Zeitschrift für wissenschaftliche Photographie, Photophysik und Photochemie, Band 59, (1965), Heft l-4, Seite 1-19
Analytical representation
of the photographic characteristic curve
accounting for the Schwarzschild-effect

Ewald Gerth
Pedagogic College Potsdam, Physical Institute, Section Isotope-Techniques
Received: August 31st, 1964

The build-up process of development specks at crystals of silver bromide in a photographic emulsion is regarded as a chain of equilibrium reactions, which are characterized in that the forward reactions are determined by the concentration of free electrons in the crystal lattice, whereas the back reactions take place due to the photoelectric effect acting directly onto the already created specks. If the intensity is low, the saturation concentration of electrons is proportional to the light intensity of the exposure. In the case of high light intensity, however, the electron concentration is proportional to the square root of the intensity.
    On the assumption that specks of the first degree are very unstable because of thermal and chemical decay and distinguished by a high power of absorption for light of special wavelengths, saturation occurs already in the first reaction step, resulting in the reduction of the order of the exposure time by one degree. The specks grow from step to step by recharging with free electrons until having accumulated at least four silver atoms, which are necessary to release the photographic development of the whole silver bromide grains.

    The reciprocity law of Bunsen and Roscoe
    E t = const          (E - intensity, t - time)
is replaced with Schwarzschild 's well-known law of blackening
    E tp = const,        (p - Schwarzschild-exponent)
which describes the long-term exposure effect. The validity of the blackening law is extended to long and short exposure times with a transition region expressed by the formula
    [(1 + εE)½ − 1] tp = const.     (ε - sensitivity coefficient)

  The analytical representation of the characteristic curve is based on the Poisson probability of the silver bromide crystals reduced by the developer to metallic silver with the average number of cell occupation as a function of the Schwarzschild-product E tp (for low intensity E and long time t only) or for the entire diapason of low and high intensity together (long and short times, resp.)
    The blackening formula (equation 32 in the original article) holds very well in the range from the toe up to the quasi-linear part of the characteristic curve but shows increasing deviation at the shoulder the more saturation of the density is reached. This is because the grain size distribution is not accounted for, which was introduced in later derivations of the characteristic curve:

Comment of the author in 2008:
The manuscript of the article was submitted to the journal Zeitschrift für wissenschaftliche Photographie, Photophysik und Photochemie (quoted: Z. wiss. Phot. 59 (1965), 1) on August 8th, 1964 in order to secure the priority for the explanation of the Schwarzschild-effect as an outcome of the kinetic process of the step-like build-up of development specks at the silver halide grains in the photographic emulsion. The article is an excerpt published in advance of the author's thesis on double exposure effects, which was defended at the Pädagogische Hochschule Potsdam on November 19th, 1965.
The text of the lecture is available in German by:  
The theses of the thesis are:  

Published in German:
P 16. Gerth E.:
Analytische Darstellung der Schwärzungskurve
unter Berücksichtigung des Schwarzschild-Effektes

Z. wiss. Phot. 59 (1965) 1-19
Abstract [pdf]:
Analytic representation of the photographic characteristic blackening curve
accounting for the Schwarzschild-effect


Last update: April 10th, 2013