# Equivalent Analysis of Thermo-Dynamic Blow-Off Impulse under X-ray Irradiation
## 4 Blackbody Spectrum and Mono-Energy X-ray  
Blackbody spectrum X-ray refers to the energy radiated outward in the form of electromagnetic waves by high temperature objects. The X-ray energy and the proportion of photon component vary with the object temperature. The radiation intensity is computed by the Plank formula.  
\[B_{\
u}=\frac{2\pi h\
u^{3}}{c^{2}}\frac{1}{e^{\frac{\
u}{2T}}-1} \tag{35}\]  
where \(h\) denotes the Plank constant (\(lt\) = 6.62 \(\times\) 10\({}^{-34}\) J-s), \(\
u\) is the photon frequency, \(c\) is the light speed, \(k\) is the Boltzmann constant (\(k\) = 1.38 \(\times\) 10\({}^{-13}\) J/K), T represents the blackbody temperature, \(h\
u\) is the photon energy (keV) and \(kT\) is the blackbody temperature (keV). Figure 3 shows black-body spectrum radiation intensity at different temperatures.  
The normalized spectrum is usually employed practically, and the formula is written as:  
\[b_{\
u}=\frac{B_{\
u}}{\sigma T^{4}}=\frac{15h\
u^{3}}{\tau^{4}k^{4}T^{4}} \frac{1}{e^{\frac{\
u}{2T}}-1} \tag{36}\]  
Figure 3: Black-body spectrum radiation intensity at different temperatures.  
Figure 2: Effect of grid size on blow-off impulse. (**a**) Mono-energetic. (**b**) Blackbody X-rays.  
The proportion of X-ray photons with the energy in the interval between \(h\
u\) and \(h\
u+\Delta h\) is expressed as:  
\[f_{\
u}=\frac{15{(h\
u)}^{3}}{\tau^{4}{(kT)}^{4}}\frac{\Delta h\
u}{e^{\frac{h \
u}{kT}}-1} \tag{37}\]  
Plank distribution reaches its peak (photon energy \(h\
u\) = 2.82 kT). The proportion of X-ray photos energy (\(h\
u\leq\) 1 kT) is less than 3.5% of the total radiation energy, and that (\(h\
u\leq\) 2.82 kT) takes up 35% of the total radiation energy, and that (\(h\
u\geq\) 10 kT) is less than 3.5% of the total radiation energy. Thus, the mean value energy of the blackbody spectrum is defined as \(h\
u\) = 3.5 kT.  
X-rays collide with nucleus and extranuclear electrons in a material; as a result, the X-ray energy will be deposited inside the material. Macroscopically, the energy deposition of X-ray abides by the Lambert's law, which is defined as:  
\[\Phi=\Phi_{0}e^{-\mu\rho\tau} \tag{38}\]  
where \(\Phi\) denotes the energy flux after X-ray passing through materials (thickness = \(\tau\)), \(\Phi_{0}\) is the incident energy fluxes, \(\rho\) is the density, \(\mu\) represents the absorption coefficients, which is related to materials and photon energy of X-ray.  
It is therefore concluded that the deposited energy of X-ray in the unit mass material at \(\delta\) is defined as:  
\[E=\frac{\Phi_{0}}{\rho\delta}\Big{(}1-e^{-\mu\rho\delta}\Big{)}e^{-\mu\rho\tau} \tag{39}\]  
With a \(\delta\), its approximate formula is expressed as:  
\[E=\Phi_{0}\mu e^{-\mu\rho\tau} \tag{40}\]  
The result of the deposition of mono-energetic photon energy is shown as above. Energy deposition requires the energy spectrum to be grouped according to the photon energy for X-ray with a certain energy spectrum (e.g., blackbody spectrum). Thus, the energy deposited in the unit mass material at \(\tau\) can be expressed as:  
\[E=\sum_{i=1}^{N}\Phi_{0i}\mu_{i}e^{-\mu_{i}\rho\tau} \tag{41}\]  
where \(i\) denotes the photon cluster number, and \(N\) is the number of clusters.