Coherent spin relaxation in molecular magnets
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A p r Coherent spin relaxation in molecular magnets
V.I. Yukalov , V.K. Henner , , and P.V. Kharebov Bogolubov Laboratory of Theoretical Physics,Joint Institute for Nuclear Research, Dubna 141980, Russia Perm State University, Perm 614000, Russia University of Louisville, Louisville 40292, KY, USA
Abstract
Numerical modelling of coherent spin relaxation in nanomagnets, formed by mag-netic molecules of high spins, is accomplished. Such a coherent spin dynamics can berealized in the presence of a resonant electric circuit coupled to the magnet. Com-puter simulations for a system of a large number of interacting spins is an efficient toolfor studying the microscopic properties of such systems. Coherent spin relaxation isan ultrafast process, with the relaxation time that can be an order shorter than thetransverse spin dephasing time. The influence of different system parameters on therelaxation process is analysed. The role of the sample geometry on the spin relaxationis investigated.
PACS : 76.20.+q, 75.30.Gw, 75.45.+j, 75.50.Xx, 76.90.+d1
Introduction
A polarized magnet placed in an external magnetic field, whose direction is opposite tothe sample magnetization, comprises a strongly nonequilibrium system. The relaxation ofspins to their equilibrium position can occur in different ways. The simplest process is theslow relaxation during the time T caused by the spin-lattice coupling [1], when the totalmagnetization tends to zero. This is an incoherent process, since T is usually much longerthan the spin dephasing time T . A straightforward way to make the spin relaxation coherentwould be by imposing a strong transverse field simultaneously driving all spins, which wouldrepresent the process of free induction [1], lasting for the time T . This, however, is a rathertrivial process, when spins are practically independent of each other.A more elaborate situation arises if the magnet is inserted into a magnetic coil of aresonant electric circuit. Then the magnetic field, induced by the coil, provides an efficientfeedback mechanism organizing the coherent motion of spins [2]. It is possible to realizesix different regimes of coherent spin relaxation in a polarized sample coupled to a resonantcircuit: Collective induction, maser generation, pure superradiance, triggered superradiance,pulsing superradiance , and punctuated superradiance . A detailed description of these regimesis given in the review articles [3–5].Collective induction and maser generation with spins were realized in a number of ex-periments [6–9]. Pure spin superradiance was, first, observed in Dubna [10,11] and laterconfirmed by the groups in St. Petersburg [12,13] and Bonn [14]. Pulsing superradiance wasdemonstrated by experiments in Z¨urich [15,16]. The regime of punctuated superradiancewas suggested in Ref. [17] and, to our knowledge, has not yet been realized experimentally.A comprehensive survey of experiments can be found in Refs. [4,5,18].It is necessary to emphasize the principal role of the resonant electric circuit, coupled tothe spin sample, for realizing the regimes of spin superradiance. This makes the fundamentaldifference between the spin superradiance and the atomic superradiance [19]. The lattercan be achieved in a resonatorless system [19–21], though a resonant cavity can enhance theeffect [22,23]. Contrary to this, the spin superradiance, occurring in the radiofrequency range,cannot be realized without a resonator, which is due to the destructive role of the dipole spininteractions and to the absence of the feedback mechanism collectivizing the spin motion.This basic difference was emphasized in Ref. [19] and thoroughly explained in Refs. [5,24,25].One should also distinguish between coherent transient effects, caused by intense alternatingexternal fields, and having much in common for optical atomic systems [20,21], gammaradiation [26–29], as well as for spin samples [30–32], and superradiant phenomena, whenthe self-organization of coherence is the basic origin of the arising superradiance [25,33,34].Here we concentrate our attention on the superradiant regime of spin motion, which re-quires the presence of the resonant electric circuit, providing the feedback mechanism for thecollective self-organization of spin motion. The regimes of free induction, collective induction,and triggered spin superradiance could be considered in the frame of the phenomenologi-cal Bloch equations supplemented by the Kirchhoff equation for the circuit [35–38]. Thesephenomenological equations, however, are not applicable for describing pure spin superra-diance, when the coherent motion of spins develops in a self-organized way from initiallychaotic spin fluctuations. The complete theory, based on the microscopic spin Hamiltonian,and describing all regimes of spin relaxation has been developed in Refs. [18,25,33,34,39]and expounded in detail in the review articles [4,5]. This theory is in good agreement with2xperiment [4,5] as well as with numerical computer simulations [40–42] accomplished fornuclear or electron spins one half, S = 1 / S ∼
10. Thesemolecules compose molecular magnets whose properties are described at length in Refs.[5,18,44,45]. For example, the molecules Mn and Fe possess the spin S = 10. Themagnetic cluster compound Mn O Br (Et dbm) has the total spin S = 12 [46]. And themagnetic molecule Cr(CNMnL) (CIO ) , where L stands for a neutral pentadentate ligand,displays the effective spin S = 27 / / s. In our case, we consider a fixed external magnetic field, with a welldefined Zeeman frequency.The avalanche time of the magnetic moment in magnetic molecules is between 10 − sand 1 s [50–57,59]. This is rather slow process, as compared to the coherent spin relaxation,which is an ultrafast process, with characteristic relaxation times about 10 − − − s[5,18,24,25,49].The peculiarities of the fast superradiant-type spin relaxation in molecular magnets canbe successfully analyzed by means of computer modelling, which, to our knowledge, hasnot yet been accomplished. This approach provides a very efficient tool for studying themicroscopic properties of spin system. And it is the aim of the present paper to describe theresults of computer simulations for analyzing the features of the fast coherent relaxation ofmolecular spins, typical of the high-spin molecular magnets.The outline of the paper is as follows. In Sec. II, we present the main definitions and3quations to be employed in our computer modelling. In Sec. III, the results for a bulksample are analyzed. Since the dipolar spin interactions are anisotropic, it is interesting tostudy the related anisotropic geometric effects for different shapes of the magnetic samples.These geometric effects are investigated for a chain of molecular spins oriented either alongthe external magnetic field or perpendicular to it and for spin planes, with the externalmagnetic field being either perpendicular to it or lying in that plane. Section IV containsconclusions. We consider a spin sample characterized by the molecular spin vectors S j = { S xj , S yj , S zj } associated with the lattice sites enumerated by the index j = 1 , , . . . , N . An externalmagnetic field is directed along the z -axis, B = B e z . (1)This defines the Zeeman frequency ω ≡ − µ ¯ h B = 2¯ h µ B B , (2)in which µ = − µ B is the electron magnetic moment, with µ B being the Bohr magneton. Ingeneral, similarly to the magnetic-resonance setup, there can exist the transverse magneticfield, directed along the x -axis, B = B e x , B = h + h cos ωt , (3)and consisting of a constant field h and an alternating field h cos ωt .Molecular magnets possess the single-site magnetic anisotropy characterized by the anisotropyparameter D , which defines the anisotropy frequency ω D ≡ (2 S − D ¯ h , (4)where S is the molecular spin. The magnetic anisotropy exists for high spins, playing animportant role, while for S = 1 / D αβij ≡ µ r ij (cid:16) δ αβ − n αij n βij (cid:17) , (5)in which r ij ≡ | r ij | , n ij ≡ r ij r ij , r ij ≡ r i − r j . For what follows, it is convenient to introduce the dipolar coefficients a ij ≡ D zzij , b ij ≡ (cid:16) D xxij − D yyij − iD xyij (cid:17) c ij ≡ (cid:16) D xxij − iD yzij (cid:17) , (6)having dimensions of energy. 4he spin sample is inserted into a magnetic coil of an electric circuit characterized bythe circuit damping γ and the circuit natural frequency ω . Moving spins generate electriccurrent in the coil, which, in turn, produces the feedback magnetic field H acting on thespins. The generated electric current is described by the Kirchhoff equation. Choosing thecoil axis along the x -axis, so that H = H e x , the Kirchhoff equation can be rewritten [33,34]as the equation for the feedback magnetic field H , dHdt + 2 γH + ω Z t H ( t ′ ) dt ′ = − πη dm x dt , (7)where the effective electromotive force in the right-hand side of Eq. (7) is produced bymoving spins, with the average magnetization m x = µ V N X j =1 < S xj > , (8) V being the sample volume. The filling factor η in the right-hand side of Eq. (7) is approx-imately equal to η = V /V c , where V c is the coil volume. For what follows, without the lossof generality, we may assume the dense filling, with η = 1. Instead of Eq. (7), we can usethe equivalent differential equation d Hdt + 2 γ dHdt + ω H = − π d m x dt , (9)in which we set η = 1.All possible attenuation mechanisms have been carefully described in Ref. [25]. Thosethat influence the spin motion are as follows. The longitudinal attenuationΓ = γ + γ ∗ (10)is the sum of the spin-lattice attenuation γ , caused by spin-phonon interactions, and ofthe polarization pump rate γ ∗ , due to a stationary nonresonant pump, if any. The totaltransverse attenuation is Γ = γ (cid:16) − s (cid:17) + γ ∗ . (11)This includes the homogeneous broadening γ , renormalized by the factor (1 − s ), appearingin the case of strongly polarized spin systems [1,25], with s being the average spin polarizationreduced to the number of spins N and to the spin value S . The last term γ ∗ is the staticinhomogeneous broadening.In order to represent the equations of spin motion in a compact form, it is convenient tointroduce the ladder spin components S − j ≡ S xj − iS yj , S + j ≡ S xj + iS yj . (12)Also, we shall use the following notation: ξ i ≡ h X j ( = i ) (cid:16) a ij S zj + c ∗ ij S − j + c ij S + j (cid:17) , i ≡ h X j ( = i ) (cid:18) c ij S zj − a ij S − j + 2 b ij S + j (cid:19) . (13)The effective force, acting on the j -spin, can be written as f j ≡ − i ¯ h µ ( B + H ) + ξ j . (14)The derivation of equations of motion for the spin variables S − j , S + j , and S zj has beendescribed in great detail in Refs. [4,5,25]. The resulting equation for S − j reads as dS − j dt = − i (cid:16) ω + ξ j − i Γ (cid:17) S − j + f j S zj + i ω D S S zj S − j . (15)The equation for S + j is conjugate to Eq. (15). And the equation for S zj is dS zj dt = − (cid:16) f + j S − j + S + j f j (cid:17) − Γ (cid:16) S zj − ζ (cid:17) , (16)where ζ is the stationary spin polarization. From Eqs. (15) and (16), with notation (12),one can always return to the evolution equations for S xj , S yj , and S zj .In numerical simulations, one treats the spins S j as classical vectors [40–42]. It is conve-nient to work with the reduced quantities characterizing the reduced transverse spin variable u ≡ SN N X j =1 S − j (17)and the reduced longitudinal spin variable s ≡ SN N X j =1 S zj . (18)The spin variables (17) and (18) characterize the collective properties of a large number ofmagnetic molecules composing the molecular magnet. The time evolution of these variablesis prescribed by the equations of motion (15) and (16). This picture of collective spin motionis a generalization of the evolution equations for a single magnetic molecule. The study ofcollective coherent effects is the main aim of the present paper.In our numerical simulations, we solve the spin evolution equations (15) and (16) for afinite number of spins N . The resonator feedback field is given by Eq. (9), with the initialconditions H (0) = 0 , ˙ H (0) = 0 , (19)where the overdot implies the time derivative of H . The spin variables S αj at the initialtime are distributed randomly over the sample, so that to obtain a prescribed value s (0) ofthe spin polarization (18), while variable (17), for sufficiently high initial spin polarization,being negligible, s (0) = s , u (0) = 0 . (20)The external magnetic field (1), with B >
0, is aligned with e z . For the initial spinpolarization s >
0, the magnetic moment of the molecular sample M (0) = N Sµ s e z = − N Sµ B s e z is opposed to e z . That is, the considered molecular magnet is prepared in a strongly nonequi-librium initial state, from which it relaxes to a stationary state.6 Results of Computer Simulation
First, we consider the coherent spin relaxation in bulk samples, where spins are locatedin lattice sites of a cubic lattice. Computer simulations are accomplished for N = 125spins. For larger values of N , the results are qualitatively similar. The periodic boundaryconditions have been imposed. Since our aim is to study the self-organized process, weset zero the transverse field B = 0. We assume that the spin sample is without defects,so that the inhomogeneous broadening is negligible, γ ∗ = 0. For low temperatures, thespin-lattice interaction in molecular magnets is very weak, with the longitudinal relaxationtime T reaching months (see review articles [5,18,44,45]), because of which the attenuationparameter Γ plays no role, and we can set Γ = 0. It is convenient to deal with dimensionlessquantities measuring all frequencies in units of γ , so that we set γ = 1. And we shallmeasure time in units of γ − , that is, in units of T .First and foremost, we have to stress the necessity of the resonant circuit. When thelatter is absent, there is no fast relaxation at all. Then there could exist only the veryslow polarization decay during the time T , which is caused by spin interactions. Thesame slow relaxation happens if there is a circuit, but there is no resonance between itsnatural frequency ω and the Zeeman frequency ω . Therefore, in what follows, we always set ω = ω . This resonance condition is necessary, though not sufficient. To realize an effectivespin reversal, it is important that ω be much larger than the anisotropy frequency (4). Thecondition ω ≫ ω D ensures that the anisotropy does not induce an effective detuning fromthe resonance [5,18,24,25].Figure 1 illustrates how the spin reversal depends on the value of the Zeeman frequency ω . The larger the latter, the more pronounced is the spin reversal.The resonator damping γ defines the resonator ringing time τ ≡ /γ , during which themagnetic sample effectively interacts with the resonator. The relation between the resonatorringing time τ and the transverse relaxation time T essentially influences the spin reversal.When τ = T , then there is a permanent exchange of energy between the spin sample andthe resonator, so that the spin polarization oscillates around zero. When τ = 0 . T , thereoccurs a well pronounced reversal, hence the value γ = 10 is optimal for the latter. Andif τ = 0 . T , then the effective interaction time between the sample and resonator istoo short to realize a good reversal of polarization. The corresponding three qualitativelydifferent cases ate shown in Fig. 2.The magnitude of spin reversal also depends on the initial polarization. The larger theinitial value s , the stronger the spin reversal, which is illustrated in Fig. 3.Magnetic anisotropy is an obstacle for the coherent spin relaxation. The larger the valueof ω D , the smaller the spin reversal, as is demonstrated in Fig. 4.Dipole spin interactions is also a factor suppressing spin coherence. This is illustratedby Fig. 5, where the behavior of spin polarization for the case with dipole interactionsis compared with that one, for which the dipole interactions are switched off by setting D αβij = 0.It is interesting that switching off the dipole interactions yields the figures that are veryclose to those obtained by the reduction of spin from S = 10 to S = 1 /
2. Thus the dashedline in Fig. 5, where γ = 10, for S = 10 can be repeated not by setting the dipole tensorto zero but by reducing the spin to S = 1 /
2. In Fig. 6, we show the behavior of spinpolarization for S = 10 and S = 1 / γ = 1. Again, switching off the dipole interactions7or S = 10 yields the dashed curve corresponding to S = 1 / z -axis, or along the feedback field, that is, along the x -axis. And weconsider the plane of spins, oriented either in the y − z plane or in the x − y plane. The resultsof computer simulation for N = 144 spins are presented in Fig. 7. These results demonstratethat the maximally efficient spin reversal happens for the chain of spins directed along the x -axis. This looks quite understandable, since the x -axis is the axis of the direction of theresonator feedback field, which is the main source of the coherent spin motion. We have accomplished computer simulations of the coherent spin relaxation in molecularmagnets with large spin. The investigation is based on the microscopic model taking intoaccount realistic dipole spin interactions and the single-site magnetic anisotropy. The systemis prepared in a strongly nonequilibrium state, with an external magnetic field opposite tothe sample magnetization.The principal point of our investigation is the presence of a resonator coupled to thesample. The later is inserted into a coil of an electric circuit, whose natural frequency is inresonance with the Zeeman frequency. Without the resonator, the coherent spin motion isimpossible. It is the resonator feedback field, which collectivizes the spin motion, making itwell correlated, hence, coherent.To realize the coherent spin relaxation, the Zeeman frequency ω has to be much largerthan the anisotropy frequency ω D . An efficient spin reversal requires that the initial spinpolarization be sufficiently high, the higher, the better, The typical spin reversal time τ isan order smaller than the transverse dephasing time T , which translates into the relation γ ∼ γ .The role of dipole spin interactions, in the presence of a resonator , is twofold, makingthe spin dynamics in a sample coupled to a resonator rather different from that happeningin a sample with no resonator feedback fields.From one side, dipole interactions influence the spin motion by making the spin reversalless pronounced. For low spins, such as S = 1 /
2, dipole interactions are less important thanfor large spins S = 10. Emphasizing the decoherence influence of the dipolar interactions,we should keep in mind that our simulations are performed for a finite number of spins. Themajority of our calculations are done for 125 spins. Because of the long range of the dipolarforces, increasing the number of spins strengthens the decohering influence of these forces.However, all qualitative results remain, as we have checked by varying the number of spinsbetween 64 and 343. Also, presenting the results in dimensionless units, as we have done,when all frequencies and attenuation parameters are normalized by the dipolar interactionstrength, makes the calculated curves for the average magnetization practically independentof the sample size. Increasing the number of interacting spins simply implies the renormal-ization of the dimensionless quantities and does not change their behavior represented in8imensionless units.At the same time, increasing the number N of spins strengthens the role of the resonatorfeedback field, which makes the process of spin coherentization faster, so that the relaxationtime, due to the coherent spin motion, depends on the number of spins as 1 /N .In this way, stronger dipole interactions, from one side, increase the transverse decoher-ence attenuation, but, from another side, they induce stronger coherence through the actionof the resonator feedback field, making the coherent relaxation faster. These two oppositeeffects, to some extent, compensate each other. Therefore, the coherent spin dynamics, oc-curring in the presence of a resonator, qualitatively does not change much under the variationof spin number.Our main concern in the present paper has been the study of spin dynamics for largespins. This is why we have done numerical simulations for S = 10. We have had no aim ofstudying the low-spin dynamics, such as that of spins one-half, since this dynamics has beenconsidered earlier. It is only to note that the low-spin dynamics is rather different that weshow its qualitative difference in one curve of Fig. 6.It is worth mentioning that for large spins S ≥ z -projection number m = − S, − S +1 , . . . , S − , S .A pair of sublevels can be treated as an effective two-level system [56,57]. Then to realize thecoherent spin relaxation, one has to tune the resonant natural frequency to the transitionfrequency between the chosen two levels. For high nuclear spins, this procedure was realizedexperimentally [15,16]. Hence, in the same way it can be realized for molecular magnets.It is important to stress that there are several principle physical differences between theexperiments with nuclear spins, described in Refs. [15,16], and the situation considered inthe present paper. In these experiments [15,16], the nuclei of Al inside the ruby crystal werestudied.
First of all, the nuclei of Al possess spins I = 5 /
2, which are not as large as wehave considered here, dealing with S = 10. Second , in the case of Al, an external resonantcircuit was tuned to the central line {− / , / } , with a fixed transition frequency ω ∼ Hz, thus, reducing the consideration to an effective two-level system with spin one-half, whilehere we always have dealt with the total spin S = 10, since we have considered the resonantcircuit tuned to the transition between − S and S . Third , as we have shown, for our high-spincase the influence of dipolar interactions is essential, while their role for an effective one-half spin system [15,16] is not of such importance.
Fourth , contrary to the case of nuclearspins, having no single-site anisotropy, the molecular magnets, we have studied, exhibitquite strong magnetic anisotropy, fundamentally distorting spin dynamics and making itprincipally different as compared with the isotropic case of nuclear spins.
Fifth , for stronglypolarized spin materials, it is necessary to take account of the saturation effect making thetotal transverse attenuation depending on the polarization level [1,25], as in Eq. (11), whilethis effect does not play role for not so strong polarization [15,16].
Sixth , in experiments[15,16], pulsing spin dynamics was analyzed, when the inversion of spin polarization waspermanently supported by constantly applied dynamic nuclear polarization with a ratherhigh pumping rate, while we studied the pure coherent spin relaxation, when there is nopermanent pumping.
Finally , we have studied here the dependence of spin relaxation onthe sample shape and orientation. Such geometric effects, to our knowledge, have not beeninvestigated. These seven factors make the spin dynamics in our case and in the case ofRefs. [15,16] principally different.When studying the geometric effects related to the sample shape and its orientation,9ifferently oriented spin chains and planes have been considered. We have found that,under the same system parameters, including the number of spins, except the sample shapeand orientation, the most efficient coherent spin relaxation, with the deepest spin reversal,happens for a chain of spins aligned with the direction of the resonator feedback field.The main aim of the present paper has been to analyze the coherent spin relaxation underwidely varying system parameters, in order to clarify the influence of different parameters onthe coherent spin motion. This should help in choosing the optimal materials for realizingsuch a coherent spin motion. Nowadays, there are plenty of magnetic materials, with widelyvarying properties, which could be used for experimentally observing the described effects.The description of the properties of various molecular magnets can be found in the reviewarticles [5,18,44,45].As an example, we can mention the most often studied molecular magnets made of themolecules Mn or Fe , whose spins are S = 10. For these materials, ω D ∼ s − . At lowtemperatures, below the blocking temperature of about 1 K, the sample can be polarized,having a very long spin-lattice relaxation time T ∼ − s. Hence γ ≡ /T ispractically negligible, γ ∼ − − − s − . Dipole spin interactions in these materials arerather strong, with γ ∼ s − . To realize the coherent spin relaxation, the externalmagnetic field B is to be sufficiently strong, such that corresponding Zeeman frequency ω ,being close to the resonator natural frequency ω , would be much larger than the anisotropyfrequency ω D . For the considered case of Mn or Fe , this requires the field B ∼
100 T.This is a strong field, though which can experimentally be reached [60]. Fortunately, thereare many other molecular magnets with a smaller anisotropy. For instance, in the case ofnanomagnets formed by the molecules Mn , whose spin is S = 12, the magnetic anisotropyis much lower, with ω D ∼ s − . Therefore the required external magnetic field is onlyabout B ∼ T (USA) and even 600 T (Japan) can be reached.In order to estimate the typical time of the coherent spin relaxation, we may notice thatthis time is an order smaller than T . The spin dephasing time in molecular magnets, suchas Mn and Fe , is due to dipole interactions yielding γ ∼ s − . Hence T ∼ − s.This means that the typical time of the coherent spin relaxation in these materials is 10 − s, which is an ultrafast process.In conclusion, it is worthwhile to mention that for many magnetic molecules the influenceof hyperfine interactions from nuclear spins could be important. These interactions resultin the appearance of an additional line broadening, which can be included in the effectiveattenuation parameters, so that the existence of the hyperfine interactions can be takeninto account by the appropriate definition of the effective attenuations, as has been done inRef. [42]. At the same time, for many molecules, typical of the large family of magneticmolecules, such as Mn and Fe , the hyperfine interactions are of the order of 10 − K, whichare much weaker than the dipolar interactions, being of the order of 0 . Acknowledgement
One of the authors (V.I.Y.) is grateful for useful discussions to B. Barbara and E.Yukalova. And the other authors (V.K.H. and P.V.K.) appreciate fruitful discussions withY.L. Raikher. Financial support under the grant of RFBR 07-02-96026 is acknowledged.11 eferences [1] A. Abraham and M. Goldman,
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9. The anisotropy frequency is ω D = 20 and the resonatordamping is γ = 10. Fig. 2 . Reduced spin polarization s , as a function of dimensionless time, for a cubiclattice, with ω = 2000 and ω D = 20, for the resonator damping γ = 1 (solid line), γ = 10(long-dashed line), and γ = 50 (short-dashed line). The sample of molecules with spin S = 10 has the initial polarization s = 0 . Fig. 3
Reduced spin polarization s , as a function of dimensionless time, for a cubiclattice, with ω = 2000, ω D = 20, γ = 10, S = 10, for different initial polarizations s = 0 . s = 0 . s = 0 . Fig. 4
Spin polarization s , as a function of dimensionless time, for a cubic lattice, with ω = 2000, γ = 10, S = 10, and for different magnetic anisotropy values characterizedby the anisotropy frequency ω D = 20 (solid line), ω D = 50 (long-dashed line), ω D = 100(short-dashed line). Fig. 5 . Spin polarization s , as a function of dimensionless time, for a cubic lattice,with ω = 2000, ω D = 20, γ = 10, and S = 10, for two different cases, when the dipoleinteractions are present (solid line) and when they are absent (dashed line). Fig. 6 . Spin polarization s , as a function of dimensionless time, for a cubic lattice, with ω = 2000, ω D = 20, γ = 1, and for different spins S = 10 (solid line) and S = 1 / Fig. 7 . Difference in the behavior of spin relaxation for different sample shapes andorientations, under the same values ω = 2000, ω D = 20, γ = 30, S = 10. The chain of spinsalong the z -axis (solid line); the chain of spins along the x -axis (long-dashed line); the plainof spins in the y − z plane (short-dashed line), and the plane of spins in the x − y plane(dashed-dotted line). 15 t -1-0.500.51 s (t) Figure 1: Reduced spin variable s , for a cubic lattice, characterizing the spin polarizationalong the z -axis, as a function of dimensionless time (measured in units of T ) for theZeeman frequencies ω = 1000 (solid line), ω = 2000 (long-dashed line), and ω = 5000(short-dashed line). The simulation is done for the molecules of spin S = 10, with thereduced initial polarization s = 0 .
9. The anisotropy frequency is ω D = 20 and the resonatordamping is γ = 10. 16 t -1-0.500.51 s (t) Figure 2: Reduced spin polarization s , as a function of dimensionless time, for a cubic lattice,with ω = 2000 and ω D = 20, for the resonator damping γ = 1 (solid line), γ = 10 (long-dashed line), and γ = 50 (short-dashed line). The sample of molecules with spin S = 10 hasthe initial polarization s = 0 .
9. 17 t -0.6-0.20.20.61 s (t) Figure 3: Reduced spin polarization s , as a function of dimensionless time, for a cubic lattice,with ω = 2000, ω D = 20, γ = 10, S = 10, for different initial polarizations s = 0 . s = 0 . s = 0 . t -1-0.500.51 s (t) Figure 4: Spin polarization s , as a function of dimensionless time, for a cubic lattice, with ω = 2000, γ = 10, S = 10, and for different magnetic anisotropy values characterizedby the anisotropy frequency ω D = 20 (solid line), ω D = 50 (long-dashed line), ω D = 100(short-dashed line). 19 t -1-0.500.51 s (t) Figure 5: Spin polarization s , as a function of dimensionless time, for a cubic lattice, with ω = 2000, ω D = 20, γ = 10, and S = 10, for two different cases, when the dipole interactionsare present (solid line) and when they are absent (dashed line).20 t -1-0.500.51 s (t) Figure 6: Spin polarization s , as a function of dimensionless time, for a cubic lattice, with ω = 2000, ω D = 20, γ = 1, and for different spins S = 10 (solid line) and S = 1 / t -1-0.500.51 s (t) Figure 7: Difference in the behavior of spin relaxation for different sample shapes and ori-entations, under the same values ω = 2000, ω D = 20, γ = 30, S = 10. The chain of spinsalong the z -axis (solid line); the chain of spins along the x -axis (long-dashed line); the plainof spins in the y − z plane (short-dashed line), and the plane of spins in the x − yy