BIBECHANA Vol. 20, No. 2, August 2023, 146–160 ISSN 2091-0762 (Print), 2382-5340 (Online) Journal homepage: http://nepjol.info/index.php/BIBECHANA Publisher:Dept. of Phys., Mahendra Morang A. M. Campus (Tribhuvan University)Biratnagar Theoretical investigation on the structural and vibrational properties of some alkali liquid metals: A pseudopotential approach Mayank H. Jani∗∗, Aditya M. Vora∗ Department of Physics, University School of Sciences, Gujarat University, Navrangpura, Ahmedabad 380 009, Gujarat, India ∗Corresponding author. Email: voraam@gmail.com ∗∗mayankjani5@gmail.com Abstract Along with a few elastic constants, a recently created pseudopotential is employed to examine the vibrational properties of some simple alkali metals, i.e. Li, Na, K, Rb and Cs. The phonon dispersion curves (PDC) are computed in the text. (ωL and ωT ), longitudinal sound velocity (vl), transverse sound velocity (vt), isothermal bulk modulus (B), modulus of rigidity (G), Young’s modulus (Y ), Debye temperature (θD) of some liquid alkali metals. The second order technique used in the current work, using Hubbard and Beeby (HB) equations, is based on pseudopotential theory. The various Percus-Yevick hard sphere (PYHS) and one-component plasma (OCP) structure factors used in the current investigation are used to construct the pair correlation function g(r). The current paper makes use of three distinct forms of local field correction functions developed by Hartree (H), Taylor (T), and Nagy (N). The current findings are found to qualitatively agree with the experimental and theoretical data that is currently available. Keywords Pseudopotential; Liquid metals, PYHS structure factor, OCP structure factor, phonon dispersion, structural properties, vibrational properties. Article information Manuscript received: May 2, 2023; Accepted: June 10, 2023 DOI https://doi.org/10.3126/bibechana.v20i2.55552 This work is licensed under the Creative Commons CC BY-NC License. https://creativecommons. org/licenses/by-nc/4.0/ 1 Introduction Since a long time, phonon dispersions and vibra- tional properties in liquids and gases have been ex- plored theoretically, through computer simulation, and empirically. The most popular and useful ex- perimental technique for determining phonon dis- persion curves is neutron scattering. Neutron scat- tering research has supplied some basic excitation in liquid metals. From Copley and et al. [1] to Cabrillo 146 http://nepjol.info/index.php/BIBECHANA voraam@gmail.com mayankjani5@gmail.com https://doi.org/10.3126/bibechana.v20i2.55552 https://creativecommons.org/licenses/by-nc/4.0/ https://creativecommons.org/licenses/by-nc/4.0/ Mayank H. Jani and Aditya M. Vora/ BIBECHANA 20 (2023) 146-160 147 et al. [2, 3], the key experimental work sheds infor- mation on the dynamical features of alkali metals. Desai et al. [4] theoretically reported the vibrational characteristics of liquid argon before this practical investigation. Theoretically, phonon dispersion curves can be obtained using force constants, however even for regular and simple systems such as cubic crystals, the number of force constants required is large. As a result, for complicated systems such as liquids, phonon models that represent the ion-electron in- teraction are critical [5]. So, since liquid metal is a highly correlated system, the random phase ap- proximation cannot produce an accurate picture of interaction. however, Hubbard and Beeby (HB) [6] has explained the vibrational properties of liquid metals quite effectively, and it may also be called a generalization of theory for the solid up to the liquid state. For the explanation of phonon in liq- uid metals, a stepwise generalized phonon theory of the crystal has been made. In the first solid state, there is no relative motion expected but in the other half, it is to be extended to the continuous system with the relative atomic motion which is called the liquid metals. To take the work based on the theo- retical formulation, the high degree physics needed which includes the kinetics with the coupling of var- ious modes [7, 8]. These excitations are density de- pendent functions so, they follow, in some manner, the structure factor variations with respect to wave numbers. The structure factor peak position falls roughly at the minima of the longitudinal frequency ωL vs. wave vector (k) plot [9]. Numerous studies on the vibrational character- istics of numerous types of materials have been published [10–12]. Thakor and his colleagues [12] used the HB method to calculate phonon dispersion curves. The same calculations were made for the metallic elements such as Na, Mg, Al and Pb. The acquired values in that study are consistent with the experimental data for Na elements. Since there are just a limited amount of experimental data points available for Al, it was not displayed. Vora [11] has published the phonon dispersion curves for liquid alkali metals after around a year. With the aid of the empty core model pseudopotential, he demon- strated how the curves are behaving consistently. The comparison with the experimental data avail- able reveals that the selection of potential may be to blame for the low agreement. Recently, Malan and Vora [12] have made another attempt to determine the PDC of liquid alkali metals. They claimed that the HB technique can be used to measure the ma- terial’s aforementioned attribute. The curves they obtained were found to be compatible with the ex- perimental data that was available [12]. Looking to the advantages of pseudopotential theory with HB approach particularly for studying the vibrational properties of studied simple liquid alkali metals are reported in the present article. For studying the vibrational properties, the structure factor S(q) and then after pair correlation func- tion g(r) plays an important role. Therefore, our motive of the present article is studying the vibra- tional properties of some simple liquid metals with Percus-Yevick hard sphere (PYHS) [13–15] and one- component plasma (OCP) [16–18] structure factor models with our own developed single parametric model pseudopotential. 2 Computational Methodology The size and locations of the longitudinal (L) and transverse (T) frequency peak positions with re- spect to the wave vector k are attempted to be determined in the current article. In the cur- rent investigation, the pseudopotential concept was used. The determined values of the pseudopo- tential parameter were applied to the newly con- structed pseudopotential. When calculating the ion-interaction, the exchange and correlation effects must also be taken into consideration. To have the highest chance of predicting the astonishing re- sults, the most appropriate exchange and correla- tion functions are advised. Here we are using two different methods for the calculation of structure factor, i.e., PYHS [13–15] and OCP [16–18]. To obtain the structure factor by OCP method, we require a pseudopotential. For the same reason, we have applied our own generated model pseudopotential to obtain the pair distribu- tion function with the help of which we will calcu- late the vibrational properties of the liquid alkali metals. The form of our newly proposed model poten- tial in real space and corresponding bare-ion form factor in the reciprocal space is given as V (r) = −3Ze2 r + Ze2 r exp ( 1− r rc ) r ≤ rc = −Ze2 r r ≥ rc (1) The Fourier transform of such model potential is given as V (q) = −4πZe2 q2Ω0 [3− q2r2c 1 + q2r2c − exp(1) + {−2 + exp(1) + exp(−1)q2r2c 1 + q2r2c } cos qrc + exp(−1)qrc 1 + qr2c sin qrc] (2) Here, Z and e are valency and electronic charge, Ω0 is atomic volume, q is wave vector, and rc is the pa- rameter of the potential, respectively. It should be noticed that this type of pseudopotential only has Mayank H. Jani and Aditya M. Vora/ BIBECHANA 20 (2023) 146-160 148 one parameter rc. Within the core region, it is con- tinuous at r = rc and becomes weaker and weaker as r decreases. Two features of this form are the Coulombic terms inside the core and the fluctuat- ing cancellation brought on by the attracting and repelling concentrations of the potential outside the core. It is thought that instead of being 0 or con- stant, the potential outside the core should change as a function of r. This demonstrates the modi- fication made to the Ashcroft empty core model. The pseudopotential parameter d can be identified using [19] as rc = 0.51RaZ −1/3 (3) The pair correlation function g(r) and the pair po- tential must be obtained in order to carry out the computations. This method generates pair corre- lation function values as a prerequisite for subse- quent calculations rather than directly extracting them from the experimental data. Therefore, the current prediction is entirely theoretical and has no connection to any earlier experimental studies. By using the generated S(q) from the PYHS [13-15] and OCP [16-18] the g(r) par distribution function for the respective reference system can be easily ob- tained. Here [20] is an equation for the g(r) utilized in this calculation. g(r) = 1 + ( 1 2π2ρr )∫ ∞ 0 q[S(q)− 1] sin(qr)dq(4) The structure factor equations of PYHS [13–15] can be derived as: S(q) = [1 + 24ξ (1− ξ)4y6 ( (1 + 2ξ) 2 y3(sin y − cos y) − 6ξ ( 1 + ξ 2 )2 y2 [ 2y sin y − ( y2 − 2 )] + ξ 2 (1 + 2ξ) 2 [ 4y3 − 24y ] sin y − ( y4 − 12y2 + 24) cos y + 24]−1 (5) And also, the structure factor equation with OCP method is mentioned below: S(q) = S0(q) 1 + ρβu∗(q)S0(q) (6) S0(q) = 1 1− ρC0(q) (7) ρC0(q) = 3Γ q4a4α2 2 [cos (qaα1) + 2 cos(qaα2) − 3 sin(qaα1) qaα1 ] (8) where, α1 and α2 are the dimensionless parame- ters and also β µ∗(q) equations were taken from the mentioned reference [16–18] for the static structure factor of the OCP reference system. Here while de- riving the static structure factor using OCP system, we have used our own developed model pseudopo- tential. 2.1 Phonon dispersion relation Finding a single model potential that can be used to precisely compute a variety of material properties is challenging. Problems arise when the physical phase shifts from solid to liquid or the other way around. In other words, not all of the properties in the liquid state can be achieved by the potential provided for the solid. We propose a new potential as stated in Eq (3) to get around this problem and evaluate the phonon dispersion curves and other vibrational properties. To compute the phonon dispersion relations of liquid metals, the most fre- quently used approach of Hubbard and Beeby [6] is adopted for the calculations. With the physical argument that the product of the static pair corre- lation function g(r) and the second derivative of the interatomic pair potential ϕ(r) is peaked near the hard sphere diameter . Hubbard and Beeby [6] have derived the equations for the longitudinal phonon frequencies ωL(q) and transverse phonon frequen- cies ωT (q) as: ωL 2(q) = ωE 2[1− 3 sin(qσ) qσ − 6 cos(qσ) (qσ) 2 + 6 sin(qσ) (qσ) 3 ] (9) and ω2 T (q) = ω2 E [ 1 + 3 cos(qσ) (qσ)2 − 3 sin(qσ) (qσ)3 ] (10) In the above mentioned equations ωE represents the maximum phonon frequency and is given as, ωE = k ∫ ∞ 0 g(r)r2ϕ ′′ (r)dr (11) with k = 4πρ 3M (12) ϕ ′′ (r) = 4Z2 r3 + Ω π2 ∫ ∞ 0 F (q)q2 [ 2 sin(qr) qr3 − 2 cos(qr) qr3 − q sin(qr) r ]dq (13) with ϕ′′(r) = ∂2ϕ(r) ∂r2 The energy wave number characteristics F (q) in the above equation can be written as, F (q) = −Ω0q 2 16π |V (q)|2 [εH(q)− 1] 1 + [εH(q)− 1] [1− f(q)] (14) Mayank H. Jani and Aditya M. Vora/ BIBECHANA 20 (2023) 146-160 149 In the above equation V (q), εH(q), F (q) are the bare ion potential, modified Hartree dielectric function and local field correction functions, respectively. The exchange and correlation screening func- tions from Hartree [21], Taylor [22], and Nagy [23] were utilized here to calculate the aforementioned properties. The Hartree screening function is purely static and it does not include any exchange and cor- relation effects. The expression for it as: f(q) = 0 (15) The Taylor local field correction function [22] is a simple formula for static electron gas dielectric function, incorporating exchange and correlation effects, which generates calculated physical prop- erties with good precession and justification. The equation for the function is follows: f(q) = q2 4k2F [ 1 + 0.1534 πkF ] (16) Nagy local field correlation function [23] is used in the static mean field approximation the result is analytic and depends upon the value of the pair- correlation function at zero inter-particle separa- tion. This analytic equation can be written as: f(q) = 1− g(0, n) + cb c2 + q2 − g(0, n) q tan−1 (q c ) (17) The parameters g(0, n)and c were taken as men- tioned in the original article [23]. 2.2 Elastic constants and parameters In addition to phonon dispersion curves, the cur- rent study has been expanded to determine a few constants and parameters required to provide a full bulk of liquid alkali metals. The usual equations are used to calculate the aforementioned parameters. In the long-wavelength limit of the frequency spectrum, the both frequencies i.e. transverse and longitudinal are proportional to the wave vectors and obey the relations. ωL ∝ q and ωT ∝ q ωL = vLq and ωT = vT q (18) where, vL and vT are the longitudinal and trans- verse sound velocities of the liquid metals respec- tively. Equations for vL and vT are given by, vL = ωE √ 3σ2 10 and vT = ωE √ σ2 10 (19) Various elastic properties are then determined by the longitudinal and transverse phonon frequencies. The bulk modulus (B), modulus of rigidity (G), Young’s modulus (Y) [24] and Debye temperature θD [24] are calculated using following expressions, BT = ρv2L− 4v2T 3 , G = ρv2T , Y = 2G(σ+1) (20) σ = 1− 2 ( v2 T v2 L ) 2− 2 ( v2 T v2 L ) (21) The Debye temperature is given in terms of both the velocities as [24],beg θD = h̄ωD kB = h̄ kB 2π [ 9ρ 4π ]1/3 [ 1 v3L + 2 v3T ]−1/3 (22) Here h̄ represents the Planck constant, kB the Boltzmann constant, ωD the Debye frequency, and ρ the number density, respectively. Here, θD shows the characteristic atomic motions in the liquid phase and is assumed to be constant over the tem- perature range of interest. For liquid metals, it is often treated as a parameter and fixed with the help of equation (14). 3 Results and Discussion The input parameters used in the present compu- tation are tabulated in Table 1 which are directly taken from [25, 26]. The parameters of the model potential are considered from Eq. (4). Table 1: Input parameters and constants Metal Z Π0 (au) T (K) ξ rc Γ Li 1 151.82 453.0 0.46 1.4649 210.75 Na 1 277.51 378.0 0.46 1.7925 206.45 K 1 529.63 343.0 0.46 2.1877 183.35 Rb 1 648.68 313.0 0.43 2.3901 187.86 Cs 1 810.06 303.0 0.43 2.5539 180.21 Mayank H. Jani and Aditya M. Vora/ BIBECHANA 20 (2023) 146-160 150 (a) (b) (c) (d) (e) Figure 1: Structure factor comparison for PYHS and OCP methods for (a) Li, (b) Na, (c) K, (d) Rb, (e) Cs Mayank H. Jani and Aditya M. Vora/ BIBECHANA 20 (2023) 146-160 151 Here, Z, Π0, T , ξ, rc and Γ are valence, atomic volume, temperature, packing fraction, pseudopo- tential parameter, and plasma component, respec- tively. The respective structure factors for the liq- uid metals Li, Na, K, Rb, and Cs are shown in Fig- ure 1. Three distinct types of techniques, including PYHS and OCP, were employed to compute the structure factors using the newly proposed pseu- dopotential. The currently available experimental structure factor data [27] and the computed struc- ture factor graphs have been compared in Figure 1. As shown in Figure 1 for the metals under study, the pseudopotential has proven to have extremely good agreement with the current experimental re- sults [27]. As can be shown, both methodologies for every ingredient offer superb agreement with exper- imental findings [27]. It can therefore be inferred that the recently created pseudopotential is partic- ularly useful in evaluating a number of liquid metal properties. All of the information we utilized in this post was obtained using the melting point temper- ature. Tables 2 and 3 compare the locations of the first and second peaks as well as the corresponding mag- nitudes of structure factors. According to Waseda [27], the hard sphere struc- ture factor shows the following features: 1. The first peak of the structural factor is sym- metrical. 2. A packing fraction ξ = 0.45 is determined to have the best agreement with experimental data at temperatures just above the melting point. 3. The ratio of the second peak’s position to the first peak, α, is roughly 1.86. Thus, the measured deviation from the hard- sphere structure factor gives useful information in the structural study of liquid metals. Therefore, from Fig. 1, it has been observed that both the structure factor models give very good qualitative data in comparison with the available experimental data [27]. From the results of the first and second peak positions of the structure factor narrated in Table 2, quite good agreement with the experimen- tal value [27] due to OCP was found in compari- son with PYHS. So, the ratio mentioned in Table 3 calculated by both the structure factor methods is found to be consistent with experimental outcomes given in the literature [27]. It has been established that both the structure factor data are qualitatively consistent with the ex- perimental findings [27]. As a result, when com- pared to the PYHS reference system, the OCP ref- erence system gives a more accurate representation of the structure of monovalent metals. According to our research, PYHS is less useful than the OCP reference system for comprehending the structural behavior of liquid metals. The results also support the study’s usage of the newly created pseudopo- tential. The pair correlation function is also an impor- tant component while examining the vibrational properties of liquid metals, as it provides the sta- tistical description of the structure of liquid metals. The g(r) has also been determined using the ear- lier estimated values of the structure factor. The phonon dispersion curves were generated using the various structure factor approaches in accordance with Eq. (1) above, and they were contrasted with the available experimental data [27]. Table 2: Comparison for the first peak positions of structure factors Metal First Peak position and Related Magnitude in S(q) Peak Position in (Å−1) Related Magnitude PYHS OCP EXPT.[27] PYHS OCP EXPT.[27] Li 2.29 2.44 2.50 2.62 2.57 2.72 Na 1.91 2.01 2.05 2.63 2.61 2.70 K 1.70 1.62 1.65 2.62 2.79 2.60 Rb 1.48 1.54 1.50 2.29 2.72 2.69 Cs 1.48 1.49 1.45 2.30 2.26 2.73 Mayank H. Jani and Aditya M. Vora/ BIBECHANA 20 (2023) 146-160 152 Table 3: Comparison for the second peak positions of structure factors Metal Second Peak position and Related Magnitude in S(q) Peak Position in (Å−1) Related Magnitude PYHS OCP EXP. [27] PYHS OCP EXP. [27] Li 4.28 4.74 4.70 1.32 1.21 1.20 Na 3.59 3.91 3.80 1.32 1.21 1.19 K 3.21 3.09 3.00 1.31 1.18 1.19 Rb 2.85 2.92 2.80 1.26 1.19 1.20 Cs 2.86 2.81 2.70 1.25 1.17 1.20 Table 4: Ratio of the structure factor S(q) Metal PYHS OCP EXP. [27] Li 1.86 1.94 1.88 Na 1.87 1.90 1.85 K 1.88 1.90 1.81 Rb 1.92 1.89 1.86 Cs 1.93 1.88 1.86 Table 5: Comparison of first peak positions and related magnitudes in g(r) for different metals. Metal Second Peak position and Related magnitude in g(r) Peak Position in (Å−1) Related Magnitude PYHS OCP EXP. [27] PYHS OCP EXP. [27] Li 2.91 3.00 3.00 2.35 2.11 2.48 Na 3.54 3.70 3.70 2.77 2.06 2.42 K 4.39 4.60 4.60 3.32 2.00 2.35 Rb 4.60 4.80 4.80 3.17 2.10 2.47 Cs 4.97 5.12 5.10 3.38 2.19 2.58 Table 6: The comparison for the second peak positions of g(r). Metal First Peak position and Related magnitude in g(r) Peak Position in (Å−1) Related Magnitude PYHS OCP EXP. [27] PYHS OCP EXP. [27] Li 5.60 5.72 5.70 0.74 1.07 1.22 Na 6.82 6.80 6.80 0.81 1.07 1.26 K 8.47 8.70 8.70 0.89 1.05 1.23 Rb 8.83 9.20 9.20 0.82 1.08 1.28 Cs 9.52 9.60 9.60 0.84 1.09 1.30 Table 7: The ratio (r1/r2) for the pair distribution function g(r). Metal PYHS OCP EXP. [27] Li 1.92 1.90 1.90 Na 1.92 1.83 1.83 K 1.92 1.81 1.81 Rb 1.91 1.91 1.91 Cs 1.92 1.88 1.88 Mayank H. Jani and Aditya M. Vora/ BIBECHANA 20 (2023) 146-160 153 (f) (g) (h) (i) (j) Figure 2: Pair correlation function comparison for PYHS and OCP methods for (f) Li, (g) Na, (h) K, (i) Rb, (j) Cs. In a system with average number density, the probability of finding another atom at a distance r from an origin atom corresponds to g(r). The function g(r) is called the pair correlation function and is frequently used for the discussion of non- crystalline systems. The information given by g(r) is only single-dimensional, but it does provide quan- titative information on non-crystalline systems [27]. Therefore, the pair correlation function is one of the most important pieces of information in this study. When compared to the PYHS technique, the pair distribution functions from various reference Mayank H. Jani and Aditya M. Vora/ BIBECHANA 20 (2023) 146-160 154 (k) (l) (m) (n) (o) Figure 3: The phonon frequencies (solid line), (dashed line) of different element using PYHS structure factor for (k) Li, (l) Na, (m) K, (n) Rb and (o) Cs. Mayank H. Jani and Aditya M. Vora/ BIBECHANA 20 (2023) 146-160 155 (p) (q) (r) (s) (t) Figure 4: The phonon frequencies (solid line), (dashed line) different elements using OCP structure factor for (p) Li, (q) Na, (r) K, (s) Rb and (t) Cs Mayank H. Jani and Aditya M. Vora/ BIBECHANA 20 (2023) 146-160 156 (a) (b) Figure 5: Metal wise comparison for ωl and ωt with PYHS method. (c) (d) Figure 6: Metal wise comparison for ωl and ωt with OCP method. systems show that the OCP method is considerably more in line with the experimental data [27]. The percentage variation in the first peak po- sition q1 of the structure factor using the PYHS model with the experimental data [27] for the given elements were found to be 1.33%–12.4%, whereas for the OCP method it was calculated as 1.33%– 2.75%. For the second peak variation q2, the PYHS model gives 0.77%–18.58%, while the OCP calculation gives a variation of 1.11%–17.21% and 0.40%–9.36%, respectively, with the experimental data [27]. For the first peak of the pair correlation function g(r), the variation found for the PYHS and OCP methods for the liquid alkali metals were 2.35%– 4.56% and 0.00%–0.39%, respectively. For the sec- ond peak variation g(r), the above-mentioned mod- els yield 0.29%–5.40% and 0.00%–0.35%, respec- tively. For all liquid alkali metals, the longitudinal vi- bration variation with regard to the wave vector, calculated using the PYHS and OCP structure fac- tors, was examined in 3 and 4. The findings of var- ious local field correction functions are displayed using colour coding. For the PYHS method, one can see that the Hartree function [21] gives the highest value for ωL(q) and ωT (q) for all the ele- ments except K because, for element K, the screen- ing function of Nagy [23] gives the least values. Also, there are experimental data available for Li, Na, and Rb [28, 29] elements, so the comparison is also shown in the same Figures 3 and 4. It shows that our obtained results are in qualitative agree- ment with the available experimental data [28, 29]. Also, for the elements, ωL(q) shows more oscilla- tory behaviour in comparison with ωT (q). In the discussion for ST (q), less oscillatory behaviour was observed, and one can see that the screening func- tion of Taylor [22] gives the maximum value for all the elements, while Nagy [23] gives the least value except for Cs. In Cs, the Hartree [21] screening function shows the least results, which are very sim- ilar to the data with Nagy [23]. Figure 6 shows that at larger q values, greater oscillatory behaviour is found in the case of ωL(q) than in the case of ωT (q) with different structure Mayank H. Jani and Aditya M. Vora/ BIBECHANA 20 (2023) 146-160 157 Table 8: Vibrational properties of liquid metals with PYHS structure factor Parameters Results Li Na K Rb Cs vL × 103cm/s H 8.3761 4.7317 3.8865 2.5546 2.0273 T 8.0555 4.0830 3.6797 2.3418 1.8110 N 5.9971 0.8279 2.2189 1.7778 1.6037 Exp. [30] - 2.51 - - - Others[12,13] 8.88, 8.61 3.77, 4.41 3.01, 2.97 1.92, 1.90 1.48, 1.44 vT × 103cm/s H 4.8359 2.7318 2.2439 1.4749 1.1704 T 4.6508 2.3573 2.1245 1.3520 1.0455 N 3.4624 0.4780 1.2810 1.0264 0.9259 Exp. [30] - 1.81 - - - Others[12,13] 5.13, 4.97 2.18, 2.54 1.74, 1.69 1.11, 1.08 0.85, 0.83 G dyne/cm2 H 1.2079 0.6928 0.4153 0.3210 0.2518 T 1.1172 0.5159 0.3723 0.2698 0.2009 N 0.6192 0.0212 0.1353 0.1554 0.1575 Others[12,31] 14.10, 13.25 4.41, 6.54 2.76, 2.59 2.01, 1.90 1.45, 1.39 B×103dyne/cm2 H 2.013 1.1547 0.6922 0.5351 0.4196 T 1.8620 0.8598 0.6205 0.4496 0.3348 T 1.0324 0.0353 0.2256 0.2591 0.2626 Exp. [12] - 5.3 - - - Others [12, 31] 2.35, 2.21 7.35, 1.09 0.46, 0.43 0.33, 0.32 0.24, 0.23 Y×1011dyne/cm2 H 3.0197 1.7321 1.0383 0.8026 0.6294 T 2.7930 1.2897 0.9307 0.6745 0.5023 N 1.5480 0.0530 0.3384 0.3887 0.3939 Others [12, 31] 3.53, 3.18 1.10, 1.64 0.69, 0.65 0.50, 0.47 0.36, 0.35 θD(K) H 566.4771 261.7193 173.3101 106.4724 78.4634 T 544.7990 225.8393 164.0891 97.6051 70.0914 N 405.5852 45.7920 98.9458 74.0991 62.0717 Others [12, 32, 33] 609.80, 579.46 250.85, 240.35 138.77, 134.45 82.82, 80.56 58.81, 58.46 factor methods of PYHS and OCP. This assures that longitudinal phonons alone, and not both, are responsible for group oscillations. The linear re- sponse exists in phonon dispersion curves at low q values but vanishes at larger q. This happens as a result of the velocity altering as q shifts. Because all of the metals under examination, due to the Hartree [21] screening function, exhibit identical os- cillating behaviour, the Hartree-Bogolyubov (HB) approach yields reasonably fruitful results. Addi- tionally, Li exhibits the maximum vibrations for all the different structure factor techniques for liquid alkali metals, but Cs gives the lowest vibrations. The vibrational properties of all the above- mentioned liquid metals are shown in Tables 8-10. The parameters include the longitudinal velocity of sound (vL), transverse velocity of sound (vT ), modulus of rigidity (G), isothermal bulk modulus (B), Young’s modulus (Y ), and Debye temperature (θD). The selected liquid alkali metals show screen- ing sensitivity in the low-momentum region. Fur- thermore, the phonons can be detected as sound waves thanks to the spectrum’s macroscopic fre- quency limit. As a result, the longitudinal and transverse sound velocities have been calculated us- ing the linear component of their dispersion curves. Both structure factor techniques’ predictions for vi- brational properties are found to be qualitatively consistent with the theoretical or experimental find- ings detailed in the literature [12]. To put it an- other way, a property essentially has the same value for all permutations of the structure factor method, but divergence is apparent when using the screening functions. One can observe that for both methods exceptionally well results in agreement with the ex- perimental values for vL and vT . Hence, PYHS and OCP structure factor techniques for calculating the isothermal bulk modulus B provide findings that are qualitatively consistent with the experimental data for all Na. Similar results were found for Li, K, Rb, and Cs as individual elements using both structure factor approaches as well as experimental data [30–35]. Mayank H. Jani and Aditya M. Vora/ BIBECHANA 20 (2023) 146-160 158 Table 9: Vibrational properties of liquid metals with OCP structure factor Parameters Results Li Na K Rb Cs vL × 103cm/s H 8.3797 4.6998 3.8307 2.5699 2.0480 T 8.0590 4.0555 3.6269 2.3559 1.8295 N 5.9996 0.8223 2.1870 1.7885 1.6202 Exp. [30] - 2.51 - - - Others [12, 31] 8.88, 8.61 3.77, 4.41 3.01, 2.97 1.92, 1.90 1.48, 1.44 vT × 103cm/s H 4.8320 2.7134 2.2117 1.4870 1.1824 T 4.6528 2.3414 2.0940 1.3601 1.0562 N 3.4639 0.4747 1.2626 1.0326 0.9354 Exp. [30] - 1.81 - - - Others [12, 31] 5.13, 4.97 2.18, 2.54 1.74, 1.69 1.11, 1.08 0.85, 0.83 G dyne/cm2 H 1.1987 0.6832 0.4043 0.3249 0.2569 T 1.1087 0.5087 0.3624 0.2730 0.2050 N 0.6145 0.0209 0.1318 0.1573 0.1608 Exp. [12] - 5.3 - - - Others [12, 31] 14.10, 13.25 4.41, 6.54 2.76, 2.59 2.01, 1.90 1.45, 1.39 B×103dyne/cm2 H 1.9979 1.1387 0.6739 0.5413 0.4282 T 1.8479 0.8479 0.6041 0.4551 0.3418 N 1.0241 0.0348 0.2196 0.2623 0.2680 Exp. [12] - 5.3 - - - Others [12, 31] 2.35, 2.21 7.35, 1.09 0.46, 0.43 0.33, 0.32 0.24, 0.23 Y×1011dyne/cm2 H 2.9968 1.7081 1.0109 0.8123 0.6424 T 2.7719 1.2719 0.9061 0.6826 0.5126 N 1.5362 0.0523 0.3295 0.3934 0.4020 Others [12, 31] 3.53, 3.18 1.10, 1.64 0.69, 0.65 0.50, 0.47 0.36, 0.35 θD(K) H 566.7200 259.9579 170.8222 107.1108 79.2655 T 545.0326 224.3194 161.7336 98.1903 70.8080 N 405.7591 45.4868 97.5255 74.5434 62.7063 Others [12, 32, 33] 609.80, 579.46 250.85, 240.35 138.77, 134.45 82.82, 80.56 58.81, 58.46 The additional theoretical data [12, 32–35] for the modulus of rigidity G for all the elements avail- able in the literature for comparison, and both structure factor approaches were determined to be appropriate to derive the same property. Similar to Young’s modulus, the statistics for all the ele- ments were accessible, which were determined to be good in comparison to the same, and the most re- cent estimated Debye temperature θD is extremely similar to the values actually obtained. With all three types of screening functions, it is discovered that the data obtained by the three structure factor methods are quite similar to one another. 4 Conclusion From the above comprehensive study, it can be con- cluded that the structural and vibrational proper- ties, i.e., phonon dispersion curves (PDC), longi- tudinal sound velocity (vL), transverse sound ve- locity (vT ), isothermal bulk modulus (B), modulus of rigidity (G), Young’s modulus (Y ), Debye tem- perature (θD) of some monovalent (Li, Na, K, Rb, Cs) liquid metals using the newly developed model potential with the two different types of structure factor methods PYHS and OCP, with three types of screening functions, namely Hartree (H), Taylor (T), and Nagy (N), are reported for the first time. It can be challenging to decide which structure fac- tor methodology is most appropriate for comput- ing the vibrational properties of liquid metals be- cause each of the three approaches employed here has its own distinct identity, significance, and lim- itations. However, from the obtained data, we can say that the HB approach calculates precise conclu- sions for phonon data with the fewest parameters possible, and the OCP method gives better results in comparison with the PYHS method. We con- clude that the vibrational properties of the afore- mentioned liquid metals may be efficiently studied by PDCs derived from the HB technique with three local field correction functions, namely H, T, and N. These formulas accurately reproduce all of the salient features of phonon dispersion curves. Hence, the presently adopted model potential is found suit- able when used in conjunction with the HB ap- proach. 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