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Journal of Physics A Mathematical and General

Mott scattering of anyons: some exact results

J. Law, M. K. Srivastava, R. K. Bhaduri, Avinash Khare

Published Feb 21, 1992• 13 citations
Publisher

Abstract

The planar differential cross section for the Mott scattering of two identical charge anyons is obtained in a closed form. This expression exhibits a marked asymmetry between forward and backward angles, and is especially simple for the semion-semion scattering. At low energies, a pronounced dip in the differential cross section due to planar Coulomb-anyonic interference appears at the forward angles, whose implication is discussed.

DOI
10.1088/0305-4470/25/4/006

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The planar differential cross section for the Mott scattering of two identical charge anyons is obtained in a closed form. This expression exhibits a marked asymmetry between forward and backward angles, and is especially simple for the semion-semion scattering. At low energies, a pronounced dip in the differential cross section due to planar Coulomb-anyonic interference appears at the forward angles, whose implication is discussed.
The exact wave functions that describe scattering of a charged particle by a confined magnetic field (Aharonov-Bohm effect) and by a Coulomb field are analyzed. It is well known that the usual procedure of finding asymptotic forms of these functions which admit a separation into a superposition of an incident plane wave and a circular or spherical scattered wave is problematic near the forward direction. It thus appears to be impossible to express the conservation of probability by means of an optical theorem of the usual kind. Both the total cross section and the forward scattering amplitude appear to be infinite. To address these difficulties we find a new representation for the asymptotic form of the Aharonov-Bohm wave function that is valid for all angles. Rather than try to define a cross section at forward angles, however, we work instead with the probability current and find that it is quite well behaved. The same is true for Coulomb scattering. We trace the usual difficulties to a nonuniformity of limits.
We treat the nonrelativistic quantum mechanical scattering of a wavepacket (for a single, charged, spinless particle) of well-resolved momentum incident on a fixed Coulomb potential. To implement this, we calculate the incoming and outgoing probability amplitudes for normalized state vectors on the basis of eigenvectors of total momentum magnitude and total angular momentum. We show that the method of partial wave analysis becomes applicable for this long range potential with the use of wavepackets. We find a relation between the finite probability and the differential cross section that is seen to hold sufficiently far from the singularity of the latter in the forward direction. The resulting differential cross section reduces to the familiar Rutherford form for scattering angles sufficiently far from the forward direction. We find that the interaction causes small time shifts in the detection probabilities. For our particular scattering geometry, we see, at low energies, a shadow zone of low probability for small angles around the forward direction, a phenomenon that cannot be predicted by the standard treatment.
Quantum-mechanical calculations of the differential cross section for the small-angle elastic scattering of heavy particles are carried out to establish more definitely the region of validity of the classical approximation. Four results are discussed: (1) The Massey—Mohr phase-shift formula corresponds to the Kennard small-angle scattering formula in the semiclassical limit. (2) The Schiff approximation for the cross section is exactly the same as the semiclassical approximation at small angles, for any central potential. (3) At very small angles the semiclassical limit for the differential cross section varies as exp (—cθ2), where c is a function of velocity for which explicit expressions are given. (4) The first quantum deviation from the classical limit, which is proportional to ℏ2, can be combined with the preceding result to give a reasonable representation of the differential cross section over the entire range of small angles for which quantum deviations are appreciable. Detailed calculations for some specific systems are made, and it is shown that Wu's misgivings over the classical interpretation of the experimental results of Amdur and co-workers are unjustified.
The one-dimensional Mott metal-insulator transition is a typical strong correlation effect triggered by the Umklapp scattering. However, in a physical system, the Umklapp scattering coexists with the normal scattering, including both forward and backward scattering, which conserves the total momentum of scattered electrons. Therefore, it is not easy to quantify the contribution of the Umklapp scattering in a Mott metal-insulator transition. To resolve this difficulty, we propose to explore these scattering processes separately. We study the contribution of each scattering process in the one-dimensional Hubbard model using the momentum-space density-matrix renormalization group (kDMRG) and bosonization methods. Our kDMRG calculation confirms that the Mott charge gap results from the Umklapp scattering, but the normal scattering processes strongly renormalize its value. Furthermore, we present a scaling analysis of the Mott charge gap in the bosonization theory and show that the interplay between the Umklapp and forward scattering dictates the charge dynamics in the half-filled Hubbard model.
The influence of coherence terms on the differential cross section for K(42P32)\mathrm{K}(4^{2}P_{\frac{3}{2}})K(42P23​​) interacting with Ar has been measured. The excitation with circularly polarized light directed perpendicularly to the scattering plane induces oscillations in the cross section for ${\ensuremath{\sigma}}^{+}$ light in contrast to ${\ensuremath{\sigma}}^{\ensuremath{-}}$. This asymmetry is verified by the measurement of the corresponding left-right asymmetry with respect to the scattering angle.