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 astrophysicistKarl Schwarzschild (1873-1916)started with a commemoration and the recognition of his outstanding scientific merits in different fields of theoretical as well as experimental physics.

With respect to his important investigation of photographic procedures in scientific research, Karl Schwarzschild is esteemed as the founder of the disciplineScientific Photography.

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^{.}t= const, established byBunsenandRoscoegenerally for all photochemical reactions producing a constant effect, has to be replaced in the case of photography with areciprocity failure lawin 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.

Thebuild-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 backward 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 region between low and high intensity undefined.

On the assumption thatcenters of the first degree are extremely unstableand 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, reducing 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 and their reduction to metallic silver.Statements and Conclusions:

- The
of the reacting centers in the crystal lattice is astep-like build-up, which can be treated analytically by means of the known methods ofkinetic processreaction kinetics.

- The
for all build-up transitions is thereaction-driving magnitude, which is caused byelectron concentrationphysical energetic influencesonto the halide crystal like the photoelectric effect, but likewise by x-ray and radioactive radiation, or even by heat and pressure.

- The
(photographic blackening density DE,t) is afunctionof the reaction-kinetic resultof a dynamic process relating to the exposure magnitudes as the independent variablesE^{.}t^{p}intensityandEtimet.- The
is a phenomenon exclusively of the physical process taking place during the exposure in the interior of the silver halide crystal - being not affected by the subsequent developing process.Schwarzschild-effectapplies only to the exposure of silver-based photomaterial to light. In case of X-ray or radioactive radiation the reciprocity law is valid. This is because of the impact of high-energetic quanta to the silver halide grains, whereby the crystal lattice is flooded with free electrons. Then all steps in the reaction chain will be passed through at once, so that no equilibrium states among the steps can occur. A grain hit by a quantum will be reduced to metallic silver as a whole.Schwarzschild's blackening formulais not confined solely to photography; it is, moreover, a prototype for a general physico-chemical law with forward and backward reactions, which come partly in equilibrium states by reducing the order and prolonging the duration of the reaction time.Schwarzschild's blackening law- A
applies also to interrupted, intermittent, double, and multible exposures, the particular ones of which need not be continuous in time but may even shrink to “transformation events”. In further generalization thegeneralized “Schwarzschild law”Schwarzschild lawdescribes various processes in nature, society, techniques, economy, and business etc., where continuousprocessesand/or discretetransformation eventsare arranged arbitrarily in acausal consecutionwith oppositely competing forward and backward transitions.

An Elementary Derivation of the Photographic Blackening Law:

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 decay and backward reactions, 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.

The magnitude n is the reaction order (called: “speck-order number”) of the inter-step transitions, the average value of which can be determined by analysing the toe of the characteristic curve. For a reaction chain consisting of 4 links the reaction order is n = 4 yielding an exponentp=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}- Schwarzschild-exponentp

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 electronsNrising 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(pages 17-19, Fig. 7) shows that also the form and the photographic densityD(E,t) are functions ofH(E,t).

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

Adapting to the square root solution, a formula (page 17, equ. (27c)) is set up for thereciprocity failurebehavior 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 + aE)^{b}− 1)t= const.^{p}

a- sensitivity coefficient

- recombination exponentb

- Schwarzschild-exponentp

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.

* * * * *

Remarks, Accomplishments, Comments− of the author in 2019:

Outlook− for subsequent results:

Since theis a function of thephotographic blackening densityDas the "exposure light quantitySchwarzschild product"

H(E,t) =φ(E)^{.}ψ(t) =E,t^{.}^{p}

it can also be used as an independent variable for a physically founded analytical representation of the characteristic curve, which leads to a practical computer-suited version of a simplified density formula (page 169, equ. 24):

D(H) = (D_{sat}/δ) (ln(1+(εH)^{n}) − ln(1+(εH)^{m}exp(−δ))

The formula contains 5 parameters:

1.D_{sat}- saturation density

2.δ- opacity density

3.n- step reaction order (toe)

4.m- step reaction order (shoulder)

5.ε- sensitivity

This is thewith toe, quasi-linear middle part and schoulder. The streching and linearisation of the curve results because of the combination of the two opposite terms in the formula.elementary formulation of the photographic characteristic function

The derivation of the formula withm = nyields a blackening function (page 171, Fig. 3) with torsionally symmetric curvatures for the regions of the toe and the shoulder. Real blackening curves, however, show asymmetrical curvatures with a flattened shoulder and a decrease of the density after going through a maximum.

A good fitting is achieved with a “toe exponent” nand a “shoulder exponent” m.A negative exponentm< 0 indicates the exposure region of the solarisation effect.

The Schwarzschild-effect relates only to the variableH(E,t) with a functional combination ofEandt. Extending the range of validity from long to short exposures, instead of the Schwarzschild productH(E,t) =Etthe term^{p}εH(E,t) = ((1+εE)^{b}− 1)tis to be inserted.^{p}

For ultrashort time and high intensity exposures the reciprocity law is confirmed (pages 2-5, equ. 2, equ. 11) because of the increasing electron concentration during the exposure and its decrease afterwards (pages 16-22, Fig. 1, Fig. 2), thus covering the whole diapason from long to short time and high to low intensity exposures:

εH(E,t) = (εEt)exp(−αt) + ((1+εE)^{b}− 1)t(1 − exp(−^{p}αt))

The(exposure light quantityHE,t) is compatible with theexposure matrixB(E,t).

Photographic exposure effects− expressed by matrices:

Regarding the kinetic character of the photographic process, the Schwarzschild productEt^{p}is represented as the, 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 quantityEyield different results if the sequence of short and of long-term partial exposures is exchanged. These^{ }tare easily formulated by thedouble exposure effects, which wasnon-commutative multiplication of matricesat thedefendedTechnical University Dresdenon April 26^{th}, 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, 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.)

Documentation− of scientific priority:

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 19^{th}, 1965. The 10 theses of the author's thesis are available in German.

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: 2019, March 25^{th}