Translation:On the Constitution of the Electron (1906)

by Walter Kaufmann (physicist)

On the Constitution of the Electron;^[1]

by W. Kaufmann.

Introduction

In the year 1881, J. J. Thomson^[2] showed that an electrically charged body (due to its magnetic field caused by its motion according to Maxwell's theory) must behave with respect to external forces, as if its mass would be increased by a certain amount depending on the magnitude of its charge and its shape. After O. Heaviside^[3] calculated the distortion of the field dragged by a charged sphere, which occurs when the velocity of the moving charge is comparable to the speed of light, it was additionally shown by Thomson^[4], that the mass-increase stated by him is not constant anymore with respect to such velocities, but grows with increasing velocity. The total inertial mass of a moving charge, had consequently to be a function of its velocity.

The observations at cathode ray gave at the beginning no such variability of inertial mass, which of course wasn't easily observable, since the easily attainable velocities were still too small to cause a noticeable mass-increase. The experimental verification of mass-variability with respect to these rays, was first achieved by H. Starke^[5] some time after the author^[6] has verified the actual mass-variability by observation of $\beta$-rays or radium.

The $\beta$-rays or radium show, as it is known, qualitatively the same properties as cathode rays; they are magnetically^[7] and electrically^[8] deflectable, and they carry an electric charge with them.

In quantitative respects, they are different from the cathode rays even under crude observation, in so far as they have a much higher penetration capability as the other ones. Now, since it is known that the penetrating power increases with velocity, the assumption was near at hand, that $\beta$-rays would have a considerable higher velocity than cathode rays. However, consequently it was also to be hoped, that the dependence of mass from velocity for $\beta$-rays can clearly be shown. This assumption was confirmed by my (still quite crude) measurements in the year 1900^[9]. The quite large mass-variability also demonstrated, that the portion of pure electromagnetic mass is very great if not overwhelming in comparison to a possible mechanic or material mass. The attempt to calculate both portions didn't lead to any correct result, since at that time, only the equation of Searle^[10] for the energy of the electron was at our disposal as a theoretical foundation, from which the "longitudinal" mass indeed could be calculated by means of the energy theorem, i.e. the relevant mass for tangential acceleration, though not the "transverse" mass with which we have to do with respect to the electric or magnetic deflection perpendicular to the original trajectory as observed by me.

After the theory was supplemented by M. Abraham^[11] by stating the strict formula for the transverse mass, the experiments have been repeated with improved setup^[12], with the result, that the measuring-results are sufficiently represented by Abraham's formula within the margin of errors, i.e. that one can consider the mass of the electron as purely electrical.

In the calculation of the field for a rapidly moving electron, Abraham made the kinematic fundamental assumption, that the field of the electron is infinitely extended into the exterior, however, into the interior it is extended up to the surface of a sphere of constant radius $a$. Within this sphere, the field shall be either zero (surface charge), or shall be decreasing according to a certain law (uniform volume charge). Thus the electron should behave the same way as a rigid sphere would behave, when Maxwell's equations for empty space are employed; in other words: the macroscopic behavior of electrons properly distributed on a certain surface or in a certain space, was formally transfered to the microscopic image of the individual electrons. The fundamental assumption on the constitution of the electron shall be denoted in the following as

rigid electron.

By the good agreement of my measurements with this theory, the question after the constitution of the electron seemed to be decided at first. Then, a work by H. A. Lorentz^[13] appeared in the year 1904, in which the attempt was made to remove the difficulties which sill existed in the optics of moving bodies, by somewhat modified fundamental assumptions on the electron and also on the molecular forces acting in-between the material body-particles. While namely the original theory of Lorentz^[14] gave an influence of second order^[15] of Earth's motion upon certain optical and electromagnetic phenomena, all attempts to experimentally demonstrate^[16] these influences, steadily led to negative results for the time being, so that the conviction became stronger that one has to modify the fundamental equations of electrodynamics, so that the "absolute velocities" related to an arbitrarily defined coordinate-system occur as computational quantities in the fundamental equations, but they have to vanish in the end results, so that the observed magnitudes would only depend on the directly observable "relative velocities" of ponderable bodies against each other.

Lorentz now showed, that one could arrive at such a result, when it is assumed that the dimensions of all physical bodies, including their individual molecules and electrons, would change their shape in a very specific way with velocity; namely, when $q$ is the velocity of the system, $c$ the speed all dimensions in the direction of motion shall be contracted in the ratio

$(1-q^{2}/c^{2})^{\frac {1}{2}}$

while the transverse directions remain unchanged.

To this geometric fundamental assumption, he still added the physical one, that all molecular forces are changing the same way with velocity as the electrostatic forces, and that the "masses" of mechanics are changing the same way as the electromagnetic mass of the electron.

From the mentioned assumptions, a total independence of all observable phenomena from the absolute velocity was consequently given.

It cannot be overlooked, that this way of proving was unsatisfactory in one respect: A quantity for calculation is employed which shall be decisive for the shape of bodies, namely the "absolute velocity", or the "velocity relative to the luminiferous aether". This quantity actually cannot be defined by us at all, since the result of the now described calculation is, that there is no means to define this velocity even by a thought experiment. We cannot even give the amount of the deformation change belonging to the observable change of relative velocity $q$, as long as we don't know the absolute velocity $q_{0}+q$. For example, if the original length of a body is $l_{0}$, it thus becomes deformed by $q_{0}$ into

$l=l_{0}\left(1-{\frac {q_{0}^{2}}{c^{2}}}\right)^{\frac {1}{2}}$

and by $q_{0}+q$ into

$l'=l_{0}\cdot \left(1-{\frac {(q_{0}+q)^{2}}{c^{2}}}\right)^{\frac {1}{2}}.$

The length change $\delta l=l'-l$ corresponding to an observable increas of velocity, is thus not only a function of $q$, but also of $q_{0}$. As we have no right to assume, that our system of fixed stars to which we relate Earth's motion, is momentarily in absolute rest in the aether, we thus actually cannot say anything about the occurring deformations.

It is now very remarkable, that, starting from quite different assumptions, Einstein^[17] recently arrived at results, which are in agreement with those of Lorentz concerning the consequences accessible to observation, though in which the previously mentioned difficulties of epistemological kind have been avoided. Einstein introduced the principle of relative motion, at least as regards translations, as a postulate. He thus places the theorem at the top, that physical phenomena observable in any rigid system, must be independent from whether the system (together with the observer) is moving relatively to any other system. From that, by an application to the propagation of light, a new definition of time and the concept of "simultaneity" for two spatially separated points immediately follows. Namely, for an observer in system $k$ moving relatively to another normal system $K$, two events happening at different points are simultaneous when two quantities become equal (for an observer resting in $K$) which are formally identical with "local time" introduced by Lorentz. Furthermore, when according to the previous postulate, all properties (including the geometric dimensions) remain unchanged for a co-moving (i.e. at rest in $k$) observer, then for an observer at rest in $K$, the "simultaneous" measurements of a structure viewed from $K$, must just be altered in a way, as it was introduced by Lorentz as a theoretical fundamental assumption. Furthermore, the total reciprocity of all laws found, i.e. the exchangeability of systems $K$ and $k$ (the "resting" and the "moving" system) will be demonstrated. The results accessible to observation are thus the same with respect to both authors; however, while Lorentz only shows that his hypotheses lead to the desired result without excluding that the same can also be achieved in another way, it is shown by Einstein, that when the desired result, namely the principle of relative motion, is placed at the top of the whole of physics, then the kinematics of the rigid body must necessarily be changed in the way stated, and that the equations of electrodynamics^[18] must assume the form stated by Lorentz.

Both authors now also give the equations of motion for an electron deformed in the given way, which essentially differ from Abraham's equations. An application of the equations to my previous measurements by Lorentz, led to the surprising result that my observations can be represented by them with the same precision, as Abraham's equations for the rigid electron.

However, it was shown that the velocity values, which had to individually attributed to the measured curve points to connect them with Lorentz's formula, are smaller by 5-7% than the values according to Abraham's formula. By that, the way was simultaneously given to decide between both theories.

When it could be achieved to determine the velocities belonging to the individual curve points (see below), independently of the electron theory employed, and directly from the constants of the experimental arrangement – in the following denoted as the "apparatus constants" –, and to compare these values with those given by the one or the other theory from the shape of the photographed curve – from the "curve constants" –, then the greater of lower degree of agreement between both systems of value, could serve as criterion for the validity of one of the two theories. Of course it didn't appear to be excluded from the outset, that maybe none of both theory gave a sufficiently precise agreement, or that also other fundamental assumptions could lead to a satisfying result.

Abraham^[19] alluded to the fact, that Lorentz's deformation of the electron requires the expenditure of work, so that one has to assume, to avoid a contradiction with the energy law, an "inner potential energy" of the electron; a pure electromagnetic foundation of the mechanics of the electron, and by that also of mechanics at all, would be proven to be impossible if new measurements should demonstrate the validity of Lorentz's theory. This conclusion would of course still be valid, when instead of the work of an unknown inner energy of the electron, an also unknown universal external pressure (following Poincaré's^[20] proposal) would by introduced, which provides the necessary compression work on the electron.

Besides that of Abraham, also a third fundamental hypothesis on the electron is free from this difficulty, which was introduced^[21] by Bucherer^[22]; he assumes that the electron is deformed at constant volume, namely in a way, so that the ratio of axis of the emerging ellipsoid always agrees with the so called "Heaviside ellipsoid"^[23]. Since the same ratio of axis also applies to Lorentz's electron, one can derive the equations given by Bucherer for mass, energy, and momentum of the electron, without further ado from the ones of Lorentz, when one introduces, instead of the invariable transverse diameter $a$ of the electron that occurs as a parameter in the equations of Lorentz, a transverse diameter depending on the velocity, so to be measured that the volume of the ellipsoid is equal to the original sphere. When differentiating with respect to velocity – when calculating the longitudinal mass – then also the variability of the transverse dimensions must of course be considered too.

Although it is immediately clear due to the unambiguity of the reasoning of Lorentz and Einstein, that Bucherer's electron cannot strictly remove the influences of the absolute velocity, it is still probable from the outset, that at least effects of second order are at least partially removed. At this time, Bucherer is still concerned with the discussion of this question. Anyway, it appeared to be appropriate to include Bucherer's electron in these considerations as well.

I anticipate the general result (described below) of the measurements:

The measurement results are not compatible with the fundamental assumption of Lorentz-Einstein. The equation of Abraham and that of Bucherer represent the measurement results equally well. A decision between the two latter theories by measuring the transverse mass of $\beta$-rays appears to be impossible for the time being.

2. Comparison with the theories of Abraham, Lorentz and Bucherer.

The function $\Phi (\beta )$ expressing the dependency of mass from velocity, has the following value (according to the three theories mentioned in the headline of this paragraph and which were closer described in the introduction):

I.	(Abraham)	$\Phi (\beta )={\frac {3}{4}}{\frac {1}{\beta ^{2}}}\left({\frac {1+\beta ^{2}}{2\beta }}\cdot lg{\frac {1+\beta }{1-\beta }}-1\right)$
II.	(Lorentz)	$\Phi (\beta )=\left(1-\beta ^{2}\right)^{-1/2}$
III.	(Bucherer)	$\Phi (\beta )=\left(1-\beta ^{2}\right)^{-1/3}$

The theory of Einstein leads to the same formula as the one of Lorentz, when one sticks to the equations of motion given by Einstein (l.c.); the somewhat deviating definition of mass for an observer co-moving with the electron given by Einstein, does not correspond to the circumstances of my observations; the deviation of his equation for the electron's mass from that of Lorentz, is thus only apparent.

In cases I. and III., the following relations between the apparatus constants, the curve constants, and $\epsilon /\mu _{0}$ are given from equations (14) to (16):

(19)	$A\cdot B=E/Mc=\beta \cdot y'/z'.\,$

(20)	$\epsilon /\mu _{0}=c/AM=Bc^{2}/E=c{\sqrt {Bc/AME}}.$

In case II., a separation of the variables is possible and one obtains as the equations of the reduced curve:

(21)	$y'^{2}=C^{2}z'^{2}+D^{2}z'^{4}\,$

where

(22)	$C=E/Mc\,$

and

$\epsilon /\mu _{0}=cC/MD=c^{2}C^{2}/ED={\frac {cC}{D}}\cdot {\sqrt {cC/ME}}.$

Equations (19) and (22) allow us to compare $AB$ or $C$ with a quantity directly calculated from the experimental arrangement. They are thus a direct means of verification of the considered theory. Equations (20) and (23) also contain the cathode-ray-value $\epsilon /\mu _{0}$, about whose amount no complete agreement seems to exist. Here it is given by different ways, under consideration of either the electric or of the magnetic field integral alone. Both values deviate the more from one another, the less equations (19) and (22) are satisfied. For a comparison with the values observed with respect to cathode rays, it seems to be the best, to uniformly distribute the possible experimental errors upon the magnetic and electric field integral, and to take the geometric average from both values; this is the meaning of the third expressions for $\epsilon /\mu _{0}$ in equations (20) and (23).

The method of least squares gives the following value of the curve constants:

I.	According to	Abraham:	A	= 2,169 · 4/3	B	= 0,08355 ·3/4
II.	"	Lorentz:	C²	= 0,02839	D²	= 0,2672
III.	"	Bucherer:	A	= 2,9337	B	= 0,06234.

The following table contains the results of calculation:

Table VIII.
Comparison of the observed with the calculated curve.
z'	y'obs.	p	y'cal.			δ			β
			I	II	III	I	II	III	I	II	III
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 0,1405 0,1667	0,5 1 1 1 1 1 1 0,25 0,25	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 -3 -15	0 +1 0 -6 -2 +4 +3 0 -11	-8 -3 0 -2 +3 +9 +5 -4 -20	0,974 0,922 0,867 0,807 0,752 0,697 0,649 0,610 0,566	0,924 0,875 0,875 0,823 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 the table it is given, first, that one cannot make a decision in favor or against a theory based on the shape of the curves alone, without consideration of the absolute values of the constants^[24]. The mean error of the individual curve points namely amounts:

For	I	: 5	Micron	(Abraham)
For„	II	: 5	"	(Lorentz)
For„	III	: 7	"	(Bucherer).

However, since in the theoretical calculation of the curve, terms of relative amount of 1-3 thousandth were neglected several times, the neglecting-amount for the average and bigger $y'$-values are thus already of the order of the average value hat was found. However, I have renounced to make a more precise calculation, since due to the great individual deviations of the single plates to one another, the omission of a plate or the introduction of other weights is actually sufficient, to cause displacements of the curve of some microns.

Thus one can at first only say as the result of the previous calculation, that all three theories represent the relative shape of the curve equally well. However, from the values for $\beta$ one can easily see, that Lorentz's theory requires totally different velocities, than the one of Bucherer or Abraham. For these two theories, the velocities are nearly identical, and what is even more remarkable: The function $\Phi (\beta )$ numerically agrees, as one can easily convince himself, with both of them (within the velocity interval given here) with a deviation of at most 2%.

The decision thus only depends on equations (19) and (22) or (20) and (23).

The values of the field integrals, calculated in the appendix p. 543ff, are:

$E=315\cdot 10^{10}$ Electromagn. Units (with 2500 volt P.D.)

$M=557,1$

Thus

$E/Mc=0,1884.\,$

On the other hand, the calculation by means of the curve constants (under consideration of the corrections due to the error of the $z$-scale mentioned on p. 513):

					Difference in perc.
I.	nach	Abraham:	AB	= 0,1817	-3,5
II.	"	Lorentz:	C	= 0,1689	-10,4
III.	"	Bucherer:	AB	= 0,1831	-2,8

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

a) For cathode rays:[25]				1,878 · 107.
b) For β-rays:					Difference in perc.
according to	Abraham	1,858	1,788	1,823	2,9
"	Lorentz	1,751	1,569	-11,6
"	Bucherer	1,833	1,780	1,808	3,7

Eventually one can also make a comparison of the different theories, by determining the constants A and B, or C and D themselves, from equations (16) be means of the measured values of M and E and the cathode-ray-value of $\epsilon /\mu _{0}$, and to calculate the curve with their aid. The calculated values of the curve are:

${\begin{array}{ll}A={\frac {c}{M}}{\frac {\mu _{0}}{\epsilon }}={\frac {3\cdot 10^{10}}{557,1\cdot 1,878\cdot 10^{7}}}&=2,867\\\\A={\frac {c}{M}}{\frac {\mu _{0}}{\epsilon }}={\frac {3\cdot 10^{10}}{557,1\cdot 1,878\cdot 10^{7}}}=2,867B={\frac {\epsilon }{\mu _{0}}}{\frac {E}{c^{2}}}={\frac {1,878\cdot 10^{7}\cdot 315\cdot 10^{10}}{9\cdot 10^{20}}}&=0,0658\\\\C=0,1884\\\\D={\frac {cC}{M}}{\frac {\mu }{\epsilon }}={\frac {E}{M^{2}}}{\frac {\mu _{0}}{\epsilon }}&=0,539.\end{array}}$

By means of the values, the curve can be calculated according to equation (18) or (21). It is given:

Table IX.
Curves calculated from the apparatus constants.
z'	y'
	Abraham	Lorentz	Bucherer
0,1 0,2 0,3 0,4 0,5	0,0191 0,0413 0,0712 0,1104 0,1595	0,0196 0,0434 0,0745 0,1144 0,1642	0,0190 0,0407 0,0696 0,1080 0,1568

The points calculated this way, are included in Plate. IV, Fig. 11. One can see on them, that the deviations of Lorentz's curve surpass the experimental errors by far. Though it shall again be alluded to the fact, that the previously done comparison is based on the assumption that Simon's value of $\epsilon /\mu _{0}$ is correct. Only the previously executed comparison of quantity $E/Mc$ is free from this assumption.

The results above, decidedly speak against the correctness of the theory of Lorentz and therefore also of the theory of Einstein; however, it we consider them as refuted, then consequently the attempt to establish the whole of physics including electrodynamics and optics, on the principle of relative motion, must be denoted as failed for the time being. A consideration of Einstein's theory shows, that if one nevertheless wants (with keeping this principle) to obtain agreement with my results, already Maxwell's equations of resting bodies must be modified, a step to which hardly anyone will decide himself for the time being.

We will rather remain at the assumption for the time being, that the physical phenomena depend on the relative motion to a very special coordinate system, which we call the absolutely resting aether. Although it is not achieved up to now, to demonstrate such an influence of motion through the aether by electrodynamic or optical experiments, the impossibility of such a demonstration may not be concluded from that.

Eventually, it seems appropriate to discuss the question after the possibility of a decision between the theories of Abraham and Bucherer that are preliminarily the only remaining ones. A decision by means of $\beta$-rays would require a considerably increased precision. However, since already now the main errors stem form the photographic plate, namely from the plate grain, concealment and distortion of the layer, a considerable magnification of the apparatus dimensions and thus also of the curves would be required at first, since the plate errors are not simultaneously increased, and as regards the layer distortion, they are even diminished. If one wants, together with the $n$-times magnification of all dimensions of the apparatus, also a $n$-times magnification of the curve, then the magnetic field must be diminished to the $n$^th part. Permanent magnets of sufficient magnitude (ca. for $n=5$), might cause difficulties and enormous costs. Thus one still has to use electromagnets, which in turn is only possible when a storage battery is at our sole disposal. To sustain the electric field completely constant, which by itself sinks to the n^th part with $n$-times plate distance and with uniform voltage, a battery of at least 3000 volt is required, which also may not used for other purposes during the expositions time. Eventually, also for measuring the magnetic field, the production of a sufficiently strong current coil would be required. With other words, an additional increase of precision in the way employed so far, would require means that surpass the current means of the institute by far.

Another way presents itself by refinement of the measurements executed by Starke^[26] with fast cathode rays. Maybe, by application of Wehnelt's gleaming oxid-cathodes, it is possible to obtain sufficiently steady discharges also with even higher voltages than 38000 volt – the upper limit up to which Starke came. Consequently it doesn't appear to be impossible to decide by which factor, ²⁄₅ or ½ or ⅓, one has to multiply the term depending on $\beta ^{2}$, to achieve precise agreement with the observed variability of $\epsilon /\mu _{0}$^[27]. However, also in this case it is about the most attainable precision in measurement and calculation, and consequently about the expenditure of extraordinary means.

---

1. Under the same title, I have already published a short excerpt of this investigation (carried out with the kind support of the Berlin Academy of Sciences) in the Berl. Ber. 45. p. 949. Nov. 1905. For those readers who are essentially interested in the general results, the following treatise provides nothing new. It contains a detailed representation of the experimental setup applied, and in the appendix also the most important measuring-protocols, to allow a verification of the numbers and error margins at any time. It furthermore contains an improved representation of the trajectory, already reported in my earlier publications on the some subject. Eventually, some things that appear as of importance to me, are additionally dealt with in the appendix, which, however, would have disturbed the context when placed in the main text. Some of the numbers reported in the excerpt had some small errors in the last decimals after renewed verification, which have been corrected here.
2. J. J. Thomson, Phil. Mag. (5) 11. p. 229. 1881.
3. O. Heaviside, Phil. Mag. April 1889.
4. J. J. Thomson, Rec. Researches p. 21.
5. H. Starke, Verh. d. D. Physik. Ges. 5. p. 241. 1903.
6. W. Kaufmann, Gött. Nachr. 1901. Heft 1; 1902. Heft 5; 1903. Heft 3. Phys. Zeitschr. 4. p. 55. 1902
7. F. Giesel, Wied. Ann. 69. p. 834. 1899; St. Meyer u. E. v. Schweidler, Phys. Zeitschr. 1. p. 90. 1899; H. Becquerel, Compt. rend. 129. p. 996. 1899.
8. H. Becquerel, Compt. rend. 130. p. 819. 1900; E. Dorn, Abh. Nat. Ges. Halle 22. p. 44. 1900.
9. W. Kaufmann, l. c.
10. G. Searle, Phil. Mag. (5) 44. p. 340. 1897.
11. M. Abraham, Gött. Nachr. 1902.
12. l. c.
13. H. A. Lorentz, Versl. Kon. Akad. v. Wet. te Amsterdam. 27. Mai 1904.
14. H. A. Lorentz, Versuch einer Theorie etc. Leiden 1895.
15. That is, an influence whose magnitude is proportional to the squared ratio $q^{2}/c^{2}$ of Earth's velocity to the speed of light
16. For literature, see. H. A. Lorentz, "Theory of Electrons" in the Enzyklopädie d. mathematischen Wissenschaften, as well as the now cited work of the same author.
17. A. Einstein, Ann. d. Phys. (4) 17, p. 891. 1905.
18. If it is presupposed, that for all relatively resting bodies, Maxwell's equations can be considered as valid.
19. M. Abraham, Theorie der Elektriz. II, Kap. 3. Leipzig 1905.
20. H. Poincaré, Compt. rend. 140. p. 1504. 1905.
21. I was informed by M. Abraham by letter, that the deformation-work with respect to Bucherer's electron is equal to zero.
22. A. Bucherer, Math. Einführung in d. Elektronentheorie p. 53. Leipzig 1904.
23. i.e., that $b:a={\sqrt {1-\beta ^{2}}}$, where $\beta =q/c$.
24. A graphical illustration of the calculated curves in the measure of Tab. IV, Fig. 11 is meaningless, since the differences would be too small to be seen with the mere eye.
25. Corrected by the value 1,865 · 10⁷ found by S. Simon (Ann. d. Phys. 69. p. 589. 1899), for ∞-small velocities. See appendix p. 548.
26. H. Starke, l. c.
27. See also appendix p. 548ff