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

The problem of self-consistent equilibrium presented by K.Lotov at EAAC-2017, 27.09.2017 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)

Equations to solve in the linear regime 3 of 19 presented by K.Lotov at EAAC-2017, 27.09.2017 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

Why the oscillation amplitude distribution is so important? 4 of 19 presented by K.Lotov at EAAC-2017, 27.09.2017 Why the oscillation amplitude distribution is so important? It determines the beam shape for a given potential well and must be chosen realistically

How we solve the problem 5 of 19 presented by K.Lotov at EAAC-2017, 27.09.2017 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

Simplifying assumptions 6 of 19 presented by K.Lotov at EAAC-2017, 27.09.2017 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)

Equilibrium amplitude distribution 7 of 19 presented by K.Lotov at EAAC-2017, 27.09.2017 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

Properties of the equilibrium beam presented by K.Lotov at EAAC-2017, 27.09.2017 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

Comparison with simulations 9 of 19 presented by K.Lotov at EAAC-2017, 27.09.2017 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)

More comparison with simulations 10 of 19 presented by K.Lotov at EAAC-2017, 27.09.2017 More comparison with simulations beam density Good agreement in the applicability area theory

More properties of equilibrium beams presented by K.Lotov at EAAC-2017, 27.09.2017 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 !

Relationship to AWAKE 12 of 19 presented by K.Lotov at EAAC-2017, 27.09.2017 Relationship to AWAKE Beam portrait (2nd half) Excited field (F)

Relationship to AWAKE Bunch shape in a self-modulated beam: 13 of 19 presented by K.Lotov at EAAC-2017, 27.09.2017 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.

Why AWAKE is so special? It is far from the benchmarked area 14 of 19 presented by K.Lotov at EAAC-2017, 27.09.2017 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

Test 1: Long term behavior of a small-amplitude plasma wave presented by K.Lotov at EAAC-2017, 27.09.2017 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:

Test 2: Growth of the seeded self-modulation process. presented by K.Lotov at EAAC-2017, 27.09.2017 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

Finally, a good agreement was reached 17 of 19 presented by K.Lotov at EAAC-2017, 27.09.2017 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 of 19 presented by K.Lotov at EAAC-2017, 27.09.2017 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, 023119 (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.

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