Full Text
ATOMIC CONSTANTS¹)
F. Dunnington, Brunswick
INTRODUCTION
The purpose of this review is: 1) to reexamine the most important experimental data whose results are based on the use of a whole series of auxiliary constants; 2) to establish which of these data are the cause of the well-known discrepancy between the values of the atomic constants; 3) to indicate what experimental investigations might lead to the resolution of the difficulties that have arisen.
A notable contradiction among the values of the three most accurately determined atomic constants—namely, the charge of the electron \(e\), its specific charge \(\frac{e}{m_0}\), and the ratio \(\frac{h}{e}\)—has for several years been the subject of discussion¹⁻⁶. Some clarity may be obtained if, alongside the three most accurately measured constants—\(e\), from X-ray diffraction; \(\frac{e}{m_0}\), from electronic and spectroscopic data; and \(\frac{h}{e}\), from the short-wavelength limit of the continuous X-ray spectrum—one considers other directly measured combinations of these constants. Recently, other combinations either have not been considered because of the lower accuracy of their values, or have been reduced to the form \(\frac{e}{m_0}\) (if the value of \(\frac{h}{e}\) is specified), or to the form \(\frac{h}{e}\) (if the value of \(\frac{e}{m_0}\) is accepted)⁷. Such a reduction confuses the problem and obscures the data that are the source of the contradiction.
In the present work, eight additional types of measurements possessing sufficient accuracy have been reexamined. The figures obtained by various methods have been analyzed with respect to the fundamental laws on which they are based.
Comparison of the results of different experiments was carried out on the Birge–Bond diagram⁸.
¹) Rev. Mod. Phys., 11, 65, 1939. Translated by A. A. Ilyina.
In general, one obtains from experiment
\[ e=f(m_0,h), \]
but the electron mass can be eliminated by means of the Rydberg formula:
\[ R_{\infty}=\frac{2\pi^2 e^4 m_0}{c h^3}. \]
This means that the results of experiment can be represented in the form \(e=Ah^n\), where \(A\) is a certain constant determined from experiment, and \(n\) is the power to which \(h\) enters the function \(f\). Taking for \(h\) some approximate value \(h_0\), one can calculate the corresponding value of \(e\), if the experimentally determined quantity \(A\) is known; it can be shown that between the value of \(e\) computed in this way and the exponent \(n\) there is a linear dependence. Birge–Bond’s method consists in constructing a graph of the values \(e_n\) as a function of \(n\).
The slope of the straight line constructed in this way determines the value of \(h\) (zero slope corresponds to \(h=h_0\); a positive slope of the straight line indicates that \(h>h_0\)). The value of \(e\) is given by the ordinate at \(n=0\).
Much labor was expended on bringing into order the system of auxiliary constants used in the present work. The conversion coefficients of the electrical units of the two international systems—namely, the system of the normal cell (which we shall denote by \(v\)) and the system of the silver voltameter (\(a\))—into absolute units were subjected to careful consideration. All sufficiently important data were treated by the method of least squares. In view of the work that has appeared since Birge’s first investigations \(^{8a}\), it is felt necessary to distinguish these two systems of international units, although the existing difference between them is fortunately very small.
All measurements that can be connected with the normal cell are described by us in the \(v\)-system. The resistance standard belongs simultaneously to both systems.
The author has recalculated the Faraday number. The most accurate value of the chemical atomic weight of silver leads to \(F=96\,493.7 \pm 0.9\) international \((a)\) coulombs, although in the work of Vinal and Bates \(^{9}\) with the iodine voltameter the number \(96510.3 \pm 68\) international \((a)\) coulombs is given. From these figures we have taken the weighted mean.
The ratio of the atomic weight obtained by the chemical method to the atomic weight measured by means of the mass spectrograph was calculated for many oxygen isotopes according to the data of the Atomic Committee of the International Union of Chemical Industry \(^{10}\). The origin of the remaining constants is indicated in Table 1. It must be emphasized that the recalculation of all experimental data considered in the present work was carried out with the aid of these auxiliary constants; in those cases where the constants used in the present article were obtained in other countries, the difference in the units employed in one country or another was taken into account \(^{11}\).
All errors are given in the form of least-square probable errors, and in the final result the larger of the two errors (internal or external) is always taken.^18
Table 1
Auxiliary constants used in the present work. The letter (B) marks Birge’s figures. (D) denotes figures obtained by the author. The literature is given at the end of the article.
| Designations | Name of constant | Value of constant |
|---|---|---|
| $F$ | Faraday number in international coulombs (D) | $96\,494{,}0 \pm 1{,}5$^1) |
| $c$ | Speed of light in cm/sec $(B^{12}—D)$ | $(2{,}99776 \pm 0{,}00015)\cdot 10^{10}$ |
| $p$ | Conversion factor from units of the international practical system to absolute ohms (D) | $1{,}000485 \pm 0{,}000007$ |
| $q$ | Conversion factor from international units $(a)$ to absolute amperes (D) | $0{,}999970 \pm 0{,}000020$ |
| $\dfrac{r}{p}$ | Conversion factor from international units $(v)$ to absolute amperes (D) | $0{,}999926 \pm 0{,}000020$ |
| $r$ | Conversion factor from international units $(v)$ to absolute volts (D) | $1{,}00041 \pm 0{,}000022$ |
| $R_{\mathrm{H}_1}$ | Rydberg constant for $\mathrm{H}^{1}$ in $\mathrm{cm}^{-1}$ $(B^{13})$ | $109\,677{,}76 \pm 0{,}05$ |
| $R_0$ | Gas constant in erg/deg/mole $(B^{13})$ | $(8{,}3136 \pm 0{,}0010)\cdot 10^{7}$ |
| $k_A$ | Ratio of mass-spectrographic atomic weights to chemical ones (D) | $1{,}000275 \pm 0{,}000020$ |
| $k_\lambda$ | Ratio of diffraction wavelengths to X-ray wavelengths (Birge^14)^2) | $1{,}00203 \pm 0{,}00002$ |
| $R_\infty$ | Rydberg constant for infinite mass in $\mathrm{cm}^{-1}$ $(B^{13})$ | $109\,737{,}42 \pm 0{,}06$ |
| $h_0$ | Arbitrarily adopted value of Planck’s constant in erg sec | $6{,}610\cdot 10^{-27}$ |
^1) This probable error does not include the voltmeter error; the latter error ($\pm 0{,}002\%$) was included in the factor $q$. The quantities $F$ and $q$ occur everywhere in our work as a product. Thus, in absolute units of the chemical system the Faraday number is equal to $(F\cdot q)\cdot 10^{-1}=9649{,}11\pm0{,}24$ CGSM and of the physical system $(F\cdot q\cdot k_A)\cdot 10^{-1}=9651{,}76\pm0{,}30$ CGSM.
^2) Robinson^15 established that the conversion factor obtained from the work of Bechlin and Söderman must be decreased by $2\cdot 10^{-4}$ times owing to the difference between Larsson’s corrected value of the wavelength of the aluminum $K_\alpha$ line and the new value found by Haglund^16. The calculation shows, [[unclear: continuation cut off]]
COMBINATIONS OF CONSTANTS OBTAINED BY THE MOST ACCURATE METHODS
A. The electron charge from diffraction measurements
The magnitude of the electron charge is obtained here in the following way: the wavelength of the characteristic X-radiation is measured with the aid of a diffraction grating. Then the Bragg angle of these same rays is determined for a calcite crystal, whence, with the aid of Bragg’s equation (corrected for refraction in calcite), the lattice constant of calcite is found. The electron charge is calculated from the constants of crystalline calcite (lattice constant, density, molecular weight), Faraday’s number, and the velocity of light.
At the basis of these calculations lies the assumption of the geometrical perfection of the calcite crystal ¹).
On the basis of a recalculation of all the factors mentioned, the author obtained the following value ²):
\[ e_0 = (4.8025 \pm 0.0004)\cdot 10^{-10}\ \mathrm{CGSE}. \]
This value is plotted on the Birge–Bond diagram (Fig. 1) as point 1 for \(n=0\). The arrows near it indicate the probable error.
B. The specific charge of the electron
In one of the preceding communications ²⁰ the author gave a review of various methods for determining \(\frac{e}{m_0}\) and their results; in doing so it was pointed out that there is a discrepancy between the values of \(\frac{e}{m_0}\) obtained from experiments with free electrons and \(\frac{e}{m_0}\) obtained from spectroscopic data. Later three new papers appeared in print on the determination of \(\frac{e^{21-23}}{m_0}\), and the final results of still another investigation on the same subject were published ²⁴. With the appearance of these works, as Birge ²⁵ has already indicated, the discrepancy between the spectroscopic and electronic data disappears. Thus, applying a more general—
however, that Birge used Larson’s number, so that if one adopts Hjalmar’s data, the conversion factor must be decreased by only \(5\cdot10^{-5}\) times for the results of Bäcklin and Söderman. The mean of their figures and Birge’s gives an insignificant difference in the mean value of the conversion factor. A work by Tyrén ¹⁷ appeared with absolute wavelengths of Al \(K_{\alpha_1,\alpha_2}\). Since his results, together with Hjalmar’s figures, give the value \(k_\lambda = 1.00202\), no noticeable changes have been indicated in \(k_\lambda\).
¹) A new work by DuMond and Bollman ¹⁹ confirms that this assumption is justified.
²) Here the following auxiliary arbitrary constants have been recalculated: molecular weight of calcite \(100.090 \pm 0.005\ \mathrm{g}\); density at \(20^\circ\mathrm{C}\) \(2.71025 \pm 0.00006\ \mathrm{g/cm^3}\); lattice period at \(20^\circ\mathrm{C}\) \((3.03566 \pm 0.00002)\times10^{-8}\ \mathrm{cm}\); unit-cell volume factor \(1.09595 \pm 0.00002\).
effective method of Fourier analysis in the consideration of fine structure, Hauston \(^{23}\) obtained a higher value of \(\dfrac{e}{m_0}\) and pointed out that the previous methods could lead to errors. Bärden \(^{22}\), measuring the refractive index of X-rays in diamond, also obtained a higher value of \(\dfrac{e}{m_0}\), in complete agreement with figures based on work with free electrons.
Fig. 1. Bärden–Bond diagram (dependence \(e_n = Ah\nu_0\)) for all known most accurate experimental data. The points plotted on the graph were calculated on the assumption of the accuracy of wavelengths determined by means of a diffraction grating. The value \(h_0\) was taken to be \(h_0 = 6.610 \cdot 10^{-27}\) erg sec. The probable error is indicated for point 10 by a circle, and in the remaining cases by arrows. For numerical data see Table 6.
The results of these studies are compared in Table 2. The figures published by the authors have been recalculated with the aid of the auxiliary constants of Table 1. Williams’s data \(^{21}\) have been replaced by the new data of Gibbs and Williams, obtained by them in continuation of the same work. In the data of Shane and Spedding \(^{26}\) the wavelengths have been reduced by us to vacuum \(^{1}\). In Shaw’s figures \(^{24}\) the charge has been recalculated into absolute electrostatic units.
The weighted mean of all ten determinations gives:
\[ \frac{e}{m_0}=(1.7591\pm 0.0002)\cdot 10^7\ \mathrm{CGSM}. \]
Here the external probable error is indicated. The fact that \(\dfrac{R_e}{R_i}\) (the ratio of the external probable error to the internal \(^{18}\)) is equal to 1.39 indicates how well the individual data agree \(^{2}\).
Substituting the value of \(\dfrac{e}{m_0}\) into Rydberg’s formula and taking \(h_0 = 6.610 \cdot 10^{-27}\) erg sec, we obtain:
\[ e_{\frac{3}{5}}=(4.7963\pm 0.0002)\cdot 10^{-10}\ \mathrm{CGSE}. \]
\(^{1}\) In recalculating all fine-structure data, Bainbridge’s \(^{27}\) data for atomic weights in the physical system were used: \(\mathrm{H}^1 = 1.00813\), \(\mathrm{H}^2 = 2.01473\), \(\mathrm{He}^4 = 4.00389\). The absolute value of the Faraday number in the physical system is given by \(F \cdot q \cdot k_A\), or \(9651.76 \pm 0.30\ \mathrm{CGSM}\).
\(^{2}\) \(\dfrac{R_e}{R_i}\) should be equal to unity if there are no systematic errors and statistical fluctuations.
This figure is plotted in Fig. 1 as point 10 for \(n=\frac{3}{5}\). The dimensions of the circle indicate the probable error.
Table 2
Summary of determinations of \(\dfrac{e}{m_0}\). The first five figures were obtained for the bound electron (from spectroscopic data), the remaining five for the free electron. No sharp discrepancy between the two groups is observed.
| Experimenter | Date | Method | \(\dfrac{e}{m_0}\cdot 10^{-7}\) CGSM |
Probable error \(\cdot 10^4\) | Weight \(=\dfrac{169}{r^2}\) |
|---|---|---|---|---|---|
| Houston \(^{28}\) \(^{1)}\) . . . | 1927 | Fine structure \(\mathrm{H^1—He^4}\) . . . . . |
1.7607 | 10 | 1.69 |
| Kinsler and Houston \(^{30}\) | 1934 | Zeeman effect . . . | 1.7571 | 7 | 3.45 |
| Shane and Spedding \(^{26}\) | 1935 | Fine structure \(\mathrm{H^1—H^2}\) . . . . . |
1.7582 | 4 | 10.57 |
| Williams \(^{21}\) . . . | 1938 | Same . . . . . . . . | 1.7580 | 4 | 10.57 |
| Houston \(^{28}\) . . . | 1938 | ” ” . . . . . . . . | 1.7593 | 5 | 6.76 |
| Perry and Chaffee \(^{31}\) | 1930 | Linear acceleration | 1.7610 | 10 | 1.69 |
| Kirchner \(^{32}\) . . . | 1932 | Same . . . . . . . . | 1.7590 | 9 | 2.09 |
| Dunnington \(^{20}\) . . | 1937 | Deflection in a magnetic field . . . | 1.7597 | 4 | 10.57 |
| Shou \(^{24}\) . . . | 1938 | Crossed fields . . . | 1.7581 | 13 | 1.00 |
| Bärden \(^{22}\) . . . | 1938 | Refraction of X-rays . . | 1.7600 | 3 | 18.79 |
| Weighted mean: 1.7591 ± 0.0002 | Weighted mean: 1.7591 ± 0.0002 | Weighted mean: 1.7591 ± 0.0002 | Weighted mean: 1.7591 ± 0.0002 | Weighted mean: 1.7591 ± 0.0002 | Weighted mean: 1.7591 ± 0.0002 |
B. \(h/e\) from the boundary of the continuous spectrum of X-rays
The experiment consists in determining the minimum voltage at which electrons can still excite X-rays that pass through a spectrometer. The latter transmits a certain line, whose wavelength is determined by a diffraction grating. Having measured the voltage \(V\) and the wavelength \(\lambda\), one can obtain the ratio \(h/e\) from the equation:
\[ \frac{hc}{\lambda}=Ve. \]
The latest and, apparently, most accurate work in this field was carried out by Du-Mond and Bollman \(^{33}\). On the isochromats they constructed (i.e., curves of X-ray intensity—electron
\(^{1)}\) Corrected by Bärden \(^{29}\), but the probable error is taken according to Houston.
potential) the authors noticed a hitherto unknown bend, appearing at several volts (by their estimate about 18) above the quantum limit. At the same time the existence is known of another bend approximately 100 V above this same limit. This maximum may affect the accuracy of determining the short-wavelength limit. Ross and Kirkpatrick \(^{34}\) showed that, with insufficient resolving power of the spectrometer used, the isochromat should turn into a straight line, and the value of the limit obtained should be lower than the true one. Thus, in evaluating these works it is extremely important to have information on the resolving power of the instruments used.
Table 3
Determination of \(\dfrac{h}{e}\) from the limit of the continuous spectrum of X-rays
| Experimentalists | Date | \(\dfrac{h}{e}\cdot 10^{17}\) CGSE | Estimate of probable error, as given by the authors | Probable error adopted in computing the mean |
|---|---|---|---|---|
| Ross and Kirkpatrick \(^{34}\) . . . . | 1934 | 1.3754 | 0.0001 | 0.0005 |
| Shaïtberger \(^{35}\) . . . . | 1935 | 1.3773 | 0.0004 | 0.0007 |
| Du-Mond and Bollman \(^{33}\) . . . | 1937 | 1.3765 | 0.0003 | 0.0003 |
| Weighted mean: \((1.3763 \pm 0.0003)\cdot 10^{-17}\) CGSE |
The data of the three works mentioned here are compared in Table 3. The following fact may cause some perplexity: Shaïtberger \(^{35}\) used a spectrometer with one crystal; the resolving power of this instrument is less than that of the instruments used in the other two works. According to the considerations given above, we should have expected lower values obtained by this investigator. However, contrary to expectation, Shaïtberger’s figures exceed the others.
Further experimental and theoretical work \(^{1)}\) is necessary in this direction, until sufficient accuracy of the results is achieved; however, large changes can hardly be expected here. We somewhat increase the probable error of the weighted mean given in Table 3. Thus the works considered in the present paragraph give for \(\dfrac{h}{e}\) the following value \(^{2)}\):
\[ \frac{h}{e}=(1.3763 \pm 0.0004)\cdot 10^{17}\ \mathrm{CGSE}. \]
\(^{1)}\) Brunner \(^{36}\) explains the bend nearest to the quantum limit by a resonance effect of extranuclear electrons.
\(^{2)}\) Let us note that the unweighted mean gives the same thing, namely:
\[ \frac{h}{e}=1.3764\cdot 10^{-17}\ \mathrm{CGSE}. \]
Substituting here the value of \(h_0\) from Table 1, we obtain:
\[ e_1=(4.8026\pm 0.0014)\cdot 10^{-10}\mathrm{CGSE}. \]
This number is represented in Fig. 1 by point 3 for \(n=1\).
OTHER COMBINATIONS OF CONSTANTS
In the eight types of experiments discussed here, the first four are connected with the mass of the electron; the others do not depend on it.
A. The ratio \(\left(\dfrac{h}{e}\right)\left(\dfrac{e}{m_0}\right)^{\frac12}\) from measurements of electron diffraction (von Friesen\(^{37}\))
In this experiment the electrons were accelerated by an electric field to a known potential, and then were diffracted by the surface of an etched galena crystal, and the angles of deviation were measured for several orders of diffraction. In this way, from the experiment it was possible to calculate the de Broglie wavelength expressed in terms of the grating constant of galena. The latter was measured in the same laboratory by Zeipel on the same crystal specimen, by the usual X-ray method.
In processing the results of this experiment von Friesen adopted the value of \(\dfrac{e}{m_0}\) and used Rydberg’s formula in order to obtain \(e\) and \(h\). Without these operations, the data of his experiment give directly the quantity
\[ \left(\frac{h}{e}\right)\left(\frac{e}{m_0}\right)^{\frac12}, \]
which, after recalculation, has the following value:
\[ \left(\frac{h}{e}\right)\left(\frac{e}{m_0}\right)^{\frac12} =(1.00084\pm 0.00058)\cdot 10^{-8}\mathrm{CGSE}. \]
The probable error has been obtained on the basis of an estimate of the experimental errors indicated by Friesen and of the errors of the auxiliary constants given in Table 1. The greatest error is connected with the measurement of the diffraction angles.
From Rydberg’s formula and \(h_0\) we obtain for the charge of the electron:
\[ e_{\frac13}=(4.7964\pm 0.0019)\cdot 10^{-10}\mathrm{CGSE}. \]
This value is plotted in Fig. 1 as point 7 for \(n=\dfrac13\). In comparing this experiment with the preceding ones, one must bear in mind the essential difference between them: the latter calculation is based on the use of a certain combination of constants (for example, the quantity \(\dfrac{e}{m_0}\) adopted by Friesen). The first group of experiments, discussed in subsection A of the preceding section, gives on the diagram points which unambiguously represent the results of measurement. This is the case when the relative position of a point (with respect to other points) does not depend on the adopted values of any combinations of constants used
in calculations¹). The use of Rydberg’s formula characterizes, of course, another method of treatment. The following graphical construction makes the point clear: in Fig. 1 a straight line drawn through point 7, representing Frisen’s results, and point 10 for \(\frac{e}{m_0}\), should give the values \(e\) and \(\frac{h}{e}\) at its intersections with the right and left scales (i.e., for \(n=0\) and \(n=1\)). If the author’s corrections are disregarded, these values should coincide with Frisen’s figures. Their dependence on \(\frac{e}{m_0}\) is evident.
B. \(\frac{h}{m_0}\) from measurements of electron diffraction
(Han³⁸, Meibohm and Rupp³⁹)
This work also consists in measuring the de Broglie wavelength for electrons; however, these measurements differ from the preceding ones in two respects. First, instead of measuring electron potentials, these experiments measured the velocities of electrons by Kirchner’s method³²; secondly, films were used as the diffracting object instead of crystals.
Han determined the lattice constant of a bismuth film by direct comparison with NaCl, whose constant was known from X-ray work. The lattice constant of the gold films used by Meibohm and Rupp was not determined by the authors. They assumed²) that the lattice constant does not differ from the constant for leaf gold, for which this quantity is known from X-ray measurements.
The experiments of these investigators give the following values:
\[ \text{Han:}\quad \frac{h}{m_0}=7.258\pm 0.022\ \text{erg sec/g}. \]
\[ \text{Meibohm and Rupp:}\quad \frac{h}{m_0}=7.289\pm 0.022\ \text{erg sec/g}. \]
The mean of the two values:
\[ \frac{h}{m_0}=7.274\pm 0.016\ \text{erg sec/g}. \]
Combining the figure obtained with \(h_0\), and using Rydberg’s formula, we obtain:
\[ e_{1/2}=(4.7972\pm 0.0026)\cdot 10^{-10}\mathrm{CGSE}. \]
This value is plotted in Fig. 1 as point 8 for \(n=1/2\).
The largest internal error is 50% greater than the external probable error. This may indicate that the authors were sufficiently cautious in estimating the inaccuracies of the experiment.
¹) The use of some definite value of \(h_0\) for constructing the Birge–Bond diagram does not contradict the consideration expressed above, since the relative positions of the points are the same for any values of \(h_0\). Changing \(h_0\) changes the vertical dimensions of the diagram, but the straight lines remain straight.
²) This assumption is quite legitimate, since the lattice constant of gold is the same for crystals of any size⁴⁰. For most of the materials studied this is not the case.
B. \( \dfrac{h}{m_0} \) from the Compton Effect
The experiment consists in determining the change in wavelength \(\Delta \lambda\) of quanta of X-rays scattered at an angle \(\theta\) in collisions with atomic electrons of a gas or solid. If the binding of the electron to the nucleus is taken into account, then to the usual equation of the Compton effect,
\[ \Delta \lambda=\left(\frac{h}{m_0 c}\right)(1-\cos\theta) \]
one must add a negative correction term proportional to \(\lambda^2\). This was justified theoretically by Ross and Kirkpatrick\(^{41}\) and by Bloch\(^{42}\), and confirmed by the experiments of Ross and Kirkpatrick\(^{43}\). The latter carried out a series of measurements fully deserving mention in the present discussion. \(\Delta \lambda\) was determined by them for three wavelengths. Carbon and beryllium were used as the scattering substances. Extrapolation of the figures they obtained to zero wavelength (for which the correction is equal to zero) gives:
\[ \frac{h}{m_0}=7.264 \pm 0.012 \ \text{erg sec}^2. \]
This value has been recalculated by us to the wavelength determined with the aid of a diffraction grating.
Using Rydberg’s formula and the value \(h_0\), we obtain:
\[ e_{1/2}=(4.7956 \pm 0.0020)\cdot 10^{-10}\ \text{CGSE}. \]
This figure is plotted in Fig. 1 as point 9 for \(n=1/2\).
G. \( \left(\dfrac{e}{m_0}\right)\left(\dfrac{e}{h}\right) \) from Ionization by X-rays
Electrons were torn out of a thin film or plate under the action of X-rays of known wavelength. The values \(H\rho\) (magnetic field \(\times\) radius of curvature) were measured for each separate group of ejected electrons. The energy of the electron must be equal to the energy of the incident photon minus the work function. The latter can be obtained by measuring the limiting wavelength of the absorbed rays for the level from which the electrons are ejected. In this equation there is a small, but nevertheless noticeable, error connected with the fact that the energy of the removed electron is obtained not quite exactly from the frequency corresponding to the absorption edge. The latter is proportional to the energy required to transfer the electron from its initial level to the first possible free level\(^{44}\). Hence, approximately, the ionization energy of the neighboring element of the periodic table must be added to the energy corresponding to the frequency of the series edge.
The outstanding work of Robinson and his collaborators\(^{45-48}\) in this field has been going on for more than 15 years. In order to avoid large errors associated with the determination of the limiting wavelength in the absorption of X-rays, and the corrections needed for this, Robinson carried out measurements with three different kinds of X-rays and used differences in energies.
The difference of the energies of photoelectrons ejected by hard or soft rays must be equal to the difference of the photon energies of these two types of rays. From the three pairs of differences obtained for Mo \(K\alpha\)—Cu \(K\alpha_1\), Mo \(K\alpha\)—Cr \(K\alpha_1\), and Cu \(K\alpha_1\)—Cr \(K\alpha_1\), the values of each initial level for each element were found. The measurements of \(H\rho\) were made with electrons of different levels for three elements—Au, Pt, and Ag.
Calculations based on a very large amount of data, after averaging, give:
\[ \left(\frac{e}{m_0}\right)\left(\frac{e}{h}\right)=(3.8220\pm0.0029)\cdot 10^{34}\ \mathrm{CGSE}. \]
Here, as before, the ratio of the wavelengths determined with the aid of a grating to the wavelengths of the Siegbahn scale is taken as \(1.00203\pm0.00002\). Corrections for the geometry of the magnetic field of the spectrograph\({}^{49}\) were unnecessary.
The probable error indicated above was still taken as the limit of the errors given by the author of the experiment (the probable error of Robinson’s experiments was of the order of one tenth of the value adopted by us). This was done in order to exclude a possible error connected with an inaccurate determination of the position of the short-wavelength edge of the line. Measurements must be made for this edge of the line, since it owes its origin to electrons that have suffered no loss of energy upon emerging from the film.
Using the value of \(h_0\) from Table 1 and applying the Rydberg formula, we obtain:
\[ e_{2/3}=(4.7953\pm0.0006)\cdot10^{-10}\ \mathrm{CGSE}. \]
Point 11 for \(n=2/3\) in Fig. 1 corresponds to this value.
Measurements of the energy of photoelectrons were also made by Kretschmer\({}^{50}\). He measured not the radius of curvature of the paths of the fastest electrons, but rather the radius of curvature corresponding to electrons with the most probable amount of energy (which corresponded to the peak of the line). From this it may be concluded that his measurements required energy corrections. Alvarez\({}^{51}\), on the basis of measurements of the thickness of one of Kretschmer’s films\({}^{1)}\), indicated a possible correction amounting from \(+0.23\) to \(+0.47\%\).
The correction for the slit width amounted to \(+0.18\%\) of the energy value\({}^{2)}\).
The total correction is therefore \(0.35+0.18=+0.53\%\).
The value
\[ \left(\frac{e}{m_0}\right)\left(\frac{e}{h}\right), \]
obtained from these experiments, was also recalculated to the wavelength determined by a diffraction grating. The corresponding \(e_n\) has the form: \(e_{2/3}=4.7922\cdot10^{-10}\ \mathrm{CGSE}\).
1) Kretschmer informed the author that, upon checking his other films, they turned out to be somewhat thinner.
2) Alvarez evidently takes the radius correction for the energy errors.
It may be noted that this value is lower by \(1/1500\) than the figure given by Robinson. However, if only the correction for the slit width is introduced into Kremser’s results, then the value of \(e_{2/3}\) already comes sufficiently close to Robinson’s figure (exceeding it only by about \(1/4400\)). Owing to the uncertainty of the energy corrections in Kremser’s work, his results have not been plotted on the diagram.
D. \(e\) from experiments with oil drops
As is known, the determination of the specific charge in this method is based on measurements of the rates of fall of oil drops in the gravitational field in the absence and in the presence of an electric field. The density of a drop is usually assumed to be the same as for macroscopic quantities. It is also assumed that, in the limit, Stokes’ law remains valid, since the product of the pressure and the radius of the drop approaches infinity.
The largest possible error of the result is connected with the error in determining the viscosity of air. A summary of the results of the most recent investigations in this field is given in Table 41. The mean value of the viscosity of air is almost half a percent greater than the Harrington figure[^57] used by Millikan[^58]. The calculated probable error of the weighted mean was \(\pm 0.9\). If the principal errors of the experiment are taken into account, it should be increased to \(\pm 1.5\).
Table 4
Most recent determinations of the viscosity of air. For comparison we recall the Harrington value: \((1822.6 \pm 1.3)\cdot 10^{-7}\) CGS
| Experimenters | Method | Date | \(\eta_{23^\circ}\cdot 10^{7}\) | Probable error | Weight |
|---|---|---|---|---|---|
| Kellström[^53] | Rotating cylinder | 1937 | 1834.9 | 2.7 | 2 |
| Houston[^54] | Same | 1937 | 1829.2 | 4.5 | 1 |
| Bond and Rigden[^55] | Capillarity | 1938 | 1830.3 | 0.7 | 1 |
| Banerji and Patnaik[^56] | Same | 1938 | 1833.3 | 2.1 | 1 |
| Weighted mean: \(1832.5 \pm 1.5\) CGS | Weighted mean: \(1832.5 \pm 1.5\) CGS | Weighted mean: \(1832.5 \pm 1.5\) CGS | Weighted mean: \(1832.5 \pm 1.5\) CGS | Weighted mean: \(1832.5 \pm 1.5\) CGS | Weighted mean: \(1832.5 \pm 1.5\) CGS |
It is difficult to explain this discrepancy between the old and the new determinations of the viscosity of air. Not one of the former figures, ob-
used by Millikan, does not approach the new ones. Nevertheless, the latter “high” values given in Table 4 were taken by us for the recalculation. This was to some extent arbitrary, although the better equipment and the newer technique of these experiments may in this case justify our choice.
Thus, Millikan’s classical experiment^59, in comparison with a higher value of the viscosity of air and with the values of the auxiliary constants accepted by us, gives
\[ e_0=(4.8059\pm 0.0052)\cdot 10^{-10}\ \mathrm{CGSE}. \]
Quite recently, Backlin and Flemberg^60 published, in a preliminary communication, the data of a similar experiment. Their experiment, carried out at atmospheric pressure, from 260 observations of the velocity on nine drops gives:
\[ e_0=(4.7941\pm 0.0089)\cdot 10^{-10}\ \mathrm{CGSE}. \]
(The calculations were made in the same way as before, using the data of Table 4 and the auxiliary constants.)
Ishida, Fukushima and Suetsugu^61 reported data from 1,000 observations of velocities for 31 drops at atmospheric pressure. Their figures, with the same calculation, give:
\[ e_0=(4.8453\pm 0.0043)\cdot 10^{-10}\ \mathrm{CGSE}, \]
which exceeds by \(\sim 0.9\%\) the two preceding figures and the value of \(e\) obtained from diffraction-grating experiments. The work of these authors was apparently done with sufficient accuracy. It may be regarded as a valuable supplementary work^62, since the measurements were made on nonspherical drops. However, in a personal communication Ishida indicated that in his work there may have been some error in the determination of the voltage and that this would subsequently be corrected. Taking all this into account, we took the quadratic mean of the data of Millikan and Backlin and obtained^1):
\[ e_0=(4.8036\pm 0.0048)\cdot 10^{-10}\ \mathrm{CGSE}. \]
In Fig. 1 this value corresponds to point 2 for \(n=0\).
E. \(\dfrac{h}{e}\) from ionization and excitation potentials
Lawrence^63 measured the ionization potentials of mercury by directing electrons into an ionization chamber and determining the minimum accelerating voltage at which ionization occurs. A magnetic velocity filter admitted electrons differing little in energy. The method of differences of potentials, used for measuring electron potentials, made it possible to avoid errors associated with contact-
^1) In order to obtain this mean and the probable error, it is necessary, before calculating the viscosity, to average the measurements with oil drops.
...potential differences, at least in the first approximation. For the ionization potential the value found was \(10.40 \pm 0.02\) int. V. If we substitute this value into the photoelectric-effect equation, together with the corresponding value of the spectroscopic term \((84178.5\ \mathrm{cm}^{-1})\) and the coefficient \(r\) (from Table 1), we obtain:
\[ \frac{h}{e} = (1.3753 \pm 0.0027)\cdot 10^{-17}\ \text{erg sec}/\mathrm{CGSE}. \]
Van Atta\({}^{64}\) measured the excitation potentials of several lines of helium, neon, and argon. The electrons scattered in the forward direction in the collision chamber were analyzed by an electrostatic velocity filter. In most of the experiments the energy of the electrons entering the chamber had a value many times greater than that required for ionization. Energy losses were found in fractions of the potential difference deflecting the electrons that excited the given line. From each measured excitation potential (converted into absolute volts), together with the wave number of the corresponding transition, it was possible to calculate the value \(h/e\). The weighted mean, calculated from five identical transitions, gives:
\[ \frac{h}{e} = (1.3753 \pm 0.0025)\cdot 10^{-17}\ \text{erg sec}/\mathrm{CGSE}. \]
In the calculations Van Atta’s data on the errors in the determination of the voltage were used\({}^{1}\). The internal probable error \(\pm 0.0017\) was taken, since it exceeded the external probable error. Owing to the presence of systematic errors (connected, for example, with surface charges on the deflecting plates\({}^{65}\) and with the fact that the central electron beam had a negative potential amounting to \(\sim 2.25\%\) of the deflecting potential), the probable error was increased to \(\pm 0.0025\).
About a year later Whiddington\({}^{66}\) and his collaborators measured the excitation potentials of the same gases. An essential difference of their apparatus was the use of a \(180^\circ\) magnetic velocity selector. The electron spectrum was obtained on an oil-sensitized photographic plate, with both the electrons of full energy and the electrons that had lost certain discrete amounts of energy producing separate lines. A clever method made it possible to avoid the need for a preliminary determination of the magnetic field and measurements of radii of curvature. This method consisted in slowing the electrons, in the absence of gas, in the ionization chamber by a known potential difference, so that this produced a calibration series of lines on the plate. The voltage difference for electrons of full energy and electrons of lower energies was measured by interpolation between neighboring calibration lines.
\({}^{1}\) No data were published concerning the errors for the transitions He \(1S_{0} — 3^{1}P\) and Ne \(^{1}S_{0} — 2p_{8}\). Comparing these errors with others and studying the current curves, one may take them to be equal to \(\pm 0.06\) and \(\pm 0.07\).
The most recent data are those of Widdington and Woodruff^67. The six lines they obtained^1), whose accuracy is exceptionally high and whose spectroscopic classification is quite reliable, give the weighted mean
\[ \frac{h}{e}=(1.3737\pm 0.0018)\cdot 10^{-17}\ \text{erg sec}/\mathrm{CGSE}. \]
Here again, the internal probable error \((\pm 0.0010)\) is twice as large as the external one. This probable error has been increased to \(\pm 0.0018\) because of possible errors connected with surface charges on the oiled plate. Widdington and Woodruff established that there is no evidence concerning the occurrence of charge on the film; however, they do not mention what they undertook in this direction. It is probable that the appearance of an equilibrium charge occurs rapidly, so that during the exposure only small changes are possible. Nevertheless, an experiment should have been carried out to ascertain the existence of this error^2).
The weighted mean of the three values from the works of Lawrence, Van Atta, and Widdington and Woodruff gives:
\[ \frac{h}{e}=(1.3745\pm 0.0013)\cdot 10^{-17}\ \text{erg sec}/\mathrm{CGSE}. \]
Here, as before, the internal probable error is twice as large as the external one. Substituting \(h_0\) here, we obtain:
\[ e_1=(4.8090\pm 0.0045)\cdot 10^{-10}\mathrm{CGSE}. \]
In Fig. 1 this value is represented by point 4 for \(n=1\).
Ж. The Stefan–Boltzmann Constant \(\sigma\)
This constant is obtained from measurements of the total radiation emitted from \(1\ \mathrm{cm}^2\) of a body heated to a definite temperature, to a receiver of known temperature. Owing to numerous experimental difficulties, it is almost impossible to obtain accurate results. Despite the errors of individual experiments, their large number makes it possible to derive a sufficiently good mean.
Ladenburg^68 calculated the value of \(\sigma\) from the results of seven investigations selected by him from twenty-four. In this selection the following conditions were taken as criteria: 1) the use of radiators and receivers sufficiently close in their properties to an absolutely black body, and 2) the presence of corrections for the absorption of radiation by water vapor and \(\mathrm{CO}_2\) in the air. A later determination of \(\sigma\)
^1) These transitions are the following:
Ne \(^{1}S_0—2^{1}P\) and \(^{1}S_0—3^{1}P\), Ne \(^{1}S_0—1s_2\), A \(^{1}S_0—2p_{10}\) and \(^{1}S_0—3s'_1\).
^2) It may be noted that although the action of surface charges probably should have shifted the line inward (toward the entrance slit), the error in the value of \(\frac{h}{e}\) must be positive or negative depending on the relative density of the lines associated with the given measurement.
was repeated by Möller[^69] with greater care. The new results of Hoare[^70] were not considered by Ladenburg1. A new compilation of the results for the determination of \(\sigma\) is given in Table 5. Using Planck’s formula, we obtain:
Table 5
Determination of the Stefan–Boltzmann constant \(\sigma\)
| Experimenters | Year | \(\sigma\) | Accuracy in % | Weight |
|---|---|---|---|---|
| Gerlach | 1916 | 5.80 | \(\pm 1\) | 1 |
| Koblentz | 1917 | 5.73 | \(\pm 1\) | 1 |
| Hoffmann | 1923 | 5.764 | \(\pm 1\) | 1 |
| Kussmann | 1924 | 5.795 | \(\pm 1\) | 1 |
| Mendenhall | 1929 | 5.79 | \(\pm 1\) | 1 |
| Möller | 1933 | 5.774 | \(\pm \tfrac{1}{2}\) | 2 |
| Weighted mean: | \multicolumn{3}{l}{\((5.775 \pm 0.022)\cdot 10^{-5}\ \mathrm{erg}\ \mathrm{cm}^{-2}\ \mathrm{sec}^{-1}\ \mathrm{grad}^{-4}\)} |
\[ \frac{e}{h^{3/4}}=(2.0778\pm0.0020)\cdot 10^{10}\mathrm{CGSE}. \]
Substitution of \(h_0\) gives:
\[ e_{3/4}=(4.8168\pm0.0046)\cdot 10^{-10}\mathrm{CGSE}. \]
This value is presented in Fig. 1 as point 6 for \(n=\,^3/_4\).
3. The radiation constant \(c_2\)
This constant occurs in Wien’s displacement law. For the last fifteen years one cannot name any new work on the determination of \(c_2\), so that one must use Bärge’s figure[^72], namely
\[ c_2=1.432\pm0.003\ \mathrm{cm\ grad}. \]
In combination with Planck’s radiation law we obtain:
\[ \frac{h}{e}=(1.3730\pm0.0029)\cdot 10^{-17}\mathrm{CGSE}, \]
or, substituting \(h_0\),
\[ e_1=(4.8145\pm0.0101)\cdot 10^{-10}\mathrm{CGSE}. \]
This value2 is plotted in Fig. 1 as point 5 for \(n=1\).
SUMMARY OF EXPERIMENTAL DATA
The results of all seventeen types of experiments that we considered above are brought together in Table 6. The experimental values found from observation for the quantity being determined are denoted by \(A\). These values of \(A\) may depend on one or several of the first ten constants of Table 1. They have been calculated on the assumption that the wavelengths determined by the diffraction method are correct. The values \(e_n\), represented by points on the Birge–Bond diagram, are placed in the last column of Table 6. They are obtained from the values \(A\) by means of the expressions given in the penultimate column. The two additional constants used for these calculations are given at the end of Table 1.
ANALYSIS AND DISCUSSION
The analysis of the data obtained will be carried out in two ways: 1) by considering the Birge–Bond diagram and 2) by applying the method of least squares.
A. Analysis of the Birge–Bond Diagram
1. Analysis of the causes of the discrepancy in the assumption of the accuracy of wavelength determinations from diffraction measurements. Considering Fig. 1, one may note not only the well-known discrepancy between the values \(e\), \(\dfrac{h}{e}\), and \(\dfrac{e}{m_0}\) (points 1, 3, and 10), but also the special and, perhaps, significant grouping of more than half the points (7 points out of 11) in the central part of the figure. The points of this group (group \(A\)) lie noticeably below the other1
This value of \(h/e\) is much closer to the directly observed value (the mean from the inverse photoelectric effect) than to that obtained by indirect means (see the solution of I and II in Table 9 and the solution I′ and II″ of Table 10). This agreement serves to some extent as an indication of the great accuracy of the first data, and consequently also that the Rydberg formula may be a source of discrepancies. This new work, indeed, deserves attention, especially because its method and theory are so excellent; however, the accuracy of the results is insufficient for the preceding confirmation to be more than a simple indication of this possibility. The analysis and discussion carried out in the present work would need to be altered only to an insignificant degree if this new work were included in the consideration.
Table 6
Summary of eleven types of measurements and of the experimental results obtained. In the fifth column of the table the dependence of these experimental quantities on the first ten auxiliary constants listed in Table 1 is indicated. \(A'\) denotes the remainder \(A\) after multiplication by the indicated constant. In the sixth column the formula for \(e_n\) is given (\(e_n\) is the quantity \(e\), representing this experiment in the Bohr–Bjerrum diagram). In the last column the values of \(e_n\) are given.
| Point | Type of measurement | Combination of measured constants | Experimental value \(A\) | Dependence of \(A\) on known constants. Formula for \(A\) | Formula for \(e_n\) | \(e_n\) for the Bohr–Bjerrum diagram, CGSE \(\cdot 10^{10}\) |
|---|---|---|---|---|---|---|
| 1 | Diffraction grating | \(e\) | \((4.8025 \pm 0.0004)\cdot 10^{-10}\) CGSE | \((Fack\,\lambda_3)A'\) | \(e\) | \(4.8025 \pm 0.0004\) |
| 2 | Oil drops | \(e\) | \((4.8036 \pm 0.0048)\cdot 10^{-10}\) CGSE | \(\left(\dfrac{c}{r}\right)A'\) | \(e\) | \(4.8036 \pm 0.0048\) |
| 3 | Limit of the continuous X-ray spectrum | \(\dfrac{h}{e}\) | \((1.3763 \pm 0.0004)\cdot 10^{-17}\) CGSE | \(\left(\dfrac{k\lambda r}{c^2}\right)A'\) \(^{1}\) | \(\left(\dfrac{1}{A}\right)h_0\) | \(4.8026 \pm 0.0014\) |
| 4 | Ionization and excitation | \(\dfrac{h}{e}\) | \((1.3745 \pm 0.0013)\cdot 10^{-17}\) CGSE | \(\left(\dfrac{r}{c^2}\right)A'\) | \(\left(\dfrac{1}{A}\right)h_0\) | \(4.8090 \pm 0.0045\) |
| 5 | Radiation constant | \(\dfrac{h}{e}\) | \((1.3730 \pm 0.0029)\cdot 10^{-17}\) CGSE | \(\left(\dfrac{R_0}{c^2Fa}\right)A'\) | \(\left(\dfrac{1}{A}\right)h_0\) | \(4.8145 \pm 0.0101\) |
| 6 | Stefan–Boltzmann constant | \(\dfrac{e}{h^{3/4}}\) | \((2.0778 \pm 0.0020)\cdot 10^{10}\) CGSE | \(\left(\dfrac{Fqc^{1/2}}{R_0}\right)A'\) | \((A)h_0^{3/4}\) | \(4.8168 \pm 0.0046\) |
| 7 | Electron diffraction \((V)\) | \(\left(\dfrac{h}{e}\right)\left(\dfrac{e}{m_0}\right)^{1/2}\) | \((1.00084 \pm 0.00058)\cdot 10^{-3}\) CGSE | \(\left[\left(\dfrac{r}{c}\right)^{1/2}k\lambda\right]A'\) \(^{2}\) | \(\left(\dfrac{cR\infty A^2}{2\pi^2}\right)^{1/3}h_0^{1/3}\) | \(4.7964 \pm 0.0019\) |
| 8 | Electron diffraction \((v)\) | \(\dfrac{h}{m_0}\) | \(7.274 \pm 0.016\) erg sec/2 | \((k\lambda)A'\) | \(\left(\dfrac{cR\infty A}{2\pi^2}\right)^{1/4}h_0^{1/3}\) | \(4.7972 \pm 0.0026\) |
| 9 | Compton effect | \(\dfrac{h}{m_0}\) | \(7.264 \pm 0.012\) erg sec/2 | \((k\lambda c)A'\) | \(\left(\dfrac{cR\infty A}{2\pi^2}\right)^{1/4}h_0^{1/2}\) | \(4.7956 \pm 0.0020\) |
| 10 | Specific charge | \(\dfrac{e}{m_0}\) | \((1.7591 \pm 0.0002)\cdot 10^7\) CGSM | Depends on the method | \(\left(\dfrac{cR\infty A}{2\pi^2}\right)^{1/5}h_0^{3/5}\) | \(4.7963 \pm 0.0002\) |
| 11 | X-ray photoelectrons | \(\left(\dfrac{e}{m_0}\right)\left(\dfrac{e}{h}\right)\) | \((3.8220 \pm 0.0029)\cdot 10^{24}\) CGSE | \(\left(\dfrac{c^3p^2}{k\lambda r}\right)A'\) \(^{3}\) | \(\left(\dfrac{cR\infty A}{2\pi^2}\right)^{1/6}h_0^{2/3}\) | \(4.7953 \pm 0.0006\) |
\(^{1}\) In all cases it was taken into account in which country the experiment had been performed.
\(^{2}\) Since it was not known which electrical standards (English or German) Frisen used, the mean value of \(r\) was taken, namely \(r = 1.000396 \pm 0.00060\). The probable error was correspondingly increased.
\(^{3}\) Robinson does not indicate that the correction \(\left(\dfrac{p}{r}\right)^2\) was taken into account. We adopted the value:
\[ \dfrac{r}{p} = 0.999941 \pm 0.000020. \]
points lying to the right and to the left, which we combine into group \(B^{1}\)). The points of the first group give a value of \(e\) of the order of \(4.796 \cdot 10^{-10}\), whereas the upper group corresponds to the value \(4.803 \cdot 10^{-10}\) CGSE.
The discrepancy between these two values lies beyond the limits of experimental errors. If this is so, then the reason for the discrepancy must be sought in the laws or equations on which the calculations of \(e\) were based. If, of the formulas used, only one contains an error, then it can be found by the following signs: 1) all points calculated on the basis of this formula, or indirectly connected with it, must be in good agreement with one another; 2) all points not connected with it must also agree well in their values; 3) a sharp discrepancy must be observed between these two groups of points.
In Table 7 all eleven types of measurements are analyzed with respect to the fundamental laws or equations used in one way or another in the calculations. The fulfillment of the first two criteria is indicated qualitatively at the bottom of the table. The third criterion is fulfilled in almost every case. It can be seen that only the groups of experiments connected with the Rydberg formula, or independent of it, exhibit all the signs indicated above.
These two groups are the above-mentioned groups \(A\) and \(B\). The experimental data indicate that the cause of the discrepancy in the values of the fundamental atomic constants is the Rydberg formula.
Another possible source of errors may be the photoelectric-effect equation. Here, however, the signs are expressed relatively more weakly, since in this case the two points connected with radiation do not coincide with the “other measurements” (i.e., those not connected with the photoelectric effect). If these insufficiently accurate radiation points are not included in the consideration (on the assumption that more accurate data will appear), or if they are included in the group connected with the photoelectric-effect equation, then, analyzing Table 7, one can see that this equation may also be regarded as a probable source of discrepancies.
The greater part of the subsequent discussion will be concerned with the question of which of these two possibilities seems more probable.
2. The effect of arbitrary changes in the Rydberg formula.
A little more than two years ago Birge\(^1\) pointed to the erroneousness of the Rydberg formula as one of the three possible causes of the contradiction between \(e\), \(\frac{e}{m_0}\), and \(h\). Later he came to the conclusion that the whole matter lies in the value of \(\frac{h}{e}\). Somewhat later DuMond\(^5\) indicated that the discrepancy disappears upon introducing into the Rydberg formula a correction term \((1-a)\), where \(a\) is the fine-structure constant.
DuMond’s reasoning is based on changes in the value of \(\frac{h}{e}\), calculated from \(\frac{e}{m_0}\) and \(e\) with the aid of the Rydberg formula. From the last—
\(^1\) The two dotted lines \(A\) and \(B\) in Fig. 1 are drawn arbitrarily.
Table 7
Analysis of eleven types of measurements with respect to the principal laws or equations on which they are based. The laws or equations used in one or another type of measurement are marked with a cross. At the end of each column it is indicated qualitatively whether a straight line can be drawn through the points marked with a cross. This serves as a test of their reliability. At the very end of each column an estimate is given of the same test for points not marked with a cross.
| Point | Type of measurement | Diffraction grating and interference rays $n\lambda=d[\cos\theta-\cos(\theta+\alpha)]$ |
Crystal grating. Bragg’s law $n\lambda=2d\sin\theta$ |
de Broglie equation $\lambda=\dfrac{h}{mv}$ |
Photoelectric equation $\dfrac{hc}{\lambda}=\dfrac{1}{2}mv^2=\Phi e$ |
Planck’s equation for radiation | Calcite crystal (geometrically perfect) $d=\left[\dfrac{Me}{2pFc\Phi(\beta)}\right]^{1/2}$ |
Expression for the Rydberg constant $R=\dfrac{2\pi^2e^4m_0}{ch^3}$ |
|---|---|---|---|---|---|---|---|---|
| 1 | Diffraction grating $(e)$ | X | X (X-ray) | X | ||||
| 2 | Oil drops $(e)$ | |||||||
| 3 | Limit of the continuous X-ray spectrum | X | X (inverse) | |||||
| 4 | Ionization and excitation | X (inverse)$^{5)}$ | ||||||
| 5 | Radiation constant$^{1)}$ | X | ||||||
| 6 | Stefan–Boltzmann constant$^{2)}$ | X | ||||||
| 7 | Electrical diffraction $(V)$ | X | X (electrons)$^{3)}$ | X | X | |||
| 8 | Electron diffraction $(v)$ | X | X (electrons) | X | X | |||
| 9 | Compton effect$^{4)}$ | X | X (X-ray) | X | ||||
| 10 | Specific charge $\dfrac{e}{m_0}$ | X | ||||||
| 11 | X-ray photoelectrons | X | X (direct) | X | ||||
| Agreement of the points marked with a cross (X) with a straight line | Extremely weak | Good | Only 2 points | Good | Only 2 points | Only 1 point | Very good | |
| The same for the remaining points | Weak | Very weak | Exceptionally weak | Very good | Exceptionally weak | Very weak | Brilliant |
$^{1)}$ The radiation constant $c_2$ occurs in Wien’s displacement law, which is obtained from Planck’s formula for radiation.
$^{2)}$ The formula for the Stefan–Boltzmann constant is obtained from Planck’s formula.
$^{3)}$ In reality the crystal is used as a transmission grating; however, the basic phenomena will be the same as in the ordinary Bragg case.
$^{4)}$ With a correction for the binding energy of the electron to the nucleus.
$^{5)}$ These experiments are inverse to the photoelectric ones in the sense that they presuppose conservation of energy in the transition of the kinetic energy of an electron into the energy of a photon $h\nu$. This process, however, proceeds in two stages: 1) excitation or ionization and 2) radiation.
of the column of Table 7, it can be seen that the changes in Rydberg’s formula have a favorable effect not only on the value of \(\dfrac{e}{m_0}\), but also on the other four measurements.
In the corrected form, Rydberg’s formula is as follows:
\[ R_{\infty}(=109\,737.42)=\frac{2\pi^2 e^4 m_0}{c h^3}\cdot\frac{1}{1+\alpha}. \]
This is equivalent to saying that the combination of constants \(\left(\dfrac{2\pi^2 e^4 m_0}{c h^3}\right)\) is increased by a certain fraction of \(\alpha\).
The fine-structure constant is not included among the eleven types of measurements in Table 7 because of the large experimental errors in its determination. The three most reliable measurements of the value \(\dfrac{1}{\alpha}\) give: \(139.9^{74}\), \(137.4\pm0.2^{75}\), \(139.3^{76}\). Their mean has the form \(138.9\pm0.6\). If this value is adopted, \(R_{\infty}\) increases by \(\dfrac{1}{138.9}\) (i.e. by \(7.2\cdot10^{-3}\)) of its original value. Recalculating the values \(e_n\) (see the last column of the table) with the aid of this corrected value of \(R_{\infty}\), we arrive at the results presented in Fig. 2. The agreement of the points seems to improve; this is especially noticeable for the following points: \(10\) (from \(\dfrac{e}{m}\)), \(1\) \((e)\), and \(3\) \(\left(\dfrac{h}{e}\right)^1\).
Fig. 2. The same diagram as in Fig. 1; however, all combinations of constants entering Rydberg’s formula have been increased by \(\alpha\) \((=7.20\cdot10^{-3})\) times. With such a count the divergence of the points in Fig. 1 is considerably reduced, although it is not completely eliminated.
Because \(R_{\infty}\) occurs in different powers\(^2\), the central group of points (from 7 to 11) not only shifted upward from its former level, but also rotated\(^3\) clockwise. In this process, points 7 and 11 turned out to be separated from a certain mean value (the dashed line) by distances 2–3 times greater than the limits of their experimental errors. (In Fig. 1 the same dashed—
\(^1\) If one takes \(\dfrac{1}{\alpha}=137\), all the points will rise by a greater distance; nevertheless, the agreement remains sufficiently good.
\(^2\) See the penultimate column of Table 6.
\(^3\) For any changes in the value of \(R_{\infty}\), the displacement of the points associated with \(R_{\infty}\) can be made quite visual if one imagines that these points are plotted on a separate flexible sheet, which is then subjected to a vertical shift, with the right edge remaining fixed. A change in the value of \(h_0\) similarly gives the same shift for all points relative to the left edge.
typical line passed through group \(B\).) This circumstance may be regarded as an argument against solving the problem by introducing changes into the Rydberg formula, or it may be considered an indication of the erroneousness of the experimental data. Of course, all these considerations emphasize the necessity of further experimental work. If it is found that what are needed are not corrections but fundamental changes in the Rydberg formula, then the final picture may change sharply.
Another contradiction, indicated in Figs. 1 and 2, is introduced by point 6. This point is connected with the experiment to determine the Stefan–Boltzmann constant. A peculiarity of this point is that the introduction of corrections for the absorption of radiation by water vapor or \(CO_2\) (which were taken into account in the experiments of recent years) shifted this point upward from its previous position, which had agreed well with the other points. The question of whether one should look here for another appreciable cause of the discrepancy, or whether Planck’s theory of radiation should be subjected to revision, remains open.
-
The effect of functional changes in the Rydberg formula. The possibility of a functional, nonlinear, change in the Rydberg formula has already been mentioned. It is necessary to emphasize that a functional change cannot affect the values of \(\frac{h}{e}\) found from the experiments represented by points 7, 8, 9, or 11 in combination with the quantity \(\frac{e}{m_0}\)¹). Graphically this means that two points must always determine a line that passes through the value \(\frac{h}{e}\) at \(n=1\). No functional changes in the Rydberg formula can make points 7—11 in Fig. 2 coincide with, or even approach, the dotted line²).
-
Analysis of the cause of the discrepancies under the assumption of the ideality of the calcite crystal. Up to now it has been assumed that the wavelength values determined with the aid of a diffraction grating are correct. The opposite assumption is that of the geometrical perfection of the crystal lattice. This thereby makes it possible, using Bragg’s law, to express the wavelengths \(\lambda\) through the constants of the calcite crystal, namely \(\lambda = Ce^{1/3}\). The Birge–Bond diagram, constructed on the basis of these calculations, without any changes in the Rydberg constant, is given in Fig. 3. Comparing it with the diagram in Fig. 1, one can note that the whole group \(A\) has contracted around point 10, representing the value of \(e\),
¹) Illustration: \(\frac{h}{e}\), found by Frisch from the data represented by point 7, and the quantity \(\frac{e}{m_0}\)—slightly below point 10—lies at the intersection of the line passing through points 7 and 10 and the ordinate for \(n=1\). This point of intersection retains its position regardless of what is done with the Rydberg formula (see the preceding note).
²) This would be an incredible case, in which functional changes would displace only those points that correspond to very small orders \(n\) (the degree \(h\)).
obtained from \(\dfrac{e}{m_0}\). Point 10 remained in place. In group \(B\) point 3 shifted\(^1\) from \(n=1\) to \(n=0.75\). The division into two groups, just as before, is clearly noticeable.
Analysis of the diagram in Fig. 3, carried out by the previous method (see Table 7), leads to the same conclusions; it must therefore be considered that nothing new can be obtained from the assumption of the geometrical perfection of calcite.
Fig. 3. A Birge–Bond diagram similar to Fig. 1, but based on the assumption of the geometrical perfection of calcite in calculating the wavelengths of X-rays. The only exception is point 1, which is also connected with wavelengths obtained from diffraction measurements. The discrepancy observed in Fig. 1 has not changed for the better in this figure.
5. Possibility of an experimental origin of the causes of the contradiction. Up to now we have proceeded from the premise that the discrepancy between groups \(A\) and \(B\) exceeds the limits of experimental errors. If this view is abandoned, the situation can be described as follows: the points of the upper group (group \(B\) in Figs. 1 and 3) are not related to one another; they are scattered over the graph on the left and on the right, and there is not a single point lying between them. On the contrary, the lower group of points (group \(A\)) constitutes something whole, and between these points, despite the entirely different methods by which they were obtained, a remarkable agreement is observed. Consequently, the lower group deserves far more confidence as regards accuracy than the upper one. If we wish to preserve the “center of gravity” of the lower group of points, situated approximately at the center of the figure, through which, in some way, the line representing the solution passes, then either points 1–2 or points 3–6 must be lowered. Considering the first of these two possibilities, it should be noted that the not entirely satisfactory state of the measurements connected with oil drops gives reason to think that further experiments may lower point 2. However, it seems much less likely that new experiments will be able noticeably to change the position—
\(^1\) The displacement is illustrated here without difficulty, which is possible in all cases: point 3 lies on the line determined by points 1 and 3 in Fig. 1. The reason that the displacement always occurs along the line determined by point 1 is that this point is connected with the use of wavelengths obtained from diffraction measurements, and with the assumption of the geometrical perfection of calcite. Consequently, each point associated with a wavelength in Fig. 1, and the corresponding point in Fig. 3, must be in mutual correspondence with point 1 (i.e., lie on the same straight line).
of point 1, since this point is connected with many experimental results used in constructing the other points1. Further, when recalculating the majority of constants connected with this point, we do not encounter indications of errors sufficient for the required lowering of its value. The first possibility therefore seems very unlikely.
The second alternative requires lowering points 3 and 6. The limits of the experimental errors of point 3 have been established sufficiently accurately from several determinations. The experimental, and perhaps also theoretical, error in the determination of two radiation constants (points 5 and 6) at present2 pushes this possibility into the background. Point 4, however, has been determined with insufficient accuracy, and further work3 in this direction is extremely necessary.
If this improbable thing nevertheless occurs and the latest investigations give increased values of \(\dfrac{h}{e}\) for points 3 and 4 (i.e., so that these points are lowered and coincide with the straight line drawn through points 1 and 10), then, of course, the main contradiction will disappear. In that case there will remain only the problem of bringing the comparatively inaccurate radiation points into agreement with the other measurements.
On the other hand, if later work confirms the “low” values of \(\dfrac{h}{e}\), i.e. points 3 and 4 occupy a higher position on the graph, then this can be interpreted in three ways: a) the results confirm the erroneousness of Rydberg’s formula; b) an error is hidden in the photoelectric-effect equation, leading to a lowered value of \(\dfrac{h}{e}\); c) the application of the photoelectric-effect equation leads to an error that lowers the value of \(\dfrac{h}{e}\). The second supposition requires such a change in the theory as must lower the points connected not only with the inverse (3 and 4), but also with the direct photoelectric effect (11), by an amount of the same order as one sixth. This will move point 11 below the mean line (1—10, Fig. 2), but not so much as to produce another undesirable discrepancy.
In contrast to this, supposition (c) leaves the position of point 11 unchanged. Thus, one must distinguish between the direct and the inverse photoelectric effect. Indeed, point 11, obtained from the direct photoelectric effect, is in excellent agreement—
in agreement with all points except the “inverse” photoelectric points 3 and 4 and the relatively inaccurate radiation points 5 and 6. However, it is impossible to accept assumptions (b) or (c), since there are no such changes in the theory that could increase the value of \(\frac{h}{e}\) for the inverse photoeffect without violating the law of conservation of energy. In short: the photon remains with “excessively large energy”1).
B. Analysis of the discrepancy by the method of least squares2.
1. Method.
In the preceding analysis we used the Birge–Bond diagram as almost the only method of graphical representation of the whole situation. However, because the Rydberg formula on which it was necessary to rely is subject to doubt, it is highly desirable that the same figures be treated by some other methods. The application3 of the method of least squares to a series of equations representing the experimental data may give some additional information concerning the causes of the discrepancies.
The method of least squares which we apply is, in general outline, as follows. The available experimental data, summed up in eleven equations (for the three unknowns \(e\), \(m_0\), and \(h\)), are given in the third and fourth columns of Table 6. In addition, we have the equation:
\[ \frac{e^4 m_0}{h^3}=(1.666564 \pm 0.000083)\cdot 10^{14}, \tag{12} \]
obtained from the Rydberg formula using the values \(R_{\infty}\) and \(c\) from Table 1. These twelve equations can be reduced to eight, using weighted means for \(e\), \(\frac{h}{e}\), and \(\frac{h}{m_0}\). The results are presented in Table 8. These equations were reduced to linear form by expansion in a Taylor series, and then the separate groups were solved simultaneously by the ordinary method of least squares4. Here the essential point is the choice of these groups; namely, the choice must be made in such a way as to obtain certain
information about the source of the contradiction. To this end, first of all, the basic solution was produced, relying on classes of observations that stood beyond any doubt. Then additional solutions were found for a system consisting of one “doubtful equation” and equations belonging to the basic solution. A number of data could
Table 8
Eight found relations of the fundamental constants and the types of experiments on which they are based. The weight assigned to each measurement is calculated by the usual least-squares formula:
$$ W=\frac{c}{r^2} $$
| Designation | Found values of the constants or their relations (in CGSE) | Weight | Type of experiment | No. of point (see Tables 6 and 7, and also all figures) |
|---|---|---|---|---|
| A | $e=(4.8025\pm0.0004)\,10^{-10}$ | 252.8 | Diffraction grating Oil drops |
1 2 |
| B | $\dfrac{h}{e}=(1.3761\pm0.0006)\cdot10^{-17}$ | 9.3 | Inverse photoeffect Radiation constant $c_2$ |
3 and 4 5 |
| C | $\dfrac{e}{h^{3/4}}=(2.0778\pm0.0020)\cdot10^{10}$ | 1.9 | Stefan–Boltzmann constant | 6 |
| D | $\left(\dfrac{h}{e}\right)\left(\dfrac{e}{m_0}\right)^{1/2}=(1.00084\pm0.00058)\times10^{-8}$ | 5.2 | Electron diffraction (voltage) | 7 |
| E | $\dfrac{h}{m_0}=7.268\pm0.010$ | 1.0 | Electron diffraction (velocity) Compton effect |
8 9 |
| F | $\dfrac{e}{m_0}=(5.2734\pm0.0007)\cdot10^{17}$ | 96.0 | Specific charge | 10 |
| G | $\left(\dfrac{e}{m_0}\right)\left(\dfrac{e}{h}\right)=(3.8220\pm0.0029)\times10^{34}$ | 3.1 | X-ray photoelectrons | 11 |
| H | $\dfrac{e^4m}{h^3}=(1.666564\pm0.000083)\times10^{14}$ | 698.0 | Rydberg formula | (12) |
be obtained by comparing those changes in the solution which were introduced into it by one or another “doubtful” equation, and also by comparing the basic solution with the directly observed values.
Let us point out the similarity and difference between our method and the method used earlier. The operation proposed by Bond82
and improved by Birge^9, consists in reducing three variables, \(e\), \(m_0\), and \(h\), to two, namely \(e\) and \(h\), by means of the Rydberg formula. The experimental relations obtained are then brought to linear form by expansion in a Taylor series and retention of the first term. Birge introduced a new parameter \(e_n=a_n h_0^n\), where \(a_n\) is a numerical constant obtained from the experimental value \(A_n\) (see the next-to-last column of Table 6), \(h_0\) is the accepted value of \(h\), and \(n\) is the power with which it enters the equation. This relation makes it possible to express the results as points on the Birge-Bond diagram, approximately corresponding to the straight line \(e_n=e+bn\), where \(e\) is the magnitude of the electron charge and \(b\) is the slope of the line, representing the solution by the method of least squares. From it the value of \(h\) can be found. The most essential feature of this method is the possibility of a graphical representation of the experimental results (see Fig. 1) and of a least-squares solution (i.e., finding the straight line most closely corresponding to the experimental points). Birge also gave a method for calculating the probable error.
Shiba^83, independently of Birge, used a method of reduction by least squares which resembles Birge’s method in that the unknown (in this case \(h\)) is eliminated by means of the Rydberg formula. The resulting equations with two unknowns, \(e\) and \(\dfrac{e}{m_0}\), are solved by the usual least-squares methods and therefore cannot be represented graphically. The probable errors were not calculated, but merely estimated.
Later Bates^78, by the same method, gave a solution for three unknowns. He devoted special attention to the calculation of the errors from the normal equations and of errors computed in such a way as to provide some information about the causes of the discrepancy.
- Reduction of the data under the assumption that the Rydberg formula and the photoelectric equation are doubtful. The Rydberg formula (equation \(H\)) and the photoelectric equation (connected with equations \(B\) and \(G\)) are two possible sources of discrepancy, as was indicated in the analysis of the Birge-Bond diagram. Mention was also made of the less important disagreement between the points connected with radiation measurements (equation \(C\) and, to a lesser degree, \(B\)). Owing to the fact that these points are in some correspondence with the photoelectric points, they have been grouped together with the latter.
It may be that the result of our reasoning would appear clearer if the radiation points were simply not included in the consideration; however, this would not noticeably affect the results of the solution. These results are brought together in Table 9. Solution I is the basic one, since it rests on classes of experiments beyond doubt. Solution II was carried out for the same classes with the addition of data connected with the Rydberg formula. In solution III all the photoelectric and radiation points are included. The differences of the solutions (see the last two columns) are insignificant for \(e\), \(m_0\), and \(\dfrac{e}{m_0}\), and com-
significantly large and opposite in sign for \(h\), \(\dfrac{h}{e}\), and \(R_\infty\). This indicates the same degree of disagreement between the photoelectric equation and Rydberg’s formula and all the remaining points. Despite the impossibility of representing completely graphically the results of the solutions obtained by this method, they can in part be illustrated with the aid of Birge–Bond diagrams. The necessary condition here is that the value of \(R_\infty\) used be found by the method of least squares, since in that case the least-square values of all combinations of constants lie on one straight line. If, further, instead of \(h_0\) one uses the least-square value for \(h\), then the straight line representing the solution will be horizontal. This has been done for solution III in Fig. 4. The coincidence of any experimental point with the straight line is, of course, a function of the distance of the corresponding circle from the dotted line, and the discrepancy may be estimated by taking the ratio of this distance to the length of the arrow (i.e., to the probable error). Points 7 and 11 lie much closer to this straight line than in Fig. 2.
Fig. 4. Birge–Bond diagram, representing, insofar as possible1, the experimental data treated by the method of least squares. The quantities connected with Rydberg’s formula are not included in this calculation (see solution III, Table 9, and III′, Table 10). The dotted line represents the solution obtained. Experimental values are indicated by circles; crosses show the results of calculating the digits by the method of least squares for the various degrees \(n\). \(R_\infty\) is calculated by the same method. The wavelengths are taken from diffraction measurements.
- Treatment of the data under the assumption that the error is connected with the inverse photoelectric effect and Rydberg’s formula. An argument against the preceding method of consideration is that photoelectric experiments are of two types: direct and inverse, and only the latter are in contradiction with the other measurements [see part A (5) of this section]. Therefore, in the series of solutions given in the present paragraph, the possible sources of errors were attributed to the inverse photoelectric equation \((B)\) and to Rydberg’s equation. As before, the radiation points were included in one group with the points obtained from the inverse photoelectric effect. However, the results would not change appreciably if these points were included in the consideration.
Table 9
Results of processing\(^{1)}\) by the method of least squares. In these calculations the possible sources of error were attributed to the Rydberg formula and to the photoelectric equation. Solution I was obtained for experimental works that do not give rise to doubt. II is the same solution, but with the addition of works based on the Rydberg formula. III is the same, with the addition of the figures of all photoelectric experiments (+ constant radiation). The errors are given in the form of probable errors, computed from the observed equations\(^{2)}\). For the designations \(A, B, C, D\), etc., see Table 8.
| Constants | Directly obtained value (mean) | Results of processing by the method of least squares | Results of processing by the method of least squares | Results of processing by the method of least squares | Differences\(^{3)}\) | Differences\(^{3)}\) |
|---|---|---|---|---|---|---|
| Solution I for: \((A, D, E, F)\) | Solution II for: \((A, D, E, F) + (H)\) | Solution III for: \((A, D, E, F) + (B, C, G)\) | \((\mathrm{II}-\mathrm{I})\cdot 10^4\) (except \(R\infty\)) | \((\mathrm{III}-\mathrm{I})\cdot 10^4\) (except \(R\infty\)) | ||
| \(e\) | \(4{,}8025\) \(\pm 4\) |
\(4{,}8025\) \(\pm 4\) |
\(4{,}8025\) \(\pm 4\) |
\(4{,}8025\) \(\pm 7\) |
\(0 \pm 4\) | \(00 \pm 7\) |
| \(m_0\) | — | \(9{,}1070\) \(\pm 14\) |
\(9{,}1070\) \(\pm 16\) |
\(9{,}1073\) \(\pm 24\) |
\(0 \pm 16\) | \(+3 \pm 24\) |
| \(h\) | — | \(6{,}6189\) \(\pm 36\) |
\(6{,}6242\) \(\pm 10\) |
\(6{,}6133\) \(\pm 34\) |
\(+53 \pm 36\) | \(-56 \pm 36\) |
| \(\dfrac{e}{m_0}\) | \(1{,}7591\) \(\pm 2\) |
\(1{,}7591\) \(\pm 4\) |
\(1{,}7591\) \(\pm 3\) |
\(1{,}7590\) \(\pm 4\) |
\(0 \pm 4\) | \(-1 \pm 4\) |
| \(\dfrac{h}{e}\) | \(1{,}3761\) \(\pm 6\) |
\(1{,}3782\) \(\pm 7\) |
\(1{,}3793\) \(\pm 1\) |
\(1{,}3771\) \(\pm 7\) |
\(+11 \pm 7\) | \(-11 \pm 7\) |
| \(R\infty\) | \(109\,737{,}42\) \(\pm 0{,}06\) |
\(110\,007\) \(\pm 177\) |
\(109\,739\) \(\pm 8\) |
\(110\,291\) \(\pm 167\) |
\(-268 \pm 177\) | \(+284 \pm 177\) |
\(^{1)}\) It must be pointed out that the value of \(e\) in solution I (and also in I′ of Table 10) is not a “least-square” value. By this they want to say that the value of \(e\) does not affect the figures obtained from experiments not connected with equation \(A\). This proposition was clearly confirmed by a remarkable diagram given by Du Mond in a personal communication to the author. The quantity \(e\) affects only equations \(C\) and \(H\), and therefore in the other equations the resulting \(e\) is the least-square value from one, two, or three equations taken.
\(^{2)}\) It must be emphasized that the probable error of the functions \(e\), \(m_0\), and \(h\) cannot be found from the probable errors of \(e\), \(m_0\), and \(h\), because the latter quantities are not independent. It is necessary to go back to the quantities entering into the solutions of Tables 9 and 10 (see Schouten\(^{30}\)). The ratios of the external probable error to the internal in the three solutions of Table 9 are respectively equal to \(0{,}004;\ 0{,}73\), and \(1{,}66\); in Table 10: \(0{,}62;\ 0{,}61\), and \(1{,}66\). The larger error was taken.
\(^{3)}\) Of the two errors cited, the larger was taken.
Such a difference between the present method of treatment and the preceding one consists only in the fact that the results of direct photoelectric measurements are here transferred from the group of unreliable works to the group of works not subject to doubt.
The results are presented in Table 10. The notation of the solutions is the same as before. Here, as before, it may be seen that the differences
Table 10
Results of treatment by the method of least squares1, when the source of discrepancies is ascribed to Rydberg’s formula and to the inverse photoelectric effect. The basic solution I′ contains no doubtful figures. II′ is computed with the addition of figures based on Rydberg’s formula, III—the same with the addition of the inverse photoelectric effect (+ constant radiation). Errors are given in the same way as in Table 9. Notation is given in Table 8.
| Constants | Observed value (mean) | Solution I′ for $(A, D, E, F, G)$ | Solution II′ for $(A, D, E, F, G) + H′$ | Solution III′ (=$\mathrm{III}$) for $(A, D, E, F, G)$ $(B, C)$ | $(\mathrm{II}'-\mathrm{I}')\cdot 10^4$ (except $R_\infty$) | $(\mathrm{III}'-\mathrm{I}')\cdot 10^4$ (except $R_\infty$) |
|---|---|---|---|---|---|---|
| $e$ | $4{,}8025$ $\pm 4$ |
$4{,}8025$ $\pm 4$ |
$4{,}8024$ $\pm 1$ |
$4{,}8025$ $\pm 7$ |
$-1 \pm 4$ | $0 \pm 7$ |
| $m_0$ | — | $9{,}1073$ $\pm 14$ |
$9{,}1071$ $\pm 14$ |
$9{,}1073$ $\pm 24$ |
$0 \pm 14$ | $0 \pm 24$ |
| $h$ | — | $6{,}6214$ $\pm 29$ |
$6{,}6242$ $\pm 10$ |
$6{,}6133$ $\pm 34$ |
$+28 \pm 29$ | $-81 \pm 34$ |
| $\dfrac{e}{m_0}$ | $1{,}7591$ $\pm 2$ |
$1{,}7591$ $\pm 4$ |
$1{,}7591$ $\pm 2$ |
$1{,}7590$ $\pm 4$ |
$0 \pm 4$ | $-1 \pm 4$ |
| $\dfrac{h}{e}$ | $1{,}3761$ $\pm 6$ |
$1{,}3787$ $\pm 7$ |
$1{,}3793$ $\pm 1$ |
$1{,}3771$ $\pm 7$ |
$+6 \pm 7$ | $-16 \pm 7$ |
| $R_\infty$ | $109\,737{,}42$ $\pm 06$ |
$109\,888$ $\pm 142$ |
$109\,741$ $\pm 8$ |
$110\,291$ $\pm 167$ |
$-147 \pm 142$ | $+403 \pm 142$ |
in the solutions for $e$, $m_0$ and $\dfrac{e}{m_0}$ are insignificant. For $h$, $\dfrac{h}{e}$ and $R_\infty$ the differences are almost three times larger when photoelectric points are added (solution III) than when points connected with Rydberg’s formula are included (solution II). This, apparently, indicates that the source of contradictions should be attributed rather to the inverse photoelectric effect than to Rydberg’s formula.
This assertion becomes still more convincing when the results of the two basic solutions (Table 11) are compared with the directly
obtained data. In neither solution does the computed value \(R_\infty\) differ from the spectroscopic value by more than the error. However, both methods of calculation lead to values of \(\dfrac{h}{e}\) exceeding the directly found values by an amount almost three times as large as the experimental error. This confirms our supposition that the source of the discrepancies is connected with the inverse photoelectric effect. The experimental errors of these works (especially those connected with point 3) are not so large that one could expect the appearance of new measurements capable of mitigating the contradiction. Consequently, the situation that has arisen requires either a new interpretation of the experimental results, or a change in the theory of the inverse photoelectric effect. The former seems more acceptable.
Table 11
| Constants | Differences \(I - Q\) | Differences \(I' - Q\) |
|---|---|---|
| \(e\) | \(0 \pm 4\) | \(0 \pm 4\) |
| \(\dfrac{e}{m_0}\) | \(0 \pm 4\) | \(0 \pm 4\) |
| \(\dfrac{h}{e}\) | \(+21 \pm 7\) | \(+26 \pm 7\) |
| \(R \infty\) | \(+270 \pm 177\) | \(+151 \pm 142\) |
In Table 11 a comparison is given of the solutions \(I\) and \(I'\), presented in Tables 9 and 10, with the directly obtained quantities \(Q\). Solution \(I'\) contains the photoelectric data omitted in solution \(I\).
Conclusions
In the present article the results of all the most important measurements of the atomic constants \(e\), \(m_0\), and \(h\), or of their combinations, have been compared. For this purpose the experimental data of various authors were treated in the following way: 1) all values of combinations of constants adopted by the authors were excluded, so that the results of the work would represent precisely what the experiment gives; 2) auxiliary constants (such as, for example, the Faraday number, conversion factors, etc.) were revised and recalculated. The experimental data treated in this way are collected in Table 6 and represented graphically in Figs. 1 and 2.
Experimental results of all types were analyzed with respect to the fundamental laws or equations on which they are based (see Table 7). A comparison of the mutual consistency of experiments connected with one or another law, and of experiments not depending on it, undoubtedly indicates that the source of the discrepancies in the atomic constants is connected with the Rydberg formula. However, if one discards the relatively inaccurate radiation measurements connected with Planck’s equation, or joins them to the results of photoelectric measurements, then an equally clear indication appears of the inaccuracy of the photoelectric-effect equation.
The correction term \(\dfrac{1}{(1-a)}\), introduced into Rydberg’s formula, noticeably reduces the discrepancy, but does not eliminate it.
An analysis of the experimental results by the method of least squares, carried out under the assumption that the possible sources of error lie either in Rydberg’s formula or in the photoelectric equation, gives no grounds for preferring one or the other assumption. This analysis rather points to the erroneousness of both equations, in view of the fact that the results obtained from Rydberg’s formula and from the photoelectric equation diverge (in opposite directions) from all the remaining experiments.
However, the same method, applied under the assumption of the erroneousness of Rydberg’s formula and of the theory or interpretation of the inverse photoelectric effect, gives more definite indications that it is precisely the latter which is the source of the contradiction. Justification for such a separation of the direct and inverse photoeffect may be seen in the fact that the results of the former agree well with the remaining experiments, whereas for the inverse photoeffect this is not observed.
Further confirmation of the indicated point of view may be found in the fact that the value \(R_\infty\), calculated from experimental results that are beyond doubt and treated by the method of least squares, agrees well with the spectroscopic values, whereas the value \(\dfrac{h}{e}\), obtained in the same way, differs from the observed value by an amount almost three times greater than the error.
The analysis carried out indicates the necessity of further experimental work in the following directions: 1) determination of \(\dfrac{h}{e}\) by any methods and especially by methods not connected with the boundary of the continuous spectrum of X-rays, since the accuracy afforded by this method is already sufficiently high; 2) determination of \(\left(\dfrac{e}{m}\right)\left(\dfrac{e}{h}\right)\) with the aid of X-ray photoelectrons; 3) determination of \(\left(\dfrac{h}{e}\right)\left(\dfrac{e}{m}\right)^{\frac{1}{2}}\) from measurements of electron diffraction; 4) determination of \(\dfrac{e}{h^{3/4}}\) from measurements of the Stefan–Boltzmann constant.
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In this part of the work the author followed Du Mond’s advice: to avoid any hints of the “finality” which is usually associated with solutions by the method of least squares and with “best” values. ↩↩↩
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In a conversation with R. A. Beth in February 1937, the author and his interlocutor came to the conclusion that it was necessary to give a solution without including the Rydberg formula. A calculation of this kind, using temporary values of the experimental results, was made by Beth 78. ↩↩
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See, for example, Palmer 79. Probable errors of functions of \(e\), \(m_0\), and \(h\) were obtained by Shoven’s method 80. Indeed, the expression standing in parentheses in equation (83) is the square of the reciprocal weight of the sought function. Hence multiplying the square root of this expression by the probable error of an observation of unit weight gives the internal probable error of the function, while multiplying the same quantity by the probable error of an unknown of unit weight gives the external probable error of the function 81. ↩