PHOTOGRAPHIC PHOTOMETRY*

Ornstein–Moll–Burger, Utrecht

1. Photographic Plates

I. Blackening.

The most important results of spectrophotometric measurements have been obtained by the photographic method. In this method the photographic plate is part of the measuring apparatus; it is therefore desirable to give at least a brief description of its basic properties.

A photographic plate, after exposure and development, exhibits blackening. This blackening is the result of the precipitation of silver in the photosensitive layer and depends on the intensity of the incident light, the exposure time, the wavelength of the light source, and the developing process. The fact that, other conditions being equal, the blackening is a function of the intensity of the incident light is the basis of photographic photometry. The measure of “blackening” is usually defined as follows. Let light from a constant source fall on a processed photographic plate, and let \(I\) denote the intensity of the light that has passed through the blackened part of the plate, while \(I_0\) denotes that through its unblackened part; the blackening is taken to be

\[ S = \lg I/I_0. \]

It is clear from the formula that \(S\) may have values from zero to infinity. The quantity thus chosen is proportional to the absorption coefficient of the silver layer and makes it possible to calculate

* The present article is a translation of several chapters of the book by Ornstein–Moll–Burger, Objektive Spektralphotometrie, Braunschweig, 1932. It sets forth mainly the methods developed or tested in the laboratory directed by Ornstein at Utrecht University, and touches only slightly on other methods. Nevertheless, the book is of great practical interest, for the services of Ornstein and his collaborators in developing methods of photographic photometry are very considerable, and acquaintance with the Utrecht experience will undoubtedly be very useful to persons working in this field.

The article is composed of a translation, with slight abridgements, of Chapters III, V, and VI, with some additions from other chapters. In addition, the translator’s notes in two or three places somewhat supplement the original text. Translation by G. S. Landsberg.

Ed.

ceteris paribus the amount of substance deposited per \(1\ \text{cm}^2\) of surface; in accordance with this, it is best suited for investigating the essence of the photographic process.

Sometimes, in order to characterize the amount of silver deposited, another quantity is introduced—“transmittance.” The transmittance \(D\) is determined by the formula:

\[ D = I/I_0 . \]

In principle, one may use any function of \(I/I_0\). In practice, in the photographic method of measuring intensities, blackening and transmittance can be applied with equal success.

In the case of a strongly blackened plate, greater accuracy in graphical representation can be achieved by using blackening \((S)\). From analogous considerations, transmittance \((D)\) should be preferred in the case of a weakly blackened plate. For the sake of simplicity, in our further discussion we shall use only one of the quantities mentioned, choosing blackening as such.

The measurement of “blackening” rests on the determination of the ratio of two intensities. The instruments by means of which such measurements are made are called microphotometers; a detailed description of them may be found, for example, in the book by Ornstein-Moll-Burger, Objektive Spektralphotometrie, Braunschweig, 1932. In those cases where measurements are concerned with the integral blackening of considerable areas of a photographic plate, especially at weak blackenings, preference should be given to the compensation method, carried out with the aid of an extinction meter[^1]. For the purposes of photographic spectral photometry, however, the microphotometer satisfies all conditions.

The ratio \(I/I_0\) is not a quantity independent of the spectral composition of the light used in the measurements. In certain investigations of the photographic process, for example in studying the size of the grains of deposited silver, this dependence plays a certain role. It may, however, be entirely disregarded in those cases where blackening serves as a measure of the light that acted on the photographic plate. Thus, in Koch’s microphotometer (photoelectric), shorter-wavelength light is used in determining blackening, whereas in Moll’s microphotometer (thermoelectric) longer wavelengths are used. The two instruments therefore give different values for one and the same blackening. This difference, however, will have no significance in determining the intensity of the source that acted on the plate. Let us note that for most plates the indicated difference is very small; nevertheless, it should be borne in mind that a change in the spectral composition of the microphotometer lamp may lead to certain errors.

2. Blackening as a Function of Intensity

For a given source and time of illumination, the blackening will, ceteris paribus, depend on the intensity \(i\) of the light, i.e., the energy falling in 1 sec. on a unit surface of the plate. If the energy emitted in 1 sec. and the magnitude of the illuminated surface are changed in one and the same ratio, then the same amount of energy will fall on \(1 \text{ cm}^2\) and, consequently, an equal amount of silver will be separated out and the blackening will be equal. This almost self-evident proposition ceases, however, to be true for surfaces of very small dimensions. We shall return to this phenomenon when we speak about defects of the plate (p. 931) and about the measurement of intensity inside a spectral line. In addition, in what follows we shall confine ourselves to the simplest case, when the intensity \(i\) remains constant during the time of illumination. The influence of interruptions of the light we shall consider later.

Fig. 1. Blackening as a function of intensity (Ilford plate developed with rodinal).

The relation characteristic of a plate between blackening and intensity can be established in the following way: different parts of the plate under investigation are illuminated so that the exposure time remains the same, while the intensities are changed in definite quantitative ratios. Of the methods by which an attenuation of the light in a definite ratio is achieved, we shall speak below. The blackening is determined with the aid of a microphotometer or, for small blackenings, by means of an extinction meter. In this way one obtains a series of values for the intensities and the corresponding blackenings, which may be compared graphically. Fig. 1 gives the curve obtained in this way for an Ilford plate (Special Rapid, H. a. D. 400), developed for 8 min. in a 0.2% solution of paramidophenol chloride (rodinal). The unit of intensity is chosen arbitrarily. As is easily seen from the figure, the curve is for the most part excessively concave with respect to the axis of intensities.

It is usually accepted to represent graphically the relation between blackening and intensity in another form, namely—the blackening is plotted as a function of \(\log i\), and not simply \(i\). Such curves we shall call “blackening curves.” The advantage of this method of graphical representation consists in the fact that

in this case the choice of the unit \(i\) is not reflected in the shape of the curve, but gives only a parallel displacement of it along the abscissa axis.

In Fig. 2 is shown the blackening curve corresponding to the data of the curve in Fig. 1. For negligibly small intensities the blackening approaches the value zero (for \(\log i=-\infty\), \(S\) is asymptotically equal to zero). The curve has a point of inflection, and its curvature near this point is small. For the given plate, in the interval of blackenings from 0.5 to 1.5 (corresponding to an intensity ratio from 1 to 6) the curve, within the limits of the errors inherent in all photographic measurements, may be regarded as a straight line. It goes without saying that in measurements there is no need to confine oneself only to this part of the curve. Since the form of the blackening curve depends on the type of plate, it is desirable to have a quantitative characteristic of the type of plate. To establish such characteristics one may use the rectilinear part of the curve. Analytically it may be represented in the form of a linear dependence between blackening and the logarithm of the intensity:

\[ S=c+\gamma\log i. \]

Fig. 2. Blackening curve (the curve of Fig. 1, recalculated to \(\log\) intensity).

This relation between blackening and intensity is known as Schwarzschild’s law. It can easily be reduced to the following form:

\[ S=\gamma\cdot\log\frac{i}{i_0}. \]

The constants \(i_0\) and \(\gamma\) have the following meaning. The quantity \(i_0\) is determined by the point of intersection of the rectilinear part of the curve with the abscissa axis and is a measure of the sensitivity of the plate. The most commonly used measure of sensitivity is the quantity inversely proportional to \(i_0\) (Hurter and Driffield, Scheiner). The quantity \(\gamma\) determines the slope of the rectilinear part of the blackening curve. If \(\gamma\) is large for a plate, then a given increase in \(\log i\) leads to a larger increment of \(S\) than in the case of small \(\gamma\). A given relative increment of \(i\) will correspond to a greater increment of blackening the larger \(\gamma\) is. A plate with a large value of \(\gamma\) is characterized,

therefore, greater contrast; hence \(\gamma\) is called the contrast factor of the plate. It should be noted that \(\gamma\) can be determined from the ratio of intensities, whereas determining \(i_0\) requires knowledge of the absolute value of the intensities.

The formula giving an analytical expression for blackening as a function of intensity has no practical application in photographic measurements; it has been derived by us in order to clarify the concepts of sensitivity and contrast factor. The relation between intensity and blackening must be studied for each individual photographic plate. We shall indicate in detail below (Chs. 2 and 3) how, with the aid of blackening curves, intensity ratios can be obtained.

3. Blackening as a Function of Exposure Time

By analogy with other photochemical processes one might expect blackening to be a function of the energy incident on a unit surface. On this basis one could expect the applicability in this case of the Bunsen–Roscoe law, which would have the form:

\[ S = f(it), \]

where \(i\) is the intensity, \(t\) the exposure time.

If this were actually the case, then studying the dependence of blackening on exposure time would give us nothing new. However, Schwarzschild showed that the Bunsen–Roscoe law is not applicable in photography. He indicated the more complex dependence

\[ S = f(it^p), \]

which is known as Schwarzschild’s law.

According to Schwarzschild, the exponent \(p\) is a constant, independent of \(i\) and \(t\), characteristic of the light-sensitive layer of the given plate. Combining Scharck’s law and Schwarzschild’s law, we obtain:

\[ S = \gamma \log \frac{it^p}{a}. \]

According to this formula, the contrast factor \(\gamma\) does not depend on the exposure time. The blackening curves for different exposure times will therefore run parallel; i.e., by a parallel displacement in the direction of the abscissa axis they can be brought into coincidence.

Detailed measurements, carried out at Schwarzschild’s suggestion and covering a wide range of intensities and exposure times, revealed deviations from Schwarzschild’s law. However, the above-mentioned parallelism of the blackening curves is a very important property in intensity measurements—

—can also be preserved in the case when the blackening depends on the product \(i \cdot t\) as on some other function of \(t\).

Measurements carried out to verify this parallelism have shown that, for Ilford chromatic plates developed with rodinal, the contrast factor \(\gamma\) remains constant at an exposure-time ratio of \(1:1000\). Only at a ratio of \(1:10000\) is some deviation from parallelism observed. When developed with glycin, already at a ratio of \(1:10\) there is a noticeable deviation from parallelism. Very often the blackening of the plate is produced under intermittent illumination. In these cases the blackening depends not only on the intensity and the time of exposure, but also on the frequency of interruption of the light, i.e. on the ratio of the times of light and darkness. These investigations are important for our question, for they show that the slopes of the blackening curves obtained with continuous and intermittent illumination may differ greatly from one another. This must always be borne in mind when working with a light source that varies in time.

4. Blackening as a function of wavelength

In order to obtain blackening curves for light of different wavelengths, one may use the method described on p. 923, modifying it so that spectra of different intensities, whose ratios are known, are obtained on the plate. Having measured the blackening for each of the wavelengths, the corresponding blackening curves are then constructed. It is also possible to expose comparatively large areas of the plate to the action of approximately monochromatic light, isolated with the aid of light filters or a monochromator. As is known, at equal intensities the blackening depends very strongly on the wavelength. To study this dependence quantitatively, one may use a light source with a known spectral distribution of energy (a standard lamp, see p. 953). Measurements carried out in this way have shown that the blackening curves for different wavelengths are not parallel. As an example, the three blackening curves shown in Fig. 3 for the wavelengths 4700, 5200, and 7000 Å may serve. It is evident from the figure that the slope of the curves for short wavelengths is smaller than for long wavelengths.

When the difference in wavelengths is small, the change in the slope of the curve may be neglected. In this case one may speak of a “relative spectral sensitivity,” independent of blackening, defining it as the reciprocal of the energy (measured in any units) producing equal blackenings. Whereas in Fig. 3 the energy for different wavelengths is measured in different units, in this case it must be expressed in one and the same units. In Fig. 4 two such...

curves of blackening. They correspond to wavelengths 4680 and 4811 Å and may be regarded as parallel. It is clearly seen that the plates are more sensitive to wavelength 4680 than to 4811 Å. For one and the same intensity the blackening in the first case is greater than in the second. The relative sensitivities can be read directly from the drawing. Equal blackenings are obtained for intensities whose ratio is 1:1.3. This ratio does not depend on the value of the blackening, since the horizontal distance between the curves (along the abscissa axis) is everywhere the same. Using the set of parallel curves, one can determine the spectral sensitivity as a function of wavelength. With a large

Fig. 3. Deviation from parallelism of blackening curves for different wavelengths.

Fig. 4. Parallel blackening curves for determining relative spectral sensitivity.

difference in wavelengths the blackening curves are not parallel and therefore this determination of spectral sensitivity loses its meaning.

But even in this case the spectral sensitivity can be determined in an analogous way; in doing so, however, it will no longer be independent of the blackening. In Fig. 5 a curve is presented of the spectral sensitivity of an Ilford panchromatic plate for the region from 3500–7000 Å, constructed for a blackening value of 0.5. Of the three maxima of the curve, two correspond to large wavelengths, while the fall toward the short-wavelength side is explained by absorption of light in the gelatin.

From the preceding it is clear that, for such a large interval of wavelengths, the corresponding curve constructed for another blackening will be noticeably different.

5. Developer

The blackening depends, other conditions being equal, on the type of developer, its concentration, and also on the temperature of the deve-

...of the developing bath. All these factors affect the contrast $\gamma$, whereas the sensitivity of the plate ($a$, and consequently also the quantity $i_0$ in the expression of Schwarzschild’s law) depends on them only slightly. With strong development, i.e. when the development time is lengthened or when the concentration of the developer is increased, the blackening increases; in doing so, for all points of the blackening curve the increase occurs in the same ratio. In accordance with this, the absolute increase in blackening proves to be greatest in the region of the greatest blackening.

Thus, under strong development the contrast factor $\gamma$ increases. With sufficiently prolonged development the blackening reaches a limit. The time during which the limit is practically reached is the shorter, the more concentrated the developer and the higher its temperature.

Fig. 5. Sensitivity curve of an Ilford panchromatic plate for blackening 0.5.

However, a developer of high concentration cannot be used, since in that case it acts nonuniformly. Thus, rodinal at a dilution of 1:5 produces spots on the plate, whereas a dilution of 1:20 gives very uniform blackening. For photographic photometry, a very important property of development is the magnitude of the contrast factor of the plate that it determines. In this respect different kinds of developer show a very considerable difference. It is difficult to obtain high contrast, for example, with glycin, whereas with paramidophenol (rodinal), at suitable concentrations and development times, good results are obtained without difficulty. To avoid fogging, it is reasonable to add potassium bromide to the developer.

However, we strongly recommend, when measuring intensities, to refrain from using it, since it makes the measurement of small intensities impossible. Fig. 6 shows the action of KBr. Its addition changes the ratio

between blackening and intensity in such a way that the curve shows an apparent threshold. Fortunately, in our measurements there is no need to deal with fogging of the plate (see p. 953).

6. Selection of the Plate

It goes without saying that when choosing a plate intended for intensity measurements, one must first of all take into account the wavelengths of the light used. For the wavelength interval from 2500 to 4700 Å, ordinary plates may be used; from 4700 to 5700 Å, orthochromatic plates are suitable,

Fig. 6. Effect of potassium bromide on the shape of the blackening curve.

and for the region 5700–7000 Å, panchromatic plates should be chosen. There are also plates made sensitive for limited spectral regions. Among these one should mention plates sensitized for green, extreme red, and ultrared rays (up to 10,000 Å). For the extreme ultraviolet, ordinary plates are unsuitable because of absorption of ultraviolet rays in the gelatin. In this case either special Schumann plates, which contain no gelatin, are used, or else ordinary plates are sensitized with a layer of a fluorescent substance. For measurements of intensities, especially in the investigation of spectral lines, Schumann plates are of little use because of their nonuniformity. Therefore we recommend sensitization. For other spectral regions in which ordinary plates are not applicable, one may use

make use of commercially available sensitized plates, which are so good that there is no need to engage in sensitization oneself.

For X-rays of relatively large wavelength, ordinary plates are used; for wavelengths of \(1\ \text{\AA}\) there are special plates.

Photographic methods of measuring intensities are especially important in the study of weak light sources. The reason for this lies in the accumulation of light action that distinguishes photographic processes. For very weak light sources the necessary exposure time may sometimes turn out to be extremely long. In order, as far as possible, to shorten this time, one has to choose the most sensitive plates. However, even with sensitive plates, when fast optical instruments are used, it is sometimes necessary to make many-hour exposures. Very sensitive plates generally have the drawback that their sensitivity is distributed unevenly over the plate and that they are easily veiled.

As we shall explain in more detail below (see p. 948), plates whose contrast factor does not depend on the illumination time and changes little with wavelength offer great advantages in measurements. In addition, it may prove convenient that the difference in the contrast factor for intermittent and continuous light be small. All these circumstances should be taken into account when choosing plates.

7. Drawbacks of the photographic plate

Examining the curve of the microphotogram of some photographic plate, we observe that it is never quite smooth, but has a zigzag appearance. These zigzags may be of twofold origin. First of all, they may be caused by foreign particles included in the gelatin layer. In an unexposed plate, after it has been developed and fixed, they are clearly visible when observed under the microscope. In microphotometry such a plate will show even larger zigzags than a plate that has been preliminarily exposed. This is explained by the fact that the silver released during exposure reduces the deflections of the galvanometer and these peaks become less noticeable. Such optical inhomogeneity of the gelatin is usually the greater, the more sensitive the plate. Low-sensitivity plates, for example lantern-slide plates, almost do not show the indicated defect.

The second cause of the zigzags consists in the precipitation of silver grains during development in the form of groups. Whereas individual silver grains, formed from a crystal of bromide silver, are too small to affect the course of the curve

microphotograms, these groups—formed accidentally or for some physico-chemical reasons—are large enough to cause zigzags on the recorded curve. And this cause affects sensitive plates to a greater degree than plates of low sensitivity. The intensity that can be detected on a plate is determined not only by its sensitivity, but also depends on the zigzags of the curve. A spectral line on the microphotographic curve is marked by a maximum. It will be the smaller, the weaker the line is, and the line will not be detectable if this maximum is smaller than the mean amplitude of the zigzags. In measurements of a weak line it should be borne in mind that the gain in sensitivity obtained by choosing a very light-sensitive plate may be completely destroyed by the inaccuracy caused by the increase of the zigzags.

Integration over the height or width of a spectral line can reduce the influence of the zigzags. In spectra poor in lines and with sufficient dispersion, integration over the width is performed automatically if the slit of the microphotometer is made wide.

A second shortcoming of the plate is the nonuniformity of its sensitivity. Even with perfectly uniform illumination and careful, prolonged development, the plate shows differences in blackening that may cause errors in intensity measurements reaching many percent. Different types of plates possess this shortcoming to different degrees. In the present case as well, this shortcoming is, in general, most noticeable in sensitive plates. We have already spoken of the influence of development. Accuracy can be increased only by many measurements carried out at different places on the plate. In this connection it is necessary to mention the property of a plate to give greater blackening at the edges than in the middle; likewise, old plates often show fog at the edges.

A third shortcoming of the plate is its inability to reproduce a large intensity gradient. If some sharply bounded region of a photographic plate is strongly illuminated, then after development it will have more or less blurred edges; the neighboring unexposed places turn out to be “infected” (the Eberhard effect). By appropriate choice of plate and development this effect can be considerably reduced. Often, for this purpose, iron oxalate is recommended as the developer. Of the ordinary developers, according to our data, glycin is very poor, while paramidophenol (Rodinal) acts much better. However, the indicated effect should always be feared, especially in photographing objects with a very fine structure, for example a line spectrum, where the error appears in a limitation of the resolving power of the plate.

Plates intended for microphotometry must be processed more carefully than is usually done. It goes without saying that care should be taken to ensure complete fixing and thorough subsequent washing of the plate. Drying must be uniform, not too rapid, in order to keep the surface of the gelatin flat. This layer is easily damaged, and every scratch on it during microphotometry may be a source of errors. It may therefore be recommended to bathe the plate after fixing in a solution of alum or formalin, for the purpose of hardening the layer. Of course, a plate treated in this way must also be protected from all kinds of damage; thus, dust on the plate should be avoided, since it leads to errors in the microphotogram curve. For this purpose, during drying the plate should be placed in a space free of dust.

II. Photographic Photometry at a Small Difference of Wavelengths

8. Principle of the Method

The simplest problem of photographic photometry consists in comparing the intensities of light of identical spectral composition. An example is the determination of the intensity of emission lines under different conditions, or absorption measurements. The methods applied in these cases are also quite suitable for spectral lines whose difference in wavelength is so small that, in practice, they give identical blackening curves. Coincidence of the blackening curves occurs even for comparatively large differences in wavelength, so that there is a whole series of important problems that can be solved by the methods described in the present chapter. These include, first of all, measurements of the intensities of the components of Zeeman and Stark splitting and of many multiplets.

The relation between the blackening of the plate and the intensity of the light that produced it depends on a very large number of factors. Therefore, from the readings of a microphotometer, which make it possible to judge the magnitude of the blackening, one still cannot draw direct conclusions about the ratio of intensities. But Hartmann had already pointed out that, for identical spectral composition of the light and identical exposure on the same plate, under identical development conditions, equal blackenings are produced by equal intensities. Therefore, by microphotometry of the blackenings one can unambiguously conclude that the intensities are equal. This proposition forms the basis of photographic photometry. To compare the intensities of two lines differing little in wavelength, one should more

weaken the stronger of them so much that the corresponding blackenings would prove identical when Hartmann’s conditions are observed. The degree of attenuation of the stronger line will then measure the ratio of the intensities of the lines being compared. The measurements are carried out in such a way that, on one plate and with the same exposure, the weaker line is photographed together with the stronger one, attenuated in a known ratio. In this case, of course, it is not necessary to achieve, by means of attenuation, exact equality of the blackenings; rather, several attenuations may be made and, after constructing the blackening curve, graphical interpolation may be applied, i.e. one may find that point at which the blackening is exactly equal to the blackening of the weaker line. The abscissa of this point will then make it possible to determine the required ratio of intensities.

Of course, the calculation could be carried out analytically, by expressing the blackening curve by a formula. But since the constants of such a formula can be established only on the basis of the measurements mentioned above, the use of the formula gives no simplification; and since the accuracy of the analytical method is, in general, not great, the graphical-empirical method should be preferred as the less laborious one.

Instead of using the lines being compared to construct the blackening curve, it is sometimes more convenient to construct it in another way. On the same plate on which the line spectrum has been photographed, “blackening marks,” whose intensity ratios are known, are photographed with the aid of an auxiliary light source. The auxiliary light source may have either a line or a continuous spectrum; it is essential only that the wavelength of the blackening marks be close to the wavelength of the lines under investigation.

9. Attenuation of Light

We shall consider methods that make it possible to attenuate light in definite quantitative ratios. These methods may be divided into two groups. The first group includes those in which the blackenings being compared are photographed successively, one after another, and in this case, of course, constancy of the light source (the source under investigation or the auxiliary source) must be ensured. The methods of the second group make it possible to obtain different blackenings with a single exposure; here it is possible to use a light source that varies in time.

In this paragraph we shall describe the methods of the first group, in which several spectra are photographed successively on one and the same plate.* Let us proceed to examine them.

* If, when photographing a continuous spectrum, there is no assurance of a strictly parallel displacement of the plate in the spectrograph, then, on the continuous spectrum, wavelength marks should be made by using well-known spectral lines.

1. Attenuation of light by absorption or scattering. For this purpose one uses a homogeneous layer of a substance absorbing or scattering the indicated wavelength. In practice, neutral attenuators are most often used, i.e., layers whose absorption depends only weakly on wavelength: a blackened photographic plate, a thin metallic layer, or smoked glass.

The advantage of a photographic plate as an attenuator is the fact that it is easy to prepare oneself for all desired gradations of blackening. For this purpose a photographic plate is exposed (lantern-slide plates are very suitable), selecting the appropriate intensity and time of illumination so as to obtain a uniform blackening over its entire surface. It is possible to prepare photographic attenuators whose transmittance in the wavelength interval from 10,000 to 3500 Å changes by only a few percent.

A substantial disadvantage of a photographic attenuator is the circumstance that its action is based in part on scattering of light, as a result of which the transmittance depends to some extent on the entire optical setup. Therefore it is desirable that its calibration and subsequent work with such an attenuator be carried out, as far as possible, under identical conditions.

Thin metallic layers can be made, by means of thermal or cathode sputtering, of any thickness, i.e., of any transmittance. Such layers are deposited on a glass or quartz plate, depending on the region of wavelengths—visible or ultraviolet light—for which they are intended.

In metallic layers light is attenuated by absorption and reflection, and their transmittance, consequently, does not depend on the optical setup. In choosing the metal one is guided by the requirements of durability of the layer, ease of preparation, and the least possible selectivity. Platinum is most often used. It can easily be deposited in a layer of any thickness on a glass or quartz plate, is well preserved in air, and is only slightly selective. Since thin layers are easily damaged by mechanical contact, they must be covered with a second plate.

Smoked glasses are commercially available, and their advantages consist in the fact that they do not deteriorate with time. The ordinary grades of such glasses possess, however, a very distinctly expressed selectivity, although to the eye they seem uncolored (gray). The transmittance of such glasses is greatest for long and short wavelengths; in the middle region between yellow and blue it gives several maxima. This circumstance makes them of little use for our purposes. In choosing an attenu—

attention should be paid to its selectivity. The degree of attenuation must be spectrally determined. For this purpose, in principle, one may use any method of measuring intensity. The determination of the attenuating capacity consists in comparing two intensities—with and without the attenuator. Such measurements may be made with a thermoelement or a photoelement connected to a monochromator. If the calibration is carried out for the ultraviolet region, then, in the absence of a sufficiently constant and strong light source, one may resort to a photographic method of measuring intensities.

2. Attenuation of light by changing the current. Instead of attenuating the light emitted by a source, one may influence the intensity of the emission itself. For lamps with electric supply this is achieved by reducing the lamp current. It is first necessary to investigate the relation between the current and the intensity of the light emitted by it for different wavelengths, for, it goes without saying, this relation depends very strongly on the wavelength. Such a calibration may be carried out by various methods.

Let us now turn to methods that are deliberately nonselective.

3. Attenuation of light by changing the distance from the source. Provided certain precautions are observed, the intensity of light decreases inversely proportional to the square of the distance. The application of this law assumes, first, that the minimum distance to the source is considerably greater (at least 10 times) than the dimensions of the source; second, that it is necessary to be completely free from light scattered by surrounding objects; third, that at any distance the light falling on the slit of the spectrograph must cover it in the same way. The last condition is difficult to fulfill if the dimensions of the source are small in comparison with the size of the slit.

It is best to vary the distance of the source from a fixed small diffusely scattering surface, which will serve to illuminate the slit of the spectrograph. Since this method is associated with a large loss of light, its implementation requires an intense light source. We have little experience in applying this method; it seems to us especially promising for the short ultraviolet region².

4. Attenuation of light by diaphragming. One may attenuate light in a known ratio by diaphragming the beam at a suitable point, for example at the lens that projects the source onto the slit of the spectrograph. If the entire lens is uniformly illuminated, then the degree of attenuation of the light can be determined directly from the dimensions of the diaphragm. In practice it is difficult to achieve uniform illumination of the lens; therefore the following is successfully used

installation. Immediately in front of (or behind) the focusing lens a rotating sector is placed, the center of which lies approximately on the optical axis of the system. Rotation of the sector corrects nonuniformities of illumination, and by changing the angle of the sector any degree of attenuation is obtained. The speed of rotation must be sufficiently high that during the exposure the sector has time to make many revolutions. Since in the present case only small nonuniformities of illumination are involved, the effect of fluctuations in intensity gives only an error of second order. To attenuate the light one may also use a metallic mesh. It should be placed in such a way that no image of it is obtained on the photographic plate. It is best to place it behind the focusing lens. To avoid reflections and diffraction from the wires of the mesh, the mesh should be carefully blackened and chosen with such a calculation that the cells are not too small. The transmittance of the mesh is rather difficult to compute from its geometrical dimensions; it should be calibrated experimentally.

5. Attenuation of light by changing the width of the slit. Another method of attenuating light that deserves attention, and that can be applied only in the case of a continuous spectrum, is changing the width of the spectrograph slit. If the slit is uniformly illuminated over its entire width, then the intensity of the continuous spectrum is directly proportional to the width of the slit. True, in this case light of several wavelengths is superposed on each region of the photographic plate, but since the sensitivity of the plate remains constant in so small a spectral interval, the result of the action of the light will be as if the light remained strictly monochromatic.

In most spectrographs the screw that separates the jaws of the slit is so precise that from its reading one can directly determine the width of the slit. In doing so, one should accurately establish the division at which the slit is still closed. If necessary, the divisions of the screw may be calibrated. In this case it is preferable not to work with a very narrow slit, since then inaccuracy in the zero reading gives a relatively large error in the measurement of the intensity. In addition, owing to diffraction, part of the light does not fall on the collimator lens, and the intensity will be less than should be expected from the reading of the slit width. The maximum width of the slit is determined, on the one hand, by the phenomenon of superposition of colors mentioned above, and on the other, by the difficulty of obtaining perfectly uniform illumination of the slit when its width is large. Asymmetry of the slit introduces a systematic error. It goes without saying that all exposures made by this method must be made with equal exposure time. To eliminate the error caused by inaccuracy of the exposure time, it should not be chosen too short, i.e., the illumination should not be very intense.

1. Attenuation of light with the aid of nicols. By passing light successively through two nicols, one may, by rotating one of them, obtain any attenuation of the light and, knowing the positions of the nicols, calculate it exactly. The chief drawback of this method is the presence of stray light. With a good arrangement the intensity of such light, transmitted through crossed nicols, is, of course, very small in comparison with the intensity transmitted by parallel nicols; but at large attenuation it may cause a quite appreciable error. It must be borne in mind that, with the usual dimensions of nicols, this method of attenuation has very little light-gathering power. Moreover, it does not fully satisfy the requirement of neutrality, since often the nicol system shows absorption in the extreme violet and ultraviolet owing to a slight coloration of the Iceland spar and of the layer of Canada balsam.

2. Attenuation by means of a rotating sector. For photometric purposes a rotating sector is sometimes used, arranged—unlike that mentioned in § 4—in such a way that it alternately opens and completely closes the path of the luminous flux. With such an arrangement the sector in essence changes the exposure time, not the intensity of the light beam. Even at a very high speed of rotation, in order to evaluate its action it is necessary to take account of the effect of interruption, which introduces great complications. Therefore one must categorically reject the use of a rotating sector in the indicated arrangement.

Similarly, the method of changing the duration of exposure, which is sometimes used in photographic photometry (see, for example, V. Henri³), cannot be recommended, since for its use it is necessary to investigate the coefficient \(pb\) in the Schwarzschild law, and this in essence requires the additional application of intensity marks.

10. Simultaneous attenuation (step attenuation)

When using a light source that varies with time, all the methods listed in the preceding paragraph are inapplicable, since they presuppose a series of photographs taken successively in time. With a varying source it is necessary to photograph the entire set of spectra of different intensity simultaneously. Such a method, moreover, gives a great saving of time, since it makes it possible to confine oneself to a single photograph instead of a series of photographs. To carry out this method, special techniques of “step attenuation,” described below, are used.

1. Step attenuation of light by absorption or scattering. To prepare a step attenuator it is necessary to prepare an entire series of light-attenuating neutral-

PHOTOGRAPHIC PHOTOMETRY

... filters of different gradations (a photographic plate, a thin metallic layer, or smoked glass, see p. 934) and arrange them one above another, as indicated in Fig. 7. The order of their arrangement may be arbitrary, which makes it possible to avoid the systematic error that might be caused by nonuniformity in the illumination of the attenuator. In order to be able to ascertain whether such nonuniformity is present, it is rational to make at least two steps of the attenuator identical. Thus, in Fig. 7 the upper and lower steps correspond to the minimum attenuation.

The attenuator made in this way should then be calibrated experimentally. The indicated attenuator may be used in various ways.

a) Direct superposition of the attenuator on the photographic plate being exposed. In the case when the measurement is made for a large interval of wavelengths, i.e. the spectrum on the plate occupies a rather large extent, the step attenuator with such an arrangement must be of large dimensions. The manufacture of such an attenuator involves considerable difficulties, so that such an arrangement cannot be recommended.

Fig. 8. Diagram of an arrangement with a step attenuator placed behind the lens.

Fig. 7. Step attenuator.

b) Arrangement of the step attenuator directly in front of the slit of the spectrograph. Since the height of the spectrograph slit is usually not great, with this method it would be very difficult to make an attenuator with a sufficient number of steps. We have never used this in principle simple arrangement, always giving preference to method “c,” introduced by Van Cittert.

c) Arrangement of the step attenuator in combination with an auxiliary lens. This arrangement is shown schematically in Fig. 8. The light source \(Q\) uniformly illuminates the lens \(L_1\). The attenuator is placed behind the lens and, by means of the lens \(L_2\), is projected onto the slit of the spectrograph \(S\). The lens \(L_1\) reduces the beam so much that it falls completely on the lens \(L_2\). The spectral lines on the photographic plate are images of the slit and, if in the spectral apparatus the image is astigmatic, then each line will represent a series of sharply bounded regions of different blackening, correspond-

...steps of the attenuator. In order that the intensity ratios of the individual photographed lines should indeed correspond to the prescribed gradations of the attenuator, it is necessary that, in the absence of the attenuator, the lines be found to have identical blackening over their entire length. For this it is necessary that the light entering the spectrograph be diaphragmed to the same degree at all points of the slit, or not diaphragmed at all.

It is necessary to test carefully whether our optical arrangement satisfies this requirement. For this purpose it is best to carry out photometry along a spectral line taken with a wide slit. A necessary, but not sufficient, check is the verification of the equality of the illumination given by identical steps of the attenuator. The lens \(L_1\) must be very well cleaned and have no scratches, since any nonuniformity on its surface leads to an uncontrolled decrease in its transmittance. If the lens is achromatic, one should bear in mind that, owing to reflection between its component parts, interference may arise, producing on the spectral line a series of bright and dark regions, which may be the cause of gross errors. The phenomenon of interference may occur in the attenuator itself if it is cemented on the outside with glass. This excellent method, unfortunately, has a substantial shortcoming, namely its low luminosity. In fact, the lens \(L_1\) is projected onto the slit of the spectrograph, and from this circular light spot the slit cuts out only an insignificant fraction. The luminosity could be considerably higher if it were possible to image a light source of small size onto the slit. In our method, however, this is not applicable. If the light source has appreciable dimensions, then the indicated shortcoming is not so significant. Likewise, in the case of a small moving source (for example, a spark), the arrangement is quite rational as regards luminosity; indeed, despite the motion of the source, the light flux that has passed through the lens \(L\) can easily remain within the limits bounded by the lens \(L_1\), so that the mean intensity incident on the slit remains constant; whereas in the case of direct imaging of the source on the slit, the light from the moving source now falls on the slit, now passes beyond its limits.

Instead of a stepped attenuator, an attenuator in the form of a wedge was formerly often used. By measuring the length of spectral lines obtained in photographing with such a wedge, one can obtain an estimate of the intensity of the lines, since the more intense the line, the longer the image observed on the plate. For exact quantitative measurements, however, this method is of little use.*

* However, Scheibe believes that such an estimate from the length of the line image can be fairly reliable. In Scheibe’s method a wedge-shaped attenuator is replaced by a rotating sector with diaphragms of such a size that various portions of the slit are subjected to exposures of different duration, which also...

The methods “b” and “c” are applicable only when working with spectrographs that give a stigmatic image, since they are based on the assumption that each point of the slit is imaged by a corresponding point on the plate.

d) Arrangement with a Rowland grating.

For an astigmatic Rowland arrangement with a concave grating, Frerichs developed a method that makes it possible, by means of a step attenuator, to photograph spectral lines with definite intensity ratios. The scheme of the method is shown in Fig. 9. On the Rowland circle there are placed the grating \(G\), \(S\)—the slit, \(P\)—the photographic plate. Each point of the slit \(S\) is imaged on the plate in the form of a vertical straight line. Therefore the horizontal steps of an attenuator situated at, or projected into, \(S\) cannot give a sharp image on the plate. But, as is known, the point \(B\) of intersection of the straight line \(GS\) and the tangent to the circle at the point \(P\) has the property of being imaged on the plate in the form of a horizontal straight line. Thus, if a horizontal step attenuator is placed at the point \(B\), the photographed lines will have a sharp stepwise structure. Just as in method “c,” it is not necessary to place the attenuator directly at the slit of the instrument, since in the present case also the attenuator must not be located at the point \(B\). Here too one may make use of a system of auxiliary lenses. The lens \(L_1\) focuses the light of the source \(Q\) on the lens \(L_2\), which projects the attenuator located at the point \(A\) into the point \(B\). In the present case as well, all the precautions mentioned above should be observed. In such an arrangement Frerichs, instead of an absorbing or scattering attenuator, used an attenuator in the form of the diaphragm shown in Fig. 10. The white areas represent openings in an opaque plate. The total area of the openings is in the ratio \(1 : 2 : 3 : 4 : 5\). Owing to astigmatism, the steps of the attenuator give images equal in size, so that the “transmission” of each step of the attenuator proves proportional to its area. Each spectral line is divided into five sharply bounded sections, the intensities of which are in the ratio \(1 : 2 : 3 : 4 : 5\).

A great advantage of such an attenuator is that its transmittance is entirely independent of wavelength and can easily be calculated from geometrical

leads to the production of images with gradually decreasing blackening. According to Scheibe, the length of the image can be reliably determined with an error not exceeding \(0.2\ \mathrm{mm}\), which leads to a fairly accurate determination of the intensity. Obviously, the results depend on the gradient of the decrease in blackening. In addition, for such measurements very good photographic plates are required (in the sense of uniformity of the light-sensitive layer). Translator’s note.

of the sizes of the apertures. Thus an optical calibration becomes unnecessary for it.

2. Stepwise attenuation by means of diaphragming. This method is characterized by the same advantage: the independence of the transmittance from the wavelength.

a) Step slit. For simultaneous stepwise attenuation of a continuous spectrum, a slit may be used whose width varies in steps in definite ratios along its length. To avoid systematic error due to inhomogeneity of the plate and nonuniform illumination of the slit, the portions of the slit of different widths are arranged arbitrarily. It is rational to make the upper and lower steps of the same maximum width. With uniform illumination of such a slit we obtain a set of continuous spectra whose intensities are proportional to the width of the corresponding step (see p. 936), and in this case the diaphragming of the light

Fig. 10. Step attenuator according to Frerichs, used in a Rowland mounting.

Fig. 9. Diagram of the arrangement of a step attenuator in a Rowland mounting according to Frerichs.

coming from the different parts of the slit must be the same in the spectrograph. When the spectra of the upper and lower portions are compared, they must give the same blackening, which can serve as a check on the correctness of the setup. A convenient form of step slit was proposed by Ello¹.

Stepwise attenuation of a continuous spectrum finds practical application in comparing intensity by the method of applying blackening marks with the aid of an auxiliary source. For the wavelength interval where an incandescent lamp can serve as a sufficiently constant source, it is more convenient to use the method of successive exposures with a varying slit width than to use a step slit. Only for wavelengths shorter than 2400 Å, the region where the intensity of an incandescent lamp is small and an arc is used as the light source, is it necessary to work with a step slit, since the burning of the arc is very unstable.

b) Hansen’s diaphragm method1. Hansen proposed a diaphragm method suitable also for line spectra. A diaphragm, similar to a step wedge, is projected by means of an optical system that includes a cylindrical lens onto the photographic plate, so that the boundaries of the steps are sharp, while the intensity of the lines is proportional to the width of the slit. The lines obtained by this method are similar to the lines obtained by method “a” (p. 938). In working by this method one must keep in mind the same precautions concerning the diaphragming of beams in the spectrograph that have been mentioned repeatedly above.

1. It is necessary to mention one more method of stepwise attenuation of line spectra, the idea of which belongs to Schwarzschild and which is reported by Gzshirprung. Monochromatic images of the shadows of an opaque object are cast onto the photographic plate, so that the intensity distribution can easily be calculated from the geometrical conditions of the arrangement. A wire of suitable thickness may serve as such a screening object.

11. Measurement of intensity without the use of an auxiliary light source

Let us return to the main subject of the present chapter: measurements of intensities when the differences in wavelengths are small. As an example we shall consider measurements in a line spectrum. The same scheme is also applicable to neighboring regions of a continuous spectrum. In order to draw conclusions about their intensities from the blackenings of photographed lines, it is necessary to know the relation between blackening and intensity, i.e., one must construct the blackening curve of our plate for the given wavelengths. For this purpose, as has already been indicated in the preceding paragraph, one may use either the line under investigation itself, or else apply blackening marks with the aid of an auxiliary light source. In this paragraph we shall consider the first method in detail. We shall dwell on the second method in the next paragraph. As an example, let us take the zinc triplet with components at 4680, 4722, 4811 Å and set ourselves the goal of determining the ratio of the intensities of these components. In Fig. 11 this zinc triplet is reproduced, each of the lines being divided, by means of a step attenuator, into six sections of different intensity that are in definite quantitative ratios. The intensity ratios, proceeding in the figure from bottom to top, are respectively equal to 100 : 72 : 61 : 34 : 16 : 100. As is easy to see from the figure, the lines in the photograph are sufficiently broad. This was achieved by deliberately widening the slit in order, as far as possible, to compensate for the error caused by inhomogeneity of the plate emulsion (see p. 930). The plate was ex—

are fastened on the microphotometer stage in such a position that the lines are vertical and the three lower sections successively enter the beam of the microphotometer. Then the stage is shifted downward, moved back, and the neighboring sections (72, 0) are photometered, etc. It goes without saying that the zero line is also recorded each time.

Fig. 11. Zinc triplet 4680, 4722, 4811 Å, taken with a step attenuator.

In this way one obtains the microphotogram curve in the form in which it is shown in Fig. 12. The three lines 4680, 4722, 4811 Å are recorded six times, corresponding to the six steps of the attenuator. Since the lines are broad, the maxima are also sufficiently broad to smooth out the error due to random “peaks” of the curve associated with inhomogeneity of the emulsion. One may, for example, using the data for the most intense line, construct the blackening curve and from the maximum blackenings of the other lines determine the ratio of their intensities. With this method we use only part of the microphotometer curve; the blackening data for the attenuated weak lines remain unused. Obviously, the accuracy of the measurements will increase if these data are brought in.

This is done as follows: on logarithmic paper the blackening curves of the three lines are plotted in such a way that

Fig. 12. Microphotogram of the zinc triplet of Fig. 11.

the abscissae corresponding to intensity 100 (72, 61, 32, 16) coincide for all the lines. In Fig. 13 these graphs are given. For practical reasons the logarithmic divisions of the paper and the points corresponding to the plate measurements are not marked in the figure. With a small difference in length the blackening curves are practically parallel, i.e. the horizontal distances between the curves along their entire length are equal.

These distances, read on the scale $\log i$, give directly

...the desired intensity ratios strongly. Owing to random errors, the experimentally obtained curves may deviate from parallelism. In such cases one should use the mean value of the distances. If, however, the distance changes strongly from the upper to the lower part of the curve, then the measurements, because of this systematic error, become unreliable and the plate should be rejected.

The great advantage of this graphical method lies in the twofold smoothing of errors in determining the intensity: first, in constructing the blackening curves (drawing smooth curves through the points), and second, in averaging the results of measuring the distance between the curves. The deviations from the value obtained by this double adjustment characterize the achieved accuracy of the method.

The exposure time should be chosen so that the region of curves for which the distance measurements are made corresponds to blackenings convenient for good measurements with a microphotometer. In Fig. 13 the distances between equal blackenings for clarity are denoted by the letters \(p\) and \(q\); below the graph they are again plotted from 100 to the left, so that the intensity ratios can be read off directly from their left edge. For the three lines 4680, 4722, and 4811 Å we find these ratios respectively equal to \(q : 100\) and \(p : 100\).

Fig. 13. Blackening curves of a zinc triplet, Fig. 12. The distances \(p\) and \(q\) give the intensity ratios.

We have chosen as an example three lines whose wavelength differences are not very small, so that the sensitivity of the plate in the interval of the three wavelengths is noticeably different. If, for example, we were making measurements for the components of Zeeman splitting, then the introduction of a correction to the obtained figures would be superfluous. For our case, however, the results obtained cannot be regarded as final; in the following chapter, using the same example, we shall indicate how to take account of the selectivity of the photographic plate. In those cases where the ratio of the intensities of the lines being compared is very large, the step attenuator must meet high requirements. If in the spectral interval under consideration there is another line of medium ...

…intensity, it is useful to include it in the measurements and to compute the desired intensity ratio by dividing the intensity readings of the auxiliary line by that of the investigated one. The relative error in measuring the intensity is equal to the absolute error in determining the distance between the blackening curves; and this latter is the greater, the greater the indicated distance.

Another difficulty arises when comparing the intensities of two very close lines with overlapping edges. Below we shall indicate which method should be used in such a case.

The method described can be successfully applied in solving various problems connected with the measurement of intensities. In the same scheme one may also carry out the determination by the method of successive photographs. Thus, in measuring the intensities of continuous spectra—for example, in absorption measurements—this procedure is the most commonly used. Indeed, in order to carry out absorption measurements, a constant source of light is required, which may also be used for calibrating the plate, i.e., for obtaining one or several blackening curves. For applying marks in this case, the simplest method is to change the slit width. If the light source is sufficiently constant, then the method of successive photographs should also be preferred for measuring line intensities, since this eliminates the need for uniform illumination of the spectrograph slit, which is always associated with well-known difficulties.

In the event that the light source is inconstant, one must resort to a step attenuator.

12. Measurement of intensities with the aid of an auxiliary source

As was already indicated above, the relation between blackening and intensity can be studied by the method of applying blackening marks with the aid of an auxiliary light source. For this purpose, the investigated spectrum and a series of spectra of definite intensities, serving as “marks” of blackening, are photographed on one plate at identical exposures. The simplest method of applying marks consists in taking a constant light source (continuous spectrum) and changing, in definite quantitative ratios, the width of the spectrograph slit. A tungsten lamp, carefully maintained at constant operating conditions, may serve as the constant source. For carrying out measurements in the ultraviolet region, the lamp may be provided with a quartz window and slightly overheated, or an arc may be used. When using an arc, one has to resort to a step attenuator in order to shorten the exp…

time of operation of the arc; in this case it is most convenient to use a step slit.

The method of widening the slit does not make it possible to vary the interval of intensities over a wide range, since a slit that is too narrow or too wide is a source of errors.

There is, however, a simple procedure that makes it possible to extend the interval of intensities. Namely, two series of blackening marks are photographed at different incandescence of the lamp, i.e., at different current strength in it. In this way two blackening curves are obtained, the lower part of one being parallel to the upper part of the other. Since the choice of the unit of intensity is arbitrary, both

Fig. 14. Below—a zinc triplet; above—“blackening marks”—continuous spectra photographed with the same slit width.

curves can be shifted so that one appears as a continuation of the other. By this procedure one obtains a single resultant curve extending over a large region of blackening.

It is also possible to use a change in the current strength in order to vary, in a known ratio, the radiation of the incandescent lamp. In this case it is necessary first, for each wavelength, to study the dependence of the intensity on the current strength. This calibration may be carried out in various ways; the most suitable for this purpose is an arrangement with a monochromator and a thermoelement*.

As was already mentioned above, when comparing two lines that differ greatly in intensity, one may introduce as an auxiliary line some line of intermediate intensity, if there is in the spectrum a suitable line of close wavelength. Po-

* It should be noted that the desired relation between the intensity and the current strength may also be calculated from the color temperature of the lamp.

To explain, as we did earlier, this method by an example. As such an example one may again consider the ratio of the intensities of three zinc lines. In Fig. 14 three lines and a series of continuous spectra are reproduced, serving as blackening marks. The lines were photographed with a wide slit, in the present case not only in order to avoid error due to accidental “peaks” of the curve, but also in view of another complicating circumstance that arises when comparing a line with a continuous spectrum, of which more will be said below. The blackening marks were obtained for the following relative slit-width values: 100 : 59 : 38 : 12 : 18 : 12 : 17. Microphotograms of the lines were obtained in the usual way and are shown in Fig. 15.

Fig. 15. Microphotometric curve of the zinc triplet of Fig. 14.

Fig. 16. Microphotometric curves of the “blackening marks” of Fig. 14, corresponding to intensities 100, 59, 38, 12, 18, 17, 7.

When considering continuous spectra we are interested in the value of the blackening of that spectral region in which the lines are located. For this purpose we take a microphotogram of the corresponding region, passing the plate through the microphotometer perpendicular to the direction of dispersion.* Fig. 16 gives the microphotogram obtained in this way for the wavelength corresponding to the middle zinc line.

We shall use it to construct the blackening curve of the plate for this wavelength. In Fig. 17 the blackening curve constructed in the usual way is shown. From the microphotogram (Fig. 15) we find the following values of the blackening maxima for the lines under study: 0.086, 0.26, 0.33. From the blackening curve we determine the corresponding intensities: 1.76, 4.6, 5.8.

* If there is no confidence that the corresponding wavelengths in the different spectra are located exactly one below another, especially when the spectral region under consideration is not characterized by sharp ends, it is necessary to place wavelength marks on the continuous spectrum and to measure the spectra in the region of interest to us in the direction of dispersion.

Assuming the value of the intensity of the strongest line to be conventionally 100, we find the corresponding values 30.5, 74, 100.

These results are in good agreement with the data obtained with a step attenuator. As we have already noted above, a correction for the selectivity of the plate should be introduced into these results. This will be discussed in the next chapter.

For most photographic plates (see p. 926) the slope of the blackening curve depends only slightly on the exposure time. It is therefore possible, without making a large error, to choose different exposure times for lines and for continuous spectra. The permissible ratio of exposure times depends on the type of plate and developer. It often reaches 100. Often, when photographing lines, the exposures turn out to be very long, and the possibility of choosing a shorter exposure for the marks gives a great saving of time. This time can with great advantage be used to repeat the same photograph, in order to eliminate any accidental errors. Since the exposure time can change the slope of the curve only slightly, these repeated photographs can be made with a different exposure. Each change in the conditions of the experiment makes it possible to eliminate one or another accidental error. In general, for a given type of photographic plate one should test to what extent the slope of the curve depends on the exposure time.

Fig. 17. Blackening curve constructed from the data of Fig. 16, and determination of the desired intensity ratios.

For very narrow lines, the blackening is due not only to the amount of energy incident on a unit surface. A blackening curve constructed from the marks of a continuous spectrum is not identical with the corresponding blackening curve of the lines. The latter is changed as a result of the “infection” effect (see p. 931). This error can be avoided only when working with broad spectral lines or when using a step attenuator.

Intermittency in the illumination also affects the slope of the blackening curve. In measurements of intensity with an intermittent light source, special measures must be taken if one is working with an auxiliary light source. By placing a suitably chosen rotating sector in the path of the light, po—

with a luminous lamp producing blackening marks, one can obtain the same number of interruptions as characterizes the source under study and thus obtain the correct blackening curves [7].

13. Blackening marks by means of multiplets of the Zeeman effect, diffraction, or interference

Along with the methods described in §§ 9 and 10, there are entirely different possibilities that make it possible to construct or to control blackening curves. To be sure, the methods described here have limited application; however, better methods are as yet unknown or require checking. The procedure described below has served to determine the intensity ratios of components of Zeeman splitting photographed without the special purpose of determining their intensities. For photographs that cannot be repeated, for example during a solar eclipse, this method represents the only possibility of deriving, subsequently, the intensity ratios, even in cases where the plate used had not been calibrated. The intensity ratios for many multiplet lines obey, as is known, simple laws; for others they are determined empirically. If multiplets of this kind fall on the plate, or if they have been specially photographed together with the lines under investigation, then, from the known intensity ratio and the measured blackenings, one can construct the blackening curve of the plate. The components of the Zeeman effect can also be used for this purpose and in general give correct results. The theoretical rules concerning the intensities of Zeeman components are better known than for multiplets, and, moreover, arbitrariness plays a smaller role in the components of Zeeman splitting.

One can calibrate the plate by photographing, in the spectral region of interest to us, some interference or diffraction pattern with a theoretically known intensity distribution. For example, a diffraction pattern from a slit would be suitable for this purpose, since the corresponding intensity distribution can be calculated from wave theory. But since the intensity of the diffraction pattern is small in comparison with the central part (the intensity of the first maximum is equal to 4.5% of the intensity of the center), this method is suitable only in those cases where the lines being compared differ greatly in intensity. It would be possible to use the intensity distribution in the central part of the diffraction pattern. In this form the method is especially convenient, since there is no need to photograph a special diffraction pattern, for every spectral line itself constitutes such a pattern.

The width and the intensity distribution within the line are обуслов—

conditioned by the smallest aperture through which the light passes, and this minimum aperture may be set by the objective itself or by the prism (grating). Of course, the distribution of intensities in the line will also depend on the width of the slit and on the manner in which it is illuminated. But this influence can be taken into account in the calculation. This method can be used to check a plate on which, although blackening marks are present, there is no certainty as to their reliability. Thus, it may happen that blackening marks made at different widths are not very reliable because their exposure is too small in comparison with the exposure of the lines. In such cases the distribution of intensity within the line itself may serve as a check. But this method is especially valuable for investigating types of plates not yet tested for intensity measurements. In doing this, all the necessary data of the spectrograph and illumination must be known. When using a diffraction grating, one can measure the intensity of the principal lines and of the “ghosts,” and use this subsequently for graduating the plate.

14. Difficulties Arising in Measurements of the Intensities of Spectral Lines

In most cases the maximum value of the blackening of a line may be regarded as the true measure of its total intensity. In principle it is more correct to determine, at every point within the line, the intensity from the blackening and to integrate the curve thus obtained. This painstaking investigation almost always proves superfluous, since the intensity-distribution curves of two lines are usually “proportional,” i.e., they can be made identical by multiplication by a certain factor. The form of the curves will depend both on the true distribution of intensity within the lines and on the properties of the spectrograph. If the form is determined by the width imposed by the apparatus, then the indicated “proportionality” is ensured. But even in those cases where this is not so, “proportionality” of the lines is sometimes present; for example, when their true width is a consequence of thermal motion (Doppler effect). If the true width is not small in comparison with the width imposed by the apparatus, and the intensity distributions of the lines are noticeably different, then it is necessary to integrate the intensity curves. Usually, for this purpose a simple approximate method is sufficient, consisting in multiplying the values of the intensity maxima by half the line width. If the distance between the lines is not too small, it is preferable to increase the line width imposed by the apparatus by widening the slit, and to take the maximum value as the measure of the intensity.

Proceeding from certain other considerations, it is also preferable to work, whenever possible, with a wide slit of the spectrograph. As we have already mentioned, this gives a broader maximum of the microphotometric curve, and this makes it possible to eliminate errors caused by the presence of zigzags in the curve.

Often both lines are so close that they partially overlap, i.e., in places their intensities are added. The resolution of both intensity curves presents certain difficulties in such a case. Suppose a microphotogram is given (Fig. 18), on the basis of which an intensity curve has been constructed in the usual way. It is required to resolve it into two separate curves. For the analysis of these curves we shall make the assumption that they are symmetrical. The left side of the left line is least of all changed as a result of the superposition of the lines. Neglecting this influence at first, we obtain an approximately correct picture by mirror reflection. The position of the axes of symmetry (shown in the figure by a dotted line), however, is to a certain degree indeterminate. The peak of the line, under the influence of the second line, is displaced somewhat to the right*. If from the resultant curve we subtract the curve constructed in this way, we can obtain the weaker line. If this latter curve is asymmetric, then in any case its right side is known more accurately. By a mirror image of this side one can construct a second approximation to the right line, and by subtracting it from the resultant one can likewise obtain a second approximation to the left curve. In general this method of approximation gives a result converging so rapidly that for the most part one may confine oneself to the first approximation.

If there is reason to expect that both lines have proportional intensity distributions, then we have the possibility of checking the final result. Usually for this it is sufficient to verify whether the half-widths of both lines are equal. In Fig. 19 this condition is not fulfilled. In our case the ratio of intensities, obtained without a detailed analysis of the given curve (Fig. 19), is equal to 100 : 42; the analysis gives the correction 100 : 27. But since the intensity distribution of the given lines is not proportional, the true ratio of the intensities of the lines is not determined by the values of the intensities of the peaks, but must be determined by integration of the curves. The final value of the sought ratio of intensities will be equal to 100 : 35. If the lines are asymmetric owing to the properties of the apparatus itself, then the analysis often proves to be quite unambiguous, if one determines the characteristic for the given apparatus (the apparatus and the type

* The distance between the lines, determined from the position of the maxima on the microphotometric curve or on the intensity curve, appears smaller than it is in reality because of the partial overlap of the curves. In visual observations of a photograph of the spectrum, or even of the lines themselves, this apparent “attraction” of the lines likewise takes place.

PHOTOGRAPHIC PHOTOMETRY

illumination) the intensity distribution for some simple isolated line.

In our example the distance between the lines is so large that a trough is observed between their peaks, so that, by the usual notation, they should be considered resolved. In the case where this is not so, the method of analysis described nevertheless makes it possible to establish the presence of two lines and to separate out their components. One often has to deal with lines that appear simple, the intensity distribution of which, however, differs greatly from that of an actually simple line. In this case too, after carrying out the analysis, one can separate the individual components of this complex. Of course, it should be remembered that deviations from the usual form may be the consequence of an actually asymmetric intensity distribution.

Fig. 18. Microphotometric curve of two partially overlapping spectral lines.

Up to now we have assumed that the blackening at any point within the line is a single-valued function of the intensity at the given point. Depending on the type of plate, the developer, and the intensity gradient, this approximation may either lie within the limits of measurement error or exceed them. For large intensities, large deviations occur more often; they were studied in detail by Eberhard, Kayser, and others. As we have said, by an appropriate choice of plates and developer one can strongly influence this effect in the desired direction. This effect chiefly interferes with the measurement of intensities within narrow lines and with the analysis of closely spaced lines. It can introduce a considerable error in the case where, in measuring the intensity of narrow lines, continuous spectra serve as standards. In the case of—

Fig. 19. Intensity curves of the lines in Fig. 18 and their analysis.

the effect of the step attenuator for lines of equal width is completely eliminated.

Often a continuous background extends between the lines being measured. It should be assumed that, at the position of the line, there is a background of the same intensity as to the right and to the left of it. The true intensity of the lines can be found by subtracting the background intensity from the total intensity at this point. It is clear that subtracting the ordinates of the corresponding microphotogram curves would give incorrect results.

A uniform background on the plate does not affect the correctness of the results. The principle of the method here is that equal intensities produce equal blackening, which, of course, remains valid also in the presence of a background. Uniform illumination of the plate, which is often obtained as a result of diffuse reflection in a spectrograph, and also in X-ray photographs, can strongly distort the results if the extraneous light is attenuated together with the light under investigation; and since the wavelength of the extraneous light is, generally speaking, different from the wavelength of the lines being measured, this error can never be completely eliminated.

We have already spoken above about all the difficulties arising from errors due to the photographic plate itself.

15. Measurement of the Distance between Spectral Lines

The measurement of the distance between spectral lines, necessary in determining their wavelength in the case of symmetrical isolated lines, presents no fundamental difficulties. Even for non-sharp lines the wavelength is determined unambiguously from the maximum of blackening, and a visual measurement of their distance can be performed with the aid, for example, of a comparator with high accuracy. For asymmetrical lines the situation is complicated by the fact that the sought wavelength can be identified either with the position of maximum intensity or with the center of gravity of the intensity curve. However, owing to asymmetry, this procedure, especially for broad lines, is not sufficiently reliable. High accuracy can be achieved only by replacing visual measurements with microphotometric ones. If it is desired to determine not the maximum but the center of gravity of the curve, then it is necessary to measure the intensity distribution within the line.

Similarly, when measuring the distance between two neighboring lines, the microphotometric method has a great advantage. As we have already noted (in the note to p. 951), in a visual determination of the distance between two neighboring lines, owing to the fact that lines with their extended edges partially over-

...are curved, we obtain exaggerated results. The microphotometric curve also shows this apparent attraction of lines. However, from such a curve one can derive the intensity curve and then, by an analytical method, both curves can be separated and their true spacing found.

16. Photographic measurement of absorption

Since in the present chapter photographic methods are considered for measuring intensity when the difference in wavelength is small, mention should also be made here of the photographic method for measuring absorption, which relates to measurements of the intensity of one and the same wavelength. In absorption measurements in the region of short wavelengths, especially when absorption depends strongly on wavelength, one has to work with a narrow slit of the spectrograph. Since this greatly reduces the intensity of the continuous spectrum, thermal methods prove unsuitable, and photographic methods should be preferred to them.

The execution of the measurement remains, in all details, the same as we described it in this chapter for the comparison of intensities. Both there and here, when calibrating the photographic plate, one must, depending on the circumstances, prefer one or another of the named methods.

Often, when using the photographic technique, blackening marks are made by means of a rotating sector, and equality of blackening is established by eye. The use of alternating light should be rejected on principle, and the visual evaluation of blackening is associated with all the unpleasant consequences of the subjective method.*

III. Photographic photometry of light of different wavelengths

17. Principle of the method

The methods described in the preceding chapter are suitable only for photometry of light of the same or nearly the same wavelength. We shall consider here a further development of this method, making it possible to compare the intensities of different wavelengths (heterochromatic photometry). The principal difficulty of heterochromatic photographic photometry lies in the specific sensitivity of the photographic plate. As is known, the blackening produced by the same energy at different wavelengths turns out to be different. Conversely: in order to obtain the same blackening of a photographic plate by means of light of different

* In the near future we shall publish a description of a very simple apparatus for applying blackening marks by means of a stepped wedge.

wavelengths, unequal amounts of energy are required. This well-known property of the photographic plate is nevertheless often overlooked, as a result of which the intensity of lines lying in spectral regions of low sensitivity of the plate is underestimated. Spectral sensitivity depends strongly on the type of plates, and even within one type it may differ noticeably for different plates. It is therefore impossible to speak of determining the spectral sensitivity of a given type of plates; each plate must be calibrated separately.

The second difficulty in heterochromatic measurements of intensity is due to the specific transmission of the optical apparatus. Thus, because of absorption in a flint-glass prism, violet lines are weakened more than blue ones; likewise, reflection causes a weakening of the light that depends on wavelength. This difference may be substantial for apparatus with many prisms, or for a grating.

Both of the above-mentioned wavelength-dependent effects can be eliminated by means of a “normal” lamp of constant source with a known spectral distribution of energy.

Our task consists in comparing the intensities of two lines. For this purpose, on the same plate on which the lines have been photographed, the spectrum of the normal lamp is also photographed, and the blackening of each of the two lines is compared with the blackening shown by the spectrum of the normal lamp for the given wavelengths. Since the distribution of intensities in this continuous spectrum is known, from the comparison of the blackenings one can draw a conclusion about the true ratio of the intensities of the two lines.

A monochromatic comparison of the intensity of each line with the corresponding part of the continuous spectrum can be carried out by one of the methods considered in the preceding chapter. For this it is necessary to photograph the line spectrum or the continuous spectrum at definite intensity ratios. The principle of the method presupposes that the light under investigation and the light of the normal lamp are weakened by the spectral apparatus to the same extent. Light beams emerging from the slit in different directions fall on different parts of the prism or grating, and consequently they may be weakened to different degrees. A correct comparison can be made only if in both cases the beams fill the spectral apparatus in the same way.

18. The normal lamp and its calibration

As a normal lamp there may be used a black body, the energy distribution of which at a known temperature is calculated according to Planck’s law. A black body is usually served by a hollow shell, i.e., a furnace, the temperature of whose walls is the same everywhere. However, in order that the intensity

radiation in the region of the spectrum acting on the photographic plate is sufficient, it is necessary to bring the furnace to a very high temperature, so that the use of a furnace satisfying the necessary conditions is both complicated and expensive. Moreover, measurements of such a high temperature with the accuracy necessary for calculating the energy distribution are rather difficult.

A simple method for realizing an almost black body at high temperature was given by Gening and Geize. They used the radiation emerging from a small aperture in a strongly heated tungsten sphere. When using such a normal lamp, difficulty arises not only in the need for an accurate temperature measurement, but also because this aperture can be regarded as black only approximately. A very inconvenient circumstance is that the black radiating surface is very small, while the adjacent parts of the tungsten sphere emit light of a different spectral composition.

For these purposes one may use any lamp, provided only that its intensity distribution over the spectrum is completely reproducible. This distribution must be determined by preliminary calibration.

It is expedient to use a tungsten lamp with a flat filament, or an ordinary gas-filled lamp, to whose terminals a constant and not very high voltage is applied. The requirements imposed on the constancy of the voltage are rather high, since a change of voltage by \(0.1\%\) already causes a noticeable change in intensity, especially for small wavelengths. In addition, attention should be paid to the fact that many minutes must pass after the lamp is lit before a stationary state of the radiation is established. The voltage must not be too high, so that the temperature of the filament does not change during use. Normal lamps suitable for measurements in the ultraviolet region must be made of quartz or be provided with a quartz window, so that the intensity transmitted by them is not too small.

We shall describe two methods of calibrating lamps.

1. The principle of the first method is as follows. By means of an auxiliary lamp and a monochromator, monochromatic images of the second slit of the apparatus are cast. By means of a thermoelement and a galvanometer, the relative intensities of these images are determined. Then, with the aid of some spectral apparatus, photographically or by another method, the intensity of the normal lamp is compared, for different wavelengths, with the intensity of the monochromatic images. In this way the selectivity of the spectral apparatus is completely eliminated. The setup usually used at the Utrecht Institute for calibrating a normal lamp is shown in Fig. 20. The auxiliary lamp \(Q\), by means of the lens \(L_1\), is projected onto the entrance slit \(S_1\)

of the double monochromator \(M\). By setting the screw \(H\) to various wavelengths, one can send different spectral regions through the exit slit \(S_3\) onto the thermopile and determine the ratios of their intensities. If the thermopile is not absolutely black, a correction must be introduced for its selective reflection. The Van Cittert double monochromator\(^9\) consists of two identical monochromators, which are arranged mirror-symmetrically with respect to the plane of the slit \(S_2\), which is at the same time the exit slit of the first and the entrance slit of the second monochromator. By moving this slit in its plane by means of the above-mentioned screw \(H\), one can set the instrument to any wavelength. This double monochromator has a number of advantages: first, high spectral purity; second, the direction of the emerging rays is the same for all wavelengths; in addition, if the slit \(S_2\) is removed, white light can be obtained from \(S_3\), whose spectral distribution is known from the above-mentioned measurements with the thermopile.

Fig. 20. Diagram of the setup for calibrating a standard lamp.

Both of the latter properties of the spectrograph are achieved by two auxiliary lenses placed on both sides of the slit \(S_2\). The lens \(L_2\) casts into \(P\) an image of the slit \(S_3\). If \(S_2\) is removed, the white image in \(P\) can be regarded as a light source of definite spectral composition. The spectrum of this source can be photographed by means of any spectrograph (\(S_4\) denotes its entrance slit). In order that rays of all wavelengths fill the aperture of the spectrograph in the same way, the light emerging from \(P_1\) is focused by means of the lens \(L_3\) onto the white diffusely reflecting surface \(W\), which serves to illuminate the slit \(S_4\) of the spectrograph.

In this way one is freed from the influence of the chromatic aberration of the lenses \(L_1\) and \(L_2\). Unfortunately, this method imposes high requirements on the achromaticity of the remaining lenses, which are difficult to satisfy completely. It is possible, however, at the cost of time, to photograph successively different regions of the spectrum without removing the slit \(S_2\), and to use the possibility of sharply setting the monochromator to any wavelength. Then the standard lamp \(N\) is placed at \(P\), and its spectrum is photographed with the aid of the same spectrograph. By placing a diaphragm in front of the white surface \(W\), one can, if desired, calibrate separate parts of the filament of the standard lamp.

By one of the methods indicated in Ch. II, the ratio of the intensities of the spectrum \(P\) and of the standard lamp is determined for a whole series of wavelengths. For the determination of wavelengths in continuous

in the spectra, it is best to superpose upon them some known, for example helium, line spectrum. Since the path of the light of both sources from \(P\) to the photographic plate is the same, the light of both sources is weakened by diaphragm stops, absorption, and reflection for all wavelengths in the same ratio. The ratio of the intensities of a given wavelength of the spectrum of the standard lamp and of the source \(P\) will in reality be the same as that given by the blackening of the photographic plate. The distribution of intensities for the spectrum of \(P\) is determined by a thermopile. From the photographic measurements the ratio of the intensities of the spectra \(P\) and \(N\) for different wavelengths is known. From these data one can calculate the distribution of intensities for the spectrum \(N\).

Into the distribution of intensity obtained in this way there also enters, however, the dispersion of the monochromator. For simplicity, we have throughout spoken of the intensity for a definite wavelength, although it is meaningful to speak of energy in a definite interval of wavelengths. Where one speaks simply of the distribution of energy in a spectrum, one should always understand the energy per unit interval of wavelengths. The energy measured by a thermoelement, however, is the energy of a definite region of wavelengths, specified by the width of the slit, the dispersion of the prism, and the focal distances of the lenses of the monochromator. This region may be either measured directly or found by differentiating the dispersion curve. The energy measured by the thermoelement in any units must be divided by this region of wavelengths in order to obtain the energy falling on a unit interval of wavelengths.

Instead of the photographic method, one can successfully use thermoelectric and photoelectric measurements of radiation for comparing \(P\) and \(N\), i.e. for calibrating the standard lamp. For this purpose, instead of the photographic plate in the spectral apparatus one must place a thermopile or a photocell and use them to carry out direct measurements. These methods may be combined and, for example, a thermoelement may be used for measuring infrared and long-wave light, and a photocell or photographic plate for short-wave light. As an example, in Table 1 we give the calibration of a standard lamp. It is a four-volt gas-filled lamp, operated at a voltage of \(2.6\ \mathrm{V}\). In the first column are placed the wavelengths in microns; in the second, the measured intensities per unit interval of wavelengths in arbitrary units. We shall dwell in more detail on the figures in columns 3 and 4.

Although the tungsten filament certainly is not a black body, one may nevertheless ask whether the relation between intensity and wavelength \(J(\lambda)\) cannot be represented in the form of Planck’s formula, by specifying a suitably chosen temperature. For

of the wavelength interval under consideration and at the temperatures usually encountered can, with sufficient approximation, be taken—

TABLE 1

Wavelengths in microns	Observed intensity values	Calculated intensity values	Difference between observed and calculated intensity values, in %
0.70	386	390	− 1
0.68	311	314	− 1
0.66	272	272	0
0.64	226	225	+ 0.5
0.62	184	183	+ 0.5
0.60	148	147	+ 1
0.58	115	115	0
0.56	88.6	87.0	+ 2
0.54	65.6	64.8	+ 1
0.52	45.9	46.8	− 2
0.50	31.6	32.7	− 3
0.48	21.3	22.0	− 3.5
0.46	14.0	14.1	− 1
0.45	10.9	10.7	+ 2
0.44	8.4	7.9	+ 6
0.43	6.4	6.3	+ 1.5
0.42	4.75	4.97	− 2.5

to replace Wien’s formula, namely to represent the quantity \(J(\lambda)\) by the following formula:

\[ J(\lambda)=c_1\lambda^{-5}e^{-c_2/\lambda T} \]

or

\[ \log[\lambda^5J(\lambda)]=\log c_1-c_2/\lambda T. \]

To verify the correctness of this relation and at the same time to establish the value of \(T\), one plots \(\log \lambda^5J(\lambda)\) as a function of \(1/\lambda\). In the example we are considering, we then find that the points lie almost on a straight line, as is required by the equation. From the slope of the straight line we find \(T=1980^\circ\). The values of \(J(\lambda)\) calculated for this temperature are collected in the third column of Table 1, while the fourth contains the percentage deviations between the measured and calculated intensity values. It should be noted that the temperature \(T\) in Wien’s formula is not the true temperature of the incandescent filament. It is a fictitious quantity—the so-called color temperature.

The expression of the intensity distribution by means of the equation given above has the advantage that the function \(J(\lambda)\) is expressed through a single constant—the color temperature. The other two constants \(c_1\) and \(c_2\) need not be determined, since \(c_1\) cancels out in all comparative measurements, while \(c_2\) is a known universal constant \((1.43\ \text{cm}\cdot e)\). Since

PHOTOGRAPHIC PHOTOMETRY

if in practice it is important to use the standard lamp at different brightnesses, then it is necessary, in the manner indicated above, to determine its color temperature for different values of the current strength and to present the relation between them in the form of a table or a graph. It is noteworthy that the relation between color temperature and current strength is linear over fairly wide limits.

1. The second method of calibrating a standard lamp is based on the use of a calibrated optical pyrometer, by means of which one can determine the black temperature of the lamp for a narrow range of wavelengths. By the black temperature of an incandescent body we mean, as is well known, the temperature which an absolutely black body would have to possess in order, in the given spectral interval, to have the same brightness.

For a lamp with a flat tungsten filament, where the observation is carried out in a direction perpendicular to the radiating surface, this problem is easily solved. The further problem consists in determining the color temperature of the standard lamp from its black temperature. The relation between black and color temperature may be taken from the data available in the literature^10 concerning the radiation of tungsten at different black temperatures in a direction perpendicular to the incandescent surface. For an ordinary gas-filled lamp this relation, however, is complicated. First, we are dealing here with radiation at different angles, and secondly, with the radiation from the inner side of the spiral and also with reflected radiation. The relation between current strength and color temperature therefore depends not only on the material, but also on the geometrical shape of the spiral. These circumstances, however, can be allowed for by calculation, and, for a given shape of the spiral, it is possible, at least approximately, to derive from the pyrometrically measured black temperature of the outer side of the spiral the desired color temperature of its total radiation. As a result of the pyrometric calibration of the standard lamp, we likewise obtain a linear relation between current strength and color temperature, from which, with the aid of Wien’s formula, the desired distribution of intensities may be calculated. With regard to a gas-filled lamp it should be noted that a noticeable difference is obtained depending on whether the total radiation also includes the ends of the spiral or whether the latter are excluded. It should therefore always be indicated whether the color-temperature data refer to the whole incandescent surface or only to its central part.

The majority of lamps with a flat filament are in this respect much more convenient, since in them the ends of the filament are bent behind the flat part and are thus screened. As pyrometric measurements have shown, with such a construction the temperature of the entire active surface is practically the same.

If one asks the question which of the two indicated meth—

...of calibration should be preferred, one must answer that each of them has its merits and shortcomings. Undoubtedly, the first method is more direct, while the second is simpler in execution. At the Utrecht Institute at the present time they use exclusively the pyrometric method, after very careful measurements by both methods led to concordant results both for the lamp with a flat filament and for the gas-filled one. Pyrometric calibration is based on visual measurements. It is therefore subjective, and only an experienced observer can achieve the desired accuracy. As far as possible, especially careful results should be widely used by employing a standard lamp for the secondary calibration of other lamps. This can be done by an objective comparison of the spectra of both lamps by one of the methods described above. In practice the most applicable is a very simple method with two filters, practiced at the Utrecht Institute.

At its basis lies the supposition that the second light source being calibrated possesses a color temperature in that spectral interval to which the calibration also refers. The higher this temperature, the richer the short-wave region will be. A comparison of the intensities of two portions of the spectrum, isolated by filters, makes it possible to determine the color temperature of the source. For this purpose the filters must be calibrated, using a standardized normal lamp whose color temperature can be varied within certain limits. In this way a relation is established between the ratio of the intensities of both sorts of radiation (isolated by the filters) and the color temperature of the standard normal lamp.

This relation remains valid for any light source for which the ratio of intensities in the indicated spectral interval can be expressed through a color temperature; consequently, it is valid also for the source being studied. It is preferable to choose a filter made of colored glasses, so that it does not change with time, and to use spectral regions differing as strongly as possible in wavelength. It is self-evident that intensity measurements may also be made with the aid of a selective apparatus, in particular with a photoelement. Recently we have been using, as normal lamps, sources with a line spectrum. For more detail on this see the author’s book[^1].

19. Measurement of intensities with the aid of a normal lamp

The measurement is carried out in such a way that the spectrum under investigation and the spectrum of the normal lamp are recorded on one and the same plate. One, at least, of these two spectra must be weakened in definite ratios. Since

usually the standard lamp is constant and gives a continuous spectrum, it is simplest to attenuate the standard spectrum by reducing the slit. In principle it is desirable that the exposure time for both spectra be approximately the same, although the permissible exposure ratio may be expressed by a small number (see p. 948). It should be borne in mind that the blackening curve for very narrow spectral lines may be modified, just as intermittency of illumination may entail a complication.

In evaluating the results it should be taken into account that the calibration of the standard lamp gives the value of the intensity per unit interval of wavelengths, whereas the blackening of the spectra under study is determined by the intensities per unit area. From the calibration curve it is therefore necessary to calculate the intensity per unit area, taking into account the dispersion of the spectral apparatus. If \(\lambda\) is the wavelength and \(x\) the distance on the plate, then one should determine the dispersion \(\frac{dx}{d\lambda}\), which can be done graphically or numerically, and divide the intensity values given by the calibration by this differential ratio. Next, one has to take into account the possible non-achromatic image of the spectrograph slit. In this case the image of the slit for one wavelength is larger than for another. In addition, for a line spectrum the intensity per unit area is inversely proportional to the square of the linear dimensions of the image, whereas for a continuous spectrum this quantity is inversely proportional to the first power. Since the measurement consists in establishing equality of blackening, i.e. equality of intensity per unit area, the obtained numbers should be multiplied by the first power of the linear magnification of the spectrograph for the various wavelengths. If the spectral line has appreciable true width, then the spectrum should be photographed with a wide slit. In this case it is necessary to determine only the values of the maxima on the blackening curve. If the lines being compared are not isolated, so that one has to work with a narrow slit, then the intensity curve of the line should be integrated. Usually one may confine oneself to an approximate method of integration, i.e. to a simple multiplication of the maximum value by the half-width of the line. The use of a standard lamp presupposes equal attenuation of the radiation of this lamp and of the source being investigated as they pass through the spectrograph. This attenuation is partly caused by diaphragming by the jaws of the slit. Such limitation of the beams for both sources can be identical only when their image is achromatic, i.e. is produced by an achromatic lens or a mirror. One must, however, bear diffraction in mind. Depending on the conditions, the treatment of blackening measurements may be carried out in various ways.

The following cases may be distinguished:

1) the blackening curves of the normal lamp and of the source under investigation run parallel, but the spectral region studied is so large that the blackening curves for different wavelengths are not parallel; 2) the blackening curves for different wavelengths and for both spectra turn out to be parallel; 3) the blackening curves for both spectra, as well as for different wavelengths, are not parallel.

1. In the first case one should apply the method indicated on p. 955, i.e., compare the blackening of each line with the blackening of the continuous spectrum for the same wavelength. Let us explain this method of measurement by an example. For this purpose let us choose the lines 4047, 4359, 5461 Å of the mercury triplet, emitted by an ordinary mercury lamp. The difference in wavelengths in this case is so great

Fig. 21. Microphotometric curves of the mercury lines 4047, 4359, 5461 Å
(for each line there are six intensity steps).

that the blackening curves of these three lines deviate considerably from parallelism. For completeness, we shall carry out the measurements by both methods, i.e., once with the use of a step attenuator, and a second time using the normal lamp as an auxiliary source and changing its intensity by the slit-widening method.

1a. Step attenuator. Using one and the same step attenuator, a line spectrum and the spectrum of the normal lamp are photographed successively on one photographic plate with the same exposure. The line spectrum is deliberately not photographed with a very narrow slit. The plate is then microphotometered. The curve of the microphotogram of all three lines is given in Fig. 21. For each of our three lines all six steps have been measured, corresponding to the relative intensities 100 : 16 : 34 : 61 : 74 : 100. From the curves the blackenings were calculated and plotted on the curves shown in Fig. 23 (mercury). In exactly the same way, the spectra of the normal lamp, taken at six different intensities, were photometered perpendicular to the na-

direction of dispersion for wavelengths 4047, 4359, 5461 Å. In Fig. 22 the curves of their microphotogram are given, and for this case the derived blackening values and the blackening curves constructed from them are shown in Fig. 23 (normal lamp).

Since the light of the mercury lamp is not continuous, the exposures for the normal and mercury lamps were approximately the same, and the lines were taken with a sufficiently wide slit; one may expect that the blackening curves for both lamps at each wavelength will run parallel. Indeed, in Fig. 23 one can be convinced of this; just as easily one notices the non-parallelism of the curves belonging to different

Fig. 22. Microphotometric curve of the normal lamp for wavelengths 4047, 4359, 5461 Å (for each wavelength 6 intensity steps).

wavelengths. From the distances between the two curves, read in the horizontal direction, the intensity ratios are found on the scale \(\log i\). For the lines 4047, 4359, 5491 Å the intensity ratios of the normal and mercury lamps are respectively \(2.30 : 2.16 : 0.90\). These numbers also give, for the unattenuated mercury lines and the unattenuated normal lamp, the ratio

Fig. 23. Blackening curves of the mercury lines 4047, 4359, 5461 Å and of the normal lamp for the same wavelengths.

intensities per unit area of the plate. The color temperature of the normal lamp was determined to be \(2910^\circ\). Hence, by means of Wien’s formula, one may calculate, for the given wavelengths, the ratio of intensities on equal spectral sections. We obtain: \(21.8:35.6:113\). It goes without saying that only the ratio of the numbers plays a role here; their absolute value has no meaning. These numbers must be corrected by taking into account the dispersion of the spectral apparatus (grating). In our case, for the three wavelengths, it is equal to \(75:74:73:72\). Thus, for the normal spectrum, the ratio of intensities per unit surface is

\[ \frac{21.8}{74}:\frac{35.6}{73}:\frac{113}{72}=29.6:48.8:157. \]

Multiplying these numbers by the ratios found above, we find for the three lines

\[ 29.6\cdot 2.30:48.8\cdot 2.16:151\cdot 0.90=68:105:141. \]

Equating the intensity of the strongest line to 100, we obtain, as the final result for the required ratio of intensities of the three mercury lines,

\[ 48:74:100. \]

Fig. 24. Microphotometric curves of three mercury lines 4047, 4359, 5461 Å.

1b. Attenuation by changing the slit width. We illustrate this method of measurement using the same light sources, spectrograph, and normal lamp (with color temperature \(2910^\circ\)) as in 1a. On the photographic plate, the line spectrum was photographed once, and the spectrum of the normal lamp seven times at different slit widths, but with exactly the same exposure. We do not reproduce here the photograph, which has an appearance analogous to that shown in Fig. 14. The lines were still photographed with a slit of sufficient width.

In Fig. 24 are given the microphotographic curves of three lines. The region of the plate between the lines of interest to us was omitted in the microphotography. For the three maximum values we obtain blackenings \(0.26:0.85:0.42\).

In Fig. 25 are shown three microphotographic curves of the normal spectrum for wavelengths 4047, 4359, 5461 Å, measured perpendicular to the direction of dispersion. The relative values of the slit width were \(10:20:5:15:40:60:100\). From these curves, blackening curves were constructed for the three wavelengths (see Fig. 26). From the figure one easily notices the different inclination of these curves. In this graphical construction, the 100% intensity values for all three wavelengths are plotted at one and the same place on the \(\log i\) scale, although they are knowingly

PHOTOGRAPHIC PHOTOMETRY

are different, and, as was derived earlier from the color temperature of the standard lamp and the dispersion of the spectral apparatus,

Fig. 25. Microphotometric curves of the spectra of the standard lamp, taken with seven different slit widths, 10, 20, 5, 15, 40, 60, 100; photometry was carried out perpendicular to the direction of dispersion.

they are in the ratio \(29.6 : 48.8 : 157\). This circumstance could have been allowed for by a parallel displacement of the curves, but in practice it is much simpler to introduce the necessary correction into the final results. Thus, without further delay, we plot the three blackening values for the three lines, each on the corresponding blackening curve of the standard lamp, and find three values of the (uncorrected) intensities: \(37.0,\ 34.3,\ 14.4\). Multiplying these numbers by \(29.6,\ 48.8,\ 157\), we obtain \(110,\ 167,\ 226\). Taking the strongest line as 100, we obtain the final result for the desired intensity ratio:
\(48 : 74 : 100\).

Fig. 26. Blackening curves constructed from the data of Fig. 25, for determining the intensity ratios of three mercury lines \(4047,\ 4359,\ 5461\ \text{Å}\).

The described methods require measuring the blackening of a continuous spectrum in as many places as there are studied

lines. Therefore, if the line spectrum is rich in lines, one has to carry out a large number of measurements and calculations. Often, however, there is an opportunity to reduce them. It often happens that, in the wavelength region of interest to us, the blackening curve of the normal spectrum has a sufficiently smooth course. In this case one may confine oneself to the necessary measurements for a few wavelengths and, on the basis of the data obtained, construct a curve of a simple form (Fig. 27), from which the blackenings for intermediate wavelengths can be found. Even when the curve has several maxima and minima, so that a comparatively large number of points is required for its construction, such graphical interpolation gives, when there is a large number of lines being compared, a considerable saving of time.

Fig. 27. Blackening as a function of wavelength in a continuous spectrum.

2. If, in the spectral region under investigation, the blackening curves corresponding to different wavelengths are parallel, then two cases may be distinguished: either the blackening curves of the line spectrum are parallel to the blackening curves of the normal lamp, or they are not parallel. In the first case one may proceed in exactly the same way as in case 1 (p. 963). The parallelism of the blackening curves for different wavelengths introduces no changes whatsoever into the method. Whether or not the condition is fulfilled is of no significance for carrying out the method described. This parallelism may, however, be used in determining the sensitivity of the plate as a function of wavelength. More precisely, in the present case it is not the sensitivity of the plate that plays a role, but the product of this quantity by the transmittance of the spectral apparatus. In what follows we shall call this product the “sensitivity of the apparatus.” Exactly as in the second case, when the curves of the line spectrum and of the normal lamp are not parallel, one may proceed according to the plan outlined above.

Let us determine the sensitivity of the apparatus as a function of wavelength for the normal lamp and assume that the same sensitivity relation is preserved for the investigated lines, although the blackening curves have a different slope, owing, for example, to discontinuity of the illumination.

As an example of such a type of intensity measurement, let us dwell on the problem already examined, concerning the determination of the intensity ratio of the zinc triplet 4680, 4722, and 4811 Å. These lines are excited by a zinc spark. Owing to the intermittency of the illumination, the blackening curves of the lines have a somewhat different slope than the curves of a normal lamp. In examining this case we neglected the dependence of the sensitivity of the apparatus on wavelength, and the intensity ratio was determined first with a step attenuator, and then with the aid of an auxiliary source. The first method gave the ratio \(31:79:100\); the second, \(35.5:79:100\). These results must now be corrected.

2a. When using a step attenuator, both the line spectrum and the spectrum of the normal lamp are photographed on one plate; both spectra are attenuated in definite ratios. Fig. 28 gives a reproduction of this photograph. The lower part had already been shown in Fig. 11 and served to obtain the numbers mentioned above. From the upper part one must find the values of the sensitivity of the apparatus for three wavelengths. For this purpose microphotograms are taken for the three lines 4680, 4722, 4811 Å perpendicular to the direction of dispersion. We do not give the microphotographic curves here; their appearance is quite analogous to the curves in Fig. 22. The blackenings determined from these microphotograms serve for constructing three blackening curves, shown in Fig. 29. If the dependence of sensitivity on wavelength is disregarded, the following intensity ratios are found from the distances between these (parallel) blackening curves: \(100:102:94\). The true intensity ratio is calculated by Wien’s formula, knowing the color temperature (\(2675^\circ\)); they are equal to \(29:31:35\). In the present case there is no need to introduce a correction for dispersion, since for these wavelengths the dispersion is practically the same. The sensitivities of the apparatus for these wavelengths are therefore related as:

Fig. 28. Zinc triplet and spectrum of a normal lamp, photographed with a step attenuator.

\[ \frac{100}{29}:\frac{102}{31}:\frac{94}{35}. \]

The final values of the intensity ratios of the three lines will be:

\[ \frac{31}{3.45} : \frac{79}{3.29} : \frac{100}{2.69}. \]

2b. When applying the method with an auxiliary source, we use the photograph given above (Fig. 14). The auxiliary source was a standard lamp with a known color temperature (2675°); therefore the blackening curves of the spectrum of the standard lamp can be used to determine the relative sensitivity of the apparatus. For this it is necessary to microphotometer the curves perpendicular to the direction of dispersion for three wavelengths: 4680, 4722, and 4811 Å. The curves of the microphotogram look like those shown in Fig. 25. With the aid of these curves, three blackening curves were constructed; their spacing gives the values of the relative sensitivity of the apparatus, in exactly the same way as was described above. Thus we obtain results which, within the limits of measurement errors, coincide with those given above. If the investigated spectrum is rich in lines, one can, with great success, find the relative sensitivity for several wavelengths, construct the sensitivity curve of the apparatus, and from it determine the relative sensitivity for any intermediate wavelength.

Fig. 29. Blackening curves of the spectrum of the standard lamp in Fig. 28.

For spectra very rich in lines, it is advisable to apply the graphical method not only for determining the sensitivity, but also for finding a number of other quantities related to wavelength by a logarithmic dependence.

3. In the case where neither the blackening curves corresponding to different wavelengths nor the blackening curves of the standard lamp and of the lines are parallel to one another, the photographic

the method for determining intensities is in general inapplicable. In practice, one must decide from case to case what small permissible deviation from parallelism is acceptable. Insufficient parallelism of the blackening curves for different wavelengths can be largely eliminated by a suitable choice of plate or by their sensitization. The nonparallelism of the blackening curves of a line spectrum and of a standard lamp is explained either by the small width of the line (p. 931) or by the discontinuity of the light. In the first case the results can be corrected by increasing the slit width, which, however, for very close lines leads to their superposition. A correction can be introduced by taking a photograph with both a narrow and a wide slit. The influence of the discontinuity of the light can be neutralized by using a rotating sector when photographing the standard lamp.

Literature

1. Moll, Kon. Akad. v. Wet. Amsterdam, März 1920.
2. See Cabannes, Ann d. Physik, 1923.
3. Proceedings of the Congress of Physicists, Leningrad, 1919.
4. Elfiot, Disser., Utrecht, 1920.
5. Hansen C., Zeits f. Phys., 29, 356, 1924.
6. Bongers A., Disser. Utrecht, 1927.
7. Burger D., Zeits. f. Phys., 54, 649, 1929.
8. Henning F. und Heuse W., Zeits. f. Phys. 16, 63, 1923.
9. Van Cittert P. H., Rev. Opt. 2, 57, 1923.
10. Forsythe W. und Worthing A., Astr. J. 61, 146, 1925.
11. Ornstein-Moll-Burger, loc. cit., p. 128.

1. Hansen. ↩