Alexander Gorn, August 2016, CERN.
gornalexander@gmail.com
Document will be updated before 01.09.2016
1:
The impact of the Earth’s
magnetic field on the electron beam trajectory
For phase one of AWAKE experiment we have the ”oblique” injection
scheme (Fig. ??). There is a problem, that the last diagnostics for the
electron beam is about 2 meters upstream before the beginning of the
plasma cell. To make sure, that the beam enters plasma section, we should
detect it on BTV screen. It seems to be easy to do without plasma, but
the Earth’s magnetic field must be accurately taken into account. Direct
measurements showed, that the magnetic field is almost uniform along all
the plasma cell, it is directed straight down with respect to the ground and
has a magnitude B0 = 0.4Gs. It means, that the electron beam, which
propagates parallely to the axis, will be reflected by this field and will not
pass trough the both irises (Fig. ??).
However, the electron beam enters plasma section at the same time
with the proton beam, which does not change its direction due to the weak
Earth’s magnetic field because of the high energy. Being at the axis, the
proton beam attracts electrons. Relativistic effects make this force weak,
but in principal it may cancel the magnetic field and capture the electrons.
We can check it for a single electron by calculating fields in its rest system.
Using Gauss’s law find electric and magnetic fields, that are generated by
the proton beam with current Ib = 50A (notice, that all the formulas are
1
obtained in CGS units):
2
2Ib 2
− r2
2σr
Er (r) =
),
σ (1 − e
cr r
2
2Ib 2
− r2
2σr
Bφ (r) = −βp σr (1 − e
),
cr
(1.1)
(1.2)
where βp = Vp /c, Vp — proton beam speed. Only the electric field acts on
the electron in its rest system, so after fields transformation
~0 = E
~ k,
E
k
~ ⊥0
E
~ ⊥0
B
~0 = B
~ k,
B
k
i
1 h~
~
~
= γ(E⊥ +
V × B ),
ch
i
1 ~
~
~
= γ(B⊥ −
V × E ),
c
we have
Er0 (r) = γe (1 − βe βp )Er (r) − γe βe B0 ,
(1.3)
where γe — electron beam relativistic factor, βe = Ve /c, Ve — electron
beam speed. For the electron with coordinate r to be in equilibrium
Er0 (r) = 0. To achieve this effect at the point r = σr , which corresponds to
approximately maximum electric field from the proton beam, B0 must be
equal to 0.28 Gs. It means, that the real Earth’s magnetic field with 0.4
Gs magnitude is too high for the electron beam to be captured.
Another option is to pass the electron beam trough the ptasma cell at
some angle to the axis. In the uniform magnetic field an electron has a
circular trajectory with the radius
RB =
pc
,
eB0
(1.4)
thus to pass the both irises the electron must have the folowing angle with
respect to the axis at the plasma cell entrance:
α≈
L
,
2RB
(1.5)
where L — plasma section length. For 16 MeV electron beam RB =
1.(3) × 103 m and α = 3.75 mrad. Of course, we should also take into
2
account fields generated by the proton beam and test our theory for realistic electron beam with nonzero emmitance. We can do it by solving the
folowing equation system numericaly:
d~r
V~ (~r) = ,
dt
p~ = γmV~ (~r),
i
d~p(~r)
e h~
~
~
= −eE − V (~r) × B .
dt
c
(1.6)
(1.7)
(1.8)
In Fig.?? and Fig.?? you can see electron trajectories without proton beam
impact and with it respectively for the electron beam with baseline parameters being focused at the first iris. As it is seen from the figures, the beam
passes trough the whole plasma cell without any losses in both cases.
3
2:
Beam loading in AWAKE
experiment
In phase two of AWAKE experiment [1, 2] it is planned to inject short
high charged electron bunch into the wakefield from the proton driver.
Basically, longitudinal wakefield profile on axis point can be described in
linear approximation as a simple function
Ewz (ξ) = −Ew cos kp ξ,
(2.1)
p
where Ew — wakefield amplitude, kp = 4πn0 e2 /m/c — plasma wave
number, n0 — unperturbed plasma density, e - elementary charge, m —
electron mass, c — speed of light, ξ = z−ct — driver comoving longitudinal
coordinate. As it is seen in the formula, wakefield is obviously nonuniform
along the electron bunch. It necessarily leads to the gain of energy spread
of the bunch during the acceleration, that of course reduces its quality.
However, a high charged bunch also creates wakefield in plasma, which
depends on its charge density profile. It was Simon van der Meer, who
suggested to use triangular beams to cancel a nonuniformity of the wakefield along the beam [3]. To make sure, let us deduce a wakefield generated
by an electron beam with a factorized charge density
ρb = −enb fk (ξ)f⊥ (r)
(2.2)
in uniform plasma column with density n0 and radius R.
Expsessions for the beam wakefields can be found from linearized
Maxwell’s equation system, equation of motion and continuity equation.
This system can be significantly simplified by an introduction of the wake4
field potential, which relates with the wakefields in the following way:
∂Φ
;
∂ξ
∂Φ
Er − Bφ = − .
∂r
Ez = −
(2.3)
(2.4)
Subsequent simplification leads to the equation on the wakefield potential
1 ∂ ∂Φ ωp2
r
− 2 Φ = 4πeδn,
r ∂r ∂r
c
(2.5)
which is basically zero order Bessel’s differential equation with a right part
is depended on plasma density disturbance
Z
ξ
δn = −nb kp f⊥ (r)
fk (ξ0 ) sin (kp (ξ − ξ0 ))dξ0 .
(2.6)
−∞
Resulting expression [4] for the wakefield potential can be factorized as
well as beam charge density (2.2). It means, that we can represent it as
a composition of a dimetional constant and functions F (ξ) and R(r), that
express longitudinal and transverse dependence respectively

r ≤ R,
mc2 nb F (kp ξ)R(kp r),
Φ(ξ, r) =
(2.7)
e n0 0,
r>R
Further on in this document we will measure the wakefield potential in units
of mc2 /c and every density and distance — in units of n0 and 1/kp respectively. Longitudinal dependence term has a simple form and shows how
p
wakefield potential oscilates along ξ with plasma density ωp = 4πn0 e2 /m
(Fig. 2.1):
Z ξ
F (ξ) =
sin (ξ − ξ0 ) fk (ξ0 )dξ0 .
(2.8)
−∞
Transverse dependence term has more complicated structure, but simple
behaviour: it monotoniously decreases from some constant value at the
5
axis point to zero at the plasma boundary (Fig. 2.2):
Z r
K0 (R)
R(r) =
I0 (r) − K0 (r)
r0 I0 (r0 )f⊥ (r0 )dr0 −
I0 (R)
0
Z R K0 (R)
−I0 (r)
r0
I0 (r0 ) − K0 (r0 ) f⊥ (r0 )dr0 .
I0 (R)
r
(2.9)
Longitudinal wakefield at the axis point can be found using (2.14)
Figure 2.1:
Figure 2.2:
Ebz = −E0
nb 0
F (ξ)R(0),
n0
(2.10)
where E0 is the wavebreaking field, which equals to 2.56 GeV for baseline
plasma density n0 = 7 · 1014 cm−3 .
Now consider a gaussian beam with a transverse size σr and a random
triangle longitudinal charge density distribution, which is injected at ξ = ξ0

2
(Aξ + B)e− 2σr r2 ,
ρb = −enb
0,
ξ0 − l ≤ ξ ≤ ξ0 ,
(2.11)
else
and find coefficients A and B, that give
Ez = Ewz + Ebz = const
6
(2.12)
along the electron beam. Using (2.10) and (2.8) we have
Ebz = E0
nb
R(0)((A + Bξ0 ) sin ξ0 − B cos ξ0 ) cos ξ−
n0
− ((A + Bξ0 ) cos ξ0 − B sin ξ0 ) sin ξ − B). (2.13)
Without loss of generality, let the coefficients of cos ξ and sin ξ equal to 1
and 0 respectively, then
A = −ξ0 cos ξ0 + sin ξ0 ,
(2.14)
B = cos ξ0
(2.15)
and
Ewz + Ebz = −Ew cos ξ + E0
nb
nb
R(0) − E0 R(0) cos ξ0 .
n0
n0
Given
nb = n0
Ew
,
E0 R(0)
(2.16)
(2.17)
we get the constant wakefield
Ez = −E0
nb
R(0) cos ξ0 .
n0
(2.18)
To achieve this effect we can not use beams of any length. It is limited
by lmax = tan ξ0 , the length, which corresponds to the triangular beam
profile. The results of LCODE [5–11] simulations for ξ0 = π/6 and l =
√
1/ 3 are shown in Fig.2.3 and Fig.2.4. It shoud be noted, that the effect
depends strongly on beam density, changing beam radius we must increase
its current equally to keep the density on the same level.
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Figure 2.3: Longitudinal wakefield with beamloading (red curve) and
without it (black curve) for Ew = 0.2E0 and corresponding beam current
(blue curve).
Figure 2.4: Longitudinal wakefield with beamloading (red curve) and
without it (black curve) for Ew = 0.05E0 and corresponding beam current
(blue curve).
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