Transcription

1 February 6, 2015

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8 According to free electron theory, the properties of a metal depends on gas of free electrons. The outer most electrons of metal atoms are also known as Valence Electrons.These valence electrons are weakly bound with the atoms. In a solid, the valence electrons interact and form a gas of electrons.the gas of electrons move with relative freedom throughout the resulting assembly of metal ions.these electrons holds the ions together.high electric and thermal conductivities, opacity, surface luster and other characteristic properties of metals can be explained on the basis of free electron theory. The alloy of different metals can be prepared because the free electrons do not belong to a particular atom-atom bonds. Formation of solid can also be explained on the basis of free electron theory.

9 When metal atoms are separate(isolated), their total energy is higher than when they are bound together.the reason for energy reduction is that each valence electron is on the average closer to one ion or another than it would be if it belonged to an isolated atom. Hence the electrons potential energy is reduced in metal crystal than in the atom.on the other hand, the kinetic energy of electron increases in the metal crystal.the valence energy levels of the metal atoms are all slightly altered by their interactions to give as many different energy levels as the total number of atoms present.the levels are so closely spaced as to form a continuous energy band. The free electrons in this band have a Fermi Dirac energy distribution in which, at 0 k, their kinetic energies range from 0K to maximum of ε F, the fermi energy.

10 H is a gas at atmospheric pressure.it can be converted into liquid or solid states.in all the states, it does not behave like metals. The reason is that both liquid and solid hydrogen at atmospheric consist of hydrogen molecules and these molecules hold their electrons so tightly and none can move freely. At high pressure, hydrogen can be converted into conducting liquid.high pressure forces H molecules so close together that their electron wave functions overlap, which allows electron to move from one molecule to another molecule. Metallic bonding occurs when the reduction in electron potential energy outbalances the increase in electron KE that accompanies it.the more valence electrons per atom, the higher the average kinetic energies of the free electrons, but without commensurate drop in their potential energy.

11 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 follows free electron theory of metals.

12 Assume the free electrons in a metal move in random directions and undergo frequent collisions.if λ is the mean free path between the collisions of a free electron, the average time τ between collisions is τ = λ v F (2) Where v F is the electron velocity that corresponds to the Fermi energy ε F, since only electron at or near the top of their energy distribution can be accelerated.electron velocity in copper v F = 2εF m = eV J/eV kg = m/s

13 The superimposed drift velocity v d due to an applied electric field is usually less than 1 mm/s.

14 Let a potential differential difference V be applied across a conductor of length L which produces an electric field E = V L in the conductor.the acceleration of the electron is given by a = F m = ee m (3) The electric field E provides drift velocity (v d ) to the free electron gas.when an electron undergoes a collision, it rebounds in an arbitrary direction.after each collision, the electron is accelerated for some time interval t before the next collision and at the end of the interval, the electron has traveled a t.when the electron has made many collisions, its average displacement will be X = a t, where t is the average of the squared time intervals. t = 2τ

15 Drift velocity is given by v d = X τ = aτ (4) Substituting the values of v d and τ, we have ( ) ( ) ee λ v d = aτ = m v F (5) If the conductor of cross sectional area A contains n free electrons per unit volume,current in the conductor is given by Substituting the value of v d, we have I = neav d (6) I = nae Eλ mv F (7)

16 Substituting the value of electric field E, we have ( ne ) ( ) λ A I = V (8) mv F L I = V R (9) where R = ( ) ( ) mvf L ne λ A (10) is the resistance of the metal conductor.the resistivity of the conductor is given by ρ = mv F ne (11) λ

17 The scattering of free electron waves in a metal is the reason for electric resistance.the scattering is caused by structural defects and by ions out of place as they vibrate. Structural defects do not depend on temperature but on the purity of the metal and on its history. The resistivities of of cold worked metals are lowered by annealing because the number of defects thereby decreased.with increase in temperature, lattice vibrations increase and thereby increase in resistivity.the resistivity of the metal is given by ρ = ρ i + ρ t (12) where ρ i depends on the concentration of defects and ρ t depends on temperature.

18 The ratio K/σ between thermal and electric conductivities is the same for all the metals and is a function of temperature. K σt = π k 3e = W.Ω/K (13)

19 Find the drift velocity v d of the free electrons in a copper wire whose cross-sectional area is A = 1mm when the wire carries a current of 1 A.Assume that each copper atom contributes one electron to the electron gas. HINT: m/s I = Q t = neav d The resistivity of copper at 20 C is ρ = Ω.m.Estimate the mean free path λ between collisions of the free electrons in copper at 20 C. Hint:λ = mv F ne ρ

20 Find the Fermi energy in copper on the assumption that each copper atom contributes one free electron to the electron gas.the density of copper is kg/m and its atomic mass is 63.5u. Hint:n = N V =. = electrons/m.. Fermi energy is given by ( ) ε F = h 3N / = J = 7.04eV 2m 8πV Derive the relation I = neavd. If the free electrons have so small a drift velocity, why does an electric appliances go on as soon as its switch is closed and not minutes or hours later? What do you mean by free electron gas? What is free electron theory of metals?

21 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 not conductor.why? How can you make liquid hydrogen conductor?give the reason for conductance. 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 ( ne ) ( ) λ A I = V mv F L

22 What are reasons for electric resistance of metals and how does it depend on temperature? What is Weidemann-Franz Law? Which energy distribution will be used to explain free electron theory?

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