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 = |B|, 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 ê1,ê2, and ê3 where ê3 points in the direction of the magnetic field, denoted as bˆ. The choice of the direction of the two unit vectors perpendicular to ˆ b, ê1 and ê2, 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 ê1 lie in the r-z plane. In these coordinates, the gyrophase, θ, is the azimuthal angle of the particle relative to ê1.[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) |qv × B| 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 −` · ê cos θ = 1 (6) |`| where the sign of θ is the same as that of −` · ê2.

2 Figure 1: ˆz out of the However, for this definition to be useful, the vectors ê1 and ê2 page. must be defined in terms of known variables. In order to define ê1 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 ˆz axis such that ˆx0 points in the ˆr direction and ˆy0 points α ˆy in the φˆ direction. Thus the first rotation angle α is defined by ê2 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 ˆy0 = φˆ axis, from ˆx now called ê2, such that ˆz points in the direction of the magnetic field, bˆ. Because of the cylindrical symmetry, B has no component in the β ê1 ˆr φˆ direction so these two rotations are always sufficient to have a basis bˆ vector defined in the bˆ 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: ê2 out of the page. These two rotations together create an invertible transformation between ˆ (ˆx, ˆy, ˆz) and the desired local coordinate system (ê1, ê2, b) given by

 B B    BxBz y z − Br   ˆe1 BBr BBr B ˆx By B ˆe2 =  − x 0  ˆy (9)    Br Br    bˆ Bx By Bz ˆz B B B

Now that ê1 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 B† u bˆ×E† du B†·E† dt = † mΓ − † dt = q † Bk Bk Bk † u ˆ † µ (11) B = B + q ∇ × b E = E − qΓ ∇B † † ˆ 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 ∂ê 1 = Ω + 1 + v bˆ · ∇ê · ê + bˆ · ∇ × v bˆ + 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] ˆ b · ∇ê1 · ê2 ≡ (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 ê1 and ê2 in terms of the magnetic field and the cylindrical symmetry this term goes to 0. Also, since time independent fields are assumed, and ê1 depends only on the ∂ê1 ˆ ˆ magnetic field, both ∂t and b · (∇ × b) are also equal to 0. Thus the gyrophase equation simplifies to dθ 1 = Ω + bˆ · (∇ × 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 perpendicular 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 θê1 + sin θê2) + X with momentum ˆ p = p⊥ sin θê1 − p⊥ cos θê2 + ub q √ where p⊥ = |q| 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. After transforming these to Cartesian coordinates, the final position and momentum of the particle are B B B B B B B x = r x z cos θ − y sin θ, y z cos θ + x sin θ, − r cos θ + X (15) BBr Br BBr Br B

4 p = BxBz p sin θ + By p cos θ + Bx u, BBr ⊥ Br ⊥ B ByBz Bx By p⊥ sin θ − p⊥ cos θ + u, (16) BBr Br B Br Bz − B p⊥ sin θ + B u

The Nab Spectrometer

From here, all analysis will be concerning specifically simulations in the Nab spectrometer. The image of the spectrometer shown here is not to scale and shows only the qualitative re- gions of the magnetic field. There is also a strong electric field localized at the top of the detector to accelerate a proton into the detector. Neutrons enter into the detector and undergo β decay in the decay volume. Defining the magnetic filter point (labeled in the picture) as z = 0, the decay volume is a cylin- der with the z coordinate in the range (−0.2477, −0.1677) and the r coordinate in the range (0, 0.0313), with units in meters. When a neutron decays in this region the proton has max energy 750 eV and the electron has a max energy of 780 keV and Figure 4: Nab Spectrometer there is no restriction on the direction of travel for a single Diagram[7] particle.[7] The desired maximum error for simulations of the Nab spectrometer is 10−4 meters. Since the path of the particle is not solvable analytically, the only reference to which the guiding center simulation can be compared is a simulation using the Lorentz force. Although this is also a numerical solution which has an associated error, it is the only com- parison available since there is no other way to calculate the error of the guiding center simulation.

Approximation Breakdown

It is important to note that the guiding center approximation is an adiabatic approximation and will break down when B changes too quickly. The approximation assumes that the magnetic moment of the particle’s circle is constant. However, when the magnetic field changes over the course of a single revolution of the particle then this assumption breaks down. So the validity of the approximation is proportional to the normalized change in the magentic field over a revolution ∆B ε ∝ (17) B However, changes in the radial component of the magnetic field is already accounted for in the guiding center equations of motion by the ∇B drift. Thus, only the change in B in the z coordinate will contribute to the breakdown of the approximation. Additionally, if the revolution has a large radius, then it is more likely to sample different fields p⊥ in a single revolution so ε ∝ r = qB as well. Thus the adiabaticity of a given system can be measured by the parameter[8] p ∂B ε = ∂z (18) qB2

5 ∂B where the magnetic fields are taken on the symmetry axis of the system and the derivative ∂z approximates the proportionality to ∆B given above. The guiding center approximation is more valid when ε is small. However, even when ε is small, the final position can still have a relatively large error since the error in the final position ∝ r∆θ for small ∆θ. Since the radius for a typical partical in the Nab spectrometer is 10−3 meters, error in the gyrophase results in a relatively large error in the final position. Moreover, the gyrophase equation of motion given above is a lower order approximation compared to the guiding center equations of motion which causes a greater effect on the error of the gyrophase for large ε.[1]

Figure 5: B and ε vs. z coordinate.

Assuming unit charge and momentum, the value of ε at point in the Nab detector can be seen in the figure in green. The red line qualitatively depicts the relative magnetic field at each point. As can be seen, ε peaks after z = 0, which is the magnetic filter point, when the field strength quickly drops, and spikes again near the end when the magnetic field changes again because the 1 magnetic field is very small here and ε ∝ B2 . In order to maximally reduce the time of a simulation the guiding center simulation must be used for as much of the trajectory as possible. However, in order to keep the final position of the particle within the given error bound, the particle must be simulated by the accurate Lorentz equation simultaion during the portions of the field for which ε is too large. So, starting a prticle in the decay volume, what is the value of ε under which the guiding center approximation is good? The answer to this question is complicated. The maximal value of ε = εmax depends on the initial position and energy of the particle in the decay volume; however, εmax does not depend on the initial direction of the particle’s momentum. The highest values for εmax are when the initial r coordinate, r0, is close to 0 and the initial energy, E0, is large. Under these conditions, εmax ≈ 0.06, using unit momentum and charge in the calculation. The value decreases when r0 increases and when E0 decreases. There is no discernable dependence of the final error on any

6 other initial conditions. As seen in Figure 5, this is a very small value for εmax compared to the max value of over 4. The region of the magnetic field for which the approximation is good, under εmax = 0.04, is marked in between the two black vertical lines: the range z = (1.079, 4.001). There is also another region where ε < εmax in the range z = (−0.2986, −0.1409). The Nab spectrometer also has an electric field and the guiding center approximation breaks down in the region of strong field. However, since the region of strong electric field is in the same area as the spike in ε near z = 5 meters, it is hard to tell how much of the error is due to the electric field. The difference between having the electric field on and off is a strong azimuthal drift of the guiding center due to the E × B term of the guiding center equations of motion.[3] The drift can be as large as 10 cm but averages around 2 cm. The error produced in this region when both fields are on is quite large, and so the approximation is not good when z > 4.001.

Results

Figure 6: Trajectory of particle using Lorentz equation and guiding center approximation.

The primary result to address is whether or not the approximation achieves its goal and a simulation under the guiding center approximation is faster than one using the Lorentz equation. In order to illustrate the diﬀerence, Figure shows the two simulations side by side. The green line shows the Lorentz equation simulation, where each point represents one time step of the simulation. Qualitatively, this trajectory is exactly as expected; it spirals around a ﬁeld line. The red path is the trajectory of the guiding center under the approximation. As can be seen, the approximation was only used in the region in which it returns answers within the given error, as determined above.

7 Qualitatively, the guiding center trajectory has many fewer time steps than the actual particle trajectory, as can be seen by comparing the number of green and red points. The average time step for the Lorentz equation simulation h ≈ 8 × 10−8 seconds whereas the average time step for the guiding center simulation h ≈ 2 × 10−6 seconds. These average step sizes are over a region where the total time elapsed is 10−5 seconds. This means that the guiding center approximation requires almost 3 orders of magnitude fewer steps in a numerical integrator. However, due to the computationally intensive substitution procedure the real time gain is slightly smaller. These calculations were done using a Runge-Kutta integrator with adaptive stepsize. The optimal stepsize is chosen by setting a bound on the difference between fourth and fifth order Runge-Kutta steps.[9] This bound has an effect on the final error of the particle’s position and was optimized such that the final position is within it’s error bound. The step sizes above are the average step sizes for each simulation using this algorithm.

Magnetic Filter

An important feature of the Nab spectrometer is the magnetic filter region, which is labeled in Figure 4. The filter acts as a mirror to particles with an insufficient proportion of momentum in the ˆz direction. We have:[2]

1 B cos2 θ = 1 − sin2 θ (19) e(V −V ) 0 1 − 0 B0 T0 which relates the initial and final angles θ0 and θ, respectively, between a particles momentum and the magnetic field at the point. The relationship depends on the initial and final values of the electric potential, V0 and V , kinetic energy, T0 and T , and magnitude of the magnetic field, B0 and B. Here, e is the charge of the particle. Take the decay volume and the filter region as the initial and final points. At both of these points, the magnetic field is approximately in the ˆz direction only. Assuming that there is no electric field between the decay volume and the filter region, which is a valid assumption in this case, this simplifies to: r pz B 2 = 1 − sin θ0 (20) p B0 From this equation, we see that there is a minimum initial angle between the momentum and the ˆz axis for which a particle can make it through the filter region with positive pz, and therefore into the upper part of the detector. Taking the ratio of magnetic fields B = 1.75 , taken from B0 4 the magnetic field as seen in Figure 5, the minimum value of cos θ0 = 0.722, which is the value at which pz → 0 in the filter. Using the Lorentz equation to simulate the trajectory of the particle numerically, the lower cutoff of cos θ0 = 0.744. The difference here is most likely due to the fact that the theoretical limit is for particles moving along the z-axis whereas the experimental result is for particlaes with a nonzero r coordinate, however the difference could also illustrate the error of the Lorentz equation simulation method. It should be noted that for any particle with initial energy E0 > 0, θ0 is the only initial condition which affects the passage of a particle through the filter region.

8 Field Calculation

One notable difference between using the Lorentz equation as equations of motion and the guiding center equations of motion is that the guiding center equations depend on spatial derivatives of the fields whereas the Lorentz equation does not. The derivatives arrise as a result of the expansion pro- cess in their derivations. Since field derivatives are needed to use the guiding center approximation, then they must be calculated. This inherently depends on the way the field itself is calculated for some given application. In this case, the fields are calculated using Legendre polynomial expansion around some point on the symmetry axis. The derivatives of the fields are given by the following equations: ∞ n 2 ∂Br X −Bn ρ 1 r (z − z ) = (z − z )2P 0 (u) + nr2P 0 (u) − 0 P 00(u) (21) 3 0 n n n ∂r n + 1 ρc ρ ρ n=0 ∞ n 2 ∂Bz X Bn ρ r = n(z − z )P (u) + P 0 (u) (22) 2 0 n n ∂z ρ ρc ρ n=0 ∞ n ∂Bz X r ρ z − z = B nP (u) − 0 P 0 (u) (23) n 2 n n ∂r ρ ρc ρ n=0 Since we can ignore the field due to the charged particle we are simulating, Maxwell’s equations give: ∂B ∂B z = r (24) ∂r ∂z The rest of the spatial derivatives go to 0 because of the cylindrical symmetry; there is no dependence on φ and Bφ = 0. In these equations, Bn are given numerical coefficients from the calculation of the field itself, ρ is the distance from the expansionn point to the field point being calculated, ρc is the radius of convergence for the expansion, z and r are the coordinates of the z−z0 field point, z0 is the coordinate of the expansion point, u = ρ , and Pn(u) are the Legendre polynomials. The values of these polynomials and their derivatives are given below: 2n−1 n−1 P0(u) = 1 P1(u) = u Pn(u) = n uPn−1(u) − n Pn−2(u) 0 0 0 2n−1 0 n 0 P0(u) = 0 P1(u) = 1 Pn(u) = n−1 uPn−1(u) − n−1 Pn−2(u) (25) 00 00 00 0 00 P0 (u) = 0 P1 (u) = 0 Pn (u) = (n + 1)Pn−1(u) + uPn−1(u) The electric field can be calculated by a similar method: ∞ n ∂Er X ρ 1 2 0 2 0 2 00 = φn+1 ns Pn(u) + u Pn(u) − s uPn (u) (26) ∂r ρc ρρc n=0 ∞ n ∂Bz X ρ s 0 = φn+1(n + 1) uPn(u) − nPn(u) (27) ∂r ρc ρρc n=0 ∞ n ∂Bz X ρ −1 2 0 = φn+1(n + 1) nuPn(u) + s Pn(u) (28) ∂z ρc ρρc n=0 ∂B ∂B r = z (29) ∂z ∂r In these equations all the variables are defined the same as above. In addition, φn is a given r numerical coefficient and s = ρ .

9 References

[1] Brizard, Alain Jean, Ph.D., Nonlinear Gyrokinetic Tokamak Physics, Princeton University, 1990, 212 pages; AAT 9012719

[2] Gl¨uck, F., Eur. Phys. J. A 23, 135-146 (2005)

[3] Jackson, J. (1962). Classical Electrodynamics. New York: John Wiley & Sons.

[4] Li, Jinxing; Qin, Hong; Pu, Zuyin; Xie, Lun; Fu, Suiyan, Physics of Plasmas 18, 052902 (2011).

[5] Littlejohn, R.G., J. Plasma Physics 29, 111 (1983).

[6] Littlejohn, R.G., Physical Review A 38, 12 (1988).

[7] R. Alarcon et al., Nab proposal update, 2011 http://nab.phys.virginia.edu/nab doe fund prop.pdf

[8] Tagare, S.G., Physical Review A 34, 2 (1986).

[9] William H. Press , Saul A. Teukolsky , William T. Vetterling , Brian P. Flannery, Numerical recipes in C (2nd ed.): the art of scientiﬁc computing, Cambridge University Press, New York, NY, 1992