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This is true for suitable linear combinations of the functions (2).
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We are using gaussian units, whereas Stratton uses mks units. Equation (4) is written for a nonmagnetic material (µ=1). The time dependence e-iωt is understood.
"Electric" and "magnetic" modes are often also called "TM" and "TE" modes, respectively.
In LiF the longitudinal optical frequency, 27defined as the frequency for which ε=1, is ΩL=ωL/ωT = 2.197 in dimensionless units, while the transverse optical frequency is ΩT=1.
The sphere must in practice remain larger than the lattice parameter.
K. L. Kliewer and R. Fuchs, Phys. Rev. 144, 495 (1966). The surface modes for a slab, shown in Fig. 2 of this reference, approach the frequency determined by ε= - 1 as kx →∞, kx being the component of the wave vector parallel to the slab. In this limit the fields become localized at the surfaces. The condition ε= - 1 also determines the frequency of the surface modes in a semi-infinite medium. Therefore, if the fields fall off fast enough inside the medium, the frequency of the surface modes does not depend on whether the solid is in the form of a sphere, a slab, or a semi-infinite medium.
Our expression for the cross sections can be obtained from those given in Sec. 9.26 of Ref. 4 by dividing by πa2 and making the replacements anr → -1/Al and bnr → -1/Bl. Equations (26), (29), and (31) in Sec. 9.26 have the wrong sign, since they give a negative extinction cross section QT.
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There is one exception to the statement that the surface modes do not give rise to visible structure at low frequencies: the unmarked plateau in the extinction cross section QT between the 1LM1 and 2LM1 peaks can be attributed to the 1S surface mode.
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