Lecture #1 Nasser S. Alzayed.
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1 Lecture #1 Nasser S. Alzayed
2 Chapter 6: Free Electro Fermi Gas Itroductio We ca uderstad may physical properties of metals, ad ot oly of the simple metals, i terms of the free electro model. Accordig to this model, the valece electros of the costituet atoms become coductio electros ad move about freely through the volume of the metal. The utility of the free electro model is greatest for properties that deped essetially o the kietic properties of the coductio electros. The iterpretatio of metallic properties p i terms of the motio of free electros was developed log before the ivetio of quatum mechaics. The classical theory had several cospicuous successes, otably the derivatio of the form of Ohm's law ad the relatio betwee the electrical ad thermal coductivity. The classical theory fails to explai the heat capacity ad the magetic susceptibility of the coductio electros. Kig Saud Uiversity, Physics Dept. Phys. 570, Nasser S. Alzayed (Nalzayed@ksu.edu.sa)
3 Classical Model: Metal is a array of positive ios with electros that are free to move through the ioic array Chapter 6: Free Electro Fermi Gas Classical Vs. Quatum Mech. model Electros are treated as a ideal eutral gas, ad their total eergy depeds o the temperature ad applied field I the absece of a electrical field, electros move with radomly distributed thermal velocities Whe a electric field is applied, electros acquire a et drift velocity i the directio opposite to the field Quatum Mech. Model: Electros are i a potetial well with ifiite barriers: They do ot leave metal, but free to move iside Electro eergy levels are discrete (quatized) ad well defied, so average eergy of electro is ot equal to (3/)k B T Electros occupy eergy levels accordig to Pauli s exclusio priciple Electros acquire additioal eergy whe electric field is applied Kig Saud Uiversity, Physics Dept. Phys. 570, Nasser S. Alzayed (Nalzayed@ksu.edu.sa)
4 Chapter 6: Free Electro Fermi Gas The model i Brief This model explais lots of properties i metals. It assumes free electros i the so called coductio bad: Example: Na 11 :Wehave 11 electros distributed das follows: 1s s p 6 3s 1 Valace electro (loosely boud) Hece, there is a free electro/atom i the 3S state Or we have oe electro/atom i the 3S coductio Bad. For a crystal of N atoms: we have N coductio electros ad N +tive Ios. Classical Theory fails to explai for C v (heat capacity) ad χ p (mageticsuc.) forthe full rage of Temperature. What is Fermi Gas? : It is a collectio of large No. of electros that are free to move but subject to Pauli Exclusio Priciple Kig Saud Uiversity, Physics Dept. Phys. 570, Nasser S. Alzayed (Nalzayed@ksu.edu.sa)
5 Chapter 6: Free Electro Fermi Gas Coductio electros i Sodium Na atoms i Na crystal overlap slightly. This leads to the fact that a valace electro is ot attached to a particular io, but belogs to all eighbourig ios at the same time. Accordigly; electrosca virtually move freely all over the crystal leadig to coductio of electricity. Kig Saud Uiversity, Physics Dept. Phys. 570, Nasser S. Alzayed (Nalzayed@ksu.edu.sa)
6 Chapter 6: Free Electro Fermi Gas Eergy Levels i 1 D Cosider a free electro gas i oe dimesio, takig accout of quatum theoryad of the Pauli priciple. A electro of mass m is cofied to a legth L by ifiite harriers. We will have to use Schrödiger Wave Equatio to solve the problem ad fid out eergy levels. H p d w ith : H w here : p i m dx d H (1) m dx is the eergy of the electro i the th. state (orbit). E. Schrödiger ( ) Kig Saud Uiversity, Physics Dept. Phys. 570, Nasser S. Alzayed (Nalzayed@ksu.edu.sa)
7 Chapter 6: Free Electro Fermi Gas Eergy Levels i 1 D Applyig Boudary Coditios for the wave fuctio: (0) 0 at borders =0 ( L ) 0 A si x satisfies the wave fuctio at boudary () or A si x L d d A cos x ad A si x dx L L dx L L A si x A si x m L L L (3) m L Kig Saud Uiversity, Physics Dept. Phys. 570, Nasser S. Alzayed (Nalzayed@ksu.edu.sa)
8 Chapter 6: Free Electro Fermi Gas Eergy Levels i 1 D Every state () ca have two electros; oe at m s =+½ ad oe at: m s = ½. If state has eergy ε ad a state m also has eergy ε : We call this degeeracy. Whe we have may electros, the eergy levels are filled from the bottom to the top. The last filled level is the Fermi level ad is deoted as: F Right: 1 D potetial well. Eergy of electro is show for lowest 3 states (=1,, ad 3) L 3 L L Kig Saud Uiversity, Physics Dept. Phys. 570, Nasser S. Alzayed (Nalzayed@ksu.edu.sa)
9 Chapter 6: Free Electro Fermi Gas Eergy Levels i 1 D Ref. Itroductory Quatum Mechaics Kig Saud Uiversity, Physics Dept. Phys. 570, Nasser S. Alzayed (Nalzayed@ksu.edu.sa)
10 Chapter 6: Free Electro Fermi Gas Fermi Surface ad Fermi Eergy levels We ca easily calculate the locatio of Fermi Level F for N electro system (eve No.): F =N F = N/ Example: N = 6 electros: =1haselectros =has electros =3haselectros( F ) Total: 6 electros F =6/=3 let F F = Fermi Eergy is the eergy of the level. l i Groud Sate for N electros: F F N (3) F (4) m L m L Kig Saud Uiversity, Physics Dept. Phys. 570, Nasser S. Alzayed (Nalzayed@ksu.edu.sa)
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