Handout 7. Properties of Bloch States and Electron Statistics in Energy Bands
|
|
- Briana McBride
- 5 years ago
- Views:
Transcription
1 Hdout 7 Popts of Bloch Stts d Elcto Sttstcs Eg Bds I ths lctu ou wll l: Popts of Bloch fuctos Podc boud codtos fo Bloch fuctos Dst of stts -spc Elcto occupto sttstcs g bds ECE 407 Spg 009 Fh R Coll Uvst Bloch Fuctos - Su Elcto gs d solutos wtt s ( d Th solutos stsf th Bloch s tho: R R s stctd to th fst BZ): d c b wtt s supposto of pl wvs s show blow fo D: A lttc vcto d cpocl lttc vcto c b wtt s: R b b b olu of th dct lttc ptv cll d th cpocl lttc fst BZ : E c b b b ECE 407 Spg 009 Fh R Coll Uvst
2 ECE 407 Spg 009 Fh R Coll Uvst Bloch Fucto Poduct Fo Epsso A Bloch fucto cospodg to th wvvcto d g bd c lws b wtt s supposto ov pl wvs th fo: c Th bov psso c b -wtt s follows: u c c Wh th fucto s lttc podc: u u c c R u R stsfs Bloch s tho R R u ot tht: ECE 407 Spg 009 Fh R Coll Uvst Allowd Wvvctos fo F-Elctos (Sofld Modl) W usd podc boud codtos: Th boud codtos dctt tht th llowd vlus of d such tht: p = 0 ± ± = 0 ± ± p = 0 ± ± Th gd pots p ut volu of -spc
3 Bloch Fuctos Podc Boud Codtos b b A vcto Dct lttc th fst BZ c b wtt s: b b b Rcpocl lttc fo D lttc wh d g fo -/ to +/: ECE 407 Spg 009 Fh R Coll Uvst Bloch Fuctos Podc Boud Codtos Fo th D cstl : A Cosd D cstl d up of ptv clls th dcto ptv clls th dcto d ptv clls th dcto olu of th t cstl s: Assug podc boud codtos ll th dctos w ust hv: ECE 407 Spg 009 Fh R Coll Uvst
4 4 ECE 407 Spg 009 Fh R Coll Uvst Bloch Fuctos Podc Boud Codtos Th podc boud codto th dcto pls: : tht cll tg s b Sc: c hv dfft tgl vlus btw / d + / Rcpocl lttc fo D lttc Dct lttc b b b b b ECE 407 Spg 009 Fh R Coll Uvst Bloch Fuctos Podc Boud Codtos Sll th podc boud codtos th dctos of d pl: & & & c hv dfft tgl vlus c hv dfft tgl vlus c hv dfft tgl vlus Rcpocl lttc fo D lttc Dct lttc b b b b b Sc -vcto th FBZ s gv s: th dfft llowd -vlus th FBZ Th s dfft llowd -vlus th FBZ s th ub of ptv clls th cstl &
5 Rcpocl lttc fo D lttc b Dst of Stts -Spc Qusto: Sc b s llowd to hv ol dsct vlus how llowd -vlus th p ut volu of th -spc? D Cs: olu of th fst BZ s: b b b b b b I ths volu th llowd -vlus Th ub of llowd -vlus p ut volu -spc : wh s th volu of th cstl ECE 407 Spg 009 Fh R Coll Uvst Dst of Stts -Spc D Cs: gth of th cstl: gth of th fst BZ s: b I th fst BZ th llowd -vlus Th ub of llowd -vlus p ut lgth -spc : D Cs: A of th cstl: A A of th fst BZ s: b b I th fst BZ th llowd -vlus Th ub of llowd -vlus p ut -spc : A ECE 407 Spg 009 Fh R Coll Uvst b b b 5
6 D Cs: Rcpocl lttc s: Stts -Spc d ub of Ptv Clls Eg ˆ b gth of th cstl: gth of th fst BZ s: b I th fst BZ th llowd -vlus Th ub of llowd -vlus p ut lgth -spc : Th llowd -vlus -spc Th llowd -vlus p g bd Th s llowd -vlus p g bd s th ub of ptv clls th t cstl ECE 407 Spg 009 Fh R Coll Uvst D Cs: Stts -Spc d ub of Ptv Clls Eg Rcpocl lttc s: b b I th fst BZ th llowd -vlus (0 ) FBZ ( 0) Th llowd -vlus p g bd Th s llowd -vlus p g bd s th ub of ptv clls th t cstl ECE 407 Spg 009 Fh R Coll Uvst 6
7 D Cs: Sttstcs of Elctos Eg Bds Suppos I wt to fd th totl ub of lctos th -th bd how should I fd t? Th pobblt tht th qutu stt of wvvcto s th -th g bd s occupd b lcto s gv b th F-Dc dstbuto: f E E K T Th th totl ub of lctos th -th bd ust qul th followg su ov ll th llowd vlus - spc th fst BZ: f ll FBZ sp Th ub of llowd -vlus p ut lgth - spc s / thfo: d f f ll FBZ f Eg E E E ECE 407 Spg 009 Fh R Coll Uvst d to fd th totl ub of lctos th -th bd f D Cs: Sttstcs of Elctos Eg Bds ll FBZ Th ub of llowd -vlus p ut -spc s: A Thfo: d f A f ll FBZ FBZ D Cs: Th ub of llowd -vlus p ut volu -spc s: Thfo: d f f ll FBZ FBZ ECE 407 Spg 009 Fh R Coll Uvst 7
8 Bd Fllg t T0K fo D lttc Suppos th ub of ptv clls = Eg Qusto: suppos w hv lctos p ptv cll How wll th bds fll up t T0K? Wh wll b th F lvl? lctos p ptv cll totl ub of lctos E f fo 4 lctos p ptv cll ub of -vlus p bd = ub of qutu stts p bd = sp Fst bd wll b copltl flld All hgh bds wll b pt Qusto: Suppos w hv lctos p ptv cll How wll th bds fll up t T0K? lctos p ptv cll totl ub of lctos Fst bd wll b copltl flld Scod bd wll b hlf flld All hgh bds wll b pt E f fo lctos p ptv cll E f fo lctos p ptv cll ECE 407 Spg 009 Fh R Coll Uvst Bd Fllg t T0K fo D lttc Suppos th ub of ptv clls = Qusto: suppos w hv lctos p ptv cll How wll th bds fll up t T0K? Wh wll b th F lvl? lctos p ptv cll totl ub of lctos ub of -vlus p bd = ub of qutu stts p bd = sp Fst bd wll b copltl flld All hgh bds wll b pt Ipott lsso: I g bd (whth D D o D) th totl ub of qutu stts vlbl s twc th ub of ptv clls th dct lttc How th bds gt flld dpds o th ub of lctos p ptv cll ECE 407 Spg 009 Fh R Coll Uvst 8
9 F Sufcs (D) d Cotous (D) Solds Eg Eg Fst g bd of D lttc E F F FBZ F ccl fo f lcto gs D Wht hpps solds wh th g bds o copl? ECE 407 Spg 009 Fh R Coll Uvst F Sufcs (D) d Cotous (D) Solds c b ˆ ˆ c cˆ Eg FBZ Fst g bd of D lttc b ˆ c F cotous fo dfft lcto dsts cospodg to th g bd show o th lft FBZ FBZ ECE 407 Spg 009 Fh R Coll Uvst 9
10 F Sufcs (D) d Cotous (D) Solds F sufc of spl cubc dct lttc show sd th fst BZ F sufc of FCC lttc show sd th fst BZ (th fgu shows th F sufc of Copp) ECE 407 Spg 009 Fh R Coll Uvst Bd Fllg t T0K fo Slco Slco: Atoc ub: 4 Elcto Cofguto: s s p 6 s p ub of lctos th outost shll: 4 Th lctos th outost shll c ov fo to to to th lttc th ot cofd to dvdul to Th gs dscbd b th g bds Th lctos th shlls cofd to dvdul tos Slco lttc s FCC Th Slco tos p ptv cll ( bss tos) ECE 407 Spg 009 Fh R Coll Uvst Th 4 lctos cotbutd b ch Slco to d so th 8 lctos p ptv cll tht vlbl to fll th g bds 0
11 Bd Fllg t T0K fo Slco Th 8 lctos p ut cll vlbl to fll th g bds Rcll tht ch g bd th ub of stts vlbl s twc th ub of ptv clls th cstl I Slco th lowst 4 g bds wll gt copltl flld t T0K d ll th hgh g bds wll b pt E f FBZ (fo FCC lttc) Slco Eg Bds ECE 407 Spg 009 Fh R Coll Uvst Eg Bds Slco Th hghst flld g bd s clld th vlc bd I slco th vlc bd s doubl dgt t ost pots th fst BZ Th lowst pt g bd s clld th coducto bd I g th vlc bd u d th coducto bd u d ot hpp t th s pot -spc (s s th cs Slco) Th lowst g of th coducto bd s clld E c d th hghst g of th vlc bd s clld E v Coducto bd lc bd E f FBZ (fo FCC lttc) Slco Eg Bds ECE 407 Spg 009 Fh R Coll Uvst
12 Eg Eg FBZ ECE 407 Spg 009 Fh R Coll Uvst
Chapter 2 Reciprocal Lattice. An important concept for analyzing periodic structures
Chpt Rcpocl Lttc A mpott cocpt o lyzg podc stuctus Rsos o toducg cpocl lttc Thoy o cystl dcto o x-ys, utos, d lctos. Wh th dcto mxmum? Wht s th tsty? Abstct study o uctos wth th podcty o Bvs lttc Fou tsomto.
More informationCBSE SAMPLE PAPER SOLUTIONS CLASS-XII MATHS SET-2 CBSE , ˆj. cos. SECTION A 1. Given that a 2iˆ ˆj. We need to find
BSE SMLE ER SOLUTONS LSS-X MTHS SET- BSE SETON Gv tht d W d to fd 7 7 Hc, 7 7 7 Lt, W ow tht Thus, osd th vcto quto of th pl z - + z = - + z = Thus th ts quto of th pl s - + z = Lt d th dstc tw th pot,,
More informationCBSE , ˆj. cos CBSE_2015_SET-1. SECTION A 1. Given that a 2iˆ ˆj. We need to find. 3. Consider the vector equation of the plane.
CBSE CBSE SET- SECTION. Gv tht d W d to fd 7 7 Hc, 7 7 7. Lt,. W ow tht.. Thus,. Cosd th vcto quto of th pl.. z. - + z = - + z = Thus th Cts quto of th pl s - + z = Lt d th dstc tw th pot,, - to th pl.
More informationHandout on. Crystal Symmetries and Energy Bands
dou o Csl s d g Bds I hs lu ou wll l: Th loshp bw ss d g bds h bs of sp-ob ouplg Th loshp bw ss d g bds h ps of sp-ob ouplg C 7 pg 9 Fh Coll Uvs d g Bds gll hs oh Th sl pol ss ddo o h l slo s: Fo pl h
More informationBoyce/DiPrima 9 th ed, Ch 7.6: Complex Eigenvalues
BocDPm 9 h d Ch 7.6: Compl Egvlus Elm Dffl Equos d Boud Vlu Poblms 9 h do b Wllm E. Boc d Rchd C. DPm 9 b Joh Wl & Sos Ic. W cosd g homogous ssm of fs od l quos wh cos l coffcs d hus h ssm c b w s ' A
More informationSchool of Aerospace Engineering Origins of Quantum Theory. Measurements of emission of light (EM radiation) from (H) atoms found discrete lines
Ogs of Quatu Thoy Masuts of sso of lght (EM adato) fo (H) atos foud dsct ls 5 4 Abl to ft to followg ss psso ν R λ c λwavlgth, νfqucy, cspd lght RRydbg Costat (~09,7677.58c - ),,, +, +,..g.,,.6, 0.6, (Lya
More informationIFYFM002 Further Maths Appendix C Formula Booklet
Ittol Foudto Y (IFY) IFYFM00 Futh Mths Appd C Fomul Booklt Rltd Documts: IFY Futh Mthmtcs Syllbus 07/8 Cotts Mthmtcs Fomul L Equtos d Mtcs... Qudtc Equtos d Rmd Thom... Boml Epsos, Squcs d Ss... Idcs,
More informationD. Bertsekas and R. Gallager, "Data networks." Q: What are the labels for the x-axis and y-axis of Fig. 4.2?
pd by J. Succ ECE 543 Octob 22 2002 Outl Slottd Aloh Dft Stblzd Slottd Aloh Uslottd Aloh Splttg Algoths Rfc D. Btsks d R. llg "Dt twoks." Rvw (Slottd Aloh): : Wht th lbls fo th x-xs d y-xs of Fg. 4.2?
More informationSection 5.1/5.2: Areas and Distances the Definite Integral
Scto./.: Ars d Dstcs th Dt Itgrl Sgm Notto Prctc HW rom Stwrt Ttook ot to hd p. #,, 9 p. 6 #,, 9- odd, - odd Th sum o trms,,, s wrtt s, whr th d o summto Empl : Fd th sum. Soluto: Th Dt Itgrl Suppos w
More informationCoordinate Transformations
Coll of E Copt Scc Mchcl E Dptt Nots o E lss Rvs pl 6, Istcto: L Ctto Coot Tsfotos Itocto W wt to c ot o lss lttv coot ssts. Most stts hv lt wth pol sphcl coot ssts. I ths ots, w wt to t ths oto of fft
More informationConvergence tests for the cluster DFT calculations
Covgc ss o h clus DF clculos. Covgc wh spc o bss s. s clculos o bss s covgc hv b po usg h PBE ucol o 7 os gg h-b. A s o h Guss bss ss wh csg s usss hs b us clug h -G -G** - ++G(p). A l sc o. Å h c bw h
More informationHomework 1: Solutions
Howo : Solutos No-a Fals supposto tst but passs scal tst lthouh -f th ta as slowss [S /V] vs t th appaac of laty alty th path alo whch slowss s to b tat to obta tavl ts ps o th ol paat S o V as a cosquc
More informationminimize c'x subject to subject to subject to
z ' sut to ' M ' M N uostrd N z ' sut to ' z ' sut to ' sl vrls vtor of : vrls surplus vtor of : uostrd s s s s s s z sut to whr : ut ost of :out of : out of ( ' gr of h food ( utrt : rqurt for h utrt
More information2sin cos dx. 2sin sin dx
Fou S & Fou To Th udtl d tht y podc ucto (.. o tht pt t ptcul tvl) c b pd t u o d co wv o dt pltud d qucy. Th c b glzd to tgl (Th Fou To), whch h wd gg pplcto o th oluto o dtl quto whch occu th Phycl cc.
More information(A) Find the sine half-range expansion of ) L L k = L. k π. 5 sin 5. sin. sin + L. sin. sin. sin (B) 5 sin 5 (C) sin. sin. sin. sin.
9-9 A Fd th hlf-g po of f < < < < f f f A B C 8 D 8 E F G H o of th bov %9 - p.-7 - b T b f d f d T f T b T d d 8 f 8 8 8 C / 9 - Fd th Fou tfom of f, f. w A w w C w w E w w G w w B w w D w w F w H o of
More informationLinear Prediction Analysis of Speech Sounds
Lr Prdcto Alyss of Sch Souds Brl Ch 4 frcs: X Hug t l So Lgug Procssg Chtrs 5 6 J Dllr t l Dscrt-T Procssg of Sch Sgls Chtrs 4-6 3 J W Pco Sgl odlg tchqus sch rcogto rocdgs of th I Stbr 993 5-47 Lr Prdctv
More information1. Stefan-Boltzmann law states that the power emitted per unit area of the surface of a black
Stf-Boltzm lw stts tht th powr mttd pr ut r of th surfc of blck body s proportol to th fourth powr of th bsolut tmprtur: 4 S T whr T s th bsolut tmprtur d th Stf-Boltzm costt= 5 4 k B 3 5c h ( Clcult 5
More informationChapter 6 Perturbation theory
Ct 6 Ptutio to 6. Ti-iddt odgt tutio to i o tutio sst is giv to fid solutios of λ ' ; : iltoi of si stt : igvlus of : otool igfutios of ; δ ii Rlig-Södig tutio to ' λ..6. ; : gl iltoi ': tutio λ : sll
More informationToday s topics. How did we solve the H atom problem? CMF Office Hours
CMF Offc ous Wd. Nov. 4 oo-p Mo. Nov. 9 oo-p Mo. Nov. 6-3p Wd. Nov. 8 :30-3:30 p Wd. Dc. 5 oo-p F. Dc. 7 4:30-5:30 Mo. Dc. 0 oo-p Wd. Dc. 4:30-5:30 p ouly xa o Th. Dc. 3 Today s topcs Bf vw of slctd sults
More informationGet Funky this Christmas Season with the Crew from Chunky Custard
Hol Gd Chcllo Adld o Hdly Fdy d Sudy Nhs Novb Dcb 2010 7p 11.30p G Fuky hs Chss Sso wh h Cw fo Chuky Cusd Fdy Nhs $99pp Sudy Nhs $115pp Tck pc cluds: Full Chss d buff, 4.5 hou bv pck, o sop. Ts & Codos
More information1. This question is about homeopathic solutions
Ju f th tl th klt ght fl t yu, ut th pt y cpltly w h qut dgd t chllgg th th typcl pp, ut yu huld tll l t ttpt th U yu ctfc kll t wk thugh th pl lgclly If yu d c tuck pt f qut, th pt ght tll ccl, d t gv
More informationERDOS-SMARANDACHE NUMBERS. Sabin Tabirca* Tatiana Tabirca**
ERDO-MARANDACHE NUMBER b Tbrc* Tt Tbrc** *Trslv Uvrsty of Brsov, Computr cc Dprtmt **Uvrsty of Mchstr, Computr cc Dprtmt Th strtg pot of ths rtcl s rprstd by rct work of Fch []. Bsd o two symptotc rsults
More informationT h e C S E T I P r o j e c t
T h e P r o j e c t T H E P R O J E C T T A B L E O F C O N T E N T S A r t i c l e P a g e C o m p r e h e n s i v e A s s es s m e n t o f t h e U F O / E T I P h e n o m e n o n M a y 1 9 9 1 1 E T
More informationChapter Linear Regression
Chpte 6.3 Le Regesso Afte edg ths chpte, ou should be ble to. defe egesso,. use sevel mmzg of esdul cte to choose the ght cteo, 3. deve the costts of le egesso model bsed o lest sques method cteo,. use
More informationCh. 22: Classical Theory of Harmonic Crystal
C. : Clssl Toy o mo Cysl gl o ml moo o o os l s ld o ls o pl ollowg:. Eqlbm Pops p o ls d Islos Eqlbm sy d Cos Egs Tml Epso d lg. Tspo Pops T pd o lo Tm Fl o Wdm-Fz Lw pody Tml Cody o Islos Tsmsso o od.
More informationQuantum Mechanics & Spectroscopy Prof. Jason Goodpaster. Problem Set #2 ANSWER KEY (5 questions, 10 points)
Chm 5 Problm St # ANSWER KEY 5 qustios, poits Qutum Mchics & Spctroscopy Prof. Jso Goodpstr Du ridy, b. 6 S th lst pgs for possibly usful costts, qutios d itgrls. Ths will lso b icludd o our futur ms..
More informationLinear System Review. Linear System Review. Descriptions of Linear Systems: 2008 Spring ME854 - GGZ Page 1
8 Sprg ME854 - Z Pg r Sym Rvw r Sym Rvw r Sym Rvw crpo of r Sym: p m R y R R y FT : & U Y Trfr Fco : y or : & : d y d r Sym Rvw orollbly d Obrvbly: fo 3.: FT dymc ym or h pr d o b corollbl f y l > d fl
More informationMajor: All Engineering Majors. Authors: Autar Kaw, Luke Snyder
Nolr Rgrsso Mjor: All Egrg Mjors Auhors: Aur Kw, Luk Sydr hp://urclhodsgusfdu Trsforg Nurcl Mhods Educo for STEM Udrgrdus 3/9/5 hp://urclhodsgusfdu Nolr Rgrsso hp://urclhodsgusfdu Nolr Rgrsso So populr
More informationLinear Algebra Existence of the determinant. Expansion according to a row.
Lir Algbr 2270 1 Existc of th dtrmit. Expsio ccordig to row. W dfi th dtrmit for 1 1 mtrics s dt([]) = (1) It is sy chck tht it stisfis D1)-D3). For y othr w dfi th dtrmit s follows. Assumig th dtrmit
More informationIn the name of Allah Proton Electromagnetic Form Factors
I th a of Allah Poto Elctoagtc o actos By : Maj Hazav Pof A.A.Rajab Shahoo Uvsty of Tchology Atoc o acto: W cos th tactos of lcto bas wth atos assu to b th gou stats. Th ct lcto ay gt scatt lastcally wth
More informationInner Product Spaces INNER PRODUCTS
MA4Hcdoc Ir Product Spcs INNER PRODCS Dto A r product o vctor spc V s ucto tht ssgs ubr spc V such wy tht th ollowg xos holds: P : w s rl ubr P : P : P 4 : P 5 : v, w = w, v v + w, u = u + w, u rv, w =
More informationLinear Prediction Analysis of
Lr Prdcto Alyss of Sch Souds Brl Ch Drtt of Coutr Scc & Iforto grg Ntol Tw Norl Uvrsty frcs: X Hug t l So Lgug g Procssg Chtrs 5 6 J Dllr t l Dscrt-T Procssg of Sch Sgls Chtrs 4-6 3 J W Pco Sgl odlg tchqus
More informationCh 1.2: Solutions of Some Differential Equations
Ch 1.2: Solutions of Som Diffrntil Equtions Rcll th fr fll nd owl/mic diffrntil qutions: v 9.8.2v, p.5 p 45 Ths qutions hv th gnrl form y' = y - b W cn us mthods of clculus to solv diffrntil qutions of
More informationThe Proof That There are no Invariabilities of Lorentz Transformations in the Interaction Theories of Micro-particle Physics
Th Poo Tht Th o Ivblts o ot Tsotos th Itcto Thos o Mco-tcl Phscs M Xoch Isttt o Iovtv Phscs Fho Ch bstct It s ov ths tht th t lst v sttos th tcto thos o co-tcl hscs tht th ot tsotos hv o vblts.. I th ol
More informationBEM with Linear Boundary Elements for Solving the Problem of the 3D Compressible Fluid Flow around Obstacles
EM wth L ou Elts o olvg th Pol o th D opssl Flu Flow ou Ostls Lut Gu o Vlsu stt hs pp psts soluto o th sgul ou tgl quto o th D opssl lu low ou ostl whh uss sopt l ou lts o Lgg tp. h sgul ou tgl quto oult
More informationChapter 3 Higher Order Linear ODEs
ht High Od i ODEs. Hoogous i ODEs A li qutio: is lld ohoogous. is lld hoogous. Tho. Sus d ostt ultils of solutios of o so o itvl I gi solutios of o I. Dfiitio. futios lld lil iddt o so itvl I if th qutio
More informationLecture 11 Waves in Periodic Potentials Today: Questions you should be able to address after today s lecture:
Lctur 11 Wvs in Priodic Potntils Tody: 1. Invrs lttic dfinition in 1D.. rphicl rprsnttion of priodic nd -priodic functions using th -xis nd invrs lttic vctors. 3. Sris solutions to th priodic potntil Hmiltonin
More informationCorrelation in tree The (ferromagnetic) Ising model
5/3/00 :\ jh\slf\nots.oc\7 Chaptr 7 Blf propagato corrlato tr Corrlato tr Th (frromagtc) Isg mol Th Isg mol s a graphcal mol or par ws raom Markov fl cosstg of a urct graph wth varabls assocat wth th vrtcs.
More informationThe Theory of Small Reflections
Jim Stils Th Univ. of Knss Dt. of EECS 4//9 Th Thory of Smll Rflctions /9 Th Thory of Smll Rflctions Rcll tht w nlyzd qurtr-wv trnsformr usg th multil rflction viw ot. V ( z) = + β ( z + ) V ( z) = = R
More informationLet's revisit conditional probability, where the event M is expressed in terms of the random variable. P Ax x x = =
L's rvs codol rol whr h v M s rssd rs o h rdo vrl. L { M } rrr v such h { M } Assu. { } { A M} { A { } } M < { } { } A u { } { } { A} { A} ( A) ( A) { A} A A { A } hs llows us o cosdr h cs wh M { } [ (
More informationKummer Beta -Weibull Geometric Distribution. A New Generalization of Beta -Weibull Geometric Distribution
ttol Jol of Ss: Bs Al Rsh JSBAR SSN 37-453 Pt & Ol htt://gss.og/.h?joljolofbsaal ---------------------------------------------------------------------------------------------------------------------------
More informationHandout 11. Energy Bands in Graphene: Tight Binding and the Nearly Free Electron Approach
Hdout rg ds Grh: Tght dg d th Nrl Fr ltro roh I ths ltur ou wll lr: rg Th tght bdg thod (otd ) Th -bds grh FZ C 407 Srg 009 Frh R Corll Uvrst Grh d Crbo Notubs: ss Grh s two dsol sgl to lr o rbo tos rrgd
More informationExtension Formulas of Lauricella s Functions by Applications of Dixon s Summation Theorem
Avll t http:pvu.u Appl. Appl. Mth. ISSN: 9-9466 Vol. 0 Issu Dr 05 pp. 007-08 Appltos Appl Mthts: A Itrtol Jourl AAM Etso oruls of Lurll s utos Appltos of Do s Suto Thor Ah Al Atsh Dprtt of Mthts A Uvrst
More informationLecture 35. Diffraction and Aperture Antennas
ctu 35 Dictin nd ptu ntnns In this lctu u will ln: Dictin f lctmgntic ditin Gin nd ditin pttn f ptu ntnns C 303 Fll 005 Fhn Rn Cnll Univsit Dictin nd ptu ntnns ptu ntnn usull fs t (mtllic) sht with hl
More informationFebruary 12 th December 2018
208 Fbu 2 th Dcb 208 Whgt Fbu Mch M 2* 3 30 Ju Jul Sptb 4* 5 7 9 Octob Novb Dcb 22* 23 Put ou blu bgs out v d. *Collctios d lt du to Public Holid withi tht wk. Rcclig wk is pik Rcclig wk 2 is blu Th stick
More informationChapter 8: Propagating Quantum States of Radiation
Quum Opcs f hcs Oplccs h R Cll Us Chp 8: p Quum Ss f R 8. lcmc Ms Wu I hs chp w wll cs pp quum ss f wus fs f spc. Cs h u shw lw f lcc wu. W ssum h h wu hs l lh qul h -c wll ssum l. Th lcc cs s fuc f l
More informationLecture 1. Probability in Quantum Mechanics
Polt pltus ctu Polt Qutu Mccs T ctl pol of qutu ccs stt s follows: If t c occu o t o pt pocss o pt ow o w t t totl polt su o ll pts? I ot wos wt t lws of polt o toc scl? Clsscll P P Fo c gu fg ullts t
More informationLuiz Leal Oak Ridge National Laboratory. of Massachusetts Institute. of Technology (MIT)
LzLl OkRdgNlLby LsPsdhNl Egg Dp f h MsshssIs f Thlgy(MIT) Csy f Lz Ll, Ok Rdg Nl Lby. Usd wh pss. NI T Idpd Tsp Eq f Φ(E,,Ωˆ ) Ωˆ. Φ + Σ Φ = dωˆ ' de'σ s (E' E, Ωˆ ' Ω)Φ(E', ',Ωˆ ) + S 4 π 0 Σ Msplsss
More information68X LOUIE B NUNN PKWYLOUIE B NUNN PKWY NC
B v Lk Wf gmt A Ix p 86 12'W 86 1'W 85 56'W 85 54'W 85 52'W 85 5'W 37 'N 68X Ggw LOI B NNN KWYLOI B NNN KWY N 36 58'N WAN p 1 36 56'N utt' v vt A S p 4 B v Lk Bg Stt Ntu v BAN 36 52'N 36 5'N p 2 Spg B
More informationAnalysis of Effects of Rebounds and Aerodynamics for Trajectory of Table Tennis Ball
Al f Effc f Ru Ac f Tjc f Tl T Bll Juk Nu Mchcl Scc Egg, Gu Schl f Egg, Ng Uv, Fu-ch, Chku-ku, Ng, J Ak Nkh Mchcl Scc Egg, Gu Schl f Egg, Ng Uv, Fu-ch, Chku-ku, Ng, J Yhku Hkw Mchcl Scc Egg, Gu Schl f
More informationNational Quali cations
PRINT COPY OF BRAILLE Ntiol Quli ctios AH08 X747/77/ Mthmtics THURSDAY, MAY INSTRUCTIONS TO CANDIDATES Cdidts should tr thir surm, form(s), dt of birth, Scottish cdidt umbr d th m d Lvl of th subjct t
More informationControl Systems. Lecture 8 Root Locus. Root Locus. Plant. Controller. Sensor
Cotol Syt ctu 8 Root ocu Clacal Cotol Pof. Eugo Schut hgh Uvty Root ocu Cotoll Plat R E C U Y - H C D So Y C C R C H Wtg th loo ga a w a ttd tackg th clod-loo ol a ga va Clacal Cotol Pof. Eugo Schut hgh
More informationStatics. Consider the free body diagram of link i, which is connected to link i-1 and link i+1 by joint i and joint i-1, respectively. = r r r.
Statcs Th cotact btw a mapulato ad ts vomt sults tactv ocs ad momts at th mapulato/vomt tac. Statcs ams at aalyzg th latoshp btw th actuato dv tous ad th sultat oc ad momt appld at th mapulato dpot wh
More information1. Improvement Resource That Can Be Discovered from Actual Site
CHECK POINTS AT ANUFACTURING FACILITY (GEBA) Th Scd d 2010/1/11 1. Iv Ruc Th C B Dcvd f Acul S I Wh T Chck Evlu l C 1.Sg Plc & Pckgg 2.Cfgu & Accuc Abl 1)A h ckd d d f hdlg? (Hgh dc dc g cc) 2)A h lcd
More informationGUC (Dr. Hany Hammad)
Lct # Pl s. Li bdsid s with ifm mplitd distibtis. Gl Csidtis Uifm Bimil Optimm (Dlph-Tchbshff) Cicl s. Pl s ssmig ifm mplitd citti m F m d cs z F d d M COMM Lct # Pl s ssmig ifm mplitd citti F m m m T
More informationIntroduction to Laplace Transforms October 25, 2017
Iroduco o Lplc Trform Ocobr 5, 7 Iroduco o Lplc Trform Lrr ro Mchcl Egrg 5 Smr Egrg l Ocobr 5, 7 Oul Rvw l cl Wh Lplc rform fo of Lplc rform Gg rform b gro Fdg rform d vr rform from bl d horm pplco o dffrl
More informationPart II, Measures Other Than Conversion I. Apr/ Spring 1
Pt II, Msus Oth hn onvsion I p/7 11 Sping 1 Pt II, Msus Oth hn onvsion II p/7 11 Sping . pplictions/exmpls of th RE lgoithm I Gs Phs Elmnty Rction dditionl Infomtion Only fd P = 8. tm = 5 K =. mol/dm 3
More informationSAMPLE CSc 340 EXAM QUESTIONS WITH SOLUTIONS: part 2
AMPLE C EXAM UETION WITH OLUTION: prt. It n sown tt l / wr.7888l. I Φ nots orul or pprotng t vlu o tn t n sown tt t trunton rror o ts pproton s o t or or so onstnts ; tt s Not tt / L Φ L.. Φ.. /. /.. Φ..787.
More informationk m The reason that his is very useful can be seen by examining the Taylor series expansion of some potential V(x) about a minimum point:
roic Oscilltor Pottil W r ow goig to stuy solutios to t TIS for vry usful ottil tt of t roic oscilltor. I clssicl cics tis is quivlt to t block srig robl or tt of t ulu (for sll oscilltios bot of wic r
More informationA Review of Dynamic Models Used in Simulation of Gear Transmissions
ANALELE UNIVERSITĂłII ETIMIE MURGU REŞIłA ANUL XXI NR. ISSN 5-797 Zol-Ios Ko Io-ol Mulu A Rvw o ls Us Sulo o G Tsssos Th vsgo o lv s lu gg g olg l us o sov sg u o pps g svl s oug o h ps. Th pupos o h ols
More informationLecture contents. Bloch theorem k-vector Brillouin zone Almost free-electron model Bands Effective mass Holes. NNSE 508 EM Lecture #9
Lctur contnts Bloch thorm -vctor Brillouin zon Almost fr-lctron modl Bnds ffctiv mss Hols Trnsltionl symmtry: Bloch thorm On-lctron Schrödingr qution ch stt cn ccommo up to lctrons: If Vr is priodic function:
More informationHelping every little saver
Spt th diffc d cut hw u c fid I c spt thigs! Hlpig v littl sv Hw d u p i? I ch Just pp it f u chs. T fid u lcl ch just visit s.c.uk/ch If u pig i chqu, it c tk ud 4 wkig ds t cl Ov th ph Just cll Tlph
More informationChapter 6. pn-junction diode: I-V characteristics
Chatr 6. -jucto dod: -V charactrstcs Tocs: stady stat rsos of th jucto dod udr ald d.c. voltag. ucto udr bas qualtatv dscusso dal dod quato Dvatos from th dal dod Charg-cotrol aroach Prof. Yo-S M Elctroc
More informationMULTI-PRODUCT INVENTORY CONTROL MODEL WITH CONSTRAINTS
spot d lcommucto Vol.7, No, 6 MUIRDU INVNRY NR MD WIH NSRAINS ug Kopytov, Fdo ss, od Gglz spot d lcommucto Isttut omoosov St., Rg, V9, tv h: 37 759. Fx: 37 766. ml: optov@ts.lv, fdotss@ts.lv Rg Ittol School
More information(A) the function is an eigenfunction with eigenvalue Physical Chemistry (I) First Quiz
96- Physcl Chmstry (I) Frst Quz lctron rst mss m 9.9 - klogrm, Plnck constnt h 6.66-4 oul scon Sp of lght c. 8 m/s, lctron volt V.6-9 oul. Th functon F() C[cos()+sn()] s n gnfuncton of /. Th gnvlu s (A)
More informationl2 l l, i.e., phase k k k, [( ). ( ). ]. l1 l l, r, 2
ISSN: 77-3754 ISO 9:8 tfd Itto Jou of Egg d Iovtv choogy (IJEI Vou 7 Iu 7 Juy 8 o dft wv gtzd duty cyd Ajy Ghot Dtt of Ad Sc Mhj Suj Ittut of choogy Dh-58 Id d ( x ( x Abtct- h ffct of dut chg fuctuto
More informationADORO TE DEVOTE (Godhead Here in Hiding) te, stus bat mas, la te. in so non mor Je nunc. la in. tis. ne, su a. tum. tas: tur: tas: or: ni, ne, o:
R TE EVTE (dhd H Hdg) L / Mld Kbrd gú s v l m sl c m qu gs v nns V n P P rs l mul m d lud 7 súb Fí cón ví f f dó, cru gs,, j l f c r s m l qum t pr qud ct, us: ns,,,, cs, cut r l sns m / m fí hó sn sí
More informationBinary Choice. Multiple Choice. LPM logit logistic regresion probit. Multinomial Logit
(c Pogsa Porchawssul, Faculty of Ecoomcs, Chulalogor Uvrsty (c Pogsa Porchawssul, Faculty of Ecoomcs, Chulalogor Uvrsty 3 Bary Choc LPM logt logstc rgrso probt Multpl Choc Multomal Logt (c Pogsa Porchawssul,
More informationKinetics. Review- Collision Theory. Reaction Rate. Page 1
Kcs Th sudy f c s. Spus cs cs h wll hpp - bu w c ll hw fs. Dmd wll spusly u gph vully. Rc mchsm- h sps by whch c ks plc. Rvw- lls Thy Pcls hv clld c. v h hd ugh Thgs h cs hs cs gh mp fs c gh cc fs c Smll
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 301 Signls & Systms Pf. Mk Fwl Discussin #1 Cmplx Numbs nd Cmplx-Vlud Functins Rding Assignmnt: Appndix A f Kmn nd Hck Cmplx Numbs Cmplx numbs is s ts f plynmils. Dfinitin f imginy # j nd sm sulting
More information, between the vertical lines x a and x b. Given a demand curve, having price as a function of quantity, p f (x) at height k is the curve f ( x,
Clculus for Businss nd Socil Scincs - Prof D Yun Finl Em Rviw vrsion 5/9/7 Chck wbsit for ny postd typos nd updts Pls rport ny typos This rviw sht contins summris of nw topics only (This rviw sht dos hv
More informationMore Statistics tutorial at 1. Introduction to mathematical Statistics
Mor Sttstcs tutorl t wwwdumblttldoctorcom Itroducto to mthmtcl Sttstcs Fl Soluto A Gllup survy portrys US trprurs s " th mvrcks, drmrs, d lors whos rough dgs d ucompromsg d to do t thr ow wy st thm shrp
More informationLecture 7 Diffusion. Our fluid equations that we developed before are: v t v mn t
Cla ot fo EE6318/Phy 6383 Spg 001 Th doumt fo tutoal u oly ad may ot b opd o dtbutd outd of EE6318/Phy 6383 tu 7 Dffuo Ou flud quato that w dvlopd bfo a: f ( )+ v v m + v v M m v f P+ q E+ v B 13 1 4 34
More informationPESIT Bangalore South Campus Hosur road, 1km before Electronic City, Bengaluru -100 Department of Basic Science and Humanities
P E PESIT Bglo South Cpus Hosu od, k bfo Elctoic Cit, Bgluu -00 Dptt of Bsic Scic d Huitis INTERNAL ASSESSMENT TEST Dt : 0/0/07 Mks: 0 Subjct & Cod : Egiig Mthtics I 5MAT Sc : ALL N of fcult : GVR,GKJ,RR,SV,NHM,DN,KR,
More informationMASSALINA CONDO MARINA
p p p p p G.. HWY. p p.. HWY... HWY. : HG, 00 T. TY, 0 Y 0 T o. B WG T T 0 okia 0 icrosoft orporatio - VY - T - T - GG - HBT - TT - T - TT T - TWT T 0 - T TT T - T TT T T T - TTY T - - T VT - K T ( H ),±
More informationC-Curves. An alternative to the use of hyperbolic decline curves S E R A F I M. Prepared by: Serafim Ltd. P. +44 (0)
An ltntiv to th us of hypolic dclin cuvs Ppd y: Sfim Ltd S E R A F I M info@sfimltd.com P. +44 (02890 4206 www.sfimltd.com Contnts Contnts... i Intoduction... Initil ssumptions... Solving fo cumultiv...
More informationChapter 1 Basic Concepts
Ch Bsc Cocs oduco od: X X ε ε ε ε ε O h h foog ssuos o css ε ε ε ε ε N Co No h X Chcscs of od: cos c ddc (ucod) d s of h soss dd of h ssocd c S qusos sd: Wh f h cs of h soss o cos d dd o h ssocd s? Wh
More information6.6 Moments and Centers of Mass
th 8 www.tetodre.co 6.6 oets d Ceters of ss Our ojectve here s to fd the pot P o whch th plte of gve shpe lces horzotll. Ths pot s clled the ceter of ss ( or ceter of grvt ) of the plte.. We frst cosder
More informationClassical Theory of Fourier Series : Demystified and Generalised VIVEK V. RANE. The Institute of Science, 15, Madam Cama Road, Mumbai
Clssil Thoy o Foi Sis : Dmystii Glis VIVEK V RANE Th Istitt o Si 5 Mm Cm Ro Mmbi-4 3 -mil ss : v_v_@yhoooi Abstt : Fo Rim itgbl tio o itvl o poit thi w i Foi Sis t th poit o th itvl big ot how wh th tio
More informationDepartment of Mathematics and Statistics Indian Institute of Technology Kanpur MSO202A/MSO202 Assignment 3 Solutions Introduction To Complex Analysis
Dpartmt of Mathmatcs ad Statstcs Ida Isttut of Tchology Kapur MSOA/MSO Assgmt 3 Solutos Itroducto To omplx Aalyss Th problms markd (T) d a xplct dscusso th tutoral class. Othr problms ar for hacd practc..
More informationThe Reign of Grace and Life. Romans 5:12-21 (5:12-14, 17 focus)
Th Rig of Gc d Lif Rom 5:12-21 (5:12-14, 17 focu) Th Ifluc of O h d ud Adolph H J o ph Smith B i t l m t Fid Idi Gdhi Ci Lu Gu ich N itz y l M d i M ch Nlo h Vig T L M uhmmd B m i o t T Ju Chit w I N h
More information3.4 Properties of the Stress Tensor
cto.4.4 Proprts of th trss sor.4. trss rasformato Lt th compots of th Cauchy strss tsor a coordat systm wth bas vctors b. h compots a scod coordat systm wth bas vctors j,, ar gv by th tsor trasformato
More informationCONIC SECTIONS. MODULE-IV Co-ordinate Geometry OBJECTIVES. Conic Sections
Conic Sctions 16 MODULE-IV Co-ordint CONIC SECTIONS Whil cutting crrot ou might hv noticd diffrnt shps shown th dgs of th cut. Anlticll ou m cut it in thr diffrnt ws, nml (i) (ii) (iii) Cut is prlll to
More informationAvailable online through
Avlble ole through wwwmfo FIXED POINTS FOR NON-SELF MAPPINGS ON CONEX ECTOR METRIC SPACES Susht Kumr Moht* Deprtmet of Mthemtcs West Begl Stte Uverst Brst 4 PrgsNorth) Kolt 76 West Begl Id E-ml: smwbes@yhoo
More informationInternational Journal of Advanced Scientific Research and Management, Volume 3 Issue 11, Nov
199 Algothm ad Matlab Pogam fo Softwa Rlablty Gowth Modl Basd o Wbull Od Statstcs Dstbuto Akladswa Svasa Vswaatha 1 ad Saavth Rama 2 1 Mathmatcs, Saaatha Collg of Egg, Tchy, Taml Nadu, Ida Abstact I ths
More informationCh. 6 Free Electron Fermi Gas
Ch. 6 lcto i Gas Coductio lctos i a tal ov fl without scattig b io cos so it ca b cosidd as if walitactig o f paticls followig idiac statistics. hfo th coductio lctos a fqutl calld as f lcto i gas. Coductio
More informationOPTICAL DESIGN. FIES fibre assemblies B and C. of the. LENS-TECH AB Bo Lindberg Document name: Optical_documentation_FIES_fiber_BC_2
OPTICAL DESIGN f h FIES fb ssmbs B d C LENS-TECH AB B Ldbg 2-4-3 Dcm m: Opc_dcm_FIES_fb_BC_2 Idc Ths p s dcm f h pc dsg f h FIES fb ssmbs B d C Th mchc dsg s shw I s shw h ssmb dwg md b Ahs Uvs Fb c Th
More informationHello! Included in this curriculum are: 1. The 10 Principles of Creativity and their Mantras -- Print this page and display it in your home.
Hll! Wl C Cv C. W v fl-fl l w l f 10 Pl f Cv. W blv lw b w Pl CHOOSE CREATIVITY w, w v f l fl! T l lv l f k w v k. Pl k f v, w w l w Fbk I. U #CCv. A fllw I @lllf w w w w Pl. Il l : 1. T 10 Pl f Cv M --
More informationChapter 5: Quantization of Radiation in Cavities and Free Space
Quu O f Ph Ol Fh R Cll vy Ch 5: Quz f R Cv F S 5 Cll ly 5 Cll Cvy ly Mxwll u f lg J 4 h lv l C fl vy W f h g f h vy Th vy u luly ll W l u h J Cvy F Mxwll u v h wv u Th v u lv h f h fu h vy I w wh h v l
More informationTHIS PAGE DECLASSIFIED IAW E
THS PAGE DECLASSFED AW E0 2958 BL K THS PAGE DECLASSFED AW E0 2958 THS PAGE DECLASSFED AW E0 2958 B L K THS PAGE DECLASSFED AW E0 2958 THS PAGE DECLASSFED AW EO 2958 THS PAGE DECLASSFED AW EO 2958 THS
More informationArea and the Definite Integral. Area under Curve. The Partition. y f (x) We want to find the area under f (x) on [ a, b ]
Are d the Defte Itegrl 1 Are uder Curve We wt to fd the re uder f (x) o [, ] y f (x) x The Prtto We eg y prttog the tervl [, ] to smller su-tervls x 0 x 1 x x - x -1 x 1 The Bsc Ide We the crete rectgles
More informationProc. of the 23rd Intl. Conf. on Parallel Processing, St. Charles, Illinois, August 1994, vol. 3, pp. 227{ Hanan Samet
P. 23 Il. C. Plll P, S. Cl, Ill, 1994, vl. 3,. 227{234 1 DT-PRE SPTI JOI GORITHMS Ek G. Hl y Gy Dv B C W, D.C. 20233 H S C S D C R I v C S Uvy Myl Cll Pk, Myl 20742 { E -lll l j l R-, l,. T l l (.., B
More informationThe real E-k diagram of Si is more complicated (indirect semiconductor). The bottom of E C and top of E V appear for different values of k.
Modr Smcoductor Dvcs for Itgratd rcuts haptr. lctros ad Hols Smcoductors or a bad ctrd at k=0, th -k rlatoshp ar th mmum s usually parabolc: m = k * m* d / dk d / dk gatv gatv ffctv mass Wdr small d /
More informationexhibitor prospectus InternatIonal art MaterIals association MARCH 4-6 DALLAS co-locating with: Denver Art Museum Colorado Convention Center
xhbo Oc IIol MIl ociio O18 DALLAS co-locg wh: Coloo Covo C Dv A Mm DALLAS IIol MIl ociio Ky bly hcho covo c h oly how c xclvly o l! O18 ch ov 700 ml by cov wh w h y oc w oc vlo w l l x bo chl c w loh xg
More informationNuclear Chemistry -- ANSWERS
Hoor Chstry Mr. Motro 5-6 Probl St Nuclar Chstry -- ANSWERS Clarly wrt aswrs o sparat shts. Show all work ad uts.. Wrt all th uclar quatos or th radoactv dcay srs o Urau-38 all th way to Lad-6. Th dcay
More informationBayesian Credibility for Excess of Loss Reinsurance Rating. By Mark Cockroft 1 Lane Clark & Peacock LLP
By Cly o c o Lo Rc Rg By M Coco L Cl & Pcoc LLP GIRO coc 4 Ac Th pp c how o v cly wgh w po- pc-v o c o lo c. Th po co o Poo-Po ol ch wh po G o. Kywo c o lo c g By cly Poo Po G po Acowlg cl I wol l o h
More informationME 501A Seminar in Engineering Analysis Page 1
St Ssts o Ordar Drtal Equatos Novbr 7 St Ssts o Ordar Drtal Equatos Larr Cartto Mcacal Er 5A Sar Er Aalss Novbr 7 Outl Mr Rsults Rvw last class Stablt o urcal solutos Stp sz varato or rror cotrol Multstp
More informationPhys Nov. 3, 2017 Today s Topics. Continue Chapter 2: Electromagnetic Theory, Photons, and Light Reading for Next Time
Phys 31. No. 3, 17 Today s Topcs Cou Chap : lcomagc Thoy, Phoos, ad Lgh Radg fo Nx Tm 1 By Wdsday: Radg hs Wk Fsh Fowls Ch. (.3.11 Polazao Thoy, Jos Macs, Fsl uaos ad Bws s Agl Homwok hs Wk Chap Homwok
More informationPH427/PH527: Periodic systems Spring Overview of the PH427 website (syllabus, assignments etc.) 2. Coupled oscillations.
Dy : Mondy 5 inuts. Ovrviw of th PH47 wsit (syllus, ssignnts tc.). Coupld oscilltions W gin with sss coupld y Hook's Lw springs nd find th possil longitudinl) otion of such syst. W ll xtnd this to finit
More informationSpin Structure of Nuclei and Neutrino Nucleus Reactions Toshio Suzuki
Spi Structur of Nucli d Nutrio Nuclus Rctios Toshio Suzuki Excittio of Spi Mods by s. Spctr DAR, DIF 3. Chrg-Exchg Rctios C, - N by iprovd spi-isospi itrctio with shll volutio Sprdig ffcts of GT strgth
More information