Theoretical Evaluation of the Escape Rate of Charged
Particles Trapped in a Potential Energy Well
Yongbin Chang and C. A. Ordonez
Department of Physics, University of North Texas, Denton, Texas 76203
Abstract. In various types of charged particle sources and traps, charged particles are temporarily trapped within a potential
energy well. In the work reported, a theoretical evaluation of the escape rate of trapped charged particles is carried out. As a
specific example, the loss rate is evaluated for trapped plasma particles that are undergoing both collisions among themselves
and collisions with particles of a different plasma species having a different temperature. Conditions are considered in which
both species are confined within a nested Penning trap.
INTRODUCTION
plies to a one component plasma interacting with itself,
as indicated below.) The objective is to obtain the escape
rate for the test particle species as a result of collisions
with the field particle species. As an example application of the theory, ion confinement in a nested Penning
trap equilibrium presented in an accompanying paper is
considered [6].
Various types of ion sources and traps produce a threedimensional electric potential well and temporarily confine ions in the well. Prior to releasing the ions, it is often
important to have good confinement of the ions to minimize ion-wall interactions and the generation of impurity
ions. Examples of ion sources and traps that produce a
three-dimensional electric potential well for ion confinement include the electron beam ion trap/source [1, 2], the
inertial electrostatic confinement approach to ion trapping [3], the Penning fusion ion trapping approach [4],
and the nested Penning trap [5, 6]. The objective of the
study reported here is to assess under what conditions
good ion confinement occurs.
Existing theory, that could be used for such predictions, was developed primarily for describing hot fusion
plasmas in magnetic mirrors [7]. The existing theory
has a number of limitations that make it unsuitable for
evaluating the confinement properties of plasma particles trapped within a three-dimensional electric potential
well. The existing theory was (1) developed for plasmas
trapped parallel to a magnetic field by a combination of
electric and magnetic gradient fields; (2) developed in the
limit that the ratio of the plasma temperature (in energy
units) to the potential energy well depth approaches zero;
and (3) developed assuming that the Coulomb logarithm
has a value larger than 10. Here we report on a theory
that applies for three-dimensional electric confinement
of plasma particles. The theory applies for any ratio of
the plasma temperature and the depth of the potential energy well within which the particles are trapped, and any
value of the Coulomb logarithm. To develop the theory,
two species of plasma particles are considered, a test particle and a field particle. (The approach nevertheless ap-
PROBABILITY FUNCTION
When a test particle is put into a Maxwellian background
of field particles, the test particle experiences Brownian motion in velocity space due to collisions. An ideal
description of the Brownian motion should be in terms
of a probability function, P(v, ∆v), which represents
the probability of a test particle with initial velocity, v,
to experience a velocity change, ∆v, due to a collision.
A derivation of the probability function, P(v, ∆v), has
been presented in our previous papers [8, 9]. Here, we
outline the derivation. The background particles have a
Maxwellian velocity distribution [10], given by
m 3/2
mF v2F
F
exp −
,
(1)
f0 (vF ) = nF
2πkT
2kT
where vF and mF are the velocity and mass of a field particle, respectively, nF and T are the density and temperature of the field particles, respectively, and k is Boltzmann’s constant. If there is a charged test particle with
velocity, v, within the field particle background, the test
particle will experience Coulomb collisions with field
particles. The collision frequency can be calculated from
Z
ν(v) =
f0 (vF )|v − vF |σR dΩdvF ,
(2)
CP680, Application of Accelerators in Research and Industry: 17th Int'l. Conference, edited by J. L. Duggan and I. L. Morgan
© 2003 American Institute of Physics 0-7354-0149-7/03/$20.00
148
0
Considering that ∆v = v −v, we can express the function
0
as C(v, v ), where v and v0 are test particle velocities
before and after a collision. This function has a symmetry
0
0
property, C(v, v ) = C(v , v). This symmetry property is
important in the proof of conservation of particle number
for a simplified linear Boltzmann collision integral [9].
where the solid differential angle is dΩ = sin θ dθ dϕ,
and the Rutherford differential scattering cross-section is
[11]
2
1
ZZF e2
σR =
.
(3)
4
4πε0 µ
4|v − vF | sin4 (θ /2)
Here Z, ZF are the charge-states of the test and field
particles, respectively; e is the unit charge; ε0 is the
permittivity of free space; µ = mmF /(m + mF ) is the
reduced mass; and m is the test particle mass.
In order to derive the probability function, Eq. (2) can
be calculated by changing the integration variables from
the velocity of the field particle, vF , to the change in
velocity of the test particle, ∆v. It is interesting to note
that the integral over scattering angle can be calculated
exactly without the usual divergence difficulty. After the
variable change and the integration over the scattering
angle, Eq. (2) can be written as [12]
Z
ν(v) =
P(v, ∆v)d∆v,
ESCAPE RATE
Assume a plasma is confined in a three-dimensional electric potential well, such as created by a nested Penning
trap [6]. Because the potential well cannot be infinitely
deep, the particles that are confined in the well have a
limited kinetic energy. Any particle that has an energy
larger than some threshold value cannot be confined.
Collisions may cause some particles to acquire enough
energy to escape the system. The purpose of this section
is to derive an escape rate from the probability function,
P(v, ∆v).
For a three-dimensional square well, we can define an
“escape” collision as a collision that causes the speed
of a test particle to be larger than ve . Here, ve is the
threshold speed for escape from the system considered.
The initial test particle speed, v, must be smaller than the
escape speed, ve . The escape frequency is calculated by
integrating the function, P(v, ∆v).
Based on the value of the collision strength (defined to
be proportional to the magnitude of the velocity change
∆v), there are three different regions: a very strong collision range (∆v ≥ v + ve ), a mixed range (ve − v ≤ ∆v ≤
v + ve ), and a weak collision range (∆v ≤ ve − v). When
the collision is in the strong collision range, the speed
of the test particle after the collision is larger than ve ,
and it will escape regardless of the direction of the velocity change, ∆v. When in the weak range, the speed of
the test particle after the collision is less than ve . Therefore, the lowest integral limit for the escape frequency
must be ∆v ≥ ve − v. In considering the mixed range,
we define χ as the angle between ∆v and v. Then, only
collisions associated with a sufficiently small value of χ
may cause the test particle speed to exceed ve . Based on
the above considerations, the escape frequency, p, can be
calculated as
(4)
where the probobility function, P(v, ∆v), is
2
ν0
∆v −5 − av∆v + vv · ∆v
th
th ∆v
e
.
P(v, ∆v) = √
( πavth )3 avth
(5)
Here a close collision frequency ν0 is defined as
ν0 = nF vth πρ 2 ;
(6)
and the interaction radius ρ is defined by
ρ=
ZZF e2
0;
8πε0 kT
(7)
0
the reduced temperature is defined as T = µT /mF ; the
mass coefficient is a = 2mF /(m + mF ); and
p the thermal
speed of the field particle is defined vth = 2kT /mF .
Two basic properties of the probability function are
easily found from Eq. (5). The factor (∆v)−5 in Eq. (5)
shows that weak collision events dominate Coulomb interactions. The probability of weak (small ∆v) collisions
is much larger than for strong (large ∆v) collisions. The
exponential factor indicates that v·∆v > 0 collisions have
less probability of occurring than v · ∆v < 0 collisions. A
test particle passing through a plasma of field particles
typically feels more collisions that decelerate its forward
motion than collisions that accelerate its forward motion.
This is the basis for the dynamic friction experienced by
a test particle.
If we multiply a test particle’s Maxwellian velocity
distribution function, f0 (v), with the probability function
Eq. (5), we define the function
C = f0 (v)P(v, ∆v).
p = p1 + p2
Z
=
∆v≥ve +v
P(v, ∆v)d∆v +
Z ∆v<ve +v
∆v>ve −v
P(v, ∆v)d∆v, (9)
where p1 represents the escape frequency in the very
strong collision range, and p2 is the contribution from
the mixed range.
If we introduce the dimensionless variables uδ =
∆v/(avth ) and u = v/vth , and use spherical coordinates,
(8)
149
we have
Z ∞ Z π Z 2π
ν0
p1 =
×u2δ
ve +v
avth
0
π 3/2
0
plasma (assumed to have a uniform density). For the integral over g(v), the upper limit has been set to ve because
no test particle has a speed larger than this magnitude. A
difficulty in calculating dN
dt is the integration over p(v).
When the speed of the test particle is close to the escape
speed, ve , the lower integral limit ve − v approaches 0,
and p(v → ve ) −→ ∞. However, we can avoid the singularity by employing a “lowered” Maxwellian velocity
distribution for the test particles [13],
h
i
2
2
gLM (v) ∝ v2 e−(v/vthT ) − e−(ve /vthT ) Θ(ve − v), (17)
2
−(uδ +u cos χ)
u−5
δ e
sin χdχdφ duδ .
(10)
After integrating over the angles φ and χ, we get
Z ∞
p 1 = ν0
ue +u
a
erf(uδ + u) − erf(uδ − u)
duδ ,
uu3δ
(11)
where ue = ve /vth . To calculate the escape frequency
from the mixed range, we integrate χ from 0 to a maximum value, χmax ,
ve +v
avth
Z
p2 =
Z χmax Z 2π
ν0
ve −v
avth
×u2δ
where vthT is the thermal speed of the test particles and Θ
is the Heaviside step function. The lowered Maxwellian
distribution makes the integral for dN
dt numerically calculable since the lowered Maxwellian distribution equals 0
when the speed of the test particle is equal to or greater
than ve . We do not use an equal sign in Eq. (17), because
the normalization constant is not included.
If the lowered Maxwellian distribution is normalized
to unity, the probability for one test particle to escape the
system per second is
0
0
−(uδ +u cos χ)2
u−5
δ e
3/2
π
sin χdχdφ duδ .
(12)
The angle χmax is determined by a relation between the
velocity of the test particle, v, velocity change, ∆v, and
minimum escape speed, ve , given by
u2 − u2 auδ
v2e − v2 − ∆v2
= e
−
.
2v∆v
2auuδ
2u
cos χmax =
Z ue
(13)
pE =
If we integrate over angle in Eq. (12), the escape frequency from the mixed range can be reduced to
erf(uδ + u) − erf(uδ + u cos χmax )
duδ .
uu3δ
(14)
The escape collision frequency for a single test particle
is
p2 = ν0
ue −u
a
Z ∞
p(u) = ν0
Z
+ν0
ue +u
a
ue −u
a
ue +u
a
erf(uδ + u) − erf(uδ − u)
duδ
uu3δ
erf(uδ + u) − erf(Auδ + uB )
δ
uu3δ
duδ ,
(15)
where A = 1 − (a/2) = m/(m + mF ), and B = (u2e −
u2 )/(2a). Considering the term u−3
δ in the integrand and
the difference of the lower integral limits of the two
integrals, the second integral of Eq. (15) is expected to
be more important than the first.
We assume that the test particle species has a nearMaxwellian velocity distribution. The escape rate of the
test particle species can be calculated by integrating over
the velocity distribution of test particles:
dN
=V
dt
Z ve
Z
p(v) f (v)dv = V
p(v)g(v)dv,
p(u)gLM (u)du,
(18)
where the escape frequency p(u) is given by Eq. (15).
We can calculate the escape time scale from τ = 1/pE .
If the total number of test particles is N, the escape rate
is then pE N, which is the number of particles that escape
from the system per second.
The escape frequency, p(u), given by Eq. (15) is expressed in terms
p of a dimensionless speed u = v/vth ,
where vth = 2kT /mF is the thermal speed of the field
particles. To calculate Eq. (18), it is more convenient to
express the lowered Maxwellian distribution in terms of
u. Next, we provide example calculations of the escape
rate for ion confinement in a nested Penning trap equilibrium presented in an accompanying paper [6].
Suppose there exists a plasma of trapped protons,
which have a uniform density of 1016 m−3 and a nearMaxwellian velocity distribution. Assume that protons
with kinetic energy greater than 3100 eV escape confinement. Assume the protons interact with an electron
plasma that has a uniform density of 8.4 × 1016 m−3 and
a Maxwellian velocity distribution associated with a temperature of 3000 eV. We can calculate the proton escape
rate due to proton-proton collisions and electron-proton
collisions for different proton temperatures. First, we calculate the proton escape rate due to electron-proton collisions when the proton temperature is 300 eV.
For the above parameters, we calculate the thermal
speed of the field particles (electrons) to be vth = 3.25 ×
107 m/s and the thermal speed of the test particles (protons) to be vthT = 2.40 × 105 m/s. The escape speed
ue +u
a
Z
0
(16)
0
where f (v) is the test particle velocity distribution function; g(v) = 4π f (v)v2 is the test particle speed distribution function; and V is the total volume of the test particle
150
TABLE 1. Escape time versus plasma
temperature
Proton
Temperature (eV)
Escape
Time (s)
300
600
900
1200
1500
1800
4.08 × 105
3.88 × 103
9.34 × 102
4.77 × 102
3.25 × 102
2.54 × 102
No. PHY-0099617 and the Texas Advanced Research
Program under Grant No. 3594-0003-2001.
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is ve = 7.71 × 105 m/s. To obtain the dimensionless
speed, u, we use the field particle thermal speed, namely,
vth = 3.25 × 107 m/s. Therefore, the escape speed normalized to the thermal speed of the field particles is ue =
0.0237. To obtain the dimensionless normalized distribution function for the test particles, we have (vth /vthT )2 =
18361.5. If we normalize the lowered Maxwellian distribution function to unity, the distribution, gLM (u), has the
form
2
gLM (u) = 5.62 × 106 u2 (e−18361.5u − 3.25 × 10−5 )
×Θ(0.0237 − u).
(19)
From Eq. (15), the escape frequency p(u) can be calculated directly. With the escape frequency p(u), and the
lowered Maxwellian distribution gLM (u), we can integrate Eq. (18) to obtain an escape time of τ = 2.04 ×
109 s. The escape time is very long, which means the escape probability is very small. The escape probability is
very small mainly due to the mass of the field particle
being small compared to the mass of a test particle. It is
very unlikely for an electron to knock a proton out of the
well during a collision for the parameters considered.
Due to the mass difference, the proton-proton collision
escape rate can be expected to be much larger than the
electron-proton collision escape rate. Thus, escape predominantly occurs due to proton-proton collisions. In Table 1, the proton escape time due to proton-proton collisions is calculated for different proton temperatures. We
find that the higher the temperature, the smaller the escape time. It can be concluded from the results presented
in Table 1 that for ions trapped in a three-dimensional
electric potential well under the conditions considered,
good ion confinement occurs when the ion temperature
multiplied by Boltzmann’s constant is much smaller than
the potential energy well in which the ions are trapped.
ACKNOWLEDGMENTS
The authors would like to thank J. R. Correa for helpful discussions. This material is based upon work supported by the National Science Foundation under Grant
151