Joseph Gastaldi

Thomas Schenk

Guillaume ; L2mp Reinhart

Holger ; Klein

Juergen Haertwig

Nathalie Mangelinck-Noël

Benjamin Grushko

Henri Nguyen-Thi

Paola Pino

Bernard Billia

Nathalie ; L2mp

Benjamin ; Fz

Pino

J Härtwig

H Nguyen Thi

J Baruchel

In situ observation of pore evolution during melting and solidification of Al-Pd-Mn quasicrystals by synchrotron X-ray radiography

Keywords: vacancies, quasicrystals, porous media in situ study, radiography

In situ observation of pore evolution during melting and solidification of Al-Pd-Mn quasicrystals by synchrotron X-ray radiography I Introduction: Since they have been successively observed in thin foil of the Al-Cu-Fe icosahedral phase [1], both at the surface and in the volume of bulk icosahedral Al-Pd-Mn grains [2] [3] and in the volume of quasicrystalline grains of some other alloys and structures [4], pores are generally admitted as important special features of quasicrystals. However as all these previous studies were carried out at room temperature, as well after interrupted [5] [6] as complete [7] [START_REF] Beeli | the 6 th International Conference on Quasicrystals[END_REF] growth, it is difficult to decide between the two hypotheses which have been put forward to explain their origin, namely the thermal vacancy condensation [7] and the hierarchical structure [START_REF] Beeli | the 6 th International Conference on Quasicrystals[END_REF]. Therefore, we have performed an "in situ" and real time observation of the pore evolution during directional solidification-melting cycles of an Al-Pd-Mn bi-grained sample, by synchrotron radiography, in order to grasp, more precisely, the origin of these intriguing species.

II Experiments:

The synchrotron experiments were carried out at the ID 19 beam line of the European Synchrotron Radiation Facility (ESRF, Grenoble, France). One thin Al-Pd-Mn sample (40x6x0.7 mm 3 ) cut in an ingot prepared at the composition which gives quasicrystal grains soon after the beginning of the solidification process (Al 72.4 Pd 20.5 Mn 7.1 ), was melted and solidified two times inside a graphite crucible held vertically in a two heating-zone furnace. This furnace, described previously in detail [START_REF] Schenk | [END_REF], is permeable to X-rays and thereby allowed us to follow the course of the solidification/melting of the investigated sample, under a 20K/cm gradient, a pressure of about 3x10 -7 mbar and pulling velocities ranging from 0 to 3.6 µm/s. Radiographs were recorded all along the processes, at the X-ray energy of 24 keV, either by using a CCD camera developed at the ESRF [10]or exposing High Resolution (<1µm) photographic plates (Ilford L4). The camera was fitted with an optics leading to an effective pixel size of 7.46 µm and a large field of view (15x15 mm 2 ). According to the high coherence of the beam delivered by the ESRF synchrotron source, phase contrasts were recorded in addition to absorption contrasts on radiographs, which increased the sensitivity to grain inhomogeneities of the technique.

III Results:

The sample, as cut in the master ingot (section II), was completely melted before being submitted to two partial solidification/melting cycles. Rather surprisingly, no pore was detected during the solidification stages at the spatial resolution allowed by our technique (≅1µm). The appearance of micrometric pores was firstly observed during melting, at about 2 mm ahead of the regressive interface, while these pores were growing (Fig. 1). Regions close to the melting interface, the external surfaces and the grain boundaries were depleted of pores.

Actually, it was impossible to visualize the nucleation stage of these pores because of the poor spatial resolution of the imaging technique used. Nevertheless, it was clearly seen that the pores were shrinking either during a resumption of growth (Fig. 2) or just being absorbed by the regressive interface (Fig. 3). The mean velocity of enlargement of pores ∆D/∆t (D diameter of pore, t time) was estimated to about five times that of their shrinking ( ≅ 0.5 µm/min against 0.1 µm/min) during re-growth and of the same order just being absorbed by the melting interface.

IV Discussion:

This "in situ" observation of pore evolution during melting and solidification disclose, for the first time, that the pores encountered in as-grown quasicrystal grains [2][3] [4],with sizes ranging from some micrometres to a few ten micrometres, do not appear during the solidification process. The fact that they were only detected when growing in a band at about two millimetres ahead of the melting interface (Fig. 1), could explain the irregular distribution of these pores noticed by A. R. Ross et al. [5]. In the same way, our observation agrees with some other results reported previously like, the absence of pore both close to the external surfaces and to grain boundaries [6], the pore morphology evolution from non-facetted to facetted (dodecahedral) [5] and the pore shrinking during post-growth grain annealing [11].

Accordingly our results confirm that pores have very likely a vacancy origin as stated by Beeli et al., but disagree with the pore formation mechanism they proposed. If the abovementioned results indicate that a vacancy supersaturation S (T) :

S (T) = C T /C (e)T
C T = actual concentration of vacancies in the grain at a temperature T, C (e)T = concentration of vacancies given by the thermal equilibrium at the same temperature T, is actually needed, the fact that pores were observed growing to such a big size at the melting temperature precludes that the invoked vacancy supersaturation stems from the destruction of the thermal equilibrium. Indeed, if a rapid calculation of the thermal vacancy supersaturation S (T) , which stands for example at 1 cm ahead of the melting interface, is made, taking into account the temperature gradient (20K/cm) and the value of ∆H f in AlPdMn given by K. Sato et al. [12] which is of 2.3 eV and supposing that all vacancies present at the melting temperature T m condense into pores at T < T m (not taking into account the grain cooling rate or taking an infinite value), it is found that this supersaturation is very low, of about 1.5, not very different from that calculated for pure Al (∆H f = 0.76 eV). At 2 mm from the melting interface (distance at which pores were observed growing) this supersaturation is only 1.08. It is not sufficient to create a chemical (osmotic) force strong enough to nucleate so large vacancy clusters (S (T) >1000) [13]. Thus the necessary vacancy supersaturation can be supposed as resulting from other processes such as growth or melting. In the past these possibilities were discarded for crystals [13]. In so far as our study points out also, that melting favours the vacancy aggregation far from the solid-liquid interface, it seems that the vacancy supersaturation is higher in the volume than close to this interface, which could exclude the melting and growth processes themselves. But, in this way, the supersaturation of vacancies could result from the very particular structure of quasicrystals which seems not exactly in a stable equilibrium state, immediately after growth [14]. This would agree with the unusual high concentration of structural vacancies inside the volume of quasicrystals reported by numerous authors [12][14] which is of about 10 -3 at Room Temperature, i.e. higher than that of thermal vacancies in crystals at the melting temperature (10 -4 ). It would also correspond to the concentration of vacancies calculated by L. Mancini et al. [3] to account for the pore density encountered in as-grown Al-Pd-Mn grains. The fact that pores became visible at a distance, not very close to the solid-liquid interface, is strongly reminiscent of the nucleation and growth of droplets observed in a constitutionally superheated transparent alloy [15]. Thus the observation of the pores shrinking at the approach of the melting front suggests a concomitant decrease of the vacancy supersaturation in the volume. This phenomenon may be connected to the increasing proximity of the vacancy-attracting solid -liquid interface.

S (T) = exp [-∆H f (1/ T m -1/ T) /

  Figure1

Figure 1 :

 1 Figure 1: Radiographs showing the size increase of pores during melting. a) Pores are growing in a band at about 2 mm from the regressing interface. b) Enlargement of the rectangular region of image a). c) Enlargement of the same region of an image recorded 12 minutes before a).

Figure 2 :

 2 Figure 2: Radiographs showing the shrinking of pores during a resumption of growth following the melting stage displayed in figure 1. a) and b) are low magnification radiographs indicating the grain re-growth. c) Enlargement of a region of image b) which corresponds to the rectangular region in figure 1a). d) Enlargement of the same region of an image recorded 50 minutes before b).

Figure 3 :

 3 Figure 3: Radiographs showing the shrinking of pores just before being absorbed by the regressing interface. a) The interface is arriving at the level of pores. b) Enlargement of the rectangular region of image a). c) Enlargement of the same region of an image recorded 45 minutes before a). 1, 2, 3 focus on pores rapidly shrinking or disappearing very close to the interface.

  Figure 3

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Acknowledgments:

The authors would like to thank M. Kleman, V. Dmitrienko, M. de Boissieu, L. Mancini, S. Agliozzo for fruitful discussions and also T. Bactivelane, R. Chagnon and P. Bernard for their technical assistance.


M. D Bernard Coqblin

M Nunez-Regueiro

Dolores

R G S Iglesias

J Roberto

F R G S Magalhaes

Núñez-Regueiro

Alba Theumann

J R Iglesias

S G Magalhães

Theory of the Kondo lattice : competition between Kondo effect and magnetic order

Keywords: magnetic order, strongly correlated electrons PACS numbers: 71.27.+a, 72.15.Qm, 75.10.Nr

We present here a review of some theoretical features of the Kondo lattice problem, which can describe anomalous experimental properties of strongly correlated electron systems with mainly Cerium, Ytterbium or Uranium. In such Kondo systems, there exist both a Kondo effect on each site and strong intersite magnetic interactions. We present firstly the general case of the Kondo lattice and the meanfield approximation. We discuss then the case of the "undercreened" Kondo lattice with a strong coexistence between the Kondo effect and a ferromagnetic order, as for example observed in Uranium compounds such as UTe. We also discuss the Kondo-Spin Glass competition with eventually an additional magnetically ordered phase (ferromagnetic or antiferromagnetic one) and we can account for phase diagrams of some Cerium and Uranium disordered alloys such as CeN i 1-x Cu x .

Finally, the introduction of a transverse magnetic field in the Kondo-Spin Glass problem yields a Quantum Critical Point and allows a better description of the phase diagrams of disordered alloys.

Introduction

It is well established that an anomalous behaviour is observed in many cerium, ytterbium, uranium or other anomalous rare-earth (Europium, Samarium, Praseodymium and Thulium) and actinide (Neptunium and Plutonium) systems. There are different anomalous behaviours observed in such systems and a very important case concerns the "intermediate valence" systems, where the number of 4f electrons is generally not an integer and varies with the external pressure or the matrix concentration, like in the famous phase diagram of Cerium metal [1,2,3,4,5,6] or in Eu(P d 1-x Au x ) 2 Si 2 alloys [7] or Y bInCu 4 [8]. Another interesting and well studied case concerns the so-called "Kondo systems", which have a valence or a number of f-electrons close to an integer, but present however an anomalous behaviour due to a strong d-f hybridization. The Kondo systems have been extensively studied experimentally and theoretically after the first explanation of the resistivity minimum of magnetic dilute alloys by Kondo [9] and the exact solution obtained within the renormalisation group technique for the single Kondo impurity [10].

In the case of a lattice, there exists a strong competition between the Kondo effect and the magnetic order in many compounds or concentrated alloys containing rare-earths or actinides. This strong interplay between the Kondo effect on each site and the magnetic ordering arising from the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between rare-earth atoms at different lattice sites is well described by the Doniach diagram [START_REF] Doniach | Proceedings of the International Conference on Valence Instabilities and Related Narrow-Band Phenomena[END_REF], which gives the variation of the Néel temperature and of the Kondo temperature with increasing antiferromagnetic intrasite exchange interaction J K between localized spins and conduction electron spins. It is well known that the ordering temperature T N is increasing initially with increasing ρJ K , then passes through a maximum and tends to zero at a critical value ρJ c K corresponding to a "Quantum Critical Point" (QCP) in the Doniach diagram. Such a behavior of T N has been experimentally observed with increasing pressure in many cerium compounds, such as CeAl 2 [START_REF] Barbara | [END_REF] or CeRh 2 Si 2 [13]. A similar effect has been observed in Y bCu 2 Si 2 [14] or in related ytterbium compounds, where the Néel temperature starts from zero at a given pressure and increases rapidly with pressure, because pressure has an opposite effect in ytterbium compounds than in cerium compounds.

The Doniach diagram presents also the great advantage of separating cerium compounds in two categories : the first one includes cerium compounds which have a mixed Kondo-magnetic character corresponding to pressures below the critical pressure of the QCP; for example, CeAl 2 has a Néel temperature of 3.8 K and also a relatively large electronic constant γ of the specific heat equal to 135 mJ/moleK 2 . The second category corresponds to cerium compounds with only a huge Kondo or heavy fermion character and for example CeAl 3 , which does not order magnetically, has an enormous value of γ = 1600mJ/moleK 2 . In the magnetically ordered compounds, there is coexistence between the magnetic order (generally antiferromagnetic like in CeAl 2 ) and the heavy fermion character, although this effect is much smaller in magnetically ordered compounds than in non magnetic ones. On the other hand, the magnetic order can be antiferromagnetic, ferromagnetic in a few cases, a classical spin glass order or a more disordered one.

The purpose of the present paper is to review recent theoretical models which can account for the difficult problem of competition or coexistence between the magnetic order and the Kondo effect which gives the heavy fermion character. In the next section, we will firstly discuss the mean field approximation and we will apply it to both the case of the Kondo effect in presence of short-range magnetic correlations and to the case of the competition between the Kondo effect and the antiferromagnetic order. Then, we will develop the case of the so-called "underscreened" Kondo model where a coexistence between the Kondo effect and the magnetic order is obtained, which can account for the magnetic properties of some uranium compounds.

Finally, we will discuss the occurrence of spin glass phases in Cerium or Uranium systems and we will describe the competition between the spin glass order, the Kondo effect and eventually a magnetically (ferromagnetic or antiferromagnetic) ordered phase.

The Kondo lattice model

We have firstly developed the Kondo-lattice model which is well appropriate to describe cerium (ytterbium) compounds corresponding to a configuration close to 4f 1 (or 4f 13 ) with one 4f electron (or one hole). The Kondo-lattice model is often studied with only the intrasite Kondo interaction. In principle, both the RKKY magnetic interaction and the Kondo effect can be obtained from the intrasite exchange term [15]. When dealing with approximation schemes, it is hard to obtain both effects starting with the intrasite interaction alone and for this reason, an explicit intersite exchange interaction is usually included in the model Hamiltonian. We will give here a very brief summary of the work presented in ref. [16,17] and obviously details can be found in these papers.

The proposed Hamiltonian of the system is, therefore :

H = kσ ε k n c kσ + J K i s i • S i + J H ij S i • S j , ( 1
)
where ε k is the energy of the conduction band, J K (> 0) is the Kondo coupling between the localized spin S i and the spin s i of a conduction electron at the same site, and J H is the interaction between nearest-neighboring localized spins. Assuming spin-1/2 localized moments, we represented them by a zerowidth f band with one electron per site, while the conduction band has width 2D and a constant density of states. We consider here the case of choosing J H to be positive, implying that intersite interactions are antiferromagnetic, as it is the case of most cerium compounds.

We now write the spin operators in fermionic representation and we impose the constraint of single-occupancy of the f level at all sites, n f i = 1. In order to discuss the Kondo effect and magnetic correlations, we define the operators :

λ iσ ≡ c † iσ f iσ , Γ ijσ ≡ f † iσ f jσ , (2)
where λ iσ describes the intrasite Kondo correlation and Γ ijσ represents an intersite correlation between two neighboring atoms. This is so because the on-site spin-spin correlation s i • S i can be written in terms of the average λ iσ λ i,-σ , and similarly for the intersite correlation using Γ ijσ . Then,with this notation we perform an extended mean-field approximation [15,18] which is presented in full detail in Ref. [16,17] and the f band of effective band width 2BD, with B given by :

B = -zJ H Γ/D, (3)
z being the number of nearest neighbors of a site.

The magnitude of the hybridization gap is directly related to λ 2 and this quantity is also a measure of the Kondo effect, as the Kondo correlation

function s i • S i is proportional to λ 2 .
The mean-field parameters λ and Γ are obtained by minimizing the total internal energy of the system at zero temperature, or the total free energy at finite temperatures [17,19]. The reference energy E 0 of the f band and We have previously treated the half-filled case, corresponding to a number of conduction electrons n = 1 [16] and then the general case of a non-integer number of conduction electrons [17]. In this previous paper, we have determined both the temperature T cor for the occurrence of short-range magnetic correlations and the Kondo temperature T K for the lattice, which is different from the one-impurity Kondo temperature T K0 obtained for J H = 0.

In the limit J H = 0 and for J K /D << 1, we obtain :

T K = C 0 D n(2 -n) exp - 1 ρJ K , ( 4
)
where C 0 is a numerical constant: C 0 = 1.1337. Eq. ( 4) gives T K depending on the band filling as n(2 -n), which is consistent with a √ n dependence when n → 0, as found in Ref. [20].

We have also extensively studied the variations of the Kondo temperature T K with J H and the conduction-band filling n [17]. We have found that T K first increases and then decreases with J H for fixed n, dropping abruptly to zero at some critical value of J H . On the other hand, for a given J H , T K decreases rapidly as n departs from half filling. Thus, we have obtained a drastic suppression of the Kondo effect with increasing antiferromagnetic intersite coupling and reducing the band-filling factor. On the other hand, the correlation temperature T cor is linear with J H independently of the value of n and the short-range magnetic correlations occur above the Kondo temperature, in good agreement with neutron diffraction experiments in Cerium systems, such as CeCu 6 and CeRu 2 Si 2 [21,22] .

We have established that the dependence with the coupling constant J K of the Kondo temperature for the lattice can be significantly different from the single-impurity one [16,17]. It is rather difficult to deduce experimentally the Kondo temperature especially in Kondo systems which are close to be magnetic and, therefore, to compare the preceding theoretical results with experimental data. However,we can deduce from experimental results in CeRh 2 Si 2 [13], CeRu 2 Ge 2 [23,24] or Ce 2 Rh 3 Ge 5 [25] that the pressure dependence of T K observed in these systems is different from the single-impurity case, in good agreement with our model.

But, on the other hand, the lattice Kondo temperature is closely following the single-impurity one in the other cerium and all ytterbium compounds. If we would like to compare cerium and ytterbium compounds, the 4f electrons are more localized in the ytterbium case than in the cerium one and it results that the intersite interaction J H must be smaller in ytterbium systems than in cerium ones. Thus, in our present model, the pressure dependence of the lattice Kondo temperature must be closer to the single-impurity one in ytterbium compounds than in cerium ones, as experimentally observed.

Further experiments are needed to better understand the conditions which can yield a Kondo temperature for the lattice much smaller than the singleimpurity one.

In this section, we have introduced the Kondo lattice problem, but in fact we have not accounted for the experimental curves of T N versus pressure which present a Quantum Critical Point (QCP). Thus, in order to better describe the behaviour of many Cerium compounds, we have studied the competition between an antiferromagnetic (AF) order and the Kondo effect within the preceding model [26]. We use again the mean-field approximation by taking two order parameters, first λ = λ iσ for describing the Kondo effect and then the average magnetization for each sub-lattice of the antiferromagnetic system. We have obtained a phase diagram showing a first-order transition from an antiferromagnetic order for small J K values to a Kondo phase for larger J K values. The transition at low temperatures between the AF phase to the Kondo one with increasing J K (or equivalently increasing pressure in Ce compounds) is a first order one without any QCP and we have, therefore, obtained a competition between these two phases and not a coexistence between them, as observed in many cerium compounds such as CeAl 2 . This mean-field model does not give any "mixed" AF-Kondo phase, but is able to give successively an antiferromagnetic order and a Kondo phase

with increasing |J K |.
Recently, the competition between the ferromagnetic order and the Kondo effect has been studied within the same Kondo lattice problem using the same mean-field approximation [27,[START_REF] Nunez-Regueiro | Proceedings of SCES04 Conference[END_REF]. The ferromagnetic order exists even for J H = 0 and the magnetic moment is maximum for small J K values.

The Kondo order parameter λ increases and the magnetic moment decreases with increasing J K , but however it remains an interval for J K where there exists a coexistence between the ferromagnetic order and the Kondo effect.

This effect is relatively small with respect to the large coexistence which is obtained for the underscreened Kondo lattice as we will see in the next section.

8

F

The underscreened Kondo lattice model

As we have seen in the previous section, there is a strong competition between the Kondo effect and the magnetic order at low temperature in cerium and ytterbium compounds [17]. On the other hand, the situation is different in uranium systems. Uranium compounds, such as UBe 13 [START_REF] Kleiman | [END_REF], URu 2 Si 2

[30], UNi 2 Al 3 and UPd 2 Al 3 [31,32] present a heavy fermion behavior characterized by large values of the electronic specific heat constant, then order antiferromagnetically, and undergo finally a transition to a superconducting state, as temperature decreases. The situation of UPt 3 is even more complex, because probably short-range magnetic correlations rather than a normal antiferromagnetic order exist [33,34]. Finally, UTe is a dense Kondo system undergoing a ferromagnetic ordering at the large Curie temperature of T c = 102K [35]. We can conclude that there is a strong coexistence between the heavy fermion character and the magnetic order in U-compounds rather than the competition generally invoked in Ce-or Yb-compounds.

Cerium and ytterbium Kondo compounds are well described by the regular Kondo effect corresponding to the 4f 1 and 4f 13 configurations, while uranium ions in such compounds have a configuration 5f n , with n = 2 or 3.

The dual nature of the 5f electrons and the possible coexistence of localized and itinerant 5f states has been developed experimentally [36] and theoretically [37] to explain the behavior of U-compounds.

We present here a theoretical model to describe the "underscreened Kondolattice" treated in the mean-field approximation [38]. This model describes 9 a lattice of ions in the 5f 2 configuration with a 5f -spin S = 1, in Kondo-like interaction with classical conduction electrons with s = 1/2 corresponding, therefore, to an underscreened situation for the f -electrons. The underscreened Kondo effect has been solved exactly for the single-impurity case [39] and studied for the case of two S = 1 impurities [40].

To describe the mean-field approximation for the underscreened Kondo model, the S = 1 localized spins are decomposed into two 1/2 spins τ f n iσ . We impose the condition that these two spins are always ferromagnetically aligned and we take only the two z-components +1 and -1 of each spin S = 1 in order to have a fermionic description of the problem with only four fermionic operators [40].

Thus, the lattice Hamiltonian is given here by:

H = k,σ k n c kσ + iσ E 0 (n f 1 iσ + n f 2 iσ ) + +J K iσ -→ s c i ( -→ τ f 1 iσ + -→ τ f 2 iσ ) + +J H i,j ( -→ τ f 1 iσ + -→ τ f 2 iσ )( -→ τ f 1 jσ + - → τ f 2 jσ ) (5)
where the first term describes the conduction band, the second term the twofold degenerate f -band without width, the third term the intra-site Kondo interaction (with J K > 0) and the fourth term the inter-site interaction between f -magnetic moments.

In fact, a more complete representation will be to take a representation of the S = 1 spins with the three components +1, -1 and 0. We are presently studying the problem with this full quantum description [41]. Then here, we use the mean-field (MF) approximation and expand the preceeding Hamiltonian according to the four following MF values:

M =< S z i >=< τ f 1 iz + τ f 2 iz > m =< s cz i > λ σ = n < f + iσn c iσ > ( 6
)
λ σ is taken independent of the site i and of n = 1, 2, but is spin-dependent.

The resulting bands are one not hybridized f -band at energy E 0 +σJ K m/2+ σJ H M z/4 and two k -f hybridized bands with spin-dependent energy:

E σ ± (k) = 1 2 ( k + E 0 + σJ K (M + m)/2 + σJ H M z/4) ± ± 1 2 ( k -E 0 + σJ K (M -m)/2 -σJ H M z/4) 2 + 2J 2 K λ 2 -σ (7)
where σ = ±1 for the two spins ↑ and ↓ and z is the number of neighboring atoms. The bands for ↑ and ↓ spins are shifted from each other. The resulting hybridization is double here than in the S = 1/2 f -spin case.

Then, the equations giving M , m, λ ↑ and λ ↓ are obtained by taking the number of f -electrons In conclusion, we emphasize the two following points : first, we obtain solutions with both a magnetization (M = 0, m = 0) coming mostly from the not hybridized band and a Kondo character (λ ↑ and λ ↓ = 0 ); this result can account for the coexistence between magnetic order and heavy fermion character observed in uranium compounds such as UTe. Second, we study the influence of the different parameters, J H which tends to give the ferromagnetic order and J K which favours the Kondo effect. A full description of the results will be published elsewhere [42].

N f = σ (n f 1 σ + n f 2 σ ) = 2

The Kondo lattice -spin glass competition

Above the QCP, there exists a very strong heavy fermion character, but several possible behaviours, i.e. the classical Fermi liquid one with eventually a reduced Kondo temperature [13,17] or different Non-Fermi-Liquid (NFL) ones, have been observed in cerium or ytterbium compounds [43,44,45].

On the other hand, the disorder can yield a Spin Glass (SG) phase in addition to the Kondo (mainly NFL) behaviour at low temperatures around the QCP in disordered cerium or uranium alloys. This is the case of the magnetic phase diagram of CeN i 1-x Cu x alloys that has been extensively studied [46,47],while the phase diagram of Ce 2 Au 1-x Co x Si 3 alloys presents the sequence of SG-AF-Kondo phases at low temperatures with increasing Cobalt concentration [48]. The three phases, AF, SG and NFL have been obtained at low temperatures for different concentrations in U Cu 5-x P d x [49] or in U 1-x La x P d 2 Al 3 [50] In fact, when temperature decreases, for alloys with 0.7 > x > 0.1, there appears a "cluster Spin Glass" phase presenting magnetic inhomogeneities, rather than a classical Spin Glass phase. At lower temperatures, there is a percolative process which increases the size of the clusters, to finally yield a ferromagnetic phase at very low temperatures. On the other hand, the transition to the Kondo phase when x decreases below roughly 0.15 is very smooth without any QCP and it seems that some magnetic clusters remain at very low temperatures with x approaching zero.

The transition between a SG phase and a Kondo one, with in addition the presence of a magnetically ordered (ferromagnetic or antiferromagnetic), has been recently studied in detail in the case of Kondo lattices. The SG-Kondo transition was firstly studied theoretically [53], then in the presence of ferromagnetic ordering [54] or antiferromagnetic ordering [55] and finally the model has been improved in order to obtain a good description of the QCP [56], by introducing a transverse magnetic field.

In the first paper on spin glass-Kondo phase transitions, we take a Hamiltonian that couples the localized spins of a Kondo lattice with an additional long range random interaction, like in the Sherrington-Kirkpatrick spin glass model [57]. The localized spins of the Kondo lattice, with localized spins S i , interact with conduction electrons and the Kondo effect is studied in the previously described mean field decoupling scheme [15,17]. The spin glass interaction is represented by the quantum Ising Hamiltonian where only interact the z-components of the localized spins [58,59,60]. The Hamiltonian of the model is:

H = k,σ k n kσ + 0 i,σ n f iσ + J K i [S + f i s - ci + S - f i s + ci ] - i,j J ij S z f i S z f j (8)
where, as usual, J K > 0.

The localized and conduction electron spins are then written in terms of fermion operators f iσ or c iσ . The energies 0 ( k ) are referred to the chemical potentials µ f (µ c ), respectively.

To describe the nature of the spin glass, the coupling J ij is taken here as an independent random variable with the gaussian distribution :

P(J ij ) = 1 J N 64π exp - (J ij + 2J 0 /N ) 2 64J 2 N . ( 9
)
The preceeding expression represents the random distribution of the exchange integrals J ij . The average value J 0 has been taken equal to zero to describe the SG-Kondo transition [53], while the case J 0 < 0 produces a complex phase diagram with spin glass, ferromagnetic, mixed (a spin glass with spontaneous magnetization) and Kondo phases [54]. Finally, the case In the first studied case [53] with J 0 = 0, we obtain three phases : At high temperatures, the "normal" phase is paramagnetic with λ = q = 0 . When temperature is lowered, for not too large values of the ratio J K /J, a SG phase, with q > 0 and λ = 0, is obtained and, for large values of the ratio J K /J, we recover the "Kondo" phase with a non-zero λ value and q = 0. The transition line from the SG phase to the Kondo phase is a first-order one and ends at

J c K at T = 0.
When the temperature is lowered, the transition temperature does not vary very much with the value of J K /J; the separation between the spin-glass and the Kondo phases departs completely from the behaviour of the single-impurity Kondo temperature T K and looks like the separation between the antiferromagnetic and Kondo phases when these two phases are considered [26]. We can also remark that we get here only "pure" Kondo or SG phases and never a mixed SG-Kondo phase with the two order parameters different from zero; this result is probably connected to the approximations used here to treat the starting Hamiltonian.

The diagram obtained theoretically for the SG-Kondo transition can give a rough account for the magnetic phase diagram observed above the Curie temperature for the CeN i 1-x Cu x [46] for small x values when there is a transition from a SG state to a Kondo state and then to the intermediate valence compound CeNi. However, the theoretical model gives an average description, while recent experiments have shown the existence of magnetic clusters and our model cannot be checked from that point of view [51,52].

There is also probably a SG-Kondo transition in CeCoGe Indeed the experimental situation is generally more complex in disordered cerium or uranium alloys and there generally exists an antiferromagnetically (AF) ordered phase in addition to the SG phase and to the Kondo one, which moreover has a Non Fermi Liquid (NFL) close to the QCP. For example, the three phases, AF, SG, and NFL have been obtained at low temperatures for different concentrations in U Cu 5-x P d x [49] or in U 1-x La x P d 2 Al 3 [50] disordered alloys. The sequence AF-NFL-SG occurs with increasing x in U Cu 5-x P d x [49] and the opposite sequence AF-SG-NFL with increasing x in U 1-x La x P d 2 Al 3 [50].

The two theoretical cases SG-Ferromagnetic-Kondo [54] and SG-AF-Kondo [55] have been studied within the same type of mean-field approximation, but here there are at least three order parameters, the spin glass order parameter q, the magnetization M (or the magnetization of each sub-lattice in the AF case) and finally the Kondo parameter λ which is generally taken as spin independent. The calculation is even more difficult in the AF case, because we are obliged to consider two sub-lattices with opposite magnetizations. We present here the results obtained for the SG-AF-Kondo phase sequence with a positive value of J 0 . There are three parameters J K , J 0 and J, but actually, J K cannot be considered as independent of J 0 and J, since an intrasite exchange term would yield both a Kondo effect and a RKKY interaction. This problem has been already discussed and we have taken the two possible choices : either we take the values of the three parameters independent from each other; or we adopt some conjecture relating J K , J 0 and J. In our theoretical approach, the variance J is kept constant and we take the following relationship between J 0 and J K :

J 0 J = α J K J 2 . ( 10
)
The quadratic choice in Eq. ( 10) is taken here in order to have a rough simulation of the fact that the RKKY interaction behaves as J 2 K , as already assumed in Ref. [16]. A phase diagram obtained for a typical value α = 0.0051 is shown in Figure 1.

The numerical solutions of the order parameters yield the sequence of phases SG-AF-Kondo shown in Figure 1. The Kondo temperature T k is decreasing with decreasing values of J K /J down to J K /J ≈ 15. From this point onwards, there is a phase transition leading to an AF order with the Néel temperature T N monotonically decreasing with J K /J. For still smaller values of J K /J (and therefore smaller values of J 0 /J), the spin glass behaviour becomes dominant. given by the eq.( 8) and of an additional term given by :

H x = -2Γ i S x f i (11)
The coupling J ij in eq.( 8) is described by a gaussian distribution of zero mean J 0 as previously used for the SG-Kondo transition [53].

Numerical solutions of the saddle-point equations allow us to draw in figure 2 the magnetic phase diagram in the J K versus T plane, using the relation Γ = αJ 2 K between the transverse field and J K , fixed values of J and D, with D/J = 12 and a value of the parameter αJ : αJ = 0.01348 [56]. Figure 2 shows three different phases : At high temperatures, the "normal" phase is paramagnetic with vanishing Kondo and spin glass order parameters, i.e. λ = q = 0. When temperature is lowered, for not too large values of the ratio J K /J, a second-order transition line is found at T = T 1 to a spin glass phase with q > 0 and λ = 0. The critical line T 1 (J K ) ends at a quantum critical point J c K1 . Then, for larger values of the ratio J K /J, there is a second order transition line starting at J c K2 to a Kondo state with λ ≥ 0. for the transverse field Γ = αJ 2 K , for fixed values of J and D, with D/J = 12, and for two values αJ = 0.01348 (solid lines) and αJ = 0.01344(dashed line).The critical (solid) second order line T 1 for low values of J K separates the paramagnetic phase (q = λ = 0) for high temperatures from the spin-glass phase (q ≥ 0, λ = 0) at low temperatures and ends at a QCP. The critical second order solid line T 2 for large values of J K separates the paramagnetic phase from the Kondo phase (λ ≥ 0, q = 0) and ends at a second QCP. The dotted line represents the "pure"(i.e. single impurity) Kondo These results are very sensitive to the value of D/J and αJ. For D/J = 12 and for example αJ = 0.01344, the two QCP disappear, as shown in figure 2, where the new transition is given by the dashed curve labelled T'. Thus, the occurrence of two QCP is extremely sensitive to the precise of value of αJ. However, the phase diagram shown in figure 2 with two very close QCP improves considerably the theoretical description of the SG-Kondo transition [56]. We can also remark that we get here only "pure" Kondo or SG phases and never a mixed SG-Kondo phase with the two order parameters different from zero, as already noticed in our mean field calculations [26].

A QCP has been observed experimentally in several cerium and uranium disordered alloys [48,49,50] and the phase diagram shown in Figure 2 improves considerably the description of the spin glass-Kondo transition with respect to Ref. [53]. However, the experimental situation appears to be really complicate, as shown for example by recent experiments on CeN i 1-x Cu x alloys [51,52]. Thus, further experimental work is certainly necessary, but however the models presented here can account roughly for the behavior of heavy fermion disordered alloys, especially in the vicinity of the quantum critical point.

Conclusion

We have presented here a brief review on recent works on the Kondo lattice problem. We have always used here the same mean field approximation for the Kondo effect, which is simple and gives correct results, but yields however a too brutal (typically mean-field-like) transition rather than the expected continuous one around the Kondo temperature. Here, we have firstly considered the case of cerium (ytterbium) compounds corresponding to a configuration close to 4f 1 (or 4f 13 ) with one 4f electron (or one hole) and then the case of uranium described here by a configuration close to 5f 2 . The "normal" Kondo effect describes the first case, while the "underscreened" Kondo effect is more appropriate for the second one. Finally, we have discussed the interaction between the Kondo effect and a disordered phase that we describe as a spin glass phase, but in fact the spin glass-Kondo problem is more complicate, as shown for example by recent experiments on CeN i 1-x Cu x alloys which have yielded a new "cluster spin glass". Finally, further experimental results and a more sophisticated theoretical approach, eventually beyond our mean-field approximation, are necessary to improve the description of anomalous rare-earth and actinide systems.

  the chemical potential µ have to be determined self-consistently in order to keep the average numbers of f and conduction electrons respectively equal 5

10 Page

 10

  and a fixed number of conduction electrons n = σ < n c σ >. The magnetic moment M arises from f -electrons and m from conduction electrons. E 0 is also expressed as a function of the four variables. The solutions for M , m, λ ↑ and λ ↓ are obtained for the Kondo (which corresponds to J K > 0) and ferromagnetic (which corresponds to J H < 0) case at T = 0 by solving the self-consistent equations.

12 http

 12 disordered alloys. Thus, a SG-Kondo transition has been observed experimentally with increasing concentration in disordered around the QCP. Recent experiments performed in CeN i 1-x Cu x alloys have shown that the phase diagram of these alloys is more complicate[51,52].

13

 13

J 0 > 14 Page

 014 0 corresponds to an antiferromagnetic phase in competition with Spin Glass and Kondo phases[55].The Kondo and spin glass transitions are both described within the mean field approximation. Using functional integrals techniques and a static, replica symmetric Ansatz for the Kondo and spin glass order parameters, a mean field expression for the free energy and the saddle point equations for the order parameters have been derived. Numerical solutions of the saddle-us to determine the Kondo order parameter λ and the spin glass (SG) one q.

  situation is also very complicate [61].

16 Page

 16

Finally, Figure 1 17 PageFigure 1 :

 1171 Figure 1: The phase diagram T /J versus J K /J showing the sequence of sg (Spin Glass), AF (Antiferromagnetism) and Kondo phases. The variance J is kept constant. The dotted line means the "pure" (i.e. single-impurity) Kondo temperature.

Figure 2 :

 2 Figure 2: Phase diagram in the T -J K plane as a function of T /J and J K /J

  temperature T K and the dashed-dotted line represents the residual magnetic moment χ c (J K ) on the line T = T 2 . The critical line (dashed) labelled by T' corresponds to αJ = 0.01344 and has no QCP.

21 Page

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  D. Eom, M. Ishikawa, J. Kitagawa and N. Takeda, J. Phys. Soc. Japan 67,

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	write λ = λ iσ for all sites and Γ = Γ ijσ for nearest-neighboring sites and
	zero otherwise. It is important to note that, in all the present mean-field
	descriptions of different Kondo lattice problems, we always use the same
	mean-field approximation, i.e. λ = λ iσ in the non magnetic case or a
	spin-dependent λ σ = λ iσ in the magnetic case, for describing the Kondo F o problem, while we use different mean-field terms to describe the different
	r parameters, like here the short-range magnetic correlations or then different
	descriptions of the magnetic moments to study different cases of magnetic P orderings. In fact, the mean-field approximation used here for the Kondo e effect has the great advantage to describe simply and correctly the main fea-e r tures of the Kondo effect, but obviously it has the disadvantage of giving a
	too brutal mean-field-like transition rather than a continuous one. R After performing this approximation, one deals with a one-electron Hamil-e w e i v tonian representing two hybridized bands : the conduction band of width 2D
	O n
	l y
	. Considering translational
	invariance and taking into account that there are no magnetic states, we can
	4

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Jean-Marie Dubois
email: dubois@mines.inpl-nancy.fr

Keywords: X-ray spectroscopy, transition-metal compounds, surface tension, ordered intermetallics, friction, electron energy states

This article attempts to understand better the origin of low friction and zero solid-solid adhesion observed on quasicrystals placed in vacuum. We present an ensemble of experimental data that allows us on the one hand to estimate the true surface energy of unoxidized quasicrystals and approximants and on the other hand to point out a correlation with electronic densities of states.

Key Words: surface energy, friction, cold-welding, density of states, quasicrystals

Introduction

That Al-based quasicrystals exhibit low friction against hard materials like Cr-steel, diamond or alumina was first evidenced in 1992 using a standard scratch test procedure [START_REF] Dubois | Useful Quasicrystals[END_REF]. This test however leads to a rather weak contrast between sliding friction measured on multi-grain quasicrystals and materials of similar bulk mechanical properties such as the Vickers hardness H v . This weak contrast is due to the tribo-oxidation of (essentially) aluminium, which obscures the friction of the bulk material underneath the oxide that forms upon sliding the indenter in ambient atmosphere and contributes to the growth of a third-body material left within the friction trace. In vacuum, the picture is much clearer as soon as the residual oxygen pressure in the atmosphere is kept below a threshold of approximately 10 -6 mbar. It is then most often possible to adjust the velocity of the sliding pin relative to the substrate in such a way that a full mono-layer of oxide can not grow during the time elapsed between two successive passages of the indenter. The oxide is then removed by the contacting indenter and essentially unoxidized surfaces come into contact (except at the very beginning of the test).

A clear difference is observed in such conditions on the friction coefficient of hard steel riding on materials of very similar hardness (and as a matter of fact, bulk and shear moduli). A series of data is presented in Fig. 1 to illustrate this standpoint. Steel riding on itself produces a friction coefficient 3 to 4 times larger than on an icosahedral Al 59 Cu 25.5 Fe 12.5 B 3 sample of quite comparable hardness. Similarly, a fretting test in vacuum shows that the adhesion force measured between a spherical hard steel pin contacting a flat fcc Cu or Al substrate is about 100 times larger than the force measured in identical conditions against icosahedral samples [START_REF] Dubois | Useful Quasicrystals[END_REF].

The present article focuses at a presentation of more experimental data recorded in the same conditions, but for a variety of quasicrystals and approximants and for binary Altransition metal (TM) compounds. On this simple experimental basis, we point out a correlation between friction as deduced from our experiment on pure TM samples and their (known) surface energy, thus demonstrating that we indeed probe the actual surface energy of each compound. We then turn to cold welding and show how the adhesive force to various indenters is linked to partial densities of states in the studied intermetallic substrates.

Experimentals

2.1. Samples A series of intermetallics based on aluminium was prepared from pure constituents using standard metallurgy techniques, followed by crushing into powder of selected granulometry (mesh size in the range 30 to 50 µm) and sintering into pellets of 20 mm in diameter and a few mm in thickness. After polishing in water down to mirror polish, drying in air and cleaning in an ultra-sound bath loaded with ethanol, these samples could be used for tribological testing. We also used Al-Pd-Mn single grains of the icosahedral phase and of the ξ'-approximant of the decagonal phase that were kindly supplied to us by M. Feuerbacher from FZ-Juelich research centre.

Experimental set-ups

Two different experimental set-ups were used, respectively for friction and fretting data. The first one is a conventional pin-on-disk system, but housed in a vacuum chamber that may be evacuated down to 10 -6 mbar or less. At this later residual pressure, the fraction of oxygen still present in the chamber is sufficient to grow one single layer of alumina every second on pure fcc Al. The linear velocity of the disk with respect to the pin may be adjusted in such a way that the time elapsed between two successive passes of the indenter is comparable or smaller to the one necessary to grow a full mono-layer. In our experiment, we used the following parameters: linear velocity (pin vs disk) = 0.05 m/s; diameter of trace = 10-12 10 -3 m; load = 2N; base pressure= 10 -6 mbar or less.

The indenter was always a fresh hard steel ball of 6 mm in diameter. After each experiment, the wear experienced by the indenter and the flat surface counterpart was examined by optical and scanning electron microscopy. In most cases, the wear was negligible and no third-body formed, as evidenced by the absence of particles spalling off the bottom of the track and the absence of debris in the trace. Counter-examples were found however, as we will discuss later in this article.

The second set-up we used comprises a fretting test mechanism, also housed in a vacuum chamber that can be evacuated to typically the same residual pressure as in the previous experiment. A fretting movement is applied to the indenter, parallel to the studied flat surface by a piezo-electric actuator. A typical run consists of 10 s of alternating movements of 30 µm amplitude at a frequency of 200 Hz. The load and curvature of the pin are taken in such a way as to probe all surfaces under identical contact pressure relative to the elastic yield of each sample. The force that opposes the opening of the gap between pin and sample is measured after each run. It is called the stick force in the following. The values reported hereafter are averages over several hundred runs.

More details about both experimental facilities may be found in a companion paper to the present article [START_REF] Sales | [END_REF].

Results

Calibration

With the view that friction in vacuum can be related to the surface energy γ S of the clean specimen, we have first studied a number of reference samples of known hardness and surface energy. These samples were chosen among Al, pure transition metals (TM = Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zr, Ta, Hf, W) and window glass, hard and mild steels, sintered alumina. At the end of the pin-on-disk test, material from steel pin has been transferred to the alumina surface, which in turn changes the relevant surface energy from that of alumina to that of iron (to a 1 st order approximation, neglecting the effect of alloying elements on γ S ) with no change of the substrate hardness.

Figure 2 shows the value of the friction coefficient µ for 3d, 4d and 5d TM elements, arranged along the x-axis according to increasing number of d+s electrons. In the same figure, the data is represented after correcting the value of µ from the ploughing contribution to friction that is assumed simply proportional to the inverse of the hardness H v , i.e. µ'=µ-α/H v (the model is therefore not valid for soft metals like Sn or Pb). A clear similarity is found with the variation of the surface energy obtained from ab-initio calculations [START_REF] Vitos | [END_REF] or from experiments [START_REF] Eustatopoulos | Wettability at High Temperatures[END_REF]. This is not true however for the metals that have the highest surface energy, i.e. Co, Hf and W. For those metals, µ' is found dramatically below the expected value, thus pointing out a severe limitation of the method, namely that unoxidized surfaces must come into contact. With high γ S values, as for W, one expects indeed that covering the surface with adsorbates at pressures in the range offered by the present instrument will occur with high stick coefficient, so that the surface is always lubricated. As a consequence, µ is artificially reduced. A simple test of the occurrence of such an artefact is achieved by introducing a minute amount of oxygen in the chamber while the pin is running. Unoxidized surfaces show an immediate drop of µ, whereas lubricated ones show only a gradual decrease, when not an increase. We could assess that all metals and reference samples, except Co, W and Hf, behave as clean surfaces in our experiments.

Accordingly, we assume that friction of hard steel in vacuum is dominated by an adhesion term towards the surface under study and contains also a ploughing contribution, so that we can write (except for too soft materials):

µ = α/H v + β W PS (1
) where α and β are fit parameters, and W PS stands for the adhesion energy between pin and solid. Considering that wear is un-important, as can be ascertained easily by microscopy inspection of the trace after the end of the test, we may replace W PS in Eq. ( 1) by the reversible adhesion energy between pin and disk. Eq. ( 1) reads then:

µ = α/H v + β(γ S + γ P -γ SP ) (2
) where γ P stands for the surface energy of the steel-made pin (≈2.4 J/m 2 ) and γ SP for the interfacial energy of the pin on the solid surface.

Since this later term is unknown, a further assumption must be made. Noticing that γ SP cannot be very different from the immersion energy of steel on a solid surface, we take the difference γ Pγ SP as zero:

µ = α/H v + βγ S (3)
When doing so, we exaggerate the result on γ S or in other words, we give an estimation of the upper limit of γ S for the various materials (which at the present time, is better that no value at all). Notice however in the following that the error on γ S can be a few tens of a percent.

A single set of α and β parameters may be adjusted (Fig. 3) in order to match calculated values of the friction coefficient to experimental ones (α = 34±0.5; β = 0.322±0.005, linear regression coefficient r = 0.96). Inverting Eq. ( 3), and within the limits of the assumptions made above, this in turn may be used to decipher a correct estimate of the true surface energy of samples of suitable hardness (above about 100 Vickers units) and surface energy (below typically 2-2.5 J/m 2 ).

Friction in vacuum

The data collected so far are assembled in Table 1. We produce no data for Al 13 Co 4 and Al-Ni compounds because the wear visible on our samples at the end of the tests was making application of this method invalid. Interestingly enough, values of γ S are found very much below the surface energy of the individual constituents for all icosahedral compounds, but also for superstructures of the β-phase in the Al-Cu system.

Fretting test

The tendency to cold-welding, as measured by the stick force, exhibited by icosahedral or high-order approximants is small, but depends on the nature of the counterpart. Whereas the stick force is sometimes found below the instrumental resolution when probed against hard steels, it is found significantly large against Al-based and Ti-based alloys [START_REF] Sales | [END_REF], but nonetheless far less than the one measured with steel against fcc Cu and fcc Al [START_REF] Dubois | Useful Quasicrystals[END_REF]. A summary of our findings for a superstructure compound of the B2-Al-Cu phase, the icosahedral Al-Cu-Fe and Al-Pd-Mn compounds, the ξ'-Al-Pd-Mn and orthorhombic O1-Al- Cr-Fe(-Mo) approximants of the decagonal phase is given in Fig. 4. We discuss the data in the next section.

Discussion

That the stick force depends on the nature of the metallic counterpart (here essentially Al or Fe) strongly suggests that it results from the formation of a band between electronic states in the sample and pin. In a very crude assumption, we shall assume it proportional to the number of sp, respectively d, states that merge with the Al, respectively Fe, states at the Fermi level (E F ) present in the Al-based and steel pins, respectively. The number of s, p and d states at E F in our samples is obtained for the very same specimens by X-ray emission spectroscopy as explained elsewhere [START_REF] Belin-Ferré | [END_REF]. The result of this simple computation is shown in Fig. 4b. Although not perfect, the matching with the experimental data given in Fig. 4a is good enough to support our assumption.

In an attempt to rationalise our estimate of the surface energy, we plot it in Fig. 5a against the number of TM s+d electrons, using the same x-axis coordinate as in Fig. 2a. Again, two clear maxima are found at 5 and 9 e -/at, respectively, and a minimum at 7 e -/at. However, the data for the icosahedral phases falls significantly below the values of γ S measured for the crystalline compounds (including compounds of nearby composition, which rules out the occurrence of an artefact like the one encountered with Co or W). The difference with respect to crystal compounds must therefore be intrinsic to the quasicrystalline structure and may be related to the formation of critical states that cannot be accounted for using the present free-electron picture. A similar trend is observed (Fig. 5b) if one plots γ S as a function of the total number of (sp+d) electrons per atom in the Al-TM alloy. Again, icosahedral compounds exhibit a smaller value of γ S compared to crystals of closely related total electron density. As expected, the overall trend observed on the figure is nevertheless a decrease of γ S with the filling of the hybrid sp-d band.

Altogether, it appears from our data that both friction and fretting in vacuum are dominated by electronic effects in our samples, and that the influence of mechanical deformation is rather limited, when not negligible.

Conclusion

To the best of our knowledge, the present approach is the first that delivers a systematic assessment of the genuine surface energy for a whole family of Al-based compounds. Although it relies on rather crude approximations, the data for γ S are valuable i) because no other method is able to probe so many different compounds in an economic way, and ii) because the values of γ S so-obtained are upper limits for the actual surface energy. We found that quasicrystals of high lattice perfection exhibit a surface energy that is smaller than that of the pure constituents by a factor 2 to 3, and falls in the range 0.5-0.8 J/m2. Conversely, compounds with a pronounced metallic character show values of γ S that are not very different from that of the constituent metals. Epitaxy of metallic films on clean quasicrystal surfaces manifests growth morphologies that are consistent with this data [6]. Friction is always found smaller on a quasicrystal of good lattice perfection. Fig. 2: Variation with the number of TM s+d electrons of the surface energy (from literature) (a) and friction coefficient µ' corrected for the ploughing contribution (b). The same trends are observed on both figures for the 3d series (solid squares), with marked maxima at 5 and 9 electrons and a minimum at 7, except for Co that is found very much below the expected value of µ'. A similarity between (a) and (b) figures is also observed for the 4d (grey diamond) and 5d (open triangles) TM series, except W that shows a too small value of µ'. For completeness of the figure, aluminium is indicated as well by a cross symbol (hence, the label sp+d on the x-axis).

  Surface energy of complex -and simple -metallic compounds as derived from friction test in vacuum J.-M. DUBOIS, +* M.-C.de WEERD, + J. BRENNER, † M. SALES, ‡ G. MOZDZEN, ‡ A. MERSTALLINGER ‡ and E. BELIN-FERRÉ x + Laboratoire de Science et Génie des Matériaux et de Métallurgie, Ecole des Mines, Parc de Saurupt, F-54042 Nancy † Austrian Centre of Competence for Tribology Research GmbH, Viktor-Kaplan Str. 2, A-2700 Wiener Neustadt ‡ Austrian Research Centre-Seibersdorf Research GmbH, A-2444 Seibersdorf x Laboratoire de Chimie Physique Matière et Rayonnement, 11 rue P. et M. Curie, F-75231 Paris Cedex 05

Fig. 1 :

 1 Figure captionsFig.1: Friction coefficient measured against hard steel in vacuum (a) and against diamond in air (b) for a variety of samples. The data is presented versus the materials hardness. Friction is always found smaller on a quasicrystal of good lattice perfection.

Fig. 3 :

 3 Fig. 3: Correlation between experimental values of the friction coefficient and values calculated according to Eq. (3). Hardness was measured, using a standard Vickers set-up and γ S data are from literature [3,4]. Small solid and open squares represent TMs, large squares are for alumina (bare: grey; open: covered with a Fe film) and window glass; the cross is for fcc Al.

Fig. 4 :

 4 Fig. 4: (a) Experimental stick force (in mN) measured in vacuum on five compounds as labelled along the vertical axis and against spherical indenters: Al-based alloy (dark grey) and Fe-based steel (light grey). (b) Stick force (in arbitrary units), calculated assuming it is proportional to the number of electrons necessary to form a sp band with the Al-based indenter (dark grey) or a d band with the Fe-based indenter (light grey).

Fig. 5 :

 5 Fig. 5: Estimated surface energy γ S versus number of s+d electrons of the TM constitutive species (a). The same γ S data is plotted in (b) against the total number of s, p and d electrons per atom in the compound. The open square is for icosahedral Al-Pd-Mn and the diamond for i-Al-Cu-Fe.

  sp+d)e -(e -/at.) sp+d)e -(e -/at.) γ S (J/m 2 ) s+d)e -(e -/at.) Nb. (sp+d)e -(e -/at.) γ S (J/m 2 )

Table 1 :

 1 Experimental results and estimated values of γ S .

	Compound	Crystal	Vickers	Friction	Estimated
	(at.% or atoms)	structure	Hardness	coefficient	γ S (J/m 2 )
			(load 0.5 N)	± 15%	± 25%
			± 8%
	Al 3 Ti	tetragonal	604	0.6	1.72
	AlTi	B2-cubic	277	0.7	1.86
	Al 3 V	tetragonal	427	0.7	1.97
	Al 9 Cr 4	γ-brass	695	0.48	1.37
	Al 8 Cr 5	γ-brass	720	0.49	1.40
	Al 11 Mn 4	monoclinic	476	0.41	1.09
	λ-Al 13 Fe 4	monoclinic	815	0.69	2.0 4
		B2-cubic	417	0.64	1.78
	Al 3 Ni	orthorhombic	662	0.45	1.27
	Al 2 Cu	tetragonal	550	0.44	1.21
	γ-Al 9 Cu 4	cubic	480	0.37	1.0
	AlCu	hexagonal	806	0.3	0.82
	oF-Al 3 Cu 4	orthorhombic	710	0.24	0.6 2
	φ-Al 10 Cu 10 Fe	cubic	650	0.32	0.9
	β-Al 55 Cu 30 Fe 15	B2-cubic	680	0.31	0.8 4
	ω-Al 70 Cu 20 Fe 10	tetragonal	640	0.75	2.2
	Al 62 Cu 25.5 Fe 12.5	icosahedral	780	0.26	0.5 5
	Al 59 B 3 Cu 25.5 Fe 12.5	icosahedral	790	0.21	0.5 4
	Al 70 Pd 20 Mn 10	icosahedral	750	0.3	0.8 2

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Acknowledgements

The present work was supported by 'ERT Quasicristaux Industriels ' from French Ministry of Research and Technology and by Growth 2000 project 'Smart Quasicrystals' of the European Union under contract n°G5RD-CT-2001-00584. We are grateful to M. Feuerbacher, Juelich, for providing single domain specimens.


Marc Heggen

Michael Feuerbacher

Metadislocation arrangements in the complex metallic alloy ξ ξ ξ ξ´-Al-Pd-Mn

Keywords: plasticity, metallic alloys, dislocations complex metallic alloy, metadislocations, metadislocation networks, ξ´-Al-Pd-Mn

Metadislocation arrangements in the complex metallic alloy ξ´-Al-Pd-Mn

Metadislocations (MDs) are novel structural defects firstly observed in the complex metallic alloys (CMAs) ξ´and ψ-Al-Pd-Mn [1]. They provide a physically useful principle of the plasticity of large-unit-cell materials. MDs comprise structural elements of both phases, ξ´as well as ψ-Al-Pd-Mn. Both phases are structurally closely related and possess the same a and b cell parameters. However, the cell parameters in the c-direction are related by a factor (τ+3) where τ ≈ 1.618 is the irrational number of the golden mean. The lattice parameters are a = 23.89 Å, b = 16.56 Å, c ξ´ = 12.56 Å, and c ψ = (τ+3) c ξ´ = 58.00 Å [2].

The core of a MD is a line defect which ideally extends along the [0 1 0] direction. It is associated to a certain number of planar defects, so-called phason planes, which are ideally (0 0 1) oriented. The term phason plane refers to the analogy to phason defects in Quasicrystals, although ξ´and ψ-Al-Pd-Mn are crystalline phases processing translational symmetry.

Different types of MDs associated with 16, 10, 6, and 4 phason planes have been observed [3]. Recently MDs associated with 2 phason half-planes were observed [4]. Different types of [1], by a modified Volterra process [5], or by calculating the lattice mismatch at the interface between the ξ´and the ψ-phase [6,7]. The Burgers vectors and numbers of phason planes of different types of MDs are compiled in Table 1. The Burgers vector moduli and the numbers of phason half-planes follow opposite sequences: a MD with a high number of phason planes possesses a small Burgers vector and vice versa. The MD associated with two phason planes possesses the largest Burgers vector of 4.80 Å. Hence, the Burgers vectors of all MDs are small fractions of the lattice parameters c ξ´ and c ψ and their local elastic strain energy is small. Therefore, in these materials, Metadislocations are energetically more favourable than perfect dislocations.

To date, the investigation of defects in ξ´-Al-Pd-Mn is limited to the structure and properties of single MDs and their associated phason planes [1,3,8,9]. However, frequently complex, network-like MD arrangements are observed. In this work the properties of MD arrangements are analyzed.

Experiments

A ξ´-Al-Pd-Mn single crystal was grown by means of the Bridgman technique. A sample of approximately 5×2×2 mm 3 in size was cut with the long axis inclined 45° to the [0 1 0] axis. The sample was uniaxially deformed in compression to 0.5 % strain. Like icosahedral Al-Pd-Mn the ξ´ phase is brittle at room temperature [10,11]. Therefore deformation experiments were performed at an elevated temperature of 700 °C.

After plastic deformation the sample was quenched on a cold metal plate and prepared for transmission electron microscopy (TEM) by standard techniques. The microstructural investigation was performed in a JEOL 4000EX electron microscope. MDs are imaged with the TEM sample oriented along the [0 1 0]-direction under bright-field Laue conditions, i.e. applying a central aperture around a symmetric arrangement of spots. Figure 4 shows a phase boundary between the ξ´-phase (upper left part of Figure 4) and the ψ-phase (lower right part of Figure 4), which appears darker than the ξ´-phase. The presence of both phases was confirmed by selected area electron diffraction patterns. The micrograph shows five MDs associated with 10, and six MDs associated with 16 phason planes, respectively. The total lattice mismatch at the phase boundary is (5 × 1.13 Å) + (6 × -0.70 Å) = 1.45 Å. The extension of the phase boundary in [0 0 1] direction is 450 nm, hence the mean residual elastic strain at the interface is 0.03 %. This is a remarkably small value. Especially, since the lattice parameters in c direction are incommensurate -they are related by a factor of (3 + τ).

Results and Discussion

MDs forming a phase boundary possess low local strain at the individual MD cores, especially when MDs with high numbers of phason planes are used. Mixing different kinds of

MDs creates a phase boundary with a low mean elastic strain at the interface, while adding up MDs of the same type and hence the same Burgers vector, adds up lattice mismatch at the phase boundary. A phase boundary with a perfect mean lattice mismatch can be created if MDs with opposite Burgers vector direction are present in the right proportion. The Burgers vectors of MDs subsequent in the series of MD types (cf. Table 1), for example MDs with 10 and 16 phason planes, are related by a factor of -τ . Hence a perfect mean lattice mismatch at the interface can be created, if these MDs are present in a proportion 1 : τ .

Conclusion

In the present work it is shown that MDs can, like dislocations in structurally simple materials, form dipoles or dissociate into partials with smaller Burgers vectors. The formation of partials leads to a decrease of the local elastic line energy and an increase of the fault plane energy, which have to be balanced in order to find the energetically most favourable configuration and geometry. Splitting of a MD into more than two partials is possible.

  Burgers vectors. They can be determined either by a Burgers circuit around the MD core

Figure 1

 1 Figure 1 shows two examples of interacting MDs. Arrows 1 and 2 point at two inversely oriented MDs sharing six phason planes. Their Burgers vectors are 1.83 and -1.83 Å, respectively (cf. Table1). Hence this defect can be interpreted as a MD dipole or as part of a

3 Figure 2 aFigure 3

 323 Figure 2 a shows a MD associated with ten phason planes (arrow 1) which is connected to a MD with six phason planes (arrow 2). They are mutually connected by six phason planes. Four phason planes are remaining, three on the right hand side and one on the bottom part of the micrograph. This situation can be interpreted in terms of MD splitting. A MD with 4 phason planes has split into an equally oriented MD with ten phason planes and an inversely oriented MD with 6 phason planes. Using Table 1 it can be shown that the Burgers vector of a MD associated with four phason planes is exactly equal to the Burgers vectors of the MD arrangement in Figure 2 a, i.e. 1.13 Å -( -1.83 Å) = 2.96 Å. During the process of MD splitting, the local elastic strains at the MD cores is reduced. On the other hand, the creation of six additional phason planes introduces additional fault plane energy. Hence the process of MD splitting has to be treated similarly as splitting of dislocations into partials in structurally simple materials. The total energy balance of all defects involved has to be considered.

Fig. 1 :

 1 FIGURE CAPTIONS

Fig. 2 :

 2 Fig. 2: Three examples of Metadislocation splitting. Arrows indicate a) a MD with ten (1) and a MD with six (2) phason planes, b) a MD with six (1) and a MD with four (2) phason planes, and c) a MD with ten (1) and two MDs with six (2 and 3) phason planes (additional numbers: see text).

Fig. 3 :

 3 Fig. 3: Metadislocation network. The orientation of MDs is indicated by triangles. Furthermore, the numbers and the arrows indicate the number of associated phason planes and the direction and length of the Burgers vector, respectively.

Fig. 4 :

 4 Fig. 4: Phase boundary between ξ´-( upper left part) and ψ-Al-Pd-Mn (lower right part). The interface is created by a stack of metadislocations.

  Figure 1 182x127mm (150 x 150 DPI)

Table 1

 1

). Hence this defect can be interpreted as a MD dipole or as part of a prismatic loop. Arrows 3 and 4 indicate a dipole of MDs associated with ten phason planes.

TABLE / TABLE CAPTION

 /CAPTION

	Number of phason planes	Burgers vector
	16	-0.70 Å
	10	1.13 Å
	6	-1.83 Å
	4	2.96 Å
	2	-4.80 Å

Table 1 :

 1 Number of phason planes and Burgers vectors of all experimentally observed MDs.The Burgers vector direction is related to MDs with phason planes oriented parallel to the [1 0 0] direction[7]. Analogously, MDs with phason planes oriented parallel to the [ 1 0 0] direction possess inverse Burgers vectors.

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M Feuerbacher

Marc Heggen

On the concept of metadislocations in complex metallic alloys

Keywords: plasticity, defects user supplied): complex metallic alloys, metadislocations Complex metallic alloys, Metadislocations, Defects, Plasticity

Metadislocations are a novel type of structural defect in complex metallic alloys.

We present an experimental characterization of metadislocation loops in ξ'-Al-Pd-Mn by means of transmission electron microscopy. Furthermore we discuss the mode of metadislocation motion (glide vs. climb) and the geometrical structure of metadislocations in other orthorhombic complex metallic alloys.

These materials are characterized by a complex atomic structure and generally possess large lattice constants. The number of atoms per unit cell is high, ranging from a hundred to some thousands. The local atomic order in CMAs is characterized by a cluster substructure.

Frequently, icosahedral-symmetric atomic arrangements, e.g. Mackay or Bergman clusters are found [1].

As direct consequence of the salient type of local order, CMAs possess a variety of physical properties, which significantly differ from those in ordinary metals (see e.g. [2]). In this paper we will address a recently discovered type of structural defect, the metadislocation.

Metadislocations have been firstly observed in the orthorhombic phase ξ'-Al-Pd-Mn [3] and for the first time provide a physically evident plastic deformation mode for large-unit-cell The different types of metadislocation and their structural properties have been investigated in some detail experimentally [4] and theoretically [5]. It is the aim of the present paper, to discuss some hitherto unregarded aspects of metadislocations, their mode of movement, and to examine the possibility of metadislocations in phases other than ξ'-Al-Pd-Mn . Å, and 70.1 Å have been reported [START_REF] Yurechko | Phase Diagrams in Materials Science[END_REF].

Metadislocations in

Metadislocations represent a novel type of defect addressing a basic issue of the plasticity of large-unit-cell materials. They are partial [0 0 1] edge dislocations, possessing, on the one hand, energetically favourable short Burgers-vector magnitudes. On the other hand, glide motion of these partials is possible without alteration of the structure (e.g. the introduction of a stacking fault or an antiphase boundary). This is due to the particular accommodation of the dislocation core into the ideal structure involving phason planes. The associated phason planes are a vital constituent of the metadislocation and hence have to be dragged along with the dislocation upon movement of the latter (see section 4). Fig. 1 0] segments of the corresponding loops. However, the above considerations hold for metadislocations as well, and hence the question for the other segments, completing loops with the [0 1 0] segments must be approached in order to obtain a complete view.

We have carried out TEM investigations on deformed ψ-Al-Pd-Mn single crystals grown by the Bridgman technique. The material was deformed in compression to 8 % strain at 725 °C. Due to the limited specimen-tilting angle in the TEM, it is experimentally impossible to turn the specimen as far as 90° to the (0 1 0) zone axis in order to confirm directly that the loops observed correspond to metadislocations. However, we have strong evidence that this is so.

First, the density of line segments along the [0 1 0] direction in Fig. 3 is consistent with the We can therefore safely conclude that the loops shown in Fig. 3 a correspond to metadislocations, and we accordingly term them metadislocation loops.

Metadislocations, usually imaged along the [0 1 0] direction [3,4,5], correspond to end-on observations of the long [0 1 0] segments. The loops are however completed by additional [1 0 0] segments. The aspect ratio of the [0 1 0] and [1 0 0] segments is very large, the lengths of the latter segments are of the order of 100 nm. This is consistent with the experimental observation that these segments are usually not observed experimentally when images are taken along the [0 1 0] direction. Note that, since metadislocation loops have [0 0 1] Burgers vectors, they are pure edge loops, i.e. they possess only edge-type and no screw type segments.

Mode of metadislocation motion

As yet, not direct observation of metadislocation motion, e.g. by in-situ tensile tests in a transmission electron microscope could be made. Therefore the question of the mode of metadislocation movement, in particular whether metadislocations motion takes place by glide or climb can to date not be resolved experimentally. We will present, however, in the following a number of arguments, which strongly indicate that metadislocations move by climb rather than glide.

The first argument, in fact, has already been presented in section 3, where it was experimentally demonstrated that metadislocation loops are pure edge loops. Edge loops can vertex jumps is necessary. The number of necessary vertex jumps is now limited to the core region and hence the number of vertex jumps can be numeralized no matter how far the associated phason planes are extended. Since each vertex jump physically corresponds to a number of local atomic jumps (see [START_REF] Beraha | [END_REF] for a correlation of atom jumps and vertex jumps in the original tiling) it is directly evident that climb motion is connected with much less effort concerning local atomic rearrangement than glide motion.

Note however, that long-range atomic transport is necessary for the climb process because, in contrast to the glide process, it is a non-conservative mode of dislocation motion. Depending on the direction of movement, either atoms or vacancies have to be moved to the dislocation core region. This fact is also directly reflected in our schematics: In Fig. 4 b, showing the glide step, each newly occurring vertex in the final hexagon lattice has a corresponding vertex in the initial lattice. This is not the case for the climb step shown in Fig. 4 c. In present depiction, the final lattice has 28 new vertices which have to be balanced against 24 vertices in the initial lattice. That is, four new vertices have to be created, or, in terms of atom positions, a number of atoms have to be transported to the core region (the exact number of which, of course, can only be calculated on the basis of an atomic model of the metadislocation core and the phason lines).

The necessity of atomic transport is not a principal objection against climb motion. Climb motion, as a non-conservative process, requires long-range atomic motion no matter whether it takes place in a complex metallic alloy, a simple metal or even a quasicrystal. For the latter two types of material it has been demonstrated that climb takes place and can represent the primary mode of dislocation motion [12,13]. The condition for the preference of climb motion for the present material, according to the above considerations is apparent

Metadislocations in other structures

Metadislocations have been reported in the literature up to now only in ξ´-Al-Pd-Mn and the related phase ψ-Al-Pd-Mn (see e.g. [3,4]). ξ´-Al-Pd-Mn is a member of a class of phases, which are frequently referred to as ε-phases. These phases have orthorhombic structures and have Mackay-type clusters as basic structural elements.

ε-phases exist in numerous Al-based alloy systems. The basic phase is frequently referred to as Al 3 Pd. To be more precise, it exists in the compositional range of about 72 to 76 at.% Al.

The ε-phases make up a phase family, the members of which possess similar a-and b-lattice parameters and various c-lattice parameters, which are related by powers of the golden mean ( )

1 5 2 1 + = τ
. The phases are frequently referred to as ε i , where i = 6, 16, 22, 28,… refers to the index (0 0 l) of the first strong reflection corresponding to the interplanar spacing of about The T-phases can also be described in terms of a lattice constructed using alternatingly oriented hexagons. Such a lattice is shown in Fig. 5 a. For the construction, a different type of hexagon than for the ε-phases is used, which we term "elongated" hexagon. The obtuse angle in the hexagons used for the description of the εand the T-phases is equal and corresponds to the internal angle of a regular pentagon. Klein et al. [18] have considered the T-phase in the system Al-Mn-Pd and have shown that, in complete analogy with the ε-phase in Al-Pd-Mn, phason lines exist. The phason lines are represented by crown-shaped tiles in the hexagon lattice (hatched area in Fig. 5 b). Accordingly, structures in analogy to ε i can be constructed as shown in Fig. 5 b. The figure shows a structure corresponding to ε 28 (or Ψ-Al-Pd-Mn).

Structures of this type have indeed been observed in Al-Cr-Fe [19] and Ga 7 Mn 5 [20] and do most probably exist in the system Al-Mn-Pd [21]. Other structures with different c-lattice parameter can be accordingly constructed introducing larger or smaller spacing between the phason planes. Let us consider a third set of orthorhombic phases, which are commonly referred to as Al 13 M 4 phases. A well known representative of this phase is Al 13 Co 4 [23], in Al 13 Fe 4 an orthorhombic high-temperature phase isostructural to Al 13 Co 4 exists (which is a replacive high-temperature variant of the monoclinic low-temperature phase) [24]. These structures can be described

  Complex metallic alloys (CMAs) represent a newly emerging field of materials science.

  Al-Pd-Mn: Recall of previous results ξ'-Al-Pd-Mn has an orthorhombic lattice (space group Pnma), the unit cell contains about 320 atoms. The cell parameters are a = 23.89 Å, b = 16.56 Å, and c = 12.56 Å [6]. Viewed along the [0 1 0] direction, a hexagon lattice composed of two types of flattened hexagon arranged in alternating orientations can be used for a schematic representation of the ideal structure as shown in Fig.1 (a). A comparison with the atomic structure of the ξ'-phase shows that the vertices of the hexagons are decorated with 52-atom clusters of local icosahedral order, the so called Pseudo Mackay Clusters. Klein et al. have discovered a salient type of defect in the ξ´-Al-Pd-Mn structure which they have termed phason lines [7]. These are linear defects extended along [0 1 0] represented in the hexagon lattice by additional tiles, a banana-shaped polygon with an attached pentagon (hatched tiles in Fig. 1 b). They can move in the structure along the [0 0 1] direction by local atomic rearrangements, represented as vertex flips in the hexagon lattice. Phason lines preferentially line up along the [1 0 0] direction, forming a (0 0 1) plane. Such planes are referred to as phason planes.ξ'-Al-Pd-Mn is the basic phase of a family of related structures with equal a-and b-but larger c-lattice parameters. The most prominent of these possesses a c-parameter of 58 Å and has ψ-Al-Pd-Mn. The structure of ψ-Al-Pd-Mn can be represented in terms of flattened hexagons as for the case of ξ'-Al-Pd-Mn with additional periodic [0 0 1] stacking of phason planes in distances of 29.0 Å (Fig.1 b). Other related structures with c = 32.4 Å, 44.9

Fig. 3 a

 3 Fig. 3 a shows a two-beam bright-field image taken close to the (1 0 3) zone axis. This

  close to the (0 1 0) zone axis in other TEM specimens taken from the same sample, and, along the latter direction we do not find any other type of dislocation. Second, the [0 0 1] Burgers vector of the dislocation loops is equal to with that of the metadislocations. Third, the dislocation loops in Fig.3a do not enclose stacking faults.

Fig. 4 c

 4 Fig. 4 c schematically shows a climb step (movement along [1 0 0] to the left) of a

0. 2

 2 nm. The basic phase Al 3 Pd has ternary extensions in a number of systems such as Al-Pd-(Mn, Fe, Co, Ru, Re,…) at approximately constant Al-content up to several percent into the ternary range. Al-Pd-Fe, for example, exists with up to 10 at.% Fe. ε-phases also exist in Al 3 Rh and the ternary extensions Al-Rh-(Cu, Ni,…). The ξ´and ψ-Al-Pd-Mn phases, in these terms, are referred to as ε 6 and ε 28 , respectively.All ε 6 -phases can be described by a lattice of alternatingly oriented flattened hexagons as shown in Fig.1a. All higher phases ε i can be described by periodic arrangements of phason planes as shown in Fig.2b. The figure corresponds to ε 28 , all other phases are represented by distances between the phason planes[START_REF] Feuerbacher | Quasicrystals[END_REF]. As a direct consequence, metadislocations can exist in all ε-phases and they can all be described by a schematic as given by Fig.1 c. The first observation of a metadislocation in an ε-phase other than ξ´and Ψ-Al-Pd-Mn has recently been made in ε-Al-Pd-Fe[START_REF] Balanetskyy | [END_REF].Let us now consider another important set of orthorhombic phases, which are based on the binary phase Al 3 Mn. The structure of Al 3 Mn was firstly approached by Taylor[START_REF] Taylor | [END_REF] and therefore this phase and the phases derived from it are referred to as T-phases. The basic Tphases has lattice parameters of a = 14.8 Å, b = 12.4 Å, and c = 12.6 Å [17]. The binary Al 3 Mn phase has known ternary extensions into the systems Al-Mn-(Pd, Co, Cr).

  With this basis, all geometrical requirements for the construction of metadislocations in the Tphases are provided. Therefore, we propose for the T-phases a metadislocation structure as schematically represented in Fig.5 c. The figure shows a metadislocation with six associated Its core is given by the dark grey tile. The Burgers-vector length can be calculated as 4 τ c b T = . Using the lattice parameter c = 12.59 Å of T-Al-Mn-Pd [18] we obtain a Burgers vector length of 1.84 Å. In complete analogy with ξ´-Al-Pd-Mn, a sequence of metadislocations of this type can exist, with two, four, six, ten, and 16 associated phason planes [4]. The corresponding Burgers vector lengths are 4.81 Å, 2.97 Å, 1.84 Å, 1.14 Å, and 0.70 Å, respectively. Note that these Burgers vector lengths are related by powers of the golden mean τ. Metadislocations, as schematically represented in Fig. 5 c, may exist in all Tphases, in the basic phases as represented by Fig. 5 a as well as in the derived phases as represented by Fig. 5 b. Metadislocations in T-phases are also discussed by Engel and Trebin [22] (this volume).

11 a

 11 using a tiling constructed by a regular pentagons and a rhombus as shown in Fig. 6 a. Proceeding in full analogy to the preceding cases, we construct phason lines by introducing an additional tile, a boat shaped heptagon * . The phason lines can be arranged along the [0 1 0] direction, forming (0 0 1) planes. Periodic stacking of these planes leads to the hypothetical structure shown in Fig. 6 b. Variants with different c-lattice parameters, of course, can be constructed by introducing larger or smaller spacing between the phason planes. The so constructed elements can be used to build another type of metadislocation for the Al 13 M 4 phases. Fig. 6 c shows a hypothetical metadislocation with six associated phason planes. Its core is given by the dark grey tile. The Burgers-vector length can be calculated as 4 τ b b AM = . Using the lattice parameter b = 12.34 Å of orthorhombic Al 13 Co 4 [23] we obtain Burgers vector length of 1.80 Å. Again, a sequence of metadislocations of this type can be constructed, with two, four, six, ten, and 16 associated phason planes. The corresponding Burgers vector lengths (again related by powers of the golden mean τ) are 4.71 Å, 2.91 Å, 1.80 Å, 1.11 Å, and 0.69 Å, respectively. Such metadislocations may exist in the basic Al 13 M 4 phases as well as in the derived phases with other b-lattice parameters.Surely, the construction of metadislocations for structures other than ξ´or Ψ-Al-Pd-Mn is rather speculative. The considerations made above merely show that the concept of metadislocations is not limited to a single phase. It can be transferred to other structures and, from a purely geometric point of view, leads to possible metadislocation structures in the corresponding phases. It is of course not said that the proposed metadislocations do exist in these materials. This will depend on many, partly unknown factors, such as the exact coexistence of phases in the phase diagram (which will determine whether or not the formation of phason planes and their periodic arrangement is energetically favourable), the energy cost of stacking faults (which will determine whether or not competing, deformation mechanisms based on conventional partials are favourable), etc.

Fig. 1 :Fig. 2 :Fig. 3 :

 123 Figure captions

Fig. 4 :Fig. 5 :Fig. 6 :

 456 Fig. 4: Schematic representation of a metadislocation (a) using a modified hexagon lattice.

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Acknowledgements

The authors thank S. Balanetskyy and B. Grushko for inspiring discussions. Carsten Thomas and Marita Schmid are thanked for performing the single-crystal growth.