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Spectral emission of moving atom exhibits always a redshift.

A renewed analysis of the H. E. Ives and G. R. Stilwell's experiment on moving hydrogen canal rays (J. Opt. Soc. Am., 1938, v. 28, 215) concludes that the spectral emission of a moving atom exhibits always a redshift which informs not the direction of the atom's motion. The conclusion is also evident from a simple energy relation: atomic spectral radiation is emitted as an orbiting electron consumes a portion of its internal energy on transiting to a lower-energy state which however has in a moving atom an additional energy gain; this results in a redshift in the emission frequency. Based on auxiliary experimental information and a scheme for de Broglie particle formation, we give a vigorous elucidation of the mechanism for deceleration radiation of atomic electron; the corresponding prediction of the redshift is in complete agreement with the Ives and Stilwell's experimental formula.

1 Introduction

Charged de Broglie particles such as the electron and the proton can be decelerated by emitting electromagnetic radiation. This occurs in all different kinds of processes, including atomic spectral emission produced in laboratory [1, 2] or from celestial processes[3], and charged particle synchrotron radiation [4, 5]. The electromagnetic radiation emission from sources of this type is in common converted from a portion of the internal energy or the mass of a de Broglie particle involved, which often involves a final state in motion, hence moving source. The associated source-motion effect has except for admitting a relativistic effect connected to high source velocity thus far been taken as no different from the ordinary Doppler effect that consists in a red- or blue-shift depending on the source is moving away or toward the observer. The ordinary Doppler effects are directly observable with moving sources of a "conventional type", like an external- field-driven oscillating electron, an automobile horn, and others, that are externally driven into oscillation which does not add directly to the mass of the source. In this paper we first (Sec. 2) examine the property, prominently an invariable redshift, of moving atom radiation as informed by the hydrogen canal ray experiment [1] of Ives and Stilwell performed at the Bell Labs in 1938 for a thorougher investigation of the associated anomalous Doppler effect then known. Combining with auxiliary experimental information and a scheme for de Broglie particle formation[6], we then elucidate (Secs. 3-5) the mechanism for spectral emission of moving atom, or in essence the underlying (relative) deceleration radiation of moving de Broglie electron, and predict Ives and Stilwell's experimental formula for redshift.

2 Indication by Ives-Stilwell's experiment on fast moving hydrogen atoms

In their experiment on fast moving hydrogen canal ray spectral emission[1], Ives and Stilwell let positively charged hydrogen ions [H.sup.+.sub.i] of mass [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and charge [q.sub.i] (i=2,3) be accelerated into a canal ray of high velocity, v, across accurately controlled electric potential V correlated with v through the work-energy relation q V = 1/2 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [v.sup.2]; or

v/c = A[square root of (V)] (1)

with c the speed of light, and A = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. For V ~ 6700-20755 volts, v ~ [10.sup.6] m/s as from (1). By neutralization and dissociation the ions are at exit converted to excited atoms that are unstable and will transit to ground state by emitting Balmer spectral lines. The wavelength, [[lambda].sub.r], of the emitted H[beta] line is then measured using diffraction grating (Fig. 1a) as a function of V. For a finite v, the spectral line produces a first-diffraction peak at P(v), at distance y(v) = PO from the center O; for a hydrogen at rest, v = 0, the line has a wavelength [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = 4861 angst. and produces a first peak at [P.sub.0], [y.sub.0] = [P.sub.0]O. These have the geometric relations: [[lambda].sub.r] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and

[FIGURE 1 OMITTED]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

[DELTA][[lambda].sub.r] being the mean displacement of the Doppler lines at a given v. The measured spectrogram, Fig. 1b, informs y - [y.sub.0] = B'[square root of (V)] with B' a constant; this combining with (2) is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If assuming

[DELTA][[lambda].sub.r]/[[lambda].sub.ro] = + v/c, (4)

then this and (3) give v/c B[square root of (V). But v/c and [square root of (V)] must satisfy (1); thus B [equivalent to] A; that is (3) writes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3')

In [1], the two variables [DELTA][[lambda].sub.r]/[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [square root of (V) are separately measured and thus given an experimental relation, shown in Fig. 10, of [1, p. 222], which agrees completely with (3'); accordingly (4) is directly confirmed. Furthermore there is a shift of center of gravity of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] from [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; or [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. With this and (4) in the first equation of (3) or similarly of (2), one gets:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

(5) gives [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] > v/c - 1/2 [(v/c).sup.2][greater than or equal to] 0; or, [[lambda].sub.r] is always elongated for |v|>0. Furthermore, (4)-(5) are obtained in [1] for both the cases where source and observer move toward and away from each other: The source velocity v is in the fixed +x-direction; waves emitted parallel with v (Fig. 2) strike on the diffraction grating D (observer 1) directly (Fig. 2b), and waves antiparallel with v (Fig. 2c) strike on mirror M (observer 2) first and are then reflected to D. That is, (5) is regardless of the direction of the vector c. Therefore from Ives and Stilwell's experiment we conclude:

[FIGURE 2 OMITTED]
 The wavelength of spectral line emitted from an atom
 in motion is always longer, or red-shifted, than from
 one at rest, irrespective if the atom is moving away
 or toward the observer; the faster the atom moves, the
 longer wavelength its spectral line is shifted to.


This apparently contrasts with the conventional Doppler effect where wavelengths will be [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1 - v/c) and [[lambda].sub.r] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1 + v/c) and show a blue or red shift according to if the source is moving toward or away from the observer.

3 Emission frequency of a moving atom

If a H atom is at rest in the vacuum, its electron, of charge -e in circular motion at velocity [u.sub.n+1] about the atomic nucleus in an excited n+1th orbit, has from quantum-mechanical solution (and also solution based on the unification scheme [6]) an eigen energy [[epsilon].sub.au.n+1] = - [[??].sup.2] / [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where n=1, 2, ... and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = [[gamma].sub.0] [M.sub.e], [[gamma].sub.0] = 1/[[1-[([u.sub.n+1]/c).sup.2].sup.-1/2] with [u.sub.n] being high (~[10.sup.6] m/s), Me the electron rest mass, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Bohr's radius (should already contain 1/[gamma]0, see below). If now the electron transits to an unoccupied nth orbit, the atom lowers its energy to [[epsilon].sub.u.n] and emits an electromagnetic wave of frequency

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

accordingly [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = 2 [pi]/[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If now the atom is moving at a velocity v in + x-direction, [(v/c).sup.2] >> 0, then in the motion direction, its orbital radius is Lorentz contracted to [a.sub.B] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] /[gamma], and its mass augmented according to Einstein to [m.sub.e] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = [[gamma][gamma].sub.0][M.sub.e] (see also the classical-mechanics solutions [6]), where [gamma] = 1 / [square root of (1-[(v/c).sup.2]]. With [a.sub.B] and [m.sub.e] for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in (6), we have [v.sub.r] = = [[epsilon].sub.u.n+1(v)-[[epsilon].sub.u.n](v) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; including in this an additional term [[delta]v.sub.r] which we will justify below to result because of an energy gain of the moving source, the spectral frequency for the n + 1 [right arrow] n transition for the moving atom then writes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

4 Atomic spectral emission scheme

We now inspect how an electron transits, from an initial n + 1th to final nth orbit in an atom moving in general, here at velocity v in + x-direction. To the initial-state electron, with a velocity [u.sub.n+1] if v = 0, the finite v of the traveling atom will at each point on the orbit project a component v cos [theta] onto [u.sub.n+1]([theta]), with [theta] in (0, 2[pi]); the average is [[??].sub.n+1] = = [[integral].sup.2[pi].sub.[theta]=0][[u.sub.n+1]+v cos [theta]]d[theta] = [u.sub.n+1]. That is, [[??].sub.n]+1 and any its derivative dynamic quantities of the stationary-state orbiting electron are not affected by v except through the second order factor [gamma](v). The situation however differs during the n+1 [right arrow] n transition which distinct features may be induced as follows:

(i) The transition ought realistically be a mechanical process in which, in each sampling, the electron comes off orbit n + 1 at a single definite location, e. g. A in Fig. 2a. That where A is located on the orbit in any sampling, is a statistic event.

(ii) The spectral radiation is a single monochromatic electromagnetic wave emitted in forward direction of the orbiting electron at the point (A) it comes off orbit n + 1, as based on observations for decelerating electron radiation in a storage ring in synchrotron experiments [4], which is no different from an orbiting atomic electron except for its macroscopic orbital size.

(iii) It follows from (i)-(ii) combined with momentum conservation condition that the transition electron coming off at A, will migrate across shortest-distance AB, perpendicular to [u.sub.n+1], to orbit n, at B if the atom is at rest, or at B' if the atom is moving at velocity v in x-direction, given after vector addition.

(iv) A stationary-state orbiting electron on orbit n* (= n+1 or n), [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], is [6] a (single) beat or de Broglie phase wave convoluted from the opposite-traveling component total waves {[[phi].sup.j.sub.kn]} generated by an oscillatory massless (vaculeon) charge - e, of wavevectors [k.sup.[dagger].sub.n*], which being Doppler shifted for the source moving at velocity [u.sup.j.sub.n+1]. An n+1 [right arrow] n transition emits the difference between the two single waves, [[psi].sub.kdn+1] and [[psi]k.sub.dn] - the emitted radiation is naturally also a single wave. And,

(v) The component total waves making up the electron beat wave at A is generated by the source in a brief time [delta]t when at A, a wave frequency ~V = 511 ke V/h [??] [10.sup.20] [s.sup.-1]; so the the time for detaching the entire radiation wave trains from the source is estimated [delta]t ~ 1/V = 8 x [10.sup.-21] s. In contrast, the orbiting period of the electron is Tod.n+1=1/id.n+1= =[([n.sub.1]).sup.2]1.5 x [10.sup.-16] s. So in time [delta]t << [T.sub.d.n+1], the electron is essentially not moved along orbit n+1 as well as path AB or AB'; hence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([??] [u.sub.n+1]) (thus c) and v are at fixed angle [theta]. Specifically if the electron comes off at [A.sub.1] and [A.sub.2] as in Fig. 2b and c, respectively, we have the approaching and receding source and observer

c \\ v and - c \\ v. (8)

The wave and dynamic variables for the nonstationary transition process would not be a simple difference between solutions of the stationary states. However, we can try to represent the process effectively using an apparent source such that:

(v.1) the total wave detached from the apparent source gives the same observed radiation as due to the actual source; and

(v.2) the apparent source in transition has the same motion as the (actual source of the) transition electron, that is, translating at the velocity v (cf. item iv) in + x-direction here.

5 A theoretical formula for the redshift

In fulfilling (v.1), the apparent source ought to be an oscillatory charge (q) executing in stationary state circular motion at velocity [u.sub.a] on orbit n+1 (insets in Fig. 2). Let first the orbit n + 1 be at rest, v =0, and so must be the apparent source as by (v.2). The apparent source generates two identical monochromatic electromagnetic waves traveling oppositely along orbit n + 1, of wavevectors [k.sup.[dagger].sub.a0] = [k.sub.a0], which superpose into a single electromagnetic wave [[psi]k.sub.a0]. On transition, the source emits the entire [[psi]k.sub.a0] in the direction parallel with [u.sub.a([theta]), by simply detaching it; thus [k.sub.a0] [equivalent to] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = 2[pi]/[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let now orbit n + 1 be in motion at velocity v in + x-direction, and so must be the apparent source. Let the source comes off orbit n + 1 at point [A.sub.1] (Fig. 2b). In a brief time at before this, the apparent source was essentially at A1 and generating two waves [[phi].sup.[dagger]] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] parallel and antiparallel with [u.sub.a], thus v; their wavelengths were owing to the source motion of v Doppler shifted, to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] 1 + v/c), and wavevectors [k.sup.[dagger].sub.a] = 2[pi]/[[lambda].sup.[dagger].sub.a] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with the Doppler shifts

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

The two waves superpose to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], being according to [6] now a single beat, or de Broglie phase wave of the moving apparent source. On transition the source detaches the entire single beat wave [[psi]k.sub.a], which is no longer "regulated" by the source and will relax into a pure electromagnetic wave [[psi]k.sub.r], but in conserving momentum, retains in the single direction parallel with [u.sub.a] thus v. Similarly, if the source exits at [A.sub.2] (Fig. 2c), a single electromagnetic wave will be emitted parallel with [u.sub.a]([A.sub.2]), or, - v x [[psi]k.sub.a] has a de Broglie wavevector given[6] by the geometric mean of 9a) and (b):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

We below aim to express the [k.sub.a.d]-effected radiation variables [k.sub.r], [v.sub.r] and [[lambda].sub.r], which being directly observable. Momentum conservation requires |[[??]k.sub.a.d]|=|[??][delta][k.sub.r]|; [k.sub.a.d] is associated with an energy gain of the apparent source, [[epsilon].sub.a.v] = ([[??]ka.d).sup.2]/2[m.sub.e], owing to its motion, and thus an energy deficit in the emitted radiation wave [[psi]k.sub.r],

[delta][[epsilon].sub.r](= [??][delta][k.sub.r]c) = - [[epsilon].sub.a.v], (11)

and accordingly momentum and frequency deficits in the emission

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

With (13) in (7), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

where [gamma] in front of [[delta]v.sub.r] is higher order thus dropped. With (14) we can further compute for the emitted wave:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

The theoretical prediction (16) for [[lambda].sub.r] above is seen to agree exactly with Ives and Stilwell's experimental formula, (5). Notice especially that the prediction gives [[delta]v.sub.r] < 0 and [delta][[lambda].sub.r] > 0 for both c || v and - c || v as follows from (11); that is, they represent always a redshift in the emission spectral line, regardless if the wave is emitted parallel or antiparallel with v.

6 Discussion

From the forgoing analysis of the direct experimental spectral data of Ives and Stilwell on hydrogen canal rays, and with the elucidation of the underlying mechanism, we conclude without ambiguity that, the spectral emission of a moving hydrogen atom exhibits always a redshift compared to that from an atom at rest; the faster the atom moves, the redder-shift it shows. This is not an ordinary Doppler effect associated with a conventional moving source, but rather is an energy deficiency resulting from the de Broglie electron kinetic energy gain in transition to a moving frame, a common feature elucidated in [7] to be exhibited by the deceleration radiation of all de Broglie particles. This redshift does not inform the direction of motion of the source (the atom).

It is on the other hand possible for an atomic spectral emission to exhibit blue shift for other reasons, for example, when the observer is moving toward the source as based on Galilean transformation. The author thanks P.-I. Johansson for his support of the research and the Studsvik Library for helping acquiring needed literature.

References

[1.] Ives H. E. and Stilwell G. R. An experimental study of the rate of a moving atomic clock. J. Opt. Soc. Am., 1938, v. 28, 215-226.

[2.] Kuhn H. G. Atomic spectra. Longmans, London, 1962.

[3.] Freedman W. L. The Hubble constant and the expansion age of the Universe. Phys. Rept., 2000, v. 333, 13-31; arXiv: astro-ph/ 9909076; Riess A. G., The case for an accelerating Universe from supernovae. Pub. of Astronom. Soc. Pacific, 2000, v. 112, 1284-1299.

[4.] Crasemann B. Synchrotron radiation in atomic physics. Can. J. Phys., 1998, v. 76, 251-272.

[5.] Winick H. and Doniach S., editors. Synchrotron radiation research. Plenum Press, New York, 1980.

[6.] Zheng-Johansson J.X. and Johansson P-I., Fwd. Lundin R., Unification of Classical, Quantum and Relativistic Mechanics and of the four forces. Nova Sci. Pub., NY, 2006; Inference of Schrodinger equation from classical-mechanics solution. Quantum Theory and Symmetries IV, ed Dobrev V. K., Heron Press, Sofia, 2006; (with Lundin R.) Cause of gravity. Prediction of gravity between charges in a dielectric medium. ibid.; also arxiv: phyiscs/0411134; arxiv physics/0411245; Bull. Am. Phys. Soc., 2006, Topics in Quantum Foundations, B40; ibid., 2004, in: Charm Quark States, D10; ibid., 2004, C1.026; Origin of mass. Mass and mass-energy equation from classical-mechanics solution. arxiv: physics/0501037; Bull. Am. Phys. Soc., 2005, New Ideas in Particle Theory, Y9; Electromagnetic radiation of a decelerating moving de Broglie particle: always a redshift, ibid., 2005, Intermediate Energy Accelerators, Radiation Sources, and New Acceleration Methods, T13; Unification of Classical and Quantum Mechanics, & the theory of relative motion. Bull. Am. Phys. Soc., 2003, General Physics, G35.01; ibid., 2004, General Theory, Y38.

7. Zheng-Johansson J.X. and Johansson P-I., Fwd. Lundin R. Inference of basic laws of Classical, Quantum and Relativistic Mechanics from first-principles classical-mechanics solutions. Nova Sci. Pub., NY, 2006.

J.X. Zheng-Johansson

Institute of Fundamental Physics Research, 611 93 Nykoping, Sweden E-mail: jxzj@iofpr.org.
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Date:Jul 1, 2006
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