Crystalline Structures The Basics
|
|
- Cornelius Johnson
- 5 years ago
- Views:
Transcription
1 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 positions plced on lttice points within the unit cell). A lttice is used in context when describing crystlline structures, mens 3-D rry of points in spce. Every lttice point must hve identicl surroundings. Unit cell: smllest repetitive volume which contins the complete lttice pttern of crystl. A unit cell is chosen to represent the highest level of geometric symmetry of the crystl structure. It s the bsic structurl unit or building block of crystl structure. 7 crystl systems in 3-D Ech crystl structure is built by stcking unit cells nd plcing objects (motifs, bsis) on the lttice points: 14 crystl lttices in 3-D, b, nd c re the lttice constnts, b, g re the interxil ngles 1
2 Metllic Crystl Structures (the simplest) Recll, tht ) coulombic ttrction between deloclized vlence electrons nd positively chrged cores is isotropic (non-directionl), b) typiclly, only one element is present, so ll tomic rdii re the sme, c) nerest neighbor distnces tend to be smll, nd d) electron cloud shields cores from ech other. For these resons, metllic bonding leds to close pcked, dense crystl structures tht mximize spce filling nd coordintion number (number of nerest neighbors). Most elementl metls crystllize in the FCC (fce-centered cubic), CC (body-centered cubic, or HCP (hexgonl close pcked) structures: Room temperture crystl structure Crystl structure just before it melts 2
3 Recll: Simple Cubic (SC) Structure Rre due to low pcking density (only -Po hs this structure) Close-pcked directions re cube edges. A hrd sphere unit cell (ech sphere represents n ion core with metllic bonding in between): Reducedsphere unit cells (8 totl: 2 x 2 x 2) Coordintion # = 6 (# of nerest neighbors) Group V elements (As, Sb nd i) ll crystllize in structures tht cn be thought of s distorted versions of the SC rrngement trigonl (rhombohedrl) 1 Three nerest-neighbors re close nd three SC: 2 bodydigonl 3 digonl re further wy. These structures re more body- 1 ccurtely described s rrngements. 3 3 (red line) 2 (green line)
4 Simple Cubic (SC) Atomic Pcking Fctor (APF) APF = Volume of toms in unit cell* Volume of entire unit cell APF for simple cubic structure = 0.52 Adpted from Fig. 3.42, Cllister & Rethwisch 3e. *ssume hrd spheres R=0.5 close-pcked directions=cube edges contins 8 x 1/8 = 1 tom/unit cell (see next slide for clcultion) toms unit cell APF = 1 volume 4 3 p (0.5) 3 tom 3 volume unit cell 4
5 Number of Lttice points (e.g. toms/ions) per Unit Cell +. N e = number of lttice points on cell edges (shred by? cells) Exmple: e = edge e 5
6 Recll: ody Centered Cubic (CC) Structure Atoms touch ech other long cube (body) digonls. --Note: All toms re identicl; the center tom is shded differently only for ese of viewing. Exmples: Cr, W, -Fe, T, Mo Coordintion # = 8 Adpted from Fig. 3.2, Cllister & Rethwisch 3e. Hrd sphere unit cell Reduced-sphere unit cell 2 toms/unit cell = 1 center + 8 corners x 1/8 Extended unit cells In ddition to the 8 N.N., there re 6 next N.N. (N.N.N.) only 15% (2/ 3) further wy. So this nerly 14 coordinte structure could lso be described s hving coordintion. 8 unit cells: 6
7 Recll: CC APF APF for body-centered cubic structure = 0.68 Q 3 Q N 2 P N Adpted from Fig. 3.2(), Cllister & Rethwisch 3e. toms O unit cell APF = R p ( 3/4 )3 3 P Close-pcked directions (cube digonls): length = 4R= 3 volume unit cell volume tom 7
8 ody Centered Tetrgonl (CT) Recll CC: or extend CC lttice: The fct tht the 6 next N.N. re so close suggests tht tetrgonl (=b c) distortions could led to the formtion of 10 nd 12 coordinte structures: -If we compress the CC structure long the c-xis, the toms long the verticl xis become N.N., i.e. CN=10 (CT) when c/= 2/ 3: -Likewise if we shrink nd b with respect to c, then the in-plne or equtoril toms become N.N., i.e., CN=12 when c/= 2. -The equtoril toms re now on the fce centers (since height of cell is = digonl of squre bse) the repet unit is now identicl to the Fce Centered cubic (FCC) or cubic close pcked (CCP): b Only Protctinium (P) crystllizes in the CT (CN=10) structure, but mny crystl structures hve CT lttice. c 8
9 Recll: Fce Centered Cubic (FCC) Structure Atoms touch ech other long fce digonls. --Note: All toms re identicl; the fce toms re shded differently only for ese of viewing. Exs.: Al, Cu, Au, Pb, Ni, Pt, Ag Coordintion # = fold coordintion of ech lttice point (sme tom) is identicl. Adpted from Fig. 3.1, Cllister & Rethwisch 3e. Hrd sphere unit cell Reduced-sphere unit cell 4 toms/unit cell = 6 fce x 1/2 + 8 corners x 1/8 2 unit cells (1 x 2) Extended unit cells 9
10 APF for fce-centered cubic structure = 0.74 which is mximum chievble APF for sme dimeter spheres, known s Kepler conjecture. Adpted from Fig. 3.1(), Cllister & Rethwisch 3e. 2 Recll: FCC APF FCC lso known s cubic close pcked (CCP) structure. Close-pcked directions: length = 4R = 2 toms unit cell APF = p ( 2/4 )3 3 volume tom volume unit cell 10
11 Recll: Hexgonl Closed Pcked (HCP) Not ll metls hve unit cells with cubic symmetry, mny metls hve HCP crystl structures. z sl plne A A y G H E F x A A A exs: Zn, Mg, -Ti (room temp) Alterntively, the unit cell of HCP my be specified s prllelepiped defined by toms lbeled A through H with J tom lying in unit cell interior (not t body center), it s t (1/3 x, 2/3 y, ½ z). CN = 12 Atoms/unit cell = 6 APF = 0.74 c/ =
12 Recll: Idel c/ rtio for HCP is A sketch of one-third of n HCP unit cell is shown: Consider the tetrhedron lbeled s JKLM, which is reconstructed s: The tom t point M is midwy between the top nd bottom fces of the unit cell--tht is MH = c/2. And, since toms t points J, K, nd M, ll touch one nother, JM=JK=2R=, where R is the tomic rdius. Furthermore, from tringle JHM, (JM) 2 =(JH) 2 +(MH) 2 or c 2 JH Now, we cn determine the JH length by considertion of tringle JKL, n equilterl tringle, cos 30 o / 2 JH 3 2 Substituting this vlue for JH in the bove expression yields or JH solving for c/ c c 12
13 Stcking of Metllic Crystl Structures How cn we stck metl toms to minimize empty spce? Which plne is more close-pcked? 2-dimensions vs. Now stck these 2-D lyers to mke 3-D structures 13
14 Stcking of HCP nd FCC Most crystl structures in this clss re described by compring them to 1 of the 2 C.N.=12 HCP or FCC It is esiest to view HCP nd FCC structures s stcked, close pcked lyers: Single closest pcked lyer: HCP Unit Cell: 3D Projection 3-D structure is formed by stcking these lyers upon one nother. Atoms in the second lyer fit into the vlleys formed by 3 toms of 1 st lyer: A A... Stcking Sequence A sites Shded spheres fit on top of the unshded lyer to mximize pcking. There re two possible positions for the 3 rd lyer: #1: if the toms tke the positions lbeled 1, then they re directly bove the toms in the 1 st lyer thus the 3 rd lyer reproduces the 1 st. If this pttern continues we hve the A A stcking sequence of the HCP structure. 2D Projection equivlent Top lyer 3 rd lyer 1 st lyer 2 nd lyer c sites Middle lyer A sites Adpted from Fig. 3.3(), Cllister & Rethwisch 3e. ottom lyer 14
15 FCC Unit Cell: FCC Stcking Sequence #2: if, on the other hnd, the toms in the 3 rd lyer occupy the positions lbeled 2, then this lyer is distinct from the 1 st nd the 2 nd. The 4 th lyer must then repet either 1 st or the 2 nd lyer. If it repets the 1 st, then we hve the AC AC stcking sequence of the FCC (CCP) structure: If it repets the 2 nd, then AC AC sequence such s in L, Nd, Pr nd Pm (recll clss7/slide 2: hc-pcked) A C equivlent A is the 1 st lyer is the 2 nd lyer C is the 3 rd lyer 3 rd lyer 1 st lyer 2 nd lyer Closed pcked plnes {111} of toms Corner hs been removed to show stcking of close pcked toms AC AC... Stcking Sequence Tringle represents (111) plne 15
16 FCC Stcking Sequence (continued) 2D Projections: C A A sites C C sites C sites 16
17 Compring HCP nd FCC FCC: The similrity of pcking in these 2 structures is noteworthy, both hve 12 CN, APF=0.74 nd identicl densities. The only significnt difference between the structures is in the stcking sequence (AC A), i.e. where the third close pcked lyer in FCC is positioned. Exmple, coblt undergoes mrtensitic trnsformtion from FCC to HCP t ~695 K. The lttice constnt of FCC Co is Å, with nerest-neighbor distnce of Å. While for HCP Co, the lttice constnts nd c re nd Equivlent Å, respectively, its nerestneighbor distnce is Å, nd the to bove but rotted rtio c/ equls Note tht A in FCC would be equivlent to A in HCP, without the third close pcked lyer. HCP:
IV. 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 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 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 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 informationPHY 140A: Solid State Physics. Solution to Midterm #1
PHY 140A: Solid Stte Physics Solution to Midterm #1 TA: Xun Ji 1 October 24, 2006 1 Emil: jixun@physics.ucl.edu Problem #1 (20pt)Clculte the pcking frction of the body-centered cubic lttice. Solution:
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 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 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 informationChapter 3: The Structure of Crystalline Solids (2)
Chpter 3: The Structure of Crstlline Solids (2) Clss Eercise Drw the unit cell structure for simple cubic (SC), bodcentered cubic (BCC), nd fce-centered cubic (FCC) lttices Give coordintion number (CN)
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 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 informationTHE SOLID STATE MODULE - 3 OBJECTIVES. Notes
The Solid Stte MODULE - 3 6 THE SOLID STATE You re wre tht the mtter exists in three different sttes viz., solid, liquid nd gs. In these, the constituent prticles (toms, molecules or ions) re held together
More information2010. Spring: Electro-Optics (Prof. Sin-Doo Lee, Rm ,
2010. Spring: Electro-Optics (Prof. Sin-Doo Lee, Rm. 301-1109, http://mipd.snu.c.kr) Opticl Wves in Crystls A. Yriv nd P. Yeh (John Wiley, New Jersey, 2003) Week Chpter Week Chpter Mr. 03 * Bsics of Crystl
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 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 informationChapter 2: Crystal Structures and Symmetry
hpter 2: rystl Structures nd Symmetry Lue, rvis Jnury 30, 2017 ontents 1 Lttice Types nd Symmetry 3 1.1 Two-Dimensionl Lttices................. 3 1.2 Three-Dimensionl Lttices................ 5 2 Point-Group
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 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 informationLecture V. Introduction to Space Groups Charles H. Lake
Lecture V. Introduction to Spce Groups 2003. Chrles H. Lke Outline:. Introduction B. Trnsltionl symmetry C. Nomenclture nd symols used with spce groups D. The spce groups E. Derivtion nd discussion of
More informationamorphous solids, liquids and gases atoms or molecules are C A indentical and all properties are same in all directions.
THE SOLID STTE 1. INTRODUCTION : Mtter cn exist in three physicl sttes nmely ; solid, liquid nd gs. Mtter consists of tiny prticles (toms, ions or molecules). If the prticles re very fr off from one nother,
More informationUniversity of Alabama Department of Physics and Astronomy. PH126: Exam 1
University of Albm Deprtment of Physics nd Astronomy PH 16 LeClir Fll 011 Instructions: PH16: Exm 1 1. Answer four of the five questions below. All problems hve equl weight.. You must show your work for
More informationPREVIOUS EAMCET QUESTIONS
CENTRE OF MASS PREVIOUS EAMCET QUESTIONS ENGINEERING Two prticles A nd B initilly t rest, move towrds ech other, under mutul force of ttrction At n instnce when the speed of A is v nd speed of B is v,
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 informationENERGY AND PACKING. Outline: MATERIALS AND PACKING. Crystal Structure
EERGY AD PACKIG Outline: Crstlline versus morphous strutures Crstl struture - Unit ell - Coordintion numer - Atomi pking ftor Crstl sstems on dense, rndom pking Dense, regulr pking tpil neighor ond energ
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 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 informationLevel I MAML Olympiad 2001 Page 1 of 6 (A) 90 (B) 92 (C) 94 (D) 96 (E) 98 (A) 48 (B) 54 (C) 60 (D) 66 (E) 72 (A) 9 (B) 13 (C) 17 (D) 25 (E) 38
Level I MAML Olympid 00 Pge of 6. Si students in smll clss took n em on the scheduled dte. The verge of their grdes ws 75. The seventh student in the clss ws ill tht dy nd took the em lte. When her score
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 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 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 informationCrystals. Fig From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
Crystls Mterils will often orgnize themselves by minimizing energy to hve long rnge order. This order results in periodicity tht determines mny properties of the mteril. We represent this periodicity by
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 informationDETERMINATION OF MECHANICAL PROPERTIES OF NANOSTRUCTURES WITH COMPLEX CRYSTAL LATTICE USING MOMENT INTERACTION AT MICROSCALE
Determintion RevAdvMterSci of mechnicl 0(009) -7 properties of nnostructures with complex crystl lttice using DETERMINATION OF MECHANICAL PROPERTIES OF NANOSTRUCTURES WITH COMPLEX CRYSTAL LATTICE USING
More informationa < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1
Mth 33 Volume Stewrt 5.2 Geometry of integrls. In this section, we will lern how to compute volumes using integrls defined by slice nlysis. First, we recll from Clculus I how to compute res. Given the
More informationForm 5 HKCEE 1990 Mathematics II (a 2n ) 3 = A. f(1) B. f(n) A. a 6n B. a 8n C. D. E. 2 D. 1 E. n. 1 in. If 2 = 10 p, 3 = 10 q, express log 6
Form HK 9 Mthemtics II.. ( n ) =. 6n. 8n. n 6n 8n... +. 6.. f(). f(n). n n If = 0 p, = 0 q, epress log 6 in terms of p nd q.. p q. pq. p q pq p + q Let > b > 0. If nd b re respectivel the st nd nd terms
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 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 informationAPPLICATIONS OF THE DEFINITE INTEGRAL
APPLICATIONS OF THE DEFINITE INTEGRAL. Volume: Slicing, disks nd wshers.. Volumes by Slicing. Suppose solid object hs boundries extending from x =, to x = b, nd tht its cross-section in plne pssing through
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 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 informationSample Exam 5 - Skip Problems 1-3
Smple Exm 5 - Skip Problems 1-3 Physics 121 Common Exm 2: Fll 2010 Nme (Print): 4 igit I: Section: Honors Code Pledge: As n NJIT student I, pledge to comply with the provisions of the NJIT Acdemic Honor
More informationa * a (2,1) 1,1 0,1 1,1 2,1 hkl 1,0 1,0 2,0 O 2,1 0,1 1,1 0,2 1,2 2,2
18 34.3 The Reciprocl Lttice The inverse of the intersections of plne with the unit cell xes is used to find the Miller indices of the plne. The inverse of the d-spcing etween plnes ppers in expressions
More informationChapter 4 Contravariance, Covariance, and Spacetime Diagrams
Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz
More informationPeriod #2 Notes: Electronic Structure of Atoms
Period # Notes: Electronic Structure of Atoms The logicl plce (for civil engineers) to begin in describing mterils is t the tomic scle. The bsic elements of the tom re the proton, the neutron, nd the electron:
More information7.6 The Use of Definite Integrals in Physics and Engineering
Arknss Tech University MATH 94: Clculus II Dr. Mrcel B. Finn 7.6 The Use of Definite Integrls in Physics nd Engineering It hs been shown how clculus cn be pplied to find solutions to geometric problems
More informationMORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)
MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give
More informationSOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 2014
SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 014 Mrk Scheme: Ech prt of Question 1 is worth four mrks which re wrded solely for the correct nswer.
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 informationMath 426: Probability Final Exam Practice
Mth 46: Probbility Finl Exm Prctice. Computtionl problems 4. Let T k (n) denote the number of prtitions of the set {,..., n} into k nonempty subsets, where k n. Argue tht T k (n) kt k (n ) + T k (n ) by
More information38 Riemann sums and existence of the definite integral.
38 Riemnn sums nd existence of the definite integrl. In the clcultion of the re of the region X bounded by the grph of g(x) = x 2, the x-xis nd 0 x b, two sums ppered: ( n (k 1) 2) b 3 n 3 re(x) ( n These
More informationSolutions to Physics: Principles with Applications, 5/E, Giancoli Chapter 16 CHAPTER 16
CHAPTER 16 1. The number of electrons is N = Q/e = ( 30.0 10 6 C)/( 1.60 10 19 C/electrons) = 1.88 10 14 electrons.. The mgnitude of the Coulomb force is Q /r. If we divide the epressions for the two forces,
More informationCONIC SECTIONS. Chapter 11
CONIC SECTIONS Chpter. Overview.. Sections of cone Let l e fied verticl line nd m e nother line intersecting it t fied point V nd inclined to it t n ngle α (Fig..). Fig.. Suppose we rotte the line m round
More information1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE
ELEMENTARY ALGEBRA nd GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the exmples, work the prolems, then check your nswers t the end of ech topic. If you don t get the nswer given, check
More information2. VECTORS AND MATRICES IN 3 DIMENSIONS
2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2-dimensionl Vectors x A point in 3-dimensionl spce cn e represented y column vector of the form y z z-xis y-xis z x y x-xis Most of the
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 informationConducting Ellipsoid and Circular Disk
1 Problem Conducting Ellipsoid nd Circulr Disk Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 (September 1, 00) Show tht the surfce chrge density σ on conducting ellipsoid,
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 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 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 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 informationCoimisiún na Scrúduithe Stáit State Examinations Commission
M 30 Coimisiún n Scrúduithe Stáit Stte Exmintions Commission LEAVING CERTIFICATE EXAMINATION, 005 MATHEMATICS HIGHER LEVEL PAPER ( 300 mrks ) MONDAY, 3 JUNE MORNING, 9:30 to :00 Attempt FIVE questions
More information10 If 3, a, b, c, 23 are in A.S., then a + b + c = 15 Find the perimeter of the sector in the figure. A. 1:3. A. 2.25cm B. 3cm
HK MTHS Pper II P. If f ( x ) = 0 x, then f ( y ) = 6 0 y 0 + y 0 y 0 8 y 0 y If s = ind the gretest vlue of x + y if ( x, y ) is point lying in the region O (including the boundry). n [ + (n )d ], then
More informationBridging the gap: GCSE AS Level
Bridging the gp: GCSE AS Level CONTENTS Chpter Removing rckets pge Chpter Liner equtions Chpter Simultneous equtions 8 Chpter Fctors 0 Chpter Chnge the suject of the formul Chpter 6 Solving qudrtic equtions
More informationChapter 2. Vectors. 2.1 Vectors Scalars and Vectors
Chpter 2 Vectors 2.1 Vectors 2.1.1 Sclrs nd Vectors A vector is quntity hving both mgnitude nd direction. Emples of vector quntities re velocity, force nd position. One cn represent vector in n-dimensionl
More informationPROPERTIES OF AREAS In general, and for an irregular shape, the definition of the centroid at position ( x, y) is given by
PROPERTES OF RES Centroid The concept of the centroid is prol lred fmilir to ou For plne shpe with n ovious geometric centre, (rectngle, circle) the centroid is t the centre f n re hs n is of smmetr, the
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 informationNotes on length and conformal metrics
Notes on length nd conforml metrics We recll how to mesure the Eucliden distnce of n rc in the plne. Let α : [, b] R 2 be smooth (C ) rc. Tht is α(t) (x(t), y(t)) where x(t) nd y(t) re smooth rel vlued
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 informationMath 520 Final Exam Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008
Mth 520 Finl Exm Topic Outline Sections 1 3 (Xio/Dums/Liw) Spring 2008 The finl exm will be held on Tuesdy, My 13, 2-5pm in 117 McMilln Wht will be covered The finl exm will cover the mteril from ll of
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 informationPre-Session Review. Part 1: Basic Algebra; Linear Functions and Graphs
Pre-Session Review Prt 1: Bsic Algebr; Liner Functions nd Grphs A. Generl Review nd Introduction to Algebr Hierrchy of Arithmetic Opertions Opertions in ny expression re performed in the following order:
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 informationl 2 p2 n 4n 2, the total surface area of the
Week 6 Lectures Sections 7.5, 7.6 Section 7.5: Surfce re of Revolution Surfce re of Cone: Let C be circle of rdius r. Let P n be n n-sided regulr polygon of perimeter p n with vertices on C. Form cone
More informationMATRICES AND VECTORS SPACE
MATRICES AND VECTORS SPACE MATRICES AND MATRIX OPERATIONS SYSTEM OF LINEAR EQUATIONS DETERMINANTS VECTORS IN -SPACE AND -SPACE GENERAL VECTOR SPACES INNER PRODUCT SPACES EIGENVALUES, EIGENVECTORS LINEAR
More informationChapter E - Problems
Chpter E - Prolems Blinn College - Physics 2426 - Terry Honn Prolem E.1 A wire with dimeter d feeds current to cpcitor. The chrge on the cpcitor vries with time s QHtL = Q 0 sin w t. Wht re the current
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 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 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 KENNESAW STATE UNIVERSITY HIGH SCHOOL MATHEMATICS COMPETITION PART I MULTIPLE CHOICE NO CALCULATORS 90 MINUTES
THE 08 09 KENNESW STTE UNIVERSITY HIGH SHOOL MTHEMTIS OMPETITION PRT I MULTIPLE HOIE For ech of the following questions, crefully blcken the pproprite box on the nswer sheet with # pencil. o not fold,
More informationLog1 Contest Round 3 Theta Individual. 4 points each 1 What is the sum of the first 5 Fibonacci numbers if the first two are 1, 1?
008 009 Log1 Contest Round Thet Individul Nme: points ech 1 Wht is the sum of the first Fiboncci numbers if the first two re 1, 1? If two crds re drwn from stndrd crd deck, wht is the probbility of drwing
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 informationAnalytically, vectors will be represented by lowercase bold-face Latin letters, e.g. a, r, q.
1.1 Vector Alger 1.1.1 Sclrs A physicl quntity which is completely descried y single rel numer is clled sclr. Physiclly, it is something which hs mgnitude, nd is completely descried y this mgnitude. Exmples
More informationChapter 12. Lesson Geometry Worked-Out Solution Key. Prerequisite Skills (p. 790) A 5 } perimeter Guided Practice (pp.
Chpter 1 Prerequisite Skills (p. 790) 1. The re of regulr polygon is given by the formul A 5 1 p P, where is the pothem nd P is the perimeter.. Two polygons re similr if their corresponding ngles re congruent
More informationPhysics 1402: Lecture 7 Today s Agenda
1 Physics 1402: Lecture 7 Tody s gend nnouncements: Lectures posted on: www.phys.uconn.edu/~rcote/ HW ssignments, solutions etc. Homework #2: On Msterphysics tody: due Fridy Go to msteringphysics.com Ls:
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 informationLoudoun Valley High School Calculus Summertime Fun Packet
Loudoun Vlley High School Clculus Summertime Fun Pcket We HIGHLY recommend tht you go through this pcket nd mke sure tht you know how to do everything in it. Prctice the problems tht you do NOT remember!
More informationTImath.com Algebra 2. Constructing an Ellipse
TImth.com Algebr Constructing n Ellipse ID: 9980 Time required 60 minutes Activity Overview This ctivity introduces ellipses from geometric perspective. Two different methods for constructing n ellipse
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 information5.04 Principles of Inorganic Chemistry II
MIT OpenCourseWre http://ocw.mit.edu 5.04 Principles of Inorgnic Chemistry II Fll 2008 For informtion bout citing these mterils or our Terms of Use, visit: http://ocw.mit.edu/terms. 5.04, Principles of
More informationSolid State Chemistry
Solid Stte Chemistry Solids re minly chrcterised by their definite shpes nd considerble mechnicl strength nd rigidity. The rigidity rises due to the bsence of the trnsltory movement of the structurl units
More informationThe Predom module. Predom calculates and plots isothermal 1-, 2- and 3-metal predominance area diagrams. Predom accesses only compound databases.
Section 1 Section 2 The module clcultes nd plots isotherml 1-, 2- nd 3-metl predominnce re digrms. ccesses only compound dtbses. Tble of Contents Tble of Contents Opening the module Section 3 Stoichiometric
More information8. Complex Numbers. We can combine the real numbers with this new imaginary number to form the complex numbers.
8. Complex Numers The rel numer system is dequte for solving mny mthemticl prolems. But it is necessry to extend the rel numer system to solve numer of importnt prolems. Complex numers do not chnge the
More informationPhysics 2135 Exam 1 February 14, 2017
Exm Totl / 200 Physics 215 Exm 1 Ferury 14, 2017 Printed Nme: Rec. Sec. Letter: Five multiple choice questions, 8 points ech. Choose the est or most nerly correct nswer. 1. Two chrges 1 nd 2 re seprted
More informationMathematics Extension 2
00 HIGHER SCHOOL CERTIFICATE EXAMINATION Mthemtics Etension Generl Instructions Reding time 5 minutes Working time hours Write using blck or blue pen Bord-pproved clcultors my be used A tble of stndrd
More informationInstructor(s): Acosta/Woodard PHYSICS DEPARTMENT PHY 2049, Fall 2015 Midterm 1 September 29, 2015
Instructor(s): Acost/Woodrd PHYSICS DEPATMENT PHY 049, Fll 015 Midterm 1 September 9, 015 Nme (print): Signture: On m honor, I hve neither given nor received unuthorized id on this emintion. YOU TEST NUMBE
More informationProblems for HW X. C. Gwinn. November 30, 2009
Problems for HW X C. Gwinn November 30, 2009 These problems will not be grded. 1 HWX Problem 1 Suppose thn n object is composed of liner dielectric mteril, with constnt reltive permittivity ɛ r. The object
More informationChapter 14. Matrix Representations of Linear Transformations
Chpter 4 Mtrix Representtions of Liner Trnsformtions When considering the Het Stte Evolution, we found tht we could describe this process using multipliction by mtrix. This ws nice becuse computers cn
More informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk bout solving systems of liner equtions. These re problems tht give couple of equtions with couple of unknowns, like: 6 2 3 7 4
More information