VACSY AND INTERPOLATION OF EXPERIMENTAL DATA

Transcription

1 chper 4 VACSY AND INTERPOLATION OF EXPERIMENTAL DATA 4. Inroducon Fro he dscusson n chper 3, we know h wo cegores of ehods re vlble o recover CSA powder perns: group one - ncoplee soropc vergng s cheved b spnnng fs bu off he gc ngle, b spnnng slowl he gc ngle or b no spnnng ll. Ths clss of echnques requres boh swf echncl oons nd sorge of he evolvng gneson, he re no suble for sples wh relvel shor T relxon e. Group - hese pproches rel on he snchronson of gc ngle spnnng nd rdo frequenc pulses. Therefore he re ver sensve o experenl perfecons such s pulse wdh nd perfec experenl preers cuse dsorons n he fnl CSA powder perns. The effor o serch for beer ehods whch cn obn undsored CSA lne-shpes hs never ended. In 99, new ehod, ered s VACSY - Vrble Angle correlon SpecroscopY, ws nroduced b Frdn e l [Frdn e l., 99] for correlng nsoropc nd soropc checl shfs of n-equvlen nucle n sold se sples. A unque feure of hs ehod s h he fnl specr re obned b processng sgnls rsng fro spnnng sple, cqured n ndependen experens s funcon of he ngle beween he xs of croscopc sple roon nd he exernl gnec feld drecon. Ths s n conrs o prevousl proposed echnques, whch re bsed on eher sudden echncl sple flppng or ul-pulse sequences. Te evoluon of he vrble ngle spnnng sgnls s deerned b dsrbuon of resonn frequences relng he soropc frequences of he spns wh her correspondng checl shf nsoropes. Fourer rnsforon of hs d resuls n wo-densonl NMR specru. The pulse sequence of VACSY experen s shown n fg4-. In VACSY, he chnge of he sple spnnng ngle s copleed beween wo ndependen experens. So, does no pu src lon on he T relxon e of he nvesged sple nd he requreens for he echncl perfornce of he probe re lso relxed. These re he predonn dvnges of VACSY coprng wh oher pproches.

2 H 3 C P(cos θ ).0 Fgure 4-: Pulse sequence of VACSY experen. In he drec denson: cross polrson beween H nd 3 C, d s cqured under H hgh power decouplng. In ndrec denson, ngle T cn be spled n n rnge correspondng o rnge of [-0.5,.0] for P (cost). 4. Theor of VACSY Suppose h one wshes o obn D NMR specru of sple, n whch unscled soropc checl shfs of dfferen ses re correled wh her correspondng full CSA powder perns. The usul D NMR pproch suggess h he spns re llowed o evolve durng e under he exclusve effecs of nsoropc nercons, followed b e n whch he spns onl process her soropc checl shf frequences. Regrdless of he w n whch hese wo nercons re sepred, hs cquson schee crees e don spce S(, ), ssoced wh ndependen nsoropc nd soropc evoluon frequences. If he e don d S(, ) s spled properl, he fnl correlon specru I(Z, Z ) cn be clculed s: ( ω ω) = S(, ) exp[ ( ω + ω ) ] dd I, (4-) Ths Fourer rnsforon provdes D NMR specru n whch slces prllel o he Z xs he soropc frequences of he resolved ses show CSA powder perns. A NMR experen lke hs, whch cn relse coplee sepron of unscled soropc nd nsoropc nercons, re hghl possble n prccl cses. However, s possble nd relvel sple o esure he e don sgnls b cqurng se of ndependen Vrble- Angle-Spnnng (VAS) NMR sgnls s ndced b he pulse sequence n fg4-. Ths s he nl de of VACSY. Le us consder sse coposed of soled spn ½ nucle, he dependence of resonnce frequenc on CS ensor orenon s gven s: -0.5

3 ;3$ 6 =3 $ 6 % <3$ 6 ; % = < ω( θ, φ) = ω so + δ (3cos θ ηsn θ cos( φ )) (4-) As shown n fg4-, when hs spn sse s spnnng n ngle E wh respec o he exernl gnec feld drecon nd n he fs spnnng rege, hs ens h he re of sple roon s lrger hn he gnude of he CSA nercon nd he effecs of e-dependen ers cn be negleced. The nsnneous precesson frequenc of he spns cn be pproxed s: ω( θ, φ) = ω = ω = ω so so so + δ + ω + ω (3cos θ ηsn θ cos(φ )) (3cos nso nso (3cos P (cos β ) β ) β ) (4-3) In hs equon Z so s he dfference beween he soropc checl shf frequenc of he nucle nd he rnser offse. Z nso s he nsoropc frequenc of he spns. P (cose) s he secondorder Legendre polnol. URWRU D[LV Fgure 4-: (lef) Relve orenon of B 0 feld n he Prncpl Vlue Sse (PAS), T descrbes he ngle beween ensor V xs nd exernl gnec feld drecon. (Rgh) Lboror co-ordne sse, exernl gnec feld B 0 s long he Z xs nd he roor spnnng xs hs ngle E wh respec o B 0. For n prculr ngle E beween he roon xs nd B 0, n equon [4-3] here s P (cose) sclng of he nsoropc checl shf nercon. A e fer he excon of he spns n he rong sple, he sgnl observed cn be wren s: S Defne: β [( )] dωdω ( ) = I( ω, ω )exp ω + ω P ( cos β), (4-4) = P (cos β ), nd = (4-5) s possble o rewre he sgnl deeced fro hs spn sse s:

4 S ( ) = I( ω, ω) exp[ ( ω + ω ) ] dωdω, (4-6) he FID sgnl S(E, ) nd he specrl I(Z, Z ) correlng he soropc checl shf nd CSA powder perns for D Fourer pr. Alhough hese wo experenl vrbles, E nd, re no cpble of explcl seprng checl shf soropc nd nsoropc nercons, he do provde prccl w of splng he (, ) spce ssoced wh he s shown n fg4-3: P=-/ P= Fgure 4-3: Dfferen pproches for correlng soropc nd nsoropc checl shfs n solds: (lef) Cresn pproch - he spn sse s llowed o evolve durng e under he effecs of he nsoropc nercon. D re subsequenl cqured s ncreen of n ndependen e preer durng whch he sse evolves under he soropc nercon. (Rgh) Mxed-densons pproch - he drecon long whch he sse evolves n he (, ) plne s long he rs orgned fro (, )=(0, 0). The splng regon s confned beween he wo ndced vlues P =-0.5 nd P =, b chngng he spnnng xs ngle E (dped fro Frdn, 99). 4.3 Mehods o pleen nerpolon or exrpolon 4.3. Norl nerpolon - Frdn procedure A norl w of evlung equon [4-6] s he fous Fs-Fourer-Trnsforon lgorh (FFT) [Chpene, 973; Ello, 98; Wll e l., 989], bu hs lgorh requres n equll spced grd of d pons for he correlon of e don d nd he frequenc specr. An obvous pproch s o djus he vlues of E nd he dwell e used n ech ndvdul cqusons, o sple ll pons on regulr grd n (, ) spce. However, hs procedure s hghl neffcen. Becuse, even for splng relvel sll d grd, wll requre lrge nuber of E vlues. Anoher ore effcen procedure s o selec regon n he

5 (, ) spce whch s suffcen o suppor he desred specrl resoluon, sple hs regon wh s n dfferen ngles E s necessr o ensure n ccure nerpolon of he e don d grd pons shown n fg4-4. φ PLQ φ PD[ 3 3FRVθPD[ PLQ W 3 3FRVθPLQ PD[ W W 3 FRV θ Fgure 4-4: Norl nerpolon procedure for obnng soropc-nsoropc correlon specr fro D VACSY NMR d. P x nd P n, he wo x of P (cose) used n he cquson, deerne wedge n (n, n ) spce. Experenl d ndced b sold dos re loced on he sold rs, he re cqured ulples of he phscl dwell e whch lso deernes he grd spcng long he n xs. Inerpolon of d vlues s crred ou for ll he grd pons fllng nsde he wedge whch re ndced b open crcles, d vlues for he grd pons fllng ousde hs wedge whch re covered whn dshed grds re se o ero. As n norl D NMR experens, he spcng (dwell e) of he e don d grds decdes he rnge of fnl VACSY specru; he ol nuber of pons decdes s specrl resoluon. Defne DW, DW s he e ncreens (dwell es) beween djcen pons long, xes. Suffcen resoluon long he Z drecon cn be ensured b seng he prccll used dwell e equl o DW for ll cqusons of dfferen ngles. The spnnng xs ngle E provdes noher freedo, ech cqured sgnl corresponds o r n he (, ) spce. For he convenence of furher hecl npulons, grd spce (n, n ) nsed of (, ) s used. In hs grd spce (n, n ), dsnces re scled b uns of DW long xs nd DW long xs. Due o he properes of.0 P (cose) -/, here re coverge lons n (, ) spce. In prccl experen, d s resrced n wedge n (n, n ) spce led on

6 he sdes wh he hghes nd lowes P (cose) vlues used, P x nd P n. The ngles I x nd I n n (n, n ) correspondng o P x nd P n wh respec o he soropc xs re gven s: n( φ n( φ x n ) = P ) = P x n DW / DW DW / DW (4-7) f N ndependen vrble ngle spnnng experens re crred ou, he ngles I whch conss of unfor splng of he (n, n ) spce re gven s: φ = φn + ( φx φn ), =,,..., N (4-8) N fro hs se of he ngles {I } he cul vlues of he ngles {E } used n N ndependen vrble ngle spnnng experens re gven s: n( ) / cos DW φ DW + β = 3 =,,..., N. (4-9) experenl experence shows h n he rnge of 0.5 P (cose) -0.5, wh he ol nuber of ndependen VAS experens less hn 30, ssfcor specr cn be obned. Ths s due o he fc h: n he rnge of.0 P (cose) 0.5, he ngle E s sll nd he col s pproxel n prllel o he exernl gnec feld drecon, herefore s effcenc o deec NMR sgnl s drscll reduced. Therefore, he sgnl nens of he VAS experens correspondng o he ngles n hs rnge s reduced nd herefore non-sgnfcn for d nerpolon. Afer he fnsh of N ndependen VAS experens, he cqured d re used o nerpole regulr D rr n he regon beween I x nd I n n (n, n ) spce. Becuse sgnls correspondng o dfferen spnnng ngles re dgsed usng n equl dwell e DW, on he grd, he posons o be nerpoled re lws flnked b wo d pons long xs. These wo pons re used o lnerl nerpole he nerede posons on he regulr grd unl regulr D NMR rr s genered. The ol nuber of pons nerpoled long he wo densons of he grd re soehow rbrr. The onl requreen s h should be lrge enough o llow he sgnl dec o ero o obn xu resoluon Norl nerpolon plus Lner Predcon One dsdvnge of he convenonl D VACSY nerpolon pproch s h s no possble o obn pure-bsorpon-ode specr becuse of phse-ws refcs whch re nheren o he experen s shown n fg4-7. The resulng loss of resoluon nd lne-shpe dsorons pede nlss of he experenl specr.

7 In D VACSY experen, seres of vrble ngle spnnng free-nducon decs re cqured nd plced ngles: [ RP β )] φ = n (cos (4-0) n (, ) spce s shown n fg4-4. Here,, defne he nsoropc nd soropc e xes, P (cose) s he second-order Legendre polnol, E s he ngle of he roon xs wh respec o he sc feld drecon, R s he ro of he nsoropc o soropc specrl wdh. Tpcll, P rnges fro -0.5 o +0.5, so he sgnl prll spns wo of he qudrns n he Fourer spce. Once he FIDs re cqured nd posoned n (, ) spce, he grd posons whn hs regon cn be nerpoled fro he experenl d pons. The res of he (, ) spce s se o ero. The phse refcs nheren o he D VACSY specru re due o he ncoplee splng n hese wo qudrns. In generl, phse refcs n n NMR specru re due o he ncoplee splng of he e-don Fourer spce. Mn procedures hve been developed o do ero order nd frs order phse correcons [Wcher e l., 989; Mong e l., 990; Vn Vls e l., 990]. In he cse of one-densonl NMR sgnl: I [ ω] S( ) = ( ω) exp dω (4-) he specrl nens dsrbuon I(Z) nd he FID sgnl S() for Fourer pr. However, he fnl experenl specru I (ω ) us ke no ccoun lne brodenng nd runcon of he sgnl. Assue h ll specrl coponens hve he se Lorenn lne brodenng, I (ω ) s he convoluon of I(Z) wh Lorenn-pon-spred funcon (PSF), P(Z) [Lee e l., 995]: I ( ω) = I( ω)* P( ω) (4-) f he sgnl covers he full Fourer spce for posve nd negve e, P(Z) s n bsorpon Lorenn lne-shpe. I (ω ) s hen spl he brodened for of I(Z). If he sgnl spns onl hlf of he Fourer spce for posve e, P(Z) s coplex vlued funcon: P( ω) = A( ω) + D( ω) λ ω (4-3) = + ω + λ ω + λ here A(Z) nd D(Z) re he bsorpon nd dsperson Lorenn lne-shpes, O s he exponenl relxon fcor. In boh cses he rel pr of P(Z) hs he se lne-shpe, he se specrl nforon s vlble fro coverng eher ll or jus hlf of he e-don Fourer spce s shown n fg4-5.

8 Fgure 4-5: One-densonl Lorenn pon-spred funcon (PSF), P(w). () Coplee sgnl cqured for posve nd negve e; P(w) s hen rel Lorenn lne-shpe. (b) Sgnl cqured onl for >0. The rel coponen s n Lorenn lne-shpe, whle he gnr coponen becoes dsperson lne-shpe (dped fro Y. K. Lee, 995). The se prncple pples o hgher densons. In he cse of wo-densonl NMR sgnl:, ) = I( ω, ω )exp[ ( ω + ω) ] dωd S( ω (4-4) he experenl specru I ω, ω ) s wo-densonl convoluon of he specrl dens ( dsrbuon I(Z,Z ) wh he D PSF funcon P(Z,Z ). If here s no runcon nd he sgnl spns he coplee Fourer spce, P(Z,Z ) s D pure bsorpon Lorenn lne-shpe. If onl hlf he Fourer spce s cqured b runcng he sgnl for <0, P(Z,Z ) s coplex vlued funcon: P( ω, ω ) = = A( ω) [ A( ω) + D( ω) ] [ A ( ω ) A ( ω ) + A ( ω ) D ( ω )] (4-5) where A (Z ) s he bsorpve coponen n Z, A (Z ) nd D (Z ) re bsorpve nd dspersve coponens n Z. The pure-bsorpve lne-shpe A (Z )A (Z ) cn be obned b splng onl hlf of he full Fourer spce. Obvousl, slr resuls wll be obned b runcng he sgnl long denson rher hn denson. In he cse of boh nd densons re runced, h s: onl one qudrn n he Fourer spce s cqured, hen boh he rel nd gnr prs of P(Z,Z ) show posve nd negve lobes due o he xng of he bsorpve nd dspersve coponens.

9 P( ω, ω ) = = [ A( ω) + D( ω) ] [ A( ω) + D( ω) ] [ A ( ω ) A ( ω ) D ( ω ) D ( ω )] + [ A ( ω ) D ( ω ) + D ( ω ) A ( ω )] nd pure bsorpon lne-shpes re no longer possble s shown n fg4-6 [Lee e l., 995]. (4-6) Fgure 4-6: Two-densonl Lorenn PSF, P(w,w ), n convenonl D NMR experens. () P(w,w ) for sgnl cqured n ll four qudrns of he e-don, P(w,w ) s rel D Lorenn lne-shpe. (b) P(w,w ) for sgnl cqured n wo of he four qudrns. The rel coponen of P(w,w ) rens D bsorpon Lorenn lne-shpe, (w ) (w ), whle he gnr coponen s xure of bsorpve nd dspersve ers. (c) P(w,w ) for sgnl cqured n onl one qudrn. P(w,w ) conns xure of bsorpve nd dspersve ers n boh he rel nd gnr coponens (dped fro Y. K. Lee, 994) Bck o he cse of VACSY, consder he PSF funcon P(Z, Z ) for he cse of R= nd +0.5 P Snce I rnges for -45 o o +45 o, he ol re of he Fourer spce covered b he VACSY sgnl s equvlen o one qudrn, so P(Z, Z ) s slr n for o P(Z,Z ) n fg4-6. The refc rdges becoe less nense when R ncreses nd lrger re of he Fourer spce conns experenl d. Unforunel hs s cheved he cos of specrl resoluon n he nsoropc denson nd ncresed nerpolon error. The specrl refcs shown n Fgure 4-7 re specl o D VACSY experens due o he unconvenonl runcon nd nerpolon of e don d. Ofen hese refcs be gnored, prculrl when he specru s doned b brod nsoropc perns. Becuse he refcs re lso brodened, hen he nerference beween dfferen ses becoes

10 neglgble. Ths explns he success of D VACSY despe he refcs nheren o he echnque. However, hese refcs cn becoe serous proble when he specru conns closel spced soropc shfs wh sll nsoropes. The rdge refcs eergng fro nrrow se nerfere wh he nsoropc perns of neghbourng ses, cusng serous lne-shpe dsorons. The reovl of such refcs becoes especll porn when here re prll overlppng or connuos dsrbuon of soropc shfs, or when ccure lneshpe nlss s requred s n he sud of prl oleculr orderng n hs work. Arefcs n D VACSY specr cn be reduced f he ssng d pons n he sgnl Fourer spce (, ) cn be exrpoled usng he experenl d [Lee e l., 995]. However, due o he lrge nuber of ssng d pons, he exrpolon echnque us nn ccurc over severl perods of he sgnl. Lner predcon wh sngulr-vlue decoposon LPSVD s one such echnque whch hs been used for exrpolon nd specrl eson n NMR [Ruledge (Ed.), 996; Brkhujsen e l., 985; Sephenson, 988; Kuresn e l., 98, K, 988]. The LPSVD ehod ssues he sgnl o be e seres represened b su of decng exponenls wh ddonl whe Gussn nose: n = M = exp n =,, ", N [( ω λ ) ( n+ δ )] d + w( n) (4-7) where, Z, O re he coplex plude, frequenc nd dpng fcor of ech exponenl er, respecvel. M s he ol nuber of exponenl coponens, d s he dwell e of he e seres sgnl, N s he ol nuber of pons n he e seres, G s n neger h specfes he shf fro he e orgn o he frs sple d pon. The generl procedure of LPSVD copuon s oulned brefl n he followng. A se of lner predcon equons n he bckwrd predcon ode s gven s: # N L # 3 N L+ " " % " L L+ # N + b0 b # bl = # N 0 L (4-8) or n copc for s: Ab = h (4-9)

11 where b s he vecor of he bckwrd LP coeffcens, A s he (N-L)uL d rx, h s he d vecor wh N-L coponens. The nuber of LP coeffcens (predcon order), L, s bounded b N-M L M, pcll s se o 0.75N [Kuresn e l., 98]. The sngulr-vlue-decoposon (SVD) of he rx A s wren s produc of hree rces [K, 988]: S = V O + A U (4-0) where V + denoes Hern conjuge; U nd V re orhogonl rces of densons (N-L) u(n-l) nd LuL respecvel. S s dgonl rx wh he sngulr vlues s s dgonl eleens { σ, k =,, ",n( L, N L) } k. O s null rx. Denong he colun vecors of he rces U nd V b { u u " } nd { v v " } predcon coeffcens re copued s: u N L v L, he bckwrd lner b = M = σ ( u + h) V (4-) where he suon l M runces he SVD soluon for b. For low nose d, M s spl he ol nuber of peks n he specru; However, f he d conns sgnfcn nose, M becoes n djusble preer [De Beer e l.,, 988; Ln e l., 993]. Once he bckwrd LP coeffcens hve been clculed, he ssng d pons n he e seres cn be exrpoled s: n = L k = b k n+ k, = (,, ", δ) n (4-) where n=-g defnes he d pon he e orgn. To ke dvnge of he sgnl-o-nose ro proveens n LPSVD, he enre e seres should be reconsruced b clculng ou he specrl preers ssoced wh he d se. The preers, Z nd D cn be obned b consrucng polnol: P * * * L ( ) = + b + b + " + b L (4-3) whch hs roos =exp [ ω + α ]. The coplex plude cn hen be obned b subsung Z nd D no equon [4-7]. Fg4-7 shows n exple of he LPSVD procedure ppled o D VACSY sulon specru, he proveen n specrl resoluon s sgnfcn.

12 Fgure 4-7: Sulon of D VACSY specr. The sulons re usng R=, d=00 us, nd hree ses wh checl shf ensors: (V xx, V, V ) = (0.3,.0,.3), (-0.5, -0.5, -.5), (-.5, -.0, -.0)(kH). () Norl phsed D VACSY specru obnng usng +0.5P-0.5 nd R=. The re of Fourer spce ousde of he shded regon s se o ero. The D specru revels he phse refcs nheren o he norl D VACSY experen. The projecon ono he Z xs elds consn funcon, wheres he projecon ono he Z xs elds he pure bsorpon soropc specru. (b) D VACSY specru fer exrpolon usng LPSVD. The re se o ero n () s exrpoled fro he nerpoled sulon d n he shded regon. LPSVD s used o exrpole he d n ech slce, prllel o he xs, o he =0 pon. he phse refcs re elned fro he D specru. The projecon ono he Z xs elds he overlp of he dfferen rceless nsoropc powder perns, whle he projecon ono he Z xs gn elds he soropc MAS specru Non-orhogonl nerpolon The nerpolon procedure nroduced n secon 4.3. cn no obn pure-bsorpve D VACSY specru, dsorons of CSA lne-shpe perns due o phse refcs re possble o be elned. The LPSVD procedure nroduced n secon 4.3. s cpble o exrpole he ssng d pons nd obn clen pure-bsorpve VACSY specru. However, f he experenl d conns sgnfcn nose hs procedure fl or gve pprenl ncorrec resuls. Here, new nerpolon procedure, ered s non-orhogonl nerpolon, s proposed b G. Hepel nd successfull pleened b us. In hs procedure, he experenl d re nerpoled o non-regulr grd bsed on wo xes, :

13 P P x n (cosθ (cosθ n x ) = P ) = P x n (4-4) nsed of regulr grd whch bsed on xes nd s shown n fg4-8: 3 3FRVθPD[ W 3 3 FRVθ PLQ PD[ PLQ W W 3 FRV θ Fgure 4-8: Non-orhogonl nerpolon pproch for obnng soropc-nsoropc correlon specr fro D VACSY NMR d. P x nd P n, he wo x of P (cose) used n he cquson, deerne wedge n (, ) spce. Experenl d ndced b sold dos re loced on he sold rs confned n he wedge. Inerpolon of d vlues s crred ou for ll he pons ndced b open crcles of non-orhogonl grd whch hs wo chrcersc xes prllel o P x nd P n respecvel. D VACSY FID d re experenll cqured b crrng ou M ndependen vrble-ngle-spnnng (VAS) experens whch corresponds o rs n he (, ) plne nd he cn be hecll expressed s: S ( θ, ) = I ( ω, ω )exp[ ( ω + ω P ( cosθ ) )] = I ( ω, ω )exp [ ( ω + ω )] dω dω dω dω (4-5) where: = 0 " n,,,,( ) P ( cosθ) =, θ = θ n, θ n + θ, θ n + θ, θ x (4-6) where: vrble n, chnges fro o N, s he ndex of d splng n he drec denson, ' s he correspondng experenl dwell e for ll M dfferen ngles. Vrble, chnges for o M, s he ndex for T ngle splng wh sep of 'T n he ndrec denson.

14 Drecl fer sgnl cquson, n rbrr VACSY FID d pon of he h ndependen experen e n, s expressed s: nd: = n co-ordnes sse { } (, ) I(, ω )exp[ ( ω n + ω P ( cos( θ + θ ) n )] FID n P np n ( ( θ + θ ) ω n ) dω dω (4-7) = P cos n (4-8) In he followng dscusson, he co-ordne of hs d pon n co-ordnes sse { } denoed s: { np, n} Now we chnge he co-ordnes sse fro {, } o { } P drecon nd long, s,, where: s long x P drecon, he re he wo bse vecors of co-ordnes sse n {, }. In co-ordnes sse { } P x P n =, =,, nd cn be expressed s followng: herefore, he rnsforon rx fro co-ordnes sse {, } o { }, s gven s: (4-9) P x P n = (4-30) Becuse he d se FID (, n) obned drecl for VAS cqusons do no fulfl he D FT requreens, he us be nerpoled o he new d se FID (, j) on he non-regulr grd pons. In co-ordnes sse { } rbrr d pon,, h s: n he non-regulr grd spce, f we focus on n FID (, j), should hve co-ordne (, j). Usng he rnsforon, rx n equon (4-30), he co-ordne of he se d pon n co-ordnes sse { } cn be wren s: ( ) j ( + j) P + ( )( P P ) P x P n P x + j P n x + j n x = = = (4-3) j j + j + j fro drec coprson of equon (4-7) nd (4-30), s es o ge h: n co-ordnes sse { } { }, :,, n rbrr d pon FID (, j) hs he followng co-ordne n sse

15 n = + j P ' = P So, n { } x + j ( + j) ( P n P x ), spce, he wo neres d pons relve o hs one hve co-ordnes: (4-3) np np + FID(, n) =, FID( +, n) = (4-33) n n where: n = + j P P P ' + In spce { } (4-34),, he vlue of hs rbrr d pon FID (, j) cn hen be nerpoled hrough hese wo nerb d pons FID (, n) nd FID ( +, n) n spce {, } b he followng equon: P P P P FID, P P P P ' ' + (, j) = FID(, n) + FID( + n) + + (4-35) In fg4-8, d pons ndced b sold crcles whch re lng on rs orgned fro (0, 0) re cqured b M ndependen VAS experens, hs D d rr does no fulfl FFT requreens. D pons whch re ndced b open crcles for non-orhogonl grds nd fulfl D FFT condons, hese re he d pons whch should be nerpoled fro experenl d b usng equon (4-34). Afer norl D FT evluon, non-orhogonl D VACSY specru s obned. Due o roon of co-ordne sse {, } relve o { }, for n ngle whch s decded b P x nd P n s shown n fg4-8, n he fnl VACSY specru he CSA powder perns re roed for he se ngle wh respec o he VACSY specru obned b he norl nerpolon procedure s shown n fg4-9c. CSI nd CSA projecons cn no be drecl ken fro he VACSY specru obned wh hs non-orhogonl pproch, reverse verson of hs non-orhogonl nerpolon procedure for he VACSY specrl d s requred o roe he fnl specru o norl drecon, h s: CSI nd CSA re prllel o Z, Z drecons respecvel. A four pons nerpolon forul s used o ccoplsh hs sooh reverse nerpolon s shown n fg4-9d, nd n he nsoropc drecon he fnl specru s scled for known fcor coprng wh he specru n fg4-9c. A drec coprson of he resuls of hese wo nerpolon pproches o suled VACSY d s presened n fg4-9. Fg4-9b shows h due o he ncoplee d splng n (, ) spce, he VACSY specru obned fro he norl nerpolon procedure cn no be dspled n phse sensve ode. A well cceped fc s h he resoluon of specru dspled n gnude ode (bsolue vlue ode) or power ode s lower hn he se

16 specru dspled n phse sensve ode when s gnr pr n no ero lke n he cse of VACSY, nd oreover he correc lne-shpe of specru s lso chnged when s dspled n gnude ode or power ode. In soe cses when good specrl resoluon s hghl desrble s n he cse of closel spced soropc checl shfs or when ccure lneshpe nlss s requred s n he nvesgon of prl oleculr orderng, s hghl desrble h he specr cn be dspled n phse sensve ode. F (pp) F (pp) Fgure 4-9: VACSY sulon specru processed b norl nerpolon procedure nd dspled n bsolue vlue ode, checl shf ensor used (V xx, V, V )=(00, 00, -00) (pp). F (pp) F (pp) Fgure 4-9b: VACSY sulon specru processed b norl nerpolon procedure nd dspled n phse sensve ode, checl shf ensor used (V xx, V, V )=(00, 00, -00) (pp).

17 F (pp) F (pp) Fgure 4-9c: VACSY sulon specru processed b non-orhogonl nerpolon procedure nd dspled n phse sensve ode, checl shf ensor used (V xx, V, V )=(00, 00, - 00) (pp). Fgure 4-9d: A four-pons reverse nerpolon pproch o roe he specru n fg4-9c no norl drecons, dspled n phse sensve ode, checl shf ensor used (V xx, V, V ) = (00, 00, -00) (pp). The dvnges of he hs non-orhogonl nerpolon ehod s h: () he CSA lneshpe dsorons re soewh less coprng wh he norl nerpolon procedure; () he fnl D VACSY specru s possble o be dspled n phse sensve ode; () coprng wh he LPSVD ehod, hs ehod cn no copleel reove phse refcs s LPSVD does, bu cn be successfull used o process nos d.

18 4.3.4 VACSY rnsforon wh egen-coordnes In ll hree bove enoned VACSY d processng ehods: nerpolon, nerpolon plus Lner Predcon nd non-orhogonl nerpolon ehods, he d vlue of he desred poson n he (, ) spce s clculed fro experenll cqured d vlues b usng vrous nerpolon or exrpolon lgorhs. Fro he prncple of nerpolon clculon, s es o undersnd h dfference lws exss beween he nerpolon resul nd he heorecl gneson evoluon. Especll when he drec dgson ndex n [,...,N] goes o bgger vlue, where he d pon o be nerpoled s que fr pr fro wo djcen cqured d pons, he error of nerpolon becoes bgger. A good nerpolon lgorh cn grel reduce hs error, bu cn no oll reove hs devon. In order o reduce he nerpolon error s uch s possble, G. Hepel proposed noher VACSY d processng pproch whch obns he fnl D VACSY correlon specru b drecl rnsforng he experenll cqured d se whou dong n nerpolon or exrpolon. Ths proposl hs been successfull ppled o process VACSY sulon/experenl d b e nd gves ssfed resuls. The procedure s ered s: VACSY rnsforon wh egen-coordnes. Fro he hecl expresson of VACSY fd d n equon (4-5), he correspondng VACSY specru cn be expressed s: ( P, ) exp[ ( ω + ω P ( θ ) )] I ( ω, ω ) d = S cos dp (4-36) denoe: p=p (cost), we hve: ( p, ) I ( ω, ω ) = S exp( ω ) d exp( ω p dp ) (4-37) hs s no norl Fourer rnsforon. Here, he frs e vrble s, correles o Z n frequenc don; he second e vrble s p ulples, correles o Z n frequenc don. In order o be clculed b copuer, he nlcl equon (4-36) hs o be convered no s correspondng dgl for. The dgson relon beween e vrbles nd frequenc vrbles re gven n he followng: ω ω k ; p P (cos θ + θ ) = p ; S( p, ) Z kn ; ω ω l ; n ; kn kl Z I ; (4-38) The coplee rnsforon s dvded no wo seps, negron for p nd for respecvel. () sep one: he negron (hs corresponds o suon n dgl for) for vrble p, he nerede resul s denoed s Z kn :

19 Z kn M = = + [ S [ S exp( ω k n p )] n 0n exp( ω k n p x ) + S Mn exp( ω k n p x )] (4-39) specl copuer progr wren n C b us s requred o relse hs pr of d clculon. The clculon resul s sored n VNMR fd d for whch cn be drecl used b VNMR sofwre for nex sep processng. () sep wo: he nerede resul of sep one Z kn s frsl ulpled b = n, hen norl D Fourer rnsforon s perfored. VNMR sofwre s used for he clculon n hs sep nd ll ler grphc npulons. The resul VACSY specru processed b hs new pproch for he se sulon d s n fg4-9 s shown n fg4-0: Fro drec coprson of fg4-0, fg4-9, fg4-9b, fg4-9c, fg4-9d, he dvnges of hs VACSY rnsforon wh egen-coordnes re: () n fg4-0, he projecon ono he soropc xs s soewh nrrower nd he sngulres of he projecon ono he nsoropc xs re shrper coprng wh he correspondng specr of fg4-9. Ths s becuse: fg4-0 s dspled n phse sensve ode whle fg4-9 n bsolue vlue ode. () n fg4-0 he D VACSY specru s no roed lke n fg4-9c, hs kes he soropc nd nsoropc projecons drecl possble. () n fg4-0, ousde he regon where specr feure s he sgnl nenses re ver low coprng wh boh fg4-9 nd fg4-9d. Ths s due o he fc h n VACSY rnsforon wh egen-coordnes, here re no nerpolon errors. However, he buer-fl-lke phse refcs surroundng he specrl feure n fg4-0 re sll presen. Ths s due o he ncoplee splng of (, ) spce s dscussed n secon4.3., unl now he onl vlble w o reove he s b perforng Lner Predcon. Fgure 4-0: VACSY sulon specru processed b VACSY rnsforon wh egencoordnes procedure nd dspled n phse sensve ode, checl shf ensor used (V xx, V, V )=(00, 00, -00) (pp).