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<p>This study investigates the elastic anisotropy of two cubic crystals, KAl(SO<sub>4</sub>)<sub>2</sub>·12H<sub>2</sub>O and Bi<sub>12</sub>SiO<sub>20</sub>, by calculating acoustic wave velocities along both principal and off-principal crystallographic directions. Using experimentally determined elastic constants, phase-velocity and slowness curves were constructed to evaluate the directional dependence of longitudinal and transverse acoustic modes. The two crystals exhibit opposite trends in anisotropy, consistent with their Zener elastic anisotropy indices of <i>A</i> = 1.201 for KAl(SO<sub>4</sub>)<sub>2</sub>·12H<sub>2</sub>O and <i>A</i> = 0.497 for Bi<sub>12</sub>SiO<sub>20</sub>. Slowness curve analysis demonstrated that the sign and magnitude of “<i>A</i>” determine whether quasi-longitudinal and quasi-transverse branches bulge outward or inward. Structural considerations revealed that the near-isotropic elasticity of KAl(SO<sub>4</sub>)<sub>2</sub>·12H<sub>2</sub>O originates from a symmetric hydrogen-bonded framework, while the pronounced anisotropy in Bi<sub>12</sub>SiO<sub>20</sub> arises from asymmetric Bi–O and Si–O bonding. These results provide a comparative understanding of cubic crystals with <i>A</i> &gt; 1 and <i>A</i> &lt; 1 and highlight the role of bonding topology in governing elastic anisotropy.</p>

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Elastic anisotropy and slowness curves in cubic crystals: KAl(SO4)2·12H2O and Bi12SiO20

  • 期刊论文

  • 发表日期：

    2026年1月9日

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  • Kwang-Sei Lee,
  • Jae-Hyeon Ko

摘要

This study investigates the elastic anisotropy of two cubic crystals, KAl(SO4)2·12H2O and Bi12SiO20, by calculating acoustic wave velocities along both principal and off-principal crystallographic directions. Using experimentally determined elastic constants, phase-velocity and slowness curves were constructed to evaluate the directional dependence of longitudinal and transverse acoustic modes. The two crystals exhibit opposite trends in anisotropy, consistent with their Zener elastic anisotropy indices of A = 1.201 for KAl(SO4)2·12H2O and A = 0.497 for Bi12SiO20. Slowness curve analysis demonstrated that the sign and magnitude of “A” determine whether quasi-longitudinal and quasi-transverse branches bulge outward or inward. Structural considerations revealed that the near-isotropic elasticity of KAl(SO4)2·12H2O originates from a symmetric hydrogen-bonded framework, while the pronounced anisotropy in Bi12SiO20 arises from asymmetric Bi–O and Si–O bonding. These results provide a comparative understanding of cubic crystals with A > 1 and A < 1 and highlight the role of bonding topology in governing elastic anisotropy.