UCLA Department of Physics and Astronomy, 405 Hilgard Ave, Los Angeles, CA 90095. Abstract. Recent proposals for using plasma wakefield accelerators in ...
The Effects of Ion Motion in Very Intense Beamdriven Plasma Wakefield Accelerators J.B. Rosenzweig, AM. Cook, M.C. Thompson, R. Yoder UCLA Department of Physics and Astronomy, 405 Hilgard Ave, Los Angeles, CA 90095
Abstract. Recent proposals for using plasma wakefield accelerators in the blowout regime as a component of a linear collider have included very intense driver and accelerating beams, which have densities many times in excess of the ambient plasma density. The electric fields of these beams are widely known to be large enough to completely expel plasma electrons from the beam path; the expelled electrons often attain relativistic velocities in the process. We examine here another aspect of this high-beam density scenario: the motion of ions. In the lowest order analysis, for both cylindrically symmetric and “flat” beams, it is seen that for the “after-burner” scenario discussed at AAC 2004 the ions completely collapse inside of the electron beam. In this case the ion density is significantly increased, with a large increase in the beam emittance expected as a result. We also examine a less severe scenario, where the ion collapse onset is expected, and new, coupled equilibria in the beam and plasma density are created.
INTRODUCTION The plasma wakefield accelerator (PWFA), driven in the blowout regime1, where the beam is denser than the ambient plasma ( n b > n 0 ), has been the subject of much recent experimental and conceptual investigation. In the blow-out regime, the plasma response is violently nonlinear, as the plasma electrons are ejected from the path of the intense driving electron beam, resulting in an electron-rarefied region. This region can be viewed as containing only ions, €and thus possesses linear electrostatic focusing fields that allow high quality propagation of both the driving2,3,4 and accelerating beams. In addition, the electron-rarefied region has superimposed upon it (TM) longitudinal electromagnetic fields, which, because the phase velocity of the axisymmetric wake is nearly the speed of light, are independent of radial offset from the axis. Thus this wake may accelerate a trailing electron beam just as a traveling wave linac, with strong transverse focusing conveniently supplied by the plasma ions3. The condition n b > n 0 is a defining characteristic of the blowout regime. Indeed, for many scenarios of interest, the self-consisted driving beam density, as well as that of the accelerating beam, greatly exceed n 0 . Under these circumstances, the electric field associated with the beam is high enough that the ions may move significantly € during the beam passage. In fact, for the parameters given by S. Lee, et al.,5 and quoted by Raubenheimer6 in his discussion of the implementation of a PWFA €
afterburner to boost the energy of linear collider at the AAC 2004 workshop, the ions should collapse, as our analysis below will show. This collapse has serious implications for the preservation of the transverse emittance of the accelerating beam, effectively negating the claimed advantage of linear transport in the blowout regime.
ION COLLAPSE IN CYLINDRICALLY SYMMETRIC BEAMS Most previous analysis of the PWFA has been carried out under the assumption of cylindrical symmetry in the beam, and therefore in the plasma response. Threedimensional effects are challenging to approach, and have been investigated in the context of electron-hose instability through time-intensive particle-in-cell and hybrid codes.7 For the purposes of our analysis here, however, we may invoke some well known approximations to give the forms of the electrostatic fields that serve to focus both the beams and the ions. The first approximation is that the net force on the beam arises only from the electrostatic fields of the ions. We assume that even when the ions are allowed to move, their currents to not give rise to appreciable electromagnetic fields. The second approximation is that the ions move under the influence of the electric field of the beam electrons, which do not “see” their own electric field because its force is nearly indentically cancelled by the beam’s self-magnetic field. As the ions will in any event remain non-relativistic, they are essentially unaffected by this magnetic field. These approximations will be useful in approaching the analysis of both 2D and 3D effects in the beam-plasma-ion interaction. TABLE 1. Beam and plasma parameters for linear collider afterburner, from Refs. 5 and 6. Bunch population N b Bunch length σ z Normalized beam energy γ Accelerating € beam emittances ε€n, x (y) Driving € beam emittance ε n,x Plasma ion density € n0 € Ion charge state Z € €
€
€
3× 1010 (driver) 1× 1010 (accelerating) 63 µm (driver) 31 µm (accelerating) 5 × 10 5 (250 GeV) € m - rad ( 4 × 10−8 m - rad ) 4 × 10−6
4 × 10−7 m - rad 2 × 1016 cm-3 1 (hydrogen) €
We will approach the analysis €of the driving and accelerating beam in different € ways. The driver, as it is not to be used directly in the linear collider experiment, may be considered to be axisymmetric, and the analysis of the beam focusing due to the ions proceeds as in Refs. 1 and 2, with the change that the ion density is allowed to vary and exceed n 0 . On the other hand, the accelerating beam must, because of the demands of the final focus (crab-crossing, beamstrahlung mitigation, etc.), have asymmetric emittances εn,x >> εn,y , and therefore asymmetric beam sizes, σ x >> σ y . In this case, € we may make use of the well-known form of the self-electric field inside of such beams, and concentrate on the relevant vertical (y) motion of the ions. The parameters assumed for driving and accelerating beam parameters in our analysis, most taken, as€in Raubenheimer’s presentation, from Ref, 5, are€given in Table 1. As a first approximation, we note that for the emittance of the axisymmetric drive beam,
we have taken the geometric mean of the accelerating beam emittances εn,x = εn,xεn,y . The intricate, unadressed question of how to create the driving and accelerating beams in close proximity (100’s of fs time delay), with such different parameters will remain unaddressed in our discussion. We have also assumed that the ionized species is € hydrogen, in order to avoid multiple ionization, and uncontrolled plasma formation inside the beam. Let us begin with the case of an axisymmetric beam, and apply our analysis to the drive beam. The matched beta-function for the electron beam in plasma is [1] (1) β eq = γ /2πre n 0 = 2γ k −1 p 2
2
−15
where the density n 0 is the ion density, and re = e /me c ≅ 2.82 ×10 m is the classical electron radius. The matched beam area (assuming cylindrical symmetry for the moment) is thus € εn,x γ , (2) σ x2 = β eq = εx = € € 2πre n 0 2πre n 0γ where εn,x is the normalized horizontal emittance, εn,x = εn,y , and we have approximated the velocity of the electron beam as c. The peak beam density is, for a bi-Gaussian beam distribution, € Nb Nb (3) n b,0 = = re n 0γ . € 3/2 2 € (2π ) σ zσ x 2πεn,xσ z The electric field associated with this density near the axis ( r < σ x ) is approximately linear in r, eN b E r = −2πen b,0 r = − re n 0γ r . (4) € εn,xσ z € The radial equation of motion for the ions is simply ZeE r Ze 2 N b ˙r˙ = =− re n 0γ r , (5) Am a Am aεn,xσ z € or in terms of ξ = z − v b t ≅ z − ct as the independent variable, Zr N r′′ = − a b re n 0γ r = −k i2 r , (6) Aεn,xσ z € € ( )′ = d /dξ , r = e 2 /m c 2 = 1.55 ×10−18 m is the classical radius of a singly where a
a
charged ion of one atomic mass unit (amu), and A is the atomic mass in amu. Equation 6€is a simple harmonic oscillator equation with spatial frequency
€
Zra N b re n 0γ . (7) Aεn,xσ z The solution of Eq. 7 is thus simply r(ξ ) = r(0) cos( k iξ ) . One can ask what the total phase advance of the ion motion is in this potential,
€
ki =
2πZra N bσ z re n 0γ . (8) Δφ =€k iΔξ ≅ k i 2πσ z = Aεn,x € If we insert in the numbers for the drive beam from Table 1, we obtain Δφ = 15.7 , where we have approximated the beam as having line-charge density (current) that is € €
€
€
uniform λb = dN /dξ = N b / 2πσ z over the length Δξ ≅ k i 2πσ z . Total collapse is Δφ = π /2 , where one expects an enormous density spike in the ion distribution; we are an order of magnitude past this disastrous increase in ion density. Note that the numbers in Table 1 are ultimately derived from Ref. 5, in which the €assumption is that the beam is round and has € 25 µm rms radial extent, whereas in our example we have self-consistently σ x ≈ 175 nm (!). The difference in ion focusing wavenumber ki is thus a factor of nearly 150 lower in Ref. 5. Even with the value Δφ ≅ 0.1 deduced for this case, the motion of the ions is not completely negligible for the scenario of Ref. 5. We note that the ion motion is of course more relevant to the € trailing, accelerating beam. Since the ions have time to move further after the drive € beam passes, the ion density perturbation is stronger inside of the accelerating beam, even before we examine the self-consistent ion motion within that beam. One may ask if it is possible to choose parameters that ameliorate the ion motion problem. The most direct method would be to use a shorter beam with less charge, and a larger emittance, as all effects reduce Δφ , if only by a square-root dependence. One may not give up beam charge without losing acceleration gradient, however. We note that the bunch length chosen is already near the state-of-the-art, and one may not be able to make it much shorter. On the other hand, the emittance may be made quite a bit larger, at the expense of ease € in manipulating the beam; for example, if the emittance is too large, one may not easily compress the beam to shorter lengths. These issues would need to be solved in the context of producing beams in the afterburner format, with the driver and accelerating bunches in very close longitudinal proximity, and having different transverse characteristics. The ion motion may also be minimized by diminishing either n 0 or γ , but Δφ depends on these parameters only in the one-fourth power, and they are not actually arbitrary, but depend on collider goals (acceleration gradient and final energy desired). Finally, we note that use of a high atomic weight species is perilous, as multiple ionization introduces ion density increases of exactly € are€seeking € to avoid. the type we
SELF-CONSISTENT BEAM-ION EQUILIBRIA We note here that the analysis used underestimates the ion motion, as it has ignored the fact that the beam becomes even denser in reaction to the increase in ion focusing due to the rise in ion density. If we assume that each beam ξ -slice, through fast betatron oscillations, equilibrates with the local ion density without emittance growth, then the beam density can be written in terms of the local ion density, N rγ € n b (ξ ) = b e n i (ξ ) . (9) 2πεn,xσ z Here, for the sake of analysis, we also have used the uniform beam approximation, ignoring both radial and longitudinal variations in the beam density. This relation may be combined with the relation governing the ion dynamics € Zr N r′′(ξ ) = −r0 a b re γn 0 = −r0 k i2 , (10) Aεn,xσ z
€
where r0 is the initial ion position. Equation 10 is trivially solved as r(ξ ) = r0 [1− 12 k i2ξ 2 ] .
(11)
This behavior is different from magnetic self-focusing of beams where the focusing is −2 € proportional to the density of the species itself, e.g. Fr ∝ n ∝ r . Here the relationship between radius and density of the species that focuses (the ions are focused by the € beam) is n b ∝ r−1 , and the simple relationship of Eq. 10 holds. 6
€
5
€
linearized self-consistent
ni/n0
4 3 2 1 0 -1.2
-1
-0.8
-0.6
-0.4
k ζ
-0.2
0
i
FIGURE 1. Predictions of self-consistent (Eq. 12) and linearized treatments of the ion density increase as a function of distance from beam head (uniform beam approximation).
It is perhaps more interesting to write this result in terms of the ion density, r0 2 n0 n i = n 0 = . r [1− 1 k 2ξ 2 ] 2 2
(12)
i
which can be compared to the linear approximation deduced from Eq. 6, r0 2 n0 . n i = n 0 = r cos 2 ( kiξ ) €
(13)
The two results in Eqs. 12 and 13 are illustrated in Fig. 1. The agreement between the self-consistent result of Eq. 12 and the linearized treatment of Eq. 13 is good up to a phase Δφ = k iξ of€about 0.8, where the density increase is roughly 50%, and one indeed may expects trouble in terms of emittance growth of the beam in any case. Thus one may use the linearize result with confidence in predicting when ion collapse can be avoided. €
ION COLLAPSE IN FLAT BEAMS The situation is more constrained for the accelerating beams, which have emittance and charge requirements set by the luminosity of the collider. With asymmetric beams inside of a pre-existing ion channel, one may assume that, ignoring the ion motion, the beam’s equilibrium β -function is given in both x and y, by Eq. 1.
€
For the case discussed by Raubenheimer, the beam sizes σ x,y are a factor of 10 different in dimension. Assuming as is customary that the beam has elliptical symmetry, the transverse electric fields ( E x and E y ) are roughly equal at the beam edges8 ( σ x and σ y ). Thus the relative ion motion that € results in density perturbation is mainly vertical. The vertical electric field inside of €the beam ( x < σ x and y < σ y ) can be again € as linear in y, € approximated € 2eN b re n 0γ 4 πen b,0 ε 2eN b re n 0γ Ey = − y =− y ≈− y, R = n,x >> 1. (14) εn,yσ z (1+ R) εn,yσ z ε€n,yεn,x εn,y (1+ R)
€
The vertical equation of motion is €
y ′′ = −
2Zra N b Aσ z
re n 0γ 2 y = −k i,y y. εn,yεn,x
(15)
The frequency of the ion motion is a factor of 2 larger than in the round beam case, if we assume for comparison as before that εn,yεn,x is equivalent to the emittance εn,x in € — this is the same as assuming the beam densities n are identical. the round beam b,0 Thus in the flat beam scenario the collapse problem is slightly worse, and the phase advance for the example of Ref. 5 is given by € € 4 πZra N bσ z re n 0γ € . (16) Δφ y ≅ k i,y 2πσ z = ≈ 9.1 A εn,yεn,x This is again unacceptably large, and should be mitigated by over an order of magnitude to give perturbations in the ion density that do not result in accelerating beam emittance growth. On the other hand, there is not much room to choose the € parameters freely in the case of the accelerating beam, as stated above, due to luminosity and final energy considerations.
ION COLLAPSE AND EMITTANCE GROWTH The emittance growth that one may expect from the process of ion collapse arises from two distinct mechanisms. The first may be extrapolated from the results given above, in that an increase in ion density along the length of the beam causes the focusing to be dependent on the longitudinal position. This mismatch between the phase space orientations of each beam “slice” can be analyzed based on the results of our self-consistent beam-ion equilbrium analysis. Unfortunately, this effect is likely much smaller than the emittance growth due to nonlinear fields within a given slice. These nonlinearities are expected to arise from the non-uniform distribution of ions that result in turn from the nonlinear fields in the beam that drive the ion collapse. This form of emittance growth is much more difficult to estimate, and should be properly treated by particle-in-cell (PIC) simulations. The self-consistent collapse of the ions and the beam under nonlinear fields will be accompanied not only by beam emittance growth, but also by its analogue, ion heating. These two effects may eventually lead to
a true nonlinear equilibrium akin to a Bennett pinch9. While it is likely that this equilibrium is stable against instabilities in cylindrical symmetry, a flat-beam version may be unstable to quadrupole perturbations. In any case, coupling between the lowemittance (y) dimension and the high emttance (x) dimension will tend to heat the lowemittance dimension.
CONCLUSIONS The problems identified in this paper that arise from ion motion in very high density, high-gradient plasma wakefield scenarios, are serious, and deserve further study. These investigations, beginning with axisymmetric PIC simulations of the parameter sets described in the above sections, are now underway. Given the severity of the ion collapse problem, however, one may anticipate that the design of a PWFAbased collider must change substantially from that deduced from Ref. 6. Two directions come to mind: the first is to lower the gradients and densities, to more moderate regime10. A more radical solution would be to investigate the removal of the ions altogether, and run the PWFA in a hollow capillary plasma, as has been suggested in any case for use in accelerating positrons [5], to completely remove the ions from the beam path. Experimental investigation of the issues involved in ion collapse would be desirable. In the E-164X experiments at the SLAC FFTB, very high gradient wakes have been observed11, at a level well in excess of the design in Ref. [5]. On the other hand, the emittance is much larger in this experiment than in an NLC design, and one may calculate that Δφ ≅ 0.4 , which is just at the onset of ion motion.
ACKNOWLEGMENTS
€ This work is supported by U.S. Dept. of Energy grant DE-FG03-92ER40693.
REFERENCES 1. J.B. Rosenzweig, B. Breizman, T. Katsouleas and J.J. Su, Phys.Rev. A 44, R6189 (1991). 2. N. Barov and J.B. Rosenzweig, Phys. Rev. E 49 4407(1994). 3. N. Barov, M.E. Conde, W. Gai, and J.B. Rosenzweig, Phys. Rev. Lett. 80, 81 (1998). 4. C. Clayton, et al., Phys. Rev. Lett. 88, 154801 (2002) 5. S. Lee, et al., Phys. Rev. ST—Accel. Beams 5, 011001 (2002). 6. T. Raubenheimer, presented at AAC 2004, http://www.bnl.gov/atf/Meetings/AAC04/plenaryabstracts/Monday/TorRaubeheimer.pdf 7. E.S. Dodd, et al., Phys. Rev. Lett. 88, 125001 (2002) 8. P. Lapostolle, IEEE Trans. Nucl. Sci. NS-18, 1101 (1971) 9. J.B. Rosenzweig, et al,, Phys. Fluids B 2, 1376 (1990). 10. J.B. Rosenzweig, et al., Nuclear Instruments and Methods A 410 532 (1998). 11. C. Joshi, presented at AAC 2004 http://www.bnl.gov/atf/Meetings/AAC04/plenaryabstracts/Monday/Chen/AAC04TALK.pdf.
