RADIATION POWER SPECTRAL DISTRIBUTION OF CHARGED PARTICLES MOVING IN A SPIRAL IN MAGNETIC FIELDS

Transcription

1 Journal of Optoeletronis and Advaned Materials Vol. 5, o. 5,, p. 4-4 RADIATIO POWER SPECTRAL DISTRIBUTIO OF CHARGED PARTICLES MOVIG I A SPIRAL I MAGETIC FIELDS A. V. Konstantinovih *, S. V. Melnyhuk, I. A. Konstantinovih Chernivtsi ational University,, Kotsyubynsky St., Chernivtsi, 5, Ukraine The expressions for the momentary and average radiation powers of the harged partiles moving on an arbitrary determined traetory in transparent isotropi medi and in vauum are studied by using the Lorentz's self-interation method. Speial attention is given to the researh of the fine struture of the synhrotron radiation spetral distribution of two eletrons moving in a spiral in vauum. The spetra of synhrotron, Cherenkov and synhrotron- Cherenkov radiations for a separate eletron are analyzed. (Reeived May 6, ; aepted August, Keywords: Cherenkov radiation, Synhrotron radiation, Synhrotron-Cherenkov radiation, Lorentz`s self-interation. Introdution The investigation of the radiation spetra of harged partiles moving in magneti fields in transparent isotropi medium and in vauum is important from the point of view of the appliations in eletronis, astrophysis, plasma physis, physis of storage rings et. [-]. Under moving harged partiles in magneti field three kinds of radiation are possible in a medium [4-6]: synhrotron, Cherenkov, and synhrotron-cherenkov ones whereas in vauum only synhrotron radiation takes plae. A question requiring further investigations is the oherene of synhrotron radiation [-]. A laser radiation is emitted when an eletron beam moves through a spiral undulator [7]. The properties of free-eletron lasers were onsidered in papers [-]. The investigations of the fine struture of synhrotron, Cherenkov, and synhrotron-cherenkov radiation spetra in vauum and in transparent media for the low-frequeny spetral range is of great interest []. The fine struture of the Cherenkov radiation spetrum in non-transparent media was investigated in papers [-]. The aim of this paper is to investigate the spetral distribution of the radiation power for the harged partiles moving along an arbitrary defined traetory using the Lorentz`s selfinteration method. Using the exat integral relationships for the spetral distribution of radiation power of two eletrons moving one after another along a spiral in vauum, the fine struture of the synhrotron radiation spetrum was investigated by means of analytial and numerial methods. The influene of the Doppler effet on the peuliarities of the radiation spetrum of a separate eletron during its motion in a spiral in transparent media and in vauum is also investigated.. Instantaneous and time-averaged radiation power of harged partiles The instantaneous radiation power of harged partiles P rad ( t medium and in vauum is expressed in [4,5] as in an isotropi transparent * Corresponding author: theormyk@hnu.v.ua

2 ' " -, 44 A. V. Konstantinovih, S. V. Melnyhuk, I. A. Konstantinovih Dir Dir rad A ( r, t ϕ ( r, t P t = r t ρ r, t dr τ t t Here ( r, t is the urrent density and ( r,t Dir some volume τ. Aording to the hypothesis of Dira [4 ], the salar ( r, t A Dir ( r, t,. ( ρ is the harge density. The integration is over ϕ and vetor potentials are defined as a half-differene of the retarded and advaned potentials: Dir ϕ = ret adv Dir ret adv ( ϕ ϕ, A = ( A A. ( After substituting ( into ( we obtain the relationship for instantaneous radiation power of harged partiles moving in isotropi transparent media as a funtion of spetral distribution W P rad ( t dωw ( t, ω, ( n ω ω sin r r ( t, ω = dr dr dt ωµ ( ω os ω( t t = where µ ( ω is the magneti permeability, ( ω π r r ( r, t ( r, t ρ( r, t ρ( r, t, (4 n ω n is the refration index, ω is the angular frequeny, and is the speed of light in vauum. The time-averaged radiation power of harged partiles is defined by the expression T rad rad P = lim P ( tdt. (5 T T T rad The values of p an be obtained after substitution of the instantaneous radiation power expressed by relationships ( and (4 into (5.. Systems of non-interating point harged partiles Let us onsider a system of point-like non-interating partiles with harges q, q,..., q and rest masses m, m,..., m moving along arbitrary traetories. Then the soure funtions of harged point-like partiles are defined as [5,5] = r, t Vl ( t ρ l ( r, t, ρ( r#, t = ρl ( r#, t, ρ l ( r, t = ql δ( r rl ( t, (6 l= l= % where r l & th ( t and V l (t are the motion law and the veloity of the l partile, respetively. By substituting relationships (6 into ( and (4 we obtain the expression for the instantaneous radiation power of harged partiles system in transparent media (magneti permeabilities, µ ( ω, and dieletri permittivities ε ( ω, are real:./ n( ω sin ω r '( *+ l t r t rad ωµ ( ω P t d dt ql q π l, = rl ( t r ( t

3 " # Radiation power spetral distribution of harged partiles moving in a spiral in magneti fields 45 os ω l. (7 ( t t V ( t V ( t n ( ω The time-averaged radiation power an be obtained from the expression T T rad rad P = lim P ( t dt = lim dt dωωµ ( ω dt T T T π T T T n( ω sin ω r t r t l ( ql q os ω t t Vl ( t V ( t = ( l, rl t r t n ω. ( ml = l m Let us onsider a system of point-like non-interating harged partiles ( ql = nl e, n moving one by one along an arbitrary defined traetory. Then the motion law and the veloity of the th l partile of this system are determined by the relationships r ( t = r ( t + t, V ( t V ( t + t p =. ( Relationships ( were obtained taking into aount that the magnitudes ml = nl m and ql = nle are linearly inluded into the movement equation of a harged partile in eletromagneti field. In this ase we obtain the averaged radiation power after substitution the expressions ( into (: where the oherene fator ( ω T rad e P = lim dt dt dωµ ( ω ωs ( ω π T T T n( ω sin ω rp ( t rp ( t os ω( t t V ( t V ( t ( rp t rp t n S is defined as S ( ω = n n { ω( t t } l l, = l ( ω, ( os. ( Relationships ( and ( obtained here may be also applied to low-length bunhs of harged partiles. For eletrons ( q l = e, m l = m the oherene fator takes the form [6,]: The oherene fator ( ω power between harmonis. S ( ω = { ω( t t } os l. ( l, = S determines a redistribution of the harged partiles radiation 4. Fine struture of the radiation spetra of two eletrons moving along a spiral in vauum Peuliarities of the radiation spetra of two eletrons moving one by one in a spiral in vauum an be investigated ombining analytial and numerial methods. The law of motion and the veloity of the eletron are given by the expressions:

4 ('& % # $ ".-, + * 46 A. V. Konstantinovih, S. V. Melnyhuk, I. A. Konstantinovih H ext r Here ( t r { ω ( t + t } i + r { ω ( t + t } + V ( t + t k = sin os, V ( t ext~ r = V ω, ω = eh Ε, ~ Ε = p + m Z, V and V are the omponents of the veloity, p ( t dr =. ( dt, the magneti intensity vetor and Ε ~ are the momentum and energy of the eletron, e and m are its harge and rest mass. After substitution of the expressions ( into ( one obtaines the time-averaged radiation power of two eletrons W ( ω = dxω S ( ω P rad = W ( ω dω, (4 sin ( x e ωη os ωx π η x V ω ω where η( x = V x + 4 sin x The oherene fator ( ω. [ V os( ωx + V ] S of two eletrons is defined as ( ω = + os( ω, (5 S t. (6 Here t = t t is the time shift of the eletrons moving along a spiral. ω = iπ / t (i=,,, the oherene fator of two eletrons (6 is At the frequenies equal to 4 and at the frequenies ω = ( i + π /( t (i=,,, the oherene fator is equal to zero. The analogous expression for the oherene fator was investigated by Bolotovskii []. From the relationships (4 and (5, after some transformations, the ontributions of separate harmonis to the averaged radiation power an be written as π rad e P = dωω sin θdθ m= [ + os{ ω( t }] m V m V J m ( q J m ( q ( V J m ( q} os δ ω θ ω + +, (7 q V ω where q = sin θ, J m ( q and J m ( q are the Bessel funtion with integer index and its ω derivative, respetively. Eah harmoni is a set of the frequenies, whih are the solution of the equation The limits of the ω V os θ mω =. ( th m harmoni are determined by the frequenies min mω ω m = V, + max mω ω m = V, (

5 Radiation power spetral distribution of harged partiles moving in a spiral in magneti fields 47 and the total radiation power emitted by a separate eletron is determined aording to [] as e ωv Pm tot =, ( V eh ext where V ω =. m Our numerial alulations of the radiation power spetral distribution were performed at ext H =Oe. For the veloities omponents V va =. m/s and =. m/s the radiation V va power spetral distributions of two eletrons in vauum depending on their loation along a spiral are shown in Figs. (urves 6. It is interesting to ompare the radiation power spetral distribution for two eletrons with the radiation power spetral distribution of a separate eletron (urve in Fig.. The radiation power of tot 6 the separate eletron in vauum P =. erg/s alulated aording to relationship ( is va int va.4 6 in good agreement to the power P = erg/s determined after integration of relationships (4 and (5. For a separate eletron the oherene fator S =. For the time differene t =.π / ω (urve in Fig. the wavelength orresponding to the basi frequeny λ = π / ω = 445m is higher by a fator of around than the distane between the eletrons. In this ase the oherene fator S ( ω = 4 and two eletrons radiate as a harged partile with the harge e and the rest mass m, i.e. by a fator of four more than a separate eletron W(ω ω -7 erg/s ω/ω Fig.. Spetral distribution of radiation power for two eletrons moving one by one in a spiral. ( V va = m/s, V va =. m/s, urves 6. Curve the radiation. 6 spetrum of a separate eletron, P tot 6 va =. erg/s P int va =. 4 erg/s. 6 Curve : t =.π / ω, P int va =. 57 erg/s. Curve : t =.π / ω, 6 P int 6 va = erg/s. Curve : t = π/ ω, P int va =. 74 erg/s, ω =. 5 rad/s, r = r = r = r = 5. m.

6 4 A. V. Konstantinovih, S. V. Melnyhuk, I. A. Konstantinovih In the ase t = π / ω the funtion of the radiation power spetral distribution has the maxima appoximately at the frequenies ω, ω and ω whereas the radiation at.5ω and.5ω is absent. 6 5 W(ω ω -7 erg/s ω/ω Fig.. Spetral distribution of radiation power for two eletrons moving one by one in a 6 spiral. Curve 4: t 4 = 4π / ω4, P int va 4 =. 55 erg/s, ω 4 =. 5 rad/s, r 4 =5. m. For the time differene t 4 = 4π / ω4 (urve 4 in Fig. we have found the maxima of the spetral distribution funtion appoximately at the frequenies ω 4,.5ω 4, ω 4,.5ω 4 and ω 4 at the frequenies.75ω 4,.5ω 4,.75ω 4,.5ω 4, while at.75ω4 the radiation is absent. For the time differenes t 5 = π / ω5 (urve 5 in Fig. and t 6 = π / ω6 (urve 6 in Fig. the radiation at the basi frequenies ω4 5 is absent. 6 5 W(ω ω -7 erg/s ω/ω Fig.. Spetral distribution of radiation power for two eletrons moving one by one in a 6 spiral. Curve 5: t 5 = π / ω5, P int va 5 =. erg/s. Curve 6: t 6 = π / ω6, 6 P int va6 =. 56 erg/s. At the basi frequeny ω the funtion of the radiation power spetral distribution of two i + π / ω eletrons is equal to zero if the time differene between them in a spiral is equal to (i=,,,.

7 Radiation power spetral distribution of harged partiles moving in a spiral in magneti fields 4 5. Spetral distribution of synhrotron-cherenkov radiation power in low-frequeny range Let us onsider the influene of the Doppler effet on synhrotron-cherenkov radiation in transparent media. The expressions for the synhrotron-cherenkov radiation power in suh a medium an be obtained starting from (. Then for the separate eletron moving in a spiral we have found [6,] W e π ( ω = dxµ where η( x = V x + sin x P rad = W ( ω dω, ( n ω ω sin η ω ω η V ω 4. ω ( x ( x os ( ωx V os( ω x + V n ( ω, ( In the ase of transparent media in low-frequeny spetral range, i.e. at ε = onst and µ =, the power of the Cherenkov radiation at retilinear motion in a medium ( n is the onstant is determined as []: e P tot h = V ωmax. ( V n For the refration index n = at the veloities V m =. 5 m/s, V m =. 4 m/s, and V m =. m/s, V m =. 46 m/s (urves 7 and in Fig. 4 the onditions for the existene of the synhrotron-cherenkov radiation are fulfilled. 6 W(ω ω -7 erg/s ω/ω Fig. 4. Spetral distribution of synhrotron-cherenkov radiation power with relative frequeny. ext (For the urves 7-: B = Gs, n =, ω 7 =. 5 rad/s, =. 75 m/s. Curve 7: V m = 5 m/s, V m =. 4 m/s,, int Pm 7 =.46 erg/s, r 7 =. 5 m. Curve : V =. m/s, V m =. 465 m/s, r = 7. m, P int m =. 46 erg/s. Curve : V m =. m/s, V m =. 5 m/s, P int m =. 6 erg/s, P tot h =. 67 erg/s, r =. 66 m. m

8 4 A. V. Konstantinovih, S. V. Melnyhuk, I. A. Konstantinovih W(ω ω -7 erg/s ω/ω Fig. 5. Spetral distribution of synhrotron-cherenkov radiation power with relative frequeny. Curve : V m =. 55 m/s, V m =. 4 m/s, r = 6. m, erg/s. Curve : m V m =. 5 m/s, V m =. 57 m/s, P int =. 666 r =. m, P int m =. 664 erg/s. Curve : V m =. 4 m/s, r =, P int =. 6 erg/s. V m =. 5 m/s, 6. m The power of the Cherenkov radiation at retilinear motion P tot h =. 67 erg/s (relation ( is in good agreement to the power of the synhrotron-cherenkov radiation P int m =. 6 erg/s alulated by applying the relationships ( and ( at the motion of the harged partile having a small ( V m =. m/s transverse veloity omponent (the absolute values of the veloities are the same. The performed high-auray alulations of relationships ( and ( for the spetral distribution of the synhrotron-cherenkov radiation power of eletrons showed that the spetral distribution at V < / n (urves 7 and in Fig. 4 essentially differs from that at V > / n (urves in Fig. 5. The analytial and numerial alulations showed that the influene of the Doppler effet on the peuliarities of the radiation power spetral distribution of the eletrons is essentially near the Cherenkov threshold. Taking into aount the frequeny dispersion, this does not hange essentially the radiation power spetral distribution in low-frequeny range but leads to some interesting peuliarities in highfrequeny spetral range [,]. 6. Conlusions The oherene fator leads to essential hanges in the radiation power spetral distribution of a system of harged partiles. In the radiation spetrum of harged partiles the Doppler effet establishes the limits between the bands of separate harmonis. The obtained spetral distributions of the power of synhrotron, Cherenkov, and synhrotron- Cherenkov radiations of the eletrons in low-frequeny spetral range an be applied to develop new soures of eletromagneti energy and for interpretation of some phenomena in eletronis, in astrophysis and plasma physis. Referenes [] C. Kunz, Properties of Synhrotron Radiation, In: Synhrotron Radiation. Springer Verlag, Berlin - Heidelberg - ew York, 7.

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