Path of calculation: dye aggregate is thermally lifted to lowest excited singlet state. Equilibrium population of this state is calculable; and an upper limit can be set on lifetime from calculable monomer fluorescence lifetime and successful quenching of fluorescence by the competing dye sensitization process. Generation of photoelectrons from excited singlet state is assumed 100% efficient. |

Grain: 1-μ cube: area = 6 × 10^{−8} cm^{2}/grain; |

Dye: monomer λ_{max} = 750 nm, ∊ = 250,000, Absorption max |

$$\mathit{\int}\u220a\frac{d\mathrm{\lambda}}{\mathrm{\lambda}}=27,200\phantom{\rule{0.2em}{0ex}}{(\text{moles}/\text{liter})}^{-1}\phantom{\rule{0.2em}{0ex}}\text{c}{\text{m}}^{-1},\text{Integrated absorption}$$ |

τ_{monomer} = 2.38 × 10^{−9} sec at 750 nm = fluorescence lifetime. |

Area occupied = 7 × 4.08_{3} Å × 3.535A = 101.0 A^{2} per molecule. |

$$\begin{array}{c}\text{Dye coverage}=\frac{6\times {10}^{-8}\phantom{\rule{0.3em}{0ex}}{\text{cm}}^{2}/\text{grain}}{1.01\times {10}^{-14}\phantom{\rule{0.3em}{0ex}}{\text{cm}}^{2}/\text{molecule}}\\ =5.94\times {10}^{6}\phantom{\rule{0.3em}{0ex}}\text{molecules per grain};\end{array}$$ |

Assume average aggregate size N = 10; |

Aggregate coverage = 5.94 × 10^{5} decamers/grain; |

Aggregate excited singlet E = 1.425 eV (8700 Å); |

Aggregate fluorescence lifetime τ_{10} = τ_{monomer}/N = 3.7 × 10^{−10} sec. This figure allows for spectral shift from monomer, and for accumulation of transition intensity in the lowest excited singlet of the decamer. |

Aggregate excited state lifetime τ on the grain | ≤ 0.01 τ_{10} |

| ≤ 3.7 × 10^{−12} sec, |

since photosensitization competes successfully with fluorescence. |

Boltzmann population factor: e^{−}(^{hv}/kT) = 2.03 × 10^{−24} at 30°C, 5.95 × 10^{−23} at 50°C, 1.18 × 10^{−21} at 70°C. But we have 5.94 ×10^{5} aggregates per grain, and these have upper-state lifetimes of ≤ 3.7 × 10^{−12} sec. |

The rate of generation of excited states |

dn_{1}/dt | = (degeneracy) × (Boltzmann population)/(life-time), |

| ≥ 3.26 × 10^{−7} excitations per grain per sec at 30°C, |

| ≥ 9.55 × 10^{−6} excitations per grain per sec at 50 °C, |

| ≥ 1.88 × 10^{−4} excitations per grain per sec at 70°C. |

But we are considering the potential accumulation of long term damage over times of the order of 1 yr (= 3.1536 × 10^{7} sec). |

dn_{1}/dt | ≥ 10.3 excitations per grain per year at 30°C, |

| ≥ 301. excitations per grain per year at 50°C, |

| ≥ 5934 excitations per grain per year at 70 °C. |

Clearly, the grain must have a mechanism for rejecting single events if thermal fog is to be avoided. This calculation has treated only one path (the path most easily calculable) to the generation of conduction electrons. These have ∼1 sec lifetime. |

The number of single excitations does not directly indicate the probability of fog via thermal exposure. However, we can estimate the probability of simultaneous existence of two thermal excitations in a grain, and consider that these will have a high probability of combining to form a stable, binary center. This center will then proceed to integrate subsequent thermal electrons and grow to developable size. At this point it is a fog center only by reason of mechanistic origin, rather than by any physical distinction from an image center. |

The lifetime of a cycling electron is roughly 1 sec in a chemically sensitized grain at room temperature. If the one year span is divided into 3 × 10^{7} lifetimes, then some number N of excitations will lead to a probability approaching 50% for simultaneous presence of two excitations. Since, statistically, every thermal electron has an opportunity a priori to coincide with any other, this probability rises as N^{2}, and the 50% probability is reached at
$N\sim {({3.10}^{7})}^{\frac{1}{2}}\sim 5.5\times {10}^{3}$ excitations. Thus at 70°C and with a chemically-sensitized lifetime of 1 sec, the aerographic emulsion considered here will fog excessively in one year. [This is like the “birthday problem”: it is advantageous to bet that two people in a room will have the same birthday when the number in the room is N > 20. Here we have 365 intervals (days) and
$20>{(365)}^{\frac{1}{2}}$ events (people), leading to a favorable probability for coincidence.] |

The limit on thermal sensitivity of non-dye-sensitized AgBr · I films is almost as severe. At I^{−} > 6 mole %, the absorption limit lies at about 500 nm. For direct thermal dissociation of a carrier, the Boltzman factor gives n_{e}^{+} = n_{h}− = e^{−}(ΔE/2RT) = 2.45 × 10^{−21} at 30°C. We do not know the density of states (degeneracy) to apply to this calculation on the mixed Br·I system, but the system is clearly close to fog under ordinary conditions. |