PHY 140A: Solid State Physics. Solution to Midterm #1
|
|
- Lawrence Parsons
- 5 years ago
- Views:
Transcription
1 PHY 140A: Solid Stte Physics Solution to Midterm #1 TA: Xun Ji 1 October 24, Emil: jixun@physics.ucl.edu
2 Problem #1 (20pt)Clculte the pcking frction of the body-centered cubic lttice. Solution: (Plese refer Problem #3 in Homework #1.) For bcc lttice, consider the body digonl, we hve 3 = 4r, thus = 4r/ (4pt) (1) Moreover, there re 8 spheres t the corners, nd 1/8 of ech is contined in the cube; there re 1 spheres t the center, which is totlly contined in the cube; thus the number of spheres in the cube is n = = (4pt) (2) 8 The volume of the cube nd the volume of ech sphere re: V c = (4pt) (3) V s = 4π 3 r (4pt) (4) Therefore, the pcking frction is: nv s 3π = = (4pt) (5) V c 8 Problem #2 (40pt)Experimentlists hve creted room-temperture superconductor, nd you ( theorist) think its structure is simple cubic with 6.0Å lttice constnt. The experimentlists hve powder Debye-Scherrer cmer with 1.5Å X-ry source. Clculte the ngulr positions φ of the first 3 diffrction rings for shre of the Nobel prize. (). Drw picture tht illustrtes the diffrction condition relting G ( reciprocl lttice vector), k (the wve vector of the incident X-ry), nd k (the wve vector of the scttered X-ry). (b). Use the picture in prt () to derive reltionship between G, k, nd the scttering ngle φ. (c). Write symbolic expression for G. (d). Use the results of prts ()-(c) to clculte the ngulr positions φ of the first 3 diffrction rings. Your nswer should be numeric, but plese leve ny nontrivil trigonometric functions, squre roots, etc. unevluted. 1
3 Solution: (). As in the Fig. :(8pt) k' G k Figure 1: Illustrtion of the diffrction condition. (b). In the cse of elstic scttering, k = k. From bove figure, the reltionship mong them re: 2 k sin φ 2 = G (8pt) (6) (c). The reciprocl lttice for sc is lso n sc. The expression of G is: G = 2π v v v (8pt) (7) (d). Since k = 2π, from bove, we hve the expression for φ: λ φ = 2 rcsin λ v1 2 + v2 2 + v (4pt) (8) 2 For the simple cubic in this problem, the reciprocl lttice is lso simple cubic. The first three rings of diffrction pek correspond to the cse (v 1, v 2, v 3 ) = (100), (110), nd (111). Substitute into Eqn. (8) with the vlue of wve length λ, nd lttice constnt, we hve: φ 1 = 2 rcsin( 1 8 ) (4pt) 2 φ 2 = 2 rcsin( 8 ) (4pt) 3 φ 3 = 2 rcsin( 8 ) (4pt) (9) Problem #3 Copper oxide lyers: The common building blocks for most high temperture superconductors re copper oxide lyers, s depicted below. Assume the distnce between 2
4 the copper toms (filled circles) is. Sor simplicity, let us lso ssume tht in the third dimension these CuO 2 lyers re simply stcked with spcing c, nd there re no other toms in the crystl. In the first pproximtion the lyers hve four-fold symmetry; the crystl is tetrgonl. (). For the crystl in Fig. 2: sketch the Brvis lttice. indicte possible set of primitive vectors. nd describe the bsis. On the digrm, indicte primitive cell. (b). In some compounds closely relted to the high-temperture superconductors one finds, t closer inspection, tht the CuO 2 lttice is ctully not flt, but tht oxygen toms re moved smll mount out of the plne ( up or down ) in n lternting fshion (see Fig. 3, + mens up nd - mens down.) On the digrm, indicte: the primitive cell. the lttice spcing. nd the bsis for the distorted crystl. (c). Wht is the reciprocl lttice the the new (distorted) Brvis lttice? Solution: Clculte G(v 1, v 2, v 3 ). Drw the reciprocl lttice plne with v 3 = 0. Indicte the primitive reciprocl lttice vector b 1 nd b 2 of the undistorted lttice on the sme drwing. Describe (qulittively) wht hppens in the reciprocl lttice, nd the X-ry diffrction pttern, s the distortion is decresed grdully to zero. (). For the crystl: Sketch the Brvis lttice. As shown in the dsh line in Fig. 2. The Brvis lttice in this 2d plne is squre structure.(2pt) Construct the coordinte s in the figure, then the primitive vectors re: 1 = x, 2 = y, 3 = cz (3pt) (10) The bsis cn be chosen s the copper tom t (0,0) nd the oxygen toms t (0, /2) nd (/2, 0), s mrked out by solid curve in the figure. 3
5 y x 2 1 Figure 2: CuO 2 plne of the superconductor. One primitive cell is the squre with four copper toms t its corners. There re mny other eqully correct choices, for exmple, the squre with copper tom t the center. Both re illustrted in the figure. (b). For the distorted crystl: The primitive cell is shown s the shded re in Fig. 3:(3pt) Now the primitive vector re: 1 = (x + y), 2 = (x y), 3 = cz (11) then the lttice spcing is: = 1 = 2 = (3pt) (12) The bsis for the distorted lttice cn be chosen s mrked out by the solid curve in the figure, tht is, the copper toms t (0, 0) nd (, 0), nd the oxygen toms t (/2, 0), (3/2, 0), (, /2), nd (, /2).(4pt) (c). For the new reciprocl lttice: 4
6 y x ' ' Figure 3: The distorted CuO 2 plne of the superconductor. From the expression of i, we hve: b 1 = (2π) ( 2 3 ) = π (x + y) b 2 = (2π) ( 2 3 ) = π (x y) (13) therefore, the reciprocl vector: b 3 = (2π) ( 2 3 ) = 2π z π G(v 1, v 2, v 3 ) = v 1 (x + y) + v π 2 (x y) + v 2π 3 z (5pt) (14) At v 3 = 0, the reciprocl lttice plne is shown s in the Fig. 4:(5pt) For the undistorted lttice, from Eqn. (10) it is esy to get: nd they re drwn in the Fig. 4.(5pt) b 1 = 2π x, b 2 = 2π y (15) In Fig. 4, the lttice points in the center of ech squre belong to the reciprocl lttice of the distorted lttice, but do not belong to the undistorted 5
7 y x b' 1 b' 2 / b 2 b 1 Figure 4: The reciprocl lttice in v 3 = 0 plne. lttice. Since ech reciprocl lttice point corresponds to diffrction pek in the X-ry diffrction pttern, s the distortion is decresed grdully to zero, the lttice points in the center of ech squre dispper, nd the diffrction pek corresponding to those points will get weker nd weker, nd finlly dispper. In other words, the structure fctor of the reciprocl lttice points in the centers of the squres goes to zero s the distortion decreses. For the peks corresponding to the rest of reciprocl lttice, which belong to both distorted nd undistorted lttice, the pek will get stronger. (5pt) 6
Point Lattices: Bravais Lattices
Physics for Solid Stte Applictions Februry 18, 2004 Lecture 7: Periodic Structures (cont.) Outline Review 2D & 3D Periodic Crystl Structures: Mthemtics X-Ry Diffrction: Observing Reciprocl Spce Point Lttices:
More informationQUB XRD Course. The crystalline state. The Crystalline State
QUB XRD Course Introduction to Crystllogrphy 1 The crystlline stte Mtter Gseous Stte Solid stte Liquid Stte Amorphous (disordered) Crystlline (ordered) 2 The Crystlline Stte A crystl is constructed by
More informationfiziks Institute for NET/JRF, GATE, IIT JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics
Solid Stte Physics JEST-0 Q. bem of X-rys is incident on BCC crystl. If the difference between the incident nd scttered wvevectors is K nxˆkyˆlzˆ where xˆ, yˆ, zˆ re the unit vectors of the ssocited cubic
More informationCrystalline Structures The Basics
Crystlline Structures The sics Crystl structure of mteril is wy in which toms, ions, molecules re sptilly rrnged in 3-D spce. Crystl structure = lttice (unit cell geometry) + bsis (tom, ion, or molecule
More informationKai Sun. University of Michigan, Ann Arbor
Ki Sun University of Michign, Ann Arbor How to see toms in solid? For conductors, we cn utilize scnning tunneling microscope (STM) to see toms (Nobel Prize in Physics in 1986) Limittions: (1) conductors
More information1.Bravais Lattices The Bravais lattices Bravais Lattice detail
1.Brvis Lttices 12.1. The Brvis lttices 2.2.4 Brvis Lttice detil The Brvis lttice re the distinct lttice types which when repeted cn fill the whole spce. The lttice cn therefore be generted by three unit
More informationMiller indices and Family of the Planes
SOLID4 Miller Indices ltest Fmily of Plnes nd Miller indices; Miller indices nd Fmily of the Plnes The geometricl fetures of the crystls represented by lttice points re clled Rtionl. Thus lttice point
More informationProblem 3: Band Structure of YBa 2 Cu 3 O 7
HW 5 SSP 601-2017. here is very relistic clcultion which uses the concepts of lttice, reciprocl spce, Brillouin zone nd tight-binding pproximtion. Go over the solution nd fill up every step nd every detil
More informationDepartment of Electrical and Computer Engineering, Cornell University. ECE 4070: Physics of Semiconductors and Nanostructures.
Deprtment of Electricl nd Computer Engineering, Cornell University ECE 4070: Physics of Semiconductors nd Nnostructures Spring 2014 Exm 2 ` April 17, 2014 INSTRUCTIONS: Every problem must be done in the
More informationIV. CONDENSED MATTER PHYSICS
IV. CONDENSED MATTER PHYSICS UNIT I CRYSTAL PHYSICS Lecture - II Dr. T. J. Shinde Deprtment of Physics Smt. K. R. P. Kny Mhvidyly, Islmpur Simple Crystl Structures Simple cubic (SC) Fce centered cubic
More informationUNIVERSITY OF OSLO. Faculty of Mathematics and Natural Sciences
UNIVERSITY OF OSLO Fculty of Mthemtics nd Nturl Sciences Midterm exm in MENA3100 Dy of exm: 19 th Mrch 2018 Exm hours: 14:30 17:30 This exmintion pper consists of 4 pges including 1 ppendix pge. Permitted
More informationAnalytical Methods for Materials
Anlyticl Methods for Mterils Lesson 7 Crystl Geometry nd Crystllogrphy, Prt 1 Suggested Reding Chpters 2 nd 6 in Wsed et l. 169 Slt crystls N Cl http://helthfreedoms.org/2009/05/24/tble-slt-vs-unrefined-se-slt--primer/
More informationLUMS School of Science and Engineering
LUMS School of Science nd Engineering PH- Solution of ssignment Mrch, 0, 0 Brvis Lttice Answer: We hve given tht c.5(î + ĵ + ˆk) 5 (î + ĵ + ˆk) 0 (î + ĵ + ˆk) c (î + ĵ + ˆk) î + ĵ + ˆk + b + c î, b ĵ nd
More informationJackson 2.26 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell
Jckson 2.26 Homework Problem Solution Dr. Christopher S. Bird University of Msschusetts Lowell PROBLEM: The two-dimensionl region, ρ, φ β, is bounded by conducting surfces t φ =, ρ =, nd φ = β held t zero
More informationName Solutions to Test 3 November 8, 2017
Nme Solutions to Test 3 November 8, 07 This test consists of three prts. Plese note tht in prts II nd III, you cn skip one question of those offered. Some possibly useful formuls cn be found below. Brrier
More informationB M S INSTITUTE OF TECHNOLOGY [Approved by AICTE NEW DELHI, Affiliated to VTU BELGAUM] DEPARTMENT OF PHYSICS. Crystal Structure
B M S INSTITUTE OF TECHNOLOGY [Approved by AICTE NEW DELHI, Affilited to VTU BELGAUM] DEPARTMENT OF PHYSICS COURSE MATERIAL SUBJECT: - Engineering Physics MODULE -IV SUBJECT CODE: - 14 PHY 1 / Crystl Structure
More informationGraduate Students do all problems. Undergraduate students choose three problems.
OPTI 45/55 Midterm Due: Februr, Grdute Students do ll problems. Undergrdute students choose three problems.. Google Erth is improving the resolution of its globl mps with dt from the SPOT5 stellite. The
More informationES.182A Topic 32 Notes Jeremy Orloff
ES.8A Topic 3 Notes Jerem Orloff 3 Polr coordintes nd double integrls 3. Polr Coordintes (, ) = (r cos(θ), r sin(θ)) r θ Stndrd,, r, θ tringle Polr coordintes re just stndrd trigonometric reltions. In
More informationWhat is solid state physics?
Wht is solid stte physics? Explins the properties of solid mterils. Explins the properties of collection of tomic nuclei nd electrons intercting with electrosttic forces. Formultes fundmentl lws tht govern
More informationSTRUCTURAL ISSUES IN SEMICONDUCTORS
Chpter 1 STRUCTURAL ISSUES IN SEMICONDUCTORS Most semiconductor devices re mde from crystlline mterils. The following gures provide n overview of importnt crystlline properties of semiconductors, like
More informationA LEVEL TOPIC REVIEW. factor and remainder theorems
A LEVEL TOPIC REVIEW unit C fctor nd reminder theorems. Use the Fctor Theorem to show tht: ) ( ) is fctor of +. ( mrks) ( + ) is fctor of ( ) is fctor of + 7+. ( mrks) +. ( mrks). Use lgebric division
More information2.57/2.570 Midterm Exam No. 1 March 31, :00 am -12:30 pm
2.57/2.570 Midterm Exm No. 1 Mrch 31, 2010 11:00 m -12:30 pm Instructions: (1) 2.57 students: try ll problems (2) 2.570 students: Problem 1 plus one of two long problems. You cn lso do both long problems,
More informationChem 130 Second Exam
Nme Chem 130 Second Exm On the following pges you will find seven questions covering vries topics rnging from the structure of molecules, ions, nd solids to different models for explining bonding. Red
More informationSTRAND B: NUMBER THEORY
Mthemtics SKE, Strnd B UNIT B Indices nd Fctors: Tet STRAND B: NUMBER THEORY B Indices nd Fctors Tet Contents Section B. Squres, Cubes, Squre Roots nd Cube Roots B. Inde Nottion B. Fctors B. Prime Fctors,
More information1 1. Crystallography 1.1 Introduction 1.2 Crystalline and Non-crystalline materials crystalline materials single crystals polycrystalline material
P g e. Crystllogrphy. Introduction Crystllogrphy is the brnch of science tht dels bout the crystl structures of elements. The crystl structures of elements re studied by mens of X-ry diffrction or electron
More information20 MATHEMATICS POLYNOMIALS
0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of
More informationChapter 1: Fundamentals
Chpter 1: Fundmentls 1.1 Rel Numbers Types of Rel Numbers: Nturl Numbers: {1, 2, 3,...}; These re the counting numbers. Integers: {... 3, 2, 1, 0, 1, 2, 3,...}; These re ll the nturl numbers, their negtives,
More informationProperties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives
Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn
More information1 ST ROUND, SOLUTIONS
ST ROUND, SOLUTIONS Problem (Lithuni) Self destructing pper ( points) Solution ( ) ( ) ( ) [Al HO OH ) ph pk lg [Al H O ( ) ( ) [Al H O OH [Al ( ).9 [Al H O.9.47 [Al ( ) ( ) ( ) [Al H O OH.9 pk ph lg.
More informationProf. Anchordoqui. Problems set # 4 Physics 169 March 3, 2015
Prof. Anchordoui Problems set # 4 Physics 169 Mrch 3, 15 1. (i) Eight eul chrges re locted t corners of cube of side s, s shown in Fig. 1. Find electric potentil t one corner, tking zero potentil to be
More informationUSA Mathematical Talent Search Round 1 Solutions Year 21 Academic Year
1/1/21. Fill in the circles in the picture t right with the digits 1-8, one digit in ech circle with no digit repeted, so tht no two circles tht re connected by line segment contin consecutive digits.
More informationR. I. Badran Solid State Physics
I Bdrn Solid Stte Physics Crystl vibrtions nd the clssicl theory: The ssmption will be mde to consider tht the men eqilibrim position of ech ion is t Brvis lttice site The ions oscillte bot this men position
More informationProblem Set 9. Figure 1: Diagram. This picture is a rough sketch of the 4 parabolas that give us the area that we need to find. The equations are:
(x + y ) = y + (x + y ) = x + Problem Set 9 Discussion: Nov., Nov. 8, Nov. (on probbility nd binomil coefficients) The nme fter the problem is the designted writer of the solution of tht problem. (No one
More informationS56 (5.3) Vectors.notebook January 29, 2016
Dily Prctice 15.1.16 Q1. The roots of the eqution (x 1)(x + k) = 4 re equl. Find the vlues of k. Q2. Find the rte of chnge of 剹 x when x = 1 / 8 Tody we will e lerning out vectors. Q3. Find the eqution
More informationEnergy Bands Energy Bands and Band Gap. Phys463.nb Phenomenon
Phys463.nb 49 7 Energy Bnds Ref: textbook, Chpter 7 Q: Why re there insultors nd conductors? Q: Wht will hppen when n electron moves in crystl? In the previous chpter, we discussed free electron gses,
More informationMath 120 Answers for Homework 13
Mth 12 Answers for Homework 13 1. In this problem we will use the fct tht if m f(x M on n intervl [, b] (nd if f is integrble on [, b] then (* m(b f dx M(b. ( The function f(x = 1 + x 3 is n incresing
More information13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS
33 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS As simple ppliction of the results we hve obtined on lgebric extensions, nd in prticulr on the multiplictivity of extension degrees, we cn nswer (in
More informationWave Phenomena Physics 15c
Wve Phenomen Physics 15c Lecture Diffrction (H&L Chpter 11) Wht We Did Lst Time! Studied interference! or more wves overlp " Amplitudes dd up " Intensity = (mplitude) does not dd up! Thin-film interference!
More informationMATH 253 WORKSHEET 24 MORE INTEGRATION IN POLAR COORDINATES. r dr = = 4 = Here we used: (1) The half-angle formula cos 2 θ = 1 2
MATH 53 WORKSHEET MORE INTEGRATION IN POLAR COORDINATES ) Find the volume of the solid lying bove the xy-plne, below the prboloid x + y nd inside the cylinder x ) + y. ) We found lst time the set of points
More informationARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac
REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b
More informationAndrew Ryba Math Intel Research Final Paper 6/7/09 (revision 6/17/09)
Andrew Ryb Mth ntel Reserch Finl Pper 6/7/09 (revision 6/17/09) Euler's formul tells us tht for every tringle, the squre of the distnce between its circumcenter nd incenter is R 2-2rR, where R is the circumrdius
More informationChapter One Crystal Structure
Chpter One Crystl Structure Drusy Qurtz in Geode Tbulr Orthoclse Feldspr Encrusting Smithsonite Peruvin Pyrite http://www.rockhounds.com/rockshop/xtl 1 Snow crystls the Beltsville Agriculturl Reserch Center
More informationGRADE 4. Division WORKSHEETS
GRADE Division WORKSHEETS Division division is shring nd grouping Division cn men shring or grouping. There re cndies shred mong kids. How mny re in ech shre? = 3 There re 6 pples nd go into ech bsket.
More informationKEY CONCEPTS. satisfies the differential equation da. = 0. Note : If F (x) is any integral of f (x) then, x a
KEY CONCEPTS THINGS TO REMEMBER :. The re ounded y the curve y = f(), the -is nd the ordintes t = & = is given y, A = f () d = y d.. If the re is elow the is then A is negtive. The convention is to consider
More informationβ 1 = 2 π and the path length difference is δ 1 = λ. The small angle approximation gives us y 1 L = tanθ 1 θ 1 sin θ 1 = δ 1 y 1
rgsdle (zdr8) HW13 ditmire (58335) 1 This print-out should hve 1 questions. Multiple-choice questions my continue on the next column or pge find ll choices before nswering. 001 (prt 1 of ) 10.0 points
More informationShape and measurement
C H A P T E R 5 Shpe nd mesurement Wht is Pythgors theorem? How do we use Pythgors theorem? How do we find the perimeter of shpe? How do we find the re of shpe? How do we find the volume of shpe? How do
More informationThe solutions of the single electron Hamiltonian were shown to be Bloch wave of the form: ( ) ( ) ikr
Lecture #1 Progrm 1. Bloch solutions. Reciprocl spce 3. Alternte derivtion of Bloch s theorem 4. Trnsforming the serch for egenfunctions nd eigenvlues from solving PDE to finding the e-vectors nd e-vlues
More informationMath 1102: Calculus I (Math/Sci majors) MWF 3pm, Fulton Hall 230 Homework 2 solutions
Mth 1102: Clculus I (Mth/Sci mjors) MWF 3pm, Fulton Hll 230 Homework 2 solutions Plese write netly, nd show ll work. Cution: An nswer with no work is wrong! Do the following problems from Chpter III: 6,
More information. Double-angle formulas. Your answer should involve trig functions of θ, and not of 2θ. sin 2 (θ) =
Review of some needed Trig. Identities for Integrtion. Your nswers should be n ngle in RADIANS. rccos( 1 ) = π rccos( - 1 ) = 2π 2 3 2 3 rcsin( 1 ) = π rcsin( - 1 ) = -π 2 6 2 6 Cn you do similr problems?
More informationStrategy: Use the Gibbs phase rule (Equation 5.3). How many components are present?
University Chemistry Quiz 4 2014/12/11 1. (5%) Wht is the dimensionlity of the three-phse coexistence region in mixture of Al, Ni, nd Cu? Wht type of geometricl region dose this define? Strtegy: Use the
More informationSection 13.1 Right Triangles
Section 13.1 Right Tringles Ojectives: 1. To find vlues of trigonometric functions for cute ngles. 2. To solve tringles involving right ngles. Review - - 1. SOH sin = Reciprocl csc = 2. H cos = Reciprocl
More informationThings to Memorize: A Partial List. January 27, 2017
Things to Memorize: A Prtil List Jnury 27, 2017 Chpter 2 Vectors - Bsic Fcts A vector hs mgnitude (lso clled size/length/norm) nd direction. It does not hve fixed position, so the sme vector cn e moved
More informationPhysics 121 Sample Common Exam 1 NOTE: ANSWERS ARE ON PAGE 8. Instructions:
Physics 121 Smple Common Exm 1 NOTE: ANSWERS ARE ON PAGE 8 Nme (Print): 4 Digit ID: Section: Instructions: Answer ll questions. uestions 1 through 16 re multiple choice questions worth 5 points ech. You
More informationMTH 4-16a Trigonometry
MTH 4-16 Trigonometry Level 4 [UNIT 5 REVISION SECTION ] I cn identify the opposite, djcent nd hypotenuse sides on right-ngled tringle. Identify the opposite, djcent nd hypotenuse in the following right-ngled
More informationReferences and Resources:
Surfce nd Interfce Science Physics 627; Chemistry 542 Lectures 4 Feb 3, 2013 Determining Surfce Structure Diffrction methods: LEED; RHEED Rel Spce: STEM References nd Resources: Woodruff nd Delchr (2 nd
More informationMaterials Analysis MATSCI 162/172 Laboratory Exercise No. 1 Crystal Structure Determination Pattern Indexing
Mterils Anlysis MATSCI 16/17 Lbortory Exercise No. 1 Crystl Structure Determintion Pttern Inexing Objectives: To inex the x-ry iffrction pttern, ientify the Brvis lttice, n clculte the precise lttice prmeters.
More informationT 1 T 2 T 3 T 4 They may be illustrated by triangular patterns of numbers (hence their name) as shown:
TOPIC 3: VISUAL EXPLANATIONS (PROOFS) (Pge references to Proof re to Bndll, P R et l, Proof in Mthemtics, KMEP, 2002). 3. The tringulr numbers form the sequence, 3, 6, 0,, 2,... T T 2 T 3 T 4 The m be
More information1. Weak acids. For a weak acid HA, there is less than 100% dissociation to ions. The B-L equilibrium is:
th 9 Homework: Reding, M&F, ch. 15, pp. 584-598, 602-605 (clcultions of ph, etc., for wek cids, wek bses, polyprotic cids, nd slts; fctors ffecting cid strength). Problems: Nkon, ch. 18, #1-10, 16-18,
More information. Double-angle formulas. Your answer should involve trig functions of θ, and not of 2θ. cos(2θ) = sin(2θ) =.
Review of some needed Trig Identities for Integrtion Your nswers should be n ngle in RADIANS rccos( 1 2 ) = rccos( - 1 2 ) = rcsin( 1 2 ) = rcsin( - 1 2 ) = Cn you do similr problems? Review of Bsic Concepts
More informationSolid State Electronics EC210 Arab Academy for Science and Technology AAST Cairo Spring 2016 Lecture 1 Crystal Structure
Solid Stte Electronics EC210 AAST Ciro Spring 2016 Lecture 1 Crystl Structure Dr. Amr Byoumi, Dr. Ndi Rft 1 These PowerPoint color digrms cn only be used by instructors if the 3 rd Edition hs been dopted
More informationLesson 8.1 Graphing Parametric Equations
Lesson 8.1 Grphing Prmetric Equtions 1. rete tle for ech pir of prmetric equtions with the given vlues of t.. x t 5. x t 3 c. x t 1 y t 1 y t 3 y t t t {, 1, 0, 1, } t {4,, 0,, 4} t {4, 0,, 4, 8}. Find
More informationragsdale (zdr82) HW2 ditmire (58335) 1
rgsdle (zdr82) HW2 ditmire (58335) This print-out should hve 22 questions. Multiple-choice questions my continue on the next column or pge find ll choices before nswering. 00 0.0 points A chrge of 8. µc
More informationDecember 4, U(x) = U 0 cos 4 πx 8
PHZ66: Fll 013 Problem set # 5: Nerly-free-electron nd tight-binding models: Solutions due Wednesdy, 11/13 t the time of the clss Instructor: D L Mslov mslov@physufledu 39-0513 Rm 11 Office hours: TR 3
More information1 Which of the following summarises the change in wave characteristics on going from infra-red to ultraviolet in the electromagnetic spectrum?
Which of the following summrises the chnge in wve chrcteristics on going from infr-red to ultrviolet in the electromgnetic spectrum? frequency speed (in vcuum) decreses decreses decreses remins constnt
More informationPHYS Summer Professor Caillault Homework Solutions. Chapter 2
PHYS 1111 - Summer 2007 - Professor Cillult Homework Solutions Chpter 2 5. Picture the Problem: The runner moves long the ovl trck. Strtegy: The distnce is the totl length of trvel, nd the displcement
More informationChapter 11. Sequence and Series
Chpter 11 Sequence nd Series Lesson 11-1 Mthemticl Ptterns Sequence A sequence is n ordered list of numbers clled terms. Exmple Pge 591, #2 Describe ech pttern formed. Find the next three terms 4,8,16,32,64,...
More informationGoals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite
Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite
More informationThe area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O
1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the x-xis etween nd is denoted y f(x) dx nd clled the
More informationPhysics 9 Fall 2011 Homework 2 - Solutions Friday September 2, 2011
Physics 9 Fll 0 Homework - s Fridy September, 0 Mke sure your nme is on your homework, nd plese box your finl nswer. Becuse we will be giving prtil credit, be sure to ttempt ll the problems, even if you
More informationKey for Chem 130 Second Exam
Nme Key for Chem 130 Second Exm On the following pges you will find questions tht cover the structure of molecules, ions, nd solids, nd the different models we use to explin the nture of chemicl bonding.
More information7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus
7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus 7.1 INTEGRAL AS NET CHANGE Notecrds from 7.1: Displcement vs Totl Distnce, Integrl s Net Chnge We hve lredy seen how the position of n oject cn e
More informationHQPD - ALGEBRA I TEST Record your answers on the answer sheet.
HQPD - ALGEBRA I TEST Record your nswers on the nswer sheet. Choose the best nswer for ech. 1. If 7(2d ) = 5, then 14d 21 = 5 is justified by which property? A. ssocitive property B. commuttive property
More informationState space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies
Stte spce systems nlysis (continued) Stbility A. Definitions A system is sid to be Asymptoticlly Stble (AS) when it stisfies ut () = 0, t > 0 lim xt () 0. t A system is AS if nd only if the impulse response
More informationMEP Practice Book ES3. 1. Calculate the size of the angles marked with a letter in each diagram. None to scale
ME rctice ook ES3 3 ngle Geometr 3.3 ngle Geometr 1. lculte the size of the ngles mrked with letter in ech digrm. None to scle () 70 () 20 54 65 25 c 36 (d) (e) (f) 56 62 d e 60 40 70 70 f 30 g (g) (h)
More information( ) ( ) Chapter 5 Diffraction condition. ρ j
Grdute School of Engineering Ngo Institute of Technolog Crstl Structure Anlsis Tkshi Id (Advnced Cermics Reserch Center) Updted Nov. 3 3 Chpter 5 Diffrction condition In Chp. 4 it hs been shown tht the
More informationExam 1 Solutions (1) C, D, A, B (2) C, A, D, B (3) C, B, D, A (4) A, C, D, B (5) D, C, A, B
PHY 249, Fll 216 Exm 1 Solutions nswer 1 is correct for ll problems. 1. Two uniformly chrged spheres, nd B, re plced t lrge distnce from ech other, with their centers on the x xis. The chrge on sphere
More information8.6 The Hyperbola. and F 2. is a constant. P F 2. P =k The two fixed points, F 1. , are called the foci of the hyperbola. The line segments F 1
8. The Hperol Some ships nvigte using rdio nvigtion sstem clled LORAN, which is n cronm for LOng RAnge Nvigtion. A ship receives rdio signls from pirs of trnsmitting sttions tht send signls t the sme time.
More informationRecitation 3: More Applications of the Derivative
Mth 1c TA: Pdric Brtlett Recittion 3: More Applictions of the Derivtive Week 3 Cltech 2012 1 Rndom Question Question 1 A grph consists of the following: A set V of vertices. A set E of edges where ech
More informationPolynomials and Division Theory
Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the
More informationGeometric Sequences. Geometric Sequence a sequence whose consecutive terms have a common ratio.
Geometric Sequences Geometric Sequence sequence whose consecutive terms hve common rtio. Geometric Sequence A sequence is geometric if the rtios of consecutive terms re the sme. 2 3 4... 2 3 The number
More information50. Use symmetry to evaluate xx D is the region bounded by the square with vertices 5, Prove Property 11. y y CAS
68 CHAPTE MULTIPLE INTEGALS 46. e da, 49. Evlute tn 3 4 da, where,. [Hint: Eploit the fct tht is the disk with center the origin nd rdius is smmetric with respect to both es.] 5. Use smmetr to evlute 3
More informationCALCULATED POWDER X-RAY DIFFRACTION LINE PROFILES VIA ABSORPTION
16 17 CALCULATED POWDER X-RAY DFFRACTON LNE PROFLES VA ABSORPTON Keji Liu nd Heifen Chen School of Mteril Science nd Engineering, Shnghi nstitute of Technology, Shnghi, Chin 2233 ABSTRACT We hve clculted
More informationSummer Work Packet for MPH Math Classes
Summer Work Pcket for MPH Mth Clsses Students going into Pre-clculus AC Sept. 018 Nme: This pcket is designed to help students sty current with their mth skills. Ech mth clss expects certin level of number
More informationMultiplying integers EXERCISE 2B INDIVIDUAL PATHWAYS. -6 ì 4 = -6 ì 0 = 4 ì 0 = -6 ì 3 = -5 ì -3 = 4 ì 3 = 4 ì 2 = 4 ì 1 = -5 ì -2 = -6 ì 2 = -6 ì 1 =
EXERCISE B INDIVIDUAL PATHWAYS Activity -B- Integer multipliction doc-69 Activity -B- More integer multipliction doc-698 Activity -B- Advnced integer multipliction doc-699 Multiplying integers FLUENCY
More informationLecture 13 - Linking E, ϕ, and ρ
Lecture 13 - Linking E, ϕ, nd ρ A Puzzle... Inner-Surfce Chrge Density A positive point chrge q is locted off-center inside neutrl conducting sphericl shell. We know from Guss s lw tht the totl chrge on
More information( dg. ) 2 dt. + dt. dt j + dh. + dt. r(t) dt. Comparing this equation with the one listed above for the length of see that
Arc Length of Curves in Three Dimensionl Spce If the vector function r(t) f(t) i + g(t) j + h(t) k trces out the curve C s t vries, we cn mesure distnces long C using formul nerly identicl to one tht we
More informationThin Film Scattering: Epitaxial Layers
Thin Film Scttering: Epitxil yers Arturs Vilionis GAM, Stnford University SIMES, SAC 5th Annul SSR Workshop on Synchrotron X-ry Scttering Techniques in Mterils nd Environmentl Sciences: Theory nd Appliction
More informationNat 5 USAP 3(b) This booklet contains : Questions on Topics covered in RHS USAP 3(b) Exam Type Questions Answers. Sourced from PEGASYS
Nt USAP This ooklet contins : Questions on Topics covered in RHS USAP Em Tpe Questions Answers Sourced from PEGASYS USAP EF. Reducing n lgeric epression to its simplest form / where nd re of the form (
More informationUnit #9 : Definite Integral Properties; Fundamental Theorem of Calculus
Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl
More informationDISCRETE MATHEMATICS HOMEWORK 3 SOLUTIONS
DISCRETE MATHEMATICS 21228 HOMEWORK 3 SOLUTIONS JC Due in clss Wednesdy September 17. You my collborte but must write up your solutions by yourself. Lte homework will not be ccepted. Homework must either
More informationIs there an easy way to find examples of such triples? Why yes! Just look at an ordinary multiplication table to find them!
PUSHING PYTHAGORAS 009 Jmes Tnton A triple of integers ( bc,, ) is clled Pythgoren triple if exmple, some clssic triples re ( 3,4,5 ), ( 5,1,13 ), ( ) fond of ( 0,1,9 ) nd ( 119,10,169 ). + b = c. For
More informationData Provided: A formula sheet and table of physical constants is attached to this paper. SOLID STATE PHYSICS
Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. DEPARTMENT OF PHYSICS AND ASTRONOMY Autumn (015) SOLID STATE PHYSICS HOURS The pper is divided into 5 questions. Answer compulsory
More informationWhat is thin film/layer?
High-esolution XD Wht is thin film/lyer? Mteril so thin tht its chrcteristics re dominted primrily by two dimensionl effects nd re mostly different thn its bulk properties Source: semiconductorglossry.com
More informationImproper Integrals. Type I Improper Integrals How do we evaluate an integral such as
Improper Integrls Two different types of integrls cn qulify s improper. The first type of improper integrl (which we will refer to s Type I) involves evluting n integrl over n infinite region. In the grph
More informationWhen a force f(t) is applied to a mass in a system, we recall that Newton s law says that. f(t) = ma = m d dt v,
Impulse Functions In mny ppliction problems, n externl force f(t) is pplied over very short period of time. For exmple, if mss in spring nd dshpot system is struck by hmmer, the ppliction of the force
More informationHW3, Math 307. CSUF. Spring 2007.
HW, Mth 7. CSUF. Spring 7. Nsser M. Abbsi Spring 7 Compiled on November 5, 8 t 8:8m public Contents Section.6, problem Section.6, problem Section.6, problem 5 Section.6, problem 7 6 5 Section.6, problem
More informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: How to identify the leding coefficients nd degrees of polynomils How to dd nd subtrct polynomils How to multiply polynomils
More informationProblem Set 3 Solutions
Chemistry 36 Dr Jen M Stndrd Problem Set 3 Solutions 1 Verify for the prticle in one-dimensionl box by explicit integrtion tht the wvefunction ψ ( x) π x is normlized To verify tht ψ ( x) is normlized,
More information13: Diffusion in 2 Energy Groups
3: Diffusion in Energy Groups B. Rouben McMster University Course EP 4D3/6D3 Nucler Rector Anlysis (Rector Physics) 5 Sept.-Dec. 5 September Contents We study the diffusion eqution in two energy groups
More information