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Scaling of the hot electron temperature and laser absorption in fast ignition

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 ). Outline. A simple energy flux model reproduces Beg’s

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Scaling of the hot electron temperature and laser absorption in fast ignition

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  1. 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)

  2. Outline • A simple energy flux model reproduces Beg’s (I2)1/3 scaling for Thot. • 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.

  3. Beg’s empirical scaling of Th(keV)=215(I182m)1/3for 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

  4. nh is the relativistic critical densityTaking the 2/3 power of this gives Eq.1or

  5. Model 2: Fully relativistic with energy and momentum balance Momentum conservation is where consistent with electron motion in a plane wave

  6. h depends on the total velocity of an electron.Transform to the axial rest-frame of the beam:Equate E0 to me0c2; 0 indicates the thermal energy in the rest frame of the beam; because transverse momenta are unaffected by the transformation

  7. In dimensionless parameters, th = eTh/mec2 and a0,th = (1+21/2a0)1/2 - 1 (2)This contrasts with the ponderomotive scaling:th = (1+a02)1/2 - 1 S.C.Wilks et al PRL(1992)69,1383Simple model of Beg scaling, Eq.1, givesth = 0.5 a02/3 (3)Eqs (2) and (3) agree to within 12% over the range 0.3<a0<300, and intersect at a0 = 0.5685 and 112.55.The total electron kinetic energy is (h - 1) = a0/21/2

  8. Various scaling laws; Beg’s empirical law is almost identical to Haines-classical and relativistic up to I = 51018 Wcm-2

  9. 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 is reduced. 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. If absorbed fraction is abs, energy conservation isI - (1-abs)I = ncpz(h-1)c2 (4)while momentum flux conservation isI/c + (1-abs)I/c = ncpz2/me (5)Define Ir = (1-abs)I; (5)c+(4) gives2I = ncpzc2[pz/mec + (h - 1)], while (5)c-(4) gives2Ir = ncpzc2[pz/mec - (h-1)], or dimensionlesslyii = 2I/ncpzc2 = pz' + h - 1 (6)ir = 2Ir/ncpzc2 = pz' - h + 1 (7)where pz' = pz/mec

  11. As before, transform the energy to the beam rest-frameE02 = E2 - pz2c2 = (hmec2)2 - pz2c2 = me2c4(h2-pz'2) = me2c402Hence Th as measured in the beam rest frame isth = 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 andth = [{1 + 21/2a0/(1+r)1/2}{1 - 21/2a0r/(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 becomesf (r )  (1 - r2)(1 - r)/(2r2) > a02 and df/dr<0 for 0<r<1

  12. Defining  as f(r)  2a02 where  > 1, th becomesth = {[1 +(1-r)/(r)][1 -(1-r)/]}1/2 - 1Using r, (0<r<1), and , ( > 1) as parameters we can also find a0 and the reflection coefficient, refl 1-abs = rThe condition refl ≤ 1 gives

  13. Table of f(r) and th() versus rr = 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1f(r) = 44.6 9.6 3.54 1.58 .75 .356 .156 .0563 .0117 0th(1.1) .265 .125 .065 .0365 .0204 .011 .0053.0021.0046 0th(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

  14. 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.

  15. 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 + a02( 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 = 3a02/4.

  16. But in an overdense plasma c/pe < /2.for a0 ≥ ~ 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 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.

  17. Relativistic collisionless skin-depth

  18. 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 2.7m.

  19. 2D effect; Magnetic field generationdue 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 the gradient of a scalar.At saturation there is pressure balance,B2/20 = nheTh = hncmec2[(1+21/2a0)1/2 -1]and h = 1+a0/21/2.E.g. I = 91019Wcm-2, ao = 8.5 gives B = 620MG(U.Wagner et al, Phys. Rev.E 70, 026401 (2004))

  20. Summary • A simple, approximate model has verified Beg’s empirical scaling law for Thot. • A fully relativistic model including photon momentum extends this to higher intensities where Thot (I2 )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 the return current, time-dependent resistivity) are needed.

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