Lecture_29 - PHYS 342 Fall Fall 2011 Lecture 29: The Free...

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PHYS 342 all 2011 Fall 2011 Lecture 29: The Free Electron Model Ron Reifenberger Birck Nanotechnology Center urdue University Purdue University Lecture 29 1
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The Problem – How to Approximate the Periodic Atomic Potential? aa a rm n ? +Ze +Ze +Ze +Ze 2
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The Free Electron Model Initially proposed by Drude - 1900 he eep lectrostatic otential ear n tom’s ucleus “ n verage” The deep electrostatic potential near an atom s nucleus on average is canceled out by the core electrons to produce a slowly varying potential between the atomic cores. A crude first approximation is a otential ell ith lat ottom: ) ? U(r) potential well with a flat bottom Free electron model ? ? ? ? () ? r eveloped to explain some of the common distinguishing Developed to explain some of the common distinguishing properties of metals: high thermal conductivity 3 good electrical conductivity low specific heats
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How many electrons in the potential well? ASSUME each atom gives up its outermost (valence) atomic electron(s) : e.g. atomic Na has 11 electrons: 1s 2 2s 2 2p 6 3s 1 . Na crystallizes in the BCC structure with a lattice constant of 0.423 nm. hat is the mass density of Na? What is the mass density of Na? 22.99 grams Na = 1 mole = 6.02 10 23 atoms 3 1 Na atom=3.82 10 -23 grams mass density volume 3 23 93 2 9 3 2 2 2 3.82 10 1 (0.423 10 ) 7.89 10 10 Na atoms grams mm c m       4 3 0.97 / g mcm
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How many “free” electrons per unit volume? The 3s 1 electron is released from the atomic core and forms a “gas” of charged particles. It becomes a 23 3 6.023 10 1 Free electrons atoms e x moles volume mole atom cm  fm g f g p . m “free” electron! 3 3 0.97 0.042 3 gm cm moles where x m    22 28 33 23 2.5 10 2.5 10 cm gm mole ee m   cm V (Compare to number density of gas atoms at STP: 2.7 10 19 per cubic cm) 5 What happens to the other 10 electrons?
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What is the correct wave equation?
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This note was uploaded on 12/08/2011 for the course PHYS 342 taught by Professor Staff during the Fall '08 term at Purdue University-West Lafayette.

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Lecture_29 - PHYS 342 Fall Fall 2011 Lecture 29: The Free...

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