Failures and successes of the free electron model

Similar documents
Phys 412 Solid State Physics. Lecturer: Réka Albert

Crystals Statics. Structural Properties. Geometry of lattices. Aug 23, 2018

3a 2. a 1 = 3a. a 2 = 3a

Keble College - Hilary 2012 Section VI: Condensed matter physics Tutorial 2 - Lattices and scattering

Introduction to Solid State Physics or the study of physical properties of matter in a solid phase

PART 1 Introduction to Theory of Solids

Crystal Structure. Dr Bindu Krishnan

MP464: Solid State Physics Problem Sheet

Crystal Structures: Bulk and Slab Calcula3ons

1 Crystal Structures. of three-dimensional crystals. Here we use two-dimensional examples to illustrate the concepts.

Nearly Free Electron Gas model - I

Solid State Physics 460- Lecture 5 Diffraction and the Reciprocal Lattice Continued (Kittel Ch. 2)

1.4 Crystal structure

MP464: Solid State Physics Problem Sheet

2. Diffraction as a means to determine crystal structure

General theory of diffraction

Condensed Matter Physics April, 8, LUMS School of Science and Engineering

Phys 460 Describing and Classifying Crystal Lattices

2. Diffraction as a means to determine crystal structure

Crystal planes. Neutrons: magnetic moment - interacts with magnetic materials or nuclei of non-magnetic materials. (in Å)

Solid State Physics Lecture 3 Diffraction and the Reciprocal Lattice (Kittel Ch. 2)

Crystal Structure SOLID STATE PHYSICS. Lecture 5. A.H. Harker. thelecture thenextlecture. Physics and Astronomy UCL

... 3, , = a (1) 3 3 a 2 = a (2) The reciprocal lattice vectors are defined by the condition a b = 2πδ ij, which gives

Bravais Lattice + Basis = Crystal Structure

Lecture Note on Crystal structures Masatsugu Sei Suzuki and Itsuko S. Suzuki Department of Physics, SUNY at Binghamton (Date: February 03, 2012)

Road map (Where are we headed?)

Introduction to crystallography The unitcell The resiprocal space and unitcell Braggs law Structure factor F hkl and atomic scattering factor f zθ

Scattering and Diffraction

Quantum Condensed Matter Physics Lecture 5

Introductory Nanotechnology ~ Basic Condensed Matter Physics ~

Reciprocal Lattice. Let A(r) be some function featuring discrete translation symmetry:

Nearly Free Electron Gas model - II

Experimental Determination of Crystal Structure

Geometry of Crystal Lattice

Crystallographic structure Physical vs Chemical bonding in solids

SOLID STATE 18. Reciprocal Space

PROBING CRYSTAL STRUCTURE

Section 10 Metals: Electron Dynamics and Fermi Surfaces

Chapter 6 Free Electron Fermi Gas

Bulk Structures of Crystals

The Solid State. Phase diagrams Crystals and symmetry Unit cells and packing Types of solid

Chapter 2. X-ray X. Diffraction and Reciprocal Lattice. Scattering from Lattices

Chap 3 Scattering and structures

3.012 Structure An Introduction to X-ray Diffraction

Introduction to Materials Science Graduate students (Applied Physics)

PH575 Spring Lecture #13 Free electron theory: Sutton Ch. 7 pp 132 -> 144; Kittel Ch. 6. 3/2 " # % & D( E) = V E 1/2. 2π 2.

Department of Physics, University of Maryland, College Park MIDTERM TEST

Chapter 6 Free Electron Fermi Gas

Physics of Condensed Matter I

UNIT I SOLID STATE PHYSICS

Introduction to Condensed Matter Physics

Lecture 11 - Phonons II - Thermal Prop. Continued

Lecture 6 - Bonding in Crystals

Unit 1 Part 2 Atomic Structure and The Periodic Table Introduction to the Periodic Table UNIT 1 ATOMIC STRUCTURE AND THE PERIODIC TABLE

7. FREE ELECTRON THEORY.

Notes on Ewald summation techniques

disordered, ordered and coherent with the substrate, and ordered but incoherent with the substrate.

Chemical Bonding Ionic Bonding. Unit 1 Chapter 2

Chapter 1. Crystal structure. 1.1 Crystal lattices

Surface Structure and Morphology 2D Crystallography

Structure of Surfaces

9/20/2017. Elements are Pure Substances that cannot be broken down into simpler substances by chemical change (contain Only One Type of Atom)

Origin of color. Gokaran Shukla. School of Physics, Trinity College, Dublin 2, Ireland (Dated: March 20, 2018)

Carrier Mobility and Hall Effect. Prof.P. Ravindran, Department of Physics, Central University of Tamil Nadu, India

High- reflec*vity high- resolu*on X- ray crystal op*cs with diamonds

Problem with Kohn-Sham equations

Minimal Update of Solid State Physics

Diamond. There are four types of solid: -Hard Structure - Tetrahedral atomic arrangement. What hybrid state do you think the carbon has?

SOLID STATE PHYSICS. Second Edition. John Wiley & Sons. J. R. Hook H. E. Hall. Department of Physics, University of Manchester

X-ray, Neutron and e-beam scattering

Handout 7 Reciprocal Space

There are four types of solid:

Made the FIRST periodic table

Introduction to Condensed Matter Physics

Remember the purpose of this reading assignment is to prepare you for class. Reading for familiarity not mastery is expected.

Analytical Methods for Materials

SCIENCE VISION INSTITUTE For CSIR NET/JRF, GATE, JEST, TIFR & IIT-JAM Web:

Chapter 2. Atomic Packing

(Re-write, January 2011, from notes of S. C. Fain Jr., L. Sorensen, O. E. Vilches, J. Stoltenberg and D. B. Pengra, Version 1, preliminary)

Crystallographic Point Groups and Space Groups

The Hamiltonian. 2.1 The Schrödinger Equation

Wave diffraction and the reciprocal lattice

Physical Chemistry I. Crystal Structure

Quantum Condensed Matter Physics Lecture 4

X-ray Diffraction. Interaction of Waves Reciprocal Lattice and Diffraction X-ray Scattering by Atoms The Integrated Intensity

Polarization is out of the plane

Introduction to Materials Science Graduate students (Applied Physics)

Rajesh Prasad Department of Applied Mechanics Indian Institute of Technology New Delhi

FYS Vår 2017 (Kondenserte fasers fysikk)

Name: SCH3U Worksheet-Trends

Introduction to Solid State Physics

materials and their properties

Structure of Crystalline Solids

Last 4 Digits of USC ID:

100% ionic compounds do not exist but predominantly ionic compounds are formed when metals combine with non-metals.

1. The arrangement of the elements from left to right in Period 4 on the Periodic Table is based on

Electronic Structure Theory for Periodic Systems: The Concepts. Christian Ratsch

Experiment 12 ELECTRON DIFFRACTION. Diffraction by Graphite 1. Diffraction by Aluminum 3. The Electron Diffraction Apparatus 5

Atomic Arrangement. Primer in Materials Spring

Physics with Neutrons I, WS 2015/2016. Lecture 11, MLZ is a cooperation between:

Transcription:

Failures and successes of the free electron model

Temperature dependence on the electric conduc1vity j e = E = ne2 m electron gas electron- phonon coupling Drude s model predicts that σ is independent of temperature unless we put in by hand that = (T ) electrical conductivity is about momentum transfer Superconductor R 1

Temperature dependence on the thermal conduc1vity electron gas phonons j Q = rt = 2 3 k 2 B Tn m METAL Trea<ng the electron gas as non- interac<ng fermions gives (only) the correct low temperature behaviour for metals. thermal conductivity is about energy transfer NON- METAL

Thermal conduc1vity of diamond

Temperature dependence on the specific heat c v = u T V 3nk B T T 3 c v = 2 2 k B T F nk B Diamond Again, only the low temperature behaviour for metals is fairly well described. For high temperatures one needs to consider heat carried by the ion cores

Hall coefficient Free electron theory predicts: - Constant Hall coefficient R H =- (nec) - 1 - Hall voltage independent of H The free electron result works well for alkali metals (Na, Li, K) but fails for Al, Mg and many more (see AM Tab. 1.4). The Hall effect is a powerful way to discover anomalous proper<es of materials! Normalised Hall coefficient for two types of high temperature super- conductors. No<ce the sign shig.

More problems Direc1on of DC current: Does not need to be parallel to the applied electric field! Too long mean free path: v F = p 2 F /m 10 100Å and mysteries Why are some materials metals and others insulators?

Main success: Widemann- Franz law T = 2 3 kb e 2 =2.44 10 8 [W Ohm K 2 ] Metal 273K 373K Mo 2.61 2.79 Pb 2.47 2.56 Pt 2.51 2.60 Sn 2.52 2.49 W 3.04 3.20 Zn 2.31 2.33 Metal 273K 373K Ag 2.31 2.37 Au 2.35 2.40 Cd 2.42 2.43 Cu 2.23 2.33 Ir 2.49 2.49 Lorentz number in 10-8 [W Ohm K - 2 ] Note: outside this temperature range the Lorentz number can depend strongly on temperature.

Review of main assump1ons 1. Free electron approxima1on The ion cores play no other role than source of collisions. Between the collisions they do not affect the electron s mo<on. Resolu1on: we will let the electrons move in a sta<c (periodic) poten<al due to a fixed array of ions rather than in free space. 2. Independent electron approxima1on electron- electron interac<ons are ignored. Resolu1on: ignore. 3. Sta1c ions The ions are also dynamics en<<es which can contribute to physical phenomena. One prominent example is heat transfer. Resolu1on: we will let the ions vibrate around their equilibrium posi<ons due to thermal fluctua<ons

CRYSTALLOGRAPHY

Defini1on of a Bravais lasce (i) Infinite array of discrete points with an arrangement and orienta<on that appears exactly the same from whichever of the points the array is viewed. (ii) All points with posi<on vectors R on the form R = n 1 a 1 + n 2 a 2 + n 3 a 3 (3D) n 1,n 2,n 3 integers not unique!

Five unique Bravais lakces in 2D Oblique Hexagonal (triangular) Rectangular Square Centred rectangular

14 unique Bravais lakces in 3D

Example: BCC (e.g. Cesium, Sodium) Primi<ve (Bravais) lakce vectors: a 1 = aˆx a 2 = aŷ a 3 = a (ˆx +ŷ +ẑ) 2 ˆx a Cs = 6.05 Å a Na = 4.23 Å ŷ ẑ

Example: FCC (e.g. Gold, Argon) Primi<ve (Bravais) lakce vectors: a 1 = a 2 (ŷ +ẑ) a 2 = a (ˆx +ẑ) 2 a 3 = a (ŷ +ˆx) 2 aau ˆx ŷ ẑ = 4.08 Å a Ar = 5.26 Å

Wigner- Seitz primi<ve cell AM: One can always choose a primi6ve cell with the full symmetry of the Bravais la;ce. By far the most common such choice is the Wigner- Seitz cell. FCC The Wigner- Seitz cell is the region of space around a la;ce point that is closer to that la;ce point than to any other la;ce point

Lakce with a basis 2D BCC Bravais primi1ve lasce vectors a 1 = aˆx, Two- point basis 0, a (ˆx +ŷ) 2 a 2 = aŷ

WAVES IN SOLIDS Experimental setup for X- ray diffrac1on Outgoing wave e ik r k = 2 ˆn0 Incoming wave e ik 0 r k 0 = 2 ˆn θ Bragg angle sample (fcc) Bragg s law gives the angles for coherent and incoherent scaqering from a crystal lakce

Reciprocal lakce AM: Consider a set of points R cons6tu6ng a Brvais la;ce and a plane wave exp(i k.r). For general k such a plane wave will, of course, not have the periodicity of the la;ce, but for special choices it will. The set of all wave vectors K that yield plane waves with the periodicity of the given Bravais la;ce is known as the reciprocal la;ce. e ik R =1 R = n 1 a 1 + n 2 a 2 + n 3 a 3

Laue condi1on (AM): Construc6ve interference will occur if the change in wave vector equals a reciprocal la;ce vector. K = k 0 k 0 = k k k K = 1 2 K Bragg plane! A geometrical interpreta<on of the Laue condi<on

Theorem (AM): For any family of la;ce planes there are reciprocal la;ce vectors that are (i) perpendicular to the planes (ii) the shortest of which have length 2π/d. Miller indices (AM): The Miller indices of a plane are the coordinates (h, k, l) of the shortest reciprocal la;ce vector to that plane.

2024 © sciencedocbox.com Privacy Policy | Terms of Service | Feedback