Larry Spruch

First experiments with the heidelberg test storage ring TSR

Abstract The Heidelberg heavy ion test storage ring TSR started operation in May 1988. The lifetimes of the ion beams observed in the first experiments can be explained by interactions with the residual gas. Multiple Coulomb scattering, single Coulomb scattering, electron capture and electron stripping are the relevant processes. Electron cooling of ions as heavy as O 8+ has been observed for the first time. With increasing particle number, the longitudinal Schottky noise spectrum becomes dominated by collective waves for cooled beams, allowing a determination of velocities of sound. After correcting for these coherent distortions fo the Schottky spectrum, the longitudinal beam temperature could be extracted. The observed longitudinal equilibrium beam temperatures increase strongly with the charge of the ions. For a cooled C 6+ beam, temperatures a factor of 120 higher were measured compared to a proton beam with the same particle number. The shrinking of the beam diameter due to electron cooling was observed with detectors which measured the profile of charge-changed ions behind a bending magnet. A strong laser-induced fluorescence was detected when storing metastable 7 Li + ions in the ring. Via the Doppler effect a very accurate measurement of the ion velocity profile could be performed. First attempts to observe laser cooling failed, probably due to heating effects from intrabeam scattering and a coupling between longitudinal and transversal motion in the beam. Several experiments under preparation are outlined.

UPPER BOUNDS ON SCATTERING LENGTHS FOR STATIC POTENTIALS

It is shown that in the zero-energy scattering of a particle by a center of force, where no bound state exists, the Kohn variational principle provides an upper bcund on the scattering length. A bound may also be obtained from Hulthens method, although with the same form of trial function the Kohn result will be lower (and therefore better) than the one obtained from the Hulthen principle. The Rubinow formulation need not provide a bound; for those calculations which were performed in this form, the results may be converted without any further calculations so that they correspond to the Kohn form, and therefore, under the circumstances considered, do give a bound. Analogous results hold for states of nonzero orbital angular momentum. Direct generalizations of the above that they correspond to the Kohn form. and therefore. under the circumstances considered, do give a bound. Analogous results hold for states of nonzero orbital angular momentum. Direct generalizations of the above results are valid for scattering by a compound system. (auth)

Long-Range (Casimir) Interactions

Normally, nonrelativistic electromagnetic theory with two-particle Coulombic interactions adequately determines the interaction potential of systems A and B if the systems are composed of particles with characteristic velocities much less than the speed of light. If, however, the time it takes light to travel between A and B exceeds a characteristic oscillation period of A or B, the way in which the potential function depends on the separation of the systems can be altered. Called the Casimir effect, it has only recently been confirmed, and it arises in physics, chemistry, and biology. It is the clearest physical manifestation of the fact that, even in a vacuum, electromagnetic fields cannot all vanish.

Retarded, or Casimir, Long‐Range Potentials

Only someone with a short‐range view could fail to be aware of the great importance of long‐range interactions. Indeed, from the late 18th century, when Coulomb discovered that the electrostatic interaction has the same 1/r2 force law that Newton had found for the gravitational interaction, until perhaps the 1930s, when the strong and weak interactions began to be understood, long‐range interactions largely were the subject of physics. By long‐range interactions I mean not only those for which the potential behaves as 1/r for all r but those whose potentials behave asymptotically as some power of 1/r. These originate in 1/r potentials and include, for example, the van der Waals 1/r6 interaction (as calculated nonrelativistically) between two spherically symmetric atoms at a large separation r, and multipole interactions between charge distributions. Long‐range potentials therefore not only play a vital role in astrophysics via Newtons law of gravitation and a significant role in nuclear physics via Coulo...

UPPER BOUND ON THE NEUTRON-DEUTERON DOUBLET SCATTERING LENGTH

Abstract The method of obtaining upper bounds on scattering lengths is applied to the determination of the n-d doublet scattering length A D . A new calculation is unnecessary; an analysis of the trial function used by Efimov in a variational estimate of A D shows that that calculation effectively provides a bound. (The Efimov trial function has the ‘inappropriate’ normalization for present purposes, but it is exceedingly unlikely that this will alter the character of the present results. This question can be readily resolved in any case.) We find that the potentials assumed by Efimov imply A D −13 cm. If these potentials are accurate (this question is still an open one), it follows that of the two experimentally allowed sets, namely A D = 0.7×10 −13 cm and A Q = 6.4×10 −13 cm (Set I), or A D = 8.3×10 −13 cm and A Q = = 2.6×10 −13 cm (Set II), Set I is correct, a conclusion consistent with some recent variational estimates of about 6×10 −13 cm for the quartet scattering length A Q . It is also shown, for the general problem, that the exact numerically determined value of the ‘static’ scattering length lies above the true value if the number of bound state solutions for the static and true problems are the same.

Identities Related to Variational Principles

The existence of a well‐known identity associated with variational principles for any scattering parameter Q, and often serving as the starting point for the development of a variational bound on Q, strongly suggests that it might be useful to construct identities associated with variational principles for quantities other than scattering parameters. An identity associated with the variational principle for the determination of inner products of the linear form g†φ, a generalization of the aforementioned identity, is presented. Here, g is a known function, and φ is an unknown function satisfying Mφ = ω and specified boundary conditions, where M is a known linear operator and ω is a known function. The generalized identity is obtained from a variational principle for g†φ, this variational principle being itself a generalization of the usual Kohn variational principle for scattering amplitudes and phase shifts. An identity associated with a variational principle for the quadratic form φ†Wφ, with φ as above ...

Bounds on the Elements of the Equivalent Network for Scattering in Waveguides. I. Theory

For a few particular waveguide problems, standard variational expressions have previously been shown to be upper or lower bounds on the quantities of interest. However, bounds have not previously been obtained for any truly three‐dimensional problem; that is, where the fields cannot be derived from a single scalar potential. An example is a three‐dimensional obstacle which contacts only one waveguide surface. As one consequence, no straightforward procedure exists for improving the approximations. Recently, Kato devised a rather general method for bounding the cotangent of the phase shift for a given angular momentum in a quantum mechanical central potential scattering problem. This method should be applicable whenever a system can be analyzed in terms of uncoupled standing waves each characterizable by one real phase shift. The method is here adapted to waveguides, including truly three‐dimensional problems. The obstacle must be symmetric about a plane perpendicular to the waveguide axis, with certain ex...

Variational principles, variational identities, and supervariational principles for wavefunctions

We develop variational principles and variational identities for bound state and continuum wavefunctions in a general context, paying particular attention to the proper choice of defining equations and boundary conditions which will lead to unique and unambiguous wavefunctions even when these functions are complex. Any functional, such as a matrix element, calculated with such a variationally determined wavefunction, will also be accurate to second order in the error of the starting choice. This provides, therefore, an alternative procedure for getting variational estimates of matrix elements to the one that already exists in the literature and we establish the equivalence of the two. Of even more interest is the possibility which now seems open of going beyond the variational principle and generating ’’supervariational’’ estimates of wavefunctions and matrix elements which are good to better than second order. We also give a simple prescription for the construction of variational identities for wavefunct...

Bounds on the Elements of the Equivalent Network for Scattering in Waveguides. II. Application to Dielectric Obstacles

The theory developed in the companion article for obtaining rigorous bounds on cotη, where η the phase shift, is applied to scattering by dielectric obstacles in rectangular waveguides. For the obstacles considered here, bounds are also obtained on the phase shifts directly and on the elements of the equivalent T network. The exact solution for a dielectric slab of finite length, which extends to the conducting boundaries of the waveguide and completely encloses the obstacle, is introduced as a convenient trial function. The permittivity of the slab is retained as a parameter which is varied to improve the bounds. In the expressions for the bounds on cotηe and cotη0, the particular obstacle configuration appears only in certain integrals of relatively simple form. Numerical results are obtained for large and small obstacles of various shapes, including some truly three‐dimensional cases. The upper and lower bounds on the phase shifts and on the elements of the equivalent circuit are found to be quite clos...

A REPORT ON SOME FEW-BODY PROBLEMS IN ATOMIC PHYSICS

Publisher Summary This chapter explains a few few-body problems in atomic physics. The two-body problem can be thought of as solvable, and considerable formal progress has been made with the introduction of the Faddeev equations. In these equations, the full wavefunction is partitioned into components, related to the pair of particles that interact first, and for other than atomic physics with its Coulomb potentials, the equations can be helpful in studies of existence and convergence questions. The channel coupling array approach can be applied to bound state and to continuum problems, and can be formulated as coupled differential or integral equations. The two-collision contribution is reduced, relative to the one-collision contribution, by the need of a second interaction, but in the two-collision process there is the saving feature that the target electron need not have a high velocity. In Rutherford scattering with the incident proton, the electron, which can be taken to be at rest initially, recoils toward the target proton where it undergoes a second Rutherford scattering and emerges with a velocity close to that of the proton.