Giới hạn áp dụng của thế Yukawa cho Plasma OCP lưu chất

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LIMITATIONS OF APPLICATION OF YUKAWA 
POTENTIAL TO FLUID OCP PLASMAS 
DO XUAN HOI*, NGUYEN THI THANH THAO** 
ABSTRACT 
After a brief introduction to the Poisson – Boltzmann equation acquired for the one-
component-plasmas (OCP), we carry out a careful treatment of the numerical data 
concerning the radial distribution function given by the Monte Carlo and the Hyper Netted 
Chain simulations for this kind of plasmas; especially, the weakly correlated ones. Based 
on some latest results for the screening potential at the near zero inter-nuclear distance, 
we propose the formulae to compute this potential by combining the Yukawa potential with 
a certain greater distance than a limit, called Debye-Hückel distance, and the Widom 
expansion with the lesser one. By this way, we show the limits of application of Yukawa 
potential to plasmas OCP. 
TÓM TẮT 
Giới hạn áp dụng của thế Yukawa cho Plasma OCP lưu chất 
Sau khi giới thiệu ngắn gọn phương trình Poisson – Boltzmann thu được cho plasma 
một thành phần (OCP), chúng tôi xử lí chi tiết các dữ liệu số liên quan đến hàm phân bố 
xuyên tâm cho bởi các mô phỏng Monte Carlo và HyperNetted Chain cho loại plasma này, 
đặc biệt là các plasma liên kết yếu. Dựa trên một vài kết quả mới nhất cho thế màn chắn ở 
khoảng cách liên hạt nhân gần bằng không, chúng tôi đề nghị các công thức tính thế màn 
chắn này bằng cách phối hợp thế Yukawa cho khoảng cách lớn hơn một giới hạn gọi là 
khoảng cách Debye-Hückel, và khai triển Widom cho khoảng cách nhỏ hơn. Bằng cách 
này, chúng tôi cũng đã chỉ ra những giới hạn áp dụng của thế Yukawa cho plasma OCP. 
1. Introduction 
The Yukawa potential was first introduced into the particle physics to describe 
the interaction between two nucleons and led to predict the existence of mesons [16]. 
However, the notion of Yukawa form potential has been widely used in from chemical 
process to others concerning the astrophysics, and especially considered as a 
generalization of Debye-Hückel (D-H) potential in the study of the effective potential 
between two ions separated by the distance R of a fluid plasma system: 
ReV
R
a
a
-
µ , (1) 
wherein a is a positive parameter characterizing the screening effect of the 
environment on the two ions under consideration. But the interaction of the form (1) 
* PhD, International University (Vietnam National University Ho Chi Minh City) 
** BSc, Luong The Vinh High School for the Gifted (Đồng Nai) 
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above can only be used with some conditions for R and for the fluidity of the plasmas, 
as shown in [5, 15]. In this work, by using new computing tools, we will suggest the 
limits of applying the Yukawa potential (1) for the one component plasmas (OCP) 
concerning the interionic distance as well as the coupling parameter. 
The content of this article will be presented in the following order: Firstly, we 
remind briefly the model used and the base of D-H theory beginning with the Poisson-
Boltzmann equation along with the specifications of applying this theory. Next, we will 
mention the latest international works related to this subject and indicate at the same 
time some useful comments for the computations in this work. The following part of 
this publication will focus on the method used for the treatment of the screening 
potential in fluid OCP and also on the new results obtained from this study. The 
conclusion is reserved for the remarks and also for the suggestions. 
2. Yukawa potential and radial distribution function for fluid OCP plasmas 
Within the scope of this work, we consider the model of OCP, that is a physical 
system at the temperature T, composed of N ions, each of Ze+ electrical charge, 
imbedded in a homogeneous medium of ZN electrons. This model is suitable for the 
study of the structure of some astrophysical objects such as the white dwarf or the 
neutron star,[9]. An OCP system may be seen as a collection of N spheres, each 
centered at one ion and having Z electrons neutralizing electrical charge. The radius of 
this ionic sphere is done by: 
1/34
3
na p
-æ ö= ç ÷è ø
, with n indicating the ion density. In order 
to measure the fluidity of such a OCP system, one uses the coupling parameter, defined 
as ( )
2Ze
akT
G = , and dense plasmas the ones which have 1G > , meaning that the 
Coulomb potential outweighs the thermal energy in magnitude. For some OCP, this 
parameter has relatively low value, for example, one has 0.76G = for brown dwarf and 
0.072 0.076G = ¸ for the solar interior. Especially, in the ICF (Inertial Confinement 
Fusion) experiments, the magnitude of G is only about 0.002 0.010¸ , [9]. In these 
fluid plasmas, the D-H theory is often used to describe the screening effect. The base of 
this theory will be briefly presented below. 
We call Rr
a
= the reduced distance and ( ) ( )
/
V Ry rV r
Ze R
= º , V(r) being the 
mean potential at each point of the system, the Poisson-Boltzmann for the OCP system 
can be described in the compact form [5]: 
2
2
( ) ( )3 1 expd y r y rr
dr r
é ùæ ö= - -Gç ÷ê úè øë û
, 
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with the limit conditions: 
0
lim ( ) 1
r
y r
®
= and lim ( ) 0
r
y r
®¥
= , expressing the interaction 
potential becomes Coulombian when two ions are near enough so that there no 
screening effect and it tends to zero when they are too far. 
Above some distance, we get this approximative equation: 
2
2
( ) 3 ( )d y r y r
dr
= G . (2) 
The solution satisfying those conditions has the expression: 
3r
DHy e
- G= , 
called Debye-Hückel solution and we have 
3r
DH
eV
r
- G
= , (3) 
a special case of Yukawa potential (1). 
At that time, the radial distribution function is described by means of this mean 
potential: 
3
/( ) exp
r
V kTDH
DH
eg r e
r
- G
- æ ö= = -Gç ÷ç ÷è ø
 (4) 
and if the screening potential is defined as the result of influence of the environment on 
the interaction between two test ions: 1( ) ( )H r V r
r
= - , we get the following 
expression for the D-H screening: 
31( )
r
DH
eH r
r
- G-
= . (5) 
According to (4), the radial distribution function is an strictly increasing function 
with respect to the distance r, in accordance with the numerical results of Monte Carlo 
simulations performed by many authors [1, 8, 13]. On the other hand, those results 
show that the behavior of this function changes to a kind of damped oscillation from 
some value of the parameter GC, signature of short-range order effect. 
3. The conditions of applications of Yukawa potential in fluid OCP plasmas 
In order to obtain the equation (2), the condition of linearization must be satisfied, 
i.e. the distance r must be greater than a certain value rDH for each G. According to [5], 
we can use the criterion: 
3
1
2 2
ry e
r r
e
- GG G
= » < to evaluate this linearization. On the 
other hand, for each value of G, with DHr r£ , the screening potential H(r) must have 
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the form of a polynomial whose degree is pair and the coefficient of r2 is 1
1
4
h = , as 
demonstrated in [10, 14]: 
2 4 6
0 1 2 3( ) ...H r h h r h r h r= - + - + (6) 
We see that the functions (5) and (6) must satisfy the continuity condition at point 
rDH for each value of G, that means: 
3
2
0
1
( )
( 1)
r
DH
i i
i DH
i
e khi r r
rH r
h r khi r r
- G
=
ì -
>ïï= í
ï - £ïî
å
 (7) 
4. Determine the Widom polynomial coefficients 
The data concerning the coefficient h0 of the polynomial (6) have been the subject 
of many discussions for its important role in the enhancement of the pycnonuclear 
reaction rate in some stellar objects with great mass density as white dwarfs, neutron 
stars, (See, for example, [3, 9, 12]). The latest MC simulations implemented by A. I. 
Chugunov et al [2] supply the value of h0 in a analytic form: 
1/2 3 31 1
0 2
2 42 1
CHU
A BA Bh
B BA
æ ö GG
= G + + +ç ÷ç ÷+ G + G + G+ Gè ø
 (8) 
with: 
1 2,7822A = , 2 98,34A = , 3 1 23 / 1,4515A A A= - = , 
1 1,7476B = - , 2 66,07B = , 3 1,12B = , và 4 65B = . 
One of the characteristics of this expression is one can obtain the asymptotic form 
1/2
0 3CHUh = G with small values of G. We recognize that the value of h0 of those two 
expressions coincide (with errors 0.3%) from 0.0032G £ , i.e. for very fluid plasma. 
In opposition to the MC simulations that give us the relatively exact of the radial 
distribution function for the dense plasmas, the HyperNetted Chain (HNC) calculations 
are more reliable for the fluid OCP systems [11]. An elaborate study of the MC and 
HNC data [1, 4, 13] show that for not too important magnitude of the coupling 
parameter: 10G £ , we can write the Widom polynomial of degree eight with the error 
about 0.2%, equivalent to that of MC simulations, that means we accept: 
4
2 4 6 8 2
0 2 3 4
0
1( ) ( 1)
4
i i
i
i
H r h r h r h r h r h r
=
= - + - + = -å . (9) 
as the expression for the screening potential for enough small interionic distances. 
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By optimizing the accordance between the polynomial (9) and the MC as well as 
the HNC data given by [1, 4, 13], one gets all the numerical values of the coefficients hi 
in (9). Especially, the numerical value of h0, presented in Table 1, can be expressed in a 
analytic form: 
5
0
1
3 ln (1 )
1 =
G
= + + G
+ G å
i
i
i
h a (10) 
with the coefficients ai given by: 
1 0,031980a = ; 2 0,232300a = ; 3 0,084350a = - ; 
4  0,011710a = ; 5   0,000579a = - . 
The error between (8) and (10) is shown in Table 1. We notice that those both 
expressions give: 00lim 3G® = Gh as we can see on the Figure 2. 
Table 1. Numerical values of h0 in function of G. The values of h0 directly 
obtained from the optimization the accordance between (6) and MC and HNC data, 
and computed from (9) are shown in the second and third columns. In the fourth 
column, we have the values of h0 according to Chugunov et al [2]. 
G h0MC 
(2) 
h0 
(3) 
h0CHU 
(4) 
(3) - (2) (4) – (2) (3) – (4) 
0,1 0,5150 0,5030 0,5050 -0,0120 -0,0100 -0,0020 
0,2 0,6615 0,6589 0,6645 -0,0029 0,0030 -0,0059 
0,5 0,8741 0,8623 0,8776 -0,0118 0,0035 -0,0152 
1 0,9586 0,9743 0,9958 0,0157 0,0372 -0,0215 
3,1748 1,0570 1,0586 1,0788 0,0016 0,0218 -0,0201 
5 1,0780 1,0735 1,0922 -0,0045 0,0142 -0,0187 
10 1,0920 1,0888 1,1007 -0,0032 0,0087 -0,0119 
20 1,0910 1,0940 1,0950 0,0030 0,0040 -0,0010 
40 1,0860 1,0882 1,0878 0,0022 0,0018 0,0004 
80 1,0810 1,0782 1,0804 -0,0028 -0,0006 -0,0022 
160 1,0750 1,0757 1,0737 0,0007 -0,0013 0,0020 
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-3 -2 -1 0 1 2 3 4 5 6
0.5
0.6
0.7
0.8
0.9
1
1.1
lnr
ho
lnG 
h 0
Figure 1. The solid line expresses the formula (10) compared 
with the dashed line for (8). The circles are values directly 
acquired from optimizing the agreement between (7) and MC and 
HNC data. 
-4 -3 -2 -1 0 1 2
0
0.5
1
1.5
2
lnG
h0
lnG 
h 0
Figure 2. The solid line expresses (10). The circles are MC and 
HNC values given in Table 1. The dashed line is the asymptotic 
behavior 3G . 
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 We can notice that another expression for h0 is also proposed for dense plasmas in 
[6]. However, the formula (10) satisfies the particular conditions for the fluid plasmas 
we shall need for the use of the Yukawa potential for this category of plasmas. 
The method mentioned above give us at the same time the numerical values for 
the other coefficients h2, h3, and h4 as seen in Table 2. 
Table 2. Numerical values of the coefficients in the Widom polynomial (9). 
G h2 h3 h4 
0,1 0,285915 0,155198 0,0298883 
0,2 0,184492 0,077716 0,0122415 
0,5 0,074081 0,0127690 0,00088438 
1 0,051772 0,0062949 0,00033008 
2 0,040241 0,0032605 0,00009693 
3,174802 0,035570 0,0020166 0,00000154 
All the numerical values of these coefficients can be found by the general analytic 
expression: 
5
0
(ln )
=
= Gå ki k
k
h b ; i = 2, 3, 4 (11) 
with values of bk shown in Table 3. A study of the variation of hi in function of G 
demonstrates that their behavior is uniformly decreasing without any unusual point. 
Table 3. The coefficients in the formula (11) computing hi. 
With the numerical values obtained from the formulae (10) and (11), we can 
compute the function of screening potential (9) and from that point, return to evaluate 
the radial distribution function g(r). The comparison of this with the MC and HNC 
results is given on Figure 3 for some values of G. The data concerning 2G = are 
quoted from [8]. We notice on the Figure 3 that the errors between the proposed 
 h2 h3 h4 
b0 0,05177 0,006295 0,0003301 
b1 -0,01518 -0,0004388 0,0005552 
b2 0,007324 0,0004114 -0,0002833 
b3 -0,02167 -0,01502 -0,002602 
b4 0,008098 0,006594 0,001285 
b5 0,005127 0,003445 0,0005493 
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analytical formulae and the simulation data are about some thousandths, equivalent to 
that of MC results. 
5. Limit rDH for each value of coupling parameter 
When one accepts that the D-H potential can only be used from interionic 
distance rDH for each G, the continuity conditions (7) will be applied for the amplitude 
of the functions: 
3
8 6 4 2
4 3 2 1 0
1( )
DH
DH
r
DH DH DH DHr r
DH
eH r h r h r h r h r h
r
- G
=
-
= = - + - + (12) 
0 0.5 1 1.5
-2
-1
0
1
2 x 10
-3
r
g(
r)
M
C
-g
(r
)
G = 0.1
g-
g H
N
C 
0 0.5 1 1.5
0
5
10
15
20 x 10
-4
r
g(
r)
M
C
-g
(r
)
G = 0.2
g-
g H
N
C 
0 0.5 1 1.5 2
0
2
4
6 x 10
-3
r
g(
r)
M
C
-g
(r
)
G = 0.5
g-
g H
N
C 
0 0.5 1 1.5 2
-5
0
5
x 10
-3
r
g(
r)M
C-
g(
r)
G = 1
g-
g M
C 
0 0.5 1 1.5 2
-5
0
5
x 10
-3
r
g(
r)
M
C
-g
(r
)
G = 2
g-
g M
C 
0 0.5 1 1.5 2
-15
-10
-5
0
5
x 10
-3
r
g(
r)
M
C
-g
(r
) G = 3.174802
g-
g M
C 
Figure 3. Errors g(r)-gMC(r) or g(r)-gHNC(r) between the radial distribution function g(r) 
deduced from (10) and (11) and MC or HNC data for each value of G. 
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to find those values rDH. An example is given on the Figure 4a and 4b for 0,5G = : We see 
that only from points with 2,01509DHr r> = , an expression of the form (5) can be 
consistent to the numerical data offered by HNC method. This remark shows that for 
the distance smaller than 2,01509DHr = , D-H potential is not suitable to describe the 
screening effect 
The common results of rDH for each value of G are presented on Table 4, which 
show more clearly the limit of application of D-H theory. 
Figure 4a. From the points whose abscissa is 
smaller than 2,01509, the D-H potential (dash 
line) must be replaced by the Widom expansion 
(solid line). The circles are HNC data. 
0 0.5 1 1.5 2 2.5 3 3.5
0.4
0.6
0.8
1
2.01509
G = 0.5
Figure 4b. At the point whose abscissa is 
2,01509, two line representing H(r) and 
HDH(r) intersect and have almost the same 
slope. 
1.9 2 2.1 2.2
0.42
0.44
0.46
0.48
0.5
G = 0.5
2.01509
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Table 4. The numerical value of the joint points between the D-H potential and 
Widom polynomial. 
G rDH 
0,1 1,29072 
0,2 1,40899 
0,5 2,01509 
1 2,09863 
2 2,12295 
The numerical values on Table 4 can be expressed by analytic function: 
1,69 0,3059arctan(3,394ln 4,156)DHr = + G + . (13) 
In order to see more [ further] the importance of the Widom polynomial in 
representing the screening potential, we can observe on the Figure 5 the variation of 
rDH with respect to G according to (13): The value of rDH increasing in function of G 
shows that the Yukawa potential expresses accurately the screening effect only for 
fluid plasmas and, and even then, this form of potential can be applied only with large 
enough distances r. 
The expression (13) presented above has a simpler form, more easily applied than 
the one proposed in [5], while the maximum error between them is only about 9% for 
0,1G = . 
It is interesting to notice that apart from the condition (12) expressing the 
continuity of the amplitude, one should verify the continuity of the slope of the two 
functions (5) and (9) as well as insure they have the same concavity at the joint point 
rDH, i.e.: 
0 0.5 1 1.5 2 2.5 3 3.5
1.4
1.6
1.8
2
2.2
rD
H
GG 
r D
H
Figure 5. The variation of rDH with respect to G. We 
see that the influence of the Yukawa potential 
decreases when the plasmas are denser. 
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3 3
7 5 3
4 3 2 12
3 3 32
6 4 2
4 3 2 12 3 2
3 1( ) 8 6 4 2
2( 1) 2 3 3( ) 56 30 12 2
DH DH
DH
DH DH DH
DH
r r
DH DH DH DHr r
DH DH
r r r
DH DH DHr r
DH DH DH
e eH r h r h r h r hr
r r r
e e eH r h r h r h r h
r r r r
- G - G
=
- G - G - G
=
ì ¶ G -
= + = - + -ï¶ï
í
¶ - G Gï = - - - = - + -ï¶î
 The more concrete calculations show that the first and second derivatives for each 
value of G of the two functions (5) and (9) at the point DHr r= have the same values 
with very small errors (the maximum error is about 1110- ). This affirms the accuracy of 
the numerical values of rDH on the Table 4 and of the expression (13). 
6. Conclusion 
We consider in detail the D-H potential, a special case of Yukawa potential, 
applied to the fluid OCP system and mention the conditions for the application of this 
theory. 
After an elaborate study of the MC and HNC data as well as of the newest 
publications, we perform the numerical calculations and suggest the formula (13) for 
the limit distance of application of the D-H potential indicating that for each value of 
the parameter G, at the interionic distances smaller than this limit, this form of potential 
should be replaced by a Widom polynomial of degree eight (9) with the coefficients 
also expressed by the analytic formulae (10) and (11). The results obtained from the 
proposed expressions have also been compared with the numerical data for the 
distribution function g(r); the discrepancy between those two values is conform to the 
expected exactitude. 
In the following works, we will especially consider the short-range order effect in 
the various plasmas, that is the onset of damped oscillations of the distribution 
function, and then, to study the apparition of this effect in function of the coupling 
parameter G. At the same time, we will focus on the determination of the value of G for 
which the Yukawa potential can accurately express the screening potential in plasmas. 
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