IACS Institutional Repository

Joshi, Mitesh

Kumar, Manjul

Gajjar, P N

Thakore, B Y

Jani, A R

IndumJ.Phys. 81(1) 137-141(2007) 
Theoretical investigation of shear viscosity of 
some less simple liquid metals 
Mitesh Joshi*, Manjul Kumar, P N Gajjar1, B Y Thakcjre and A R Jani 
Department of Physics, Sardar Patel University, Vallabh Vidyanagar-388 120, Gujarat, India 
department of Physics, University of School of Sciences, Gujarat University, 
Ahmedabad-380 009, Gujarat, India 
E-mail mjjoshi2004(tf>yahoo com 
Abstract : Theoretical investigation of the temperature dependent Shear Viscosities are reported for some less simple 
liquid metals of group ll-VI Our well established model potential with recent local field correction functions of Ichimaru-Utsumi 
ib used to derive the proper mtenonic pair interactions Charged Hard Sphere reference system is employed to compute the 
structural data for these less simple liquid metals Results are in good agreement when compared with the available 
experimental and other theoretical findings This reveals qualitative applicability of the model potential to define atomic 
transport properties 
Keywords : Shear viscosity, pseudopotential, structure factor 
PACS Nos. : 66 20 +d, 83 60 Fg. 61 25 Mv 
1. Introduction 
One of the most exciting properties of a liquid is its shear viscosity. The study of 
viscosity of liquid metals is important for various metallurgical, industrial and biophysical 
applications. In the recent years considerable efforts have been done to study the 
shear viscosity of liquid [1-8]. In the course of theoretical investigation Molecular 
Dynamics (MD) also provides substantial understanding of microscopical physical 
mechanism underlying the dynamics of liquids [3,9]. Besides this, the distribution 
function as developed by Rice and Allnatt [10] is very convenient for calculation of 
shear viscosity due to its simple form. 
The present study includes calculation of shear viscosity of liquid Mg, Zn, Ga In, 
Sn, and Pb metals. Since all these metals are hybridizing in a similar fashion and 
more or less close to cf-shell elements, except Mg. It is interesting to focus our 
attention in the computation of atomic transport properties of above listed metals of 
group II to VI. We have used the effective pair potential based on well established 
model potential [11-13] which has proved its ability previously by providing excellent 
results of structure factor and its dependent properties, 
^responding Author © 2007 IACS 
138 MiteshJoshi, ManjulKumar, PNGajjar,BYThakoreandARjani 
2. Form factor and pair potential 
The essential ingredients in calculation of the shear viscosity are pair potential and pair 
correlation function. Proper incorporation of model potential in describing these two 
properties are important. 
The single parametric model potential explaining the proper electron-ion interaction 
used in the present investigation in r-space (in Rydberg units) is given by [11], 
W{r) = 0 
2Z} 
r 
1-expj 
r<Rr 
r>Rr 0) 
This model potential is the modified version of Ashcroft's empty core model potential 
[14]. 
The Effective interionic pair potential is given by 
V<0»- 1--/^(Q)sin(gr)c*7 , (2) 
where FN(q), the normalized energy wave number characteristics is expressed as 
Fw«fl« 
{<fWB(q)f 
AnZn [1-2-1 r 1 " [l-G(q)J (3) 
Here, WB(q) is the Fourier transform of the corresponding bare-ion form factor in the 
momentum space [11] while Z, q and rc are respectively valency, magnitude of the 
momentum transfer vector and the parameter of the potential. In eq. 3, e(q) denotes 
the dielectric function and G(q) takes care for the local field correction functions. The 
local field correction due to Ichimaru-Utsumi (IU) [15] is used for the proper incorporation 
of the exchange and correlation among the conduction electrons in dielectric screening. 
3. Structure factor and pair correlation function 
Within a linear screening approximation, the structure factor of a liquid metal in 
charged hard sphere (CHS) reference system is represented as 
Sp(q) 
[l + pPV(q)S0(q)]' 
The necessary details and the expression is found in [13]. The expression for 
the radial distribution function based on structure factor is given by 
\-^}lq{8lq)-^(v)<* (5) 
V "JO 
S(Q)-, (4) 
ff(f) = 1 + 
4. Shear viscosity 
The shear viscosity of hard sphere liquid is be written as [16] 
no=nn + n, + nk. (6) 
The first term on right hand side is the contribution of the hard part of the 
potential expressed by 
Theoretical investigation of shear viscosity of some less simple liquid metals 139 
n* = 
KTo < T ) K T 1 (7) 2p(«r)L" 5 "~']~y Sn I £2 ] 
m which <r is the hard sphere diameter and Dy, Q, 4 are constants defined in [16] 
The contribution of soft part of the potential in shear viscosity is 
AnMir 7
 4 
n
*-~ « « K — I ' 
^d2v 4dv o v •» oy .
 v , 
-
+ 7 ¥ 9(f)df 
The kinetic contribution may be written as 
nt = ]5KT]u*°m^/(Bg{a) fl + - 5«s 
(8) 
(9) 4nMg(a)j 
g(a) in the above expression is the specific value of pair correlation function at hard 
sphere diameter, hence the final expression of shear viscosity is 
rj = Cs(a)n0 . (10) 
C\(j) is the scaling correction term which takes into account the effect of multiple 
scattering. 
5. Results and discussion 
The constants and parameters used in the present computations for the elements of 
group II—VI viz. Mg, Zn, In, Sn, Ga, and Pb are tabulated in Table 1. 
Table 1 . Input parameters. 
Element T(K) Go (a.u.)3 ni [16] rc (a.u.) 
Mg 
Zn 
Ga 
Sn 
Pb 
953 
1063 
1t53 
723 
833 
933 
323 
473 
823 
1073 
433 
573 
773 
973 
523 
573 
973 
1173 
613 
823 
1023 
176.754 
177.904 
178.851 
105.975 
107.531 
108.971 
128.415 
130.780 
135.828 
139.293 
182.964 
185.873 
189.991 
194.590 
191.860 
192.694 
203.301 
203.613 
217.771 
219.123 
221.453 
0.4525 
0.4361 
0.4237 
0.4534 
0.4410 
0.4300 
0.4219 
0.3977 
0.3466 
0.3142 
0.4470 
0.4286 
0.4036 
0.3801 
0.4266 
0.4212 
0.3800 
0.3610 
0.4540 
0.4047 
0.3699 
1.2651 
1.2986 
1.3088 
1.8613 
1.8704 
1.8787 
1.6902 
1.7005 
1.7221 
1.7366 
1.8113 
1.8208 
1.8342 
1.8489 
1.6921 
1.6945 
1.7250 
1.7259 
1.0491 
1.0559 
1.0640 
140 Mitesh Josh/, Man/ul Kumar, P N Ga/jar, B Y Thakore and A R jm 
The resulting pair potential is shown in Figure 1. It is illustrated from Figure 1 
that the depth of the pair potential is minimum for Zn and maximum for Pb while 
depths of rest of the elements considered here are lie in between. The distance of 
position of first minimum is found be higher for Pb due to higher atomic volume and 
valancy. 
ao 
- 2 0 
p 
r x 
t 
F 
t 
F 
t 
r 
t 
r 
bu-u 
T— sn I 
,\\ ,n V - - P b 
\i \ \ 
\ V ^-n 
V N--Xf \ \ y/ V-w'C-' 
. l U l . l i U l l l l l l l l l l l l l l l l i l l l l 
40 45 55 60 65 70 75 40 45 50 55 60 65 70 75 
r (a u) r (a u) 
Figure 1. Effective pair potential for liquid metals Mg, Zn, Ga t Sn, In and Pb 
The results obtained for the shear viscosity are narrated in Table 2 with available 
experimental results at temperatures near and above melting. Shear viscosity of liquid 
metals can also be estimated by using different formalism as done by Zahid et al [7] 
They have used the self-consistent Variational Modified Hypernetted Chain (VMHNC) 
integral equation based on Omstein-Zernike equation 
Table 2. Shear viscosity of liquid metals 
Element T(K) Shear viscosity rj (in cP) 
Present Expt [16] Others [7] 
Mg 
Zn 
Ga 
Sn 
Pb 
953 
1063 
1153 
723 
833 
933 
323 
473 
823 
1073 
433 
573 
773 
973 
523 
573 
973 
1173 
613 
823 
1023 
1 02 
0 99 
0 95 
3 34 
2 76 
2 08 
1 66 
1 48 
1 18 
0 91 
1 60 
1 49 
1 38 
1 21 
1 66 
1 64 
1 54 
1 44 
2.02 
1.94 
1 85 
111 
3 85 
2 14 
1 89 
1 85 
2 38 
1 12 
3 64 
0 98 
1 78 
1 61 
1 97 
Theoretical investigation of shear viscosity of some less simple liquid metals 141 
It is immediately evident from the present study of the pair potential and pair 
correlation function derived from the CHS model that the computed values of shear 
viscosity match qualitatively well with the experimental and other theoretical findings. It 
is concluded here that attractive part and oscillation on the higher q values are not so 
significant in pair potential In the case of shear viscosity; of liquid Ga, the deviation 
is very large which may be due to the structural discrepancy. It is also observed that 
with increase in temperature the calculated shear viscosity decreases. 
The estimated results are free from fitting procedures! and confirm the applicability 
of established model potential as well as CHS method for predicting the proper 
structural behaviour and shear viscosity of the group II to VI elements. 
References 
[1] M Shimoji and I Itami Atomic transport in Liquid metais (Switzerland: Trans. Tech. Pub. Ltd.) (1986) 
[2] M Shimoji Liquid metals (London : Academic Press) (1977) 
[3] J F Wax, R Albaki and J L Brettonet Phys. Rev. B62 14818 (2000) 
[4] R N Singh and F Sommer Phys. Chem. Liq. 36 17 (1998) 
[5] S M Osman, I AH and R N Singh J. Phys.: Condensed Matter 14 8415 (2002) 
[6] S K Sharma and K Tankeshwar J. Phys.: Condensed Matter 14 8415 (2002) 
[7] F Zahid, G M Bhuiyan, S Sultana, M A Khalique, RI M A Rashid, S M Mujibur Rahman Phys. Stat. Sol. (b) 
215 987(1999) 
[8] G M Bhuiyan, I Ali and S M Mujibur Rahman Physica B 334 147 (2003) 
[9] MMG Alemany,C Rey andLJGullego J. Chem. Phys. 1095175(1998) 
[10] S A Rice and A R Allnatt J. Chem. Phys. 34 2144 (1961) 
[11] P N Gajjar, B Y Thakore, J S Luhar and A R Jani Acta. Phys. Pol. A 86 369 (1994) 
[12] P B Thakor, V N Patel, B Y Thakore, P N Gajjar and A R Jani Indian J .Pure Appi Phys. 42 684 (2004) 
[13] P B Thakor, P N Gajjar and A R Jani Commun. Phys. 1415 (2004) 
[14] N W AShcroft Phys. Letts. 23 48 [1966] 
[15] S Ichimaru and K Utsumi Phys. Rev. B24 3220 (1981) 
[16] Y Waseda The Structure of Non-crysttalline Materials (New York : McGraw-Hill) (1980) 


English

Theoretical Investigation of Shear Viscosity of Some Less Simple Liquid Metals

Yadav, Jayprakash

Rafique, S M

Indian J. Phys. 81(1) 161-165(2007) 
# 
Superconducting parameters of metals and alloys: 
HFP technique 
Jayprakash Yadav1 and S M Rafique2* 
JKhunt, Konapatti, Supaul-852 214, Bihar, India 
2P G Department of Physics, T M Bhagalpur University, BhagaIpur-812 007, Bihar, India 
E-mail sm jrafq(«Jyahoo com 
Abstract : The superconducting parameters of metals and alloys have been studied within the framework of 
BCS theory via McMillan [1] technique for different metals and alloys using Harrisons First Principles (HFP) 
pseudopotential. This technique is being used for the first time for this investigation although it is more rigorous 
and based on sound theoretical background In the present paper we have studied the electron phonon coupling 
strength A, Coulomb pseudopotential ft*, superconducting transition temperature Tc of MgAI and ZnAI binary 
alloys The results are compared with available experimental and theoretical data giving a reasonable agreement 
and providing an insight into the systems under investigation 
Keywords : Superconductivity, pseudopotential, alloys 
PACS No. : 74 20.-2 
1. Introduction 
In the present era the study of Superconductivity (SC) of metals and alloys occupies 
the centre stage in the field of condensed matter physics. The theoretical work in this 
field got impetus since the proposition of BCS theory which was later developed to 
suite within the framework of pseudopotentials by McMillan [1], Allen and Cohen [2] 
and Eliashberg [3]. The theory was extended to binary alloys by Allen and Dynes [4]. 
Scattered attempts have been made to apply this technique using model potentials or 
other techniques [6-10]. 
A study of the existing literature reveals that one of the more rigorous and 
theoretically sound technique viz, Harrison's first principle (HFP) pseudopotential [11] 
has not been used for the purposa although it ha9^ been successfully used in the past 
few decades to explain the atomic, electronic and other properties of metals and alloys 
[12-15]. This has encouraged us to apply this technique to study the superconducting 
transition parameters of binary alloys MgAI and ZnAI. 
Corresponding Author © ^007 IACS 
162 Jayprakash Yadav andSM Rafique 
2. Formalism 
The parameters under investigation are the electron-phonon coupling strength (EPCS) A, 
the Coulomb coupling strength //* and the SC transition temperature Tc represented by 
n 
where 
" (2) 
L f <*? . (3) 
^'nkF [ qe*{q) 
7 c «A.exp 0
 1.45 
1.04Q + A) 
I A-/T(1 + 0.62A)I (4) 
In these equations, kB is the Boltzmann's constant, m* is the effective electronic mass, 
M the atomic mass, Z the valence, /cF the Fermi wave vector, 6D the Debye 
temperature, q the phonon wave vector, w(k,q) the screened form factor, e*{q) the 
modified Hartree dielectric function using Vashista-Singwi (V-S) form of exchange 
correlation [16] and EF the Fermi energy. 
The input parameters (X) of alloys are evaluated through 
XAQ « (1 - c)XA + cXB, (5) 
where A and B are constituent metals, c is the concentration of B and X represents 
the atomic volume Q0, valence Z, atomic mass Mt Fermi wave vector kf of the 
species. The Debye temperature of the alloy is given by 
( ^ ' M T S O T (6> 
The electron-phonon coupling strength X involves the screened form factor given by 
^(^Q) = (v-'^vJ+varf)/fMQ) + [l-G(Q)K/fMQ) + Wf?l (7) 
where v*'b, v£ \%and vfq are the valence charge and core electron potential, conduction 
band-core exchange potential, conduction electron potential and screening potential 
respectively and WR is the repulsive potential, G(q) is the exchange correlation function 
which is given by 
G( r i j )«A[ l -e"a ? f ] where rj = q//fF (8) 
Superconducting parameters of metals and alloys: HFP technique 163 
in the V-S form of exchange-correlation [17]. e*(rj) is the modified Hartree dielectric 
function given by 
6*(f j)-[ l- f l ( l?)][ f i«(l?)- l ]+1 f 
where 
£„(77) = 1 + 
T T M 
1 + — - i - / ? 7 
4r] 
2 + r7 
2-77 
(9) 
(10) 
Is the Hartree dielectric function. The parameter e*(rj) is alao involved in the expression 
for the Coulomb coupling ju*. 
3. Computation 
The computation of the form factor w(k,q) and the modified Hartree dielectric function 
r*(q) is carried on through eqs. (7) and (8), respectively. The core energy eigenvalues 
*•„, needed for the computation of w(/c,g) is taken from Herman-Skillman program [17] 
and the input parameters are presented in Table 1. From these computed w(k%q) of 
constituent metals the form factor of the alloys and other constants are obtained 
through eq. (5) and the Debye temperature {0D)AB is obtained through eq. (6). After 
these computations we proceed to compute the superconducting parameters through 
eqs. (1) to (4). The results thus obtained have been presented in Table 1 for 
comparison with experimental observations. 
Table 1. SC parameters of binary alloys AB. 
Alloy (AB) S< 
MgAI 0.3 
MgAI 0.5 
ZnAI 0.3 
ZnAI 0.5 
arameten 
mb 
X 
n* 
Tc 
mb 
A 
M' 
Tc 
mb 
k 
M* 
Tc 
mb 
A 
M* 
Tr> 
5 
1.0000 
0.2777 
0.0773 
0.2300 
1.0000 
0.3022 
0.0767 
0.4613 
1.0000 
0.3480 
0.0753 
0.9702 
1.0000 
0.3531 
0.0752 
1.1131 
Present 
1.2780 
0.3550 
0.0773 
1.2900 
1.4650 
0.4427 
0.0767 
3.6400 
1.0200 
0.3549 
0.0753 
1.1400 
1.2500 
0.4414 
0.0752 
3.1600 
0.9180 
0.2549 
0.0773 
0.1000 
1.0200 
0.3082 
0.0767 
0.5316 
0.9360 
0.3257 
0.0753 
0.6900 
0.9800 
0.3460 
0.0752 
0.9900 
Expt, 
0.84 
0.84 
0.5-O.84 
0.5-0.84 
164 Jayprakash Yadav andSM Rafique 
4. Result and observation 
The computed SC parameters A, //* and Tc, alongwith the band mass mb of the alloys 
MgAI and ZnAI treated as effective mass rrf have been presented in Table 1 for two 
different concentration c = 0.3 and c = 0.5 (equiatomic composition). Considering the 
free electron behavior of the constituent metals band mass mb = 1 has been taken 
initially for the above alloys. Two other band masses have also been evaluated for the 
binary alloy, as provided in this table. Other theoretical data of A and v* or Tc is not 
available for comparison. Hence, only the experimental value of Tc has been given in 
the table. 
An inspection of Table 1 brings out the fact that for MgAI with c = 0.3 and 
mb = 1.278 we get A = 0.355 and Tc = 1.29 against the experimental value 0.84 K. 
With c = 0.5 and mb ~ 1.02 we get A = 0.3082 and Tc = 0.53. These are in qualitative 
agreement with experiment. For ZnAI with c = 0.3 and mb = 0.936 we get A = 0.3257 
and Tc = 0.69 which lies within the experimental range 0.5-0.84 K. However, with 
c = 0.5 and mb - 0.980 we get A = 0.346 and Tc = 0.99 which is slightly higher than 
the experimental range. 
5. Conclusion 
Harrison's first principle (HFP) pseudopotential technique has been used for the first 
time to study the superconducting parameters of binary alloys MgAI and ZnAI with 
reasonable success. The present investigation shows that the transitin temperature Tc 
is quite sensitive to the electron-phonon coupling strength A. The Coulomb pseudopotential 
//* is in general of the order of 0.1 as for pure metals. The accuracy of A and Tc 
mainly depends upon the correctness of the form factor which, in turn, hinges upon the 
input parameters such as core energy eigenvalues, Xor-exchange parameter, opw 
parameter and the form of exchange-correlation function. 
References 
[1) W L McMillan Phys. Rev. 167 331 (1968) 
[2] P B Allen and M L Cohen Phys. Rev. 187 525 (1969) 
[3) G M Eliashberg Soviet Phys. JETP11 696 (1960); 12 1000 (1961) 
[4] P B Allen and R C Dynes Phys. Rev. B12 905 (1975) 
[5J H Khan, V Singh and K S Sharma Indian J. Pure Appl. Phys. 31 628 (1993) 
[6] V Singh, H Khan and K S Sharma Indian J. Pure Appl. Phys. 32 915 (1994) 
[7] P P Singh Bull. Mater Sci. 26 131 (2003) 
[8] O M Gartonde, P Modak, R S Rao and B K Godwal Bull. Mater Sci. 26 137 (2003) 
[9] S Sharma, Haniph Khan and K S Sharma Indian J. Pure Appl. Phys. 41 301 (2003) 
[10J AM Vora, M H Patel, P N Gajjarand A R Jani Pramana58 849 (2002) 
[11] R S Day, F Sun and P H Cutler Phys. Rev. A19 328 (1979) 
[12] J Hafner Phys. Stat So/. (b)56 579 (1973); 57 101 (1973); 57 497 (1973) 
Superconducting parameters of metals and alloys: HFP technique 165 
[ 13 j J A Morlarty Phys. Rev. B16 2537 (1977) 
[14] M M Khan and S M Rafique Phys. Stat. Sol. (b)139 K131 (1987) 
[15] K K Mukerjee, S M Rafique, M M Khan and S Bagchi Phys. Stat. Sol, (b)147 K51 (1988) 
[16] P Vashishta and K S Singwi Phys. Rev. B6 875 (1972) 
[17] P Herman and J Skillman Atomic Structure Calculation (Englewood Cliff, N.J.: Prentice Hall) (1963) 


Superconducting Parameters of Metals and Alloys : HFP Technique

From Editor's Desk

Indian J. Phys. 81 (2), 187-188 (2007) 
ImSRI mm 
Vol.81 • No. 2 • February 2007 
CONTENTS 
Front Title 181 
Advisory Board and Editors 183-184 
From Editor's Desk 185 
Front Contents 187-188 
Review 
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Variation of stratospheric ozone with relative humidity and sharp 
depletion of absolute humidity associated with Nor' wester over Kolkata 
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N SHANTA SINGH, W SHAMBHUNATH SINGH, SH DORENDRAJIT SINGH 
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Thermal and magnetic properties of hydrothermally synthesized 
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|c| U Fano and A R P Rao Atomic Collisions andSpactra (New York Academic) Vol 1, Ch 2. Sec 4, p 25 
(1986) 
[7) T Atsumi, T Isihara, M Koyama and M Matsuzawa Phvs Rev A42 6391 (1990) 
| l l | T Le-Brun. M Lavoliee and P Morin X-ray and Inner Shell Processes (AIP Coni. Proc. 215) (eds) T A 
Canson, M O Krause and S Manson (New York . AIP) p 846 (1990) 
114| SB Hansen, K B MacAdam and L G Gray J 2th Int Conf. on Atomic Physics (Ann. Arbor) Abstracts px-
12(1990) 
115| H Pauly Atomic and Molecular Beams Methods (eds) G Scoles, D Bassi, U Buck and D Laine (London 
Oxford University Press) p 83 (1988) 
[19] W Fntsch (private communication) (1988) 
|2I1 KB MacAdam (to be published) (1991) 
[2^ J T Roy PhD Thesis (University of Calcutta, India) (1992) 
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