, given by. , I y. and I z. , are self adjoint, meaning that the adjoint of the operator is equal to the operator. This follows as A.

Transcription

1 Further relaxation 6. ntrodution As resented so far the theory is aable of rediting the rate of transitions between energy levels i.e. it is onerned with oulations. The theory is thus erfetly aetable for rediting the rate onstants for relaxation of longitudinal magnetization ("T ") but is inaable of dealing with transverse magnetization ("T "). To do this it is neessary to onsider the evolution of oherenes under the random hamiltonian. A onvenient way of roeeding is to look at the relaxation behaviour of oerators. This fits in well with the Solomon equations an aroah already used for disussing the dynamis of z-magnetization and oulations. The oerator method is less well suited to disussing the relaxation behaviour of individual lines but it will be seen that with some modifiations is an be adated to this use. This Setion starts with a brief revision of the roerties of oerators. 6. Proerties of oerators 6.. Matrix reresentations Oerators were introdued in Setion. as the way in whih observable quantities are reresented in quantum mehanis. t is sometimes onvenient to think of oerators as matries and the matrix reresentation of an oerator an be formed in a artiular basis set of wavefuntions (Setions.. and.7.). The ijth element (meaning the element in row i and olumn j) of the matrix reresentation of an oerator A is A ij given by Aij = φ * i A φ j d τ φ i A φ j where φ i is a basis funtion. The same oerator will have different matrix reresentations in different sets of basis funtions. f a matrix has the roerty that A ji = A * the matrix is said to be hermetian. Oerators whose matrix reresentations are hermetian are alled hermetian oerators. Examles of suh oerators are the familiar angular momentum oerators x y and z. The adjoint of an oerator A A is defined in the following way * ( A ) = A ij ji Hermetian oerators suh as x are self adjoint meaning that the adjoint of the oerator is equal to the oerator. This follows as A * ji = Aij for a hermetian oerator and so A = A. ij ij The raising and lowering oerators + and are not hermetian a fat that an readily be areiated from the matrix reresentations: ij 6

2 = + i = + x y x iy 0 = = These oerators are the adjoints of one another: + = = + Again this is readily areiated from the matrix reresentations. The trae of an oerator is equal to the sum of its diagonal elements { }= Tr A 6.. Basis sets of oerators The density oerator σ an be exanded in a basis of oerators B i : bb bb bb i A ii σ = + + K = where the b i are numerial oeffiients. A basis set of wavefuntions an be desribed as being orthonormal (Setion..) whih means that they have the roerties φi φj = 0 if i j; and = if i = j For oerators orthogonality is defined using the trae. Two oerators B i and B j are orthogonal if Tr{ BB i j }= 0 To omute the trae it would be neessary to find the matrix reresentations of B i and B j multily these together to give another matrix and then add u the diagonal elements of the resulting matrix. The oerators x y and z are all orthogonal to one another. Normalization also involves omutation of the trae of the adjoint of an oerator with itself Tr{ BB i i }= β i Sometimes a basis set of oerators are hosen so that β i = and sometimes the oerators are hosen so that β i is the same for eah oerator but not neessarily =. For examle: { }= { } { }=. Tr Tr = Tr x x y y z z The set of rodut oerators for two sins ( x x z et.) all have β i = ; the fator of in the oerator roduts with two sin oerators is needed to kee the β i the same for all the oerators. 6.. Commutators The ommutator of two oerators is written [AB] and defined as follows [ A B]= AB BA f A and B are numbers or simle funtions then the order in whih they aear does not matter. For examle = and x y = y x. Suh simle funtions are said to ommute; the ommutator between them is zero i i i 6

3 [ xy ]= xy yx= 0 However not all oerators are simle funtions and so they do not neessarily ommute with one another. n artiular the angular momentum oerators obey the following ommutation relations x y and z. [ ]= [ ]= [ ]= i i i x y z y z x z x y The seond ommutator an be found from the first by yli ermutation of the indies: x y y z z x; likewise the third an be found from the seond by further yli ermutation. Using these relationshis and the definitions of + and the following an be derived [ ]= [ ]= [ ]= + z + z + z Oerators for different sins always ommute with one another and an oerator always ommutes with itself. For examle [ x z]= [ x z z x z z]= Commutator relations The following roerties are often useful [ AB ]= [ BA ] [ PA B]= P[ A B] if P ommutes with A and B A+ B C A C B C [ ]= [ ]+ [ ] 6... Oerator roduts The ommutator relations given in Setion 6.. for x y and z aly for any sin. For sin only the following relationshis also aly = i = i and for yli ermutations of x y and z x y z y x z = = = E where Eis the unit oerator or matrix x y z There are also similar seial relationshis for the raising and lowering oerators. = ( ) + = = 0 + = + z + Sine [ x y]= iz and x y= iz it follows that = [ ] = [ ] x y x y y x y x These relationshis are also valid for yli ermutations of x y and z. f these relationshis aly (i.e. for sin half) then it follows that [ AP BQ]= 0 if A and B both ommute with P and Q An examle of the latter is when A and B are oerators of sin and P and Q are oerators of sin [ xy yz]= 0 Another seial ase whih will be of use is z 6

4 AP AQ A P Q P Q A or rovided A ommutes with P and Q. 6 [ ]= [ ]= [ ] = x y z 6. Oerator equations The time evolution of oherenes is onveniently handled using the density matrix introdued in Setion.7. Further it is useful to make an oerator exansion of this density matrix as this fits in well with the rodut oerator aroah (Setion ) used to desribed many multile ulse exeriments. n addition this oerator aroah is losely related to the Solomon equations introdued in Setion 5... The starting oint is the equation of motion of the density matrix (or oerator) dσ () t = ih [ σ () t] where the square braket indiates the ommutator [ A B]= AB BA As before the hamiltonian onsists of a stati art H 0 and a random time deendent art H (t) H = H0 + H() t A transformed density oerator σ T and transformed time deendent hamiltonian H T () t are defined as T T σ ()= t ex( iht) σ() t ex( iht) H ()= t ex( ihth ) () t ex( iht) t an be shown that the equation of motion of σ T is () = () () T dσ t T T ih [ t σ t] [] This transformed reresentation is alled the interation reresentation. t is useful as σ T just evolves under the ation of the random hamiltonian; the evolution due to the large stati hamiltonian has been fatored out. t will be seen that the influene of H 0 aears as a simle hase evolution of H T (). t Equation [] an be solved to seond order in the erturbation reresented by the random hamiltonian to give () = () [ ] T dσ t T T T H t [ H ( t τ ) σ ( t ) ] dτ [] 0 There are various aroximations involved in this derivation all of whih rely on the fat that the random hamiltonian reresents a small erturbation and that the theory is to be used for times longer than the orrelation time τ. Equation [] is the master equation for the motion of the density oerator. However this equation has one defet: it redits that at long times the density oerator goes to zero. This is inorret at long times the density oerator must go to its equilibrium value whih is not zero but orresonds to the equilibrium oulation distribution. The reason for this defet is

5 idential to that disussed in Setion 5.. it omes from the failure to treat the lattie as a quantum objet. The solution to this roblem is either to treat that lattie roerly (whih will be rather omlex) or simly to relae σ T by (σ T σ eq ) where σ eq is the equilibrium density oerator. n all that follows it will be assumed that this relaement is made at the end of the alulation. 6.. Oerator exansion To develo Eq. [] into a useful form H (t) is exanded aording to q H t F t A q ( q) = () where the F (q) (t) are satial funtions and the A (q) are sin oerators. This exansion allows for the ossibility that there are several different terms with different satial fators in the random hamiltonian. Sine ollisions ause the moleule to randomly hange its orientation the satial funtions beome random funtions of time. The oerators A (q) are hosen so that they evolve under the stati hamiltonian aording to = 0 0 q q q ex ih t A ex ih t ex iω t A [] where the frequeny ω (q) is harateristi of the oerator A (q). For examle in the ase of a single sin for whih H 0 = ω 0 z the oerator + evolves in the way given by Eq. []: ex( ih t) ex( ih t)= ex( iω t) with ω (q) = ω 0. Likewise and z evolve at frequenies +ω 0 and 0 resetively. n more omlex ases it may be that A (q) is a sum of oerators all of whih have the same satial funtion F (q) (t) but all of whih do not evolve at the same frequeny under the transformation reresented by Eq. []. For examle in the diolar interation there is a set of oerators S + z z S + S whih all have the same satial deendene but whih under the stati hamiltonian H 0 = ω z + ω S S z evolve at frequenies 0 ω +ω S and ω ω S resetively. To oe with situation suh as these Eq. [] is modified to ( q) ( ex ih t A ex ih t ex i q ) ( q) ( 0 ) 0 ω t A [] = where the sum over allows for the ossibility that the A (q) is a sum of ( q oerators A ) ( q eah with its own assoiated frequeny ω ). The hamiltonian must be hermetian so it follows that for eah oerator with a ositive index q there must be one with a negative index with the two oerators being related by ( q) ( q) [ A ] = A [5] where the dagger ( ) indiates the adjoint. For examle 6 5

6 6 6 [ ] = [ z] = + Likewise for eah F (q) there is an assoiated F ( q) related by F ( q) = F (q)*. t ( q) ( q) follows from these definitions that ω = ω. Substituting the exansion of Eq. [] into the master equation (Eq. []) gives after onsiderable maniulation T dσ t where J q () = () q' q ' q' ( q) ( q' ) T ( A A σ t Jq q ω q ' ) ( [ [ ]] ( ) ( i( ω q ) ( + ω q ' ) ' ' ' ' ) t) ex ω is the Fourier transform of the orrelation funtion G q = ( ) J ω G τ iωτ dτ q q' q q' ex and where the orrelation funtion is defined as q q' = () z q' [6] ( τ ) q q' G τ F t F t τ [7] 6.. Simlifiation by negleting ross orrelation and non-seular terms The relaxation behaviour redited by Eq. [6] deends on the orrelation funtion between all ossible interations i.e. all ossible airs qq'. However as has been seen above it is often the ase or an be assumed to be the ase that there is no orrelation between the flutuations of different terms in the random hamiltonian; in other words ross orrelation an be ignored. f ross orrelation is indeed ignored then the only non-zero orrelation funtions are ones with the same index q for both funtions i.e. () ( τ ) q q * F t F t By definition F ( q) = F (q)* this non zero orrelation funtion an be written () ( τ ) ( ) q q F t F t whih from Eq. [7] is G q q' with q' = q i.e. G qq n Eq. [6] there is a omlex exonential whih auses a hase osillation q q' ( ( + ω ' ) t) ex i ω q q f ross orrelation is ignored q' = q and as ω = ω this term beomes q q q q ex i ω ( ω ) ( ( + ' ) t)= ex( i( ω ω ' ) t) ( q) ( q) f ω and ω ' differ signifiantly then this terms auses a raid hase osillation so the term will not ontribute to relaxation simly as it is onstantly hanging sign. The terms whih will ontribute are those with ω( q) ( q) = ω ' for whih there is no hase osillation. This ondition is only met if = '. Terms whih satisfy this restrition are termed seular ontributions (as oosed to the remainder whih are non-seular). Thus retaining only the seular terms and ignoring ross orrelation simlifies Eq. [6] to

7 T dσ t q q T [ A [ A ( σ t ]] ) Jqq( ω ( q) ' ) q () = () [8] 6.. Relaxation of individual oerators As was desribed in Setion 6.. it is ommon to exand the density oerator is exanded in terms of a basis of oerators B i σ()= t b() t B i where the oeffiients b i arry the time deendene. Evolution under ulses and delays auses the oerators to transform into one another. t would be useful to have a similar way of handling the effet of relaxation on the oerators. n suh an aroah eah oerator would relax in a harateristi way erhas being transferred to other oerators. The theory would redit these rate onstants for self relaxation and for transfer to other oerators. Suh a way of treating the motion of oerators under the influene of relaxation has already been introdued as the Solomon equations. For examle the equation = 0 d z z 0 0 R ( z z ) σ S( Sz Sz ) says that z relaxes with a rate onstant R and that S z is transferred to z with a rate onstant σ S. The aim is to use the master equation Eq. [8] to rodue similar equations for any oerator B i. Suose that the density oerator just ontains a single oerator B with oeffiient b (t). To use Eq. [8] the density oerator has to be transformed to the interation reresentation σ T ()= t ex( iht 0 ) σ() t ex( iht 0 ) = ex( ih0t) b() t Bex( ih0t) = b() t B ex( iω t) where it has been assumed that under this transformation the oerator B aquires a hase modulation at a harateristi frequeny ω. The exonential term an be merged into the oeffiient by defining so that ()= () b' t b t ex iω t ()= () σ T t b' t B The right-hand side of Eq. [8] beomes i [ [ ]] ( ( q A A b t B J ) ' qq ω ) ( q ) q '() q The effet of the double ommutator will in general be to transform the oerator B into other oerators; there will be some oeffiient for this transfer deending on the details of the oerators and the setral densities. n general the result an be written i 6 7

8 () + () + () + K b ' tb b' tb b ' tb where j is the oeffiient for the transfer of oerator to oerator j. So Eq. [8] beomes d ( b' () t B)= b' () t B + b' () t B + b' () t B + K = b' () t B j j j However in general the density oerator will not start out as just onsisting of a single oerator B but will be a sum of many oerators σ T ()= t b' () tb + b' () tb + b' () tbk = b' () t B i i i Eah of these oerators an be transformed into any other oerator by the ation of the double ommutators and if as above the oeffiient for transfer of oerator B j into oerator B i is ji the result is d bi' () t Bi b' t B b' t B b' t B b' tb b' tb b' tb + b ' () tb + b' () tb + b' tb = b' () tb i = () + () + () + + () + () + () + i j ji j i () + K K K This begins to look like a Solomon equation but is rather omlex as it has a sum of derivatives on the left. To ik out just one of these say that of the oerator B k both left and right hand sides are multilied by the oerator B k and the trae taken d j bj' () t Tr{ Bk Bj} jibj' t Tr Bk Bi = () { } The oerators are assumed to be orthogonal and to have a onstant normalization fator β: Tr { B Bq}= 0 if q = β if = q The β anel and so the differential equation beomes d ( bk' () t )= jkbj' () t [9] j This is a Solomon-tye equation. What it says is that if the rate of hange of the "amount" of oerator B k b k (t) deends on the amounts of all other oerators B j b j (t) resent. The rate onstant for transfer of oerator B j into oerator B k is given jk. The rate onstant jk is simly found by omuting i j [ ] q q q jk = Tr ( Bk ) A [ A ' Bj ] Jqq ω β ( ) [0] q 6 8

9 n words what this means that starting with a single oerator B j it is seen how this oerator is transformed into other oerators by the double ommutators; from the resulting set of oerators just the omonent of B k is iked out. A knowledge of the jk rate onstants gives a omlete set of Solomon equations and so a omlete desrition of the relaxation behaviour of the set of oerators. n ratie therefore the theory is used to alulate these jk rather than the relaxation of an arbitrary density oerator Time deendene Equation [9] is written in the interation reresentation with the oeffiients b i '(t) having a hase osillation due to the evolution of the oerators under the stati hamiltonian. ()= () b' t b t ex iω t [] i i i The aim is now to searate the evolution due to relaxation from that due to evolution under the stati hamiltonian. Starting from Eq. [9] the oeffiients b i '(t) are relaed by those from Eq. [] d ( bk' () t )= jkbj' () t j d ( bk() t ex( iωkt) )= jkbj() t ex( iω jt) j dbk () t ex( iωkt)+ bk() t iωkex( iωkt)= jkbj() t ex( iω jt) j dbk () t = bk () tiωk + jkbj() tex ( i( ω j ωk ) t) [] j Equation [] an be interreted in the following way. The first term on the right desribes the hase evolution of the oerator B k due to the stati hamiltonian; it has nothing to do with relaxation. The sum of terms on the right gives the rate of relaxation-indued transfer from oerator B j to B k with rate onstant jk. However eah term on the right inludes a hase osillation at (ω j ω k ); suh terms will not be effetive at transferring B j to B k unless (ω j ω k ) = 0 so that this hase osillation goes away. Unless this is the ase the sign of the relaxation terms will hange raidly (muh more raidly than the rather slow rate of relaxation) and so the net effet will be zero. For examle suose that the two oerators involved are + and S + ; the harateristi frequenies are therefore ω and ω S (the two Larmor frequenies) and as these are not the same any relaxation indued transfer between the two oerators whih in rinile might take lae will atually have no net effet. n ontrast the two oerators z and S z whih both have harateristi frequenies of zero an have relaxation indued transfer between them (just as has been seen in the Solomon equations). So Eq. [] an be written 6 9

10 dbk t bk t iω k j' kbj' t [] j' where the sum over j' is taken to inlude only those terms for whih ω j' = ω k. Conentrating on just the relaxation arts of the equation it is ossible to write Eq. [] in terms of the oerators rather than their oeffiients as was done in the ase of the Solomon equations dbk = jk ' Bj' 6 0 () = () + () j' As before it is imortant to remember that this equation is just a short hand it is not the oerators that are hanging with time but their ontributions to the density oerator. 6.. Sueroerators Often it is aetable to think of relaxation and evolution under the stati hamiltonian as being searate even though in reality they are learly taking lae at the same time. However there are oasions on whih it is essential to onsider the two roesses ating at the same time. Essentially this is what a omlete set of equations like Eq. [] (for all oerators B k ) will do. A onvenient way of exressing suh equations is to use the idea of sueroerators. The starting oint is the equation of motion of the density oerator: dσ () t = ih [ σ () t] For suh an equation it is usual to think of σ and H as being matries exressed in some basis of wavefuntions. However if σ is exressed as a linear ombination of oerators as in the rodut oerator formalism it is then natural to think of σ as a (olumn) vetor of the oeffiients the individual oerators. The equation of motion is then written dσ () t ˆ = ihσ () t [] where Ĥ is the hamiltonian sueroerator; this an be thought of as a matrix. Formally the sueroerator is defined as a ommutator: ˆ Hσ = [ H σ]= Hσ σh f H is time indeendent the solution to Eq. [] is σ()= t ˆ ex iht σ ( 0) Relaxation an be added to Eq. [] by introduing a relaxation sueroerator ˆΓ dσ() t ih ˆ σ t ˆΓ = () { σ () t σ } eq where ˆΓ is defined via

11 [ [ ' ]] q qq ( ) q ˆ T T Γσ t A ( q ) A q σ t J ω ()= () n the ase that σ is written as a vetor of oerators ˆΓ will be a matrix with the elements jk given by Eq. [0]. 6. Examle alulations 6.. One sin with random field The simlest ase to onsider is a single sin exeriening a random field in the x y and z diretions given by the hamiltonian H t ω t ω t ω t = () + () + () x x y y z z t will be onvenient to re-exress the x and y oerators in terms of the raising and lowering oerators to give + + H t ω t ω t ω t ()= () [ + ]+ () [ ]+ () x y i z z [ x y ]+ [ x()+ y() ]+ z() z = () () + ω t iω t ω t iω t ω t From this three searate terms are identified q A (q) F (q) (t) ω (q) 0 z ω z () t 0 + ωx() t iωy() t ω ωx()+ t iωy() t +ω where the stati hamiltonian H 0 has been taken as ω z so that ex( ih0t) ± ex( ih0t)= ex( miω t) ± ex( ih0t) zex( ih0t)= z For eah value of q there is just one term so the index in Eq. [8] is not needed. The orrelation funtion G (τ) is () ( ) G( τ)= F () t F ( t + τ) = ω () t iω () t ω ( t τ)+ iω ( t + τ) [ ][ + ] x y x y [ x x+ x() y( + ) y() x( + )+ y() y( + τ )] = ω () t ω t + τ iω t ω t τ iω t ω t τ ω t ω t f ross orrelation between the x and y omonents is ignored then the ensemble averages of the seond and third terms is zero. The first and seond terms will be written So that ω () t ω ( t + τ)= ω g( τ) ω ( t) ω ( t + τ)= ω g( τ) x x x y y y The orresonding setral density is G J [ x y] ( τ)= ω + ω g τ [ x y] ( ω)= ω + ω j ω 6

12 The orrelation funtion G (τ) is the same as G (τ). G 00 (τ) is similarly given by ( 0) ( 0) G00( τ)= F () t F ( t + τ) = ωz() t ωz( t + τ) = ωz g( τ) and so J00( ω)= ωz j( ω). The aroah will be to start with Eq. [0] [ ] q q q jk = Tr ( Bk ) A [ A ' Bj ] Jqq ω β ( ) [0] q and to ut eah oerator B j in turn into this relationshi. To start with just the art in the round brakets will be evaluated Motion of z The term to alulate is 6 [ ] ( ( q ) qq ω ) ( q) q [ ' z] A A J q where q = 0 + ; the index is not used. The following terms are found = [ [ ]] = [ ] = = [ [ ]] [ ] = [ ] ω = [ ] = q 0 : z z z J00 0 z 0 J q : J ( ω )= J ( ω )= J ω z z + [ ] ( ) J ω zj ω + q : z J The double ommutators generate no new oerators so all that haens is that z relaxes on its own. t is not therefore neessary to go through the formal alulation of the trae. Rather the oeffiient 00 an be iked out as: { + } 00 = J ω J ω = J( ω) J( ω) = [ ωx + ωy] j( ω) [ ωx + ωy] j( ω) = [ ωx + ωy] j( ω) where is has been assumed that the setral densities at ±ω 0 are the same. The Solomon equation is therefore dz = 00z = R z z where R z the longitudinal relaxation rate onstant is Rz = [ ω x + ω y] j( ω ) As exeted the rate of longitudinal relaxation deends on the size of transverse fields and the setral density at the Larmor frequeny.

13 Motion of + The term to alulate this time is ( q) [ ] qq ( ω ) ( q) [ ( q ) ' + ] A A J q where q = 0 + ; the index is not used. The following terms are found [ z z + ] 00=[ z + ] 00= 00 + q 0 : J 0 J 0 J 0 = [ ] = [ ] = [ ] [ ] = [ ]= q : J ω [ + + ] ( )= [ + z] = ( ω ) q : J ω J ω J + The double ommutators generate no new oerators so as before all that haens is that + relaxes on its own. The oeffiient that is the rate onstant for this roess an be extrated by insetion as [ 00 ω ] J 0 J = + = ω j( 0) ω + ω j ω [ ] z x y The Solomon-tye equation is therefore d+ = + = R t + where R t the transverse relaxation rate onstant is [ ] Rt = ωz j( 0)+ ωx + ωy j ω 6... nterretation t is interesting to omare the longitudinal and transverse relaxation rate onstants Rz = [ ω x + ω y] j( ω ) 0 [ ] = + Rt = ω j( 0)+ ω + ω j ω0 ω j 0 R z x y z z Longitudinal relaxation only deends on the setral density at the Larmor frequeny but transverse relaxation deends on the setral density both at the Larmor frequeny and at zero frequeny. The two ontributions to transverse relaxation are quite distint. Longitudinal random fields give a ontribution whih deends on the setral density at zero frequeny; this is alled the seular ontribution as no energy hange is involved. Transverse random fields give a ontribution whih deends on the setral density at the Larmor frequeny; this is alled the non-seular ontribution as hanges of energy are involved as the sins fli. Furthermore longitudinal relaxation is only brought about by transverse magneti fields but both transverse and longitudinal magneti fields give rise to transverse relaxation. The longitudinal fields are assoiated with the setral density at zero frequeny. The seular art of transverse relaxation 6

14 Reall that transverse magnetization is due to the resene of a oherene in the system; this oherene has a hase osillation at the Larmor frequeny and this in turn is what gives rise to the measured reession of transverse magnetization at the Larmor frequeny. Eah sin exerienes the alied magneti field along the z-diretion lus any loal random field ω z (t). As a result the frequeny of the hase osillation of the ontribution to the oherene from eah sin has a sread of values aross the samle. As time roeeds therefore these individual ontributions get out of hase with one another and so the oherene (and hene the transverse magnetization) deays. For suh a mehanism to be effetive at dehasing oherene it is not neessary for there to be any time variation in the fields along the z- diretion all that is required is that there is a distribution of fields aross the samle. However the dehasing aused by a omletely stati distribution of fields is not lassed as relaxation sine the dehasing ould be reversed by the aliation of a 80 refousing ulse. ndeed this kind of dehasing is exatly that whih ours when a field gradient ulse is alied. Relaxation is a dissiative irreversible roess whose effets annot be undone with ulses. So for a distribution of fields in the z-diretion to be effetive at ausing transverse relaxation these fields must be time deendent so that their effet annot be reversed by a refousing ulse. The time deendene has a different role in transverse and longitudinal relaxation. n the latter the time deendene is needed to ause transitions in the former it is needed to inhibit refousing. The seular art of transverse relaxation deends on j(0) whih = τ. So as the orrelation time gets shorter the rate of transverse relaxation dereases. To understand why this is it is useful to imagine first that the sins are frozen and not moving. The distribution of fields aross the samle will give rise to a range of different Larmor frequenies and as a result the line observed in the setrum will be broad. Now suose that the sins start to move; they will jum from osition to osition exeriening a different field eah time (just like hemial exhange but between very many sites). As the juming rate beomes omarable with the original linewih the line will start to narrow (again just like exhange narrowing). What is haening is that there are now so many jums that all the sins are beginning to see the same average frequeny. As the jums beome muh faster than the original linewih the line is narrowed drastially; however there is still a residual wih whih deends on the rate of juming and the original linewih for the stati arrangement. For tyial NMR samles the linewih of the frozen samle would be rather large (tens of khz or more). However tyial orrelation times for liquids are in the s to ns range; these times are an indiation of the time between jums. With suh fast motion omared to the frozen linewih the line observed in the liquid is very muh narrower than in the frozen samle. Nevertheless the extent of narrowing deends on the rate of 6

15 juming and this is why the relaxation rate onstant is roortional to τ. The smaller τ the faster the juming and the narrower the line. The non-seular art of transverse relaxation Longitudinal relaxation is assoiated with hanges in oulations and hene transitions between sin states. Suh roesses involve a transfer of energy between the sins and the lattie. t is also lear that a transition from one sin state to another will ause an interrution in the hase of the ontribution of a artiular sin to the overall oherenes. Thus suh transitions also lead to transverse relaxation. The non-seular ontribution to the transverse rate onstant is R z. The half arises beause one sin fliing auses the oulation differene and hene the z-magnetization to hange by two units. Sin flis are thus twie as effetive at ausing longitudinal relaxation as they are at ausing transverse relaxation. 6.. Variation with orrelation times For simliity it will be assumed that n whih ase ω = ω = ω = ω R x y z Rz ωr j ω Rt ωr j 0 ωr j ω The simlest model is to assume that the orrelation funtion is exonential g(t) = ex( τ/τ ); this gives a lorentzian setral density = = + τ j( ω)= + ωτ The relaxation rate onstants are therefore τω R τ Rz = Rt = ωr τ + + ωτ + ωτ Several imortant oints an be drawn from these relationshis. The first is that the rate of transverse relaxation is always greater than or equal to that of longitudinal relaxation. This follows as the differene is learly always ositive. R Rt R z = τωωτ [5] + ωτ 6... Extreme narrowing The extreme narrowing or fast motion limit is when the orrelation time is so short that ω 0 τ <<. f this is the ase then + ωτ 0 and the setral density is just τ at all frequenies. The rate onstants for longitudinal and transverse relaxation are equal in this limit: R t ex. narrow = = τω ex. narrow R z As the orrelation time inreases out of the extreme narrowing limit the transverse relaxation rate onstant beomes greater than the longitudinal R 6 5

16 rate onstant as an be seen from Eq. [5]. For a 500 MHz setrometer the extreme narrowing limit imlies a orrelation time of muh less than 00 s. A small moleule in a nonvisous solvent might have a orrelation time of 0 s or less whih would lae the moleule easily in the extreme narrowing limit Sin diffusion The sin diffusion or slow motion limit is when ω 0 τ >> so that the setral density at zero frequeny is muh larger that that at the Larmor frequeny. n suh a ase the transverse relaxation rate onstant is dominated by the term whih deends on j(0) R t sin diff. = ωτ R The longitudinal rate onstant goes on getting smaller and smaller as the orrelation time inreases in rinile beoming vanishingly small in the limit ω 0 τ >>. There is thus a strong ontrast in the behaviour of the longitudinal and transverse rates. This differene omes about beause longitudinal relaxation requires motion at the Larmor frequeny but transverse relaxation an be aused either by motion at zero frequeny or at the Larmor frequeny. As was noted in Setion 5.. for a given frequeny the setral density is a maximum when ω 0 τ. Therefore the longitudinal relaxation rate onstant is a maximum when the orrelation time is suh that ω 0 τ. R t R z ( ωτ 0 ) The longitudinal and transverse relaxation rate onstants as a funtion of the orrelation time for fixed Larmor frequeny. For short orrelation times the two rate onstants are equal but as the orrelation time inreases the longitudinal rate onstant goes through a maximum and then falls away. n ontrast the transverse rate onstant ontinues to inrease. 6.. More than one sin with random field Suose that there are two sins eah exeriening a random field so that the hamiltonian is H t ω t ω t ω t ω ts ω ts ω ts = () + () + () + () + () + () whih an be written x x y y z z Sx x Sy y Sz z + () ( ) z S S S z + () 0 0 H ()= t F + F + F + S F + S F + F S 6 6

17 where [ ] ( ± ) S z S x y 0 S S S F ω t F ω t m iω t = () = () () The sin oerators in the hamiltonian are the only ones whih an affet sins oerators in the density oerator and likewise for S sin oerators. So the set of double ommutators [ ] ( ( q ) qq ) ( q) q [ ' ] A A α Sβ J ω q where α and S β are any and S sin oerators an be searated into two arts ( S A q ) A q β α J ' q q [ ] qq( ω ( ) ) [ ] Sq [ ] Sqq( ω ( ) ) q ( A Sq ) A S q α ' Sβ J + [ ] [6] The sub and suersrit labels and S on the oerators setral densities and frequenies distinguish those for the sins from those for the S sins. As an examle of using this searation suose that the relaxation behaviour of the oerator rodut + S z is required. From Eq. [6] [ ] q qq( ) + ( ) ω S A q A q ( ) z ' J q A Sq A S q + ( ) ( ) ( + [ [ Sz ]] J Sq ) ' Sqq ( ω ) q The first braket has already been alulated in the revious setion when the relaxation of + was onsidered; likewise the seond braket was also alulated above when the relaxation of S z was onsidered. So the relaxation rate onstants for + S z is just the sum of these two terms RS ( + z) = R ( + ) + RS ( z) ( ) ( ) ( S) ( S) = ( z ) j+ ( x ) + ( y ) j x y j S + + ω 0 ω ω ω ω ω ω n fat this aroah will work for any oerator with any number of terms in it. The overall relaxation rate onstant will be the sum of the rate onstants for eah individual oerator. Note that with an oerator suh as + S z the result is a mixture of longitudinal and transverse relaxation rates. A seond examle is a double quantum oherene: + S + RS ( + + ) = R ( + ) + RS ( + ) ( ) ( ) = ( z ) j+ ( x ) + ( y ) j ω 0 ω ω ω ( S ) ( S) ( S) + ( z ) j+ ( x ) + ( y ) j S ω 0 ω ω ω [ ] The relaxation does not deend on the setral density at the sum of the Larmor frequenies of and S. 6 7

18 x φ z The diole interation deends on the angles θ and φ whih are made by the vetor joining the two nulei. The stati magneti field is assumed to be along the z-axis. θ y 6.. Two sins with diolar interation The diole relaxation mehanism was desribed in Setion 5..; essentially one sin gives rise to a magneti field at a seond sin due to the magneti diole that the former ossesses. The interation deends on the distane between the two sins r and the angles that the vetor between them makes to the diretion of the alied magneti field. These angles are the usual θ and φ used in sherial olar oordinates and illustrated oosite. The diole hamiltonian an be searated into five terms with index q = 0 ± ± q A (q) F (q) [ sinθosθex m φ ] b[ sin θex miφ ] 0 S z z S + + S + b os θ ± S S ± S z ± + ± z b i ± ± where S b = µγγ 0 h πr in whih γ and γ S are the gyromagneti ratios. The angles and the distane an in rinile all hange randomly with time; it will be assumed that r is fixed and that the angles hange due to moleular tumbling. t is lear from the resene of oerators suh as z and + that the hamiltonian will give rise to longitudinal and transverse relaxation. However as will be seen the really imortant thing about the diole interation is that it gives rise to ross relaxation as was desribed in Setion 5... The details of how this omes about will be seen in this setion. The oerators A (q) evolve under the influene of the stati hamiltonian at harateristi frequenies as given in Eq. []. ( q) ( ex ih t A ex ih t ex i q ) ( q) ( 0 ) 0 ω t A [] = Taking the stati hamiltonian to be ω z + ω S S z the oerators A (q) searate out ( q into the A ) ( q eah with its assoiated frequeny ω ) aording to q ( q) A ( q) ω 0 0 S z z 0 +0 S + ω + ω S 0 S + ω ω S ± S z ± ±ω S ± S z ± ±ω 6 8

19 ± S ± ± m (ω + ω S ) The notation q = ±0 is used to indiate that the oerator with q = +0 is the adjoint of the one with q = 0 as required by Eq. [5]. The oerators with q = evolve at only one frequeny so the index is not required for these; likewise for q = Correlation funtions f the alulation is restrited to seular ontributions (Setion 6..) the required orrelation funtions are ( ) = () ( ) q q Gqq τ F t F t τ and these will be written as = () () = q q q Gqq τ F t F t g τ F g τ where g(τ) is the redued orrelation funtion. The average of the square of F (q) is omuted by averaging over the ensemble. f it is assumed that all angles are equally likely i.e. all orientations are equally robable then this averaging is equivalent to integrating over all angles θ and φ π π ( q) ( q) ( q) F = F F sinθdθdφ π θ = 0 φ = 0 where sinθdθd φ is the volume element in sherial olar o-ordinates. The division by π is needed for normalization as For q = 0 the integral is π π sinθdθdφ = π θ = 0 φ = 0 ( 0) F = b( ) b( ) π os θ os θ sinθdθdφ = 5 b π θ = 0 φ = 0 π The integral is tedious to evaluate by hand Mathematia or some similar rogram makes short work of it though. Similar alulations give the following results for the orrelation funtions and setral densities. q F ( q) J qq ω 0 5 b 5 ± 0 ± 0 b 0 b 0 b j ω b j( ω) b j( ω) From now on for brevity the setral densities J qq J q ( ω). will be written as ω 6 9

20 6... Relaxation of z The relaxation behaviour of the z oerator is determined by evaluating [ ] ( ( q ) q ω ) ( q) q [ ' z] A A J q where the restrition to seular terms has been assumed. The setral densities J q ( ω) have been omuted in the revious setion so the roblem redues to evaluating the double ommutator for all the relevant oerators. n evaluating the ommutators the results given in Setion 6.. will be useful. n addition the following ommutators will be needed: [ ]= [ ]= + S S S S + + z z + + S z Sz The following table shows the evaluation of all the double ommutators q 0 0 S S +0 6 S + S + z 0 6 S S z S S ( q) q [ A [ A ( ' z] ] A q ) [ Q result ] [ z z [ z z z] ] S z z 0 [ ] 0 [ [ ]] 6 [ S + S + ] 6 ( z Sz) [ + [ + ]] 6 [ S + S + ] 6 ( z + Sz) z + [ z z] [ S z + 0] 0 + z [ z z] [ S + z S z] z z [ z + z] [ S z 0] 0 z [ + z z] [ S z S + z] z + + [ ] [ S + + S ] ( z + Sz) [ + + ] [ S S + + ] ( z + Sz) [ ] [ ] [ ] [ ] [ ] [ ] S S S S S S S S z S S z The final result is therefore 6 ( z Sz) J ( ω + ωs)+ 6 ( z Sz) J 0 0( + ω ωs) + J z ( ω)+ J z ( ω) + ( z + Sz) J ( ω + ωs)+ ( z + Sz) J ( ω ω S ) where the aroriate setral densities have been inserted. Assuming that J ( ω)= J ( ± ω)= J ( ± ω) gives q q q q q { ( + )} ( ω ω ) ( ω ω ) J ω ω J ω J ω ω z S S { } Sz J + S 6 J0 S nserting the exliit exressions for the setral densities this gives 6 0

21 { ( + )} { 0 ( ω ω ) 6 5 ( ω ω )} z{ 0 ( ω ωs)+ 0 ( ω)+ 0 ( ω + ωs) } j( ω ω ) j( ω ω ) b j ω ω j ω j ω ω z S S S j + j z S S = j j j b z{ 0 S 0 S } [7] S + An exatly analogous alulation for the oerator S z gives { ( + )} ( ω ω ) ( ω ω ) S j ω ω j ω j ω ω b z S S S { } z 0 j + S 0 j S b whih is easily obtained by swaing the indies and S. [8] From Eq. [7] it is lear that the term multilying z is the relaxation rate onstant for z and the term multilying S z is the rate onstant for transfer from S z to z. Likewise in Eq. [8] the first term is the relaxation rate onstant for S z and the seond is the term for transfer of z to S z. These equations imly that z and S z both relax on their own and are interonverted by relaxation aording to the air of differential equations dz dsz = R z σssz = RS S z σsz where { + + ( + )} + + ( + ) R = j ω ω j ω j ω ω b S 0 S 0 S 0 S { } R = 0 j ω ωs 0 j ω 0 j ω ωs b σs = { 0 j( ω + ωs) 0 j( ω ωs) } b These are reisely the Solomon equations found in Setion 5... The differene is that it now it has been ossible to identify the ontributions to the self and ross relaxation rate onstants. Alternatively these equations an be written in terms of the J q (ω) as { + + ( + )} + + ( + ) R = J ω ω J ω J ω ω 6 0 S S { } { } RS = 6 J0 ω ωs J ωs J ω ωs σ = J ( ω + ω ) J ( ω ω ) S S 6 0 S [9] nterretation Referring to Eq. [9] it is seen that the setral densities J (ω S ) J 0 (ω ω S ) and J (ω + ω S ) all ontribute to the self relaxation of the and S sin. However only the latter two setral densities ontribute to the ross relaxation rate onstant. These two setral densities are assoiated with the terms in the hamiltonian S ± m alled the fli-flo terms as one sin flis one way and the other flis the other and the terms S ± ± alled the fli-fli and flo-flo terms. Diole relaxation is almost unique in having these terms resent and hene giving rise to ross relaxation. This is the reason why the resene of a nulear Overhauser effet whih requires there to be ross relaxation an almost always be assoiated with diolar relaxation. 6

22 Writing in the simlest setral density funtion gives the following exression for σ S σ S τ = 0 + ( ω + ω ) τ 0 + τ ω ω τ S S µ 0h π S γ γ 6 r f and S are both the same nulear seies say rotons it is safe to assume that (ω + ω S ) = ω and (ω ω S ) = 0 giving the simler exression σ S τ µ = τ 0h 0 +ωτ 0 π S γ γ 6 r The first thing to notie about this is that the ross-relaxation rate onstant is roortional to /r 6. Transient NOE exeriments an be used to measure the ross-relaxation rate onstant (see Setion 5..) and so rovided the orrelation time is known it is ossible to determine the distane. Tyially the orrelation time is found from indeendent exeriments suh as the measurement of the arbon- relaxation times and NOE enhanements. Alternatively relative distanes in the same moleule an simly be determined from the ratio of ross-relaxation rate onstants 6 σ r S PQ = 6 σ PQ rs This will work rovided it is valid to assume that the motion of both sin airs S and P Q an be modelled by a single orrelation time (i.e. a rigid moleule). Note that as only diolar relaxation ontributes to the rossrelaxation rate onstant it does not matter if other mehanisms are ating. These will alter the self relaxation rate onstants but not the rossrelaxation rate onstants. n the extreme narrowing limit ω S τ << and so σ S simlifies to ext. narrow µ γ γ S µ γ γ S σs = τ τ = τ 0 0h 0h 0 π 6 r π 6 r This is ositive resulting in a ositive NOE enhanement i.e. the sin reeiving the enhanement will have its magnetization inreased above the equilibrium value. This is tyial of the NOE in small moleules. n the sin diffusion limit the setral density at ω beomes negligible and so σ sin diff. S = µ τ 0h 0 π S γ γ 6 r This is negative giving rise to negative NOE enhanements i.e. the sin reeiving the enhanement will have its magnetization redued when omared to the equilibrium value. The self relaxation rate is R = τ τ µ 0h τ 0 + ωτ 0 ( ω + ωs) τ + π whih simlifies in the two limits to: S γ γ 6 r 6

23 R extr. narrow = { τ } µ 0h π S γ γ 6 r sin diff. S R = µ 0h γ γ τ π 6 0 r n Setion 5... it was shown that the steady-state NOE enhanement is σ η = S SS R f the relaxation is urely due to the diole interation between and S then this enhanement is + in the extreme narrowing limit and in the sin diffusion limit. Somewhere in between the NOE enhanement goes to zero when the two terms in the ross-relaxation rate onstant and equal i.e. τ µ γ γ S σ S = τ 0h 0 ωτ π r 6τ 5 whih is zero when = τ ie ωτ =.. + ωτ At a Larmor frequeny of 500 MHz this zero rossing orresonds to a orrelation time of about 0.5 ns. Suh a orrelation time is harateristi of a medium-sized moleule suh as a short etide or oligo-saharide dissolved in water Relaxation of z S z As in the revious setion the roess starts by evaluating [ ] ( ( q A A S J ) q ω ) ( q) q [ ' z z] q Following through the same roedure as before gives { } S J( ω )+ J( ω ) z z S there is no transfer to z or S z. The differential equation is thus d ( S z z) = RS ( zsz ) where { } R J ω J = + ( ω ) S S = { 0 j( ωs)+ 0 j( ω) } b τ τ = + µ 0h 0 + ωτ S 0 ωτ π + Note that in the sin diffusion limit this rate tends to zero. S γ γ 6 r 6... Relaxation of + Evaluating 6

24 gives 6 ( q) [ ] q( ω ) ( q) [ ( q ) ' + ] A A J q +{ 8 0 S 0( S)+ ( )+ ( + S) } J ( 0)+ J ( ω )+ J ω ω J ω J ω ω There are no transfer terms to other oerators. The differential equation is thus d+ t = R + where t R = { 8 J0( 0)+ J( ωs)+ J0( ω ωs)+ J( ω)+ J( ω + ωs) } = { 8 J0( 0)+ J( ωs) }+ { 6 J0( ω ωs)+ J( ω)+ J( ω + ωs) } = { 8 J0( 0) + J( ω S )}+ R Just as in the random field ase the transverse relaxation rate onstant searates into two arts. The seond art is just half of the longitudinal rate onstant R. The first art has a term whih deends on the setral density at zero frequeny and a term whih deends on the setral density at the Larmor frequeny of sin S. This latter term is a little unusual. t says that the relaxation of transverse magnetization of sin is affeted by the setral density at the Larmor frequeny of sin S. The interretation of this is that this latter setral density is assoiated with roesses whih involve the fliing of sin S (terms suh as z S ± ). As is (diolar) ouled to S when the sin state of S hanges the field seen by the sin hanges and so the reession frequeny of the sin oherene hanges. These sin flis thus give rise to a onstantly varying loal field at sin whih auses loss of hase oherene amongst the sins whih are ontributing to the oherene. The term is seular from the oint of view of the sin as no energy is transferred to or from it. n terms of the redue setral densities the relaxation rate onstant is { + + ( + )} t R = j( 0)+ j( ω )+ j ω ω j ω j ω ω b 0 0 S 0 S 0 0 S Transverse ross relaxation n ontrast to the relaxation behaviour of z and S z there is no ross relaxation transfer between + and S + ; i.e. no transverse ross relaxation. Suh a transfer would ome about through the following double ommutators q= 0 : [ S z z [ S + + ]]= [ S z z S z + ]= 8 S+ [ S + [ S z z + ]]= [ S z z S + z]= 8 S+ q =± : [ Sz [ zs+ + ]]= [ Sz S + + ]= S+ [ S z + [ S z + ]]=[ S z + S z z]= S+ However none of these terms ontribute as they are non-seular. For

25 examle in the first ommutator the frequeny assoiated with z S z is 0 whereas that assoiated with S + is ( ω + ω S ). The resulting term is therefore raidly osillating and thus ineffetive at ausing relaxation. There are two ases in whih these terms beome seular. Firstly when ω = ω S i.e. when the two sins are degenerate. Seondly when it aears that the two sins have the same Larmor frequeny. The first ase is not of muh interest as it would be imossible to detet the transfer of magnetization between degenerate sins. The latter ase is imortant as it is ossible to make it aear that two sins have the same Larmor frequeny by sin loking their magnetization. The simlest sin loking exeriment involves a non-seletive 90 ulse about the y-axis followed by a eriod during whih a strong radiofrequeny field is alied along the x-axis. The field is suffiiently strong that for both sins (whih have different offsets) the effetive field lies very lose to the x-axis. The magnetization whih is along the axis of the alied radiofrequeny field (here x) exerienes no rotation from that field; viewed in the laboratory frame the magnetization is reessing at the same rate as the rotating field. n the rotating frame both aear stati. The magnetization is said to be sin loked. Sine the magnetization from the two sins does not diverge while the sin-loking field is on it aears as if they have the same offset or Larmor frequeny. Thus transverse ross-relaxation an take lae. The results of the ross relaxation an be observed by removing the sin-loking field and then observing the FD in the usual way. The exeriments used to observe suh transverse NOE enhanements as they are alled are generally referred to as ROESY exeriments (or sometime CAMELSPN). Both one- and two-dimensional versions of the exeriment are available. Using the above double ommutators the ross-relaxation rate onstant is omuted as { } tr σs = J( ω )+ J ( ) or in terms of the redued setral densities tr σs = { 0 j( ω )+ 0 j( 0) } b The imortant oint about this rate onstant is that for all values of the orrelation time it is always ositive. Therefore the NOE enhanements are always ositive and there is no range of orrelation times for whih the enhanement rosses zero. ROESY has therefore roved to be a useful tehnique for obtaining distane information on moleules for whih beause of their orrelation times the onventional NOE is lose to zero. (a) (b) S (y) 90 (y) τ m x τ m x One-dimensional ROESY exeriment. The ombination of the seletive 80 ulse and the 90 ulse in (a) generates a state in whih the sin is along x and the S sin is along x. During the eriod of sin loking transverse ross relaxation an take lae. Sequene (b) generates the referene setrum without ross relaxation; it is subtrated from (a) to give the usual differene setrum. τ t t mix Two-dimensional ROESY exeriment whih uses a eriod of sin loking during the mixing time. Off-resonane ROESY n ratie it is diffiult to use a suffiiently strong sin loking field that the sin loking axis is in the transverse lane. Rather the axis will by tilted u into the xz-lane. The tile angle of the effetive field θ deends on the offset Ω and the radio frequeny field strength ω. Ω θ ω Ω eff 6 5

26 ω ω Ω tanθ = sinθ = os θ = Ωeff = ω + Ω Ω Ω Ω eff When the sin loking field is alied omonents of the magnetization whih are not arallel to the field are quikly dehased and lost. Only the art along the sin lok axis need be onsidered. Magnetization whih is sin loked about an axis tilted at angle θ will have a transverse omonent roortional to sinθ this relaxes at the aroriate rate for transverse magnetization and a longitudinal omonent roortional to osθ this relaxes at the longitudinal rate. The rate onstant for the relaxation of sin loked magnetization thus deends on the rate onstants for both transverse and longitudinal magnetization and the angle θ. n an ROE exeriment the two sins whih are ross-relaxing one another will have different offsets and thus have different tilt angles θ and θ S. t is the transverse omonents whih undergo transverse ross relaxation and the longitudinal omonents whih undergo normal ross relaxation. To work out the effetive ross relaxation rate of the sin loked magnetization it is thus neessary to onsider the two omonents searately. z (a) eff (b) S S b θ d a θ S x Geometri onstrutions for the ase where the sin loking axes of and S are tilted to different extents. Case (a) is for transverse and (b) for longitudinal relaxation. Referring to diagram (a) the transverse omonent of labelled a is roortional to sinθ ; this omonent ross relaxes with S with a rate onstant σ t. Any S magnetization generated in this roess aears along the x-axis but only the omonent along the sin loking axis of S will survive. This omonent labelled b is roortional to sinθ S. The effetive transverse ross relaxation rate is thus sinθ sinθ S σ t. Diagram (b) shows the longitudinal omonents. is the longitudinal omonent of the magnetization and is roortional to osθ. This omonent ross relaxes with rate onstant σ l to give S magnetization. The omonent of this magnetization along the S sin lok axis d is roortional to osθ S. The effetive longitudinal ross relaxation rate onstant is thus osθ osθ S σ l. Overall therefore the ross relaxation rate onstant of the sin loked magnetization is given by σ tilted = sinθ sinθ S σ t + osθ osθ S σ l Using a similar geometri onstrution it is ossible to show that the self 6 6

27 relaxation rate onstant is and likewise for S. R tilted = sin θ R t + os θ R l Relaxation of + S z Evaluating gives ( q) [ A A S ] Jq( ω ) ( q) [ ( q ) ' + z] q { ( + )} S J( 0)+ J ω ω J ω J ω ω + z S S there is no transfer to other terms. The differential equation is thus where ds + z = R + Sz S { + + ( + )} R S = J J S J J + z 8 0( 0)+ 0 ω ω ω ω ωs = j( 0)+ j ω ω j ω j ω ω b { 0 0 ( S)+ 0 ( )+ 0 ( + S) } t is interesting to omare this with the relaxation time onstant for + { + + ( + )} t R = J ( 0)+ J ( ω )+ J ω ω J ω J ω ω 8 0 S 0 S S t is immediately lear that the antihase term relaxes more slowly than the inhase term; the differene is the term J( ω S) whih does not ontribute to the relaxation of the antihase term. The effet of this term was ommented on in the revious setion. There is another imortant feature of the differene between the relaxation of inhase and antihase terms. For the antihase term any additional longitudinal relaxation of the S sin R addn. S ontributes diretly to the relaxation of the oerator rodut + S z ; the reasons for this were disussed in Setion 6... R = diole R + addn. R + Sz + Sz S Suh longitudinal relaxation of sin S does not ontribute to the relaxation of +. As suh extra relaxation of S is often found in ratie it is usually the ase that antihase terms relax more quikly than inhase terms Relaxation of multile quantum terms + z Double quantum Evaluating gives ( q) [ A A S ] Jq( ω ) ( q) [ ( q ) ' + + ] q S J( ω )+ J( ω )+ J( ω + ω ) + +{ S S } 6 7