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$$ \begin{split} & \frac{{{{\text{d}}^2}X}}{{{\text{d}}{s^2}}} + \left( {\frac{{p'_0}}{{{p_0}}}} \right)\frac{{{\text{d}}X}}{{{\text{d}}s}} + k_x^2(s)X - \frac{{K{u_0}{\text{π }}{\lambda _3}}}{{{l^2}}}{G_{311}}(X,Y,{u_0}T)X - \\& \qquad {\left( \frac{\delta }{{l{p_0}}}\right)^2}\frac{{\varepsilon _x^2}}{{{X^3}}} = 0 \text{,} \end{split} $$ (1) $$ \frac{{{{\text{d}}^2}Y}}{{{\text{d}}{s^2}}} + \left( {\frac{{p'_0}}{{{p_0}}}} \right) \frac{{{\text{d}}Y}}{{{\text{d}}s}} + k_y^2(s)Y - \frac{{K{u_0}{\text{π }} {\lambda _3}}}{{{l^2}}}{G_{311}}(X,Y,{u_0}T) Y - {( \frac{\delta }{{l{p_0}}} )^2} \frac{{\varepsilon _y^2}}{{{Y^3}}} = 0 \text{,} $$ (2) $$ \begin{split} \\ & \frac{{{{\text{d}}^2}T}}{{{\text{d}}{s^2}}} + 3\left( {\frac{{p'_0}}{{{p_0}}}} \right)\frac{{{\text{d}}T}}{{{\text{d}}s}} + k_t^2(s)T - \frac{{K{u_0}{\text{π }}{\lambda _3}}}{{{l^2}}}{G_{311}}(X,Y,{u_0}T)T - \\&\qquad {\left( { \frac{\delta }{{l{p_0}u_0^2}} } \right)^2}\frac{{\varepsilon _t^2}}{{{T^3}}} = 0 \text{,} \end{split} $$ (3) $$ {G_{lmn}}(X,Y,Z) = \frac{3}{2}\int_0^\infty {\frac{{{\text{d}}s}}{{{{({x^2} + s)}^{l/2}}{{({y^2} + s)}^{m/2}}{{({z^2} + s)}^{n/2}}}}} \text{,} $$ (4) 其中:X和Y分别代表水平和垂直的RMS束流尺寸;T代表RMS纵向相宽;
$ {p_0} $ 代表参考粒子的动量;$p'_0$ 代表距离的微分;$ k_x^2,k_y^2,k_t^2 $ 代表横纵向外聚焦力;$ {\varepsilon _x},{\varepsilon _y},{\varepsilon _{\text{t}}} $ 代表三个方向的归一化RMS发射度。$l = c/\omega$ ,$\delta = mc$ ,其中$ c $ 代表真空中的光速;$ \omega $ 代表高频角频率;${u_0} = {\gamma _0}{\beta _0}$ ;$ K $ 是与空间电荷力相关的导流系数:$$ K=\frac{qI}{2\pi {\varepsilon }_{0}{p}_{0}{\upsilon }_{0}^{2}{\gamma }_{0}^{2}}, $$ 其中:
$ I $ 代表平均电流;$ q $ 代表粒子的电荷态;$ {\varepsilon _0} $ 是真空介电常数;$ {\upsilon _0} $ 表示参考粒子速度;$ {\gamma _0} $ 代表参考粒子的相对论因子。$ k_t^2 $ 代表纵向聚焦[8];$ {E_{\text{s}}} $ 代表同步粒子的电场;$ T $ 代表渡越时间因子,表征腔体的加速效率;$ {\varphi _{\text{s}}} $ 代表同步相位;$ {\gamma _{\text{s}}} $ 代表同步粒子的相对论因子;$ {\sigma _0} $ 代表纵向相移:$$ {k}_{t}^{2}=\dfrac{2\pi q{E}_{\text{s}}T\mathrm{sin}(-{\varphi }_{\text{s}})}{m{c}^{2}{\beta }_{\text{s}}^{3}{\gamma }_{\text{s}}^{3}{\lambda}}, $$ $$ {\sigma }_{0}={k}_{t}L=\sqrt{\dfrac{2\pi q{E}_{\text{s}}T\mathrm{sin}(-{\varphi }_{\text{s}})}{m{c}^{2}{\beta }_{\text{s}}^{3}{\gamma }_{\text{s}}^{3}{\lambda}}}L, $$ 由上述公式可以看出,对一个确定的加速结构,纵向相移与加速梯度和同步相位呈正比,即腔体加速梯度越高,相移变化速度越快。
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摘要: 空间电荷驱动的包络不稳定性是引起强流超导质子直线加速器束流损失的重要因素,零流强下的周期相移小于90度是目前国际普遍遵循的强流超导质子直线加速器设计原则,这一设计原则是在不考虑加速的情况下提出的。本论文通过动力学仿真模拟,对加速条件下的周期性不稳定性进行了分析,结果表明,合理选择聚焦变化速度和加速梯度可以减弱包络不稳定度的影响,从而可以突破物理设计周期相移小于90度的严格限制,提高加速器的加速效率。Abstract: Beam envelope instability driven by space charge is an important factor causing the beam loss of intense beam superconducting linear accelerator. Zero-current period phase advance less than 90 degrees is the design principle of strong current superconducting linear accelerator generally followed in the world, and this design principle is proposed without considering acceleration. In this paper, the periodic instability under the condition of acceleration effect is analyzed by dynamic simulation. The results show that reasonable choice of focus change rate and acceleration gradient can reduce the influence of envelope instability, so that the strict limit of physical design period phase advance of less than 90 degrees can be broken and the acceleration efficiency of the accelerator can be improved.
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Key words:
- superconducting linear accelerator /
- high current /
- space charge /
- envelope instability
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