February 6, 2015

According to free electron theory, the properties of a metal depends ongas of free electrons.

The outer most electrons of metal atoms are also known as”Valence Electrons”.These valence electrons are weakly boundwith the atoms.

In a solid, the valence electrons interact and form a ”gas” ofelectrons.The gas of electrons move with relative freedomthroughout the resulting assembly of metal ions.These electronsholds the ions together.High electric and thermal conductivities,opacity, surface luster and other characteristic properties of metalscan be explained on the basis of free electron theory.

The alloy of different metals can be prepared because the freeelectrons do not belong to a particular atom-atom bonds.

Formation of solid can also be explained on the basis of free electrontheory.

When metal atoms are separate(isolated), their total energy ishigher than when they are bound together.The reason for energyreduction is that each valence electron is on the average closer toone ion or another than it would be if it belonged to an isolatedatom.

Hence the electrons potential energy is reduced in metal crystalthan in the atom.On the other hand, the kinetic energy of electronincreases in the metal crystal.The valence energy levels of themetal atoms are all slightly altered by their interactions to give asmany different energy levels as the total number of atomspresent.The levels are so closely spaced as to form a continuousenergy band.

The free electrons in this band have a Fermi Dirac energydistribution in which, at 0 k, their kinetic energies range from 0Kto maximum of 𝜖𝐹 , the fermi energy.

𝐻 is a gas at atmospheric pressure.It can be converted into liquidor solid states .In all the states, it does not behave like metals.

The reason is that both liquid and solid hydrogen at atmosphericconsist of hydrogen molecules and these molecules hold theirelectrons so tightly and none can move freely.

At high pressure, hydrogen can be converted into conductingliquid.High pressure forces 𝐻 molecules so close together thattheir electron wave functions overlap, which allows electron tomove from one molecule to another molecule.

Metallic bonding occurs when the reduction in electron potentialenergy outbalances the increase in electron KE that accompaniesit.The more valence electrons per atom, the higher the averagekinetic energies of the free electrons, but without commensuratedrop in their potential energy.

𝐼 =𝑉

𝑅(1)

Here, R is the resistance of the conductor, depends on its dimensions,composition, and temperature, but is independent of V.Ohm’s law fol-lows free electron theory of metals.

Assume the free electrons in a metal move in random directions and un-dergo frequent collisions.If 𝜆 is the mean free path between the collisionsof a free electron, the average time 𝜏 between collisions is

𝜏 =𝜆

𝑣𝐹(2)

Where 𝑣𝐹 is the electron velocity that corresponds to the Fermi energy𝜖𝐹 , since only electron at or near the top of their energy distribution canbe accelerated.Electron velocity in copper

𝑣𝐹 =

√2𝜖𝐹𝑚

=

√2 × 7.04𝑒𝑉 × 1.60 × 10−𝐽/𝑒𝑉

9.11 × 10−𝑘𝑔= 1.57 × 10𝑚/𝑠

The superimposed drift velocity 𝑣𝑑 due to an applied electric field isusually less than 1 mm/s.

Let a potential differential difference V be applied across a conductor oflength L which produces an electric field 𝐸 = 𝑉

𝐿 in the conductor.Theacceleration of the electron is given by

𝑎 =𝐹

𝑚=

𝑒𝐸

𝑚(3)

The electric field E provides drift velocity (𝑣𝑑) to the free electron gas.Whenan electron undergoes a collision, it rebounds in an arbitrary direc-tion.After each collision, the electron is accelerated for some time interval∆𝑡 before the next collision and at the end of the interval, the electronhas traveled

𝑎∆𝑡.When the electron has made many collisions, its av-

erage displacement will be 𝑋 = 𝑎∆𝑡, where ∆𝑡 is the average of the

squared time intervals.

∆𝑡 = 2𝜏

Drift velocity is given by

𝑣𝑑 =𝑋

𝜏= 𝑎𝜏 (4)

Substituting the values of 𝑣𝑑 and 𝜏 , we have

𝑣𝑑 = 𝑎𝜏 =

(𝑒𝐸

𝑚

)(𝜆

𝑣𝐹

)(5)

If the conductor of cross sectional area A contains n free electrons perunit volume,current in the conductor is given by

𝐼 = 𝑛𝑒𝐴𝑣𝑑 (6)

Substituting the value of 𝑣𝑑, we have

𝐼 =𝑛𝐴𝑒𝐸𝜆

𝑚𝑣𝐹(7)

Substituting the value of electric field E, we have

𝐼 =

(𝑛𝑒𝜆

𝑚𝑣𝐹

)(𝐴

𝐿

)𝑉 (8)

𝐼 =𝑉

𝑅(9)

where

𝑅 =

(𝑚𝑣𝐹𝑛𝑒𝜆

)(𝐿

𝐴

)(10)

is the resistance of the metal conductor.The resistivity of the conductoris given by

𝜌 =𝑚𝑣𝐹𝑛𝑒𝜆

(11)

The scattering of free electron waves in a metal is the reason forelectric resistance.The scattering is caused by structural defectsand by ions out of place as they vibrate.

Structural defects do not depend on temperature but on the purityof the metal and on its history.

The resistivities of of cold worked metals are lowered by annealingbecause the number of defects thereby decreased.With increase intemperature, lattice vibrations increase and thereby increase inresistivity.The resistivity of the metal is given by

𝜌 = 𝜌𝑖 + 𝜌𝑡 (12)

where 𝜌𝑖 depends on the concentration of defects and 𝜌𝑡 dependson temperature.

The ratio 𝐾/𝜎 between thermal and electric conductivities is the samefor all the metals and is a function of temperature.

𝐾

𝜎𝑇=

𝜋𝑘

3𝑒= 2.45 × 10−𝑊.Ω/𝐾 (13)

Find the drift velocity 𝑣𝑑 of the free electrons in a copper wirewhose cross-sectional area is 𝐴 = 1𝑚𝑚 when the wire carries acurrent of 1 A.Assume that each copper atom contributes oneelectron to the electron gas.HINT:7.4 × 10−𝑚/𝑠

𝐼 =𝑄

𝑡= 𝑛𝑒𝐴𝑣𝑑

The resistivity of copper at 20𝐶 is 𝜌 = 1.72 × 10−Ω.𝑚.Estimatethe mean free path 𝜆 between collisions of the free electrons incopper at 20𝐶.Hint:𝜆 = 𝑚𝑣𝐹

𝑛𝑒𝜌

Find the Fermi energy in copper on the assumption that eachcopper atom contributes one free electron to the electron gas.Thedensity of copper is 8.94 × 10𝑘𝑔/𝑚 and its atomic mass is 63.5u.

Hint:𝑛 = 𝑁𝑉 = .×

. ×.×− = 8.48 × 108𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠/𝑚

Fermi energy is given by

𝜖𝐹 =ℎ

2𝑚

(3𝑁

8𝜋𝑉

)/

= 1.13 × 10−18𝐽 = 7.04𝑒𝑉

Derive the relation 𝐼 = 𝑛𝑒𝐴𝑣𝑑.

If the free electrons have so small a drift velocity, why does anelectric appliances go on as soon as its switch is closed and notminutes or hours later?

What do you mean by free electron gas?

What is free electron theory of metals?

Explain the formation of solid on the basis of free electron theory?

What is the condition for the formation of metallic bond.

Hydrogen belong to group 1 in periodic table an still it is notconductor.Why?

How can you make liquid hydrogen conductor?Give the reason forconductance.

What is drift velocity?

What do you mean by fermi energy?

What d you mean by electron velocity at fermi energy?

Use free electron theory to derive Ohm’s law i.e.prove that

𝐼 =

(𝑛𝑒𝜆

𝑚𝑣𝐹

)(𝐴

𝐿

)𝑉

What are reasons for electric resistance of metals and how does itdepend on temperature?

What is Weidemann-Franz Law?

Which energy distribution will be used to explain free electrontheory?

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