Dislocation motions are studied in NaCl crystals exposed to an EPR impact under crossed ultralow magnetic fields, the Earth field BEarth ∼ 50 µT, and the AC pumping field of the amplitude 2.5 µT in the frequency band of 6 kHz to 2.1 MHz. Mean dislocation paths form a spectrum of multiple peaks at definite resonance frequencies. Dislocation motions are supposed to be caused by spin-dependent transformation of impurity centers, which, in turn, provide depinning of dislocations and relaxation of their structure. The observed spectrum is attributed to specific features of the hyperfine EPR at the applied field BEarth small as compared with the crystalline local fields Bloc created by nuclei of Cl ligands surrounding the Ca pinning centers. The developed theory well describes peak positions in the observed frequency spectrum.
I. INTRODUCTION
Manifestations of magnetic influence in the plasticity of solids are not at all reduced to trivial ponderomotive effects. Being typical for ferromagnetic media, they are ordinarily very weak in nonmagnetic materials where the magnetic field causes rather low energy impact. For instance, the field 1 T in such media creates the magnetic energy ∼10−3 eV per unit cell.1 Meanwhile, in this case, there is an alternative much more efficient phenomenon—the so-called magnetoplastic effect (MPE) based on the completely different mechanism. The magnetic field initiates spin evolution in paramagnetic impurity centers to the metastable state eliminating the spin exclusion of definite electron transitions. The latter, in turn, initiates some transformations of the center structure. Here, there is a close similarity with the mechanism of magnetic influence on chemical reactions and other spin-dependent effects.1–5
In solids, such magneto induced transformation of impurities may reduce their pinning interaction with dislocations. As a result, dislocations become more mobile and could move under internal stresses from other dislocations: the system relaxes to a configuration with less energy. Such particular process was first observed6 on NaCl crystals. The samples with freshly introduced dislocations of known positions indicated by preliminary etching of the crystal surface were exposed to the static magnetic field B for some time t. After exposure, the repeated etching showed that dislocations moved on macroscopic distances l ∝ B2t ∼10–100 µm. There was no loading or other impact on the samples apart from their magnetic treatment.
This was practically the first observation of MPE. Later, the phenomenon was studied at many independent laboratories in different crystals (dielectrics, ferroelectrics, semiconductors, and nonmagnetic metals) for individual dislocations and on a macroplastic level with measuring active deformation or loading, creep, internal friction, and so on (see review articles7–11). In ionic dielectrics (e.g., NaCl), a strong acceleration of MPE intensity was found11–13 (up to 102–103 times) when the magnetic exposure was accompanied by the electric field, even of rather low magnitude (∼1–10 kV/m). The electric exposure without magnetic impact caused no effect.
Apart from in situ effects when the process occurs only under the magnetic field, there were found memory effects when the preliminary magnetic exposure initiates post-factum changes. In this case, the exposure transforms the state of some point defects that perturb the equilibrium of the entire defect system and cause its diffusive self-organization. Thus, the crystal properties vary after some delay and after mounting to some maximum changes usually tend back to the initial state. Typically, the measurable property was a microhardness of the material.14–17 Later on, it was widely recognized that magnetic exposure transforms not only mechanical characteristics but also other physical properties sensitive to the state of point defect subsystem (e.g., electric properties of ferroelectrics and ferroelastics,18–22 conductance of semiconductors,23 etc.).
Theoretically, spin effects in defect reactions were studied in frames of different approaches including both formal quantum mechanical analysis24,25 (often in terms of being rather far from results of observations) and formulating simple physical models26–28 for interpretation of experimental data and even their predictions. For instance, in Ref. 27, the authors have predicted the resonance form of MPE in the EPR conditions. The processes of spin evolution in impurity centers under magnetic fields were also considered in Refs. 29–31 with the very nontrivial physical interpretation of magnet induced dislocation depinning from such centers. Kinetics of dislocation motions in the MPE conditions were investigated by analytical study combined with computer simulation.32
Local magnetic fields are produced by nuclear spins of the paramagnetic center itself and/or of atoms (ligands) surrounding the center. The resonance frequency (2) is independent of the magnitude of Bloc being strongly sensitive to its direction (3). According to Ref. 47, directions of the field Bloc in crystals correlate with their elements of symmetry.
In all three crystals studied in Refs. 46 and 47, we observed single resonance peaks characterized by formula (2) with the factor g ≈ 2. However, in the classical EPR effect, the local fields Bloc ≪ B ordinarily cause the so-called hyperfine splitting of resonance lines. In some crystals, the number of such lines might be rather large due to ligands involvement.48,49 In particular, it occurs in crystals SrF2(Ni)50 (due to the ligands of the nuclei of F atoms around impurities Ni).
In a recent paper,51 the discussed EPR spectrum under the Earth's magnetic field was studied by measuring the microhardness of crystal NaCl(Ni) in the range of frequencies 1.1–2.2 MHz. The spectrum contains the two groups of resonance peaks, each of them consisting of nine peaks. The resonant frequencies in groups, and , with high accuracy relate to each other as (n = 1, …, 9). These results satisfy Eq. (2) at θI = 0 and θII = π/4 (with g → gn), which relates to the directions and , that is, the first direction is along the fourfold symmetry axis and the second along the twofold symmetry axis of the crystal.
In this paper, we study the similar spectrum of resonances related not to a memory effect of the magnetic impact on the microhardness of crystal, but to an in situ phenomenon of magnetoplasticity associated with relaxation motion of individual dislocations in NaCl(Ca) crystals where microhardness is not at all magnetosensitive. It will be shown that the spectrum of resonance dislocation paths exists in a much more wide frequency range ν = 6 kHz–2.1 MHz and contains more than 50 peaks. Their interpretation will require substantial extension of physical grounds for phenomenological equation [Eq. (2)].
II. EXPERIMENTAL PROCEDURE AND RESULTING SPECTRUM
This study was performed using NaCl single crystals grown in LOMO using the Kyropoulos method with the yield point of 500 kPa and the impurity content (primarily Ca) of no more than 10 ppm. These crystals contain the Ca impurity as the ion Ca2+ replacing the couple of Na+ cations (for neutrality). It is well known52 that such impurity center is not magnetoactive and cannot manifest itself in ordinary EPR or in macroscopic measurements of microhardness under the magnetic impact. This is why, the study51 was performed using the other crystal NaCl(Ni). However, as explained in Refs. 11 and 29–32, falling in the dislocation core, the ion Ca2+ transforms into the paramagnetic center Ca+, which can participate in the magnetoplasticity based on individual dislocations.
The conditions for the low-frequency EPR-type resonance were realized in the setup schematically shown in Fig. 1. The Earth field measured just in place of the sample position was of 49.97 µT and made an angle of 29.5° with the vertical direction. The alternating pump field was produced by a sinusoidal current flowing through the rectilinear conductor in the screened coaxial chamber. The amplitude of the pump field was 2.5 µT, its direction was chosen to be orthogonal to the Earth field, ⊥BEarth, and its frequency varied in the range of 6 kHz to 2.1 MHz. During an exposure for 5 min, a sample always was oriented so that one of its faces was parallel to the plane of fields (BEarth, ) (Fig. 2). In our experiments, we used the orientation of samples relative to the Earth field BEarth||[001].
(a) Scheme of the setup. (b) Geometry of the experiment in the crossed Earth’s magnetic field BEarth and AC pumping field : (1) the central wire with an alternating current in a coaxial chamber and (2) the position of the sample at which ⊥BEarth.
(a) Scheme of the setup. (b) Geometry of the experiment in the crossed Earth’s magnetic field BEarth and AC pumping field : (1) the central wire with an alternating current in a coaxial chamber and (2) the position of the sample at which ⊥BEarth.
Schematic plot of dislocation paths li in the two slip planes {110} along the directions [011] and under the Earth magnetic field BEarth||[001] and the pump field ||[010]. The shown flat-bottomed etch pits relate to the dislocation positions before exposition and the sharp-pointed pits relate to those after the experiment.
Schematic plot of dislocation paths li in the two slip planes {110} along the directions [011] and under the Earth magnetic field BEarth||[001] and the pump field ||[010]. The shown flat-bottomed etch pits relate to the dislocation positions before exposition and the sharp-pointed pits relate to those after the experiment.
Both the path l and the number N of mobile dislocations arise due to the resonance treatment of the sample and the resulting relaxation of the dislocation system. They represent two different characteristics of the given resonance. The related peaks in the spectra should be of identical (or at least close) frequencies; however, their amplitudes appear to be absolutely non-correlated being determined by relaxation kinetics. One should note that sometimes the large mean dislocation path l not necessarily relates to a large effect. This happens when, in the ratio L/N, the number N of moved dislocations is small. Physically, such situation occurs when the number of magnetically transformed pinning centers is not so large and most dislocations do not start during the exposure, but some parts of them take up unstable positions and even small perturbations can drive them to rather large displacements li. That is why, in Fig. 3 that plots the peaks of resonances l(ν), we will combine them with the dependence N(ν). To be exact, we will use the dimensionless paths equal to the ratios of the displacements l to the mean distance between dislocations .
Spectra of resonance dislocation paths (a) and number N of mobile dislocations (b) under the Earth magnetic field vs the frequency ν of the pumping AC field .
Spectra of resonance dislocation paths (a) and number N of mobile dislocations (b) under the Earth magnetic field vs the frequency ν of the pumping AC field .
Figure 3 demonstrates the results of our measurements of dislocation mean paths and the number N of moved dislocations at different frequencies ν of the pump field in the range of 6 kHz to 2.1 MHz. Both the dependences represent large sets of peaks with identical or rather close positions of maxima determining the resonance frequencies. In further matchings with the theory, we will consider frequencies of the spectrum. However, the amplitudes of the alternative spectrum N(ν) also contain quite useful information.
Comparing spectra and N in Fig. 3, one can see different combinations of peak amplitudes: (1) large l and small N, (2) small l and large N, (3) both large, and (4) both small. The first situation (see, e.g., frequencies 1.44, 1.25, 0.52, 0.38 MHz) was already mentioned above. It relates to a weak reaction of dislocations on the magnetic impact, and the long paths of several of them might be a manifestation of instability. The second and third variants (1.38, 1.30, 0.98, 0.81, 0.78, 0.73, 0.21, 0.15, 0.12, 0.06 MHz) really point to strong resonance effects. In this case, small mean paths of dislocations might just occur due to a large number of active mobile dislocations providing effective relaxation at sufficiently short paths. Finally, the fourth case of both small paths trivially means evidently weak effect (see, e.g., the region of 0.3–0.5 MHz). However, one should stress that the weakness of the effect at some frequency does not exclude an existence of resonance.
Here, we have to declare that a small part of the presented spectrum (the peaks in the region of 1.20–1.45 MHz and the range of lowest frequencies ν < 0.5 MHz) were earlier experimentally studied in our group (in the scale ). The first of them was published43 without any interpretation. The low frequency peaks were attributed44 to different positions of pinning centers in the dislocation core. As we now understand, this explanation is correct for a small range of very low frequencies ν < 0.13 MHz, which includes only the three last peaks of the spectrum. Below, we will show that, practically, the collection of peaks in Fig. 3 is a consequence of fundamental properties of anisotropic hyperfine interactions in the low magnetic field BEarth ≪ Bloc.
III. THEORETICAL MODEL AND ANALYSIS
A. Spin reactions under magnetic impact
As was mentioned above, the dislocation pinning in our NaCl crystals is mainly provided by Ca impurities that are present in the bulk as Ca2+ ions replacing in the lattice a pair of Na+ ions for local electro neutrality. It means that such center must contain in addition to a substitutional impurity also a Na vacancy: Ca2+V−. It is evident that such centers do not contain unpaired electrons and should be magnetically insensitive. Meanwhile, the experiments7,11 clearly showed that NaCl(Ca) crystals under the magnetic field are very sensitive to Ca concentration. In fact, it is generally accepted11,29–32 that, in the dislocation core, the ion Ca2+ could capture the electron being transformed into a magnetosensitive ion Ca+. It might happen in the following way. When the dislocation approaches the impurity center, the ion Na+ closest to the vacancy V− and situated at the extra-plane edge of the dislocation will be pushed out into the vacancy hole by the enormous pressure in the dislocation core. Simultaneously, for local electro neutrality, the neighbored in the edge ion Cl− should give its electron to Ca2+. This happens without any magnetic impact due to mechanical pressure in the core and for minimization of the local Coulomb energy in the ionic lattice. Switching on the magnetic field somehow might cause the depinning because we do observe dislocation motion.
The mechanism of magnetoplasticity suggested by Buchachenko29 looks to be rather probable, the more so, that it was supported by the authors of the studies of the magnetoplasticity in CaF2 crystals.53,54 However, this version appears to be in contradiction with some experimental facts. For instance, according to our observations,7,11–13,32 replacing the Ca impurity in NaCl crystal by Pb leads to the radical change of the effect sign: magnetoplasticity is transformed into magnetic hardening, i.e., the pinning stress on dislocations under the magnetic field increases. Why the quite similar transformations Ca2+ → Ca+ and Pb2+ → Pb+ provide opposite effects on the pinning stress?
The other argument in favor of this model is the experimental fact established in Golovin’s group.55 They moved dislocations in NaCl crystals by means of short strong pulses of the magnetic field and found that the second pulse provided the magnetoplastic effect only after the pause of ∼5–10 min since the first pulse. This means that the first pulse transformed the pinning centers into long-lived metastable complexes. It might be complexes like CaCl2. It is evident that the Ca+ ion would return to the initial state Ca2+V− not in several min but almost immediately after dislocation has moved from this position and the mechanical pressure on the center has weakened.
Anyway, both discussed versions are not more than hypotheses. Fortunately, for our further theoretical description of the observed spectrum, it will be sufficient to suppose that we deal with singlet-triplet or triplet-singlet transitions of any type.
Our physical model will be based on the anisotropic hyperfine interaction of Ca pinning centers with local magnetic fields Bloc created by the nuclear spins of atoms surrounding these paramagnetic impurities. It is essential that Ca centers, similar to Ni impurities in NaCl(Ni) studied in Ref. 51, do not manifest any noticeable anisotropic hyperfine effect.48 Therefore, in both cases, it is natural to attribute the observed multiple peaks in spectra to the impact of Cl nuclei ligands around the impurity centers. These nuclei are characterized by the total spin I = 3/2 and manifest substantial anisotropic hyperfine interaction.48 Possibly just for this reason, the observed spectra of two different effects created in NaCl crystals by two different impurity centers are so similar (though, of course, not identical). In particular, for the spectrum of dislocation paths, we will again come to the nine empirical g-factors (gn). We stress that, in a cubic NaCl crystal, gn-factors may be considered as scalar parameters independent of orientation.48,49 As we will see, this does not cancel our anisotropic effects.
B. Theoretical model of hyperfine interaction in our system
For the considered range of small external magnetic field B ≪ Bloc, the quantum mechanical description of the hyperfine interaction has its specifics. In this case, one should again quantize angle moment (J) and nuclear spin (I) but not independently. Instead, their sum F = J + I is formed and its projection on the arbitrary z axis must be quantized in accordance with some rules.
Hyperfine energy spectrum in the range of low magnetic fields B ≪ Bloc for the system .
Hyperfine energy spectrum in the range of low magnetic fields B ≪ Bloc for the system .
The ∓ for relates to the negative in (12), which, in turn, corresponds to the inequality > (see also Fig. 4).
It is remarkable that, under this condition, the resulting Zeeman energies (together with positions of our resonances) are independent of magnitudes of the local fields . In our case, we deal with the magnetic field B = BEarth = 50 µT, which certainly is much less than the local fields. Indeed, substituting into (13) the estimate A ∼ 100 MHz for Cl,48 one obtains ∼ 102BEarth. This means that our approximation (19) might bring to troubles only for θ close to 90°. Below, we will meet this situation for several peaks in the lowest frequency part of our spectrum.
The 12 energy levels shown in Fig. 4 are clearly not the total energy spectrum. One should take into account that there is quite a group of mentioned above gn-factors and a series of possible orientations of local crystalline fields making different angles θ with B. As was mentioned at the beginning of this subsection, there are nine gn-factors that will be found in the experiment below. Here, let us discuss possible orientations of the local magnetic fields .
IV. COMPARISON OF THEORY WITH EXPERIMENTAL SPECTRUM
In Fig. 4, at each energy level, there presented the list of possible spin states (I, ) related to the given state . Only two levels, and , are characterized by pure states, i.e., by definite pairs of spin states (3/2, 1) and (−3/2, −1), respectively. Transitions between these pure levels are highly forbidden. The other levels relate to mixed quantum states and may participate in any transitions with other levels including any one of the pure levels. Of course, we are interested only in low-energy transitions inside the separate “fans” I, II, or III related to Zeeman resonances. All transitions of this type contain singlet-triplet and/or triplet-singlet transformations. Thus, in the considered theory, both singlet-triplet transitions (5) (the Buchachenko model) and triplet–singlet transitions (6) (our model) should create the same spectrum.
Setgn . | |||||||||
---|---|---|---|---|---|---|---|---|---|
. | 1.84 . | 1.77 . | 1.64 . | 1.55 . | 1.49 . | 1.47 . | 1.43 . | 1.39 . | 1.36 . |
2.06 | 1.98 | 1.84 | 1.74 | 1.67 | 1.64 | 1.60 | 1.56 | 1.52 | |
1.457 | 1.400 | 1.301 | 1.230 | 1.181 | 1.160 | 1.131 | 1.103 | 1.075 | |
1.44 | 1.40 | 1.30 | 1.25 | 1.18 | 1.14 | 1.11 | 1.07 | ||
1.189 | 1.143 | 1.062 | 1.004 | 0.964 | 0.947 | 0.924 | 0.901 | 0.878 | |
1.18 | 1.14 | 1.07 | 1.00 | 0.98 | 0.93 | 0.91 | 0.87 | ||
1.545 | 1.485 | 1.380 | 1.305 | 1.252 | 1.230 | 1.200 | 1.170 | 1.140 | |
1.56 | 1.48 | 1.38 | 1.30 | 1.25 | 1.18 | 1.14 | |||
1.092 | 1.047 | 0.972 | 0.919 | 0.884 | 0.870 | 0.834 | 0.827 | 0.806 | |
1.09 | 1.04 | 0.98 | 0.93 | 0.89 | 0.87 | 0.84 | 0.81 | ||
0.892 | 0.857 | 0.797 | 0.753 | 0.723 | 0.710 | 0.693 | 0.675 | 0.658 | |
0.89 | 0.87 | 0.81 | 0.75 | 0.73 | 0.68 | ||||
1.030 | 0.990 | 0.920 | 0.870 | 0.835 | 0.820 | 0.800 | 0.780 | 0.760 | |
1.04 | 0.98 | 0.91 | 0.87 | 0.84 | 0.81 | 0.78 | 0.75 | ||
0.735 | 0.700 | 0.643 | 0.615 | 0.594 | 0.58 | 0.566 | 0.552 | 0.544 | |
0.73 | 0.68 | 0.65 | 0.63 | 0.60 | 0.58 | 0.56 | |||
0.600 | 0.566 | 0.525 | 0.502 | 0.485 | 0.473 | 0.462 | 0.450 | 0.433 | |
0.60 | 0.56 | 0.52 | 0.49 | 0.47 | 0.45 | 0.41 | |||
0.515 | 0.495 | 0.460 | 0.435 | 0.420 | 0.410 | 0.400 | 0.390 | 0.380 | |
0.52 | 0.49 | 0.47 | 0.45 | 0.41 | 0.38 | ||||
0.368 | 0.346 | 0.332 | 0.307 | 0.297 | 0.290 | 0.283 | 0.276 | 0.269 | |
0.38 | 0.34 | 0.32 | 0.30 | 0.28 | 0.25 | ||||
0.297 | 0.283 | 0.266 | 0.251 | 0.242 | 0.237 | 0.230 | 0.225 | 0.219 | |
0.30 | 0.28 | 0.25 | 0.21 | ||||||
0.687 | 0.660 | 0.613 | 0.580 | 0.560 | 0.547 | 0.533 | 0.520 | 0.507 | |
0.68 | 0.65 | 0.60 | 0.58 | 0.56 | 0.52 | 0.49 | |||
0.486 | 0.467 | 0.433 | 0.410 | 0.396 | 0.387 | 0.377 | 0.368 | 0.358 | |
0.49 | 0.47 | 0.45 | 0.41 | 0.38 | 0.34 | ||||
0.397 | 0.381 | 0.354 | 0.335 | 0.323 | 0.316 | 0.308 | 0.300 | 0.293 | |
0.41 | 0.38 | 0.34 | 0.32 | 0.30 | |||||
0.343 | 0.328 | 0.306 | 0.290 | 0.280 | 0.273 | 0.266 | 0.260 | 0.253 | |
0.34 | 0.32 | 0.30 | 0.28 | 0.25 | |||||
0.243 | 0.232 | 0.216 | 0.205 | 0.198 | 0.193 | 0.188 | 0.184 | 0.179 | |
0.25 | 0.21 | 0.17 | |||||||
0.198 | 0.189 | 0.177 | 0.167 | 0.162 | 0.158 | 0.154 | 0.150 | 0.146 | |
0.21 | 0.17 | 0.15 | |||||||
0.859 | 0.825 | 0.767 | 0.725 | 0.696 | 0.683 | 0.667 | 0.649 | 0.633 | |
0.87 | 0.84 | 0.78 | 0.73 | 0.68 | 0.65 | 0.63 | |||
0.607 | 0.583 | 0.542 | 0.513 | 0.492 | 0.483 | 0.472 | 0.459 | 0.448 | |
0.60 | 0.58 | 0.52 | 0.49 | 0.47 | 0.45 | ||||
0.495 | 0.476 | 0.443 | 0.419 | 0.402 | 0.394 | 0.385 | 0.375 | 0.365 | |
0.49 | 0.47 | 0.45 | 0.41 | 0.38 |
Setgn . | |||||||||
---|---|---|---|---|---|---|---|---|---|
. | 1.84 . | 1.77 . | 1.64 . | 1.55 . | 1.49 . | 1.47 . | 1.43 . | 1.39 . | 1.36 . |
2.06 | 1.98 | 1.84 | 1.74 | 1.67 | 1.64 | 1.60 | 1.56 | 1.52 | |
1.457 | 1.400 | 1.301 | 1.230 | 1.181 | 1.160 | 1.131 | 1.103 | 1.075 | |
1.44 | 1.40 | 1.30 | 1.25 | 1.18 | 1.14 | 1.11 | 1.07 | ||
1.189 | 1.143 | 1.062 | 1.004 | 0.964 | 0.947 | 0.924 | 0.901 | 0.878 | |
1.18 | 1.14 | 1.07 | 1.00 | 0.98 | 0.93 | 0.91 | 0.87 | ||
1.545 | 1.485 | 1.380 | 1.305 | 1.252 | 1.230 | 1.200 | 1.170 | 1.140 | |
1.56 | 1.48 | 1.38 | 1.30 | 1.25 | 1.18 | 1.14 | |||
1.092 | 1.047 | 0.972 | 0.919 | 0.884 | 0.870 | 0.834 | 0.827 | 0.806 | |
1.09 | 1.04 | 0.98 | 0.93 | 0.89 | 0.87 | 0.84 | 0.81 | ||
0.892 | 0.857 | 0.797 | 0.753 | 0.723 | 0.710 | 0.693 | 0.675 | 0.658 | |
0.89 | 0.87 | 0.81 | 0.75 | 0.73 | 0.68 | ||||
1.030 | 0.990 | 0.920 | 0.870 | 0.835 | 0.820 | 0.800 | 0.780 | 0.760 | |
1.04 | 0.98 | 0.91 | 0.87 | 0.84 | 0.81 | 0.78 | 0.75 | ||
0.735 | 0.700 | 0.643 | 0.615 | 0.594 | 0.58 | 0.566 | 0.552 | 0.544 | |
0.73 | 0.68 | 0.65 | 0.63 | 0.60 | 0.58 | 0.56 | |||
0.600 | 0.566 | 0.525 | 0.502 | 0.485 | 0.473 | 0.462 | 0.450 | 0.433 | |
0.60 | 0.56 | 0.52 | 0.49 | 0.47 | 0.45 | 0.41 | |||
0.515 | 0.495 | 0.460 | 0.435 | 0.420 | 0.410 | 0.400 | 0.390 | 0.380 | |
0.52 | 0.49 | 0.47 | 0.45 | 0.41 | 0.38 | ||||
0.368 | 0.346 | 0.332 | 0.307 | 0.297 | 0.290 | 0.283 | 0.276 | 0.269 | |
0.38 | 0.34 | 0.32 | 0.30 | 0.28 | 0.25 | ||||
0.297 | 0.283 | 0.266 | 0.251 | 0.242 | 0.237 | 0.230 | 0.225 | 0.219 | |
0.30 | 0.28 | 0.25 | 0.21 | ||||||
0.687 | 0.660 | 0.613 | 0.580 | 0.560 | 0.547 | 0.533 | 0.520 | 0.507 | |
0.68 | 0.65 | 0.60 | 0.58 | 0.56 | 0.52 | 0.49 | |||
0.486 | 0.467 | 0.433 | 0.410 | 0.396 | 0.387 | 0.377 | 0.368 | 0.358 | |
0.49 | 0.47 | 0.45 | 0.41 | 0.38 | 0.34 | ||||
0.397 | 0.381 | 0.354 | 0.335 | 0.323 | 0.316 | 0.308 | 0.300 | 0.293 | |
0.41 | 0.38 | 0.34 | 0.32 | 0.30 | |||||
0.343 | 0.328 | 0.306 | 0.290 | 0.280 | 0.273 | 0.266 | 0.260 | 0.253 | |
0.34 | 0.32 | 0.30 | 0.28 | 0.25 | |||||
0.243 | 0.232 | 0.216 | 0.205 | 0.198 | 0.193 | 0.188 | 0.184 | 0.179 | |
0.25 | 0.21 | 0.17 | |||||||
0.198 | 0.189 | 0.177 | 0.167 | 0.162 | 0.158 | 0.154 | 0.150 | 0.146 | |
0.21 | 0.17 | 0.15 | |||||||
0.859 | 0.825 | 0.767 | 0.725 | 0.696 | 0.683 | 0.667 | 0.649 | 0.633 | |
0.87 | 0.84 | 0.78 | 0.73 | 0.68 | 0.65 | 0.63 | |||
0.607 | 0.583 | 0.542 | 0.513 | 0.492 | 0.483 | 0.472 | 0.459 | 0.448 | |
0.60 | 0.58 | 0.52 | 0.49 | 0.47 | 0.45 | ||||
0.495 | 0.476 | 0.443 | 0.419 | 0.402 | 0.394 | 0.385 | 0.375 | 0.365 | |
0.49 | 0.47 | 0.45 | 0.41 | 0.38 |
The magnitudes of gn calculated from (27) are presented as the first line in Table I. Under it, the table contains the set (26) being experimental and simultaneously the basis of our theory. After these two lines, Table I is arranged in the following way. Each two-line section shows, on top, the theoretical nine resonance frequencies given by (25) and (26) and, below, the corresponding experimental data closest to the upper magnitudes. The first such pair is and , the second is and , and further according the order in (25).
As can be seen from (24) and (25), some of the theoretical resonances are very close to each other. For instance, (identically), , , etc. Even more coincidences occur for differing numbers n. Therefore, some theoretical frequencies find themselves inside a common peak, i.e., these resonances are not resolved experimentally. As a result, the set contains only eight resonance peaks. The same occurs with the sets , , and . The sets , , , , and contain seven resonance peaks, and the sets , , and , only six of them, and so on, down to three peaks in two sets, and , are related to lowest frequencies. This behavior appears to be quite natural for the spectrum (Fig. 3), which shrinks with decreasing frequency. It is evident that the more dense the spectrum, the less its resolution.
There is also another rather typical manifestation of the discussed closeness of the resonance frequencies (25). According to Table I, some experimental frequencies are often simultaneously present in several sets. In fact, this feature means that many of the experimental peaks represent a mixture of several close resonances. For example, the experimental peak at 1.14 MHz is formed by the resonances described by the theoretical frequencies (25).
Table II gives the complete characterization of all peaks in our spectrum attributing each of them to a definite mixture of specific resonances from the list (25). Here, we again use for shortness more compact notation: for . In addition, the table also shows for each peak its amplitudes in the scale and compares its experimental frequency with the arithmetical mean of theoretical frequencies of constituent resonances. It appears to be important that these compared frequencies for almost all peaks turn out to be very close to each other.
Frequencies of peaks (Fig. 3) in comparison with those of arithmetical average of related theoretical constituent resonances and the list of these constituents and amplitudes of experimental peaks.
No. . | νexp (MHz) . | (MHz) . | Constituents . | . |
---|---|---|---|---|
1 | 2.06 | 2.06 | A1 | 1.04 |
2 | 1.98 | 1.98 | A2 | 0.70 |
3 | 1.84 | 1.84 | A3 | 0.69 |
4 | 1.74 | 1.74 | A4 | 0.67 |
5 | 1.67 | 1.67 | A5 | 0.62 |
6 | 1.64 | 1.64 | A6 | 0.75 |
7 | 1.60 | 1.60 | A7 | 0.76 |
8 | 1.56 | 1.553 | A8, B1 | 0.82 |
9 | 1.52 | 1.520 | A9 | 0.92 |
10 | 1.48 | 1.485 | B2 | 0.56 |
11 | 1.44 | 1.457 | 1.24 | |
12 | 1.40 | 1.400 | 0.79 | |
13 | 1.38 | 1.380 | B3 | 1.29 |
14 | 1.30 | 1.303 | 1.14 | |
15 | 1.25 | 1.237 | 1.09 | |
16 | 1.18 | 1.185 | 0.52 | |
17 | 1.14 | 1.143 | 0.77 | |
18 | 1.11 | 1.103 | 0.62 | |
19 | 1.09 | 1.092 | 0.57 | |
20 | 1.07 | 1.069 | 0.56 | |
21 | 1.04 | 1.039 | 0.61 | |
22 | 1.00 | 1.004 | 0.59 | |
23 | 0.98 | 0.975 | 0.96 | |
24 | 0.93 | 0.933 | 0.41 | |
25 | 0.91 | 0.915 | 0.67 | |
26 | 0.89 | 0.888 | 0.52 | |
27 | 0.87 | 0.867 | 0.71 | |
28 | 0.84 | 0.830 | 0.74 | |
29 | 0.81 | 0.806 | 0.88 | |
30 | 0.78 | 0.774 | C8, G3 | 0.89 |
31 | 0.75 | 0.757 | 0.60 | |
32 | 0.73 | 0.728 | 1.28 | |
33 | 0.68 | 0.685 | 0.76 | |
34 | 0.65 | 0.651 | 0.89 | |
35 | 0.63 | 0.624 | 0.91 | |
36 | 0.60 | 0.603 | 0.92 | |
37 | 0.58 | 0.581 | 0.98 | |
38 | 0.56 | 0.556 | 0.78 | |
39 | 0.52 | 0.524 | 0.94 | |
40 | 0.49 | 0.493 | 0.55 | |
41 | 0.47 | 0.468 | 0.51 | |
42 | 0.45 | 0.445 | 0.55 | |
43 | 0.41 | 0.413 | 0.60 | |
44 | 0.38 | 0.382 | 0.79 | |
45 | 0.34 | 0.344 | 0.54 | |
46 | 0.32 | 0.321 | 0.46 | |
47 | 0.30 | 0.299 | 0.43 | |
48 | 0.28 | 0.279 | 0.58 | |
49 | 0.25 | 0.250 | 0.89 | |
50 | 0.21 | 0.206 | 0.91 | |
51 | 0.17 | 0.174 | 0.78 | |
52 | 0.15 | 0.150 | 0.82 | |
53 | 0.12 | 0.93 | ||
54 | 0.06 | 0.95 | ||
55 | 0.03 | 1.08 |
No. . | νexp (MHz) . | (MHz) . | Constituents . | . |
---|---|---|---|---|
1 | 2.06 | 2.06 | A1 | 1.04 |
2 | 1.98 | 1.98 | A2 | 0.70 |
3 | 1.84 | 1.84 | A3 | 0.69 |
4 | 1.74 | 1.74 | A4 | 0.67 |
5 | 1.67 | 1.67 | A5 | 0.62 |
6 | 1.64 | 1.64 | A6 | 0.75 |
7 | 1.60 | 1.60 | A7 | 0.76 |
8 | 1.56 | 1.553 | A8, B1 | 0.82 |
9 | 1.52 | 1.520 | A9 | 0.92 |
10 | 1.48 | 1.485 | B2 | 0.56 |
11 | 1.44 | 1.457 | 1.24 | |
12 | 1.40 | 1.400 | 0.79 | |
13 | 1.38 | 1.380 | B3 | 1.29 |
14 | 1.30 | 1.303 | 1.14 | |
15 | 1.25 | 1.237 | 1.09 | |
16 | 1.18 | 1.185 | 0.52 | |
17 | 1.14 | 1.143 | 0.77 | |
18 | 1.11 | 1.103 | 0.62 | |
19 | 1.09 | 1.092 | 0.57 | |
20 | 1.07 | 1.069 | 0.56 | |
21 | 1.04 | 1.039 | 0.61 | |
22 | 1.00 | 1.004 | 0.59 | |
23 | 0.98 | 0.975 | 0.96 | |
24 | 0.93 | 0.933 | 0.41 | |
25 | 0.91 | 0.915 | 0.67 | |
26 | 0.89 | 0.888 | 0.52 | |
27 | 0.87 | 0.867 | 0.71 | |
28 | 0.84 | 0.830 | 0.74 | |
29 | 0.81 | 0.806 | 0.88 | |
30 | 0.78 | 0.774 | C8, G3 | 0.89 |
31 | 0.75 | 0.757 | 0.60 | |
32 | 0.73 | 0.728 | 1.28 | |
33 | 0.68 | 0.685 | 0.76 | |
34 | 0.65 | 0.651 | 0.89 | |
35 | 0.63 | 0.624 | 0.91 | |
36 | 0.60 | 0.603 | 0.92 | |
37 | 0.58 | 0.581 | 0.98 | |
38 | 0.56 | 0.556 | 0.78 | |
39 | 0.52 | 0.524 | 0.94 | |
40 | 0.49 | 0.493 | 0.55 | |
41 | 0.47 | 0.468 | 0.51 | |
42 | 0.45 | 0.445 | 0.55 | |
43 | 0.41 | 0.413 | 0.60 | |
44 | 0.38 | 0.382 | 0.79 | |
45 | 0.34 | 0.344 | 0.54 | |
46 | 0.32 | 0.321 | 0.46 | |
47 | 0.30 | 0.299 | 0.43 | |
48 | 0.28 | 0.279 | 0.58 | |
49 | 0.25 | 0.250 | 0.89 | |
50 | 0.21 | 0.206 | 0.91 | |
51 | 0.17 | 0.174 | 0.78 | |
52 | 0.15 | 0.150 | 0.82 | |
53 | 0.12 | 0.93 | ||
54 | 0.06 | 0.95 | ||
55 | 0.03 | 1.08 |
In Fig. 6, to avoid the mess, we ignore the weakest peaks focusing attention on more meaningful resonances, in contrast to Table II which presents the full characterization of all observed peaks independently of their amplitudes.
Spectrum of normalized dislocation paths with attribution of resonance peaks to theoretical classification (24) and (25). The short-cut notation is explained in the text. Parts (a) and (b) of the spectrum are given on different scales for the convenience of reading.
We note that our comparison of the experimental spectrum with the developed theory is finished at the frequency 0.15 MHz minimal in our theoretical prediction in Table I. However, there are three more experimental peaks in the spectrum at 0.12, 0.06, and 0.03 MHz being less than this minimum. In fact, their interpretation is rather natural. Our model was based on the hypothesis that local magnetic fields in the crystal should be oriented along symmetry axes, particularly along the axis 4||⟨100⟩. We considered only the direction [001]||BEarth because the two other directions [100] and [010] are orthogonal to BEarth, which provides cos 90° = 0. However, if some distortions of the lattice occur in the vicinity of the dislocation core, then the directions [100] and [010] become almost orthogonal to BEarth, i.e., the angle (3) should be θ = 90° − Δθ, where Δθ is the small angle of the order of several degrees. In this situation, cos θ becomes small but does not vanish and provides small frequencies below minimum.
This idea was earlier introduced in our paper44 where the distortions in the core were estimated based on the computer model of the edge dislocation core structure in NaCl crystal. It was found that, depending on the position of the impurity Ca in the dislocation core, the angle Δθ of inclination of the directions ⟨100⟩ from the perfect orientation might be both very small (say, less than 1°) and rather large (up to ∼20°). We interpreted all peaks of dislocation paths of the observed spectrum studied in Ref. 44 at ν < 0.5 MHz in accordance with their related positions in the dislocation core (see the remark at the end of Sec. II). Now, we understand that such interpretation fits only for three last peaks at 0.12, 0.06, and 0.03 MHz. In terms of our present theory, they relate to distortion angles Δθ ∼ 1°–4°. Unfortunately, we cannot yet estimate these angles with more accuracy neither theoretically [see condition (19)], nor experimentally because these three peaks for sure are degenerate being a mixture of quite several resonances. A quantitative analysis of this part of the spectrum will require measurements made under a much higher external field than BEarth. We have got yet only preliminary results corresponding to the peak of lowest frequency 0.03 MHz in Fig. 3. The check experiments show that this peak at the external field 10BEarth splits into nine separate peaks. It appears to be a good argument in favor of our phenomenological interpretation presented above.
It is seen from (29) that the inclinations of found angles from 45° are really small being less than 1°.
Note that the results in (29) might also indicate the important correlation. The two considered directions [011] and [] are not equivalent: the first one is parallel to the slip plane of the dislocation, and the other one is parallel to its extra-plane. It is well known that the first direction is only slightly distorted as compared with the second one (see Fig. 7). Thus, the two pairs in (29), and , could relate to those two expectations.
Schematic structure of the edge dislocation core in the plane (010) of NaCl crystal.
Schematic structure of the edge dislocation core in the plane (010) of NaCl crystal.
Anyway, both pairs demonstrate rather small inclinations. Apparently, this experimental result again shows that the impurities “choose” the positions in dislocation cores relating to a minimal curvature of the lattice from rather wide options. Probably, this particular “choice” corresponds to the minimal energy of the local structure. In fact, when dislocation comes close to the impurity atom, the latter finds itself under the great stress field impact, which possibly provides some optimization of the local structure.
As shown in Fig. 6, there are many peaks that are highly likely to have a fine structure. This is a good reason for future experimental study. Here, we can discuss the other aspect of the problem related to the peaks of the mentioned above “quartet” between 1.25 and 1.44 MHz. Here, there are a series of experimental points in non-monotonic positions that look like overtones (at 1.32, 1.36, and 1.42 MHz) and might be an indication of some fine structure. Fortunately, in our initial study43 of this “quartet,” we have checked this suspicion by means of a rise in the static field from BEarth ≈ 50 to 165 µT which provided the abrupt increase in resolution. This check experiment has shown that there was no fine structure and the “quartet” remained a “quartet.” In fact, this proves to be a nice experimental corroboration of the above theory not indicating at those frequencies any peaks. All the rest peaks in the spectrum (Fig. 3) look to be described by the presented theory.
V. CONCLUSIONS
In this paper, we studied the phenomenon determined by the hyperfine interaction between electron and nuclei spins in paramagnetic impurities being simultaneously pinning centers on dislocations in NaCl(Ca) crystals. In the absence of an external magnetic field, these spins form the combined state Fα and the common energy spectrum Wα. Formally, the electron spin could be considered as being under the local magnetic field Bloc created by the nuclei spin. In these terms, they describe the phenomenon called the EPR of zero field.48 Under such nuclei “protection,” the electron spin is not subjected to thermal chaotization. It is well known that nuclear spins rather slowly react to thermal vibrations. This is why, our spectrum contains peaks at very small frequencies down to ν ∼10 kHz.
In classic EPR studies, they use large fields B ≫ Bloc when the hyperfine interaction manifests itself in rather weak splitting of standard resonance lines. Often, there is a problem to observe such splitting. For instance, in order to observe,50 in SrF2(Ni) crystals, the hyperfine splitting of the Ni+ lines caused by the ligands of F nuclei, the authors had to use the helium temperature and the preliminary x-ray irradiation of samples. They obtained five or nine split-lines depending on the sample orientation in the magnetic field. In our case, we observed 55 hyperfine peaks at room temperature (Fig. 3). The difference in numbers proves to be quite clear. Indeed, in the range of large fields B ≫ Bloc, most of transitions between energy levels are forbidden. The allowed transitions are limited by the condition: , ΔI = 0. In our case of the low field BEarth, almost all energy levels relate to the mixed quantum states of electron and nuclei spins when the above selection rules are inapplicable.
As was shown in this paper, at B ≪ Bloc, the theoretical description of such spectra proves to be rather straightforward. The above-developed theory was reduced to the extension for anisotropic hyperfine interaction of the approach49 valid for the isotropic case. As a result, we obtained the first strict explanation of anisotropic dependence of resonance frequencies ν ∝ cos θ (2), (3), which was earlier established empirically39,42,44 and on a semi-intuitive level.47,51 The observed maxima in Fig. 3 in this theory obtained physical interpretation. Each of them was supplied (Fig. 6 and Table II) by particular information in terms of concrete transitions between the quantum energy states of the spectrum that was theoretically established.
The results of this paper contain useful findings not only in the aspect of hyperfine interaction. The mechanical properties of crystal under EPR impact also may be important. The observation itself of 55 resonance peaks of dislocation paths appears to be quite unexpected and nontrivial. The obtained spectrum allows one to specify conditions of strongest effect by choosing the frequencies related to high amplitudes in Fig. 3 simultaneously in both scales, and N. The other new conclusion relates to positions of point defects in the dislocation core. Our analysis of experimental data showed that impurities situate in positions with low curvature of the latter. Probably, huge stress in the core allows the system to optimize the local structure.
ACKNOWLEDGMENTS
The authors were grateful to E. A. Petrzhik and S. A. Minyukov for helpful discussions and to E. A. Stepantsov for the help in annealing the crystals. This work was performed within the State assignment of the Federal Scientific Research Center “Crystallography and Photonics” of the Russian Academy of Sciences.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
M. V. Koldaeva: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). V. I. Alshits: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal).
DATA AVAILABILITY
The data that support the findings of this study are available within the article.