The Shpe of the P Dstbuton Functon. Vlentn Levshov nd.f. Thope Deptment of Phscs & stonom nd Cente fo Fundmentl tels Resech chgn Stte Unvest Sgnfcnt pogess n hgh-esoluton dffcton epements on powde smples hs been cheved n ecent es. We pesent some new developments of theoetcl methods fo the clculton of the p dstbuton functon tht gves mpoved geement wth epement. The ole of the epementl esoluton functon n p dstbuton clcultons wll be dscussed; togethe wth compson wth hgh-qult mesuement on led. n ppomte epesson fo the non-gussn pek shpe tht should be obseved n the p dstbuton functon n hghl nsotopc powde mtels s deved. Fnll we dscuss the mpotnce of ncludng ll mult-phonon pocesses n the compson of theoetcl models wth epementl esults.
The ole of the epementl esoluton functon n p dstbuton clcultons Epementl PDF: G π q[ S( q) ] sn( q) dq q[ S( q) ] G sn( q)d I S ( q), Nf N ( q) q e I q f q Sctteng ntenst, f ( q) G s not n ect Foue tnsfomton of [ S( q) ] tomc fom fcto. q snce the dt n epement cn be collected onl ove fnte sctteng momentum nge [, Q m ]. Thee s fnte esoluton n epementl mesuements of I ( q). Theoetcl PDF: ep G π π π ρ o Ths fom does not tke nto ccount fnte esoluton nd fnte sctteng momentum nge n epementl mesuements. The coecton to the theoetcl PDF due to fnte nge cn be mde though convoluton functon (ws mde befoe). Two peks n epementl S ( q) cn be esolved s septe peks f the dstnce between them s bgge then ( q). Thee poposseed ffom ffo epementl esoluton functon s: ( q q ) F q q, ep δ q Then: q π q q ( q) S( q ) F( q, q ) dq S c It s ssumed hee tht S ( q ) s pue theoetcl ntenst nd S c ( q ) s ntenst tht cn be comped wth epementl mesuements. Let:
S c q q ( q) F e d S( q ) F e d c Then t cn be shown tht: F c F( ) C( ) d C(, ) ( ) δ, e π δ Now f we ssume tht eve tom bngs ts own gussn nto the gussn should be tnsfomed ccodng to: ( ) G then ths F ep ( ) ( ) F c F C, d ep π ( ) π δ Thus the wdeness of the peks n F c s nceses wth compe wth the cse of F. ut snce peks e now one cn substtute δ wth δ. The fnl epesson fo p dstbuton functon hs the fom: ( ) ep δ G c π ρ o π π ( δ ) In ode to tke nto ccount the fnte nge of momentum ove whch sctteng ntenst I ( q) ws mesued t s lso necess to convolute ths epesson wth convoluton functon. Fnll: ( ) { q ( )} { q ( m sn m )} d sn G G ( c ) ep π We compe the esults of ou clcultons wth esults of hgh qult mesuements on led. In ode to obtn theoetcl PDF t ptcul tempetue t s necess to clculte. Clcultons wee pefomed n the fme of Kkwood model. The method of clculton ws dscussed ele (). Compson wth Epement. The pmetes wee chosen n ode to obtn the best geement on the foth fgue. ( δ )
Thus fst thee fgues shows how the geement nceses when the effects of fnte nge nd fnte esolutons e ncluded septel nd when the e ncluded togethe. The ed cuve on ll fgues epesents the esults of epement. Ths cuve s the sme on ll fou fgues. The blue dotted cuves shows the esults of clcultons when dffeent effects lke fnte nge of ntecton o fnte esoluton o both e tken nto ccount. Yes-mens: wee tken nto ccount No-mens: wee not tken nto ccount Fg. Fg. Fg. Fg. Fnte nge No Yes No Yes Fnte esoluton No No Yes Yes Fg. Fg. G() G() 8 8 Rdus () 8 8 Rdus ()
Fg. Fg. G() G() 8 8 Rdus () 8 8 Rdus ()
Non-Gussn Pek Shpe of P Dstbuton Functon Let suppose tht we consde cstl nd the equlbum poston of n tom s (,, ) wth espect to the cente tom. toms vbte ne the equlbum postons. The pobblt tht the tom wll be found n the pont (,, ) s gven b ( ) ( ) ( ) P,, ep ( π ) Whee coodntes (,, ) nd (,, ) e gven n the fme of pncpl es, whee the mt of dsplcements u u α β s dgonl ( u ). We wnt to fnd the PDF (e.g. P ) of hghl nsotopc ( ) powde mtels. Ele when PDF ws clculted t ws ssumed tht nd tht ( ) P old ep π Nowds when epementl technques fo PDF mesuement ws mpoved sgnfcntl one clculted unde cn epect to see the dffeence between mesued PDF nd the PDF ssumpton bove. In ode to fnd PDF of hgl nstopc powde metels one should pefom the ngul vege: P new ep dω ( π ) It s es to show tht n sotopc cse when ( ) ths vege leds to (ect esult): ( ) ( ) ( ) P new ep ep ep π π Ths esult cn be ewtten s: ( ) P new ep. π Thus one cn see tht f peks e ve now then the dffeence between P old nd P new s smll. ut thee s possblt tht n hgh qult mesuements ths dffeence cn be seen nd P old. P new cn gve bette geement between theo nd epement then In nsotopc cse ( P old ) dffeence between el pek shpe nd ts gussn ppomton cn be even bgge. We deved ppomte epesson (epnson) fo the pek
shpe n nsotopc cse. In mn cses ths epesson gves sgnfcntl bette geement wth el shpe then fome gussn ppomton. Summ of deved fomuls Ψ Ψ ep P π f f f f f Ψ Ψ H f n n n ε γ β τ α R R R R R!! β τ α!! ε γ
8 9 7 8 8 9 9 Some Emples
.8 Ect Gussn Fomul.8.79 Pobblt P()......77.7..7.8.9. Dstnce
.7. Ect Gussn Fomul.. Pobblt P()....7.....7.8.9.. Dstnce
.8 Ect Gussn Fomul Some Emples.79.77 Pobblt P()......7.7.....7. 7 8 Dstnce
Z gussn Fomul Ect... s s s....9.98..
ultphonon contbutons to peks n the PDF. If the toms n cstl would not vbte then the PDF of the cstl should consst of set of δ -functons. Pek bodenng occus due to the toms/lttce vbtons. Hee we dscuss the ole of dffeent multphonon contbutons to the pek bodenng. The wdth of the PDF pek s cn be clculted ectl wthn the hmonc ppomton b the epesson whee W ρ j @ E q j dqe Cj q π q q @ j je q q j W q j j u. / u. / λ @ E / λ @ E j j C q e e e e e e q s Debe-Wlle, j ( j uj ), j j, uj u j u pmete. It cn be shown tht π µ π nd λ s epnson ρ s equl to j / µ e µ µ 8? D /? D / / e µ 8? D /? D j? j D /, µ j λ? jd / Whee: 8 µ 8 / 8... Whee the fst tem s the Gussn fom the gg sctteng (wth ts ssocted Debe Wlle fcto). The tem lne n µ s the one phonon contbuton, the tem n µ s the two phonon contbuton etc..
.7.. PDF f f f f f.. g... Fgue. Plot of dffeent multphonon contbutons s shown shown usng pmete µ? jd /. whch s esonble vlue.. The totl pek s n ed, the gg contbuton n blue, the sum of the gg nd one-phonon contbuton n geen, nd the one phonon contbuton n geen, the two phonon contbuton n puple, nd the thee phonon contbuton n blue.
The pek s Gussn f onl the gg contbuton s counted o f ll the contbutons e counted. Othewse the shpe s moe comple. Fgue Showng how the pek n the PDF cn go negtve wth onl. g PDF g g g g..8.... the one nd two phonon peces dded. We hve lso checked tht summng up to the phonon pt lso goes negtve so t s lkel tht ll tems e needed fo convegence. Fgues bove show tht ll multphonon contbutons cn be mpotnt fo the detemnton of pek shpe n p dstbuton functon.