Research Project: Kusurları ve Katışıklıkları Olan Eğri Yüzey Üzerindeki Skaler Parçacıkların Saçılması
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Contributors
Funders
ID
TB.00324
Authors
Mostafazadeh, Ali
Faculty Member
Publications
Geometric scattering in the presence of line defects
(Springer, 2021) Mostafazadeh, Ali; Bui, Hai Viet; Seymen, Sema; Department of Mathematics; Department of Physics; Yes; College of Sciences
A nonrelativistic scalar particle moving on a curved surface undergoes a geometric scattering whose behavior is sensitive to the theoretically ambiguous values of the intrinsic and extrinsic curvature coefficients entering the expression for the quantum Hamiltonian operator. This suggests using the scattering data to settle the ambiguity in the definition of the Hamiltonian. It has recently been shown that the inclusion of point defects on the surface enhances the geometric scattering effects. We perform a detailed study of the geometric scattering phenomenon in the presence of line defects for the case that the particle is confined to move on a Gaussian bump and the defect(s) are modeled by delta function potentials supported on a line or a set of parallel lines normal to the scattering axis. In contrast to a surface having point defects, the scattering phenomenon associated with this system is generically geometric in nature in the sense that for a flat surface, the scattering amplitude vanishes for all scattering angles ? except ?= ? and ?- ?, where ? is the angle of incidence. We show that the presence of the line defects amplifies the geometric scattering due to the Gaussian bump. This amplification effect is particularly strong when the center of the bump is placed between two line defects.
Geometric scattering of a scalar particle moving on a curved surface in the presence of point defects
(Elsevier, 2019) Mostafazadeh, Ali; Bui, Hai Viet; Department of Mathematics; Department of Physics; Graduate School of Sciences and Engineering; Yes; College of Sciences; GRADUATE SCHOOL OF SCIENCES AND ENGINEERING
A nonrelativistic scalar particle that is constrained to move on an asymptotically flat curved surface undergoes a geometric scattering that is sensitive to the mean and Gaussian curvatures of the surface. A careful study of possible realizations of this phenomenon in typical condensed matter systems requires dealing with the presence of defects. We examine the effect of delta-function point defects residing on a curved surface S. In particular, we solve the scattering problem for a multi-delta-function potential in plane, which requires a proper regularization of divergent terms entering its scattering amplitude, and include the effects of nontrivial geometry of S by treating it as a perturbation of the plane. This allows us to obtain analytic expressions for the geometric scattering amplitude for a surface consisting of one or more Gaussian bumps. In general the presence of the delta-function defects enhances the geometric scattering effects.
Scattering due to geometry: case of a spinless particle moving on an asymptotically flat embedded surface
(American Physical Society (APS), 2018) Mostafazadeh, Ali; Oflaz, Neslihan; Ahmady, Mehrdad; Department of Mathematics; Department of Physics; Graduate School of Sciences and Engineering; Yes; College of Sciences; GRADUATE SCHOOL OF SCIENCES AND ENGINEERING
A nonrelativistic quantum mechanical particle moving freely on a curved surface feels the effect of the nontrivial geometry of the surface through the kinetic part of the Hamiltonian, which is proportional to the Laplace-Beltrami operator, and a geometric potential, which is a linear combination of the mean and Gaussian curvatures of the surface. The coefficients of these terms cannot be uniquely determined by general principles of quantum mechanics but enter the calculation of various physical quantities. We examine their contribution to the geometric scattering of a scalar particle moving on an asymptotically flat embedded surface. In particular, having in mind the possibility of an experimental realization of the geometric scattering in a low-density electron gas formed on a bumped surface, we determine the scattering amplitude for arbitrary choices of the curvature coefficients for a surface with global or local cylindrical symmetry. We also examine the effect of perturbations that violate this symmetry and consider surfaces involving bumps that form a lattice.
Exact solution of the two-dimensional scattering problem for a class of delta-function potentials supported on subsets of a line
(Institute of Physics (IOP) Publishing, 2018) Mostafazadeh, Ali; Loran, Farhang; Department of Mathematics; Department of Physics; Yes; College of Sciences
We use the transfer matrix formulation of scattering theory in twodimensions (2D) to treat the scattering problem for a potential of the form v(x, y) = ζ δ(ax + by)g(bx − ay) where ζ, a, and b are constants, δ(x) is the Dirac δ function, and g is a real- or complex-valued function. We map this problem to that of v(x, y) = ζ δ(x)g(y) and give its exact (nonapproximate) and analytic (closed-form) solution for the following choices of g(y): (i) a linear combination of δ functions, in which case v(x, y) is a finite linear array of 2D δ functions; (ii) a linear combination of eiαny with αn real; (iii) a general periodic function that has the form of a complex Fourier series. In particular we solve the scattering problem for a potential consisting of an infinite linear periodic array of 2D δ functions. We also prove a general theorem that gives a sufficient condition for different choices of g(y) to produce the same scattering amplitude within specific ranges of values of the wavelength λ. For example, we show that for arbitrary real and complex parameters, a and z, the potentials z ¬∞ n=−∞ δ(x)δ(y − an) and a−1zδ(x)[1 + 2 cos(2πy/a)] have the same scattering amplitude for a < λ ¬ 2a.
