Wissenschaftliche Zeitschift der Paedagogischen Hochschule Potsdam, 1966, Band 10 Nr. 3, S. 399-410
On the theoretical interpretation
of Schwarzschild's law of blackening -
with a recognition of the founder of Scientific Photography:
Karl Schwarzschild

by Ewald Gerth
Pedagogic College Potsdam, Physical Institute, Section Isotope-Techniques

Keywords: Schwarzschild effect formula equation, reciprocity failure relation, theoretical explanation derivation, photographic characteristic density curve, crystal lattice defects, inter-lattice free electrons and defectelectrons, reaction kinetics system process equilibrium, matrices and tensors

The lecture given at the Astrophysical Observatory Potsdam on occasion of the 50th anniversary of the death of the famous astrophysicist Karl Schwarzschild (1873-1916), who is regarded as the founder of the discipline Scientific Photography, started with a commemoration and the recognition of his outstanding scientific merits in different fields of theoretical as well as experimental physics.
The observation of stars using photographic plates required reliable reduction methods and therefore the investigation of the photographic blackening function. Schwarzschild realized by means of long-time exposures of stellar objects, that the efficiency of the exposure declines with exposure time; that means: the product of light intensity E and time t, the so-called reciprocity law Et = const, established by Bunsen and Roscoe generally for all photochemical reactions, has to be replaced in the case of photography with a law in the form Et p = const, where p is an exponent within the limits 0.7 < p < 1.

Schwarzschild's law is an analytical formulation of an empirically found result - without interpretation of the underlying physics. Similar attempts with a broader range of validity were made by Abney, Miethe, Michalke, Scheiner, Englisch, Kron, and others. The author shows that the physics of crystals, by accounting for the photoelectric effect and the creation of inter-lattice free electrons and defect-electrons interacting with silver ions, can give a reasonable interpretation and even an analytical derivation of Schwarzschild's and Kron's formulae.

The new concept is related to research on the photographic primary process by Gurney, Berg, Mott, and Mitchell and based on investigations of the photoelectric properties of model halide crystals by Pohl, which is completed with the process-like character of the emergence of development centers in the crystal lattice during the exposure to light.
The build-up process of development centers, arising from structural lattice defects in crystals of silver bromide embedded 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 thermal and chemical decay as well as the photoelectric effect acting directly onto the already created centers, consisting of conglomerated silver atoms. If the light 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 centers of the first degree are extremely unstable and distinguished by a high power of absorption for light of special wavelengths, such first-degree centers will easily be destroyed so that saturation occurs already in the first reaction step, resulting in the reduction of the order of the exposure time by one degree. After the decay of almost all primarily created centers, those remaining will gradually grow by successively adding silver ions and recharging with free electrons. Only such centers, which have accumulated at least four silver atoms, are capable of releasing the photographic development of the silver bromide grains.

Schwarzschild's well-known photographic blackening law

    E t p = const.        (p - Schwarzschild-exponent)

describes only the long-term exposure effect.
After the new concept, the validity of this law is extended to long and short exposure times with a transition region expressed by a formula

    [(1 + εE)½ − 1] t p = const.     (ε - sensitivity coefficient)

The new blackening formula is to be proved using the "historical" measurements made by Kron.

Comment − of the author in 2016:
The manuscript of the article was submitted to the journal Wissenschaftliche Zeitschrift der Pädagogischen Hochschule Potsdam [quoted: Wiss. Z. Paed. Hochsch. Potsdam, 10, 3 (1966), 399-410] 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 centers in silver halide grains of the photographic emulsion. The article is a special excerpt of the author's doctoral thesis on double exposure effects, which was defended at the Pädagogische Hochschule Potsdam on November 19th, 1965.
The 10 theses of the author's thesis are available in German.

Outlook − for subsequent results:
Since the photographic blackening density is a monotone function of the Schwarzschild product E t p, it can also be used for the analytical representation of the photographic characteristic curve, which leads to a practical computer-suited version of a four-parameter density formula.
Regarding the kinetic character of the photographic process, the Schwarzschild product E t p is represented as the exposure matrix, from which Schwarzschild's blackening law can be conclusively derived.
The matrix formulation of the Schwarzschild-product gives a reasonable explanation for the fact that double exposures of equal light quantity E.t yield different results if the sequence of short and of long-term partial exposures is exchanged. These double exposure effects are easily formulated by the non-commutative multiplication of matrices, which was defended at the Technical University Dresden in 1972.
A more comprehensive explanation of the non-commutativity of double exposures was given in an excerpt of the postdoctoral thesis by the author and in a monograph published in 2013:
Book: Analytical representation of the kinetics of speck build-up in the photographic process. (Fulltext in German) , ISBN 978-3-8316-4299-1, Herbert Utz Verlag GmbH, Munich, 2013
Further, we mention that a relation like Schwarzschild's blackening law as well as the non-commutativity of exposure sequences is valid not only for photography but it is - in matrix formulation - a general kinetic law of dynamic processes, as it is shown in the examples of radiative energy transfer, radioactive nuclide conversion chains, oscillation systems, or pharmacokinetic reactions, comprised in a monograph:
Book: Seven articles on reaction kinetics. (Fulltext in German) , ISBN 978-3-8316-4403-2, Herbert Utz Verlag GmbH, Munich, 2014

Generalization − of kinetic processes:

The treatment of kinetic processes with matrices enables a linear solution of the problem. Concerning the photographic process, however, already the reactions of electrons and defect-electrons are nonlinear. Including also the reactions of higher than linear order in the reactions system, the formulation by means of matrices does not suffice; then, a comprehensive analytical representation of the reaction system is achieved by means of tensors.
The analytic representation of the photographic process by reaction tensors is a first attempt to comprise the entire reaction system in a general way. In this case, photography is used as a prototype for dynamic processes with all manner of interconnecting relations and reaction orders. Matrices are defined as special tensors of second order. Nevertheless, matrix algebra can be used advantageously in most cases for its mathematical convenience, if the reaction systems are to be linearized approximately - as given for short time intervals and continued matrix multiplication.
(Computer programming of tensor reactions has been produced by the author.)

Original Publications − published in German:

Ewald Gerth:
Zur theoretischen Deutung des Schwarzschildschen Schwaerzungsgesetzes -
mit einer Wuerdigung des Begruenders der Wissenschaftlichen Photographie:
Karl Schwarzschild 1873-1916

Wiss. Z. PH Potsdam 10 (1966) 339-410

Abstract [pdf]:
On the theoretical interpretation of Schwarzschild's law of blackening -
with a recognition of the founder of Scientific Photography:
Karl Schwarzschild (1873 - 1916)

The derivations are taken mostly from:

Ewald Gerth:
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: 2016, May 11th
(The day of the 100th anniversary of Schwarzschild's death)