Symmetry, phase states and dynamics of magnets with spin s=1
The results of investigations of magnets with spin s=1 are presented. The analysis of the possible symmetry of exchange interactions and its relationship with the magnetic degrees of freedom was done. We formulate the dynamics of normal non-equilibrium states. The generalization of the Bloch equatio...
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| Zitieren: | Symmetry, phase states and dynamics of magnets with spin s=1 / M.Y. Kovalevsky, O.A. Kotelnikova // Вопросы атомной науки и техники. — 2012. — № 1. — С. 316-320. — Бібліогр.: 11 назв. — англ. |
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| citation_txt | Symmetry, phase states and dynamics of magnets with spin s=1 / M.Y. Kovalevsky, O.A. Kotelnikova // Вопросы атомной науки и техники. — 2012. — № 1. — С. 316-320. — Бібліогр.: 11 назв. — англ. |
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| description | The results of investigations of magnets with spin s=1 are presented. The analysis of the possible symmetry of exchange interactions and its relationship with the magnetic degrees of freedom was done. We formulate the dynamics of normal non-equilibrium states. The generalization of the Bloch equations is obtained and the effect of magnetic field on the spectral characteristics is considered. The influence of dissipative processes is investigated and the relaxation fluxes corresponding to the exchange symmetry of the magnetic Hamiltonian are obtained.
Представлены результаты исследований магнетиков со спином 1. Дан анализ возможной симметрии обменных взаимодействий и ее связь с магнитными степенями свободы. Сформулирована динамика нормальных неравновесных состояний. Получено обобщение уравнения Блоха и изучено влияние магнитного поля на спектральные характеристики. Рассмотрено влияние диссипативных процессов и найдены релаксационные потоки, обусловленные обменной симметрией магнитного гамильтониана.
Представлені результати досліджень магнетиків зі спіном 1. Дано аналіз можливої симетрії обмінних взаємодій та їх зв'язок з магнітними ступенями свободи. Сформульована динаміка нормальних нерівноважних станів. Отримано узагальнення рівняння Блоха і вивчено вплив магнітного поля на спектральні характеристики. Розглянуто вплив дисипативних процесів і знайдені релаксаційні потоки, зумовлені обмінної симетрією магнітного гамільтоніана.
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SYMMETRY, PHASE STATES AND DYNAMICS OF
MAGNETS WITH SPIN s=1
M.Y. Kovalevsky 1,2∗and O.A. Kotelnikova 2
1National Science Center “Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine
2Belgorod State University, 308015, Belgorod, Russia
(Received November 7, 2011)
The results of investigations of magnets with spin s = 1 are presented. The analysis of the possible symmetry
of exchange interactions and its relationship with the magnetic degrees of freedom was done. We formulate the
dynamics of normal non-equilibrium states. The generalization of the Bloch equations is obtained and the effect of
magnetic field on the spectral characteristics is considered. The influence of dissipative processes is investigated and
the relaxation fluxes corresponding to the exchange symmetry of the magnetic Hamiltonian are obtained.
PACS: 75.10Dg, 75.25+z
1. INTRODUCTION
The Landau-Lifshitz equation [1] defines the evolu-
tion of magnets in terms of the spin vector. This
equation is well justified for the spin s=1/2 and
used for studying the static and dynamic properties
of magnetic insulators [2]. Discovery of quadrupole
states and synthesizing of high-spin molecules have
required clarifying the ideology of the macroscopic
description of magnets with a spin s > 1/2 [3]. An
additional stimulus came from the investigating of
Bose-Einstein condensates of neutral atoms with a
nonzero spin [4, 5]. For them the realization the mag-
netic states with higher symmetries of ordering, as
compared with SO(3) symmetry, is possible. Using
technology of optical lattices it will be possible, in
principle, to construct magnetic materials with new
physical properties at low temperatures.
Two points must be kept in mind regarding the
development of notions pertaining to a abridge de-
scription of nonequilibrium magnetic states. The first
is the need to extend the degrees of freedom in mag-
netic systems with spin s > 1/2. For pure quan-
tum states these degrees of freedom are associated
with the number of parameters characterizing the
one-particle spin states. The normalization condition
and the freedom to choose the wave-function phase
lead to Npure (s) = 4s independent parameters for
spin s. In the case of mixed quantum states because
of the hermiticity and the normalization condition
for density matrix, the number of such parameters
is Nmix (s) = 4s (s+ 1). The other important point
in generalizing the macroscopic description of mag-
nets is related to the notion of normal and degen-
erate equilibrium states for quantum objects. Nor-
mal states correspond to a paramagnetic state. The
other states of magnets are states with spontaneously
broken symmetry. Depending on the pattern of sym-
metry breaking due to the nature of the order para-
meter, adequately treating the system also requires
extending the set of macroscopic parameters. Sev-
eral options of dynamical behavior with different full
sets of parameters of the abridge description can be
realized in the s = 1 magnets. The set of these pa-
rameters essentially depends on the symmetry of the
Hamiltonian and the symmetry of equilibrium states,
which may not coincide in general.
2. SYMMETRY OF THE EXCHANGE
HAMILTONIAN AND EQUILIBRIUM
STATES
The symmetry of the exchange Hamiltonian and the
equilibrium states allows one to find a set of ther-
modynamic parameters describing the macroscopic
magnetic states. To formulate these symmetry prop-
erties, we use the construction of the Gibbs statisti-
cal operator, which is not part of Hamiltonian me-
chanics. We give the necessary mathematical for-
mulation and physical clarifications regarding the
use of the terms “normal” and “degenerate” equi-
librium states below, using the language of quan-
tum mechanics. In the case SO(3) symmetry the
exchange interaction Hamiltonian and normal equi-
libriums described by the Gibbs statistical operator
ŵ (Y ) = exp (Ω − Yaγ̂a) satisfy the equalities[
Ĥ, Ŝα
]
= 0,
[
ŵ, Σ̂α (Y)
]
= 0. (1)
The generalized operator of spin moment is intro-
duced here as
Σ̂α (Y) ≡ Ŝα + SY
α , SY
α ≡ −iεαβγYβ
∂
∂Yγ
. (2)
∗Corresponding author E-mail address: mikov51@mail.ru
316 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2012, N 1.
Series: Nuclear Physics Investigations (57), p. 316-320.
It acts in the Hilbert space and in the space of
thermodynamic forces Ya = (Y0, Yα). The thermo-
dynamic forces determine the temperature Y −1
0 ≡ T
and the internal magnetic field −Yα/Y0 ≡ hα and
conjugate with the motion integrals γ̂a ≡ Ĥ, Ŝα. The
thermodynamic potential is can be determined from
the condition of normalization Spŵ = 1. Relations
(1) mean that the Hamiltonian and the equilibrium
are invariant under unitary transformations of homo-
geneous spin rotation Û = exp iθαΣ̂α (Y), whose gen-
erator is the operator (2). Degenerate equilibriums
have a symmetry lower than that of the Hamiltonian,
with
[
ŵ, Σ̂α (Y)
]
�= 0. In the case of degenerate equi-
librium states their description requires to use the
quasiaverages conception [6, 7]. It hence follows that
the equilibrium states depend on the unitary transfor-
mation parameters: ŵ = ŵ (Y, θα). Table 1 shows the
relationship of the symmetry properties of the Hamil-
tonian and the equilibrium states with the number of
magnetic degrees of freedom for magnets with spin
s = 1/2.
Now we consider the magnets with spin s = 1.
This case, as seen from Table 2, is much more com-
plicated. In addition to the well-known SO(3) sym-
metry, for spin s=1 it is possible to implement the
SU(3) symmetry. Along with a vector order para-
meter a tensor order parameter may exist and also
more diverse ways of symmetry breaking of the equi-
librium state may be realized. A number of mag-
netic degrees of freedom increases. The statistical
operator of normal equilibriums of magnets with the
SU(3) symmetry has a similar form. In addition to
the Hamiltonian, a set of additive integrals of motion
γ̂a ≡
(
Ĥ, Ĝαβ
)
contain the matrix operator Ĝαβ =∫
d3xĝαβ (x). Here, following [8], we introduce the
tensor density operator ĝαβ (x) ≡ ψ̂+
α (x) ψ̂β (x) −
δαβψ̂
+
γ (x) ψ̂γ (x) /3 in terms of the Bose field cre-
ation and annihilation operators ψ̂+
α (x) , ψ̂α (x). The
SU(3) symmetry of normal equilibriums is formulated
similarly according to relations (1) and (2). For this,
we introduce the operator
Ĝαβ (Y) ≡ Ĝαβ +GY
αβ ,
GY
αβ ≡ Yαλ
∂
∂Yβλ
− Yλβ
∂
∂Yλα
. (3)
Table 1. Normal and degenerate equilibrium states of magnets with spin s = 1/2
The symmetry
of Hamiltonian
The symmetry
of the equilib-
rium state
The symme-
try group
Magnetic de-
grees of free-
dom
The number of
magnetic de-
grees of freedom
Order
parameter
[
Ĥ, Ŝα
]
= 0
[
ŵ, Σ̂α (Y)
]
= 0 SO(3) sα 3 -[
Ĥ, Ŝα
]
= 0
[
ŵ, Σ̂α (Y)
]
�= 0 SO(3) broken sα Rαβ 6 vector
Table 2. Normal and degenerate equilibrium states of magnets with spin s = 1
The symmetry
of Hamiltonian
The symmetry of
the equilibrium
state
The symme-
try group
Magnetic de-
grees of free-
dom
The number of
magnetic de-
grees of freedom
Order
parameter
[
Ĥ, Ŝα
]
= 0[
Ĥ, Q̂αβ
]
�= 0
[
ŵ, Σ̂α (Y)
]
= 0 SO(3) sα 3 —
[
Ĥ, Ŝα
]
= 0[
Ĥ, Q̂αβ
]
�= 0
[
ŵ, Σ̂α (Y)
]
�= 0 SO(3) broken sα Rαβ 6 vector
[
Ĥ, Ŝα
]
= 0[
Ĥ, Q̂αβ
]
�= 0
[
ŵ, Σ̂α (Y)
]
�= 0 SO(3) broken sα qαβ 8 tensor
[
Ĥ, Ĝαβ
]
= 0
[
ŵ, Ĝαβ (Y)
]
= 0 SU(3) gαβ 8 —[
Ĥ, Ĝαβ
]
= 0
[
ŵ, Ĝαβ (Y)
]
�= 0 SU(3) broken gαβ Rαβ 11 vector[
Ĥ, Ĝαβ
]
= 0
[
ŵ, Ĝαβ (Y)
]
�= 0 SU(3) broken gαβ Δαβ 16 tensor
317
This operator acts in a Hilbert space and in the
space of thermodynamic parameters. Using this gen-
erator, we can write property of SU(3) symmetry
for Hamiltonian and equilibrium states. Operator
Ĝαβ (Y) satisfies the relations
[
Ĝαβ (Y) , Ĝμν (Y)
]
=
Ĝαν (Y) δβμ − Ĝμβ (Y) δαν . The SU(3) symmetry
conditions for the Hamiltonian and normal equilib-
riums then become
[
Ĥ, Ĝαβ
]
= 0,
[
ŵ, Ĝαβ (Y)
]
= 0.
These formulas mean that the Hamiltonian and the
equilibrium are invariant under homogeneous lin-
ear transformation Û = exp iθαβĜβα (Y), whose
generator is the operator in (3). In the case of
spontaneous symmetry breaking (degenerate states),[
ŵ, Ĝαβ (Y)
]
�= 0, which results in an additional de-
pendence of the equilibrium on the parameters of
the unitary transformation ŵ = ŵ (Y, θαβ). Table 2
shows the relationship of the symmetry properties of
the Hamiltonian and the equilibrium states with the
number of magnetic degrees of freedom for magnets
with spin s = 1.
3. NONEQUILIBRIUM PROCESSES
In [9] the Poisson brackets for the Hermitian ma-
trix ĝ (x) were obtained:
i{gαβ (x) , gγρ (x′)}
= (−gαρ (x) δγβ + gγβ (x) δαρ) δ (x−x′) . (4)
This matrix is related to the quadrupole matrix
qαβ (x) and spin density sα (x) by relation: gαβ (x) ≡
qαβ (x) − iεαβγsγ (x) /2. Dynamics of normal non-
equilibrium states with spin 1 is described by the
Hamiltonian, which is a functional of matrix ĝ (x):
H = H (ĝ (x)). Using standard Hamiltonian formal-
ism, we obtain the equations of nonlinear dynamics
for the matrix
ˆ̇g (x) = i
[
ĝ (x) ,
δĤ (g)
δg (x)
]
, (5)
which generalizes the Landau-Lifshitz equation for
the considered magnets. In the case of SU(3) sym-
metry of the Hamiltonian a set of integrals of motion
consists of the exchange Hamiltonian and the matrix
Gαβ : γa ≡ (H,Gαβ) =
∫
d3xζa (x), {H, γa} = 0.
Here ζa (x) = ε (x) , gαβ (x) are densities of the addi-
tive integrals of motion (a = 0, αβ). Using the rep-
resentation of the flux densities of additive integrals
of motion [10], we obtain the dynamic equations, re-
flecting the conservation laws in the differential form
ε̇ (x) = −∇kqk (x) ,
˙̂g (x) = −∇k ĵk (x) ,
(6)
qk(x) ≡ ς
(0)
0k (x) =
=
1
2
∫
d3x′x′k
1∫
0
dλ{ε(x + λx′), ε(x − (1 − λ)x′)},
ĵk(x) ≡ ς̂
(0)
k (x) =
=
∫
d3x′x′k
1∫
0
dλ{ĝ(x + λx′), ε(x − (1 − λ)x′)},
where qk (x) is the energy flux density and ĵk (x) is
the flux density corresponding to the conserved quan-
tity Ĝ. Taking into account the equality (4), from (5),
(6) we obtain expressions for the flux densities of the
additive integrals of motion
ĵk = i
[
ĝ,
∂ε̂
∂∇kg
]
, qk = Sp
δĤ
δg
ĵk. (7)
Consider the homogeneous dynamics of the mag-
netic medium in an external constant magnetic field.
Hamiltonian V (h), describing such an interacting
medium, in its simplest form can be written as
V (h) ≡ SpĜĥ = V1 (h) + V2 (h), where the term
linear in magnetic field
V1 (h) = −ihαεαβγGβγ = −hαSα (8)
is the Zeeman interaction. Term
V2 (h) = Qαβhβα, hαβ ≡ hαhβ − δαβh
2/3 + iεαβγhγ
(9)
is squared in the magnetic field. In the absence of
spatial inhomogeneities and of term V2 (h) for matrix
gαβ, according (4), (5), (8) we obtain the dynamic
equation ġαβ = hσ (gαρεσβρ − εσραgρβ). Hence, sep-
arating the symmetric and antisymmetric parts, we
obtain the equations
ṡα = εαβγsβhγ , q̇αβ = hσ (qαρεσβρ − εσραqρβ) .
(10)
The first of them is the Bloch equation, which de-
scribes the spin dynamics. The second one is the
equation of motion for the quadrupole matrix. Ob-
viously, in equilibrium, the spin is directed along the
magnetic field sα||hα, quadrupole matrix is uniaxial
and has the form qαβ = q (nαnβ − δαβ/3), where unit
vector nα ≡ hα/h. The solution of the first equation
in (10) leads to two spin-wave spectra ω = 0 and
ω = h. The solution of the second equation in (10)
leads to three quadrupole spectra of waves: ω = 0,
ω = h, ω = 2h. Consider now the effect of interaction
V2 (h) on dynamics of the system. The corresponding
equation for the matrix gαβ has the form ˆ̇g = i
[
ĝ, ĥ
]
.
This implies the following dynamical equations for
the density matrix of spin and quadrupole:
ṡα = 2εαβρqγρhβγ ,
q̇αβ = sσhρ (hβεαρσ + hαεβρσ) . (11)
318
Stationary solutions of these equations lead to
the condition of collinearity of the vectors sα||hα.
The quadrupole matrix still has the form qαβ =
q (nαnβ − δαβ/3), where there are two possible solu-
tions for the unit vector nα = hα/h and nα⊥hα. The
solution of (11) leads to the two spectra of collective
excitations: ω = 0, ω = h2.
Let us consider the relaxation processes in mag-
netic materials with spin s=1. For this we use the ap-
proach of [11], where the dissipative Poisson brackets
were introduced and the relaxation equations for the
dynamics of condensed matter were obtained. The
equations of motion for the densities of additive inte-
grals of motion can be written as
ζ̇a (x) ≡ {ζa (x) , H} − T0{ζa (x) ,Σ}D. (12)
Here Σ =
∫
d3xs (x) is the entropy and T0 is a con-
stant having the dimensionality of temperature. Re-
active Poisson bracket describes the dynamics of the
system in the adiabatic approximation, while the dis-
sipative bracket – the relaxation processes. Dissipa-
tive brackets are symmetric and satisfy the Leibnitz
identity
{A,B}D = {B,A}D,
{A,BC}D = {A,B}DC +B{A,C}D.
For the densities of additive integrals of motion,
using (4) and (6), (7), we obtain the Poisson brackets:
{ζa (x) , ζb (x′)} = −iδa,αβδb,γρδ (x − x′)×
× (gγβ (x) δαρ − gαρ (x) δγβ)+
+
[
δa0ζ
(0)
bk (x) + δb0ζ
(0)
ak (x′)
]
∇′
kδ (x − x′)
(13)
The right side of the bracket is represented in terms of
densities and the corresponding fluxes of the additive
integrals of motion. The explicit form of the dissi-
pative Poisson brackets can be expressed in terms of
the dissipation function, which under consideration
for magnets has the form
R ≡ 1
2
∫
d3x∇kYa (x) Iak,bl (x)∇lYb (x)
=
∫
d3xr (x). (14)
Here Ya (x) = δΣ/δζa (x) are thermodynamic forces
conjugate to the additive integrals of motion, Iak;bl
are generalized kinetic coefficients, which satisfy the
Onsager principle of the kinetic coefficients symmetry
Iak,bl = Ibl,ak. Since the matrix ĝ is traceless, then we
have the additional relations Iααk,bl = 0, Iak,γγl = 0.
Taking into account [11] we obtain expression
{ζa (x) , ζb (x′)}D ≡ −δ2R/δYa (x) δYb (x′)
= − 1
T0
∇k∇′
l (Iak,bl (x) δ (x− x′)) . (15)
Accounting for the relaxation processes leads to
the equations of dynamics for the densities of addi-
tive integrals of motion
ς̇a (x)=−∇k
(
ζ
(0)
ak (x) + ζ
(1)
ak (x)
)
≡ LR
a (x)+LD
a (x) ,
(16)
where we obtain
LD
a (x) = −T0
∫
d3x′
δΣ
δζb (x′)
{ζa (x) , ζb (x′)}
D
.
The equations (13), (15), (16) yield the dynamic
equation for the entropy density
ṡ (x) = −∇kj
(1)
sk (x) + I (x) , (17)
where j(1)sk = Yaζ
(1)
ak is the flux density of entropy and
I = ζ
(1)
ak ∇kYa is the entropy production. Taking into
account formulas (14), (17), we see that the dissipa-
tion function is associated with the densities of the
dissipative flow of the additive integrals of motion
equation LD
a (x) = −∇kζ
(1)
ak (x) = δR/δYa (x). In
the exchange approximation the tensor structure of
the generalized transport coefficients is such that the
spatial and spin indices are not mixed and there is no
preferred direction in configuration space. Therefore,
Iak,bl = δklIab. In this case, for the dissipative flux
densities of additive integrals of motion we obtain the
expressions
j
(1)k
αβ = −Dαβ∇kT − σαβ,γρ∇khργ ,
q
(1)
k = − (κ+ hβαDαβ)∇kT−
−TDαβ∇khβα − σαβ,γρhβα∇khργ .
(18)
Coefficients of thermal conductivity κ, magnetic ther-
modiffusion Dαβ and magnetic diffusion σαβ,γρ are
associated with generalized kinetic coefficients by the
relations: Iαβ,0 = T 2Dαβ + Thγρσαβ,ργ , Iαβ,γρ =
Tσαβ,ργ , I0,0 = T 2κ+ 2T 2hγρDργ + Thβαhγρσαβ,ργ .
Account now for the specific structure of the trans-
port coefficients for the paramagnetic state, where in
equilibrium gαβ = 0 and hαβ = 0. The expressions
for the tensor of kinetic coefficients become simplified
and take the form
σαβ,γρ = σ (δαρδβγ − δαγδβρ) /4+
+σ′ (δαγδβρ + δαρδβγ − 2
3δαβδγρ
)
/2 ,
Dαβ = 0. (19)
Here σ, σ′ are, respectively, the spin diffusion coeffi-
cient and diffusion of the quadrupole matrix. As a
result, we obtain the flux density of the matrix gαβ
and energy density
j
(1)k
αβ = iσεαβγ∇khγ/2 − σ′∇kh
s
αβ ,
q
(1)
k = −κ∇kT ,
(20)
where ha
αβ ≡ −iεαβγhγ , hs
αβ ≡ (hαβ + hβα) /2. From
(17) - (20) the expressions for the dissipative flux and
entropy production follow:
j
(1)k
s = − κ
T ∇kT, I = κ
T 2 (∇kT )2 +
+ σ
T (∇khα)2 + σ′
T
(
∇kh
s
αβ
)2
≥ 0.
Positivity of entropy production is ensured by the in-
equalities κ ≥ 0, σ ≥ 0, σ′ ≥ 0.
319
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| id | nasplib_isofts_kiev_ua-123456789-107380 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T15:27:19Z |
| publishDate | 2012 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Kovalevsky, M.Y. Kotelnikova, O.A. 2016-10-19T07:01:48Z 2016-10-19T07:01:48Z 2012 Symmetry, phase states and dynamics of magnets with spin s=1 / M.Y. Kovalevsky, O.A. Kotelnikova // Вопросы атомной науки и техники. — 2012. — № 1. — С. 316-320. — Бібліогр.: 11 назв. — англ. 1562-6016 PACS: 75.10Dg, 75.25+z https://nasplib.isofts.kiev.ua/handle/123456789/107380 The results of investigations of magnets with spin s=1 are presented. The analysis of the possible symmetry of exchange interactions and its relationship with the magnetic degrees of freedom was done. We formulate the dynamics of normal non-equilibrium states. The generalization of the Bloch equations is obtained and the effect of magnetic field on the spectral characteristics is considered. The influence of dissipative processes is investigated and the relaxation fluxes corresponding to the exchange symmetry of the magnetic Hamiltonian are obtained. Представлены результаты исследований магнетиков со спином 1. Дан анализ возможной симметрии обменных взаимодействий и ее связь с магнитными степенями свободы. Сформулирована динамика нормальных неравновесных состояний. Получено обобщение уравнения Блоха и изучено влияние магнитного поля на спектральные характеристики. Рассмотрено влияние диссипативных процессов и найдены релаксационные потоки, обусловленные обменной симметрией магнитного гамильтониана. Представлені результати досліджень магнетиків зі спіном 1. Дано аналіз можливої симетрії обмінних взаємодій та їх зв'язок з магнітними ступенями свободи. Сформульована динаміка нормальних нерівноважних станів. Отримано узагальнення рівняння Блоха і вивчено вплив магнітного поля на спектральні характеристики. Розглянуто вплив дисипативних процесів і знайдені релаксаційні потоки, зумовлені обмінної симетрією магнітного гамільтоніана. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases Symmetry, phase states and dynamics of magnets with spin s=1 Симметрия, фазовые состояния и динамика магнетиков со спином s=1 Симетрія, фазовий стан та динаміка магнетиків зі спіном s=1 Article published earlier |
| spellingShingle | Symmetry, phase states and dynamics of magnets with spin s=1 Kovalevsky, M.Y. Kotelnikova, O.A. Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases |
| title | Symmetry, phase states and dynamics of magnets with spin s=1 |
| title_alt | Симметрия, фазовые состояния и динамика магнетиков со спином s=1 Симетрія, фазовий стан та динаміка магнетиків зі спіном s=1 |
| title_full | Symmetry, phase states and dynamics of magnets with spin s=1 |
| title_fullStr | Symmetry, phase states and dynamics of magnets with spin s=1 |
| title_full_unstemmed | Symmetry, phase states and dynamics of magnets with spin s=1 |
| title_short | Symmetry, phase states and dynamics of magnets with spin s=1 |
| title_sort | symmetry, phase states and dynamics of magnets with spin s=1 |
| topic | Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases |
| topic_facet | Section E. Phase Transitions and Diffusion Processes in Condensed Matter and Gases |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/107380 |
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