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Fast thermonuclear ignition with two nested high current lower voltage – high voltage lower current magnetically insulated transmission lines
Physics Letters A 318 (2003) 570–573 www.elsevier.com/locate/pla
Fast thermonuclear ignition with two nested high current lower voltage – high voltage lower current magnetically insulated transmission lines F. Winterberg University of Nevada, USA Received 27 August 2003; accepted 24 September 2003 Communicated by F. Porcelli
Abstract Fast thermonuclear ignition with a high gain seems possible with two Marx generators feeding two nested magnetically insulated transmission lines, one delivering a high current lower voltage pulse for compression and confinement, and one delivering a high voltage lower current pulse for fast ignition. With an input energy conceivably as small as 100 kJ the gain can be as large as 103 . The concept not only would be by orders of magnitude less expensive than laser compression and fast ignition schemes, but because of the large gain with a small yield also be more suitable for a thermonuclear reactor. 2003 Elsevier B.V. All rights reserved. PACS: 28.52.-s Keywords: Fast thermonuclear ignition; Magnetic insulation
About 35 years ago [1] I had shown that the ignition of thermonuclear microexplosions should be possible by the bombardment of a small solid DT target with an intense (108 A, 107 V) relativistic electron beam drawn from a large Marx generator, and it was found that breakeven would require an energy of about 10 MJ, to be delivered in about 10−8 seconds to a less than cm-size target. More specifically, it was proposed to place the target inside a hollowed-out metallic tamp, with the beam energy dissipated in the target by the electrostatic two-stream instability.
E-mail address: [email protected] (F. Winterberg). 0375-9601/$ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2003.09.063
Here I will show that at a fraction of the cost of the laser fast ignition concept [2], a high gain thermonuclear detonation wave can be ignited with two much smaller Marx generators, one with a high current lower voltage for compression and confinement, and one with a high voltage lower current for fast ignition. The concept is explained in Fig. 1. The high current ( 107 A) lower voltage (∼ 106 V) pulse for compression and confinement (lasting ∼ 10−8 s), drawn from the first Marx generator, passes over a thin metallic liner containing solid DT, with the inner wall of a liquid lithium vortex acting at the same time as a return current conductor and fast neutron absorber of the outer magnetically insulated transmission
F. Winterberg / Physics Letters A 318 (2003) 570–573
Fig. 1. The two magnetically insulated transmission lines with the relativistic electron beam from the high voltage (V ∼ 107 V) lower current (I ∼ 105 A) line bombarding the DT compressed inside a liner by a high current (I0 ∼ 107 A) lower voltage (V0 ∼ 106 V) ∼ 10−8 s lasting pulse flowing over the liner.
line, while the high voltage ( 107 V) lower current (∼ 105 A) pulse drawn from the second Marx generator feeds a second magnetically insulated transmission line nested inside the outer line and ending in a cathode tip emitting an intense relativistic electron beam lasting ∼ 10−9 s. Focused onto the end of the DT cylinder inside the liner the electron beam is stopped over a short distance dissipating its energy inside a small volume. In the presence of the strong magnetic field set up by the large current passing over the DT containing liner, a thermonuclear detonation wave is ignited at the end of the liner propagating down the liner with supersonic speed [3]. As an example we assume a liner with a radius r0 = 10−2 cm. With a current equal to I = 107 A flowing over the liner, the magnetic field at the surface of the liner is H = 2 × 108 G, exerting a pressure H 2 /8π = 1.6 × 1015 dyn/cm2 onto the DT. As for exploding wires, the temperature of a metallic liner is at these energy densities determined by the Stefan–Boltzmann law with temperatures in excess of ∼ 106 K (kT ∼ 10−10 erg), with the DT assuming the same temperature by heat conduction from the liner. Equating for example, the DT plasma pressure 2nkT at a temperature of 4 × 106 K with the magnetic pressure H 2 /8π one finds that n 30n0 , where n0 = 5 × 1022 cm−3 is the particle number density of solid DT.
571
At a magnetic field strength of 2 × 108 G the Larmor radius of the DT fusion reaction α particles is rL 1.3 × 10−3 cm, almost 10 times smaller than the radius of the liner. Accordingly, the magnetic field acts like a mirror confining the charged fusion products within the DT cylinder, a necessary condition for thermonuclear burn. In a three-dimensional geometry the condition for thermonuclear burn requires that ρr 1 g/cm2 . While in three dimensions only the fraction 1/6 of the charged fusion products goes into the direction of the burn wave, this fraction is 1/2, or three times larger in one dimension established in the presence of a strong azimuthal magnetic field. This changes the condition ρr 1 g/cm2 valid in three dimensions into ρz (1/3) g/cm2 valid in one dimension. With the 30 fold compression of the DT, making its density equal to ρ 3 g/cm3 , then requires that z 0.1 cm. For ignition a cylinder of length z = 0.1 cm and radius r0 = 10−2 cm (prior to compression) containing 2n0 πr02 z 3 × 1018 DT nuclei (and electrons) must be heated to T ∼ 108 K (kT ∼ 10−8 erg), requiring an energy of ∼ 3 × 1010 erg = 3 kJ, to be delivered in 10−9 s by a 107 V, 3 × 105 A intense relativistic electron beam. The ignition energy has to be delivered in a time not larger than the bremsstrahlungs loss time √ τR 3 × 1011 T /n (1) which for T 108 K and n = 30n0 is equal to τR 3 × 10−9 s, or about three times longer. In addition, the power flux of the electron beam φin = I V
(2)
must balance the power flux of the DT ablated from the end of the DT cylinder: φout = 2n
Mv 2 v 2 1 πr = nMv 3 πr 2 , 2 6 6
(3)
where v is the nondirectional ablation velocity, with the fraction 1/6 going in one direction, M the mass of the DT nuclei and r the radius of the DT cylinder. With nr 2 = n0 r02 where n0 , r0 are the number density and radius prior to the compression of the DT cylinder by the magnetic field, one can write for (3) 1 φout = ρv 3 πr02 , 6
(4)
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F. Winterberg / Physics Letters A 318 (2003) 570–573
where ρ = n0 M = 0.21 g/cm3 is the density of solid DT. Equating (4) with (2) one has IV 1 = ρv 3 . 2 6 πr0
(5)
For the example I = 3 × 105 A, V = 107 V and r0 = 10−2 cm, one has I V /πr02 1023 erg/(cm2 s), and one finds that v 108 cm/s with the kinetic temperature of the DT ions equal to T = Mv 2 /3k 108 K
(6)
that is about equal the ignition temperature of the DT thermonuclear reaction. The stopping length of the electron beam in solid DT by the collective electron plasma two-stream instability is [4]: λ=
1.4 cγ , ωp ε1/3
(7)
where ε = nb /n0 , ωp = 4πn0 e2 /m0 , γ = (1 − v 2 /c2 )−1/2 with nb the electron number density in the electron beam, and with the relativistic factor γ taking into account the longitudinal electron mass m = γ 3 m0 . For a beam radius equal to r0 = 10−2 cm, a beam current of I = 3 × 105 A, one has nb 1017 cm−3 , and for 10 MeV electrons with γ 20 one finds that λ 10−2 cm. By comparison, the noncollective classical stopping power length of relativistic electrons in matter of density ρ is λ∗ =
1 (0.543 E0 − 0.16) cm, ρ
(8)
where E0 is the electron energy in MeV. For 30 fold compressed DT one has ρ 6.3 g/cm3 , and for E0 = 10 MeV, one finds that λ = 0.7 cm, still short enough to satisfy the ρz (1/3) g/cm2 condition for detonation with an ignition energy less than 100 kJ. The rapid dissipation of the electron beam energy leads in the presence of a strong transverse magnetic field to a collisionless shock with a shock-thickness of the order of the ion gyroradius [5]. At a temperature of 108 K and a transverse magnetic field of ∼ 108 G, generated by the 107 A current, the gyroradius is of the order ∼ 10−4 cm. All three numbers show that the stopping length is short enough to ignite a thermonuclear detonation wave.
The current density of the field emitted electrons from the cathode tip of the inner high voltage transmission line is [6] j = 1.55 × 10−8 E 2 /W × exp −6.9 × 107W 3/2 /E A/cm2 , (9) where E is the electric field at the cathode in V/cm and W the work function in eV. For a semispherical cathode tip of radius r one has E ∼ = V /r, where V is the electric potential of the cathode. If W = 4.4 eV as for tungsten, one has for the total current emitted from a semispherical surface area 2πr 2 I = 2.2 × 106 V 2 exp −6.4 × 108r/V A. (10) For the example V = 107 V, r = 0.1 cm one finds I = 3.5 × 105 A, about equal the Alfvén current. The magnetic field of this current at r = 0.1 cm is H = 7 × 105 G, with a magnetic pressure H 2 /8π 2 × 1010 dyn/cm2 , at the limit of the tensile strength for the material of the cathode tip. The radial force on an electron in a partially neutralizing background plasma of density n and in the presence of an externally applied axial magnetic field Hz is [7] F = e (1 − f )Er − βHφ − eβ⊥ Hz , β = v/c,
β⊥ = vφ /c,
(11)
where f = n/nb . Since Er = 2πnb er = Hφ /β one has Hφ 1 − eβ⊥ Hz . F =e 2 −f γ β
(12)
(13)
For β 1, β⊥ 1 this force becomes attractive if f=
n 1 Hz 2− nb γ Hφ
(14)
with the beam focused down to a smaller radius. Without an axial magnetic field it becomes attractive if n 1 (14a) . nb γ 2 For nb = 1017 cm−3 and γ = 20, one has n 2.5 × 1014 cm3 which is a low density background plasma. With an axial magnetic field but no background
F. Winterberg / Physics Letters A 318 (2003) 570–573
Fig. 2. Focusing of the relativistic electron beam onto the DT cylinder by a magnetic mirror field setup by a magnetic solenoid.
plasma it becomes attractive if 1 Hz 2. Hφ γ
(14b)
For a current of ∼ 3 × 105 A at a radius r0 = 10−2 cm, Hφ 7 × 106 G, and for γ = 20, one has Hz 2 × 104 G, which can be easily established with ordinary electromagnets setting up a magnetic mirror field as shown in Fig. 2. In the DT cylinder of radius r0 and length l there are πr02 ln0 DT nuclei releasing the energy Eout = πr02 l(n0 /2)εf , where εf = 17.6 MeV = 2.8 × 10−5 erg is the DT nuclear reaction energy. For r0 = 10−2 cm and n0 = 5 × 1022 cm−3 one finds Eout = 2.2 × 107l J. This energy has to be compared with the input energy, mainly determined by the energy input from by the high current lower voltage Marx generator. Assuming that the current is I = 107 A
573
at a voltage of V = 106 V lasting for 10−8 s, one has Ein = 105 J. With l 5 cm, Eout 100 MJ and the gain Eout /Ein 103 , is the yield of 100 MJ is comfortably low for a thermonuclear reactor, where most of the energy is released in energetic neutrons which dissipate their energy in the lithium vortex, breeding tritium at the same time. To make sure that the high burn rate of 100% is actually reached one must have (according to the Lawson criterion) nτ 1014 s/cm3 . For n = 30, n0 = 1.5 × 1024 cm−3 and τ = 10−8 s one has nτ = 1.5 × 1016 s/cm3 , more than 100 times larger than the critical Lawson value. The remaining problem of the magnetohydrodynamic pinch instabilities of the liner can be solved by a special corrugated form of the liner surface inducing axial and rotational shear flow [8].
References [1] F. Winterberg, Phys. Rev. 174 (1968) 212. [2] M. Tabak, et al., Phys. Plasmas 1 (1994) 1626. [3] F. Winterberg, Atomkernenergie-Kerntechnik 39 (1981) 181; F. Winterberg, Atomkernenergie-Kerntechnik 39 (1981) 265; F. Winterberg, Z. Naturforsch. A 58 (2003) 197. [4] O. Buneman, Phys. Rev. 115 (1959) 503. [5] L. Davis, R. Lüst, A. Schlüter, Z. Naturforsch. A 13 (1958) 916. [6] W. Finkelnburg, Structure of Matter, Academic Press, New York, 1964. [7] F. Winterberg, Phys. Plasmas 2 (1995) 733. [8] F. Winterberg, Z. Naturforsch. A 54 (1999) 459.
Fast thermonuclear ignition with two nested high current lower voltage – high voltage lower current magnetically insulated transmission lines F. Winterberg University of Nevada, USA Received 27 August 2003; accepted 24 September 2003 Communicated by F. Porcelli
Abstract Fast thermonuclear ignition with a high gain seems possible with two Marx generators feeding two nested magnetically insulated transmission lines, one delivering a high current lower voltage pulse for compression and confinement, and one delivering a high voltage lower current pulse for fast ignition. With an input energy conceivably as small as 100 kJ the gain can be as large as 103 . The concept not only would be by orders of magnitude less expensive than laser compression and fast ignition schemes, but because of the large gain with a small yield also be more suitable for a thermonuclear reactor. 2003 Elsevier B.V. All rights reserved. PACS: 28.52.-s Keywords: Fast thermonuclear ignition; Magnetic insulation
About 35 years ago [1] I had shown that the ignition of thermonuclear microexplosions should be possible by the bombardment of a small solid DT target with an intense (108 A, 107 V) relativistic electron beam drawn from a large Marx generator, and it was found that breakeven would require an energy of about 10 MJ, to be delivered in about 10−8 seconds to a less than cm-size target. More specifically, it was proposed to place the target inside a hollowed-out metallic tamp, with the beam energy dissipated in the target by the electrostatic two-stream instability.
E-mail address: [email protected] (F. Winterberg). 0375-9601/$ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2003.09.063
Here I will show that at a fraction of the cost of the laser fast ignition concept [2], a high gain thermonuclear detonation wave can be ignited with two much smaller Marx generators, one with a high current lower voltage for compression and confinement, and one with a high voltage lower current for fast ignition. The concept is explained in Fig. 1. The high current ( 107 A) lower voltage (∼ 106 V) pulse for compression and confinement (lasting ∼ 10−8 s), drawn from the first Marx generator, passes over a thin metallic liner containing solid DT, with the inner wall of a liquid lithium vortex acting at the same time as a return current conductor and fast neutron absorber of the outer magnetically insulated transmission
F. Winterberg / Physics Letters A 318 (2003) 570–573
Fig. 1. The two magnetically insulated transmission lines with the relativistic electron beam from the high voltage (V ∼ 107 V) lower current (I ∼ 105 A) line bombarding the DT compressed inside a liner by a high current (I0 ∼ 107 A) lower voltage (V0 ∼ 106 V) ∼ 10−8 s lasting pulse flowing over the liner.
line, while the high voltage ( 107 V) lower current (∼ 105 A) pulse drawn from the second Marx generator feeds a second magnetically insulated transmission line nested inside the outer line and ending in a cathode tip emitting an intense relativistic electron beam lasting ∼ 10−9 s. Focused onto the end of the DT cylinder inside the liner the electron beam is stopped over a short distance dissipating its energy inside a small volume. In the presence of the strong magnetic field set up by the large current passing over the DT containing liner, a thermonuclear detonation wave is ignited at the end of the liner propagating down the liner with supersonic speed [3]. As an example we assume a liner with a radius r0 = 10−2 cm. With a current equal to I = 107 A flowing over the liner, the magnetic field at the surface of the liner is H = 2 × 108 G, exerting a pressure H 2 /8π = 1.6 × 1015 dyn/cm2 onto the DT. As for exploding wires, the temperature of a metallic liner is at these energy densities determined by the Stefan–Boltzmann law with temperatures in excess of ∼ 106 K (kT ∼ 10−10 erg), with the DT assuming the same temperature by heat conduction from the liner. Equating for example, the DT plasma pressure 2nkT at a temperature of 4 × 106 K with the magnetic pressure H 2 /8π one finds that n 30n0 , where n0 = 5 × 1022 cm−3 is the particle number density of solid DT.
571
At a magnetic field strength of 2 × 108 G the Larmor radius of the DT fusion reaction α particles is rL 1.3 × 10−3 cm, almost 10 times smaller than the radius of the liner. Accordingly, the magnetic field acts like a mirror confining the charged fusion products within the DT cylinder, a necessary condition for thermonuclear burn. In a three-dimensional geometry the condition for thermonuclear burn requires that ρr 1 g/cm2 . While in three dimensions only the fraction 1/6 of the charged fusion products goes into the direction of the burn wave, this fraction is 1/2, or three times larger in one dimension established in the presence of a strong azimuthal magnetic field. This changes the condition ρr 1 g/cm2 valid in three dimensions into ρz (1/3) g/cm2 valid in one dimension. With the 30 fold compression of the DT, making its density equal to ρ 3 g/cm3 , then requires that z 0.1 cm. For ignition a cylinder of length z = 0.1 cm and radius r0 = 10−2 cm (prior to compression) containing 2n0 πr02 z 3 × 1018 DT nuclei (and electrons) must be heated to T ∼ 108 K (kT ∼ 10−8 erg), requiring an energy of ∼ 3 × 1010 erg = 3 kJ, to be delivered in 10−9 s by a 107 V, 3 × 105 A intense relativistic electron beam. The ignition energy has to be delivered in a time not larger than the bremsstrahlungs loss time √ τR 3 × 1011 T /n (1) which for T 108 K and n = 30n0 is equal to τR 3 × 10−9 s, or about three times longer. In addition, the power flux of the electron beam φin = I V
(2)
must balance the power flux of the DT ablated from the end of the DT cylinder: φout = 2n
Mv 2 v 2 1 πr = nMv 3 πr 2 , 2 6 6
(3)
where v is the nondirectional ablation velocity, with the fraction 1/6 going in one direction, M the mass of the DT nuclei and r the radius of the DT cylinder. With nr 2 = n0 r02 where n0 , r0 are the number density and radius prior to the compression of the DT cylinder by the magnetic field, one can write for (3) 1 φout = ρv 3 πr02 , 6
(4)
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F. Winterberg / Physics Letters A 318 (2003) 570–573
where ρ = n0 M = 0.21 g/cm3 is the density of solid DT. Equating (4) with (2) one has IV 1 = ρv 3 . 2 6 πr0
(5)
For the example I = 3 × 105 A, V = 107 V and r0 = 10−2 cm, one has I V /πr02 1023 erg/(cm2 s), and one finds that v 108 cm/s with the kinetic temperature of the DT ions equal to T = Mv 2 /3k 108 K
(6)
that is about equal the ignition temperature of the DT thermonuclear reaction. The stopping length of the electron beam in solid DT by the collective electron plasma two-stream instability is [4]: λ=
1.4 cγ , ωp ε1/3
(7)
where ε = nb /n0 , ωp = 4πn0 e2 /m0 , γ = (1 − v 2 /c2 )−1/2 with nb the electron number density in the electron beam, and with the relativistic factor γ taking into account the longitudinal electron mass m = γ 3 m0 . For a beam radius equal to r0 = 10−2 cm, a beam current of I = 3 × 105 A, one has nb 1017 cm−3 , and for 10 MeV electrons with γ 20 one finds that λ 10−2 cm. By comparison, the noncollective classical stopping power length of relativistic electrons in matter of density ρ is λ∗ =
1 (0.543 E0 − 0.16) cm, ρ
(8)
where E0 is the electron energy in MeV. For 30 fold compressed DT one has ρ 6.3 g/cm3 , and for E0 = 10 MeV, one finds that λ = 0.7 cm, still short enough to satisfy the ρz (1/3) g/cm2 condition for detonation with an ignition energy less than 100 kJ. The rapid dissipation of the electron beam energy leads in the presence of a strong transverse magnetic field to a collisionless shock with a shock-thickness of the order of the ion gyroradius [5]. At a temperature of 108 K and a transverse magnetic field of ∼ 108 G, generated by the 107 A current, the gyroradius is of the order ∼ 10−4 cm. All three numbers show that the stopping length is short enough to ignite a thermonuclear detonation wave.
The current density of the field emitted electrons from the cathode tip of the inner high voltage transmission line is [6] j = 1.55 × 10−8 E 2 /W × exp −6.9 × 107W 3/2 /E A/cm2 , (9) where E is the electric field at the cathode in V/cm and W the work function in eV. For a semispherical cathode tip of radius r one has E ∼ = V /r, where V is the electric potential of the cathode. If W = 4.4 eV as for tungsten, one has for the total current emitted from a semispherical surface area 2πr 2 I = 2.2 × 106 V 2 exp −6.4 × 108r/V A. (10) For the example V = 107 V, r = 0.1 cm one finds I = 3.5 × 105 A, about equal the Alfvén current. The magnetic field of this current at r = 0.1 cm is H = 7 × 105 G, with a magnetic pressure H 2 /8π 2 × 1010 dyn/cm2 , at the limit of the tensile strength for the material of the cathode tip. The radial force on an electron in a partially neutralizing background plasma of density n and in the presence of an externally applied axial magnetic field Hz is [7] F = e (1 − f )Er − βHφ − eβ⊥ Hz , β = v/c,
β⊥ = vφ /c,
(11)
where f = n/nb . Since Er = 2πnb er = Hφ /β one has Hφ 1 − eβ⊥ Hz . F =e 2 −f γ β
(12)
(13)
For β 1, β⊥ 1 this force becomes attractive if f=
n 1 Hz 2− nb γ Hφ
(14)
with the beam focused down to a smaller radius. Without an axial magnetic field it becomes attractive if n 1 (14a) . nb γ 2 For nb = 1017 cm−3 and γ = 20, one has n 2.5 × 1014 cm3 which is a low density background plasma. With an axial magnetic field but no background
F. Winterberg / Physics Letters A 318 (2003) 570–573
Fig. 2. Focusing of the relativistic electron beam onto the DT cylinder by a magnetic mirror field setup by a magnetic solenoid.
plasma it becomes attractive if 1 Hz 2. Hφ γ
(14b)
For a current of ∼ 3 × 105 A at a radius r0 = 10−2 cm, Hφ 7 × 106 G, and for γ = 20, one has Hz 2 × 104 G, which can be easily established with ordinary electromagnets setting up a magnetic mirror field as shown in Fig. 2. In the DT cylinder of radius r0 and length l there are πr02 ln0 DT nuclei releasing the energy Eout = πr02 l(n0 /2)εf , where εf = 17.6 MeV = 2.8 × 10−5 erg is the DT nuclear reaction energy. For r0 = 10−2 cm and n0 = 5 × 1022 cm−3 one finds Eout = 2.2 × 107l J. This energy has to be compared with the input energy, mainly determined by the energy input from by the high current lower voltage Marx generator. Assuming that the current is I = 107 A
573
at a voltage of V = 106 V lasting for 10−8 s, one has Ein = 105 J. With l 5 cm, Eout 100 MJ and the gain Eout /Ein 103 , is the yield of 100 MJ is comfortably low for a thermonuclear reactor, where most of the energy is released in energetic neutrons which dissipate their energy in the lithium vortex, breeding tritium at the same time. To make sure that the high burn rate of 100% is actually reached one must have (according to the Lawson criterion) nτ 1014 s/cm3 . For n = 30, n0 = 1.5 × 1024 cm−3 and τ = 10−8 s one has nτ = 1.5 × 1016 s/cm3 , more than 100 times larger than the critical Lawson value. The remaining problem of the magnetohydrodynamic pinch instabilities of the liner can be solved by a special corrugated form of the liner surface inducing axial and rotational shear flow [8].
References [1] F. Winterberg, Phys. Rev. 174 (1968) 212. [2] M. Tabak, et al., Phys. Plasmas 1 (1994) 1626. [3] F. Winterberg, Atomkernenergie-Kerntechnik 39 (1981) 181; F. Winterberg, Atomkernenergie-Kerntechnik 39 (1981) 265; F. Winterberg, Z. Naturforsch. A 58 (2003) 197. [4] O. Buneman, Phys. Rev. 115 (1959) 503. [5] L. Davis, R. Lüst, A. Schlüter, Z. Naturforsch. A 13 (1958) 916. [6] W. Finkelnburg, Structure of Matter, Academic Press, New York, 1964. [7] F. Winterberg, Phys. Plasmas 2 (1995) 733. [8] F. Winterberg, Z. Naturforsch. A 54 (1999) 459.