In the present paper, we have used Ziman’s approach and transition matrix (t-matrix) approach to study the electrical resistivities of bcc liquid metals. By carrying out this study, we have verified the validity of our proposed pseudopotential extracted from generalized pseudopotential theory (GPT). Our theoretical results agree well with experimental results. Also, it has been verified that for transition metals tmatrix approach is more realistic and physically sound than Ziman approach

Bhatt, N.K.

Gohel, V.B.

Kumar, P.

Nag, A.V.

Tripathi, D.S.

Vyas, P.R.

SocioEconomic Challenges Journal (Sumy State University)

JOURNAL OF NANO- AND ELECTRONIC PHYSICS ЖУРНАЛ НАНО- ТА ЕЛЕКТРОННОЇ ФІЗИКИ 
Vol. 12 No 2, 02020(3pp) (2020) Том 12 № 2, 02020(3cc) (2020) 
 
 
2077-6772/2020/12(2)02020(3) 02020-1  2020 Sumy State University 
The Comparative Study of Electrical Resistivity of bcc Liquid Transition Metals 
 
A.V. Nag1,*, P. Kumar2, D.S. Tripathi3, N.K. Bhatt4, P.R. Vyas3, V.B. Gohel3 
 
1 Department of Physics, L.J.I.A.S., Gujarat University, S.G. Highway, Ahmedabad 382210 Gujarat, India 
2 Department of Science & Humanities, Government Polytechnic, Dahod 389151 Gujarat, India 
3 Department of Physics, School of Science, Gujarat University, Ahmedabad 380009 Gujarat, India 
4 Department of Physics, M.K. Bhavnagar University, Bhavnagar 394001 Gujarat, India 
 
(Received 15 February 2020; revised manuscript received 15 April 2020; published online 25 April 2020) 
 
In the present paper, we have used Ziman’s approach and transition matrix (t-matrix) approach to 
study the electrical resistivities of bcc liquid metals. By carrying out this study, we have verified the validi-
ty of our proposed pseudopotential extracted from generalized pseudopotential theory (GPT). Our theoreti-
cal results agree well with experimental results. Also, it has been verified that for transition metals t-
matrix approach is more realistic and physically sound than Ziman approach. 
 
Keywords: Pseudopotential, Liquid metal resistivity, Transition metal, T-matrix approach. 
 
DOI: 10.21272/jnep.12(2).02020 PACS numbers: 72.15.Cz, 71.22. + i, 61.25.Mv 
 
 
                                                                
* aditin706@gmail.com  
1. INTRODUCTION 
 
Over the past few years there have been several the-
oretical and experimental investigations on electronic 
transport properties of liquid metals in order to under-
stand its structure and atomic interactions occurring in 
the similar solid state metals [1-6]. These studies can 
help to establish relation between the electronic state 
theory and liquid state theory in metals as liquid state 
shows dual behavior, metallic as well as fluidic. Pseudo-
potential approach has been an important area of re-
search for investigations of nearly all physical proper-
ties such as static, dynamic and electronic properties of 
metallic solids, liquids, alloys and glasses [7, 8]. This 
approach is found to be simple in computation, less 
complicated and physically more transparent in compar-
ison with first principles method [9]. The t-matrix and 
Ziman’s approach requires pseudopotential phenome-
nology. It can be used to study the electrical resistivity 
of bcc liquid transition metals which can further explain 
the electron-ion interaction. According to the past stud-
ies, it has been established that t-matrix is more suita-
ble and can provide accurate information about the in-
teractions occurring in liquid metals whereas Ziman’s 
approach, that is based on electron-ion interactions, is 
not applicable to transition metals due to the strong 
resonant scattering caused by empty d states [10]. 
In present communication, we have computed electri-
cal resistivity of molybdenum, tungsten, niobium, chro-
mium and iron (Mo, W, Nb, Cr and Fe) at a given tem-
perature using Ziman and t-matrix approach and have 
proposed a transition metal pseudopotential (TMPP) by 
using generalized pseudopotential theory (GPT). 
The present paper is organized in a given manner: In 
section 2 we have described necessary mathematical 
equation to calculate liquid metal resistivity. Section 3 is 
devoted for comparison of our computed results with 
other theoretical data and experimental findings. We 
have also discussed merits and demerits of present ap-
proach in this section. The paper is concluded with con-
cluding remarks in section 4. 
2. THEORY 
 
We use t-matrix of pseudopotential to calculate the 
relevant scattering cross-section. The transition matrix for 
the scattering on the energy shell can be written as [11] 
 
t(k,k)=
     
  (    )
 
 
 
   [∑ (    )      (  ) 
   (  )  (    ) ], (1) 
 
The t-matrix is normalized to atomic volume Ω and 
it has dimension of energy. The values of phase shifts 
(∆0, ∆1, ∆2) are calculated using method described in 
reference [1]. The electrical resistivity computed using 
t-matrix approach takes the following form 
 
    
   
      
∫     ( )
 
 
 | (    )|    , (2) 
 
where y = q/kF. Here, we have used Percus-Yevick 
equation for S(q) with packing fraction  = 0.45. 
The form of bare-ion pseudopotential, directly ex-
tracted from generalized pseudopotential theory (GPT), 
with having three parameters core radius (rc), radius of 
d-electron (rd) and hybridization parameter () has the 
following form in q-space (in a.u.) [12] 
 
     (q) = 
 
 
 [
    
  
        
 
[      
 ] 
], (3) 
 
Z is the valency and Ω is the atomic volume at melting 
temperature. To obtain screened pseudopotential we 
have adopted method suggested by Wallace [13]. The 
present calculation consist of exchange and correlation 
function due to Hubbard and Sham [14, 15]. 
 
3. RESULTS AND DISCUSSION 
 
In present work, we have proposed a simple scheme to 
determine pseudopotential parameter in which three 
parameters are reduced to two. Here, rd/rc, i.e. ratio of d-
electron along with the core radius is kept within the 
range of 1.1 to 1.5 and hybridization parameter   for all 
metals is tuned in such a manner that first zero (V(q0) = 0)  
 
 A.V. NAG, P. KUMAR ET AL. J. NANO- ELECTRON. PHYS. 12, 02020 (2020) 
 
 
02020-2 
are found between 1 kF and 1.5 kF. The values of the in-
put parameters used in the calculation of pseudopotential 
are tabulated in Table 1. 
Table 2 exhibits the computed values of phase shifts 
∆0, ∆1 and ∆2 (in radian) at q = 2kF. In addition, it shows 
the calculated values of liquid metal resistivity using 
Ziman approach and t-matrix approach compared with 
the experimental findings and other theoretical values. 
From the detailed analysis in Table 2, we observe that 
our results are in good agreement with the experimental 
values at a given temperature. In addition, it has been 
found that our results for molybdenum, niobium, chromi-
um and iron are far more superior in comparison with the 
theoretical values obtained by Ononiwu [10] and Waseda 
[11]. We would like to elaborate these studies in detail. 
Waseda have used muffin tin potential to determine 
phase shifts at fermi energy and by using these phase 
shifts, he has constructed t-matrix form factor as a func-
tion of q/kF. Also by utilizing experimental structural 
data, he has calculated the potential. Ononiwu have used 
Ziman’s theory updated by Evan’s et al. (for details see 
[1]) to study the transport properties of liquid transition 
metals. In nearly free electron approximation, he has 
simulated the effect of the d-band resonance. Here, t-
matrix form factor replaces the ordinary V-matrix pseu-
dopotential form factor. 
In comparison with both the theoretical researchers, 
we find that present form of pseudopotential with simple 
method to determine potential parameter yields better 
results. One more thing emerging from the present study 
is that the values of t-matrix resistivity of all the five 
metals molybdenum, tungsten, niobium, chromium and 
iron are far better than nearly free electron Ziman ap-
proach. Such observation is in agreement with Bhatia et 
al. [1] who have used both the approaches to carry out 
comparative study of liquid metal resistivities for 16 
transition metals. 
For further detailed explanation of the given calcula-
tion, we have also presented the variation of ∆0, ∆1, ∆2 up 
to q = 2kF (up to fermi surface) for chromium and iron in 
Fig. 1 and Fig. 2, respectively. During computation, we 
observe that the maximum value of the phase shift for 
each metal is found to be less than   radian. Such ob-
servation goes in the favor of application of the pertur-
bation expansion and born approximation. The main 
ingredient for the calculation of t-matrix resistivity is t-
matrix form factor t(k,k) (see Eq. (1)). We have also ex-
amined the behavior of t-matrix form factor t(k,k) for all 
the five metals and presented the behavior of t(k,k) for 
Cr and Fe in Fig. 3 and Fig. 4, respectively. 
 
Table 1 – The input parameters and pseudopotental parameters (in a.u.). The quantity shown in parenthesis represents temper-
ature (in C) at which atomic volume (Ω) is used 
 
Metal Z Ω   rc rd 
Mo 6 115.27 (2617) 63 0.227 0.255 
W 6 116.47 (3410) 60 0.275 0.3063 
Nb 5 137.02 (2741) 52 0.2725 0.3025 
Cr 3 93.04 (1900) 35 0.25636 0.282 
Fe 3 89.35 (1550) 24 0.215 0.2365 
 
Table 2 – The computed results of phase shifts ∆0, ∆1 and ∆2 (in radian) and liquid metal resistivity ρziman and ρt (in µΩcm) 
 
Metal ∆0 ∆1 ∆2 ρziman ρt ρexp Other theoretical results 
Mo 2.3529 2.0233 0.9135 32.97 91.05 97 [16] 94 [10] 
W 2.7415 1.9374 0.8938 40.96 81.92 131 [16] 96 [10] 
Nb 2.6938 1.7071 0.7284 56.27 96.11 92.97 [17] 110 [10] 
Cr 1.5412 1.0655 0.4421 14.04 77.52 80 [11] 
56 [10];  
120 [11] 
Fe 1.9075 1.0483 0.4146 40.27 137.88 140 [11] 
92 [10];  
182 [11] 
 
 
 
 
Fig. 1 – The variation of phase shift for Cr 
 
 
Fig. 2 – The variation of phase shift for Fe 
 THE COMPARATIVE STUDY OF ELECTRICAL RESISTIVITY … J. NANO- ELECTRON. PHYS. 12, 02020 (2020) 
 
 
02020-3 
 
 
Fig. 3 – The variation of t-matrix form factor for Cr 
 
 
 
Fig. 4 – The variation of t-matrix form factor for Fe 
 
It is interesting to note that the behavior of t-matrix 
form factors are identical to screened pseudopotential 
V(q). From the present study, we can observe one inter-
esting observation that the first zero of t-matrix form 
factor for all the metals (Mo, W, Nb, Cr and Fe) are 
found near Fermi surface (within the range (1-1.2) kF). 
By taking the limit of Eq. (1) as q → 0, the values of the 
t-matrix form factors at q = 0 can be found. This limit-
ing value can be represented by following form in the 
atomic unit: 
 
t(k,k)= 
  
   
[           
                          ]. (4) 
 
4. CONCLUSIONS 
 
The current study establishes that the local pseudo-
potential method extracted from the generalized pseu-
dopotential theory (GPT) can help in understanding the 
transport properties of bcc liquid transition metals. 
From the comparative studies, we observe that the val-
ues of the liquid metal resistivities for transition metals 
t-matrix approach is satisfactory and far more realistic 
than nearly free electron (NFE) Ziman approach occur-
ring due to weak scattering. In addition, it has been 
found that our results agree very well with the experi-
mental investigations compared to other theoretical val-
ues. Encouraged by present method, we would like to 
extend it for the study of other bcc metals and their al-
loys. 
 
ACKNOWLEDGEMENTS 
 
On behalf of all authors, the corresponding author 
states that there is no conflict of interest. Authors are 
thankful for the computational facilities developed at 
the Department of Physics, Gujarat University, Ahmed-
abad by using financial assistance of (i) Department of 
Sciences and Technology (DST), New Delhi through the 
DST-FIST (Level 1) project (SR/FST/PSI-001/2006) (ii) 
University Grant Commission (UGC), New Delhi 
through DRS SAP (AP-I) project (F.530/10/DRS/2020), 
(iii) Department of Sciences and Technology (DST), 
New Delhi through the DST-FIST project (SR/FST/PSI-
198/2014). 
 
 
REFERENCES 
 
1. K.G. Bhatia, N.K. Bhatt, P.R. Vyas, V.B. Gohel, Pramana-
J. Phys. 92(2) 28, 1 (2019). 
2. A.B. Patel, N.K. Bhatt, B.Y. Thakore, P.R. Vyas, A.R. Jani, 
Phys. Chem. Liq. 52, 471 (2014). 
3. A.B. Patel, N.K. Bhatt, B.Y. Thakore, P.R. Vyas, A.R. Jani, 
Mol. Phys. 112, 2000 (2014). 
4. P.B. Thakor, Y.A. Sonvane, A.R. Jani, Phys. Chem. Liq. 47 
No 6, 653 (2009). 
5. J.G. Gasser, J. Phys. Cond. Matter. 20, 114103 (2008). 
6. K.G. Bhatia, N.K. Bhatt, P.R. Vyas, V.B. Gohel, J. Nano- 
Electron. Phys. 10, 04011 (2018). 
7. P. Kumar, N.K. Bhatt, P.R. Vyas, V.B. Gohel, Chin. Phys. B 
25(11), 116401 (2016). 
8. P. Kumar, N.K. Bhatt, P.R. Vyas, V.B. Gohel, Phys. C. 561, 
71 (2019). 
9. N.K. Bhatt, P.R. Vyas, A.R. Jani, Philos. Mag. 90(12), 1599 
(2010). 
10. J.S. Ononiwu, phys. stat. sol. b 177, 413 (1993). 
11. Y. Waseda, The structure of non-crystalline materials (New 
York: McGraw-Hill International book: 1980). 
12. C.V. Pandya, P.R. Vyas, T.C. Pandya, N. Rani, V.B. Gohel, 
Phys. B: Condens. Matter. 307(14), 138 (2001). 
13. D.C. Wallace, Thermodynamics of crystals (New York: Dover’s 
Publication: 1998). 
14. J. Hubbard, Proc. Roy. Soc. A 243, 336 (1957). 
15. P. Kumar, N.K. Bhatt, P.R. Vyas, V.B. Gohel, Int. J. Thermo-
phys. 38(6), 87 (2017). 
16. P.D. Desai, T.K. Chu, H.M. James, C.Y. Ho, J. Phys. Chem. 
Ref. Data 13 No 4, 1069 (1984). 
17. A. Cezairliyan, J.L. McClure, Int. J. Thermophys. 8 No 6, 803 
(1987). 
 
 


English

The Comparative Study of Electrical Resistivity of bcc Liquid Transition Metals

Electronic Sumy State University Institutional Repository

JOURNAL OF NANO- AND ELECTRONIC PHYSICS ЖУРНАЛ НАНО- ТА ЕЛЕКТРОННОЇ ФІЗИКИ Vol. 12 No 2, 02020(3pp) (2020) Том 12 № 2, 02020(3cc) (2020)   2077-6772/2020/12(2)02020(3) 02020-1  2020 Sumy State University The Comparative Study of Electrical Resistivity of bcc Liquid Transition Metals  A.V. Nag1,*, P. Kumar2, D.S. Tripathi3, N.K. Bhatt4, P.R. Vyas3, V.B. Gohel3  1 Department of Physics, L.J.I.A.S., Gujarat University, S.G. Highway, Ahmedabad 382210 Gujarat, India 2 Department of Science & Humanities, Government Polytechnic, Dahod 389151 Gujarat, India 3 Department of Physics, School of Science, Gujarat University, Ahmedabad 380009 Gujarat, India 4 Department of Physics, M.K. Bhavnagar University, Bhavnagar 394001 Gujarat, India  (Received 15 February 2020; revised manuscript received 15 April 2020; published online 25 April 2020)  In the present paper, we have used Ziman’s approach and transition matrix (t-matrix) approach to study the electrical resistivities of bcc liquid metals. By carrying out this study, we have verified the validi-ty of our proposed pseudopotential extracted from generalized pseudopotential theory (GPT). Our theoreti-cal results agree well with experimental results. Also, it has been verified that for transition metals t-matrix approach is more realistic and physically sound than Ziman approach.  Keywords: Pseudopotential, Liquid metal resistivity, Transition metal, T-matrix approach.  DOI: 10.21272/jnep.12(2).02020 PACS numbers: 72.15.Cz, 71.22. + i, 61.25.Mv                                                                   * aditin706@gmail.com  1. INTRODUCTION  Over the past few years there have been several the-oretical and experimental investigations on electronic transport properties of liquid metals in order to under-stand its structure and atomic interactions occurring in the similar solid state metals [1-6]. These studies can help to establish relation between the electronic state theory and liquid state theory in metals as liquid state shows dual behavior, metallic as well as fluidic. Pseudo-potential approach has been an important area of re-search for investigations of nearly all physical proper-ties such as static, dynamic and electronic properties of metallic solids, liquids, alloys and glasses [7, 8]. This approach is found to be simple in computation, less complicated and physically more transparent in compar-ison with first principles method [9]. The t-matrix and Ziman’s approach requires pseudopotential phenome-nology. It can be used to study the electrical resistivity of bcc liquid transition metals which can further explain the electron-ion interaction. According to the past stud-ies, it has been established that t-matrix is more suita-ble and can provide accurate information about the in-teractions occurring in liquid metals whereas Ziman’s approach, that is based on electron-ion interactions, is not applicable to transition metals due to the strong resonant scattering caused by empty d states [10]. In present communication, we have computed electri-cal resistivity of molybdenum, tungsten, niobium, chro-mium and iron (Mo, W, Nb, Cr and Fe) at a given tem-perature using Ziman and t-matrix approach and have proposed a transition metal pseudopotential (TMPP) by using generalized pseudopotential theory (GPT). The present paper is organized in a given manner: In section 2 we have described necessary mathematical equation to calculate liquid metal resistivity. Section 3 is devoted for comparison of our computed results with other theoretical data and experimental findings. We have also discussed merits and demerits of present ap-proach in this section. The paper is concluded with con-cluding remarks in section 4. 2. THEORY  We use t-matrix of pseudopotential to calculate the relevant scattering cross-section. The transition matrix for the scattering on the energy shell can be written as [11]  t(k,k)=       (    )      [∑ (    )      (  )    (  )  (    ) ], (1)  The t-matrix is normalized to atomic volume Ω and it has dimension of energy. The values of phase shifts (∆0, ∆1, ∆2) are calculated using method described in reference [1]. The electrical resistivity computed using t-matrix approach takes the following form               ∫     ( )   | (    )|    , (2)  where y = q/kF. Here, we have used Percus-Yevick equation for S(q) with packing fraction  = 0.45. The form of bare-ion pseudopotential, directly ex-tracted from generalized pseudopotential theory (GPT), with having three parameters core radius (rc), radius of d-electron (rd) and hybridization parameter () has the following form in q-space (in a.u.) [12]       (q) =    [               [       ] ], (3)  Z is the valency and Ω is the atomic volume at melting temperature. To obtain screened pseudopotential we have adopted method suggested by Wallace [13]. The present calculation consist of exchange and correlation function due to Hubbard and Sham [14, 15].  3. RESULTS AND DISCUSSION  In present work, we have proposed a simple scheme to determine pseudopotential parameter in which three parameters are reduced to two. Here, rd/rc, i.e. ratio of d-electron along with the core radius is kept within the range of 1.1 to 1.5 and hybridization parameter   for all metals is tuned in such a manner that first zero (V(q0) = 0)    A.V. NAG, P. KUMAR ET AL. J. NANO- ELECTRON. PHYS. 12, 02020 (2020)   02020-2 are found between 1 kF and 1.5 kF. The values of the in-put parameters used in the calculation of pseudopotential are tabulated in Table 1. Table 2 exhibits the computed values of phase shifts ∆0, ∆1 and ∆2 (in radian) at q = 2kF. In addition, it shows the calculated values of liquid metal resistivity using Ziman approach and t-matrix approach compared with the experimental findings and other theoretical values. From the detailed analysis in Table 2, we observe that our results are in good agreement with the experimental values at a given temperature. In addition, it has been found that our results for molybdenum, niobium, chromi-um and iron are far more superior in comparison with the theoretical values obtained by Ononiwu [10] and Waseda [11]. We would like to elaborate these studies in detail. Waseda have used muffin tin potential to determine phase shifts at fermi energy and by using these phase shifts, he has constructed t-matrix form factor as a func-tion of q/kF. Also by utilizing experimental structural data, he has calculated the potential. Ononiwu have used Ziman’s theory updated by Evan’s et al. (for details see [1]) to study the transport properties of liquid transition metals. In nearly free electron approximation, he has simulated the effect of the d-band resonance. Here, t-matrix form factor replaces the ordinary V-matrix pseu-dopotential form factor. In comparison with both the theoretical researchers, we find that present form of pseudopotential with simple method to determine potential parameter yields better results. One more thing emerging from the present study is that the values of t-matrix resistivity of all the five metals molybdenum, tungsten, niobium, chromium and iron are far better than nearly free electron Ziman ap-proach. Such observation is in agreement with Bhatia et al. [1] who have used both the approaches to carry out comparative study of liquid metal resistivities for 16 transition metals. For further detailed explanation of the given calcula-tion, we have also presented the variation of ∆0, ∆1, ∆2 up to q = 2kF (up to fermi surface) for chromium and iron in Fig. 1 and Fig. 2, respectively. During computation, we observe that the maximum value of the phase shift for each metal is found to be less than   radian. Such ob-servation goes in the favor of application of the pertur-bation expansion and born approximation. The main ingredient for the calculation of t-matrix resistivity is t-matrix form factor t(k,k) (see Eq. (1)). We have also ex-amined the behavior of t-matrix form factor t(k,k) for all the five metals and presented the behavior of t(k,k) for Cr and Fe in Fig. 3 and Fig. 4, respectively.  Table 1 – The input parameters and pseudopotental parameters (in a.u.). The quantity shown in parenthesis represents temper-ature (in C) at which atomic volume (Ω) is used  Metal Z Ω   rc rd Mo 6 115.27 (2617) 63 0.227 0.255 W 6 116.47 (3410) 60 0.275 0.3063 Nb 5 137.02 (2741) 52 0.2725 0.3025 Cr 3 93.04 (1900) 35 0.25636 0.282 Fe 3 89.35 (1550) 24 0.215 0.2365  Table 2 – The computed results of phase shifts ∆0, ∆1 and ∆2 (in radian) and liquid metal resistivity ρziman and ρt (in µΩcm)  Metal ∆0 ∆1 ∆2 ρziman ρt ρexp Other theoretical results Mo 2.3529 2.0233 0.9135 32.97 91.05 97 [16] 94 [10] W 2.7415 1.9374 0.8938 40.96 81.92 131 [16] 96 [10] Nb 2.6938 1.7071 0.7284 56.27 96.11 92.97 [17] 110 [10] Cr 1.5412 1.0655 0.4421 14.04 77.52 80 [11] 56 [10];  120 [11] Fe 1.9075 1.0483 0.4146 40.27 137.88 140 [11] 92 [10];  182 [11]     Fig. 1 – The variation of phase shift for Cr   Fig. 2 – The variation of phase shift for Fe  THE COMPARATIVE STUDY OF ELECTRICAL RESISTIVITY … J. NANO- ELECTRON. PHYS. 12, 02020 (2020)   02020-3   Fig. 3 – The variation of t-matrix form factor for Cr    Fig. 4 – The variation of t-matrix form factor for Fe  It is interesting to note that the behavior of t-matrix form factors are identical to screened pseudopotential V(q). From the present study, we can observe one inter-esting observation that the first zero of t-matrix form factor for all the metals (Mo, W, Nb, Cr and Fe) are found near Fermi surface (within the range (1-1.2) kF). By taking the limit of Eq. (1) as q → 0, the values of the t-matrix form factors at q = 0 can be found. This limit-ing value can be represented by following form in the atomic unit:  t(k,k)=      [                                     ]. (4)  4. CONCLUSIONS  The current study establishes that the local pseudo-potential method extracted from the generalized pseu-dopotential theory (GPT) can help in understanding the transport properties of bcc liquid transition metals. From the comparative studies, we observe that the val-ues of the liquid metal resistivities for transition metals t-matrix approach is satisfactory and far more realistic than nearly free electron (NFE) Ziman approach occur-ring due to weak scattering. In addition, it has been found that our results agree very well with the experi-mental investigations compared to other theoretical val-ues. Encouraged by present method, we would like to extend it for the study of other bcc metals and their al-loys.  ACKNOWLEDGEMENTS  On behalf of all authors, the corresponding author states that there is no conflict of interest. Authors are thankful for the computational facilities developed at the Department of Physics, Gujarat University, Ahmed-abad by using financial assistance of (i) Department of Sciences and Technology (DST), New Delhi through the DST-FIST (Level 1) project (SR/FST/PSI-001/2006) (ii) University Grant Commission (UGC), New Delhi through DRS SAP (AP-I) project (F.530/10/DRS/2020), (iii) Department of Sciences and Technology (DST), New Delhi through the DST-FIST project (SR/FST/PSI-198/2014).   REFERENCES  1. K.G. Bhatia, N.K. Bhatt, P.R. Vyas, V.B. Gohel, Pramana-J. Phys. 92(2) 28, 1 (2019). 2. A.B. Patel, N.K. Bhatt, B.Y. Thakore, P.R. Vyas, A.R. Jani, Phys. Chem. Liq. 52, 471 (2014). 3. A.B. Patel, N.K. Bhatt, B.Y. Thakore, P.R. Vyas, A.R. Jani, Mol. Phys. 112, 2000 (2014). 4. P.B. Thakor, Y.A. Sonvane, A.R. Jani, Phys. Chem. Liq. 47 No 6, 653 (2009). 5. J.G. Gasser, J. Phys. Cond. Matter. 20, 114103 (2008). 6. K.G. Bhatia, N.K. Bhatt, P.R. Vyas, V.B. Gohel, J. Nano- Electron. Phys. 10, 04011 (2018). 7. P. Kumar, N.K. Bhatt, P.R. Vyas, V.B. Gohel, Chin. Phys. B 25(11), 116401 (2016). 8. P. Kumar, N.K. Bhatt, P.R. Vyas, V.B. Gohel, Phys. C. 561, 71 (2019). 9. N.K. Bhatt, P.R. Vyas, A.R. Jani, Philos. Mag. 90(12), 1599 (2010). 10. J.S. Ononiwu, phys. stat. sol. b 177, 413 (1993). 11. Y. Waseda, The structure of non-crystalline materials (New York: McGraw-Hill International book: 1980). 12. C.V. Pandya, P.R. Vyas, T.C. Pandya, N. Rani, V.B. Gohel, Phys. B: Condens. Matter. 307(14), 138 (2001). 13. D.C. Wallace, Thermodynamics of crystals (New York: Dover’s Publication: 1998). 14. J. Hubbard, Proc. Roy. Soc. A 243, 336 (1957). 15. P. Kumar, N.K. Bhatt, P.R. Vyas, V.B. Gohel, Int. J. Thermo-phys. 38(6), 87 (2017). 16. P.D. Desai, T.K. Chu, H.M. James, C.Y. Ho, J. Phys. Chem. Ref. Data 13 No 4, 1069 (1984). 17. A. Cezairliyan, J.L. McClure, Int. J. Thermophys. 8 No 6, 803 (1987).   

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The Comparative Study of Electrical Resistivity of bcc Liquid Transition Metals

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