Wissenschaftliche Zeitschrift 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, intralattice free electrons and defect electrons, reaction kinetics system process equilibrium, matrices and tensors

Extended Abstract

Thelecturegiven at the Astrophysical Observatory Potsdam on occasion of the 50^{th}anniversary of the death of the famous astrophysicist Karl Schwarzschild (1873-1916), who is regarded as the founder of the disciplineScientific Photography, started with a commemoration and the recognition of his outstanding scientific merits in different fields of theoretical as well as experimental physics.

Theobservation of starsusingphotographic platesrequired reliable reduction methods and therefore the investigation of the photographic blackening function.Schwarzschildrealized 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 intensityEand timet, the so-calledreciprocity law E= const, established by^{.}tBunsenandRoscoegenerally for all photochemical reactions, has to be replaced in the case of photography with a law in the formE^{.}t= const, where^{ p}pis an exponent within the limits 0.7 <p< 1.

Schwarzschild's lawis an analytical formulation of an empirically found result - without interpretation of the underlying physics. Similar attempts with a broader range of validity were made byAbney, 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 intralattice 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.

Schwarzschild'sblackening lawis regarded as the most criticalhard touchstonefor the consistence of any theory of the photographic process with the physical reality.

The presenttheoretical conceptis related to research on the photographic primary process byGurney, Berg, Mott,andMitchelland based on investigations of the photoelectric properties of model halide crystals byPohl,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 case of high light intensity, however, the electron concentration is proportional to the square root of the intensity − leaving the transition between low and high intensity regions undefined.

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.

Thestep-like build-upof the reacting centers in the crystal lattice is akinetic process, which can be treated analytically by means of the known methods of reaction kinetics.

Thereaction-driving magnitudefor all build-up transitions is theelectron concentration, which is caused by physical energetic influences onto the halide crystal like the photoelectric effect, but likewise by x-ray and radioactive radiation, or even by pressure.

Asimple formulationofSchwarzschild's blackening lawon the above-outlined concept according to the intralattice micro-processes is obtained already on the very plausible grounds, that the probabilityPof the transition from one step to the next one is proportional to the electron concentrationNas well as to the reaction timet:

P~N t

For n successive transitions, the chain rule in probability theory is valid, yielding the n-fold product (N t)^{n}:

P ~ N^{n}t^{n}

On the assumption that the first step of the transition chain is very unstable because of back reactions as well as thermal and chemical decay, then equilibrium takes place reducing the order of time by one step:

P ~ N^{n}t^{n-1}

This is the fundamental shape ofSchwarzschild's formula for constant effect withP= const

N t^{(n-1)/n}= const.

For a reaction chain consisting of 4 links the exponent isp=¾= 0.75.

If the electron concentrationNis proportional to the light intensityEduring the exposure, then there followsSchwarzschild's well-known photographic blackening law

(E t= const,^{ p}p - Schwarzschild-exponent)

which describes only the long-term exposure effect.

According to the new theoretical concept, the validity of this law is extended to long and short exposure times by accounting for theelectron concentrationin the crystal lattice as the essential reacting medium. The amount rate of free electrons rising during the exposure to light is proportional to the light intensityE, expressed by the reaction velocity dN/dt=ηE, which is diminished by the capture of electrons in trapsαNand the recombination of electrons with defect electrons as a bimolecular reactionβN^{ 2}:

dN/dt=ηE−αN−βN^{2}

In the state of equilibrium the reaction velocity is dN/dt= 0. Then the root of the quadratic equation is the saturation electron concentration

N_{sat}= (α/2β)((1 + aE)^{½}− 1)

with the sensitivity coefficient

a = (4βη)/(α^{2}) .

The square root exponent½is the least value for bimolecular reactions. However, the recombination of electrons with defect electrons does not exclude higher-molecular reactions. Trimolecular reactionsγN^{ 3}occur, when two competing electrons are attracted and captured by one defect electron.

The mathematical difficulties of an analytical solution of the cubic equation

ηE−αN−βN^{2}−γN^{3}= 0

forces to simplify the analytic expression of the root. Neglecting small terms gives an adequate approximation to the form of the square root with a root exponentb=^{2}/3.

The abundance of free electrons in the crystal lattice increases in the course of the exposition to light with the quantity ofH = Et. The investigation of the measuring data ofKron's catenaria-likereciprocity curves(article, pages 17-19, Fig. 7) shows that also the form and the photographic density are functions ofH = Et.

Summarizing and approximating all influences onto the recombination process, the root exponent½has to be varied. UsingKron's data (article, page 20, table, Fig. 8), we find for this exponent an increase from½to¾.

Adapting to the square root solution, a formula is set up for the reciprocity behavior of the photographic material with a root exponent 0.5<b< 1, which describes the transition region of the exposure from low to high intensities:

((1 + a(a - sensitivity coefficient,E)^{b}− 1)t= const.^{p}b- recombination exponent)

The new blackening formula enlargesSchwarzschild's blackening law extending it to a wide region of exposure times and replacesKron's"catenaria-like reciprocity relations". The formula is proved using the"historical"measurements made byKron.

Comment− of the author in 2016:

The manuscript of the article was submitted to the journalWissenschaftliche 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 theSchwarzschild-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 thePädagogische Hochschule Potsdamon 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, 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.Et^{ p}

Regarding the kinetic character of the photographic process, the Schwarzschild productis represented as theEt^{ p}, from which Schwarzschild's blackening law can be conclusively derived.exposure matrix

The matrix formulation of the Schwarzschild-product gives a reasonable explanation for the fact that double exposures of equal light quantityyield different results if the sequence of short and of long-term partial exposures is exchanged. TheseE^{ }tare easily formulated by thedouble exposure effects, which wasnon-commutative multiplication of matricesdefended 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

Generalization− of kinetic processes:

Schwarzschild's photographic blackening law as well as the non-commutativity of exposure sequences is a, as it is shown in the examples of radiative energy transfer, radioactive nuclide conversion chains, oscillation systems, or pharmacokinetic reactions, comprised in a monograph:general kinetic law of dynamic processes

Book:Seven articles on reaction kinetics (Fulltext in German), ISBN 978-3-8316-4403-2, Herbert Utz Verlag GmbH, Munich, 2014

Reaction kinetics− using tensor algebra:

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 derWissenschaftlichen 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 11^{th}

(The day of the 100^{th}anniversary of Schwarzschild's death)