Impurity-impurity pair correlation function and paramagnetic-to-ferromagnetic transition in the random Ising model
In this Monte Carlo study we concentrated on the influence of non-magnetic impurities arranged as the lines with random orientation on paramagnetic-to-ferromagnetic phase transition in the 3D Ising model. Special emphasize is given to the long-distance decay of the impurity-impurity pair correlation...
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| Cite this: | Impurity-impurity pair correlation function and paramagnetic-to-ferromagnetic transition in the random Ising model / D. Ivaneyko, B. Berche, Yu. Holovatch, Ja. Ilnytskyi // Вопросы атомной науки и техники. — 2007. — № 3. — С. 372-375. — Бібліогр.: 12 назв. — англ. |
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Ivaneyko, D. Berche, B. Holovatch, Yu. Ilnytskyi, Ja. 2017-01-07T18:50:55Z 2017-01-07T18:50:55Z 2007 Impurity-impurity pair correlation function and paramagnetic-to-ferromagnetic transition in the random Ising model / D. Ivaneyko, B. Berche, Yu. Holovatch, Ja. Ilnytskyi // Вопросы атомной науки и техники. — 2007. — № 3. — С. 372-375. — Бібліогр.: 12 назв. — англ. 1562-6016 PACS: 05.10.Ln, 64.60.Fr, 75.10.Hk https://nasplib.isofts.kiev.ua/handle/123456789/111038 In this Monte Carlo study we concentrated on the influence of non-magnetic impurities arranged as the lines with random orientation on paramagnetic-to-ferromagnetic phase transition in the 3D Ising model. Special emphasize is given to the long-distance decay of the impurity-impurity pair correlation function. It is shown that for the lattice sizes considered (L=10-96) and for the two different impurity distributions (purely random and mutually avoiding lines) the function is governed by the power law of 1/ra with an universal exponent a≈2. This result supports our findings about the numerical values of the critical exponents governing magnetic phase transition in the 3D Ising model with long-range-correlated disorder. Oбговорюються результати досліджень методом Монте-Карло впливу немагнітних домішок у вигляді ліній з випадковою орієнтацією на фазовий перехід парамагнетик-феромагнетик в тривимірній моделі Ізинга. Особливу увагу приділено загасанню на великих відстанях парної кореляційної функції домішка-домішка. Для розглянених розмірів граток (L=10…96) і для двох типів розподілу домішків (лінії, що перетинаються і лінії, що не перетинаються) показано, що функція має степеневий вигляд 1/ra з універсальним показником a≈2. Цей результат підтверджує отримані нами числові значення критичних показників магнітного фазового переходу в тривимірній моделі Ізинга з далекосяжно-cкорельованим безладом. Oбсуждаются результаты исcледований методом Монте-Карло влияния немагнитных примесей в виде линий со случайной ориентацией на фазовый переход парамагнетик-ферромагнетик в трехмерной модели Изинга. Особое внимание уделено угасанию на больших расстояниях парной корреляционной функции примесь-примесь. Для обсуждаемых размеров решеток (L=10…96) и для двух типов распределений примесей (пересекающиеся и непересекающиеся линии) показано, что функция подчиняется закону 1/ra с универсальным показателем a≈2. Этот результат поддерживает полученные нами ранее данные о числовых значениях критических показателей магнитного фазового перехода в трехмерной модели Изинга со скоррелированным на больших расстояниях беспорядком. Results discussed here were presented at the 2nd International Conference on Quantum Electrodynamics and Statistical Physics (Kharkiv, 19-23 September, 2006). Yu. H. deeply acknowledges Yurij Slyusarenko for his kind hospitality during stay in Kharkiv. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Phase transformations in condensed matter Impurity-impurity pair correlation function and paramagnetic-to-ferromagnetic transition in the random Ising model Парна кореляційна функція домішка-домішка і перехід парамагнетик–феромагнетик в невпорядкованій моделі Ізинга Парная корpеляционная функция примесь-примесь и переход парамагнетик–ферромагнетик в неупорядоченной модели Изинга Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Impurity-impurity pair correlation function and paramagnetic-to-ferromagnetic transition in the random Ising model |
| spellingShingle |
Impurity-impurity pair correlation function and paramagnetic-to-ferromagnetic transition in the random Ising model Ivaneyko, D. Berche, B. Holovatch, Yu. Ilnytskyi, Ja. Phase transformations in condensed matter |
| title_short |
Impurity-impurity pair correlation function and paramagnetic-to-ferromagnetic transition in the random Ising model |
| title_full |
Impurity-impurity pair correlation function and paramagnetic-to-ferromagnetic transition in the random Ising model |
| title_fullStr |
Impurity-impurity pair correlation function and paramagnetic-to-ferromagnetic transition in the random Ising model |
| title_full_unstemmed |
Impurity-impurity pair correlation function and paramagnetic-to-ferromagnetic transition in the random Ising model |
| title_sort |
impurity-impurity pair correlation function and paramagnetic-to-ferromagnetic transition in the random ising model |
| author |
Ivaneyko, D. Berche, B. Holovatch, Yu. Ilnytskyi, Ja. |
| author_facet |
Ivaneyko, D. Berche, B. Holovatch, Yu. Ilnytskyi, Ja. |
| topic |
Phase transformations in condensed matter |
| topic_facet |
Phase transformations in condensed matter |
| publishDate |
2007 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
Парна кореляційна функція домішка-домішка і перехід парамагнетик–феромагнетик в невпорядкованій моделі Ізинга Парная корpеляционная функция примесь-примесь и переход парамагнетик–ферромагнетик в неупорядоченной модели Изинга |
| description |
In this Monte Carlo study we concentrated on the influence of non-magnetic impurities arranged as the lines with random orientation on paramagnetic-to-ferromagnetic phase transition in the 3D Ising model. Special emphasize is given to the long-distance decay of the impurity-impurity pair correlation function. It is shown that for the lattice sizes considered (L=10-96) and for the two different impurity distributions (purely random and mutually avoiding lines) the function is governed by the power law of 1/ra with an universal exponent a≈2. This result supports our findings about the numerical values of the critical exponents governing magnetic phase transition in the 3D Ising model with long-range-correlated disorder.
Oбговорюються результати досліджень методом Монте-Карло впливу немагнітних домішок у вигляді ліній з випадковою орієнтацією на фазовий перехід парамагнетик-феромагнетик в тривимірній моделі Ізинга. Особливу увагу приділено загасанню на великих відстанях парної кореляційної функції домішка-домішка. Для розглянених розмірів граток (L=10…96) і для двох типів розподілу домішків (лінії, що перетинаються і лінії, що не перетинаються) показано, що функція має степеневий вигляд 1/ra з універсальним показником a≈2. Цей результат підтверджує отримані нами числові значення критичних показників магнітного фазового переходу в тривимірній моделі Ізинга з далекосяжно-cкорельованим безладом.
Oбсуждаются результаты исcледований методом Монте-Карло влияния немагнитных примесей в виде линий со случайной ориентацией на фазовый переход парамагнетик-ферромагнетик в трехмерной модели Изинга. Особое внимание уделено угасанию на больших расстояниях парной корреляционной функции примесь-примесь. Для обсуждаемых размеров решеток (L=10…96) и для двух типов распределений примесей (пересекающиеся и непересекающиеся линии) показано, что функция подчиняется закону 1/ra с универсальным показателем a≈2. Этот результат поддерживает полученные нами ранее данные о числовых значениях критических показателей магнитного фазового перехода в трехмерной модели Изинга со скоррелированным на больших расстояниях беспорядком.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/111038 |
| citation_txt |
Impurity-impurity pair correlation function and paramagnetic-to-ferromagnetic transition in the random Ising model / D. Ivaneyko, B. Berche, Yu. Holovatch, Ja. Ilnytskyi // Вопросы атомной науки и техники. — 2007. — № 3. — С. 372-375. — Бібліогр.: 12 назв. — англ. |
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| fulltext |
IMPURITY-IMPURITY PAIR CORRELATION FUNCTION
AND PARAMAGNETIC-TO-FERROMAGNETIC TRANSITION
IN THE RANDOM ISING MODEL
D. Ivaneyko1, B. Berche2, Yu. Holovatch3, and Ja. Ilnytskyi4
1Ivan Franko National University of Lviv, 79005, Lviv, Ukrain,
e-mail: ivaneiko@ktf.franko.lviv.ua;
2Laboratore de Physique des Matériaux, Université Henri Poincaré, Nancy 1, 54506,
Vandœuvre les Nancy, Cedex, France;
e-mail: berche@lpm.u-nancy.fr;
3Institute for Condensed Matter Physics, National Acad. Sci. of Ukraine, 79011 Lviv, Ukraine;
Institut für Theoretische Physik, Johannes Kepler Universität Linz, 4040, Linz, Austria;
e-mail: hol@icmp.lviv.ua;
4Institute for Condensed Matter Physics, National Acad. Sci. of Ukraine, 79011, Lviv, Ukraine;
e-mail: iln@icmp.lviv.ua
In this Monte Carlo study we concentrated on the influence of non-magnetic impurities arranged as the lines
with random orientation on paramagnetic-to-ferromagnetic phase transition in the 3D Ising model. Special empha-
size is given to the long-distance decay of the impurity-impurity pair correlation function. It is shown that for the
lattice sizes considered (L=10-96) and for the two different impurity distributions (purely random and mutually
avoiding lines) the function is governed by the power law of 1/ra with an universal exponent a≈2. This result sup-
ports our findings about the numerical values of the critical exponents governing magnetic phase transition in the
3D Ising model with long-range-correlated disorder.
PACS: 05.10.Ln, 64.60.Fr, 75.10.Hk
The ferromagnetic phase transition in the three-
dimensional (3D) Ising model with quenched long-
range-correlated impurities is governed by the critical
exponents that are different of both those for the pure
3D Ising model and for the 3D Ising model with uncor-
related impurities [1-5]. In particular, we are interested
in the case, when the impurity-impurity pair correlation
function h(r) decays at large separations r accordings to
a power law [1]:
arrh /1)( ≈ , . (1) ∞→r
Besides purely academic interest, a reason behind such
a choice is that the power law decay (1) allows a direct
geometrical interpretation. Indeed, for integer a it corre-
sponds to the lines (at a=D-1) or the planes (at a=D-2)
of impurities of random orientation [1]. Moreover, non-
integer a is sometimes treated in terms of impurities
fractal dimension [6].
Whereas both analytical and numerical studies agree
on the fact that the 3D Ising model with long-range cor-
related disorder (i.e. when a<D) possesses a new uni-
versality class [1-5], the numerical values of the critical
exponents are found to be rather different. Indeed, the
renormalization group (RG) estimates of Ref. [1] gave
the values of the exponents in the first order of ε=4-D,
δ=4-a-expansion. Furthermore, the first-order result for
the correlation length critical exponent ν=2/a was con-
jectured to be an exact one [1]. On contrary, the second-
order RG calculations of Ref. [2] lead to non-trivial
dependence of exponents on the correlation parameter
a. MC simulations also split into two groups. First, the
random Ising model with power-law correlated non-
magnetic impurities was considered in Ref. [3]. There,
the impurities were simulated in two different ways: (i)
as point-like particles, correlated according to (1) with
correlation exponent a=2 and (ii) as the lines of random
orientation. An outcome of the simulations favoured
theoretical predictions of Ref. [1]. Indeed, for the impu-
rity concentration p=0.2 the correlation length and pair
correlation function exponents were estimated by means
of combination of Wolff and Swendsen-Wang algo-
rithms as ν=1.012(10) and η=0.043(4) [3], whereas the
theoretical estimate of Ref. [1] reads: ν(a=2)=1, η=0.
However, the following MC simulation [4] questioned
results of Refs. [1,3]. Two sets of estimates for the ex-
ponents obtained there by using different algorithms at
p=0.2 read: ν=0.719(22), β=0.375(45) (short-time criti-
cal dynamics with Metropolis algorithm), and
ν=0.710(10), γ=1.441(15), β=0.362(20) (finite-size
scaling with Wolff algorithm). In turn, these results
support a theoretical estimate of Ref. [2]: ν=0.7151
(note however, that η= - 0.0205 in [2]). As a possible
reason for the discrepancy with Refs. [1,3] the authors
of Ref. [4] mention that they implemented a mutual
avoidance condition on the lines of impurities, whereas
it was not the case in the simulations of Ref. [3].
To resolve such a bias, we performed MC simula-
tions of the 3D Ising model with the impurities arranged
as lines of random orientation [5]. Our estimates for the
exponents differ from the results of the two numerical
simulations performed so far [3,4] and are in favour of a
non-trivial dependency of the critical exponents on the
peculiarities of long-range correlations. Moreover, we
have analysed both previously considered cases of
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2007, N3 (2), p. 372-375. 372
purely random and mutually avoiding impurity lines
distributions. No difference was found within the error
bars (see below for more details).
One more question remained unanswered in the
above-mentioned context. Namely, in the numerical
simulations performed so far [3-5] it was tacitly as-
sumed that the impurity-impurity pair correlation func-
tion power law asymtotics holds for the randomly ori-
ented impurity lines with an exponent a=2. Although, it
is certainly true for an infinite system, it is not obvious
that such behaviour holds for the finite-size systems
considered during simulations. Therefore, the goal of
this paper is to supply the MC simulations of the phase
transition in magnetic subsystem by simultaneous con-
trol of the structural properties of the impurities.
We consider a 3D Ising model with non-magnetic
sites arranged in a form of randomly oriented lines. The
Hamiltonian reads:
< >
= − ∑ i j i j
ij
H J c c S S , (2)
where the summation is over the nearest neighbour
sites of a s.c. lattice of linear size L, J>0 is the interac-
tion constant, Ising spins Si= ± 1, and ci = 0,1 is the
occupation number for the i-th site. Non-magnetic sites
(ci = 0) are located along the lines and quenched in a
fixed configuration. To ensure an isotropic distribution
of lines, we take their number to be the same along each
axis.
We performed the MC simulations by means of the
Wolff cluster algorithm [7] using histogram reweighting
technique [8] imposing periodic boundary conditions,
measuring system magnetisation, energy, Binder cumu-
lant, and magnetic susceptibility at the critical tempera-
ture for the lattices of varying sizes L=10… 96 and ap-
plying finite-size scaling technique to extract the values
of the critical exponents. The impurity concentration
was taken to be p=0.2 both to adhere previous MC
simulations [3,4] as well as to minimize possible correc-
tion-to-scaling effects. The presence of quenched disor-
der leads to two different types of averaging to be per-
formed: besides the Boltzmann averaging, the observ-
ables are to be averaged with respect to different disor-
der realizations. To perform the averaging over differ-
ent disorder configurations we generated 104 lattice
samples for the sizes L=10… 48 and 103 samples for
L=64,96. To accomplish the Boltzmann averaging for
each disorder realization, a run of 250τE Monte Carlo
steps (MCS) was performed for system equilibration
with following 20000 MCS for further calculations at
L=10… 32. For the larger lattice sizes, L=48… 96 the
number of MCS steps was 103τE (τE being energy auto-
correlation time) [9]. Further details of our simulations
are reported in [5].
As has already been outlined in the introduction, we
are interested in the analysis of two different variants of
non-magnetic sites distribution: in the first one the lines
of impurities are randomly oriented and some of them
may intersect (from now on we call such situation ‘dis-
tribution A’); whereas in the second variant we impose
a mutual avoidance condition on the randomly oriented
lines (‘distribution B’, correspondingly). Theoretically,
these two distributions may lead to different critical
behaviours, as far as only the distribution A corresponds
to the impurities in the form of lines, whereas the
distribution B may result in objects of a different
dimension. However it is not obvious that such effect
may show up and cause any influence on the critical
behaviour for the systems of sizes considered in the
simulation. Indeed, the results for the critical exponents
we obtain in the simulations for the above two
distributions read [5]:
A: ν=0.864(10), β=0.519(11), γ=1.555(26); (3)
B: ν=0.872(19), β=0.522(16), γ=1.450(39). (4)
The exponents ν and β obtained do agree within the
confidence interval whereas exponent γ differs within
3%. Already the above numbers enabled us to arrive at
the conclusion [5] that the difference between the re-
sults of simulations performed in Ref. [3] and Ref. [4]
can not be caused solely by the difference between im-
purity line distributions A and B.
To complete an analysis of the paramagnetic-to-
ferromagnetic phase transition that has led to the esti-
mates (3), (4) for the critical exponents, we studied the
behaviour of the impurity-impurity pair correlation
function h(r) for the distributions A and B for the lat-
tices considered. To this end, we define h(r) in terms of
the radial distribution function g(r)
, (5) 1)()( −= rgrh
where g(r) is given by [10]:
3 3
( ) .
4 (( ) )
3
=
π + δ −
g r
r r r
( )< >n r (6)
In (6), <n(r)> means average number of non-
magnetic sites which lie within distance δr of a sphere
of radius r that contains a non-magnetic site in the ori-
gin. Note that for the diluted system (6) is to be normal-
ized by the non-magnetic component concentration p.
By applying Eq. (6) for each disorder realization it is
straightforward to find the impurity-impurity pair corre-
lation function by counting the number of non-magnetic
sites that lie within the distance interval [r, r+δr] from
the given one. Then, the resulting histogram is to be
averaged over different disorder realizations (different
samples). The number of samples was taken N=1000 for
the lattice sizes L=10-64 and N=100,50 for the lattices
with L=96,128, respectively. Note, that for a given sam-
ple the statistics is enriched by placing in turn each non-
magnetic site at the origin. This increases the number of
data points for the largest lattice sizes L=128 by pL3≈
105. In Figs. 1 and 2 we give a typical outcome of the
analysis: configurationally averaged impurity-impurity
pair correlation function h(r) for one of the lattice sizes
(L=96) and for the impurity distributions A and B. Fit to
a power-law (1) is shown by a solid line.
It is appropriate to make several comments before pass-
ing to numerical estimates of the exponents governing
such fits. For the sake of theoretical analysis, one is
interested in the behaviour of h(r) for large r. In the
373
simulations, r is limited by the linear lattice size
L.However, because of the system finite size and a
presence of periodic boundary conditions, the cutoff
distance for the interparticle correlations is L/2. There-
fore, although the theoretical estimates of Refs. [1,2] are
intended for the systems where (1) holds in asymptotics,
in the finite-size systems with randomly distributed
lines the power-law decay (1) is observed in the wide
region of distances, starting already from r = 1 (note
that at the origin h(r) = –1 by definition (5)).
Fig. 1. Impurity-impurity pair correlation function
h(r) for the distribution A and lattice size L=96. Solid
line (red online) shows a power-law fit (1) with an ex-
ponent a=2.05 (see Table)
Fig. 2. Impurity-impurity pair correlation function
h(r) for the distribution B and lattice size L=96. Solid
line (red online) shows a power-law fit (1) with an ex-
ponent a=2.06 (see Table 1)
Table 1 summarizes our results of the fits of the im-
purity-impurity pair correlation function h(r) to the
power law for lattice sizes L = 10-128 and for impurity
lines distributions A and B. There are several conclu-
sions that follow from these data:
(i) the power-law behaviour of h(r) is confirmed for
the lattice sizes used in the simulations;
(ii) distributions A and B are characterised by the
same (within the confidence interval) value of the ex-
ponent a;
(iii) the value of the exponent a within the confi-
dence interval does not vary with L.
The last fact is of crucial importance as far as it jus-
tifies application of the finite-size-scaling technique
used to extract the values of critical exponents that gov-
ern paramagnetic-to-ferromagnetic phase transition.
Indeed, if a were found to vary with L an application of
this technique would be questioned, as far as in this case
one faces an analysis of systems which are differently
correlated for each L. Note also that the correlation pa-
rameter is a≈2, and the deviations found are of order of
several percents and do not suffice to cause changes in
the critical exponents values within the confidence in-
terval of data given in (3), (4).
The values of exponent a for different lattice sizes L and
for two different impurity lines distributions. Distribu-
tion A: lines are allowed to intersect;
Distribution B: mutually avoiding lines
L Distribution A Distribution B
10 2.16 ± 0.18 2.17 ± 0.20
12 2.08 ± 0.17 2.10 ± 0.18
16 2.07 ± 0.13 2.08 ± 0.14
24 2.04 ± 0.10 2.06 ± 0.10
32 2.03 ± 0.08 2.06 ± 0.10
48 2.04 ± 0.07 2.05 ± 0.08
64 2.04 ± 0.06 2.05 ± 0.06
96 2.05 ± 0.06 2.06 ± 0.06
128 2.05 ± 0.06 2.06 ± 0.06
Together with the results for critical exponents (3),
(4) the data presented in Table bring about the fact that
the 3D Ising model with long-range-correlated disorder
in the form of non-magnetic impurity lines of random
orientation belongs to the new universality class. The
values of the exponent obtained certainly differ from
those of the pure 3D Ising model (ν = 0.630(1), β =
0.3265(15), γ = 1.237(3) [11]) as well as from the 3D
Ising model with uncorrelated impurities (ν = 0.68(2), β
= 0.35(1), γ = 1.34(1) [12]). However, our results differ
from previous MC estimates [3,4] of the exponents for
3D Ising model with long-range-correlated disorder (see
the numbers given at the beginning of this paper). The
reason for the discrepancy remains unclear. Further-
more, for the lattice sizes considered the constraint of
mutual avoidance imposed on the impurity lines appears
to be an irrelevant one.
Results discussed here were presented at the 2nd In-
ternational Conference on Quantum Electrodynamics
and Statistical Physics (Kharkiv, 19-23 September,
2006). Yu. H. deeply acknowledges Yurij Slyusarenko
for his kind hospitality during stay in Kharkiv.
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ПАРНАЯ КОРPЕЛЯЦИОННАЯ ФУНКЦИЯ ПРИМЕСЬ-ПРИМЕСЬ И ПЕРЕХОД
ПАРАМАГНЕТИК–ФЕРРОМАГНЕТИК В НЕУПОРЯДОЧЕННОЙ МОДЕЛИ ИЗИНГА
Д. Иванейко, Б. Берш, Ю. Головач, Я. Ильницкий
Обсуждаются результаты исследований методом Монте-Карло влияния немагнитных примесей в виде
линий со случайной ориентацией на фазовый переход парамагнетик-ферромагнетик в трехмерной модели
Изинга. Особое внимание уделено угасанию на больших расстояниях парной корреляционной функции
примесь-примесь. Для обсуждаемых размеров решеток (L=10…96) и для двух типов распределений приме-
сей (пересекающиеся и непересекающиеся линии) показано, что функция подчиняется закону 1/ra с универ-
сальным показателем a≈2. Этот результат поддерживает полученные нами ранее данные о числовых значе-
ниях критических показателей магнитного фазового перехода в трехмерной модели Изинга со скоррелиро-
ванным на больших расстояниях беспорядком.
ПАРНА КОРЕЛЯЦІЙНА ФУНКЦІЯ ДОМІШКА-ДОМІШКА І ПЕРЕХІД ПАРАМАГНЕТИК–
ФЕРОМАГНЕТИК В НЕВПОРЯДКОВАНІЙ МОДЕЛІ ІЗИНГА
Д. Іванейко, Б. Берш, Ю. Головач, Я. Ільницький
Oбговорюються результати досліджень методом Монте-Карло впливу немагнітних домішок у вигляді
ліній з випадковою орієнтацією на фазовий перехід парамагнетик-феромагнетик в тривимірній моделі Ізин-
га. Особливу увагу приділено загасанню на великих відстанях парної кореляційної функції домішка-
домішка. Для розглянених розмірів граток (L=10…96) і для двох типів розподілу домішків (лінії, що пере-
тинаються і лінії, що не перетинаються) показано, що функція має степеневий вигляд 1/ra з універсальним
показником a≈2. Цей результат підтверджує отримані нами числові значення критичних показників магніт-
ного фазового переходу в тривимірній моделі Ізинга з далекосяжно-cкорельованим безладом.
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