Solutions to the problems in Chapter 6 and 7
|
|
- Abel Cameron
- 5 years ago
- Views:
Transcription
1 Solutions to the probles in Chapter 6 and Pressure of a Feri gas at zero teperature The nuber of electrons N and the internal energy U, inthevoluev,are N = V D(ε)f(ε)dε, U = V εd(ε)f(ε)dε, () The Feri distribution function f(ε) and the density of states (per unit volue) D(ε) are f(ε) = e β(ε µ) +, D(ε) =A 3 ε, A = π 3. () At zero teperature f(ε) =θ( ε), = kf ( /3, k F = 3π n) (3) Introducing the step function and D(ε) =A ε in (), we ay deterine U() and an alternative expression for A: N = V εf A εdε = 3 VAε3/ F A = 3n ε 3/ F, U() = V 5 Aε5/ F = 3 5 N () i.e. U kf V /3 and the pressure at zero teperature is P = U() = U V 3 V = 5 n = (3π ) /3 n 5/3 (5) 5 in sharp contrast to a classical gas, where the zero-teperature pressure is zero. 6. Density of states in low diensions The density of states in k-space is D k =/(π) d in d diensions (6.3) D(ε) = [d k]δ(ε ε k ) D k δ(ε ε k )d k = (π) d δ(ε ε k )d k () Introducing spherical coordinates in d diensions and k = k, then d dk, d = k = πk dk, d = () πk dk, d =3 The factor in the one-diensional case appears because the one-diensional wave vector ay assue both positive and negative values, whereas k per definition. ε k = ε k = k ( ) / k = εk, dk = dε ε k (3) k ) / D(ε) = ( δ(ε ε π k ) ε k (π) (π) 3 εk π δ(ε ε k ) π ε k δ(ε ε k ) ( ) / ε k ( ) / ε k π ε /, d = π, d = 3 π 3 ε/, d =3
2 The probles of st week ( electrons ) 6.5 Feri pancakes Thin layer of Ag: L x = L y = L = 6 ÅandL z = d. The density of electrons is the sae as the density of atos (one conduction electron per Ag ato), i.e. n =5.6 c 3 =.56 Å 3, according to the properties given in the periodic table on the front page of Marder. The wave function of the free electrons should vanish at the boundaries z =andz = d. This condition is fulfilled if assuing the one-electron wave function to be ψ(x, y, z) e i(k x x+k y y) sin (pqz), q = π, p =,,... () d The electron states are characterized by the two-diensional wave vector k = (k x,k y, ) and the band index p. The eigenenergies are (k = k x + k y ): ε p k = (k + p q ) ε = q () At T =(T T F ), as assued iplicitly in the exercise, the occupied states are all those with energies saller than. The nuber of electrons in the pth band, N p, is zero if ε p >. In the opposite case: N p = L kp (π) πk dk = L k p π (3) where k p is the Feri wave nuber of the pth band, i.e. the largest value of k of occupied states in the pth band, as deterined by ε pkp = ( kp + p q ) = ε F kp = ε ( ) F p q = q εf p () ε (a) In the first case d =. Åorq = π/d =.766 Å,and ε = q = ( ) ev =.369 ev (5).677 Assuing the Feri energy to be saller than the lowest energy of the (p =) band, <ε =ε, then only the (p = ) states are occupied, in which case N has to be equal the total nuber N of free electrons: N = L dn = N = L k π k = πd n =.66 Å (6) Introducing this result in (3) we get the Feri energy and the band width W of the occupied states: = ε k = ε [ ] (k /q) + =7.99 ev, W = ε =5.75 ev (7) which is in accordance with our starting assuption of <ε =.9 ev. These results ay be copared with that obtained for bulk Ag: (bulk) = W (bulk) = ( 3π /3 n) =5.5 ev ()
3 Solid state physics II ( electrons ) 3 (b) In the case of d =. Å, q = π/d =.33 Å and ε =.559 ev. In order to deterine in this case, we need to include ore bands than the lowest one. The nuber of bands turns out to be 3, and [( ) ( ) ( )] N = N +N +N 3 πd n = q εf εf + εf + 3 (9) ε ε ε Solving this equation with respect to /ε,weget /ε =.53, which is larger than 3 but saller than, in accordance with the assuption that all electrons are found in the three lowest bands. Hence the results are: =6. ev, W = ε =5.9 ev () 6 p Band energy (ev) Band energy (ev) Wave nuber k [r.l.u.] 6 k 5 p 3 The two first figures show the energy bands as functions of k in the cases (a) and (b), where the unit of k is the length of a reciprocal lattice vector π/a [r.l.u.]. The last figure shows the Feri energy (and the band width) as a function of the nuber of atoic layers of Ag. The crystal structure of Ag is fcc with the lattice paraeter a =.9 Å. This is very nearly the thickness d assued in (a), i.e. this case corresponds to a fil with two atoic layers of Ag atos. Feri energy and band width (ev) k Wave nuber k [r.l.u] 6 W 6 Nuber of Ag layers (d/a) k k Bulk value
4 The probles of st week ( electrons ) Solution to HS s proble Heat capacity of a two-diensional electron gas GaAs/AlGaAs heterostructure (see Marder Section 9.5): n = c, ε = k, =.67 e () The present situation corresponds to the case (a) of the previous proble 6.5 in Marder, i.e. the gas is purely two-diensional in the sense that only the (p =) band needs to be considered, and k is a two-diensional vector with the length k = kx + ky. The ost iportant quantity is the Feri energy, which is deterined by evaluating N at zero teperature: n = N A = D k d k = kf k k F (π) πk dk = k F π k F = πn (a) The sae result is obtained by using that, according to proble 6. or equation (6.35), D(ε) = /π in the two-diensional case: n = εf D(ε) dε = π = nπ or k F = πn (b) Introducing the nubers, and using that k F =Å corresponds to a Feri energy =3. ev when the ass is e,thenweget ( k F =.793 Å.793 ), = 3. ev = 3.5 ev (3).67 This Feri energy corresponds to a Feri teperature T F = /k B =.5 K. ) T = K is uch saller than the Feri teperature and the heat capacity ay be deterined by the leading order expression (6.77) c V = π 3 D()k BT = π 6 T T F k Fk B, as D( )= π = In the case of a saple with the area A =c the result is k F π = k F πk B T F () C V (el) = Ac V = A π 6.5 ( c ) J/K =. 3 J/K (5) The explicit result is C V (el) = Aπ kb T/(3 ). Hence, the sall value of C V is not due to the low electron density n but to the two-diensionality of the syste and the sall effective ass. ) In order to estiate the phonon contribution to the heat capacity we shall use the Debye odel (Section 3.3. in Marder). The Debye teperature of GaAs is Θ D = 3 K (which value is not uch change when soe of the Ga ions are replaced by Al ions). Using (3.7) in Marder and x e x π (e x dx = ) 5 ( ) 3 c V = π T 5 n A k B (T Θ Θ D ) (6) D
5 Solid state physics II ( electrons ) 5 According to Table.5 (page 7) in Marder, GaAs has the zincblende structure ( diaond structure) with the lattice paraeter a = 5.63 Å. In this structure there are atos per unit cell, and the density of atos is n A =/a 3 =. c 3. [The ean atoic ass (periodic table) is ( )/ = 7.3 u, iplying a ass density ρ =5.3 g c 3 ]. Assuing V =c 3 and T = K, the phonon contribution becoes C V (ph) = π ( ) 3 J/K = J/K (7) 3 which is uch larger than the electronic contribution. Utilizing that C V (el) T and C V (ph) T 3, the teperature T at which the two contributions are equal, is deterined by (T in K). 3 T = T 3 T =. K () A ore fair coparison would be to consider a fil of thickness µ, i.e. A =c and V = c 3,inwhichcaseT =. K,ateperaturewithin an accessible range (however, the reduction of the size of the saple akes it ore difficult to deterine the heat capacity). Solutions to the probles in Chapter 7 7. Norals to surfaces r =(x,x,x 3 )= s(t) is the paraetrization of a curve lying within the surface defined by f( r) =ε. Since f( s(t)) is a constant ε, the derivative of this function is : d dt f( s(t)) = α f ds α x α dt = f d s dt = () Because s(t) ay be any arbitrary curve lying within the surface, the sae is true for the curve tangent d s(t)/dt, and () is only generally valid if f is noral to the surface. 7.3 Van Hove singularities (a) The proble becoes the sae as the one considered in proble 7. if aking the replaceents k r and ε n k f( r), hence k ε n k is perpendicular to the energy surface defined by ε n k = ε. (b) The energy is assued to be ε n k = ε ax k, and in the two-diensional case D(ε) = [d k]δ(ε ε n k )= (π) πk δ(ε ε ax + k ) dk = δ(ε ε π ax + k )d(k )= () π θ(ε ax ε) The density of states is zero if ε>ε ax and /(π) whenε<ε ax. (c) In the three diensional case, the result is D(ε) = [d k]δ(ε ε n k )= (π) 3 πk δ(ε ε ax + k ) dk = π kδ(ε ε ax + k )d(k )= εax π εθ(ε ax ε) ()
Chemistry 432 Problem Set 11 Spring 2018 Solutions
1. Show that for an ideal gas Cheistry 432 Proble Set 11 Spring 2018 Solutions P V 2 3 < KE > where is the average kinetic energy of the gas olecules. P 1 3 ρ v2 KE 1 2 v2 ρ N V P V 1 3 N v2 2 3 N
More informationpoints Points <40. Results of. Final Exam. Grade C D,F C B
Results of inal Exa 5 6 7 8 9 points Grade C D, Points A 9- + 85-89 7-8 C + 6-69 -59 < # of students Proble (che. equilibriu) Consider the following reaction: CO(g) + H O(g) CO (g) + H (g) In equilibriu
More informationPHY 171. Lecture 14. (February 16, 2012)
PHY 171 Lecture 14 (February 16, 212) In the last lecture, we looked at a quantitative connection between acroscopic and icroscopic quantities by deriving an expression for pressure based on the assuptions
More informationReading from Young & Freedman: For this topic, read the introduction to chapter 25 and sections 25.1 to 25.3 & 25.6.
PHY10 Electricity Topic 6 (Lectures 9 & 10) Electric Current and Resistance n this topic, we will cover: 1) Current in a conductor ) Resistivity 3) Resistance 4) Oh s Law 5) The Drude Model of conduction
More informationPhys463.nb. Many electrons in 1D at T = 0. For a large system (L ), ΕF =? (6.7) The solutions of this equation are plane waves (6.
â â x Ψn Hx Ε Ψn Hx 35 (6.7) he solutions of this equation are plane waves Ψn Hx A exphä n x (6.8) he eigen-energy Εn is n (6.9) Εn For a D syste with length and periodic boundary conditions, Ψn Hx Ψn
More information1 (40) Gravitational Systems Two heavy spherical (radius 0.05R) objects are located at fixed positions along
(40) Gravitational Systes Two heavy spherical (radius 0.05) objects are located at fixed positions along 2M 2M 0 an axis in space. The first ass is centered at r = 0 and has a ass of 2M. The second ass
More informationChem/Biochem 471 Exam 3 12/18/08 Page 1 of 7 Name:
Che/Bioche 47 Exa /8/08 Pae of 7 Please leave the exa paes stapled toether. The forulas are on a separate sheet. This exa has 5 questions. You ust answer at least 4 of the questions. You ay answer ore
More informationKeywords: Estimator, Bias, Mean-squared error, normality, generalized Pareto distribution
Testing approxiate norality of an estiator using the estiated MSE and bias with an application to the shape paraeter of the generalized Pareto distribution J. Martin van Zyl Abstract In this work the norality
More information1 Bounding the Margin
COS 511: Theoretical Machine Learning Lecturer: Rob Schapire Lecture #12 Scribe: Jian Min Si March 14, 2013 1 Bounding the Margin We are continuing the proof of a bound on the generalization error of AdaBoost
More informationE0 370 Statistical Learning Theory Lecture 6 (Aug 30, 2011) Margin Analysis
E0 370 tatistical Learning Theory Lecture 6 (Aug 30, 20) Margin Analysis Lecturer: hivani Agarwal cribe: Narasihan R Introduction In the last few lectures we have seen how to obtain high confidence bounds
More informationProblem Set 2. Chapter 1 Numerical:
Chapter 1 Nuerical: roble Set 16. The atoic radius of xenon is 18 p. Is that consistent with its b paraeter of 5.15 1 - L/ol? Hint: what is the volue of a ole of xenon atos and how does that copare to
More informationPh 20.3 Numerical Solution of Ordinary Differential Equations
Ph 20.3 Nuerical Solution of Ordinary Differential Equations Due: Week 5 -v20170314- This Assignent So far, your assignents have tried to failiarize you with the hardware and software in the Physics Coputing
More informationA Simple Regression Problem
A Siple Regression Proble R. M. Castro March 23, 2 In this brief note a siple regression proble will be introduced, illustrating clearly the bias-variance tradeoff. Let Y i f(x i ) + W i, i,..., n, where
More informationMODIFICATION OF AN ANALYTICAL MODEL FOR CONTAINER LOADING PROBLEMS
MODIFICATIO OF A AALYTICAL MODEL FOR COTAIER LOADIG PROBLEMS Reception date: DEC.99 otification to authors: 04 MAR. 2001 Cevriye GECER Departent of Industrial Engineering, University of Gazi 06570 Maltepe,
More informationHW 6 - Solutions Due November 20, 2017
Conteporary Physics I HW 6 HW 6 - Solutions Due Noveber 20, 2017 1. A 4 kg block is attached to a spring with a spring constant k 200N/, and is stretched an aount 0.2 [5 pts each]. (a) Sketch the potential
More informationChapter 4: Hypothesis of Diffusion-Limited Growth
Suary This section derives a useful equation to predict quantu dot size evolution under typical organoetallic synthesis conditions that are used to achieve narrow size distributions. Assuing diffusion-controlled
More informationAn Approximate Model for the Theoretical Prediction of the Velocity Increase in the Intermediate Ballistics Period
An Approxiate Model for the Theoretical Prediction of the Velocity... 77 Central European Journal of Energetic Materials, 205, 2(), 77-88 ISSN 2353-843 An Approxiate Model for the Theoretical Prediction
More informationDarcy s law describes water flux determined by a water potential gradient: q w = k dψ dx
6 Hydraulic Properties The soil hydraulic properties are of utost iportance for a large variety of soil processes and in particular for soil water flow. The two ost coon hydraulic properties are the soil
More informationFeature Extraction Techniques
Feature Extraction Techniques Unsupervised Learning II Feature Extraction Unsupervised ethods can also be used to find features which can be useful for categorization. There are unsupervised ethods that
More informationClassical systems in equilibrium
35 Classical systes in equilibriu Ideal gas Distinguishable particles Here we assue that every particle can be labeled by an index i... and distinguished fro any other particle by its label if not by any
More information12 Towards hydrodynamic equations J Nonlinear Dynamics II: Continuum Systems Lecture 12 Spring 2015
18.354J Nonlinear Dynaics II: Continuu Systes Lecture 12 Spring 2015 12 Towards hydrodynaic equations The previous classes focussed on the continuu description of static (tie-independent) elastic systes.
More informationChapter 6 1-D Continuous Groups
Chapter 6 1-D Continuous Groups Continuous groups consist of group eleents labelled by one or ore continuous variables, say a 1, a 2,, a r, where each variable has a well- defined range. This chapter explores:
More informationPhysics 139B Solutions to Homework Set 3 Fall 2009
Physics 139B Solutions to Hoework Set 3 Fall 009 1. Consider a particle of ass attached to a rigid assless rod of fixed length R whose other end is fixed at the origin. The rod is free to rotate about
More informationAN EFFICIENT CLASS OF CHAIN ESTIMATORS OF POPULATION VARIANCE UNDER SUB-SAMPLING SCHEME
J. Japan Statist. Soc. Vol. 35 No. 005 73 86 AN EFFICIENT CLASS OF CHAIN ESTIMATORS OF POPULATION VARIANCE UNDER SUB-SAMPLING SCHEME H. S. Jhajj*, M. K. Shara* and Lovleen Kuar Grover** For estiating the
More informationSOLUTIONS. PROBLEM 1. The Hamiltonian of the particle in the gravitational field can be written as, x 0, + U(x), U(x) =
SOLUTIONS PROBLEM 1. The Hailtonian of the particle in the gravitational field can be written as { Ĥ = ˆp2, x 0, + U(x), U(x) = (1) 2 gx, x > 0. The siplest estiate coes fro the uncertainty relation. If
More informationMachine Learning Basics: Estimators, Bias and Variance
Machine Learning Basics: Estiators, Bias and Variance Sargur N. srihari@cedar.buffalo.edu This is part of lecture slides on Deep Learning: http://www.cedar.buffalo.edu/~srihari/cse676 1 Topics in Basics
More informationPAP342-Solid State Physics I Solution 09/10 Semester 2
PAP342-Solid State Physics I Solution 09/10 Seester 2 Wang Shengtao May 10, 2010 Question 1. (a) A scheatic showing the position of the Feri level related to the (b) band edges can be found in [Kittel]
More informationMeasuring Temperature with a Silicon Diode
Measuring Teperature with a Silicon Diode Due to the high sensitivity, nearly linear response, and easy availability, we will use a 1N4148 diode for the teperature transducer in our easureents 10 Analysis
More informationNon-Parametric Non-Line-of-Sight Identification 1
Non-Paraetric Non-Line-of-Sight Identification Sinan Gezici, Hisashi Kobayashi and H. Vincent Poor Departent of Electrical Engineering School of Engineering and Applied Science Princeton University, Princeton,
More information1 The properties of gases The perfect gas
1 The properties of gases 1A The perfect gas Answers to discussion questions 1A. The partial pressure of a gas in a ixture of gases is the pressure the gas would exert if it occupied alone the sae container
More informationMass Spectrum and Decay Constants of Conventional Mesons within an Infrared Confinement Model
Mass Spectru and Decay Constants of Conventional Mesons within an Infrared Confineent Model Gurjav Ganbold (BLTP, JINR; IPT MAS (Mongolia)) in collaboration with: T. Gutsche (Tuebingen) M. A. Ivanov (Dubna)
More informationHee = ~ dxdy\jj+ (x) 'IJ+ (y) u (x- y) \jj (y) \jj (x), V, = ~ dx 'IJ+ (x) \jj (x) V (x), Hii = Z 2 ~ dx dy cp+ (x) cp+ (y) u (x- y) cp (y) cp (x),
SOVIET PHYSICS JETP VOLUME 14, NUMBER 4 APRIL, 1962 SHIFT OF ATOMIC ENERGY LEVELS IN A PLASMA L. E. PARGAMANIK Khar'kov State University Subitted to JETP editor February 16, 1961; resubitted June 19, 1961
More informationWork, Energy and Momentum
Work, Energy and Moentu Work: When a body oves a distance d along straight line, while acted on by a constant force of agnitude F in the sae direction as the otion, the work done by the force is tered
More informationCh 12: Variations on Backpropagation
Ch 2: Variations on Backpropagation The basic backpropagation algorith is too slow for ost practical applications. It ay take days or weeks of coputer tie. We deonstrate why the backpropagation algorith
More informationScattering and bound states
Chapter Scattering and bound states In this chapter we give a review of quantu-echanical scattering theory. We focus on the relation between the scattering aplitude of a potential and its bound states
More informationSolutions to Problems in Condensed Matter Physics 2
Solutions to Problems in Condensed Matter Physics Textbook: Michael P. Marder, Condensed Matter Physics, Wiley, Contents page Chapter 6 and 7 electrons 1 Chapter 11 phonons 6 Chapter 1 elasticity 8 Chapter
More informationModel Fitting. CURM Background Material, Fall 2014 Dr. Doreen De Leon
Model Fitting CURM Background Material, Fall 014 Dr. Doreen De Leon 1 Introduction Given a set of data points, we often want to fit a selected odel or type to the data (e.g., we suspect an exponential
More informationKinetic Molecular Theory of. IGL is a purely empirical law - solely the
Lecture -3. Kinetic Molecular Theory of Ideal Gases Last Lecture. IGL is a purely epirical law - solely the consequence of experiental obserations Explains the behaior of gases oer a liited range of conditions.
More informationSpine Fin Efficiency A Three Sided Pyramidal Fin of Equilateral Triangular Cross-Sectional Area
Proceedings of the 006 WSEAS/IASME International Conference on Heat and Mass Transfer, Miai, Florida, USA, January 18-0, 006 (pp13-18) Spine Fin Efficiency A Three Sided Pyraidal Fin of Equilateral Triangular
More informationDispersion. February 12, 2014
Dispersion February 1, 014 In aterials, the dielectric constant and pereability are actually frequency dependent. This does not affect our results for single frequency odes, but when we have a superposition
More informationChaotic Coupled Map Lattices
Chaotic Coupled Map Lattices Author: Dustin Keys Advisors: Dr. Robert Indik, Dr. Kevin Lin 1 Introduction When a syste of chaotic aps is coupled in a way that allows the to share inforation about each
More informationKinetic Molecular Theory of Ideal Gases
Lecture -3. Kinetic Molecular Theory of Ideal Gases Last Lecture. IGL is a purely epirical law - solely the consequence of experiental obserations Explains the behaior of gases oer a liited range of conditions.
More informationLecture #8-3 Oscillations, Simple Harmonic Motion
Lecture #8-3 Oscillations Siple Haronic Motion So far we have considered two basic types of otion: translation and rotation. But these are not the only two types of otion we can observe in every day life.
More informationExtension of CSRSM for the Parametric Study of the Face Stability of Pressurized Tunnels
Extension of CSRSM for the Paraetric Study of the Face Stability of Pressurized Tunnels Guilhe Mollon 1, Daniel Dias 2, and Abdul-Haid Soubra 3, M.ASCE 1 LGCIE, INSA Lyon, Université de Lyon, Doaine scientifique
More informationHALF-LIFE TIME FOR VOC EMISSION AND SORPTION OF POROUS BUILDING MATERIALS
HAF-IFE TIME FOR VOC EMISSION AND SORPTION OF POROUS BUIDING MATERIAS C.-S. ee is Ph.D. Candidate, Departent of Building, Civil, and Environental Engineering; F. Haghighat is Professor, Departent of Building,
More informationProjectile Motion with Air Resistance (Numerical Modeling, Euler s Method)
Projectile Motion with Air Resistance (Nuerical Modeling, Euler s Method) Theory Euler s ethod is a siple way to approxiate the solution of ordinary differential equations (ode s) nuerically. Specifically,
More informationA Note on the Applied Use of MDL Approximations
A Note on the Applied Use of MDL Approxiations Daniel J. Navarro Departent of Psychology Ohio State University Abstract An applied proble is discussed in which two nested psychological odels of retention
More informationMolecular interactions in beams
Molecular interactions in beas notable advanceent in the experiental study of interolecular forces has coe fro the developent of olecular beas, which consist of a narrow bea of particles, all having the
More informationPROBLEM SOLUTIONS. g, recognizing that 2 is a dimensionless. 1.1 Substituting dimensions into the given equation T 2. constant, we have.
PROBLEM SOLUTIONS 1.1 Substituting diensions into the given equation T g, recognizing that is a diensionless constant, we have T g L or T T T LT Thus, the diensions are consistent. 1. (a) Fro x = Bt, we
More informationLecture 12: Waves in periodic structures
Lecture : Waves in periodic structures Phonons: quantised lattice vibrations of a crystalline solid is: To approach the general topic of waves in periodic structures fro a specific standpoint: Lattice
More informationProbability Distributions
Probability Distributions In Chapter, we ephasized the central role played by probability theory in the solution of pattern recognition probles. We turn now to an exploration of soe particular exaples
More informationKinetic Theory of Gases: Elementary Ideas
Kinetic Theory of Gases: Eleentary Ideas 17th February 2010 1 Kinetic Theory: A Discussion Based on a Siplified iew of the Motion of Gases 1.1 Pressure: Consul Engel and Reid Ch. 33.1) for a discussion
More informationOptical Properties of Plasmas of High-Z Elements
Forschungszentru Karlsruhe Techni und Uwelt Wissenschaftlishe Berichte FZK Optical Properties of Plasas of High-Z Eleents V.Tolach 1, G.Miloshevsy 1, H.Würz Project Kernfusion 1 Heat and Mass Transfer
More informationNumerically repeated support splitting and merging phenomena in a porous media equation with strong absorption. Kenji Tomoeda
Journal of Math-for-Industry, Vol. 3 (C-), pp. Nuerically repeated support splitting and erging phenoena in a porous edia equation with strong absorption To the eory of y friend Professor Nakaki. Kenji
More informationQ5 We know that a mass at the end of a spring when displaced will perform simple m harmonic oscillations with a period given by T = 2!
Chapter 4.1 Q1 n oscillation is any otion in which the displaceent of a particle fro a fixed point keeps changing direction and there is a periodicity in the otion i.e. the otion repeats in soe way. In
More informationChapter 1: Basics of Vibrations for Simple Mechanical Systems
Chapter 1: Basics of Vibrations for Siple Mechanical Systes Introduction: The fundaentals of Sound and Vibrations are part of the broader field of echanics, with strong connections to classical echanics,
More informationUfuk Demirci* and Feza Kerestecioglu**
1 INDIRECT ADAPTIVE CONTROL OF MISSILES Ufuk Deirci* and Feza Kerestecioglu** *Turkish Navy Guided Missile Test Station, Beykoz, Istanbul, TURKEY **Departent of Electrical and Electronics Engineering,
More informationPhysics 525, Condensed Matter Homework 5 Due Tuesday, 7 th November 2006
Physics 55, Condensed Matter Hoework 5 Due Tuesday, 7 th Noveber 6 Jacob Lewis Bourjaily Proble : Phonon Spectru of a Diatoic One-Diensional Crystal Consider a one-diensional, diatoic crystal coposed of
More informationIn this chapter we will start the discussion on wave phenomena. We will study the following topics:
Chapter 16 Waves I In this chapter we will start the discussion on wave phenoena. We will study the following topics: Types of waves Aplitude, phase, frequency, period, propagation speed of a wave Mechanical
More informationThe Weierstrass Approximation Theorem
36 The Weierstrass Approxiation Theore Recall that the fundaental idea underlying the construction of the real nubers is approxiation by the sipler rational nubers. Firstly, nubers are often deterined
More informationKinetic Theory of Gases: Elementary Ideas
Kinetic Theory of Gases: Eleentary Ideas 9th February 011 1 Kinetic Theory: A Discussion Based on a Siplified iew of the Motion of Gases 1.1 Pressure: Consul Engel and Reid Ch. 33.1) for a discussion of
More informationSF Chemical Kinetics.
SF Cheical Kinetics. Lecture 5. Microscopic theory of cheical reaction inetics. Microscopic theories of cheical reaction inetics. basic ai is to calculate the rate constant for a cheical reaction fro first
More informationResults of Final Exam
Results of Fial Exa # of studets 1 3 4 5 6 7 8 9 Grade Poits A >15 + 1-14 75-94 C + 7-74 C 45-69 D 35-44 F
More informationMassachusetts Institute of Technology Quantum Mechanics I (8.04) Spring 2005 Solutions to Problem Set 4
Massachusetts Institute of Technology Quantu Mechanics I (8.04) Spring 2005 Solutions to Proble Set 4 By Kit Matan 1. X-ray production. (5 points) Calculate the short-wavelength liit for X-rays produced
More informationBiostatistics Department Technical Report
Biostatistics Departent Technical Report BST006-00 Estiation of Prevalence by Pool Screening With Equal Sized Pools and a egative Binoial Sapling Model Charles R. Katholi, Ph.D. Eeritus Professor Departent
More informationCHEM 481 Assignment 3 Answers
CHEM 481 Assignent Answers. In the standard notation of closest-packing of spheres, the letters A, B and C refer to close-packed layers. Which of the following sequences describe closest-packing in diensions,
More informationIn chapter 3, when we discussed metallic bonding, the primary attribute was that electrons are delocalized.
Lecture 15: free electron gas Up to this point we have not discussed electrons explicitly, but electrons are the first offenders for most useful phenomena in materials. When we distinguish between metals,
More informationOBJECTIVES INTRODUCTION
M7 Chapter 3 Section 1 OBJECTIVES Suarize data using easures of central tendency, such as the ean, edian, ode, and idrange. Describe data using the easures of variation, such as the range, variance, and
More informationExperimental Design For Model Discrimination And Precise Parameter Estimation In WDS Analysis
City University of New York (CUNY) CUNY Acadeic Works International Conference on Hydroinforatics 8-1-2014 Experiental Design For Model Discriination And Precise Paraeter Estiation In WDS Analysis Giovanna
More informationNational 5 Summary Notes
North Berwick High School Departent of Physics National 5 Suary Notes Unit 3 Energy National 5 Physics: Electricity and Energy 1 Throughout the Course, appropriate attention should be given to units, prefixes
More informationPhysics 201, Lecture 15
Physics 0, Lecture 5 Today s Topics q More on Linear Moentu And Collisions Elastic and Perfect Inelastic Collision (D) Two Diensional Elastic Collisions Exercise: Billiards Board Explosion q Multi-Particle
More information3.8 Three Types of Convergence
3.8 Three Types of Convergence 3.8 Three Types of Convergence 93 Suppose that we are given a sequence functions {f k } k N on a set X and another function f on X. What does it ean for f k to converge to
More informationBlock designs and statistics
Bloc designs and statistics Notes for Math 447 May 3, 2011 The ain paraeters of a bloc design are nuber of varieties v, bloc size, nuber of blocs b. A design is built on a set of v eleents. Each eleent
More informationCrystallization of Supercooled Liquid Elements Induced by Superclusters Containing Magic Atom Numbers Abstract: Keywords: 1.
Crystallization of Supercooled Liquid Eleents Induced by Superclusters Containing Magic Ato Nubers Robert F. Tournier, CRETA /CNRS, Université Joseph Fourier, B.P. 166, 804 Grenoble cedex 09, France. E-ail:
More informationAn Approach to Conic Sections
n pproach to Conic ections Jia. F. Weng 1998 1 Introduction. The study of conic sections can be traced back to ancient Greek atheaticians, usually to pplonious (ca. 220-190 bc) [2]. The nae conic section
More informationDynamics of Bass Reflex Loudspeaker Systems (3)
MCAP5E Dynaics of Bass Refle Loudspeaker Systes (3) Deriving Equations of Motion of Bass Refle Speaker Systes 3. Deriving Equations of Motion Shigeru Suzuki Released in May 5th, 8 in Japanese Released
More informationElectromagnetic Waves
Electroagnetic Waves Physics 4 Maxwell s Equations Maxwell s equations suarize the relationships between electric and agnetic fields. A ajor consequence of these equations is that an accelerating charge
More informationThe Hydrogen Atom. Nucleus charge +Ze mass m 1 coordinates x 1, y 1, z 1. Electron charge e mass m 2 coordinates x 2, y 2, z 2
The Hydrogen Ato The only ato that can be solved exactly. The results becoe the basis for understanding all other atos and olecules. Orbital Angular Moentu Spherical Haronics Nucleus charge +Ze ass coordinates
More information1 Statistics of volumes, swept by spheroidal particles, in a turbulent flow.
1 Statistics of volues, swept by spheroidal particles, in a turbulent flow. B. Grits*, M. Pinsky, and A. Khain Institute of Earth Science, The Hebrew University of Jerusale 1. INTRODUCTION Collisions between
More informationANALYSIS OF HALL-EFFECT THRUSTERS AND ION ENGINES FOR EARTH-TO-MOON TRANSFER
IEPC 003-0034 ANALYSIS OF HALL-EFFECT THRUSTERS AND ION ENGINES FOR EARTH-TO-MOON TRANSFER A. Bober, M. Guelan Asher Space Research Institute, Technion-Israel Institute of Technology, 3000 Haifa, Israel
More informationProc. of the IEEE/OES Seventh Working Conference on Current Measurement Technology UNCERTAINTIES IN SEASONDE CURRENT VELOCITIES
Proc. of the IEEE/OES Seventh Working Conference on Current Measureent Technology UNCERTAINTIES IN SEASONDE CURRENT VELOCITIES Belinda Lipa Codar Ocean Sensors 15 La Sandra Way, Portola Valley, CA 98 blipa@pogo.co
More informationChapter 11: Vibration Isolation of the Source [Part I]
Chapter : Vibration Isolation of the Source [Part I] Eaple 3.4 Consider the achine arrangeent illustrated in figure 3.. An electric otor is elastically ounted, by way of identical isolators, to a - thick
More informationSoft Computing Techniques Help Assign Weights to Different Factors in Vulnerability Analysis
Soft Coputing Techniques Help Assign Weights to Different Factors in Vulnerability Analysis Beverly Rivera 1,2, Irbis Gallegos 1, and Vladik Kreinovich 2 1 Regional Cyber and Energy Security Center RCES
More informationEgyptian Mathematics Problem Set
(Send corrections to cbruni@uwaterloo.ca) Egyptian Matheatics Proble Set (i) Use the Egyptian area of a circle A = (8d/9) 2 to copute the areas of the following circles with given diaeter. d = 2. d = 3
More informatione = n 1 ( ) 3 [ m 3] = n [ m 3] n
Magnetospheric Physics - Hoework Solutions, /7/4 7. Plasa definition Can a plasa be aintained at teperatures of T e K Hint: Calculate the density liit using the plasa paraeter and explain your result).
More informationare equal to zero, where, q = p 1. For each gene j, the pairwise null and alternative hypotheses are,
Page of 8 Suppleentary Materials: A ultiple testing procedure for ulti-diensional pairwise coparisons with application to gene expression studies Anjana Grandhi, Wenge Guo, Shyaal D. Peddada S Notations
More information6 Free Electron Fermi Gas
6 Free Electron Feri Gas 6.. Electrons in a etal 6... Electrons in one ato One electron in an ato (a hydrogen-lie ato): the nucleon has charge Z e, where Z is the atoic nuber, and there is one electron
More informationUNCERTAINTIES IN THE APPLICATION OF ATMOSPHERIC AND ALTITUDE CORRECTIONS AS RECOMMENDED IN IEC STANDARDS
Paper Published on the16th International Syposiu on High Voltage Engineering, Cape Town, South Africa, 2009 UNCERTAINTIES IN THE APPLICATION OF ATMOSPHERIC AND ALTITUDE CORRECTIONS AS RECOMMENDED IN IEC
More informationA method to determine relative stroke detection efficiencies from multiplicity distributions
A ethod to deterine relative stroke detection eiciencies ro ultiplicity distributions Schulz W. and Cuins K. 2. Austrian Lightning Detection and Inoration Syste (ALDIS), Kahlenberger Str.2A, 90 Vienna,
More information2. Electric Current. E.M.F. of a cell is defined as the maximum potential difference between the two electrodes of the
2. Electric Current The net flow of charges through a etallic wire constitutes an electric current. Do you know who carries current? Current carriers In solid - the electrons in outerost orbit carries
More informationHolographic Description of 2-Dimensional Quantum Black Holes
Author: Facultat de Física, Universitat de Barcelona, Diagonal 645, 0808 Barcelona, Spain. Advisor: Crisitano Gerani Abstract: In this work we obtain and analyse a black hole solution on the 1-brane of
More informationANALYSIS ON RESPONSE OF DYNAMIC SYSTEMS TO PULSE SEQUENCES EXCITATION
The 4 th World Conference on Earthquake Engineering October -7, 8, Beijing, China ANALYSIS ON RESPONSE OF DYNAMIC SYSTEMS TO PULSE SEQUENCES EXCITATION S. Li C.H. Zhai L.L. Xie Ph. D. Student, School of
More informationSupplementary Information for Design of Bending Multi-Layer Electroactive Polymer Actuators
Suppleentary Inforation for Design of Bending Multi-Layer Electroactive Polyer Actuators Bavani Balakrisnan, Alek Nacev, and Elisabeth Sela University of Maryland, College Park, Maryland 074 1 Analytical
More informationCONTINUOUS THERMODYNAMICS FINITE DIFFUSION MODEL FOR MULTICOMPONENT FUEL SPRAY EVAPORATION
CONTINUOUS THERMODYNAMICS FINITE DIFFUSION MODEL FOR MULTICOMPONENT FUEL SPRAY EVAPORATION Dongyao Wang 1 and Chia-fon F. Lee Departent of Theoretical and Applied Mechanics 1 Departent of Mechanical and
More informationIn this chapter, we consider several graph-theoretic and probabilistic models
THREE ONE GRAPH-THEORETIC AND STATISTICAL MODELS 3.1 INTRODUCTION In this chapter, we consider several graph-theoretic and probabilistic odels for a social network, which we do under different assuptions
More informationChapter 2: Introduction to Damping in Free and Forced Vibrations
Chapter 2: Introduction to Daping in Free and Forced Vibrations This chapter ainly deals with the effect of daping in two conditions like free and forced excitation of echanical systes. Daping plays an
More informationLecture Frontier of complexity more is different Think of a spin - a multitude gives all sorts of magnetism due to interactions
Lecture 1 Motivation for course The title of this course is condensed atter physics which includes solids and liquids (and occasionally gases). There are also interediate fors of atter, e.g., glasses,
More informationMultiscale Entropy Analysis: A New Method to Detect Determinism in a Time. Series. A. Sarkar and P. Barat. Variable Energy Cyclotron Centre
Multiscale Entropy Analysis: A New Method to Detect Deterinis in a Tie Series A. Sarkar and P. Barat Variable Energy Cyclotron Centre /AF Bidhan Nagar, Kolkata 700064, India PACS nubers: 05.45.Tp, 89.75.-k,
More informationThe Lagrangian Method vs. other methods (COMPARATIVE EXAMPLE)
The Lagrangian ethod vs. other ethods () This aterial written by Jozef HANC, jozef.hanc@tuke.sk Technical University, Kosice, Slovakia For Edwin Taylor s website http://www.eftaylor.co/ 6 January 003 The
More informationResearch in Area of Longevity of Sylphon Scraies
IOP Conference Series: Earth and Environental Science PAPER OPEN ACCESS Research in Area of Longevity of Sylphon Scraies To cite this article: Natalia Y Golovina and Svetlana Y Krivosheeva 2018 IOP Conf.
More information