This page presents tools to calculate and visualize the vibration eigenmodes of anisotropic objects. The calculation method is described in Nanomaterials 11, 1838 (2021). The shape of the objects is delimited by superellipses, superquadratics or superellipsoids.
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 |
Here are the calculation steps which are common to all the calculators:
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}+z^{n} 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 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.
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.
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.
spherical | cubic | tetragonal | orthorhombic |
---|---|---|---|
SO(3) | O_{h} | D_{4h} | D_{2h} |
SPH,ℓ=0 | A_{1g} | A_{1g} | A_{g} |
SPH,ℓ=1 | T_{1u} | A_{2u} | B_{1u} |
E_{u} | B_{2u} | ||
B_{3u} | |||
SPH,ℓ=2 | E_{g} | A_{1g} | A_{g} |
B_{1g} | A_{g} | ||
T_{2g} | B_{2g} | B_{1g} | |
E_{g} | B_{2g} | ||
B_{3g} | |||
SPH,ℓ=3 | A_{2u} | B_{1u} | A_{u} |
T_{1u} | A_{2u} | B_{1u} | |
E_{u} | B_{2u} | ||
B_{3u} | |||
T_{2u} | B_{2u} | B_{1u} | |
E_{u} | B_{2u} | ||
B_{3u} | |||
SPH,ℓ=4 | A_{1g} | A_{1g} | A_{g} |
E_{g} | A_{1g} | A_{g} | |
B_{1g} | A_{g} | ||
T_{1g} | A_{2g} | B_{1g} | |
E_{g} | B_{2g} | ||
B_{3g} | |||
T_{2g} | B_{2g} | B_{1g} | |
E_{g} | B_{2g} | ||
B_{3g} | |||
⋮ | ⋮ | ⋮ | ⋮ |
spherical | cubic | tetragonal | orthorhombic |
---|---|---|---|
SO(3) | O_{h} | D_{4h} | D_{2h} |
TOR,ℓ≠0 | |||
TOR,ℓ=1 | T_{1g} | A_{2g} | B_{1g} |
E_{g} | B_{2g} | ||
B_{3g} | |||
TOR,ℓ=2 | E_{u} | A_{1u} | A_{u} |
B_{1u} | A_{u} | ||
T_{2u} | B_{2u} | B_{1u} | |
E_{u} | B_{2u} | ||
B_{3u} | |||
TOR,ℓ=3 | A_{2g} | B_{1g} | A_{g} |
T_{1g} | A_{2g} | B_{1g} | |
E_{g} | B_{2g} | ||
B_{3g} | |||
T_{2g} | B_{2g} | B_{1g} | |
E_{g} | B_{2g} | ||
B_{3g} | |||
TOR,ℓ=4 | A_{1u} | A_{1u} | A_{u} |
E_{u} | A_{1u} | A_{u} | |
B_{1u} | A_{u} | ||
T_{1u} | A_{2u} | B_{1u} | |
E_{u} | B_{2u} | ||
B_{3u} | |||
T_{2u} | B_{2u} | B_{1u} | |
E_{u} | B_{2u} | ||
B_{3u} | |||
⋮ | ⋮ | ⋮ | ⋮ |
The following table shows the correspondence between the vibrations in an isotropic circular cylinder and those of tetragonal and orthorhombic cylinders.
isotropic circular | tetragonal | |
---|---|---|
C_{∞v}, D_{∞h} (Q=0) | C_{4v} | D_{4h} (Q=0) |
m ≡ 0 mod 4 (0, 4, 8, …) |
A_{1} | A_{1g} |
A_{2u} | ||
A_{2} | A_{1u} | |
A_{2g} | ||
m ≡ 2 mod 4 (2, 6, 10, …) |
B_{1} | B_{1g} |
B_{2u} | ||
B_{2} | B_{1u} | |
B_{2g} | ||
m ≡ 1 mod 2 (1, 3, 5, …) |
E (×2) | E_{g} (×2) |
E_{u} (×2) |
isotropic circular | orthorhombic | |
---|---|---|
C_{∞v}, D_{∞h} (Q=0) | C_{2v} | D_{2h} (Q=0) |
m ≡ 0 mod 2 (0, 2, 4, …) |
A_{1} | A_{g} |
B_{1u} | ||
A_{2} | A_{u} | |
B_{1g} | ||
m ≡ 1 mod 2 (1, 3, 5, …) |
B_{1} | B_{2g} |
B_{3u} | ||
B_{2} | B_{2u} | |
B_{3g} |
Let us try these calculators to check their validity by comparing with a few cases for which analytic solutions exist.
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:
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.
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).
Here are a few links to reproduce some published results:
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.
Table 1A | cubique | générique |
Table 1B | cubique | générique |
Table 1C | cubique | générique |