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APPENDIX 1 Algebra of -l/2 and Spin-l Operators

It is frequently convenient to work with the matrix representation of spin operators in the eigenbase of the Zeeman Hamiltonian. Some results for spin-1/2 and spin-l systems are given in this Appendix. Eigenvectors Eigenvectors are represented as column matrices (kets) and row matrices (bras), while operators are square matrices. The Ill() and 1m states for spin-t/2 are represented by:

Ill() == G) and 1{3) == G) (1) while, for spin 1:

(2) 11) ~ m' 10) ~ m' and 1-1) ~ m

Operators The matrix representation of spin operators in this eigenbase may be obtained by applying the results of angular theory:

IzII, m) = mil, m)

I+II,m) = J(I - m)(I + m + 1) II,m + 1), 1+ = Ix + iIy (3) LII,m) = J(I + m)(I - m + 1)II,m -1), L = Ix - iIy J2 II,m) =1(1 + l)II,m). Operators 351

The matrix representation of the 12 is thus 1(1 + 1) times the unit matrix of (21 + 1). The other operators are given by:

Spin 1/2 Spin 1

(4)

L G~)

1 0 0) 12 ~(l 0) 2 ( 0 1 0 4 0 1 001

The element ij of these matrices represents (iIOlj), Ii) being the ket corre• sponding to m = mi (ml = I, m2 = I - 1, m3 = I - 2, ... ). The matrix representation of eigenvectors and operators in a composite spin system is obtained by working in the appropriate direct product space. Thus, for two spins 1/2,

laa) ~ G)®G) ~ m

(5)

Ipa) ~ (~)®G) ~ m

For such an IS spin system, we have in the direct product space: 352 Appendix 1. Matrix Algebra of Spin-lj2 and Spin-l Operators

la) IP) (al (0 2Ix = (PI 1 ~ )@ ls= G ~)@(~ ~) laa) lap) IPa) IPP) (aal 0 1 (aPi 0 0 = (pal 0 0 (6) (PPI l! 0 ~J la) IP) 2S - 1 x (al ( 0 1) - C ~ x - I @ (PI 1 0 - 0 ~)@G ~) laa) lap) IPa) IPP} (aal 1 0 (aPI 0 0 (7) = (pal 0 0 (PPI ~~ 0 !J Note that the order of the spins is chosen arbitrarily; the choice made, however, must be maintained right through. The direct product of two ma• trices A and B is given by: Ai:iB A12B ...... Ai:nBj C=A@B= .: .: (8) l . . Ani B An2B ...... Ann B The direct product of an (m x m) matrix with an (n x n) matrix is thus an (mn x mn) matrix. Note that the direct product of matrices is therefore very different from . Also listed below are the matrix representations of some higher powers of spin operators. These results may be checked by usual matrix multiplication. S~nlp S~nl ~ (H J ~ ( ~ ~ -~) -1 0 1 !(1 0) 4 0 1 (H ~) (9) Operators 353

Spin 1/2 Spin 1 0 [Ix, IyJ+ 0 0 i -i) (~ 0 -1 [Iy, IzJ + 0 0 fi (! -1 !) 1 [Iz,IxJ+ 0 1 (0 0 fl~ -1 -!) Various recursions then follow for i = x, y, z:

If = iI7- 2 (I = 1/2, n ~ 2) (10) If = 17- 2 (I = 1, n ~ 3) For spin 1, the following relations also hold, with i = x, y, z:

IJ)i =0 (i of. j) (11 ) IJf + If Ii = Ii (i of. j) In general, for a spin I, we have the Cayley identity: [Iz - IJ[Iz - (I - I)J[Iz - (I - 2)J ... [Iz + IJ =0 and

I~I+l) = I~I+l) = 0; (12) /-i ) I~ ( . Il . (Iz - j) I~= 0, i = 1, 2, ... , I. J= -(/-,) Finally, we give below the matrix representation of various bilinear operator products for a two-spin-l/2 situation. The results may be verified by matrix multiplication, working with the representation of the individual single-spin operators in the direct product space of the two spins 1/2.

0 0 0 1 0 1 i [ 0 0 1 IxSx = 4 Y 1 0 IxS =4 0 -1 0 -~J ~~ 0 0 ~J 1 0 0 0 0 0 0 i 0 -1 1 0 1 IySx = 4 IySy = 4 0 -~1 0 -~J I~ 0 0 U0 0 354 Appendix 1. Matrix Algebra of Spin-1/2 and Spin-I Operators

0 0 1 0 1 0 -1 0 1 0 0 IzSx = (13) IzSz = 4 ~ 0 -1 4 0 0 ~ 0 0 ~J ~ 0 -1 -!J -1 0 0 1 i 1 0 0 1 0 0 IzSy = 4 ~o~ IxSz = 4 0 0 0 0 -] 0 -1 ~J l~ -1 0 0 -1 i 0 0 0 IySz = 4 ~ 0 0 [ -1 0 ~J From the above, it is clear that IxSx, IySy, IxSy, and IySx represent zero,- and double-quantum coherences in the spin-I/2 IS system. Operator representations for an IS system with I = 1, S = 1/2 involve (6 x 6) matrices. For example, 0 1 0) (0 1 j2Ix = (1 0 I ® Is = 1 0 o 1 0 0 I

11a) lIP) lOa) lOP) 1-1a) 1-1P) (lal 0 0 0 0 o (lPI 0 0 0 1 0 o (Oal 1 0 0 0 o (14) =(OPI 0 1 0 0 0 1 ( -lal 0 0 0 0 o ( -lPI 0 0 0 0 o

(1 0 0) (15) 2Sx = II ® G~) ~ ~ ~ ~ 09 (~ ~)

0 0 0 0 1 0 0 0 0 0 0 0 0 (16) 0 0 1 0 0 0 0 ~1 0 0 0 0 0 0 0 1 ~j Operators 355

0 0 -1 0 0 0 0 0 0 -1 0 0 1 0 0 0 -1 0 j2Iy = i (17) 0 1 0 0 0 -1 0 0 1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 2Sy= i (18) 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1= (19) z 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 2Sz= (20) 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 APPENDIX 2 The Hausdorff Formula

It is frequently necessary to evaluate expressions of the form:

RA = exp( -iXt) A exp(iXt) which describe the evolution of an operator A for t , under the action of a time-independent Hamiltonian x. The Baker-Campbell-Hausdorff for• mula consolidates the series expansion of exponential operators to evaluate such operator evolutions:

RA == exp (-iXt) A exp (iXt) (it)2 = A - (it) [X, A] + T![X, [X,A]] (1) (it)3 - 3! [X, [X, [X, A]]] + ...

Note that this formula involves successive with the Hamiltonian. An NMR problem has a well-defined, finite dimensional operator space associated with it and so the number of independent commutators is limited and the series expansion quickly leads to an operator recursion. It may be noted especially that it is crucial to work with the appropriate set of linearly independent operators, and to insure that successive commutators are expressed in this basis set, so that the operator recursions are not lost sight of. Suitable basis set operators for problems involving spin-l/2 and spin-l systems have been discussed in Chapter 1. We discuss below briefly some cases of interest. Appendix 2. The Hausdorff Formula 357

CASE 1. [£', A] = O. It follows that RA = A since all the commutators vanish. In other words, A is invariant to evolution under a Hamiltonian with which it commutes. For example consider a spin-1 system, with £' = tv; - 1(1 + 1)) and A = [/x,!y] + . [£', A] ~ [1;, Ixly + Iylx] = [1;, Ixly] + [1;, Iylx]

= ilz(l; - In + i(l; - I;)Iz - iIAI; - I;) - iU; - I;)Iz (2)

= 2i(lAI; - I;) + (I; - I;)Iz) = 2i(lz - Iz) = 0 where we have employed the properties of spin-1 operators discussed in Appendix 1, and some properties of commutators to be discussed later in this Appendix. Thus, RA = [1x'!y]+.

CASE 2. [£', A] = aA.

In this case, RA = A exp ( - iat). As an example, consider £' = M z and A = I + = Ix + ily • Then, [£', A] = iMy + Mx = L\A. Thus, RA = 1+ exp (-iL\t).

CASE 3. [£', A] = B, [£', B] = kA, A =1= aB. This has been termed a problem of order 2. We find,

RA = ( A + (i~{2 kA + (i~t k2A + ... ) _ (itB + (i~;3 kB + ... )

2 4 2 3 _ A t k + t k _ ) _ iB (t _ t k + ) - (1 _ 2! 4! ... 3! ... (3)

= A cos}kt - ~iB sin}kt

Problems of order 2 have been encountered frequently in Chapters 1 and

3 and the corresponding equations of , R A , have been given there. We consider here a few representative examples.

(1) A = Ix, £' = 2nJIzSz.

[£', A] = i2nJIySz = B; (4)

Thus, RA = Ix cos nJt + 2IySz sin nJt. (5) (2) A = Ix, £' = M z· [£',A] = iMy = B; (6)

Thus RA = Ix cos M + Iy sin M. (7) 358 Appendix 2. The Hausdorff Formula

Problems of order higher than two are encountered, for example, in AMXN systems evolving under a strong Hamiltonian as discussed in Chapter 1 and lead to multiple frequencies of evolution. Algebra We give here some of the standard results of commutator algebra that are useful in evaluating the Hausdorff formula for specific Yf and A. These expressions follow from the definition of commutators:

[A, B] = (AB - BA) (10) We find: [A,B] = -[B,A] [A,kB] = k[A,B], k being a constant. [AB, C] = A[B, C] + [A, C]B (11)

n-l [An,B] = L Am[A,B]An- m- 1, n ~ 1 m=O APPENDIX 3 Fourier Transformation

In this Appendix we briefly discuss some important properties of the Fourier and related issues of sampling, and sensitivity and resolution enhance• ment by exponential multiplication. We also touch upon the dynamic range problem in FT NMR.

The Fourier Transform A time domain f(t) with period T ---+ 00 may be represented in the basis set of exponential functions {einwot} (wo = 2n/T), which goes over into {e iwt } as the period T ---+ 00. We thus have:

f(t) = - 1 fco F(w)e iwt dw (1) 2n -ex)

the inverse Fourier transform (FT) of F(w), with

F(w) = f~co f(t)e-iwt dt (2)

the Fourier Transform of f(t). Note that these prescriptions for f(t) and its Fourier transform F(w) correspond to an exponential Fourier analysis of a periodic function, in the limit of the period T ---+ 00. We denote an FT pair by f(t) <-> F(w). F(w) is in general a complex function:

F(w) = IF(w)le i6(w) (3)

If f(t) is a real function of t, the of F(w), i.e., F*(w) = F( -w). Thus

F( -w) = IF(w)le-i6(w) (4) 360 Appendix 3. Fourier Transformation

The magnitude spectrum IF(w)1 is an even function of w, while the phase spectrum 8( w) is odd. Several properties of the FT follow from the definition, considering the FT pair f(t) +-> F(w).

1.

F(t) +-> 2nf( - w)

If f(t) is an even function, we have F(t) +-> 2nf(w) 2. Linearity

for arbitrary constants a 1 and a 2 • 3. Scaling

f(at) +->~ F (~) lal a for a real constant a. 4. Frequency shifting

5. Time shifting

6. Time differentiation and integration df -+->iwF(w) dt if the FT of df/dt exists; and

t 1 f f(t) dt +-> -:-F(w) -00 lW if F(w)/w is bounded at w = O. 7. Frequency differentiation dF - itf(t) +->• dw and

8. Convolution Appendix 3. Fourier Transformation 361

Also, 1 fa) !l(t)!2(t)<-+ 2n -00 Fl(v)F2(w - v)dv (5) 1 == - [Fl (w) * F2 (w)] 2n Convolutions obey the commutative, associative, and distributive laws. We list below some useful FT pairs:

f(t) F(w)

1 27[b(w) 6(£) 1 u(t) nb(w) + (l/iw) e-atu(t) (a + iwtl cos wot n[b(w - wo) + 6(w + wo)] sin wot in[b(w + wo) - 6(w - wo) cos wot u(t) ~{b(w - wo) + 6(w + wo)] + (iw/(w6 - ( 2 )) sin wot u(t) (wo/(w6 - ( 2 )) + ;i [6(w - wo) - 6(w + wo)] e-at sin wot u(t) wo/[(a + iW)2 + W6] GT(t) rSa(wr/2)

(wo/2n)Sa( Wo t12) Gwo (w) M2- -,,2W 2/2 e -t2 /2(J2 (Jy",-ne

'i) bT(t) == L b(t - 11 T) w0 6",0(w) == wOn=toc 6(w - I1Wo) ( Wo = 2;) n= -cc e-a1tl 20(0 2 + ( 2 )

Notes 1. The step function u(t) = 1, t > ° = 0, t < ° 2. The Dirac delta function d ()(t) = lim-(uAt)) ,,~odt . 1 = hm - [u(t) - u(t - a)], a~O a where ua(t) is a step function with rise time t = a.

f~w 6(t) dt = 1 362 Appendix 3. Fourier Transformation

3. The sampling function

sa(~,) = sin (w,/2)/(w,/2)

4. The gate function GAt) = 1, It I < ,/2

= 0, It I > ,/2

It is of particular interest for FT NMR to consider the time-domain functions SI (t) = u(t)e-at sin wot and S2(t) = u(t)e-at cos wot, (6) which represent a general FID (a ~ 0). We find from the definitions that their Fourier transforms are given by:

SI (t) ...... f~oo u(t)e-at sin wote- iwt dt

(7)

a a - ~ [a2 + (w _ WO)2 - a2 + (W + WO)2 ] The real part, therefore, corresponds to a Lorentzian dispersion signal at Wo, whereas the imaginary part is a Lorentzian absorption at Wo (full width at half height is equal to a/n Hertz), ignoring the responses at - wo' On the other hand:

l[ a a ] (8) = 2: a2 + (w - wo)2 + a2 + (w + WO)2

i [ (w - wo) (w + wo) ] - 2: a2 + (w - WO)2 + a2 + (w + WO)2

In this case, the real part corresponds to an absorption at Wo, and the imaginary part corresponds to a dispersion. These results have been employed throughout the book in discussions of lineshapes. In the above expressions, a = (T2*tl. Resolution enhancement may be achieved by mUltiplying SI (t) and S2(t) by exp (bt) (b ~ 0), leading to a linewidth of (a - b)/n Hz; sensitivity is reduced in this process because the later, noisy portions of the FID are weighted in preference to the initial portion, which has the maximum signal content. Sensitivity enhancement, on the other hand, is achieved by multiplying SI (t) and S2(t) by exp( -bt), (b ~ 0), leading to a preferential weighting of the signal-rich initial portion of the FID relative to the noisy FID tail. From the Dynamic Range Problem 363 expressions for the FT of S1 (t) and Sz(t), however, it is clear that in this process resolution is lost, the resulting linewidth being (a + b)/n Hz. Sampling In actual practice, most FT NMR spectrometers perform a discrete Fourier transform of the FID to retrieve the frequency spectrum. To this end, the FID, which is an analog voltage waveform, is sampled at regular intervals and digitized for computer handling. The sampling frequency must be sufficiently high to reproduce the frequency spectrum with fidelity. In fact for a band• limited signal with no frequency components above Vm Hz, the sampling frequency should be Vs ~ 2vm to produce the correct spectrum. This result of "Nyquist sampling theorem" may be proved as follows for a function f(t)

which is sampled every T seconds, generating f(t)6T (t), where: co 6T (t) = L 6(t - nT) (9) n=: -00

For f(t) +4 F(w), we have:

f(t)6T(t) +4 F(w) * Ws 6w ,(W) (10) Employing the frequency shift property, we find:

F(w) * 6ws(w) = F(w) * L 6(w - nw,) n

1 w ~ !(t)6T (t) +4_ L F(w - nw,), (12) T n=-oo

which is the function F(w) repeating itself every Ws radians per ; this periodic repetition occurs without overlap as long as Ws ~ 2wm. This implies that the sampling interval (also called dwell time) 2n 1 1 T=-=-~- (13) WS Vs 2vm

If this condition is violated, the overlap of successive F(w - nws) leads to a frequency Vs + ~V being represented in the spectrum as Vs - ~v, which is known as aliasing. Aliased signals in FT NMR spectra may be identified readily because they occur with wrong phases which cannot be corrected by the zero- and first-order phase correction routines. Dynamic Range Problem The detection of very weak signals in the presence of very strong ones is a problem that is often encountered in practical NMR, e.g., in the study of 364 Appendix 3. Fourier Transformation the lH NMR ofbiomolecules in D20. The small percentage ofHDO even in a highly deuterated sample of D2 0 can cause considerable problems. In CW NMR, the problem may sometimes be avoided by not exciting the HDO signal at all; in pulsed methods of excitation however, this requires special approaches, such as tailored excitation, described in Chapter 2. In the presence of very strong resonances, weak resonances in an FID may be detected only if the digitizer resolution and computer word length are sufficiently high. In the frequency domain, single-scan spectra can have a dynamic range as high as ± 40,000: 2 with a 12-bit digitizer (analog to digital converter, or ADC). The dynamic range increases further with a 16-bit digitizer and can be as high as ± 280,000: 2. In order to utilize the available dynamic range, however, the computer word length should be sufficiently high. For instance, the dynamic range of ± 40,000: 2 cannot be availed if the computer word length is only 16 bits (± 32 K). APPENDIX 4 Dipolar Relaxation

The magnetic interaction between nuclear was the subject of Chapter 6, where the line broadening induced by it in solids was discussed, together with means to counter it. Some consequences of cross-relaxation arising from -dipole interactions in liquids and solids were also described in Chapter 2 and Chapter 6, in the context of sensitivity enhancement by incoherent transfer. In this Appendix, we summarize some basic results for relaxation under dipolar coupling. The Hamiltonian describing dipolar interaction between two spins I and S is given by:

+ sin2 B sin2cp1vSy + cos2 BlzSz + sin2Bcoscpsincp(/xSy + IySx) + sin B cos B sin cp(/xSz + IzSx)

+ sin Bcos Bcos cp(/ySZ + IzSy))]

= ('hYSh2)(A + B + C + D + E + F) (2) 4n2 r3 with 366 Appendix 4. Dipolar Relaxation

A = Iz SAl - 3 cos2 8) B = -t(I+ S- + r S+)(1 - 3 cos 2 8)

C = -~(I+ Sz + IzS+)sin8cos8e-i

The structure of this Hamiltonian insures zero-, single-, and double-quantum connectivities. We express Yf.J as a product of a spin operator part and a function of spatial coordinates:

·ffa(t) = OJ!(t) where random lead to the time dependence in f(t). Employing the standard results of time-dependent perturbation theory to compute the proba• bility of relaxation transitions induced by Yf.J(t) between a pair oflevels Ii) and Ij) of the two-spin system, we have, assuming for the fluctuations in Yf.J a stationary random process:

(4)

where G(r) is an autocorrelation function and is generally a rapidly decaying function of r. The transition between levels i andj under random fluctuations in Yf.J(t) is thus proportional to the spectral density J(w) of the fluctuations at wij = (Ei - Ej ); J(w) is the Fourier transform of the correlation function:

G(r) = f(t)f(t + r) (5) the statistical avarage of f(t)f(t + r). Three distinct are to be calculated for zero-, single-, and double• quantum processes, as indicated above. The probabilities are all products of the square of the appropriate matrix element, with the function of spatial coordinates averaged suitably. We find for I = S = 1/2: Appendix 4. Dipolar Relaxation 367

Wo '" l6(1 - 3 cos 2 e)2 J(W 23 ) = 116 X ~J(W23)

WlI '" (6 sin 2 e cos 2 e J(w12 ) = 196 X lsJ(w12 ) (6)

W2 '" (6 sin 4 e J(W 14 ) = 196 X 18SJ(W14) with the spatial coordinates isotropically averaged over the unit sphere. When dealing with white spectral conditions for J(w), i.e., in the extreme narrowing

limit, we therefore have: Wo: 2W1 : W2 = 1: 3: 6. For homonuclear inter• actions in the "black" spectral limit, on the other hand W1 = W2 = 0, and Wo alone is nonzero. In general therefore the spectral density at W = WI' WS , IWI - wsl and (WI + ws) are all relevant parameters contributing to relaxation under dipolar coupling; for Tz relaxation J(O) contributes additionally via the A(t) term of ~(t). In many practical situations the correlation function G(r) may be char• acterized with a single time constant which is the correlation time re: (7) J(W) now takes the form:

J(w) = f:", exp(-lrl/rJexp(-iwr)dr (8)

A plot of J(w) versus w is given in Fig. AA.1 for three situations: Wo < l/ro Wo '" l/re and Wo > l/ro Wo being the Larmor frequency. The nuclear Overhauser effect may be derived at once from the relaxation and level diagram (Fig. AA.2) for two spins 1/2, employing the above

discussion. Notice that observing I spins while saturating S, WlI contri-

J (W)

Intermediate 't (w rv l/'t ) i t----+-_ C 0 c

___ w

Figure A.4.1. Spectral density J(w) versus frequency w. 368 Appendix 4. Dipolar Relaxation

Figure A.4.2. diagram and relaxation probabilities in a system of two spins 1/2.

butes to establishment of Boltzmann populations, while W2 and Wo lead to nonequilibrium populations of opposite . This leads at once to the enhancement factor:

When I = 13C and S = 1H, for example, upon saturating the proton reso• nances, the dipole coupled 13C'S attain a polarization which is three times the Boltzmann equilibrium quantity. A negative NOE ( ~ 5) results for 15N dipolar coupled to (l = 15N, S = 1H), since 15N has a negative magnetogyric ratio. In practical situations, the dipolar interaction between spins mayor may not be the dominant mechanism of relaxation; in case other mechanisms such as spin- interaction, anisotropy, etc., contribute sig• nificantly to the establishment of Boltzmann equilibrium, observed nuclear Overhauser enhancements will be below the maximum derived above: <1z) - (10) 'Is (W2 - Wo) (11 ) <10> 11 (Wo + 2WlI + Wz + W O) where W O is the transition probability under other relaxation mechanisms. In the homo nuclear situation, we find: Appendix 4. Dipolar Relaxation 369

(1~ (10 ) 1· h h· II· . - >- = 2" III t e w lte spectra lmlt (10 ) (12) = - 1 in the "black" spectral limit It is interesting to note that in the isotropic phase, negative NOE's can result for a pair of nuclei with the same sign of y, if Tc is above a threshold value, provided the larger y is not more than 2.38 times the smaller y. This again corresponds to the situation that Wo > Wz, i.e., J((1)23) » J((1)14). APPENDIX 5 Magnus Expansion and the Average Hamiltonian Theory

The of the of the system under the influence of the internal Hamiltonians is governed by the von Neumann equation:

a(t) = -i[Yf,a(t)] (1) When the Hamiltonian is time independent then the above equation integrates to:

a(t) = U(t)a(O)U-1(t) with:

U(t) = exp( -iYft) (2) However, when the Hamiltonian is not constant throughout the time interval of interest but can be assumed to change with a proviso that when the total interval is divided into n subintervals of duration, l' '2' ... , Tn' the cor• responding Hamiltonians ~(k = 1, ... , n) are constant in the kth interval we can write U(t) as:

U(t) = exp ( - iYl;', Tn) exp ( - iYl;',-l 'n-d ... exp ( - iYf;.' d (3) provided that all the ~'s commute. If, on the other hand, the Hamiltonian varies continuously we have to express U(t) in integral form as:

U(t) = T exp( -i rYf(t')dt') (4) where T is the Dyson time ordering operator, which orders the operators of higher time arguments in the expanded exponential to the left. Magnus expansion is the tool used to express the operator U(te), (this replaces U(t), when dealing with cyclic pulse sequences such as W AHUHA-4, Appendix 5. Magnus Expansion and the Average Hamiltonian Theory 371

MREV -8, see Chapter 6) in the form of exp ( - iFtJ The form of F is readily found when we expand V(te) in a power series (under conditions of actions corresponding to infinitesimal ):

(5)

If the time intervals are so divided as to make ~ 'k a small quantity of first order we can rewrite the above equation as:

+ 2(£'2£'1 '2'1 + £'3£'2'3'2 + ...

Yt. Yt. )] ( - i)3 [~3 3 £3 3 (6) + n n-1'n'n-l +~ 1 '1 + 2 '2

+ ,··3(£'2.1t;2'2,i + £Z2£'1'~'1 + ... ) + 6(£'3£'2£'1 '3'2'1 + £'4£'3£'2'4'3'2 + ... ) ... + ... ]

The time ordering operator makes sure that all operators are arranged in ascending order from right to left. We said we wanted to express V(tJ as exp( -iFtJ We now make the assumption that F can be written as a sum of Hamiltonians of different order of magnitude (similar to perturbation order); i.e., F = .if + yf(1) + yf(2) + ... (7) This leads to:

exp (-iFtJ = exp (- itAyf + yf(1) + yf(2) + ... ))

= 1 + (-itJ(yf + yf(1) + yf(2) + ... )

+ ( - iU2 [(yff + yf yf(1) + yf(1) yf + (yf(1»)2 2! (8) +.ifyf(2) + yf(2) yf + yf(2) yf(1) + ... ]

+ (-itY [(yf)3 + (yf)2 yf(1) + yf(1)(yf)2 3! + yf yf(1) yf + (yf(1»)3 + ... ] 372 Appendix 5. Magnus Expansion and the Average Hamiltonian Theory

Grouping terms that are correct to a given order, the above equation can be rewritten: _ _ (-it)2_ exp ( - iFtJ = 1 - itc:!lf - iteJlf(1) + --t-(J'ff (9) _ it if(Z) + (- itJ2 [ifif(1) + il(1) if] + ... c 2! - we shall now equate the terms in the above equation to those given in the expansion of U(tJ, leading to:

if = l/tAJIfl 'I + :!Ifz'z + :!If3'3 + ... )

if(1) =(-i/(2te))([£;,:!IfI ]'2'1 + [:!If3,:!IfI]'3'1 (10)

+ [Jlf3 ,:!Ifz], 3' z + ... ) etc.

H is obvious that teif must be a quantity offirst order, tcif(1), of second order, etc. As an example, if the :!If takes only two values during a cycle such that during 0 to te/2, ~'I = A and during tel2 to to K 2 , 2 = B, then Magnus expansion leads to:

te'.:If = A + B tcif(!) = ( - i/2) [B, A] (11)

tcif(2) = (l/12){[B,[B,A]] + [[B,A],A]} From which it follows that, for noncommuting operators A and B,

exp( -iB)exp( -iA) = exp {-i(A + B) - 1/2[B,A] (12) + (i/12)([B, [B, A]] + [[B, A], A]) + ... } which is the well known Baker-Campbell-Hausdorff formula. In the limit 'k ..... 0: if = (1 /tJ Ie il(t) dt (13) where if is the average Hamiltonian correct to first order. Likewise,

if(1) = (- i/2U Ie dtz I2 dt! [if(tz), <¥f(t!)] (14)

For a symmetrical cycle of time-dependent Hamiltonians such that:

if(t) = if(tc - t) v-! (tJ = exp (iK! '!) exp (iKz, 2) ... exp (iJlf" 'II) (15) = exp ( - i£" 'II) exp ( - iJlf,,_! 'II-d' .. exp ( - i.if!, d

= U(-tJ Appendix 5. Magnus Expansion and the Average Hamiltonian Theory 373

For such cycles it is straightforward to show that: £'(~) = ( - 1)~ £'(~) (16) and hence: £'(~) = 0 for k odd. If, on the other hand, we consider antisymmetrical cycles cor• responding to the equality:

£'(t) = - £'(t - tJ then:

U(tJ = exp(iK1rdexp(iKzrz)'" x ···exp(iJt,l-lrn-l)exp(i~r.) = U-1(tJ i.e.,

Therefore: (17) Applying the average Hamiltonian theory to the Carr-Purcell sequence corresponding to 90~-r-180~-2r-180~2r, we can see that the sequence is cyclic with cycle time 4r. It is used mainly to suppress the applied inhomogeneity term:

(18)

where ABo~ is the deviation of the static field from its mean value at the kth nucleus. In the interaction representation I~z toggles between I~z and - I~z so that for integer cycle times chemical shift evolution and field inhomogeneities are removed. Under [) pulse approximation the Hamiltonian commutes at all times and hence no contribution from £,(n) (n > 1) exists. For finite pulse

width t w ,

(19)

For the phase alternated sequence -r-(}_y-2r-(}y-r- first-order average Hamiltonian is given by: £' = (1/2)(Yfx + ~) (20)

when () = 7[/2. For dipolar interaction:

£'0 = (1/2)(Kox + Koz) = -(1/2)Koy (21) and for chemical shift interaction: 374 Appendix 5. Magnus Expansion and the Average Hamiltonian Theory

where:

(k) _ 1 1<101>- j2(-lkX + I kz ) (22) with the chemical shifts being scaled by a factor (j2r1 . The cycle is sym• metrical and hence all :/t(k), k = odd vanishes. The leading even-order cor• rection term is given by:

2 ,i(2) = - ~ L(jk + ,1)3 (2lkz - IkJ (23) 12 k

When the pulses are 0 pulses (0 =1= n/2), - 1 yt = - L [Ikz(l + cos 0) - Ikx sin 0] (24) 2 k and the corresponding scale factor can be shown to be Icos (8/2)1. The phase• alternated pulse sequence is used to scale down chemical shifts uniformly. APPENDIX 6 Representation of Spin Hamiltonians

All interactions that we come across in magnetic resonance • dipolar coupling, chemical shift, spin-spin coupling, quadrupole coupling, etc.,-can be written in compact form using the tensor notation in terms of irreducible spherical tensor operators because the spin-dependent and coordinate-dependent parts transform as second-rank . The spin operators in the and the various are very similar and it is possible to construct equivalents by a systematic substitution of variables by the corresponding spin components using the equivalent operator formalism. The position operators do commute with each other, while the spin operators need not and to avoid any complication out of this we follow Rose's definition of a tensor operator. The mth component of an lth rank tensor is given by: (1) where the operator (I' V)l operates on the position coordinates of the spherical harmonic 1I.m and systematically replaces them by the appropriate spin com• ponent. The spherical harmonics are given by: / Y; (() ) = (_1)(m+ 1ml)/2 [(21 + 1)(l-lml)!]1 2 l.m ,cp 4n(l + Iml)! (2) X PI,m(cos())exp(imcp)

where PI.m(cos ()) are the associated Legendre . Equation (1) auto• matically gives for first-rank tensor elements: 376 Appendix 6. Tensor Representation of Spin Hamiltonians

Tl.1 = -(3/4)I+/fi

Tt,o = (3/4)10 (3)

Tt,-t = (3/4)L/fi where 10 = 1z , 1+ = Ix + iIy , and L = Ix - iIy • The higher rank tensors can be generated from the first-rank tensors by the recursion :

T, = (4n)t/2 [1!(21 + 1)!!Jt/2 I.m 3 4n (4) x LC(l,l- 1,1; -/1,/1 + m) x 1't-l,J1+mTt,-J1 J1 where C( ...) are the Clebsch-Gordon coefficients. Thus T2 ,0 is given by: 4n (15)t/2 T2 ,0=3 2n [C(1,1,2;-1,1)Tt,t Tt,-t

+ C(1,1,2;O,O)Tt ,oTt ,0 (5)

+ C(1, 1,2; 1,-1)Tt,-1 Tl,lJ The second-rank tensors are thus derived to be:

2 2 T2 ,0 = if4n~(3Io - I ) (6)

(7) 5 2 T2 '-+2 = jg~I+8n -

The nine components 7;j of a Cartesian second-rank tensor {T} can be decomposed into a :

To = tTr {T} (8) and an antisymmetric first-rank tensor: (9) of which there will be three components with zero and a traceless second-rank tensor;

T 2 : 7;} = ~(7;j + 1j;) - t Tr {T} (10) ha ving five components. Thus any of the nine elements can be given by: (11) Just as rotation in coordinate space can be represented by Euler rotation Appendix 6. Tensor Representation of Spin Hamiltonians 377 matrices R(cxfly) such that:

T:q = R( cx fly)7;, qR-1( cx fly) (12) the corresponding irreducible spherical tensor components are governed for coordinate space rotation by the Wigner rotation matrices g;~q(cxflY) (see Table A.6.1). For second-rank tensors one need consider only irreducible spheriral tensor operators up to order 2. Hamiltonians can be expressed as the product of two irreducible tensors Ak and Tk with components Akq , 7;,p so that:

+k +k Ak'Tk = L (-l)qA kq 7;,_q= L (-l)QAk - q7;,q (13) q=-k q=-k Any Hamiltonian involving spin interactions can be expressed as:

JIf = X . A . Y = L AijXi }j (14) i.j where X and Yare vectors and A is a 3 x 3 matrix. In terms of irreducible spherical tensor operators,

2 +k JIf = L L (-I)q Akq 7;,-q (15) k=O q=-k with:

and:

(16) 1 T20 = j6(3 'f"z - (Txx + Tyy + 'f"z))

1 T2 ±1 = +:2(Txz + 'f"x ± i(Tyz + 'f"y))

The corresponding elements of the A tensor are given by replacing T by spin operators. For symmetric tensors,

,rr = Aoo Too + Azo T20 - (AZ,-l TZI + AZI TZ - 1 ) (17) !,;.) -...J 00

Table A.6.1. Wigner D;"m(rx,{J,y)

'0> 2 o -1 -2 '0(1) x :l 0- (1 + cos f3)2 e2i(d y) 1 + cos f3 sin f3ei(2y +a) 1 - cos f3 sin f3ei(2y-a) (1 - cos f3)2 e2i(y-a) >;. 2 j!sin 2 f3e2iY 4 2 4 ?'

1 + cos f3 sin f3ei(2a+y) 1 - cos f3] i(a+y) [ 1 ~ f3 _ cos2 f3}i(y-a) 1 - cos f3 sin f3ei(y-2a) ""'l [ cos2 f3 - 2 e f3eiY ;os (1) . 2 !ssin2 2 :l on 3 cos2 f3 - 1 o.., -sm. 2f3e2ia (3 . 2f3 ia -fi sin 2f3e -ia -sm. 2f3 e -2ia o Is - './Ssm e Is :;d 8 2 8 (1) './S 1 - cosf3]e_i(a+y) '0 1 - cos f3 sin f3ei(2a- y) [_I_+_;_O_s_f3_ cos2 f3]ei(a- y ) 2 iy 2 1 + cos f3 sin f3e -i(2a+y) @ -1 . 2 - Issin f3e- [cos f3 - 2 on 2 (1) :l (1 - cos f3)2 e (a-y) (1 cos f3)2 (a+ ) 2i 1 - cos f3 sin f3ei(a-2 y) 2iy 1 + cos f3 sin f3e- i(a+2Y) + e- 2i y E -2 . 2 !ssin 2 f3e- . 2 o· 4 4 :l -.o '0Vl ::i' ::c: I'» [ 8' :l ~. :l on Appendix 6. Tensor Representation of Spin Hamiltonians 379 with:

A2± I = =+= (Axz ± iAyz ) For dipolar (D) and quadrupolar (Q) interaction the corresponding elements are:

A IO =Al±l = 0

A 20 = (3/j6}Dzz (18)

and:

T20 = (1/ j6)(3IzSz - I· S) 1 T2+1- = =+=-(/zS+ 2 - + I+S - z} (19)

T2±2 = (1/2}I±S± Table A.6.2 gives the various interactions in terms of spin operators cor• responding to the ~.m's. Table A.6.3 summarizes the transformation properties under spin operator space rotation (cf., Chapter 6, multiple-pulse line-narrowing sequences) of operators that are not rejected by truncation (i.e., that correspond to first• order in perturbation). As can be seen I -spin homonuclear bilinear spin operators and single spin

Table A.6.2. 7;m's Corresponding to Various Interactions Interaction A IaA~ Too Tzo TZ±1 TZ±2 1 Shielding CS I~Bp nBo 0 A3I~Bo fili+Bo2 - 1 1 Spin rotation SR I~Jp Ii .J j6(3/~Jo - Ii. J) fi(/~IJo + nJ±d Ii±IJ ±1 1 Quadrupole Q (Ii)2 - (Ii)2) I~/~ ~(3(l~f fi(li+l/~2 - + 1~/i+l) - (I'±lf

Dipole indirect D 1 . . 1 i k i k I~/~ Ii'Ik -(3/' Ik - I'· Ik) --=(/+1 10 + 10I+d Ii±II~1 Spin-spin J j6 0 0 J2 -- 380 Appendix 6. Tensor Representation of Spin Hamiltonians

Table A.6.3. Transformation of 11m's in I-spin Space Transforms in I-spin space according to: T,)' Interaction 10 After truncation 1= m= Cs Shielding r00 I~Bo o

T CS o 20 A"J IbBo Quadrupolar T2~ ~(3(I~)2 - W)2) 2 o

Direct and indirect T,lo Ii'Ik o o between Ii and Ik TD.J _1_(3Ii Ik _ Ii . Ik) 2 o 20 fi 0000 Direct and indirect T,lo I~S~ o between Ii and Sk TO'] AIiSk o 20 3 00

quadrupolar operators are identical in their transformation properties in both full space and I spin subspace. However, operators that are linear in I space transform differently in full space and I -spin subspace and this aids in selective averaging by multiple-pulse sequences. Selected Bibliography

Some Fundamental Papers 1. F. Bloch, Phys. Rev. 70, 460 (1946). 2. F. Bloch, W.W. Hansen, and M. Packard, Phys. Rev. 70,474 (1946). 3. N. Bloembergen, E.M. Purcell, and R.V. Pound, Phys. Rev. 73, 679 (1948). 4. E.L. Hahn, Phys. Rev. 80,580 (1950). 5. 1.H. Van Vleck, Phys. Rev. 74,1168 (1948). 6. E.L. Hahn and D.E. Maxwell, Phys. Rev. 88, 1070 (1952). 7. N.F. Ramsay, Phys. Rev. 78, 699 (1950). 8. H.S. Gutowsky, D.W. McCall, and c.P. Slichter, J. Chon. Phys. 21, 279 (1953). 9. 1. Solomon, Phys. Rev. 99, 559 (1955). 10. H.Y. Carr and E.M. Purcell, Phys. Rev. 94, 630 (1954). II. U. Fano, Rev. Mod. Phys. 29, 74 (1957). 12. E.R. Andrew, A. Bradbury, and R.G. Eades, Arch. Sci. Geneva, 11,223 (1958). 13.1.1. Lowe and R.E. Norberg, Phys. Rev. 107,46 (1957). 14. A. Abragam and W.G. Proctor, Phys. Rev. 109, 1441 (1958). 15. c.P. Slichter and w.c. Holton, Phys. Rev. 122, 1701 (1961). 16. U. Haeberlen and 1.S. Waugh, Phys. Rev. 175,453 (1968). 17. F.M. Lurie and c.P. Slichter, Phys. Rev. 123, AI 108 (1964). 18. S.R. Hartmann and E.L. Hahn, Phys. Rev. 128, 2042 (1962). 19. A. Pines, M.G. Gibby and 1.S. Waugh, J. Chem. Phys. 56,1776 (1972). 20. W.P. Aue, E. Bartholdi and R.R. Ernst, J. Chem. Phys. 64, 2229 (1976). 21. R. Freeman and G.A. Morris, Bull Magn. Reson. 1,5 (1979). 22. K.1. Packer and K.M. Wright, Mol. Phys. 50, 797 (1983). 23. O.W. Sorensen, G. Eich, M.H. Levitt, G. Bodenhausen, and R.R. Ernst, Prog. NMR Spectroscopy, 16, 163 (1983). 24. G. Bodenhausen, Prog. NMR Spectroscopy, 14, 137 (1981). 25. D.P. Weitekamp, Adv. Magn. Resonance, 11, III (1983). 382 Selected Bibliography

Books and Monographs I. E.R. Andrew: Nuclear Magnetic Resonance, Cambridge University Press, Cam• bridge, 1955. 2. A. Abragam: The Principles of Nuclear , Clarendon Press, Oxford, 1961. 3. J .A. PopIe, W. G. Schneider, and H.J. Bernstein: High Resolution Nuclear Magnetic Resonance, McGraw-Hili, New York, 1957. 4. M.E. Rose: Elementary Theory of , Wiley, New York, 1957. 5. E.P. Wigner: Theory and its Applications to the Quantum of Atomic Spectra, Academic Press, New York, 1959. 6. CP. Slichter: Principles ofMagnetic Resonance, Springer-Verlag, New York, 1980. 7. F.W. Wehrli and T. Wirthlin: Interpretation ofCarbon-13 NMR Spectra, Heyden and Son Ltd., London, 1976. 8. T.C Farrar and E.D. Becker: Pulse and Fourier Transform NMR, Academic Press New York, 1983. 9. U. Haeberlen: High Resolution NMR in Solids-Selective Averaging, Academic Press, New York, 1976. 10. M. Goldman: Spin Temperature and Nuclear Magnetic Resonance in Solids, Oxford University Press, Oxford, 1970. 11. M. Mehring: Principles of High Resolution NMR in Solids, Springer-Verlag, New York,1983. 12. Ad Bax: Two Dimensional Nuclear Magnetic Resonance in Liquids, Delft University Press, D. Reidel Publishing Co. Dordrecht, 1982. 13. D. Shaw, Fourier Transform NMR Spectroscopy: Elsevier, Amsterdam, 1976. 14. P. Mansfield and P.G. Morris: NMR Imaging in Biomedicine, Academic Press, New York, 1982. 15. K. Roth: NM R Tomography and Spectroscopy in Medicine, an Introduction, Springer-Verlag, New York, 1984.

Coherence Transfer I. O. W. Sorensen and R.R. Ernst, 1. Magn. Reson. 51, 477 (1983). 2. H. Bildsoe, S. Donstrup, H.J. Jakobsen, and O.W. Sorensen, 1. Magn. Reson. 53, 154 (1983), and 55,347 (1983). 3. M.R. Bendall, D.T. Pegg, J.R. Wesener, and H. Gunther, 1. Magn. Reson. 59, 223 (1984). 4. G.C Chingas, A.N. Garroway, W.B. Moniz, and R.D. Bertrand, J. Amer. Chem. Soc. 102,2526 (1980). 5. G.C Chingas, A.N. Garroway, R.D. Bertrand, and W.B. Moniz, 1. Magn. Reson. 35,283 (1979). 6. D.B. Zax, A. Bielecki, K.W. Zilm and A. Pines, Chem. Phys. Lett. 106, 550 (1984). 7. O.W. Sorensen, R. Freeman, T. Frenkiel, T.H. Mareci, and R. Schuck, J. Magn. Reson. 46, 180 (1982). 8. P.J. Hore, E.R.P. Zinderweg, K. Nicolay, K. Dijkstra, and R. Kaptein, J. Amer. Chem. Soc. 104,4286 (1982). 9. O.W. Sorensen, M.H. Levitt, and R.R. Ernst, 1. Magn. Reson. 55,104 (1983). 10. P.J. Hare, R.M. Scheck, and R. Kaptein, J. Magn. Reson. 52, 339 (1983). II. O.W. Sorensen, M.H. Levitt, and R.R. Ernst, 1. Magn. Reson. 55, 104 (1983). 12. M.H. Levitt and R.R. Ernst, Chem. Phys. Lett. 100, 119 (1983). 13. O.W. Sorensen, M. Rance, and R.R. Ernst, 1. Magn. Reson, 56, 527 (1984). 14. O.W. Sorensen, U.B. Sorensen, and H.J. Jakobsen, 1. Magn. Reson. 59, 332 (1984). Selected Bibliography 383

15. N. Chandrakumar and S. Subramanian, J. Magn. Resol1. 62, 346 (1985). 16. N. Chandrakumar, 1. Magn. Reson. 63, 202 (1985). 17. N. Chandrakumar, G.V. Visalakshi, D. Ramaswamy, and S. Subramanian, J. Magn. Reson. 67, 307 (1986).

Two-Dimensional NMR in Liquids 1. D.L. Turner and R. Freeman, J. Magn. Reson. 29, 587 (1978). 2. K. Nagayama, P. Bachmann, K. Wuthrich, and R.R. Ernst, J. Magn. Reson. 31, 133 (1978). 3. H. Kessler, H. Oschkinot, O.W. S0rensen, H. Kogler, and R.R. Ernst, J. Magn. Reson. 55, 329 (1983). 4. A.A. Maudsley, L. Muller, and R.R. Ernst, J. Magn. Reson. 28,463 (1977). 5. G.A. Morris, J. Magn. Reson. 44, 277 (1981). 6. M.H. Levitt, O.W. S0rensen, and R.R. Ernst, Chem. Phys. Lett. 94,540 (1983). 7. A. Bax and R. Freeman, J. Magn. Reson. 44, 542 (1981). 8. M. Rance. O.W. S0rensen, G. Bodenhausen, G. Wagner, R.R. Ernst, and K. Wuthrich, Biochem. Biophys. Res. Comm. 117,479 (1983). 9. Anil Kumar, R.V. Hosur, and K. Chandrasekhar, J. Magn. Reson. 60,143 (1984). 10. G. Eich, G. Bodenhausen, and R.R. Ernst, J. Amer. Chon. Soc. 104, 3731 (1982). 11. A. Bax, J. Magn. Reson. 53, 149, (1983). 12. H. Kogler, O.W. S0rensen, G. Bodenhausen, and R.R. Ernst, J. Magn. Reson. 55, 157 (1983). 13. L. Braunschweiler and R.R. Ernst, J. Magn. Reson. 53, 521 (1983). 14.1. leener, B.H. Meier, P. Bachmann, and R.R. Ernst, J. Chem. Phys. 71, 4546 (1979). 15. S. Macura and R.R. Ernst, Mol. Phys. 41, 95 (1980). 16. S. Macura, K. Wuthrich, and R.R. Ernst, J. Magn. Reson. 46, 269 (1982). 17. C.A.G. Haasnoot, F.l.M. van de Ven, and C.W. Hilbers, J. Magn. Reson. 56, 343 (1984). 18. G. Bodenhausen, G. Wagner, M. Rance, O.W. S0rensen, K. Wiithrich, and R.R. Ernst, J. Magn. Reson. 59, 542 (1984). 19. G. Bodenhausen and R.R. Ernst, J. Magn. Reson. 45, 367 (1981). 20. N. Kurihara, O. Kamo, M. Umeda, K. Sato, K. Hyakuna, and K. Nagayama, J. Magn. Reson. 65, 405 (1985). 21. K. Nagayama, Analytical , 1,95 (1985).

Multiple-Qualltum Spectroscopy 1. T.H. Mareci and R. Freeman, J. Magn. Reson. 48, 158 (1982). 2. L. Braunschweiler, G. Bodenhausen, and R.R. Ernst, Mol. Phys. 48,535 (1983). 3. T.H. Mareci and R. Freeman, J. Magn. Reson. 51, 531 (1983). 4. Y.S. Yen and D.P. Weitekamp, J. Magn. Reson. 47, 476 (1982). 5. L. Muller, J. Magn. Reson. 59, 326 (1984). 6. G. Drobny, A Pines, S. Sinton, D.P. Weitekamp, and D. Wemmer, Symp. Faraday Soc. 13,49 (1979). 7. D.P. Weitekamp, Adv. Magn. Reson. 11, III (1983). 8. D.P. Weitekamp, 1.R. Garbow, 1.B. Murdoch, and A. Pines, J. Amer. Chem. Soc. 103,3578 (1981). 9. W.S. Warren, S. Sinton, D.P. Weitekamp, and A. Pines, Phys. Rev. Lett. 43, 1791 (1979). 384 Selected Bibliography

10. A.N. Garroway, 1. Baum, M.G. Munowitz, and A. Pines, J. Magn. Reson. 60, 337 (1984). II. S. Vega, T.N. Shattuck, and A. Pines, Phys. Rev. Lett. 37, 43 (1976). 12. S. Vega, T.N. Shattuck, and A. Pines, Phys. Rev. A22, 638 (1980). 13. P. Brunner, M. Reinhold, and R.R. Ernst, J. Chern. Phys. 73, 1086 (1980).

High Resolution NMR o/Solids 1. 1. Schaefer, E.O. Stejskal, and R. Buchdahl, Macrornolecules, 10, 384 (1977). 2.1. Hcrzfeld and A.E. Berger, J. Chern. Phys. 73, 6024 (1980). 3. D. Ellett, U. Haeberlen, and 1.S. Wuagh, J. Arner. Chern. Soc. 92, 411 (1970). 4. U. Haeberlen and 1.S. Waugh, Phys. Rev. 175,453 (1968). 5. W.K. Rhim, D.O. Elleman and R.W. Vaughan, J. Chern. Phys. 59, 3740 (1973), and 58, 1772 (1973). 6. W.K. Rhim, D.O. Elleman, L.B. Schreiber, and R.W. Vaughan, J. Chern. Phys. 60, 4595 (1974). 7. N. Chandrakumar, D. Ramaswamy, and S. Subramanian, J. Magn. Reson. 43,345 (1983). 8. A. Pines. M.G. Gibby, and 1.S. Wuagh, J. Chern. Phys. 56, 1776 (1972). 9. E.O. Stejskal, 1. Schaefer, and 1.S. Wuagh, J. Magn. Reson. 59, 569 (1973). 10. W.T. Dixon, J. Chern. Phys. 77, 1806 (1982). II. R.G. Griffin, G. Bodenhausen, R.H. Habcrkorn, T.H. Huang, M. Munowitz, R. Osredkar, OJ. Ruben, R.E. Stark, and H. van Willigan, Phil, Trans. Royal. Soc. A299, 547 (1981). 12. R.K. Hester, 1.L. Ackerman, B.L. Neff, and 1.S. Waugh, Phys. Rev. Lett. 36,1081 (1976). 13. M. Linder, A. Hohener, and R.R. Ernst, J. Chern. Phys. 73, 4959 (1980). 14. W.P. Aue. 0.1. Ruben, and R.G. Griffin, J. Magn. Reson. 43, 472 (1981). 15. P. Caravatti, G. Bodenhausen, and R.R. Ernst, Chern. Phys. Lett. 89, 363 (1982). 16. P. Caravatti, L. Braunschweiler, and R.R. Ernst, Chern. Pllys. Lett. 100, 305 (1983). 17. D.P. Weitekamp, A. Bielecki, D. Zax, K. Zilm, and A. Pines, Phys. Rev. Lett. 50, 1807 (1983). 18. D.B. Zax, A. Bielecki, K.W. Ziltn, and A. Pines, Chern. Phys. Lett. 106, 550 (1984). 19. T. Terao, T. Fujii, T. Onodera, and A. Saika, Chern. Phys. Lett. 107, 145 (1984).

Cornposite Pulses and Imaging I. M.H. Levitt, R. Freeman, and T. Frankiel in "Advances in Magnetic Resonance" (l.S. Waugh, Ed.), Vol. II, Academic Press, New York, 1983. 2. R. Tycko, H.M. Cho, E. Schneider, and A. Pines, J. Magn. Reson. 61, 60 (1985). 3. R.E. Gordon, P.E. Hanley, D. Shaw, D.G. Gadian, G.K. Radda, P. Styles, P.l. Bore, and L. Chan, , 287, 736 (1980). 4. J.J.H. Ackerman, T.H. Grove, G.G. Wong, D.G. Gadian, and G.K. Radda, Nature, 283, 167 (1980). 5. M.R. Bendall, R.E. Gordon, J. Magn. Reson, 53, 365 (1983). 6. M. Garwood, T. Schleich, G.B. Matson, and G. Acosta, J. Magn. Reson. 60, 268 ( 1984). Index

Absorption, 5 Coherences, representation of Accordion spectroscopy, 216f single quantum, 8 Adiabatic zero- and double quantum, 8 demagnetization in the rotating Coherence transfer, 67f frame ADRF, 297 pathways,2lS remagnetization in the rotating Commutators, 9 frame ARRF, 297 Commutator algebra, 358 Anticommutators, 9 Composite pulses, 332f Antiphase , 9, 30f with retrogade compensation, 338 Average Hamiltonian, 27Sf, 370f Composite pulse decoupling, 335f for MREV -S, 2S4 Correlation function, 48, 367 for WAHUHA-4, 2S1 CPMAS,307f Averaging CRAMPS,305f coordinate space, in, 266f CTEF, 245f, 247 spin-operator-space, in, 273f CYCLOPS, 112, 121, 331

Baker-Campbell-Hausdorff formula, DANTE,64 10, 356, 372 Decoupler calibration, 79f BLEW-12,294 Decoupling BLEW-4S, 294 artifacts in, 337 BR-24,292f heteronucIear, inverse-gated, 5If BR-52,293 heteronucIear, gated, 5lf homonucIear, 53 off-resonance, 51 Chemical shift scale factors, Density operator, 6 284f DEPT,86f Coherences DEPr,8S , Sf DEPr·, SSt' unobservable, Sf DEPT GL, 90 386 Index

Depth pulses, 343f J-cross polarization Dipolar relaxation, 48f, 200f, 365f AJCP, 105f Dipolar spectra in solids, 313f CJCP,97f Double quantum filtering, llOf JCP,95f DOUBTFUL, 115 PCJCP,96f Dynamic range, 61, 363 RJCP, 102f under mismatch, 99f J-scaling, 58f Evolution under J-spectroscopy, 41f anisotropic strong coupling, 20 dipolar coupling, 22 isotropic strong coupling, 21 Larmor off-set, chemical shift, Bo inhomo- frequency, 2 genity, 15f , 4 passive coupling, 19f Lineshapes in solids, 264f quadrupole coupling, 23 rf pulses, 14 week coupling, 18 Magic angle spinning MAS, Evolution of equivalent spins under 266f dipolar coupling, 23 Magnetization scalar coupling, 18 longitudinal, 7 transverse, 7 Magnus expansion, 279, 370f Free induction decay FID, 24f, 34f MODEPT,91 Fourier transform FT, 26, 359f Modulation, phase and amplitude, double FT, 126f 128f MQ cross-polarization, 255f MQ excitation, 228f Hamiltonian order selective, 249f average, 278f, 370f MQ filters, 116f external, 262f MQ relaxation, 259 internal, 262f MQ spectroscopy, 226f Hartmann-Hahn condition, 95, 298 CIDNP, in, 242f High-resolution NMR of solids, 260f imaging, 251f heteronuclear, 237f heteronuclear relay, 236f Imaging, 338f selective detection, 244f INADEQUATE, llOf solids, heteronuclear, 258f INEPT,70f spin-I, 253f INEPr,75 MREV-8,280f INEPT GL, 90f MUltiple-pulse line-narrowing, INEPT-INADEQUATE, 114f 273f INEPT with 76 sPin>~, anomalies, 31, 32 Internal Hamiltonians, effect of rota• patterns, 26, 31, 32 tion on, 262f Irreducible spherical tensor operators, 375f Nuclear Overhauser effect NOE, 50, Isotropic mixing, 106f 367f Index 387

Operator basis set Spin temperature, 295 . I 8 Superposition spm-Z' coherent, 6 spin-I, 8 incoherent, 6 two spins- Y2, 8 Surface coils, 339f Operators Susceptibility exponential, 10 Curie, 3 matrix representation of 8, 350f rf,5

PASS, 310 Tailored excitation, 65 Phase anomalies, 27 Time evolution, II Polarization transfer, 68f Topical imaging, 340f rotating frame, in, 95f TOSS, 313 variable flip angles, with, 85 TPPI, 238f, 245 Power spectrum, 48, 367 TSCTES, 247f , 38, 143, 238, 277f Two dimensional INADEQUATE, Pulse errors, 291f, 332 230f Two dimensional NMR, 124f cross-section projection theorem, Quadrature detection, 330f 145f display cosmetics, 132f nomenclature, 134f Resonance,S Two dimensional NMR. liquids, cor• Rf inhomogeneity effects, 332f related, 154f RINEPT,73f COSY, 155f Rotating frame,S flip angle effects, 170f doubly, 12 FOCSY, 184 tilted doubly, 57 heteronuclear, 155f zeugmatography, 347f homonuclear, 164f mUltiplet separated hetero, 163f MQ filtered, 173f Sampling theorem, 363 SECSY, 179f, 213 Selective population inversion SPI, 69 super COSY, 177f SEMINA, 114f z-filtered SECSY, 182f SEMUT,82f Two dimensional NMR, liquids, J re- SEMUT GL, 84 solved, 137f SHRIMP, 106 hetronuclear, 137f Sideband suppression, 310f homonuclear, 144f Signal-to-noise ratio, 67f MQ filtered, 150 in 2 D, 222f strong coupling effects, 151f SINEPT,78 Two dimensional NMR, liquids, Solvent suppression, 61f NOESY and chemical ex• , 35f change, 193f Carr Purcell sequence, 39 artifacts in, 206f CPMG sequence, 40 COCONOESY,209f SEFT,44f zz peaks in, 210f Spin filters, 116f Two dimensional relayed COSY, 184f Spin tickling, 55 homonuclear, 184f 388 index

Two diti1eilsional relayed COSY (cont.) UPT,92 hetemnuclear, 186f upr, 92 low-pass l-filtered hetero, 188f TOCSY, 190 Two dimensional NMR, solids, 256f WAHUHA-4,280f CPMAS,318f Wigner rotation matrices, 268, 378 heteronuclear correlation, 321 f hOll1ojwclear correlation, 324f magic angle flip SASS, 321 Zero-field NMR, 108[, 325f magic angle hopping, 320f Zero-quantum spectroscopy, 240f separation in solids, 314f homonuclear, 240 SLF,3ISf heteronuclear, 242