In this review paper on heavy ion inertial fusion (HIF), the state-of-the-art scientific results are presented and discussed on the HIF physics, including physics of the heavy ion beam (HIB) transport in a fusion reactor, the HIBs-ion illumination on a direct-drive fuel target, the fuel target physics, the uniformity of the HIF target implosion, the smoothing mechanisms of the target implosion non- uniformity and the robust target implosion. The HIB has remarkable preferable features to release the fusion energy in inertial fusion: in particle accelerators HIBs are generated with a high driver efficiency of ~ 30-40%, and the HIB ions deposit their energy inside of materials. Therefore, a requirement for the fusion target energy gain is relatively low, that would be ~50-70 to operate a HIF fusion reactor with the standard energy output of 1GW of electricity. The HIF reactor operation frequency would be ~10~15 Hz or so. Several- MJ HIBs illuminate a fusion fuel target, and the fuel target is imploded to about a thousand times of the solid density. Then the DT fuel is ignited and burned. The HIB ion deposition range is defined by the HIB ions stopping length, which would be ~1 mm or so depending on the material. Therefore, a relatively large density-scale length appears in the fuel target material. One of the critical issues in inertial fusion would be a spherically uniform target compression, which would be degraded by a non-uniform implosion. The implosion non-uniformity would be introduced by the Rayleigh-Taylor (R-T) instability, and the large density-gradient-scale length helps to reduce the R-T growth rate.
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