# Equivalent Analysis of Thermo-Dynamic Blow-Off Impulse under X-ray Irradiation
## 5 Equivalent Analysis of Mono-Energy and Blackbody Spectrum X-ray
The equal impulse relation should be adopted according to the actual data, as shown in Figure 5.  
Figure 4: Comparison between the Mono-energetic 3 keV and 1 keV Black-Body Spectra. The blow-off impulse result between 5 keV mono-energetic and 1.4 keV blackbody X-rays is given in Figure 6, in which the blow-off impulse monotonously increased with energy flux in a certain range of flux. This trend is different from that in Figure 2, due to the difference in energy deposition profile caused by different X-ray energy spectrum distributions. The impulse of 5 keV mono-energetic X-ray was greater than that of 1.4 keV blackbody X-ray, and such difference increased with the increase in energy injection, the maximum difference was nearly 150 Pa\(\cdot\)s.  
The results of blow-off impulse of two types of X-ray were analyzed, and the equivalent blow-off relation was given. The equivalent formula is written as:  
\[I_{kT=1.4}=\frac{2}{3}I_{hv=5}\,\mathrm{Pa}\cdot\mathrm{s} \tag{45}\]  
Figure 6 presents the blow-off impulse results of the numerical and 1.4 keV blackbody spectrum from the equal energy flux relation, complying with the range of energy flux. The  
Figure 5: Equal impulse relation of 1 keV blackbody spectrum under anti-radiation reinforcement parameter. Figure 6: Comparison between the 5 keV mono-energetic and 1.4 keV blackbody spectra. energy flux equivalent relationship of equal impulse relation for 5 keV mono-energetic and 1.4 keV blackbody spectrum was given. The formula is written as:  
\[\Phi_{kT=1.4}=1.6\Phi_{hv=5}-40\,\mathrm{J/cm^{2}} \tag{46}\]  
Figure 7 also gives the numerical results of 5 keV mono-energetic and 1.4 keV blackbody spectrum, as well as the fitting curve of equal blow-off impulse and equal energy flux relation. They were consistent within 500 J/cm\({}^{2}\), revealing that the two relationships were self-consistent and complete. Figure 7 also shows the relationship between the numerical calculation results and the X-ray blow-off impulse of 1 keV blackbody spectrum calculated by two formulas within the radiation reinforcement energy injection range, and the results are satisfactory. Figures 8 and 9, respectively, show equal-energy fluence curves on the impulse plane and equal-impulse curves on the energy injection plane obtained by the equal-energy fluence relationship and equal-impulse relationship. Even though the curves are basically a straight line, the equal energy flux curve and equal impulse curve should be a non-linear curve, and mathematical expressions were presented. In fact, the equal-energy fluence curve and equal-impulse curve are a nonlinear curve, with their mathematical expressions determined by numerical calculation or experimental results or both. The point above the curve shows that the energy flux of the blackbody spectrum is greater than the mono-energetic X-ray under the corresponding impulse, whereas the point on the curve equals to the blackbody spectrum and the mono-energetic. The blow-off impulse of the blackbody spectrum is greater than that of the mono-energetic in the upper region of the equal impulse curve, exactly equal to them on the curve, but it is opposite underneath the curve. It lays the basis for blow-off impulse equivalence of the equal blow-off impulse relation and the equal energy flux relation, and they are mutually complemented and confirmed. It reflects the law of equivalence from different angles and aspects in the study on blow-off impulse equivalence, and the characteristics of self-consistency and completeness are presented. The equal blow-off impulse relation and the equal energy flux relation can be exploited as the method of equivalent blow-off impulse, which will be critical to the equivalent simulation of X-ray irradiation blow-off impulse. Figure 7: Comparison between 5 keV mono-energetic and 1.4 keV blackbody X-rays under anti-radiation reinforcement parameter.