Random deposition: a mirror is plated by evaporating a gold...

Random deposition: a mirror is plated by evaporating a gold electrode in vaccum by passing an electric current. The gold atoms fly off in all directions, and a portion of them sticks to the glass (or to other gold atoms already on the glass plate). Assume that each column of deposited atoms is independent of neighboring columns, and that the average deposition rate is $d$ layers per second. (a) What is the probability of $m$ atoms deposited at a site after a time $t$ ? What fraction of the glass is not covered by any gold atoms? (b) What is the variance in the thickness?

Random deposition: a mirror is plated by evaporating a gold electrode in vaccum by passing an electric current. The gold atoms fly off in all directions, and a portion of them sticks to the glass (or to other gold atoms already on the glass plate). Assume that each column of deposited atoms is independent of neighboring columns, and that the average deposition rate is $d$ layers per second.
(a) What is the probability of $m$ atoms deposited at a site after a time $t$ ? What fraction of the glass is not covered by any gold atoms?
(b) What is the variance in the thickness?

Key Concepts

Variance of the Poisson Distribution

A key property of the Poisson distribution is that its variance is equal to its mean. In the context of random deposition, the variance in the number of atoms (or the thickness) at a site after time t is equal to d*t, indicating that as the average deposition increases, the spread or fluctuation in deposition thickness increases in equal measure.

Probability of Zero-Events (Exponential Decay)

The probability that no atoms are deposited at a site in time t, corresponding to the fraction of the glass remaining uncovered, is given by the zero-event probability of the Poisson distribution. This probability is exp(-d*t), emphasizing that the uncovered fraction decays exponentially as the deposition time increases.

Poisson Process

The Poisson process models the random, independent events occurring at a constant average rate over time. In the context of deposition, it is assumed that each site receives atoms independently and randomly with a mean rate, making the Poisson process a natural model for describing how cells accumulate deposited atoms.

Poisson Distribution

The Poisson distribution gives the probability of a certain number of events occurring within a fixed interval of time when events happen independently at a constant average rate. Here, it is used to calculate the probability that exactly m atoms are deposited at one site over a time interval t, using the formula P(m) = (d*t)^m * exp(-d*t) / m!, where d is the deposition rate.

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