Vibrations of anisotropic objects

Calculators

Generic calculators exist for objects with a piecewise-superquadratic shape and for infinite cylinders with a piecewise-superelliptic cross-section. However, using one of the following symmetry-adapted calculators is recommended.

Finite objects
cubic	tetragonal	orthorhombic

Infinite cylinders
tetragonal	orthorhombic

Usage

Here are the calculation steps which are common to all the calculators:

• define the object
  • d_□: total length along the given direction except for the generic calculators for which the total length is the sum of the + and - lengths.
  • form factors n_□>0. Faces become flat when n_□ tend to infinity.
• define the elasticity (mass density ρ and stiffness tensor C_ij)
• choose the mode symmetry for symmetry-adapted calculators (irreducible representation)
• define the wavevector Q or the wavelength λ for the infinite cylinders
• Run!
• click on a line in the result table to display the corresponding vibration

The shape of the finite objects or the cross-section of the infinite cylinders and their vibrations are displayed using three.js. It is possible rotate the view (left mouse button or button+A keyboard key), zoom (middle button or button+S key) or pan (right button or button+D key). When visualizing the vibrations, the surface can be displayed in 4 different ways: using a texture (default), with colors depending on the direction of the displacement or in blue.

The displacements are expressed in a basis consisting of x^l+y^m+zⁿ functions with l+m+n≤N_max. Calculations with the generic calculators can take a long time when N_max is large. N_max=15 for symmetry-adapted calculators and N_max=20 when the checkbox next to Run! is checked () or absent.

Only one mode is shown for degenerate irreducible representations (E and T modes). Displaying all the degenerate modes is possible using the orthorhombic calculator because there are no degenerate modes in this case.

Method

The calculations use the method introduced by W. M. Visscher, A. Migliori, T. M. Bell and R. A. Reinert in J. Acoust. Soc. Am. 90, 2154 (1991) which was extended to handle superquadratic and superellipsoid shapes in Nanomaterials 11, 1838 (2021).

N. Nishiguchi, Y. Ando et M. N. Wybourne, J. Phys.: Condens. Matter 9 5751 (1997) extended this approach to infinite cylinders. It is extended to handle superellipse cross-sections.

The work by E. Mochizuki in J. Phys. Earth 35, 159 (1987) is used to speed up the calculations for objects with symmetries. It makes it possible to increase accuracy by using the symmetry of the vibrations (irreducible representations).

The implementation (source) is based on the GNU Scientific Library and emscripten so that the calculations can be performed by a modern web browser.

Symmetry

Finite objects

The tables below show how the degeneracies are lifted when lowering the symmetry starting from the vibrations of an isotropic sphere. They were prepared using the O_h→D_4h and D_4h→D_2h(C'₂) correlation tables. For the piecewiese superquadric shape, in the general case such objects have no symmetry operations except identity. Therefore all their vibrations share the same irreducible representation.

Infinite cylinders

The following table shows the correspondence between the vibrations in an isotropic circular cylinder and those of tetragonal and orthorhombic cylinders.

Tutorial

Let us try these calculators to check their validity by comparing with a few cases for which analytic solutions exist.

Isotropic sphere

Let us calculate the vibrations of a sphere of diameter 10 nm made of “isotropic gold” (v_l=3330 and v_t=1250 m/s). The lowest frequency for a spheroidal mode with ℓ=2 is 105.78 GHz as obtained with the isotropic calculator referenced above.

According to the table, these 5 modes (m=-2, -1, 0, 1 and 2) transform into 2 A_g + B_1g + B_2g + B_3g in the orthorhombic symmetry. Let us use the orthorhombic calculator for an identical sphere by choosing:

• spheroid with d_x=d_y=d_z=10 nm
• mass density ρ=19.283 g/cm³
• C₁₁=C₃₃=ρ*v_l²=213.8272587 GPa
• C₄₄=C₆₆=ρ*v_t²=30.1296875 GPa
• C₁₂=C₁₃=C₁₁-2*C₄₄=153.5678837 GPa

For the A_g irreducible representation, we obtain 2 modes at 105.784 GHz. One additional mode at the same frequency is obtained for B_1g, B_2g and B_3g. These are the 5 fundamental ℓ=2 spheroidal modes.

Right rectangular prism with cubic elasticity

According to R. D. Mindlin in J. Appl Phys 27, 1462 (1956), a right rectangular prism with sides of length a et b along x and y has vibrations with frequencies $\omega =\frac{m\pi }{2a}\sqrt{\frac{C_{11}-C_{12}}{\rho }}$ if $\frac{a}{m}=\frac{b}{n}$, m and n being integers.

For a cube made of cubic gold with side length 10 nm, $\omega =\frac{m\pi }{2a}\sqrt{\frac{C_{11}-C_{12}}{\rho }}\approx m\times 61.317$ GHz.

We can reproduce these vibrations with the cubic calculator. For odd m, these modes have the A_2g et E_g irreducible representations and have frequencies 61.317, 183.951 and 306.587 GHz. For even m, they are the T_1g modes at 122.634 and 245.268 GHz. The frequencies for the next overtones differ a bit between both calculation methods.

For a square cuboid made of cubic gold with side length 10 nm along x and y and an arbitrary length along z (let us choose 6 nm for example), we obtain the same frequencies as before with the tetragonal calculator for B_1g at 61.317 and 183.951 GHz and for A_2g at 122.634 and 245.268 GHz. These irreducible representation agree with the table above.

Similarly, for a right rectangular prism made of cubic gold with side length 10, 15 and 6 nm along x, y and z, noting that a/2=b/3, we obtain a B_3u mode at 122.634 GHz with the orthorhombic calculator. The symmetry is different in this case (in particular the parity) because m=2 and n=3 while we had m=n before with a=b.

Note that the last eigenmodes obtained with the tetragonal and orthorhombic calculators do not depend on the length along z. As a result, eigenmodes with the same frequencies and irreducible representations exists also for the corresponding infinite cylinders at Q=0: B_1g and A_2g (tetragonal) and B_3u (orthorhombic).

Comparison with published results

Here are a few links to reproduce some published results:

• Table 1 (aluminum potato) from J. Acoust. Soc. Am. 90, 2154 (1991). Note that the frequencies reported in the paper were scaled to match the experimental ones. To achieve the same effect with the calculator, the mass density is modified (≈2.7430 instead of 2.7065 g/cm³).
• Tables 5 to 13 of Journal of Sound and Vibration 311 (2008) 184–199 (2008). The sizes are chosen so that the frequencies in GHz match the reduced frequencies in the paper. For the pyramids, the lattice structure of the cubic materials were rotated by 45° around the z axis so that the sides of the base of the pyramids are aligned along [100] as in the paper. In each case, a link using the generic calculator is provided. A link using a symmetry-adapted calculator is also provided for the spheres and cubes.
• Figures S3 and S5 (polystyrene sphere and spheroid) from Phys. Rev. Lett. 128, 048003 (2022) (Supplemental material).
• Table 1: NPr, NPc, NPcs and NPrs (gold rounded nanocubes) from J. Phys. Chem. C 126, 3606 (2022).
• Something bigger? What about a white dwarf with cubic elasticity as in AIP Advances 11, 105111 (2021)?

  Table 1A	cubique	générique
  Table 1B	cubique	générique
  Table 1C	cubique	générique

  The frequencies differ a bit because the exact parameters used in the article are unknown.