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Dynamical properties of stable icosahedral alloys
Journal of Non-Crystalline Solids 156-158 (1993) 872-881 North-Holland
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Dynamical properties of stable icosahedral alloys Jens-Boie Suck 1 Institut Max yon L a u e - P a u l Langevin, B P 156, F38042 Grenoble c~dex 9, France
Results from theoretical investigations of the vibrational properties of quasicrystals are compared with those obtained so far for the atomic dynamics of stable icosahedral alloys by neutron inelastic scattering, especially using as an example the m o s t recently studied icosahedral A1-Pd-Mn system. It is concluded that although many of the dynamical properties to be expected on the basis of the theoretical investigations of three-dimensional quasicrystals have been found in experiments done with stable icosahedral alloys, some very important questions concerning properties specific for quasiperiodic lattices like the existence of gaps or localization still await a final answer.
1. Introduction
First, the atomic dynamics of metastable icosahedral alloys have been studied [1], which are obtained by the same methods as those used to produce metallic glasses [2]. These methods normally introduce such a large amount of disorder in the atomic structure, that it is difficult to decide whether the model of the icosahedral glass [3] or a strongly defective quasiperiodic lattice is the appropriate description for the atomic structure of these alloys. The atomic dynamics (like the electronic properties) are considerably influenced by this disorder [4], which cannot be removed by an anneal because of the metastability of the icosahedral phase. For stable icosahedral alloys, for which the disorder introduced by the production methods can be removed and the phase can be stabilized by annealing, or which can be obtained directly by slow cooling from the melt as monodomain grains produced by Bridgeman or Czochralski
i On leave from: Kernforschungszentrum Karlsruhe, Institut fiir Nukleare Festk6rperphysik, P.O.B. 3640, D-7500 Karlsruhe, Germany. Correspondence to." Dr J.-B. Suck, Institute Laue-Langevin, Avenue des Martyrs, BP 156, F-38042 Grenoble c6dex, France. Tel: + 33 76 20 71 35. Telefax. + 33 76 48 39 06.
techniques or as polydomain ingots, the icosahedral glass model is inappropriate and one of the two tiling models (with or without long-range order in the atomic positions) should apply [5]. From all the models set up in order to describe the atomic structure of icosahedral (octahedral, decagonal or dodecagonal) alloys, this model of a quasiperiodic lattice, of the quasicrystal (QC), (with or without matching rules) makes the most definite predictions concerning the physical properties of these alloys.
2. Theoretical predictions for the atomic dynamics on the basis of the quasicrystalline model
2.1. Hydrodynamic limit In the hydrodynamic limit of long wavelength and low frequencies, the existence of three acoustic phonon modes, corresponding to the three translational degrees of freedom in the three-dimensional subspace, Eli, of the six-dimensional space, in which the periodic icosahedral crystal image is embedded, and three phason modes, corresponding to the three degrees of freedom in the three-dimensional subspace, E ± , orthogonal or perpendicular to Ell, was predicted. The high symmetry of the icosahedral lattice leads to only
0022-3093/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
J.-B. Suck / Dynamical properties of stable icosahedral alloys
two phonon elastic constants (one for longitudinal and one for transverse modes) and two phason elastic constants plus one related to phonon-phason interaction [6,7]. Thus for small momentum transfers and frequencies one should expect isotropic longitudinal and isotropic transverse acoustic modes. They also suggest that phasons should be frozen at ambient temperatures [7]. 2.2. One-dimensional models Most analytical and numerical calculations were carried out for one-dimensional systems, Fibonacci chains or modulated Fibonacci chains. Generally these calculations predict critical wavefunctions (neither extended nor exponentially localized) and singular continuous spectra (Cantor sets). Critical wavefunctions are the consequence of the competition between the aperiodicity of the lattice, which leads to localization, and the long-range orientational order and the effect of the infinite short-range repetition of each local pattern in a quasiperiodic lattice (Conway's theorem), which both favour extended modes [8]. Critical modes seem to have been observed recently as modes on surfaces, which had been corrugated in a Fibonacci sequence [9]. However, depending on the potential parameters or a generalization of the Fibonacci sequence [10], extended eigenstates (with well defined wavevector) and corresponding continuous spectra or localized states with discrete spectra are also found [11], Extensive investigations on one-dimensional quasiperiodic systems have been done by Benoit and collaborators calculating the harmonic response directly from the dynamical matrix via the spectral moment methods [12]. These investigations have shown that phonon dispersion curves should be observable by neutron inelastic scattering near the strongest Bragg peaks, and pseudo Brillouin zones should exist around these F points. The dispersion should be observable until the Brillouin zone under consideraation 'interacts' with the one from the next strong Bragg peak. The observed phonon intensities should be directly proportional to the intensity of the Bragg peak at the F point. Besides acoustic modes with
873
dispersion, dispersionless optic modes are predicted by these calculations. A main point raised in all these investigations is the existence of gaps in the phonon dispersion curves and the DOS of one-dimensional systems, which, if they existed as gaps or as pseudogaps in the spectra of three-dimensional QC also, would lead to very structured DOS for these systems. For two-dimensional QC, Penrose tilings and Fibonacci superlattices have been investigated and similar results have been obtained as for one-dimensional QC [13-15]. 2.3. Three-dimensional models For three-dimensional QC, numerical investigations have mainly been done for Ammann type of tilings with just one atom per vertex and only one degree of freedom [16] or more realistically admitting atomic motions in all three directions [17]. In the first case, a single band DOS has been found with very sharp structures, which are partly due to the fact that for quasicrystals the rank r is larger than the dimension, d, [18] (e.g., r = 6 in the case of a 3d icosahedral quasilattice) and thus more basis vectors in the reciprocal space lead to more van Hove singularities than in the case of periodic crystals. Most of these structures disappear as soon as all three degrees of motional freedom are admitted [17]. A slightly different approach was taken most recently using the tiling of Danzer decorated with three different sorts of atom [19]. The calculations are based on a matrix continued fraction for the larger systems and exact diagonalization of the dynamical matrix for a smaller cluster. These latter calculations give a rather smooth one-band DOS which looks very similar to the one-band DOS determined experimentally so far. Finally the existence of localized and propagating modes in a realistic model of the icosahedral quasicrystal (Al Zn)agMg32 [20] was investigated using recursion methods and a priori psuedopotentials for these s-p bounded metals [21,22]. Localized modes (defined from their participation ratio) were found in several narrow frequency regions distributed over about one third of the full frequency range near the low-energy
874
J.-B. Suck / Dynamical properties of stable icosahedral alloys
end and over about the same frequency interval at the high-energy end of the frequency spectrum. These localized modes could be related to local topological frustration due to deviations from ideal icosahedral packing induced during the relaxation of the initial tiling with perfect icosahedral symmetry under the realistic atomic pair potentials [23]. Thus localization here is caused by local properties of the lattice and not by its long-range aperiodicity! The same model was used to calculate the spectral functions in order to search for propagating modes [22]. Longitudinal and transverse acoustic modes were found in the vicinity of strong Bragg peaks, which had been assigned to the F points of nearly isotropic pseudo Brillouin zones [23]. From the initial slopes of the dispersion curves, the authors find elastic isotropy for longitudinal and for transverse acoustic modes (i.e. only two elastic constants as expected for icosahedral symmetry). The system is isotropic also with respect to the polarization of the transverse acoustic modes. The dispersions of these latter modes meet with a horizontal slope at the pseudo Brillouin zone boundary. Since the atomic dynamics of a QC, because of its aperiodicity, are described by an infinite number of coupled equations of motion, all numerical results mentioned above have been obtained for approximations of QC: either for finite embedded clusters or for rational approximants, i.e., periodic crystals with large quasiperiodic unit cells. Crystals of this type in fact have been found and their atomic dynamics have been investigated in at least one case [25,26]. As the number of atoms in the unit cell increases very rapidly with increasing order of approximant, presently the dynamics of rational approximants up to 2/1 can be diagonalized exactly (688 atoms per unit cell), and for higher approximants (up to the 5 / 3 with 12380 atoms per unit cell) recursion techniques have been used to determine the local density of states (LDOS). In all cases, the interactions have been limited to harmonic interaction of nearest neighbours. With the exception of the computer simulation [21,22], the precise character of the modes is difficult to determine in these investigations.
3. Results from neutron inelastic scattering
3.1 Experimental methods In neutron inelastic scattering experiments [27] one determines the intensity I(ko, k) of the scattered neutrons as a function of the wavevector of the neutron before (k 0) and after scattering (k) either subsequently for each energy, hto, and each momentum transfer, hQ, on a triple-axis spectrometer (TAS), if one has a monodomain grain of the alloy and can therefore determine the phonon branches in chosen directions in the reciprocal space of the crystal, or by measuring a broad range of hto and hQ values at the same time on a time-of-flight (TOF) spectrometer using a polydomain sample. From I(ko, k), the double differential scattering cross-section and the total dynamic structure factor, S(Q, to), is determined for a sample containing n different elements (since the stable icosahedral alloys are all ternary alloys, n = 3):
S(Q, to)
4~k 0 d2o" ~rSCk dO d E
(1)
Here o,so is the total scattering cross-section (coherent plus incoherent) of the sample material and k 0 and k are the moduli of the wave-vector of the neutron before and after scattering. The measured S(Q, to) is the weighted sum of the partial dynamic structure factors Sij(Q, to) plus the self part of the structure factors SiS(Q, to) if the sample scatters neutrons incoherently also: n
o-ScS(Q, to) =
4.rr)" bibjcicjSij( O , to) i,j n
+ ]~_~o'iincciSS(O , to).
(2)
i
For most of the icosahedral alloys the incoherent scattering cross-sections are small. Thus the total dynamic structure factor contains the information on the collective atomic vibrations (phonons). The single particle motions of the atoms are reflected in the generalized vibrational density of states (GVDOS), G(to), which is the normalized weighted sum of the partial density of states, gi(to), (the Fourier transform of the velocity auto-
J.-B. Suck / Dynamical properties of stable icosahedral alloys
correlation function) of the element i in the alloy: S sc
G(~o) = E e
_2W
i
o-/sc
l ci-77gi(to)/~_,e-ZWici
i
Mi
i
•
Mi
(3) Here e - 2 w , ci ' tri~c and M i are the Debye-Waller factor, the atomic concentration, the scattering cross-section and the mass of the element i in the alloy, respectively. The GVDOS can be determined directly from the second sum in eq. (2), if the first sum is zero (purely incoherent scatterer). In most cases, and
'
i
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Q=115 ~
this applies especially also to the stable icosahedral alloys discussed here, which are all coherent scatterers, the first sum is dominant and one therefore has to determine the mean value of the scattered intensity for each frequency over reciprocal space in order to average to zero the coherence effects in the scattering (for details see ref.
[27,33]). 3.2 Phonon Dispersion of Al7o.sPd21Mns. 5
Up to now, the atomic dynamics of three stable icosahedral alloys have been investigated: 0 0
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Fig. 1. Constant q scans of transverse and longitudinal phonons measured at room temperature with the triple-axis spectrometer 1T in different directions in the reciprocal space of an icosahedral A170.sPd21Mn8.5 monograin between strong elastic reflections at the F points of the corresponding pseudo Brillouin zones [32]. The transverse phonon in fig. l(a) has a resolution-limited width up to the zone boundary at q = 0.35. For higher energies and above q = 0.5 ,~- 1, the phonons broaden rapidly (b).
J.-B. Suck / Dynamicalproperties of stable icosahedral alloys
876
determined; also in one case the phason hopping at high temperature has been studied [30,31]. As an example, the results of the most recent investigation of a 'monodomain' grain of A170.5Pd21Mn8.s obtained by Bridgeman techniques [32] and of a polydomain sample of m171Pd19Mn10obtained by slow cooling of the melt [33] will be discussed. The condition of formation of the stable icosahedral alloy has been studied extensively [34-37]. The alloy can be obtained in the icosahedral phase around 20 at.% Pd and 8 to 10 at.% Mn by conventional casting or by rapid quenching and remains stable during heat treatment up to the temperature of fusion. Likewise, the structure of
AlxLiTCu z [24-26], AlxCuyFe z [28-31] and AlxPdyMn z [32,33]. In the sample preparation, the relative concentrations of the elements play a crucial role in order to obtain a well ordered, purely icosahedral sample [34]. x, y and z vary very little from one sample to the other if they contain the same elements. However, monodomain grains and polydomain sample sometimes do not have absolutely identical nominal concentrations. For all three stable icosahedral alloys, the low-energy phonon branches [25,26,28,33], the generalized vibrational density of states and the total dynamic structure factor [24,29,32] have been
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Fig. 2. Dispersion curves of longitudinal and transverse phonons measured with a monograin of Ai70.sPd2zMns. 5 [32]. The initial slopes of the dispersions are the same for either transverse phonons or longitudinal phonons independent of the direction of propagation (fivefold axis, twofold axis). In the lower part of the figure (c), a nearly dispersionless branch is observed above the longitudinal acoustic branch which has been attributed to optic modes. No phonons could be measured above 4 THz (16.53 meV).
J.-B. Suck / Dynamical properties of stable icosahedral alloys
icosahedral A1-Pd-Mn has been investigated intensively with powder diffraction [38] and fourcircle X-ray and neutron diffraction on monograins [39]. In the six-dimensional description, the lattice is face centered but one can regard it as a superstructure of a primitive lattice. This is due to the strong chemical ordering, which was realized from the very beginning, when the icosahedral alloy was first discovered [40,35]. The 'monodomain' grain was investigated at room temperature on the thermal neutron tripleaxis spectrometer 1T at the Orph6e reactor in Saclay. Constant-Q scans in different symmetry directions of the reciprocal lattice between strong elastic peaks at the F point of the pseudo Brillouin zone were measured. Figure l(a) shows three scans of a transverse phonon propagating along a fivefold axis. The phonon can be easily followed up to the boundary of the pseudo Brillouin zone at q = 0.35 ,~,-1. This boundary can be clearly observed in the experiment. The corresponding dispersion curves are shown in fig. 2. All dispersion curves of longitudinal modes have the same initial slope regardless of where in reciprocal space and in which symmetry direction they have been measured. The same argument applies (with a different slope) to the transverse modes. In fig. 2(c) also a nearly dispersionless branch at 4 THz (1 THz--4.13 meV) is seen, which was attributed to optic type of vibrations. Below this branch, the dispersion curve of the LA modes flattens rather quickly at higher energy. This has been interpreted by the authors as being due to some tendency towards localization more pronounced than in periodic crystals. No phonons could be measured with frequencies larger than 4 THz. As can be seen from fig. l(b), this is largely due to the fact that the phonons broaden considerably at higher energy transfers. The low-energy phonons at q < 0.5 on the contrary are sharp and only broadened by the resolution function of the spectrometer.
3.3. Total dynamic structure factor of Al71Pd w Mn lo If one wants to investigate the total dynamic structure factor at higher energy transfers than
877
done in the TAS experiment, then one has to use large polydomain samples and TOF techniques. Polydomain samples have the disadvantage of giving only orientationally averaged information as there is no preferred orientation for the randomly oriented grains. For icosahedral alloys, because of their isotropic elastic behaviour, this disadvantage is probably less severe than for other less isotropic crystalline material as far as the acoustic modes are concerned. The second disadvantage is the fact that the structure of the sample cannot be investigated by a diffraction method as detailed as that possible for a monodomain grain. The isocahedral structure of the sample material, obtained by slow cooling from the melt without subsequent annealing, was tested by neutron powder diffraction [33], and the stability of the phase was investigated using differential scanning calorimetry [41]. The experiment was done using the cold-neutron time-focusing TOF spectrometer IN6 at the HFR of the Institut Laue-Langevin with an incident energy of 4.7 meV. From the TOF spectra, the total dynamic structure factor, S(Q, to), was determined. Details of the
Fig. 3. 59 sections through the total dynamic structure factor of icosahedral A171PdlgMn10, measured with a polydomain sample at 280 K on the cold-neutron time-of-flight spectrometer IN6. Only the foot of the elastic peaks is shown in order to augment the inelastic (neutron energy gain) part of S(Q, to). The maxima of the structures observed in S(Q, to) at constant Q have been used to determine the dispersion shown in fig. 4.
878
Suck / Dynamicalproperties of stable icosahedralalloys
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neutrons), which have not been subtracted, have no peak structure. Only a few acoustic modes are possibly observed at energies < 10 meV. There is a dominant band near 12 meV and a second band of this type is seen near 15 meV. The lower band shows appreciable dispersion only in the neighbourhood of the strongest D e b y e - S c h e r r e r peaks near 25, 29 and 30.5 nm -1. Some further dispersionless bands at 18, 21.5, 24.5 and 35 meV are seen in a rather limited range of momentum transfers.
40
Fig. 4. Energies corresponding to the maxima and shoulders in the total dynamic structure factor of icosahedral A171Pd19Mn10 in dependence on the corresponding momentum transfer, hQ. Apart from the energy region below 10 meV, where the excitations are most likely to be acoustic modes, the energy bands observed at 12 and 15 meV show appreciable dispersion only near the strongest DebyeScherrer lines in the diffraction pattern (marked by arrows, the length of which corresponds to their relative intensity). At higher energies, dispersionless branches are found in a restricted region of the extended zone scheme only. The lines enclose the dynamical space of the experiment. Open symbols: weak structures; filled symbols: strong, well defined structures. Different symbols have been used to indicate different energy regions. ~
experiment and the data evaluation will be given elsewhere [41]. 59 cuts through S(Q, o~) are shown in fig. 3. A detailed analysis of the spectra at each momentum transfer allows the energy corresponding to the maxima and shoulders in the spectra to be determined. The resulting 'dispersion curves' are shown in fig. 4. In this derivation of the dispersion curves, it has been assumed that the multi-phonon contributions to S(Q, o~) (and likewise the intensity due to multiply scattered
In order to determine the G V D O S for the same polydomain AI71Pdl9M10sample, an experiment was performed at the thermal neutron T O F spectrometer IN4 at the H F R of the Institut L a u e - L a n g e v i n in Grenoble, with a sample temperature of 120 K (to keep the multiphonon contribution small) and an incident energy of 68.8 meV. The high incident energy guarantees an effective averaging of the coherence effects, a small error only due to sampling in parts of Brillouin zones [33] and the sampling of phonon frequencies mainly where the paramagnetic scattering from the sample is small [41]. Details of the experiment, the data evaluation and the corrections applied are given elsewhere [33]. The G V D O S consists of two broad main bands centred near 16 and 31 meV with a shallow pseudo-gap near 21 meV. A very weak third band is indicated in the pseudo-gap at 22.5 meV. In the main band at low energy, a very weak shoulder is indicated near 14 meV. For energies > 25 meV, the statistical accuracy of the data is insufficient to resolve any fine structure in the main band at higher energies. In order to investigate finer structures, the G V D O S was also determined from the weighted sum of all neutron energy gain spectra measured with cold neutrons on IN6 (see section 3.3.). The resolution near hw = 0 was nearly a factor of 40 better than on IN4, which is very important in the search for gaps or a pseudo-gap structure in the low-energy region of the GVDOS. Details of this experiment and the evaluation of the data will be given elsewhere [41]. Because the very good reso-
J.-B. Suck / Dynamical properties of stable icosahedral alloys
0.040 0.032 "~0.024 3
.~ 0.016 ¢...9 0.O08j
0(5
10
20,.h~
[me3~j 4b
50
Fig. 5. Generalized vibrational density of states of icosahedral A17tPd19Mn10 determined from the weighted sum of all TOF spectra measured at 280 K at the cold-neutron time-focusing spectrometer IN6. A splitting of the first main band into two subbands, 2.5 meV apart, is indicated at 13 meV.
lution at hoo = 0 (the energy transfer with the optimal focusing condition) becomes rapidly worse at higher energy transfers, the discussion will be focussed on the low-energy band. Figure 5 shows the resulting GVDOS. It is essentially identical within experimental errors to the GVDOS determined with thermal neutrons on IN4 [33], apart from an energy shift of both bands by ~ 1.2 meV to lower energies which can partly be explained by the difference in the resolution functions of the two spectrometers. In place of the very weak shoulder at the low-energy band mentioned above, now a clear splitting of this band into two 'subbands', 2.5 meV apart, is indicated.
4. Discussion
4.1 Phonon dispersion Even though no special effort has been put into the determination of the phonon linewidth with the help of a high-resolution measurement for icosahedral AIPdMn, the intense and resolution-limited transverse phonons shown in fig. l(a) suggest that these excitations have a relatively long lifetime and can travel far, on the scale of phonons in crystals, especially if they are compared with the relatively broad excitations measured so far in the other stable icosahedral alloys
879
[25,26,28]. A reason for the existence of these well defined excitations in A I - P d - M n may be the high degree of chemical (and topological) order, which leads to a strain-free quasiperiodic lattice [40]. From the theoretical point of view, one would expect that the tendency towards localized modes decreases going from one-dimensional to three-dimensional QC especially at lower frequencies [8]. For the reasons mentioned above, the dispersion curves obtained from the total dynamic structure factor and from the TAS experiment can only be compared in the upper energy region of the TAS results. Here the energy bands observed at 3 THz in fig. 2(b) and the flat part of the longitudinal phonon branch in figs. 2(a) and (c) coincide very reasonably with the extended band of excitations seen in the total structure factor at 12 meV (3 THz = 12.4 meV). The same conclusion is valid for the dispersionless branches seen in figs. 2(b) and (c) and above 15 meV in fig. 4. The short dispersionless branches, which were also found earlier in the dispersion curves obtained from the total dynamic structure factor of AICuFe [29], are possibly so limited in the extended zone scheme due to pseudo Brillouin zone 'interaction'. Dispersionless branches in the higher-energy region of this type were also obtained most recently from the calculated onephonon spectral functions of a 5/3 approximant of icosahedral (mlZn)49Mg32 [22]. Whether these modes are extended or not cannot be decided at present, but certainly there is a larger tendency for localization at higher frequencies as all theoretical investigations have shown so far. Indications of a tendency towards localization have also been found in some phonon branches above the acoustic modes in a comparison of the atomic dynamics of I- and R-AILiCu [26]. The isotropy of all acoustic branches, which was found in all experimental and all theoretical investigations, is due to the high symmetry of the icosahedral point group and is not specific to any quasicrystaUine model (and is therefore also of no great help in the effort to rule out experimentally one or the other model). Consequently, this isotropy is also found in periodic approximants
880
J.-B. Suck / Dynamicalproperties of stable icosahedral alloys
such as the R-phase of A1LiCu [26]. This general similarity of the dynamical properties of icosahedral phases and their rational approximants had been found in theoretical investigations of one-dimensional quasicrystals [42] and experimentally [24] before. No clear experimental results exist so far, which prove the existence of the gaps predicted by the one-dimensional models discussed above, even though indications may have been found in some cases [25].
4.2 Vibrational density of states Comparing the GVDOS of AI71PD19Mnl0 in fig. 5 with the dispersion curves in figs. 2 and 4, one realizes that the 'splitting' of the main band at lower energies is caused by the gap between the band at approximately 12 meV and above 14 meV. It is the first time that any finer structure has been observed in the GVDOS of an icosahedral alloy. Apart from this finer structure in one of the main bands, the GVDOS shown in fig. 5 is very representative for all other GVDOS measured so far for ternary stable icosahedral alloys: two to three broad main bands with nearly no structure [1,4,24,29,33]. No gaps have been detected so far. Either they do not exist, or the gap structure is much finer than resolvable with a resolution of 125 i~eV.
5. Conclusions
The most detailed information obtained so far on the vibrational properties of icosahedral alloys, has been obtained for the three best known stable icosahedral alloys AI-Li-Cu, A1-Cu-Fe and AI-Pd-Mn. The dynamics of this latter alloy were discussed here as an example. Generally the results obtained for one of these ternary stable icosahedral alloys are quite representative for all three of them: well defined propagating modes (acoustic and possibly also optic phonons) can be measured in pseudo Brillouin zones at frequencies up to about 4 THz. At higher frequencies, these excitations become rather broad. The elastic isotropy of the icosahe-
dral alloys (and of their rational approximants as far as they have been investigated) has been found in all cases. It is related to the high symmetry of the icosahedral point group and not to the quasiperiodicity of the lattice. Indications for localization of modes at higher energies have been found, but there is as yet no clear experimental proof. Up to now, this would be the only striking difference in the dynamical properties of icosahedral alloys and their periodic higher order rational approximants. Likewise, the existence or not of a hierarchy of gaps in the dispersion branches and in the density of states of icosahedral alloys, as they are typical for one-dimensional quasi-periodic lattices with critical wavefunctions, remains an open question from the experimental point of view. In fact, all measured generalized vibrational densities of states show that such gaps do not exist in the GVDOS or that they cause a fine structure which is not resolvable with a resolution of about 0.1 meV. Experimentally one finds rather simply structured GVDOS (and total dynamic structure factors) similar but often less structured than that of other ternary alloys with periodic crystal lattices [29]. Thus, most of the special dynamical properties assumed to be characteristic for three-dimensional quasiperiodic lattices have still to be detected in more refined investigations. It is a pleasure to acknowledge the help obtained from H. Breitenstein in the group of Professor H.J. Giintherodt at the University of Basel in casting the A I - P d - M n alloy, and the fruitful discussions with M. de Boissieu, J.M. Dubois and Ch. Janot. The author is grateful to M. Boudard for providing the material for figs. 1 and 2.
References [1] J.-B. Suck, in: Quasicrystalline Materials, ed. Ch. Janot and J.M. Dubois (World Scientific, Singapore, 1988) p. 337. [2] See, for example, Ch. Janot and J.M. Dubois, J. Phys. F. 18 (1988) 2303. [3] D. Schechtman and I.A. Blech, Metall. Trans. 16A (1985) 1005; P.W. Stephens and A.I. Goldman, Phys. Rev. Lett. 56 (1986) 1168; 57 (1986) 2331, 2770.
J.-B. Suck / Dynamical properties of stable icosahedral alloys [4] J.-B. Suck, in: Phonons 89, ed. S. Hunklinger, W. Ludwig and G. Weiss (World Scientific, Singapore, 1990) p. 397. [5] For a recent review, see: D.P. Di Vincenzo and P.J. Steinhardt, eds., Quasicrystals, the State of the Art, (World Scientific, Singapore, 1991). [6] P. Bak, Phys. Rev. B32 (1985) 5764. [7] T.C. Lubensky, S. Ramaswamy and J. Toner, Phys. Rev. B32 (1985) 7444. [8] M. Komoto and B. Sutherland, Phys. Rev. B34 (1986) 3849. [9] J.P. Desideri, L. Macon and D. Sornette, Phys. Rev. Lett. 63 (1989) 390; Phys. Rev. B44 (1991) 6755. [10] H. B6ttger and G. Kasner, Phys. Status Solidi (b)162 (1990) 489. [11] e.g.W. Xu and Z. Zheng, J. Phys.: Condens. Matter 1 (1989) 9533 (fig. 2). [12] C. Benoit, G. Poussigue and A. Azougarh, J. Phys.: Condens. Matter 2 (1990) 2519; C. Benoit, J. Phys.: Condens. Matter 1 (1989) 335; J. Phys C20 (1987) 765. [13] M. Klenin, Z. Phys. B65 (1986) 1. [14] J.A. Ashraff, J.M. Luck and R.B. Stinchcombe, Phys. Rev. B41 (1990) 4314. [15] J.H. Patel and D. Sherrington, Phys. Rev. B40 (1989) 1185. [16] T. Janssen, in: Quasicrystalline Materials, ed. Ch. Janot and J.M. Dubois (World Scientific, Singapore, 1988) p. 327. [17] J. Los and T. Janssen, J. Phys.: Condens. Matter 2 (1990) 9553; J. Los, T. Janssen and F. G~ihler, J. Non-Cryst. Solids, in press. [18] T. Janssen, Phys. Rep. 168 (1988) 57. [19] G. Kasner and H. B6ttger, J. Non-Cryst. Solids, in press. [20] J. Hafner and M. Krajcl, Europhys. Lett. 13 (1990) 335. [21] J. Hafner and M. Krajcl, Phys. Rev. B (1993). [22] J. Hafner and M. Krajcl, Europhys. Lett., in press. [23] J. Hafner and M. Krajcl, Phys. Rev. Lett. 68 (1992) 2321. [24] J.-B. Suck, Ch. Janot, M. de Boissieu and B. Dubost, in: Phonons 89, ed. S. Hunklinger, W. Ludwig and G. Weiss (World Scientific, Singapore, 1990) p. 576. [25] A.I. Goldman, C. Stassis, R. Bellissent, H. Moudden, N. Pyka and F.W. Gayle, Phys. Rev. B43 (1991) 8763.
881
[26] A.I. Goldman, C. Stassis, M. de Boissieu, R. Currat, Ch. Janot, R. Bellissent, H. Moudden and F.W. Gayle, J. Non-Cryst. Solids, in press. [27] J.B. Such and H. Rudin, in: Glassy Metals II, Topics in Applied Physics, Vol. 53, ed. H. Beck and H.J. Giintherodt (Springer, Heidelberg, 1983) p. 217. [28] M. Quilichini, G. Heger, B. Hennion, S. Lefebvre and A. Quivy, J. Phys. (Paris) 51 (1990) 1785. [29] T. Klein, G. Pares, J.-B. Suck, G. Fourcaudot and F. Cyrot-Lackmann, J. Non-Cryst. Solids 153&154 (1993) 562. See also J.-B. Suck, J. Non-Cryst. Solids 150 (1992) 231. [30] G. Coddens, R. Bellissent, Y Calvayrac and J.P. Ambroise, Europhys. Lett. 16 (1991) 271. [31] G. Coddens, Y. Calvayrac and R. Bellissent, J. Non-Cryst. Solids, in press. [32] M. de Boissieu, M. Boudard, H. Moudden, M. Quilichini, R. Bellissent, B. Hennion, R. Currat, A.I. Goldman and Ch. Janot, J. Non-Cryst. Solids, in press. [33] J.B. Suck, J. Non-Cryst. Solids 150 (1992) 573. [34] A.P. Tsai, H.S. Chen, A. Inoue and T. Masumoto, Phys. Rev. B43 (1991) 8782. [35] A.P. Tsai, A. Inoue, Y. Yokoyama and T. Masumoto, Mater. Trans Jpn. Inst. Met. 31 (1990) 98. [36] A. Waseda, H. Morioka, K. Kimura and H. Ino, Philos. Mag. Lett. 65 (1992) 25. [37] C. Dong, J.M. Dubois, M. de Boissieu and Ch. Janot, J. Phys.: Condens. Matter 2 (1990) 6339. [38] M. Boudard, M. de Boissieu, Ch. Janot, J.M. Dubois and C. Dong, Philos. Mag. Lett. 64 (1991) 197. [39] M. Boudard, M. de Boissieu, Ch. Janot, G. Heger, C. Beeli, H.N. Nissen, H. Vincent, M. Audier and J.M. Dubois, J. Non-Cryst. Solids 150 (1992) 5. [40] A.P. Tsai, A. Inoue, Y. Yokoyama and T. Masumoto, Philos. Mag. Lett. 61 (1990) 9. [41] J.-B. Suck, to be published. [42] K. Machida and M. Fugita, J. Phys. Soc. Jpn. 55 (1986) 1799.
JOURNa L or
~
~0IlI~
Dynamical properties of stable icosahedral alloys Jens-Boie Suck 1 Institut Max yon L a u e - P a u l Langevin, B P 156, F38042 Grenoble c~dex 9, France
Results from theoretical investigations of the vibrational properties of quasicrystals are compared with those obtained so far for the atomic dynamics of stable icosahedral alloys by neutron inelastic scattering, especially using as an example the m o s t recently studied icosahedral A1-Pd-Mn system. It is concluded that although many of the dynamical properties to be expected on the basis of the theoretical investigations of three-dimensional quasicrystals have been found in experiments done with stable icosahedral alloys, some very important questions concerning properties specific for quasiperiodic lattices like the existence of gaps or localization still await a final answer.
1. Introduction
First, the atomic dynamics of metastable icosahedral alloys have been studied [1], which are obtained by the same methods as those used to produce metallic glasses [2]. These methods normally introduce such a large amount of disorder in the atomic structure, that it is difficult to decide whether the model of the icosahedral glass [3] or a strongly defective quasiperiodic lattice is the appropriate description for the atomic structure of these alloys. The atomic dynamics (like the electronic properties) are considerably influenced by this disorder [4], which cannot be removed by an anneal because of the metastability of the icosahedral phase. For stable icosahedral alloys, for which the disorder introduced by the production methods can be removed and the phase can be stabilized by annealing, or which can be obtained directly by slow cooling from the melt as monodomain grains produced by Bridgeman or Czochralski
i On leave from: Kernforschungszentrum Karlsruhe, Institut fiir Nukleare Festk6rperphysik, P.O.B. 3640, D-7500 Karlsruhe, Germany. Correspondence to." Dr J.-B. Suck, Institute Laue-Langevin, Avenue des Martyrs, BP 156, F-38042 Grenoble c6dex, France. Tel: + 33 76 20 71 35. Telefax. + 33 76 48 39 06.
techniques or as polydomain ingots, the icosahedral glass model is inappropriate and one of the two tiling models (with or without long-range order in the atomic positions) should apply [5]. From all the models set up in order to describe the atomic structure of icosahedral (octahedral, decagonal or dodecagonal) alloys, this model of a quasiperiodic lattice, of the quasicrystal (QC), (with or without matching rules) makes the most definite predictions concerning the physical properties of these alloys.
2. Theoretical predictions for the atomic dynamics on the basis of the quasicrystalline model
2.1. Hydrodynamic limit In the hydrodynamic limit of long wavelength and low frequencies, the existence of three acoustic phonon modes, corresponding to the three translational degrees of freedom in the three-dimensional subspace, Eli, of the six-dimensional space, in which the periodic icosahedral crystal image is embedded, and three phason modes, corresponding to the three degrees of freedom in the three-dimensional subspace, E ± , orthogonal or perpendicular to Ell, was predicted. The high symmetry of the icosahedral lattice leads to only
0022-3093/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
J.-B. Suck / Dynamical properties of stable icosahedral alloys
two phonon elastic constants (one for longitudinal and one for transverse modes) and two phason elastic constants plus one related to phonon-phason interaction [6,7]. Thus for small momentum transfers and frequencies one should expect isotropic longitudinal and isotropic transverse acoustic modes. They also suggest that phasons should be frozen at ambient temperatures [7]. 2.2. One-dimensional models Most analytical and numerical calculations were carried out for one-dimensional systems, Fibonacci chains or modulated Fibonacci chains. Generally these calculations predict critical wavefunctions (neither extended nor exponentially localized) and singular continuous spectra (Cantor sets). Critical wavefunctions are the consequence of the competition between the aperiodicity of the lattice, which leads to localization, and the long-range orientational order and the effect of the infinite short-range repetition of each local pattern in a quasiperiodic lattice (Conway's theorem), which both favour extended modes [8]. Critical modes seem to have been observed recently as modes on surfaces, which had been corrugated in a Fibonacci sequence [9]. However, depending on the potential parameters or a generalization of the Fibonacci sequence [10], extended eigenstates (with well defined wavevector) and corresponding continuous spectra or localized states with discrete spectra are also found [11], Extensive investigations on one-dimensional quasiperiodic systems have been done by Benoit and collaborators calculating the harmonic response directly from the dynamical matrix via the spectral moment methods [12]. These investigations have shown that phonon dispersion curves should be observable by neutron inelastic scattering near the strongest Bragg peaks, and pseudo Brillouin zones should exist around these F points. The dispersion should be observable until the Brillouin zone under consideraation 'interacts' with the one from the next strong Bragg peak. The observed phonon intensities should be directly proportional to the intensity of the Bragg peak at the F point. Besides acoustic modes with
873
dispersion, dispersionless optic modes are predicted by these calculations. A main point raised in all these investigations is the existence of gaps in the phonon dispersion curves and the DOS of one-dimensional systems, which, if they existed as gaps or as pseudogaps in the spectra of three-dimensional QC also, would lead to very structured DOS for these systems. For two-dimensional QC, Penrose tilings and Fibonacci superlattices have been investigated and similar results have been obtained as for one-dimensional QC [13-15]. 2.3. Three-dimensional models For three-dimensional QC, numerical investigations have mainly been done for Ammann type of tilings with just one atom per vertex and only one degree of freedom [16] or more realistically admitting atomic motions in all three directions [17]. In the first case, a single band DOS has been found with very sharp structures, which are partly due to the fact that for quasicrystals the rank r is larger than the dimension, d, [18] (e.g., r = 6 in the case of a 3d icosahedral quasilattice) and thus more basis vectors in the reciprocal space lead to more van Hove singularities than in the case of periodic crystals. Most of these structures disappear as soon as all three degrees of motional freedom are admitted [17]. A slightly different approach was taken most recently using the tiling of Danzer decorated with three different sorts of atom [19]. The calculations are based on a matrix continued fraction for the larger systems and exact diagonalization of the dynamical matrix for a smaller cluster. These latter calculations give a rather smooth one-band DOS which looks very similar to the one-band DOS determined experimentally so far. Finally the existence of localized and propagating modes in a realistic model of the icosahedral quasicrystal (Al Zn)agMg32 [20] was investigated using recursion methods and a priori psuedopotentials for these s-p bounded metals [21,22]. Localized modes (defined from their participation ratio) were found in several narrow frequency regions distributed over about one third of the full frequency range near the low-energy
874
J.-B. Suck / Dynamical properties of stable icosahedral alloys
end and over about the same frequency interval at the high-energy end of the frequency spectrum. These localized modes could be related to local topological frustration due to deviations from ideal icosahedral packing induced during the relaxation of the initial tiling with perfect icosahedral symmetry under the realistic atomic pair potentials [23]. Thus localization here is caused by local properties of the lattice and not by its long-range aperiodicity! The same model was used to calculate the spectral functions in order to search for propagating modes [22]. Longitudinal and transverse acoustic modes were found in the vicinity of strong Bragg peaks, which had been assigned to the F points of nearly isotropic pseudo Brillouin zones [23]. From the initial slopes of the dispersion curves, the authors find elastic isotropy for longitudinal and for transverse acoustic modes (i.e. only two elastic constants as expected for icosahedral symmetry). The system is isotropic also with respect to the polarization of the transverse acoustic modes. The dispersions of these latter modes meet with a horizontal slope at the pseudo Brillouin zone boundary. Since the atomic dynamics of a QC, because of its aperiodicity, are described by an infinite number of coupled equations of motion, all numerical results mentioned above have been obtained for approximations of QC: either for finite embedded clusters or for rational approximants, i.e., periodic crystals with large quasiperiodic unit cells. Crystals of this type in fact have been found and their atomic dynamics have been investigated in at least one case [25,26]. As the number of atoms in the unit cell increases very rapidly with increasing order of approximant, presently the dynamics of rational approximants up to 2/1 can be diagonalized exactly (688 atoms per unit cell), and for higher approximants (up to the 5 / 3 with 12380 atoms per unit cell) recursion techniques have been used to determine the local density of states (LDOS). In all cases, the interactions have been limited to harmonic interaction of nearest neighbours. With the exception of the computer simulation [21,22], the precise character of the modes is difficult to determine in these investigations.
3. Results from neutron inelastic scattering
3.1 Experimental methods In neutron inelastic scattering experiments [27] one determines the intensity I(ko, k) of the scattered neutrons as a function of the wavevector of the neutron before (k 0) and after scattering (k) either subsequently for each energy, hto, and each momentum transfer, hQ, on a triple-axis spectrometer (TAS), if one has a monodomain grain of the alloy and can therefore determine the phonon branches in chosen directions in the reciprocal space of the crystal, or by measuring a broad range of hto and hQ values at the same time on a time-of-flight (TOF) spectrometer using a polydomain sample. From I(ko, k), the double differential scattering cross-section and the total dynamic structure factor, S(Q, to), is determined for a sample containing n different elements (since the stable icosahedral alloys are all ternary alloys, n = 3):
S(Q, to)
4~k 0 d2o" ~rSCk dO d E
(1)
Here o,so is the total scattering cross-section (coherent plus incoherent) of the sample material and k 0 and k are the moduli of the wave-vector of the neutron before and after scattering. The measured S(Q, to) is the weighted sum of the partial dynamic structure factors Sij(Q, to) plus the self part of the structure factors SiS(Q, to) if the sample scatters neutrons incoherently also: n
o-ScS(Q, to) =
4.rr)" bibjcicjSij( O , to) i,j n
+ ]~_~o'iincciSS(O , to).
(2)
i
For most of the icosahedral alloys the incoherent scattering cross-sections are small. Thus the total dynamic structure factor contains the information on the collective atomic vibrations (phonons). The single particle motions of the atoms are reflected in the generalized vibrational density of states (GVDOS), G(to), which is the normalized weighted sum of the partial density of states, gi(to), (the Fourier transform of the velocity auto-
J.-B. Suck / Dynamical properties of stable icosahedral alloys
correlation function) of the element i in the alloy: S sc
G(~o) = E e
_2W
i
o-/sc
l ci-77gi(to)/~_,e-ZWici
i
Mi
i
•
Mi
(3) Here e - 2 w , ci ' tri~c and M i are the Debye-Waller factor, the atomic concentration, the scattering cross-section and the mass of the element i in the alloy, respectively. The GVDOS can be determined directly from the second sum in eq. (2), if the first sum is zero (purely incoherent scatterer). In most cases, and
'
i
'
u
Q=115 ~
this applies especially also to the stable icosahedral alloys discussed here, which are all coherent scatterers, the first sum is dominant and one therefore has to determine the mean value of the scattered intensity for each frequency over reciprocal space in order to average to zero the coherence effects in the scattering (for details see ref.
[27,33]). 3.2 Phonon Dispersion of Al7o.sPd21Mns. 5
Up to now, the atomic dynamics of three stable icosahedral alloys have been investigated: 0 0
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Fig. 1. Constant q scans of transverse and longitudinal phonons measured at room temperature with the triple-axis spectrometer 1T in different directions in the reciprocal space of an icosahedral A170.sPd21Mn8.5 monograin between strong elastic reflections at the F points of the corresponding pseudo Brillouin zones [32]. The transverse phonon in fig. l(a) has a resolution-limited width up to the zone boundary at q = 0.35. For higher energies and above q = 0.5 ,~- 1, the phonons broaden rapidly (b).
J.-B. Suck / Dynamicalproperties of stable icosahedral alloys
876
determined; also in one case the phason hopping at high temperature has been studied [30,31]. As an example, the results of the most recent investigation of a 'monodomain' grain of A170.5Pd21Mn8.s obtained by Bridgeman techniques [32] and of a polydomain sample of m171Pd19Mn10obtained by slow cooling of the melt [33] will be discussed. The condition of formation of the stable icosahedral alloy has been studied extensively [34-37]. The alloy can be obtained in the icosahedral phase around 20 at.% Pd and 8 to 10 at.% Mn by conventional casting or by rapid quenching and remains stable during heat treatment up to the temperature of fusion. Likewise, the structure of
AlxLiTCu z [24-26], AlxCuyFe z [28-31] and AlxPdyMn z [32,33]. In the sample preparation, the relative concentrations of the elements play a crucial role in order to obtain a well ordered, purely icosahedral sample [34]. x, y and z vary very little from one sample to the other if they contain the same elements. However, monodomain grains and polydomain sample sometimes do not have absolutely identical nominal concentrations. For all three stable icosahedral alloys, the low-energy phonon branches [25,26,28,33], the generalized vibrational density of states and the total dynamic structure factor [24,29,32] have been
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Fig. 2. Dispersion curves of longitudinal and transverse phonons measured with a monograin of Ai70.sPd2zMns. 5 [32]. The initial slopes of the dispersions are the same for either transverse phonons or longitudinal phonons independent of the direction of propagation (fivefold axis, twofold axis). In the lower part of the figure (c), a nearly dispersionless branch is observed above the longitudinal acoustic branch which has been attributed to optic modes. No phonons could be measured above 4 THz (16.53 meV).
J.-B. Suck / Dynamical properties of stable icosahedral alloys
icosahedral A1-Pd-Mn has been investigated intensively with powder diffraction [38] and fourcircle X-ray and neutron diffraction on monograins [39]. In the six-dimensional description, the lattice is face centered but one can regard it as a superstructure of a primitive lattice. This is due to the strong chemical ordering, which was realized from the very beginning, when the icosahedral alloy was first discovered [40,35]. The 'monodomain' grain was investigated at room temperature on the thermal neutron tripleaxis spectrometer 1T at the Orph6e reactor in Saclay. Constant-Q scans in different symmetry directions of the reciprocal lattice between strong elastic peaks at the F point of the pseudo Brillouin zone were measured. Figure l(a) shows three scans of a transverse phonon propagating along a fivefold axis. The phonon can be easily followed up to the boundary of the pseudo Brillouin zone at q = 0.35 ,~,-1. This boundary can be clearly observed in the experiment. The corresponding dispersion curves are shown in fig. 2. All dispersion curves of longitudinal modes have the same initial slope regardless of where in reciprocal space and in which symmetry direction they have been measured. The same argument applies (with a different slope) to the transverse modes. In fig. 2(c) also a nearly dispersionless branch at 4 THz (1 THz--4.13 meV) is seen, which was attributed to optic type of vibrations. Below this branch, the dispersion curve of the LA modes flattens rather quickly at higher energy. This has been interpreted by the authors as being due to some tendency towards localization more pronounced than in periodic crystals. No phonons could be measured with frequencies larger than 4 THz. As can be seen from fig. l(b), this is largely due to the fact that the phonons broaden considerably at higher energy transfers. The low-energy phonons at q < 0.5 on the contrary are sharp and only broadened by the resolution function of the spectrometer.
3.3. Total dynamic structure factor of Al71Pd w Mn lo If one wants to investigate the total dynamic structure factor at higher energy transfers than
877
done in the TAS experiment, then one has to use large polydomain samples and TOF techniques. Polydomain samples have the disadvantage of giving only orientationally averaged information as there is no preferred orientation for the randomly oriented grains. For icosahedral alloys, because of their isotropic elastic behaviour, this disadvantage is probably less severe than for other less isotropic crystalline material as far as the acoustic modes are concerned. The second disadvantage is the fact that the structure of the sample cannot be investigated by a diffraction method as detailed as that possible for a monodomain grain. The isocahedral structure of the sample material, obtained by slow cooling from the melt without subsequent annealing, was tested by neutron powder diffraction [33], and the stability of the phase was investigated using differential scanning calorimetry [41]. The experiment was done using the cold-neutron time-focusing TOF spectrometer IN6 at the HFR of the Institut Laue-Langevin with an incident energy of 4.7 meV. From the TOF spectra, the total dynamic structure factor, S(Q, to), was determined. Details of the
Fig. 3. 59 sections through the total dynamic structure factor of icosahedral A171PdlgMn10, measured with a polydomain sample at 280 K on the cold-neutron time-of-flight spectrometer IN6. Only the foot of the elastic peaks is shown in order to augment the inelastic (neutron energy gain) part of S(Q, to). The maxima of the structures observed in S(Q, to) at constant Q have been used to determine the dispersion shown in fig. 4.
878
Suck / Dynamicalproperties of stable icosahedralalloys
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neutrons), which have not been subtracted, have no peak structure. Only a few acoustic modes are possibly observed at energies < 10 meV. There is a dominant band near 12 meV and a second band of this type is seen near 15 meV. The lower band shows appreciable dispersion only in the neighbourhood of the strongest D e b y e - S c h e r r e r peaks near 25, 29 and 30.5 nm -1. Some further dispersionless bands at 18, 21.5, 24.5 and 35 meV are seen in a rather limited range of momentum transfers.
40
Fig. 4. Energies corresponding to the maxima and shoulders in the total dynamic structure factor of icosahedral A171Pd19Mn10 in dependence on the corresponding momentum transfer, hQ. Apart from the energy region below 10 meV, where the excitations are most likely to be acoustic modes, the energy bands observed at 12 and 15 meV show appreciable dispersion only near the strongest DebyeScherrer lines in the diffraction pattern (marked by arrows, the length of which corresponds to their relative intensity). At higher energies, dispersionless branches are found in a restricted region of the extended zone scheme only. The lines enclose the dynamical space of the experiment. Open symbols: weak structures; filled symbols: strong, well defined structures. Different symbols have been used to indicate different energy regions. ~
experiment and the data evaluation will be given elsewhere [41]. 59 cuts through S(Q, o~) are shown in fig. 3. A detailed analysis of the spectra at each momentum transfer allows the energy corresponding to the maxima and shoulders in the spectra to be determined. The resulting 'dispersion curves' are shown in fig. 4. In this derivation of the dispersion curves, it has been assumed that the multi-phonon contributions to S(Q, o~) (and likewise the intensity due to multiply scattered
In order to determine the G V D O S for the same polydomain AI71Pdl9M10sample, an experiment was performed at the thermal neutron T O F spectrometer IN4 at the H F R of the Institut L a u e - L a n g e v i n in Grenoble, with a sample temperature of 120 K (to keep the multiphonon contribution small) and an incident energy of 68.8 meV. The high incident energy guarantees an effective averaging of the coherence effects, a small error only due to sampling in parts of Brillouin zones [33] and the sampling of phonon frequencies mainly where the paramagnetic scattering from the sample is small [41]. Details of the experiment, the data evaluation and the corrections applied are given elsewhere [33]. The G V D O S consists of two broad main bands centred near 16 and 31 meV with a shallow pseudo-gap near 21 meV. A very weak third band is indicated in the pseudo-gap at 22.5 meV. In the main band at low energy, a very weak shoulder is indicated near 14 meV. For energies > 25 meV, the statistical accuracy of the data is insufficient to resolve any fine structure in the main band at higher energies. In order to investigate finer structures, the G V D O S was also determined from the weighted sum of all neutron energy gain spectra measured with cold neutrons on IN6 (see section 3.3.). The resolution near hw = 0 was nearly a factor of 40 better than on IN4, which is very important in the search for gaps or a pseudo-gap structure in the low-energy region of the GVDOS. Details of this experiment and the evaluation of the data will be given elsewhere [41]. Because the very good reso-
J.-B. Suck / Dynamical properties of stable icosahedral alloys
0.040 0.032 "~0.024 3
.~ 0.016 ¢...9 0.O08j
0(5
10
20,.h~
[me3~j 4b
50
Fig. 5. Generalized vibrational density of states of icosahedral A17tPd19Mn10 determined from the weighted sum of all TOF spectra measured at 280 K at the cold-neutron time-focusing spectrometer IN6. A splitting of the first main band into two subbands, 2.5 meV apart, is indicated at 13 meV.
lution at hoo = 0 (the energy transfer with the optimal focusing condition) becomes rapidly worse at higher energy transfers, the discussion will be focussed on the low-energy band. Figure 5 shows the resulting GVDOS. It is essentially identical within experimental errors to the GVDOS determined with thermal neutrons on IN4 [33], apart from an energy shift of both bands by ~ 1.2 meV to lower energies which can partly be explained by the difference in the resolution functions of the two spectrometers. In place of the very weak shoulder at the low-energy band mentioned above, now a clear splitting of this band into two 'subbands', 2.5 meV apart, is indicated.
4. Discussion
4.1 Phonon dispersion Even though no special effort has been put into the determination of the phonon linewidth with the help of a high-resolution measurement for icosahedral AIPdMn, the intense and resolution-limited transverse phonons shown in fig. l(a) suggest that these excitations have a relatively long lifetime and can travel far, on the scale of phonons in crystals, especially if they are compared with the relatively broad excitations measured so far in the other stable icosahedral alloys
879
[25,26,28]. A reason for the existence of these well defined excitations in A I - P d - M n may be the high degree of chemical (and topological) order, which leads to a strain-free quasiperiodic lattice [40]. From the theoretical point of view, one would expect that the tendency towards localized modes decreases going from one-dimensional to three-dimensional QC especially at lower frequencies [8]. For the reasons mentioned above, the dispersion curves obtained from the total dynamic structure factor and from the TAS experiment can only be compared in the upper energy region of the TAS results. Here the energy bands observed at 3 THz in fig. 2(b) and the flat part of the longitudinal phonon branch in figs. 2(a) and (c) coincide very reasonably with the extended band of excitations seen in the total structure factor at 12 meV (3 THz = 12.4 meV). The same conclusion is valid for the dispersionless branches seen in figs. 2(b) and (c) and above 15 meV in fig. 4. The short dispersionless branches, which were also found earlier in the dispersion curves obtained from the total dynamic structure factor of AICuFe [29], are possibly so limited in the extended zone scheme due to pseudo Brillouin zone 'interaction'. Dispersionless branches in the higher-energy region of this type were also obtained most recently from the calculated onephonon spectral functions of a 5/3 approximant of icosahedral (mlZn)49Mg32 [22]. Whether these modes are extended or not cannot be decided at present, but certainly there is a larger tendency for localization at higher frequencies as all theoretical investigations have shown so far. Indications of a tendency towards localization have also been found in some phonon branches above the acoustic modes in a comparison of the atomic dynamics of I- and R-AILiCu [26]. The isotropy of all acoustic branches, which was found in all experimental and all theoretical investigations, is due to the high symmetry of the icosahedral point group and is not specific to any quasicrystaUine model (and is therefore also of no great help in the effort to rule out experimentally one or the other model). Consequently, this isotropy is also found in periodic approximants
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J.-B. Suck / Dynamicalproperties of stable icosahedral alloys
such as the R-phase of A1LiCu [26]. This general similarity of the dynamical properties of icosahedral phases and their rational approximants had been found in theoretical investigations of one-dimensional quasicrystals [42] and experimentally [24] before. No clear experimental results exist so far, which prove the existence of the gaps predicted by the one-dimensional models discussed above, even though indications may have been found in some cases [25].
4.2 Vibrational density of states Comparing the GVDOS of AI71PD19Mnl0 in fig. 5 with the dispersion curves in figs. 2 and 4, one realizes that the 'splitting' of the main band at lower energies is caused by the gap between the band at approximately 12 meV and above 14 meV. It is the first time that any finer structure has been observed in the GVDOS of an icosahedral alloy. Apart from this finer structure in one of the main bands, the GVDOS shown in fig. 5 is very representative for all other GVDOS measured so far for ternary stable icosahedral alloys: two to three broad main bands with nearly no structure [1,4,24,29,33]. No gaps have been detected so far. Either they do not exist, or the gap structure is much finer than resolvable with a resolution of 125 i~eV.
5. Conclusions
The most detailed information obtained so far on the vibrational properties of icosahedral alloys, has been obtained for the three best known stable icosahedral alloys AI-Li-Cu, A1-Cu-Fe and AI-Pd-Mn. The dynamics of this latter alloy were discussed here as an example. Generally the results obtained for one of these ternary stable icosahedral alloys are quite representative for all three of them: well defined propagating modes (acoustic and possibly also optic phonons) can be measured in pseudo Brillouin zones at frequencies up to about 4 THz. At higher frequencies, these excitations become rather broad. The elastic isotropy of the icosahe-
dral alloys (and of their rational approximants as far as they have been investigated) has been found in all cases. It is related to the high symmetry of the icosahedral point group and not to the quasiperiodicity of the lattice. Indications for localization of modes at higher energies have been found, but there is as yet no clear experimental proof. Up to now, this would be the only striking difference in the dynamical properties of icosahedral alloys and their periodic higher order rational approximants. Likewise, the existence or not of a hierarchy of gaps in the dispersion branches and in the density of states of icosahedral alloys, as they are typical for one-dimensional quasi-periodic lattices with critical wavefunctions, remains an open question from the experimental point of view. In fact, all measured generalized vibrational densities of states show that such gaps do not exist in the GVDOS or that they cause a fine structure which is not resolvable with a resolution of about 0.1 meV. Experimentally one finds rather simply structured GVDOS (and total dynamic structure factors) similar but often less structured than that of other ternary alloys with periodic crystal lattices [29]. Thus, most of the special dynamical properties assumed to be characteristic for three-dimensional quasiperiodic lattices have still to be detected in more refined investigations. It is a pleasure to acknowledge the help obtained from H. Breitenstein in the group of Professor H.J. Giintherodt at the University of Basel in casting the A I - P d - M n alloy, and the fruitful discussions with M. de Boissieu, J.M. Dubois and Ch. Janot. The author is grateful to M. Boudard for providing the material for figs. 1 and 2.
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