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8Scientific Notes of Taurida National V. I. Vernadsky University
Series : Physics and Mathematics Sciences. Volume 27 (66). 2014. No. 2. P. 65-69
UDK 537.9
THREE-PULSE NMR ECHO IN CoCh^O Ryabushkin D. S., Sapiga A. V., Solovyev A. V.
Taurida National V. I. Vernadsky University, 4 Vernadsky Ave., Simferopol 95007, Crimea, Russia E-mail: druabushkinia crimea. edu, sapisa av@mail.ru
The features of three-pulse echo formed by series 90°x — t1 — 90°x — t2 — 90°x — t are investigated - shape of the signal, dependence of the response amplitude on time, the time of the echo maximum formation, the temperature dependence of the echo. It is shown good agreement of theoretical and experimental results for hydrogen containing systems with dipole-dipole interaction and internal molecular mobility. Crystals of CoCh^HzO were chosen as the experimental sample.
Keywords: nuclear magnetic resonance, three-pulse echo, temperature dependence. PACS: 75.30. ±m
INTRODUCTION
Three-pulse series 90°x — t1 — 90°x — t2 — 90°x — t is used in practice of nuclear magnetic resonance (NMR) for restoration of the initial part of free induction decay (FID) hidden by "dead time". As is known just this part of FID defines moments of NMR line shape. Up to present properties of the three-pulse echo itself were not investigated.
In given paper the system was considered to have Gaussian distribution for description the random fields on resonating nuclei and Markov's model of mobility. Such a model hasn't universal character but allows to cover a lot of samples. In given case tiny pink crystals of CoCl2-6H2O were chosen for experiment.
1. THEORY
The series 90°x — t1 — 90°x — t2 — 90°x — t is represented at Fig. 1. Time intervals between the first and the second pulses is ti , between the second and the third ones is ¿2. Time after the last pulse is indicated as t. The pulses act like rotation operators.
echo
t2
x
x
x
0
t
Fig. 1. Formation of echo in series 90°x — — 90°x — t2 — 90°x — t.
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For evaluation the echo shape the theoretical method based on formalism of density matrix was used [1-4]. In accordance with that the signal of the echo is written as
Tr(p(t) • Iy)
V (t) =-^, (1)
Tr( I 2)
y
where p(t) is the operator of density matrix at the moment t, Iy is the operator of y-component of the sample's spin in rotating system of axes.
The action of each 90°x pulse has the result of exchange the components of total spin accordingly the rule: Ix ^ Ix, Iy ^ - Iz, Iz ^ Iy. The operator of density matrix is evaluated from Liouville equation:
|P = iP, H ], (2)
ot
where hH is the Hamiltonian of dipole-dipole interaction that depends on time because of internal molecular mobility.
Choosing Gaussian distribution for description the random fields on resonating nuclei and Markov's model of mobility, have
/ \ - It1"|
(a(t')a(t")) = M2 + AM2 • exp(-J-1). (3)
Tc
Here M2 is the second moment of a NMR line narrowed by mobility, AM2 is the difference of the second moments of absorption lines in rigid and fast-moving systems, Tc is the time of correlation (average time of standing in given lattice configuration). Evaluations give the next final result:
V(t) = exp J -1My (t - (ty - O)21 • exp(-AMT (t + tl +12 - 3 + exp(--) +
I 2 J Tc Tc
exp(--tL) + exp(-+ (exp(-t +tl +12) - exp(- h+L)) • (exp (t-L+t2) + (4)
Tc Tc Tc Tc Tc
1 - 2 • exp(-+ (exp(-^) - exp(-• (exp( A.) -1)))
T T T T
c c c c
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2. EXPERIMENT
Signals of the echo formed in CoCh^^O are shown Fig. 2. Easy to see good agreement of the experiment and theory.
Amplitude, r.u.
1,0- A
0.8- \ I tl
0,6- ft \ \\
0.4- \\
0,2- \
0.0- • I
Experiment (t1=15 us, t2=20 us)
Theory
1-1-1-1-1-'-n-1-r
0 20 40 60 SO 100 120 140 160 180 200
t, Us
Amplitude, r.u 1,00,80,6 -0,4 -0,2-
0.0
Experiment (t1=15 us, t2=35 us)
Theory
50
100
150
200 t, us
Fig. 2. Signals of the echo formed in CoCh^^O. 67
Temperature dependence of the echo amplitude is shown at Fig. 3.
Amplitude, r.u.
1 .0
0 .8
a)
0.2
0 .6
0 .4
0
0
10
20
t/Tc
Fig. 3. Temperature dependence of the echo amplitude. AM2 Tc2 = 0.5, 1.0, 3.0 for lines a), b), c) correspondingly, t = ti = t2/2.
The shape of temperature dependence of the echo amplitude is typical for various pulse sequences and may be explained by the next way.
While the lattice is rigid any magnetic moment is bound to a definite point with given local magnetic field. As a result, synchronism in moving of the moments is kept during action of pulses and between applied pulses. In such case conditions for forming the echo are the most favorable. As soon as the sample is being heated, the moments begin to visit points with different local fields, synchronism breaks and amplitude of the echo decreases. At enough high temperature the amplitude begins to grow again because movements of the magnetic moments are so intensive that they don't have time to react for changing the position in the lattice. All the moments feel averaged field and it means that conditions for forming the echo appear again.
CONCLUSIONS
The formula (4) for the series 90°x — ti — 90°x — t2 — 90°x — t allows to describe responses of many particle systems with Gaussian distribution of the random fields on nuclei and Markov's process of mobility. Good quantitative accordance of theory and
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experiment is observed for time dependence of the echo. The temperature dependence of the echo contains typical minimum.
References
1. J. S. Waugh, Spin echoes and Thermodynamics. In: Pulsed Magnetic Resonance : NMR, ESR and Optics, (Clarendon Press, Oxford, 1992).
2. D. Fenzke, W. Rinck, H. Schneider, I Specialized Colloque AMPERE, 156 (1973)
3. N. Sergeev, Acta Physica, Zeszyty naukowe US, No. 9, 23 (1998).
4. D. S. Ryabushkin, A. V. Sapiga, Ye. S. Redka, Scientific Notes of Taurida National V. I. Vernadsky University, Ser. Physics and Mathematics Sciences 26 (65), No. 2, 125 (2013).
Рябушюн Д. С. Трех1мпульсное луна ЯМР у СоСЬ'бШО / Д. С. Рябушюн, А. В. Canira, А. В. Солов'йов // Вчеш записки Тавршського нацюнального уншерситету iMeHi В. I. Вернадського. Серiя : Фiзико-математичнi науки. - 2014 - Т. 27 (66), № 2. - С. 65-69.
Дослщжено особливост трехiмпульсного луна-сигналу сформованого cepieKi 90°х — ^ — 90°х — t2 — 90°х — t - форма сигналу, залежшсть ампттуди вщгуку вщ часу, момент формування максимуму луна-сигналу, температурна залежшсть луна-сигналу. Показано добра яюсна i кшьюсна згода теоретичних i експериментальних результат^ для водневомгсних систем з диполь-дипольною взаeмодieю i внутршньо! молекулярно! рухливютю. В якостi експериментального зразка обраш кристали C0CI26H2O.
KnwHoei слова: ядерний магнiтний резонанс, трехiмпульсний луна-сигнал, температурна залежшсть.
Рябушкин Д. С. Трехимпульсное эхо ЯМР в CoCh-бШО / Д. С. Рябушкин, А. В. Сапига, А. В. Соловьев // Ученые записки Таврического национального университета имени В. И. Вернадского. Серия : Физико-математические науки. - 2014. - Т. 27 (66), № 2. - С. 65-69. Исследованы особенности трехимпульсного эха, формируемого серией 90°х — ^ — 90°х — t2 — 90°х — t -форма сигнала, зависимость амплитуды отклика от времени, момент формирования максимума эха, температурная зависимость эха. Показано хорошее качественное и количественное согласие теоретических и экспериментальных результатов для водородосодержащих систем с диполь-дипольным взаимодействием и внутренней молекулярной подвижностью. В качестве экспериментального образца выбраны кристаллы C0CI26H2O.
Ключевые слова: ядерный магнитный резонанс, трехимпульсное эхо, температурная зависимость.
Список литературы
1. Waugh J. S. Spin echoes and Thermodynamics. In: Pulsed Magnetic Resonance : NMR, ESR and Optics / J. S. Waugh ; ed. D. M. S. Bagguely. - Oxford : Clarendon Press, 1992. - 174 p.
2. Fenzke D. Measurement of the second moment in NMR using instationary methods / D. Fenzke, W. Rinck, H. Schneider // I Specialized Colloque AMPERE. - 1973. - P. 156-158.
3. Sergeev N. Magic echoes in NMR of solids with thermal motion / N. Sergeev // Acta Physica, Zeszyty naukowe US. - 1998. - No 9 - P. 23-34.
4. Ryabushkin D.S. NMR magic echo in natrolite / D. S. Ryabushkin, A. V. Sapiga, Ye. S. Redka // Scientific Notes of Taurida National V. I. Vernadsky University, Series : Physics and Mathematics Sciences - 2013. - Vol. 26 (65), No 2 - P. 125.
Received 14 September 2014.
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