Die Elektroynamik es rotierenen Elektrons, Zeit. f. Phys. 37 (196), 43-6. The electroynamics of rotating electrons. By J. Frenkel 1 ) in Leningra. (Receive on May 196) Translate by D. H. Delphenich The Uhlenbeck-Gousmit conception of the rotating electron will be employe, following Thomas, for the presentation of the equations of motion in a given electromagnetic fiel by means of special relativity. The electron will thus be treate simply as a point whose magnetic properties are couple with a well-efine six-vector ( moment tensor ). In this way, one arrives at the explanation that Thomas alreay gave for the origin of the anomalous Zeeman effect in a more thorough an rigorous way. In conclusion, the electromagnetic fiel that is generate by a rotating electron will be etermine, an this will suggest the possibility that the structure of the atomic nucleus is inuce mainly by the magnetostatic interaction between electrons an protons. 1. Introuction. Uhlenbeck an Gousmit ) have recently applie, to great effect, the concept of rotating quantize electrons that was alreay propose by H. A. Compton to the problem of the multiplet structure of the spectral term in the omain of optics an Röntgen rays. They starte from the fact that in a coorinate system S, in which a electron that circles a nucleus is at rest, an aitional magnetic fiel strength arises: H = 1 [v E]. (1) c In this, v means the translational velocity of the electron relative to the coorinate system S that is fixe in the nucleus, an E is the electric fiel strength that prevails relative to this system an is generate by the nucleus. If one now ascribes a proper magnetic moment m to the electron then the magnetic fiel strength (1) must correspon to an aitional magnetic energy: U = m H = v m E c. (1a) Uhlenbeck an Gousmit have now shown that the structure of the optical an Röntgen multiplet terms are explaine immeiately when one ascribes the values 1, 3, 5, etc., to the azimuthal quantum number (k), in agreement with the Lané normalization, 1 ) International Eucation Boar Fellow for 196. ) Nature 117, 64, Feb. 0, 196.
Frenkel The electroynamics of rotating electrons. an as the resulting relativistic correction) for the thermal energy to the mean value of the extra magnetic energy (1a) uner the assumption that m equals one-half the Bohr magneton an that the ratio of the magnetic moment m to the corresponing mechanical e impulse moment has the same value cm (e < 0 is the charge of the electron, m 0 is its 0 mass, c is the velocity of light) as it oes for the orbital motion. In other wors, the impulse moment of the proper rotation must therefore also be set equal to one-half the orinary elementary Bohr value h/π. The concept of rotating electrons makes it further possible to give a complete explanation for the anomalous Zeeman effect (in which, e.g., the noteworthy Paschen- Back effect of the hyrogen lines is mae unerstanable) when the atomic hull of the previous Sommerfel-Lané schema is replace with the proper rotation of the electron. However, one must, while preserving the previous value 1 h for the impulse moment of π the electron, ascribe a magnetic moment that is twice as large m = m, so it equals a complete Bohr magneton. The ratio of the two moments viz., the magnetic an the mechanical ones must then equal: e κ = cm, () by assumption, in contraiction to the previous assumption that necessarily arises for the explanation of the multiplet structure. 0. The Thomas theory. Thomas 1 ) sought to give a resolution of the contraiction on the grouns of the following relativistic argument: One consiers the electron at two successive points t = t an t = t + t. Let the corresponing rest systems that result from S by a Lorentz transformation without rotation be S an S. It may now be easily shown that S can be obtaine from S irectly by an infinitesimal Lorentz transformation that correspons to an infinitesimal relative velocity v = ɺv t (ɺv = acceleration), an likewise an infinitesimal rotation of the coorinate axes that is given (approximately) by the vector: 1 w = [ ɺvv ] t. (3) c Now, accoring to Thomas, the temporal change in the impulse moment of the electron m / k must be etermine by the usual ifferential equation: 1 ) Nature, April 10, 196, pp. 514. The manuscript of this paper was corially mae available to me by Dr. W. Pauli at the en of February, an this gave rise to my own paper.
Frenkel The electroynamics of rotating electrons. 3 where: m = [m H ], t κ m m w m = t κ t κ t κ (4a) (4b) means the rate of change of the vector m / κ relative to a coorinate system that goes from S to S in a time t by means of a translational acceleration ɺv an a rotational velocity w / t. m0 If one replaces κ with the value () an observe that (in the first approximation) v e ɺ = E then, from (1), one has: an it results from (4a) an (4b) that: w m 1 v t κ = + E c m = 1 [m H ], t m 1 = [m H ]. (5) κ In this way, we obtain an equation in the usual form (4) when we introuce the apparent moment: m = 1 m in place of the actual magnetic moment m, an then replace the ratio κ with: κ = κ. Thus, (5) becomes: m = [m H ], (5a) t κ in agreement with (4). We then see that from the stanpoint of the usual theory the temporal change in the impulse moment of the electron correspons to a rotational force whose moment f = [m H ] equals one-half the actual rotational moment f = [m H ]. Corresponingly, for the consieration of the change in energy that is inuce by this rotational work, one must compute with an apparent magnetic energy U = 1 (m H ) = (m H ).
Frenkel The electroynamics of rotating electrons. 4 Let it be remarke that the result above relates to the case in which no true magnetic fiel is present; i.e., when the magnetic fiel strength vanishes in the nuclear coorinate system S. If this fiel strength H is non-zero then one must replace equation (5) with the following general equation: or: t m 1 = [m H ] + [m H], (5) κ m = [m H ] + [m H]. t κ The total magnetic energy is therefore expresse by the sum: U = 1 (m H ) = (m H). (6a) The following objection can be raise against the Thomas argument. First, one is ealing with the impulse moment an the magnetic moment of the electron as invariant quantities, which is certainly incorrect, since three-imensional vectors must transform in a certain way uner a Lorentz transformation. Secon, this theory relates exclusively to the rotational motion of the electrons. It shoul then follow that the complete magnetic moment, not one-half of it, is also appropriate to the translational motion in the case of H = 0, from the usual expression for the riving force (m gra) H. There is no proof that in the absence of an external magnetic fiel the precession velocity of the electron axes an that of the path plane are equal such that the resulting impulse moment woul remain constant in quantity an irection. In the sequel, we woul like exhibit the precise equations of motion for the rotating electron by a consequently four-imensional representation (in the sense of special relativity, just like Thomas) of the usual three-imensional equations. Thus, one obtains a complete resolution of the contraiction that was suggeste in 1 between the explanation for the multiplet structure an the Zeeman effect. In particular, it yiels that the Thomas equation (5) etermines, not the actual, but the mean, secular variation of the magnetic moment; i.e., it is only correct when one replaces / t m an H with the corresponing mean values 1 ). 3. The moment tensor. We will ignore any sort of consierations regaring the structure of the electron from the outset an simply treat it as a point whose properties are characterize by certain scalar, vector, an tensor quantities. We especially regar its magnetic properties in such a way that the given of the threeimensional vector of the magnetic moment m is funamentally insufficient for its 1 ) From a written communication of Pauli that I receive in connection with my paper, Thomas has evelope the same theory as the one presente below, inepenently of myself. [Rem. by the eitor]
Frenkel The electroynamics of rotating electrons. 5 complete characterization, since a three-imensional vector must only be consiere to be the spatial part (i.e., projection) of a four-imensional vector (viz., a four-vector) or an anti-symmetric tensor (viz., a six-vector). As is known, the magnetic fiel strength H represents the spatial part of the electromagnetic fiel tensor F = F (, = 1,, 3, 4), whose temporal part etermines the electric fiel strength E accoring to the schema: F3 F31 F1 F14 F4 F34. (I) H1 H H3 ie1 ie ie3 Corresponingly, we woul like to efine the magnetic moment of the electron m as the spatial part of an anti-symmetric tensor = by means of the schema: 3 31 1 14 4 34, (II) m1 m m3 + ip1 + ip + ip3 where p 1, p, p 3 are the spatial components of a three-imensional vector p that is analogous to the electric moment of a ipole 1 ). We woul like to etermine this vector from the conition that it shoul vanish (p = 0) in the coorinate system S in which the electron is instantaneously at rest. It then follows, in an arbitrary coorinate system S relative to which the electron has the translational velocity v, from the known transformation formulas for the quantities (II) an (I), that: v p = m c. (7) One can also erive this result inepenently of the formulas above in the following way ): Let x be the coorinates of the electron an the time multiplie by ic (ict = x 4 ) relative to the system S. We construct the four-imensional vector xɺ (the summation sign for equal inex pairs will always omitte in what follows) from an xɺ = x / t, where = t 1 v / c means the proper time of the electron. The components of this vector xɺ vanish in the rest system S, so one has x ɺ 1 = x ɺ = x ɺ 3 = 0, an from our assumption, 14 = 4 = 34 = 0. However, it follows from this that for any other coorinate system S, the equations: xɺ = 0 (7a) 1 ) This analogy will be clarifie later on. ) From a remark by W. Pauli.
Frenkel The electroynamics of rotating electrons. 6 are fulfille, which express the vanishing of the vector above. If one substitutes the corresponing three-imensional expressions for an xɺ then for all = 1,, 3 one gets the spatial components of the vector: while for = 4: c 1 v / c v m c p, i xɺ = ( vp ). 1 v / c The vanishing of the secon expression follows immeiately from the vanishing of the first one i.e., equation (7). As is known, one may efine the following two invariant scalar quantities by means of the tensor components = : an m p = 1 (m p) = i( 3 14 + 31 4 + 1 34 ). Thus, ue to (7) (i.e., since p = 0), one has: an (m p) = m p = 0 m p = m v m c = m. (8) The last equation etermines the inepenence of the magnetic moment of the electron from its translational velocity v. One can escribe it in the form: m = 1 v / c, where v means the component of v that is perpenicular to m; m = is the magnitue of the magnetic moment in the rest system. 4. The temporal variation of the moment tensor. We now introuce the fourimensional quantities that correspon to the magnetic energy (m H) = m H, an the magnetic rotational moment [m H]; i.e., the vector or anti-symmetric tensor with the components m H m H. The four-imensional extension of the energy function is obviously the scalar:
Frenkel The electroynamics of rotating electrons. 7 U = 1 F = (m H) (p E). (9) The corresponing extension for the rotational moment is given, as one easily recognizes, by the anti-symmetric four-imensional tensor (i.e., the six-vector): with the spatial part: an the temporal part: f = F F (10) (f 3, f 31, f 1 ) = [m H] + [p E] (10a) i(f 14, f 4, f 34 ) = [m H] [p E]. (10b) We efine the impulse moment of the electron to be the spatial part of the tensor: 1 κ with κ = e / cm 0. The simplest four-imensional extension of the ifferential equation (4) for the temporal variation of woul then rea: i.e.: an ɺ κ ɺm = [m H] + [p E] κ ɺp = [p H] [m E], κ = f ; (11) (11a) (11b) where the ot means ifferentiation with respect to proper time. Equations (11a) an (11b) can be simultaneously satisfie only in the case where the vectors m an v are (a priori) inepenent of each other. However, the relation (7) must, in fact, exist between them, which means that equations (11a), (11b) are incompatible. Now, it is easy to moify the general equation (11) in such a way that the conition (7a) is fulfille. To that en, we introuce an initially ineterminate four-imensional vector a an efine the invariant scalar: a xɺ = 1 ( a xɺ a xɺ ), (1) which vanishes, from (7a). We a this scalar to the energy function U; i.e., replace the former with: U = 1 ( F + a xɺ a xɺ ) = 1 F. (1a) We corresponingly replace the tensor f with:
Frenkel The electroynamics of rotating electrons. 8 i.e.: = F F, (1b) f f = f + a ( xɺ xɺ ), (1c) an the equation of motion (11) with: or, when written out in full: ɺ κ = f, (13) ɺ κ = F F + a ( xɺ xɺ ). (13a) We now etermine the vector a in such a way that this equation is in harmony with the relation (7a). Moreover, it inee follows from (13a), with consieration for (7a) an the ientity relation: xɺ xɺ = c, that: ɺ xɺ = 1 ɺɺ x = F xɺ a xɺ xɺ = ( F xɺ + a c ), κ κ or ɺɺ x + F xɺ + a c = 0. κ One thus infers that: a = 1 ( κ ) F x ɺ ɺɺ x. (14) κc Inepenently of this expression for a, one gets from (13a), with consieration of (7a): 1 κ ɺ ɺ = F F = F = 0 (ue to the anti-symmetric character of F ); i.e.: or 1 = 0, = m p = = const. (15) This formula shows that the magnetic moment of the electron (as assesse in a rest system ) can, in fact, be quantize. If its magnitue were not constant then one coul not speak of it being quantize.
Frenkel The electroynamics of rotating electrons. 9 As is known, the equations of motion of a non-magnetic electron rea: or, with e m c = κ: 0 mɺɺ 0x = e F c xɺ, ɺɺ x = κ F xɺ. (15a) If one neglects the force that arises from the magnetic moment in comparison to the v Lorentz force e E + H c, which correspons to the four-vector e F xɺ, one will c have, from (15) an (15a): a 0. (15b) In this approximation i.e., upon neglecting the perturbation to the translational motion of the electron that is implie by the magnetic force one can thus etermine its rotational motion i.e., the temporal variation of the vector m by way of the simple equations (11) or (11a). v If one substitutes p = m c in (11), using (7), then one has: ɺm [m H] + κ v m c E. (16) We now consier the case in which the electron moves aroun the nucleus in a weak external magnetic fiel H. In a still larger egree of approximation (viz., by neglecting the terms that are quaratic in 1/c), one can then set: m0 v E ~. e t (16a) With that, the secon term on the right-han sie of (16) assumes the form: m0 v [ ] ec vm t. We woul now like to compute the mean value of this expression for the unperturbe motion. One has (for the unperturbe motion!): [ [ ] ] x vm v = [ ] v v vm + x m x v = 0.
Frenkel The electroynamics of rotating electrons. 10 One further has the ientity: [ ] v v v + + vm x m x v v x m = 0. From this, it follows that: [ ] v vm x = 1 v m v x, or, from (16a) an (1): v m E 1 [ [ ]] c c m Ev = 1 [ mh ]. (16b) The secular variation of the vector m is then etermine in the approximation above from the equation: 1 m [m H] + 1 κ t [ mh ]. (17) This is the correcte Thomas equation (6). 5. Derivation of the equations of motion from Hamilton s principle. We will now carry out a more rigorous erivation of the ifferential equation (13) for the rotational motion of the electron on the basis of Hamilton s principle. With that, we will, at the same time, obtain the precise ifferential equation for the translational motion. We thus set, as usual: δ L = 0, (18) with the supplementary conitions: xɺ = c, xɺ = 0. (18a) (18b) We then write the Lagrangian function in the form: e 1 L = ϕ xɺ + T + F, (19) c where T * means the kinetic energy of the rotational motion. We consier this energy, in the context of the usual three-imensional mechanics, as a function of the angular velocity, which we will characterize by the anti-symmetric tensor ω = ω. We then set, by efinition:
Frenkel The electroynamics of rotating electrons. 11 δt * = κ δω. (19a) In orer to etermine the variation of, we next observe the corresponing operation in orinary mechanics. The work one by the magnetic torque [m H] for a virtual infinitesimal rotation δw is equal to the inner prouct (δw [m H]). On the other han, it must be equal to the increase in magnetic energy δ( m H) = (δm, H). One then has (δm, H) = (δw, [m H]) or (δm, H) = ([δw, m] H), an as a result: δm = [δw, m]. The corresponing four-imensional variational formula must be erive in the same way that formula (10) is erive from the three-imensional expression for the rotational moment [m H]. If one then introuces the four-imensional anti-symmetric rotation tensor δω, whose spatial part is equal to the vector δw, then one has: δ = δω δω. (13b) It is self-explanatory that the quantities δω (like the δw) o not represent exact ifferentials i.e., there is no angle coorinate Ω that correspons to the coorinates x (so one has an anholonomic system). Nevertheless, along with the relations: δɺ x = δ x, (0) one must obviously also have the corresponing commutation relations for δω an Ω = ω ; i.e.: By means of the formulas above an the relations: δω = δ Ω. (0a) we get: δl = δϕ = δf = ϕ δ x F δ x, ϕɺ =, Fɺ = ϕ xɺ, F x ɺ, e ϕ e ϕ e ɺ xɺ δ x xɺ δ x + ϕ δ x δω c c c κ
Frenkel The electroynamics of rotating electrons. 1 or since: one has: 1 F 1 + δ Ω + δ x + F ( δ Ω δ Ω ), κ ϕ ϕ = F, e 1 F δl = F x ɺ + δ x c + 1 ɺ e + F F δω + ϕδ x + δω. κ c κ Likewise, from (18a) an (18b), with the aition of the unetermine Lagrange multipliers λ an a ( = 1,, 3, 4): an a δ ( xɺ ) = λxɺ δ xɺ = δ x ( λxɺ ) + ( λxɺ δ x ) = 0 1 ( a δ x ) δ x ( a ) + δ Ω a ( xɺ xɺ ) = 0. With the usual assumption that the variations δx, δω vanish at the bounaries of the integral (18), it then follows from (18), (18a), an (18b) (by aing the above expressions an setting the coefficients of δx an δω equal to zero) that: an ( λ xɺ + a ) = e 1 F F x ɺ + (1) c 1 κ ɺ = F F + a ( xɺ xɺ ). The latter equation agrees with (13a); the former one is the generalization of the usual equation of motion (15a) for a non-magnetic electron. We corresponingly set: λ = m 0 + λ, (1a) where λ means an aitional term that is inepenent of the magnetic moment of the electron. After performing the ifferentiation on the left-han sie of (1), we get, from (15): λ ɺɺ x + ɺ λ xɺ + aɺ + ɺ a = κ m 0 c a + 1 F.
Frenkel The electroynamics of rotating electrons. 13 From this, it follows by multiplication by xɺ, ue to the relations an a xɺ = 0, that: or: c ɺ λ + ɺ a x ɺ = 1 F c λ ɺ = From (1a), (1b), an (13), we have: x ɺ = 1 F, 1 1 F ɺ ( F + a xɺ a xɺ ). 1 ɺ ( F + a xɺ a xɺ ) = 1 ɺ F = κ ( F F ) F κ = ( F F F F ) = κ F F = 0, ɺ = c, xɺɺɺ x = 0 x ue to the anti-symmetric character of the tensor. As a consequence, one has: 1 λ = F. (1b) c The increase in the mass m 0 is then equal to the relativistic magnetic energy of the electrons (relative to the nucleus an other particles that generate the fiel F ), ivie by the square of the velocity of light. One can interpret the expression a in (1) as the -component of the aitional impulse that originates in the absolute energy of the electron; i.e., the kinetic energy of its rotation. By substituting (1a) in (1), this yiels, ue to (15): ( λ xɺ + a ) = c m 0 κ a + 1 F. () One can use this equation only for the approximate etermination of a. Moreover, when one neglects the left-han sie of () (since c m 0 κ = e c), one inee has: a = 1 F ec. (a) 6. The translational motion of the rotating electron in an atom. From equation (1), it follows that:
Frenkel The electroynamics of rotating electrons. 14 or x ( λ x a ) x ( λ x a ) ɺ + ɺ + = 1 Fρσ Fρσ ρσ x x, { λ ( x xɺ x xɺ ) + a ( x x )} = 1 Fρσ F ρσ ρσ x x a ( xɺ xɺ ). (3) This equation can be regare as the generalization of the law of areas ; i.e., the usual formula for the rate of change of the orinary impulse moment of the translational r motion m 0 r. This impulse moment will then be replace by the anti-symmetric tensor: I = l ( x xɺ x xɺ ) + a ( x x ), (3a) whose spatial part agrees with m 0 [ r r ɺ] in the first approximation. Let it be further remarke that the secon term on the right-han sie of (3) is equal an opposite to the corresponing aitional term in formula (13a) for the rate of change of the impulse moment of the rotational motion. If one sets: κ = i then, from (13a) an (3), the sum of both moments becomes: (i + I ) = F F + where U means the relative energy: x U = 1 F. U x U, (3b) We now consier the case in which the electron moves in a raially-symmetric electric fiel E = ψ(r) r in the absence of an (external) magnetic fiel. In this case, one has U = (p E) = ψ(p r), an as a result, for a, b = 1,, 3: x U x U = ψ(x p x p ) = E p E p. The resulting angular momentum, which correspons to the spatial part of the tensor on the right-han sie of (3b), will then be equal to zero ([p E] + [E p]) = 0).
Frenkel The electroynamics of rotating electrons. 15 It follows from this that the resulting impulse moment of the electron in the case consiere must remain constant in magnitue an irection. As we alreay saw above [equation (15)], the magnitue of the tensor, an consequently, also i, is constant in time. In the first approximation (viz., by neglecting the terms that are quaratic in 1/e) one can, as a result, regar the magnitue of the impulse moment of the rotational motion i = κ m as constant in time. If one enotes the impulse moment of the translational motion (i.e., the spatial part of the tensor I ) by F then it follows, ue to the conition i + F = const., that the magnitue of F also remains constant, an that both vectors i an F precess aroun their resultants with the same angular velocity. This result is very essential for atomic mechanics, since otherwise one coul not quantize the impulse moment of an atom. If one replaces the impulse moments i an F with the corresponing magnetic moments m = κ i an M = κ F then one sees that the sum m + M = κ (i + F) + κ i oes not represent a constant vector. Inee, the magnitue of this vector remains constant, but its irection must precess aroun the atomic axis with the aforementione angular velocity 1 ). The angular velocity may not be simply etermine; however, formula (17) shows that its mean value agrees with the orinary Larmor velocity of the electron path in an external magnetic fiel H. 7. The electromagnetic fiel of a rotating electron. If one consiers the electron as a point charge an ignores its magnetic fiel then one can represent its electromagnetic fiel by the formula: x k x k ϕ = π i = τ S π i (4) S for the components of the four-potential. The integration is thus taken along a close curve in the complex τ -plane. [τ is the proper time of the electron, S = ( x x ) is its four-imensional istance from the origin x ; k = e] ). If this curve encloses only one pole of the integran, namely, the pole that correspons to the real roots of the equation R c(t t ) = 0: [R = 3 ( x x ) ], = 1 then one gets the well-known Liénar-Wiechert formula for the retare potential of a moving point charge by fining the resiue: 1 ) Such that no secular variation of m + M appears. ) Cf., my paper Zur elektroynamik punktförmiger Elektronen, ZS. f. Phys. 3, 518, 195.
Frenkel The electroynamics of rotating electrons. 16 x k 1 ϕ = k ( S ) r t = t c. (4a) We woul now like to etermine the aitional electric fiel in a completely analogous way that is inuce by the rotation of the electron; i.e., its magnetic moment. The corresponing part of the four-potential ψ must obviously be representable in terms of the moment tensor an the four-vector ( x x ) by means of a complex integration of the same type as (4). Furthermore, since ψ must be a linear vector function of following Ansatz:, we come to the Q ψ = ( x x ) f ( S) π i, (5) where Q is a proportionality coefficient an f(s) means an initially unknown function of S. For the etermination of this function, we substitute (5) into the ifferential equation: This yiels: an furthermore: f ( x x ) f ( x x ) λ ψ = 0. 4 = 1 f = f + ( x x ) = f f + ( x x ),, f x = f S x x, S f x = f ( x x ) f S ( x x ) +, 3 S S S S an it follows that: ψ = i.e.: 4 = 1 Q 5 ( ) f f x x π i + S S S = 0; f S 5 f + = 0. S S One will then have f = 1 / S 4, an from (5):
Frenkel The electroynamics of rotating electrons. 17 Q ( x x ) ψ = 4 π i. (5a) S As one easily sees, the supplementary conition: ϕ = 0 4 = 1 is fulfille ue to the fact that the integration path is close. For the etermination of the coefficients Q, we consier the simplest case of an electron at rest with a moment m (p = 0) that is constant in magnitue an irection. In this case, when one replaces the integration path with the imaginary axis 1 ), one gets: i.e.: t = t + i Q t ψ = + ( x x ) πi [ R c ( t t) ] t = t i + Q ( x 4 x4 ) = ( x x ) π c ; [ R + ( x x ) ] Q ( x x ) ψ =. 3 4cR 4 4 If one thinks of the vector R as pointing from the electron to the origin (R = x x ) an observes that ϕ 1, ϕ, ϕ 3 mean the components of the vector potential A then one has: Q [ R m] A = 3 4c R an ϕ 4 = i ϕ = 0. The formula above for A agrees with the usual expression for the vector potential of an elementary current with the moment m when one sets: Q = 4c. (5b) In the general case of an arbitrarily moving electron, one can calculate the integral (5a) by fining the resiues. We therefore assume that we are ealing with the retare potential; i.e., the resiue relative to the real pole: R 0 c(t t 0 ) = 0. 1 ) loc. cit., pp. 53.
Frenkel The electroynamics of rotating electrons. 18 If one introuces the orinary (complex) time t as the inepenent variable, in place of the proper time τ (where = an it follows that as t t 0 : t 1 v / c ), then this yiels: S 4 = [R c (t t ) ] = [R + c(t t )] [R c (t t )], i.e.; R c(t t ) = [ R c ( t t )] ( t t 0) t 1 ( ), c v = c R t t 0 S 4 = c [R + c(t t )] v 1 R ( t t 0), c where v R means the projection of the velocity of the electron in the R 0 -irection at the time point t = t 0. From (5a), we then get: Q 1 F ( t ) ψ = t ( v πi, t t 0) R c 1 c with the abbreviation: F a (t ) = ( x x ) 1 v / c [ R + c( t t )] Since the function F(t ) remains non-zero for t = t 0, it is known that one has:. 1 F ( t ) πi ( t t ) t = 0 F ( t ) t t = t0 It then results from (5b) that: ψ = 4 ( x x ) 1 v / c v t [ R c( t t )] R + c 1 c t = t0. By performing the ifferentiation, this yiels, ue to the conition means of the relation c ( t t 0) = R 0 : x t = 0, by
Frenkel The electroynamics of rotating electrons. 19 1 x x v ( x x ) R ψ = 1 + 3, (6) v R t c R R c 1 c where the inex 0 is omitte, an we have set: = 1 v / c, (6a) to abbreviate. In the case of an electron at rest with time-varying components of the moment tensor, since 4 = i p = 0, (6) reuces to: A [ ɺmR ] [ mr] = 3 cr + R ϕ = 0. (7) Let it be remarke that, from (15), the magnitue of the magnetic moment m must then remain constant 1 ). Formula (7) above can be applie to the case of an electron that oes not move to rapily as the zeroth-orer approximation. We woul not like to go into the calculation of the electric an magnetic fiel strengths, which follows with no ifficulty from the usual formulas E = 1 A gra ϕ, H = rot A. c t In conclusion, we woul now like to prove the following fact: If the electrons are ascribe a magnetic moment with the magnitue of a Bohr magneton then for istances < 10 11 cm. their magnetic interaction, which is known to be inversely proportional to the fourth power of istance, shoul outweigh their electric (Coulomb) repulsion. This magnetic interaction can alreay make known the value of the screening constant for the inner electrons for heavy atoms. In the atomic nucleus, however, it must be a million times larger than the electrostatic forces. If one ascribes an impulse moment of the same magnitue as the electron s to the protons, an corresponingly a magnetic moment that is 000 times smaller, then their magnetic interaction with each other an with the electrons shoul also strongly outweigh the electrostatic interaction. It thus seems justifie to assert that the structure of the atomic nucleus is practically inepenent of the electric charges of the electrons an protons, an must be primarily inuce by its magnetostatic interactions (in conjunction with the usual quantum conitions). This gives that, e.g., an electron an a proton at a istance of 5 10 13 cm can remain in static 1 ) If the moment tensor were not subject to the conition x = 0 then the following two terms woul enter into the expression above for A: an woul be, moreover: ɺp p +, cr R ( Rpɺ ) ( pr ) ϕ = + 3. cr R
Frenkel The electroynamics of rotating electrons. 0 equilibrium. However, this equilibrium woul be unstable relative to the orientation of the magnetic axes of the two particles. If one thus assumes that the electron orbits aroun the nucleus then this woul yiel, in aition to the orinary first-quantize path with a raius 0.55 10 8 cm, a secon first-quantize path of raius 3 10 14 cm that is require by the magnetic attraction, where the electric attraction woul seem to be a weak perturbing force. The quantities above work quite well for the measurements of the simplest nucleus. However, one may not conceal the complication here that the electron mass, ue to its large velocity, increases to perhaps a thousan times the orinary value, which will be, in part, compensate by the ecrease in the mutual potential energy. I hope to treat this question more thoroughly in a later communication. In conclusion, I woul like to express my eepest thanks to Dr. W. Pauli for proviing the impetus for this paper an many worthwhile consultations. I must further warmly thank Prof. P. Langevin an my frien G. Krutkow for some suggestions (an also the latter for reviewing the manuscript). Hamburg-Nizza, April 196.