Classical Lifetime of a Bohr Atom

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1 1 Poblem Classical Lifetime of a Boh Atom James D. Olsen and Kik T. McDonald Joseph Heny Laboatoies, Pinceton Univesity, Pinceton, NJ 85 (Mach 7, 5) In the Boh model of the hydogen atom s gound state, the electon moves in a cicula obit of adius a = m aound the poton, which is assumed to be igidly fixed in space. Since the electon is acceleating, a classical analysis suggests that it will continuously adiate enegy, and theefoe the adius of the obit would shink with time. Consideations such as these in 193 by J.J. Thomson, Phil. Mag. 6, 673 (193), 1 led him to note that thee is no adiation if the chage is distibuted in space so as to fom steady cuents. While a spinning shell o ing of chage povides a model fo the magnetic moment of an atom, such chage configuations povide no estoing foce against displacements of the cente of the shell/ing fom the nucleus. The (continuous) chage distibution must extend all the way to the nucleus if thee is to be any possibility ofclassical electostatic stability. A vesion of these insights was incopoated in Thomson s (not entiely self-consistent) model of the atom as a kind of plum pudding whee the nucleus had a continuous, extended chage distibution in which moe pointlike electons wee embedded. In this context, Ruthefod s measuements of α-paticle scatteing, which showed that the nucleus was compact, came as something of a supise, and e-opened the doo to models such as that of Boh in which pointlike electons obited pointlike nuclei, and adiation was suppessed by a quantum ule. Note that the model of the atom that emeged following Schödinge contains some classically ageeable featues if (gound states of) atoms ae not to adiate: The electon in an atom is consideed to have a spatially extended wave function that extends to the oigin. The electic cuent associated with the electon is steady, and hence would not adiate if this wee a classical cuent. a) Assuming that the electon is always in a nealy cicula obit and that the ate of adiation of enegy is sufficiently well appoximated by classical, nonelativistic electodynamics, how long is the fall time of the electon, i.e., the time fo the electon to spial into the oigin? b) The chage distibution of a poton has a adius of about 1 15 m, so the classical calculation would be modified once the adius of the electon s obit is smalle than this. But even befoe this, modifications may be equied due to elativistic effects. Based on the analysis of pat a), at what adius of the electon s obit would its velocity be, say.1c, wheec is the speed of light, such that elativistic coections 1 Seveal subsequent authos have claimed the existence of adiationless obital motion of classcial chages. Howeve, these claims all appea to have defects. See P. Peale, Absence of Radiationless Motions of Relativistically Rigid Classical Electon, Found. Phys. 7, 931 (1977), 1

2 become significant? What faction of the electon s fall time emains accoding to pat a) when the velocity of the electon eaches.1c? c) Do the elativistic coections incease o decease the fall time of the electon? It suffices to detemine the sign of the leading coection as the adial velocity of the adiating electon appoaches the speed of light. A question closely elated to the pesent one is whethe the ate of decay of the obit of a binay pulsa system due to gavitational adiation is inceased o deceased by specialelativistic coections as the obital velocity becomes elativistic. Solution a) The dominant enegy loss is fom electic dipole adiation, which obeys the Lamo fomula (in Gaussian units), dt = P E1 = e a, (1) 3c 3 whee a is the acceleation of the electon. Fo an electon of chage e and (est) mass m in an obit of adius about a fixed nucleus of chage +e, the adial component of the nonelativistic foce law, F = m a, tells us that e = m vθ a m, () in the adiabatic appoximation that the obit emains nealy cicula at all times. In the same appoximation, a θ a, i.e., a a, and hence, 3 dt = e6 3 m c = 3 3 m c 3. (3) whee = e /m c = m is the classical electon adius. The total nonelativistic enegy (kinetic plus potential) is, using eq. (), U = e + 1 m v = e = m c. () Equating the time deivative of eq. () to eq. (3), we have o Hence, dt = ṙm c = 3 ṙ = m c 3, (5) d 3 dt = 3 c. (6) 3 = a 3 ct. (7)

3 The time to fall to the oigin is t fall = a3 c. (8) With = manda = m, t fall = s. This is of the ode of magnitude of the lifetime of an excited hydogen atom, whose gound state, howeve, appeas to have infinite lifetime. b) The velocity v of the electon has components v =ṙ = c, (9) 3 using eq. (6), and v θ = θ e = m = c, (1) accoding to eq. (). The azimuthal velocity is much lage than the adial velocity so long as. Hence, v/c v θ /c equals.1 when /.1, o 1. When = 1 the time t is given by eq. (7) as t = a3 3 c, (11) so that t fall t t fall = 3 a 3 = ( ) (1) Fo completeness, we ecod othe kinematic facts in the adiabatic appoximation. The angula velocity θ follows fom eq. (1) as The second time deivatives ae thus θ = c. (13) 3 = 8 3 ṙc = c, θ = ṙc = 5 c. (1) The components of the acceleation ae a = θ = c c c = θ, (15) a θ = ṙ θ + θ = c a. (16) 3

4 c) We now examine the leading elativistic coections to the nonelativistic analysis of pat a). Fist, we ecall that the lab fame ate of adiation by an acceleated chage obeys the Lamo fomula (1) povided we use the acceleation in the instantaneous est fame athe than in the lab fame. This is tue because both and dt tansfom like the time components of a fou-vecto, so thei atio is invaiant. In the adiabatic appoximation, the acceleation is tansvese to the velocity. That is, v v θ fom eqs. (9) and (1), while a a θ fom eqs. (15) and (16). Theefoe, a = γ a, (17) whee a is the lab-fame acceleation, a is the acceleation in the instantaneous est fame, and γ = 1/ (1 v /c )=1/ 1 β. Equation (17) holds because a = d l /dt,anddl = dl fo motion tansvese to the velocity of the electon, while the time-dilation is dt = dt/γ. Thus, the ate of adiation of enegy by a elativistic obiting electon is dt = P E1 = e a 3c 3 = γ e a 3c 3. (18) [If the acceleation wee paallel to the velocity, a = γ 3 a sincenowtheewouldalso be the Loentz contaction, dl = dl /γ.] The adiabatic obit condition () fo a elativistic electon becomes e = γm vθ a = γm γm v. (19) This can be thought of as the tansfom of the est-fame elation ee = dp /dt upon noting that E = γe since the electic field is tanvese to the velocity, dt = dt/γ, and dp = dp = γm dv. Combining eq. (18) with the fist fom of eq. (19), we have We also ewite eq. (19) as dt = γ e 6 3m c 3 = 3 γ 3 m c 3. () e m c = = γ v c = γβ γ (1 1γ ), (1) and hence, γ = γ γ 1=, () + + = (3)

5 The total lab-fame enegy is now ( U = γm c e = γ ) ( m c 1 ) + m 8 c, () using eq. (3). Then, dt ( ) ( ṙm 3 c = 1 ) ṙm c = 3 3 γ m c 3, (5) using eq. (). Finally, ṙ 3 γ c 1 ( 3 c 1+ 3 ), (6) which is lage than the nonelativistic esult (6) by the facto /. Hence, the fall time of the electon is deceased by the elativistic coections. We note that the elativistic coections inceased the ate of adiation, and deceased the facto A in the elation /dt = Aṙ. Hence, both of these coections lead to an incease in the adial velocity ṙ, and to a decease in the coesponding fall time of the electon. 5

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