Surface Modes in Small Spheres

In the preceding paragraphs we considered a homogeneous sphere. Let us now examine what happens when a homogeneous core sphere is uniformly coated with a mantle of different composition. Again, the condition for excitation of the first-order surface mode can be obtained from electrostatics. In Section 5.4 we derived an expression for the polarizability of a small coated sphere the condition for excitation of the Frohlich mode follows by setting the denominator of (5.36) equal to zero ... 

Ions in the lattice of a solid can also partake in a collective oscillation which, when quantized, is called a phonon. Again, as with plasmons, the presence of a boundary can modify the characteristics of such lattice vibrations. Thus, the infrared surface modes that we discussed previously are sometimes called surface phonons. Such surface phonons in ionic crystals have been clearly discussed in a landmark paper by Ruppin and Englman (1970), who distinguish between polariton and pure phonon modes. In the classical language of Chapter 4 a polariton mode is merely a normal mode where no restriction is made on the size of the sphere pure phonon modes come about when the sphere is sufficiently small that retardation effects can be neglected. In the language of elementary excitations a polariton is a kind of hybrid excitation that exhibits mixed photon and phonon behavior. 

For any real material, the frequency at which (12.27) is satisfied is complex—the surface modes are virtual. However, its real part is approximately the frequency where the cross sections have maxima, provided that the imaginary part is small compared with the real part. We shall denote this frequency by us. For a sphere, o>s is the Frohlich frequency wF. If used intelligently, always keeping in mind its limitations, (12.27) is a guide to the whereabouts of peaks in extinction spectra of small ellipsoidal particles but it will not necessarily lead to the exact frequency. 

In this section we compare the theory of the preceding two sections with experimental measurements of infrared extinction by small particles. Comparisons between experiment and theory for spheres of various solids, most notably alkali halides and magnesium oxide, have been published in the scientific literature many of these papers are cited in this chapter. In most of this work, however, there is an arbitrary normalization of theory and experiment, which tends to hide discrepancies. For this reason, most theoretical calculations in this section are compared with mass-normalized extinction measurements. The new measurements presented here were made in the Department of Physics at the University of Arizona. A group of solids was selected to illustrate different aspects of surface modes. Results on amorphous quartz (Si02) particles, for example, illustrate the agreement between experi-... 

Several predicted features of infrared surface mode absorption by small spheres are verified by the experimental results shown in Fig. 12.13. The frequency of peak absorption by spheres is shifted an appreciable amount from what it is in the bulk solid the e" curve peaks at 1070 cm", whereas the peak of the small-sphere absorption is at 1111 cm-1, very close to the frequency where e is — 2em (— 4.6 for a KBr matrix). The absorption maximum (absorption is nearly equal to extinction for these small particles) is very strong Qabs for a 0.1-jum particle is about 7 at the Frohlich frequency. 

The observed darkening of the indium slides results from a shift of the absorption peak because of the coating on the particles. Because of the cumbersomeness of the expressions for coated ellipsoids (Section 5.4) this shift can be understood most easily by appealing to (12.15), the condition for surface mode excitation in a coated sphere. For a small metallic sphere with dielectric function given by the Drude formula (9.26) and coated with a nonabsorbing material with dielectric function c2, the wavelength of maximum absorption is approximately... 

Semi-reflective and polished samples are also probed in reflection mode. As the SNR of the spectra relies heavily on collection of the reflected light back into the IR objective, it is important that these samples have a smooth, flat surface and are correctly oriented. Samples with smooth surfaces that are not flat can be mounted into a micro-goiniometer, so as to adjust the tilt of the sample with respect to the incoming beam. Even simpler, samples can also be pressed into a small sphere of putty so that the sample surface is parallel to the microscope stage. 

Sphere. A complete description of the coupling of an electromagnetic wave and the eigenmodes of an isolated sphere of any size, given by polariton theory based on Mie s formalism (Section 1.10), indicates that all modes of a sphere-shaped crystal are radiative [293, 298], These modes are called surface modes since their origin lies in the finite size of the sample [297]. For very small spheres, there is only the lowest order surface mode (the Frohlich mode), which is neither transverse nor longitudinal [293]. Its frequency (the Frohlich frequency) is given by... 

This group of tests has already been referred to in order to emphasize the multifunctional nature of hardness. There are several modes depending upon whether a ball, cone, or pyramid is forced into the ceramic surface. In all cases the load per unit area of impression is given as the measure of hardness. Results here are more variable than might seem necessary because different test methods use different unit areas. For example, in the Brinell test where a small sphere is used to indent the surface the hardness is calculated from the contact area, not the area in the plane of the surface which would seem to be a more directly measured and calculated variable. Thus... 

Several additional features of the model are noteworthy. First, it is possible to build it without straining chemical bonds or causing unfavorable steric interactions. The polyamine chain is sufficiently long to reach around a cluster, but it is not so long or so bulky as to cause excessive crowding near the surface of the cluster. The spaces at this surface between the polyamine chains (Fig. 11) are likely binding sites for small apolar molecules, since such molecules can be bound at the interface or partially penetrate into the domain of the hydrocarbon sphere in response to favorable apolar interactions. In the model shown in Fig. 11, three bound p-nitrophenyl caproate molecules have also been included to illustrate possible modes of binding. An arrow points to one of these small molecules. 

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