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J order and cross polarization in scalar-coupled spin systems
JOURNAL
OF MAGNETIC
41, 268-278 (1980)
RESONANCE
J Order and Cross Polarization in Scalar-Coupled Spin Systems KENNETH School
of Chemical
Sciences,
J. PACKER*
University
AND KEVIN
of East A&a,
Norwich,
M. WRIGHT NR47TJ,
Norfolk,
United
Kingdom
Received December 5, 1979; revised March 17, 1980 The phenomenon of “J order” in scalar-coupled spin systems is discussed, analogies being drawn with states of dipolar and quadrupolar order. In particular the creation of a state of J order via the X region of an AX spin system by means of adiabatic demagnetization in the rotating frame (ADRF) is considered in detail. Equations are derived describing the ADRF process, as are equations describing the transfer of order from the J-ordered state to A-spin transverse magnetization in the presence of a resonant rf field near the A-spin Larmor frequency. Comparisons are made with pulse methods for producing the same J-ordered state and polarization transfer. The ability of ADRF to selectively produce a state of J order for a given value of the coupling constant.! is considered and some pulse sequences are described briefly which are also capable ofJ-selective excitation ofl-ordered states. Some possible general areas of application of these experiments are considered. INTRODUCTION
The phenomenon of “dipolar order” is well known in the NMR properties of solids (I -5). For the experimentally more usual situation of high-field conditions (B, % Blocal) there are a number of techniques which may be used to transfer the dominant Zeeman order existing at thermal equilibrium (i.e., M,) into dipolar order. These include adiabatic demagnetization in the rotating frame (ADRF) (6) and pulse methods (5). In recent papers we examined the concept of quadrupole order in a situation in which the internal spin Hamiltonian comprised single-spin quadrupole terms only (7-9), and in this paper we look at some aspects of the concept of “J-ordered” states, that is, states in which Zeeman order has been transformed into order in scalar spin-spin couplings. The following section presents a qualitative discussion of the nature of order in spin interactions while the succeeding two sections present detailed analyses of the process of ADRF for an AX spin system and the use of the J-ordered state so produced for polarization transfer between the A and X nuclei. The last section discusses certain ways in which these ideas may be utilized; in particular, suggestions are made for pulse sequences which produce J-ordered states selectively for chosen values of J. INTERNAL
Under high-field conditions
ORDER IN SPIN SYSTEMS
a spin system often has a Hamiltonian
HT = Hz + Hs + HsL(I), * To whom correspondence
should be addressed. 268
002222364/80/14026~11$02.00/0
Copyright 0 1960 by Academic Press, Inc. All rights of reproduction in any form reserved.
of the form
HI
J ORDER IN SCALAR-COUPLED
SPINS
269
b
FIG. 1. The NMR absorption spectrum for a spin system (a) in a state of Zeeman order and (b) in a state of Internal order in the secular spin interaction, Hs, which is taken to determine the lineshape.
where HZ represents the large Zeeman coupling of the spins to the external field II,,; HSL(f), the fluctuating interactions responsible for spin-lattice relaxation; and H:j, a secular spin-coupling interaction (dipolar, quadrupolar, scalar, etc.). At thermal equilibrium the spin-density matrix in the high-temperature limit has the form PI P m c [l - aeq(Hz + H,)], where &q = (kT,)-’
with TL the lattice temperature.
Since for high-field conditions Zeeman in character. The objective of such procedures as ADRF is to transfer the Zeeman order to the secular term Hs to give a spin-density matrix
H,: % Hs, the order in such a system is overwhelmingly
P 0: [1
-
aHz
-
PHsl
[31
inwhicha=Oand I/3] b Iareql.Th is represents a state of high order in the spin interaction Hs and such states have certain basic similarities independent of the nature of Hs. Thus, following Jeener et al. (4,lO) and Anderson and Hartmann (6), Fig. 1 illustrates the general features of the NMR spectra for a system in a state of (a) Zeeman order and (b) internal (Hs) order. The essential feature to note is that a state of order in the internal spin interaction corresponds to a correlation of “fast” with “up” spin states and of “slow” with “down” spin states, where “fast” and “slow” indicate whether the spins contribute to the spectrum on the high- or low-frequency side of its center and “up” and “down” refer, as usual, to the spin orientation with respect to B,. Thus, if we consider the case of an AX spin system where Hs takes the form Hs = JZAZX z 27
[41
then a state of “.I order” in, say, the X region of the spectrum corresponds to one component of the X doublet being in absorption, the other in emission. Such a state may be produced by pulse techniques (1 I ) (although it generally has not been
270
PACKER
AND
WRIGHT
thought of in terms of J order) or by ADRF, section. ADRF
IN AN
AX
which is dealt with in the following
SPIN
SYSTEM
The process of ADRF is carried out by spin locking, in this case say, the X region of the spectrum with an rf field of amplitude wIx B J and then reducing this amplitude to a value much less than J at an appropriately slow rate. This is illustrated in Fig. 2. The calculation of the behavior of the system under this irradiation is best carried out by expressing the equations of motion in an interaction representation such that p* = R-‘pR,
H* = R-‘HR,
PI
+ wxZ:)t
PI
where R = expl(o,Z,A
with oA and w, frequencies in the vicinity of the A and X Larmor frequencies, respectively. They will normally be chosen to be the irradiating frequencies. The spin-density matrix and Hamiltonian in this doubly rotating frame then take the form p = I%[ 1 + E*z; + ExZ$], %? = JZAZx z z - til,Z&,
- OI&,,
where eA/x = {fi~A,xBO/kTr,} and where wIA (=yJIIA) and wIx (=yxBIx) amplitudes of the irradiating fields at the A and X frequencies. An unselective 90: pulse of oIx transforms the density matrix to p(o) = 1/[1 + E,& + +I$].
[71 are the
PI
The spin-locking rf field is now applied along the y axis of the X-spin rotating frame and is slowly reduced to zero. During this process the Hamiltonian has the form X = JZ,“Zf - o,,(t)Z;.
[91
Neglecting relaxation effects and using the usual equation of motion for the density matrix it can be shown, given the initial conditions expressed by Eq. [81 and using the Zeeman basis, that at time t the density matrix has the form
time +
FIG. 2. The irradiation pulse which starts with
procedure an amplitude
involved 9 J/2y
in ADRF. A 90, pulse is immediately followed (spin locking) and is then reduced adiabatically
by a E,, to zero.
J ORDER IN SCALAR-COUPLED
(1 + 42 -f(t)>
MO
{s(t) - Wt)l
”
271
SPINS
+ ih(
0
0
+ ‘$-ct)l
0
0
p(t, = f
3
0
0
(1 - E.412 + f(t)>
0
0
{-g(t)
where the functionsf(t),
- h(t)}
{-g(t) {I - EAl2 -f(t))
[lOI
+ ih(t)}
g(t), and h(t) satisfy the initial conditions h(0) = -EJ2,
f(O) = g(O) = 0,
IllI
and the equations of motion f(t)
= -@1xdQ,
g(t) = o&-(t)
+ (J/2)h(&
h(t) = -(J/2)g(t).
[Ql
In general, these equations are very difficult to solve for a time-dependent oIx. However, an approximate solution, appropriate to the ADRF process, can be found by considering the meaning of Eq. [lo] in physical terms. The quantity I consists of two 2 x 2 submatrices which describe the motion of two magnetization vectors M+ and M- (corresponding to 1,” values +i/ and --1/, respectively) whose components are given by M:(t)
= -K(t)
= (yxWg(tL
M;(t)
= M,(t)
= -(yxh/4)h(t),
M:(t)
= -M;(t)
= -(yxh/4)f(t).
1131
Comparing Eqs. [12] and [13] it is found that M+ and M- precess in effective magnetic fields B$ with components (0, Blx, ~Ji2-y~) (12) (see Fig. 3). As Blx is reduced to zero, the BQ tip toward the 7z directions respectively in the X-spin rotating frame. Consider the motion of M-. In order that it follow B; as it moves toward the +z axis, it is necessary that the angle 0 between B; and M- remain approximately constant at its initially very small value given by tan 9 = J/2w,,(O).
u41
This can only be so if the rate of precession, n(t), of M- about B; is much greater than the rate of change of direction of B;, i.e., k(t). Since tan a = J/2ax(t) and fi2(t) = o?,(t) + J2/4,
u51
272
PACKER AND WRIGHT
---Py
FIG. 3. The conditions for ADRF in one region of an AX spin system viewed in the rotating frame at exact resonance. The two lines of the doublet spectrum correspond to the magnetizations M+ and Mwhich precess about instantaneous effective fields, Bz, as shown (12). In order that the process of reduction ofl3, to zero be adiabatic, the angle f3must remain small as the angle cuchanges from 0 -+ n/2.
it follows that the necessary condition
is
10IX 1 Q 2cl3/J.
WI
This is analogous to the conditions required for adiabatic rapid passage (23). It is readily shown that, under the conditions of Eq. [16], the solutions of the equations of motion become
f(t) = 4wm(~) ExJ
[(Ih,(O)- ax(t) cos 41,
g(r) = - - EXJ sin 4, 4fw)
[I71 where @J(t) =
t n(t’)dt’. I0
It can be seen from these equations that when wIx = 0, M- and M+ are aligned approximately along +z and -z, respectively, this being the J-ordered state consistent with Fig. 1 and the qualitative discussion given earlier. As pointed out in that discussion, such a state may also be produced by pulse methods, an example of which is the so-called INERT experiment of Freeman and Morris (II ) consisting of the sequence 90~-r/2-180X/180A-7/2-W,X, which gives a result identical to that of ADRF if Jr = T. It should be noted that ADRF only works when the frequency wx lies within the X doublet. The effect of an off-resonance condition may be considered by introducing an offset term AxZ,Xinto the Hamiltonian with Ax = (wx - -yxBO). The vectors
J ORDER
IN
SCALAR-COUPLED
SPINS
273
M+ and M- then experience different effective fields with components in the ox rotating frame of (0, Blx, -(J/2yx + Ax/y,)) and (0, Blx, J/2y, - Ax/-&, respectively. It can be seen that, if lAxI > ]J/2 1, the final effective fields (i.e., when B 1X= 0) will both lie along the +z or -z axes, thus serving to produce only Zeeman order. ADRF thus has an element of selectivity, which is discussed in more detail in the last section. Some of these concepts have also been discussed recently by Garroway and Chingas (14). POLARIZATION
TRANSFER
FROM
J-ORDERED
STATES
In the NMR of solids the use of cross-polarization double-resonance techniques is well established (15). In particular, polarization transfer between different spin species via their rotating frame spin-locked Zeeman reservoirs, mediated by dipolar couplings when the so-called Hartmann-Hahn matching condition is satisfied (16), is widely used for sensitivity enhancement (17). Similarly, as originally discussed by Hartmann and Hahn (16) and more recently by Bertrand e? al. (18), similar processes may occur in liquids where the cross polarization is mediated by the scalar coupling. An alternative polarization transfer process in solids is first, to ADRF one spin species, thus producing a highly ordered dipolar state, and then to tra.nsfer this order directly to the second spin species by applying to it a resonant rf held with an amplitude of the same order as the dipolar couplings (17, 19). In this section we analyze the equivalent process for J-coupled spins; i.e., starting with a J-ordered state for the X spins in an AX system we investigate the effect of applying to the A spins an rf field which is off resonance by an amount AA. The Hamiltonian during the irradiation period is given by
[I81
X = JZ*Z* 2 2 + AAZ,A- o ‘AI*Y,
where AA = wA - yABo. The initial density matrix for the J-ordered
state is
p(o) = %[ 1 + EAz; - 2E,Z,AIf].
[I91
From the equations of motion it is found that the density matrix at time t, in the Zeeman basis, has the form
(1 - F+(O) 0
0 (1 + F-(f)1
1 G+(f) + iH+(Ol
0
0
{G-(t) + l-H_(r)}
p(t) = (G+(f) - iH+(Ol 0
0
0
(1 + F+(f)1
{G-(t) - iH_(t)}
0
(1 - F-(t))
where F+(f) =
(AA + J/212+ W?Acos R ~ fi:
nz,
+
I
3
7 rw
274
PACKER
F-(t)
=
WRIGHT
(6x + EA) (AA - JW 2 QZ i w1A
G+(t) = n G-(t)
AND
(EX
-
EA)
sin
R
2
+
= - z
+ 4~ cos a-t , R1 I c
+ 9
(EA i Ex)~sin n-t, a+?
H+(t) = - $
(eX - EA)(AA + J/2) sin2 2
,
+ H-(t)
R-t
(EA + +)(AA - J/2) sin2 2
= 2
WI
,
and where and
Cl: = 03, + (AA f J/2)2
The magnetization
components
WA = yAB,A.
at time t can then be shown to be
M:(t)
= M;(t)
= M,X(t) = 0
WI
and z WIA@A
+ fit
M;(t)
3/z)
sin2
WI
sin R-r,
R,t
2
= + y(z
+ 1)(‘AA--;2)2
+ $cos
n-r)
, [23c]
where M,A = ~Ah2B,/4kTL. Equation [22] follows as the state of the X spins at t = 0 is a l-ordered state, having no observable magnetization and thus, ignoring relaxation, there is no possibility of the development of an X magnetization. Equation [23], on the other hand, describes the precession of the Me doublet components about their effective fields in the A-spin rotating frame and this is illustrated in Fig. 4. As is shown there, the two doublet components of the A-spin spectrum have relative intensities of (1 + yXlyA)(M,f-/2) and (1 - yXlyA)(M@) in the J-ordered state. These components precess in the rotating frame about effective fields (a-/y,) and (a+/~*), respectively. In particular, the behavior of M,A(t) is of interest as it constitutes the development of a magnetization along the rf field axis (chosen arbitrarily as y) which represents a polarization transfer from the J-ordered state to the A-spin
J ORDER IN SCALAR-COUPLED
275
SPINS
FIG. 4. The physical meaning of Eqs. [23] (see text) is illustrated above. The M,A doublet components, A- and A+, (which have magnitudes (1 + yx/y,)M$/2 and (1 - yx/y,)Mt/2, respectively, in the J-ordered state) precess about their effective fields, (K/y*) and (a+/~~), in the A-spin rotating frame. The x, y, and z components of MA, as given in Eqs. [23], can be obtained at any time by projecting A- and A+ onto the appropriate axis.
rotating frame Zeeman state, a form of cross polarization. From Eq. [23b] it can be seen that, for the on-resonance case (AA = O), the magnitude of M:(t) depends on the factor { Jw,,lC12} which has a maximum value of unity when the HartmannHahn type of matching condition ( 17), wIA = J/2, is satisfied. Under this condition the initial X-spin Zeeman order, converted into J order by the ADRF process, undergoes oscillatory conversion into A-spin transverse magnetization and back to X-spin J order at a frequency J/2 1/Z. Such a process of transfer of order between arc internally and an externally generated set of levels is a well-known phenomenon (20). An alternative to this low-level irradiation or contact pulse method of polarization transfer is simply an unselective 90” pulse at wA (II). Such a 90” pulse will immediately transfer the J order into A-spin transverse magnetization independent of the value of J in which the X-spin order is stored. The low-amplitude crosspolarization pulse, on the other hand, allows a degree of discrimination between different J values since (a) the maximum amplitude transferred depends on {.IwlA/CIz> and (b) the frequency of the oscillatory transfer, 1R, is given by (& + J2/4)1’z. SELECTIVE
EXCITATION
OF J-ORDERED
STATES AND POLARIZATION
TRANSFER
In the investigation of complex molecules it is often useful to be able to selectively enhance one feature or group of features in its NMR spectrum. The use of the 967- 180 spin-echo sequence to reveal the presence of doublets or triplets in the ‘I-I spectra of proteins (21) is such an example. In this section we briefly outline some possible uses of J order. In particular we make suggestions for ways of selectively exciting such a state for a chosen value of J and, hence, via polarization transfer, to selectively excite the spectrum of
276
PACKER
AND
WRIGHT
coupled spins. As an example consider an AX spin system such as a 13C-lH methine group in a complex molecule. We could aim to selectively excite a state of J order in the proton region for this group and others with the same J coupling and then observe the 13C spectrum of only such groups by polarization transfer from the J-ordered state, gaining a factor (Y~~/Y,:,~)in sensitivity in the process. In the following we confine our comments to AX spin systems but the general ideas are capable of extension, though with differences in detail, to other systems such as AX,,. J-Selective Excitation
of J Order
ADRF. As was pointed out above, ADRF, as a means of producing J order, is by its nature selective. If the irradiating frequency does not lie within the X doublet, then only Zeeman order is generated on reducing the rf field to zero. Thus, if in a spectrum we had two doublets arising from AX systems with different J values, then we could produce J order from one of these and Zeeman order from the other by carrying out ADRF with an irradiating frequency set to be inside one doublet but outside the other, which is possible in principle as long as 6x, the chemical shift difference for the two X doublets, satisfies the condition 26x > AJ, where AJ is the difference in the two AX J couplings. Pulse sequences. An alternative to ADRF for the J-selective excitation of J-ordered states is to devise appropriate sequences of unselective rf pulses. Two such possibilities are (I) 90,~7-(-8,-4T-),-, where no, = 90,, and (II) 90,-T-(-90,-FGP-90-,-2~-)-90,-, where FGP = field gradient pulse. For simplicity these have been written assuming that the irradiating frequency is exactly the center of the X-doublet spectrum. Suitable practical modifications are possible for the general case of off-resonance irradiation. These involve the application of 180” pulses to both A and X spins at appropriate times (ZZ) and the modified sequences are shown in Fig. 5. The basic principle of sequence I is that a single pulse may be made frequency selective by replacing it with a series of shorter pulses of the same total effect (22). We have shown that if a spin interaction, Hs, is characterized by a number of frequencies (e.g., a powder quadrupole pattern, a number of J couplings, etc.) and if a pulse sequence 90,-7-O@ may be devised which produces a certain desired state of the spin system for a given value of such a frequency, then the sequence may be made selective with regard to that frequency of spin interaction by replacing the O8 pulse by a -(-&,-47-),sequence such that n& = Om. The full details of this will be dealt with elsewhere but we note, in the context of this paper, that if we had two AX systems with different J values, then application of sequence I, with the condition 57 = r satisfied for one of the J values, would selectively excite a state of J order for the AX system having that J value. The degree of selectivity increases with n. The principle of sequence II is rather different and works only for states for which the density matrix is diagonal. It relies on the fact that a pulse of B0 gradient can
J ORDER IN SCALAR-COUPLED
9g
-z2
leOAx 7
ey" ,
-r_-2
-
SPINS
211
WIAX
27
__
p-27
n
FIG. 5. The pulse sequences for J-selective excitation ofJ-ordered states for the general case of an off-resonance irradiation. The only difference from the on-resonance cases, I and II, described in the texl , is the addition of simultaneous 180” pulses at A and X frequencies (labeled 1809 at the times indicated. These have the effect of reversing the time evolution of the transverse magnetization of, say, the X spins due to the resonance offset but leavingunchangedthe evolution due to theJ coupling. The hatched pulse in sequence II is the field gradient pulse for destroying off-diagonal elements of the density matrix as described in the text.
be used to effectively destroy all off-diagonal elements. Consider again the case of two AX spin systems with differingJ values,J1 and.J,, and assume that the conditionJIT = n holds. Following the 90,~r-90, pulse pair at the start of sequence II, all the Zeeman order for the X region of the system with J, is converted into J, order. For the other AX system, only part of the original Zeeman order is converted into Jz order, the rest remaining as transverse magnetization. The pulse of B,, gradient destroys this and then the subsequent 90-, pulse converts the J order (for both J, and J.& back into transverse magnetization where, after another interval 27, the components of the J1 transverse magnetization are again antiphase along thex axis in the rotating frame and able to be totally returned toJ, order by the next 90, pulse. The transverse order corresponding to Jz, however, has, as in the first interval, only precessed through some angle different from 180” again and the 90, pulse followed by the field gradient pulse further attenuates the order associated with J2. This is repeated n times with the selectivity increasing with n. Selective Polarization
Transfer
Once a state of J order has been selectively excited then the order associated with it may be transferred to the other, A spins. This may be accomplished by a 90A pulse or by a cross-polarization contact pulse of amplitude J/2, as discussed ea.rlier. The 90* pulse is simple and quick; on the other hand the contact pulse is additionally selective since the maximum amplitude of J order transferred is
278
PACKER
(Joh,liY)
AND
WRIGHT
sin2 (M2).
If we consider the case of two AX spin systems with J values J’ and J”, where J” = fJ’, and imagine that we have selectively excited a state of J order for J’ but may also have created some forJ”, then if we cross polarize for a time t = (21’2n-/J’) with an t-f field of amplitude WIA = J’/2, the relative magnitudes of the M:(t) for the two AX spin systems are 1:{2fl(l
+ f”)}
sin2 {( 1 + f2)1~27r/2(2)1’2}Q,
where Q is the ratio of the J order in J” and J’ prior to cross polarization. Thus, with this method of polarization transfer, a further attenuation and hence selectivity is obtained. We are currently developing an experimental program to investigate these phenomena and their applications. ACKNOWLEDGMENT The Science Research one of us (K.M.W.).
Council
of the United
Kingdom
is thanked
for
a research
studentship
for
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Il. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.
A. J. J. J. M.
G. ANDERSON, Phys. Rev. 125, 1517 (1962). JEENER, H. EISENDRATH, AND R. VAN STEENWINKEL, Phys. Rev. A 133, 478 (1964). JEENER, R. DE Bors, AND P. BROEKAERT, Phys. Rev. A 139, 1959 (1965). JEENER AND P. BROEKAERT, Phys. Rev. 157,232 (1%7). GOLDMAN, “Spin Temperature and Nuclear Magnetic Resonance in Solids,” Chap. 2, Oxford Univ. Press, Oxford, 1970. A. G. ANDERSON AND S. R. HARTMANN, Phys. Rev. 128,2023 (1%2). S. B. AHMAD AND K. J. PACKER, Mol. Phys. 37,47 (1979). S. B. AHMAD AND K. J. PACKER, Mol. Phys. 37, 59 (1979). K. J. PACKER, Mol. Phys. 39, 15 (1980). H. EISENDRATH, W. STONE, AND J. JEENER, Phys. Rev. B 17, 47 (1978). R. FREEMAN AND G. A. MORRIS, J. Am. Chem. Sot. 101,760 (1979). A. L. BLOOM AND J. N. SHOOLERY, Phys. Rev. 97, 1261 (1955). J. G. POWLES, Proc. Phys. Sot. London 71,497 (1958). A. N. GARROWAY AND G. C. CHINGAS, J. Map. Reson. (1980). M. MEHRING, in “NMR Basic Principles and Progress” (P. Diehl, E. Fluck, and R. Kosfeld, Eds.), Vol. 11, p. 1, 1976. S. R. HARTMANN AND E. L. HAHN, Phys. Rev. 128, 2042 (1962). A. PINES, M. G. GIBBY, AND J. S. WAUGH, J. Chem. Phys. 59, 569 (1973). R. D. BERTRAND, W. B. MONIZ, A. N. GARROWAY, AND G. C. CHINGAS,.~. Am. Chem. Sot. 100, 5227 (1978). A. PINES, in “Proceedings, First Specialised Colloque Ampere” (J. W. Hennel, Ed.), p. 165, Institute of Nuclear Physics, Krak6w, 1973. R. L. STROMBOTNE AND E. L. HAHN, Phys. Rev. 133, 1616 (1964). I. D. CAMPBELL, C. M. DOBSON, R. G. RATCLIFFE, AND R. J. P. WILLIAMS,J. Mugn. Reson. 31, 341 (1978). G. A. MORRIS AND R. FREEMAN, J. Magn. Reson. 29,433 (1978).
OF MAGNETIC
41, 268-278 (1980)
RESONANCE
J Order and Cross Polarization in Scalar-Coupled Spin Systems KENNETH School
of Chemical
Sciences,
J. PACKER*
University
AND KEVIN
of East A&a,
Norwich,
M. WRIGHT NR47TJ,
Norfolk,
United
Kingdom
Received December 5, 1979; revised March 17, 1980 The phenomenon of “J order” in scalar-coupled spin systems is discussed, analogies being drawn with states of dipolar and quadrupolar order. In particular the creation of a state of J order via the X region of an AX spin system by means of adiabatic demagnetization in the rotating frame (ADRF) is considered in detail. Equations are derived describing the ADRF process, as are equations describing the transfer of order from the J-ordered state to A-spin transverse magnetization in the presence of a resonant rf field near the A-spin Larmor frequency. Comparisons are made with pulse methods for producing the same J-ordered state and polarization transfer. The ability of ADRF to selectively produce a state of J order for a given value of the coupling constant.! is considered and some pulse sequences are described briefly which are also capable ofJ-selective excitation ofl-ordered states. Some possible general areas of application of these experiments are considered. INTRODUCTION
The phenomenon of “dipolar order” is well known in the NMR properties of solids (I -5). For the experimentally more usual situation of high-field conditions (B, % Blocal) there are a number of techniques which may be used to transfer the dominant Zeeman order existing at thermal equilibrium (i.e., M,) into dipolar order. These include adiabatic demagnetization in the rotating frame (ADRF) (6) and pulse methods (5). In recent papers we examined the concept of quadrupole order in a situation in which the internal spin Hamiltonian comprised single-spin quadrupole terms only (7-9), and in this paper we look at some aspects of the concept of “J-ordered” states, that is, states in which Zeeman order has been transformed into order in scalar spin-spin couplings. The following section presents a qualitative discussion of the nature of order in spin interactions while the succeeding two sections present detailed analyses of the process of ADRF for an AX spin system and the use of the J-ordered state so produced for polarization transfer between the A and X nuclei. The last section discusses certain ways in which these ideas may be utilized; in particular, suggestions are made for pulse sequences which produce J-ordered states selectively for chosen values of J. INTERNAL
Under high-field conditions
ORDER IN SPIN SYSTEMS
a spin system often has a Hamiltonian
HT = Hz + Hs + HsL(I), * To whom correspondence
should be addressed. 268
002222364/80/14026~11$02.00/0
Copyright 0 1960 by Academic Press, Inc. All rights of reproduction in any form reserved.
of the form
HI
J ORDER IN SCALAR-COUPLED
SPINS
269
b
FIG. 1. The NMR absorption spectrum for a spin system (a) in a state of Zeeman order and (b) in a state of Internal order in the secular spin interaction, Hs, which is taken to determine the lineshape.
where HZ represents the large Zeeman coupling of the spins to the external field II,,; HSL(f), the fluctuating interactions responsible for spin-lattice relaxation; and H:j, a secular spin-coupling interaction (dipolar, quadrupolar, scalar, etc.). At thermal equilibrium the spin-density matrix in the high-temperature limit has the form PI P m c [l - aeq(Hz + H,)], where &q = (kT,)-’
with TL the lattice temperature.
Since for high-field conditions Zeeman in character. The objective of such procedures as ADRF is to transfer the Zeeman order to the secular term Hs to give a spin-density matrix
H,: % Hs, the order in such a system is overwhelmingly
P 0: [1
-
aHz
-
PHsl
[31
inwhicha=Oand I/3] b Iareql.Th is represents a state of high order in the spin interaction Hs and such states have certain basic similarities independent of the nature of Hs. Thus, following Jeener et al. (4,lO) and Anderson and Hartmann (6), Fig. 1 illustrates the general features of the NMR spectra for a system in a state of (a) Zeeman order and (b) internal (Hs) order. The essential feature to note is that a state of order in the internal spin interaction corresponds to a correlation of “fast” with “up” spin states and of “slow” with “down” spin states, where “fast” and “slow” indicate whether the spins contribute to the spectrum on the high- or low-frequency side of its center and “up” and “down” refer, as usual, to the spin orientation with respect to B,. Thus, if we consider the case of an AX spin system where Hs takes the form Hs = JZAZX z 27
[41
then a state of “.I order” in, say, the X region of the spectrum corresponds to one component of the X doublet being in absorption, the other in emission. Such a state may be produced by pulse techniques (1 I ) (although it generally has not been
270
PACKER
AND
WRIGHT
thought of in terms of J order) or by ADRF, section. ADRF
IN AN
AX
which is dealt with in the following
SPIN
SYSTEM
The process of ADRF is carried out by spin locking, in this case say, the X region of the spectrum with an rf field of amplitude wIx B J and then reducing this amplitude to a value much less than J at an appropriately slow rate. This is illustrated in Fig. 2. The calculation of the behavior of the system under this irradiation is best carried out by expressing the equations of motion in an interaction representation such that p* = R-‘pR,
H* = R-‘HR,
PI
+ wxZ:)t
PI
where R = expl(o,Z,A
with oA and w, frequencies in the vicinity of the A and X Larmor frequencies, respectively. They will normally be chosen to be the irradiating frequencies. The spin-density matrix and Hamiltonian in this doubly rotating frame then take the form p = I%[ 1 + E*z; + ExZ$], %? = JZAZx z z - til,Z&,
- OI&,,
where eA/x = {fi~A,xBO/kTr,} and where wIA (=yJIIA) and wIx (=yxBIx) amplitudes of the irradiating fields at the A and X frequencies. An unselective 90: pulse of oIx transforms the density matrix to p(o) = 1/[1 + E,& + +I$].
[71 are the
PI
The spin-locking rf field is now applied along the y axis of the X-spin rotating frame and is slowly reduced to zero. During this process the Hamiltonian has the form X = JZ,“Zf - o,,(t)Z;.
[91
Neglecting relaxation effects and using the usual equation of motion for the density matrix it can be shown, given the initial conditions expressed by Eq. [81 and using the Zeeman basis, that at time t the density matrix has the form
time +
FIG. 2. The irradiation pulse which starts with
procedure an amplitude
involved 9 J/2y
in ADRF. A 90, pulse is immediately followed (spin locking) and is then reduced adiabatically
by a E,, to zero.
J ORDER IN SCALAR-COUPLED
(1 + 42 -f(t)>
MO
{s(t) - Wt)l
”
271
SPINS
+ ih(
0
0
+ ‘$-ct)l
0
0
p(t, = f
3
0
0
(1 - E.412 + f(t)>
0
0
{-g(t)
where the functionsf(t),
- h(t)}
{-g(t) {I - EAl2 -f(t))
[lOI
+ ih(t)}
g(t), and h(t) satisfy the initial conditions h(0) = -EJ2,
f(O) = g(O) = 0,
IllI
and the equations of motion f(t)
= -@1xdQ,
g(t) = o&-(t)
+ (J/2)h(&
h(t) = -(J/2)g(t).
[Ql
In general, these equations are very difficult to solve for a time-dependent oIx. However, an approximate solution, appropriate to the ADRF process, can be found by considering the meaning of Eq. [lo] in physical terms. The quantity I consists of two 2 x 2 submatrices which describe the motion of two magnetization vectors M+ and M- (corresponding to 1,” values +i/ and --1/, respectively) whose components are given by M:(t)
= -K(t)
= (yxWg(tL
M;(t)
= M,(t)
= -(yxh/4)h(t),
M:(t)
= -M;(t)
= -(yxh/4)f(t).
1131
Comparing Eqs. [12] and [13] it is found that M+ and M- precess in effective magnetic fields B$ with components (0, Blx, ~Ji2-y~) (12) (see Fig. 3). As Blx is reduced to zero, the BQ tip toward the 7z directions respectively in the X-spin rotating frame. Consider the motion of M-. In order that it follow B; as it moves toward the +z axis, it is necessary that the angle 0 between B; and M- remain approximately constant at its initially very small value given by tan 9 = J/2w,,(O).
u41
This can only be so if the rate of precession, n(t), of M- about B; is much greater than the rate of change of direction of B;, i.e., k(t). Since tan a = J/2ax(t) and fi2(t) = o?,(t) + J2/4,
u51
272
PACKER AND WRIGHT
---Py
FIG. 3. The conditions for ADRF in one region of an AX spin system viewed in the rotating frame at exact resonance. The two lines of the doublet spectrum correspond to the magnetizations M+ and Mwhich precess about instantaneous effective fields, Bz, as shown (12). In order that the process of reduction ofl3, to zero be adiabatic, the angle f3must remain small as the angle cuchanges from 0 -+ n/2.
it follows that the necessary condition
is
10IX 1 Q 2cl3/J.
WI
This is analogous to the conditions required for adiabatic rapid passage (23). It is readily shown that, under the conditions of Eq. [16], the solutions of the equations of motion become
f(t) = 4wm(~) ExJ
[(Ih,(O)- ax(t) cos 41,
g(r) = - - EXJ sin 4, 4fw)
[I71 where @J(t) =
t n(t’)dt’. I0
It can be seen from these equations that when wIx = 0, M- and M+ are aligned approximately along +z and -z, respectively, this being the J-ordered state consistent with Fig. 1 and the qualitative discussion given earlier. As pointed out in that discussion, such a state may also be produced by pulse methods, an example of which is the so-called INERT experiment of Freeman and Morris (II ) consisting of the sequence 90~-r/2-180X/180A-7/2-W,X, which gives a result identical to that of ADRF if Jr = T. It should be noted that ADRF only works when the frequency wx lies within the X doublet. The effect of an off-resonance condition may be considered by introducing an offset term AxZ,Xinto the Hamiltonian with Ax = (wx - -yxBO). The vectors
J ORDER
IN
SCALAR-COUPLED
SPINS
273
M+ and M- then experience different effective fields with components in the ox rotating frame of (0, Blx, -(J/2yx + Ax/y,)) and (0, Blx, J/2y, - Ax/-&, respectively. It can be seen that, if lAxI > ]J/2 1, the final effective fields (i.e., when B 1X= 0) will both lie along the +z or -z axes, thus serving to produce only Zeeman order. ADRF thus has an element of selectivity, which is discussed in more detail in the last section. Some of these concepts have also been discussed recently by Garroway and Chingas (14). POLARIZATION
TRANSFER
FROM
J-ORDERED
STATES
In the NMR of solids the use of cross-polarization double-resonance techniques is well established (15). In particular, polarization transfer between different spin species via their rotating frame spin-locked Zeeman reservoirs, mediated by dipolar couplings when the so-called Hartmann-Hahn matching condition is satisfied (16), is widely used for sensitivity enhancement (17). Similarly, as originally discussed by Hartmann and Hahn (16) and more recently by Bertrand e? al. (18), similar processes may occur in liquids where the cross polarization is mediated by the scalar coupling. An alternative polarization transfer process in solids is first, to ADRF one spin species, thus producing a highly ordered dipolar state, and then to tra.nsfer this order directly to the second spin species by applying to it a resonant rf held with an amplitude of the same order as the dipolar couplings (17, 19). In this section we analyze the equivalent process for J-coupled spins; i.e., starting with a J-ordered state for the X spins in an AX system we investigate the effect of applying to the A spins an rf field which is off resonance by an amount AA. The Hamiltonian during the irradiation period is given by
[I81
X = JZ*Z* 2 2 + AAZ,A- o ‘AI*Y,
where AA = wA - yABo. The initial density matrix for the J-ordered
state is
p(o) = %[ 1 + EAz; - 2E,Z,AIf].
[I91
From the equations of motion it is found that the density matrix at time t, in the Zeeman basis, has the form
(1 - F+(O) 0
0 (1 + F-(f)1
1 G+(f) + iH+(Ol
0
0
{G-(t) + l-H_(r)}
p(t) = (G+(f) - iH+(Ol 0
0
0
(1 + F+(f)1
{G-(t) - iH_(t)}
0
(1 - F-(t))
where F+(f) =
(AA + J/212+ W?Acos R ~ fi:
nz,
+
I
3
7 rw
274
PACKER
F-(t)
=
WRIGHT
(6x + EA) (AA - JW 2 QZ i w1A
G+(t) = n G-(t)
AND
(EX
-
EA)
sin
R
2
+
= - z
+ 4~ cos a-t , R1 I c
+ 9
(EA i Ex)~sin n-t, a+?
H+(t) = - $
(eX - EA)(AA + J/2) sin2 2
,
+ H-(t)
R-t
(EA + +)(AA - J/2) sin2 2
= 2
WI
,
and where and
Cl: = 03, + (AA f J/2)2
The magnetization
components
WA = yAB,A.
at time t can then be shown to be
M:(t)
= M;(t)
= M,X(t) = 0
WI
and z WIA@A
+ fit
M;(t)
3/z)
sin2
WI
sin R-r,
R,t
2
= + y(z
+ 1)(‘AA--;2)2
+ $cos
n-r)
, [23c]
where M,A = ~Ah2B,/4kTL. Equation [22] follows as the state of the X spins at t = 0 is a l-ordered state, having no observable magnetization and thus, ignoring relaxation, there is no possibility of the development of an X magnetization. Equation [23], on the other hand, describes the precession of the Me doublet components about their effective fields in the A-spin rotating frame and this is illustrated in Fig. 4. As is shown there, the two doublet components of the A-spin spectrum have relative intensities of (1 + yXlyA)(M,f-/2) and (1 - yXlyA)(M@) in the J-ordered state. These components precess in the rotating frame about effective fields (a-/y,) and (a+/~*), respectively. In particular, the behavior of M,A(t) is of interest as it constitutes the development of a magnetization along the rf field axis (chosen arbitrarily as y) which represents a polarization transfer from the J-ordered state to the A-spin
J ORDER IN SCALAR-COUPLED
275
SPINS
FIG. 4. The physical meaning of Eqs. [23] (see text) is illustrated above. The M,A doublet components, A- and A+, (which have magnitudes (1 + yx/y,)M$/2 and (1 - yx/y,)Mt/2, respectively, in the J-ordered state) precess about their effective fields, (K/y*) and (a+/~~), in the A-spin rotating frame. The x, y, and z components of MA, as given in Eqs. [23], can be obtained at any time by projecting A- and A+ onto the appropriate axis.
rotating frame Zeeman state, a form of cross polarization. From Eq. [23b] it can be seen that, for the on-resonance case (AA = O), the magnitude of M:(t) depends on the factor { Jw,,lC12} which has a maximum value of unity when the HartmannHahn type of matching condition ( 17), wIA = J/2, is satisfied. Under this condition the initial X-spin Zeeman order, converted into J order by the ADRF process, undergoes oscillatory conversion into A-spin transverse magnetization and back to X-spin J order at a frequency J/2 1/Z. Such a process of transfer of order between arc internally and an externally generated set of levels is a well-known phenomenon (20). An alternative to this low-level irradiation or contact pulse method of polarization transfer is simply an unselective 90” pulse at wA (II). Such a 90” pulse will immediately transfer the J order into A-spin transverse magnetization independent of the value of J in which the X-spin order is stored. The low-amplitude crosspolarization pulse, on the other hand, allows a degree of discrimination between different J values since (a) the maximum amplitude transferred depends on {.IwlA/CIz> and (b) the frequency of the oscillatory transfer, 1R, is given by (& + J2/4)1’z. SELECTIVE
EXCITATION
OF J-ORDERED
STATES AND POLARIZATION
TRANSFER
In the investigation of complex molecules it is often useful to be able to selectively enhance one feature or group of features in its NMR spectrum. The use of the 967- 180 spin-echo sequence to reveal the presence of doublets or triplets in the ‘I-I spectra of proteins (21) is such an example. In this section we briefly outline some possible uses of J order. In particular we make suggestions for ways of selectively exciting such a state for a chosen value of J and, hence, via polarization transfer, to selectively excite the spectrum of
276
PACKER
AND
WRIGHT
coupled spins. As an example consider an AX spin system such as a 13C-lH methine group in a complex molecule. We could aim to selectively excite a state of J order in the proton region for this group and others with the same J coupling and then observe the 13C spectrum of only such groups by polarization transfer from the J-ordered state, gaining a factor (Y~~/Y,:,~)in sensitivity in the process. In the following we confine our comments to AX spin systems but the general ideas are capable of extension, though with differences in detail, to other systems such as AX,,. J-Selective Excitation
of J Order
ADRF. As was pointed out above, ADRF, as a means of producing J order, is by its nature selective. If the irradiating frequency does not lie within the X doublet, then only Zeeman order is generated on reducing the rf field to zero. Thus, if in a spectrum we had two doublets arising from AX systems with different J values, then we could produce J order from one of these and Zeeman order from the other by carrying out ADRF with an irradiating frequency set to be inside one doublet but outside the other, which is possible in principle as long as 6x, the chemical shift difference for the two X doublets, satisfies the condition 26x > AJ, where AJ is the difference in the two AX J couplings. Pulse sequences. An alternative to ADRF for the J-selective excitation of J-ordered states is to devise appropriate sequences of unselective rf pulses. Two such possibilities are (I) 90,~7-(-8,-4T-),-, where no, = 90,, and (II) 90,-T-(-90,-FGP-90-,-2~-)-90,-, where FGP = field gradient pulse. For simplicity these have been written assuming that the irradiating frequency is exactly the center of the X-doublet spectrum. Suitable practical modifications are possible for the general case of off-resonance irradiation. These involve the application of 180” pulses to both A and X spins at appropriate times (ZZ) and the modified sequences are shown in Fig. 5. The basic principle of sequence I is that a single pulse may be made frequency selective by replacing it with a series of shorter pulses of the same total effect (22). We have shown that if a spin interaction, Hs, is characterized by a number of frequencies (e.g., a powder quadrupole pattern, a number of J couplings, etc.) and if a pulse sequence 90,-7-O@ may be devised which produces a certain desired state of the spin system for a given value of such a frequency, then the sequence may be made selective with regard to that frequency of spin interaction by replacing the O8 pulse by a -(-&,-47-),sequence such that n& = Om. The full details of this will be dealt with elsewhere but we note, in the context of this paper, that if we had two AX systems with different J values, then application of sequence I, with the condition 57 = r satisfied for one of the J values, would selectively excite a state of J order for the AX system having that J value. The degree of selectivity increases with n. The principle of sequence II is rather different and works only for states for which the density matrix is diagonal. It relies on the fact that a pulse of B0 gradient can
J ORDER IN SCALAR-COUPLED
9g
-z2
leOAx 7
ey" ,
-r_-2
-
SPINS
211
WIAX
27
__
p-27
n
FIG. 5. The pulse sequences for J-selective excitation ofJ-ordered states for the general case of an off-resonance irradiation. The only difference from the on-resonance cases, I and II, described in the texl , is the addition of simultaneous 180” pulses at A and X frequencies (labeled 1809 at the times indicated. These have the effect of reversing the time evolution of the transverse magnetization of, say, the X spins due to the resonance offset but leavingunchangedthe evolution due to theJ coupling. The hatched pulse in sequence II is the field gradient pulse for destroying off-diagonal elements of the density matrix as described in the text.
be used to effectively destroy all off-diagonal elements. Consider again the case of two AX spin systems with differingJ values,J1 and.J,, and assume that the conditionJIT = n holds. Following the 90,~r-90, pulse pair at the start of sequence II, all the Zeeman order for the X region of the system with J, is converted into J, order. For the other AX system, only part of the original Zeeman order is converted into Jz order, the rest remaining as transverse magnetization. The pulse of B,, gradient destroys this and then the subsequent 90-, pulse converts the J order (for both J, and J.& back into transverse magnetization where, after another interval 27, the components of the J1 transverse magnetization are again antiphase along thex axis in the rotating frame and able to be totally returned toJ, order by the next 90, pulse. The transverse order corresponding to Jz, however, has, as in the first interval, only precessed through some angle different from 180” again and the 90, pulse followed by the field gradient pulse further attenuates the order associated with J2. This is repeated n times with the selectivity increasing with n. Selective Polarization
Transfer
Once a state of J order has been selectively excited then the order associated with it may be transferred to the other, A spins. This may be accomplished by a 90A pulse or by a cross-polarization contact pulse of amplitude J/2, as discussed ea.rlier. The 90* pulse is simple and quick; on the other hand the contact pulse is additionally selective since the maximum amplitude of J order transferred is
278
PACKER
(Joh,liY)
AND
WRIGHT
sin2 (M2).
If we consider the case of two AX spin systems with J values J’ and J”, where J” = fJ’, and imagine that we have selectively excited a state of J order for J’ but may also have created some forJ”, then if we cross polarize for a time t = (21’2n-/J’) with an t-f field of amplitude WIA = J’/2, the relative magnitudes of the M:(t) for the two AX spin systems are 1:{2fl(l
+ f”)}
sin2 {( 1 + f2)1~27r/2(2)1’2}Q,
where Q is the ratio of the J order in J” and J’ prior to cross polarization. Thus, with this method of polarization transfer, a further attenuation and hence selectivity is obtained. We are currently developing an experimental program to investigate these phenomena and their applications. ACKNOWLEDGMENT The Science Research one of us (K.M.W.).
Council
of the United
Kingdom
is thanked
for
a research
studentship
for
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