In the present communication, we have proposed a simple method using which two parameters of Kumar’s pseudopotential can be reduced to effectively single parameter. Further the reliability of our proposed method is examined by carrying out electron dispersion curves (E(k) → k), Fermi energy (EF) and density of states at Fermi energy (N(EF)) for sixteen liquid metals. In absence of any experimental or theoretical results of aforesaid properties, our results will throw some light in the determination of interactions persisting in the liquid state of transition metals

Bhatia, K.G.

Bhatt, N.K.

Gohel, V.B.

Vyas, P.R.

SocioEconomic Challenges Journal (Sumy State University)

JOURNAL OF NANO- AND ELECTRONIC PHYSICS ЖУРНАЛ НАНО- ТА ЕЛЕКТРОННОЇ ФІЗИКИ 
Vol. 10 No 4, 04011(4pp) (2018) Том 10 № 4, 04011(4cc) (2018) 
 
 
2077-6772/2018/10(4)04011(4) 04011-1  2018 Sumy State University 
The Theoretical Study of Electron Dispersion of Some Liquid Metals Using Transition Metal 
Model Potential (TMMP) 
 
Kamaldeep G. Bhatia1, N.K. Bhatt2, P.R. Vyas3, V.B. Gohel3 
 
1 Department of Physics, L.J.I.E.T., Gujarat Technological University, Ahmedabad- 380009, India 
2 Department of Physics, M. K. Bhavnagar University, Bhavnagar- 394001, Gujarat, India 
3 Department of Physics, School of Science, Gujarat University, Ahmedabad- 380009, Gujarat, India 
 
(Received 11 February 2018; revised manuscript received 14 August 2018; published online 25 August 2018) 
 
In the present communication, we have proposed a simple method using which two parameters of Ku-
mar’s pseudopotential can be reduced to effectively single parameter. Further the reliability of our pro-
posed method is examined by carrying out electron dispersion curves (E(k) → k), Fermi energy (EF) and 
density of states at Fermi energy (N(EF)) for sixteen liquid metals. In absence of any experimental or theo-
retical results of aforesaid properties, our results will throw some light in the determination of interactions 
persisting in the liquid state of transition metals. 
 
Keywords: Pseudopotential, Liquid metals, Electron dispersion, Density of states. 
 
DOI: 10.21272/jnep.10(4).04011 PACS numbers: 72.15.Cz, 71.22. + i , 61.25.Mv 
 
 
1. INTRODUCTION 
 
The pseudopotential formalism in its different form 
(local and non-local) has been used successfully for the-
oretical understanding of almost all the physical prop-
erties (static, dynamic and transport) for solid and liq-
uid metals, metallic alloys and metallic glasses with 
good degree of success [1-4]. The reason behind such 
applicability of pseudopotential is due to its physical 
transparency and computational simplicity. Recently, it 
has been observed that local form of the pseudopoten-
tial can be used successfully for the theoretical study of 
thermodynamic properties of metals at extreme envi-
ronment (high temperatures and high pressures). In 
case of metals and their alloys, first principles methods 
are more powerful but at the same time less transpar-
ent, computationally lengthy and time consuming. The 
first principles or ab initio methods has a limitation for 
the study of physical properties of liquid state (for more 
details See Ref. [5]). The study of physical properties at 
liquid density has a great importance in many areas of 
science like fusion and fission reactors, geophysics, as-
trophysics etc. Transport properties of liquid metals 
provide better understanding about nature of the in-
teractions persisting among atoms at melting tempera-
ture. For such study one requires two main ingredients, 
first electron ion interaction called pseudopotential and 
structure factor which gives information about atomic 
distributions. The later can be measured with the help 
of X-ray scattering experiments or with the help of the-
oretical approaches.  
Looking to the importance of study of liquid state 
properties, we in the present study calculate total energy 
as a function of propagation vector k   known as electron 
dispersion for some liquid transition and rare earth met-
als. In order to understand electron ion interaction we 
use Kumar’s [6] pseudopotential with two parameters. 
During literature survey we find that the theoretical 
study of electron dispersion is very rare and limited for 
simple liquid metals. The form of Kumar’s pseudopoten-
tial in real space is as following [6], 
 
 
 
 
 
2
2
2
      for 0  r ,
        for 
       .
,
for
ion c
m c
ion c m
m
ion m
Ze r
V r r
r r
Ze
V r r r r
r
Ze
V r r r
r

  

  

 
 (1.1) 
 
where Z is the valency. In the region r  rc, there is no 
complete cancellation of potential but it decreases line-
arly and in the region rc to rm it remains constant. It 
possesses pure coulombic behavior for the region out-
side rm. The potential between region rc to rm accounts 
hybridization effects because in case of d-band metals 
there is no complete cancellation of potential within 
this range. 
 
2. THEORY 
 
The Fourier transform of Eq. (1.1) gives bare-ion 
pseudopotential in momentum space which has the 
following form [6],  
 
 
 
2 2
2 2
2 4
2
3
4 4
cos cos 2 sin 2cos
4
[( cos cos ) (sin sin )].
ion m c c c c c
m c
c c m m m c
m
Ze Ze
V q qr q r qr qr qr qr Z
q q r r
Ze
qr qr qr qr qr qr
q r
 

 
 
        
 

 (2.1) 
 
The bare-ion pseudopotential is screened by usual 
procedure, the screened pseudopotential 
V(q)  (Vion(q))/ε(q). Here ε(q) is the modified Hartree 
dielectric function. Due to the weakness of the pseudo-
potential, one can use conventional perturbation theory 
to calculate total energy as a sum of ground state, first 
and second order corrections to the energy. In case of 
liquid state, the analogy of the total energy can be un-
 KAMALDEEP G. BHATIA, N.K. BHATT, P.R. VYAS, V.B. GOHEL J. NANO- ELECTRON. PHYS. 10, 04011 (2018) 
 
 
04011-2 
derstood in terms of screened pseudopotential V(q) and 
structure factor S(q) as [7],  
 
 
       
2
22 2
2 2 22
| |2
- ,
2 -
q
S q V qS q V qђ k m
E k
m ђ qk k q
 
 
   
  
 (2.2) 
    
2 2
- 0 ,
2
ђ k
k
m
     (2.3) 
 
where, 
 
      
2
0
3 2
  |   ln | ,
8 2F F
Z k q
k S q V q q dq
k E k k q
 
 

  (2.4) 
 
S(q) is the structure factor. In the present study, we 
have used the hard sphere solution of the Percus-
Yevick equation with packing fraction   0.45 [5]. 
 
      
2
0
3
0 | | .
2 F F
Z
S q V q dq
k E k

    (2.5) 
 
3. RESULTS AND DISCUSSION 
 
We have proposed simple scheme in which we have 
approximated rm by experimental values of atomic radii 
for all the elements [9]. It has been observed in many 
studies that the first zero of the screened pseudopoten-
tial is found near the Fermi surface at metallic densi-
ties. Keeping this in our mind we have adjusted rc to 
get first zero near Fermi surface for all the metals at 
liquid densities. The values of parameters along with 
the atomic volume at melting temperature are tabulat-
ed in Table 1. 
As we have discussed earlier, the main ingredient 
for the calculation of electron dispersion curves i.e. plot 
of (k) – (0) against k is screened pseudopotential 
V(q). For the completeness of the present calculations 
we have presented variations of V(q)s for some transi-
tion metals (Cr, Fe and Co) in Fig. 1. From such plots, 
the first zeroes are observed near the Fermi surface, 
also behaviour of V(q)s are wiggles free upto 8kF. In the 
present calculation the behaviour of pseudopotential at 
higher values of q are most important and without us-
ing any extra damping factor near or about 8kF pseu-
dopotential becomes zero. It has also been found during 
literature survey that the use of artificial damping fac-
tor with screened pseudopotential has no physical sig-
nificance. 
The plots of E(k)  (k) – (0) against k, calculated 
using set of equations (2.2-2.5) are plotted in Fig.(2-7) 
for all sixteen metals. For the study of electron disper-
sion (k) should be calculated precisely. In the expres-
sion of (k) (See Eq. 2.4), logarithmic term which de-
pends upon q as well k has highly oscillatory behaviour 
even at large values of q and k. As a result, in order to 
achieve proper convergence we have evaluated such 
integration with upper limit upto 80kF in step of 0.1kF. 
The order of E(k) is are same as those obtained by Vora 
[7]. In absence of experimental as well as other theoreti-
cal results, the variations of E(k) are acceptable. 
Further, in order to verify predictivity of the pre-
sent pseudopotential we have calculated Fermi energy 
 
Table 1 – The values of Input Parameters: atomic volume Ω 
(in atomic unit) at melting temperature (in K shown in paran-
thesis) taken from Waseda [8]). The Pseudopotential parame-
ters rc and rm  (in atomic unit) 
 
 
 
 
Fig. 1 – Variations of pseudopotential formfactors against q/kF 
for Cr, Fe and Co 
 
EF and Density of States at Fermi energy N(EF). 
The Fermi energy can be evaluated with k  kF in Eq. 
(2.3), 
 
    
2 2
0 .
2
F
F F
k
E k
m
      (3.1) 
Metal Ω 
Valency 
 Z 
Pseudopotential 
parameter 
rc rm 
Cr 
93.04 
(2173) 
3 1.90 3.4965 
Fe 
89.35 
(1833) 
3 2.60 3.2508 
Co 
85.94 
(1823) 
2 2.75 3.1563 
Ni 
85.29 
(1773) 
2 2.40 3.0618 
Pd 
113.72 
(1863) 
2 2.20 3.3831 
Pt 
117.07 
(2053) 
2 1.19 3.4587 
Si 
121.71 
(1733) 
4 1.70 2.7594 
Ge 
146.52 
(1253) 
4 1.50 2.8728 
Sn 
200.44 
(973) 
4 1.20 3.2508 
Pb 
228.98 
(1023) 
4 1.90 3.4209 
Sb 
211.09 
(933) 
5 2.10 2.8917 
La 
259.81 
(1243) 
3 2.60 5.1786 
Ce 
232.83 
(1143) 
3 2.55 5.1030 
Gd 
254.12 
(1603) 
3 2.85 4.8006 
Eu 
367.93 
(1103) 
2 3.50 4.8384 
Yb 
309.96 
(1123) 
2 2.35 4.5360 
 THE THEORETICAL STUDY OF ELECTRON DISPERSION… J. NANO- ELECTRON. PHYS. 10, 04011 (2018) 
 
 
04011-3 
 
 
Fig. 2 – Electron Dispersion for Cr, Fe and Co 
 
 
 
Fig. 3 – Electron Dispersion for Ni, Pd and Pt 
 
 
 
Fig. 4 – Electron Dispersion for Si and Ge 
 
 
 
Fig. 5 – Electron Dispersion for Sn, Pb and Sb 
 
 
 
Fig. 6 – Electron Dispersion for La, Ce and Gd 
 
 
 
Fig. 7 – Electron Dispersion for Eu and Yb 
 
Such values of EF are displayed in Table 2. The 
Density of States at Fermi energy N(EF) can be calcu-
lated as,  
 
( 1)
2
2
( )
) .(
F
F
F
k k
k E
N E
kn


 
  
  
 (3.2) 
 
Table 2 – The calculated values of Fermi energy EF ( 10 – 12 erg) 
and density of states at Fermi level N(EF) ( 1012 (energy-atom) – 1) 
 
 
We could not find experimental as well as theoretical 
results for N(EF) during literature survey. The numerical 
Metal EF N(EF) 
Cr 19.391 1.4221 
Fe 19.624 1.4278 
Co 16.247 1.2074 
Ni 16.413 1.2017 
Pd 13.545 1.4460 
Pt 13.097 1.4234 
Si 18.671 1.9348 
Ge 16.543 2.1509 
Sn 13.347 2.6100 
Pb 12.123 2.9712 
Sb 14.260 3.1046 
La 9.697 2.8298 
Ce 10.475 2.6137 
Gd 9.684 2.8882 
Eu 6.299 3.3473 
Yb 6.947 2.8218 
 KAMALDEEP G. BHATIA, N.K. BHATT, P.R. VYAS, V.B. GOHEL J. NANO- ELECTRON. PHYS. 10, 04011 (2018) 
 
 
04011-4 
results of N(EF) are important for the understanding of 
many liquid state properties like free energy due to elec-
tron gas, entropy, thermal conductivity etc. 
 
4. CONCLUSION 
 
The present form of the local pseudopotential with 
proper form of exchange and correlation function is 
used for less studied physical properties like electron 
dispersion, Fermi energy and Density of States at Fer-
mi energy for many transition and rare earth liquid 
metals for the first time. In absence of experimental 
and theoretical results, presently obtained results fa-
vours the qualities like validity, transferability and 
capability of the local form of pseudopotential along-
with the procedure proposed by us to determine pseu-
dopotential parameters for the study of liquid state 
properties (near melting point). Encouraged by such 
study we would like to extend present form of pseudo-
potential with the same set of parameters for the study 
of liquid state properties like phonon dispersion, densi-
ty of states, elastic constants and many thermodynamic 
functions.  
 
ACKNOWLEDGMENTS 
 
Authors are thankful for the computational facili-
ties developed at the Department of Physics, Gujarat 
University, Ahmedabad 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 Commis-
sion (UGC), New Delhi through DRS SAP (AP-I) pro-
ject (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. C.V. Pandya, P.R. Vyas, T.C. Pandya, V.B. Gohel, Physica 
B: Conden. Matter. 307, 138 (2001). 
2. A.B. Patel, N.K. Bhatt, B.Y. Thakore, P.R. Vyas, 
A.R. Jani, Phys. Chem. Liquid. 52, 471 (2014).  
3. A.M. Vora, Cent. Eur. J. Phys. 6(2), 238 (2008).  
4. P.N. Gajjar, A.M. Vora, A.R. Jani, Modern Phys. Lett. B 18 
No 12-13, 573 (2004). 
5. K.G. Bhatia, N.K. Bhatt, P.R. Vyas, V.B. Gohel, AIP Con-
ference Proc. 1665, 110014 (2015).  
6. J. Kumar, Solid State Commun. 21, 945 (1977). 
7. A.M. Vora, Phys. Chem. Liquid. 47 No 6, 663 (2009). 
8. Y. Waseda, The structure of non-crystalline materials 
(New York: McGraw-Hill International Book: 1980). 
9. chemglobe.org. 
 


The Theoretical Study of Electron Dispersion of Some Liquid Metals Using Transition Metal Model Potential (TMMP)

Electronic Sumy State University Institutional Repository

JOURNAL OF NANO- AND ELECTRONIC PHYSICS ЖУРНАЛ НАНО- ТА ЕЛЕКТРОННОЇ ФІЗИКИ Vol. 10 No 4, 04011(4pp) (2018) Том 10 № 4, 04011(4cc) (2018)   2077-6772/2018/10(4)04011(4) 04011-1  2018 Sumy State University The Theoretical Study of Electron Dispersion of Some Liquid Metals Using Transition Metal Model Potential (TMMP)  Kamaldeep G. Bhatia1, N.K. Bhatt2, P.R. Vyas3, V.B. Gohel3  1 Department of Physics, L.J.I.E.T., Gujarat Technological University, Ahmedabad- 380009, India 2 Department of Physics, M. K. Bhavnagar University, Bhavnagar- 394001, Gujarat, India 3 Department of Physics, School of Science, Gujarat University, Ahmedabad- 380009, Gujarat, India  (Received 11 February 2018; revised manuscript received 14 August 2018; published online 25 August 2018)  In the present communication, we have proposed a simple method using which two parameters of Ku-mar’s pseudopotential can be reduced to effectively single parameter. Further the reliability of our pro-posed method is examined by carrying out electron dispersion curves (E(k) → k), Fermi energy (EF) and density of states at Fermi energy (N(EF)) for sixteen liquid metals. In absence of any experimental or theo-retical results of aforesaid properties, our results will throw some light in the determination of interactions persisting in the liquid state of transition metals.  Keywords: Pseudopotential, Liquid metals, Electron dispersion, Density of states.  DOI: 10.21272/jnep.10(4).04011 PACS numbers: 72.15.Cz, 71.22. + i , 61.25.Mv   1. INTRODUCTION  The pseudopotential formalism in its different form (local and non-local) has been used successfully for the-oretical understanding of almost all the physical prop-erties (static, dynamic and transport) for solid and liq-uid metals, metallic alloys and metallic glasses with good degree of success [1-4]. The reason behind such applicability of pseudopotential is due to its physical transparency and computational simplicity. Recently, it has been observed that local form of the pseudopoten-tial can be used successfully for the theoretical study of thermodynamic properties of metals at extreme envi-ronment (high temperatures and high pressures). In case of metals and their alloys, first principles methods are more powerful but at the same time less transpar-ent, computationally lengthy and time consuming. The first principles or ab initio methods has a limitation for the study of physical properties of liquid state (for more details See Ref. [5]). The study of physical properties at liquid density has a great importance in many areas of science like fusion and fission reactors, geophysics, as-trophysics etc. Transport properties of liquid metals provide better understanding about nature of the in-teractions persisting among atoms at melting tempera-ture. For such study one requires two main ingredients, first electron ion interaction called pseudopotential and structure factor which gives information about atomic distributions. The later can be measured with the help of X-ray scattering experiments or with the help of the-oretical approaches.  Looking to the importance of study of liquid state properties, we in the present study calculate total energy as a function of propagation vector k   known as electron dispersion for some liquid transition and rare earth met-als. In order to understand electron ion interaction we use Kumar’s [6] pseudopotential with two parameters. During literature survey we find that the theoretical study of electron dispersion is very rare and limited for simple liquid metals. The form of Kumar’s pseudopoten-tial in real space is as following [6],      222      for 0  r ,        for        .,forion cm cion c mmion mZe rV r rr rZeV r r r rrZeV r r rr      (1.1)  where Z is the valency. In the region r  rc, there is no complete cancellation of potential but it decreases line-arly and in the region rc to rm it remains constant. It possesses pure coulombic behavior for the region out-side rm. The potential between region rc to rm accounts hybridization effects because in case of d-band metals there is no complete cancellation of potential within this range.  2. THEORY  The Fourier transform of Eq. (1.1) gives bare-ion pseudopotential in momentum space which has the following form [6],     2 22 22 4234 4cos cos 2 sin 2cos4[( cos cos ) (sin sin )].ion m c c c c cm cc c m m m cmZe ZeV q qr q r qr qr qr qr Zq q r rZeqr qr qr qr qr qrq r             (2.1)  The bare-ion pseudopotential is screened by usual procedure, the screened pseudopotential V(q)  (Vion(q))/ε(q). Here ε(q) is the modified Hartree dielectric function. Due to the weakness of the pseudo-potential, one can use conventional perturbation theory to calculate total energy as a sum of ground state, first and second order corrections to the energy. In case of liquid state, the analogy of the total energy can be un- KAMALDEEP G. BHATIA, N.K. BHATT, P.R. VYAS, V.B. GOHEL J. NANO- ELECTRON. PHYS. 10, 04011 (2018)   04011-2 derstood in terms of screened pseudopotential V(q) and structure factor S(q) as [7],           222 22 2 22| |2- ,2 -qS q V qS q V qђ k mE km ђ qk k q        (2.2)     2 2- 0 ,2ђ kkm     (2.3)  where,        203 2  |   ln | ,8 2F FZ k qk S q V q q dqk E k k q    (2.4)  S(q) is the structure factor. In the present study, we have used the hard sphere solution of the Percus-Yevick equation with packing fraction   0.45 [5].        2030 | | .2 F FZS q V q dqk E k    (2.5)  3. RESULTS AND DISCUSSION  We have proposed simple scheme in which we have approximated rm by experimental values of atomic radii for all the elements [9]. It has been observed in many studies that the first zero of the screened pseudopoten-tial is found near the Fermi surface at metallic densi-ties. Keeping this in our mind we have adjusted rc to get first zero near Fermi surface for all the metals at liquid densities. The values of parameters along with the atomic volume at melting temperature are tabulat-ed in Table 1. As we have discussed earlier, the main ingredient for the calculation of electron dispersion curves i.e. plot of (k) – (0) against k is screened pseudopotential V(q). For the completeness of the present calculations we have presented variations of V(q)s for some transi-tion metals (Cr, Fe and Co) in Fig. 1. From such plots, the first zeroes are observed near the Fermi surface, also behaviour of V(q)s are wiggles free upto 8kF. In the present calculation the behaviour of pseudopotential at higher values of q are most important and without us-ing any extra damping factor near or about 8kF pseu-dopotential becomes zero. It has also been found during literature survey that the use of artificial damping fac-tor with screened pseudopotential has no physical sig-nificance. The plots of E(k)  (k) – (0) against k, calculated using set of equations (2.2-2.5) are plotted in Fig.(2-7) for all sixteen metals. For the study of electron disper-sion (k) should be calculated precisely. In the expres-sion of (k) (See Eq. 2.4), logarithmic term which de-pends upon q as well k has highly oscillatory behaviour even at large values of q and k. As a result, in order to achieve proper convergence we have evaluated such integration with upper limit upto 80kF in step of 0.1kF. The order of E(k) is are same as those obtained by Vora [7]. In absence of experimental as well as other theoreti-cal results, the variations of E(k) are acceptable. Further, in order to verify predictivity of the pre-sent pseudopotential we have calculated Fermi energy  Table 1 – The values of Input Parameters: atomic volume Ω (in atomic unit) at melting temperature (in K shown in paran-thesis) taken from Waseda [8]). The Pseudopotential parame-ters rc and rm  (in atomic unit)     Fig. 1 – Variations of pseudopotential formfactors against q/kF for Cr, Fe and Co  EF and Density of States at Fermi energy N(EF). The Fermi energy can be evaluated with k  kF in Eq. (2.3),      2 20 .2FF FkE km      (3.1) Metal Ω Valency  Z Pseudopotential parameter rc rm Cr 93.04 (2173) 3 1.90 3.4965 Fe 89.35 (1833) 3 2.60 3.2508 Co 85.94 (1823) 2 2.75 3.1563 Ni 85.29 (1773) 2 2.40 3.0618 Pd 113.72 (1863) 2 2.20 3.3831 Pt 117.07 (2053) 2 1.19 3.4587 Si 121.71 (1733) 4 1.70 2.7594 Ge 146.52 (1253) 4 1.50 2.8728 Sn 200.44 (973) 4 1.20 3.2508 Pb 228.98 (1023) 4 1.90 3.4209 Sb 211.09 (933) 5 2.10 2.8917 La 259.81 (1243) 3 2.60 5.1786 Ce 232.83 (1143) 3 2.55 5.1030 Gd 254.12 (1603) 3 2.85 4.8006 Eu 367.93 (1103) 2 3.50 4.8384 Yb 309.96 (1123) 2 2.35 4.5360  THE THEORETICAL STUDY OF ELECTRON DISPERSION… J. NANO- ELECTRON. PHYS. 10, 04011 (2018)   04011-3   Fig. 2 – Electron Dispersion for Cr, Fe and Co    Fig. 3 – Electron Dispersion for Ni, Pd and Pt    Fig. 4 – Electron Dispersion for Si and Ge    Fig. 5 – Electron Dispersion for Sn, Pb and Sb    Fig. 6 – Electron Dispersion for La, Ce and Gd    Fig. 7 – Electron Dispersion for Eu and Yb  Such values of EF are displayed in Table 2. The Density of States at Fermi energy N(EF) can be calcu-lated as,   ( 1)22( )) .(FFFk kk EN Ekn      (3.2)  Table 2 – The calculated values of Fermi energy EF ( 10 – 12 erg) and density of states at Fermi level N(EF) ( 1012 (energy-atom) – 1)   We could not find experimental as well as theoretical results for N(EF) during literature survey. The numerical Metal EF N(EF) Cr 19.391 1.4221 Fe 19.624 1.4278 Co 16.247 1.2074 Ni 16.413 1.2017 Pd 13.545 1.4460 Pt 13.097 1.4234 Si 18.671 1.9348 Ge 16.543 2.1509 Sn 13.347 2.6100 Pb 12.123 2.9712 Sb 14.260 3.1046 La 9.697 2.8298 Ce 10.475 2.6137 Gd 9.684 2.8882 Eu 6.299 3.3473 Yb 6.947 2.8218  KAMALDEEP G. BHATIA, N.K. BHATT, P.R. VYAS, V.B. GOHEL J. NANO- ELECTRON. PHYS. 10, 04011 (2018)   04011-4 results of N(EF) are important for the understanding of many liquid state properties like free energy due to elec-tron gas, entropy, thermal conductivity etc.  4. CONCLUSION  The present form of the local pseudopotential with proper form of exchange and correlation function is used for less studied physical properties like electron dispersion, Fermi energy and Density of States at Fer-mi energy for many transition and rare earth liquid metals for the first time. In absence of experimental and theoretical results, presently obtained results fa-vours the qualities like validity, transferability and capability of the local form of pseudopotential along-with the procedure proposed by us to determine pseu-dopotential parameters for the study of liquid state properties (near melting point). Encouraged by such study we would like to extend present form of pseudo-potential with the same set of parameters for the study of liquid state properties like phonon dispersion, densi-ty of states, elastic constants and many thermodynamic functions.   ACKNOWLEDGMENTS  Authors are thankful for the computational facili-ties developed at the Department of Physics, Gujarat University, Ahmedabad 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 Commis-sion (UGC), New Delhi through DRS SAP (AP-I) pro-ject (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. C.V. Pandya, P.R. Vyas, T.C. Pandya, V.B. Gohel, Physica B: Conden. Matter. 307, 138 (2001). 2. A.B. Patel, N.K. Bhatt, B.Y. Thakore, P.R. Vyas, A.R. Jani, Phys. Chem. Liquid. 52, 471 (2014).  3. A.M. Vora, Cent. Eur. J. Phys. 6(2), 238 (2008).  4. P.N. Gajjar, A.M. Vora, A.R. Jani, Modern Phys. Lett. B 18 No 12-13, 573 (2004). 5. K.G. Bhatia, N.K. Bhatt, P.R. Vyas, V.B. Gohel, AIP Con-ference Proc. 1665, 110014 (2015).  6. J. Kumar, Solid State Commun. 21, 945 (1977). 7. A.M. Vora, Phys. Chem. Liquid. 47 No 6, 663 (2009). 8. Y. Waseda, The structure of non-crystalline materials (New York: McGraw-Hill International Book: 1980). 9. chemglobe.org.  

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The Theoretical Study of Electron Dispersion of Some Liquid Metals Using Transition Metal Model Potential (TMMP)

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