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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 Beg’s (I 2 ) 1/3 scaling for T hot. A fully relativistic “black-box” model including momentum conservation extends this to higher intensities. The effect of reflected laser light from the electrons is added, leading to an upper limit on reflectivity as a function of intensity. The relativistic motion of an electron in the laser field confirms the importance of the skin-depth.

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Beg’s empirical scaling of T h (keV)=215(I 18 2 m ) 1/3 for 70 < T h < 400keV & 0.03 < I 18 < 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

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n h is the relativistic critical density Taking the 2/3 power of this gives Eq.1 or

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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

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h depends on the total velocity of an electron. Transform to the axial rest-frame of the beam: Equate E 0 to m e 0 c 2 ; 0 indicates the thermal energy in the rest frame of the beam; because transverse momenta are unaffected by the transformation

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In dimensionless parameters, t h = eT h /m e c 2 and a 0, t h = (1+2 1/2 a 0 ) 1/2 - 1 (2) This contrasts with the ponderomotive scaling: t h = (1+a 0 2 ) 1/2 - 1 S.C.Wilks et al PRL(1992)69,1383 Simple model of Beg scaling, Eq.1, gives t h = 0.5 a 0 2/3 (3) Eqs (2) and (3) agree to within 12% over the range 0.3
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"name": "In dimensionless parameters, t h = eT h /m e c 2 and a 0, t h = (1+2 1/2 a 0 ) 1/2 - 1 (2) This contrasts with the ponderomotive scaling: t h = (1+a 0 2 ) 1/2 - 1 S.C.Wilks et al PRL(1992)69,1383 Simple model of Beg scaling, Eq.1, gives t h = 0.5 a 0 2/3 (3) Eqs (2) and (3) agree to within 12% over the range 0.3

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Various scaling laws; Beg’s empirical law is almost identical to Haines- classical and relativistic up to I = 5 10 18 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 T hot is reduced. The accelerating electrons will form a moving mirror, but the return cold electrons ensure that the net J z, 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 = n c p z ( h -1)c 2 (4) while momentum flux conservation is I/c + (1- abs )I/c = n c p z 2 /m e (5) Define I r = (1- abs )I; (5) c+(4) gives 2I = n c p z c 2 [p z /m e c + ( h - 1)], while (5) c-(4) gives 2I r = n c p z c 2 [p z /m e c - ( h -1)], or dimensionlessly i i = 2I/n c p z c 2 = p z ' + h - 1 (6) i r = 2I r /n c p z c 2 = p z ' - h + 1 (7) where p z ' = p z /m e c

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As before, transform the energy to the beam rest-frame E 0 2 = E 2 - p z 2 c 2 = ( h m e c 2 ) 2 - p z 2 c 2 = m e 2 c 4 ( h 2 -p z ' 2 ) = m e 2 c 4 0 2 Hence T h as measured in the beam rest frame is t h = eT h /m e c 2 = 0 - 1 = [( h +p z ')( h - p z ')] 1/2 - 1 = [(1+i i )(1- i r )] 1/2 - 1 Use (6) and (7) to eliminate p z ' to give i i +i r =2p z '. Define r = i r /i i ; then i i = 2 1/2 a o (1+r) -1/2 and t h = [{1 + 2 1/2 a 0 /(1+r) 1/2 }{1 - 2 1/2 a 0 r/(1+r) 1/2 }] 1/2 - 1 (8) This becomes Eq (2) for r = 0, and for r > 0, t h is reduced. The condition t h > 0 becomes f (r ) (1 - r 2 )(1 - r)/(2r 2 ) > a 0 2 and df/dr<0 for 0

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Defining as f(r) 2 a 0 2 where > 1, t h becomes t h = {[1 +(1-r)/(r )][1 -(1-r)/ ]} 1/2 - 1 Using r, (0 1) as parameters we can also find a 0 and the reflection coefficient, refl 1- abs = r The condition refl ≤ 1 gives

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Table of f(r) and t h ( ) 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 t h (1.1).265.125.065.0365.0204.011.0053.0021.0046 0 t h (1.2).44.202.108.0607.0341.0184.0089.0035.0077 0 t h (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). t h is tabulated for 3 values of where > 1

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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 10 19 Wcm -2. The low T hot and low reflectivity are advantageous to fast ignition, but require further experimental verification, additional physics in the theory, and simulations.

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Relativistic motion of an electron in a plane e.m. wave In a plane polarized e.m.wave (E x,B y ) of arbitrary form in vacuum an electron starting from rest at E x =0 will satisfy p z =p x 2 /2mc A wave E 0 sin( t-kz) and proper time gives x/c = a 0 (s - sin s) z/c = a 0 2 ( 3s/4 - sin s + 8 -1 sin 2s) t = s + a 0 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 = 3a 0 2 /4.

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But in an overdense plasma c/ pe < /2 . for a 0 ≥ ~ 1 an electron will traverse a distance greater than the skin depth without seeing even a quarter of a wavelength, i.e. the electron will not attain the full ponderomotive potential, before leaving the interaction region. Thus it can be understood why the T hot 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

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Sweeping up the precursor plasma Assuming a precursor density n = n pr exp(-z/z 0 ) with energy content 1.5n pr eTz 0 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 / (cn pr m i )] 1/2 For I = 10 23 Wm -2, n pr = 10 27 m -3, m i = 27m p, this gives 2.7 10 6 m/s, i.e. in 1ps plasma moves only 2.7 m.

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2D effect; Magnetic field generation due to localised photon momentum deposition: An E z electric field propagates into the solid accelerating the return current. It has a curl, unlike the ponderomotive force which is the gradient of a scalar. At saturation there is pressure balance, B 2 /2 0 = n h eT h = h n c m e c 2 [(1+2 1/2 a 0 ) 1/2 -1] and h = 1+a 0 /2 1/2. E.g. I = 9 10 19 Wcm -2, a o = 8.5 gives B = 620MG (U.Wagner et al, Phys. Rev.E 70, 026401 (2004))

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Summary A simple, approximate model has verified Beg’s empirical scaling law for T hot. A fully relativistic model including photon momentum extends this to higher intensities where T hot (I 2 ) 1/4. Electrons leave the collisionless skin depth in less than a quarter-period for a o 2 > 1. Including reflected light deposits more photon momentum, lowers T hot, and restricts the reflectivity at high intensity. Precursor plasma can change the scaling law. More data, more physics (e.g. inclusion of E z to drive the return current, time-dependent resistivity) are needed.

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