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Concepts in Materials Science I The Need for Quantum Mechanics in Materials Science VBS/MRC Need for Quantum Mechanics – 0 Concepts in Materials Science I Some Experimental Facts Regarding Metals Dulong-Petit Law (Kittel) High-Temperature Molar Specific Heat = 3R Atoms - Classical Oscillators VBS/MRC Need for Quantum Mechanics – 1 Concepts in Materials Science I Some Experimental Facts Regarding Metals Wiedemann-Franz Law: Ratio of thermal (κ) to electrical conductivities (σ) depends linearly on T κ/σ = (Const)T, (Const) ≈ 2.3 × 10−8 watt-ohm/K2 (Ashcroft-Mermin) VBS/MRC Need for Quantum Mechanics – 2 Concepts in Materials Science I Some Basics Electron charge (e), mass(m), number density(n) dp Newtons’ Law = F (p–momentum, F –force, dt t–time) Force on a charged particle (q), (q = −e for electron, E, B–electric and magnetic fields) p × B) F = q(E + m p Current density j = −en m Conductivities: j = σE (Electrical), VBS/MRC q = −κ∇T (Thermal) Need for Quantum Mechanics – 3 Concepts in Materials Science I Drudé Theory Early 19th century Electrons treated as classical particles Electrons collide with atoms (and other electrons) About 1023 electrons (How to do this???) What do you expect? VBS/MRC Need for Quantum Mechanics – 4 Concepts in Materials Science I Drudé Model How to handle all the electrons? Relaxation time approximation: τ – time in which an electron will definitely undergo a collision Drudé Equation dt (p(t) + F dt) p(t + dt) = 1− τ | {z } prob. no coll. dp p =⇒ = − +F dt τ VBS/MRC Need for Quantum Mechanics – 5 Concepts in Materials Science I Drudé Model Over a long time tl , electrons attain drift velocity and do not accelerate Thus, 1 tl Z 0 1 tl Z tl dp dt = dt =0 tl p dt = pd 0 Drudé Equation gives drift velocity: pd = τ F τ =⇒ v d = F m VBS/MRC (drift velocity) Need for Quantum Mechanics – 6 Concepts in Materials Science I Drudé Model Drift velocity in an electric field (F = −eE) τe vd = − E m Current density ne2 τ E j = −nev d = m Electrical conductivity ne2 τ σ= m VBS/MRC Need for Quantum Mechanics – 7 Concepts in Materials Science I So What? The theory has a parameter τ How to calculate τ ? Well, we don’t know yet! And...so, what? Calculate relaxation times from conductivity measurements What do you think it will be? VBS/MRC Need for Quantum Mechanics – 8 Concepts in Materials Science I Relaxation Times Calculated relaxation times (Ashcroft-Mermin) VBS/MRC Need for Quantum Mechanics – 9 Concepts in Materials Science I Thermal Conductivity Energy of an electron at temperature T , E[T ] = 32 kB T T(x−vx τ ) T(x+vxτ ) vx τ vx τ A (one-D) body with a temperature gradient Magnitude of velocity (speed) in x direction = vx n Heat flux from left to right = vx E[T (x − vx τ )] 2 n Heat flux from right to left = vx E[T (x + vx τ )] 2 VBS/MRC Need for Quantum Mechanics – 10 Concepts in Materials Science I Thermal Conductivity contd. Net heat flux towards positive x axis ∂E n vx (E[T (x − vx τ )] − E[T (x + vx τ )]) = nvx q = 2 ∂T 2 3nkB τ T ∂T 2 ∂E ∂T =− = −n vx |{z} |{z} ∂T ∂x 2m ∂x kB T 3 m 2 kB ∂T − vx τ ∂x Thermal conductivity: 2 Tτ 3nkB κ= 2m VBS/MRC Need for Quantum Mechanics – 11 Concepts in Materials Science I And now, Wiedemann-Franz! Ratio of thermal to electrical conductivity κ 3 = σ 2 kB e 2 T It is linear in T ! It is independent of the metal! What about the constant (Lorentz number)? 3 2 kB e 2 = 1.11 × 10−8 watt-ohm/K2 Expt. value ≈ 2.3 × 10−8 watt-ohm/K2 ! Celebrations! VBS/MRC Need for Quantum Mechanics – 12 Concepts in Materials Science I There is just one more thing... Dulong-Petit say specific heat is 3R One mole of univalent metal contains one mole of ions and one mole of electrons Ionic specific heat = 3R 3 Electronic specific heat = R (ideal gas) 2 9 Total specific heat = R 2 9 6= 3 for usual values of 2, 3, 9! 2 Ok, turn the music down! VBS/MRC Need for Quantum Mechanics – 13 Concepts in Materials Science I Hall Effect B y Ey z x j x Electric field applied in the x direction – current flows jx Magnetic field B applied in the z direction An electric field Ey develops in the y-direction VBS/MRC Need for Quantum Mechanics – 14 Concepts in Materials Science I Hall Effect contd. Hall coefficient RH Ey = jx B Drudé value of Hall coefficient D RH 1 =− ne (prove this!) Independent of relaxation time! VBS/MRC Need for Quantum Mechanics – 15 Concepts in Materials Science I What about experiments? Ratio of theoretical and experimental Hall coefficients (Ashcroft-Mermin) Ok for some metals, but suggests that electrons are positively charged in some metals! Oh, God! Stop the party! VBS/MRC Need for Quantum Mechanics – 16 Concepts in Materials Science I Conclusions...and then... Classical theory is able to explain Wiedemann-Franz (Success!) Fails with specific heat (Ok, lets see!) Fails miserably with Hall coefficients (Surprise!) Well, we DO NEED Quantum Mechanics! VBS/MRC Need for Quantum Mechanics – 17