用振丝法测量加速器磁体的场质量--偏微分方程求解的实例研究
Measuring the Field Quality in Accelerator Magnets with the Oscillating-Wire Method -- a Case Study for Solving Partial Differential Equations
摘要：
加速器磁铁的磁场强度和磁轴的测量通常采用单拉丝法。导线连接端子处的积分电压是与由移位的导线描绘出的表面相关联的通量的量度。拉伸的金属丝也可以用远低于共振频率的交流电来激励。因此，可以通过利用线振荡幅度、积分场和电流幅度之间的线性关系来测量多极场误差。这种技术是求解偏微分方程的一个很好的例子，或者更准确地说，一维和二维的边值问题。特别是，在加速器磁体的孔径中的字段由拉普拉斯方程，这导致一个边界值的问题，通过确定一系列的本征函数的系数从测量的字段组件或线振荡幅度域边界上解决。张紧弦的振荡是一维非齐次波动方程的一个例子。由于只考虑了线运动的线性化方程和磁体的积分场谐波，因此，导线系统的非线性表征产生了对该方法的不确定性（和限制）的反馈。
英文摘要：
The single stretched-wire method is commonly used to measure the magnetic field strength and magnetic axis in an accelerator magnet. The integrated voltage at the connection terminals of the wire is a measure for the flux linked with the surface traced out by the displaced wire. The stretched wire can also be excited with an alternating current well below the resonance frequency. It is thus possible to measure multipole field errors by making use of the linear relationship between the wire-oscillation amplitude, integrated field, and current amplitude. This technique is a good example for solving partial differential equations, or more precisely, boundary value problems in one and two dimensions. In particular, the field in the aperture of accelerator magnets is governed by the Laplace equation, which leads to a boundary-value problem that is solved by determining the coefficients in the series of eigenfunctions from measurements of the field components or wire-oscillation amplitudes on the domain boundary. The oscillation of the taut string is an example of a one-dimensional, in-homogenous wave equation. The metrological characterization of the oscillating-wire system yield the feedback on the uncertainties (and limitations) of the method, as only the linearized equations of the wire motion and the integrated field harmonics of the magnet are considered.
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