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The Sommerfeld Theory of Metal

1) The Sommerfeld theory of metals uses the quantum mechanical concept of the Fermi-Dirac distribution to describe the electron gas in metals. 2) It introduces the Fermi energy and Fermi momentum/velocity to characterize the highest occupied electron state in the metal. 3) Key predictions include the specific heat of metals being proportional to temperature, in contrast to the classical prediction, and electron densities varying little below the Fermi temperature down to absolute zero.

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209 views11 pages

The Sommerfeld Theory of Metal

1) The Sommerfeld theory of metals uses the quantum mechanical concept of the Fermi-Dirac distribution to describe the electron gas in metals. 2) It introduces the Fermi energy and Fermi momentum/velocity to characterize the highest occupied electron state in the metal. 3) Key predictions include the specific heat of metals being proportional to temperature, in contrast to the classical prediction, and electron densities varying little below the Fermi temperature down to absolute zero.

Uploaded by

firdous
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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The sommerfeld theory of metal

In Drude’s Model, electron gas density n = N/V in equilibrium


temperature is given by Maxwell-Boltzman distribution

This predicts a contribution of the specific heat of metal 3/2KB per electron
energy, which is not observed.

1
From Quantum theory: Pauli exclusion principle
leads to Fermi-Dirac distribution

For Temperature less than 103 K

For most metals electron densities vary a little below To down to T = 0 K.


2
Ground state property of the electron gas
• Electrons don’t interact with one another (independent electron
approximation
• Electron energy levels in volume V, fill up levels according to Pauli exclusion
principle
• A single electron can be considered by a wave function (r), with two
possible orientations, spin up/spin down.

• Assuming no interaction between electrons


(r) associated with energy level satisfies the time independent Schrodinger
equation

3
In a metal with a cubic shape and volume V, the side L = V1/3

Boundary conditions

(r) vanishes when r at the surface of the cube

On a line from 0 – L (this give a standing wave solution)

k(r)=   𝒌.𝒓

With energy K any positive independent


vector 4
The normalized condition

The significance of the vector k can be seen in (r) as an eigenstate of the


momentum operator

With eigenvalue

ℏ 𝒊𝒌.𝒓 k 𝒊𝒌.𝒓
𝒊 𝒓

A electron in a level k(r) has a definite momentum p = k


and a velocity v = P/m = k/m
5
K is also a wave vector in a plane wave
K is constant in any perpendicular plane, hence k.r = constant
and periodic along r

The boundary conditions are satisfied by the general


wave function when

In three dimensional space with Cartesian axes kx , ky , kz (k-space)

The allowed wave vectors are given by multiple of


6
Large number of k vectors are allowed
in k space on the scale

approximation of allowed points

The volume of k-space/Volume


of space per point

Number of allowed k values


per unit volume K- space density of levels

7
To build up N electron ground state
1. We assume electrons are not interacting
2. We may place at most one electron in each single electron level (Pauli exclusion
principle)
3. With wave vector k, there are two associated election levels with spin up and
down, with either values or -
4. For instance, at k = 0 we place two electrons with energy = 0.

5. One electron energy is proportional to


6. For a large number of k values, the occupied region of volume V is a sphere a

radius (F or Fermi), the volume is


7. Then the number of allowed valued of k within the sphere is

8
Each allowed k-value has two one-electron levels, then

The electronic density n = N/V is given by

The ground state of N-electron system is formed by occupying


all single particle levels with k 

The surface of the sphere that separate the occupied from unoccupied
levels is called Fermi surface.

9
The momentum of electron of highest occupied level is called
Fermi momentum 𝑭 = 𝑭

Fermi Energy

Fermi velocity

From

and

In terms of
10
The Fermi Velocity is

From classical mechanics for classical gas


At T = 0, V = 0
Even at room temperature v is in the order of 107 cm/sec.

The Fermi energy can also be written as

Since in the order of


unity
11

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