Calculation method of beam dynamics in multi-gap accelerating cavity

Abstract

The invention relates to a beam dynamics calculation method for a multi-gap accelerating cavity, comprising: dividing the longitudinal phase space at the entrance of a current element into a plurality of grids, and providing a transverse beam matrix corresponding to each grid; calculating the corresponding position of each grid in the longitudinal phase space at the exit of the current element according to the position of each grid in the longitudinal phase space at the entrance of the current element; calculating the transverse transmission matrix of the current element for the grid according to the position of each grid in the longitudinal phase space at the entrance and exit of the current element, and then calculating the transverse beam matrix of the grid at the exit of the current element according to the transverse transmission matrix and the transverse beam matrix of the grid at the entrance of the current element; repeating the above steps to determine the transverse beam matrix of each grid at the exit of a downstream element, and finally calculating the total transverse beam matrix according to the transverse beam matrices of each grid at the exit of the nth element.

Description

Technical Field

[0001] The invention relates to a beam dynamics calculation method for a multi-gap accelerating cavity. Background Art

[0002] Common multi-gap accelerating cavities include Alvarez drift tube linear accelerators, interdigital (IH: interdigital-H) drift tube linear accelerators, crossbar (CH: crossbar-H) drift tube linear accelerators, etc. These accelerators can accelerate proton or heavy ion beams and are commonly used in the field of proton and heavy ion accelerators. They are often used as injectors for synchrotron accelerators. The main working principle of the multi-gap accelerating cavity is to use the longitudinal electric field near the axis of the cavity to accelerate the beam.

[0003] The drift tube linear accelerator is composed of two or more hollow cylindrical drift tube electrodes arranged in a cylindrical resonant cavity along the beam forward direction. High-frequency power is provided in the cylindrical resonant cavity, and the high-frequency electric field generated between the drift tube electrodes is used to accelerate the charged particles (for example, protons or carbon ions, etc.) along the beam forward direction. The drift tube electrodes are designed so that when the direction of the high-frequency electric field is opposite to the forward direction of the positive ion beam, the charged particles move exactly within the drift tube electrodes to avoid being decelerated by the electric field.

[0004] The main purpose of beam dynamics calculation in multi-gap accelerating cavity is to calculate the beam envelope, beam emittance and other parameters in the cavity. The accuracy of the calculation results is related to the quality of the design results, and the speed of the calculation determines the speed of the optimization design.

[0005] Phase space is a multidimensional space represented by generalized coordinates and generalized momentum. The longitudinal phase space refers to the phase-energy phase space of the beam motion direction at the Z position, that is, Phase space, transverse phase space refers to the phase space in the horizontal x direction and the vertical y direction perpendicular to the beam motion direction at the Z position, which is the position-momentum phase space in the horizontal and vertical directions, that is, the x-x' and y-y' phase spaces. In the beam dynamics calculation, each accelerating gap, focusing and defocusing magnet, solenoid and drift segment of the multi-gap accelerating cavity are regarded as different "elements".

[0006] At present, the mainstream beam dynamics calculation methods for multi-gap accelerating cavities in the world are transmission matrix calculation, single particle simulation calculation and multi-particle simulation calculation.

[0007] The transfer matrix is ​​calculated by the element entrance beam matrix Σ and the transfer matrix R, through Σ'=RΣR TCalculate the beam matrix at the exit of the computing element. Its calculation speed is fast, but due to the nonlinear effect in the acceleration gap and the different transverse forces acting on particles at different longitudinal phase space positions, the calculation of the transfer matrix cannot well calculate the change of beam emittance in the acceleration cavity.

[0008] Single-particle simulation calculation represents a single particle by its horizontal and vertical positions, momenta, phases, and energy dispersions at the entrance position of the element in the beam movement direction. The horizontal and vertical positions, momenta, phases, and energy dispersions of the particle at the exit position of the element in the beam movement direction are calculated through the dynamic equation. This method can only represent the movement trajectory of a single particle and cannot represent the dynamic situation of the entire beam.

[0009] Multi-particle simulation calculation represents the beam by multiple single particles. By calculating the movement trajectories of each single particle, the movement trajectory of the entire beam is obtained, and dynamic results such as emittance are analyzed and calculated for the entire beam. Generally, when the number of single particles is greater than 10,000, the calculation is relatively accurate, but it takes a long time for the optimization design.

[0010] Among them, the longitudinal phase space refers to the phase-energy phase space in the beam movement direction z, that is, the P-W phase space, and the transverse phase space refers to the horizontal x and vertical y phase spaces perpendicular to the beam movement direction, which are the position-momentum phase spaces in the horizontal and vertical directions, that is, the x-x' and y-y' phase spaces. In beam dynamics calculation, each acceleration gap, focusing and defocusing magnet, and drift tube of the multi-gap acceleration cavity are regarded as different "elements".

[0011] To achieve the rapidity and accuracy of the beam dynamics calculation results, it is necessary to improve the existing beam dynamics calculation method and find a more rapid and accurate beam dynamics calculation method for the multi-gap acceleration cavity. Summary of the Invention

[0012] To solve the above technical problems, the present invention provides a beam dynamics calculation method for a multi-gap acceleration cavity. Using each acceleration gap, quadrupole magnet, solenoid, and drift section of the multi-gap acceleration cavity as the elements of the multi-gap acceleration cavity, this beam dynamics calculation method is used to calculate the beam particle distribution at the exit of the nth element based on the known beam particle distribution at the entrance of the pth element, where 1 ≤ p < n ≤ N, and N is the total number of elements. The characteristics of this beam dynamics calculation method include the following steps:

[0013] Step S0: Use the pth element as the current element;

[0014] Step S1: Divide the longitudinal phase space at the entrance of the current element into several grids and give the corresponding transverse beam matrices for each grid;

[0015] Step S2: according to the position of each grid at the entrance of the current element in the longitudinal phase space, respectively calculating the position of the grid at the exit of the current element in the longitudinal phase space;

[0016] Step S3: when the current element is not a drift section, according to the corresponding positions of each grid at the entrance and exit of the current element in the longitudinal phase space, calculate the modified transverse transmission matrix of the current element for the grid, and then calculate the transverse beam matrix of the grid at the exit of the current element according to the modified transverse transmission matrix and the transverse beam matrix of the grid at the entrance of the current element; when the current element is a drift section, use the transverse beam matrix of the grid at the entrance of the current element as the transverse beam matrix of the grid at the exit of the current element;

[0017] Step S4: Determine whether the current element is the nth element. If the determination result is no, use the transverse beam matrix at the outlet of the current element as the transverse beam matrix at the inlet of the downstream element adjacent to the current element, and then repeat steps S1 to S3 with the downstream element as the current element to determine the transverse beam matrix of each grid at the outlet of the downstream element; if the determination result is yes, calculate the total transverse beam matrix based on the transverse beam matrices of each grid at the outlet of the current element.

[0018] The beam envelope and beam emittance of the multi-gap accelerating cavity can be determined according to the total transverse beam matrix calculated in step S4.

[0019] Different from the calculation method that simply uses the transmission matrix, the beam dynamics calculation method of the multi-gap accelerating cavity of the present invention first divides the grid in the longitudinal phase space, and then uses the calculation method of the transmission matrix to calculate the transverse beam matrix at the exit of each element corresponding to different grids, and finally combines the transverse beam matrices of each grid to calculate the total transverse beam matrix, which can give the change of emittance and envelope. This beam dynamics calculation method based on the grid model can ensure the calculation speed, and by using the grid-by-grid calculation method in the longitudinal phase space, it can reduce the longitudinal nonlinearity and improve the accuracy of the calculation.

[0020] The beam dynamics calculation method of the multi-gap accelerating cavity according to the present invention calculates the beam particle distribution at the outlet of the nth element according to the known beam particle distribution at the inlet of the pth element. When p=1 and n=N, that is, the beam particle distribution at the outlet of the multi-gap accelerating cavity is calculated based on the known beam particle distribution at the inlet of the multi-gap accelerating cavity.

[0021] This method uses the transfer matrix calculation method in the calculation of the transverse beam dynamics of each grid, but divides the beam distribution into grids in the longitudinal phase space, so that the calculation results are close to those of the multi-particle simulation calculation. In the calculation of the position of each grid in the longitudinal phase space, it is equivalent to using a single particle to characterize the position of the grid in the longitudinal phase space. The single particle simulation calculation method is used to calculate the motion trajectory of the element entrance and element exit, but in the transverse phase space, the transfer matrix calculation method is used. Figuratively speaking, it is equivalent to the grid at the entrance "penetrating" these elements through the transmission matrix corresponding to each element to transform to the exit of the last element. The number of grids divided in the longitudinal phase space by this method can be less than 100, which is much less than the number of single particles required for multi-particle simulation calculation. Therefore, this method combines the characteristics of the fast calculation speed of the transfer matrix calculation method and the single particle simulation calculation, and can achieve the accuracy close to that of the multi-particle simulation calculation under the premise of fast calculation speed.

[0022] The multi-gap accelerating cavity is preferably an Alvarez drift tube linear accelerator ( Figure 1 ) or interdigital drift tube linear accelerator ( Figure 2 In the figure, reference numeral 1 is a vacuum radio frequency resonant cavity, 2 is an accelerating gap, 3 is a drift tube, 4 is a support rod, and 5 is a ridge structure. For simplicity, a quadrupole magnet is not shown.

[0023] The step size of the grid used to divide the longitudinal phase space at the entrance of each element can be selected arbitrarily. According to an implementation form of the beam dynamics calculation method of the multi-gap acceleration cavity of the present invention, for the p-th element to the n-th element, in step S1, the longitudinal phase space is divided into grids according to a uniform fixed step size between the maximum and minimum values ​​of the phase and energy. However, it is obviously also conceivable that, for one or some of the elements, the longitudinal phase space is divided into grids according to a non-uniform step size between the maximum and minimum values ​​of the phase and energy. Alternatively, the longitudinal phase space is divided into grids according to a uniform step size between the maximum and minimum values ​​of the phase and energy, but for each element, the step size of the grid division is not fixed but different. The change in step size selection makes the longitudinal phase space have a more subdivided grid in the key area at the entrance or exit of the key element compared to other areas. Although it will appropriately increase the complexity of the calculation, it is desirable in some cases.

[0024] Preferably, in step 2, the position of the grid at the outlet of the current element in the longitudinal phase space is calculated according to the following formula:

[0025] (1) When the element is an accelerating gap,

[0026]

[0027] (2) When the element is a solenoid, a drift segment, a focusing quadrupole magnet or a defocusing quadrupole magnet,

[0028]

[0029] W2=W1

[0030] Among them, the effective accelerating voltage of the accelerating gap is V, the charge of the accelerated particle is q, the element length is s, and the position of the grid at the entrance of the current element in the longitudinal phase space is The position of the grid at the outlet of the current element in the longitudinal phase space is The relativistic speed of the accelerated particle at the grid (i, j) at the entrance of the current element is The wavelength in the cavity λ=c / f, f is the cavity frequency, and m is 1 for the Alvarez drift tube linear accelerator and 0.5 for the interdigital drift tube linear accelerator.

[0031] Preferably, when performing step 3:

[0032] (1) When the element is an accelerating gap, the modified lateral transfer matrix is ​​calculated according to the following formula:

[0033]

[0034] (2) When the element is a focusing quadrupole magnet, the modified transverse transmission matrix is ​​calculated according to the following formula:

[0035]

[0036] (3) When the element is a defocused quadrupole magnet, the modified transverse transfer matrix is ​​calculated according to the following formula:

[0037]

[0038] Where M is the mass of the accelerated particle, q is the charge of the accelerated particle, c is the speed of light in a vacuum, V is the effective accelerating voltage of the accelerating gap, the magnet gradient is G, the element length is s,

[0039] Among them, the position of the grid of the inlet of the current element (n≥1) in the longitudinal phase space is The corresponding Lorentz factor is The relativistic speed of the particle is At the exit of the current element, the position of the grid in the longitudinal phase space is The corresponding Lorentz factor is The relativistic speed of the particle is

[0040] in, Among them, β0=(β1+β2) / 2, γ0=(γ1+γ2) / 2, and the wavelength in the cavity is λ=c / f, where f is the cavity frequency. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] The embodiments of the present invention are explained below in conjunction with the accompanying drawings. In the accompanying drawings:

[0042] Figure 1 It is a schematic diagram of the multi-gap accelerating chamber of the Alvarez type drift tube linear accelerator;

[0043] Figure 2 It is a schematic diagram of the multi-gap accelerating cavity of the interdigital drift tube linear accelerator;

[0044] Figure 3 is a schematic diagram of the grid division of the longitudinal phase space, where the solid line is the ellipse of the beam distribution in the longitudinal phase space, the dotted line is the divided grid, and ρ i,j is the particle density of the grid;

[0045] Figure 4 A flow chart of a beam dynamics calculation method for a multi-gap accelerating cavity is schematically shown;

[0046] Figure 5 The comparison of the emittance between the present calculation method and the transfer matrix calculation and the multi-particle simulation calculation is shown;

[0047] Figure 6 The phase space comparison between the results of this calculation method and multi-particle simulation calculation is shown. DETAILED DESCRIPTION

[0048] Figure 4 The flowchart of the beam dynamics calculation method of the multi-gap accelerating cavity is schematically shown. In the calculation, the entrance of the accelerator is selected as the starting position of the calculation, and the exit of the accelerator is selected as the end position of the calculation, so the accelerated particles pass through N elements in the accelerator.

[0049] First, the beam particle distribution at the accelerator entrance (position 0) is determined, and the longitudinal phase space (i.e., The phase space is divided into several grids (i, j) in a uniform and fixed step between the maximum and minimum values ​​of phase and energy. The transverse beam matrix corresponding to different grids is given according to the beam particle distribution at the accelerator entrance. To reflect the influence of different longitudinal positions on the transverse beam dynamics. Figure 3 is a schematic diagram of the grid division of the longitudinal phase space, where the solid line is the ellipse of the beam distribution in the longitudinal phase space, the dotted line is the divided grid, and ρ i,j is the particle density of the grid.

[0050] The transverse beam matrix corresponding to the grid (i, j) at position 0 is:

[0051]

[0052] Secondly, according to the corresponding positions of each grid (i, j) at the entrance of the current element in the longitudinal phase space, the corresponding positions of these grids at the exit of the current element in the longitudinal phase space are calculated.

[0053] The position of the grid (i, j) at the entrance of the current element in the longitudinal phase space is Its Lorentz factor is The relativistic speed is The position of the grid in the vertical space at the exit of the current element is

[0054] If the element is an accelerating gap, the element length is 0, and the effective accelerating voltage of the accelerating gap is V, then

[0055]

[0056] If the element is a drift segment, a quadrupole magnet, etc., and the element length is s, then

[0057]

[0058] W2=W1

[0059] Then, according to the position of each grid (i, j) at the entrance and exit of the current element in the longitudinal phase space, the modified lateral transmission matrix of the current element for the grid (i, j) is calculated: That is, the modified transverse transfer matrix corresponding to the beam of the grid is given, and the transverse forces acting on the grids at different longitudinal phase space positions are calculated respectively.

[0060] For different components, the correction method is different:

[0061] If the element is an accelerating gap, the element length is 0, the effective accelerating voltage of the accelerating gap is V, and the modified transverse transfer matrix is

[0062]

[0063] Among them, β0=(β1+β2) / 2, γ0=(γ1+γ2) / 2,

[0064] It is the result of correction based on the position of the vertical grid.

[0065] If the element is a focusing quadrupole magnet, the magnet gradient is G, the element length is s, and the modified transfer matrix is

[0066]

[0067] If the element is a defocused quadrupole magnet, the modified transfer matrix is

[0068]

[0069] in, It is the result of correction based on the position of the vertical grid.

[0070] If the element is a drift segment, the lateral transfer matrix does not need to be corrected.

[0071] Then, according to the modified transverse transmission matrix and the transverse beam matrix of each grid (i, j) at the entrance of the current element The respective transverse beam matrices of these grids at the exit of the current element are calculated according to the following formula:

[0072]

[0073] Thus, the lateral beam matrix of each grid (i, j) at the exit of the current element is obtained: If the current element is not the last element to be calculated, the transverse beam matrix at the exit of the current element is used as the transverse beam matrix at the entrance of the next element, and the above steps are repeated with the next element as the current element until the current element is the last element. At this time, the total transverse beam matrix can be calculated based on the transverse transmission matrices of different grids, so that the beam envelope, emittance and other parameters can be given.

[0074] That is, for the longitudinal phase space grid (i, j) divided at position 0, the transverse beam matrix at the exit is calculated element by element until the transverse beam matrix of different grids is calculated at the exit of the last element. The superposition forms an approximate lateral beam distribution Σ n , to obtain the final beam dynamics results.

[0075]

[0076] in, p and q represent x, x′, y, y′.

[0077] Figure 5The comparison of the emittance results between the present calculation method, the transfer matrix calculation, and the multi-particle simulation calculation is shown. The accelerating cavity is 4 meters long and the number of elements is 170. Among them, the solid line is the result calculated by the present calculation method using 16 grids, the dotted line is the result of the transfer matrix calculation, and the dashed line is the result of the multi-particle simulation calculation (using 100,000 single particles). The present calculation method takes 0.16 s, the transfer matrix calculation takes 0.1 s, and the multi-particle simulation calculation takes 17 s.

[0078] Figure 6 The phase space comparison between the present calculation method and the multi-particle simulation calculation results is given. Figure 6 The phase space comparison between the present calculation method and the multi-particle simulation calculation results is shown. Each ellipse of different line types respectively represents the result obtained by the transfer matrix of a grid passing through each element, and they jointly enclose an envelope graph that is very approximate to the result of the multi-particle simulation calculation.

[0079] Although only the application situation of the method of the present invention for solving the beam current distribution at the exit of the multi-gap accelerating cavity when calculating the beam current distribution at the entrance of the known multi-gap accelerating cavity is given in the above text, it is not limited thereto. As described in the summary of the invention, it can start the calculation at the entrance of any element (the p-th element) of the multi-gap accelerating cavity as long as the beam current distribution condition at the entrance of this element is known; it can be calculated element by element downstream until the beam current distribution condition at the exit of another arbitrary element (the n-th element) of the multi-gap accelerating cavity is obtained, where 1 ≤ p < n ≤ N, and N is the total number of elements of the multi-gap accelerating cavity.

[0080] Although the embodiments of the present invention have been shown and described, those of ordinary skill in the art can understand that various changes, modifications, substitutions, variations, and any combinations can be made to these embodiments without departing from the principles and purposes of the present invention. The scope of the present invention is defined by the claims and their equivalents.

[0081] List of reference numerals

[0082] 1 Vacuum radio frequency resonator

[0083] 2 Accelerating gap

[0084] 3 Drift tube

[0085] 4 Support rod

[0086] 5 Ridge structure

Claims

1. A beam dynamics calculation method for a multi-gap accelerating cavity, wherein each accelerating gap, quadrupole magnet, solenoid and drift segment of the multi-gap accelerating cavity are used as elements of the multi-gap accelerating cavity, and the beam dynamics calculation method is used to calculate the beam particle distribution at the outlet of the nth element according to the known beam particle distribution at the entrance of the pth element, wherein: 1 ≤ p < n ≤ N, where N is the total number of the elements. It is characterized in that the beam dynamics calculation method includes the following steps: Step S0: Take the p-th element as the current element; Step S1: Divide the longitudinal phase space at the entrance of the current element into several grids, and give the transverse beam matrix corresponding to each grid; Step S2: Calculate the position of each grid in the longitudinal phase space at the exit of the current element according to the position of each grid in the longitudinal phase space at the entrance of the current element; Step S3: When the current element is not a drift section, calculate the corrected transverse transfer matrix of the current element for each grid according to the positions of each grid in the longitudinal phase space at the entrance and exit of the current element, and then calculate the transverse beam matrix of each grid at the exit of the current element according to the corrected transverse transfer matrix and the transverse beam matrix of each grid at the entrance of the current element; when the current element is a drift section, use the transverse beam matrix of each grid at the entrance of the current element as the transverse beam matrix of each grid at the exit of the current element; Step S4: Determine whether the current element is the n-th element. If the judgment result is no, use the transverse beam matrix at the exit of the current element as the transverse beam matrix at the entrance of the downstream element adjacent to the current element, and then use the downstream element as the current element to repeat steps S1 to S3 to determine the transverse beam matrix of each grid at the exit of the downstream element; if the judgment result is yes, calculate the total transverse beam matrix according to the transverse beam matrices of each grid at the exit of the current element.

2. The beam dynamics calculation method of a multi-gap accelerating cavity according to claim 1, characterized in that: The multi-gap accelerating cavity is an Alvarez-type drift tube linear accelerator or an interdigital drift tube linear accelerator.

3. The beam dynamics calculation method of a multi-gap accelerating cavity according to claim 2, characterized in that: For the p-th element to the n-th element, in step S1, divide the longitudinal phase space into the grids at a uniform fixed step between the maximum and minimum values of the phase and energy.

4. The beam dynamics calculation method of a multi-gap accelerating cavity according to claim 2, characterized in that: For the p-th element to the n-th element, in step S1, divide the longitudinal phase space into the grids at a non-uniform or non-fixed step between the maximum and minimum values of the phase and energy, so that the longitudinal phase space has more subdivided grids in the key areas at the entrance or exit of the key elements than in other areas.

5. The beam dynamics calculation method of a multi-gap accelerating cavity according to claim 3 or 4, characterized in that: Calculate the position of the grid in the longitudinal phase space at the exit of the current element according to the following formula in step S2: (1) When the element is an accelerating gap, (2) When the element is a solenoid, a drift section, a focusing quadrupole magnet or a defocusing quadrupole magnet, Among them, the effective accelerating voltage of the accelerating gap is V, the charge of the accelerated particle is q, the element length is s, and the position of the grid at the entrance of the current element in the longitudinal phase space is The position of the grid at the outlet of the current element in the longitudinal phase space is The relativistic speed of the accelerated particle at the grid (i, j) at the entrance of the current element is The wavelength in the cavity λ = c / f, f is the cavity frequency, m is 1 for the Alvarez drift tube linear accelerator and 0.5 for the interdigital drift tube linear accelerator, where γ1 is the corresponding Lorentz factor and satisfies Where M is the mass of the accelerating particle and c is the speed of light in a vacuum.

6. The beam dynamics calculation method of a multi-gap accelerating cavity according to claim 3 or 4, characterized in that: When performing step S3, (1) When the element is an accelerating gap, calculate the corrected transverse transfer matrix according to the following formula: (2) When the element is a focusing quadrupole magnet, calculate the corrected transverse transfer matrix according to the following formula: (3) When the element is a defocusing quadrupole magnet, calculate the corrected transverse transfer matrix according to the following formula: where M is the mass of the accelerated particle, q is the charge of the accelerated particle, c is the speed of light in vacuum, V is the effective accelerating voltage of the accelerating gap, the magnet gradient is G, and the element length is s. Among them, the position of the grid at the entrance of the current element in the longitudinal phase space is The corresponding Lorentz factor is The relativistic speed of the particle is At the exit of the current element, the position of the grid in the longitudinal phase space is The corresponding Lorentz factor is The relativistic speed of the particle is in, Among them, β0=(β1+β2) / 2, γ0=(γ1+γ2) / 2, and the wavelength in the cavity is λ=c / f, where f is the cavity frequency.

7. The beam dynamics calculation method for a multi-gap accelerating cavity according to any one of claims 1 to 4, characterized in that: The beam envelope and beam emittance of the multi-gap accelerating cavity are determined according to the total transverse beam matrix calculated in step S4.

8. The beam dynamics calculation method for a multi-gap accelerating cavity according to any one of claims 1 to 4, characterized in that: The number of the grids is less than 100.

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