Esther Belin-Ferré

Jean-Marie Marie Dubois

Experimental evidence of the formation of d-like states near the Fermi energy in Complex Metallic Alloys

Keywords: electronic density of states, electronic structure user supplied): complex alloys, electronic structure, d-states, Fermi level complex alloys, electronic structure, d-states, Fermi level

published or not. The documents may come

Experimental evidence of the formation of d-like states near the Fermi energy in Complex Metallic Alloys

Introduction

The variety of solid structures that go from simple, close packed and periodic to non-periodic arrangements questions about why Nature builds complex systems. Knowledge of their electronic structure is one of the means that may help to approach an answer. The electronic structure can be investigated by various means among which Soft X-ray Spectroscopy techniques. For the valence band (VB), Soft X-ray Emission Spectroscopy (SXES) probes separately the distribution of partial and local occupied states around each chemical species in the solid, averaging data over the various atomic sites. At a given energy, the intensity of the emitted x-rays is proportional to the probed partial density of states (DOS). Therefore, comparison of the shapes of the distribution curves for a given spectral character around a specific component in various materials informs about changes of the electronic interactions from one compound to another.

The SXES technique was extensively applied to the investigation of the VB structure in series of Al and (or) Mg-based compounds, ranging from conventional to complex systems, including quasicrystals and approximants, with special attention paid to Al s,d and Mg s,d distributions in the pure fcc Al and hcp Mg metals, respectively. In this article, we analyse new binary compounds and revisit ternary systems (Al-Mg-Zn) with better experimental accuracy than previously done [1]. s and d states are obtained together. Because transition probabilities favour d states with respect to s ones, we focus at those states and come to the conclusion that complexity of the crystal structure is linked to the occurrence of d-like states close to the Fermi energy.

To the best of our knowledge, our present data bring, for the first time, direct experimental confirmation of the formation of localised states at the edge of the VB of compounds based on sp elements. To achieve this conclusion, we first locate d-like states on the binding energy scale in Al-TM and Mg-TM samples (TM stands for transition metals). These states form due to sp-d hybridization and are readily observed on the partial DOS of the sp-element. Then we focus at Al-Mg alloys, and show that they exhibit a pronounced d-like peak just below the Fermi energy, although they are formed of pure sp elements.

Experimental

Except for two meta-stable compounds, we have revisited previous data [1] especially positioning of the Fermi energy E F on the X-ray transition energy scale, may be found in Ref. [START_REF] Belin-Ferré | [END_REF].

In strong contrast to photoemission spectroscopy, which investigates the total DOS (but requires knowledge of cross sections for a better understanding of the total DOS), SXES delivers only a partial DOS corresponding to a single species. Therefore, each spectral curve refers to the atom that produces the signal. We insist on this: the spectral character pointed out in a series of SXES experiments belongs to the source atom, in this study either Al or Mg, and not to the other alloy constituents. SXES intensities need to be expressed in arbitrary units. To this end, the most frequently applied scheme in the SXES community is a normalization to maximum intensity, instead of a normalisation to the total integrated intensity, which as a matter of facts makes little difference. After normalisation, intensities obtained from various samples, but in the same experimental conditions, may be compared.

Results

Locating d-like states in Al-TM based alloys

Figure 1 shows three sets of distribution curves in the binding energy scale representing the VB of stable Al 2 Cu, Al 3 Fe and Al 3 Ni compounds. The data for Al 2 Cu, in line with our previous results, may be summarised as follows: the Cu 3d states lie in the middle of the VB (they do not contribute to the intensity at E F ) whereas the Al 3p and 3s,d sub-bands overlap and are separated in two parts located on each side of the maximum of the Cu 3d states curve. DOS calculations have shown that Al s states extend between 4 and 11 eV below E F whereas Al d states are hybridized to Al p states in the upper part of the VB below E F [4]. This is also true for many other systems including approximants of quasicrystals, for which VB DOS calculations [4,5] have pointed out that Al 3s,d states close to E F are essentially of d character.

In the structurally complex Al 3 Fe binary alloy, from E F towards the centre of the VB, one finds first Fe 3d states, then Al 3p states that overlap over about 3eV the Al 3s,d states, and then Al 3s,d states alone. This picture of the VB in Al 3 Fe has been found to hold true in Al-TM-based quasicrystals and their approximants [START_REF] Belin-Ferré | [END_REF]5]. It has been argued that the stability of these complex structures is to a large extent due to the interaction near E F between the TM d states and Al spd states through sp-d hybridisation effects [6].

In Al 3 Ni, another structurally complex alloy, Ni plays the same role as Cu in Al 2 Cu. As a result, a significant proportion of Al states lies in a narrow energy range of about 1 eV below E F . According to DOS calculations available for Al-MT systems, these states have a localised, or for short a d-like, character.

Conclusively at this stage, we have shown experimentally that in stable complex structures d-like states arising from hybridization effects of Al states with TM states are present close to E F , at the top of the VB.

Alloys based solely on sp elements

Fcc Al and hcp Mg are free-electron metals with not only sp electronic states, but also a faint proportion of d-like states located close to their VB edge. Actually, according to Papaconstantopoulos [START_REF] Papaconstantopoulos | Handbook of the band structure of elemental solids[END_REF], the VB of Al contains 1.55 s and 0.07 d electronic states whereas in hcp Mg, there are 0.77 s and 0.11 d electronic states. The other electronic states of the VB are of character p. The two Al and Mg 3s,d distributions curves are displayed in Figure 2. In both cases, below the Fermi energy, in the energy range of the d-like states, exists an intense and narrow peak that arises from a many-body effect following the x-ray emission process [START_REF] Nozières | [END_REF]. This peak is much more narrow in Mg than in Al. Qualitative comparison of Al-or Mg-based alloys with the respective pure element is possible The presence of a sharp narrow peak just below E F in such an alloy would reveal a free-electron-like behaviour. However, in the pure metals, the occurrence of the many-body peak obscures the contribution of the d-like states, therefore, we shall not make any comparison to the pure metals in the following and restrict ourselves to studying the compounds only. The compounds for which we have investigated the Al and/or Mg 3s,d distributions are listed in Table 1, namely the stable -Al 3 Mg 2 so-called Samson phase [9], one metastable quasiperiodic counterpart of composition Al 61 Mg 39 (also noted Mg-61at% Al in the original article [10] which the Reader may consult for details about the crystal aperiodic structure and that is quoted QP-Al-Mg hereafter), the Laves phase Zn 2 Mg [11], the stable crystalline phases Mg 39 . 5 Al 30 Zn 30.5 , Mg 39 . 5 Al 40 Zn 20.5 , Mg 39 . 5 Al 50 Zn 10.5 and a metastable quasicrystalline phase of nominal composition Mg 43 Al 15 Zn 42 [12].

For all the compounds, the Al 3s,d as well as the Mg 3s,d curves consist in a broad peak near E F (denoted P) followed by a large marked bump whose maximum intensity is set around 4 eV below P (Figures 3 and4). This emphasises a complete redistribution of the 3s,d states in the compounds.

In an attempt to put this observation on a more quantitative basis, we have measured the energy extent W of the base of peak P in each of the compounds. This was done by extrapolating first the part of the curve corresponding to the bump on the low energy side of peak P, and second by measuring the energy distance between this point and the point on the other side of peak P at the same intensity. The values are given in Table 1 together with the height H (intensity difference between maximum intensity of the peak and its base line) and the approximate area A=WxH/2 of the peak P. It is worth insisting at this point that this approximation of the true area is by far sufficient for the present purpose, although it does obviously not offer a rigorous measure of the actual area. The extent of the spectral distribution of the peak, of the order of 0.9 L W/2 L1.2 eV, signals a well defined peak and therefore the occurrence of localized, or equivalently d-like, states. In this table, we have reported the Al 3s,d data obtained from a sample of icosahedral Mg 43 Al 15 Zn 42 phase because, although of rather bad quality due to experimental constrains, it follows the same trend as found for the Mg 3s,d states. From the values obtained for area A in Table 1, it is clear that the number of d-like states originating from Al or Mg atoms in stable crystalline compounds is significantly larger than in metastable quasiperiodic or quasicrystalline counterparts. This difference, in our opinion, is related to the occurrence of a significant degree of disorder in the metastable compounds, as can be assessed from the width of x-ray diffraction lines (not shown here), whereas stable compounds show less, but not absent, disorder like in all stable complex metallic alloys studied so far. For Mg d states, the values corresponding to the two Mg-Al-Zn cubic approximant alloys only slightly differ from each other, which gives confidence in our approximation. However, a more meaningful quantitative comparison between systems whose atomic structures are strongly different still remains a very difficult task and certainly cannot be achieved without theoretical modelling. near, and just below, the Fermi energy in these complex compounds, and especially in the very complex -Al 3 Mg 2 phase. They are far more abundant in stable than in metastable counterparts, although similarities are exhibited in the local atomic environments. This issue suggests that localised states should form at E F for a complex compound to be stable. At this point, it is clear that more work is needed, especially on Mg-based compounds, to ascertain our findings and possibly put our conclusions on a more quantitative basis.

Conclusion

  Our new SXES investigation of the Al 3s,d and Mg 3s,d partial DOS in a series of binary Al-and Mgbased compounds of varying complexity has focused at the s,d spectral distribution in a narrow range below the Fermi energy. It has featured that a significant amount of states with a d-like character exists

Figure 1 :

 1 Figure 1: From bottom to top, valence band of stable Al 3 Ni , Al 3 Fe and Al 2 Cu.(Al 3p states: full line, Al 3s,d states: dotted line. Stars and triangles are for 3d distribution curves of Cu, Fe and Ni, respectively.

Figure 2 :

 2 Figure 2: From bottom to top, 3s,d electronic distribution curves in hcp Mg and fcc Al.

Figure 3 :

 3 Figure 3: From bottom to top, Al 3s,d electronic distribution curves in -Al 3 Mg 2 , QP-Al-Mg, Mg 39 . 5 Al 30 Zn 30.5 and icosahedral Mg 43 Al 15 Zn 42 .

Figure 4 :

 4 Figure 4: From bottom to top: Mg 3s,d electronic distribution curves in -Al 3 Mg 2 , QP-Al-Mg, the two crystalline alloys Mg 39 . 5 Al 40 Zn 20.5 and Mg 39 . 5 Al 50 Zn 10.5 , as-quenched icosahedral Mg 43 Al 15 Zn 42 and in the Zn 2 Mg Laves phase.

  by producing new stable samples synthesized according to new techniques set up by Brunet et al. on the one hand and Feuerbacher et al. on the other (see Ref.2 for details). All stable samples were cut in the shape of bulk slabs of about 1cm 2 cross section. They were carefully polished down to mirror polish and placed perpendicular to the electron beam axis in the SXES spectrometer. Details about SXES measurements,

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Table 1 Al

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http://mc.manuscriptcentral.com/pm-pml Philosophical Magazine & Philosophical Magazine Letters

Acknowledgments

Pr. U. Mizutani (Nagoya Univ. & Toyota Res. Centre, Japan) is warmly acknowledged for providing the series of Al-Mg-Zn specimens. Financial support from the European Commission under contract NMP3-CT-2005-500140 (CMA European Network of Excellence) is also gratefully acknowledged.


Francoise Hippert
email: francoise.hippert@inpg.fr

Jean-Jacques J Prejean

Magnetism in Quasicrystals

Keywords: magnetic structure, magnetic transitions, magnetism, quasicrystals, transport properties magnetic structure, magnetic transitions, magnetism

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Introduction

Magnetic moments only exist in a minority of the numerous discovered QCs. Nevertheless magnetism in quasicrystals is a fascinating field of investigation, obviously in the case of RM gZn QCs with R a magnetic rare earth, but also in the case of Al-based QCs containing M n. In many QCs, such as AlLiCu, GaM gZn, ..., the absence of magnetic moments is a trivial observation since none of their constituting elements is expected to be magnetic. The case of Al-based QCs with transition metals was more striking. The N i and Co are found non magnetic 1 , which suggests that the electronic structure of QCs does not favor moment formation. But M n localized moments have been systematically detected since the pioneering work of Hauser et al [1] published in 1986, very soon after the discovery of the first QCs in the AlM n system [2]. The magnetic susceptibility χ, deduced from the linear response of the magnetization to a small static field, was found to obey a Curie-Weiss law: χ ≈ C/(T + θ) above 5 K in icosahedral AlM n and AlSiM n QCs with M n content from 14 to 22%, which proves the existence of local moments. A peak in the χ ac (T ) obtained using a small ac field (frequency ≈ a few Hz) was also observed at a temperature T f < 5 K. This proves the presence of magnetic interactions between the local moments. A spin glass behavior was proposed. A quite intriguing fact was the weakness of the Curie constant C, found much smaller than calculated if all the M n atoms are magnetic, each carrying a spin S close to 5/2. The Curie constant is proportional to N xS(S + 1) with N the total number of atoms and x the concentration of local moments. Soon after, in another pioneering work [3], Warren et al demonstrated by observing a 55 M n nuclear magnetic resonance (NMR) that most of the M n sites do not carry a local moment, which should explain the small Curie constant. Subsequently, the same features (small Curie terms, non-magnetic M n and χ ac peak) were observed in the successively discovered QC phases: The decagonal metastable AlM n [4], the stable icosahedral AlP dM n ( [5,6] and references therein) and also the decagonal AlP dM n phases [7,8]. Recently magnetic moments have been observed in ZnM Sc QCs [9] with M = M n and F e 2 , Co and N i being found non magnetic. Magnetism of M n in Al-based QCs motivated numerous experimental and theoretical works in order to answer several fundamental questions. The first series of questions concerns the occurrence of magnetism on a M n site: First of all, why most of the M n atoms are non magnetic? Which is the spin value of the magnetic M n? Are the latter located in some foreign magnetic phases? If they belong to the QC phase, are they located on specific sites, intrinsic to the structure, or is the moment formation induced by a local defect in the quasiperiodic potential? At the present time, it is possible to draw several major conclusions from systematic experimental studies of AlM n and AlP dM n QCs. The M n atoms which carry a moment have a large spin value: S ≈ 5/2. They do belong to the QC phase and their concentration is very low, from a few % down to 10 -4 . They are presumably randomly distributed in the quasiperiodic structure and their concentration depends drastically on the alloy composition. Thus the presence of a moment on a M n site could originate in a neighboring local defect in the quasiperiodic structure. These results have motivated theoretical studies aimed at understanding the parameters governing moment formation on M n atoms in Al-based QCs and related crystalline phases. Another series of questions concern the nature of the magnetic interactions between the few M n moments and that of the magnetic state below T f . First, from the above results, one concludes that the magnetic moments do not form a quasiperiodic array. So it is not a surprise that the magnetic order is not specific to the QC structure: A standard spin glass transition is found to occur at T f . It originates in RKKY (Ruderman-Kittel-Kasuya-Yosida) interactions, i.e. mediated by the polarization of the spins of the conduction electrons. The interaction between two moments alternates in sign rapidly with the relative distance r of the two moments. Thus it can be either ferromagnetic (F) or antiferromagnetic (AF) according to the r value. This feature associated to a random spatial distribution of the moments are the key ingredients to form a spin glass state [11], such as that one observed in dilute CuM n alloys where a few M n atoms (then all magnetic) are randomly distributed in a periodic Cu matrix. All these points will be developed in Section 2. It could not be imagined at the beginning that these studies would have fruitful consequences in other fields. Indeed, they allowed a better understanding of the unconventional transport properties of quasicrystals (Section 3). Besides, AlP dM n QCs provided the rare opportunity to allow a comprehensive study of the Kondo and RKKY interactions, which is a fundamental problem in magnetism, independently of the QC study (Section 4).

If the disordered repartition of magnetic M n in Al-based QCs does not allow the existence of a specific magnetic order, a quite different situation occurs in the QCs containing magnetic rare earth R atoms, either in the RM gZn or in the RM gCd system. Since all the R atoms are magnetic, these QCs allow to investigate the nature of magnetic correlations on a quasiperiodic lattice (section 5), a fundamental issue in the domain of magnetism in QCs [12]. Up to now only short-range correlations have been observed [13], the exact nature of the spin glass like transition [14,15] remaining a debated question.

2 M n magnetism in Al-based QC Fig. 1 illustrates two main features exhibited by all the Al-based QCs containing M n: An ac-susceptibility peak and a Curie-like temperature dependence of χ(T ) at T > T f 1 . All these phases exhibit a very small Curie constant and a behavior at low temperature which suggests the occurrence of a spin glass state: For instance, a broad bump of the specific heat at a temperature around T f , [4,6,[16][17][18] and magnetic hysteresis features at T < T f [17,19]. These observations were important but could not help to know the concentration of M n atoms which are magnetic, their location, their spin value or the existence of a true spin glass transition. In order to clear up these issues, additional works had to be performed, as presented below.

F

Spin glass transition and deduction of the M n spin value

It is the thorough knowledge of the spin glass properties, acquired well before the QC discovery [11,20], which helped to clarify the nature of the M n magnetism in the Al-based QCs. Many features often claimed to be characteristic of spin glasses can be indeed observed in systems which are not spin glasses. For instance, small ferromagnetic particles (iron grains dispersed in an alumina matrix) exhibit precisely such features [21]: A sharp maximum in the χ ac at a temperature T f , the frequency dependence of T f observed in spin glasses and a magnetization below T f depending on time logarithmically and on the magneto-thermal history. The magnetic hysteresis was even shown to exist sizably in a region of the (H, T )-plane limited by a line obeying the law proposed theoretically by de Almeida and Thouless for mean field Ising spin glasses [11]. For this system, the above properties originate in a large distribution of blocking temperatures, while the Curie-Weiss temperature θ reflects the role of the dipolar interactions between the grain moments. So, to differentiate a spin glass from the latter system requires more elaborate studies. The more convincing one is the study of the temperature dependence of the so-called non linear susceptibilities above T f , i.e. of the temperature dependence of the A i s involved in the series expansion of the magnetization M (H) in terms of odd powers of H:

M (H, T ) = C × H T + θ -A 3 × [ H T + θ ] 3 + A 5 × [ H T + θ ] 5 -..., (1)
In spin glasses, θ T f in the temperature decade above T f . This originates in some compensation of the effects of the F and AF interactions (θ = 0 corresponds to an equal weight of these interactions). The interaction effects manifest themselves clearly in the terms non linear in field in Eq.1. A true phase transition is characterized by the divergence of a correlation length ξ(T ) at a temperature T c . In the spin glass case, this results in a divergence of A 3 (T ), A 5 (T ), ..., which obey power laws of ξ(T ) [11]. In contrast, temperature independent A i s indicate the lack of divergence of any ξ(T ), which is the case for a collection of fine magnetic grains [21]. So the goal is to determine experimentally the A i values. The principle of the experimental procedure is simple. At each temperature T i (from high temperature down to T f ), M (H) is measured, which allows to deduce the prefactor of H (i.e. the linear susceptibility χ) and those of H 3 , H 5 in the series expansion of M (Eq. 1). (The measurement fields have to be sufficiently low so that the subsequent H 2n+1 terms are negligible.) This allows to calculate the values of A 3 (T i ) and A 5 (T i ) once the θ value has been deduced from the temperature dependence of χ. Actually, the study of A 3 and A 5 is difficult since one has to measure the very initial deviations of M from its dominant term χH. In practice, this requires an unusual relative precision (much better than 10 -3 ) on M , H, T at low temperature and in weak field conditions [20,22]. The work [22] performed on a melt spun i -Al 73 M n 21 Si 6 QC sample (for which θ = 0) showed that A 3 (T ) and A 5 (T ) vary by orders of magnitude when the temperature is ) of a crystalline dilute Cu 99% M n 1% alloy (from [20]) and of an icosahedral AlM nSi QC (from [22]). On this figure, only the T -dependence is significant: The C, A 3 and A 5 data have been arbitrarily vertically shifted to be presented altogether. The lines are fits used to deduce the critical exponents.

decreased only by a small factor, exhibiting the same qualitative behavior as in the periodic dilute alloy CuM n (Fig. 2). Quantitatively, they could be accurately analyzed in terms of power laws of T /(T -T c ) (The T c , here 5.42 K, is slightly smaller than T f like in CuM n). The critical exponents were found identical to those for CuM n [20] and many other 3D spin glasses.

The second challenge was to determine the spin value S of the magnetic moments and their concentration x. Of course, this cannot be done from the only measurement of the term linear in field of M (H) which yields the Curie constant and thus only the value of the product xS(S + 1). One can expect to get an independent determination of x and S from the analysis of the M (H) curvature. But the analysis of M (H) is not simple. It is because the magnitude of the magnetic interactions energy scale compare with the value of µ B H (µ B H/k B = 5.4 K for H = 8 Tesla available in standard magnetometers). These interactions are the RKKY ones responsible for the spin glass transition (T f ≈ a few Kelvin) and the Kondo one (energy scale T K ≈ 3.3 K in AlP dM n QCs: See Section 4). However, in the non linear susceptibility study close to the transition, a careful quantitative comparison of the A 3 /C ratio 1 in AlM nSi QC and CuM n (where the spin value is known) could provide the value of the spin value in the QC: S ≈ 3.7 [22]. This rather large value of S (> 5/2 expected at maximum) could originate in the presence of ferromagnetic pairs of 5/2-M n spins involved in the transition. From S and the Curie constant, one found a moment concentration equal to 0.27% whereas the concentration of M n is 21%. Other works, taking into account the RKKY interactions effects in the analysis of M (H), concluded also a rather large value of the M n-spin, about 5/2 [17,19,[START_REF] Fukamichi | Physical Properties of Quasicrystals[END_REF]. We note that all these determinations of S were carried when assuming implicitly a bimodal distribution of the spin values: either 0 or a finite value S = 0.

More on the fraction of magnetic M n atoms

From the previous estimate of the M n spin value and the measured Curie constants, one deduces that, at maximum, a few % of the M n atoms are magnetic. Besides, NMR studies, the very early one of Warren et al. on metastable AlM n QCs [3] and those subsequently performed on stable AlP dM n QCs [24,25] conclude that most M n atoms do not carry a magnetic moment (S = 0). A precise determination of the proportion of non magnetic M n atoms cannot be obtained because of the uncertainty on line intensity 1 C, A 3 , A 5 are all proportional to N x but their dependence on S is different. So the ratio A 3 /C depends on S but not on x. Using classic magnetic moments µ instead of quantum spins S, one has:

C = N xµ 2 /3k B , A 3 = N xa 3 (T )µ 4 /k 3 B and thus A 3 /C ∝ µ 2 .
For a paramagnet with no interactions, a 3 is constant, equal to 1/45 as given by the series expansion of the Langevin function, i.e. the classic limit for the Brillouin function. For 3D spin glasses, a 3 (T ) = b 3 [T /(T -Tc)] γ where b 3 is of constant value, found experimentally to be close to 1/45 in standard M n-based spin glasses [20,22].

Thus A 3 /C ≈ [T /(T -Tc)] γ µ 2 /15k 2
B , which allows to deduce the value of µ once those of the transition temperature Tc and the critical exponent γ have been extracted from the temperature dependence of A 3 (T ). measurements in NMR experiments. So, one can imagine the existence of other M n atoms which would be magnetic (hence not contributing to the NMR 55 M n signal) and would not contribute to the measured Curie constant. This should be the case of two M n moments interacting via a strong AF interaction or more complex arrangements of correlated M n spins, provided that their resulting spin is zero. The method which has to be used for detecting such spins depends on their fluctuation time. If the latter is long, these spins can be detected in specific heat experiments since they give rise to a magnetic hyperfine contribution C hyp = C N /T 2 to the specific heat. Indeed, C N takes into account all the localized moments (concentration x * ): C N ∝ N x * S 2 , whatever their magnetic interactions and relative orientations, provided that the spins are frozen at the time scale (a few seconds) of the experiment. (Because of its T -2 dependence, C hyp is only measurable at very low temperature, practically below 300 mK, and thus includes the contribution of the spins frozen in the spin glass interactions when T f 300 mK.) The value of the product x * S 2 deduced in this way has been systematically found to be very small and, so, could be measurable only for the more magnetic AlM n and AlP dM n QC samples [4,16,17]. It is precisely the case where T f 300 mK and where almost all the spins contributing to the Curie constant are involved in the spin glass phase (almost no Kondo spins: See section 4). One found a value of x * S 2 well comparing with the Curie constant measured on the same QC samples. This discards the hypothesis of additional frozen AF pairs or clusters. Hence if strongly correlated groups of spin do exist, they must fluctuate (as a whole) at any temperature including below the spin glass transition, which implies that they must be entirely decoupled from the spins contributing to the Curie constant and the spin glass transition. Their existence remains a quite open question. 2 In summary, a very few Mn atoms appear to carry a magnetic moment in AlM n and AlP dM n QCs. In the next section, the origin of these moments will be discussed as well as the parameters governing the moment formation.

About the moment formation

The first question is to know the location of the very few magnetic M n atoms. First, they do not cluster in fine grains, as shown from the existence of a spin glass transition. Secondly, they do belong to the QC structure as proved from their effect on several bulk properties. Indeed, i) the anomalous Hall effect is directly related to the M n magnetization in decagonal T-AlM n phases [4], ii) the electrical conductivity of AlP dM n single grains increases linearly with the concentration of magnetic M n atoms (See ref. [26] and below Section 3) and iii) all 27 Al nuclear spins are sensitive to the conduction electron spin polarization induced by the localized M n moments, as shown by several NMR studies [24,25]. Consequently, the challenge was to understand the conditions for moment formation in the QC structure.

In icosahedral AlM nSi and decagonal T-AlM n phases [4,19,22], as well as in the first synthesized AlP dM n single grains (containing about 9 at% M n) [16,17], the concentration of magnetic M n atoms, although sample dependent, was always close to 1%. This could suggest the existence of specific M n sites in the QC structure with environments favouring moment formation. Later on, the possibility to grow AlP dM n single grains containing slightly smaller M n amounts (about 8 at% M n) gave the opportunity to observe that the moment concentration could range actually over two decades, from 10 -4 to about 1 % for the studied samples [5,26]. Then a systematic study of AlP dM n QCs of various composition, focusing in particular on 14 different QC pieces cut in Czochralski grown single grains, could be performed [5,27]. Note that for such studies, it is important to avoid grain boundaries which could favor spurious M n moment formation, thus clouding the issue when the moment concentration is very low. Although each QC piece had been previously annealed at 800 • C in order to improve the structural homogeneity, the moment concentration was found to depend (within a factor about 2) on subsequent thermal treatments [5,27]. More precisely, the cooling rate after a heating at 800 • C was found to be a pertinent parameter: The 2 Inelastic neutron scattering should be a quite suitable tool in order to detect such correlated spins through the measurement of time and space spin correlations. One of the present author (JJP) has carried such experiments in collaboration with L.P. Regnault (CEA Grenoble) on spectrometers IN 22 and IN 8 at Institut Laue Langevin (ILL-Grenoble). No magnetic signal could be detected up to now in the Q range where it should appear if one assumes that the correlated M n spins are located on vertices of M n icosahedra with identical orientation, in agreement with structural models (See Section 2.3). slower the cooling rate, the smaller the moment concentration (Fig. 3). Besides, for a given cooling process, the moment concentration depends strongly on the alloy composition: See Fig. 3. By contrast the exact structural state (the so called F , F 2 , F 2M phases) was found to have no influence [5].

One observes that the concentration of magnetic M n atoms is minimum for a composition about 7.75% of M n, 21.9% of P d and increases drastically when the M n as well as the P d amount changes (Fig. 3). These results allowed to clear up the origin of the large scattering of the magnetization amplitudes of AlP dM n samples reported in the literature [5]. They suggest that magnetic M n in AlP dM n QCs are not located on particular sites but instead most probably associated with defective local arrangements of the various chemical species in the neighbouring of the M n atom which becomes magnetic. One can conjecture the existence of an ideal composition for which no localized moments would be present, provided that the structure is entirely relaxed (See below).

This conclusion is different from that reached from the study of crystalline AlM n and AlP dM n phases. In the hexagonal µAl 4 M n phase, most of the M n atoms are non magnetic but experiments and theoretical calculations both suggest the presence of magnetic moment on a given M n crystallographic site [28,29]. The same conclusion has been reached from the study of the 1/1-AlP dM nSi approximant [30]. In these two cases, moment formation occurs on particular sites, intrinsic to the structure. Note that these two crystalline phases, as well as the M n-rich T-AlP dM n, are exceptions. No M n atoms are magnetic in most of the crystalline AlM n and AlP dM n phases, for instance in Al 6 M n, Al 12 M n, β -Al 9 M n 3 Si, the 1/1 approximant α -Al 73 Si 10 M n 17 , the M n poor T-AlP dM n phase or the ξ -AlP dM n phase [18,30].

So, the question is: Why are most M n atoms non magnetic ? Answering this question requires to combine an accurate description of the quasicrystalline structure and criteria for moment formation. All spin polarized band structure calculations [30,31] show that the presence of a pseudo-gap in the local Mn-DOS impedes the formation of a local moment (S = 0). This is indeed the case of many M n sites in QCs and approximants. Another result of these calculations is that magnetic moments exist on a small fraction of the Mn sites in QCs [31]. But, of course, the validity of this prediction strongly depends on that of the structural model used. In contrast, experiments and ab initio calculations show that most Mn atoms carry a moment in AlMn and AlPdMn liquids [32,33]. It could be tempting to try to establish a correlation between the local environment of a given M n atom (number, nature and distances of its first neighbours) and its magnetic state. However this approach is too incomplete. M n -M n interactions, mediated by the conduction electrons over large distances (5 Å and more), have to be taken into account [29,34]. The interaction energy Φ M n-M n between two M n atoms depends on the magnetic moments m 1 and m 2 of the

M n-M n (r, m 1 , m 2 ) = a(r) + (b(r)/2)(m 2 1 + m 2 2 ) + c(r)m 1 .m 2 .
(The development is correct for moments smaller than 3µ B ). a(r), b(r) and c(r) have a sign oscillating with r and a magnitude slowly decreasing with r, so that their range extends up to about 10 Å. If both m 1 and m 2 are null, the only contribution to the energy is a(r), the potential energy between two non magnetic M n atoms (when they are not nearest neighbors) [35]. The c(r) term corresponds to the usual RKKY interaction between moments. The existence of the b(r) term has strong consequences on the moment formation. Even if one of the atom of the pair is non magnetic, the interaction energy depends on the existence or not of a moment on the other atom. Indeed one finds a deep minimum of Φ M n-M n for r = 4.8 Å when the two M n atoms are not magnetic. But moment formation on only one of the atoms would increase this minimum energy by 200 K (Fig. 4) and hence is not favorable. The same result, but even more marked, is found if the two atoms are magnetic [29]. These results have to be related with those of structural models, for instance that one for i-AlPdMn (and i-AlCuFe) recently published [36]. In this model, constructed with no fitting parameters and exhibiting a minimum number of different chemical configurations for the main atomic aggregates, the M n atoms are found to form a remarkable network of icosahedra, connected by bridges in form of squares (Fig. 4). Strikingly the only first neighbor distance for M n atoms on this network is precisely equal to 4.8 Å, which could explain the absence of magnetism in AlPdMn QCs. Note that in Ref. [36], the stoichiometry of an ideal i-AlPdMn phase is proposed to be Al 70.36 P d 21.35 M n 8.29 , not far from the composition with the minimum moment concentration (See above).

In summary, the theoretical works explain why Mn atoms are not magnetic in the QC structure, unless local deviations from the ideal structure, necessarily present for concentrations different from the ideal one, induce moment formation.

Interplay between magnetism and electronic transport

The transport behavior of the first discovered QCs, which were metastable, was comparable with that of very resistive periodic or amorphous alloys. The situation changed radically when stable QCs were discovered. In 1990, it was shown that melt spun AlCuRu samples could exhibit a resistivity ρ reaching unexpected values, from 5000 to 30000 µΩcm at 4 K depending on the composition, thus 15 to 100 times larger than in the most resistive standard alloys [37]. Soon after, other stable QC phases were shown to exhibit similar behavior, for instance AlCuF e [38] and AlP dM n [39]. For all these phases, the conductivity σ was shown to be sample dependent and to increase with the temperature. But, strikingly, the σ(T ) curves for different samples in a given system appear to be all parallel with one another (above 100 K) exactly as the ρ(T ) curves of simple metals with different amounts of impurities. Another puzzling feature was x at different constant temperature (inset), from [40,41]. The curves are guides for the eye. Right : Same for AlP dM n QCs [26], except that i) x denotes the actual moment concentration and not the total M n concentration and ii) the ordinate is the conductivity σ instead of the resistivity ρ.

a σ decrease with improvement of the structural quality, in strong contrast with the simple metal case where the residual resistivity decreases with the defect concentration. A quite useful tool to understand the intrinsic character (i.e. in the absence of defects) of this very particular electronic diffusion is precisely the study of the disturbance of this diffusion caused by local defects in the quasiperiodic potential. Here, one faces the same challenge as for the resistivity in standard metals decades ago. Unfortunately, for a long time, such a quantitative study for QCs could not be carried out, due to the impossibility to find a way for detecting a very small concentration of local defects. The understanding of the moment formation in the AlP dM n QCs gives a solution, at least in this phase. Indeed, a local defect in the quasiperiodic potential can be detected if it induces a moment on a neighboring M n site. So, to obtain the concentration of such defects amounts to a measurement of the Curie constant which yields the moment (and thus defect) concentration. This hypothesis has been tested for a series of AlP dM n QC single grains by combining measurements of the conductivity and magnetization on the very same samples in the same annealed states [26]. A very simple relation between the conductivity σ and x has been found above 100 K: σ = σ 0 (x) + δσ(T ) where the residual conductivity σ 0 (x) increases linearly with x. As for other stable QC phases, δσ(T ) is sample independent. At lower temperatures, one observes an upturn of σ(T ), exhibiting a Log(T ) behavior and of amplitude also linear in x. So, this feature has been attributed to the Kondo effect (which results in a Log(T )-upturn of the resistivity at low temperature in simple metals). Thus, in QCs, the conductivity (instead of the resistivity) obeys a Mathiessen rule, i.e. is a sum of independent contributions:

σ = σ 0 (x) + δσ(T ) + xK[Log(T )], σ 0 (x) = σ 00 + ax (2)
This explains the striking similarity between σ(T, x) of the QC and ρ(T, x) of the very diluted Cu 1-x M n x , shown in Fig. 5. Such a σ law in QCs implies that the transport between two scattering events is not ballistic as shown in Ref. [START_REF] Trambly De Laissardière | this issue and[END_REF]. Interesting enough, this law appeared to apply only for samples exhibiting a vanishing concentration of defects, namely below 5 10 -4 , i.e. for defects distant by more than 30 Å apart from one another. In view of the theoretical results for transport [43] and structural models (Section 2.3) for AlP dM n QCs, one can conjecture that the low value of the conductivity is due to an electronic confinment by the M n icosahedra. Besides, the increase of σ with the defect concentration could be related to the rapid decrease of this electronic localization in presence of a defect [43].

In conclusion, the thorough understanding of magnetism in AlP dM n QCs allowed to state quantitative T(K) T f (K)

! emu/g ! K (T K =1.2K) ! K1 (T K1 =3.35K) + ! K2 (T K2 =0 .
x, x 2

Al Pd Mn -QC Cu 1-x Mn x Figure 6. Left: The χ(1/T ) curvature for AlP dM n QCs increases when the moment concentration decreases from that of sample R down to that of sample B -b, found to be 100 times less magnetic than sample R (For sample labels: See Ref. [5]). Right: χ(T ) of sample B -b in a semilog diagram and two fits using the susceptibility law for the n = 2S Kondo model. A single T K law well accounts for the data from 2 K to 100 K, yielding T K = 1.2 K, but is unable to account for additional data obtained over a third decade, here from 0.1 to 2 K. In contrast the χ(T ) can be well analyzed as the sum of two independent contributions over the whole temperature range: χ(T ) = χ K1 (T, T K1 ) + χ K2 (T, T K2 ) with T K1 = 3.35 K and T K2 = 0.1 K [44]. Insert: Dependence of T f on x 2 compared to that on the moment concentration in the crystalline spin glass CuM n. The lines are power fits, yielding the same exponent value: 0.7 for AlP dM n QCs and CuM n alloys.

laws for the very unconventional conductivity behavior in QCs.

4 Kondo vs RKKY interactions for the few magnetic M n

Here, we enter upon a subject somewhat apart from the QC topic. As shown below, the AlP dM n QCs of very low moment concentration exhibit the characteristics necessary for a thorough study of the competition between the RKKY and the Kondo interactions. Because most of the standard dilute alloys do not present such interesting characteristics, it was worth undertaking this study [44], in order to bring information in a debated field in magnetism physics. Let us recall that the RKKY interaction J ij between moments i and j, r ij apart, decreases with the distance as 1/r 3 ij . So, when the moment concentration is vanishing, most local moments undergo almost vanishing J ij couplings. Then, the Kondo interaction (energy scale: T K ), which is a single moment interaction with the conduction electron spins, may become dominant. For intermediate moment concentrations, the problem of the RKKY × Kondo competition becomes relevant. Theoretically, this problem remains a challenge in contrast with the two limiting cases (pure RKKY, pure Kondo) which are (now) well understood. The difficulty comes from the very same origin of the RKKY and Kondo interactions, i.e. the antiferromagnetic coupling between the spin of the local moment and the spins of the conduction electrons on the moment site.

The only mentioned magnetic interactions in the first discovered Al-based QCs with M n were RKKY like, due to the magnitude of the moment concentration, about 1% (Section 2.3). But once AlP dM n QC single grains of much lower moment concentration could be synthesized, the Kondo interactions could be detected. Let us recall several Kondo properties. The Kondo coupling results in the formation of a non magnetic singlet at low temperature, which induces a continuous negative curvature in the χ(1/T ) curve from the high temperatures down to 0 K where χ reaches a finite value χ(T = 0) = 0. In addition, in simple metals, the Kondo interaction is responsible for a characteristic Log(T ) upturn in the low temperature resistivity. The same features are observed in the less magnetic AlP dM n QCs, except that it is the conductivity which exhibits the Log(T ) upturn (Section 3). When plotting the susceptibility of AlP dM n QCs vs. 1/T (Fig. 6), one observes that χ is 1/T -linear (Curie behavior) above T f for the more magnetic sample denoted R (Here, we use the notations of Ref. [5]). Hence the R sample is a pure RKKY spin glass. But the other χ(1/T ) curves exhibit continuous curvatures which are more and more pronounced when the sample is less and less magnetic, thus when approaching the single moment limit, i.e. the pure Kondo case [44]. A Kondo fit of the χ(T ) of the less magnetic sample B -b performed in the 2 K -100 K temperature range provides a T K equal to 1.2 K. The competition Kondo × RKKY manifests itself in two phenomena. First, the χ(1/T ) curvature attributed to the Kondo effect evolves progressively when one changes the moment concentration from that of sample R, with pure RKKY couplings down to that of sample B -b, with presumably dominating Kondo couplings (Fig. 6). Second, for the very same sample, one observes the Kondo effect from the χ(1/T ) curvature at large temperature and the RKKY manifestation, i.e. a χ ac (T ) peak, at low temperature. So AlP dM n QCs meet with all the characteristics suitable for the Kondo × RKKY competition study, difficult to get in standard alloys because of the solubility limit (case of F e in Cu) and/or the too low T K value (5 mK for M n in Cu). In the AlP dM n QC case, the T K lies in the Kelvin range and it is possible to get samples of moment concentration varying over two decades (Section 2.3). The recently published study performed on 22 samples [44] could answer to several questions concerning the competition effects. For instance, it was shown that the susceptibility does not reach the expected constant value at low temperature. This can be interpreted as the effect of the existence of two very different T K values: The larger one (T K1 ) accounts for the high temperature χ(1/T ) curvature and the smaller one (T K2 ) accounts for the lack of saturation of χ(T ) at low temperature (Fig. 6). More generally, when using very general Kondo physics arguments, and thus carrying out the data analysis with no preconceived model, a very robust and simple result was found: The magnetization M (H, T ) is a sum of two contributions. One can write:

M (H, T ) = [N gµ B S] x 1 P 1 (H, T, T K1 = 3.35 K) + [N gµ B S] x 2 P 2 (H, T, T K2 ≈ 0) (3)
where the x i s and the P i s have the dimension of a moment concentration and polarization respectively. The first contribution, of T K1 = 3.35 K, behaves as a pure Kondo magnetization, whereas the second contribution, of T K2 ≈ 0, appears to be purely RKKY like, directly responsible for the spin glass characteristics. In particular, it was shown that T f follows a power law of x 2 : T f ∝ x 0.7 2 over the two available decades of x 2 , exactly as it does for the archetypical pure spin glass Cu 1-x M n x : T f ∝ x 0.7 , involving the same exponent value (Fig. 6). Here, the surprise comes from the domain of validity of Eq.3: Indeed, the later holds from the lowest moment concentrations (where x 1

x 2 , i.e. with Kondo dominant) up to the concentration where the RKKY couplings dominate fully the magnetization (i.e. when x 2

x 1 ) and thus cannot be considered as a perturbation. For more details, one can refer to Ref. [44]. Note that, in view of the present Kondo analysis, the prefactor of the Log(T ) term in Eq. 2 for the conductivity should be x 1 instead of x. However, only the less magnetic samples exhibit the transport behavior illustrated in Fig. 5 and for these samples x 2

x 1 , so that x can be identified with x 1 [44]. In summary, the diversity of the temperature dependence of the magnetization in Al-based QCs containing M n is fully understood. It results from the sample dependent relative weight of Kondo and RKKY interactions. Besides, the QC study allowed to enlighten on the effects of the Kondo × RKKY competition from the Kondo dominant regime to the RKKY dominant regime.

R-Mg-Zn QCs

In the F-type RM gZn and P-type RM gCd QCs, with R = T b, Dy, Ho, Gd in concentration around 9%, a Curie-Weiss law and a spin glass like freezing are observed (see [13] and references therein) . The measured Curie constants agree with those calculated assuming a free R 3+ effective moment. Any significant crystal field effect is therefore undetectable from the susceptibility data. In RM gZn QCs, a peak in the ac susceptibility at T f (5.8 K for R = T b, 3.6 K for R = Dy ) and magnetization irreversibilities below T f suggest a spin glass ordering [14], the R moments being coupled via RKKY interactions. The case of the more recently discovered RM gCd QCs is less clear : a broad transition is observed in single grains but several anomalies occur in polycrystalline samples [13,15]. Hence we will mainly focus in the following on the much more studied RM gZn QCs in which the existence of a spin glass transition is strongly supported by the observation of an anomaly in the third order susceptibility (A 3 term with the notations of section 2.1) above T f and by the fact that T f is proportional to x 0.66 in Y 1-x T b x M gZn and Y 1-x Gd x M gZn QCs, where the magnetic rare earths Tb or Gd are substituted by non magnetic Y [14]. Such a law, with this exponent value, is common to all RKKY spin glasses and also observed in AlP dM n QCs (Fig. 6). However, a striking feature is that the Curie-Weiss temperature Θ is i) positive (using the definition χ = C/(T + Θ)), indicating dominant AF interactions between the R moments, ii) much larger than T f . The ratio Θ/T f is close to 4 in all studied cases (R = T b, Dy, Ho and Er). This feature is reminiscent of the situation encountered in the geometrically frustrated periodic systems [START_REF] Ramirez | Handbook on Magnetic Materials[END_REF]. The possibility of an unusual spin freezing in T bM gZn has indeed been suggested [START_REF] Dolinsek | [END_REF].

The most fascinating properties of RM gZn QCs were revealed from neutron diffraction experiments [13,47]. In HoM gZn, a strong magnetic diffuse scattering, due to short-range spin correlations (extending over about 10 Å), is observed below 5 K, i.e. definitively above T f = 1.95 K. The magnetic diffuse scattering occurs outside the intense nuclear Bragg peaks, indicating AF short-range order, and exhibits an isosahedral symmetry. It can be accounted for by assuming short range spin correlations in 6D with a magnetic vector modulation (3/4,0,0,1/2,3/4,1/2). In the real space, most Ho moments are located on a network of edgesharing dodecahedra (with edge length equal to 5.5 Å). The analysis of the spin correlations, assuming classical spins, on a single dodecahedron leads to diffuse scattering patterns close to the observed ones, despite a simplified hypothesis on exchange bounds.

At the present time, one has an unified view of the M n magnetism in Al-based QCs. The few moments are randomly distributed in the structure. One can assume that the moment formation originates in defects which locally break the ideal quasiperiodic potential, the same defects which affect drastically the electronic transport. In the case of QCs containing rare earth atoms R, there are still many unresolved issues. More information is needed on the dynamics of the correlations, although recent studies did tackle this point [48,49]. Determining the nature of the freezing probably requires a careful analysis of the critical behaviour above T f , as done for AlM n QCs (section 2.1). Finally an open and much debated question is why one has not yet observed long-range AF order in a real QC. From a theoretical point of view, no symmetrybased argument forbids the existence of a quasiperiodic long-range magnetic order [12]. Indeed theoretical models of magnetism on frustrated and non frustrated quasiperiodic tilings may exhibit long-range AF order [50][51][52]. However in these models, only first neighbor interactions are usually considered whereas, in all magnetic QCs known up to now, the interactions are of the RKKY type, i.e. long ranged and of variable sign. This may be an important point [52]. One can try to sketch the main points required for a fruitful comparison of models and experiments. Interactions beyond first neighbors have to be taken into account in models. Frustration obviously plays a central role both in models and in real QCs. It could interesting to compare QCs with the much studied periodic frustrated systems, such as the pyrochlore compounds which contain R tetrahedra. In these systems, frustration induces a rich variety of ground states [START_REF] Ramirez | Handbook on Magnetic Materials[END_REF], including non ordered ones (some of them exhibiting short range spin correlations [53]). Another point is to determine the crystal field effects in real QCs containing rare earths in order to know the nature of the ground state above T f . Note that frustration on a quasiperiodic lattice may induce non colinearity for three or two dimensional spins [51]. Hence it is important to know whether spins in QCs are Ising or Heisenberg like. Finally, when studying magnetic properties of any new QC system, it is necessary to know precisely its structure and especially whether the magnetic atoms form a quasiperiodic array. Knowledge of the structure will in addition allows to quantify the importance of the geometrical frustration. In the case of transition elements, an additional essential question is to know whether they all carry a moment. If not and if the magnetic atoms do not occupy well defined sites forming a quasiperiodic array, nothing more than a usual spin glass order is expected in presence of RKKY interactions, as it is indeed the case in AlP dM n QCs. This question is open in the case of the recently discovered ZnM nSc and ZnF eSc QCs [9].

Figure 1 .

 1 Figure 1. Susceptibility vs. temperature for several AlM n and AlP dM n QCs. Left: ac-susceptibility at low temperature showing a peak at T f ; Right: dc-susceptibility above T f .

Figure 2 .

 2 Figure 2. Temperature dependence of C, A 3 , A 5 (a.u.) of a crystalline dilute Cu 99% M n 1% alloy (from[20]) and of an icosahedral AlM nSi QC (from[22]). On this figure, only the T -dependence is significant: The C, A 3 and A 5 data have been arbitrarily vertically shifted to be presented altogether. The lines are fits used to deduce the critical exponents.

Figure 3 .

 3 Figure 3. Left : Influence of the thermal treatment on the magnetic susceptibility of a i-Al 69.8 P d 21.8 Al 8.4 QC single grain. Starting from state 1, obtained from a slowly cooling from 800 • C at a rate of 10 • C/hour, the concentration of magnetic M n atoms increases by a factor 2 after a heating at 800 • C followed by a rapid cooling at a rate of 150 • C/mn (state 2). A new heating at 800 • C followed by a slow cooling at a rate of 10 • C/hour (state 3) allows to recover the same moment concentration as in the initial state 1. Right: Composition dependence of the magnetic M n concentration (deduced from the Curie constant assuming S = 5/2): Relative values are given, 1 being assigned to the less magnetic sample studied. The P d and M n concentration are measured by XWDS and ICP-OES. All the samples are in the same annealing state: Cooled from 800 • C at a cooling rate of 10 • C/hour.

Figure 4 .

 4 Figure 4. Left: M n -M n interaction of two M n atoms as a function of their relative distance r. The interaction depends on the M n moments m 1 and m 2 (By courtesy of G. Trambly de Laissardière). Right: piece of the M n network deduced in the model of Ref. [36] (By courtesy of M. Quiquandon and D. Gratias)

4 Figure 5 .

 45 Figure 5. Left : Resistivity vs. T for dilute Cu 1-x M nx of different Mn concentration x and vs. x at different constant temperature(inset), from[40,41]. The curves are guides for the eye. Right : Same for AlP dM n QCs[26], except that i) x denotes the actual moment concentration and not the total M n concentration and ii) the ordinate is the conductivity σ instead of the resistivity ρ.

  More generally avoiding any kind of disorder is a essential in order to have a chance to observe AF order in a real QC.

Conclusion

Acknowledgments

We would like to thank M. Audier, C. Berger, Y. Calvayrac, J.C. Lasjaunias, E. Lhotel, V. Simonet, A. Sulpice and G. Trambly de Laissardière who brought major contributions to many of the works cited in the present paper. We are also grateful to S. Bonnevault, D. Gratias, D. Mayou and M. Quiquandon for many fruitful discussions.


Marie-Genevieve Barthes-Labrousse

Jean-Marie Dubois

Keywords: metallic alloys, quasicrystalline alloys user supplied): complex metallic alloys, properties, applications Quasicrystals, complex metallic alloys, properties, applications

HAL is

Introduction

Complex Metallic Alloys (CMAs) are compounds essentially consisting of more than two (often three or four) metal components at comparable quantities. They are characterised by giant unit cells containing up to thousands atoms, a structure based on well-defined (often icosahedral) clusters and some amount of disorder due to the fact that icosahedra do not fill in three dimensional space. Although the first structural studies of CMAs date back to the 1920s [1] very little attention has been initially paid to the practical usefulness of these materials until the discovery of QCs, in 1982 [2], re-activated the interest for these materials. Although aperiodic, QCs exhibit short range order structures similar to CMAs. Attractive and sometimes unexpected properties were then evidenced, opening the way for technological applications [3][4][START_REF] Gibbons | Physical Properties of Quasicrystals[END_REF][START_REF] Dubois | [END_REF][START_REF] Dubois | Useful quasicrystals[END_REF].

In order to be of technological relevance, a material must not only present challenging properties, but also be easy to produce in the desired shape (bulk, powder or coating), stable in operating conditions, of low cost and its constituents must be free of any known toxicity. In addition, it must benchmark already existing devices. It is not that easy to fulfil these criteria in full and only a few devices, mainly based on the mechanical properties of these materials (high hardness and/or low friction coefficient), are commercially available so far. However, based on their physical and chemical properties, CMAs and QCs may prove suitable for a number of other applications.

A few selected examples of QCs and CMAs most promising properties will be presented in this paper together with trends for possible applications. Special emphasis will be put on the electronic structure of CMAs and QCs, in particular on the appearance of a pseudo-gap in the density of states at the Fermi energy, which plays a specific role in the stabilisation of these alloys and in most of their properties.

The pseudo-gap at the Fermi energy

At equilibrium, the stability of a given structure is induced by the minimisation of the free energy F = U-TS. The internal energy U mainly depends on atomic size ratios, electronegativity differences and a band structure term. As many complex metallic phases only have small differences in electronegativity and atomic sizes, the band structure term dominates the stabilisation conditions. CMAs and QCs can thus be considered as a special class of Hume-Rothery alloys [START_REF] Friedel | [END_REF]. In particular, for Al-based complex alloys, the combination of the Hume-Rothery scattering of electron waves by Bragg planes and sp-d hybridization effects leads to the opening of a well-defined pseudo-gap in the Al 3p partial density of states at the energy of the Fermi level. The first experimental evidence of the existence of such a pseudo-gap was obtained on quasicrystalline AlMn thin films [START_REF] Traverse | Quasicrystalline Materials[END_REF] and was subsequently confirmed for a large number of quasicrystals and complex alloys. In particular, a detailed study of the variation of the density of Al 3p states at Fermi level as a function of valence electron per atom e/a (electronic concentration) for a number of AlCuFe intermetallics has shown that n(E F ) is much lower in the compounds based on the icosahedral structure (quasicrystalline or approximants) than in simple -cubic CsCl-type phases or tetragonal -Al 7 Cu 2 Fe crystals [10; 11] (see Figure 1). In short, a deep pseudo-gap seems to be characteristic of icosahedral short range order and to increase with structural complexity.

Insert Figure 1 The pseudo-gap is not only playing an important role in the thermodynamic stability of the complex materials but it has also important consequences on some of the most exciting properties of these alloys, such as transport properties. In addition it must be noted that the lowering of the density of states close to the Fermi edge is maintained on the CMAs and QCs smooth (sputtered and annealed) surfaces [12] and surface properties such as friction and wetting will thus also be strongly influenced by the electronic structure.

Optical conductivity and solar selective absorbers

The optical conductivity response ( ) of QCs is characterized by very low values and the absence of a Drude peak at low frequencies, together with the appearance of a peak conductivity in the visible range [13][14][15][16][17][18][19]. The absence of the Drude peak can be interpreted in the frame of the generalised Drude law proposed by Mayou [20]:

1 2 2 1 ) 1 2 ( ) ( ) ( ~ + = i A E n e
where ) ( ~ the complex conductivity, n(E) the total density of states, the gamma function, the diffusion exponent, A a constant, the frequency, the lifetime of the valence electrons.

The shape of ( ) (hence the existence of a Drude peak) is fixed by the value of which varies from =1 for simple metals to values close to =0 for pure icosahedral samples.

The optical conductivities ( ) of a mixture of two orthorhombic approximants of the decagonal O 1 /O 2 -AlCrFe phase and of a pure rhombohedral -AlCrFe brass phase have been recently investigated [21] and compared with those of the icosahedral AlCuFeB phase (i-AlCuFeB) in a broad range of frequencies . It has been shown that, in all complex alloys of large (or infinite) unit cell, the optical conductivity remains very weak at low frequencies. In addition, both the i-AlCuFeB and the O 1 /O 2 -AlCrFe are characterized by the absence of the Drude peak whereas the -brass phase exhibits a clear Drude peak albeit several orders of magnitude weaker than in pure aluminium (see Figure 2). Moreover, as shown in Figure 2, for mixtures of orthorhombic approximants and -brass phase (x % Al-Cr-Fe means x % -brass phase), the metallic character increases with the proportion of the -brass phase. In other words, the larger the unit cell the lower the conductivity at low frequencies.

Insert Figure 2 In view of their high absorbance in the solar wavelength region and their high reflectance at longer wavelength, CMAs and QCs soon appeared as good candidates for use as solar selective absorbers. Extensive studies have been performed by Eisenhammer et al. on AlCuFe thin films and cermets [22][START_REF] Eisenhammer | Quasicrystals[END_REF][START_REF] Eisenhammer | [END_REF][25][26][START_REF] Eisenhammer | Quasicrystals:Preparation, Properties and Applications[END_REF]. Devices consisting of sandwich systems of Al 2 O 3 /AlCuFe/Al 2 O 3 films deposited on copper films on Al 2 O 3 coated silicon were produced, showing 90% solar absorbance and low thermal emittance. Reasonable chemical and thermal stabilities in air were observed up to 400°C making these materials appropriate candidates for high temperature applications. However, rapid degradation in humid conditions as well as diffusion of copper through the Al 2 O 3 /AlCuFe sandwich at higher temperatures still are serious limitations to the usefulness of these devices in extreme conditions (i.e. beyond house ware).

Thermal conductivity and thermal barriers

The first evidence of the peculiar behaviour of the thermal conductivity in QCs was given in a pioneer work performed by Kuo and co-workers on Al 65 Ni 20 Co 15 and Al 62 Si 3 Cu 20 Co 15 decagonal crystals [START_REF] Zhang | [END_REF]. These authors have shown that the thermal conductivity of these 2D quasicrystals was abnormally low compared to that of their metallic constituents. Moreover, the values of measured perpendicular to the stacking axe (i.e. along the aperiodic direction) were an order of magnitude lower than in the periodic direction. The thermal conductivity can be expressed as the sum of a lattice contribution due to structural scattering of phonons lat and of the electronic contribution el :

) ( ) ( ) ( T T T el lat + =
The electronic contribution, which can be described by a modified Wiederman-Franz law [29], is negligible at low temperature where is dominated by umklapp scattering of phonons and phason-phonon interactions (see Figure 3a). Above room temperature the electronic contribution becomes comparable to the lattice contribution (see Figure 3b) but is hindered by the reduced density of states at the Fermi energy. It is worth noticing that the value of the thermal conductivity at room temperature ( 1Wm -1 K -1 ) is close to the value corresponding to pure zirconia and orders of magnitude lower than that of simple metals (200 Wm -1 K -1 for Al and 400 Wm -1 K -1 for Cu).

Insert Figure 3 The low thermal conductivity exhibited by these materials, combined with a thermal expansion close to metals and good corrosion and oxidation resistance made them interesting candidates for applications as barriers coatings for heat insulation. In particular coatings consisting of a AlCoFeCr approximant of the decagonal phase have been extensively studied for potential applications as coatings for components of aeronautical and power generation turbines [30; 31]. Excellent results were obtained in terms of thermal and corrosion properties. However, a strong interdiffusion between the coating and the substrate was observed at high temperatures (above 950°C), which drastically hinders potential applications to aeronautic devices, even though some diffusion barriers proved reasonably efficient to slow the interdiffusion effect. Nevertheless it is very likely that the excellent thermal properties exhibited by CMAs and QCs will find applications in less demanding fields in terms of temperature range.

Catalysis and hydrogen production

The interest in catalytic properties of CMAs arises from the fact that they often contain transition metals as typical constituents (which are good catalysts and can eventually be cheap raw materials!). In addition their brittleness turns to be an advantage as they can easily be crushed into powders. Extensive studies of the activity of AlCuFe alloys toward steam reforming of methanol (CH 3 OH + H 2 O 3H 2 + CO 2 ) have been performed by Tsai and co-workers [START_REF] Tsai | [END_REF][33][34][35] in pressure and temperature conditions reflecting those used in catalysis. Various compositions were evaluated based on the production of hydrogen gas. As these materials are covered with an aluminium passive layer, leaching treatments similar to that applied to Raney Al-alloys based catalysts, were followed by reduction in a hydrogen atmosphere before carrying the steam reforming of methanol experiments. Following such treatments, the prepared catalysts can be viewed as copper and copper oxide particles located on the surface of a complex metallic alloy core. The rate of hydrogen production was found to be comparable to that of the presently available commercial Cubased catalysts. Furthermore, higher production rates were obtained at high temperature for the AlCuFe quasicrystalline particles compared to conventional Cu-Raney catalysts (see Figure 4). This increase in the performance has been ascribed to a better dispersion of the copper metal in the porous aluminium oxide layer which is formed on the quasicrystalline catalyst surface and to the inhibition of sintering of the copper particles due to the presence of iron. However, as the catalytic activity and stability of the quasicrystalline Al-Cu-Fe phase is much higher than that of other Al-Cu-Fe phases (e.g. Al 55 Cu 25 Fe 12 beta-and Al 2 Cu theta-phases) specific interactions between copper particles and the quasicrystalline support leading to stabilisation of the copper clusters were also invoked [35]. Although it is not yet clear which specific property determines the good catalytic performance of the quasicrystalline phase [36], it seems that the catalytic activity could be related to a size effect that gives to Cu an atomic-like character [37].

Insert Figure 4

Friction under vacuum and cold welding

It was recognized soon after the discovery of QCs that the friction coefficient of these materials against antagonists like hard steel is significantly reduced compared with the friction coefficient measured in comparable conditions on conventional metallic alloys of similar mechanical characteristics [38]. Hence, reduced friction was for long taken as one of the best example of potential application of QCs and CMAs [3; 7]. However, the friction coefficient observed in air is only low in a transient regime [39]. It increases with time during the test and many artifacts due to tribo-oxidation are appearing.

Investigations performed in vacuum have shown that the friction coefficient strongly varies with the oxidation state of the surface [40][41][42]. In addition, on clean surfaces, the friction coefficient is lower for QCs and CMAs than for conventional metals and alloys [40][41][42][43][44][45][46]. As an example, a recent AFM investigation performed on a clean 2-fold decagonal d-Al-Ni-Co surface clearly evidenced the strong anisotropy of the friction force between the periodic 10-fold and the aperiodic 2-fold directions [47], the latter direction exhibiting much lower friction.

A comparison of the friction coefficients measured under vacuum in similar conditions on hard steel, sintered alumina and icosahedral Al 59 B 3 Cu 25.5 Fe 12.5 using a pin-on-disk test (hard steel ball) is shown on Figure 5. The most striking feature in this figure is the constant value observed for the AlBCuFe phase even for long riding distances, which contrasts with the strong variations observed for alumina and hard steel. These variations have been ascribed to the appearance of wear debris which either originate from the hard steel substrate and can form a lubricating third body by sticking to the pin surface (friction on hard steel) or originate from the pin and stick to the alumina surface which is ultimately covered with steel (friction on alumina). The low and constant value observed in the case of i-AlBCuFe indicates that wear is reduced and that steel does not stick on the substrate [42].

Insert Figure 5 Further experiments carried out on various materials (AlCuFe and AlPdMn quasicrystals, approximants of the decagonal phase like -Al 13 and friction [42; 44]. These results can be of primary importance for applications such as hindrance of cold welding in aerospace or vacuum industries.

Conclusion

Due to their remarkable diversity of properties, metallic materials will remain of utmost technological importance in the coming years. Although the industrial development of CMAs and QCs has remained quite limited so far, a number of interesting potential applications can be foreseen based on the specific properties of these materials. A few examples of these properties and potential applications have been briefly described in this paper. However, this description is far from being exhaustive and a number of other applications can also be foreseen, such as hydrogen storage, high-performance alloys, metal and polymer composites, thermo-power generation, quantum dots…. Furthermore, it must be pointed out that perhaps the most promising future for QCs and CMAs is the possibility they can offer to combine properties which are mutually excluding in conventional materials (e.g. metallic electric conductivity combined with low thermal conductivity, good light absorption and high temperature stability, high metallic hardness and low wetting by polar liquids…). However it must be kept in mind that, in order to successfully compete with other materials such as polymers and ceramics, QCs and CMAs must offer superior solutions at competitive costs together with environmental harmlessness and low energy consumption during production, service and recycling. In addition, due to the intrinsic brittleness of these materials at low temperatures, many applications are foreseen in the form of surface coatings. Therefore tayloring thin films or coatings with reproducible characteristics, stability and low cost in large quantities is a major issue for industrial success. Finally, due to the huge number of possible combinations between the 80 possible constitutive metals, it is likely that a high throughput combinatorial approach will be necessary for identifying the most promising materials.

  Figure 1

  Fe 4 , orthorhombic O-and -brass AlCrFe compounds, Al-based intermetallic compounds like -Al 7 Cu 2 Fe, -Al 10 Cu 10 Fe, -Al 50 Cu 35 Fe 15 , -Al 4 Cu 9 and o-Al 3 Cu 4 , pure Al and pure Cu) have shown that the friction coefficient scales with the inverse of the hardness of the material and with the density of d-states at the Fermi energy which can interact with the d-states of the ironcontaining rider. A low density of d-states at the Fermi energy leads to low interaction hence low adhesion

Figure 1 .

 1 Figure captionsFigure 1. Variation of the density of Al 3p states n(E F ) as a function of electronic concentration in fcc Al (cross), -cubic crystal (open squares), -Al 7 Cu 2 Fe crystals (open dot), AlCuFe approximants (solid squares) and icosahedral Al 62 Cu 25.5 Fe 12.5 (solid dot). From [10].

Figure 2 .

 2 Figure 2. Optical conductivity at low frequency for an icosahedral AlCuFeB phase (i-AlCuFeB) and mixtures of AlCrFe orthorhombic approximants of the decagonal phase and -brass phase (0%-AlCrFe means pure O 1 /O 2 -AlCrFe and 100% means pure -brass phase). Courtesy V. Demange, Nancy.

Figure 3 .

 3 Figure 3. Total thermal conductivity (solid circles) and lattice contribution (open circles) at a) low temperature in i-Al 62 Cu 25.5 Fe 12.5 and b) above room temperature in Al 59 B 3 Cu 25.5 Fe 12.5 . After [7].

Figure 4 .

 4 Figure 4. H 2 production rate for the steam reforming of methanol as a function of the reaction temperature for Cu-based catalysts leached in NaCO3 aq. for a) Raney Cu (commercial Cu-based catalyst); b) quasicrystalline Al-Cu-Fe crushed by wet milling in ethanol and leached at 323K. From [34].

Figure 5 .

 5 Figure 5. Friction coefficient in vacuum against a hard steel ball observed during a pin-on-disk test for hard steel, alumina and i-Al 59 B 3 Cu 25.5 Fe 12.5 . After [42].

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Acknowledgements

The authors gratefully thank Dr. Valérie Demange for the kind provision of figure 2. This review was undertaken within the activities of the 6 th Framework EU Network of Excellence "Complex Metallic Alloys" (Contract No. NMP3-CT-2005-500140).


Marc Heggen

Lothar Houben

Michael Feuerbacher

Metadislocations in the

Keywords: dislocation structures, phase changes, plasticity, TEM user supplied): complex metallic alloys, metadislocation

HAL is

Metadislocations are novel defects in complex metallic alloys (CMAs) and were firstly observed in the phases ´-and -Al-Pd-Mn [1], and more recently in -Al-Pd-Fe [2]. The core of a metadislocation is a line defect which ideally extends along the [0 1 0] direction of the ´-phase lattice. It is associated to a certain number of planar defects, so-called phason planes, which are ideally (0 0 1) oriented. The existence of metadislocations in other groups of CMAs, which are not isostructural or directly structurally related to ´-and -Al-Pd-Mn, was proposed [3]. One of these phases is the orthorhombic phase Al 13 Co 4 .

In a previous work on single crystalline o-Al 13 Co 4 plastic deformation experiments and microstructural investigations were performed by means of transmission electron microscopy (TEM) [4]. These investigations on plastically deformed samples revealed a high density of [ [6,7,8,9,10]. The tiling description of m-Al 13 Co 4 (figure 1 b) is characterized by a parallel arrangement of the pentagon and rhomb elements.

In order to compare both tilings a cubic basis using the edge length of the tiles (4.7 Å) is used in the remainder of this paper. In terms of this basis the (1 0 0) projection of the orthorhombic cell is spanned by the vectors ( 1 + 0) and (0 2 + ) and that of the monoclinic cell by ( 1 + 0) and (1 2 + ), where = 2 1 5 + is the number of the golden mean. § 3. Experimental Al 13 Co 4 single grains of the order of 1 to 2 cm 3 were grown by means of the Bridgman technique. The quality of the material was characterized by means of phase-contrast optical microscopy, scanning electron microscopy, and TEM. Cuboid shaped samples were plastically deformed using a modified Zwick Z050 testing machine in uniaxial compression at constant strain rate of 10 -5 s -1 . Samples were deformed at 650 °C and 800 °C to about 6 % strain. After deformation, the samples were rapidly unloaded, quenched on a cold metal plate to preserve their microstructural state, and prepared for TEM. The microstructural investigations were carried out using a JEOL 4000EX transmission electron microscope and a FEI TITAN 80-300 transmission electron microscope equipped with an aberration corrector for the probe-forming lens. Highangle annular dark-field (HAADF) images were taken at 300 kV at a probe semiangle of 25 mrad and a detector inner collection angle of 70 mrad. § 4. Microstructural investigation were observed. Saito and coworkes [8] identified (0 0 1) planar defects in as-cast Al 13 Co 4 and identified them, by means of high-resolution TEM and X-ray diffraction, as monoclinic slabs in an orthorhombic matrix. Further examples and references on planar defects and twinning in phases which are structurally related to Al 13 Co 4 are found, e.g., in [11].

Let us analyze the structure of the dislocation in figure 3 in more detail. The tiling description cannot be applied directly to the high-resolution TEM micrograph in the same obvious way like to the HAADF micrograph. However, the configuration of unit cells around the dislocation is clearly observable and will be used to develop a tiling model of the dislocation core. In figure 4 a the tiling for o-and m-Al 13 Co 4 is applied to the lattice structure around the dislocation core. The dark grey polygon represents the core of the dislocation. To match the strained structure, the tiles are slightly distorted around the core. A Burgers circuit is performed around the dislocation core (figure 4 a) and transferred to an undistorted tiling (figure 4 b). The latter tiling reveals a closure failure and displays the Burgers vector of the dislocation (white arrow). It corresponds to the vector 1/ 3 12.3 Å [0 1 0] = 2.9 Å [0 1 0] in terms of the orthorhombic basis. We note that the determination of the Burgers vector is not dependent on a specific geometric description of the dislocation core in terms of the tiling. The same Burgers vector is determined by performing a Burgers circuit, shown in figure 3, around the dislocation core using the unit cell representation.

The dislocation observed in the present work is an edge dislocation which moves in terms of glide [4]. Upon movement it creates not a conventional stacking fault but a structurally modified slab of the m-Al 13 Co 4 phase. The structural transition partially compensates the large lattice distortion of the inserted half plane and yields a Burgers vector which is much smaller than the respective lattice constant. This mechanism resembles shear transformations observed in Laves phases where moving dislocations may introduce structurally modified slabs [12][13][14]. In the C14 Laves phase Cr 2 Hf, for instance, dislocation motion creates a slab of the cubic C15 Laves phase within the C14 matrix [14]. Furthermore, the dislocations observed in the present work are closely related to metadislocations having the same Burgers vector [3]. The main difference, however, is related to the planar defects associated with them. Metadislocations are necessarily associated to a set of planar defects, so-called phason planes [1]. While the dislocations observed in the present paper are connected to (0 0 1) planar defects, the associated phason planes of the proposed metadislocations are (0 1 0) planes [3]. Therefore the mode of dislocation motion is different.

While metadislocations move by means of pure climb, the present dislocations move by glide.

In conclusion, despite the difference in the mode of dislocation motion, the dislocations observed in the present work show clear similarities with metadislocations. This observation suggests an extension of the term metadislocation to a more general concept including metadislocation glide: A metadislocation is a line defect with a Burgers vector which is a 1/ nfraction (n = 1, 2, 3,…) of the corresponding lattice constant. It is associated with a local phase transformation to a phase which is structurally closely related to the matrix. By this means, the irrational Burgers vector fraction is accommodated to the lattice. The metadislocation core region comprises structural features of both phases. This definition involves the dislocations observed in CMAs Al 13 Co 4 (this work), ´-and -Al-Pd-Mn [1], and -Al-Pd-Fe [2].

  1 0 0] dislocations terminating (0 0 1) planar defects. Contrast-extinction experiments disclosed that the Burgers vectors of dislocations and the displacement vector of the associated planar defects are parallel to the [0 1 0] direction. It was concluded that dislocation glide on (0 0 1) planes is the basic mechanism of plastic deformation. Here, we present a more detailed microstructural investigation on the structure of dislocations in o-Al 13 Co 4 by means of scanning transmission electron microscopy (STEM) and TEM. The microstructural investigation aims at a comparison with the structure of metadislocations. Results on the basis of a tiling approach are discussed. § 2. The structure of Al 13 Co 4 Al 13 Co 4 is an orthorhombic phase (denoted o-Al 13 Co 4 ) with space group Pmn2 1 and lattice parameters a = 8.2 Å, b = 12.3 Å, and c = 14.5 Å [5,6]. Structure investigations were performed by several groups [e.g. 6,7]. The main structural feature of o-Al 13 Co 4 are pair-connected pentagonal prismatic channels extending along the [1 0 0] direction [6]. Within the (1 0 0) plane, the o-Al 13 Co 4 structure can be matched by a tiling consisting of regular pentagons and rhombs [8], depicted in figure 1 a. The edges of the tiles correspond to atomic columns, which are predominately occupied by cobalt atoms. The orthorhombic unit cell is superimposed onto the tiling. The structure of monoclinic phase m-Al 13 Co 4 is homeotypic to that of monoclinic Al 13 Fe 4 , and to o-Al 13 Co 4

Figure 2

 2 Figure 2 shows a HAADF micrograph of a deformed Al 13 Co 4 sample along the [1 0 0] direction. Bright dots in the micrograph correspond to atomic columns which are preferentially occupied by cobalt atoms. They can be matched by the o-Al 13 Co 4 tiling. Columns occupied by aluminium are not visible. We have imaged stacking faults with (0 0 1) fault plane in edge-on orientation (marked by arrows). They are not conventional planar defects involving a displacement at a sharp interface, but structurally modified slabs of nanometre thickness. The structure within the slabs can be matched by a tiling corresponding to the m-Al 13 Co 4 rather than that of the o-Al 13 Co 4 phase.

Figure 3

 3 Figure 3 shows a high-resolution TEM micrograph of a dislocation (central part of the image) in end-on orientation. The dislocation terminates a (0 0 1) planar defect, i.e. a structurally modified slab of m-Al 13 Co 4 , extending to the right-hand side of the image. The outline of the o-Al 13 Co 4 unit cells is depicted by white rectangles, which can be used to cover the undistorted o-Al 13 Co 4 crystal regions. Rhomboids representing the unit cell of the monoclinic Al 13 Co 4 phase are used to cover the structurally modified slab. Analysing the structure around the dislocation core region, the micrograph clearly shows the presence of three unit cells of o-Al 13 Co 4 directly above the dislocation core and two unit cells below the dislocation core, i.e. one additional (0 1 0)half plane, extending to the upper region of the image, is inserted at the dislocation core. The defect displayed in Figure3represents a typical dislocation, terminating a planar defect in deformed Al 13 Co 4 . Similar dislocations, which can be described using the same arrangement of rectangular and rhomboidal structural elements have been observed at a high density.

Fig. 1

 1 FIGURE CAPTIONS

Fig. 2

 2 Fig.2 High-angle annular dark field micrograph of a deformed Al 13 Co 4 sample along the [1 0 0] direction. White circles correspond to cobalt columns. Tiling representations are superimposed. Structurally modified slabs corresponding to the m-Al 13 Co 4 rather than that of the o-Al 13 Co 4 phase can be observed (arrows and broken lines).

Fig. 3 Fig. 4

 34 Fig.3 High-resolution micrograph of a dislocation (centre part of the image) in edge-on orientation terminating a (0 0 1) planar defect (right hand side of the image). White rectangles and rhomboids indicate the unit cell of o-and m-Al 13 Co 4 .

  Tiling representation of the orthorhombic (a) and monoclinic (b) Al13Co4 structure using pentagon and rhomb elements. The unit cells are superimposed to the tilings. A projection of the orthorhombic Al13Co4 structure according to Ref. [6] is given. 705x257mm (72 x 72 DPI) annular dark field micrograph of a deformed Al13Co4 sample along the [1 0 0] direction. White circles correspond to cobalt columns. Tiling representations are superimposed. Structurally modified slabs corresponding to the m-Al13Co4 rather than that of the o-Al13Co4 phase can be observed (arrows and broken lines). 521x437mm (72 x 72 DPI) micrograph of a dislocation (centre part of the image) in edge-on orientation terminating a (0 0 1) planar defect (right hand side of the image). White rectangles and rhomboids indicate the unit cell of o-and m-Al13Co4. 564x395mm (72 x 72 DPI) Structural model of the dislocation in figure 4 in terms of the tiling description. (b) Enlargement of the dislocation core in terms of an unstrained tiling disclosing the Burgers vector (arrow) of the dislocation. 564x204mm (72 x 72 DPI)

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Acknowledgments

This work was supported by the Deutsche Forschungsgemeinschaft (PAK 36), and the 6 th Framework EU Network of Excellence "Complex Metallic Alloys" (Contract No. NMP3-CT-2005-500140).