Global spill control in RF-knockout slow-extraction
全球泄漏控制在 RF 击倒慢抽提中

Received 25 August 2003; accepted 14 November 2003
2003 年 8 月 25 日收到；2003 年 11 月 14 日接受

(C) 2003 Elsevier B.V. All rights reserved.
(版权) 2003 Elsevier B.V. 保留所有权利。

PACS:

Keywords: RF-knockout slow-extraction; Synchrotron; Heavy ion therapy; Flat spill; Scanning irradiation
关键词：RF 敲除慢抽取；同步辐射；重离子治疗；平板泄漏；扫描照射

The time structure of extracted beams has been improved in synchrotron rings employing resonant slow-extraction [1-4]. Such improvements have also been achieved in the RF-knockout slowextraction method [5] with amplitude modulation (AM) and frequency modulation (FM) [6-8] at the HIMAC [9,10] synchrotron. RF-knockout
提取束流的时间结构在采用谐振慢提取的同步辐射环中得到了改善。在 HIMAC 同步辐射环中，也通过 RF-knockout 慢提取方法以及幅度调制（AM）和频率调制（FM）取得了这样的改进。

extraction has an advantage for quick beam on/ off within several s compared with other slow-extraction methods using a magnetic element. Owing to such a quick response, this method has achieved respiration-gated irradiation [12] during the cancer therapy at HIMAC. In RF-knockout extraction, the beam ripple on the kHz order has been successfully suppressed by using the separated function method. In this method, a transverse RF-field with a mono-frequency is additionally applied for extraction, while the RF-field with a frequency bandwidth is applied to diffuse particles located deep inside of the separatrices.
提取相对于其他使用磁元件的慢提取方法具有快速开/关束流的优势，在几秒钟内完成。由于这种快速响应，该方法已在 HIMAC 的癌症治疗中实现了呼吸门控照射[12]。在 RF-击穿提取中，通过使用分离功能方法成功抑制了 kHz 级别的束流波动。在这种方法中，除了应用具有单一频率的横向 RF 场进行提取外，还应用具有频率带宽的 RF 场来扩散位于分离面深处的粒子。

In RF-knockout extraction, on the other hand, the global spill structure has not yet been sufficiently controlled, although it has been strongly required for medical and other applications. A flat spill is expected to greatly contribute, especially for the beam scanning irradiation method [13-15]. For this purpose, we have proposed a scheme to realize a flat spill through optimizing the AM function of the transverse RFfield. In this scheme, a simple model of the extraction process, in which the radial distribution of particles with diffusion by the RF-knockout is assumed to be the Reyleigh distribution [16], was used to obtain a new AM function for a flat spill. Using this model, the global spill structure was analyzed to determine the parameter relevant to the diffusion process by RF-knockout. It was verified in both the simulation and the experiment at the HIMAC synchrotron that a flat spill within can be provided by applying the new AM function obtained by the scheme. Cooperating with the feedback system, finally, a flat spill was provided to be less than in fluctuation, while a kHz order ripple was suppressed by the separated function method.
在 RF-knockout 提取中，另一方面，全局溢出结构尚未得到充分控制，尽管医疗和其他应用强烈要求。预计平坦的溢出将在特别是对于束扫描照射方法[13-15]有很大贡献。为此，我们提出了通过优化横向 RF 场的 AM 函数来实现平坦溢出的方案。在这个方案中，提出了一个提取过程的简单模型，假设通过 RF-knockout 扩散的粒子的径向分布为 Reyleigh 分布[16]，用于获得一个新的用于平坦溢出的 AM 函数。利用这个模型，分析了全局溢出结构，以确定与 RF-knockout 扩散过程相关的参数。在 HIMAC 同步加速器的模拟和实验中验证了通过该方案获得的新 AM 函数可以提供一个 内的平坦溢出。最终，与反馈系统合作，提供了一个波动小于 的平坦溢出，同时通过分离函数方法抑制了 kHz 级的纹波。

This paper describes the details of the simulation and experimental results for global spill control in RF-knockout extraction.
本文描述了 RF 击穿提取中全局泄漏控制的模拟和实验结果的细节。

The radial distribution function of particles in normalized phase-space ( ) can be expressed by using the Reyleigh distribution function as
粒子在归一化相空间中的径向分布函数可以用 Reyleigh 分布函数表示为

where is the standard deviation of the Reyleigh distribution and the root-mean-square emittance. Concerning the analogy of RF-knockout slow extraction, as shown in Fig. 1, we define the boundary of the separatrix as . In this model, it is
其中 是 Reyleigh 分布的标准偏差， 是均方根发射度。关于 RF-knockout 慢抽取的类比，如图 1 所示，我们将分离界定为 。在这个模型中，它是

Fig. 1. Radial distribution in the normalized phase-space for considering the simple extraction model.
图 1. 考虑简单提取模型的归一化相空间中的径向分布。

assumed that particles having a larger radial amplitude than are extracted from the ring. With increasing the number of turns (from to in Fig. 1), further, increases through diffusion by the transverse RF-field in RF-knockout slow extraction. Thus, the number of the extracted particles, (hatched area in Fig. 1), can be expressed as
假设比 更大径向振幅的粒子从环中提取出来。随着转数的增加（从图 1 中的 到 ），通过横向 RF 场在 RF 击穿慢抽取中扩散， 进一步增加。因此，提取出的粒子数量 （图 1 中的阴影区域）可以表示为

where is the total number of particles in the ring before extraction. Thus, the time structure of the extracted beam can be represented as
其中 是提取前环中的粒子总数。因此，提取束流的时间结构可以表示为

.

Concerning the diffusion process in the RFknockout extraction, the value can be estimated as a function of the turn number ( ) in the linearized condition [7] as
关于 RF 敲除提取中的扩散过程， 值可以被估计为线性化条件下的转数（ ）的函数[7]

where is the horizontal emittance ( ), the beta-function at the kicker electrode, the amplitude of the transverse RF-field represented in units of the kick angle, the decimal part of the tune corresponding to the frequency of the transverse RF-field and that of the tune.
其中 是 的水平发射度（ ）， 是踢腿电极处的贝塔函数， 是以踢角单位表示的横向射频场幅度， 是对应于横向射频场频率的共振频率的小数部分， 是共振频率的小数部分。

When we apply a transverse RF-field with a bandwidth of , Eq. (4) should be averaged over
当我们应用带有 带宽的横向 RF 场时，方程（4）应该进行平均处理
all components of the frequency as
频率的所有组件

where is the revolution frequency. It was assumed that is proportional to the number of turns, even in the case with nonlinearity caused by the extraction sextupoles. When the kick-angle amplitude ( ) has a time dependence, using the constant related to the bandwidth of the RF-knockout, the growth of the value can be written as
其中 是革命频率。假设 与转数成正比，即使在由提取六极场引起的非线性情况下也是如此。当踢角幅度（ ）具有时间依赖性时，使用与 RF 击穿带宽相关的常数 ， 值的增长可以写成

.

Under a constant kick angle of the RF-knockout, Eq. (6) can be written as
在 RF 击倒的恒定踢角下，方程（6）可以写成

where is the initial value.
其中 是初始 值。

Using Eq. (6), the time structure of the extracted beam can be estimated through Eq. (3). Thus, when the value is obtained, the spill can be estimated for any case of the AM for the transverse RF-field.
使用方程（6），可以通过方程（3）估计提取束流的时间结构。因此，当获得 值时，可以估计横向射频场的任何情况下的泄漏。

In order to obtain a flat spill and to extract all of the particles in the ring, further, Eq. (3) should be kept constant during extraction as
为了获得平坦的溢出并提取环中的所有粒子，此外，在提取过程中，应保持方程（3）恒定。

const.  常数。

where is the duration of the extraction. It should be noted that a very small number of particles, , have already been extracted from the ring at in this model. Therefore, the number of extracted particles in each turn can be written as Eq. (8).
其中 是提取的持续时间。值得注意的是，在这个模型中，已经从环中提取了非常少量的粒子 在 。因此，每个转弯提取的粒子数量可以写成方程（8）。

From Eqs. (3), (6) and (8), the new AM function for a flat spill was obtained as
从方程（3），（6）和（8）中，得到了平面溢出的新 AM 函数。

where 哪里

In order to verify the feasibility of the simple model and the scheme for a flat spill, a simulation was carried out using the particle-tracking code [17]. The parameters for the simulation are summarized in Table 1. It should be noted that the original single-knockout method [8], in which the frequency of the transverse RF-field is swept linearly during a cycle of the FM to cover the tune spread in the separatrix, was employed for simplifying the simulations.
为了验证简单模型和平面泄漏方案的可行性，使用粒子跟踪代码[17]进行了模拟。模拟的参数总结在表 1 中。值得注意的是，为了简化模拟，采用了原始的单击方法[8]，即在 FM 周期内线性扫描横向 RF 场的频率，以覆盖分离子中的调谐扩展。

When the parameters in Eqs. (3) and (6) were obtained, the spill structure could be estimated through an analytical approach. Both parameters and can be estimated analytically, but not for the value because of the amplitude dependence of the tune in the separatrix and the bandwidth of the RF-knockout. Thus, the value is estimated by fitting the spill structure under a constant kick angle. When we apply a constant kick angle, , the time structure can be easily obtained by Eqs. (3)
当在方程（3）和（6）中获得参数 时，可以通过分析方法估计溢出结构。参数 和 都可以通过分析估计，但不能对 值进行估计，因为分离子中的共振频率和 RF-knockout 的带宽依赖于振幅。因此， 值是通过拟合在恒定踢角下的溢出结构来估计的。当我们应用恒定踢角 时，时间结构可以通过方程（3）轻松获得。

Table 1 表 1

Main parameters of the simulation
模拟的主要参数

and (7) as 和（7）作为

The spill structure obtained by the simulation was fitted by using this formula in order to obtain the value. The simulation was carried out for three different kick angles of the RF-knockout as 0.5 , 1.0 , and . As a result, the value was estimated to be 312 by averaging the values obtained by fitting each spill. It is noted that the ripple more than 100 Hz is neglected in this fitting. Both parameters, and , were analytically estimated to be and , respectively. This value corresponds to the radius of a circle, which has the same area as the separatrix ( mrad) of reference momentum. Further, corresponds to the initial emittance of mrad. The spill structure obtained by the simulation and the analytical approach with the above parameters (Eq. (11) with ) is shown in Fig. 2. The spill structure estimated by the analytical approach is in good agreement with the simulation result.
通过使用这个公式拟合模拟得到的溢出结构，以获得 值。模拟分别进行了三种不同的 RF-knockout 踢角度为 0.5、1.0 和 。结果，通过对每个溢出进行拟合并取平均值，估计得到 值为 312。值得注意的是，在这个拟合中忽略了超过 100 赫兹的纹波。两个参数 和 分别被分析估计为 和 。这个 值对应于一个圆的半径，其面积与参考动量的分离线（ 毫弧度）相同。此外， 对应于 毫弧度的初始发射度。通过上述参数（方程（11）与 ）得到的模拟和分析方法估计的溢出结构如图 2 所示。分析方法估计的溢出结构与模拟结果吻合良好。

Since the value was obtained, we can estimate the spill structure for any case of the AM for a transverse RF-field by using Eqs. (3) and (6). As one example, a simulation was carried out for the case of the linear AM function routinely used for therapy at the HIMAC synchrotron. The kick angle of the transverse RF-field is linearly swept from 0.60 to during extraction having a duration of 1.60 s . As can be seen in Fig. 3, the spill structure estimated by the analytical approach is in good agreement with the simulation result. In order to characterize the spill structure, and to compare it, two types of fluctuation ratio are defined as described in the Appendix. In this case of a linear AM simulation, the deviation from the analytical estimation was estimated to be , and the flatness . Since the binwidth was set to be 100 ms in this calculation to estimate the fluctuation ratio of the global spill structure, only a time structure of less than 10 Hz was considered.
自从获得了 值以来，我们可以使用方程（3）和（6）来估计横向 RF 场的任何情况下的 AM 的溢出结构。作为一个例子，对于在 HIMAC 同步加速器常规用于治疗的线性 AM 函数的情况进行了模拟。横向 RF 场的脉冲角度在 1.60 秒的提取过程中从 0.60 线性扫过到 。如图 3 所示，通过分析方法估计的溢出结构与模拟结果吻合良好。为了表征溢出结构并进行比较，定义了两种波动比率，如附录所述。在这种线性 AM 模拟情况下，从分析估计中估计的偏差 为 ，平坦度 。由于在此计算中将 binwidth 设置为 100 毫秒以估计全局溢出结构的波动比率，因此仅考虑了小于 10 Hz 的时间结构。

Fig. 2. Comparison of the spill structure between the simulation (dashed line) and the analytical approach (solid line) with a constant amplitude of the RF-knockout. The kick angle amplitude is (a) 0.5 , (b) 1.0 , and (c) .
图 2. 模拟（虚线）和分析方法（实线）之间的溢出结构比较，具有恒定 RF 击倒的幅度。踢角幅度为（a）0.5，（b）1.0，（c） 。

Using Eqs. (8) and (9) with the above parameters, we can estimate the new AM function for the flat spill, as shown in Fig. 4. In this calculation, in Eqs. (9) and (10) should be chosen to have a larger value than the desired extraction duration for the following reason. Choosing to be equal to the desired extraction duration, the amplitude of the RF-knockout finally reaches up to infinity at the end of the spill, which is an unacceptable value for practical use. Therefore, was set to be
使用上述参数和方程（8）和（9），我们可以估计平面泄漏的新 AM 函数，如图 4 所示。在这个计算中，方程（9）和（10）中的 应选择比所需的提取持续时间更大的值，原因如下。选择 等于所需的提取持续时间，RF-knockout 的幅度最终在泄漏结束时达到无穷大，这对实际使用是不可接受的值。因此， 被设置为

Fig. 3. Comparison of the spill structure between the simulation (dashed line) and the analytical approach (solid line) with the linear AM function.
图 3. 使用线性 AM 函数比较模拟（虚线）和分析方法（实线）之间的溢出结构。

Fig. 4. AM function of the RF-knockout as a function of the time. The solid line is the new AM function of the RF-knockout obtained by Eqs. (8) and (9); the dashed line is the linear AM function used for the simulation shown in Fig. 3.
图 4. RF 敲除的 AM 功能随时间变化的函数。实线是通过方程（8）和（9）获得的 RF 敲除的新 AM 功能；虚线是用于图 3 中显示的模拟的线性 AM 功能。

1.68 s in the calculation, while the desired extraction duration was set to be 1.60 s . Thus, it was estimated by the analytical approach that more than of the particles will be extracted from the ring, because the desired extraction duration is shorter by only than . Using this new AM function, the simulation was carried out as shown in Fig. 5. Though an extraction efficiency of almost in this simulation is different from
1.68 秒的计算时间，而期望的提取持续时间设定为 1.60 秒。因此，通过分析方法估计超过 的粒子将从环中提取出来，因为期望的提取持续时间仅比 短 。使用这个新的 AM 函数，模拟如图 5 所示进行。尽管在这个模拟中提取效率几乎为 ，与

Fig. 5. Comparison of the spill structure between the simulation (dashed line) and analytical approach (solid line) with the AM function obtained by Eqs. (8) and (9).
图 5. 模拟（虚线）和分析方法（实线）之间的溢出结构比较，使用由方程（8）和（9）获得的 AM 函数。

the analytical estimation of , the flatness of the spill was considerably improved by the new AM function compared with the case of a linear AM function. The and values were estimated to be .
对于 的分析估计，与线性 AM 函数相比，新的 AM 函数显着改善了溢出的平整度。 估计 和 值为 。

The experimental condition was similar to the simulation summarized in Table 1. The initial emittance before extraction was measured to be mrad by a non-destructive beam-profile monitor [18], which was the same value as in the simulation. At the HIMAC synchrotron, the flattop duration is kept at around 2 s under an operation period of 3.3 s to provide carbon ions with an energy of . A transverse RFfield was applied along with a beam gate signal of 1.6 s . The beam was bunched by a longitudinal RF-field with an amplitude of 4 kV . The momentum distribution of the circulating beam was measured to be around at by observing the Schottky spectrum of the debunched beam, which was taken into account in the simulation. The beam spill was measured by using the existing beam-spill monitor [6], which consists of a plastic scintillator with a thickness of 0.2 mm , a
实验条件与表 1 中总结的模拟类似。在提取之前，通过非破坏性束流剖面监视器[18]测得初始发射度为 mrad，与模拟值相同。在 HIMAC 同步加速器中，保持平顶持续时间约为 2 秒，在 3.3 秒的运行周期内提供碳离子能量 。横向 RF 场与 1.6 秒的束流门信号一起应用。束流通过具有 4 kV 幅度的纵向 RF 场成束。通过观察去束束流的 Schottky 谱，测得循环束流的动量分布约为 在 处，这在模拟中得到考虑。使用现有的束流溢出监视器[6]测量束流溢出，该监视器由厚度为 0.2 毫米的塑料闪烁体组成。
photomultiplier, and a preamplifier. The preamplifier has a frequency response of 10 kHz .
光电倍增管和前置放大器。前置放大器具有 10 kHz 的频率响应。

A block diagram of the RF-knockout system at the HIMAC synchrotron, including the newly installed feedback system, is shown in Fig. 6. The signals for the AM and FM are produced respectively by two function generators (Sony Tektronix AFG2020). They are fed into the other one (Hewlett-Packard HP3314A) in order to control the central frequency and amplitude of the transverse RF-field. As the result of the simulation, the new AM function obtained by Eqs. (8) and (9) is efficient to produce a flat spill, but not sufficient when using only the analytical approach. Thus, a feedback system was installed to obtain a suitable AM function, and to compare it with the analytical one. The feedback signal is generated by summing up two signals: one is the inverse signal of the beam ripple, which has a frequency band of less than 100 Hz by a low-pass filter; the other is a DC voltage corresponding to the intensity from a DC current transformer (DCCT) in order to settle the beam ripple independently of intensity fluctuations of at the maximum in each operation cycle. Finally, the feedback signal is added into the AM function generated by AFG2020.
在 HIMAC 同步加速器上的 RF-knockout 系统的框图，包括新安装的反馈系统，如图 6 所示。 AM 和 FM 的信号分别由两台函数发生器（Sony Tektronix AFG2020）产生。它们被馈送到另一台（惠普 HP3314A）中，以控制横向 RF 场的中心频率和幅度。作为模拟的结果，由方程（8）和（9）得到的新 AM 函数能够有效产生平坦的溢出，但仅使用分析方法时不足。因此，安装了反馈系统以获得合适的 AM 函数，并将其与分析函数进行比较。反馈信号是通过将两个信号相加而生成的：一个是束流波动的反向信号，通过低通滤波器具有低于 100 Hz 的频带；另一个是与直流电流互感器（DCCT）的强度对应的直流电压，以独立于每个操作周期中的强度波动解决束流波动。最后，反馈信号被添加到由 AFG2020 生成的 AM 函数中。

Using a new AM function obtained by the analytical approach for a flat spill, the spill structure was measured as shown in Fig. 7(a). The measured spill structure was in good agreement with the simulation result, as shown in Fig. 5. The flatness of the spill was at , which was also consistent with the simulation result. In combination with the feedback system, a flat spill was obtained, as shown in Fig. 7(b). The flatness of the spill was considerably improved to be at . In this case, the original single-knockout method [8] was employed to be the same as the simulation. In order to suppress any kHz order ripple, furthermore, the separated function method [8] was employed in the same way as the AM function, as shown in Fig. 8(a). There was no difference in the global spill structure, while the ripple of kHz order was successfully suppressed. Finally, as shown in Fig. 8(b), a smooth spill was obtained by the feedback system.
使用通过分析方法获得的新 AM 功能对平面溢出进行测量，如图 7(a)所示。测得的溢出结构与模拟结果相符，如图 5 所示。溢出的平整度在 处为 ，这也与模拟结果一致。结合反馈系统，获得了一个平整的溢出，如图 7(b)所示。溢出的平整度显著提高至 处的 。在这种情况下，采用了原始的单击方法[8]，与模拟相同。此外，为了抑制任何 kHz 级别的纹波，还采用了与 AM 功能相同的分离函数方法[8]，如图 8(a)所示。全局溢出结构没有差异，而 kHz 级别的纹波成功被抑制。最后，如图 8(b)所示，通过反馈系统获得了平滑的溢出。

In the case of the feedback, on the other hand, the amplitude of the RF-knockout was slightly corrected by the feedback system over the whole extraction duration. The feedback signal, which
在反馈的情况下，另一方面，RF 去除的振幅在整个提取持续时间内被反馈系统略微校正。反馈信号，其中

Fig. 6. Block diagram of the RF-knockout system.
图 6. RF 去除系统的模块图。

(a) （a）

(b)

Fig. 7. (a) Spill structure by using the AM function obtained by Eqs. (8) and (9) without a feedback system, (b) with a feedback system. From the bottom of each figure, the spill structure, the AM signal, the transverse RF-field, and the circulating intensity (DCCT).
图 7. (a) 使用由方程（8）和（9）获得的 AM 函数的溢流结构，没有反馈系统，(b) 有反馈系统。从每个图的底部，溢流结构，AM 信号，横向 RF 场和循环强度（DCCT）。

represents the difference between the AM in the feedback and in the analytical approach, is shown in Fig. 9. As can be clearly observed in Fig. 9, the AM in the feedback is slightly larger than that in the analytical approach over the whole extraction duration. Although the amplitude difference is not very large, this scheme with the simple model cannot supply an extremely flat spill within without a feedback system. Thus, the reason for the deviation from the model was considered to be as follows. From the simulation result shown in Fig. 10, it was found that the radial distribution of
代表反馈中的 AM 与分析方法中的 AM 之间的差异如图 9 所示。如图 9 中可以清楚地观察到，反馈中的 AM 略大于整个提取持续时间内分析方法中的 AM。尽管振幅差异不是很大，但这种简单模型无法在没有反馈系统的情况下提供一个极其平坦的泄漏。因此，偏离模型的原因被认为是如下所述。从图 10 中显示的模拟结果中发现，径向分布是

(a) （a）

(b)

Fig. 8. Cooperating with the separate function method. See the caption of Fig. 7.
图 8. 与分离功能方法合作。请参阅图 7 的标题。

Fig. 9. Measured feedback signal represented in units of the kick angle during extraction.
图 9. 在提取过程中以踢角单位表示的测量反馈信号。

Fig. 10. Simulation result of the radial distribution. The solid line was obtained by the analytical approach and dashed line obtained by a simulation. In this case, the kick angle was kept constant at .
图 10. 径向分布的模拟结果。实线是通过分析方法获得的，虚线是通过模拟获得的。在这种情况下，踢角保持在 不变。

the particles deviated from a Reyleigh distribution through a diffusion process by RF-knockout. We could observe that the diffusion of particles deeply inside of the separatrix was slower than the analytical estimation. From this phenomenon, it is considered that the diffusion rate has a dependence on the normalized betatron amplitude. The betatron-tune shift for each particle consists of a phase shift by the sextupole kick and the chromaticity. Since the phase shift in one turn by the sextupole depends on the betatron amplitude [19], the tune for each turn can be schematically shown in Fig. 11. A particle on a stable trajectory near the boundary of the separatrix has a wider tune oscillation width ((a) in Fig. 11) than that of the particle deeply inside of the separatrix, while both of them experience tune oscillation ((b) in Fig. 11) according to the synchrotron oscillation through the chromaticity. Since the magnitude of the RF-knockout frequency component is constant independently of the frequency within the bandwidth, therefore, the diffusion speed increases with increasing the radial betatron amplitude. The Reyleigh distribution cannot be conserved during extraction, unless the frequency spectrum is optimized. Further, the effect of synchrotron oscillation, which contributes to the extraction process through the horizontal chromaticity, is neglected in this model.
粒子通过 RF 击穿的扩散过程偏离了 Reyleigh 分布。我们观察到，在分离子内部，粒子的扩散速度比解析估计要慢。从这种现象来看，扩散速率取决于归一化贝塔振幅。每个粒子的贝塔振幅变化包括六极踢和色散引起的相位变化。由于六极在一个周转中的相位变化取决于贝塔振幅，因此每个周转的频率可以在图 11 中示意显示。在分离子边界附近的稳定轨道上的粒子具有比深入分离子内部的粒子更宽的频率振荡幅度（图 11 中的 a），而它们都根据色散通过色散引起的同步辐射振荡经历频率振荡（图 11 中的 b）。由于 RF 击穿频率分量的幅度与频率带宽内的频率无关，因此扩散速度随着径向贝塔振幅的增加而增加。 Reyleigh 分布在提取过程中无法保持不变，除非频谱被优化。此外，通过水平色散贡献于提取过程的同步辐射振荡效应在该模型中被忽略。

Fig. 11. Schematic drawing of the tune shift for each turn.
图 11. 每个转弯的音调变化的示意图。

A scheme with a simple model of extraction to obtain the AM function for a flat spill in the RFknockout extraction method was proposed. In this scheme, the parameter relevant to the diffusion process was obtained by analyzing the spill structure. It was verified in both the simulation and the experiment at the HIMAC synchrotron that the flatness of the spill within was obtained by using a new AM function. A flat spill of was finally delivered with the assistance of a feedback system. Although the simple model slightly differs from the real extraction process, this technique will greatly contribute to synchrotron rings employing RF-knockout slow-extraction in cooperation with the separate function method.
提出了一种简单的提取模型方案，用于获取 RFknockout 提取方法中平面泄漏的 AM 函数。在这个方案中，通过分析泄漏结构获得了与扩散过程相关的参数。在 HIMAC 同步加速器的模拟和实验中验证了通过使用新的 AM 函数获得了 内泄漏的平坦度。最终在反馈系统的帮助下交付了 的平面泄漏。尽管简单模型与真实提取过程略有不同，但这种技术将对采用 RF-knockout 慢提取与分离功能方法合作的同步加速器环起到很大作用。

The authors would like to express their thanks to the members of Accelerator Engineering Corporation for their skilful operation of the HIMAC accelerator complex. They are also grateful to the
作者们要感谢加速器工程公司的成员对 HIMAC 加速器复合体的熟练操作。他们也感激
other members of the Department of Accelerator Physics and Engineering at NIRS for warm support. This work was carried out as a part of Research Project with Heavy Ions at NIRSHIMAC.
日本放射线医学研究所加速器物理与工程系的其他成员对我的热情支持。这项工作是作为日本放射线医学研究所重离子研究项目的一部分进行的。

The two types of the fluctuation ratio are defined as follows. One is a standard deviation of the fluctuation from the analytical estimation as
波动比率的两种类型定义如下。一种是从分析估计中的波动的标准偏差。

where is the turn number of the extraction duration. The other, which is defined as the flatness, is the standard deviation of the fluctuation from the average as
其中 是提取持续时间的轮数。另一个定义为平坦度，是从平均值波动的标准偏差

where the brackets mean averaging over the extraction duration.
括号表示在提取持续时间内进行平均。

[1] P. Strolin, CERN 69-6, 1969.
P. Strolin，CERN 69-6，1969。

[2] R. Cappi, Ch. Steinbach, CERN/PS/OP 80-10, 1980.

[3] J.C. Ciret, Extraction du faisceau de Saturne II par accélération bétatronique le Gephyrotron, GERMA 76.02/1E-117, 1976.
J.C. Ciret，通过 Gephyrotron 的 bétatron 加速提取 Saturne II 束流，GERMA 76.02/1E-117，1976。

[4] K. Noda, A. Itano, H. Ogawa, M. Kanazawa, M. Kumada, E. Takada, M. Torikoshi, S. Minohara,
[4] 野田克, 伊藤安, 小川宏, 金澤真, 熊田真, 高田恵, 鳥越正, 蓑原慎。
S. Sato, S. Yamada, N. Araki, M. Tadokoro, K. Hiramoto, K. Sato, Proceedings of the 16th RCNP Osaka International Symposium on Multi-GeV HighPerformance Accelerator and Related Technology, Osaka, 1997, pp.171-178.
S. Sato, S. Yamada, N. Araki, M. Tadokoro, K. Hiramoto, K. Sato, 1997 年大阪 RCNP 第 16 届国际多 GeV 高性能加速器及相关技术研讨会论文集，第 171-178 页。

[5] M. Tomizawa, M. Yoshizawa, K. Chida, J. Yoshizawa, Y. Arakaki, R. Nagai, A. Mizobuchi, A. Noda, K. Noda, M. Kanazawa, A. Ando, H. Muto, T. Hattori, Nucl. Instr. and Meth. A 326 (1993) 399.

[6] K. Noda, M. Kanazawa, A. Itano, E. Takada, M. Torikoshi, N. Araki, J. Yoshizawa, K. Sato, S. Yamada, H. Ogawa, H. Itoh, A. Noda, M. Tomizawa, M. Yoshizawa, Nucl. Instr. and Meth. A 374 (1996) 269.

[7] K. Noda, T. Furukawa, S. Shibuya, M. Muramatsu, T. Uesugi, M. Kanazawa, M. Torikoshi, E. Takada, S. Yamada, Nucl. Instr. and Meth. A 492 (2002) 241.

[8] K. Noda, T. Furukawa, S. Shibuya, T. Uesugi, M. Muramatsu, M. Kanazawa, E. Takada, S. Yamada, Nucl. Instr. and Meth. A 492 (2002) 253.

[9] Y. Hirao, H. Ogawa, S. Yamada, Y. Sato, T. Murakami, A. Kitagawa, K. Sato, A. Itano, M. Kumada, E. Takada, M. Kanazawa, K. Noda, M. Sudou, K. Kawachi, F. Soga, M. Endo, T. Kanai, S. Minohara, H. Koyama-Ito, T. Kohno, Nucl. Phys. A 538 (1992) 541c.
[9] 平尾裕，小川宏，山田慎，佐藤洋，村上隆，北川明，佐藤健，伊藤亚，熊田美，高田恵，金泽真，野田健，须藤美，川内健，曽我富，遠藤美，金井隆，蓑原秀，小山伊藤，小野亨，核物理学 A 538（1992）541c。

[10] S. Yamada, Proceedings of the 2nd APAC, Beijing, China, 2001, pp. 829-833.
S. Yamada，第 2 届 APAC 会议论文集，中国北京，2001 年，第 829-833 页。

[11] T. Furukawa, K. Noda, E. Urakabe, M. Muramatsu, M. Kanazawa, K. Maeda, Nucl. Instr. and Meth. A 503 (2003) 485 .
[11] T.古川，K.野田，E.浦边，M.村松，M.金泽，K.前田，Nucl. Instr. and Meth. A 503 (2003) 485 .

[12] S. Minohara, T. Kanai, M. Endo, K. Noda, M. Kanazawa, Int. J. Radiat. Oncol. Biol. Phys. 47 (4) (2000) 1097.
S. Minohara, T. Kanai, M. Endo, K. Noda, M. Kanazawa, Int. J. Radiat. Oncol. Biol. Phys. 47 (4) (2000) 1097.

[13] W.T. Chu, B.A. Ludewigt, T.R. Renner, Rev. Sci. Instrum. 64 (8) (1993) 2055.

[14] P. Holy, CERN/PS 98-064 (DI).
P. Holy，CERN/PS 98-064（DI）。

[15] Th. Haberer, W. Becher, D. Schardt, G. Kraft, Nucl. Instr. and Meth. A 330 (1993) 296.
Th. Haberer, W. Becher, D. Schardt, G. Kraft, Nucl. Instr. and Meth. A 330 (1993) 296.

[16] H. Bruck, Accelerateurs circulaires de particules, Presses Universitaires de France, Paris, 1966.
H. Bruck，粒子环形加速器，法国大学出版社，巴黎，1966 年。

[17] T. Furukawa, K. Noda, Nucl. Instr. and Meth. A 489 (2002) 59 .
T. Furukawa, K. Noda, Nucl. Instr. and Meth. A 489 (2002) 59 .

[18] T. Honma, D. Ohsawa, K. Noda, T. Iwashima, H.Y. Ogawa, Y. Sano, E. Takada, S. Yamada, Nucl. Instr. and Meth. A 490 (2002) 435.
[18] 本间隆，大泽大，野田圭，岩岛隆，小川浩之，佐野阳一，高田英，山田茂，核仪器和方法 A 490（2002）435。

[19] E. Wilson, CERN accelerator school, CERN 94-01, pp. 239-252.
E. Wilson，欧洲核子研究中心加速器学校，CERN 94-01，第 239-252 页。