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Charged Particle Simulation Under the Guiding Center Approximation Charged Particle Simulation Under the Guiding Center Approximation Jonathan Brown December 20, 2011 Motivation Motion of a charged particle in electric and magnetic fields is governed by the Lorentz force:[3] F = q(E + v × B) (1) where q is the charge and v is the velocity. By this force, charged particles spiral around magnetic field lines where the component of the velocity perpendicular to the magnetic field is entirely in the circular motion while the component parallel to the magnetic field is not affected by the Lorentz force. Thus the radius of the spiral can be derived as[3] γmv r = ? (2) qB 1 where B = jBj, m is the mass, γ = q 2 is the Lorentz factor, and v? is the component of the 1− v c2 velocity perpendicular to the magnetic field. If a particles momentum is small compared to the magnetic field then its gyration radius will be small. Furthermore, electrons can also have relativistic velocities under this condition. Because of these fast, tight spirals, a numerical integrator demands a small step size to properly account for the quickly changing motion. This results in a much slower simulation of the charged particle. The simulation speed can be increased by ignoring the spiral itself and approximating the particles motion by looking only at the motion of the center of the circle the particle inscribes. This is called the guiding center approximation. Motion of the center of the circle is caused by velocity parallel to the magnetic field, the electric field, change in the magnitude of the magnetic field, and curvature of magnetic field lines.[3] Here, cylindrical symmetry will be assumed for both the electric and magnetic fields. Substitution Procedure There are several steps to substitute the particle simulation using the Lorentz force equations of motion with a simulation using the guiding center EOM. First, the initial position of the center must be found from the initial position of the particle. Then equations of motion governing the guiding center must replace the Lorentz force equations of motion. Finally, the final position of the particle must be extrapolated from the final position of the center. 1 There is an inherent loss of information under the initial transformation from the particle position to the center of its circle since all points on the circle map to the same center point. Thus there is no unique inverse transformation for the end of the simulation to find the final position of the particle from the final position of the guiding center. This information can be saved in the gyrophase of the particle. Understanding the gyrophase requires the introduction of a new local coordinate system centered at the guiding center. Define a right-handed coordinate system centered at the guiding center with basis unit vectors ^e1,^e2, and ^e3 where ^e3 points in the direction of the magnetic field, denoted as b^. The choice of the direction of the two unit vectors perpendicular to ^ b, ^e1 and ^e2, which lie in the plane of the circle, represents a choice of gauge which does not affect the end result.[6] As will be shown later, the optimal choice for cylindrically symmetric E and B fields is to have ^e1 lie in the r-z plane. In these coordinates, the gyrophase, θ, is the azimuthal angle of the particle relative to ^e1.[6] The introduction of the gyrophase may seem counterproductive given the motivation of the guiding center approximation. Even though the simulation requires the gyrophase, the goal of the approximation is still achieved. The quick oscillations in each momentum coordinate under the Lorentz equation demands a small step size. The gyrophase, however, can be tracked by an integrator more easily even with a larger step size since it increases linearly. Initial conditions Since the particle moves in a circle by the Lorentz force, the vector connecting the particle position to the center of the circle has the same direction as the acceleration and must have the magnitude of the radius of the circle. This vector is given by qv × B ` = r (3) jqv × Bj The only portion of the momentum of the particle that is important for the simulation is the component parallel to the magnetic field since the perpendicular component is only important in the gyration of the particle, motion which the guiding center approximation eliminates. Under the guiding center approximation, the magnetic moment of the gyration circle of the particle is a conserved quantity, and is given by[5] p2 µ = ? (4) 2mB where p? is the component of the momentum perpendicular to the magnetic field. Since the radius also depends on p?, the radius can be found from the conserved magnetic moment and depends only on the magnetic field. r2mµ r = (5) q2B The final initial condition needed is that of the gyrophase so that the transformation to the center is invertible. The gyrophase is defined by −` · ^e cos θ = 1 (6) j`j where the sign of θ is the same as that of −` · ^e2. 2 Figure 1: ^z out of the However, for this definition to be useful, the vectors ^e1 and ^e2 page. must be defined in terms of known variables. In order to define ^e1 in the r-z plane, it is easiest to define the local coordinate system by two rotations from Cartesian coordinates. The first rotation is about the ^z axis such that ^x0 points in the ^r direction and ^y0 points α ˆy in the φ^ direction. Thus the first rotation angle α is defined by ˆe2 cos α = Bx sin α = By (7) ˆr Br Br α If Br = 0, then Bx = By = 0 and so the limit must be taken of these ratios to define α. The second rotation is about the ^y0 = φ^ axis, from ˆx now called ^e2, such that ^z points in the direction of the magnetic field, b^. Because of the cylindrical symmetry, B has no component in the β ˆe1 ˆr φ^ direction so these two rotations are always sufficient to have a basis bˆ vector defined in the b^ direction. This rotation angle β, again from the β cylindrical symmetry, is defined by the magnetic field. ˆz Bz Br cos β = B sin β = B (8) Figure 2: ^e2 out of the page. These two rotations together create an invertible transformation between ^ (^x; ^y; ^z) and the desired local coordinate system (^e1; ^e2; b) given by 2 B B 3 2 3 BxBz y z − Br 2 3 ^e1 BBr BBr B ^x By B ^e2 = 6 − x 0 7 ^y (9) 4 5 4 Br Br 5 4 5 b^ Bx By Bz ^z B B B Now that ^e1 is defined in terms of the magnetic field, the gyrophase can be defined in terms of the magnetic field B2` − B (B ` + B ` ) cos θ = r z z x x y y (10) rBBr and the sign of the gyrophase is the same as that of `xBy − `yBx . Br Br Figure 3: Representation of local coordinates and definition of gyrophase θ. The guiding center is at the origin and the particle is at point P . 3 Equations of Motion The relativistic equations of motion for the guiding center motion are given by[4] dX By u b^×Ey du By·Ey dt = y mΓ − y dt = q y Bk Bk Bk y u ^ y µ (11) B = B + q r × b E = E − qΓ rB y y ^ Bk = B · b q u 2 2µB where X is the position of the guiding center, Γ = 1 + mc + mc2 is the Lorentz factor, and u = p? is the component of the momentum perpendicular to the magnetic field. The equation governing the evolution of the gyrophase is[1] dθ @^e 1 = Ω + 1 + v b^ · r^e · ^e + b^ · r × v b^ + u (12) dt @t k 1 2 2 k E qB where Ω = − Γm is the relativistic cyclotron frequency, vk is the component of the velocity parallel E×B to the magnetic field, and uE = B2 . In covariant notation, assuming the Einstein summation convention, the term[6] ^ b · r^e1 · ^e2 ≡ (e1)i;jbj(e2)i (13) where the semicolon denotes a covariant derivative and repeated indices are summed over. However, due to the definition of ^e1 and ^e2 in terms of the magnetic field and the cylindrical symmetry this term goes to 0. Also, since time independent fields are assumed, and ^e1 depends only on the @^e1 ^ ^ magnetic field, both @t and b · (r × b) are also equal to 0. Thus the gyrophase equation simplifies to dθ 1 = Ω + b^ · (r × u ) (14) dt 2 E Final Conditions The final position of the particle can be found from the position of the guiding center and the gyrophase while the final momentum can be found by combining the component parallel to the magnetic field, which is a result of integrating its equation of motion, and the component perpen- dicular to the magnetic field, which depends on the conserved magnetic moment and the magnetic field. The final position of the particle is found from the position of the guiding center and the gyrophase. In the local coordinate system, the final position is located at x = r (cos θ^e1 + sin θ^e2) + X with momentum ^ p = p? sin θ^e1 − p? cos θ^e2 + ub q p where p? = jqj 2mµB. The sign of p? depends on the sign of the charge of the paricle and is determined by the Lorentz force which causes positively charged particles to spiral clockwise and negatively charged particles to spiral counterclockwise.
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