Duesseldorf T(Hot)

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


4/20

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


5/20

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


8/20

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.


10/20

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


18/20

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.

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