Scattering rates arising from the interactions of electrons with bulk longitudinal optical (LO) phonon modes in a hollow cylinder are calculated as functions of the inner radius and the uniform axial applied magnetic field. Now, the specific nature of electron-phonon interactions mainly depends on the character of the energy spectrum of electrons. As is well known, in cylindrical quantum wires, the application of a parallel magnetic field lifts the double degeneracy of the non-zero azimuthal quantum number states; m≠0\documentclass[12pt]{minimal}
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\begin{document}$$m \not = 0$$\end{document}; irrespective of all electron’s radial quantum number l states. In fact, this Zeeman splitting is such that the m<0\documentclass[12pt]{minimal}
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\begin{document}$$m < 0$$\end{document} electron’s energy subbands initially decrease with the increase of the parallel applied magnetic field. In a solid cylinder, the lowest-order; {l=1;m=0\documentclass[12pt]{minimal}
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\begin{document}$$l = 1;\,m = 0$$\end{document}} subband is always the ground state. In a hollow cylinder, however, as the axial applied magnetic field is increased, the electron’s energy subbands take turns at becoming the ground state; following the sequence {m=0,-1,-2...-N}\documentclass[12pt]{minimal}
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\begin{document}$$\lbrace m=0,-1,-2...~ -N\rbrace$$\end{document} of azimuthal quantum numbers. Furthermore, in a hollow cylinder, in general, the electron’s energy separations between any two subbands are less than the LO phonon energy except for exceptionally high magnetic fields, and some highest-order quantum number states. In view of this, the discussion of the energy relaxation here is focused mainly on intrasubband scattering of electrons and only within the lowest-order {l=1;m=0\documentclass[12pt]{minimal}
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\begin{document}$$l = 1;\,m = 0$$\end{document}} electron’s energy subband. The intrasubband scattering rates are found to be characterized by shallow minima in their variations with the inner radius, again, for a fixed outer radius. This feature is a consequence of a balance between two seemingly conflicting effects of the electron’s confinement by the inner and outer walls of the hollow cylinder. First; increased confinement of the charge carriers generally leads to the enhancement of the rates. Second; the presence of a hole in a hollow cylinder leads to a significant suppression of the scattering rates. The intrasubband scattering rates also show a somewhat parabolic increase in their variations with the applied magnetic field; an increase which is more pronounced in a relatively thick hollow cylinder.
