CERN – ACCELERATORS AND TECHNOLOGY SECTOR CERN-ATS-2012-120 STUDY OF MULTIPOLAR RF KICKS FROM THE MAIN DEFLECTING MODE IN COMPACT CRAB CAVITIES FOR LHC
CERN – ACCELERATORS AND TECHNOLOGY SECTOR CERN-ATS-2012-120 STUDY OF MULTIPOLAR RF KICKS FROM THE MAIN DEFLECTING MODE IN COMPACT CRAB CAVITIES FOR LHC
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CERN – 加速器和技术部门 CERN-ATS-2012-120 LHC 紧凑蟹腔中主偏转模式的多极射频冲击研究
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发表时间：
2012
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通讯作者：
R. Tomás
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作者：
Barranco García;R. Calaga;R. Maria;M. Giovannozzi;A. Grudiev;R. Tomás
在高亮度大型强子对撞机项目的框架内，正在设计一个蟹状腔系统。由于一方面的横向空间限制和另一方面的RF频率要求，蟹腔的设计必须紧凑。这导致蟹状腔形状远离轴向对称，因此，主偏转模式的高阶多极分量是非零的。本文计算了LHC紧凑型CC中主偏转模产生的多极RF反冲。他们相比，多极误差在LHC磁铁。的RF踢上的束流动力学的可能的影响进行了研究，通过分析估计。在国际粒子加速器会议（IPAC'12）-2012年5月20日至25日，N。美国奥尔良，瑞士日内瓦，2012年5月大型强子对撞机紧凑蟹形腔中主偏转模的多极射频反冲研究 * J. Barranco García，R.卡拉加河De Maria，M. Giovanjani，A.格鲁季耶夫河Tomás CERN，日内瓦，瑞士摘要在高亮度大型强子对撞机计划的框架内，正在设计一个螃蟹腔（CC）系统。由于一方面的横向空间限制和另一方面的RF频率要求，蟹腔的设计必须紧凑。这导致蟹状腔形状远离轴向对称，因此，主偏转模式的高阶多极分量是非零的。本文计算了LHC紧凑型CC中主偏转模产生的多极RF反冲。他们相比，多极误差在LHC磁铁。在高亮度大型强子对撞机（LHC）的框架下，我们正在设计一个蟹状腔（CC）系统。由于一方面的横向空间限制和另一方面的RF频率要求，蟹腔的设计必须紧凑。这导致蟹状腔形状远离轴向对称，因此，主偏转模式的高阶多极分量是非零的。本文计算了LHC紧凑型CC中主偏转模产生的多极RF反冲。他们相比，多极误差在LHC磁铁。的RF踢上的束流动力学的可能的影响进行了研究，通过分析估计。多极RF反冲计算LHC CC原型的几何形状在图1中，显示了LHC CC的三个原型的几何形状：（左）脊波导腔（RWCAV）[1];（中）四分之一波谐振腔（QWCAV）[2];（右）4杆谐振腔（4 RCAV）[3]。他们利用不同的方法使400 MHz的偏转腔如此紧凑，以至于它适合LHC相互作用区域的两个束管之间的可用空间[4]。然而，在所有三种情况下，腔的几何形状是轴向非对称的，从而产生主偏转模式的高阶多极分量。在轴对称的偏转腔中，主偶极模（m=1）只有偶极变化的电磁场~exp（单位φ），其中n=m=1。然而，对于图1所示的LHC CC原型，情况并非如此，这是由于腔体形状与轴对称腔体形状的强烈偏离。在这种情况下，根据腔对称性，存在所有多极分量n ≥ 1。为了简化数学计算，我们假设所有腔相对于水平面XZ对称，并且以偶极反冲在X方向上的方式定向。图1所示的所有三个腔都是如此。在这种情况下，所有偏斜分量为零，方位角依赖性为~cos（nφ）。此外，假设带电粒子e以光速c平行于Z轴运动，则作用在其上的垂直洛伦兹力可表示为：π kick z kick Hu Z E E F π kick π 0其中Z 0为真空阻抗，uz为Z方向的单位矢量，cz j kick cz j kick e H E E/ / ;分别是粒子坐标系中的电场和磁场，而E和H是腔坐标系中相应的电场和磁场。这些场可以表示为多极之和，类似于对磁体所做的[5]。然后，垂直洛仑兹力被表示为其多极分量的和）（n F：）] sin（）cos（[）（），，（1 1 1）（n n u r z F z r F
A crab cavity (CC) system is under design in the framework of the High Luminosity LHC project. Due to transverse space constraints on one hand and the RF frequency requirements on the other hand, the design of the crab cavities has to be compact. This results in the crab cavity shape being far from axially symmetric and, as a consequence, higher order multipolar components of the main deflecting mode are non-zero. In this paper, multipolar RF-kicks from the main deflecting mode are calculated in the compact CC for LHC. They are compared to the multipolar error in LHC magnets. The possible influence of the RF-kicks on the beam dynamics has been investigated by means of analytical estimates. Presented at the International Particle Accelerator Conference (IPAC’12) – May 20-25, 2012, N. Orleans, USA Geneva, Switzerland, May 2012 STUDY OF MULTIPOLAR RF KICKS FROM THE MAIN DEFLECTING MODE IN COMPACT CRAB CAVITIES FOR LHC* J. Barranco García, R. Calaga, R. De Maria, M. Giovannozzi, A. Grudiev, R. Tomás CERN, Geneva, Switzerland Abstract A crab cavity (CC) system is under design in the framework of the High Luminosity LHC project. Due to transverse space constraints on one hand and the RF frequency requirements on the other hand, the design of the crab cavities has to be compact. This results in the crab cavity shape being far from axially symmetric and, as a consequence, higher order multipolar components of the main deflecting mode are non-zero. In this paper, multipolar RF-kicks from the main deflecting mode are calculated in the compact CC for LHC. They are compared to the multipolar error in LHC magnets. The possible influence of the RF-kicks on the beam dynamics has been investigated by means of analytical estimates.A crab cavity (CC) system is under design in the framework of the High Luminosity LHC project. Due to transverse space constraints on one hand and the RF frequency requirements on the other hand, the design of the crab cavities has to be compact. This results in the crab cavity shape being far from axially symmetric and, as a consequence, higher order multipolar components of the main deflecting mode are non-zero. In this paper, multipolar RF-kicks from the main deflecting mode are calculated in the compact CC for LHC. They are compared to the multipolar error in LHC magnets. The possible influence of the RF-kicks on the beam dynamics has been investigated by means of analytical estimates. MULTIPOLAR RF KICK CALCULATION Geometry of the LHC CC prototypes In Fig 1, geometry of the three prototypes of the LHC CC are shown: (left) Ridged Waveguide cavity (RWCAV) [1]; (middle) Quarter Wave resonator cavity (QWCAV) [2]; (right) 4 rod resonator cavity (4RCAV) [3]. They exploit different ways to make a 400 MHz deflecting cavity so compact that it fits in the space available between two beam pipes of the LHC interaction regions [4]. Nevertheless, in all three cases, geometry of the cavity is axially non-symmetric giving rise to higher order multipolar components of the main deflecting mode. Electro-magnetic field representation of the main deflecting mode In an axially symmetric deflecting cavity, the main dipole mode (m=1) has only dipolar variation of the electromagnetic field ~exp(inφ), where n=m=1. It is not the case, however, for the LHC CC prototypes shown in Fig. 1 due to strong deviation of the cavity shape from the axially symmetric one. In this case, all multipolar components n ≥ 1 are present depending on the cavity symmetry. In order to simplify the math we assume that all cavities are symmetric with respect to the horizontal plane XZ and are oriented in such a way that dipolar kick is in X direction. This is true all three cavities shown in Fig 1. In this case, all skew components are zero and azimuthal dependence is ~cos(nφ). Furthermore, assuming that the particle of charge e moves parallel to the Z-axis with the speed of light c, the perpendicular Lorentz force acting on it can be expressed as follows: kick z kick H u Z E e F 0 where Z0 is vacuum impedance, uz the unit vector in Z direction, c z j kick c z j kick e H H e E E / / ; are the electric and magnetic fields in the particle frame, respectively, while E and H are the corresponding electric and magnetic fields in the cavity frame. These fields can be expressed as a sum of multipoles similarly to what is done for magnets [5]. Then, the perpendicular Lorentz force is expressed as a sum over its multipolar components ) (n F : )] sin( ) cos( [ ) ( ) , , ( 1 1 ) ( n u n u r z F z r F