The Anisotropic Glassy Properties of Decagonal Quasicrystals

Abstract

We use an extended version of the standard tunneling model to explain the anisotropic sound absorption in decagonal quasicrystals. The glassy properties are determined by an ensemble of two level systems (TLSs), arbitrarily oriented. The TLS is characterized by a 3 × 3 symmetric tensor, [𝑇], which couples to the strain field, [𝑆], through a 3 × 3 × 3 × 3 tensor of coupling constants, [𝑅]. The structure of [𝑅] reflects the symmetry of the quasicrystal. We also analyze the probability distributions of the elements of [𝑇] in this particular model for a better understanding of the characteristics of “isotropic” and “anisotropic” distributions of the ensemble of TLSs. We observe that the distribution of the elements is neither simple nor intuitive and therefore it is difficult to guess it a priory, using qualitative arguments based on the symmetry properties.

1. Introduction

Despite almost four decades of study of the glassy properties materials [1–4], the nature of the two-level systems (TLS), the ubiquitous hypothetical microscopic entities that are held responsible for these properties, is still not known. As a general picture, it is accepted that they represent dynamic defects, which are atoms or groups of atoms that tunnel from one minimum energy configuration to another. But these atoms are in general not identified, and even in the cases when they are identified (like in crystals with defects), the TLS spectrum cannot be obtained based on a microscopic model.

The study of solids with anisotropic glassy properties is especially interesting since this brings additional information about the TLSs and requires a critical perspective on the standard tunneling model (STM).

To identify the origin of the TLS, the thermal and acoustical properties of a number of crystalline systems with defects [5–13] and quasicrystals [14–18] at low temperatures have been investigated thoroughly. As expected, the glass like properties in both the disordered crystalline system and in the quasicrystals were revealed. However, a crucial difference in comparison with ordinary amorphous solids which possess TLS excitations was detected. Namely, pronounced anisotropy in internal friction was clearly marked. As a consequence of that, the baffling physical problem about the possible origin of the anisotropy appeared. Indeed, there are two competitive ways for the explanation. First, the effect is explained by an anisotropy in the distribution over the “orientations” of the TLSs in the ensemble. The interaction between the TLS and the elastic field is described by the Hamiltonian 𝐻_𝐼=12𝛿00−𝛿, (1) where 𝛿≡2𝛾_𝑖𝑗𝑆_𝑖𝑗 and [𝑆] is the strain field of the phonon—we assume everywhere summation over the repeated indices. The symmetric second rank tensor [𝛾] characterizes the TLS and its “deformability” under elastic strain. The anisotropy of the physical properties is a reflection of the values taken by the elements of [𝛾], which are determined by the lattice symmetries. Bert et al. made a conjecture regarding these values to recover the anisotropic sound attenuation rates that they observed in decagonal quasicrystals [18].

In the second approach [19–21]—the one that we will employ in this paper—the TLS is characterized by a 3 × 3 symmetric tensor [𝑇] and the coupling between [𝑇] and [𝑆] is made through a forth rank tensor of coupling constants denoted by [𝑅]. Explicitly, 𝛾_𝑖𝑗≡𝑅_{𝑘𝑙𝑖𝑗}𝑇_𝑘𝑙 and 𝛿≡[𝑇]^𝑡∶[𝑅]∶[𝑆]. The elements of [𝑇] are determined by a unit vector 𝐭̂, which is the direction of the TLS, whereas the structure of [𝑅] is determined by the symmetries of the host material. In this model, even if the TLSs are isotropically oriented, the anisotropy of the system is imposed by the properties of the tensor of coupling constants.

We applied the second approach to the crystalline materials of different symmetries with embedded TLSs, assuming the isotropy of the orientations of the TLSs [22–25] and we calculated the attenuation of ultrasound waves of different polarizations and propagating in different crystallographic directions. This model describes not only in a simple way the asymmetries of the glassy properties but also allows us to make predictions about the relative attenuation rates of sound propagating in different directions.

In this paper, we apply the model of [19] to the attenuation of ultrasound waves in quasicrystals and we obtain the attenuation rates along different crystallographic directions expressed in terms of the components of the tensor of coupling constants. We apply our calculations to the experimental results of [18], which enable us to calculate the relations between some of the components of the coupling constants tensor.

In order to better understand the distribution of the TLSs in isotropic and anisotropic materials, we calculate the distribution of the elements of [𝑇] for an isotropic distribution of unit vectors 𝐭̂.

2. The Anisotropy of the Glassy Properties

Let us introduce the notations by presenting briefly the model. The Hamiltonian of the free TLS is 𝐻_TLS=Δ2𝜎_𝑧−Λ2𝜎_𝑥≡12Δ−Λ−Λ−Δ, (2) where Δ is called the asymmetry of the potential and Λ the tunnel splitting. The eigenvalues of 𝐻_TLS are ±𝜖/2, where 𝜖≡Δ²+Λ² is the excitation energy of this TLS. The ground state will be denoted by |↓⟩ and the excited state by |↑⟩. The interaction Hamiltonian of the TLS with the strain field is given by (1).

The parameters Δ and Λ are distributed with the probability 𝑃(Δ,Λ)=𝑃₀/Λ, where 𝑃₀ is a constant. If expressed in terms of 𝜖 and 𝑢≡Λ/𝜖, the probability distribution becomes 𝑃(𝜖,𝑢)=𝑃₀/(𝑢√1−𝑢²).

As usual, we work in the abbreviated subscript notations and write [𝑆] and [𝛾] as six-dimensional vectors: 𝐒=(𝑆₁₁,𝑆₂₂,𝑆₃₃,2𝑆₂₃,2𝑆₁₃,2𝑆₁₂)^𝑡 and 𝜸=(𝛾₁₁,𝛾₂₂,𝛾₃₃,𝛾₂₃,𝛾₁₃,𝛾₁₂)^𝑡, where the superscript 𝑡 denotes the transpose of a matrix or a vector.

As stated before, 𝛾_𝑖𝑗=𝑅_{𝑘𝑙𝑖𝑗}𝑇_𝑘𝑙. Assuming that the TLS is characterized by a direction in space, 𝐭̂, the tensor 𝐓 is formed of the components of 𝐭̂ [19]. In abbreviated subscript notations 𝐓≡(𝑡²_𝑥,𝑡²_𝑦,𝑡²_𝑥,2𝑡_𝑦𝑡_𝑧,2𝑡_𝑧𝑡_𝑥,2𝑡_𝑥𝑡_𝑦)^𝑇, 𝑅_{𝑖𝑗𝑘𝑙} becomes 𝑅_𝐼𝐽, and 𝜸≡[𝑅]^𝑡⋅𝐓 [19]. The structure of [𝑅] is determined by the symmetries of the lattice, since 𝛿 is a scalar and it should be invariant under coordinates transformations, whereas [𝑅] should be invariant under the symmetry transformations that leaves the lattice invariant [19].

The absorption rate of a phonon—with wavenumber 𝐤 and polarization 𝜎—by a TLS is [19, 21, 22, 24] Γ_𝐤𝜎𝐭̂=2𝜋ℏΛ²𝑛_𝐤𝜎𝜖²𝐓^𝑡⋅[𝑅]⋅𝐒_𝐤𝜎²𝛿(𝜖−ℏ𝜔). (3) The main characteristic of the TLS-phonon interaction is contained in the quantity 𝑀_𝐤,𝜎(𝐭)̂≡𝐓^𝑡⋅[𝑅]⋅𝐒_𝐤𝜎, which bears an intrinsic anisotropy through the matrix [𝑅], on which the symmetries of the lattice are imposed.

The average scattering rate of a phonon by the ensemble of TLSs is obtained by averaging Γ_𝐤𝜎(𝐭)̂ over 𝜖, Λ, and 𝐭̂. In this way we get the total phonon absorption rate: 𝜏⁻¹_𝐤𝜎=2𝜋ℏ𝑃₀𝑀_𝐤𝜎𝐭̂²𝑛_𝐤𝜎𝜖tanh2𝑘_B𝑇, (4) which may be put into the standard form 𝜏⁻¹_𝐤,𝜎=2𝜋ℏ𝑃₀𝛾²𝐤_,𝜎𝑁²𝑘²𝑛_𝐤𝜎𝜖tanh2𝑘_B𝑇, (5) where 𝑁=ℏ/(2𝑉𝜌𝜔) is the normalization constant of the phonon mode and 𝛾𝐤^,𝜎≡⟨|𝑀_𝐤𝜎(𝐭)̂|²⟩^1/2/(𝑁𝑘) is the (average) phonon-TLS coupling constant. We make the typical assumption that 𝐭̂ is isotropically oriented and the anisotropy is induced only by the properties of [𝑅].

As shown in [19], the structure of [𝑅] should be similar to that of the elastic stiffness constants, [𝑐]. The quasicrystal analyzed in [17] has decagonal symmetry and in this case [𝑅] should have the form [18, 26–28], [𝑅]=𝑟₁₁𝑟₁₂𝑟₁₃𝑟000₁₂𝑟₁₁𝑟₁₃𝑟000₁₃𝑟₁₃𝑟₃₃000000𝑟₄₄000000𝑟₄₄000000𝑟₆₆, (6) where 𝑟₆₆=(𝑟₁₁−𝑟₁₂)/2—the axis 𝑧 is taken along the tenfold axis. The structure of both, [𝑅] and [𝑐], is similar in decagonal quasicrystals and in hexagonal lattices.

Like in hexagonal lattices, in the decagonal quasicrystals, we can have pure longitudinal and transversal waves propagating in all the three directions, 𝑥, 𝑦, and 𝑧.

The coupling constants, 𝛾𝐤_,𝜎 are similar to those calculated for hexagonal lattices in [24]:𝛾²_𝑘𝐱_,𝑙=2𝑟²₁₁+𝑟²₁₂+𝑟²₁₃+𝑟₁₁+𝑟₁₂+𝑟₁₃²15=𝛾²_𝑘𝐲_,𝑙,𝛾²_𝑘𝐳̂_,𝑙=8𝑟²₁₃+4𝑟₁₃𝑟₃₃+3𝑟²₃₃,𝛾15_2𝑘𝐲_,𝐱_,𝑡=𝛾²_𝑘𝐱_,𝐲_,𝑡=𝑟₁₁−𝑟₁₂²=154𝑟²₆₆,𝛾15_2𝑘𝐱_,𝐳̂_,𝑡=𝛾²_𝑘𝐲_,𝐳̂_,𝑡=𝛾²_𝑘𝐳̂_,𝐱_,𝑡=𝛾²_𝑘𝐳̂_,𝐲_,𝑡=4𝑟²₄₄,15 (7a)(7b)(7c)(7d)

where by 𝑙 and 𝑡, we refer to longitudinal and transversal polarizations, respectively.

While for the longitudinal waves, the direction of polarization is obvious, for the transversal waves, the direction of polarization is indicated by the second unit vector in the subscript of 𝛾 in (7c) and (7d).

Due to the isotropy condition in the decagonal plane, the coupling constants of the phonons propagating in this plane are independent of the direction of propagation if they have similar polarization.

Comparing (7c) and (7d) with the results of Bert et al. [18], we obtain 𝑃𝛾²_‖𝑃𝛾²_⟂≡𝛾²_𝑘𝐱_,𝐲_,𝑡𝛾²_𝑘𝐱_,𝐳̂_,𝑡=𝑟₁₁−𝑟₁₂2𝑟₄₄²=𝑟₆₆𝑟₄₄²≈4.2, (8) where 𝑃𝛾²_‖ and 𝑃𝛾²_⟂ are obvious notations from [18].

To find the ranges of the coupling constants, we rewrite (7a) and (7b) as𝛾²_𝑘𝐱_,𝑙=𝛾²_𝑘𝐲_,𝑙=𝑟²₁₃4𝑟80𝑟¹¹₁₃+1²+4𝑟₁₂𝑟₁₃+1²+234𝑟₁₁𝑟₁₃4𝑟+1¹²𝑟₁₃+1+403,𝛾²_𝑘𝐳̂_,𝑙=𝑟²₁₃3𝑟45𝑟³³₁₃+2²+20. (9a)(9b)

Since the function 𝑓(𝑎,𝑏)=𝑎²+𝑏²+2𝑎𝑏/3 satisfies 𝑓(𝑎,𝑏)≥𝑓(0,0)=0 for any 𝑎 and 𝑏, we find from (9a) and (9b) that 𝛾²_𝑘𝐱_,𝑙=𝛾²_𝑘𝐲_,𝑙≥𝑟²₁₃6,𝛾²_𝑘𝐳̂_,𝑙≥4𝑟²₁₃9. (10)The minimum values of the coupling constants, 𝛾²_𝑘𝐱_,𝑙 and 𝛾²_𝑘𝐳̂_,𝑙, for given 𝑟₁₃ (10), are obtained for 𝑟₁₁=𝑟₁₂=−𝑟₁₃/4 and 𝑟₃₃=−2𝑟₁₃/3. The ratio between these minimum values is 𝛾²_𝑘𝐱^,𝑙_min𝛾²_𝑘𝐳̂_,𝑙min=38 (11) and is independent of 𝑟₁₁, 𝑟₁₂, 𝑟₁₃, and 𝑟₃₃.

Another inequality exists for the ratio 𝛾²_𝑘𝐱_,𝑙/𝛾²_𝑘𝐲_,𝐱_,𝑡. If we denote 𝑋=𝑟₁₂/(𝑟₁₁−𝑟₁₂) and 𝑌≡𝑟₁₃/(𝑟₁₁−𝑟₁₂), then 𝛾²_𝑘𝐱_,𝑙𝛾²_𝑘𝐲_,𝐱_,𝑡=3+8𝑋²+3𝑌²+4𝑋𝑌+8𝑋+2𝑌≥1. (12)The limit value of the inequality (12), 𝛾²_𝑘𝐱_,𝑙=𝛾²_𝑘𝐲_,𝐱_,𝑡, is reached for 𝑋_min=−1/2 and 𝑌_min=0.

Relations (11) and (12) are tests for the model, since they should be valid independent of the concrete values of the coupling constants.

3. The Components of the Tensors

The anisotropies of the glassy properties of a disordered system are determined by the properties of the tensor 𝜸. In our model, the “orientations” of the TLSs are defined by the unit vectors 𝐭̂ and the symmetries of the host material are incorporated into the coupling constant tensor, [𝑅]. These determine the distributions of the elements of the tensors 𝐓 and 𝜸. This is not possible in the STM. To understand the difficulties to define by general, qualitative arguments the “anisotropy” of 𝜸 in the STM, we calculate in this section the probability distribution of the elements of our tensor 𝐓, under the assumption that the unit vectors 𝐭̂ are isotropically oriented. We will see that neither these elements have all the same distribution of probability, nor the distributions are constant for the ranges of these variables.

We define 𝐭̂ by the Euler angles, 𝜃 and 𝜙: 𝑡_𝑥=sin𝜃cos𝜙, 𝑡_𝑦=sin𝜃sin𝜙, and 𝑡_𝑧=cos𝜃. Keeping 𝑡_𝑧 and 𝜙 as variables, we write 𝐓^𝑡=[(1−𝑡²_𝑧)cos²𝜙,(1−𝑡²_𝑧)sin²𝜙,𝑡²_𝑧,2𝑡_𝑧1−𝑡²_𝑧sin𝜙, 2𝑡_𝑧1−𝑡²_𝑧cos𝜙,(1−𝑡²_𝑧)sin(2𝜙)].

The variables 𝑡_𝑧 and 𝜙 are uniformly distributed in the intervals [0,1] and [0,2𝜋], respectively. We denote their probability distribution by 𝑃𝐭(̂𝑡_𝑧,𝜙)=1/(4𝜋). Then the probability distribution in the variables 𝑇_𝐼  (𝐼=1,…,6) and 𝜙 is 𝑃_𝐼,𝜙𝑇_𝐼=1,𝜙𝜕𝑡4𝜋^𝑧,𝜙𝜕𝑇_𝐼=1,𝜙𝜕𝑇4𝜋^𝐼,𝜙𝜕𝑡_𝑧,𝜙⁻¹. (13)

The simplest of all is eventually 𝑃_3,𝜙𝑇₃=1,𝜙4𝜋𝑇₃, (14) which, if integrated over 𝜙, gives 𝑃₃𝑇₃=12𝑇₃. (15)

For the components 𝑇₁ and 𝑇₂, we obtain the probability distributions𝑃_1,𝜙𝑇₁=1,𝜙𝜃4𝜋cos²𝜙−𝑇₁1−𝑇₁/cos²𝜙cos²𝜙,𝑃_2,𝜙𝑇₂=1,𝜙𝜃4𝜋sin²𝜙−𝑇₂1−𝑇₂/sin²𝜙sin²𝜙, (16a)(16b)

where 𝜃(𝑥) is the Heaviside step function. An extra factor of 2 appears in (16a) and (16b) because 𝑇₁ and 𝑇₂ are even functions of 𝑡_𝑧.

For an isotropic distribution of 𝐭̂, the components 𝑇₁, 𝑇₂, and 𝑇₃ are equivalent, and using (15) and (16a) and (16b), we obtain the identity (Figure 1, solid line) 𝑃₁(𝑇)=𝑃₂(𝑇)=1𝜋^1√_𝑇𝑑𝑥𝑥𝑥²−𝑇₁1−𝑥²≡12√𝑇. (17)

Figure 1 

The probability distributions of the components of the tensor 𝐓: 𝑃1(𝑇)=𝑃2(𝑇)=𝑃3(𝑇)=1/(2√𝑇), for 𝑇∈(0,1] (solid line), and 2𝑃4(|𝑇|)=2𝑃5(|𝑇|)=2𝑃6(|𝑇|), for 𝑇∈[−1,0)∪(0,1] (dashed line).

The components 𝑇₄, 𝑇₅, and 𝑇₆ are also equivalent to isotropic 𝐭̂. For example,𝑃_6,𝜙𝑇₆=1,𝜙𝜃−𝑇4𝜋sin2𝜙₆1−𝑇₆,/sin2𝜙sin2𝜙 (18a)where 𝑇₆/sin(2𝜙)=1−𝑡²_𝑧≥0. From (18a) we get 𝑃₆(𝑇)=12𝜋¹_|𝑇|𝑑𝑥𝑥(𝑥−𝑇)1−𝑥²=𝑃₄(𝑇)=𝑃₅(𝑇), (18b)(18c)

where 𝑇∈[−1,1] and 𝑃₄(|𝑇|)=𝑃₅(|𝑇|)=𝑃₆(|𝑇|) (Figure 1, dashed line).

We notice that lim_𝑇→1𝑃_𝑇6(𝑇)=1/(2√2), whereas in 0, 𝑃_𝑇6(𝑇) has a logarithmic divergence.

The equalities (18c) are obtained using the equivalence between the components 𝑇₄, 𝑇₅, and 𝑇₆.

We observe that the isotropic distribution of the TLS orientations does not correspond to a constant distribution of the values of the components of 𝐓, nor to equal distributions of these values. Therefore it is not straightforward to draw conclusions regarding the relations between the distributions of the values of 𝜸_𝐼, based only on general arguments.

4. Conclusions

In this paper we describe the anisotropy of the glassy properties of the decagonal quasicrystals in the model of [19]. We show that the glassy properties of these quasicrystals are similar to those of hexagonal disordered lattices [24] and we obtained the TLS-phonon average coupling constants, 𝛾²_𝐤,𝜎, which are dependent on the phonon's propagation direction, 𝐤, and polarization, 𝜎. We apply the results to the experimental data of [18] and we obtain the ratio (𝑟₆₆/𝑟₄₄)²≈4.2, where 𝑟₄₄ and 𝑟₆₆ are components of the tensor of coupling constants.

In order to better understand the characteristics of “isotropic” and “anisotropic” distributions of TLSs, we calculated the probability distributions of the elements of the tensor 𝐓—which describes the TLS—under the assumption that the directions of the ensemble of TLSs, defined by the unit vectors 𝐭̂, are isotropically oriented. We observe that the distributions of the elements of 𝐓 are rather complicated and therefore they cannot be found in the STM, simply based on qualitative, general arguments.

Acknowledgments

The work was supported by the Romanian National Authority for Scientific Research projects PN-II-ID-PCE-2011-3-0960 and PN09370102/2009. The travel support from the Romania-JINR Dubna collaboration project Titeica-Markov and project N4063 is gratefully acknowledged.