Radiofrequency irradiation schemes and methods of design and display for use in performing nuclear magnetic resonance spectroscopy

[0002] Benefit is claimed under 35 U.S.C. 119(e) for the following United States provisional applications:

[0003] The context of the present work can best be described as a conventional NMR system. Such a conventional system is described in U.S. Pat. No. 4,742,303 incorporated herein by reference. The invention concerns the use of coupling modulation, manifested by a system of (two or more) spin-half nuclear spins I and S (i.e., an IS spin-half system), during RF (radiofrequency) irradiation of one of the spins, in various types of NMR (nuclear magnetic resonance) experiments.

[0004] The terminology used in this disclosure is as commonly used in the NMR literature and examples may be found in the publications cited. The names of the IS spinstates are as used by Sorensen, Eich, Levitt, Bodenhausen and Ernst (Progress in NMR Spectroscopy, 16, 163 (1983)) for the product operator (PO) formalism. However, unlike Sorensen et al we group all the one-spin product operators, S_i(i=x, y, or z), together and describe them as in-phase states or magnetization, and we describe all the two-spin product operators, 2S_iI_j(i=x, y, or z; j=x, y, or z) as antiparallel spinstates or antiparallel magnetization. A second difference, which will be expanded further in this disclosure, is that Sorensen et al describe spinstates as observable only if they provide detectable magnetization during signal acquisition. In this disclosure we are concerned with the actual measurable state of an IS system at any time during an RF pulse or sequence of pulses. We consider a state to be measurable if maximum detectable signal can be obtained after a hard on-resonance pulse on the S spins and/or a hard on-resonance pulse on the I spins. Thus the orthogonal PO states are convenient descriptors of the total state because the total state can be resolved into a linear combination of the PO states, and each PO state is measurable after a hard on-resonance 90° pulse on the S and/or I spins.

[0005] For an IS J-coupled system at the beginning of a 1D (one-dimensional) decoupled acquisition, it has been considered impossible to produce detectable signal from pure antiparallel spinstates, 2S_yI_j(j=x, y, or z), because it has been assumed that the antiparallel components self cancel (e.g. see Levitt, Bodenhausen and Ernst, Journal of Magnetic Resonance 53, 443 (1983)). However, we find that a decoupler field of constant amplitude and frequency, applied to the I spins while observing the S spins, establishes an additional channel for communication between I and S which produces, via J-mediated effects, a coupled time evolution yielding directly detectable in-phase transverse magnetization, S_x. This mechanism provides an initial basis for the embodiments of the invention. The S-spin signal produced whilst continuously irradiating the I spins may appear as sideband or centerband intensity depending on the initial antiparallel spinstates. Alternatively, discrete I-spin irradiation (RF pulses) may enable 100% transformation between different antiparallel and in-phase spinstates. An exact QM (quantum mechanical) analysis of this behaviour, generalized for all possible intial conditions, provides a means to calculate and design new selective RF pulses, and sequences of RF pulses. Several fast and efficient methods for characterizing the Insensitive I-spin channel of a spectrometer by observing the sideband signals with the Sensitive S-spin channel are described. These various methods of design and display comprise embodiments of the invention.

[0006] The detailed QM study of CW (continuous wave) decoupling of I spins in an IS spin system, by Anderson and Freeman (Journal of Chemical Physics 37, 85 (1962)), and other pioneering work noted by these authors, provided an analytical solution for the amplitude and frequency of the decoupled S_xsignal starting with pure in-phase magnetization, S_x, of the observed nucleus. However, the S_xspinstate is just one of 15 possible IS product operator spinstates that can arise from irradiation of either the I or S spins. In general, any of the 15 PO spinstates can be produced beginning with any single PO spinstate so there are 15×15=225 analytical equations that completely describe an IS spin system subject to RF irradiation. The theoretical basis of the invention required the derivation of the other 224 equations, and all the embodiments of the invention for an IS spin system use the entire set or subsets of these 225 equations.

[0007] In a first general embodiment of the invention, a method is obtained to allow the rapid calculation of and determination of the result of a complex series of RF pulses and time delays (a pulse sequence) and thus can be used to design RF pulse sequences or the equipment that delivers RF pulse sequences or to determine the exact result of a pulse sequence. For example, the invention can be used to display the mechanism of RF pulse sequences for each time increment throughout the sequence for the purposes of education, software/hardware debugging, or the discovery of new pulse sequences.

[0008] In a second specific embodiment of the invention, continuous RF irradiation generates sideband signals whose frequency depends on the amplitude of the irradiation and thus the power and homogeneity of the radiofrequency source. This information may be used to assess or improve the design of the NMR probe that produces and detects the radiofrequency energy.

[0009] In a third general embodiment of the invention, RF pulses of low power interconvert orthogonal PO (product operator) spinstates of the IS system. This interconversion is frequency selective and thus may be used to discriminate between spins in differing chemical environments that consequently resonate at different frequencies.

[0010] Whilst all the examples and illustrations of the invention presented in this specification are in the context of J (or scalar) coupling between the two types of nuclear spin, the theoretical basis of the invention is also valid for dipolar coupling, thus permitting the invention to be applied to two or more dipolar-coupled spins in solids or partially-aligned chemical systems.

[0011] For simplicity all the examples in this specification are in terms of two single spins I and S, but the invention also applies to multiple I and S spins as in an I_mS_nspin system. Provided the RF irradiation is applied to only one type of spin at any one time, the invention is equally applicable to homonuclei or heteronuclei. In this context, the distinguishing feature of different types of homonuclei is that they must resonate at different NMR frequencies. Since the invention is applicable to RF irradiation of only one type of spin at any one time, the behaviour of more than two types of coupled spins can be determined by sequential application of the invention to each pair of spin types, I and S.

[0012] Although common aspects of NMR spectroscopy such as pulsed field gradients and nuclear spin relaxation do not form a part of the invention, such aspects may, nevertheless, be included within calculations of RF pulse sequences as expressed in the first general embodiment of the invention. The new art comprises a means of rapidly calculating the effect of RF pulses inclusive of coupling modulation, and thus a new fundamental means of analysing NMR experiments. Other common attributes such as gradients and relaxation can be added to this fundamental approach using conventional protocols.

[0013] TABLE 1 lists expansion coefficients of the Pauli basis set

[0014] TABLE 2 translates these coefficients in terms of Cayley-Klein parameters.

[0015] TABLE 3 tabulates the product operator coefficients for IS_nsystems.

[0016] TABLE 4 lists the arguments of TABLE 3 in terms of the functions in TABLE 2.

[0017] TABLE 5 provides standard matrix results for an arbitrary classical rotation.

[0018] TABLE 6 lists combinations of classical rotations applicable to irradiation of the I spins.

[0019] FIG. 1 shows the classical rotation of a magnetization vector of an S spin around an effective RF field B_e.

[0020] FIG. 2 pictures some classical rotations of an S spin coupled to an I spin and illustrates the various possible types of PO spinstates for an IS spin system.

[0021] FIG. 3 establishes the experimentally-measurable I-spin magnetizations resulting from initial in-phase S-spin magnetization.

[0022] FIG. 4 describes some nonclassical rotations for on-resonance RF irradiation of an I spin subsequent to excitation of the S spin beginning with the S_xproduct operator spinstate.

[0023] FIG. 5 graphs experimental results for the nonclassical rotation pictured in FIG. 4 for an RF field strength of B₁=J/2.

[0024] FIG. 6 shows in diagrammatic matrix form the 225 terms for an IS spin system subject to irradiation of the I spins.

[0025] FIG. 7 demonstrates the RF frequency calibration of the I-spin decoupler channel.

[0026] FIG. 8 illustrates an amplitude calibration of the I-spin decoupler channel.

[0027] FIG. 9 provides ID spectra showing the extent of RF inhomogeneity for a typical I-spin decoupler channel.

[0028] FIG. 10 displays profiles of RF amplitude, obtained from 2D images, along the sample axis, for both I-spin and S-spin channels.

[0029] FIG. 11 presents the results of calculations of constant amplitude 180^Jand 90^Jfrequency selective pulses.

[0030] FIG. 12 provides examples of shaped J pulses of increasing bandwidth.

[0031] FIG. 13 exemplifies a shaped J pulse with increased bandwidth and squarer frequency offset profile.

[0032] These various embodiments of the invention derive from our solutions for the time-dependent density operator, ρ(t), resulting from CW decoupling (RF of constant amplitude, phase, and frequency) applied to the various initial configurations of an IS spin system. We have calculated the evolution of the total density operator for the spin system, which includes the observed spectrum emphasized by earlier treatments (Anderson and Freeman, Journal of Chemical Physics 37, 85 (1962); Waugh, Journal of Magnetic Resonance 49, 517 (1982); comprehensive review by Shaka and Keeler, Progress in NMR Spectroscopy, 19, 47 (1986)) as a small fraction of the complete treatment summarized here. The desired PO form of the solution, i.e. the separation of ρ(t) into the magnitude of the pure PO spinstates, arises as a result of factoring the time-dependent density operator into a linear combination of I-spin angular momentum operators times a sum of S-spin operators. The coefficients of individual spinstate terms, listed in the following as analytical expressions, are derived from 2×2 matrices and do not require diagonalization of large matrices, ensuring the calculations remain efficient and no approximations are required. In consequence, the entire set of 225 solutions for an IS spin system, for example, can be written down in explicit form and yield exact, completely definable, analytical equations. These 225 equations provide a basis for the invention.

[0033] The various forms of the invention are not, however, restricted to a coupled IS spin system. The following theory is generalized to a coupled I_mS_nspin system and thus various embodiments of the invention may be generalized to a coupled I_mS_nspin system.

[0034] Complete information on the state of an NMR system at a given time is contained in the density operator for the system. The time evolution of the density operator is determined by the Hamiltonian for the system. Both of these terms will be defined more fully in the following. The density operator can be usefully represented as a linear combination of any set of basis operators, as described by Søorensen, Eich, Levitt, Bodenhausen, and Ernst (Progress in NMR Spectroscopy 16, 163-192 (1983)) for the particular set of orthogonal Cartesian product operators. Other basis sets are useful (e.g., Sanctuary, Journal of Chemical Physics, 64, 4352-4361 (1976); Nakashima and McClung, Journal of Magnetic Resonance, 70, 187-203 (1986); Bain, Progress in NMR Spectroscopy 20, 295-315 (1988)), and the transformations between basis sets is straightforward. The method described in the following calculates the density operator and provides exact analytical solutions in terms of the Cartesian product operators. However, the method is general and applies to any basis set for the density operator. The analytical solutions obtained can be readily transformed to any desired basis set.

[0035] In the following, the density operator solution is provided in Eq. [38] for a general I_mS_nsystem for initial states composed of transverse S-spin operators (i.e., x or y components). The solution is compact, being composed of terms derived for simpler spin systems. In order to make clear the final, most general result, the formalism is first described in some detail for an IS_nsystem, which provides the basis for the extensions which follow.

[0036] The Hamiltonian. In units of angular frequency, the effective RF field in the rotating frame of the I spins is

[0037] ω_e(t)=ω_rf(t)[cos φ(t){circumflex over (x)}+sin φ(t)ŷ]+δ(t){circumflex over (z)}  [1]

[0038] which encompasses any desired modulation of the amplitude (ω_rf), phase (φ), and frequency offset (δ), where offset is measured from the center of the I-spin sweep width. For the following QM threory, we use the convention of right-hand rotations about any general ω=−γB, so that, for example, an on-resonance RF pulse applied along the +x axis in the rotating frame produces, for positive γ, a right-hand rotation about −x. The RF phase φ is always the phase of ω_rfin this convention, whereas the phase of the RF field depends on the sign of γ. The time dependent

[0039] Hamiltonian for an IS_nsystem can be written in the form $\begin{array}{cc}\mathscr{H}\left(t\right)={\omega }_e\left(t\right)·I+I_z\sum _{i=1}^n{S}_{\mathrm{iz}},& \left[2\right]\end{array}$

[0040] where J is 2π times the coupling, J, in Hz and S_izis the z component of the i^thS spin. Each operator is implicitly a direct product of the form V¹{circle around (x)}V^S1{circle around (x)} . . . {circle around (x)}V^Snon for the (n+1) two-dimensional operators from each vector space of the individual I- and S-spin operators. Since operators from the different spaces commute, we will not necessarily maintain this particular ordering of the operators in what follows. The projection operators P_i^± onto the spin-up (+) or spin-down (−) states in the S_isubspace are written in terms of S_izand the identity operator, E_i, as

[0041] P_i^±=1/2E_i±S_iz,  [3]

[0042] so that

[0043] E_i=P_i⁺+P_i⁻  [4]

[0044] and

[0045] S_iz=1/2(P_i⁺−P_i⁻).  [5]

[0046] In the vector space of an IS₃system, for instance, we represent I_zS_1zas 1/2I_z(P₁⁺−P₁⁻)E₂E₃.

[0047] The Hamiltonian, recast in terms of these operators, is $\begin{array}{cc}\mathscr{H}\left(t\right)={\omega }_e\left(t\right)·I\prod _{i=1}^n\left(P_i^{+}+P_i^{-}\right)+/2I_z\sum _{i=1}^n\left(P_i^{+}-P_i^{-}\right)\prod _{j\ne i}^n\left(P_j^{+}+P_j^{-}\right).& \left[6\right]\end{array}$

[0048] Performing the multiplications and grouping like terms yields a more concise expression. Define

[0049] ω_q(t)=ω_e(t)+qJ/2{circumflex over (z)}  [7]

[0050] as the equivalent effective field seen by an uncoupled I-spin at a resonance offset δ+qJ/2, oriented at an angle θ_qfrom the z axis given by

[0051] tan θ_q=ω_rf/[δ+qJ/2].  [8]

[0052] Then, for example, with n=2, $\begin{array}{cc}\begin{array}{ccc}{\mathscr{H}}_{\left(n=2\right)}& =& \left({\omega }_c·I+2/2I_z\right){P_1}^{+}{P_2}^{+}+\left({\omega }_c·I+0/2I_z\right)({P_1}^{+}{P}_2^{-}+\\ & & {P_1}^{-}{P_2}^{+})+{\omega }_c·I-2/2I_z){P_1}^{-}{P_2}^{-}\\ & =& {\omega }_{\left(+2\right)}·I{P_1}^{+}{P_2}^{+}+{\omega }_{\left(0\right)}·I({P_1}^{+}{P}_2^{-}+{P_1}^{-}{P_2}^{+})+{\omega }_{\left(-2\right)}·\\ & & I{P_1}^{-}{P_2}^{-}\end{array}& \left[9\right]\end{array}$

[0053] More generally, for n spin-1/2 S operators, the maximum total S spin is n/2, with z components n/2, (n/2−1), . . . , −n/2. For each product p₁^q. . . P_n^qthat appears in H, as illustrated above, the sum of the angular momenta q_i=±1 (in units of half-integral spin) associated with the individual P_i^± is $\begin{array}{cc}{q}_s=\sum _{i=1}^n{q}_i,& \left[10\right]\end{array}$

[0054] which equals the coefficient of J/2. We therefore write $\begin{array}{cc}\mathscr{H}\left(t\right)=\sum _{q_1,\dots ,q_n-\pm 1}{\omega }_{q_s}\left(t\right)·I\prod _{j=1}^n{P}_j^{q_j},& \left[11\right]\end{array}$

[0055] which merely says that for an IS_nsystem, the values of q_srange from +n to −n, with Δq_s=2. Each ω_q·I is multiplied by all possible ways of generating a product of n different projection operators with total spin equal to q_s, for a total of 2ⁿterms, as illustrated previously for the n=2 case in Eq. [9]. For n=3, there are terms ω_±3·I P₁^±P₂^±P₃^± and three terms for ω_±1involving two P^± and one P^≠. The Hamiltonian for an IS system written in this form consists of the two comparatively simple terms ω_±1·I P₁^±. The basic constructs used in Eq. [11] and, hence the simple algorithm described above, appear frequently as elements in the calculations which follow. Differences in the scalar couplings are readily incorporated by changing the component q_sJ/2 in each ω_qdefined by Eq. [7] to $\sum _{j=1}^{\stackrel{.}{n}}{q}_j{1}_{}/2.$

[0056] Thus, ω₊₁in a term such as ω₊₁·I P₁⁺P₂⁻P₃⁺ would become ω_e+(J₁−J₂+J₃)/2 {circumflex over (z)}, for example.

[0057] The Time Evolution Operator. The RF generated by the spectrometer is normally a digitized approximation of a continuous waveform in which the RF amplitude, phase, and frequency are constant during time increments that sequentially comprise the overall pulse sequence. During individual RF pulses, these time increments are most commonly intervals of fixed length Δt. Thus, the Hamiltonian is time-independent at each time t_k=kΔt, and the exact expression for time-development operator or propagator U(t_k,t_k−1) between times t_k−1and t_kis simply exp[−i H(t_k) Δt]. In Eq. [11], the direct product of the n projection operators in each term sets one of the diagonal elements of the resulting 2ⁿ×2ⁿprojection operator equal to one and all other elements equal to zero. We define ω_q^k=ω_q(t_k) and unit vector {circumflex over (ω)}_q^k=ω_q^k/|ω_q^k| with components determined by Eq. [7], together with a rotation angle

[0058] β_q^k=1/2ω_q^kΔt.  [12]

[0059] The propagator in matrix form is composed of the 2×2 elements $\begin{array}{cc}\begin{array}{c}{U}_q\left(t_k,t_{k-1}\right)=\exp \left[-{\mathrm{i\omega }}_q^k·I\Delta t\right]\\ =\cos {\beta }_q^k-2i\left({\hat{\omega }}_q^k·I\right)\sin {\beta }_q^k\\ =\left(\begin{array}{cc}{a}_q^k& b_q^k\\ -b_q^{k*}& a_q^{k*}\end{array}\right)\end{array}& \left[13\right]\end{array}$

[0060] along the diagonal, where

[0061] a_q^k=cos β_q^k−i{circumflex over (ω)}_q,z^ksin β_q^k

[0062] b_q^k=−({circumflex over (ω)}_q,y^k+i{circumflex over (ω)}_q,x^k)sin β_q^k  [14]

[0063] are the standard Cayley-Klein parameters for a rotation by 2β_q^kabout {circumflex over (ω)}_q^k, and (*) represents complex conjugation. For the n=2 example, the diagonal elements are U₂, U₀, U₀, and U₋₂, with multiplicities determined by the number of ways to combine n projection operators with total spin quantum number q. Starting at time t₀, the propagator to any later time t_kis obtained by successive application of the propagators for each of the k intervals, and the block-diagonal structure of U ensures that each block will also be a concatenation of the U_q, so that $\begin{array}{cc}U\left(t_k,t_0\right)=\sum _{q_1,\dots ,q_n=\pm 1}{U}_{q_s}\left(t_k,t_0\right)\prod _{j=1}^n{P}_j^{q_j},& \left[15\right]\end{array}$

[0064] similar to Eq.[11]. For a truly continuous RF waveform, this expression for the propagator would be an approximation that can be made as accurate as desired by decreasing the interval Δt. We will use the abbreviated notation a_qand b_qto describe either the incremental form U_q(t_k, t_k−1) or the total effective propagator U_q(t_k, t₀) in the subspace labeled by q when the distinction is clear from the context in which they are used.

[0065] The Density Operator. The state of the system at time t is embodied in the density operator, which evolves from an initial state ρ(t₀) according to the relation

[0066] ρ(t)=U(t, t₀)ρ(t₀)U^†(t, t₀).  [16]

[0067] We consider the set of four initial conditions

[0068] ρ_ν(t₀)ε{S_x, 2S_yI_x, 2S_yI_z} (ν=0, 1, 2, 3)  [17]

[0069] composed of transverse S-spin operators and assign the label ν sequentially to members of the set. We write the x_jcomponent of the I spin as I_xfor (x₁, x₂, x₃)=(x, y, z) and define I_x=1/2E, giving ρ₀(t₀)=2S_xI_xto complete the symmetry of the set. As will be shown in the following, the set is closed for an IS system in the sense that there is no evolution to any state that is not a member of the set. Each S-spin operator is a sum Σ_j=1ⁿS_j, which is implicit in the notation. We also employ the usual raising and lowering operators

[0070] S_j^±=S_jx±iS_jy,  [18]

[0071] which can be obtained by interchanging the two columns of P_j^± in the matri_xrepresentation.

[0072] In the calculation of ρ_ν(t), the product Uρ_ν(t₀) produces terms P_j^±S_jx=(1/2)S_j^± and P_j^±S_jy=∓(i/2)S_j^±. The operators (P_j^±)^† in the adjoint operator U^† are equal to P_j^±, giving terms P_j^±P_j^∓=0, S_j^±P_j^±=0, and S_j^±P_j^∓=S_j^±. Thus, the only nonzero terms in Eq. [16] are those containing factors of the form P_j^±S_jxP_j^∓=(1/2)S_j^± and P_j^±S_jyP_j^∓=∓(i/2)S_j^±. This reduces the density operator to an expression similar to Eq. [9], with the product of projection operators in each term replaced by the sum of unique products in which a single P_j^± is converted to S_j^±. As an example using the initial condition ν=2 for the IS₂case,

[0073] U=P₁⁺P₂⁺U₍₊₂₎+(P₁⁺P₂⁻+P₁⁻P₂⁺)U₍₀₎+P₁⁻P₂⁻U₍₋₂₎

[0074] ρ₂(t₀)=2I_y(S_1y+S_2y)

[0075] U^†=P₁⁺P₂⁺U₍₊₂₎^†+(P₁⁺P₂⁻+P₁⁻P₂⁺)U₍₀₎^†+P₁⁻P₂⁻U₍₋₂₎^†.  [19]

[0076] Picking out the nonzero terms from the product Uρ₂(t₀)U^† and using the relation S⁻=(S⁺)^† results in terms of the form

[0077] Z=(S₁⁺P₂⁺+P₁⁺S₂⁺)U₍₊₂₎2I_yU₍₀₎^†+(S₁⁺P₂⁻+P₁⁻S₂⁺)U₍₀₎2I_yU₍₋₂₎^†,  [20]

[0078] to give

[0079] $\begin{array}{cc}\begin{array}{c}{\rho }_2\left(t\right)=-i/2\left[Z-Z^{†}\right]\\ =\mathrm{Im}\left[Z\right].\end{array}& \left[21\right]\end{array}$

[0080] For the other initial conditions, the I-spin operator becomes I_x in the expression for Z. In addition, for initial S_x(ν=0), the terms P_j^±S_jxP_j^∓=(1/2)S_j^± that arise in the calculation of ρ₀(t) give only positive coefficients to produce terms of the form Z+Z^†, resulting in the real part of Z instead of the imaginary part.

[0081] Therefore, for general n, the simple algorithm that generates the equation for ρ_ν(t) is (i) construct the terms U_(q)2I_xU^†_(q−2) for q=n, n−2, . . . , −(n−2) and then (ii) multiply each term by all combinations of a single S_i⁺ (i=1, . . . ,n) and n−1 operators P_j^± (j≠i) with total angular momentum equal to q. This procedure is described more formally by the expression $\begin{array}{cc}{Z}_v\left(t\right)=\sum _{i=1}^n\sum _{q_1,\dots \underset{q_i\ne -1}{,q_n}=\pm 1}\left[U_{\left(q_s\right)}2I_{x_v}{U}_{\left(q_s-2\right)}^{†}\right]\left[S_i^{q_i}\prod _{j\ne i}^n{P}_j^{q_j}\right],& \left[22\right]\end{array}$

[0082] giving $\begin{array}{cc}{\rho }_v\left(t\right)=\{\begin{array}{cc}\mathrm{Re}\left[Z_v\right]& v=0\left(\mathrm{Initial}\mathrm{in}\text{-}\mathrm{phase}{S}_x\right)\\ \mathrm{Im}\left[Z_v\right]& v\ne 0\left(\mathrm{Initial}2S_y{I}_{x_v}\mathrm{coherence}\right)\end{array}& \left[23\right]\end{array}$

[0083] The bracketed expressions in Eq. [22] provide, in turn, the I-spin and S-spin constituents of ρ_ν(t).

[0084] The solution for the density operator can thus be reduced to calculating the general products U_q2I_xU_r^†. Using Eq.[13] for the effective propagator in each subspace, these can be expanded as linear combinations of the Pauli basis set, 2I, as $\begin{array}{cc}{U}_{q}2I_{x_v}{U}_r^{†}=\sum _{\mu =0}^3{\alpha }_{v\mu }^{\left(q,r\right)}2I_{x_{\mu }}& \left[24\right]\end{array}$

[0085] to obtain the simple expressions for the expansion coefficients [α]_μ listed in TABLE 1 as a function of the indices ν, q, and r. TABLE 2 translates these coefficients to their expressions in terms of the C-K parameters.

[0086] The solution for the density operator is completed using Eq. [18] for S_i⁺ and Eq. [3] for P_j^± to perform the multiplications in Eq.[22], keeping only the real terms for initial in-phase magnetization or the imaginary terms for coherences 2S_yI_xat the start of the RF irradiation. As an illustration, we consider the solution ρ_ν(t) that evolves when RF is applied to the I spins in an IS₃system that is initially in the ν=0 state, ν₀(t₀)=S_x. According to Eq. [23], ρ₀(t)=Re[Z₀], and Z₀is easily constructed by analogy to the IS₂example given in Eq. [21] for ν=2 (see, for example, the text preceding Eq. [22], which formally defines the algorithm). The elements α_0μ for general (q, r) listed in row ν=0 of TABLE 1 determine the components I_x in Z₀. Since α₀₀is real, only terms involving S_ixand S_jz, from the products S_i⁺P_j^± are associated with 2I_x=E. Since the α_0j(j=1, 2, 3) are imaginary, Re [Z₀] associates similar terms involving S_iyand S_jzwith 2I_x. In the interest of brevity, we write out fully only the terms involving S_ixto obtain $\begin{array}{cc}\begin{array}{c}{\rho }_0\left(t\right)\propto \mathrm{Re}\left[A_0^{\left(3,1\right)}+2A_0^{\left(1,-1\right)}+A_0^{\left(-1,-3\right)}\right]\sum _{i=1}^n{\left(S_i\right)}_x+\\ \mathrm{Re}\left[A_0^{\left(3,1\right)}-A_0^{\left(-1,-3\right)}\right]2[S_{1x}\left(S_{2z}+S_{3z}\right)+\\ S_{2x}\left(S_{1z}+S_{3z}\right)+S_{3x}\left(S_{1z}+S_{2z}\right)]+\\ \mathrm{Re}\left[A_0^{\left(3,1\right)}-2A_0^{\left(1,-1\right)}+A_0^{\left(-1,-3\right)}\right]4[S_{1x}{S}_{2z}{S}_{3z}+S_{2x}{S}_{1z}{S}_{3z}+\\ S_{3x}{S}_{1z}{S}_{2z}]+\left\{2S_y{I}_{x_j}+\dots \mathrm{terms}\right\}\end{array}& \left[25\right]\end{array}$

[0087] Coefficients of the 2S_yI_xterms have the same functional form shown above for the S_xterms, but are constructed from other elements.

[0088] Thus, there are n levels of product operator terms for an IS_nsystem, depending on the number of S-spin operators in the product. The functions F_iⁿ(Ω) listed in row i, column n of TABLE 3 multiply terms consisting of products of i S-spin operators, with i≦n. The F_iⁿwith i>n are zero, so the solutions for ρ_ν(t) become progressively simpler for smaller n. For an IS₂system, terms involving S_3xand S_3zare eliminated, and the coefficients change to the form given in column 2 of TABLE 3. For an IS system, there are no terms involving S_2xand S_2z, and the coefficients change to the form given in column 1 of TABLE 3. The particular arguments Ω_νμ of the functions depend on the initial condition, denoted by ν, and on the component 2I_x in the product. They are listed in TABLE 4 for general (q, r) in terms of A_ν and B_ν tabulated earlier in TABLE 2. For example, the terms involving operator 2I_xin Eq. [25] (n=3) require Ω₀₁of TABLE 4 and the three functions 𝑓_i³of Ω₀₁=Im[B₀] in column 1 of TABLE 3. We merely take the terms involving 2I_xwhich are written out fully as functions of Ω₀₀^(q,r)=Re [A₀^(q,r)] and change I_xto I_x, S_xto S_y, and Re[A₀] to −Im[B₀]. The remaining terms are generated similarly using the other elements in row zero of TABLE 4. Solutions for the evolution of initial states labeled by index ν in Eq. [17] are constructed similarly using row v of TABLE 4.

[0089] The final result for general n and initial condition ν can be written succinctly as $\begin{array}{cc}{\rho }_v\left(t\right)=\frac{1}{2^{n-1}}\sum _{i=1}^n\left(S_{i,x}\left\{f_1^n\left({\Omega }_{\mathrm{v0}}\right)+2\sum _{j\ne i}^n{S}_{j,z}\left[f_2^n\left({\Omega }_{\mathrm{v0}}\right)+2\sum _{\begin{array}{c}k>j\\ k\ne i\end{array}}^n{S}_{k,z}{f}_3^n\left({\Omega }_{\mathrm{v0}}\right)\right]\right\}+2\sum _{r=1}^3{I}_{x_r}{S}_{i,y}\left\{f_1^n\left({\Omega }_{\mathrm{vr}}\right)+2\sum _{j\ne i}^n{S}_{j,z}\left[f_2^n\left({\Omega }_{\mathrm{vr}}\right)+2\underset{\begin{array}{c}k>j\\ k\ne i\end{array}}{\sum ^n}{S}_{k,z}{f}_3^n\left({\Omega }_{\mathrm{vr}}\right)\right]\right\}\right).& \left[26\right]\end{array}$

[0090] As an example, Eq. [25] was provided to show the explicit correspondence between terms involving S_i,xin Eq. [26] for an IS₃system starting with initial S_x(ν=0).

[0091] The observed signal, <S_x(t)>∝Tr [S_xρ_ν(t)], is proportional to F_iⁿ(QO), since only the S_x²terms have nonzero trace. The other terms give the relative proportions of the different coherences that have evolved. Alternatively, the expansion in Eq.[24] can be performed just as readily in terms of either I^± or the related spherical tensor basis if these forms of the solution are required, or the desired transformations between bases can be performed on the final result.

[0092] The previous results can be extended to obtain the solutions for an I_mS_nsystem. We write the x_jcomponent of the i^thI spin as I_i,x. For the i^thI spin, Eq. [11] becomes $\begin{array}{cc}\mathscr{H}\left(t\right)=-\sum _{q_1,\dots ,q_n=\pm 1}{\omega }_{q_s}·I_i\prod _{j=1}^n{P}_j^{q_j},& \left[27\right]\end{array}$

[0093] so that H(t)=Σ_i=1^mH_i. Although the Hamiltonian no longer exhibits the simple 2×2 block-diagonal structure of the IS_nsystems, the individual H_icommute among themselves. In addition, all the terms in H_icommute, since each term contains a unique product of projection operators, which guarantees the commutator has at least one product of the form P_j^±P_j^∓=0 for each term. The propagator is still written as in Eq. [15], but now $\begin{array}{cc}{U}_{\left(q_s\right)}\left(t_k,t_0\right)=\prod _{i=1}^m{U}_{i\left(q_s\right)}\left(t_k,t_0\right).& \left[28\right]\end{array}$

[0094] The evolution of the density operator, described in Eqs. [16] through [23], follows immediately, with a minor modification to Eq. [22] for Z_ν. Define the function Υ(m,q) as the product $\begin{array}{cc}{\rm Y}\left(m,q\right)=\prod _{k=1}^m{U}_{k\left(q\right)}{U}_{k\left(q-2\right)}^{†}\mathrm{and}\mathrm{define}& \left[29\right]\\ {\rm Y}\left(m,x_j,q\right)=\sum _{k=1}^m\left(U_{k\left(q\right)}2I_{k,x_j}{U}_{k\left(q-2\right)}^{†}\prod _{l\ne k}^m{U}_{l\left(q\right)}{U}_{l\left(q-2\right)}^{†}\right).\mathrm{Then}& \left[30\right]\\ Z_v\left(t\right)->\{\begin{array}{cc}\sum _{i=1}^n\sum _{\begin{array}{c}{q}_1,\dots ,q_n=\pm 1\\ q_i\ne -1\end{array}}{\rm Y}\left(m,q_s\right)S_i^{q_i}\prod _{j\ne i}^n{P}_j^{q_j}& v=0\\ \sum _{i=1}^n\sum _{\begin{array}{c}{q}_1,\dots ,q_n=\pm 1\\ q_i\ne -1\end{array}}{\rm Y}\left(m,x_j,q_s\right)S_i^{q_i}\prod _{j\ne i}^n{P}_j^{q_j}& v\ne 0\end{array}& \left[31\right]\end{array}$

[0095] and the final solution for the density operator is still given by Eq.[23].

[0096] Single S-Spin. For in-phase S_xat the start of the I-spin irradiation, ρ₀(t) is obtained from Z₀in Eq. [31] as

[0097] ρ₀(t)=Re[S⁺Υ(m, +1)]=Re[(S_xiS_y)Υ(m, +1)].  [32]

[0098] According to Eq. [24] defining the general matrix α_νμ^(q,r)listed in TABLE 1, Υ(m, q) is all possible products of m elements from row zero, so that, most generally, $\begin{array}{cc}{\rm Y}\left(m,q\right)=2^m\sum _{{\mu }_1,\dots ,{\mu }_m=0}^3\left[{\alpha }_{0_{{\mu }_1}}^{\left(q,q-2\right)}\dots {\alpha }_{0_{{\mu }_m}}^{\left(q,q-2\right)}\right]\left[I_{{\mu }_1}\dots I_{{\mu }_m}\right],& \left[33\right]\end{array}$

[0099] where α₀₀is pure real and α_0j(j=1, 2, 3) is pure imaginary.

[0100] The terms multiplying S_xin Eq. [32] for ρ₀(t) must be the real part of Σ(m,+1), so the coefficient of S_x=2S_xI_xis immediately seen to be (α₀₀^1,−1)^m=[Re(A₀^1,−1)]^m. Operators of the form 4S_xI_xI_xresult from multiplying two elements from columns one through three for an I₂S system and using 2I_xfor the required third element in an I₃S system. Terms multiplying iS_ymust be pure imaginary to produce a real result, leading to operators of the form 2S_yI_xand, for I₃S, additional operators 8S_xI_xI_xI_x. The m=1 result for an IS system is given in Eq. [26] as functions of the pure real elements Ω_νμ in TABLE 4, where α_0j=−iΩ_0jand α_j0=iΩ_j0(j=1, 2, 3). More generally, we can select the terms from Eq. [33] that satisfy the above requirements to define the resulting coefficients of the S_xand S_yoperators in terms of general indices (q,r) as $\begin{array}{cc}\begin{array}{c}{c}_{0,x}^{\left(q,r\right)}={\left[{\Omega }_{00}^{\left(q,r\right)}\right]}^m-\frac{m-1}{2^{m-2}}{4\left[{\Omega }_{00}^{\left(q,r\right)}\right]}^{m-2}\sum _{k,l=1}^3{\Omega }_{0k}^{\left(q,r\right)}{\Omega }_{0l}^{\left(q,r\right)}\left(\sum _{p=1}^{m-1}\sum _{s>p}^m{I}_{p,x_k}{I}_{s,x_l}\right)\\ c_{0,y}^{\left(q,r\right)}={2\left[{\Omega }_{00}^{\left(q,r\right)}\right]}^{m-1}\sum _{k=1}^3{\Omega }_{0k}^{\left(q,r\right)}{I}_{x_k}-\frac{1}{2}\left(m-1\right)\left(m-2\right)\\ 8\sum _{k,t,p=1}^3{\Omega }_{0k}^{\left(q,r\right)}{\Omega }_{0l}^{\left(q,r\right)}{\Omega }_{0p}^{\left(q,r\right)}{I}_{1,x_k}{I}_{2,x_l}{I}_{3,x_p}\mathrm{for}m=1,2,3.\end{array}& \left[34\right]\end{array}$

[0101] The density operator resulting from the initial state 2S_yI_x, with $I_{x_j}=\sum _{i=1}^m{I}_{i,x_j},$

[0102] is obtained from Z_jin Eq. [31] as

[0103] ρ_j(t)=Im[S⁺Σ(m,x_j, +1)].  [35]

[0104] From Eq. [30] defining Σ(m,x_j,q), there is a factor U₍₊₁₎2I_i,xU₍₋₁₎^† given by row j of TABLE 1 for a single I spin, multiplied by U₍₊₁₎U₍₋₁₎^† for the remaining m−1 spins. This process is repeated for each I spin contained in the sum over the individual spins, and the final result must be imaginary. Compared to row zero of TABLE 1, the elements which are either pure real or pure imaginary in rows 1-3 are interchanged, so the solutions for ρ_j(t) can be taken from the ρ₀(t) results by adding all combinations in Eq. [34] that replace a single element from row zero with one from row j. The coefficients of the S_xand S_yoperators can then be written in general form for this case as $\begin{array}{cc}{c}_{j,x}^{\left(q,r\right)}=m{{\Omega }_{\mathrm{j0}}^{\left(q,r\right)}\left({\Omega }_{00}^{\left(q,r\right)}\right)}^{m-1}-\frac{m-1}{2^{m-2}}\times 4\sum _{k,l=1}^3\left[\left(m-2\right){\Omega }_{\mathrm{j0}}^{\left(q,r\right)}{\Omega }_{0k}^{\left(q,r\right)}{\Omega }_{0l}^{\left(q,r\right)}+{\left({\Omega }_{00}^{\left(q,r\right)}\right)}^{m-2}\left({\Omega }_{\mathrm{jk}}^{\left(q,r\right)}{\Omega }_{0l}^{\left(q,r\right)}+{\Omega }_{0k}^{\left(q,r\right)}{\Omega }_{\mathrm{jl}}^{\left(q,r\right)}\right)\right]\times \left[\sum _{p=1}^{m-1}\sum _{s>p}^m{I}_{p,x_k}{I}_{s,x_l}\right]c_{j,y}^{\left(q,r\right)}=2\sum _{k=1}^3\left[{\left({\Omega }_{00}^{\left(q,r\right)}\right)}^{m-1}{\Omega }_{\mathrm{jk}}^{\left(q,r\right)}+\left(m-1\right){\left({\Omega }_{00}^{\left(q,r\right)}\right)}^{m-2}{\Omega }_{\mathrm{j0}}^{\left(q,r\right)}{\Omega }_{0k}^{\left(q,r\right)}\right]I_{x_k}-\frac{1}{2}\left(m-1\right)\left(m-2\right)\times 8\sum _{k,l,p=1}^3\left({\Omega }_{\mathrm{jk}}^{\left(q,r\right)}{\Omega }_{0l}^{\left(q,r\right)}{\Omega }_{0p}^{\left(q,r\right)}+{\Omega }_{0k}^{\left(q,r\right)}{\Omega }_{\mathrm{jl}}^{\left(q,r\right)}{\Omega }_{0p}^{\left(q,r\right)}+{\Omega }_{0k}^{\left(q,r\right)}{\Omega }_{0l}^{\left(q,r\right)}{\Omega }_{\mathrm{jp}}^{\left(q,r\right)}\right)I_{1,x_k}{I}_{2,x_l}{I}_{3,x_p}& \left[36\right]\end{array}$

[0105] so that, combining these results,

[0106] ρ_ν(t)=c_ν,x^{(1, −1)}S_x+c_ν,y^{(1, −1)}S_y  [37]

[0107] for an I_mS system starting with the initial conditions denoted by ν=0, . . . ,3 of Eq. [17]. The expressions given in Eqs. [34] and [36] for the coefficients show explicitly which I-spin operators contribute as the number, m, of I spins increases.

[0108] Multiple S-Spins. For n>1, terms of the form S_iII_jP_jin Eq. [31] result in the same products of S-spin operators as for the IS_ncase. Noting that the terms involving components of the total I-spin operator for m I spins are included in the expressions for c_ν,xand c_ν,yderived above, we can immediately write from Eq. [26] $\begin{array}{cc}{\rho }_v\left(t\right)=\frac{1}{2^{n-1}}\sum _{i=1}^n\left(S_{i,x}\left\{f_1^n\left(c_{v,x}\right)+2\sum _{j\ne i}^n{S}_{j,z}\left[f_2^n\left(c_{v,x}\right)+2\sum _{\begin{array}{c}k>j\\ k\ne i\end{array}}^n{S}_{k,z}{f}_3^n\left(c_{v,x}\right)\right]\right\}+2S_{i,y}\left\{f_1^n\left(c_{v,y}\right)+2\sum _{j\ne i}^n{S}_{j,z}\left[f_2^n\left(c_{v,y}\right)+2\sum _{\begin{array}{c}k>j\\ k\ne i\end{array}}^n{S}_{k,z}{f}_3^n\left(c_{v,y}\right)\right]\right\}\right),& \left[38\right]\end{array}$

[0109] using the functions 𝑓_iⁿfrom TABLE 3. Variations in the scalar couplings are handled as for the IS_ncase, with an additional modification of ω_qfor each I spin, leading to separate versions of TABLE 4 labeled by the I spin and corresponding modifications of Eqs. [34] and [36].

[0110] The following provide analytical solutions for the irradiation of the I spins and the evolution of the I-spin magnetization and associated spinstates, involving only longitudinal S_z, that were not considered in the foregoing. The formalism shows immediately that these can be viewed as classical rotations about effective fields that incorporate the coupling, which allows the solutions to be written down by inspection. (In the subsequent reduction of theory to practice, we switch the spin-labelling convention to irradiation of the S spins for spinstates involving only longitudinal I_zfor these classical rotations.)

[0111] The propagator is still given by Eq. [15], and we consider the set of si_xinitial states

[0112] ρ(t₀)ε{I_x, 2I_xS_z}.   [39]

[0113] As for the initial conditions listed in Eq. [17] of the previous section, the members of this set only evolve, for an IS system, to members within the set. The time development of the system is given by $\begin{array}{cc}\rho \left(t\right)=U\left(t,t_0\right)\rho \left(t_0\right)U^{†}\left(t,t_0\right)=\sum _{q_1,\dots ,q_n=\pm 1}{U}_{q_s}\rho \left(t_0\right)U_{q_s}^{†}\prod _{j=1}^n{P}_k^{q_j},& \left[40\right]\end{array}$

[0114] which follows from P_k^±P_k^∓=0 and P_k^±(2S_z)=P_k^±. The sums and products that appear in Eq. [40] are parsed in the discussion surrounding Eq. [11]. Thus, the solution for the density operator again reduces to calculating products of the form U_qI_xU_r^†, similar to Eq. [22], and we could proceed with the expansion in terms of the Pauli basis as in the previous section. However, Eq. [22] was obtained for initial states involving transverse S-spin operators, which, in turn, forced r=q−2. This mixing of propagators associated with different allowed values for the magnetic quantum number of the total S-spin angular momentum has no classical analogue. In the present case, where r=q, each term of the form U_qI_xU_q^† in Eq. [40] can be immediately recognized as the transformation that generates a classical rotation of each component of the quantum-mechanical vector operator I about ω_q, and the solutions can be obtained by inspection using standard matrix results for a 3D rotation about an arbitrary axis. The desired rotation matrix R_qis listed in TABLE 5 in terms of the RF phase angle φ, the polar angle θ_qgiven in Eq. [8], and rotation angle β_qof Eq. [12]. The rotated I-spin components are then multiplied by the specified products of the projection operators P_k^±=1/2E_k±S_kz.

[0115] For n=1, there are product operator terms involving both no S-spins and a single S_z, so that

[0116] 2I_x→[R₍₊₁₎+R₍₋₁₎]I_x+[R₍₊₁₎−R₍₋₁₎]2I_xS_z(n=1),  [41]

[0117] where the rotations R_(±1)act only on the I-spin operator. For n=2, there are rotations {R_(±2), R₍₀₎} and an additional product S_1zS_2z, which, for n=3, expands to include S_1zS_3zand S_2zS_3zplus a term involving three S-spins, S_1zS_2zS_3ztogether with rotations {R_(±3), R_(±1)} acting on the given I-spin operator. The functions g_νⁿ(R)of the specific rotation matrices which give the coefficients of these operators are tabulated in TABLE 6, where (as for the functions 𝑓_iⁿ) the superscript n is the number of S-spins in the system and the subscript ν=0, . . . ,n gives the number of S-spins in the particular product operator term. The results, again, are of similar structure to Eq. [26], and we obtain, for ρ_j(t₀)=I_x, $\begin{array}{cc}\begin{array}{c}{\rho }_j\left(t\right)=\frac{1}{2^n}(g_0^n\left(R\right)+2\sum _{i=1}^n{S}_{i,z}\\ \left\{g_1^n\left(R\right)+2\sum _{k>i}^n{S}_{k,z}\left[g_2^n\left(R\right)+2\sum _{l>k}^n{S}_{l,z}{g}_3^n\left(R\right)\right]\right\})I_{x_j}\end{array}& \left[42\right]\end{array}$

[0118] This equation can be used for the initial state 2I_xS_zby multiplying all R₍₀₎, R₍₋₁₎, and R₍₋₁₎in TABLE 6 by minus one, which follows from the relation 2P_j^±S_j,z=±P_j^± that arises in the derivation.

[0119] For isotropic J-coupling, additional I-spins produce no effect other than increasing the initial polarization of I. The subspace propagators U_(q)in Eq. [13] become a product composed of individual propagators for each I-spin, as in Eq. [28], and the previous rotations become rotations of the total I-spin for each component $I_{x_j}=\sum _{i=1}^m{I}_{i,x_j}.$

[0120] Different scalar couplings J_ISmerely result in separate rotations of individual I-spin components determined by each U_i(q).

[0121] It is well known that the torque on a magnetization vector of a single spin, subject to continuous RF irradiation of constant amplitude and phase, is constant and proportional to the effective field, B_e, where B_eis the resultant of the RF field, B₁, and the resonance offset, ΔH (Hz). Defining B₁as the reciprocal 360° pulse time on resonance

[0122] B_e²=B₁²+ΔH²,  [43]

[0123] or equivalently, as pictured in FIG. 1, the tilt angle, α, of the effective field relative to the xy plane of the rotating reference frame is defined as

[0124] sin α=ΔH/B_e, or cos α=B₁/B_e.  [44]

[0125] On resonance, the magnetization will rotate around the RF field by an angle

[0126] θ=2πB₁t,  [45]

[0127] where t is the length of time the RF is applied and off resonance the rotation is increased to

[0128] θ=2πB_et.  [46]

[0129] An arbitrary rotation can be described in terms of a 3×3 rotation matrix to calculate the final orthogonal components of the S magnetization, S_j, from the initial components, S_j⁰: $\begin{array}{cc}\left[\begin{array}{c}{S}_x\\ S_y\\ S_z\end{array}\right]=\left[\begin{array}{ccc}{f}_{\mathrm{xx}}& f_{\mathrm{xy}}& f_{\mathrm{xz}}\\ f_{\mathrm{yx}}& f_{\mathrm{yy}}& f_{\mathrm{yz}}\\ f_{\mathrm{zx}}& f_{\mathrm{zy}}& f_{\mathrm{zz}}\end{array}\right]\left[\begin{array}{c}{S}_x^o\\ S_y^o\\ S_z^o\end{array}\right],& \left[47\right]\end{array}$

[0130] which in terms of linear combinations is $\begin{array}{cc}{S}_i=\sum _j{f}_{\mathrm{ij}}{S}_j^o,& \left[48\right]\end{array}$

[0131] where i and j take the values x, y, z. The axis of rotation is the direction of B_e, and in addition to the tilt, α, the direction of this axis in 3D space is also determined by the phase of the RF energy. In NMR, the phase of the applied RF irradiation is commonly x, y, −x, or −y (i.e. 0°, −90°, −180, or −270°), but any phase angle, β, relative to zero phase along the x axis can be used. This phase angle is also depicted in FIG. 1.

[0132] Beginning with the simple 3×3 matrices for rotations about the x, y and z axes, Bendall and Pegg (Magnetic Resonance in Medicine 2, 91 (1985); Journal of Magnetic Resonance 63, 494 (1985)) showed with basic mechanics that the matrix for a pulse of angle θ′ about an axis which is rotated a phase angle β from the x axis and is tilted upwards by an angle α from the xy plane as in FIG. 1, is $\begin{array}{cc}f\left({\theta }^{\prime }\left[\beta \right],\alpha \right)=\left[\begin{array}{ccc}{f}_A{\cos }^2\beta +f_D{\sin }^2\beta & f_B-f_G\sin 2\beta & f_C\cos \beta +f_E\sin \beta \\ -f_B-f_G\sin 2\beta & f_A{\sin }^2\beta +f_D{\cos }^2\beta & -f_C\sin \beta +f_E\cos \beta \\ f_C\cos \beta -f_E\sin \beta & -f_C\sin \beta -f_E\cos \beta & f_F\end{array}\right],& \left[49\right]\end{array}$

[0133] where

[0134] 𝑓_A=cos²α+sin²α cos θ′,  [50]

[0135] 𝑓_B=sin α sin θ′,  [51]

[0136] 𝑓_C=sin2α sin²(θ′/2),  [52]

[0137] 𝑓_D=cos θ′,  [53]

[0138] 𝑓_E=cos α sin θ′,  [54]

[0139] 𝑓_F=sin²α+cos²α cos θ′,  [55]

[0140] and

[0141] 𝑓_G=cos²α sin²(θ′/2)=(𝑓_A−𝑓_D)/2.  [56]

[0142] The sign convention used in the following in this disclosure is that clockwise rotations are positive and the x axis is clockwise of the y axis as in FIG. 1. Changing the sign of either of these changes the sign of all sine terms and thus all sinβ and sin2β terms in matrix [49] as well as the sign of 𝑓_cand 𝑓_E. The rotation of a magnetization vector, S_yinitially along the z axis (i. e. initially S_z⁰), around B_eis illustrated in FIG. 1 along with the resolution of this unit vector into its orthogonal components, S_x, S_y, and S_z. The orthogonality of these components ensures, via the Pythagorus theorem, that their root-sum-of-squares is the S vector of unit magnitude.

[0143] For RF of x phase (β=0), $\begin{array}{cc}f\left({\theta }^{\prime }\left[x\right],\alpha \right)=\left[\begin{array}{ccc}{f}_A& f_B& f_C\\ -f_B& f_D& f_E\\ f_C& -f_E& f_F\end{array}\right].& \left[57\right]\end{array}$

[0144] In the absence of an RF field, B_e=ΔH, and all the terms in [57] are zero except 𝑓_F=1, 𝑓_A=𝑓_D=cos[2πΔHt_D] and 𝑓_B=sin[2πΔHt_D], which expresses the interchange of S_xand S_ymagnetization via chemical-shift precession during time delay t_D.

[0145] Phase cycling a single rectangular pulse can be analyzed by changing β to β±n*90° in matrix [49], and summing or subtracting the corresponding 𝑓_ijterms in the resulting matrices depending on the sign of the receiver phase. This eliminates some of the 𝑓_ijterms in the final 3×3 matrix and thus some orthogonal components of magnetization, and demonstrates how the results of pulse sequences can be changed by phase cycling single pulses—examples are given by M. R. Bendall and D. T. Pegg, Magnetic Resonance in Medicine 2, 91 (1985); Journal of Magnetic Resonance 63, 494 (1985); 66, 546 (1986). Most instances of the use of pulsed field gradients in NMR methods eliminates particular 𝑓_ijterms and thus mimics or replaces the use of phase cycling for this purpose. Consequently, these applications of pulsed gradients can be analyzed quantitatively in the same way as phase cycling.

[0146] By digitizing shaped pulses into small time increments, each having constant attributes of amplitude and phase, matrix [49] can be used to calculate the time course and result of any frequency/amplitude modulated pulse (since phase is the integral of frequency) by repetitive matrix multiplication. But the result of multiplying 3×3 matrices is a 3×3 rotation matrix, so the magnetization of a spin at a particular resonance offset has undergone an overall single rotation about some axis in 3D space and the logic concerning phase cycling applies to the overall complex pulse. Thus, for example, if the 𝑓_xyterm is eliminated for a rectangular pulse by phase alternation, it will also be eliminated for the complex pulse. These straightforward algorithms also apply to the 4×4 and 6×6 J-rotation matrices developed below.

[0147] Matrix [49] is a general analytical solution to the Bloch equations, neglecting relaxation—the only input from quantum mechanics is that there is an initial Boltzmann distribution producing a net magnetization. These analytical solutions for a single spin, derived from basic mechanics, must also be found in the foregoing QM derivation, and the corresponding QM solutions are listed in TABLE 5. Substituting for the different angle symbols shows that the nine terms in TABLE 5 are identical to those in matrix [49]. Thus, the effect of RF irradiation on a single spin can be determined from basic mechanics as a classical rotation around the effective field B_e. Expressing a general rotation in terms of a rotation matrix (e.g. [49]) directly provides the magnitude of the three possible orthogonal PO spinstates, S_x, S_yand S_z.

[0148] The first step in any pulse sequence applied to a coupled IS spin system is the excitation of the S spins prior to any RF on the I spins. The foregoing QM analysis shows that this irradiation involves exchange between the six spinstates, S_x, S_y, S_z, 2S_xI_z, 2S_yI_z, and 2S_zI_z, and indicates that these can be viewed as classical rotations. In the following it is shown that a 6×6 rotation matrix can be derived from 3×3 rotation matrix [49] using classical principles and that an exact vector model is applicable.

[0149] The basis of this classical vector model is that one half of the S spins, S⁺, are coupled to I spins aligned with the +z axis (and rotate around an effective field B_e⁺), and the other half, denoted by S⁻, are coupled to I spins aligned with the −z axis (and rotate around B_e⁻). This is the only additional input from quantum mechanics. The magnitudes of these effective fields are

[0150] (B_e^±)²=B₁²+[ΔH±(J/2)]²,  [58]

[0151] the tilt angles are

[0152] sin α^±=[ΔH±(J/2)]/B_e^±, or cos α^±=B₁/B_e^±.   [59]

[0153] and the rotation angles are given by

[0154] θ^±=2πB_e^±t,  [60]

[0155] in parallel with Eqs. [43] to [46]. On resonance, the magnitude of B_e⁺ and B_e⁻ are identical, so B_e^±=B_e^J=[B₁²+(J/2)²]^0.5will be used.

[0156] An on-resonance S_zto 2S_xI_zconversion is shown in FIG. 2ato 2cto illustrate the two effective fields, B_e⁺ and B_e⁻, two of the possible spinstates, and the rotation of S⁺ and S⁻ around B_e⁺ and B_e⁻, respectively. The clockwise rotation of S⁻ about B_e⁻ is displayed more readily in FIG. 2bas an anticlockwise rotation around −B_e⁻. Matrix [49] can be applied to the individual rotations of S⁺ and S⁻, and the magnitudes of the si_xspinstates are determined by noting that in FIG. 2a,S_zis the sum of S_z⁺ (the component of S⁺ along z) and S_z⁻, whereas in FIG. 2c,the antiparallel state, 2S_xI_z, is given by S_x⁺ minus S_x⁻. Thus, scaling all vectors relative to unit positive magnitude, the orthogonal spinstates are given by

[0157] S_i=0.5(S_i⁺+S_i⁻)  [61]

[0158] and

[0159] 2S_iI_z=0.5(S_i⁺−S_i⁻),  [62]

[0160] where i takes the values x, y, z as usual.

[0161] Accordingly, the terms in the 6×6 matrices for the interconversion of the six spinstates can be written down by inspection since they are given by the sums and differences of the terms in the 3×3 matrices, [49] or [57], for rotations around B_e⁺ and B_e⁻. In general, listing the spinstates in the order, S_x, S_y, S_z, 2S_xI_z, 2S_yI_z, and 2S_zI_z, the 6×6 matrices will be $\begin{array}{cc}f\left({\theta }^{\pm }\left[\beta \right],{\alpha }^{\pm }\right)=0.5*\left[\begin{array}{cc}\mathrm{SUM}\mathrm{of}3\times 3& \mathrm{DIFFERENCE}\mathrm{of}3\times 3\\ \mathrm{matrix}\mathrm{terms}& \mathrm{matrix}\mathrm{terms}\\ & \\ \mathrm{DIFFERENCE}& \mathrm{SUM}\end{array}\right],& \left[63\right]\end{array}$

[0162] where the diagonally opposite quadrants are identical. For example, defining 𝑓^± as the appropriate member of Eqs. [50] to [56] with arguments α^± and θ^±, the top-left term for the most general case resulting from matrix [49] is (𝑓_A⁺+𝑓_A⁻)cos²β+(𝑓_D⁺+𝑓_D⁻)sin²β, which simplifies to 𝑓_A⁺+𝑓_A⁻ for the 6×6 matrix resulting from Eq. [57] for RF of a single phase x: $\begin{array}{cc}2.0*f\left({\theta }^{\pm }\left[x\right],{\alpha }^{\pm }\right)=\left[\begin{array}{cccccc}{f}_A^{+}+f_A^{-}& f_B^{+}+f_B^{-}& f_C^{+}+f_C^{-}& f_A^{+}-f_A^{-}& f_B^{+}-f_B^{-}& f_C^{+}-f_C^{-}\\ -f_B^{+}-f_B^{-}& f_D^{+}+f_D^{-}& f_E^{+}+f_E^{-}& -f_B^{+}+f_B^{-}& f_D^{+}-f_D^{-}& f_E^{+}-f_E^{-}\\ f_C^{+}+f_C^{-}& -f_E^{+}-f_E^{-}& f_F^{+}+f_F^{-}& f_C^{+}-f_C^{-}& -f_E^{+}+f_E^{-}& f_F^{+}-f_F^{-}\\ f_A^{+}-f_A^{-}& f_B^{+}-f_B^{-}& f_C^{+}-f_C^{-}& f_A^{+}+f_A^{-}& f_B^{+}+f_B^{-}& f_C^{+}+f_C^{-}\\ -f_B^{+}+f_B^{-}& f_D^{+}-f_D^{-}& f_E^{+}-f_E^{-}& -f_B^{+}-f_B^{-}& f_D^{+}+f_D^{-}& f_E^{+}+f_E^{-}\\ f_C^{+}-f_C^{-}& -f_E^{+}+f_E^{-}& f_F^{+}-f_F^{-}& f_C^{+}+f_C^{-}& -f_E^{+}-f_E^{-}& f_F^{+}+f_F^{-}\end{array}\right],& \left[64\right]\end{array}$

[0163] The six spinstates are orthogonal and thus the root-sum-of-squares of the terms in the rows or columns of any of the 6×6 matrices derived as for matrix [64] must be unity in the same sense that the Pythagorus theorem applies to the 3D components of a single vector via the 3×3 rotation matrix as in FIG. 1. That is, beginning with a spinstate of unit magnitude, the total of the spinstates or vector sum must always be unity at any time independent of the complexity of the RF pulse applied.

[0164] The corresponding rotation matrices for more than one coupled I spin can also be written down by inspection and the number of coupled S spins does not affect the progress of these classical rotations. For example, for an I₃S_ngroup the total S spin magnetization splits into four components rotating around four effective fields which are the resultants of B₁, offset, and coupling fields ±J/2 and ±3J/2. In the forgoing QM analysis it was assumed that the I spins rather than the S spins were irradiated during classical I_mS_nrotations. With this change of labels, analytical terms for these various rotation matrices can be written down from TABLE 6, giving, for example, exactly the same analytical expressions as those in matrix [64].

[0165] Bazzo and Boyd in the Journal of Magnetic Resonance 79, 568 (1988) undertook a limited QM analysis to obtain the exchange between the six spinstates, S_x, S_y, S_z, 2S_xI_z, 2S_yI_z, and 2S_zI_z, via coupling during selective pulses on the S spins in an IS system and summarized their analytical formulae in a 6×6 matrix identical to matrix [64] except for a different sign convention. However, neither Bazzo and Boyd, nor any other authors, have determined the most general analytical expressions obtainable from matrix [49], nor did these or any other authors consider I_mS_nsystems, or research exact analytical expressions for any nonclassical rotations, or consider the total set of 225 possible analytical expressions required to determine all 15 PO spinstates for an IS system after arbitrary RF irradiation of one type of spin.

[0166] Off resonance the rotations of S⁺ and S⁻ are unequal, occurring about effective fields of different magnitude which are not symmetrically tilted with respect to the xy plane as they are in FIG. 2a.For large frequency offsets or for large B₁, the difference between B_e⁺ and B_e⁻ approaches zero, S⁺ and S⁻ follow identical paths, and J can be ignored. When J=0, or when B₁or ΔH is large, the top right and bottom left quadrants of matrix [64] are all zeros and the other two quadrants reduce to 3×3 matrix [57]. There is no exchange of magnetization between S_iand 2S_iI_z—this is the normal high power approximation. If B₁=5J, failure to include ±J/2 in B_e^± produces an error of 0.5% in B_e^± on resonance and a maximum error of 5% at about 10J off resonance. Thus, the high power approximation is good for classical rotations for B₁>5J.

[0167] On resonance at lower power, B_e^±=B_e^J, α⁺=−α⁻=α, so equation [64] reduces to $\begin{array}{cc}f\left({\theta }^{\prime }\left[x\right],\alpha \right)=\left[\begin{array}{cccccc}{f}_A& 0& 0& 0& f_B& f_C\\ 0& f_D& f_E& -f_B& 0& 0\\ 0& -f_E& f_F& f_c& 0& 0\\ 0& f_B& f_C& f_A& 0& 0\\ -f_B& 0& 0& 0& f_D& f_E\\ f_C& 0& 0& 0& -f_E& f_F\end{array}\right],& \left[65\right]\end{array}$

[0168] with θ′=2πB_e^Jt. When both B₁and ΔH are zero, all the terms in the 6×6 matrices are zeros except 𝑓_A=𝑓_D=cos[πJt_D] and 𝑓_B=sin[πJt_D], which corresponds to free J-coupled precession interconverting S_x⇄2S_yI_z, and S_y⇄2S_xI_zduring delay t_D. As noted for 3×3 rotation matrices, the elimination of some orthogonal spinstates by phase cycling can be analyzed by adding or subtracting 6×6 matrices for RF of different phase, and the results will be true for single rectangular pulses or shaped pulses simulated by repetitive matrix multiplication.

[0169] The theoretical basis of all previous studies concerning excitation pulses are exactly explicable in terms of small subsets of the analytical expressions contained in matrix [64], and the Selective Population Transfer method (Chalmers, Pachler and Wessels, Journal of Magnetic Resonance, 15, 415 (1974); Jakobsen, Kanyha and Brey, Journal of Magnetic Resonance, 54, 134 (1983); Bildsoe, Journal of Magnetic Resonance, 27, 393 (1977); and references contained therein) is an example. As another example, the 𝑓_cterm in the third column of matrix [65] corresponds to the on-resonance transformation of S_zto 2S_xI_zdiscovered by Brondeau and Canet (Journal of Magnetic Resonance 47, 419 (1982)), and illustrated in FIG. 2ato 2c.From Eq. [52], the transformation occurs in 100% yield when B₁=J/2 (sin2α=1) and the rotation is 180° for a pulse length of t=(2^0.5j)⁻¹. The 𝑓_cterm in the fourth column of Eq. [65] determines the inverse transformation, 2S_xI_z→S_z, not previously described. The 𝑓_cterms in the first and last columns of matrix [65] reveal the new transformations, S_x⇄2S_zI_z, and the forward conversion is illustrated in FIG. 2dleading to FIG. 2e.

[0170] The time course of the spinstate transformations in FIG. 2 clearly demonstrate the contribution from J coupling during the pulse, a contribution which arises from the angle between B_e⁺ and B_e⁻. We will call the above matrices that include the effect of J modulation, “J-rotation matrices”, and we have named the corresponding RF pulses, “J pulses”, or more specifically 180^Jor 90^Jdepending on the extent of the ideal rotation on resonance—the 90^Jvariety arises from nonclassical rotations as described in the following. These 180^Jor 90^Jpulses are frequency selective and in the following we will also show that either a selective 90^Jor a selective 180^Jpulse of any frequency bandwidth can take the place of any combination of any hard 90° pulse and a consecutive (2J)⁻¹free precession delay in any IS pulse sequence.

[0171] The results of any NMR experiment are observed by detecting rotating nuclear magnetization vectors in the sample. These vectors evolve during an NMR pulse sequence and this evolution could be followed by stopping the pulse sequence at any point in its time course and measuring the magnetizations present at that point in time. For the purpose of establishing the various embodiments of the invention we introduce a specific definition of the measurability of the state of an NMR system: A state is measurable if maximum detectable signal can be observed directly, or is observable after a hard on-resonance pulse on the S spins and/or on the I spins.

[0172] Within this definition, all pure orthogonal IS spinstates, and any mix of these spinstates, are measurable. Furthermore, the I⁺, I⁻, S⁺ and S⁻ vectors comprising each spinstate are measurable. For example, the 2S_xI_zstate (FIG. 2c) is measured as directly-observable antiphase S⁺ and S⁻ signal at +J/2 and −J/2 respectively, whose magnitude sums to unity, compared to in-phase signal at ±J/2 of total unit magnitude for S_x. The same applies to 2S_yI_zand S_yexcept for a phase shift of 90°. In contrast, potentially-observable S_zand 2S_zI_z(FIG. 2e) may be measured by applying a hard 90[S] pulse in the first case to detect in-phase signal, or either a 90[S] or 90[I] pulse in the second case to observe the antiphase S signal or the antiphase I signal respectively. (These hard 90° pulses must lie well within the high power approximation discussed above in relation to matrix [64] and in the following in relation to Eqs. [86] and [87]). Similarly, antiphase I signal from 2S_xI_zand 2S_yI_zare measurable after a 90[S] pulse of y or x phase, respectively, and a 90[I] pulse of any phase. The transverse spinstates, e.g. 2S_yI_xas in FIG. 2f,can be measured by a hard 90° pulse of appropriate phase on one spin and observation of the antiphase signal from the other spin.

[0173] Mixed spinstates, as in FIG. 2b,are the vector sums of individual spinstates. These, and in particular the resultant magnetization vectors S⁺ and S⁻, can be measured using fractional hard pulses of incremented phase. Assuming no knowledge of the prior experiment, two signals at ±J/2 will be initially detected. Then, concentrating on the signal at +J/2, both the pulse angle and phase of a hard S pulse at +J/2 could be varied by trial-and-error until pulse angle, φ, phase, φ, gives a maximum signal (S⁺ transverse) and pulse angle, 90—φ, phase, 180—φ, gives zero signal (S⁺ along z). The 3D orientation of S⁺ is then known unambiguously, and the procedure could be repeated at −J/2 to determine S⁻. For arbitrary conditions, the experimentally-measured orientations of S⁺ and S⁻ can be resolved into the six possible spinstates, or the QM or vector calculation of the spinstates can be summed to determine S⁺ and S⁻. Thus there is a straightforward match between quantum mechanics, the vector model of FIG. 1 and FIG. 2, and what is measurable.

[0174] However, the vector model illustrated in FIG. 2 ignores what is measurable in terms of any I-spin magnetization vectors. This question is resolved as in FIG. 3 for the simple experiment of free J-coupled precession for a total time of (2J)⁻¹(corresponding to column one of matrix [65] when both B₁and ΔH are zero (𝑓_c=0)). At time zero, FIG. 3a,directly after a 90[S,−y] pulse (90° pulse on the S spins of −y phase), the spinstate is pure S_xand no I signal can be detected after application of a 90[S, x];90[I] pulse pair. Accordingly, there are no I-spin vectors along ±z. Part way through the evolution as in FIG. 3b,the I-spin vectors have grown in proportion to −sinπJt (=−sin γ in FIG. 3b) reaching unit magnitude when S⁺ and S⁻ are antiphase along ±y. Accordingly, experimental measurement of the antiphase I signal after 90[S, x];90[I] returns a total intensity of −sin πJt in accordance with the transformation of S_xto −2S_yI_zgiven by the −sin πJt term in matrix [65] when both B₁and ΔH are zero. Addition of the evolving nature of the detectable I-spin magnetization, depicted in FIG. 3, to the vector model of FIG. 2, provides an exact correspondence with all aspects of the 6×6 J-rotation matrices. The measurable I-spin signal (the magnitude of I⁺ plus I⁻) is the vector sum (root-sum-of-squares) of all the antiphase 2S_iI_z(i=x, y, and z) states, whereas the total S-spin signal (the magnitude of S⁺ plus S⁻) is the vector sum of all six states. Stating this in alternate form, S⁺ and S⁻ are of constant amplitude, whereas I⁺ and I⁻ vary as sine of half the angle subtended by S⁺ and S⁻. The orientation of S⁺, for example, is given by the vector sum of the S⁺ portions of each pure PO spinstate, and the orientation and magnitude of I⁺ is obtainable from the vector sum of the I⁺ parts of all the pure antiparallel states. Thus the experimental result of any S-spin irradiation is determined by the 3D orientation and magnitude of S⁺, S⁻, I⁺ and I⁻ and these attributes are predictable from the 6×6 J-rotation matrix.

[0175] The resulting pictorial vector model unambiguously illustrates the evolution of the measurable I-spin vectors. The addition of this I-spin evolution (as depicted in FIG. 3) to vector diagrams for classical S-spin rotations (as exemplified in FIG. 2band 2d), provides an exact correspondence with all aspects of the 6×6 J-rotation matrices and completes the vector model of classical rotations. In the following, nonclassical IS-spin rotations that interchange four orthogonal spinstates are addressed. We have been unable to deduce the associated analytical equations on the basis of linear vector rotations. However, the general rule (and its sub-rules) that the measurable I and S-spin vectors correspond to the vector sum of the four PO states holds true, and provides a minimal but adequate representation. Although the resulting vector descriptions are not readily predictive in all cases, because they depend on non-linear rates of rotation, they are nevertheless helpful in many circumstances.

[0176] Thus, in terms of an IS spin-half system, the logical steps that establish the rigor of graphical (vector) representations of nuclear spin magnetizations are:

[0177] The result of any NMR experiment is detected via the oscillating electric currents induced in a coil by rotating macroscopic nuclear magnetic fields (magnetizations) in the sample;

[0178] Calculations using the QM density matrix for any time point in an NMR pulse sequence must exactly predict these observable magnetizations;

[0179] The same QM calculations, or equivalently our explicit equations that are general solutions of the density matrix, also determine the proportions of all the PO states existing at any time point in a sequence;

[0180] The four individual magnetization vectors that comprise a pure PO state (two are of zero magnitude for an in-phase PO state) can be measured directly (i.e. observed), or they can be measured after application of hard 90° pulses;

[0181] For mixed spinstates, the vector sums of these four individual vectors provide the total measurable magnetizations under all conditions and at any time.

[0182] A picture of the rotation and fluctuating magnitude of these vectors in 3D space throughout a pulse sequence provides a rigorous description of the NMR experiment.

[0183] The universality claimed for such vector descriptions is justified by the finding that all previous studies on an IS spin system are subsumed within the overall picture, including previous vector models (e.g. Pegg, Bendall and Doddrell, Journal of Magnetic Resonance, 44, 238 (1981); Bendall, Pegg and Doddrell, Journal of Magnetic Resonance, 45, 8 (1981); Bendall, Journal of Magnetic Resonance, A 116, 46 (1995); Skinner and Bendall, Journal of Magnetic Resonance, 134, 315 (1998); and the common form of the PO formalism (Sorensen, Eich, Levitt, Bodenhausen and Ernst, Progress in NMR Spectroscopy, 16, 163 (1983)).

[0184] Important embodiments of the invention utilize the finding that these vectors are literal or real in all circumstances: They can be calculated from QM or measured by experiment. This finding is valid for the classical rotations described in the foregoing, and the nonclassical rotations discussed in the following.

[0185] The foregoing discussion provides the theoretical basis of the invention and describes known classical rotations of nuclear spin magnetizations, plus some new classical rotations that can be discovered from the general theoretical analysis presented here. In this disclosure we further show that nonclassical rotations are induced for both spins in a coupled IS spin system when the I spin is irradiated after excitation of the S spin. This contrasts with the classical rotations describable in terms of 3×3 rotation matrices, or 6×6 J-rotation matrices. Some of the embodiments of the invention arise from determining the exact behaviour of the I and S spins during these purely quantum-mechanical nonclassical evolutions.

[0186] In an IS-spin system, when the I-spin is irradiated after excitation of the S spin, the foregoing exact QM analysis shows that such an NMR experiment interchanges four orthogonal spinstates. From TABLES 1 to 4 an explicit 4×4 J-rotation matrix can be written for these interconversions. Listing the spinstates in the order S_x, 2S_yI_x, 2S_yI_y, and 2S_yI_z, the most general matrix is given by 𝑓(θ^±[β],α^±)= $\begin{array}{cc}f\left({\theta }^{\pm }\left[\beta \right],{\alpha }^{\pm }\right)=0.5*\left[\begin{array}{cccc}{f}_1& f_2\cos \beta +f_3\sin \beta & f_2\sin \beta -f_3\cos \beta & f_4\\ -f_2\cos \beta +f_3\sin \beta & f_5+f_6\cos 2\beta & f_6\sin 2\beta +f_7& f_8\cos \beta -f_9\sin \beta \\ -f_2\sin \beta -f_3\cos \beta & f_6\sin 2\beta -f_7& f_5-f_6\cos 2\beta & f_8\sin \beta +f_9\cos \beta \\ -f_4& f_8\cos \beta +f_9\sin \beta & f_8\sin \beta -f_9\cos \beta & f_{10}\end{array}\right],& \left[66\right]\end{array}$

[0187] which, for an RF pulse of x phase, simplifies to $\begin{array}{cc}f\left({\theta }^{\pm }\left[x\right],{\alpha }^{\pm }\right)=0.5*\left[\begin{array}{cccc}{f}_1& f_2& -f_3& f_4\\ -f_2& f_5+f_6& f_7& f_8\\ -f_3& -f_7& f_5-f_6& f_9\\ -f_4& f_8& -f_9& f_{10}\end{array}\right],& \left[67\right]\end{array}$

[0188] where

[0189] 𝑓₁=(1+cos[α⁺−α⁻])cos0.5[θ⁺−θ⁻]+(1−cos[α⁺−α⁻])cos0.5[θ⁺+θ⁻],  [68]

[0190] 𝑓₂=(cos α⁺+cos α⁻)sin0.5[θ⁺−θ⁻]+(cos α⁺−cos α⁻)sin0.5[θ⁺+θ⁻],  [69]

[0191] 𝑓₃=sin[α⁺−α⁻](cos0.5[θ⁺−θ⁻]−cos0.5[θ⁺+θ⁻]),  [70]

[0192] 𝑓₄=(sin α⁺+sin α⁻)sin0.5[θ⁺−θ⁻]+(sin α⁺−sin α⁻)sin0.5[θ⁺+θ⁻],  [71]

[0193] 𝑓₅=(1−sin α⁺sin α⁻)cos0.5[θ⁺−θ⁻]+(1+sinα⁺sinα⁻)cos0.5[θ⁺+θ⁻],  [72]

[0194] 𝑓₆=cosα⁺cosα⁻(cos0.5[θ⁺−θ⁻]−cos0.5[θ⁺+θ⁻]),  [73]

[0195] 𝑓₇=(sin α⁺−sin α⁻)sin0.5[θ⁺−θ⁻]+(sin α⁺+sin α⁻)sin0.5[θ⁺+θ⁻],  [74]

[0196] 𝑓₈=sin[α⁺+α⁻](cos0.5[θ⁺−θ⁻]−cos0.5[θ⁺+θ⁻]),  [75]

[0197] 𝑓₉=(cos α⁺−cos α⁻)sin0.5[θ⁺−θ⁻]+(cos α⁺+cos α⁻)sin0.5[θ⁺+θ⁻],  [76]

[0198] 𝑓₁₀=(1−cos[α⁺+α⁻])cos0.5[θ⁺−θ⁻]+(1+cos[α⁺+α⁻])cos0.5[θ⁺+θ⁻],  [77]

[0199] and sin α^±, cos α^±, and θ^± are defined in Eqs. [58] to [60]. On resonance, θ⁺=θ⁻=θ′, α⁻=−α⁺=−α, so the 4×4 matrix reduces to $\begin{array}{cc}f\left({\theta }^{\prime }\left[x\right],\alpha \right)=\left[\begin{array}{cccc}{f}_A& 0& -f_C& f_B\\ 0& 1& 0& 0\\ -f_C& 0& f_F& f_E\\ -f_B& 0& -f_E& f_D\end{array}\right],& \left[78\right]\end{array}$

[0200] where the terms are given by Eqs. [50] to [55] and θ′=2πB_et in which B_e=B_e^±=B_e^Jas for the discussion following Eqs. [58] to [60]. The angle α is analogous to the residual on-resonance tilt of the effective field shown in FIG. 2 for classical rotations, arising from the ±J/2 coupling fields, although, as described in the following, no classical rotations can be found around these individual tilted B_e^± fields.

[0201] In the absence of an RF field, J-rotation matrix [78] reduces to $\begin{array}{cc}f\left(t\right)=\left[\begin{array}{cccc}\cos \left[\pi Jt\right]& 0& 0& \sin \left[\pi Jt\right]\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ -\sin \left[\pi Jt\right]& 0& 0& \cos \left[\pi Jt\right]\end{array}\right].& \left[79\right]\end{array}$

[0202] Columns one and four describe free J-coupled precession interconverting S_x⇄2S_yI_z, whereas columns two and three correspond to the well known invariance of the transverse antiparallel 2S_yI_xand 2S_yI_yspinstates during time delays (Bendall, Pegg, and Doddren, Journal of Magnetic Resonance, 46, 43 (1982); 52, 81 (1983)).

[0203] For large B₁, the high power limit, or large ΔH, matrix [67] reduces to $\begin{array}{cc}f\left({\theta }^{\prime }\left[x\right].\alpha \right)=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& f_A& f_B& f_C\\ 0& -f_B& f_D& f_E\\ 0& f_C& -f_E& f_F\end{array}\right],& \left[80\right]\end{array}$

[0204] where J has been eliminated as insignificant and B_ein θ′=2πB_et is given by Eq. [43].

[0205] For the nonclassical rotations induced by RF on the I-spins, governed by the 4×4 matrices [66], [67] and [78], a simple predictive vector model as in FIG. 2 is not possible, but the orientation and magnitude of S⁺, S⁻, I⁺ and I⁻, may still be calculated in agreement with experiment and a reasonably simple vector description results, especially for ideal conditions such as on-resonance irradiation.

[0206] Such a vector description is shown for an ideal case in FIG. 4ato 4cfor initial S_x, B₁=J/2, and RF of x phase applied on resonance to the I spins for a time of (2^0.5J)⁻¹. In this case the S⁺ and S⁻ vectors precess 90° in the xy plane during the same time that the I⁺ and I⁻ vectors rotate from the z axis to the y axis. The rotation of the S-spin vectors results entirely from J coupling, whereas the rotation of the I-spin vectors is a combination of J and RF modulation. In consequence we will refer to this new kind of pulse as a “90^J” pulse. The axis of rotation of the I spins is the x axis. It is not tilted as it was for the classical rotations depicted in FIG. 2 and this can be proven by experimental measurement or from the observation that there is no 2S_yI_xmagnetization indicated in matrix [78] since the rotation lies entirely in the zy plane. Neither of the rotation angles shown in FIG. 4a,γ or φ, evolve linearly with time since cos γ=𝑓_Aand cos φ=𝑓_B. The 100% transformation of S_xto 2S_yI_yvia some intermediate 2S_yI_zdescribed in FIG. 4ato 4c,is given by the terms in the first column of matrix [78] when B₁=J/2, and the reverse transformation corresponds to the third column. The more complex formation of some 2S_yI_xoff resonance is given by the non-zero 𝑓₂term in matrix [67].