Assignment 10-Chapter 19

Assignment 10-Chapter 19 - Problem 3 For an electron gas in...

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Assignment # 10. Due on Wed, March 30. Problem 1. An atomic nucleus can be roughly modeled as a gas of nucleus with a number density of 0.18 fm -3 (1 fm=10 -15 m). Because nucleons come in two different types (protons and neutrons), each with spin ½, each spatial wavefunction can hold four nucleons. (a) Calculate the Fermi energy of this system, in MeV. (b) Calculate the Fermi temperature, and comment on the result. Problem 2 . Consider an electron gas in which essentially all of the electrons are highly relativistic ( ε >>mc 2 ), so that their energies are ε =pc (p is the momentum, c the light speed). (a) Show that the Fermi energy ε F (the chemical potential at T=0) is given by ε F =hc(3N/8 π V) 1/3 . (b) Find the total energy of this system in terms of N and ε F . (c) Find the average energy of each electron in the system.
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Unformatted text preview: Problem 3. For an electron gas in a metal, (a) Prove that the average energy per electron is (3/5) ε F at T=0 by making a direct calculation of U e (0)/N. (b) Prove that the average speed of the electron at T=0 is (3/4)v F , where the Fermi speed v F is defined by ε F =(1/2)mv F 2 . Problem 4 . For the metal aluminum (Al), each Al atom contributes three electrons to the conduction band (the free electron gas). The density of aluminum is 2.69x10 3 kgm-3 and its atomic number is 27. (a) Calculate the Fermi energy ε F . (b) Show that at T=300 K, the chemical potential differs from ε F by less than 0.01%. (c) Calculate the electronic contribution to the specific heat capacity of aluminum at T=300 K and compare to the contribution of lattice vibrations....
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