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A new approach to spin diffusion
JOURNAL
OF MAGNETIC
RESONANCE
l&219-228
(1973)
A New Approach to Spin Diffusion HAKAN
WENNERSTR~M
Division of Physical Chemistry 2, The Lund Institute of Technology, Chemical Center, P.O.B.740, S-220 07 Lund 7, Sweden Received February 16, 1973 A partially new way of viewing the nuclear spin diffusion phenomenon is presented. The basic idea is that in a strongly coupled spin system the eigenfunctions are sums of products of individual spin functions; and, consequently, a perturbation at an individual spin couples a large number of eigenstates. This in turn implies that the total spin system can reach equilibrium through a local coupling to the lattice. This conclusion is proved by a detailed calculation using conventional relaxation theory. The calculation also gives the conditions when a single experimental TI is to be expected. INTRODUCTION
In a famous paper from 1949, Bloembergen (I) showed that the nuclear spin-lattice relaxation of spins with I = l/2 in diamagnetic ionic crystals was often caused by paramagnetic impurities present in low concentrations. At first sight this seemed to be a surprising conclusion since only a small fraction of the nuclear spins could be so near the unpaired electron that the interaction between nuclear and electron spin was substantial. However, Bloembergen explained that, due to the dipolar couplings between the nuclear spins, the nuclear magnetization could diffuse to the vicinity of the electron spin, where the relaxation takes place. He showed that this transport process obeys a diffusion equation and called the phenomenon “spin diffusion.” (This should not be confused with ordinary diffusion in space of nuclei having a spin.) Later the Bloembergen work was extended by several authors (2). Their work was concentrated on the problem of solving the diffusion equation given by Bloembergen under various conditions. One of the main results of these calculations was that. one can distinguish between two rate-determining factors for the nuclear relaxation, First the relaxation rate can be determined by the rate of energy transfer to the lattice; but second, if this energy transfer is very efficient, the actual spin diffusion can be ratelimiting (3). It has been suggested that spin diffusion is of practical importance in other cases than the one treated by Bloembergen. Such cases are proton relaxation in solid alkanes (4) and polymers (5), where the relaxation is caused by nuclear dipole-dipole couplings. Recently it has also been suggested that spin diffusion takes place in proteins in solutions (6) and in liquid crystals (7). In all these cases the fact that a single TI is observed for the protons in spite of the fact that the sample contains a number of chemically different protons has been explained by spin diffusion. The aim of the present work is to reexamine the concept of spin diffusion. This is done both from the point of intuitive physical understanding and from the point of a more Copyright 0 1973 by Academic Press, Inc. Ail rights of reproduction in any form reserved. Printed in Great Britain
219
220
WENNERSTRGM
rigorous treatment with the use of the relaxation theory not available at the time of loembergen’s work. BASIC
The spin Hamiltonian
THEORY
for nuclear spins with I = l/2 can rather generally be written
The first term in Eq. [I] is the ordinary Zeeman term. The second term represents couplings not averaged to zero by the molecular motion. In isotropic liquids this is the spinspin coupling. In anisotropic solutions and in solids this term also contains dipoledipole interactions. Jy is a function of the lattice coordinates of nuclei i andj. The A ii’s are components of irreducible tensor operators working on the spin functions of nuclei i andj. These two terms in [l] are time-independent and it is convenient to define
The third term in [I], H,(t), contains the time dependent interactions having zero mean value. This term causes the spin-lattice relaxation. The decomposition of ~9 into HO + H,(t) is uniquely defined only when the motion of the lattice is either fast or slow relative to the inverse of the interaction strength, in frequency units, represented by H,(t). This difficulty will be neglected henceforth, but it should be remembered that, this point needs consideration especially when the relaxation is caused by paramagnetic impurities in solids. If the coupling constants are small compared to the Larmor frequency only those terms in HO where the spin operator A: have zero projection quantum number 4 need to be considered. This is the so called secular approximation (8, p. 104). Furthermore the following treatment will be specialized to the case of only dipole-dipole coupling terms in U,. The derived results are, however, equally valid for the spin-spin coupling case. If one studies only one type of spins, say protons, the chemical shift differences are normally negligible compared to the largest dipole coupling constants. With these simplifications HO becomes Ho = cogT z: i- 2 J’j(3 co? eij - 1) (21; 1; - +(I; ri + 1: 11)). i>j
C3!
Here J’j is the dipole-dipole coupling constant between spins i and j and is proportional to the inverse cube of the distance between the spins. The angle Bij is that between the vector from nucleus i to nucleus j and the direction of the magnetic field. In the following, for convenience, a random field relaxation mechanism will be assumed. In a molecule fixed coordinate system H,(t) can then be written H,(t) = 2 H;(t) = 1 F;& I 1
where Fi is the strength of the random field at nucleus i. A transformation operators into the lab-fixed coordinate system gives H,(t) = 2 Ft Ai D$‘(Qi(t>> i.q
L41
of the spin !51
NEW
APPROACH
TO SPIN
221
DIFFUSION
where 06:’ is a first rank Wigner rotation matrix element (9) and Q(t) is the time dependent eulerian angle describing the transformation. For a discussion of the random field relaxation mechanism see Ref. (IO). Since the Hamiltonian has now been established it is only a technical problem to predict the time variation of for example the M, magnetization from given initial conditions. The technical problems in the present case are, however, considerable. THE
BLOEMBERGEN
EXPLANATION
OF
SPIN
DIFFUSION
Using the Hamiltonian [3] it is possible to describe the Bloembergen view on spin diffusion in a slightly more detailed manner. Of all the terms in [3] only the ones containing (Zi 42 + Z! Z$ do not commute with Z,i.Let for example Zi Z! operate on a product of individual spin eigenfunctions pi and clj of Z,i and Z; 1; z_j. pi &
=
cli pj.
El
This has been described as a spin flip-flop. Tf the chemical shifts of i and j are equal this process is not associated with an energy transfer to the lattice. It can on the other hand be described as a fiow of “magnetization” from spin j to spin i. Since the spins occupy different positions in space, the spin flip-flop process thus provides a mechanism for rnaintaining internal equilibrium within the spin system even when its coupling to the lattice is local in nature. This is, in short, the traditional explanation of spin diffusion as for instance given by Abragam (8, p. 137). The Zeeman term gives the zero order wave functions in this treatmerrt. The terms containing the spin couplings and the coupling to the lattice ZZr(t), are treated as perturbations that give the time dependent effects. An obvious and more exact method would be to include the spin coupling term in ZZOas in Eq. [3] and let H,(t) be the only perturbation. This is the standard method of treating magnetic relaxation in liquids. The physical interpretation of spin diffusion based on this latter approach will now be given. AN
ALTERNATIVE
EXPLANATION
OF
SPIN
DIFFUSION
The spin-lattice relaxation of a. nuclear spin system can be understood as caused by transitions between the different eigenstates of the spin system induced by the perturbation H,(t). The eigenfunctions of the Hamiltonian [3] consist in general of sums of products of spin functions for the individual spins. A consequence of this is that a transition between two eigenstates can not be identified with a transition of a particular spin. Thus even if the perturbation H,(t) only affects one single spin, the spin-lattice relaxation is not local in nature, since the eigenstates are not. As an example take a strongly coupled two spin system (Ii = l/2). The eigenfunctions are /l,l)===aa!; /l,O) = 1/2’(c@+ba); /l,- l)=p/? and jO,O)= l/2*(@-@) where the eigenvalues Z and m of the total spin 1’ = (I1 + 1’)’ and its z-component Z, = Z,” + Z: have been given. Let this spin system be coupled to the lattice through a random magnetic field at the first spin only, i.e.,
222
WENNERSTRijM
The spin operators are Ai, = “Fl/P’l:;
A0 = I;.
PI
The operators 1: have nonvanishing matrix elements between states differing in m by one, while I,l connects states with equal m-value. Since only transitions between levels with different m-value change the MZ-magnetization, one can neglect the effect of 1: to a first approximation in the calculation of T,. A simple calculation shows that all the nonzero matrix elements involving the operators 1: and 12 are equal except for sign. A consequence of this is that the relaxation is equally efficient for all the eigenstates and a single T1 will be observed. The difference in sign for the matrix elements is of no importance since the transition probabilities are proportional to the square of the matrix elements. This example clearly shows that in this formulation there is no physical spin diffusion associated with the relaxation process. The single T1 obtained is a simple consequence of the fact that the perturbation H,(t) affects all the eigenstates with equal strength. Although it is not possible to generalize the formal part of the arguments given above to systems containing more than two spins the physical interpretation suggests that the observation of a single Tl in a larger spin system has the same explanation as the one given above. Symmetry arguments can be used to predict if one or several T,‘s are to be expected for a spin system. This is done by considering how the constants of the motion CJ ([Q,HJ = 0) attain equilibrium. From the nature of the perturbation H,(t) it is often possible to conclude if there exists a constant of the motion which reaches equilibrium faster than the other ones. In for example a weakly coupled system all the individual 1: are constants of the motion. If the coupling to the lattice is different for the various spins in the system one should expect a particular Tl for each spin. In a strongly coupled system on the other hand the different Ii, are not constants of the motion and one has no reason for assigning different Tl’s for the different spins even if the coupling to the lattice is asymmetrical. THE
LONGITUDINAL
RELAXATION
IN
A MULTISPIN
SYSTEM
It is possible to formalize the physical arguments given in the previous section using conventional relaxation theory. This will now be done to show in more detail when and to what approximation a single Tl is to be expected. The problem is to calculate the time variation of the M,-magnetization, which is the property one observes experimentally. This problem is conveniently solved in the present case using the Abragam operator method. Equation [VIII:311 of Ref. (8) is
where CJis the deviation from equilibrium of the density operator for the spin system. The asterisk indicate that the operators should be taken in the interaction representation, e.g., CJ*= exp(iHO t) (Texp(-iH0 t).
NEW
APPROACH
223
TO SPIN DIFFUSION
The deviation from equilibrium of the expectation value of the total M,-magnetization is given by M, = y(Zz) = y Tr (q*Z,) = y Tr (crZ=); z, = 2 z;* u 31 i The third equality in [ll] follows since Z, commutes with NO as given in Eq. [3]. The time derivative of the expectation value (I,) can be obtained from Eq. [9] by multiplying it by Z, and then taking the trace.
m
= - .cTr (a*@> <<[I-r?(t- 4, [fJ%))), &II) dz. 0 Using Eqs. [5] and [S] it is convenient to consider one term of the commutator for a random field at two arbitrary spins i and j: (([Hf*(t
WI in [Ii]
- T), [exp(iZZo t) F~(D~~ (QJt)) Zi - l/2+ 0::) (!C!,(t))Z$ + l/2+ OF& (Q](t)) Zi} exp(-iZ& t))), 2 Z:]] = Jij exp(iZZ, t)
Ff/2[exp(-i&Z0
r)(({-D:t)(C&(t - z)) Z: + DFio (C&G- 7)) ZJ + 2* D$j) (&(t - T)) I:) exp(i& r), ZI$L)(Q,(t)) Z$ + D(_l:((Oi(t))))Z!] exp(-iZZo t).
WI Under conditions of strong narrowing (II), i.e., when the relevant correlation times -t, are shorter than the inverse of the largest coupling constants, one can set, in keening with the secular approximation, exp(-iZZo 7) Zi exp(iHo z) = exp(+io, 2) Zi exp(-iZZo z) Z: exp( iHo z) = Zt T < z,. With these approximations
K
and
Eq. [13] becomes
4cll
= 6,,(Gi(-T) (exp(io, 7) + exp(-iw, 7)) Zk - l/2* Gh(-z) I!,? + l/2% Gh-,(-T) ZF>, fl51
where G’;(T)= G";(-T) = ((-Dl'd(ai(t))D~~o(szi(t GIFT= D$)(Qi(t))D$l~o(Qi(t + z))Ff.
+ z))))F$
The terms in Eq. [15] that contain ZT and Z? give contributions that oscillate at the Larmor frequency and can consequently be neglected. Equation [ 121is now simplified to ; (0
= - F JXOO) a>,
1161
J:(o) = fm exp(-iwr) G:(T) dz. 1171 --al In the derivation of Eq. [16] no assumptions about the strength of the couplings within the spin system have been made. In a weakly coupled system the spins are distinguish-
224
WENNERSTRijM
able and in general (I:) # (Zi). For a strongly coupled system such as the one described by the Hamiltonian [3] this conclusion is no longer valid. Instead it is possible to show that (Ii) = (ZJ;} for all i and j to a good approximation under certain circumstances. Consider (Zf) = Tr (Zi o).
I181
It is convenient to evaluate the trace in the basis given by the eigenvectors jrZm> of Ho in Eq. [3]. The quantity Z is the eigenvalue of the total spin and m the eigenvalue of Z,. The label 11distinguishes between independent vectors with the same I. It is possible to write the eigenvectors in this way since Z2 and Z, commute with Ho. If Ho has a nondegenerate eigenvalue spectrum the given basis is uniquely defined, and the equilibrium density matrix is diagonal. Furthermore the spin-lattice relaxation depends only on changes in these diagonal elements of the density matrix and the off-diagonal eiements do not couple to the diagonal ones to a good approximation. One can thus for simplicity assume, that the nonequilibrium initial state is described by a diagonal density matrix in the given basis. These arguments also imply that the density matrix will remain diagonal throughout the relaxation process. It should be stressed that these conclusions are not valid in the presence of degeneracies. The trace in Eq. [lS] is now Tr (Zkcr) = c (ylmll~lqlm) urn
(ylmjolqlm).
[19]
For the case of two spins, which was treated in the previous section, two eigenvectors did not have the same pair of quantum numbers Z and m. An immediate consequence of this is that the matrix elements (yZm1151qZm) are independent of i and Eq. [16] predicts a single T1 given by l/T, = 1/2(1/T: + l/T;).
PO1
Here 7-i is the relaxation time of spin i if it were independent of the other spins. For a multispin system, however, the problem is more complicated. In this case there are in general several independent eigenvectors with the same land m quantum numbers. The diagonal matrix elements of Zf are then not independent of i, but the sum ; (yZmlZ~jyZm) is. This is seen from the fact that the sum is the trace for a given mvalue, and the operator Zj does not couple states with different m-value. The sumis thus independent of the basis chosen as long as the m-value is given and can conveniently be evaluated in the direct product basis formed by the individual spin eigenfunctions. In that basis it is evident that the sum is independent of i. If now all diagonal density matrix elements (ylrnlolqlrn) are independent of r~ and Z it follows from Eq. 1191 that (Z;) = (Zj,) independent of i and j. Equation [16] then gives l/T, = l/N : l/T; i=l
WI
where N is the number of spins. It still remains to show that actually (yZmlo/qZm) is independent of r and Z to finally prove Eq. [21]. This step requires the calculation of the individual diagonal density matrix elements, which could be done by the use of the Redfield theory (II), but will be done here directly from Eq. [9]. With the use of Eqs. [5], [S], and [14], a diagonal density matrix and the fact that the only nonzero contributions are obtained when 14 = 0
NEW
APPROACH
TO SPIN DIFFUSION
225
in the expansion of H,(t), one has the following expression for a diagonal matrix element of Eq. [9]: $i@rnl~l
qh> = - C 1/4JS(o,) (glrn~[l&
[IL, o]]
I
+ [IL, [I:, c]][ifIm> + 1/2.@0) (qlrni [Ii, [I:, a]][qIm) = - T l/2Jf(oo){(?~mlalvIm> - ~~llW+~l~l~l~ - I>/’
x
+ 1)) +J~(0){1/4(yImjalvllm)
-g$I I(y~mI~~ly~~~m)12(y~~~mloly,I;m)).
WI
In obtaining Eq. [22] the assumption that the random fields at two nuclei are uncorrelated has also been made. The term proportional toJ,(o,) in Eq. [22] describes transitions between states with the m-value differing by one and contains those terms that accomplishes the longitudinal relaxation. The terms proportional to J,,(O) are, however, more interesting in the present context. These terms describe transitions between states of equal m-value in such a way that if not all the (yIm /G./rZm> are equal for a given M there is a driving force towards equality. This requires of course that not all (qZm 11:/y 1I, m) are zero for I# 1, or y # y~~.In a strongly coupled system these terms are not all zero if there are no constants of the motion besides I’, I,, and H,, that are taken from equilibrium during the experiment. An example of this will be shown in the next section, From Eq. [22] one can then say that especially in the nonextreme narrowing case when ~~(0) + J,(o,) the diagonal density matrix elements with the same m-value are equal to a good approximation (cf. Ref. (12)). Even when Jo(O) =J,(o,,) the elements are equal to a fair approximation. Equation [21] should be approximately valid and the deviations from an exponential decay of the M,-magnetization is expected to consist of a faster decay at short times. This latter conclusion follows from the general behavior of coupled differential equations like Eq. [22]. AN
ILLUSTRATIVE
EXAMPLE
To illustrate the accuracy of the approximations made in the previous section, Eq. [22] will now be solved for a particular three-spin system. The effect of extra constants of the motion will then be shown. Take as an example a strongly coupled threespin system with the coupling constants Jlz =J13 #Jz3. Let this system be relaxed by a random field at spin 1 only. The eigenvectors with positive m-value are (13, p. 276) 4; j 312, 312) = am, 3;/3/2, l/2) = l/2/3@@ 2; / 1; l/2, l/2) = l/1/6(& 1; 12; l/2, l/2) = l/d2(a@
+ cl@ t jhx), + @a + 2paa), - &).
The other four eigenvectors are obtained by applying Corio’s spin inversion operator (1.3, p. 155). Denote these by the corresponding negative labels -4, -3, -2, -1 and denote the diagonal density matrix elements by a single index corresponding to the eigenvectors. Only the differences (gi - G-J are required to calculate the Mz-magnetization. The system of coupled differential equations constructed from Eq. [22] for the derivatives
226
WENNERSTRk
(d/dt)(~~ - OJ is closed due to the symmetry of the eigenvectors towards spin inversion. An explicit calculation gives
fcG4 - G-4)= -~:(~,)/2((~,
- a-4) - i/3(0, - gw3)- 213h - G-~)), E23d
;(cr3 - ab3) = -J:(o,)/2(13/9(0,
- (~-3) + 2/9(az - ~-2)
- l/3(0, - 0-J) - 2J:tO)/9((0, - c-3) - (~2 - Q)), ;(G’
- oe2)= -J;(o,)/2(8/9(o,
- a-,)+
2/'9(0, - G-~)
- 213(~, - Go)) - 2JA(0)19((~ -a-,)
g (a, - G-1) = -J:(w,)
L23bl
- to3 - ~-,a
E23d
(G.1- O-l).
Equation [23d] shows that (rrl - aPI) is uncoupled to the other matrix elements. The reason for this is that (I2 + 13)’ is a constant of the motion. Equation [23d] can be integrated directly and gives an exponential term with the time constant -l/Ji(w,). The other three coupled equations give rise to three exponential terms with the inverse time constants -J:(cc?‘,) : -J:(o,)/18(7
+ 4J;(O)/J;(w,)+
(13 + 2OJ~(O)/'J~(w,)+ I~(J~(O)/J:(W~))~)~~~.
With initial conditions given by a 180”-pulse the time dependence of the M,-magnetization is given for (a) extreme narrowing J:(O) = Ji(w,) : M,(t) = MO - 2&/7{exp(-J’(O) where MO is the equilibrium
t) + 6 exp(-2/9J”(O) t)}
magnetization;
and (b) J;(O) * J:(o,):
M,(t) = MO -2M,/9(exp(-J:(w,)
t) + 8 exp(-1/4J:(w,)
t)}.
1241
P51
The same result is obtained by setting (crS- K~) equal to (c2 - gP2) in Eq. [23] as expected. Finally if JIZ #J,, so that (crI - gml) couples to the other elements in Eq. [23] and nonextreme narrowing so that (c3 - c-~) = (c2 - c--2) = (rrl - awl) one gets M,(t) = MO - 2Mo exp(-1/3J:(w,)
t).
WI
This last example corresponds to the approximation made in the previous section in the derivation of Eq. [21]. Equations [24-261 illustrate the errors made in these approximations. Even at extreme narrowing and with an extra constant of the motion the decay of the M,-magnetization goes essentially as a single exponential? though with a small but detectable correction for small t. The difference between Eqs. 1241 and [25] illustrates what happens when one goes to nonextreme narrowing. There exist differences but they are small. A final conclusion is that the approximations made in the previous section are justified, under the conditions given, for a three-spin system. COMPARISON
WITH
EXPERIMENTS
From the calculations in the two preceding sections one can predict that for systems described by the Hamiltonian [3] the decay of the Ilrl,-magnetization towards equilibrium is essentially described by a single exponential with the time constant given by Eq. [21]. For small times a deviation from the single exponential might be observable.
NEW
APPROACH
TO SPIN DIFFUSION
227
This is especially so in the extreme narrowing region. These simple predictions will now be compared with experiments. The proportionality between N and Tl in Eq. [Zl] has been experimentally verified in the case of solids containing paramagnetic impurities (3,I4) and in solid alkanes (4). For solids where the relaxation is caused by paramagnetic impurities an extra complication arises, however. Nuclear spins near the paramagnetic ion not only experience a strong fluctuating field but also an extra static field, which causes a shift. This shift can very well be of the same order of magnitude as the nuclear dipole-dipole couplings (15). These nuclear spins are then only intermediately strongly coupled to the other spins. The consequence of this is that in addition to the slow exponential relaxation, a faster relaxation should also be present at shorter times. This has been observed (3) but was interpreted as diffusion limited relaxation. In the present formulation diffusion limited relaxation is interpreted as the necessary appearance of fast decaying exponential terms due to the coupled equations in Eq. [22], as explained above. Which of the two factors mentioned actually gives the observed deviation from a single exponential is not clear. If the relaxation time of the electron spin is long compared to the inverse of the nuclear Larmor frequency, the shift effect is expected to be the most important one. In liquid crystals, where the molecular motion is fast, a single T1 is generally observed. The liquid crystalline phases are anisotropic, which results in static dipole-dipole couplings that are usually greater than the shift differences. All the protons in an alkyl chain in such a phase can consequently be expected to form a strongly coupled system. If on the other hand the alkyl chain is broken by an ether linkage, for example, the dipole couplings over this linkage could be small, resulting in two weakly interacting spin systems. In such a case two Tl’s might be observed. Neglecting this difficulty the fast motion in the liquid crystal still indicates that for short times a nonexponential decay could be observed. If this is not the case it can be explained in two ways. First, the relaxation interaction is of the same order of magnitude at all the protons and, second, the relaxation is due to nuclear dipole-dipole interactions, which introduces an extra coupling term in Eq. [22]. Either of these factors would favor the observation of a single Tl. The suggestion that spin diffusion is of importance for protons in proteins in solution (6) is questionable in the light of the present calculation. In general the shifts are larger than the spin-spin coupling constants within the amino acid side-chains and in particular little coupling exists between side-chains. The total spin systems is not described by a Ramiltonian of type [3]. If nevertheless a single Tl is observed this can be explained by the factors : (a) The strength of the relaxation interaction and its time dependence do not differ in orders of magnitude from proton to proton. (b) The dipole-dipole relaxation mechanism introduces a coupling between the relaxation of the different spin systems (see, e.g., (8, Eq. [VIII:87])) that tends to give the total system a single Tl. (c) Since usually Jo(O) > J1(oO) in these solutions even a weak coupling can couple diagonal density matrix elements with equal m-value in Eq. 1221. This fact together with the changes in absolute shift should be considered when the frequency dependence of T1 is interpreted. lYnthis short survey only the simplest aspect of the relation between experimenrs and
,228
WENNERSTRGM
the presented theory have been outlined. A complete analysis of the often compiicated experiments is beyond the scope of this paper. CONCLUDING
REMARKS
In the present paper a new way of viewing spin diffusion has been proposed. The basic idea is that in a strongly coupled spin system the individual spins have lost- their identity in the sense that a transition between two eigenstates can not be identified with a transition of a single spin. A consequence of this is that a perturbation at a single spin affects all eigenstates. This is shown to lead to a single Tl for the total spin system under certain circumstances. A single Tl is predicted, at strong narrowing, when the coupling constants are much greater than the shift differences and J,(O)/J,(w,) s 1 and no essential constants of the motion except 12, I,, and Ho are present. Even when nof all these conditions are satisfied a single experimental T, is often expected. The present approach to spin diffusion might seem very different from the one given by Bloembergen. This is in fact not the case. The main difference is that the present method is basically one degree less approximative than Bloembergen’s method in which a diffusion equation is employed. If one uses the Bloembergen diffusion equation as a starting point for a derivation, some of the results will be artifacts of the approximation without any physical relevance. REFERENCES
I. N. BLQEMBERGEN,Physica 15,386 (1949). 2. 6. R. KHUTSISH;VILI,“Progress in Low Temperature Physics,” Vol. VII, p. 375 (C. J. Gorrer, Ed.), North Holland, Amsterdam, 1970. 3. W. E. BLUMBERG,P&s. Rev. 119,79 (1960). 4. J. E. ANDERSONAND W. P. SLIGHTER,J. Phys. Chem. 69,3099 (1965). 5. D. W. MCCALL AND D. C. DOUGLASS, Polymer 4,433 (1963). 6. R. KIMMICH AND F. NOACK, Bet-. Bunsenges. Phys. Chem. 75,269 (1971). 7. J. CHARVOLIN AND P. RIGNY, J. Magn. Resonance 4,40 (1971); or E. OLDFIELD AE;DD. CHAPMAX, FEBS Letters 23,285 (1972). 8. A. ABRAGAM, “The Principles of Nuclear Magnetism,” Clarendon Press, Oxford, 1961. Ye M. E. ROSE, “Elementary Theory of Angular Momentum,” Wiley, New York, 1957, 10. A. KUMAR AND B. D. N. RAO, J. Magn. Resonance 8,l (1972). il. R. A. HOFFMAN, “Advances in Magnetic Resonance” (J. S. Waugh, Ed.), Vol. 4, p. 96, Academic Press, New York, 1970. 12. J. E. ANDERSON AND K. J. Lru, J. Chem. Phys. 49,285O (1968). 13. P. L. CORIO, “Structure of High Resolution NMR Spectra,” p. 267, Academic Press, New York, 1966. 14. D. TSE AND I. J. LOWE, Phys. Rev. 166,292 (1968). 15. A. CARRINGTON AND A. S. MCLACHLAN, “Introduction to Magnetic Resonance,” p. 222, Harper and Rowe, New York, 1967.
OF MAGNETIC
RESONANCE
l&219-228
(1973)
A New Approach to Spin Diffusion HAKAN
WENNERSTR~M
Division of Physical Chemistry 2, The Lund Institute of Technology, Chemical Center, P.O.B.740, S-220 07 Lund 7, Sweden Received February 16, 1973 A partially new way of viewing the nuclear spin diffusion phenomenon is presented. The basic idea is that in a strongly coupled spin system the eigenfunctions are sums of products of individual spin functions; and, consequently, a perturbation at an individual spin couples a large number of eigenstates. This in turn implies that the total spin system can reach equilibrium through a local coupling to the lattice. This conclusion is proved by a detailed calculation using conventional relaxation theory. The calculation also gives the conditions when a single experimental TI is to be expected. INTRODUCTION
In a famous paper from 1949, Bloembergen (I) showed that the nuclear spin-lattice relaxation of spins with I = l/2 in diamagnetic ionic crystals was often caused by paramagnetic impurities present in low concentrations. At first sight this seemed to be a surprising conclusion since only a small fraction of the nuclear spins could be so near the unpaired electron that the interaction between nuclear and electron spin was substantial. However, Bloembergen explained that, due to the dipolar couplings between the nuclear spins, the nuclear magnetization could diffuse to the vicinity of the electron spin, where the relaxation takes place. He showed that this transport process obeys a diffusion equation and called the phenomenon “spin diffusion.” (This should not be confused with ordinary diffusion in space of nuclei having a spin.) Later the Bloembergen work was extended by several authors (2). Their work was concentrated on the problem of solving the diffusion equation given by Bloembergen under various conditions. One of the main results of these calculations was that. one can distinguish between two rate-determining factors for the nuclear relaxation, First the relaxation rate can be determined by the rate of energy transfer to the lattice; but second, if this energy transfer is very efficient, the actual spin diffusion can be ratelimiting (3). It has been suggested that spin diffusion is of practical importance in other cases than the one treated by Bloembergen. Such cases are proton relaxation in solid alkanes (4) and polymers (5), where the relaxation is caused by nuclear dipole-dipole couplings. Recently it has also been suggested that spin diffusion takes place in proteins in solutions (6) and in liquid crystals (7). In all these cases the fact that a single TI is observed for the protons in spite of the fact that the sample contains a number of chemically different protons has been explained by spin diffusion. The aim of the present work is to reexamine the concept of spin diffusion. This is done both from the point of intuitive physical understanding and from the point of a more Copyright 0 1973 by Academic Press, Inc. Ail rights of reproduction in any form reserved. Printed in Great Britain
219
220
WENNERSTRGM
rigorous treatment with the use of the relaxation theory not available at the time of loembergen’s work. BASIC
The spin Hamiltonian
THEORY
for nuclear spins with I = l/2 can rather generally be written
The first term in Eq. [I] is the ordinary Zeeman term. The second term represents couplings not averaged to zero by the molecular motion. In isotropic liquids this is the spinspin coupling. In anisotropic solutions and in solids this term also contains dipoledipole interactions. Jy is a function of the lattice coordinates of nuclei i andj. The A ii’s are components of irreducible tensor operators working on the spin functions of nuclei i andj. These two terms in [l] are time-independent and it is convenient to define
The third term in [I], H,(t), contains the time dependent interactions having zero mean value. This term causes the spin-lattice relaxation. The decomposition of ~9 into HO + H,(t) is uniquely defined only when the motion of the lattice is either fast or slow relative to the inverse of the interaction strength, in frequency units, represented by H,(t). This difficulty will be neglected henceforth, but it should be remembered that, this point needs consideration especially when the relaxation is caused by paramagnetic impurities in solids. If the coupling constants are small compared to the Larmor frequency only those terms in HO where the spin operator A: have zero projection quantum number 4 need to be considered. This is the so called secular approximation (8, p. 104). Furthermore the following treatment will be specialized to the case of only dipole-dipole coupling terms in U,. The derived results are, however, equally valid for the spin-spin coupling case. If one studies only one type of spins, say protons, the chemical shift differences are normally negligible compared to the largest dipole coupling constants. With these simplifications HO becomes Ho = cogT z: i- 2 J’j(3 co? eij - 1) (21; 1; - +(I; ri + 1: 11)). i>j
C3!
Here J’j is the dipole-dipole coupling constant between spins i and j and is proportional to the inverse cube of the distance between the spins. The angle Bij is that between the vector from nucleus i to nucleus j and the direction of the magnetic field. In the following, for convenience, a random field relaxation mechanism will be assumed. In a molecule fixed coordinate system H,(t) can then be written H,(t) = 2 H;(t) = 1 F;& I 1
where Fi is the strength of the random field at nucleus i. A transformation operators into the lab-fixed coordinate system gives H,(t) = 2 Ft Ai D$‘(Qi(t>> i.q
L41
of the spin !51
NEW
APPROACH
TO SPIN
221
DIFFUSION
where 06:’ is a first rank Wigner rotation matrix element (9) and Q(t) is the time dependent eulerian angle describing the transformation. For a discussion of the random field relaxation mechanism see Ref. (IO). Since the Hamiltonian has now been established it is only a technical problem to predict the time variation of for example the M, magnetization from given initial conditions. The technical problems in the present case are, however, considerable. THE
BLOEMBERGEN
EXPLANATION
OF
SPIN
DIFFUSION
Using the Hamiltonian [3] it is possible to describe the Bloembergen view on spin diffusion in a slightly more detailed manner. Of all the terms in [3] only the ones containing (Zi 42 + Z! Z$ do not commute with Z,i.Let for example Zi Z! operate on a product of individual spin eigenfunctions pi and clj of Z,i and Z; 1; z_j. pi &
=
cli pj.
El
This has been described as a spin flip-flop. Tf the chemical shifts of i and j are equal this process is not associated with an energy transfer to the lattice. It can on the other hand be described as a fiow of “magnetization” from spin j to spin i. Since the spins occupy different positions in space, the spin flip-flop process thus provides a mechanism for rnaintaining internal equilibrium within the spin system even when its coupling to the lattice is local in nature. This is, in short, the traditional explanation of spin diffusion as for instance given by Abragam (8, p. 137). The Zeeman term gives the zero order wave functions in this treatmerrt. The terms containing the spin couplings and the coupling to the lattice ZZr(t), are treated as perturbations that give the time dependent effects. An obvious and more exact method would be to include the spin coupling term in ZZOas in Eq. [3] and let H,(t) be the only perturbation. This is the standard method of treating magnetic relaxation in liquids. The physical interpretation of spin diffusion based on this latter approach will now be given. AN
ALTERNATIVE
EXPLANATION
OF
SPIN
DIFFUSION
The spin-lattice relaxation of a. nuclear spin system can be understood as caused by transitions between the different eigenstates of the spin system induced by the perturbation H,(t). The eigenfunctions of the Hamiltonian [3] consist in general of sums of products of spin functions for the individual spins. A consequence of this is that a transition between two eigenstates can not be identified with a transition of a particular spin. Thus even if the perturbation H,(t) only affects one single spin, the spin-lattice relaxation is not local in nature, since the eigenstates are not. As an example take a strongly coupled two spin system (Ii = l/2). The eigenfunctions are /l,l)===aa!; /l,O) = 1/2’(c@+ba); /l,- l)=p/? and jO,O)= l/2*(@-@) where the eigenvalues Z and m of the total spin 1’ = (I1 + 1’)’ and its z-component Z, = Z,” + Z: have been given. Let this spin system be coupled to the lattice through a random magnetic field at the first spin only, i.e.,
222
WENNERSTRijM
The spin operators are Ai, = “Fl/P’l:;
A0 = I;.
PI
The operators 1: have nonvanishing matrix elements between states differing in m by one, while I,l connects states with equal m-value. Since only transitions between levels with different m-value change the MZ-magnetization, one can neglect the effect of 1: to a first approximation in the calculation of T,. A simple calculation shows that all the nonzero matrix elements involving the operators 1: and 12 are equal except for sign. A consequence of this is that the relaxation is equally efficient for all the eigenstates and a single T1 will be observed. The difference in sign for the matrix elements is of no importance since the transition probabilities are proportional to the square of the matrix elements. This example clearly shows that in this formulation there is no physical spin diffusion associated with the relaxation process. The single T1 obtained is a simple consequence of the fact that the perturbation H,(t) affects all the eigenstates with equal strength. Although it is not possible to generalize the formal part of the arguments given above to systems containing more than two spins the physical interpretation suggests that the observation of a single Tl in a larger spin system has the same explanation as the one given above. Symmetry arguments can be used to predict if one or several T,‘s are to be expected for a spin system. This is done by considering how the constants of the motion CJ ([Q,HJ = 0) attain equilibrium. From the nature of the perturbation H,(t) it is often possible to conclude if there exists a constant of the motion which reaches equilibrium faster than the other ones. In for example a weakly coupled system all the individual 1: are constants of the motion. If the coupling to the lattice is different for the various spins in the system one should expect a particular Tl for each spin. In a strongly coupled system on the other hand the different Ii, are not constants of the motion and one has no reason for assigning different Tl’s for the different spins even if the coupling to the lattice is asymmetrical. THE
LONGITUDINAL
RELAXATION
IN
A MULTISPIN
SYSTEM
It is possible to formalize the physical arguments given in the previous section using conventional relaxation theory. This will now be done to show in more detail when and to what approximation a single Tl is to be expected. The problem is to calculate the time variation of the M,-magnetization, which is the property one observes experimentally. This problem is conveniently solved in the present case using the Abragam operator method. Equation [VIII:311 of Ref. (8) is
where CJis the deviation from equilibrium of the density operator for the spin system. The asterisk indicate that the operators should be taken in the interaction representation, e.g., CJ*= exp(iHO t) (Texp(-iH0 t).
NEW
APPROACH
223
TO SPIN DIFFUSION
The deviation from equilibrium of the expectation value of the total M,-magnetization is given by M, = y(Zz) = y Tr (q*Z,) = y Tr (crZ=); z, = 2 z;* u 31 i The third equality in [ll] follows since Z, commutes with NO as given in Eq. [3]. The time derivative of the expectation value (I,) can be obtained from Eq. [9] by multiplying it by Z, and then taking the trace.
m
= - .cTr (a*@> <<[I-r?(t- 4, [fJ%))), &II) dz. 0 Using Eqs. [5] and [S] it is convenient to consider one term of the commutator for a random field at two arbitrary spins i and j: (([Hf*(t
WI in [Ii]
- T), [exp(iZZo t) F~(D~~ (QJt)) Zi - l/2+ 0::) (!C!,(t))Z$ + l/2+ OF& (Q](t)) Zi} exp(-iZ& t))), 2 Z:]] = Jij exp(iZZ, t)
Ff/2[exp(-i&Z0
r)(({-D:t)(C&(t - z)) Z: + DFio (C&G- 7)) ZJ + 2* D$j) (&(t - T)) I:) exp(i& r), ZI$L)(Q,(t)) Z$ + D(_l:((Oi(t))))Z!] exp(-iZZo t).
WI Under conditions of strong narrowing (II), i.e., when the relevant correlation times -t, are shorter than the inverse of the largest coupling constants, one can set, in keening with the secular approximation, exp(-iZZo 7) Zi exp(iHo z) = exp(+io, 2) Zi exp(-iZZo z) Z: exp( iHo z) = Zt T < z,. With these approximations
K
and
Eq. [13] becomes
4cll
= 6,,(Gi(-T) (exp(io, 7) + exp(-iw, 7)) Zk - l/2* Gh(-z) I!,? + l/2% Gh-,(-T) ZF>, fl51
where G’;(T)= G";(-T) = ((-Dl'd(ai(t))D~~o(szi(t GIFT= D$)(Qi(t))D$l~o(Qi(t + z))Ff.
+ z))))F$
The terms in Eq. [15] that contain ZT and Z? give contributions that oscillate at the Larmor frequency and can consequently be neglected. Equation [ 121is now simplified to ; (0
= - F JXOO) a>,
1161
J:(o) = fm exp(-iwr) G:(T) dz. 1171 --al In the derivation of Eq. [16] no assumptions about the strength of the couplings within the spin system have been made. In a weakly coupled system the spins are distinguish-
224
WENNERSTRijM
able and in general (I:) # (Zi). For a strongly coupled system such as the one described by the Hamiltonian [3] this conclusion is no longer valid. Instead it is possible to show that (Ii) = (ZJ;} for all i and j to a good approximation under certain circumstances. Consider (Zf) = Tr (Zi o).
I181
It is convenient to evaluate the trace in the basis given by the eigenvectors jrZm> of Ho in Eq. [3]. The quantity Z is the eigenvalue of the total spin and m the eigenvalue of Z,. The label 11distinguishes between independent vectors with the same I. It is possible to write the eigenvectors in this way since Z2 and Z, commute with Ho. If Ho has a nondegenerate eigenvalue spectrum the given basis is uniquely defined, and the equilibrium density matrix is diagonal. Furthermore the spin-lattice relaxation depends only on changes in these diagonal elements of the density matrix and the off-diagonal eiements do not couple to the diagonal ones to a good approximation. One can thus for simplicity assume, that the nonequilibrium initial state is described by a diagonal density matrix in the given basis. These arguments also imply that the density matrix will remain diagonal throughout the relaxation process. It should be stressed that these conclusions are not valid in the presence of degeneracies. The trace in Eq. [lS] is now Tr (Zkcr) = c (ylmll~lqlm) urn
(ylmjolqlm).
[19]
For the case of two spins, which was treated in the previous section, two eigenvectors did not have the same pair of quantum numbers Z and m. An immediate consequence of this is that the matrix elements (yZm1151qZm) are independent of i and Eq. [16] predicts a single T1 given by l/T, = 1/2(1/T: + l/T;).
PO1
Here 7-i is the relaxation time of spin i if it were independent of the other spins. For a multispin system, however, the problem is more complicated. In this case there are in general several independent eigenvectors with the same land m quantum numbers. The diagonal matrix elements of Zf are then not independent of i, but the sum ; (yZmlZ~jyZm) is. This is seen from the fact that the sum is the trace for a given mvalue, and the operator Zj does not couple states with different m-value. The sumis thus independent of the basis chosen as long as the m-value is given and can conveniently be evaluated in the direct product basis formed by the individual spin eigenfunctions. In that basis it is evident that the sum is independent of i. If now all diagonal density matrix elements (ylrnlolqlrn) are independent of r~ and Z it follows from Eq. 1191 that (Z;) = (Zj,) independent of i and j. Equation [16] then gives l/T, = l/N : l/T; i=l
WI
where N is the number of spins. It still remains to show that actually (yZmlo/qZm) is independent of r and Z to finally prove Eq. [21]. This step requires the calculation of the individual diagonal density matrix elements, which could be done by the use of the Redfield theory (II), but will be done here directly from Eq. [9]. With the use of Eqs. [5], [S], and [14], a diagonal density matrix and the fact that the only nonzero contributions are obtained when 14 = 0
NEW
APPROACH
TO SPIN DIFFUSION
225
in the expansion of H,(t), one has the following expression for a diagonal matrix element of Eq. [9]: $i@rnl~l
qh> = - C 1/4JS(o,) (glrn~[l&
[IL, o]]
I
+ [IL, [I:, c]][ifIm> + 1/2.@0) (qlrni [Ii, [I:, a]][qIm) = - T l/2Jf(oo){(?~mlalvIm> - ~~llW+~l~l~l~ - I>/’
x
+ 1)) +J~(0){1/4(yImjalvllm)
-g$I I(y~mI~~ly~~~m)12(y~~~mloly,I;m)).
WI
In obtaining Eq. [22] the assumption that the random fields at two nuclei are uncorrelated has also been made. The term proportional toJ,(o,) in Eq. [22] describes transitions between states with the m-value differing by one and contains those terms that accomplishes the longitudinal relaxation. The terms proportional to J,,(O) are, however, more interesting in the present context. These terms describe transitions between states of equal m-value in such a way that if not all the (yIm /G./rZm> are equal for a given M there is a driving force towards equality. This requires of course that not all (qZm 11:/y 1I, m) are zero for I# 1, or y # y~~.In a strongly coupled system these terms are not all zero if there are no constants of the motion besides I’, I,, and H,, that are taken from equilibrium during the experiment. An example of this will be shown in the next section, From Eq. [22] one can then say that especially in the nonextreme narrowing case when ~~(0) + J,(o,) the diagonal density matrix elements with the same m-value are equal to a good approximation (cf. Ref. (12)). Even when Jo(O) =J,(o,,) the elements are equal to a fair approximation. Equation [21] should be approximately valid and the deviations from an exponential decay of the M,-magnetization is expected to consist of a faster decay at short times. This latter conclusion follows from the general behavior of coupled differential equations like Eq. [22]. AN
ILLUSTRATIVE
EXAMPLE
To illustrate the accuracy of the approximations made in the previous section, Eq. [22] will now be solved for a particular three-spin system. The effect of extra constants of the motion will then be shown. Take as an example a strongly coupled threespin system with the coupling constants Jlz =J13 #Jz3. Let this system be relaxed by a random field at spin 1 only. The eigenvectors with positive m-value are (13, p. 276) 4; j 312, 312) = am, 3;/3/2, l/2) = l/2/3@@ 2; / 1; l/2, l/2) = l/1/6(& 1; 12; l/2, l/2) = l/d2(a@
+ cl@ t jhx), + @a + 2paa), - &).
The other four eigenvectors are obtained by applying Corio’s spin inversion operator (1.3, p. 155). Denote these by the corresponding negative labels -4, -3, -2, -1 and denote the diagonal density matrix elements by a single index corresponding to the eigenvectors. Only the differences (gi - G-J are required to calculate the Mz-magnetization. The system of coupled differential equations constructed from Eq. [22] for the derivatives
226
WENNERSTRk
(d/dt)(~~ - OJ is closed due to the symmetry of the eigenvectors towards spin inversion. An explicit calculation gives
fcG4 - G-4)= -~:(~,)/2((~,
- a-4) - i/3(0, - gw3)- 213h - G-~)), E23d
;(cr3 - ab3) = -J:(o,)/2(13/9(0,
- (~-3) + 2/9(az - ~-2)
- l/3(0, - 0-J) - 2J:tO)/9((0, - c-3) - (~2 - Q)), ;(G’
- oe2)= -J;(o,)/2(8/9(o,
- a-,)+
2/'9(0, - G-~)
- 213(~, - Go)) - 2JA(0)19((~ -a-,)
g (a, - G-1) = -J:(w,)
L23bl
- to3 - ~-,a
E23d
(G.1- O-l).
Equation [23d] shows that (rrl - aPI) is uncoupled to the other matrix elements. The reason for this is that (I2 + 13)’ is a constant of the motion. Equation [23d] can be integrated directly and gives an exponential term with the time constant -l/Ji(w,). The other three coupled equations give rise to three exponential terms with the inverse time constants -J:(cc?‘,) : -J:(o,)/18(7
+ 4J;(O)/J;(w,)+
(13 + 2OJ~(O)/'J~(w,)+ I~(J~(O)/J:(W~))~)~~~.
With initial conditions given by a 180”-pulse the time dependence of the M,-magnetization is given for (a) extreme narrowing J:(O) = Ji(w,) : M,(t) = MO - 2&/7{exp(-J’(O) where MO is the equilibrium
t) + 6 exp(-2/9J”(O) t)}
magnetization;
and (b) J;(O) * J:(o,):
M,(t) = MO -2M,/9(exp(-J:(w,)
t) + 8 exp(-1/4J:(w,)
t)}.
1241
P51
The same result is obtained by setting (crS- K~) equal to (c2 - gP2) in Eq. [23] as expected. Finally if JIZ #J,, so that (crI - gml) couples to the other elements in Eq. [23] and nonextreme narrowing so that (c3 - c-~) = (c2 - c--2) = (rrl - awl) one gets M,(t) = MO - 2Mo exp(-1/3J:(w,)
t).
WI
This last example corresponds to the approximation made in the previous section in the derivation of Eq. [21]. Equations [24-261 illustrate the errors made in these approximations. Even at extreme narrowing and with an extra constant of the motion the decay of the M,-magnetization goes essentially as a single exponential? though with a small but detectable correction for small t. The difference between Eqs. 1241 and [25] illustrates what happens when one goes to nonextreme narrowing. There exist differences but they are small. A final conclusion is that the approximations made in the previous section are justified, under the conditions given, for a three-spin system. COMPARISON
WITH
EXPERIMENTS
From the calculations in the two preceding sections one can predict that for systems described by the Hamiltonian [3] the decay of the Ilrl,-magnetization towards equilibrium is essentially described by a single exponential with the time constant given by Eq. [21]. For small times a deviation from the single exponential might be observable.
NEW
APPROACH
TO SPIN DIFFUSION
227
This is especially so in the extreme narrowing region. These simple predictions will now be compared with experiments. The proportionality between N and Tl in Eq. [Zl] has been experimentally verified in the case of solids containing paramagnetic impurities (3,I4) and in solid alkanes (4). For solids where the relaxation is caused by paramagnetic impurities an extra complication arises, however. Nuclear spins near the paramagnetic ion not only experience a strong fluctuating field but also an extra static field, which causes a shift. This shift can very well be of the same order of magnitude as the nuclear dipole-dipole couplings (15). These nuclear spins are then only intermediately strongly coupled to the other spins. The consequence of this is that in addition to the slow exponential relaxation, a faster relaxation should also be present at shorter times. This has been observed (3) but was interpreted as diffusion limited relaxation. In the present formulation diffusion limited relaxation is interpreted as the necessary appearance of fast decaying exponential terms due to the coupled equations in Eq. [22], as explained above. Which of the two factors mentioned actually gives the observed deviation from a single exponential is not clear. If the relaxation time of the electron spin is long compared to the inverse of the nuclear Larmor frequency, the shift effect is expected to be the most important one. In liquid crystals, where the molecular motion is fast, a single T1 is generally observed. The liquid crystalline phases are anisotropic, which results in static dipole-dipole couplings that are usually greater than the shift differences. All the protons in an alkyl chain in such a phase can consequently be expected to form a strongly coupled system. If on the other hand the alkyl chain is broken by an ether linkage, for example, the dipole couplings over this linkage could be small, resulting in two weakly interacting spin systems. In such a case two Tl’s might be observed. Neglecting this difficulty the fast motion in the liquid crystal still indicates that for short times a nonexponential decay could be observed. If this is not the case it can be explained in two ways. First, the relaxation interaction is of the same order of magnitude at all the protons and, second, the relaxation is due to nuclear dipole-dipole interactions, which introduces an extra coupling term in Eq. [22]. Either of these factors would favor the observation of a single Tl. The suggestion that spin diffusion is of importance for protons in proteins in solution (6) is questionable in the light of the present calculation. In general the shifts are larger than the spin-spin coupling constants within the amino acid side-chains and in particular little coupling exists between side-chains. The total spin systems is not described by a Ramiltonian of type [3]. If nevertheless a single Tl is observed this can be explained by the factors : (a) The strength of the relaxation interaction and its time dependence do not differ in orders of magnitude from proton to proton. (b) The dipole-dipole relaxation mechanism introduces a coupling between the relaxation of the different spin systems (see, e.g., (8, Eq. [VIII:87])) that tends to give the total system a single Tl. (c) Since usually Jo(O) > J1(oO) in these solutions even a weak coupling can couple diagonal density matrix elements with equal m-value in Eq. 1221. This fact together with the changes in absolute shift should be considered when the frequency dependence of T1 is interpreted. lYnthis short survey only the simplest aspect of the relation between experimenrs and
,228
WENNERSTRGM
the presented theory have been outlined. A complete analysis of the often compiicated experiments is beyond the scope of this paper. CONCLUDING
REMARKS
In the present paper a new way of viewing spin diffusion has been proposed. The basic idea is that in a strongly coupled spin system the individual spins have lost- their identity in the sense that a transition between two eigenstates can not be identified with a transition of a single spin. A consequence of this is that a perturbation at a single spin affects all eigenstates. This is shown to lead to a single Tl for the total spin system under certain circumstances. A single Tl is predicted, at strong narrowing, when the coupling constants are much greater than the shift differences and J,(O)/J,(w,) s 1 and no essential constants of the motion except 12, I,, and Ho are present. Even when nof all these conditions are satisfied a single experimental T, is often expected. The present approach to spin diffusion might seem very different from the one given by Bloembergen. This is in fact not the case. The main difference is that the present method is basically one degree less approximative than Bloembergen’s method in which a diffusion equation is employed. If one uses the Bloembergen diffusion equation as a starting point for a derivation, some of the results will be artifacts of the approximation without any physical relevance. REFERENCES
I. N. BLQEMBERGEN,Physica 15,386 (1949). 2. 6. R. KHUTSISH;VILI,“Progress in Low Temperature Physics,” Vol. VII, p. 375 (C. J. Gorrer, Ed.), North Holland, Amsterdam, 1970. 3. W. E. BLUMBERG,P&s. Rev. 119,79 (1960). 4. J. E. ANDERSONAND W. P. SLIGHTER,J. Phys. Chem. 69,3099 (1965). 5. D. W. MCCALL AND D. C. DOUGLASS, Polymer 4,433 (1963). 6. R. KIMMICH AND F. NOACK, Bet-. Bunsenges. Phys. Chem. 75,269 (1971). 7. J. CHARVOLIN AND P. RIGNY, J. Magn. Resonance 4,40 (1971); or E. OLDFIELD AE;DD. CHAPMAX, FEBS Letters 23,285 (1972). 8. A. ABRAGAM, “The Principles of Nuclear Magnetism,” Clarendon Press, Oxford, 1961. Ye M. E. ROSE, “Elementary Theory of Angular Momentum,” Wiley, New York, 1957, 10. A. KUMAR AND B. D. N. RAO, J. Magn. Resonance 8,l (1972). il. R. A. HOFFMAN, “Advances in Magnetic Resonance” (J. S. Waugh, Ed.), Vol. 4, p. 96, Academic Press, New York, 1970. 12. J. E. ANDERSON AND K. J. Lru, J. Chem. Phys. 49,285O (1968). 13. P. L. CORIO, “Structure of High Resolution NMR Spectra,” p. 267, Academic Press, New York, 1966. 14. D. TSE AND I. J. LOWE, Phys. Rev. 166,292 (1968). 15. A. CARRINGTON AND A. S. MCLACHLAN, “Introduction to Magnetic Resonance,” p. 222, Harper and Rowe, New York, 1967.