Inertial plasma confinement in a miniature magnetic bottle induced by circularly polarized laser light

Physics LettersA 180 (1993) 132—136 North-Holland

PHYSICS LETTERS A

Inertial plasma confinement in a miniature magnetic bottle induced by circularly polarized laser light E. Kolka, S. Eliezer and Y. Paiss Plasma Physics Group, SOREQ N.R.C., Yavne 70600, Israel Received 28 February 1993; accepted for publication 28 June 1993 Communicated by M. Porkolab

A megagauss magnetic field generated by circularity polarized laser light is used to get confinement ofa plasma contained in a good conductor vessel. Inthis scheme the inertial confinement is supported by the magnetic forces and the Lawson criterion for a DT plasma might be achieved.

A new concept of inertial plasma confinement in a miniature magnetic bottle induced by circularly polarized laser light (CPLL) is suggested in this paper. Generation of a megagauss magnetic field by CPLL was recently suggested [1], B(Gauss) / A =

2 X 10 1 0~Wcm

—2

1021 cm

—~

(l)

where B is the magnetic field, A is the laser wavelength, ‘L is the laser intensity and n is the plasma density. For example, taking .2. = 1.06 ~.tm,as for a neodymium laser, and n= 1021 cm3, ‘L= 1016 WI cm2, gives B= 2 MG. These megagauss magnetic fields can be used for magnetic insulation. A hybrid scheme of inertial and magnetic confinement was proposed [2,3]. For example a plasma with a density of the order of 102! cm3 can be confined by inertia of a heavy metallic container while its heat is insulated by a self-generated magnetic field of the order of 1 MG. The basic structure of the plasma container [2,3] consists of a spherical metallic shell coated from inside with a solid DT fuel. Plasma is produced by ablation of the solid fuel caused by an injected laser (or particle) beam through a hole. The laser creates a toroidal magnetic field as a result of the current loop produced by the ejected hot electrons (a grad n x grad T process). The 132

inertial confinement time is increased by a reduction of the sound speed, partly from the larger atomic mass of the shell and partly by the reduction of the shell temperature due to the thermal insulation of the magnetic field. The new concept presented in this paper relies upon the hybrid use of inertial and magnetic confinements and megagauss field generation by CPLL. The schematic structure of the suggested configuration is as follows: a DT plasma is created inside a cylindrical or a spherical heavy conductor (or superconductor) shell with a hole. The plasma is irradiated by an intense circularly polarized laser beam. The CPLL creates a toroidal current in the plasma which in turn induces an opposite current in the wall (during the plasma current creation). The currents induce poloidal magnetic fields inside and outside the plasma (see fig. 1) in addition to the toroidal magnetic field formed by the grad nXgrad T mechanism. The plasma is heated resonantly by the CPLL from a temperature of 1 to 5 keV during a time of a few nanoseconds. After the laser is turned off a process of expansion and diffusion of the magnetic field lines begins both into the walls and inside the plasma. The typical time for the diffusion of the magnetic field to a length ‘B in a matter with conductivity a is given by ~ 4~/2/c2 (2) B

B

0375-9601 1931$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.

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PHYSICS LETTERS A

~C.P.L.L.

last feature can improve significantly the stability and confinement of the hot plasma. Moreover, due to the existence of both toroidal and poloidal fields many plasma instabilities are reduced. This paper does not describe in detail the plasma

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30 August 1993

can be exploding created bywire means capillarydischarge [4],a by an (orofa foil) or directly by formation process. However, an [51 appropriate plasma laser beam sorption mechanism [2]. between the electrons and the The CPLL heats the plasma also by a resonant ab-

~

circularly polarized photons. It was calculated [1] that the efficiency of absorption of the CPLL gyrating photon angular momentum by the electrons is

________

given by (3)

2LL,

Le = (~0pe/0))

n (plasma)

where Le is the electron angular momentum, LL is the photon angular momentum, Wpe is the electron plasma frequency, and w is the laser angular frequency. From eq. (3) follows that the power absorbed by the electrons We is

BO

r

Fig. 1. A description of the suggested scheme. CPLL is the circularly polarized laser light, j~is the plasma current and j is the induced current into the vessel wall. B~and B~° are the poloidal magnetic fields for the plasma density n.

where c is the speed oflight. For copper walls at room temperature as well as for a 5 keY DT plasma the diffusion time is more than 100 ns for a scale length of20 JIm. Thus the magnetic field exists long enough after the laser is turned off. This magnetic field reduces the heat conduction, and therefore the hydrodynamics the inertialbyconfinement time of the mehot plasma is of determined the cold matter (the tallic walls). A new feature ofthis proposal is the existence of the magnetic field after the laser is turned off. For this reason one might get a more economic fusion device in comparison with the previous hybi-id proposal. Another advantage of this new configuration is the strong magnetic field between the plasma and the walls that can make the pressure profile on the walls as a function of time smoother. This

(4)

We~(Wpe/W)2WL,

where WL is the laser power. Therefore at the critical density (w~= w) the resonant absorption efficiency is 50%. The high absorption efficiency of the CPLL makes it possible to heat the plasma to the high ternperatures such as 5 keY necessary for fusion. The electric field of a spatial Gaussian CPLL that propagates in the z direction is E= E

1 (e~+ ie~)exp (

2 —

iwt + ikz) exp(

—

)‘

2L (5)

where E 1 is the electric field component in the x and y directions, e~and e~are unit vectors, k and L are the CPLL wave number and Gaussian width respec2= tivelyThe andelectrons the electric amplitude is and I El the 2E~. satisfyfield the law of motion continuity equation mn [ôv/ôt+ (v•grad )vJ = —grad P— enE,

(6)

ôn/ôt+div(nv)=0,

(7)

where m, e, n and v are the electron mass, charge, number density and velocity respectively and P is the pressure. The collisions are neglected in eq. (6) since 133

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the electron collision frequency Ve ~z w/2it for the plasma parameters considered. The grad P term on the right hand side of eq. (6) can be neglected assuming

where 1= mm (/~.,Ia); /T and /,, are the scale length of the temperature and density gradients. Equation (8) is satisfied for an electron temperature of a few keY and a CPLL i-radiance of the order of 1016 W/cm2. Linearizing eqs. (6) and (7) one gets for the second order perturbed toroidal 2” 3E2current (a j r 1 e2w3 ~ n~(r) 6(r) ~ ex~—~ m

(

no(r))

—

L2

‘

(9)

where r is the cylindrical coordinate in the x—y plane and n 0(r) is the zero order electron number density determined during the plasma formation. Assuming a Gaussian zero order density with the the same width 2/2L2) toroidal as the laser current is n0 ( r) = n0 exp ( — r

(

0r

(10)

m2w37JT~

For simplicity the magnetic field is calculated for a cylindrical configuration. The magnetic field of a thin cylindrical shell of radius a, height 2h and width 2) ~a in induced by a current density 18 (statampere/cm cylindrical coordinates whose origin is in the center of the shell is B~(j 0,r, z)=~ Cr

—h—z h—1z

2+~2] 2~

+E(K2) (a—r)2+ç~2J a2+r2+’~2“~d

B~(j

f

(11)

a2—r2—c~2’\ 2 + ~2) d~ (12) (a r) 2), E(K2) are the complete elliptic intewhere grals ofK(K the first and second kind and —

134

J b

B~( r, z) =

B~[/~( ~), r, z] d~, (15) o where b is the vessel’s inner radius and B~,B~,j 8are given in eqs. (11), (12) and (10) respectively. Equations (15)current. describe the magnetic field induced (14) by theand plasma During the laser pulse the plasma current increases from zero to a saturation value given in eq. (10). As a consequence a current j~is induced into the wall (see fig. 1). The wall current is produced in such a way that the penetration of the magnetic field into the wall is avoided. This statement is accurate for a superconducting shell, diffuses howeverinto forthe a conducting the magnetic field wall with a vessel time constant given in eq. (2). The total magnetic field B~( r, z) was calculated numerically assuming a wall current (_ r_b) (r>b),

(16)

j~(r)=j~exp where r~is the typical penetration length of the current into the wall. totalcurrents magneticis field induced by the plasma and The the wall calculated as a function of j~’.This parameter is determined by demanding a negligible magnetic field for r> b. In

mm.toThe sen be: peak appropriate density plasma 1022 cm3, parameters a Gaussian were profile choa width L=0.5 mm, the vessel’s inner radius

current r~=10 The ~im and halfpenetration height are 1 parameter mm and 3 was mm taken respectively. implying a peak wall current j~’=4.4x108 A/cm2.

1/2

X(K(K2) +E(ic2)

4ar (a+r)2+~2

o

with

2+~2] 0,r, z) = ./~a —h—z 2 c [(a+r)

—

(14)

following laserW/cm2 parameters: 0.35 ~m, irradiance 1018 and awavelength Gaussian L0.5 fig. 2 B~(r,z=0) is calculated (solid width line) for the

1/2

[(a+r)

h—z

K

b

B~(r,z)=$B~Uo(~),r,z] d~,

l.5r2) e3E2 n

j6(r)=—~exP\~~V

X(_K(K2)

The total magnetic field induced by the plasma current is

(8)

kBT/1<
30 August 1993

(13)

The magnetic field changes direction at r

The magnetic fluxes through 0 ~ r ~ r 0=0.55 mm. 01 (GOut) and equal magnitudes fig. 2). The plasmaand expands r01 ~r
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electron collision time. The classic heat confinement time is then given by

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2~106 1.55106

a ~ ~

~—.—

5x~S 6 1x10

(18)

tHfllH/Ke.

-

For a typical time of 100 ns one gets a diffusion scale

—

-

-

5~1Q~—

L....

—ixio6 _______________________________ I I 0 0.05 0.1 0.15 0.2

r01

r02

b

r(cm) -~

Fig. 2. The poloidal magnetic field forz=0 (the center ofa cylindrical vessel) as calculated numerically is described by the solid line. The dash-dotted and the dotted curves are two-step approximations for different times.

length inertial(/H) confinement ofis tiner 100 b/Cs time. ~m. The An approximation third scalefor is the inertial time where b istime the vessel inner radius and C~is the speed of sound in cold matter. The ICF hydrodynamics is determined by cold matter because of the reduction of the heat conduction by the magnetic field. Taking C 5=4 km/s and a yessd radius R = 1 mm one gets an inertial confinement time of 125 ns. An economical fusion reactor must satisfy the “bookkeeping” requirement (19) 1/D~ lab, ~h and lth are the efficiencies of the where driver, the laser absorption, the energy transport from the absorbed laser to the DT fuel and the thermodynamic cycle respectively. 0 is the fraction of the X~GF?/D?1abflh?/1hflecøl

schematic description is given in fig. 2: the calculated magnetic field profile is approximated by a constant field in(solid each curve) region (dash-dotted curve) with the same fluxes cJ~,and ~ During the plasma expansion one can see the increase in the magnetic field in the region r 01 ~r1 keY) DT plasma we find that for 100 ns the magnetic diffusion scale length (‘B) ~S less than 20 ~m. The second is the typical time for diffusion of a thermal wave from the plasma into the wall. The thermal diffusion is reduced due to thermal insulation of the magnetic field. The heat conductivity of the electrons perpendicular to the magnetic field is given by [2] 1C~= 4.7flpe2/Te, (17) where p~is the electron Larmor radius and ; is the

,

burned fuel and ~lec is an economic factor. GF is the intrinsic gain defined as the ratio of the nuclear fusion output to the internal fuel energy and x is the fraction of the fuel heated to fusion temperature by the driver (without the alpha heating). Equation (19) should be satisfied all (with ICF schemes. The main differences between for ICF high compression) and the present proposal (without compression) are the values of i~h and x. In our scheme 17h is close to unity while for a typical ICF ~b 0.1, therefore the necessary compression fora spark ignition scheme in ICF (x—’ 0.1) is not required in our configuration (x—~1). For a MIICF reactor with an electrical output energy of 100 Mi per pulse and a repetition rate of 10 Hz (1 GW output electrical power) one needs a laser with an energy of about 3 MJ. In summary, a new concept ofinertial plasma confinement in a miniature magnetic bottle is suggested. Our scheme uses circularly polarized laser light (CPLL) in order to create large magnetic fields inside a good conductor vessel containing the plasma. The Lawson criteria for a DT plasma might be satisfied for plasma densities of the order of 5 x 1021 cm3 and confinement times of 20 ns. 135

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References [I] S. Eliezer, Y. Paiss and H. Strauss, Phys. Lett. A 164 (1992) 416. [2] A. Hasegawa, K. Nishihara, H. Daido, M. Fujita, R. Ishizaki, F. Miki, K. Mima, M. Murakami, S. Nakai, K. Terai and C. Yanamaka, Nucl. Fusion 28 (1988) 369.

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Daido, F. Miki, M. Fuhita, K. Sawai, H. Fujita, Y. Kitagawa, S. Nakai and C. Yamanaka, Phys. Rev. Lett. 56 (1986) 846. [4] A. Zigler, M. Kishenevsky, M. Givon, E. Yarkoni and B. Arad, Phys.Rey.A35 (1987) 4446. [51R.A. LondonandM.D. Rosen, Phys. FIuids29 (1986) 3813.