1. 2. Crosschecking computer codes for AWAKE
1. Radial equilibrium of ultrarelativistic particle beams in plasma wakefield accelerators
2. Crosschecking computer codes for AWAKE
Konstantin Lotov
Budker Institute of Nuclear Physics SB RAS, Novosibirsk, Russia
Novosibirsk State University, Novosibirsk, Russia
AWAKE Collaboration

2. The problem of self-consistent equilibrium
presented by K.Lotov at EAAC-2017,
The problem of self-consistent equilibrium
Particle beams in the plasma must reach the radial equilibrium well before they get depleted or accelerated, we want to know the final equilibrium state
The problem is difficult, because:
There are no collisions between beam particles, so the established equilibrium is not a thermal one (not Gaussian)
The focusing force depends on the beam shape
Equations allow for a variety of equilibrium states
(Attempts to solve since 1990s)
(No problem for the blowout regime)

3. Equations to solve in the linear regime
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presented by K.Lotov at EAAC-2017,
Equations to solve in the linear regime
It drives the wakefield:
We have a beam:
The potential well shape
determines the beam shape
We can iteratively find the solution for any (reasonable)
– oscillation amplitude distribution of beam particles

4. Why the oscillation amplitude distribution is so important?
4 of 19
presented by K.Lotov at EAAC-2017,
Why the oscillation amplitude distribution is so important?
It determines the beam shape for a given potential well
and must be chosen realistically

5. How we solve the problem
5 of 19
presented by K.Lotov at EAAC-2017,
How we solve the problem
We simulate some “clean” case and observe the equilibrium state
Understand which simplifying assumptions can we make
Develop the (semi)-analytical theory
Compare the results with simulations, understand the applicability area of the theory

6. Simplifying assumptions
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presented by K.Lotov at EAAC-2017,
Simplifying assumptions
The plasma response is linear (otherwise no analytical solution)
The initial beam emittance is negligibly small (from paper review)
The initial radial profile of the beam is Gaussian (most common case)
Equilibrium beam density and wakefield potential are separable, the equilibrium radial profiles are the same at all cross-sections (from simulations)
The equilibrium is simply related with the initial amplitude distribution (simulations)

7. Equilibrium amplitude distribution
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presented by K.Lotov at EAAC-2017,
Equilibrium amplitude distribution
Look at simulations:
The relationship between the initial amplitude (ra0) and the equilibrium amplitude (ra) is the same in a major part of the beam
We take the initial amplitude distribution (corresponds to the Gaussian beam),
Calculate the equilibrium amplitude distribution
Start iterations, find equilibrium beam shape and potential well profile

8. Properties of the equilibrium beam
presented by K.Lotov at EAAC-2017,
Properties of the equilibrium beam
Beam shape:
strongly peaked near the axis
singular behaviour (1/r)
(the smaller the initial emittance, the higher on-axis density)
not Gaussian
equilibrium
equilibrium with initial amplitude distribution
initial Gaussian
Potential well:
funnel-shaped
(not usual parabolic)
Er = const up to the axis
(no usual linear decrease)
(may be important for ionization by the beam field)
several times deeper than for a Gaussian beam
initial Gaussian
equilibrium

9. Comparison with simulations
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presented by K.Lotov at EAAC-2017,
Comparison with simulations
Excellent agreement for the leading half of the bunch (no vertical scale adjustment)
What is different with the trailing part?
(after some studies we find)
It is accelerated
Our theory is good for drivers
and not for witnesses
beam rms radius
simulations
theory
(½ of initial value)
beam emittance
(60% of rough estimate)

10. More comparison with simulations
10 of 19
presented by K.Lotov at EAAC-2017,
More comparison with simulations
beam density
Good agreement in the applicability area
theory

11. More properties of equilibrium beams
presented by K.Lotov at EAAC-2017,
More properties of equilibrium beams
Betatron frequency
Commonly used estimate
(for Gaussian beam):
For most beam particles, the betatron frequency is much higher than the commonly used estimate
Transverse momentum distribution
Natural scale:
Not Gaussian !

12. Relationship to AWAKE 12 of 19
presented by K.Lotov at EAAC-2017,
Relationship to AWAKE
Beam portrait (2nd half)
Excited field (F)

13. Relationship to AWAKE Bunch shape in a self-modulated beam:
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presented by K.Lotov at EAAC-2017,
Relationship to AWAKE
Bunch shape in a self-modulated beam:
close to the equilibrium one,
different from Gaussian
These are not AWAKE baseline simulations, but an “easy” case for self-modulation studies.
The reason is AWAKE itself is extremely difficult for detailed simulations.

14. Why AWAKE is so special? It is far from the benchmarked area
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presented by K.Lotov at EAAC-2017,
Why AWAKE is so special? It is far from the benchmarked area
Two orders of magnitude jump in window length.
Several acknowledged codes initially produced different results,
so AWAKE became a testing area for PWFA codes,
several of them were finally improved or tuned for consistent
simulations of long-term dynamics
benchmarked area
So AWAKE became a testing area for PWFA codes

15. Test 1: Long term behavior of a small-amplitude plasma wave
presented by K.Lotov at EAAC-2017,
Special tests were developed to make sure the codes simulate basic physical effects correctly
Test 1: Long term behavior of a small-amplitude plasma wave
The period must be close to 2π,
no phase shift after 500 oscillations:
The amplitude must remain constant:

16. Test 2: Growth of the seeded self-modulation process.
presented by K.Lotov at EAAC-2017,
Special tests were developed to make sure the codes simulate basic physical effects correctly
Test 2: Growth of the seeded self-modulation process.
The growth rate and saturation amplitude must be the same

17. Finally, a good agreement was reached
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presented by K.Lotov at EAAC-2017,
Finally, a good agreement was reached
Same instability growth rates
Same saturation fields
Same dynamics of injected electrons
Same beam loading effects
Same final energy spectra of accelerated electrons:
Codes participated in AWAKE studies and cross-checks:
LCODE 2d PIC quasistatic – fast, now used for parameter scans and predicting results of experimental diagnostics
LCODE 2d fluid quasistatic – very fast, earlier studies at lower beam populations
OSIRIS 2d PIC – used for cross-checks and for studies at low plasma densities
VLPL 3d PIC – fundamental studies at lower plasma densities
VLPL 3d PIC quasistatic – fast, recently developed, studies of non-axisymmetric problems
QuickPIC 2d – fast quasistatic, used for benchmarking only
If the experiment will confirm predictions made with the codes, then we can safely use these codes for attacking new parameter areas (higher energies, longer propagation distances, HEP applications)

18. 18 of 19
presented by K.Lotov at EAAC-2017,
To conclude:
Now we know the self-consistent equilibrium state of the particle beam in its own wakefield
It is quite unusual
This will help in making theoretical studies
and in explaining experimental observations
More details: Phys.Plasmas 24, (2017)
Even old and reliable codes need cross-checks when entering new parameter areas
Several codes agree between themselves in AWAKE-like studies, and any new code willing to enter this area must pass basic tests before its results can be taken seriously.
Experiments in new parameter areas are important not only for new physics, but also as ultimate tests for codes. These tests make next steps into new unexplored areas possible.

19. Thank you