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3. Coulomb blockade in gold nanoparticles. The capacitance of a nanoparticle is given as 𝐢 = πœ‹πœ‹πœ€0 π‘Ÿ 𝑑 (π‘Ÿ + 𝑑). where πœ€ is the dielectric constant of the ligand shell of thickness d, and r is the core radius. (a) For a single-nanoparticle STM experiment, the charging energy, π‘ˆ = 𝑒2 2𝐢 . In order to observe the Coulomb blockade, thermally-driven electron tunneling must be prevented, which is the requirement π‘˜π΅π‘‡ β‰ͺ π‘ˆ. Practically, this requirement meansπ‘˜π΅π‘‡π‘šπ‘šπ‘š β‰ˆ π‘ˆ/25, where π‘‡π‘šπ‘šπ‘š is the maximum temperature at which the Coulomb blockade is observable. Find an expression for π‘‡π‘šπ‘šπ‘š in terms of π‘Ÿ, 𝑑 and πœ€. (b) Consider octanethiolate-stabilized Au nanoparticles, for which πœ€ = 0.8 and 𝑑 = 1.0 inn. Plot π‘‡π‘šπ‘šπ‘š vs π‘Ÿ for radii from 1 nm to 10 nm. (c) For which radius is π‘‡π‘šπ‘šπ‘š room temperature (293 K)? (An estimate from your graph is OK.)

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