Exact relativistic plasma dispersion functions
On the ground of the theory of singular integrals with Cauchy kernel the exact plasma dispersion functions (PDFs) are introduced and studied. Those PDFs make more exact the weakly relativistic PDFs and generalize them on the case of arbitrary plasma temperature. На основі теорії сингулярних інтеграл...
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| Опубліковано в: : | Вопросы атомной науки и техники |
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| Дата: | 2005 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2005
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Exact relativistic plasma dispersion functions / S.S. Pavlov, F. Castejon // Вопросы атомной науки и техники. — 2005. — № 2. — С. 55-57. — Бібліогр.: 13 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859678656428769280 |
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| author | Pavlov, S.S. Castejon, F. |
| author_facet | Pavlov, S.S. Castejon, F. |
| citation_txt | Exact relativistic plasma dispersion functions / S.S. Pavlov, F. Castejon // Вопросы атомной науки и техники. — 2005. — № 2. — С. 55-57. — Бібліогр.: 13 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | On the ground of the theory of singular integrals with Cauchy kernel the exact plasma dispersion functions (PDFs) are introduced and studied. Those PDFs make more exact the weakly relativistic PDFs and generalize them on the case of arbitrary plasma temperature.
На основі теорії сингулярних інтегралів з ядром Коші вводяться і досліджуються точні плазмові дисперсійні функції (ПДФ), уточнюючі слабо релятивістські ПДФ і узагальнюючі їх на випадок довільної температури.
На основе теории сингулярных интегралов с ядром Коши вводятся и исследуются плазменные дисперсионные функции (ПДФ), уточняющие слабо релятивистские ПДФ и обобщающие их на случай произвольной температуры.
|
| first_indexed | 2025-11-30T16:51:54Z |
| format | Article |
| fulltext |
EXACT RELATIVISTIC PLASMA DISPERSION FUNCTIONS
S.S. Pavlov1 and F. Castejon2
1 Institute of Plasma Physics, NSC KhIPT, Akademicheskaya 1, 61108, Kharkov, Ukraine;
2 Asociación EURATOM-CIEMAT PARA Fusión, 28040, Madrid, Spain
On the ground of the theory of singular integrals with Cauchy kernel the exact plasma dispersion functions (PDFs) are
introduced and studied. Those PDFs make more exact the weakly relativistic PDFs and generalize them on the case of
arbitrary plasma temperature.
PACS: 52.27.Ny
1. INTRODUCTION
Plasma waves have a wide range of applications.
Essential to each of the applications is knowledge of the
dielectric properties of the plasma. Analytical treatment of
those properties leads to expressions for the dielectric
tensor in terms of plasma dispersion functions (PDFs).
As the non-relativistic PDF, +−= 1)[exp()( 2zzW
∫
z
dtti
0
2 ])exp()/2( π , introduced into the theory of plasma
and tabulated in complex region in the works [1,2],
respectively, (or PDF, )()( zWizZ π= , introduced in the
work [3]), so and the weakly relativistic PDFs, introduced
in the work [4], are approximate ones, since have been
derived through some approximations in the parameter
μ/1 ( ,2
0 )/( TVc=μ 00 / mTVT = , is the rest mass of
electron).
0m
Problems of Atomic Science and Technology. Series: Plasma Physics (11). 2005. № 2. P. 55-57 55
The primary purpose of the present work is
introducing the exact relativistic PDFs, which makes
more exact the non-relativistic PDF and the weakly
relativistic PDFs and generalize them on the case of
arbitrary plasma temperature. It is achieved on the ground
of a deep connection between the theory of plasma waves
and the theory of Cauchy type integrals.
2. INTRODUCING OF THE EXACT
RELATIVISTIC PDFs
Starting from the 1st integral form of Trubnikov’s
plasma dielectric tensor, that neglects ion dynamics, one
can write it in Cartesian coordinates with z -axis directed
along static magnetic field in the next equivalent form [5]
( ) ×⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−= ∑∑
∞
=
∞
−∞=
k
k
kn
n
n
p
jiji k
C
nK 2!2!
1
2 0
,
2
2
λλ
μ
μ
ω
ω
δε
( ) ( )( ) , (1) μωτπ ,,, //
,
c
rkn
nij nNkn ij Ω+
where ω and pω are an angular wave and plasma
frequencies, ( )μ2K is Macdonald function of second
order, ωckN //// = is longitudinal refractive index, =Ωc
)( 0 cmeB is the electron cyclotron frequency,
( ) ( )[ ]
( ) ( )!!2
!2!1
, knkn
knn
C
k
kn ++
+−
= ,
λ
π
2n
xx = , ( )
λ
ππ
knn
iyxxy
+
=−= ,
λ
ππ n
zxxz == , ( )( )
( )λλ
π
122
1222
−+
−++
+=
kn
knknkn
yy ,
( )
λ
ππ
knn
izyyz
+
−=−= , 1=zzπ ,
( )( ) ( ) ×⎟
⎠
⎞
⎜
⎝
⎛
+
=Ω
+
+
knr
c
rkn
n kn
nN
ij
ij
2!
,,
2
//
, μμμωτ
( )
∫ ∫
∞
∞−
∞
+
+
−+−
+
⎟
⎠
⎞
⎜
⎝
⎛ ++
0
2
//1
////
122
//
n
kn
kn
xpr
x
pxx
dxeeppd ij
β
μμ . (2)
In the expression (2) )( 0cmpp = is normalized
momentum, ( ) ( )2
//
22
// 11 pppx +−++= ⊥ , ( ) −+= 2
//1 pnβ
ωcnpN Ω−//// and 0==== yyyxxyxx rrrr , == zxxz rr
, 1== zyyz rr 2=zzr .
Bortnatici and Ruffina shown that for the case
anti-Hermitian parts of the functions,
0≥n
( ) ( ) ( ), can be
exactly expressed in terms of modified spherical Bessel
functions [6]. Obviously, the same result takes place and
for functions,
μωτπμμω μ ,,2,, //
0,
//
0
23 c
q
ncq nNenNF Ω=Ω+
( ) ( )( ) /,,, //
0,
//
0
23 ωτμω c
q
ncq nNnNZ Ω=Ω+
))(2( 2 μK , that appear in the expression (1) since they
differ from functions, ( μω,,//
0
23 cq nNF Ω+ ) , by the factor
not depending of two first arguments. In first, let us
generalize these results on the case of arbitrary harmonic
number, , using some facts from the theory of Cauchy
type integrals [7,8]. Then on the same way we’ll give the
exact analytical expressions for the Hermitian parts of the
functions,
n
( )μω,,//
0
23 cq nNZ Ω+
and define these whole
functions as the exact relativistic PDFs.
Using in the inside integral of the expression (2) the
first of the Sokhotskii-Plemelj formulas,
( ) ( ) ( )
∫
+∞
∞−
+
−
+=
t
dP
i
ttF
τ
ττϕ
π
ϕ
2
1
2
,
( ) ( ) ( )
∫
+∞
∞−
−
−
+−=
t
dP
i
ttF
τ
ττϕ
π
ϕ
2
1
2
, (3)
corresponding to the Landau rule of passing the pole, and
passing to the arguments used by Robinson [9-11]:
2// μpx = , ( )ωμ cnz Ω−= 1 , 22
//Na μ= one can, at once,
receive
( ) ( )
×
−
=
+−
+ !)(2
21,,Im
2
21
23 qK
ezaZ
ij
ij
rqzr
q μμ
πμ
μ
( )∫
+
−
⎥⎦
⎤
⎢⎣
⎡ −−+−−
x
x
q
xa dxxazxxaze )2(22
2212212 21
μ , (4)
where limits of integration, , can be obtained from the
condition of appearing the pole in the inside integral of
the expression (2):
±x
0≤nβ . Then for the limits
of integration equal
10 // <≤ N
( ) ±−=± μβ zax 1[ 21
])2(2 μzza +− and for ones equal , 1// >N +∞=+x
( ) ])2(1[ 221 μμβ zzazax +−−−=− ; here for shortness
was used denoting ( )a2−= μμβ . The direct integration in
the expression (4) gives anti-Hermitian parts of
functions ( μ,,23 zaZ ijr
q+
56
) . So, for the case 0=jir and
10 // <≤ N
( ) ×
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜
⎝
⎛ +−
−=
+
+
21
2
2
2123
0
23
)2(
)(2
,,Im
q
q a
zza
K
zaZ μ
μμ
βπμ
( )[ ] ⎟
⎠
⎞
⎜
⎝
⎛ +−−− + )2(22exp 221
21 μβμβ zzaaIaz q , ( ),(5) ∗< az
( ) 0,,Im 0
23 ≡+ μzaZq , ( ), (6) ∗≥ az
here )(21 xIq+ is modified Bessel function of half-integer
order and ( )βμ 11−=∗a . By the similar way for 0=jir
and 1// >N
( ) ×
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜
⎝
⎛ +−−
−=
+
+
21
2
2
21
0
23
)2(
)(2
)(,,Im
q
q a
zza
K
zaZ μ
μμ
πβμ
( )[ ] ⎟
⎠
⎞
⎜
⎝
⎛ +−−−− + )2(22exp 221
21 μβμβ zzaaKaz q , (7)
for arbitrary z ; here )(21 xK q+ is Mac Donald function of
half-integer order.
It is known [7, 8] that for the density of Cauchy
integral, ( )tϕ , satisfied to the Holder condition and
integrated along the real axis in the Cauchy sense real and
imagine parts of the boundary function,
, are mutually single significantly
connected by the Hilbert transforms
)()()( tivtutF +=+
( ) ( )
∫
+∞
∞− −
=
t
duP
i
tiv
τ
ττ
π
1 , ( ) ( )
∫
+∞
∞− −
=
t
divP
i
tu
τ
ττ
π
1 . (8)
Here and everywhere letter P before integrals shows that
integrals are understood in the sense of principle value.
Consequently, implying ( μ,,Im)( 0
23 zaZtv q+= ) one can use
the second transform to express real parts of the boundary
functions, ( μ,,0
23 zaZ q+ ) , through their imagine parts. Then
for the case one can receive 10 // <≤ N
( )
( )
P
aK
ezaZ q
aa
q 21
2
)2(2121
0
23
)(2
,,Re +
−−−
+
∗
=
μμ
βπμ
μβ
∫
∞+
∗
−
+
+
+−
⎟
⎠
⎞
⎜
⎝
⎛ ++++
0
221
21
212 )2(2))2((
zau
dueaaIa u
q
q βμττβμττ
. (9)
In correspondence with the Sokhotskii-Plemely formula,
giving the right sign (as in the weakly relativistic case) of
anti-Hermitian parts of the functions, ( ) , and
corresponding to passing of the contour of integration
above the pole, one can receive the exact integral form for
the whole those functions
μ,,0
23 zaZq+
( )
( )
×=
+
−
+ 21
2
0
23
)(2
,, qq
aK
e
zaZ
μμ
πβ
μ
βμ
∫
∞+
−
+
+
+−−
⎟
⎠
⎞
⎜
⎝
⎛ ++
0
21
21
21
)1(
/1)2/((2)/1)2/(((
zu
dueuuaIuu u
q
q
βμ
βμββμ β
.
(10)
For the case using the expression (7) for anti-
Hermitian parts of the functions,
1// >N
( ) , by the
similar way one can obtain
( ) [ ]
( ) ×
−−
−−= ++ 21
2
0
23
)(2
2exp,, qq
aK
azaZ
μπμ
μβ
βμ
( )
∫
∞
∞−
+
+
−
⎟
⎠
⎞
⎜
⎝
⎛ +−−+−
z
daaKa q
q
τ
ττβμττβμττ exp)2(2))2(( 221
21
212
,
(11)
where the contour of integration is passing below the
pole.
Integral forms (10), (11) define analytically the exact
relativistic PDFs for the cases and ,
respectively.
10 // <≤ N 1// >N
3. ANALYTICAL PROPERTIES OF THE
EXACT PDFs
In first, let us study the integral form (10) defining the
exact relativistic PDFs for the case 10 // <≤ N that is
interesting from the point of view of the EC waves
description in magnetized plasma; for shortness of writing
from now on we are missing arguments of these
functions. Using the recurrent formula for modified
Bessel functions, )()/2()()( 11 zIzzIzI ννν ν−= −+ , one can
receive
=+−−++ +++
0
21
20
23
0
25 )]2([)21( qqq ZzzaZqaZ μββ
( )
×
−
−
21
2 )(22 q
aK
e
μμ
πβ
μ
β βμ
(12)
∫
∞ . −
−
−∗ ⎟
⎠
⎞
⎜
⎝
⎛ ++−
0
21
21
21 /1)2/((2)/1)2/(()(( dueuuaIuuua u
q
q ββμββμ
Both integrals in the expression (12), using the known
integral [12]
×
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜
⎝
⎛
−
=++
+
−
∞+
∫
21
22
2
0
2
2
)()(
ν
ν
ν
ν
ν
π cp
zcdxexzxcIxzx px
⎟
⎠
⎞
⎜
⎝
⎛ −+
22
21
2/
2
cpzKe pz
ν , ( cp ReRe > ; 1Re −>ν , π<zarg ),
can be integrated
( )
( )
)(
)(
2)(
)(2 20
21
21
21
21
2
μ
μ
β
μμ
πβ β
βμ
K
K
duesaIs
aK
e qu
q
q
q
=∫
∞
−
−
−
−
−
,
(13)
( )
( ) =−∫
∞
−
−
−∗
−
−
0
21
21
21
21
2
2))((
)(2
duesaIsua
aK
e u
q
q
q
β
βμ
β
μμ
πβ
)(
)()(
2
1
μ
μμ
μ
K
KK qq −
− + , (14)
here, for shortness, noting βμ /1)2/(( += uus was used.
Thus, from (12) we receive a recurrent relation for the
exact PDFs
=+−−++ +++
0
21
20
23
0
25 )]2([)21( qqq ZzzaZqaZ μββ
)(/)]()()1[(
2 21 μμμμβ KKKz qq ++− . (15)
For the case , that is interesting for the
description of ICR waves in relativistic plasma, by the
similar way one can receive the same recursive relation
(15). Only in this case it is necessary to use the recurrent
formula for Mac Donald functions,
1// >N
+= −+ )()( 11 zKzK νν
)()/2( zKz νν , and the integral
μ,,0
23 zaZ q+
One can verify the formulas for the 1st and 2nd derivatives
of the exact PDFs in the parameter z ×
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
−
=⎟
⎠
⎞
⎜
⎝
⎛ +⎟
⎠
⎞
⎜
⎝
⎛ +
+
∞
∞−
∫
21
22
2222 2
ν
ν
ν
ν
π
bp
z
pdxezxpKzx bx
⎟
⎠
⎞
⎜
⎝
⎛ −+
22
21 bpzKν , ( 1Re −>ν )
])1([)( 0
1
00
−−+=′ qqq ZzZZ μβ ,
{ }0
2
20
1
020 )1()](1)1(2[)( −− −++−+=′′ qqqq ZzZzZZ μβμμβ . (20)
Excepting PDFs, and from the expressions
(15), (20) one can obtain the usual linear deferential
equation in the parameter
0
1−qZ 0
1−qZ
z for the exact PDFs
that follows by integrating in the parameter from the
known integral [13]
p
∫
∞
+− ⎟
⎠
⎞
⎜
⎝
⎛ −
−
=
0
22
1
22
22
bpzK
bp
pzchbxdxe zxp , ( bp ReRe > ;
). 0Re >z
−′′+−− )))(2/()(( 02
qZzzaz μμ
+′+−−+−+++−− )))](2/()(32()2())2/()((2[ 022
qZzzqqazzaz μμμμβ
++−−+−+++−− 022 ))]2/()(32()2())2/()([( qZzzqqazzz μμμμβ
0)2/(])([)( 2
3
2523
32 =−−− −− KKzKz qq μμμμβ , (21)
Then instead integrals (13), (14) one can obtain the
integrals
[ ]
( )
( ) ( )∫
∞
∞−
−
−
−
=−
−−
− ττββ
μπμ
μβ
β dsaKs
aK
a
q
q
q
exp2)(
)(2
2exp 21
21
21
21
2
)(
)(
2 μ
μ
K
Kq , (16)
that also may be useful for their study.
At last, from the formulas (4) by differentiation in the
parameter a it follows the identity
)(/ 0
1
0 ′′= +qq ZdadZ ,
[ ]
( )
( ) ( )∫
∞
∞−
−
−
−
=−
−−
− ττββ
μπμ
μβ
β dsaKs
aK
a
q
q
q
exp2)(
)(2
2exp 21
21
21
21
2
)(
)()(
2
1
μ
μμ
μ
K
KK qq −
− + . (17)
connecting derivatives of these PDFs in the parameter
and
a
z that coincides with the respective identity for the
weakly relativistic PDFs.
REFERENCES
Here, for shortness, noting )2(2 μττ +−= as was used. 1. G.V. Gordeev// JETPh (24). 1953, p.445 (in Russian).
The integrals (13), (14) and (16), (17) are moments on the
density in the respective Cauchy integrals defining the
exact PDFs, 0
21+qZ , and have the next physical sense: the
integrals (13), (16) are the densities of resonance
electrons in the respective Dopler spectral line of
absorption, the integrals (14), (17) define z -coordinate
of this spectral line.
2. V.N. Faddeeva and N.M. Terentjev. Tabulation of the
function ])exp(21)[exp()(
0
22 dttizzW
z
∫+−= π of complex
argument. Moscow. 1954 (in Russian).
3. B.D. Fried and S.D. Conte. The plasma dispersion
function. New York: Academic Press. 1961.
4. I.P. Shkarofsky// Phys. Fluids (9), 1966, p.561.
5. M. Brambilla. Kinetic theory of plasma waves,
homogeneous plasmas. Oxford: Oxford University Press.
1998.
So, from all those integrals it follows the next theorem:
z-coordinates of Dopler absorption spectral lines and
densities of resonance electrons in these lines don’t
depend from the direction of waves propagation and are
defined by the moments (13), (14), (16), (17) of anti-
Hermitian parts of the exact PDFs.
6. M. Bortnatici, U. Ruffina// IL Nuovo Cimento, vol. 6D,
1985, N.3, p.231.
7. N.I. Muskhelishvili. Singular integral equations. /3 ed.,
Moscow, 1968 (in Russian). Those integrals have also and the mathematical sense:
(13), (16) are first and (14), (17) second coefficients in
asymptotic expansion of the PDF, 0
21+qZ ,
8. F.D. Gakhov. Boundary problems. / 2 ed., Moscow,
1963 (in Russian).
9. P.A. Robinson// J. Math. Phys. 1986, 27(5), p.1206.
=+
−
−= +
+ ...
)(
)()(
)(
)(
2
2
1
2
0
21 zK
KK
zK
K
Z qqq
q μ
μμ
μ
μ
μ
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+++ ...1
2
211
0 z
A
z
A
A
z
qq
q .
10. P.A. Robinson// J. Math. Phys. 1987, 28(5), p.1203.
11. P.A. Robinson// J. Math. Phys. 1989, 30(11), p.2484.
12. A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev.
Integrals and series (Special functions). Moscow:
“Science”. 1983 (in Russian).
(18)
Substituting the expression (18) into the recursive relation
(15) one can receive recurrent formulas for calculating an
arbitrary coefficient in this expansion through first two
coefficients
13. A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev.
Integrals and series (Elementary functions). Moscow:
“Science”. 1981 (in Russian).
q
n
q
n
q
n
q
n
q
n AaAAqaAA 12
1
2
2
2 )21( −−
+
−
+
− +−++= βββ . (19)
ТОЧНЫЕ РЕЛЯТИВИСТСКИЕ ПЛАЗМЕННЫЕ ДИСПЕРСИОННЫЕ ФУНКЦИИ
С.С. Павлов и Ф. Кастехон
На основе теории сингулярных интегралов с ядром Коши вводятся и исследуются плазменные дисперсионные
функции (ПДФ), уточняющие слабо релятивистские ПДФ и обобщающие их на случай произвольной
температуры.
ТОЧНІ РЕЛЯТИВІСТСЬКІ ПЛАЗМОВІ ДИСПЕРСІЙНІ ФУНКЦІЇ
С.С. Павлов і Ф. Кастехон
На основі теорії сингулярних інтегралів з ядром Коші вводяться і досліджуються точні плазмові дисперсійні
функції (ПДФ), уточнюючі слабо релятивістські ПДФ і узагальнюючі їх на випадок довільної температури.
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EXACT RELATIVISTIC PLASMA DISPERSION FUNCTIONS
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| id | nasplib_isofts_kiev_ua-123456789-79343 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-11-30T16:51:54Z |
| publishDate | 2005 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Pavlov, S.S. Castejon, F. 2015-03-31T13:49:55Z 2015-03-31T13:49:55Z 2005 Exact relativistic plasma dispersion functions / S.S. Pavlov, F. Castejon // Вопросы атомной науки и техники. — 2005. — № 2. — С. 55-57. — Бібліогр.: 13 назв. — англ. 1562-6016 PACS: 52.27.Ny https://nasplib.isofts.kiev.ua/handle/123456789/79343 On the ground of the theory of singular integrals with Cauchy kernel the exact plasma dispersion functions (PDFs) are introduced and studied. Those PDFs make more exact the weakly relativistic PDFs and generalize them on the case of arbitrary plasma temperature. На основі теорії сингулярних інтегралів з ядром Коші вводяться і досліджуються точні плазмові дисперсійні функції (ПДФ), уточнюючі слабо релятивістські ПДФ і узагальнюючі їх на випадок довільної температури. На основе теории сингулярных интегралов с ядром Коши вводятся и исследуются плазменные дисперсионные функции (ПДФ), уточняющие слабо релятивистские ПДФ и обобщающие их на случай произвольной температуры. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Basic plasma physics Exact relativistic plasma dispersion functions Точні релятивістські плазмові дисперсійні функції Точные релятивистские плазменные дисперсионные функции Article published earlier |
| spellingShingle | Exact relativistic plasma dispersion functions Pavlov, S.S. Castejon, F. Basic plasma physics |
| title | Exact relativistic plasma dispersion functions |
| title_alt | Точні релятивістські плазмові дисперсійні функції Точные релятивистские плазменные дисперсионные функции |
| title_full | Exact relativistic plasma dispersion functions |
| title_fullStr | Exact relativistic plasma dispersion functions |
| title_full_unstemmed | Exact relativistic plasma dispersion functions |
| title_short | Exact relativistic plasma dispersion functions |
| title_sort | exact relativistic plasma dispersion functions |
| topic | Basic plasma physics |
| topic_facet | Basic plasma physics |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/79343 |
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