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 B̃(ν) 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.

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 lB2t ∼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 

The predicted, in Ref. 27, resonance MPE was indeed found experimentally in the EPR scheme of magnetic exposure of samples NaCl33–36 and Si37,38 under the crossed magnetic fields, static (B) and alternating (B̃). Most of these experiments were carried out under standard frequency ν ∼ 10 GHz. The resonance was found for the ordinary static field B ∼ 0.3 T in accordance with the classical EPR condition,
(1)
where h is the Planck constant, β is the Bohr magneton, and g ≈ 2 is the Lander factor. As usual, the effect was maximal for the orientation BB̃ and disappeared when it tended to B||B̃.34 No anisotropy of the effect in NaCl crystal with respect to its orientation was declared,33–36 neither for dislocations nor for microhardness.
As was found in our group, the similar EPR-type effect occurs for ultralow magnetic fields when the Earth field BEarth ≈ 50 µT serves for the static component and the pump field amplitude is B̃ ∼ 2–3 µT. The in situ effect for individual dislocations was experimentally studied39–45 in NaCl crystals for both harmonic and pulsed41 pumping. The intensity of low-frequency effects in NaCl crystals was comparable with previous observations33–36 for standard EPR parameters. However, in contrast to the standard conditions, the low-frequency resonance turned out to be very specific in its properties. The main new feature of the effect was its very strong anisotropy regarding the sample orientation relative to the crossed magnetic fields. In particular, dislocations parallel to the field BEarth manifested rather weak effect.39 The largest amplitudes of resonance paths l occurred for dislocations orthogonal to both fields BEarthB̃. In this case for the orientation BEarth||[001], Eq. (1) (with B = BEarth) again determines the resonance frequency of the pump field B̃. However, the variations of the sample orientation by its rotating about the dislocation direction [100] by the angle θ relative to the unchanged fields led to regular changes in the resonance frequency proportional to cos θ,42 
(2)
The memory effect on microhardness of ZnO, triglycine sulfate (TGS), and potassium acid phthalate (KAP) crystals was also observed at the same low fields EPR scheme being described by Eq. (1) for symmetrical orientations of samples relative the Earth field46 and by Eq. (2) for their rotation angle θ in the plane of magnetic fields BEarthB̃.47 This dependence (2) was interpreted as a result of the combined action of two magnetic fields BEarth and the local crystalline field Bloc under the condition BEarthBloc. In these terms,
(3)

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 BlocB 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, νnI and νnII, with high accuracy relate to each other as νnIIνnI/2 (n = 1, …, 9). These results satisfy Eq. (2) at θI = 0 and θII = π/4 (with ggn), which relates to the directions BlocIBEarth[001] and BlocII[011], 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)].

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 B̃ 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, B̃BEarth, 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, B̃) (Fig. 2). In our experiments, we used the orientation of samples relative to the Earth field BEarth||[001].

FIG. 1.

(a) Scheme of the setup. (b) Geometry of the experiment in the crossed Earth’s magnetic field BEarth and AC pumping field B̃: (1) the central wire with an alternating current in a coaxial chamber and (2) the position of the sample at which B̃BEarth.

FIG. 1.

(a) Scheme of the setup. (b) Geometry of the experiment in the crossed Earth’s magnetic field BEarth and AC pumping field B̃: (1) the central wire with an alternating current in a coaxial chamber and (2) the position of the sample at which B̃BEarth.

Close modal
FIG. 2.

Schematic plot of dislocation paths li in the two slip planes {110} along the directions [011] and [011̄] under the Earth magnetic field BEarth||[001] and the pump field B̃||[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.

FIG. 2.

Schematic plot of dislocation paths li in the two slip planes {110} along the directions [011] and [011̄] under the Earth magnetic field BEarth||[001] and the pump field B̃||[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.

Close modal
The samples of approximate dimensions 3 × 3 × (5–7) mm were chipped out the NaCl(Ca) single crystal along the {100} cleavage planes. They were annealed and underwent chemical polishing before experiments. Fresh dislocations were introduced in a sample by a weak impact just before the magnetic exposure. As was earlier established,11 in this case, most of them prove to be rectilinear being directed along the orientations ⟨100⟩. The density ρm of the mobile fresh dislocations was of the same order of magnitude as the whole dislocation density ρ ∼ 104 cm−2. We watched motions of edge dislocations in their slip planes of the system {110}. In this study, we measured displacements of only dislocations orthogonal to both cross fields BEarthB̃ manifesting the maximum effect. The positions of dislocations before and after the exposure were determined by the selective etching method (Fig. 2). From these measurements, the related mean paths l were found
(4)
where N is the number of moved dislocations.

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 lρ equal to the ratios of the displacements l to the mean distance between dislocations 1/ρ.

FIG. 3.

Spectra of resonance dislocation paths lρ (a) and number N of mobile dislocations (b) under the Earth magnetic field vs the frequency ν of the pumping AC field B̃.

FIG. 3.

Spectra of resonance dislocation paths lρ (a) and number N of mobile dislocations (b) under the Earth magnetic field vs the frequency ν of the pumping AC field B̃.

Close modal

Figure 3 demonstrates the results of our measurements of dislocation mean paths lρ and the number N of moved dislocations at different frequencies ν of the pump field B̃ 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 lρ spectrum. However, the amplitudes of the alternative spectrum N(ν) also contain quite useful information.

Comparing spectra lρ 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 lρ). 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 BEarthBloc.

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.

There are two versions of the mechanism of depinning. The first idea was introduced by Buchachenko.29 He supposed that the radical electron pair that will change its initial S-state after switching on the magnetic field is the same pair that was perturbed simultaneously with the Ca2+ ion. The captured electron has two profitable positions: at the Ca+ due to the Coulomb action and back at the Cl ion because of the exchange interaction. Thus, it oscillates between these positions with a large frequency until the pair stays in the S-state. Under this condition, the lifetime of the Ca+ ion is not sufficient for the depinning. However, the magnetic field may transform this state into the triplet one that will stop oscillations because, in T-state, the electron cannot be localized at the Cl atom. Meanwhile, the transformed Ca+ ion is quite similar to a normal Na+ ion and the energy barrier for dislocation depinning might be substantially less than for Ca2+. Thus, in the given version, the key spin transition of the radical pair for the depinning is
(5)

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?

In Refs. 11 and 32, we supposed the other scenario of magnetic impact on the Ca+ ion. It is possible that the captured electron at Ca+ has no exchange interaction with its previous coupled electron in Cl0 (e.g., because of the distance). In fact, an electron jump was caused by the combination of enormous mechanical stresses and Coulomb interaction. This state not necessarily should provide depinning of dislocation. On the other hand, the electron at Ca+ has the other five neighbors, Cl ions, and can form the exchange interaction with the nearest electron from this group. The new pair would be in one of the triplet states. Switching the magnetic field could provide the transition of this pair to the S-state,
(6)
in which the Ca+ ion may capture the second electron and become the Ca0 atom. Thus, now, we get three neutral atoms in the dislocation core excluded from the Coulomb interaction. They might react into the covalent molecule Cl0 + Ca0 + Cl0 = CaCl2. This molecule may have radically distinct properties from the PbCl2. For instance, the first one might occur more flexible that would facilitate depinning, while the other one might become more rigid that would vice versa trap the dislocation. Thus, it could be no wonder that the NaCl(Ca) crystal manifests magnetoplasticity and the NaCl(Pb) – magnetic hardening.

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.

For the considered range of small external magnetic field BBloc, 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.

To be exact, for the free atom or ion, the angle moment J is in turn the sum of the total orbital moment (L) and electron spin angle moment (S): J = L + S. In our case, it appears more adequate to apply the approach of the theory of crystalline field where the moment J is replaced by the phenomenological effective spin S̃ to be found experimentally. Taking into account that we deal with the radical electron pair with total spin 1 and the Cl ligands with total nuclear spin 3/2, it is natural to use in further considerations the parameters
(7)
We start from the characterization of the energy levels Wα of hyperfine interaction at B = 0. In terms of (7), the corresponding possible projections of F on the local field Bloc are equal,49 
(8)
The energy levels Wα (α = I, II, III) corresponding to (7) and (8) are given by the following expression:49 
(9)
where A is the hyperfine interaction constant.
Now, let us estimate the local fields Blocα related to the energies Wα (9). It can be done by the formal expressing Wα in terms of Zeeman energies,
(10)
where μFα is the magnetic dipole moment precessing about the magnetic field Blocα,
(11)
and the gFα-factor is equal,49 
(12)
Combining Eqs. (8)(12), one finds the three levels of the local magnetic field,
(13)
Thus, the magnitudes of the local fields differ in proportion 3:5:15.
Each of three energy levels (9) at B = 0 is (2Fα + 1) multiple degenerates, i.e., 6-, 4-, and 2-multiple, respectively. Switching on the weak magnetic field B removes the degeneracies and splits the energy levels (Fig. 4). If the applied and local fields are parallel (B||Bloc), the stationary component of Fα is again parallel to the z axis of quantization (F̄α||B||z) and the additional Zeeman energy may be again found based on Eqs. (10) and (11),49 
(14)
Here, quantizing Fzα in (14) takes into account the mentioned above splitting of levels,
(15)
FIG. 4.

Hyperfine energy spectrum in the range of low magnetic fields BBloc for the system S̃=1,I=3/2.

FIG. 4.

Hyperfine energy spectrum in the range of low magnetic fields BBloc for the system S̃=1,I=3/2.

Close modal

The ∓ for FzIII relates to the negative gFIII in (12), which, in turn, corresponds to the inequality WZIII(1/2) > WZIII(+1/2) (see also Fig. 4).

Let us suppose now that the magnetic field B is not parallel to Bloc being as before small. In this case, it is convenient to choose a quantization axis z along the effective field Beff = Bloc + B. The precession of the vectors μFα and Fα in (11) occurs about this new direction. Now, the Zeeman energy is determined by the scalar product of the stationary component of Fα with the field B directed not along z,
(16)
The problem can be easily solved due to the occurrence of the small parameter B/Bloc, which provides a small change in the quantization z axis orientation for any direction of B. We introduce the two auxiliary magnetic fields defined by BB + B with BBloc and BBloc (Fig. 5). Let us find the key scalar product in (16),
(17)
where the angle (in rad) γB/Bloc ≪ 1. With (17), the approximate extension of Eq. (14) for arbitrary orientation of the external field B is reduced to the replacement in (14),
(18)
with the angle θ = ∠(B, Bloc) [see Fig. 5 and Eq. (3)]. This approximation remains valid until
(19)
FIG. 5.

Scheme of the layout of vectors Bloc, B = B + B, Beff, and F̄α.

FIG. 5.

Scheme of the layout of vectors Bloc, B = B + B, Beff, and F̄α.

Close modal

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 Blocα. 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 Blocα ∼ 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.

Now, we are ready to describe theoretically all hyperfine energy levels related to our results,
(20)
Here, the subscripts on the left relate equations corresponding to appropriate levels in Fig. 4; the first subscript in all pairs corresponds to the upper sign in ± on the right. The intervals Δα between neighboring levels in each group of terms are equidistant and do not contain A,
(21)

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 Blocα.

The previous experiments in our group47,51 have shown that their orientations are directly associated with symmetry directions in the crystal. In the case of NaCl(Ni),51 these were directions of symmetry axes 4 (⟨100⟩) and 2 (⟨110⟩). As will be shown, in our spectrum, the direction along axis 3 (⟨111⟩) also manifests itself but at frequencies beyond the range of studies in Ref. 51. Below, we will consider the angle θ to be acute because of ± in (20). Thus, with our permanent orientation BEarth||4 ([001]), the three possible directions of the local fields Blocα along symmetry axes 4, 2, and 3 will, respectively, relate to
(22)
Strictly speaking, relations (22) are also approximate because the directions ⟨100⟩, ⟨110⟩, and ⟨111⟩ in the vicinity of dislocations might be somewhat distorted. However, as we will see, the effect of these distortions on our spectrum proves to be reasonably small. Thus, the projections B in (20) and (21) may be considered in the three possible variants,
(23)

Certainly, each realization of (23) must create its spectrum of the type shown in Fig. 4.

In Fig. 4, at each energy level, there presented the list of possible spin states (I, S̃) related to the given state Fzα. Only two levels, WaI and WfI, 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.

Thus, in terms of the presented theory, the resonance frequencies (at given gn and B|| = BEarth) are equal in downward order,
(24)
Of course, alongside with (24), there are also two other sets of frequencies related to fields B̂ and B in (23). It is convenient to express all the frequencies of our spectrum in their ratios to νnA with the same number n,
(25)
Thus, by choosing some definite set of experimental frequencies νnA, one can predict the whole spectrum using relations (25). Comparison of the result with experimental data will show how lucky was the choice. Taking into account the number of peaks in Fig. 3, the criterion looks to be rather convincing. Following Ref. 51, we checked the variant of the first nine resonances in Fig. 3 to the side of decreasing frequencies (in MHz),
(26)
The comparison of Eqs. (25) and (26) with experiments presented in Table I appear to be quite satisfactory. Our alternative attempts of variating the number of frequencies in the basic list (26) have made convincingly worse agreement with the experiment. Thus, we can accept the theoretical arrangement based on the list (26). This makes it possible also to find the set gn by substitution (26) into the first equation in (24),
(27)
TABLE I.

Theoretical resonance frequencies (νthA,B,, ν̂thA,B,, and νthA,B,) of sets (25) and related close positions (νexpA,B,, ν̂expA,B,, and νexpA,B,) of experimental peaks in Fig. 3 (all frequencies in MHz).

Setgn
1.841.771.641.551.491.471.431.391.36
νthA=4ΔnI=νexpA 2.06 1.98 1.84 1.74 1.67 1.64 1.60 1.56 1.52 
ν̂thAνexpA/2 1.457 1.400 1.301 1.230 1.181 1.160 1.131 1.103 1.075 
ν̂expA 1.44 1.40 1.30 1.25 1.18 1.14 1.11 1.07 
νthAνexpA/3 1.189 1.143 1.062 1.004 0.964 0.947 0.924 0.901 0.878 
νexpA 1.18 1.14 1.07 1.00 0.98 0.93 0.91 0.87 
νthB=νexpA34 1.545 1.485 1.380 1.305 1.252 1.230 1.200 1.170 1.140 
νexpB 1.56 1.48 1.38 1.30 1.25 1.18 1.14 
ν̂thBνthB/2 1.092 1.047 0.972 0.919 0.884 0.870 0.834 0.827 0.806 
ν̂expB 1.09 1.04 0.98 0.93 0.89 0.87 0.84 0.81 
νthBνthB/3 0.892 0.857 0.797 0.753 0.723 0.710 0.693 0.675 0.658 
νexpB 0.89 0.87 0.81 0.75 0.73 0.68 
νthC=νexpA24 1.030 0.990 0.920 0.870 0.835 0.820 0.800 0.780 0.760 
νexpC 1.04 0.98 0.91 0.87 0.84 0.81 0.78 0.75 
ν̂thÇνthC/2 0.735 0.700 0.643 0.615 0.594 0.58 0.566 0.552 0.544 
ν̂expC 0.73 0.68 0.65 0.63 0.60 0.58 0.56 
νthCνexpC/3 0.600 0.566 0.525 0.502 0.485 0.473 0.462 0.450 0.433 
νexpC 0.60 0.56 0.52 0.49  0.47  0.45 0.41 
νthD=νexpA14 0.515 0.495 0.460 0.435 0.420 0.410 0.400 0.390 0.380 
νexpD 0.52 0.49 0.47 0.45 0.41 0.38 
ν̂thDνthD/2 0.368 0.346 0.332 0.307 0.297 0.290 0.283 0.276 0.269 
ν̂expD 0.38 0.34  0.32 0.30 0.28 0.25 
νthDνthD/3 0.297 0.283 0.266 0.251 0.242 0.237 0.230 0.225 0.219 
νexpD 0.30 0.28   0.25   0.21 
νthE=νexpA13 0.687 0.660 0.613 0.580 0.560 0.547 0.533 0.520 0.507 
νexpE 0.68 0.65 0.60 0.58 0.56 0.52 0.49 
ν̂thE=νthE/2 0.486 0.467 0.433 0.410 0.396 0.387 0.377 0.368 0.358 
ν̂expE 0.49 0.47 0.45 0.41 0.38 0.34 
νthE=νthE/3 0.397 0.381 0.354 0.335 0.323 0.316 0.308 0.300 0.293 
νexpE 0.41 0.38 0.34 0.32 0.30 
νthF=νexpA16 0.343 0.328 0.306 0.290 0.280 0.273 0.266 0.260 0.253 
νexpF 0.34 0.32 0.30  0.28  0.25 
ν̂thFνthF/2 0.243 0.232 0.216 0.205 0.198 0.193 0.188 0.184 0.179 
ν̂expF 0.25  0.21   0.17 
νthFνthF/3 0.198 0.189 0.177 0.167 0.162 0.158 0.154 0.150 0.146 
νexpF 0.21  0.17   0.15 
νthG=νexpA512 0.859 0.825 0.767 0.725 0.696 0.683 0.667 0.649 0.633 
νexpG 0.87 0.84 0.78 0.73  0.68  0.65 0.63 
ν̂thG=νthG/2 0.607 0.583 0.542 0.513 0.492 0.483 0.472 0.459 0.448 
ν̂expG 0.60 0.58 0.52 0.49 0.47 0.45 
νthG=νthG/3 0.495 0.476 0.443 0.419 0.402 0.394 0.385 0.375 0.365 
νexpG 0.49 0.47 0.45 0.41   0.38 
Setgn
1.841.771.641.551.491.471.431.391.36
νthA=4ΔnI=νexpA 2.06 1.98 1.84 1.74 1.67 1.64 1.60 1.56 1.52 
ν̂thAνexpA/2 1.457 1.400 1.301 1.230 1.181 1.160 1.131 1.103 1.075 
ν̂expA 1.44 1.40 1.30 1.25 1.18 1.14 1.11 1.07 
νthAνexpA/3 1.189 1.143 1.062 1.004 0.964 0.947 0.924 0.901 0.878 
νexpA 1.18 1.14 1.07 1.00 0.98 0.93 0.91 0.87 
νthB=νexpA34 1.545 1.485 1.380 1.305 1.252 1.230 1.200 1.170 1.140 
νexpB 1.56 1.48 1.38 1.30 1.25 1.18 1.14 
ν̂thBνthB/2 1.092 1.047 0.972 0.919 0.884 0.870 0.834 0.827 0.806 
ν̂expB 1.09 1.04 0.98 0.93 0.89 0.87 0.84 0.81 
νthBνthB/3 0.892 0.857 0.797 0.753 0.723 0.710 0.693 0.675 0.658 
νexpB 0.89 0.87 0.81 0.75 0.73 0.68 
νthC=νexpA24 1.030 0.990 0.920 0.870 0.835 0.820 0.800 0.780 0.760 
νexpC 1.04 0.98 0.91 0.87 0.84 0.81 0.78 0.75 
ν̂thÇνthC/2 0.735 0.700 0.643 0.615 0.594 0.58 0.566 0.552 0.544 
ν̂expC 0.73 0.68 0.65 0.63 0.60 0.58 0.56 
νthCνexpC/3 0.600 0.566 0.525 0.502 0.485 0.473 0.462 0.450 0.433 
νexpC 0.60 0.56 0.52 0.49  0.47  0.45 0.41 
νthD=νexpA14 0.515 0.495 0.460 0.435 0.420 0.410 0.400 0.390 0.380 
νexpD 0.52 0.49 0.47 0.45 0.41 0.38 
ν̂thDνthD/2 0.368 0.346 0.332 0.307 0.297 0.290 0.283 0.276 0.269 
ν̂expD 0.38 0.34  0.32 0.30 0.28 0.25 
νthDνthD/3 0.297 0.283 0.266 0.251 0.242 0.237 0.230 0.225 0.219 
νexpD 0.30 0.28   0.25   0.21 
νthE=νexpA13 0.687 0.660 0.613 0.580 0.560 0.547 0.533 0.520 0.507 
νexpE 0.68 0.65 0.60 0.58 0.56 0.52 0.49 
ν̂thE=νthE/2 0.486 0.467 0.433 0.410 0.396 0.387 0.377 0.368 0.358 
ν̂expE 0.49 0.47 0.45 0.41 0.38 0.34 
νthE=νthE/3 0.397 0.381 0.354 0.335 0.323 0.316 0.308 0.300 0.293 
νexpE 0.41 0.38 0.34 0.32 0.30 
νthF=νexpA16 0.343 0.328 0.306 0.290 0.280 0.273 0.266 0.260 0.253 
νexpF 0.34 0.32 0.30  0.28  0.25 
ν̂thFνthF/2 0.243 0.232 0.216 0.205 0.198 0.193 0.188 0.184 0.179 
ν̂expF 0.25  0.21   0.17 
νthFνthF/3 0.198 0.189 0.177 0.167 0.162 0.158 0.154 0.150 0.146 
νexpF 0.21  0.17   0.15 
νthG=νexpA512 0.859 0.825 0.767 0.725 0.696 0.683 0.667 0.649 0.633 
νexpG 0.87 0.84 0.78 0.73  0.68  0.65 0.63 
ν̂thG=νthG/2 0.607 0.583 0.542 0.513 0.492 0.483 0.472 0.459 0.448 
ν̂expG 0.60 0.58 0.52 0.49 0.47 0.45 
νthG=νthG/3 0.495 0.476 0.443 0.419 0.402 0.394 0.385 0.375 0.365 
νexpG 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 νnA (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 ν̂thA and ν̂expA, the second is νthA and νexpA, 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, νnC=νnC (identically), ν̂nEνnG0.24νnA, νnEν̂nG0.29νnA, 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 ν̂expA contains only eight resonance peaks. The same occurs with the sets νexpA, ν̂expB, and νexpC. The sets νexpB, ν̂expC, νexpC, νexpE, and νexpG contain seven resonance peaks, and the sets νexpB, ν̂expD, and ν̂expG, only six of them, and so on, down to three peaks in two sets, ν̂expF and νexpF, 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 Â67, A2, B9 described by the theoretical frequencies ν̂6,7A,ν2A,ν9B (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: A2 for ν2A. In addition, the table also shows for each peak its amplitudes in the scale lρ 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.

TABLE II.

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)νth̄ (MHz)Constituentslρ
2.06 2.06 A1 1.04 
1.98 1.98 A2 0.70 
1.84 1.84 A3 0.69 
1.74 1.74 A4 0.67 
1.67 1.67 A5 0.62 
1.64 1.64 A6 0.75 
1.60 1.60 A7 0.76 
1.56 1.553 A8, B1 0.82 
1.52 1.520 A9 0.92 
10 1.48 1.485 B2 0.56 
11 1.44 1.457 Â1 1.24 
12 1.40 1.400 Â2 0.79 
13 1.38 1.380 B3 1.29 
14 1.30 1.303 Â3, B4 1.14 
15 1.25 1.237 Â4, B56 1.09 
16 1.18 1.185 Â5, A1, B78 0.52 
17 1.14 1.143 Â67, A2, B9 0.77 
18 1.11 1.103 Â8 0.62 
19 1.09 1.092 B̂1 0.57 
20 1.07 1.069 Â9, A3 0.56 
21 1.04 1.039 B̂2,C1 0.61 
22 1.00 1.004 A4 0.59 
23 0.98 0.975 A5, B̂3,C2 0.96 
24 0.93 0.933 A6, B̂4 0.41 
25 0.91 0.915 A78, C3 0.67 
26 0.89 0.888 B̂5,B1 0.52 
27 0.87 0.867 A9,B̂6,B2,C4,G1 0.71 
28 0.84 0.830 B̂78,C5,G2 0.74 
29 0.81 0.806 B̂9,B3, C67 0.88 
30 0.78 0.774 C8, G3 0.89 
31 0.75 0.757 B4,C9 0.60 
32 0.73 0.728 B5,Ĉ1,G4 1.28 
33 0.68 0.685 B69,Ĉ2,E1,G57 0.76 
34 0.65 0.651 Ĉ3,E2,G8 0.89 
35 0.63 0.624 Ĉ4,G9 0.91 
36 0.60 0.603 Ĉ5,C1,E3,Ĝ1 0.92 
37 0.58 0.581 Ĉ6,E4,Ĝ2 0.98 
38 0.56 0.556 Ĉ79,C2,E56 0.78 
39 0.52 0.524 C3,D1,E78,Ĝ34 0.94 
40 0.49 0.493 C45,D2,E9,Ê1,Ĝ56,G1 0.55 
41 0.47 0.468 C67,D3,Ê2,Ĝ7,G2 0.51 
42 0.45 0.445 C8,D4,Ê3,Ĝ89,G3 0.55 
43 0.41 0.413 C9,D57,Ê4,E1,G4 0.60 
44 0.38 0.382 D89,D̂1,Ê58,E2,G59 0.79 
45 0.34 0.344 D̂2,E34,F1 0.54 
46 0.32 0.321 D̂34,E56, F2 0.46 
47 0.30 0.299 D̂5,D1,E79, F34 0.43 
48 0.28 0.279 D̂67,D2, F57 0.58 
49 0.25 0.250 D̂89,D37, F89,F̂12 0.89 
50 0.21 0.206 D89, F̂36,F12 0.91 
51 0.17 0.174 F̂79,F36 0.78 
52 0.15 0.150 F79 0.82 
53 0.12   0.93 
54 0.06   0.95 
55 0.03   1.08 
No.νexp (MHz)νth̄ (MHz)Constituentslρ
2.06 2.06 A1 1.04 
1.98 1.98 A2 0.70 
1.84 1.84 A3 0.69 
1.74 1.74 A4 0.67 
1.67 1.67 A5 0.62 
1.64 1.64 A6 0.75 
1.60 1.60 A7 0.76 
1.56 1.553 A8, B1 0.82 
1.52 1.520 A9 0.92 
10 1.48 1.485 B2 0.56 
11 1.44 1.457 Â1 1.24 
12 1.40 1.400 Â2 0.79 
13 1.38 1.380 B3 1.29 
14 1.30 1.303 Â3, B4 1.14 
15 1.25 1.237 Â4, B56 1.09 
16 1.18 1.185 Â5, A1, B78 0.52 
17 1.14 1.143 Â67, A2, B9 0.77 
18 1.11 1.103 Â8 0.62 
19 1.09 1.092 B̂1 0.57 
20 1.07 1.069 Â9, A3 0.56 
21 1.04 1.039 B̂2,C1 0.61 
22 1.00 1.004 A4 0.59 
23 0.98 0.975 A5, B̂3,C2 0.96 
24 0.93 0.933 A6, B̂4 0.41 
25 0.91 0.915 A78, C3 0.67 
26 0.89 0.888 B̂5,B1 0.52 
27 0.87 0.867 A9,B̂6,B2,C4,G1 0.71 
28 0.84 0.830 B̂78,C5,G2 0.74 
29 0.81 0.806 B̂9,B3, C67 0.88 
30 0.78 0.774 C8, G3 0.89 
31 0.75 0.757 B4,C9 0.60 
32 0.73 0.728 B5,Ĉ1,G4 1.28 
33 0.68 0.685 B69,Ĉ2,E1,G57 0.76 
34 0.65 0.651 Ĉ3,E2,G8 0.89 
35 0.63 0.624 Ĉ4,G9 0.91 
36 0.60 0.603 Ĉ5,C1,E3,Ĝ1 0.92 
37 0.58 0.581 Ĉ6,E4,Ĝ2 0.98 
38 0.56 0.556 Ĉ79,C2,E56 0.78 
39 0.52 0.524 C3,D1,E78,Ĝ34 0.94 
40 0.49 0.493 C45,D2,E9,Ê1,Ĝ56,G1 0.55 
41 0.47 0.468 C67,D3,Ê2,Ĝ7,G2 0.51 
42 0.45 0.445 C8,D4,Ê3,Ĝ89,G3 0.55 
43 0.41 0.413 C9,D57,Ê4,E1,G4 0.60 
44 0.38 0.382 D89,D̂1,Ê58,E2,G59 0.79 
45 0.34 0.344 D̂2,E34,F1 0.54 
46 0.32 0.321 D̂34,E56, F2 0.46 
47 0.30 0.299 D̂5,D1,E79, F34 0.43 
48 0.28 0.279 D̂67,D2, F57 0.58 
49 0.25 0.250 D̂89,D37, F89,F̂12 0.89 
50 0.21 0.206 D89, F̂36,F12 0.91 
51 0.17 0.174 F̂79,F36 0.78 
52 0.15 0.150 F79 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.

FIG. 6.

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.

FIG. 6.

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.

Close modal

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.

On the other hand, we have sufficient data in order to estimate distortions occurring in the close-packed directions along ⟨110⟩. Until now, we supposed that these distortions can be neglected and with this assumption put the corresponding angle between magnetic fields equal θ̂=45°. Let us check this assumption. According to Fig. 6 and Table II, there are four single (resolved) peaks Â1, Â2, Â8, and B̂1 related to this direction. In our terms, experimental estimate of the angle θnA,B is determined by the following equation:
(28)
Substituting here the data from Table I, one can easily estimate the corresponding angles,
(29)

It is seen from (29) that the inclinations of found angles θnA,B 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 [01̄1] 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), θ2Aθ1B and θ1Aθ8A, could relate to those two expectations.

FIG. 7.

Schematic structure of the edge dislocation core in the plane (010) of NaCl crystal.

FIG. 7.

Schematic structure of the edge dislocation core in the plane (010) of NaCl crystal.

Close modal

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.

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 BBloc 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 BBloc, most of transitions between energy levels are forbidden. The allowed transitions are limited by the condition: ΔS̃=±1, 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 BBloc, 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, lρ 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.

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.

The authors have no conflicts to disclose.

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).

The data that support the findings of this study are available within the article.

1.
Y. B.
Zel’dovich
,
A. L.
Buchachenko
, and
E. L.
Frankevich
, “
Magnetic-spin effects in chemistry and molecular physics
,”
Usp. Fiz. Nauk
155
,
3
(
1988
) [Sov. Phys. Usp. 31, 385 (1988)].
2.
A. L.
Buchachenko
, “
Chemistry on the border of two centuries—Achievements and prospects
,”
Usp. Khimii
68
,
99
(
1999
) [Russ. Chem. Rev. 68, 85 (1999)].
3.
K. M.
Salikhov
,
Y. N.
Molin
,
A. L.
Buchachenko
, and
R. Z.
Sagdeev
,
Spin Polarization and Magnetic Effects in Radical Reaction
(
Elsevier
,
Amsterdam
,
1984
).
4.
U. E.
Steiner
and
T.
Ulrich
, “
Magnetic field effects in chemical kinetics and related phenomena
,”
Chem. Rev.
89
,
51
(
1989
).
5.
A. L.
Buchachenko
,
Magneto-Biology and Medicine
(
Nova Science Publishers
,
New York
,
2014
).
6.
V. I.
Alshits
,
E. V.
Darinskaya
,
T. M.
Perekalina
, and
A. A.
Urusovskaya
, “
Dislocation motion in NaCl crystals under static magnetic field
,”
Fiz. Tverd. Tela
29
,
467
(
1987
) [Sov. Phys.-Solid State 29, 265 (1987)].
7.
V. I.
Alshits
,
E. V.
Darinskaya
,
M. V.
Koldaeva
, and
E. A.
Petrzhik
, “
Magnetoplastic effect: Basic properties and physical mechanisms
,”
Kristallografiya
48
,
826
(
2003
) [Crystallogr. Rep. 48, 768 (2003)].
8.
A. A.
Urusovskaya
,
V. I.
Alshits
,
A. E.
Smirnov
, and
N. N.
Bekkauer
, “
The influence of magnetic effects on the mechanical properties and real structure of nonmagnetic crystals
,”
Kristallografiya
48
,
855
(
2003
) [Crystallography Reports 48, 796 (2003)].
9.
Y. I.
Golovin
, “
Magnetoplasticity of solids
,”
Fiz. Tverd. Tela
46
,
769
(
2004
) [Phys. Solid State 46, 789 (2004)].
10.
R. B.
Morgunov
, “
Spin micromechanics in the physics of plasticity
,”
Usp. Fiz. Nauk
174
,
131
(
2004
) [Phys.-Usp. 47, 125 (2004)].
11.
V. I.
Alshits
,
E. V.
Darinskaya
,
M. V.
Koldaeva
, and
E. A.
Petrzhik
, “
Magnetoplastic effect in nonmagnetic crystals
,” in
Dislocations in Solids
, edited by
J. P.
Hirth
(
Elsevier
,
Amsterdam
,
2008
), Vol.
14
, pp.
333
438
.
12.
V. I.
Alshits
,
E. V.
Darinskaya
,
M. V.
Koldaeva
, and
E. A.
Petrzhik
, “
Electric stimulation of magnetoplasticity and magnetic hardening in crystals
,”
Pis'ma Zh. Eksp. Teor. Fiz.
88
,
500
(
2008
) [JETP Lett. 88, 428 (2008)].
13.
V. I.
Alshits
,
E. V.
Darinskaya
,
M. V.
Koldaeva
, and
E. A.
Petrzhik
, “
Electric amplification of the magnetoplastic effect in nonmagnetic crystals
,”
J. Appl. Phys.
105
,
063520
(
2009
).
14.
Y. I.
Golovin
,
R. B.
Morgunov
,
D. V.
Lopatin
, and
A.
Baskakov
, “
Influence of a strong magnetic field pulse on NaCl crystal microhardness
,”
Phys. Status Solidi A
160
,
R3
(
1997
).
15.
E. V.
Darinskaya
,
E. A.
Petrzhik
,
Y. M.
Ivanov
,
S. A.
Erofeeva
, and
M. R.
Raukhman
, “
Magnetostimulated softening and hardening of semiconductors
,”
Phys. Status Solidi C
2
,
1873
(
2005
).
16.
R. B.
Morgunov
and
A. L.
Buchachenko
, “
Magnetic field response of NaCl:Eu crystal plasticity due to spin-dependent Eu2+ aggregation
,”
Phys. Rev. B
82
,
014115
(
2010
).
17.
E. V.
Darinskaya
,
M. V.
Koldaeva
,
V. I.
Alshits
,
I. M.
Pritula
, and
A. E.
Voloshin
, “
Relaxation kinetics of the microhardness of KDP crystals after their exposure to a magnetic field
,”
Pis'ma Zh. Eksp. Teor. Fiz.
108
,
236
(
2018
) [JETP Lett. 108, 231 (2018)].
18.
E. A.
Petrzhik
,
E. S.
Ivanova
, and
V. I.
Alshits
, “
Changes in the microhardness and dielectric permittivity of TGS crystals after their exposure to a static magnetic field or ultralow crossed fields in the EPR scheme
,”
Izv. Ross. Akad. Nauk. Ser. Fiz.
78
,
1305
(
2014
) [Bull. Russ. Acad. Sci., Phys. Ser. 78, 1052 (2014)].
19.
E. D.
Yakushkin
, “
Dielectric response of a uniaxial ferroelectric in a magnetic field
,”
Pis’ma Zh. Eksp. Teor. Fiz.
99
,
483
(
2014
) [JETP Lett. 99, 415 (2014)].
20.
R. V.
Gainutdinov
,
E. S.
Ivanova
,
E. A.
Petrzhik
,
A. K.
Lashkova
, and
T. R.
Volk
, “
Magnetic memory effects in triglycine sulfate ferroelectric crystals
,”
Pis'ma Zh. Eksp. Teor. Fiz.
106
,
84
(
2017
) [JETP Lett. 106, 97 (2017)].
21.
E. D.
Yakushkin
and
V. A.
Sandler
, “
Phase transition in CsHSO4 ferroelastic in a magnetic field
,”
Pis'ma Zh. Eksp. Teor. Fiz.
113
,
348
(
2021
) [JETP Lett. 113, 352 (2021)].
22.
E. D.
Yakushkin
, “
Polarization switching of a Rochelle salt single crystal in a magnetic field
,”
Pis'ma Zh. Eksp. Teor. Fiz.
117
,
598
(
2023
) [JETP Lett. 117, 593 (2023)].
23.
I. S.
Volchkov
,
V. M.
Kanevskii
, and
M. D.
Pavlyuk
, “
Effect of weak magnetic fields on the electric properties of CdTe crystals
,”
Pis'ma Zh. Eksp. Teor. Fiz.
107
,
276
(
2018
) [JETP Lett. 107, 269 (2018)].
24.
V. I.
Belyavsky
and
M. N.
Levin
, “
Spin effects in defect reactions
,”
Phys. Rev. B
70
,
104101
(
2004
).
25.
V. I.
Belyavsky
,
M. N.
Levin
, and
N. J.
Olson
, “
Defect-induced lattice magnetism: Phenomenology of magnetic-field-stimulated defect reactions in nonmagnetic solids
,”
Phys. Rev. B
73
,
054429
(
2006
).
26.
M.
Molotskii
and
V.
Fleurov
, “
Spin effects in plasticity
,”
Phys. Rev. Lett.
78
,
2779
(
1997
).
27.
M.
Molotskii
and
V.
Fleurov
, “
Manifestations of hyperfine interaction in plasticity
,”
Phys. Rev. B
56
,
10809
(
1997
).
28.
M. I.
Molotskii
, “
Theoretical basis for electro- and magnetoplasticity
,”
Mater. Sci. Eng.: A
287
,
248
(
2000
).
29.
A. L.
Buchachenko
, “
Effect of magnetic field on mechanics of nonmagnetic crystals: The nature of magnetoplasticity
,”
Zh. Eksp. Teor. Fiz.
129
,
909
(
2006
) [JETP 102, 795 (2006)].
30.
A. L.
Buchachenko
, “
The physical kinetics of magnetoplasticity of diamagnetic crystals
,”
Zh. Eksp. Teor. Fiz.
132
,
827
(
2007
) [JETP 105, 722 (2007)].
31.
R. B.
Morgunov
and
A. L.
Buchachenko
, “
Magnetoplasticity and magnetic memory in diamagnetic solids
,”
Zh. Eksp. Teor. Fiz.
136
,
505
(
2009
) [JETP 109, 434 (2009)].
32.
V. I.
Alshits
,
E. V.
Darinskaya
,
M. V.
Koldaeva
,
R. K.
Kotowski
,
E. A.
Petrzhik
, and
P.
Tronczyk
, “
Dislocation kinetics in nonmagnetic crystals: A look through a magnetic window
,”
Usp. Fiz. Nauk
187
,
327
(
2017
) [Phys.-Usp. 60, 305 (2017)].
33.
Y. I.
Golovin
,
R. B.
Morgunov
,
V. E.
Ivanov
,
S. E.
Zhulikov
, and
A. A.
Dmitrievskii
, “
Electron paramagnetic resonance in a subsystem of structural defects as a factor in the plasticization of NaCl crystals
,”
Pis'ma Zh. Eksp. Teor. Fiz.
68
,
400
(
1998
) [JETP Lett. 68, 426 (1998)].
34.
Y. I.
Golovin
,
R. B.
Morgunov
,
V. E.
Ivanov
, and
A. A.
Dmitrievskii
, “
Radio-frequency paramagnetic resonance spectra, detected from dislocation displacement in NaCl single crystals
,”
Fiz. Tverd. Tela
41
,
1778
(
1999
) [Phys. Solid State 41, 1631 (1999)].
35.
Y. I.
Golovin
,
R. B.
Morgunov
,
V. E.
Ivanov
, and
A. A.
Dmitrievskii
, “
Softening of ionic crystals as a result of a change in the spin states of structural defects under paramagnetic resonance conditions
,”
Zh. Eksp. Teor. Fiz.
117
,
1080
(
2000
) [JETP 90, 939 (2000)].
36.
Y.
Golovin
,
R.
Morgunov
, and
A.
Baskakov
, “
Magnetoresonant softening of solids
,”
Mol. Phys.
100
,
1291
(
2002
).
37.
Y. A.
Osip’yan
,
R. B.
Morgunov
,
A. A.
Baskakov
,
A. M.
Orlov
,
A. A.
Skvortsov
,
E. N.
Inkina
, and
J.
Tanimoto
,
Pis'ma Zh. Eksp. Teor. Fiz.
79
,
158
(
2004
) [JETP Lett. 79, 126 (2004)].
38.
M. V.
Badylevich
,
V. V.
Kveder
,
V. I.
Orlov
, and
Y. A.
Ossipyan
, “
Spin-resonant change of unlocking stress for dislocations in silicon
,”
Phys. Status Solidi C
2
,
1869
(
2005
).
39.
V. I.
Alshits
,
E. V.
Darinskaya
,
V. A.
Morozov
,
V. M.
Kats
, and
A. A.
Lukin
, “
ESR in the Earth’s magnetic field as a cause of dislocation motion in NaCl crystals
,”
Pis'ma Zh. Eksp. Teor. Fiz.
91
,
97
(
2010
) [JETP Lett. 91, 91 (2010)].
40.
V. I.
Alshits
,
E. V.
Darinskaya
,
V. A.
Morozov
,
V. M.
Kats
, and
A. A.
Lukin
, “
Resonant dislocation motions in NaCl crystals under EPR conditions in the Earth’s magnetic field with a radio-frequency pump field
,”
Fiz. Tverd. Tela
53
,
2010
(
2011
) [Phys. Solid State 53, 2117 (2011)].
41.
V. I.
Alshits
,
E. V.
Darinskaya
,
V. A.
Morozov
,
V. M.
Kats
, and
A. A.
Lukin
, “
Resonant dislocation motion in NaCl crystals in the EPR scheme in the Earth’s magnetic field with pulsed pumping
,”
Fiz. Tverd. Tela
55
,
2176
(
2013
) [Phys. Solid State 55, 2289 (2013)].
42.
V. I.
Alshits
,
E. V.
Darinskaya
,
M. V.
Koldaeva
, and
E. A.
Petrzhik
, “
Anisotropic resonant magnetoplasticity of NaCl crystals in the Earths magnetic field
,”
Fiz. Tverd. Tela
55
,
318
(
2013
) [Phys. Solid State 55, 358 (2013)].
43.
V. I.
Alshits
,
M. V.
Koldaeva
,
E. A.
Petrzhik
,
S. A.
Minyukov
,
E. V.
Darinskaya
,
D. E.
Kaputkin
, and
E. K.
Naimi
, “
Quartet of resonance peaks of dislocation displacements in NaCl crystals during their magnetic processing in the low-frequency EPR scheme
,”
Pis'ma Zh. Eksp. Teor. Fiz.
98
,
33
(
2013
) [JETP Lett. 98, 28 (2013)].
44.
V. I.
Alshits
,
M. V.
Koldaeva
,
E. A.
Petrzhik
,
A. Y.
Belov
, and
E. V.
Darinskaya
, “
Determination of the positions of impurity centers in a dislocation core in a NaCl crystal from magnetoplasticity spectra
,”
Pis'ma Zh. Eksp. Teor. Fiz.
99
,
87
(
2014
) [JETP Lett. 99, 82 (2014)].
45.
V. I.
Alshits
,
E. V.
Darinskaya
,
M. V.
Koldaeva
, and
E. A.
Petrzhik
, “
Resonance magnetoplasticity in ultralow magnetic fields
,”
Pis'ma Zh. Eksp. Teor. Fiz.
104
,
362
(
2016
) [JETP Lett. 104, 353 (2016)].
46.
V. I.
Alshits
,
E. V.
Darinskaya
,
M. V.
Koldaeva
, and
E. A.
Petrzhik
, “
Change in the microhardness of nonmagnetic crystals after their exposure to the Earth’s magnetic field and AC pump field in the EPR scheme
,”
Fiz. Tverd. Tela
54
,
305
(
2012
) [Phys. Solid State 54, 324 (2012)].
47.
V. I.
Alshits
,
M. V.
Koldaeva
, and
E. A.
Petrzhik
, “
Anisotropy of the resonance transformation of the microhardness of crystals after their exposure in the EPR scheme in the Earth’s magnetic field
,”
Pis'ma Zh. Eksp. Teor. Fiz.
107
,
650
(
2018
) [JETP Lett. 107, 618 (2018)].
48.
J. A.
Weil
and
J. R.
Bolton
,
Electron Paramagnetic Resonance: Elementary Theory and Practical Applications
(
Wiley-Interscience
,
New York
,
2007
).
49.
A.
Abragam
and
B.
Bleaney
,
Electron Paramagnetic Resonance of Transition Ions
(
Oxford University
,
London
,
1970
), Vol.
1
.
50.
P. J.
Alonso
,
J. C.
González
,
H. W.
den Hartog
, and
R.
Alcala
, “
EPR study of Ni+ centers in SrF2
,”
Phys. Rev. B
27
,
2722
(
1983
).
51.
E. A.
Petrzhik
and
V. I.
Alshits
, “
Magnetically induced resonance change in the microhardness of NaCl crystals
,”
Pis'ma Zh. Eksp. Teor. Fiz.
113
,
678
(
2021
) [JETP Lett. 113, 646 (2021)].
52.
V. A.
Zakrevskii
,
V. A.
Pakhotin
, and
A. V.
Shul’diner
, “
On the possible effect of a magnetic field on the breaking of mechanically loaded covalent chemical bonds
,”
Fiz. Tverd. Tela
44
,
1990
(
2002
) [Phys. Solid State 44, 2083 (2002)].
53.
Y.
Guo
,
Y. J.
Lee
,
Y.
Zhang
,
A.
Sorkin
,
S.
Manzhos
, and
H.
Wang
, “
Effect of a weak magnetic field on ductile–brittle transition in micro-cutting of single-crystal calcium fluoride
,”
J. Mater. Sci. Technol.
112
,
96
113
(
2022
).
54.
Y.
Guo
,
J.
Zhan
,
W. F.
Lu
, and
H.
Wang
, “
Mechanism in scratching of calcium fluoride with magneto-plasticity
,”
Int. J. Mech. Sci.
263
,
108768
(
2024
).
55.
Y. I.
Golovin
,
R. B.
Morgunov
,
S. E.
Zhulikov
,
V. A.
Kiperman
, and
D. A.
Lopatin
, “
Dislocations used to probe the defect state of an ionic crystal lattice excited by a pulsed magnetic field
,”
Fiz. Tverd. Tela
39
,
634
(
1997
) [Phys. Solid State 39, 554 (1997)].