Journal of the Korean Physical Society, Vol. 32, No. 1, January 1998, pp. 30∼34
Relativistic Cyclotron Motion in a Polarized Electric Field Sang Wook Kim, Duck-Hee Kwon and Hai-Woong Lee∗ Department of Physics, Korea Advanced Institute of Science and Technology, Taejon 305-701 (Received 20 June 1997) The relativistic cyclotron motion in a uniform magnetic field and in a transverse time-periodic electric field is studied, with particular attention to qualitative differences in the nature of the motion under linearly and circularly polarized electric fields. Theoretical analysis based on the Lagrangian formulation of the cyclotron motion allows a direct comparision for the two polarizations and confirms the earlier observation that the cyclotron motion is always integrable when the electric field is circularly polarized, but is not under a linearly polarized electric field.
I. INTRODUCTION
in mutually orthogonal uniform magnetic and oscillating electric fields is integrable when the electric field is circularly polarized [20]. On the other hand, the relativistic cyclotron motion can become chaotic if the electric field is linearly polarized [8]. Thus, the question concerning the role played by the polarization state of the electric field in cyclotron motion naturally arises. The main pupose of the present work is to provide a theoretical analysis of the effect of the polarization state of the electric field upon the integrability of cyclotron motion. We formulate the problem in such a way that the two cases of linearly and circularly polarized electric fields can be compared directly and analyzed in a unified fashion. Also considered is the cyclotron motion in the nonrelativistic limit. It is shown that nonrelativistic cyclotron motion is integrable, regardless of whether the electric field is circularly or linearly polarized.
The study of the relativistic dynamics of charged particles moving in electric and magnetic fields is of prime importance in accelerator and plasma physics, where such devices as the cyclotron, tokamak, and free-electron laser are frequently encountered [1,2]. The recent surge of interest in this study stems partly from theoretical and experimental findings that such particles are capable of exhibiting chaotic behavior. Systems that have recently been found to exhibit chaos include electrons undergoing relativistic cyclotron motion [3–8], electrons moving at relativistic velocities through a wiggler in a free-electron laser device [9–15], and particles oscillating at relativistic velocities under the influence of a harmonic or nonharmonic potential [16,17]. These studies are not just of academic interest; identifying the cause of and finding a way of suppressing or controlling such “relativistic chaos” is directly linked to the practical problem of improving the performance of the device being considered, whether it be the cyclotron, the tokamak or the free-electron laser. Relativistic cyclotron motion is a subject of interest also in atomic physics, as it occurs, for example, in the Penning trap. Despite the fact that one usually deals with only weakly relativistic electrons in the Penning trap, some interesting relativistic effects, such as bistable hysteresis, were observed to occur in the cyclotron motion there [18,19].
II. THEORY OF CYCLOTRON MOTION Let us consider a particle of mass m and charge q moving in a uniform magnetic field B = B0 eˆz and a timeperiodic electric field E assumed to be orthogonal to B. From here on the vector eˆi denotes a unit vector along the i direction. Below, we analyze the motion of the particle for the cases when the electric field is circularly polarized and when it is linearly polarized. In both cases, no force exits along the z direction; thus, the motion can be considered to occur in the xy plane. We use Gaussian units throughout.
Recent investigations of relativistic cyclotron motion have revealed that the qualitative nature of the motion can be quite different, depending upon the polarization state of the electric field. In particular, it was shown that the relativistic motion of a charged particle moving ∗
1. Circularly Polarized Electric Field
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Relativistic Cyclotron Motion in a Polarized Electric Field – Sang Wook Kim et al.
When the electric field is circularly polarized, it can be written as E = E0 cos wt eˆx + E0 sin wt eˆy .
(1)
The scalar and the vector potentials, φ and A, can be taken to be φ = −xE0 cos wt − yE0 sin wt,
(2)
yB0 xB0 eˆx + eˆy . 2 2
(3)
A=−
Although our main interest is in the relativistic motion, we first consider the motion in the nonrelativistic limit. The Lagrangian in the nonrelativistic region is given by q L = T − qφ + A · v c
(4)
where T is the kinetic energy. For the particle being considered, we can immediately write L =
1 m(x˙ 2 + y˙ 2 ) + qE0 x cos wt 2 qB0 +qE0 y sin wt − (y x˙ − xy). ˙ 2c
(5)
The canonical momenta are then given by px = mx˙ −
qB0 y, 2c
py = my˙ +
qB0 x, 2c
and the Hamiltonian by 2 2 1 qB0 1 qB0 H = px + y + py − x 2m 2c 2m 2c −qE0 x cos wt − qE0 y sin wt. Hamilton’s equations of motion take the form dx 1 qB0 = px + y , dt m 2c dy 1 qB0 = py − x , dt m 2c dpx qB0 qB0 py − = x + qE0 cos wt, dt 2mc 2c dpy qB0 qB0 =− px + y + qE0 sin wt. dt 2mc 2c
(6)
(7)
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the Lagrangian given by Eq. (5) is invariant under the transformation ε x → x + εy, y → y − εx, (14) t→t− , w the third constant of motion can be determined as H px y − py x + = const. (15) w With the constants of motion as found above, one can show that the nonrelativistic motion being considered is integrable [20]. We now wish to consider the relativistic motion for which the Lagrangian is given by r v2 q 2 L = −mc 1 − 2 − qφ + A · v. (16) c c For the particle being considered, the Lagrangian becomes r x˙ 2 + y˙ 2 2 L = −mc 1 − + qE0 x cos wt c2 qB0 +qE0 y sin wt − (y x˙ − xy), ˙ (17) 2c and the canonical momenta r x˙ 2 + y˙ 2 qB0 px = mx/ ˙ 1− − y, 2 c 2c r x˙ 2 + y˙ 2 qB0 py = my/ ˙ 1− + x. (18) 2 c 2c The Hamiltonian can easily be obtained as r qB0 2 2 qB0 2 2 H = m2 c4 + px + y c + py − x c 2c 2c −qE0 x cos wt − qE0 y sin wt. (19) Hamilton’s equations of motion now become
(8)
(9)
(10)
(11)
Equations (8)-(10) immediately suggest that we have two constants of motion: px −
qB0 qE0 y− sin wt = const, 2c w
(12)
py +
qB0 qE0 x+ cos wt = const. 2c w
(13)
Following Bourdier et al. [20], a third constant of motion can be found using Noether’s theorem. Noting that
2 0 (px + qB dx 2c y)c =q , (20) dt 0 2 c2 +(p − qB0 x)2 c2 m2 c4 +(px + qB y) y 2c 2c 2 0 (py − qB dy 2c x)c , (21) =q dt 0 2 c2 +(p − qB0 x)2 c2 m2 c4 +(px + qB y) y 2c 2c
0 qB0 c(py − qB dpx 2c x) = q dt qB0 0 2 2 2 2 2 m2 c4 +(px + qB 2c y) c +(py − 2c x) c
+qE0 cos wt,
(22)
0 qB0 c(px + qB dpy 2c y) = − q dt qB0 0 2 2 2 2 2 m2 c4 +(px + qB 2c y) c +(py − 2c x) c
+qE0 sin wt.
(23)
As in the nonrelativistic case, we immediately see from Eqs. (20)-(23) that there are two constants of motion
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Journal of the Korean Physical Society, Vol. 32, No. 1, January 1998
given by Eqs. (12) and (13). At the same time, the Lagrangian of Eq. (17) is invariant under the transformation of Eq. (14), which indicates that Eq. (15) is valid here. We can, thus, conclude that the relativistic motion, as well as the nonrelativistic motion, is integrable if the electric field is circularly polarized.
2. Linearly Polarized Electric Field
φ = −rE0 sin θ cos wt,
We now assume that the electric field is linearly polarized and choose the direction of polarization to be the y direction, E = E0 cos wt eˆy .
Rather than trying to find a third contant of motion, we found it convenient to go to a rotating frame of reference and to look at the motion in that frame. In order to describe the motion in a rotating frame, we need first to express Eqs. (26)-(28) in terms of the cylindrical coordinates r and θ. Since the scalar and the vector potentials are given in terms of the cylindrical coordinates by
The scalar potential can be taken as
while the vector potential A is still given by Eq. (3). Let us first consider the nonrelativistic motion. The Lagrangian, the canonical momenta, and the Hamiltonian can easily be obtained as L=
1 qB0 m(x˙ 2 + y˙ 2 ) + qE0 y cos wt − (y x˙ − xy),(26) ˙ 2 2c
qB0 y, px = mx˙ − 2c
H =
1 qB0 px + y 2m 2c −qE0 y cos wt,
qB0 py = my˙ + x, 2c 2
+
1 2m
(27)
and Hamilton’s equations of motion as dx 1 qB0 = px + y , dt m 2c dy 1 qB0 = py − x , dt m 2c dpx qB0 qB0 = py − x , dt 2mc 2c dpy qB0 qB0 =− px + y + qE0 cos wt. dt 2mc 2c
qB0 y = const, 2c
qB0 qE0 py + x− sin wt = const. 2c w
H =
pθ = mr2 θ˙ +
qB0 2 r , 2c
2 1 1 qB0 2 pr 2 + p − r θ 2m 2mr2 2c −qE0 r sin θ cos wt.
(38)
(39)
We now consider a frame of reference rotating clockwise with an angular velocity of, say, Ω. The Lagrangian in this rotating frame can be expressed as
(28)
1 2 (r˙ + r2 (θ˙ − Ω)2 ) + qE0 r sin (θ − Ωt) cos wt 2 qB0 2 ˙ + r (θ − Ω). (40) 2c
Equation (40) takes a particularly simple form if we choose (29)
(30)
Ω=
qB0 , 2mc
(41)
which corresponds to the Larmor frequency, i.e., the angular frequency of the rotational motion of the charged particle due to the action of the magnetic field B = B0 eˆz . Substituting Eq. (41) into Eq. (40), we obtain
(31)
1 1 L = m(r˙ 2+r2 θ˙2 )− mr2 Ω2+qE0 r sin (θ−Ωt) cos wt.(42) 2 2
(32)
Going back to the Cartesian coordinates in this rotating frame, Eq. (42) becomes
It then is immediately clear that two constants of motion are given by px −
1 qB0 2 ˙ m(r˙ 2 + r2 θ˙2 ) + qE0 r sin θ cos wt + r θ,(37) 2 2c
pr = mr, ˙
L =
2 qB0 py − x 2c
(36)
we obtain L=
(25)
B0 r eˆθ , 2
A=
(24)
φ = −yE0 cos wt
(35)
(33) (34)
In this case, however, the Lagrangian given by Eq. (26) is not invariant under the transformation of Eq. (14).
L =
1 1 m(x˙ 2 + y˙ 2 ) − m(x2 + y 2 )Ω2 2 2 +qE0 (y cos Ωt cos wt−x sin Ωt cos wt).
(43)
Equation (43) indicates that, when viewed in the rotating frame of reference with Ω given by Eq. (41), the motion being considered appears as driven harmonic oscillations in the x and the y directions, with the amplitude of the driving force of frequency of w being modulated at a frequency of Ω. The motion viewed in this rotating frame
Relativistic Cyclotron Motion in a Polarized Electric Field – Sang Wook Kim et al.
is, thus, regular, and so is the motion in the laboratory frame. Finally, we wish to consider the relativistic motion. The Lagrangian of the particle is given in the laboratory frame by r x˙ 2 + y˙ 2 qB0 2 +qE0 y cos wt− (y x−x ˙ y).(44) ˙ L = −mc 1− c2 2c As in the relativistic case, this Lagrangian is not invariant under the tranformation of Eq. (14); thus, we again attempt to view the motion in a rotating frame of reference. Expressing first the Lagrangian in the larboratory frame in terms of cylindrical coordinates, we have s r˙ 2 +r2 θ˙2 qB0 2 ˙ L = −mc2 1− +qE0 r sin θ cos wt+ r θ.(45) 2 c 2c In a frame rotating at a frequency of, say, Ω, Eq. (45) becomes s r˙ 2 + r2 (θ˙ − Ω)2 L = −mc2 1 − c2 qB0 2 ˙ +qE0 r sin (θ−Ωt) cos wt+ r (θ−Ω). (46) 2c Unlike the case of the nonrelativistic motion, no particular value of Ω exists for which Eq. (46) reduces to a simple form. Mathematically, this is due to the fact that the kinetic-energy term in the Lagrangian involves a more complicated expression in the relativistic case. Physically, this originates from the fact that the rotational frequency of the particle in the presence of a constant magnetic field B0 in the relativistic case is given by Ω=
qB0 2γmc
(47)
and is no longer a constant; it depends on p the velocity of the particle through the quantity γ = 1/ 1 − v 2 /c2 . One can, thus, conclude that no rotating frame of reference exists in which the motion of the particle appears “simple”. It, therefore, is not surprising that the relativistic cyclotron motion for the case when the electric field is linearly polarized can display chaotic behavior.
III. CONCLUSION AND DISCUSSION In conclusion, we have shown that the relativistic motion of a charged particle moving in a uniform magnetic field and a time-periodic electric field is integrable when the electric field is circularly polarized, but is not integrable when the electric field is linearly polarized. The corresponding nonrelativistic motion, however, is integrable, regardless of whether the electric field is circularly polarized or linearly polarized. In the relativistic case, one, thus, sees that the qualitative nature of the
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motion is different, depending on the polarization state of the electric field. One can perhaps gain some physical insight into this polarization effect by considering the motion in a rotating frame of reference. For the case of a circularly polarized electric field, one may choose to view the particle motion in the frame rotating with the electric field at frequency w. The particle motion is then essentially equivalent to that in the presence of static electric and magnetic fields. For the case of a linearly polarized electric field, there, of course, does not exist a frame rotating with the electric field. Nevertheless, if the motion remains nonrelativistic, one may choose to view the motion in a frame rotating at the Larmor frequency Ω = qB0 /2mc . In this rotating frame, the electric field is still time dependent, but our analysis in Sec. II shows that in the rotating frame, the combined effect of the electric and the magnetic fields can be represented by forced harmonic oscillations. If the motion is relativistic, however, the rotational frequency Ω = qB0 /2γmc in the presence of a constant magnetic field B0 is no longer constant, and the above argument for the nonrelativistic case breaks down. Finally, we mention that there is no rigorous way of proving that a given system is nonintegrable, except to show via numerical computation that the system exhibits chaotic behavior. A computation of Poincare phase-space maps, which was performed recently on a particle undergoing relativistic cyclotron motion, has clearly indicated the occurrence of chaotic behavior when the electric field is linearly polarized [8]. On the other hand, when the electric field is circularly polarized, it can be rigorously proven that the system is integrable by finding the constants of motion, as was done earlier [20]. The significance of the present work lies in the fact that the two cases of linearly and circularly polarized electric fields are analyzed within the same framework, which allows one to compare directly the two cases and to understand clearly the role played by the polarization state of the electric field. ACKNOWLEDGMENTS This research was supported by the Korea Science and Engineering Foundation (KOSEF) under Grant No. 9610202-011-2, by the Ministry of Science and Technology (MOST) of Korea, and by a Korea Advanced Institute of Science and Technology (KAIST) research grant. REFERENCES [1] C. J. Joshi and P. B. Corkum, Physics Today, January, 36 (1995). [2] T. Tajima and J. M. Dawson, Phys. Rev. Lett. 43, 267 (1979). [3] A. A. Chernikov, T. Tel, G. Vattay and G. M. Zaslavsky, Phys. Rev. A40, 4072 (1989). [4] Y. Nomura, Y. H. Ichikawa and W. Horton, Phys. Rev. A45, 1103 (1992).
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