J. Electroanal. Chem., 391 (1985) 41-75 75 reiveslE aiouqeS ,.A.S ennasuaL - detnirP ni ehT sdnalrehteN REPLY TO COMMENTS BY ALMOND AND WEST ON Na &ALUMINA IMMITTANCE DATA ANALYSIS .J SSOR DLANODCAM dna .B.G KOOC Department fo ,scisyhP ytrsreumU of North Caroltna, Chapel CNHill, 27514 ).A.S.U( devieceR( ht03 yraunaJ ;5891 ni lanif mrof ht52 lirpA )5891 TCARTSBA eW nigeb yb gnizirammus eht dnuorgkcab fo eht tneserp .ysrevortnoc esuaceB eht tseW/dnomlA atad no aN animula-I/ era eht ylno dehsilbup langis-llams ca stluser rof eht wol erutarepmet egnar morf 38 K ot 151 ,K yeht era ylralucitrap tnatropmi dna evresed luferac dna lluf .sisylana eW tsrif e htssucsid atad ,sevlesmeht gnitniop tuo niatrec secneicifed ni ,meht dna neht erapmoc eht tahwemos evitcejbus sisylana sdohtem deyolpme yb dnomlA dna tseW htiw eht erom evitcejbo seno ew evah .desu ,txeN ew ssucsid etairporppa gnittif sledom dna wohs taht eht ledom desu yb dnomlA dna tseW si tnelaviuqe ot eno gnol desu ni eht cinoi rotcudnoc .dleif eW neht enimaxe eht sisylana dna noitaterpretni fo gnittif stluser ni emos .liated ehT niam lanigiro noitubirtnoc fo dnomlA dna tseW ni siht aera si rieht etelpmoc noitacifitnedi fo a retemarap pw ppa gnirae ni rieht gnittif ledom htiw eht egareva yllamreht detavitca gnippoh ,ycneuqerf .Hy ruO deliated noitanimaxe fo siht noitpmussa setactdni taht ereht ,si os ,raf on gnorts laciteroeht sisab rof ti dna on ylluf yhtrowtsurt latnemirepxe ecnedive rof ti .rehtie eW evah dezylana-er eht atad rof lla enin serutarepmet neewteb 38 K dna 151 K yb xelpmoc raenilnon tsael serauqs gnittif dna dnif taht retteb stif naht ew reilrae deniatbo rof eht eerht tsehgih serutarepmet era ,elbissop gnivig stluser ni siht egnar hcum resolc ot eht reilrae seno fo dnomlA dna .tseW .sselehtreveN lamreht noitavitca stolp llits tibihxe ,gnorts llew denifed skaerb dna seitiunitnocsid neewteb 201 K dna 011 ,K elbissop ecnedive rof a ekil-ssalg noitisnart ni siht .lairetam ,yllaniF rof nosirapmoc htiw erutuf stnemerusaem dna stluser rof aN animula-p ni eht wol erutarepmet ,emiger ew evah detalubat lla ruo gnittif .setamitse NOITCUDORTNI nI siht eton we dnopser ot the stnemmoc of dnomlA dna tseW ]l[ (hereafter )W/A no emos of the stluser of ruo tnecer paper no Na animula-?/ [2]. nI rieht stnemmoc W/A llac otni noitseuq emos of the kooC/dlanodcaM (hereafter M/C) sisylana sdohtem dna .stluser hguohtlA these srettam lliw eb addressed ni liated ,niereh we eveileb that a short laitini noissucsid of emos of the dnuorgkcab of the rettam lliw edivorp a lufesu evitcepsrep no .ti ,tsriF ew hsiw ot knaht Dr. ynohtnA tseW ecno erom [2] for yrev yldnik gniylppus the 1977 W/A K151-K38 Na animula& data ot .su nehW we deton that these data had neeb desu yb iagN dna mortS [3], dna that the suoirav reilrae sesylana of the data yb ,dnomlA ,tseW dna srekrow-oc ,demees ni ruo ,noinipo ot evael gnihtemos 03.30%/58/8270-2200 0 5891 reiveslE aiouqeS .A.S 85 ot eb ,derised eno of su (J.R.M.) detseuqer the data morf tseW ni March 1983 [4], retfA emos yranimilerp sisylana of the deviecer data was deirrac ,tuo Dr. tseW was detivni [4] ni enuJ 1984 ot eb a rohtua-oc of a paper we hoped ot etirw no the sisylana of these data. nI a rettel morf tseW deviecer ni rebmevoN 1984 after the paper had neeb yllaitnesse detelpmoc [5], he detcele ton ot eb a .rohtua-oc ,dnoceS we hsiw ot knaht dnomlA dna tseW for rieht tnecer stnemmoc ,]l[ hcihw evah detalumits su ot enimaxe-er ruo sisylana dna evah del ot the lufesu stluser dessucsid .woleb eW eveileb that a ylper ot the W/A stnemmoc elb~illetni dna elbaulav ot the lareneg reader seriuqer emos noissucsid of the data ni ,noitseuq erom nosirapmoc of the sisylana sdohtem desu yb W/A dna ,C/M dna rehtruf noissucsid dna noisnetxe of past gnittif .stluser hcuS noissucsid si detneserp ni the gniwollof work. THE Na B-ALUMINA DATA ehT data esirpmoc enin sets of langis-llams ac ecnattimda/ecnadepmi stluser yllareneg gnidnetxe morf 2ol ot 7 x lo6 Hz dna gninnaps the erutarepmet egnar morf 83 K ot 151 .K ehT lairetam was elgnis latsyrc Na animula-P nworg-tlem yb noinU ;edibraC sti yrtemoihciots-non noitcarf was nwonknu tub ti ylbaborp llef ni the egnar of 0.2 ot 0.3. tseW dna sih srekrow-oc evah reilrae dehsilbup data stolp dna noissucsid of emos of these data sets ni at tsael the nevele papers denoitnem ni refs. .61-6 htiW siht ynam papers detoved at tsael ni part ot the emas data, there ,si ni ruo ,noinipo elbaredisnoc palrevo gnoma ynam of .meht dnomlA dna tseW evah denoitnem [ 131 a = 2 K error ni rieht yllanimon 113 K data set. sihT stseggus that there yam eb siht hcum erutarepmet error ni lla enin sets. tuB ni noitidda ot elbissop smelborp htiw erutarepmet lortnoc there appear ot eb other ycarucca smelborp htiw the W/A data [2]. esehT smelborp are detartsulli ni the D-3 evitcepsrep stolp of .giF 1 for the 83 K W/A data. erugiF al shows that the tsewol ycneuqerf tniop tsegral( 1 )eulavlP si yltnerappa tnetsisnocni htiw the others. tI si tnatropmi ot eton that siht ylamona does ton show pu at lla ni the lanoitnevnoc stolp of - ”Z .sv )v(gol dna ’Z .sv ,)v(gol nwohs as snoitcejorp ni .giF !)a(l Here 2 = 2 + iZ” dna Y si the tnemerusaem .ycneuqerf nehW the suludom noitcnuf M = M’ + iM” = (iuC,)Z si dettolp for these emas data, two other smelborp appear hcihw were ton tnerappa ni the 2 .stolp Here ,C si the ecnaticapac of the ytpme gnirusaem llec dna u = .~2 ,tsriF we see that M’ reaches a mumixam dna neve snigeb ot decrease at the tsehgih ,seicneuqerf etiuq yrartnoc ot the roivaheb expected morf yna elbanosaer .ledom ,dnoceS na elbaicerppa hctilg appears at etaidemretni seicneuqerf where the srotnemirepxe detfihs morf eno epyt of gnirusaem egdirb ot .rehtona ,yltnedivE the noitarbilac of the two segdirb was tnetsisnocni raen rieht revo-ssorc .seicneuqerf niagA eciton that siht ,hctilg dna neve the suolamona M’ roivaheb at hgih -neuqerf ,seic does ton show pu ni lla the noitcejorp .stolp esehT dna other ralimis stluser tseggus that lla the data dluohs ton eb detsurt at the tsewol dna tsehgih seicneuqerf dna that, ni ,noitidda tahwemos suolamona stluser yam ylbissop eb expected morf gnittif of data hguorht the hctilg .noiger 59 Fig. 1. lanoisnemid-eerhT evitcepsrep stolp fo the 83 K W/A aN animula-lf .atad (a) ehT conjugate ecnadepmi ;*Z (b) the suludom ,noitcnuf M. NOSIRAPMOC OF SISYLANA SDOHTEM erehT era eerht tcnitsid stcepsa ot this .cipot ehT tsrif slaed htiw what ledom ot ;esu eht dnoces sevlovni woh tseb ot niatbo ledom retemarap setamitse yb -tnorfnoc gni ledom and data; and eht driht sevlovni eht tsom etairporppa yaw ot rehtruf ezylana and terpretni eht retemarap setamitse deniatbo morf data .gnittif eW shall discuss eseht srettam ni .nrut 60 Model selection erugiF 2 shows the tnelaviuqe tiucric ledom we proposed dna desu ni ruo lanigiro gnittif [2]. Here a tnatsnoc phase tnemele (CPE) ecnattimda si nevig yb Y = ,“)wi(,A where ,A dna n are tnedneped-erutarepmet .stnatsnoc sA denoitnem reilrae [2], sC dna R, are klub ;stnemele ni ,tsartnoc ,EPC si ylekil ot eb detaicossa htiw ecafretni effects. tnatsnoC phase esnopser was tsrif dessucsid yb ekcirF [17] dna retal yb eloC dna eloC [18]. A tnecer paper sredisnoc etairporppa noitaziretemarap of the EPC [19]. tI osla stniop tuo that the noitanibmoc of a EPC dna na laedi rotsiser ni lellarap sdael ot a depressed elcricimes arc (the CRAZ )esnopser nehw the ecnadepmi of the noitanibmoc si dettolp ni the xelpmoc .enalp hcuS esnopser was tsrif proposed yb eloC dna eloC [18] ni 1941 for xelpmoc cirtceleid tnatsnoc stolp dna yb eniavaR dna teuquoS [20] for gnitcudnoc smetsys ni 1973. stI ecnadepmi yam eb ,nettirw for ,elpmaxe as Z ZARC=RB/[l+(iWT)n]~ where T = ( RBAO)‘ln. ehT yratinu R,, 7, n noitaziretemarap si netfo dnuof roirepus ot the etisopmoc R,, A,, n noitaziretemarap of .giF 2, see noissucsid .woleb Now W/A evah yllaitnesse desu .giF 2 tuohtiw ,EPC ni rieht gnittif ro/dna ;snoissucsid suht yeht take on tnuocca of ycneuqerf-wol ecafretni effects, -ucitrap ylral tnatropmi for rieht erutarepmet-rehgih data sets. More ,ylesicerp W/A evah desu ylno the laer part of the ecnattimda of the CRAZ ledom hcihw( sedulcni the etercsid EPC-rotsiser )noitanibmoc ni refs. 9, 11, dna 12. yehT evah nevig the lluf CRAZ ,ecnattimda sulp that of the lacirtemoeg ,ecnaticapac ,sC ni refs. 10, 13, 15, dna 16. hguohtlA yeht won noitnem ni rieht stnemmoc ]l[ that rieht reilrae sisylana deyolpme a elgnis ,EPC enon of rieht snoitauqe was so deifitnedi .161-9[ nI ref. 10 yeht wrote rieht detubirtsid tnemele ecnattimda as Y = w”[ A + ,]Bi neht dettif-evruc ot niatbo A dna B retemarap setamitse ,yletarapes dna did ton eton that A dna B are ton tnednepedni tub tsum eb detcennoc yb the sremarK-ginorK snoitaler [21] ni Fig. 2. Fitting tiucric desu .niereh ehT lellarap elements R, dna ,,,,Z yam be decalper htiw the CRAZ ecnadepmi .nqe( l), htiw on change ni llarevo tiucric ecnadepmi ro fit. 61 ecnanosnoc htiw eht EPC ,ledom ,.e.i B = A .)2/an(nat nI retal krow ,]31[ eht K-K noitcennoc is deedni ,detic tub n, ,A and B era llits detalubat as etarapes .seititnauq atad-ledoM fitting As denoitnem ni rieht stnemmoc ,]l[ A/W evah yllapicnirp desu rieht noitcepsni“ ”dohtem ot niatbo retemarap setamitse morf rieht data. hguohtlA on deliated noitpircsed fo this dohtem has neeb dedivorp yb A/W, ti smees ot evlovni gnittif eht laer trap ylno fo eht ecnattimda ot eht data ni redro ot niatbo setamitse fo ,R and eht owt tnemele-detubirtsid )EPC( .sretemarap hcuS ,gnittif hcihw ylraelc seod ton dleiy Cs ,setamitse is ylbaborp lacihparg no( gol-gol )stolp rehtar than yb snaem fo raenilnon tsael squares. tI erofereht sevlovni evitcejbus .stnemele yB ,tsartnoc ew evah desu xelpmoc raenilnon tsael squares ]22[ )SLNC( ot tif eht laer and yranigami parts fo eht ecnadepmi ro ecnattimda fo eht eritne tiucric fo .giF 1 ylsuoenatlumis revenehW‘.]2[ lacitcarp ew evah dewolla all tnednepedni -emarap sret fo eht tiucric ot eb ,eerf gnidleiy setamitse fo meht all hcihw tcelfer all eht data, ton just a trap fo .ti etoN( that ,A ,B and n fo eht A/W mrof fo eht EPC era ton all ,tnednepedni as dessucsid .)evoba ,rehtruF SLNC gnittif is an evitcejbo ,erudecorp nehw all sretemarap era eerf and ecnegrevnoc ot an lautca tsael squares noitulos is .deniatbo On this basis enola ti is roirepus ot yna etamixorppa evitcejbus ;erudecorp erom yltnatropmi its noituloser is hcus that sessecorp esohw evitceffe emit stnatsnoc era esolc rehtegot nac yllausu eb llew detarapes .]22[ ,yllaniF ti sdleiy an evitcejbo erusaem fo eht llarevo ssendoog fo ,tif eht gnittif standard noitaived ,ru and, ni ,noitidda sdael ot lufesu setamitse fo eht evitaler standard snoitaived fo eht -emarap ret ,setamitse gniwohs yletaidemmi hcihw era tnatropmi and ,hcihw fi ,yna era ylroop denimreted yb eht data. Analysis and interpretation of atad fitting results nI this noitces ew lliw tcirtser ,noitnetta ni redro ot lellarap eht A/W ,hcaorppa ot just eht ,R and ,EPC ro( ZARC) sretemarap fo .giF 1. At eht ecnattimda ,level eno nac neht etirw morf eht ,tiucric Y, = ,G + ’n),7ai(gG = BG + ’“)oi(,A )2( erehw ,G = ’;R and ,A = .ln),~(gG ehT xelpmoc ,ytivitcudnoc ,)o(u is just gY,, erehw g = ,,A/,1 eht llec .tnatsnoc ereH ,I is eht edortcele noitarapes ni eht gnirusaem llec and ,A is eht edortcele .aera oT niatbo an noitauqe rof eht laer trap fo ,)w(’a .e.i ,)o(a A/W did ton start morf noitaredisnoc fo ZARC ro EPC stiucric tub edam emos gnitseretni evitanretla assumptions .]9[ yehT detrats htiw an laciripme noitauqe rof eht yranigami trap fo eht cirtceleid ,ytilibitcepsus ,”x nevig yb rehcsnoJ ]42,32[ 1 ( ~-”JJ+ + ]‘-hn),W/W( x” :0 )3( 26 erehw ,,o was deifitnedi yb rehcsnoJ ylno as a citsiretcarahc yllamreht detavitca .ycneuqerf tI nac ylidaer eb nwohs ]52[ that this tros fo roivaheb sevlovni at eht lluf xelpmoc cirtceleid tnatsnoc level just owt s’EPC ni .lellarap ehT laer trap fo )o(a gnidnopserroc ot .nqe )3( yam eb nettirw ]9[ )w(;u = K [ *nJ%-;O + ],#Qn-$& )4( erehw K is a retemarap hcihw lliw eb dessucsid .retal ,txeN A/W edam a pael fo eht .noitanigami hguohtlA rehcsnoJ detcirtser his lanoitcarf tnenopxe seulav ot eht egnar 0 < n < 1, A/W tes n, = 0 and ,tn = n, ruo( .)noitaton noitauqE )4( thus secuder ot eW ees that this tluser is fo yltcaxe eht lareneg mrof fo eht nwonk-gnol ZARC ledom ro EPC-rotsiser ;noitanibmoc rof ,elpmaxe erapmoc eht laer trap fo .nqe )2( htiw 1““])2/s,n(SOC[17 = wp .‘ tI was thus yrassecennu rof A/W ot ekovni eht laciripme rehcsnoJ noitauqe at all. ehT ZARC, desoporp reilrae than s’rehcsnoJ noitauqe ,evoba has gnol neeb desu rof sisylana fo gnitcudnoc ,slairetam just as its cirtceleid ,golana eht eloC-eloC ledom ,]81[ has neeb desu ni eht ralopid lairetam .aera nehW eno Serapnroc eht laer trap fo .nqe )2( htiw .nqE ,)5( eno sniatbo eht snoitauqe ,ro = gK-‘G, (6) po = ’“/‘])2/rl,n(ces[‘;7 = {/aG[ A, ”“{})2/r7,n(soc )7( As detniop tuo yb A/W, eseht stluser establish a noitaler neewteb eht cd tnenopmoc fo )w(’u and eht o > 0 ca .tnenopmoc tuB litnu po ro( ),7 is ,deterpretni eht pihsnoitaler is ylno a noit~etemaraper and is fo on erom eulav than that tnerehni ni .nqe .)2( ehT tsom laicurc pets ni eht A/W hcaorppa is rieht detaeper-semit-ynam -ressa noit that wp is eht yllamreht detavitca gnippoh .etar sihT noitpmussa is detcennoc htiw rieht esu fo a klaw-modnar noisuffid noisserpxe detic yb snigguH ]62[ ot dleiy a alumrof rof eht cd trap fo ,)o(;a .Oru ehT tluser yam eb nettirw erehw nv is denifed as eht egareva pmuj ycneuqerf and ,ul sevlovni eht reirrac noitartnecnoc and rehto sretemarap fo eht noitom .]62,2[ nehW eht ,,iu fo .nqe )8( is tes lauqe ot ,,MK morf .nqe )5( eno sniatbo SK ~~~/~v()~/~Kt )9( dnomlA and tseW edam eht ad coh noitpmussa that ,ry = nv and evag on laciteroeht noitacifitsuj rof this .noisulcnoc nehT and w,, setamitse yam eb detaluclac morf K. eht retemarap setamitse deniatbo morf eht data gnittif flesti and neht derapmoc htiw 36 .snoitatcepxe ,yllaniF eton that neve fi eno sekam the ssel tnegnirts noitpmussa wp = ,uv,u where a, si a ytilanoitroporp factor hcihw thgim eb yllamreht detavitca ,flesti K tonnac eb detaluclac tuohtiw tnednepedni egdelwonk of .,u hguohtlA snoitauqe yllaitnesse the emas as .nqe (8) evah neeb desu ni the past for Na animula-P [26] the ssenetairporppa of the ytilauqe or = nv smees ton ot evah neeb deredisnoc suht far ni the .erutaretil ecniS siht noitpmussa appears ot eb the lapicnirp lanigiro tnemele ni the ynam W/A papers no the sisylana of rieht Na animula-P data, dna ecnis ti was osla desu yllacitircnu yb su htiw ylno emos noitacifidom ni ref. 2 na( devorpmi noisserpxe for ,K was nevig ni ref. 2, na noisserpxe hcihw smees ot eb tahwemos erom etairporppa yllacificeps for Na animula-p naht the lanigiro lareneg noisuffid noisserpxe of snigguH [26]), ti sevresed emos rehtruf .yniturcs nI a ylriaf tnecer paper no dc dna ac noitcudnoc ni gnippoh smetsys [27] lliH dna rehcsnoJ :etats ehT“ noitisop htiw regard ot the noitaterpretni of dc gnippoh data ,si therefore, that at tneserp we are yllaitnesse elbanu ot tnuocca ylevitatitnauq for tsom snoitavresbo ni smret of sretemarap that are nwonk yletarapes morf other secruos . . . “. suoiverP work no the dc noitcudnoc melborp stseggus that edom-cinomrahna nonohp effects yam etubirtnoc ot a,,, tub hcus effects are ton dedulcni ni the snigguH noisserpxe of .nqe (8). lliH dna rehcsnoJ etirw a(w) ni the mrof )w(:u+)T(,u=)w(’a (10) dna rehcsnoJ has netfo proposed that )o(‘,u a a", as ni .snqe (2) dna (5). ,yllaniF lliH dna rehcsnoJ eveileb that )w(‘,u dluohs syawla edulcni emos noitubirtnoc morf the ecittal flesti [24,27]. No hcus noitubirtnoc appears ni the )w(‘,u mret of .nqe (5) nehw .snqe (8) dna (9) are desu htiw o,, = .nv ehT evoba snoitaredisnoc tseggus that there ,si so far, on gnorts laciteroeht noitacifitsuj for the wp = uv W/A noitpmussa dna perhaps neve emos ytniatrecnu ni the ssenetairporppa for animula-p of the snigguH mrof of .nqe (8). nI etips of the tnerappa kcal of a gnorts lacisyhp sisab for the noitpmussa ~a = ,uv ti si of tseretni ot erapmoc the stluser dnuof yb W/A dna C/M gnisu siht .noitpmussa nO gnimussa that or, si yllamreht ,detavitca W/A etirw 151,31-11,9[ wp = w, exp[ - E,/kT] (11) where we si nekat ot eb na evitceffe tpmetta ycneuqerf for gnippoh dna E, si deifitnedi as the noitavitca ygrene for cinoi .noitcudnoc ,ylraelC E, si yllautca na yplahtne for the wp process. Now W/A dnuof (131 a eulav of we = 1.2 x ’-s2’01 morf the ,sisylana as ,evoba of rieht 113 K data, a eulav hcihw serapmoc etiuq llew htiw that of tuoba 2 X ’-s2’01 deniatbo yb tcerid tnemerusaem ni the far arfni red. hguohtlA the evoba tnemeerga smees ot retslob the case for or = ,Hv ti dluohs eb derebmemer that ni order ot niatbo na etamitse of we etarucca setamitse of op are .dedeen tuB as .nqe (7) shows, wp tsum eb detaluclac morf niatrecnu setamitse of ,,G ,,A dna .,n nI ,ralucitrap the W/A A seulav ylesolc( detaler ot ),A are stated ot evah ylno *20% ycarucca [9]. ,ylevitanretlA wp nac ylno eb ,deniatbo niaga 46 ,yletamixorppa morf the dnoces mrof of .nqe (5) nehw the Y, of .nqe (2) nac eb detalosi morf the latot Y of the metsys dna nehw sti laer part nac eb nwohs ot eb ylesolc lanoitroporp ni sti ycneuqerf ecnedneped ot the )o(;a of .nqe (5). htiW ylno %02-~ ycarucca of the A ,retemarap W/A nac ylecracs eb dias ot evah enod .siht Now w, flesti si a prefactor ni .nqe ,)ll( dna ti si ylsuoiroton tluciffid ot niatbo a lufgninaem eulav of na laitnenopxe prefactor sselnu the data are yllanoitpecxe .etarucca dlanodcaM dna Cook [2] deniatbo or setamitse morf those of ,,G ,,A dna ,n dna neht devired oe dna E, setamitse morf raenilnon tsael serauqs gnittif ot .nqe (11) yltcerid ton( yb gnisu na yranidro tsael serauqs tif ot (gol or), a erudecorp hcihw secudortni saib of sti .)nwo ehT stluser deniatbo were we 3 (6.82 k 3.57) X ’-s“’01 for T G 102 K dna we s (7.1 & 8.3) X ’-s5’01 for 110 < T G 151 .K sA eno nac see, rehtien of these stluser si raen 2 X *’ol -s ’ ,tub erom ,yltnatropmi rieht seitniatrecnu are so great that yeht tonnac eb deterpretni as dilav snoitciderp ni yna case. Yet these stluser were deniatbo morf the tseb stif of the data elbaliava at the .emit tI suht appears that litnu ylbaicerppa erom etarucca data naht that of W/A are elbaliava for Na ,animula-p ti lliw ton eb elbissop ot show ylbacoviuqenu that yna w, devired ylevitcejbo morf the data si deedni esolc ot 2 X *IOl .‘-s tI si ylekil that ylbarapmoc great ytniatrecnu ni oe seulav dna the ytniatrecnu ni the Pw = nv noitpmussa ylppa as llew ot the other 161-41,21[ gnippoh noitcudnoc slairetam dezylana yb .W/A hguohtlA ti smees raelc that the case for a laciremun etamitse of oe devired morf the lacirtcele data hcihw si esolc ot the tnednepedni far arfni red eulav sniamer ,nevorpnu ti si ton ni fact yrassecen ot niatbo we ot evorp Pw = .nv enO ylno sdeen ot niatbo nv yltnednepedni dna neht show that at tnereffid serutarepmet Pw s .nv Now .giF 6 of the W/A stnemmoc ]l[ deedni shows that Pw devired morf ecnattimda stnemerusaem appears ot agree yltnellecxe htiw ”nv“ seulav devired morf cinosartlu lacinahcem noitaxaler .stnemerusaem Case ?edam Not !ylirassecen ,tsriF W/A evah stated ...“:]21[ the tnerappa gnippoh ycneuqerf detacidni yb the lacirtcele suludom ssol peak si ton yllareneg a erusaem of the eurt gnippoh ycneuqerf dna the good tnemeerga htiw lacinahcem noitaxaler stluser ni Na animula-P (see ref. 8) si tahwemos ”.suotiutrof nI retal work [13], W/A evah detniop tuo that the seicneuqerf of the peaks of ”Z- dna M” sevruc yllausu reffid ,ylbaicerppa -dneped gni no .n ,rehtruF rehtien of these peak snoitisop sdnopserroc ylpmis ot wP. ehT or, seulav of .giF 6 gnimoc morf lacirtcele stnemerusaem were devired morf yllacihparg dettif Y’ data, ton morf peak snoitisop at .lla esehT suoirav stnemetats mees ot dael ot a suoiruc ,ycnetsisnocni eno hcihw yam eb dezirammus as :swollof (a) cinosartlu wr’s dna kaep-suludom wr’s are ni good tnemeerga (ref. ;)8 )b( cinosartlu wr’s dna Y’ wr’s are ni good tnemeerga .giF( 6 of Ref. ;)1 (c) ,tub kaep-suludom or’s dna Y’ or’s are tnereffid dna do ton erusaem the emas .ytitnauq nA evitanretla yaw ot niatbo na etamitse of Pw morf langis-llams ac -erusaem ,stnem eno hcihw takes lla the data otni ,tnuocca si ot esu .nqe (7) htiw gnittif setamitse of ,G dna ,,A = ,A cos( )2/r?,“ (12) 65 Here ,,A si the ytitnauq demret A ni the W/A work. sihT was the approach desu ni ruo reilrae paper ;]2[ ti si ton raelc that ti si the eno desu yb W/A at lla the enin tnereffid tnemerusaem .serutarepmet tI si tnatropmi ot ezisahpme that the wp devired morf siht approach, or neve morf .nqe (5) si ton the emas as the eno deniatbo morf the peak - 2” .eulav morF .nqe ,)l( eno sdnif that the peak eulav of - c,*G srucco at apm z 17 ’- = ,%‘])2/a,n(soc[ 03) ,suhT ylno as ,n ,-- 0 do the two s’pu approach .ytilauqe Here we evah desu a tpircsbus of eno ot refer ot the CRAZ or ,R-EPC stnemele of .giF 1. morF a lacigol tniop of ,weiv fi the laciripme .nqe (1) seilppa ti smees hcum erom elbisualp ot take w mp lauqe ot nv naht ot set wp = uv as W/A evah .enod tuB there llits sniamer hcum ytniatrecnu ni gnittes rehtie ,_,,pu or wp lauqe ot .nv ecniS the lacirtcele dna lacinahcem snoitaxaler are detaicossa htiw tnereffid lacisyhp processes ni a gnippoh rotcudnoc hcus as Na ,animula-p there si on ,ecnarussa tuohtiw delliated laciteroeht sesylana of htob processes, that op or ,,,pit si the emas ytitnauq as that deniatbo morf lacinahcem noitaxaler peaks or that rehtie of the rates devired morf lacirtcele or lacinahcem stnemerusaem si yltcaxe lauqe ot .nv ecniS -idutignol lan cinosartlu sevaw are delpuoc ot the Na animula-?/ ecittal dna ot the gnitcudnoc snoi ni na yleritne tnereffid noihsaf naht are gnitanretla lacirtcele ,stnerruc hcihw are yltcerid detaicossa htiw cinoi ,noitom ti dluow deedni eb etiuq gnisirprus fi htob processes devlovni uv na( yllacisnirtni ,gnitcudnoc cinoi )ytitnauq yltcerid ni hcus a noihsaf that ti dluoc eb deniatbo yltcerid morf rehtie lacinahcem or lacirtcele stnemerusaem of the dnik dessucsid .evoba tA the tneserp ,emit ti suht appears that neve fi lla lacirtcele rate setamitse were dnuof ot eb ylesolc lauqe ot the lacinahcem ,seno a noitautis llits ni ,obmil the stluser dluow eb ylekilnu ot lauqe .uv NEW FI’ITING RESULTS ecniS W/A evah detcerid laiceps noitnetta ]l[ ot ruo T = 151 K gnittif ,stluser we started ruo sisylana-er of the W/A data htiw the lanigiro 151 K data set. A xelpmoc enalp ecnadepmi tolp of siht data set si detneserp ni .giF 3. nI order ot niatbo a lufgninaem SLNC tif ot data gnivah hcus ,ytiralugerri we detanimile three :stniop the tsewol ycneuqerf ,eno eno ni the hctilg ,noiger dna the txen ot the tsal eno at the tsehgih ycneuqerf dne of the .evruc ruO laitini SLNC gnittif stluser detseggus that ti dluow eb elbarised ot tifer lla enin W/A data sets, ton tsuj the three esoht,tsehgih where we ylsuoiverp desu detalopartxe ,A seulav [2]. ,eroferehT we dettolp lla enin sets ekil that ni .giF 3 dna detanimile the worst ralugerri dna gniyltuo stniop erofeb .gnittif ehT rebmun of stniop detanimile deirav morf zero ot .evif ruO wen gnittif ot the 151 ,K 141 ,K dna 132 K data sets was dnuof ot eb elbissop tuohtiw gnisu ,detalopartxe dexif seulav of A, as ni the reilrae work. llA sretemarap were nekat free ot ,yrav dna we deniatbo ru seulav ylhguor three semit rellams naht those of the lanigiro !stif Here a, si the detamitse dradnats noitaived of the data tif .slaudiser ,yltnedivE ruo reilrae erutarepmet-rehgih stluser were ton eurt 66 0 . _ 0.00 30000.0 60000.0 Zf/ R Fig. 3. xelpmoC enalp ecnadepmi tolp fo the lanigiro W/A 151 K .atad least erauqs seno tub devlovni lacol rather naht etulosba .aminim sihT ,ytilibissop syawla a regnad ni evitareti raenilnon tsael serauqs ,gnittif si hcum ssel ylekil nehw lla sretemarap are free as evoba dna nehw good stif are .deniatbo ,suhT we eveileb that ruo tnerruc stluser are lla least serauqs .seno eW hsiw ot knaht W/A ecno erom for gnitalumits su ot enimaxe-er dna evorpmi ruo reilrae gnittif .stluser eW eveileb ruo tnerruc stluser are the tsom evitcejbo seno yltnerruc elbaliava for the W/A data. ,rehtruF as we llahs see, yeht are yllausu hcum resolc ot the tif-evitcejbus seno deniatbo reilrae yb .W/A Before gnissucsid the etelpmoc gnittif retemarap etamitse stluser ni emos ,liated ti si elihwhtrow gnirapmoc emos of the stluser of ruo wen 151 K tif htiw those deniatbo gnisu s’W/A detamitse retemarap seulav for siht erutarepmet [9,10]. hcuS nosirapmoc si deirrac tuo ni .sgiF .6-4 erehW there are yrev egral seicnapercsid neewteb the W/A snoitciderp dna the data, these esira esuaceb W/A did ton edulcni the ,EPC tnemele of .giF 2 ni rieht .sisylana nehW ti si ,dedulcni as ni ruo ,stluser eno sees tnellecxe tnemeerga neewteb the data dna the tif stluser revo yllautriv the eritne ycneuqerf .egnar hguohtlA the stluser of .giF 5 are dedulcni ni the noitcejorp senalp of .giF 4, we evah detneserp meht here, revo the detimil ycneuqerf egnar where the W/A tif ,seilppa ni order ot etartsulli no a regral elacs the secnereffid neewteb the W/A dna C/M .stif erugiF 6, hcihw dluohs eb derapmoc ot .sgiF 3 dna 4 of the W/A stnemmoc ,]l[ shows the nosirapmoc at the gol ecnattimda