Sampling the Brillouin-zone:

Size: px
Start display at page:

Download "Sampling the Brillouin-zone:"

Transcription

1 Sampling the Brillouin-zone: Andreas EICHLER Institut für Materialphysik and Center for Computational Materials Science Universität Wien, Sensengasse 8, A-090 Wien, Austria b-initio ienna ackage imulation A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page

2 Overview introduction k-point meshes Smearing methods What to do in practice A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 2

3 Introduction For many properties (e.g.: density of states, charge density, matrix elements, response functions,... ) integrals (I) over the Brillouin-zone are necessary: I ε Ω BZBZ F ε δ ε nk ε dk To evaluate computationally integrals weighted sum over special k-points Ω BZBZ ω ki k A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 3

4 k k function f k k-points meshes - The idea of special points Chadi, Cohen, PRB 8 (973) with complete lattice symmetry introduce symmetrized plane-waves (SPW): A m k R Cm eıkr sum over symmetry-equivalent R C m Cm develope f SPW shell of lattice vectors in Fourier-series (in SPW) f f0 k m f m A m A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 4

5 k 0 m k evaluate integral (=average) over Brillouin-zone f 2π Ω 3 BZ f dk with: Ω 2π 3 BZ A m dk f f0 2 taking n k-points with weighting factors ω k so that n i ω ki A m ki 0 m N f = weighted sum over k-points for variations of f that can be described within the shell corresponding to C N. A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 5

6 Monkhorst and Pack (976): Idea: equally spaced mesh in Brillouin-zone. Construction-rule: k prs u p b u r b 2 usb3 u r 2rq r 2q r r 2 qr b i q r reciprocal lattice-vectors determines number of k-points in r-direction A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 6

7 k k3 k Example: quadratic 2-dimensional lattice q q2 4 6 k-points only 3 inequivalent k-points ( IBZ) 4k 4k 2 8k ω ω2 ω ½ 0 b k k k b IBZ BZ Ω BZ BZ F dk 4 F k 4 F k2 2 F A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 7

8 k 2πnk more Fourier-components Interpretation: representation of functionf E on a discrete equally-spaced mesh N n 0 a n cos -½ density of mesh 0 ½ k higher accuracy Common meshes : Two choices for the center of the mesh centered on Γ ( Γbelongs to mesh). centered around Γ. (can break symmetry!!) A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 8

9 Algorithm: calculate equally spaced-mesh shift the mesh if desired apply all symmetry operations of Bravaislattice to all k-points extract the irreducible k-points (IBZ) calculate the proper weighting A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 9

10 " Smearing methods Problem: in metallic systems Brillouin-zone integrals over functions that are discontinuous at the Fermi-level. Fourier-components high dense grid is necessary. Solution: replace step function by a smoother function. Example: bandstructure energy with: Θ ω k ε nk Θ nk x ε nk µ x 0 0 x0 E ω k ε nk f nk ε nkµ! -½ 0 ½ k necessary: appropriate function f f equivalent to partial occupancies. A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 0

11 $ % Fermi-Dirac function f ε nk µ exp ε nkµ!# consequence: energy is no longer variational with respect to the partial occupacies f. F E n S f n 2 S f f ln f F free energy. new variational functional - defined by (). 3 kb T f ln f S f entropy of a system of non-interacting electrons at a finite temperature T. smearing parameter. can be interpreted as finite temperature via (3). calculations at finite temperature are possible (Mermin 965) A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page

12 , ( )( * ( )( 4 2 $ & ' $ Consistency: 9 (fn % F E n S fn S f kbt f ln f f ln f fn E fn S f E fn εn exp F µ n fn N S εn fn ln εnµ exp µ ln f fn fn ε nk-µ. + µ 0 ln 0 f n 0 f % ln f f A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 2

13 / & ' Gaussian smearing broadening of energy-levels with Gaussian function. f becomes an integral of the Gaussian function: f ε nk µ 2 erf ε nk µ no analytical inversion of the error-function erf exists entropy and free energy cannot be written in terms of f. S µ ε 2 π exp 0 µ ε 2 has no physical interpretation. variational functional F differs from E forces are calculated as derivatives of the variational quantity (F not necessarily equal to forces at E ). A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 3

14 2 4 ( Improvement: extrapolation to (0. F 32E 0 γ2 F E S S F 2γ E E 2E 2Ê 0 0 γ 2 2 F E A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 4

15 Method of Methfessel and Paxton (989) Idea: expansion of stepfunction in a complete set of orthogonal functions term of order 0 = integral over Gaussians generalization of Gaussian broadening with functions of higher order. A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 5

16 F E f 0 f N S N x x x with: 2 f0 A n H N : x erf 2 A NH 2N n!4 n x N m n 4 π x A m H 2m ex 2 x e x 2 Hermite-polynomial of order N advantages: deviation of F from E 0 only of order 2+N in extrapolation for (0 usually not necessary, but also possible: E 0 2Ê N 2 N A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 6

17 The significance of N and MP of order N leads to a negligible error, if X degree 2N around ε F. ε is representable as a polynomial of linewidth can be increased for higher order to obtain the same accuracy entropy term (S n SN f n ) describes deviation of F from E. if S 5few mev then Ê 32F 2E 32E 0. forces correct within that limit. in practice: smearings of order N= or 2 are sufficient A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 7

18 Linear tetrahedron method Idea:. dividing up the Brillouin-zone into tetrahedra 2. Linear interpolation of the function to be integrated X n within these tetrahedra 3. integration of the interpolated function X n A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 8

19 ad. How to select mesh for tetrahedra map out the IBZ b b use special points b b A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 9

20 k j ad 2. interpolation Seite von X n j k c j k X n j... k-points A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 20

21 k k k j ad 3. k-space integration: simplification by Blöchl (993) remapping of the tetrahedra onto the k-points ω n j Ω BZΩBZ dkc j f ε n effective weights ωn j for k-points. k-space summation: ω n j X n n j A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 2

22 Drawbacks: tetrahedra can break the symmetry of the Bravaislattice at least 4 k-points are necessary Γ must be included linear interpolation under- or overestimates the real curve A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 22

23 µ Corrections by Blöchl (993) Idea: linear interpolation under- or overestimates the real curve for full-bands or insulators these errors cancel for metals: correction of quadratic errors is possible: δω kn T 40 D T ε F 4 j ε jn ε kn j D T corners (k-point) of the tetrahedront DOS for the tetrahedron T at ε F. A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 23

24 Result: best k-point convergence for energy forces: with Blöchl corrections the new effective partial occupancies do not minimize the groundstate total energy variation of occupancies ω nk w.r.t. the ionic positions would be necessary with US-PP and PAW practically impossible A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 24

25 Convergence tests (from P.Blöchl, O. Jepsen, O.K. Andersen, PRB 49,6223 (994).) bandstructure energy of silicon: conventional LT -method vs. LT+Blöchl corrections bandstructure energy vs. k-point spacing : A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 25

26 What to do in practice energy/dos calculations: linear tetrahedron method with Blöchl corrections ISMEAR=-5 calculation of forces: semiconductors: Gaussian smearing (ISMEAR=0; SIGMA=0.) metals : Methfessel-Paxton (N= or 2) always: test for energy with LT+Blöchl-corr. in any case: careful checks for k-point convergence are necessary A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 26

27 The KPOINTS - file: > k-points for a metal 2> 0 3> Gamma point 4> > st line: comment 2nd line: 0 ( automatic generation) 3rd line: Monkhorst or Gammapoint (centered) 4th line: mesh parameter 5th line: (shift) A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 27

28 mesh parameter determine the number of intersections in each direction longer axes in real-space shorter axes in k-space less intersections necessary for equally spaced mesh Consequences: molecules, atoms (large supercells) is enough. Γ surfaces (one long direction 2-D Brillouin-zone) x for the direction corresponding to the long direction. y typical values (never trust them!): metals: semiconductors: /atom /atom A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 28

29 b ; b G Example - real-space/ reciprocal cell 4 4 k k k k 2 k 3 ACBED 0 IBZ BZ b : KMLEN 0 IBZ BZ b F doubling the cell in real space halves the reciprocal cell zone boundary is folded back to Γ same sampling is achieved with halved mesh parameter A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 29

30 OQP Example - hexagonal cell RTSVU Z[]\ ^`_ before symmetrization after WTXVY shifted to atbvc in certain cell geometries (e.g. hexagonal cells) even meshes break the symmetry symmetrization results in non equally distributed k-points Gamma point centered mesh preserves symmetry A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 30

31 Convergence tests with respect to and number of k-points in the IBZ G.Kresse, J. Furthmüller, Computat. Mat. Sci. 6, 5 (996). A. EICHLER, SAMPLING THE BRILLOUIN-ZONE Page 3

DFT in practice : Part I. Ersen Mete

DFT in practice : Part I. Ersen Mete plane wave expansion & the Brillouin zone integration Department of Physics Balıkesir University, Balıkesir - Turkey August 13, 2009 - NanoDFT 09, İzmir Institute of Technology, İzmir Outline Plane wave

More information

Hands on Session II:

Hands on Session II: Hands on Session II: Robert LORENZ Institut für Materialphysik and Center for Computational Material Science Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria b-initio ienna ackage imulation R.

More information

Accuracy and Validation of Results

Accuracy and Validation of Results Accuracy and Validation of Results Georg KRESSE Institut für Materialphysik and Center for Computational Material Science Universität Wien, Sensengasse 8 4, A-1090 Wien, Austria b-initio ienna ackage imulation

More information

Reciprocal Space and Brillouin Zones in Two and Three Dimensions As briefly stated at the end of the first section, Bloch s theorem has the following

Reciprocal Space and Brillouin Zones in Two and Three Dimensions As briefly stated at the end of the first section, Bloch s theorem has the following Reciprocal Space and Brillouin Zones in Two and Three Dimensions As briefly stated at the end of the first section, Bloch s theorem has the following form in two and three dimensions: k (r + R) = e 2 ik

More information

The nearly-free electron model

The nearly-free electron model Handout 3 The nearly-free electron model 3.1 Introduction Having derived Bloch s theorem we are now at a stage where we can start introducing the concept of bandstructure. When someone refers to the bandstructure

More information

Analysis, post-processing and visualization tools

Analysis, post-processing and visualization tools Analysis, post-processing and visualization tools Javier Junquera Andrei Postnikov Summary of different tools for post-processing and visualization DENCHAR PLRHO DOS, PDOS DOS and PDOS total Fe, d MACROAVE

More information

Types of Elements

Types of Elements chapter : Modeling and Simulation 439 142 20 600 Then from the first equation, P 1 = 140(0.0714) = 9.996 kn. 280 = MPa =, psi The structure pushes on the wall with a force of 9.996 kn. (Note: we could

More information

Semiconductor Physics

Semiconductor Physics 10p PhD Course Semiconductor Physics 18 Lectures Nov-Dec 2011 and Jan Feb 2012 Literature Semiconductor Physics K. Seeger The Physics of Semiconductors Grundmann Basic Semiconductors Physics - Hamaguchi

More information

A Beginner s Guide to Materials Studio and DFT Calculations with Castep. P. Hasnip (pjh503@york.ac.uk)

A Beginner s Guide to Materials Studio and DFT Calculations with Castep. P. Hasnip (pjh503@york.ac.uk) A Beginner s Guide to Materials Studio and DFT Calculations with Castep P. Hasnip (pjh503@york.ac.uk) September 18, 2007 Materials Studio collects all of its files into Projects. We ll start by creating

More information

DIGITAL FOREX OPTIONS

DIGITAL FOREX OPTIONS DIGITAL FOREX OPTIONS OPENGAMMA QUANTITATIVE RESEARCH Abstract. Some pricing methods for forex digital options are described. The price in the Garhman-Kohlhagen model is first described, more for completeness

More information

Lecture 3: Electron statistics in a solid

Lecture 3: Electron statistics in a solid Lecture 3: Electron statistics in a solid Contents Density of states. DOS in a 3D uniform solid.................... 3.2 DOS for a 2D solid........................ 4.3 DOS for a D solid........................

More information

Point Lattices in Computer Graphics and Visualization how signal processing may help computer graphics

Point Lattices in Computer Graphics and Visualization how signal processing may help computer graphics Point Lattices in Computer Graphics and Visualization how signal processing may help computer graphics Dimitri Van De Ville Ecole Polytechnique Fédérale de Lausanne Biomedical Imaging Group dimitri.vandeville@epfl.ch

More information

The Central Equation

The Central Equation The Central Equation Mervyn Roy May 6, 015 1 Derivation of the central equation The single particle Schrödinger equation is, ( H E nk ψ nk = 0, ) ( + v s(r) E nk ψ nk = 0. (1) We can solve Eq. (1) at given

More information

Hands on Session I: Georg KRESSE

Hands on Session I: Georg KRESSE Hands on Session I: Georg KRESSE Institut für Materialphysik and Center for Computational Material Science Universität Wien, Sensengasse 8, A-1090 Wien, Austria b-initio ienna ackage imulation G. KRESSE,

More information

PURE MATHEMATICS AM 27

PURE MATHEMATICS AM 27 AM SYLLABUS (013) PURE MATHEMATICS AM 7 SYLLABUS 1 Pure Mathematics AM 7 Syllabus (Available in September) Paper I(3hrs)+Paper II(3hrs) 1. AIMS To prepare students for further studies in Mathematics and

More information

PURE MATHEMATICS AM 27

PURE MATHEMATICS AM 27 AM Syllabus (015): Pure Mathematics AM SYLLABUS (015) PURE MATHEMATICS AM 7 SYLLABUS 1 AM Syllabus (015): Pure Mathematics Pure Mathematics AM 7 Syllabus (Available in September) Paper I(3hrs)+Paper II(3hrs)

More information

An Overview of the Finite Element Analysis

An Overview of the Finite Element Analysis CHAPTER 1 An Overview of the Finite Element Analysis 1.1 Introduction Finite element analysis (FEA) involves solution of engineering problems using computers. Engineering structures that have complex geometry

More information

SCHWEITZER ENGINEERING LABORATORIES, COMERCIAL LTDA.

SCHWEITZER ENGINEERING LABORATORIES, COMERCIAL LTDA. Pocket book of Electrical Engineering Formulas Content 1. Elementary Algebra and Geometry 1. Fundamental Properties (real numbers) 1 2. Exponents 2 3. Fractional Exponents 2 4. Irrational Exponents 2 5.

More information

Homework set 1: posted on website

Homework set 1: posted on website Homework set 1: posted on website 1 Last time: Interpolation, fitting and smoothing Interpolated curve passes through all data points Fitted curve passes closest (by some criterion) to all data points

More information

Finite Elements for 2 D Problems

Finite Elements for 2 D Problems Finite Elements for 2 D Problems General Formula for the Stiffness Matrix Displacements (u, v) in a plane element are interpolated from nodal displacements (ui, vi) using shape functions Ni as follows,

More information

Mesh Discretization Error and Criteria for Accuracy of Finite Element Solutions

Mesh Discretization Error and Criteria for Accuracy of Finite Element Solutions Mesh Discretization Error and Criteria for Accuracy of Finite Element Solutions Chandresh Shah Cummins, Inc. Abstract Any finite element analysis performed by an engineer is subject to several types of

More information

Mean value theorem, Taylors Theorem, Maxima and Minima.

Mean value theorem, Taylors Theorem, Maxima and Minima. MA 001 Preparatory Mathematics I. Complex numbers as ordered pairs. Argand s diagram. Triangle inequality. De Moivre s Theorem. Algebra: Quadratic equations and express-ions. Permutations and Combinations.

More information

Numerical Analysis An Introduction

Numerical Analysis An Introduction Walter Gautschi Numerical Analysis An Introduction 1997 Birkhauser Boston Basel Berlin CONTENTS PREFACE xi CHAPTER 0. PROLOGUE 1 0.1. Overview 1 0.2. Numerical analysis software 3 0.3. Textbooks and monographs

More information

MATHEMATICAL METHODS FOURIER SERIES

MATHEMATICAL METHODS FOURIER SERIES MATHEMATICAL METHODS FOURIER SERIES I YEAR B.Tech By Mr. Y. Prabhaker Reddy Asst. Professor of Mathematics Guru Nanak Engineering College Ibrahimpatnam, Hyderabad. SYLLABUS OF MATHEMATICAL METHODS (as

More information

Lecture 2: Semiconductors: Introduction

Lecture 2: Semiconductors: Introduction Lecture 2: Semiconductors: Introduction Contents 1 Introduction 1 2 Band formation in semiconductors 2 3 Classification of semiconductors 5 4 Electron effective mass 10 1 Introduction Metals have electrical

More information

Notes on Factoring. MA 206 Kurt Bryan

Notes on Factoring. MA 206 Kurt Bryan The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor

More information

degrees of freedom and are able to adapt to the task they are supposed to do [Gupta].

degrees of freedom and are able to adapt to the task they are supposed to do [Gupta]. 1.3 Neural Networks 19 Neural Networks are large structured systems of equations. These systems have many degrees of freedom and are able to adapt to the task they are supposed to do [Gupta]. Two very

More information

Two Topics in Parametric Integration Applied to Stochastic Simulation in Industrial Engineering

Two Topics in Parametric Integration Applied to Stochastic Simulation in Industrial Engineering Two Topics in Parametric Integration Applied to Stochastic Simulation in Industrial Engineering Department of Industrial Engineering and Management Sciences Northwestern University September 15th, 2014

More information

Finite Element Formulation for Plates - Handout 3 -

Finite Element Formulation for Plates - Handout 3 - Finite Element Formulation for Plates - Handout 3 - Dr Fehmi Cirak (fc286@) Completed Version Definitions A plate is a three dimensional solid body with one of the plate dimensions much smaller than the

More information

This is the 11th lecture of this course and the last lecture on the topic of Equilibrium Carrier Concentration.

This is the 11th lecture of this course and the last lecture on the topic of Equilibrium Carrier Concentration. Solid State Devices Dr. S. Karmalkar Department of Electronics and Communication Engineering Indian Institute of Technology, Madras Lecture - 11 Equilibrium Carrier Concentration (Contd.) This is the 11th

More information

ME6130 An introduction to CFD 1-1

ME6130 An introduction to CFD 1-1 ME6130 An introduction to CFD 1-1 What is CFD? Computational fluid dynamics (CFD) is the science of predicting fluid flow, heat and mass transfer, chemical reactions, and related phenomena by solving numerically

More information

Introduction to General and Generalized Linear Models

Introduction to General and Generalized Linear Models Introduction to General and Generalized Linear Models General Linear Models - part I Henrik Madsen Poul Thyregod Informatics and Mathematical Modelling Technical University of Denmark DK-2800 Kgs. Lyngby

More information

Express Introductory Training in ANSYS Fluent Lecture 1 Introduction to the CFD Methodology

Express Introductory Training in ANSYS Fluent Lecture 1 Introduction to the CFD Methodology Express Introductory Training in ANSYS Fluent Lecture 1 Introduction to the CFD Methodology Dimitrios Sofialidis Technical Manager, SimTec Ltd. Mechanical Engineer, PhD PRACE Autumn School 2013 - Industry

More information

Analysis of Multi-Spacecraft Magnetic Field Data

Analysis of Multi-Spacecraft Magnetic Field Data COSPAR Capacity Building Beijing, 5 May 2004 Joachim Vogt Analysis of Multi-Spacecraft Magnetic Field Data 1 Introduction, single-spacecraft vs. multi-spacecraft 2 Single-spacecraft data, minimum variance

More information

PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 4: LINEAR MODELS FOR CLASSIFICATION

PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 4: LINEAR MODELS FOR CLASSIFICATION PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 4: LINEAR MODELS FOR CLASSIFICATION Introduction In the previous chapter, we explored a class of regression models having particularly simple analytical

More information

CHAPTER 12 MOLECULAR SYMMETRY

CHAPTER 12 MOLECULAR SYMMETRY CHAPTER 12 MOLECULAR SYMMETRY In many cases, the symmetry of a molecule provides a great deal of information about its quantum states, even without a detailed solution of the Schrödinger equation. A geometrical

More information

Solid State Theory Physics 545

Solid State Theory Physics 545 Solid State Theory Physics 545 CRYSTAL STRUCTURES Describing periodic structures Terminology Basic Structures Symmetry Operations Ionic crystals often have a definite habit which gives rise to particular

More information

Math 215 HW #6 Solutions

Math 215 HW #6 Solutions Math 5 HW #6 Solutions Problem 34 Show that x y is orthogonal to x + y if and only if x = y Proof First, suppose x y is orthogonal to x + y Then since x, y = y, x In other words, = x y, x + y = (x y) T

More information

Linear Threshold Units

Linear Threshold Units Linear Threshold Units w x hx (... w n x n w We assume that each feature x j and each weight w j is a real number (we will relax this later) We will study three different algorithms for learning linear

More information

Lecture 7 - Meshing. Applied Computational Fluid Dynamics

Lecture 7 - Meshing. Applied Computational Fluid Dynamics Lecture 7 - Meshing Applied Computational Fluid Dynamics Instructor: André Bakker http://www.bakker.org André Bakker (2002-2006) Fluent Inc. (2002) 1 Outline Why is a grid needed? Element types. Grid types.

More information

Visualization and post-processing tools for Siesta

Visualization and post-processing tools for Siesta Visualization and post-processing tools for Siesta Andrei Postnikov Université Paul Verlaine, Metz CECAM tutorial, Lyon, June 22, 2007 Outline 1 What to visualize? 2 XCrySDen by Tone Kokalj 3 Sies2xsf

More information

Lecture 2. Surface Structure

Lecture 2. Surface Structure Lecture 2 Surface Structure Quantitative Description of Surface Structure clean metal surfaces adsorbated covered and reconstructed surfaces electronic and geometrical structure References: 1) Zangwill,

More information

Christfried Webers. Canberra February June 2015

Christfried Webers. Canberra February June 2015 c Statistical Group and College of Engineering and Computer Science Canberra February June (Many figures from C. M. Bishop, "Pattern Recognition and ") 1of 829 c Part VIII Linear Classification 2 Logistic

More information

1 The Brownian bridge construction

1 The Brownian bridge construction The Brownian bridge construction The Brownian bridge construction is a way to build a Brownian motion path by successively adding finer scale detail. This construction leads to a relatively easy proof

More information

An Explicit Solution of the Cubic Spline Interpolation for Polynomials

An Explicit Solution of the Cubic Spline Interpolation for Polynomials An Explicit Solution of the Cubic Spline Interpolation for Polynomials Byung Soo Moon Korea Atomic Energy Research Institute P.O. Box105, Taeduk Science Town Taejon, Korea 305-600 E-mail:bsmoon@@nanum.kaeri.re.kr

More information

The properties of an ideal Fermi gas are strongly determined by the Pauli principle. We shall consider the limit: µ >> k B T βµ >> 1,

The properties of an ideal Fermi gas are strongly determined by the Pauli principle. We shall consider the limit: µ >> k B T βµ >> 1, Chapter 3 Ideal Fermi gas The properties of an ideal Fermi gas are strongly determined by the Pauli principle. We shall consider the limit: µ >> k B T βµ >>, which defines the degenerate Fermi gas. In

More information

Interpolating and Approximating Implicit Surfaces from Polygon Soup

Interpolating and Approximating Implicit Surfaces from Polygon Soup Interpolating and Approximating Implicit Surfaces from Polygon Soup 1. Briefly summarize the paper s contributions. Does it address a new problem? Does it present a new approach? Does it show new types

More information

Genetic Algorithms commonly used selection, replacement, and variation operators Fernando Lobo University of Algarve

Genetic Algorithms commonly used selection, replacement, and variation operators Fernando Lobo University of Algarve Genetic Algorithms commonly used selection, replacement, and variation operators Fernando Lobo University of Algarve Outline Selection methods Replacement methods Variation operators Selection Methods

More information

- in a typical metal each atom contributes one electron to the delocalized electron gas describing the conduction electrons

- in a typical metal each atom contributes one electron to the delocalized electron gas describing the conduction electrons Free Electrons in a Metal - in a typical metal each atom contributes one electron to the delocalized electron gas describing the conduction electrons - if these electrons would behave like an ideal gas

More information

Solid State Theory. (Theorie der kondensierten Materie) Wintersemester 2015/16

Solid State Theory. (Theorie der kondensierten Materie) Wintersemester 2015/16 Martin Eckstein Solid State Theory (Theorie der kondensierten Materie) Wintersemester 2015/16 Max-Planck Research Departement for Structural Dynamics, CFEL Desy, Bldg. 99, Rm. 02.001 Luruper Chaussee 149

More information

Calculation of Eigenmodes in Superconducting Cavities

Calculation of Eigenmodes in Superconducting Cavities Calculation of Eigenmodes in Superconducting Cavities W. Ackermann, C. Liu, W.F.O. Müller, T. Weiland Institut für Theorie Elektromagnetischer Felder, Technische Universität Darmstadt Status Meeting December

More information

Free Electron Fermi Gas (Kittel Ch. 6)

Free Electron Fermi Gas (Kittel Ch. 6) Free Electron Fermi Gas (Kittel Ch. 6) Role of Electrons in Solids Electrons are responsible for binding of crystals -- they are the glue that hold the nuclei together Types of binding (see next slide)

More information

CAD and Finite Element Analysis

CAD and Finite Element Analysis CAD and Finite Element Analysis Most ME CAD applications require a FEA in one or more areas: Stress Analysis Thermal Analysis Structural Dynamics Computational Fluid Dynamics (CFD) Electromagnetics Analysis...

More information

LECTURE NOTES: FINITE ELEMENT METHOD

LECTURE NOTES: FINITE ELEMENT METHOD LECTURE NOTES: FINITE ELEMENT METHOD AXEL MÅLQVIST. Motivation The finite element method has two main strengths... Geometry. Very complex geometries can be used. This is probably the main reason why finite

More information

820446 - ACMSM - Computer Applications in Solids Mechanics

820446 - ACMSM - Computer Applications in Solids Mechanics Coordinating unit: 820 - EUETIB - Barcelona College of Industrial Engineering Teaching unit: 737 - RMEE - Department of Strength of Materials and Structural Engineering Academic year: Degree: 2015 BACHELOR'S

More information

Coefficient of Potential and Capacitance

Coefficient of Potential and Capacitance Coefficient of Potential and Capacitance Lecture 12: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay We know that inside a conductor there is no electric field and that

More information

DYNAMIC ANALYSIS OF THICK PLATES SUBJECTED TO EARTQUAKE

DYNAMIC ANALYSIS OF THICK PLATES SUBJECTED TO EARTQUAKE DYNAMIC ANALYSIS OF THICK PLATES SUBJECTED TO EARTQUAKE ÖZDEMİR Y. I, AYVAZ Y. Posta Adresi: Department of Civil Engineering, Karadeniz Technical University, 68 Trabzon, TURKEY E-posta: yaprakozdemir@hotmail.com

More information

AN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS

AN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS AN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS Revised Edition James Epperson Mathematical Reviews BICENTENNIAL 0, 1 8 0 7 z ewiley wu 2007 r71 BICENTENNIAL WILEY-INTERSCIENCE A John Wiley & Sons, Inc.,

More information

Some general points about bandstructure

Some general points about bandstructure Handout 5 Some general points about bandstructure 5.1 Comparison of tight-binding and nearly-free-electron bandstructure Let us compare a band of the nearly-free-electron model with a one-dimensional tight-binding

More information

Topic 4. Chemical bonding and structure

Topic 4. Chemical bonding and structure Topic 4. Chemical bonding and structure There are three types of strong bonds: Ionic Covalent Metallic Some substances contain both covalent and ionic bonding or an intermediate. 4.1 Ionic bonding Ionic

More information

CS 688 Pattern Recognition Lecture 4. Linear Models for Classification

CS 688 Pattern Recognition Lecture 4. Linear Models for Classification CS 688 Pattern Recognition Lecture 4 Linear Models for Classification Probabilistic generative models Probabilistic discriminative models 1 Generative Approach ( x ) p C k p( C k ) Ck p ( ) ( x Ck ) p(

More information

Continuous Groups, Lie Groups, and Lie Algebras

Continuous Groups, Lie Groups, and Lie Algebras Chapter 7 Continuous Groups, Lie Groups, and Lie Algebras Zeno was concerned with three problems... These are the problem of the infinitesimal, the infinite, and continuity... Bertrand Russell The groups

More information

Group Theory and Chemistry

Group Theory and Chemistry Group Theory and Chemistry Outline: Raman and infra-red spectroscopy Symmetry operations Point Groups and Schoenflies symbols Function space and matrix representation Reducible and irreducible representation

More information

Optical Properties of Solids. Claudia Ambrosch-Draxl Chair of Atomistic Modelling and Design of Materials University Leoben, Austria

Optical Properties of Solids. Claudia Ambrosch-Draxl Chair of Atomistic Modelling and Design of Materials University Leoben, Austria Optical Properties of Solids Claudia Ambrosch-Draxl Chair of Atomistic Modelling and Design of Materials University Leoben, Austria Outline Basics Program Examples Outlook light scattering dielectric tensor

More information

Numerical Methods Lecture 5 - Curve Fitting Techniques

Numerical Methods Lecture 5 - Curve Fitting Techniques Numerical Methods Lecture 5 - Curve Fitting Techniques Topics motivation interpolation linear regression higher order polynomial form exponential form Curve fitting - motivation For root finding, we used

More information

Model Order Reduction for Linear Convective Thermal Flow

Model Order Reduction for Linear Convective Thermal Flow Model Order Reduction for Linear Convective Thermal Flow Christian Moosmann, Evgenii B. Rudnyi, Andreas Greiner, Jan G. Korvink IMTEK, April 24 Abstract Simulation of the heat exchange between a solid

More information

Calculation of Eigenfields for the European XFEL Cavities

Calculation of Eigenfields for the European XFEL Cavities Calculation of Eigenfields for the European XFEL Cavities Wolfgang Ackermann, Erion Gjonaj, Wolfgang F. O. Müller, Thomas Weiland Institut Theorie Elektromagnetischer Felder, TU Darmstadt Status Meeting

More information

A python pre-processing module for Chimera assembly. S. Péron, C. Benoit ONERA, CFD and Aeroacoustics Dept.

A python pre-processing module for Chimera assembly. S. Péron, C. Benoit ONERA, CFD and Aeroacoustics Dept. A python pre-processing module for Chimera assembly S. Péron, C. Benoit ONERA, CFD and Aeroacoustics Dept. Outline of talk Motivation Principle of the Chimera assembly Basic examples Implementation in

More information

ADVANCED LINEAR ALGEBRA FOR ENGINEERS WITH MATLAB. Sohail A. Dianat. Rochester Institute of Technology, New York, U.S.A. Eli S.

ADVANCED LINEAR ALGEBRA FOR ENGINEERS WITH MATLAB. Sohail A. Dianat. Rochester Institute of Technology, New York, U.S.A. Eli S. ADVANCED LINEAR ALGEBRA FOR ENGINEERS WITH MATLAB Sohail A. Dianat Rochester Institute of Technology, New York, U.S.A. Eli S. Saber Rochester Institute of Technology, New York, U.S.A. (g) CRC Press Taylor

More information

Introduction to the Finite Element Method

Introduction to the Finite Element Method Introduction to the Finite Element Method 09.06.2009 Outline Motivation Partial Differential Equations (PDEs) Finite Difference Method (FDM) Finite Element Method (FEM) References Motivation Figure: cross

More information

Diploma Plus in Certificate in Advanced Engineering

Diploma Plus in Certificate in Advanced Engineering Diploma Plus in Certificate in Advanced Engineering Mathematics New Syllabus from April 2011 Ngee Ann Polytechnic / School of Interdisciplinary Studies 1 I. SYNOPSIS APPENDIX A This course of advanced

More information

Advanced Higher Mathematics Course Assessment Specification (C747 77)

Advanced Higher Mathematics Course Assessment Specification (C747 77) Advanced Higher Mathematics Course Assessment Specification (C747 77) Valid from August 2015 This edition: April 2016, version 2.4 This specification may be reproduced in whole or in part for educational

More information

Efficient Curve Fitting Techniques

Efficient Curve Fitting Techniques 15/11/11 Life Conference and Exhibition 11 Stuart Carroll, Christopher Hursey Efficient Curve Fitting Techniques - November 1 The Actuarial Profession www.actuaries.org.uk Agenda Background Outline of

More information

Introduction to CFD Analysis

Introduction to CFD Analysis Introduction to CFD Analysis Introductory FLUENT Training 2006 ANSYS, Inc. All rights reserved. 2006 ANSYS, Inc. All rights reserved. 2-2 What is CFD? Computational fluid dynamics (CFD) is the science

More information

Volume visualization I Elvins

Volume visualization I Elvins Volume visualization I Elvins 1 surface fitting algorithms marching cubes dividing cubes direct volume rendering algorithms ray casting, integration methods voxel projection, projected tetrahedra, splatting

More information

Electrical properties of Carbon Nanotubes

Electrical properties of Carbon Nanotubes Electrical properties of Carbon Nanotubes Kasper Grove-Rasmussen Thomas Jørgensen August 28, 2000 1 Contents 1 Preface 3 2 Introduction to Carbon Nanotubes 4 3 Single wall Carbon Nanotubes 5 4 Reciprocal

More information

Digital Imaging and Multimedia. Filters. Ahmed Elgammal Dept. of Computer Science Rutgers University

Digital Imaging and Multimedia. Filters. Ahmed Elgammal Dept. of Computer Science Rutgers University Digital Imaging and Multimedia Filters Ahmed Elgammal Dept. of Computer Science Rutgers University Outlines What are Filters Linear Filters Convolution operation Properties of Linear Filters Application

More information

jorge s. marques image processing

jorge s. marques image processing image processing images images: what are they? what is shown in this image? What is this? what is an image images describe the evolution of physical variables (intensity, color, reflectance, condutivity)

More information

Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum

Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum UNIT I: The Hyperbolic Functions basic calculus concepts, including techniques for curve sketching, exponential and logarithmic

More information

An-Najah National University Faculty of Engineering Industrial Engineering Department. Course : Quantitative Methods (65211)

An-Najah National University Faculty of Engineering Industrial Engineering Department. Course : Quantitative Methods (65211) An-Najah National University Faculty of Engineering Industrial Engineering Department Course : Quantitative Methods (65211) Instructor: Eng. Tamer Haddad 2 nd Semester 2009/2010 Chapter 5 Example: Joint

More information

STA 4273H: Statistical Machine Learning

STA 4273H: Statistical Machine Learning STA 4273H: Statistical Machine Learning Russ Salakhutdinov Department of Statistics! rsalakhu@utstat.toronto.edu! http://www.cs.toronto.edu/~rsalakhu/ Lecture 6 Three Approaches to Classification Construct

More information

POLYNOMIAL AND MULTIPLE REGRESSION. Polynomial regression used to fit nonlinear (e.g. curvilinear) data into a least squares linear regression model.

POLYNOMIAL AND MULTIPLE REGRESSION. Polynomial regression used to fit nonlinear (e.g. curvilinear) data into a least squares linear regression model. Polynomial Regression POLYNOMIAL AND MULTIPLE REGRESSION Polynomial regression used to fit nonlinear (e.g. curvilinear) data into a least squares linear regression model. It is a form of linear regression

More information

Physics 551: Solid State Physics F. J. Himpsel

Physics 551: Solid State Physics F. J. Himpsel Physics 551: Solid State Physics F. J. Himpsel Background Most of the objects around us are in the solid state. Today s technology relies heavily on new materials, electronics is predominantly solid state.

More information

Customer Training Material. Lecture 2. Introduction to. Methodology ANSYS FLUENT. ANSYS, Inc. Proprietary 2010 ANSYS, Inc. All rights reserved.

Customer Training Material. Lecture 2. Introduction to. Methodology ANSYS FLUENT. ANSYS, Inc. Proprietary 2010 ANSYS, Inc. All rights reserved. Lecture 2 Introduction to CFD Methodology Introduction to ANSYS FLUENT L2-1 What is CFD? Computational Fluid Dynamics (CFD) is the science of predicting fluid flow, heat and mass transfer, chemical reactions,

More information

A logistic approximation to the cumulative normal distribution

A logistic approximation to the cumulative normal distribution A logistic approximation to the cumulative normal distribution Shannon R. Bowling 1 ; Mohammad T. Khasawneh 2 ; Sittichai Kaewkuekool 3 ; Byung Rae Cho 4 1 Old Dominion University (USA); 2 State University

More information

Nonlinear Iterative Partial Least Squares Method

Nonlinear Iterative Partial Least Squares Method Numerical Methods for Determining Principal Component Analysis Abstract Factors Béchu, S., Richard-Plouet, M., Fernandez, V., Walton, J., and Fairley, N. (2016) Developments in numerical treatments for

More information

Statistical Machine Learning

Statistical Machine Learning Statistical Machine Learning UoC Stats 37700, Winter quarter Lecture 4: classical linear and quadratic discriminants. 1 / 25 Linear separation For two classes in R d : simple idea: separate the classes

More information

Section 3: Crystal Binding

Section 3: Crystal Binding Physics 97 Interatomic forces Section 3: rystal Binding Solids are stable structures, and therefore there exist interactions holding atoms in a crystal together. For example a crystal of sodium chloride

More information

Back to Elements - Tetrahedra vs. Hexahedra

Back to Elements - Tetrahedra vs. Hexahedra Back to Elements - Tetrahedra vs. Hexahedra Erke Wang, Thomas Nelson, Rainer Rauch CAD-FEM GmbH, Munich, Germany Abstract This paper presents some analytical results and some test results for different

More information

These slides follow closely the (English) course textbook Pattern Recognition and Machine Learning by Christopher Bishop

These slides follow closely the (English) course textbook Pattern Recognition and Machine Learning by Christopher Bishop Music and Machine Learning (IFT6080 Winter 08) Prof. Douglas Eck, Université de Montréal These slides follow closely the (English) course textbook Pattern Recognition and Machine Learning by Christopher

More information

FINITE DIFFERENCE METHODS

FINITE DIFFERENCE METHODS FINITE DIFFERENCE METHODS LONG CHEN Te best known metods, finite difference, consists of replacing eac derivative by a difference quotient in te classic formulation. It is simple to code and economic to

More information

Accuracy of the coherent potential approximation for a onedimensional array with a Gaussian distribution of fluctuations in the on-site potential

Accuracy of the coherent potential approximation for a onedimensional array with a Gaussian distribution of fluctuations in the on-site potential Accuracy of the coherent potential approximation for a onedimensional array with a Gaussian distribution of fluctuations in the on-site potential I. Avgin Department of Electrical and Electronics Engineering,

More information

Module 6 : PHYSICS OF SEMICONDUCTOR DEVICES Lecture 34 : Intrinsic Semiconductors

Module 6 : PHYSICS OF SEMICONDUCTOR DEVICES Lecture 34 : Intrinsic Semiconductors Module 6 : PHYSICS OF SEMICONDUCTOR DEVICES Lecture 34 : Intrinsic Semiconductors Objectives In this course you will learn the following Intrinsic and extrinsic semiconductors. Fermi level in a semiconductor.

More information

Symmetry and Molecular Structures

Symmetry and Molecular Structures Symmetry and Molecular Structures Some Readings Chemical Application of Group Theory F. A. Cotton Symmetry through the Eyes of a Chemist I. Hargittai and M. Hargittai The Most Beautiful Molecule - an Adventure

More information

Exploratory Data Analysis

Exploratory Data Analysis Exploratory Data Analysis Johannes Schauer johannes.schauer@tugraz.at Institute of Statistics Graz University of Technology Steyrergasse 17/IV, 8010 Graz www.statistics.tugraz.at February 12, 2008 Introduction

More information

1. χ 2 minimization 2. Fits in case of of systematic errors

1. χ 2 minimization 2. Fits in case of of systematic errors Data fitting Volker Blobel University of Hamburg March 2005 1. χ 2 minimization 2. Fits in case of of systematic errors Keys during display: enter = next page; = next page; = previous page; home = first

More information

Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c

Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c Lesson Outline BIG PICTURE Students will: manipulate algebraic expressions, as needed to understand quadratic relations; identify characteristics

More information

Chapter 3 RANDOM VARIATE GENERATION

Chapter 3 RANDOM VARIATE GENERATION Chapter 3 RANDOM VARIATE GENERATION In order to do a Monte Carlo simulation either by hand or by computer, techniques must be developed for generating values of random variables having known distributions.

More information

arxiv:hep-th/0507236v1 25 Jul 2005

arxiv:hep-th/0507236v1 25 Jul 2005 Non perturbative series for the calculation of one loop integrals at finite temperature Paolo Amore arxiv:hep-th/050736v 5 Jul 005 Facultad de Ciencias, Universidad de Colima, Bernal Diaz del Castillo

More information

SOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties

SOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 3 Fall 2008 III. Spaces with special properties III.1 : Compact spaces I Problems from Munkres, 26, pp. 170 172 3. Show that a finite union of compact subspaces

More information