Paramagnetic resonance in electron-impurity systems

The equation for the magnetization is obtained on the basis of the kinetic equation for an isotropic distribution function of electrons scattering on massive impurity centers in the presence of magnetic and electric fields. The analytical solution of the Cauchy problem for a given initial distributi...

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Datum:	2001
1. Verfasser:	Ivanchenko, E.A.
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Sprache:	English
Veröffentlicht:	Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2001
Schriftenreihe:	Вопросы атомной науки и техники
Schlagworte:	Kinetic theory
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spelling	irk-123456789-800422015-04-10T03:03:00Z Paramagnetic resonance in electron-impurity systems Ivanchenko, E.A. Kinetic theory The equation for the magnetization is obtained on the basis of the kinetic equation for an isotropic distribution function of electrons scattering on massive impurity centers in the presence of magnetic and electric fields. The analytical solution of the Cauchy problem for a given initial distribution of the magnetization under conditions of paramagnetic resonance is obtained. The estimated dynamic frequency shift of the forced precession has nonlocal and nonlinear dependence on the nonuniform distribution of the initial magnetization. The dynamic frequency shift of the free precession has only nonlocal character. Time and space dependence of the internal field is obtained. All results are expressed in terms of the initial distribution of the magnetization without specifying its functional form and in terms of the propagation function. These results may be used for analysis of spin diffusion in natural and manmade materials and also in magnetometry. 2001 Article Paramagnetic resonance in electron-impurity systems / E.A. Ivanchenko // Вопросы атомной науки и техники. — 2001. — № 6. — С. 321-325. — Бібліогр.: 11 назв. — англ. 1562-6016 PACS: 33.35.Cv, 75.40.Gb, 76.30.–v, 76.60.Jx, 76.60.-k http://dspace.nbuv.gov.ua/handle/123456789/80042 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Kinetic theory Kinetic theory
spellingShingle	Kinetic theory Kinetic theory Ivanchenko, E.A. Paramagnetic resonance in electron-impurity systems Вопросы атомной науки и техники
description	The equation for the magnetization is obtained on the basis of the kinetic equation for an isotropic distribution function of electrons scattering on massive impurity centers in the presence of magnetic and electric fields. The analytical solution of the Cauchy problem for a given initial distribution of the magnetization under conditions of paramagnetic resonance is obtained. The estimated dynamic frequency shift of the forced precession has nonlocal and nonlinear dependence on the nonuniform distribution of the initial magnetization. The dynamic frequency shift of the free precession has only nonlocal character. Time and space dependence of the internal field is obtained. All results are expressed in terms of the initial distribution of the magnetization without specifying its functional form and in terms of the propagation function. These results may be used for analysis of spin diffusion in natural and manmade materials and also in magnetometry.
format	Article
author	Ivanchenko, E.A.
author_facet	Ivanchenko, E.A.
author_sort	Ivanchenko, E.A.
title	Paramagnetic resonance in electron-impurity systems
title_short	Paramagnetic resonance in electron-impurity systems
title_full	Paramagnetic resonance in electron-impurity systems
title_fullStr	Paramagnetic resonance in electron-impurity systems
title_full_unstemmed	Paramagnetic resonance in electron-impurity systems
title_sort	paramagnetic resonance in electron-impurity systems
publisher	Національний науковий центр «Харківський фізико-технічний інститут» НАН України
publishDate	2001
topic_facet	Kinetic theory
url	http://dspace.nbuv.gov.ua/handle/123456789/80042
citation_txt	Paramagnetic resonance in electron-impurity systems / E.A. Ivanchenko // Вопросы атомной науки и техники. — 2001. — № 6. — С. 321-325. — Бібліогр.: 11 назв. — англ.
series	Вопросы атомной науки и техники
work_keys_str_mv	AT ivanchenkoea paramagneticresonanceinelectronimpuritysystems
first_indexed	2025-07-06T03:58:52Z
last_indexed	2025-07-06T03:58:52Z
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fulltext	PARAMAGNETIC RESONANCE IN ELECTRON-IMPURITY SYSTEMS E.A. Ivanchenko National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine The equation for the magnetization is obtained on the basis of the kinetic equation for an isotropic distribution function of electrons scattering on massive impurity centers in the presence of magnetic and electric fields. The analytical solution of the Cauchy problem for a given initial distribution of the magnetization under conditions of paramagnetic resonance is obtained. The estimated dynamic frequency shift of the forced precession has nonlocal and nonlinear dependence on the nonuniform distribution of the initial magnetization. The dynamic frequency shift of the free precession has only nonlocal character. Time and space dependence of the internal field is obtained. All results are expressed in terms of the initial distribution of the magnetization without specifying its functional form and in terms of the propagation function. These results may be used for analysis of spin diffusion in natural and manmade materials and also in magnetometry. PACS: 33.35.Cv, 75.40.Gb, 76.30.–v, 76.60.Jx, 76.60.-k 1. INTRODUCTION The system of electrons, interacting among themselves and with motionless potential impurity centres randomly distributed in uniform external fields is described by a distribution function f , obeying the kinetic equation [1,2] fLLff c q fqffwif t ee+= ∂ ∂ + ∂ ∂+ ∂ ∂++ ∂ ∂ − p Bv p E x v ],[ ],[ (1) where ),( tff xp≡ is the distribution function of the electrons, which is a matrix in electron spin space; q and p v p ∂ ∂ = e are the electron charge and velocity, respectively; L and eeL are the electron-impurity and electron-electron collision integrals; B is the magnetic field, E is the static electric field; Bσµ 0−=w ( 0µ is the Bohr magneton, σ are the Pauli matrices). We assume that massive charged impurities whose kinetics is not considered here form neutralizing electrical background. We shall define the distribution function of electrons on energy e [3] ( ) ( ) ( ) ( ) ( ) , 2 ,,x1,x, 3 3 p ppp π = −δ ρ == ∫ pddV eetfdV e ften (2) where ( ) =eρ ( )∫ − pp eedV δ is the electron density of states, and the brackets mean averaging defined by the formula (2). It follows from Eq. (2), that ,0=fL where ( ) ( ) .],[',' p BvBB ∂ ∂−=+= c qLLLL (3) Indeed, the electron-impurity collision integral has the form ( ) ( ) ( ) ( )( )∫ −−= ppppp ppp ffeewdVNLf ''' ',2 δπ , and hence 0=Lf . Here N is the impurity density, ( )',ppw is the probability per unit time of electron scattering on the impurity centre. As 0 2 = ∂ ∂ = ∂ ∂ ki ikl i k ikl pp e p v pεε , hence the mean ( ) ( )∫ − ∂ ∂= ∂ ∂ ppp Bv eef p BvdV e f i lkikl δ ρ ε )(1],[ , and after integration by parts we have 0=fL . The operator L has property ( ) ( )BB −= +LL , (4) where + means the conjugate operation, defined by the formula ( ) yxyx ,, ≡ . By virtue of the definition of the operators L and 'L we have ( ) ( )LyxyLx ,, = , ( )( ) ( )( )yLxyxL BB −= ',,' , i. e. Eq. (4) is valid. As the result of averaging Eq. (1) we obtain the equation for distribution function ( ) :,, ten x ( ) ( )( )kk k k ee je ee qE j x fLnwin t ρ ρ ∂ ∂ − ∂ ∂−=+ ∂ ∂ − 1 ],[ (5) PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 321-325. 65 To close this equation one has to express the current kj in terms of n . It can be done, if the frequency of electron-impurity collisions 1− eimpτ is much greater than the frequencies of electron-electron collisions 1− eeτ , and if the times t large in comparison with the corresponding relaxation time eimpτ , the electron distribution function becomes some functional of n , e. i., the electron distribution function becomes independent of the electron momentum direction due to the collisions of electrons with impurities. On this basis it is possible to show, that in the linear approximation with respect to the gradients and on electric field the diffusion current is [4] ( )     ∂ ∂+ ∂ ∂−= n e qEn x Dj i i kik B , (6) where ( ) ikki vvLD , 1− =B is the diffusion coefficient of electrons in a magnetic field having the property, ( ) ( )BB −= ikki DD , which follows from (4). In case of an isotropic electron dispersion relation pp ee = we get ( ) ( ) ceimpc c eimp ciiklcikkiki d bbbdD Ω== + = ++= τω ω τ ωεωδ ,, 13 , 2 2 2 B Bb v (7) The cyclotron frequency cΩ is equal to p v c Bq c =Ω , ( ) ( )∫         −−≡− '\|\| ' 1',2 '' 1 pp pppp ppp eewdVNeimp δπτ . (8) Equation (5) and (6) determine a closed equation for the distribution function ( )ten ,, x , which is isotropic with respect to the moments [4]. 2. MACROSCOPIC EQUATION FOR MAGNETIZATION We define the macroscopic density of the electron magnetic moment ( ) −= 321 ,, MMMM magnetization by the formula ( ) ( )∫= tendVSpt ,, 2 12, 0 xxM pσµ . (9) In view of the relation for Pauli matrixes ljkljkkj i σεσσσσ 2=− the kinetic equation (5) after multiplication by σµ 0 , taking the trace on the spin variables, and integration over pdV takes the form 0],[2 0 = ∂ ∂++ ∂ ∂ ik k ii I xt MBM µ , (10) where the flux density of the electron magnetic moment is equal to ∫         ∂ ∂+ ∂ ∂−≡ e nqE x nDdVSpI p p kpiik pσµ 2 12 0 . (11) In order to obtain the closed equation for the magnetization we assume that function Dkp is smooth over е, therefore it is possible to take it out under integral sign. We integrate the second term in (11) by parts on е, and assume that the surface terms are small at е=0, and at e=eF, where eF is the Fermi energy. This approximation allows us to write down the equation for the magnetization as [ ] 0 2 ,2 2 0 =        ∂ ∂− ∂∂ ∂ −+ ∂ ∂ M MBM k p Fpk kp x E e q xx D t µ (12) Equation (12) without allowance of spatial non- homogeneity corresponds to the Bloch equation, and forms the basis of the theory of paramagnetic resonance. The account of nonhomogeneity is carried out in Refs. [1,5] without concrete description of the character of the diffusion mechanism. The nonlinear equation describing a collision dynamics of magnetization in the absence of external fields is obtained in Ref. [6] The purpose of the present work is to study the magnetization dynamics of electron-impurity systems on the basis of Eq. (12) under conditions of paramagnetic resonance. 3. PARAMAGNETIC RESONANCE IN ELECTRON-IMPURITY SYSTEMS We consider the magnetization behaviour in the case, when the external magnetic field in Eq. (12) consists of two terms B=B0+h(t), where B0 is the static field, and h(t) is the alternating field. To find the solution of Eq. (12) we shall develop the scheme described by Bar'yakhtar and Ivanov in Ref. [7]. For this purpose, we shall present the solution for the magnetization as an expansion in powers of the amplitude of the external alternating magnetic field ( ) ( ) ( )tt k k ,, 0 xmxM ∑ ∞ == . (13) After substituting Eq. (13) into Eq. (12) we have an infinite system of equations for ( )km ( ) ( ) ( ) ( ) ( ) ],[2 ],[2 1 0 0 −− =−+ ∂ ∂ k kkk t D t mh mmBm µ µ , (14) ( ) ,0,...;2,1,0 1 ≡= −mk ( )B,0,0=B , 66 ( ) ( ) ( ) ( ) .)1 2 22 1( 2 3 1221 2 2 2 2 2 2 2 z E e q y EE e q x EE e q zyx dD c F c F c F cF ∂ ∂+− ∂ ∂−− ∂ ∂+ − ∂ ∂++ ∂ ∂+ ∂ ∂≡ ω ωω ω (15) At first we find the solution ( )0m of the Cauchy problem with the help of the change of dependent variables, ( ) ( ) ( ) ( ) ( ) ( )3,2,1 ,2, 03 2 3 0 = = ∫ −− i tmkedtm i i i kx kxπ (16) For the Fourier-components ( ) ( )ti ,0 km we get a system of differential equations of the first order in time. This system is easily solved. Carrying out return transformation, ( ) ( ) ( ) ( ) ( )tmexdtm i i i ,''2, 0'3 2 3 0 xk kx∫−= π (17) we find the solution for magnetization in the form of free precession in constant fields, ( ) ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ))','sin' ,'cos'(,', , 3 3 0 xxx xxxx xm mtm tmtxgd t ϕ ϕ +Ω− +Ω = ∫ (18) on the set of the known initial data of the form ( ) ( ) ( ) ( ) ( ) ( ) ( )( )xxxxx xm 3 0 ,sin,cos 0, mmm t ϕϕ − == (19) with the propagation function equal to ( ) ( ) ( ) ( ) ( ) ], 2 1 212 ' 222 ' 222 'exp[ 2 1 1 1,', 2 3 2 1 2 2 1 2 2 12 2 21 3 2 1 2 1 2           + + + −−        − +− −        + +−−         + = td E e q td zztd EE e q td yy td EE e q td xx td tg F c F Fc F c FF F c FFFc ω ω ω ω πω xx (20) where ( )2 2 0 13 ,2 c eimpF F v dB ω+ τ =µ=Ω and Fν is the Fermi velocity. This function obeys the equation 0=− ∂ ∂ Dgg t (21) and also has the properties ( ) ( ) +→ −= 0 ,',',lim t tg xxxx δ (22) ( ) ( ) +→ −= ∂ ∂ 0 .',',lim t Dtg t xxxx δ (23) An important property of the propagation function is that it satisfies the Smolukhowski-Chapman- Kolmogoroff equation ( ) ( ) ( )x,''x,'x,''x,'x,'x,''3 tgtgttgxd =−∫ . (24) The particular solution ( )1m of Eq. (14) with the right hand side ( ) ( ) ( )],,[2 0 0 tt xmhµ− can be obtained, using the semigroup property of the function ( )xx ,',tg (24): ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )[ ] ( ) ( )( ) ( ) ,]''''' ,',''2 ,, '' 3321 3 0 ' 0 1 2 1 1 xx xx xx x methmtihth tgxdedti timtm ti t tti ϕ µ +Ω− −Ω −+ × =+ ∫∫ (25) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )( )[ ].''cos'''sin' ',',''2, 21 3 0 0 1 3 xx xxxx ϕϕ µ +Ω++Ω ×= ∫∫ tthtth mtgxddttm t (26) It is seen from the formulae (25), (26), that the magnetization at the time t is determined by the field ( )th at all previous moments of time, starting the moment of inclusion. We choose left rotation for external alternating magnetic field, which is perpendicular to the static field 0B , ( ) ( )0,sin,cos ttht ωω −=h , h is the amplitude of the field, ω is the frequency of the alternating magnetic field. Since at the paramagnetic resonancе Ω=ω , we find from formulae (25), (26), that the particular solution for the magnetization linear in the field approximation is the forced precession, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ),,, , 2 sin,, , 2 cos,, 1 1 3 0 31 1 2 0 31 1 1 ttAtm tttmtm tttmtm xx xx xx ω πω πω =      −Ω−=      −Ω= (27) ( ) ( ) ( ) ( ) h mtgxdtA 01 3 2 ,'sin',',', µω ϕ = ≡ ∫ xxxxx (28) ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2000, №2. Серия: Ядерно-физические исследования (36), с. 3-6. 67 lagging in phase behind in the phase of the alternating magnetic field by 2 π . Having continued the procedure of iteration, it is possible to sum up series on t1ω and find the general exact solution for the magnetization dynamics (12) at paramagnetic resonance: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ,)1cos,sin, ,cossin,cos1, ,sinsin,cos1,( , ,,,, 1 0 31 1 0 31 1 0 31 0 −+ Ω+− Ω+− = += ttmttA tttmttA tttmttA t ttt ωω ωω ωω xx xx xx xm xmxmxM (29) At 0=h this solution transforms into ( ) ( ) ( )tt ,, 0 xmxM = , see Eq. (18). As is seen from the solution (29), there is no divergence in time. Finally we come to the conclusion, that the solution of the Cauchy problem of the Eq. (12) with the initial distribution (19) in the class of square integrable functions is completely determined by the propagation function and the shape of the sample, i. e., by the integration volume. For an unbounded medium under conditions of paramagnetic resonance the solution of the Cauchy problem takes the form of Eq. (29). The magnetization projection ( )tM ,3 x oscillates. This fact implies that a inverse population occurs in the system considered. For the analysis of the forced precession we write the solution (29) for 21 , MM as ( ) ( ) ( ) ( ) ,sin, ,cos, 2 1 φ φ +Ω−= +Ω= tatM tatM x x (30) where the local amplitude and phase of precession are equal to ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) .'cos',',', , , sin,cos,, ,,sin,cos, , 3 1 1 1 0 31 2 1 2 1 0 31 xxxxx x xxx xxx x ϕ ωωφ ωω mtgxdtA tA ttmttAarctgt tAttmttA ta ∫≡ −= ++− = (31) Expanding the phase ( )t,xφ (31) with respect to t and restricting ourselves to the term linear in t , we get, in view of property (23), the local dynamic shift of the forced frequency 'Ω with respect to Larmor precession Ω , ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )( )},sin ]'cos'''[ cos]'sin''' {[10,' ,...0,', 31 3 3 312 xxx xxxx xxxxxx x x x xxx mm mDxd mmDxd m m tt ωϕ ϕδ ϕϕδ ω ϕφ − −− − +−=Ω +Ω+= ∫ ∫ (32) the cause of which has the meaning of the internal field at the point x (analog of Suhl-Nakamura field [8] in a paramagnetic medium). This field depends on initial nonuniform magnetization distribution at all points, that is, it has nonlocal character. Without nonlocality being taken into account, this shift is proportional to the amplitude of the forced field and has the simple form, ( ) ( ) ( ) ( )x xxx m mh ϕ µ cos 20,' 3 0−=Ω (33) As it is seen from the formula (33), this shift depends non-linearly on initial distribution. This result coincides with that of Ref. [8] in view of heterogeneity. It follows from formula (32) that the dynamic shift of the free precession ( )0,' xfreeΩ is completely nonlocal: ( ) ( ) ( )( ) ( ) ( ) ( )( )∫ −−− =Ω 'sin'''1 0, 3 ' xxxxx x x ϕϕδ mDxd m free (34) Now it is obvious, that in the general case the dependence of dynamic shift on time and co-ordinates is ( ) ( ) ( ) t tt xxx ϕφ −=Ω ,,' . (35) We find the maximal amplitude of the forced precession maxa from the condition 03 =M , i. e., ( ) 0cossin 1 0 31 =+ tmtA ωω (36) After substituting (36) in (31), we get ( ) , sin 2 1 1 2 20 3max A t ma += ω (37) and the ( )kt are determined by the solution of Eq. (36), which can be written in equivalent form as ( ) ( ) A m arctgtSin 0 3 1 ,0 ==+ δδω (38) In the simplest case we find: 4 πδ −≈ , ...,2,1,0, 41 =+≈ kkt ππω ( ) ( ) ,..., 4 5, 4 1 1 1 0 ω π ω π ≈≈ tt ( ) ( ) ( )0 0 0 3 max 3 tt ma tt = ≈= . (39) Decaying bursts of precession amplitude maxa are observed. The general exact solution of the equation (12) is given in the Appendix. 4. CONCLUSIONS The evolution dynamics in the system of electrons and impurities placed in static electrical and magnetic fields is investigated under the influence of a alternating magnetic field under conditions of paramagnetic resonance. The general formulas for all three magnetization components in their evolutionary interrelation are obtained, since experimental engineering allows one to measure these components [9]. The behaviour of forced precession is theoretically investigated. The dynamic shift of the frequency of 68 paramagnetic resonance caused by a nonuniform distribution of initial magnetization is found. All results are expressed in terms of the initial magnetization distribution and a propagation function. The results obtained are applied to the analysis of spin diffusion in natural and manmade materials [10,11] and also in magnetometry [9]. ACKNOWLEDGMENT The author thanks Prof. S.V. Peletminskii for the interest in the research and for useful discussion. APPENDIX If the mismatch ω−Ω=∆ , i. e. the difference between Larmor precession Ω and the frequency of the alternating magnetic field is not equal to zero, the general exact solution of the equation (12) has the form: ( ) ( ) ( ) ( ) ,,,,,, 0 ttt ∆+=∆ xmxmxM ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ), cos1 sin1cos ,cossincossin sincos1sin sincossincos ,cos cos1 sin sincossinsincos sinsincos(,, 211 0 3 2 10 3 11 21 20 31 1 0 3 1 21 20 31 111 0 3 1 γ γ ω γ γ ω γ ωγ γ ω γ γωγ γ ωωγ γ ω γ ω γ γ ωγ γ ωγ γ ω ωγ γ ω γ tAm tAtm tAtAttA ttAmtA tAttmtA ttAmtA tAtAttAtA ttmtAt − ∆+∆ ++− Ω+Ω+∆ −    −∆−∆−∆ +−    +−     − ∆−∆−∆ −Ω+Ω−∆− +    +−=∆xm where .22 1 ∆+= ωγ The dynamic shift is ( ) ( ) ( ) ,,,,,' t tt xxx ϕφ −∆=∆Ω where ( ) ( ) ( )( ) ( ) . cos1sin sinsin cos ,, 21 20 311 1 0 31 γ γ ω γ γ γ γ γ γ ωγ φ tAmtAA tAtmtA arctg t − ∆−∆+∆− ∆+− =∆x REFERENCES 1. V.P. Silin. Kinetics of paramagnetic phenomena // Soviet Zh. Eksperim. i Teor. Fiz. 1956, v. 30, p. 421-422. 2. M.Ia. Azbel', V.I. Gerasimenko and I.M. Lif- shitz. Paramagnetic resonance and nuclear polarization in metals // Soviet Zh. Eksperim. i Teor. Fiz. 1957, v. 32, p. 1212-1225. 3. L.V. Keldush. Theory of impact ionization in semiconductors // Soviet Zh. Eksperim. i Teor. Fiz. 1965, v. 48, p. 1692-1707. 4. E.A. Ivanchenko, V.V. Krasil'nikov, S.V. Pe- letminskii. Kinetic equation for isotropic distribution function and relaxation of electrons in electric field // Fiz. Met. i Metallovedyenie. 1984, v. 57, p. 441-449 (in Russian). 5. G.D. Gaspari. Bloch Equation for Conduction-Electron Spin Resonance // Phys. Rev. 1966, v. 151, p. 215-219. 6. T.L. Andreeva, P.L. Rubin.Dynamic of magnetic momentum in Bolzman polarized gas // Zh. Eksperim. i Teor. Fiz. 2000, v. 118, p. 877-884. 7. V. Baryakhtar, B. Ivanov, Modern magnetism. 1986, M.: «Nauka», 176 p. 8. М.I. Kurkin, Е.А. Тurov. NMR in magnetically ordered materials and its applications. 1990, M.: “Nauka”, 248 p. 9. N.M. Pomerantsev, V.M. Ryzhov, G. V. Skrotskii. Physical basis of quantum magnetometry. 1972, M.: “Nauka”, 448 p. 10. K.R. Brownstein and C.E. Tarr. Importance of classical diffusion in NMR studies of water in biological cells // Phys. Rev. 1979, v. 19 A, p. 2446-2453. 11. Yi-Qiao Song. Detection of the High of Spin Diffusion in Porous Media // Phys. Rev. Lett. 2000, v. 85, p. 3878-3881. ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2000, №2. Серия: Ядерно-физические исследования (36), с. 3-6. 69 E.A. Ivanchenko National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine PACS: 33.35.Cv, 75.40.Gb, 76.30.–v, 76.60.Jx, 76.60.-k 1. INTRODUCTION 2. MACROSCOPIC EQUATION FOR MAGNETIZATION

Paramagnetic resonance in electron-impurity systems

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