Translation:On the Constitution of the Electron (1905)

by Walter Kaufmann (physicist)

On the Constitution of the Electron.

By W. Kaufmann.

Submitted by Mr. Warburg.

In some earlier reports ^[1] I have shown, that the mass of the electron moving in the $\beta$-rays of radium, considerably grows with increased approximation to the speed of light, and that the law of growth is in agreement with the formulas calculated by Abraham^[2]. The presupposition made by Abraham reads:

The electron is to be considered as a rigid sphere, uniformly charged over its surface or its volume.

The good agreement of my previous measurements with the consequences drawn from this assumption by application of the Maxwell-Lorentz equations, at first seem to justify the conclusion that the question after the constitution of the electron is definitely solved.

A recent investigation by H. A. Lorentz^[3], however, led to the surprising result that an agreement with my previous experiments can be achieved using totally different assumptions on the electron. Lorentz namely showed, that the difficulties which prevented the explanation of the negative results of certain experiments aimed to demonstrate an influence (of second and higher order) of Earth's motion upon optical and electric phenomena, can be completely removed when it is assumed that all bodies (including the electrons) are deformed in a very specific way in their motion through the aether.

If $q$ is the velocity of the moving system, $c$ the speed of light, and if $\beta =q/c$, then all dimensions parallel to the direction of motion shall be contracted in the ratio $(1-\beta ^{2})^{\tfrac {1}{2}}$, the dimensions normal to the direction of motion shall remain unchanged.

By this dimension change, as shown by Lorentz, any influence of absolute motion upon optical and electromagnetic phenomena will be strictly removed up to any order.

Although the equation for the dependence of the electromagnetic mass of the electron, as it was obtained by Lorentz using his assumption of the deformable electron, very much differs from Abraham's equation for the rigid electron, it was possible that my experimental results could be suited to the equations of Lorentz as well. The only difference was, that the value of $\beta$ for one and the same curve point was smaller by 5 to 7 percent according to Lorentz, and that for the relation

${\frac {\epsilon }{\mu _{0}}}={\frac {\mathsf {Charge}}{{\mathsf {Mass\ for}}:\ \beta =0}}$.

a smaller value was obtained as well.

Now, since the $\beta$-values in my previous measurements weren't determined from the apparatus constants, but using Abraham's formula from the method of least squares so that a connection as good as possible was obtained, it followed that a decision was impossible since the mean error of the observed curve against the calculated one was not essentially different even when applying the formula of Lorentz.

The decision between the two mentioned theories and probably also other theories on the constitution of the electron, was only possible when it was achieved to calculate the two constants of the obtained curve (the number of curve constants is steadily equal to 2) in absolute measure from the dimensions of the apparatus and the strength of the deflecting magnetic or electric field, and to compare these "apparatus constants" with the "curve constants".

From the insignificance of the differences in the values for $\beta$ and for ${\frac {\epsilon }{\mu _{0}}}$ it was given, however, that for achieving this goal we had to execute the observations with far larger precision than before. After the (for this purpose) necessary means were given to me in the kindest manner by the Royal academy of sciences at Berlin, to which I express my greatest gratitude here, I have carried out the experiments in the course of last year, and I shortly report in the following about the general experimental arrangement and the results; a more extended report with a more precise specification of the experimental arrangement, and with the results of all auxiliary measurements etc. shall appear in the Annalen der Physik.

The experimental arrangement was in principle the same as earlier; the main differences consisted in the following:

Instead of the electro magnet used earlier, two superimposed and very strong horseshoe magnets were employed, generating a field which was homogeneous within $\pm 2$ along the ray-path, and in consequence of the high age of the magnet it was completely constant (H ca. 140 C. G. S. E.).

The small apparatus containing the grain of radium, the condenser plates serving for the generation of the electric field, and the photographic plate, had ca. the same dimensions as earlier (the whole ray-path = 4cm). All parts, however, were produced with the greatest precision, and the photographic plate was (especially for my experiments) poured upon mirror glass, so that the length of the ray-paths in terms of their single parts, and the "field integrals" relevant for the deflection, could be safely determined to ¹⁄₁₀₀mm or to a fraction of one percent. The distance of both condenser plates (that were smoothly cut) was determined by pressing them through two isolated screws against four quartz-plates slided in-between, which were cut from a single plan-parallel plate, and whose thickness was determined as 1.242 mm ± 1μ by the company Zeiß.

The course of the electric field along the ray-path was determined, by producing (with 29times magnification) a copy of the whole condenser apparatus, in which the field distribution was determined by means of a oscillating metal mirror of 5mm diameter by observing the oscillation period.

The potential difference on the actual apparatus was produced by means of a high-voltage battery; the maximal voltage amounted ca. 1600 volt. By an automatic seesaw, two Leyden jars connected with the condenser plates where brought to a constant voltage of +V or -V, so that the potential difference of the plates amounted 2V, where V is the available battery voltage. The latter was measured before and after every exposition (which lasted ca. two days) by means of a compensating apparatus and a Weston cell.

The strength of the magnetic field was measured along the whole ray-path by means of a small induction coil, and was compared (for one point) with the field of a precisely measured wire-coil traversed by a current. The current within it, was measured by a precise ammeter whose indications were tested by means of a Weston cell, the electrical resistance, and a compensating apparatus.

For the dimensions of the curves, a small Abbe comparator was used, upon whose slide a second slide was installed at right angles to its direction of motion, which carried the photographic plate and the glass-micrometer (divided in ¹⁄₁₀mm) that was laid within. As setting-mark, a six-times magnifying microscope located at the focal plane was used, i.e. a sphere scratched upon a glass-plate of 0.2mm diameter. The magnetic deflection was read from the glass-micrometer, and the electric ones from the measuring rod of the main-slide; the screws only served for interpolation.

5 plates were measured altogether, where every single point represents the average from 10 measurements each. All curves were reduced upon the same ordinate measuring-rod (corresponding to a voltage of 2500 volt at the condenser plates) and consequently unified to a single curve. Even though the average error of any point (when calculated from the setting errors) only amounted 2 to 4 microns, the deviations of the curve points (stemming from the different plates) from a steady curve going through all of them, were considerably greater up to 30 microns. The reason for this deviation appears to be on the one hand as stemming from the strong blurring of the plates by diffused radiation that displaced the main line of the curve in an irregular way when it is of spatially variable intensity, and on the other hand from a distortion of the photographic layer which (as special experiments have shown) irregularly varies from plate to plate, and which can amount up to ca. ½ percent in the vicinity of the boundary, however, which is surely small in the parts relevant for this measurement, and which always let the observed deflections to appear as too great.

Therefore – which seems to be admissible regarding the weak curvature of the curve –, by unification of (on average) ca. 5 points (belonging to the different plates) to a center of gravity under consideration of the weights of the single points, a balanced curve consisting of 9 points was generated, which was (as far as possible) free from the individual deviations of the single curves. The curve obtained in this way, was the basis for the comparison of the different theories.

==General theory of the curve.==

Let^[4] $\epsilon$ be the charge of the electron in electromagnetic measure, $\mu$ its mass at velocity $q$, $\mu _{0}$ its mass when the velocity is very small. Furthermore, let $z$ and $q$ the magnetic and electric deflections; eventually let $E$ and $M$ be the "electric" and "magnetic field integral" reduced to proportionality with the deflectabilities; i.e. two quantities equal to the mean field strength with a factor each dependent on one of the dimensions of the apparatus. At last let $\beta =q/c$, where $c$ is the speed of light. It is

$\mu =\mu _{0}\cdot \phi (\beta ),\,$

where $\phi (\beta )$ is a function of velocity, whose form is different depending on the basic assumption concerning the constitution of the electron. Then it is:

1.	$z'={\frac {\epsilon }{\mu q}}M={\frac {\epsilon }{\mu \beta }}\cdot {\frac {M}{c}}={\frac {\epsilon }{\mu _{0}}}{\frac {M}{c}}\cdot {\frac {1}{\beta \phi (\beta )}}$

2.	$y'={\frac {\epsilon }{\mu q^{2}}}E={\frac {\epsilon }{\mu \beta ^{2}}}\cdot {\frac {E}{c^{2}}}={\frac {\epsilon }{\mu _{0}}}{\frac {E}{c^{2}}}\cdot {\frac {1}{\beta ^{2}\phi (\beta )}}$

If one sets for abbreviation:

3.	${\frac {\epsilon }{\mu _{0}}}{\frac {M}{c}}=1/A$

4.	${\frac {\epsilon }{\mu _{0}}}\cdot {\frac {E}{c^{2}}}=B$,

5.	$1/\beta \phi (\beta )=u,\,$,

6.	$1/\beta ^{2}\phi (\beta )=v=f(u),\,$,

thus one can write:

7.	$y=B\cdot v=B\cdot f(A\cdot z'),\,$.

Thus a auxiliary table is calculated, which gives the $v$-values corresponding to a most tight row of $u$-values, then (when the constants $A$ and $B$ are known) the corresponding $y'$ by equation (7.) can be compared to every $z'$. It is easily achieved to find approximate values for both "curve constants" $A$ and $B$, and then to calculate the improvements still to be made, by the method of least squares.

The curve constants determined in this way, can be compared with those values partly given from the dimensions of the apparatus alone, and partly from the connection with the value ${\tfrac {\epsilon }{\mu _{0}}}$ found with respect to cathode rays. The more or less precise degree of agreement then decides in favor or against the concerning theory.

The following three theories on the constitution of the electrons will be mutually compared:

1. Rigid electron (Abraham),

2. deformable electron (Lorentz),

3. deformable electron (Bucherer).

No. 1 and 2 are already mentioned; Bucherer^[5] assumes that the electron is (like that of Lorentz) deformed into an oblate ellipsoid with axis-ratio $(1-\beta ^{2})^{-{\tfrac {1}{2}}}:1$ (so called "Heaviside-Ellipsoid"), however, the perpendicular dimensions are increased and simultaneously the volume remains constant.

The function $\phi \beta$ expressing the dependence of mass from velocity, has the following value according to the mentioned three theories:

${\begin{array}{ll}1.&\phi (\beta )={\frac {3}{4}}{\frac {1}{\beta ^{2}}}\cdot \left({\frac {1+\beta ^{2}}{2\beta }}\cdot \lg {\frac {1+\beta }{1-\beta }}-1\right)\\\\2.&\phi (\beta )=(1-\beta ^{2})^{-{\frac {1}{2}}}\\\\3.&\phi (\beta )=(1-\beta ^{2})^{-{\frac {1}{3}}}\end{array}}$

Eventually, also a theory of electrodynamics recently published by A. Einstein^[6] has to be mentioned, which leads to consequences formally identical to those of Lorentz's theory, and consequently to which the second equation applies as well.

In the cases (1.) and (3) the following relations between the apparatus constants, the curve constants and ${\tfrac {\epsilon }{\mu _{0}}}$ are given from the equations (1.) to (3.):

8.	${\frac {\epsilon }{\mu _{0}}}=c/AM+Bc^{2}/E=c\cdot (Bc/AME)^{\frac {1}{2}}$

9.	$AB=E/Mc=\beta \cdot y'/z'.\,$

In case (2.) (Lorentz-Einstein) a separation of the variables is possible, so that one obtains as the trajectory:

10.	$y'^{2}=C^{2}z'^{2}+D^{2}z'^{2},\,$

where

11.	$C=E/Mc,\,$

and

12.	${\frac {\epsilon }{\mu _{0}}}=cC/MD=c^{2}C^{2}/DE=cC/D\cdot (cC/ME)^{\frac {1}{2}}$

The application of the method of least squares gives as values of the curve constants for the balanced curve:

according to	Abraham:	A=2,169 · 4⁄3	B=0,08355 · ¾,
according to	Lorentz:	C²=0,02839	D²=0,2672,
according to	Bucherer:	A=2,9337	B=0,06234.

The following table I contains a comparison of the observed and the calculated electric deflections, where the magnetic deflections are considered as independent variables. All numbers are denoted as centimeter, only the deviations are given in micron.

Table I

$z'$[7]	$y'$ beob.	Gewicht	$y'$ ber.			δ in μ			β
			(1)	(2)	(3)	(1)	(2)	(3)	(1)	(2)	(3)
0,1350 0,1919 0,2400 0,2890 0,3359 0,3832 0,4305 0,4735 0,5252	0,0246 0,0376 0,0502 0,0545 0,0811 0,1001 0,1205 0,1404 0,1606	½ 1 1 1 1 1 1 ¼ ¼	0,0251 0,0377 0,0502 0,0649 0,0811 0,0995 0,1201 0,1408 0,1682	0,0246 0,0375 0,0502 0,0651 0,0813 0,0997 0,1202 0,1405 0,1678	0,0254 0,0379 0,0502 0,0647 0,0808 0,0992 0,1200 0,1409 0,1687	-5 -1 0 -4 0 +6 +4 -4 -16	-0 +1 0 -6 -2 +4 +3 -1 -12	-8 -3 0 -2 +3 +9 +5 -5 -21	0,947 0,922 0,867 0,807 0,752 0,697 0,649 0,610 0,566	0,924 0,875 0,823 0,765 0,713 0,661 0,616 0,579 0,527	0,971 0,919 0,864 0,805 0,750 0,695 0,647 0,608 0,564

From this table it follows, that from the shape of the table alone, without consideration of the absolute values of the constants, it is hardly possible to make a decision in favor of one or the other theory. The average error of the single curve points amounts:

For (1): 5 micron (Abraham),
For (2): 5 micron (Lorentz),
For (3): 7 micron (Bucherer).

A final decision is only possible by comparison of the curve constants with the apparatus constants and the value of the cathode ray from ${\tfrac {\epsilon }{\mu _{0}}}$.

There, the third of equations (8.) and (12.) gives, as it can easily be seen, the average between the values calculated from the first and second ones, which only deviate from each other as far as equation (9.) is not precisely satisfied.

The measurement of the electric and magnetic fields gave:

E = 315 · 10¹⁰ El. Magn. Units (at a potential difference of 2500 volt),

M = 557

Thus it is:

E/Mc = 0,1884

On the other hand, the calculation by means of the curve constants, gives according to (9) and (11):

		Diff. (curve-apparatus-constant)
according to Abraham:	AB=0,1817	-3,5 percent
according to Lorentz:	C=0,1689	-10,4 percent
according to Bucherer:	AB=0,1831	-2,8 percent

The differences simultaneously represent the deviation between the $\beta$-values derived from the curve constants and the apparatus constants for any curve point.

The values for ${\tfrac {\epsilon }{\mu _{0}}}$ are:

a) Cathode rays:^[8] 1,885 · 10⁷

b) β-rays:

		Diff. (β-cathode rays)
according to Abraham:	1,858; 1,788; 1,823	-3,3 percent
according to Lorentz:	1,751; 1,569; 1,660	-11,9 percent
according to Bucherer:	1,833; 1,780; 1808	-4,0 percent

The preceding results decidedly speak against the validity of the fundamental assumption of Lorentz and therefore also of Einstein. If one consequently considers this fundamental assumption as disproved, then the attempt to establish the whole of physics (including electrodynamics and optics) upon the principle of relative motion, has to be denoted as failed for the time being.

A decision between the theories of Abraham and Bucherer is impossible for the time being, and seems to be unreachable at all by observations of the kind previously described, due to the far-reaching numeric agreement of the values of $\phi (\beta )$. Whether Bucherer's formula accomplishes the same for the optics of moving bodies in terms of possible observations, as that of Lorentz, has still to be investigated.

---

1. Gött. Nachr. 1901, Heft 1; 1902, Heft 5; 1903, Heft 3; Phys. Zeitschr. 4, 55, 1902.
2. Gött. Nachr. 1902, Heft 1; Ann. d. Phys. (4) 10, 105, 1903.
3. Versl. Akad. Amsterdam, 27. Mai 1904.
4. See the treatise of the author and of C. Runge cited above, Gött. Nachr. 1903, Heft 5.
5. Mathematische Einführung in die Elektronentheorie, Leipzig 1904.
6. Ann.d. Phys. (4)17, 891; 1905.
7. A subsequent comparison of the glass micrometer with the slide-measuring-rod of the comparator gave an error of the magnitude of a scale mark of -2.5 thousandth. The $z'$-values are thus too high by this amount, the constants $A$ too low. The corresponding corrections are already included in the end-results reported below.
8. S. Simon, Ann. d. Phys. (3) 69, 589, 1899. The value stated there of ${\frac {1}{\mu }}=1,865\cdot 10^{7}$, extrapolated to $\infty$-small velocities, gives the previous number.