Germany 2) Max-Born-Institut für Nichtlineare Optik und Kurzzeitspektroskopie, D Berlin,

Size: px
Start display at page:

Download "Germany 2) Max-Born-Institut für Nichtlineare Optik und Kurzzeitspektroskopie, D Berlin,"

Transcription

1 Crystl struture of polymeri rbon nitride nd determintion of its proess-temperture-indued modifitions T. Tyborski, 1, 2 C. Mershjnn, 1, 3 S. Orthmnn, 1 F. Yng, 1 M.-Ch. Lux-Steiner, 1 nd Th. Shedel-Niedrig 1 1) Helmholtz-Zentrum Berlin für Mterilien und Energie GmbH, D Berlin, Germny 2) Mx-Born-Institut für Nihtlinere Optik und Kurzzeitspektroskopie, D Berlin, Germny ) 3) Universität Rostok, D Rostok, Germny Bsed on the rrngement of two-dimensionl melon, we onstrut unit ell for polymeri rbon nitride (PCN) synthesized vi therml polyondenstion whose theoretil diffrtion powder pttern inludes ll mjor fetures mesured in X-ry diffrtion. With the help of tht unit ell, we desribe the proesstemperture-indued rystllogrphi hnges in PCN tht our within temperture intervl between 510 C nd 610 C. We lso disuss further potentil modifitions of the unit ell for PCN. It is found tht both trizine nd heptzine-bsed g-c 3 N 4 n only ount for minor phses within the investigted synthesis produts. Keywords: polymeri rbon nitride, g-c 3 N 4, therml polyondenstion, rystl struture Polymeri rbon nitride (PCN) stnds for very remrkble lss of mterils. It revels phototlyti tivity, both s pristine powder nd s nnorystlline film in heterojuntion photothode. 1 3 Furthermore, PCN only onsists of bundnt elements, n be synthesized vi strightforwrd, slble proess nd offers high hemil nd therml stbility. 4 Light-indued hydrogen relese on the bsis of PCN in n idi queous solution under the presene of n eletron donor hs been demonstrted. 1 At the sme time, synthesis-prmeter nd thus rystl struture dependene of the phototlyti tivity s well s of the photophysis of PCN is doumented. 1,4 6 These fetures ombined in PCN estblish fr-rnging interest on tht lss of mterils, espeilly with respet to sustinble hydrogen prodution. 7 The synthesis of PCN vi bulk therml polyondenstion is hrterized by both the retion-temperturedependent hydrogen ontent nd the degree of polymeriztion within temperture rnge of bout 510 C up to 610 C. 4,8 10 The therml polyondenstion itself is ontinuous proess tht revels severl dominnt phses t ertin tempertures, suh s melmine t 250 C or melem t 450 C. 4,9 While some of these dominnt phses n be exmined from rystllogrphi point of view, the rystllogrphi struture of PCN hs only been desribed insuffiiently. 11 X-ry diffrtion (XRD) dt of PCN revel preferentilly two mjor, slightly retiontemperture-dependent reflexes t 2Θ 27.3 (3.26 Å) nd 2Θ 13.2 (6.70 Å). These reflexes re understood s diffrtion t grphite-like sheets for the former nd s diffrtion t periodi struturl feture within the sheets for the ltter diffrtion ngle. 4 The grphite-like sheets exhibit pores nd onsist of polymerized heptzine units. 4,12,13 However, so fr distint unit ell with onrete spe group for PCN tht ombines the listed struturl spets nd is in good greement with the mesured XRD powder ptterns hs not been identified. The usul wy to determine the rystl struture of ertin mteril on the bsis of its powder pttern is exeuted s follows (1) The unit ell is determined by indexing the peks of the powder pttern. (2) The powder pttern is deomposed into integrted intensities I h,k,l. (3) The spe group is ssigned bsed on the systemti bsene of severl I h,k,l. (4) The phse problem is solved, e.g. with the help of the Ptterson Method. (5) The struture is refined, typilly with Rietveld refinement. This proper struture nlysis ssumes high purity s well s high rystllinity of the investigted mteril whih in the se of PCN synthesized vi therml polyondenstion n both not be fforded. 2,4 This is the point where subjetive judgements nd fesible rguments ome to the fore in order to overome the steps (1)-(3) while proper refinement does not stnd hne. In this work, we ontrst melon-bsed unit ells with trizine - nd heptzine-bsed unit ells nd ompre their theoretil X-ry diffrtion ptterns with mesured ptterns from PCN. Our findings of tht nlysis suggest tht the mjor phse of PCN is bsed on melon sheets s well s tht g-c 3 N 4 n only ount for minor phse within the investigted synthesis produts. We lso disuss geometril modifitions of the melon-bsed PCN unit ell nd demonstrte their influene on X-ry diffrtion ptterns. Furthermore, we use the melon-bsed PCN unit ell in order to desribe rystllogrphi hnges within PCN tht ome long with higher proess tempertures. Detils onerning the rystllogrphi strutures n be found in the supporting informtion. ) Eletroni mil: tybo@physik.fu-berlin.de

2 2 A. EXPERIMENTAL SECTION Synthesis of PCN Diyndimide (Aldrih, 99 %, bout 2.5 g) ws heted to tempertures between 490 C nd 610 C (in 20 -steps) t rte of 25 min 1 in ermi ruible within muffle furne under ontinuous N 2 gs flow. In first proessing step, the therml tretment lsts for five minutes fter whih the resulting produts were grounded. In seond proessing step, the grounded mterils were heted to the respetive proessing tempertures for nother three hours under N 2 gs flow, fter wht they were llowed to ool down pssively in the furne. Struturl hrteriztion Struturl phse nlysis of the PCN powders ws performed by X-ry diffrtometry (Bruker AXS D8, Cu Kα - line) in Θ 2Θ onfigurtion. Approximtely 100 mg of eh PCN powder ws nlyzed for one hour between 2Θ = 10 nd 2Θ = 90. The resulting powder ptterns hve been bkround-orreted with n exponentil funtion in order to minimize the influene of diffuse sttering. Sine they do not exhibit nrrow reflexes, the powder ptterns hve been smoothed for the purpose of emphsizing their fetures with respet to the noise. Struture nlysis Eh disussed theoretil unit ell ws nlyzed in numeril X-ry diffrtion experiment. These nlysis were performed with the progrm Dimond 3.2i nd reviewed with the progrm PowderCell The following prmeters (djustble in the softwre) were used: X-ry lbortory soure with wvelength λ = Å, 2Θ min = 10 nd 2Θ mx = 90, tivted Lorentz ftor, tivted Polriztion ftor nd disbled displement ftor. Depending on the prtiulr nlyzed unit ell, the softwre lultes the orresponding struture ftors nd thus the orresponding diffrtion pttern. The physil bkground of tht lultion is the independent-tom model (IAM), in whih the eletroni hrge density is exlusively loted t the toms. All disussed numeril vlues for reflex positions or seprtion distnes suh s d s or d hve been mesured with the progrm Dimond 3.2i on the bsis of the orresponding tomi positions nd rystl prmeters. The men seprtion distne d is defined s d = i=1 d i in whih d i re the seprtion distnes of the (five) prtiulr pirs of nitrogen toms tht re loted fe to fe with eh other long the heptzine hins (figure 5). B. Grphiti rbon nitride g-c 3N 4 A trizine-bsed struture for g-c 3 N 4 ws theoretilly suggested by Teter nd Hemley in Next to the unit ell (upper inset), figure 1 shows the theoretil diffrtion pttern of tht mteril in ombintion with mesured XRD pttern of PCN. This omprison revels fundmentl struturl differenes between PCN nd trizine-bsed g-c 3 N 4. Espeilly the intense reflexes of PCN t diffrtion ngles smller thn 2Θ = 20 nnot be ttributed to g-c 3 N 4. Similrities n be found for the g-c 3 N 4 -reflexes (011) t 2Θ = 25.40, (002) t 2Θ = nd (020) t 2Θ = FIG. 1. Top: Unit ell for trizine-bsed g-c 3N 4 with spe group P 6m2 (187), = Å, = Å nd 14 toms (8N(blue) +6C(grey)) together with the diffrtion pttern of PCN (blk urve) nd the most intense diffrtion reflexes of n idel trizine-bsed g-c 3N 4-rystl (blue lines). 12 Bottom: Struture of heptzine-bsed g-c 3N 4 synthezised by Bojdys et l. with spe group P 6 3m (185), = Å, = Å nd its most intense diffrtion reflexes (grey lines). 21 Bojdys et l. hve demonstrted n ionotherml synthesis route for substne they lso ll g-c 3 N Their mteril (lower inset figure 1) exhibits the lrgest rystllites known so fr for ny grphite-like rbon nitride, showing dimeters of bout 200 nm. The orresponding struture nlysis revels the hexgonl spe group P6 3 m (185). However, XRD nlysis lso indites tht this is diverse mteril ompred to both the trizinebsed g-c 3 N 4 nd the PCN synthesized vi therml polyondenstion whih neessittes the onsidertion of different unit ell for PCN. C. Unit ell for PCN We onstrut unit ell for PCN on the bsis of poly(minoimino)heptzine [C 6 N 7 (NH 2 )(NH)] n, lso known s melon, whose struture hs been nlyzed by

3 3 Wng et l. nd Lotsh et l.1,10 The reltive positions of the toms within the melon sheet were extrted digitlly from figure 12 in Ref. 10 with the id of the imge dt proessing progrm ImgeJ 1.43u.10,22,23 In this proedure, the entrl points of the illustrted grey nd blk dots served s the tomi oordintes for nitrogen nd rbon. We dded oordintes for the missing hydrogen toms. With these tomi oordintes, we onstrut theoretil unit ell for PCN. The reltive lignment of the toms in the unit ell n be modified mthemtilly suh tht, for exmple, bukling or different stking motifs n be introdued. Sine the grphil melon representtion in Ref. 10 nd the extrtion of the tomi oordintes re inherently orrelted with minor unertinties, we initilly ssume the lowest possible symmetry (spe group P 1) in our unit ell. The ell prmeters hve been hosen to be in line with the mesured XRD dt of our PCN smples s well s with the findings from Lotsh et l.10 In this wy, we suggest unit ell for PCN synthesized vi therml polyondenstion of diyndimide with the following prmeters: spe group P 1, = 16.2 A, b = 12.1 A, = A nd α = β = γ = 90, in whih nd γ exhibit proess-temperture dependene (figure 4 nd figure 5). Figure 2 shows unit ell for A-A-stked PCN with flt melon sheets. The orresponding theoretil diffrtion pttern is opposed to mesured powder pttern of PCN in the upper prt of figure 2. This omprison revels tht espeilly the strong reflexes (2 10), (210) t 2Θ = nd (001) t 2Θ = mth the mesured powder pttern with respet to their ngulr positions nd intensities persusively. The evident pek between the mjor reflexes n be understood s diffrtion t the hkl-plnes (3 10) nd (310) t 2Θ = The pirs (2 10), (210) nd (3 10), (310) re eh overlpping reflexes. Three modifitions need to be disussed in the further proedure ompred to flt A-A stked PCN: bukling within singulr melon sheets, different stking motifs nd the influene of the proess temperture.4 1. Bukling Compred to infinite C6 N8 sheets, in whih the energeti dvntge of sinusoidl bukling with n mplitude of 0.7 A prllel to the plne norml is lulted to be 11 kj mol 1 per neighboring N-N pir, the melon sheets in PCN very likely feture relted bukling.4,24 hkl-plne (101) hkl-plne (001) ds b hkl-plne (101) b hkl-plne (210) hkl-plne (210) FIG. 2. Trilini unit ell for PCN with flt melon sheets, n A-A-stking motif nd 72 toms (36N + 24C + 12H). Bottom: PCN ltties long -diretion nd with view into the stked melon sheets, eh with highlighted hkl-plnes. Top: Mesured XRD pttern (blk urve) nd the most intense diffrtion reflexes of the orresponding idel PCN (blue lines). FIG. 3. Trilini unit ell for PCN with sinusoidlly bukled melon sheets (prllel to --plne, 0.7 A mplitude) nd n A-A-stking motif. Bottom: PCN lttie with view into the stked melon sheets, highlighted hkl plnes nd trilini unit ell. Top: Mesured XRD pttern (blk urve) together with the most intense theoretil diffrtion reflexes of the orresponding PCN lttie (blue lines). In the speil se of sinusoidl bukling prllel to the --plne, s demonstrted in figure 3, ssoited fetures in the diffrtion pttern re rising reflexes (1 01)

4 4 nd (101), both t 2Θ = Emerging reflexes (0 11) nd (011), t 2Θ = 28.21, re the result of sinusoidl bukling prllel to the b--plne (not shown). A sinusoidl bukling prllel to the -diretion nd long the -b-digonl uses wek emerging ( 101),(101) - nd (0 11),(011) - s well s (111),( 111),(1 11),( 1 11) - reflexes (not shown). The orresponding bukling-indued inrese of N-N-distnes between nerest neighbors in one heptzine unit ounts for roughly Å, the distne d (figure 5) between the in-plne heptzine hins inreses by bout Å while the intrplnr N- N-distnes of seond nerest neighbors extend by bout Å. Comprble, minor hnges in the theoretil diffrtion ptterns n be observed for slightly vried bukling mplitudes. 2. Stking motifs If one ssumes the stking of the melon sheets in PCN to be of the form A-B, the (001)-reflex is repled by (002)-reflex in the orresponding diffrtion pttern nd the mount of toms per unit ell is doubled, generlly. A displement of every seond melon sheet 0.5 long the -xis leds to diffrtion pttern tht fetures forbidden reflexes ( 310) nd (310) only (not shown). In exhnge reloted nd enhned qud reflex (311),( 311),(3 11),( 3 11) ppers t 2Θ = (not shown). In the se of displement 0.5 b long the b-xis, the reflexes ( 210) nd (210) vnish wheres strong qud reflex (211),( 211),(2 11),( 2 11) emerges t 2Θ = (not shown). A ombintion of these two displements ( b) lso leds to vnishing (2 10),(210) - nd rising (211),( 211),(2 11),( 2 11) reflexes (not shown). 3. Proess-temperture-indued influene on the rystl struture of PCN FIG. 4. Mesured XRD ptterns of PCN synthesized t 510 C (top) nd t 610 C (bottom), together with the most intense theoretil reflexes of flt, A-A-stked PCN with two different unit ell ngles γ. Figure insets show n enlrgement of the peks t 2Θ = 13.2 nd 2Θ = 27.2, respetively, for three different proess tempertures. Inresing the proess temperture of PCN leds to signifint hnges in the mesured diffrtion ptterns. 4,21 As n be seen in figure 4, the higher the proess temperture, the weker nd broder is the pek t 2Θ = At the sme time the pek t 2Θ = 27.2 is shifted towrds lrger diffrtion ngles. While the wekening nd brodening of the 2Θ = pek is lerly observble, one n only observe slight shift towrds higher diffrtion ngles for the 2Θ = pek. Next to the speified hnges, deresing FWHM of the pek from roughly 1.6 to 1.3 is observble. These proess-temperture-indued hnges n be understood on the bsis of the trilini PCN suggestion. During the therml polyondenstion, the heptzine units begin to hemilly ondenste t tempertures of round 510 C, thus forming PCN. Further heting to tempertures of round 610 C redues the hydrogen ontent vi mmoni relese, pointing to n enhned rystllinity. 4 An enhned rystllinity long the -xis is lerly observble in the shift of the pek nd its redued FWHM, inditing n inresed number of stked melon sheets per rystllite s well s redued stking distne d s (figure 2). Neessrily orrelted to the deresing distne between the sheets is the deresing distne of prtiulr nitrogen toms. Sine ll nitrogen toms in PCN exhibit n unbound pir of eletrons, lrge N-N-distnes re energetilly benefiil. So one

5 5 single melon sheet distorts its two-dimensionl ordering suh, tht the inner nitrogen distnes inrese. For tht reson, the ngle γ of PCN s trilini unit ell is slightly shifted to vlues < 90, s shown in figure 5. FIG. 5. Left: Sheme of the temperture-indued distortion of PCN s rystl struture together with setion of distorted unit ell (γ = 87 ) nd the men seprtion distne between two heptzine hins d. Note tht the sheme is illustrted exessively for n enhned lrity. Right: Reltion between γ, d, ds nd the proess temperture (in flt, A-A-stked PCN). While the stking distne ds stys unhnged with respet to deresing ngle γ, the heptzine units re slightly distorted nd the men seprtion distne d between the heptzine hins is inresed from 3.11 A to 3.19 A with slope of A / C. This distortion influenes the theoretil diffrtion pttern of PCN suh, tht the overlpping reflexes (2 10), (210) nd (3 10), (310) re displed ginst eh other (figure 4). A very similr behvior n be observed for n enlrged ngle γ. But enlrging γ is orrelted to deresed distnes d, whih mkes this kind of distortion unplusible. Slightly vrying the ngles α nd β ends up in only minorly hnged theoretil diffrtion ptterns. The disussed results n be found for the flt s well s for bukled PCN. D. DISCUSSION As mentioned before, the pplied independent-tom model restrits the eletron density t the toms. Thus, the eletron density is desribed by the spherilly verged density of the isolted toms.25 The model works very onviningly for solely hevy toms in the unit ell. This is beuse in tht se, the vlene shells re only minor prt of the totl eletron density t whih the X-ry photons re sttered. It ffords unuries for light toms. For exmple, the bond length of ovlent C-H bond is typilly underestimted within the IAM piture sine the hydrogens eletron density is signifintly distributed over the bond.25,26 Nevertheless, the IAM hs been the bsis of X-ry struture nlysis sine its ineption.25 Espeilly for PCN, whose X-ry diffrtion ptterns do not llow n dvned struture nlysis, the IAM offers n estblished tool in order to determine unit ell for PCN. We hve lso nlyzed the influene of tomi disple- ment ftors on the diffrtion ptterns of PCN. For tht purpose, we hve used unit ell for flt, A-A-stked PCN nd implemented n isotropi displement ftor of 0.04 A 2 for eh tom, ording to the melem rystl investigted by Ju rgens et l. nd A. Sttler et l.9,11 The orresponding findings re slightly vried totl reflex intensities nd intensity rtios. For exmple, the (001)reflex intensity is deresed by bout 15 % nd the sum intensity of (210) nd (2 10) by bout 4 %. This results in modified intensity rtio of (I[(210)] + I[(2 10)]) /I[(001)] from pproximtely 0.41 for disbled displement ftors (figure 2) to 0.46 for enbled displement ftors (not shown). The proposed unit ell for PCN nd its modifitions do not llow inversion symmetry. However, Lotsh et l. desribe melons in-plne symmetry with the plne group p2gg, whih inludes 2-fold rottions, glide-refletions nd trnsltions.10 In order to onserve the 2-fold rottionl symmetry within the melon sheets nd thus inversion symmetry within the PCN unit ells, the following proedure my be pplied: One ould onstrut PCN unit ell with the disussed ell prmeters, but only inlude hlf of the toms per unit ell, suh tht n inversion mps the missing toms to the pproprite tomi positions. The orresponding spe group, whih llows this symmetry opertion, is the spe group P 1. The theoretil diffrtion ptterns for inversionsymmetri PCN (spe group P 1 ) re very similr to those of PCN without inversion symmetry (spe group P 1). For exmple, the sum intensity of the (210) nd (2 10)-reflex is deresed by bout 4 % for n inversion symmetry in flt, A-A-stked PCN, ompred to the non-inversion-symmetri se, while the diffrtion ngle stys unhnged. We propose trilini rystl lttie for PCN, sine this is the only rystl system for whih 6= b 6= nd γ 6= 90 whih is neessry for the desription of the disussed proess-temperture-indued modifitions in PCN. Vrying α nd β leds to shifted (001) - nd (210),(2 10)-reflexes in the diffrtion ptterns towrds higher diffrtion ngles nd modified intensities. For α = 85 or α = 95 nd untouhed residul lttie prmeters, the (210) - nd (2 10)-reflexes re displed by bout 2Θ = 0.02, wheres the (001)-reflex is displed by bout 2Θ = For β = 85 or β = 95 the (210) - nd (2 10)-reflexes re displed by bout 2Θ = 0.03, wheres the (001)-reflex is displed by bout 2Θ = Seyfrth et l. hve disussed some possible rrngements of stked melon sheets bsed on theoretil lultions. Their findings led to signifint reltive displements.27 Implementing these displements into the unit ell for flt PCN results in theoretil diffrtion ptterns tht re ll not suffiient to reonstrut the mesured ones (see supporting informtion). The only treble hint for grphiti rbon nitride gc3 N4 within the synthesized PCN is the wek but omnipresent pek round 2Θ = 44 in the mesured diffr-

6 6 tion ptterns (figure 1). This one n be relted to the (200)-reflex of trizine-bsed g-c 3 N 4 or to the (300)- reflex of heptzine-bsed g-c 3 N 4 (ppendix). Thus, inor se, g-c 3 N 4 is only minor outome of the therml polyondenstion of diyndimide. This is lso onsistent with the reported hydrogen ontent of 1 2 wt% in the synthesized PCN, sine the theoretil vlue for its hydrogen ontent is expeted to be 1.27 wt%. 4,21 One finds synthesis produts for proess tempertures below 510 C or bove 610 C whose mesured diffrtion ptters devite signifintly from the ones of PCN. 4,5,21 This is minly used by onsiderble mount of nonpolymerized melem within the synthesis produts for proess tempertures below 510 C nd degrdtion of PCN for proess tempertures bove 610 C. 4,9,11 The proess-temperture dependene of the photophysis of PCN n be extended to tempertures from bout 400 C to 610 C. The reson for tht is the building unit C 6 N 10 H x, melem in the non-polymerized nd heptzine in the polymerized stte, whih is omnipresent in tht temperture rnge nd in whih the photophysis is found to tke ple. 6,9 These units re more densely pked for higher proess tempertures, thus enhning the eletroni intertion between them whih influenes PCN s optil properties. 6 Furthermore, we suggest the spe group P 6(174) for heptzine-bsed g-c 3 N 4. From rystllogrphi point of view, this is the only hexgonl spe group tht inludes the symmetry opertions for whih the sum of Wykoff multipliities is identil to the generted toms per heptzine-bsed g-c 3 N 4 unit ell. A-A-stked s well s A-B-stked heptzine-bsed g-c 3 N 4 is thinkble (see supporting informtion). To onlude our nlysis bout the rystl struture of PCN nd the ssoited disussion, we sum up our findings s follows. We expet PCN to minly onsist of bukled, A-A-stked PCN nd/or flt, A-A-stked PCN with proess-temperture-dependent unit ell prmeters nd γ. For these rystl strutures, the strongest ordne between the mesured diffrtion ptterns of PCN nd the orresponding theoretil diffrtion ptterns n be found. Nevertheless, due to the speifi hrter of the mesured diffrtion ptterns with its brod peks, we n only drw qulittive piture. In tht piture, we n not expliitely exlude ny of the disussed potentil forms of PCN. We rther ssume the PCN to be onglomerte of mny rystllites with slightly different rystl strutures nd/or defets, in whih bukled nd flt, A-A-stked PCN is the dominnt phse. The dimeters of the rystllites re pproximtely 15 nm, estimted with the Sherrer eqution on the bsis of the reflexes (210) nd (001) of flt PCN. 28,29 We lso expet n morphous ontent in the PCN minly resulting in non-speifi diffrtion intensity between the two mjor peks. Furthermore, PCN seems to be the result of severl other synthesis routes. 30,31 We hve shown tht trilini rystl struture (P 1 or P 1) with = 16.2 Å, b = 12.1 Å, 3.26 Å Å, α = β = 90 nd 87 γ 90 bsed on bukled nd flt melon sheets n explin ll mjor fetures of the diffrtion ptterns of PCN. The strong mesured pek t 13.2 is identified to be superposition of the reflexes ( 210) nd (210), the mesured pek t 17.9 is ssigned to superposition of the reflexes ( 310) nd (310), while the mesured pek t 27.2 n be relted to the reflexes (001) or (002). Different bukling nd stking motifs re probble rystllogrphi devitions tht re inluded within the frmework of trilini PCN. An inresing proess temperture leds to more dense pking long the -xis nd lso to distortion of the melon sheets. The unit ell ngle γ is redued to vlue of bout 87, leding to non-overlpping ( 210),(210) nd ( 310),(310) reflexes for high proess tempertures, nd thus to brodening of the respetive peks in the mesured XRD ptterns. We hve ssumed hexgonl unit ell P 6(174) for heptzine-bsed g-c 3 N 4 nd illustrted tht the heptzine-bsed s well s the trizine-bsed g- C 3 N 4 n only be minor synthesis outome of the therml polyondenstion of diyndimide or relted retnts. ACKNOWLEDGMENTS The uthors would like to thnk F. Zmponi, M. Wörner nd T. Elsässer from the Mx-Born-Institute für Nihtlinere Optik und Kurzzeitspektroskopie for professionl dvies nd the oppurtunity to finish this work. Finnil support by the Germn Bundesministerium für Bildung und Forshung (exellene luster projet Light2Hydrogen, grnt No. 03IS2071F) is grtefully knowledged. 1 X. C. Wng, K. Med, A. Thoms, K. Tknbe, G. Xin, J. M. Crlsson, K. Domen, nd M. Antonietti, Nture Mterils 8, 76 (2009). 2 F. Yng, M. Lublow, S. Orthmnn, C. Mershjnn, T. Tyborski, M. Rusu, S. Kubl, A. Thoms, R. Arrigo, M. Häveker, nd T. Shedel-Niedrig, ChemSusChem 5, 1227 (2012). 3 F. Yng, V. Kuznietsov, M. Lublow, C. Mershjnn, A. Steigert, J. Kler, A. Thoms, nd T. Shedel-Niedrig, J. Mter. Chem. A 1 (2013). 4 A. Thoms, A. Fisher, F. Goettmnn, M. Antonietti, J.-O. Müller, R. Shlögl, nd J. M. Crlsson, J. Mter. Chem. 18, 4893 (2008). 5 T. Tyborski, C. Mershjnn, S. Orthmnn, F. Yng, M.-C. Lux- Steiner, nd T. Shedel-Niedrig, J. Phys.: Condens. Mtter 24, (2012). 6 C. Mershjnn, T. Tyborski, S. Orthmnn, F. Yng, K. Shwrzburg, M. Lublow, M.-C. Lux-Steiner, nd T. Shedel- Niedrig, Phys. Rev. B 87, (2013). 7 Y. Wng, X. Wng, nd M. Antonietti, Angew. Chem. Int. Ed. 51, 68 (2012). 8 B. V. Lotsh nd W. Shnik, Chem. Eur. J. 13, 4956 (2007). 9 B. Jürgens, E. Irrn, J. Senker, P. Kroll, H. Müller, nd W. Shnik, J. Am. Chem. So. 125, (2003). 10 B. V. Lotsh, M. Döblinger, J. Sehnert, L. Seyfrth, J. Senker, O. Oekler, nd W. Shnik, Chem. Eur. J. 13, 4969 (2007). 11 A. Sttler nd W. Shnik, Z. Anorg. Allg. Chem. 632 (2006).

7 7 12 D. M. Teter nd R. J. Hemley, Siene 271, 53 (1996). 13 E. Kroke nd M. Shwrz, Coord. Chem. Rev. 248, 493 (2004). 14 H. M. Rietveld, J. Appl. Crsyt. 2, 65 (1969). 15 H. M. Rietveld, At Cryst. 22, 151 (1967). 16 W. I. F. Dvid, K. Shnklnd, L. B. MCusker, nd C. Berloher, Struture Determintion from Powder Diffrtion Dt (OXFORD SCIENCE Publitions, 2002). 17 H. Putz nd K. Brndenburg, Crystl Impt, Kreuzherrenstr. 102, Bonn, Germny, 18 G. Bergerhoff, M. Berndt, nd K. Brndenburg, J. Res. Ntl. Inst. Stnd. Tehnol. 101, 221 (1996). 19 W. T. Pennington, J. Appl. Cryst. 32, 1028 (1999). 20 W. Krus nd G. Nolze, Federl Institute for Mterils Reserh nd Testing, Rihrd-Willstätter-Str. 11, Berlin, Germny. 21 M. J. Bojdys, J.-O. Müller, M. Antonietti, nd A. Thoms, Chem. Eur. J. 14, 8177 (2008). 22 W. Rsbnd, ImgeJ, U. S. Ntionl Institutes of Helth, Bethesd, Mrylnd, USA, ( ). 23 M. Abrmoff, P. Mglhes, nd S. Rm, Biophotonis Interntionl 11, 36 (2004). 24 J. Sehnert, K. Berwinkel, nd J. Senker, J. Phys. Chem. B 111, (2007). 25 P. Coppens, X-Ry Chrge Densities nd Chemil Bonding (Oxford University Press, In., 1997). 26 G. Vidl-Vlt nd J.-P. Vidl, At Cryst. A 48, 46 (1992). 27 L. Seyfrth, J. Seyfrth, B. V. Lotsh, W. Shnik, nd J. Senker, Phys. Chem. Chem. Phys. 12, 2227 (2010). 28 P. Sherrer, Göttinger Nhrihten 2, 98 (1918). 29 M. Birkholz, Thin film nlysis by X-ry sttering (Wiley-VCH, 2005). 30 J. Liu, T. Zhng, Z. Wng, G. Dwson, nd W. Chen, J. Mter. Chem. 21, (2011). 31 F. Dong, Y. Sun, L. Wu, M. Fu, nd Z. Wu, Ctl. Si. Tehnol. 2, 1332 (2012).

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

Bravais lattices and crystal systems

Bravais lattices and crystal systems 3 Brvis ltties nd rystl systems 3. Introdution The definitions of the motif, the repeting unit of pttern, nd the lttie, n rry of points in spe in whih eh point hs n identil environment, hold in three dimensions

More information

GM1 Consolidation Worksheet

GM1 Consolidation Worksheet Cmridge Essentils Mthemtis Core 8 GM1 Consolidtion Worksheet GM1 Consolidtion Worksheet 1 Clulte the size of eh ngle mrked y letter. Give resons for your nswers. or exmple, ngles on stright line dd up

More information

(h+ ) = 0, (3.1) s = s 0, (3.2)

(h+ ) = 0, (3.1) s = s 0, (3.2) Chpter 3 Nozzle Flow Qusistedy idel gs flow in pipes For the lrge vlues of the Reynolds number typilly found in nozzles, the flow is idel. For stedy opertion with negligible body fores the energy nd momentum

More information

Electromagnetism Notes, NYU Spring 2018

Electromagnetism Notes, NYU Spring 2018 Eletromgnetism Notes, NYU Spring 208 April 2, 208 Ation formultion of EM. Free field desription Let us first onsider the free EM field, i.e. in the bsene of ny hrges or urrents. To tret this s mehnil system

More information

Activities. 4.1 Pythagoras' Theorem 4.2 Spirals 4.3 Clinometers 4.4 Radar 4.5 Posting Parcels 4.6 Interlocking Pipes 4.7 Sine Rule Notes and Solutions

Activities. 4.1 Pythagoras' Theorem 4.2 Spirals 4.3 Clinometers 4.4 Radar 4.5 Posting Parcels 4.6 Interlocking Pipes 4.7 Sine Rule Notes and Solutions MEP: Demonstrtion Projet UNIT 4: Trigonometry UNIT 4 Trigonometry tivities tivities 4. Pythgors' Theorem 4.2 Spirls 4.3 linometers 4.4 Rdr 4.5 Posting Prels 4.6 Interloking Pipes 4.7 Sine Rule Notes nd

More information

SECOND HARMONIC GENERATION OF Bi 4 Ti 3 O 12 FILMS

SECOND HARMONIC GENERATION OF Bi 4 Ti 3 O 12 FILMS 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

More information

Table of Content. c 1 / 5

Table of Content. c 1 / 5 Tehnil Informtion - t nd t Temperture for Controlger 03-2018 en Tble of Content Introdution....................................................................... 2 Definitions for t nd t..............................................................

More information

Comparing the Pre-image and Image of a Dilation

Comparing the Pre-image and Image of a Dilation hpter Summry Key Terms Postultes nd Theorems similr tringles (.1) inluded ngle (.2) inluded side (.2) geometri men (.) indiret mesurement (.6) ngle-ngle Similrity Theorem (.2) Side-Side-Side Similrity

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

THE INFLUENCE OF MODEL RESOLUTION ON AN EXPRESSION OF THE ATMOSPHERIC BOUNDARY LAYER IN A SINGLE-COLUMN MODEL

THE INFLUENCE OF MODEL RESOLUTION ON AN EXPRESSION OF THE ATMOSPHERIC BOUNDARY LAYER IN A SINGLE-COLUMN MODEL THE INFLUENCE OF MODEL RESOLUTION ON AN EXPRESSION OF THE ATMOSPHERIC BOUNDARY LAYER IN A SINGLE-COLUMN MODEL P3.1 Kot Iwmur*, Hiroto Kitgw Jpn Meteorologil Ageny 1. INTRODUCTION Jpn Meteorologil Ageny

More information

(a) A partition P of [a, b] is a finite subset of [a, b] containing a and b. If Q is another partition and P Q, then Q is a refinement of P.

(a) A partition P of [a, b] is a finite subset of [a, b] containing a and b. If Q is another partition and P Q, then Q is a refinement of P. Chpter 7: The Riemnn Integrl When the derivtive is introdued, it is not hrd to see tht the it of the differene quotient should be equl to the slope of the tngent line, or when the horizontl xis is time

More information

SECTION A STUDENT MATERIAL. Part 1. What and Why.?

SECTION A STUDENT MATERIAL. Part 1. What and Why.? SECTION A STUDENT MATERIAL Prt Wht nd Wh.? Student Mteril Prt Prolem n > 0 n > 0 Is the onverse true? Prolem If n is even then n is even. If n is even then n is even. Wht nd Wh? Eploring Pure Mths Are

More information

On the Scale factor of the Universe and Redshift.

On the Scale factor of the Universe and Redshift. On the Sle ftor of the Universe nd Redshift. J. M. unter. john@grvity.uk.om ABSTRACT It is proposed tht there hs been longstnding misunderstnding of the reltionship between sle ftor of the universe nd

More information

, g. Exercise 1. Generator polynomials of a convolutional code, given in binary form, are g. Solution 1.

, g. Exercise 1. Generator polynomials of a convolutional code, given in binary form, are g. Solution 1. Exerise Genertor polynomils of onvolutionl ode, given in binry form, re g, g j g. ) Sketh the enoding iruit. b) Sketh the stte digrm. ) Find the trnsfer funtion T. d) Wht is the minimum free distne of

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

Novel Fiber-Optical Refractometric Sensor Employing Hemispherically-Shaped Detection Element

Novel Fiber-Optical Refractometric Sensor Employing Hemispherically-Shaped Detection Element Novel Fier-Optil Refrtometri Sensor Employing Hemispherilly-Shped Detetion Element SERGEI KHOTIAINTSEV, VLADIMIR SVIRID Deprtment of Eletril Engineering, Fulty of Engineering Ntionl Autonomous University

More information

Review Topic 14: Relationships between two numerical variables

Review Topic 14: Relationships between two numerical variables Review Topi 14: Reltionships etween two numeril vriles Multiple hoie 1. Whih of the following stterplots est demonstrtes line of est fit? A B C D E 2. The regression line eqution for the following grph

More information

1 This question is about mean bond enthalpies and their use in the calculation of enthalpy changes.

1 This question is about mean bond enthalpies and their use in the calculation of enthalpy changes. 1 This question is out men ond enthlpies nd their use in the lultion of enthlpy hnges. Define men ond enthlpy s pplied to hlorine. Explin why the enthlpy of tomistion of hlorine is extly hlf the men ond

More information

First compression (0-6.3 GPa) First decompression ( GPa) Second compression ( GPa) Second decompression (35.

First compression (0-6.3 GPa) First decompression ( GPa) Second compression ( GPa) Second decompression (35. 0.9 First ompression (0-6.3 GP) First deompression (6.3-2.7 GP) Seond ompression (2.7-35.5 GP) Seond deompression (35.5-0 GP) V/V 0 0.7 0.5 0 5 10 15 20 25 30 35 P (GP) Supplementry Figure 1 Compression

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

University of Sioux Falls. MAT204/205 Calculus I/II

University of Sioux Falls. MAT204/205 Calculus I/II University of Sioux Flls MAT204/205 Clulus I/II Conepts ddressed: Clulus Textook: Thoms Clulus, 11 th ed., Weir, Hss, Giordno 1. Use stndrd differentition nd integrtion tehniques. Differentition tehniques

More information

Green s Theorem. (2x e y ) da. (2x e y ) dx dy. x 2 xe y. (1 e y ) dy. y=1. = y e y. y=0. = 2 e

Green s Theorem. (2x e y ) da. (2x e y ) dx dy. x 2 xe y. (1 e y ) dy. y=1. = y e y. y=0. = 2 e Green s Theorem. Let be the boundry of the unit squre, y, oriented ounterlokwise, nd let F be the vetor field F, y e y +, 2 y. Find F d r. Solution. Let s write P, y e y + nd Q, y 2 y, so tht F P, Q. Let

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

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 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. 1 PYTHAGORAS THEOREM 1 1 Pythgors Theorem In this setion we will present geometri proof of the fmous theorem of Pythgors. Given right ngled tringle, the squre of the hypotenuse is equl to the sum of the

More information

The Emission-Absorption of Energy analyzed by Quantum-Relativity. Abstract

The Emission-Absorption of Energy analyzed by Quantum-Relativity. Abstract The mission-absorption of nergy nlyzed by Quntum-Reltivity Alfred Bennun* & Néstor Ledesm** Abstrt The uslity horizon llows progressive quntifition, from n initil nk prtile, whih yields its energy s blk

More information

LECTURE 14. Dr. Teresa D. Golden University of North Texas Department of Chemistry

LECTURE 14. Dr. Teresa D. Golden University of North Texas Department of Chemistry LECTURE 14 Dr. Teres D. Golden University of North Texs Deprtment of Chemistry Quntittive Methods A. Quntittive Phse Anlysis Qulittive D phses by comprison with stndrd ptterns. Estimte of proportions of

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

Translation symmetry, Space groups, Bloch functions, Fermi energy

Translation symmetry, Space groups, Bloch functions, Fermi energy 9/7/4 Trnsltion symmetry, Spe groups, Bloh funtions, Fermi energy Roerto Orlndo Diprtimento di Chimi Università di Torino Vi ietro Giuri 5, 6 Torino, Itly roerto.orlndo@unito.it Outline The rystllogrphi

More information

INTEGRATION. 1 Integrals of Complex Valued functions of a REAL variable

INTEGRATION. 1 Integrals of Complex Valued functions of a REAL variable INTEGRATION NOTE: These notes re supposed to supplement Chpter 4 of the online textbook. 1 Integrls of Complex Vlued funtions of REAL vrible If I is n intervl in R (for exmple I = [, b] or I = (, b)) nd

More information

April 8, 2017 Math 9. Geometry. Solving vector problems. Problem. Prove that if vectors and satisfy, then.

April 8, 2017 Math 9. Geometry. Solving vector problems. Problem. Prove that if vectors and satisfy, then. pril 8, 2017 Mth 9 Geometry Solving vetor prolems Prolem Prove tht if vetors nd stisfy, then Solution 1 onsider the vetor ddition prllelogrm shown in the Figure Sine its digonls hve equl length,, the prllelogrm

More information

Chapter 2. Typology of Polymers

Chapter 2. Typology of Polymers Chpter 2. Typology of Polymers 2.1 Types of bonds in Polymers 1. primry ovlent nd metlli bonds 2. hydrogen bonding 3. dipole intertion 2nd fore 4. dispersion fore (vn der Wls fore) 5. ioni bond e.g.) polyproltone,

More information

Studies on Neon irradiated amorphous carbon using X-ray Diffraction. technique

Studies on Neon irradiated amorphous carbon using X-ray Diffraction. technique Studies on Neon irrdited morphous rbon using X-ry Diffrtion tehnique A.Srkr *, K. Dsgupt #, P. Brt *@, P. Mukherjee *, D. Sthiymoorthy # * Vrible Energy Cylotron Centre, 1/AF Bidhn Ngr, Kolkt 700 064,

More information

State space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies

State space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies Stte spce systems nlysis (continued) Stbility A. Definitions A system is sid to be Asymptoticlly Stble (AS) when it stisfies ut () = 0, t > 0 lim xt () 0. t A system is AS if nd only if the impulse response

More information

Lecture Notes No. 10

Lecture Notes No. 10 2.6 System Identifition, Estimtion, nd Lerning Leture otes o. Mrh 3, 26 6 Model Struture of Liner ime Invrint Systems 6. Model Struture In representing dynmil system, the first step is to find n pproprite

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

Intermediate Math Circles Wednesday 17 October 2012 Geometry II: Side Lengths

Intermediate Math Circles Wednesday 17 October 2012 Geometry II: Side Lengths Intermedite Mth Cirles Wednesdy 17 Otoer 01 Geometry II: Side Lengths Lst week we disussed vrious ngle properties. As we progressed through the evening, we proved mny results. This week, we will look t

More information

Generalization of 2-Corner Frequency Source Models Used in SMSIM

Generalization of 2-Corner Frequency Source Models Used in SMSIM Generliztion o 2-Corner Frequeny Soure Models Used in SMSIM Dvid M. Boore 26 Mrh 213, orreted Figure 1 nd 2 legends on 5 April 213, dditionl smll orretions on 29 My 213 Mny o the soure spetr models ville

More information

Section 3.6. Definite Integrals

Section 3.6. Definite Integrals The Clulus of Funtions of Severl Vribles Setion.6 efinite Integrls We will first define the definite integrl for funtion f : R R nd lter indite how the definition my be extended to funtions of three or

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

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

Tutorial Worksheet. 1. Find all solutions to the linear system by following the given steps. x + 2y + 3z = 2 2x + 3y + z = 4.

Tutorial Worksheet. 1. Find all solutions to the linear system by following the given steps. x + 2y + 3z = 2 2x + 3y + z = 4. Mth 5 Tutoril Week 1 - Jnury 1 1 Nme Setion Tutoril Worksheet 1. Find ll solutions to the liner system by following the given steps x + y + z = x + y + z = 4. y + z = Step 1. Write down the rgumented mtrix

More information

] dx (3) = [15x] 2 0

] dx (3) = [15x] 2 0 Leture 6. Double Integrls nd Volume on etngle Welome to Cl IV!!!! These notes re designed to be redble nd desribe the w I will eplin the mteril in lss. Hopefull the re thorough, but it s good ide to hve

More information

8 THREE PHASE A.C. CIRCUITS

8 THREE PHASE A.C. CIRCUITS 8 THREE PHSE.. IRUITS The signls in hpter 7 were sinusoidl lternting voltges nd urrents of the so-lled single se type. n emf of suh type n e esily generted y rotting single loop of ondutor (or single winding),

More information

Lecture Summaries for Multivariable Integral Calculus M52B

Lecture Summaries for Multivariable Integral Calculus M52B These leture summries my lso be viewed online by liking the L ion t the top right of ny leture sreen. Leture Summries for Multivrible Integrl Clulus M52B Chpter nd setion numbers refer to the 6th edition.

More information

Acceptance Sampling by Attributes

Acceptance Sampling by Attributes Introduction Acceptnce Smpling by Attributes Acceptnce smpling is concerned with inspection nd decision mking regrding products. Three spects of smpling re importnt: o Involves rndom smpling of n entire

More information

Learning Objectives of Module 2 (Algebra and Calculus) Notes:

Learning Objectives of Module 2 (Algebra and Calculus) Notes: 67 Lerning Ojetives of Module (Alger nd Clulus) Notes:. Lerning units re grouped under three res ( Foundtion Knowledge, Alger nd Clulus ) nd Further Lerning Unit.. Relted lerning ojetives re grouped under

More information

912 o C 1400 o C 1539 o C α iron γ iron δ iron. liquid iron BCC FCC BCC

912 o C 1400 o C 1539 o C α iron γ iron δ iron. liquid iron BCC FCC BCC Polymorphism or Allotropy Mny elements or ompounds exist in more thn one rystlline form under different onditions of temperture nd pressure. This phenomenon is termed polymorphism nd if the mteril is n

More information

Formula for Trapezoid estimate using Left and Right estimates: Trap( n) If the graph of f is decreasing on [a, b], then f ( x ) dx

Formula for Trapezoid estimate using Left and Right estimates: Trap( n) If the graph of f is decreasing on [a, b], then f ( x ) dx Fill in the Blnks for the Big Topis in Chpter 5: The Definite Integrl Estimting n integrl using Riemnn sum:. The Left rule uses the left endpoint of eh suintervl.. The Right rule uses the right endpoint

More information

System Validation (IN4387) November 2, 2012, 14:00-17:00

System Validation (IN4387) November 2, 2012, 14:00-17:00 System Vlidtion (IN4387) Novemer 2, 2012, 14:00-17:00 Importnt Notes. The exmintion omprises 5 question in 4 pges. Give omplete explntion nd do not onfine yourself to giving the finl nswer. Good luk! Exerise

More information

Maintaining Mathematical Proficiency

Maintaining Mathematical Proficiency Nme Dte hpter 9 Mintining Mthemtil Profiieny Simplify the epression. 1. 500. 189 3. 5 4. 4 3 5. 11 5 6. 8 Solve the proportion. 9 3 14 7. = 8. = 9. 1 7 5 4 = 4 10. 0 6 = 11. 7 4 10 = 1. 5 9 15 3 = 5 +

More information

Jackson 2.26 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell

Jackson 2.26 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell Jckson 2.26 Homework Problem Solution Dr. Christopher S. Bird University of Msschusetts Lowell PROBLEM: The two-dimensionl region, ρ, φ β, is bounded by conducting surfces t φ =, ρ =, nd φ = β held t zero

More information

NEW CIRCUITS OF HIGH-VOLTAGE PULSE GENERATORS WITH INDUCTIVE-CAPACITIVE ENERGY STORAGE

NEW CIRCUITS OF HIGH-VOLTAGE PULSE GENERATORS WITH INDUCTIVE-CAPACITIVE ENERGY STORAGE NEW CIRCUITS OF HIGH-VOLTAGE PULSE GENERATORS WITH INDUCTIVE-CAPACITIVE ENERGY STORAGE V.S. Gordeev, G.A. Myskov Russin Federl Nuler Center All-Russi Sientifi Reserh Institute of Experimentl Physis (RFNC-VNIIEF)

More information

Global alignment. Genome Rearrangements Finding preserved genes. Lecture 18

Global alignment. Genome Rearrangements Finding preserved genes. Lecture 18 Computt onl Biology Leture 18 Genome Rerrngements Finding preserved genes We hve seen before how to rerrnge genome to obtin nother one bsed on: Reversls Knowledge of preserved bloks (or genes) Now we re

More information

Development of Failure Probability Analysis Method for. Concrete Piers of Multi-span Continuous Bridges using

Development of Failure Probability Analysis Method for. Concrete Piers of Multi-span Continuous Bridges using Development o Filure Probbility Anlysis Method or Conrete Piers o Multi-spn Continuous Bridges using the Probbilisti Cpity Spetrum Method Je Shin CHOI, Je Kwn KIM ABSTRACT When erthqukes our, strutures

More information

TIME AND STATE IN DISTRIBUTED SYSTEMS

TIME AND STATE IN DISTRIBUTED SYSTEMS Distriuted Systems Fö 5-1 Distriuted Systems Fö 5-2 TIME ND STTE IN DISTRIUTED SYSTEMS 1. Time in Distriuted Systems Time in Distriuted Systems euse eh mhine in distriuted system hs its own lok there is

More information

MATH Final Review

MATH Final Review MATH 1591 - Finl Review November 20, 2005 1 Evlution of Limits 1. the ε δ definition of limit. 2. properties of limits. 3. how to use the diret substitution to find limit. 4. how to use the dividing out

More information

Math 8 Winter 2015 Applications of Integration

Math 8 Winter 2015 Applications of Integration Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl

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

HS Pre-Algebra Notes Unit 9: Roots, Real Numbers and The Pythagorean Theorem

HS Pre-Algebra Notes Unit 9: Roots, Real Numbers and The Pythagorean Theorem HS Pre-Alger Notes Unit 9: Roots, Rel Numers nd The Pythgoren Theorem Roots nd Cue Roots Syllus Ojetive 5.4: The student will find or pproximte squre roots of numers to 4. CCSS 8.EE.-: Evlute squre roots

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

Application of the theory of compound cores for the assessment of stress pattern in the cross section of a strengthened beam column

Application of the theory of compound cores for the assessment of stress pattern in the cross section of a strengthened beam column IOP Conferene Series: Mterils Siene nd Engineering PAPER OPEN ACCESS Applition of the theory of ompound ores for the ssessment of stress pttern in the ross setion of strengthened bem olumn To ite this

More information

Part 4. Integration (with Proofs)

Part 4. Integration (with Proofs) Prt 4. Integrtion (with Proofs) 4.1 Definition Definition A prtition P of [, b] is finite set of points {x 0, x 1,..., x n } with = x 0 < x 1

More information

Section 1.3 Triangles

Section 1.3 Triangles Se 1.3 Tringles 21 Setion 1.3 Tringles LELING TRINGLE The line segments tht form tringle re lled the sides of the tringle. Eh pir of sides forms n ngle, lled n interior ngle, nd eh tringle hs three interior

More information

THE ASYMMETRY OF COASTAL WATER LEVEL RESPONSE TO LANDFALLING HURRICANES SIMULATED BY A THREE-DIMENSIONAL STORM SURGE MODEL

THE ASYMMETRY OF COASTAL WATER LEVEL RESPONSE TO LANDFALLING HURRICANES SIMULATED BY A THREE-DIMENSIONAL STORM SURGE MODEL THE ASYMMETRY OF COASTAL WATER LEVEL RESPONSE TO LANDFALLING HURRICANES SIMULATED BY A THREE-DIMENSIONAL STORM SURGE MODEL Mhun Peng *, Lin Xie nd Leonrd J. Pietrfes Deprtment of Mrine, Erth nd Atmospheri

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

Probability. b a b. a b 32.

Probability. b a b. a b 32. Proility If n event n hppen in '' wys nd fil in '' wys, nd eh of these wys is eqully likely, then proility or the hne, or its hppening is, nd tht of its filing is eg, If in lottery there re prizes nd lnks,

More information

Magnetically Coupled Coil

Magnetically Coupled Coil Mgnetilly Coupled Ciruits Overview Mutul Indutne Energy in Coupled Coils Liner Trnsformers Idel Trnsformers Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver..3 Mgnetilly Coupled Coil i v L

More information

1 Probability Density Functions

1 Probability Density Functions Lis Yn CS 9 Continuous Distributions Lecture Notes #9 July 6, 28 Bsed on chpter by Chris Piech So fr, ll rndom vribles we hve seen hve been discrete. In ll the cses we hve seen in CS 9, this ment tht our

More information

1 This diagram represents the energy change that occurs when a d electron in a transition metal ion is excited by visible light.

1 This diagram represents the energy change that occurs when a d electron in a transition metal ion is excited by visible light. 1 This igrm represents the energy hnge tht ours when eletron in trnsition metl ion is exite y visile light. Give the eqution tht reltes the energy hnge ΔE to the Plnk onstnt, h, n the frequeny, v, of the

More information

THE PYTHAGOREAN THEOREM

THE PYTHAGOREAN THEOREM THE PYTHAGOREAN THEOREM The Pythgoren Theorem is one of the most well-known nd widely used theorems in mthemtis. We will first look t n informl investigtion of the Pythgoren Theorem, nd then pply this

More information

AP CALCULUS Test #6: Unit #6 Basic Integration and Applications

AP CALCULUS Test #6: Unit #6 Basic Integration and Applications AP CALCULUS Test #6: Unit #6 Bsi Integrtion nd Applitions A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS IN THIS PART OF THE EXAMINATION. () The ext numeril vlue of the orret

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

DETERMINATION OF MECHANICAL PROPERTIES OF NANOSTRUCTURES WITH COMPLEX CRYSTAL LATTICE USING MOMENT INTERACTION AT MICROSCALE

DETERMINATION OF MECHANICAL PROPERTIES OF NANOSTRUCTURES WITH COMPLEX CRYSTAL LATTICE USING MOMENT INTERACTION AT MICROSCALE Determintion RevAdvMterSci of mechnicl 0(009) -7 properties of nnostructures with complex crystl lttice using DETERMINATION OF MECHANICAL PROPERTIES OF NANOSTRUCTURES WITH COMPLEX CRYSTAL LATTICE USING

More information

Journal of Chemical and Pharmaceutical Research, 2013, 5(12): Research Article

Journal of Chemical and Pharmaceutical Research, 2013, 5(12): Research Article Avilble online www.jopr.om Journl of Chemil nd Phrmeutil Reserh, 2013, 5(12):1283-1288 Reserh Artile ISSN : 0975-7384 CODEN(USA) : JCPRC5 Study on osion resistne of zin lloy oting of mehnil plting by eletrohemil

More information

Supporting Information

Supporting Information tom-thik Interlyer Mde of VD-Grown Grphene Film on Seprtor for dvned thium-sulfur tteries Zhenzhen Du 1, hengkun Guo 2, njun Wng 3, jun Hu 1, Song Jin 1, Timing Zhng 1, Honghng Jin 1, Zhiki Qi 1, Sen Xin

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

Electronic Supplementary Information (ESI) for:

Electronic Supplementary Information (ESI) for: Eletroni Supplementry Mteril (ESI) for RSC Advnes. This journl is The Royl Soiety of Chemistry 2015 Eletroni Supplementry Informtion (ESI) for: Novel physio-hemil mehnism of the mutgeni tutomeristion of

More information

Final Exam Review. [Top Bottom]dx =

Final Exam Review. [Top Bottom]dx = Finl Exm Review Are Between Curves See 7.1 exmples 1, 2, 4, 5 nd exerises 1-33 (odd) The re of the region bounded by the urves y = f(x), y = g(x), nd the lines x = nd x = b, where f nd g re ontinuous nd

More information

Computer-Aided Design

Computer-Aided Design Computer-Aided Design 4 (29) 792 8 Contents lists vilble t SieneDiret Computer-Aided Design journl homepge: www.elsevier.om/lote/d Feture-bsed rystl onstrution in omputer-ided nno-design Cheng Qi Yn Wng

More information

Dynamics of grain boundary motion coupled to shear deformation: An analytical model and its verification by molecular dynamics

Dynamics of grain boundary motion coupled to shear deformation: An analytical model and its verification by molecular dynamics PHYSICAL REVIEW B 78, 6416 28 Dynmis of grin boundry motion oupled to sher deformtion: An nlytil model nd its verifition by moleulr dynmis V. A. Ivnov* nd Y. Mishin Deprtment of Physis nd Astronomy, George

More information

Math 1B, lecture 4: Error bounds for numerical methods

Math 1B, lecture 4: Error bounds for numerical methods Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the

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

Spacetime and the Quantum World Questions Fall 2010

Spacetime and the Quantum World Questions Fall 2010 Spetime nd the Quntum World Questions Fll 2010 1. Cliker Questions from Clss: (1) In toss of two die, wht is the proility tht the sum of the outomes is 6? () P (x 1 + x 2 = 6) = 1 36 - out 3% () P (x 1

More information

ANALYSIS AND MODELLING OF RAINFALL EVENTS

ANALYSIS AND MODELLING OF RAINFALL EVENTS Proeedings of the 14 th Interntionl Conferene on Environmentl Siene nd Tehnology Athens, Greee, 3-5 Septemer 215 ANALYSIS AND MODELLING OF RAINFALL EVENTS IOANNIDIS K., KARAGRIGORIOU A. nd LEKKAS D.F.

More information

Discrete Structures Lecture 11

Discrete Structures Lecture 11 Introdution Good morning. In this setion we study funtions. A funtion is mpping from one set to nother set or, perhps, from one set to itself. We study the properties of funtions. A mpping my not e funtion.

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

Chapter 0. What is the Lebesgue integral about?

Chapter 0. What is the Lebesgue integral about? Chpter 0. Wht is the Lebesgue integrl bout? The pln is to hve tutoril sheet ech week, most often on Fridy, (to be done during the clss) where you will try to get used to the ides introduced in the previous

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 Fun Gme Properties Arrow s Theorem Arrow s Impossiility Theorem Leture 12 Arrow s Impossiility Theorem Leture 12, Slide 1 Rep Fun Gme Properties Arrow s Theorem Leture Overview 1 Rep 2 Fun Gme 3 Properties

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

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

Learning Partially Observable Markov Models from First Passage Times

Learning Partially Observable Markov Models from First Passage Times Lerning Prtilly Oservle Mrkov s from First Pssge s Jérôme Cllut nd Pierre Dupont Europen Conferene on Mhine Lerning (ECML) 8 Septemer 7 Outline. FPT in models nd sequenes. Prtilly Oservle Mrkov s (POMMs).

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

The influence of 2,2 -dipyridyl on non-formaldehyde electroless copper plating

The influence of 2,2 -dipyridyl on non-formaldehyde electroless copper plating Eletrohimi At 9 () 1789 179 The influene of, -dipyridyl on non-formldehyde eletroless opper plting Jun Li, Hrley Hyden, Pul A. Kohl Shool of Chemil & Biomoleulr Engineering, Georgi Institute of Tehnology,

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

Chemical Equilibrium

Chemical Equilibrium Chpter 16 Questions 5, 7, 31, 33, 35, 43, 71 Chemil Equilibrium Exmples of Equilibrium Wter n exist simultneously in the gs nd liquid phse. The vpor pressure of H O t given temperture is property ssoited

More information

MATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1. 1 [(y ) 2 + yy + y 2 ] dx,

MATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1. 1 [(y ) 2 + yy + y 2 ] dx, MATH3403: Green s Funtions, Integrl Equtions nd the Clulus of Vritions 1 Exmples 5 Qu.1 Show tht the extreml funtion of the funtionl I[y] = 1 0 [(y ) + yy + y ] dx, where y(0) = 0 nd y(1) = 1, is y(x)

More information

A Non-parametric Approach in Testing Higher Order Interactions

A Non-parametric Approach in Testing Higher Order Interactions A Non-prmetri Approh in Testing igher Order Intertions G. Bkeerthn Deprtment of Mthemtis, Fulty of Siene Estern University, Chenkldy, Sri Lnk nd S. Smit Deprtment of Crop Siene, Fulty of Agriulture University

More information