Search Results (16 total)

Using field matching techniques, closed form analytic expressions for the transverse impedance and for the shielding effectiveness of a smooth cylindrical beam pipe of arbitrary thickness are presented. In the limit of thick and thin beam pipes the well-known expressions are reproduced. The transverse transmission coefficient is compared with the longitudinal one that has been obtained in our previous work [A. M. Al-Khateeb, O. Boine-Frankenheim, R. W. Hasse, and I. Hofmann, Phys. Rev. E 71, 026501 (2005).]. The results are applied to the heavy ion synchrotron SIS 18 and to the planned SIS 100 at GSI. In both machines the stainless steel beam pipe in the dipole sections is much thinner than the skin depths at the revolution frequency and, therefore, the impedance value and the transmission are of concern.

The longitudinal coupling impedance of a cylindrical beam pipe for arbitrary relativistic γ₀ and mode frequency is obtained analytically for finite wall conductivity and finite wall thickness. Closed form expressions for the electromagnetic fields excited by a beam perturbation are derived analytically. General expressions for the resistive-wall impedance in the presence of a metallic shield and for the rf shielding effectiveness of the beam pipe have been obtained and then compared with approximate expressions. The results are applied to the GSI synchrotron SIS, where the thickness of the vacuum chamber in the dipole magnets is much smaller than the skin depth at injection energy.

We derive four rigorous conditions for the stability of Coulomb strings in circular storage rings. These criteria are well met by the existing data from experiments in SIS, ESR, and CRYring but not by the NAP-M experiment. We calculate the potential of the joint transverse zigzag excitation and the longitudinal motion against each other of a string of charged particles as a function of their amplitudes and with the linear density as parameter. This potential exhibits a saddle point in amplitude space which, if overcome, destroys the order of the string. The conditions of stability are derived from the position and height of the saddle point which are fairly independent of the linear density. Our findings confirm the supposition that only the Coulomb interaction in the immediate vicinity of very close encounters of particles is important for the existence of strings.

Longitudinal ballistic and collective beam echoes with diffusion effects are investigated theoretically. In the presence of the space-charge impedance, the collective echo amplitude is obtained as a closed form expression. In contrast to the ballistic case, the collective echo amplitude consists of one maximum at time t_echo. The echo amplitude grows up and damps down with a rate proportional to the Landau damping rate of space-charge waves. The effect of weak diffusion is found to modify the ballistic and the collective echo amplitudes in the same manner. This effect of diffusion was confirmed using a “noiseless,” grid-based simulation code. As a first application the amount of numerical diffusion in our simulation code was determined using the echo effect.

We explain that the anomalous frequency shifts of very close masses obtained in the high precision mass measurement experiments in the ESR storage ring result from the locking of Coulomb interacting strings of ions. Here two concentric strings which run horizontally close to each other are captured into a single string if their thermal clouds overlap and give up their identity.

We verify theoretically that the anomalous longitudinal temperature reduction of strongly electron cooled heavy ions in the Experimental Storage Ring of GSI at very low density is explained by the fact that there is no intrabeam scattering and that the particles by their small relative velocity and their Coulomb repulsion cannot pass each other anymore. At the achievable momentum spreads, Coulomb order is reached at particle distances of the order of centimeters. It is also shown that under the given experimental conditions in the proton NAP-M experiment of 1980 intrabeam heating counteracts Coulomb order.

With the help of molecular dynamics we study the properties of a string of ions that is confined by an external harmonic potential and that initially is cold in the beam direction but warm in the transverse one. Under the influence of the mutual Coulomb repulsion, intrabeam scattering leads to a flow of energy from the transverse to the longitudinal direction. Depending on the density and the initial transverse temperature T_⊥0, three regimes can be distinguished: a fast transient increase of the longitudinal temperature T_⊥0 in a fraction of a betatron period, a slow linear increase in some ten betatron periods, and a fast exponential increase followed by complete thermal equilibration. If T_⊥0 is large, linear order is lost and thermal equilibrium is postponed to large times. Fourier analyses show that the energy flows into the long-wavelength modes of the collective string excitations. A comparison of calculated heating rates with achievable cooling rates reveals that maintaining linear order in a storage ring is marginal.

With the help of molecular-dynamics computer simulations, we study the equilibrium configurations of systems of N=2–5000 strongly correlated charged particles under the influence of a radial harmonic external confining force and their mutual Coulomb forces. The temperature is well below the crystallization point; i.e., the ratio of Coulomb to kinetic energy is as large as Γ=10⁹. The particles arrange in concentric spherical shells with approximately constant intershell distances. On the surfaces plane hexagonal structures are well pronounced. The calculated radii, occupation numbers, and energies per particle are compared with results of classical geometrical and shell models with homogeneously charged shells corrected for hexagonal surface occupation. The closed-shell particle numbers also agree well with those of multilayer icosahedra. From the computer simulations we extract a Madelung (excess) energy of -0.8926, which is close to the theoretical value of the shell model corrected for plane hexagonal surfaces, -0.8923, but larger than the one of the infinite geometrical lattice, -0.8944, and of the bcc value of -0.8959. Surface-energy effects are positive and of the order of N^-1/3.

Excess energies are calculated for cylindrical Coulomb crystals at very low temperature and various densities as an approximation of a cooled ion beam in a storage ring or of cold trapped ions. Even for the large six-shell structures it is found that the plane hexagonal lattice on the surface is still dominating–with an excess (Madelung) energy of -0.892 28–over the expected infinite bcc structure. The results of molecular-dynamics computer simulations are compared with results of a classical shell model which takes into account the constant intershell distances and hexagonal surface occupation.

One-particle, one-hole and two-particle, two-hole level densities are calculated in the local-density, i.e., Thomas-Fermi, approach. For the harmonic oscillator this leads to analytical expressions which are in excellent agreement with the exact quantum results.

The long-mean-free path nuclear fluid dynamics is extended to include damping. First the damping stress is derived from the solution of the Boltzmann equation for a breathing spherical container filled with a Fermi gas. Then the corresponding damping force is incorporated into Euler equations of motion and energies and widths of low lying collective resonances are computed as eigenfrequencies of a vibrating nucleus under surface tension and Coulomb potential as well as the high lying isoscalar giant resonances as eigenfrequencies of an elastic nucleus. Maximum damping is obtained if the particle frequency approximately resonates with the wall frequency. Theoretical results are compared with experimental data and future improvements are indicated.

NUCLEAR STRUCTURE Nuclear giant resonances; calculated widths of isoscalar giant resonances. Elastic vibrations, Boltzmann equation, collision term, long-mean-free path dissipation.

The collisionless Vlasov equation is solved by a moment expansion and truncated after the fourth moments. This scheme yields fluid dynamical equations for the eigenvibrations of a nucleus which are beyond the usual elastic equations. These Euler equations are solved exactly and also approximately for the normal parity (electric, surface) modes 2⁺, first excited 2⁺, 3^-, 4⁺, the abnormal parity (magnetic, twist) modes 1⁺, 2^-, and the compression (breathing and squeezing) modes 0⁺, 1^-. Without adjustable parameters, agreement of the resulting giant resonance energies with experimental data, where available, is reasonably good. Since the corrections due to the inclusion of the third and fourth moments enhance the energies by up to 35%, the moment expansion coverges only slowly.

NUCLEAR STRUCTURE Giant resonances; electric, magnetic, and compression modes; elastic vibrations; calculated energies.

Sine-Gordon kinks under the influence of a constant force do not travel along the classical trajectory. It is shown here that solitons and kinks which obey one of the nonlinear Schrödinger equations and which are also subject to a constant force, on the contrary, do travel along the classical trajectory.

Although the method of dual coordinates has been employed in the literature to quantize the damped harmonic oscillator, one cannot reproduce the proper classical limit by this method. This is shown by constructing operators which have the correct damped behavior and obtaining coherent states as their eigenstates. These states turn out to be nonnormalizable; this is shown to be due to the unphysical nature of the Hamiltonian employed in the method of dual coordinates.

It has been proven by explicit construction that there exists a quantum harmonic potential (equally spaced energy levels) which does not belong to the class of classical harmonic potentials (frequency independent of energy). In this paper we prove the converse, also by example, that all classical harmonic potentials are not necessarily quantum harmonic potentials. Thus all classical harmonic potentials are not quantum harmonic potentials and vice versa.

The dynamic model of asymmetric fission used in calculating the mass and kinetic energy distributions of the fragments from thermal-neutron-induced fission of ²³⁵U is employed here to calculate these distributions for fission at higher excitation energies and for other heavy nuclei at low excitation energies. The model itself is investigated with respect to the semiphenomenological shell correction. It is found that the mass distribution of the fragments in the reaction ²³⁵U(n,f), which for thermal neutrons is strongly peaked at the heavy-fragment mass 132, goes over into a symmetric mass distribution at a compound-nuclear temperature of about 6 MeV.