The equation of state and symmetry energy of low density nuclear matter

The symmetry energy of nuclear matter is a fundamental ingredient in the investigation of exotic nuclei, heavy-ion collisions and astrophysical phenomena. A recently developed quantum statistical (QS) approach that takes the formation of clusters into account predicts low density symmetry energies far above the usually quoted mean field limits. A consistent description of the symmetry energy has been developed that joins the correct low-density limit with values calculated from quasi-particle approaches valid near the saturation density. The results are confronted with experimental values for free symmetry energies and internal symmetry energies, determined at sub-saturation densities and temperatures below 10 MeV using data from heavy-ion collisions. There is very good agreement between the experimental symmetry energy values and those calculated in the QS approachComment: 16 pages, 10 figures. arXiv admin note: text overlap with arXiv:0908.234

Closed Analytical Expression for the Electric Field Profile in a Loaded RF Structure with Arbitrarily Varying vgv_{g} and R′/QR'/Q

The design of a detuned and damped accelerating structure implies variations in the geometry which induce in turn a variation of the group velocity v^g and of the impedance per unit length R', divided by the quality factor Q. The resulting differential equation for the longitudinal electric field (fundamental mode) contains coefficients that depend on the distance z along the structure. This report describes a possible method to solve this nonlinear, first order differential equation analytically and how to obtain approximate closed algebraic forms, by using the sequence of Gauss integration methods. Analytical expressions of the longitudinal field profile in a loaded or unloaded accelerating section is deduced for both linear and arbitrary variations of v^g and R'/Q = Q. Simple relations between the average field and the field at the entrance of the structure E(0) make it possible to provide the dependence of the field function E(z) on the design value for ? and on the structure parameters. The results are in good agreement with the direct numerical integration. Applications are presented for particular structure designs

Multibunch BNS Damping and Wakefield Attenuation in High Frequency Linacs

In high frequency linacs, where the wakefields are strong, the stability of a train of bunches is critical. It was therefore important for the Compact Linear Collider study (CLIC) to investigate numerically and theoretically this question. Basically, two methods of controlling beam break up have been considered; firstly a multibunch generalization of the BNS damping principle and secondly the attenuation of the long-range fields as it results from damping or staggered tuning of the accelerating sections. Simulation codes have been written for both checking the theoretical predictions and investigating the requirements associated with a possible application to the CLIC main linac

New theory of single bunch stability in a linac with quadrupole displacements

The analytical treatment previously described by Guingard and Hagel (1998) has been extended to include the important effect of magnetic quadrupole transverse displacements, the chromatic variation of the magnetic focusing, the energy spread along the bunch and possible microwave quadrupoles, the last two in relation to BNS damping The analytical treatment previously described has been extended to include the important effect of magnetic quadrupole transverse displacements, the chromatic variation of the magnetic focusing, the energy spread along the bunch and possible microwave quadrupoles, the last two in relation to BNS damping. Both, the longitudinal and transverse equations of motion are solved, the second by using the perturbation method with partial expansions developed for this theory. The localized nature of the quadrupole displacements is preserved by using thin lenses and the super-position principle for the kick effects. The causality principle applied to the downstream beam oscillations due to the kicks is introduced via Heaviside functions. The treatment presented provides formulae for the tuneshift in the bunch and first-order solutions for the transverse beam off-sets within the bunch. It presents a break-through in the recent efforts to solve the problem of the bunch stability theoretically, with realistic beam and linac models

Analytical Treatment of Single Bunch Transverse Dynamics in Linacs with Wakefields

In this paper we present an analytical treatment of the equation of motion of single bunch particles traveling in the linac of a linear collider, in the presence of wakefields. Using a somewhat simplified model, the equation is solved for a Gaussian distribution of charge, a linear variation of the wakefield along the bunch, a smooth focusing and in the absence of acceleration. The specificity of the method consists of preventing the appearance of artificial secular terms and keeping at any stage the intrinsic tune-shift that characterizes the problem and stabilizes the motion. Hence, a first order perturbation becomes sufficient provided that a non-standard perturbation expansion, specially developed for this analysis, be however applied. Solutions for the particle off-sets within the bunch are obtained and their contributions to the effective emittance calculated. The results explain the observed features of beam breakup and BNS damping, and reproduce the bunch dipole-oscillations visible on the animated graphics after simulations with the code MUSTAFA. This treatment provides in addition a closed expression for the tune shift along the bunch and confirms the existence of an optimum BNS damping setting which differs from autophasing in a single bunch mode

Analytical study of the conjecture rule for the combination of multipole effects in LHC

This paper summarizes the analytical investigation done on the conjecture law found by tracking for the effect on the dynamic aperture of the combination of two multipoles of various order. A one-dimensional model leading to an integrable system has been used to find closed formulae for the dynamic aperture associated with a fully distributed multipole. The combination has then been studied and the resulting expression compared with the assumed conjecture law. For integrated multipoles small with respect to the focusing strength, the conjecture appears to hold, though with an exponent different from the one expected by crude reasoning

MUSTAFA: a tool for numerical simulations of the beam behavior in a linac

MUSTAFA stands for MUltibunch Simulation and Tracking Algorithm for Future linear Accelerators and provides an environment for tracking a multibunch beam accelerated in the linac of a collider. It is made of three parts: i) an interactive utility MBUNCH which allows to run the simulations, edit the input data, systematically vary some parameters, launch statistics on misalignment realisations and select the outputs, ii) the tracking program itself MBTR that includes magnetic focusing, acceleration, wakefields, misalignments and trajectory correction, and iii) a graphcs facility called MOVIE th at provides an animated representation of the transverse motion within each bunch of a train, either in the geometric space or in the phase space. While MBTR is written in FORTRAN, the other two codes are written in QUICKBASIC (MS V4.5). They run on PC Pentium computers in a stand-alone mode, independent of any network. Hence MUSTAFA is easily transportable and offers a unique graphics facility. It has been extensively used for multibunch simulations in the Compact Linear Collider scheme (CLIC) as well as for demonstrations of the beam behaviour in different stability or beam break-up condition

Beam Characteristics Versus Cavity Models in CLIC

The luminosity requested for linear colliders with center-of-mass energies exceeding 500 GeV, implies a multibunch operation be considered in order to limit the RF power consumption. In the Compact Linear Collider (CLIC), though the repetition rate is high, a train of at least 20 bunches is necessary to obtain the performance needed for the experiments. Since at high RF frequency the wakefields are large, beam break-up is critical and stability simulations have been carried out for different pre-estimated models of detuned and/or damped cavities. The results obtained give indications about the wakefield level that should not be exceeded along the train, to avoid significant emittance growth. They also show the sensitivity to some specific parameters and the dependence on the scaling of focusing with energy. Eventually, they are used as guide lines for accelerating structure development and as a basis for a possible set of CLIC parameters

Theory of single bunch stability and dynamics in linacs with strong wakefields and misalignments

The basic method we propose in order to solve analytically the equation of motion of a relativistic single-bunch travelling in a linac, in the presence of wakefields, has been summarised in a preceding report [1]. The extended treatment presented here includes the quadrupole transverse displacements, the chromatic variation of the magnetic focusing, the energy spread along the bunch and possible microwave quadrupoles. It deals with a Gaussian distribution of charge, linear variation of the wakefields within the bunch and smooth focusing. The energy is assumed to be constant in linac sectors, but increases from one sector to the next to simulate acceleration. The longitudinal and transverse equations of motion are solved,the second by using the perturbation method with partial expansions developed for this theory. The localised nature of the misalignment kicks and their superposition property are preserved by using thin lenses. The causality of the downstream oscillations due to these kicks is introduced via Heaviside functions. These ideas make it possible to build an analytical model for quadrupole misalignments and correlated displacements due to trajectory corrections. The resulting theory provides algebraic expressions for BNS damping, tune shifts, transverse off-sets and emittance dilution. It represents a significant break-through complementing the simulations and reproducing the oscillations observed numerically