Longitudinal beam dynamics with phase slip in race-track microtrons

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Abstract

Implementation of low-energy injection schemes in race-track microtron (RTM) designs requires a better understanding of the longitudinal beam dynamics. Unlike the high-energy case a low-energy beam slips in phase with respect to the accelerating field phase so that the standard notion of synchronous particle is not applicable. In the article, we generalize the concept of synchronous particle for the case of non-relativistic energies. An analytic approach for the description of the synchronous phase slip is developed and explicit, though approximate, formulas which allow to determine the equilibrium injection phase and to fix the parameters of the accelerator are derived. The approximation can be improved in a systematic way by calculating higher-order corrections. The precision of the analytic approach is checked by direct numerical computations and is shown to be quite satisfactory. Explicit examples of injection schemes and fixing of RTM global parameters are presented. We also address the issue of stability of synchrotron oscillations around the generalized synchronous trajectory and introduce the notion of critical energy.

Introduction

Race-track microtron (RTM) is a specific type of electron accelerator combining properties of linear accelerator (LINAC) and circular machine [1], [2]. For applications in which a modest beam power at a relatively high beam energy is required this type of particle accelerators allows to get pulsed and continuous beams in the most cost and energy effective way with the most optimal dimensions of the machine. Main features of the longitudinal beam dynamics in RTMs, in particular small width of the region of stable phase oscillations and nonlinear resonances, are essentially the same as in case of the classical microtron [3], [4], [5], [6]. However, the existence of a drift space between the bending magnets and effect of the fringe field in these magnets lead to an energy-dependent phase slip. This phenomenon affects the width of the machine acceptance and complicates the choice of initial beam parameters such as the injection energy and the injection phase. Till now the main approach to fixing these parameters is by means of numerical simulations of the phase motion using special codes [7], [8]. One of the main difficulties in developing effective analytic techniques of RTM beam dynamics resides in non-applicability of standard perturbation methods, successfully used in beam dynamics calculations in synchrotrons and storage rings, due to a large energy gain per turn and a low number of turns of the beam in RTMs.

Recent intensive developments of accelerator technologies open new possibilities of applications of RTMs for cargo inspection of containers with detection of elemental composition of their content, production of short-life isotopes, some types of radiation therapy, etc. In designing of an RTM for each particular application one has to optimize machine parameters, in particular the RF wavelength, synchronous phase, energy gain per turn, harmonic number increase, etc. in order to achieve the most effective operation of the accelerator. For this, one needs better understanding of beam dynamics in such machines, therefore the development of analytic methods of RTM beam dynamics becomes an actual and important problem of the physics of this type of accelerators. Such methods would allow to analyze and to control analytically the dependence of machine characteristics on the choice of its main parameters, and also to give an initial approximation of the values of these parameters for their further more precise determination by numerical simulations.

In the RTM design certain parameters of the machine (magnetic field in the end-magnets, energy gain in the accelerating structure (AS), length of the drift space, etc.) are adjusted in such a way that the condition of resonance acceleration is fulfilled for a reference synchronous particle with the relativistic factor β=1, and as a result this particle enters the AS at the fixed synchronous phase ϕs. However, in many designs the electrons after the injection are not ultra-relativistic and have β<1, so that at the first orbits the resonance condition cannot be fulfilled and the beam phase at the AS slips, i.e. changes from turn to turn. As it is shown below the phase slip over an orbit is essentially(2π)×2lλ(1β¯-1)where l is the distance between the end magnets, λ is the free space wavelength of the accelerating field and β¯ is an average value of the relativistic factor over the orbit. Thus, the effect is not negligible and should be taken into account in RTMs with (a) low-energy injection; (b) long distance between the end magnets, and (c) short wavelength. There are a few factors in RTM designs which may require long distance between the end magnets, one of them is the use of an AS with a low gradient, a well-known example is the normal conducting continuous wave LINAC with an accelerating gradient ∼1 MeV/m of the MAMI facility [9]. As an example of design with non-negligible phase-slip effect due to a short wavelength, we can mention a compact 12 MeV RTM with λ≈5 cm proposed in Ref. [10]. An approach to reduce the phase-slip effect, described in Ref. [11], is to reflect the beam in the end magnet with the fringe field of special profile in the longitudinal direction [12] and accelerate it in the AS in the opposite direction, thus increasing the energy at the beginning of the beam recirculation. This method works well in case of pulsed RTMs with λ≈10 cm, solving at the same time the problem of the LINAC bypass, but not in the above-mentioned examples.

Till now all analytical estimates of the main RTM parameters are obtained using the standard approach based on a simple formalism (see, for example, Ref. [2] and also Section 2.1 of the present article) which assumes that the particles of the beam move in the drift space at the speed of light and which does not take into account the fringe-field effect in the end magnets. As we argued above, for many RTMs such approximation does not give satisfactory description of their beam physics.

In the present paper, we develop an analytic method of description of the phase motion in RTMs taking into account the phase slip in the drift space between the end magnets (but in the absence of the fringe field). This method allows to determine analytically, though approximately, a synchronous phase trajectory and main parameters of the beam. The obtained results are compared with results of numerical simulations. In our analysis, we will assume a zero-length acceleration gap and consider the stationary regime with constant amplitude of the accelerating field in the AS resonant cavities.

The article is organized as follows. In Section 2, we describe the RTM model, introduce the notion of non-relativistic synchronous particle, develop the formalism and present a solution of the equations of phase motion taking into account the phase-slip effect. In Section 3, we introduce the notion of critical energy and discuss the stability of synchrotron oscillations around the synchronous particle trajectory. In Section 4, the analytic description of the synchronous particle is applied to the calculation of the injection phase in RTMs and tuning in resonance (adjusting the drift space length in this case). Also a method for numerical determination of the injection phase is described and a comparison between the analytic and numerical results is given. Section 5 contains some concluding remarks.

Section snippets

Equilibrium phase

Let us consider the longitudinal phase motion of electrons in an electron RTM with the magnetic field induction B in the end magnets, separation l between the magnets (straight section length), and the maximum energy gain ΔEmax in the AS or LINAC (see Fig. 1). In this article, we neglect effects of the magnet fringe field on the phase motion and model the LINAC by an infinitely thin accelerating gap, i.e. we suppose that the energy increase ΔE of a particle passing through the LINAC is

Critical energy and stability of oscillations

In the previous section, we found the general synchronous trajectory as a deviation from the ultrarelativistic (asymptotic) synchronous one given by formulas (8), (12). The longitudinal phase oscillations, Eqs. (34), (36), were also described with respect to the same reference trajectory, the condition of their stability being given by inequalities (35). Such description is practical if the deviations ψn and wn are small, i.e. the expansion parameter εn is small enough, which means that the

Calculation of the synchronous particle injection phase and RTM parameters

Let us describe an analytic, though approximate, method for determining the initial conditions for the synchronous trajectory corresponding to an asymptotic synchronous phase ϕs and asymptotic energy En,s.

Suppose that the geometrical and physical RTM parameters λ, μ, ν, l, ΔEmax, ϕs are fixed. One has to check that for a given asymptotic energy En,s at the nth orbit they are consistent with relationEn,s=ΔEsν(μ+(n-1)ν-2lλ)following from Eqs. (12), (13). In this case, the particle phase dynamics

Summary and discussion

We developed an analytic approach to the analysis of the beam phase motion in RTMs which takes into account the phase-slip effect in the drift space. We introduced the notion of non-relativistic synchronous particle and found its phase trajectory analytically though approximately (see Eqs. (41), (42)). As the energy grows, this synchronous trajectory approaches monotonously the ultra-relativistic synchronous trajectory known in the literature. The synchronous trajectory corresponds to the

Acknowledgements

The authors would like to thank B.S. Ishkhanov for his interest in the work and valuable comments and Juan Pablo Rigla for checking some calculations. The work was supported by Grant 08-02-00273-a of the Russian Fund for Basic Research and Grants PCI2005-A7-0284 and FIS2006-07016 of the Spanish Ministry of Science and Education.

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Cited by (3)

  • Stability of the phase motion in race-track microtrons

    2017, Physica D: Nonlinear Phenomena
    Citation Excerpt :

    Once the beam gets the final design energy, it is deflected by an extraction magnet and is directed towards the accelerator beam exit. We end this section by stressing that the model of longitudinal oscillations around the synchronous trajectory for non-ultra-relativistic beams is more complicated [14]. Fig. 2 shows the basic skeleton from which an expert reader can deduce the main dynamical changes.

1

On leave of absence from the Institute of Nuclear Physics, Moscow State University, 119992 Moscow, Russia.

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