## Probability of print out as a function of grain radius

Silverhalide Emulsions / Chemistry.
Din
Posts: 262
Joined: Thu Mar 12, 2015 4:47 pm

### Probability of print out as a function of grain radius

I recently had an occasion to wonder what the probability of print out was if a silver halide emulsion were to be bombarded by photons of some frequency, ν. This can be radiation of any frequency, including visible light or uv. The calculation assumes temporal coherence, but it can be adapted to partial coherence by the FT of the wave packet.

Anyway, I begin by assuming that photons are bombarding grains composed of silver bromide. The number of 'hits' is then the number of events where a photon hits a silver bromide molecule, causing an actinic reaction, which will reduce the bromide to elemental silver. This probability depends on the ratio of the area of the silver bromide molecules to the area being bombarded. To keep it simple, I assume unit area of emulsion.

So, the number of actinic events by photons impinging on the grain, N('hits') is :

N('hits') = N(inc)*n(tar)*σ
where:
N(inc) is the number of photons/unit area hitting the grains
n(tar) is the number of molecules of silver bromide per unit area of emulsion
σ is the cross sectional area of the silver bromide molecules in the grains .

The number of silver bromide molecules is the grain density ρ multiplied by the number of molecules/grain. So,
n(tar) σ= ρ(r²(grain)/r²(bromide))*πr²(bromide)
= πρ'r²(grain)

Here, the number of bromide molecules, targets, in the grain is the grain area (πr²(grain)) divided by the molecule's area (πr²(bromide), and therefore, the number of targets multiplied by the area of each target can be derived. It's a little crude because I haven't taken in a fill factor. Following Avagadro, we may take a fill factor into consideration by multiplying by 2/3

We can get the photon flux by the intensity of the radiation, I, since
I = N(photons)*h*ν

So, finally, the number of actinic events N('hits') is:

N('hits') = (I/(h*ν))*(πρ'r²(grain))

As you can see, the probability of print out depends on the grain density and the square of the grain radius. Therefore the smaller the grain radius, the fewer the actinic events, but, the larger the grain density and/or the beam intensity, the greater is the number of actinic events. However, because the probability is proportional to the square of the grain radius, this factor will dominate.