Constraint on foil temperature

When a solid-state stripper is utilized, its lifetime is one of the main concerns. The lifetime of a foil stripper is mainly determined by accumulated radiation damage and evaporation of the foil. From the first adoption of solid-state strippers, researchers have been making efforts to estimate the lifetime of strippers quantitatively. Among them, Levedev drew a physical picture of carbon foils’ failure under irradiation and derived the formulae to estimate the lifetime of carbon foils due to radiation damage [42]. Based on Levedev’s theory, “stripper lifetime utility” was integrated into the program LISE++ since Version 8.3.6 [43], which utilizes Eq. (15) in Ref. [42] to calculate the lifetime of carbon foils due to radiation damages. In the following updated versions, k₁’s dependence on the atomic number of the projectile has been introduced to reproduce experimental data for projectiles in a wide region, and then Equation 15 in Ref. [42] has been transformed as follows $\begin{array}{l}t=k_1\left(Z_1\right)K_d^{-\frac{5}{4}}\exp \left(-\frac{k_2}{T}\right), \\ k_1\left(Z_1\right)=k_{10}\text{exp}\left(-k_{11}{Z}_1\right)\end{array}$ (4) where t is the foil lifetime dominated by the radiation damage; K_d is the rate of atom displacement, and its expression can be found in Ref. [42], Z₁ is the atomic number of the projectile; k₂ is a characteristic constant related to the foil material and its default value for carbon is 870, and the default values of k₁₀ and k₁₁ are 50 and -0.07, respectively. With the modification of k₁, the calculated results are in better agreement with the experimental results in a wide region (from Ne to U).

Compared with some semi-empirical treatments during the derivation of Eq. (4), the calculation of the foil evaporation could be more accurate. Although the sublimation temperature of the carbon is approximately 3900 K, evaporation occurs for a carbon foil bombarded by intensive ion beams even if the temperature is much lower than 3900 K. The evaporation rate increases with the temperature as follows [44, 45] ${\log }_{10}\left(\text{d}m/\text{d}t\right)={\log }_{10}{P}_{\text{atm}}-0.5{\log }_{10}T-2.187,$ (5) where dm/dt is the evaporation rate in (g/cm²)/s; T is the temperature of the carbon foil in K; P_atm is the vapor pressure of carbon in the atmosphere and is also a function of T. ${\log }_{10}{P}_{\text{atm}}=-37.3\left(1000/T\right)+8.16$ (6)

According to the research of Lebedev et al. [42], the failure of a carbon foil is dominated by irradiation damage at the temperature below 2500 K and by evaporation and sublimation at the temperature beyond 2500 K, in which case the lifetime of the carbon foil is generally only a few hours or even shorter. Here, we set a limit of 2200 K on the foil temperature, corresponding to a maximum evaporation rate of 10^-3 (μg/cm²)/s.

Non-stationary thermal analysis of foil bombarded by pulsed ion beams

To evaluate the lifetime of the stripper foil in terms of heating effect, the temperature evolution of the foil was calculated with the code Ansys with a non-stationary thermal model. The following equations can describe the heat conduction in the foil [46]. $\begin{array}{l}\frac{1}{\alpha c_{\text{p}}}\frac{\partial T}{\partial t}={\nabla }^{2}T+\frac{1}{k{t}_{\text{foil}}}\left[P-2{\sigma }_0\epsilon \left(T^4-T_0^4\right)\right], \\ \alpha =k/\rho c_{\text{p}}\left({\text{m}}^2/\text{s}\right) \end{array}$ (7) where T is the temperature of the foil; T₀=295 K is the ambient temperature; t is the time; α is the thermal diffusivity of carbon; ρ=2000 kg/m³ is the density of carbon; σ₀=5.67×10^-8 W(m^-2·K^-4) is the Stefan-Boltzmann constant, ε is the emissivity of carbon and is set to 0.8 in the simulation; c_p and k are the specific heat capacity and heat conduction coefficient of carbon and are also functions of the temperature T [46]. $\begin{array}{l}{c}_{\text{p}}=12.7+2.872T-0.00145T^2+3.12\times {10}^{-7}{T}^3-2.38\times {10}^{-11}{T}^{4}J/\left(\text{kg K}\right)\\ k=241.54-0.241T+1.088\times {10}^{-4}{T}^2-2.144\times {10}^{-8}{T}^3+1.531\times {10}^{-12}{T}^{4}W/\left(\text{m k}\right)\end{array}$ (8)

P in Eq. (6) represents the power density deposited in the foil by the pulsed ion beams and can be expressed as follows. $P=\frac{j}{q}\left(\frac{dE}{dx}\right)t_{\text{foil}}{\text{ W/m}}^2$ (9) where j is the electrical current density, and q is the charge quantity of the incident ions, the product of the ion charge state and the elementary charge. Here, we assume the transverse beam distribution is 2-dimensional Gaussian. $j\left(x,y\right)=\frac{I}{2\pi {\sigma }_x{\sigma }_y}\text{exp}\left(-\frac{{\left(x-x_0\right)}^2}{2{\sigma }_x^2}\right)\text{exp}\left(-\frac{{\left(y-y_0\right)}^2}{2{\sigma }_y^2}\right){\text{A/m}}^2$ (10) where I is the total beam intensity, x₀ and y₀ are the coordinates of the beam center on the carbon foil, and σ_x and σ_y are the rms beam spot size in the horizontal and vertical directions, respectively. In the simulation, the dependence of the stopping power $\left(\frac{\text{d}E}{\text{d}x}\right)$ on the energy was ignored because the energy loss of the beam is much lower compared with the incident energy, and the variation of the foil thickness with the temperature was not considered, either.

With the non-stationary model, the evolution of the foil temperature for the beam parameters listed in Table 1 (1-emA beam intensity in 1-ms pulse duration with the repetition frequency of 1 Hz) was simulated by the code Ansys. The values of σ_x and σ_y were set to 2.45 mm, which are the baseline design values for the injection beamline at the position of the stripper. The maximum foil temperature for U and Xe beams are 4188.3 and 2460.6 K, respectively, resulting in very short lifetimes of the strippers due to evaporation, which is unacceptable.

Synthesis of foil temperature, emittance growth, and number of injection turns

One important fact has been ignored in the preceding temperature calculation. In the simulation, we took the beam pulse duration of 1 ms. This value corresponds to the injection turns of 100 for the BRing [47, 48], derived based on the emittance of 5 π mm·mrad, i.e., for the unstripped ion beams¹. Meanwhile, emittance growth is inevitable when an ion beam traverses a stripper owing to small-angle scattering of the projectile ions off the target atoms. Taking account of the emittance growth during stripping, the number of injection turns for stripped ion beams cannot achieve that for unstripped ion beams; hence, the beam pulse duration of 1 ms is unnecessarily long.

To determine the beam’s appropriate pulse duration, the stripped beam’s emittance needs to be known such that the number of injection turns could be estimated correspondingly. The emittance for the beam traversing foil can be calculated by the Monte Carlo codes, such as SRIM and GEANT4. Here, the GEANT4-10.06.p02 was utilized to calculate the emittance of the stripped ion beam. In Fig. 4, we present the simulated beam distributions in the horizontal/vertical phase space for the U beam just exiting the C-foil stripper of 1 mg/cm² calculated with GEANT4 and that for the beam just before the stripper.

Fig. 4Full-screen View

(Color online) Particle distributions in transverse phase spaces for U beam (a) before stripping (b) after stripping. Emittance is 6-rms emittance

As stated in Refs. [1] and [28], the two-plane painting injection scheme will be applied for the BRing, in which the number of injection turns can be estimated by the following empirical equation [49-51]. $N_{\text{inj}-\text{turn}}=\alpha \frac{A_x{A}_y}{{\epsilon }_x{\epsilon }_y}$ (11) where A_x=200 π mm·mrad and A_y=100 π mm·mrad are the acceptances of the BRing in the horizontal and vertical phase spaces, respectively; ε_x and ε_y are the horizontal and vertical emittances (6 rms) of the beam, respectively, and α is the dilution factor, the value of which is in the range of 0.1-0.125 [51]. Here, the value of α is set as 0.125. Taking the emittance of 12 π mm·mrad for the stripped U beam into Equation 10, one can obtain the injection turn number of 18; hence, the beam pulse duration of 0.184 ms, corresponding to 18 revolution periods for the stripped U ions in the BRing, is long enough. The shortened beam pulse duration reduces the thermal load by one beam pulse and the consequent decrease of the maximum foil temperature to 1350.4 K, much lower than the temperature constraint, 2200 K. The beam pulse duration, even the nominal one-1 ms, is too short compared with the thermal diffusion in a carbon foil; hence, the energy deposited by the beam could be taken to be instantaneous. That is why the maximum temperature correlates directly with the beam pulse duration.

The injection turn number for the stripped U beam is much lower than the designed value due to the unavoidable emittance growth during stripping. Comparing the phase space distributions before and after stripping, one can find that there is nearly no change in the beam envelope and that the emittance growth is completely dominated by the increase of the beam divergence. It is easy to understand: the stripper thickness is too thin to demonstrate the change of the particle transverse positions, while the angle straggling due to the small-angel scattering superposed on the initial beam angular spread makes the change of the beam divergence non-negligible. Thus, the angular spread and the emittance of the stripped beams can be calculated with the following analytic expressions. ${\sigma }_{x^{\prime }\left(y^{\prime }\right)-f}=\sqrt{{\sigma }_{x^{\prime }\left(y^{\prime }\right)-i}^2+{\sigma }_{\text{sct}-\text{ang}}^2}=\sqrt{\frac{{\epsilon }_{x\left(y\right)-\text{rms}-i}}{{\beta }_{x\left(y\right)-i}}+{\sigma }_{\text{sct}-\text{ang}}^2}$ (12) ${\epsilon }_{x\left(y\right)-\text{rms}-f}={\sigma }_{x\left(y\right)}{\sigma }_{x^{\prime }\left(y^{\prime }\right)-f}=\sqrt{{\epsilon }_{x\left(y\right)-\text{rms}-i}^2+{\sigma }_{x\left(y\right)}^2{\text{σ}}_{\text{sct}-\text{ang}}^2}$ (13) where ε_x(y)-rms is the horizontal/vertical rms emittance; β_x(y) is the horizontal/vertical beta function; σ_x(y) and σ_x’(y’) are the horizontal/vertical rms beam size and the horizontal/vertical rms angular spread, respectively; the i and f in the subscripts denote the indexes for the beam before and after stripping, respectively, and σ_sct-ang is the rms angle straggling of the ions introduced by the small-angle scatterings in the stripper. The holding of the second equality of Eq. (12) was based on the precondition of the beam with upright beam ellipses in the transverse phase spaces, i.e., the transverse alpha functions α_x and α_y equaling 0. This is not just for convenience. More importantly, for a beam with fixed emittance and beam size, the emittance growth due to stripping gets to the minimum when the beam has upright ellipses in the transverse phase spaces [52]. The angle straggling σ_sct-ang due to the small-angle scattering is correlated to the mass and energy of the projectile, the mass of the target atom, and the stripper thickness. It is irrelevant to the initial incident angle. It can be calculated by the codes SRIM or GEANT4, and it is 0.68 mrad for the U beam. Taking the value of σ_sct-ang into Eq. (13), one can get the rms emittance of 1.987 π mm·mrad after stripping for the U beam, consistent with the results presented in Fig. 4. The emittances after stripping were also calculated for the Xe and Kr beams with Eq. (13) and GEANT4 as for the U beam presented in Fig. 4. The results by the two approaches also agree. The consistency confirms the validity of the formulae. From Eq. (13), we can get an important conclusion that for a certain initial emittance, the larger the beam size on the stripper, the larger the emittance growth induced by stripping because the angle straggling due to the small-angle scatterings in the stripper is irrelevant to the initial beam emittance and envelope. Therefore, the number of injection turns can be enhanced by shrinking the beam size on the stripper.

Substituting Eq. (13) into Eq. (11), one can get the necessary beam pulse duration for injecting the stripped ions as follows. $\text{τ}\approx \frac{1}{36}\text{α}{\tau }_{\text{r}}\frac{A_x{A}_y}{{\epsilon }_{x-\text{rms}-f}{\epsilon }_{y-\text{rms}-f}}=\frac{1}{36}\alpha {\tau }_{\text{r}}\frac{A_x{A}_y}{\sqrt{\left({\epsilon }_{x-\text{rms}-i}^2+{\sigma }_x^2{\text{σ}}_{\text{sct}-\text{ang}}^2\right)\left({\epsilon }_{y-\text{rms}-i}^2+{\sigma }_y^2{\text{σ}}_{\text{sct}-\text{ang}}^2\right)}}$ (14) where τ is the necessary beam pulse duration, and τ_r is the revolution period for the ions considered. Taking the circumference of the BRing, 569 m, into account, the revolution period is 10.2, 7.69, and 8.08 μs for the stripped U, Xe, and Kr ions, respectively. The factor $\frac{1}{36}$ in Equation 14 was introduced because the beam emittances utilized here are the rms emittances. Although shrinking the beam size on the stripper can enhance the number of injection turns, it will also increase the stripper temperature because of the increasing particle density in one beam pulse. Therefore, caution must be paid when reducing the beam size because the stripper temperature will approach and even exceed the constraint at some beam sizes. If the beam size is further decreased beyond the critical point, with which the stripper temperature exceeds the temperature constraint, the current intensity must be reduced correspondingly. As discussed above, the energy deposited by the beam could be treated as instantaneous. Thus, the maximum foil temperature is positively correlated to the ion density at the beam center during one beam pulse, no matter how long the beam pulse duration is. That is to say, the maximum foil temperature could be kept under the constraint provided that the ion density was kept below a certain value. Therefore, the highest pulsed beam intensity tolerable for the foil is proportional to the beam size and inversely proportional to the beam pulse duration. Taking the expression of the necessary beam pulse duration, Eq. (14), we can get the relationship between the highest tolerable beam intensity and the beam size expressed in a semi-quantitative form. $I_{\text{max}}\propto {\sigma }_x{\sigma }_y\sqrt{{\epsilon }_{x-i}^2+{\sigma }_x^2{\sigma }_{\text{sct}-\text{ang}}^2}\sqrt{{\epsilon }_{y-i}^2+{\sigma }_y^2{\sigma }_{\text{sct}-\text{ang}}^2}$ (15)

Some factors are ignored in the above discussion, such as the dependence of the heat capacity and the heat conduction on the temperature, the change of the temperature gradient due to the change of the beam size, and the high-order terms in Eq. (7). Nevertheless, the particle number injected into the ring can be estimated roughly by combining Eqs. (14) and (15). $N_{\text{inj}}={\kappa }_{\text{str}}\frac{I_{\text{max}}}{q}\tau \propto {\sigma }_x{\sigma }_y$ (16) where κ_str is the stripping efficiency, which has been given in Sect. 3 for the typical ions. Now, based on Eqs. (11), (13), (15), and (16), the final emittance, the injection turn number, the highest tolerable beam intensity, and the number of the particles injected into the ring are plotted against the beam size for the U beam in Fig. 5 (a). With the plots, it is easy to find appropriate working points. The number of the injected ions increases with the beam size, although the injection turn number decreases because of the increased emittance. This is because the tolerable beam intensity increases with the beam size nearly quadratically, i.e., faster than the decreasing trend of the injection turn number. But it has to be mentioned that the beam intensity could not keep increasing with the beam size, and hence neither did the injected ion number, given the designed beam intensity of 1 emA. The beam size, with which the tolerable beam intensity gets to 1 emA, is indicated with the green dot-dash line in Fig. 5 (a). The trend of the injected ion number for the beam size beyond this critical point, provided that the beam intensity is kept at 1 emA, is presented with the dashed magenta line.

Fig. 5Full-screen View

(Color online) Relationships between emittance after stripping (black solid lines), number of injection turns (red solid lines), highest tolerable beam intensity (blue solid lines), number of injected particles (magenta solid lines) and rms beam size: (a) for U beam, (b) for Xe beam and (c) for Kr beam. Green dot-dash lines indicate the beam size beyond which the highest tolerable beam intensity exceeds 1 emA, designed value for iLinac. Magenta dash lines present trend of injected ion number when input beam current is kept at 1 emA

It must be emphasized that the plots in Fig. 5 are only utilized to quickly evaluate the working points: for a certain beam size, how much will the emittance be after stripping and how many stripped ions can be injected into the ring roughly? In calculating the injected particle number, the beam loss along the beamline from the stripper to the injection point and during the injection was not considered. Therefore, to get the accurate number of particles that can be accumulated in the ring, detailed simulation and optimization procedures, such as the work in Refs. [47] and [53], are needed.

Except for the accumulated ion number in the ring, the accurate foil temperature must also be calculated with the thermal analysis presented in Sect. 5.2. Here, we choose two beam sizes: one is the smallest size presented in Fig. 5, 1 mm, and the other is the beam size of 1.89 mm, with which the highest tolerable beam intensity achieves 1 emA. The temperature evolution at the hot spot simulated by Ansys for these two U beam sizes is presented in Fig. 6 (a) with their respective highest tolerable beam intensities indicated in the legends and the corresponding emittances after stripping, injection turn numbers, beam pulse durations, and injected particle numbers listed in Table 3. With the temperature evolution, the evaporation rate at the hot spot averaged over one second, i.e., the repetition period of the beam pulses, is 6 × 10^-9 (μg/cm²)/s for both beam sizes, because the temperature evolution in the two cases is the same. However, the peak temperature is slightly different.

Fig. 6Full-screen View

Temperature evolution at hot spot simulated by Ansys: (a) for U beam size of 1 and 1.89 mm, (b) for Xe beam size of 1 and 1.54 mm, (c) for Kr beam size of 1 and 1.35 mm

Similarly, the dependence of the corresponding parameters on the beam size was also derived for the Xe and Kr beams and plotted in Figs. 5 (b) and (c). The temperature evolution at the hot spot for two beam sizes for each ion species is presented in Figs. 6 (b) and (c). The corresponding parameters for these two beam sizes are listed in Table 3.

With the beam parameters in Table 3, the foil lifetime due to radiation damages calculated by LISE++ is 25.85, 622.35, and 2110.94 hrs for U, Xe and Kr beams. The results for the two beam sizes are the same because the averaged particle flux density is nearly the same in the two cases.