# Equivalent Analysis of Thermo-Dynamic Blow-Off Impulse under X-ray Irradiation
## 5 Equivalent Analysis of Mono-Energy and Blackbody Spectrum X-ray  
The mono-energetic X-ray refers to an X-ray spectrum containing only one energy component. Given the characteristics of blackbody radiation, the simulation of the X-ray radiating effect of blackbody spectrum with several characteristic energies is of practical significance. 3 keV and 5 keV can more easily generate mono-energetic X-rays in experiments, while producing K-layer radiation of argon and titanium. Furthermore, the peak energy of 3 keV approaches to 2.8 keV blackbody spectrum, while 5 keV is close to the mean value energy of 1.4 keV blackbody spectrum (4.9 keV). Thus, it is more practical to consider that 3 keV and 5 keV mono-energetic X-rays are equated with the X-ray impulse of 1 keV and 1.4 keV blackbody spectra. Accordingly, mono-energetic X-rays of 3 keV and 5 keV, and the blackbody X-rays of 1 keV and 1.4 keV were introduced in this study. There were two calculated mono-energetic and blackbody X-rays, respectively. The range of injection was 50-500 J/cm\({}^{2}\), the X-ray was a rectangular pulse of 0.1 s in width. Figure 3 presents the impulse results of 3 keV mono-energetic and 1 keV blackbody X-ray in different energy fluxes. As suggested from the results, the blow-off impulse rises monotonously with the energy flux in a certain range. Overall, the blow-off impulse of 3 keV mono-energetic is larger than 1 keV black body X-ray, whereas that of 1 keV black  
body is higher than that of 3 keV mono-energetic in the case of high injection. Such result suggests a feasible way to simulate 1 keV blackbody X-ray irradiated blow-off impulse by 3 keV mono-energetic X-ray, and the numerical results of two types of X-ray irradiated blow-off impulse were analyzed, and some numerical correlations were identified. On that basis, a simple numerical relationship of blow-off is given. The equivalent formula is written as:  
\[I_{kT=1}=\frac{6}{5}I_{hv=3}-60\ \mathrm{Pa\cdot s} \tag{42}\]  
where \(I\) denotes the blow-off impulse, \(kT\) is the black body temperature and \(hv\) is mono-energetic. This formula expresses the numerical relationship of blow-off impulse in the identical energy flux, which is termed as the equal energy flux relation. Figure 4 also draws the comparison between the fitting and numerical results. It is inconvenient to apply the spectrum directly to equivalence and formulate the equivalent principle, though the equal energy flux relation is significantly simple. Thus, the relationships of two blow-off impulse dates were delved, and the relationship of energy flux in the identical blow-off impulse was given, which is termed as the equal blow-off impulse relation. The equivalent formula is written as:  
\[\Phi_{kT=1}=0.85\Phi_{hv=3}+70\ \mathrm{J/cm^{2}} \tag{43}\]  
where \(\Phi\) denotes the energy flux. This formula gives the energy flux of 1 keV blackbody spectrum X-ray in the identical blow-off impulse, and the equivalent relationship can be determined. Figure 3 also illustrates the equal blow-off impulse relation, and the two curves show consistency above 200 \(\mathrm{J/cm^{2}}\), and a big difference was identified in low energy flux. From equal energy flux relation and equal blow-off impulse relation, it is suggested that the relations were self-consistent and complete above 200 \(\mathrm{J/cm^{2}}\). Overall, the range of anti-radiation reinforcement parameter was 120-300 \(\mathrm{J/cm^{2}}\), so this study placed more emphasis on the mentioned results in this range. To solve the error problem of low energy flux, a new equal-impulse relation is fitted for the anti-radiation reinforcement energy injection quantity index. The equivalent formula is presented as:  
\[\Phi_{kT=1}=\Phi_{hv=3}+35\ \mathrm{J/cm^{2}} \tag{44}\]  
The X-ray blow-off impulse of 1 keV blackbody spectrum given by the equation is very close to the numerical calculation result, whereas the results were very different in the high energy flux above 300 \(\mathrm{J/cm^{2}}\).