water adding dye partial mixing homogenization time

Size: px
Start display at page:

Download "water adding dye partial mixing homogenization time"

Transcription

1 iffusion iffusion is a process of mass transport that involves the movement of one atomic species into another. It occurs by ranom atomic jumps from one position to another an takes place in the gaseous, liqui, an soli state for all classes of materials. water aing ye partial mixing time homogenization What is iffusion? iffusion is material transport by atomic motion. Inhomogeneous materials can become homogeneous by iffusion. For an active iffusion to occur, the temperature shoul be high enough to overcome energy barriers to atomic motion.

2 iffusion Mechanisms. There are two main mechanisms of iffusion of atoms in a crystalline lattice: the vacancy or substitutional mechanism the interstitial mechanism Atoms move from concentrate regions to less concentrate regions. Vacancy iffusion. To jump from lattice site to lattice site, atoms nee energy to break bons with neighbors, an to cause the necessary lattice istortions uring jump. This energy comes from the thermal energy of atomic vibrations (E av ~ kt). Materials flow (the atom) is opposite the vacancy flow irection.

3 Interstitial iffusion: Interstitial iffusion is generally faster than vacancy iffusion because boning of interstitials to the surrouning atoms is normally weaker an there are many more interstitial sites than vacancy sites to jump to. Requires small impurity atoms (e.g. C, H, O) to fit into interstices in host. Generation of Point efects Point efects are cause by: 1. Thermal energy X efect n efect n site * E C exp[ ( efect )] kt ln[ Xefect ] lnceefect/ kt Ln[X] E efect /k 1/T

4 Example If, at 400 o C, the concentration of vacancies in aluminum is.3 x 10-5, what is the excess concentration of vacancies if the aluminum is quenche from o C to room temperature? What is the number of vacancies in one cubic μm of quenche aluminum? Given, E s 0.6 ev ; k 86. x 10-6 ev/k, ; r Al nm

5 iffusion Flux: The flux of iffusing atoms, J, is use to quantify how fast iffusion occurs. The flux is efine as either in number of atoms iffusing through unit area an per unit time (e.g., atoms/m -secon) or in terms of the mass flux - mass of atoms iffusing through unit area per unit time, (e.g., kg/m -secon). J M At J 1 δm A δt (Kg m - s -1 ); where M is the mass of atoms iffusing through the area A uring time t. in Area A out δc δx Flux is proportional to the concentration graient an the iffusion coefficient, (m /s), by Fick s first law: J δc δx Steay-State iffusion Flux oes not change with time Concentration profile concentration graient is maintaine constant. Concentration is expresse in terms of mass of iffusing species per unit volume of soli (kg/m 3 ) Negative sign inicates irection of graient It is the riving force [m /s (kg/m 3 )/m] kg/(m s)

6 δc δx c x B B c x A A J C C1 C or J x Δx Fick s First Law of iffusion Where J: the number of atom iffusing own the concentration graient per secon per unit area, unit: atoms/cm s C: the concentration of molecules (or the number of iffuse molecules per unit volume), unit: atoms/cm 3 x: atomic jump istance : iffusion coefficient, unit: cm /s [ ] cm J i units? s C x g cm 4, ( i x, y, z ) J i [ units ] g s cm

7 Example: (Fick s 1 st Law) : A thin plate of BCC Fe, T1000K Oxiizing ensity of Fe: ρ 7.9g/cm CO/CO 3 atmosphere Fe cm /s at 1000K Calculate: the number of carbon atoms transport to t0.1cm back surface per secon through an area of 1cm carbon concentration: C 1 0.wt%; C 0% Solution: wt% ρ The concentration of carbon (atoms/cm 3 Fe ): C N A C C J 1 0.% 7.9g / cm 1.01g / mol C x atoms / cm C1 C t atoms / cm s atoms / mol cm A atoms / cm / s 0.1cm C 3

8 iffusivity -- the proportionality constant between flux an concentration graient epens on: Type of boning iffusion mechanism. Substitutional vs interstitial. Temperature. Type of crystal structure of the host lattice. Interstitial iffusion easier in BCC than in FCC. Type of crystal imperfections. (a) iffusion takes place faster along grain bounaries than elsewhere in a crystal. (b) iffusion is faster along islocation lines than through bulk crystal. (c) Excess vacancies will enhance iffusion. Concentration of iffusing species.

9 iffusion coefficient epens on the temperature ln e o ln RT o - R T is the iffusivity or iffusion Coefficient (m / sec ) o is the prexponential factor or iffusion constant (m / sec ) is the activation energy for iffusion (joules / mole ) R is the gas constant ( joules / (mole eg) ) T is the absolute temperature ( K in Kelvin ) /R

10

11

12 Non Steay State iffusion iffusion flux an the concentration graient at some particular point in a soli vary with time, with a net accumulation of epletion of the iffusing species resulting Fick s secon law apples (when is inepenent of composition)

13 Fick s n Law x A Fick s n Law: C high J in J out C low C t V ( J in J out ) A VA x The rate of change of the number of atoms in the slice V The rate that atoms entering the slice the rate that atoms leaving the slice C t ( J in x J out ) A V C x J x C x C t C x

14 C t x C In wors: The rate of change of composition at position x with time, t, is equal to the rate of change of the prouct of the iffusivity,, times the rate of change of the concentration graient, C x /x, with respect to istance, x. Solutions to the E are possible when physically meaningful bounary conitions are specifie Particularly important solution semi-infinite soli in which surface concentrations are constant, iffusing species is usually a gas, an the partial pressure is maintaine at a constant value Secon orer ifferential equations are nontrivial an ifficult to solve. Consier iffusion in from a surface where the concentration of iffusing species at the surface is always constant. This solution applies to gas iffusion into a soli as in carburization of steels or oping of semiconuctors. Bounary Conitions For t 0, C C o at 0 < x For t > 0 C C S at x 0 an C C o at x

15 where C C - C - C C S surface concentration 1- C o initial uniform bulk concentration x s o o erf x t C x concentration of element at istance x from surface at time t x istance from surface iffusivity of iffusing species in host lattice t time erf error function erf (x/ t) is the Gaussian error function this is like a continuous probability ensity function from 0 to x/ t

16 The equation below emonstrates the relationship between concentration, position, an time C x being a function of the imensionless parameter x/ t may be etermine at any time an position if the parametes C o, C x, an are known C C x s - C - C o o 1 - erf x t Special Case esire to achieve some specific concentration of solute, C 1 in an alloy, then C C x s - C - C o o constant x t constant

17

18 Example The carburization of a steel gear at a temperature of 1000 o C in gaseous CO/CO mixture, took 10hours. How long will take to carburize the steel gear to attain similar concentration conitions at 100 o C? For C in γ iron 0. exp{ / T} cm /s

19 Example: (Fick s n Law) etermine the time it takes to obtain a carbon concentration of 0.4% at epth 0.01cm beneath the surface of an iron bar at 1000 o C. The initial concentration of carbon in the iron bar is 0.0% an the surface concentration is maintaine at 0.40%. The Fe has FCC structure an the iffusion coefficient is 14,000 J / mol RT 10-5 m /s exp( ). Known: T1000 o C, epth x 0.01cm, C X 0.4% C O 0.%, C S 0.4% 10-5 m /s 14,000 exp( J / mol ) R J/K RT Fin: time t?

20 Solution: 14, K 10-5 m /s exp ( ) K m /s cm /s C C X S C C O O erf(z) 0.8, where z erf(z) erf x t x t erf ( z) erf ( z1) erf ( z ) erf ( z ) 1 z z z1 z z

21 x z t (0.01/1.81) t [x / ( 0.906)] / 104s 1.73min C0 + Cs C( xeff, t), C0 + Cs C C 0 C C C C s 0 s 0 t 1.73min. Effective penetration istance: x eff (for 50% of concentration) C0 ( Cs C0) / 0.5 Cs C0 C C erf C C Fick s n eff Law: ( ) s 0 x t erf (0.5) 0.5 x eff t

22 Effective penetration istance In general, for most iffusion problems x eff γ t where γ: a geometry-epenent parameter γ 1 for a flat plate γ for cyliners

23 Thermal iffusion of Impurities into Silicon The ability to moify the properties of a semiconuctor through the aition of controlle amounts of impurity atoms is an important aspect of silicon evice an IC manufacture. There are two principal methos which are use to introuce impurities into silicon, thermal iffusion an ion implantation. We will iscuss the basic equations escribing the impurity profiles below the surface of the wafer using the thermal iffusion metho. Thermal iffusion is a high temperature process where the opant atoms are eposite on to or near the surface of the wafer from the gas phase. Wafers can be batch-processe in furnaces. The impurity profile or istribution is etermine mainly by the iffusion temperature an time, an ecreases monotonically from the surface. The maximum concentration of a particular iffusing impurity is always foun at the surface.

24 The impurity concentration C(x,t) as a function of epth below the wafer surface, x, an iffusion time, t is etermine from Fick's iffusion law; is the iffusion coefficient an varies markely from one impurity to another; some impurities iffuse quickly through silicon (fast iffusants), while others move more slowly (slow iffusants). of impurities in silicon. epens on the temperature of iffusion an can be expresse in the generalize form as (T) o exp(-e A / k B T) where o is the iffusion coefficient extrapolate to infinite temperature an E A is an activation energy (usually quote in ev). Thus, a plot of log (T) (µm / hr) vs 1/T (K -1 ) will give a straight line with slope E A.

25 iffusion: Smaller atoms iffuse more reaily than big ones, an iffusion is faster in open lattices or in open irections Self-iffusion coefficients for Ag epen on the iffusion path. In general the iffusivity if greater through less restrictive structural regions grain bounaries, islocation cores, external surfaces.

26 Example (A)For an ASTM grain size of 6, approximately how many grains woul there be per square inch at a magnification of 100? (B)The iffusion coefficients for copper in aluminum at an o C are 4.8x10-14 an 5.3x10-13 m s -1, respectively. etermine the approximate time at o C that will prouce the same iffusion results (in terms of concentration of Cu at some specific point in Al) as a 10 hour heat treatment at o C. (C) For the problem (B) compute the activation energy for the iffusion of Cu in Al.

27 (A) This problem asks that we compute the number of grains per square inch for an ASTM grain size of 6 at a magnification of 100x. All we nee o is solve for the parameter N in the equation below, inasmuch as n 6. Thus N n grains/in (B) Fick s secon law, as it is esire to achieve some specific concentration conitions. t cons tan t t t t t ( x10 m. s )( 10hours) ( x10 m s ) 110.4hours

28 (C) Using the equation an RT RT RT RT RT o RT o RT o RT o e e e e e e e ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ln ln ln ln 1 1 ln ln ln mol kj K K s m x s m x K mol J T T R T T R RT RT

Introduction To Materials Science FOR ENGINEERS, Ch. 5. Diffusion. MSE 201 Callister Chapter 5

Introduction To Materials Science FOR ENGINEERS, Ch. 5. Diffusion. MSE 201 Callister Chapter 5 Diffusion MSE 201 Callister Chapter 5 1 Goals: Diffusion - how do atoms move through solids? Fundamental concepts and language Diffusion mechanisms Vacancy diffusion Interstitial diffusion Impurities Diffusion

More information

DIFFUSION IN SOLIDS. IE-114 Materials Science and General Chemistry Lecture-5

DIFFUSION IN SOLIDS. IE-114 Materials Science and General Chemistry Lecture-5 DIFFUSION IN SOLIDS IE-114 Materials Science and General Chemistry Lecture-5 Diffusion The mechanism by which matter is transported through matter. It is related to internal atomic movement. Atomic movement;

More information

Applications of First Order Equations

Applications of First Order Equations Applications of First Orer Equations Viscous Friction Consier a small mass that has been roppe into a thin vertical tube of viscous flui lie oil. The mass falls, ue to the force of gravity, but falls more

More information

5-4 Electrostatic Boundary Value Problems

5-4 Electrostatic Boundary Value Problems 11/8/4 Section 54 Electrostatic Bounary Value Problems blank 1/ 5-4 Electrostatic Bounary Value Problems Reaing Assignment: pp. 149-157 Q: A: We must solve ifferential equations, an apply bounary conitions

More information

Diffusion. Diffusion. Diffusion in Solid Materials

Diffusion. Diffusion. Diffusion in Solid Materials Atoms movements in materials Diffusion Movement of atoms in solids, liquids and gases is very important Eamples: Hardening steel, chrome-plating, gas reactions, Si wafers.. etc. We will study: Atomic mechanisms

More information

CHAPTER 4 IMPERFECTIONS IN SOLIDS PROBLEM SOLUTIONS

CHAPTER 4 IMPERFECTIONS IN SOLIDS PROBLEM SOLUTIONS 4-1 CHAPTER 4 IMPERFECTIONS IN SOLIDS PROBLEM SOLUTIONS Vacancies and Self-Interstitials 4.1 In order to compute the fraction of atom sites that are vacant in copper at 1357 K, we must employ Equation

More information

Thermal conductivity of graded composites: Numerical simulations and an effective medium approximation

Thermal conductivity of graded composites: Numerical simulations and an effective medium approximation JOURNAL OF MATERIALS SCIENCE 34 (999)5497 5503 Thermal conuctivity of grae composites: Numerical simulations an an effective meium approximation P. M. HUI Department of Physics, The Chinese University

More information

Model for Dopant and Impurity Segregation During Vapor Phase Growth

Model for Dopant and Impurity Segregation During Vapor Phase Growth Mat. Res. Soc. Symp. Proc. Vol. 648, P3.11.1-7 2001 Materials Research Society Moel for Dopant an Impurity Segregation During Vapor Phase Growth Craig B. Arnol an Michael J. Aziz Division of Engineering

More information

Calculus in the AP Physics C Course The Derivative

Calculus in the AP Physics C Course The Derivative Limits an Derivatives Calculus in the AP Physics C Course The Derivative In physics, the ieas of the rate change of a quantity (along with the slope of a tangent line) an the area uner a curve are essential.

More information

inflow outflow Part I. Regular tasks for MAE598/494 Task 1

inflow outflow Part I. Regular tasks for MAE598/494 Task 1 MAE 494/598, Fall 2016 Project #1 (Regular tasks = 20 points) Har copy of report is ue at the start of class on the ue ate. The rules on collaboration will be release separately. Please always follow the

More information

Diffusion. Diffusion = the spontaneous intermingling of the particles of two or more substances as a result of random thermal motion

Diffusion. Diffusion = the spontaneous intermingling of the particles of two or more substances as a result of random thermal motion Diffusion Diffusion = the spontaneous intermingling of the particles of two or more substances as a result of random thermal motion Fick s First Law Γ ΔN AΔt Γ = flux ΔN = number of particles crossing

More information

Critical Size and Particle Growth

Critical Size and Particle Growth Critical Size an article Growth rof. Sotiris E. ratsinis article Technology Laboratory Department of Mechanical an rocess Engineering, ETH Zürich, Switzerlan www.ptl.ethz.ch 1 Nucleation-Conensation A

More information

Table of Common Derivatives By David Abraham

Table of Common Derivatives By David Abraham Prouct an Quotient Rules: Table of Common Derivatives By Davi Abraham [ f ( g( ] = [ f ( ] g( + f ( [ g( ] f ( = g( [ f ( ] g( g( f ( [ g( ] Trigonometric Functions: sin( = cos( cos( = sin( tan( = sec

More information

Related Rates. Introduction. We are familiar with a variety of mathematical or quantitative relationships, especially geometric ones.

Related Rates. Introduction. We are familiar with a variety of mathematical or quantitative relationships, especially geometric ones. Relate Rates Introuction We are familiar with a variety of mathematical or quantitative relationships, especially geometric ones For example, for the sies of a right triangle we have a 2 + b 2 = c 2 or

More information

N = N A Pb A Pb. = ln N Q v kt. = kt ln v N

N = N A Pb A Pb. = ln N Q v kt. = kt ln v N 5. Calculate the energy for vacancy formation in silver, given that the equilibrium number of vacancies at 800 C (1073 K) is 3.6 10 3 m 3. The atomic weight and density (at 800 C) for silver are, respectively,

More information

Module 16. Diffusion in solids II. Lecture 16. Diffusion in solids II

Module 16. Diffusion in solids II. Lecture 16. Diffusion in solids II Module 16 Diffusion in solids II Lecture 16 Diffusion in solids II 1 NPTEL Phase II : IIT Kharagpur : Prof. R. N. Ghosh, Dept of Metallurgical and Materials Engineering Keywords: Micro mechanisms of diffusion,

More information

Steady-state diffusion is diffusion in which the concentration of the diffusing atoms at

Steady-state diffusion is diffusion in which the concentration of the diffusing atoms at Chapter 7 What is steady state diffusion? Steady-state diffusion is diffusion in which the concentration of the diffusing atoms at any point, x, and hence the concentration gradient at x, in the solid,

More information

Interfacial Defects. Grain Size Determination

Interfacial Defects. Grain Size Determination Interfacial Defects 4.27 For an FCC single crystal, would you expect the surface energy for a (00) plane to be greater or less than that for a () plane? Why? (Note: You may want to consult the solution

More information

Lectures - Week 10 Introduction to Ordinary Differential Equations (ODES) First Order Linear ODEs

Lectures - Week 10 Introduction to Ordinary Differential Equations (ODES) First Order Linear ODEs Lectures - Week 10 Introuction to Orinary Differential Equations (ODES) First Orer Linear ODEs When stuying ODEs we are consiering functions of one inepenent variable, e.g., f(x), where x is the inepenent

More information

IMPLICIT DIFFERENTIATION

IMPLICIT DIFFERENTIATION IMPLICIT DIFFERENTIATION CALCULUS 3 INU0115/515 (MATHS 2) Dr Arian Jannetta MIMA CMath FRAS Implicit Differentiation 1/ 11 Arian Jannetta Explicit an implicit functions Explicit functions An explicit function

More information

Chapter 9 Method of Weighted Residuals

Chapter 9 Method of Weighted Residuals Chapter 9 Metho of Weighte Resiuals 9- Introuction Metho of Weighte Resiuals (MWR) is an approimate technique for solving bounary value problems. It utilizes a trial functions satisfying the prescribe

More information

Unified kinetic model of dopant segregation during vapor-phase growth

Unified kinetic model of dopant segregation during vapor-phase growth PHYSICAL REVIEW B 72, 195419 2005 Unifie kinetic moel of opant segregation uring vapor-phase growth Craig B. Arnol 1, * an Michael J. Aziz 2 1 Department of Mechanical an Aerospace Engineering an Princeton

More information

2-7. Fitting a Model to Data I. A Model of Direct Variation. Lesson. Mental Math

2-7. Fitting a Model to Data I. A Model of Direct Variation. Lesson. Mental Math Lesson 2-7 Fitting a Moel to Data I BIG IDEA If you etermine from a particular set of ata that y varies irectly or inversely as, you can graph the ata to see what relationship is reasonable. Using that

More information

Convergence of Random Walks

Convergence of Random Walks Chapter 16 Convergence of Ranom Walks This lecture examines the convergence of ranom walks to the Wiener process. This is very important both physically an statistically, an illustrates the utility of

More information

TMA 4195 Matematisk modellering Exam Tuesday December 16, :00 13:00 Problems and solution with additional comments

TMA 4195 Matematisk modellering Exam Tuesday December 16, :00 13:00 Problems and solution with additional comments Problem F U L W D g m 3 2 s 2 0 0 0 0 2 kg 0 0 0 0 0 0 Table : Dimension matrix TMA 495 Matematisk moellering Exam Tuesay December 6, 2008 09:00 3:00 Problems an solution with aitional comments The necessary

More information

Calculus and optimization

Calculus and optimization Calculus an optimization These notes essentially correspon to mathematical appenix 2 in the text. 1 Functions of a single variable Now that we have e ne functions we turn our attention to calculus. A function

More information

Lecture 1: Atomic Diffusion

Lecture 1: Atomic Diffusion Part IB Materials Science & Metallurgy H. K. D. H. Bhadeshia Course A, Metals and Alloys Lecture 1: Atomic Diffusion Mass transport in a gas or liquid generally involves the flow of fluid (e.g. convection

More information

Solution to the exam in TFY4230 STATISTICAL PHYSICS Wednesday december 1, 2010

Solution to the exam in TFY4230 STATISTICAL PHYSICS Wednesday december 1, 2010 NTNU Page of 6 Institutt for fysikk Fakultet for fysikk, informatikk og matematikk This solution consists of 6 pages. Solution to the exam in TFY423 STATISTICAL PHYSICS Wenesay ecember, 2 Problem. Particles

More information

HW #3. Concentration of nitrogen within the surface (at the location in question), C B = 2 kg/m3

HW #3. Concentration of nitrogen within the surface (at the location in question), C B = 2 kg/m3 HW #3 Problem 5.7 a. To Find: The distance from the high-pressure side at which the concentration of nitrogen in steel equals 2.0 kg/m 3 under conditions of steady-state diffusion. b. Given: D = 6 10-11

More information

18 EVEN MORE CALCULUS

18 EVEN MORE CALCULUS 8 EVEN MORE CALCULUS Chapter 8 Even More Calculus Objectives After stuing this chapter you shoul be able to ifferentiate an integrate basic trigonometric functions; unerstan how to calculate rates of change;

More information

d dx But have you ever seen a derivation of these results? We ll prove the first result below. cos h 1

d dx But have you ever seen a derivation of these results? We ll prove the first result below. cos h 1 Lecture 5 Some ifferentiation rules Trigonometric functions (Relevant section from Stewart, Seventh Eition: Section 3.3) You all know that sin = cos cos = sin. () But have you ever seen a erivation of

More information

Lecture 5. Symmetric Shearer s Lemma

Lecture 5. Symmetric Shearer s Lemma Stanfor University Spring 208 Math 233: Non-constructive methos in combinatorics Instructor: Jan Vonrák Lecture ate: January 23, 208 Original scribe: Erik Bates Lecture 5 Symmetric Shearer s Lemma Here

More information

Math 342 Partial Differential Equations «Viktor Grigoryan

Math 342 Partial Differential Equations «Viktor Grigoryan Math 342 Partial Differential Equations «Viktor Grigoryan 6 Wave equation: solution In this lecture we will solve the wave equation on the entire real line x R. This correspons to a string of infinite

More information

To understand how scrubbers work, we must first define some terms.

To understand how scrubbers work, we must first define some terms. SRUBBERS FOR PARTIE OETION Backgroun To unerstan how scrubbers work, we must first efine some terms. Single roplet efficiency, η, is similar to single fiber efficiency. It is the fraction of particles

More information

6. Friction and viscosity in gasses

6. Friction and viscosity in gasses IR2 6. Friction an viscosity in gasses 6.1 Introuction Similar to fluis, also for laminar flowing gases Newtons s friction law hols true (see experiment IR1). Using Newton s law the viscosity of air uner

More information

APPROXIMATE SOLUTION FOR TRANSIENT HEAT TRANSFER IN STATIC TURBULENT HE II. B. Baudouy. CEA/Saclay, DSM/DAPNIA/STCM Gif-sur-Yvette Cedex, France

APPROXIMATE SOLUTION FOR TRANSIENT HEAT TRANSFER IN STATIC TURBULENT HE II. B. Baudouy. CEA/Saclay, DSM/DAPNIA/STCM Gif-sur-Yvette Cedex, France APPROXIMAE SOLUION FOR RANSIEN HEA RANSFER IN SAIC URBULEN HE II B. Bauouy CEA/Saclay, DSM/DAPNIA/SCM 91191 Gif-sur-Yvette Ceex, France ABSRAC Analytical solution in one imension of the heat iffusion equation

More information

19 Eigenvalues, Eigenvectors, Ordinary Differential Equations, and Control

19 Eigenvalues, Eigenvectors, Ordinary Differential Equations, and Control 19 Eigenvalues, Eigenvectors, Orinary Differential Equations, an Control This section introuces eigenvalues an eigenvectors of a matrix, an iscusses the role of the eigenvalues in etermining the behavior

More information

Math Notes on differentials, the Chain Rule, gradients, directional derivative, and normal vectors

Math Notes on differentials, the Chain Rule, gradients, directional derivative, and normal vectors Math 18.02 Notes on ifferentials, the Chain Rule, graients, irectional erivative, an normal vectors Tangent plane an linear approximation We efine the partial erivatives of f( xy, ) as follows: f f( x+

More information

Schrödinger s equation.

Schrödinger s equation. Physics 342 Lecture 5 Schröinger s Equation Lecture 5 Physics 342 Quantum Mechanics I Wenesay, February 3r, 2010 Toay we iscuss Schröinger s equation an show that it supports the basic interpretation of

More information

The Standard Atmosphere. Dr Andrew French

The Standard Atmosphere. Dr Andrew French The Stanar Atmosphere Dr Anrew French 1 The International Stanar Atmosphere (ISA) is an iealize moel of the variation of average air pressure an temperature with altitue. Assumptions: The atmosphere consists

More information

Extinction, σ/area. Energy (ev) D = 20 nm. t = 1.5 t 0. t = t 0

Extinction, σ/area. Energy (ev) D = 20 nm. t = 1.5 t 0. t = t 0 Extinction, σ/area 1.5 1.0 t = t 0 t = 0.7 t 0 t = t 0 t = 1.3 t 0 t = 1.5 t 0 0.7 0.9 1.1 Energy (ev) = 20 nm t 1.3 Supplementary Figure 1: Plasmon epenence on isk thickness. We show classical calculations

More information

(ii).conversion from 0 C to Fahrenheit:- 0 C= 5 9. (F- 32) (ii).conversion from Fahrenheit to 0 C:- F= 9 5 C + 32 Relation between different scales:-

(ii).conversion from 0 C to Fahrenheit:- 0 C= 5 9. (F- 32) (ii).conversion from Fahrenheit to 0 C:- F= 9 5 C + 32 Relation between different scales:- Thermal properties of matter Heat: - Heat is a form of energy transferre between two (or more) systems or a system an its surrounings by virtue of temperature ifference. **Conventionally, the heat energy

More information

LINEAR DIFFERENTIAL EQUATIONS OF ORDER 1. where a(x) and b(x) are functions. Observe that this class of equations includes equations of the form

LINEAR DIFFERENTIAL EQUATIONS OF ORDER 1. where a(x) and b(x) are functions. Observe that this class of equations includes equations of the form LINEAR DIFFERENTIAL EQUATIONS OF ORDER 1 We consier ifferential equations of the form y + a()y = b(), (1) y( 0 ) = y 0, where a() an b() are functions. Observe that this class of equations inclues equations

More information

Unit 5: Chemical Kinetics and Equilibrium UNIT 5: CHEMICAL KINETICS AND EQUILIBRIUM

Unit 5: Chemical Kinetics and Equilibrium UNIT 5: CHEMICAL KINETICS AND EQUILIBRIUM UNIT 5: CHEMICAL KINETICS AND EQUILIBRIUM Chapter 14: Chemical Kinetics 14.4 & 14.6: Activation Energy, Temperature Depenence on Reaction Rates & Catalysis Reaction Rates: - the spee of which the concentration

More information

Final Exam Study Guide and Practice Problems Solutions

Final Exam Study Guide and Practice Problems Solutions Final Exam Stuy Guie an Practice Problems Solutions Note: These problems are just some of the types of problems that might appear on the exam. However, to fully prepare for the exam, in aition to making

More information

Physics 2212 K Quiz #2 Solutions Summer 2016

Physics 2212 K Quiz #2 Solutions Summer 2016 Physics 1 K Quiz # Solutions Summer 016 I. (18 points) A positron has the same mass as an electron, but has opposite charge. Consier a positron an an electron at rest, separate by a istance = 1.0 nm. What

More information

Exam 2 Review Solutions

Exam 2 Review Solutions Exam Review Solutions 1. True or False, an explain: (a) There exists a function f with continuous secon partial erivatives such that f x (x, y) = x + y f y = x y False. If the function has continuous secon

More information

Experiment 2, Physics 2BL

Experiment 2, Physics 2BL Experiment 2, Physics 2BL Deuction of Mass Distributions. Last Upate: 2009-05-03 Preparation Before this experiment, we recommen you review or familiarize yourself with the following: Chapters 4-6 in Taylor

More information

Summary: Differentiation

Summary: Differentiation Techniques of Differentiation. Inverse Trigonometric functions The basic formulas (available in MF5 are: Summary: Differentiation ( sin ( cos The basic formula can be generalize as follows: Note: ( sin

More information

SYNCHRONOUS SEQUENTIAL CIRCUITS

SYNCHRONOUS SEQUENTIAL CIRCUITS CHAPTER SYNCHRONOUS SEUENTIAL CIRCUITS Registers an counters, two very common synchronous sequential circuits, are introuce in this chapter. Register is a igital circuit for storing information. Contents

More information

'HVLJQ &RQVLGHUDWLRQ LQ 0DWHULDO 6HOHFWLRQ 'HVLJQ 6HQVLWLYLW\,1752'8&7,21

'HVLJQ &RQVLGHUDWLRQ LQ 0DWHULDO 6HOHFWLRQ 'HVLJQ 6HQVLWLYLW\,1752'8&7,21 Large amping in a structural material may be either esirable or unesirable, epening on the engineering application at han. For example, amping is a esirable property to the esigner concerne with limiting

More information

JUST THE MATHS UNIT NUMBER DIFFERENTIATION 2 (Rates of change) A.J.Hobson

JUST THE MATHS UNIT NUMBER DIFFERENTIATION 2 (Rates of change) A.J.Hobson JUST THE MATHS UNIT NUMBER 10.2 DIFFERENTIATION 2 (Rates of change) by A.J.Hobson 10.2.1 Introuction 10.2.2 Average rates of change 10.2.3 Instantaneous rates of change 10.2.4 Derivatives 10.2.5 Exercises

More information

Oxide growth model. Known as the Deal-Grove or linear-parabolic model

Oxide growth model. Known as the Deal-Grove or linear-parabolic model Oxide growth model Known as the Deal-Grove or linear-parabolic model Important elements of the model: Gas molecules (oxygen or water) are incident on the surface of the wafer. Molecules diffuse through

More information

3-D FEM Modeling of fiber/matrix interface debonding in UD composites including surface effects

3-D FEM Modeling of fiber/matrix interface debonding in UD composites including surface effects IOP Conference Series: Materials Science an Engineering 3-D FEM Moeling of fiber/matrix interface eboning in UD composites incluing surface effects To cite this article: A Pupurs an J Varna 2012 IOP Conf.

More information

ELECTRON DIFFRACTION

ELECTRON DIFFRACTION ELECTRON DIFFRACTION Electrons : wave or quanta? Measurement of wavelength an momentum of electrons. Introuction Electrons isplay both wave an particle properties. What is the relationship between the

More information

collisions of electrons. In semiconductor, in certain temperature ranges the conductivity increases rapidly by increasing temperature

collisions of electrons. In semiconductor, in certain temperature ranges the conductivity increases rapidly by increasing temperature 1.9. Temperature Dependence of Semiconductor Conductivity Such dependence is one most important in semiconductor. In metals, Conductivity decreases by increasing temperature due to greater frequency of

More information

Integration Review. May 11, 2013

Integration Review. May 11, 2013 Integration Review May 11, 2013 Goals: Review the funamental theorem of calculus. Review u-substitution. Review integration by parts. Do lots of integration eamples. 1 Funamental Theorem of Calculus In

More information

Problem Set 2: Solutions

Problem Set 2: Solutions UNIVERSITY OF ALABAMA Department of Physics an Astronomy PH 102 / LeClair Summer II 2010 Problem Set 2: Solutions 1. The en of a charge rubber ro will attract small pellets of Styrofoam that, having mae

More information

EMA5001 Lecture 2 Interstitial Diffusion & Fick s 1 st Law. Prof. Zhe Cheng Mechanical & Materials Engineering Florida International University

EMA5001 Lecture 2 Interstitial Diffusion & Fick s 1 st Law. Prof. Zhe Cheng Mechanical & Materials Engineering Florida International University EMA500 Lecture Interstitial Diffusion & Fick s st Law Prof. Zhe Cheng Mechanical & Materials Engineering Florida International University Substitutional Diffusion Different possibilities Exchange Mechanis

More information

This section outlines the methodology used to calculate the wave load and wave wind load values.

This section outlines the methodology used to calculate the wave load and wave wind load values. COMPUTERS AND STRUCTURES, INC., JUNE 2014 AUTOMATIC WAVE LOADS TECHNICAL NOTE CALCULATION O WAVE LOAD VALUES This section outlines the methoology use to calculate the wave loa an wave win loa values. Overview

More information

Separation of Variables

Separation of Variables Physics 342 Lecture 1 Separation of Variables Lecture 1 Physics 342 Quantum Mechanics I Monay, January 25th, 2010 There are three basic mathematical tools we nee, an then we can begin working on the physical

More information

The derivative of a function f(x) is another function, defined in terms of a limiting expression: f(x + δx) f(x)

The derivative of a function f(x) is another function, defined in terms of a limiting expression: f(x + δx) f(x) Y. D. Chong (2016) MH2801: Complex Methos for the Sciences 1. Derivatives The erivative of a function f(x) is another function, efine in terms of a limiting expression: f (x) f (x) lim x δx 0 f(x + δx)

More information

MATH 205 Practice Final Exam Name:

MATH 205 Practice Final Exam Name: MATH 205 Practice Final Eam Name:. (2 points) Consier the function g() = e. (a) (5 points) Ientify the zeroes, vertical asymptotes, an long-term behavior on both sies of this function. Clearly label which

More information

23 Implicit differentiation

23 Implicit differentiation 23 Implicit ifferentiation 23.1 Statement The equation y = x 2 + 3x + 1 expresses a relationship between the quantities x an y. If a value of x is given, then a corresponing value of y is etermine. For

More information

Implicit Differentiation

Implicit Differentiation Implicit Differentiation Thus far, the functions we have been concerne with have been efine explicitly. A function is efine explicitly if the output is given irectly in terms of the input. For instance,

More information

Non-Equilibrium Continuum Physics TA session #10 TA: Yohai Bar Sinai Dislocations

Non-Equilibrium Continuum Physics TA session #10 TA: Yohai Bar Sinai Dislocations Non-Equilibrium Continuum Physics TA session #0 TA: Yohai Bar Sinai 0.06.206 Dislocations References There are countless books about islocations. The ones that I recommen are Theory of islocations, Hirth

More information

Extrinsic Point Defects: Impurities

Extrinsic Point Defects: Impurities Extrinsic Point Defects: Impurities Substitutional and interstitial impurities Sol solutions, solubility limit Entropy of ing, eal solution model Enthalpy of ing, quasi-chemical model Ideal and regular

More information

TEST 2 (PHY 250) Figure Figure P26.21

TEST 2 (PHY 250) Figure Figure P26.21 TEST 2 (PHY 250) 1. a) Write the efinition (in a full sentence) of electric potential. b) What is a capacitor? c) Relate the electric torque, exerte on a molecule in a uniform electric fiel, with the ipole

More information

Qubit channels that achieve capacity with two states

Qubit channels that achieve capacity with two states Qubit channels that achieve capacity with two states Dominic W. Berry Department of Physics, The University of Queenslan, Brisbane, Queenslan 4072, Australia Receive 22 December 2004; publishe 22 March

More information

Conservation Laws. Chapter Conservation of Energy

Conservation Laws. Chapter Conservation of Energy 20 Chapter 3 Conservation Laws In orer to check the physical consistency of the above set of equations governing Maxwell-Lorentz electroynamics [(2.10) an (2.12) or (1.65) an (1.68)], we examine the action

More information

Lecture XII. where Φ is called the potential function. Let us introduce spherical coordinates defined through the relations

Lecture XII. where Φ is called the potential function. Let us introduce spherical coordinates defined through the relations Lecture XII Abstract We introuce the Laplace equation in spherical coorinates an apply the metho of separation of variables to solve it. This will generate three linear orinary secon orer ifferential equations:

More information

The total derivative. Chapter Lagrangian and Eulerian approaches

The total derivative. Chapter Lagrangian and Eulerian approaches Chapter 5 The total erivative 51 Lagrangian an Eulerian approaches The representation of a flui through scalar or vector fiels means that each physical quantity uner consieration is escribe as a function

More information

arxiv: v1 [physics.flu-dyn] 8 May 2014

arxiv: v1 [physics.flu-dyn] 8 May 2014 Energetics of a flui uner the Boussinesq approximation arxiv:1405.1921v1 [physics.flu-yn] 8 May 2014 Kiyoshi Maruyama Department of Earth an Ocean Sciences, National Defense Acaemy, Yokosuka, Kanagawa

More information

Math 2163, Practice Exam II, Solution

Math 2163, Practice Exam II, Solution Math 63, Practice Exam II, Solution. (a) f =< f s, f t >=< s e t, s e t >, an v v = , so D v f(, ) =< ()e, e > =< 4, 4 > = 4. (b) f =< xy 3, 3x y 4y 3 > an v =< cos π, sin π >=, so

More information

A. Incorrect! The letter t does not appear in the expression of the given integral

A. Incorrect! The letter t does not appear in the expression of the given integral AP Physics C - Problem Drill 1: The Funamental Theorem of Calculus Question No. 1 of 1 Instruction: (1) Rea the problem statement an answer choices carefully () Work the problems on paper as neee (3) Question

More information

Physics 505 Electricity and Magnetism Fall 2003 Prof. G. Raithel. Problem Set 3. 2 (x x ) 2 + (y y ) 2 + (z + z ) 2

Physics 505 Electricity and Magnetism Fall 2003 Prof. G. Raithel. Problem Set 3. 2 (x x ) 2 + (y y ) 2 + (z + z ) 2 Physics 505 Electricity an Magnetism Fall 003 Prof. G. Raithel Problem Set 3 Problem.7 5 Points a): Green s function: Using cartesian coorinates x = (x, y, z), it is G(x, x ) = 1 (x x ) + (y y ) + (z z

More information

A SIMPLE ENGINEERING MODEL FOR SPRINKLER SPRAY INTERACTION WITH FIRE PRODUCTS

A SIMPLE ENGINEERING MODEL FOR SPRINKLER SPRAY INTERACTION WITH FIRE PRODUCTS International Journal on Engineering Performance-Base Fire Coes, Volume 4, Number 3, p.95-3, A SIMPLE ENGINEERING MOEL FOR SPRINKLER SPRAY INTERACTION WITH FIRE PROCTS V. Novozhilov School of Mechanical

More information

Differentiation ( , 9.5)

Differentiation ( , 9.5) Chapter 2 Differentiation (8.1 8.3, 9.5) 2.1 Rate of Change (8.2.1 5) Recall that the equation of a straight line can be written as y = mx + c, where m is the slope or graient of the line, an c is the

More information

Electromagnet Gripping in Iron Foundry Automation Part II: Simulation

Electromagnet Gripping in Iron Foundry Automation Part II: Simulation www.ijcsi.org 238 Electromagnet Gripping in Iron Founry Automation Part II: Simulation Rhythm-Suren Wahwa Department of Prouction an Quality Engineering, NTNU Tronheim, 7051, Norway Abstract This paper

More information

arxiv:cond-mat/ v1 [cond-mat.mtrl-sci] 29 Sep 1997

arxiv:cond-mat/ v1 [cond-mat.mtrl-sci] 29 Sep 1997 Dopant Spatial Distributions: Sample Inepenent Response Function An Maximum Entropy Reconstruction D. P. Chu an M. G. Dowsett Department of Physics, University of Warwick, Coventry CV4 7AL, UK arxiv:con-mat/970933v

More information

PD Controller for Car-Following Models Based on Real Data

PD Controller for Car-Following Models Based on Real Data PD Controller for Car-Following Moels Base on Real Data Xiaopeng Fang, Hung A. Pham an Minoru Kobayashi Department of Mechanical Engineering Iowa State University, Ames, IA 5 Hona R&D The car following

More information

Chapter 4. Electrostatics of Macroscopic Media

Chapter 4. Electrostatics of Macroscopic Media Chapter 4. Electrostatics of Macroscopic Meia 4.1 Multipole Expansion Approximate potentials at large istances 3 x' x' (x') x x' x x Fig 4.1 We consier the potential in the far-fiel region (see Fig. 4.1

More information

Electronic Devices and Circuit Theory

Electronic Devices and Circuit Theory Instructor s Resource Manual to accompany Electronic Devices an Circuit Theory Tenth Eition Robert L. Boylesta Louis Nashelsky Upper Sale River, New Jersey Columbus, Ohio Copyright 2009 by Pearson Eucation,

More information

Related Rates. Introduction

Related Rates. Introduction Relate Rates Introuction We are familiar with a variet of mathematical or quantitative relationships, especiall geometric ones For eample, for the sies of a right triangle we have a 2 + b 2 = c 2 or the

More information

Homework 7 Due 18 November at 6:00 pm

Homework 7 Due 18 November at 6:00 pm Homework 7 Due 18 November at 6:00 pm 1. Maxwell s Equations Quasi-statics o a An air core, N turn, cylinrical solenoi of length an raius a, carries a current I Io cos t. a. Using Ampere s Law, etermine

More information

θ x = f ( x,t) could be written as

θ x = f ( x,t) could be written as 9. Higher orer PDEs as systems of first-orer PDEs. Hyperbolic systems. For PDEs, as for ODEs, we may reuce the orer by efining new epenent variables. For example, in the case of the wave equation, (1)

More information

Calculus BC Section II PART A A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS

Calculus BC Section II PART A A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS Calculus BC Section II PART A A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS. An isosceles triangle, whose base is the interval from (0, 0) to (c, 0), has its verte on the graph

More information

Physics 2212 GJ Quiz #4 Solutions Fall 2015

Physics 2212 GJ Quiz #4 Solutions Fall 2015 Physics 2212 GJ Quiz #4 Solutions Fall 215 I. (17 points) The magnetic fiel at point P ue to a current through the wire is 5. µt into the page. The curve portion of the wire is a semicircle of raius 2.

More information

EE 330 Lecture 12. Devices in Semiconductor Processes. Diodes

EE 330 Lecture 12. Devices in Semiconductor Processes. Diodes EE 330 Lecture 12 evices in Semiconuctor Processes ioes Review from Last Lecture http://www.ayah.com/perioic/mages/perioic%20table.png Review from Last Lecture Review from Last Lecture Silicon opants in

More information

CHEM 211 THE PHYSICAL PROPERTIES OF GASES, LIQUIDS AND SOLUTIONS UNIT 2 THE PHYSICAL PROPERTIES OF LIQUIDS AND SOLUTIONS

CHEM 211 THE PHYSICAL PROPERTIES OF GASES, LIQUIDS AND SOLUTIONS UNIT 2 THE PHYSICAL PROPERTIES OF LIQUIDS AND SOLUTIONS Course Title: Basic Physical Chemistry I Course Coe: CHEM211 Creit Hours: 2.0 Requires: 111 Require for: honours Course Outline: How oes the vapour pressure of a liqui vary with temperature? What is the

More information

Lecture contents. Metal-semiconductor contact

Lecture contents. Metal-semiconductor contact 1 Lecture contents Metal-semiconuctor contact Electrostatics: Full epletion approimation Electrostatics: Eact electrostatic solution Current Methos for barrier measurement Junctions: general approaches,

More information

Alpha Particle scattering

Alpha Particle scattering Introuction Alpha Particle scattering Revise Jan. 11, 014 In this lab you will stuy the interaction of α-particles ( 4 He) with matter, in particular energy loss an elastic scattering from a gol target

More information

An inductance lookup table application for analysis of reluctance stepper motor model

An inductance lookup table application for analysis of reluctance stepper motor model ARCHIVES OF ELECTRICAL ENGINEERING VOL. 60(), pp. 5- (0) DOI 0.478/ v07-0-000-y An inuctance lookup table application for analysis of reluctance stepper motor moel JAKUB BERNAT, JAKUB KOŁOTA, SŁAWOMIR

More information

1 dx. where is a large constant, i.e., 1, (7.6) and Px is of the order of unity. Indeed, if px is given by (7.5), the inequality (7.

1 dx. where is a large constant, i.e., 1, (7.6) and Px is of the order of unity. Indeed, if px is given by (7.5), the inequality (7. Lectures Nine an Ten The WKB Approximation The WKB metho is a powerful tool to obtain solutions for many physical problems It is generally applicable to problems of wave propagation in which the frequency

More information

Analytic Scaling Formulas for Crossed Laser Acceleration in Vacuum

Analytic Scaling Formulas for Crossed Laser Acceleration in Vacuum October 6, 4 ARDB Note Analytic Scaling Formulas for Crosse Laser Acceleration in Vacuum Robert J. Noble Stanfor Linear Accelerator Center, Stanfor University 575 San Hill Roa, Menlo Park, California 945

More information

CURRENT ELECTRICITY Q.1

CURRENT ELECTRICITY Q.1 CUENT EECTCTY Q. Define Electric current an its unit.. Electric Current t can be efine as the time rate of flow of charge in a conuctor is calle Electric Current. The amount of flow of charge Q per unit

More information

Modeling and analysis of hydrogen permeation in mixed proton electronic conductors

Modeling and analysis of hydrogen permeation in mixed proton electronic conductors Chemical Engineering Science 58 (2003 1977 1988 www.elsevier.com/locate/ces Moeling an analysis of hyrogen permeation in mixe proton electronic conuctors Lin Li a;b;1, Enrique Iglesia a;b; a Department

More information

Math 1271 Solutions for Fall 2005 Final Exam

Math 1271 Solutions for Fall 2005 Final Exam Math 7 Solutions for Fall 5 Final Eam ) Since the equation + y = e y cannot be rearrange algebraically in orer to write y as an eplicit function of, we must instea ifferentiate this relation implicitly

More information

Characterization of dielectric barrier discharge in air applying current measurement, numerical simulation and emission spectroscopy

Characterization of dielectric barrier discharge in air applying current measurement, numerical simulation and emission spectroscopy Characterization of ielectric barrier ischarge in air applying current measurement, numerical simulation an emission spectroscopy Priyaarshini Rajasekaran, ikita Bibinov an Peter Awakowicz nstitute for

More information

Calculus of Variations

Calculus of Variations 16.323 Lecture 5 Calculus of Variations Calculus of Variations Most books cover this material well, but Kirk Chapter 4 oes a particularly nice job. x(t) x* x*+ αδx (1) x*- αδx (1) αδx (1) αδx (1) t f t

More information