The Peak Shape of the Pair Distribution Function

Transcription

1 The Peak Shape of the Pa Dstbuton Functon V. Levashov, M.F. Thope and M.Le Phscs & Astonom Depatment Mchgan State Unvest East Lansng, MI 4884 Sgnfcant pogess n X-a and neuton dffacton epements on powde samples has been acheved n ecent eas. Hgh-esoluton data nowadas make the compason of theoetcal calculatons wth epemental measuements to a hghe degee of accuac. Because of ths, small sstematc eos that wee gnoed befoe can lead to the notceable dsageement that can now be obseved. It was shown [] that n measuements of Pa Dstbuton Functon (PDF) fom powde samples, the postons of the peaks shfted and that the measued atomc dstances ae bgge then the actual one. It was also shown that the shape of the peaks s not gaussan as t often assumed. It was also ponted out that effect s elatvel small. Hee we pesent futhe developments of the wok []. We have deved an appomate epesson fo the non-gaussan peak shape. It s the most pobable to see ths effect n hghl ansotopc mateals. Estmates fo the elatve se of the effect wee made. In some specal cases ths s a sgnfcant coecton, but usuall t s not.

2 We calculated the Radal Dstbuton Functon (RDF) fo buckball and benene molecules usng the Gaussan98 [] pogam wth AM and SNDO sem-empcal methods. Due to the hghl ansotopc stuctue of those molecules one could epect that t wll be possble to see n the RDF non-gaussan behavo of the peaks. Howeve t was shown that effect s ve small fo buckballs. Fo the benene molecule, the effect s qute lage fo one peak. Usng the calculated esults fo the adal dstbuton functon we calculated pa dstbuton functon of the fullete cstal, and a compason wth the epement esulta of Bllnge, Petkov and Yavas was made. The ole of the fnte esoluton measuements of epement s dscussed. It s shown that coecton to the theoetcal calculatons due to the fnte esoluton of the measuements sgnfcantl mpoves the ageement wth epement. *Wok Suppoted n pat b the DOE unde gant # DE FG97ER4565

3 Non-Gaussan Peak Shape of Pa Dstbuton Functon Let suppose that we consde a cstal and the equlbum poston of an atom s (,, ) wth espect to the cente atom. Atoms vbate nea the equlbum postons. The pobablt that the atom found at poston (,, ) s gven b P (,, ) Coodnates ( ) ( ) ( ) ep ( π ) 3, (, ) and (,, ) aes, whee the mat of dsplacements We want to fnd the ( P() wll be ae gven n the fame of pncpal u αuβ s dagonal ( u ). fo a hghl ansotopc ) powde mateals. Eale, when PDF was calculated, t was assumed that P gauss and that () ep ( ) π Nowadas when epemental technques fo PDF measuement was mpoved sgnfcantl one can epect to see the dffeence between measued PDF and the PDF above. P ngauss () calculated unde assumpton 3

4 R Pefomng of angula aveage s equvalent to the fndng the mass that s lng on the sphee of adus. On aveage due to vbatons the mass of eve atom s dstbuted wth some pobablt ove ts own ellpse. It was shown that the mass dstbuton n eve ellpse s the poduct of the thee gaussans n Catesan coodnates. The sphee of adus R and oentaton. R cuts the ellpses n a wa that depends on the ellpse poston 5

5 To fnd PDF of hgl anstopc powde meteals one should pefom the angula aveage: ) P () ( ) ( ) ( ) ep dω 3 ( π ) It s eas to show that n sotopc case when ( ) ths aveage leads to (eact esult): P so () ep ( ) ( ) ( ) π ep π ep Ths esult can be ewtten as: ( ) () P so ep π (D.A.Dmtov et al. []). If the peaks ae naow the dffeence between and s small. But thee s a possblt that n hgh qualt measuements ths dffeence can be seen and P ngauss () () theo and epement then. P gauss can gve bette ageement between P gauss () (). P ngauss In ansotopc case ( ) dffeence between eal peak shape and ts gaussan appomaton can be even bgge. We deved appomate epesson (epanson) fo the peak shape n ansotopc 6

6 case. In man cases ths epesson gves sgnfcantl bette ageement wth eal shape then fome gaussan appomaton. Summa of deved fomulas () ( ) () () Ψ Σ Ψ Σ Σ Σ P ngauss ep ) π () () () () () () () () ( ) n n n a a H f f a f a f a f a f a,,,,, Σ Σ Ψ Ψ Dstance Pobablt P() Gaussan Fomula Eact 7

7 It follows fom the fomulas above that the bggest devatons fom the gaussan peak shape should occu n case of stong ansotopc mateals. The futhe s the peak fom the ogn the bette ou appomaton woks. On anothe hand the best esoluton s usuall acheved on the fst peak that s the closest to the ogn. Eamples Eact cuve s obtaned b dect numecal ntegaton of (). Gaussan cuve s the cuve that epesents the eponental pat of the deved fomula wthout coecton tems n backets. Ths coesponds to the appomaton that was used befoe. Fomula cuve shows the appomaton of eact cuve gven b (). Pobablt P() 4 3 Eact Gaussan Fomula Dstance 8

8 Non-Gaussan Peak Shape of Ansotopc Molecules Smple wa to check how mpotant the coectons to the pevousl used gaussan lne shape ae s to calculate Radal Dstbuton Functon (RDF) and PDF of ansotopc molecules. In fact all molecules ae ansotopc and that dffeence between molecules and cstallne solds s ve mpotant fo ou case. We consde molecules of fulleene and benene. One can epect that n fulleene the vbatons of the cabon atoms n decton paallel and pependcula to the suface have sgnfcantl dffeent ampltudes. The same can be tue fo n plane and out of plane atomc vbatons n benene. In ode to calculate the RDF of the molecules we have to calculate aveage (elatve) dsplacements of the atoms (,, ) due to the vbatons. The mat of (elatve) aveage dsplacements U can be calculated f the egenfequences and egenvectos of molecula vbatons ae known. Ou devaton shows that: e h ( ) ( ) α β jα jβ α jβ jα β u u j u u j n α β ω m m j mm j mm j e e e e e e e 9

9 Whee ( u u j ) α ( u u j ) β s the mat of elatve atomc dsplacements. ω ae the fequences of molecula vbatons. ae othogonal and nomaled Catesan ( α ) components of the atomc dsplacements that coespond to the egenfequnc ω. m s the mass of atom and Bose-Ensten dstbuton functon. The mat of atomc dsplacements can be dagonaled and ts egenvalues ae the squaes of,,. Then the coodnates of the atoms n the fame whee U s dagonal can be found. Afte that the applcaton of out fomulas s staghtfowad. In ode to calculates egenfequences and egenvectos of molecules vbatons we used Gaussan98 pogam and two sem empcal methods: AM and CNDO. In the tables below we show the values of coodnates and aveage atomc dsplacements fo the seveal neaest atoms fo the molecules of fulleene and benene. Fulleene at 3K. All dstances ae n angstoms. R,, E E ,, E E E- R,, E E-3,, E E E- R,, E E-,, E E E- Benene at 3K. All dstances ae n angstoms (Fom Cabon) R to H,, E E,, E e α

10 R to C,,,, R to H,,,, R to C,,,, E E E E E E E-5.E E E-.7769E- One can see that t s the most pobable to see the fst peak n benene snce t s close to the ogn and t s caused b the most ansotopc vbatons of the neaest hdogen atom. 4 Fullete at 3K Reduced Radal Dstbuton Functon Theo AM, gaussan AM, non gaussan o Dstance (A) As one can see the coectons due to non-gaussan appomaton ae ve small hee. 8

11 Pa Dstbuton Functon Of Fullete Cstal In ode to make the compason wth epement one should calculate the Pa Dstbuton Functon (PDF) defned as: [ ρ() ρ ] G( ) 4π ρ o s the aveage denst of the mateal t s equal to eo fo the case of a sngle solated molecule. The ρ () obes nomalaton condton: ( ) N R ( R) 4π ρ( )d Whee N R s the numbe of atoms nsde the sphee of adus R. Thus fo the sngle molecule we have: G( ) 4π 4π P new o () Z Z P () P new s the new deved functon that should substtute old gaussan appomaton. In case of X-a scatteng Z o and Z stand fo the numbe of electons n the cental atom and atom. The same functon can be plotted fo the old gaussan appomaton n assumpton that Σ. o new Net two pctues show the calculated and benen. G() fo the case of fulleen One can see that the ole of the coectons n case of small. C 6 s ve In case of benene thee s sgnfcant effect that coesponds to the Cabon-Hdogen peak. Unfotunatel the ntenst s smalle then ntenst of Cabon-Cabon Peaks.

12 7 Benene C 6 H 6, T3K Reduced Radal Dstbuton Functon (Cabon) AM Gaussan Non Gaussan o Dstance (A) Hee the coectons due to non-gaussan appomaton ae bgge, but the ae elated to the small fst peak. Hdogen s one of the atoms that cause the fst peak. That s wh t should be had to measue n the X-a dffacton epements. 3

13 Pa Dstbuton Functon of the Fullete Cstall. The molecules of fulleen fom fcc cstal. At 3 K molecules otales aound thee centes. It s mpotant to compae the esults of out calculatons wth the epemental measuements [3]. Let suppose that we st on a patcula atom that belongs to some of the fulleene molecules. Then we can see that atoms that belong to the same molecule as the atom on whch we ae sttng ae n moe o less fed postons wth espect to us. On anothe hand due to the otatons of fulleene molecule we see the smooth dstbuton of the atoms mass ove the sufaces of the othes molecule. 4

14 Thus the contbuton to ρ () fom the othe molecules can be modeled as the RDF of two sphecal shells wth contnuous dstbuton of mass. Thus ρ () consst of two pats contbuton fom the atoms n the same molecule ρ mol () and contbuton fom the othes molecules (coelatons) ρ co (). ρ () ρ mol () ρ () I dscussed above onl the fst contbuton. Fo the detals of second contbuton ou can to the M.Le s poste. 4 co PDF of the fullete. Fullete at 3K Pa Dstbuton Functon G() Theo AM CNDO Gaussan Resoluton Epement o Dstance (A) 5

15 Fullete at 3K 4 Pa Dstbuton Functon G() Theo AM CNDO Gaussan Resoluton Epement o Dstance (A) The ageement on the pctues above can be mpoved f one wll take nto account the fnte esoluton n epemental measuements of the scatteng ntenst. 6

16 The Pa Dstbuton Functon of Fullete wth coecton to the fnte esoluton n the measuements of scatteng ntenst. 4 Fullete at T 3K Pa Dstbuton Functon G() Theo AM CNDO Peak Boadenng Epement o Dstance (A) 7

17 Refeences. D.A.Dmtov, H. Rode, and A.R.Bshop, Peak postons and shapes n neuton pa coelaton functon fom powde hghl ansotopc cstals. axv.og e-pnt achve. Gaussan 98, Revson A.7, M. J. Fsch, G. W. Tucks, H. B. Schlegel, G. E. Scusea, M. A. Robb, J. R. Cheeseman, V. G. Zakewsk, J. A. Montgome, J., R. E. Statmann, J. C. Buant, S. Dappch, J. M. Mllam, A. D. Danels, K. N. Kudn, M. C. Stan, O. Fakas, J. Tomas, V. Baone, M. Coss, R. Camm, B. Mennucc, C. Pomell, C. Adamo, S. Clffod, J. Ochtesk, G. A. Petesson, P. Y. Aala, Q. Cu, K. Mookuma, D. K. Malck, A. D. Rabuck, K. Raghavacha, J. B. Foesman, J. Coslowsk, J. V. Ot, A. G. Baboul, B. B. Stefanov, G. Lu, A. Lashenko, P. Psko, I. Komaom, R. Gompets, R. L. Matn, D. J. Fo, T. Keth, M. A. Al-Laham, C. Y. Peng, A. Nanaakkaa, C. Gonale, M. Challacombe, P. M. W. Gll, B. Johnson, W. Chen, M. W. Wong, J. L. Andes, C. Gonale, M. Head-Godon, E. S. Replogle, and J. A. Pople, Gaussan, Inc., Pttsbugh PA,