Ideal Shear Strength of a Quantum Crystal

Using path-integral Monte Carlo simulations, we compute the ideal shear strength (ISS) on the basal plane of hcp ${}^4\mathrm{He}$. The failure mode upon reaching the ISS limit is characterized by the homogeneous nucleation of a stacking fault and it is found to be anisotropic, consistent with Schmid’s law of resolved shear stress. Comparing the ISS of hcp ${}^4\mathrm{He}$ to a large set of classical crystals shows that it closely fits the approximately universal modified Frenkel model of ideal strength. In addition to giving quantitative stress levels for the homogeneous nucleation of extended defects in hcp ${}^4\mathrm{He}$, our findings lend support to assumptions in the literature that inherently classical models remain useful for the description of mechanical behavior in quantum crystals.

Figure 1

Typical shear stress response as a function of strain. Upper (red) and lower (blue) irregular lines denote stress components parallel and perpendicular to direction of applied strain, respectively. Dashed black line represents linear elastic response as determined in Ref. [42]. Solid smooth (black) line represents sinusoidal fit to data. Corresponding ideal strain and stress values are denoted by the square.

Figure 2

Snapshots of centroid (circles) and bead (dots) positions of two adjacent atomic basal planes at different stages of deformation. (a) Zero deformation. (b) Immediately after discontinuity in shear stress.

Figure 3

Shear directions and anisotropy of the ISS in the basal plane. a) Black (in groups of three) and red (in groups of six) arrows describe the “easy” and “hard” shearing directions, respectively, for the two distinct atomic planes, labeled $A$ and $B$. Unoccupied position is shown by dashed circle labeled $C$. b) Polar plot of the minimum ISS as a function of shear direction. Diamonds (red) depict results for the system containing $N=180$ atoms at $T=1\mathrm{K}$ for $\dot{d}=1\times {10}^{-6}\AA$ per step and $\tau =1/40{\mathrm{K}}^{-1}$. Blue (at $0°$) and green (at $30°$) squares represent results for $N=480$ at $T=0.5\mathrm{K}$ for $\dot{d}=2\times {10}^{-6}\AA$ per step and $\tau =1/40\mathrm{K}$ for the hard and easy directions, respectively. Error bars are smaller than symbol sizes.

Figure 4

Shearability for the easy direction as a function of temperature. Data points represent minimum shearabilities obtained from sets of 15 independent deformation simulations. Error bars represent standard deviations of respective data sets. Dashed line serves as a guide to the eye. Inset displays converged low-temperature limit of the shear stress-strain curve at $T=0.5\mathrm{K}$. Data point shows estimated shearability and ISS for hcp ${}^4\mathrm{He}$ at a mmolar volume of $21{\mathrm{cm}}^3$.

Figure 5

Relation between the ISS scaled by the shear modulus and shearability for solids characterized by different crystal structures and bonding types. Line describes Frenkel model (see text). All data points, except for ${}^4\mathrm{He}$ and ${\mathrm{Ti}}_3\mathrm{Nb}$, represent the strain-controlled values reported in Ref. [3]. Different symbols correspond to distinct crystal structures. Blue (all data points left of $\beta \text{−}{\mathrm{S}\mathrm{i}}_3{\mathrm{N}}_4$, except ${}^4\mathrm{He}$) and red (five rightmost) symbols represent metallic and covalent crystals, respectively. Green filled upward triangle (second data point from the left) represents present PIMC++ result for hcp ${}^4\mathrm{He}$. Result for ${\mathrm{Ti}}_3\mathrm{Nb}$ is based on data from Ref. [52].