Continuum elastic sphere vibrations as a model for low-lying optical modes in icosahedral quasicrystals

Abstract

The nearly dispersionless, so-called "optical" vibrational modes observed by inelastic neutron scattering from icosahedral Al-Pd-Mn and Zn-Mg-Y quasicrystals are found to correspond well to modes of a continuum elastic sphere that has the same diameter as the corresponding icosahedral basic units of the quasicrystal. When the sphere is considered as free, most of the experimentally found modes can be accounted for, in both systems. Taking into account the mechanical connection between the clusters and the remainder of the quasicrystal allows a complete assignment of all optical modes in the case of Al-Pd-Mn. This approach provides support to the relevance of clusters in the vibrational properties of quasicrystals.

Full text

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arXiv:cond-mat/0405649v2 [cond-mat.mtrl-sci] 30 May 2005
Continuum elastic sphere vibrations as a model for
low-lying optical modes in icosahedral quasicrystals
E. Duval† , L. Saviot‡ , A. Mermet† , D. B. Murray§
† Laboratoire de Physicochimie des Mat´ eriaux Luminescents, Universit´ e Lyon I -
UMR-CNRS 5620 43, boulevard du 11 Novembre 69622 Villeurbanne Cedex, France
‡ Laboratoire de Recherche sur la R´ eactivit´ e des Solides, Universit´ e de Bourgogne -
UMR-CNRS 5613, 9 avenue Alain Savary, BP 47870, 21078 Dijon Cedex, France
§ Department of Physics and Astronomy, Okanagan University College, 3333
University Way, Kelowna, British Columbia, Canada V1V 1V7
E-mail: lucien.saviot@u-bourgogne.fr
Abstract.
The nearly dispersionless, so-called “optical” vibrational modes observed
by inelastic neutron scattering from icosahedral Al-Pd-Mn and Zn-Mg-Y quasicrystals
are found to correspond well to modes of a continuum elastic sphere that has the same
diameter as the corresponding icosahedral basic units of the quasicrystal. When the
sphere is considered as free, most of the experimentally found modes can be accounted
for, in both systems. Taking into account the mechanical connection between the
clusters and the remainder of the quasicrystal allows a complete assignment of all
optical modes in the case of Al-Pd-Mn. This approach provides support to the relevance
of clusters in the vibrational properties of quasicrystals.
Submitted to: J. Phys.: Condens. Matter
PACS numbers: 71.23.Ft, 63.22.+m, 6146.+w
1. Introduction
Quasicrystals are long-range ordered materials whose diffraction patterns present
symmetries incompatible with translational invariance [1, 2]. Most structural analyses
of icosahedral phases have shown that they can be described as a quasiperiodic packing
of group of atoms or “clusters” with a local icosahedral symmetry (e.g. pseudo-Mackay
icosahedra).
Experimental electron density maps of crystal approximants have shown that more
electrons are localized on clusters than between clusters [3], supporting this idea of
cluster building blocks. From the vibrational point of view, the experimental evidence
for the resulting expected phonon confinement is less clear. Inelastic neutron scattering
(as well as inelastic X-ray scattering [4]) on several icosahedral phases revealed that
the excitation spectrum can be split into two regimes: acoustic and optical [5, 6, 7].

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Continuum calculations for optical modes in icosahedral quasicrystals
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The acoustic regime is characterised by well defined longitudinal and transverse modes
which display a linear dispersion in the low wavevector q range. The high energy part
of the excitation spectrum displays somewhat broadened bands (∼4 meV width) of
dispersionless excitations or optical modes. The cross-over between the two regimes is
abrupt and occurs for wavevectors q between 3.5 and 6 nm−1, i.e. for a wavelength of
the order of Dcl(Dcl: cluster diameter). This shows up as a rapid broadening of the
acoustic modes and an intermixing of phonon modes in the different branches. Although
no gap opening has been observed, the energy position of the optical bands matches the
crossing of the acoustic branch with pseudo-zone boundaries as defined by Niizeki [8].
These observations are in general agreement with the theoretical calculations of Hafner
and Krajˇ ci for icosahedral quasicrystals [9] although these calculations do not provide
a clear description of the optical modes.
The low lying energy range of the so-called optical modes, which features acoustic-
optical crossing points in the dispersion curves, suggests some substantial interaction
with the acoustic modes. Along these lines, a hybridization scheme between acoustic
and optical modes has recently been proposed [10]. In this latter work, the so-called
optical modes correspond to modes which are confined in the nanometric clusters.
Following the idea that the clusterlike nature of quasicrystals is crucial for both
sound waves [10] and electronic properties, we propose a detailed identification of the so-
called optical modes in quasicrystals, based on the approach of fundamental vibrational
modes of nanospheres. The first kind of analysis that we apply (free sphere model,
section 3) is to consider clusters as the elemental structural entities of quasicrystals.
In this way, we can account for the low lying optical modes, since the model gives
frequencies which compare well with those experimentally found both for i-Al-Pd-
Mn and i-Zn-Mg-Y. These modes are spheroidal and/or torsional vibrations of the
corresponding clusters. Although crude, this description turns out to already provide
satisfactory estimates of the optical mode frequencies in a model without any freely
adjustable parameters. In a second stage (sphere weakly coupled to a matrix, section 4),
the cluster picture is given further realistic input by considering the vibrational coupling
with a surrounding matrix, to tentatively account for cluster/cluster interactions and/or
clusters overlapping. This latter model is found to match more completely with the
experimental observations, particularly with respect to the damping parameters of the
considered modes.
2. Vibrations of nanospheres and scattering of acoustic waves
For almost twenty years, studies have been carried out on the vibrational modes of
approximately spherical nanometric clusters or nanocrystals embedded in glasses or in
macroscopic crystals using Raman or Brillouin light scattering [11, 12, 13, 14, 15, 16].
For these embedded nanocrystal systems, the observed low-lying modes correspond well
to those of a free continuous-medium nanosphere of density ρcl, longitudinal speed of
sound vL
cl, transverse speed of sound vT
cland diameter Dcl. The vibrational modes of

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Continuum calculations for optical modes in icosahedral quasicrystals
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such a sphere were studied for the first time by Lamb [17]. The modes of a free sphere
are characterized by a polarization index p that denotes either spheroidal (SPH) or
torsional (TOR) modes, the usual quantum numbers ℓ and m of spherical harmonics
and the harmonic index n. Among the spheroidal modes the simplest are the spherical
(ℓ = 0), dipolar (ℓ = 1) and quadrupolar (ℓ = 2) modes. The mode frequency νpℓmn
is inversely proportional to the diameter Dcland proportional to the (longitudinal or
transverse) sound speed vpin the material of the nanosphere:
νpℓmn=Spℓnvp
Dcl
(1)
where vTOR= vT
numerical values of Spℓnwere found long ago [17] for a free vibrating sphere.
In a recent work [18, 19], it was shown that a sphere embedded inside a matrix
scatters acoustic waves more efficiently when their frequencies match certain values
which turn out to be generally very close to the free sphere eigenfrequencies. More
precisely, it is possible to calculate the position and width of these resonances
using the so-called Complex Frequency Model (CFM). The interesting point is that
these resonances do not depend on the nature of the incident acoustic waves (plane
waves, spherical waves, longitudinal, transverse, ...). Therefore, we can expect these
resonances to be important even when many spheres are involved. This is similar to
the “hard” and “soft” scatterers picture invoked to interpret sound mode broadening in
quasicrystals [10].
In the following, we will apply this continuum mechanical model to very small
clusters (less than 2 nm diameter). Also the application of continuum type boundary
conditions at the interface between the icosahedral cluster and the rest of the quasicrystal
is an idealization. A more accurate treatment would need to consider an atomic level
description of the cluster and its interface [20]. However, despite these limitations, our
numerical results are in good agreement with experiments.
In order to illustrate the applicability of nanosphere mode analysis to quasicrystals,
we first consider the results of the free sphere model (because it requires fewer
parameters) for Al-Pd-Mn and Zn-Mg-Y. Then, in order to account for a more realistic
situation and a more complete vibrational pattern, we examine the case where Al-Pd-Mn
clusters are weakly coupled to a surrounding matrix.
cl, vSPH = vL
cland Spℓnis a constant for modes of type (p,ℓ,n). The
3. Free sphere model
3.1. Al-Pd-Mn
The shape of the icosahedral clusters in Al-Pd-Mn is nearly-spherical. The number of
atoms per cluster (≃ 51) can be considered as sufficient to approximate the lowest-energy
confined modes by those of a continuum sphere. Using the approach described in [18],
the frequencies of the vibrational modes were calculated for a continuous-medium free
nanosphere with a diameter equal to the size of the cluster, i.e. Dcl= 1.0 nm [21, 22, 23].

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Continuum calculations for optical modes in icosahedral quasicrystals
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Table 1. Calculated vibrational energies for a free cluster of Al-Pd-Mn of diameter
Dcl= 1.0 nm. Corresponding experimentally observed optical modes are indicated in
superscripts.
Spheroidal
E (meV)
Torsional
E (meV)
ℓ = 0
n = 0
n = 1
n = 0
n = 1
n = 0
n = 1
22.8O4
52.2
16.3O3
32.7
12.2O2
23.1O4
ℓ = 126.6
42.0
11.5O2
32.9
ℓ = 2
Both longitudinal and transverse sound speeds in the clusters were approximated with
the bulk sound speeds in the Al-Pd-Mn quasicrystal, i.e. vL
3500 m/s [24]. Table 1 displays the so-obtained frequency values for the fundamental
(n = 0) and the first harmonic (n = 1) of each type of mode (SPH or TOR) and angular
momentum ℓ.
The optical modes of Al-Pd-Mn quasicrystals observed by inelastic neutron
scattering [6] have the following approximate energies: EO1≃ 7 meV, EO2≃ 12 meV,
EO3≃ 16 meV and EO4≃ 24 meV, with hardly any dependence on wavevector q. The
results of the free sphere model calculations presented in table 1 permit the following
assignments:
cl= 6500 m/s and vT
cl=
O2: the EO2 ≃ 12 meV mode corresponds to the spheroidal and torsional ℓ = 2
fundamental modes. Both of these modes are five-fold degenerate.
O3: the EO3 ≃ 16 meV mode corresponds to the triply degenerate dipolar ℓ = 1
spheroidal mode.
O4: the EO4≃ 24 meV mode corresponds to the spherical ℓ = 0 mode and also to the
five-fold degenerate quadrupolar ℓ = 2,n = 1 spheroidal mode.
It turns out that the free sphere calculations are able to provide an accurate
assignment for all of the optical mode energies except for the lowest one (O1) (as
described in section 4, calculations taking into account the cluster-matrix interaction
allows the “missing” mode O1 to be accounted for). For each of these modes, we might
expect the inelastically scattered intensities to scale with the corresponding total number
of modes, taking into account their degeneracy. Such tentatively appears to be the case
when examining the reported lineshapes [6]. However, to make a detailed comparison
would require a quantitative analysis of the inelastic scattering intensities of individual
modes, which is beyond the scope of this work.
In order to further assess the relevance of the free sphere description to the low
energy optical modes in quasicrystalline structures, we now examine the case of Zn-Mg-
Y.

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Continuum calculations for optical modes in icosahedral quasicrystals
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Table 2. Calculation of vibrational energies for a free sphere of Zn-Mg-Y of diameter
Dcl= 1.2 nm.
Spheroidal
E (meV)
Torsional
E (meV)
ℓ = 0
n = 0
n = 1
n = 0
n = 1
n = 0
n = 1
12.4O2
31.6
10.8O2
20.7
8.9O1
15.5O3
ℓ = 1 19.6O3
31.0
8.5O1
24.3
ℓ = 2
3.2. Zn-Mg-Y
The icosahedral quasicrystal Zn-Mg-Y has clusters of diameter Dcl= 1.2 nm [7]. We
calculated the energies of a free spherical cluster having the same diameter (Dcl= 1.2
nm) as the Zn-Mg-Y clusters and the same sound speeds as bulk Zn-Mg-Y : vL
and vT
Once again, the calculated vibrational energies are found to compare well with
those of the optical modes measured through inelastic neutron scattering, i.e. EO1≃ 8
meV, EO2≃ 12 meV and EO3≃ 17 meV [7] :
cl= 4800
cl= 3100 m/s. [7] The results are given in table 2.
O1: the EO1 ≃ 8 meV mode corresponds to the five-fold degenerate spheroidal and
torsional ℓ = 2 modes (whose frequencies are usually almost the same),
O2: the EO2 ≃ 12 meV mode corresponds to the spherical ℓ = 0 and the spheroidal
dipolar ℓ = 1 modes,
O3: the EO3 ≃ 17 meV mode corresponds to the five-fold degenerate quadrupolar
ℓ = 2, n = 1 and the triply degenerate torsional ℓ = 1 modes.
Much like the results obtained for Al-Pd-Mn, the free sphere model is found to
provide a sound description of all the optical modes in Zn-Mg-Y. The above assignments
are restricted to the low energy range (E < 30 meV) where modes were unambiguously
identified experimentally.
It is worth noting that from group theoretical selection rules [25], only the spherical
and spheroidal quadrupolar modes can be observed by Raman scattering, as experiments
confirm (these selection rules can be broken for anisotropic materials or under resonant
excitation). No such selection rules exist for inelastic neutron scattering. Besides, it
should be noted that the degeneracies of the aforementioned modes (in any case for
ℓ ≤ 2) are not lifted by lowering the symmetry from spherical to icosahedral. This
provides a supplementary justification for using spheres to approximate the icosahedral
clusters.
The present calculation predicts undamped vibrational modes whereas the observed
excitations are somewhat broadened (∼4 meV). This may be at least partly attributed
to a variation of actual quasicrystal cluster masses, since structural studies showed that