Direct-drive heavy ion beam inertial confinement fusion: a review, toward our future energy source

ABSTRACT Direct-drive heavy ion beam (HIB) inertial confinement fusion (ICF), or HIF would be a promising future energy source for society. Particle accelerators produce HIBs with precise particle energies, pulse lengths and pulse shapes with high energy efficiencies of ~30-40%. Higher energy driver efficiency means that a lower fusion energy output is required to construct a HIF power station to supply ~1 GW of electricity. A HIF power station could use about 4 to 5 MJ of HIB energy per shot at a shot rate of ~10 Hz. This review is focused on the direct-drive scheme in HIF. In direct-drive fuel target HIBs deposit their energy into a shell surrounded by a denser tamping outer layer. The DT (Deuterium-Tritium) fusion fuel, with a total mass of several mg, must be compressed to about one thousand times solid density to reduce the input driver energy and to achieve an adequate burn fraction. High-density compression is a major challenge in ICF, requiring that non-uniformity in driver energy deposition be kept lower than a few percent. The axis of an HIB can be made to oscillate sufficiently rapidly to improve the uniformity of energy deposition. Graphical abstract

In Ref [11], we have reviewed HIF physics in 2016, and it was oriented to specialists in HIF. After that, new remarkable scientific advances have been achieved in HIF studies, including ion source in accelerators, accelerator science, cluster ion beam science and technology, beam manipulation science, control of fuel target injection, instability control by heavy ion beam, fuel target uniformity mitigation, etc. The present review paper is prepared to present the overall achievements to date in HIF studies in a concise manner for scientists not only in HIF but also in other disciplines.
Heavy ions may penetrate a thin heavy tamping layer, and deposit the particle energy near the stop position inside the material beyond the heavy layer. Figure 1 shows a schematic figure of this energy deposition profile in the middle layer by a dotted red curve in Figure 1. Near the ion stop region, the Coulomb collision becomes strong [27][28][29][30], and the HIB energy deposition peak appears in the inner region as shown in Figure 1. If the mass of the middle layer and the thermal energy of the materials are negligible, the input HIB energy E d is transferred to the kinetic energy of both the right layer E KM ¼ 1 2 MV 2 and the left layer E Km ¼ 1 2 mv 2 . Based on the energy conservation and the momentum balance, the energy efficiency η imp � E Km =E d ¼ 1 The energy efficiency η imp shows the ratio of the kinetic energy E Km of the left layer and the input driver energy E d . In ICF, η imp is implosion efficiency. When the right heavy mass M becomes large, the efficiency η imp becomes large. In HIF direct-drive fuel target is designed to follow this fact to realize a high implosion efficiency η imp .
Driver efficiency η d determines a requirement for fusion energy gain G. Fusion energy gain G is defined by fusion energy output E fusion divided by input driver energy E d . The high HIB driver efficiency η d of ~30-40% allows a lower fusion gain to construct a fusion power plant. Figure 2 shows energy balance in ICF reactor systems. In Figure 2 the driver system is assumed to be HIB accelerators, though the overall system is common in ICF. The electric energy E in goes to accelerators to generate HIBs E d with the driver efficiency η d . ICF fuel target releases fusion energy output E fusion ¼ GE d .
The thermal energy E fusion is converted to electricity η th GE d with efficiency η th . A fraction f of electricity η th GE d is recirculated to accelerators: ð Þ is supplied to our society. The high driver efficiency relaxes the fusion energy gain requirement to construct a fusion power plant, which delivers ~1 GW of electricity to society [11]. HIB illuminates a three-layer target from the right in this example. The interaction of beam ions with materials is almost explained by the Coulomb collision, and the HIB energy deposition profile shown by a dotted red line has a peak (the Bragg peak) near the stop region. In Figure 1 the heavy ions stop in the middle layer. Whenm < M, the energy conversion becomes efficient:η imp � M=ðM þ mÞ: In this example, we ignore the mass of the middle layer and the thermal energy in the materials.
The flexibility of HIB accelerators relaxes the requirements for driver HIB parameters in order to deliver sufficient fusion energy output. For example, HIB axis can be made to oscillate with a high frequency, for example, ~several hundred MHz -1 GHz. The HIB axis oscillation or wobbling behavior helps reduce HIBs illumination non-uniformity on a fusion fuel target. The D (deuterium)-T (tritium) fusion fuel should be compressed to about a thousand times solid density to reduce the input driver energy and to achieve an adequate burn fraction [2,3]. The idea of the hollow sphere implosion and fuel compression was proposed in Refs. [8,32]. First the driver energy E d deposited is effectively converted to the kinetic energy of the imploding DT fuel. Figure 3(a) shows an example structure of a hollow shell fuel target in HIF. The spherical target outer radius would be several mm. Near the target center, the imploding DT fuel is stagnated and compressed. The DT fuel kinetic energy is converted to the DT fuel thermal energy, and the DT fuel is ignited. In order to reach the high density, the fuel target implosion non-uniformity should be kept lower than a few percent [2,3,33,34]. The implosion non-uniformity of spherical fuel target comes from driver illumination non-uniformity, imperfect sphericity of fuel target, target injection alignment error into a fusion reactor, etc [35][36][37][38][39][40][41]. The high-density fuel compression is a major challenge in ICF. HIB accelerators also provide a unique feature of the HIB axis wobbling behavior, which mitigates the HIBs illumination non-uniformity [11,31,42,43]. Multiple driver beams irradiate an ICF fuel target. The HIB illumination scheme is important to obtain a uniform energy deposition on fuel target material [44][45][46]. Energy cycle of ICF reactor system. In Figure 2, energy driver is assumed to be HIB accelerators, though the overall system is general in ICF.
In Section 2, the concept of heavy ion inertial fusion (HIF) is presented, and the characteristics of HIF are introduced. The present international research projects are also introduced, relating to HIF and high energy density physics (HEDP) based on HIB. Section 3 summarizes the heavy ion interaction with materials. The HIBs illumination scheme is also discussed. In Section 4 application topics relating to HEDP based on HIB are introduced briefly. Section 5 is devoted to describe HIF energy release from fuel target. In this review we focus on direct-drive spherical target in HIF. In Section 6 we conclude the HIF study review.

Concept and characteristics of heavy ion inertial fusion
Research projects are first presented in HIF and HIB-based HEDP researches. The FAIR (Facility for Antiprotons and Ion Research) project has been started at Darmstadt, Germany [19]. FAIR is oriented to basic physics to understand the structure of matter, the evolution of the universe, etc., including HIF, HEDP and plasma physics. At Berkeley, CA, U. S. A. an experimental device NDCX II works on HIB accelerator physics including HIB space charge neutralization [18,47]. The HIAF (High Intensity heavy ion Accelerator Facility) project in China has been planned for HIF and HEDP studies [20][21][22][48][49][50]. HIAF construction started in 2018 in Huizhou City, China. The maximum energy and power in one beam pulse  Figure 3(a). The driver energy deposited creates a pressure peak in Al, and multiple shock waves drive the inner DT fuel, which is accelerated to ,3 À 5 � 10 5 m=s: During the implosion and the stagnation period, the DT fuel is compressed to about one thousand times solid density. Then at the peak compression near the target center the DT fuel is ignited and burned [31]. could reach 100 kJ and 1 TW, respectively. With the high power of the heavy ion beam, a large volume of a material would be heated to ~10 eV [21,49,50]. Researches on HIF have been conducted at a large number of research labs and universities in Japan, Germany, China, France, U.S.A., Russia, Italy, Spain, Kazakhstan, etc. [11,12,16,51]. Now the concept and characteristics of HIF and HIB are presented. When HIF reactor system is focused, one of the merits of HIF would be the high driver efficiency η d as presented in the introduction. In Figure 2 one can easily derive a relation of a requirement for the fusion energy gain G ¼ E fusion =E d , to sustain the fusion power plant: G > 1= f η d η th À � . The electricity generation efficiency η th would be ~40% from thermal energy to electricity. The fraction f of electricity generated goes to operate the HIF system, and f should be smaller than 1. The HIB accelerator driver efficiency η d is ~30-40% from electricity to the HIB particle energy. In conclusion, the requirement for the fusion energy output gain G should be G > ,50 À 70 in HIF, if we assume f ,0:12 À 0:15. The low gain requirement helps relax the fusion reactor design parameters. The HIB driver input energy E d per shot would be ~4-5 MJ. In order to release electricity of ~1GJ/s for one power plant, the reactor operation rate would be ~10-15 Hz in HIF.
In nuclear fusion, two ions are fused to release fusion energy, for example, D þ T ! n þ α þ 17:6MeV. DT fusion has a relatively larger reaction cross section among other fusion reactions, and the DT fusion is the present target reaction in fusion studies. The DT fusion cross-section peaks around 64 keV [3,52]. In ICF DT fuel mass of several mg is contained in a fusion target capsule. The DT fuel should be compressed to around a thousand times solid density to reduce the input driver energy and to achieve an adequate burn fraction. At the same time the DT fuel must be heated to the extremely high temperature of ~5 keV at ignition time. The fuel compression is performed during the fuel implosion and the stagnation. At the final stage of the stagnation phase, the DT fuel is compressed further and ignited. In order to realize the complex process in the fusion reactions, an example target structure is shown in Figure 3(a). The HIB driver energy is mainly deposited in the Al layer in Figure 3(a). The example target structure in Figure 3(a) is designed based on the idea in Figure 1 and the relating considerations. The heavy outer layer acts as a tamper, and the middle layer is the HIBs energy absorber. A pressure peak appears in the Al layer, and the pressure gradient in Al pushes the inner DT fuel inward. The acceleration magnitude is roughly ~10 13 m/s 2 , and during a few tens of ns the DT fuel is accelerated to ~3 À 5 � 10 5 m/s. Figure 3(b) shows an example r À t diagram during the DT implosion. The DT fuel is gradually accelerated by multiple shock waves. A typical HIBs input pulse shape is introduced in Section 5. During the DT fuel compression, it is favorable to keep the DT temperature low, and the adiabatic compression is ideal. However, the successive shock waves appear in the implosion phase, because the implosion speed of ~3 À 5 � 10 5 m/s is comparable or larger than the DT fuel sound speed. Therefore, we need to keep the adiabat as low as possible. The 'adiabat' α adiabat in ICF is the isentrope parameter, and is defined by the ratio of the fuel pressure p and the adiabatic pressure p adiabat , which is the pressure derived from the initial pressure by the adiabatic compression: α adiabat ¼ p=p adiabat [3]. During the implosion and stagnation periods, the DT fuel is compressed to about a thousand times solid density from the initial DT liquid density. The fusion products of α particles in the DT reactions deposit their energy inside the DT fuel [2,3]. The Lawson criterion is rewritten in ICF for the DT density-radius product of ρR as follows: ρR > ,2 À 5kg/m 2 [2,3]. The burning fraction θ burn of the DT fuel is estimated by θ burn ,ρR= 60 þ ρR ð Þ. In our studies below, almost ρR,10kg=m 2 . DT fuel stable compression is a key issue in ICF. In the NIF target experiments at Livermore, CA, U.S.A., they have succeeded to compress the DT fuel to a few thousands times solid density in ICF [53,54]. They did not reach a perfect ignition of the fuel. However, the achievement of the high density is remarkable in ICF. On the other hand, the DT fuel met material mix, that would be induced by the R-T instability. The R-T instability and material mix are caused by implosion non-uniformity, driver illumination non-uniformity, etc. [53,54]. The DT fuel implosion stability should be also studied carefully even in HIF. The HIB driver efficiency η d is relatively high in HIF, and driver input energy of ~5MJ would be available by conventional accelerator technology.
In HIF one HIB has ~10 15 ions. Promising and stable ion source is laser ion source [55][56][57][58], since it delivers heavy ions with a high brightness. In the laser ion source system, an intense laser pulse is focused on a solid target placed in a vacuum vessel. The laser energy is used to ionize the target material, and an ablation plasma is produced. Since the laser irradiation period is short and typically less than a few tens of nanoseconds, the heated plasma does not have time to expand during the laser pulse. When the plasma plume reaches the ion extraction acceleration gap, heavy ions are extracted. Recently, a solenoid magnetic field is proposed to enhance and control the beam current density at the plasma plume travelling section. The HIB current is also adjusted stably by changing the strength of the solenoid magnetic field within a proper range [55,58]. Heavy ions should be repetitively extracted, accelerated, manipulated and transported to the fuel target, which is injected and aligned in a fusion reactor. The laser ion source would be appropriate to the repetitive operation [55,58] in HIF.
The final HIB pulse length should be about a few tens of ns. An advantage of heavy ions is the low space charge effect during the HIB acceleration, manipulation and transport. In the present HIF reactor designs, the candidates of the ion species include Pb + , CS + and Au + . In this review, we focus on Pb + ions. One of the merits of HIB accelerators is stable and repetitive operations with precise beam controllability. HIB accelerators have control capabilities of HIB focal position, pulse length, particle energy and axis manipulation. The HIB flexibility also relaxes the requirement of the HIBs illumination uniformity on the fuel target. This point is again discussed in Sections 3 and 5. Studies suggest that at least 32 HIBs are required to reduce the HIBs illumination non-uniformity on direct-drive fuel target [46]. The HIBs illumination scheme is also important to release the fusion energy (see Figure 4(a)), and is discussed in Section 3.
In ICF reactor, DT fuel target is repetitively injected and precisely aligned. When a spherical DT fuel is injected from outside a reactor, the DT fuel pellet is accelerated to a high speed of ~100 m/s to avoid the DT fuel melting by the reactor exhaust reactor gas [36][37][38][39][40][41]. Figure 4(b) shows a schematic diagram for the fuel target injection into a reactor chamber. As shown in Figure 3(a), DT fuel is filled in fuel capsule. The DT fuel is cooled down to be a liquid, and the DT fuel layer is formed as shown in Figure 3(a) in the hollow fuel capsule. The stable fabrication of the fuel capsules is also important for the ICF reactor system [59][60][61][62][63]. When the cryo-target in Figure 3(a) is employed, the most outer layer of Pb becomes a superconductor [35][36][37] in a cryostat. The cryo-target is transferred to an injection gun, by which the cryo-target is injected into reactor. The traveling time of the fuel target injected is about 0.05 ~ 0.06 s [36], which is estimated by the traveling distance (~5-6 m) divided by the target injection speed of ~100 m/s. During the time interval of about 0.05 ~ 0.06 s, it was found that the superconducting state of the cryo-target is not melted [36]. The results in Refs. [35][36][37] demonstrate that the target trajectory and speed are controlled by magnetic force. The HIF direct-drive target in Figure 3(a) is designed to have the heavy outer layer which becomes a superconducting state. This point would be also a merit of the HIF target. The outer superconducting shell would help reduce the target alignment error in reactor chamber in HIF [35][36][37].
Between accelerator exit and fuel target aligned at a reactor chamber center, HIBs are transported. The distance would be ~5-10 m, depending on the reactor chamber radius (see Figure 4(b)). Issues in HIF also include HIF reactor design, tritium breeding and handling, first wall configuration, control of exhaust reactor gas, activation of the reactor structure, etc. For example, accelerators should be protected from the reactor exhaust backward gas [64]. DT fusion products are neutrons and alpha particles. Therefore, the fusion products would be relatively easy to handle. However, the reactor structure, which would be made of Al, is activated by 14.1 MeV neutrons. The management of activated structures should be considered to keep our environment clean.
HIB final transport near and inside the reactor chamber has been also intensively studied [65][66][67]. It was considered that charged HIB is transported in a vacuum to fuel target, because heavy ions are not seriously influenced by their net charge. Almost ballistically heavy ions can be transported between accelerator exit to fuel target in reactor center. In order to enhance the HIB controllability, charge neutralization of HIB has been studied well. Experimentally at LBNL and theoretically at Princeton University and Japan, HIB space charge neutralization, focusing and manipulation are studied [65][66][67][68][69][70][71][72]. Plasma electrons are supplied to neutralize the HIB charge from a gas plasma preformed in space or at the material surface by active or passive discharges [65][66][67][68][69][70][71][72]. HIB instabilities were studied inside reactor gas, and the results show that HIB is rather stable during the transport in reactor gas against the two-stream and filamentation instabilities [67,73,74].

Heavy ion interaction with material
Heavy ion interaction is explained rather well by the Coulomb collisions with target material electrons and ions [28,30]. The stopping power E stop of a target is the sum of those of target nuclei E nuc , target bound electrons E boudÀ e , free electrons E freeÀ e and target ions E ion [28][29][30]75,76]. One can find each term in Refs. [11,28,30]. The stopping power comes mostly from the Coulomb collisions. When a projectile ion speed is slow, the stopping power becomes large from the nature of the Coulomb collision. When fusion target material is heated by HIBs illumination, usually the energy absorber layer in Figure 3(a) is heated to ~200-300 eV, and the pressure peak appears there. When material temperature becomes high, material-free electrons contribute to the stopping power. At the same time, plasma electron waves are also excited and the free-electron stopping power is slightly enhanced compared with that by the bound electrons. The stopping power enhancement induces the ion stopping range shortening [29,77,78]. The ionization degree of swift heavy ions changes as they move though materials. The ionization degree would be called the effective charge [28,30]. In the review, numerical studies include all the effects on the stopping power. The energy deposition peak, that is, the Bragg peak (see Figure 1), appears near the stop point, which is designed to be localized in the energy deposition layer of the fuel target in Figure 3(a). All HIB ions deposit their energy inside the target material, and the target structure in Figure 3(a) becomes appropriate for HIF direct-drive fuel target. In Review, we focus on direct-drive fuel target. Energy deposition profile is predictable and well defined. In addition, the typical density gradient scale length L would be about several hundreds of μm around the pressure gradient peak, where the Rayleigh-Taylor instability appears during the fuel implosion. Figure 5 presents a schematic diagram of density and pressure profiles of imploding spherical fuel target near the pressure peak. The solid curve shows the density profile, and the dotted line the pressure profile. In HIF target implosion, the HIB stopping range is relatively large and ~500-700μm, depending on ion particle energy and target materials. The pressure Figure 5. Schematic diagram of density and pressure profiles of imploding spherical fuel target. The solid curve shows the density profile, and the dotted line the pressure profile. In HIF target implosion, HIB stopping range is relatively large and ~ 500-700μm, depending on the ion particle energy. The pressure gradient scale length is long as well, and the density gradient scale length L is also large ~several hundreds of μm or so. gradient scale length is long as well, and the density gradient scale length L is also large ~several hundred micrometers or so. When the density gradient scale length L is large, the growth rate γ reduction would be expected [79,80]: γ ¼ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi gk= 1 þ kL ð Þ p . Here g is the implosion acceleration, and k the wave number. Therefore, short wavelength modes may be stabilized or mitigated in HIF, and the mode with k,2π=L would be dangerous in HIF. The large density scale length of L is also one of the characteristic features of HIB interaction with matters.
HIBs deposit the energy volumetrically. The Pb + heavy ion energy would be ~8-10 GeV in HIF, and the stopping range is roughly ~0.5-0.7 mm or so, depending on target design. HIBs interaction with material should be considered in three dimensions [46,[81][82][83]. HIBs illumination scheme is also important to obtain a low non-uniformity of HIBs driver energy deposition [11,46,[84][85][86]. Figure 6 shows (a) a typical example of 32 HIBs illumination scheme onto a spherical fuel target [11,44,46] and (b) the HIBs illumination RMS non-uniformity versus the HIBs number employed. Figure 6(b) presents that the minimal HIBs number would be 32 to obtain a low driver beam illumination non-uniformity of a few percent [33,34]. In Figure 6, we employ Pb + ion HIBs with the mean particle energy of 8 GeV. The beam radius at the entrance of reactor chamber wall R en is 35 mm, and the reactor chamber radius is assumed to be 5 m. The beam particle density distribution is the Gaussian one. The longitudinal temperature of HIB ions is 100 MeV. The beam transverse emittance ε r is 3.5 mm mrad [84]. The target temperature increases linearly during HIB deposition from 0.025 eV to 300 eV in this case in Figure 6(b). We employ a Pb + Al pellet structure with 4 mm external radius in Figure 3(a). The Pb mass density is 11.3 g/cm 3 . In the pellet structure, the outer Pb layer thickness is 0.06 mm. The results on the HIBs illumination nonuniformity are found in Ref [84]., including the mode analyses.
In addition to single heavy ions as HIF driver, it would be valuable to mention cluster ion beam for ICF. We presently see a reevaluation of the cluster ion beam (CIB) approach to ICF, including cluster ion acceleration, cluster fragmentation and target correlated stopping of cluster ions [87][88][89][90]. The CIB drive HIF demonstrates a potential for ICF through the enhanced and correlated ion stopping together with a lower q/A [87,88], which may relax the accelerator design further, compared with that for the conventional HIB accelerators. At the same time, the cluster ion source and the stable cluster acceleration should be further studied experimentally [90].

High energy density physics with heavy ion beam
Besides HIF studies, research on high-energy density physics (HEDP) [91,92] has been explored based on HIB to investigate high temperature, high density and high pressure, which normally cannot be attained in nature on earth [12,24,[93][94][95][96][97][98][99]. In HIB-based HEDP, the target state of matter is in a temperature of about ~1 eV, a pressure of ~GPa or more and about solid density. HEDP contributes widely from fundamental physics to applications to understand plasmas, materials at extreme states, space, nuclear physics and engineering, radiation matter interaction, etc. HEDP research topics include non-ideal equation of state (EOS), strongly coupled plasma, nonideal electrical and heat conduction [100,101], phase transitions at extreme states [96][97][98][99], radiation-rich plasmas and other fundamental phenomena in plasmas. HEDP is also closely relating to ICF.
Here we introduce interesting HEDP studies based on HIB. Reference [102] shows an interesting study plan to investigate a dense and warm matter. A target material is compressed by a diamond anvil cell, and the compressed material is illuminated by an HIB. HIB deposits its energy uniformly in a relatively large volume in high density.
Other HEDP studies are conducted and planned at the existing and constructing HIB facilities of FAIR [19], HIAF [20][21][22][48][49][50], NDCX II [18,47] and an induction accelerator at High Energy Accelerator Research Organization (KEK), Tsukuba, Japan [103]. At FAIR, GSI, Darmstadt, EOS and HEDP studies are planned at various material states with phase transition, driven by HIB [97,98,[104][105][106][107][108][109][110][111][112][113][114][115]. A magnetized plasma compression [24] is also planned at FAIR, and an annular HIB illuminates a cylindrical hollow target. One of the research purposes of the magnetized plasma is to focus the HIB [24], as a part of the HEDgeHOB Collaboration at FAIR [104]. HEDP studies are being performed under the experiments LAPLAS (Laboratory Planetary Sciences) and HIHEX (Heavy Ion Heating and Expansion) at HEDgeHOB at FAIR with collaborators including scientists in Darmstadt, Germany, Univ. de Castilla-La Mancha, Spain and others [98,[105][106][107][108][109][110][111][112][113][114][115]. An annular HIB is created by a HIB wobbler, in which the HIB axis is oscillated at a frequency of ~1 GHz to create the axial and azimuthal symmetry of the HED state in the cylindrical target [24--26,107,115]. HIAF is planned to be operated in 2024 in China. Studies at HIAF include EOS, material phase transitions, hydrodynamic behavior of matter, ion interaction with materials in the HED state [22]. At KEK, Japan, HEDP and ICF based on a cluster ion beam (CIB) are being considered [12,[87][88][89][90]103,116]. For example, Si clusters are efficiently formed in an ellipsoidal-shaped cell with a Si crystal at the focal point of the cell. The Si crystal is illuminated by an intense laser, Si clusters are formed and confined in the cell by a shock wave created in the cell [116]. The KEK induction accelerator accelerates giant Si clusters successfully [12,103,117]. In Refs. [87][88][89]118,119], cluster ion-based ICF is studied, though further studies are needed, including size distribution control of cluster ions, stable cluster ion acceleration and transport, cluster ion beam interaction with matter, etc. Relating to cluster ion interaction with solid material, correlation effect among ions of one cluster would introduce an enhancement of ion stopping power [87][88][89]. On the other hand, a peculiar ion-overtaking behavior between two Si ions of one Si cluster was also found in a Si-cluster solid interaction theoretically [90]. Near future experiments are needed to confirm the stopping power enhancement and physics in cluster ion interaction with material.

Energy release in heavy ion inertial fusion
In this section, we present fusion fuel target implosion and fusion energy release in HIF. A typical HIBs input pulse shape is shown in Figure 7(a). The foot pulse in Figure 7(a) kicks the DT fuel by a weak initial shock wave, and the main pulse with the higher intensity accelerates the DT fuel by the pressure gradient explained in Figure 5. The input HIBs energy is converted to the DT fuel kinetic energy first. DT fuel is accelerated to ~3 � 10 5 m/s. An example time history of the DT fuel implosion speed is presented in Figure 7 (b). In this example case, the fuel outer radius is 4 mm, the Pb layer thickness is 30μm, the energy absorber is the 460μm thick Al, and the DT fuel of 2.68 mg is confined in the fuel target capsule in Figure 3(a). The DT inner surface is at 3.43 mm, and the initial densities of Pb and Al are the solid density. The initial target temperature was set to be 0.1 eV. When the 32 HIBs irradiate the fuel target with the illumination scheme in Figure 6(a) [44,84], the simulation result in Figure 7(b) was obtained [31]. A twodimensional (2D) fluid implosion code was used to analyze the implosion dynamics [81][82][83]120]. The precise HIBs energy deposition profile was also computed together with the target plasma behavior [31,120].   Figure 6(a). The multiple HIBs introduce a small illumination nonuniformity (see Figure 8(a)). The initial HIBs illumination non-uniformity is about 2% in this example case, and the fusion energy gain is 62.4. number of the fusion energy gain G in HIF. The HIBs illumination nonuniformity was ~2% in Figure 8. During the DT fuel implosion and stagnation, the initial small implosion non-uniformity grows as shown in Figure 8 (b,c) during the implosion. Even with the evident non-uniformity, the DT fuel is successfully ignited and burnt. Figure 9 shows the mass density distributions for this example case. In Figure 9(b,c) the implosion nonuniformity also appears in the density distributions.
Here we revisit and estimate requirement for implosion non-uniformity. ρRat the stagnation decides the fusion energy output. When a small implosion nonuniformity appears, ρR decreases from the perfect uniform ρR ð Þ 0 . The total DT mass is fixed during the implosion, and ρR is proportional to 1=R 2 . Therefore, On the other hand, nonuniformity δα of implosion acceleration α would be estimated by where r 0 is the fuel initial radius and η the density compression ratio of the DT fuel. The density compression ratio η may be about 1000 in HIF. We obtain the relation of In ICF reactor power plant, the degradation threshold of ρR ð Þ= ρR ð Þ 0 would be about 0.5 ~ 0.8, and δα=α should be less than about 3 ~ 4% [33,34]. Figure 10 shows the relation between δα=α and ρR ð Þ= ρR ð Þ 0 . The results above and in Figure 10 reproduce the results obtained in Refs. [33,34]. When the density compression ratio becomes higher, the nonuniformity requirement becomes stringent.
As presented in Sections 1 and 2, HIB accelerators have flexibility to control the HIB axis position. HIB accelerators can oscillate the HIB axis with a high frequency ~100 MHz-1 GHz [24][25][26]. The axis-oscillating or wobbling HIB also provides a unique tool to smooth the HIBs illumination non-uniformity in the HIF target implosion [31,42,43,46]. The wobbling HIBs are applied to smooth HIBs illumination non-uniformity in fuel target implosion.
The idea of the non-uniformity smoothing by the wobblers comes from feedforward control theory [31,[121][122][123][124][125]. Figure 11 shows the initial perturbation growth in an unstable plasma system. At t ¼ Δt, the perturbation grows, and in Figure 11 at the same time another perturbation, which has the inverse phase, is added to compensate the perturbation growth. The perturbation is mitigated even in plasma systems. In HIF target implosion, multiple HIBs illuminate fuel target, and HIBs introduce a small non- Figure 11. Perturbation mitigation mechanism. In an unstable plasma, a perturbation grows in time. At t ¼ Δt, if another perturbation with the inverse phase is applied to the system, the overall perturbation superimposed is mitigated, similar to the feedforward control. uniformity. If HIBs axes are oscillated or wobbled in time, the HIBs illumination non-uniformity would be mitigated.
If a single mode k ¼ 2π=λ is introduced in a plasma, the amplitude a of the perturbation becomes a ¼ a 0 e ikxþγt . For stable system γ is negative, and for unstable system γ shows the instability growth rate. Here λ is the wave length. An ideal dynamic smoothing mechanism is shown in Figure 11 [122][123][124][125]. In heavy ion accelerators HIB axis can be oscillated or wobbled in a controlled manner to realize the dynamic smoothing of HIB illumination non-uniformity [24][25][26].
A superimposed perturbation may be expressed for a physical quantity F at t ¼ τ as follows: Here we assume a uniform oscillation in time for the perturbed driver. In Equation (1) the amplitude is described by δF, Ω shows the wobbling frequency of the driving beam, and Ωτ is the phase shift of the perturbations superimposed. The integrated perturbation at t is derived as follows: When γ � 0, the system is stable and Equation (2) shows a simple dynamic smoothing of the perturbations. When Ω > > γ j j, the amplitude reduction ratio becomes γ j j=Ω [122][123][124]. Even for Ω ffi γ j j we can still expect the significant mitigation. At this point, it should be noted that the integrated perturbation amplitude is mitigated well, but the growth rate γ of the instability does not change. The result in Equation (2) suggests that the wobbling frequency Ω should be high or comparable compared with the instability growth rate of γ for the effective mitigation of the perturbation amplitude. The dynamic smoothing or mitigation mechanism is rather simple and easy to apply to instability mitigations in plasmas and fluids [43,[122][123][124][125]. Especially the Rayleigh-Taylor (R-T) instability tends to appear during the DT fuel implosion in ICF. When the dynamic mitigation mechanism is applied to the R-T instability, it was found that the R-T instability growth is mitigated well [31,122,123].
The dynamic smoothing mechanism was applied to the HIF target implosion to smooth the HIBs illumination non-uniformity by the spirally axis-oscillating HIBs [31,42]. Figure 12 shows a schematic diagram of the HIB rotation or wobbling behavior. In Refs. [31,42], 32 HIBs axes oscillate spirally from the initial axis positions shown in Figure 6(a). After around one rotation, each HIB rotates along the circle trajectory. The HIBs with the simple circularly oscillation induce a large initial imprint of the illumination uniformity [42]. The spirally oscillating or wobbling HIBs smooth the HIBs illumination non-uniformity well [42]. It is expected that the HIBs wobbling motion realizes the dynamic smoothing and mitigation mechanism in HIF target implosion.
Including the wobbling motion of the 32 HIBs [120], another 2D implosion simulation was performed [31], and Figure 13(a) shows the ion temperature distribution at t=29 ns. Compared with the result in Figure 13(b) without the HIBs wobbling, it is apparent that the spatial non-uniformity is smoothed by the wobbling HIBs. For both the cases in Figure 13(a, b) the difference is just the HIBs wobbling motion in the computation conditions. In this case, the HIBs wobbling frequency is 424 MHz. Figure 14 shows the mode analyses of the DT fuel ion temperature at t=25 ns for the cases (a) with and (b) without the wobbling HIBs. The mode analyses are performed based on the Legendre function P n , and the mode n shows the mode of the Legendre function [126] in Figure 14.
Relating to Equation (2) and Figure 11 on the dynamic mitigation and smoothing mechanism, we also checked the phase of the implosion acceleration non-uniformity. The implosion acceleration distributions are presented in Figure 15. During the HIBs half rotation between 6 turns and 6.5 turns of the rotating HIBs, the phase of the implosion acceleration nonuniformity of the DT fuel is clearly inversed. The result means that the phase Initially the HIB center stays at the optimal position to reduce the initial imprint of the illumination non-uniformity. After around one rotation, each HIB axis tends to a circle rotation. The HIB wobbling behavior contributes to smooth the illumination non-uniformity, and mitigates the implosion non-uniformity. The dynamic mitigation mechanism would be realized by the HIBs wobbling motion.
of the implosion acceleration non-uniformity is controlled externally by the driver HIBs wobbling behavior. From Equation (2) and the relating theory of the dynamic control mechanism, the HIBs illumination non-uniformity is smoothed, and the R-T instability would be also mitigated. The fusion output gain G becomes 81 for this case with the wobbling frequency of 424 MHz, though without the wobbling behavior G was 62.4. The detail parameter studies are also presented in Ref [31,127]., including a relation between G and Ω of the wobbling frequency.  Relating to how to produce the wobbling HIBs [24,25], precision studies are required to investigate the effect of the wobbling error on the implosion uniformity in the future.
Here other topics relating to fuel target physics are briefly summarized. DT fuel target alignment error in a fusion reactor is also one of the issues in ICF. As shown in Figure 4(b), ICF target is injected into reactor chamber, and the research results show that the target alignment error should be less than 100 ~ 120μm in HIF [11,31,46]. The target injection control technique into a reactor has been also studied well to reduce the target alignment error [35][36][37], as shown in Section 2. In Refs. [35][36][37] the control method was proposed theoretically to reduce the target alignment error in ICF. In the near future we need experimental verifications on the target alignment. In this review, we assume each fuel target contains the same amount of D and T [11,128]. In addition to DT reactions, DD reactions also contribute to the T creation in DT fuel target [1][2][3]52]. The results in Ref [128] show that 20 ~ 30% reduction of tritium amount in DT pellet still produces a sufficient fusion energy output in DT ICF reactor. In other words, T amount can be reduced significantly without a notable reduction in DT fusion energy output. In this review, we focus on direct-drive fuel target scheme in HIF. HIBs deposit their energy in a target energy absorber volumetrically. For the 8 GeV Pb+ ion, the stopping range is ~500-700μm in metal target materials. Figure 15. Implosion acceleration distributions of DT fuel along the azimuthal direction with the HIBs wobbling behavior. The implosion acceleration non-uniformity is clearly controlled by the wobbling motion. During the half rotation of the wobbling HIBs, the phase of the implosion acceleration non-uniformity is inversed. The results demonstrate that the dynamic mitigation mechanism works well to smooth the HIB illumination non-uniformity and to mitigate the implosion non-uniformity. SOURCE: Ref [31]., doi.org/10.1038/s41598-019-43,221-7.
If heavy ion particle energy can be reduced significantly or correlation effect on cluster ion stopping power is enhanced [87][88][89][90]129], HIB ion stopping range would be shortened. In this case, it may be expected to increase the target material temperature and the radiation temperature, and the radiation energy would dominate the energy transfer in the target energy absorbing layer and in the implosion phase. Indirect drive fuel targets have been also studied in HIF [130,131]. The radiation effect on the implosion dynamics was also studied even in a direct-drive HIF target [132], in which a low-density foam was inserted in the HIBs energy absorber to enhance the radiation energy conversion from HIBs energy inside the target material. The inserted foam creates a wider density valley, in which radiation is confined and radiation transport in the target lateral direction contributes to smooth HIBs illumination non-uniformity additionally [132].

Summary and perspectives
The present status of HIF research activities is introduced and discussed concisely in review. HIB accelerators may open a road to our future fusion energy release. The most remarkable merit in HIF is the high energy efficiency of HIB accelerator, which relaxes the ICF requirements to release fusion energy. The requirement for fusion energy gain is relaxed to be ~50-70. The high controllability of HIB accelerators is another merit. The HIF fuel target structure, which has a relatively thick energy absorber and creates a long density gradient, contribute to the relatively uniform target implosion. In Review all the merits and demerits in HIF are presented and discussed, as well as the recent scientific results and the recent international projects relating to HIB.
DT fusion targets at fusion reactor center would be imploded, ignited and burnt with the repetition rate of 10 ~ 15 Hz, and would supply ~GW electricity to society in the future. Many researchers and scientists have contributed to HIF studies toward our future fusion energy source. Currently, it seems that fatal flaws have not been found in the HIF fusion scenario, though experimental research is required further.
At present new accelerators are being built in the world: HIAF near Hong Kong in China and FAIR at Darmstadt, Germany, as well as the KEK induction accelerator. The activities planned in accelerator projects are oriented to basic physics and HEDP, as well as fundamental studies toward HIF. We need to continue the HIF related studies toward our future stable energy source.