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CONSERVATION OF MAGNETIC MOMENT OF CHARGED PARTICLES IN STATIC ELECTROMAGNETIC FIELDS V.Е. Moiseenko1,2, M.A. Surkova1, O. Ågren2 1 Institute of Plasma Physics, NSC “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine; 2 Uppsala University, Ångström laboratory, SE-751 21 Uppsala, Sweden E-mail: [email protected] In the report, corrections to the magnetic moment invariant for a charged particle motion are calculated, and the equation defining magnetic moment variation in time is derived. PACS: 52.65.Сс 1. INTRODUCTION The magnetic moment of the charged particle moving in the electric and magnetic fields is μ= mv 2⊥ , 2B (1) B(B ⋅ v ) v || = are the where and v ⊥ = v − v || B2 perpendicular and parallel to the steady magnetic field components of the particle velocity. The magnetic moment is an approximate invariant of motion, when the motion is adiabatic and the fields vary slowly at the particle gyro-center. The accuracy of the expression for the magnetic moment is of low-order with respect to the adiabaticity parameter, i.e. the ratio λ = ρ L / L of the particle Larmor radius to the characteristic scale of non-uniformity, and a higher order invariant, which to leading order is the magnetic moment, needs to be consistent with other independent invariants. Corrections for the invariant for several particular cases are calculated in Refs. [1-6]. However, the equation for the evolution of μ is not derived there. 2. ANALYTICAL TREATMENT To describe the particle motion in static electric and magnetic fields which are slowly varying in space, the Newton’s and Lorentz force equations are analyzed: m dv 1 ⎡ ⎤ = e ⎢E + (v × B )⎥ , dt c ⎣ ⎦ dx = v. dt (2) (3) The particle velocity and the equations of motion are projected onto the unitary orthogonal vector triplet aligned to the magnetic field: (A × B ) × B , (4) e1 = BA×B e2 = A×B , A×B e|| = 56 B , B (5) (6) where A is an arbitrary constant in space vector ( ∇A = 0 ). The electric field is equal to: E = −∇ϕ . (7) Here ϕ is the scalar electric potential. Energy conservation reads: ε= mv 2 + eϕ = const . 2 (8) The equation for parallel velocity can be written in the following form: dv || dt = v ⋅ (v ⋅ ∇ ) B e B + E⋅ . B m B (9) Using the above formula the equation for the variation of the magnetic moment (1) could be derived: mv mv 2 dμ ⎛B ⎞ = − 2 || v ⊥ ⋅ (v ⊥ ⋅ ∇ )B − 2|| v ⊥ ⋅ ⎜ ⋅ ∇ ⎟B + dt B B ⎝B ⎠ mv 2⊥ (v ⋅ ∇ ) + e v ⊥ ⋅ E . + 2 B (10) In non-uniform fields the motion is mainly Larmor rotation with slowly varying parallel and perpendicular guiding center velocities. For this reason the right-hand side of the equation (10) contains slowly varying parts and parts oscillating with the gyro-frequency and its harmonics. The particle perpendicular velocity could be represented as: v ⊥ = v (⊥1) + δv (⊥0 ) + δv (⊥2 ) , (11) where v (⊥1) is the term responsible for the Larmor rotation, δv (⊥0 ) -drift in inhomogeneous magnetic field, δv (⊥2) unharmonicity, the term describing Larmor circle deformation. The upper index values could be explained in the following way: '0' is associated with the nonoscillating motion, '1' describes the fundamental cyclotron harmonic, '2' stands for second cyclotron harmonic terms. Following (9), the parallel part of the particle velocity is v || = v ||(0 ) + δv ||(1) , (12) PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2011. № 1. Series: Plasma Physics (17), p. 56-58. where v ||(0 ) is the major part of the parallel velocity, and δT2 = − δv||(1) is the first order correction which is the fundamental harmonic oscillation. − ∂Bx ∂B y ∂By ∂By ∂Bx ∂Bx − + + ∂x ∂x ∂x ∂y ∂y ∂x + ∂By ∂Bx ∂ 2 By ∂ 2 Bx ⎤ − Bz + Bz ⎥, ∂y ∂y ∂x∂z ∂y∂z ⎦ 3. ORDERING TERMS To provide the same order contribution to the magnetic moment corrections, the oscillating terms should be one order higher in the adiabaticity parameter than the slowly ~ ~ f fL , ∫ fdt ~ , where f is varying terms, since ∫ fdt ~ v || ωH the oscillating function, f - the slowly varying function. The expression (10) could be represented in following form: dμ = T1 + T2 + T3 + T4 , dt mv T1 = − 2 || v ⊥ ⋅ (v ⊥ ⋅ ∇ )B , B mv||2 ⎛B ⎞ T2 = − 2 v ⊥ ⋅ ⎜ ⋅ ∇ ⎟B , B ⎝B ⎠ mv 2⊥ T3 = (v ⋅ ∇ ) , 2 v ⋅E T4 = e ⊥ . B mv 2⊥ v ||2 ⎛ ∂Bz ∂B y ∂Bz ∂Bx ⎞ ⎜ ⎟, − ∂y ∂z ⎟⎠ B 3ω c ⎜⎝ ∂x ∂z δT4 = 0 . δT3 = δT = (15) 4 ∑ δT i =1 = (14) d δμ i + δTi , dt = mv v 2 ⎛ ∂B ∂B ⎞ T01 = − || 2 ⊥ ⎜⎜ x + y ⎟⎟ , ∂y ⎠ 2 B ⎝ ∂x T02 = 0 , mv|| v 2⊥ ∂Bz , 2 B 2 ∂z T04 = 0 . T03 = − ⎛ B ⋅∇ × B ⎞ B ⋅ ∇⎜ ⎟. B2 2 B ωc ⎝ ⎠ ( 27 ) 3 The contributions to the magnetic moment from the oscillating terms are: δμ1 = (17) mv|| v x v y ⎛ ∂B y ∂Bx ⎜ − 2 B 2ωc ⎜⎝ ∂y ∂x ⎞ ⎟⎟ − ⎠ mv|| (v 2x - v 2y ) ⎛ ∂B y ∂Bx ⎞ ⎜ ⎟, − + 4 B 2ωc ⎜⎝ ∂x ∂y ⎟⎠ mv 2 ⎛ ∂B ∂B ⎞ δμ 2 = 2 || ⎜⎜ v x y − v y x ⎟⎟ , B ωc ⎝ ∂z ∂z ⎠ (18) δμ3 = mv ⊥2 B 2ωc ⎛ ∂Bz ∂B ⎞ ⎜⎜ v x − v y z ⎟⎟ , ∂y ∂x ⎠ ⎝ e (v y Ex - v x E y ) . Bωc δμ 4 = (28) (29) (30) (31) The cumulative expression for the contributions of the oscillating terms to the magnetic moment can be written as: 4 (19) (20) (21) (22) The sum of the zero-order terms nullify because ∇ ⋅ B = 0 . This is the sign of the magnetic moment conservation. The slowly-varying first-order terms are: δT1 = 0 , (26) = i m v 2⊥ v ||2 (16) where T0i are zero-order slowly varying terms, δμ i is associated with the contribution to the magnetic moment for oscillating terms, δTi - slowly varying first-order terms. These terms are calculated using the energy conservation law (8) and the equation for the parallel velocity (9). The zero-order slowly varying terms are: (25) m v 2⊥ v ||2 ⎛ ∂ B z ∂ ⎞ (∇ × B ) z + Bz (∇ × B ) z ⎟ = ⎜− 2 B 3ω c ⎝ ∂ z ∂z ⎠ In each of these terms the contributions of different orders are separated: Ti = T0i − (24) The slowly varying first-order terms could be written in coordinate-independent form: (13) where mv ⊥2 v||2 ⎡ ∂Bz ∂B y ∂B ∂Bx − −2 z ⎢2 ∂y ∂z 2 B 3ωc ⎣ ∂x ∂z δμ = ∑ δμ i = i =1 ∂B y ∂Bx v z 2 ⎡ 2 ⎢ v x v y v z ∂x − 2 (v x − v y ) ∂x + ⎣ ∂B ∂B v + v y (v 2x + v 2y ) z − z (v 2x − v 2y ) x − ∂x ∂y 2 =− m 2ω B B 2 − vxv yvz ∂B y ∂y − v x (v 2x + v 2y ) − ∂B mv ⎛ ∂Bx −v x y ⎜vy ∂z ∂z ωB B ⎝ − ∂ϕ ⎞ e ⎛ ∂ϕ ⎜⎜ v y ⎟. − vx ∂y ⎟⎠ ω B B ⎝ ∂x 2 z 2 ∂Bz ⎤ − ∂y ⎥⎦ (32) ⎞ ⎟⎟ − ⎠ (23) 57 The terms containing the drift velocity may be separated: δμ + mv ⋅ v d m =− 2ω B B 2 B ∂Bx ⎡ ⎢ v x v y v z ∂x − ⎣ ∂B vz 2 (33) − (v x − v 2y ) y − 2 ∂x ∂B ⎤ ∂B v − z (v 2x − v 2y ) x − v x v y v z y ⎥, 2 ∂y ∂y ⎦ Finally, the equation for the corrected magnetic moment in coordinate-independent form reads: 2 2 d μ m v ⊥ v || ⎛ B ⋅∇ × B ⎞. = B ⋅ ∇⎜ ⎟ 2 dt 2 B ωc B2 ⎝ ⎠ (37) The right-hand side of this equation determines the slow variation of the magnetic moment in time, and is associated with the magnetic field vorticity in currentcarrying plasma. 4. CONCLUSIONS where ∇ϕ × B mcv 2⊥ (B × ∇B) + + B2 2eB 3 mcv ||2 [B × (B ⋅ ∇)B] + eB 4 v d = −c (34) is the drift velocity. The residual terms in (33), which stand for the second harmonic oscillations, can be grouped as: δμ + mv ⋅ v d mv z =− B 4ω B B 2 + ⎧ ⎡∂ ⎨ v y ⎢ ( v x Bx + v y B y ) + ⎩ ⎣ ∂x ⎤ ∂ ( v y Bx − v x B y ) ⎥ + ∂y ⎦ ⎡ ∂ + v x ⎢ − ( v x Bx + v y B y ) + ⎣ ∂y + (35) ∂ ⎤⎫ ( v y Bx − v x B y ) ⎥ ⎬. ∂x ⎦⎭ REFERENCES Finally, the expression for the corrected magnetic moment becomes: mv || ⎧ mv 2⊥ mv ⋅ v d − + B ⋅ [ v × ∇ ( v ⋅ B) + B 4ω B B 3 ⎩ 2B μ =⎨ ⎫ + ( v ⋅ ∇)( v × B) v || ( v × ∇B )]⎬. ⎭ In the report, the adiabatic motion of charged particles in static electromagnetic fields is analyzed. The equation for the corrected magnetic moment is obtained in coordinateindependent form. The derived local corrections to the magnetic moment invariant are oscillating and are associated with the particle drift. They have no influence on conservation of the magnetic moment in average, but they make the higher order magnetic moment invariant consistent with the other invariants such as the generalized momentum. The right-hand side of the equation determines the slow variation of the magnetic moment in time, and is associated with the magnetic field vorticity in currentcarrying plasma. The corrections to the magnetic moment invariant are consistent with the standard expressions for the first order drift and parallel motion of the guiding center. (36) Here is μ = μ + δμ and the first term in (36) is the standard expression for the magnetic moment. 1. M.D. Kruskal. Prinston Univ. Rep. PM-S-33, 1958. 2. C.S. Gardner // Phys. Fluids. 1966, v. 9, 1997. 3. A.M. Dykhne, V.L. Pokrovsky // Soviet Physics JETP. 1961, v. 12, p. 264. 4. A.M. Dykhne, A.V. Chaplik // Soviet Physics JETP. 1961, v. 13, p. 465. 5. V.M. Balebanov, N.N. Semashko // Nuclear Fusion. 1967, v. 7, p. 207. 6. R.J. Hastie, G.B. Hobbs, J.B. Taylor // Plasma Physics and Controlled Nuclear Fusion Research/ Vienna: «IAEA», 1969, v. 1, p. 389-401. Article received 28.10.10 СОХРАНЕНИЕ МАГНИТНОГО МОМЕНТА ЗАРЯЖЕННОЙ ЧАСТИЦЫ В СТАТИЧЕСКИХ ЭЛЕКТРОМАГНИТНЫХ ПОЛЯХ В.Е. Моисеенко, М.А. Суркова, O. Ågren Pассчитаны поправки к магнитному моменту заряженной частицы, а также получено уравнение для скорректированного магнитного момента. ЗБЕРЕЖЕННЯ МАГНІТНОГО МОМЕНТУ ЗАРЯДЖЕНОЇ ЧАСТИНКИ В СТАТИЧНИХ ЕЛЕКТРОМАГНІТНИХ ПОЛЯХ В.Є. Моісеєнко, М.О. Суркова, O. Ågren Pозраховано поправки до магнітного моменту зарядженої частинки, також отримано рівняння для скоректованого магнітного моменту. 58