Plasma is a partially ionized gas consisting of free negative electrons, positive ions and neutral atoms. When we heat a liquid, we can see more and more vapors created as the temperature rises till reaching the boiled point. After that, all liquid molecules are turned into gas molecules. If we continue to heat the gas, some atoms or molecules will eventually get ionized. Thus in addition to the three thermodynamic states, plasma is sometimes referred to as the fourth state of matter. Because the ionized energy of atoms is of the order 10 eV, plasma is usually created in a very high temperature. But in fact, the atoms still have chances to be ionized at the room temperature due to the tail of thermal distribution. The amount of ionization is very rare so that we can not feel the plasma around us. To estimate the portion of ionization in thermal equilibrium, we can use Saha equation[30]
ni
nn ≈ 2.4 × 1015T3/2
ni e−Ui/kT, (2.1)
where ni and nn are the densities of ionized atoms and of neutral atoms re-spectively, T is the gas temperature in unit of K, and Ui is the ionization energy of the gas. If we take the room temperature T=300K, gas density nn≈ 3 × 1019cm−3, andUi≈ 14.5 eV for nitrogen, we can predict the fraction of ionization
ni
nn+ni ≈ ni
nn ≈ 10−122 which is extremely low.
2.2 Definition of Plasma
Not any ionized gas can be called a plasma. As mentioned above, there is al-ways some small fraction of ionization in any gas. So the plasma is defined from its most important properties, collectiveness and the quasi-neutrality. Since plasma contains charged particles, the moves of these charges can generate local concentrations of positive or negative charges which give rise to electric fields.
The motions of charges also generate currents and then the magnetic fields.
These fields are long-range and could affect the motions of other charged par-ticles far away. To see the effect, let us imagine two small charged regions of plasma separated by a distancer. Even if the Coulomb interaction between the two individual charged particles diminishes as 1/r2, for a given solid angle, one region can feel a total force from the other region with volume increasing as r3. Therefore, elements of plasma can experience a force on one another even at large distances. By ”collectiveness” we mean that plasma motions depend not only on local conditions but on the state of the plasma in remote regions as well.
The quasi-neutrality comes from a fundamental characteristic of the plasma, which is the capability to shield out electric potentials that are applied to it. Suppose we set up an electric field by inserting a ball charged with posi-tive charges, the ball would naturally attract an electron cloud with the same amount of positive charges surrounded. If we assume an electron distribution
which follows the Boltzman’s equation so that
ne(Φ) =n0eeΦ/kBT (2.2)
where Φ, is the potential associated with the slight separation of electrons and ions, n0 is the plasma density at Φ = 0 and T is the electron temperature.
Since an ion is 1800 times heavier than an electron, the ion background can be regarded as motionless. Therefore the ion densityni, where the subscripti denotes the ion background, is approximately equal to the plasma density n0. Considering only the one-dimensional case, the Poisson equation turns into with Eq. (2.2)
∂2Φ
∂z2 = 4πen0
eeΦ/kBT − 1
. (2.3)
In the region where|eΦ/kBT | 1, we can expand the exponential to the first order,
that gives the solution of Φ
Φ = Φ0e−|z|/λD (2.5)
with the characteristic lengthλD defined as λD= the velocity of electron thermal motion. λD is called Debye length, named after the Dutch physicist Peter Debye. If the plasma is cold, T = 0, then λD = 0 and the shielding is perfect. It allows no electric field being presented outside the electron cloud. However if T = 0 , λD is accordingly finite. The potential will be no longer perfectly shielded but decay exponentially with the distance.
Because of the shielding, the distant particles will not feel the existence of the charged ball in the plasma. Therefore for remaining the quasi-neutrality, the condition for a plasma isλD L, where L is the plasma size. In addition, the Debye shielding itself is actually a statistic concept. Thus for the validity of Debye shielding, we should compute the number of particles in a Debye sphere g and require
g ≡ n4
3πλ3D= 1380T3/2n1/2≫ 1, (2.7)
wheren = ne=ni is the plasma density andg is called the plasma parameter.
Combining the two conditions, we can make the criteria for plasma 1) λ L.
2) g ≫ 1.
3) ωτ > 1 .
where ω is the frequency of typical plasma collision and τ is the mean time between collisions. Finally the item 3 requests a low collision rate for plasma.
2.3 Dynamics of Plasma
2.3.1 Fluid Description
Since a typical plasma density might be a huge number of ion-electron pairs per cm−3, it is impossible to deal with each plasma particle. Fortunately, the majority of plasma presents a macroscopic behavior. So we are able to treat plasma as fluids, composed of electrons , ions and neutral atoms. As a result, the motion of individual particle is neglected and only the averaging motion is taken into account. The plasma fluid containing an additional electromagnetic effect is different from an ordinary fluid. Such effect leads to the complexity of plasma, and the varieties of phenomena could occur in a plasma.
In plasma, Maxwell’s equations can tell us how E and B are associated with a given state of the plasma. To maintain the self-consistency, we include equations that describe the plasma response to the E and B field such that
∇ · E = 4πe(ni− ne) = 4πρ
∇ × E = −∂B
∂t
∇ · B = 0
c∇ × B = 4πe(niui− neue) +∂E
∂t = 4πJ +∂E
∂t
whereρ and J are the charge density and charge current given by the plasma, and u is the fluid velocity of from averaging the total velocity in the fluid unit.
These E and B fields above also act back on the plasma species, therefore the equation of motion regarding the electromagnetic force is described,
mjnj
wherej = i, e stands for fluid of ions and electrons respectively. The mj is the mass of the fluid element. The equation above is in Eulerian representation, dealing with the time and space derivatives separately. Sometimes we describe the fluid in either Eulerian (the coordinate scheme) or Lagragian(the co-moving scheme) representations. The relation between the two representations is
d
The second term on the right hand side is called the convective term. Finally combining the above equations and the continuity equation, we obtain the com-plete set of fluid equations
∇ · E = 4π conti-nuity equation and the last equation is the equation of state, withC a constant andγ = Cp/Cv the ratio of specific heats.
2.3.2 Kinetic Description
Beside the fluid theory, the alternative way to describe plasma is the kinetic theory. In most cases, the fluid equations can solve the plasma problems with acceptable good accuracy. But for some special cases, such as the instabilities, the fluid treatment will be inadequate. Thus, we directly look at the distribution
function fj(r, v, t) for each plasma species (here v is the individual velocity).
By knowing the distribution function, we are able to derive the macroscopic physical variables from integrating the function over all velocity spaces. This treatment is called kinetic theory.
The time evolution of distribution functionfj(r, v, t) is govern by the Boltzmann equation, where (δfj/δt)c is the collision term. The plasma density can be obtained from
nj(r, t) =
vfj(r, v, t)d3v, and the average velocity uj is given by
uj(r, t) =
vvfj(r, v, t)d3v,
vfj(r, v, t)d3v
If the plasma is collisionless, the collision term vanishes and Eq. (2.9) takes the form
∂f
∂t + v· ∇f + q
m(E +v
c × B) · ∇vf = 0, (2.10) where we drop the subscriptj. This is called the Vlasov equation, most com-monly studied in the kinetic theory. Regardless of the collision term, the zero moment of Eq. (2.9) is obtained by integrating over the velocity space,
∂f and the next moment is obtained by multiplyingmv to the equation and inte-grating over v such that
m
Taking the above two moments of the Boltzmann’s equation leads to the con-tinuity equation Eq. (2.8e) and the equation of motion of fluid(2.8f). We show that the fluid theory can be derived from the kinetic theory; therefore the fluid and kinetic representations of plasma are equivalent.
2.4 Waves in the Plasma
We have already established the complete set of fluid equations for plasma. To solve these equations, we introduce the perturbation theory describing a small deviation of physical quantities to their equilibrium state. These quantities can be decomposed into the equilibrium solution plus a small perturbation. After taking the Fourier expansion, the perturbations are transformed into a super-position of sinusoidal oscillations in different frequencies. As a result the fluid equations Eq. (2.8a) to (2.8g) can be linearized in (ω, k) space and the result-ing equations are easier to solve. In this section, I review the physics of plasma oscillation and electromagnetic wave in plasma for demonstrating the technique for solving the fluid equations.
2.4.1 Plasma Oscillation
When the electrons in plasma are displaced by some perturbations from a uni-form background of ions, electric fields are built up in such a direction as to restore the neutrality of the plasma by pulling the electrons back to their origi-nal positions. Because of their inertia, the electrons will overshoot and oscillate around their equilibrium positions. The process is known as the plasma oscilla-tion, also known as Langmuir wave, with a characteristic frequencyωp[31]. To derive the expression for the plasma frequencyωp, we solve the fluid equation in the simplest case that (1)B = 0, there is no magnetic field; (2) T = 0, hence
∇p = 0, there are no thermal motions; (3) background ions are fixed; (4) the electron motions only take place along to z direction. The fluid equations in this simplification are obtained,
mne
∂ve
∂t + (ve· ∇)ve
= −eneE (2.13a)
∂n
∂t +∇ · (neve) = 0 (2.13b)
∇ · E = 4πe(ni− ne), (2.13c)
with electron fluid velocity uereplaced by vefor convenience. Since the plasma is assumed to be slightly perturbed, the variables in the equations can be sepa-rated into two parts: the equilibrium part, subscripted by 0, and the
perturba-tion part by 1 that
ne = n0+n1, ve = v0+ v1, E = E0+ E1.
If the plasma is initially stationary, we have the equilibrium state∇n0= v0= E0= 0. The fluid equations are then given by
m∂v1
∂t = −eE1, (2.14a)
∂n1
∂t +n0∇ · v1 = 0, (2.14b)
∇ · E1 = −4πen1, (2.14c)
where the convective term
(v· ∇)v = (v1· ∇)v1
vanishes due to the higher order perturbation. For plasma oscillation, it is assumed that the quantities oscillate sinusoidally and,
n1 = n1ei(kz−ωt) v1 = v1ei(kz−ωt)ˆz E1 = E1ei(kz−ωt)zˆ.
So the time derivative ∂/∂t can be replaced by −iω and the space derivative
∇ can be replaced by ik in the equations. Then the differential equations Eq. (2.14a) to (2.14c) are linearized such that,
−imωv1=−eE1 (2.15a)
−iωn1+n0ikv1= 0 (2.15b)
ik · E1=−4πen1. (2.15c)
where E1, which is associated with the plasma oscillation, is an electrostatic field along thek direction. We can rewrite Eq. (2.15a) by applying Eq. (2.15b) and (2.15c) as
ω2−4πn0e2 m
v1= 0. (2.16)
Eq. (2.16) is the dispersion relation for the plasma oscillation. Because there is nok dependence in this expression, the plasma oscillation does not depend on the wavelength. Hence the phase velocity defined asω/k and the group velocity defined as ∂ω/∂k are both zero. When v1 is finite, a non trivial solution for Eq. (2.16) requires terms in the parentheses to be 0. Therefore the frequency
ω = ωp≡
4πn0e2 m
1/2
. (2.17)
is defined as the plasma frequency. Numerically, with the known physical para-meter numbers, one can make the approximate formula
fp≈ 9000√ n0
which only depends on the plasma density. So far the treatments are all done in cold plasma case (T = 0). For warn plasma (T = 0), the pressure term
∇p should be taken into account in Eq. (2.13a). The dispersion relation then becomes to be 1 in the isothermal sate, is the adiabatic constant for the pressure term Eq. (2.8g). So that the plasma oscillation starts to propagate asymptotically with electron sound speed. Such wave is called the electron acoustic wave.
2.4.2 Electromagnetic Wave in Plasma
Next we study the case of electromagnetic waves in plasma. When an elec-tromagnetic wave travels through a plasma, its associated elecelec-tromagnetic field shall push the charged particles from their original states and the resulting plasma motions will induce the currents that contribute back to the fields them-selves. As a consequence, the dispersion relation of the electromagnetic wave in the plasma contains the plasma effect. If there is no magnetic background field, B0 = E0 = 0, the electric and magnetic fields in plasma, denoted by E1 and B1, are related to each other according to the Maxwell equations
∇ × E1=−1
where the term (4π/c)J1 ≡ −(4π/c)en0v1 is the plasma current. Taking the time derivative on Eq. (2.20), we combine the above two equations and obtain
c2∇ × (∇ × E1) = 4πen0∂v1
∂t +∂E1
∂t . (2.21)
Here v1 is directly related to the oscillating electric field because the second force term v1/c × B1 in Eq. (2.8f) is neglected since it is of the second order.
Assuming a plane wave varying as exp[i(kz − ωt)], the electron velocity v1 is given by
−iωv1=−e
mE1. (2.22)
We then rewrite Eq. (2.21) as
−c2k× (k × E1) =−i4πen0ωv1− iωE1. (2.23)
Substituting Eq. (2.22) to Eq. (2.23), we obtain
−k(k · E1) +k2E1=−4πe2n0
mc2 E1+ω2
c2E1. (2.24)
We note that k· E1= 0 because the wave is transverse. Then the equation can be rearranged as
, which leads to the dispersion relation for electromagnetic wave in unmagnetized plasma.
ω2=ωp2+k2c2. (2.26)
We can calculate the phase velocity vph by using this dispersion relation and obtain
vph= ω
k = c
1−ωωp > c. (2.27)
The phase velocity is real only whenω > ωp. Therefore a threshold of frequency exists for the electromagnetic wave to penetrate into the plasma. If the wave has frequency ω < ωp, it will be reflected by the plasma surface and decays exponentially in the plasma within a skin depth defined asc/ωp. This concept
is the working principle for radio station to transmit the signals. Similarly, The group velocity can be calculated as
vg=∂ω
∂k = c2
vph. (2.28)
Clearlyvg is always smaller that the speed of lightc.
2.5 Plasma Wakefield Acceleration
It has been 30 years since T. Tajima and J. Dawson proposed plasma as an accelerator[24], to transfer electromagnetic energy from a laser pulse into the kinetic energy of the accelerated electron by letting the short laser pulse excites large-amplitude plasma waves. In fact the ”plasma wave” we call here is the plasma oscillation but having a phase velocity exactly equal to the pulse speed.
An electrostatic field relate to this plasma wave is called plasma wakefield. If its phase velocity is closed to the speed of light, a test particle with similar velocity injected to its accelerating phase can surf on the wave and continually gain energy from it. The characteristic accelerating gradient for the plasma wakefield isG = eEwb =mcωp∼
n0[cm]−3 Since the mechanism provides a great accelerating gradient which can accelerate charged particles to very high energy in a short distance, it is very attractive to accelerator physics, plasma physics and astrophysics.
Since then there has been several reviews discussing about the plasma based accelerators [32, 33, 34]. So far there have been three plasma wakefield accelera-tors utilizing laser pulses: laser wakefield accelerator(LWFA), plasma beat-wave accelerator(PBWA) and self-modulated LWFA (SM-LWFA)[32]. In the PBWA [24, 35], two long pulse laser beams with frequencies differed by ωp are used to resonantly excite the plasma wave. This method was first proposed as an alternative to the laser wakefield accelerator because of the lack of technol-ogy for generating ultra-intense picosecond laser pulses at that time. The last one SM-LWFA is somewhat similar to LWFA with a single short pulse but operated at higher density[36, 37, 38]. Therefore SM-LWFA involves a longer length that L > λp and slightly larger laser power P than the critical power Pc = 17ω2/ω2p for relativistic optical guiding. In the high density regime, the pulse becomes self-modulated at the plasma period due to the self-modulation
instability [39, 40, 41]. Then the plasma wave is generated coinciding with the modulated regime. Instead of utilizing laser as driving pulses, Chen et al in 1985 proposed another way to excite the plasma wake by using the charged particle beams [25]. The charged particle which moves relativistically generates the quasi-perpendicular electric field in the lab frame and the magnetic field as well according to the relativistic dynamics. Therefore the charged particle beams behave similar to the laser pulse and the dynamics of plasma wakefield for the two schemes was also shown to be similar by Ruth and Chen[42].
However, either laser beam or charged particle beam is the external impulse and could not be found in the astrophysical environment. Motivated by the ultrahigh energy cosmic ray acceleration issue, Chen, Tajima and Takahashi in 2002 proposed the third type of plasma wakefield acceleration invoking Alfven-shocks as the driving pulses. Different from the laser and the particle beam, Alfven wave is a medium wave which only exist with the support from plasma.
Therefore the wakefield driven by Alfven wave is more relevant to the astro-physical settings. F. Y. Chang et al. [29] extended the concept to the high frequency mode (whistler wave). According to Maxwell’s equations, these waves have theB component exceeding the E component since their phase velocities are less than the speed of light. We categorize such wave as ”magnetowave”.
In the following chapters, I will discuss the plasma wakefield in magnetized plasma and introduce the theory of magnetowave induced plasma wakefield ac-celeration (MPWA) in both linear and nonlinear regimes. I will also present a self-consistent plasma simulation which is performed to validate this theory.
Chapter 3
Plasma Wakefield in Magnetized Plamsa I
We have studied the dispersion relation of electromagnetic wave traveling in plasma (Eq. (2.26)). Once the plasma is imposed a background magnetic field, the electromagnetic wave presents various different modes at arbitrary angles to the external magnetic field. Among that, we concentrate on the modes parallel to the external magnetic field for our purpose to ensure the linear acceleration that minimizes the energy loss. With the parallel background magnetic field B0, the electromagnetic wave becomes circularly polarized and its dispersion relation is given by
ω2=k2c2+ ωip2
1± ωic/ω+ ωp2
1∓ ωc/ω, (3.1)
where the upper (lower) signs denote the right-hand (left-hand) circularly polarized waves. ωc=eB0/mc is the electron cyclotron frequency and the sub-scripti denotes the ion species. Each polarization has two real solutions with high and low frequency branches and both have a frequency cutoff which forms a forbidden gap for wave propagation. Figure 3.1 exhibits the solution of all possible modes and the light curve in vacuum (dashed line) is superimposed.
Above the light curve, there are two curves labeled L and R waves to stand for the left-handed and right-handed circularly polarized electromagnetic waves respectively. Whereas the two solutions below the light curve are the whistler
0 1 2 3 4 5 0
1 2 3 4 5
ckΩp
ΩΩp
L wave R wave
whistler wave
ion cyclron wave
ΩcΩp2
Figure 3.1: The full solutions of Eq. (3.1) withωc/ωp= 2. The two curves (R and L waves) above the light curve (dashed line) would havevph > c and the two curves (whistler wave and ion cyclotron wave) below the light curve would havevph< c.
wave and the ion cyclotron wave, having a lower phase velocity than the speed of light. We call such waves the ”magnetowaves” because of their exceedingB components in all reference frames. To explain the production of UHECR, Chen et al proposed Alfven shocks as the driving pulses for plasma wakefield. Since the Alfven wave is an ion wave, having very low frequency and low phase veloc-ity, it was the first idea of magnetowave induced plasma wakefield acceleration (MPWA).
In fact the non-relativistic plasma wakefield in magnetized plasma was first studied by P. K. Shukla[43] in 1994. Shukla introduced the ponderomotive force from a circularly polarized electromagnetic pulse that is applicable for all frequency range to excite the plasma wakefield. However he only addressed the upper branch issue (the laser case) in his calculation. For R and L waves which have frequenciesω ωc, the dispersion relation Eq. (3.1) can be reduced to that in unmagnetized plasma. Therefore the background magnetic field doesn’t play a significant role to the wakefield excitation. Whereas the wakefield induced by the wave with ω < ωc will greatly determined by the ratio of ωc/ω. In the
following two chapters we will discuss the physics of wakefield induced in the
following two chapters we will discuss the physics of wakefield induced in the