Keplerian motion of particles with permanent electric dipole within cylindrical coaxial electrodes

TL;DR

This paper investigates trapping particles with permanent electric dipoles in cylindrical electrostatic fields, revealing Keplerian orbits and establishing criteria for stable trapping using dimensionless parameters and numerical methods.

Contribution

It introduces the first detailed analysis of Keplerian orbits for electric dipoles in cylindrical electrodes and develops a numerical interpolation method for complex cases.

Findings

Keplerian orbits exist for certain dipole particles within cylindrical electrodes.

A dimensionless criterion for successful trapping on closed orbits is derived.

A numerical interpolation method for local electric field maps is proposed.

Abstract

A research on a possibility of trapping a particle with permanent electric dipole in an electrostatic field has been conducted. For cylindrical coaxial electrodes, Keplerian orbits for some particles were revealed. The exact criterion of successful trapping on a closed orbit within the electrodes is expressed in dimensionless parameters. For more complicated cases where the exact solution is unknown, a useful tool for numerical solution -- local field map interpolation with continuous first derivatives -- is constructed.

Equations

$${\vec{\mu }}_{e(\vec{E})}={\mu }_{e(\vec{E})}\frac{\vec{E}}{E}=\left({\vec{\mu }}_e\cdot \vec{E}\right)\cdot \frac{\vec{E}}{E^2},$$

$${\mu }_{e(\vec{E})}=\frac{3}{2}e{a}_{0}np,$$

$$W_p={\mu }_{e(\vec{E})}E\left(={\mu }_{e(\vec{E})}\left|\vec{E}\left(\vec{R}\right)\right|\right),$$

$$\vec{F}=-\vec{\nabla }{W}_p$$

$${\Phi }_{\mathrm{in}}=\Phi \left(r_{\mathrm{in}}\right)$$

$${\Phi }_{\mathrm{out}}=\Phi \left(r_{\mathrm{out}}\right).$$

$$0=\frac{{\partial }^2\Phi }{\partial r^2}+\frac{1}{r}\cdot \frac{\partial \Phi }{\partial r},$$

$$\Phi =\frac{{\Phi }_{\mathrm{out}}\cdot \ln \frac{r}{r_{\mathrm{in}}}-{\Phi }_{\mathrm{in}}\cdot \ln \frac{r}{r_{\mathrm{out}}}}{\ln \frac{r_{\mathrm{out}}}{r_{\mathrm{in}}}}.$$

$$\vec{E}=\left(E_r,E_{\phi },E_z\right)=\left(-\frac{\partial \Phi }{\partial r},-\frac{1}{r}\cdot \frac{\partial \Phi }{\partial \phi },-\frac{\partial \Phi }{\partial z}\right)=\left(-\frac{{\Phi }_{\mathrm{out}}-{\Phi }_{\mathrm{in}}}{\ln \frac{r_{\mathrm{out}}}{r_{\mathrm{in}}}}\cdot \frac{1}{r},0,0\right)$$

$$E=\left|\vec{E}\right|=\left|\frac{{\Phi }_{\mathrm{out}}-{\Phi }_{\mathrm{in}}}{\ln \frac{r_{\mathrm{out}}}{r_{\mathrm{in}}}}\right|\cdot \frac{1}{r}$$

$$W_p={\mu }_{e(\vec{E})}E={\mu }_{e(\vec{E})}\left|\frac{{\Phi }_{\mathrm{out}}-{\Phi }_{\mathrm{in}}}{\ln \frac{r_{\mathrm{out}}}{r_{\mathrm{in}}}}\right|\cdot \frac{1}{r}.$$

$$W_p=-\frac{\tilde{\kappa }}{r},\tilde{\kappa }>0$$

$${\vec{v}}_0={\vec{v}}_{0(xy)}+{\vec{v}}_{0(z)}$$

$$L=Mvr\sin \beta =M{v}_{0(xy)}{r}_0\sin {\beta }_0=L_0;$$

$$\frac{\frac{L_0^2}{m\tilde{\kappa }}}{r}=1+\sqrt{1+\frac{2W_0{L}_0^2}{m{\tilde{\kappa }}^2}}\cdot \cos \phi ,$$

$$\rho \equiv \frac{r_{\mathrm{in}}}{r_0}\left(<1\right),$$

$$R\equiv \frac{r_{\mathrm{out}}}{r_0}\left(>1\right),$$

$${\eta }_0\equiv \frac{|W_{p0}|}{W_{k0}}\left(>0\right).$$

$$\frac{\frac{2r_0}{{\eta }_0}{\sin }^2{\beta }_0}{r}=1+\sqrt{1+4\left(\frac{1}{{\eta }_0^2}-\frac{1}{{\eta }_0}\right){\sin }^2{\beta }_0}\cdot \cos \phi .$$

$$\rho <\frac{\frac{2}{{\eta }_0}{\sin }^2{\beta }_0}{1+\sqrt{1+4\left(\frac{1}{{\eta }_0^2}-\frac{1}{{\eta }_0}\right){\sin }^2{\beta }_0}},$$

$$R>\frac{\frac{2}{{\eta }_0}{\sin }^2{\beta }_0}{1-\sqrt{1+4\left(\frac{1}{{\eta }_0^2}-\frac{1}{{\eta }_0}\right){\sin }^2{\beta }_0}}.$$

$${\rho }^2-{\rho }^2{\eta }_0<{\sin }^2{\beta }_0-{\eta }_0\rho .$$

$${\rho }^2-{\rho }^2{\eta }_0+{\eta }_0\rho <{\sin }^2{\beta }_0\le 1$$

$${\eta }_0<\frac{1}{\rho (1-\rho )}{\sin }^2{\beta }_0-\frac{\rho }{1-\rho }.$$

$${\eta }_0<\frac{1}{\rho (1-\rho )}{\sin }^2{\beta }_0-\frac{\rho }{1-\rho }$$

$${\eta }_0>-\frac{1}{R(R-1)}{\sin }^2{\beta }_0+\frac{R}{(R-1)}.$$

$$E(X)=\frac{E_{n+1}-E_n}{\Delta X}\left(X-n\Delta X\right)+E_n,\forall X\in I_n.$$

$$E(X)=\left\{\begin{array}[]{l}a_{L}X+b_{L},\hskip 49.79231ptX\in\left<X_{n},X_{n}+\frac{\Delta X}{4}\right)\\ aX^{3}+bX^{2}+cX+d,X\in\left<X_{n}+\frac{\Delta X}{4},X_{n+1}-\frac{\Delta X}{4}\right)\\ a_{R}X+b_{R},\hskip 46.94687ptX\in\left<X_{n+1}-\frac{\Delta X}{4},X_{n+1}\right)\end{array}\right.,$$

$$a_L=\frac{X_{n+1}-X_{n-1}}{2\Delta X}$$

$$E_n=a_L{X}_n+b_L$$

Full text

Keplerian motion of particles with permanent electric

dipole within cylindrical coaxial electrodes

Michal Špaček and Vojtěch Petráček

Department of Physics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Břehová 7, 115 19 Prague 1, Czech Republic

19.2.2019

A research on a possibility of trapping a particle with permanent electric dipole in an electrostatic field has been conducted. For cylindrical coaxial electrodes, Keplerian orbits for some particles were revealed. The exact criterion of successful trapping on a closed orbit within the electrodes is expressed in dimensionless parameters. For more complicated cases where the exact solution is unknown, a useful tool for numerical solution – local field map interpolation with continuous first derivatives – is constructed.

1 Permanent electric dipole – simplified dynamics

It has been suggested [1, 2] that for moderate external fields and non-relativistic velocities, the dynamics of a particle with a permanent electric dipole111A hydrogen atom in o proper quantum state, a water or ozone molecule etc. $\vec{\mu}_{e}$ can take rather simple form. If the dipole of a particle at $\vec{R}$ is considered to align with the local electric intensity $\vec{E}=\vec{E}\left(\vec{R}\right)$ promptly, the torque equation is effectively eliminated, and what is left is a force on a point-like body. As a matter of fact, it is not the whole dipole what aligns with $\vec{E}$ but its projection

[TABLE]

where $E\equiv|\vec{E}|$ and $\mu_{e(\vec{E})}$ denotes the projection’s magnitude which is quantized. In the case of a hydrogen atom, for instance, these discrete values are

[TABLE]

where $e$ is an elementary charge, $a_{0}\left(=\frac{\hbar c}{\alpha m_{e}c^{2}}\right)$ stands for the Bohr’s radius and $n$ and $p$ are the principal and the so called parabolic quantum numbers respectively [3, 4]222Moreover, as stated in [4], the semiclassical interpretation of the alignment is the precession of $\vec{\mu}_{e}$ about $\vec{E}$.. The potential energy of the particle is given as

[TABLE]

where $W$ stands for energy to distinguish it from the electric field. The force on the particle therefore is

[TABLE]

which is apparently (contrary to a simple charged particle) proportional to the field’s derivatives $\frac{\partial E_{i}}{\partial X_{j}}$.

2 Cylindrical electrodes

Let us have a vacuum region between two coaxial electrodes. In cylindrical coordinates (with $z$ coordinate along their joint axis), $r=r_{\mathrm{in}}$ and $r=r_{\mathrm{out}}$ simply represent the surfaces of the electrodes. The boundary conditions of the region are

[TABLE]

[TABLE]

The highly symmetrical setup – $\frac{\partial\Phi}{\partial z}=0$ and $\frac{\partial\Phi}{\partial\varphi}=0$ – reduces the Poisson equation $\Delta\Phi=0$ to

[TABLE]

the unique solution to which (with the boundary conditions (2.1a) and (2.1b) applied) is

[TABLE]

Since it only depends on $r=\sqrt{X^{2}+Y^{2}}$, the electric field in cylindrical coordinates follows as

[TABLE]

which implies

[TABLE]

and eventually

[TABLE]

Since the projections of $\mu_{e(\vec{E})}$ can be both positive and negative, there also can always be particles with potential energy

[TABLE]

which formally matches the potential energy leading to Keplerian orbits.

3 Kepler-like motion

Potential energy (2.7) depends on (cylindrical) radial distance $r$ from the $z$ axis. Ordinary Kepler potential has a point-like source whereas the force $\vec{F}=-\vec{\nabla}W_{p}=\left(-\frac{\tilde{\kappa}}{r^{2}},0,0\right)$ always points perpendicular to the $z$ axis – the source of the force is a line. Equivalently – in true Kepler potential, the force $\vec{F}$ on a particle, its initial position $\vec{R}_{0}$333The origin coincides with the potential centre of symmetry, as usual. and initial velocity $\vec{v}_{0}$ lie in one plane each time, but this is not met in the cylindrical region. However, it could be quickly fixed if the initial velocity is decomposed to directions perpendicular and parallel to the $z$ axis:

[TABLE]

There is no force in the direction parallel to $\vec{v}_{0(z)}$ and the motion is uniform with the velocity $v_{0(z)}$ that way – therefore in the frame where the parallel motion vanishes, $\tilde{\vec{v}}_{0}=\vec{v}_{0(xy)}$ is in the same plane (perpendicular to $z$) as $\tilde{\vec{R}}_{0}=\vec{r}_{0}$ and $\vec{F}=\vec{F}(r)$. The motion is hereby effectively separated into a Keplerian motion in the plane perpendicular to $z$ and uniform translation of that plane along $z$.

For $v_{0(z)}=0$, the angular momentum $L=L_{0}$ is conserved; it can be expressed with radial distance, velocity and direction in the initial time:

[TABLE]

$M$ stands for the particle’s mass and $\beta_{0}\in\left<0,\pi\right>$ is the angle between initial position vector $\vec{r}_{0}$ and initial velocity $\vec{v}_{0(xy)}$. Representation of the quantities is provided in the Figure 1.

Adapted in this way, the problem is straightforward as in any textbook on mechanics and results in an equation of the trajectory in cylindrical coordinates

[TABLE]

where $W_{0}=W_{k0}+W_{p0}=W_{k0}-\left|W_{p0}\right|$ denotes the initial total energy444The form with absolute values holds for $W_{p}<0$ which only leads to closed orbits. ($W_{k0}=\frac{1}{2}Mv_{0(xy)}^{2}$) and the square root is the conic sections’ eccentricity555Only $0\leq e<1$ can result in particle’s successful trapping..

4 Criterion of trapping

There are many parameters of the motion – mass $M$, initial position $r_{0}$, initial velocity $v_{0(xy)}$, initial direction of motion $\beta_{0}$, electrodes’ parameters $\Phi_{\mathrm{in}}$, $r_{\mathrm{in}}$, $\Phi_{\mathrm{out}}$, $r_{\mathrm{out}}$, dipole projection (in case of a hydrogen atom given by two quantum numbers) – at least nine of them. They can be, however, effectively reduced to four dimensionless quantities to test whether the particle stays inside the region for an unlimited time or not: these are the already introduced $\sin\beta_{0}$ and

[TABLE]

[TABLE]

[TABLE]

Only particles with $W_{p}<0$ are taken into account. Equation (3.3) transforms into

[TABLE]

The fundamental condition for successful trapping is $r\in\left(r_{\mathrm{in}},r_{\mathrm{out}}\right)$ or $r_{\mathrm{in}}<r_{\mathrm{min}}\leq r_{\mathrm{max}}<r_{\mathrm{out}}$ where $r_{\mathrm{min}}$ and $r_{\mathrm{max}}$ are the ellipse’s666Circle, as a special case, is also included. apsides. In terms of $R$, $\rho$, $\eta_{0}$, $\sin\beta_{0}$ follows

[TABLE]

[TABLE]

If the square root in the first relation is expressed and then both sides raised to the second power, a simple inequality is obtained:

[TABLE]

This could be understood in two ways:

[TABLE]

[TABLE]

The first implies $\eta_{0}\leq 1+\frac{1}{\rho}$, while the second one puts a restriction on initial conditions in terms both $\eta_{0}$ and $\sin\beta_{0}$:

[TABLE]

The same procedure applied on (4.3b) results in $\eta_{0}\geq 1+\frac{1}{R}$ and

[TABLE]

The requirement of $0\leq e<1$ does not bring any more restrictions. The criterion of a particle being trapped on a closed orbit within the electrodes is therefore given by (4.6), (4.7) and naturally $0\leq\sin\beta_{0}\leq 1$ and represents an area in $\sin\beta_{0}$-$\eta_{0}$ diagram as depicted in the Figure 2.

5 Trap specifics beyond Keplerian potential

The parameters $\eta_{0}$ and $\sin\beta_{0}$ do not describe the state of a particle in a unique way – for a trapped particle with given $r_{0}$, $v_{0(xy)}$ and $\sin\beta_{0}$, there are up to four distinct $\vec{v}_{0(xy)}$. It should also be noted that $\rho$, $R$ and $\eta_{0}$ are not fully independent – for example, radial pulsation of the inner electrode surface also changes the potential which is not the case in gravity.

If a particle with a permanent dipole is inserted between the electrodes through the external electrode777The same holds for the internal, of course. non-tangentially ($r_{0}=r_{\mathrm{out}}\Rightarrow R=1$, $\sin\beta_{0}\neq 1$), it surely hits one of the electrode elsewhere. The trap is therefore suitable for particles produced inside it – for example through charge exchange of ions which obey Lorentz force888The motivation of the analysis is connected to production of antihydrogen or Rydberg matter. – or for particles transported along the $z$ axis. Appropriate perturbation of the electrodes’ boundaries $r=r_{\mathrm{const}}\rightarrow r=r(z)$, in order to trap the particles in the $z$ direction as well, has not been sufficiently explored yet.

6 Numerical approach to electric dipole in electrostatic field

According to symmetry of the given field and the coordinate system, there are up to nine non-zero functions $\frac{\partial E_{i}}{\partial X_{j}}\left(\vec{R}\right)$ on which the force on the particle is dependent. If the analytic form of the field is not known, it is often expressed in the form of a field map in discrete points in space, which has to be interpolated to determine the field in any point where the particle is located. For simplicity, let us examine one-dimensional map of a scalar ($E=E(X)$) in equidistant points $X_{n}=n\Delta X,n\in\left\{N_{i},...,N_{f}\right\},N_{i},N_{f}\in\mathbb{Z}$. Splining the whole map is both lengthy and error-prone, it should be performed locally. However, as it is usually $\vec{E}$ what is mapped, simple linear interpolation

[TABLE]

does not keep the first derivative $\frac{\partial E}{\partial X}$ continuous. Since the force on a dipole particle is proportional to first space derivatives of the (from the field map interpolated) electric field, it leads to non-physical results which resemble apsis precession or perturbing particle’s plane of orbit; example is in Figure 3.

A satisfactory solution in $\left<X_{n},X_{n+1}\right)$999These are neighbouring points in the map: $X_{n+1}-X_{n}=\Delta X$ is a specific cubic interpolation if not only $\left(X_{n},E_{n}\right)$ and $\left(X_{n+1},E_{n+1}\right)$ but also $\left(X_{n-1},E_{n-1}\right)$ and $\left(X_{n+2},E_{n+2}\right)$ are applied. In $\left<X_{n},X_{n}+\frac{\Delta X}{4}\right)$ a straight line101010The lines right in $\left<X_{n},X_{n}+\frac{\Delta X}{4}\right)$ and in $\left<X_{n+1}-\frac{\Delta X}{4},X_{n+1}\right)$ make the evaluation of parameters in (6.2) less complicated and, moreover, half of the interpolation completely avoids the higher polynomial. is constructed to pass through the $\left(X_{n},E_{n}\right)$ point with the slope given by $\left(X_{n-1},E_{n-1}\right)$ and $\left(X_{n+1},E_{n+1}\right)$ as $\frac{X_{n+1}-X_{n-1}}{2\Delta X}$. A line in $\left<X_{n+1}-\frac{\Delta X}{4},X_{n+1}\right)$ is found likewise. The middle half of the interval is eventually interpolated with a polynomial of a degree up to three which connects to the two lines so that the whole function

[TABLE]

is continuous in $\left(X_{n},X_{n+1}\right)$, as well as its first derivative. The coefficients in (6.2) are given by simultaneous equations

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

the solution of which is

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

What makes the interpolated derivative continuous even in the field map vertices $X_{n}$ is that the derivative from left and right are both given by the same formula and by the same values. Example of an interpolation in two neighbouring intervals is in Figure 4.

For a multivariate function, the interpolation has to be carried out multiple times, first for proper points on lines connecting the map vertices, and subsequently in the given point – this is the common Particle in Cell algorithm.

7 Conclusion

In a model of a point-like torque-less particle with permanent electric dipole, a pair of cylindrical coaxial electrodes was found to act on the particle in a way that it undergoes a Keplerian orbit in a special frame. The exact conditions when the particle stays within the electrodes for an unlimited amount of time has been derived and the result discussed. A local interpolation of a field map, the first derivative of which is continuous, has been constructed and its inevitability for numerical simulations of electric dipole particles in electric field given by a field map was justified.