Photochemistry Core (Quantum Scale)

When photons strike a silver halide crystal (AgX), a photolytic reaction is triggered:

1.1 Photoelectron Excitation

$$ \text{AgX} + h\nu \xrightarrow{k_{\text{photo}}} \text{Ag}^+ + \text{e}^- + \text{X}^0 $$

Where:

$h\nu$ is the photon energy ($\nu$ is frequency)
$k_{\text{photo}}$ is the photochemical reaction rate constant, following the quantum efficiency model: $$ k_{\text{photo}} = \eta \cdot \sigma_{\text{abs}} \cdot \Phi $$
- $\eta$: Quantum efficiency (≈0.6-0.9)
- $\sigma_{\text{abs}}$: Crystal absorption cross-section
- $\Phi$: Photon flux (photons/(cm²·s))

1.2 Latent Image Center Formation

Released electrons are trapped by silver ions, forming silver atom clusters:

$$ \text{Ag}^+ + \text{e}^- \xrightarrow{k_{\text{trap}}} \text{Ag}^0 $$

Critical Condition: A stable latent image center is formed when a single crystal surface accumulates 5-10 silver atoms:

$$ n_{\text{Ag}} \geq n_{\text{critical}} \approx 5-10 $$

This process can be modeled using a Poisson distribution:

$$ P(n_{\text{Ag}} \geq n_{\text{crit}}) = 1 - \sum_{k=0}^{n_{\text{crit}}-1} \frac{\lambda^k e^{-\lambda}}{k!} $$

Where $\lambda = \eta \cdot N_{\text{photons}}$ is the expected number of electrons.

Exposure and Density Mapping (Macro Scale)

2.1 Exposure Definition

$$ E = I \cdot t $$

$I$: Irradiance (lux)
$t$: Exposure time (s)
$E$: Exposure (lux·s)

2.2 Optical Density

The ability of silver grains to block light after development is quantified by Optical Density $D$:

$$ D = \log_{10}\left(\frac{I_0}{I_t}\right) = -\log_{10}(T) $$

$I_0$: Incident light intensity
$I_t$: Transmitted light intensity
$T = I_t/I_0$: Transmittance

Characteristic Curve (Hurter-Driffield Curve)

This is the core mathematical model of film imaging, describing the S-shaped relationship between $D$ and $\log_{10}E$:

3.1 Piecewise Linear Model (Engineering Simplification)

$$ D(\log E) = \begin{cases} D_{\text{min}} + \frac{\log E - \log E_{\text{toe}}}{\gamma_{\text{toe}}} & \text{Toe (Underexposed)} \ D_{\text{min}} + \gamma \cdot (\log E - \log E_0) & \text{Linear Region (Normal Exposure)} \ D_{\text{max}} - \frac{\log E_{\text{shoulder}} - \log E}{\gamma_{\text{shoulder}}} & \text{Shoulder (Overexposed)} \end{cases} $$

Key Parameters:

$\gamma$: Contrast coefficient (slope of the linear region) $$ \gamma = \frac{\Delta D}{\Delta \log E} = \frac{D_2 - D_1}{\log E_2 - \log E_1} $$
Dynamic Range: $\text{DR} = \log_{10}(E_{\text{max}}/E_{\text{min}}) \approx 2.0$ (corresponding to 100:1 brightness ratio)