Impurity spin in normal stochastic field: basic model of magnetic resonance Текст научной статьи по специальности «Физика»

ISSN 2072-5981

aänetic Resonance in Solids

Electronic Journal

Volume 18, Issue 2 Paper No 16201, 1-8 pages 2016

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Kanazava) Murat Tagirov (KFU, Kazan) Dmitrii Tayurskii (KFU, Kazan) Valentin Zhikharev (KNRTU, Kazan)

In Kazan University the Electron Paramagnetic Resonance (EPR) was discovered by Zavoisky E.K. in 1944.

Impurity spin in normal stochastic field: basic model of magnetic resonancef

F.S. Dzheparov, D.V. Lvov Institute for Theoretical and Experimental Physics, B. Cheremushkinskaya 25, Moscow 117258, Russia National Research Nuclear University "MEPhl", Kashirskoe shosse 31, Moscow 115409, Russia

E-mail: dzheparov@itep.ru, lvov@itep.ru

(Received December 10, 2016; accepted December 17, 2016)

Famous Anderson-Weiss-Kubo model of magnetic resonance is reconsidered in order to bridge existing gaps in its applications for solutions of fundamental problems of spin dynamics and theory of master equations. The model considers the local field fluctuations as one-dimensional normal random process. We refined the conditions of applicability of perturbation theory to calculate the spin depolarization. It is shown that for very slow fluctuations the behavior of the longitudinal magnetization is simply related to the correlation function of the local field. The effect could be checked by the experimental studies of magnetic resonance in quasi-Ising paramagnets.

PACS: 45.10.Hj, 76.60.-k, 76.30.-v, 75.75.Jn, 05.40.-a

Keywords: magnetic resonance, spin relaxation, normal stochastic field, longitudinal correlation function, Anderson-Weiss-Kubo model

Spin system with the Hamiltonian (written in rotating frame)

H = H 0 (t) + #1, H = axlx, H 0(t) = (A + ®l (t)) Iz = HI + HI (t) (1)

is one of most important basic models for studies in spin dynamics. Here la is spin operator, A is the detuning from the resonance, a1 represents magnitude of the rotating field, and wl (t) corresponds to time dependent local field, produced by surrounding substance. Famous Anderson-Weiss-Kubo (AWK) model [1] considers w1(t) as a normal stationary stochastic process with the correlation

function (w,(t)al(t1))n = M2k(-111), M2 = (a>2) , it means, that for any reasonable function a(t) the moment-generating functional is of the form

O(t,t1,[a]) = ^exp(ij dra(r)a>t(r)= expM2j dr1dr2a(r1)a(r2)K(r1 -r2)j. (2)

Here averaging is fulfilled on distribution of random trajectories wl(t).

The model was created to explain the "narrowing of the resonance line by motion" using Gaussian or simple exponential /r(| t|), but it was successful for explaining of the line shape for impurity beta-active nuclei [2, 3] without observable differences between precise experimental data and theoretical predictions for more realistic /r(|t|). It was adopted to describe two- and multi-spin transitions [3-5]

with successful incorporation both static and dynamic correlations of local fields on impurity spins [5, 6]. The model found important application in description of the electron spin echo [7] and it produces a kernel for modern theory of spin dynamics in magnetically diluted systems [8, 9].

An important application of the AWK model consists in derivation of the applicability conditions for perturbation theory for small a1 in calculation of longitudinal correlation function [5, 10, 11]

f This paper is prepared on base of invited lecture at XIX International Youth Scientific School "Actual problems of magnetic resonance and its application", Kazan, 24 - 28 October 2016 and it is published after additional MRSej reviewing.

F (t) = ( IJZ (0) / (II) = (( V, (t ))„) m /{I2\ . (3)

Here (A) = Tr(A) / Tr (i) for any relevant operator A, that corresponds to averaging with infinite spin temperature. The function F (t) is proportional to observable value of the operator Iz (t), if the initial state of the density matrix p(t) is of standard form

p =p(t = 0) =-i-il + 1, , Po = Tr (I,p(t = 0)). (4)

( 3 P ^ Tr (1) ^ I (I +1)

Indeed, p0 doesn't depend on wl, therefore

J, (t) = {Tr (Ip(t))) = (Tr (Iz (t )p(t = 0))) = F (t) po. (5)

The advantage of the model is the realistic smooth time dependence of local field contrary to known exactly solvable models with hopping evolution of (t). It is usually expected [1, 10, 11], that if A = 0, then the simplest conditions s1 = R0T2 1, and s2 = R0rc 1 produce F(t) = exp(-R0t). Here

R0 = ^2, T2 =j; dt exp (-M2 \[dT(t -t)k(t) ), Tc =J0" dtK(t), (6)

Nothing is known for slow smooth motion, when R0Tc » 1.

We will indicate below, that for smooth co, (t) there exist logarithmical correction to the condition with s2, similar to the correction, indicated previously for two-spin flip-flop transitions [5]. Further we will construct the solution for very slow smooth motions, which is valid in the main order in s1 = o2T ■ 1. It has the form

2

F (t) = — arcsin K(t). n

1. Very fast fluctuations - 8-correlated local fields

For the sake of brevity below t > 0. If tc = 0 then correlation of local field can be written as

C (0c (^ = T2 8(t -11), (7)

T 2

and the phase evolutions at different times are independent. It means that for t < t1 < t' we have

0(t, t',[«]) = 0(t, t1,[a])0(t1, t',[«]). (8)

The quantum Liouville equation for density matrix p(t)

d

—p = -i [H, p] = -iLp = -i (L0 +A)p, (9)

LoP = [Ho, p^ LlP = [Hi, P], can be rewritten in the integral form

-i J dTL0(T) rt -i JdTL0(T)

p(t) = e J0 p0 -i J dze Jt 1 L1p(r). (10)

J 0

Here and below we will use superoperator formalism, basic information about which can be found, for example, in the textbook [12]. The superoperator L0(t) commutes with itself for different times

L0(t1)L0(t2) = L0(t2)L0(t1), that admits to use simple exponential in (10) instead of chronological ordering, required in absence of the commutativity, and, as a consequence, the multipliers with phase evolution in (10) can be calculated exactly:

i

(11)

T - ' V

V

U(t-O = ^exp(-i|4idrL0(r)^ = exp - iAlzx + T-(lzx)2j|t-1]

New superoperator lX here is produced from usual spin operator lz according to the standard rule lZ f = [lz, f ] , where f is arbitrary operator.

Averaging the Eq. (10) we have, as in Ref. [13],

{P(t))n = (exp(-i\'drL0(r)^ p0 -ij0tdt^exp(-ifdrl^r^ Lx{p(tl)). (12)

The relation (8) together with r > t1 was applied here. As a result of Eqs. (11) and (12) we have for t > 0

a

= -(iAX +R (x)2 +L )p)) n. (13)

Last equation is equivalent to Bloch's equations in absence of longitudinal spin-lattice relaxation

a a i

aJz =[" X J ]z , J =[" X J l,y - Y2 Jxy , " ^M) (14)

where Ja(t) = Tr (la{p(t)) n) are average values of spin operators. Excluding orthogonal components Jx and Jy with initial condition Jxy (t = 0) we arrive to the equation for polarization along z-axis

a 2 pt (iA-ir)r

—Jz = -a12 Re f dreV T 1 Jz (t - r), (15)

at

which is exact for 5-correlated process (7). We see that for small a>1 the derivative a/z / at ~ a12 is small. Therefore the variation of Jz (t -r) during the times r, important for the factor exp ((iA -1 / T2 )r) in the integrand of Eq. (15), is negligible, and for t > T2 we can replace the Eq. (15) by

- Jz = -a2Re f dreV j J (t) = -R(A) J (t). R (A) = a g (A), (16)

at 0

g(A) = Re j" dteiAt-1^ =^2-

^ f f-" 2^ ^(1 + A2T22)

Here normalized resonance line shape g(A) is introduced; it is a Fourier transform of free induction decay, which has simple exponential form F0(t) = exp(-|t|/T) for 5-correlated local field.

Comparing Eqs. (15) and (16) we see, that a1 is small, if, at least, R ■ 1/T2. To refine the condition we can retain in (16) next, linear in r term of Jz (t - r) expansion

a 2 r" (iA-Tr 1 r ( a | a

-Jz =-®2Re f drr (t)-r-Jz (t)| = -Rt + 3-J, (17)

at v at j at

2 f" (iA—— )r 2 T22 (1 -(AT2 )2 )

3 = Rea12 j dre[ T2> r = a12—i-|3|< R(A)T2 < RT =e1, R = R(A = 0) .

0 (1 + (AT2)2)

It is evident, that condition of smallness of c1 received the form || ■ 1. For A = 0 new condition coincides with previous s1 = R0T2 = co\TT ■ 1, but with increasing of A it can be less restrictive.

2. Fast fluctuation of local fields and the perturbation theory

Modern perturbation theory consists of two different, but connected parts: obtaining the effective Hamiltonian and derivation of master equation, see [14] for example. Master equation is an equation for important part of the density matrix p(t), which is sufficient for calculation of necessary observables. To derive it according to projection technique of Nakajima-Zwanzig we can introduce the projection superoperator P, which separates the important part:

PP = (PD)n = (P n )D. (18)

Here index D separates the part, diagonal in representation of eigenstates of Iz, i.e., if I, |m) = m|m), then (n\Pd\m) = Smn{m\p\m).

Multiplication of Liouville Eq.(9) on P and P = 1 - P produces

J^Pp = -iPL (p + P) p, JtPP = -PL (P + P) P.

Solving second equation with initial condition Pp0 = 0 and substituting the solution into the first equation, we receive a master equation

J ft

—Pp = -j drM(r)Pp(t -r), (19)

ot J0

M (r) = PLTT exp (-i \ldsPL(s) ~P ) Lf.

Here T exp(- • •) is the standard chronological exponential. It is taken into account here, that the projectors (18) obey the relations PLP = 0, PLP = PL^P = PL,, PLP = PL.P = Lf. The memory kernel M(t) has second order in Lj ~ c1. As a consequence the main order master equation is of the form

J-Pp^dr^A exp(-i|W0(s))L^j Pp(r)j = -j'(drM0(t -r)Pp(r). (20)

It is taken into account here, that L0(t1)L0(t2) = L0(t2)L0(t1) (therefore Texp is not necessary), and action of L0 on nondiagonal operators produces nondiagonal operator as well (therefore P is not necessary).

Substituting here L0 (t) = ol (t)I* and L = c1 Iyx, after straightforward transformations we obtain that the polarization satisfies the equation

JP = -C j'QdrTr {([[I,, I+], I_]( e'A(t ^M) n + [[, I_], I+]( e-A(t r*',r)) n) Pp(r)j,

where <p(t,r) = jdsa¡(s). The function F0(t -r) = ^exp(jq>(t,r)) = ^exp(-iq(t,r)) represents,

evidently, free induction decay.

Now, after calculation of the commutators, we have

= -c2 jorcos (A(t-r))F0(t-r)J, (r) = -j[drW0{r)Jz (t-r), (21)

W° (r) = a2 cos ( Ar) F (r) = Tr ((M0 (r)lz) / Tr (l,2).

The Eq. (21) is similar to Eq. (19), it can be transformed to local in time (Markov) form for small a1 by the same way, expanding upper limit of the integral to infinity and replacing Jz (t - r) by Jz (t). As a result for t > T2 we obtain

= -R(A)J(t), R(A) = f"dt W°(t) = a2g(A), g(A) = — dtelAiF°(t). (22)

at J0 2-j

Direct application of the definition (2) produces famous relation for the free induction decay within the AWK theory:

F0(t) = (e*(t,°))n = exp(-2 ¡'drjr2 (a, (r1 )a (r2))nj = exp(-M2{°'dr(t-r)(r)). (23)

First condition of applicability of the Eq. (22) can be received again by retaining in (21) the term, linear in r

JJ =-®f f0" dr cos (Ar) F°(r) ( J (t)-rjJ (t) jj = -R (A) J , (24)

3 = a210°dt • tcos (At) F°(t) < a12dt • tF°(t) ~ a12T22 = R°T2. (25)

Therefore new term is negligible if 3 ■ 1, and condition s1 ■ 1 is sufficient, but the condition 3 < R(A)T2 is not fulfilled here, contrary to Eqs. (17), because R(Arc ^ ") decays exponentially (as a Fourier transform of a smooth function), while 3(A ^<x>) ~ A 2.

To obtain second condition of the applicability of perturbation theory we should calculate next term M1 (t) of the expansion of the memory kernel (19) in powers of a1. It is of the form

M1(t) = -|°idsduPL1U°(t, s)PL1U°(s, u)PL1U0(u,0)PL1P, U°(t, s) = exp (-i |'drL°(r)). (26)

This term produces correction W1(t) to the memory function W°(t) in the Eq. (21)

W (t) = Tr (lzM1 (t)lz) / Tr (l2) = -a41°'ds£du (S1(t - s, s - u,u) - S°(t - s)S° (u)), (27)

where S° (t-s) = (Ree^s^ and S1(t-s,s -u,u) = (Ree^s)Ree*"^), together with correction of the saturation rate R in the Eq. (22):

i"

° dt W1(t). (28)

Here and below the case A = 0 is discussed only. After transformations we obtain

4

R1 = R1(+) + R1(-), R±) = dtdsdue-Q(t)-Q(u) ((t,s,u) -1), (29)

Q(t) = ~2M2 f'^dudv k(u -v)= M2f'^du(t -u) k(u), (30)

¥(t, s, u) = M2 dt' |°u du }k (t'+ s + u'). (31)

For preliminary qualitative understanding we should recognize, that essential range on t and u of the integrand (29) is of order T2, while its duration on 5 has the order tc , because with increasing of 5 we have S1(t,5 u) ^ S0(t)S0(u) (or ¥(t,5 u) ^ 0) that produces a rough estimation

R ~ <T22t .

1 12 c

If tc T2, then this estimation is sufficient, but opposite relation tc » T2 ~ M21/2 requires more detailed analysis. Below, following to Refs. [5] and [3], we apply rather general form of the local field correlation function

*(t) = (( +to )/ ((t2 + T22t )1/2 +tO ))3/2. (32)

This relation includes all existing qualitative information about correlation function: existence of smooth quadratic in time evolution at t < T2T , its transformation into linear in time dependence at

T2T < t < to with consequent transformation to 3d-diffusional asymptotics ^(t) ~ t-3/2 at t» to . It is

evident, that tc ~ T2T + to here.

Substituting this correlation function in (29) - (31) we obtain second condition of applicability of the Eq. (22)

£ = R / Ro ~ Rotc (1 + (T2TT0) / t2c ln2 (T2T / (T22to)))1/2 « 1, (33)

that for T2T ~ to ~ tc is equivalent to

£ = RJ Ro ~ Rt (1 + ln2 (tc / T2 ))1/2 « 1. (34)

Similar condition was derived in Ref. [5] for two-spin cross-relaxation transitions, that is natural, because two-spin cross-relaxation problem can be reduced to one-spin evolution with the Hamiltonian (1), see for example [15].

3. Very slow evolution of local fields

If the local field evolves very slowly, then according to the adiabatic theorem of Landau-Majorana-Stuckelberg-Zener [16] the projection J(t)Q(t)/ Q(t) of the spin J(t) on the effective field

Q(t) = (<,0,< (t)) is adiabatic invariant. The case A = 0 is discussed below only. We can suppose that tc » T2 «(/ (2M2)) 2 < and introduce the time of averaging Tav for which tc » Tav » T2. The evolution starts from initial state p0 (4) and after the short time Tav Jz (t = Tav) = « p0. Here and below upper bar indicates averaging during the time Tav . Later we will omit the difference (p0 - JZ) / p0 ~ <T2 1 and use JZ0 = p0. In most of time < (t)| »<, and, as a result of the adiabatic theorem, at this time Jz (t) = ±p0 corresponding to the sign of < (t). Therefore in these conditions

F(t) = ^ ^ = {sign(< (t)) • 5ign(< (0)))n. (35)

Further calculation should be based on main definition (2). Using the Fourier-transformation, we have

5ign(x) = f™ ^ • eiqxs(q), 5(q) = f™ dxe-iqx-/Xsign(x) = , / ^ +0. (36)

J-™ 2n J-™ / + q2

Substituting this Fourier decomposition into (35), we have

F(t) = f™ dqd1.5*(q1)5(q )/e-'q1<(t)+q<(0)\ = (2^)2 /n

J—M (2*)

It is evident from (36) and (37) that

d

S (qXqgexp

M.

—2(q2 + q2 — 2qxq2K(t))

(37)

dK(t )

F (t ) = 4M2 f

J —<

1 :fexp

(2*)

M (qi2 + q22 — ))

*(i—^2(t ) )

(38)

1/2

= — arcsm^ *

(t ).

(39)

Integration of the last equation produces final result

F(t) = 1 + 2 4

n (1 - y2)

Estimation of corrections to the relation (39) and elucidation of its range of applicability requires much more complex calculations, than the transformation from (35) to (39). It is natural as well as receiving of main order master equation (22) was much simpler, then derivation of the conditions of its applicability (33) and (34).

Standard condition of applicability of the adiabatic theorem on one trajectory requires

Q(t)

Sadi = maX

-n(t )

/ co1 1, n(t ) =

Q(t )

Direct calculation produces

1 dal

®2 dt

(40)

(41)

and this value is realized at a, = 0. According to the adiabatic theorem [16] decrease of the polarization during one passage of the range a, ~ a1 is Snad = exp (-n / sad1) 1, that produces

Snad = (à,ad}n = (eXP (—* / £ad1 ))

exp

1—1/2

27*

o1 t T2

1/3 \

(42)

where T2 =(n/ (2M2)) and r' = p2K(t = 0)/ dt^ ~ tc. It was taken into account here that distribution of drnl / dt is Gaussian with i(dal / dt)2^ =-M2d4(t = 0)/ dt2 . The relation (42) indicates, that the condition of applicability of the results (39) and (42) receives the form Snad ■ 1 or ((27n / 4)®14r'2 T22)-1/3 ■ 1.

Frequency of passages of the local field a, (t) near the value a, = 0 is Wnad ~ 1/ rc, therefore nonadiabatic losses of polarization should follow the law exp (SnadWnadt) for SnadWnadt < 1 at least. Accounting it, we get

F(t) = -arcsin^(t) * eXP (—$nadWnadt).

*

(43)

The spin-lattice relaxation should be considered separately of course.

We expect, that the relation (43) can be useful in studies of quasi-Ising spin systems by the magnetic resonance of impurity spins and in quantum information processing, therefore additional theoretical and experimental studies are necessary.

4. Conclusions

The model (1) is very important in spin dynamics and physical kinetics. Content of the lecture bridges existing gaps in known textbooks and, we hope, will give new possibility for applications of the magnetic resonance.

Sad1 =

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