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Methods for Determining Atomic Constants1
F. Kirchner, Kiel
I. Introduction
The precise determination of the electron charge and of Planck’s constant belongs among the most important tasks of atomic physics. It is therefore understandable that, over time, a whole series of physicists have expended much effort in devising ever new methods for the experimental measurement of these constants and in continually increasing the accuracy of their determination. And when the author, who belongs to the number of these physicists, was asked to give a survey of the results obtained in this field and at the same time to present a picture of the accuracy achieved at the present time, he saw clearly from the very outset the difficulties that arise here—with greater clarity than anyone who has not himself worked in this field. Indeed, the successful realization of a “precision measurement” is connected not only with the development of an appropriate and technically successful arrangement of the experiments, not only with a thorough processing of the most careful observations and reports and with a conscientious determination of the final result, but also with a critical assessment of the reliability of this result. It would be very simple to evaluate the accuracy if it depended only on “random errors of measurement,” to which the statistical theory of errors is applicable.
Unfortunately, however, random errors of measurement usually play a secondary role in comparison with other sources of error, which cannot be encompassed mathematically. As a result, in the final analysis the accuracy of the final result obtained generally cannot be measured or calculated—it must be “estimated,” and the result of such an estimate depends not only on the true accuracy of the experiment alone, but also on the observer’s experience and self-criticism. Remarks of this kind, of a personal character, which, however, are essential in all experiments that aim to bring results to the limit of the possibilities of observation and measurement, are very rarely encountered in the publications themselves, since in these publica-
usually strive for maximum “efficiency,” and, moreover, there is no doubt that the description of a difficult precision experiment in a journal—even when the editors do not compel the author to shorten the article—can never give a complete picture of all sources of unreliability. Therefore there is no possibility of checking the predominantly subjective data concerning the influence on the result of error sources that are not amenable to statistical accounting.
A further difficulty consists in the fact that data on accuracy have different meanings for different observers from the very outset. One observer seeks to indicate his “limits of error” and his “maximum error” in such a way that it is clear that the true value of the sought quantity lies within these limits; another gives only the “probable” error of the observation, computed by Gauss’s method of least squares, which says nothing about the suitability of the method but is merely a measure of the reproducibility of the result obtained by means of the given special method; finally, a third occupies an intermediate position between the two extreme possibilities just mentioned. (The “probable” error is 0.6745 of the “mean” error in the Gaussian theory of errors. The probability that the true error is greater than the “probable” one is equal to 1:2.)
Thus, the uncertainty hidden in statements of accuracy greatly hampers the critical comparison of measurement results carried out by different authors and, in most cases, by different methods. If it turns out that different results do not agree with one another, then, of course, one must not simply “average” the deviations and indicate the result as the “most probable”; rather, it is necessary to wait until new measurement results appear that are sufficiently reliable to remove the difficulty that has arisen. But even in the case when a whole series of measurement results by different observers, within the limits of their own “error bounds,” agree with one another in a particularly reasonable way, and one may be certain that they all arose independently of one another, it still cannot be concluded from this that the remaining deviations of the individual results from one another are due only to random errors and that, therefore, the statistical theory of errors may be applied. Thus it cannot always be regarded as irreproachable when, by mechanical averaging of different individual measurements of this kind, a “most probable” value is obtained; likewise, taking into account weights corresponding to the indicated errors cannot remove this difficulty, since the authors’ assessment of the errors is subjective. Moreover, a considerable overestimation of the true accuracy and reliability of the result occurs when, according to the rules of the statistical theory of errors, the “probable” error of the averaged “most probable” value is calculated and attached to the final result—as has unfortunately become customary recently—since there are no prerequisites for applying the Gaussian theory here at all.
II. THE ELECTRON CHARGE
We shall begin our survey by considering the determination of the electron charge for the following reasons: first, because only the elementary electric charge can be measured separately—all the other atomic constants always occur in combinations either with the elementary charge or with one another; second, because absolute measurements of X-ray wavelengths, which at present give a more satisfactory value of the elementary charge than do direct measurements, are so precise that, when discussing measurements of \(\frac{h}{e}\) from the short-wavelength limit of the X-ray spectrum and other measurements into which X-ray wavelengths enter, it is expedient to regard these wavelengths as known with the same certainty as the other auxiliary constants entering into these measurements (for example, the velocity of light).
- The oil-drop method. For an entire generation of physicists, Millikan’s determination of the elementary electric charge served as a model of work of the highest degree of thoroughness and masterful precision.
The value obtained by Millikan, \(e = 4.774 \cdot 10^{-10}\) CGSE, for the last two decades occupied an honored place in all physics textbooks and handbooks. As the greatest possible error, Millikan\(^{76}\), on the basis of his careful measurements over many years, completed in 1916, indicated 1 per mille; this maximum error, \(e = (4.774 \pm 0.005)\cdot 10^{-10}\) CGSE, was considered generally accepted. Millikan’s authority and the extremely high precision of measurement claimed by him led to a very peculiar situation: whereas Millikan’s oil-drop experiment was repeated countless times in all physics laboratories, for 15 years no one seriously undertook a critical check of all sources of error in Millikan’s determination of \(e\). R. T. Birge\(^{20}\) performed only a recalculation on the basis of changed values of the speed of light and on the basis of the difference between the international and absolute volt, and Millikan\(^{77}\) himself in 1930 gave the newly recalculated value \(e = (4.770 \pm 0.005)\cdot 10^{-10}\) CGSE. Moreover, when in 1928, thanks to E. Backlin’s\(^{3}\) pioneering work on the “Absolute measurement of X-ray wavelengths,” certain doubts arose about the correctness of Millikan’s value of \(e\), since Backlin computed from his measurements a significantly larger value, many physicists still long adhered to the old value, and in the critical survey\(^{20}\) of the most probable values of physical quantities that appeared in 1929, Backlin’s “suspiciously high” value was rejected, among other reasons, on the grounds that it had been obtained in pioneering work and that it was quite probable that this pioneering work contained various unforeseen systematic errors. Almost another 8 years passed before it was shown beyond any doubt that, on the contrary, it was precisely in Millikan’s classic work that there was an unexpected systematic error, which
many times greater than the accepted limit of error of this work: namely, the unreliability of the value of the coefficient of internal friction of air, which, as is known, enters to the power \(3/2\) in the result of determining \(e\) by Millikan’s method. In his final determination of \(e\), Millikan relied only on Harrington’s\(^{53}\) measurements from Chicago. These measurements, in Millikan’s opinion, were at that time the only ones in terms of their reliability and accuracy, which he estimated at \(0.05\%\); the careful measurements of Vogel from Berlin\(^{98}\) and Gil from Halle\(^{49}\), which (see Table 1) gave a value \(0.5\%\) larger, were rejected by Millikan as less reliable. The Harrington value accepted by Millikan was then, for 16 years, regarded without criticism as correct. Only in 1932 did K. Shiba\(^{91}\) draw attention to the fact that this value was probably too small. In the interval between 1936 and 1938, several new measurements of the coefficient of viscosity of air were carried out by various authors; in part the rotating-cylinder method was used, and in part the capillary method. The results of the various new measurements are compared in Table 1 and in Fig. 1 with the former values.
Fig. 1. Old and new measurements of the coefficient of viscosity of air
Table 1
Measurements of the coefficient of internal friction of air
| Author | Method | \(\eta_{23^\circ}\cdot 10^4\) CGSM |
|---|---|---|
| Vogel\(^{98}\) 1913 | Oscillation method | 1.833\(^{1}\) |
| Gil\(^{49}\) 1914 | Capillary | 1.831\(^{1}\) |
| Harrington\(^{53}\) 1916 | Rotating cylinder | 1.8227 |
| Kellström\(^{61}\) 1936 | Rotating cylinder | 1.8349 |
| Bond\(^{31,32}\) 1936 | Capillary | 1.8338 |
| Houston\(^{58}\) 1937 | Rotating cylinder | 1.8292 |
| Rigden\(^{81}\) 1938 | Capillary | 1.8303 |
| Banerji and Pattanaik\(^{10}\) 1938 | Capillary | 1.8333 |
All the new figures lie considerably above Harrington’s figure; specifically, the difference ranges from 3.6 per mille (Houston) to 6.7 per—
\(^{1}\) Although the results of Vogel and Gil are mentioned in Millikan’s classic work, in the numerous new works in which the question of the correct value of \(\eta\) is discussed there are no indications that these results, contrary to Millikan’s view at the time, are correct.
mille (Kellström). But precisely the measurements of Kellström and Houston, which were undoubtedly carried out with the greatest care and which, perhaps, of all the new measurements are the most satisfactory, still differ from one another by 3.1 per mille (since $\eta$ enters into the formula for $e$ to the power $3/2$, this difference corresponds to a difference in the values of $e$ of 4.6 per mille, i.e., five times the previously accepted limit of error in the determination of $e^1$).
As the mean of the results of the new measurements one obtains $\eta_{23^\circ}=1.832\cdot10^{-4}$—a brilliant confirmation of the old measurements of Vogel (1.833) and Gill (1.831). At the same time, with a value of $\eta$ larger by $0.50\%$, Millikan’s number for $e$ increases by 7.5 per mille, i.e., to $4.805\cdot10^{-10}$ CGSE. Millikan himself,^78 who recently expressed his opinion on this question, considers Houston’s value the most acceptable $(\eta=1.8292\pm0.0045)$ and therefore calculated from his old measurements $e=4.796\cdot10^{-10}$ CGSE.
Measurements carried out in 1925 by Mattauch^74 by Millikan’s method led, with the old value used by Millikan, $\eta=1.8227\cdot10^{-4}$, to the result $e=4.758\cdot10^{-10}$; if, however, one takes for $\eta$ the value 1.832 according to Vogel and Gill, this gives $e=4.793\cdot10^{-10}$. Further, in 1936 Bäcklin and Flemberg^9 published a brief communication on a new determination of $e$ by the oil-drop method. In processing their measurements they took as the correct value of $\eta$ Kellström’s number $1.8349\pm0.0027$ and obtained $e=4.800\cdot10^{-10}$ CGSE; with $\eta=1.832$, the measurements of Bäcklin and Flemberg lead to the result $e=4.788\cdot10^{-10}$ CGSE. The limit of accuracy of the measurements of Bäcklin and Flemberg is apparently the same as Millikan’s.
2. Absolute measurement of X-ray wavelengths. Let us now turn to the absolute measurements of the wavelengths of X-rays, which, as has already been mentioned, gave the impetus to the just-discussed revision of the direct measurements of the elementary electric charge and which at present have indirectly (see subsection 5 of this section) made it possible to obtain a more accurate value of $e$ than the direct measurements by the oil-drop method. The foundation for this important branch of modern measuring technique was provided: on the one hand, by X-ray spectroscopy with crystal gratings, which was brought to special precision chiefly by Siegbahn and his school, and on the other hand, by the fact, discovered by Compton and Doan, that at a strongly glancing incidence of X-rays, i.e., in the region of total reflection, diffraction spectra of X-rays can be obtained with the aid of ordinary reflection gratings. The latter method was first developed by E. Bäcklin at Siegbahn’s Institute, and then by J. Bearden in America, and was turned by them into a precision method. The principle of this method is shown in Fig. 2. X-rays issuing from an X-ray tube are first subjected to preliminary decomposition into a spectrum with the aid of a crystal grating $K$ (calcite); the preliminarily decomposed ray passes through the slit $S$ and falls on the reflection grating $G$, and from there, after specular reflection at the “glancing angle” $\varphi$
onto the photographic plate \(P\). On both sides of the reflected zero-order ray (\(C\) in Fig. 2), zero-order diffraction spectra appear at an angle \(\theta_n\) with respect to \(C\); here \(\theta_n\), exactly as in the case of an optical reflection grating, satisfies the condition
\[ n\lambda=d\{\cos\varphi-\cos(\varphi+\theta_n)\} =2d\sin\frac{2\varphi+\theta_n}{2}\sin\frac{\theta_n}{2}. \]
The difference in the arrangement of the experiments of Bäcklin\(^{3-8}\) and Bearden\(^{11,12,15,16}\) is essentially as follows: in order to fix as accurately as possible the distance \(R\), needed for measuring angles, Bäcklin placed in front of the grating \(G\) a steel knife edge, by means of which, just as in Siegbahn’s X-ray spectrograph, the part of the grating reflecting the beam was limited to the portion situated in front of the knife edge.
Fig. 2. Arrangement of the experiment for absolute measurements of the wavelength of X-rays (after Bäcklin\(^{3}\))
Further, in order to facilitate the measurement of angles, Bäcklin used rather long-wavelength radiation (the aluminum \(K\alpha\) line, \(\lambda \simeq 8.3\,\text{\AA}\)) so as to make the diffraction angles as large as possible. Bearden, on the other hand, worked with copper \(K\alpha\) radiation (\(\lambda \simeq 1.5\,\text{\AA}\)) and chromium radiation (\(\lambda \simeq 2.3\,\text{\AA}\)). In order to remove the objection that, when a small part of the reflecting grating is used, small local irregularities in the ruling of the grating may lead to appreciable errors, in his final experiments he placed the grating between two crystals of a double spectrometer in such a way that almost the entire surface of the grating was used. In this case the second crystal of the spectrometer served to measure the angle of the rays reflected by the grating; this crystal was mounted on the table of a goniometer with precision graduation, so that readings could be made with an accuracy of up to \(0.1\) sec. The gratings used by both investigators were glass gratings, ruled shallowly but carefully, with 100–300 divisions per \(1\ \text{mm}\); Bearden’s grating, moreover, was coated with a thin layer of gold. Adjustment was carried out by the usual methods employed in precise optical measurements; it required exceptional care, since absolute measurements were involved.
A check on the direct absolute measurements of Bäcklin and Bearden was made possible by the experiments of Söderman\(^{94}\) and Tyrén\(^{96}\), carried out at Siegbahn’s Institute. In these experiments, the X-ray lines of aluminum \(K\)-radiation were recorded simultaneously with optical lines in the extreme ultraviolet of other elements, which made it possible to tie in directly with the optical scale of wavelengths. Concave gratings were used; Tyrén’s grating had 576 lines per \(1\ \text{mm}\) and a radius of curvature of \(471\ \text{cm}\). In Fig. 3
Fig. 3.
Visible labels in the figure include ionic designations such as \( \mathrm{Be\,IV} \), \( \mathrm{O\,VI} \), \( \mathrm{O\,V} \), \( \mathrm{C\,VI} \), \( \mathrm{N\,V} \), \( \mathrm{B\,V} \), and \( \mathrm{N\,IV} \), along with wavelength-scale markings approximately from 15 to 105.
one of the spectra obtained by Tyrén is shown, on which one can see the aluminum \(K\)-lines in orders II—XII simultaneously with numerous standard optical lines of highly ionized elements Be, B, C, and O.
The random errors of measurement among different authors are of the order of tenths per mille, and sometimes even less. Backlin gives as the mean error \(\pm 0.12\) per mille for 56 measurements, of which 32 lie within the limits of the mean error. Bearden gives a probable error of 0.03 per mille, which corresponds to a mean error of 0.05 per mille; his maximum deviation from the mean is 0.4 per mille. Tyrén obtains, as the mean error of 68 individual measurements, 0.04 per mille.
The magnitude of the systematic error is estimated differently by different authors. The excellent agreement, immediately adjoining the optical wavelength scale, of the measurement results of Söderman and Tyrén with the measurements of Backlin and Bearden (cf. below) makes it possible to conclude with sufficient reliability that the systematic error in no case exceeds the random ones.
The results of the measurements are as follows. Backlin obtained for the aluminum \(K\alpha\)-line \(8.3395 \ \text{Å}\), Tyrén—for the same line \(8.3397 \ \text{Å}\), Söderman—\(8.3401 \ \text{Å}\). Bearden obtained for the copper \(K\alpha\)-line \(1.5406 \ \text{Å}\). His numerous measurements, carried out both by the photographic method and by the ionization method, and also with a double spectrometer, are compared with the measurements of Backlin and Söderman in Table 2. In
Table 2 (Bearden\(^{15}\))
| Cu \(K\beta\) | Cu \(K\alpha\) | Cr \(K\beta\) | Cr \(K\alpha\) | Al \(K\alpha\) | |
|---|---|---|---|---|---|
| Bearden (1929) | 0,24 (10) | 0,25 (10) | |||
| Bearden (1931) | 0,241 (26) | 0,229 (46) | 0,239 (16) | 0,245 (28) | |
| Bearden (1931) | 0,234 (4) | 0,250 (11) | 0,250 (15) | 0,255 (27) | |
| Bearden (1931) | 0,264 (30) | 0,257 (49) | 0,253 (3) | 0,254 (5) | |
| Bearden (1931) | 0,246 (41) | 0,234 (73) | 0,235 (32) | 0,239 (51) | |
| Bearden (1931) | 0,259 (49) | 0,250 (82) | 0,256 (44) | 0,255 (67) | |
| Bearden (1931) | 0,239 (11) | 0,244 (16) | 0,240 (3) | 0,240 (4) | |
| Backlin (1935) | 0,249 (56) | ||||
| Söderman (1935) | Cu \(K\alpha_1\) | 0,255 (9) | |||
| Bearden (1935) | 0,245 (6) | ||||
| Bearden (1935) | 0,261 (6) | ||||
| Mean: 0,249‰ with a probable error of 0,0016‰. |
this case the actually measured wavelengths are not indicated, but only the percentage deviations from the corresponding values of the crystal-spectroscopic scale are given, in which for the copper \(K\alpha\)-line \(1.53671 \ \text{Å}\) is adopted. In parentheses are added the numbers of individual measurements from which each of the values given in the table was determined. The crystal-spectroscopic scale adopted by Bearden is shifted
relative to the Siegbahn scale generally accepted in X-ray spectroscopy, where the wavelength of copper \(K\alpha\) is taken to be \(1.537395\ \text{Å}\), by \(0.045\%\) toward shorter wavelengths; the deviations should be reduced by the corresponding amount if the wavelengths are referred to the Siegbahn scale. As the result of Bearden’s averaging of his own results and those of Bechlin and Söderman, it is therefore found that the true wavelengths are \(0.203\%\) greater than the wavelengths on the Siegbahn scale. The most recent measurements of Tyrén, which were not taken into account in the averaging, give, in comparison with the new measurements of Hetlund\(^{52}\), made with a crystal lattice, a difference of \(0.202\%\). This result is thus in excellent agreement with that indicated above.
As is known, the Siegbahn scale is established in such a way that the “effective” lattice constant of rock salt, which is to be substituted into the simple Bragg formula without taking refraction into account, for \(18^\circ\text{C}\) and first order is assumed equal to \(2.81400 \cdot 10^{-8}\ \text{cm}\). For the effective lattice constant of calcite, crystallospectroscopic measurements in this case give, according to Siegbahn, \(3.02904 \cdot 10^{-8}\ \text{cm}\). If it is now required, from the diffraction angle measured on a crystal and from the absolute wavelength, to determine the absolute constant of the crystal, then it must be taken into account that in Bragg reflection from a crystal there is also refraction; in view of this, to calculate the “true” lattice constant it is necessary to use, instead of the simple Bragg formula, the modified one:
\[ n\lambda = 2d \sin \vartheta \left(1 - \frac{\delta}{\sin^{2}\vartheta}\right), \]
where \(\delta\) is the difference between the index of refraction and unity. Therefore the true lattice constant of calcite at \(18^\circ\text{C}\) is obtained somewhat larger (namely, by \(0.135\) per mille) than the effective constant; its value is \(3.02945 \cdot 10^{-8}\ \text{cm}\) (on Bearden’s wavelength scale the corresponding value will be \(3.02810 \cdot 10^{-8}\ \text{cm}\)). Finally, for the absolute lattice constant of calcite one obtains:
\(3.02945 \cdot 1.00203 = 3.02810 \cdot 1.00248 = 3.03560 \cdot 10^{-8}\ \text{cm}\).
- What influence on the density can deviations from the ideal crystal lattice have? The calculation just presented meets no objections and may be considered completely correct. If, however, with the aid of the absolute lattice constant found in this way one calculates the volume of the unit cell in order, from this, with the aid of the molecular weight and density, in turn to find Avogadro’s number and then the elementary electric charge, the question arises how legitimate it is to ascribe the same density to the crystal throughout its entire thickness. Indeed, Bragg reflection occurs in an extremely thin surface layer, of the order of magnitude of approximately \(10^{-4}\)—\(10^{-6}\ \text{cm}\), and this means that, generally speaking, no conclusions can be drawn in this way about the lattice constant in the interior of the crystal. This question arises all the more naturally since the data concerning the intensity and angular
The widths of Bragg reflection for most crystals have long since led to the necessity of admitting a so-called “mosaic structure,” i.e. to the conclusion that many crystals which outwardly appear impeccably regular are in reality constructed not from a completely homogeneous lattice, as would have to be expected for an ideal crystal, but from numerous negligibly small blocks. It is evident that these small blocks are arranged not parallel, but at very small angles to one another. Unfortunately, very little is known experimentally about this undoubtedly existing “mosaic structure” of crystals. It was therefore very difficult to judge whether it is, and if it is, to what extent, the cause of the discrepancy found by Bechlin and Baerden between the absolute lattice constant and the constant calculated with the aid of the Millikan value of \(e\) and of the value of Avogadro’s number following from it.
If the lattice constant in the thin surface layer, on which the Bragg reflection of the first order chiefly depends, differed noticeably from the constant in deeper layers, then this difference would show up in reflections of higher orders, since under these conditions the angle of incidence becomes smaller and the depth of penetration greater. According to Allison’s measurements\(^1\), however, this is not observed up to the fifth order, for which the depth of penetration is approximately \(1 \cdot 10^{-3}\ \mathrm{cm}\), whereas for the first order it may be taken to be approximately \(5 \cdot 10^{-5}\ \mathrm{cm}\). Moreover, according to the especially careful measurements of Du Mond and Bollman\(^ {41}\), even in the case where the Bragg reflection occurs inside a crystalline plate, i.e. at an even considerably greater distance from the surface, the angle of reflection agrees with that obtained for reflection at the surface within the limits of the achieved accuracy of measurement, namely \(0.1\) per mille. In these investigations one very curious phenomenon was discovered which at first appeared to be an interference: when rays were passed through a thin plate of Iceland spar polished on both sides, it turned out that the crystal reflects nonuniformly throughout its depth, but the parts adjacent to the surface, both on the entrance and on the exit side, reflect more strongly than the internal parts. This phenomenon is evidently due to structural changes of the surface caused by polishing; Du Mond and Bollman achieved a reduction of the difference in reflection intensity by etching the surface. The observed increase in the reflection intensity at a mechanically treated surface, according to Du Mond and Bollman, arises as follows: the interval of angles within which the crystal reflects appreciably, in the case of a good crystal of Iceland spar (calcite), which, as is known, behaves with fairly good approximation as an “ideal crystal,” is very small; it amounts to several seconds of arc. Consequently, of all the intensity issuing from a definite point of the anticathode, only a negligible fraction is reflected by the calcite crystal. If now the calcite lattice, under the action of the mechanical treatment of the crystal
at the surface is disturbed, then new crystallites, oriented somewhat differently, arise at various places on the surface, so that rays emerging from each point of the anticathode encounter regions on the crystal surface where additional reflection occurs.
In this way, under certain experimental conditions, an increase in the total reflected intensity may in fact be obtained.
Thou[^95] pointed out that the calculation of \(N\) and \(e\) from the absolute lattice constant and the density with the aid of such crystals as, for example, \(\mathrm{CaCO_3}\) and \(\mathrm{NaCl}\), which behave very differently with respect to the mosaic structure established by X-ray spectroscopy, should lead to substantially different results if the mosaic structure has any noticeable influence on the result. However, Thou found that for all accurately measured crystals—\(\mathrm{CaCO_3}\), \(\mathrm{NaCl}\), \(\mathrm{KCl}\), diamond—the same result is obtained.
A further important investigation on the question of whether it is possible, from the absolute lattice constant measured on large crystals, to calculate Avogadro’s number and the elementary electric charge was carried out by Du Mond and Bollman[^39] on powdered Iceland spar. The crystallites used were sufficiently small (\(10^{-4}\ \mathrm{cm}\) and less) and therefore participated in the reflection with their entire volume. On one and the same powder, the density was first measured by means of a pycnometer; then, in a Seemann–Bohlin spectrograph, the interference angle was accurately measured for a whole series of lattice planes. From this, finally, the spacing was calculated for planes parallel to the cleavage plane. This spacing proved to be \(d = 3.02823\ \text{\AA}\) on the Siegbahn scale, whereas on large crystals the spacing measured and calculated on the same scale is \(d = 3.02904\). The difference (0.27 per mille) lies within the accuracy of measurements in interference experiments with powders. Likewise, the density measurement of the powder used for the interference experiments gave a result agreeing, within the accuracy of the method (0.18 per mille), with the value obtained for large crystals (\(\rho = 2.71022\) for the powder and \(\rho = 2.71030\) for a large crystal of Iceland spar).
Thus, within the limits of the accuracy achieved in the experiments of Du Mond and Bollman, both the lattice constant and the density of powdered calcite measured by ordinary methods, for particle sizes from \(10^{-4}\) to \(10^{-5}\ \mathrm{cm}\), coincide with the lattice constant and density of large crystals used in X-ray spectroscopic investigations. Even if the small deviations obtained are regarded as real and the probable error of the density determination is added to them, even then the product \(d^{3}\rho\), entering into the formula for the calculation of \(N\) and \(e\) (cf. below), differs for the powder by no more than 1 per mille from the same product for a large crystal.
However, if lattice irregularities on the scale corresponding to the size of the powder particles were present, one would rather expect differences in the density than in the value of the lattice constant.
Still, it cannot be considered with complete certainty that some irregularities in the lattice do not exist over distances of \(10^{-5}\) cm and smaller. In fact, although such small particles were undoubtedly present in the powder used by Du Mond and Bollman and prepared by grinding in an agate mortar, they could nevertheless hardly have affected the intensity of the interference. The reason for this is that, although for the X-rays used (nickel \(K\alpha\)) the extinction, i.e., the attenuation due to reflection, is very small for particles of size less than \(10^{-4}\), the intensity of the radiation reflected from an individual particle decreases in proportion to the number of atoms, i.e., in proportion to the cube of the linear dimensions of the particle. In order to establish whether irregularities in the lattice structure exist in the submicroscopic region, the corresponding investigations must be carried out with considerably smaller particles.
At the author’s suggestion, Boersch\(^{33}\) carried out a series of comparative measurements with electron rays in the range of particle sizes of the order of \(10^{-6}\) cm. In this case, unlike other investigators, Boersch could not detect any deviations from the properties of large crystals.
- Calculation of the Avogadro constant. The present state of the question may be summarized as follows: there is no doubt that irregularities are normally found in the structure of a lattice, which account, for example, for the phenomena of mosaic structure known in X-ray spectroscopy. However, so far there are no experimental indications that these irregularities can cause a noticeable difference in the spatial distribution of mass in a large crystal as compared with a very small, i.e., ideal, crystal. On the contrary, for particles up to \(10^{-4}\) cm in size it has been experimentally proved that the lattice constant and the density, within 1 per mille, are the same as in a large crystal.
Furthermore, it has been experimentally proved that the lattice constant inside a large crystal is the same as at its surface. Therefore, at present there is no reason to doubt that the calculation of the Avogadro constant, and consequently also of the charge of the electron, by means of absolute measurements of the wavelength of X-rays is legitimate. Since the elementary cell of Iceland spar is a rhombohedron, the angle \(\beta\) of the rhombohedron also enters into the expression for the volume of the elementary cell. This latter can be measured both goniometrically on a large crystal and interferometrically with the aid of X-rays. Du Mond and Bollman obtained for this angle of the rhombohedron, with crystalline powder, exactly the same value as had been measured for large crystals.
The Avogadro constant is obtained by dividing the mass \(M\) of a mole of \(\mathrm{CaCO_3}\) by the mass of an individual molecule, the latter having to be expressed in terms of the lattice volume per molecule and the density \(\rho\). Thus one obtains
\[ N=\frac{M}{2d^3\Phi(\beta)\rho}. \]
Assuming
\[ \begin{aligned} M&=100.078\pm0.005,\\ d_{18^\circ}&=3.0356\pm0.0001\ \text{\AA}\quad(d_{20^\circ}=3.03566\ \text{\AA}),\\ \Phi(\beta)&=1.0960\pm0.0005,\\ \rho_{20^\circ}&=2.7103\pm0.0004^{1}), \end{aligned} \]
we obtain for Avogadro’s constant:
\[ N=(6.022\pm0.005)\cdot10^{23}. \]
In estimating the accuracy of this value, we follow Tiren2. The indicated error of the final result is obtained in the following way by adding the individual errors:
| Maximum inaccuracy of the lattice constant | 0.033 per mille |
| × 3 = | 0.1 per mille |
| Inaccuracy in the determination of the density \(\rho\) | 0.15 » |
| Inaccuracy of \(\Phi(\beta_1)\) | 0.5 » |
| Maximum total error | 0.75 per mille |
In this calculation, of course, no account has been taken of the inaccuracy associated with possible deviations from the ideal structure of the crystal. If one assumes that the experiments of Du Mond and Bollman certainly exclude deviations greater than 1 per mille, then in the most unfavorable case the maximum total error should be taken as 1.75 per mille. Thus, strictly speaking, one should have written
\[ N=(6.022\pm0.011)\cdot10^{23}. \]
- Calculation of the electron charge. To calculate the elementary electric charge it is necessary to divide the total charge carried during electrolysis by 1 mole of a monovalent substance by Avogadro’s constant. The total charge carried by one mole—the Faraday constant—is obtained, as is known, by dividing the atomic weight of silver (107.88) by the amount of silver deposited by a current equal to 1 in unit time (by definition, \(0.001118\) g). If the international coulomb is taken equal to the absolute coulomb, then from this one obtains (cf. Henning and Eger3)
\[ F=96494\ \text{coulomb}/\text{mole}= \]
\[ =9649.4\ \mathrm{CGSE}/\text{mole}. \]
R. T. Birge, however, takes the ratio \(q\) of the international to the absolute unit of current as not equal to 1, but to \(0.99995^{20}\) or \(0.99993^{27}\); under this condition one obtains
\[ F=9648.9\ \mathrm{CGSM}/\text{mole}\quad \text{or}\quad 9648.7\ \mathrm{CGSE}/\text{mole}. \]
In a recently published survey by Wensel^99, on the basis of measurements newly carried out at the American Bureau of Standards^37, it is even assumed^1)
\[ q=0.99986. \]
In this case the Faraday constant is reduced to
\[ F=9648.0\ \mathrm{CGSM}/\text{mole}. \]
Wensel considers, however, that the value of this constant should not be derived solely from measurements with the silver voltameter, since it is not entirely certain that the measured quantity of deposited silver really corresponds to the quantity associated with the measured electric charge. He therefore also takes into account measurements made in the same Bureau of Standards with an iodine voltameter^37, which lead to the result
\[ F=9650.0\ \mathrm{CGSM}/\text{mole}, \]
and proposes as the final figure the mean of both numbers:
\[ F=9694.0\ \mathrm{CGSM}/\text{mole}= \]
\[ =2.8926\cdot 10^{4}\ \mathrm{CGSE}/\text{mole}. \]
We therefore adopt
\[ F=(9649\pm 1)\ \mathrm{CGSE}/\text{mole}= \]
\[ =(2.8926\pm 0.0003)\cdot 10^{14}\ \mathrm{CGSE}/\text{mole}. \]
Hence one obtains
\[ e=\frac{2.8926\cdot 10^{14}}{6.022\cdot 10^{23}} =(4.803\pm 0.004)\cdot 10^{-10}\ \mathrm{CGSE}. \]
Here too the error limit should be increased by another 1 per mille, i.e. to 0.009, if one takes into account the remaining uncertainty concerning possible deviations from the ideal structure of the crystal. Nevertheless, the value of \(e\) calculated in this way should be regarded as considerably more accurate than the value obtained by any other method, including the oil-drop method. Among the many other methods which lead to the magnitude of the elementary charge by partly quite different routes, mention should be made here only of the counting of \(\alpha\)-particles and the measurement of the total charge carried by them, the measurement of fluctuations in amplifier tubes caused by the atomic nature of the electron current, and a whole series of mechanical and optical phenomena connected with fluctuations (diffusion, Brownian motion, sedimentation equilibrium of colloidal particles, opalescence), which make it possible to calculate Avogadro’s number. All these methods
^1) Denington (l. c.) points out that two different conversion factors should be used for the unit of current, namely: 0.99997 for the system of the normal element and 0.99993 for the system of the “silver voltameter.” For the Faraday constant he gives \(F=(9649.11\pm 0.24)\ \mathrm{CGSM}/\text{mole}\).
lead to results that agree, within their inherent more or less significant errors, with the number given above.
6. Mass of the hydrogen atom. To determine the mass \(m_{\mathrm H}\) of the hydrogen atom, we must first of all convert Faraday’s constant \(F\) to the physical scale of masses. Indeed, whereas on the chemical scale the chemical atomic weight of oxygen, which, as is known, in addition to atoms of mass 16 also contains atoms with masses (in round numbers) 17 and 18, is taken to be equal to 16.0000, on the physical mass scale only the mass of the isotope \(O^{16}\) is taken as 16.0000. Therefore the physical unit of mass is smaller than the chemical one in the ratio corresponding to the fraction of all three oxygen isotopes in the mixture. According to Birge\(^{27}\), this ratio, on the basis of the measured isotopic composition \(O^{16}:O^{17}:O^{18}=99.76:0.04:0.20^{102}\), should be taken equal to 1.00027. For Faraday’s constant one therefore obtains, on the physical scale,
\[ 9649\cdot 1.00027=9651.5\ \mathrm{CGSM}. \]
The Avogadro number obtained from this by means of the formula \(N=\dfrac{Fc}{e}\), referred to the physical scale of masses, is smaller by exactly 0.27 per mille than the number given above, referred to the chemical scale of masses. In order to compute the mass of the hydrogen atom, one should divide the mass of the hydrogen isotope \(H^1=1.00813\) by Avogadro’s number referred to the physical scale of masses,
\[ m_{\mathrm H}=\frac{1.00813}{6.022\cdot 10^{23}\cdot 1.00027}\ \mathrm{g}=1.673_{6}\cdot 10^{-24}\ \mathrm{g}. \]
Further, for \(H^1=1.00813\) (from \(F=9651.5\)) one obtains
\[ \frac{e}{m_{\mathrm H}}=9573.8\ \mathrm{CGSM}. \]
III. THE SPECIFIC CHARGE OF THE ELECTRON
There is hardly any other physical problem for whose solution, over a number of years, so much labor and experimental skill have been expended by numerous investigators as for the determination of the specific charge of the electron. However, of the numerous methods that in the course of time were developed and tested for determining \(\dfrac{e}{m}\), only those will be considered here by means of which genuinely precise measurements have been carried out during the last ten years.
As is known, the various methods of determining \(\dfrac{e}{m}\) may be divided into two groups: to one group belong methods in which one deals with free electrons and studies their behavior in electric and magnetic fields; the other group of methods is applied to electrons bound in the atomic shell, and uses
emission of light by these electrons or their interaction with light. At the time when Birge published the above-mentioned detailed review of the probable values of the physical constants, the results obtained by the methods of the one group and of the other differed so greatly from one another that it seemed necessary to adopt two different values of \(\frac{e}{m}\): the “spectroscopic” value \((1.761 \pm 0.001)\cdot 10^{7}\) and \(\frac{e}{m}\) from experiments with the deflection of electrons, \((1.769 \pm 0.002)\cdot 10^{7}\ \mathrm{CGSM}/\mathrm{g}\). This difficulty, especially strongly emphasized by Birge, undoubtedly gave impetus to a whole series of new measurements carried out in subsequent years by various methods, which are discussed below. The results of these measurements agree with one another so well that the difficulty indicated must now be regarded as completely removed.
A. Measurements on Free Electrons
7. Determination of \(\frac{e}{m}\) by Kirchner’s method.
Let us first consider the methods for determining \(\frac{e}{m}\) for free electrons. Among them, at present the most accurate are those in which the time of flight of electrons over a definite segment is compared with the period of oscillation of rapid electric oscillations. This principle was first used by F. Kirchner\(^{64,65,67}\) for precision measurements. A narrow beam of cathode rays of the same velocity, selected by diaphragms, enters (cf. Fig. 4), after passing through the slit or aperture \(D_{1}\), the capacitor \(K_{1}\), to which a rapidly alternating field of electric oscillations is applied.
Fig. 4. Arrangement of the experiment for direct measurement of velocity (F. Kirchner\(^{64,65}\)).
As a result, the cathode ray is deflected to one side or the other in step with the electric oscillations. The broadly opened fan of cathode rays at \(P_{2}\) is stopped by a screen that has in its middle (at \(D_{2}\)) a slit or aperture. Thus, through \(D_{2}\) only those cathode rays can pass which passed through the capacitor \(K_{1}\) at the moment when the electric field in it was zero. Since the cathode ray oscillating back and forth meets \(D_{2}\) every half-period, a short current pulse passes through \(D_{2}\) every half-period and then enters the capacitor \(K_{2}\). The same alternating field as on \(K_{1}\) is applied to this latter capacitor. Consequently, the electrons passing through \(D_{2}\) are deflected to one side or the other depending on what field was in \(K_{2}\) at the moment the electrons passed through. As a result, generally speaking, two deflected ...
images of the slit \(D_2\), arranged symmetrically with respect to the undeflected one. The distance between the two deflected beams reaches its greatest value if, when the beam passes through \(K_2\), it finds there the maximum value of the field strength. Conversely, the two beams merge into one when the time needed by the particles of the cathode rays to go from \(K_1\) to \(K_2\) is exactly equal to the period of oscillation or to an integral multiple of it, since in this case the particles that have passed through the capacitor \(K_1\) at the moment when the field in it was equal to zero find in the capacitor \(K_2\) a field that is likewise equal to zero. The accuracy of adjustment to this exceptional case is extraordinarily great; indeed, a change in velocity \(\Delta v\) causes on the fluorescent screen a deflection \(\Delta a\), which is expressed as follows:
\[ \Delta a=\frac{e}{m}E_0\omega lss'\frac{\Delta v}{v^4} \]
(here \(\frac{e}{m}\) is the specific charge of the electron in electromagnetic units, \(E_0\) is the maximum field strength in the capacitors—in the same units, \(\omega\) is the angular frequency of the oscillations, \(l\) is the length of the capacitors, \(s\) is the distance between them, \(s'\) is the distance between the second capacitor and the fluorescent screen, and \(v\) is the velocity of the cathode rays).
For electrons of approximately \(2000\ \mathrm{V}\), the error of adjustment, as is easily calculated from the formula given, can without particular difficulty be reduced to a few hundred-thousandths. Since, further, the frequency of the oscillations and the distance between the capacitors can be measured with an accuracy of up to \(0.1\) per mille, the absolute velocity of the cathode rays can be unerringly found with an accuracy of up to several ten-thousandths. In order to find the specific charge, it is necessary to perform one more measurement, for example, a measurement of the accelerating potential through which the electrons must pass before they enter the apparatus for measuring the velocity. From the energy equation
\[ eV=m_0c^2\left(\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}-1\right) \]
with known velocity \(v\), the specific charge is immediately obtained:
\[ \frac{e}{m_0}=\frac{c^2}{V}\left(\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}-1\right). \]
The voltage measurement can be carried out by the compensation method, by comparison with a standard cell, with an accuracy of up to \(1\) per mille. It is necessary, however, to take into account the possibility that the actual velocity of the electrons, owing to the presence of a contact potential difference, differs from that measured from the potential by an amount of the order of \(1\ \mathrm{V}\).
This difficulty can be avoided in two ways: either by trying to determine the contact potential directly and introducing the corresponding correction, or by carrying out measurements of the velocity with one and the same arrangement, but at two substantially different voltages. The first approach was used by the author of this survey in processing his measurements of 1930; attempts to achieve the goal by the second method have recently been undertaken, at the author’s suggestion, by Berg.
In explaining the principle of the measurement it was assumed that the beam entering capacitor \(K_2\) passes through capacitor \(K_1\) at the moment when the field there is equal to zero. Since, however, electrons require a finite interval of time in order to pass through capacitor \(K_1\), at their entry into and exit from the capacitor the electric field differs somewhat from zero. As a consequence, those electrons which leave the capacitor without changing direction experience a small parallel displacement and may fail to pass through the diaphragm in front of the second capacitor. In order to enter the second capacitor at all, they must be deflected through a corresponding small angle, i.e. the change in sign of the electric field in the capacitor must occur somewhat earlier than the electron passes through the middle of the capacitor. This “lead time” can be calculated from the experimental data; it is equal to \(\Delta t = \dfrac{l^2}{12vs}\).
If the electrons passed through the second capacitor with the same lead time, then they would pass through the capacitor without deflection and would meet the fluorescent screen not on the axis, but on the continuations of the two oblique rays entering capacitor \(K_2\). In order for both beams to meet on the axis, i.e. in order for both images on the screen to coincide, the beams must pass through capacitor \(K_2\) with a lead time increased or decreased by \(\Delta t\), depending on whether the deflection occurs in the same direction as in the first capacitor or in the opposite direction. Thus, the measured distance \(s\) between the centers of the two capacitors is in fact traversed by the electron not in the time \(T/2\), but in the interval
\[ \frac{T}{2}+\frac{l^2}{12vs} \]
or
\[ \frac{T}{2}-\frac{l^2}{12vs}, \]
depending on whether the leads to the two capacitors are “parallel” or “crossed.” The correction for the finite length of the capacitors which must be introduced into the final result of the velocity measurement is therefore \(\dfrac{l^2}{12s^2}\), i.e. for \(l = 3\ \mathrm{cm}\) and \(s = 50\ \mathrm{cm}\) (the figures correspond to the actual conditions of the author’s experiment) 0.3 per mille. Since in calculating \(e/m\) the square of the velocity enters the formula, the correction to the final result is 0.6 per mille.
The result of the measurements carried out in 1930 is as follows:
\[ \frac{e}{m} = (1.7598 \pm 0.0025)\cdot 10^7 \mathrm{CGSM}. \]
The indicated “maximum error” of 1.4 per mille is composed as follows:
| Source of error | Value |
|---|---|
| Maximum error in measuring the distance of the condensers | 0.1 per mille |
| Maximum error in determining the frequency of oscillations (piezo-quartz standard) | 0.1 ” |
| Both of these measurements enter into the final result quadratically, i.e. \(0.2 + 0.2\) | 0.4 per mille |
| Maximum error in measuring the resistance and potential | 0.1 ” |
| Uncertainty of the correction for the potential drop across the filament | 0.1 ” |
| Maximum error in measuring the voltage, due to the contact potential | 0.8 ” |
| Maximum total error, equal to the sum of all the individual errors | 1.4 per mille |
In 1931 an attempt was made to reduce the uncertainty connected with the contact potential by measuring the characteristics in the region of the space charge. These measurements led to a correction for the space potential of 1.3 V; the final result was therefore reduced to
\[ \frac{e}{m}=(1.7585\pm 0.0012)\cdot 10^7\ \mathrm{CGSM} \]
with the error reduced to 0.7 per mille. A new series of measurements, made at approximately the same voltage, gives, if the indicated correction for the contact potential is taken into account,
\[ \frac{e}{m}=(1.7590\pm 0.0015)\cdot 10^7. \]
This result, obtained almost ten years ago, can still be considered correct. It is true that in subsequent years, in new measurements of \(\frac{e}{m}\) by other methods, results different from the one indicated and not mutually identical were obtained; however, the very latest values of \(\frac{e}{m}\) again agree, within the limits of error, with the author’s results given above.
Perry and Chaffee\({}^{80}\), using the same method, determined \(\frac{e}{m}\) in the voltage interval 10–20 kV. The resolving power of the apparatus they used was 10 times less than in the author’s experiments described above, and, moreover, it was further reduced by the use of focusing coils. Nevertheless, Perry and Chaffee achieved remarkable accuracy; in the final result they obtained
\[ \frac{e}{m}=(1.761\pm 0.001)\cdot 10^7\ \mathrm{CGSM}. \]
- Determination of \(\frac{e}{m}\) by Dunnington’s method. Another method, in which, as in the preceding one, the time of flight...
electrons is compared with the period of electrical oscillations, was proposed by E. Lawrence and used by F. Dunnington^42,43 as a precision method. In this method, however, the measurement of the velocity is combined not with a measurement of voltage, but with magnetic deflection along a circular path fixed by slits.
The arrangement of Dunnington’s experiment is shown schematically in Fig. 5. During one half-period of the alternating field applied between 1 and 2, electrons from the filament are accelerated by the latter and pass through slit 2. Their velocity is different and depends on the voltage that exists between the filament and the slit at the instant of acceleration. A uniform magnetic field, perpendicular to the plane of the drawing, deflects the electrons. Of these deflected electrons only those can pass through slits 3, 4, 5 and reach the collector whose circular path has a definite radius determined by the position of the slits. Before the collector, namely between 5 and 1, the same field that served to accelerate the electrons will now retard them. The frequency of the alternating field and its voltage are chosen so that the time of flight around the circle for a part of the electrons is approximately equal to an integral period.
Fig. 5. Diagram of Dunnington’s experiment^42,43
For each value of the magnetic-field strength, only electrons of one definite velocity can pass through the system of slits. The essence of the experiment is that (at a constant frequency of the alternating electric field) one finds that value of the magnetic-field strength at which the flight time of the electrons passing through the system of slits is equal to the period of the electric field. In this case the electrons approaching the collector each time encounter a retarding field of the same magnitude as the accelerating one, and therefore cannot enter the collector. For any other value of the magnetic field, the electrons passing through the system of slits encounter a retarding field that is either greater or smaller in magnitude than the accelerating field, so that half of the electrons with the velocity corresponding to the given magnetic field always reach the collector. Thus the measurement consists in determining, for a given frequency \(\nu\), the magnetic-field strength \(H_\rho\) at which the current to the collector reaches a minimum. If the angle described by the electrons is \(\theta\), then
\[ \frac{mv}{\rho}=eH_\rho \quad \text{and} \quad v=\rho\theta\nu, \]
whence
\[ \frac{e}{m}=\frac{\theta\nu}{H_\rho}. \]
Fig. 6 shows that the minimum is characterized by unusual sharpness. However, determining the angle \(\theta\) presents some difficulty. The “effective” angle \(\theta\) cannot be identified with the “instrumental” angle \(\theta\) for the following reasons:
-
Since finite intervals of time are required for the acceleration and deceleration of the electrons, the time during which the electrons traverse their circuit with velocity \(v\) is never equal to the period, but is always somewhat less than it.
-
The field-free space does not extend from the exit slit to the entrance slit, but has a smaller extent, since the accelerating and decelerating fields penetrate through the corresponding slits (when the acceleration takes place over a segment \(d = 1.6\) mm and the slit width is \(\Delta y = 0.25\) mm, the field at a distance \(0.1d\) from the slit is still \(10\%\) of its maximum value).
Fig. 6. Current behavior near the minimum (Dunnington\({}^{43}\))
The calculation of the “effective angle \(\theta_i\)” for a finite slit width runs into insurmountable mathematical difficulties; it is therefore carried out only for the case \(\Delta y = 0\), i.e., for a slit width equal to 0. Therefore the resonant value of the magnetic field as a function of the slit width must be determined experimentally and extrapolated to zero. If terms \(< 1 : 10^6\) are neglected, the calculation gives
\[ \theta_{\mathrm{eff}} = \frac{d_f + \frac{4}{3}d}{r} = \theta_f + \frac{4}{3}\theta, \]
where \(d_f\) is the path length from slit 2 (Fig. 5) to slit 5 (about 60 cm), \(d\) is the distance between slits 1 and 2 and, correspondingly, 5 and 6 (\(\sim 0.16\) mm), and \(\theta_f\) and \(\theta\) are the corresponding angles.
The measuring chamber was constructed with maximum precision. In order to reduce as much as possible the influence of surface charges on the path of the electrons, the inside of the measuring chamber was coated with a thin layer of gold; in the part of the chamber directly adjacent to the hot cathode, it proved necessary to renew the gold coating frequently, since in the part subjected to the action of cathode rays, surface charges formed especially easily on thin insulating layers. For measuring the angle \(\theta\), a special additional device was designed, as also for measuring the radius. Although the latter does not enter into the formula for calculating \(e/m\), it is nevertheless necessary to know it in order to calculate the magnetic field. The magnetic field was produced
by two Helmholtz coils, each of which consisted of nineteen sections of 21 turns each; their mean radius was approximately 20 cm. The absolute values of the field constants of these coils were determined by comparison with a carefully constructed solenoid. The construction and dimensions of the solenoid, as well as the calibration of the Helmholtz coils, are described in great detail by Dunnington.
In order to ascertain possible disturbing influences on the final result, special series of experiments were carried out. Among such influences, in addition to the surface charges already mentioned, which may affect the trajectory of the electrons, are: an asymmetric form of the “resonance minimum,” i.e., of the decisive curve of the receiver current as a function of the current in the Helmholtz coils; the possible influence of the energy of the electrons on the value of \(\frac{e}{m_0}\); the influence of the space charge in front of the hot cathode on the accelerating field; and, finally, the influence of the slit width. Dunnington\({}^{43}\) carefully discussed all sources of error; the following two tables give a survey of the magnitude of the probable errors which, according to the author’s estimate, enter into the determination of \(\frac{e}{m}\).
As the final result Dunnington gives
\[ \frac{e}{m_0}=1.7597\cdot 10^7\ \mathrm{CGSM} \]
with a probable error of \(0.0004\cdot 10^7\ \mathrm{CGSM}\). [Combining this value with the nine previously found during the preceding ten years, Dunnington obtained, as the “most probable value,”
\[ \frac{e}{m}=(1.7584\pm 0.0003)\cdot 10^7\ \mathrm{CGSM}.] \]
Table 3
Probable error in the determination of the magnetizing current in the case of resonance (Dunnington\({}^{43}\))
| Source of error | Probable error \(r\cdot 10^5\) |
|---|---|
| Observational error in measuring the current | 0.9 |
| Inaccuracy of the correction for slit width | 1.2 |
| Influence of the probable error in determining the coil constants | 0.1 |
| Inaccuracy of the voltage of the standard cell | 5.0 |
| Probable error of the comparison resistance | 2.5 |
| Ohmic error and thermoelectric voltage of the potentiometer | 0.5 |
| \(\sqrt{\sum r^2}\) | 5.8 |
| Allowance for constant errors, including extrapolation, etc. | 14.2 |
| Total probable error | 20.0 |
From Tables 3 and 4 it is seen that the most substantial source of error in this method is associated with the measurement of the magnetic field, despite
nevertheless, the probable error in determining the Helmholtz-coil constants is only 0.06 per mille!
In comparing the error limits of this method with the error limits of other methods for determining \(\frac{e}{m}\), one should bear in mind that here one is dealing with probable errors and that the total error is not obtained by adding the individual errors.
Table 4
Probable error in \(\dfrac{e}{m_0}\), estimated from the various components entering into the determination of this quantity
| Source of error | Probable error \(r \cdot 10^5\) |
|---|---|
| Determination of the angle \(\theta\) | 0.8 |
| Frequency \(\nu\) | 0.6 |
| Helmholtz-coil constant | 6.0 |
| Additional inaccuracy in the constants of the coils owing to a probable error of 0.5 per mille in the measurement of the radius of the electron trajectory | 0.7 |
| Measurement of the magnetizing current | 20.0 |
| Final value of the probable error in \(\dfrac{e}{m_0}\) \(\sqrt{\sum r^2}\) | 20.9 |
but by extracting the square root of the sum of the squares. (The difference in the determination of the total error will, of course, be the smaller, the more strongly the total error depends on any one particularly significant error.) If all the partial errors are added and it is assumed that the error limits are approximately four times the probable error, then the numbers given by Dunnington give a maximum error limit for the final result of 1.2 per mille.
Thus, the accuracy of the final result here is approximately of the same order of magnitude as in the determination of \(\frac{e}{m}\) by means of the measurement of velocity and voltage (cf. p. 328). This must be pointed out because recently, by comparing the error limits of one method with the probable error of another method, the impression has repeatedly been created that the accuracy of Dunnington’s measurements is considerably higher than the accuracy of other determinations of \(\frac{e}{m}\).
9. Hedicke’s measurements by Busch’s method. The following precision method for determining the specific charge, which was indicated as early as 1922 by Busch\(^{34}\), is based on the focusing action
of a longitudinal magnetic field on a weakly diverging beam of cathode rays. As is known, rays emerging from a diaphragm describe, in a uniform longitudinal magnetic field, cylindrical helical lines whose pitch is determined by the formula
\[ l=\frac{2\pi v\cos\alpha}{\frac{e}{m}H}. \]
Thus, if all the electrons emerging from the diaphragm fly out at the same angle \(\alpha\) to the axis, i.e., to the magnetic lines of force, and if, in addition, they all have the same velocity \(v\), then for such electrons \(l=\mathrm{const}\), and all of them, after advancing by a distance equal to the pitch of the helix, meet again at one point. The measurement therefore consists in finding such a magnetic-field strength \(H\) at which the diaphragm from which the rays emanate is sharply imaged at a definite distance \(l\), for example on a fluorescent screen. To determine \(\frac{e}{m}\), of course, one more measurement is necessary—say, a measurement of the accelerating voltage that the electrons must traverse before they pass through the diaphragm.
Precision measurements by this method had already long ago been carried out by F. Wolf\({}^{101}\). These measurements, however, gave for \(\frac{e}{m}\) a value considerably larger than all subsequent measurements, namely:
\[ \frac{e}{m}=1.7679\pm0.0018\ \mathrm{CGSM}. \]
Since no sources of error can be found in Wolf’s work that might explain so large a discrepancy, E. Hedicke\({}^{51}\), at the Institute of the author, in 1935–1938 carried out a new measurement of the specific charge by this method. In these measurements the maximum sharpness of the image was determined not only by observations on a fluorescent screen, but also by photographing the image of the diaphragm. The final result was:
\[ \frac{e}{m_0}=1.7586\cdot 10^7\ \mathrm{CGSM} \]
with a maximum error of 1.3 per mille. The maximum error of the final result is composed of the following parts:
| Total inaccuracy of the length measurement (in measurements on the coil this inaccuracy enters quadratically into the final result) | 0.4 per mille |
| Inaccuracy of the voltage measurement and possible influence of contact potential | 0.4 » |
| Influence of the limits of error on the measurement of the coil current (0.1 per mille) | 0.2 » |
| Inaccuracy of setting to maximum image sharpness | 0.3 » |
| Maximum error of the final result | 1.3 per mille |
10. Shaw’s Determination of \(\dfrac{e}{m}\) by the Method of Crossed Electric and Magnetic Fields
Finally, quite recently Shaw\(^{90}\) published measurements of \(\dfrac{e}{m}\) by a method in which the focusing properties of crossed electric and magnetic fields are used. The principle of the arrangement of the experiment is as follows: electrons from an incandescent filament pass through a slit \(0.03\ \text{mm}\) wide and \(2\ \text{mm}\) long into a cylindrical condenser. In addition to the electric field of the condenser, a homogeneous magnetic field perpendicular to it also acts on these electrons. Under these conditions, for electrons whose orbits are almost circular, focusing is obtained after passing through an angle of \(127^\circ 17'\). A gold wire of diameter \(0.045\ \text{mm}\) serves as the receiver. For the case of optimal focusing the relation holds:
\[ \frac{e}{m}=\frac{4V}{H^{2}\rho^{2}\lg \frac{r_{2}}{r_{1}}}, \]
where \(V\) is the voltage on the cylindrical condenser, \(H\) is the magnetic-field strength, \(\rho\) is the radius of the electron path, and \(r_{2}\) and \(r_{1}\) are the radii of the condenser plates. As a result of his measurements Shaw gives:
\[ \frac{e}{m}=(1.7571 \pm 0.0013)\cdot 10^{7}\ \mathrm{CGSM}. \]
The indicated error was computed simply by the method of least squares from fourteen separate measurements; Shaw gives no detailed discussion of possible sources of error. According to a personal communication in a letter to the author, Shaw intends to improve the measurements still further and only after that to give final indications regarding the accuracy achieved. The result cited should also be recalculated from international to absolute electromagnetic units; it is therefore raised to
\[ (1.7581 \pm 0.0013)\cdot 10^{7}\ \text{abs. CGSM}. \]
In the work just described, Shaw expresses a judgment concerning the reliability of the results of Kirchner and Dunnington. We shall dwell briefly on this judgment here, since it is untenable. In fact, in Shaw’s experimental arrangement, with the given electric and magnetic fields, exact focusing occurs only for electrons with a definite volt velocity, the magnitude of which can be calculated from the field strengths and the constants of the apparatus. Shaw found that the calculated velocity does not coincide with the accelerating voltage between the incandescent filament and the narrow slit, but, depending on the magnitude of the volt velocity, is smaller by 10, 15, or 20 V. Shaw ascribes this phenomenon to polarization charges on the slit and believes that the existence of the “polarization effect” observed by him constitutes a serious difficulty for my measurements of \(\dfrac{e}{m}\), which—
which were carried out at still higher accelerating voltages than those used in his experiments. This view, however, is based on a misconception. In order to show this, let us consider in more detail the experimental conditions under which the effect observed by Shaw arises.
It has long been known that in a high vacuum “polarization charges” can arise on metallic surfaces if thin insulating layers are deposited on the metals (the point here is hydrocarbons; for example, condensing vapors of fatty substances, and under certain circumstances also water films). These polarization charges become especially significant when cathode rays, or rays consisting of other charged particles, fall upon the corresponding surfaces, i.e., for example, in the case where cathode rays pass through a narrow slit. As a consequence, a retarding field can in fact arise in front of such a slit when a negative polarization charge is formed on the surface of the slit. But if an electron has passed through the slit, it will then be accelerated by the polarization charge, and in doing so it will, of course, regain the same amount of energy as it lost in overcoming the retarding field of those same polarization charges. Therefore Shaw’s opinion that his observations can be reduced to the retarding action of such surface charges is untenable. The difference in volt velocity established by him is more likely explained by some inelastic collisions of the electrons. The fact that the differences in volt velocity observed by Shaw are measured by a jump of 5 V compels one to suppose that the matter is one of inelastic collisions with mercury atoms (resonance voltage 4.9 V). In this case, perhaps, there is no need at all to assume the presence of an atmosphere of mercury vapors; the existence of a thin surface layer of weakly adsorbed Hg atoms on the gold surface of the slit is sufficient. In any case, the criticism which Shaw directs at my determinations of \(\frac{e}{m}\) is entirely unfounded.
B. Spectroscopic methods for determining \(\frac{e}{m}\)
Spectroscopic methods for determining the specific charge of the electron are based on the measurement of very closely spaced spectral lines, i.e., on the measurement of differences of wavelengths.
- Determination of \(\frac{e}{m}\) by Kinsler and Houston from the Zeeman effect. The first method is based on the “classical theory” of the so-called normal Zeeman effect, according to which, in the so-called normal doublet splitting, the difference of wavelengths is expressed by the formula
\[ \Delta \lambda=\frac{e}{m}\cdot \frac{H\lambda^2}{4\pi c}. \]
The most accurate measurements by this method were made in 1934 by Kinsler and Houston62,63. Their results, obtained from the splitting of various lines of zinc, cadmium, helium, and neon, are given in the following table:
| $\dfrac{e}{m}\cdot 10^7$ | |
|---|---|
| Zinc . . . . | $1.7570 \pm 0.0009$ |
| Cadmium . . . | $1.7570 \pm 0.0008$ |
| Helium . . . | $1.7564 \pm 0.0009$ |
| Neon . . . . | $1.7580 \pm 0.0014$ |
Assigning weights corresponding to the error limits of the different measurements, Kinsler and Houston obtained from these as the mean
\[ \frac{e}{m}=1.7570\cdot 10^7\ \mathrm{CGSM} \]
with a probable error of $0.0007\cdot 10^7\ \mathrm{CGSM}$.
12. Determination of $\dfrac{e}{m}$ from the difference of the Rydberg constant for D and H and He$^+$ and H. The second method does not, strictly speaking, give directly the specific charge of the electron, but its mass, and moreover in units of the atomic-weight scale $(O=16)$. It is based on measuring the differences of the wavelengths of corresponding lines in the spectra of hydrogen and ionized helium, or of ordinary hydrogen (H) and heavy hydrogen (D). According to the Bohr–Sommerfeld theory of spectral lines, owing to the proper motion of the nucleus, the Rydberg constant, which determines the magnitude of the spectral term, depends on the ratio of the electron mass to the nuclear mass. If by $R_\infty$ we denote the value that the Rydberg constant would have under the condition that the mass of the nucleus is infinitely large in comparison with the mass of the electron, then the Rydberg constants for H and D will be
\[ R_{\mathrm H}=\frac{R_\infty}{1+\dfrac{m}{m_{\mathrm H}}},\qquad R_{\mathrm D}=\frac{R_\infty}{1+\dfrac{m}{m_{\mathrm D}}} \]
(here $m_{\mathrm H}$ and $m_{\mathrm D}$ denote the masses of the atomic nuclei!).
Combining the two equalities, we obtain
\[ \frac{m}{m_{\mathrm H}}= \frac{(R_{\mathrm D}-R_{\mathrm H})(m_{\mathrm D}+m)} {R_{\mathrm H}\cdot (m_{\mathrm D}-m_{\mathrm H})}. \]
Thus, the measurement directly gives the “atomic weight” of the electron. Since, however, $F=Ne$, $m_{\mathrm H}=\dfrac{\mathrm H}{N}-m$, $m_{\mathrm D}=\dfrac{\mathrm D}{N}-m$, then with the aid of the preceding formula one obtains:
\[ \frac{e}{m}=\frac{F}{Nm}= \frac{F\cdot R_{\mathrm H}(m_{\mathrm D}-m_{\mathrm H})} {(R_{\mathrm D}-R_{\mathrm H})(m_{\mathrm D}-m)m_{\mathrm H}} = F\frac{R_{\mathrm H}(\mathrm D-\mathrm H-Nm)} {(R_{\mathrm D}-R_{\mathrm H})(\mathrm D-Nm)(\mathrm H-Nm)}. \]
Here there is no need to know the quantities \(Nm\) on the right-hand side with especially great accuracy, since the quantities in question are corrections. An analogous formula holds for the calculation of \(\frac{e}{m}\) from the values of the Rydberg constant for hydrogen and ionized helium.
The difficulty of this method consists in the fact that, in order to determine the required differences of wavelengths, one has to use not simple lines, but the components of a doublet, the lines of which cannot be completely separated, i.e., partly overlap, and in which, moreover, there is an exactly unknown number of weaker components. Recently several new determinations have been made on the red hydrogen doublet. Shen and Spedding\(^{69}\), as well as Gibbs and Williams\(^{48,100}\), assumed that the ratio of the intensities of the components of the doublet is the same for both isotopes of hydrogen; as a result, according to Bearden’s\(^{27}\) new recalculation, one obtains:
\[ \begin{array}{rcl} \text{From the data of Shen and Spedding} & \dfrac{e}{m} &= 1.7581 \cdot 10^{7}\ \mathrm{CGSM} \\ \text{” \quad ” \quad Williams} & \dfrac{e}{m} &= 1.7579 \cdot 10^{7}\ \text{”} \\ \text{with probable error of} && 0.0004 \cdot 10^{7}\ \text{”} \end{array} \]
In a special investigation Williams studied the question of whether the distribution of intensity in the doublet depends on the conditions of the discharge; he could establish no dependence. Further, in contradiction to Shen and Spedding, but in agreement with the earlier results of other investigators, he found that the true splitting of the doublet is approximately \(2\%\) less than could be expected from theory.
Especially careful investigations of the fine structure of hydrogen lines were carried out by V. Houston and his collaborators. In order to avoid the uncertainty which arises in ordinary measurement of lines as a result of partial overlap with other bright lines or with weaker components, Houston\(^{57}\) developed a new method, in which interferometric photographs of the doublet are processed by means of Fourier analysis. The practical application of this method amounts to using the Fourier coefficients found to try, by trial, to determine the correct parameters of the lines. On the basis of this new method of Houston’s, Robinson\(^{86}\) recently again determined the specific charge of the electron from the difference of wavelengths in the spectra of H and D. His result is as follows:
\[ \frac{e}{m}=1.7592 \cdot 10^{7}\ \mathrm{CGSM} \]
with a probable error of \(0.0005 \cdot 10^{7}\ \mathrm{CGSM}\).
Finally, Chu\(^{35}\), in the same Institute, again measured the Rydberg constant of singly ionized helium; the processing of his observations gave
\[ \frac{e}{m}=1.7596 \cdot 10^{7}\ \mathrm{CGSM}. \]
Houston correctly pointed out that the good agreement of these values with the results of “direct” determinations of \(\dfrac{e}{m}\) by other methods gives independent proof of the correctness of the numerical value of Faraday’s constant.
- Determination of \(\dfrac{e}{m}\) by Bearden from the refraction of X-rays. In conclusion we must also point to determinations of \(\dfrac{e}{m}\) from the refraction of X-rays. This method, proposed by Bearden, may also be called spectroscopic. If Drude’s dispersion theory is applied to the refraction of X-rays, then for the \(\delta\)-difference between unity and the refractive index one obtains the expression
\[ \delta = n \cdot \frac{e^2}{m}\cdot \frac{\lambda^2}{2\pi} \sum N_s(1 + A), \]
where \(n\) is the number of molecules per unit volume, \(\lambda\) is the wavelength, \(N_s\) is the number of molecules with frequency \(s\), and \(A\) is a correction factor (small compared with unity!), depending on the strength of the binding of the electrons and on the absorption of X-rays.
If \(n\) is replaced by \(\dfrac{N\rho}{M}\) and \(Ne\) by Faraday’s constant \(F\), then one obtains
\[ \frac{e}{m} = \frac{\delta \cdot 2\pi M} {\lambda^2 \cdot \rho F \cdot \sum N_s(1 + A)} . \]
Measurements of the refractive index suitable for such treatment were carried out by Bearden and Shaw\(^{18}\) for the Cu \(K\alpha\)-line and quartz as the refracting medium, and recently by Bearden\(^{17}\) with the same line and a large diamond (dimensions \(9 \times 9 \times 3\) mm). The path of the rays used in the latter work is shown in Fig. 7. The X-ray beam strikes, at a small angle \(\alpha\), the face of the diamond prism. The angle \(\alpha\), however, is slightly greater than the limiting angle of total internal reflection for Cu \(K\alpha\). As a result, the Cu \(K\alpha\) radiation, being refracted, enters the prism and, after leaving it, falls on the photographic plate at \(C\), whereas the longer-wavelength radiation of the continuous spectrum undergoes total reflection at \(A\) and consequently strikes the photographic plate at \(B\). Finally, by removing the prism, one obtains at point \(D\) the trace of the incident primary beam. The distances \(DC\) and \(DB\) determine the angle of refraction and the position of the prism; the refractive index is determined from the known formula
\[ n = 1 - \delta = \frac{\cos \alpha}{\cos(\alpha - \beta)} . \]
Fig. 7. Scheme of the experiment for measuring the refractive index of X-rays (determination of \(\dfrac{e}{m}\) by Bearden\(^{17}\)).
As a result of measuring twenty-five photographs, Bearden\(^{17}\) obtained \(\delta = 9.2244 \cdot 10^{-9}\) with a probable error of 0.1 per mille; the maximum deviation of individual values from the mean is 0.5 per mille. In calculating the sum \(N_s(1 + A)\), which was carried out by semiclassical, semi-quantum methods, the formula includes, in addition to the wavelength of the critical absorption edge, also the exponent in the absorption law. Bearden took it as 2.75. Since, however, the sum can be calculated only for half-integer or integer exponents in the absorption law, the desired value has to be found by interpolation between the values for the exponents 2.5 (6.0194) and 3.0 (6.0131); one obtains
\[ N_s(1 + A) = 6.0163 \pm 0.0006. \]
The value thus found was further confirmed by the fact that the application of Henle’s wave-mechanical approximate calculation leads precisely to the value 6.016. In addition, the final result includes the molecular weight of carbon, which was taken as \(12.0148 \pm 0.0005\), the density of diamond, which Bearden determined specially and found to be \(3.51536 \pm 0.00004\), Faraday’s constant \(F = 96513 \pm 8\), and the absolute wavelength of Cu \(K\alpha\), equal to
\[ \lambda = (1.39220 \pm 0.00003)\,\text{\AA}. \]
With these numbers Bearden obtained
\[ \frac{e}{m} = 1.7601 \cdot 10^7 \ \mathrm{CGSM} \]
with a probable error of \(0.0003 \cdot 10^7\) CGSM.
The experiments of Bearden and Shaw with quartz\(^{18}\), however, gave a considerably lower value, namely \(1.757 \cdot 10^7\) CGSM; the deviation is probably due to the unreliability of the calculation of the sum \(N_s(1 + A)\), which in the case of quartz is significantly larger than in the case of diamond.
The accuracy of Bearden’s measurements on diamond, as well as of the other spectroscopic measurements of \(\frac{e}{m}\), is of the same order of magnitude as the accuracy of determining \(\frac{e}{m}\) on free electrons.
14. Summary of the Results of New Determinations of \(\frac{e}{m}\)
The results of determinations of \(\frac{e}{m}\) have several times been compared by various investigators. Unfortunately, in doing so an excessive accuracy was indicated, since in the processing it was assumed that the errors of the various measurements could be regarded as random and, consequently, by means of the Gaussian method of calculating errors, one could determine not only the “most probable” mean value, but also the “probable” error of this mean value. This, however, as was already pointed out in the introduction, is inadmissible, since the matter concerns chiefly so-called “systematic” sources of error, and not random errors of measurement.
In Fig. 8 we give a comparison of the results of all the new determinations of \(\dfrac{e}{m}\); within their true error limits—about 1 per mille—they agree with one another quite satisfactorily.
Fig. 8. Results of the determination of \(\dfrac{e}{m}\) over the last 10 years
Thus\(^1\) it may be concluded that the true value of \(\dfrac{e}{m}\) lies with sufficient reliability between \(1.758\) and \(1.760 \cdot 10^7\), and that it probably is about \(1.7590 \cdot 10^7\). We therefore write
\[ \frac{e}{m}=(1.7590 \pm 0.0015)\cdot 10^7 \ \mathrm{CGSM}. \]
Combining this value with \(e=4.803\cdot 10^{-10}\) CGSE, we obtain the rest mass of the electron
\[ m=9.108\cdot 10^{-28}\ \mathrm{g}. \]
IV. DETERMINATION OF \(\dfrac{h}{e}\)
Detailed reviews of various methods for determining Planck’s constant \(h\) are available, on the one hand, in P. Ladenburg’s article in the Handbuch der Physik \({}^{72}\), and, on the other, in the already mentioned compilation of the most probable values of the general physical constants by R. T. Birge \({}^{20}\), and finally in the recently published review by Wensel \({}^{99}\), devoted to radiation constants. The most accurate of all the methods is the determination of \(\dfrac{h}{e}\) from the short-wavelength limit of the continuous X-ray spectrum; only by means of this method has a series of new determinations of \(h\) been made during the last ten years. The significance of the other methods \(\left(\dfrac{h}{e}\right.\) from the photoelectric effect, \(\dfrac{h}{e}\) from ionization and resonance potentials, \(\dfrac{h}{e}\) from Wien’s displacement law, \(\dfrac{h}{e^{4/3}}\) from the Stefan–Boltzmann radiation law and Planck’s radiation formula\()\) consists chiefly in the fact that all these methods give values of \(h\) which agree within the corresponding errors.
\(^1\) Birge (Phys. Rev., 55, 584, 1939), however, holds the opinion that there is a real difference between the “spectroscopic value” \(\dfrac{e}{m}\) \((1.7583 \pm 0.0002)\) and the value of \(\dfrac{e}{m}\) for free electrons \((1.75988 \pm 0.00017)\), and he assigns his own measurements to the latter group.
- Determination of \(\dfrac{h}{e}\) from the short-wavelength limit of the bremsstrahlung spectrum. New determinations of \(\dfrac{h}{e}\) have been carried out exclusively by the “isochromat” method developed by E. Wagner. The method consists, as is known, in determining the intensity of a definite wavelength as a function of the voltage on the X-ray tube, whereas Duane and his collaborators, conversely, investigated the distribution of intensity at the short-wavelength limit of the continuous spectrum at constant tube voltage.
The voltage \(V\) at which a definite wavelength \(\lambda\) first appears is related to this wavelength by the simple relation
\[ eV=h\nu=h\frac{c}{\lambda}. \]
As long as absolute measurements of X-ray wavelengths did not exist, it was natural to express the wavelength entering this formula, using Bragg’s relation, through the grating constant \(d\), and to calculate \(d\) from the macroscopically measured density, Avogadro’s number, and the molecular weight of the corresponding crystal by means of the formula
\[ d^{3}=\frac{M}{2N\Phi(\beta)\rho} \]
(cf. p. 320). If, further, in this formula Avogadro’s number \(N\) is replaced by the Faraday constant and the elementary charge, and it is assumed that the Faraday constant is known with such accuracy that it can be combined with the speed of light into one known constant factor, then in the end the following formula is obtained for the wavelength of the short-wavelength limit:
\[ \frac{h}{e^{4/3}}=\mathrm{const}\cdot V\cdot \sin\vartheta. \]
Thus, according to this conception, the measured voltage \(V\), together with the reflection angle \(\vartheta\) of the X-ray spectrometer, gives the ratio
\[ \frac{h}{e^{4/3}}. \]
However, in the opinion of the author of the present article, this method of calculation is no longer sufficiently justified, since the absolute values of X-ray wavelengths, as shown on p. 316, are now known with at least the same reliability as the numerical value of the Faraday constant. Therefore, at present one may consider that measurements of the short-wavelength limit of the X-ray spectrum give directly the ratio \(\dfrac{h}{e}\), namely:
\[ \frac{h}{e}=\frac{2dV\sin\vartheta}{c}, \]
where for \(d\) one must now, of course, take the absolute value of the constant of the crystal used. New measurements of the short-wavelength limit in recent years have been made by Feder\(^{44}\) and Scheitberger\(^{88}\) in Germany and by Kirkpatrick and Ross\(^{70,71}\), as well as by DuMond and Bollman\(^{40}\), in America.
Fedders’s and Shaitberger’s works were both carried out at the Würzburg Institute. The arrangement of the experiment in the two works was essentially the same; only Fedders used, as the voltage source, a high-voltage storage battery, while Shaitberger worked with a transformer, the current of which was rectified by valve tubes and smoothed by capacitors.
Fig. 9. Determination of \(\dfrac{h}{e}\) by the “isochromat” method: the intensity of a definite X-ray spectral line as a function of the voltage on the tube (Shaitberger’s measurements\(^ {88}\))
Some of the ionization curves obtained by Shaitberger as a function of the voltage on the tube are reproduced in Fig. 9. Shaitberger points out
Fig. 10. Isochromat according to the measurements of Kirkpatrick and Ross\(^ {7,11}\)
that the greatest error in his determination of the critical voltage is due to the inaccuracy in determining the travel of the electrometer. As the limit of error for the final value of \(\dfrac{h}{e}\), Shaitberger takes 0.5 per mille. This includes:
| Inaccuracy of the voltage measurement | 0.2 per mille |
| Inaccuracy of the correction for the width of the line | 0.03 » |
| Error due to fluctuations of the temperature of the crystal | 0.07 » |
A very good crystal of rock salt was used as the spectrometer crystal. The measurements were made at four wavelengths (\(WL\beta_1\), \(WL\gamma_1\), Pt \(L\gamma_1\), and Pb \(L\gamma_1\)).
Kirkpatrick and Ross worked with a double spectrometer with two calcite crystals and with the wavelength Ag \(K\alpha\); one of their ionization curves is shown in Fig. 10. As the probable error
for the final value of \(\dfrac{h}{e}\) they give 0.1 per mille; this includes, among other things:
Statistical probable error of 11 separate measurements . . . . . . . . . . . . . 0.05 per mille
Probable error of the voltage measurement . . . . . . . . . . . . . . . . . . . . 0.02 »
Exceptionally careful measurements were carried out by Du Mond and Bollman\(^ {40}\). They likewise used a double spectrometer with calcite crystals and worked with the molybdenum \(K\beta_{2}\) line and the tungsten \(L\beta_{1}\) line. They investigated in particular the influence of limited resolving power. In fact, owing to the limited resolving power of the spectrometer, even with the best collimation of the rays, what is selected is not a strictly monochromatic beam, but a more or less broad region of the spectrum, the intensity of which falls off on both sides approximately as the square of the distance (Fig. 11). Therefore, however great the resolving power of the spectrometer may be, as measured by the half-width of its spectral curve, a certain part of the radiation with wavelengths separated by a distance several times greater than the half-width from the mean wavelength nevertheless always passes through the spectrometer. If now—as is, of course, the case in measurements at the short-wavelength limit—the distant wavelengths have a considerably greater intensity than the mean wavelength to which the spectrometer is set, then the limited resolving power may become a serious source of errors. Du Mond and Bollman achieved a reduction of this error by using, as anticathode, a thin layer of tungsten, obtained by evaporating the incandescent filament onto a massive nickel substrate. Kulenkampff showed that the slope of the isochromats for a given anticathode material is proportional to the atomic number of the element of which this anticathode consists. Therefore, in the case of a thin layer with a higher atomic number, the isochromat at first, i.e. immediately at the limit, must rise steeply upward, and then, at a distance from the limit of as many volts as the energy loss of the electrons in the thin layer amounts to, it must abruptly change its slope and proceed more gently.
To the measured voltage there were made, just as by Kirschbaum and Ross, the following two corrections: 1) the voltage drop in the incandescent filament (3 V); 2) the work function of the electrons (4.5 V). Both of these corrections must be added to the measured voltage, as was shown in detail by Du Mond and Bollman.
The finite width of the spectral region transmitted by the spectrometer, considered above, entails that the measured isochromats do not begin at once at an acute angle to the axis of abscissas, but, depending on the magnitude of the voltage, approach it more or less asymptotically. Du Mond and Bollman showed that the method adopted before them for determining the point of onset of an isochromat, consisting in drawing a tangent to it and extending it to its intersection with the axis of abscissas,
leads to unimpeachable results only when the isochromat, at some distance from the short-wavelength limit, actually proceeds rectilinearly. This, generally speaking, is not the case. In fact, it was already known from earlier observations that isochromats, at some distance from the limit, exhibit a “break,” and DuMond and Bollman found one more break in the immediate vicinity of the limit (Fig. 11).
Fig. 11. Determination of \(\dfrac{h}{e}\) according to DuMond and Bollman\(^{40}\)
the limit (Fig. 11). It is therefore necessary, in processing the results, to take into account the smoothing of the course of the isochromats owing to the finite width of the spectral region transmitted by the spectrometer (and, to a slight degree, also owing to the finite width of the voltage of the thermoelectron source). DuMond and Bollman give an analytical treatment
effect of smoothing; they discuss various methods of solution based on an analytic treatment, but give preference to the following graphical method: from the measured isochromat the first and second derivatives are determined and curves are plotted for them; the sharply expressed cusps of these curves make it possible to find the position of the true boundary with great accuracy. The final results of DuMond and Bollman are presented in Fig. 11. Here the measurements with both X-ray wavelengths are plotted on one diagram in approximately the same way as, in the study of the photoeffect, the measured maximum electron energy is plotted as a function of the wavelength of light. In the left part of the figure are presented, for both applied voltages, the spectral regions transmitted by the spectrometer; the greater width at the higher voltage is naturally explained by the stronger smoothing of the corresponding isochromat. The boundary of the upper isochromat (\(\mathrm{Mo}\ K_{\beta 2}\)), determined from the first (51 in Fig. 11) and from the second (52) derivative, lies, together with the boundary of the lower isochromat (\(\mathrm{WL}_{\beta 1}\)) found in an analogous manner, on the straight line passing through the origin of coordinates
\[ \frac{h}{e} = 1.3762 \cdot 10^{-17}\ \mathrm{erg}\cdot\mathrm{sec}/\mathrm{CGSE}. \]
As for the points of intersection \(W_1\) and \(W_2\) with the two voltage axes, we shall return to them below on p. 354. In Table 5 we give a comparison of the results of various determinations of \(\frac{h}{e}\) from the short-wavelength boundary of the continuous spectrum.
Table 5
| Author | \(\frac{h}{e}\cdot 10^{17}\) |
|---|---|
| Feder | 1.3766 |
| Kirkpatrick and Ross | 1.3756 |
| Schäfer | 1.3783 |
| DuMond and Bollman | 1.3762 |
The results of Feder, Kirkpatrick and Ross, and Schäfer have been converted to the system of absolute X-ray wavelengths so that they could be compared with the results of DuMond and Bollman; in addition, to the results of Feder and Schäfer a correction of \(4.5\ \mathrm{V}\) has been introduced for the work function of electrons from the incandescent filament (cf. above, p. 342).
As has already been mentioned, the latest experiments of DuMond and Bollman, both from the point of view of experimental arrangement and in the accuracy of the treatment of the results, should be regarded as the most satisfactory. However, their final value
\[ \frac{h}{e} = 1.3762 \cdot 10^{-17}, \]
as will be shown on p. 354, does not agree with the system of the remaining values of the atomic constants, whereas Schäfer’s value still satisfies this requirement. The limits of error in these experiments are approximately about 1 per mille.
16. Survey of the results of determinations of \(\frac{h}{e}\) by other methods. In the following table we give a survey of the results of some determinations of \(\frac{h}{e}\) by other methods. These determinations
however, owing to their lesser accuracy, as has already been mentioned, decisive significance cannot be ascribed to them. In this connection we use the numbers given in Wensel’s recently published review^99.
Table 6
| Author | Year | Method | $\dfrac{h}{e}\cdot 10^{17}$ |
|---|---|---|---|
| Millikan^75 | 1916 | Photoelectric effect | $1.3777 \pm 0.006$ |
| Olpin^79 | 1930 | Photoelectric effect | $1.3748 \pm 0.006$ |
| Lawrence^73 | 1926 | Ionization potential | $1.3753 \pm 0.004$ |
| Mean^99 | 1899–1922 | Wien displacement law | $1.3769 \pm 0.005$ |
| Mean^99 | 1898–1919 | Stefan–Boltzmann radiation law | $1.3775 \pm 0.004$ |
| Wensel and Roeser^99 | 1939 | Radiation constant from optical-pyrometric measurements | $1.3771 \pm 0.004$ |
For a discussion of these methods one should consult the reviews by Birge^20 and Wensel^99, in which there are also detailed bibliographic references. The result of all these measurements, within the limits of experimental error, agrees with the results of determinations of $\dfrac{h}{e}$ from the short-wavelength limit of the bremsstrahlung spectrum. It is noteworthy, however, that these results are only slightly lower than the values that would be expected on the basis of the most reliable present-day values of $e$, $\dfrac{e}{m}$, etc., and the Rydberg relation (cf. Fig. 14 and p. 354).
V. FURTHER COMBINATIONS OF $h$, $m$, AND $e$
17. Determination of $\dfrac{h}{m}$ from the Compton effect and from the de Broglie wavelength and velocity. For determining the ratio $\dfrac{h}{m}$ there are two methods. One is based on the theory of the Compton effect, according to which the change in wavelength in the scattering of hard X-rays through an angle of $90^\circ$ must be equal to $\Delta\lambda=\dfrac{h}{mc}$; the second method, proposed by Kirchner^66, is based on the combination of a measurement of the electron wavelength $\left(\lambda=\dfrac{h}{mv}\right)$ with a measurement of velocity.
The most accurate measurements by the first method were carried out by Ross and Kirkpatrick^87. They worked with the $K\beta$ lines of molybdenum (0.631 Å), silver (0.496 Å), and tin (0.435 Å). In photographing the scattering spectrum of these lines on carbon they established that the change in wavelength in the scattering process is not
independent of the wavelength, as follows from the elementary theory of the Compton effect, but increases as the wavelength decreases.
In Fig. 12 the measured Compton shifts for \(90^\circ\) are plotted as a function of \(\lambda^2\); through the three measured points one can draw a straight line whose intersection with the ordinate axis, according to Ross and Kirkpatrick, gives the desired value \(\dfrac{h}{mc}\) for wavelength \(\lambda = 0\). (The measured points for beryllium are so scattered that they do not give convincing proof of the correctness of this extrapolation.) If it is assumed that the extrapolation is valid, then the intersection with the ordinate axis gives \(\Delta \lambda_{90^\circ} = \dfrac{h}{mc} = 0.02415\ \text{Å}\); multiplying by \(c\), we obtain from this \(\dfrac{h}{m} = 7.240\). This quantity must still be converted to the absolute scale of wavelengths, since Ross and Kirkpatrick referred their measurements to the wavelength scale of the crystal lattice; in this way one finally obtains \(\dfrac{h}{m} = 7.255\). On the basis of Fig. 12 it may be assumed that the accuracy of this result is \(0.3\%\). (Ross and Kirkpatrick state in the text of their paper that the point of intersection with the ordinate axis of the straight line passing through the points measured on carbon lies at \(0.02418 \pm 0.00004\ \text{Å}\); however, from their drawing it is not clear where this number comes from. It would give \(\dfrac{h}{m} = 7.264\).)
Fig. 12. Dependence of the Compton shift on wavelength (Ross and Kirkpatrick\(^{87}\))
The second method is based on simpler principles: it requires only measurement of the wavelength and direct measurement of the velocity; their combination gives the desired quantity \(\dfrac{h}{m}\). Unfortunately, up to now there exists only one single flawless measurement by this method; it was carried out by Gnan\(^{50}\) at the author’s suggestion. In order to obtain good diffraction patterns on passing through thin crystalline layers, Gnan worked with electrons whose velocity lay around \(0.3—0.4\,c\). The diffraction patterns—Debye–Scherrer diagrams were involved—were measured photometrically. Unfortunately, because of the high velocity of the cathode rays and with the experimental means used, it was not possible to achieve very high accuracy.
If for the bismuth grating used for the diffraction measurements one takes the constant \(2d = 4.5353\ \text{Å}\) in the crystal-lattice scale found by Jette and Foote\(^{60}\), and then converts it to the absolute scale, then Gnan’s measurements give \(\dfrac{h}{m} = 7.267\). As the limit of error of this result Gnan gives \(0.4\%\).
18. Stern von Friesen’s work on determining \(\frac{h}{m}\cdot\frac{h}{e}\) from the de Broglie wavelength and the discharge voltage
Very precise measurements of the wavelength of cathode rays were carried out by Stern von Friesen\({}^{45}\) in Uppsala. His cathode rays were very finely collimated by two slits of 0.03 and 0.016 mm, placed at a distance of 10.5 cm; as a diffraction grating he used a very good crystal of galena, which had been etched with hot concentrated hydrochloric acid.
Fig. 13. Spectra of electron beams of different volt speeds, obtained with the aid of a crystal (Stern von Friesen\({}^{45}\))
In Fig. 13 several diffraction spectra obtained by him with this crystal are reproduced. Measurement of these spectra, after recalculating the lattice constant on an absolute scale, gives the wavelength
\[ \lambda=\frac{h}{mv}. \]
If at first we do not take into account the change of electron mass as a function of velocity, then measurement of the voltage gives
\[ \frac{mv^{2}}{2}=eV \quad \text{or} \quad mv=\sqrt{2meV}. \]
If this is substituted into the formula for the wavelength, one obtains
\[ \lambda=\frac{h}{\sqrt{em}}\cdot\frac{h}{\sqrt{2V}} \quad \text{or} \quad \frac{h}{m}\cdot\frac{h}{e}=2V\lambda^{2}. \]
When taking into account the dependence of mass on velocity, one must put
\[ \lambda=\frac{h}{\sqrt{2em_{0}V\left(1+\frac{eV}{2m_{0}c^{2}}\right)}} ; \]
as a consequence of this, to the right-hand side of the expression for \(\frac{h}{m}\cdot\frac{h}{e}\) there is appended the factor
\[ 1+\frac{eV}{2m_{0}c^{2}}, \]
in which, however, there is no need to know \(\frac{e}{m}\) with very great precision.
Stern von Friesen treated the results of his measurements in another way. From the measured de Broglie wave he calculated, with the aid of the Rydberg formula and under the assumption that \(\frac{e}{m}=1.7592\cdot 10^{7}\), the charge of the electron \(e\) and the quantum of action \(h\). Taking \(R=109737\), \(c=2.998\cdot 10^{10}\), and for \(\frac{e}{m}\) the value indicated above, \(1.7594\cdot 10^{7}\), Stern von Friesen obtained:
\[ e=(4.7963\pm 0.005)\cdot 10^{-10}\ \mathrm{CGSE}, \]
\[ h=6.610\cdot 10^{-27}\ \mathrm{erg\cdot sec}. \]
The processing by the method indicated earlier gives
\[ \frac{h}{m}\cdot \frac{h}{e}=1.0017\cdot 10^{-16}. \]
The maximum error limit with this method of processing gives a less favorable picture than in Friesen’s calculation of \(e\). The various sources of error have been discussed in detail by Sten von Friesen.
Error in the measurement of the wavelength
| Source | Error |
|---|---|
| Inaccuracy in measuring the distance crystal—plate | 0.3 per mille |
| Inaccuracy in measuring the lattice constant | 0.3 » |
| Inaccuracy in measuring the spectra | 0.4 » |
Error in the measurement of the voltage
| Source | Error |
|---|---|
| Standard cell | 0.1 per mille |
| Potentiometer | 0.1 » |
| Comparison resistance | 0.2 » |
| High-ohmic resistance | 0.3 » |
| Temperature coefficient of the resistance | 0.1 » |
| Contact potentials | 0.1 » |
Since in the final result the wavelength enters squared, while the voltage enters only to the first power, the error in the measurement of the wavelength has the predominant influence. If, together with Sten von Friesen, the total error in the measurement of the voltage is taken to be 0.5 per mille, then the maximum total error of \(\dfrac{h}{m}\cdot \dfrac{h}{e}\) is found to be equal to 2.5 per mille.
19. Determination of \(\dfrac{e}{m}\cdot \dfrac{e}{h}\) by measuring the magnetic deflection of photoelectrons liberated by X-rays. When X-rays of frequency \(\nu\) fall upon a substance, they, as is known, liberate photoelectrons whose energy is equal to the product \(h\nu\) minus the work of liberation of the electron. Since in this process electrons of the inner shells are predominantly liberated, the work of detachment is simply equal to the product of \(h\) by the frequency of the corresponding absorption edge (for example, the \(K\)-absorption edge, etc.). Thus one obtains
\[ \frac{mv^{2}}{2}=h(\nu-\nu_{K,L,\ldots}). \]
Maurice de Broglie was the first to show that a spectrum of photoelectrons can be obtained if the photoelectrons liberated by X-rays from thin layers are made to describe a semicircle in a magnetic field and are then received on a photographic plate. For the magnetic deflection of electrons in a homogeneous magnetic field perpendicular to the electron trajectory, as is known, the relation holds
\[ \frac{mv}{\rho}=eH. \]
The magnetic spectra of electrons were studied by various investigators—most thoroughly by Robinson. For determination either for
for checking the numerical values of the atomic constants, the magnetic spectra of electrons were used, on the one hand, by Kretschmer^69, and on the other by Robinson and his collaborators^82–85. Kretschmer worked with the \(K\)-radiation of molybdenum and used his measurements to calculate \(\frac{e}{m}\), taking the value of \(\frac{h}{e}\) from measurements of the short-wavelength limit of X-ray spectra (assuming \(\frac{h}{e}=1.3737\cdot 10^{-17}\), he obtained \(\frac{e}{m}=1.7570\cdot 10^7\) CGSM). Robinson and his collaborators used the \(K\)-radiation of Cr, Cu, and Mo and, thanks to this, were able to eliminate certain inaccuracies due to insufficiently precise data on absorption edges. If the value of \(\nu\) from the energy equation is substituted into the equation for magnetic deflection and written somewhat differently, one obtains
\[ \frac{e}{m}\cdot\frac{e}{h}=\frac{2(\nu-\nu_{K,\ldots})}{\rho^2 H^2}. \]
If we regard the absolute wavelengths (and together with them the frequencies appearing in the formula) as known very accurately, then on the right-hand side there remain only the measurable quantities \(\rho\) and \(H\); the measurements with photoelectrons described here therefore give \(\frac{e}{m}:\frac{h}{e}\). Such a treatment of the observations gives, according to Robinson^85,
\[ \frac{\dfrac{e}{m}}{\dfrac{h}{e}}=(3.822\pm0.004)\cdot 10^{34}, \]
if the charge is expressed in electrostatic units.
VI. RELATIONS BETWEEN ATOMIC CONSTANTS AND THEIR COMBINATIONS
In addition to the relations between constants that have already been discussed in considering individual measurements, the fundamental spectroscopic relation—the so-called Rydberg formula—plays a very important role. The Rydberg constant, as is well known, characterizes the lowest energy term of the hydrogen atom; it can therefore be determined directly from spectroscopic observations of the position of the series limit or, with great precision, calculated from measurements of the wavelengths of various lines of one and the same series. In this way one obtains
\[ R_{\mathrm H}=109\,677.759. \]
According to the Bohr–Sommerfeld theory, the Rydberg number
\[ R_\infty = R_{\mathrm H}\cdot\frac{m_{\mathrm H}+m}{m_{\mathrm H}} \]
for an atom whose nuclear mass is very large in comparison with the mass of the electron is expressed by the following formula:
\[ R_\infty=\frac{2\pi^2 e^5}{h^3 c^2\cdot \dfrac{e}{m}} = \frac{2\pi^2 e^2}{c^2\left(\dfrac{h}{e}\right)^3\cdot \dfrac{e}{m}}. \]
As long as there was still doubt as to which of the values of \(e\) was correct—the Millikan number or the value obtained from absolute measurements of X-ray wavelengths—it seemed desirable to substitute into this formula the value \(\dfrac{h}{e}\), found from the short-wavelength limit of the continuous X-ray spectrum, and \(\dfrac{e}{m}\) from measurements with cathode rays; this would give a completely independent value of \(e\). This path was tried in 1932 by the author,^8 where the then available values of the absolute wavelengths of the short-wavelength limit were taken as correct. Taking \(R_{\infty}=109737.4\) and \(\dfrac{e}{m}=1.7585\cdot 10^{7}\) CGSM, the author obtained from the measurements of Duane and his collaborators \(\left(\dfrac{h}{e}=1.3787\cdot 10^{-17}\right)\):
\[ e=4.798\cdot 10^{-10}\ \mathrm{CGSM}, \]
and from Feder’s measurements \(\left(\dfrac{h}{e}=1.3755\cdot 10^{-17}\right)\):
\[ e=4.782\cdot 10^{-10}\ \mathrm{CGSE}. \]
Thus, a clear choice between the X-ray value \((4.80\cdot 10^{-10})\) and Millikan’s value \(e=4.77\cdot 10^{-10}\) proved impossible. In order to make this choice, Kirkpatrick and Ross undertook the new determination of \(\dfrac{h}{e}\) mentioned on p. 341. They obtained an even smaller value of \(\dfrac{h}{e}\) than Feder; moreover, they introduced significant corrections into the number obtained by Duane, as a result of which this number became smaller than Feder’s result, and from this they concluded that the Millikan value of \(e\) is correct, while the value calculated from the absolute wavelength of X-rays is too large. However, this conclusion, as is now known, is untenable; probably, on the contrary, the error should be sought in the determination of \(\dfrac{h}{e}\) (cf. below, p. 354).
Bond^28 and Birge^21 discussed in detail the relation between \(h\) and \(e\) by means of a graphical method. Since the Rydberg formula was also used in this procedure, it was necessary to reckon only with the difficulty already mentioned of bringing the measured values of \(e\), \(\dfrac{h}{e}\), and \(\dfrac{e}{m}\)^1) into the mutual agreement required by the Rydberg formula. Bond^29,30 proposed eliminating this difficulty by assuming that the measured value of \(\dfrac{e}{m}\) is not the true specific charge of the electron, but \(\dfrac{136}{137}\) of the true value. He tried
^1) An especially detailed and careful discussion, with the aid of Birge–Bond diagrams, was recently given by Dunnington: F. Dunnington, Rev. Mod. Phys., 11, 65, 1939 (see Uspekhi fizich. nauk 23, 162, 1940. Ed.).
substantiate this assumption by reference to Eddington’s theory. Bond’s proposal was put up for discussion by Birge^23; however, in Birge’s opinion, one should also take into account the possibility that Rydberg’s formula is inaccurate and, perhaps, requires a correction of the order of magnitude \(136/137\).
To these broad conclusions Sten von Friesen^46 opposed the cautious point of view of an experimentalist. He proposed, on the basis of the results of measurements carried out before 1936, that the following numbers be regarded as correct:
\[ e = (4.800 \pm 0.005)\cdot 10^{-10}\ \mathrm{CGSE}, \]
\[ \frac{e}{m} = (1.7585 \pm 0.002)\cdot 10^{7}\ \mathrm{CGSM}_{1/2}, \]
\[ h = (6.610 \pm 0.015)\cdot 10^{-27}\ \mathrm{erg\cdot sec}. \]
Against this system of values Birge^26 raised the objection that these numbers do not exactly satisfy Rydberg’s formula. However, Sten von Friesen replied to this^47 that Rydberg’s formula is satisfied by his values, although not exactly, but within the indicated errors.
Another way to eliminate the mentioned difficulty was proposed by K. Shiba^91–93. From the very beginning he rejects determinations of \(\frac{h}{e}\) as unsuitable, without, however, giving fully convincing grounds for this, and then combines the remaining determinations with one another in such a way that \(h\) is eliminated. In this way he obtains, in addition to the two results of direct measurements \(e = 4.8030\cdot 10^{-10}\) and \(\frac{e}{m} = 1.7590\cdot 10^{7}\), the following combinations of both these quantities:
\[ e^{2}\cdot \left(\frac{e}{m}\right)^{3} = 1.2525\cdot 10^{3}, \]
\[ e^{-1}\cdot \left(\frac{e}{m}\right)^{4} = 1.9950\cdot 10^{38}, \]
\[ e\cdot \left(\frac{e}{m}\right)^{2} = 1.4843\cdot 10^{5}, \]
\[ e^{2}\cdot \left(\frac{e}{m}\right)^{-1} = 1.3004\cdot 10^{-26}. \]
From this, by means of the method of least squares, he finds as the “most probable values”:
\[ e = 4.8025\cdot 10^{-10}\ \mathrm{CGSE}, \]
\[ \frac{e}{m} = 1.7589\cdot 10^{7}\ \mathrm{CGSM}_{1/2}, \]
\[ h = 6.625\cdot 10^{-27}\ \mathrm{erg\cdot sec}. \]
The best possibility for surveying all the individual measurements of the fundamental atomic constants is provided by the method of representation proposed by Du Mond^38. Du Mond first of all records the results of various methods for determining atomic constants in
in the following form:
\[ e=A_1 \quad (1) \]
\[ e^{-1}h=A_2 \quad (2) \]
\[ e^{-\frac12}m^{-\frac12}h=A_3 \quad (3) \]
\[ m^{-1}h=A_4 \quad (4) \]
\[ em^{-1}=A_5 \quad (5) \]
\[ e^2m^{-1}h^{-1}=A_6 \quad (6) \]
\[ e^4mh^{-3}=A_7 \quad (7) \]
If now \(e_0\), \(m_0\), \(h_0\) are arbitrarily fixed values near the true value, then the relative deviation \(x_e\) of the measured value \(e\) from this fixed value is equal to
\[ x_e=\frac{e-e_0}{e_0} \]
and, correspondingly:
\[ x_m=\frac{m-m_0}{m_0} \]
and
\[ x_h=\frac{h-h_0}{h_0}. \]
In an analogous manner, the relative deviations of the measured constants \(A_i\) from the corresponding values of these constants \(A_{i0}\), calculated with the aid of the arbitrarily fixed \(e_0\), \(m_0\), and \(h_0\), can be written in the form
\[ a_i=\frac{A_i-A_{i0}}{A_{i0}}. \]
In a three-dimensional rectangular coordinate system with axes \(x_e\), \(x_m\), \(x_h\), each of equations (1)—(7) is represented by a surface which, within a small region around \(x_e=x_m=x_h=0\), can with sufficient accuracy be replaced by a tangent plane. By means of a Taylor expansion, the following relations are obtained as the equations of the tangent planes corresponding to equations (1)—(7):
\[ x_e=a_1 \quad (1) \]
\[ x_h-x_e=a_2 \quad (2) \]
\[ x_h-\frac12 x_e-\frac12 x_m=a_3 \quad (3) \]
\[ x_h-x_m=a_4 \quad (4) \]
\[ x_e-x_m=a_5 \quad (5) \]
\[ 2x_e-x_m-x_h=a_6 \quad (6) \]
\[ x_m+4x_e-3x_h=a_7 \quad (7) \]
Each of these planes has a different orientation, fixed by the coefficients \(x_i\), i.e., by the powers of \(e\), \(m\), and \(h\) entering into the definition. A change in the measured value \(A_i\) causes a displacement of the corresponding plane in the direction of its normal, so that the plane remains parallel to its initial position. If three (or more) planes intersect at one point \(x_e\), \(x_m\), \(x_h\), then they correspond to one and the same values of \(e\), \(m\), and \(h\).
The five planes corresponding to equations (2)—(6) are parallel to a certain common axis; the direction of this common axis is such that it forms equal angles with the three positive axes \(x_e\), \(x_m\), and \(x_h\). The displacements which these five planes undergo when the measured values \(A_i\) change can therefore be represented without distortion by the straight lines of intersection of these planes with a plane perpendicular to their common axis. In the drawing, constructed-
in this way (Fig. 14)\(^1\), different parallel lines correspond to different values \(A_i\), obtained by each particular method, whereas different methods correspond to different directions of the intersection lines. If three or more intersection lines pass through one point, this means for the corresponding methods that their results are in agreement with one another, i.e., that they lead to a common system of values \(e, m, h\).
Fig. 14. Comparison of the results of measuring the numerical values of atomic constants (DuMond\({}^{38}\))
The planes not yet taken into account, (1) \(x_e = a_1\) and (7) \(x_m + 4x_e - 3x_h = a_7\), which are not parallel to the common axis of the remaining planes, intersect along a straight line inclined to the plane of the drawing in Fig. 14; this straight line is projected onto the plane of the drawing. It is clear that the measurement results represented by equations (1) and (7) are in agreement with the various results, pre-
\(^1\) The drawing has been modified in comparison with DuMond’s original drawing; the following have been added to it: the value of \(\frac{e}{m}\) obtained by Bärlein, \(\frac{h}{m}\)—from Gnan’s measurements, and \(\frac{h}{m}\)—according to Kirkpatrick and Ross (cf. p. 346).
composed equations (2) to (6), only in the case where the just-mentioned line of intersection passes through the common point of intersection of the lines corresponding to equations (2)—(6).
For some methods their limits of error are indicated; for the more accurate methods, by means of which a whole series of measurements has been carried out by various investigators, the errors are not indicated. In the latter case the difference between the individual values already indicates the degree of reliability of the corresponding result.
The drawing shows with exceptional clarity that, with the standard values adopted by Du Mond,
\(e_0 = 4.8029 \cdot 10^{-10}\),
\(m_0 = 9.11096 \cdot 10^{-28}\),
\(h_0 = 6.62602 \cdot 10^{-27}\)
(and correspondingly:
\(\dfrac{e}{m} = 1.7585 \cdot 10^7\),
\(\dfrac{h}{m} = 7.27258\),
\(\dfrac{h}{e} = 1.37959 \cdot 10^{-17}\)),
all the separate measurements of the various combinations of constants are in very good agreement with one another, except the determinations of \(\dfrac{h}{e}\) from the short-wavelength limit of the continuous X-ray spectrum and the determinations of \(\dfrac{h}{m}\) from the Compton effect. In the latter case the deviation lies within rather broad limits of error. Among the separate measurements of the group \(\dfrac{h}{e}\), in any case, Schaitberger’s measurements may still be considered to agree with the cited system of standard values; however, the deviation of the most recent values of Du Mond and Bollman is too great for it to be ascribed to any intelligible error. The deviation is in such a direction that a definite X-ray line of frequency \(\nu\) appears already at a lower voltage on the tube than could have been expected from the standard value of \(\dfrac{h}{e}\) (Fig. 14). The voltage at which the lines measured by Du Mond and Bollman should first appear is denoted in Fig. 11 by \(W_1\) and, correspondingly, \(W_2\); at 20,000 V it is lower by 70 V, and at 10,000 V higher by 35 V, than the observed minimum voltage. Both “theoretical” values of the voltage lie so far from the true voltage boundary that the deviation can in no case be reduced to any trivial error of observation. It is very difficult to find a plausible explanation for this discrepancy. In fact, it is necessary to explain why the electrons incident on the anticathode (or some of them) possess 35—70 V more energy in comparison with that corresponding to the accelerating voltage. Undoubtedly, the X-rays coming from the anticathode liberate by the photoelectric effect from the cathode a certain number of electrons, which in part have a considerably greater initial energy. However, it is unknown why these photoelectrons should have initial energies precisely of 35—70 V, and, moreover, their number is altogether insufficient to explain the observed effect, as was shown by the special experiments of Du Mond and Bollman. Another possible explanation, discussed by Du Mond and Bollman, consists
in the assumption that on the surface of the anticathode there is formed a thin layer, insulated from the mass of the metal, which under the influence of the secondary radiation becomes positively charged and therefore imparts an additional acceleration to the electrons falling on the anticathode. However, such a charge would have to depend on the strength of the primary current, whereas the magnitude of the “displacement,” according to the data of Du Mond and Bollman, definitely does not depend on the current strength in the X-ray tube.
In conclusion we shall give a summary of the numerical values of the atomic constants which, on the basis of modern measurements, may be regarded as reliable. From the combination of measurements of absolute X-ray wavelengths with crystal-spectroscopic measurements, one obtains for the lattice constant of calcite
\[ d = 3.0356 \pm 0.0001\ \text{\AA}. \]
On the basis of the assumption that the calcite lattice is the lattice of an ideal crystal (cf. on this point p. 318), this gives
\[ N = (6.022 \pm 0.005)\cdot 10^{23} \]
(\(N\) here refers to 1 mole on the chemical scale of atomic weights!). Taking \(F = (9649 \pm 1)\ \mathrm{CGSE} = 2.8926\cdot 10^{14}\ \mathrm{CGSE}\) per mole, we obtain
\[ e = (4.803 \pm 0.004)\cdot 10^{-10}\ \mathrm{CGSE}. \]
After recalculating Avogadro’s constant to the physical mass scale, one obtains
\[ m_{\mathrm{H}}=\frac{\mathrm{H}^{1}}{N\cdot 1.00027} =\frac{1.00813}{6.022\cdot 10^{23}\cdot 1.00027} =1.6736\cdot 10^{-24}\ \mathrm{g}. \]
For \(\mathrm{H}^{1}=1.00813\) one further obtains
\[ \frac{e}{m_{\mathrm{H}}} =\frac{F\cdot 1.00027}{1.00813} =(9573.8 \pm 1)\ \mathrm{CGSE}. \]
Combining this with
\[ \frac{e}{m}=1.759\cdot 10^{7}\ \mathrm{CGSE}, \]
we obtain
\[ \frac{m_{\mathrm{H}}}{m}=1837 \quad\text{and}\quad m=9.108\cdot 10^{-28}\ \mathrm{g}. \]
Finally, one should indicate the value of \(h\) which follows from the accepted values \(e=4.803\cdot 10^{-10}\) and \(\frac{e}{m}=1.759\cdot 10^{7}\ \mathrm{CGSE}\), by substitution in the Rydberg formula \(R_{\infty}=109\,737\):
\[ h = 6.626\cdot 10^{-27}\ \mathrm{erg}\cdot\mathrm{sec}. \]
This is contradicted, however, by the value \(h=6.614\cdot 10^{-27}\ \mathrm{erg}\cdot\mathrm{sec}\) as the result of combining the most precise measurements of \(\frac{h}{e}\) from the short-wavelength limit of the X-ray spectrum with the value \(e=4.803\cdot 10^{-10}\ \mathrm{CGSE}\).
As yet there is no satisfactory explanation for the peculiar position occupied by the definitions of $\frac{h}{e}$ in the system of the other measurements of atomic constants. But precisely this situation creates an especially strong stimulus for the development of new methods and for the further increase in the accuracy of measurements, so that in the coming years further successes in this field may be expected.
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