Physics Letters A 343 (2005) 181–189 www.elsevier.com/locate/pla
Fusion energy in degenerate plasmas Pablo T. León, Shalom Eliezer 1 , José M. Martínez-Val ∗ Institute of Nuclear Fusion, ETSII-UPM, Spain Received 22 April 2005; received in revised form 18 May 2005; accepted 27 May 2005 Available online 9 June 2005 Communicated by F. Porcelli
Abstract In inertial confinement fusion (ICF), a high density, low temperature plasma can be obtained during the compression phase, so minimizing the energy needed for compression. If the final temperature reached is low enough, the electrons of the plasma can be degenerate. In this case, bremsstrahlung emission is strongly suppressed and ignition temperature becomes lower than in classical plasmas, which offers a new design window for ICF. Fusion ignition can then be triggered by an additional energy beam. The main difficulties to produce degenerate plasmas are the compression energy and the compression performance needed for it. Besides that, the low specific heat of degenerate electrons (as compared to classical values) is also a problem because of the rapid heating of the plasma. The main contribution of the Letter is to show that the plasma degeneracy lowers the ignition temperature for DT plasmas, but it does not increase the target energy gain. Some numerical results are given on that. In the case of proton–boron 11 plasmas, the densities have to be extremely high in order to reduce the ignition temperature, but even so the energy gains remain rather low. 2005 Elsevier B.V. All rights reserved. PACS: 52.50.Lp; 52.57.Kk; 52.58.-c Keywords: Degenerate plasmas; Fusion ignition; Inertial confinement
1. Introduction and scope
* Corresponding author. Mailing address: ETS Ingenieros Industriales, J. Gutiérrez Abascal, 2, 28006 Madrid, Spain. E-mail addresses:
[email protected],
[email protected] (J.M. Martínez-Val). 1 Visiting Professor from Soreq, NRC, Israel.
0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.05.079
In inertial confinement fusion, high densities are required to obtain high gains [1,2]. In the fast ignition scheme [3,4], high density–low temperature plasmas can be obtained during the compression phase [5–9], so minimizing the energy needed for compression. If the temperature is low enough, the electrons of the plasma can be degenerate [10]. In this
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case, bremsstrahlung emission is highly suppressed [11–15], because Pauli’s exclusion principle forbids many electron energy transitions after photon emission. As bremsstrahlung is the main energy loss mechanism from the fusion plasma, the temperature needed for ignition decreases. In the fast ignition scheme, a small portion of the compressed target is suddenly heated by an external beamlet of energy, and the fusion burning wave is launched from there to the rest of precompressed plasma. The small volume heated is called the ignitor. There are different mechanisms for energy deposition in the ignitor (laser induced fast electrons, ions, plasma jets, plasma impact, etc. [16–23]). One of the most appealing mechanisms to trigger fast ignition is plasma block impact [24–27] but practical configurations for accelerating the plasma blocks are still under discussion. To take advantage of the degenerate plasma, the electron temperature has to be below Fermi energy for enough time. This means that heating mechanisms have to deposit the energy to the ions in the plasma, not to the electrons. That is the reason why ions have been chosen in this study as heating mechanism. The energy deposited in electrons and ions depends of the stopping power in both species. In degenerate plasmas, the equation that defines the stopping power of ions is different from the one governing classical plasmas. As will be seen later in this Letter, for highly degenerate electrons, the equation of the stopping power is nearly
independent on density. Since the stopping power of ions behaves as the classical expression, it mainly depends on density. In very high densities plasmas, the energy of the incoming ions goes to the plasma ions and not to the plasma electrons, in agreement with early analysis on this subject [28]. When the ions temperature is higher than the electron temperature, energy from collisional phenomena goes from ions to electrons. If electrons are degenerate, the energy exchange term between ions and electrons is different from the classical one. For very high densities, the ignition temperature of the plasma is lower than classical predictions, but the compression energy is high. However, for moderate densities, the electrons temperature increases very fast (due to the energy deposition of the external ions), so the plasma becomes classical rapidly (very low specific heat of degenerate electrons), and the ignition temperature is similar to the classical predictions. A potential way for the practical realization of the scheme proposed in this Letter is depicted in Fig. 1. In the left-hand side image, a typical ICF target is depicted. In this case, a solid target is considered, instead a hollow one, because of the need of suppressing strong shock-waves and strong shocks between material layers in the compression process, which must be as close to isentropic as possible. (In Ref. [29, pp. 182– 183], some calculations are given about the features of this type of compression, and the results are very promising to achieve very high densities.) A close-to-
Fig. 1. Implosion scheme for obtaining a very high density at the end of the compression phase, followed by a fast ignition system based on the interaction of particle beamlets into a fraction of the compressed target. The left-hand side image shows the interaction of the ablation driver beams with the initial target. The right image shows the fast ignition onset, once the central core of the target has been compressed.
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Fig. 2. Reference temperatures for reaching fully degenerate DT plasmas (they correspond approximately to one tenth Fermi energy, Ef0 ). If the plasma temperature is much higher than the corresponding reference value, the plasma becomes classical.
isentropic compression is the fundamental guideline for tailoring the target implosion, which must be totally different than the implosion required to create a central hot spark at the end of the compression phase, in order to have a self-triggered ignition. It is worth noting that in our case (which follows the fast ignition model [3,4]) ignition must be triggered by some additional energy beamlets. This is depicted in the righthand side image in Fig. 1. Moreover, in order to reach the very high densities needed to have a degenerate plasma, the implosion process must suppress all types of mechanisms preheating the fuel. In this context, it is worth remembering the experiments done some years ago at Osaka University, in the so-called pusherless implosion scheme, leading to stagnation free targets at the end of the compression phase [5–7]. Those experiments were discontinued because they were not relevant to produce ignition from a central hot spark, but they showed the way to get very high compressions, showing degeneracy effects (in particular, a lengthening of the range of the charged particles generated within the deuterated fuel by fusion reactions). The importance of avoiding preheating can be seen in Fig. 2, where the level of the plasma temperature is depicted vs. density in order to have full degeneracy. If the temperature becomes much higher than the corresponding reference value, the plasma becomes classical. Another relevant magnitude is the plasma pressure (Fig. 3), that must be reached by the implosion forces (driven by the external ablation process, that must be properly tailored in order to produce a shock-less compression, as close as possible to an isentropic evolu-
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Fig. 3. Plasma (DT) pressure in the degenerate regime, with a temperature equal to Fermi energy (Ef0 /10). This pressure expresses the value that has to be reached in the compression phase to attain degenerate regimes.
tion). This evolution obviously depends on the initial state of the target, and the initial preheating. If we presume that the target is at 300 K (and a density slightly lower than that of frozen DT, for instance, 0.1 g/cc) the specific entropy would be about 13 J/g K. If such a target could be compressed isentropically up to 1000 g/cc, its temperature would be around 55 eV, a value much lower than the corresponding Fermi energy, 1.4 keV. Hence, there is room enough for allowing some preheating during the compression phase. Pressure to drive this compression process would have to be over 140 Gbar (Fig. 3). Uniformity requirements on the spherical shape of the driving pulses would be very challenging, as in any case in ICF. However, the scheme presents the advantage of not needing the generation of a central hot spark to trigger ignition (because there is some contradiction between compressing and heating, in the sense that is very difficult to compress hot matter). On the other hand, hydrodynamic instabilities must be prevented, but this is not a particular requirement of this scheme, but a general one. In our case, where a solid target is the first option to be considered, the central core of the target (the fuel) must undergo a radius reduction by a factor 20 in order to achieve a compression factor of 8000, which is the value needed to reach 1000 g/cc in the final density. These values are not very far from the general design windows of the targets in main-stream ICF research. Of course, the time profile of the driving pulse would have to be adjusted to launch a continuous train of micro shock-waves of
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increasing intensity, in order to produce the isentropic compression needed in this case. The analysis of these problems lies outside of the scope of this Letter, aimed at studying the inherent features of degenerate plasmas in the context of ICF targets triggered by fast ignition.
Node 2: Volume annexed to the ignitor
dEe dt
= Pie − Pb + f ηPf − Phe − Pme node 2
2. Fast ignition scheme. Energy equations Numerical simulations of this study were based on a two-node energy code. The first node represents the ignitor. The size of the ignitor is defined by the external ion beam range. The second node represents a surrounding volume that encloses the ignitor, and the volume is characterized by the alpha particle range (see Fig. 4). 2.1. Energy equations The governing equations are the following ones: Node 1: Ignitor dEe = Pie − Pb + f ηPfα + ηd Pign dt node 1 − Phe − Pme , (1) dEi 3 d(kTi ) = ni dt node 1 2 dt = −Pie + f (1 − η)Pfα + f Pfn + (1 − ηd )Pign − Pmi .
(2)
dEi dt dEi dt
+ Phe + f ηPfα + Pme + ξ Pb ignitor ,
(3)
= −Pie + f (1 − η)Pf − Pmi + f Pfn node 2
= + f (1 − η)Pfα + Pfn + Pmi ignitor . node 2 (4)
The energy of the electrons in the ignitor depends on several energy terms: • the energy that comes from the ions (Pie ); • bremsstrahlung losses (Pb ); • the energy that deposit the alpha particles due to fusion reactions (PF ). This energy goes partially to electrons, partially to ions, according to a parameter η which depends on temperature and density. The energy is deposited partially in the ignitor, and partially in the node 2, according to a parameter f which depends on the node sizes and the alpha particle range; • the energy deposited by the external ion beam (Pign ) (partly to electrons, partly to ions ηd );
Fig. 4. Fast ignition model for the FINE (fast ignition nodal energy) code.
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Fig. 5. Static ignition temperature for pB11 plasma, for different densities, as a function of plasma temperature. The static ignition temperature is defined as the plasma temperature when fusion power (PF : described by A1 , B1 and C1 ) equals bremsstrahlung radiation power (WB : described by A2 , B2 and C2 ). Bremsstrahlung lines show a T 2 dependence for degenerate plasmas, and a T 1/2 dependence for classical ones.
• the losses of heat conduction in the plasma (Phe ); • mechanical losses (expansion of the ignitor), Pme . The energy of the ions in the ignitor varies with time due to: • the energy exchange with electrons; • the energy deposited by alpha particles; • the energy deposited by the neutrons from fusion reactions (Pfn ) (that in the case of high density plasmas, it is not negligible), and can go either to the ignitor or to the second node, depending on a parameter f ; • the energy deposited by the external beam; and • the mechanical losses (Pmi ). In node 2, the terms of the energy equations are similar to those of node 1, taking into account the energy from the ignitor that goes to this volume. That is the reason why the volume of node 2 is characterized by the alpha particle mean free path. The source
energy terms from the ignitor are within the square brackets. All the equations have to be defined for classical and degenerate plasmas. For the analysis to be valid, the transition from degenerate to classical plasmas has to be smooth. Some of the equations have a perfect transition from degenerate to classical plasmas (for example, bremsstrahlung equations [11,12]). Fermi– Dirac statistics include the classical results when Fermi energy goes to −∞. Hence, if the equation is obtained with Fermi–Dirac statistics, the classical behaviour is well defined, and transition from classical to degenerate plasmas is smooth [29]. In Fig. 5, both the fusion reaction power and bremsstrahlung emission are depicted for the proton–boron 11 fusion reaction, for different densities. The crossing point of both curves for the same density is the ignition temperature. The advantage of degenerate plasmas is clear, because the ignition temperature becomes lower than the classical value.
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2.2. Stopping power and internal plasma heating During the heating process by an external ion beam, the electrons of the plasma have to maintain the degeneracy in order to have a low ignition temperature. That means that the energy has to go to ions, and not to electrons, and that the energy exchange of ions and electrons due to the temperature difference has to be as low as possible. The equations of the stopping power of projectiles in the electrons and ions of the plasma are [30–35]: (a) Electrons 1 dE −4 m2e α 2 v1 D = ln ΛRPA , dx e 3π h¯ 3 1 + exp(−η) (5) D ln ΛRPA = 1 + exp(−η) ∞ 1 k3 ; × dk (6) (k 2 + k02 ) exp(− h¯ 2 k 2 ) − η 8m·kTe 0
(b) Ions dE −4π α 2 ln(Λ) = ni , dx i mi v12 q1 q 2 α= , 4πε0
(7) (8)
where me is electron mass, v1 is the velocity of the projectile, and the corresponding subindexes are ‘1’ for projectile, and ‘2’ for the different species of the plasma (electrons or ions). The expression ln(Λ) is the classical Coulomb logarithm. For very high density, and low temperature, the last term of Eq. (5) disappears, (kTe EF ), and finally the expression is −4 m2e α 2 v1 D dE = ln ΛRPA . (9) dx e 3π h¯ 3 This corresponds to a well established result in the literature of [24]. This expression is independent of the density of the electrons in the plasma. That means that for high densities, the stopping power of ions is higher that for electrons. For this result to be true, the density has to be extremely high, more than 106 g/cm3 for DT. This density is too high for any fusion energy scenario. For example, for a 1 mg of DT, the total
energy from fusion is 340 MJ. The compression energy to reach 106 g/cm3 in DT is 3.24 MJ. The gain, taking into account all the processes, would be negligible. As will be shown in next section, part of the energy of the projectiles goes to electrons, and depending of the Fermi energy of the plasma (which is a function of density and temperature), the plasma will be classical or degenerate when ignition is attempted.
3. Numerical simulations The equations defining the energy conservation equations have been implemented in a code, named FINE (Fast Ignition Nodal Energy) [36,37]. The code can obtain the ignition temperature of a degenerate or classical plasma, and the behaviour of the plasmas during the heating process, with the fast ignition technique, using a two-node geometry (Fig. 4). The main advantage of the code is the smooth transition between the simulation of degenerate and classical plasmas. Results are shown in Figs. 6 and 7, for different DT stoichiometric plasma densities, and different projectile energies. The initial temperature of electrons and ions in the compressed plasma is EF0 /10, to start with highly degenerate plasma. The ignition temperature is defined as the temperature which must be achieved in the ions in the ignitor to launch a fusion burning wave in the precompressed target (second node), once the external igniting beam has been switched off. Before obtaining this ignition temperature, an analysis of the minimum beam power needed to reach ignition has been done for each density (depending on the energy of the projectile). The first clear result is that the initial hypothesis was correct (Fig. 7), and the ignition temperature is much lower for very high density degenerate plasmas. In the case of 105 g/cm3 DT plasma density, the ignition temperature is much smaller (1.5 keV) than the one predicted for classical plasmas (near 4.5 keV). In this case, Fermi energy is very high, much higher than the ignition temperature, so the plasma is well degenerate during all the heating process. When the density of the plasma decreases, the ignition temperature increases. For a plasma density of 103 g/cm3 , the ignition temperature is near the classi-
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Fig. 6. Compression energy (left-hand side scale) and ignition energy (right-hand side scale) for 1 mg DT compressed target.
Fig. 7. Gain and ignition temperature for degenerate 1 mg DT target (after compression) as a function of density. Initial plasma temperature given by kTe = EF0 /10.
cal value. In this case, the Fermi energy is below the ignition temperature. The degenerate plasma increases its temperature very rapidly, due to the small specific heat of those plasmas, so the behaviour is very similar to the classical case. In inertial fusion energy, the most important parameter is not the ignition temperature, but the gain, which must be defined in terms of the fusion energy released. This value cannot be calculated with the code.
The code can only take into account two nodes (ignitor and surrounding volume enclosing the ignitor). The rest of the target is not modelled. Two nodes are enough to check whether a fusion burning wave is launched in the degenerate plasma or not, but not to calculate the fusion energy released in the total target. This magnitude depends of the total areal density of the compressed target. In our analysis, a well-known theoretical formula has been used to calculate the fu-
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sion energy, through a factor ‘b’ which is the fraction of the compressed target mass that actually undergoes fusion burn-up [38,39]
b=
0 1/3 ρ 2/3 ( 3m ρR 4π ) . = ρR + 6 ρ 2/3 ( 3m0 )1/3 + 6( g 2 ) 4π cm
(10)
Target gain is thus defined as
G=
bEfusion , Ecomp + Eign
(11)
where the Eign is the energy deposited in the ignitor by the external ion beam, and Ecomp is the energy of the compressed plasma. In classical plasmas, the latter is a function of temperature only, for a given mass. In the degenerate region (low electron temperatures), the compression energy is not a function of temperature, but a function of density. For a given mass and electron temperature of the plasma, compression energy increases with density. For example, the energy needed to compress a DT plasma to 103 g/cm3 and 1 keV temperature, is smaller than the energy needed to compress the same plasma up to a density of 104 g/cm3 , and Tplasma = 0 K. The compression energy decreases with lower densities, but the fraction of fuel burned increases with density, for a given fuel mass. For high density plasmas, gain decreases with higher densities, due to high compression energies in degenerate plasmas (as a function of density, and not of temperature). In classical plasmas, and for a given fuel mass, the compression energy depends of the temperature of the fuel, so the problem changes drastically. In the fast ignition scheme, the problem of low densities is the size of the ignitor (the smaller the density, the higher the size of the ignitor). In the limit, for very low densities (ρ = 0.2 g/cm3 ) and to maintain the product ρR constant, the mass needed is too high (in this case, more than 20 kg). So, there is an optimum in the middle of low and high compression ranges, which depends of many factors (for example, electron and ion temperature before compression, Fermi energy of the plasma, etc.).
4. Proton–boron 11 plasmas Proton–boron 11 plasmas present the advantage of minimizing the neutron yield [13,14]. The reaction is p + B11 → 3α(+8.7 MeV).
(12)
The problem of this reaction is that the reactivity σ v is much smaller than the DT case, and the ignition temperature is much larger. Ignition temperature has been defined in the literature as the plasma temperature for which the fusion energy equals the bremsstrahlung energy. This is not an exact definition, but it can be taken as a “static” ignition temperature. The actual ignition temperature must take into account the temperature difference between electrons and ions in the plasma, the heating process of the ignitor, the mechanical losses and so forth. Nevertheless, the static ignition temperature can point out the order of magnitude of the temperature needed to start the fusion onset. In the case of pB11 plasma, results were shown in Fig. 5 for different plasma temperatures and densities. For a density of ne = 1027 e/cc, which corresponds to a pB11 plasma density of 3,333 g/cm3 , the fusion power line crosses the bremsstrahlung line at the classical temperature, 100 keV. For densities above ne = 1029 e/cc (333,333 g/cm3 ), the fusion line crosses the bremsstrahlung line at a lower “static” ignition temperature. For example, for 3.3 × 107 g/cm3 , the ignition temperature decrease to near 20 keV. But the densities needed to decrease the ignition temperature are extremely high. For 3.3 × 107 g/cm3 density, the compression energy is 10.51 MJ/mg, and the maximum fusion energy that can be obtained is 69.6 MJ/mg. Therefore the practical gain would be negligible. Although some previous calculations had pointed out that the actual ignition temperature could be lowered by high density effects and thermal decoupling between ions and electrons [40,41], the main conclusion is that the energy gain cannot reach high values.
5. Conclusions In inertial fusion targets following the fast ignition scheme, degenerate plasmas could be obtained, if the compression phase undergoes an evolution close to the isentropic one. In those plasmas, bremsstrahlung
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radiation emission would be minimized. A main consequence of this fact is that the ignition temperature decreases as the compressed plasma density increases, which is a positive feature for this scheme. However, the energy needed for compression becomes very high in the degenerate case, due to Pauli’s exclusion principle. Energy density of the degenerate plasmas is a function of density, not as in classical plasmas, where it is a function of temperature. Because of that, the target fusion energy gain decreases very much when increasing the density, which is very negative for this scheme. Moreover, if densities are not high enough (higher than 103 g/cc for DT) the plasma becomes classical in a few picoseconds after starting the igniting process, and the behaviour of the plasma becomes similar to the classical case, with the same ignition temperature.
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