Studying structural properties of rubidium and cesium liquid metals using an effective hard-core Yukawa potential

Accepted Manuscript Studying structural properties of rubidium and cesium liquid metals using an effective hard-core Yukawa potential Hossein Nikoofard, Leila Hajiashrafi PII:

S0378-3812(15)30148-5

DOI:

10.1016/j.fluid.2015.09.043

Reference:

FLUID 10785

To appear in:

Fluid Phase Equilibria

Received Date: 21 June 2015 Revised Date:

19 September 2015

Accepted Date: 21 September 2015

Please cite this article as: H. Nikoofard, L. Hajiashrafi, Studying structural properties of rubidium and cesium liquid metals using an effective hard-core Yukawa potential, Fluid Phase Equilibria (2015), doi: 10.1016/j.fluid.2015.09.043. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Graphical Abstract

EXP PRESENT WORK Exp. Cal.

2.5

1.5

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S(k)

2.0

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3.0

0.5

0.0 0

2

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1.0

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Studying structural properties of rubidium and cesium liquid metals using an effective hard-core Yukawa potential

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Hossein Nikoofard* and Leila Hajiashrafi

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Faculty of Chemistry, Shahrood University, Shahrood 36199-95161, Iran

*Corresponding author:

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Tel.: +98 23 3239 5441; Fax: +98 23 3239 5441

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E-mail: [email protected]

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Abstract In this work, we investigate the applicability of a modification of the random phase approximation (RPA) theory with an effective hard-core Yukawa potential to the static structure factor for the rubidium

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and cesium liquid metals. Based upon the perturbation theory, we assume that the core contribution of the direct correlation function is related to the geometric effects via a linear form for a hard-sphere fluid, in which the molecular diameter is obtained at any temperature and, its tail contribution is related to the

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long-range intermolecular interactions of the system via a linear form. We use the long-range Yukawa potential in the modeling of liquid alkali metals (LAMs) with the decay parameter λ = 1.8 and a well

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depth potential that is state-dependent. The linear isotherm regularity (LIR) equation of state (EOS) is applied to obtain the long-range interactions. The results obtained show that the proposed approach for LAMs can predict the behavior of the structure factor, S(k), at a wide range of k values with a good accuracy, as compared to the experimental data, over the whole liquid states. The interaction model used

including the S(0) values.

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is also successful in the presentation of the Ornstein-Zernike (OZ) behavior of S(k) at the low-k region

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Keywords: Liquid alkali metals (LAMs), Random phase approximation (RPA); Structure factor; Linear

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isotherm regularity (LIR) equation of state (EOS).

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1. Introduction Alkali metals, in the liquid state, are promising for both the fundamental and technological applications due to their physical and thermodynamic properties such as high heats

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of vaporization, extended liquid range, and thermal conductivities, which make them good heat transfer fluids in the reactors operating at high temperatures and at high energy rates, as high as nuclear power reactors. Liquid alkali metals (LAMs) are widely under investigation for use in

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the nuclear energy, emission electronics, new power-intensive chemical current sources, medicine, and a number of other fields [1–7]. These applications require having the knowledge

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of high-temperature thermodynamic and structural properties of LAMs. As the critical point is approached, the conductivity drops sharply, thus showing a metal-nonmetal transition. Near the critical point, LAMs gradually turn to insulators, and their thermodynamic behaviors would be different from those at low temperatures and high densities [8,9]. In this regard, a quantitative

condensed matter.

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description of the structural properties of LAMs is an important subject in the theory of

Alkali metals, in comparison with normal fluids, do not follow the law of corresponding

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state, in which the interactions between particles at the solid, liquid, and gaseous phases are supposed to have the same form. In fact, the interatomic interactions in these fluids are strongly

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dependent on the thermodynamic state of the system [10,11]. The structure of a solid metal is usually considered to be a collection of ions, fixed in a solid matrix, and thus creating an ionic lattice. A conclusion is that the valence electrons are delocalized over the lattice, and hence have little correlations with the core electrons [12]. For liquid densities near the melting point, the interatomic pair potentials of alkali metals are the same as those in the solid state, and may be described by the pseudo-potential obtained from the first-order perturbation theory based on the

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nearly free electron (NFE) model. This potential consists of a direct coulomb potential (repulsion) between the ions plus the attraction of each ion to the screening charge distribution induced by the pseudo-potential of the second ion [13,14]. This effective pair potential describes

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the interatomic potential and structure of the metals with a relativity high accuracy, only when the density of the fluid is more than twice the critical density [15]. For lower densities, the free electron belongs to its ion, and thus the NFE model fails to give a reasonable pair potential,

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especially near the critical point. In these states, alkali metal fluids consist of neutral atoms, molecules, and some small clusters, and the interaction potentials between these atoms and

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molecules may be described, in some cases, by a simple potential without an oscillating tail [16,17]. With a further decrease in density in the gaseous phase, the fluid is in an atomic from, and the interactions between these atoms are van der Waals forces, and may be considered as the Lennard–Jones (LJ)-type pair potentials [17]. In fact, the interatomic interactions for alkali

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metals change from a screened columbic potential to the LJ-type interactions with decrease in density. Therefore, the interatomic interactions at moderate densities are very complex, and there is no theoretical method available to obtain them.

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Fortunately, studies on the experimental structure factor, S(k), of alkali metals provide an important source of information for the interatomic interactions. The structure factor at low-k

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values is very sensitive to the intermolecular interactions at the long-range limit [18], whereas the short-range potential has a central role in the large k values [19]. In our previous works, using the linear isotherm regularity (LIR) equation of state (EOS), the accuracy and role of different parts of the effective pair potential functions have been extensively studied for nonmetal fluids [20,21]. However, a number of EOSs and different potential functions have been used to predict and reproduce the thermodynamic properties of ALMs [22-24]. Some of these attempts are

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restricted to a limited range of temperatures and pressures, and the low-k values for S(k), and the results obtained show different degrees of accuracy. In this work, based on a modification of the random phase approximation, we investigated the structural properties of the liquid alkali metals

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rubidium and cesium in term of the direct correlation function (DCF) and structure factor. The Yukawa potential function was used to model the long-range attractions. In our proposed approach, the contribution of all interactions was considered via a definition for the effective pair

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potential, whose intermolecular parameters vary with the thermodynamic states. Using the LIR equation of state, we evaluated the behavior of S(k) for rubidium and cesium over the whole

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liquid ranges within the temperature range of 500-1000 K.

2. Theory 2.1. Equation of state

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It has been found that the linear isotherm regularity equation of state (LIR-EOS) is derived as the following form:

(1)

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( Z − 1 ) v 2 = A + Bρ 2

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where Z is the compressibility factor, ρ = 1/v is the number density, and A and B are the temperature-dependent parameters; the details can be found elsewhere [25]. This regularity is experimentally valid for all types of fluids including polar, non-polar, hydrogen-bonded, and quantum fluids, for the densities greater than the Boyle density, and the temperatures less than twice the Boyle temperature [26-29]. The temperature dependency of the LIR parameters can be considered as:

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A′ RT

(2)

B = B ′′ +

B′ RT

(3)

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A = A′′ −

where A' and B' are related to the attraction and repulsion terms of the applied potential function, respectively. A'' is related to the non-ideal thermal pressure, and B'' is a constant that is equal to

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zero for simple dense fluids. The experimental pvT data can be used to determine the A and B parameters in the LIR equation of state for the Rb and Cs liquid metals over the whole liquid

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range.

2.2. Direct correlation function and structure factor

A more common route toward the structure factor is via the perturbation theory and is known as the random phase approximation (RPA). The RPA in the theory of simple liquids

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suggests a combination of a linear function related to the long-range attraction interaction, φ(r), with the representation of the direct correlation function, c(r), of the reference system owing to

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the short-range repulsion interaction in the form of c0(r):

r
c (r ), c RPA (r ) =  0 − βϕ (r ),

(4)

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r≥d

where β = 1/(kBT) is the inverse temperature and d is the hard-core diameter, in which the Mayer cluster function was used for a reasonable choice of d(T) at any temperature T [30]. In this work, we approximated c0(r) by the Henderson-Grundke correction to the direct correlation function of the hard-sphere fluids [31]: c0 (r ) = c HS (r ) (1 − 0.127 ρ 2 d 6 )

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r
(5)

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where cHS(r) is DCF of the hard-sphere reference fluid, which is expressed using the PercusYevick (PY) equation for hard-sphere fluids [32]. In the general case, the pair potential with a

∞,  w(r ),

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hard-core, φHC(r), can be written as: r
ϕ HC (r ) = 

(6)

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where w(r) is a pair potential in the long-range interactions. Here, on account of the Yukawa potential playing an important role in classical simple liquid metals and combined versatility and

w(r ) = −ε

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simplicity, we selected it to describe the long-range interactions in ALMs:

exp[−λ (r / d − 1)] , r/d

r≥d

(7)

where ε is the well depth and λ is a decay parameter that measures the intermolecular attraction

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range. We may account for the contribution of many-body interactions using an effective pair intermolecular potential, weff(r), which is dependent on the thermodynamic state. Therefore, the final expression for DCF is:

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c HS (r )(1 − 0.127 ρ 2 d 6 ), c( r ) =  − βweff (r ),

r
(8)

In order to predict the behavior of S(k) over different thermodynamic states, we used the Ornstein-Zernike (OZ) equation, as:

S (k ) = (1 − ρ c(k )) −1

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(9)

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where c(k) is the Fourier transform (FT) of c(r). In all the calculations used to predict S(k), we need to have the molecular parameters in DCF (Eq. 8). In this way, the well-known

 ∂P  B r = β   T = 1 − 4πρ  ∂ρ 

∫

∞ 0

c ( r ) r 2 dr

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thermodynamic relationship for the reduced bulk modulus Br can be used as: (10)

where P is the pressure. Based on the perturbation approximation in Eq. (8), it is convenient to

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split the reduced bulk modulus into long- and short-range terms, as:

d ∞  ∂P  T = 1 − 4πρ  ∫ c0 ( r ) r 2 dr + ∫ [ − β w( r ) eff ]r 2 dr   0  d  ∂ρ 

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β 

(11)

where the first and second integrals represent the contributions of the core and tail of DCF to Br, respectively. Using the LIR equation of state for Br, we obtained:

(12)

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 ∂P  3 A′ 2 5B′ 4  T = 1 + 3 A′′ρ 2 − ρ + 5 B ′′ρ 4 + ρ  RT RT  ∂ρ 

β 

Besides, a comparison between Equations (11) and (12) shows that the molecular parameters in

d ∞ 3 A′ 2 5B′ 4 ρ + 5 B ′′ρ 4 + ρ = 1 − 4πρ  ∫ c0 ( r ) r 2 dr + ∫ [ − β weff ( r )]r 2 dr   0  d RT RT

(13)

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1 + 3 A′′ρ 2 −

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DCF are related to the LIR parameters as:

The equation obtained (Eq. 13) shows that the DCF parameters ε and λ can be calculated using the LIR equation of state. It was found that the Yukawa potential (Eq. 7) with λ = 1.8 has very often been used as a prototype of an analytically solvable model of simple fluids. In this way, Pini et al. [33] have applied a thermodynamically self-consistent Ornstein-Zernike approximation (SCOZA) to the HC-Yukawa fluids, and obtained very good overall

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thermodynamics and a remarkably accurate critical point with λ = 1.8. Furthermore, in our previous work [34], we have successfully predicted the structure factor for the monatomic fluids by application of the λ = 1.8 in the Yukawa potential model. Here, we took the same value for λ

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in order to simplify the calculations. The parameter ε, which is T- and ρ-dependent, can be determined using Eq. (13). It is expected that all the approximations used in our approach to obtain c(r) may be exaggerated in the FT of c(r). Therefore, one can investigate the accuracy of

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the proposed approach for the rubidium and cesium liquids via prediction of the behavior of S(k)

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at different densities and temperatures.

3. Results and discussion

The experimental pvT data [35,36] for the Rb and Cs liquids was used to construct Eq. (1) at several isotherms. Fig. 1 shows the perfect linear behavior of (Z-1)/ρ2 vs. ρ2 over the whole Rb

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liquid range. The linear behavior is quantified by R2 > 0.999, where R2 is the linear correlation coefficient. The LIR parameters A and B vary with temperature, and their temperature dependency is demonstrated in Fig. 2. The results obtained in this investigation for both Rb and

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Cs is summarized in Table 1. In fact, the linear behavior observed in Figs. 1 and 2 demonstrates

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the validity of the LIR equation of state for the thermodynamic states studied here. We obtained the intermolecular parameters d and ε for the liquid rubidium and also liquid

cesium using the procedure mentioned in section 2.2 at different thermodynamic states. The results obtained were tabulated in Table 2. Then the behavior of c(r) vs. reduced distance, r, for liquid rubidium along an isotherm, T = 900 K, and also a reduced isochor, ρ* = 0.745, were plotted in Fig. 3. We know that for a real fluid, c(r) has a negative part for r < 1, rises steeply around r = 1, and decreases exponentially. We can observe in Fig. 3(up) that the negative

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contribution of c(r) becomes important when the density increases. We can also observe in Fig. 3(down) that the negative contribution of c(r) does not change when the temperature increases, while the tail contribution of c(r) decreases when the temperature increases. A similar result was

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observed for cesium over the whole liquid range within the temperature range 500-1000 K.

We demonstrated the behavior of S(k) for Rb at two different thermodynamic states in

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Fig. 4. The calculated results show that a reasonably qualitative behavior of S(k) is generated, and the number and height of the bumping vary correctly corresponding to the thermodynamic

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states, which is expected [37]. According to Fig. 4(B), our calculations are capable of predicting the Ornstein–Zernike behavior of S(k) at low-k values with a minimum S(k) value over the low densities. This behavior verifies that the long-range attraction forces have correctly been taken into account in the tail contribution of the DCF model. The same behavior can be observed for the liquid cesium. In order to confirm the accuracy of our calculations, we examined the

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application of the model for the prediction of S(k) for the liquid Rb at the dense region. Fig. 5 shows a comparison between the calculated values for S(k) at a wide range of k values for rubidium at T = 313 K and ρ = 17.50 mol/L and the experimental data taken from Ref. 38. It is

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interesting that our calculations, carried out at the melting point region of liquid Rb can present

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qualitatively the behavior of S(k). As shown in Fig. 6, our calculations are in a good agreement with the experimental data, especially at the long-wavelength limit region (low-k). However, they show some deviations at the k values larger than the position of the first peak, which may be due to the contribution of the reference part of DCF that cannot include all the short-range interactions in the system. It should be noted that this simple model without any complications may predict the critical behavior of S(k) just by using an effective pair potential instead of using the realistic two- and three-body interactions. In this way, in order to investigate our model at a

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point close to the melting point, we calculated the behavior of S(k) for cesium at T = 303 K and ρ = 14.10 moll/L, and compared it with the experimental data, taken from Ref. 39, (Fig. 6). As shown in this figure, our calculations are in a good agreement with the experimental results at a

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wide range of k values.

However, our calculations can predict the S(k) values at a wide range of k values; the

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prediction of the low-k behavior of S(k) is a considerable result. It is well-known that the main feature of S(k) for large values of k is almost completely determined by the short-range repulsive

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part of the interatomic forces, the range of which develops with density [40]. However, the behavior of S(k) at low-k values is correctly determined only by using the long-range attractive forces. In Fig. 7, we presented the low-k behavior of S(k) for rubidium at T = 900 K and ρ = 14.20 moll/L, and compared this behavior with the experimental data taken from Ref. 41. It can be seen that, in general, our results are in a good agreement with the experimental data in this

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region. Furthermore, we predicted the low-k behavior of the structure factor for the liquid cesium at T = 773 K and ρ = 12.26 mol/L, and compared this behavior with the experimental data taken from Ref. 41 (Fig. 8). This figure obviously indicates that the agreement between our results and

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the experimental data is so good, and that the low-k behavior of S(k) is reasonable. Therefore, we

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may conclude that the effective pair potential Yukawa used in this model is suitable to predict S(k) for the alkali metal liquids for the low-k values in our examined range of temperatures and densities.

In addition, the long-wavelength limit of the structure factor, S(0), calculated for Rb and

Cs at points close to their melting points using the equation of state LIR, was compared with that for the RPA and extended random phase approximation (ERPA) theories, and the experimental values taken form Ref. 39. The results obtained are summarized in Table 3. According to this

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table, the calculated results lead to an improved S(0) value for these dense thermodynamic states. It is known that the magnitude of S(0) depends strongly upon the form of the long-range part of

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the interatomic potential corresponding to the state-dependent molecular parameters.

4. Conclusion

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This work provided a modification of the RPA theory for the alkali metal liquids using an effective hard-core Yukawa potential over the whole liquid ranges. We can predict the structure

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factor, S(k), for the Rb and Cs liquids via an analytical expression for DCF based on the combination of the LIR equation of state. In this model, the core contribution of DCF was related to the co-volume and geometric effects, and the effective Yukawa potential was used as the longrange attraction forces to its tail contribution. The results obtained show that the LIR equation of

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state can predict the behavior of S(k) for the alkali metal liquids at a wide range of k values with a good accuracy, as compared to the experimental data. The calculations successfully reproduced the Ornstein–Zernike behavior of S(k) in the small wave number region. The results obtained

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indicated that the effective hard-core Yukawa potential can describe the interatomic potential and structure of metals with a relativity high accuracy over the considered liquid range. Despite its

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simplicity, the proposed approach provides the results for S(k) at small-k values of LAMs, which are closest to those obtained by other theories in the predicted and experimental measurements of S(k) for the k → 0 limit.

Acknowledgement

We acknowledge the financial support of Shahrood University for this work.

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[10] M. Moosavi, S. Sabzevari, Fluid Phase Equilib. 329 (2012) 63-70. [11] F. Yosefi, H. Karimi, Z. Gandomkar, Fluid Phase Equilib. 370 (2014) 43-49.

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[12] M.H. Ghatee, M. Bahadori, J. Phys. Chem. B 105 (2001) 11256-11263. [13] G. Kahl, J. Hafner, J. Phys. Rev. A 29 (1984) 3310-3319. [14] G.A. Mansoori, C. Jedrzejek, N.H. Shah, M. Blander,Chemical Metallurgys-A Tribute to Carl Wagner, Gokcen N A, The Metallurgical Society of AIME, New York, 1981, pp233-240. [15] R. Evans, T.J. Sluckin, J. Phys. C: Solid State Phys. 14 (1981) 3137-3153. [16] S. Munejiri, F. Shimojo, K. Hoshino, M. Watabe, J. Non-Cryst. Solids 205 (1996) 278-281.

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[28] J.S. Rolison, F.L. Swinton, Liquids and Liquid Mixtures, Butterworth, London, 1982. [29] G. Parsafar, N. Farzi, B. Najafi, Int. J. Thermophys. 18 (1997) 1197-1216. [30] H.C. Andersen, D. Chandler, J.D. Weeks, Adv. Chem. Phys. 34 (1976) 105-153.

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[33] D. Pini, G. Stell, N.B. Wilding, Mol. Phys. 95 (1998) 483-494. [34] H. Nikoofard, A. Amin, J. Phys. Soc. Jap. 82 (2013) 084602-6. [35] N.B. Vargaftik, V.F. Kozhevnikov, V.A. Alekseev, J. Eng. Phys. 35 (1978) 1415-1422. [36] N.B. Vargaftik, V.A. Alekseev, V.F. Kozhevnikov, Y.F. Ryzhkov, V.G. Stepanov, J. Eng. Phys. 35 (1978) 1361-1369. [37] C.A. Croxton, Introduction to Liquid State Physics, Wiley, New York, 1978, pp83–93.

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[38] J.N. Herrera, P.T. Cummings, H. Ruaiz- Estrada, Mol. Phys. 96 (1999) 835-847. [39] R. Evans, W. Schirmacker, J. Phys. C: Solid State Phys. 11 (1978) 2437-2452. [40] R. Evans, T.J. Sluckin, J. Phys. C: Solid State Phys. 14 (1981) 2569-2580.

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[41] G. Franz, W. Freyland, W. Glaser, F. Hensel, E. Schneider, J. Phys. Coll. Suppl. 41 (1980)

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194-197.

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Tables

Table 1

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Parameters used to obtain LIR equation of state for Rb and Cs liquids. A"

A'

B''

(L2/mol2)

(L3.atm/mol3)

(L4/mol4)

(L5.atm/mol5)

Rb

-6.26*10-4

2.13

4.38*10-5

5.61*10-3

Cs

-1.13

2.86

2.68*10-3

3.84*10-2

B'

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Liquids

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Table 2

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Calculated values for molecular parameters d and ε for Rb and Cs liquids in different thermodynamic states.

ε /kT

Liquids

T(K)

ρ (mol/L)

d (A)

Rb

350

16.246

4.40

Rb

900

14.533

4.27

Rb

1400

12.021

3.65

-0.294

Cs

500

13.079

4.73

-1.122

Cs

700

Cs

900

-0.948

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-1.057

4.60

-0.832

11.424

4.45

-0.594

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13.738

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Table 3

Comparison of calculated values for S(0) for alkali metal liquids at near

T (K)

ρ (nm-3)

RPA

ERPA

Rb

312

10.4

0.027

0.028

Cs

303

8.3

0.028

0.028

LIR

Exp.

0.022

0.022

0.025

0.024

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Liquids

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melting point with other theories and experimental data [39].

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Figure captions

Fig. 1. Plots of (Z-1)/ρ2 vs. ρ2 for liquid Rb in several isotherms over whole liquid range.

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Fig. 2 Plots of LIR parameters A and B as a linear function of T -1 for Rb at temperature range of

500-1000 K (up and down, respectively).

Fig. 3. Behavior of c(r) vs. r (reduced distance) for fluid Rb along T = 900 K with several

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densities (up panel), and along ρ* = 0.745 with several temperatures (down panel).

Fig. 4. Behavior of calculated S(k) vs. kσ for Rb at different thermodynamic states: (A) T = 350

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K and ρ* = 0.813, and (B) T = 900 K and ρ* = 0.681.

Fig. 5. Comparison between calculated S(k) values and experimental data [38] for liquid Rb at T

= 313 K and ρ = 17.50 mol/L.

Fig. 6. Comparison between calculated S(k) values and experimental data [39] for liquid Cs at T

TE D

= 303 K and ρ = 14.10 mol/L.

Fig. 7. Comparison between calculated S(k) values at low-k region and experimental data [41]

for Rb at T = 900 K and ρ = 14.20 mol/L.

EP

Fig. 8. Comparison between calculated S(k) values at low-k region and experimental data [41]

AC C

for Cs at T = 773 K and ρ = 12.26 mol/L.

19

ACCEPTED MANUSCRIPT

0.002

RI PT

2

-2

(Z-1)/ρ2 (L mol )

0.000

-0.002

SC

500K 600K 700K 800K 900K 1000K

-0.006 180

200

220

M AN U

-0.004

240

260

AC C

Fig. 1

EP

TE D

ρ2 (mol2 L-2 )

20

280

300

320

ACCEPTED MANUSCRIPT

-0.025

RI PT

-0.030

-0.040

SC

2

-2

A (L mol )

-0.035

-0.045

-0.055 0.010

0.012

0.014

M AN U

-0.050

0.016

0.018

0.020

0.022

0.024

0.026

1/RT(mol L-1atm-1)

2.0

4

EP

1.4

TE D

1.6

-4

B (L mol ) *10

4

1.8

AC C

1.2

1.0 0.010

0.012

0.014

0.016

0.018

0.020

1/RT(mol L-1atm-1)

Fig. 2

21

0.022

0.024

0.026

ACCEPTED MANUSCRIPT

RI PT

0

SC

c(r)

-10

-20

-30 0

1

M AN U

14.53 (mol/L) 15.16 (mol/L) 15.58 (mol/L)

2

3

r/σ

4

TE D

2 0 -2

-6

EP

c(r)

-4

-8

-10

350 K 550 K 750 K

AC C

-12 -14

0

1

2

r/σ

Fig. 3

22

3

ACCEPTED MANUSCRIPT

3.0

(A)

RI PT

2.5

S(k)

2.0

1.5

SC

1.0

0.0 0

10

M AN U

0.5

20

30

40

50

60

kσ

2.5

TE D

2.0

(B)

S(k)

1.5

EP

1.0

0.5

AC C

0.0

0

10

20

30

kσ

Fig. 4

23

40

50

60

ACCEPTED MANUSCRIPT

3.0 EXP Experimental PRESENT WORK Calculated

RI PT

2.5

1.5

SC

S(k)

2.0

1.0

0.0 0

2

M AN U

0.5

4

6

AC C

Fig. 5

EP

TE D

k (A-1 )

24

8

10

ACCEPTED MANUSCRIPT

RI PT

3.0

Experimental Calculated

2.5

1.5

SC

S(k)

2.0

1.0

0.0 0

2

M AN U

0.5

4

6

-1

AC C

Fig. 6

EP

TE D

k (A )

25

8

10

ACCEPTED MANUSCRIPT

RI PT

Experimental Calculated

0.5

0.3

SC

S(k)

0.4

0.2

0.0 0.0

0.2

0.4

M AN U

0.1

0.6

0.8

-1

AC C

Fig. 7

EP

TE D

k (A )

26

1.0

1.2

1.4

ACCEPTED MANUSCRIPT

RI PT

0.8

Experimental Calculated

0.4

SC

S(k)

0.6

0.0 0.0

0.2

0.4

M AN U

0.2

0.6

0.8

-1

AC C

Fig. 8

EP

TE D

k (A )

27

1.0

1.2

1.4

ACCEPTED MANUSCRIPT Highlights

A modification of the RPA theory applied to the rubidium and cesium liquid metals.

•

The linear isotherm regularity EOS was used to present the direct correlation function.

•

Long-range interactions modeled by a pair Yukawa potential with λ = 1.8.

•

Predictions for the structure factor compared with the simulation and experimental data.

AC C

EP

TE D

M AN U

SC

RI PT

•