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    Lecture - 18-The Free-Electron Theory: Metals Are of Great Importance in Our Daily Lives

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    Lecture - 18-The Free-Electron Theory: Metals Are of Great Importance in Our Daily Lives

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    Lecture - 18-The Free-Electron Theory: Metals Are of Great Importance in Our Daily Lives

    Uploaded by

    Znar Saeed

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    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
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    Lecture -18-

    The Free-electron theory


    1. Introduction.
    Metals are of great importance in our daily lives.
    Iron is used in automobiles, copper in electrical
    wiring, silver and gold as jewelry. Metals are
    characterized by common physical properties: great
    physical strength, high density, good electrical and
    thermal conductivities, and high optical reflectivity,
    which is responsible for their characteristic bright
    appearance.
    2. Conduction electrons
    What are the conduction electrons?
    A valence electron really belongs to the whole
    crystal, since it can move readily from one ion to its
    neighbor, and then the neighbor's neighbor, and so
    on. This mobile electron, which is called a valence
    electron in a free atom, becomes a conduclion
    electron in a solid.
    In Na the number of conduction electrons is the
    same as the number of atoms, and the same is true
    for K, and also for the noble
    metals Cu, Ag, Au, all of which are monovalent.In
    divalent metals-such as Be, Mg, Zn, and Cd-the
    number of electrons is twice the number of atoms,
    and so on.
    We can find the electron concentration by this
    equation.
    = ( ) NA
    Where:
    is the number of electrons per unit volume,
    is the atomic valence,
    NA is Avogadro's number,
    ρ is density of the substance.
    2. The Free-electron gas
    In the free-electron model, the conduction electrons
    are assumed to be completely free, except for a
    potential at the surface.
    According to this model, the conduction electrons
    move about inside the specimen without any
    collisions, except for an occasional reflection from
    the surface ( see figure below) . Because of this, we
    speak of a free-electron gas.
    Q1/ What is the reason for the weakness of the
    interaction between the conduction electrons?
    There are actually two reasons.
    1. According to the Pauli exclusion principle, electrons
    of parallel spins tend to stay away from each
    other.
    2. even if their spins are opposite, electrons tend
    to stay away from each other, in order to minimize
    the energy of the system. If two electrons come
    very close to each other, the coulomb potential
    energy becomes highly large, and this violates the
    tendency of the electron system to have the lowest
    possible energy.

    Q2/ Why the interaction between the ions core


    and conduction electrons appears to be
    weak?
    when an electron passes an ion, its velocity
    increases rather rapidly in the ion's neighborhood,
    due to the decrease in the potential. Because of this,
    the electron spends only a small fraction of its time
    near the ion, where the potential is strong.
    Q3/ What is the different between the free-
    electron gas and the ordinary gas?
    1. Free-electron gas is charged, while in ordinary
    gases the molecules are mostly neutral. So We
    may thus think of free-electron gas is thus
    actually similar to a plasma.
    2. The concentration of electrons in metals is
    large:(1029) el./m3. By contrast, the ordinary gas
    has about 1025 molecules/m3.

    3. Static Electrical conductivity(Drude calculation


    The law of electrical conduction in metals-Ohm's
    law- is:
    I=V/R ……….. 1
    Suppose that L and A are, respectively, the length
    and cross section of the wire; then
    J=I/A …………. 2

    ……… 3

    R= ……….4
    : is the electric field , and is the electrical
    resistivity.
    …………5 , J= …………6

    which is the form of Ohm's law.


    The velocity of the electron ( is:

    is a constant called the collision time.


    This, then, is the steady-state velocity of the
    electron. It is opposite to because the charge on
    the electron is negative.

    The current density J can be calculated from eq. 7

    J= ……..8
    Comparing (8) with Ohm's law (6), one
    finds the following expression for the conductivity.

    σ = ne μd …..10
    where μd = = d / ……..11
    μd: is drift mobility , it's unit m2 V-1S-1
    Note: μd is – Ve for electrons
    & is + Ve for holes
    =𝜆m / sth = 𝜆m /(3kBT/m)½
    substitute in Eq. 9 , we get:

    σ = ne2 𝜆m / (3m kBT )½ = const. ( T-½) …………..16

    i.e T-½ (according to Drude calculations)

    Drude couldn't justify the experimental result (σ T-1 ) , nor could he


    explain the observed low temp. behavior of σ which varies with T-5
    before becoming constant.

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