contrast-enhanced enlargement of the boxed area.
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The occurrence of such a persisting slip-trace contrast in the wake of moving dislocations is well known for all kinds of metals (see e.g. Ref. [14]) and is due to the lattice displacement at the surface of the specimen foil. The second type of contrast in the wake of the dislocation seen in Fig. 4 has a completely different appearance. It shows a fringe contrast with three bright internal fringes, and is visible only within a range of about 200 nm behind the dislocation line as marked by the arrowheads in the enlarged and contrast-enhanced view of the boxed area. A direct measurement of the time before the contrast has disappeared for a given specimen position crossed by the dislocation yields a value of 0.3 s for the temperature of 750 C. Fig. 5 shows an observation made at 720 C. A dislocation moves rapidly at about 2.5 · 106 m s1 from the lower left to the upper right corner of the images crossing a dark persistent dislocation trace. No contrast corresponding to the dislocation line can be seen due to its rapid motion – the time resolution of the video system is too low. The fringe contrast in the wake of the moving dislocation can clearly be seen. A disappearance time of 1.8 s was determined. The vanishing behavior of the contrast can be characterized by analyzing the grayscale in the wake of the moving dislocation on still images of the video sequence. We find a behavior that can be described by Eq. (1) with time constants of the order of 0.1–1 s, i.e. much smaller than those found for the lower temperatures. For all temperatures investigated, the time until the fringe contrast has completely disappeared, td, was determined. Fig. 6 shows td plotted logarithmically against the inverse absolute temperature. We conducted in situ heat treatments in the temperature range 560–750 C. The leftmost data points for the high-temperature range, 750 and 720 C, were obtained by direct observation of vanishing fringe contrast in the wake of moving dislocations as shown in Figs. 4 and 5. The other data points, for the low-temperature range 560–640 C, were obtained by heat
Fig. 6. Arrhenius representation of the temperature dependence of the time until the fringe contrast completely disappeared, td.
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treatment of planar fault fringe contrast present in the as-grown material as shown in Figs. 1 and 2. The results of all experiments conducted follow perfectly a linear behavior in the Arrhenius-type representation of Fig. 6. 4. Discussion 4.1. General discussion We have investigated the contrast dynamics of planar faults in icosahedral Al–Pd–Mn quasicrystals. In the lowtemperature range (560–640 C) we studied planar faults in as-grown material, and, at more elevated temperatures of 720–750 C, we studied faults, the creation of which by moving dislocations was observed in situ. The visual appearance of the faults as well as the time and temperature dependence of their contrast are compatible in the low- and high-temperature ranges. This fact provides strong evidence that all faults investigated are of the same physical origin, that is, they are all created by moving dislocations. The plane normals of the planar faults investigated were in all cases found to be parallel to twofold quasilattice directions. For the faults in the as-grown material, we have determined the displacement vectors and the Burgers vectors of the terminating dislocations to be parallel to the plane normals. Hence, the corresponding dislocation motion takes place by pure climb. Corresponding observations indicating that pure climb is a prominent mode of dislocation movement in icosahedral Al–Pd–Mn were previously made by other authors [15,16]. The fact that dislocation movement in the icosahedral structure leads to the introduction of a planar fault can be straightforwardly related to the quasiperiodicity of the material. In any ordered non-periodic structure, the motion of a dislocation for strictly geometrical reasons must alter the structure in its wake: due to the lack of translational symmetry, the Burgers vector of a dislocation cannot be translationally invariant and hence, upon its movement, a planar defect must be created. The nature of the defect created for the case of quasicrystals was first described in a theoretical treatment by Socolar et al. [1]. Those authors demonstrated that an agglomeration of local matching-rule violations is introduced in the plane swept over by a dislocation. In other words, a moving dislocation in a quasicrystal creates a plane of phason defects. This finding was confirmed by molecular dynamics simulations [17] and corresponds to dislocation simulations for icosahedral Al–Pd– Mn [18]. Such a fault plane, consisting of a dense agglomeration of individual phason defects, is in the following referred to as a ‘‘phason wall’’, a nomenclature introduced by Mikulla et al. [19] and also used in Ref. [15]. The faults characterized in the present paper can be clearly distinguished from other types of planar fault in icosahedral quasicrystals, i.e. stacking faults [20], antiphase boundaries [21], or decagonal precipitates [22]. The latter can directly be ruled out, as they necessarily possess fivefold habit planes. Antiphase boundaries can be ruled out
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as they obey very specific imaging conditions [21], i.e. they are only visible if superstructure reflections are used for imaging, which is not the case for the faults in this study. Imaged under Bragg contrast conditions, the present faults show a characteristic fringe contrast, a characteristic behavior under variation of imaging conditions, and possess twofold habit planes. Previous experimental characterizations of these faults have revealed their phason-type nature. Wang et al. [12] performed contrast extinction analyses as well as high-resolution TEM studies of the faults in edge-on orientation in material subjected to different degrees of plastic deformation. Among other things, they directly imaged the matching-rule violations along lattice planes perpendicular to the plane of dislocation motion. The authors concluded that the fringe contrast is due to planes of phason defects introduced into the structure by moving dislocations. Caillard et al. [15] carried out fringe contrast analysis experiments on planar fault defects in the same material. By means of a dedicated evaluation procedure these authors could unambiguously demonstrate that the faults were phason walls. Upon annealing, the contrast of the planar faults vanishes gradually without involving any movement of the bounding dislocations. In agreement with our previous conclusions, this rules out ordinary stacking faults, antiphase boundaries, or vacancies condensed in planes as an origin of the contrast observed. The removal of these defects would necessarily require movement of the bounding dislocations. The gradual vanishing shows that the faults consist of an agglomeration of individual defects that can be dissolved by essentially independent movement of the latter. The exponential disappearance of the contrast (Fig. 3) shows that the underlying mechanism is a firstorder reaction, which indicates a diffusive dissolution of the faults. We can therefore draw the following conclusions: (1) The planar faults investigated are created by moving dislocations. (2) The faults are phason walls, i.e. planar agglomerations of individual phason defects. (3) Upon annealing the faults undergo dissolution by diffusion of the individual phasons. It should be noted that (3) verifies the anticipation of a diffusive phason-relaxation mechanism made on the basis of hydrodynamic considerations in a continuum elastic picture [1]. Further insight into the origin of the fringe contrast and the mechanism of fading out of the contrast is expected from simulations being carried out by Gratias and Beauchesne [23]. 4.2. Activation parameters For all temperatures investigated, the time until the fringe contrast completely disappeared, td, was determined.
Fig. 6 shows td plotted logarithmically against the inverse absolute temperature. The behavior can, as for ordinary diffusion processes, be described by an Arrhenius-type equation: 1 t1 d ¼ t d0 expðDH =k B T Þ;
ð2Þ
where DH is the activation enthalpy, t1 d0 is the frequency constant of the process, and kB is the Boltzmann constant. The disappearance of the fault contrast in the TEM images is thus based on a thermally activated diffusion process. A linear fit (solid line) to the data points yields the activation parameters: DH ¼ ð4:28 0:17Þ eV; t1 d0
¼ 4 10
211
1
s :
ð3aÞ ð3bÞ
The creation of phason walls by moving dislocations implies the introduction of strong phason density gradients into the structure – a high density is narrowly located in the plane of dislocation movement while the average phason density in the bulk is much lower. Since an increase in the concentration of phason defects raises the free energy, this gradient leads to a driving force for a net flux of phason defects into the bulk. Hence, if the temperature is high enough, diffusion of the phason defects away from the plane of dislocation movement will set in. In TEM the fringe contrast of the initial contrast observed under twobeam conditions occurs due to a phase shift imposed on the electrons crossing the defect relative to those crossing the specimen in undistorted regions. Therefore, a broadening of the defect region leads to weakening of the contrast, which eventually, when diffusion has proceeded such that no considerable defect gradient is left in the material, entirely vanishes. The parameters determined (Eq. (3)) can therefore be understood as the activation parameters for the diffusion of phason defects in icosahedral Al–Pd–Mn. We should point out that the activation parameters found for the diffusion of phason defects are high in comparison with diffusion parameters found for ordinary metals. For the latter, diffusion or self-diffusion processes take place by vacancy or interstitial mechanisms and activation enthalpies of the order of 1–2 eV and pre-exponential frequency factors close to the Debye frequency (e.g. of the order 1013 s1) are found [24]. The parameters are, however, compatible with processes involving collective jumps of numerous atoms. It is well known that such diffusion mechanisms lead to high activation enthalpies and preexponential factors. This is the case, for example, for selfdiffusion in Si and Ge, for which activation enthalpies of about 5 and 3 eV, respectively, and significantly larger pre-exponential factors than for metals were observed [25]. Also, diffusion processes in metallic glasses are known to show high activation enthalpies approaching values up to about 4 eV and pre-exponential terms showing large variations over 30 orders of magnitude (see e.g. Refs. [26,27]). In these materials, the activation parameters are also generally interpreted in terms of collective atomic
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jumps. Diffusion mechanisms involving collective atom jumps were indeed also discussed for quasicrystals by Coddens et al. [28], who found evidence for correlated simultaneous phason jumps in quasielastic neutron scattering studies of icosahedral Al–Pd–Mn. Furthermore, in structure models based on embedding in a higher-dimensional hyperspace, a common and approved approach for quasicrystal structure description [29,30], collective atomic jumps occur as a direct consequence of the specific decoration of the atomic surfaces in many structural models [31,32]. There are two previous studies in the literature in which the phason diffusion parameters for icosahedral Al–Pd–Mn were estimated. Feuerbacher et al. [7] performed internal friction studies on Al–Pd–Mn single quasicrystals. In that study, two major internal friction peaks were observed, and were attributed to the occurrence two different phason relaxation mechanisms. The high-temperature peak, as in the present study, was interpreted in terms of collective phason jumps. The activation enthalpy for the underlying process was determined as DH = 4.0 eV and the authors found a frequency constant of 3.0 · 1024 s1. These parameters compare well to those obtained in the present study. More recent experiments on the dynamics of phason fluctuations by means of coherent X-ray scattering were performed by Francoual et al. [9]. The activation energy of the long-wavelength phason fluctuations was estimated from temperature-dependent measurements of the time evolution of diffuse scattering patterns. Those authors obtained a value of 3 eV, which, as for the values obtained in the present study, is a rather high value compared to diffusion parameters of ordinary metals. The time evolution found by Francoual et al. is dominated by an exponential decay. For 650 C they find a characteristic time constant of about 60 s. These results also fit very well to those obtained in the present study. The diffusion coefficient D for the phason diffusion propffiffiffiffiffiffiffi ffi cess can be estimated using w ¼ 2Dt [33], where w, the width of a Gaussian distribution of the diffusing species after a diffusion time t, is a measure of the diffusion distance. The contrast of the planar faults in TEM is assumed to vanish as soon as the phasons have diffused a certain distance into the bulk, such that the distribution of phasons is too wide to introduce a defined phase shift to the electron beam. If, say, the diffusion distances are of the order of half the specimen thickness, the contrast should disappear. Usual specimen thicknesses d were determined to amount to about 200 nm [34]. Setting w = d/2 and entering our experimental value for td0, we obtain a rough estimate of the diffusion constant, D0, of 2 · 107 m2 s1. This value is much higher than those obtained for diffusion processes in crystalline metals, which show diffusion constants of the order of 105 m2 s1 with little spread. However, for metallic glasses values of this order of magnitude are frequently observed [27,35,36]. Note that our results differ considerably from those of tracer diffusion studies of icosahedral Al–Pd–Mn [4],
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revealing DH and D0 values of the order of 2 eV and 103 m2 s1, respectively. In these experiments the longrange transport of tracer atoms is measured, which may be a fundamentally different physical process than the diffusion of phason defects considered in the present study. The diffusion parameters obtained in the tracer diffusion studies are, however, comparable to the diffusion parameters found for ordinary metals, and, indeed, the tracer diffusion studies in icosahedral Al–Pd–Mn were interpreted in terms of a conventional vacancy mechanism. 4.3. Consequences on plasticity and dislocation mobility The introduction of phason walls by moving dislocations is a geometrical necessity, which costs energy and therefore leads to a corresponding flow stress contribution. For icosahedral Al–Pd–Mn, values for the latter were estimated to be of the order of 100 MPa [37,38], which is a considerable portion of the experimentally observed flow stress values [39]. Even though at high temperatures the diffusion of the phason defects into the bulk takes place very rapidly, the energy for the initial creation of the planar fault nevertheless has to be expended. Our experiments show that the diffusive dissolution of the phason wall takes place in an independent subsequent step, decoupled from the dislocation propagation process. This is clearly seen in the case of the low-temperature experiments of the present study, where the creation of the faults took place right after the growth of the quasicrystals, whereas their diffusive dissolution happened during our heat treatment. Note furthermore that the timescales of td and the dislocation velocities are uncoupled. The comparison of the sequences shown in Figs. 4 and 5 reveal that in the latter the dislocation velocity is much larger while td is shorter. The diffusive dissolution process and therefore td depend only on temperature, whereas the dislocation velocity is a function of temperature and stress. As a consequence, the flow stress contribution due to the creation of a phason wall is not only present at low temperatures but also at high temperatures. While the introduction of phason defects is an athermal process, the diffusional step is not. At very low temperatures the kinetics of the diffusion process is low, so that the phason defects introduced essentially remain in the planes of dislocation motion, which may contribute to the low-temperature brittleness of the material. At high temperatures, in contrast, the phason diffusion kinetics becomes so fast that no permanent defect planes are created. The defects immediately diffuse away from the dislocation, which is then surrounded by a ‘‘cloud’’ of phason defects. Indeed, in post-mortem studies of icosahedral Al–Pd–Mn plastically deformed in the ductile high-temperature range, residual fault planes have never been observed [39]. Nevertheless, phason defects are continuously produced during dislocation motion and hence plastic deformation leads to their accumulation in the structure. This was confirmed experimentally by the electron diffraction
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studies of Franz et al. [40] and has considerable effects on the macroscopic plastic behavior of icosahedral quasicrystals. The continuous accumulation of phason defects leads to a decreasing resistance against dislocation mobility, which, in combination with fast dislocation annihilation [41], is a major ingredient of the characteristic strain-softening phenomenon of icosahedral quasicrystals [39]. In conclusion, we have presented the first direct experimental investigation of the dynamics of phason defects in icosahedral quasicrystals and determined the thermodynamic activation parameters of phason diffusion. We have discussed the consequences of our findings of phason dynamics on dislocation mobility and plasticity of icosahedral quasicrystals. Acknowledgements The authors thank C. Thomas for the growth of the single quasicrystals, Dr. F. Mompiou for fruitful discussions, and Prof. H. Mehrer for critical reading of the manuscript. The experiments were carried out during the stay of one of the authors (M.F.) at the CEMES-CNRS in Toulouse, France, the support and hospitality of which is kindly acknowledged. The work was supported by the Deutsche Forschungsgemeinschaft. References [1] Socolar JES, Lubensky TC, Steinhardt PJ. Phys Rev B 1986;34:3345. [2] Levine D, Lubensky TC, Ostlund S, Ramaswamy S, Steinhardt PJ. Phys Rev Lett 1985;54:1520. [3] Kalugin PA, Katz A. Europhys Lett 1993;21:921. [4] Zumkley Th, Mehrer H, Freitag K, Wollgarten M, Tamura N, Urban K. Phys Rev B 1996;54:R6815. [5] Blu¨her R, Scharwa¨chter P, Frank W, Kronmu¨ller H. Phys Rev Lett 1998;80:1014. [6] Gavilano JL, Mushkolaj S, Ott HR, Apih T, Dolinsek J, Dubois JM, et al. Physica B 2000;284:1167. [7] Feuerbacher M, Weller M, Diehl J, Urban K. Philos Mag Lett 1996;74:81. [8] Matsuo S, Takano H, Ishimasa T. Solid State Commun 1997;102:575. [9] Francoual S, Livet F, De Boissieu M, Yakhou F, Bley F, Le´toublon A, et al. J Phys Rev Lett 2003;91:225501.
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