Self-consistent non-stationary theory of multipactor in DLA structures

O. V. Sinitsyn, G. S. Nusinovich, T. M. Antonsen, Jr. and R. Kishek

13th Advanced Accelerator Concepts Workshop, July 27th-August 2nd 2008, Santa Cruz, CA

Outline

• Introduction

• Previous work: 1-D model of multipactor

• Non-stationary 2D-model of multipactor

• Simulation results

• Summary and future work

Introduction• Multipactor (MP) may occur in many situations: one- and two-

surface MP, resonant and poly-phase MP, on the surface of metals and dielectrics etc.

• Below we consider only dielectric loaded accelerator (DLA) structures.

• The starting point for our work is experimental and theoretical studies of such structures jointly done by Argonne National Lab and Naval Research Lab (J. G. Power et al., PRL, 92, 164801, 2004).

• In the theoretical model developed during those studies, the space charge field Edc due to the total number of particles is taken into account as a parameter. We offer a simple non-stationary model where the DC field is taken into account self-consistently.

Previous work: 1-D model of multipactor• Some basic mechanisms of multipactoring can be understood by considering

1-D radial motion of electrons. Such motion near the surface of dielectric can be described by a simple equation of electron motion:

• is the rf phase at the instant of emission, is the axial wavelength.

• The DC field acting on the electron with radial coordinate r is created by charges located at r’ < r. Correspondingly, it can be determined as

When the height of electron trajectories is much smaller than the radius of the dielectric we may neglect the cylindricity and calculate DC field as

sinddc

z

rmr eE eA t

2 /z zk

0

4 r

dc

eE r n r dr

r

4h

dc

y

E e n y dy

Equation of electron motion in normalized variables for this case can be written as:

Here , ,

,

, , where nmax is the maximum electron density.

1D-model (cont.)

2

22

sinh

y

d yt n y dy

dt

/y c y t t ( / ) /d zeA mc r

2 2 2max4 / /e n m max/n n n

•Equation should be supplemented by initial conditions at the dielectric surface for the particle coordinate y: and particle velocity: .

•We assume that initially all electrons are uniformly distributed over the emission phases.

• Initial velocities of true secondaries, which are emitted from the dielectric surface in the process of multipactoring are randomly distributed in the interval which corresponds to kinetic energies from 0 to 20 eV.

0( ) 0y t 0

0/t t

dy dt

1-D model (cont.)

Simulations were done for the parameters shown in the figures, Vmax= 250 eV.

Shortcomings of the 1-D model

• Particles with ‘high altitude’– Depending on initial conditions, some electron sheets during their

motion are not affected or affected by very weak dc field and may be subject to altitude growth without limits. What can we do about it?

– We tried to artificially limit their motion by setting the highest possible height for a sheet. However, subsequent charge accumulation at this fixed level affected the saturation conditions of the process.

• Neglecting the effects of cylindricity– We assumed that most of the particles would stay close to the dielectric

surface where the radial dependence of the rf field amplitude can be neglected. Later, our calculations demonstrated that particles may travel distances comparable with the radius of the vacuum region. Therefore, real trajectories may differ significantly from the ones calculated within the 1-D model.

2-D model of multipactor in a DLA structure• Equations of particle motion:

• RF field components for TE01-mode:

2

, ,

, ,

rdc r rf z rf

r

zz rf r rf

vdv qE E v B

dt r mdv v v

dt rdv q

E v Bdt m

( ), 0 1

( ), 1 1

1

( )0 0, 1 1

1

Re (| | )

Re (| | )| |

Re (| | )| |

z

z

z

i t k zz rf

i t k zzr rf

i t k zrf

E AI k r e

kE i AI k r e

k

B i AI k r ek

Here r and φ are radial and azimuthal coordinates of the particle, vr, vφ and vz are its radial, azimuthal and axial velocities, respectively.

The dc electric field, Edc, acting on the particle is created by cylindrical layers of charge and can be calculated by using the Gauss law.

2-D model of multipactor (cont.)• DC electric field:

Here rd is the radius of vacuum/dielectric interface, σn0 is the initial surface charge density of n-th cylindrical layer of charge. Summation is done over all layers with rn< r.

• If the operation is close to the experimental parameters (f = 11.424 GHz, rd = 5 mm, rw = 7.185 mm, εrd = 9.4), then , and . Also, when we may neglect the z-motion of the particle and reduce the set of equations of particle motion to

Here we have used normalized variables and parameters -normalized wave amplitude and

-plasma parameter; wn is the normalized weight of n-th layer.

( )

010

( )N t

ddc n

n

rE r

r

1 1 1(| | ) 0.5 | |I k r k r 1| | 0.058dk r 0 1(| | ) 1I k r

zv c

2 ( )0 2

31

0

2

1sin( )

N tr

nn

r

vdvw r t

dt r r

drv

dtvd

dt r

0 0/ , , / , /d r r d dr r r t t v v r v v r

2 2 2 20 0| | / /d pe m r

/ 2eA mc

2-D model of multipactor (cont.)• Each time a macro-particle leaves the surface it is assigned a random initial energy

and emission angle according to the following PDFs:

Here Eom corresponds to the peak of the energy distribution.

• We used Vaughan’s empirical model to compute the secondary emission yield for particles with impact energies Vi and impact angle :

Here and are the parameters for the impact angle equal to 0, i.e. normal to the surface, ks is the “smoothness factor” for the surface.

20 0 / 2E mv

e0 0/0

0 20

1( ) , ( ) cos( ),

2 2 2mE E

e e em

Ef E e g

E

i2 2

max max 0 max max 01 , 1 ,2

s i s ik kV V

1.35max

max 0.35max

1 exp (1.844 / )1.379

(1.844 / )i

i

V V

V V

max 0max 0V

Model implementation

Particles uniformly

distributed over initial emission phases.

Each particle has

random initial

velocity and emission

angle

Solve equations of motion for one t-

step

Check particles for

surface impact

Get particle impact energy and calculate secondary emission

yield. wn+1 = wn*δ

Check particle weight

Sort particles by height

Calculate dc field

acting on each

particle

wn<= wmin

Eliminate ‘dead’

particles

Get data for total dc field, particle

location and energy

histograms

Split particle

wn >= wmax

‘Kill’ particle

t = tn?

Assign random velocities and emission angles to the particles

END, save data files

Time loop

Initial number of particles in the simulations Ninit= 1000. Initial weight of each particle winit= 1/Ninit.

Impact = true

wmin < wn < wmax

yes

no

Impact = false

t >= tfinal?

yes

no

Simulation results

• Sample particle trajectories in the presence of a) weaker and b) stronger dc field. Rf field amplitude is the same for both cases.

a) b)

Simulation results (cont.)

α=0.2 corresponds to A=50 MV/m in these simulations, δmax0 = 4. (For alumina, Vmax0= 350-1300 eV and δmax0= 1.5-9 ).

Simulation results (cont.)

Increase in the rf amplitude did not bring to significant changes in the results. α=0.4 corresponds to A = 100 MV/m.

Simulation results (cont.)

Effect of the change of Vmax0.

Results (cont.)

• Some ‘exotic’ results.

Summary and future work• We have developed a 2D model of multipactor in dielectric-loaded

accelerator structures. The model allows to analyze the effect for a reasonably small set of parameters.

• Simulations were done for constant rf amplitude. The effect of the rf pulse shape should be analyzed.

• We need to verify our results with experimental data.

Acknowledgement

• This work has been supported by the US Department of Energy (DoE).