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Afterburner Simulations without tunneling ionization

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					Particle-in-Cell Simulations of Tunneling Ionization
     Effects in Plasma Wakefield Accelerators
                                      David Bruhwiler
                                     Tech-X Corporation




      Tech-X Corporation
      5541 Central Ave., Suite 135
      Boulder, Colorado 80301
      http://www.techxhome.com
                                      Acknowledgements
•     Collaborators:
        – D. Dimitrov, J.R. Cary, E. Esarey, W.P. Leemans, R. Giacone
•     Financial support from the US Department of Energy:
        – DE-FG03-99ER82903 (Tech-X Corp., SBIR program)
        – DE-FG03-20ER83557 (Tech-X Corp., SBIR program)
        – SciDAC (Scientific Discovery through Advanced Computing)
              » “Advanced Computing for 21st Century Accelerator Science & Technology”
              » DE-FG02-01ER41178
        – DE-FG03-95ER40926 (U. Colorado)
        – DE-AC03-76SF00098 (LBNL & use of NERSC)
•     Many helpful scientific discussions:
        –   C. Birdsall, B. Blue, P. Chen, C. Clayton, S. Deng, D. Gordon
        –   B. Hafizi, M. Hogan, R. Hubbard, C. Joshi, G. Joyce, T. Katsouleas
        –   P. Mardahl, K. Marsh, W. Mori, P. Muggli, J. Ng, B. Shadwick
        –   G. Shvets, F. Tsung, D. Umstadter, J. Verboncoeur, J. Wurtele
•     Help with code development & visualization:
        – R. Busby, K. Luetkemeyer, P. Messmer, P. Stoltz



    February 18, 2003 – PIC simulations of tunneling ionization in plasma accelerators – 2nd ORION Workshop   p. 2
                                         Motivation

• Why PIC simulations of ionization in plasma accelerators?
     – Plasma-based accelerators show great promise for the future
     – PIC simulations play a key role, together with theory & experiments
     – Ionization of neutral gas by particles & fields is ubiquitous
• Ionization by fields and electrons are both relevant
     – relativistic impact ionization by both beam and wake electrons
           » elastic electron-neutral scattering can also affect wake dynamics
     – Field-induced tunneling ionization can be much more dramatic
• Laser pulses often must ionize a neutral gas
     – Leads to blue-shifting, pump depletion, can enhance instabilities, …
• Exciting suite of PWFA experiments underway at SLAC
     – E-157 & E-162 (completed), E-164 (approved) and E-164’ (proposed)
     – “Plasma afterburner” or “Energy doubler” concept for SLC
     – Wake fields and beam self-fields can tunnel ionize Li, Cs



 February 18, 2003 – PIC simulations of tunneling ionization in plasma accelerators – 2nd ORION Workshop   p. 3
        The OOPIC (Object-Oriented PIC) Code
• Developed in C++ at UC Berkeley, beginning in 1992
     – Tech-X, UC Berkeley collaboration began in 1998
     – CU Boulder & LBNL joined collaboration in 1999
• 2-D (x-y & r-z), time-explicit, fully electromagnetic
     – Includes Monte Carlo collision (MCC) models
           » Electron-neutral collisions (scattering, excitation, ionization)
           » Similar code structures used to handle tunneling ionization
     – Uses MPI for parallel computing
     – Has a nice GUI for both MS Windows & Unix/Linux
     – Post processing tools (HDF5, IDL, Python) for viz & analysis
• OOPIC is on-line: www.techxhome.com/products/oopic/
• We’re using OOPIC for LWFA & PWFA simulations
• The GUI makes it easy to generate movies:  PWFA

                                                                                              Movie

 February 18, 2003 – PIC simulations of tunneling ionization in plasma accelerators – 2nd ORION Workshop   p. 4
        Parameters for 2-D (r,z) “Afterburner” Simulations
                 (without tunneling ionization)

• 50 GeV e- drive beam
    – σr=20 µm & σz=63 µm
    – 2x1010 e- in the bunch
• r,z are normalized to λe=282 µm
    – e- density is ne=1.4x1016 cm-3
•   Grid size is ∆z=∆r=7.5 µm
•   34,000 beam ptcls (80 per cell)
•   56,000 plasma ptcls (7 per cell)
•   Moving window is used
    – Particles & fields removed at left
    – Cold plasma enters at right




     February 18, 2003 – PIC simulations of tunneling ionization in plasma accelerators – 2nd ORION Workshop   p. 5
Peak fields exceed 16 GV/m                                 tunneling ionization will occur

 •     r,z are normalized to λe=282 µm
 •     fields are normalized to E0=11.4 GV/m
         – cold, nonrelativistic wavebreaking field
         – indicates a nonlinear plasma response
 •     Peaks are separated by λe




     February 18, 2003 – PIC simulations of tunneling ionization in plasma accelerators – 2nd ORION Workshop   p. 6
                   Tunneling Ionization Probability Rate
    •      Adiabatic and quasi-classical approximations are used to calculate the
           probability rate for tunneling ionization:
             – Keldish, Sov. Phys. JETP 20, 1307 (1965).
             – Ammosov, Delone and Krainov, Sov. Phys. JETP 64, 1191 (1986).
    •      Result is an ionization probability rate
             –   Taken from Eq. (1) of ADK paper
             –   Presented here in convenient units
             –   W=1 would imply 100% ionization in 1 s, assuming constant E
             –   This is the model implemented in OOPIC:

                    15 4 ξ i [eV ]        ξ [eV ]                                                 ξ i3 2 [eV ] 
                                                                              2 n*−1
                                                                                          
  [ ]
                                 n*                            32
W s −1    ≈ 1.52 x10                20.5
                                                            i
                                                                                       exp − 6.83               
                       n * Γ(2n *)       E [GV / m] 
                                                     
                                                                                          
                                                                                                  E [GV / m]   

                         E is the local electric field magnitude
                         ξi is the ionization energy
                         n* ≈ 3.69 Z / ξi1/2 [eV] is the “effective” principal quantum number
                         Z is the charge state of the resulting ion (Z=1 for neutral atom)
         February 18, 2003 – PIC simulations of tunneling ionization in plasma accelerators – 2nd ORION Workshop   p. 7
                      ADK Implementation – details
•     Details: here we tabulate ξi, Z and n* for H, He, Li and Cs:
              Table I. Ionization energy and related quantities for H, He, Li and Cs

              Atom/Ion       ξi [eV]         Z          n*        Ecrit [GV/m]
                H             13.6           1        1.00            75.3
                He            24.5           1        0.746           182.
                He+           54.4           2        1.00            602.
                Li            5.39           1        1.59            18.7
                Li+           75.5           2        0.848           985.
                Cs            3.89           1        1.87            11.5
                Cs+           25.1           2        1.47            189.




    February 18, 2003 – PIC simulations of tunneling ionization in plasma accelerators – 2nd ORION Workshop   p. 8
Tunneling Ionization for PWFA Concepts with Li

• What is the relevance to the SLAC experiments?
     – E-157 / E-162 had peak fields of ~1 GV/m or less
     – Afterburner would have fields ~15 GV/m or greater
• When will tunneling ionization occur?
• Specializing the ionization rate eq. for Li yields:

                                       [ ]
                                WLi s −1
                                                   3.60 x10 21
                                                 ≈ 2.18
                                                  E [GV / m]
                                                                   − 85.5. 
                                                                   E [GV / m] 
                                                               exp            
                                                                              

• We apply this equation to the following scenario:
     – σz~100 µm and L=6σz
     – Ttransit~2x10-12 s, PIC time step δt~2x10-14 s

 February 18, 2003 – PIC simulations of tunneling ionization in plasma accelerators – 2nd ORION Workshop   p. 9
 Ionization will dominate Afterburner wake dynamics

• Plot shows fractional ionization of Li for our assumed parameters
• This thought experiment leads to two conclusions
   – For peak fields less than 3 GV/m, ionization is not a concern
   – Peak fields greater than 6 GV/m will cause 100% ionization in some regions
                                   1.E+00
          Fraction of ionized Li




                                   1.E-02


                                                                                         time step
                                   1.E-04
                                                                                         transit time


                                   1.E-06



                                   1.E-08
                                            2   3                4                   5                   6
                                                    Peak Electric field (GV/m)

     February 18, 2003 – PIC simulations of tunneling ionization in plasma accelerators – 2nd ORION Workshop   p. 10
                                    What about Cs and other neutral gasses?
•                    Plot shows fractional ionization in a single PIC time step
                                  – 100% ionization for Cs occurs when fields exceed 3.2 GV/m
                                      » For Li, fields of 6 GV/m are required
                                      » More conventional gasses, like H and He, require much higher fields
                                  – 100% ionization of Cs+ occurs when fields exceed 50 GV/m
                                      » This could be seen in E-164x or an afterburner
                                      » The fields required for ionization of Li+, 400 GV/m, are less likely
                                  1.E+00
    Fraction of Ionized Species




                                  1.E-01
                                                                                                               Cs -> Cs+
                                                                                                               Li -> Li+
                                  1.E-02                                                                       H -> H+
                                                                                                               Cs+ -> Cs++
                                                                                                               He -> He+
                                  1.E-03                                                                       Li+ -> Li++


                                  1.E-04
                                           1                10                    100                 1000
                                                            Peak Electric field (GV/m)
    February 18, 2003 – PIC simulations of tunneling ionization in plasma accelerators – 2nd ORION Workshop         p. 11
Afterburner with tunneling ionization: the wake is disrupted
 •     r,z are normalized to λe=89 µm
         – ne=1.4x1016 cm-3; nLi=1.26x1017 cm-3
         – ntotal = nLi+ne = 1.4x1017 cm-3 = 10*ne
 •     fields are normalized to E0=36.1 GV/m
         – cold, nonrelativistic wavebreaking field
         – indicates a linear plasma response
 •     Peaks are separated by λe
 •     The usual plasma wake has been completely disrupted




     February 18, 2003 – PIC simulations of tunneling ionization in plasma accelerators – 2nd ORION Workshop   p. 12
No preionized plasma, with the correct neutral Li density
•     Variables r & z are normalized to λe=282 µm
        – neutral Li density is nLi=1.4x1016 cm-3
        – thus, the peak e- density is ne=nLi
        – up to 27 Li+/e- pairs are created in each cell where ionization occurs
•     The fields are normalized to E0=11.4 GV/m
        – λe scale length is still visible
        – but fields are small and dominated by noise
•     The gas is not ionized early enough for the wake to form correctly
        – Self-field amplitudes at head of e- drive bunch must be increased




    February 18, 2003 – PIC simulations of tunneling ionization in plasma accelerators – 2nd ORION Workshop   p. 13
A shorter (higher-density) e- drive beam recovers first peak in Ez
   •     50 GeV e- drive beam is now half as long
           – σr=20 µm & σz=30 µm                                                                        PWFA
           – 2x1010 e- in the bunch
   •     Variables r & z are normalized to λe=282 µm
           – neutral Li density is nLi=1.4x1016 cm-3                             With TI
           – thus, the peak e- density is ne=nLi
           – up to 27 Li+/e- pairs are created in each cell where ionization occurs
   •     The fields are normalized to E0=11.4 GV/m
           – first peak in Ez is greater than E0, which indicates a nonlinear response
           – the wake then rapidly loses coherence




       February 18, 2003 – PIC simulations of tunneling ionization in plasma accelerators – 2nd ORION Workshop   p. 14
                                        Conclusions

• Tunneling ionization will dominate the wake dynamics for
  high-field PWFA concepts, such as the plasma afterburner

• A sufficiently high-density e- drive beam can produce a
  strong wake with no pre-ionization
     – Allows future PWFA experiments to dispense with pre-ionization
     – Will reduce the cost and complexity of these experiments
     – Places a greater burden on the linac driver




 February 18, 2003 – PIC simulations of tunneling ionization in plasma accelerators – 2nd ORION Workshop   p. 15

				
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