Multifrequency resonances in multiple-pulse NMR on a spin-1Õ2 system
G. B. Furman, S. D. Goren, V. M. Meerovich, and V. L. Sokolovsky
Department of Physics, Ben-Gurion University, Be’er-Sheva, Israel
G. E. Kibrik and A. Yu. Polyakov
Department of Physics, Perm State University, Perm, Russia
Received 30 May 2003; published 18 December 2003
We have observed multifrequency resonances in a system with a spin 1/2 located in dc magnetic field and
irradiated simultaneously by a multiple-pulse radio frequency sequence and a low-frequency field swept in the
range 0– 80 kHz. The used excitation scheme allowed us to measure the effective field of the radio frequency
sequence. A peculiarity of this scheme is that the intensity of the resonance lines decreases slowly with the
mode number. The theoretical description of the effect is presented using both the rotating frame approximation
and the Floquet theory. Both approaches give identical results at the calculation of the resonance frequencies,
transition probabilities, and shifts of resonance frequency. The calculated magnetization vs the frequency of the
low-frequency field agrees well with the obtained experimental data. The multifrequency spectra give a way
for studying slow atomic motion in solids.
DOI: 10.1103/PhysRevA.68.063410 PACS numbers: 33.25.+k
I. INTRODUCTION
One of the most effective and promising high-resolution
nuclear magnetic resonance NMR techniques for the study
of solids is a multiple-pulse radio frequency RF action
1,2. The multiple-pulse methods allow one to remove dipo-
lar broadening from a resonance line in solids thus, increas-
ing by several orders the sensitivity of the NMR spectros-
copy in the study of weak interaction. These methods are
very effective in the study of the spin-lattice relaxation pro-
cesses due to a slow atomic motion. Usually the theoretical
description of multiple-pulse experiments is based on the
construction of the effective time-independent Hamiltonian
by using the conditions for periodicity and cyclicity of the
pulsed action 1,2. Then the dynamics of a spin system sub-
jected by pulsed RF fields is presented in an equivalent form
as the motion of nuclear spins in a constant effective field H
e
3. The magnitude and direction of this effective field are
determined by parameters of the multiple-pulse sequence. An
experimental measurement of the value of the effective field
is important for the confirmation of this theoretical model.
It is reasonable to suggest that an additional field with an
angular frequency close to
e
= H
e
should cause reso-
nance absorption of energy ( is the gyromagnetic ratio of
nuclei. Spin-echo signals observed between RF pulse se-
quence would allow us to determine H
e
as well as to obtain
the information on slow atomic motion that is not available
from the traditional high-frequency NMR.
With this in mind, we have studied experimentally reso-
nance transitions in the nuclear-spin system subjected by a
simultaneous action of a multiple-pulse RF sequence and an
additional low-frequency LF field with an angular fre-
quency . The results of our experiments described in the
following section have shown that resonance transitions were
observed not only at the frequency close to
0
=
e
, but also
at frequencies close to
n
given by the expression
n
=|
e
2 n / t
c
| , n =1,2, . . . 1
where t
c
is the period of the multiple-pulse RF sequence.
Multiple resonance modes of higher orders have been de-
tected by microwave spectroscopy 4, molecular beam tech-
nique 5, optical pumping 6, and observed previously in
NMR experiments 7–11. However, the amplitude of these
resonances decreased abruptly with the mode order of the
resonance. As distinct from this, the amplitude of the reso-
nances observed in our experiments decreased slowly and the
resonances of higher orders were well observable.
Because the nuclear-spin system possesses a set of the
resonance frequencies, the relaxation measurements per-
formed on one resonance frequency
n
can give the infor-
mation on oscillations of atoms on all the frequencies from
the spectrum determined by Eq. 1. It moves us to compre-
hensive experimental and theoretical study of this system.
The theoretical treatment of NMR phenomena is usually
based on three approaches: i a semiclassical mathematical
approach 12; ii a second quantization method 13,14;
and iii the Floquet theory 15.
The semiclassical mathematical approach 12, where the
field is considered as a classical system and the atomic sys-
tem as a quantum one, has allowed one to explain a series of
experimentally observed phenomena. This approach is quite
natural if to take into account that the average number of
photons in a mode of the periodic field is extremely great.
The main method used in the framework of the semiclassical
approach is the so-called ‘‘rotating frame approximation,’’
keeping exactly just the terms that are resonant. The remain-
ing nonresonant terms are considered as a perturbation.
Intrinsic inconsistency of the semiclassical approach is
obviated in the framework of the secondary quantization
method 13,14. Treating the RF field as photons, the evolu-
tion of the united system ‘‘atom+field’’ so called ‘‘dressed’’
atom is described by the Hamiltonian which is independent
of time, and its investigation turns out simpler than solving
the Schro
¨
dinger equation with the time-dependent Hamil-
tonian. With the time-independent Hamiltonian, one can de-
fine energy levels of the physical system. Each of these lev-
PHYSICAL REVIEW A 68, 063410 2003
1050-2947/2003/686/0634109/$20.00 ©2003 The American Physical Society 68 063410-1