Martensitic transformations in the body-centered cubic $\beta$-phase (Im$\overline{3}$m) of zirconium are studied using first-principles calculations, group-theoretical methods, and symmetry analysis. Phonon dispersion relations in the $\beta$-phase calculated within the harmonic approximation predicted an unstable phonon at wave vector $\frac{2\pi }{\mathrm{a}}[\frac{1}{2},\frac{1}{2},0]$(N) and a soft phonon at wave vector $\frac{2\pi }{\mathrm{a}}[\frac{2}{3},\frac{2}{3},\frac{2}{3}]$($\mathrm{\Lambda }$). The symmetry of the unstable phonon is the same as the symmetry of the N${}_4^{-}$ irreducible representation, and the symmetry of the soft phonon is the same as the symmetry of the ${\mathrm{\Lambda }}_1$ irreducible representation. Martensitic transformations are simulated considering two steps. Frozen phonon calculations are used to determine the first step, i.e., the transformation of the $\beta$-phase to an intermediate phase due to phonon motion. Structure relaxation is used to determine the second step, i.e., the transformation of the intermediate phase to the final phase. The unstable N${}_4^{-}$ phonon transforms the $\beta$-phase into an intermediate orthorhombic phase (Cmcm), which further transforms to a hexagonal close packed $\alpha$-phase ($\mathrm{P}{6}_3$/mmc) after structure relaxation. The soft ${\mathrm{\Lambda }}_1$ phonon transforms the $\beta$-phase into an intermediate trigonal phase (P$\overline{3}$m1), which further transforms to a hexagonal close packed $\omega$-phase (P6/mmm) after structure relaxation. The intermediate phase space group (Cmcm/P$\overline{3}$m1) is a common subgroup of the parent phase ($\beta$) space group and the final phase ($\alpha /\omega$) space group. Therefore, the martensitic transformations in zirconium are reconstructive transformations of the second kind. Symmetry characterization of the martensitic transformations is also presented.

中文翻译：

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锆中β相的马氏体转变

体心立方中的马氏体转变 $\beta$-相（Im$\overline{3}$m) 锆的研究使用第一性原理计算、群论方法和对称性分析。声子色散关系$\beta$-在谐波近似内计算的相位预测了波矢量处的不稳定声子 $\frac{2\pi }{\mathrm{一种}}[\frac{1}{2},\frac{1}{2},0]$(N) 和波矢量处的软声子 $\frac{2\pi }{\mathrm{一种}}[\frac{2}{3},\frac{2}{3},\frac{2}{3}]$($\mathrm{\Lambda }$）。不稳定声子的对称性与 N 的对称性相同${}_4^{-}$ 不可约表示，软声子的对称性与软声子的对称性相同 ${\mathrm{\Lambda }}_1$不可约的表示。考虑两个步骤来模拟马氏体转变。冻结声子计算用于确定第一步，即变换$\beta$由于声子运动，-相到中间相。结构松弛用于确定第二步，即中间相到最终相的转变。不稳定的 N${}_4^{-}$ 声子变换 $\beta$-相变成中间正交相（Cmcm），进一步转变为六方密堆积 $\alpha$-阶段 （$\mathrm{磷}{6}_3$/mmc) 结构松弛后。柔软的${\mathrm{\Lambda }}_1$ 声子变换 $\beta$-相变成中间三角相（P$\overline{3}$m1)，进一步转化为六边形密堆积 $\omega$结构松弛后的 - 相 (P6/mmm)。中间相空间群 (Cmcm/P$\overline{3}$m1) 是母相 ($\beta$) 空间群和最后阶段 ($\alpha /\omega$) 空间群。因此，锆中的马氏体转变是第二类重构转变。还介绍了马氏体转变的对称特征。