The Peak Shape of the Pair Distribution Function
|
|
- Oswald Jessie Hawkins
- 6 years ago
- Views:
Transcription
1 The Peak Shape of the Pa Dstbuton Functon V. Levashov, M.F. Thope and M.Le Phscs & Astonom Depatment Mchgan State Unvest East Lansng, MI 4884 Sgnfcant pogess n X-a and neuton dffacton epements on powde samples has been acheved n ecent eas. Hgh-esoluton data nowadas make the compason of theoetcal calculatons wth epemental measuements to a hghe degee of accuac. Because of ths, small sstematc eos that wee gnoed befoe can lead to the notceable dsageement that can now be obseved. It was shown [] that n measuements of Pa Dstbuton Functon (PDF) fom powde samples, the postons of the peaks shfted and that the measued atomc dstances ae bgge then the actual one. It was also shown that the shape of the peaks s not gaussan as t often assumed. It was also ponted out that effect s elatvel small. Hee we pesent futhe developments of the wok []. We have deved an appomate epesson fo the non-gaussan peak shape. It s the most pobable to see ths effect n hghl ansotopc mateals. Estmates fo the elatve se of the effect wee made. In some specal cases ths s a sgnfcant coecton, but usuall t s not.
2 We calculated the Radal Dstbuton Functon (RDF) fo buckball and benene molecules usng the Gaussan98 [] pogam wth AM and SNDO sem-empcal methods. Due to the hghl ansotopc stuctue of those molecules one could epect that t wll be possble to see n the RDF non-gaussan behavo of the peaks. Howeve t was shown that effect s ve small fo buckballs. Fo the benene molecule, the effect s qute lage fo one peak. Usng the calculated esults fo the adal dstbuton functon we calculated pa dstbuton functon of the fullete cstal, and a compason wth the epement esulta of Bllnge, Petkov and Yavas was made. The ole of the fnte esoluton measuements of epement s dscussed. It s shown that coecton to the theoetcal calculatons due to the fnte esoluton of the measuements sgnfcantl mpoves the ageement wth epement. *Wok Suppoted n pat b the DOE unde gant # DE FG97ER4565
3 Non-Gaussan Peak Shape of Pa Dstbuton Functon Let suppose that we consde a cstal and the equlbum poston of an atom s (,, ) wth espect to the cente atom. Atoms vbate nea the equlbum postons. The pobablt that the atom found at poston (,, ) s gven b P (,, ) Coodnates ( ) ( ) ( ) ep ( π ) 3, (, ) and (,, ) aes, whee the mat of dsplacements We want to fnd the ( P() wll be ae gven n the fame of pncpal u αuβ s dagonal ( u ). fo a hghl ansotopc ) powde mateals. Eale, when PDF was calculated, t was assumed that P gauss and that () ep ( ) π Nowadas when epemental technques fo PDF measuement was mpoved sgnfcantl one can epect to see the dffeence between measued PDF and the PDF above. P ngauss () calculated unde assumpton 3
4 R Pefomng of angula aveage s equvalent to the fndng the mass that s lng on the sphee of adus. On aveage due to vbatons the mass of eve atom s dstbuted wth some pobablt ove ts own ellpse. It was shown that the mass dstbuton n eve ellpse s the poduct of the thee gaussans n Catesan coodnates. The sphee of adus R and oentaton. R cuts the ellpses n a wa that depends on the ellpse poston 5
5 To fnd PDF of hgl anstopc powde meteals one should pefom the angula aveage: ) P () ( ) ( ) ( ) ep dω 3 ( π ) It s eas to show that n sotopc case when ( ) ths aveage leads to (eact esult): P so () ep ( ) ( ) ( ) π ep π ep Ths esult can be ewtten as: ( ) () P so ep π (D.A.Dmtov et al. []). If the peaks ae naow the dffeence between and s small. But thee s a possblt that n hgh qualt measuements ths dffeence can be seen and P ngauss () () theo and epement then. P gauss can gve bette ageement between P gauss () (). P ngauss In ansotopc case ( ) dffeence between eal peak shape and ts gaussan appomaton can be even bgge. We deved appomate epesson (epanson) fo the peak shape n ansotopc 6
6 case. In man cases ths epesson gves sgnfcantl bette ageement wth eal shape then fome gaussan appomaton. Summa of deved fomulas () ( ) () () Ψ Σ Ψ Σ Σ Σ P ngauss ep ) π () () () () () () () () ( ) n n n a a H f f a f a f a f a f a,,,,, Σ Σ Ψ Ψ Dstance Pobablt P() Gaussan Fomula Eact 7
7 It follows fom the fomulas above that the bggest devatons fom the gaussan peak shape should occu n case of stong ansotopc mateals. The futhe s the peak fom the ogn the bette ou appomaton woks. On anothe hand the best esoluton s usuall acheved on the fst peak that s the closest to the ogn. Eamples Eact cuve s obtaned b dect numecal ntegaton of (). Gaussan cuve s the cuve that epesents the eponental pat of the deved fomula wthout coecton tems n backets. Ths coesponds to the appomaton that was used befoe. Fomula cuve shows the appomaton of eact cuve gven b (). Pobablt P() 4 3 Eact Gaussan Fomula Dstance 8
8 Non-Gaussan Peak Shape of Ansotopc Molecules Smple wa to check how mpotant the coectons to the pevousl used gaussan lne shape ae s to calculate Radal Dstbuton Functon (RDF) and PDF of ansotopc molecules. In fact all molecules ae ansotopc and that dffeence between molecules and cstallne solds s ve mpotant fo ou case. We consde molecules of fulleene and benene. One can epect that n fulleene the vbatons of the cabon atoms n decton paallel and pependcula to the suface have sgnfcantl dffeent ampltudes. The same can be tue fo n plane and out of plane atomc vbatons n benene. In ode to calculate the RDF of the molecules we have to calculate aveage (elatve) dsplacements of the atoms (,, ) due to the vbatons. The mat of (elatve) aveage dsplacements U can be calculated f the egenfequences and egenvectos of molecula vbatons ae known. Ou devaton shows that: e h ( ) ( ) α β jα jβ α jβ jα β u u j u u j n α β ω m m j mm j mm j e e e e e e e 9
9 Whee ( u u j ) α ( u u j ) β s the mat of elatve atomc dsplacements. ω ae the fequences of molecula vbatons. ae othogonal and nomaled Catesan ( α ) components of the atomc dsplacements that coespond to the egenfequnc ω. m s the mass of atom and Bose-Ensten dstbuton functon. The mat of atomc dsplacements can be dagonaled and ts egenvalues ae the squaes of,,. Then the coodnates of the atoms n the fame whee U s dagonal can be found. Afte that the applcaton of out fomulas s staghtfowad. In ode to calculates egenfequences and egenvectos of molecules vbatons we used Gaussan98 pogam and two sem empcal methods: AM and CNDO. In the tables below we show the values of coodnates and aveage atomc dsplacements fo the seveal neaest atoms fo the molecules of fulleene and benene. Fulleene at 3K. All dstances ae n angstoms. R,, E E ,, E E E- R,, E E-3,, E E E- R,, E E-,, E E E- Benene at 3K. All dstances ae n angstoms (Fom Cabon) R to H,, E E,, E e α
10 R to C,,,, R to H,,,, R to C,,,, E E E E E E E-5.E E E-.7769E- One can see that t s the most pobable to see the fst peak n benene snce t s close to the ogn and t s caused b the most ansotopc vbatons of the neaest hdogen atom. 4 Fullete at 3K Reduced Radal Dstbuton Functon Theo AM, gaussan AM, non gaussan o Dstance (A) As one can see the coectons due to non-gaussan appomaton ae ve small hee. 8
11 Pa Dstbuton Functon Of Fullete Cstal In ode to make the compason wth epement one should calculate the Pa Dstbuton Functon (PDF) defned as: [ ρ() ρ ] G( ) 4π ρ o s the aveage denst of the mateal t s equal to eo fo the case of a sngle solated molecule. The ρ () obes nomalaton condton: ( ) N R ( R) 4π ρ( )d Whee N R s the numbe of atoms nsde the sphee of adus R. Thus fo the sngle molecule we have: G( ) 4π 4π P new o () Z Z P () P new s the new deved functon that should substtute old gaussan appomaton. In case of X-a scatteng Z o and Z stand fo the numbe of electons n the cental atom and atom. The same functon can be plotted fo the old gaussan appomaton n assumpton that Σ. o new Net two pctues show the calculated and benen. G() fo the case of fulleen One can see that the ole of the coectons n case of small. C 6 s ve In case of benene thee s sgnfcant effect that coesponds to the Cabon-Hdogen peak. Unfotunatel the ntenst s smalle then ntenst of Cabon-Cabon Peaks.
12 7 Benene C 6 H 6, T3K Reduced Radal Dstbuton Functon (Cabon) AM Gaussan Non Gaussan o Dstance (A) Hee the coectons due to non-gaussan appomaton ae bgge, but the ae elated to the small fst peak. Hdogen s one of the atoms that cause the fst peak. That s wh t should be had to measue n the X-a dffacton epements. 3
13 Pa Dstbuton Functon of the Fullete Cstall. The molecules of fulleen fom fcc cstal. At 3 K molecules otales aound thee centes. It s mpotant to compae the esults of out calculatons wth the epemental measuements [3]. Let suppose that we st on a patcula atom that belongs to some of the fulleene molecules. Then we can see that atoms that belong to the same molecule as the atom on whch we ae sttng ae n moe o less fed postons wth espect to us. On anothe hand due to the otatons of fulleene molecule we see the smooth dstbuton of the atoms mass ove the sufaces of the othes molecule. 4
14 Thus the contbuton to ρ () fom the othe molecules can be modeled as the RDF of two sphecal shells wth contnuous dstbuton of mass. Thus ρ () consst of two pats contbuton fom the atoms n the same molecule ρ mol () and contbuton fom the othes molecules (coelatons) ρ co (). ρ () ρ mol () ρ () I dscussed above onl the fst contbuton. Fo the detals of second contbuton ou can to the M.Le s poste. 4 co PDF of the fullete. Fullete at 3K Pa Dstbuton Functon G() Theo AM CNDO Gaussan Resoluton Epement o Dstance (A) 5
15 Fullete at 3K 4 Pa Dstbuton Functon G() Theo AM CNDO Gaussan Resoluton Epement o Dstance (A) The ageement on the pctues above can be mpoved f one wll take nto account the fnte esoluton n epemental measuements of the scatteng ntenst. 6
16 The Pa Dstbuton Functon of Fullete wth coecton to the fnte esoluton n the measuements of scatteng ntenst. 4 Fullete at T 3K Pa Dstbuton Functon G() Theo AM CNDO Peak Boadenng Epement o Dstance (A) 7
17 Refeences. D.A.Dmtov, H. Rode, and A.R.Bshop, Peak postons and shapes n neuton pa coelaton functon fom powde hghl ansotopc cstals. axv.og e-pnt achve. Gaussan 98, Revson A.7, M. J. Fsch, G. W. Tucks, H. B. Schlegel, G. E. Scusea, M. A. Robb, J. R. Cheeseman, V. G. Zakewsk, J. A. Montgome, J., R. E. Statmann, J. C. Buant, S. Dappch, J. M. Mllam, A. D. Danels, K. N. Kudn, M. C. Stan, O. Fakas, J. Tomas, V. Baone, M. Coss, R. Camm, B. Mennucc, C. Pomell, C. Adamo, S. Clffod, J. Ochtesk, G. A. Petesson, P. Y. Aala, Q. Cu, K. Mookuma, D. K. Malck, A. D. Rabuck, K. Raghavacha, J. B. Foesman, J. Coslowsk, J. V. Ot, A. G. Baboul, B. B. Stefanov, G. Lu, A. Lashenko, P. Psko, I. Komaom, R. Gompets, R. L. Matn, D. J. Fo, T. Keth, M. A. Al-Laham, C. Y. Peng, A. Nanaakkaa, C. Gonale, M. Challacombe, P. M. W. Gll, B. Johnson, W. Chen, M. W. Wong, J. L. Andes, C. Gonale, M. Head-Godon, E. S. Replogle, and J. A. Pople, Gaussan, Inc., Pttsbugh PA,
The Shape of the Pair Distribution Function.
The Shpe of the P Dstbuton Functon. Vlentn Levshov nd.f. Thope Deptment of Phscs & stonom nd Cente fo Fundmentl tels Resech chgn Stte Unvest Sgnfcnt pogess n hgh-esoluton dffcton epements on powde smples
More informationScalars and Vectors Scalar
Scalas and ectos Scala A phscal quantt that s completel chaacteed b a eal numbe (o b ts numecal value) s called a scala. In othe wods a scala possesses onl a magntude. Mass denst volume tempeatue tme eneg
More informationRigid Bodies: Equivalent Systems of Forces
Engneeng Statcs, ENGR 2301 Chapte 3 Rgd Bodes: Equvalent Sstems of oces Intoducton Teatment of a bod as a sngle patcle s not alwas possble. In geneal, the se of the bod and the specfc ponts of applcaton
More informationPhysics 11b Lecture #2. Electric Field Electric Flux Gauss s Law
Physcs 11b Lectue # Electc Feld Electc Flux Gauss s Law What We Dd Last Tme Electc chage = How object esponds to electc foce Comes n postve and negatve flavos Conseved Electc foce Coulomb s Law F Same
More information24-2: Electric Potential Energy. 24-1: What is physics
D. Iyad SAADEDDIN Chapte 4: Electc Potental Electc potental Enegy and Electc potental Calculatng the E-potental fom E-feld fo dffeent chage dstbutons Calculatng the E-feld fom E-potental Potental of a
More informationEngineering Mechanics. Force resultants, Torques, Scalar Products, Equivalent Force systems
Engneeng echancs oce esultants, Toques, Scala oducts, Equvalent oce sstems Tata cgaw-hll Companes, 008 Resultant of Two oces foce: acton of one bod on anothe; chaacteed b ts pont of applcaton, magntude,
More informationA. Thicknesses and Densities
10 Lab0 The Eath s Shells A. Thcknesses and Denstes Any theoy of the nteo of the Eath must be consstent wth the fact that ts aggegate densty s 5.5 g/cm (ecall we calculated ths densty last tme). In othe
More informationChapter I Matrices, Vectors, & Vector Calculus 1-1, 1-9, 1-10, 1-11, 1-17, 1-18, 1-25, 1-27, 1-36, 1-37, 1-41.
Chapte I Matces, Vectos, & Vecto Calculus -, -9, -0, -, -7, -8, -5, -7, -36, -37, -4. . Concept of a Scala Consde the aa of patcles shown n the fgue. he mass of the patcle at (,) can be epessed as. M (,
More informationPhysics 1501 Lecture 19
Physcs 1501 ectue 19 Physcs 1501: ectue 19 Today s Agenda Announceents HW#7: due Oct. 1 Mdte 1: aveage 45 % Topcs otatonal Kneatcs otatonal Enegy Moents of Ineta Physcs 1501: ectue 19, Pg 1 Suay (wth copason
More informationCorrespondence Analysis & Related Methods
Coespondence Analyss & Related Methods Ineta contbutons n weghted PCA PCA s a method of data vsualzaton whch epesents the tue postons of ponts n a map whch comes closest to all the ponts, closest n sense
More informationChapter Fifiteen. Surfaces Revisited
Chapte Ffteen ufaces Revsted 15.1 Vecto Descpton of ufaces We look now at the vey specal case of functons : D R 3, whee D R s a nce subset of the plane. We suppose s a nce functon. As the pont ( s, t)
More informationPHY126 Summer Session I, 2008
PHY6 Summe Sesson I, 8 Most of nfomaton s avalable at: http://nngoup.phscs.sunsb.edu/~chak/phy6-8 ncludng the sllabus and lectue sldes. Read sllabus and watch fo mpotant announcements. Homewok assgnment
More informationSet of square-integrable function 2 L : function space F
Set of squae-ntegable functon L : functon space F Motvaton: In ou pevous dscussons we have seen that fo fee patcles wave equatons (Helmholt o Schödnge) can be expessed n tems of egenvalue equatons. H E,
More informationThermodynamics of solids 4. Statistical thermodynamics and the 3 rd law. Kwangheon Park Kyung Hee University Department of Nuclear Engineering
Themodynamcs of solds 4. Statstcal themodynamcs and the 3 d law Kwangheon Pak Kyung Hee Unvesty Depatment of Nuclea Engneeng 4.1. Intoducton to statstcal themodynamcs Classcal themodynamcs Statstcal themodynamcs
More informationCSJM University Class: B.Sc.-II Sub:Physics Paper-II Title: Electromagnetics Unit-1: Electrostatics Lecture: 1 to 4
CSJM Unvesty Class: B.Sc.-II Sub:Physcs Pape-II Ttle: Electomagnetcs Unt-: Electostatcs Lectue: to 4 Electostatcs: It deals the study of behavo of statc o statonay Chages. Electc Chage: It s popety by
More information3.1 Electrostatic Potential Energy and Potential Difference
3. lectostatc Potental negy and Potental Dffeence RMMR fom mechancs: - The potental enegy can be defned fo a system only f consevatve foces act between ts consttuents. - Consevatve foces may depend only
More informationALL QUESTIONS ARE WORTH 20 POINTS. WORK OUT FIVE PROBLEMS.
GNRAL PHYSICS PH -3A (D. S. Mov) Test (/3/) key STUDNT NAM: STUDNT d #: -------------------------------------------------------------------------------------------------------------------------------------------
More informationUNIT10 PLANE OF REGRESSION
UIT0 PLAE OF REGRESSIO Plane of Regesson Stuctue 0. Intoducton Ojectves 0. Yule s otaton 0. Plane of Regesson fo thee Vaales 0.4 Popetes of Resduals 0.5 Vaance of the Resduals 0.6 Summay 0.7 Solutons /
More informationP 365. r r r )...(1 365
SCIENCE WORLD JOURNAL VOL (NO4) 008 www.scecncewoldounal.og ISSN 597-64 SHORT COMMUNICATION ANALYSING THE APPROXIMATION MODEL TO BIRTHDAY PROBLEM *CHOJI, D.N. & DEME, A.C. Depatment of Mathematcs Unvesty
More informationCapítulo. Three Dimensions
Capítulo Knematcs of Rgd Bodes n Thee Dmensons Mecánca Contents ntoducton Rgd Bod Angula Momentum n Thee Dmensons Pncple of mpulse and Momentum Knetc Eneg Sample Poblem 8. Sample Poblem 8. Moton of a Rgd
More informationPhysics 207 Lecture 16
Physcs 07 Lectue 6 Goals: Lectue 6 Chapte Extend the patcle odel to gd-bodes Undestand the equlbu of an extended object. Analyze ollng oton Undestand otaton about a fxed axs. Eploy consevaton of angula
More informationPHYS Week 5. Reading Journals today from tables. WebAssign due Wed nite
PHYS 015 -- Week 5 Readng Jounals today fom tables WebAssgn due Wed nte Fo exclusve use n PHYS 015. Not fo e-dstbuton. Some mateals Copyght Unvesty of Coloado, Cengage,, Peason J. Maps. Fundamental Tools
More informationRemember: When an object falls due to gravity its potential energy decreases.
Chapte 5: lectc Potental As mentoned seveal tmes dung the uate Newton s law o gavty and Coulomb s law ae dentcal n the mathematcal om. So, most thngs that ae tue o gavty ae also tue o electostatcs! Hee
More informationIntegral Vector Operations and Related Theorems Applications in Mechanics and E&M
Dola Bagayoko (0) Integal Vecto Opeatons and elated Theoems Applcatons n Mechancs and E&M Ι Basc Defnton Please efe to you calculus evewed below. Ι, ΙΙ, andιιι notes and textbooks fo detals on the concepts
More informationUNIVERSITÀ DI PISA. Math thbackground
UNIVERSITÀ DI ISA Electomagnetc Radatons and Bologcal l Inteactons Lauea Magstale n Bomedcal Engneeng Fst semeste (6 cedts), academc ea 2011/12 of. aolo Nepa p.nepa@et.unp.t Math thbackgound Edted b D.
More informationMULTIPOLE FIELDS. Multipoles, 2 l poles. Monopoles, dipoles, quadrupoles, octupoles... Electric Dipole R 1 R 2. P(r,θ,φ) e r
MULTIPOLE FIELDS Mutpoes poes. Monopoes dpoes quadupoes octupoes... 4 8 6 Eectc Dpoe +q O θ e R R P(θφ) -q e The potenta at the fed pont P(θφ) s ( θϕ )= q R R Bo E. Seneus : Now R = ( e) = + cosθ R = (
More informationDYNAMICS VECTOR MECHANICS FOR ENGINEERS: Kinematics of Rigid Bodies in Three Dimensions. Seventh Edition CHAPTER
Edton CAPTER 8 VECTOR MECANCS FOR ENGNEERS: DYNAMCS Fednand P. Bee E. Russell Johnston, J. Lectue Notes: J. Walt Ole Teas Tech Unvest Knematcs of Rgd Bodes n Thee Dmensons 003 The McGaw-ll Companes, nc.
More information8 Baire Category Theorem and Uniform Boundedness
8 Bae Categoy Theoem and Unfom Boundedness Pncple 8.1 Bae s Categoy Theoem Valdty of many esults n analyss depends on the completeness popety. Ths popety addesses the nadequacy of the system of atonal
More informationComplex atoms and the Periodic System of the elements
Complex atoms and the Peodc System of the elements Non-cental foces due to electon epulson Cental feld appoxmaton electonc obtals lft degeneacy of l E n l = R( hc) ( n δ ) l Aufbau pncple Lectue Notes
More informationGenerating Functions, Weighted and Non-Weighted Sums for Powers of Second-Order Recurrence Sequences
Geneatng Functons, Weghted and Non-Weghted Sums fo Powes of Second-Ode Recuence Sequences Pantelmon Stăncă Aubun Unvesty Montgomey, Depatment of Mathematcs Montgomey, AL 3614-403, USA e-mal: stanca@studel.aum.edu
More informationExam 1. Sept. 22, 8:00-9:30 PM EE 129. Material: Chapters 1-8 Labs 1-3
Eam ept., 8:00-9:30 PM EE 9 Mateal: Chapte -8 Lab -3 tandadzaton and Calbaton: Ttaton: ue of tandadzed oluton to detemne the concentaton of an unknown. Rele on a eacton of known tochomet, a oluton wth
More informationTest 1 phy What mass of a material with density ρ is required to make a hollow spherical shell having inner radius r i and outer radius r o?
Test 1 phy 0 1. a) What s the pupose of measuement? b) Wte all fou condtons, whch must be satsfed by a scala poduct. (Use dffeent symbols to dstngush opeatons on ectos fom opeatons on numbes.) c) What
More informationCOMPLEMENTARY ENERGY METHOD FOR CURVED COMPOSITE BEAMS
ultscence - XXX. mcocd Intenatonal ultdscplnay Scentfc Confeence Unvesty of skolc Hungay - pl 06 ISBN 978-963-358-3- COPLEENTRY ENERGY ETHOD FOR CURVED COPOSITE BES Ákos József Lengyel István Ecsed ssstant
More informationPHYS 705: Classical Mechanics. Derivation of Lagrange Equations from D Alembert s Principle
1 PHYS 705: Classcal Mechancs Devaton of Lagange Equatons fom D Alembet s Pncple 2 D Alembet s Pncple Followng a smla agument fo the vtual dsplacement to be consstent wth constants,.e, (no vtual wok fo
More information1. A body will remain in a state of rest, or of uniform motion in a straight line unless it
Pncples of Dnamcs: Newton's Laws of moton. : Foce Analss 1. A bod wll eman n a state of est, o of unfom moton n a staght lne unless t s acted b etenal foces to change ts state.. The ate of change of momentum
More informationEvent Shape Update. T. Doyle S. Hanlon I. Skillicorn. A. Everett A. Savin. Event Shapes, A. Everett, U. Wisconsin ZEUS Meeting, October 15,
Event Shape Update A. Eveett A. Savn T. Doyle S. Hanlon I. Skllcon Event Shapes, A. Eveett, U. Wsconsn ZEUS Meetng, Octobe 15, 2003-1 Outlne Pogess of Event Shapes n DIS Smla to publshed pape: Powe Coecton
More information2/24/2014. The point mass. Impulse for a single collision The impulse of a force is a vector. The Center of Mass. System of particles
/4/04 Chapte 7 Lnea oentu Lnea oentu of a Sngle Patcle Lnea oentu: p υ It s a easue of the patcle s oton It s a vecto, sla to the veloct p υ p υ p υ z z p It also depends on the ass of the object, sla
More informationThe Forming Theory and the NC Machining for The Rotary Burs with the Spectral Edge Distribution
oden Appled Scence The Fomn Theoy and the NC achnn fo The Rotay us wth the Spectal Ede Dstbuton Huan Lu Depatment of echancal Enneen, Zhejan Unvesty of Scence and Technoloy Hanzhou, c.y. chan, 310023,
More informationRotary motion
ectue 8 RTARY TN F THE RGD BDY Notes: ectue 8 - Rgd bod Rgd bod: j const numbe of degees of feedom 6 3 tanslatonal + 3 ota motons m j m j Constants educe numbe of degees of feedom non-fee object: 6-p
More informationReview of Vector Algebra and Vector Calculus Operations
Revew of Vecto Algeba and Vecto Calculus Opeatons Tpes of vaables n Flud Mechancs Repesentaton of vectos Dffeent coodnate sstems Base vecto elatons Scala and vecto poducts Stess Newton s law of vscost
More informationMachine Learning. Spectral Clustering. Lecture 23, April 14, Reading: Eric Xing 1
Machne Leanng -7/5 7/5-78, 78, Spng 8 Spectal Clusteng Ec Xng Lectue 3, pl 4, 8 Readng: Ec Xng Data Clusteng wo dffeent ctea Compactness, e.g., k-means, mxtue models Connectvty, e.g., spectal clusteng
More information19 The Born-Oppenheimer Approximation
9 The Bon-Oppenheme Appoxmaton The full nonelatvstc Hamltonan fo a molecule s gven by (n a.u.) Ĥ = A M A A A, Z A + A + >j j (883) Lets ewte the Hamltonan to emphasze the goal as Ĥ = + A A A, >j j M A
More informationRotating Disk Electrode -a hydrodynamic method
Rotatng Dsk Electode -a hdodnamc method Fe Lu Ma 3, 0 ente fo Electochemcal Engneeng Reseach Depatment of hemcal and Bomolecula Engneeng Rotatng Dsk Electode A otatng dsk electode RDE s a hdodnamc wokng
More informationDynamics of Rigid Bodies
Dynamcs of Rgd Bodes A gd body s one n whch the dstances between consttuent patcles s constant thoughout the moton of the body,.e. t keeps ts shape. Thee ae two knds of gd body moton: 1. Tanslatonal Rectlnea
More informationChapter 2. A Brief Review of Electron Diffraction Theory
Chapte. A Bef Revew of Electon Dffacton Theoy 8 Chapte. A Bef Revew of Electon Dffacton Theoy The theoy of gas phase electon dffacton s hadly a new topc. t s well establshed fo decades and has been thooughly
More informationiclicker Quiz a) True b) False Theoretical physics: the eternal quest for a missing minus sign and/or a factor of two. Which will be an issue today?
Clce Quz I egsteed my quz tansmtte va the couse webste (not on the clce.com webste. I ealze that untl I do so, my quz scoes wll not be ecoded. a Tue b False Theoetcal hyscs: the etenal quest fo a mssng
More information3. A Review of Some Existing AW (BT, CT) Algorithms
3. A Revew of Some Exstng AW (BT, CT) Algothms In ths secton, some typcal ant-wndp algothms wll be descbed. As the soltons fo bmpless and condtoned tansfe ae smla to those fo ant-wndp, the pesented algothms
More information, the tangent line is an approximation of the curve (and easier to deal with than the curve).
114 Tangent Planes and Linea Appoimations Back in-dimensions, what was the equation of the tangent line of f ( ) at point (, ) f ( )? (, ) ( )( ) = f Linea Appoimation (Tangent Line Appoimation) of f at
More informationChapter 23: Electric Potential
Chapte 23: Electc Potental Electc Potental Enegy It tuns out (won t show ths) that the tostatc foce, qq 1 2 F ˆ = k, s consevatve. 2 Recall, fo any consevatve foce, t s always possble to wte the wok done
More informationMultistage Median Ranked Set Sampling for Estimating the Population Median
Jounal of Mathematcs and Statstcs 3 (: 58-64 007 ISSN 549-3644 007 Scence Publcatons Multstage Medan Ranked Set Samplng fo Estmatng the Populaton Medan Abdul Azz Jeman Ame Al-Oma and Kamaulzaman Ibahm
More informationChapter 13 - Universal Gravitation
Chapte 3 - Unesal Gataton In Chapte 5 we studed Newton s thee laws of moton. In addton to these laws, Newton fomulated the law of unesal gataton. Ths law states that two masses ae attacted by a foce gen
More informationCSU ATS601 Fall Other reading: Vallis 2.1, 2.2; Marshall and Plumb Ch. 6; Holton Ch. 2; Schubert Ch r or v i = v r + r (3.
3 Eath s Rotaton 3.1 Rotatng Famewok Othe eadng: Valls 2.1, 2.2; Mashall and Plumb Ch. 6; Holton Ch. 2; Schubet Ch. 3 Consde the poston vecto (the same as C n the fgue above) otatng at angula velocty.
More informationVibration Input Identification using Dynamic Strain Measurement
Vbaton Input Identfcaton usng Dynamc Stan Measuement Takum ITOFUJI 1 ;TakuyaYOSHIMURA ; 1, Tokyo Metopoltan Unvesty, Japan ABSTRACT Tansfe Path Analyss (TPA) has been conducted n ode to mpove the nose
More informationPhysics 2A Chapter 11 - Universal Gravitation Fall 2017
Physcs A Chapte - Unvesal Gavtaton Fall 07 hese notes ae ve pages. A quck summay: he text boxes n the notes contan the esults that wll compse the toolbox o Chapte. hee ae thee sectons: the law o gavtaton,
More informationThe geometric construction of Ewald sphere and Bragg condition:
The geometic constuction of Ewald sphee and Bagg condition: The constuction of Ewald sphee must be done such that the Bagg condition is satisfied. This can be done as follows: i) Daw a wave vecto k in
More informationNuclear and Particle Physics - Lecture 20 The shell model
1 Intoduction Nuclea and Paticle Physics - Lectue 0 The shell model It is appaent that the semi-empiical mass fomula does a good job of descibing tends but not the non-smooth behaviou of the binding enegy.
More informationChem 453/544 Fall /08/03. Exam #1 Solutions
Chem 453/544 Fall 3 /8/3 Exam # Solutions. ( points) Use the genealized compessibility diagam povided on the last page to estimate ove what ange of pessues A at oom tempeatue confoms to the ideal gas law
More information7/1/2008. Adhi Harmoko S. a c = v 2 /r. F c = m x a c = m x v 2 /r. Ontang Anting Moment of Inertia. Energy
7//008 Adh Haoko S Ontang Antng Moent of neta Enegy Passenge undego unfo ccula oton (ccula path at constant speed) Theefoe, thee ust be a: centpetal acceleaton, a c. Theefoe thee ust be a centpetal foce,
More informationSUPPLEMENTARY INFORMATION
SUPPLEMENTARY INORMATION do: 10.1038/nPHYS140 Eneg gaps and zeo-feld quantum Hall effect n gaphene b stan engneeng. Gunea, M. I. Katsnelson, A. K. Gem I. Let us eplan fst how the two dmensonal elastct
More informationMath Section 4.2 Radians, Arc Length, and Area of a Sector
Math 1330 - Section 4. Radians, Ac Length, and Aea of a Secto The wod tigonomety comes fom two Geek oots, tigonon, meaning having thee sides, and mete, meaning measue. We have aleady defined the six basic
More informationCOLLEGE OF FOUNDATION AND GENERAL STUDIES PUTRAJAYA CAMPUS FINAL EXAMINATION TRIMESTER /2017
COLLEGE OF FOUNDATION AND GENERAL STUDIES PUTRAJAYA CAMPUS FINAL EXAMINATION TRIMESTER 1 016/017 PROGRAMME SUBJECT CODE : Foundaton n Engneeng : PHYF115 SUBJECT : Phscs 1 DATE : Septembe 016 DURATION :
More informationPhysics 202, Lecture 2. Announcements
Physcs 202, Lectue 2 Today s Topcs Announcements Electc Felds Moe on the Electc Foce (Coulomb s Law The Electc Feld Moton of Chaged Patcles n an Electc Feld Announcements Homewok Assgnment #1: WebAssgn
More informationThe Greatest Deviation Correlation Coefficient and its Geometrical Interpretation
By Rudy A. Gdeon The Unvesty of Montana The Geatest Devaton Coelaton Coeffcent and ts Geometcal Intepetaton The Geatest Devaton Coelaton Coeffcent (GDCC) was ntoduced by Gdeon and Hollste (987). The GDCC
More informationLASER ABLATION ICP-MS: DATA REDUCTION
Lee, C-T A Lase Ablaton Data educton 2006 LASE ABLATON CP-MS: DATA EDUCTON Cn-Ty A. Lee 24 Septembe 2006 Analyss and calculaton of concentatons Lase ablaton analyses ae done n tme-esolved mode. A ~30 s
More informationTHE TIME-DEPENDENT CLOSE-COUPLING METHOD FOR ELECTRON-IMPACT DIFFERENTIAL IONIZATION CROSS SECTIONS FOR ATOMS AND MOLECULES
Intenatonal The Tme-Dependent cence Pess Close-Couplng IN: 9-59 Method fo Electon-Impact Dffeental Ionzaton Coss ectons fo Atoms... REVIEW ARTICE THE TIME-DEPENDENT COE-COUPING METHOD FOR EECTRON-IMPACT
More informationA Study about One-Dimensional Steady State. Heat Transfer in Cylindrical and. Spherical Coordinates
Appled Mathematcal Scences, Vol. 7, 03, no. 5, 67-633 HIKARI Ltd, www.m-hka.com http://dx.do.og/0.988/ams.03.38448 A Study about One-Dmensonal Steady State Heat ansfe n ylndcal and Sphecal oodnates Lesson
More information3.1 Random variables
3 Chapte III Random Vaiables 3 Random vaiables A sample space S may be difficult to descibe if the elements of S ae not numbes discuss how we can use a ule by which an element s of S may be associated
More informationTian Zheng Department of Statistics Columbia University
Haplotype Tansmsson Assocaton (HTA) An "Impotance" Measue fo Selectng Genetc Makes Tan Zheng Depatment of Statstcs Columba Unvesty Ths s a jont wok wth Pofesso Shaw-Hwa Lo n the Depatment of Statstcs at
More informationCOORDINATE SYSTEMS, COORDINATE TRANSFORMS, AND APPLICATIONS
Dola Bagaoo 0 COORDINTE SYSTEMS COORDINTE TRNSFORMS ND PPLICTIONS I. INTRODUCTION Smmet coce of coodnate sstem. In solvng Pscs poblems one cooses a coodnate sstem tat fts te poblem at and.e. a coodnate
More informationLab 10: Newton s Second Law in Rotation
Lab 10: Newton s Second Law in Rotation We can descibe the motion of objects that otate (i.e. spin on an axis, like a popelle o a doo) using the same definitions, adapted fo otational motion, that we have
More informationLarge scale magnetic field generation by accelerated particles in galactic medium
Lage scale magnetc feld geneaton by acceleated patcles n galactc medum I.N.Toptygn Sant Petesbug State Polytechncal Unvesty, depatment of Theoetcal Physcs, Sant Petesbug, Russa 2.Reason explonatons The
More informationRotational Kinematics. Rigid Object about a Fixed Axis Western HS AP Physics 1
Rotatonal Knematcs Rgd Object about a Fxed Axs Westen HS AP Physcs 1 Leanng Objectes What we know Unfom Ccula Moton q s Centpetal Acceleaton : Centpetal Foce: Non-unfom a F c c m F F F t m ma t What we
More informationASTR415: Problem Set #6
ASTR45: Poblem Set #6 Cuan D. Muhlbege Univesity of Mayland (Dated: May 7, 27) Using existing implementations of the leapfog and Runge-Kutta methods fo solving coupled odinay diffeential equations, seveal
More informationInstantaneous velocity field of a round jet
Fee shea flows Instantaneos velocty feld of a ond et 3 Aveage velocty feld of a ond et 4 Vtal ogn nozzle coe Developng egon elf smla egon 5 elf smlaty caled vaables: ~ Q ξ ( ξ, ) y δ ( ) Q Q (, y) ( )
More informationTEST-03 TOPIC: MAGNETISM AND MAGNETIC EFFECT OF CURRENT Q.1 Find the magnetic field intensity due to a thin wire carrying current I in the Fig.
TEST-03 TPC: MAGNETSM AND MAGNETC EFFECT F CURRENT Q. Fnd the magnetc feld ntensty due to a thn we cayng cuent n the Fg. - R 0 ( + tan) R () 0 ( ) R 0 ( + ) R 0 ( + tan ) R Q. Electons emtted wth neglgble
More informationME 210 Applied Mathematics for Mechanical Engineers
Tangent and Ac Length of a Cuve The tangent to a cuve C at a point A on it is defined as the limiting position of the staight line L though A and B, as B appoaches A along the cuve as illustated in the
More informationOne-dimensional kinematics
Phscs 45 Fomula Sheet Eam 3 One-dmensonal knematcs Vectos dsplacement: Δ total dstance taveled aveage speed total tme Δ aveage veloct: vav t t Δ nstantaneous veloct: v lm Δ t v aveage acceleaton: aav t
More informationWaves Basics. April 2001 Number 17
Apl 00 Numbe 7 Waes Bascs Ths Factsheet wll ntoduce wae defntons and basc popetes. Types of wae Waes may be mechancal (.e. they eque a medum such as o to popagate) o electomagnetc (whch popagate n a acuum)
More informationgravity r2,1 r2 r1 by m 2,1
Gavtaton Many of the foundatons of classcal echancs wee fst dscoveed when phlosophes (ealy scentsts and atheatcans) ted to explan the oton of planets and stas. Newton s ost faous fo unfyng the oton of
More informationEnergy in Closed Systems
Enegy n Closed Systems Anamta Palt palt.anamta@gmal.com Abstact The wtng ndcates a beakdown of the classcal laws. We consde consevaton of enegy wth a many body system n elaton to the nvese squae law and
More informationLecture 5 Single factor design and analysis
Lectue 5 Sngle fcto desgn nd nlss Completel ndomzed desgn (CRD Completel ndomzed desgn In the desgn of expements, completel ndomzed desgns e fo studng the effects of one pm fcto wthout the need to tke
More informationChapter 8. Linear Momentum, Impulse, and Collisions
Chapte 8 Lnea oentu, Ipulse, and Collsons 8. Lnea oentu and Ipulse The lnea oentu p of a patcle of ass ovng wth velocty v s defned as: p " v ote that p s a vecto that ponts n the sae decton as the velocty
More informationPhysics 2020, Spring 2005 Lab 5 page 1 of 8. Lab 5. Magnetism
Physics 2020, Sping 2005 Lab 5 page 1 of 8 Lab 5. Magnetism PART I: INTRODUCTION TO MAGNETS This week we will begin wok with magnets and the foces that they poduce. By now you ae an expet on setting up
More informationPart V: Velocity and Acceleration Analysis of Mechanisms
Pat V: Velocty an Acceleaton Analyss of Mechansms Ths secton wll evew the most common an cuently pactce methos fo completng the knematcs analyss of mechansms; escbng moton though velocty an acceleaton.
More informationV. Principles of Irreversible Thermodynamics. s = S - S 0 (7.3) s = = - g i, k. "Flux": = da i. "Force": = -Â g a ik k = X i. Â J i X i (7.
Themodynamcs and Knetcs of Solds 71 V. Pncples of Ievesble Themodynamcs 5. Onsage s Teatment s = S - S 0 = s( a 1, a 2,...) a n = A g - A n (7.6) Equlbum themodynamcs detemnes the paametes of an equlbum
More informationContact, information, consultations
ontact, nfomaton, consultatons hemsty A Bldg; oom 07 phone: 058-347-769 cellula: 664 66 97 E-mal: wojtek_c@pg.gda.pl Offce hous: Fday, 9-0 a.m. A quote of the week (o camel of the week): hee s no expedence
More informationUnits, Physical Quantities and Vectors
What s Phscs? Unts, Phscal Quanttes and Vectos Natual Phlosoph scence of matte and eneg fundamental pncples of engneeng and technolog an epemental scence: theo epement smplfed (dealed) models ange of valdt
More informationChapter 3 Vector Integral Calculus
hapte Vecto Integal alculus I. Lne ntegals. Defnton A lne ntegal of a vecto functon F ove a cuve s F In tems of components F F F F If,, an ae functon of t, we have F F F F t t t t E.. Fn the value of the
More information1 Dark Cloud Hanging over Twentieth Century Physics
We ae Looking fo Moden Newton by Caol He, Bo He, and Jin He http://www.galaxyanatomy.com/ Wuhan FutueSpace Scientific Copoation Limited, Wuhan, Hubei 430074, China E-mail: mathnob@yahoo.com Abstact Newton
More informationAPPLICATION OF MAC IN THE FREQUENCY DOMAIN
PPLICION OF MC IN HE FREQUENCY DOMIN D. Fotsch and D. J. Ewins Dynamics Section, Mechanical Engineeing Depatment Impeial College of Science, echnology and Medicine London SW7 2B, United Kingdom BSRC he
More informationPhysics 1: Mechanics
Physcs : Mechancs Đào Ngọc Hạnh Tâm Offce: A.503, Emal: dnhtam@hcmu.edu.vn HCMIU, Vetnam Natonal Unvesty Acknowledgment: Sldes ae suppoted by Pof. Phan Bao Ngoc Contents of Physcs Pat A: Dynamcs of Mass
More informationElectrostatics (Electric Charges and Field) #2 2010
Electic Field: The concept of electic field explains the action at a distance foce between two chaged paticles. Evey chage poduces a field aound it so that any othe chaged paticle expeiences a foce when
More informationMachine Learning 4771
Machne Leanng 4771 Instucto: Tony Jebaa Topc 6 Revew: Suppot Vecto Machnes Pmal & Dual Soluton Non-sepaable SVMs Kenels SVM Demo Revew: SVM Suppot vecto machnes ae (n the smplest case) lnea classfes that
More informationEvaluation of Various Types of Wall Boundary Conditions for the Boltzmann Equation
Ealuaton o Vaous Types o Wall Bounday Condtons o the Boltzmann Equaton Chstophe D. Wlson a, Ramesh K. Agawal a, and Felx G. Tcheemssne b a Depatment o Mechancal Engneeng and Mateals Scence Washngton Unesty
More information11) A thin, uniform rod of mass M is supported by two vertical strings, as shown below.
Fall 2007 Qualifie Pat II 12 minute questions 11) A thin, unifom od of mass M is suppoted by two vetical stings, as shown below. Find the tension in the emaining sting immediately afte one of the stings
More informationDetermining solar characteristics using planetary data
Detemining sola chaacteistics using planetay data Intoduction The Sun is a G-type main sequence sta at the cente of the Sola System aound which the planets, including ou Eath, obit. In this investigation
More informationStellar Astrophysics. dt dr. GM r. The current model for treating convection in stellar interiors is called mixing length theory:
Stella Astophyscs Ovevew of last lectue: We connected the mean molecula weght to the mass factons X, Y and Z: 1 1 1 = X + Y + μ 1 4 n 1 (1 + 1) = X μ 1 1 A n Z (1 + ) + Y + 4 1+ z A Z We ntoduced the pessue
More information16 Modeling a Language by a Markov Process
K. Pommeening, Language Statistics 80 16 Modeling a Language by a Makov Pocess Fo deiving theoetical esults a common model of language is the intepetation of texts as esults of Makov pocesses. This model
More informationAnalytic Evaluation of two-electron Atomic Integrals involving Extended Hylleraas-CI functions with STO basis
Analytic Evaluation of two-electon Atomic Integals involving Extended Hylleaas-CI functions with STO basis B PADHY (Retd.) Faculty Membe Depatment of Physics, Khalikote (Autonomous) College, Behampu-760001,
More informationPattern Analyses (EOF Analysis) Introduction Definition of EOFs Estimation of EOFs Inference Rotated EOFs
Patten Analyses (EOF Analyss) Intoducton Defnton of EOFs Estmaton of EOFs Infeence Rotated EOFs . Patten Analyses Intoducton: What s t about? Patten analyses ae technques used to dentfy pattens of the
More information