In single-crystal x-ray crystallography both isotropic and anisotropic displacement parameters $U_{ij}$ of thermal ellipsoids have dimension of square angstrom ($Å^2$) as follows from the definition of Debye–Waller factor ($T$ is dimensionless):
For isotropic approximation (conditions of Bragg's law): $$T_\text{iso} = 8\pi^2U\left(\frac{\sin \Theta}{\lambda}\right)^2 \to [U] = Å^2 \tag{1}$$
For anisotropic displacement parameters (ADPs): $$T_\text{ani} = 2\pi^2 \sum_{i=1\\j=1}^3{H[i]H[j]U_{ij}e^\star[i]e^\star[j]} \to [U] = Å^2 \tag{2}$$ where $H$ is reflex's index; $e^\star$ is length of the reciprocal lattice basis vector.
In both cases $U_1$, $U_2$ and $U_3$ ($U_{ij}$ in general) are basically showing contribution of oscillation along three principal (orthogonal) axes and are measured in $Å^2$. So, is there any physical meaning that can be attributed to square angstrom in this case? At first glance, it seems rather confusing that a parameter that seems to represent a linear quantity is measured in square units of length.
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