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, non-commutative double exposures, X-ray and radioactive radiation, transformation of matrices and tensors, generalization Schwarzschild law

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 (described by Mees, pages 101-153) 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 in an endoenergetic reaction.
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 external physical 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 by endoenergetic reactions. 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.

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.
The statistics bundles and averages a multiplicity of parallel consecutive reaction chains, making the even transition numbers of the reaction orders odd.

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 exposure 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 shoulder. The stretching and linearization 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 solarization 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, and its composition of differently riped and mixed emulsions. For computing the formula with its algorithm is to be used as a standard function.
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).

Avoidance − of the Schwarzschild-effect:
Since its detection by Abney (1893), the reciprocity failure of silver-bound photographic materials is a very unwanted phenomenon, for it diminishes the sensitivity and makes any forecast of exposures uncertain. Therefore, early endeavors were made to overcome this phenomenon.
Schwarzschild (1900, first page, below) relates in his announcement on the deviations from the law of reciprocity to A. Schellen (Thesis 1898, Rostock, Germany), who found − contradictory to Schwarzschild's own results − “the law of reciprocity exactly confirmed for the same Schleussner plates”, although there was made a preliminary exposure.

The explanation is given easily by the theory: The first steps in the reaction chain are filled during the preliminary exposure (page 129, Fig. 9), so that only one step has to be brought on to make the specks developable. Then, the effective reaction order is n = 1 and the Schwarzschild-exponent is p = 1.

Methods to get reciprocity and higher sensitivity of the photoplates:

  • Pre-exposure: A preliminary uniform exposure of the photolayer to light promotes sensitivity and reciprocity for the following main exposure.
    1. Normal exposures with light intensities comparable with that of the main exposure have only little effect on the sensitivity but increase the fog.
    2. Short exposures with high light intensity like flashes produce abundantly free electrons in the crystal lattice, thus promoting forward reactions and suppressing backward reactions, thereby impeding the emergence of a reaction equilibrium. The second exposure starts with a vast amount of sub-image centers ready to become development centers by only one step.
    3. Spectral exposure. In the spectral region of visible light, no difference of any color effect is observed. This is because the colored light produces only different quantities of free electrons, which react similarly in the reaction chain. However, in direction to shorter wavelengths, the ultraviolet light produces increasingly more free electrons. Going still further to electro-magnetic radiation up to weak X-rays, then reciprocity for the following second exposure is reached.
    4. X-ray exposure. A pre-exposure with X-rays makes the post-exposure − as the main exposure − completely free of the reciprocity failure. The exposure of photographic material to radioactive radiation (α-, β-, γ-rays) obeys strictly to the reciprocity law.
  • Hypersensitization, which consists in an uniform preliminary physical and/or chemical treatment of the photographic layer. The information is stored in the the layer by the second exposure.
    The main sensitization methods are:
    1. Ripening: In the ripening process of the emulsion most of the grains grow up to the reaction step before the development capability is reached.
    2. Physical sensitization: The photolayer is exposed to heat (baking method) and/or pressure in order to let the specks grow to the first undercritical order. The growth of the specks will be stopped by low temperature. Then, the plates are stored in a refrigerator until/during the exposure.
    3. Chemical sensitization: During/after the ripening process of the emulsion or by bathing of the photoplates, chemical substances (compounds with mercury, gold, sulphur, selenium, tellurium etc.) will be admixed, which adhere to the undercritical steps making them sensitive.
    A further method of increasing the sensitivity of the photolayer is:
  • Latensification: The effectiveness of an exposure can be increased by a post-exposure with uniform light and/or physical as well as chemical treatments until or during the following development.
    The first exposure is the main informative one. The posterior exposure/treatment brings undercritical steps on the developable level but increases the fog. Because of the build-up reactions during the first exposure, reciprocity is not reached.
  • Other photography systems (like the digital photography), which do not develop an equilibrium between neighboring reaction steps, are free of the reciprocity failure and do not show the Schwarzschild-effect.

    All these procedures and methods represent double exposures with light or other energetic influences and are on principle non-commutative.

    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

    Other kinetic processes − using matrix algebra:
    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 in photography − 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 expansion of the matrix-based analytical description of the photographic micro-process to tensors stands for more precise consideration of different influences but does not mean another quality. The exposure-matrix goes over to an exposure-tensor and the multiplication of matrices will be carried out by tensors − always non-commutatively.
    The analytic representation of the photographic process by reaction tensors 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.

    Schwarzschild's photographic blackening law
    is a prototype for causally consecutive processes
    •   Schwarzschild's blackening law is not confined solely to photography; it is, moreover, a general physicochemical law with forward and backward reactions, which come partly in equilibrium states by reducing the order and prolonging or shortening the duration of the reaction time.
    •   In the course of the reaction process the compartments of the system will be created and redistributed accumulating the results.
    •   The general Schwarzschild law does not require absolutely any reaction chain; it holds, moreover, also for networks with branching and separation by inner and outer regions of conjuncted compounds with differently acting influences.
    •   The transition of the compound system from one state to a following one is performed − schematically, analytically, and numerically − by tensor transformation.
    •   The exchange of process periods with different transition parameters is generally non-commutative.
    •   A “generalized Schwarzschild law” applies also to interrupted, intermittent, double, and multiple processes, the particular ones of which need not be continuous in time but may even shrink to “transformation events”.
    •   Tensor-determined processes exist functionally in reality with all physical, natural, and logical consequences independently of the possibility to calculate them mathematically.
    •   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.
    •   The mathematical treatment of of reaction systems by means of tensor algebra becomes more and more complicated the higher the order and the more numerous the compartments are.
          A good way out of this dilemma is resorting to probability calculation, because every transition from a certain condition to another one corresponds statistically in a group of compartments to a probability factor.
    •   Usually, forward and backward reactions take place simultaneously. A single transition from one stage to the next one fulfills a reciprocity law as the probability product of the generalized driving power P multiplied by the reaction time t
          P . t = const .
      If for a sequence of consecutive transitions the balance of build-up and build-down is biased, then the product power .time yielding a constant work-effect deviates from reciprocity.
      According to the multiplication rule of consecutive probabilities and with the transition numbers for both magnitudes m and n,
      yielding a reciprocity criterion by
          m/n <=> 1,
      there results the characteristic product of exponentials:
          Pm . tn = const
      This is the General Schwarzschild-Law !
    •   Conclusion:
      Schwarzschild's famous blackening law
      found and explored at photographic plates
      is valid for all kinds of causally consecutive processes!

    Documentation − of scientific priority:
    The manuscript of the article was submitted to the journal Wissenschaftliche Zeitschrift der Paedagogischen 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 Paedagogische 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 Schwaerzungskurve
    unter Beruecksichtigung 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: 2020, February 20th (20.02.2020)