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60ISSN 2072-5981 doi: 10.26907/mrsej
aänetic Resonance in Solids
Electronic Journal
Volume 21 Issue 1 Paper No 19107 1-10 pages
2019
doi: 10.26907/mrsej-19107
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Established and published by Kazan University Endorsed by International Society of Magnetic Resonance (ISMAR) Registered by Russian Federation Committee on Press (#015140),
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Vadim Atsarkin (Institute of Radio Engineering and Electronics, Moscow) Yurij Bunkov (CNRS, Grenoble) Mikhail Eremin (KFU, Kazan) David Fushman (University of Maryland, College Park) Hugo Keller (University of Zürich,
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Yoshio Kitaoka (Osaka University,
Osaka)
Boris Malkin (KFU, Kazan) Alexander Shengelaya (Tbilisi State University, Tbilisi) Jörg Sichelschmidt (Max Planck Institute for Chemical Physics of Solids, Dresden) Haruhiko Suzuki (Kanazawa University, Kanazava) Murat Tagirov (KFU, Kazan) Dmitrii Tayurskii (KFU, Kazan) Valentine Zhikharev (KNRTU,
Kazan)
* In Kazan University the Electron Paramagnetic Resonance (EPR) was discovered by Zavoisky E.K. in 1944.
Contributions of different parts of dipole-dipole interactions to relaxation of multiple quantum NMR coherences in a single crystal of calcium fluorapatite^
G.A. Bochkin1, E.B. Fel'dman1, I.D. Lazarev1'2'*, S.G. Vasil'ev1 institute of Problems of Chemical Physics, 142432 Chernogolovka, Russia 2Lomonosov Moscow State University, Faculty of Physics and Chemistry,
119991 Moscow, Russia * E-mail: the.ilia.lazarev@gmail.com
(Received December 18, 2018; accepted December 19, 2018; published February 8, 2019)
Dipolar relaxation of multiple quantum (MQ) nuclear magnetic resonance (NMR) coherences on the evolution period of the MQ NMR experiment is investigated experimentally and theoretically in a single crystal of calcium fluorapatite. The theoretical approach is based on the zz model when the flip-flop part of the dipole-dipole interactions is not taken into account. The analytical results obtained in the zz model are compared with the numerical calculations for a chain consisting of 12 spins. The role of heteronuclear interactions of 19F and 31P nuclear spins in the dipolar relaxation of MQ NMR coherences is also investigated. A comparison of the experimental data and the theoretical predictions is discussed.
PACS: 76.30.-v, 68.65.-k.
Keywords: multiple quantum NMR, multiple quantum coherence, multiple quantum dynamics, dipolar relaxation, fermions, zz model, heteronuclear interactions.
1. Introduction
Multiple quantum (MQ) nuclear magnetic resonance (NMR) [1] not only creates MQ coherences but also allows to investigate their relaxation [2]. MQ NMR coherences emerge on the preparation period of the MQ NMR experiment, and their relaxation occurs on the evolution period [1]. Dipolar relaxation of MQ NMR coherences can be considered as the simplest model of deco-herence of many-qubit coherent clusters [3]. An investigation of decoherence in such clusters is very important for quantum devices (in particular, for quantum computers).
Relaxation of MQ NMR coherences in one-dimensional systems was studied with the second moments of the line shapes of MQ coherences of different orders [2,4]. Recently an investigation of the dipolar relaxation of MQ NMR coherences in spin chains was performed [5] using the zz model, in which the flip-flop part of the dipole-dipole interactions [6] is neglected on the evolution period of the MQ NMR experiment. However, the applicability of the zz model to the considered problem was not clear until now. In this article, we attempt to clarify this question using a numerical solution of the problem for systems with 6-18 spins.
The experimental investigation of MQ NMR dynamics and relaxation were performed on a single crystal of calcium fluorapatite Cas(PO4)3F [2]. The structure of this crystal contains parallel chains of 19F atoms (nuclear spin 1) arranged along the c-axis. The distance between neighboring chains is about three times larger than the distance between adjacent fluorine atoms in a chain [7,8]. As a result, the system can be considered as a set of isolated spin chains.
^This paper was selected at XX International Youth Scientific School "Actual problems of magnetic resonance and its application", Kazan, 24-29 September 2018. The MRSej Editors, Prof. M.S. Tagirov and Prof. V.A. Zhikharev, are responsible for the publication.
An isolated spin chain is the underlying model for the theoretical description of the spin dynamics on the preparation period of the MQ NMR experiment [1]. Every F atom in calcium fluorapatite is surrounded by three P atoms at the same distance. All these four atoms lie in the same plane perpendicular to the chains of F atoms and, accordingly, to the c-axis. The arrangement of atoms in fluorapatite is shown in Fig. 1 (the image was prepared using VESTA software [9]). The specially tailored multi-pulse sequence which irradiates 19F spins on the preparation period of the MQ NMR experiment [1] eliminates the het-eronuclear interactions of 31P (s = 1) and 19F spins. However, such averaging does not occur on the evolution period of the experiment [1] since the irradiation is absent. The role of the het-eronuclear interactions in the dipolar relaxation of MQ NMR coherences is not clear so far.
The main goal of the present paper is an examination of the applicability of the zz model to the study of the dipolar relaxation of MQ NMR coherences in one-dimensional systems and the investigation of the contribution of the heteronu-clear interactions to the dipolar relaxation of MQ NMR coherences in the MQ NMR experiments in
Figure 1. Part of the crystal structure of the fluorapatite (Cas(PO4)3F) showing surroundings of the fluorine atoms. Oxygen atoms have been removed for clarity. Fluorine atoms are equally spaced (rFF = 344 pm) and arranged in columns along the c-axis of the crystal. Every fluorine atom is surrounded by three equidistant phosphorus atoms (rFP = 367.3 pm) situated in vertexes of equilateral triangles in the planes perpendicular to the c-axis.
a single crystal of calcium fluorapatite.
The paper is organised as follows. The brief description of the theory [5] of the dipolar relaxation of MQ NMR coherences in one-dimensional systems is given in Section 2. The het-eronuclear contribution to the dipolar relaxation of MQ NMR coherences of the zeroth and second orders is investigated in Section 3. A comparison of the theoretical results with the obtained experimental data for a single crystal of calcium fluorapatite is presented in Section 4. In the conclusions (Section 5), we briefly list our main results. In the Appendix we give justification of an application of the zz model [5] for a description of the dipolar relaxation of the MQ NMR coherence of the zeroth order.
2. Dipolar relaxation of MQ NMR coherences in one-dimensional systems
The MQ NMR experiment [1] consists of four successive periods: preparation (t), free evolution (t), mixing (t), and detection. The sequence of radio-frequency pulses, irradiating the spin system on the preparation period, consists of a basic cycle with eight resonance (for 19F nuclei) n pulses (Fig. 2). The irradiation leads to the emergence of even order MQ NMR coherences. The mixing period repeats the preparation period except for the | phase shift of all pulses in order to convert MQ coherences into Z-magnetization, which is further transformed into observable transverse magnetization by the | pulse on the detection period. The periods of the MQ NMR experiment are repeated many times with the phase increment of the radio-frequency pulses, irradiating the spin system on the preparation period, at each repetition. The Fourier
Figure 2. Scheme of the MQ NMR experiment; t is the duration of the preparation and mixing periods; t is the duration of the free evolution period. The basic cycle of the multipulse sequence, consisting of eight 2-pulses with the duration tp separated by delays A and A' = 2A + tp is repeated m times for obtaining the necessary time t = 12m(A + tp); m is a natural number.
transform of the transverse magnetization with respect to the phase increment of the radio-frequency pulses on the preparation period yields the intensities of the MQ coherences as a function of their orders. Exact solutions for MQ NMR dynamics of one-dimensional systems demonstrate [10, 11] that, with an initial equilibrium state, only zero and double quantum coherences are produced. The density matrix of the system p(t) at the time moment t of the preparation period can be presented as follows [10,11]
p(t ) = P0 (t )+ P2 (t )+ P-2 (t ), (1)
where p(t) (i = 0, -2, 2) describes the MQ NMR coherence of order i. For a spin chain with an even number N of spins
Po (t ) =
- ^ e-2iDTcos k ( ^ (-1) ¥ 2+'-2 sin(kl) sin(k1/)If
+ k V l = 1,3,...,N-1
l' = 1,3,...,N-1
E (-1)^21+1'-2 sin(fcZ) sin(k1/)If
l=2,4,...,N l' =2,4 ,...,N
P2(t) =
- ^ e-2iDTcosk ^ (-1) ii^21+1'-2 sin(kl) sin(k1/)If
+ k l = 1,3,...,N-1
l' =2,4,...,N
P-2(t) =
- N+T E e2iDT COS k E (-1)^21+1'-2 sin(fci)sin(fc1/)If ...If-! I- If ...If- I-, (4)
+ k l = 2,4,...,N
l' = 1,3,...,N-1
. . . If-11+11 . . . If-1I- + ...If-! I- If ...If-i I^ , (2)
...If-i I+If ...If-i I+, (3)
where k = N+1 (n = 1,2,..., N), D is the dipolar coupling constant between nearest neighbors in the spin chain, im is the z-projection of the spin angular momentum operator and /+, im are the raising and lowering operators of spin m.
Using equations (2), (3), (4), one can obtain very simple expressions for the experimentally observed intensities of MQ NMR coherences of the zeroth (G0(t)) and plus/minus second (G±2(t)) orders [11]:
Go(t) = N E cos2 (2Dt cos(k)), (5)
k
G±2(t) = 2N E sin2 (2Dtcos(k)). (6)
2N k
Theoretical expressions are in a good agreement with the experimental data obtained for a single crystal of calcium fluorapatite [12]. Thus, we use the density matrix p(t) at the end of the preparation period as the initial state for the relaxation process which occurs the evolution period of the MQ NMR experiment.
3. Dipolar relaxation of MQ NMR coherences of the zeroth and second orders in heteronuclear systems
The Fig. 3 shows the interaction constants of the fluorine spin in the chain with its environment at different angles. It is seen that the main contribution to the dipole-dipole relaxation will give the interaction between the next nearest neighbors in the chain and the interaction of the spin of fluorine with the nearest three phosphorus atoms. A significant contribution of heteronuclear interactions is observed in the vicinity of the magic angle for FF interaction in the chain. Relaxation of MQ NMR coherences is caused by the secular (with respect to the external magnetic field
18000 16000 14000 12000 t 10000
w
O" 8000 6000 4000 2000 0
a 0 is an angle between c-axis of crystal and the external magnetic field
Figure 3. Homo- and heteronuclear contributions to the dipolar coupling constant of fluorine spins at different orientation of the crystal relatively to the external magnetic field. DFF is the dipolar interaction constant between adjacent fluorine spins in the chain. JFP is the mean value of the dipolar interaction constant between the fluorine spin in the chain and the nearest phosphorus spins. JFP2 is the average value of the constant of the dipole interaction between the spin of fluorine in the chain and the spin of the phosphorus on the second order of the proximity. DFF2 and DFF3 are the mean value of the dipole interaction constant of the spin of fluorine in the chain with the surrounding fluorines of the second and third order of proximity, respectively.
0 9 18 27 36 45 54 63 72 81 90
directed along the z axis) dipole-dipole interactions (DDI). For a linear chain of 19F spins, when every 19F spin interacts with three 31P spins (s = 2), the DDI Hamiltonian is given by
H
dz
EAj
i<j
3
+2 E E zz '
i j=1
(7)
where Dj is the coupling constant of fluorine spins i and j, Di,i+1 = D; Jj (i = 1,...,N; j = 1,2,3) is the dipolar coupling constant of fluorine spin i and phosphorus spin j, and Sj is the z-projection of the angular spin moment of phosphorus spin j (j = 1,2,3) interacting with fluorine spin i (i = 1,2,..., N). In order to simplify an investigation of the relaxation process with the Hamiltonian (7) we restrict ourselves to the zz part of (7), neglecting its flip-flop part:
3
Hzz = 2 £ D jIf If + 2 ^ ^ JjIfSf. (8)
i<j i j=1
In the Appendix we show that this procedure is justified in the approximation of the nearest neighbor interactions and N ^ 1 for calculations of the intensity of MQ NMR coherence of the zeroth order.
Since we have a system consisting of 4N spins (N fluorines and 3N phosphoruses), the intensity fo(t, t) of the MQ NMR coherence of the zeroth order during the evolution period at the time moment t is
Tr [e-iHdZipo(r)eiHdz4po(r)]
fq(t'î) =
Tr
N
(1z)2 ® Ep
(9)
where = ^ If is the total z-projection of the angular momentum of F spins, EP is the i=1
unit operator of dimension 23N acting on 31P spins, and the trace of the tensor product [13] of operators If and EP is
Tr [42 <g> EP = TrF(/z2) Trp(Ep) = N2n-223N = N24N_2.
(10)
Indices F and P means the subspaces in which the traces should be calculated.
Analogously, the intensities of the MQ NMR coherences of the plus/minus second order on the evolution period at the time moment t are given by
f2(t't) = f_2 (t, t) =
Tr [e_iHdzîP2(t)eiHdz1 p_2(t)]
Tr
(1z)2 ® Ep
(11)
The following commutation relations are useful for further calculations: _i E DnfcinIZt , i £ Dnfc/,zift £ Dmni/,
e
1 ± e n = k
-'m °
= e n=(m) (12)
e n=1 zme n=1 =e n=1 Zm. (13)
Using the commutation relations (12), (13), the expression (9) for the intensity of the MQ NMR
coherence of the zeroth order, and Eq. (2) for pq(t), one can find that
6
4 II cos(Jmt) Fo(T'f)= m=A, x
2i E Jmn /m i 2i E ^mn1)^ Smn ^ ~F2i E JmntSmn i
=i Zm e n=i = e n=i
N (N + 1)2 / \
^ n cos [(Al' - Ai)i] ( X) sin(kl) sin(kZ')e
\i = i' /
— 2iDr cos fc
(14)
or
M'=2,4,...,N
Taking into account the heteronuclear interactions leads to emergence of a factor
6
H cos(Jmt) in the expression (14) for the intensity of the MQ NMR coherence of the ze-
m=1
roth order. The reason is as follows. The density matrix p0(r) contains one lowering and one raising operators for two different 19F spins. Every 19F spin interacts with "its own" three 31P
spins. By taking the trace in the subspace of spin Sj (j = 1,2, 3) we obtain the multiplier
6
cos(Jjt). The total contribution contains 2 x 3 such multipliers, i.e. H cos(Jmt).
m=1
Analogously, using the commutation relations (12), (13), the expression (11) for the intensity of the MQ NMR coherence of the plus/minus second order, and Eqs. (3), (4) for p2(r), p-2(r), one can find that
6
4 II cos(Jmt)
F±2(T,t) = - N=(N +1)2 X , ,
( ) I w \2
E H cos [(Da' + Dii)t] E sin(kl) sin(fc1')e-2iDr cos k . (15)
l=1,3,...,N-1 l'=2,4,...,N
i=l
\i=l' )
k
4. A comparison of the theoretical and experimental data
The experiments were performed on a Bruker Avance III spectrometer with the static magnetic field of 9.4 T (the corresponding frequency on 19F nuclear spins is 376.6 MHz). The DDI coupling constant D between neighboring 19F spins inside a chain in a single crystal of calcium fluorapatite is 16.4 x 103 s-1 for the distance of 344 pm between them [2] and chains oriented along the external magnetic field. The heteronuclear DDI coupling constant J between spins of 19F and 31P nuclei for the distance rFP = 367.3 pm between them [7] is
YFYpft I1 - 3cos2(m3)sin2(a)\
Jm = 1^1-2-) (m '2'...'6)' (16)
where yf, Yp are the gyromagnetic ratios of nuclei 19F, and 31P. In Eq. (16) a is an angle between vector connecting nuclei 19F and 31P and the external magnetic field.
In Fig. 4 the relaxation decay of the experimental intensities of the MQ NMR coherence of the zeroth order on the evolution period for the duration of the preparation period t and the theoretical result calculated using Eq. (14) are presented for N = 100. For comparison, the theoretical results obtained without taking into account of the heteronuclear interactions are also presented. One can see that accounting for the heteronuclear interactions significantly
improves the agreement of the theoretical results with the experimental data.
(n)
We investigated numerically the dependence of the second moment M2 of the line shape of the MQ NMR coherence of order n (n = 0, ±2) on the number of spins for both the zz model (M"2n)(Hzz)) and the interaction (M"2n)(Hdz)) in homonuclear systems. In Fig. 5 the ratios
of M:L)(Hzz) (n = 0, ±2) are presented for 6 ^ 18 spins. One can see that the ratio M2<0}Hzz) M(2n)(Hdz) v ' 7 p P M2(0)(Hdz)
increases with the growth of the number N of spins. We proved in Appendix that the flipflop part of does not contribute to the M(0) in the approximation of the nearest neighbor
interactions for N ^ 1. At the same time the flip-flop part of the DDI is significant for the second (±2)
moment M^ ) of the line shape of the MQ NMR coherence of the plus/minus second order.
0.8 0.6 0.4 0.2 0
1
0.8 0.6 0.4 0.2 0
* EXPERIMENT
--FF only
-FF and FP
:/2 = = 204 us
0 50
t, us
0
100
= 0 = = 201.6 us
50 t, us
100
1
0.8 0.6
0.2 0
1
0.8 0.6 0.4 0.2 0
:/2 = = 306 us
0 50
t, us
100
= 0 = = 252 us
0
50 t, us
* *
100
Figure 4. The experimental and theoretical intensities of the MQ NMR coherence of the zeroth order on the evolution period of the MQ NMR experiment for N = 100 spins in the chain and the duration of the preparation period (t) specified above axes. The solid line is the theoretical results obtained with Eq. (14); the dashed line is the theoretical results obtained without accounting for the heteronuclear interactions [5]. The points are the experimental values of the intensity of the MQ NMR coherence of the zeroth order for a quasi-one-dimensional chain of 19F nuclear spins in a single crystal of calcium fluorapatite.
0.8 0.75 0.7
tq
s cj
^ 0.65 0.6 0.55
o n = 0
* n = 2
* *
10
15
N
20
M (n)(H )
Figure 5. The dependence of the ratio of the second moments M2,n)^H^ for the line shapes of the MQ NMR coherences of orders n = 0, 2 on the number of spins for the system with Hzz and Hdz interactions
on the spin chain length (circles denote the ratio
M(0)(Hzz )
M(2)(Hzz ) >
—,
(0)(Hdz )
stars denote the ratio —2
M
22)(Hdz )'
DT = 2.
5
Thus, the possible sources of discrepancy between the theoretical predictions and experimental data for the intensity of the MQ NMR coherence of the zeroth order are various kinds of experimental imperfections.
In Fig. 6 the relaxation decay of the experimental intensity of the MQ NMR coherence of the second order on the evolution period for the duration of the preparation period t and the theoretical results calculated using Eq. (15) for N = 100 are presented. The theoretical results obtained without the contribution of the heteronuclear interactions are also presented. Although the agreement between the experimental and theoretical results is improved when the heteronuclear contributions are taken into account, discrepancy between the theoretical and experimental data is bigger than in the previous case. It is possible that the discrepancy is mostly due to neglecting the flip-flop part of the DDI in the zz model. Indeed, one can see from Fig. 5 that the flip-flop part of the DDI is getting more significant with increasing number of spins. We performed numerical calculation of the intensity of the MQ NMR coherence of the second order for a system consisting of 12 spins. Fig. 6 demonstrates that even for this system with a small number of spins the numerical results are in better agreement with the experimental data than the predictions of the zz model.
* EXPERIMENT
--FF only
-FF and FP
0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
:/2 = = 204 us
:/2 = = 306 us
0 50
t, us
0
100
= 0 = = 201.6 us
50 t, us
100
0.25
0.2 0.15 0.1 0.05 0
0.35
0.3
0.25**,
0.2
0.15
0.1
0.05
0 0
0 50
t, us
, = 0 = = 252 us
100
50 t, us
100
Figure 6. The experimental and theoretical intensities of the MQ NMR coherence of the second order on the evolution period of the MQ NMR experiment for N = 100 spins in the chain and the duration of the preparation period specified above axis. The solid line is the theoretical results obtained with Eq. (15); the dased-line is the theoretical results obtained without an account of the heteronuclear interactions [5]. The points are experimental values of the intensity of the MQ NMR coherence of the second order for a quasi-one-dimensional chain of 19F nuclear spins in a single crystal of calcium fluorapatite.
5. Conclusion
We investigated an influence of the heteronuclear interactions on the dipolar relaxation of MQ NMR coherences of the zeroth and plus/minus second orders in one-dimensional systems. Although the heteronuclear interactions do not affect the dynamics of MQ NMR coherences on the preparation period due to their averaging by the irradiating multipulse sequence [1], they are significant on the evolution period where the irradiation is absent. We also investigated the accuracy of the zz model for studying the dipolar relaxation of MQ NMR coherences in one-dimensional systems.
Accounting for the heteronuclear interactions and the flip-flop part of the DDI improves the agreement of the theoretical predictions with the experimental data and allows us to estimate the experimental errors in the complex MQ NMR experiments.
Appendix. Independence of the second moment M^ of the line shape of MQ NMR coherence of order 0 on the flip-flop part
We show that the second moment of the line shape of the MQ NMR coherence of the zeroth order [4] for the homonuclear system
M (0) = Tr {[po(T ),Hdz ][Hdz ,Po(T )]}
M = N2N-2Fo(T, 0) (17)
is determined only by the zz part Hzz of Hdz in the approximation of the nearest neighbors for N » 1. Indeed, the density matrix p0(t) for N » 1 is [10]
Po(t) = 2 Ecos[2Dtsin(k)](1 - 2a+ak), (18)
k
where a+, ak are the fermion operators. The flip-flop part Hff of Hdz in the approximation of the nearest neighbors
Hff = - f Ejj+i+'-j+i) (19)
j
can be expressed via the fermion operators using the Jordan-Wigner transformation [14] as
Hff = -d E cos(k) a+ak. (20)
k
One can see from Eqs. (18), (20) that
[Hff, po(t)]=0. (21)
Since Hdz = Hzz + Hff, the assertion is proved.
Acknowledgments
This work is supported by the Russian Foundation for Basic Research (grant No. 16-03-00056) and the Program of the Presidium of RAS No. 5 "Electron Spin Resonance, Spin-Dependent Electron Effects and Spin Technologies'! The experimental measurements are carried out using the equipment of "Analytical center of collective use of IPCP RAS'.
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