Quantum and thermal fluctuations of a thin elastic plate

We consider a Hamiltonian description of the vibrations of a clamped, elastic circular plate. The Hamiltonian of this system features a potential energy with two distinct contributions: one that depends on the local mean curvature of the plate and a second that depends on its Gaussian curvature. We quantize this model using a complete, orthonormal set of eigenfunctions for the clamped, vibrating plate. The resulting quanta are the flexural phonons of the thin circular plate. As an application, we use this quantized description to calculate the fluctuations in the displacement of the plate for arbitrary temperature. We compare the fluctuation profile with that from an elastic membrane under tension. At low temperature, we find that while both profiles have a circular ring of local maxima, the ring in the membrane profile is much more pronounced and sharper. We also note that with increasing temperature the plate profile develops two additional rings of extrema.

Figure 1

Sketch of an elastic plate (radius $a$, thickness $h$) in equilibrium.

Figure 2

Plot of selected low-frequency eigenmodes $w_{mn}\left(\mathbf{r}\right)$ for the vibrating plate.

Figure 3

Plot of the thermal average of the squared displacement (scaled) ${\langle u^2\left(r\right)\rangle }_T/{\langle u^2\left(0\right)\rangle }_T$ versus (reduced) radius $r/a$ for a silicon nitride plate with ${\mathrm{\Theta }}_c=80$ mK (with ${\mathrm{\Theta }}_c\equiv \hslash {\omega }_c/k_B$; top row) and an elastic membrane under tension (bottom row). Both have a radius of $a=1\mu \mathrm{m}$. The plot is obtained by a partial sum of Eq. (23). The truncated series includes over 3200 terms and is converged to better than 1%.

Figure 4

Plot of the thermal average of the squared displacement of the center of the plate (scaled) ${\langle u^2\left(0\right)\rangle }_T/u_0^2$ versus temperature $T$ (K) for a silicon nitride plate (top). Derivative with respect to temperature of the average squared displacement at the center versus temperature $T$ (bottom). We take ${\mathrm{\Theta }}_D=850$ K (Ref. [19]), ${\mathrm{\Theta }}_c=100$ mK (solid line), 300 mK (dashed line), and 800 mK (dot-dashed line). The zero-point fluctuation scale is set by $u_0^2\equiv \sqrt{\frac{{\hslash }^2}{64{\pi }^2\sigma D}}$.