Microscopic description of relaxation of impurity magnetic atoms in spiral magnetics
Kinetics of the impurity magnetic atoms in spiral magnetic is researched. The spectrum of the excitations of the impurity spin is obtained in approximation of weakly coupling. The equation for impurity density matrix up to second order of the perturbation theory is obtained. For impurity spin s=1/2...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
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| Zitieren: | Microscopic description of relaxation of impurity magnetic atoms in spiral magnetics / A.P. Ivashin // Вопросы атомной науки и техники. — 2001. — № 6. — С. 326-328. — Бібліогр.: 10 назв. — англ. |
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| citation_txt | Microscopic description of relaxation of impurity magnetic atoms in spiral magnetics / A.P. Ivashin // Вопросы атомной науки и техники. — 2001. — № 6. — С. 326-328. — Бібліогр.: 10 назв. — англ. |
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| container_title | Вопросы атомной науки и техники |
| description | Kinetics of the impurity magnetic atoms in spiral magnetic is researched. The spectrum of the excitations of the impurity spin is obtained in approximation of weakly coupling. The equation for impurity density matrix up to second order of the perturbation theory is obtained. For impurity spin s=1/2 the kinetic coefficients in terms of correlation functions are found.
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MICROSCOPIC DESCRIPTION OF RELAXATION
OF IMPURITY MAGNETIC ATOMS IN SPIRAL MAGNETICS
A.P. Ivashin
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
e-mail: ivashin@kipt.kharkov.ua
Kinetics of the impurity magnetic atoms in spiral magnetic is researched. The spectrum of the excitations of the
impurity spin is obtained in approximation of weakly coupling. The equation for impurity density matrix up to sec-
ond order of the perturbation theory is obtained. For impurity spin σ=1/2 the kinetic coefficients in terms of correla-
tion functions are found.
PACS: 76.90.+d; 34.10.+x
1. INTRODUCTION
The kinetics of impurity spin was considered in
many papers [1-4]. This paper deals with the kinetics of
weakly coupled impurity spin in magnetic dielectric,
which is considered as a thermostat. The thermostat has
a spiral structure of magnetic ordering. Objects with a
spiral magnetic structure are systems with a sponta-
neously broken symmetry, in which the symmetry of the
statistical equilibrium state is lower than the symmetry
of the Hamiltonian. Such state is not translationally in-
variant and does not exhibit invariance to spin rotations
around the spiral axes. The method of quasi-averages is
a convenient tool for analyzing such structures [5-7]. In
this paper, using the reduced description method [8], the
equation for impurity spin matrix was obtained. From
this equation in main approximation the energy spec-
trum of impurity spin was found. In last section we de-
scribe the evolution of impurity spin σ=1/2. We define
the kinetic coefficients, which are expressed in terms
correlation function of the thermostat.
2. KINETIC EQUATION FOR IMPURITY
DENSITY MATRIX
The equilibrium state of spiral magnetic is described
by statistical operator
−−−Ω= )(exp( 0
z
Bm ShHYW νν
.)).))(exp((0 сhthxpiSY Bll
l
+−− +∑ ν
(1)
where: Hm is the Hamiltonian of matrix; 1
0
−Y is the
temperature; BghhB 0µ+= , h is the magnetic
bias field; mceo 2=µ is a Bohr magneton; g' is the
gyromagnetic ratio for matrix spin; B is the magnetic
field, which is directed parallel to ax Z; p is the spiral
vector; νΩ is the thermodynamic potential, which is
found from the normalizing condition 1=νSpW
(trace is calculated only on thermostat states).
The conditions, which are assumed from the struc-
ture of operator (1), are as follows
[ ] 0, ≠zSW , [ ] 0, ≠mHW , [ ] 0)(, =− z
Bm ShHW .
It means, that statistical operator W depends on time.
The statistical operator also has such space symmetry
WSapiWUUSapi z
aa
z =−+ )exp()exp(
,
where the operator of the translation aU satisfies the fol-
lowing condition
alala SUSU
+
+ = , IUU aa =+
,
a- pitch of lattice, I - unit operator.
In our case the equilibrium averages of spin components
of matrix are as follows
= SS z , { }))((exp thxpiSS Bl −= ⊥
± ,
where ASpWA
V
νν ∞→→
≡ limlim
0 is the quasi averages value
of operator A
; ⊥S is the module of the transversal com-
ponent of spin; S is the longitudinal component of
spin.
We suppose, that the impurity concentration is very
low, thus the Hamiltonian of system is as follows
VgBBHH z
m +−= σµ 0)( ,
))(
2
1( +−
⊥ ++Σ−= σσσ z
l
z
ll
zz
lzll
SSISIV ,
where VBSgHBH z
mm +−= '
0)( µ Hamiltonian of spi-
ral magnetic in magnetic field, V Hamiltonian of the ex-
change interaction between the magnetic matrix and the
impurity; zgBσµ 0− Hamiltonian of the impurity; g is
gyromagnetic ratio of the impurity; Izl, lI⊥ are the longi-
tudinal and transversal integrals of interaction between
impurity spin σ and matrix spin Sl.
From the Liouville equation
[ ]Hti
t
t ),()( ρρ =
∂
∂
(2)
we obtain the equation for density matrix ρSpw =
,M
t
w =
∂
∂
[ ]HiSpM ,ρ= (3)
where M is the impurity collision integral.
The impurity matrix w satisfies equation 1=trw , tr
is a trace on impurity state. We assume [8], that the sta-
tistical operator is the functional of impurity density ma-
trix and phase thxp B−= ϕ , when t >> 0τ (τ0 is the time
of relaxation in the thermostat)
{ })(),( ttw ϕρρ → .
70 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 326-328.
It means, that
.),(),(),( ..
w
w
ww
t
w
δ
ϕδ ρ+ϕ
δ ϕ
ϕδ ρ=
∂
ϕρ∂
(4)
Substituting this equality into Liouville equation (2) and
assuming
Bh−≈
.
ϕ in main approximation, we obtain
jointly with (3) the equations set for impurity density
matrix w(t). As boundary condition we use the asymp-
totic condition for operator ),( ϕρ w
wWiHwiH )()exp(),()exp(
0
00 ϕτϕρτ τττ →− > >− , (5)
where )exp()exp( 00
zz gBiwgBiw σµσµτ −=− ,
z
B
z
m ShgBHH −−= σµ 00 .
Thus the Liouville equation jointly with (4) is as follows
[ ]VHiMM
w
w
V +=+ 00 ,)(),( ρ
δ
ϕδ ρ
,
where [ ]zwiM σ,0 Ω−= , [ ]ViSpMV ,ρ≡ , gB0µ=Ω .
We take into account, that [ ]z
B Sihw ,),( .
ρϕ
δ ϕ
ϕδ ρ = . As a
result we obtain the integral equation for ),( ϕρ w in
common with (5)
[ ] ττ
τ
δ
δ ρρτϕϕρ 00
0
,)(),( iH
ww
V
iH eM
w
ViedwWw −
∞− →
∫
−
−+=
This equation and the equation (2) form the closed set of
the equations for impurity density matrix w (t).
The zero and the first approximation for ρ(w,ϕ) equal
correspondently to
wWw )(),()0( ϕϕρ = , (6a)
[ ] ττ
τ
ϕτϕρ 00 ,)(),(
0
)1( iH
ww
iH eVVwWedw −
→
∞−
−
−= ∫ , (6b)
Here VSpWV )(ϕ≡ .
The collision integral subject to (6a), (6b) is as follows
correspondently
[ ] [ ]VwiVWiSpM ,),,()0()1( == ϕρ ,
[ ] [ ][ ]VVVwWSpdViSpM ,,)(- ,
0
)1()2( −== ∫
∞−
τϕτρ (7)
ττ
τ
00 iHiH VeeV −= .
As a result the kinetic equation for w up to the second
order on interaction take on form
)2()1()0( MMMw ++= ,
where M(0), M(1), M(2) are defined by formulae (7).
In order to calculate V we use the unitary transforma-
tion
+= ppWUUW ~ , )exp( ∑−≡
l
z
llp SxpiU .
Then the calculation of the averages in spiral mag-
netic can be obtained with the operator
.)).)exp(()(exp(~
00 сhtihSYShHYW Bl
l
z
B
p
m +−−−−Ω= +∑νν
where +≡ pmp
p
m UHUH
.
This statistical operator will be the space homoge-
nous operator. In order to get rid of dependence on time
in the statistical operator we must turn into rotating sys-
tem of coordinates. It means, that we must consider the
operator
tSihtSih z
B
z
B WeeW −=' , aSpWa '= .
Taking these notes into account and using the reduced
description method [8], the equation for impurity densi-
ty matrix in the described condensed matter was ob-
tained:
−−= ],[0 αα σω nwiw
.).]),()[((
0
chwKd +− ∫
∞−
βαα β στσττ . (8)
Frequency 0ω and ort n are defined by formulae:
2
~
2
0
'
0
~
0 ))(())0(( ⊥⊥ ++++= SpIgBBghSI z µµω ,
+++
= ⊥⊥
0
0
'
0
~
0
~
))0((
,0,)(
ω
µµ
ω
gBBghSISpIn
z ,
)(
~
pI z , )(
~
pI ⊥ are the Fourier components of the longi-
tudinal )(xI z and transversal )(xI⊥ exchange integrals;
Kαβ is the spin correlation function
( )∫ ∑
∞−
−=
0
,
)()()(),()()()(
li
eieiei xSxSxSxSxIxIdK βαβαβαα β τττ ,
⊥±+= III zz ααα δδ , τ
α
τ
α στσ 00)( iHiH ee −= .
Here the averaging is performed with homogeneous and
time independent statistical operator of magnetic matrix.
3. ENERGY SPECTRUM OF
ANY IMPURITY SPIN IN MAGNETIC
The equations (8) allow us to find the spectrum of
any impurity spin in magnetic in main approximation.
We perform the turn in spin space so as
zUnU σσ αα =+ ,
where yieU α σ
α = is the unit operator of turn, tan
zx nn /=α .
In a result we obtain the equation for impurity density
matrix w~
[ ]ztwitw σω ),(~)(~
0−= , (9)
+= αα UtwUtw )()(~ .
The Hermit operator w~ is characterized by the set of
the matrix elements
nwmwmn
~~ ≡ , σσ +−= ,..., nm ,
where m is the eigenvector of the operator zσ ,
,mmmz =σ σσ ≤≤− m .
The equation (9) in terms of the matrix elements is as
follows
)(0
~)()(~
tmnmn wnmitw −−= ω .
From this equation we define the energy spectrum of the
impurity spin.
)(0 nm −= ωω , σσ +−= ,...,, nm .
The spectrum we obtain has equidistant character.
This spectrum is nondegenerated spectrum, when
71
σσ −== nm , and σσ =−= nm , . When nm = the
repetition factor of the degenerate equals 12 +σ .
4. KINETIC COEFFICIENTS FOR IMPURI-
TY SPIN 2/1=σ
It is very difficult to solve equation (8) in general
case. Now we pay main attention to the case, when the
impurity spin is equal to 1/2. In this case:
)1)(2/1( iiPw τ+= , where iτ is the Pauli matrix and P
is the polarization vector. From (8) we get the equation
for a motion of polarization vector
βα βγββα β γ
α −−ωε−=
∂
∂
PDP)Dn(
t
P S
2
1
0 (10)
α βα β γα ε DD = , )DD(D S
β αα βα β +=
2
1 ,
)Re( α βλ λα βα β δ KKD −= .
The first item in round brackets describes the precession
of the polarization vector. The coefficients αD define
the shift and the thin structure of levels, and the coeffi-
cients SDα β – their width.
Now we will find the solutions of the equations (10)
in form
t)exp(i~ ωP .
In main approximation in the interaction between the
matrix and the impurity spin we have obtained disper-
sion equation
0)( 2
0
2 =− ωωω .
From this equation we have two solutions: 0=ω and
0ωω = . Next approximation gives frequency shift δ ϖ
and damping decrement γ .
The solution for 0=ω is as follows:
01 =δ ω , S
z
S
zx
S
x DnDnnDn 33
2
1311
2
1 2 ++=γ .
And the solution for 0ωω = is
312 DnDn zx −−=δ ω , ))(2/1( 12 γγ −= S
iiD .
These results correspond the results obtained in [9].
Now we consider the particular case, corresponding
to the case of a magnetic with “light axes” ordering
type. For this case we have 0,0 ≠=± zSS . For fun-
damental frequency 0=ω we have
01 =δ ω , SD331 =γ .
For frequency 0ωω = correspondently
32 D−=δ ω , ))(2/1( 22112
SS DD +=γ .
The results correspond the results obtained in [10].
Using the Goldstein – Primakov form for matrix
spin we can find the temperature dependence of the ob-
tained decrements. In quadratic approximation by f (the
magnon distribution function) we have only decrement
2γ , which is not equal to zero:
)1()(~)0(Re 2
2 qpqp
pq
qpzz ffIK +−== ∑ − εεδπγ .
Performing the integration we get in spin-wave approxi-
mation
=
)0(
ln
)2(
~ 2
4
2
2 εθθπ
γ kTkTI
cc
,
where Bgapp cp
'2)()( µθεε +=≡ is the energy of mag-
non, θ c is the Curie energy, )0(~~ II = is the zero Fouri-
er component of the longitudinal exchange integral.
ACKNOWLEDGMENT
The author would like to thank M.Yu. Kovalevsky
for helpful discussion of the results.
REFERENCES
1. T.L. Andreeva, P.L. Rubin. Dynamics of magnetic
momentum in Boltzman polarized gas subject to dis-
sipation // JETF. 2000, v. 118, p. 877-884.
2. A.E. Meyerovich, J.N. Naish, J.R. Overs-Bradly and
A. Stepaniants. Zero-temperature relaxation in spin
polarized Fermi systems // Fiz. Nizk. Temp. 1997,
v. 23, № 5/6, p. 553-563 (in Russian).
3. V.F. Los. Kinetics of impurity spin in magnetic //
FNT. 1981, v. 7, № 3, p. 336-349 (in Russian).
4. E.G. Petrov. Kinetic equations for dynamical sys-
tem, placed in condensed matter and strong external
field // TMF. 1986, v. 68, № 1, p. 117-127 (in Rus-
sian).
5. N.N. Bogolubov. Quasiaverages values in problems
of statistical mechanics. Preprint D-781 DUBNA, 1961.
6. M.Yu. Kovalevsky, S.V. Peletminsky, Yu.V. Slu-
sarenko. Thermodynamics and kinetics of spiral
magnetic structure and method of quasiaverage val-
ues // TMF. 1988, v. 74, p. 281-295 (in Russian).
7. M.Yu. Kovalevsky, S.V. Peletminsky, A.A. Ro-
zhkov. Green functions of spiral magnetic in hydro-
dynamic and quasi-particle approximation // TMF.
1988, v. 75, p. 86-99 (in Russian).
8. A.I. Akhieser, S.V. Peletminsky. Methods of statisti-
cal physics. M.: “Nauka”, 1977, p. 368.
9. A.P. Ivashin, M.Yu. Kovalevsky. Kinetics of impu-
rity magnetic atoms in spiral magnetic. Preprint KPTI
89-8, M: CNIIAtominform, 1989, p. 16 (in Russian).
10.A.P. Ivashin, V.D. Tsukanov. Kinetics of spin mag-
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ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2000, №2.
Серия: Ядерно-физические исследования (36), с. 3-6.
72
A.P. Ivashin
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
2. KINETIC EQUATION FOR IMPURITY DENSITY MATRIX
4. KINETIC COEFFICIENTS FOR IMPURITY SPIN
acknowledgment
REFERENCES
|
| id | nasplib_isofts_kiev_ua-123456789-80043 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T15:46:02Z |
| publishDate | 2001 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Ivashin, A.P. 2015-04-09T16:26:04Z 2015-04-09T16:26:04Z 2001 Microscopic description of relaxation of impurity magnetic atoms in spiral magnetics / A.P. Ivashin // Вопросы атомной науки и техники. — 2001. — № 6. — С. 326-328. — Бібліогр.: 10 назв. — англ. 1562-6016 PACS: 76.90.+d; 34.10.+x https://nasplib.isofts.kiev.ua/handle/123456789/80043 Kinetics of the impurity magnetic atoms in spiral magnetic is researched. The spectrum of the excitations of the impurity spin is obtained in approximation of weakly coupling. The equation for impurity density matrix up to second order of the perturbation theory is obtained. For impurity spin s=1/2 the kinetic coefficients in terms of correlation functions are found. The author would like to thank M.Yu. Kovalevsky for helpful discussion of the results. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Kinetic theory Microscopic description of relaxation of impurity magnetic atoms in spiral magnetics Микроскопическое описание релаксации примесных магнитных атомов в спиральных магнетиках Article published earlier |
| spellingShingle | Microscopic description of relaxation of impurity magnetic atoms in spiral magnetics Ivashin, A.P. Kinetic theory |
| title | Microscopic description of relaxation of impurity magnetic atoms in spiral magnetics |
| title_alt | Микроскопическое описание релаксации примесных магнитных атомов в спиральных магнетиках |
| title_full | Microscopic description of relaxation of impurity magnetic atoms in spiral magnetics |
| title_fullStr | Microscopic description of relaxation of impurity magnetic atoms in spiral magnetics |
| title_full_unstemmed | Microscopic description of relaxation of impurity magnetic atoms in spiral magnetics |
| title_short | Microscopic description of relaxation of impurity magnetic atoms in spiral magnetics |
| title_sort | microscopic description of relaxation of impurity magnetic atoms in spiral magnetics |
| topic | Kinetic theory |
| topic_facet | Kinetic theory |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/80043 |
| work_keys_str_mv | AT ivashinap microscopicdescriptionofrelaxationofimpuritymagneticatomsinspiralmagnetics AT ivashinap mikroskopičeskoeopisanierelaksaciiprimesnyhmagnitnyhatomovvspiralʹnyhmagnetikah |