Plasma Excited by Radionuclides

Electron density is determined using the transport equation. The electron diffusion in plasma is similar to the free diffusion of gas molecules [13]. $\frac{\text{d}{n}_{\text{e}}}{\text{d}t}-D_{\text{a}}{\nabla }^2{n}_{\text{e}}=G(\overrightarrow{{\displaystyle r}}\mathrm{,}t)-L\mathrm{,}$ (4) where n_e is the electron density, $G(\overrightarrow{{\displaystyle r}}\mathrm{,}t)$ is the source function, L describes electrons recombining with positive ions (L_r) and mainly attached by neutral particles (La), and Da is the bipolar diffusion coefficient, which is approximately twice the diffusion coefficient of positive ions if the temperatures of the different particles in the plasma are approximately the same. The diffusion coefficient of positive ions is proportional to T^1.5/p [14], and its experimental values for different gases at 300 K and 1 mmHg are listed in table 0. Subsequently, the bipolar diffusion coefficient can be easily determined using the pressure (p) and temperature (T) in the MF microsphere shells. In the shells, the temperature (T) is approximately the same as the atmospheric temperature and pressure (p=N_gk_BT). For a given atmospheric temperature, the pressure (T) in the shells is linearly and uniquely mapped onto the PRSM.

The source function presented in Eq. 4 generates free electrons mainly when the gas molecules are ionized by α particles. The average number of electron–ion pairs liberated when an α particle travels across a distance l within gas is [17] $n_{\text{e}-{\text{i}}^{+}}=\frac{\langle \text{d}E/\text{d}x\rangle \cdot l}{W}\mathrm{,}$ (5) where W is the average electron–ion creation potential energy of gas molecules, as listed in Table 1, and $\langle \text{d}E/\text{d}x\rangle$ is the average energy loss of an α particle, which can be estimated using the Bethe–Bloch equation [17, 18]. We calculated the deposited energy $(\langle \text{d}E/\text{d}x\rangle \cdot l)$ directly using the Geant4 program. The source function $G(\overrightarrow{{\displaystyle r}}\mathrm{,}t)$ can be obtained from $n_{e-i^{+}}$ if the α particle energy and per area radioactivity (PAR) of the radioactive material are given [9].

Ion–electron recombination is proportional to the density of electrons and positive ions. In weakly ionized plasma, the density of electrons and positive ions are approximately equal. $L_r=\alpha n_e{n}_{i^{+}}\approx \alpha n_e^2\mathrm{,}$ (6) where α is the recombination coefficient given by α = 2.7 × 10^-13/(k_BT_P)^3/4 cm³s^-1[9, 19], and the unit of k_BT_P is eV.

Because negative ions produced at the electron attachment are not stable[9, 10, 13], the process of electrons attached by neutral molecules (L_a) can be ignored here.

Substituting $G(\overrightarrow{{\displaystyle r}}\mathrm{,}t)$ and Lr into equation 4, $\frac{\text{d}{n}_{\text{e}}}{\text{d}t}-D_{\text{a}}{\nabla }^2{n}_{\text{e}}=G(\overrightarrow{{\displaystyle r}}\mathrm{,}t)-\alpha n_{\text{e}}^2\mathrm{.}$ (7) Solving Eq. 7, we can obtain the steady-state distribution of the electron density in the MF microsphere shell. The parameters affecting electron density include the atmospheric temperature, encapsulated gas type, pressure in the shell, α particle energy, PAR of the radioactive material, and position of the spherical shell relative to the radioactive material.

Reflection Coefficient of Electrically Large Target

When coating the PF on an electrically large target, its interaction with an EMW can be described as a multi-layer structure with the EMW, as shown in Fig. 2. The incident layer (on the left) is air, and the target (on the right) is always replaced with a perfect electrical conductor (PEC). From the incident layer to the PEC layer, all layers are numbered from 0 to m+1, where 0 is the incident layer and m+1 is the PEC. When the EMW propagates through the multi-layer film, the fields in every layer satisfy the following relations via the transfer matrix method (TMM) [20]: $\left[\begin{array}{c}{E}_0\\ H_0\end{array}\right]={\displaystyle \prod _{k=0}^m}\left[\begin{array}{cc}\cos {\delta }_k& i{z}_k\sin {\delta }_k\\ \frac{i\sin {\delta }_k}{z_k}& \cos {\delta }_k\end{array}\right]\cdot \left[\begin{array}{c}{E}_{m+1}\\ H_{m+1}\end{array}\right]\mathrm{,}$ (8) where δk is the phase factor of the k-th layer, given by equation 9. ${\delta }_k=\frac{2\pi }{\lambda }{n}_k{d}_k\cos {\theta }_k=\frac{2\pi d_k}{\lambda }\sqrt{{ϵ}_k{\mu }_k-n_0^2{{\displaystyle \sin }}^2{\theta }_0}\mathrm{,}$ (9) where dk is the thickness of the k-th layer, θ₀ is the incident angle of the incident wave on the incident layer, λ is the wavelength in vacuum, nk is the refractive index of the k-th layer, and ϵk and μk are the equivalent permittivity and permeability of the k-th layer, respectively. zk is the impedance; for a TM wave, $z_k^{\text{p}}=\sqrt{\frac{{\mu }_k}{{ϵ}_k}}\cdot \cos {\theta }_k=\frac{\sqrt{n_k^2-n_0^2{{\displaystyle \sin }}^2{\theta }_0}}{{ϵ}_k}$, and for a TE wave, $z_k^{\text{p}}=\sqrt{\frac{{\mu }_k}{{ϵ}_k}}\cdot \frac{1}{\cos {\theta }_k}=\frac{{\mu }_k}{\sqrt{n_k^2-n_0^2{{\displaystyle \sin }}^2{\theta }_0}}$. The relationship between θk and θ₀ is derived from Snell's law.

Fig. 2Full-screen View

Schematic of the propagation of an electromagnetic wave in multi-layer film

According to the effective medium theory [22, 23], the equivalent permittivity (ϵk) and permeability (μk) depend on the effective medium mass of each layer. $\{\begin{array}{c}{ϵ}_k={ϵ}_{\text{r}}V+{ϵ}_{\text{MF}}(1-V)\\ {\mu }_k={\mu }_{\text{r}}V+{\mu }_{\text{MF}}(1-V)\end{array}$ (10) where μ_MF and ${ϵ}_{\text{MF}}$ are the permeability and permittivity of the MF material, μ_r and ${ϵ}_{\text{r}}$ are the permeability and permittivity of plasma, and V is the volume fraction of plasma in the k-th layer.

By defining the interface impedance as Zk = Ek/Hk, it can be derived from Eq. 8 that $Z_k=\frac{Z_{k+1}+z_k\cdot \tanh (i{\delta }_k)}{Z_{k+1}\cdot \tanh (i{\delta }_k)+z_k}\cdot z_k\mathrm{.}$ (11) Because the (m+1)-th layer is a PEC, Z_m+1=0, and therefore the first interface impedance can be obtained using the iteration method. As we know, the field on the incident layer is the superposition of the incident and reflected waves ($E_0=E^{\text{i}}+E^{\text{r}}$, $H_0=H^{\text{i}}+H^{\text{r}}=(E^{\text{i}}-E^{\text{r}})/z_0$), and by applying the continuity boundary condition at the first interface ($E_0=E_1$, $H_0=E_1/Z_1$), one gets $\{\begin{array}{c}{E}_1=E^{\text{i}}+E^{\text{r}}\\ \frac{E_1}{Z_1}=\frac{E^{\text{i}}-E^{\text{r}}}{z_0}\end{array}\mathrm{.}$ (12) Then, the EMW reflection coefficient of the electrically large target coated with PF can be directly obtained. $R_{\text{dB}}=20\cdot \lg \left(\frac{E^{\text{r}}}{E^{\text{i}}}\right)=20\cdot \lg \left(\frac{Z_1-z_0}{Z_1+z_0}\right)\mathrm{.}$ (13) The parameters that affect the attenuation of the reflected EMW include the atmospheric temperature, encapsulated gas type, pressure in the shells, α particle energy, PAR of the radioactive material, number of SPFs (thickness of PF), and the frequency, incident angle, and polarization of the incident EMW.

Preventing Radiation Leakage

To prevent radiation leakage, we analyzed the travel distance (range) of α particles in the PF. Because the thickness of the PF must be increased (decreased) with an increase (decrease) in the number of MFLs, we use “Range Layer” to describe the maximum number of MFLs penetrated by α particles, which mainly depends on the α particle energy and PF properties. For example, the properties of one of the constructed MF microsphere shells were as follows: the outer diameter was approximately 50 μm, the wall thickness was approximately 5 μm, the compressive strength was 10 atm, the density was 1.1 g/cm³, and the refractive index was 1.6 [21]. Two pure α decay radionuclides (²¹⁰Po and ²⁴²Cm) were considered because their specific activities were 3891.9 Ci/g and 3324.3 Ci/g, respectively. Their relative activities were significantly larger than those of other pure α decay radionuclides. The energy of the α particle emitted by ²¹⁰Po was 5.301 MeV(100%), whereas that by ²⁴²Cm was 6.069 MeV(26%) and 6.112 MeV(74%). Their Bragg peaks in the MF are given in Fig. 3. The range of the 5.301 MeV α particle was approximately 40 μm, and that of the 6.069 or 6.112 MeV α particle was approximately 50 μm. To obtain the same range in gaseous He, Ne, Ar, or CO₂, densities of 1.0 g/cm³, 1.55 g/cm³, 1.90 g/cm³, and 1.30 g/cm³, respectively, should be set. When the atmospheric temperature was 300 K, according to the state equation, their corresponding pressures were 6150 atm, 1891 atm, 1171 atm, and 727 atm, which are significantly larger than the pressure used in practice. Therefore, the leakage of α particles was mostly prevented by the MF.

Fig. 3Full-screen View

Bragg peaks of α particles within the MF

Although the encapsulated gases barely prevented the leakage of α particles, the different energy losses of the α particles affected the range layer. To obtain specified range layers with different radionuclides and gases at an atmospheric temperature of 300 K, the required pressures in the microsphere shell are listed in Table 2. Among them, the minimum value was also significantly larger than the compressive strength of the MF microsphere shell (10 atm). Therefore, in the PF, the range layer of the α particles emitted by ²¹⁰Po or ²⁴²Cm was only 4/5 MFLs, respectively. Although the radiation of the radioactive material was isotropic, leakage prevention of the emitted α particles could be realized by sandwiching them in the SPF, as shown in Fig. 1. The thickness of the SPF constructed with ²¹⁰Po or ²⁴²Cm was only 8/10 MFLs (approximately 0.4/0.5 mm).

Electron Density in the Microsphere shell

From the above, it was found that the electron density in the MF microsphere shell was affected by the atmospheric temperature, encapsulated gas type, pressure in the shell, α particle energy, PAR of the radioactive material, and position of the microsphere shell relative to the radioactive material. It appears that no matter how much these parameters changed, the contour lines of the electron density were approximately parallel to the temperature axis and perpendicular to the other, as shown in Fig. 4. Therefore, the electron density was almost independent of atmospheric temperature.

Fig. 4Full-screen View

(Color online) Electron density in the microsphere shell near the radioactive material. The radionuclide is ²¹⁰Po, the encapsulated gas is CO₂, and (a) the PAR is 1 mCi/cm²; (b) the PRSM is 1 atm.

The change in the electron density contours in Fig. 4(b) indicates that the electron density increased with increasing PAR. By taking the logarithms of the electron density and PAR, a linear correlation was obtained, that is, (n_e2/(n_e1) = k·lg(A₂/A₁), as shown in Fig. 5. After fitting, the value of k was approximately 0.5, and the deviation was less than 0.001, whereas other parameters changed.

Fig. 5Full-screen View

Electron density vs. PAR while the shells are near the radioactive material, the PRSM is 1 atm, the radionuclide is ²¹⁰Po, and (a) the encapsulated gas is CO₂; (b) the PRSM is 1 atm.

The relative position of the MF microsphere shell to the radioactive material leads to a change in the electron density in the shell with the pressure in the shell. The electron density in the shells located outside the range layer increased with increasing pressure in the shells, but decreased with increasing speed, as shown in Fig. 4(a). At the same shell pressure, the electron densities for different gases, listed from the largest to smallest values, were CO₂, Ar, Ne, and He, as shown in Fig. 6(a). The electron density in the shells located at the range layer first increased and then decreased with increasing PRSM, as shown in Fig. 6(b), which was easily observed when the gas was CO₂ or Ar yet was almost unobserved when the gas was He or Ne. From Eq. (5), the electron density was proportional to the deposition energy of α particles in the gas. With increasing PRSM, the deposition energy of α particles in Ar or CO₂ gas increased. However, the deposition energy of α particles in the shells located at the range layer decreased with increasing PRSM when the PRSM was greater than a certain value because most of the energy of the α particles was lost in the MFLs near the radionuclide material. Therefore, the trends of the Ar and CO₂ curves first rose and then fell. Meanwhile, because the energy loss of the α particles in CO₂ or Ar was larger than that in He or Ne, this phenomenon was almost unobserved when the gas was one of the latter.

Fig. 6Full-screen View

Electron density vs. PRSM while the radionuclide is ²⁴²Cm, the PAR is 50 mCi/cm², and the shells are (a) near the radioactive material; (b) located at the range layer.

Radar Echo Attenuation

According to the above analysis, the reflected EMW attenuation was dependent on the atmospheric temperature, encapsulated gas type, pressure in the shell, α particle energy, PAR of the radioactive material, number of SPFs (thickness of the PF), and the frequency, incident angle, and polarization of the incident EMW.

Open plasma stealth technology is highly atmosphere dependent; however, only atmospheric temperature can affect an enclosed device, as previously analyzed. As shown in Fig. 7, the reflected curves under different atmospheric temperatures almost completely overlapped. This shows that the enclosed plasma stealth technology excited by radionuclides was barely affected by the atmospheric temperature; therefore, it effectively avoided the dependence on atmosphere.

Fig. 7Full-screen View

Reflected power vs. frequency at different atmospheric temperatures while the radionuclide is ²⁴²Cm, the PAR is 50 mCi/cm², the encapsulated gas is CO₂, the PRSM is 3 atm, and the SPF number is 50.

For open plasma stealth technology, only air can be considered for the gas; however, we can choose any type of gas to be encapsulated in the enclosed device. Furthermore, we can obtain a similar attenuation with different gases by adjusting only the radioactivity. This means that for stealth performance, the changing of the gas type and the changing of the PAR of the radioactive material are equivalent, and clearly the changing of the PAR is more flexible. If we pick the latter as the control parameter, the choice of gas will only affect the cost of the radioactive material. For example, when the gas is CO₂, 16.7%, 64.3%, and 90.7% of ²⁴²Cm will be saved compared with Ar, Ne, and He, respectively.

When the PAR of the radioactive material was small (such as, 5 mCi/cm²), EMW attenuation greater than 10 dB was located in the low-frequency domain, but its bandwidth was narrow (widest is 2.5–4.6 GHz). However, as the PAR increased, the widest bandwidth of EMW attenuation greater than 10 dB will gradually widened, while its upper and lower frequency points moved toward the higher side along the frequency axis. As shown in Fig. 8, when the PAR of the radioactive material was 0.5 Ci/cm², the widest bandwidth of attenuation over 10 dB was 4.9–283 GHz, which was significantly wider than when the PAR was 5 mCi/cm²; however, the attenuation of the EMWs whose frequencies were below 4.9 GHz was less than 10 dB.

Fig. 8Full-screen View

(Color online) Reflected power while the SPF number is 50, the radionuclide is ²⁴²Cm, the encapsulated gas is CO₂, and the PAR is (a) 5 mCi/cm²; (b) 0.5 Ci/cm².

When the PF was thinner (such as, 25 SPFs), the EMW attenuation greater than 10 dB was located in the low-frequency domain, but its bandwidth was narrow (widest was 5.5–10 GHz). Comparing Figs. 9a, b, with increasing PF thickness, the largest bandwidth of EMW attenuation greater than 10 dB gradually increased, while the upper frequency point moved toward the higher side and the lower point moved toward the lower side along the frequency axis. This was different with increasing PAR.

Fig. 9Full-screen View

(Color online) Reflected power while the radionuclide is ²⁴²Cm, the encapsulated gas is CO₂, the PAR is 50 mCi/cm², and the SPF number is (a) 25; (b) 50.

Therefore, we could increase the EMW attenuation in the high-frequency domain via enriching the PAR of the radioactive material or thickening the PF; however, in the low-frequency domain only the latter method could be achieved. Regardless of whether we increase the PAR or thicken the PF, the cost of the radioactive material and adjusted pressure in the shells should be considered. As given in Fig. 10, two different devices with a broad attenuation bandwidth of 2–50 GHz, which covers most common frequencies used in radar, were obtained by modifying the PAR of the radioactive material and the thickness of the PF. The encapsulated gas was CO₂. The radioactive materials were 50 mCi/cm² of ²⁴²Cm and ²¹⁰Po, the thicknesses were 2.5 cm (blue dotted line) and 2.4 cm (black solid line), respectively, and the corresponding areal densities were 0.991 g/cm² and 0.951 g/cm², respectively. The light weight and self-provided plasma of this method perfectly suit small-satellite radar stealth technology, while near-Earth satellites utilized in wars, such as Musk's “Star Chain,” are currently being used in the Russia–Ukraine conflict.

Fig. 10Full-screen View

Optimized reflected power with different radionuclides while the encapsulated gas is CO₂.

As we know, the change in the reflected EMW attenuation with incident angle differs as the polarization changes. However, the changing of polarization or incident angle cannot significantly narrow the attenuation bandwidth. For example, as shown in Fig. 11, as the incident angle increased in the range 0–45° and the other condition was the optimized ²⁴²Cm (blue dotted line in Fig. 10), the maximum attenuation peak of the TE wave increased, whereas that of the TM wave decreased; however, the bandwidth of EMW attenuation greater than 10 dB was almost unchanged. This shows that even for networked radar, this method will provide the target with excellent stealth performance.

Fig. 11Full-screen View

(Color online) Reflected power of TE (a) and TM (b) waves with different incident angles and the optimized condition of ²⁴²Cm in Fig. 10.