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Let's say I have a Silica nano-particle trapped with an optical tweezer in high vacuum (pressure ~ $10^{-6} \mathrm{mbar}$). I know the molar mass of the background gas and its temperature.

How would you calculate the scattering cross-section with the background gas? Or equally, the number of gas collisions per second?

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I would say the cross-section is just

$\sigma = 4 \pi R^2$ with $R$ being the radius of the sphere. The scattering rate would then be

$\gamma = \sigma n \bar{v}$ with n the particle density and $\bar{v}$ the mean velocity of the gas particles.

Correct?

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