Spin counting experiments in the dipolar-ordered state

Spin counting experiments in the dipolar-ordered state

H. Cho, D. G. Cory, and C. Ramanathan

Citation: The Journal of Chemical Physics 118, 3686 (2003); doi: 10.1063/1.1538244 View online: http://dx.doi.org/10.1063/1.1538244

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Spin counting experiments in the dipolar-ordered state

H. Cho, D. G. Cory, and C. Ramanathan^a)

Department of Nuclear Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 共Received 21 October 2002; accepted 25 November 2002兲

The dipolar-ordered state (␳⬀H_d) can be prepared from a state of Zeeman equilibrium at high field by adiabatically removing the Zeeman field. It is also possible to produce dipolar-ordered states in the rotating frame using two well-known methods—adiabatic demagnetization in the rotating frame and the Jeener–Broekaert 共JB兲 pulse pair. Using multiple quantum nuclear magnetic resonance techniques, we illustrate the transient dynamics of the spin system following the JB pulse pair and its evolution to a dipolar-ordered state over a period of approximately 60␮s in a single crystal of CaF₂ oriented along the 关100兴 direction, and approximately 100␮s when the crystal is oriented along the关110兴direction. By encoding the coherence numbers in an orthogonal basis共the x basis兲 to the Zeeman basis, we show that the dipolar-ordered state is a two spin correlated state. The observed ratio between double quantum and zero quantum coherences in the x basis confirms the presence of the I_i^⫹I^⫺_j ⫹I_i^⫺I_j^⫹共flip–flop兲terms in the experimentally prepared dipolar-ordered state.

© 2003 American Institute of Physics. 关DOI: 10.1063/1.1538244兴

I. INTRODUCTION

Multiple quantum 共MQ兲 nuclear magnetic resonance 共NMR兲 techniques in solids have generally been used to study the size and dimensionality of localized spin clusters as well as to probe the many-body dynamics of a solid state spin system.^1–13 However, these techniques cannot be used to study spin dynamics under a Hamiltonian that conserves coherence number. For example, during an FID the system evolves under the secular dipolar Hamiltonian, and the z ba- sis coherence number of the spins remains⫾1, but the num- ber of correlated spins changes. However, Tomaselli et al.,¹⁴ using an extension of the earlier work by Suter and Pearson,¹⁵ recently encoded multiple quantum coherences in a y basis to characterize the evolution of multi-spin order in polarization and coherence echoes.

Another situation where standard MQ techniques are of limited utility is in the study of the dipolar-ordered state, as this state is not encoded by usual MQ techniques. Dipolar ordered states are eigenstates of the spin Hamiltonian, and hence are constants of the motion. They decay on a time scale T_1D, as the spin system interacts with the external en- vironment or lattice. Dipolar ordered states 共in the rotating frame兲 can be prepared by performing an adiabatic demag- netization in the rotating frame共ADRF^16,17兲, or using a pair of phase shifted RF pulses.¹⁸ While the system is in the dipolar-ordered state at the end of the ADRF process 共by definition兲, following a Jeener–Broekaert pulse pair the spin system is in a transient state, and only evolves into a state of dipolar-order on a timescale of the order of T₂.¹⁸Emid et al.

have shown that the spin system contains multiple quantum coherences immediately following the second pulse.^19,20

In this paper we use an improved version of the tech- nique originally suggested by Suter and Pearson¹⁵ to inves- tigate the nature of the state prepared using these two meth-

ods, as well as explore the transient response of the spin system following a Jeener–Broekaert pulse pair and its evo- lution to a dipolar-ordered state. We measure the coherence numbers of the dipolar-ordered state in both the x and z basis, performed by observing the response of the system to a col- lective rotation of the spins about the appropriate axis,²¹and use this information to characterize the state.

II. THEORY

In a strong magnetic field (B₀zˆ), an N spin-1/2 system has 2^N stationary states. These can be classified according to the magnetic quantum number, M_z⫽兺im_zi⫽(n_兩⫹1/2典

⫺n_兩⫺1/2典)/2, where m_zi⫽⫾1/2 is the eigenvalue of the ith spin in the system, and the energy eigenvalue corresponding to M_z is E_z⫽⫺␥បB₀M_z. For nondegenerate stationary states there are 2^2N⫺1 possible transitions between any two levels. The difference in M_z values between the coupled states is referred to as the coherence number.

While these coherences refer to transitions between lev- els, it is useful to discuss multiple quantum coherences for states of a system. When the state is expressed in the eigen- basis of the system, the presence of a nonzero matrix element 具^zi兩␳兩z_j典, indicates the presence of an n quantum coherence, where n⫽M_z(z_j)⫺M_z(z_i). Since M_z⫽兺im_zi is a good quantum number, we use a collective rotation about the axis of quantization, 兺iI_zⁱ, to characterize it

具^zi兩exp

冉

^⫺i␾

^兺

^{i Izi}

冊

^␳exp

冉

^⫹i␾

^兺

^{i Izi}

冊

^兩zj典

⫽exp共in␾兲具^zi兩␳兩z_j典^. 共1兲 In the dipolar-ordered state the density operator of the spin system corresponds to the dipolar Hamiltonian. In the high temperature approximation,

␳D⫽exp共⫺H_D⁰/kT_s兲 Z ⬇1

Z

冉

^1⫺kTHD0s

冊

^,

共2兲 Z⫽Tr兵^exp共⫺HD

0/kT_s兲其^,

a兲Author to whom correspondence should be addressed. Electronic mail:

sekhar@mit.edu

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where

H_D⁰⫽

兺

_i⬍j ^{di j}

再

^{2IziIzj⫺12共Izi⫹Izj⫺⫹Izi⫺Izj⫹兲}

冎

^,

共3兲 d_{i j}⫽␥²ប²

r_{i j}³

冉

^{1⫺3 cos2 2共␪i j兲}

冊

H_D⁰ is the secular dipolar Hamiltonian, T_s is the spin tem- perature, ␥ is the gyromagnetic ratio, r is the distance be- tween spins,␪ is the angle between the z axis and the inter- nuclear vector. As the dipolar-ordered state only contains population and zero quantum terms in the z basis, its struc- ture cannot be revealed by z basis encoding.

While these coherences have a physical meaning in the eigenbasis of the system, a generalized coherence number reports on the response of the system to any collective rota- tion of the spins 共about the x axis, for example兲. This is equivalent to a requantization of the spins along the axis of rotation. The transformation between the z and x bases can be summarized as follows:²²

I_z⫽⫺i

2共I_x^⫹⫺I_x^⫺兲, I_z^⫹⫽I_x⫹ i

2共I_x^⫹⫹I_x^⫺兲,

共4兲 I_z^⫺⫽I_x⫺i

2共I_x^⫹⫹I_x^⫺兲.

Table I shows the secular dipolar Hamiltonian 共and corre- spondingly the dipolar-ordered state兲 expressed in the two bases. The presence of both zero and double quantum terms in the x basis representation provides additional observable information about the state of the spin system.

Consider the experiments in Fig. 1, illustrating the x and z basis measurements. The propagator U_A corresponds to the creation of the dipolar-ordered state, while U_Brepresents the conversion of the dipolar-ordered state back to single-spin single-quantum terms that are observable. The final density operator in the z basis experiment is

␳f⫽U_B exp共iH_d⁰␦兲exp共⫺i␾^Iz兲exp共iH_d⁰t₁兲U_A␳0U_A^†

⫻exp共⫺iH_d⁰t₁兲exp共i␾^Iz兲exp共⫺iH_d⁰␦兲U_B^† 共5兲 while that in the x basis experiment is

␳f⫽U_B exp共iH_d⁰␦兲exp共⫺i␾^Ix兲exp共iH_d⁰t₁兲U_A␳0U_A^†

⫻exp共⫺iH_d⁰t₁兲exp共i␾^Ix兲exp共⫺iH_d⁰␦兲U_B^† 共6兲 The operator A corresponds to the application of a Jeener-Broekaert pulse pair, or an ADRF sequence. The op- erator B corresponds to an adiabatic remagnetization in the rotating frame共ARRF兲or a 45° pulse to convert the dipolar- ordered state into observable magnetization. Fourier trans- forming the two-dimensional time domain data with respect to␾ yields the coherence number in the appropriate basis.

III. EXPERIMENTAL RESULTS AND DISCUSSION The pulse sequences used in the various experiments are shown in Figs. 2共a兲–2共c兲. The experiments were performed at 2.35 T, using a Bruker Avance spectrometer and a home- built probe. The sample used was a single crystal of calcium fluoride, CaF₂, with T₁ ⬃ 7 s.

In the coherence encoding experiment, the phases (␾⁾ were incremented from zero to 4␲ ^with⌬␾⫽␲/8 to encode up to eight quantum coherences in every experiment. A fixed

TABLE I. Coherence numbers and selection rules of the secular dipolar Hamiltonian in the z and x bases.

z basis x basis

Dipolar Hamiltonian 兺i⬍jdi j兵^2IziIz j⫺¹2(I_i^⫹I_j^⫺⫹I_i^⫺I^⫹_j)其 ⫺¹2兺i⬍jdi j兵^2IxiIx j⫺¹2(I_xi^⫹I_{x j}^⫺⫹I_xi^⫺I_{x j}^⫹)其

⫺³4兺i⬍jdi j兵Ixi⫹Ix j⫹⫹Ixi⫺Ix j⫺其

Coherence number selection rule 0 0,⫾2

FIG. 1. Generalized x and z basis encoding experiments.

FIG. 2. Pulse sequences for x and z basis encoding of the dipolar-ordered state.共a兲JB creation and 45^opulse readout for simultaneous z and x basis encoding,共b兲ADRF creation and ARRF readout for x basis encoding,共c兲JB creation and ARRF readout for x basis encoding. The 48 pulse sequence was used to suppress the evolution of the internal Hamiltonian between the two

␲^{/2 pulses}共Ref. 23兲. We assume that relaxation effects are negligible during the sequence.

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time point, corresponding to a maximum intensity signal in the echo, was sampled for each value of ␾ and the data Fourier transformed with respect to␾ to yield the coherence numbers.

A. Estimation of the dipolar-ordered state

In order to estimate the state of the system, the two dimensional coherence encoding experiment shown in Fig.

2共a兲 was performed. By incrementing the phases ␾ ^and ␩ independently, it is possible to simultaneously encode the system in both the z and x bases, and observe correlations between them.^15,21The pulse spacing (␶) in the JB sequence was set to 13␮^{s, and t}1 was set to 1 ms. It is assumed that all transients have decayed away by this time.

The 48 pulse sequence is used to suppress evolution of the internal Hamiltonian共the dipolar interaction兲between the 90^o pulses in the x basis experiment.²³The propagator of the 48 pulse sequence is the identity operator, if relaxation ef- fects can be neglected over the course of the pulse sequence.

It is necessary to separate the two ␲/2 pulses in time, in order to avoid phase switching transients. The spacing be- tween the pulses in the sequence was set to 1.5␮s. The delay

␦ was set to 20␮^s.

Figure 3 shows the coherences observed for the dipolar- ordered state in a 2D experiment where the coherences in the two bases are correlated. The dipolar-ordered state is ob- served to contain zero quantum terms in the z basis, and zero and double quantum terms in the x basis. The highest coher- ence order observed was⫾2 in the x basis experiment. The absence of single quantum order in the x basis data illustrates the symmetry of the state under exchange of two spins.

Given the axial symmetry of the secular dipolar Hamiltonian, a generic two spin zero quantum state is

␳⫽

兺

_i⬍j ^{兵aIziIzj⫹b共Izi⫹Izj⫺⫹Izi⫺Izj⫹兲其. 共7兲}

Transforming this state to the x basis yields

␳⫽

兺

_i⬍j

再

^{14共⫺a⫹2b兲共Ixi⫹Ixj⫹⫹Ixi⫺Ixj⫺兲⫹2bIxIx}

⫹1

4共a⫹2b兲共I_x^i⫹I_x^j⫺⫹I_x^i⫺I_x^j⫹兲

冎

^{. 共8兲}

We expect the ⫹2 and ⫺2 coherences to have the same intensity, and the ratio of the double quantum coherence to the zero quantum coherence to be

R⫽⫺a⫹2b

6b⫹2a . 共9兲

We performed one dimensional x basis encoding experi- ments in order to measure the ratio between the zero and double quantum terms, following the ADRF and JB se- quences. The pulse sequences for this experiment are shown in Figs. 2共b兲 and 2共c兲. Adiabatic remagnetization in the ro- tating frame 共ARRF兲 was used to refocus the state in both cases these experiments, in order to ensure that the full di- polar state was refocused.

The pulse spacing (␶) in the JB sequence was set to 13

␮^{s, and t}1 varied from 1 ms to 5 s. A 3 ms long hyperbolic secant pulse, corresponding to an adiabatic half passage of the amplitude, was used for both the ADRF共decreasing am- plitude兲and the ARRF共increasing amplitude兲. A delay of 20

␮s was inserted before the ARRF to minimize transient ef- fects.

Figure 4 shows the experimentally obtained ratio using both the JB and ADRF sequences as a function of t₁. While the dipolar-ordered state decays with a time constant of T_1D,

FIG. 3. Results of the correlated 2D experiment that simultaneously encodes z and x basis coherences for the dipolar-ordered state. The dipolar state is observed to contain only zero quantum coherences in the z basis, and both zero and double quantum coherences in the x basis.

FIG. 4. Ratio between x basis double and zero quantum signals as a function of t1. As the dipolar-ordered state decays, the intensity of the double and zero quantum orders decays. However, the ratio is maintained at⬃1.5. The errors were estimated from the variance of the noise in a signal free region of the spectrum. The increase in the error with longer evolution time is due to the deteriorating signal-to-noise ratio as the signal decays.

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the ratio between the two coherences remains constant at 1.5 in both cases. When this is set equal to R, we get a⫽⫺4b, indicating that the state measured is

␳⫽

兺

_i⬍j ^a

再

^{IziIzj⫺14共Izi⫹Izj⫺⫹Izi⫺Izj⫹兲}

冎

^共10兲

which is exactly the state shown in Table I.

B. Zeeman contamination inT_1Dmeasurements T_1Dmeasurements are done exclusively on resonance in order to avoid possible contamination of the data by encoded Zeeman magnetization. However, it can be difficult to find the exact resonance condition in strongly coupled dipolar solids such as single crystal CaF₂, and impossible to be si- multaneously on resonance everywhere in a powder. Emid et al. designed an offset-independent method to measure di- polar relaxation times, by summing the results of two experi- ments using a ␲^{/4 and a} ⫺3␲/4 pulse to convert dipolar- order back to Zeeman order.²⁴The difference in pulse lengths can be a problem in strongly coupled dipolar systems, as the evolution under a long pulse becomes non-negligible. In the x basis measurements presented here, the Zeeman term (兺iI_zⁱ) is encoded into single quantum coherence while the dipolar-ordered state is encoded into zero and double quan- tum coherence. Figure 5 shows the x basis coherences mea- sured following a JB pulse pair关Fig. 2共c兲兴when the experi- ment is performed on and off resonance by 21 kHz.

This clean separation of the two terms allows very accu- rate T_1Dmeasurement, without any T₁contamination. Figure 6 shows the relaxation data for dipolar-ordered states created by the JB and ADRF methods. The measured T_1Dwas essen- tially identical for the zero and double quantum x basis co- herences in both sets of experiments (255⫾3.1 ms and 254

⫾3.1 ms for the zero and double quantum terms, respec-

tively, in the ADRF experiment, and 254⫾2.9 ms for both the zero and double quantum terms in the JB experiment兲as expected, as dipolar relaxation rates should not depend on the method used to prepare the state.

C. Initial transient dynamics for JB sequence

Emid et al.^19,20 have previously shown that multiple quantum coherences are contained in the transverse magne- tization following the JB sequence. By performing the JB experiment off-resonance they were able to detect the pres- ence of single, double, and triple quantum coherences during the initial transient regime following the␲/4 pulse. Using a combination of JB experiments with different phases, they were also able to selectively observe the even and odd co- herences in this initial transient regime as well in a sample of l-alanine powder, in which the NH₃ group had been deuterated²⁰. Their experimental data showed the evolution of the transients out to about 60␮^s.

We used the pulse sequence shown in Fig. 2共a兲to study the transient response of the spin system following a JB pulse pair, and the evolution of the system to a state of di- polar order. Figure 7 shows the transient response of the spin system following a JB sequence, for a single crystal of cal- cium fluoride oriented along the关110兴direction with respect to the external field 共the crystal was actually between the 关110兴and关111兴directions兲. Immediately after the␲^{/4 pulse,} the largest intensity is present in the z⫽0, x⫽⫾2 term, but zero, single and double quantum z basis terms correlated to

FIG. 5. Pulse sequence in Fig. 2共c兲was used to encode the x basis coher- ences共a兲on resonance, and共b兲21 kHz off resonance. t1was set to 5 ms in both experiments.

FIG. 6. Decay of x basis double and zero quantum signals in the共a兲ADRF and共b兲JB experiments. The measured decay constant was 255⫾3.1 ms and 254⫾3.1 ms for the zero and double quantum terms, respectively, in the ADRF experiment, and 254⫾2.9 ms for both the zero and double quantum terms in the JB experiment. The pulse sequences in Figs. 2共b兲and 2共c兲were used. The 90° pulse was 1.5␮s and a 4 ms hyperbolic secant pulse was used for the ADRF and ARRF共decaying and growing respectively兲. The error bars shown were obtained from the variance of the noise in a signal free region of the spectrum. The errors in T1Dmeasurement were observed from a standard propagation of errors, following a linear fit.

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zero, single, double, and even a little triple quantum x basis contributions are observed. As the spin system evolves under the secular dipolar Hamiltonian, the coherence distribution appears to expand in the x basis and contract in the z basis.

This is expected as the Hamiltonian is zero quantum selec- tive in the z basis, and contains both zero and double quan- tum selective terms in the x basis. The intensity of the triple quantum grows indicating the presence of three spin terms.

The presence of four spin terms have also been observed albeit at low intensity. These coherences then decay and at around 100␮s, the system appears to have reached a dipolar- ordered state.

Figures 8共a兲 and 8共b兲 show the FIDs measured for a single crystal of calcium fluoride at two orientations共a兲the 关100兴direction and共b兲the 关110兴direction. Figures 8共c兲and 8共d兲show the projection of the measured 2D data onto the z axis for the two orientations, and Figs. 8共e兲and 8共f兲show the corresponding projection on the x axis. When the crystal is aligned along the 关110兴 direction, the dipolar couplings be- tween the nearest-neighbor fluorine spins becomes small. In the z basis, the double quantum term is seen to decay while the single quantum term initially grows, reaches a maximum around 30 ␮s and then decays. The intensity of the zero quantum term remains constant, as expected. In the x basis the single quantum term is seen to peak around 20 ␮^{s and} the triple quantum term to peak around 45␮s, while the zero and double quantum terms are seen to oscillate before set- tling down to constant values. The dipolar-ordered state ap- pears to be created after approximately 100 ␮s. When the crystal is oriented along the关100兴direction the transient sig- nals decay much faster as expected, as the dipolar coupling between the nearest neighbor spins is the strongest. The dipolar-ordered state appears to be established within ap- proximately 60 ␮s. As the data plotted are absolute values, the curves do not go down to zero, and the limiting values are determined by the noise floor.

IV. CONCLUSION

We have been able to illustrate the nature of the dipolar- ordered state by simultaneously encoding the state in the x and z bases. We have also been able to show the dynamical evolution of the spin system to the state of quasiequilibrium represented by the dipolar-ordered state, following the appli- cation of a Jeener–Broekaert pulse pair. Such techniques open up the possibility of experimentally investigating the details of spin dynamics in a strongly dipolar coupled spin system.

ACKNOWLEDGMENTS

The authors thank P. Capperallo and G. S. Boutis for helpful discussions and the NSF and DARPA DSO for finan- cial support.

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