Paul L. Csonka

General nondynamical formalism for reactions with particles of arbitrary spin. Number of form factors. Parity conservation

Abstract The general nondynamical formalism for reactions involving particles of arbitrary spins, discussed in several previous papers, is further developed. The number of form factors in the M -matrix is determined under parity conservation as well as other conservation laws, and from this it is derived that many conservation laws have no nondynamical tests of validity in reactions involving spin-zero particles only. The factorization of the M -matrix into irreducible constituents is developed for parity conserving reactions and it is shown how the compound observable can be written down in terms of observables and so-called pseudoobservables of the constituent reactions. It is then demonstrated that for parity conserving reactions, an unpolarized and uncorrelated particle is in general equivalent to a fictitious spin-zero particle possessing both positive and negative parity (dual particle). The structure of observables in terms of the type of traces they contain is studied. The number of terms in a compound observable is derived. It is shown that the number of observables is a parity conserving reaction is always larger than the number of independent bilinear combinations of form factors. Thus, there are always linear relations among the observables. The number and character of these relations is explored and it is shown that they also lead to the complete list of parity experiments, i.e., experiments from which we can determine the intrinsic parity of one of the particles in the reaction, knowing the intrinsic parities of the other participating reactions but knowing nothing about the dynamics of the reaction. It is shown that the observables can be divided into subclasses, a given subclass containing observables which are different linear combinations of the same bilinear combinations of form factors. Each such bilinear combination appears in one and only one subclass.

GENERAL NONDYNAMICAL FORMALISM FOR REACTIONS WITH PARTICLES OF ARBITRARY SPIN: ROTATION INVARIANCE.

Abstract The purpose of this paper is to give a detailed and practical prescription for calculating physical observables from form factors (i.e., invariant amplitudes) of the M -matrix for an arbitrary reaction containing an arbitrary number of particles having arbitrary spins. The treatment is fully relativistic. No invariance principles have been assumed (except rotational invariance in three-space) and the discussion is valid whether or not any of the usual invariance principles (parity conservation, time-reversal invariance, etc.) hold. The prescription is based on those properties of such reactions which have already been treated in the literature, as well as on several remarks that have not been discussed before. Among the latter are the factorizability of the M -matrix in spin-space, the treatment of nonsquare spin matrix tensors, and the general evaluation of traces of products of spin tensors. The expansion of the M -matrix in terms of spin-momentum tensors is explained for the general case, and a similar discussion is given for the density matrix. From these the observables are calculated, and the mathematical background of such calculations are treated in appendices. Some of the results of this paper have been used in several past papers of ours, others will appear in papers completed and soon to be published. A table containing the coefficients of the bilinear products of form factors in the various experimental observables for various reactions containing plausible spins will soon be available also as an unpublished report, as it was calculated by a computer program utilizing the procedure outlined in this paper. The program will also be available to calculate, as the need arises, any reactions not contained in the table.

An introduction to x-coefficients and a tabulation of their values**

X -coefficients, as defined here, can be used to simplify the calculation of particle polarizations (vector polarizations as well as higher-order, so called, tensor polarizations) and spin correlations. If not more than two particles have nonzero spin in the reaction under study, then all polarizations and correlations can be expressed as linear combinations of X -coefficients, and only the numbers multiplying them depend on the form factors (invariant amplitudes). If three or four particles have nonzero spin, then bilinear combinations of X -coefficients appear, etc. The X -coefficients do not depend on the dynamics of the reaction and can be tabulated once and for all. They do depend on the spins of the participating particles and are given in this table for those spin values which are the most important ones from a practical point of view.

Radiated power from short period undulators with a large magnet gap

The total power Ptot radiated by an electron beam passing through an undulator is studied as a function of the undulator wavelength λu, while the magnet gap g remains fixed. As expected, for λu≫g, the Ptot falls off exponentially for decreasing λu, but for smaller λu the falloff is less steep and, in particular, for λu→0, Ptot is asymptotically proportional to λu.

Active spectral narrowing by dynamical optical means

Dynamical optics can be used to alter the frequency spectrum of electromagnetic radiation. This can increase the intensity in a given narrow-frequency band. The method is useful when one deals with spectral purity λ/Δλ > 105. In addition, the method can also be used to shift the average frequency of electromagnetic radiation such as x rays.

Coherence saturation for beams of energetic photons

A new method is proposed to generate transversely coherent energetic photon beams. It does not rely on stimulated photon emission, therefore, is not subject to the same technological and cost limitations which affect bound‐state x‐ray lasers and free‐electron x‐ray lasers. The method leaves unchanged the photon source; it increases transverse coherence by dynamical optical means, i.e., using nonstationary optical elements. It is well suited to first‐order interference experiments (e.g., x‐ray holography) particularly with synchrotron radiation sources.