Chem 253A. Crystal Structure. Chem 253B. Electronic Structure
|
|
- Piers Martin
- 5 years ago
- Views:
Transcription
1 Chem 53, UC, Bereley Chem 53A Crystl Structure Chem 53B Electroic Structure Chem 53, UC, Bereley 1
2 Chem 53, UC, Bereley Electroic Structures of Solid Refereces Ashcroft/Mermi: Chpter 1-3, 8-10 Kittel: chpter 6-9 Gerste: Chpter 7, 11 Burdett: chpter 1-3 Hoffm: p1-0 Chem 53, UC, Bereley
3 Chem 53, UC, Bereley Si NW Chem 53, UC, Bereley 3
4 Chem 53, UC, Bereley Chem 53, UC, Bereley Crrier Mobility Mometum gied durig the me free flight ee v d * m vd ee m* Mometum lost i collisio Drift velocity Mobility: the rtio of the drift velocity over the pplied electric field v d E e m* 4
5 Chem 53, UC, Bereley Chem 53, UC, Bereley Idepedet Electros Free Electro Approimtio 5
6 Chem 53, UC, Bereley Chem 53, UC, Bereley 6
7 7 Chem 53, UC, Bereley ( m E From 1D to 3D: z z y y z y A r si si si ( ] ( ( [( ( z z y y m E Chem 53, UC, Bereley Periodic Boudry Coditio ( ( E F K F K ( ( r r m Solutio: trvelig ple wve ep( i A Where: m E (
8 Chem 53, UC, Bereley Normliztio: 3D Periodic Boudry Coditio * Aep( i r dr 1 A ep( i rep( i r dr A V V: uit cell volume V 1/ ep( i r de Broglie wvelegth Chem 53, UC, Bereley Eergy Eigevlue: Momet Opertor: E( m pˆ i r pˆ ( r i ( m y z ( r ( r r Aep( i r E F Mometum Eigevlue: K F K p 8
9 9 Chem 53, UC, Bereley With periodic boudry coditios: 1 z z y y i i i e e e z z z y y y D spce: Are per poit: y 3D spce: Are per poit: V z y 3 8 A regio of spce of volume will coti: llowed vlues ( V V Chem 53, UC, Bereley Reciprocl ttice ( c b c b ( c b c b ( c b b c Reciprocl lttice is lwys oe of 14 Brvis ttice.
10 Chem 53, UC, Bereley K spce desity of level: V 3 8 No-iterctig electros: Puli eclusio priciple Ech wve vector two electroic level (spi up/dow Fermi wve vector: F Volume eclosed by the Fermi surfce: F Chem 53, UC, Bereley # of llowed sttes withi: F V 3 8 F 6 V # of electros N: N 3 F 3 V Electroic desity: N V 3 F 3 F ( 3 1/3 10
11 Chem 53, UC, Bereley Free & idepedet electro groud stte: Fermi wve vector Eclosed Fermi sphere F ( 3 1/3 Fermi Surfce Fermi Mometum Fermi eergy Fermi velocity p E v F F F F m p F F / m* Chem 53, UC, Bereley Estimtio bsed o coductio electro desity: V N 1 4 r s 3 F E F F m F (9 / 4 r s E 1/3 F 1.9 r s 50.1eV ( rs / 0 Rdius of sphere where volume Equls to the volume per Coductio electro -3, for my metl Fermi eergy for metllic elemets: ev Fermi temperture: EF TF 10 K ( r / B s 0 11
12 Chem 53, UC, Bereley Chem 53, UC, Bereley Desity of Sttes The umber of orbitls/sttes per uit eergy rge D( E dn de E N 3 N ( m m V V E 3 3/ ( /3 dn V m 3/ D( E ( E de 1/ 1
13 Chem 53, UC, Bereley Qutum Cofiemet d Dimesiolity Chem 53, UC, Bereley Fermi-Dirc distributio: f ( E ep[( E 1 E / F B T ] 1 13
14 Chem 53, UC, Bereley Chem 53, UC, Bereley 14
15 Chem 53, UC, Bereley Chem 53, UC, Bereley 15
16 Chem 53, UC, Bereley Nerly Free Electro Model Addig smll perturbtio by the periodic potetil of the ioic cores E F K F K E( m Chem 53, UC, Bereley Periodic Boudry Coditio m ( r ( r ( ( Solutio: trvelig ple wve Aep( i Where: E F K F K E( m 16
17 Chem 53, UC, Bereley 3D Periodic Boudry Coditio Normliztio: Aep( i r * dr 1 A ep( i rep( i r dr A V V: uit cell volume V 1/ ep( i r de Broglie wvelegth Chem 53, UC, Bereley Periodic Potetils d Bloch's Theorem V ( r V ( r R R ttice vector Bloch s theorem: the eigesttes of the Hmiltoi bove c be chose to hve the form of ple wve times fuctio with the periodicity of the Brvis ttice. Bloch Wvefuctio: ir e ( r u( r V u( r u( r R periodic prt of Bloch fuctio 17
18 Chem 53, UC, Bereley Brgg reflectio of electro wves i crystl is the cuse of the eergy gp. First Brgg reflectio: Other gp: Chem 53, UC, Bereley 18
19 Chem 53, UC, Bereley Reciprocl ttice d R ' ' R 1 b 3 c e ikr 1 i e ' R( K ' 1 ue Coditio Reciprocl lttice vector For ll R i the Brvis ttice Chem 53, UC, Bereley For 1D ttice: Reciprocl lttice vector: ' K Diffrctio Coditio: 1 K C be eteded to 3D 19
20 Chem 53, UC, Bereley Brgg reflectio of electro wves i crystl is the cuse of the eergy gp. First Brgg reflectio: First Brilloui Zoe Other gp: Chem 53, UC, Bereley Wiger-Seitz cell 0
21 Chem 53, UC, Bereley The wvefuctio t re ot trvelig wve of free electros: ep( i ep( i Isted: equl prts of the wves trvelig to the left d right A wve trvels either to the left or to the right is stdig wve. Chem 53, UC, Bereley Two differet stdig wves: ( ep( i ep( i cos ( ep( i ep( i si Probbility desity: 1
22 Chem 53, UC, Bereley Pile electro betwee the core ioshigher eergy Pile electro o the core ioslower eergy Chem 53, UC, Bereley Eteded zoe scheme reduced zoe scheme
23 Chem 53, UC, Bereley Fermi Surfce E( m E F K F K Chem 53, UC, Bereley For divlet elemets: free electro model 3
24 Chem 53, UC, Bereley Chem 53, UC, Bereley For erly free electro: 1. Iterctio of electro with periodic potetil opes gp t zoe boudry. Almost lwys Fermi surfce will itersect zoe Boudries perpediculrly. 3. The totl volume eclosed by the Fermi surfce depeds oly o totl electro cocetrtio, ot o iterctio 4
25 Chem 53, UC, Bereley Alli Metl N, Cs: sphericl Fermi surfce r r 0.4 Al. Erth metl: Be, Mg:: erly sphericl Fermi surfce D cse r r 0.56 Chem 53, UC, Bereley 5
26 Chem 53, UC, Bereley Chem 53, UC, Bereley 6
27 Chem 53, UC, Bereley Chem 53, UC, Bereley 7
28 Chem 53, UC, Bereley Chem 53, UC, Bereley 8
29 Chem 53, UC, Bereley 9
Chem 253B. Crystal Structure. Chem 253C. Electronic Structure
Chem 5, UC, Berele Chem 5B Crstl Structure Chem 5C Electroic Structure Chem 5, UC, Berele 1 Chem 5, UC, Berele Electroic Structures of Solid Refereces Ashcroft/Mermi: Chpter 1-, 8-10 Kittel: chpter 6-9
More informationis completely general whenever you have waves from two sources interfering. 2
MAKNG SENSE OF THE EQUATON SHEET terferece & Diffrctio NTERFERENCE r1 r d si. Equtio for pth legth differece. r1 r is completely geerl. Use si oly whe the two sources re fr wy from the observtio poit.
More informationParticle in a Box. and the state function is. In this case, the Hermitian operator. The b.c. restrict us to 0 x a. x A sin for 0 x a, and 0 otherwise
Prticle i Box We must hve me where = 1,,3 Solvig for E, π h E = = where = 1,,3, m 8m d the stte fuctio is x A si for 0 x, d 0 otherwise x ˆ d KE V. m dx I this cse, the Hermiti opertor 0iside the box The
More informationLecture 38 (Trapped Particles) Physics Spring 2018 Douglas Fields
Lecture 38 (Trpped Prticles) Physics 6-01 Sprig 018 Dougls Fields Free Prticle Solutio Schrödiger s Wve Equtio i 1D If motio is restricted to oe-dimesio, the del opertor just becomes the prtil derivtive
More informationUsing Quantum Mechanics in Simple Systems Chapter 15
/16/17 Qutiztio rises whe the loctio of prticle (here electro) is cofied to dimesiolly smll regio of spce qutum cofiemet. Usig Qutum Mechics i Simple Systems Chpter 15 The simplest system tht c be cosidered
More informationContent: Essential Calculus, Early Transcendentals, James Stewart, 2007 Chapter 1: Functions and Limits., in a set B.
Review Sheet: Chpter Cotet: Essetil Clculus, Erly Trscedetls, Jmes Stewrt, 007 Chpter : Fuctios d Limits Cocepts, Defiitios, Lws, Theorems: A fuctio, f, is rule tht ssigs to ech elemet i set A ectly oe
More informationSchrödinger Equation Via Laplace-Beltrami Operator
IOSR Jourl of Mthemtics (IOSR-JM) e-issn: 78-578, p-issn: 39-765X. Volume 3, Issue 6 Ver. III (Nov. - Dec. 7), PP 9-95 www.iosrjourls.org Schrödiger Equtio Vi Lplce-Beltrmi Opertor Esi İ Eskitşçioğlu,
More informationElectronic Structure in Periodic Systems. b a
1. Isolted vs. periodic systems How to model thigs tht re ot moleculr, like metl or semicoductor or the surfce of somethig? Could crete very lrge chuk, or cluster, of the mteril, but this c be both expesive
More informationPostulates of quantum mechanics
Postultes of qutum mechics P1. The stte of qutum mechicl sstem is completel specified b wvefuctio Ψ(q,t) P. For ever mesurble propert of the sstem, there eists correspodig opertor i QM (mesuremet i the
More informationClassical Electrodynamics
A First Look t Qutum Phsics Clssicl Electrodmics Chpter Boudr-Vlue Prolems i Electrosttics: I 11 Clssicl Electrodmics Prof. Y. F. Che Cotets A First Look t Qutum Phsics.1 Poit Chrge i the Presece of Grouded
More informationIntrinsic Carrier Concentration
Itrisic Carrier Cocetratio I. Defiitio Itrisic semicoductor: A semicoductor material with o dopats. It electrical characteristics such as cocetratio of charge carriers, deped oly o pure crystal. II. To
More informationPoint 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 information,... are the terms of the sequence. If the domain consists of the first n positive integers only, the sequence is a finite sequence.
Chpter 9 & 0 FITZGERALD MAT 50/5 SECTION 9. Sequece Defiitio A ifiite sequece is fuctio whose domi is the set of positive itegers. The fuctio vlues,,, 4,...,,... re the terms of the sequece. If the domi
More informationPROGRESSIONS AND SERIES
PROGRESSIONS AND SERIES A sequece is lso clled progressio. We ow study three importt types of sequeces: () The Arithmetic Progressio, () The Geometric Progressio, () The Hrmoic Progressio. Arithmetic Progressio.
More informationis continuous at x 2 and g(x) 2. Oil spilled from a ruptured tanker spreads in a circle whose area increases at a
. Cosider two fuctios f () d g () defied o itervl I cotiig. f () is cotiuous t d g() is discotiuous t. Which of the followig is true bout fuctios f g d f g, the sum d the product of f d g, respectively?
More informationV. Mironov, J.P.M. Beijers
NUMERICAL MODELING OF ION PRODUCTION IN ECRIS BY USING THE PARTICLE-IN-CELL METHOD V. Miroov, J.P.M. Beijers Kerfysisch Verseller Istituut, Uiversity of Groige, The Netherlds 1 1. Itroductio. KVI A-ECRIS
More informationTopic 4 Fourier Series. Today
Topic 4 Fourier Series Toy Wves with repetig uctios Sigl geertor Clssicl guitr Pio Ech istrumet is plyig sigle ote mile C 6Hz) st hrmoic hrmoic 3 r hrmoic 4 th hrmoic 6Hz 5Hz 783Hz 44Hz A sigle ote will
More informationNational Quali cations SPECIMEN ONLY
AH Ntiol Quli ctios SPECIMEN ONLY SQ/AH/0 Mthemtics Dte Not pplicble Durtio hours Totl mrks 00 Attempt ALL questios. You my use clcultor. Full credit will be give oly to solutios which coti pproprite workig.
More informationHIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/4 UNIT (COMMON) Time allowed Two hours (Plus 5 minutes reading time)
HIGHER SCHOOL CERTIFICATE EXAMINATION 999 MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/ UNIT (COMMON) Time llowed Two hours (Plus 5 miutes redig time) DIRECTIONS TO CANDIDATES Attempt ALL questios. ALL questios
More informationSemiconductor Statistical Mechanics (Read Kittel Ch. 8)
EE30 - Solid State Electroics Semicoductor Statistical Mechaics (Read Kittel Ch. 8) Coductio bad occupatio desity: f( E)gE ( ) de f(e) - occupatio probability - Fermi-Dirac fuctio: g(e) - desity of states
More informationb a 2 ((g(x))2 (f(x)) 2 dx
Clc II Fll 005 MATH Nme: T3 Istructios: Write swers to problems o seprte pper. You my NOT use clcultors or y electroic devices or otes of y kid. Ech st rred problem is extr credit d ech is worth 5 poits.
More informationThere are 7 crystal systems and 14 Bravais lattices in 3 dimensions.
EXAM IN OURSE TFY40 Solid State Physics Moday 0. May 0 Time: 9.00.00 DRAFT OF SOLUTION Problem (0%) Itroductory Questios a) () Primitive uit cell: The miimum volume cell which will fill all space (without
More informationElectrical Resistance
Electrical Resistace I + V _ W Material with resistivity ρ t L Resistace R V I = L ρ Wt (Uit: ohms) where ρ is the electrical resistivity Addig parts/billio to parts/thousad of dopats to pure Si ca chage
More informationStudents must always use correct mathematical notation, not calculator notation. the set of positive integers and zero, {0,1, 2, 3,...
Appedices Of the vrious ottios i use, the IB hs chose to dopt system of ottio bsed o the recommedtios of the Itertiol Orgiztio for Stdrdiztio (ISO). This ottio is used i the emitio ppers for this course
More informationMulti-Electron Atoms-Helium
Multi-lecto Atos-Heliu He - se s H but with Z He - electos. No exct solutio of.. but c use H wve fuctios d eegy levels s sttig poit ucleus sceeed d so Zeffective is < sceeig is ~se s e-e epulsio fo He,
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 informationSPH3UW Unit 7.5 Snell s Law Page 1 of Total Internal Reflection occurs when the incoming refraction angle is
SPH3UW Uit 7.5 Sell s Lw Pge 1 of 7 Notes Physis Tool ox Refrtio is the hge i diretio of wve due to hge i its speed. This is most ommoly see whe wve psses from oe medium to other. Idex of refrtio lso lled
More information17 Phonons and conduction electrons in solids (Hiroshi Matsuoka)
7 Phoos ad coductio electros i solids Hiroshi Matsuoa I this chapter we will discuss a miimal microscopic model for phoos i a solid ad a miimal microscopic model for coductio electros i a simple metal.
More informationNational Quali cations AHEXEMPLAR PAPER ONLY
Ntiol Quli ctios AHEXEMPLAR PAPER ONLY EP/AH/0 Mthemtics Dte Not pplicble Durtio hours Totl mrks 00 Attempt ALL questios. You my use clcultor. Full credit will be give oly to solutios which coti pproprite
More informationMATH 104 FINAL SOLUTIONS. 1. (2 points each) Mark each of the following as True or False. No justification is required. y n = x 1 + x x n n
MATH 04 FINAL SOLUTIONS. ( poits ech) Mrk ech of the followig s True or Flse. No justifictio is required. ) A ubouded sequece c hve o Cuchy subsequece. Flse b) A ifiite uio of Dedekid cuts is Dedekid cut.
More informationALGEBRA II CHAPTER 7 NOTES. Name
ALGEBRA II CHAPTER 7 NOTES Ne Algebr II 7. th Roots d Rtiol Expoets Tody I evlutig th roots of rel ubers usig both rdicl d rtiol expoet ottio. I successful tody whe I c evlute th roots. It is iportt for
More informationChapter Real Numbers
Chpter. - Rel Numbers Itegers: coutig umbers, zero, d the egtive of the coutig umbers. ex: {,-3, -, -,,,, 3, } Rtiol Numbers: quotiets of two itegers with ozero deomitor; termitig or repetig decimls. ex:
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 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 (2016) SOLID STATE PHYSICS 2 HOURS Instructions: The pper is divided into 5 questions.
More informationExercises and Problems
HW Chapter 4: Oe-Dimesioal Quatum Mechaics Coceptual Questios 4.. Five. 4.4.. is idepedet of. a b c mu ( E). a b m( ev 5 ev) c m(6 ev ev) Exercises ad Problems 4.. Model: Model the electro as a particle
More informationMA123, Chapter 9: Computing some integrals (pp )
MA13, Chpter 9: Computig some itegrls (pp. 189-05) Dte: Chpter Gols: Uderstd how to use bsic summtio formuls to evlute more complex sums. Uderstd how to compute its of rtiol fuctios t ifiity. Uderstd how
More informationSolids - types. correlates with bonding energy
Solids - types MOLCULAR. Set of sigle atoms or molecules boud to adjacet due to weak electric force betwee eutral objects (va der Waals). Stregth depeds o electric dipole momet No free electros poor coductors
More informationF x = 2x λy 2 z 3 = 0 (1) F y = 2y λ2xyz 3 = 0 (2) F z = 2z λ3xy 2 z 2 = 0 (3) F λ = (xy 2 z 3 2) = 0. (4) 2z 3xy 2 z 2. 2x y 2 z 3 = 2y 2xyz 3 = ) 2
0 微甲 07- 班期中考解答和評分標準 5%) Fid the poits o the surfce xy z = tht re closest to the origi d lso the shortest distce betwee the surfce d the origi Solutio Cosider the Lgrge fuctio F x, y, z, λ) = x + y + z
More informationUNIVERSITY OF BRISTOL. Examination for the Degrees of B.Sc. and M.Sci. (Level C/4) ANALYSIS 1B, SOLUTIONS MATH (Paper Code MATH-10006)
UNIVERSITY OF BRISTOL Exmitio for the Degrees of B.Sc. d M.Sci. (Level C/4) ANALYSIS B, SOLUTIONS MATH 6 (Pper Code MATH-6) My/Jue 25, hours 3 miutes This pper cotis two sectios, A d B. Plese use seprte
More informationWaves in dielectric media. Waveguiding: χ (r ) Wave equation in linear non-dispersive homogenous and isotropic media
Wves i dieletri medi d wveguides Setio 5. I this leture, we will osider the properties of wves whose propgtio is govered by both the diffrtio d ofiemet proesses. The wveguides re result of the ble betwee
More informationEXERCISE a a a 5. + a 15 NEETIIT.COM
- Dowlod our droid App. Sigle choice Type Questios EXERCISE -. The first term of A.P. of cosecutive iteger is p +. The sum of (p + ) terms of this series c be expressed s () (p + ) () (p + ) (p + ) ()
More informationName: A2RCC Midterm Review Unit 1: Functions and Relations Know your parent functions!
Nme: ARCC Midterm Review Uit 1: Fuctios d Reltios Kow your pret fuctios! 1. The ccompyig grph shows the mout of rdio-ctivity over time. Defiitio of fuctio. Defiitio of 1-1. Which digrm represets oe-to-oe
More informationWhy study large deviations? The problem of estimating buer overow frequency The performance of many systems is limited by events which have a small pr
Why study lrge devitios? The problem of estimtig buer overow frequecy The performce of my systems is ited by evets which hve smll probbility of occurrig, but which hve severe cosequeces whe they occur.
More informationOffice: JILA A709; Phone ;
Office: JILA A709; Phoe 303-49-7841; email: weberjm@jila.colorado.edu Problem Set 5 To be retured before the ed of class o Wedesday, September 3, 015 (give to me i perso or slide uder office door). 1.
More information1. (25 points) Use the limit definition of the definite integral and the sum formulas to compute. [1 x + x2
Mth 3, Clculus II Fil Exm Solutios. (5 poits) Use the limit defiitio of the defiite itegrl d the sum formuls to compute 3 x + x. Check your swer by usig the Fudmetl Theorem of Clculus. Solutio: The limit
More information n. A Very Interesting Example + + = d. + x3. + 5x4. math 131 power series, part ii 7. One of the first power series we examined was. 2!
mth power series, prt ii 7 A Very Iterestig Emple Oe of the first power series we emied ws! + +! + + +!! + I Emple 58 we used the rtio test to show tht the itervl of covergece ws (, ) Sice the series coverges
More informationPOWER SERIES R. E. SHOWALTER
POWER SERIES R. E. SHOWALTER. sequeces We deote by lim = tht the limit of the sequece { } is the umber. By this we me tht for y ε > 0 there is iteger N such tht < ε for ll itegers N. This mkes precise
More informationExponents and Radical
Expoets d Rdil Rule : If the root is eve d iside the rdil is egtive, the the swer is o rel umber, meig tht If is eve d is egtive, the Beuse rel umber multiplied eve times by itself will be lwys positive.
More informationLEVEL I. ,... if it is known that a 1
LEVEL I Fid the sum of first terms of the AP, if it is kow tht + 5 + 0 + 5 + 0 + = 5 The iterior gles of polygo re i rithmetic progressio The smllest gle is 0 d the commo differece is 5 Fid the umber of
More informationLecture 6. Semiconductor physics IV. The Semiconductor in Equilibrium
Lecture 6 Semicoductor physics IV The Semicoductor i Equilibrium Equilibrium, or thermal equilibrium No exteral forces such as voltages, electric fields. Magetic fields, or temperature gradiets are actig
More informationRemarks: (a) The Dirac delta is the function zero on the domain R {0}.
Sectio Objective(s): The Dirc s Delt. Mi Properties. Applictios. The Impulse Respose Fuctio. 4.4.. The Dirc Delt. 4.4. Geerlized Sources Defiitio 4.4.. The Dirc delt geerlized fuctio is the limit δ(t)
More informationThe Reimann Integral is a formal limit definition of a definite integral
MATH 136 The Reim Itegrl The Reim Itegrl is forml limit defiitio of defiite itegrl cotiuous fuctio f. The costructio is s follows: f ( x) dx for Reim Itegrl: Prtitio [, ] ito suitervls ech hvig the equl
More informationWe will begin by supplying the proof to (a).
(The solutios of problem re mostly from Jeffrey Mudrock s HWs) Problem 1. There re three sttemet from Exmple 5.4 i the textbook for which we will supply proofs. The sttemets re the followig: () The spce
More informationEigenfunction Expansion. For a given function on the internal a x b the eigenfunction expansion of f(x):
Eigefuctio Epsio: For give fuctio o the iterl the eigefuctio epsio of f(): f ( ) cmm( ) m 1 Eigefuctio Epsio (Geerlized Fourier Series) To determie c s we multiply oth sides y Φ ()r() d itegrte: f ( )
More informationThe evaluation of P, and T from these formulae indeed requires that the energy E be expressed as a function of the quantities N, V and S.
d dq, dq d d d, d d d d, e evlutio of, d from tese formule ideed requires tt te eerg be epressed s fuctio of te qutities, d. f (,,) is sould, i priciple, be possible oce is kow s fuctio of, d. f (,, )
More informationf(bx) dx = f dx = dx l dx f(0) log b x a + l log b a 2ɛ log b a.
Eercise 5 For y < A < B, we hve B A f fb B d = = A B A f d f d For y ɛ >, there re N > δ >, such tht d The for y < A < δ d B > N, we hve ba f d f A bb f d l By ba A A B A bb ba fb d f d = ba < m{, b}δ
More informationPhysics 525, Condensed Matter Homework 3 Due Tuesday, 16 th October 2006
Physics 55 Condensed Mtter Homework Due Tuesdy 6 th October 6 Jcob Lewis Bourjily Problem : Electron in Wek Sinusoidl Potentil Consider n electron moving in one-dimensionl periodic potentil Ur = cosπr/.
More informationLimit of a function:
- Limit of fuctio: We sy tht f ( ) eists d is equl with (rel) umer L if f( ) gets s close s we wt to L if is close eough to (This defiitio c e geerlized for L y syig tht f( ) ecomes s lrge (or s lrge egtive
More informationCHAPTER 2: Boundary-Value Problems in Electrostatics: I. Applications of Green s theorem
CHAPTER : Boudr-Vlue Problems i Electrosttics: I Applictios of Gree s theorem .6 Gree Fuctio for the Sphere; Geerl Solutio for the Potetil The geerl electrosttic problem (upper figure): ( ) ( ) with b.c.
More information334 MATHS SERIES DSE MATHS PREVIEW VERSION B SAMPLE TEST & FULL SOLUTION
MATHS SERIES DSE MATHS PREVIEW VERSION B SAMPLE TEST & FULL SOLUTION TEST SAMPLE TEST III - P APER Questio Distributio INSTRUCTIONS:. Attempt ALL questios.. Uless otherwise specified, ll worig must be
More informationGRAPHING LINEAR EQUATIONS. Linear Equations. x l ( 3,1 ) _x-axis. Origin ( 0, 0 ) Slope = change in y change in x. Equation for l 1.
GRAPHING LINEAR EQUATIONS Qudrt II Qudrt I ORDERED PAIR: The first umer i the ordered pir is the -coordite d the secod umer i the ordered pir is the y-coordite. (, ) Origi ( 0, 0 ) _-is Lier Equtios Qudrt
More informationApproximations of Definite Integrals
Approximtios of Defiite Itegrls So fr we hve relied o tiderivtives to evlute res uder curves, work doe by vrible force, volumes of revolutio, etc. More precisely, wheever we hve hd to evlute defiite itegrl
More informationLecture #1 Nasser S. Alzayed.
Lecture #1 Nasser S. Alzayed alzayed@ksu.edu.sa Chapter 6: Free Electro Fermi Gas Itroductio We ca uderstad may physical properties of metals, ad ot oly of the simple metals, i terms of the free electro
More informationNotes 17 Sturm-Liouville Theory
ECE 638 Fll 017 Dvid R. Jckso Notes 17 Sturm-Liouville Theory Notes re from D. R. Wilto, Dept. of ECE 1 Secod-Order Lier Differetil Equtios (SOLDE) A SOLDE hs the form d y dy 0 1 p ( x) + p ( x) + p (
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 informationPHYS-3301 Lecture 10. Wave Packet Envelope Wave Properties of Matter and Quantum Mechanics I CHAPTER 5. Announcement. Sep.
Aoucemet Course webpage http://www.phys.ttu.edu/~slee/3301/ PHYS-3301 Lecture 10 HW3 (due 10/4) Chapter 5 4, 8, 11, 15, 22, 27, 36, 40, 42 Sep. 27, 2018 Exam 1 (10/4) Chapters 3, 4, & 5 CHAPTER 5 Wave
More informationProbability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers Roy D. Yates and David J.
Probbility d Stochstic Processes: A Friedly Itroductio for Electricl d Computer Egieers Roy D. Ytes d Dvid J. Goodm Problem Solutios : Ytes d Goodm,4..4 4..4 4..7 4.4. 4.4. 4..6 4.6.8 4.6.9 4.7.4 4.7.
More informationReview of the Riemann Integral
Chpter 1 Review of the Riem Itegrl This chpter provides quick review of the bsic properties of the Riem itegrl. 1.0 Itegrls d Riem Sums Defiitio 1.0.1. Let [, b] be fiite, closed itervl. A prtitio P of
More informationChapter 30: Reflection and Refraction
Chpter 30: Reflectio d Refrctio The ture of light Speed of light (i vcuum) c.9979458 x 0 8 m/s mesured ut it is ow the defiitio Michelso s 878 Rottig Mirror Experimet Germ Americ physicist A.A. Michelso
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 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 information0 otherwise. sin( nx)sin( kx) 0 otherwise. cos( nx) sin( kx) dx 0 for all integers n, k.
. Computtio of Fourier Series I this sectio, we compute the Fourier coefficiets, f ( x) cos( x) b si( x) d b, i the Fourier series To do this, we eed the followig result o the orthogolity of the trigoometric
More information: : 8.2. Test About a Population Mean. STT 351 Hypotheses Testing Case I: A Normal Population with Known. - null hypothesis states 0
8.2. Test About Popultio Me. Cse I: A Norml Popultio with Kow. H - ull hypothesis sttes. X1, X 2,..., X - rdom smple of size from the orml popultio. The the smple me X N, / X X Whe H is true. X 8.2.1.
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 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 informationForce and Motion. Force. Classifying Forces. Physics 11- Summer /21/01. Chapter 4 material 1. Forces are vector quantities!
Force d Motio Cocept of Force Newto s hree Lws ypes of Forces Free body lysis Equilibrium Noequilibrium Frictio Problem Solvig Force A Force is push or pull tht is exerted o object by some other object.
More informationIntroduction to Solid State Physics
Itroductio to Solid State Physics Class: Itegrated Photoic Devices Time: Fri. 8:00am ~ 11:00am. Classroom: 資電 206 Lecturer: Prof. 李明昌 (Mig-Chag Lee) Electros i A Atom Electros i A Atom Electros i Two atoms
More informationMAHESH TUTORIALS SUBJECT : Maths(012) First Preliminary Exam Model Answer Paper
SET - GSE tch : 0th Std. Eg. Medium MHESH TUTILS SUJET : Mths(0) First Prelimiry Exm Model swer Pper PRT -.. () like does ot exist s biomil surd. () 4.. 6. 7. 8. 9. 0... 4 (c) touches () - d () -4 7 (c)
More informationDoped semiconductors: donor impurities
Doped semicoductors: door impurities A silico lattice with a sigle impurity atom (Phosphorus, P) added. As compared to Si, the Phosphorus has oe extra valece electro which, after all bods are made, has
More information2a a a 2a 4a. 3a/2 f(x) dx a/2 = 6i) Equation of plane OAB is r = λa + µb. Since C lies on the plane OAB, c can be expressed as c = λa +
-6-5 - - - - 5 6 - - - - - - / GCE A Level H Mths Nov Pper i) z + z 6 5 + z 9 From GC, poit of itersectio ( 8, 9 6, 5 ). z + z 6 5 9 From GC, there is o solutio. So p, q, r hve o commo poits of itersectio.
More informationLinford 1. Kyle Linford. Math 211. Honors Project. Theorems to Analyze: Theorem 2.4 The Limit of a Function Involving a Radical (A4)
Liford 1 Kyle Liford Mth 211 Hoors Project Theorems to Alyze: Theorem 2.4 The Limit of Fuctio Ivolvig Rdicl (A4) Theorem 2.8 The Squeeze Theorem (A5) Theorem 2.9 The Limit of Si(x)/x = 1 (p. 85) Theorem
More informationEECS130 Integrated Circuit Devices
EECS130 Itegrated Circuit Devices Professor Ali Javey 9/04/2007 Semicoductor Fudametals Lecture 3 Readig: fiish chapter 2 ad begi chapter 3 Aoucemets HW 1 is due ext Tuesday, at the begiig of the class.
More informationFree electron gas. Nearly free electron model. Tight-binding model. Semiconductors
Electroic Structure Drude theory Free electro gas Nearly free electro model Tight-bidig model Semicoductors Readig: A/M 1-3,8-10 G/S 7,11 Hoffma p. 1-0 106 DC ELECTRICAL CONDUCTIVITY A costat electric
More informationPHYS-3301 Lecture 9. CHAPTER 5 Wave Properties of Matter and Quantum Mechanics I. 5.3: Electron Scattering. Bohr s Quantization Condition
CHAPTER 5 Wave Properties of Matter ad Quatum Mecaics I PHYS-3301 Lecture 9 Sep. 5, 018 5.1 X-Ray Scatterig 5. De Broglie Waves 5.3 Electro Scatterig 5.4 Wave Motio 5.5 Waves or Particles? 5.6 Ucertaity
More informationUnit 1. Extending the Number System. 2 Jordan School District
Uit Etedig the Number System Jord School District Uit Cluster (N.RN. & N.RN.): Etedig Properties of Epoets Cluster : Etedig properties of epoets.. Defie rtiol epoets d eted the properties of iteger epoets
More informationa b c d e f g h Supplementary Information
Supplemetary Iformatio a b c d e f g h Supplemetary Figure S STM images show that Dark patters are frequetly preset ad ted to accumulate. (a) mv, pa, m ; (b) mv, pa, m ; (c) mv, pa, m ; (d) mv, pa, m ;
More informationRecurrece reltio & Recursio 9 Chpter 3 Recurrece reltio & Recursio Recurrece reltio : A recurrece is equtio or iequlity tht describes fuctio i term of itself with its smller iputs. T T The problem of size
More informationLecture 9: Diffusion, Electrostatics review, and Capacitors. Context
EECS 5 Sprig 4, Lecture 9 Lecture 9: Diffusio, Electrostatics review, ad Capacitors EECS 5 Sprig 4, Lecture 9 Cotext I the last lecture, we looked at the carriers i a eutral semicoductor, ad drift currets
More informationForce and Motion. Force
Force d Motio Cocept of Force Newto s hree Lws ypes of Forces Free body lysis Equilibrium Noequilibrium Frictio Problem Solvig Force A Force is push or pull tht is exerted o object by some other object.
More informationDiffraction: Real Samples Powder Method
Diffrctio: Rel Smples Powder Method Diffrctio: Rel Smples Up to this poit we hve bee cosiderig diffrctio risig from ifiitely lrge crystls tht re stri free d behve like idelly imperfect mterils ( x-rys
More informationMAS221 Analysis, Semester 2 Exercises
MAS22 Alysis, Semester 2 Exercises Srh Whitehouse (Exercises lbelled * my be more demdig.) Chpter Problems: Revisio Questio () Stte the defiitio of covergece of sequece of rel umbers, ( ), to limit. (b)
More informationGraphing Review Part 3: Polynomials
Grphig Review Prt : Polomils Prbols Recll, tht the grph of f ( ) is prbol. It is eve fuctio, hece it is smmetric bout the bout the -is. This mes tht f ( ) f ( ). Its grph is show below. The poit ( 0,0)
More informationUNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION
School Of Distce Eductio Questio Bk UNIVERSITY OF ALIUT SHOOL OF DISTANE EDUATION B.Sc MATHEMATIS (ORE OURSE SIXTH SEMESTER ( Admissio OMPLEX ANALYSIS Module- I ( A lytic fuctio with costt modulus is :
More informationTheorem 5.3 (Continued) The Fundamental Theorem of Calculus, Part 2: ab,, then. f x dx F x F b F a. a a. f x dx= f x x
Chpter 6 Applictios Itegrtio Sectio 6. Regio Betwee Curves Recll: Theorem 5.3 (Cotiued) The Fudmetl Theorem of Clculus, Prt :,,, the If f is cotiuous t ever poit of [ ] d F is tiderivtive of f o [ ] (
More informationFACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures
FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING Lectures MODULE 0 FURTHER CALCULUS II. Sequeces d series. Rolle s theorem d me vlue theorems 3. Tlor s d Mcluri s theorems 4. L Hopitl
More informationBC Calculus Review Sheet
BC Clculus Review Sheet Whe you see the words. 1. Fid the re of the ubouded regio represeted by the itegrl (sometimes 1 f ( ) clled horizotl improper itegrl). This is wht you thik of doig.... Fid the re
More informationThe Atmosphere. Atmospheric composition Measures of concentration Atmospheric pressure Barometric law
The Atmosphere Atmospheric compositio Mesures of cocetrtio Atmospheric pressure Brometric lw Literture coected with tody s lecture: Jcob, chpter 1-2 Exercises: 1:1 1:6; 2:1 2:4 The Atmosphere The tmosphere
More information( a n ) converges or diverges.
Chpter Ifiite Series Pge of Sectio E Rtio Test Chpter : Ifiite Series By the ed of this sectio you will be ble to uderstd the proof of the rtio test test series for covergece by pplyig the rtio test pprecite
More informationJEE-2015 : Advanced Paper 1 Answers and Explanations
CODE 5 JEE-5 : Advced Pper Aswers d Epltios Physics Chemistry Mthemtics A,C 8 A 8 5 A,D,C 7 B,C B,D 5 5 C,D A,C A 5 B,C B,D D 5 B,D 5 5 D 5 5 A 5 55 A,B 7 C C 5 A,D 7 7 B 7 7 A,B,C 7 8 57 A,C 8 8 A,B,C
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 information