Treat sodium as a free-electron metal with and

PHYSICS · Multi-part · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Dielectric and optical properties of solids · Dielectric and optical properties of solids

Solid State Physics, Ch Collective and coupled excitations: plasmons, polaritons and polarons

Problem

Treat sodium as a free-electron metal with εb=1\varepsilon_b=1 and ℏωp=5.9 eV\hbar\omega_p=5.9\,\mathrm{eV}, so that εm(ω)=1−ωp2/ω2\varepsilon_m(\omega)=1-\omega_p^2/\omega^2 with damping neglected. ((a)) A flat sodium surface fills z<0z<0, with vacuum (εd=1\varepsilon_d=1) above it. Taking the electrostatic potential ϕ=A+eiqxe−qz\phi=A_+e^{iqx}e^{-qz} above and ϕ=A−eiqxeqz\phi=A_-e^{iqx}e^{qz} below, show from the boundary conditions that a surface mode exists only when εm+εd=0\varepsilon_m+\varepsilon_d=0, and find its energy ℏωsp\hbar\omega_{sp}. ((b)) Find the resonance energy of a small sodium sphere in vacuum, using the condition εm+2εd=0\varepsilon_m+2\varepsilon_d=0. ((c)) At ℏω=3.0 eV\hbar\omega=3.0\,\mathrm{eV}, use the retarded dispersion relation q=(ω/c)εmεd/(εm+εd)q=(\omega/c)\sqrt{\varepsilon_m\varepsilon_d/(\varepsilon_m+\varepsilon_d)} to find qq in units of ω/c\omega/c, and explain why light arriving from the vacuum side cannot excite this mode at any angle. ((d)) The light is instead brought in through a glass prism of refractive index 1.501.50 held just above the surface. At what angle of incidence inside the glass does the light's wavevector along the surface match qq?

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