1.2 Conventional Model
1.2.2 Limitations
With the successful application to supernova remnants for cosmic ray around the knee (1015) eV, which has been confirmed via the x-ray observation[23], DSA is conventionally considered as the possible solution of high energy cosmic ray.
However DAS relies on the random collisions of the high energy particle against magnetic field domains or the shock media. They restrict the accelerating time from the shock lateral size and the strength of background magnetic field, and therefore the maximum energy gain. At very high energy, the collision process in magnetic turbulence necessarily induces severe synchrotron radiation loss, which is proportional to the fourth power ofγ. Compiling above limitations, DAS has difficulties to explain UHECRs. Evidently, novel acceleration mechanisms that can avoid some of the difficulties faced by these conventional models should not be overlooked.
1.3 Plasma Wakefield Acceleration as a Possible Mechanism
Plasma wakefield accelerators[24, 25] are known to possess two salient features:
(i) The plasma can support an extremely high ”acceleration gradient,” i.e., energy gain per unit distance, which does not depend (inversely) on the parti-clesinstantaneous energy or momentum. This is essential to avoid the gradual decrease of efficiency in reaching ultrahigh energies. (ii) The acceleration field is collinear to the particle momentum. Therefore, bending of the trajectory is not necessary in this mechanism. This helps to minimize inherent energy losses that would be severe at ultrahigh energies.
So motivated by these considerations, it was proposed that UHECR can be produced from the plasma wakefield excited in astrophysical setting[26]. In-stead of using laser or charged beam which does not exist in astrophysical en-vironments, Chen et al invoked Alfven shocks as the driving sources to excite plasma wakefields. This idea of using shocks to excite plasma wakefield has at-tracted several astrophysical plasma physicists [27, 28]. Chen et al showed that the power-law spectrum is accounted for the stochastic encounters between the particles and the randomly generated wakefields. Using the short gamma ray burst(GRB) as the working source, Chen et al obtained the maximum accel-erating gradient and predicted the event rate. However, in that paper, their estimation of accelerating gradient was based upon the theory of laser wake-field acceleration without taking the background magnetic effect into account.
Furthermore, this concept has never been validated through computer simula-tion. Thus, we develop a new mechanism of plasma wakefield (the magnetowave induced plasma wakefield acceleration (MPWA) invoking the high frequency and high speed whistler mode as driving pulse, and confirm this concept via computer simulations[29]. The magnetowave with phase velocity vph < c has component|B| > |E| in nature. On the other hand the laser and charged beam have|E| > |B|.
In this thesis, we will discuss the complete theory of MPWA and its appli-cation to UHECR. The content is the following: in Chap.2, the basic plasma physics is viewed to give the way for subsequent discussions. The last section of Chap. 2 introduces the different types of the plasma wakeifled accelerator. In Chap. 3, we start looking at the plasma wakefield acceleration in magnetized plasma. With ω ωc, the magnetic field effect can be ignored. In Chap.4, we focus on the driving pulse withω < ωc (MPWA) and introduce a MPWA condition. The theory of MPWA in linear and nonlinear regimes under the MPWA condition are presented and the limitation of MPWA is also discussed.
In Chap. 5, the particle in cell(PIC) code to produce MPWA is introduced. We show that the simulation results are in good agreements with the theoretical prediction. Finally with the theoretical model established, we apply this mech-anism to explain the UHECR acceleration. In Chap. 6 we obtain the power-law spectrum from the stochastic process of the wakefield acceleration and estimate the accelerating gradient provided by AGN jets. The summary and conclusion are presented in Chap. 7.
Chapter 2
Basic Concept of Plasma
It is known that 99 percent of visible matter in the universe is in plasma state.
All the astrophysical objects, such as stars, relativistic jets, accretion disks, etc.., are made of plasma. Therefore a though understanding of plasma physics could lead to an understanding of 99 percent of the visible universe. Based on that, the idea of plasma wakefield acceleration for UHECR is therefore possible.
In this chapter I will briefly introduce the basic concept of plasma. The cgs unit system is used in the following treatment.
2.1 What is Plasma
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