Phonon–Phason Coupling Strength in a Tsai-Type Ag–In–Yb Icosahedral Quasicrystal

Abstract

Owing to the quasiperiodic order, one independent term in the elasticity of quasicrystals (phonon–phason coupling) is characteristic of quasicrystals. It is not seen in conventional crystals and has continued to be a significant subject in the research field of quasicrystals. Recently, a novel method was applied to a Mackay-type Al–Pd–Mn icosahedral quasicrystal to prove the existence of phonon–phason coupling and to evaluate its strength based on the elasticity of quasicrystals. This study applied the method to a Tsai-type Ag–In–Yb icosahedral system to evaluate its phonon–phason coupling strength. We applied phonon strain to the quasicrystal at a temperature with active phason, and the induction of phason strain was successfully detected and evaluated using powder X-ray diffraction. We evaluated the phonon–phason coupling elastic constant of the Ag–In–Yb icosahedral quasicrystal to be 0.17 ± 0.04 GPa by quantitatively comparing the measured induced phason strain with our calculation results.

Fig. 2 Phason momentum (|G_⊥|) dependence of the full width at half maximum of the X-ray diffraction peaks ((Δq)_FWHM) for the as-grown sample, the samples with the compressed stress of 75 and 100 MPa, and the sample subjected to the same heat treatment but without compression (0 MPa). Reciprocal vector (|G_|||) dependences of (Δq)_FWHM for these samples are shown in the inset.

1. Introduction

Quasicrystals (QCs) have a unique type of ordered structure characterized by disallowed crystallographic rotational symmetry and quasiperiodic translational order.^1–3) Owing to the quasiperiodic translational order, QCs have a peculiar type of elastic degrees of freedom called phason degrees of freedom, in addition to phonon degrees of freedom. With the lowest-order approximation, the elastic free energy (f_el) of the QCs per unit volume can be expressed as a function of the phonon (conventional) strain u_ij and phason strain w_ij, f_el comprising the following three independent terms:   

\begin{equation} f_{\text{el}}(u_{ij},w_{ij}) = f_{u - u}(u_{ij}) + f_{w - w}(w_{ij}) + f_{u - w}(u_{ij},w_{ij}), \end{equation} (1)

where f_u−u and f_w−w represent the pure phonon, and pure phason elasticities, respectively, f_u−w represents the phonon-phason coupling. The terms f_w−w and f_u−w in eq. (1) appear only for QCs, whereas f_u−u arises also for all conventional solid crystals. The phason elasticity, represented by f_w−w, has attracted much attention and many studies have so far been conducted, because it is closely related to the fundamental question of what is the physical origin of the stability of QCs,⁴⁾ and also because it plays an important role in the emergence of characteristic physical properties such as the breakdown of Dulong–Petit’s law in the specific heat.^5,6) In contrast, f_u−w has not been extensively discussed in the literature. Additionally, f_u−w significantly influences the strain field around dislocations in QCs, internal friction, and the shape of neutron or X-ray diffuse scattering of QCs; therefore, it is considered a significant subject in the research of QCs.

Koschella et al.⁷⁾ and Zhu et al.⁸⁾ evaluated the elastic coupling constant of f_u−w for the model QC structures of decagonal and icosahedral systems. They introduced phason strain into QCs to construct a series of crystal approximants (CAs) and measured their phonon strain to evaluate the coupling constant. Edagawa et al.^9–11) applied this method to real QC alloys for experimental evaluation. However, CAs are generally different phases from the QCs; therefore, the method using CAs cannot directly probe the f_el of QCs. This indicated that the estimated values may not be accurate. Recently, a new approach, which directly probes the f_el of QCs, was proposed for the quantitative evaluation of the coupling constant and was applied to a single crystal of the Al–Pd–Mn icosahedral QC, by which the existence of f_u−w was demonstrated.¹²⁾ The icosahedral QC phases discovered so far can be classified into three groups based on the type of atomic cluster that constitutes the QC structure: Mackay-type, Bergman-type, and Tsai-type. The Al–Pd–Mn icosahedral QC belongs to the Mackay type.

In this paper, this approach was applied to a Tsai-type Ag–In–Yb icosahedral system. Phonon strain was applied to single crystals of the Ag–In–Yb icosahedral QC at an elevated temperature with active phason. The induction of phason strain was successfully observed by powder X-ray diffraction, and the X-ray diffraction peaks were simulated based on the elasticity of the QCs. The magnitude of the induced phason strain was quantitatively determined by comparing the measured X-ray diffraction peaks with the simulated results, from which the coupling constant of f_u−w was evaluated.

2. Experimental Procedures and Calculation Method

Pure metallic elements with a nominal composition of Ag₄₂In₄₂Yb₁₆ were placed in a conical-bottom alumina crucible with a diameter of 16 mm and sealed in a silica tube under an argon atmosphere. The silica tube was slowly pulled down in the furnace at a tube speed of 0.8 mm/h. In this way, large single crystals of the Ag–In–Yb icosahedral QC were grown by using the Bridgman method. The crystallographic orientation of the grown single crystals were characterized using back-scattering X-ray Laue diffraction measuring instruments, and 2 × 2 × 5 mm³ rectangular samples were formed with the long axis of the sample along one of the 2-fold axes of the icosahedral QC. Compression tests were performed at 573 K using an automatic material-testing machine equipped with a heating furnace. Constant elastic stress was maintained along the long axis of the sample over time. The samples were then quenched to room temperature while maintaining the same stress. Subsequently, the stress was removed. Finally, the samples were powdered and subjected to X-ray diffraction measurements using Cu Kα radiation at 40 kV and 200 mA.

The calculation method is described in detail in our previous work.¹²⁾ We summarize the equations used in our calculation as follows. Based on the generalized elasticity of the icosahedral QC, when an elastic stress σ is applied along [001] (one of the 2-fold axes of the icosahedral QC), the applied phonon strain $u_{ij}^{\text{appl}}$ can be expressed as follows:   

\begin{equation} u_{ij}^{\text{appl}} = \begin{bmatrix} - \nu \varepsilon & 0 & 0\\ 0 & - \nu \varepsilon & 0\\ 0 & 0 & \varepsilon \end{bmatrix} , \end{equation} (2)

where ε = σ/E (E is Young’s modulus) and ν is Poisson’s ratio. This type of phonon strain brings about the symmetry breaking from the icosahedral point group I_h to an orthorhombic point group D_2h. When $u_{ij}^{\text{appl}}$ is given, by taking the partial derivative of phason strain w_ij for f_el, the corresponding phason strain $w_{ij}^{\text{ind}}$ can be calculated as follows:   

\begin{align} & w_{ij}^{\text{ind}} = M \begin{bmatrix} \tau & 0 & 0\\ 0 & -\dfrac{1}{\tau} & 0\\ 0 & 0 & - 1 \end{bmatrix} ,\\ & M = K_{3}\varepsilon (1 + \nu)/(K_{1} - 4K_{2}/3) \end{align} (3)

where K₁ and K₂ are the phason elastic constants in f_w−w in eq. (1), and the coupling constant was represented by K₃. The induced phason strain $w_{ij}^{\text{ind}}$ in eq. (3) gives the symmetry breaking I_h → D_2h, agreeing with that induced by the applied phonon strain $u_{ij}^{\text{appl}}$, in eq. (2). In this study, we calculated the diffraction peaks using $w_{ij}^{\text{ind}}$ in eq. (3) and compared them with the experimental peaks to evaluate K₃.

3. Results and Discussion

Figure 1 shows the X-ray diffraction spectrum at scattering angles 2θ from 20 to 80° for the as-grown sample of Ag–In–Yb icosahedral QC, where the Cu Kα₂ component was numerically removed from the raw data. All peaks can be indexed as icosahedral phase. These peak widths are close to the X-ray instrumental resolution limit, indicating that the single-phase samples’ structural quality is excellent.

Fig. 1

X-ray diffraction spectrum at scattering angles 2θ from 20° to 80° for the as-grown sample of Ag–In–Yb icosahedral quasicrystal, where the Cu Kα₂ component was removed numerically from the raw data.

Figure 2 shows |G_⊥| (the magnitude of the phason momentum) dependences of the full width at half maximum intensities of the X-ray diffraction peaks (Δq)_FWHM for the as-grown sample, the samples with compressed stresses of 75 and 100 MPa for 1 h at 573 K, and the sample processed with the same heat treatment as compressed samples at 573 K but without compression (0 MPa). No clear |G_⊥| dependence of (Δq)_FWHM can be exhibited for the as-grown sample and the sample without compression at 573 K. In contrast, the (Δq)_FWHM of samples with compressed stresses of 75 and 100 MPa at 573 K have a clear upward trend with increasing |G_⊥|, where the slope becomes more significant with increasing stresses. Because no oxidation or phase changes can be detected in all compressed samples, the broadening of X-ray peaks is attributable to the induction of the phason strain. Meanwhile, there is no noticeable |G_||| (the magnitude of the reciprocal vector, that is, scattering vector q-value (q ≡ 4π sin θ/λ)) dependence of (Δq)_FWHM for all these samples, as shown in the inset. These results indicate that when applying phonon strain to the Ag–In–Yb icosahedral QC, the phason strain was induced successfully, and the induced phason strain was positively dependent on the applied phonon strain, proving the existence of f_u−w. We note here that the phonon strain was removed upon the removal of the compressed stresses on samples and that the phason one was “frozen” at room temperature.

Fig. 2

Phason momentum (|G_⊥|) dependence of the full width at half maximum of the X-ray diffraction peaks ((Δq)_FWHM) for the as-grown sample, the samples with the compressed stress of 75 and 100 MPa, and the sample subjected to the same heat treatment but without compression (0 MPa). Reciprocal vector (|G_|||) dependences of (Δq)_FWHM for these samples are shown in the inset.

Then, based on the elasticity of the QCs, we simulated the powder X-ray diffraction peaks, in which the phason strain $w_{ij}^{\text{ind}}$ in eq. (3) is induced. The simulation method was described in detail in our previous paper¹²⁾ and is reviewed in the following section. The induced phason strain $w_{ij}^{\text{ind}}$ leads to the shifting of Bragg spots in the 3D reciprocal space: $\mathbf{G}_{\| } \to \mathbf{G}'_{\| }$, as follows:   

\begin{equation} \mathbf{G}'_{\|} = \mathbf{G}_{\|} - \mathbf{G}_{\bot} \cdot w_{ij}^{\text{ind}}. \end{equation} (4)

Generally, Bragg spots {G_||} with the same |G_||| in reciprocal space constitute a single peak in the powder X-ray diffraction spectrum. By the induced phason strain, they shift in different directions, causing the peak to broaden, deform, and/or split. The new q-value, that is, the $|\mathbf{G}'_{\| }|$ of each Bragg spot, was calculated using eq. (4). Then, Lorenz functions with a constant (Δq)_FWHM = 0.10 nm⁻¹ were placed at these q-positions of $|\mathbf{G}'_{\| }|$, and the vertical bars in Fig. 4 represent the $|\mathbf{G}'_{\| }|$ of the Bragg spots. Here, (Δq)_FWHM = 0.10 nm⁻¹ was determined using the average value of (Δq)_FWHM for the measured X-ray peaks of the as-grown sample. The simulated peaks were constructed as the sum of these Lorenz functions. The X-ray diffraction peaks, which have a relatively high intensity in Fig. 1, were simulated using the above method. The |G_⊥| dependence of (Δq)_FWHM for the simulated peaks was calculated using eq. (4) and was obtained for various |M|, as shown in Fig. 3. Comparing Figs. 2 and 3, we estimated |M| ≈ 0.015 ± 0.005 for 75 MPa compressed stress and |M| ≈ 0.02 ± 0.005 for 100 MPa compressed stress.

Fig. 3

Phason momentum (|G_⊥|) dependence of the full width at half maximum of the X-ray peaks ((Δq)_FWHM) for simulated X-ray diffraction peaks for various |M|.

The comparison of the measured and simulated X-ray diffraction peaks for the 100 MPa (top), 75 MPa (middle) compressed samples, and the as-grown sample (bottom) are shown in Fig. 4, while M = −0.02, −0.015 and 0 were used in our simulations, respectively. The consistency between the measured and simulated peaks was quite good. For the Tsai-type Ag–In–Yb icosahedral system, the values of E and ν have never been reported. However, the Ag–In–Yb icosahedral QC is isostructural with the Tsai-type Cd–Yb icosahedral QC, and their melting points are nearly equal. Thus, it may be assumed that the Lamé constants λ and μ of the Ag–In–Yb icosahedral QC are approximately equal to those of the Cd–Yb icosahedral QC, which have been previously reported to be 35.28 and 25.28 GPa, respectively.¹³⁾ Therefore, we obtain E = μ(3λ + 2μ)/(λ + μ) ≈ 65 GPa and ν = λ/2(λ + μ) ≈ 0.29, respectively. Then, the values of K₁ and K₂ for Ag–In–Yb icosahedral QC have been previously reported to be 24 MPa and 5.7 MPa, respectively.¹⁴⁾ For our compressed stress σ = 100 MPa and ε = −σ/E ≈ −1.53 × 10⁻³, substituting K₁, K₂, ε, ν, and M = −0.02 into the M = K₃ε(1 + ν)/(K₁ − 4K₂/3) in eq. (3), we obtain the absolute value of the phonon–phason coupling constant, |K₃| ≈ 0.17 ± 0.04 GPa. The value of |K₃| is in good agreement with previous results of internal friction measurements. It quantitatively explains the internal friction peaks previously observed for the Ag–In–Yb icosahedral phase.¹⁴⁾ The absolute value of the evaluated phonon–phason coupling constant |K₃| for the Ag–In–Yb icosahedral QC in the present study and other previously reported |K₃| of icosahedral QCs are presented in Table 1.

Fig. 4

Measured (black) and simulated (red) X-ray diffraction peaks for the 100 MPa (top), 75 MPa (middle) compressed samples, and as-grown sample (bottom), while M = −0.02, −0.015, and 0 were used in our simulations, respectively.

Table 1 The absolute value of the phonon–phason coupling constant |K₃| of icosahedral QCs.

4. Conclusions

The method used in our previous work to evaluate the phonon–phason coupling strength of a Mackay-type Al–Pd–Mn icosahedral system¹²⁾ was applied to evaluate the coupling strength of a Tsai-type Ag–In–Yb icosahedral system. In our experiment, phonon strain was applied to single crystals of Ag–In–Yb icosahedral QC at a temperature with active phason. The induced phason strain was successfully observed and measured by X-ray diffraction measurements, providing clear evidence of phonon–phason coupling in the Tsai-type Ag–In–Yb icosahedral system. The powder X-ray diffraction peaks were simulated based on the elasticity of the QCs. The induced phason strain was quantitatively evaluated by comparing the measured X-ray peaks with our simulated peaks. The absolute value of the phonon–phason coupling elastic constant |K₃| was determined to be 0.17 ± 0.04 GPa. The value of |K₃| is in good agreement with previous results of internal friction experiments for an Ag–In–Yb icosahedral QC.

Acknowledgments

This work was supported by JST, CREST Grant Number JPMJCR22O3 Japan, and KAKENHI Grant-in-Aid (No. JP19H05821) from the Japan Society for the Promotion of Science (JSPS). This work was also supported by the Light Metal Educational Foundation, Inc., under funds for the Encouragement and Promotion of Research, Study, and Education.

REFERENCES

• 1)   D.  Levine and  P.J.  Steinhardt: Phys. Rev. B 34 (1986) 596–616. doi:10.1103/PhysRevB.34.596
• 2)   J.E.S.  Socolar and  P.J.  Steinhardt: Phys. Rev. B 34 (1986) 617–647. doi:10.1103/PhysRevB.34.617
• 3)  P.J. Steinhardt and S. Ostlund: The Physics of Quasicrystals, (World Scientific, 1987).
• 4)  C.L. Henley: Quasicrystals: The State of the Art, ed. by D.P. DiVincenzo and P.J. Steinhardt, (World Scientific, Singapore, 1991) pp. 459–560.
• 5)   S.  Tamura,  K.  Fukushima,  Y.  Tokumoto,  Y.  Takagiwa and  K.  Edagawa: Mater. Trans. 62 (2021) 356–359. doi:10.2320/matertrans.MT-MB2020004
• 6)   K.  Fukushima,  H.  Suyama,  Y.  Tokumoto,  Y.  Kamimura,  Y.  Takagiwa and  K.  Edagawa: J. Phys. Commun. 5 (2021) 085002. doi:10.1088/2399-6528/ac1875
• 7)   U.  Koschella,  F.  Gaeler,  J.  Roth and  H.-R.  Trebin: J. Alloy. Compd. 342 (2002) 287–290. doi:10.1016/S0925-8388(02)00194-9
• 8)   W.J.  Zhu and  C.L.  Henley: Europhys. Lett. 46 (1999) 748–754. doi:10.1209/epl/i1999-00328-9
• 9)   K.  Edagawa: Philos. Mag. Lett. 85 (2005) 455–462. doi:10.1080/09500830500321191
• 10)   K.  Edagawa and  Y.G.  So: Philos. Mag. 87 (2007) 77–95. doi:10.1080/14786430600891352
• 11)   K.  Edagawa: Philos. Mag. 87 (2007) 2789–2798. doi:10.1080/14786430701264178
• 12)   J.  Zhang,  Y.  Kamimura,  Y.  Tokumoto and  K.  Edagawa: Philos. Mag. 102 (2022) 1461–1480. doi:10.1080/14786435.2022.2052376
• 13)   J.  Schreuer,  W.  Steurer,  T.A.  Lograsso and  D.  Wu: Philos. Mag. Lett. 84 (2004) 643–653. doi:10.1080/09500830512331329132
• 14)   K.  Handa,  Y.G.  So,  Y.  Kamimura,  R.  Tamura and  K.  Edagawa: Mater. Trans. 55 (2014) 754–757. doi:10.2320/matertrans.M2013448
• 15)   Y.G.  So,  S.  Sato,  K.  Edagawa,  T.  Mori and  R.  Tamura: Phys. Rev. B 80 (2009) 224204. doi:10.1103/PhysRevB.80.224204
• 16)   P.S.  Spoor,  J.D.  Maynard and  A.R.  Kortan: Phys. Rev. Lett. 75 (1995) 3462–3465. doi:10.1103/PhysRevLett.75.3462
• 17)   R.  Sterzel,  C.  Hinkel,  A.  Haas,  A.  Langsdorf,  G.  Bruls and  W.  Assmus: Europhys. Lett. 49 (2000) 742–747. doi:10.1209/epl/i2000-00213-1