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We show that for nonresonant excitation of a quantum system, with an NMR spectrum consisting of two inhomogeneously broadened lines, by a combination of an extended and a short delta-shaped pulse, in the free-precession and spin-echo signals after the second pulse we observe the effect of doubling of the additional emission signals. We establish that the nature of this effect is due to zero beats arising when the variable oscillation frequency of the magnetic moments of the material matches the detuning of the pulse carrier frequency from the resonant frequencies of the spectral lines. Within the theory obtained, we propose a formula which we can use, if we know the times at which the additional emission signals arise, to very accurately determine the frequency shift of the lines in the complex spectrum of the material. The theoretical results agree well with experimental data on generation of multiple NMR signals in toluene, the spectrum of which consists of two non-overlapping lines.Key words: free precession and spin echo signals after the second pulse, inhomogeneous broadening, nuclear magnetic resonance, additional NMR emission signals, chemical shift, toluene.Introduction. In experimental observation of nonsteady-state coherent phenomena in condensed media, with the aim of determining the relaxation times for a quantum system, typically short (compared with the relaxation times) excitation pulses are used [1][2][3]. This circumstance automatically leads to small areas for the corresponding pulses, and makes it possible to neglect the effect of relaxation processes on the quantum system during the time of action of the pulses. In cases when the relaxation times are long, satisfying this condition is no longer critical and, as shown in [4,5], the use of long pulses (with large pulse areas) can reveal some fine features of the coherent response, for example the multiple structure of the nuclear spin echo in ferromagnets [6] and photon echo in dielectric crystals [7].Such conditions for excitation of coherent responses are interesting mainly due to the following. During excitation of a quantum system by an extended electromagnetic pulse, periodic variation in the populations of both levels occurs, leading to oscillations of the magnetic moments (nutational oscillations) with Larmor frequencies matching the carrier frequency of the external pulsed field [2,3]. In other words, during action of the pulse, the two-level atom absorbing and then emitting the energy of the variable magnetic field periodically makes a transition from the ground state to the excited state (and vice versa) with Rabi frequency ω 1 = γH 1 , where γ is the gyromagnetic ratio, H 1 is the amplitude of the external variable magnetic field. Consequently, the induced emission also will change with the same frequency. If the line is inhomogeneously broadened, then the magnetic moments with Larmor frequencies not matching the carrier frequency of the pulse will undergo nonresonant excitation and will not take part in the energy absorption process ...

We have obtained an analytical expression for nuclear precession and nuclear echo signals generated in magnetically ordered materials upon resonant excitation of the nuclear subsystem by two pulses of identical amplitude but different durations. We show that in a nuclear subsystem with inhomogeneous broadening of the spectroscopic transition and an inhomogeneous gain distribution, the two-pulse precession and echo signals are split into four and nine components respectively. We have analytically established a correlation between the macroscopic parameters of the components of the two-pulse signals (relative amplitudes, signal formation times) and the microscopic parameters of the magnetically ordered media (inhomogeneous half-width of the spectral line, half-width of the gain distribution function, average gain). The theoretically calculated formation times for the components of the nuclear precession and nuclear echo signals agree with the experimental data obtained for the alloy FeNiCo (70% Co).Key words: nuclear magnetic resonance, two-pulse precession and echo signals, magnetically ordered FeNiCo alloys, inhomogeneous broadening of a spectroscopic transition, inhomogeneous gain distribution.Introduction. The procedure of averaging the observable quantity (polarizability, magnetization) along the contour of the inhomogeneously broadened line is traditional in echo spectroscopy [1][2][3]. Also in some cases the need arises for additional averaging connected with the spread in the values of the transition operator matrix elements [2,3]. For example, in an ensemble of two-level resonant systems, the orientations of the transition dipole moments can be different [2]. In this case, the observable quantity must be averaged over the distribution function of the angles between the directions of the external variable electromagnetic field and the transition dipole moment.In ferromagnets and ferrimagnets, which are combined under the general name "magnetically ordered media", besides inhomogeneous broadening of the NMR line we need to take into account the inhomogeneous gain distribution [3,4]. The nature of the gain distribution involves the following. Magnetically ordered media are characterized by a branched domain structure with a certain direction of the magnetic moment in each domain. Due to shielding of the nucleus by the electron shell of the atom, the external variable magnetic field acts on the nucleus through the electronic subsystem via the hyperfine interaction [3,4]. Therefore the amplitude of the NMR signal will be determined to a large extent by the amplitude of the oscillations of the electronic moments. Such a situation is described by introducing the gain, bearing in mind that the amplitude of the external variable magnetic field at the nucleus in a ferromagnet is amplified compared with the case of the direct action of a variable field on the nucleus [4]. In the general case, there is a spread in the gain over the volume of the sample, since the amplitudes of the field are different within the domain...

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An analytical expression for a single-pulse nuclear echo signal in magnetically ordered materials has been obtained taking into account inhomogeneous broadening of the spectroscopic transition and non-uniform distribution of the enhancement factor. The signal has been shown to result from superimposing the nuclear magnetic moment oscillations of the same amplitude and phase that arise upon termination of a radio-frequency pulse. The cause for the effective suppression of oscillations of nuclear magnetic moments in the initial phase of the free precession signal has been determined analytically. The single-pulse echo signal amplitude has been found as a function of the external variable magnetic field strength, the exciting pulse duration, the inhomogeneous NMR line broadening, and the distribution width of the enhancement factor. The results have been compared with experimental data observed in a Co 2 MnSi ferromagnetic polycrystalline sample.Keywords: nuclear magnetic resonance, single-pulse ferromagnetic echo signal, Co2MnSi-like ferromagnetic polycrystalline substances, inhomogeneous broadening of spectroscopic transition, inhomogeneous distribution of enhancement factor.Introduction. Inhomogeneous broadening of spectroscopic transitions plays a decisive role in formulating the responses of a quantum system to a pulsed radio-frequency (RF) exciting field in non-steady-state magnetic spectroscopy [1]. The principal sources of inhomogeneous broadening are heterogeneities in the external controlling constant magnetic field, dipole-dipole couplings between spins of different nature, etc.[1]. Besides these sources, static scattering of internal stresses in addition to the presence of point or extended defects leading to inhomogeneous distribution of electric fields within the bulk sample must be considered in the solid state [1]. Therefore, phase inversion of magnetic spin system oscillations in an actual substance can occur through several inhomogeneous broadening mechanisms.Coherent NMR responses in magnetically ordered media are formed not only by inhomogeneous broadening of a spectroscopic transition but also by enhancement of the RF field on nuclei that occurs due to hyperfine coupling between electronic and nuclear subsystems [2,3]. The RF field enhancement factor in nuclei within domains and at their boundaries differs greatly in magnitude because of the presence of the domain structure in ferromagnetics [2, 3]. As a result, magnetic moments of nuclei will have different oscillation frequencies during the action of the variable magnetic field pulse [4]. The superposition of such magnetic moment oscillations generates a ferromagnetic singlepulse echo signal (FSE) in magnetically ordered media after the RF pulse field is switched off [5,6]. Although the theoretical treatment of this effect has been well studied [7][8][9], the role of the inhomogeneous distribution of the enhancement factor in formation of the FSE signal remains unclear. The proposed mechanism [10] of formation of a single-pulse echo in spin sy...

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