81Hamiltonian r L 81 I1 and I2 areseparatedbyadistance r z I1I2 xyq r thentheenergyofinteractionbetweenthetwomagneticdipoles 1 and 2 isgivenbyE0 4 1 2 ID: 338695
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' &$ % 8{NMRInteractions:DipolarCoupling 8.1Hamiltonian ~ =\r ~L : (8.1) ~^I1 and ~^I2 areseparatedbyadistance r , z I1I2 xyq r ' &$ % thentheenergyofinteractionbetweenthetwomagneticdipoles, ~1 and ~2 isgivenbyE=0 4[( ~1 ~2 )r 3 3( ~1 ~r)( ~2 ~r)r 5] (8.2) where0isthepermeabilityconstantandisequalto410 7kg:m:s 2:A 2(whereAisforAmperes).Ifthetwodipoleshavethesameorientationwithrespectto~r,then rqmB01m2 ( ~1 ~r)( ~2 ~r)=( ~1 ~2 )r2cos2 (8.3) andthus, ' &$ % E=0 4( ~1 ~2 )r 3[1 3cos2]: (8.4) where ~1 ~2 = 12 sincetheyareparallel.Thereforetheenergyhasanangulardependence: NOTE: Thatthe rstnullintheenergyoccursat=1 p (3)or54.7degrees.Thisangleisknownasthemagicangle.TheHamiltonianisthengivenbyreplacingthevectors ~1 and ~2 bytheircorrespondingoperators\r1h ~^I1 and\r2h ~^I2 inequation8.2above,i.e. ' &$ % ^HD=0 \r1\r2 h2 163[(~^I1~^I2)r 3 3(~^I1~r)(~^I2~r)r 5]: (8.5) ThisequationrepresentsthefulldipolarHamiltonian.Wecanwritethevector~rintermsofpolarcoordinates,i.e.~r=(rx;ry;rz)=(rsincos ;rsinsin ;rcos) (8.6) andthereforerewritethedipolarHamiltonianas^HD=0 \r1\r2 h2 163r3[A+B+C+D+E+F] (8.7) whereA=^I1z^I2z(1 3cos2)B= 1 4h^I+1^I 2+^I 1^I+2i(1 3cos2)C= 3 2h^I+1^I2z+^I1z^I+2isincose i D= 3 2h^I 1^I2z+^I1z^I 2isincosei E= 3 4^I+1^I+2sin2e 2i F= 3 4^I 1^I 2sin2e2i ' &$ % (8.8) ThesetermsoftheHamiltoniancanberepresentedasamatrixoftheform(A+B+C+D+E+F)=0BBBBB@termsfromA:::C:::C:::E:::D:::A:::B:::C:::D:::B:::A:::C:::F:::D:::DtermsfromA1CCCCCA (8.9) TheeigenvaluesofthismatrixgivetheenergylevelsofthedipolarHamiltonianinzeromagnetic eld. abba B E,F bb C,DC,DC,DC,D AAAA Inthepresenceofanexternalmagnetic eld ~B0 , ' &$ % however,manyofthehigherordertermscanbeneglected,i.e.ifthefollowingconditionisvalid:j01 02j 0 \r1\r2 h2 163r3 (8.10) Thisapproximationiscalledasecularapproximation.Solet'shavealookatwhichofthetermsabove(A,B,...,F)remainandwhichcanbeneglected: A A,writteninthebasissetof ; ; ; ,hasdiagonalelements: jjAj &#x-277;&#x.592;=Ajkjk (8.11) Thesetermsarealwayspresent. B Bhaselementsbetween and jBj &#x-277;&#x.592;= jBj &#x-277;&#x.592;= 1 4(1 3cos2) (8.12) When01=02,i.e.inthehomonuclearcase,theapproximationabovedoesnotholdandtheBtermmustbekept. ' &$ % CandD Thesetermsconnectlevelsseparatedby1,i.e.singlequantum.Forhigh elds,theycanbeneglected. EandF Thesetermsconnectlevelsseparatedby2,i.e.doublequantum.Forhigh elds,theycanbeneglected.ThustheHamiltonianinthecaseofhomonuclearcoupling(betweenlikespins)is^HD= 0 \r1\r2 h2 163r31 2(3cos2 1)[3^I1z^I2z ~^I1~^I2] (8.13) whereasinthecaseofheteronuclearcoupling,itisgivenbyHD= 0 \r1\r2 h2 163r3(3cos2 1)^I1z^I2z: (8.14) Theconstantterminfrontofequations8.13and8.14arethehomonuclearandheteronucleardipolarcouplingconstants,respectively.Typicalvaluesfor1H,13C,and15Nare: ' &$ % IS= 0\rI\rSh 82rIS3(inHz) (8.15) with\r1H=42:5759106Hz:T 1\r13C=10:7054106Hz:T 1\r15N=4:3142106Hz:T 10=410 7N:A 2h=1:0546210 34J:s (8.16) ThusHH:r3=120000Hz:A3CC:r3=7500Hz:A3NN:r3=1200Hz:A3HC:r3=30000Hz:A3HN:r3=12000Hz:A3CN:r3=3000Hz:A3 (8.17) Notethatsincethestaticmagnetic eldliesalongthez-axisinthe gureonpage1,thedipolar ' &$ % interactionhasanorientationaldependencewithrespectto ~B0 givenbytheexpression3cos2 1.Thismanifestsitselfbyadependenceoftheobserveddipolarsplittingontheorientationofthecrystalliteinasinglecrystalintheprobe(recalltheorientationdependenceoftheCSAforsinglecrystals).Forapowdersample,aPakepattern,whichisthesumofthespectraofindividualcrystalliteswhicharerandomlydistributedinthesample,isobserved.Themaximumsplittingwhichcanbeobservedis3IIforthehomonuclearcaseand2ISfortheheteronuclearcase. ' &$ % ref:R.E.Wasylishen,EncyclopediaofNMR,GrantandHarris(eds.) 8.2SphericalTensorNotation Asmentionedpreviously,manyofthetermsinthespinsphericaltensorsarethesameforthescalarcouplingasforthedipolarinteraction:T10=0 ' &$ % T11=0T20=1 p 6(3IzSz IS)T21=1 2(IzS+ISz)T22=1 2IS (8.18) Note,thathereIchosetowritethetwospinsasIandSinsteadofI1andI2,asabove.Bothnotationsareequivalent.Thechoicedependsonyou.Thespatialpartsare:APAS20=p 6ISA21=0A22=0 (8.19) whereIS= 0 4\rI\rSh r3IS (8.20) isthedipolarcouplingconstant.Asbefore,wecantransformthespatialpartintoany ' &$ % arbitraryframe:A20=r 3 2IS(3cos2 1)A21=3 2ISsin(2 )i\rA22=3 2IS(sin2 )i\r: (8.21) 8.3MeasuringtheDipolarSplitting Thedipolarinteractioncanbemeasuredinanumberofways.AswiththeCSA,themethodsuseddependonthestateofthesample.Forapowder,forinstance,onecanobtaindipolarinformationfromthepowderpatterndirectly,asshowninb): ' &$ % ref:R.E.Wasylishen,EncyclopediaofNMR,GrantandHarris(eds.)Thedipolarinformationcanbeextractedby ttingthespectratoextractthedipolarsplitting.Again,aswiththemethodstodeterminetheCSA,thismethodof ttingspectraislimitedtothecaseswherethelinesarenotoverlapped.Otherwise,two-dimensionalmethodssuchasthe\separatedlocal eld"experiment(SLF)orthe\polarizationinversionspinexchangeatthemagicangle"experiment(PISEMA)canbeused.Bothoftheseexperimentsworkwellonpowdersaswellas ' &$ % singlecrystals. chemicalshift(ppm)ref:A.Ramamoorthy,S.J.Opella,SolidStateNMR,4,387-392(1995). chemicalshift(ppm)ref:A.Ramamoorthy,C.H.Wu,S.J.Opella,J.Magn.Reson.,140,131-40(1999). ' &$ % Formoredetailsondipolarspectroscopyseealso: 1. M.Engelsberg,EncyclopediaofNMR,GrantandHarris(eds.). 2. K.Schmidt-Rohr,H.W.Spiess,MutlidimensionalSolid-StateNMRandPolymers,AcademicPress,SanDiego,CA,1994. 8.4ImportanceoftheDipolarInteraction 1. Insolution,thoughthedipolarinteractionisaveraged(becauseall'saresampled),itstillplaysaroleincross-relaxationandisusedinNOESYspectroscopy-moreonthislater.Relaxation. 2. Insolids,thedipolarinteractionisusedtogetdistanceandorientationalinformation:e.g.REDORRotational-EchoDoubleResonanceisaMASexperimentformeasuringdistancebetweentwoselectivelylabelledsite(e.g.13Cand15N).MASaveragesthedipolarinteractionbetweenthese ' &$ % two,sothepulsesequenceworkstoreintroducethisinteractionbybreakingtheaveraging: DataiscollectedforanincreasingnumberofrotorcyclesN. ' &$ % e.g.PISEMAUsingboththeorientationaldependenceofthechemicalshiftandthatofthedipolar ' &$ % interaction,itispossibletodeterminethethree-dimensionalstructureofanalignedproteinusingthePISEMApulsesequence.