Look how far we’ve come…

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OutlineReading: Livingston, Chapter 12.1-12.3

• Quantum Mechanical Free Electron Theory of Metals• Filling the Energy Levels with Free Electrons• Density of States• Fermi Function• Maxwell-Boltzmann Approximation

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Quantum Mechanical Free Electron Theory of Metals

fixed lattice and “free”electron gas

BUT this time apply quantum

mechanics!

“particle in a box”

L

http://www.youtube.com/watch?v=dyX5I_io7bg

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Filling the Energy Levels with Free Electrons

E

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Filling the Energy Levels with Free Electrons (cont.)

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Density of States

3/2

3

2

83

2

=

LN

mhEF π

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Example: Using the Density of States to Find Average Properties

What is the average energy of the electrons in a 3D free electron metal at T=0 K?

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Fermi Function: Probability of Occupation for T > 0

At T = 0E<EF allowed states full, F(E) = 1E>EF allowed states empty, F(E) = 0

At T > 0SOME e- excited to HIGHER energy statesE<EF some allowed states emptyE>EF some allowed states full

At ANY TF(E=EF) = 1/2

TkEE BFeEF /)(1

1)( −+=

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Maxwell-Boltzmann Approximation

TkEE BFeEF /)(1

1)( −+= TkEE BFeEF /)()( −−≈

TkEE BFeEF /)(1)( −−≈

TkEE BFeEF /)()(1 −≈−Density of OCCUPIED states:

)()( EFEZ

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Measuring EF: X-Ray Emission Spectroscopy

e- X-ray

E (eV)

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Measuring EF: Photoelectron Spectroscopye-

photons

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Example: Photoelectron SpectroscopyA metal is bombarded with ultraviolet photons of 12 eV energy. The velocities of the photoelectrons emitted from the conduction band range from 1.19 x 106 m/s to 1.74 x 106 m/s. What is the Fermi energy of the metal? What is the work function?

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Thermionic EmissionThermionic Emission: Electrons emitted from a metal due to resistive heating

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Review Questions

1. What is k-space?2. What is the Fermi Energy at T=0?3. What function describes the distribution of allowed energy states?4. What is the procedure for finding the density of states?5. How is the 3D density of states related to energy?6. What is the Fermi-Dirac distribution in words?7. How do you find the probability that a state is Unoccupied?8. What is the probability that a state is filled above the Fermi energy

at T=0?9. What is the probability of occupation at the Fermi level?10. How do you find the density of occupied states?11. Describe two ways to measure the Fermi energy of a metal.

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Important Equations

3/2

3

2

83

2

=

LN

mhEF π

TkEE BFeEF /)(1

1)( −+=

2/13

2/33 )2(4)( Eh

mLEZ π=

Kb EhE −= υ

TkEE BFeEF /)()( −−≈

TkEE BFeEF /)(1)( −−≈

TkEE BFeEF /)()(1 −≈−

oK hhE υυ −=