A $5 \mathrm{~cm}$ long metal rod $\mathrm{AB}$ was initially at a uniform temperature of $T_0{ }^{\circ} \mathrm{C}$. Thereafter, temperature at both the ends are maintained at $0{ }^{\circ} \mathrm{C}$. Neglecting the heat transfer from the lateral surface of the rod, the heat transfer in the rod is governed by the one-dimensional diffusion equation $\frac{\partial T}{\partial t}=D \frac{\partial^2 T}{\partial x^2}$, where $D$ is the thermal diffusivity of the metal, given as $1.0 \mathrm{~cm}^2 / \mathrm{s}$.
The temperature distribution in the rod is obtained as
$$T(x, t)=\sum_{n=1,3,5 \ldots}^{\infty} C_n \sin \frac{n \pi x}{5} e^{-\beta n^2 t}$$
where $x$ is in cm measured from A to B with $x=0$ at $\mathrm{A}, t$ is in $\mathrm{s}, C_n$ are constants in ${ }^{\circ} \mathrm{C}, T$ is in ${ }^{\circ} \mathrm{C}$, and $\beta$ is in $\mathrm{s}^{-1}$.
The value of $\beta$ (in $\mathrm{s}^{-1}$, rounded off to three decimal places) is__________________.