106 unifying physics of accelerators, lasers and plasma
Plasma frequency in Hz is
f p ∼ 9000 n 1/2 where plasma
density n is in cm −3 .
Maximum accelerating field
in plasma eE max ≈
−3 1/2
1 GeV/cm · n/10 18 cm
.
from Chapter 1 regarding the use of the AS-TRIZ approach
to post-facto analyze the invention of plasma acceleration.
6.1.1 Maximum field in plasma
By using plasma as an accelerating medium, we can remove
the limitation of the accelerating gradient relating to the material’s damage threshold. The maximum field in plasma will
still be limited, but by other factors.
Let us look again at plasma oscillation, as illustrated in
Fig. 4.21, and briefly recall that this diagram allowed us to
estimate the plasma frequency ω p . Assuming that a fraction
of the charges is shifted by distance x, we selected an integral contour that enclosed the displaced charges, and we then
equated, according to Maxwell’s equations, the surface integral of electric field — E · dS to the volume integral of charge
density — ρdV /ε 0 , to obtain the electric field created by the
shifted charges E = nex/ε 0 , which creates the restoring force.
The equation of motion F = md 2 x/dt 2 = −eE = −ne 2 x/ε 0 then
gave us the oscillation frequency ω 2 = ne 2 /(ε 0 m), which, with
p
use of 4πε 0 r e = e 2 /(m e c 2 ), we rewrote as: ω p
2 = 4πnc 2 r e — the
angular plasma frequency.
Very similar calculations allow us to estimate the maximum accelerating field in plasma. Imagine that the plasma
oscillation in Fig. 4.21 is excited by a charged object moving
with velocity c. In the case where a total charge separation is
achieved in plasma, the maximal field is estimated assuming
c
x ∼ λ p ∼
(6.1)
ω p
which results in the following for the maximum field
nec
mcω p
E max ∼
=
(6.2)
ε 0 ω p
e
or equivalently
ω p
eE max � mc
2
(6.3)
c
We can use the practical formula f p ∼ 9000 n 1/2 where n is
defined in cm −3 to obtain a formula for the maximum possible accelerating field in plasma:
eV 1/2
−3
eE max ≈ 1
· n
cm
(6.4)
cm
This means that 1 GeV/cm accelerating gradient can be
achieved for plasma of 10 18 cm −3 density.
Theoretical predictions, made back in 1979, of the principal feasibility of such large accelerating gradients were an
essential driving force towards the development of plasma
acceleration technology.
Plasma frequency in Hz is
f p ∼ 9000 n 1/2 where plasma
density n is in cm −3 .
Maximum accelerating field
in plasma eE max ≈
−3 1/2
1 GeV/cm · n/10 18 cm
.
from Chapter 1 regarding the use of the AS-TRIZ approach
to post-facto analyze the invention of plasma acceleration.
6.1.1 Maximum field in plasma
By using plasma as an accelerating medium, we can remove
the limitation of the accelerating gradient relating to the material’s damage threshold. The maximum field in plasma will
still be limited, but by other factors.
Let us look again at plasma oscillation, as illustrated in
Fig. 4.21, and briefly recall that this diagram allowed us to
estimate the plasma frequency ω p . Assuming that a fraction
of the charges is shifted by distance x, we selected an integral contour that enclosed the displaced charges, and we then
equated, according to Maxwell’s equations, the surface integral of electric field — E · dS to the volume integral of charge
density — ρdV /ε 0 , to obtain the electric field created by the
shifted charges E = nex/ε 0 , which creates the restoring force.
The equation of motion F = md 2 x/dt 2 = −eE = −ne 2 x/ε 0 then
gave us the oscillation frequency ω 2 = ne 2 /(ε 0 m), which, with
p
use of 4πε 0 r e = e 2 /(m e c 2 ), we rewrote as: ω p
2 = 4πnc 2 r e — the
angular plasma frequency.
Very similar calculations allow us to estimate the maximum accelerating field in plasma. Imagine that the plasma
oscillation in Fig. 4.21 is excited by a charged object moving
with velocity c. In the case where a total charge separation is
achieved in plasma, the maximal field is estimated assuming
c
x ∼ λ p ∼
(6.1)
ω p
which results in the following for the maximum field
nec
mcω p
E max ∼
=
(6.2)
ε 0 ω p
e
or equivalently
ω p
eE max � mc
2
(6.3)
c
We can use the practical formula f p ∼ 9000 n 1/2 where n is
defined in cm −3 to obtain a formula for the maximum possible accelerating field in plasma:
eV 1/2
−3
eE max ≈ 1
· n
cm
(6.4)
cm
This means that 1 GeV/cm accelerating gradient can be
achieved for plasma of 10 18 cm −3 density.
Theoretical predictions, made back in 1979, of the principal feasibility of such large accelerating gradients were an
essential driving force towards the development of plasma
acceleration technology.