Formation of hot spots at end-on pre-compressed isochoric fuels for fast ignition

It is crucial to form a hot spot that can drive a self-heating nuclear burn propagation into the surrounding cold fuel in ignition process of laser fusion. In this paper, we discuss the formation of a hemispherical hot spot located at the edge of an end-on precompressed isochoric fuel instead of a central spherical hot spot surrounded by cold fuel in a corona plasma shell. This configuration leads to a new energy loss mechanism named hot spot collapse, which originates from mass loss of the hot spot through the interface with the vacuum. A semi-analytical model is proposed including α particle induced burning, hot spot collapse and other energy loss processes. Then a modified ignition criterion is derived. The results show that the competition between the hot spot collapse and the burn propagation is crucial in the formation of the hot spot. The formation of such a hot spot at end-on precompressed isochoric fuels would require a somewhat higher initial temperature of Th=9 keV for a hot spot with an initial areal density ρd=0.6gcm−2 . Simulations are performed with a 3D hybrid particle-in-cell/fluid code for the heating process by fast electrons and a 3D radiation hydrodynamics code for the burning process. Simulation results agree with the model. Our optimized result shows a 10 ps heating laser pulse with an energy as low as 39 kJ can lead to a fast ignition in the ideal case. This study provides an effective reference for design and evaluation of experiments planned for fast ignition schemes.

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Introduction
In controlled nuclear fusion reactions, the Lawson criterion was proposed as the necessary condition for a self-heating thermonuclear reaction [1].In 1974, Brueckner and Jorna [2] derived the burn propagation speed based on the works by Fraley et al [3] and Krokhin and Rozanov [4] and discovered the self-regulating behavior of the hot spot in the central ignition schemes.Since then, researchers have studied the effects of various mechanisms on ignition and burn propagation processes [5][6][7][8].Recently, in order to evaluate the experimental progress of the national ignition facility, various ignition criteria with experimentally measurable parameters have also been proposed [9][10][11][12][13].Ignition is defined as the state where the α-particles generated in the hot spot starts to deposit their energies inside the hot spot and also at the boundary of the hot spot, creating a burning wave into the surrounding cold fuel [13].
Apart from the studies for isobaric models [14], attention has also been paid to the formation of hot spots in isochoric model [15].With the limitation of current laser technologies and heating efficiency, formation of a hot spot, which can start a sustainable burn propagation into the surrounding cold fuel, is a critical issue for isochoric fast ignition schemes [15][16][17].Based on 1D analysis and simulation, Atzeni obtained the isochoric ignition condition, T h > 12 keV and ρr h > 0.5 g cm −2 , and the ignition energy E isoch ig ⩾ 72 ( ρ 100 g cm −3   ) −2 kJ by using a simple equation E ig = (4/3) π ρr 3 h C V T h , where C V is the specific heat [18].The formula is subsequently modified to the widely used form of E ig = 140 ( ρ 100 g cm −3   ) − 1.85 kJ by 2D simulations and the ignition window in the energy, power and intensity space is discussed [19].Recent studies claim the ignition energy can be as low as 1 kJ when ρr h ≈ 0.36 g cm −2 , T ≈ 20 keV, ρ ⩾ 700 g cm −2 with non-thermal soft x-rays as the heating source [20,21].Most previous studies on the isochoric fast ignition model assumed hot spots in the spherical center of the fuel.However, it is difficult to ignite such a hot spot by fast electrons, which have to transport through rather large corona shell via hole-boring or ignition cones before reaching the high-density core, leading to a rather poor transport efficiency.Although Atzeni considers the eccentricity and cylindrical shape of the hot spot in the simulation [19], the effects of these circumstances and the possible changes in physical mechanisms dominating hot spot evolution have not been fully discussed.
It becomes important to investigate the formation of a hot spot in a more realistic configuration with the progress of fast ignition schemes, such as the cone-in-shell fast-ignition [22,23], indirect-drive fast-ignition [24] and double-cone ignition (DCI) [25][26][27].In the DCI scheme, drive laser beams first drive the fuels embedded in the two head-on cones, leading to a head-on collision of two high speed plasma jets from the tips of the cones.During the ejecting process, the corona shells of the plasma jets in the radial direction could be peeled off, leading to an end-on isochoric distribution of high-density fuel plasma with sharp edges in the heating direction.Energetic particles, such as fast electrons generated by interaction of relativistic picosecond laser pulses with the inner bottom of a heating cone inserted into the high-density plasma, are subsequently guided from the tip of the cone to heat the plasma directly.This would mitigate the scattering of fast electrons by the conventional corona plasma shell outside of the high-density core and considerably increases the transport efficiency.A hot spot with nearly hemispherical geometry (see figure 1(a)) is then created at the edge of the high-density plasma, generating a burning wave propagating from the hot spot to the whole fuel.The position of the hot spot leads to an interface between the hot spot and vacuum.Such interface results in a new energy loss mechanism named hot spot collapse, which refers to the internal energy carried away by the direct mass loss of the hot spot through the interface.This new energy loss is fundamentally different from the mechanical work in previous studies [28], which originates from mechanical work done by the hot spot during expansion rather than mass loss.A sustainable burn propagation requires that the burn propagation speed is faster than the fuel collapse speed.
In the paper, we focus on the formation process of a hot spot located at the edge of a pre-compressed isochoric plasma and investigate the burn propagation considering the effect of the hot spot collapse.The paper is organized as follows: in section 2, a semi-analytical model is established with the effect of the hot spot collapse, and the modified ignition conditions are given.In section 3, we use O-SUKI-N 3D [29] to analyze the burn propagation, and use HEETS [30] to discuss the impact of heating time on the fast-heating laser energy.In section 4, we conclude our results.

Semi-analytical model of isochoric burn propagation
In this section, we first discuss several physical processes related to the burn propagation in a pre-compressed isochoric plasma.The results are then integrated into the semi-analytical model.

Physical mechanisms related to power balance
Energy conservation equation of the hot spot can be written as where e and M are the specific energy and the mass of the hot spot.The right side of equation (1) shows the α-heating power, the loss power due to radiation cooling and hot spot collapse, respectively.In some literature [12,13,31,32], there is an electron thermal conduction term in the right hand of equation (1).However, since the energy conducted into the cold fuel is recirculated into the hot spotwhen the burning front propagates into the cold fuel, the thermal conduction is not the loss.Besides, because the thermal conductivity is proportional to T 5/2 , the energy loss by the thermal conduction could be neglected in the cold fuel [6,7,11,33,34].The α-heating is the most important energy source for selfheating of deuterium-tritium (DT) fuels.For an equimolar DT plasma, the DT reaction rate per unit volume is described as , where ρ is the plasma density, ⟨σv⟩ is the average reactivity and m f = 2.5m p [35].When the temperature is lower than 20 keV, the relationship between α-particle kinetic energy and the transport distance [36] is as follows: where ρ is density in g cm −3 , T e is the electron temperature in units of keV, transport distance x in µm, ln Λ αe is the Coulomb logarithm for collisions between α particles and electrons [37].Since the hot spot is located at the edge, a fraction of the α-particles can escape directly into vacuum.So, the α-heating power per unit volume is , where E α = 3.5 MeV and f α is the fraction of energy carried by the α-particles escaping from the vacuum interface.For a fully ionized DT plasma, the power radiated per unit volume is ) [28].Recent studies show that the radiation energy can be reabsorbed by the hot spot or by the ablating cold fuel [10,11], so we assume f rad is the fraction of reabsorbed radiation energy.The radiation energy loss power per unit volume is When the hot spot is formed at the edge of the fuel, it loses a part of mass and energy directly into the vacuum, which is estimated by the rarefaction wave.The rarefaction wave front velocity and the burning front velocity compete in the temporal evolution of the mass of the burning plasma.For an equimolar DT plasma, the collapse speed is estimated by the isothermal sound speed c = √ 2k B T/m f .The relationship between the burn propagation speed and isothermal sound speed is v burn /c = 1.37 × 10 16 ⟨σv⟩ s cm −3 [2].Then the temporal evolution of mass follows: where S h−c is the area of the interface between hot spot and the cold fuel.During the collapsing time of the DT hot spot, since the density in rarefaction wave region does not drop to zero instantaneously, fusion reaction can still occur in this region.Thus, we introduce collapse loss factor f c < 1, where f c is defined according to equation ( 5) There is an analytical solution for one-dimensional rarefaction wave expanding into vacuum [28].For the rarefaction wave region, it is estimated as follows: ] . ( For DT plasmas, γ = 5/3, we obtain

Semi-analytical model of burn propagation
In isochoric fast ignition model, heating process by fast electrons would start from the edge of the fuel, making a small part (hot spot) rapidly reach fusion conditions.The burning wave propagates from the edge to the fuel center, while the hot spot collapses from the edge into the vacuum.The condition for a sustainable burn propagation of a hot spot (ignition condition) is determined by the competition between the burn propagation and the hot spot collapse.
In this paper, we consider a preassembled model consisted of a hemispherical hot spot and semi-infinite fuel as shown in figure 1(b).Assuming at the end of fast electron heating, the hemispherical hot spot region has the initial radius R 0 and the initial temperature T 0 .The temperature of the cold fuel is T c .Both the hot spot and cold fuel densities are ρ.The burning wave starts at the interface between the hot spot and cold fuel and propagates into the cold fuel with the speed of v burn .The rarefaction wave front speeds are the isothermal sound speeds c h and c c for the hot spot and cold fuels.Since the hot spot temperature is higher than the cold fuel temperature, c h > c c .
Following the analysis in section 2.1, the energy conservation equation of the hot spot is where N h is the total number of particles in the hot spot, V is the volume of hemispherical hot spot, T h is the average hot spot temperature, W α is the α-particle deposition power per unit volume and W c is the averaged energy loss power per unit volume by the hot spot collapse, referring to the energy carried away by the lost mass of the hot spot.To simplify the analytical derivation, we further make the following assumptions: The heating pulse duration is much shorter than the burning characteristic time and the energy of α-particles propagating into the cold fuel is recirculated into the hot spot.We also assume that the relaxation time between α-particles and electrons is short enough compared with the fluid characteristic time, and the electron and ion temperatures are equal.The propagation distance of the rarefaction wave is denoted as l, the radius of the hot spot during the burn propagation is R, the distance between the burning wave front and rarefaction wave front is d (see figure 1(c)).The number of particles loss per unit time by the hot spot collapse is dN loss where n is the ion number density in the hot spot.The energy loss power per unit volume by the hot spot collapse is obtained after the volume averaging: where f c is the collapse energy loss factor mentioned in section 2.1.The radiation loss power per unit volume satisfies equation (3).Then we obtain the energy conservation equation: It should be noted that this equation is for a hemispherical hot spot at the fuel edge.V is the hemispherical volume.f α is the fraction of escaping alpha energy from the bottom of the hemisphere.The hot spot collapse term of −f c 3nc h T h d only appears when the hot spot is hemispherical and at the edge of the compressed fuel.The temporal evolution of hot spot mass is calculated in hemispherical geometry, as shown in equation ( 4).Letting the left side of equation ( 7) be zero, we obtain the power balance per unit volume equation of the burn propagation in the isochoric fast ignition model, Since f α and f rad are functions of temperature and areal density, it can be observed from equation ( 8) that the hot spot temperature and the areal density are still the critical physical quantities that determine the power balance.Now the areal density is defined as ρd.Equation ( 7) is solved numerically, assuming T c = 1 keV, ρ = 300 g cm −3 , and f rad = 0.2.Using equation ( 2), f α in each time step is calculated by the Monte Carlo method: A sufficient number of α particles are put in the hemispherical hot spot and their positions and emission angles are set randomly.Then the number and energy of α-particles escaping from the bottom surface are calculated to obtain f α .Taking different initial conditions, we obtain the hot spot trajectories on the ρd − T phase diagram and ignition conditions as shown in figure 2(a).The hot spot evolutions are divided into four types.
Type I: When the temperature and areal density of the hot spot are relatively low, as shown by the curves a and b, the hot spot quenches.Although the temperatures exceed 5 keV, the hot spot collapse leads to a rapid decrease in the hot spot volume and the α-particles generated in the hot spot are insufficient to maintain the temperature.
Type II: When the initial hot spot temperature is relatively high, as shown by the curves c and d, the burning wave propagates rapidly into the cold fuel.Because the burn propagation speed is faster than the collapse speed, a large amount of cold fuel is ignited and joins the hot spot.So, the hot spot mass increases.At the early stage of burn propagation, the newly ignited fuel reduces the average temperature of the hot spot slightly.However, as hot spot mass and areal density increase, the α-particle generation rate and the energy deposition fraction in the hot spot increase accordingly.As a result, the hot spot temperature rises and a sustainable burn propagation is achieved.
Type III: When the initial areal density and temperature of the hot spot are appropriate, as shown by the curve e, the areal density and temperature of the hot spot increase simultaneously, and the burning wave propagates steadily.From the trajectories, it is observed that there is a band region on the phase diagram.If the hot spot goes in this region, the temperature and the areal density will increase simultaneously.
Type IV: When the hot spot temperature is relatively low but the areal density is relatively high, as shown by the curves f and g, the speed of the hot spot collapse into the vacuum is faster than the burn propagation speed at first, so the areal density decreases.However, the high areal density gives the hot spot sufficient time to generate sufficient α-particles to raise the hot spot temperature.Consequently, the burn propagation speed exceeds the collapse speed.Then the hot spot volume and temperature grow simultaneously, and a sustainable burn propagation is achieved finally.For an increased areal density, lower hot spot temperature is required for ignition.
The curves c-g present the self-regulating behavior against the hot spot temperature and the areal density.When the temperature is relatively high, α-particles have a larger deposition range and mainly deposit in the cold fuel.The burn propagation speed is faster than the collapse speed.The rapid burn propagation leads to a large amount of cold fuel added into the hot spot.In this case, the areal density increases rapidly, and the temperature increases slowly or decreases, until the temperature and areal density are appropriate.When the temperature is relatively low, α-particles have a smaller deposition range and mainly deposit their energy in the hot spot.8) and the light grey area is the ignition area given by semi-analytical model represented by equation ( 7).(b) Comparison of modified ignition conditions and ignition conditions for central hot spherical spots obtained from figure 4.4 in [28] (assuming ρd = ρ h r h ).The modified ignition conditions are fitted from numerical solution data points (magenta cross).E ig = 2 3 π d 3 ρC V T for ρ = 300 g cm −3 is plotted.The minimum E ig and a more feasible condition are marked with red pentagram (ρd = 0.438 g cm −2 , T = 17.9 keV, E ig = 4.0 kJ) and blue diamond (ρd = 0.6 g cm −2 , T = 9 keV, E ig = 5.2 kJ).
So, the burn propagation speed is smaller than the collapse speed, and the areal density decreases and the temperature increases.When the areal density and temperature are appropriate, both the areal density and temperature increase simultaneously.The self-regulating behavior between the hot spot areal density and temperature reflects the competition between the burn propagation and the hot spot collapse.
The modified ignition conditions are fitted with R 2 = 0.9995 as The ignition energy E ig = 2 3 π d 3 ρC V T for igniting conditions are also plotted in figure 2(b).Atzeni's ignition conditions [28] is also plotted with the assumption of ρ h r h = ρd for comparison.However, we must emphasize such comparison is not rigorous, since the horizontal axes have different meanings.The new required temperature for ignition is stricter than central spherical hot spots primarily for two reasons.First, the mass of a hemispherical hot spot is a half of the mass of a central spherical hot spot for ρ h r h = ρd.Second, the new energy loss term of the hot spot collapse is considered.When the areal density is sufficiently high, the required temperature approaches the ideal ignition temperature of 4.3 keV, because the hot spot collapse has less effect as the areal density increases (S bottom /V h ∝ 1/R) and the radiation loss becomes dominant.For ρ = 300 g cm −2 , the minimum ignition energy E ig = 4.0 kJ occurs when ρd = 0.438 g cm −2 , T = 17.9 keV.However, it is difficult to produce such an extreme condition with fast electron beams.A more feasible condition is ρd = 0.6 g cm −2 , T = 9 keV, and the corresponding E ig = 5.2 kJ.
From the analyses above, we obtain the following conclusions: In the burn propagation process of the isochoric fast ignition, the hot spot collapse is an important source of energy loss, and it must be compensated by higher fusion reaction rate.Therefore, when the areal density is relatively low, the hot spot temperature required for ignition is higher than that of the central ignition schemes.The larger areal density reduces the required temperature significantly.Due to the competition between the hot spot collapse and the burn propagation, the hot spot exhibits a self-regulating behavior.Our model also presents the initial temperature required for the sustainable burn propagation at different areal densities, and the corresponding E ig are given.These analytical results are also reproduced by the following simulation results.

Simulations of burn propagation
In the paper, we use the O-SUKI-N 3D [29] radiation hydrodynamics simulation code to study the burning wave propagation in detail.The impact of heating time on the fast-heating laser energy is discussed using a 3D hybrid particle-in-cell (PIC)/fluid code HEETS [30].
3D simulations of the pre-assembled model in section 2.2 are performed.Figure 3 shows the evolution of density and temperature for a hot spot with an initial radius of 20 µm, an initial temperature of 8 keV (figures 3(a)-(c)) and 9 keV (figures 3(d)-(f )).The simulation length in (x, y, z) directions is 200 µm, the total mesh number is 200 × 200 × 200, the region z ⩾ 100 µm is filled with DT fuel, the region z < 100µm is vacuum.The fuel density is 300 g cm −3 and the cold fuel temperature T c = 1 keV.The hemispherical hot spot has the z-axis as the symmetry axis.The initial electron temperature and radiation temperature are set to be equal to the ion temperature, and the three temperatures are allowed to evolve separately.The boundary conditions are free out-flow conditions.From figure 3, it is observed that the motion of the hot spot plasma is mainly consisted of two parts in the first 10 ps, a rarefaction wave driven by the collapse from the left vacuum interface, and a shock wave driven by the pressure difference between the hot spot and the cold fuel at the semispherical surface.Due to the spherical shock wave propagating outwards into the cold fuel, the density behind the shock front also decreases.At the beginning, fusion reactions occur unaffectedly at the central region of the hot spot, because it takes a period of time for the effects of both rarefaction and shock waves to reach the center.Consequently, the plasma in the central region reaches high temperature, leading to a jellyfishlike structure in both the density and temperature distributions.With the development of rarefaction and shock waves, the complex structures of temperature and density are homogenized.The hot spot becomes a nearly elliptical shape at 45 ps.When the initial temperature is low, the α-heating power is not sufficient to heat the surrounding cold fuel or compensate the energy loss caused by mechanisms such as the hot spot collapse.The hot spot temperature can hardly increase and even decreases, so the burning wave is quenched, as shown in figure 3(c).When the initial temperature exceeds the required temperature for sustainable burn propagation, as shown in figure 3(f ), the hot spot temperature increases continuously and drives the burning wave to propagate into the cold fuel steadily.Figures 3(c) and (f ) also show that the burn propagation speed of T h0 = 9 keV is apparently faster than that of T h0 = 8 keV.
The comparison of ion temperature distributions at t = 50 ps is shown in figure 4. For the case of T h0 = 9 keV, the burn wave propagates much faster and the maximum temperature exceeds 40 keV, indicating a rapid and sustainable burn propagation.However, for the case of T h0 = 8 keV, the maximum temperature has dropped from 15 keV at 10 ps to less than 10 keV at 50 ps and the burn propagation tends to quench.
Two series of simulations are performed to obtain a global characteristic between the burn propagation sustainability and the initial parameters of the hot spot.The initial radii are set to 20 µm and 30 µm, corresponding to the areal density ρd 0 = 0.6 g cm −2 and 0.9 g cm −2 , respectively.The initial temperature range covers 5 − 15 keV with a step of 1 keV.The normalized fusion output energy of each case is plotted in figure 5.For the same areal density, when the initial temperature rises to a critical temperature, the fusion output increases by two orders of magnitude, marking a sustainable burn propagation into the semi-infinite cold fuel.We simulate  the burn process up to 70 ps, and the fusion output energy saturates.It is interesting to note that the simulations give the same required temperature (ignition temperature) for sustainable burn propagation as the prediction of the semi-analytic model in section 2. We also plot the time history of hot spot average temperature on the two sides of ignition temperature in figure 5(b).When the initial hot spot temperature is lower than the critical temperature, it first increases slightly and then decreases, marking the burn quenching.When the initial temperature is higher than or equal to the ignition temperature, the hot spot temperature increases continuously and maintains a high growth rate at the end of the simulation.The burning wave propagates steadily into the cold fuel.
To compare the simulation results with the results of the semi-analytical model, we also plot the hot spot trajectories on ρd − T phase diagram in figure 6(a).The hot spot is defined as the region where T i > 5 keV.The areal density is calculated along z-axis and the temperature is the average ion temperature in the hot spot.
It is observed that the simulation results agree well with the semi-analytical results.The simulations also reproduce the self-regulating behaviors presented in section 2.2.The asymptotic behavior to the ideal ignition temperature of 4.3 keV is also observed for large areal density.These phenomena confirm the importance of the competition between the burn propagation and the hot spot collapse.
However, some of the trajectories are somehow different.For example, trajectories a and b in figure 6(a) show a short increase in temperature at first.Trajectory c, on the other hand, shows simultaneous increase in areal density and temperature at the beginning.All the three trajectories show the unpredicted temperature increase in the initial phase.The main reason for the slight discrepancy is that the semi-analytical model only uses simple average temperature and ignores the internal structure of the hot spot.As mentioned in the analysis of figure 3, the temperature distribution of hot spot forms a jellyfish-like structure in the initial phase.The plasma in the central region reaches high temperature and raises the average temperature.Figure 5(b) also shows this phenomenon.Besides, in the simulations, the rate of areal density increase is slower than the semi-analytical model.The reason is that the deposition range of α-particles is very short in the high-density shock front.Therefore, the α-particles cannot pass through the shock front and heat the cold fuel outside the shock.This feature helps to quickly heat the new fuel as it enters the hot spot, which benefits the burn propagation in general.
The fast-heating laser pulse energy for ignition depends on the heating time τ (the heating laser pulse duration) [19].The 3D hybrid PIC/fluid code HEETS [30] is used to simulate the relativistic electron beams (REBs) fast heating process with τ ranging from 3 ps to 10 ps.Because HEETS assumes that background ions do not move, the ps laser pulses are limited to 10 ps to avoid significant hydrodynamic effects.The heating time range can be provided by some advanced laser facilities for integrated fast ignition experiments, including the Shenguang II upgrade laser facility (SG-II-U), which is currently under construction for the DCI experiments in the near future.The longer heating time is discussed in [19], which shows the hydrodynamic effects would play a role in the heating process.The total energy of the heating laser E ps is taken from 35 kJ to 70 kJ.The heating laser pulses are injected into plasmas with ρ = 300 g cm −3 , with a Gaussian spatial profile, the divergence angle 30 • , and the full width at half maximum of intensity (FWHM) of 20 µm.We assume the energy transfer efficiency from the heating laser pulse to REB is η = 50% and the REB temperature is determined by Beg's law [38,39].An example result is shown in figure 1(a).Our results for different E ps and for τ = 3 ps, 5 ps and 10 ps are shown in figure 6(b).The heating laser energy E ps required for ignition increases as the heating time decreases, primarily because of the following two reasons: Firstly, for the same E ps , the heating laser intensity rises for shorter heating time, then the higher intensity leads to a higher REB temperature.Consequently, the average range of the fast electrons increases.Therefore, the heated fuel mass and the hot spot ρd increase.As a result, the higher E ig is required (see figure 2(b)).Secondly, the energy deposition rate from fast electrons to background electrons is much faster than that from fast electrons to ions.Consequently, the electron temperature is higher compared with the ion temperature (see figure 6(b) inset).The higher laser intensity increases T e and enlarges the difference between T i and T e , so that more energy is wasted by radiation during the electron-ion temperature relaxation process.Our optimized result indicates that a 10 ps laser pulse with an energy as low as 39 kJ, corresponding to laser power 3.9 PW, leads to a successful ignition of the hot spot.However, it should be noted that the energy requirement is calculated for the ideal case shown in this section and should be considered as a lower limit.The heating energy required could increase in practical experiments due to a lower heating efficiency of fast electrons.The decrease in efficiency could be caused by several factors, including different spatial profile and energy spectrum of fast electrons and the possible hydrodynamic effects during the 10 ps heating pulse.The fast electrons could be scattered by the corona edge (although much shorter than the corona shell generated by conventional central implosions).The hydrodynamic effects could become more important that should not be neglected in the HEETS simulation.Since the practical experiments are complicated, there could be more factors beyond the above discussion that may further affect the heating energy requirements.

Conclusions
In this paper, we have discussed the burn propagation of a hemispherical hot spot at the edge of an isochoric fuel instead of central hot spots in previous studies.This more realistic configuration leads to a new energy loss mechanism named hot spot collapse, which originates from mass loss of hot spot through the interface with the vacuum.In this work, a semianalytical model is proposed and the modified ignition criteria are obtained.The results present that the hot spot collapse is an important source of energy loss, which should be compensated by a faster fusion reaction rate via increasing the initial temperature.The competition between the burn propagation and the hot spot collapse plays a decisive role in the formation of the hot spot required for a sustainable burn propagation.The model shows that a larger areal density could significantly reduce the required temperature.For the areal density ρd = 0.6 g cm −2 defined in our model, the required temperature is 9 keV.Our simulations of three-dimensional radiation hydrodynamics code show that the ignition conditions and self-regulating behavior are consistent well with the model.Using the 3D hybrid PIC/fluid code, the effects of heating time (⩽10 ps) are discussed on the required ps laser energy E ps .The E ps required for ignition increases for a shorter heating time.Our optimized result shows a 10 ps heating laser pulse with an energy as low as 39 kJ can lead to a fast ignition in the ideal case.
It should be noted that the model is relatively simple.However, the predicted conditions for a sustainable burn propagation are consistent with the simulation results.Both the analytical and numerical results present the key importance of competition between the hot spot collapse and the burn propagation for hot spots in the end-on isochoric fast ignition model.This model can provide an effective reference for design and evaluation of experiments of fast ignition schemes.

Figure 1 .
Figure 1.(a) A simulation result of electron-beam fast heating process by HEETS with the energy of ps laser Eps = 40 kJ, FWHM= 20 µm and pulse length τ = 10 ps.(b) Schematic of fast ignition and preassembled model of a hemispherical hot spot in a semi-infinite fuel.(c) Schematic of the burn propagation and the hot spot collapse after a certain time.Here d is the distance between the burning wave front and rarefaction wave front, l is the propagation distance of the rarefaction wave and R is the radius of the hot spot during the burn propagation.

Figure 2 .
Figure 2. (a) Hot spot trajectories on the ρd − T phase diagram with different initial conditions.Different line styles refer to different evolution types.Filled (void) circles refer to sustainable (unsustainable) burn propagation.The dark grey area is the self-heating area calculated by equation (8) and the light grey area is the ignition area given by semi-analytical model represented by equation (7).(b) Comparison of modified ignition conditions and ignition conditions for central hot spherical spots obtained from figure 4.4 in[28] (assuming ρd = ρ h r h ).The modified ignition conditions are fitted from numerical solution data points (magenta cross).E ig = 2 3 π d 3 ρC V T for ρ = 300 g cm −3 is plotted.The minimum E ig and a more feasible condition are marked with red pentagram (ρd = 0.438 g cm −2 , T = 17.9 keV, E ig = 4.0 kJ) and blue diamond (ρd = 0.6 g cm −2 , T = 9 keV, E ig = 5.2 kJ).

Figure 3 .
Figure 3. Density (upper half of each figure) and ion temperature (lower half of each figure) evolutions for a hemispherical hot spot with the initial radius R h0 = 20 µm and the initial areal density ρd 0 = 0.6 g cm −2 .(a)-(c): with the hot spot initial temperature T h0 = 8 keV.(d)-(f ): with the hot spot initial temperature T h0 = 9 keV.

Figure 5 .
Figure 5. (a) Normalized fusion energy output during 70 ps with different initial conditions of the hemispherical hot spot.(b) Time history of the hot spot average temperature on the two sides of ignition temperature, 9 keV and 8 keV.The initial areal density ρd 0 = 0.6 g cm −2 .

Figure 6 .
Figure 6.(a) Simulated hot spot trajectories on ρd − T phase diagram.The hot spot is defined as the region where T i > 5 keV.Areal density is calculated along z axis, and temperature is the average ion temperature in the hot spot.Filled(void) circles and solid(dashed) curves refer to a sustainable(unsustainable) burn propagation.Shaded areas are the same as figure 2. (a).(b) Fast heating simulation results of ps laser pulse τ = 3 ps, 5 ps, 10 ps and different laser energy by HEETS.The triangular (59 kJ), diamond (46 kJ) and pentagram (39 kJ) represent the minimum energy required for ignition.Inset: temporal evolution of averaged T i and Te in region where (T i + Te) /2 > 5 keV.Eps = 46 kJ, τ = 5 ps, intensity FWHM = 20 µm.