Downloaded By: [Georgia Technology Library] At: 16:30 24 May 2007 Crepnigato.Eal [email protected] Email: author. *Corresponding Li by properties anharmonic by dynamics the lattice of and and properties [5] Smith elastic by Parlinski the spectra and of phonon Jochym Previous calculations [3], recent relatives. Graham been and close have [2], Chang There Houska are by by the point and measurements with expansion constants melting thermal elastic HfC compounds includes known ZrC the interstitial highest on and work refractory the experimental superconductors highly have conventional of that Nb(N,C) family materials The large [1]. a structure of NaCl-type one is ZrC Introduction 1. 2507–2519 2007, June 17, No. 87, Vol. Magazine Philosophical h eut fteemaueet r eeal ngo gemn ihprevious with data agreement expansion thermal good K. expansion the 15–1600 in analyzed thermal We range generally calculations. report are recent temperature we with measurements and the paper these measurements over of this results measurements In The [7]. factor Balluffi the Debye–Waller using and and formation Simmons vacancy carbon of for method energies activation determining of intention eue eto ifato omauetetemlepnino r ihthe with ZrC of expansion thermal the measure to diffraction neutron used We .C LAWSON* C. A. eutdmntae h motneo namncefcsi eemnn the determining last in This effects material. refractory anharmonic highly of this importance point. of with melting the the ZrC with K 3700 agreement of demonstrates good of point in result K, melting point 4000 of the have melting estimate estimate We an measured to obtain calculations. and measurements rule recent our Lindemann the good of with in results and are the measurements dependence measurements these used temperature of previous the results the takes with The with that account. agreement into CTE analyzed stiffness the been elastic for the have procedure of new data a from The and ZrC factors diffraction. of neutron factors using Debye–Waller Gru and K 1600 constants to lattice 12 the measured have We hra xaso n tmcvbain fzirconium of vibrations atomic and expansion Thermal ¨ esneuto fsaeuiga xsigpoeuefrteDebye–Waller the for procedure existing an using state of equation neisen o lmsNtoa aoaoy o lms M855 USA 87545, NM Alamos, Los Laboratory, National Alamos Los ( eevd1 eebr20;acpe nrvsdfr 2Jnay2007 January 12 form revised in accepted 2006; December 13 Received ron ainlLbrtr,Agne L649 USA 60439, IL Argonne, Laboratory, National Argonne SN17–45pitIS 4864 online 1478–6443 print/ISSN 1478–6435 ISSN { z hoSaeUiest,Clmu,O 31,USA 43210, OH Columbus, University, State Ohio eateto aeil cec n Engineering, and Science Materials of Department eateto aeil cec n Engineering, and Science Materials of Department os tt nvriy os,I 32,USA 83725, ID Boise, University, State Boise y aeil cec n ehooyDivision, Technology and Science Materials x y , nes usdNurnSuc Division, Source Neutron Pulsed Intense .P BUTT P. D. , O:10.1080/14786430701227548 DOI: http://www.tandf.co.uk/journals abd o10 K 1600 to carbide hlspia Magazine Philosophical yz .W RICHARDSON W. J. , ß 07Tyo Francis & Taylor 2007 x tal et n ULI JU and ) [6]. . tal et { [4]. . Downloaded By: [Georgia Technology Library] At: 16:30 24 May 2007 icnu abd odro atcedaee 20–30 diameter particle of powder carbide Zirconium Experiment 2. paper. this to appendix the in eetrbnso PD hra h o-eprtr esrmnsprovided measurements low-temperature the whereas with GPPD, data of banks detector and 20 made between temperatures were to heated measurements tube alumina High-temperature an 1300 with K. 300 furnace helium special National and a cycle Argonne using closed 15 at a using (IPNS) between made source refrigerator were neutron measurements pulsed temperature Low intense Laboratory. the at (GPPD) ometer disks. the of centre the from cut were ZrO of reduction carbothermic Mie–Gru the on based procedure new a with 2508 hn10pmoye.Dsso prxmt iesos9 mdiameter mm 90 dimensions less approximate contain of to combustion Disks and oxygen. analysis ppm LECO 100 of than combination a by determined was hc eepeae yhtpesn h odra 2100 at powder the pressing hot by prepared were thick novnetyln ewe 0ad300 and 20 between long inconveniently eprtrswr eemndwt tPR hroope n r eivdt be to believed are and thermocouples to Pt–PtRh accurate with determined were Temperatures airto icniut t30K h piie aaeesaetevibrational the with an expansion, are on the agreement parameters based thermal accommodate optimized was fair of to The data modified characteristic and K. constant appendix, 300 temperature the lattice at [2] from ZrC discontinuity measurements (A3) are the calibration equation They of earlier temperature. of analysis versus optimization with The plotted [6]. ZrC agreement of calculations constants good lattice the in shows 1 Figure results Experimental 3. of (GSAS) system analysis and structure [11]. general Diffraction of Dreele the [8–10]. dependence Von diffraction with and temperature neutron Larson analyzed the pulsed of were by analysis data obtained the factors for Debye–Waller developed technique analysis hoeia est.Dfrcinsmlso prxmt iesos5 dimensions approximate of samples Diffraction density. theoretical eprtr,adti a eoe nteaayi rcdr.Asecond A procedure. temporary room analysis by 900 introduced at the at was calibration in furnace results diffractometer the factor removed of Debye–Waller the was failure the in in this discontinuity discontinuity and small temperature, a inevitably, n parameter a and ifato esrmnswr aeo h eea ups odrdiffract- powder purpose general the on made were measurements Diffraction h eino h unc emte esrmnswt nythe only with measurements permitted furnace the of design The nahlu topee stetemltm-osato hsfraeis furnace this of time-constant thermal the As atmosphere. helium a in C 90 5 C. n h ihrresolution higher the and hti hrceitco h eprtr eedneo the of dependence temperature the of characteristic is that 2 ,adti a comdtdi h standard the in accommodated was this and C, narfatr ea unc.Tepwe material powder The furnace. metal refractory a in .C Lawson C. A. ,dt eentcletdi htrange. that in collected not were data C, ¨ esneuto fsaeta sexplained is that state of equation neisen tal et 148 .  h Gru the , eetrbns hr was, There banks. detector m a rprdvia prepared was m ,40 s onear to psi 4000 C, ¨ esnconstant neisen 5 0mm 30 5mm 15 90 3 , Downloaded By: [Georgia Technology Library] At: 16:30 24 May 2007 hra xaso eotdb 6.I h nlssw aeue h value the used have we analysis calculated the the plotted In also have [6]. we by comparison, the For reported and B expansion, 2. thermal expansion figure the in calculate thermal to shown used is are parameters result fitted The 1. figure to expansion thermal tm lte esstmeaueuigasadr eiiinof definition standard a using temperature versus plotted atoms constant. lattice low-temperature the from derived stiffness: shown Also fit. the Li of of error points the calculated and (A3) and equation [2] to Houska fit of ZrC, points of experimental constants Lattice the are 1. Figure ¼ ¼ .() h ro in error The 4.9(5). iue3sostefte ensur hra ipaeet fteZ n C and Zr the of displacements thermal mean-square fitted the shows 3 Figure h euto h piiainfrZCis ZrC for optimization the of result The 2.23 hra xaso n tmcvbain fzroimcriet 60K 1600 to carbide zirconium of vibrations atomic and expansion Thermal 10 12 ye cm dynes   oee,teerro h i sqiesal ssonin shown as small, quite is fit the of error the However, . 2 slre elcigteisniiiyo h ihtemperature high the of insensitivity the reflecting large, is ae rmteltrtr 2 ] n au of value a and 4], [2, literature the from taken ¼ u 2 1 B u @ @ x 2 T B þ P u 3 þ y 2  þ CTE @ @ 1 1 u nV n z 2 ¼ 2(0 and K 627(30) : T : tal et h u 2 [6]. . i : ¼ .22 with 1.42(2) 2509 V ð ð 2 1 m Þ Þ Downloaded By: [Georgia Technology Library] At: 16:30 24 May 2007 2510 eto ifato.As hw steaeaedslcmn uv acltdb Li by calculated curve displacement average the is shown Also diffraction. neutron iue3 ensur tmctemldslcmn ( displacement thermal atomic Mean-square 3. Figure to fit a from derived ZrC for Li by (CTE) calculated expansion curve thermal the is of shown coefficient Also Volume (A3). equation 2. Figure .C Lawson C. A. tal et h . u tal et 2 i o radCi r esrdby measured ZrC in C and Zr for ) [6]. . tal et [6]. . Downloaded By: [Georgia Technology Library] At: 16:30 24 May 2007 ecso the of dences nraoal gemn ihtevleo 2 on rmtetemlexpansion. thermal the from found K 627 of value the with agreement reasonable in needn ftemperature: of independent o h aa u h ey oe isaebte,a eemndb h values the model by determined Einstein as the better, tested are also fits we model Zr; Debye and the C but reduced both data, of for C used the is for model Debye The eprtrs r ut ifrn,bttesrn osat r bu h same, the about are constants spring of the ratio: independent the but by different, is shown subsequently as quite was carbon are refinement of temperatures, this fraction and 1.00(1), atom is off. the value switched average that the found temperature; we refinements, osat n hi eprtr eedneoe h ag –0 .Te made They K. 4–300 range elastic crystal the single over the dependence 1. obtain table temperature to in methods their shown ultrasonic and also used constants [3] is Graham temperature and Debye–Waller Chang reported his constants; lattice dependent: o h vrg Gru average the For h vrg ey eprtr sfudfo h vrg pigconstant: spring average the from found is temperature Debye average The osat n ey–alrfcoso r otevr ihtmeaueof lattice temperature the high measure very to the diffraction to X-ray ZrC used of [2] factors Houska present Debye–Waller 1700 the 1. with and compared table constants and shown in are investigators results previous by obtained Data work previous with Comparison the 4. and ZrC, of expansion thermal the for responsible contribute. itself not do by atoms is carbon atoms Zr Gru the atomic two the between elce hsfauei iwo h ag ro of error large the have of we but view capacities, in heat the feature of this dependences temperature neglected the track to Gru dependent atomic The ecndtrieaoi Gru atomic determine can We h w tmcvbainlfeunis smaue yteDebye the by measured as frequencies, vibrational atomic two The .W eemndtetemlepninfrhsdt yalna i othe to fit linear a by data his for expansion thermal the determined We C. hra xaso n tmcvbain fzroimcriet 60K 1600 to carbide zirconium of vibrations atomic and expansion Thermal  2 Zr  o icnu,teDbetmeauei oehttemperature- somewhat is temperature Debye the zirconium, For . : ¼ ¨  esncntnsms npicpeb oehttemperature somewhat be principle in must constants neisen 484(18) ¼ ¨ esncntn ehv [12]: have we constant neisen   ¼ 0 ¼ þ Zr  cT C C 0.038(9) ¨ Zr esncntnssosta h namncmto of motion anharmonic the that shows constants neisen Zr Zr o abn efind we carbon, For . p p ð ð T  Zr T M ffiffiffiffiffiffiffiffiffi  M ffiffiffiffiffiffiffiffiffi Zr  Þþ Þþ ¼ C Zr Zr ¨ p p C esncntnsfo h eprtr depen- temperature the from constants neisen T þ ¼ þ C M ffiffiffiffiffiffiffiffi M ffiffiffiffiffiffiffiffiffi ;frcro,teteDbetmeaueis temperature Debye the the carbon, for K;  C C p 201)K na al tg fthe of stage early an In K. 1220(11)  C Zr c C ð Zr Zr T M ffiffiffiffiffiffiffiffi C C ¼ p Þ ð C T ¼ M ffiffiffiffiffiffiffiffi 0 Þ : 91 3 C : 4 ð ¼ 4 2 1 ð 9 Þ 680 : Þ Zr : C Zr ð ¼ ¼ 26 h infcn difference significant The . .Frzroim efind we zirconium, For 0. 1 Þ : K, 7 ð 5 Þ : 2511 ð ð ð ð 4 5 3 6 Þ Þ Þ Þ Downloaded By: [Georgia Technology Library] At: 16:30 24 May 2007 2512

Table 1. Thermoelastic properties of ZrC at 1000 K.

BB1 dB/Dt CTE (dynes cm2 )(K1 )(K1 ) Â (K)

Calculation [5] 2.3 1012 Lawson C. A. Calculation [6] 21.7 106 1.38 747 Elastic (at 300 K) [3] 2.22 1012 6.25 105 19.2 106 2.8 714 X-ray lattice constants [2] 22.4 106 a 6 Neutron lattice constants 22.5 10 1.42(2) 4.9(5) ÂCTE ¼ 627(30)

X-ray al et Debye–Waller [2] 1.6 614–0.021 T Neutron . a Debye–Waller  ¼ 1.7 ÂZr ¼ 484 0.038 T ÂC ¼ 1220(11) Â ¼ 680(26) Heat capacity [13] 495(10) aPresent work. Downloaded By: [Georgia Technology Library] At: 16:30 24 May 2007 siae T ymkn ierftt hi rp ftemlepnin We expansion. thermal of graph necessary have their for We to provide expansion. fit value thermal to linear a a a K report determined 4–500 making not by also did range CTE but a the data, estimated elastic in their to measurements corrections expansion thermal alrfcosfrtezroimadcro tm,adteeaei excellent in are of their definition these divided our have and with we 3; accordance atoms, figure in in carbon three shown by as and values results, zirconium experimental our the with agreement for factors Waller oprsno u eut o u ie nteapni,t eut rmthe from results to appendix, the in reliable. be given of not Cu. values may for that results show literature, our of comparison eedneo h lsi osat.CagadGaa 3 lorpre value a reported also [3] Smith Graham temperature. and Debye Chang elastic constants. the for elastic the of dependence oprsno u atc osat ihtoeotie yHuk 2 n the and [2] Houska be by a would obtained includes 1 Li actual) those Figure of credible. with than be results constants lower to expansion large lattice (measured thermal too our us 10% to of seems order comparison error an the such of and required, error temperature the a Li for by value calculated comparison. a for derived included is and [13] expansion temperature thermal the Debye get the to measured able Gru we were order: they Li these, reverse [3]. From dependences. Graham volume Gru and with their Chang agreement and the frequencies good by obtained confirmed obtained and and Jocym Smith results that spectrum of spectra. spectrum phonon the Their measured in the the scattering. found calculated were [5] neutron anomalies Parlinski no using and measurements, experimentally ultrasonic ZrC of erytietevlefor value the q twice executing nearly atoms atomic carbon cages. light zirconium the the their of inside but by think vibrations therefore given frequency agreement, may high interpretation We general heat the carbon-like. are and to in above According Debye–Waller are structure. for finer DOSs used Li much model The show Debye analyses. calculations the from data thermal DOS capacity Debye for the average with temperature an together Li Debye to by leads measured and this the shown, with already lower have agrees expansion. much we the Debye–Waller that with As the accordance temperature carbon. in but higher, for capacity, considerably mass is heat the carbon with low-temperature for agreement good temperature from in is temperature Zr for temperature Debye Debye–Waller neutron The 1. table ¼ tal et ¨ ¨ h al hw htormaue hra xaso sabthge hnteone the than higher bit a is expansion thermal measured our that shows table The The [5] Parlinski and Jocym by calculated was (DOS) states of density vibrational The in reported temperatures Debye the among disparity considerable is There ( esncntn.Alo hs eut r olce ntbe1 calorimetric A 1. table in collected are in results proceeded these work of present All The constant. expansion. neisen thermal the and constant neisen @ ln 6,nal l h rqece eo 0Tzaezroimlk n eryall nearly and zirconium-like are THz 10 below frequencies the all nearly [6], . / hra xaso n tmcvbain fzroimcriet 60K 1600 to carbide zirconium of vibrations atomic and expansion Thermal @ ausi al r ngnrlygo gemn.Tevleof value The agreement. good generally in are 1 table in values tal et ln V 6.I iue4w hwtevbainlDSfo Li from DOS vibrational the show we 4 figure In [6]. . ) T h ouevraino h Gru the of variation volume the , tal et 6.I hsdsrpnywr nteeprmna side, experimental the on were discrepancy this If [6]. . B n qain()sget ag au for value large a suggests (1) equation and , q tal et tal et 1 bandfo hra xaso measurements expansion thermal from obtained ( dB/dT 4.Tecluae ukmdlsare with agrees modulus bulk calculated The [4]. . 6.Li [6]. . tal. et rmtergah ftetemperature the of graphs their from ) tal et 4 eemndtepoo spectrum phonon the determined [4] tal et 6 locluae h Debye– the calculated also [6] . h u 6 acltdtephonon the calculated [6] . 2 ¨ esncntn.However, constant. neisen i . tal et 2513 [6] . is Downloaded By: [Georgia Technology Library] At: 16:30 24 May 2007 etake We 00K u hi siaenget namncefcs n hyueasomewhat a use the they by and limited rule.) effects, Lindemann is anharmonic the neglects of carbides formulation estimate interstitial different their of but family K, (Li 4000 stiffness. important the of this dependence temperature in point melting on fZC ttrsotta ti eyipratt aetetemperature the take to important very is it that account: out into temperature interatomic turns melting Debye the the the the It estimate when of of to dependence melts 0.08) ZrC. 17] [16, (about material of previously can fraction derived a we point small equations basis, that used a have physical states is We established distance. vibration which more thermal high-melting rule, a [15]. of of with melting K family amplitude 4160 theory Lindemann a binary, melting a of the a for member apply point of a melting absence is known the highest ZrC In the [14], has K 3700 HfC materials. of point point melting a With point melting high and rule Lindemann 5. Li by data. calculated diffraction (DOS) neutron states from of derived density Phonon DOS model 4. Figure 2514 eedn,te ehv nipii qainfor equation implicit an have we and then criterion, dependent, Lindemann the to according melting for (7) equation In ueial.I ssle xlctyb Lawson by explicitly solved is It numerically. %hge hnteosre au.Inrn h eprtr eedneof dependence temperature the Ignoring value. observed the than higher 8% get T ntebsso h aapeetdi iue3 eestimate we 3, figure in presented data the of basis the On m ¼ 60K hc smr hntieteeprmna au.I per htthe that appears It value. experimental the twice than more is which K, 9600 f ¼ .8 ssgetdb iue6o asn[7.If [17]. Lawson of 6 figure by suggested as 0.08, f stertoo irtoa mltd oitrtmcdsac required distance interatomic to amplitude vibrational of ratio the is T melt .C Lawson C. A. ¼ f 2

2 = 3 tal et 3 h mk  2 tal et tal et B 6 loetmt etn on of point melting a estimate also [6] . Â . [16]. . 2 : tal et T melt

6 oprdt h Debye the to compared [6] . steaoi volume. atomic the is T htcnb solved be can that m ¼  00K hc is which K, 4000 stemperature is  ,we ð 7 Þ Downloaded By: [Georgia Technology Library] At: 16:30 24 May 2007 ilb rsne ndti n aiae yapiaint aafor data to that application data by expansion validated thermal Gru and the for with detail analyzed procedure often are in data new expansion presented Thermal a copper. be developed will have We copper to application and analysis expansion Thermal Appendix: under Science, BES-Materials Energy, of Laboratory, Department National U.S. Argonne W-31-109-ENG-38. at the Contract source National benefited by neutron has Energy funded pulsed work is of intense The which the Department Science. of of States use Office references. the United the the from the of and the one on Administration by with Security discussions help sponsored Nuclear illuminating for is Lashley for Jason research Ledbetter Dr. This Hassel and properties Dr. elastic thank of topic to pleased are We more Acknowledgments K. 3700 much to ZrC are of point melting atoms the atomic particular limiting in zirconium two for behaviour, responsible the anharmonic is the This softening, of atoms. elastic carbon dependence that the light temperature reported the indicates than previously The anharmonic with theory. factors agreement existing good Debye–Waller with in are and diffraction data neutron similar by with measured semimetals. work other should maybe form and HfC potential finite- TiC, the minimal the to for fitted that This quality we without surprising since constants. well, too predicted force properties, not be a a harmonic probably can in ZrC seek new is of accuracy to it properties high vibrational that, fitting is achieving temperature said of potential Having luxury in this be the property. of than minimal certain could form rather philosophy improvement description, a The potential 0th-order significant properties. robust sought that present other expect some they the deteriorating not set, with do we data achieved energies), DFT parameters. cohesive of avoid and number small least to the a Furthermore, with interactions, strength’’. over-fitting angular the ‘‘hopping of of the representation in problem potential, 2nd-moment interactions the the like angular looks with still form but potential overall the where potential, ucinlter DT acltos hste eeoe a density in developed lattice, ZrC. they the in Thus of atom C calculations. properties or Zr (DFT) the the theory of fit displacement functional distribution small spatial a adequately the by at induced to looks forces one of when infeasible strong quite was indeed are it interactions Angular found ‘‘hopping Zr), isotropic they with pure form strengths’’, of atom) Li (embedded that potential by of 2nd-moment studied simple were one-third a consequences Using about whose is interactions covalent conductivity strong electrical with but (its semimetal a is ZrC summary and Discussion 6. ntepeetwr,w on htltiecntnsadDbeWle factors Debye–Waller and constants lattice that found we work, present the In constants force DFT to already (fitted parameters of number small the to Due hra xaso n tmcvbain fzroimcriet 60K 1600 to carbide zirconium of vibrations atomic and expansion Thermal pseudo ¨ esnequation neisen 2nd-moment tal et 2515 [6]. . Downloaded By: [Georgia Technology Library] At: 16:30 24 May 2007 B ouu n Gru and the modulus example, for expansion thermal Si, the in drive temperatures. structure materials; low open all at the negative for from arising work modes not vibrational extra will assumptions Our capacity. fsae[8 19]: [18, state of 2516 hsgvsaqartceuto for equation quadratic a gives This with cln h constant the scaling rnvrevbain 2,2]ta oseeg.I esrdha aaiydt are data capacity heat Gru measured thermodynamic If usual energy. the does then than available, 21] [20, vibrations transverse oehtfo h calorimetric the from somewhat o nw navne easm htteeqatte r ie yteDbemodel Debye the by given are quantities these that that that assume and we found advance, in is temperature. known not it of function materials, weak real a for rather is made but is constant analysis this When temperature where h Gru the httetemlepnini rprinlto proportional is expansion thermal the that h quantity The ie by given ¼ sarsl ftevlm hneidcdb h hra xaso,tebulk the expansion, thermal the by induced change volume the of result a As ic h eprtr aito fteitra nryadha aaiyi usually is capacity heat and energy internal the of variation temperature the Since B 0 (1 P ¨ esnconstant. neisen stepressure, the is ssrcl osatwt eprtr.W loatcpt that anticipate also We temperature. with constant strictly is  T D and ) steAnderson–Gru the is aigtetmeauedrvtv tcntn oue tfollows it volume, constant at derivative temperature the Taking . ¨ ð esncntn aywt eprtr codn to according temperature with vary constant neisen T @ @ ytertoo h esrdha aaiyt h ey heat Debye the to capacity heat measured the of ratio the by V ¼ T , U  U , P 0 suulytknfo h ey oe ihcharacteristic with model Debye the from taken usually is steitra nryand energy internal the is (1 ¼ ¼ , þ Þ¼ V B 0  0 0 ð 0 .C Lawson C. A. qT q B ð 1 ð D 1 1 1 0 þ P ð þ ¨ eas h hra xaso eed oeon more depends expansion thermal the because ,where ), 1 esncntn.Tetemlepnini now is expansion thermal The constant. neisen q q ihtesolution the with p ¼ @ @ ¼ P T 1 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 T T B 0 nV n BV Þ Þ þ C ¼ T Þ C C 4 @ @ Þ q T B : T ¨ V : tal et esncntn a ecmue by computed be can constant neisen C U 2 , and B C v P V i.e. , T 0 ð : ð . T 0 1 B q ,  0 0 r eie by: defined are ð C 1 Þ sacntn,nwkonas known now constant, a is T 0 C = T B Þ 0 V T 0 Þ : sntprecisely not is  a differ may ð ð A1 A2 Þ Þ Downloaded By: [Georgia Technology Library] At: 16:30 24 May 2007 novstepesr derivative pressure the involves atc r vial o aaaayi.Gvnthat Given analysis. data for available are tactics þ nerlo qain(A2): equation of integral the by derived of set copper a Given for (CTE) [22]. Hahn expansion of thermal data the of to (A3) coefficient equation Volume of application A1. Figure hra xaso,oeestimates one expansion, thermal for negligible is (A4) equation in term last The n calculates and h i hw nfgr 1i ut odcnieigtesmlct ftemodel. the of simplicity the considering copper. good for [22] quite Hahn is of A1 gives data figure optimization high-quality The in the shown against fit (A3) The equation test now We oe ntbeA.I oe ,a xeietlvlefor value experimental an 2, model In A1. table in 1 model h ieaueare literature the , . tro temperature. room at 0.2  eedn nhwmc skoni dac bu ie aeil several material, given a about advance in known is much how on Depending nadto otedfnto of definition the to addition In and hra xaso n tmcvbain fzroimcriet 60K 1600 to carbide zirconium of vibrations atomic and expansion Thermal o atc osat,teotmzto a edn ihaftt the to fit a with done be can optimization the constants, lattice for ; V ð T q , esstmeauevle,aftt hseuto a eotmzdfor optimized be can equation this to fit a values, temperature versus and   , D , ¼ ysmlaeu ouino qain A)ad(4.Ti is This (A4). and (A1) equations of solution simultaneous by Þ¼ 4 2]and [23] K 343  CTE V K 0 ¼ 0 ! ¼ 1 2.()and 324.1(5) K þ 0 K þ ¼ Z hr sarltosi 1]between [19] relationship a is there , 0 0 1 ( T rmSaysvrino h ey oe [25] model Debye the of version Stacy’s from @ B 1 / ¼ q @ P .8 [24]. 1.985 p ): 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ @ @ 1n 1n .9() with 1.993(2), 4 C T V 2 V 4 C and ð  T : ,  n o uhstevalue the has Cu for and Þ r eemndfo the from determined are = B 0 V ¼ stknfo the from taken is 0 .02.Vle in Values 4.00(2). d : and q ð ð 2517 that A4 A3 Þ Þ Downloaded By: [Georgia Technology Library] At: 16:30 24 May 2007 References altogether. r .22 . . 1.1 3.8 2.8 1.7 3.7 4.9(5) 1.42(2) 2 1 ZrC ZrC u11932 .92 . . . 1.4 0.0 3.9 4.19(2) 5.4 5.5 literature, 4.9 3.99(2) e d c 3.99(2) b 1.993(2) a 3 2 1 Cu Cu Cu 2518 1]AC asn .Mrie,JA Roberts, J.A. Martinez, B. Lawson, A.C. [16] 1]DP utadTC alc r,J m ea.Soc. Ceram. Am. J. Sr., Moffatt, Wallace W.G. T.C. [15] and Gmelin, Butt E. D.P. Weber, [14] W. Roedhammer, P. LAUR [13] Report Laboratory National Alamos Wallace, Los D.C. Dreele, [12] Von R.B. and Martinez, B. Larson Roberts, A.C. A. [11] J. Lawson, A.C. [10] aeavlefor value a have ple oes13t oprad12t r.W idta h eie ausfor values derived the that find have We we ZrC. A1 to table 1–2 and In copper circumstance. to by 1–3 suggested models be applied may routes Other respectively. etk neprmna au for value experimental an take we 1]AC asn hl a.B Mag. Phil. Lawson, A.C. [17] hn n rhm[3]. Graham and Chang [29]. (calculation) K 300 Tolpadi, ail n mt [26]. Smith and Daniels hn n utrn[30]. Hultgren and Chang Holtzapfel 8 ..Lwo,A ilas ..Goldstone, J.A. Williams, A. Lawson, A.C. [8] Goldschmidt, H.J. [1] 9 ..Lwo,J .Glsoe .Cort, B. Goldstone, A. J. Lawson, A.C. [9] 4 ..Sih .WkbysiadM otle,in Mostoller, M. and Wakabayashi Phys. Appl. N. J. 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A. K T K 0 Peu rs,NwYr,16) .88. p. 1967), York, New Press, (Plenum 25 4 n s qain A)ad(4 for (A4) and (A1) equations use and 0 a  5 (1964). 359 tal. et tal. et K tal. et n oaodueo h ey model Debye the of use avoid to and tal. et tal. et 37 93 0 .Ce.Phys. Chem. J. , 117 tal et 78(1966). 3778 hl a.B Mag. Phil. , 15 02(2003). 9072 hl a.B Mag. Phil. , .Ls-omnMetals Less-common J. , ,in K K 6 (2000). 265 2(1960). 52 calc 0 . 0 rmeuto A) nmdl3, model In (A4). equation from 76 uecnutvt nd n f-Band and d- in Superconductivity ciiePoesn:Mtosand Methods Processing: Actinide Gnu ulsig Schenectady, Publishing, (Genium 49(1993). 1409 4.19(2) WrdSinii,NwJersey, New Scientific, (World liter e 80 82 64 d d 3(2000). 53 87(2002). 1837 8 (1976). 581 4.7 calc 167 q calc 5 (1991). 353 2.1 . 3.4 0.2 0.2 q and q liter c b , Downloaded By: [Georgia Technology Library] At: 16:30 24 May 2007 2]C Kittel, C. [23] 3]YA hn n .Hlge,J hs Chem. Phys. J. Hultgren, R. and Chang F Y.A. Phys. [30] J. Tolpadi, S. [29] Phys. Appl. J. Hahn, A. T. [22] 2]OL nesn .Py.Ce.Solids Chem. Phys. J. Anderson, O.L. [20] Anderson, O.L. [19] 2]WB ozpe,M ati n .Sees .Py.Ce.Rf Data Ref. Chem. Phys. Res. J. Press. Sievers, High W. Holzapfel, and W.B. Hartwig [28] Rev. M. Phys. Holzapfel, Smith, W.B. C.S. [27] and Daniels W.B. [26] Voc D L. Phys. [25] J. White, G.K. [24] Rev. Phys. Gibbons, D.F. [21] Poirier, J. [18] www.itl.nist.gov/div898/handbook/pmd/section6/pmd641.htm). nvriyPes e ok 1996). York, New Press, University 1991). Cambridge, hra xaso n tmcvbain fzroimcriet 60K 1600 to carbide zirconium of vibrations atomic and expansion Thermal ˇ do ..PiiradGD rc,A.Miner. Am. Price, G.D. and Poirier J.P. adlo, nrdcint oi tt Physics State Solid to Introduction nrdcint h hsc fteErhsInterior Earth’s the of Physics the to Introduction qain fSaeo oisfrGohsc n eai Science Ceramic and Geophysics for Solids of State of Equations 4 6 18(1974). 2138 00(1973). 2070 112 3 (1959). 136 41 16 06(90 Dt r vial niea:http:// at: online available are (Data (1970) 5096 1(1998). 81 12 1(1959). 41 111 t d Wly e ok 96,p 126. p. 1996), York, New (Wiley, ed. 7th , 1 (1958). 713 69 12(1965). 4162 85 9 (2000). 390 CmrdeUiest Press, University (Cambridge 30 1 (2001). 515 (Oxford 2519