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

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) 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 discipline Scientific Photography.

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 E.t = const, established by Bunsen and Roscoe generally for all photochemical reactions producing a constant effect, has to be replaced in the case of photography with a reciprocity failure law in the form E.t 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 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's blackening law is regarded as the most critical hard touchstone for the consistence of any theory of the photographic process with the physical reality.

The present theoretical 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 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 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, 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 step-like build-up of the reacting centers in the crystal lattice is a kinetic process, which can be treated analytically by means of the known methods of reaction kinetics.
  •   The reaction-driving magnitude for all build-up transitions is the electron 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 heat and pressure.
  •   The photographic blackening density D(E,t) is a function of the reaction-kinetic result E.tp of a dynamic process relating to the exposure magnitudes as the independent variables intensity E and time t.
  •   The Schwarzschild-effect 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's blackening formula applies 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 law is 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.
  •   A generalized “Schwarzschild law” 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 the Schwarzschild law describes various processes in nature, society, techniques, economy, and business etc., where continuous processes and/or discrete transformation events are arranged arbitrarily in a causal consecution with oppositely competing forward and backward transitions.

An Elementary Derivation of the Photographic Blackening Law:

A simple formulation of Schwarzschild's blackening law on the above-outlined concept according to the intralattice micro-processes is obtained already on the very plausible grounds, that the probability P of the transition from one step to the next one is proportional to the electron concentration N as well as to the reaction time t :

    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 nt 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 nt n-1

This is the fundamental shape of Schwarzschild's formula for constant effect with P = 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 exponent p =  ¾ = 0.75.
If the electron concentration N is proportional to the light intensity E during the exposure, then there follows Schwarzschild's well-known photographic blackening law

    E t p = const,           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 the electron concentration in the crystal lattice as the essential reacting medium. The amount rate of free electrons N rising during the exposure to light is proportional to the light intensity E, expressed by the reaction velocity dN/dt = ηE, which is diminished by the capture of electrons in traps αN and the recombination of electrons with defect electrons as a bimolecular reaction βN 2 :

    dN/dt = ηEαNβN 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

    Nsat = (α /2β)((1 + aE)½ − 1)

with the sensitivity coefficient

    a = (4 βη)/(α2)  .

The square root exponent ½ is the least value for bimolecular reactions. However, the recombination of electrons with defect electrons does not exclude higher-molecular reactions. Trimolecular reactions γN 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

    ηEαNβN 2γN 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 exponent b = 2/3.

The abundance of free electrons in the crystal lattice increases in the course of the exposition to light with the quantity of H = Et. The investigation of the measuring data of Kron's catenaria-like reciprocity curves (pages 17-19, Fig. 7) shows that also the form and the photographic density D(E,t) are functions of H(E,t).
Summarizing and approximating all influences onto the recombination process, the root exponent  ½  has to be altered. Using Kron's data (page 20, table, Fig. 8), we find for this exponent an increase from  ½  to  ¾.

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

a - sensitivity coefficient
b - recombination exponent
p - Schwarzschild-exponent

The new blackening formula enlarges Schwarzschild's blackening law extending it to a wide region of exposure times and replaces Kron's “catenaria-like reciprocity relations”. The formula is proved using the “historical” measurements made by Kron.

    *         *         *         *         *

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

Outlook − for subsequent results:
Since the photographic blackening density D is a function of the exposure light quantity as the "Schwarzschild product"
    H(E,t) = φ(E).ψ(t) = E.tp,
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) = (Dsat/δ) (ln(1+(εH)n) ln(1+(εH)nexp(−δ))
The formula contains 4 parameters:
    1. Dsat - saturation density
    2. δ - opacity density
    3. ε - sensitivity
    4. n - step reaction order (toe)
This is the elementary formulation of the photographic characteristic function with toe, quasi-linear middle part, and schoulder. The streching and linearisation of the curve results from the combination of the two contrariwise to each other by −δ shifted terms in the formula.
The derivation of the formula yields a blackening function (page 171, Fig. 3) with torsionally symmetric curvatures for the regions of the toe and the shoulder. Real blackening curves, however, have asymmetrical curvatures with a flattened shoulder and a decrease of the density after going through a maximum, caused by the continued reaction up to higher but no more developable steps in the reaction chain.
The last reaction affects especially the shoulder of the blackening curve and is determined by a fifth reaction order parameter
    5. m - step reaction order (shoulder),
describing the solarisation effect, which can be added to the elementary formula as a factor with a descending exponential function
    exp(-m εH).
Anyhow, sufficient fitting is achieved by terms like the above formula accounting for the size and form distribution of the silver halide grains in the emulsion and the radiative transfer through the photographic layer during the exposure.
The Schwarzschild-effect relates only to the variable H(E,t) with a functional combination of E and t. Extending the range of validity from long to short exposures, instead of the Schwarzschild product
    H(E,t) = Etp
the term
    εH(E,t) = ((1+εE)b − 1)tp
is to be inserted.
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)tp(1 − exp(−αt))
The exposure light quantity H(E,t) is compatible with the exposure matrix B(E,t).

Photographic exposure effects − expressed by matrices:
Regarding the kinetic character of the photographic process, the Schwarzschild product Etp 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 on April 26th, 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 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

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 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.

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: 2019, May 20th