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Bulk/Volume Plasmon are collective oscillations of free electrons with a longitudinal wave, which are induced in metals by an electron beam or a charged-particle. The volume plasmon is excited also in semiconductors and insulators. Its oscillation energy (plasmon energy) is proportional to the square root of the free electron density (valence electron density for semiconductors and insulators). The volume plasmon is directly observed as a peak in an electron energy-loss spectrum. It should be noted that the volume plasmon cannot be excited nor can be observed with a light wave or a transverse wave.

Exitation of Volume Plasmon
When a high-speed electron beam is incident on a solid metal, the Coulomb force formed by the beam induces a density change (compression) in the homogeneously distributed free electrons in the metal. With the induced Coulomb force as a driving force, a longitudinal oscillation wave of the free electrons with a specific frequency is created. Quantization of this collective motion of the free electrons is called volume plasmon. The energy of the volume plasmon, $$E_p$$, is expressed by the following equation:

$$E_p=\hbar\omega_p=\hbar\sqrt{\frac{4\pi e^2 N}{m}}$$

Here, $$\omega_p$$ is the angular frequency of the plasma oscillation（plasma frequency）. $$\hbar=\frac{h}{2\pi}$$ is Planck' s constant. e, m, and N respectively express the elementary charge of an electron, the mass of an electron, and the density of the free electrons. The plasmon energy is proportional to the square root of the free electron density in a metal.

Volume Plasmon in Semiconductors and Insulators
Although the volume plasmon is originally considered for free electrons, it is excited also in semiconductors and insulators as a collective oscillation of the whole valence electrons. The valence electrons vibrate collectively against positive ion cores. The energy or frequency of the plasmon is calculated by substituting the density of the valence electrons into the above equation.

Plasmon Energy
The charge density of Al is N = 1.8 × 10^23 e/cm3. The plasmon energy is calculated to be 15.7 eV, showing a good agreement with an experimental value of 15.0 eV. For monovalent metals (Li, Na, etc.), the calculated energies agree well with the energies experimentally obtained.

For diamond (insulator), the plasmon energy is calculated to be 31 eV using the valence electron density (N = 7.0 × 10^23 e/cm3), showing a rather good agreement with an experimental energy of 34 eV. In the cases of ionic crystals such as LiF, NaCl etc., their plasmon energies calculated using the valence electron densities well reproduce the energies experimentally obtained. Table shows the plasmon energies for various materials.

However, for a material in which interband transitions strongly occur close to the expected plasmon energy, the experimental plasmon energy can be largely deviated from the expected energy. For example, in the case of Ag, the plasmon energy expected from the valence electron density of N=0.59×10^23 e/cm3 is 9.0 eV but the experimental plasmon energy is observed at 3.9 eV due to a strong interband transition of the d orbital electrons at 4.0 eV.

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