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7/31/2019 Duesseldorf T(Hot)
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Scaling of the hot electron
temperature and laser absorption
in fast ignition
Malcolm Haines
Imperial College, London
Collaborators: M.S.Wei, F.N.Beg (UCSD, La Jolla) and
R.B.Stephens (General Atomics, San Diego)
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Outline
A simple energy flux model reproduces Begs
(I2)1/3 scaling for Thot.
A fully relativistic black-box model includingmomentum conservation extends this to higherintensities.
The effect of reflected laser light from the
electrons is added, leading to an upper limit onreflectivity as a function of intensity.
The relativistic motion of an electron in the laserfield confirms the importance of the skin-depth.
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Begs empirical scaling of
Th(keV)=215(I18
2
m)
1/3
for 70 < Th < 400keV & 0.03 < I18 < 6 can be
found from a simple approximate model:
Assume that I is absorbed, resulting in a non-relativistic inward energy flux of electrons:
and
Relativistic quiver motion gives
I1
2
nhmevh3
vosc
c
eE0
mec a
0as
vosc
c1
vh 2eTh /me 1/2
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nh is the relativistic critical density
Taking the 2/3 power of this gives Eq.1
or
nh nc 4
2
me
0e22
I22me
2c3a0
2
0e2
2
1
2
42mea0
0e2
2
me2eTh
me
3/2
Thmec
2
2ea0
2/ 3
Th(keV) 230(I18m
2)1/ 3
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Model 2: Fully relativistic with energy and
momentum balance
Momentum conservation is
where
consistent with electron motion in a plane wave
I nhme (h 1)vzc2
ncpz (h 1)c2
I
c nh pzvz
ncpz2
me
mehvz pz
pz
mec
pz h 1
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h depends on the total velocity of an electron.
Transform to the axial rest-frame of the beam:
Equate E0 to me0c2; 0 indicates the thermal
energy in the rest frame of the beam; becausetransverse momenta are unaffected by the
transformation
E0
2 E2 pz2c2 me
2c4
1pzmec
2
pz2
me2c2
me
2c41
2pzmec
eTh mec2(0 1) mec
21
2
mec
meI
ncc
1/ 2
1/2
1
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In dimensionless parameters, th = eTh/mec2 and a0,
th = (1+2
1/2
a0)
1/2
- 1 (2)
This contrasts with the ponderomotive scaling:
th = (1+a02)1/2 - 1 S.C.Wilks et alPRL(1992)69,1383
Simple model of Beg scaling, Eq.1, gives
th = 0.5 a02/3 (3)
Eqs (2) and (3) agree to within 12% over the range
0.3
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Various scaling laws; Begs empirical law is almost identical to Haines-
classical and relativistic up to I = 51018 Wcm-2
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Model 3: Addition of reflected or back-
scattered laser light
When light is reflected, twice the photon momentum is
deposited on the reflecting medium; thus the electrons
will be more beam-like, and we will find that Thot isreduced.
The accelerating electrons will form a moving mirror,but the return cold electrons ensure that the net Jz, and
thus the mean axial velocity of the interacting electrons
is zero.
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If absorbed fraction is abs, energy conservation is
I - (1-abs
)I = nc
pz
(h
-1)c2 (4)
while momentum flux conservation is
I/c + (1-abs)I/c = ncpz2/me (5)
Define Ir= (1-abs)I; (5)c+(4) gives2I = ncpzc
2[pz/mec + (h - 1)], while (5)c-(4) gives
2Ir= ncpzc2[pz/mec - (h-1)], or dimensionlessly
ii = 2I/ncpzc2 = pz' + h - 1 (6)ir= 2Ir/ncpzc
2 = pz' - h + 1 (7)
where pz' = pz/mec
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As before, transform the energy to the beam rest-frame
E02 = E2 - pz
2c2 = (hmec2)2 - pz
2c2
= me2c4(h
2-pz'2) = me
2c402
Hence Th as measured in the beam rest frame is
th = eTh/mec2 = 0 - 1 = [(h+pz')(h- pz')]
1/2 - 1
= [(1+ii)(1- ir)]
1/2
- 1Use (6) and (7) to eliminate pz' to give ii+ir=2pz'.
Define r = ir/ii ; then ii = 21/2ao(1+r)
-1/2 and
th = [{1 + 21/2
a0/(1+r)1/2
}{1 - 21/2
a0r/(1+r)1/2
}]1/2
- 1 (8)This becomes Eq (2) for r = 0, and for r > 0, th is reduced.
The condition th > 0 becomes
f (r ) (1 - r2)(1 - r)/(2r2) > a02 and df/dr
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Defining as f(r) 2a02 where > 1, th becomes
th = {[1 +(1-r)/(r)][1 -(1-r)/]}1/2 - 1
Using r, (0
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Table of f(r) and th() versus r
r = 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
f(r) = 44.6 9.6 3.54 1.58 .75 .356 .156 .0563 .0117 0
th(1.1) .265 .125 .065 .0365 .0204 .011 .0053.0021.0046 0
th(1.2) .44 .202 .108 .0607 .0341 .0184 .0089.0035.0077 0th(2) .739 .342 .187 .107 .0607 .0328 .0159 .0062 .0014 0
For a given value of (intensity) f( r) must be larger than
this, leading to a restriction on r (reflectivity). th is
tabulated for 3 values of where > 1
a02
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Restriction of the fraction of laser light
reflected or back-scattered
For a given value of (i.e. intensity) f(r) must be
larger than this which then leads to a restriction on
the fraction of light reflected.For example we require r < 0.1 for = 45,
i.e. I = 6 1019 Wcm-2.
The low Thot
and low reflectivity are advantageous
to fast ignition, but require further experimental
verification, additional physics in the theory, and
simulations.
a02
a02
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Relativistic motion of an electron in a plane
e.m. wave
In a plane polarized e.m.wave (Ex,By) of arbitrary form in
vacuum an electron starting from rest at Ex=0 will satisfy
pz=px2/2mc
A wave E0sin(t-kz) and proper time gives
x/c = a0 (s - sin s)
z/c = a02( 3s/4 - sin s + 8-1 sin 2s)t = s + a0
2( 3s/4 - sin s + 8-1 sin 2s)
in a full period of the wave as seen by the moving
electron i.e. s=2, forward displacement is z = 3a0
2/4.
s dt/
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But in an overdense plasma c/pe < /2.
for a0 ~ 1 an electron will traverse a
distance greater than the skin depth withoutseeing even a quarter of a wavelength, i.e. the
electron will not attain the full ponderomotive
potential, before leaving the interactionregion.
Thus it can be understood why the Thot scaling
leads to a lower temperature.However if there is a significant laser prepulse
leading to an under-dense precursor plasma,
electrons here will experience the full field.
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Relativistic collisionless skin-depth
Jx ncriteca0 (1 coss) 1
0
By
zme
e0
a0
zsins
z c
a02 3s
5
5! a
0
a0/z
s 802
a02
p2
1/ 6
0.963c
p
p
2/ 3
a0
1/ 3
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Sweeping up the precursor plasma
Assuming a precursor density n = nprexp(-z/z0) with
energy content 1.5npreTz0 per unit area.
Using an equation of motion
dv/dt = - p + (I/c)
The velocity of the plasma during the high intensity
pulse I when p is negligible is
z/t [ I / (cnprmi)]1/2
For I = 1023 Wm-2, npr= 1027 m-3, mi = 27mp, this
gives 2.7 106 m/s, i.e. in 1ps plasma moves only
2.7m.
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2D effect; Magnetic field generation
due to localised photon momentum deposition:An Ez electric field propagates into the solid
accelerating the return current. It has a curl,
unlike the ponderomotive force which is thegradient of a scalar.
At saturation there is pressure balance,
B2/20
= nheT
h=
hn
cm
ec2[(1+21/2a
0)1/2 -1]
and h = 1+a0/21/2.
E.g. I = 91019Wcm-2, ao = 8.5 gives B = 620MG(U.Wagneret al, Phys. Rev.E 70, 026401 (2004))
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Summary
A simple, approximate model has verified Begs empiricalscaling law for Thot.
A fully relativistic model including photon momentum
extends this to higher intensities where Thot (I2 )1/4. Electrons leave the collisionless skin depth in less than a
quarter-period for ao2 > 1.
Including reflected light deposits more photon momentum,
lowers Thot, and restricts the reflectivity at high intensity. Precursor plasma can change the scaling law.
More data, more physics (e.g. inclusion of Ez to drive thereturn current, time-dependent resistivity) are needed.