Magnetization dynamics of electron-impurity systems at paramagnetic resonance
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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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1301552018-02-09T03:03:49Z Magnetization dynamics of electron-impurity systems at paramagnetic resonance Ivanchenko, E.A. Низкотемпеpатуpный магнетизм 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 exact solutions are found for magnetization dynamics in samples having the forms of an ellipsoid of revolution and a cylinder, as well. The influence of magnetic exchange on the surface of the cylinder (III boundary value problem) is taken into account. The dependence of the magnetization on the static electric field is exponentially decreasing with time, with an exponent proportional to the electric field squared times the diffusion coefficient. For a fixed instant of time the magnetization depends nonlocally on the magnitude and direction of the electric field. The estimated dynamic shift of the forced precession has a nonlocal and nonlinear dependence on the nonuniform distribution of the initial magnetization. This shift is estimated with boundary conditions taken into account and is caused by a field similar to the Suhl-Nakamura field in a paramagnetic medium. The dynamic shift of the free precession has only a nonlocal character. The 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, which depends on the shape of the sample. These results may be used for analysis of spin diffusion in natural and manmade materials and also in magnetometry. 2002 Article Magnetization dynamics of electron-impurity systems at paramagnetic resonance / E.A. Ivanchenko // Физика низких температур. — 2002. — Т. 28, № 2. — С. 168-175. — Бібліогр.: 12 назв. — англ. 0132-6414 PACS: 33.35.+r, 75.40.Gb, 76.30.-v, 76.60.Jx, 76.60.-k http://dspace.nbuv.gov.ua/handle/123456789/130155 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Низкотемпеpатуpный магнетизм Низкотемпеpатуpный магнетизм |
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Низкотемпеpатуpный магнетизм Низкотемпеpатуpный магнетизм Ivanchenko, E.A. Magnetization dynamics of electron-impurity systems at paramagnetic resonance Физика низких температур |
| 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 exact solutions are found for magnetization dynamics in samples having the forms of an ellipsoid of revolution and a cylinder, as well. The influence of magnetic exchange on the surface of the cylinder (III boundary value problem) is taken into account. The dependence of the magnetization on the static electric field is exponentially decreasing with time, with an exponent proportional to the electric field squared times the diffusion coefficient. For a fixed instant of time the magnetization depends nonlocally on the magnitude and direction of the electric field. The estimated dynamic shift of the forced precession has a nonlocal and nonlinear dependence on the nonuniform distribution of the initial magnetization. This shift is estimated with boundary conditions taken into account and is caused by a field similar to the Suhl-Nakamura field in a paramagnetic medium. The dynamic shift of the free precession has only a nonlocal character. The 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, which depends on the shape of the sample. 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 |
Magnetization dynamics of electron-impurity systems at paramagnetic resonance |
| title_short |
Magnetization dynamics of electron-impurity systems at paramagnetic resonance |
| title_full |
Magnetization dynamics of electron-impurity systems at paramagnetic resonance |
| title_fullStr |
Magnetization dynamics of electron-impurity systems at paramagnetic resonance |
| title_full_unstemmed |
Magnetization dynamics of electron-impurity systems at paramagnetic resonance |
| title_sort |
magnetization dynamics of electron-impurity systems at paramagnetic resonance |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2002 |
| topic_facet |
Низкотемпеpатуpный магнетизм |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/130155 |
| citation_txt |
Magnetization dynamics of electron-impurity systems at paramagnetic resonance / E.A. Ivanchenko // Физика низких температур. — 2002. — Т. 28, № 2. — С. 168-175. — Бібліогр.: 12 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT ivanchenkoea magnetizationdynamicsofelectronimpuritysystemsatparamagneticresonance |
| first_indexed |
2025-07-09T12:59:05Z |
| last_indexed |
2025-07-09T12:59:05Z |
| _version_ |
1837174306488975360 |
| fulltext |
Fizika Nizkikh Temperatur, 2002, v. 28, No. 2, p.168–175Ivanchenko Å. À.Magnetization dynamics of electron–impurity systems at paramagnetic resonanceIvanchenko Å. À.Magnetization dynamics of electron–impurity systems at paramagnetic resonance
Magnetization dynamics of electron–impurity systems
at paramagnetic resonance
Å. À. Ivanchenko
National Science Center «Kharkov Institute of Physics and Technology»,
Institute for Theoretical Physics, 1 Akademicheskaya Str., 61108 Kharkov, Ukraine
E-mail: yevgeny@kipt.kharkov.ua
Received September 27, 2001
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 exact solutions are found for magnetization dynamics in samples having the
forms of an ellipsoid of revolution and a cylinder, as well. The influence of magnetic
exchange on the surface of the cylinder (III boundary value problem) is taken into account.
The dependence of the magnetization on the static electric field is exponentially decreasing
with time, with an exponent proportional to the electric field squared times the diffusion
coefficient. For a fixed instant of time the magnetization depends nonlocally on the
magnitude and direction of the electric field. The estimated dynamic shift of the forced
precession has a nonlocal and nonlinear dependence on the nonuniform distribution of the
initial magnetization. This shift is estimated with boundary conditions taken into account
and is caused by a field similar to the Suhl–Nakamura field in a paramagnetic medium. The
dynamic shift of the free precession has only a nonlocal character. The 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, which depends on the shape of the sample. These results may be used
for analysis of spin diffusion in natural and manmade materials and also in magnetometry.
PACS: 33.35.+r, 75.40.Gb, 76.30.–v, 76.60.Jx, 76.60.–k
1. Introduction
A system of electrons, interacting among them-
selves and with motionless potential impurity cen-
ters randomly distributed in uniform external fields
is described by a distribution function f obeying the
kinetic equation [1,2]
∂
∂t
f + i[w,f]− + v
∂
∂x
f + eE
∂
∂p
f +
+
q
c
[v,B]
∂
∂p
f = Lf + Leef , (1)
where f ≡ fp(x,t) is the distribution function of the
electrons, which is a matrix in the electron spin
space; q and v = ∂ep/∂p are the electron charge and
velocity, respectively; L and Lee are the electron–
impurity and electron–electron collision integrals;
B is the magnetic field, E is the static electric field;
w = − µ0σσσσB (µ0 is the Bohr magneton and σσσσ are the
Pauli matrices). We assume that massive charged
impurities, whose kinetics is not considered here,
form a neutralizing electrical background.
We shall define the distribution function of elec-
trons over energy e [3]
n(e,x,t) = 〈f〉 =
1
ρ(e)
∫ dVpfp(x,t)δ(e − ep) ,
dVp =
d3p
(2π)3
,
(2)
© Å. À. Ivanchenko, 2002
where
ρ(e) = ∫ dVpδ(e − ep)
is the electron density of states, and the brackets
mean the averaging defined by formula (2). It
follows from Eq. (2) that 〈L
__
f〉 = 0, where
L
__
= L + L′(B) , L′(B) = −
q
c
[v,B]
∂
∂p
. (3)
Indeed, the electron–impurity collision integral has
the form
(Lf)(p) = 2πN ∫ dVp′ w(p,p′)δ(ep − ep′)(fp′ − fp) ,
and hence 〈Lf〉 = 0. Here N is the impurity den-
sity, w(p,p′) is the probability per unit time of
electron scattering on the impurity center. As
εikl(∂vk/∂pi) = εikl(∂
2ep/∂pi pk) = 0, hence the mean
〈[v,B]
∂
∂p
f
〉 = εikl
1
ρ(e)
∫ dVp vk Bl
∂
∂pi
f
δ(e − ep) ,
and after integration by parts we have 〈L
__
f〉 = 0.
The operator L
__
has the property
L
__
(B) = L
__
+(− B) , (4)
where + means the conjugate operation, defined by
the formula (x,y) ≡ 〈x,y〉. By virtue of the definition
of the operators L and L′, we have (Lx,y) = (x,Ly),
(L′(B)x,y) = (x,L′(−B)y), i.e., Eq. (4) is valid.
As the result of averaging Eq. (1), we obtain the
equation for the distribution function n(e,x,t):
∂
∂t
n + i[w,n]− = 〈Leef〉 −
∂
∂xk
jk − qEk
1
ρ(e)
∂
∂e
(ρ(e)jk) .
(5)
To close this equation one has to express the current
jk in terms of n. This can be done, if the frequency
of electron–impurity collisions τe imp
−1 is much grea-
ter than the frequencies of electron–electron colli-
sions τee
−1 and if the times t are large in comparison
with the corresponding relaxation time τe imp , the
electron distribution function becomes some func-
tional of n, i.e., 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 electric field the diffusion current
is [4]
jk = − Dki(B)
∂
∂xi
n + qEi
∂
∂e
n
, (6)
where Dki(B) = 〈L
__
−1vk,vi〉 is the diffusion coeffi-
cient of electrons in a magnetic field having the
property Dki(B) = Dik(− B), which follows from (4).
In case of an isotropic electron dispersion rela-
tion ep = e|p| we get
Dki = d[δki + bk bi ωc
2
τe imp + εikl bi ωc] ,
d =
τe imp v
2
3(1 + ωc
2)
, b =
B
|B|
, ωc = τe impΩc .
(7)
The cyclotron frequency Ωc is equal to
Ωc = (q|v|B)/c|p|,
τe imp
−1 ≡ 2πN ∫ dVp′ w(p,p′)δ(ep − ep′)
1 −
pp′
|pp′ |
.
(8)
Equations (5) and (6) together determine a closed
equation for the distribution function n(e,x,t),
which is isotropic with respect to the moments [4].
2. Ìacroscopic equation for magnetization
We define the macroscopic density of the elec-
tron magnetic moment M = (M1,M2,M3), the mag-
netization, by the formula
M(x,t) = 2µ0 Sp
1
2
σσσσ ∫ dVp n(e,x,t) . (9)
In view of the relation for Pauli matrixes
σj σk − σk σj = 2iεjkl σl , the kinetic equation (5)
after multiplication by µ0 σσσσ, taking the trace on the
spin variables, and integration over dVp takes the
form
∂
∂t
Mi + 2µ0[B,M]i +
∂
∂xk
Iik = 0 , (10)
where the flux density of the electron magnetic
moment is equal to
Magnetization dynamics of electron–impurity systems at paramagnetic resonance
Fizika Nizkikh Temperatur, 2002, v. 28, No. 2 169
Iik ≡ − 2µ0 Sp
1
2
σσσσi ∫ dVp Dkp
∂n
∂xp
+ qEp
∂n
∂e
.
(11)
In order to obtain a closed equation for the magne-
tization we assume that the function Dkp is smooth
over e, and it is therefore possible to take it out
from under the integral sign. We integrate the
second term in (11) by parts on e, and assume that
the surface terms are small at e = 0 and at e = eF ,
where eF is the Fermi energy. This approximation
allows us to write down the equation for the mag-
netization as
∂
∂t
M + 2µ0[B,M] − Dkp
∂2
∂xk∂xp
−
e
2eF
Ep
∂
∂xk
M = 0 .
(12)
Equation (12) without allowance for spatial inho-
mogeneity corresponds to the Bloch equation and
forms the basis of the theory of paramagnetic reso-
nance. The incorporation of inhomogeneity is carried
out in Refs. 1, 5 without specifying 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 sys-
tems on the basis of Eq. (12) under conditions of
paramagnetic resonance with boundary conditions
taken into account.
3. Paramagnetic resonance in
electron–impurity systems
We consider the magnetization behavior 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 alter-
nating 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 [see Eq. (12)] as an
expansion in powers of the amplitude of the exter-
nal alternating magnetic field:
M(x,t) = ∑
k=0
∞
m(k)(x,t) . (13)
After substituting Eq. (13) into Eq. (12) we
have an infinite system of equations for m(k)
∂
∂t
m(k) + 2µ0[B,m(k)] − Dm(k) = − 2µ0[h(t),m(k−1)] ,
k = 0, 1, 2, ... ; m(−1) ≡ 0 , B = (0,0,B) ,
(14)
D ≡ dF
∂2
∂x2
+
∂2
∂y2
+ (1 + ωc
2)
∂2
∂z2
−
q
2eF
(E1 + ωcE2)
∂
∂x
−
−
q
2eF
(E2 − ωc E1) ∂
∂y
−
q
2eF
E3(1 + ωc
2) ∂
∂z
. (15)
At first we find the solution m(0) of the Cauchy
problem with the help of the change of dependent
variables
mi
(0)(x,t) = (2π)−3/2 ∫ d3k e−ikx mi
(0)(k,t)
(i = 1, 2, 3) .
(16)
For the Fourier-components mi
(0)(k,t) we get a
system of differential equations of first order in
time. This system is easily solved. Carrying out the
inverse transformation,
mi
(0)(k,t) = (2π)−3/2 ∫ d3x′ eikx′ mi
(0)(x,t) , (17)
we find the solution for the magnetization in the
form of free precession in constant fields,
m(0)(x,t) = ∫ d3x′ g(t,x′,x) ×
×
m(x′) cos (Ωt+ϕ(x′)), − m(x′) sin (Ωt+ϕ(x′)), m3(x′)
(18)
on the set of the known initial data of the form
m(0)(x,t = 0) =
m(x) cos ϕ(x), − m(x) sin ϕ(x), m3(x)
,
(19)
with a propagation function equal to
Å. À. Ivanchenko
170 Fizika Nizkikh Temperatur, 2002, v. 28, No. 2
g(t,x′,x) =
1
(1 + ωc
2)1/2
1
2(πdF t)1/2
3
exp
−
x′ − x
2√dF t
+
q
2eF
E1 + ωcE2
2
√dFt
2
−
−
y′ − y
2√dFt
+
q
2eF
E2 − ωcE1
2
√dFt
2
−
z′ − z
2(1 + ωc
2)1/2√dFt
+
q
2eF
(1 + ωc
2)1/2E3
2
√dFt
2
, (20)
Ω = 2µ0 B , dF =
vF
2 τe imp
3(1 + ωc
2)
,
where vF is the Fermi velosity.
This function obeys the equation
∂
∂t
g − Dg = 0 (21)
and also has the properties
lim
t → 0+
g(t,x′,x) = δ(x′ − x) , (22)
lim
t → 0+
∂
∂t
g(t,x′,x) = Dδ(x′ − x) . (23)
An important property of the propagation function is that it satisfies the Smolukhowski–Chapman–Kolmo-
goroff equation
∫ d3x′ g(t − t′,x′,x)g(t′,x′′ ,x) = g(t,x′′ ,x) . (24)
The particular solution m(1) of Eq. (14) with the right-hand side − 2µ0 [h(t),m(0)(x,t)] can be obtained, using
the semigroup property of the function g(t,x′,x) (24):
m1
(1)(x,t) + im2
(1)(x,t) = ∫
0
t
dt ′ eiΩ(t′−t) ∫ d3x′ g(t,x′,x)[(h1(t′) + ih2(t′))m3(x′) − h3(t′) e−i(Ωt′+ϕ(x′))m(x′)] , (25)
m3
(1)(x,t) = ∫
0
t
dt′ ∫ d3x′ g(t,x′,x)m(x′)[h1(t′) sin (Ωt′ + ϕ(x′)) + h2(t′) cos (Ωt′ + ϕ(x′))] . (26)
It is seen from formulas (25), (26), that the magne-
tization at the time t is determined by the field
h(t) at all previous times, starting at the instant it
is turned on.
We choose left rotation for the external alternat-
ing magnetic field, which is perpendicular to the
static field B0 , h(t) = h(cos ωt,− sin ωt, 0), h is the
amplitude of the field, and ω is the frequency of the
alternating magnetic field. Since at the paramag-
netic resonancå ω = Ω, we find from formulas (25),
(26) that the particular solution for the magnetiza-
tion in the approximation linear in the field is the
forced precession
m1
(1)(x,t) = ω1tm3
(0)(x,t) cos (Ωt − π/2) ,
m2
(1)(x,t) = − ω1tm3
(0)(x,t) sin (Ωt − π/2) , (27)
Magnetization dynamics of electron–impurity systems at paramagnetic resonance
Fizika Nizkikh Temperatur, 2002, v. 28, No. 2 171
m3
(1)(x,t) = ω1tA(x,t) ,
A(x,t) ≡ ∫ d3x′ g(t,x′,x)m(x′) sin ϕ(x′) , ω1 = 2µ0 h,
(28)
lagging in phase behind the phase of the alternating
magnetic field by π/2.
Having continued the procedure of iteration, we
can sum the series on ω1t and find the general exact
solution for the magnetization dynamics (12) at
paramagnetic resonance:
M(x,t) = m(0)(x,t) + m(x,t) , (29)
m(x,t) =
=
[A(x,t)(1 − cos ω1t) + m3
(0)(x,t) sin ω1t] sin Ωt,
[A(x,t)(1 − cos ω1t) + m3
(0)(x,t) sin ω1t] cos Ωt,
A(x,t) sin ω1t + m3
(0)(x,t)(cos ω1t − 1)
.
At h = 0 this solution transforms into M(x,t) =
= m(0)(x,t); see Eq. (18). As is seen from the solu-
tion (29), there is no divergence in time. Finally we
come to the conclusion that the solution of the
Cauchy problem of Eq. (12) with the initial dis-
tribution (19) in the class of square-integrable func-
tions 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 so-
lution of the Cauchy problem takes the form of
Eq. (29).
The magnetization projection M3(x,t) oscillates.
This fact implies that a population inversion occurs
in the system considered.
For the analysis of the forced precession we write
the solution (29) for M1 , M2 as
M1(x,t) = a cos (Ωt + φ) ,
M2(x,t) = − a sin (Ωt + φ) ,
(30)
where the local amplitude and phase of precession
are equal to
a(x,t) =
= √ [− A(x,t) cos ω1t + m3
(0)(x,t) sin ω1t]
2 + A1
2(x,t) ,
φ(x,t) = arctg
A(x,t) cos ω1t − m3
(0)(x,t) sin ω1t
A1(x,t)
,
A1(x,t) ≡ ∫ d3x′ g(t,x′,x)m(x′) cos ϕ(x′) .
(31)
Expanding the phase φ(x,t) (31) in a series with
respect to t and restricting ourselves to the term
linear in t, we get, in view of the property (23), the
local dynamic shift of the forced frequency Ω′ with
respect to the Larmor precession frequency Ω,
φ(x,t) = ϕ(x) + Ω′(x,0)t + ... ,
Ω′(x,0) =
1
m2(x)
− ω1m3(x) +
+ ∫ d3x′ (Dδ(x − x′))m(x′) sin ϕ(x′)
m(x) cos ϕ(x) −
−
∫ d3x′ (Dδ(x − x′))m(x′) cos ϕ(x′)
×
×
m(x) sin ϕ(x) − ω1m3(x)
, (32)
the cause of which has the meaning of the internal
field at the point x (an analog of the Suhl–Naka-
mura field [8] in a paramagnetic medium, coordi-
nated with the boundary conditions). This field
depends on an initial nonuniform magnetization
distribution at all points of the sample and the
shape of the sample, that is, it has a nonlocal
character. Without nonlocality being taken into
account, this shift is proportional to the amplitude
of the forced field and has the simple form,
Ω′(x,0) = − 2µ0 h
m3(x) cos ϕ(x)
m(x)
. (33)
As it is seen from the formula (33), this shift
depends nonlinearly on the 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 Ω′free(x,0) is
completely nonlocal:
Ω′free(x,0) =
= −
1
m(x)
∫ d3x′ (Dδ(x−x′))m(x′) sin (ϕ(x) − ϕ(x′)).
(34)
Å. À. Ivanchenko
172 Fizika Nizkikh Temperatur, 2002, v. 28, No. 2
Now it is obvious that in the general case the
dependence of the dynamic shift on time and coor-
dinates is
Ω′(x,t) =
φ(x,t) − ϕ(x)
t
. (35)
We find the maximal amplitude of the forced pre-
cession amax from the condition M3 = 0, i.e.,
A sin ω1t + m3
(0) cos ω1t = 0 . (36)
After substituting (36) in (31), we get
amax =
m3
(0)2
sin2 ω1t
+ A1
2
1/2
, (37)
and the times t(k) are determined by the solution
of Eq. (36), which can be written in equivalent
form as
sin (ω1t + δ) = 0 , δ = arctg
m3
(0)
A
. (38)
In the simplest case we find δ ≈ − π/4, ω1t ≈ kπ +
+ π/4, k = 0, 1, 2,...
t(0) ≈
π
4ω1
, t(1) ≈
5π
4ω1
, ... ,
(39)
at=t(0)
max ≈ √3 |m3
(0)|t=t(0) .
Decaying bursts of precession amplitude amax are
observed.
To take into account the particular shape of the
sample it is necessary to use the eigenfunctions of
the Laplace operator in cylindrical, spherical, or
other coordinates as well. The general formulas
(29) and the ones related to them retain their form
under replacement of the propagation function and
volume of integration. The propagation functions
for the cylinder and ellipsoid of revolution are given
in Appendix 1. The general exact solution of the
equation (12) is given in Appendix 2.
4. Conclusions
The dynamics of the evolution of a system of
electrons and impurities placed in static electric and
magnetic fields is investigated under the influence
of an alternating magnetic field under conditions of
paramagnetic resonance. The general formulas for
all three magnetization components in their evo-
lutionary interrelation are obtained with the shape
of the sample (cylinder, ellipsoid of revolution)
taken into account, since experimental engineering
allows one to measure these components [9]. The
behavior of forced precession in samples is theoreti-
cally investigated. The dynamic shift of the fre-
quency of paramagnetic resonance caused by a non-
uniform distribution of initial magnetization is
found. All results are expressed in terms of the
initial magnetization distribution and a propagation
function dependent on the shape of the sample. The
results obtained are applied to the analysis of spin
diffusion in natural and manmade materials [10,11]
and also in magnetometry [9].
Acknowledgements
The author thanks Prof. S. V. Peletminskii for
interest in the research and for useful discussion.
Appendix 1
Cylinder (III boundary value problem) Ref. 12
E = (0,0,E3) ,
x = r cos ϕ , y = r sin ϕ ,
z
√ 1 + ωc
2
= u ;
0 ≤ r ≤ r0 , 0 ≤ ϕ ≤ 2π , 0 ≤ z ≤ l , 0 ≤ t ≤ ∞ ,
d3x′ = r′ dr′dϕ′du′ .
The boundary conditions are
∂
∂z
M − h1M
z = 0
= 0 ,
∂
∂z
M + h2M
z = l
= 0 ,
∂
∂r
M + H3M
z = l
= 0 ,
where h1 , h2 , H3 are the surface magneto-ex-
change factors.
The initial conditions are
Mt = 0 =
m(x) cos ϕ(x), − m(x) sin ϕ(x), m3(x)
.
The propagation function is
Magnetization dynamics of electron–impurity systems at paramagnetic resonance
Fizika Nizkikh Temperatur, 2002, v. 28, No. 2 173
g(t,r ′,ϕ′,u′,r,ϕ,u) =
4
πr0
2
exp
−
qE3
4eF
√ 1 + ωc
2 (u′ − u) −
(1 + ωc
2)q2E3
2
16eF
2
dF t
×
× ∑
n=0; m,k = 1
∞
l +
(H1H2 + νm
2 )(H1 + H2)
(H1
2 + νm
2 )(H2
2 + νm
2 )
−1
1 +
r0
2H3 − n2
(µk
(n))2
−1
Jn
−2(µk
(n))Jn
µk
(n)
r0
r
Jn
µk
(n)
r0
r ′
×
× exp
−
µk
(n)
r0
2
+ νm
2
dF t
(εn
−1 cos nϕ cos nϕ′ + sin nϕ sin nϕ′) sin (νm u + zm) sin (νm u′ + zm) ,
εn = 2 if n = 0 and εn = 1 if n ≠ 0, Jn are the Bessel functions, µk
(n), νm are the positive roots according to the
equations
µk
(n)J′(µk
(n)) + r0H3J(µk
(n)) = 0 , ctg
νm l
√ 1 + ωc
2
=
νm
2 − H1H2
νm(H1 + H2)
,
zm = arctg
νm
H1
, H1 = √ 1 + ωc
2
h1 −
qE3
4eF
, H2 = √ 1 + ωc
2
h2 +
qE3
4eF
.
Ellipsoid of revolution
x = r cos θ sin ϕ , y = r cos θ cos ϕ , u =
z
√ 1 + ωc
2
= r cos θ ,
0 ≤ r ≤ r0 , 0 ≤ ϕ ≤ 2π , 0 ≤ θ ≤ π , d3x′ = r ′2 sin θ′ dr ′dθ′dϕ′ .
The boundary conditions are [M(x,t)]r = r
0
= 0.
The initial conditions are Mt = 0 = (m(x) cos ϕ(x), − m(x) sin ϕ(x), m3(x)). The propagation function is
g(t,r′,θ′,ϕ′,r,θ,ϕ) =
=
1
πr0
2
exp
−
q(E1 + ωcE2)(x′ − x)
4eF
−
q(E2 − ωcE1)(y′ − y)
4eF
−
qE3√ 1 + ωc
2 (u′ − u)
4eF
−
(1 + ωc
2)q2E2dF t
16eF
2
×
× ∑
n = 0, m = 1
∞
∑
k = 1
n
εk
−1 cos kϕ cos kϕ′ + sin kϕ sin kϕ′
(n + k)!
(2n + 1)(n − k)!
J′n+1⁄2(µm
(n))
2
exp
− (µm
(n)/r0)2dF t
×
×
Jn+1⁄2(µm
(n)r/r0)Jn+1⁄2(µm
(n)r ′/r0)
√rr ′
Pn,k(cos θ)Pn,k(cos θ′) ,
Jn+1⁄2 are the Bessel functions of half-integral order, Pn,k are the associated Legendre functions, and µm
(n) are
the positive roots of the equation Jn+1⁄2(µm
(n)) = 0.
Å. À. Ivanchenko
174 Fizika Nizkikh Temperatur, 2002, v. 28, No. 2
Appendix 2
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:
M(x,∆,t) = m(0)(x,t) + m(x,∆,t) ,
m(x,∆,t) =
− A cos γt +
ω1
γ
m3
(0) sin γt
sin ωt + A1 cos ωt −
∆
γ
A1 sin γt sin ωt − A1 cos Ωt + A sin Ωt −
−
∆
γ
A sin γt − (ω1∆m3
(0) − ∆2A1)
1 − cos γt
γ2
cos ωt,
− A cos γt +
ω1
γ
m3
(0) sin γt
cos ωt − A1 sin ωt +
+
∆
γ
A sin γt − (ω1∆m3
(0) − ∆2A1)
1 − cos γt
γ2
sin ωt − ∆
γ
A1 sin γt cos ωt + A1 sin Ωt + A cos Ωt,
m3
(0)(cos γt − 1) +
ω1
γ
A sin γt + (∆2m3
(0) + ω1∆A1)
1 − cos γt
γ2
,
where γ = √ω1
2 + ∆2 .
The dynamic shift is Ω′(x,∆,t) =
φ(x,∆,t) − ϕ(x)
t
, where
φ(x,∆,t) = arctg
A cos γt − ω1m3
(0)(sin γt/γ) + ∆A1(sin γt/γ)
A1 − ∆A(sin γt/γ) + (ω1∆m3
(0) − ∆2A1)(1 − cos γt)/γ2
.
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