Pages 2-33 of this pdf file are Tycko's lecture slides from January 8, Pages are notes about quantum mechanics, NMR, homonuclear

Transcription

1 Pags -33 of hs pdf fl ar Tycko's lcur slds from January 8, 8. Pags ar nos abou quanum mchancs, NM, homonuclar rcouplng, and rlad hngs, prpard orgnally n 8 and subsqunly corrcd/amndd.

2 nroducon o h (Quanum Mchancal) Thory of Sold Sa NM. How do w dscrb h sa of a nuclar spn sysm?. Angular momnum and roaons 3. How do w rprsn h nracons of nuclar spns ha drmn NM sgnals? 4. How do w calcula NM sgnals? 5. Nuclar spn nracons ha affc sold sa NM (dpndnc on srucur, ornaon, moon) 6. Manpulaon of spn nracons by rado frquncy pulss and sampl spnnng (Avrag Hamlonan hory; rcouplng) (. Tycko, 8 Wnr School on Bomolcular Sold Sa NM)

3 How do w dscrb h sa of a nuclar spn sysm? n quanum mchancs, h sa of a sysm a m can b rprsnd by h absrac symbol (), whch s calld a sa vcor. Sa vcors hav "compl conjugas", whch ar wrn as (). Sa vcors hav a "lngh", dfnd as h nnr produc () (). Th nnr produc rprsns h projcon of on sa vcor ono anohr sa vcor. f =, h wo sas ar orhogonal o on anohr. No ha = *. For any sysm, hr s a s of sas ha rprsns all possbl condons of h sysm. Such a s s calld a bass. (n fac, hr ar many dffrn bass for a gvn sysm.) For a spn / nuclus (.g., H, 3C, 5N, 3P), h bass can b { +, }. n ohr words, h wo sas n whch h spn s hr "ponng up" or "ponng down". Ths mans ha any sa of h spn / nuclus s som lnar combnaon of + and. n ohr words, () = a() + + b(), whr a() and b() ar compl numbrs. Th bass sas ar usually chosn o b "orhonormal", manng ha + + = = and + =. Also, a + b =, so ha () () =.

4 How do w dscrb h sa of a nuclar spn sysm? For a many spn sysm, w can choos h bass o b { m m m 3...m N }, whr m j s h "drcon" of h j h spn. f all of h nucl ar spn /, h bass conans N sa vcors. Ths choc of bass s calld h "drc produc" bass. Th ovrall sa of h many spn sysm can hn b wrn as s () a () m m...m N m,m,...,m N s m,m,...,m N n sandard quanum mchancs, h squard magnuds of h coffcns n such a lnar combnaon rprsn h probabls of bng n ach ndvdual sa. f you wr o ask for h probably ha h many spn sysm a m was n h sa whr m = +/, m = +/, m 3 = /, m 4 = /, c., h answr would b a +,+,,,.... Th rlav phass of h coffcns ar also mporan. For ampl, n a on spn sysm, h sas = a + + b and = a + + b ar dffrn. f a = b = /, hn s a sa ha "pons" along +, whl s a sa ha "pons" along +y. Th dscrpon gvn so far appls whn all of h coffcns hav prcsly dfnd magnuds and phass, so ha hr s a unqu lnar combnaon of bass sas ha fully dscrbs h ovrall sa of h spn sysm. Such a sa s calld a "pur sa". Ths may happn n ohr aras of scnc, bu dos no happn n NM.

5 How do w dscrb h sa of a nuclar spn sysm? Wha happns n NM s ha w hav many cops of nomnally dncal spn sysms, all of whch conrbu smulanously o our NM sgnals, bu ach of whch s n a dffrn quanum mchancal sa. Th "many cops" could b h ndvdual molculs whn our sampl, or hy could b dffrn nanomr scal pcs whn a largr chunk of maral. Th dffrn cops hav dffrn valus of hr compl coffcns n h lnar combnaon of bass sas on h prvous sld. f w knw h probably of havng ach spcfc choc of compl coffcns, hn w could calcula NM sgnals from ach spcfc choc, and add up h rsuls from many, many such calculaons o g h oal NM sgnal. Ths would b prfcly corrc, bu would b vry nconvnn and laborous. A mor convnn, concs, and racabl approach s o us a dnsy opraor (or dnsy mar) dscrpon of h sa of h nuclar spn sysm. Th dnsy opraor approach allows us o avrag ovr h many cops of dncal spn sysms. Bu s no h sam as smply akng h avrag quanum mchancal sa. s () a () m m...m a () m m...m m,m,...,m N m,m,...,m N m,m,...,m s m,m,...,m s N N N N s m,m,...,m N s b m,m,...,m N N () m m...m s Ths s sll a pur sa.

6 grsson: wha s an opraor? An opraor O s somhng ha ransforms a sa o anohr sa ', wrn as h quaon O = '. n quanum mchancs and NM, w ar mosly concrnd wh lnar opraors, whch hav h propry ha O(a + b ) = ao + bo = ' + '. Som, bu no all, opraors hav nvrss, rprsnd by O, such ha OO = O O =. Hr, h symbol "" mans h "dny opraor", whch has h propry ha = for any sa. Som opraors hav "gnsas", whch ar spcal sas ha ar ransformd no hmslvs, mulpld by a numbr ha s calld h "gnvalu". So for an gnsa of O, O =. n gnral, s a compl numbr. All lnar opraors hav "adjons", symbolzd by O. Adjons ar lk compl conjugas. f O = ', hn O = '. And (co) = c*o. Two dffrn opraors usually do no commu wh on anohr, manng ha O O O O. Th opraor [O,O ] O O O O s calld h "commuaor of O wh O ". f wo opraors do commu, hn hy hav h sam gnsas (unlss hr ar "dgnra" gnsas,.., groups of gnsas wh h sam gnvalu). f wo opraors hav h sam gnsas, hn hy commu (whn oprang on hr gnsas).

7 grsson: wha s an opraor? n quanum mchancs and NM, w ar parcularly nrsd n wo spcal yps of lnar opraors, namly "Hrman" opraors and "unary" opraors. Hrman opraors hav a compl s of gnsas, so ha h gnsas can b usd as a bass o rprsn any ohr sa of h sysm. n ohr words, any sa can b wrn as a lnar combnaon of h gnsas. Morovr, h gnvalus ar ral numbrs. Consqunly, f A s Hrman, hn A = A. A s calld "slf adjon". Hrman opraors ar usd o dscrb nrgs (.., rms n h nuclar spn Hamlonan) and ohr obsrvabl quans (.g., angular momnum componns). Th fac ha Hrman opraors hav ral gnvalus mpls ha obsrvabl quans ar ral numbrs. Unary opraors hav h propry ha hr adjons ar hr nvrss. n ohr words, f B s unary, hn B = B. And vc vrsa. Consqunly, unary opraors prsrv nnr producs of pars of sas. f B = ' and B = ', hn ' ' =. Unary opraors ar usd o dscrb h voluon n m of a quanum mchancal sa, as n () = U() (), whr U() s h "m voluon opraor". Unary opraors ar also usd o dscrb roaons and smlar ransformaons. So quanum mchancally, an rf puls s a unary opraor.

8 grsson: marcs and vcors Somms w can calcula NM sgnals by usng h absrac symbols for opraors and sas dscussd so far. Mor frqunly, sgnals ar calculad n compur programs by usng marcs and vcors. Convrng h absrac symbols o marcs and vcors rqurs us o choos a spcfc bass s, for ampl h drc produc bass { m m m 3...m N }. f h bass sas ar wrn smply as k, wh k =,,...,N: N A sa a k s rprsnd by an N dmnsonal column vcor, wh lmns a k. k k s rprsnd by an N dmnsonal row vcor, wh lmns a k *. An opraor O s rprsnd by an NN mar, wh h numbr j O k n row j and column k. A Hrman mar has ral numbrs along h dagonal (j = k). Off dagonal lmns on oppos sds of h dagonal ar compl conjugas of on anohr, bcaus j O k = k O j* for a Hrman opraor. For a unary mar, rows and columns ar orhonormal: N N N * * OqkOrk qok rok qok ko r qoo r qoo r q r q,r k k k N N N N * * OkqOkr k O q k O r k O q r O k r O k k O q r O O q r q r,q k k k k

9 How do w dscrb h sa of a nuclar spn sysm? Now back o h dnsy opraor as a dscrpon of a "md sa" n NM,.., a sampl ha conans many cops of h sam spn sysm, ach n a dffrn quanum mchancal sa, and all conrbung smulanously o h NM sgnals: Th dnsy opraor () s dfnd o b h avrag of ()(), whr ab s h our produc of sas a and b. () () () An our produc of wo sas s an opraor, bcaus can b appld o som ohr sa accordng o ( ab ) = ab = a, wh = b. Such opraors ar somms calld "projcon opraors". N f w prss h sa n a bass { k} as () a () k, hn h dnsy opraor s k N N N N * * k k' k k' k k' k k' () a () k a () k' a ()a () k k' * n hs bass, h dnsy mar has h numbr () a ()a () n row k and column k'. k kk ' k k ' agonal lmns of h dnsy mar (k=k') ar h populaon of sa k, n ohr words h ovrall probably of fndng h sysm n hs sa, avragd ovr h "many cops". Off dagonal lmns (kk') ar calld "cohrncs". A non zro cohrnc mans ha hr s a non random rlaonshp bwn h ampluds of sas k and k'.

10 How do w dscrb h sa of a nuclar spn sysm? Th dnsy opraor sounds complcad, bu acually maks calculaons of sgnals rlavly smpl, spcally whn h nal "md sa" of h sysm s somhng smpl. f h sysm has quanum mchancal nrgy lvls E k (k =,,...,N) and f h sysm s nally a hrmal qulbrum, hn h rlav populaons of h nrgy lvls ar gvn by Bolzmann facors, so kk () p( E k ), whr = /k B T. A hrmal qulbrum, cohrncs bwn dffrn nrgy sas ar assumd o b zro, so kk' () = for k k'. n gnral, f h Hamlonan of h sysm (.., h nrgy opraor) s H, hn a hrmal qulbrum, () p( H). Th dnsy mar lmns hn hav hs proprs. n a hgh fld NM prmn, h largs par of h Hamlonan s h Zman nracon wh h larg rnal fld along z. For a homonuclar sysm wh NM frquncy, hs s H Z = S z, whr S z s h z componn of spn angular momnum. s hrfor a vry good appromaon o us () p( H Z ) = p( S z ). s also ru ha <<, cp a vry, vry low mpraurs. Thrfor, () S z. Th "" dos no conrbu o NM sgnals. So n sgnal calculaons, w can say () S z Ths corrsponds o an nal sa wh h nuclar spns wakly polarzd along h rnal fld.

11 Angular momnum and roaons Th "spn" of a nuclus s h oal angular momnum of h nuclus (n s ground sa). Angular momnum n quanum mchancs s rprsnd by opraors such as S, S y, and S z. Ths ar h, y, and z componns of h angular momnum vcor opraor S. Angular momnum componns do no commu wh on anohr, whch mans ha spn sas can b gnsas of only on angular momnum componn, mos commonly S z. Th commuaors ar [S,S y ] = S z and [S y,s z ] = S and [S z,s ] = S y (wh Planck's consan h bar = ). Ths commuaon rlaons dca h proprs of angular momnum opraors, and any ohr s of opraors wh h sam commuaon rlaons wll bhav n h sam way as angular momnum opraors. Each angular momnum componn dos commu wh S = S +S y +S z. So NM acv nucl hav a quanum numbr s, whch dfns h gnvalu of S o b s(s+), and hy can smulanously hav gnvalus of S z, rangng from s o +s n ncrmns of. Ths can b provn from h commuaon rlaons. s convnn o dfn spn "rasng" and "lowrng" opraors S + = S + S y and S = S S y. f m s an gnsa of S z wh gnvalu m (.., S z m = m m), hn S + m and S m ar gnsas of S z wh gnvalus m+ and m. Ecp ha, f m = s, hn S + m =. And f m = s, hn S m =.

12 Angular momnum and roaons n gnral: S m s(s ) m(m ) m and S m s(s ) m(m ) m Ths rlaons lls us h mar lmns of S + and S. n h bass m, dagonal lmns ar zro. Th only non zro lmns ar on spac off h dagonal. Usng S = (S + + S )/ and S y = (S + S )/, w can hn consruc marcs for S and S y. For a spn / nuclus, h "Paul spn marcs" ar: / / / S Sy Sz / / / For a par of spn / nucl, S = S + S, S y = S y + S y, and S z = S z + S z. How can w wr hs as marcs n h 4 dmnsonal drc produc bass { ++, +, +, }? W can hnk of such hngs as S = S + S, whr h "nsor produc" mans: aa ba ab bb a b A B ca da cb db c d C ac bc a b cc dc c d n hs way, marcs for S, S y, and S z can b consrucd for arbrarly larg spn sysms, for ampl n an NM smulaon program.

13 Angular momnum and roaons Angular momnum opraors ar "gnraors" of roaons. Ths mans ha h unary opraor () = p( S) s a roaon opraor. Spcfcally, () roas angular momna around an as gvn by h drcon of h vcor, by a roaon angl gvn by h magnud of. For ampl, () = p( S ) s a roaon around by. W apply hs roaon o sas accordng o ' = (). W apply hs roaon o opraors accordng o O' = ()O () = ()O ( ). can b shown ha: S S ( )S ( ) S S coss sn y y y z and smlar rlaons for roaons around y and z. Ths s wha on pcs for roaons n a rgh handd as sysm. Ths rlaons can b provn (laborously) by pandng p( S ) and p(s ) n Taylor srs and usng h commuaon rlaons o rarrang h rsulng prssons. z y

14 A fw nrsng facs abou roaons: Angular momnum and roaons Any produc of roaon opraors s quvaln o a sngl roaon opraor for som vcor. n ohr words, hr s always som n roaon as and n roaon angl. Any roaon or produc of roaons can b prssd as z () y () z () for som choc of "Eulr angls",,. Producs of roaon opraors can ofn by smplfd or rarrangd by rcks such as h followng: ( /) z( /) y( /) z( /) ( /) z( /) y( /) z( /) z( /) z( /) ( / ) ( / ) ( / ) ( / ) ( / ) ( / ) z y z z z ( /) ( /) z( /) z( /) () z Th hrd ln follows from h scond ln bcaus of h gnral rlaon, for any unary opraor U and any opraor A, ha A UAU U U Ths s provn asly by pandng A as a Taylor srs. n fac, for any funcon F, UF(A)U = F(UAU ).

15 How do w rprsn h nracons ha drmn NM sgnals? nracons of nuclar spns wh on anohr, wh rnal flds, and wh hr chmcal/maral nvronmn ar rprsnd by h Hamlonan opraor H(), whch s a Hrman opraor. Th gnvalus of H() ar h nsananous nrgy lvls of h spn sysm. Bcaus H() s Hrman, always has a compl s of gnsas, whch could hrfor b usd as a bass. Th Hamlonan gnrally s a sum of rms, ach rprsnng on yp of nracon. n hgh fld NM, h largs rm s h Zman nracon wh h rnal magnc fld along z, H Z = S z, whr s h Larmor frquncy n radans pr scond (agan assumng Planck's consan h bar =, so ha nrgs ar masurd n rad/s). (f hr ar wo dffrn yps of spns, calld S and for ampl, hn H Z = S S z + z.) nracons of spns wh rf pulss ar rprsnd (n h laboraory fram) by rms of h form H rf () = cos( rf )S, whr s h rf amplud, s h rf phas, and rf s h rf frquncy, or "carrr frquncy". n h roang fram, H rf = (S cos + S y sn) as wll b shown lar. Ohr rms, such as rms rprsnng spn spn couplngs, ar conand n an "nrnal Hamlonan" H n (), wh a m dpndnc ha may ars from sampl spnnng or molcular moons. als of hs rms wll b dscussd lar. So usually h Hamlonan can b wrn as H() = H Z + H rf () + H n ().

16 How do w rprsn h nracons ha drmn NM sgnals? Th Hamlonan drmns how h sa of h sysm changs wh m. n ohr words, h Hamlonan s h gnraor of m voluon, jus as angular momnum opraors ar gnraors of roaons. Tm voluon s dscrbd by a unary m voluon opraor U(), whch ransforms sas accordng o ()=U() () and ransforms dnsy opraors accordng o ()=U()()U(). d Th Schrodngr quaon of sandard quanum mchancs says () H() () d d f ()=U() () for any nal sa (), hn mus b ru ha U() H()U() d Ths s h Schrodngr quaon for U(). n h spcal cas ha H() = H for all, h soluon s U() = p( H ). f H() s "pcws consan", qual o H, H,..., H M for succssv m prods of lngh,,..., M, hn h ovrall voluon opraor up o m = M s U() = p( H M M )...p( H ) p( H ) No ha h ordr of rms n hs produc s vry mporan, bcaus n gnral [H p,h q ]. And no ha h nvrs voluon opraor (for gong backwards n m) has h oppos ordr: U() = p(h ) p(h )...p(h M M )

17 How do w rprsn h nracons ha drmn NM sgnals? n h spcal cas ha h Hamlonan dos commu wh slf a dffrn ms, s OK o add h ponns: U() = p[ (H M M H +H )] and U() = p[(h M M H +H )] f H() s a gnral funcon of m, no ncssarly pcws consan, and [H(),H(')] = for any wo ms and ', hs bcoms U() p[ H( ')d '] and U() p[ H( ')d '] f H() s a gnral funcon of m and [H(),H(')], w can wr U() Tp[ H( ')d '] whr "T wh h arrow on op" s h "yson m ordrng opraor". n pracc, hs s jus fancy noaon for h da of dvdng no many shor m prods, durng whch H() s narly consan, and rprsnng U() by H H H H H U()... M M M H

18 How do w calcula NM sgnals? n quanum mchancal rms, sgnals ar "obsrvabls", and obsrvabl quans ar rprsnd by Hrman opraors. Th "pcaon valu" of an obsrvabl O n a pur sa () a m s o() = () O (). f () happns o b an gnsa of O wh gnvalu, hn o() = () O () = () () =. f () s a lnar combnaon of gnsas n wh gnvalus n, such as () N n a () n n, hn N N N * * n' n n' n n n n n,n' n,n' n o() a () a () n' O n a () a () n' n a () So o() s a wghd avrag of h gnvalus of O, wghd by h rlav populaons of h gnsas. For a md sa, h pcaon valu bcoms o() () O (), whr h bar ovr N h op agan mans an avrag ovr "many cops". Usng h fac ha n n for any orhonormal bass { n}, w can prss o() as: n N N N o() () n n O () n O () () n n O () () n n n n N no () n Tr{O ()} n whr Tr{A} s h "rac" of A, qual o h sum of h dagonal lmns of A (n any bass).

19 How do w calcula NM sgnals? n NM, sgnals ars from oscllang magnc flu n h rf col (annna) of our NM prob, du o prcssng nuclar spn magnzaon. Magnzaon s proporonal o spn angular momnum (hrough h gyromagnc rao). Th NM sgnals F lab () (n h laboraory fram) ar hn proporonal o on of h ransvrs angular momnum componns, S. F () Tr{S ()} Tr{S U() ()U() } Tr{S U()S U() } Thrfor, lab z As you may know, NM sgnals ar apparnly dcd n h roang fram, and w masur boh and y componns of h spn magnzaon, or "ral" and "magnary" componns. Whr dos hs com from? Th ransformaon no a roang fram n NM s a spcal cas of a mor gnral ransformaon of rfrnc sysm, whch s dscrbd n quanum mchancs by som unary opraor A(). f sas n h orgnal rfrnc sysm ar (), hn sas n h nw rfrnc sysm ar '() = A() (). Smlarly, '() = A()()A(). Wha ar h Hamlonan and h voluon opraor n h nw rfrnc sysm? H'() s h ransformd d d da() d () '() [A() () ] () A()[ ] Hamlonan. No ha d d d d ncluds an ra rm, da() () A()H() () somms calld a d fcous fld (or a gaug da() fld). U'() s h A() A()H()A() '() H'() '() d voluon opraor gnrad by H'().

20 How do w calcula NM sgnals? Th roang fram n NM s h ransformaon brough abou A() = p(s z rf ), n ohr words by a roaon of spn angular momna abou z a h rf carrr frquncy (oppos o h drcon of prcsson). f H() = H Z + H rf () + H n () n h laboraory fram, wh H Z = S z and H rf () = cos( rf )S, hn Szrf d( ) H'() ( S ) [ cos( )S ] H () d rfsz Sz cos( rf )(S cos rf Sy sn rf ) H n () ( rf)sz (cosrfcossn rf sn )(S cosrf Sy sn rf ) H n () S (S coss sn ) H () Szrf Szrf Szrf Szrf Szrf Szrf Szrf z rf n z y n Th las ln follows by akng h m avrags of rms lk cos rf cos rf and cos rf sn rf, whch ar / and zro, rspcvly. Ths s h "roang fram appromaon" or h "roang wav appromaon". s h rsonanc offs. Th rf rm now looks h way w pc. Mor wll b sad abou H n () lar.

21 How do w calcula NM sgnals? Th voluon opraor n h roang fram s U'(), gnrad by H'(). Bu h voluon opraor n h laboraory fram can now b wrn as U()=A() U'()=p( S z rf )U'(). urnng o h prsson for h NM sgnals: F () Tr{S U()S U() } Tr{S ( )U'()S U'() ( )} lab z z rf z z rf Tr{ z( rf)s z( rf)u'()szu'() } Tr{(Scos rf Sysnrf)U'()SzU'() } Tr{S U'()S U'() }cos Tr{S U'()S U'() }sn z rf y z rf F ral ()cos rf F mag ()sn rf NM spcromrs ar consrucd o masur sparaly h cos rf and sn rf rms, so n h nd w g h pcd ral and magnary sgnals n h roang fram, whr prcsson around z occurs a frquncy, whr rf pulss look lk roaons around or y for = or = /, and whr h nrnal spn nracons ar changd o. H n ()

22 Nuclar spn nracons ha affc sold sa NM H n () conans magnc dpol dpol nracons (H ), ansoropc chmcal shfs (H CSA ), scalar couplngs (H J ), and lcrc quadrupol couplngs (H Q ). Ohr nracons ar mporan n sold sa physcs, bu ar no rlvan o bomolcular sold sa NM of nonparamagnc sampls. For now, w wll focus on H and H CSA. pol dpol nracon bwn wo magnc momns and, wh = S: H 3( μ r)( μ r) μ μ 3( S r)( S r) r r r S S r 3 r m 3 ( ) Y m(, )T m m Angls and spcfy h drcon of h nrnuclar vcor n h currn as sysm. Th nrnuclar dsanc s r. sphrcal harmonc funcons Y (3cos ) 6 Y sncos Y sn rrducbl nsor opraors T (3SzS z SS ) 6 [SzS z (S S S S )] 6 T (SzS SzS ) T S S

23 Nuclar spn nracons ha affc sold sa NM rrducbl nsor opraors (and sphrcal harmoncs) ransform among hmslvs undr roaons, whch maks hm usful for undrsandng how roaons affc h spn nracons. n parcular, f a nw as sysm 'y'z' s rlad o an orgnal as sysm yz by Eulr angls,,, hn T m n h orgnal as sysm urns no Ř()T m n h nw as sysm, wh: ( )T (,, )T () m m'm m' m' whr m'm () s a "Wgnr roaon mar lmn", dfnd by S Sz y Sz (, ) sm ' sm (s) m'm for a spn s parcl. (,, ) d ( ) (s) m' (s) m m'm m'm Also,, whr d () m'm s a "rducd Wgnr roaon mar lmn". rrducbl nsors hav h mporan propry ha S z S z T m T (No: us Ř abov o rprsn a "passv" roaon, manng ha w chang h ornaon of h as sysm and hn ask "Wha dos T m look lk n h nw as sysm?" Ths s h oppos of kpng h as sysm fd, roang T m, and hn askng "Wha has T m urnd no?", whch would b an "acv" roaon.) m m

24 Nuclar spn nracons ha affc sold sa NM n a non spnnng sampl (and whou molcular moons), H s consan. f hr ar no ohr nracons cp H Z, h full roang fram Hamlonan s HS H () z n Szrf Szrf z 3 m Szrf Szrf z 3 m m r m S H S ( ) Y (, ) T 3 m mrf S ( ) Y (, ) T z 3 m m r m 3 (3cos ) z 3 z 3 z z SS S Y (, )T S (3S S ) r r Ths s h famlar homonuclar dpol dpol couplng. Th dpol dpol couplng Hamlonan s "runcad" by h larg rnal fld along z, so ha only h par of H ha commus wh S z affcs NM sgnals. n hs ramn, runcaon occurs bcaus p( m rf ) avrags o zro n h roang fram unlss m =. So only h T rm survvs.

25 Nuclar spn nracons ha affc sold sa NM n h hronuclar cas, spns and hav vry dffrn valus of, so (3cos ) HSz S z (3S 3 zs z SS) r ( ) ( ) (3cos ) (Sz S z) (Sz S z) (3S 3 zs z SS ) r ( ) ( ) (Sz S z) (Sz S z) (3cos ) [S 3 zs z (S S S S )] r f w ransform o a fram of rfrnc dfnd by A() = p(h dff ), whr H dff s h rm proporonal o S z S z, w fnd ha h "flp flop" rm n h dpol dpol couplng (.., h rm nvolvng S + S +S S + ) looks rapdly oscllaory. Ths rm hn avrags ou. Transformng back, w g HS S (3cos )S S r z z 3 z z Ths s h famlar hronuclar dpol dpol couplng. s appropra whnvr h dffrnc bwn NM frquncs of spns and graly cds h srngh of h dpol dpol couplng.

26 Tha was for a non spnnng sampl. Nuclar spn nracons ha affc sold sa NM W saw ha a larg rnal fld along z runcas h dpol dpol couplng. Bu for a spnnng sampl, h drcon of h rnal fld rlav o h sampl s changng wh m, so w can no runca H rgh away. For magc angl spnnng (MAS) prmns, w mus frs drmn how H s affcd by spnnng, bfor runcang. To do hs, w ransform from a "molcul fd" as sysm, o a "roor fd" as sysm, and hn o h laboraory as sysm (whr h rnal fld s along z). Thn w can runca h couplng. y z molcul fd arbrary (powdr parn) m roor fd y z ld and spnnng z laboraory y

27 3 r 3 3 r m'',m',m 3 r 3 Nuclar spn nracons ha affc sold sa NM 3 m',m (m' ) 3 H ( ) Y (, )T m',m m 3 m m r m (,, arbrary) ( ) Y (, ) d ( )T m m m' () m m', m m' (,, ) m m m m' () m' () ( ) Y (, ) d ( ) d ( )T m m', m m'',m' m m'' runca m m m'( ) ( ) Y (, ) d ( )d ( )T m m', m,m' m No ha d (),(± m ) =. So H conans only rms ha osclla a and, and hrfor avrags o zro a h magc angl. (Unlss w also apply rf puls squncs)

28 Nuclar spn nracons ha affc sold sa NM H BS Ansoropc chmcal shfs hav h form, whr s a 3 X 3 CSA CSA mar. Ths mans ha any componn of h magnc fld B coupls o any componn of h spn angular momnum S. Bu n h prncpal as sysm of h CSA mar, whr hs mar s purly dagonal: H ( B S B S B S ) CSA y y y z z z ( B S B S B S ) BS y y 3 z z so so ( y z ) 3 s h soropc shf (or shldng), and 3 H CSA n s prncpal as sysm can hn b rwrn as: H [(B S B S B S ) (B S B S ) CSA 3 z z y y y y so 3 6T (T T ) 3 33 so BS BS Thn MAS can b analyzd n h sam way as for dpol dpol couplngs o g H CSA n h laboraory fram. Agan, h ansoropc par oscllas a and, bu wh an ra rm du o h asymmry paramr = ( )/ 3. Th soropc par bcoms so B z S z.

29 Manpulaon of spn nracons by rf pulss and sampl spnnng Puls squncs, oghr wh sampl roaon, can b usd o chang h form of spn nracons and o urn varous nracons on and off. As a smpl ampl, consdr a homonuclar wo spn sysm wh a dpol dpol couplng and wh chmcal shfs, subjcd o h followng puls squnc: 8 X 8 X / / Th voluon opraor for hs squnc s U( ) ( ) ( ) whr H and H CSA ar now akn o b n hr hgh fld, runcad forms. U( ) (H H CSA ) / (H H CSA ) / Ths can b rwrn as CSA CSA [ ( )(H H ) ( )] / (H H ) / (H H ) / (H H ) / CSA CSA whch looks lk h voluon opraor for: H H CSA H H CSA

30 Manpulaon of spn nracons by rf pulss and sampl spnnng Snc H s proporonal o (3S z S z S S ) and H Z s proporonal o S z + S z : H H and H H Z Z Srcly spakng, [H +H Z,H H Z ], so s no prcsly ru ha (H H ) / (H H ) / (H H H H ) / CSA CSA CSA CSA H U( ) Howvr, hs s a good appromaon as long as s small (compard wh h nvrs of h dpol dpol couplng srngh and h chmcal shf dffrnc). So h n ffc of hs smpl puls squnc s o mak h Hamlonan n h scond / prod dffrn from h Hamlonan n h frs / prod. And h pulss hmslvs vansh. f s suffcnly small, h ovrall voluon opraor s drmnd by an "ffcv" Hamlonan, gvn (o a lows ordr appromaon) by h mavrag of h Hamlonan. Ths s ssnally how avrag Hamlonan hory (AHT) works. n hs smpl ampl, h ffcv Hamlonan s jus H. H Z s avragd ou by h pulss.

31 Manpulaon of spn nracons by rf pulss and sampl spnnng Th mor absrac dscrpon of AHT s as follows: Consdr a roang fram Hamlonan of h form H() = H n ()+H F (). H() gnras h voluon opraor U(). H F () by slf gnras h voluon opraor U F (), whch s a roaon opraor, snc rf pulss produc roaons. magn ha h rf puls squnc consss of a block of pulss, wh lngh c calld h "cycl m", ha s rpad many ms. W ar nrsd n h voluon opraor for on compl cycl U( c ). W ar also nrsd n puls squncs whr h n roaon ovr on block s zro (or a mulpl of ), so ha h pulss alon would hav no n ffc a mulpls of c. Thus, U F ( c ) =. W ransform o a fram of rfrnc dfnd by A() = U F () and rcall h followng rsul for h Hamlonan n h ransformd fram: du () d F H'() U F() U F() H()U F()

32 Manpulaon of spn nracons by rf pulss and sampl spnnng du () can b shown ha F U F() H F() d Thrfor, H'() U () H ()U () H '() F n F n Th ovrall voluon opraor bcoms U() = U F ()U' n (), so ha U( c ) = U' n ( c ). Thrfor, c c U( ) Tp[ H '(')d'] f H n' () commus wh slf a all ms, hn c U( ) p[ H '( ')d '] p( H ' ) c n av c n c H ' H '(')d' av c n Mor gnrally, U( c ) can b prssd as p( H ff c ), wh h ffcv Hamlonan rprsnd by a "Magnus panson": () () () Hff H H H... () H H av ' () c H d ' [H ' ( '), H ' ( '')] c ' n n

33 Thngs o rmmbr: Manpulaon of spn nracons by rf pulss and sampl spnnng AHT appls only o puls squncs ha ar "prodc and cyclc", n ohr words rpv and wh U( c ) =. So, for ampl, s no usd much n soluon NM, whr such puls squncs ar rar. AHT only provds nformaon abou h ffcv Hamlonan, and hnc h NM sgnals, a mulpls of h cycl m c. So, for ampl, dos no provd nformaon abou spnnng sdbands, or ohr ffcs du o varaons whn h cycl m. AHT works wll only whn / c s larg compard o h sz of H n (or whn H n' () commus wh slf a all ms). AHT has bn vry mporan n h dvlopmn of modrn sold sa NM.

34 Homonuclar polar couplng n Sold Sa NM: Analyss wh Avrag Hamlonan Thory (Lcur nos orgnally prpard for h frs Wnr School on Bomolcular Sold Sa NM, Sow, Vrmon, January -5, 8. Subsqunly ndd n Jun 8, and corrcd and ndd n May. Corrcd agan n January 8.) obr Tycko Buldng 5, oom Naonal nsus of Halh Bhsda, M 89-5 phon: ; -mal: robry@mal.nh.gov fnon of homonuclar rcouplng Puls squncs ha cra non-zro ffcv (.., avrag) dpol-dpol couplngs among lk spns (.g., 3 C- 3 C couplngs) durng magc-angl spnnng (MAS). Movaons for homonuclar rcouplng (paral ls) () o masur dsancs bwn lk nucl () o produc crosspaks bwn lk nucl n or 3 MAS NM spcra (3) o prm doubl-quanum flrng, for slcv obsrvaon of NM sgnals arsng from pars or groups of dpol-coupld nucl (4) o prm spn polarzaon ransfrs, as rqurd for varous ohr srucural chnqus (.g., "nsor corrlaon" chnqus) Why ar puls squncs ncssary? MAS s usually rqurd for suffcn rsoluon and snsvy n sold sa NM of unornd sysms. MAS producs narrow lns by avragng ou ansoropy of chmcal shfs and magnc dpol-dpol couplngs. couplng squncs ar ndd o rsor hs nracons.

35 3 C NM spcra of unformly 5 N, 3 C-labld L-valn powdr, oband n a 4. T magnc fld a h ndcad MAS frquncs. Ouln. lvan quanum mchancal prncpls. Usful mahmacal dns and rcks. Nuclar spn nracons undr MAS V. Avrag Hamlonan Thory n smpl rms V. Homonuclar dpolar rcouplng mchansms A. la-funcon puls squncs B. Connuous rf rradaon C. Fn-puls rcouplng squncs. Chmcal-shf-drvn rcouplng V. Symmry prncpls for rcouplng squncs Appnd: rvaon of m-dpndn dpol-dpol couplng undr MAS

36 3 SCLAMES:. THE PESENTATON OF POLA ECOUPLNG TECHNQUES AN OTHE TOPCS N THESE NOTES S MOTVATE SOLELY BY PEAGOGCAL CONSEATONS. MANY USEFUL TECHNQUES AN BLLANT EAS AE OMTTE, AN MANY MPOTANT PAPES AE NOT CTE. THE GOAL OF THESE NOTES S SOLELY TO SUMMAZE THE THEOETCAL/MATHEMATCAL BACKGOUN EQUE FO AN UNESTANNG OF POLA ECOUPLNG AN ELATE TECHNQUES N SOL STATE NM.. THESE NOTES MAY CONTAN MSTAKES. PLEASE LET ME KNOW F YOU NOTCE ANYTHNG THAT SEEMS TO BE NCOECT.

37 4. lvan quanum mchancal prncpls f a spn sysm s n a sngl, wll-dfnd sa, ha sa s rprsnd by a sa vcor (). For ampl, a sysm of hr spn-/ nucl could b n h sa () a m =. Th voluon of () wh m s drmnd by h Schrodngr quaon: d () H() () (.) d whr H() s h Hamlonan opraor (n angular frquncy uns), whch conans rms ha rprsn ach of h nuclar spn nracons. f H() s consan (.., H() = H), hn Eq. (.) has h soluon () H () (.) f H() s no consan, h soluon o Eq. (.) s () U() () (.3a) U() Tp{ d'h(')} (.3b) whr T s h yson m-ordrng opraor. U() s h voluon opraor. f h m nrval from o s dvdd no N nrvals wh lnghs j durng whch h Hamlonan s Hj, Eq. (.3b) s shor-hand for H N N H N N H H U()... (.3c) whch s smply an nson of Eq. (.). Also, Eqs. (.) and (.3a) mply d U() H()U() (.4) d Sgnals n quanum mchancs ar "pcaon valus" of Hrman opraors, valuad accordng o S A () () A () (.5) () and () ar calld "k" and "bra" vcors.

38 5 n acual calculaons, Eq. (.5) would b valuad by choosng a compl bass of sas for h sysm, { n>}, ha sasfs n n' n,n'. would b rprsnd as a column vcor wh lmns n (). would b rprsnd by a row vcor wh lmns n () *. A would b a mar wh lmns n A m n h n h row and m h column. () () n NM, w don' usually hav spn sysms n sngl, wll-dfnd sas. Thrfor, w us dnsy opraors nsad of sa vcors. Th dnsy opraor s dfnd as ( ) () () (.6) whr h bar rprsns a wghd avrag ovr h spn sas ha ar prsn n h sampl. can b shown ha Eq. (.) mpls ha () sasfs h quaon d d () [ H(), ()] (.7) whr [A,B] AB - BA mans h commuaor of opraor A and opraor B. Eq. (.7) mpls () U() ()U() (.8) W usually assum ha h nal condon () bfor applyng our puls squnc s proporonal o h sum of h z componns of spn angular momnum for h rlvan nucl,.., ( ) z zk. Ths s appropra a normal mpraurs whn h spns ar a k hrmal qulbrum n a srong magnc fld along z. Sgnals ar () S A Tr{A()} (.9) whr Tr{B} s h rac of opraor B, dfnd as Tr {B} n B n f { n>} s a bass of n sas as dscussd abov. Convnonal NM sgnals ar proporonal o h ransvrs componns of spn angular momnum. n h roang fram, h NM sgnals hav ral and magnary pars, proporonal o k and y yk. Ths ar usually combnd no k k on compl sgnal S(), whch s hn S() S ral Tr{ ()} Tr( ()} Tr{ ()} Tr{ () S U() mag z U() () y } (.)

39 6 whr = y and U() s h voluon opraor for h nuclar spn sysm, rsulng from a combnaon of nracons wh rf pulss and nrnal spn nracons.. Usful mahmacal dns and rcks f A, B, and C ar normal quanum mchancal opraors, hn Tr{ABC} = Tr{CAB} (.) f h opraor A has an nvrs A -, such ha AA - =, hn Tr{B} = Tr{ABA - } A B A ABA (.) (.3) f, y, and z ar h opraors for h, y, and z componns of spn angular momnum, hn roaons of spn angular momnum (for ampl, by rf pulss) ar prssd mahmacally by quaons such as cos sn (.4a) y y z cos sn (.4b) z z y Th sam quaons hold f h followng subsuons ar mad: { y, y z, and z } or { z, y, and z y}. Ths ar cyclc prmuaons of, y, and z. s ofn usful o rprsn spn angular momna by rasng and lowrng opraors, whch hav h proprs y (.5a) z z (.5b) Whn wo opraors A and B sasfy AB = BA, hs opraors ar sad o commu wh on anohr. n gnral, quanum mchancal opraors do no commu wh on anohr. n ohr words, [A,B] AB BA. Ths mans hy can no b prmud whou changng h rsul. Howvr, f B has an nvrs B -, hn prmuaons can b accomplshd (.., h ordr or groupng of A and B can b rarrangd) by usng h followng rck: AB BA' (.6a) A' B AB (.6b)

40 7 Thrfor, f you hav a s of N opraors {Aj} and a s of N nvrbl opraors {Bj}, by applyng Eqs. (.6a,b) rpadly, you can show ha ANBNANB NA NBN...ABAB BNBNB N...BB AN'AN'A N'...A 'A' (.7a) A k k k k ' B B...B B B A B B B...B B (.7b) k k k Eqs. (.7a,b) ar on of h kys o undrsandng Avrag Hamlonan Thory and rcouplng squncs, as shown blow. Ths quaons show ha all of h opraors {Bj} can b pulld o on sd, lavng h opraors {Aj} on h ohr sd, bu n h alrd form {Aj }. As mnond abov, acual calculaons or numrcal smulaons ar usually prformd by choosng a compl bass of sas for h sysm, { n>}, ha sasfs n n' n,n' and rprsnng h dnsy opraor, Hamlonan, and ohr opraors as N X N marcs (whr N s now h numbr of sas n h bass s). For an opraor A, h numbr n h n h row and m h column would b n A m. n gnral, n A m s a compl numbr. Th adjon k of A s anohr opraor A wh mar lmns n A m m A n *. A Hrman opraor s on for whch A = A, or n A m m A n *. n quanum mchancs, H,, and all opraors ha rprsn obsrvabl quans ar Hrman. A unary opraor s on for whch A - = A, whch mpls ha N n A m m A n' * n,n'. Quanum mchancal voluon opraors U() ar unary. m n gnral, f A s Hrman, hn A and -A ar unary. Angular momnum opraors, y, and z (bu no + and -) ar Hrman, so roaon opraors (.g., ) ar unary. For non-commung opraors, ( ABC) C B A (.8a) ( ABC) C B A (.8b) Accordng o Eqs. (.8), w mus rvrs h ordr of noncommung opraors whn w ak h adjon or nvrs of a produc of hs opraors. n gnral, f [A,B], hn A B A+B B A (or A B (A+B) B A f w ar concrnd wh makng unary opraors from Hrman opraors A and B). Howvr, f boh A and B ar vry small, hn h followng appromaons can b mad: A B AB B A (.9)

41 8 On h ohr hand, f [A,B] =, Eqs. (.9) ar acly ru vn f A and/or B ar no small.. Nuclar spn nracons undr MAS NM prmns ar prformd n h roang fram. For prsn purposs, w assum ha h nuclar spn Hamlonan n h roang fram conans only four rms, rprsnng homonuclar dpol-dpol couplngs, chmcal shf ansoropy, soropc chmcal shfs, and nracons wh rf pulss: H() H () H () H H () (.) CSA CS F A. Magnc dpol-dpol couplng undr MAS For a par of spns and, h dpol-dpol couplng undr MAS can b prssd as: H() [A(, )cos( ) B(, )sn( ) C(, )cos( ) (, )sn( )] (3z z ) (.) Ths s h runcad dpol-dpol couplng,.., h par for whch [ z,h()], whr s h NM frquncy (n rad/s) and z s h Zman nracon wh h larg rnal sac fld along z (whch vanshs n h roang fram). Th full dpol-dpol couplng ncluds ohr rms (s h Appnd), whch normally don affc hgh-fld NM spcra drcly bcaus hy do no commu wh h vry srong Zman nracon, bu do conrbu o spn rlaaon. Eq. (.) s on usful way o prss H(), bu hr ar ohrs (s Scon V, for ampl). Th MAS frquncy s /. Th angls,, ar Eulr angls ha rla h ornaon of a parcular molcul whn h MAS roor o an as sysm ha s fd wh rspc o h roor: z'' ( ) y'' ( ) z' ' ' '' ( ) y' y'' z' z'' (.3) n Eq. (.3), {,y,z } ar h molcul-fd as and {,y,z } ar h roor-fd as. z () and y () ar roaons of h as abou z and y by angl. Ths as sysms ar dpcd blow:

42 9 Schmac dpcon of an MAS roor, showng roor-fd as {,y,z } and randomly ornd molculs wh molcul-fd as {,y,z }. Th wo as sysms ar rlad by roaons by Eulr angls, as n Eq. (.3), whch ar dffrn for dffrn molculs. Th MAS roor roas abou s z as. Th coffcns A(,), B(,), C(,), and (,) n Eq. (.) ar proporonal o / 3, whr s h gyromagnc rao and s h nrnuclar dsanc. W shall no worry abou h dald funconal form of hs coffcns n h analyss of rcouplng squncs prsnd blow. S h Appnd for a compl drvaon of Eq. (.), ncludng gnral prssons for hs coffcns. Th mamum valu of h oal coffcn of ( 3zz ) s 3 /. For 3 C spn pars, hs s 7.59 khz whn =. Å. No ha h dpndnc on h Eulr angl always appars as +. Ths s bcaus molculs wh dffrn valus of (bu h sam and ) dffr n hr ornaons by a roaon abou z'', whch s h MAS roaon as. So, as h sampl spns, molculs wh dffrn valus of ar road no on anohr, hus havng h sam coffcn of 3 ) a dffrn valus of. ( z z No ha h m avrag of H() s zro undr MAS. No also ha H() conans rms ha osclla a and rms ha osclla a. Fnally, no ha H() s a zro-quanum opraor. Ths mans ha H() has non-zro mar lmns only bwn sas ha hav h sam oal z componn of angular momnum. For a sysm of wo spn-/ nucl, h only non-zro mar lmns ar (3z z ) (3z z ) / (.4a)

43 (3z z ) (3z z ) / (.4b) Th dpol-dpol couplng for a sac sampl s oband by sng o zro n Eq. (.). coupld dpol-dpol nracons gnrally hav dffrn ornaon dpndncs han sac couplngs (.., dffrn dpndncs on,,), and can b zro-quanum, on-quanum, or woquanum opraors (or a mur of hs). B. Chmcal shf ansoropy and soropc chmcal shf For on spn, HCSA () [A'(, ) cos( ) B' (, )sn( ) C' (, ) cos( ) ' (, )sn( )]z (.5a) HCS z (.5b) Th soropc chmcal shf s dfnd hr o b h m-ndpndn par of h dffrnc bwn h acual NM frquncy of spn and h rf carrr frquncy rf. Undr MAS, h chmcal shf ansoropy (CSA) has h sam yp of m-dpndnc as h dpol-dpol couplng (.., rms ha osclla a and ), bu h opraor par s smply z. An mporan dffrnc s ha bu H () H () (.6a) HCSA() HCSA() (.6b) Ths allows chmcal shfs and dpol-dpol couplngs o b affcd dffrnly by puls squncs, spcally rcouplng squncs. C. ado-frquncy pulss H F () ()[ cos() sn ()] (.7) Th rf amplud s (). Th rf phas s (). y Whn a puls squnc conans shor pulss wh ampluds ha graly cd h srnghs of dpol-dpol and chmcal shf nracons and h MAS frquncy, h pulss can b rad as nsananous roaons of spn angular momna. Ths s h dla-funcon puls lm. n hs lm, h roaon (.., h voluon opraor) producd by a puls of lngh p s h opraor

44 (, ) ( cos y sn ) z z ( ) ( ) ( ) z z ( ) ( ) (.7) whr = p s h puls flp angl. n gnral, h n roaon producd by a puls squnc alon (.., gnorng H(), HCSA(), and HCS) s UF () Tp{ d' (')[ cos(') sn (' )]} y (.8) f all pulss n a puls squnc ar phas-shfd by, hn h n roaon bcoms U F (; ) Tp{ Tp{ [ z T p{ z U d' (')[ z d' d' z F (;) cos( (') ) (')[ (')[ y cos(') cos(') sn( (') )]} y y sn (')] sn (')]} z ]} z (.9) Smlarly, f w nclud all four Hamlonan rms, h voluon opraor for h puls squnc s U() Tp{ d'[ (')[ cos(') sn (')] H (') H (') H ]} y CSA CS (.) and a phas shf changs h voluon opraor o U(; ) Tp{ Tp{ [ z d'[ (')[ z U(;) z cos( (') ) d'[ (')[ y cos(') sn( (') )] H y sn (')] H (') H (') H CSA CSA (') H (') H CS ] CS z (.) Eq. (.) s vald bcaus H(), HCSA(), and HCS all commu wh z. f hs wr no ru (.., f w wr no n h hgh-fld lm), h ffc of an ovrall rf phas shf could b mor complcad. Anohr usful opraon s an rf phas rvrsal, manng () -(). Th ffc of a phas rvrsal s o chang h voluon opraor o ]} ]}

45 U' () T p{ T p{ T p{ T p{ d'[ d' d' U' '() d'[ (')[ (')[ [ [[ cos( (' )) cos (' ) (')[ (')[ y y cos (' ) sn (' )] H y cos (' ) sn( (' ))] H y sn (' )] (') H sn (' )] H (') H CSA H CSA (') H (') H (') H CSA (') H CS CSA ] CS (') H ] (') H Thus, a phas rvrsal s quvaln o changng h sgn of chmcal shfs and roang h rsulng voluon opraor U () by 8 around. V. Avrag Hamlonan Thory n smpl rms CS CS ]] ] (.) AHT [Habrln and Waugh, Phys. v. 75, 453 (968)] s a mahmacal formalsm ha allows us o analyz how puls squncs affc nrnal spn nracons,. AHT s parcularly usful n h drvaon and analyss of puls squncs ha conss of a block of rf rradaon ha s rpad many ms. Ths s prcsly h suaon ha arss n dpolar rcouplng prmns. f h rf block has lngh c (calld h cycl m), hn AHT s applcabl whn h followng condons ar m: H H H U ( ) H () (V.a) F c F ( c ) H() (V.b) CSA ( c ) HCSA () (V.c) F ( ) (V.) c H c, HCSA c, HCS c (V.3) Eqs. (V.) says ha h rf puls squnc s prodc and h nrnal spn nracons ar also prodc. n rcouplng squncs, hs mans c should b a mulpl of. Eq. (V.) says ha h n roaon producd by h rf block s zro (or somms a mulpl of ). Th rf block s hn calld a cycl. Eq. (V.3) says ha c s shor nough ha h dpol-dpol and chmcal shf nracons can no produc a larg chang n h sa of h spn sysm n on cycl. Ths allows a prurbaon hory approach such as AHT o b mployd, basd on Eq. (.9). AHT dpnds on changng h pcur n whch w vw h voluon of h spn sysm from h usual roang fram (n whch HF, H, HCSA, and HCS oghr drmn h spn voluon) o a nw fram of rfrnc n whch h rf pulss no longr appar drcly. nsad,

46 3 h rf pulss caus addonal m dpndncs n H, HCSA, and HCS. Ths nw fram of rfrnc s calld h nracon rprsnaon wh rspc o HF. Mahmacally, h nracon rprsnaon works as follows: f U() s h voluon opraor for h nr roang fram Hamlonan, w dfn a nw voluon opraor ha sasfs U() U ()U ~ F (). Thn w ask, Wha s h nw Hamlonan ha corrsponds o d Ths s asly calculad, usng h gnral rlaons U() H()U() [s Eq. (.4)] and d d [U() ] U() H() [s Eqs (.8) and rcall ha U() s unary and H() s Hrman; d also, h adjon of h numbr s ]. U ~ () U ~ ()? d d U ~ () U U d [U d d [U d F F () F () F () () H [H U()] F ] U() U ()U() U () H CSA F F () () () H CS d d [H ]U U() F F () H ()U ~ () () H CSA () H CS ]U() (V.4) Thrfor, H ~ () wh H ~ () CSA () CS (V.5) H ~ H ~ H ~ F F H ~ CSA () UF () HCSA ()UF () H ~ CS UF () HCSU F () () U () H ()U () (V.6a) (V.6b) (V.6c) So h dpol-dpol and chmcal shf Hamlonan rms n h nracon rprsnaon ar h sam as n h usual roang fram, bu wh hr spn opraor pars road by UF () (No ha hs roaon s h nvrs of U F (), so h ordr of h pulss n a puls squnc s rvrsd and h sgn of h flp angls s changd. Ths s vry mporan n AHT calculaons.) F () U acs on h spn opraor pars of H(), HCSA(), and HCS(), makng hs spn opraor pars m-ndpndn. n rcouplng chnqus, h m-dpndnc of h spn opraor pars nducd by h rf pulss nrfrs wh h spaal m dpndnc from MAS, n gnral prvnng H ~ () and/or H ~ CSA () from avragng o zro..

47 4 Bcaus of Eq. (V.), U( ) U ~ c ( c). Bcaus of Eq. (V.), Eq. (V.3), w can approma U ~ ( c ) by N U(N c ) U ~ ( c ). Bcaus of c U ~ ( c) p{ dh ~ ()} H ~ (V.7) av c wh av c dh ~ ( c ). Thrfor, as long as w car only abou h sa of h spn sysm a mulpls of c (and no n h mddl of h rf blocks), hn s suffcn o calcula h avrag Hamlonan n h nracon rprsnaon. To a good appromaon, h NM sgnals wll b drmnd by alon. H ~ H ~ av V. Homonuclar dpolar rcouplng mchansms A. la-funcon puls squncs Consdr h followng smpl puls squnc, calld AMA [s. Tycko and G. abbagh, Chm. Phys. L. 73, 46 (99)] 3 : How do w us AHT o calcula wha hs dos o homonuclar dpol-dpol couplngs and chmcal shfs? Bcaus h squnc consss of dla-funcon pulss, U F () s pcws-consan:, / U F (), (V.), Abbrvang Acos( ) Bsn( ) Ccos( ) sn( ) from Eq. (.) by (A,B,C,), h nracon rprsnaon Hamlonans ar

48 5 H ~ H ~ H ~ (A,B,C,) (3z () (A,B,C,) (3 y (A,B,C,) (3 CSA CS z y z z ), ), ), (A',B',C',') z, () (A',B',C',') y, (A',B',C','), z, () y,, z z (V.) (V.3a) (V.3b) Th avrag Hamlonans ar H ~,av 3( y y C [sn( 4 z A B z){ [sn( ) sn( )] [cos( ) cos( ) sn( )] [cos( ) cos( )]} 4 (V.4) A' B' H ~ CSA,av (y z){ [sn( ) sn( )] [cos( ) cos( )] C' ' [sn( ) sn( )] [cos( ) cos( )]} 4 4 (V.5a) H ~ CS,av [( ) y z z ] (V.5b) So boh h dpol-dpol couplng and h CSA ar rcoupld, and h soropc chmcal shf s alrd. No ha only h 3 z z par of H() conrbus o h rcoupld dpol-dpol Hamlonan. Ths s a gnral rul, bcaus h par s no affcd by h rf pulss. Now consdr a longr AMA puls squnc wh addonal 8 pulss: )]

49 6 For hs squnc U H ~ H ~ H ~ F () 3 /, /,,,,, (A,B,C,) (3z z ), (A,B,C,) (3 y y ), (A,B,C,) (3z z ), () (A,B,C,) (3z z ), (A,B,C,) (3 y y ), (A,B,C,) (3z z ), CSA CS (A',B',C',') z, (A',B',C',') y, (A',B',C',') z, () (A',B',C',') z, (A',B',C',') y, (A',B',C',') z, z, y, z, () z, y, z, (V.6) (V.7) (V.8a) (V.8b)

50 7 H ~,av Now, s h sam as bfor (bcaus s no affcd by h addonal 8 pulss), bu H ~ H ~ (bcaus h 8 pulss chang h sgns of and H ~ CS() CSA,av CS, av H ~ () H ~ CSA () n h scond roor prod). pol-dpol couplngs ar now slcvly rcoupld. Through h dpndnc on,, and, H ~,av dpnds on molcular ornaon n h MAS roor. polar lnshaps ar hn powdr parns. Th lnshap dpnds on -, bcaus h A, B, C, and rms hav dffrn rlav magnuds for dffrn valus of -. n h lm ha ( ) /, h lnshap rsmbls h Pak doubl parn of a sac sampl for a wo-spn sysm. No ha h opraor par of H ~, av undr AMA s dffrn from h opraor par of H(). Undr h AMA squncs dpcd abov, zro-quanum and doubl-quanum rms. H ~,av conans boh Smulad dpolar powdr parns for wo-spn 3 C sysm wh mamum splng =.5 khz ( =.48 Å) Puls squncs usd n AMA prmns [Chm. Phys. L. 73, 46 (99)] Eprmnal dpolar powdr parns (Fourr ransform wh rspc o ) for ( 3 CH3)C(OH)SO3Na powdr, for whch =.5 Å.

51 8 B. Connuous rf rradaon Consdr h followng puls squnc, calld Q-HOO [s N.C. Nlsn, H. Bldso, H.J. Jakobsn, and M.H. Lv, J. Chm. Phys., 85 (994)] 4 : n ohr words, h cycl hr s a 9y-36-9-y squnc, wh dla-funcon 9 pulss and a vry long, wak 36 puls. For hs squnc, / y UF () / y / (V.9) for < <. Th nracon rprsnaon Hamlonans ar hn H ~ () (A,B,C,) [3( cos sn )( cos sn y y ) ] (A,B,C,) [3( cos sn ( )sn cos y y y y ] 3 (A,B,C,) [ (( y y)cos ( y y) (y y)sn ) ] 3 (A,B,C,) [ (( )cos ( )sn ( )) ] 4 (V.) H ~ () (A',B',C',') ( cos sn CSA y ) (V.a) H ~ () ( cos sn CS y ) (V.b) Eq. (V.) maks us of h dns cos ( cos) / and sn ( cos) /, as wll as = + y. Alhough H ~ () conans boh doubl-quanum and zro-quanum opraor rms, only h doubl-quanum rms osclla n m. Thrfor, only h doubl-quanum rms ar rcoupld. Usng h rlaons k n d cos cos k,n,

52 9 k n d sn sn k,n, and k n d sn cos for ngrs k and n, s sraghforward o show ha H ~ H ~ and ha CSA,av CS, av 3 H ~,av [( )(A cos Bsn ) ( )( Asn Bcos )] (V.) 8 Thus, h Q-HOO squnc cras a pur doubl-quanum rcoupld Hamlonan (assumng ha h AHT appromaon s vald and ha h pulss ar prfc). Th avrag Hamlonan conans wo doubl-quanum rms, whos magnuds dpnd on h Eulr angls,,. nrsngly, h ovrall magnud of s ndpndn of h angl. couplng H ~,av squncs wh hs propry ar calld -ncodd. Ths propry causs h dpolar powdr parn lnshap undr Q-HOO and ohr -ncodd squncs o hav wo sharp horns (s blow), whch mans ha h sgnal dcay undr s srongly oscllaory. Ths s H ~,av good for quanav masurmns of nrnuclar dsancs, and also for doubl-quanum flrng ffcncs. Smulad and prmnal Q- HOO daa for a 3 C- 3 C par wh.54 Å [from Nlsn al, J. Chm. Phys., 85 (994)].

53 C. Fn-puls rcouplng squncs Consdr h followng squnc, calld fn-puls ado-frquncy-rvn couplng or fpf [s Y. sh, J. Chm. Phys. 4, 8473 () and A.E. Bnn, C.M. nsra, J.M. Grffhs, W.G. Zhn, P.T. Lansbury, and.g. Grffn, J. Chm. Phys. 8, 9463 (998)] 5,6 : Th cycl m s 4, hr s on 8 puls n ach roor prod, and h puls lngh s a sgnfcan fracon of h roor prod. Th rf amplud durng ach puls s p. For hs puls squnc, p p )/ 3 ( p p )/ ( p p )/ ( p p / F 4,3 3,3 3,,,,,, () U y y y p y y y p y p y p (V.3) Usng h dny z y, Eq. (V.3) can b smplfd:

54 U F () y y ( ( ) / ) /, 3 /, z, p ( ) / p p z, p p p,,, p p z z,3 4 3 p 3 p p p (V.4) gnorng h rm (whch s no rcoupld, as pland abov), h nraconrprsnaon dpol-dpol Hamlonan s hn (A,B,C,) [3(z cos y sn )(z cos y sn )], p p p p p (A,B,C,) (3z z), p ( ) ( ) ( ) ( ) (A,B,C,) [3( z cos sn )(z cos sn )], p p p p p (A,B,C,) (3z z), p H ~ () ( ) ( ) ( ) ( ) (A,B,C,) [3(z cos y sn )(z cos y sn )], p p p p p (A,B,C,) (3 z z), p 3 ( 3) ( 3) ( 3) ( 3) (A,B,C,) [3(z cos sn )(z cos sn )],3 3 p p p p p (A,B,C,) (3z z),3 p 4 Ths can b rwrn as (V.5a) H ~ () (A, (A, (A, B,C, ) B,C, ) B,C, ) (A, B,C, ) [3( z z cos y y sn (z y y z )sn cos )], p p p p p (A, B,C, ) (3z z ), p ( ) ( ) ( ) ( ) [3( z z cos sn (z z )sn cos )], p p p p p (A, B,C, ) (3 z z ), p ( ) ( ) ( ) ( ) [3( z z cos y y sn (z y y z )sn cos )], p p p p p (A, B,C, ) (3 z z ), p 3 ( 3 ) ( 3 ) ( 3 ) ( 3 ) [3( z z cos sn (z z )sn cos )],3 3 p p p p p (A, B,C, ) (3 z z ),3 p 4 (V.5b) No ha h sgns of z+z and zy+yz ar rvrsd n h hrd and fourh roor prods, rlav o h frs and scond roor prods. Ths s a consqunc of h choc of

55 phass n h fpf squnc [calld XY-4 phass 7 ], and causs hs sngl-quanum rms o cancl ou n h avrag Hamlonan. Evaluang h ngrals n h avrag Hamlonan, w fnd (omng many nrmda sps) H ~,av 4 4 {4 {4 d H ~ d (A, B, C, ) (3 p 3 { ( 4 3 6( [3 z z p p () d (A, B, C, ) (3 p [sn( ) [sn( ) ] d H ~ () z z 3 ) 4 y y )sn d H ~ d (A, B, C, ) (3 p p } p () p d H ~ z z () cos ) p 3 3 p d H ~ ()} 3 p ) sn ]A [cos( p ) cos ]B ( 4 ) 3 p ) sn ]C [cos( p ) cos ]} 6( ) (V.6) Mraculously (or as a consqunc of symmry, as dscrbd blow), h avrag dpol-dpol Hamlonan undr h fpf squnc s a zro-quanum opraor wh h sam opraor form as h dpol-dpol couplng n a non-spnnng sampl. Ths has usful consquncs n cran applcaons, for ampl by allowng das ha wr orgnally dvlopd for NM of sac solds o b appld n MAS prmns 8,9. Undr fpf, s also ru ha H ~ H ~. CSA,av CS, av n Eq. (V.6) s h rf amplud durng h 8 pulss. No ha H ~, avvanshs as and p (.., n h dla-funcon puls lm). As shown blow, a dffrn rcouplng mchansm, ladng o a dffrn avrag dpol-dpol Hamlonan, coms no play whn h coupld spns hav larg chmcal shf dffrncs. Ths chmcal-shf-drvn rcouplng mchansm dos no dsappar n h dla-funcon puls lm.

56 3. Chmcal-shf-drvn rcouplng. oaonal rsonanc Now consdr h cas whr wo dpol-coupld spns hav a larg dffrnc n hr soropc chmcal shfs. gnor chmcal shf ansoropy for now. Th nuclar spn Hamlonan undr MAS s hn H() z ( ( z z z )( ) H z z ( () ) z ( z ) H () )( z z ) H () (V.7) whr and ar h sum and dffrnc of h wo chmcal shfs (acually, rsonanc offss). Now go no an nracon rprsnaon wh rspc o h chmcal shf rms. f U() s h voluon opraor for H(), hs nracon rprsnaon s dfnd by U() U CS U ~ () U ()U ~ () (V.8a) CS () p{ [ p[ Tp{ ( ( d' H ~ z z z (')} z ) ( )]p[ z ( z z )]} z )] (V.8b) (V.8c) H ~ () UCS () H()UCS () p[ ( )]H () p[ z z (z )] z (A, B, C, ) p[ ( )](3 ) p[ z z z z (z )] z (A, B, C, ) p[ ( )][ ( )]p[ z z z z (z z)] (A, B, C, ) (z z) (A, B, C, ) p[ ( )]( )]p[ z z (z z)] (V.9) Eq. (V.9) uss h facs ha [(z+z),h()] =, so h par of h chmcal shf dos no affc H ~ (), and ha [(z-z),zz] =, so h zz par of H() s no affcd by h chmcal shfs. Also, Eq. (V.9) uss h dny z z ( ). Now, usng h dny H ~ () (A, B,C, ) (A, B,C, ) z z ( ) z z ( ) z z (A,, Eq. (V.9) bcoms B,C, ) ( ) (A, B,C, ) [(cos sn ) (cos sn ) ]

57 4 (V.) Eq. (V.) shows ha h zz par of h dpol-dpol couplng s no rcoupld by h chmcal shf dffrnc, bu h flp-flop par can b rcoupld f or Ths ar h n = and n = roaonal rsonanc condons -4. A h roaonal rsonanc condons, dpol-dpol Hamlonan a roaonal rsonanc s H ~,av H ~ () s prodc wh prod and AHT can b appld. Th avrag [(A B) 4,n (C ),n ] [(A B) 4,n (C ),n ] for n = or n =. No ha hs s a zro-quanum opraor, bu s no h sam as h zroquanum opraor crad by h fpf squnc.. (V.) n a sysm wh many 3 C-labld ss, roaonal rsonanc allows pars of spns wh parcular chmcal shf dffrncs o b rcoupld slcvly 5,6, as shown blow. Howvr, unlss h MAS frquncy s vard durng h puls squnc, h couplngs can no b swchd on and off. Thrfor, ohr approachs o frquncy-slcv homonuclar dpolar rcouplng ha mploy rf pulss and do no dpnd on bng acly on roaonal rsonanc hav bn dvlopd by svral groups C NM spcra of U- 5 N, 3 C-alann powdr, rcordd a.8 MHz, wh MAS frquncs ndcad by dashd lns. A h roaonal rsonanc condons, h rcoupld lns bcom dpolar powdr parns, whl ohr lns rman sharp. Clos o roaonal rsonanc condons (C and ), h man NM lns of srongly coupld pars hb apparn shfs on h ordr of.5 ppm, whch can complca accura masurmns of chmcal shfs n unformly labld sampls.

58 5. ado-frquncy-rvn couplng Consdr h Hamlonan n Eq. (V.7), bu wh an addonal HF() rm ha corrsponds o h followng puls squnc, wh m bng an arbrary posv ngr: Ths squnc, calld SEA 3 or mor commonly F 6,4, can b analyzd by frs ransformng h Hamlonan no an nracon rprsnaon wh rspc o HF(), and hn no an nracon rprsnaon wh rspc o h chmcal shfs. Followng h sam prncpls as usd abov for ohr rcouplng squncs, h Hamlonan n h nracon rprsnaon wh rspc o HF() s H ~ () ( ( ( z z z z z z ) ) ) ( ( ( z z z z z z ) H ) H ) H (), (), c (), 3 / 4 c c 3 / 4 / 4 c c / 4 (V.) H() s no affcd drcly by h dla-funcon 8 pulss, bu h chmcal shfs chang sgn bwn h wo 8 pulss. Thrfor, h soropc chmcal shfs now appar o b m-dpndn. n h scond nracon rprsnaon, h dpol-dpol couplng bcoms H ~ p[ ( z z)]h () p[ (z z)], c () p[ ( c c z z)( )]H () p[ (z z)( )], c / 4 p[ ( z z)( c)]h () p[ ( z z)( c)], 3 c / 4 3 / 4 c (V.) gnorng h zz par of H(), whch as shown abov s no rcoupld, Eq. (V.) can b wrn as H ~ () (A, B, C, ) [ (A, B, C, ) [ ( / ) ( ) [ c (A, B, C, ) ( / ) (c ) c c ], / 4 3 / 4 (V.3) ], c ], 3 c c / 4 / 4 c c / 4 c

59 6 Th avrag Hamlonan s hn sn(m / ) sn(m / ) H ~ m [( ) (A cos Bsn ),av (C cos sn )]( ) m ( ) m (4 ) (V.4) No ha h avrag dpol-dpol Hamlonan, valuad n h doubl nracon rprsnaon dscrbd abov, s non-zro for narly all valus of h chmcal shf dffrnc. Ths s bcaus h 8 pulss, spacd m apar, forc h flp-flop rm n () H ~ () H ~ o b prodc wh prod m. For narly all valus of, has non-zro Fourr componns a and/or. Howvr, H ~,av as. Thus, h rcouplng mchansm for F n h dla-funcon puls lm (.., whn h 8 pulss ar vry shor compard wh ) s qualavly dffrn from h rcouplng mchansm n h fn-puls lm (.., whn h 8 pulss occupy a sgnfcan fracon of h roor prod). n acual prmns, h phass of h 8 pulss ar usually chosn o follow an XY-4 or hgh XY-n phas parn 7, bcaus XY-n phas parns compnsa for rf nhomogny, rsonanc offss, and ohr mprfcons n h 8 pulss. Ths maks h F chnqu (and h fpf vrson) qu robus and usful n many prmnal suaons. V. Symmry prncpls for rcouplng squncs A. Lv s C squncs Malcolm Lv and hs collagus hav dvlopd an approach o h dvlopmn of rcouplng squncs ha rls on gnral symmry proprs of puls squncs, whch lad o slcon ruls ha rval whch yps of nracons can b rcoupld by a squnc wh a gvn symmry. Squncs blongng o wo dsnc symmry classs hav bn dscrbd. Th frs class ncluds "C" squncs, comprsd of rf blocks (calld C lmns) ha produc no n roaon of spn angular momna 5,6. Th gnral form for a C squnc s: n ohr words, h C squnc conans N rpons of h C lmn, wh ovrall rf phas shfs ha ncras n uns of, and wh a oal cycl m of n roor prods. Th phas

60 7 ncrmn sasfs / N. N, n, and ar posv ngrs. Th symmry s rprsnd by h symbol CNn. To analyz h ffc of a CNn squnc wh AHT, on consdrs a gnral nuclar spn Hamlonan undr MAS ha s a sum of rms of h form Hm () Am T n h roang fram [.., bfor ransformng o an nracon rprsnaon wh rspc o HF()]. Am s a funcon of h Eulr angls,, dscussd abov. m s -, -,, or, and s anohr posv ngr. T s h "lmn of an rrducbl nsor opraor of rank " ha commus wh h oal spn angular momnum z. Whou gong no h dals of rrducbl nsor opraors, hs mans T s an opraor ha s a mmbr of a s of + opraors {T}, wh bng an ngr ha sasfs -. For dpol-dpol couplngs, = and h rlvan s of opraors s T T T (3 ) m ( ) (V.) 6 z z z z Thus, T s a -quanum opraor. For soropc and ansoropc chmcal shfs, h rlvan opraors hav =. mporan proprs of rrducbl nsor opraors nclud: z T z T (V.a) T T ( ) z T ( ) T (V.b) (V.c) Elmns of h s of opraors {T} ar ransformd no on anohr by roaons of spn angular momnum (.., by rf pulss n NM prmns). Thrfor, n h nracon rprsnaon wh rspc o h frs C lmn, h nuclar spn Hamlonan s a sum of rms of h form H ~ A ~ m () () m m T n h nrval < < n/n. Thr ar 4(+) such rms. Th nracon rprsnaon Hamlonan for h k h C lmn mus hn b a sum of rms of h form H ~ A ~ m () () (k ) (k ) m m ( z T z ) (V.3) n h nrval (k-)n/n < < kn/n, akng no accoun h ffc of h ovrall rf phas shf as n Eqs. (.8-.). Th avrag Hamlonan for h k h C lmn wll hn b a sum of rms of h form

61 8 H ~ m,av N (k) n T kn r / N (k)n N n / N d H ~ (k) m m () (k)n / N n r / N da ~ m () m (V.4) Eq. (V.4) maks us of Eq. (V.a) and h fac ha A ~ m () s prodc, wh prod qual o n/n (h lngh of on C lmn). Th oal avrag Hamlonan, for h nr cycl m n, s oband by summng ovr conrbuons from all C lmns. Th oal avrag Hamlonan wll hn conan 4(+) rms of h form H ~ m,av T T T N n N n N n k N n n { r r (k) m / N / N da ~ da ~ m m () () (k)n m m N k N k / N n r / N (k) m da ~ m () (k)n (k)( mn)/ N m / N } (V.5) For such a rm o b non-zro, h sum ovr k mus b non-zro. Bu urns ou ha h followng rlaon s always ru, for any posv ngr N: N k (k)q / N N, q NZ, q NZ (V.6) whr Z s som ohr ngr. So h quany - mn mus b zro or anohr ngr mulpl of N for h avrag Hamlonan undr a C squnc o conan non-zro -quanum rms ha ars from MAS-nducd oscllaons a frquncy m. Ths s h slcon rul for CNn squncs. Th C7 and POST-C7 rcouplng squncs 7,8 ar good ampls. For hs squncs, N = 7, n =, and =, whch mply h slcon rul ha -m =, 7, 4, c. Thus, rms wh = and m = or = - and m = -ar rcoupld (corrspondng o doubl-quanum dpolar rcouplng). Trms wh = and ar no rcoupld, corrspondng o h absnc of CSA rcouplng and h absnc of boh zro-quanum and on-quanum dpolar rcouplng. (call ha m has only h valus and, bu no, undr MAS.) C7 and POST-C7 dffr n h choc of h C lmn slf, whch s br compnsad for rsonanc offss n h POST-C7 cas.

62 9 B. Lv s squncs Th scond symmry class consdrd by Lv and coworkrs ncluds h "" squncs 9,3, whch hav h gnral form Th squnc consss of N rpons of an " lmn" n a cycl m n. N s an vn ngr. Th rf pulss n h lmn produc a n roaon by 8 around. Th vrson alrnas wh h - vrson. For h vrson, all rf pulss n h lmn ar phasshfd by / N. For h - vrson, all rf pulss n h lmn ar frs rvrsd n sgn, hn phas-shfd by -. Th symmry s rprsnd by h symbol Nn. Analyss of squncs s smlar o analyss of C squncs, bu h fac ha on lmn producs a n 8 roaon mans ha T and T- rms mus b rad oghr [s Eq. (V.b)]. For h Hamlonan o b Hrman, hs mus occur n h nracon rprsnaon as rms of h form H ~ A ~ m () () T ( ) A ~ m m() * m m T [so ha H ~ m H ~ () m() ; s Eq. (V.c)]. Consdrng only on m, combnaon nally, h voluon opraor for h lmn, sarng a =, s U Tp{ n / N d[h F () (A m m * m m A )T ]} (V.7a) Thn, n h AHT appromaon and consdrng only on m,, combnaon, n / N m * m U p[ d m () T ( ) m() T )] (V.7b) (A ~ n gong from Eq. (V.7a) o Eq. (V.7b), h fac ha HF() producs a n roaon by 8 around has bn usd. n h nracon rprsnaon wh rspc o HF(), T bcoms a sum of rms of h form c()t + (-) c() * T-. n Eq. (V.7b), A ~ m() Amc(). A ~

63 3 Trms proporonal o also prsn. m T and m T ar no shown plcly, bu of cours ar Th corrspondng quaons for h phas-rvrsd lmn, sarng a =, ar [s Eq. (.)] U ' Tp{ n / N d[ m m * m m H () (A A )T ]} F n / N m * m U' { p[ ( ) d (A ~ m () T ( ) A ~ m() T )]} n / N m * m p[ d (A ~ m () T ( ) A ~ m() T)] (V.8a) (V.8b) Eqs. (V.7) and (V.8) show ha U and U dffr by chang of T and T- n h nracon rprsnaon. ncludng roaons abou z o accoun for h phas shfs, h voluon opraor for h frs - par s n / N m m U z p{ d[a ~ n / N m () T ( ) A ~ m()* T]} z n / N m m z p{ d[a ~ m () T ( ) A ~ m()* T ]} z n / N m m z p{ d[a ~ n / N m () T ( ) A ~ m()* T]} n / N m m z p{ d[a ~ m () T ( ) A ~ m()* T ]} z n / N mn / N m mn / N m z p{ d[a ~ m () T ( ) A ~ m()* T]} n / N m m z p{ d[a ~ m () T ( ) A ~ m()* T ]} z n / N mn / N m mn / N m z p{ d( ) [A ~ m () T ( ) A ~ m()* T ]} n / N m m z p{ d[a ~ z m () T ( ) A ~ m()* T ]} 4 n / N mn / N m 3 mn / N m 3 z p{ d( ) [A ~ m () T ( ) A ~ m()* T ]} n / N m m p{ d[a ~ m () T ( ) A ~ m()* T ]} Th oal voluon opraor s (V.9) (N / ) N Uoal U ~ ' (k)u ~ z (k) (V.) k wh U ~ (k) and U ~ '(k) dfnd by

64 3 U ~ n / N d[a ~ 4mnk / N m (k) p{ () (4k) T ( ) A ~ m (V. a) 4mnk / N m (4k ) ()* T ]} m U ~ n / N [A ~ mn(k ) / N m (k) p{ d( ) () (4k3) ' T ( ) A ~ m (V.b) mn(k ) / N m (4k 3) m ()* T ]} N z No ha N n Eq. (V.) s a mulpl of, so h opraor has no ffc. Th avrag Hamlonan for h nr squnc s oband by summng h ponns n U(k) and U (k) ovr all valus of k and dvdng by n. Thrfor, h coffcn of T n h avrag Hamlonan arsng from MAS-nducd oscllaons a m s proporonal o n / N d[a ~ (N / ) (N / ) m () / N m ] p[4(mn )k / N] ( ) p[4(mn )(k / ) / N] k k n / N d[a ~ N N m () / N m ] p[(mn )k / N] ( ) p[(mn )k / N] k k (vn) (odd) n / N d[a ~ N m () / N k m ] ( ) p[(mn )k / N] k (odd and vn) (V.) Eq. (V.) shows ha f s vn, h coffcn of T s zro (.., no rcouplng) unlss mn - s an vn mulpl of N/. f s odd, h coffcn of T s zro unlss mn - s an odd mulpl of N/. Ths ar h slcon ruls for Nn squncs. Many ampls of Nn rcouplng squncs hav bn rpord. Th fpf squnc dscrbd abov s a vry smpl ampl, for whch n = 4, N = 4, and =. Th symmry slcon ruls ndca ha dpol-dpol couplngs ( = ) can b rcoupld n a zro-quanum ( = ) form, wh m = or m =. Chmcal shfs ( = ; = or ) can no b rcoupld. Ths s n agrmn wh h dald calculaons dscrbd abov. C. Cyclc m dsplacmn symmry and consan-m rcouplng Anohr usful symmry propry of rcouplng squncs nvolvs hr bhavor whn all pulss ar cyclcally dsplacd n m whn h cycl m c = n. n h pcur blow, a block of pulss (or dlays) of lngh, calld P, s dsplacd by, causng h rmanng block P o mov from h nd o h bgnnng of h cycl:

65 3 Wha s h ffc of hs cyclc dsplacmn on a rcoupld Hamlonan? f h roaons producd by rf puls blocks P and P ar U() and U(), wh m masurd from h bgnnng of ach block, hn h oal roaon bfor h cyclc dsplacmn s UF(): U(), U F() (V.3) U( )U ( ), n Th oal roaon afr h cyclc dsplacmn s UC(): U(), U C() (V.4) U( )U( ), n n h nracon rprsnaon wh rspc o h orgnal puls squnc, a Hamlonan rm m of h form H m() A m T undr MAS bcoms (rcallng ha + = n) H ~ m () A A m m m m U ( A m m ) U ( () ) m m U() m ' U( ) U A U T T T (') U (), U ( T U U (), )U ( ), n (')U ( ), ' (V.5) wh = +. n h nracon rprsnaon wh rspc o h cyclcally dsplacd puls squnc, h sam Hamlonan rm bcoms

66 33 Ĥ m () A A m m m m U A m m ) U ( m Am m '' U U ( ( () ) () ) U U T T T ('') U U (), U ( )U (), T U ('')U ( ), n ( ), '' (V.6) wh = +. n boh cass, h avrag Hamlonan s h ngral of h nracon rprsnaon Hamlonan ovr m from o n, dvdd by n. Accordng o Eqs. (V.5) and (V.6), h ngral n boh cass s h sum of wo ngrals, ovr m nrvals of lngh m and. callng ha m and U( )U( ), can b shown ha h ngrands for h cyclcally dsplacd puls squnc ar qual o h ngrands for h orgnal puls squnc afr roaon by U() - = U() and mulplcaon by m. Thus, m U ( ) H ~ U ( ) (V.7) Ĥ m,av m,av Eq. (V.7) summarzs h ffc of a cyclc m dsplacmn on h avrag Hamlonan for an arbrary rcouplng squnc. Why s hs usful? As an ampl, consdr a gnral rcouplng squnc ha nds n wo prods of lngh /3. urng hs fnal wo prods, hr no pulss ar appld or h appld pulss produc a n roaon of. For such a squnc, wo succssv cyclc dsplacmns by m/3 4m /3 /3 mulply h avrag Hamlonan by and. For m = and m =, h oal avrag Hamlonan for h puls squnc oband by concanang h orgnal m /3 squnc wh h wo dsplacd vrsons wll b zro, bcaus + + 4m / 3. Ths provds a smpl mans of crang a consan-m dpolar rcouplng chnqu 3, as shown blow usng cyclcally dsplacd vrsons of h fpf squnc:

67 Consrucon of a consan-m dpolar rcouplng squnc from h fpf squnc. urng h k prod, dpolar rcouplng by h A, B, and C blocks cancls du o Eq. (V.7). Only h k3 prod has a n rcouplng ffc. Thus, by dcrmnng k and ncrmnng k3 whl kpng k + k3 consan, h ffcv rcouplng prod can b ncrasd from o k(k + k3). For all valus of, h oal fpf prod s k(k + k3) and h oal numbr of pulss s consan. Thus, sgnal dcay du o spn rlaaon, ncompl proon dcouplng, and puls mprfcons s mnmzd. Th dpndnc of NM sgnals on s du prmarly o dpol-dpol couplngs rahr han hs ranous ffcs, allowng h rcouplng daa o b analyzd n a quanav mannr. [from J. Chm. Phys. 6, 6456 (7)] 34