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Physical Review A

Loss of wave-packet coherence in stationary scattering experiments

James M. Feagin, Leigh R. Hargreaves

Published Sep 24, 2013• 19 citations
Publisher

Abstract

We consider the cross section for a scattering reaction averaged over a statistical ensemble of incident beam wave packets. We generalize a result known from neutron diffraction that static experiments involving stationary scattering via conservative interactions and steady beam currents cannot distinguish the wave-packet nature of the incident beam. Thus we interpret the results of recent experiments on the loss of interference effects in proton scattering from molecular hydrogen as due to the weak collimation of the incident beam rather than alterations to beam wave packets.

DOI
10.1103/physreva.88.032705

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We consider the cross section for a scattering reaction averaged over a statistical ensemble of incident beam wave packets. We generalize a result known from neutron diffraction that static experiments involving stationary scattering via conservative interactions and steady beam currents cannot distinguish the wave-packet nature of the incident beam. Thus we interpret the results of recent experiments on the loss of interference effects in proton scattering from molecular hydrogen as due to the weak collimation of the incident beam rather than alterations to beam wave packets.
The projectile beam coherence effects occurring in ion-atom collisions are analyzed on the basis of the recent theory of Karlovets et al. [Phys. Rev. A 92, 052703 (2015)] developed for the elastic scattering of wave packets of particles off a potential field. This theory is generalized to estimate the loss of coherence for inelastically scattered projectiles in ionizing collisions. The results obtained by the suggested model are compared with experimental data for the ionization of hydrogen atoms and molecules by 75-keV proton impact. Significantly improved agreement is observed between the theory and experiment.
The elastic scattering of spinless particles from a spherically symmetric stationary target is considered. The main idea is to describe the incident beam of particles as a certain statistical ensemble of one-particle wave packets, and to define the differential cross section in terms of the average scattering of the particles in this ensemble. The physical significance of this mode of description is discussed, and arguments are presented as to why it should be regarded as a realistic description of typical scattering experiments. The discussion is general, and not restricted to potential scattering. It is based on the notion of the S matrix, and the most basic ideas of the S matrix are discussed. A simple derivation of the partial wave expansion is given within the framework of S matrix theory. The aim of this paper is mainly pedagogical, and the final results obtained are old and well established.
In the analysis of neutron scattering measurements of condensed matter structure, it normally suffices to treat the incident and scattered neutron beams as if composed of incoherent distributions of plane waves with wavevectors of different magnitudes and directions which are taken to define an instrumental resolution. However, despite the wide-ranging applicability of this conventional treatment, there are cases in which the wave function of an individual neutron in the beam must be described more accurately by a spatially localized packet, in particular with respect to its transverse extent normal to its mean direction of propagation. One such case involves the creation of orbital angular momentum (OAM) states in a neutron via interaction with a material device of a given size. It is shown in the work reported here that there exist two distinct measures of coherence of special significance and utility for describing neutron beams in scattering studies of materials in general. One measure corresponds to the coherent superposition of basis functions and their wavevectors which constitute each individual neutron packet state function whereas the other measure can be associated with an incoherent distribution of mean wavevectors of the individual neutron packets in a beam. Both the distribution of the mean wavevectors of individual packets in the beam as well as the wavevector components of the superposition of basis functions within an individual packet can contribute to the conventional notion of instrumental resolution. However, it is the transverse spatial extent of packet wavefronts alone that determines the area within which a coherent scattering process can occur in the first place. This picture is shown to be consistent with standard quantum theory. It is also demonstrated that these two measures of coherence can be distinguished from one another experimentally.
Two methodological difficulties of the quantum theory of collisions are considered. The first is the undesirable interference of the incident and scattered waves in the stationary approach to scattering. The second concerns the nonstationary approach to the theory of collisions of the type a + b → c + d. In order to calculate the cross section one uses the matrix element 〈cd|S|ab〉 of the S-matrix. The element is proportional to δ-function expressing the energy conservation. The corresponding probability |〈cd|S|ab〉|2 contains δ2, which is mathematically meaningless. The known regular way to overcome the difficulty seems to be unsatisfactory. In this paper, both the problems are resolved using wave packets of incident particles.
The conventional theory of neutron beams interacting with many-body systems treats the beam as a classical system, i.e. with its dynamical variables appearing in the quantum dynamics of the scattering process not as operators but only as c-numbers. Moreover, neutrons are described with plane waves, i.e. the concept of neutron's (finite) coherence length is here irrelevant. The same holds for electron, atom or X-ray scattering. This simplification results in the full decoupling of the probe particle's dynamics from the quantum dynamics of the scatterer---a well-known fact also reflected in the standard formalism of time-correlation functions (see textbooks). Making contact with modern quantum theoretical approaches (e.g., quantum entanglement, 'which-path detection' versus interference, von Neumann measurement, Week Values, etc.) new observable effects may be exposed and/or predicted, for instance, a momentum-transfer deficit and an intensity deficit in neutron scattering from protons of hydrogen-containing samples. 7
Two methodological troubles of the quantum theory of collisions are considered. The first is the undesirable interference of the incident and scattered waves in the stationary approach to scattering. The second concerns the nonstationary approach to the theory of collisions of the type a+b→c+da+b\to c+da+b→c+d. In order to calculate the cross section one uses the matrix element $$ of the SSS-matrix. The element is proportional to δδδ-function expressing the energy conservation. The corresponding probability ∣<cd∣S∣ab>∣2|< cd|S|ab>|^2∣<cd∣S∣ab>∣2 contains δ2δ^2δ2 which is mathematically senseless. The known regular way to overcome the difficulty seems to be unsatisfactory. In this paper, both the troubles are resolved using wave packets of incident particles.