SECOND HARMONIC GENERATION OF Bi 4 Ti 3 O 12 FILMS

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "SECOND HARMONIC GENERATION OF Bi 4 Ti 3 O 12 FILMS"

Transcription

1 SECOND HARMONIC GENERATION OF Bi 4 Ti 3 O 12 FILMS IN-SITU PROBING OF DOMAIN POLING IN Bi 4 Ti 3 O 12 THIN FILMS BY OPTICAL SECOND HARMONIC GENERATION YANIV BARAD, VENKATRAMAN GOPALAN Mterils Reserh Lortory nd Dept. of Mterils Siene nd Engineering, Pennsylvni Stte University, University Prk, PA 16802; We quntittively trk in rel-time, the hnges in domin sttistis with eletri field poling, of ferroeletri Bi 4 Ti 3 O 12 thin film using optil seond hrmoni genertion s proe. The ferroeletri hysteresis loop is extrted from these optil mesurements using theoretil model. The model lso yields rtios of intrinsi nonliner oeffiients suh s d 11 /d 12 =-3.54±0.31, d 26 /d 11 =0.4±0.03, nd optil irefringene n -n =0.079± Keywords: Feroeletris, Bismuth titnte, Thin films, Seond Hrmoni Genertion INTRODUCTION Ferroeletri Bismuth Titnte, Bi 4 Ti 3 O 12, whih elongs to the Aurivillius phses, is of interest in nonvoltile memory due to exellent ftigue resistne during repeted polriztion reversls with eletri field. [1],[2] The spontneous polriztion in monolini unit of Bi 4 Ti 3 O 12 hs omponents long oth the - nd - rystllogrphi diretions where - forms the mirror plne (010). Both - nd - omponents of the polriztion n e independently reversed, thus

2 Y. BARAD ET. AL. resulting in four different lsses of domin wlls nd 18 wll onfigurtions. [3] All these onfigurtions re not redily distinguishle in thin film y onventionl x-ry diffrtion or trnsmission eletron mirosopy. In reent pper, we showed how proing the seond hrmoni genertion (SHG) response of Bi 4 Ti 3 O 12 film with omplex domin mirostruture n provide mny of these domin distintions in quntittive mnner. [4] We present here the results of eletri field poling of Bi 4 Ti 3 O 12 thin film on SrTiO 3 sustrte. The ferroeletri hysteresis loop n e diretly extrted from the optil mesurements. As we show here, the poling experiments revel not only the mgnitude of domin mirostruturl is ut lso the sign of the is. DOMAIN STRUCTURE IN THE FILM The Bi 4 Ti 3 O 12 thin film studied here ws grown on SrTiO 3 (001) sustrte using moleulr em epitxy (MBE) s previously reported in detil. [5] The lttie prmeters of the ui SrTiO 3 (001) sustrte, = Å losely mth long its digonls of 2, with the lttie prmeters = Å nd = Å of the monolini Bi 4 Ti 3 O 12. (The other lttie prmeters re = Å, nd β = ). [6] The lttie plnes -, -, nd - of Bi 4 Ti 3 O 12 re respetively denoted s (100), (010) nd the (001) plnes. As reported efore, the epitxil reltionship is SrTiO 3 (001)[110]//Bi 4 Ti 3 O 12 (001)[100]. In this onfigurtion, there re eight possile domin onfigurtions of the Bi 4 Ti 3 O 12 s shown shemtilly in Fig. 1. Eh of the possile domins hs monolini unit ell, whih devites only slightly from the orthorhomi unit ell. The polriztion xis forms n ngle of ~ 4.5 from the rystllogrphi -xis in the - (010) plne. The film thikness is ~0.1µm.

3 SECOND HARMONIC GENERATION OF Bi 4 Ti 3 O 12 FILMS Bsed on the ove disussion, we define four lsses of domin vrints X+, X-, Y+ nd Y- ording to whether the -omponent of polriztion points in the SrTiO 3 [110], [1 10], [110], or [110] diretions, or lterntively (+x, -x, +y, -y) diretions in Fig. 1, respetively. In the following setions, we desrie how the hnges in the domin sttistis of X+, X-, Y+, nd Y- domins n e distinguished y seond hrmoni genertion mesurements. Domin Vrints in Bi 4 Ti 3 O 12 (001) film on SrTiO 3 (001) sustrte Y+ X- X+ - z Y- y, SrTiO 3 [110] x, SrTiO 3 [110] FIGURE 1. Ferroeletri domins vrints in the Bi 4 Ti 3 O 12 film on SrTiO 3 (001) sustrte. SECOND HARMONIC GENERATION MEASUREMENTS Detils of the experimentl setup hve een disussed in n erlier pulition. [4] The fundmentl em from 10 Hz Q-swithed Nd:YAG lser (λ = 1064 nm), nd pproximtely 1.2 mm is pssed through the film in norml inidene. The input polriztion of infrred light is rotted using hlf-wve plte. At the output, the seond hrmoni genertion (SHG) signl (λ=532nm) is seprted nd deteted for polriztions prllel to two in-

4 Y. BARAD ET. AL. plne SrTiO 3 <110> diretions in the sustrte (x- or y- xes in Fig. 1). An externl eletri field is pplied to the smple using gold eletrodes sputtered on the smple surfe with gp of ~ 3 mm etween them, to ssure of enough lerne for the em. The SHG signl I 2ω is mesured s funtion of input polriztion ngle, θ of the fundmentl. In generl, different re on the film gives slightly different polr plots. However, ll these plots n e nlyzed within the sme theoretil frmework given in our previous work. [4] Figure 2 presents results of typil SHG mesurement. y y I 2ω x FIGURE 2. The polr plots of SHG intensity, I 2ω, s funtion of input polriztion ngle θ of the fundmentl, in Bi 4 Ti 3 O 12 thin film on SrTiO 3 (001) sustrte. () y-polrized SHG, nd () x-polrized SHG, where x,y xes re shown in Fig. 1. Cirles re experiment nd solid line is the theoretil fit sed on Eq. 1. The experimentl results of Fig. 2 n e fit to the following theoretil eqution:

5 SECOND HARMONIC GENERATION OF Bi 4 Ti 3 O 12 FILMS I j 2ω = K 1, j (sin 2 θ + K 2,j os 2 θ ) 2 + K 3, j sin 2 2θ + K 4,j (sin 2 θ + K 2,j os 2 θ )sin 2θ (1) where j=x,y denote the output polriztion of the mesured SHG signl, nd the eight K i,j prmeters (i=1,2,3,4) re experimentlly determined from nonliner urve fitting s shown in Fig. 2. These prmeters re then used to lulte the following intrinsi mteril prmeters, s desried in Ref. [4]: d 11 /d 12 =-3.54±0.31, d 26 /d 11 =0.4±0.03, nd n -n =0.079±0.015, where d ij re seond order nonliner optil oeffiients for seond hrmoni genertion, nd n nd n re refrtive indies long rystllogrphi nd xes of Bi 4 Ti 3 O 12. In ddition, new mirostruturl informtion n e extrted from the mesured K ij prmeters. If the light therefore psses through thikness frtion t X+ of the X+ domin nd t X- of the X- domin, in n re da X of the X+/- domins, then the net thikness frtion, δa x, defined s A x = A X 0 s A y = (t X+ t X )da X. Similrly, onsidering the Y+ nd Y- domins in n re da y of the growth plne, the net thikness frtion, δa y, is defined A Y 0 (t Y + t Y )da Y. In words, δa x is proportionl to how muh more (or less) frtion of X+ domin exists in the proe re versus the X- domins. Similrly, δa y is proportionl to how muh more (or less) frtion of Y+ domin exists in the proe re versus the Y- domins. These reltionships reflets the net destrutive interferene of the seond hrmoni eletri fields reted y X+ (Y+) nd X-(Y-) domins due to the π phse shift etween the two fields. This nlysis therefore ssumes omplete phse orreltion, whih implies tht the seond hrmoni response of ll domin vrints re phse orrelted. This ssumption is justified in our present se sine the domin sizes of X+ nd X- vrints re of the order of

6 nm in the film growth plne, whih is less thn the wvelength of light. [4]

7 Y. BARAD ET. AL. From the K ij prmeters mesured from Fig. 2, we n lso extrt the rtio, δa y /δa x =2.25±0.05. In ddition, using stndrd rystl of LiTO 3 whose solute d ij oeffiients re known, we n lulte the solute mgnitudes of (δa y d 11 ) 2 ~10-3 nd (δa x d 11 ) 2 ~4.4x10-4. If the solute vlue of d 11 were known for Bi 4 Ti 3 O 12 (whih it isn t), one ould then lulte the tul mirostruturl ises δa y nd δa x. ELECTRIC FIELD POLING OF Bi 4 Ti 3 O 12 THIN FILMS Figure 3 shows the vrition of the mirostruturl is (δa y d 11 ) 2 s funtion of pplied voltge. (Note tht this proe re of the film ws different from tht proed in Fig. 2). FIGURE 3. The vrition of mirostruturl is (δa y d 11 ) 2 (see text for definition) (extrted from SHG mesurements) s funtion of eletri field pplied long the ±y xis in Fig. 1.

8 SECOND HARMONIC GENERATION OF Bi 4 Ti 3 O 12 FILMS The ove in-situ eletri field poling ws performed y pplying slow tringulr voltge wveform of 100 mhz frequeny ross two surfe gold eletrode pds, 3mm prt. The eletri field diretion ws long the y- xis with referene to Fig. 1. Sine δa y is proportionl to the net frtion of Y+ to Y- domins in the proe re, it mkes sense tht pplying negtive voltge (field in y diretion) inreses the frtion of Y- domins, nd positive voltge inreses the frtion of Y+ domins. At the minimum, the two domin types re pproximtely equl in mgnitude. One lso noties n symmetry in domin poling etween positive nd negtive field diretions, inditing tht n intrinsi domin is exists where Y- domins re more undnt thn Y+ domins t zero-field. This mirostruturl is is not unique to ll of the film re, ut is rther very lol feture within the proe re of the em, inditing the sensitivity of the SHG mesurements to the lol domin mirostruture. In onlusion, we hve developed seond hrmoni genertion s sensitive nd quntittive proe of the lol domin sttistis within the proe re. It n distinguish ntiprllel domins (180 domins) whih is diffiult y onventionl X-ry nd eletron mirosopy. The SHG mesurements n e performed in-situ during eletri field poling of the film. The ferroeletri hysteresis loop n then e diretly extrted from the study giving quntittive estimtes for re frtions of different domins vrints in the proe re of the film. Aknowledgements This work ws supported y the NSF wrds nd Referenes [1] C. A-Pz de rujo, J. D. Cuhiro, L. D. MMilln, M. C. Sott, nd J. F. Sott, Nture (London) 374, 627 (1995).

9 Y. BARAD ET. AL. [2] R. E. Newnhm, R. W. Wolfe, nd J. F. Dorrin, Mter. Res. Bull. 6, 1029 (1971). [3] S. E. Cummins, nd L. E. Cross, J. Appl. Phys., 39, 2268 (1968). [4] Y. Brd, J. Lettieri, C. D. Theis, D. G. Shlom, V. Gopln J. C. Jing, X. Q. Pn, J. Appl. Phys. 89, 1387 (2001). [5] C. D. Theis, J. Yeh, D. G. Shlom, M. E. Hwley, G. W. Brown, J. C. Jing, nd X. Q. Pn, Appl. Phys. Lett. 72, 2817 (1998). [6] A. D. Re, At Cryst. B46, 474 (1990).

Project 6: Minigoals Towards Simplifying and Rewriting Expressions

Project 6: Minigoals Towards Simplifying and Rewriting Expressions MAT 51 Wldis Projet 6: Minigols Towrds Simplifying nd Rewriting Expressions The distriutive property nd like terms You hve proly lerned in previous lsses out dding like terms ut one prolem with the wy

More information

1.3 SCALARS AND VECTORS

1.3 SCALARS AND VECTORS Bridge Course Phy I PUC 24 1.3 SCLRS ND VECTORS Introdution: Physis is the study of nturl phenomen. The study of ny nturl phenomenon involves mesurements. For exmple, the distne etween the plnet erth nd

More information

Symmetrical Components 1

Symmetrical Components 1 Symmetril Components. Introdution These notes should e red together with Setion. of your text. When performing stedy-stte nlysis of high voltge trnsmission systems, we mke use of the per-phse equivlent

More information

Math 32B Discussion Session Week 8 Notes February 28 and March 2, f(b) f(a) = f (t)dt (1)

Math 32B Discussion Session Week 8 Notes February 28 and March 2, f(b) f(a) = f (t)dt (1) Green s Theorem Mth 3B isussion Session Week 8 Notes Februry 8 nd Mrh, 7 Very shortly fter you lerned how to integrte single-vrible funtions, you lerned the Fundmentl Theorem of lulus the wy most integrtion

More information

Reflection Property of a Hyperbola

Reflection Property of a Hyperbola Refletion Propert of Hperol Prefe The purpose of this pper is to prove nltill nd to illustrte geometrill the propert of hperol wherein r whih emntes outside the onvit of the hperol, tht is, etween the

More information

Numbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point

Numbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point GCSE C Emple 7 Work out 9 Give your nswer in its simplest form Numers n inies Reiprote mens invert or turn upsie own The reiprol of is 9 9 Mke sure you only invert the frtion you re iviing y 7 You multiply

More information

Calculus Cheat Sheet. Integrals Definitions. where F( x ) is an anti-derivative of f ( x ). Fundamental Theorem of Calculus. dx = f x dx g x dx

Calculus Cheat Sheet. Integrals Definitions. where F( x ) is an anti-derivative of f ( x ). Fundamental Theorem of Calculus. dx = f x dx g x dx Clulus Chet Sheet Integrls Definitions Definite Integrl: Suppose f ( ) is ontinuous Anti-Derivtive : An nti-derivtive of f ( ) on [, ]. Divide [, ] into n suintervls of is funtion, F( ), suh tht F = f.

More information

SUPPLEMENTARY INFORMATION

SUPPLEMENTARY INFORMATION SUPPLEMENTARY INFORMATION doi: 1.138/nnno.29.451 Aove-ndgp voltges from ferroelectric photovoltic devices S. Y. Yng, 1 J. Seidel 2,3, S. J. Byrnes, 2,3 P. Shfer, 1 C.-H. Yng, 3 M. D. Rossell, 4 P. Yu,

More information

Matrices SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics (c) 1. Definition of a Matrix

Matrices SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics (c) 1. Definition of a Matrix tries Definition of tri mtri is regulr rry of numers enlosed inside rkets SCHOOL OF ENGINEERING & UIL ENVIRONEN Emple he following re ll mtries: ), ) 9, themtis ), d) tries Definition of tri Size of tri

More information

NON-DETERMINISTIC FSA

NON-DETERMINISTIC FSA Tw o types of non-determinism: NON-DETERMINISTIC FS () Multiple strt-sttes; strt-sttes S Q. The lnguge L(M) ={x:x tkes M from some strt-stte to some finl-stte nd ll of x is proessed}. The string x = is

More information

m m m m m m m m P m P m ( ) m m P( ) ( ). The o-ordinte of the point P( ) dividing the line segment joining the two points ( ) nd ( ) eternll in the r

m m m m m m m m P m P m ( ) m m P( ) ( ). The o-ordinte of the point P( ) dividing the line segment joining the two points ( ) nd ( ) eternll in the r CO-ORDINTE GEOMETR II I Qudrnt Qudrnt (-.+) (++) X X - - - 0 - III IV Qudrnt - Qudrnt (--) - (+-) Region CRTESIN CO-ORDINTE SSTEM : Retngulr Co-ordinte Sstem : Let X' OX nd 'O e two mutull perpendiulr

More information

for all x in [a,b], then the area of the region bounded by the graphs of f and g and the vertical lines x = a and x = b is b [ ( ) ( )] A= f x g x dx

for all x in [a,b], then the area of the region bounded by the graphs of f and g and the vertical lines x = a and x = b is b [ ( ) ( )] A= f x g x dx Applitions of Integrtion Are of Region Between Two Curves Ojetive: Fin the re of region etween two urves using integrtion. Fin the re of region etween interseting urves using integrtion. Desrie integrtion

More information

Section 4.4. Green s Theorem

Section 4.4. Green s Theorem The Clulus of Funtions of Severl Vriles Setion 4.4 Green s Theorem Green s theorem is n exmple from fmily of theorems whih onnet line integrls (nd their higher-dimensionl nlogues) with the definite integrls

More information

QUADRATIC EQUATION. Contents

QUADRATIC EQUATION. Contents QUADRATIC EQUATION Contents Topi Pge No. Theory 0-04 Exerise - 05-09 Exerise - 09-3 Exerise - 3 4-5 Exerise - 4 6 Answer Key 7-8 Syllus Qudrti equtions with rel oeffiients, reltions etween roots nd oeffiients,

More information

6.3.2 Spectroscopy. N Goalby chemrevise.org 1 NO 2 CH 3. CH 3 C a. NMR spectroscopy. Different types of NMR

6.3.2 Spectroscopy. N Goalby chemrevise.org 1 NO 2 CH 3. CH 3 C a. NMR spectroscopy. Different types of NMR 6.. Spetrosopy NMR spetrosopy Different types of NMR NMR spetrosopy involves intertion of mterils with the lowenergy rdiowve region of the eletromgneti spetrum NMR spetrosopy is the sme tehnology s tht

More information

6.5 Improper integrals

6.5 Improper integrals Eerpt from "Clulus" 3 AoPS In. www.rtofprolemsolving.om 6.5. IMPROPER INTEGRALS 6.5 Improper integrls As we ve seen, we use the definite integrl R f to ompute the re of the region under the grph of y =

More information

A Mathematical Model for Unemployment-Taking an Action without Delay

A Mathematical Model for Unemployment-Taking an Action without Delay Advnes in Dynmil Systems nd Applitions. ISSN 973-53 Volume Number (7) pp. -8 Reserh Indi Publitions http://www.ripublition.om A Mthemtil Model for Unemployment-Tking n Ation without Dely Gulbnu Pthn Diretorte

More information

Introduction to Olympiad Inequalities

Introduction to Olympiad Inequalities Introdution to Olympid Inequlities Edutionl Studies Progrm HSSP Msshusetts Institute of Tehnology Snj Simonovikj Spring 207 Contents Wrm up nd Am-Gm inequlity 2. Elementry inequlities......................

More information

Lesson 2: The Pythagorean Theorem and Similar Triangles. A Brief Review of the Pythagorean Theorem.

Lesson 2: The Pythagorean Theorem and Similar Triangles. A Brief Review of the Pythagorean Theorem. 27 Lesson 2: The Pythgoren Theorem nd Similr Tringles A Brief Review of the Pythgoren Theorem. Rell tht n ngle whih mesures 90º is lled right ngle. If one of the ngles of tringle is right ngle, then we

More information

Problems set # 3 Physics 169 February 24, 2015

Problems set # 3 Physics 169 February 24, 2015 Prof. Anhordoqui Problems set # 3 Physis 169 Februry 4, 015 1. A point hrge q is loted t the enter of uniform ring hving liner hrge density λ nd rdius, s shown in Fig. 1. Determine the totl eletri flux

More information

(See Notes on Spontaneous Emission)

(See Notes on Spontaneous Emission) ECE 240 for Cvity from ECE 240 (See Notes on ) Quntum Rdition in ECE 240 Lsers - Fll 2017 Lecture 11 1 Free Spce ECE 240 for Cvity from Quntum Rdition in The electromgnetic mode density in free spce is

More information

Lecture 27: Diffusion of Ions: Part 2: coupled diffusion of cations and

Lecture 27: Diffusion of Ions: Part 2: coupled diffusion of cations and Leture 7: iffusion of Ions: Prt : oupled diffusion of tions nd nions s desried y Nernst-Plnk Eqution Tody s topis Continue to understnd the fundmentl kinetis prmeters of diffusion of ions within n eletrilly

More information

6.3.2 Spectroscopy. N Goalby chemrevise.org 1 NO 2 H 3 CH3 C. NMR spectroscopy. Different types of NMR

6.3.2 Spectroscopy. N Goalby chemrevise.org 1 NO 2 H 3 CH3 C. NMR spectroscopy. Different types of NMR 6.. Spetrosopy NMR spetrosopy Different types of NMR NMR spetrosopy involves intertion of mterils with the lowenergy rdiowve region of the eletromgneti spetrum NMR spetrosopy is the sme tehnology s tht

More information

PAIR OF LINEAR EQUATIONS IN TWO VARIABLES

PAIR OF LINEAR EQUATIONS IN TWO VARIABLES PAIR OF LINEAR EQUATIONS IN TWO VARIABLES. Two liner equtions in the sme two vriles re lled pir of liner equtions in two vriles. The most generl form of pir of liner equtions is x + y + 0 x + y + 0 where,,,,,,

More information

u( t) + K 2 ( ) = 1 t > 0 Analyzing Damped Oscillations Problem (Meador, example 2-18, pp 44-48): Determine the equation of the following graph.

u( t) + K 2 ( ) = 1 t > 0 Analyzing Damped Oscillations Problem (Meador, example 2-18, pp 44-48): Determine the equation of the following graph. nlyzing Dmped Oscilltions Prolem (Medor, exmple 2-18, pp 44-48): Determine the eqution of the following grph. The eqution is ssumed to e of the following form f ( t) = K 1 u( t) + K 2 e!"t sin (#t + $

More information

Non Right Angled Triangles

Non Right Angled Triangles Non Right ngled Tringles Non Right ngled Tringles urriulum Redy www.mthletis.om Non Right ngled Tringles NON RIGHT NGLED TRINGLES sin i, os i nd tn i re lso useful in non-right ngled tringles. This unit

More information

QUADRATIC EQUATION EXERCISE - 01 CHECK YOUR GRASP

QUADRATIC EQUATION EXERCISE - 01 CHECK YOUR GRASP QUADRATIC EQUATION EXERCISE - 0 CHECK YOUR GRASP. Sine sum of oeffiients 0. Hint : It's one root is nd other root is 8 nd 5 5. tn other root 9. q 4p 0 q p q p, q 4 p,,, 4 Hene 7 vlues of (p, q) 7 equtions

More information

Forces on curved surfaces Buoyant force Stability of floating and submerged bodies

Forces on curved surfaces Buoyant force Stability of floating and submerged bodies Stti Surfe ores Stti Surfe ores 8m wter hinge? 4 m ores on plne res ores on urved surfes Buont fore Stbilit of floting nd submerged bodies ores on Plne res Two tpes of problems Horizontl surfes (pressure

More information

ENERGY AND PACKING. Outline: MATERIALS AND PACKING. Crystal Structure

ENERGY 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 information

I 3 2 = I I 4 = 2A

I 3 2 = I I 4 = 2A ECE 210 Eletril Ciruit Anlysis University of llinois t Chigo 2.13 We re ske to use KCL to fin urrents 1 4. The key point in pplying KCL in this prolem is to strt with noe where only one of the urrents

More information

H (2a, a) (u 2a) 2 (E) Show that u v 4a. Explain why this implies that u v 4a, with equality if and only u a if u v 2a.

H (2a, a) (u 2a) 2 (E) Show that u v 4a. Explain why this implies that u v 4a, with equality if and only u a if u v 2a. Chpter Review 89 IGURE ol hord GH of the prol 4. G u v H (, ) (A) Use the distne formul to show tht u. (B) Show tht G nd H lie on the line m, where m ( )/( ). (C) Solve m for nd sustitute in 4, otining

More information

2.1 ANGLES AND THEIR MEASURE. y I

2.1 ANGLES AND THEIR MEASURE. y I .1 ANGLES AND THEIR MEASURE Given two interseting lines or line segments, the mount of rottion out the point of intersetion (the vertex) required to ring one into orrespondene with the other is lled the

More information

( ) as a fraction. Determine location of the highest

( ) as a fraction. Determine location of the highest AB/ Clulus Exm Review Sheet Solutions A Prelulus Type prolems A1 A A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f( x) Set funtion equl to Ftor or use qudrti eqution if qudrti Grph to

More information

H 4 H 8 N 2. Example 1 A compound is found to have an accurate relative formula mass of It is thought to be either CH 3.

H 4 H 8 N 2. Example 1 A compound is found to have an accurate relative formula mass of It is thought to be either CH 3. . Spetrosopy Mss spetrosopy igh resolution mss spetrometry n e used to determine the moleulr formul of ompound from the urte mss of the moleulr ion For exmple, the following moleulr formuls ll hve rough

More information

Chapter 4 Contravariance, Covariance, and Spacetime Diagrams

Chapter 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 information

m A 1 1 A ! and AC 6

m A 1 1 A ! and AC 6 REVIEW SET A Using sle of m represents units, sketh vetor to represent: NON-CALCULATOR n eroplne tking off t n ngle of 8 ± to runw with speed of 6 ms displement of m in north-esterl diretion. Simplif:

More information

Chem Homework 11 due Monday, Apr. 28, 2014, 2 PM

Chem Homework 11 due Monday, Apr. 28, 2014, 2 PM Chem 44 - Homework due ondy, pr. 8, 4, P.. . Put this in eq 8.4 terms: E m = m h /m e L for L=d The degenery in the ring system nd the inresed sping per level (4x bigger) mkes the sping between the HOO

More information

STRAND J: TRANSFORMATIONS, VECTORS and MATRICES

STRAND J: TRANSFORMATIONS, VECTORS and MATRICES Mthemtics SKE: STRN J STRN J: TRNSFORMTIONS, VETORS nd MTRIES J3 Vectors Text ontents Section J3.1 Vectors nd Sclrs * J3. Vectors nd Geometry Mthemtics SKE: STRN J J3 Vectors J3.1 Vectors nd Sclrs Vectors

More information

f (x)dx = f(b) f(a). a b f (x)dx is the limit of sums

f (x)dx = f(b) f(a). a b f (x)dx is the limit of sums Green s Theorem If f is funtion of one vrible x with derivtive f x) or df dx to the Fundmentl Theorem of lulus, nd [, b] is given intervl then, ording This is not trivil result, onsidering tht b b f x)dx

More information

4 VECTORS. 4.0 Introduction. Objectives. Activity 1

4 VECTORS. 4.0 Introduction. Objectives. Activity 1 4 VECTRS Chpter 4 Vectors jectives fter studying this chpter you should understnd the difference etween vectors nd sclrs; e le to find the mgnitude nd direction of vector; e le to dd vectors, nd multiply

More information

Biaxial Minerals Descriptions

Biaxial Minerals Descriptions Bixil Minerls Desriptions Olivine Pyroxenes Orthopyroxene Clinopyroxene Amphiole Hornlende Atinolite Mis Biotite, musovite, hlorite Feldsprs Plgiolse Miroline, ortholse, snidine Feldsprs Tetosilites -Si:O=

More information

Lecture 5: Crystal planes and Miller Indices

Lecture 5: Crystal planes and Miller Indices Leture Notes on Struture of Mtter y Mohmmd Jellur Rhmn, Deprtment of Physis, BUET, Dhk-000 Leture 5: Crystl plnes nd Miller Indies Index system for rystl diretions nd plnes Crystl diretions: Any lttie

More information

The Ellipse. is larger than the other.

The Ellipse. is larger than the other. The Ellipse Appolonius of Perg (5 B.C.) disovered tht interseting right irulr one ll the w through with plne slnted ut is not perpendiulr to the is, the intersetion provides resulting urve (oni setion)

More information

Core 2 Logarithms and exponentials. Section 1: Introduction to logarithms

Core 2 Logarithms and exponentials. Section 1: Introduction to logarithms Core Logrithms nd eponentils Setion : Introdution to logrithms Notes nd Emples These notes ontin subsetions on Indies nd logrithms The lws of logrithms Eponentil funtions This is n emple resoure from MEI

More information

Physics 505 Homework No. 11 Solutions S11-1

Physics 505 Homework No. 11 Solutions S11-1 Physis 55 Homework No 11 s S11-1 1 This problem is from the My, 24 Prelims Hydrogen moleule Consider the neutrl hydrogen moleule, H 2 Write down the Hmiltonin keeping only the kineti energy terms nd the

More information

DEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS

DEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS 3 DEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS This chpter summrizes few properties of Cli ord Algebr nd describe its usefulness in e ecting vector rottions. 3.1 De nition of Associtive

More information

Waveguide Guide: A and V. Ross L. Spencer

Waveguide Guide: A and V. Ross L. Spencer Wveguide Guide: A nd V Ross L. Spencer I relly think tht wveguide fields re esier to understnd using the potentils A nd V thn they re using the electric nd mgnetic fields. Since Griffiths doesn t do it

More information

ILLUSTRATING THE EXTENSION OF A SPECIAL PROPERTY OF CUBIC POLYNOMIALS TO NTH DEGREE POLYNOMIALS

ILLUSTRATING THE EXTENSION OF A SPECIAL PROPERTY OF CUBIC POLYNOMIALS TO NTH DEGREE POLYNOMIALS ILLUSTRATING THE EXTENSION OF A SPECIAL PROPERTY OF CUBIC POLYNOMIALS TO NTH DEGREE POLYNOMIALS Dvid Miller West Virgini University P.O. BOX 6310 30 Armstrong Hll Morgntown, WV 6506 millerd@mth.wvu.edu

More information

POLYPHASE CIRCUITS. Introduction:

POLYPHASE CIRCUITS. Introduction: POLYPHASE CIRCUITS Introduction: Three-phse systems re commonly used in genertion, trnsmission nd distribution of electric power. Power in three-phse system is constnt rther thn pulsting nd three-phse

More information

8.3 THE HYPERBOLA OBJECTIVES

8.3 THE HYPERBOLA OBJECTIVES 8.3 THE HYPERBOLA OBJECTIVES 1. Define Hperol. Find the Stndrd Form of the Eqution of Hperol 3. Find the Trnsverse Ais 4. Find the Eentriit of Hperol 5. Find the Asmptotes of Hperol 6. Grph Hperol HPERBOLAS

More information

Solutions to Assignment 1

Solutions to Assignment 1 MTHE 237 Fll 2015 Solutions to Assignment 1 Problem 1 Find the order of the differentil eqution: t d3 y dt 3 +t2 y = os(t. Is the differentil eqution liner? Is the eqution homogeneous? b Repet the bove

More information

Lecture 1 - Introduction and Basic Facts about PDEs

Lecture 1 - Introduction and Basic Facts about PDEs * 18.15 - Introdution to PDEs, Fll 004 Prof. Gigliol Stffilni Leture 1 - Introdution nd Bsi Fts bout PDEs The Content of the Course Definition of Prtil Differentil Eqution (PDE) Liner PDEs VVVVVVVVVVVVVVVVVVVV

More information

Proving the Pythagorean Theorem

Proving the Pythagorean Theorem Proving the Pythgoren Theorem W. Bline Dowler June 30, 2010 Astrt Most people re fmilir with the formul 2 + 2 = 2. However, in most ses, this ws presented in lssroom s n solute with no ttempt t proof or

More information

Vectors. Chapter14. Syllabus reference: 4.1, 4.2, 4.5 Contents:

Vectors. Chapter14. Syllabus reference: 4.1, 4.2, 4.5 Contents: hpter Vetors Syllus referene:.,.,.5 ontents: D E F G H I J K Vetors nd slrs Geometri opertions with vetors Vetors in the plne The mgnitude of vetor Opertions with plne vetors The vetor etween two points

More information

Thermodynamics. Question 1. Question 2. Question 3 3/10/2010. Practice Questions PV TR PV T R

Thermodynamics. Question 1. Question 2. Question 3 3/10/2010. Practice Questions PV TR PV T R /10/010 Question 1 1 mole of idel gs is rought to finl stte F y one of three proesses tht hve different initil sttes s shown in the figure. Wht is true for the temperture hnge etween initil nd finl sttes?

More information

New Expansion and Infinite Series

New Expansion and Infinite Series Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University

More information

Correct answer: 0 m/s 2. Explanation: 8 N

Correct answer: 0 m/s 2. Explanation: 8 N Version 001 HW#3 - orces rts (00223) 1 his print-out should hve 15 questions. Multiple-choice questions my continue on the next column or pge find ll choices before nswering. Angled orce on Block 01 001

More information

β 1 = 2 π and the path length difference is δ 1 = λ. The small angle approximation gives us y 1 L = tanθ 1 θ 1 sin θ 1 = δ 1 y 1

β 1 = 2 π and the path length difference is δ 1 = λ. The small angle approximation gives us y 1 L = tanθ 1 θ 1 sin θ 1 = δ 1 y 1 rgsdle (zdr8) HW13 ditmire (58335) 1 This print-out should hve 1 questions. Multiple-choice questions my continue on the next column or pge find ll choices before nswering. 001 (prt 1 of ) 10.0 points

More information

#6A&B Magnetic Field Mapping

#6A&B Magnetic Field Mapping #6A& Mgnetic Field Mpping Gol y performing this lb experiment, you will: 1. use mgnetic field mesurement technique bsed on Frdy s Lw (see the previous experiment),. study the mgnetic fields generted by

More information

Arrow s Impossibility Theorem

Arrow s Impossibility Theorem Rep Voting Prdoxes Properties Arrow s Theorem Arrow s Impossiility Theorem Leture 12 Arrow s Impossiility Theorem Leture 12, Slide 1 Rep Voting Prdoxes Properties Arrow s Theorem Leture Overview 1 Rep

More information

Partial Derivatives. Limits. For a single variable function f (x), the limit lim

Partial Derivatives. Limits. For a single variable function f (x), the limit lim Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the right-hnd side limit equls to the left-hnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles

More information

http:dx.doi.org1.21611qirt.1994.17 Infrred polriztion thermometry using n imging rdiometer by BALFOUR l. S. * *EORD, Technion Reserch & Development Foundtion Ltd, Hif 32, Isrel. Abstrct This pper describes

More information

where the box contains a finite number of gates from the given collection. Examples of gates that are commonly used are the following: a b

where the box contains a finite number of gates from the given collection. Examples of gates that are commonly used are the following: a b CS 294-2 9/11/04 Quntum Ciruit Model, Solovy-Kitev Theorem, BQP Fll 2004 Leture 4 1 Quntum Ciruit Model 1.1 Clssil Ciruits - Universl Gte Sets A lssil iruit implements multi-output oolen funtion f : {0,1}

More information

CALCULATED POWDER X-RAY DIFFRACTION LINE PROFILES VIA ABSORPTION

CALCULATED POWDER X-RAY DIFFRACTION LINE PROFILES VIA ABSORPTION 16 17 CALCULATED POWDER X-RAY DFFRACTON LNE PROFLES VA ABSORPTON Keji Liu nd Heifen Chen School of Mteril Science nd Engineering, Shnghi nstitute of Technology, Shnghi, Chin 2233 ABSTRACT We hve clculted

More information

For the flux through a surface: Ch.24 Gauss s Law In last chapter, to calculate electric filede at a give location: q For point charges: K i r 2 ˆr

For the flux through a surface: Ch.24 Gauss s Law In last chapter, to calculate electric filede at a give location: q For point charges: K i r 2 ˆr Ch.24 Guss s Lw In lst hpter, to lulte eletri filed t give lotion: q For point hrges: K i e r 2 ˆr i dq For ontinuous hrge distributions: K e r 2 ˆr However, for mny situtions with symmetri hrge distribution,

More information

Algebra 2 Semester 1 Practice Final

Algebra 2 Semester 1 Practice Final Alger 2 Semester Prtie Finl Multiple Choie Ientify the hoie tht est ompletes the sttement or nswers the question. To whih set of numers oes the numer elong?. 2 5 integers rtionl numers irrtionl numers

More information

along the vector 5 a) Find the plane s coordinate after 1 hour. b) Find the plane s coordinate after 2 hours. c) Find the plane s coordinate

along the vector 5 a) Find the plane s coordinate after 1 hour. b) Find the plane s coordinate after 2 hours. c) Find the plane s coordinate L8 VECTOR EQUATIONS OF LINES HL Mth - Sntowski Vector eqution of line 1 A plne strts journey t the point (4,1) moves ech hour long the vector. ) Find the plne s coordinte fter 1 hour. b) Find the plne

More information

T b a(f) [f ] +. P b a(f) = Conclude that if f is in AC then it is the difference of two monotone absolutely continuous functions.

T b a(f) [f ] +. P b a(f) = Conclude that if f is in AC then it is the difference of two monotone absolutely continuous functions. Rel Vribles, Fll 2014 Problem set 5 Solution suggestions Exerise 1. Let f be bsolutely ontinuous on [, b] Show tht nd T b (f) P b (f) f (x) dx [f ] +. Conlude tht if f is in AC then it is the differene

More information

The Riemann-Stieltjes Integral

The Riemann-Stieltjes Integral Chpter 6 The Riemnn-Stieltjes Integrl 6.1. Definition nd Eistene of the Integrl Definition 6.1. Let, b R nd < b. ( A prtition P of intervl [, b] is finite set of points P = { 0, 1,..., n } suh tht = 0

More information

Supporting Online Material for

Supporting Online Material for Correction: 1 December 007 www.sciencemg.org/cgi/content/full/318/5857/1750/dc1 Supporting Online Mteril for Mott Trnsition in VO Reveled by Infrred Spectroscopy nd Nno- Imging M. M. Qzilbsh,* M. Brehm,

More information

Line Integrals and Entire Functions

Line Integrals and Entire Functions Line Integrls nd Entire Funtions Defining n Integrl for omplex Vlued Funtions In the following setions, our min gol is to show tht every entire funtion n be represented s n everywhere onvergent power series

More information

This enables us to also express rational numbers other than natural numbers, for example:

This enables us to also express rational numbers other than natural numbers, for example: Overview Study Mteril Business Mthemtis 05-06 Alger The Rel Numers The si numers re,,3,4, these numers re nturl numers nd lso lled positive integers. The positive integers, together with the negtive integers

More information

set is not closed under matrix [ multiplication, ] and does not form a group.

set is not closed under matrix [ multiplication, ] and does not form a group. Prolem 2.3: Which of the following collections of 2 2 mtrices with rel entries form groups under [ mtrix ] multipliction? i) Those of the form for which c d 2 Answer: The set of such mtrices is not closed

More information

2. VECTORS AND MATRICES IN 3 DIMENSIONS

2. 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 information

QUB XRD Course. The crystalline state. The Crystalline State

QUB 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 information

Standard Trigonometric Functions

Standard Trigonometric Functions CRASH KINEMATICS For ngle A: opposite sine A = = hypotenuse djent osine A = = hypotenuse opposite tngent A = = djent For ngle B: opposite sine B = = hypotenuse djent osine B = = hypotenuse opposite tngent

More information

Shape and measurement

Shape 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 information

Math 100 Review Sheet

Math 100 Review Sheet Mth 100 Review Sheet Joseph H. Silvermn December 2010 This outline of Mth 100 is summry of the mteril covered in the course. It is designed to be study id, but it is only n outline nd should be used s

More information

The International Association for the Properties of Water and Steam. Release on the Ionization Constant of H 2 O

The International Association for the Properties of Water and Steam. Release on the Ionization Constant of H 2 O IAPWS R-7 The Interntionl Assocition for the Properties of Wter nd Stem Lucerne, Sitzerlnd August 7 Relese on the Ioniztion Constnt of H O 7 The Interntionl Assocition for the Properties of Wter nd Stem

More information

Energy Bands Energy Bands and Band Gap. Phys463.nb Phenomenon

Energy 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 information

Non-Linear & Logistic Regression

Non-Linear & Logistic Regression Non-Liner & Logistic Regression If the sttistics re boring, then you've got the wrong numbers. Edwrd R. Tufte (Sttistics Professor, Yle University) Regression Anlyses When do we use these? PART 1: find

More information

Polarimetric Target Detector by the use of the Polarisation Fork

Polarimetric Target Detector by the use of the Polarisation Fork Polrimetri rget Detetor y the use of the Polristion For Armndo Mrino¹ hne R Cloude² Iin H Woodhouse¹ ¹he University of Edinurgh, Edinurgh Erth Oservtory (EEO), UK ²AEL Consultnts, Edinurgh, UK POLinAR009

More information

Dense Coding, Teleportation, No Cloning

Dense Coding, Teleportation, No Cloning qitd352 Dense Coding, Teleporttion, No Cloning Roert B. Griffiths Version of 8 Ferury 2012 Referenes: NLQI = R. B. Griffiths, Nture nd lotion of quntum informtion Phys. Rev. A 66 (2002) 012311; http://rxiv.org/rhive/qunt-ph/0203058

More information

Figure XX.1.1 Plane truss structure

Figure XX.1.1 Plane truss structure Truss Eements Formution. TRUSS ELEMENT.1 INTRODUTION ne truss struture is ste struture on the sis of tringe, s shown in Fig..1.1. The end of memer is pin juntion whih does not trnsmit moment. As for the

More information

More Properties of the Riemann Integral

More Properties of the Riemann Integral More Properties of the Riemnn Integrl Jmes K. Peterson Deprtment of Biologil Sienes nd Deprtment of Mthemtil Sienes Clemson University Februry 15, 2018 Outline More Riemnn Integrl Properties The Fundmentl

More information

Proportions: A ratio is the quotient of two numbers. For example, 2 3

Proportions: A ratio is the quotient of two numbers. For example, 2 3 Proportions: rtio is the quotient of two numers. For exmple, 2 3 is rtio of 2 n 3. n equlity of two rtios is proportion. For exmple, 3 7 = 15 is proportion. 45 If two sets of numers (none of whih is 0)

More information

Conducting Ellipsoid and Circular Disk

Conducting 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 information

Section 5.1 #7, 10, 16, 21, 25; Section 5.2 #8, 9, 15, 20, 27, 30; Section 5.3 #4, 6, 9, 13, 16, 28, 31; Section 5.4 #7, 18, 21, 23, 25, 29, 40

Section 5.1 #7, 10, 16, 21, 25; Section 5.2 #8, 9, 15, 20, 27, 30; Section 5.3 #4, 6, 9, 13, 16, 28, 31; Section 5.4 #7, 18, 21, 23, 25, 29, 40 Mth B Prof. Audrey Terrs HW # Solutions by Alex Eustis Due Tuesdy, Oct. 9 Section 5. #7,, 6,, 5; Section 5. #8, 9, 5,, 7, 3; Section 5.3 #4, 6, 9, 3, 6, 8, 3; Section 5.4 #7, 8,, 3, 5, 9, 4 5..7 Since

More information

QUADRATIC EQUATIONS OBJECTIVE PROBLEMS

QUADRATIC EQUATIONS OBJECTIVE PROBLEMS QUADRATIC EQUATIONS OBJECTIVE PROBLEMS +. The solution of the eqution will e (), () 0,, 5, 5. The roots of the given eqution ( p q) ( q r) ( r p) 0 + + re p q r p (), r p p q, q r p q (), (d), q r p q.

More information

A Formulary for Mathematics

A Formulary for Mathematics A Formulry for Mthemtis A olletion of the Formuls, Fts nd Figures often needed in mthemtis These re some of the pges of the first rough drft of ooklet whih hs now een pulished It is in hndier A5 size,

More information

Alg. Sheet (1) Department : Math Form : 3 rd prep. Sheet

Alg. Sheet (1) Department : Math Form : 3 rd prep. Sheet Ciro Governorte Nozh Directorte of Eduction Nozh Lnguge Schools Ismili Rod Deprtment : Mth Form : rd prep. Sheet Alg. Sheet () [] Find the vlues of nd in ech of the following if : ) (, ) ( -5, 9 ) ) (,

More information

Chapter 9 Definite Integrals

Chapter 9 Definite Integrals Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished

More information

Farey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University

Farey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions

More information

Available online at ScienceDirect. Procedia Engineering 120 (2015 ) EUROSENSORS 2015

Available online at  ScienceDirect. Procedia Engineering 120 (2015 ) EUROSENSORS 2015 Aville online t www.sienediret.om SieneDiret Proedi Engineering 10 (015 ) 887 891 EUROSENSORS 015 A Fesiility Study for Self-Osillting Loop for Three Degreeof-Freedom Coupled MEMS Resontor Fore Sensor

More information

Investigations on Power Quality Disturbances Using Discrete Wavelet Transform

Investigations on Power Quality Disturbances Using Discrete Wavelet Transform I J E E E Interntionl Journl of Eletril, Eletronis ISSN No. (Online): 77-66 nd omputer Engineering (): 47-53(13) Investigtions on Power Qulity Disturnes Using Disrete Wvelet Trnsform hvn Jin, Shilendr

More information

Simulation of Eclipsing Binary Star Systems. Abstract

Simulation of Eclipsing Binary Star Systems. Abstract Simultion of Eclipsing Binry Str Systems Boris Yim 1, Kenny Chn 1, Rphel Hui 1 Wh Yn College Kowloon Diocesn Boys School Abstrct This report briefly introduces the informtion on eclipsing binry str systems.

More information

Designing Information Devices and Systems I Fall 2016 Babak Ayazifar, Vladimir Stojanovic Homework 6. This homework is due October 11, 2016, at Noon.

Designing Information Devices and Systems I Fall 2016 Babak Ayazifar, Vladimir Stojanovic Homework 6. This homework is due October 11, 2016, at Noon. EECS 16A Designing Informtion Devices nd Systems I Fll 2016 Bk Ayzifr, Vldimir Stojnovic Homework 6 This homework is due Octoer 11, 2016, t Noon. 1. Homework process nd study group Who else did you work

More information

Homework Solution - Set 5 Due: Friday 10/03/08

Homework Solution - Set 5 Due: Friday 10/03/08 CE 96 Introduction to the Theory of Computtion ll 2008 Homework olution - et 5 Due: ridy 10/0/08 1. Textook, Pge 86, Exercise 1.21. () 1 2 Add new strt stte nd finl stte. Mke originl finl stte non-finl.

More information

5.4 The Quarter-Wave Transformer

5.4 The Quarter-Wave Transformer 3/4/7 _4 The Qurter Wve Trnsformer /.4 The Qurter-Wve Trnsformer Redg Assignment: pp. 73-76, 4-43 By now you ve noticed tht qurter-wve length of trnsmission le ( = λ 4, β = π ) ppers often microwve engeerg

More information

13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS

13.3 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 information